TPTP Problem File: SLH0702^1.p

%------------------------------------------------------------------------------
% File     : SLH0000^1 : TPTP v8.2.0. Released v8.2.0.
% Domain   : Archive of Formal Proofs
% Problem  :
% Version  : Especial.
% English  :

% Refs     : [Des23] Desharnais (2023), Email to Geoff Sutcliffe
% Source   : [Des23]
% Names    : FOL_Seq_Calc3/0011_Completeness/prob_00198_008246__12396224_1 [Des23]

% Status   : Theorem
% Rating   : ? v8.2.0
% Syntax   : Number of formulae    : 2111 ( 666 unt; 812 typ;   0 def)
%            Number of atoms       : 3872 (1286 equ;  15 cnn)
%            Maximal formula atoms :   12 (   2 avg)
%            Number of connectives : 15464 ( 220   ~;  48   |; 445   &;13004   @)
%                                         (   0 <=>;1747  =>;   0  <=;   0 <~>)
%            Maximal formula depth :   22 (   7 avg)
%            Number of types       :  145 ( 144 usr)
%            Number of type conns  : 3439 (3439   >;   0   *;   0   +;   0  <<)
%            Number of symbols     :  672 ( 668 usr;  24 con; 0-4 aty)
%            Number of variables   : 4864 ( 820   ^;3747   !; 297   ?;4864   :)
% SPC      : TH0_THM_EQU_NAR

% Comments : This file was generated by Isabelle (most likely Sledgehammer)
%            2023-01-19 15:28:50.028
%------------------------------------------------------------------------------
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    comp_A7552847697395116572m_rule: ( abstra4572470701423150929m_rule > produc164195504107695125m_rule ) > ( abstra4572470701423150929m_rule > abstra4572470701423150929m_rule ) > abstra4572470701423150929m_rule > produc164195504107695125m_rule ).

thf(sy_c_Fun_Ocomp_001t__List__Olist_It__Syntax__Ofm_J_001t__List__Olist_It__Syntax__Ofm_J_001t__Product____Type__Oprod_It__List__Olist_It__Syntax__Ofm_J_Mt__List__Olist_It__Syntax__Ofm_J_J,type,
    comp_l1868218281415653180ist_fm: ( list_fm > list_fm ) > ( produc1996495991257130529ist_fm > list_fm ) > produc1996495991257130529ist_fm > list_fm ).

thf(sy_c_Fun_Ocomp_001t__List__Olist_It__Syntax__Ofm_J_001t__Product____Type__Oprod_It__List__Olist_It__Syntax__Ofm_J_Mt__List__Olist_It__Syntax__Ofm_J_J_001t__Product____Type__Oprod_It__List__Olist_It__Syntax__Ofm_J_Mt__List__Olist_It__Syntax__Ofm_J_J,type,
    comp_l2016392514137073923ist_fm: ( list_fm > produc1996495991257130529ist_fm ) > ( produc1996495991257130529ist_fm > list_fm ) > produc1996495991257130529ist_fm > produc1996495991257130529ist_fm ).

thf(sy_c_Fun_Ocomp_001t__Product____Type__Oprod_It__List__Olist_It__Syntax__Ofm_J_Mt__List__Olist_It__Syntax__Ofm_J_J_001t__List__Olist_It__Syntax__Ofm_J_001t__Abstract____Completeness__Otree_It__Product____Type__Oprod_It__List__Olist_It__Syntax__Ofm_J_Mt__List__Olist_It__Syntax__Ofm_J_J_J,type,
    comp_P8076707111405491611ist_fm: ( produc1996495991257130529ist_fm > list_fm ) > ( abstra5042247676209264507ist_fm > produc1996495991257130529ist_fm ) > abstra5042247676209264507ist_fm > list_fm ).

thf(sy_c_Fun_Ocomp_001t__Product____Type__Oprod_It__List__Olist_It__Syntax__Ofm_J_Mt__List__Olist_It__Syntax__Ofm_J_J_001t__List__Olist_It__Syntax__Ofm_J_001t__Product____Type__Oprod_It__List__Olist_It__Syntax__Ofm_J_Mt__List__Olist_It__Syntax__Ofm_J_J,type,
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thf(sy_c_Fun_Ocomp_001t__Product____Type__Oprod_It__List__Olist_It__Syntax__Ofm_J_Mt__List__Olist_It__Syntax__Ofm_J_J_001t__Product____Type__Oprod_It__List__Olist_It__Syntax__Ofm_J_Mt__List__Olist_It__Syntax__Ofm_J_J_001t__Abstract____Completeness__Otree_It__Product____Type__Oprod_It__Product____Type__Oprod_It__List__Olist_It__Syntax__Ofm_J_Mt__List__Olist_It__Syntax__Ofm_J_J_Mt__Syntax__Orule_J_J,type,
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thf(sy_c_Fun_Ocomp_001t__Product____Type__Oprod_It__List__Olist_It__Syntax__Ofm_J_Mt__List__Olist_It__Syntax__Ofm_J_J_001t__Product____Type__Oprod_It__List__Olist_It__Syntax__Ofm_J_Mt__List__Olist_It__Syntax__Ofm_J_J_001t__Product____Type__Oprod_It__Product____Type__Oprod_It__List__Olist_It__Syntax__Ofm_J_Mt__List__Olist_It__Syntax__Ofm_J_J_Mt__Syntax__Orule_J,type,
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thf(sy_c_Fun_Ocomp_001t__Product____Type__Oprod_It__List__Olist_It__Syntax__Ofm_J_Mt__List__Olist_It__Syntax__Ofm_J_J_001t__Product____Type__Oprod_It__Product____Type__Oprod_It__List__Olist_It__Syntax__Ofm_J_Mt__List__Olist_It__Syntax__Ofm_J_J_Mt__Product____Type__Oprod_It__List__Olist_It__Syntax__Ofm_J_Mt__List__Olist_It__Syntax__Ofm_J_J_J_001t__Product____Type__Oprod_It__Product____Type__Oprod_It__List__Olist_It__Syntax__Ofm_J_Mt__List__Olist_It__Syntax__Ofm_J_J_Mt__Syntax__Orule_J,type,
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thf(sy_c_Fun_Ocomp_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_001t__Nat__Onat_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J,type,
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    comp_P4378417847848123130m_rule: ( produc164195504107695125m_rule > produc1996495991257130529ist_fm ) > ( abstra4572470701423150929m_rule > produc164195504107695125m_rule ) > abstra4572470701423150929m_rule > produc1996495991257130529ist_fm ).

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thf(sy_c_Fun_Ocomp_001t__Product____Type__Oprod_It__Product____Type__Oprod_It__List__Olist_It__Syntax__Ofm_J_Mt__List__Olist_It__Syntax__Ofm_J_J_Mt__Syntax__Orule_J_001t__Syntax__Orule_001t__Abstract____Completeness__Otree_It__Product____Type__Oprod_It__Product____Type__Oprod_It__List__Olist_It__Syntax__Ofm_J_Mt__List__Olist_It__Syntax__Ofm_J_J_Mt__Syntax__Orule_J_J,type,
    comp_P1643750715206316048m_rule: ( produc164195504107695125m_rule > rule ) > ( abstra4572470701423150929m_rule > produc164195504107695125m_rule ) > abstra4572470701423150929m_rule > rule ).

thf(sy_c_Fun_Ocomp_001t__Product____Type__Oprod_It__Product____Type__Oprod_It__List__Olist_It__Syntax__Ofm_J_Mt__List__Olist_It__Syntax__Ofm_J_J_Mt__Syntax__Orule_J_001t__Syntax__Orule_001t__Product____Type__Oprod_It__Product____Type__Oprod_It__List__Olist_It__Syntax__Ofm_J_Mt__List__Olist_It__Syntax__Ofm_J_J_Mt__Syntax__Orule_J,type,
    comp_P422132551275928404m_rule: ( produc164195504107695125m_rule > rule ) > ( produc164195504107695125m_rule > produc164195504107695125m_rule ) > produc164195504107695125m_rule > rule ).

thf(sy_c_Fun_Ocomp_001t__Product____Type__Oprod_It__Product____Type__Oprod_It__List__Olist_It__Syntax__Ofm_J_Mt__List__Olist_It__Syntax__Ofm_J_J_Mt__Syntax__Orule_J_001t__Syntax__Orule_001t__Product____Type__Oprod_It__Syntax__Orule_Mt__Product____Type__Oprod_It__List__Olist_It__Syntax__Ofm_J_Mt__List__Olist_It__Syntax__Ofm_J_J_J,type,
    comp_P4888803072877744336ist_fm: ( produc164195504107695125m_rule > rule ) > ( produc4630866025709511057ist_fm > produc164195504107695125m_rule ) > produc4630866025709511057ist_fm > rule ).

thf(sy_c_Fun_Ocomp_001t__Product____Type__Oprod_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_Mt__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_001t__Product____Type__Oprod_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_Mt__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J,type,
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thf(sy_c_Fun_Ocomp_001t__Product____Type__Oprod_It__Syntax__Orule_Mt__Product____Type__Oprod_It__List__Olist_It__Syntax__Ofm_J_Mt__List__Olist_It__Syntax__Ofm_J_J_J_001t__Product____Type__Oprod_It__List__Olist_It__Syntax__Ofm_J_Mt__List__Olist_It__Syntax__Ofm_J_J_001t__Product____Type__Oprod_It__Product____Type__Oprod_It__List__Olist_It__Syntax__Ofm_J_Mt__List__Olist_It__Syntax__Ofm_J_J_Mt__Syntax__Orule_J,type,
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thf(sy_c_Fun_Ocomp_001t__Product____Type__Oprod_It__Syntax__Orule_Mt__Product____Type__Oprod_It__List__Olist_It__Syntax__Ofm_J_Mt__List__Olist_It__Syntax__Ofm_J_J_J_001t__Syntax__Orule_001t__Product____Type__Oprod_It__Product____Type__Oprod_It__List__Olist_It__Syntax__Ofm_J_Mt__List__Olist_It__Syntax__Ofm_J_J_Mt__Syntax__Orule_J,type,
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thf(sy_c_Fun_Ocomp_001t__Product____Type__Oprod_It__Syntax__Orule_Mt__Syntax__Orule_J_001t__Syntax__Orule_001t__Product____Type__Oprod_It__Product____Type__Oprod_It__List__Olist_It__Syntax__Ofm_J_Mt__List__Olist_It__Syntax__Ofm_J_J_Mt__Syntax__Orule_J,type,
    comp_P2548450393677710116m_rule: ( produc780299800278082923e_rule > rule ) > ( produc164195504107695125m_rule > produc780299800278082923e_rule ) > produc164195504107695125m_rule > rule ).

thf(sy_c_Fun_Ocomp_001t__Syntax__Orule_001t__Product____Type__Oprod_It__Syntax__Orule_Mt__Syntax__Orule_J_001t__Product____Type__Oprod_It__Product____Type__Oprod_It__List__Olist_It__Syntax__Ofm_J_Mt__List__Olist_It__Syntax__Ofm_J_J_Mt__Syntax__Orule_J,type,
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thf(sy_c_Fun_Ocomp_001t__Syntax__Orule_001t__Syntax__Orule_001t__Product____Type__Oprod_It__Product____Type__Oprod_It__List__Olist_It__Syntax__Ofm_J_Mt__List__Olist_It__Syntax__Ofm_J_J_Mt__Syntax__Orule_J,type,
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thf(sy_c_Fun__Def_Opair__leq,type,
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thf(sy_c_Fun__Def_Opair__less,type,
    fun_pair_less: set_Pr8693737435421807431at_nat ).

thf(sy_c_Groups_Ominus__class_Ominus_001t__Nat__Onat,type,
    minus_minus_nat: nat > nat > nat ).

thf(sy_c_Groups_Ominus__class_Ominus_001t__Set__Oset_It__Syntax__Orule_J,type,
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thf(sy_c_If_001t__Abstract____Completeness__Otree_It__Product____Type__Oprod_It__Product____Type__Oprod_It__List__Olist_It__Syntax__Ofm_J_Mt__List__Olist_It__Syntax__Ofm_J_J_Mt__Syntax__Orule_J_J,type,
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thf(sy_c_If_001t__FSet__Ofset_It__Abstract____Completeness__Otree_It__Product____Type__Oprod_It__Product____Type__Oprod_It__List__Olist_It__Syntax__Ofm_J_Mt__List__Olist_It__Syntax__Ofm_J_J_Mt__Syntax__Orule_J_J_J,type,
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thf(sy_c_If_001t__Product____Type__Oprod_It__List__Olist_It__Syntax__Ofm_J_Mt__List__Olist_It__Syntax__Ofm_J_J,type,
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thf(sy_c_If_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J,type,
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thf(sy_c_If_001t__Stream__Ostream_It__Product____Type__Oprod_It__Product____Type__Oprod_It__List__Olist_It__Syntax__Ofm_J_Mt__List__Olist_It__Syntax__Ofm_J_J_Mt__Syntax__Orule_J_J,type,
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thf(sy_c_If_001t__Stream__Ostream_It__Syntax__Orule_J,type,
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thf(sy_c_List_OBleast_001t__Nat__Onat,type,
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thf(sy_c_List_Oabort__Bleast_001t__Nat__Onat,type,
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thf(sy_c_List_Olist_Oset_001t__Abstract____Completeness__Otree_It__Product____Type__Oprod_It__Product____Type__Oprod_It__List__Olist_It__Syntax__Ofm_J_Mt__List__Olist_It__Syntax__Ofm_J_J_Mt__Syntax__Orule_J_J,type,
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thf(sy_c_List_Olist_Oset_001t__Product____Type__Oprod_It__List__Olist_It__Syntax__Ofm_J_Mt__List__Olist_It__Syntax__Ofm_J_J,type,
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thf(sy_c_List_Olist_Oset_001t__Product____Type__Oprod_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_Mt__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J,type,
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thf(sy_c_List_Olist_Oset_001t__Syntax__Ofm,type,
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thf(sy_c_List_Olist_Oset_001t__Syntax__Orule,type,
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thf(sy_c_List_OremoveAll_001t__Syntax__Ofm,type,
    removeAll_fm: fm > list_fm > list_fm ).

thf(sy_c_Nat_OSuc,type,
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thf(sy_c_Nat__Bijection_Oprod__decode__aux,type,
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thf(sy_c_Nat__Bijection_Oprod__decode__aux__rel,type,
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thf(sy_c_Option_Ooption_ONone_001t__FSet__Ofset_It__Product____Type__Oprod_It__List__Olist_It__Syntax__Ofm_J_Mt__List__Olist_It__Syntax__Ofm_J_J_J,type,
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    member8686995742288324264ist_fm: produc4630866025709511057ist_fm > set_Pr2241365836748572487ist_fm > $o ).

thf(sy_c_member_001t__Syntax__Ofm,type,
    member_fm: fm > set_fm > $o ).

thf(sy_c_member_001t__Syntax__Orule,type,
    member_rule: rule > set_rule > $o ).

thf(sy_v_p____,type,
    p: fm ).

thf(sy_v_steps,type,
    steps: stream727092118206550309m_rule ).

thf(sy_v_t____,type,
    t: tm ).

thf(sy_v_thesis____,type,
    thesis: $o ).

% Relevant facts (1275)
thf(fact_0__C_K_C,axiom,
    member_fm @ ( uni @ p ) @ ( treeA @ steps ) ).

% "*"
thf(fact_1_rule_Oinject_I4_J,axiom,
    ! [X71: tm,X72: fm,Y71: tm,Y72: fm] :
      ( ( ( uniL @ X71 @ X72 )
        = ( uniL @ Y71 @ Y72 ) )
      = ( ( X71 = Y71 )
        & ( X72 = Y72 ) ) ) ).

% rule.inject(4)
thf(fact_2__092_060open_062_092_060exists_062k_O_AenabledAtStep_A_IUniL_At_Ap_J_A_Ishd_A_Isdrop_Ak_Asteps_J_J_092_060close_062,axiom,
    ? [K: nat] :
      ( abstra3055540779341473792ist_fm
      @ ^ [R: rule,S: produc1996495991257130529ist_fm,Ss: fset_P661503646757059847ist_fm] :
          ( ( eff @ R @ S )
          = ( some_f2162114139285431020ist_fm @ Ss ) )
      @ ( uniL @ t @ p )
      @ ( produc6879501374131015971m_rule @ ( shd_Pr7235097944458474089m_rule @ ( sdrop_7224736112439592940m_rule @ K @ steps ) ) ) ) ).

% \<open>\<exists>k. enabledAtStep (UniL t p) (shd (sdrop k steps))\<close>
thf(fact_3__092_060open_062_092_060And_062r_An_O_AenabledAtStep_Ar_A_Ishd_A_Isdrop_An_Asteps_J_J_A_092_060Longrightarrow_062_A_092_060exists_062k_O_Asnd_A_Ishd_A_Isdrop_Ak_A_Isdrop_An_Asteps_J_J_J_A_061_Ar_092_060close_062,axiom,
    ! [R2: rule,N: nat] :
      ( ( abstra3055540779341473792ist_fm
        @ ^ [R: rule,S: produc1996495991257130529ist_fm,Ss: fset_P661503646757059847ist_fm] :
            ( ( eff @ R @ S )
            = ( some_f2162114139285431020ist_fm @ Ss ) )
        @ R2
        @ ( produc6879501374131015971m_rule @ ( shd_Pr7235097944458474089m_rule @ ( sdrop_7224736112439592940m_rule @ N @ steps ) ) ) )
     => ? [K: nat] :
          ( ( produc7165828336582415457m_rule @ ( shd_Pr7235097944458474089m_rule @ ( sdrop_7224736112439592940m_rule @ K @ ( sdrop_7224736112439592940m_rule @ N @ steps ) ) ) )
          = R2 ) ) ).

% \<open>\<And>r n. enabledAtStep r (shd (sdrop n steps)) \<Longrightarrow> \<exists>k. snd (shd (sdrop k (sdrop n steps))) = r\<close>
thf(fact_4_sdrop__szip,axiom,
    ! [N: nat,S1: stream_rule,S2: stream_rule] :
      ( ( sdrop_2199339672054592340e_rule @ N @ ( szip_rule_rule @ S1 @ S2 ) )
      = ( szip_rule_rule @ ( sdrop_rule @ N @ S1 ) @ ( sdrop_rule @ N @ S2 ) ) ) ).

% sdrop_szip
thf(fact_5_sdrop__szip,axiom,
    ! [N: nat,S1: stream8299795917829157543ist_fm,S2: stream_rule] :
      ( ( sdrop_7224736112439592940m_rule @ N @ ( szip_P1977448745965526924m_rule @ S1 @ S2 ) )
      = ( szip_P1977448745965526924m_rule @ ( sdrop_4442373711808556042ist_fm @ N @ S1 ) @ ( sdrop_rule @ N @ S2 ) ) ) ).

% sdrop_szip
thf(fact_6_sdrop__szip,axiom,
    ! [N: nat,S1: stream727092118206550309m_rule,S2: stream_rule] :
      ( ( sdrop_6392754500180887844e_rule @ N @ ( szip_P553291425819358518e_rule @ S1 @ S2 ) )
      = ( szip_P553291425819358518e_rule @ ( sdrop_7224736112439592940m_rule @ N @ S1 ) @ ( sdrop_rule @ N @ S2 ) ) ) ).

% sdrop_szip
thf(fact_7_sdrop__szip,axiom,
    ! [N: nat,S1: stream_rule,S2: stream727092118206550309m_rule] :
      ( ( sdrop_1764812713384251940m_rule @ N @ ( szip_r2254114822203693110m_rule @ S1 @ S2 ) )
      = ( szip_r2254114822203693110m_rule @ ( sdrop_rule @ N @ S1 ) @ ( sdrop_7224736112439592940m_rule @ N @ S2 ) ) ) ).

% sdrop_szip
thf(fact_8_sdrop__szip,axiom,
    ! [N: nat,S1: stream727092118206550309m_rule,S2: stream727092118206550309m_rule] :
      ( ( sdrop_7373388980841566196m_rule @ N @ ( szip_P2499414959592755846m_rule @ S1 @ S2 ) )
      = ( szip_P2499414959592755846m_rule @ ( sdrop_7224736112439592940m_rule @ N @ S1 ) @ ( sdrop_7224736112439592940m_rule @ N @ S2 ) ) ) ).

% sdrop_szip
thf(fact_9_sdrop__szip,axiom,
    ! [N: nat,S1: stream_rule,S2: stream3588992565182678257e_rule] :
      ( ( sdrop_3185404920971892658e_rule @ N @ ( szip_r730181393530711434e_rule @ S1 @ S2 ) )
      = ( szip_r730181393530711434e_rule @ ( sdrop_rule @ N @ S1 ) @ ( sdrop_2199339672054592340e_rule @ N @ S2 ) ) ) ).

% sdrop_szip
thf(fact_10_sdrop__szip,axiom,
    ! [N: nat,S1: stream3588992565182678257e_rule,S2: stream_rule] :
      ( ( sdrop_2433707926279816502e_rule @ N @ ( szip_P3067794746305893590e_rule @ S1 @ S2 ) )
      = ( szip_P3067794746305893590e_rule @ ( sdrop_2199339672054592340e_rule @ N @ S1 ) @ ( sdrop_rule @ N @ S2 ) ) ) ).

% sdrop_szip
thf(fact_11_sdrop__szip,axiom,
    ! [N: nat,S1: stream_list_fm,S2: stream_list_fm] :
      ( ( sdrop_4442373711808556042ist_fm @ N @ ( szip_list_fm_list_fm @ S1 @ S2 ) )
      = ( szip_list_fm_list_fm @ ( sdrop_list_fm @ N @ S1 ) @ ( sdrop_list_fm @ N @ S2 ) ) ) ).

% sdrop_szip
thf(fact_12_sdrop__szip,axiom,
    ! [N: nat,S1: stream_rule,S2: stream8299795917829157543ist_fm] :
      ( ( sdrop_2468034597186633064ist_fm @ N @ ( szip_r8549736553299883584ist_fm @ S1 @ S2 ) )
      = ( szip_r8549736553299883584ist_fm @ ( sdrop_rule @ N @ S1 ) @ ( sdrop_4442373711808556042ist_fm @ N @ S2 ) ) ) ).

% sdrop_szip
thf(fact_13_sdrop__szip,axiom,
    ! [N: nat,S1: stream3588992565182678257e_rule,S2: stream3588992565182678257e_rule] :
      ( ( sdrop_7075696550712280400e_rule @ N @ ( szip_P4443206486938582938e_rule @ S1 @ S2 ) )
      = ( szip_P4443206486938582938e_rule @ ( sdrop_2199339672054592340e_rule @ N @ S1 ) @ ( sdrop_2199339672054592340e_rule @ N @ S2 ) ) ) ).

% sdrop_szip
thf(fact_14_sdrop__simps_I1_J,axiom,
    ! [N: nat,S3: stream8299795917829157543ist_fm] :
      ( ( shd_Pr772355297128350925ist_fm @ ( sdrop_4442373711808556042ist_fm @ N @ S3 ) )
      = ( snth_P2624749845746815685ist_fm @ S3 @ N ) ) ).

% sdrop_simps(1)
thf(fact_15_sdrop__simps_I1_J,axiom,
    ! [N: nat,S3: stream3588992565182678257e_rule] :
      ( ( shd_Pr4573454438745646103e_rule @ ( sdrop_2199339672054592340e_rule @ N @ S3 ) )
      = ( snth_P8487267234472522255e_rule @ S3 @ N ) ) ).

% sdrop_simps(1)
thf(fact_16_sdrop__simps_I1_J,axiom,
    ! [N: nat,S3: stream1878305877988316353m_rule] :
      ( ( shd_Pr3564872132220007m_rule @ ( sdrop_1764812713384251940m_rule @ N @ S3 ) )
      = ( snth_P7967347330282200671m_rule @ S3 @ N ) ) ).

% sdrop_simps(1)
thf(fact_17_sdrop__simps_I1_J,axiom,
    ! [N: nat,S3: stream2312013117288958913e_rule] :
      ( ( shd_Pr4631506658928855911e_rule @ ( sdrop_6392754500180887844e_rule @ N @ S3 ) )
      = ( snth_P3371917080224060767e_rule @ S3 @ N ) ) ).

% sdrop_simps(1)
thf(fact_18_sdrop__simps_I1_J,axiom,
    ! [N: nat,S3: stream6494289010434245521m_rule] :
      ( ( shd_Pr2340755956080993207m_rule @ ( sdrop_7373388980841566196m_rule @ N @ S3 ) )
      = ( snth_P8950227884416379311m_rule @ S3 @ N ) ) ).

% sdrop_simps(1)
thf(fact_19_sdrop__simps_I1_J,axiom,
    ! [N: nat,S3: stream727092118206550309m_rule] :
      ( ( shd_Pr7235097944458474089m_rule @ ( sdrop_7224736112439592940m_rule @ N @ S3 ) )
      = ( snth_P1712579846118164081m_rule @ S3 @ N ) ) ).

% sdrop_simps(1)
thf(fact_20_sdrop__simps_I1_J,axiom,
    ! [N: nat,S3: stream_rule] :
      ( ( shd_rule @ ( sdrop_rule @ N @ S3 ) )
      = ( snth_rule @ S3 @ N ) ) ).

% sdrop_simps(1)
thf(fact_21_prod_Oexpand,axiom,
    ! [Prod: produc164195504107695125m_rule,Prod2: produc164195504107695125m_rule] :
      ( ( ( ( produc6879501374131015971m_rule @ Prod )
          = ( produc6879501374131015971m_rule @ Prod2 ) )
        & ( ( produc7165828336582415457m_rule @ Prod )
          = ( produc7165828336582415457m_rule @ Prod2 ) ) )
     => ( Prod = Prod2 ) ) ).

% prod.expand
thf(fact_22_prod_Oexpand,axiom,
    ! [Prod: produc1996495991257130529ist_fm,Prod2: produc1996495991257130529ist_fm] :
      ( ( ( ( produc1501393135466168645ist_fm @ Prod )
          = ( produc1501393135466168645ist_fm @ Prod2 ) )
        & ( ( produc4588648349897876871ist_fm @ Prod )
          = ( produc4588648349897876871ist_fm @ Prod2 ) ) )
     => ( Prod = Prod2 ) ) ).

% prod.expand
thf(fact_23_prod__eqI,axiom,
    ! [P: produc164195504107695125m_rule,Q: produc164195504107695125m_rule] :
      ( ( ( produc6879501374131015971m_rule @ P )
        = ( produc6879501374131015971m_rule @ Q ) )
     => ( ( ( produc7165828336582415457m_rule @ P )
          = ( produc7165828336582415457m_rule @ Q ) )
       => ( P = Q ) ) ) ).

% prod_eqI
thf(fact_24_prod__eqI,axiom,
    ! [P: produc1996495991257130529ist_fm,Q: produc1996495991257130529ist_fm] :
      ( ( ( produc1501393135466168645ist_fm @ P )
        = ( produc1501393135466168645ist_fm @ Q ) )
     => ( ( ( produc4588648349897876871ist_fm @ P )
          = ( produc4588648349897876871ist_fm @ Q ) )
       => ( P = Q ) ) ) ).

% prod_eqI
thf(fact_25_exE__realizer_H,axiom,
    ! [P2: rule > produc1996495991257130529ist_fm > $o,P: produc164195504107695125m_rule] :
      ( ( P2 @ ( produc7165828336582415457m_rule @ P ) @ ( produc6879501374131015971m_rule @ P ) )
     => ~ ! [X: produc1996495991257130529ist_fm,Y: rule] :
            ~ ( P2 @ Y @ X ) ) ).

% exE_realizer'
thf(fact_26_exE__realizer_H,axiom,
    ! [P2: list_fm > list_fm > $o,P: produc1996495991257130529ist_fm] :
      ( ( P2 @ ( produc4588648349897876871ist_fm @ P ) @ ( produc1501393135466168645ist_fm @ P ) )
     => ~ ! [X: list_fm,Y: list_fm] :
            ~ ( P2 @ Y @ X ) ) ).

% exE_realizer'
thf(fact_27_prod__eq__iff,axiom,
    ( ( ^ [Y2: produc164195504107695125m_rule,Z: produc164195504107695125m_rule] : ( Y2 = Z ) )
    = ( ^ [S: produc164195504107695125m_rule,T: produc164195504107695125m_rule] :
          ( ( ( produc6879501374131015971m_rule @ S )
            = ( produc6879501374131015971m_rule @ T ) )
          & ( ( produc7165828336582415457m_rule @ S )
            = ( produc7165828336582415457m_rule @ T ) ) ) ) ) ).

% prod_eq_iff
thf(fact_28_prod__eq__iff,axiom,
    ( ( ^ [Y2: produc1996495991257130529ist_fm,Z: produc1996495991257130529ist_fm] : ( Y2 = Z ) )
    = ( ^ [S: produc1996495991257130529ist_fm,T: produc1996495991257130529ist_fm] :
          ( ( ( produc1501393135466168645ist_fm @ S )
            = ( produc1501393135466168645ist_fm @ T ) )
          & ( ( produc4588648349897876871ist_fm @ S )
            = ( produc4588648349897876871ist_fm @ T ) ) ) ) ) ).

% prod_eq_iff
thf(fact_29_sndsp_Ocases,axiom,
    ! [P: produc164195504107695125m_rule,A: rule] :
      ( ( basic_5782820696336509243m_rule @ P @ A )
     => ( A
        = ( produc7165828336582415457m_rule @ P ) ) ) ).

% sndsp.cases
thf(fact_30_sndsp_Ocases,axiom,
    ! [P: produc1996495991257130529ist_fm,A: list_fm] :
      ( ( basic_7394098686315630381ist_fm @ P @ A )
     => ( A
        = ( produc4588648349897876871ist_fm @ P ) ) ) ).

% sndsp.cases
thf(fact_31_sndsp_Osimps,axiom,
    ( basic_5782820696336509243m_rule
    = ( ^ [P3: produc164195504107695125m_rule,A2: rule] :
          ( A2
          = ( produc7165828336582415457m_rule @ P3 ) ) ) ) ).

% sndsp.simps
thf(fact_32_sndsp_Osimps,axiom,
    ( basic_7394098686315630381ist_fm
    = ( ^ [P3: produc1996495991257130529ist_fm,A2: list_fm] :
          ( A2
          = ( produc4588648349897876871ist_fm @ P3 ) ) ) ) ).

% sndsp.simps
thf(fact_33_fm_Oinject_I3_J,axiom,
    ! [X4: fm,Y4: fm] :
      ( ( ( uni @ X4 )
        = ( uni @ Y4 ) )
      = ( X4 = Y4 ) ) ).

% fm.inject(3)
thf(fact_34_sndsp_Ointros,axiom,
    ! [P: produc164195504107695125m_rule] : ( basic_5782820696336509243m_rule @ P @ ( produc7165828336582415457m_rule @ P ) ) ).

% sndsp.intros
thf(fact_35_sndsp_Ointros,axiom,
    ! [P: produc1996495991257130529ist_fm] : ( basic_7394098686315630381ist_fm @ P @ ( produc4588648349897876871ist_fm @ P ) ) ).

% sndsp.intros
thf(fact_36_enabled__def,axiom,
    ! [R2: rule,S3: produc1996495991257130529ist_fm] :
      ( ( abstra3055540779341473792ist_fm
        @ ^ [R: rule,S: produc1996495991257130529ist_fm,Ss: fset_P661503646757059847ist_fm] :
            ( ( eff @ R @ S )
            = ( some_f2162114139285431020ist_fm @ Ss ) )
        @ R2
        @ S3 )
      = ( ? [Sl: fset_P661503646757059847ist_fm] :
            ( ( eff @ R2 @ S3 )
            = ( some_f2162114139285431020ist_fm @ Sl ) ) ) ) ).

% enabled_def
thf(fact_37_assms_I2_J,axiom,
    ( abstra8647779336736900908ist_fm
    @ ^ [R: rule,S: produc1996495991257130529ist_fm,Ss: fset_P661503646757059847ist_fm] :
        ( ( eff @ R @ S )
        = ( some_f2162114139285431020ist_fm @ Ss ) )
    @ rules
    @ steps ) ).

% assms(2)
thf(fact_38_option_Oinject,axiom,
    ! [X2: fset_P661503646757059847ist_fm,Y22: fset_P661503646757059847ist_fm] :
      ( ( ( some_f2162114139285431020ist_fm @ X2 )
        = ( some_f2162114139285431020ist_fm @ Y22 ) )
      = ( X2 = Y22 ) ) ).

% option.inject
thf(fact_39_assms_I1_J,axiom,
    ( abstra5415202973855822231ist_fm
    @ ^ [R: rule,S: produc1996495991257130529ist_fm,Ss: fset_P661503646757059847ist_fm] :
        ( ( eff @ R @ S )
        = ( some_f2162114139285431020ist_fm @ Ss ) )
    @ rules
    @ steps ) ).

% assms(1)
thf(fact_40_treeA__snth,axiom,
    ! [P: fm,Steps: stream727092118206550309m_rule] :
      ( ( member_fm @ P @ ( treeA @ Steps ) )
     => ? [N2: nat] : ( member_fm @ P @ ( set_fm2 @ ( produc1501393135466168645ist_fm @ ( produc6879501374131015971m_rule @ ( shd_Pr7235097944458474089m_rule @ ( sdrop_7224736112439592940m_rule @ N2 @ Steps ) ) ) ) ) ) ) ).

% treeA_snth
thf(fact_41_snd__swap,axiom,
    ! [X3: produc164195504107695125m_rule] :
      ( ( produc4514744107061996309ist_fm @ ( produc2641608828089375343m_rule @ X3 ) )
      = ( produc6879501374131015971m_rule @ X3 ) ) ).

% snd_swap
thf(fact_42_snd__swap,axiom,
    ! [X3: produc4630866025709511057ist_fm] :
      ( ( produc7165828336582415457m_rule @ ( produc9213896635423732003ist_fm @ X3 ) )
      = ( produc4228417144610596823ist_fm @ X3 ) ) ).

% snd_swap
thf(fact_43_snd__swap,axiom,
    ! [X3: produc1996495991257130529ist_fm] :
      ( ( produc4588648349897876871ist_fm @ ( produc7936585717479832313ist_fm @ X3 ) )
      = ( produc1501393135466168645ist_fm @ X3 ) ) ).

% snd_swap
thf(fact_44_fst__swap,axiom,
    ! [X3: produc4630866025709511057ist_fm] :
      ( ( produc6879501374131015971m_rule @ ( produc9213896635423732003ist_fm @ X3 ) )
      = ( produc4514744107061996309ist_fm @ X3 ) ) ).

% fst_swap
thf(fact_45_fst__swap,axiom,
    ! [X3: produc164195504107695125m_rule] :
      ( ( produc4228417144610596823ist_fm @ ( produc2641608828089375343m_rule @ X3 ) )
      = ( produc7165828336582415457m_rule @ X3 ) ) ).

% fst_swap
thf(fact_46_fst__swap,axiom,
    ! [X3: produc1996495991257130529ist_fm] :
      ( ( produc1501393135466168645ist_fm @ ( produc7936585717479832313ist_fm @ X3 ) )
      = ( produc4588648349897876871ist_fm @ X3 ) ) ).

% fst_swap
thf(fact_47_RuleSystem__Defs_Oenabled__def,axiom,
    ( abstra3055540779341473792ist_fm
    = ( ^ [Eff: rule > produc1996495991257130529ist_fm > fset_P661503646757059847ist_fm > $o,R: rule,S: produc1996495991257130529ist_fm] :
        ? [X5: fset_P661503646757059847ist_fm] : ( Eff @ R @ S @ X5 ) ) ) ).

% RuleSystem_Defs.enabled_def
thf(fact_48_RuleSystem__Defs_Oenabled_Ocong,axiom,
    abstra3055540779341473792ist_fm = abstra3055540779341473792ist_fm ).

% RuleSystem_Defs.enabled.cong
thf(fact_49_szip_Osimps_I1_J,axiom,
    ! [S1: stream_list_fm,S2: stream_list_fm] :
      ( ( shd_Pr772355297128350925ist_fm @ ( szip_list_fm_list_fm @ S1 @ S2 ) )
      = ( produc381145313068854617ist_fm @ ( shd_list_fm @ S1 ) @ ( shd_list_fm @ S2 ) ) ) ).

% szip.simps(1)
thf(fact_50_szip_Osimps_I1_J,axiom,
    ! [S1: stream_nat,S2: stream_nat] :
      ( ( shd_Pr4260400998323988397at_nat @ ( szip_nat_nat @ S1 @ S2 ) )
      = ( product_Pair_nat_nat @ ( shd_nat @ S1 ) @ ( shd_nat @ S2 ) ) ) ).

% szip.simps(1)
thf(fact_51_szip_Osimps_I1_J,axiom,
    ! [S1: stream6724221391990029191at_nat,S2: stream6724221391990029191at_nat] :
      ( ( shd_Pr8412153233960533267at_nat @ ( szip_P4314639285670189082at_nat @ S1 @ S2 ) )
      = ( produc6161850002892822231at_nat @ ( shd_Pr4260400998323988397at_nat @ S1 ) @ ( shd_Pr4260400998323988397at_nat @ S2 ) ) ) ).

% szip.simps(1)
thf(fact_52_szip_Osimps_I1_J,axiom,
    ! [S1: stream727092118206550309m_rule,S2: stream727092118206550309m_rule] :
      ( ( shd_Pr2340755956080993207m_rule @ ( szip_P2499414959592755846m_rule @ S1 @ S2 ) )
      = ( produc5927390650430071747m_rule @ ( shd_Pr7235097944458474089m_rule @ S1 ) @ ( shd_Pr7235097944458474089m_rule @ S2 ) ) ) ).

% szip.simps(1)
thf(fact_53_szip_Osimps_I1_J,axiom,
    ! [S1: stream727092118206550309m_rule,S2: stream_rule] :
      ( ( shd_Pr4631506658928855911e_rule @ ( szip_P553291425819358518e_rule @ S1 @ S2 ) )
      = ( produc4383365014245906419e_rule @ ( shd_Pr7235097944458474089m_rule @ S1 ) @ ( shd_rule @ S2 ) ) ) ).

% szip.simps(1)
thf(fact_54_szip_Osimps_I1_J,axiom,
    ! [S1: stream_rule,S2: stream8299795917829157543ist_fm] :
      ( ( shd_Pr2478396429205514213ist_fm @ ( szip_r8549736553299883584ist_fm @ S1 @ S2 ) )
      = ( produc7063755442767259331ist_fm @ ( shd_rule @ S1 ) @ ( shd_Pr772355297128350925ist_fm @ S2 ) ) ) ).

% szip.simps(1)
thf(fact_55_szip_Osimps_I1_J,axiom,
    ! [S1: stream_rule,S2: stream727092118206550309m_rule] :
      ( ( shd_Pr3564872132220007m_rule @ ( szip_r2254114822203693110m_rule @ S1 @ S2 ) )
      = ( produc6084188410630241011m_rule @ ( shd_rule @ S1 ) @ ( shd_Pr7235097944458474089m_rule @ S2 ) ) ) ).

% szip.simps(1)
thf(fact_56_szip_Osimps_I1_J,axiom,
    ! [S1: stream_rule,S2: stream_rule] :
      ( ( shd_Pr4573454438745646103e_rule @ ( szip_rule_rule @ S1 @ S2 ) )
      = ( produc1769597910952761123e_rule @ ( shd_rule @ S1 ) @ ( shd_rule @ S2 ) ) ) ).

% szip.simps(1)
thf(fact_57_szip_Osimps_I1_J,axiom,
    ! [S1: stream8299795917829157543ist_fm,S2: stream_rule] :
      ( ( shd_Pr7235097944458474089m_rule @ ( szip_P1977448745965526924m_rule @ S1 @ S2 ) )
      = ( produc491467635432902671m_rule @ ( shd_Pr772355297128350925ist_fm @ S1 ) @ ( shd_rule @ S2 ) ) ) ).

% szip.simps(1)
thf(fact_58_pickEff,axiom,
    ! [R2: rule,S3: produc1996495991257130529ist_fm] :
      ( ( abstra3055540779341473792ist_fm
        @ ^ [R: rule,S: produc1996495991257130529ist_fm,Ss: fset_P661503646757059847ist_fm] :
            ( ( eff @ R @ S )
            = ( some_f2162114139285431020ist_fm @ Ss ) )
        @ R2
        @ S3 )
     => ( ( eff @ R2 @ S3 )
        = ( some_f2162114139285431020ist_fm
          @ ( abstra458490714785879965ist_fm
            @ ^ [R: rule,S: produc1996495991257130529ist_fm,Ss: fset_P661503646757059847ist_fm] :
                ( ( eff @ R @ S )
                = ( some_f2162114139285431020ist_fm @ Ss ) )
            @ R2
            @ S3 ) ) ) ) ).

% pickEff
thf(fact_59_prod_Oinject,axiom,
    ! [X1: produc1996495991257130529ist_fm,X2: rule,Y1: produc1996495991257130529ist_fm,Y22: rule] :
      ( ( ( produc491467635432902671m_rule @ X1 @ X2 )
        = ( produc491467635432902671m_rule @ Y1 @ Y22 ) )
      = ( ( X1 = Y1 )
        & ( X2 = Y22 ) ) ) ).

% prod.inject
thf(fact_60_prod_Oinject,axiom,
    ! [X1: list_fm,X2: list_fm,Y1: list_fm,Y22: list_fm] :
      ( ( ( produc381145313068854617ist_fm @ X1 @ X2 )
        = ( produc381145313068854617ist_fm @ Y1 @ Y22 ) )
      = ( ( X1 = Y1 )
        & ( X2 = Y22 ) ) ) ).

% prod.inject
thf(fact_61_prod_Oinject,axiom,
    ! [X1: nat,X2: nat,Y1: nat,Y22: nat] :
      ( ( ( product_Pair_nat_nat @ X1 @ X2 )
        = ( product_Pair_nat_nat @ Y1 @ Y22 ) )
      = ( ( X1 = Y1 )
        & ( X2 = Y22 ) ) ) ).

% prod.inject
thf(fact_62_prod_Oinject,axiom,
    ! [X1: product_prod_nat_nat,X2: product_prod_nat_nat,Y1: product_prod_nat_nat,Y22: product_prod_nat_nat] :
      ( ( ( produc6161850002892822231at_nat @ X1 @ X2 )
        = ( produc6161850002892822231at_nat @ Y1 @ Y22 ) )
      = ( ( X1 = Y1 )
        & ( X2 = Y22 ) ) ) ).

% prod.inject
thf(fact_63_prod_Oinject,axiom,
    ! [X1: rule,X2: produc1996495991257130529ist_fm,Y1: rule,Y22: produc1996495991257130529ist_fm] :
      ( ( ( produc7063755442767259331ist_fm @ X1 @ X2 )
        = ( produc7063755442767259331ist_fm @ Y1 @ Y22 ) )
      = ( ( X1 = Y1 )
        & ( X2 = Y22 ) ) ) ).

% prod.inject
thf(fact_64_old_Oprod_Oinject,axiom,
    ! [A: produc1996495991257130529ist_fm,B: rule,A3: produc1996495991257130529ist_fm,B2: rule] :
      ( ( ( produc491467635432902671m_rule @ A @ B )
        = ( produc491467635432902671m_rule @ A3 @ B2 ) )
      = ( ( A = A3 )
        & ( B = B2 ) ) ) ).

% old.prod.inject
thf(fact_65_old_Oprod_Oinject,axiom,
    ! [A: list_fm,B: list_fm,A3: list_fm,B2: list_fm] :
      ( ( ( produc381145313068854617ist_fm @ A @ B )
        = ( produc381145313068854617ist_fm @ A3 @ B2 ) )
      = ( ( A = A3 )
        & ( B = B2 ) ) ) ).

% old.prod.inject
thf(fact_66_old_Oprod_Oinject,axiom,
    ! [A: nat,B: nat,A3: nat,B2: nat] :
      ( ( ( product_Pair_nat_nat @ A @ B )
        = ( product_Pair_nat_nat @ A3 @ B2 ) )
      = ( ( A = A3 )
        & ( B = B2 ) ) ) ).

% old.prod.inject
thf(fact_67_old_Oprod_Oinject,axiom,
    ! [A: product_prod_nat_nat,B: product_prod_nat_nat,A3: product_prod_nat_nat,B2: product_prod_nat_nat] :
      ( ( ( produc6161850002892822231at_nat @ A @ B )
        = ( produc6161850002892822231at_nat @ A3 @ B2 ) )
      = ( ( A = A3 )
        & ( B = B2 ) ) ) ).

% old.prod.inject
thf(fact_68_old_Oprod_Oinject,axiom,
    ! [A: rule,B: produc1996495991257130529ist_fm,A3: rule,B2: produc1996495991257130529ist_fm] :
      ( ( ( produc7063755442767259331ist_fm @ A @ B )
        = ( produc7063755442767259331ist_fm @ A3 @ B2 ) )
      = ( ( A = A3 )
        & ( B = B2 ) ) ) ).

% old.prod.inject
thf(fact_69_swap__simp,axiom,
    ! [X3: produc1996495991257130529ist_fm,Y3: rule] :
      ( ( produc2641608828089375343m_rule @ ( produc491467635432902671m_rule @ X3 @ Y3 ) )
      = ( produc7063755442767259331ist_fm @ Y3 @ X3 ) ) ).

% swap_simp
thf(fact_70_swap__simp,axiom,
    ! [X3: list_fm,Y3: list_fm] :
      ( ( produc7936585717479832313ist_fm @ ( produc381145313068854617ist_fm @ X3 @ Y3 ) )
      = ( produc381145313068854617ist_fm @ Y3 @ X3 ) ) ).

% swap_simp
thf(fact_71_swap__simp,axiom,
    ! [X3: nat,Y3: nat] :
      ( ( product_swap_nat_nat @ ( product_Pair_nat_nat @ X3 @ Y3 ) )
      = ( product_Pair_nat_nat @ Y3 @ X3 ) ) ).

% swap_simp
thf(fact_72_swap__simp,axiom,
    ! [X3: product_prod_nat_nat,Y3: product_prod_nat_nat] :
      ( ( produc7225117575323628663at_nat @ ( produc6161850002892822231at_nat @ X3 @ Y3 ) )
      = ( produc6161850002892822231at_nat @ Y3 @ X3 ) ) ).

% swap_simp
thf(fact_73_swap__simp,axiom,
    ! [X3: rule,Y3: produc1996495991257130529ist_fm] :
      ( ( produc9213896635423732003ist_fm @ ( produc7063755442767259331ist_fm @ X3 @ Y3 ) )
      = ( produc491467635432902671m_rule @ Y3 @ X3 ) ) ).

% swap_simp
thf(fact_74_prod_Ocollapse,axiom,
    ! [Prod: product_prod_nat_nat] :
      ( ( product_Pair_nat_nat @ ( product_fst_nat_nat @ Prod ) @ ( product_snd_nat_nat @ Prod ) )
      = Prod ) ).

% prod.collapse
thf(fact_75_prod_Ocollapse,axiom,
    ! [Prod: produc859450856879609959at_nat] :
      ( ( produc6161850002892822231at_nat @ ( produc3213797794245857475at_nat @ Prod ) @ ( produc6408287024330202629at_nat @ Prod ) )
      = Prod ) ).

% prod.collapse
thf(fact_76_prod_Ocollapse,axiom,
    ! [Prod: produc4630866025709511057ist_fm] :
      ( ( produc7063755442767259331ist_fm @ ( produc4228417144610596823ist_fm @ Prod ) @ ( produc4514744107061996309ist_fm @ Prod ) )
      = Prod ) ).

% prod.collapse
thf(fact_77_prod_Ocollapse,axiom,
    ! [Prod: produc164195504107695125m_rule] :
      ( ( produc491467635432902671m_rule @ ( produc6879501374131015971m_rule @ Prod ) @ ( produc7165828336582415457m_rule @ Prod ) )
      = Prod ) ).

% prod.collapse
thf(fact_78_prod_Ocollapse,axiom,
    ! [Prod: produc1996495991257130529ist_fm] :
      ( ( produc381145313068854617ist_fm @ ( produc1501393135466168645ist_fm @ Prod ) @ ( produc4588648349897876871ist_fm @ Prod ) )
      = Prod ) ).

% prod.collapse
thf(fact_79_snth__szip,axiom,
    ! [S1: stream_rule,S2: stream_rule,N: nat] :
      ( ( snth_P8487267234472522255e_rule @ ( szip_rule_rule @ S1 @ S2 ) @ N )
      = ( produc1769597910952761123e_rule @ ( snth_rule @ S1 @ N ) @ ( snth_rule @ S2 @ N ) ) ) ).

% snth_szip
thf(fact_80_snth__szip,axiom,
    ! [S1: stream_rule,S2: stream727092118206550309m_rule,N: nat] :
      ( ( snth_P7967347330282200671m_rule @ ( szip_r2254114822203693110m_rule @ S1 @ S2 ) @ N )
      = ( produc6084188410630241011m_rule @ ( snth_rule @ S1 @ N ) @ ( snth_P1712579846118164081m_rule @ S2 @ N ) ) ) ).

% snth_szip
thf(fact_81_snth__szip,axiom,
    ! [S1: stream727092118206550309m_rule,S2: stream_rule,N: nat] :
      ( ( snth_P3371917080224060767e_rule @ ( szip_P553291425819358518e_rule @ S1 @ S2 ) @ N )
      = ( produc4383365014245906419e_rule @ ( snth_P1712579846118164081m_rule @ S1 @ N ) @ ( snth_rule @ S2 @ N ) ) ) ).

% snth_szip
thf(fact_82_snth__szip,axiom,
    ! [S1: stream727092118206550309m_rule,S2: stream727092118206550309m_rule,N: nat] :
      ( ( snth_P8950227884416379311m_rule @ ( szip_P2499414959592755846m_rule @ S1 @ S2 ) @ N )
      = ( produc5927390650430071747m_rule @ ( snth_P1712579846118164081m_rule @ S1 @ N ) @ ( snth_P1712579846118164081m_rule @ S2 @ N ) ) ) ).

% snth_szip
thf(fact_83_snth__szip,axiom,
    ! [S1: stream8299795917829157543ist_fm,S2: stream_rule,N: nat] :
      ( ( snth_P1712579846118164081m_rule @ ( szip_P1977448745965526924m_rule @ S1 @ S2 ) @ N )
      = ( produc491467635432902671m_rule @ ( snth_P2624749845746815685ist_fm @ S1 @ N ) @ ( snth_rule @ S2 @ N ) ) ) ).

% snth_szip
thf(fact_84_snth__szip,axiom,
    ! [S1: stream_list_fm,S2: stream_list_fm,N: nat] :
      ( ( snth_P2624749845746815685ist_fm @ ( szip_list_fm_list_fm @ S1 @ S2 ) @ N )
      = ( produc381145313068854617ist_fm @ ( snth_list_fm @ S1 @ N ) @ ( snth_list_fm @ S2 @ N ) ) ) ).

% snth_szip
thf(fact_85_snth__szip,axiom,
    ! [S1: stream_nat,S2: stream_nat,N: nat] :
      ( ( snth_P9042041890044241317at_nat @ ( szip_nat_nat @ S1 @ S2 ) @ N )
      = ( product_Pair_nat_nat @ ( snth_nat @ S1 @ N ) @ ( snth_nat @ S2 @ N ) ) ) ).

% snth_szip
thf(fact_86_snth__szip,axiom,
    ! [S1: stream6724221391990029191at_nat,S2: stream6724221391990029191at_nat,N: nat] :
      ( ( snth_P9098630209796499211at_nat @ ( szip_P4314639285670189082at_nat @ S1 @ S2 ) @ N )
      = ( produc6161850002892822231at_nat @ ( snth_P9042041890044241317at_nat @ S1 @ N ) @ ( snth_P9042041890044241317at_nat @ S2 @ N ) ) ) ).

% snth_szip
thf(fact_87_snth__szip,axiom,
    ! [S1: stream_rule,S2: stream8299795917829157543ist_fm,N: nat] :
      ( ( snth_P6179250367719980013ist_fm @ ( szip_r8549736553299883584ist_fm @ S1 @ S2 ) @ N )
      = ( produc7063755442767259331ist_fm @ ( snth_rule @ S1 @ N ) @ ( snth_P2624749845746815685ist_fm @ S2 @ N ) ) ) ).

% snth_szip
thf(fact_88_old_Oprod_Oexhaust,axiom,
    ! [Y3: produc164195504107695125m_rule] :
      ~ ! [A4: produc1996495991257130529ist_fm,B3: rule] :
          ( Y3
         != ( produc491467635432902671m_rule @ A4 @ B3 ) ) ).

% old.prod.exhaust
thf(fact_89_old_Oprod_Oexhaust,axiom,
    ! [Y3: produc1996495991257130529ist_fm] :
      ~ ! [A4: list_fm,B3: list_fm] :
          ( Y3
         != ( produc381145313068854617ist_fm @ A4 @ B3 ) ) ).

% old.prod.exhaust
thf(fact_90_old_Oprod_Oexhaust,axiom,
    ! [Y3: product_prod_nat_nat] :
      ~ ! [A4: nat,B3: nat] :
          ( Y3
         != ( product_Pair_nat_nat @ A4 @ B3 ) ) ).

% old.prod.exhaust
thf(fact_91_old_Oprod_Oexhaust,axiom,
    ! [Y3: produc859450856879609959at_nat] :
      ~ ! [A4: product_prod_nat_nat,B3: product_prod_nat_nat] :
          ( Y3
         != ( produc6161850002892822231at_nat @ A4 @ B3 ) ) ).

% old.prod.exhaust
thf(fact_92_old_Oprod_Oexhaust,axiom,
    ! [Y3: produc4630866025709511057ist_fm] :
      ~ ! [A4: rule,B3: produc1996495991257130529ist_fm] :
          ( Y3
         != ( produc7063755442767259331ist_fm @ A4 @ B3 ) ) ).

% old.prod.exhaust
thf(fact_93_surj__pair,axiom,
    ! [P: produc164195504107695125m_rule] :
    ? [X: produc1996495991257130529ist_fm,Y: rule] :
      ( P
      = ( produc491467635432902671m_rule @ X @ Y ) ) ).

% surj_pair
thf(fact_94_surj__pair,axiom,
    ! [P: produc1996495991257130529ist_fm] :
    ? [X: list_fm,Y: list_fm] :
      ( P
      = ( produc381145313068854617ist_fm @ X @ Y ) ) ).

% surj_pair
thf(fact_95_surj__pair,axiom,
    ! [P: product_prod_nat_nat] :
    ? [X: nat,Y: nat] :
      ( P
      = ( product_Pair_nat_nat @ X @ Y ) ) ).

% surj_pair
thf(fact_96_surj__pair,axiom,
    ! [P: produc859450856879609959at_nat] :
    ? [X: product_prod_nat_nat,Y: product_prod_nat_nat] :
      ( P
      = ( produc6161850002892822231at_nat @ X @ Y ) ) ).

% surj_pair
thf(fact_97_surj__pair,axiom,
    ! [P: produc4630866025709511057ist_fm] :
    ? [X: rule,Y: produc1996495991257130529ist_fm] :
      ( P
      = ( produc7063755442767259331ist_fm @ X @ Y ) ) ).

% surj_pair
thf(fact_98_prod__cases,axiom,
    ! [P2: produc164195504107695125m_rule > $o,P: produc164195504107695125m_rule] :
      ( ! [A4: produc1996495991257130529ist_fm,B3: rule] : ( P2 @ ( produc491467635432902671m_rule @ A4 @ B3 ) )
     => ( P2 @ P ) ) ).

% prod_cases
thf(fact_99_prod__cases,axiom,
    ! [P2: produc1996495991257130529ist_fm > $o,P: produc1996495991257130529ist_fm] :
      ( ! [A4: list_fm,B3: list_fm] : ( P2 @ ( produc381145313068854617ist_fm @ A4 @ B3 ) )
     => ( P2 @ P ) ) ).

% prod_cases
thf(fact_100_prod__cases,axiom,
    ! [P2: product_prod_nat_nat > $o,P: product_prod_nat_nat] :
      ( ! [A4: nat,B3: nat] : ( P2 @ ( product_Pair_nat_nat @ A4 @ B3 ) )
     => ( P2 @ P ) ) ).

% prod_cases
thf(fact_101_prod__cases,axiom,
    ! [P2: produc859450856879609959at_nat > $o,P: produc859450856879609959at_nat] :
      ( ! [A4: product_prod_nat_nat,B3: product_prod_nat_nat] : ( P2 @ ( produc6161850002892822231at_nat @ A4 @ B3 ) )
     => ( P2 @ P ) ) ).

% prod_cases
thf(fact_102_prod__cases,axiom,
    ! [P2: produc4630866025709511057ist_fm > $o,P: produc4630866025709511057ist_fm] :
      ( ! [A4: rule,B3: produc1996495991257130529ist_fm] : ( P2 @ ( produc7063755442767259331ist_fm @ A4 @ B3 ) )
     => ( P2 @ P ) ) ).

% prod_cases
thf(fact_103_Pair__inject,axiom,
    ! [A: produc1996495991257130529ist_fm,B: rule,A3: produc1996495991257130529ist_fm,B2: rule] :
      ( ( ( produc491467635432902671m_rule @ A @ B )
        = ( produc491467635432902671m_rule @ A3 @ B2 ) )
     => ~ ( ( A = A3 )
         => ( B != B2 ) ) ) ).

% Pair_inject
thf(fact_104_Pair__inject,axiom,
    ! [A: list_fm,B: list_fm,A3: list_fm,B2: list_fm] :
      ( ( ( produc381145313068854617ist_fm @ A @ B )
        = ( produc381145313068854617ist_fm @ A3 @ B2 ) )
     => ~ ( ( A = A3 )
         => ( B != B2 ) ) ) ).

% Pair_inject
thf(fact_105_Pair__inject,axiom,
    ! [A: nat,B: nat,A3: nat,B2: nat] :
      ( ( ( product_Pair_nat_nat @ A @ B )
        = ( product_Pair_nat_nat @ A3 @ B2 ) )
     => ~ ( ( A = A3 )
         => ( B != B2 ) ) ) ).

% Pair_inject
thf(fact_106_Pair__inject,axiom,
    ! [A: product_prod_nat_nat,B: product_prod_nat_nat,A3: product_prod_nat_nat,B2: product_prod_nat_nat] :
      ( ( ( produc6161850002892822231at_nat @ A @ B )
        = ( produc6161850002892822231at_nat @ A3 @ B2 ) )
     => ~ ( ( A = A3 )
         => ( B != B2 ) ) ) ).

% Pair_inject
thf(fact_107_Pair__inject,axiom,
    ! [A: rule,B: produc1996495991257130529ist_fm,A3: rule,B2: produc1996495991257130529ist_fm] :
      ( ( ( produc7063755442767259331ist_fm @ A @ B )
        = ( produc7063755442767259331ist_fm @ A3 @ B2 ) )
     => ~ ( ( A = A3 )
         => ( B != B2 ) ) ) ).

% Pair_inject
thf(fact_108_prod__cases3,axiom,
    ! [Y3: produc859450856879609959at_nat] :
      ~ ! [A4: product_prod_nat_nat,B3: nat,C: nat] :
          ( Y3
         != ( produc6161850002892822231at_nat @ A4 @ ( product_Pair_nat_nat @ B3 @ C ) ) ) ).

% prod_cases3
thf(fact_109_prod__cases3,axiom,
    ! [Y3: produc4630866025709511057ist_fm] :
      ~ ! [A4: rule,B3: list_fm,C: list_fm] :
          ( Y3
         != ( produc7063755442767259331ist_fm @ A4 @ ( produc381145313068854617ist_fm @ B3 @ C ) ) ) ).

% prod_cases3
thf(fact_110_mem__Collect__eq,axiom,
    ! [A: fm,P2: fm > $o] :
      ( ( member_fm @ A @ ( collect_fm @ P2 ) )
      = ( P2 @ A ) ) ).

% mem_Collect_eq
thf(fact_111_mem__Collect__eq,axiom,
    ! [A: rule,P2: rule > $o] :
      ( ( member_rule @ A @ ( collect_rule @ P2 ) )
      = ( P2 @ A ) ) ).

% mem_Collect_eq
thf(fact_112_mem__Collect__eq,axiom,
    ! [A: produc1996495991257130529ist_fm,P2: produc1996495991257130529ist_fm > $o] :
      ( ( member8102475879199740618ist_fm @ A @ ( collec7231575776666803724ist_fm @ P2 ) )
      = ( P2 @ A ) ) ).

% mem_Collect_eq
thf(fact_113_mem__Collect__eq,axiom,
    ! [A: produc859450856879609959at_nat,P2: produc859450856879609959at_nat > $o] :
      ( ( member8206827879206165904at_nat @ A @ ( collec7088162979684241874at_nat @ P2 ) )
      = ( P2 @ A ) ) ).

% mem_Collect_eq
thf(fact_114_Collect__mem__eq,axiom,
    ! [A5: set_fm] :
      ( ( collect_fm
        @ ^ [X6: fm] : ( member_fm @ X6 @ A5 ) )
      = A5 ) ).

% Collect_mem_eq
thf(fact_115_Collect__mem__eq,axiom,
    ! [A5: set_rule] :
      ( ( collect_rule
        @ ^ [X6: rule] : ( member_rule @ X6 @ A5 ) )
      = A5 ) ).

% Collect_mem_eq
thf(fact_116_Collect__mem__eq,axiom,
    ! [A5: set_Pr7058068377845519745ist_fm] :
      ( ( collec7231575776666803724ist_fm
        @ ^ [X6: produc1996495991257130529ist_fm] : ( member8102475879199740618ist_fm @ X6 @ A5 ) )
      = A5 ) ).

% Collect_mem_eq
thf(fact_117_Collect__mem__eq,axiom,
    ! [A5: set_Pr8693737435421807431at_nat] :
      ( ( collec7088162979684241874at_nat
        @ ^ [X6: produc859450856879609959at_nat] : ( member8206827879206165904at_nat @ X6 @ A5 ) )
      = A5 ) ).

% Collect_mem_eq
thf(fact_118_prod__induct3,axiom,
    ! [P2: produc859450856879609959at_nat > $o,X3: produc859450856879609959at_nat] :
      ( ! [A4: product_prod_nat_nat,B3: nat,C: nat] : ( P2 @ ( produc6161850002892822231at_nat @ A4 @ ( product_Pair_nat_nat @ B3 @ C ) ) )
     => ( P2 @ X3 ) ) ).

% prod_induct3
thf(fact_119_prod__induct3,axiom,
    ! [P2: produc4630866025709511057ist_fm > $o,X3: produc4630866025709511057ist_fm] :
      ( ! [A4: rule,B3: list_fm,C: list_fm] : ( P2 @ ( produc7063755442767259331ist_fm @ A4 @ ( produc381145313068854617ist_fm @ B3 @ C ) ) )
     => ( P2 @ X3 ) ) ).

% prod_induct3
thf(fact_120_RuleSystem__Defs_OSaturated_Ocong,axiom,
    abstra8647779336736900908ist_fm = abstra8647779336736900908ist_fm ).

% RuleSystem_Defs.Saturated.cong
thf(fact_121_RuleSystem__Defs_OpickEff_Ocong,axiom,
    abstra458490714785879965ist_fm = abstra458490714785879965ist_fm ).

% RuleSystem_Defs.pickEff.cong
thf(fact_122_RuleSystem__Defs_Oepath_Ocong,axiom,
    abstra5415202973855822231ist_fm = abstra5415202973855822231ist_fm ).

% RuleSystem_Defs.epath.cong
thf(fact_123_RuleSystem__Defs_OpickEff,axiom,
    ! [Eff2: rule > produc1996495991257130529ist_fm > fset_P661503646757059847ist_fm > $o,R2: rule,S3: produc1996495991257130529ist_fm] :
      ( ( abstra3055540779341473792ist_fm @ Eff2 @ R2 @ S3 )
     => ( Eff2 @ R2 @ S3 @ ( abstra458490714785879965ist_fm @ Eff2 @ R2 @ S3 ) ) ) ).

% RuleSystem_Defs.pickEff
thf(fact_124_fst__conv,axiom,
    ! [X1: nat,X2: nat] :
      ( ( product_fst_nat_nat @ ( product_Pair_nat_nat @ X1 @ X2 ) )
      = X1 ) ).

% fst_conv
thf(fact_125_fst__conv,axiom,
    ! [X1: product_prod_nat_nat,X2: product_prod_nat_nat] :
      ( ( produc3213797794245857475at_nat @ ( produc6161850002892822231at_nat @ X1 @ X2 ) )
      = X1 ) ).

% fst_conv
thf(fact_126_fst__conv,axiom,
    ! [X1: rule,X2: produc1996495991257130529ist_fm] :
      ( ( produc4228417144610596823ist_fm @ ( produc7063755442767259331ist_fm @ X1 @ X2 ) )
      = X1 ) ).

% fst_conv
thf(fact_127_fst__conv,axiom,
    ! [X1: produc1996495991257130529ist_fm,X2: rule] :
      ( ( produc6879501374131015971m_rule @ ( produc491467635432902671m_rule @ X1 @ X2 ) )
      = X1 ) ).

% fst_conv
thf(fact_128_fst__conv,axiom,
    ! [X1: list_fm,X2: list_fm] :
      ( ( produc1501393135466168645ist_fm @ ( produc381145313068854617ist_fm @ X1 @ X2 ) )
      = X1 ) ).

% fst_conv
thf(fact_129_fst__eqD,axiom,
    ! [X3: nat,Y3: nat,A: nat] :
      ( ( ( product_fst_nat_nat @ ( product_Pair_nat_nat @ X3 @ Y3 ) )
        = A )
     => ( X3 = A ) ) ).

% fst_eqD
thf(fact_130_fst__eqD,axiom,
    ! [X3: product_prod_nat_nat,Y3: product_prod_nat_nat,A: product_prod_nat_nat] :
      ( ( ( produc3213797794245857475at_nat @ ( produc6161850002892822231at_nat @ X3 @ Y3 ) )
        = A )
     => ( X3 = A ) ) ).

% fst_eqD
thf(fact_131_fst__eqD,axiom,
    ! [X3: rule,Y3: produc1996495991257130529ist_fm,A: rule] :
      ( ( ( produc4228417144610596823ist_fm @ ( produc7063755442767259331ist_fm @ X3 @ Y3 ) )
        = A )
     => ( X3 = A ) ) ).

% fst_eqD
thf(fact_132_fst__eqD,axiom,
    ! [X3: produc1996495991257130529ist_fm,Y3: rule,A: produc1996495991257130529ist_fm] :
      ( ( ( produc6879501374131015971m_rule @ ( produc491467635432902671m_rule @ X3 @ Y3 ) )
        = A )
     => ( X3 = A ) ) ).

% fst_eqD
thf(fact_133_fst__eqD,axiom,
    ! [X3: list_fm,Y3: list_fm,A: list_fm] :
      ( ( ( produc1501393135466168645ist_fm @ ( produc381145313068854617ist_fm @ X3 @ Y3 ) )
        = A )
     => ( X3 = A ) ) ).

% fst_eqD
thf(fact_134_snd__eqD,axiom,
    ! [X3: nat,Y3: nat,A: nat] :
      ( ( ( product_snd_nat_nat @ ( product_Pair_nat_nat @ X3 @ Y3 ) )
        = A )
     => ( Y3 = A ) ) ).

% snd_eqD
thf(fact_135_snd__eqD,axiom,
    ! [X3: product_prod_nat_nat,Y3: product_prod_nat_nat,A: product_prod_nat_nat] :
      ( ( ( produc6408287024330202629at_nat @ ( produc6161850002892822231at_nat @ X3 @ Y3 ) )
        = A )
     => ( Y3 = A ) ) ).

% snd_eqD
thf(fact_136_snd__eqD,axiom,
    ! [X3: rule,Y3: produc1996495991257130529ist_fm,A: produc1996495991257130529ist_fm] :
      ( ( ( produc4514744107061996309ist_fm @ ( produc7063755442767259331ist_fm @ X3 @ Y3 ) )
        = A )
     => ( Y3 = A ) ) ).

% snd_eqD
thf(fact_137_snd__eqD,axiom,
    ! [X3: produc1996495991257130529ist_fm,Y3: rule,A: rule] :
      ( ( ( produc7165828336582415457m_rule @ ( produc491467635432902671m_rule @ X3 @ Y3 ) )
        = A )
     => ( Y3 = A ) ) ).

% snd_eqD
thf(fact_138_snd__eqD,axiom,
    ! [X3: list_fm,Y3: list_fm,A: list_fm] :
      ( ( ( produc4588648349897876871ist_fm @ ( produc381145313068854617ist_fm @ X3 @ Y3 ) )
        = A )
     => ( Y3 = A ) ) ).

% snd_eqD
thf(fact_139_snd__conv,axiom,
    ! [X1: nat,X2: nat] :
      ( ( product_snd_nat_nat @ ( product_Pair_nat_nat @ X1 @ X2 ) )
      = X2 ) ).

% snd_conv
thf(fact_140_snd__conv,axiom,
    ! [X1: product_prod_nat_nat,X2: product_prod_nat_nat] :
      ( ( produc6408287024330202629at_nat @ ( produc6161850002892822231at_nat @ X1 @ X2 ) )
      = X2 ) ).

% snd_conv
thf(fact_141_snd__conv,axiom,
    ! [X1: rule,X2: produc1996495991257130529ist_fm] :
      ( ( produc4514744107061996309ist_fm @ ( produc7063755442767259331ist_fm @ X1 @ X2 ) )
      = X2 ) ).

% snd_conv
thf(fact_142_snd__conv,axiom,
    ! [X1: produc1996495991257130529ist_fm,X2: rule] :
      ( ( produc7165828336582415457m_rule @ ( produc491467635432902671m_rule @ X1 @ X2 ) )
      = X2 ) ).

% snd_conv
thf(fact_143_snd__conv,axiom,
    ! [X1: list_fm,X2: list_fm] :
      ( ( produc4588648349897876871ist_fm @ ( produc381145313068854617ist_fm @ X1 @ X2 ) )
      = X2 ) ).

% snd_conv
thf(fact_144_prod_Oswap__def,axiom,
    ( produc9213896635423732003ist_fm
    = ( ^ [P3: produc4630866025709511057ist_fm] : ( produc491467635432902671m_rule @ ( produc4514744107061996309ist_fm @ P3 ) @ ( produc4228417144610596823ist_fm @ P3 ) ) ) ) ).

% prod.swap_def
thf(fact_145_prod_Oswap__def,axiom,
    ( product_swap_nat_nat
    = ( ^ [P3: product_prod_nat_nat] : ( product_Pair_nat_nat @ ( product_snd_nat_nat @ P3 ) @ ( product_fst_nat_nat @ P3 ) ) ) ) ).

% prod.swap_def
thf(fact_146_prod_Oswap__def,axiom,
    ( produc7225117575323628663at_nat
    = ( ^ [P3: produc859450856879609959at_nat] : ( produc6161850002892822231at_nat @ ( produc6408287024330202629at_nat @ P3 ) @ ( produc3213797794245857475at_nat @ P3 ) ) ) ) ).

% prod.swap_def
thf(fact_147_prod_Oswap__def,axiom,
    ( produc2641608828089375343m_rule
    = ( ^ [P3: produc164195504107695125m_rule] : ( produc7063755442767259331ist_fm @ ( produc7165828336582415457m_rule @ P3 ) @ ( produc6879501374131015971m_rule @ P3 ) ) ) ) ).

% prod.swap_def
thf(fact_148_prod_Oswap__def,axiom,
    ( produc7936585717479832313ist_fm
    = ( ^ [P3: produc1996495991257130529ist_fm] : ( produc381145313068854617ist_fm @ ( produc4588648349897876871ist_fm @ P3 ) @ ( produc1501393135466168645ist_fm @ P3 ) ) ) ) ).

% prod.swap_def
thf(fact_149_epath__sdrop,axiom,
    ! [Steps: stream727092118206550309m_rule,N: nat] :
      ( ( abstra5415202973855822231ist_fm
        @ ^ [R: rule,S: produc1996495991257130529ist_fm,Ss: fset_P661503646757059847ist_fm] :
            ( ( eff @ R @ S )
            = ( some_f2162114139285431020ist_fm @ Ss ) )
        @ rules
        @ Steps )
     => ( abstra5415202973855822231ist_fm
        @ ^ [R: rule,S: produc1996495991257130529ist_fm,Ss: fset_P661503646757059847ist_fm] :
            ( ( eff @ R @ S )
            = ( some_f2162114139285431020ist_fm @ Ss ) )
        @ rules
        @ ( sdrop_7224736112439592940m_rule @ N @ Steps ) ) ) ).

% epath_sdrop
thf(fact_150_prod_Oexhaust__sel,axiom,
    ! [Prod: product_prod_nat_nat] :
      ( Prod
      = ( product_Pair_nat_nat @ ( product_fst_nat_nat @ Prod ) @ ( product_snd_nat_nat @ Prod ) ) ) ).

% prod.exhaust_sel
thf(fact_151_prod_Oexhaust__sel,axiom,
    ! [Prod: produc859450856879609959at_nat] :
      ( Prod
      = ( produc6161850002892822231at_nat @ ( produc3213797794245857475at_nat @ Prod ) @ ( produc6408287024330202629at_nat @ Prod ) ) ) ).

% prod.exhaust_sel
thf(fact_152_prod_Oexhaust__sel,axiom,
    ! [Prod: produc4630866025709511057ist_fm] :
      ( Prod
      = ( produc7063755442767259331ist_fm @ ( produc4228417144610596823ist_fm @ Prod ) @ ( produc4514744107061996309ist_fm @ Prod ) ) ) ).

% prod.exhaust_sel
thf(fact_153_prod_Oexhaust__sel,axiom,
    ! [Prod: produc164195504107695125m_rule] :
      ( Prod
      = ( produc491467635432902671m_rule @ ( produc6879501374131015971m_rule @ Prod ) @ ( produc7165828336582415457m_rule @ Prod ) ) ) ).

% prod.exhaust_sel
thf(fact_154_prod_Oexhaust__sel,axiom,
    ! [Prod: produc1996495991257130529ist_fm] :
      ( Prod
      = ( produc381145313068854617ist_fm @ ( produc1501393135466168645ist_fm @ Prod ) @ ( produc4588648349897876871ist_fm @ Prod ) ) ) ).

% prod.exhaust_sel
thf(fact_155_exI__realizer,axiom,
    ! [P2: nat > nat > $o,Y3: nat,X3: nat] :
      ( ( P2 @ Y3 @ X3 )
     => ( P2 @ ( product_snd_nat_nat @ ( product_Pair_nat_nat @ X3 @ Y3 ) ) @ ( product_fst_nat_nat @ ( product_Pair_nat_nat @ X3 @ Y3 ) ) ) ) ).

% exI_realizer
thf(fact_156_exI__realizer,axiom,
    ! [P2: product_prod_nat_nat > product_prod_nat_nat > $o,Y3: product_prod_nat_nat,X3: product_prod_nat_nat] :
      ( ( P2 @ Y3 @ X3 )
     => ( P2 @ ( produc6408287024330202629at_nat @ ( produc6161850002892822231at_nat @ X3 @ Y3 ) ) @ ( produc3213797794245857475at_nat @ ( produc6161850002892822231at_nat @ X3 @ Y3 ) ) ) ) ).

% exI_realizer
thf(fact_157_exI__realizer,axiom,
    ! [P2: produc1996495991257130529ist_fm > rule > $o,Y3: produc1996495991257130529ist_fm,X3: rule] :
      ( ( P2 @ Y3 @ X3 )
     => ( P2 @ ( produc4514744107061996309ist_fm @ ( produc7063755442767259331ist_fm @ X3 @ Y3 ) ) @ ( produc4228417144610596823ist_fm @ ( produc7063755442767259331ist_fm @ X3 @ Y3 ) ) ) ) ).

% exI_realizer
thf(fact_158_exI__realizer,axiom,
    ! [P2: rule > produc1996495991257130529ist_fm > $o,Y3: rule,X3: produc1996495991257130529ist_fm] :
      ( ( P2 @ Y3 @ X3 )
     => ( P2 @ ( produc7165828336582415457m_rule @ ( produc491467635432902671m_rule @ X3 @ Y3 ) ) @ ( produc6879501374131015971m_rule @ ( produc491467635432902671m_rule @ X3 @ Y3 ) ) ) ) ).

% exI_realizer
thf(fact_159_exI__realizer,axiom,
    ! [P2: list_fm > list_fm > $o,Y3: list_fm,X3: list_fm] :
      ( ( P2 @ Y3 @ X3 )
     => ( P2 @ ( produc4588648349897876871ist_fm @ ( produc381145313068854617ist_fm @ X3 @ Y3 ) ) @ ( produc1501393135466168645ist_fm @ ( produc381145313068854617ist_fm @ X3 @ Y3 ) ) ) ) ).

% exI_realizer
thf(fact_160_conjI__realizer,axiom,
    ! [P2: nat > $o,P: nat,Q2: nat > $o,Q: nat] :
      ( ( P2 @ P )
     => ( ( Q2 @ Q )
       => ( ( P2 @ ( product_fst_nat_nat @ ( product_Pair_nat_nat @ P @ Q ) ) )
          & ( Q2 @ ( product_snd_nat_nat @ ( product_Pair_nat_nat @ P @ Q ) ) ) ) ) ) ).

% conjI_realizer
thf(fact_161_conjI__realizer,axiom,
    ! [P2: product_prod_nat_nat > $o,P: product_prod_nat_nat,Q2: product_prod_nat_nat > $o,Q: product_prod_nat_nat] :
      ( ( P2 @ P )
     => ( ( Q2 @ Q )
       => ( ( P2 @ ( produc3213797794245857475at_nat @ ( produc6161850002892822231at_nat @ P @ Q ) ) )
          & ( Q2 @ ( produc6408287024330202629at_nat @ ( produc6161850002892822231at_nat @ P @ Q ) ) ) ) ) ) ).

% conjI_realizer
thf(fact_162_conjI__realizer,axiom,
    ! [P2: rule > $o,P: rule,Q2: produc1996495991257130529ist_fm > $o,Q: produc1996495991257130529ist_fm] :
      ( ( P2 @ P )
     => ( ( Q2 @ Q )
       => ( ( P2 @ ( produc4228417144610596823ist_fm @ ( produc7063755442767259331ist_fm @ P @ Q ) ) )
          & ( Q2 @ ( produc4514744107061996309ist_fm @ ( produc7063755442767259331ist_fm @ P @ Q ) ) ) ) ) ) ).

% conjI_realizer
thf(fact_163_conjI__realizer,axiom,
    ! [P2: produc1996495991257130529ist_fm > $o,P: produc1996495991257130529ist_fm,Q2: rule > $o,Q: rule] :
      ( ( P2 @ P )
     => ( ( Q2 @ Q )
       => ( ( P2 @ ( produc6879501374131015971m_rule @ ( produc491467635432902671m_rule @ P @ Q ) ) )
          & ( Q2 @ ( produc7165828336582415457m_rule @ ( produc491467635432902671m_rule @ P @ Q ) ) ) ) ) ) ).

% conjI_realizer
thf(fact_164_conjI__realizer,axiom,
    ! [P2: list_fm > $o,P: list_fm,Q2: list_fm > $o,Q: list_fm] :
      ( ( P2 @ P )
     => ( ( Q2 @ Q )
       => ( ( P2 @ ( produc1501393135466168645ist_fm @ ( produc381145313068854617ist_fm @ P @ Q ) ) )
          & ( Q2 @ ( produc4588648349897876871ist_fm @ ( produc381145313068854617ist_fm @ P @ Q ) ) ) ) ) ) ).

% conjI_realizer
thf(fact_165_surjective__pairing,axiom,
    ! [T2: product_prod_nat_nat] :
      ( T2
      = ( product_Pair_nat_nat @ ( product_fst_nat_nat @ T2 ) @ ( product_snd_nat_nat @ T2 ) ) ) ).

% surjective_pairing
thf(fact_166_surjective__pairing,axiom,
    ! [T2: produc859450856879609959at_nat] :
      ( T2
      = ( produc6161850002892822231at_nat @ ( produc3213797794245857475at_nat @ T2 ) @ ( produc6408287024330202629at_nat @ T2 ) ) ) ).

% surjective_pairing
thf(fact_167_surjective__pairing,axiom,
    ! [T2: produc4630866025709511057ist_fm] :
      ( T2
      = ( produc7063755442767259331ist_fm @ ( produc4228417144610596823ist_fm @ T2 ) @ ( produc4514744107061996309ist_fm @ T2 ) ) ) ).

% surjective_pairing
thf(fact_168_surjective__pairing,axiom,
    ! [T2: produc164195504107695125m_rule] :
      ( T2
      = ( produc491467635432902671m_rule @ ( produc6879501374131015971m_rule @ T2 ) @ ( produc7165828336582415457m_rule @ T2 ) ) ) ).

% surjective_pairing
thf(fact_169_surjective__pairing,axiom,
    ! [T2: produc1996495991257130529ist_fm] :
      ( T2
      = ( produc381145313068854617ist_fm @ ( produc1501393135466168645ist_fm @ T2 ) @ ( produc4588648349897876871ist_fm @ T2 ) ) ) ).

% surjective_pairing
thf(fact_170_Saturated__sdrop,axiom,
    ! [Steps: stream727092118206550309m_rule,N: nat] :
      ( ( abstra8647779336736900908ist_fm
        @ ^ [R: rule,S: produc1996495991257130529ist_fm,Ss: fset_P661503646757059847ist_fm] :
            ( ( eff @ R @ S )
            = ( some_f2162114139285431020ist_fm @ Ss ) )
        @ rules
        @ Steps )
     => ( abstra8647779336736900908ist_fm
        @ ^ [R: rule,S: produc1996495991257130529ist_fm,Ss: fset_P661503646757059847ist_fm] :
            ( ( eff @ R @ S )
            = ( some_f2162114139285431020ist_fm @ Ss ) )
        @ rules
        @ ( sdrop_7224736112439592940m_rule @ N @ Steps ) ) ) ).

% Saturated_sdrop
thf(fact_171_enabled__ex__taken,axiom,
    ! [Steps: stream727092118206550309m_rule,R2: rule] :
      ( ( abstra5415202973855822231ist_fm
        @ ^ [R: rule,S: produc1996495991257130529ist_fm,Ss: fset_P661503646757059847ist_fm] :
            ( ( eff @ R @ S )
            = ( some_f2162114139285431020ist_fm @ Ss ) )
        @ rules
        @ Steps )
     => ( ( abstra8647779336736900908ist_fm
          @ ^ [R: rule,S: produc1996495991257130529ist_fm,Ss: fset_P661503646757059847ist_fm] :
              ( ( eff @ R @ S )
              = ( some_f2162114139285431020ist_fm @ Ss ) )
          @ rules
          @ Steps )
       => ( ( abstra3055540779341473792ist_fm
            @ ^ [R: rule,S: produc1996495991257130529ist_fm,Ss: fset_P661503646757059847ist_fm] :
                ( ( eff @ R @ S )
                = ( some_f2162114139285431020ist_fm @ Ss ) )
            @ R2
            @ ( produc6879501374131015971m_rule @ ( shd_Pr7235097944458474089m_rule @ Steps ) ) )
         => ? [K: nat] :
              ( ( produc7165828336582415457m_rule @ ( shd_Pr7235097944458474089m_rule @ ( sdrop_7224736112439592940m_rule @ K @ Steps ) ) )
              = R2 ) ) ) ) ).

% enabled_ex_taken
thf(fact_172_treeB__snth,axiom,
    ! [P: fm,Steps: stream727092118206550309m_rule] :
      ( ( member_fm @ P @ ( treeB @ Steps ) )
     => ? [N2: nat] : ( member_fm @ P @ ( set_fm2 @ ( produc4588648349897876871ist_fm @ ( produc6879501374131015971m_rule @ ( shd_Pr7235097944458474089m_rule @ ( sdrop_7224736112439592940m_rule @ N2 @ Steps ) ) ) ) ) ) ) ).

% treeB_snth
thf(fact_173_epath__Pre__sdrop_I1_J,axiom,
    ! [Steps: stream727092118206550309m_rule,P2: nat,Ts: list_tm,M: nat] :
      ( ( abstra5415202973855822231ist_fm
        @ ^ [R: rule,S: produc1996495991257130529ist_fm,Ss: fset_P661503646757059847ist_fm] :
            ( ( eff @ R @ S )
            = ( some_f2162114139285431020ist_fm @ Ss ) )
        @ rules
        @ Steps )
     => ( ( member_fm @ ( pre @ P2 @ Ts ) @ ( set_fm2 @ ( produc1501393135466168645ist_fm @ ( produc6879501374131015971m_rule @ ( shd_Pr7235097944458474089m_rule @ Steps ) ) ) ) )
       => ( member_fm @ ( pre @ P2 @ Ts ) @ ( set_fm2 @ ( produc1501393135466168645ist_fm @ ( produc6879501374131015971m_rule @ ( shd_Pr7235097944458474089m_rule @ ( sdrop_7224736112439592940m_rule @ M @ Steps ) ) ) ) ) ) ) ) ).

% epath_Pre_sdrop(1)
thf(fact_174_epath__Pre__sdrop_I2_J,axiom,
    ! [Steps: stream727092118206550309m_rule,P2: nat,Ts: list_tm,M: nat] :
      ( ( abstra5415202973855822231ist_fm
        @ ^ [R: rule,S: produc1996495991257130529ist_fm,Ss: fset_P661503646757059847ist_fm] :
            ( ( eff @ R @ S )
            = ( some_f2162114139285431020ist_fm @ Ss ) )
        @ rules
        @ Steps )
     => ( ( member_fm @ ( pre @ P2 @ Ts ) @ ( set_fm2 @ ( produc4588648349897876871ist_fm @ ( produc6879501374131015971m_rule @ ( shd_Pr7235097944458474089m_rule @ Steps ) ) ) ) )
       => ( member_fm @ ( pre @ P2 @ Ts ) @ ( set_fm2 @ ( produc4588648349897876871ist_fm @ ( produc6879501374131015971m_rule @ ( shd_Pr7235097944458474089m_rule @ ( sdrop_7224736112439592940m_rule @ M @ Steps ) ) ) ) ) ) ) ) ).

% epath_Pre_sdrop(2)
thf(fact_175_BNF__Greatest__Fixpoint_Osubst__Pair,axiom,
    ! [P2: nat > nat > $o,X3: nat,Y3: nat,A: product_prod_nat_nat] :
      ( ( P2 @ X3 @ Y3 )
     => ( ( A
          = ( product_Pair_nat_nat @ X3 @ Y3 ) )
       => ( P2 @ ( product_fst_nat_nat @ A ) @ ( product_snd_nat_nat @ A ) ) ) ) ).

% BNF_Greatest_Fixpoint.subst_Pair
thf(fact_176_BNF__Greatest__Fixpoint_Osubst__Pair,axiom,
    ! [P2: product_prod_nat_nat > product_prod_nat_nat > $o,X3: product_prod_nat_nat,Y3: product_prod_nat_nat,A: produc859450856879609959at_nat] :
      ( ( P2 @ X3 @ Y3 )
     => ( ( A
          = ( produc6161850002892822231at_nat @ X3 @ Y3 ) )
       => ( P2 @ ( produc3213797794245857475at_nat @ A ) @ ( produc6408287024330202629at_nat @ A ) ) ) ) ).

% BNF_Greatest_Fixpoint.subst_Pair
thf(fact_177_BNF__Greatest__Fixpoint_Osubst__Pair,axiom,
    ! [P2: rule > produc1996495991257130529ist_fm > $o,X3: rule,Y3: produc1996495991257130529ist_fm,A: produc4630866025709511057ist_fm] :
      ( ( P2 @ X3 @ Y3 )
     => ( ( A
          = ( produc7063755442767259331ist_fm @ X3 @ Y3 ) )
       => ( P2 @ ( produc4228417144610596823ist_fm @ A ) @ ( produc4514744107061996309ist_fm @ A ) ) ) ) ).

% BNF_Greatest_Fixpoint.subst_Pair
thf(fact_178_BNF__Greatest__Fixpoint_Osubst__Pair,axiom,
    ! [P2: produc1996495991257130529ist_fm > rule > $o,X3: produc1996495991257130529ist_fm,Y3: rule,A: produc164195504107695125m_rule] :
      ( ( P2 @ X3 @ Y3 )
     => ( ( A
          = ( produc491467635432902671m_rule @ X3 @ Y3 ) )
       => ( P2 @ ( produc6879501374131015971m_rule @ A ) @ ( produc7165828336582415457m_rule @ A ) ) ) ) ).

% BNF_Greatest_Fixpoint.subst_Pair
thf(fact_179_BNF__Greatest__Fixpoint_Osubst__Pair,axiom,
    ! [P2: list_fm > list_fm > $o,X3: list_fm,Y3: list_fm,A: produc1996495991257130529ist_fm] :
      ( ( P2 @ X3 @ Y3 )
     => ( ( A
          = ( produc381145313068854617ist_fm @ X3 @ Y3 ) )
       => ( P2 @ ( produc1501393135466168645ist_fm @ A ) @ ( produc4588648349897876871ist_fm @ A ) ) ) ) ).

% BNF_Greatest_Fixpoint.subst_Pair
thf(fact_180_sndI,axiom,
    ! [X3: product_prod_nat_nat,Y3: nat,Z2: nat] :
      ( ( X3
        = ( product_Pair_nat_nat @ Y3 @ Z2 ) )
     => ( ( product_snd_nat_nat @ X3 )
        = Z2 ) ) ).

% sndI
thf(fact_181_sndI,axiom,
    ! [X3: produc859450856879609959at_nat,Y3: product_prod_nat_nat,Z2: product_prod_nat_nat] :
      ( ( X3
        = ( produc6161850002892822231at_nat @ Y3 @ Z2 ) )
     => ( ( produc6408287024330202629at_nat @ X3 )
        = Z2 ) ) ).

% sndI
thf(fact_182_sndI,axiom,
    ! [X3: produc4630866025709511057ist_fm,Y3: rule,Z2: produc1996495991257130529ist_fm] :
      ( ( X3
        = ( produc7063755442767259331ist_fm @ Y3 @ Z2 ) )
     => ( ( produc4514744107061996309ist_fm @ X3 )
        = Z2 ) ) ).

% sndI
thf(fact_183_sndI,axiom,
    ! [X3: produc164195504107695125m_rule,Y3: produc1996495991257130529ist_fm,Z2: rule] :
      ( ( X3
        = ( produc491467635432902671m_rule @ Y3 @ Z2 ) )
     => ( ( produc7165828336582415457m_rule @ X3 )
        = Z2 ) ) ).

% sndI
thf(fact_184_sndI,axiom,
    ! [X3: produc1996495991257130529ist_fm,Y3: list_fm,Z2: list_fm] :
      ( ( X3
        = ( produc381145313068854617ist_fm @ Y3 @ Z2 ) )
     => ( ( produc4588648349897876871ist_fm @ X3 )
        = Z2 ) ) ).

% sndI
thf(fact_185_eq__snd__iff,axiom,
    ! [B: nat,P: product_prod_nat_nat] :
      ( ( B
        = ( product_snd_nat_nat @ P ) )
      = ( ? [A2: nat] :
            ( P
            = ( product_Pair_nat_nat @ A2 @ B ) ) ) ) ).

% eq_snd_iff
thf(fact_186_eq__snd__iff,axiom,
    ! [B: product_prod_nat_nat,P: produc859450856879609959at_nat] :
      ( ( B
        = ( produc6408287024330202629at_nat @ P ) )
      = ( ? [A2: product_prod_nat_nat] :
            ( P
            = ( produc6161850002892822231at_nat @ A2 @ B ) ) ) ) ).

% eq_snd_iff
thf(fact_187_eq__snd__iff,axiom,
    ! [B: produc1996495991257130529ist_fm,P: produc4630866025709511057ist_fm] :
      ( ( B
        = ( produc4514744107061996309ist_fm @ P ) )
      = ( ? [A2: rule] :
            ( P
            = ( produc7063755442767259331ist_fm @ A2 @ B ) ) ) ) ).

% eq_snd_iff
thf(fact_188_eq__snd__iff,axiom,
    ! [B: rule,P: produc164195504107695125m_rule] :
      ( ( B
        = ( produc7165828336582415457m_rule @ P ) )
      = ( ? [A2: produc1996495991257130529ist_fm] :
            ( P
            = ( produc491467635432902671m_rule @ A2 @ B ) ) ) ) ).

% eq_snd_iff
thf(fact_189_eq__snd__iff,axiom,
    ! [B: list_fm,P: produc1996495991257130529ist_fm] :
      ( ( B
        = ( produc4588648349897876871ist_fm @ P ) )
      = ( ? [A2: list_fm] :
            ( P
            = ( produc381145313068854617ist_fm @ A2 @ B ) ) ) ) ).

% eq_snd_iff
thf(fact_190_eq__fst__iff,axiom,
    ! [A: nat,P: product_prod_nat_nat] :
      ( ( A
        = ( product_fst_nat_nat @ P ) )
      = ( ? [B4: nat] :
            ( P
            = ( product_Pair_nat_nat @ A @ B4 ) ) ) ) ).

% eq_fst_iff
thf(fact_191_eq__fst__iff,axiom,
    ! [A: product_prod_nat_nat,P: produc859450856879609959at_nat] :
      ( ( A
        = ( produc3213797794245857475at_nat @ P ) )
      = ( ? [B4: product_prod_nat_nat] :
            ( P
            = ( produc6161850002892822231at_nat @ A @ B4 ) ) ) ) ).

% eq_fst_iff
thf(fact_192_eq__fst__iff,axiom,
    ! [A: rule,P: produc4630866025709511057ist_fm] :
      ( ( A
        = ( produc4228417144610596823ist_fm @ P ) )
      = ( ? [B4: produc1996495991257130529ist_fm] :
            ( P
            = ( produc7063755442767259331ist_fm @ A @ B4 ) ) ) ) ).

% eq_fst_iff
thf(fact_193_eq__fst__iff,axiom,
    ! [A: produc1996495991257130529ist_fm,P: produc164195504107695125m_rule] :
      ( ( A
        = ( produc6879501374131015971m_rule @ P ) )
      = ( ? [B4: rule] :
            ( P
            = ( produc491467635432902671m_rule @ A @ B4 ) ) ) ) ).

% eq_fst_iff
thf(fact_194_eq__fst__iff,axiom,
    ! [A: list_fm,P: produc1996495991257130529ist_fm] :
      ( ( A
        = ( produc1501393135466168645ist_fm @ P ) )
      = ( ? [B4: list_fm] :
            ( P
            = ( produc381145313068854617ist_fm @ A @ B4 ) ) ) ) ).

% eq_fst_iff
thf(fact_195_fstI,axiom,
    ! [X3: product_prod_nat_nat,Y3: nat,Z2: nat] :
      ( ( X3
        = ( product_Pair_nat_nat @ Y3 @ Z2 ) )
     => ( ( product_fst_nat_nat @ X3 )
        = Y3 ) ) ).

% fstI
thf(fact_196_fstI,axiom,
    ! [X3: produc859450856879609959at_nat,Y3: product_prod_nat_nat,Z2: product_prod_nat_nat] :
      ( ( X3
        = ( produc6161850002892822231at_nat @ Y3 @ Z2 ) )
     => ( ( produc3213797794245857475at_nat @ X3 )
        = Y3 ) ) ).

% fstI
thf(fact_197_fstI,axiom,
    ! [X3: produc4630866025709511057ist_fm,Y3: rule,Z2: produc1996495991257130529ist_fm] :
      ( ( X3
        = ( produc7063755442767259331ist_fm @ Y3 @ Z2 ) )
     => ( ( produc4228417144610596823ist_fm @ X3 )
        = Y3 ) ) ).

% fstI
thf(fact_198_fstI,axiom,
    ! [X3: produc164195504107695125m_rule,Y3: produc1996495991257130529ist_fm,Z2: rule] :
      ( ( X3
        = ( produc491467635432902671m_rule @ Y3 @ Z2 ) )
     => ( ( produc6879501374131015971m_rule @ X3 )
        = Y3 ) ) ).

% fstI
thf(fact_199_fstI,axiom,
    ! [X3: produc1996495991257130529ist_fm,Y3: list_fm,Z2: list_fm] :
      ( ( X3
        = ( produc381145313068854617ist_fm @ Y3 @ Z2 ) )
     => ( ( produc1501393135466168645ist_fm @ X3 )
        = Y3 ) ) ).

% fstI
thf(fact_200_pred__equals__eq2,axiom,
    ! [R3: set_Pr1008144964186165195m_rule,S4: set_Pr1008144964186165195m_rule] :
      ( ( ( ^ [X6: produc1996495991257130529ist_fm,Y5: rule] : ( member4220325220686508332m_rule @ ( produc491467635432902671m_rule @ X6 @ Y5 ) @ R3 ) )
        = ( ^ [X6: produc1996495991257130529ist_fm,Y5: rule] : ( member4220325220686508332m_rule @ ( produc491467635432902671m_rule @ X6 @ Y5 ) @ S4 ) ) )
      = ( R3 = S4 ) ) ).

% pred_equals_eq2
thf(fact_201_pred__equals__eq2,axiom,
    ! [R3: set_Pr7058068377845519745ist_fm,S4: set_Pr7058068377845519745ist_fm] :
      ( ( ( ^ [X6: list_fm,Y5: list_fm] : ( member8102475879199740618ist_fm @ ( produc381145313068854617ist_fm @ X6 @ Y5 ) @ R3 ) )
        = ( ^ [X6: list_fm,Y5: list_fm] : ( member8102475879199740618ist_fm @ ( produc381145313068854617ist_fm @ X6 @ Y5 ) @ S4 ) ) )
      = ( R3 = S4 ) ) ).

% pred_equals_eq2
thf(fact_202_pred__equals__eq2,axiom,
    ! [R3: set_Pr1261947904930325089at_nat,S4: set_Pr1261947904930325089at_nat] :
      ( ( ( ^ [X6: nat,Y5: nat] : ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X6 @ Y5 ) @ R3 ) )
        = ( ^ [X6: nat,Y5: nat] : ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X6 @ Y5 ) @ S4 ) ) )
      = ( R3 = S4 ) ) ).

% pred_equals_eq2
thf(fact_203_pred__equals__eq2,axiom,
    ! [R3: set_Pr8693737435421807431at_nat,S4: set_Pr8693737435421807431at_nat] :
      ( ( ( ^ [X6: product_prod_nat_nat,Y5: product_prod_nat_nat] : ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ X6 @ Y5 ) @ R3 ) )
        = ( ^ [X6: product_prod_nat_nat,Y5: product_prod_nat_nat] : ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ X6 @ Y5 ) @ S4 ) ) )
      = ( R3 = S4 ) ) ).

% pred_equals_eq2
thf(fact_204_pred__equals__eq2,axiom,
    ! [R3: set_Pr2241365836748572487ist_fm,S4: set_Pr2241365836748572487ist_fm] :
      ( ( ( ^ [X6: rule,Y5: produc1996495991257130529ist_fm] : ( member8686995742288324264ist_fm @ ( produc7063755442767259331ist_fm @ X6 @ Y5 ) @ R3 ) )
        = ( ^ [X6: rule,Y5: produc1996495991257130529ist_fm] : ( member8686995742288324264ist_fm @ ( produc7063755442767259331ist_fm @ X6 @ Y5 ) @ S4 ) ) )
      = ( R3 = S4 ) ) ).

% pred_equals_eq2
thf(fact_205_fm_Oinject_I1_J,axiom,
    ! [X21: nat,X22: list_tm,Y21: nat,Y222: list_tm] :
      ( ( ( pre @ X21 @ X22 )
        = ( pre @ Y21 @ Y222 ) )
      = ( ( X21 = Y21 )
        & ( X22 = Y222 ) ) ) ).

% fm.inject(1)
thf(fact_206_fm_Odistinct_I9_J,axiom,
    ! [X21: nat,X22: list_tm,X4: fm] :
      ( ( pre @ X21 @ X22 )
     != ( uni @ X4 ) ) ).

% fm.distinct(9)
thf(fact_207_eff__preserves__Pre_I2_J,axiom,
    ! [A5: list_fm,B5: list_fm,R2: rule,Ss2: fset_P661503646757059847ist_fm,A6: list_fm,B6: list_fm,P2: nat,Ts: list_tm] :
      ( ( ( eff @ ( produc7165828336582415457m_rule @ ( produc491467635432902671m_rule @ ( produc381145313068854617ist_fm @ A5 @ B5 ) @ R2 ) ) @ ( produc6879501374131015971m_rule @ ( produc491467635432902671m_rule @ ( produc381145313068854617ist_fm @ A5 @ B5 ) @ R2 ) ) )
        = ( some_f2162114139285431020ist_fm @ Ss2 ) )
     => ( ( fmembe3381613331217039976ist_fm @ ( produc381145313068854617ist_fm @ A6 @ B6 ) @ Ss2 )
       => ( ( member_fm @ ( pre @ P2 @ Ts ) @ ( set_fm2 @ B5 ) )
         => ( member_fm @ ( pre @ P2 @ Ts ) @ ( set_fm2 @ B6 ) ) ) ) ) ).

% eff_preserves_Pre(2)
thf(fact_208_eff__preserves__Pre_I1_J,axiom,
    ! [A5: list_fm,B5: list_fm,R2: rule,Ss2: fset_P661503646757059847ist_fm,A6: list_fm,B6: list_fm,P2: nat,Ts: list_tm] :
      ( ( ( eff @ ( produc7165828336582415457m_rule @ ( produc491467635432902671m_rule @ ( produc381145313068854617ist_fm @ A5 @ B5 ) @ R2 ) ) @ ( produc6879501374131015971m_rule @ ( produc491467635432902671m_rule @ ( produc381145313068854617ist_fm @ A5 @ B5 ) @ R2 ) ) )
        = ( some_f2162114139285431020ist_fm @ Ss2 ) )
     => ( ( fmembe3381613331217039976ist_fm @ ( produc381145313068854617ist_fm @ A6 @ B6 ) @ Ss2 )
       => ( ( member_fm @ ( pre @ P2 @ Ts ) @ ( set_fm2 @ A5 ) )
         => ( member_fm @ ( pre @ P2 @ Ts ) @ ( set_fm2 @ A6 ) ) ) ) ) ).

% eff_preserves_Pre(1)
thf(fact_209_epath__eff,axiom,
    ! [Steps: stream727092118206550309m_rule,Ss2: fset_P661503646757059847ist_fm] :
      ( ( abstra5415202973855822231ist_fm
        @ ^ [R: rule,S: produc1996495991257130529ist_fm,Ss: fset_P661503646757059847ist_fm] :
            ( ( eff @ R @ S )
            = ( some_f2162114139285431020ist_fm @ Ss ) )
        @ rules
        @ Steps )
     => ( ( ( eff @ ( produc7165828336582415457m_rule @ ( shd_Pr7235097944458474089m_rule @ Steps ) ) @ ( produc6879501374131015971m_rule @ ( shd_Pr7235097944458474089m_rule @ Steps ) ) )
          = ( some_f2162114139285431020ist_fm @ Ss2 ) )
       => ( fmembe3381613331217039976ist_fm @ ( produc6879501374131015971m_rule @ ( shd_Pr7235097944458474089m_rule @ ( stl_Pr950425576149878629m_rule @ Steps ) ) ) @ Ss2 ) ) ) ).

% epath_eff
thf(fact_210_fstOp__def,axiom,
    ( bNF_fs2143400664727635963e_rule
    = ( ^ [P4: produc1996495991257130529ist_fm > rule > $o,Q3: rule > rule > $o,Ac: produc164195504107695125m_rule] : ( produc491467635432902671m_rule @ ( produc6879501374131015971m_rule @ Ac ) @ ( bNF_pi580457772512444745e_rule @ P4 @ Q3 @ ( produc6879501374131015971m_rule @ Ac ) @ ( produc7165828336582415457m_rule @ Ac ) ) ) ) ) ).

% fstOp_def
thf(fact_211_fstOp__def,axiom,
    ( bNF_fs6361025107161210196ist_fm
    = ( ^ [P4: list_fm > list_fm > $o,Q3: list_fm > list_fm > $o,Ac: produc1996495991257130529ist_fm] : ( produc381145313068854617ist_fm @ ( produc1501393135466168645ist_fm @ Ac ) @ ( bNF_pi3539088406881798022ist_fm @ P4 @ Q3 @ ( produc1501393135466168645ist_fm @ Ac ) @ ( produc4588648349897876871ist_fm @ Ac ) ) ) ) ) ).

% fstOp_def
thf(fact_212_sndOp__def,axiom,
    ( bNF_sn6617447237779055481m_rule
    = ( ^ [P4: produc1996495991257130529ist_fm > produc1996495991257130529ist_fm > $o,Q3: produc1996495991257130529ist_fm > rule > $o,Ac: produc164195504107695125m_rule] : ( produc491467635432902671m_rule @ ( bNF_pi3438825987790746729m_rule @ P4 @ Q3 @ ( produc6879501374131015971m_rule @ Ac ) @ ( produc7165828336582415457m_rule @ Ac ) ) @ ( produc7165828336582415457m_rule @ Ac ) ) ) ) ).

% sndOp_def
thf(fact_213_sndOp__def,axiom,
    ( bNF_sn5292590578203848342ist_fm
    = ( ^ [P4: list_fm > list_fm > $o,Q3: list_fm > list_fm > $o,Ac: produc1996495991257130529ist_fm] : ( produc381145313068854617ist_fm @ ( bNF_pi3539088406881798022ist_fm @ P4 @ Q3 @ ( produc1501393135466168645ist_fm @ Ac ) @ ( produc4588648349897876871ist_fm @ Ac ) ) @ ( produc4588648349897876871ist_fm @ Ac ) ) ) ) ).

% sndOp_def
thf(fact_214_per__rules_H,axiom,
    ! [R2: rule,A5: list_fm,B5: list_fm,A6: list_fm,B6: list_fm,R4: rule,Ss3: fset_P661503646757059847ist_fm] :
      ( ( abstra3055540779341473792ist_fm
        @ ^ [R: rule,S: produc1996495991257130529ist_fm,Ss: fset_P661503646757059847ist_fm] :
            ( ( eff @ R @ S )
            = ( some_f2162114139285431020ist_fm @ Ss ) )
        @ R2
        @ ( produc381145313068854617ist_fm @ A5 @ B5 ) )
     => ( ~ ( abstra3055540779341473792ist_fm
            @ ^ [R: rule,S: produc1996495991257130529ist_fm,Ss: fset_P661503646757059847ist_fm] :
                ( ( eff @ R @ S )
                = ( some_f2162114139285431020ist_fm @ Ss ) )
            @ R2
            @ ( produc381145313068854617ist_fm @ A6 @ B6 ) )
       => ( ( ( eff @ R4 @ ( produc381145313068854617ist_fm @ A5 @ B5 ) )
            = ( some_f2162114139285431020ist_fm @ Ss3 ) )
         => ( ( fmembe3381613331217039976ist_fm @ ( produc381145313068854617ist_fm @ A6 @ B6 ) @ Ss3 )
           => ( R4 = R2 ) ) ) ) ) ).

% per_rules'
thf(fact_215_image2__eqI,axiom,
    ! [B: nat,F: fm > nat,X3: fm,C2: nat,G: fm > nat,A5: set_fm] :
      ( ( B
        = ( F @ X3 ) )
     => ( ( C2
          = ( G @ X3 ) )
       => ( ( member_fm @ X3 @ A5 )
         => ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ B @ C2 ) @ ( bNF_Gr5714924575499503436at_nat @ A5 @ F @ G ) ) ) ) ) ).

% image2_eqI
thf(fact_216_image2__eqI,axiom,
    ! [B: nat,F: rule > nat,X3: rule,C2: nat,G: rule > nat,A5: set_rule] :
      ( ( B
        = ( F @ X3 ) )
     => ( ( C2
          = ( G @ X3 ) )
       => ( ( member_rule @ X3 @ A5 )
         => ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ B @ C2 ) @ ( bNF_Gr4883198214526954049at_nat @ A5 @ F @ G ) ) ) ) ) ).

% image2_eqI
thf(fact_217_image2__eqI,axiom,
    ! [B: list_fm,F: fm > list_fm,X3: fm,C2: list_fm,G: fm > list_fm,A5: set_fm] :
      ( ( B
        = ( F @ X3 ) )
     => ( ( C2
          = ( G @ X3 ) )
       => ( ( member_fm @ X3 @ A5 )
         => ( member8102475879199740618ist_fm @ ( produc381145313068854617ist_fm @ B @ C2 ) @ ( bNF_Gr2680958839322950764ist_fm @ A5 @ F @ G ) ) ) ) ) ).

% image2_eqI
thf(fact_218_image2__eqI,axiom,
    ! [B: list_fm,F: rule > list_fm,X3: rule,C2: list_fm,G: rule > list_fm,A5: set_rule] :
      ( ( B
        = ( F @ X3 ) )
     => ( ( C2
          = ( G @ X3 ) )
       => ( ( member_rule @ X3 @ A5 )
         => ( member8102475879199740618ist_fm @ ( produc381145313068854617ist_fm @ B @ C2 ) @ ( bNF_Gr6421903352100228449ist_fm @ A5 @ F @ G ) ) ) ) ) ).

% image2_eqI
thf(fact_219_image2__eqI,axiom,
    ! [B: produc1996495991257130529ist_fm,F: fm > produc1996495991257130529ist_fm,X3: fm,C2: rule,G: fm > rule,A5: set_fm] :
      ( ( B
        = ( F @ X3 ) )
     => ( ( C2
          = ( G @ X3 ) )
       => ( ( member_fm @ X3 @ A5 )
         => ( member4220325220686508332m_rule @ ( produc491467635432902671m_rule @ B @ C2 ) @ ( bNF_Gr4411798878337052732m_rule @ A5 @ F @ G ) ) ) ) ) ).

% image2_eqI
thf(fact_220_image2__eqI,axiom,
    ! [B: produc1996495991257130529ist_fm,F: rule > produc1996495991257130529ist_fm,X3: rule,C2: rule,G: rule > rule,A5: set_rule] :
      ( ( B
        = ( F @ X3 ) )
     => ( ( C2
          = ( G @ X3 ) )
       => ( ( member_rule @ X3 @ A5 )
         => ( member4220325220686508332m_rule @ ( produc491467635432902671m_rule @ B @ C2 ) @ ( bNF_Gr6058931800003792391m_rule @ A5 @ F @ G ) ) ) ) ) ).

% image2_eqI
thf(fact_221_image2__eqI,axiom,
    ! [B: nat,F: produc1996495991257130529ist_fm > nat,X3: produc1996495991257130529ist_fm,C2: nat,G: produc1996495991257130529ist_fm > nat,A5: set_Pr7058068377845519745ist_fm] :
      ( ( B
        = ( F @ X3 ) )
     => ( ( C2
          = ( G @ X3 ) )
       => ( ( member8102475879199740618ist_fm @ X3 @ A5 )
         => ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ B @ C2 ) @ ( bNF_Gr330866185133063785at_nat @ A5 @ F @ G ) ) ) ) ) ).

% image2_eqI
thf(fact_222_image2__eqI,axiom,
    ! [B: product_prod_nat_nat,F: fm > product_prod_nat_nat,X3: fm,C2: product_prod_nat_nat,G: fm > product_prod_nat_nat,A5: set_fm] :
      ( ( B
        = ( F @ X3 ) )
     => ( ( C2
          = ( G @ X3 ) )
       => ( ( member_fm @ X3 @ A5 )
         => ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ B @ C2 ) @ ( bNF_Gr62440225311546218at_nat @ A5 @ F @ G ) ) ) ) ) ).

% image2_eqI
thf(fact_223_image2__eqI,axiom,
    ! [B: product_prod_nat_nat,F: rule > product_prod_nat_nat,X3: rule,C2: product_prod_nat_nat,G: rule > product_prod_nat_nat,A5: set_rule] :
      ( ( B
        = ( F @ X3 ) )
     => ( ( C2
          = ( G @ X3 ) )
       => ( ( member_rule @ X3 @ A5 )
         => ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ B @ C2 ) @ ( bNF_Gr8059785081440213727at_nat @ A5 @ F @ G ) ) ) ) ) ).

% image2_eqI
thf(fact_224_image2__eqI,axiom,
    ! [B: rule,F: fm > rule,X3: fm,C2: produc1996495991257130529ist_fm,G: fm > produc1996495991257130529ist_fm,A5: set_fm] :
      ( ( B
        = ( F @ X3 ) )
     => ( ( C2
          = ( G @ X3 ) )
       => ( ( member_fm @ X3 @ A5 )
         => ( member8686995742288324264ist_fm @ ( produc7063755442767259331ist_fm @ B @ C2 ) @ ( bNF_Gr1760714648816633584ist_fm @ A5 @ F @ G ) ) ) ) ) ).

% image2_eqI
thf(fact_225_prover__def,axiom,
    ( prover
    = ( abstra178918799166095863ist_fm
      @ ^ [R: rule,S: produc1996495991257130529ist_fm,Ss: fset_P661503646757059847ist_fm] :
          ( ( eff @ R @ S )
          = ( some_f2162114139285431020ist_fm @ Ss ) )
      @ rules ) ) ).

% prover_def
thf(fact_226_treeB__sdrop,axiom,
    ! [N: nat,Steps: stream727092118206550309m_rule] : ( ord_less_eq_set_fm @ ( treeB @ ( sdrop_7224736112439592940m_rule @ N @ Steps ) ) @ ( treeB @ Steps ) ) ).

% treeB_sdrop
thf(fact_227_ssubst__Pair__rhs,axiom,
    ! [R2: produc1996495991257130529ist_fm,S3: rule,R3: set_Pr1008144964186165195m_rule,S5: rule] :
      ( ( member4220325220686508332m_rule @ ( produc491467635432902671m_rule @ R2 @ S3 ) @ R3 )
     => ( ( S5 = S3 )
       => ( member4220325220686508332m_rule @ ( produc491467635432902671m_rule @ R2 @ S5 ) @ R3 ) ) ) ).

% ssubst_Pair_rhs
thf(fact_228_ssubst__Pair__rhs,axiom,
    ! [R2: list_fm,S3: list_fm,R3: set_Pr7058068377845519745ist_fm,S5: list_fm] :
      ( ( member8102475879199740618ist_fm @ ( produc381145313068854617ist_fm @ R2 @ S3 ) @ R3 )
     => ( ( S5 = S3 )
       => ( member8102475879199740618ist_fm @ ( produc381145313068854617ist_fm @ R2 @ S5 ) @ R3 ) ) ) ).

% ssubst_Pair_rhs
thf(fact_229_ssubst__Pair__rhs,axiom,
    ! [R2: nat,S3: nat,R3: set_Pr1261947904930325089at_nat,S5: nat] :
      ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ R2 @ S3 ) @ R3 )
     => ( ( S5 = S3 )
       => ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ R2 @ S5 ) @ R3 ) ) ) ).

% ssubst_Pair_rhs
thf(fact_230_ssubst__Pair__rhs,axiom,
    ! [R2: product_prod_nat_nat,S3: product_prod_nat_nat,R3: set_Pr8693737435421807431at_nat,S5: product_prod_nat_nat] :
      ( ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ R2 @ S3 ) @ R3 )
     => ( ( S5 = S3 )
       => ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ R2 @ S5 ) @ R3 ) ) ) ).

% ssubst_Pair_rhs
thf(fact_231_ssubst__Pair__rhs,axiom,
    ! [R2: rule,S3: produc1996495991257130529ist_fm,R3: set_Pr2241365836748572487ist_fm,S5: produc1996495991257130529ist_fm] :
      ( ( member8686995742288324264ist_fm @ ( produc7063755442767259331ist_fm @ R2 @ S3 ) @ R3 )
     => ( ( S5 = S3 )
       => ( member8686995742288324264ist_fm @ ( produc7063755442767259331ist_fm @ R2 @ S5 ) @ R3 ) ) ) ).

% ssubst_Pair_rhs
thf(fact_232_RuleSystem__Defs_OmkTree_Ocong,axiom,
    abstra178918799166095863ist_fm = abstra178918799166095863ist_fm ).

% RuleSystem_Defs.mkTree.cong
thf(fact_233_Collect__restrict,axiom,
    ! [X7: set_rule,P2: rule > $o] :
      ( ord_less_eq_set_rule
      @ ( collect_rule
        @ ^ [X6: rule] :
            ( ( member_rule @ X6 @ X7 )
            & ( P2 @ X6 ) ) )
      @ X7 ) ).

% Collect_restrict
thf(fact_234_Collect__restrict,axiom,
    ! [X7: set_Pr7058068377845519745ist_fm,P2: produc1996495991257130529ist_fm > $o] :
      ( ord_le2055072402069232929ist_fm
      @ ( collec7231575776666803724ist_fm
        @ ^ [X6: produc1996495991257130529ist_fm] :
            ( ( member8102475879199740618ist_fm @ X6 @ X7 )
            & ( P2 @ X6 ) ) )
      @ X7 ) ).

% Collect_restrict
thf(fact_235_Collect__restrict,axiom,
    ! [X7: set_Pr8693737435421807431at_nat,P2: produc859450856879609959at_nat > $o] :
      ( ord_le3000389064537975527at_nat
      @ ( collec7088162979684241874at_nat
        @ ^ [X6: produc859450856879609959at_nat] :
            ( ( member8206827879206165904at_nat @ X6 @ X7 )
            & ( P2 @ X6 ) ) )
      @ X7 ) ).

% Collect_restrict
thf(fact_236_Collect__restrict,axiom,
    ! [X7: set_fm,P2: fm > $o] :
      ( ord_less_eq_set_fm
      @ ( collect_fm
        @ ^ [X6: fm] :
            ( ( member_fm @ X6 @ X7 )
            & ( P2 @ X6 ) ) )
      @ X7 ) ).

% Collect_restrict
thf(fact_237_prop__restrict,axiom,
    ! [X3: rule,Z3: set_rule,X7: set_rule,P2: rule > $o] :
      ( ( member_rule @ X3 @ Z3 )
     => ( ( ord_less_eq_set_rule @ Z3
          @ ( collect_rule
            @ ^ [X6: rule] :
                ( ( member_rule @ X6 @ X7 )
                & ( P2 @ X6 ) ) ) )
       => ( P2 @ X3 ) ) ) ).

% prop_restrict
thf(fact_238_prop__restrict,axiom,
    ! [X3: produc1996495991257130529ist_fm,Z3: set_Pr7058068377845519745ist_fm,X7: set_Pr7058068377845519745ist_fm,P2: produc1996495991257130529ist_fm > $o] :
      ( ( member8102475879199740618ist_fm @ X3 @ Z3 )
     => ( ( ord_le2055072402069232929ist_fm @ Z3
          @ ( collec7231575776666803724ist_fm
            @ ^ [X6: produc1996495991257130529ist_fm] :
                ( ( member8102475879199740618ist_fm @ X6 @ X7 )
                & ( P2 @ X6 ) ) ) )
       => ( P2 @ X3 ) ) ) ).

% prop_restrict
thf(fact_239_prop__restrict,axiom,
    ! [X3: produc859450856879609959at_nat,Z3: set_Pr8693737435421807431at_nat,X7: set_Pr8693737435421807431at_nat,P2: produc859450856879609959at_nat > $o] :
      ( ( member8206827879206165904at_nat @ X3 @ Z3 )
     => ( ( ord_le3000389064537975527at_nat @ Z3
          @ ( collec7088162979684241874at_nat
            @ ^ [X6: produc859450856879609959at_nat] :
                ( ( member8206827879206165904at_nat @ X6 @ X7 )
                & ( P2 @ X6 ) ) ) )
       => ( P2 @ X3 ) ) ) ).

% prop_restrict
thf(fact_240_prop__restrict,axiom,
    ! [X3: fm,Z3: set_fm,X7: set_fm,P2: fm > $o] :
      ( ( member_fm @ X3 @ Z3 )
     => ( ( ord_less_eq_set_fm @ Z3
          @ ( collect_fm
            @ ^ [X6: fm] :
                ( ( member_fm @ X6 @ X7 )
                & ( P2 @ X6 ) ) ) )
       => ( P2 @ X3 ) ) ) ).

% prop_restrict
thf(fact_241_szip_Osimps_I2_J,axiom,
    ! [S1: stream727092118206550309m_rule,S2: stream727092118206550309m_rule] :
      ( ( stl_Pr2200982372225425851m_rule @ ( szip_P2499414959592755846m_rule @ S1 @ S2 ) )
      = ( szip_P2499414959592755846m_rule @ ( stl_Pr950425576149878629m_rule @ S1 ) @ ( stl_Pr950425576149878629m_rule @ S2 ) ) ) ).

% szip.simps(2)
thf(fact_242_szip_Osimps_I2_J,axiom,
    ! [S1: stream727092118206550309m_rule,S2: stream_rule] :
      ( ( stl_Pr830518440239794027e_rule @ ( szip_P553291425819358518e_rule @ S1 @ S2 ) )
      = ( szip_P553291425819358518e_rule @ ( stl_Pr950425576149878629m_rule @ S1 ) @ ( stl_rule @ S2 ) ) ) ).

% szip.simps(2)
thf(fact_243_szip_Osimps_I2_J,axiom,
    ! [S1: stream_rule,S2: stream727092118206550309m_rule] :
      ( ( stl_Pr5425948690297933931m_rule @ ( szip_r2254114822203693110m_rule @ S1 @ S2 ) )
      = ( szip_r2254114822203693110m_rule @ ( stl_rule @ S1 ) @ ( stl_Pr950425576149878629m_rule @ S2 ) ) ) ).

% szip.simps(2)
thf(fact_244_szip_Osimps_I2_J,axiom,
    ! [S1: stream_rule,S2: stream_rule] :
      ( ( stl_Pr4904654673476115995e_rule @ ( szip_rule_rule @ S1 @ S2 ) )
      = ( szip_rule_rule @ ( stl_rule @ S1 ) @ ( stl_rule @ S2 ) ) ) ).

% szip.simps(2)
thf(fact_245_szip_Osimps_I2_J,axiom,
    ! [S1: stream8299795917829157543ist_fm,S2: stream_rule] :
      ( ( stl_Pr950425576149878629m_rule @ ( szip_P1977448745965526924m_rule @ S1 @ S2 ) )
      = ( szip_P1977448745965526924m_rule @ ( stl_Pr5027778045650968273ist_fm @ S1 ) @ ( stl_rule @ S2 ) ) ) ).

% szip.simps(2)
thf(fact_246_subset__Collect__iff,axiom,
    ! [B5: set_rule,A5: set_rule,P2: rule > $o] :
      ( ( ord_less_eq_set_rule @ B5 @ A5 )
     => ( ( ord_less_eq_set_rule @ B5
          @ ( collect_rule
            @ ^ [X6: rule] :
                ( ( member_rule @ X6 @ A5 )
                & ( P2 @ X6 ) ) ) )
        = ( ! [X6: rule] :
              ( ( member_rule @ X6 @ B5 )
             => ( P2 @ X6 ) ) ) ) ) ).

% subset_Collect_iff
thf(fact_247_subset__Collect__iff,axiom,
    ! [B5: set_Pr7058068377845519745ist_fm,A5: set_Pr7058068377845519745ist_fm,P2: produc1996495991257130529ist_fm > $o] :
      ( ( ord_le2055072402069232929ist_fm @ B5 @ A5 )
     => ( ( ord_le2055072402069232929ist_fm @ B5
          @ ( collec7231575776666803724ist_fm
            @ ^ [X6: produc1996495991257130529ist_fm] :
                ( ( member8102475879199740618ist_fm @ X6 @ A5 )
                & ( P2 @ X6 ) ) ) )
        = ( ! [X6: produc1996495991257130529ist_fm] :
              ( ( member8102475879199740618ist_fm @ X6 @ B5 )
             => ( P2 @ X6 ) ) ) ) ) ).

% subset_Collect_iff
thf(fact_248_subset__Collect__iff,axiom,
    ! [B5: set_Pr8693737435421807431at_nat,A5: set_Pr8693737435421807431at_nat,P2: produc859450856879609959at_nat > $o] :
      ( ( ord_le3000389064537975527at_nat @ B5 @ A5 )
     => ( ( ord_le3000389064537975527at_nat @ B5
          @ ( collec7088162979684241874at_nat
            @ ^ [X6: produc859450856879609959at_nat] :
                ( ( member8206827879206165904at_nat @ X6 @ A5 )
                & ( P2 @ X6 ) ) ) )
        = ( ! [X6: produc859450856879609959at_nat] :
              ( ( member8206827879206165904at_nat @ X6 @ B5 )
             => ( P2 @ X6 ) ) ) ) ) ).

% subset_Collect_iff
thf(fact_249_subset__Collect__iff,axiom,
    ! [B5: set_fm,A5: set_fm,P2: fm > $o] :
      ( ( ord_less_eq_set_fm @ B5 @ A5 )
     => ( ( ord_less_eq_set_fm @ B5
          @ ( collect_fm
            @ ^ [X6: fm] :
                ( ( member_fm @ X6 @ A5 )
                & ( P2 @ X6 ) ) ) )
        = ( ! [X6: fm] :
              ( ( member_fm @ X6 @ B5 )
             => ( P2 @ X6 ) ) ) ) ) ).

% subset_Collect_iff
thf(fact_250_subset__CollectI,axiom,
    ! [B5: set_rule,A5: set_rule,Q2: rule > $o,P2: rule > $o] :
      ( ( ord_less_eq_set_rule @ B5 @ A5 )
     => ( ! [X: rule] :
            ( ( member_rule @ X @ B5 )
           => ( ( Q2 @ X )
             => ( P2 @ X ) ) )
       => ( ord_less_eq_set_rule
          @ ( collect_rule
            @ ^ [X6: rule] :
                ( ( member_rule @ X6 @ B5 )
                & ( Q2 @ X6 ) ) )
          @ ( collect_rule
            @ ^ [X6: rule] :
                ( ( member_rule @ X6 @ A5 )
                & ( P2 @ X6 ) ) ) ) ) ) ).

% subset_CollectI
thf(fact_251_subset__CollectI,axiom,
    ! [B5: set_Pr7058068377845519745ist_fm,A5: set_Pr7058068377845519745ist_fm,Q2: produc1996495991257130529ist_fm > $o,P2: produc1996495991257130529ist_fm > $o] :
      ( ( ord_le2055072402069232929ist_fm @ B5 @ A5 )
     => ( ! [X: produc1996495991257130529ist_fm] :
            ( ( member8102475879199740618ist_fm @ X @ B5 )
           => ( ( Q2 @ X )
             => ( P2 @ X ) ) )
       => ( ord_le2055072402069232929ist_fm
          @ ( collec7231575776666803724ist_fm
            @ ^ [X6: produc1996495991257130529ist_fm] :
                ( ( member8102475879199740618ist_fm @ X6 @ B5 )
                & ( Q2 @ X6 ) ) )
          @ ( collec7231575776666803724ist_fm
            @ ^ [X6: produc1996495991257130529ist_fm] :
                ( ( member8102475879199740618ist_fm @ X6 @ A5 )
                & ( P2 @ X6 ) ) ) ) ) ) ).

% subset_CollectI
thf(fact_252_subset__CollectI,axiom,
    ! [B5: set_Pr8693737435421807431at_nat,A5: set_Pr8693737435421807431at_nat,Q2: produc859450856879609959at_nat > $o,P2: produc859450856879609959at_nat > $o] :
      ( ( ord_le3000389064537975527at_nat @ B5 @ A5 )
     => ( ! [X: produc859450856879609959at_nat] :
            ( ( member8206827879206165904at_nat @ X @ B5 )
           => ( ( Q2 @ X )
             => ( P2 @ X ) ) )
       => ( ord_le3000389064537975527at_nat
          @ ( collec7088162979684241874at_nat
            @ ^ [X6: produc859450856879609959at_nat] :
                ( ( member8206827879206165904at_nat @ X6 @ B5 )
                & ( Q2 @ X6 ) ) )
          @ ( collec7088162979684241874at_nat
            @ ^ [X6: produc859450856879609959at_nat] :
                ( ( member8206827879206165904at_nat @ X6 @ A5 )
                & ( P2 @ X6 ) ) ) ) ) ) ).

% subset_CollectI
thf(fact_253_subset__CollectI,axiom,
    ! [B5: set_fm,A5: set_fm,Q2: fm > $o,P2: fm > $o] :
      ( ( ord_less_eq_set_fm @ B5 @ A5 )
     => ( ! [X: fm] :
            ( ( member_fm @ X @ B5 )
           => ( ( Q2 @ X )
             => ( P2 @ X ) ) )
       => ( ord_less_eq_set_fm
          @ ( collect_fm
            @ ^ [X6: fm] :
                ( ( member_fm @ X6 @ B5 )
                & ( Q2 @ X6 ) ) )
          @ ( collect_fm
            @ ^ [X6: fm] :
                ( ( member_fm @ X6 @ A5 )
                & ( P2 @ X6 ) ) ) ) ) ) ).

% subset_CollectI
thf(fact_254_stream_Oexpand,axiom,
    ! [Stream: stream727092118206550309m_rule,Stream2: stream727092118206550309m_rule] :
      ( ( ( ( shd_Pr7235097944458474089m_rule @ Stream )
          = ( shd_Pr7235097944458474089m_rule @ Stream2 ) )
        & ( ( stl_Pr950425576149878629m_rule @ Stream )
          = ( stl_Pr950425576149878629m_rule @ Stream2 ) ) )
     => ( Stream = Stream2 ) ) ).

% stream.expand
thf(fact_255_stream_Oexpand,axiom,
    ! [Stream: stream_rule,Stream2: stream_rule] :
      ( ( ( ( shd_rule @ Stream )
          = ( shd_rule @ Stream2 ) )
        & ( ( stl_rule @ Stream )
          = ( stl_rule @ Stream2 ) ) )
     => ( Stream = Stream2 ) ) ).

% stream.expand
thf(fact_256_stream_Ocoinduct,axiom,
    ! [R3: stream727092118206550309m_rule > stream727092118206550309m_rule > $o,Stream: stream727092118206550309m_rule,Stream2: stream727092118206550309m_rule] :
      ( ( R3 @ Stream @ Stream2 )
     => ( ! [Stream3: stream727092118206550309m_rule,Stream4: stream727092118206550309m_rule] :
            ( ( R3 @ Stream3 @ Stream4 )
           => ( ( ( shd_Pr7235097944458474089m_rule @ Stream3 )
                = ( shd_Pr7235097944458474089m_rule @ Stream4 ) )
              & ( R3 @ ( stl_Pr950425576149878629m_rule @ Stream3 ) @ ( stl_Pr950425576149878629m_rule @ Stream4 ) ) ) )
       => ( Stream = Stream2 ) ) ) ).

% stream.coinduct
thf(fact_257_stream_Ocoinduct,axiom,
    ! [R3: stream_rule > stream_rule > $o,Stream: stream_rule,Stream2: stream_rule] :
      ( ( R3 @ Stream @ Stream2 )
     => ( ! [Stream3: stream_rule,Stream4: stream_rule] :
            ( ( R3 @ Stream3 @ Stream4 )
           => ( ( ( shd_rule @ Stream3 )
                = ( shd_rule @ Stream4 ) )
              & ( R3 @ ( stl_rule @ Stream3 ) @ ( stl_rule @ Stream4 ) ) ) )
       => ( Stream = Stream2 ) ) ) ).

% stream.coinduct
thf(fact_258_stream_Ocoinduct__strong,axiom,
    ! [R3: stream727092118206550309m_rule > stream727092118206550309m_rule > $o,Stream: stream727092118206550309m_rule,Stream2: stream727092118206550309m_rule] :
      ( ( R3 @ Stream @ Stream2 )
     => ( ! [Stream3: stream727092118206550309m_rule,Stream4: stream727092118206550309m_rule] :
            ( ( R3 @ Stream3 @ Stream4 )
           => ( ( ( shd_Pr7235097944458474089m_rule @ Stream3 )
                = ( shd_Pr7235097944458474089m_rule @ Stream4 ) )
              & ( ( R3 @ ( stl_Pr950425576149878629m_rule @ Stream3 ) @ ( stl_Pr950425576149878629m_rule @ Stream4 ) )
                | ( ( stl_Pr950425576149878629m_rule @ Stream3 )
                  = ( stl_Pr950425576149878629m_rule @ Stream4 ) ) ) ) )
       => ( Stream = Stream2 ) ) ) ).

% stream.coinduct_strong
thf(fact_259_stream_Ocoinduct__strong,axiom,
    ! [R3: stream_rule > stream_rule > $o,Stream: stream_rule,Stream2: stream_rule] :
      ( ( R3 @ Stream @ Stream2 )
     => ( ! [Stream3: stream_rule,Stream4: stream_rule] :
            ( ( R3 @ Stream3 @ Stream4 )
           => ( ( ( shd_rule @ Stream3 )
                = ( shd_rule @ Stream4 ) )
              & ( ( R3 @ ( stl_rule @ Stream3 ) @ ( stl_rule @ Stream4 ) )
                | ( ( stl_rule @ Stream3 )
                  = ( stl_rule @ Stream4 ) ) ) ) )
       => ( Stream = Stream2 ) ) ) ).

% stream.coinduct_strong
thf(fact_260_sdrop__stl,axiom,
    ! [N: nat,S3: stream8299795917829157543ist_fm] :
      ( ( sdrop_4442373711808556042ist_fm @ N @ ( stl_Pr5027778045650968273ist_fm @ S3 ) )
      = ( stl_Pr5027778045650968273ist_fm @ ( sdrop_4442373711808556042ist_fm @ N @ S3 ) ) ) ).

% sdrop_stl
thf(fact_261_sdrop__stl,axiom,
    ! [N: nat,S3: stream3588992565182678257e_rule] :
      ( ( sdrop_2199339672054592340e_rule @ N @ ( stl_Pr4904654673476115995e_rule @ S3 ) )
      = ( stl_Pr4904654673476115995e_rule @ ( sdrop_2199339672054592340e_rule @ N @ S3 ) ) ) ).

% sdrop_stl
thf(fact_262_sdrop__stl,axiom,
    ! [N: nat,S3: stream1878305877988316353m_rule] :
      ( ( sdrop_1764812713384251940m_rule @ N @ ( stl_Pr5425948690297933931m_rule @ S3 ) )
      = ( stl_Pr5425948690297933931m_rule @ ( sdrop_1764812713384251940m_rule @ N @ S3 ) ) ) ).

% sdrop_stl
thf(fact_263_sdrop__stl,axiom,
    ! [N: nat,S3: stream2312013117288958913e_rule] :
      ( ( sdrop_6392754500180887844e_rule @ N @ ( stl_Pr830518440239794027e_rule @ S3 ) )
      = ( stl_Pr830518440239794027e_rule @ ( sdrop_6392754500180887844e_rule @ N @ S3 ) ) ) ).

% sdrop_stl
thf(fact_264_sdrop__stl,axiom,
    ! [N: nat,S3: stream6494289010434245521m_rule] :
      ( ( sdrop_7373388980841566196m_rule @ N @ ( stl_Pr2200982372225425851m_rule @ S3 ) )
      = ( stl_Pr2200982372225425851m_rule @ ( sdrop_7373388980841566196m_rule @ N @ S3 ) ) ) ).

% sdrop_stl
thf(fact_265_sdrop__stl,axiom,
    ! [N: nat,S3: stream727092118206550309m_rule] :
      ( ( sdrop_7224736112439592940m_rule @ N @ ( stl_Pr950425576149878629m_rule @ S3 ) )
      = ( stl_Pr950425576149878629m_rule @ ( sdrop_7224736112439592940m_rule @ N @ S3 ) ) ) ).

% sdrop_stl
thf(fact_266_sdrop__stl,axiom,
    ! [N: nat,S3: stream_rule] :
      ( ( sdrop_rule @ N @ ( stl_rule @ S3 ) )
      = ( stl_rule @ ( sdrop_rule @ N @ S3 ) ) ) ).

% sdrop_stl
thf(fact_267_treeA__sdrop,axiom,
    ! [N: nat,Steps: stream727092118206550309m_rule] : ( ord_less_eq_set_fm @ ( treeA @ ( sdrop_7224736112439592940m_rule @ N @ Steps ) ) @ ( treeA @ Steps ) ) ).

% treeA_sdrop
thf(fact_268_epath_Ocoinduct,axiom,
    ! [X7: stream727092118206550309m_rule > $o,X3: stream727092118206550309m_rule] :
      ( ( X7 @ X3 )
     => ( ! [X: stream727092118206550309m_rule] :
            ( ( X7 @ X )
           => ? [Steps2: stream727092118206550309m_rule,Sl2: fset_P661503646757059847ist_fm] :
                ( ( X = Steps2 )
                & ( member_rule @ ( produc7165828336582415457m_rule @ ( shd_Pr7235097944458474089m_rule @ Steps2 ) ) @ ( sset_rule @ rules ) )
                & ( fmembe3381613331217039976ist_fm @ ( produc6879501374131015971m_rule @ ( shd_Pr7235097944458474089m_rule @ ( stl_Pr950425576149878629m_rule @ Steps2 ) ) ) @ Sl2 )
                & ( ( eff @ ( produc7165828336582415457m_rule @ ( shd_Pr7235097944458474089m_rule @ Steps2 ) ) @ ( produc6879501374131015971m_rule @ ( shd_Pr7235097944458474089m_rule @ Steps2 ) ) )
                  = ( some_f2162114139285431020ist_fm @ Sl2 ) )
                & ( ( X7 @ ( stl_Pr950425576149878629m_rule @ Steps2 ) )
                  | ( abstra5415202973855822231ist_fm
                    @ ^ [R: rule,S: produc1996495991257130529ist_fm,Ss: fset_P661503646757059847ist_fm] :
                        ( ( eff @ R @ S )
                        = ( some_f2162114139285431020ist_fm @ Ss ) )
                    @ rules
                    @ ( stl_Pr950425576149878629m_rule @ Steps2 ) ) ) ) )
       => ( abstra5415202973855822231ist_fm
          @ ^ [R: rule,S: produc1996495991257130529ist_fm,Ss: fset_P661503646757059847ist_fm] :
              ( ( eff @ R @ S )
              = ( some_f2162114139285431020ist_fm @ Ss ) )
          @ rules
          @ X3 ) ) ) ).

% epath.coinduct
thf(fact_269_epath_Ointros,axiom,
    ! [Steps: stream727092118206550309m_rule,Sl3: fset_P661503646757059847ist_fm] :
      ( ( member_rule @ ( produc7165828336582415457m_rule @ ( shd_Pr7235097944458474089m_rule @ Steps ) ) @ ( sset_rule @ rules ) )
     => ( ( fmembe3381613331217039976ist_fm @ ( produc6879501374131015971m_rule @ ( shd_Pr7235097944458474089m_rule @ ( stl_Pr950425576149878629m_rule @ Steps ) ) ) @ Sl3 )
       => ( ( ( eff @ ( produc7165828336582415457m_rule @ ( shd_Pr7235097944458474089m_rule @ Steps ) ) @ ( produc6879501374131015971m_rule @ ( shd_Pr7235097944458474089m_rule @ Steps ) ) )
            = ( some_f2162114139285431020ist_fm @ Sl3 ) )
         => ( ( abstra5415202973855822231ist_fm
              @ ^ [R: rule,S: produc1996495991257130529ist_fm,Ss: fset_P661503646757059847ist_fm] :
                  ( ( eff @ R @ S )
                  = ( some_f2162114139285431020ist_fm @ Ss ) )
              @ rules
              @ ( stl_Pr950425576149878629m_rule @ Steps ) )
           => ( abstra5415202973855822231ist_fm
              @ ^ [R: rule,S: produc1996495991257130529ist_fm,Ss: fset_P661503646757059847ist_fm] :
                  ( ( eff @ R @ S )
                  = ( some_f2162114139285431020ist_fm @ Ss ) )
              @ rules
              @ Steps ) ) ) ) ) ).

% epath.intros
thf(fact_270_epath_Osimps,axiom,
    ! [A: stream727092118206550309m_rule] :
      ( ( abstra5415202973855822231ist_fm
        @ ^ [R: rule,S: produc1996495991257130529ist_fm,Ss: fset_P661503646757059847ist_fm] :
            ( ( eff @ R @ S )
            = ( some_f2162114139285431020ist_fm @ Ss ) )
        @ rules
        @ A )
      = ( ? [Steps3: stream727092118206550309m_rule,Sl: fset_P661503646757059847ist_fm] :
            ( ( A = Steps3 )
            & ( member_rule @ ( produc7165828336582415457m_rule @ ( shd_Pr7235097944458474089m_rule @ Steps3 ) ) @ ( sset_rule @ rules ) )
            & ( fmembe3381613331217039976ist_fm @ ( produc6879501374131015971m_rule @ ( shd_Pr7235097944458474089m_rule @ ( stl_Pr950425576149878629m_rule @ Steps3 ) ) ) @ Sl )
            & ( ( eff @ ( produc7165828336582415457m_rule @ ( shd_Pr7235097944458474089m_rule @ Steps3 ) ) @ ( produc6879501374131015971m_rule @ ( shd_Pr7235097944458474089m_rule @ Steps3 ) ) )
              = ( some_f2162114139285431020ist_fm @ Sl ) )
            & ( abstra5415202973855822231ist_fm
              @ ^ [R: rule,S: produc1996495991257130529ist_fm,Ss: fset_P661503646757059847ist_fm] :
                  ( ( eff @ R @ S )
                  = ( some_f2162114139285431020ist_fm @ Ss ) )
              @ rules
              @ ( stl_Pr950425576149878629m_rule @ Steps3 ) ) ) ) ) ).

% epath.simps
thf(fact_271_epath_Ocases,axiom,
    ! [A: stream727092118206550309m_rule] :
      ( ( abstra5415202973855822231ist_fm
        @ ^ [R: rule,S: produc1996495991257130529ist_fm,Ss: fset_P661503646757059847ist_fm] :
            ( ( eff @ R @ S )
            = ( some_f2162114139285431020ist_fm @ Ss ) )
        @ rules
        @ A )
     => ~ ( ( member_rule @ ( produc7165828336582415457m_rule @ ( shd_Pr7235097944458474089m_rule @ A ) ) @ ( sset_rule @ rules ) )
         => ! [Sl4: fset_P661503646757059847ist_fm] :
              ( ( fmembe3381613331217039976ist_fm @ ( produc6879501374131015971m_rule @ ( shd_Pr7235097944458474089m_rule @ ( stl_Pr950425576149878629m_rule @ A ) ) ) @ Sl4 )
             => ( ( ( eff @ ( produc7165828336582415457m_rule @ ( shd_Pr7235097944458474089m_rule @ A ) ) @ ( produc6879501374131015971m_rule @ ( shd_Pr7235097944458474089m_rule @ A ) ) )
                  = ( some_f2162114139285431020ist_fm @ Sl4 ) )
               => ~ ( abstra5415202973855822231ist_fm
                    @ ^ [R: rule,S: produc1996495991257130529ist_fm,Ss: fset_P661503646757059847ist_fm] :
                        ( ( eff @ R @ S )
                        = ( some_f2162114139285431020ist_fm @ Ss ) )
                    @ rules
                    @ ( stl_Pr950425576149878629m_rule @ A ) ) ) ) ) ) ).

% epath.cases
thf(fact_272_subset__antisym,axiom,
    ! [A5: set_fm,B5: set_fm] :
      ( ( ord_less_eq_set_fm @ A5 @ B5 )
     => ( ( ord_less_eq_set_fm @ B5 @ A5 )
       => ( A5 = B5 ) ) ) ).

% subset_antisym
thf(fact_273_subsetI,axiom,
    ! [A5: set_rule,B5: set_rule] :
      ( ! [X: rule] :
          ( ( member_rule @ X @ A5 )
         => ( member_rule @ X @ B5 ) )
     => ( ord_less_eq_set_rule @ A5 @ B5 ) ) ).

% subsetI
thf(fact_274_subsetI,axiom,
    ! [A5: set_Pr7058068377845519745ist_fm,B5: set_Pr7058068377845519745ist_fm] :
      ( ! [X: produc1996495991257130529ist_fm] :
          ( ( member8102475879199740618ist_fm @ X @ A5 )
         => ( member8102475879199740618ist_fm @ X @ B5 ) )
     => ( ord_le2055072402069232929ist_fm @ A5 @ B5 ) ) ).

% subsetI
thf(fact_275_subsetI,axiom,
    ! [A5: set_Pr8693737435421807431at_nat,B5: set_Pr8693737435421807431at_nat] :
      ( ! [X: produc859450856879609959at_nat] :
          ( ( member8206827879206165904at_nat @ X @ A5 )
         => ( member8206827879206165904at_nat @ X @ B5 ) )
     => ( ord_le3000389064537975527at_nat @ A5 @ B5 ) ) ).

% subsetI
thf(fact_276_subsetI,axiom,
    ! [A5: set_fm,B5: set_fm] :
      ( ! [X: fm] :
          ( ( member_fm @ X @ A5 )
         => ( member_fm @ X @ B5 ) )
     => ( ord_less_eq_set_fm @ A5 @ B5 ) ) ).

% subsetI
thf(fact_277_dual__order_Orefl,axiom,
    ! [A: set_fm] : ( ord_less_eq_set_fm @ A @ A ) ).

% dual_order.refl
thf(fact_278_dual__order_Orefl,axiom,
    ! [A: nat] : ( ord_less_eq_nat @ A @ A ) ).

% dual_order.refl
thf(fact_279_order__refl,axiom,
    ! [X3: set_fm] : ( ord_less_eq_set_fm @ X3 @ X3 ) ).

% order_refl
thf(fact_280_order__refl,axiom,
    ! [X3: nat] : ( ord_less_eq_nat @ X3 @ X3 ) ).

% order_refl
thf(fact_281_subset__code_I1_J,axiom,
    ! [Xs: list_rule,B5: set_rule] :
      ( ( ord_less_eq_set_rule @ ( set_rule2 @ Xs ) @ B5 )
      = ( ! [X6: rule] :
            ( ( member_rule @ X6 @ ( set_rule2 @ Xs ) )
           => ( member_rule @ X6 @ B5 ) ) ) ) ).

% subset_code(1)
thf(fact_282_subset__code_I1_J,axiom,
    ! [Xs: list_P5616295576739893671ist_fm,B5: set_Pr7058068377845519745ist_fm] :
      ( ( ord_le2055072402069232929ist_fm @ ( set_Pr8767716839810916150ist_fm @ Xs ) @ B5 )
      = ( ! [X6: produc1996495991257130529ist_fm] :
            ( ( member8102475879199740618ist_fm @ X6 @ ( set_Pr8767716839810916150ist_fm @ Xs ) )
           => ( member8102475879199740618ist_fm @ X6 @ B5 ) ) ) ) ).

% subset_code(1)
thf(fact_283_subset__code_I1_J,axiom,
    ! [Xs: list_P8469869581646625389at_nat,B5: set_Pr8693737435421807431at_nat] :
      ( ( ord_le3000389064537975527at_nat @ ( set_Pr5518436109238095868at_nat @ Xs ) @ B5 )
      = ( ! [X6: produc859450856879609959at_nat] :
            ( ( member8206827879206165904at_nat @ X6 @ ( set_Pr5518436109238095868at_nat @ Xs ) )
           => ( member8206827879206165904at_nat @ X6 @ B5 ) ) ) ) ).

% subset_code(1)
thf(fact_284_subset__code_I1_J,axiom,
    ! [Xs: list_fm,B5: set_fm] :
      ( ( ord_less_eq_set_fm @ ( set_fm2 @ Xs ) @ B5 )
      = ( ! [X6: fm] :
            ( ( member_fm @ X6 @ ( set_fm2 @ Xs ) )
           => ( member_fm @ X6 @ B5 ) ) ) ) ).

% subset_code(1)
thf(fact_285_RuleSystem__Defs_Oepath_Ocoinduct,axiom,
    ! [X7: stream8299795917829157543ist_fm > $o,X3: stream8299795917829157543ist_fm,Rules: stream_list_fm,Eff2: list_fm > list_fm > fset_list_fm > $o] :
      ( ( X7 @ X3 )
     => ( ! [X: stream8299795917829157543ist_fm] :
            ( ( X7 @ X )
           => ? [Steps2: stream8299795917829157543ist_fm,Sl2: fset_list_fm] :
                ( ( X = Steps2 )
                & ( member_list_fm @ ( produc4588648349897876871ist_fm @ ( shd_Pr772355297128350925ist_fm @ Steps2 ) ) @ ( sset_list_fm @ Rules ) )
                & ( fmember_list_fm @ ( produc1501393135466168645ist_fm @ ( shd_Pr772355297128350925ist_fm @ ( stl_Pr5027778045650968273ist_fm @ Steps2 ) ) ) @ Sl2 )
                & ( Eff2 @ ( produc4588648349897876871ist_fm @ ( shd_Pr772355297128350925ist_fm @ Steps2 ) ) @ ( produc1501393135466168645ist_fm @ ( shd_Pr772355297128350925ist_fm @ Steps2 ) ) @ Sl2 )
                & ( ( X7 @ ( stl_Pr5027778045650968273ist_fm @ Steps2 ) )
                  | ( abstra6579460235205289861ist_fm @ Eff2 @ Rules @ ( stl_Pr5027778045650968273ist_fm @ Steps2 ) ) ) ) )
       => ( abstra6579460235205289861ist_fm @ Eff2 @ Rules @ X3 ) ) ) ).

% RuleSystem_Defs.epath.coinduct
thf(fact_286_RuleSystem__Defs_Oepath_Ocoinduct,axiom,
    ! [X7: stream5128679079952569136_fm_fm > $o,X3: stream5128679079952569136_fm_fm,Rules: stream_fm,Eff2: fm > produc1996495991257130529ist_fm > fset_P661503646757059847ist_fm > $o] :
      ( ( X7 @ X3 )
     => ( ! [X: stream5128679079952569136_fm_fm] :
            ( ( X7 @ X )
           => ? [Steps2: stream5128679079952569136_fm_fm,Sl2: fset_P661503646757059847ist_fm] :
                ( ( X = Steps2 )
                & ( member_fm @ ( produc6738438136904575596_fm_fm @ ( shd_Pr7265661315679113588_fm_fm @ Steps2 ) ) @ ( sset_fm @ Rules ) )
                & ( fmembe3381613331217039976ist_fm @ ( produc1217498474891896878_fm_fm @ ( shd_Pr7265661315679113588_fm_fm @ ( stl_Pr2715999368752424048_fm_fm @ Steps2 ) ) ) @ Sl2 )
                & ( Eff2 @ ( produc6738438136904575596_fm_fm @ ( shd_Pr7265661315679113588_fm_fm @ Steps2 ) ) @ ( produc1217498474891896878_fm_fm @ ( shd_Pr7265661315679113588_fm_fm @ Steps2 ) ) @ Sl2 )
                & ( ( X7 @ ( stl_Pr2715999368752424048_fm_fm @ Steps2 ) )
                  | ( abstra9155489748554225612ist_fm @ Eff2 @ Rules @ ( stl_Pr2715999368752424048_fm_fm @ Steps2 ) ) ) ) )
       => ( abstra9155489748554225612ist_fm @ Eff2 @ Rules @ X3 ) ) ) ).

% RuleSystem_Defs.epath.coinduct
thf(fact_287_RuleSystem__Defs_Oepath_Ocoinduct,axiom,
    ! [X7: stream727092118206550309m_rule > $o,X3: stream727092118206550309m_rule,Rules: stream_rule,Eff2: rule > produc1996495991257130529ist_fm > fset_P661503646757059847ist_fm > $o] :
      ( ( X7 @ X3 )
     => ( ! [X: stream727092118206550309m_rule] :
            ( ( X7 @ X )
           => ? [Steps2: stream727092118206550309m_rule,Sl2: fset_P661503646757059847ist_fm] :
                ( ( X = Steps2 )
                & ( member_rule @ ( produc7165828336582415457m_rule @ ( shd_Pr7235097944458474089m_rule @ Steps2 ) ) @ ( sset_rule @ Rules ) )
                & ( fmembe3381613331217039976ist_fm @ ( produc6879501374131015971m_rule @ ( shd_Pr7235097944458474089m_rule @ ( stl_Pr950425576149878629m_rule @ Steps2 ) ) ) @ Sl2 )
                & ( Eff2 @ ( produc7165828336582415457m_rule @ ( shd_Pr7235097944458474089m_rule @ Steps2 ) ) @ ( produc6879501374131015971m_rule @ ( shd_Pr7235097944458474089m_rule @ Steps2 ) ) @ Sl2 )
                & ( ( X7 @ ( stl_Pr950425576149878629m_rule @ Steps2 ) )
                  | ( abstra5415202973855822231ist_fm @ Eff2 @ Rules @ ( stl_Pr950425576149878629m_rule @ Steps2 ) ) ) ) )
       => ( abstra5415202973855822231ist_fm @ Eff2 @ Rules @ X3 ) ) ) ).

% RuleSystem_Defs.epath.coinduct
thf(fact_288_RuleSystem__Defs_Oepath_Ocoinduct,axiom,
    ! [X7: stream812778442145099016ule_fm > $o,X3: stream812778442145099016ule_fm,Rules: stream_fm,Eff2: fm > abstra4572470701423150929m_rule > fset_A5396818216090053761m_rule > $o] :
      ( ( X7 @ X3 )
     => ( ! [X: stream812778442145099016ule_fm] :
            ( ( X7 @ X )
           => ? [Steps2: stream812778442145099016ule_fm,Sl2: fset_A5396818216090053761m_rule] :
                ( ( X = Steps2 )
                & ( member_fm @ ( produc8096733172076065128ule_fm @ ( shd_Pr849711437766469422ule_fm @ Steps2 ) ) @ ( sset_fm @ Rules ) )
                & ( fmembe1635983029874454474m_rule @ ( produc8194852094735076902ule_fm @ ( shd_Pr849711437766469422ule_fm @ ( stl_Pr2064664776838053682ule_fm @ Steps2 ) ) ) @ Sl2 )
                & ( Eff2 @ ( produc8096733172076065128ule_fm @ ( shd_Pr849711437766469422ule_fm @ Steps2 ) ) @ ( produc8194852094735076902ule_fm @ ( shd_Pr849711437766469422ule_fm @ Steps2 ) ) @ Sl2 )
                & ( ( X7 @ ( stl_Pr2064664776838053682ule_fm @ Steps2 ) )
                  | ( abstra1364558218721404774m_rule @ Eff2 @ Rules @ ( stl_Pr2064664776838053682ule_fm @ Steps2 ) ) ) ) )
       => ( abstra1364558218721404774m_rule @ Eff2 @ Rules @ X3 ) ) ) ).

% RuleSystem_Defs.epath.coinduct
thf(fact_289_RuleSystem__Defs_Oepath_Ocoinduct,axiom,
    ! [X7: stream7755889244956737277e_rule > $o,X3: stream7755889244956737277e_rule,Rules: stream_rule,Eff2: rule > abstra4572470701423150929m_rule > fset_A5396818216090053761m_rule > $o] :
      ( ( X7 @ X3 )
     => ( ! [X: stream7755889244956737277e_rule] :
            ( ( X7 @ X )
           => ? [Steps2: stream7755889244956737277e_rule,Sl2: fset_A5396818216090053761m_rule] :
                ( ( X = Steps2 )
                & ( member_rule @ ( produc4652457655725677917e_rule @ ( shd_Pr1235285222789368611e_rule @ Steps2 ) ) @ ( sset_rule @ Rules ) )
                & ( fmembe1635983029874454474m_rule @ ( produc7822628117027650331e_rule @ ( shd_Pr1235285222789368611e_rule @ ( stl_Pr8695508334200742183e_rule @ Steps2 ) ) ) @ Sl2 )
                & ( Eff2 @ ( produc4652457655725677917e_rule @ ( shd_Pr1235285222789368611e_rule @ Steps2 ) ) @ ( produc7822628117027650331e_rule @ ( shd_Pr1235285222789368611e_rule @ Steps2 ) ) @ Sl2 )
                & ( ( X7 @ ( stl_Pr8695508334200742183e_rule @ Steps2 ) )
                  | ( abstra5823804033175701339m_rule @ Eff2 @ Rules @ ( stl_Pr8695508334200742183e_rule @ Steps2 ) ) ) ) )
       => ( abstra5823804033175701339m_rule @ Eff2 @ Rules @ X3 ) ) ) ).

% RuleSystem_Defs.epath.coinduct
thf(fact_290_RuleSystem__Defs_Oepath_Ocoinduct,axiom,
    ! [X7: stream3409308193418444653ist_fm > $o,X3: stream3409308193418444653ist_fm,Rules: stream8299795917829157543ist_fm,Eff2: produc1996495991257130529ist_fm > produc1996495991257130529ist_fm > fset_P661503646757059847ist_fm > $o] :
      ( ( X7 @ X3 )
     => ( ! [X: stream3409308193418444653ist_fm] :
            ( ( X7 @ X )
           => ? [Steps2: stream3409308193418444653ist_fm,Sl2: fset_P661503646757059847ist_fm] :
                ( ( X = Steps2 )
                & ( member8102475879199740618ist_fm @ ( produc1639249265332130309ist_fm @ ( shd_Pr4587271278095355155ist_fm @ Steps2 ) ) @ ( sset_P3431374123360452149ist_fm @ Rules ) )
                & ( fmembe3381613331217039976ist_fm @ ( produc5894184038225292483ist_fm @ ( shd_Pr4587271278095355155ist_fm @ ( stl_Pr8700568296466786583ist_fm @ Steps2 ) ) ) @ Sl2 )
                & ( Eff2 @ ( produc1639249265332130309ist_fm @ ( shd_Pr4587271278095355155ist_fm @ Steps2 ) ) @ ( produc5894184038225292483ist_fm @ ( shd_Pr4587271278095355155ist_fm @ Steps2 ) ) @ Sl2 )
                & ( ( X7 @ ( stl_Pr8700568296466786583ist_fm @ Steps2 ) )
                  | ( abstra5061764961602482947ist_fm @ Eff2 @ Rules @ ( stl_Pr8700568296466786583ist_fm @ Steps2 ) ) ) ) )
       => ( abstra5061764961602482947ist_fm @ Eff2 @ Rules @ X3 ) ) ) ).

% RuleSystem_Defs.epath.coinduct
thf(fact_291_RuleSystem__Defs_Oepath_Ocoinduct,axiom,
    ! [X7: stream1622999302952468147at_nat > $o,X3: stream1622999302952468147at_nat,Rules: stream8372878641218411373at_nat,Eff2: produc859450856879609959at_nat > produc1996495991257130529ist_fm > fset_P661503646757059847ist_fm > $o] :
      ( ( X7 @ X3 )
     => ( ! [X: stream1622999302952468147at_nat] :
            ( ( X7 @ X )
           => ? [Steps2: stream1622999302952468147at_nat,Sl2: fset_P661503646757059847ist_fm] :
                ( ( X = Steps2 )
                & ( member8206827879206165904at_nat @ ( produc2750861588394444363at_nat @ ( shd_Pr5062688428016131033at_nat @ Steps2 ) ) @ ( sset_P9164763103296375419at_nat @ Rules ) )
                & ( fmembe3381613331217039976ist_fm @ ( produc2934448030062000137at_nat @ ( shd_Pr5062688428016131033at_nat @ ( stl_Pr1564049449699658205at_nat @ Steps2 ) ) ) @ Sl2 )
                & ( Eff2 @ ( produc2750861588394444363at_nat @ ( shd_Pr5062688428016131033at_nat @ Steps2 ) ) @ ( produc2934448030062000137at_nat @ ( shd_Pr5062688428016131033at_nat @ Steps2 ) ) @ Sl2 )
                & ( ( X7 @ ( stl_Pr1564049449699658205at_nat @ Steps2 ) )
                  | ( abstra1311857326046938941ist_fm @ Eff2 @ Rules @ ( stl_Pr1564049449699658205at_nat @ Steps2 ) ) ) ) )
       => ( abstra1311857326046938941ist_fm @ Eff2 @ Rules @ X3 ) ) ) ).

% RuleSystem_Defs.epath.coinduct
thf(fact_292_RuleSystem__Defs_Oepath_Ocoinduct,axiom,
    ! [X7: stream642806101564963573m_rule > $o,X3: stream642806101564963573m_rule,Rules: stream727092118206550309m_rule,Eff2: produc164195504107695125m_rule > produc1996495991257130529ist_fm > fset_P661503646757059847ist_fm > $o] :
      ( ( X7 @ X3 )
     => ( ! [X: stream642806101564963573m_rule] :
            ( ( X7 @ X )
           => ? [Steps2: stream642806101564963573m_rule,Sl2: fset_P661503646757059847ist_fm] :
                ( ( X = Steps2 )
                & ( member4220325220686508332m_rule @ ( produc2016132375285593905m_rule @ ( shd_Pr1407159933805374393m_rule @ Steps2 ) ) @ ( sset_P4979403222752902913m_rule @ Rules ) )
                & ( fmembe3381613331217039976ist_fm @ ( produc8602611229794383091m_rule @ ( shd_Pr1407159933805374393m_rule @ ( stl_Pr8578586445745536693m_rule @ Steps2 ) ) ) @ Sl2 )
                & ( Eff2 @ ( produc2016132375285593905m_rule @ ( shd_Pr1407159933805374393m_rule @ Steps2 ) ) @ ( produc8602611229794383091m_rule @ ( shd_Pr1407159933805374393m_rule @ Steps2 ) ) @ Sl2 )
                & ( ( X7 @ ( stl_Pr8578586445745536693m_rule @ Steps2 ) )
                  | ( abstra6058711572139910855ist_fm @ Eff2 @ Rules @ ( stl_Pr8578586445745536693m_rule @ Steps2 ) ) ) ) )
       => ( abstra6058711572139910855ist_fm @ Eff2 @ Rules @ X3 ) ) ) ).

% RuleSystem_Defs.epath.coinduct
thf(fact_293_RuleSystem__Defs_Oepath_Ocoinduct,axiom,
    ! [X7: stream7989562319868115861ist_fm > $o,X3: stream7989562319868115861ist_fm,Rules: stream8299795917829157543ist_fm,Eff2: produc1996495991257130529ist_fm > abstra4572470701423150929m_rule > fset_A5396818216090053761m_rule > $o] :
      ( ( X7 @ X3 )
     => ( ! [X: stream7989562319868115861ist_fm] :
            ( ( X7 @ X )
           => ? [Steps2: stream7989562319868115861ist_fm,Sl2: fset_A5396818216090053761m_rule] :
                ( ( X = Steps2 )
                & ( member8102475879199740618ist_fm @ ( produc3674566753905654281ist_fm @ ( shd_Pr563299216415026649ist_fm @ Steps2 ) ) @ ( sset_P3431374123360452149ist_fm @ Rules ) )
                & ( fmembe1635983029874454474m_rule @ ( produc2646333476271629515ist_fm @ ( shd_Pr563299216415026649ist_fm @ ( stl_Pr6727363551237990101ist_fm @ Steps2 ) ) ) @ Sl2 )
                & ( Eff2 @ ( produc3674566753905654281ist_fm @ ( shd_Pr563299216415026649ist_fm @ Steps2 ) ) @ ( produc2646333476271629515ist_fm @ ( shd_Pr563299216415026649ist_fm @ Steps2 ) ) @ Sl2 )
                & ( ( X7 @ ( stl_Pr6727363551237990101ist_fm @ Steps2 ) )
                  | ( abstra8066765647771630831m_rule @ Eff2 @ Rules @ ( stl_Pr6727363551237990101ist_fm @ Steps2 ) ) ) ) )
       => ( abstra8066765647771630831m_rule @ Eff2 @ Rules @ X3 ) ) ) ).

% RuleSystem_Defs.epath.coinduct
thf(fact_294_RuleSystem__Defs_Oepath_Ocoinduct,axiom,
    ! [X7: stream1819640557982756571at_nat > $o,X3: stream1819640557982756571at_nat,Rules: stream8372878641218411373at_nat,Eff2: produc859450856879609959at_nat > abstra4572470701423150929m_rule > fset_A5396818216090053761m_rule > $o] :
      ( ( X7 @ X3 )
     => ( ! [X: stream1819640557982756571at_nat] :
            ( ( X7 @ X )
           => ? [Steps2: stream1819640557982756571at_nat,Sl2: fset_A5396818216090053761m_rule] :
                ( ( X = Steps2 )
                & ( member8206827879206165904at_nat @ ( produc601515061966782543at_nat @ ( shd_Pr618815079712798623at_nat @ Steps2 ) ) @ ( sset_P9164763103296375419at_nat @ Rules ) )
                & ( fmembe1635983029874454474m_rule @ ( produc567330280010208273at_nat @ ( shd_Pr618815079712798623at_nat @ ( stl_Pr2369401607900929691at_nat @ Steps2 ) ) ) @ Sl2 )
                & ( Eff2 @ ( produc601515061966782543at_nat @ ( shd_Pr618815079712798623at_nat @ Steps2 ) ) @ ( produc567330280010208273at_nat @ ( shd_Pr618815079712798623at_nat @ Steps2 ) ) @ Sl2 )
                & ( ( X7 @ ( stl_Pr2369401607900929691at_nat @ Steps2 ) )
                  | ( abstra3157339599784851765m_rule @ Eff2 @ Rules @ ( stl_Pr2369401607900929691at_nat @ Steps2 ) ) ) ) )
       => ( abstra3157339599784851765m_rule @ Eff2 @ Rules @ X3 ) ) ) ).

% RuleSystem_Defs.epath.coinduct
thf(fact_295_pred__subset__eq2,axiom,
    ! [R3: set_Pr1008144964186165195m_rule,S4: set_Pr1008144964186165195m_rule] :
      ( ( ord_le4891813983447887534rule_o
        @ ^ [X6: produc1996495991257130529ist_fm,Y5: rule] : ( member4220325220686508332m_rule @ ( produc491467635432902671m_rule @ X6 @ Y5 ) @ R3 )
        @ ^ [X6: produc1996495991257130529ist_fm,Y5: rule] : ( member4220325220686508332m_rule @ ( produc491467635432902671m_rule @ X6 @ Y5 ) @ S4 ) )
      = ( ord_le5052251716133888043m_rule @ R3 @ S4 ) ) ).

% pred_subset_eq2
thf(fact_296_pred__subset__eq2,axiom,
    ! [R3: set_Pr7058068377845519745ist_fm,S4: set_Pr7058068377845519745ist_fm] :
      ( ( ord_le1774903256965159270t_fm_o
        @ ^ [X6: list_fm,Y5: list_fm] : ( member8102475879199740618ist_fm @ ( produc381145313068854617ist_fm @ X6 @ Y5 ) @ R3 )
        @ ^ [X6: list_fm,Y5: list_fm] : ( member8102475879199740618ist_fm @ ( produc381145313068854617ist_fm @ X6 @ Y5 ) @ S4 ) )
      = ( ord_le2055072402069232929ist_fm @ R3 @ S4 ) ) ).

% pred_subset_eq2
thf(fact_297_pred__subset__eq2,axiom,
    ! [R3: set_Pr1261947904930325089at_nat,S4: set_Pr1261947904930325089at_nat] :
      ( ( ord_le2646555220125990790_nat_o
        @ ^ [X6: nat,Y5: nat] : ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X6 @ Y5 ) @ R3 )
        @ ^ [X6: nat,Y5: nat] : ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X6 @ Y5 ) @ S4 ) )
      = ( ord_le3146513528884898305at_nat @ R3 @ S4 ) ) ).

% pred_subset_eq2
thf(fact_298_pred__subset__eq2,axiom,
    ! [R3: set_Pr8693737435421807431at_nat,S4: set_Pr8693737435421807431at_nat] :
      ( ( ord_le5604493270027003598_nat_o
        @ ^ [X6: product_prod_nat_nat,Y5: product_prod_nat_nat] : ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ X6 @ Y5 ) @ R3 )
        @ ^ [X6: product_prod_nat_nat,Y5: product_prod_nat_nat] : ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ X6 @ Y5 ) @ S4 ) )
      = ( ord_le3000389064537975527at_nat @ R3 @ S4 ) ) ).

% pred_subset_eq2
thf(fact_299_pred__subset__eq2,axiom,
    ! [R3: set_Pr2241365836748572487ist_fm,S4: set_Pr2241365836748572487ist_fm] :
      ( ( ord_le8079975967087607008t_fm_o
        @ ^ [X6: rule,Y5: produc1996495991257130529ist_fm] : ( member8686995742288324264ist_fm @ ( produc7063755442767259331ist_fm @ X6 @ Y5 ) @ R3 )
        @ ^ [X6: rule,Y5: produc1996495991257130529ist_fm] : ( member8686995742288324264ist_fm @ ( produc7063755442767259331ist_fm @ X6 @ Y5 ) @ S4 ) )
      = ( ord_le6285472588696295335ist_fm @ R3 @ S4 ) ) ).

% pred_subset_eq2
thf(fact_300_less__eq__set__def,axiom,
    ( ord_less_eq_set_rule
    = ( ^ [A7: set_rule,B7: set_rule] :
          ( ord_less_eq_rule_o
          @ ^ [X6: rule] : ( member_rule @ X6 @ A7 )
          @ ^ [X6: rule] : ( member_rule @ X6 @ B7 ) ) ) ) ).

% less_eq_set_def
thf(fact_301_less__eq__set__def,axiom,
    ( ord_le2055072402069232929ist_fm
    = ( ^ [A7: set_Pr7058068377845519745ist_fm,B7: set_Pr7058068377845519745ist_fm] :
          ( ord_le8518180823940269308t_fm_o
          @ ^ [X6: produc1996495991257130529ist_fm] : ( member8102475879199740618ist_fm @ X6 @ A7 )
          @ ^ [X6: produc1996495991257130529ist_fm] : ( member8102475879199740618ist_fm @ X6 @ B7 ) ) ) ) ).

% less_eq_set_def
thf(fact_302_less__eq__set__def,axiom,
    ( ord_le3000389064537975527at_nat
    = ( ^ [A7: set_Pr8693737435421807431at_nat,B7: set_Pr8693737435421807431at_nat] :
          ( ord_le7858099551454983350_nat_o
          @ ^ [X6: produc859450856879609959at_nat] : ( member8206827879206165904at_nat @ X6 @ A7 )
          @ ^ [X6: produc859450856879609959at_nat] : ( member8206827879206165904at_nat @ X6 @ B7 ) ) ) ) ).

% less_eq_set_def
thf(fact_303_less__eq__set__def,axiom,
    ( ord_less_eq_set_fm
    = ( ^ [A7: set_fm,B7: set_fm] :
          ( ord_less_eq_fm_o
          @ ^ [X6: fm] : ( member_fm @ X6 @ A7 )
          @ ^ [X6: fm] : ( member_fm @ X6 @ B7 ) ) ) ) ).

% less_eq_set_def
thf(fact_304_shd__sset,axiom,
    ! [A: stream_fm] : ( member_fm @ ( shd_fm @ A ) @ ( sset_fm @ A ) ) ).

% shd_sset
thf(fact_305_shd__sset,axiom,
    ! [A: stream8299795917829157543ist_fm] : ( member8102475879199740618ist_fm @ ( shd_Pr772355297128350925ist_fm @ A ) @ ( sset_P3431374123360452149ist_fm @ A ) ) ).

% shd_sset
thf(fact_306_shd__sset,axiom,
    ! [A: stream8372878641218411373at_nat] : ( member8206827879206165904at_nat @ ( shd_Pr8412153233960533267at_nat @ A ) @ ( sset_P9164763103296375419at_nat @ A ) ) ).

% shd_sset
thf(fact_307_shd__sset,axiom,
    ! [A: stream_rule] : ( member_rule @ ( shd_rule @ A ) @ ( sset_rule @ A ) ) ).

% shd_sset
thf(fact_308_shd__sset,axiom,
    ! [A: stream727092118206550309m_rule] : ( member4220325220686508332m_rule @ ( shd_Pr7235097944458474089m_rule @ A ) @ ( sset_P4979403222752902913m_rule @ A ) ) ).

% shd_sset
thf(fact_309_stl__sset,axiom,
    ! [X3: fm,A: stream_fm] :
      ( ( member_fm @ X3 @ ( sset_fm @ ( stl_fm @ A ) ) )
     => ( member_fm @ X3 @ ( sset_fm @ A ) ) ) ).

% stl_sset
thf(fact_310_stl__sset,axiom,
    ! [X3: produc1996495991257130529ist_fm,A: stream8299795917829157543ist_fm] :
      ( ( member8102475879199740618ist_fm @ X3 @ ( sset_P3431374123360452149ist_fm @ ( stl_Pr5027778045650968273ist_fm @ A ) ) )
     => ( member8102475879199740618ist_fm @ X3 @ ( sset_P3431374123360452149ist_fm @ A ) ) ) ).

% stl_sset
thf(fact_311_stl__sset,axiom,
    ! [X3: produc859450856879609959at_nat,A: stream8372878641218411373at_nat] :
      ( ( member8206827879206165904at_nat @ X3 @ ( sset_P9164763103296375419at_nat @ ( stl_Pr5010465336888994071at_nat @ A ) ) )
     => ( member8206827879206165904at_nat @ X3 @ ( sset_P9164763103296375419at_nat @ A ) ) ) ).

% stl_sset
thf(fact_312_stl__sset,axiom,
    ! [X3: rule,A: stream_rule] :
      ( ( member_rule @ X3 @ ( sset_rule @ ( stl_rule @ A ) ) )
     => ( member_rule @ X3 @ ( sset_rule @ A ) ) ) ).

% stl_sset
thf(fact_313_stl__sset,axiom,
    ! [X3: produc164195504107695125m_rule,A: stream727092118206550309m_rule] :
      ( ( member4220325220686508332m_rule @ X3 @ ( sset_P4979403222752902913m_rule @ ( stl_Pr950425576149878629m_rule @ A ) ) )
     => ( member4220325220686508332m_rule @ X3 @ ( sset_P4979403222752902913m_rule @ A ) ) ) ).

% stl_sset
thf(fact_314_snth__sset,axiom,
    ! [S3: stream_fm,N: nat] : ( member_fm @ ( snth_fm @ S3 @ N ) @ ( sset_fm @ S3 ) ) ).

% snth_sset
thf(fact_315_snth__sset,axiom,
    ! [S3: stream8299795917829157543ist_fm,N: nat] : ( member8102475879199740618ist_fm @ ( snth_P2624749845746815685ist_fm @ S3 @ N ) @ ( sset_P3431374123360452149ist_fm @ S3 ) ) ).

% snth_sset
thf(fact_316_snth__sset,axiom,
    ! [S3: stream8372878641218411373at_nat,N: nat] : ( member8206827879206165904at_nat @ ( snth_P9098630209796499211at_nat @ S3 @ N ) @ ( sset_P9164763103296375419at_nat @ S3 ) ) ).

% snth_sset
thf(fact_317_snth__sset,axiom,
    ! [S3: stream_rule,N: nat] : ( member_rule @ ( snth_rule @ S3 @ N ) @ ( sset_rule @ S3 ) ) ).

% snth_sset
thf(fact_318_snth__sset,axiom,
    ! [S3: stream727092118206550309m_rule,N: nat] : ( member4220325220686508332m_rule @ ( snth_P1712579846118164081m_rule @ S3 @ N ) @ ( sset_P4979403222752902913m_rule @ S3 ) ) ).

% snth_sset
thf(fact_319_subrelI,axiom,
    ! [R2: set_Pr1008144964186165195m_rule,S3: set_Pr1008144964186165195m_rule] :
      ( ! [X: produc1996495991257130529ist_fm,Y: rule] :
          ( ( member4220325220686508332m_rule @ ( produc491467635432902671m_rule @ X @ Y ) @ R2 )
         => ( member4220325220686508332m_rule @ ( produc491467635432902671m_rule @ X @ Y ) @ S3 ) )
     => ( ord_le5052251716133888043m_rule @ R2 @ S3 ) ) ).

% subrelI
thf(fact_320_subrelI,axiom,
    ! [R2: set_Pr7058068377845519745ist_fm,S3: set_Pr7058068377845519745ist_fm] :
      ( ! [X: list_fm,Y: list_fm] :
          ( ( member8102475879199740618ist_fm @ ( produc381145313068854617ist_fm @ X @ Y ) @ R2 )
         => ( member8102475879199740618ist_fm @ ( produc381145313068854617ist_fm @ X @ Y ) @ S3 ) )
     => ( ord_le2055072402069232929ist_fm @ R2 @ S3 ) ) ).

% subrelI
thf(fact_321_subrelI,axiom,
    ! [R2: set_Pr1261947904930325089at_nat,S3: set_Pr1261947904930325089at_nat] :
      ( ! [X: nat,Y: nat] :
          ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X @ Y ) @ R2 )
         => ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X @ Y ) @ S3 ) )
     => ( ord_le3146513528884898305at_nat @ R2 @ S3 ) ) ).

% subrelI
thf(fact_322_subrelI,axiom,
    ! [R2: set_Pr8693737435421807431at_nat,S3: set_Pr8693737435421807431at_nat] :
      ( ! [X: product_prod_nat_nat,Y: product_prod_nat_nat] :
          ( ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ X @ Y ) @ R2 )
         => ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ X @ Y ) @ S3 ) )
     => ( ord_le3000389064537975527at_nat @ R2 @ S3 ) ) ).

% subrelI
thf(fact_323_subrelI,axiom,
    ! [R2: set_Pr2241365836748572487ist_fm,S3: set_Pr2241365836748572487ist_fm] :
      ( ! [X: rule,Y: produc1996495991257130529ist_fm] :
          ( ( member8686995742288324264ist_fm @ ( produc7063755442767259331ist_fm @ X @ Y ) @ R2 )
         => ( member8686995742288324264ist_fm @ ( produc7063755442767259331ist_fm @ X @ Y ) @ S3 ) )
     => ( ord_le6285472588696295335ist_fm @ R2 @ S3 ) ) ).

% subrelI
thf(fact_324_pred__subset__eq,axiom,
    ! [R3: set_rule,S4: set_rule] :
      ( ( ord_less_eq_rule_o
        @ ^ [X6: rule] : ( member_rule @ X6 @ R3 )
        @ ^ [X6: rule] : ( member_rule @ X6 @ S4 ) )
      = ( ord_less_eq_set_rule @ R3 @ S4 ) ) ).

% pred_subset_eq
thf(fact_325_pred__subset__eq,axiom,
    ! [R3: set_Pr7058068377845519745ist_fm,S4: set_Pr7058068377845519745ist_fm] :
      ( ( ord_le8518180823940269308t_fm_o
        @ ^ [X6: produc1996495991257130529ist_fm] : ( member8102475879199740618ist_fm @ X6 @ R3 )
        @ ^ [X6: produc1996495991257130529ist_fm] : ( member8102475879199740618ist_fm @ X6 @ S4 ) )
      = ( ord_le2055072402069232929ist_fm @ R3 @ S4 ) ) ).

% pred_subset_eq
thf(fact_326_pred__subset__eq,axiom,
    ! [R3: set_Pr8693737435421807431at_nat,S4: set_Pr8693737435421807431at_nat] :
      ( ( ord_le7858099551454983350_nat_o
        @ ^ [X6: produc859450856879609959at_nat] : ( member8206827879206165904at_nat @ X6 @ R3 )
        @ ^ [X6: produc859450856879609959at_nat] : ( member8206827879206165904at_nat @ X6 @ S4 ) )
      = ( ord_le3000389064537975527at_nat @ R3 @ S4 ) ) ).

% pred_subset_eq
thf(fact_327_pred__subset__eq,axiom,
    ! [R3: set_fm,S4: set_fm] :
      ( ( ord_less_eq_fm_o
        @ ^ [X6: fm] : ( member_fm @ X6 @ R3 )
        @ ^ [X6: fm] : ( member_fm @ X6 @ S4 ) )
      = ( ord_less_eq_set_fm @ R3 @ S4 ) ) ).

% pred_subset_eq
thf(fact_328_sset__sdrop,axiom,
    ! [N: nat,S3: stream8299795917829157543ist_fm] : ( ord_le2055072402069232929ist_fm @ ( sset_P3431374123360452149ist_fm @ ( sdrop_4442373711808556042ist_fm @ N @ S3 ) ) @ ( sset_P3431374123360452149ist_fm @ S3 ) ) ).

% sset_sdrop
thf(fact_329_sset__sdrop,axiom,
    ! [N: nat,S3: stream3588992565182678257e_rule] : ( ord_le4515388646595166059e_rule @ ( sset_P4203794550272384383e_rule @ ( sdrop_2199339672054592340e_rule @ N @ S3 ) ) @ ( sset_P4203794550272384383e_rule @ S3 ) ) ).

% sset_sdrop
thf(fact_330_sset__sdrop,axiom,
    ! [N: nat,S3: stream1878305877988316353m_rule] : ( ord_le6218097786609347643m_rule @ ( sset_P8020534884870570447m_rule @ ( sdrop_1764812713384251940m_rule @ N @ S3 ) ) @ ( sset_P8020534884870570447m_rule @ S3 ) ) ).

% sset_sdrop
thf(fact_331_sset__sdrop,axiom,
    ! [N: nat,S3: stream2312013117288958913e_rule] : ( ord_le6651805025909990203e_rule @ ( sset_P3425104634812430543e_rule @ ( sdrop_6392754500180887844e_rule @ N @ S3 ) ) @ ( sset_P3425104634812430543e_rule @ S3 ) ) ).

% sset_sdrop
thf(fact_332_sset__sdrop,axiom,
    ! [N: nat,S3: stream6494289010434245521m_rule] : ( ord_le3684904531561281547m_rule @ ( sset_P270791984657859871m_rule @ ( sdrop_7373388980841566196m_rule @ N @ S3 ) ) @ ( sset_P270791984657859871m_rule @ S3 ) ) ).

% sset_sdrop
thf(fact_333_sset__sdrop,axiom,
    ! [N: nat,S3: stream_rule] : ( ord_less_eq_set_rule @ ( sset_rule @ ( sdrop_rule @ N @ S3 ) ) @ ( sset_rule @ S3 ) ) ).

% sset_sdrop
thf(fact_334_sset__sdrop,axiom,
    ! [N: nat,S3: stream727092118206550309m_rule] : ( ord_le5052251716133888043m_rule @ ( sset_P4979403222752902913m_rule @ ( sdrop_7224736112439592940m_rule @ N @ S3 ) ) @ ( sset_P4979403222752902913m_rule @ S3 ) ) ).

% sset_sdrop
thf(fact_335_sset__sdrop,axiom,
    ! [N: nat,S3: stream_fm] : ( ord_less_eq_set_fm @ ( sset_fm @ ( sdrop_fm @ N @ S3 ) ) @ ( sset_fm @ S3 ) ) ).

% sset_sdrop
thf(fact_336_sset__induct,axiom,
    ! [Y3: fm,S3: stream_fm,P2: fm > stream_fm > $o] :
      ( ( member_fm @ Y3 @ ( sset_fm @ S3 ) )
     => ( ! [S6: stream_fm] : ( P2 @ ( shd_fm @ S6 ) @ S6 )
       => ( ! [S6: stream_fm,Y: fm] :
              ( ( member_fm @ Y @ ( sset_fm @ ( stl_fm @ S6 ) ) )
             => ( ( P2 @ Y @ ( stl_fm @ S6 ) )
               => ( P2 @ Y @ S6 ) ) )
         => ( P2 @ Y3 @ S3 ) ) ) ) ).

% sset_induct
thf(fact_337_sset__induct,axiom,
    ! [Y3: produc1996495991257130529ist_fm,S3: stream8299795917829157543ist_fm,P2: produc1996495991257130529ist_fm > stream8299795917829157543ist_fm > $o] :
      ( ( member8102475879199740618ist_fm @ Y3 @ ( sset_P3431374123360452149ist_fm @ S3 ) )
     => ( ! [S6: stream8299795917829157543ist_fm] : ( P2 @ ( shd_Pr772355297128350925ist_fm @ S6 ) @ S6 )
       => ( ! [S6: stream8299795917829157543ist_fm,Y: produc1996495991257130529ist_fm] :
              ( ( member8102475879199740618ist_fm @ Y @ ( sset_P3431374123360452149ist_fm @ ( stl_Pr5027778045650968273ist_fm @ S6 ) ) )
             => ( ( P2 @ Y @ ( stl_Pr5027778045650968273ist_fm @ S6 ) )
               => ( P2 @ Y @ S6 ) ) )
         => ( P2 @ Y3 @ S3 ) ) ) ) ).

% sset_induct
thf(fact_338_sset__induct,axiom,
    ! [Y3: produc859450856879609959at_nat,S3: stream8372878641218411373at_nat,P2: produc859450856879609959at_nat > stream8372878641218411373at_nat > $o] :
      ( ( member8206827879206165904at_nat @ Y3 @ ( sset_P9164763103296375419at_nat @ S3 ) )
     => ( ! [S6: stream8372878641218411373at_nat] : ( P2 @ ( shd_Pr8412153233960533267at_nat @ S6 ) @ S6 )
       => ( ! [S6: stream8372878641218411373at_nat,Y: produc859450856879609959at_nat] :
              ( ( member8206827879206165904at_nat @ Y @ ( sset_P9164763103296375419at_nat @ ( stl_Pr5010465336888994071at_nat @ S6 ) ) )
             => ( ( P2 @ Y @ ( stl_Pr5010465336888994071at_nat @ S6 ) )
               => ( P2 @ Y @ S6 ) ) )
         => ( P2 @ Y3 @ S3 ) ) ) ) ).

% sset_induct
thf(fact_339_sset__induct,axiom,
    ! [Y3: rule,S3: stream_rule,P2: rule > stream_rule > $o] :
      ( ( member_rule @ Y3 @ ( sset_rule @ S3 ) )
     => ( ! [S6: stream_rule] : ( P2 @ ( shd_rule @ S6 ) @ S6 )
       => ( ! [S6: stream_rule,Y: rule] :
              ( ( member_rule @ Y @ ( sset_rule @ ( stl_rule @ S6 ) ) )
             => ( ( P2 @ Y @ ( stl_rule @ S6 ) )
               => ( P2 @ Y @ S6 ) ) )
         => ( P2 @ Y3 @ S3 ) ) ) ) ).

% sset_induct
thf(fact_340_sset__induct,axiom,
    ! [Y3: produc164195504107695125m_rule,S3: stream727092118206550309m_rule,P2: produc164195504107695125m_rule > stream727092118206550309m_rule > $o] :
      ( ( member4220325220686508332m_rule @ Y3 @ ( sset_P4979403222752902913m_rule @ S3 ) )
     => ( ! [S6: stream727092118206550309m_rule] : ( P2 @ ( shd_Pr7235097944458474089m_rule @ S6 ) @ S6 )
       => ( ! [S6: stream727092118206550309m_rule,Y: produc164195504107695125m_rule] :
              ( ( member4220325220686508332m_rule @ Y @ ( sset_P4979403222752902913m_rule @ ( stl_Pr950425576149878629m_rule @ S6 ) ) )
             => ( ( P2 @ Y @ ( stl_Pr950425576149878629m_rule @ S6 ) )
               => ( P2 @ Y @ S6 ) ) )
         => ( P2 @ Y3 @ S3 ) ) ) ) ).

% sset_induct
thf(fact_341_nle__le,axiom,
    ! [A: nat,B: nat] :
      ( ( ~ ( ord_less_eq_nat @ A @ B ) )
      = ( ( ord_less_eq_nat @ B @ A )
        & ( B != A ) ) ) ).

% nle_le
thf(fact_342_le__cases3,axiom,
    ! [X3: nat,Y3: nat,Z2: nat] :
      ( ( ( ord_less_eq_nat @ X3 @ Y3 )
       => ~ ( ord_less_eq_nat @ Y3 @ Z2 ) )
     => ( ( ( ord_less_eq_nat @ Y3 @ X3 )
         => ~ ( ord_less_eq_nat @ X3 @ Z2 ) )
       => ( ( ( ord_less_eq_nat @ X3 @ Z2 )
           => ~ ( ord_less_eq_nat @ Z2 @ Y3 ) )
         => ( ( ( ord_less_eq_nat @ Z2 @ Y3 )
             => ~ ( ord_less_eq_nat @ Y3 @ X3 ) )
           => ( ( ( ord_less_eq_nat @ Y3 @ Z2 )
               => ~ ( ord_less_eq_nat @ Z2 @ X3 ) )
             => ~ ( ( ord_less_eq_nat @ Z2 @ X3 )
                 => ~ ( ord_less_eq_nat @ X3 @ Y3 ) ) ) ) ) ) ) ).

% le_cases3
thf(fact_343_order__class_Oorder__eq__iff,axiom,
    ( ( ^ [Y2: set_fm,Z: set_fm] : ( Y2 = Z ) )
    = ( ^ [X6: set_fm,Y5: set_fm] :
          ( ( ord_less_eq_set_fm @ X6 @ Y5 )
          & ( ord_less_eq_set_fm @ Y5 @ X6 ) ) ) ) ).

% order_class.order_eq_iff
thf(fact_344_order__class_Oorder__eq__iff,axiom,
    ( ( ^ [Y2: nat,Z: nat] : ( Y2 = Z ) )
    = ( ^ [X6: nat,Y5: nat] :
          ( ( ord_less_eq_nat @ X6 @ Y5 )
          & ( ord_less_eq_nat @ Y5 @ X6 ) ) ) ) ).

% order_class.order_eq_iff
thf(fact_345_ord__eq__le__trans,axiom,
    ! [A: set_fm,B: set_fm,C2: set_fm] :
      ( ( A = B )
     => ( ( ord_less_eq_set_fm @ B @ C2 )
       => ( ord_less_eq_set_fm @ A @ C2 ) ) ) ).

% ord_eq_le_trans
thf(fact_346_ord__eq__le__trans,axiom,
    ! [A: nat,B: nat,C2: nat] :
      ( ( A = B )
     => ( ( ord_less_eq_nat @ B @ C2 )
       => ( ord_less_eq_nat @ A @ C2 ) ) ) ).

% ord_eq_le_trans
thf(fact_347_ord__le__eq__trans,axiom,
    ! [A: set_fm,B: set_fm,C2: set_fm] :
      ( ( ord_less_eq_set_fm @ A @ B )
     => ( ( B = C2 )
       => ( ord_less_eq_set_fm @ A @ C2 ) ) ) ).

% ord_le_eq_trans
thf(fact_348_ord__le__eq__trans,axiom,
    ! [A: nat,B: nat,C2: nat] :
      ( ( ord_less_eq_nat @ A @ B )
     => ( ( B = C2 )
       => ( ord_less_eq_nat @ A @ C2 ) ) ) ).

% ord_le_eq_trans
thf(fact_349_order__antisym,axiom,
    ! [X3: set_fm,Y3: set_fm] :
      ( ( ord_less_eq_set_fm @ X3 @ Y3 )
     => ( ( ord_less_eq_set_fm @ Y3 @ X3 )
       => ( X3 = Y3 ) ) ) ).

% order_antisym
thf(fact_350_order__antisym,axiom,
    ! [X3: nat,Y3: nat] :
      ( ( ord_less_eq_nat @ X3 @ Y3 )
     => ( ( ord_less_eq_nat @ Y3 @ X3 )
       => ( X3 = Y3 ) ) ) ).

% order_antisym
thf(fact_351_order_Otrans,axiom,
    ! [A: set_fm,B: set_fm,C2: set_fm] :
      ( ( ord_less_eq_set_fm @ A @ B )
     => ( ( ord_less_eq_set_fm @ B @ C2 )
       => ( ord_less_eq_set_fm @ A @ C2 ) ) ) ).

% order.trans
thf(fact_352_order_Otrans,axiom,
    ! [A: nat,B: nat,C2: nat] :
      ( ( ord_less_eq_nat @ A @ B )
     => ( ( ord_less_eq_nat @ B @ C2 )
       => ( ord_less_eq_nat @ A @ C2 ) ) ) ).

% order.trans
thf(fact_353_order__trans,axiom,
    ! [X3: set_fm,Y3: set_fm,Z2: set_fm] :
      ( ( ord_less_eq_set_fm @ X3 @ Y3 )
     => ( ( ord_less_eq_set_fm @ Y3 @ Z2 )
       => ( ord_less_eq_set_fm @ X3 @ Z2 ) ) ) ).

% order_trans
thf(fact_354_order__trans,axiom,
    ! [X3: nat,Y3: nat,Z2: nat] :
      ( ( ord_less_eq_nat @ X3 @ Y3 )
     => ( ( ord_less_eq_nat @ Y3 @ Z2 )
       => ( ord_less_eq_nat @ X3 @ Z2 ) ) ) ).

% order_trans
thf(fact_355_linorder__wlog,axiom,
    ! [P2: nat > nat > $o,A: nat,B: nat] :
      ( ! [A4: nat,B3: nat] :
          ( ( ord_less_eq_nat @ A4 @ B3 )
         => ( P2 @ A4 @ B3 ) )
     => ( ! [A4: nat,B3: nat] :
            ( ( P2 @ B3 @ A4 )
           => ( P2 @ A4 @ B3 ) )
       => ( P2 @ A @ B ) ) ) ).

% linorder_wlog
thf(fact_356_dual__order_Oeq__iff,axiom,
    ( ( ^ [Y2: set_fm,Z: set_fm] : ( Y2 = Z ) )
    = ( ^ [A2: set_fm,B4: set_fm] :
          ( ( ord_less_eq_set_fm @ B4 @ A2 )
          & ( ord_less_eq_set_fm @ A2 @ B4 ) ) ) ) ).

% dual_order.eq_iff
thf(fact_357_dual__order_Oeq__iff,axiom,
    ( ( ^ [Y2: nat,Z: nat] : ( Y2 = Z ) )
    = ( ^ [A2: nat,B4: nat] :
          ( ( ord_less_eq_nat @ B4 @ A2 )
          & ( ord_less_eq_nat @ A2 @ B4 ) ) ) ) ).

% dual_order.eq_iff
thf(fact_358_dual__order_Oantisym,axiom,
    ! [B: set_fm,A: set_fm] :
      ( ( ord_less_eq_set_fm @ B @ A )
     => ( ( ord_less_eq_set_fm @ A @ B )
       => ( A = B ) ) ) ).

% dual_order.antisym
thf(fact_359_dual__order_Oantisym,axiom,
    ! [B: nat,A: nat] :
      ( ( ord_less_eq_nat @ B @ A )
     => ( ( ord_less_eq_nat @ A @ B )
       => ( A = B ) ) ) ).

% dual_order.antisym
thf(fact_360_dual__order_Otrans,axiom,
    ! [B: set_fm,A: set_fm,C2: set_fm] :
      ( ( ord_less_eq_set_fm @ B @ A )
     => ( ( ord_less_eq_set_fm @ C2 @ B )
       => ( ord_less_eq_set_fm @ C2 @ A ) ) ) ).

% dual_order.trans
thf(fact_361_dual__order_Otrans,axiom,
    ! [B: nat,A: nat,C2: nat] :
      ( ( ord_less_eq_nat @ B @ A )
     => ( ( ord_less_eq_nat @ C2 @ B )
       => ( ord_less_eq_nat @ C2 @ A ) ) ) ).

% dual_order.trans
thf(fact_362_antisym,axiom,
    ! [A: set_fm,B: set_fm] :
      ( ( ord_less_eq_set_fm @ A @ B )
     => ( ( ord_less_eq_set_fm @ B @ A )
       => ( A = B ) ) ) ).

% antisym
thf(fact_363_antisym,axiom,
    ! [A: nat,B: nat] :
      ( ( ord_less_eq_nat @ A @ B )
     => ( ( ord_less_eq_nat @ B @ A )
       => ( A = B ) ) ) ).

% antisym
thf(fact_364_Orderings_Oorder__eq__iff,axiom,
    ( ( ^ [Y2: set_fm,Z: set_fm] : ( Y2 = Z ) )
    = ( ^ [A2: set_fm,B4: set_fm] :
          ( ( ord_less_eq_set_fm @ A2 @ B4 )
          & ( ord_less_eq_set_fm @ B4 @ A2 ) ) ) ) ).

% Orderings.order_eq_iff
thf(fact_365_Orderings_Oorder__eq__iff,axiom,
    ( ( ^ [Y2: nat,Z: nat] : ( Y2 = Z ) )
    = ( ^ [A2: nat,B4: nat] :
          ( ( ord_less_eq_nat @ A2 @ B4 )
          & ( ord_less_eq_nat @ B4 @ A2 ) ) ) ) ).

% Orderings.order_eq_iff
thf(fact_366_order__subst1,axiom,
    ! [A: set_fm,F: set_fm > set_fm,B: set_fm,C2: set_fm] :
      ( ( ord_less_eq_set_fm @ A @ ( F @ B ) )
     => ( ( ord_less_eq_set_fm @ B @ C2 )
       => ( ! [X: set_fm,Y: set_fm] :
              ( ( ord_less_eq_set_fm @ X @ Y )
             => ( ord_less_eq_set_fm @ ( F @ X ) @ ( F @ Y ) ) )
         => ( ord_less_eq_set_fm @ A @ ( F @ C2 ) ) ) ) ) ).

% order_subst1
thf(fact_367_order__subst1,axiom,
    ! [A: set_fm,F: nat > set_fm,B: nat,C2: nat] :
      ( ( ord_less_eq_set_fm @ A @ ( F @ B ) )
     => ( ( ord_less_eq_nat @ B @ C2 )
       => ( ! [X: nat,Y: nat] :
              ( ( ord_less_eq_nat @ X @ Y )
             => ( ord_less_eq_set_fm @ ( F @ X ) @ ( F @ Y ) ) )
         => ( ord_less_eq_set_fm @ A @ ( F @ C2 ) ) ) ) ) ).

% order_subst1
thf(fact_368_order__subst1,axiom,
    ! [A: nat,F: set_fm > nat,B: set_fm,C2: set_fm] :
      ( ( ord_less_eq_nat @ A @ ( F @ B ) )
     => ( ( ord_less_eq_set_fm @ B @ C2 )
       => ( ! [X: set_fm,Y: set_fm] :
              ( ( ord_less_eq_set_fm @ X @ Y )
             => ( ord_less_eq_nat @ ( F @ X ) @ ( F @ Y ) ) )
         => ( ord_less_eq_nat @ A @ ( F @ C2 ) ) ) ) ) ).

% order_subst1
thf(fact_369_order__subst1,axiom,
    ! [A: nat,F: nat > nat,B: nat,C2: nat] :
      ( ( ord_less_eq_nat @ A @ ( F @ B ) )
     => ( ( ord_less_eq_nat @ B @ C2 )
       => ( ! [X: nat,Y: nat] :
              ( ( ord_less_eq_nat @ X @ Y )
             => ( ord_less_eq_nat @ ( F @ X ) @ ( F @ Y ) ) )
         => ( ord_less_eq_nat @ A @ ( F @ C2 ) ) ) ) ) ).

% order_subst1
thf(fact_370_order__subst2,axiom,
    ! [A: set_fm,B: set_fm,F: set_fm > set_fm,C2: set_fm] :
      ( ( ord_less_eq_set_fm @ A @ B )
     => ( ( ord_less_eq_set_fm @ ( F @ B ) @ C2 )
       => ( ! [X: set_fm,Y: set_fm] :
              ( ( ord_less_eq_set_fm @ X @ Y )
             => ( ord_less_eq_set_fm @ ( F @ X ) @ ( F @ Y ) ) )
         => ( ord_less_eq_set_fm @ ( F @ A ) @ C2 ) ) ) ) ).

% order_subst2
thf(fact_371_order__subst2,axiom,
    ! [A: set_fm,B: set_fm,F: set_fm > nat,C2: nat] :
      ( ( ord_less_eq_set_fm @ A @ B )
     => ( ( ord_less_eq_nat @ ( F @ B ) @ C2 )
       => ( ! [X: set_fm,Y: set_fm] :
              ( ( ord_less_eq_set_fm @ X @ Y )
             => ( ord_less_eq_nat @ ( F @ X ) @ ( F @ Y ) ) )
         => ( ord_less_eq_nat @ ( F @ A ) @ C2 ) ) ) ) ).

% order_subst2
thf(fact_372_order__subst2,axiom,
    ! [A: nat,B: nat,F: nat > set_fm,C2: set_fm] :
      ( ( ord_less_eq_nat @ A @ B )
     => ( ( ord_less_eq_set_fm @ ( F @ B ) @ C2 )
       => ( ! [X: nat,Y: nat] :
              ( ( ord_less_eq_nat @ X @ Y )
             => ( ord_less_eq_set_fm @ ( F @ X ) @ ( F @ Y ) ) )
         => ( ord_less_eq_set_fm @ ( F @ A ) @ C2 ) ) ) ) ).

% order_subst2
thf(fact_373_order__subst2,axiom,
    ! [A: nat,B: nat,F: nat > nat,C2: nat] :
      ( ( ord_less_eq_nat @ A @ B )
     => ( ( ord_less_eq_nat @ ( F @ B ) @ C2 )
       => ( ! [X: nat,Y: nat] :
              ( ( ord_less_eq_nat @ X @ Y )
             => ( ord_less_eq_nat @ ( F @ X ) @ ( F @ Y ) ) )
         => ( ord_less_eq_nat @ ( F @ A ) @ C2 ) ) ) ) ).

% order_subst2
thf(fact_374_order__eq__refl,axiom,
    ! [X3: set_fm,Y3: set_fm] :
      ( ( X3 = Y3 )
     => ( ord_less_eq_set_fm @ X3 @ Y3 ) ) ).

% order_eq_refl
thf(fact_375_order__eq__refl,axiom,
    ! [X3: nat,Y3: nat] :
      ( ( X3 = Y3 )
     => ( ord_less_eq_nat @ X3 @ Y3 ) ) ).

% order_eq_refl
thf(fact_376_linorder__linear,axiom,
    ! [X3: nat,Y3: nat] :
      ( ( ord_less_eq_nat @ X3 @ Y3 )
      | ( ord_less_eq_nat @ Y3 @ X3 ) ) ).

% linorder_linear
thf(fact_377_ord__eq__le__subst,axiom,
    ! [A: set_fm,F: set_fm > set_fm,B: set_fm,C2: set_fm] :
      ( ( A
        = ( F @ B ) )
     => ( ( ord_less_eq_set_fm @ B @ C2 )
       => ( ! [X: set_fm,Y: set_fm] :
              ( ( ord_less_eq_set_fm @ X @ Y )
             => ( ord_less_eq_set_fm @ ( F @ X ) @ ( F @ Y ) ) )
         => ( ord_less_eq_set_fm @ A @ ( F @ C2 ) ) ) ) ) ).

% ord_eq_le_subst
thf(fact_378_ord__eq__le__subst,axiom,
    ! [A: nat,F: set_fm > nat,B: set_fm,C2: set_fm] :
      ( ( A
        = ( F @ B ) )
     => ( ( ord_less_eq_set_fm @ B @ C2 )
       => ( ! [X: set_fm,Y: set_fm] :
              ( ( ord_less_eq_set_fm @ X @ Y )
             => ( ord_less_eq_nat @ ( F @ X ) @ ( F @ Y ) ) )
         => ( ord_less_eq_nat @ A @ ( F @ C2 ) ) ) ) ) ).

% ord_eq_le_subst
thf(fact_379_ord__eq__le__subst,axiom,
    ! [A: set_fm,F: nat > set_fm,B: nat,C2: nat] :
      ( ( A
        = ( F @ B ) )
     => ( ( ord_less_eq_nat @ B @ C2 )
       => ( ! [X: nat,Y: nat] :
              ( ( ord_less_eq_nat @ X @ Y )
             => ( ord_less_eq_set_fm @ ( F @ X ) @ ( F @ Y ) ) )
         => ( ord_less_eq_set_fm @ A @ ( F @ C2 ) ) ) ) ) ).

% ord_eq_le_subst
thf(fact_380_ord__eq__le__subst,axiom,
    ! [A: nat,F: nat > nat,B: nat,C2: nat] :
      ( ( A
        = ( F @ B ) )
     => ( ( ord_less_eq_nat @ B @ C2 )
       => ( ! [X: nat,Y: nat] :
              ( ( ord_less_eq_nat @ X @ Y )
             => ( ord_less_eq_nat @ ( F @ X ) @ ( F @ Y ) ) )
         => ( ord_less_eq_nat @ A @ ( F @ C2 ) ) ) ) ) ).

% ord_eq_le_subst
thf(fact_381_ord__le__eq__subst,axiom,
    ! [A: set_fm,B: set_fm,F: set_fm > set_fm,C2: set_fm] :
      ( ( ord_less_eq_set_fm @ A @ B )
     => ( ( ( F @ B )
          = C2 )
       => ( ! [X: set_fm,Y: set_fm] :
              ( ( ord_less_eq_set_fm @ X @ Y )
             => ( ord_less_eq_set_fm @ ( F @ X ) @ ( F @ Y ) ) )
         => ( ord_less_eq_set_fm @ ( F @ A ) @ C2 ) ) ) ) ).

% ord_le_eq_subst
thf(fact_382_ord__le__eq__subst,axiom,
    ! [A: set_fm,B: set_fm,F: set_fm > nat,C2: nat] :
      ( ( ord_less_eq_set_fm @ A @ B )
     => ( ( ( F @ B )
          = C2 )
       => ( ! [X: set_fm,Y: set_fm] :
              ( ( ord_less_eq_set_fm @ X @ Y )
             => ( ord_less_eq_nat @ ( F @ X ) @ ( F @ Y ) ) )
         => ( ord_less_eq_nat @ ( F @ A ) @ C2 ) ) ) ) ).

% ord_le_eq_subst
thf(fact_383_ord__le__eq__subst,axiom,
    ! [A: nat,B: nat,F: nat > set_fm,C2: set_fm] :
      ( ( ord_less_eq_nat @ A @ B )
     => ( ( ( F @ B )
          = C2 )
       => ( ! [X: nat,Y: nat] :
              ( ( ord_less_eq_nat @ X @ Y )
             => ( ord_less_eq_set_fm @ ( F @ X ) @ ( F @ Y ) ) )
         => ( ord_less_eq_set_fm @ ( F @ A ) @ C2 ) ) ) ) ).

% ord_le_eq_subst
thf(fact_384_ord__le__eq__subst,axiom,
    ! [A: nat,B: nat,F: nat > nat,C2: nat] :
      ( ( ord_less_eq_nat @ A @ B )
     => ( ( ( F @ B )
          = C2 )
       => ( ! [X: nat,Y: nat] :
              ( ( ord_less_eq_nat @ X @ Y )
             => ( ord_less_eq_nat @ ( F @ X ) @ ( F @ Y ) ) )
         => ( ord_less_eq_nat @ ( F @ A ) @ C2 ) ) ) ) ).

% ord_le_eq_subst
thf(fact_385_linorder__le__cases,axiom,
    ! [X3: nat,Y3: nat] :
      ( ~ ( ord_less_eq_nat @ X3 @ Y3 )
     => ( ord_less_eq_nat @ Y3 @ X3 ) ) ).

% linorder_le_cases
thf(fact_386_order__antisym__conv,axiom,
    ! [Y3: set_fm,X3: set_fm] :
      ( ( ord_less_eq_set_fm @ Y3 @ X3 )
     => ( ( ord_less_eq_set_fm @ X3 @ Y3 )
        = ( X3 = Y3 ) ) ) ).

% order_antisym_conv
thf(fact_387_order__antisym__conv,axiom,
    ! [Y3: nat,X3: nat] :
      ( ( ord_less_eq_nat @ Y3 @ X3 )
     => ( ( ord_less_eq_nat @ X3 @ Y3 )
        = ( X3 = Y3 ) ) ) ).

% order_antisym_conv
thf(fact_388_in__mono,axiom,
    ! [A5: set_rule,B5: set_rule,X3: rule] :
      ( ( ord_less_eq_set_rule @ A5 @ B5 )
     => ( ( member_rule @ X3 @ A5 )
       => ( member_rule @ X3 @ B5 ) ) ) ).

% in_mono
thf(fact_389_in__mono,axiom,
    ! [A5: set_Pr7058068377845519745ist_fm,B5: set_Pr7058068377845519745ist_fm,X3: produc1996495991257130529ist_fm] :
      ( ( ord_le2055072402069232929ist_fm @ A5 @ B5 )
     => ( ( member8102475879199740618ist_fm @ X3 @ A5 )
       => ( member8102475879199740618ist_fm @ X3 @ B5 ) ) ) ).

% in_mono
thf(fact_390_in__mono,axiom,
    ! [A5: set_Pr8693737435421807431at_nat,B5: set_Pr8693737435421807431at_nat,X3: produc859450856879609959at_nat] :
      ( ( ord_le3000389064537975527at_nat @ A5 @ B5 )
     => ( ( member8206827879206165904at_nat @ X3 @ A5 )
       => ( member8206827879206165904at_nat @ X3 @ B5 ) ) ) ).

% in_mono
thf(fact_391_in__mono,axiom,
    ! [A5: set_fm,B5: set_fm,X3: fm] :
      ( ( ord_less_eq_set_fm @ A5 @ B5 )
     => ( ( member_fm @ X3 @ A5 )
       => ( member_fm @ X3 @ B5 ) ) ) ).

% in_mono
thf(fact_392_subsetD,axiom,
    ! [A5: set_rule,B5: set_rule,C2: rule] :
      ( ( ord_less_eq_set_rule @ A5 @ B5 )
     => ( ( member_rule @ C2 @ A5 )
       => ( member_rule @ C2 @ B5 ) ) ) ).

% subsetD
thf(fact_393_subsetD,axiom,
    ! [A5: set_Pr7058068377845519745ist_fm,B5: set_Pr7058068377845519745ist_fm,C2: produc1996495991257130529ist_fm] :
      ( ( ord_le2055072402069232929ist_fm @ A5 @ B5 )
     => ( ( member8102475879199740618ist_fm @ C2 @ A5 )
       => ( member8102475879199740618ist_fm @ C2 @ B5 ) ) ) ).

% subsetD
thf(fact_394_subsetD,axiom,
    ! [A5: set_Pr8693737435421807431at_nat,B5: set_Pr8693737435421807431at_nat,C2: produc859450856879609959at_nat] :
      ( ( ord_le3000389064537975527at_nat @ A5 @ B5 )
     => ( ( member8206827879206165904at_nat @ C2 @ A5 )
       => ( member8206827879206165904at_nat @ C2 @ B5 ) ) ) ).

% subsetD
thf(fact_395_subsetD,axiom,
    ! [A5: set_fm,B5: set_fm,C2: fm] :
      ( ( ord_less_eq_set_fm @ A5 @ B5 )
     => ( ( member_fm @ C2 @ A5 )
       => ( member_fm @ C2 @ B5 ) ) ) ).

% subsetD
thf(fact_396_equalityE,axiom,
    ! [A5: set_fm,B5: set_fm] :
      ( ( A5 = B5 )
     => ~ ( ( ord_less_eq_set_fm @ A5 @ B5 )
         => ~ ( ord_less_eq_set_fm @ B5 @ A5 ) ) ) ).

% equalityE
thf(fact_397_subset__eq,axiom,
    ( ord_less_eq_set_rule
    = ( ^ [A7: set_rule,B7: set_rule] :
        ! [X6: rule] :
          ( ( member_rule @ X6 @ A7 )
         => ( member_rule @ X6 @ B7 ) ) ) ) ).

% subset_eq
thf(fact_398_subset__eq,axiom,
    ( ord_le2055072402069232929ist_fm
    = ( ^ [A7: set_Pr7058068377845519745ist_fm,B7: set_Pr7058068377845519745ist_fm] :
        ! [X6: produc1996495991257130529ist_fm] :
          ( ( member8102475879199740618ist_fm @ X6 @ A7 )
         => ( member8102475879199740618ist_fm @ X6 @ B7 ) ) ) ) ).

% subset_eq
thf(fact_399_subset__eq,axiom,
    ( ord_le3000389064537975527at_nat
    = ( ^ [A7: set_Pr8693737435421807431at_nat,B7: set_Pr8693737435421807431at_nat] :
        ! [X6: produc859450856879609959at_nat] :
          ( ( member8206827879206165904at_nat @ X6 @ A7 )
         => ( member8206827879206165904at_nat @ X6 @ B7 ) ) ) ) ).

% subset_eq
thf(fact_400_subset__eq,axiom,
    ( ord_less_eq_set_fm
    = ( ^ [A7: set_fm,B7: set_fm] :
        ! [X6: fm] :
          ( ( member_fm @ X6 @ A7 )
         => ( member_fm @ X6 @ B7 ) ) ) ) ).

% subset_eq
thf(fact_401_equalityD1,axiom,
    ! [A5: set_fm,B5: set_fm] :
      ( ( A5 = B5 )
     => ( ord_less_eq_set_fm @ A5 @ B5 ) ) ).

% equalityD1
thf(fact_402_equalityD2,axiom,
    ! [A5: set_fm,B5: set_fm] :
      ( ( A5 = B5 )
     => ( ord_less_eq_set_fm @ B5 @ A5 ) ) ).

% equalityD2
thf(fact_403_subset__iff,axiom,
    ( ord_less_eq_set_rule
    = ( ^ [A7: set_rule,B7: set_rule] :
        ! [T: rule] :
          ( ( member_rule @ T @ A7 )
         => ( member_rule @ T @ B7 ) ) ) ) ).

% subset_iff
thf(fact_404_subset__iff,axiom,
    ( ord_le2055072402069232929ist_fm
    = ( ^ [A7: set_Pr7058068377845519745ist_fm,B7: set_Pr7058068377845519745ist_fm] :
        ! [T: produc1996495991257130529ist_fm] :
          ( ( member8102475879199740618ist_fm @ T @ A7 )
         => ( member8102475879199740618ist_fm @ T @ B7 ) ) ) ) ).

% subset_iff
thf(fact_405_subset__iff,axiom,
    ( ord_le3000389064537975527at_nat
    = ( ^ [A7: set_Pr8693737435421807431at_nat,B7: set_Pr8693737435421807431at_nat] :
        ! [T: produc859450856879609959at_nat] :
          ( ( member8206827879206165904at_nat @ T @ A7 )
         => ( member8206827879206165904at_nat @ T @ B7 ) ) ) ) ).

% subset_iff
thf(fact_406_subset__iff,axiom,
    ( ord_less_eq_set_fm
    = ( ^ [A7: set_fm,B7: set_fm] :
        ! [T: fm] :
          ( ( member_fm @ T @ A7 )
         => ( member_fm @ T @ B7 ) ) ) ) ).

% subset_iff
thf(fact_407_subset__refl,axiom,
    ! [A5: set_fm] : ( ord_less_eq_set_fm @ A5 @ A5 ) ).

% subset_refl
thf(fact_408_Collect__mono,axiom,
    ! [P2: fm > $o,Q2: fm > $o] :
      ( ! [X: fm] :
          ( ( P2 @ X )
         => ( Q2 @ X ) )
     => ( ord_less_eq_set_fm @ ( collect_fm @ P2 ) @ ( collect_fm @ Q2 ) ) ) ).

% Collect_mono
thf(fact_409_subset__trans,axiom,
    ! [A5: set_fm,B5: set_fm,C3: set_fm] :
      ( ( ord_less_eq_set_fm @ A5 @ B5 )
     => ( ( ord_less_eq_set_fm @ B5 @ C3 )
       => ( ord_less_eq_set_fm @ A5 @ C3 ) ) ) ).

% subset_trans
thf(fact_410_set__eq__subset,axiom,
    ( ( ^ [Y2: set_fm,Z: set_fm] : ( Y2 = Z ) )
    = ( ^ [A7: set_fm,B7: set_fm] :
          ( ( ord_less_eq_set_fm @ A7 @ B7 )
          & ( ord_less_eq_set_fm @ B7 @ A7 ) ) ) ) ).

% set_eq_subset
thf(fact_411_Collect__mono__iff,axiom,
    ! [P2: fm > $o,Q2: fm > $o] :
      ( ( ord_less_eq_set_fm @ ( collect_fm @ P2 ) @ ( collect_fm @ Q2 ) )
      = ( ! [X6: fm] :
            ( ( P2 @ X6 )
           => ( Q2 @ X6 ) ) ) ) ).

% Collect_mono_iff
thf(fact_412_Collect__subset,axiom,
    ! [A5: set_rule,P2: rule > $o] :
      ( ord_less_eq_set_rule
      @ ( collect_rule
        @ ^ [X6: rule] :
            ( ( member_rule @ X6 @ A5 )
            & ( P2 @ X6 ) ) )
      @ A5 ) ).

% Collect_subset
thf(fact_413_Collect__subset,axiom,
    ! [A5: set_Pr7058068377845519745ist_fm,P2: produc1996495991257130529ist_fm > $o] :
      ( ord_le2055072402069232929ist_fm
      @ ( collec7231575776666803724ist_fm
        @ ^ [X6: produc1996495991257130529ist_fm] :
            ( ( member8102475879199740618ist_fm @ X6 @ A5 )
            & ( P2 @ X6 ) ) )
      @ A5 ) ).

% Collect_subset
thf(fact_414_Collect__subset,axiom,
    ! [A5: set_Pr8693737435421807431at_nat,P2: produc859450856879609959at_nat > $o] :
      ( ord_le3000389064537975527at_nat
      @ ( collec7088162979684241874at_nat
        @ ^ [X6: produc859450856879609959at_nat] :
            ( ( member8206827879206165904at_nat @ X6 @ A5 )
            & ( P2 @ X6 ) ) )
      @ A5 ) ).

% Collect_subset
thf(fact_415_Collect__subset,axiom,
    ! [A5: set_fm,P2: fm > $o] :
      ( ord_less_eq_set_fm
      @ ( collect_fm
        @ ^ [X6: fm] :
            ( ( member_fm @ X6 @ A5 )
            & ( P2 @ X6 ) ) )
      @ A5 ) ).

% Collect_subset
thf(fact_416_RuleSystem__Defs_Oepath_Ocases,axiom,
    ! [Eff2: list_fm > list_fm > fset_list_fm > $o,Rules: stream_list_fm,A: stream8299795917829157543ist_fm] :
      ( ( abstra6579460235205289861ist_fm @ Eff2 @ Rules @ A )
     => ~ ( ( member_list_fm @ ( produc4588648349897876871ist_fm @ ( shd_Pr772355297128350925ist_fm @ A ) ) @ ( sset_list_fm @ Rules ) )
         => ! [Sl4: fset_list_fm] :
              ( ( fmember_list_fm @ ( produc1501393135466168645ist_fm @ ( shd_Pr772355297128350925ist_fm @ ( stl_Pr5027778045650968273ist_fm @ A ) ) ) @ Sl4 )
             => ( ( Eff2 @ ( produc4588648349897876871ist_fm @ ( shd_Pr772355297128350925ist_fm @ A ) ) @ ( produc1501393135466168645ist_fm @ ( shd_Pr772355297128350925ist_fm @ A ) ) @ Sl4 )
               => ~ ( abstra6579460235205289861ist_fm @ Eff2 @ Rules @ ( stl_Pr5027778045650968273ist_fm @ A ) ) ) ) ) ) ).

% RuleSystem_Defs.epath.cases
thf(fact_417_RuleSystem__Defs_Oepath_Ocases,axiom,
    ! [Eff2: fm > produc1996495991257130529ist_fm > fset_P661503646757059847ist_fm > $o,Rules: stream_fm,A: stream5128679079952569136_fm_fm] :
      ( ( abstra9155489748554225612ist_fm @ Eff2 @ Rules @ A )
     => ~ ( ( member_fm @ ( produc6738438136904575596_fm_fm @ ( shd_Pr7265661315679113588_fm_fm @ A ) ) @ ( sset_fm @ Rules ) )
         => ! [Sl4: fset_P661503646757059847ist_fm] :
              ( ( fmembe3381613331217039976ist_fm @ ( produc1217498474891896878_fm_fm @ ( shd_Pr7265661315679113588_fm_fm @ ( stl_Pr2715999368752424048_fm_fm @ A ) ) ) @ Sl4 )
             => ( ( Eff2 @ ( produc6738438136904575596_fm_fm @ ( shd_Pr7265661315679113588_fm_fm @ A ) ) @ ( produc1217498474891896878_fm_fm @ ( shd_Pr7265661315679113588_fm_fm @ A ) ) @ Sl4 )
               => ~ ( abstra9155489748554225612ist_fm @ Eff2 @ Rules @ ( stl_Pr2715999368752424048_fm_fm @ A ) ) ) ) ) ) ).

% RuleSystem_Defs.epath.cases
thf(fact_418_RuleSystem__Defs_Oepath_Ocases,axiom,
    ! [Eff2: rule > produc1996495991257130529ist_fm > fset_P661503646757059847ist_fm > $o,Rules: stream_rule,A: stream727092118206550309m_rule] :
      ( ( abstra5415202973855822231ist_fm @ Eff2 @ Rules @ A )
     => ~ ( ( member_rule @ ( produc7165828336582415457m_rule @ ( shd_Pr7235097944458474089m_rule @ A ) ) @ ( sset_rule @ Rules ) )
         => ! [Sl4: fset_P661503646757059847ist_fm] :
              ( ( fmembe3381613331217039976ist_fm @ ( produc6879501374131015971m_rule @ ( shd_Pr7235097944458474089m_rule @ ( stl_Pr950425576149878629m_rule @ A ) ) ) @ Sl4 )
             => ( ( Eff2 @ ( produc7165828336582415457m_rule @ ( shd_Pr7235097944458474089m_rule @ A ) ) @ ( produc6879501374131015971m_rule @ ( shd_Pr7235097944458474089m_rule @ A ) ) @ Sl4 )
               => ~ ( abstra5415202973855822231ist_fm @ Eff2 @ Rules @ ( stl_Pr950425576149878629m_rule @ A ) ) ) ) ) ) ).

% RuleSystem_Defs.epath.cases
thf(fact_419_RuleSystem__Defs_Oepath_Ocases,axiom,
    ! [Eff2: fm > abstra4572470701423150929m_rule > fset_A5396818216090053761m_rule > $o,Rules: stream_fm,A: stream812778442145099016ule_fm] :
      ( ( abstra1364558218721404774m_rule @ Eff2 @ Rules @ A )
     => ~ ( ( member_fm @ ( produc8096733172076065128ule_fm @ ( shd_Pr849711437766469422ule_fm @ A ) ) @ ( sset_fm @ Rules ) )
         => ! [Sl4: fset_A5396818216090053761m_rule] :
              ( ( fmembe1635983029874454474m_rule @ ( produc8194852094735076902ule_fm @ ( shd_Pr849711437766469422ule_fm @ ( stl_Pr2064664776838053682ule_fm @ A ) ) ) @ Sl4 )
             => ( ( Eff2 @ ( produc8096733172076065128ule_fm @ ( shd_Pr849711437766469422ule_fm @ A ) ) @ ( produc8194852094735076902ule_fm @ ( shd_Pr849711437766469422ule_fm @ A ) ) @ Sl4 )
               => ~ ( abstra1364558218721404774m_rule @ Eff2 @ Rules @ ( stl_Pr2064664776838053682ule_fm @ A ) ) ) ) ) ) ).

% RuleSystem_Defs.epath.cases
thf(fact_420_RuleSystem__Defs_Oepath_Ocases,axiom,
    ! [Eff2: rule > abstra4572470701423150929m_rule > fset_A5396818216090053761m_rule > $o,Rules: stream_rule,A: stream7755889244956737277e_rule] :
      ( ( abstra5823804033175701339m_rule @ Eff2 @ Rules @ A )
     => ~ ( ( member_rule @ ( produc4652457655725677917e_rule @ ( shd_Pr1235285222789368611e_rule @ A ) ) @ ( sset_rule @ Rules ) )
         => ! [Sl4: fset_A5396818216090053761m_rule] :
              ( ( fmembe1635983029874454474m_rule @ ( produc7822628117027650331e_rule @ ( shd_Pr1235285222789368611e_rule @ ( stl_Pr8695508334200742183e_rule @ A ) ) ) @ Sl4 )
             => ( ( Eff2 @ ( produc4652457655725677917e_rule @ ( shd_Pr1235285222789368611e_rule @ A ) ) @ ( produc7822628117027650331e_rule @ ( shd_Pr1235285222789368611e_rule @ A ) ) @ Sl4 )
               => ~ ( abstra5823804033175701339m_rule @ Eff2 @ Rules @ ( stl_Pr8695508334200742183e_rule @ A ) ) ) ) ) ) ).

% RuleSystem_Defs.epath.cases
thf(fact_421_RuleSystem__Defs_Oepath_Ocases,axiom,
    ! [Eff2: produc1996495991257130529ist_fm > produc1996495991257130529ist_fm > fset_P661503646757059847ist_fm > $o,Rules: stream8299795917829157543ist_fm,A: stream3409308193418444653ist_fm] :
      ( ( abstra5061764961602482947ist_fm @ Eff2 @ Rules @ A )
     => ~ ( ( member8102475879199740618ist_fm @ ( produc1639249265332130309ist_fm @ ( shd_Pr4587271278095355155ist_fm @ A ) ) @ ( sset_P3431374123360452149ist_fm @ Rules ) )
         => ! [Sl4: fset_P661503646757059847ist_fm] :
              ( ( fmembe3381613331217039976ist_fm @ ( produc5894184038225292483ist_fm @ ( shd_Pr4587271278095355155ist_fm @ ( stl_Pr8700568296466786583ist_fm @ A ) ) ) @ Sl4 )
             => ( ( Eff2 @ ( produc1639249265332130309ist_fm @ ( shd_Pr4587271278095355155ist_fm @ A ) ) @ ( produc5894184038225292483ist_fm @ ( shd_Pr4587271278095355155ist_fm @ A ) ) @ Sl4 )
               => ~ ( abstra5061764961602482947ist_fm @ Eff2 @ Rules @ ( stl_Pr8700568296466786583ist_fm @ A ) ) ) ) ) ) ).

% RuleSystem_Defs.epath.cases
thf(fact_422_RuleSystem__Defs_Oepath_Ocases,axiom,
    ! [Eff2: produc859450856879609959at_nat > produc1996495991257130529ist_fm > fset_P661503646757059847ist_fm > $o,Rules: stream8372878641218411373at_nat,A: stream1622999302952468147at_nat] :
      ( ( abstra1311857326046938941ist_fm @ Eff2 @ Rules @ A )
     => ~ ( ( member8206827879206165904at_nat @ ( produc2750861588394444363at_nat @ ( shd_Pr5062688428016131033at_nat @ A ) ) @ ( sset_P9164763103296375419at_nat @ Rules ) )
         => ! [Sl4: fset_P661503646757059847ist_fm] :
              ( ( fmembe3381613331217039976ist_fm @ ( produc2934448030062000137at_nat @ ( shd_Pr5062688428016131033at_nat @ ( stl_Pr1564049449699658205at_nat @ A ) ) ) @ Sl4 )
             => ( ( Eff2 @ ( produc2750861588394444363at_nat @ ( shd_Pr5062688428016131033at_nat @ A ) ) @ ( produc2934448030062000137at_nat @ ( shd_Pr5062688428016131033at_nat @ A ) ) @ Sl4 )
               => ~ ( abstra1311857326046938941ist_fm @ Eff2 @ Rules @ ( stl_Pr1564049449699658205at_nat @ A ) ) ) ) ) ) ).

% RuleSystem_Defs.epath.cases
thf(fact_423_RuleSystem__Defs_Oepath_Ocases,axiom,
    ! [Eff2: produc164195504107695125m_rule > produc1996495991257130529ist_fm > fset_P661503646757059847ist_fm > $o,Rules: stream727092118206550309m_rule,A: stream642806101564963573m_rule] :
      ( ( abstra6058711572139910855ist_fm @ Eff2 @ Rules @ A )
     => ~ ( ( member4220325220686508332m_rule @ ( produc2016132375285593905m_rule @ ( shd_Pr1407159933805374393m_rule @ A ) ) @ ( sset_P4979403222752902913m_rule @ Rules ) )
         => ! [Sl4: fset_P661503646757059847ist_fm] :
              ( ( fmembe3381613331217039976ist_fm @ ( produc8602611229794383091m_rule @ ( shd_Pr1407159933805374393m_rule @ ( stl_Pr8578586445745536693m_rule @ A ) ) ) @ Sl4 )
             => ( ( Eff2 @ ( produc2016132375285593905m_rule @ ( shd_Pr1407159933805374393m_rule @ A ) ) @ ( produc8602611229794383091m_rule @ ( shd_Pr1407159933805374393m_rule @ A ) ) @ Sl4 )
               => ~ ( abstra6058711572139910855ist_fm @ Eff2 @ Rules @ ( stl_Pr8578586445745536693m_rule @ A ) ) ) ) ) ) ).

% RuleSystem_Defs.epath.cases
thf(fact_424_RuleSystem__Defs_Oepath_Ocases,axiom,
    ! [Eff2: produc1996495991257130529ist_fm > abstra4572470701423150929m_rule > fset_A5396818216090053761m_rule > $o,Rules: stream8299795917829157543ist_fm,A: stream7989562319868115861ist_fm] :
      ( ( abstra8066765647771630831m_rule @ Eff2 @ Rules @ A )
     => ~ ( ( member8102475879199740618ist_fm @ ( produc3674566753905654281ist_fm @ ( shd_Pr563299216415026649ist_fm @ A ) ) @ ( sset_P3431374123360452149ist_fm @ Rules ) )
         => ! [Sl4: fset_A5396818216090053761m_rule] :
              ( ( fmembe1635983029874454474m_rule @ ( produc2646333476271629515ist_fm @ ( shd_Pr563299216415026649ist_fm @ ( stl_Pr6727363551237990101ist_fm @ A ) ) ) @ Sl4 )
             => ( ( Eff2 @ ( produc3674566753905654281ist_fm @ ( shd_Pr563299216415026649ist_fm @ A ) ) @ ( produc2646333476271629515ist_fm @ ( shd_Pr563299216415026649ist_fm @ A ) ) @ Sl4 )
               => ~ ( abstra8066765647771630831m_rule @ Eff2 @ Rules @ ( stl_Pr6727363551237990101ist_fm @ A ) ) ) ) ) ) ).

% RuleSystem_Defs.epath.cases
thf(fact_425_RuleSystem__Defs_Oepath_Ocases,axiom,
    ! [Eff2: produc859450856879609959at_nat > abstra4572470701423150929m_rule > fset_A5396818216090053761m_rule > $o,Rules: stream8372878641218411373at_nat,A: stream1819640557982756571at_nat] :
      ( ( abstra3157339599784851765m_rule @ Eff2 @ Rules @ A )
     => ~ ( ( member8206827879206165904at_nat @ ( produc601515061966782543at_nat @ ( shd_Pr618815079712798623at_nat @ A ) ) @ ( sset_P9164763103296375419at_nat @ Rules ) )
         => ! [Sl4: fset_A5396818216090053761m_rule] :
              ( ( fmembe1635983029874454474m_rule @ ( produc567330280010208273at_nat @ ( shd_Pr618815079712798623at_nat @ ( stl_Pr2369401607900929691at_nat @ A ) ) ) @ Sl4 )
             => ( ( Eff2 @ ( produc601515061966782543at_nat @ ( shd_Pr618815079712798623at_nat @ A ) ) @ ( produc567330280010208273at_nat @ ( shd_Pr618815079712798623at_nat @ A ) ) @ Sl4 )
               => ~ ( abstra3157339599784851765m_rule @ Eff2 @ Rules @ ( stl_Pr2369401607900929691at_nat @ A ) ) ) ) ) ) ).

% RuleSystem_Defs.epath.cases
thf(fact_426_RuleSystem__Defs_Oepath_Osimps,axiom,
    ( abstra6579460235205289861ist_fm
    = ( ^ [Eff: list_fm > list_fm > fset_list_fm > $o,Rules2: stream_list_fm,A2: stream8299795917829157543ist_fm] :
        ? [Steps3: stream8299795917829157543ist_fm,Sl: fset_list_fm] :
          ( ( A2 = Steps3 )
          & ( member_list_fm @ ( produc4588648349897876871ist_fm @ ( shd_Pr772355297128350925ist_fm @ Steps3 ) ) @ ( sset_list_fm @ Rules2 ) )
          & ( fmember_list_fm @ ( produc1501393135466168645ist_fm @ ( shd_Pr772355297128350925ist_fm @ ( stl_Pr5027778045650968273ist_fm @ Steps3 ) ) ) @ Sl )
          & ( Eff @ ( produc4588648349897876871ist_fm @ ( shd_Pr772355297128350925ist_fm @ Steps3 ) ) @ ( produc1501393135466168645ist_fm @ ( shd_Pr772355297128350925ist_fm @ Steps3 ) ) @ Sl )
          & ( abstra6579460235205289861ist_fm @ Eff @ Rules2 @ ( stl_Pr5027778045650968273ist_fm @ Steps3 ) ) ) ) ) ).

% RuleSystem_Defs.epath.simps
thf(fact_427_RuleSystem__Defs_Oepath_Osimps,axiom,
    ( abstra9155489748554225612ist_fm
    = ( ^ [Eff: fm > produc1996495991257130529ist_fm > fset_P661503646757059847ist_fm > $o,Rules2: stream_fm,A2: stream5128679079952569136_fm_fm] :
        ? [Steps3: stream5128679079952569136_fm_fm,Sl: fset_P661503646757059847ist_fm] :
          ( ( A2 = Steps3 )
          & ( member_fm @ ( produc6738438136904575596_fm_fm @ ( shd_Pr7265661315679113588_fm_fm @ Steps3 ) ) @ ( sset_fm @ Rules2 ) )
          & ( fmembe3381613331217039976ist_fm @ ( produc1217498474891896878_fm_fm @ ( shd_Pr7265661315679113588_fm_fm @ ( stl_Pr2715999368752424048_fm_fm @ Steps3 ) ) ) @ Sl )
          & ( Eff @ ( produc6738438136904575596_fm_fm @ ( shd_Pr7265661315679113588_fm_fm @ Steps3 ) ) @ ( produc1217498474891896878_fm_fm @ ( shd_Pr7265661315679113588_fm_fm @ Steps3 ) ) @ Sl )
          & ( abstra9155489748554225612ist_fm @ Eff @ Rules2 @ ( stl_Pr2715999368752424048_fm_fm @ Steps3 ) ) ) ) ) ).

% RuleSystem_Defs.epath.simps
thf(fact_428_RuleSystem__Defs_Oepath_Osimps,axiom,
    ( abstra5415202973855822231ist_fm
    = ( ^ [Eff: rule > produc1996495991257130529ist_fm > fset_P661503646757059847ist_fm > $o,Rules2: stream_rule,A2: stream727092118206550309m_rule] :
        ? [Steps3: stream727092118206550309m_rule,Sl: fset_P661503646757059847ist_fm] :
          ( ( A2 = Steps3 )
          & ( member_rule @ ( produc7165828336582415457m_rule @ ( shd_Pr7235097944458474089m_rule @ Steps3 ) ) @ ( sset_rule @ Rules2 ) )
          & ( fmembe3381613331217039976ist_fm @ ( produc6879501374131015971m_rule @ ( shd_Pr7235097944458474089m_rule @ ( stl_Pr950425576149878629m_rule @ Steps3 ) ) ) @ Sl )
          & ( Eff @ ( produc7165828336582415457m_rule @ ( shd_Pr7235097944458474089m_rule @ Steps3 ) ) @ ( produc6879501374131015971m_rule @ ( shd_Pr7235097944458474089m_rule @ Steps3 ) ) @ Sl )
          & ( abstra5415202973855822231ist_fm @ Eff @ Rules2 @ ( stl_Pr950425576149878629m_rule @ Steps3 ) ) ) ) ) ).

% RuleSystem_Defs.epath.simps
thf(fact_429_RuleSystem__Defs_Oepath_Osimps,axiom,
    ( abstra1364558218721404774m_rule
    = ( ^ [Eff: fm > abstra4572470701423150929m_rule > fset_A5396818216090053761m_rule > $o,Rules2: stream_fm,A2: stream812778442145099016ule_fm] :
        ? [Steps3: stream812778442145099016ule_fm,Sl: fset_A5396818216090053761m_rule] :
          ( ( A2 = Steps3 )
          & ( member_fm @ ( produc8096733172076065128ule_fm @ ( shd_Pr849711437766469422ule_fm @ Steps3 ) ) @ ( sset_fm @ Rules2 ) )
          & ( fmembe1635983029874454474m_rule @ ( produc8194852094735076902ule_fm @ ( shd_Pr849711437766469422ule_fm @ ( stl_Pr2064664776838053682ule_fm @ Steps3 ) ) ) @ Sl )
          & ( Eff @ ( produc8096733172076065128ule_fm @ ( shd_Pr849711437766469422ule_fm @ Steps3 ) ) @ ( produc8194852094735076902ule_fm @ ( shd_Pr849711437766469422ule_fm @ Steps3 ) ) @ Sl )
          & ( abstra1364558218721404774m_rule @ Eff @ Rules2 @ ( stl_Pr2064664776838053682ule_fm @ Steps3 ) ) ) ) ) ).

% RuleSystem_Defs.epath.simps
thf(fact_430_RuleSystem__Defs_Oepath_Osimps,axiom,
    ( abstra5823804033175701339m_rule
    = ( ^ [Eff: rule > abstra4572470701423150929m_rule > fset_A5396818216090053761m_rule > $o,Rules2: stream_rule,A2: stream7755889244956737277e_rule] :
        ? [Steps3: stream7755889244956737277e_rule,Sl: fset_A5396818216090053761m_rule] :
          ( ( A2 = Steps3 )
          & ( member_rule @ ( produc4652457655725677917e_rule @ ( shd_Pr1235285222789368611e_rule @ Steps3 ) ) @ ( sset_rule @ Rules2 ) )
          & ( fmembe1635983029874454474m_rule @ ( produc7822628117027650331e_rule @ ( shd_Pr1235285222789368611e_rule @ ( stl_Pr8695508334200742183e_rule @ Steps3 ) ) ) @ Sl )
          & ( Eff @ ( produc4652457655725677917e_rule @ ( shd_Pr1235285222789368611e_rule @ Steps3 ) ) @ ( produc7822628117027650331e_rule @ ( shd_Pr1235285222789368611e_rule @ Steps3 ) ) @ Sl )
          & ( abstra5823804033175701339m_rule @ Eff @ Rules2 @ ( stl_Pr8695508334200742183e_rule @ Steps3 ) ) ) ) ) ).

% RuleSystem_Defs.epath.simps
thf(fact_431_RuleSystem__Defs_Oepath_Osimps,axiom,
    ( abstra5061764961602482947ist_fm
    = ( ^ [Eff: produc1996495991257130529ist_fm > produc1996495991257130529ist_fm > fset_P661503646757059847ist_fm > $o,Rules2: stream8299795917829157543ist_fm,A2: stream3409308193418444653ist_fm] :
        ? [Steps3: stream3409308193418444653ist_fm,Sl: fset_P661503646757059847ist_fm] :
          ( ( A2 = Steps3 )
          & ( member8102475879199740618ist_fm @ ( produc1639249265332130309ist_fm @ ( shd_Pr4587271278095355155ist_fm @ Steps3 ) ) @ ( sset_P3431374123360452149ist_fm @ Rules2 ) )
          & ( fmembe3381613331217039976ist_fm @ ( produc5894184038225292483ist_fm @ ( shd_Pr4587271278095355155ist_fm @ ( stl_Pr8700568296466786583ist_fm @ Steps3 ) ) ) @ Sl )
          & ( Eff @ ( produc1639249265332130309ist_fm @ ( shd_Pr4587271278095355155ist_fm @ Steps3 ) ) @ ( produc5894184038225292483ist_fm @ ( shd_Pr4587271278095355155ist_fm @ Steps3 ) ) @ Sl )
          & ( abstra5061764961602482947ist_fm @ Eff @ Rules2 @ ( stl_Pr8700568296466786583ist_fm @ Steps3 ) ) ) ) ) ).

% RuleSystem_Defs.epath.simps
thf(fact_432_RuleSystem__Defs_Oepath_Osimps,axiom,
    ( abstra1311857326046938941ist_fm
    = ( ^ [Eff: produc859450856879609959at_nat > produc1996495991257130529ist_fm > fset_P661503646757059847ist_fm > $o,Rules2: stream8372878641218411373at_nat,A2: stream1622999302952468147at_nat] :
        ? [Steps3: stream1622999302952468147at_nat,Sl: fset_P661503646757059847ist_fm] :
          ( ( A2 = Steps3 )
          & ( member8206827879206165904at_nat @ ( produc2750861588394444363at_nat @ ( shd_Pr5062688428016131033at_nat @ Steps3 ) ) @ ( sset_P9164763103296375419at_nat @ Rules2 ) )
          & ( fmembe3381613331217039976ist_fm @ ( produc2934448030062000137at_nat @ ( shd_Pr5062688428016131033at_nat @ ( stl_Pr1564049449699658205at_nat @ Steps3 ) ) ) @ Sl )
          & ( Eff @ ( produc2750861588394444363at_nat @ ( shd_Pr5062688428016131033at_nat @ Steps3 ) ) @ ( produc2934448030062000137at_nat @ ( shd_Pr5062688428016131033at_nat @ Steps3 ) ) @ Sl )
          & ( abstra1311857326046938941ist_fm @ Eff @ Rules2 @ ( stl_Pr1564049449699658205at_nat @ Steps3 ) ) ) ) ) ).

% RuleSystem_Defs.epath.simps
thf(fact_433_RuleSystem__Defs_Oepath_Osimps,axiom,
    ( abstra6058711572139910855ist_fm
    = ( ^ [Eff: produc164195504107695125m_rule > produc1996495991257130529ist_fm > fset_P661503646757059847ist_fm > $o,Rules2: stream727092118206550309m_rule,A2: stream642806101564963573m_rule] :
        ? [Steps3: stream642806101564963573m_rule,Sl: fset_P661503646757059847ist_fm] :
          ( ( A2 = Steps3 )
          & ( member4220325220686508332m_rule @ ( produc2016132375285593905m_rule @ ( shd_Pr1407159933805374393m_rule @ Steps3 ) ) @ ( sset_P4979403222752902913m_rule @ Rules2 ) )
          & ( fmembe3381613331217039976ist_fm @ ( produc8602611229794383091m_rule @ ( shd_Pr1407159933805374393m_rule @ ( stl_Pr8578586445745536693m_rule @ Steps3 ) ) ) @ Sl )
          & ( Eff @ ( produc2016132375285593905m_rule @ ( shd_Pr1407159933805374393m_rule @ Steps3 ) ) @ ( produc8602611229794383091m_rule @ ( shd_Pr1407159933805374393m_rule @ Steps3 ) ) @ Sl )
          & ( abstra6058711572139910855ist_fm @ Eff @ Rules2 @ ( stl_Pr8578586445745536693m_rule @ Steps3 ) ) ) ) ) ).

% RuleSystem_Defs.epath.simps
thf(fact_434_RuleSystem__Defs_Oepath_Osimps,axiom,
    ( abstra8066765647771630831m_rule
    = ( ^ [Eff: produc1996495991257130529ist_fm > abstra4572470701423150929m_rule > fset_A5396818216090053761m_rule > $o,Rules2: stream8299795917829157543ist_fm,A2: stream7989562319868115861ist_fm] :
        ? [Steps3: stream7989562319868115861ist_fm,Sl: fset_A5396818216090053761m_rule] :
          ( ( A2 = Steps3 )
          & ( member8102475879199740618ist_fm @ ( produc3674566753905654281ist_fm @ ( shd_Pr563299216415026649ist_fm @ Steps3 ) ) @ ( sset_P3431374123360452149ist_fm @ Rules2 ) )
          & ( fmembe1635983029874454474m_rule @ ( produc2646333476271629515ist_fm @ ( shd_Pr563299216415026649ist_fm @ ( stl_Pr6727363551237990101ist_fm @ Steps3 ) ) ) @ Sl )
          & ( Eff @ ( produc3674566753905654281ist_fm @ ( shd_Pr563299216415026649ist_fm @ Steps3 ) ) @ ( produc2646333476271629515ist_fm @ ( shd_Pr563299216415026649ist_fm @ Steps3 ) ) @ Sl )
          & ( abstra8066765647771630831m_rule @ Eff @ Rules2 @ ( stl_Pr6727363551237990101ist_fm @ Steps3 ) ) ) ) ) ).

% RuleSystem_Defs.epath.simps
thf(fact_435_RuleSystem__Defs_Oepath_Osimps,axiom,
    ( abstra3157339599784851765m_rule
    = ( ^ [Eff: produc859450856879609959at_nat > abstra4572470701423150929m_rule > fset_A5396818216090053761m_rule > $o,Rules2: stream8372878641218411373at_nat,A2: stream1819640557982756571at_nat] :
        ? [Steps3: stream1819640557982756571at_nat,Sl: fset_A5396818216090053761m_rule] :
          ( ( A2 = Steps3 )
          & ( member8206827879206165904at_nat @ ( produc601515061966782543at_nat @ ( shd_Pr618815079712798623at_nat @ Steps3 ) ) @ ( sset_P9164763103296375419at_nat @ Rules2 ) )
          & ( fmembe1635983029874454474m_rule @ ( produc567330280010208273at_nat @ ( shd_Pr618815079712798623at_nat @ ( stl_Pr2369401607900929691at_nat @ Steps3 ) ) ) @ Sl )
          & ( Eff @ ( produc601515061966782543at_nat @ ( shd_Pr618815079712798623at_nat @ Steps3 ) ) @ ( produc567330280010208273at_nat @ ( shd_Pr618815079712798623at_nat @ Steps3 ) ) @ Sl )
          & ( abstra3157339599784851765m_rule @ Eff @ Rules2 @ ( stl_Pr2369401607900929691at_nat @ Steps3 ) ) ) ) ) ).

% RuleSystem_Defs.epath.simps
thf(fact_436_RuleSystem__Defs_Oepath_Ointros,axiom,
    ! [Steps: stream8299795917829157543ist_fm,Rules: stream_list_fm,Sl3: fset_list_fm,Eff2: list_fm > list_fm > fset_list_fm > $o] :
      ( ( member_list_fm @ ( produc4588648349897876871ist_fm @ ( shd_Pr772355297128350925ist_fm @ Steps ) ) @ ( sset_list_fm @ Rules ) )
     => ( ( fmember_list_fm @ ( produc1501393135466168645ist_fm @ ( shd_Pr772355297128350925ist_fm @ ( stl_Pr5027778045650968273ist_fm @ Steps ) ) ) @ Sl3 )
       => ( ( Eff2 @ ( produc4588648349897876871ist_fm @ ( shd_Pr772355297128350925ist_fm @ Steps ) ) @ ( produc1501393135466168645ist_fm @ ( shd_Pr772355297128350925ist_fm @ Steps ) ) @ Sl3 )
         => ( ( abstra6579460235205289861ist_fm @ Eff2 @ Rules @ ( stl_Pr5027778045650968273ist_fm @ Steps ) )
           => ( abstra6579460235205289861ist_fm @ Eff2 @ Rules @ Steps ) ) ) ) ) ).

% RuleSystem_Defs.epath.intros
thf(fact_437_RuleSystem__Defs_Oepath_Ointros,axiom,
    ! [Steps: stream5128679079952569136_fm_fm,Rules: stream_fm,Sl3: fset_P661503646757059847ist_fm,Eff2: fm > produc1996495991257130529ist_fm > fset_P661503646757059847ist_fm > $o] :
      ( ( member_fm @ ( produc6738438136904575596_fm_fm @ ( shd_Pr7265661315679113588_fm_fm @ Steps ) ) @ ( sset_fm @ Rules ) )
     => ( ( fmembe3381613331217039976ist_fm @ ( produc1217498474891896878_fm_fm @ ( shd_Pr7265661315679113588_fm_fm @ ( stl_Pr2715999368752424048_fm_fm @ Steps ) ) ) @ Sl3 )
       => ( ( Eff2 @ ( produc6738438136904575596_fm_fm @ ( shd_Pr7265661315679113588_fm_fm @ Steps ) ) @ ( produc1217498474891896878_fm_fm @ ( shd_Pr7265661315679113588_fm_fm @ Steps ) ) @ Sl3 )
         => ( ( abstra9155489748554225612ist_fm @ Eff2 @ Rules @ ( stl_Pr2715999368752424048_fm_fm @ Steps ) )
           => ( abstra9155489748554225612ist_fm @ Eff2 @ Rules @ Steps ) ) ) ) ) ).

% RuleSystem_Defs.epath.intros
thf(fact_438_RuleSystem__Defs_Oepath_Ointros,axiom,
    ! [Steps: stream727092118206550309m_rule,Rules: stream_rule,Sl3: fset_P661503646757059847ist_fm,Eff2: rule > produc1996495991257130529ist_fm > fset_P661503646757059847ist_fm > $o] :
      ( ( member_rule @ ( produc7165828336582415457m_rule @ ( shd_Pr7235097944458474089m_rule @ Steps ) ) @ ( sset_rule @ Rules ) )
     => ( ( fmembe3381613331217039976ist_fm @ ( produc6879501374131015971m_rule @ ( shd_Pr7235097944458474089m_rule @ ( stl_Pr950425576149878629m_rule @ Steps ) ) ) @ Sl3 )
       => ( ( Eff2 @ ( produc7165828336582415457m_rule @ ( shd_Pr7235097944458474089m_rule @ Steps ) ) @ ( produc6879501374131015971m_rule @ ( shd_Pr7235097944458474089m_rule @ Steps ) ) @ Sl3 )
         => ( ( abstra5415202973855822231ist_fm @ Eff2 @ Rules @ ( stl_Pr950425576149878629m_rule @ Steps ) )
           => ( abstra5415202973855822231ist_fm @ Eff2 @ Rules @ Steps ) ) ) ) ) ).

% RuleSystem_Defs.epath.intros
thf(fact_439_RuleSystem__Defs_Oepath_Ointros,axiom,
    ! [Steps: stream812778442145099016ule_fm,Rules: stream_fm,Sl3: fset_A5396818216090053761m_rule,Eff2: fm > abstra4572470701423150929m_rule > fset_A5396818216090053761m_rule > $o] :
      ( ( member_fm @ ( produc8096733172076065128ule_fm @ ( shd_Pr849711437766469422ule_fm @ Steps ) ) @ ( sset_fm @ Rules ) )
     => ( ( fmembe1635983029874454474m_rule @ ( produc8194852094735076902ule_fm @ ( shd_Pr849711437766469422ule_fm @ ( stl_Pr2064664776838053682ule_fm @ Steps ) ) ) @ Sl3 )
       => ( ( Eff2 @ ( produc8096733172076065128ule_fm @ ( shd_Pr849711437766469422ule_fm @ Steps ) ) @ ( produc8194852094735076902ule_fm @ ( shd_Pr849711437766469422ule_fm @ Steps ) ) @ Sl3 )
         => ( ( abstra1364558218721404774m_rule @ Eff2 @ Rules @ ( stl_Pr2064664776838053682ule_fm @ Steps ) )
           => ( abstra1364558218721404774m_rule @ Eff2 @ Rules @ Steps ) ) ) ) ) ).

% RuleSystem_Defs.epath.intros
thf(fact_440_RuleSystem__Defs_Oepath_Ointros,axiom,
    ! [Steps: stream7755889244956737277e_rule,Rules: stream_rule,Sl3: fset_A5396818216090053761m_rule,Eff2: rule > abstra4572470701423150929m_rule > fset_A5396818216090053761m_rule > $o] :
      ( ( member_rule @ ( produc4652457655725677917e_rule @ ( shd_Pr1235285222789368611e_rule @ Steps ) ) @ ( sset_rule @ Rules ) )
     => ( ( fmembe1635983029874454474m_rule @ ( produc7822628117027650331e_rule @ ( shd_Pr1235285222789368611e_rule @ ( stl_Pr8695508334200742183e_rule @ Steps ) ) ) @ Sl3 )
       => ( ( Eff2 @ ( produc4652457655725677917e_rule @ ( shd_Pr1235285222789368611e_rule @ Steps ) ) @ ( produc7822628117027650331e_rule @ ( shd_Pr1235285222789368611e_rule @ Steps ) ) @ Sl3 )
         => ( ( abstra5823804033175701339m_rule @ Eff2 @ Rules @ ( stl_Pr8695508334200742183e_rule @ Steps ) )
           => ( abstra5823804033175701339m_rule @ Eff2 @ Rules @ Steps ) ) ) ) ) ).

% RuleSystem_Defs.epath.intros
thf(fact_441_RuleSystem__Defs_Oepath_Ointros,axiom,
    ! [Steps: stream3409308193418444653ist_fm,Rules: stream8299795917829157543ist_fm,Sl3: fset_P661503646757059847ist_fm,Eff2: produc1996495991257130529ist_fm > produc1996495991257130529ist_fm > fset_P661503646757059847ist_fm > $o] :
      ( ( member8102475879199740618ist_fm @ ( produc1639249265332130309ist_fm @ ( shd_Pr4587271278095355155ist_fm @ Steps ) ) @ ( sset_P3431374123360452149ist_fm @ Rules ) )
     => ( ( fmembe3381613331217039976ist_fm @ ( produc5894184038225292483ist_fm @ ( shd_Pr4587271278095355155ist_fm @ ( stl_Pr8700568296466786583ist_fm @ Steps ) ) ) @ Sl3 )
       => ( ( Eff2 @ ( produc1639249265332130309ist_fm @ ( shd_Pr4587271278095355155ist_fm @ Steps ) ) @ ( produc5894184038225292483ist_fm @ ( shd_Pr4587271278095355155ist_fm @ Steps ) ) @ Sl3 )
         => ( ( abstra5061764961602482947ist_fm @ Eff2 @ Rules @ ( stl_Pr8700568296466786583ist_fm @ Steps ) )
           => ( abstra5061764961602482947ist_fm @ Eff2 @ Rules @ Steps ) ) ) ) ) ).

% RuleSystem_Defs.epath.intros
thf(fact_442_RuleSystem__Defs_Oepath_Ointros,axiom,
    ! [Steps: stream1622999302952468147at_nat,Rules: stream8372878641218411373at_nat,Sl3: fset_P661503646757059847ist_fm,Eff2: produc859450856879609959at_nat > produc1996495991257130529ist_fm > fset_P661503646757059847ist_fm > $o] :
      ( ( member8206827879206165904at_nat @ ( produc2750861588394444363at_nat @ ( shd_Pr5062688428016131033at_nat @ Steps ) ) @ ( sset_P9164763103296375419at_nat @ Rules ) )
     => ( ( fmembe3381613331217039976ist_fm @ ( produc2934448030062000137at_nat @ ( shd_Pr5062688428016131033at_nat @ ( stl_Pr1564049449699658205at_nat @ Steps ) ) ) @ Sl3 )
       => ( ( Eff2 @ ( produc2750861588394444363at_nat @ ( shd_Pr5062688428016131033at_nat @ Steps ) ) @ ( produc2934448030062000137at_nat @ ( shd_Pr5062688428016131033at_nat @ Steps ) ) @ Sl3 )
         => ( ( abstra1311857326046938941ist_fm @ Eff2 @ Rules @ ( stl_Pr1564049449699658205at_nat @ Steps ) )
           => ( abstra1311857326046938941ist_fm @ Eff2 @ Rules @ Steps ) ) ) ) ) ).

% RuleSystem_Defs.epath.intros
thf(fact_443_RuleSystem__Defs_Oepath_Ointros,axiom,
    ! [Steps: stream642806101564963573m_rule,Rules: stream727092118206550309m_rule,Sl3: fset_P661503646757059847ist_fm,Eff2: produc164195504107695125m_rule > produc1996495991257130529ist_fm > fset_P661503646757059847ist_fm > $o] :
      ( ( member4220325220686508332m_rule @ ( produc2016132375285593905m_rule @ ( shd_Pr1407159933805374393m_rule @ Steps ) ) @ ( sset_P4979403222752902913m_rule @ Rules ) )
     => ( ( fmembe3381613331217039976ist_fm @ ( produc8602611229794383091m_rule @ ( shd_Pr1407159933805374393m_rule @ ( stl_Pr8578586445745536693m_rule @ Steps ) ) ) @ Sl3 )
       => ( ( Eff2 @ ( produc2016132375285593905m_rule @ ( shd_Pr1407159933805374393m_rule @ Steps ) ) @ ( produc8602611229794383091m_rule @ ( shd_Pr1407159933805374393m_rule @ Steps ) ) @ Sl3 )
         => ( ( abstra6058711572139910855ist_fm @ Eff2 @ Rules @ ( stl_Pr8578586445745536693m_rule @ Steps ) )
           => ( abstra6058711572139910855ist_fm @ Eff2 @ Rules @ Steps ) ) ) ) ) ).

% RuleSystem_Defs.epath.intros
thf(fact_444_RuleSystem__Defs_Oepath_Ointros,axiom,
    ! [Steps: stream7989562319868115861ist_fm,Rules: stream8299795917829157543ist_fm,Sl3: fset_A5396818216090053761m_rule,Eff2: produc1996495991257130529ist_fm > abstra4572470701423150929m_rule > fset_A5396818216090053761m_rule > $o] :
      ( ( member8102475879199740618ist_fm @ ( produc3674566753905654281ist_fm @ ( shd_Pr563299216415026649ist_fm @ Steps ) ) @ ( sset_P3431374123360452149ist_fm @ Rules ) )
     => ( ( fmembe1635983029874454474m_rule @ ( produc2646333476271629515ist_fm @ ( shd_Pr563299216415026649ist_fm @ ( stl_Pr6727363551237990101ist_fm @ Steps ) ) ) @ Sl3 )
       => ( ( Eff2 @ ( produc3674566753905654281ist_fm @ ( shd_Pr563299216415026649ist_fm @ Steps ) ) @ ( produc2646333476271629515ist_fm @ ( shd_Pr563299216415026649ist_fm @ Steps ) ) @ Sl3 )
         => ( ( abstra8066765647771630831m_rule @ Eff2 @ Rules @ ( stl_Pr6727363551237990101ist_fm @ Steps ) )
           => ( abstra8066765647771630831m_rule @ Eff2 @ Rules @ Steps ) ) ) ) ) ).

% RuleSystem_Defs.epath.intros
thf(fact_445_RuleSystem__Defs_Oepath_Ointros,axiom,
    ! [Steps: stream1819640557982756571at_nat,Rules: stream8372878641218411373at_nat,Sl3: fset_A5396818216090053761m_rule,Eff2: produc859450856879609959at_nat > abstra4572470701423150929m_rule > fset_A5396818216090053761m_rule > $o] :
      ( ( member8206827879206165904at_nat @ ( produc601515061966782543at_nat @ ( shd_Pr618815079712798623at_nat @ Steps ) ) @ ( sset_P9164763103296375419at_nat @ Rules ) )
     => ( ( fmembe1635983029874454474m_rule @ ( produc567330280010208273at_nat @ ( shd_Pr618815079712798623at_nat @ ( stl_Pr2369401607900929691at_nat @ Steps ) ) ) @ Sl3 )
       => ( ( Eff2 @ ( produc601515061966782543at_nat @ ( shd_Pr618815079712798623at_nat @ Steps ) ) @ ( produc567330280010208273at_nat @ ( shd_Pr618815079712798623at_nat @ Steps ) ) @ Sl3 )
         => ( ( abstra3157339599784851765m_rule @ Eff2 @ Rules @ ( stl_Pr2369401607900929691at_nat @ Steps ) )
           => ( abstra3157339599784851765m_rule @ Eff2 @ Rules @ Steps ) ) ) ) ) ).

% RuleSystem_Defs.epath.intros
thf(fact_446_Saturated__def,axiom,
    ! [Steps: stream727092118206550309m_rule] :
      ( ( abstra8647779336736900908ist_fm
        @ ^ [R: rule,S: produc1996495991257130529ist_fm,Ss: fset_P661503646757059847ist_fm] :
            ( ( eff @ R @ S )
            = ( some_f2162114139285431020ist_fm @ Ss ) )
        @ rules
        @ Steps )
      = ( ! [X6: rule] :
            ( ( member_rule @ X6 @ ( sset_rule @ rules ) )
           => ( abstra7252176601000738572ist_fm
              @ ^ [R: rule,S: produc1996495991257130529ist_fm,Ss: fset_P661503646757059847ist_fm] :
                  ( ( eff @ R @ S )
                  = ( some_f2162114139285431020ist_fm @ Ss ) )
              @ X6
              @ Steps ) ) ) ) ).

% Saturated_def
thf(fact_447_fsubsetI,axiom,
    ! [A5: fset_P661503646757059847ist_fm,B5: fset_P661503646757059847ist_fm] :
      ( ! [X: produc1996495991257130529ist_fm] :
          ( ( fmembe3381613331217039976ist_fm @ X @ A5 )
         => ( fmembe3381613331217039976ist_fm @ X @ B5 ) )
     => ( ord_le2064643713053750439ist_fm @ A5 @ B5 ) ) ).

% fsubsetI
thf(fact_448_fsubsetI,axiom,
    ! [A5: fset_A5396818216090053761m_rule,B5: fset_A5396818216090053761m_rule] :
      ( ! [X: abstra4572470701423150929m_rule] :
          ( ( fmembe1635983029874454474m_rule @ X @ A5 )
         => ( fmembe1635983029874454474m_rule @ X @ B5 ) )
     => ( ord_le6951374404268211425m_rule @ A5 @ B5 ) ) ).

% fsubsetI
thf(fact_449_snth__sset__smerge,axiom,
    ! [Ss2: stream_stream_fm,N: nat,M: nat] : ( member_fm @ ( snth_fm @ ( snth_stream_fm @ Ss2 @ N ) @ M ) @ ( sset_fm @ ( smerge_fm @ Ss2 ) ) ) ).

% snth_sset_smerge
thf(fact_450_snth__sset__smerge,axiom,
    ! [Ss2: stream8183306005803988909ist_fm,N: nat,M: nat] : ( member8102475879199740618ist_fm @ ( snth_P2624749845746815685ist_fm @ ( snth_s4276058539312847179ist_fm @ Ss2 @ N ) @ M ) @ ( sset_P3431374123360452149ist_fm @ ( smerge8067294617398470191ist_fm @ Ss2 ) ) ) ).

% snth_sset_smerge
thf(fact_451_snth__sset__smerge,axiom,
    ! [Ss2: stream7445237232777090803at_nat,N: nat,M: nat] : ( member8206827879206165904at_nat @ ( snth_P9098630209796499211at_nat @ ( snth_s7872458624752591889at_nat @ Ss2 @ N ) @ M ) @ ( sset_P9164763103296375419at_nat @ ( smerge8841642037904234357at_nat @ Ss2 ) ) ) ).

% snth_sset_smerge
thf(fact_452_snth__sset__smerge,axiom,
    ! [Ss2: stream_stream_rule,N: nat,M: nat] : ( member_rule @ ( snth_rule @ ( snth_stream_rule @ Ss2 @ N ) @ M ) @ ( sset_rule @ ( smerge_rule @ Ss2 ) ) ) ).

% snth_sset_smerge
thf(fact_453_snth__sset__smerge,axiom,
    ! [Ss2: stream3016090547575141045m_rule,N: nat,M: nat] : ( member4220325220686508332m_rule @ ( snth_P1712579846118164081m_rule @ ( snth_s8878883179916095873m_rule @ Ss2 @ N ) @ M ) @ ( sset_P4979403222752902913m_rule @ ( smerge3869575255966183559m_rule @ Ss2 ) ) ) ).

% snth_sset_smerge
thf(fact_454_i_Oepath_Ocases,axiom,
    ! [Eff3: list_fm > list_fm > option_fset_list_fm,Rules: stream_list_fm,A: stream8299795917829157543ist_fm] :
      ( ( abstra6579460235205289861ist_fm @ ( abstra256862739949234434ist_fm @ Eff3 ) @ Rules @ A )
     => ~ ( ( member_list_fm @ ( produc4588648349897876871ist_fm @ ( shd_Pr772355297128350925ist_fm @ A ) ) @ ( sset_list_fm @ Rules ) )
         => ! [Sl4: fset_list_fm] :
              ( ( fmember_list_fm @ ( produc1501393135466168645ist_fm @ ( shd_Pr772355297128350925ist_fm @ ( stl_Pr5027778045650968273ist_fm @ A ) ) ) @ Sl4 )
             => ( ( abstra256862739949234434ist_fm @ Eff3 @ ( produc4588648349897876871ist_fm @ ( shd_Pr772355297128350925ist_fm @ A ) ) @ ( produc1501393135466168645ist_fm @ ( shd_Pr772355297128350925ist_fm @ A ) ) @ Sl4 )
               => ~ ( abstra6579460235205289861ist_fm @ ( abstra256862739949234434ist_fm @ Eff3 ) @ Rules @ ( stl_Pr5027778045650968273ist_fm @ A ) ) ) ) ) ) ).

% i.epath.cases
thf(fact_455_i_Oepath_Ocases,axiom,
    ! [Eff3: fm > produc1996495991257130529ist_fm > option6029102822600979917ist_fm,Rules: stream_fm,A: stream5128679079952569136_fm_fm] :
      ( ( abstra9155489748554225612ist_fm @ ( abstra5048403109753943183ist_fm @ Eff3 ) @ Rules @ A )
     => ~ ( ( member_fm @ ( produc6738438136904575596_fm_fm @ ( shd_Pr7265661315679113588_fm_fm @ A ) ) @ ( sset_fm @ Rules ) )
         => ! [Sl4: fset_P661503646757059847ist_fm] :
              ( ( fmembe3381613331217039976ist_fm @ ( produc1217498474891896878_fm_fm @ ( shd_Pr7265661315679113588_fm_fm @ ( stl_Pr2715999368752424048_fm_fm @ A ) ) ) @ Sl4 )
             => ( ( abstra5048403109753943183ist_fm @ Eff3 @ ( produc6738438136904575596_fm_fm @ ( shd_Pr7265661315679113588_fm_fm @ A ) ) @ ( produc1217498474891896878_fm_fm @ ( shd_Pr7265661315679113588_fm_fm @ A ) ) @ Sl4 )
               => ~ ( abstra9155489748554225612ist_fm @ ( abstra5048403109753943183ist_fm @ Eff3 ) @ Rules @ ( stl_Pr2715999368752424048_fm_fm @ A ) ) ) ) ) ) ).

% i.epath.cases
thf(fact_456_i_Oepath_Ocases,axiom,
    ! [Eff3: rule > produc1996495991257130529ist_fm > option6029102822600979917ist_fm,Rules: stream_rule,A: stream727092118206550309m_rule] :
      ( ( abstra5415202973855822231ist_fm @ ( abstra5459341389059251418ist_fm @ Eff3 ) @ Rules @ A )
     => ~ ( ( member_rule @ ( produc7165828336582415457m_rule @ ( shd_Pr7235097944458474089m_rule @ A ) ) @ ( sset_rule @ Rules ) )
         => ! [Sl4: fset_P661503646757059847ist_fm] :
              ( ( fmembe3381613331217039976ist_fm @ ( produc6879501374131015971m_rule @ ( shd_Pr7235097944458474089m_rule @ ( stl_Pr950425576149878629m_rule @ A ) ) ) @ Sl4 )
             => ( ( abstra5459341389059251418ist_fm @ Eff3 @ ( produc7165828336582415457m_rule @ ( shd_Pr7235097944458474089m_rule @ A ) ) @ ( produc6879501374131015971m_rule @ ( shd_Pr7235097944458474089m_rule @ A ) ) @ Sl4 )
               => ~ ( abstra5415202973855822231ist_fm @ ( abstra5459341389059251418ist_fm @ Eff3 ) @ Rules @ ( stl_Pr950425576149878629m_rule @ A ) ) ) ) ) ) ).

% i.epath.cases
thf(fact_457_i_Oepath_Ocases,axiom,
    ! [Eff3: fm > abstra4572470701423150929m_rule > option7479661550464326865m_rule,Rules: stream_fm,A: stream812778442145099016ule_fm] :
      ( ( abstra1364558218721404774m_rule @ ( abstra7119683891065068131m_rule @ Eff3 ) @ Rules @ A )
     => ~ ( ( member_fm @ ( produc8096733172076065128ule_fm @ ( shd_Pr849711437766469422ule_fm @ A ) ) @ ( sset_fm @ Rules ) )
         => ! [Sl4: fset_A5396818216090053761m_rule] :
              ( ( fmembe1635983029874454474m_rule @ ( produc8194852094735076902ule_fm @ ( shd_Pr849711437766469422ule_fm @ ( stl_Pr2064664776838053682ule_fm @ A ) ) ) @ Sl4 )
             => ( ( abstra7119683891065068131m_rule @ Eff3 @ ( produc8096733172076065128ule_fm @ ( shd_Pr849711437766469422ule_fm @ A ) ) @ ( produc8194852094735076902ule_fm @ ( shd_Pr849711437766469422ule_fm @ A ) ) @ Sl4 )
               => ~ ( abstra1364558218721404774m_rule @ ( abstra7119683891065068131m_rule @ Eff3 ) @ Rules @ ( stl_Pr2064664776838053682ule_fm @ A ) ) ) ) ) ) ).

% i.epath.cases
thf(fact_458_i_Oepath_Ocases,axiom,
    ! [Eff3: rule > abstra4572470701423150929m_rule > option7479661550464326865m_rule,Rules: stream_rule,A: stream7755889244956737277e_rule] :
      ( ( abstra5823804033175701339m_rule @ ( abstra3964157444165176280m_rule @ Eff3 ) @ Rules @ A )
     => ~ ( ( member_rule @ ( produc4652457655725677917e_rule @ ( shd_Pr1235285222789368611e_rule @ A ) ) @ ( sset_rule @ Rules ) )
         => ! [Sl4: fset_A5396818216090053761m_rule] :
              ( ( fmembe1635983029874454474m_rule @ ( produc7822628117027650331e_rule @ ( shd_Pr1235285222789368611e_rule @ ( stl_Pr8695508334200742183e_rule @ A ) ) ) @ Sl4 )
             => ( ( abstra3964157444165176280m_rule @ Eff3 @ ( produc4652457655725677917e_rule @ ( shd_Pr1235285222789368611e_rule @ A ) ) @ ( produc7822628117027650331e_rule @ ( shd_Pr1235285222789368611e_rule @ A ) ) @ Sl4 )
               => ~ ( abstra5823804033175701339m_rule @ ( abstra3964157444165176280m_rule @ Eff3 ) @ Rules @ ( stl_Pr8695508334200742183e_rule @ A ) ) ) ) ) ) ).

% i.epath.cases
thf(fact_459_i_Oepath_Ocases,axiom,
    ! [Eff3: produc1996495991257130529ist_fm > produc1996495991257130529ist_fm > option6029102822600979917ist_fm,Rules: stream8299795917829157543ist_fm,A: stream3409308193418444653ist_fm] :
      ( ( abstra5061764961602482947ist_fm @ ( abstra4222672346834617088ist_fm @ Eff3 ) @ Rules @ A )
     => ~ ( ( member8102475879199740618ist_fm @ ( produc1639249265332130309ist_fm @ ( shd_Pr4587271278095355155ist_fm @ A ) ) @ ( sset_P3431374123360452149ist_fm @ Rules ) )
         => ! [Sl4: fset_P661503646757059847ist_fm] :
              ( ( fmembe3381613331217039976ist_fm @ ( produc5894184038225292483ist_fm @ ( shd_Pr4587271278095355155ist_fm @ ( stl_Pr8700568296466786583ist_fm @ A ) ) ) @ Sl4 )
             => ( ( abstra4222672346834617088ist_fm @ Eff3 @ ( produc1639249265332130309ist_fm @ ( shd_Pr4587271278095355155ist_fm @ A ) ) @ ( produc5894184038225292483ist_fm @ ( shd_Pr4587271278095355155ist_fm @ A ) ) @ Sl4 )
               => ~ ( abstra5061764961602482947ist_fm @ ( abstra4222672346834617088ist_fm @ Eff3 ) @ Rules @ ( stl_Pr8700568296466786583ist_fm @ A ) ) ) ) ) ) ).

% i.epath.cases
thf(fact_460_i_Oepath_Ocases,axiom,
    ! [Eff3: produc859450856879609959at_nat > produc1996495991257130529ist_fm > option6029102822600979917ist_fm,Rules: stream8372878641218411373at_nat,A: stream1622999302952468147at_nat] :
      ( ( abstra1311857326046938941ist_fm @ ( abstra6247935586015979962ist_fm @ Eff3 ) @ Rules @ A )
     => ~ ( ( member8206827879206165904at_nat @ ( produc2750861588394444363at_nat @ ( shd_Pr5062688428016131033at_nat @ A ) ) @ ( sset_P9164763103296375419at_nat @ Rules ) )
         => ! [Sl4: fset_P661503646757059847ist_fm] :
              ( ( fmembe3381613331217039976ist_fm @ ( produc2934448030062000137at_nat @ ( shd_Pr5062688428016131033at_nat @ ( stl_Pr1564049449699658205at_nat @ A ) ) ) @ Sl4 )
             => ( ( abstra6247935586015979962ist_fm @ Eff3 @ ( produc2750861588394444363at_nat @ ( shd_Pr5062688428016131033at_nat @ A ) ) @ ( produc2934448030062000137at_nat @ ( shd_Pr5062688428016131033at_nat @ A ) ) @ Sl4 )
               => ~ ( abstra1311857326046938941ist_fm @ ( abstra6247935586015979962ist_fm @ Eff3 ) @ Rules @ ( stl_Pr1564049449699658205at_nat @ A ) ) ) ) ) ) ).

% i.epath.cases
thf(fact_461_i_Oepath_Ocases,axiom,
    ! [Eff3: produc164195504107695125m_rule > produc1996495991257130529ist_fm > option6029102822600979917ist_fm,Rules: stream727092118206550309m_rule,A: stream642806101564963573m_rule] :
      ( ( abstra6058711572139910855ist_fm @ ( abstra2934389420647555978ist_fm @ Eff3 ) @ Rules @ A )
     => ~ ( ( member4220325220686508332m_rule @ ( produc2016132375285593905m_rule @ ( shd_Pr1407159933805374393m_rule @ A ) ) @ ( sset_P4979403222752902913m_rule @ Rules ) )
         => ! [Sl4: fset_P661503646757059847ist_fm] :
              ( ( fmembe3381613331217039976ist_fm @ ( produc8602611229794383091m_rule @ ( shd_Pr1407159933805374393m_rule @ ( stl_Pr8578586445745536693m_rule @ A ) ) ) @ Sl4 )
             => ( ( abstra2934389420647555978ist_fm @ Eff3 @ ( produc2016132375285593905m_rule @ ( shd_Pr1407159933805374393m_rule @ A ) ) @ ( produc8602611229794383091m_rule @ ( shd_Pr1407159933805374393m_rule @ A ) ) @ Sl4 )
               => ~ ( abstra6058711572139910855ist_fm @ ( abstra2934389420647555978ist_fm @ Eff3 ) @ Rules @ ( stl_Pr8578586445745536693m_rule @ A ) ) ) ) ) ) ).

% i.epath.cases
thf(fact_462_i_Oepath_Ocases,axiom,
    ! [Eff3: produc1996495991257130529ist_fm > abstra4572470701423150929m_rule > option7479661550464326865m_rule,Rules: stream8299795917829157543ist_fm,A: stream7989562319868115861ist_fm] :
      ( ( abstra8066765647771630831m_rule @ ( abstra1645840942258654514m_rule @ Eff3 ) @ Rules @ A )
     => ~ ( ( member8102475879199740618ist_fm @ ( produc3674566753905654281ist_fm @ ( shd_Pr563299216415026649ist_fm @ A ) ) @ ( sset_P3431374123360452149ist_fm @ Rules ) )
         => ! [Sl4: fset_A5396818216090053761m_rule] :
              ( ( fmembe1635983029874454474m_rule @ ( produc2646333476271629515ist_fm @ ( shd_Pr563299216415026649ist_fm @ ( stl_Pr6727363551237990101ist_fm @ A ) ) ) @ Sl4 )
             => ( ( abstra1645840942258654514m_rule @ Eff3 @ ( produc3674566753905654281ist_fm @ ( shd_Pr563299216415026649ist_fm @ A ) ) @ ( produc2646333476271629515ist_fm @ ( shd_Pr563299216415026649ist_fm @ A ) ) @ Sl4 )
               => ~ ( abstra8066765647771630831m_rule @ ( abstra1645840942258654514m_rule @ Eff3 ) @ Rules @ ( stl_Pr6727363551237990101ist_fm @ A ) ) ) ) ) ) ).

% i.epath.cases
thf(fact_463_i_Oepath_Ocases,axiom,
    ! [Eff3: produc859450856879609959at_nat > abstra4572470701423150929m_rule > option7479661550464326865m_rule,Rules: stream8372878641218411373at_nat,A: stream1819640557982756571at_nat] :
      ( ( abstra3157339599784851765m_rule @ ( abstra3953117623887779576m_rule @ Eff3 ) @ Rules @ A )
     => ~ ( ( member8206827879206165904at_nat @ ( produc601515061966782543at_nat @ ( shd_Pr618815079712798623at_nat @ A ) ) @ ( sset_P9164763103296375419at_nat @ Rules ) )
         => ! [Sl4: fset_A5396818216090053761m_rule] :
              ( ( fmembe1635983029874454474m_rule @ ( produc567330280010208273at_nat @ ( shd_Pr618815079712798623at_nat @ ( stl_Pr2369401607900929691at_nat @ A ) ) ) @ Sl4 )
             => ( ( abstra3953117623887779576m_rule @ Eff3 @ ( produc601515061966782543at_nat @ ( shd_Pr618815079712798623at_nat @ A ) ) @ ( produc567330280010208273at_nat @ ( shd_Pr618815079712798623at_nat @ A ) ) @ Sl4 )
               => ~ ( abstra3157339599784851765m_rule @ ( abstra3953117623887779576m_rule @ Eff3 ) @ Rules @ ( stl_Pr2369401607900929691at_nat @ A ) ) ) ) ) ) ).

% i.epath.cases
thf(fact_464_i_Oepath_Osimps,axiom,
    ! [Eff3: list_fm > list_fm > option_fset_list_fm,Rules: stream_list_fm,A: stream8299795917829157543ist_fm] :
      ( ( abstra6579460235205289861ist_fm @ ( abstra256862739949234434ist_fm @ Eff3 ) @ Rules @ A )
      = ( ? [Steps3: stream8299795917829157543ist_fm,Sl: fset_list_fm] :
            ( ( A = Steps3 )
            & ( member_list_fm @ ( produc4588648349897876871ist_fm @ ( shd_Pr772355297128350925ist_fm @ Steps3 ) ) @ ( sset_list_fm @ Rules ) )
            & ( fmember_list_fm @ ( produc1501393135466168645ist_fm @ ( shd_Pr772355297128350925ist_fm @ ( stl_Pr5027778045650968273ist_fm @ Steps3 ) ) ) @ Sl )
            & ( abstra256862739949234434ist_fm @ Eff3 @ ( produc4588648349897876871ist_fm @ ( shd_Pr772355297128350925ist_fm @ Steps3 ) ) @ ( produc1501393135466168645ist_fm @ ( shd_Pr772355297128350925ist_fm @ Steps3 ) ) @ Sl )
            & ( abstra6579460235205289861ist_fm @ ( abstra256862739949234434ist_fm @ Eff3 ) @ Rules @ ( stl_Pr5027778045650968273ist_fm @ Steps3 ) ) ) ) ) ).

% i.epath.simps
thf(fact_465_i_Oepath_Osimps,axiom,
    ! [Eff3: fm > produc1996495991257130529ist_fm > option6029102822600979917ist_fm,Rules: stream_fm,A: stream5128679079952569136_fm_fm] :
      ( ( abstra9155489748554225612ist_fm @ ( abstra5048403109753943183ist_fm @ Eff3 ) @ Rules @ A )
      = ( ? [Steps3: stream5128679079952569136_fm_fm,Sl: fset_P661503646757059847ist_fm] :
            ( ( A = Steps3 )
            & ( member_fm @ ( produc6738438136904575596_fm_fm @ ( shd_Pr7265661315679113588_fm_fm @ Steps3 ) ) @ ( sset_fm @ Rules ) )
            & ( fmembe3381613331217039976ist_fm @ ( produc1217498474891896878_fm_fm @ ( shd_Pr7265661315679113588_fm_fm @ ( stl_Pr2715999368752424048_fm_fm @ Steps3 ) ) ) @ Sl )
            & ( abstra5048403109753943183ist_fm @ Eff3 @ ( produc6738438136904575596_fm_fm @ ( shd_Pr7265661315679113588_fm_fm @ Steps3 ) ) @ ( produc1217498474891896878_fm_fm @ ( shd_Pr7265661315679113588_fm_fm @ Steps3 ) ) @ Sl )
            & ( abstra9155489748554225612ist_fm @ ( abstra5048403109753943183ist_fm @ Eff3 ) @ Rules @ ( stl_Pr2715999368752424048_fm_fm @ Steps3 ) ) ) ) ) ).

% i.epath.simps
thf(fact_466_i_Oepath_Osimps,axiom,
    ! [Eff3: rule > produc1996495991257130529ist_fm > option6029102822600979917ist_fm,Rules: stream_rule,A: stream727092118206550309m_rule] :
      ( ( abstra5415202973855822231ist_fm @ ( abstra5459341389059251418ist_fm @ Eff3 ) @ Rules @ A )
      = ( ? [Steps3: stream727092118206550309m_rule,Sl: fset_P661503646757059847ist_fm] :
            ( ( A = Steps3 )
            & ( member_rule @ ( produc7165828336582415457m_rule @ ( shd_Pr7235097944458474089m_rule @ Steps3 ) ) @ ( sset_rule @ Rules ) )
            & ( fmembe3381613331217039976ist_fm @ ( produc6879501374131015971m_rule @ ( shd_Pr7235097944458474089m_rule @ ( stl_Pr950425576149878629m_rule @ Steps3 ) ) ) @ Sl )
            & ( abstra5459341389059251418ist_fm @ Eff3 @ ( produc7165828336582415457m_rule @ ( shd_Pr7235097944458474089m_rule @ Steps3 ) ) @ ( produc6879501374131015971m_rule @ ( shd_Pr7235097944458474089m_rule @ Steps3 ) ) @ Sl )
            & ( abstra5415202973855822231ist_fm @ ( abstra5459341389059251418ist_fm @ Eff3 ) @ Rules @ ( stl_Pr950425576149878629m_rule @ Steps3 ) ) ) ) ) ).

% i.epath.simps
thf(fact_467_i_Oepath_Osimps,axiom,
    ! [Eff3: fm > abstra4572470701423150929m_rule > option7479661550464326865m_rule,Rules: stream_fm,A: stream812778442145099016ule_fm] :
      ( ( abstra1364558218721404774m_rule @ ( abstra7119683891065068131m_rule @ Eff3 ) @ Rules @ A )
      = ( ? [Steps3: stream812778442145099016ule_fm,Sl: fset_A5396818216090053761m_rule] :
            ( ( A = Steps3 )
            & ( member_fm @ ( produc8096733172076065128ule_fm @ ( shd_Pr849711437766469422ule_fm @ Steps3 ) ) @ ( sset_fm @ Rules ) )
            & ( fmembe1635983029874454474m_rule @ ( produc8194852094735076902ule_fm @ ( shd_Pr849711437766469422ule_fm @ ( stl_Pr2064664776838053682ule_fm @ Steps3 ) ) ) @ Sl )
            & ( abstra7119683891065068131m_rule @ Eff3 @ ( produc8096733172076065128ule_fm @ ( shd_Pr849711437766469422ule_fm @ Steps3 ) ) @ ( produc8194852094735076902ule_fm @ ( shd_Pr849711437766469422ule_fm @ Steps3 ) ) @ Sl )
            & ( abstra1364558218721404774m_rule @ ( abstra7119683891065068131m_rule @ Eff3 ) @ Rules @ ( stl_Pr2064664776838053682ule_fm @ Steps3 ) ) ) ) ) ).

% i.epath.simps
thf(fact_468_i_Oepath_Osimps,axiom,
    ! [Eff3: rule > abstra4572470701423150929m_rule > option7479661550464326865m_rule,Rules: stream_rule,A: stream7755889244956737277e_rule] :
      ( ( abstra5823804033175701339m_rule @ ( abstra3964157444165176280m_rule @ Eff3 ) @ Rules @ A )
      = ( ? [Steps3: stream7755889244956737277e_rule,Sl: fset_A5396818216090053761m_rule] :
            ( ( A = Steps3 )
            & ( member_rule @ ( produc4652457655725677917e_rule @ ( shd_Pr1235285222789368611e_rule @ Steps3 ) ) @ ( sset_rule @ Rules ) )
            & ( fmembe1635983029874454474m_rule @ ( produc7822628117027650331e_rule @ ( shd_Pr1235285222789368611e_rule @ ( stl_Pr8695508334200742183e_rule @ Steps3 ) ) ) @ Sl )
            & ( abstra3964157444165176280m_rule @ Eff3 @ ( produc4652457655725677917e_rule @ ( shd_Pr1235285222789368611e_rule @ Steps3 ) ) @ ( produc7822628117027650331e_rule @ ( shd_Pr1235285222789368611e_rule @ Steps3 ) ) @ Sl )
            & ( abstra5823804033175701339m_rule @ ( abstra3964157444165176280m_rule @ Eff3 ) @ Rules @ ( stl_Pr8695508334200742183e_rule @ Steps3 ) ) ) ) ) ).

% i.epath.simps
thf(fact_469_i_Oepath_Osimps,axiom,
    ! [Eff3: produc1996495991257130529ist_fm > produc1996495991257130529ist_fm > option6029102822600979917ist_fm,Rules: stream8299795917829157543ist_fm,A: stream3409308193418444653ist_fm] :
      ( ( abstra5061764961602482947ist_fm @ ( abstra4222672346834617088ist_fm @ Eff3 ) @ Rules @ A )
      = ( ? [Steps3: stream3409308193418444653ist_fm,Sl: fset_P661503646757059847ist_fm] :
            ( ( A = Steps3 )
            & ( member8102475879199740618ist_fm @ ( produc1639249265332130309ist_fm @ ( shd_Pr4587271278095355155ist_fm @ Steps3 ) ) @ ( sset_P3431374123360452149ist_fm @ Rules ) )
            & ( fmembe3381613331217039976ist_fm @ ( produc5894184038225292483ist_fm @ ( shd_Pr4587271278095355155ist_fm @ ( stl_Pr8700568296466786583ist_fm @ Steps3 ) ) ) @ Sl )
            & ( abstra4222672346834617088ist_fm @ Eff3 @ ( produc1639249265332130309ist_fm @ ( shd_Pr4587271278095355155ist_fm @ Steps3 ) ) @ ( produc5894184038225292483ist_fm @ ( shd_Pr4587271278095355155ist_fm @ Steps3 ) ) @ Sl )
            & ( abstra5061764961602482947ist_fm @ ( abstra4222672346834617088ist_fm @ Eff3 ) @ Rules @ ( stl_Pr8700568296466786583ist_fm @ Steps3 ) ) ) ) ) ).

% i.epath.simps
thf(fact_470_i_Oepath_Osimps,axiom,
    ! [Eff3: produc859450856879609959at_nat > produc1996495991257130529ist_fm > option6029102822600979917ist_fm,Rules: stream8372878641218411373at_nat,A: stream1622999302952468147at_nat] :
      ( ( abstra1311857326046938941ist_fm @ ( abstra6247935586015979962ist_fm @ Eff3 ) @ Rules @ A )
      = ( ? [Steps3: stream1622999302952468147at_nat,Sl: fset_P661503646757059847ist_fm] :
            ( ( A = Steps3 )
            & ( member8206827879206165904at_nat @ ( produc2750861588394444363at_nat @ ( shd_Pr5062688428016131033at_nat @ Steps3 ) ) @ ( sset_P9164763103296375419at_nat @ Rules ) )
            & ( fmembe3381613331217039976ist_fm @ ( produc2934448030062000137at_nat @ ( shd_Pr5062688428016131033at_nat @ ( stl_Pr1564049449699658205at_nat @ Steps3 ) ) ) @ Sl )
            & ( abstra6247935586015979962ist_fm @ Eff3 @ ( produc2750861588394444363at_nat @ ( shd_Pr5062688428016131033at_nat @ Steps3 ) ) @ ( produc2934448030062000137at_nat @ ( shd_Pr5062688428016131033at_nat @ Steps3 ) ) @ Sl )
            & ( abstra1311857326046938941ist_fm @ ( abstra6247935586015979962ist_fm @ Eff3 ) @ Rules @ ( stl_Pr1564049449699658205at_nat @ Steps3 ) ) ) ) ) ).

% i.epath.simps
thf(fact_471_i_Oepath_Osimps,axiom,
    ! [Eff3: produc164195504107695125m_rule > produc1996495991257130529ist_fm > option6029102822600979917ist_fm,Rules: stream727092118206550309m_rule,A: stream642806101564963573m_rule] :
      ( ( abstra6058711572139910855ist_fm @ ( abstra2934389420647555978ist_fm @ Eff3 ) @ Rules @ A )
      = ( ? [Steps3: stream642806101564963573m_rule,Sl: fset_P661503646757059847ist_fm] :
            ( ( A = Steps3 )
            & ( member4220325220686508332m_rule @ ( produc2016132375285593905m_rule @ ( shd_Pr1407159933805374393m_rule @ Steps3 ) ) @ ( sset_P4979403222752902913m_rule @ Rules ) )
            & ( fmembe3381613331217039976ist_fm @ ( produc8602611229794383091m_rule @ ( shd_Pr1407159933805374393m_rule @ ( stl_Pr8578586445745536693m_rule @ Steps3 ) ) ) @ Sl )
            & ( abstra2934389420647555978ist_fm @ Eff3 @ ( produc2016132375285593905m_rule @ ( shd_Pr1407159933805374393m_rule @ Steps3 ) ) @ ( produc8602611229794383091m_rule @ ( shd_Pr1407159933805374393m_rule @ Steps3 ) ) @ Sl )
            & ( abstra6058711572139910855ist_fm @ ( abstra2934389420647555978ist_fm @ Eff3 ) @ Rules @ ( stl_Pr8578586445745536693m_rule @ Steps3 ) ) ) ) ) ).

% i.epath.simps
thf(fact_472_i_Oepath_Osimps,axiom,
    ! [Eff3: produc1996495991257130529ist_fm > abstra4572470701423150929m_rule > option7479661550464326865m_rule,Rules: stream8299795917829157543ist_fm,A: stream7989562319868115861ist_fm] :
      ( ( abstra8066765647771630831m_rule @ ( abstra1645840942258654514m_rule @ Eff3 ) @ Rules @ A )
      = ( ? [Steps3: stream7989562319868115861ist_fm,Sl: fset_A5396818216090053761m_rule] :
            ( ( A = Steps3 )
            & ( member8102475879199740618ist_fm @ ( produc3674566753905654281ist_fm @ ( shd_Pr563299216415026649ist_fm @ Steps3 ) ) @ ( sset_P3431374123360452149ist_fm @ Rules ) )
            & ( fmembe1635983029874454474m_rule @ ( produc2646333476271629515ist_fm @ ( shd_Pr563299216415026649ist_fm @ ( stl_Pr6727363551237990101ist_fm @ Steps3 ) ) ) @ Sl )
            & ( abstra1645840942258654514m_rule @ Eff3 @ ( produc3674566753905654281ist_fm @ ( shd_Pr563299216415026649ist_fm @ Steps3 ) ) @ ( produc2646333476271629515ist_fm @ ( shd_Pr563299216415026649ist_fm @ Steps3 ) ) @ Sl )
            & ( abstra8066765647771630831m_rule @ ( abstra1645840942258654514m_rule @ Eff3 ) @ Rules @ ( stl_Pr6727363551237990101ist_fm @ Steps3 ) ) ) ) ) ).

% i.epath.simps
thf(fact_473_i_Oepath_Osimps,axiom,
    ! [Eff3: produc859450856879609959at_nat > abstra4572470701423150929m_rule > option7479661550464326865m_rule,Rules: stream8372878641218411373at_nat,A: stream1819640557982756571at_nat] :
      ( ( abstra3157339599784851765m_rule @ ( abstra3953117623887779576m_rule @ Eff3 ) @ Rules @ A )
      = ( ? [Steps3: stream1819640557982756571at_nat,Sl: fset_A5396818216090053761m_rule] :
            ( ( A = Steps3 )
            & ( member8206827879206165904at_nat @ ( produc601515061966782543at_nat @ ( shd_Pr618815079712798623at_nat @ Steps3 ) ) @ ( sset_P9164763103296375419at_nat @ Rules ) )
            & ( fmembe1635983029874454474m_rule @ ( produc567330280010208273at_nat @ ( shd_Pr618815079712798623at_nat @ ( stl_Pr2369401607900929691at_nat @ Steps3 ) ) ) @ Sl )
            & ( abstra3953117623887779576m_rule @ Eff3 @ ( produc601515061966782543at_nat @ ( shd_Pr618815079712798623at_nat @ Steps3 ) ) @ ( produc567330280010208273at_nat @ ( shd_Pr618815079712798623at_nat @ Steps3 ) ) @ Sl )
            & ( abstra3157339599784851765m_rule @ ( abstra3953117623887779576m_rule @ Eff3 ) @ Rules @ ( stl_Pr2369401607900929691at_nat @ Steps3 ) ) ) ) ) ).

% i.epath.simps
thf(fact_474_i_Oepath_Ointros,axiom,
    ! [Steps: stream8299795917829157543ist_fm,Rules: stream_list_fm,Sl3: fset_list_fm,Eff3: list_fm > list_fm > option_fset_list_fm] :
      ( ( member_list_fm @ ( produc4588648349897876871ist_fm @ ( shd_Pr772355297128350925ist_fm @ Steps ) ) @ ( sset_list_fm @ Rules ) )
     => ( ( fmember_list_fm @ ( produc1501393135466168645ist_fm @ ( shd_Pr772355297128350925ist_fm @ ( stl_Pr5027778045650968273ist_fm @ Steps ) ) ) @ Sl3 )
       => ( ( abstra256862739949234434ist_fm @ Eff3 @ ( produc4588648349897876871ist_fm @ ( shd_Pr772355297128350925ist_fm @ Steps ) ) @ ( produc1501393135466168645ist_fm @ ( shd_Pr772355297128350925ist_fm @ Steps ) ) @ Sl3 )
         => ( ( abstra6579460235205289861ist_fm @ ( abstra256862739949234434ist_fm @ Eff3 ) @ Rules @ ( stl_Pr5027778045650968273ist_fm @ Steps ) )
           => ( abstra6579460235205289861ist_fm @ ( abstra256862739949234434ist_fm @ Eff3 ) @ Rules @ Steps ) ) ) ) ) ).

% i.epath.intros
thf(fact_475_i_Oepath_Ointros,axiom,
    ! [Steps: stream5128679079952569136_fm_fm,Rules: stream_fm,Sl3: fset_P661503646757059847ist_fm,Eff3: fm > produc1996495991257130529ist_fm > option6029102822600979917ist_fm] :
      ( ( member_fm @ ( produc6738438136904575596_fm_fm @ ( shd_Pr7265661315679113588_fm_fm @ Steps ) ) @ ( sset_fm @ Rules ) )
     => ( ( fmembe3381613331217039976ist_fm @ ( produc1217498474891896878_fm_fm @ ( shd_Pr7265661315679113588_fm_fm @ ( stl_Pr2715999368752424048_fm_fm @ Steps ) ) ) @ Sl3 )
       => ( ( abstra5048403109753943183ist_fm @ Eff3 @ ( produc6738438136904575596_fm_fm @ ( shd_Pr7265661315679113588_fm_fm @ Steps ) ) @ ( produc1217498474891896878_fm_fm @ ( shd_Pr7265661315679113588_fm_fm @ Steps ) ) @ Sl3 )
         => ( ( abstra9155489748554225612ist_fm @ ( abstra5048403109753943183ist_fm @ Eff3 ) @ Rules @ ( stl_Pr2715999368752424048_fm_fm @ Steps ) )
           => ( abstra9155489748554225612ist_fm @ ( abstra5048403109753943183ist_fm @ Eff3 ) @ Rules @ Steps ) ) ) ) ) ).

% i.epath.intros
thf(fact_476_i_Oepath_Ointros,axiom,
    ! [Steps: stream727092118206550309m_rule,Rules: stream_rule,Sl3: fset_P661503646757059847ist_fm,Eff3: rule > produc1996495991257130529ist_fm > option6029102822600979917ist_fm] :
      ( ( member_rule @ ( produc7165828336582415457m_rule @ ( shd_Pr7235097944458474089m_rule @ Steps ) ) @ ( sset_rule @ Rules ) )
     => ( ( fmembe3381613331217039976ist_fm @ ( produc6879501374131015971m_rule @ ( shd_Pr7235097944458474089m_rule @ ( stl_Pr950425576149878629m_rule @ Steps ) ) ) @ Sl3 )
       => ( ( abstra5459341389059251418ist_fm @ Eff3 @ ( produc7165828336582415457m_rule @ ( shd_Pr7235097944458474089m_rule @ Steps ) ) @ ( produc6879501374131015971m_rule @ ( shd_Pr7235097944458474089m_rule @ Steps ) ) @ Sl3 )
         => ( ( abstra5415202973855822231ist_fm @ ( abstra5459341389059251418ist_fm @ Eff3 ) @ Rules @ ( stl_Pr950425576149878629m_rule @ Steps ) )
           => ( abstra5415202973855822231ist_fm @ ( abstra5459341389059251418ist_fm @ Eff3 ) @ Rules @ Steps ) ) ) ) ) ).

% i.epath.intros
thf(fact_477_i_Oepath_Ointros,axiom,
    ! [Steps: stream812778442145099016ule_fm,Rules: stream_fm,Sl3: fset_A5396818216090053761m_rule,Eff3: fm > abstra4572470701423150929m_rule > option7479661550464326865m_rule] :
      ( ( member_fm @ ( produc8096733172076065128ule_fm @ ( shd_Pr849711437766469422ule_fm @ Steps ) ) @ ( sset_fm @ Rules ) )
     => ( ( fmembe1635983029874454474m_rule @ ( produc8194852094735076902ule_fm @ ( shd_Pr849711437766469422ule_fm @ ( stl_Pr2064664776838053682ule_fm @ Steps ) ) ) @ Sl3 )
       => ( ( abstra7119683891065068131m_rule @ Eff3 @ ( produc8096733172076065128ule_fm @ ( shd_Pr849711437766469422ule_fm @ Steps ) ) @ ( produc8194852094735076902ule_fm @ ( shd_Pr849711437766469422ule_fm @ Steps ) ) @ Sl3 )
         => ( ( abstra1364558218721404774m_rule @ ( abstra7119683891065068131m_rule @ Eff3 ) @ Rules @ ( stl_Pr2064664776838053682ule_fm @ Steps ) )
           => ( abstra1364558218721404774m_rule @ ( abstra7119683891065068131m_rule @ Eff3 ) @ Rules @ Steps ) ) ) ) ) ).

% i.epath.intros
thf(fact_478_i_Oepath_Ointros,axiom,
    ! [Steps: stream7755889244956737277e_rule,Rules: stream_rule,Sl3: fset_A5396818216090053761m_rule,Eff3: rule > abstra4572470701423150929m_rule > option7479661550464326865m_rule] :
      ( ( member_rule @ ( produc4652457655725677917e_rule @ ( shd_Pr1235285222789368611e_rule @ Steps ) ) @ ( sset_rule @ Rules ) )
     => ( ( fmembe1635983029874454474m_rule @ ( produc7822628117027650331e_rule @ ( shd_Pr1235285222789368611e_rule @ ( stl_Pr8695508334200742183e_rule @ Steps ) ) ) @ Sl3 )
       => ( ( abstra3964157444165176280m_rule @ Eff3 @ ( produc4652457655725677917e_rule @ ( shd_Pr1235285222789368611e_rule @ Steps ) ) @ ( produc7822628117027650331e_rule @ ( shd_Pr1235285222789368611e_rule @ Steps ) ) @ Sl3 )
         => ( ( abstra5823804033175701339m_rule @ ( abstra3964157444165176280m_rule @ Eff3 ) @ Rules @ ( stl_Pr8695508334200742183e_rule @ Steps ) )
           => ( abstra5823804033175701339m_rule @ ( abstra3964157444165176280m_rule @ Eff3 ) @ Rules @ Steps ) ) ) ) ) ).

% i.epath.intros
thf(fact_479_i_Oepath_Ointros,axiom,
    ! [Steps: stream3409308193418444653ist_fm,Rules: stream8299795917829157543ist_fm,Sl3: fset_P661503646757059847ist_fm,Eff3: produc1996495991257130529ist_fm > produc1996495991257130529ist_fm > option6029102822600979917ist_fm] :
      ( ( member8102475879199740618ist_fm @ ( produc1639249265332130309ist_fm @ ( shd_Pr4587271278095355155ist_fm @ Steps ) ) @ ( sset_P3431374123360452149ist_fm @ Rules ) )
     => ( ( fmembe3381613331217039976ist_fm @ ( produc5894184038225292483ist_fm @ ( shd_Pr4587271278095355155ist_fm @ ( stl_Pr8700568296466786583ist_fm @ Steps ) ) ) @ Sl3 )
       => ( ( abstra4222672346834617088ist_fm @ Eff3 @ ( produc1639249265332130309ist_fm @ ( shd_Pr4587271278095355155ist_fm @ Steps ) ) @ ( produc5894184038225292483ist_fm @ ( shd_Pr4587271278095355155ist_fm @ Steps ) ) @ Sl3 )
         => ( ( abstra5061764961602482947ist_fm @ ( abstra4222672346834617088ist_fm @ Eff3 ) @ Rules @ ( stl_Pr8700568296466786583ist_fm @ Steps ) )
           => ( abstra5061764961602482947ist_fm @ ( abstra4222672346834617088ist_fm @ Eff3 ) @ Rules @ Steps ) ) ) ) ) ).

% i.epath.intros
thf(fact_480_i_Oepath_Ointros,axiom,
    ! [Steps: stream1622999302952468147at_nat,Rules: stream8372878641218411373at_nat,Sl3: fset_P661503646757059847ist_fm,Eff3: produc859450856879609959at_nat > produc1996495991257130529ist_fm > option6029102822600979917ist_fm] :
      ( ( member8206827879206165904at_nat @ ( produc2750861588394444363at_nat @ ( shd_Pr5062688428016131033at_nat @ Steps ) ) @ ( sset_P9164763103296375419at_nat @ Rules ) )
     => ( ( fmembe3381613331217039976ist_fm @ ( produc2934448030062000137at_nat @ ( shd_Pr5062688428016131033at_nat @ ( stl_Pr1564049449699658205at_nat @ Steps ) ) ) @ Sl3 )
       => ( ( abstra6247935586015979962ist_fm @ Eff3 @ ( produc2750861588394444363at_nat @ ( shd_Pr5062688428016131033at_nat @ Steps ) ) @ ( produc2934448030062000137at_nat @ ( shd_Pr5062688428016131033at_nat @ Steps ) ) @ Sl3 )
         => ( ( abstra1311857326046938941ist_fm @ ( abstra6247935586015979962ist_fm @ Eff3 ) @ Rules @ ( stl_Pr1564049449699658205at_nat @ Steps ) )
           => ( abstra1311857326046938941ist_fm @ ( abstra6247935586015979962ist_fm @ Eff3 ) @ Rules @ Steps ) ) ) ) ) ).

% i.epath.intros
thf(fact_481_i_Oepath_Ointros,axiom,
    ! [Steps: stream642806101564963573m_rule,Rules: stream727092118206550309m_rule,Sl3: fset_P661503646757059847ist_fm,Eff3: produc164195504107695125m_rule > produc1996495991257130529ist_fm > option6029102822600979917ist_fm] :
      ( ( member4220325220686508332m_rule @ ( produc2016132375285593905m_rule @ ( shd_Pr1407159933805374393m_rule @ Steps ) ) @ ( sset_P4979403222752902913m_rule @ Rules ) )
     => ( ( fmembe3381613331217039976ist_fm @ ( produc8602611229794383091m_rule @ ( shd_Pr1407159933805374393m_rule @ ( stl_Pr8578586445745536693m_rule @ Steps ) ) ) @ Sl3 )
       => ( ( abstra2934389420647555978ist_fm @ Eff3 @ ( produc2016132375285593905m_rule @ ( shd_Pr1407159933805374393m_rule @ Steps ) ) @ ( produc8602611229794383091m_rule @ ( shd_Pr1407159933805374393m_rule @ Steps ) ) @ Sl3 )
         => ( ( abstra6058711572139910855ist_fm @ ( abstra2934389420647555978ist_fm @ Eff3 ) @ Rules @ ( stl_Pr8578586445745536693m_rule @ Steps ) )
           => ( abstra6058711572139910855ist_fm @ ( abstra2934389420647555978ist_fm @ Eff3 ) @ Rules @ Steps ) ) ) ) ) ).

% i.epath.intros
thf(fact_482_i_Oepath_Ointros,axiom,
    ! [Steps: stream7989562319868115861ist_fm,Rules: stream8299795917829157543ist_fm,Sl3: fset_A5396818216090053761m_rule,Eff3: produc1996495991257130529ist_fm > abstra4572470701423150929m_rule > option7479661550464326865m_rule] :
      ( ( member8102475879199740618ist_fm @ ( produc3674566753905654281ist_fm @ ( shd_Pr563299216415026649ist_fm @ Steps ) ) @ ( sset_P3431374123360452149ist_fm @ Rules ) )
     => ( ( fmembe1635983029874454474m_rule @ ( produc2646333476271629515ist_fm @ ( shd_Pr563299216415026649ist_fm @ ( stl_Pr6727363551237990101ist_fm @ Steps ) ) ) @ Sl3 )
       => ( ( abstra1645840942258654514m_rule @ Eff3 @ ( produc3674566753905654281ist_fm @ ( shd_Pr563299216415026649ist_fm @ Steps ) ) @ ( produc2646333476271629515ist_fm @ ( shd_Pr563299216415026649ist_fm @ Steps ) ) @ Sl3 )
         => ( ( abstra8066765647771630831m_rule @ ( abstra1645840942258654514m_rule @ Eff3 ) @ Rules @ ( stl_Pr6727363551237990101ist_fm @ Steps ) )
           => ( abstra8066765647771630831m_rule @ ( abstra1645840942258654514m_rule @ Eff3 ) @ Rules @ Steps ) ) ) ) ) ).

% i.epath.intros
thf(fact_483_i_Oepath_Ointros,axiom,
    ! [Steps: stream1819640557982756571at_nat,Rules: stream8372878641218411373at_nat,Sl3: fset_A5396818216090053761m_rule,Eff3: produc859450856879609959at_nat > abstra4572470701423150929m_rule > option7479661550464326865m_rule] :
      ( ( member8206827879206165904at_nat @ ( produc601515061966782543at_nat @ ( shd_Pr618815079712798623at_nat @ Steps ) ) @ ( sset_P9164763103296375419at_nat @ Rules ) )
     => ( ( fmembe1635983029874454474m_rule @ ( produc567330280010208273at_nat @ ( shd_Pr618815079712798623at_nat @ ( stl_Pr2369401607900929691at_nat @ Steps ) ) ) @ Sl3 )
       => ( ( abstra3953117623887779576m_rule @ Eff3 @ ( produc601515061966782543at_nat @ ( shd_Pr618815079712798623at_nat @ Steps ) ) @ ( produc567330280010208273at_nat @ ( shd_Pr618815079712798623at_nat @ Steps ) ) @ Sl3 )
         => ( ( abstra3157339599784851765m_rule @ ( abstra3953117623887779576m_rule @ Eff3 ) @ Rules @ ( stl_Pr2369401607900929691at_nat @ Steps ) )
           => ( abstra3157339599784851765m_rule @ ( abstra3953117623887779576m_rule @ Eff3 ) @ Rules @ Steps ) ) ) ) ) ).

% i.epath.intros
thf(fact_484_i_Oepath_Ocoinduct,axiom,
    ! [X7: stream8299795917829157543ist_fm > $o,X3: stream8299795917829157543ist_fm,Rules: stream_list_fm,Eff3: list_fm > list_fm > option_fset_list_fm] :
      ( ( X7 @ X3 )
     => ( ! [X: stream8299795917829157543ist_fm] :
            ( ( X7 @ X )
           => ? [Steps2: stream8299795917829157543ist_fm,Sl2: fset_list_fm] :
                ( ( X = Steps2 )
                & ( member_list_fm @ ( produc4588648349897876871ist_fm @ ( shd_Pr772355297128350925ist_fm @ Steps2 ) ) @ ( sset_list_fm @ Rules ) )
                & ( fmember_list_fm @ ( produc1501393135466168645ist_fm @ ( shd_Pr772355297128350925ist_fm @ ( stl_Pr5027778045650968273ist_fm @ Steps2 ) ) ) @ Sl2 )
                & ( abstra256862739949234434ist_fm @ Eff3 @ ( produc4588648349897876871ist_fm @ ( shd_Pr772355297128350925ist_fm @ Steps2 ) ) @ ( produc1501393135466168645ist_fm @ ( shd_Pr772355297128350925ist_fm @ Steps2 ) ) @ Sl2 )
                & ( ( X7 @ ( stl_Pr5027778045650968273ist_fm @ Steps2 ) )
                  | ( abstra6579460235205289861ist_fm @ ( abstra256862739949234434ist_fm @ Eff3 ) @ Rules @ ( stl_Pr5027778045650968273ist_fm @ Steps2 ) ) ) ) )
       => ( abstra6579460235205289861ist_fm @ ( abstra256862739949234434ist_fm @ Eff3 ) @ Rules @ X3 ) ) ) ).

% i.epath.coinduct
thf(fact_485_i_Oepath_Ocoinduct,axiom,
    ! [X7: stream5128679079952569136_fm_fm > $o,X3: stream5128679079952569136_fm_fm,Rules: stream_fm,Eff3: fm > produc1996495991257130529ist_fm > option6029102822600979917ist_fm] :
      ( ( X7 @ X3 )
     => ( ! [X: stream5128679079952569136_fm_fm] :
            ( ( X7 @ X )
           => ? [Steps2: stream5128679079952569136_fm_fm,Sl2: fset_P661503646757059847ist_fm] :
                ( ( X = Steps2 )
                & ( member_fm @ ( produc6738438136904575596_fm_fm @ ( shd_Pr7265661315679113588_fm_fm @ Steps2 ) ) @ ( sset_fm @ Rules ) )
                & ( fmembe3381613331217039976ist_fm @ ( produc1217498474891896878_fm_fm @ ( shd_Pr7265661315679113588_fm_fm @ ( stl_Pr2715999368752424048_fm_fm @ Steps2 ) ) ) @ Sl2 )
                & ( abstra5048403109753943183ist_fm @ Eff3 @ ( produc6738438136904575596_fm_fm @ ( shd_Pr7265661315679113588_fm_fm @ Steps2 ) ) @ ( produc1217498474891896878_fm_fm @ ( shd_Pr7265661315679113588_fm_fm @ Steps2 ) ) @ Sl2 )
                & ( ( X7 @ ( stl_Pr2715999368752424048_fm_fm @ Steps2 ) )
                  | ( abstra9155489748554225612ist_fm @ ( abstra5048403109753943183ist_fm @ Eff3 ) @ Rules @ ( stl_Pr2715999368752424048_fm_fm @ Steps2 ) ) ) ) )
       => ( abstra9155489748554225612ist_fm @ ( abstra5048403109753943183ist_fm @ Eff3 ) @ Rules @ X3 ) ) ) ).

% i.epath.coinduct
thf(fact_486_i_Oepath_Ocoinduct,axiom,
    ! [X7: stream727092118206550309m_rule > $o,X3: stream727092118206550309m_rule,Rules: stream_rule,Eff3: rule > produc1996495991257130529ist_fm > option6029102822600979917ist_fm] :
      ( ( X7 @ X3 )
     => ( ! [X: stream727092118206550309m_rule] :
            ( ( X7 @ X )
           => ? [Steps2: stream727092118206550309m_rule,Sl2: fset_P661503646757059847ist_fm] :
                ( ( X = Steps2 )
                & ( member_rule @ ( produc7165828336582415457m_rule @ ( shd_Pr7235097944458474089m_rule @ Steps2 ) ) @ ( sset_rule @ Rules ) )
                & ( fmembe3381613331217039976ist_fm @ ( produc6879501374131015971m_rule @ ( shd_Pr7235097944458474089m_rule @ ( stl_Pr950425576149878629m_rule @ Steps2 ) ) ) @ Sl2 )
                & ( abstra5459341389059251418ist_fm @ Eff3 @ ( produc7165828336582415457m_rule @ ( shd_Pr7235097944458474089m_rule @ Steps2 ) ) @ ( produc6879501374131015971m_rule @ ( shd_Pr7235097944458474089m_rule @ Steps2 ) ) @ Sl2 )
                & ( ( X7 @ ( stl_Pr950425576149878629m_rule @ Steps2 ) )
                  | ( abstra5415202973855822231ist_fm @ ( abstra5459341389059251418ist_fm @ Eff3 ) @ Rules @ ( stl_Pr950425576149878629m_rule @ Steps2 ) ) ) ) )
       => ( abstra5415202973855822231ist_fm @ ( abstra5459341389059251418ist_fm @ Eff3 ) @ Rules @ X3 ) ) ) ).

% i.epath.coinduct
thf(fact_487_i_Oepath_Ocoinduct,axiom,
    ! [X7: stream812778442145099016ule_fm > $o,X3: stream812778442145099016ule_fm,Rules: stream_fm,Eff3: fm > abstra4572470701423150929m_rule > option7479661550464326865m_rule] :
      ( ( X7 @ X3 )
     => ( ! [X: stream812778442145099016ule_fm] :
            ( ( X7 @ X )
           => ? [Steps2: stream812778442145099016ule_fm,Sl2: fset_A5396818216090053761m_rule] :
                ( ( X = Steps2 )
                & ( member_fm @ ( produc8096733172076065128ule_fm @ ( shd_Pr849711437766469422ule_fm @ Steps2 ) ) @ ( sset_fm @ Rules ) )
                & ( fmembe1635983029874454474m_rule @ ( produc8194852094735076902ule_fm @ ( shd_Pr849711437766469422ule_fm @ ( stl_Pr2064664776838053682ule_fm @ Steps2 ) ) ) @ Sl2 )
                & ( abstra7119683891065068131m_rule @ Eff3 @ ( produc8096733172076065128ule_fm @ ( shd_Pr849711437766469422ule_fm @ Steps2 ) ) @ ( produc8194852094735076902ule_fm @ ( shd_Pr849711437766469422ule_fm @ Steps2 ) ) @ Sl2 )
                & ( ( X7 @ ( stl_Pr2064664776838053682ule_fm @ Steps2 ) )
                  | ( abstra1364558218721404774m_rule @ ( abstra7119683891065068131m_rule @ Eff3 ) @ Rules @ ( stl_Pr2064664776838053682ule_fm @ Steps2 ) ) ) ) )
       => ( abstra1364558218721404774m_rule @ ( abstra7119683891065068131m_rule @ Eff3 ) @ Rules @ X3 ) ) ) ).

% i.epath.coinduct
thf(fact_488_i_Oepath_Ocoinduct,axiom,
    ! [X7: stream7755889244956737277e_rule > $o,X3: stream7755889244956737277e_rule,Rules: stream_rule,Eff3: rule > abstra4572470701423150929m_rule > option7479661550464326865m_rule] :
      ( ( X7 @ X3 )
     => ( ! [X: stream7755889244956737277e_rule] :
            ( ( X7 @ X )
           => ? [Steps2: stream7755889244956737277e_rule,Sl2: fset_A5396818216090053761m_rule] :
                ( ( X = Steps2 )
                & ( member_rule @ ( produc4652457655725677917e_rule @ ( shd_Pr1235285222789368611e_rule @ Steps2 ) ) @ ( sset_rule @ Rules ) )
                & ( fmembe1635983029874454474m_rule @ ( produc7822628117027650331e_rule @ ( shd_Pr1235285222789368611e_rule @ ( stl_Pr8695508334200742183e_rule @ Steps2 ) ) ) @ Sl2 )
                & ( abstra3964157444165176280m_rule @ Eff3 @ ( produc4652457655725677917e_rule @ ( shd_Pr1235285222789368611e_rule @ Steps2 ) ) @ ( produc7822628117027650331e_rule @ ( shd_Pr1235285222789368611e_rule @ Steps2 ) ) @ Sl2 )
                & ( ( X7 @ ( stl_Pr8695508334200742183e_rule @ Steps2 ) )
                  | ( abstra5823804033175701339m_rule @ ( abstra3964157444165176280m_rule @ Eff3 ) @ Rules @ ( stl_Pr8695508334200742183e_rule @ Steps2 ) ) ) ) )
       => ( abstra5823804033175701339m_rule @ ( abstra3964157444165176280m_rule @ Eff3 ) @ Rules @ X3 ) ) ) ).

% i.epath.coinduct
thf(fact_489_i_Oepath_Ocoinduct,axiom,
    ! [X7: stream3409308193418444653ist_fm > $o,X3: stream3409308193418444653ist_fm,Rules: stream8299795917829157543ist_fm,Eff3: produc1996495991257130529ist_fm > produc1996495991257130529ist_fm > option6029102822600979917ist_fm] :
      ( ( X7 @ X3 )
     => ( ! [X: stream3409308193418444653ist_fm] :
            ( ( X7 @ X )
           => ? [Steps2: stream3409308193418444653ist_fm,Sl2: fset_P661503646757059847ist_fm] :
                ( ( X = Steps2 )
                & ( member8102475879199740618ist_fm @ ( produc1639249265332130309ist_fm @ ( shd_Pr4587271278095355155ist_fm @ Steps2 ) ) @ ( sset_P3431374123360452149ist_fm @ Rules ) )
                & ( fmembe3381613331217039976ist_fm @ ( produc5894184038225292483ist_fm @ ( shd_Pr4587271278095355155ist_fm @ ( stl_Pr8700568296466786583ist_fm @ Steps2 ) ) ) @ Sl2 )
                & ( abstra4222672346834617088ist_fm @ Eff3 @ ( produc1639249265332130309ist_fm @ ( shd_Pr4587271278095355155ist_fm @ Steps2 ) ) @ ( produc5894184038225292483ist_fm @ ( shd_Pr4587271278095355155ist_fm @ Steps2 ) ) @ Sl2 )
                & ( ( X7 @ ( stl_Pr8700568296466786583ist_fm @ Steps2 ) )
                  | ( abstra5061764961602482947ist_fm @ ( abstra4222672346834617088ist_fm @ Eff3 ) @ Rules @ ( stl_Pr8700568296466786583ist_fm @ Steps2 ) ) ) ) )
       => ( abstra5061764961602482947ist_fm @ ( abstra4222672346834617088ist_fm @ Eff3 ) @ Rules @ X3 ) ) ) ).

% i.epath.coinduct
thf(fact_490_i_Oepath_Ocoinduct,axiom,
    ! [X7: stream1622999302952468147at_nat > $o,X3: stream1622999302952468147at_nat,Rules: stream8372878641218411373at_nat,Eff3: produc859450856879609959at_nat > produc1996495991257130529ist_fm > option6029102822600979917ist_fm] :
      ( ( X7 @ X3 )
     => ( ! [X: stream1622999302952468147at_nat] :
            ( ( X7 @ X )
           => ? [Steps2: stream1622999302952468147at_nat,Sl2: fset_P661503646757059847ist_fm] :
                ( ( X = Steps2 )
                & ( member8206827879206165904at_nat @ ( produc2750861588394444363at_nat @ ( shd_Pr5062688428016131033at_nat @ Steps2 ) ) @ ( sset_P9164763103296375419at_nat @ Rules ) )
                & ( fmembe3381613331217039976ist_fm @ ( produc2934448030062000137at_nat @ ( shd_Pr5062688428016131033at_nat @ ( stl_Pr1564049449699658205at_nat @ Steps2 ) ) ) @ Sl2 )
                & ( abstra6247935586015979962ist_fm @ Eff3 @ ( produc2750861588394444363at_nat @ ( shd_Pr5062688428016131033at_nat @ Steps2 ) ) @ ( produc2934448030062000137at_nat @ ( shd_Pr5062688428016131033at_nat @ Steps2 ) ) @ Sl2 )
                & ( ( X7 @ ( stl_Pr1564049449699658205at_nat @ Steps2 ) )
                  | ( abstra1311857326046938941ist_fm @ ( abstra6247935586015979962ist_fm @ Eff3 ) @ Rules @ ( stl_Pr1564049449699658205at_nat @ Steps2 ) ) ) ) )
       => ( abstra1311857326046938941ist_fm @ ( abstra6247935586015979962ist_fm @ Eff3 ) @ Rules @ X3 ) ) ) ).

% i.epath.coinduct
thf(fact_491_i_Oepath_Ocoinduct,axiom,
    ! [X7: stream642806101564963573m_rule > $o,X3: stream642806101564963573m_rule,Rules: stream727092118206550309m_rule,Eff3: produc164195504107695125m_rule > produc1996495991257130529ist_fm > option6029102822600979917ist_fm] :
      ( ( X7 @ X3 )
     => ( ! [X: stream642806101564963573m_rule] :
            ( ( X7 @ X )
           => ? [Steps2: stream642806101564963573m_rule,Sl2: fset_P661503646757059847ist_fm] :
                ( ( X = Steps2 )
                & ( member4220325220686508332m_rule @ ( produc2016132375285593905m_rule @ ( shd_Pr1407159933805374393m_rule @ Steps2 ) ) @ ( sset_P4979403222752902913m_rule @ Rules ) )
                & ( fmembe3381613331217039976ist_fm @ ( produc8602611229794383091m_rule @ ( shd_Pr1407159933805374393m_rule @ ( stl_Pr8578586445745536693m_rule @ Steps2 ) ) ) @ Sl2 )
                & ( abstra2934389420647555978ist_fm @ Eff3 @ ( produc2016132375285593905m_rule @ ( shd_Pr1407159933805374393m_rule @ Steps2 ) ) @ ( produc8602611229794383091m_rule @ ( shd_Pr1407159933805374393m_rule @ Steps2 ) ) @ Sl2 )
                & ( ( X7 @ ( stl_Pr8578586445745536693m_rule @ Steps2 ) )
                  | ( abstra6058711572139910855ist_fm @ ( abstra2934389420647555978ist_fm @ Eff3 ) @ Rules @ ( stl_Pr8578586445745536693m_rule @ Steps2 ) ) ) ) )
       => ( abstra6058711572139910855ist_fm @ ( abstra2934389420647555978ist_fm @ Eff3 ) @ Rules @ X3 ) ) ) ).

% i.epath.coinduct
thf(fact_492_i_Oepath_Ocoinduct,axiom,
    ! [X7: stream7989562319868115861ist_fm > $o,X3: stream7989562319868115861ist_fm,Rules: stream8299795917829157543ist_fm,Eff3: produc1996495991257130529ist_fm > abstra4572470701423150929m_rule > option7479661550464326865m_rule] :
      ( ( X7 @ X3 )
     => ( ! [X: stream7989562319868115861ist_fm] :
            ( ( X7 @ X )
           => ? [Steps2: stream7989562319868115861ist_fm,Sl2: fset_A5396818216090053761m_rule] :
                ( ( X = Steps2 )
                & ( member8102475879199740618ist_fm @ ( produc3674566753905654281ist_fm @ ( shd_Pr563299216415026649ist_fm @ Steps2 ) ) @ ( sset_P3431374123360452149ist_fm @ Rules ) )
                & ( fmembe1635983029874454474m_rule @ ( produc2646333476271629515ist_fm @ ( shd_Pr563299216415026649ist_fm @ ( stl_Pr6727363551237990101ist_fm @ Steps2 ) ) ) @ Sl2 )
                & ( abstra1645840942258654514m_rule @ Eff3 @ ( produc3674566753905654281ist_fm @ ( shd_Pr563299216415026649ist_fm @ Steps2 ) ) @ ( produc2646333476271629515ist_fm @ ( shd_Pr563299216415026649ist_fm @ Steps2 ) ) @ Sl2 )
                & ( ( X7 @ ( stl_Pr6727363551237990101ist_fm @ Steps2 ) )
                  | ( abstra8066765647771630831m_rule @ ( abstra1645840942258654514m_rule @ Eff3 ) @ Rules @ ( stl_Pr6727363551237990101ist_fm @ Steps2 ) ) ) ) )
       => ( abstra8066765647771630831m_rule @ ( abstra1645840942258654514m_rule @ Eff3 ) @ Rules @ X3 ) ) ) ).

% i.epath.coinduct
thf(fact_493_i_Oepath_Ocoinduct,axiom,
    ! [X7: stream1819640557982756571at_nat > $o,X3: stream1819640557982756571at_nat,Rules: stream8372878641218411373at_nat,Eff3: produc859450856879609959at_nat > abstra4572470701423150929m_rule > option7479661550464326865m_rule] :
      ( ( X7 @ X3 )
     => ( ! [X: stream1819640557982756571at_nat] :
            ( ( X7 @ X )
           => ? [Steps2: stream1819640557982756571at_nat,Sl2: fset_A5396818216090053761m_rule] :
                ( ( X = Steps2 )
                & ( member8206827879206165904at_nat @ ( produc601515061966782543at_nat @ ( shd_Pr618815079712798623at_nat @ Steps2 ) ) @ ( sset_P9164763103296375419at_nat @ Rules ) )
                & ( fmembe1635983029874454474m_rule @ ( produc567330280010208273at_nat @ ( shd_Pr618815079712798623at_nat @ ( stl_Pr2369401607900929691at_nat @ Steps2 ) ) ) @ Sl2 )
                & ( abstra3953117623887779576m_rule @ Eff3 @ ( produc601515061966782543at_nat @ ( shd_Pr618815079712798623at_nat @ Steps2 ) ) @ ( produc567330280010208273at_nat @ ( shd_Pr618815079712798623at_nat @ Steps2 ) ) @ Sl2 )
                & ( ( X7 @ ( stl_Pr2369401607900929691at_nat @ Steps2 ) )
                  | ( abstra3157339599784851765m_rule @ ( abstra3953117623887779576m_rule @ Eff3 ) @ Rules @ ( stl_Pr2369401607900929691at_nat @ Steps2 ) ) ) ) )
       => ( abstra3157339599784851765m_rule @ ( abstra3953117623887779576m_rule @ Eff3 ) @ Rules @ X3 ) ) ) ).

% i.epath.coinduct
thf(fact_494_eff__S,axiom,
    ! [S3: produc1996495991257130529ist_fm,R2: rule,Sl3: fset_P661503646757059847ist_fm,S5: produc1996495991257130529ist_fm] :
      ( ( member8102475879199740618ist_fm @ S3 @ top_to728902761071950545ist_fm )
     => ( ( member_rule @ R2 @ ( sset_rule @ rules ) )
       => ( ( ( eff @ R2 @ S3 )
            = ( some_f2162114139285431020ist_fm @ Sl3 ) )
         => ( ( fmembe3381613331217039976ist_fm @ S5 @ Sl3 )
           => ( member8102475879199740618ist_fm @ S5 @ top_to728902761071950545ist_fm ) ) ) ) ) ).

% eff_S
thf(fact_495_minWait__least,axiom,
    ! [N: nat,Rs: stream_rule,S3: produc1996495991257130529ist_fm] :
      ( ( abstra3055540779341473792ist_fm
        @ ^ [R: rule,S: produc1996495991257130529ist_fm,Ss: fset_P661503646757059847ist_fm] :
            ( ( eff @ R @ S )
            = ( some_f2162114139285431020ist_fm @ Ss ) )
        @ ( shd_rule @ ( sdrop_rule @ N @ Rs ) )
        @ S3 )
     => ( ord_less_eq_nat
        @ ( abstra738319790293016300ist_fm
          @ ^ [R: rule,S: produc1996495991257130529ist_fm,Ss: fset_P661503646757059847ist_fm] :
              ( ( eff @ R @ S )
              = ( some_f2162114139285431020ist_fm @ Ss ) )
          @ Rs
          @ S3 )
        @ N ) ) ).

% minWait_least
thf(fact_496_UNIV__I,axiom,
    ! [X3: fm] : ( member_fm @ X3 @ top_top_set_fm ) ).

% UNIV_I
thf(fact_497_UNIV__I,axiom,
    ! [X3: produc859450856879609959at_nat] : ( member8206827879206165904at_nat @ X3 @ top_to8454300755403694231at_nat ) ).

% UNIV_I
thf(fact_498_UNIV__I,axiom,
    ! [X3: produc1996495991257130529ist_fm] : ( member8102475879199740618ist_fm @ X3 @ top_to728902761071950545ist_fm ) ).

% UNIV_I
thf(fact_499_UNIV__I,axiom,
    ! [X3: rule] : ( member_rule @ X3 @ top_top_set_rule ) ).

% UNIV_I
thf(fact_500_UNIV__eq__I,axiom,
    ! [A5: set_fm] :
      ( ! [X: fm] : ( member_fm @ X @ A5 )
     => ( top_top_set_fm = A5 ) ) ).

% UNIV_eq_I
thf(fact_501_UNIV__eq__I,axiom,
    ! [A5: set_Pr8693737435421807431at_nat] :
      ( ! [X: produc859450856879609959at_nat] : ( member8206827879206165904at_nat @ X @ A5 )
     => ( top_to8454300755403694231at_nat = A5 ) ) ).

% UNIV_eq_I
thf(fact_502_UNIV__eq__I,axiom,
    ! [A5: set_Pr7058068377845519745ist_fm] :
      ( ! [X: produc1996495991257130529ist_fm] : ( member8102475879199740618ist_fm @ X @ A5 )
     => ( top_to728902761071950545ist_fm = A5 ) ) ).

% UNIV_eq_I
thf(fact_503_UNIV__eq__I,axiom,
    ! [A5: set_rule] :
      ( ! [X: rule] : ( member_rule @ X @ A5 )
     => ( top_top_set_rule = A5 ) ) ).

% UNIV_eq_I
thf(fact_504_UNIV__witness,axiom,
    ? [X: fm] : ( member_fm @ X @ top_top_set_fm ) ).

% UNIV_witness
thf(fact_505_UNIV__witness,axiom,
    ? [X: produc859450856879609959at_nat] : ( member8206827879206165904at_nat @ X @ top_to8454300755403694231at_nat ) ).

% UNIV_witness
thf(fact_506_UNIV__witness,axiom,
    ? [X: produc1996495991257130529ist_fm] : ( member8102475879199740618ist_fm @ X @ top_to728902761071950545ist_fm ) ).

% UNIV_witness
thf(fact_507_UNIV__witness,axiom,
    ? [X: rule] : ( member_rule @ X @ top_top_set_rule ) ).

% UNIV_witness
thf(fact_508_RuleSystem_OminWait_Ocong,axiom,
    abstra738319790293016300ist_fm = abstra738319790293016300ist_fm ).

% RuleSystem.minWait.cong
thf(fact_509_RuleSystem__Defs_Osaturated_Ocong,axiom,
    abstra7252176601000738572ist_fm = abstra7252176601000738572ist_fm ).

% RuleSystem_Defs.saturated.cong
thf(fact_510_UNIV__def,axiom,
    ( top_to728902761071950545ist_fm
    = ( collec7231575776666803724ist_fm
      @ ^ [X6: produc1996495991257130529ist_fm] : $true ) ) ).

% UNIV_def
thf(fact_511_UNIV__def,axiom,
    ( top_top_set_rule
    = ( collect_rule
      @ ^ [X6: rule] : $true ) ) ).

% UNIV_def
thf(fact_512_top__greatest,axiom,
    ! [A: set_Pr7058068377845519745ist_fm] : ( ord_le2055072402069232929ist_fm @ A @ top_to728902761071950545ist_fm ) ).

% top_greatest
thf(fact_513_top__greatest,axiom,
    ! [A: set_rule] : ( ord_less_eq_set_rule @ A @ top_top_set_rule ) ).

% top_greatest
thf(fact_514_top__greatest,axiom,
    ! [A: set_fm] : ( ord_less_eq_set_fm @ A @ top_top_set_fm ) ).

% top_greatest
thf(fact_515_top_Oextremum__unique,axiom,
    ! [A: set_Pr7058068377845519745ist_fm] :
      ( ( ord_le2055072402069232929ist_fm @ top_to728902761071950545ist_fm @ A )
      = ( A = top_to728902761071950545ist_fm ) ) ).

% top.extremum_unique
thf(fact_516_top_Oextremum__unique,axiom,
    ! [A: set_rule] :
      ( ( ord_less_eq_set_rule @ top_top_set_rule @ A )
      = ( A = top_top_set_rule ) ) ).

% top.extremum_unique
thf(fact_517_top_Oextremum__unique,axiom,
    ! [A: set_fm] :
      ( ( ord_less_eq_set_fm @ top_top_set_fm @ A )
      = ( A = top_top_set_fm ) ) ).

% top.extremum_unique
thf(fact_518_top_Oextremum__uniqueI,axiom,
    ! [A: set_Pr7058068377845519745ist_fm] :
      ( ( ord_le2055072402069232929ist_fm @ top_to728902761071950545ist_fm @ A )
     => ( A = top_to728902761071950545ist_fm ) ) ).

% top.extremum_uniqueI
thf(fact_519_top_Oextremum__uniqueI,axiom,
    ! [A: set_rule] :
      ( ( ord_less_eq_set_rule @ top_top_set_rule @ A )
     => ( A = top_top_set_rule ) ) ).

% top.extremum_uniqueI
thf(fact_520_top_Oextremum__uniqueI,axiom,
    ! [A: set_fm] :
      ( ( ord_less_eq_set_fm @ top_top_set_fm @ A )
     => ( A = top_top_set_fm ) ) ).

% top.extremum_uniqueI
thf(fact_521_subset__UNIV,axiom,
    ! [A5: set_Pr7058068377845519745ist_fm] : ( ord_le2055072402069232929ist_fm @ A5 @ top_to728902761071950545ist_fm ) ).

% subset_UNIV
thf(fact_522_subset__UNIV,axiom,
    ! [A5: set_rule] : ( ord_less_eq_set_rule @ A5 @ top_top_set_rule ) ).

% subset_UNIV
thf(fact_523_subset__UNIV,axiom,
    ! [A5: set_fm] : ( ord_less_eq_set_fm @ A5 @ top_top_set_fm ) ).

% subset_UNIV
thf(fact_524_i_Oenabled__def,axiom,
    ! [Eff3: rule > produc1996495991257130529ist_fm > option6029102822600979917ist_fm,R2: rule,S3: produc1996495991257130529ist_fm] :
      ( ( abstra3055540779341473792ist_fm @ ( abstra5459341389059251418ist_fm @ Eff3 ) @ R2 @ S3 )
      = ( ? [X5: fset_P661503646757059847ist_fm] : ( abstra5459341389059251418ist_fm @ Eff3 @ R2 @ S3 @ X5 ) ) ) ).

% i.enabled_def
thf(fact_525_i_OSaturated__def,axiom,
    ! [Eff3: rule > produc1996495991257130529ist_fm > option6029102822600979917ist_fm,Rules: stream_rule,Steps: stream727092118206550309m_rule] :
      ( ( abstra8647779336736900908ist_fm @ ( abstra5459341389059251418ist_fm @ Eff3 ) @ Rules @ Steps )
      = ( ! [X6: rule] :
            ( ( member_rule @ X6 @ ( sset_rule @ Rules ) )
           => ( abstra7252176601000738572ist_fm @ ( abstra5459341389059251418ist_fm @ Eff3 ) @ X6 @ Steps ) ) ) ) ).

% i.Saturated_def
thf(fact_526_i_OpickEff,axiom,
    ! [Eff3: rule > produc1996495991257130529ist_fm > option6029102822600979917ist_fm,R2: rule,S3: produc1996495991257130529ist_fm] :
      ( ( abstra3055540779341473792ist_fm @ ( abstra5459341389059251418ist_fm @ Eff3 ) @ R2 @ S3 )
     => ( abstra5459341389059251418ist_fm @ Eff3 @ R2 @ S3 @ ( abstra458490714785879965ist_fm @ ( abstra5459341389059251418ist_fm @ Eff3 ) @ R2 @ S3 ) ) ) ).

% i.pickEff
thf(fact_527_eqfelem__imp__iff,axiom,
    ! [X3: produc1996495991257130529ist_fm,Y3: produc1996495991257130529ist_fm,A5: fset_P661503646757059847ist_fm] :
      ( ( X3 = Y3 )
     => ( ( fmembe3381613331217039976ist_fm @ X3 @ A5 )
        = ( fmembe3381613331217039976ist_fm @ Y3 @ A5 ) ) ) ).

% eqfelem_imp_iff
thf(fact_528_eqfelem__imp__iff,axiom,
    ! [X3: abstra4572470701423150929m_rule,Y3: abstra4572470701423150929m_rule,A5: fset_A5396818216090053761m_rule] :
      ( ( X3 = Y3 )
     => ( ( fmembe1635983029874454474m_rule @ X3 @ A5 )
        = ( fmembe1635983029874454474m_rule @ Y3 @ A5 ) ) ) ).

% eqfelem_imp_iff
thf(fact_529_if__split__fmem2,axiom,
    ! [A: produc1996495991257130529ist_fm,Q2: $o,X3: fset_P661503646757059847ist_fm,Y3: fset_P661503646757059847ist_fm] :
      ( ( fmembe3381613331217039976ist_fm @ A @ ( if_fse3714977293194272717ist_fm @ Q2 @ X3 @ Y3 ) )
      = ( ( Q2
         => ( fmembe3381613331217039976ist_fm @ A @ X3 ) )
        & ( ~ Q2
         => ( fmembe3381613331217039976ist_fm @ A @ Y3 ) ) ) ) ).

% if_split_fmem2
thf(fact_530_if__split__fmem2,axiom,
    ! [A: abstra4572470701423150929m_rule,Q2: $o,X3: fset_A5396818216090053761m_rule,Y3: fset_A5396818216090053761m_rule] :
      ( ( fmembe1635983029874454474m_rule @ A @ ( if_fse3370020817956878651m_rule @ Q2 @ X3 @ Y3 ) )
      = ( ( Q2
         => ( fmembe1635983029874454474m_rule @ A @ X3 ) )
        & ( ~ Q2
         => ( fmembe1635983029874454474m_rule @ A @ Y3 ) ) ) ) ).

% if_split_fmem2
thf(fact_531_if__split__fmem1,axiom,
    ! [Q2: $o,X3: produc1996495991257130529ist_fm,Y3: produc1996495991257130529ist_fm,B: fset_P661503646757059847ist_fm] :
      ( ( fmembe3381613331217039976ist_fm @ ( if_Pro3930376587665744871ist_fm @ Q2 @ X3 @ Y3 ) @ B )
      = ( ( Q2
         => ( fmembe3381613331217039976ist_fm @ X3 @ B ) )
        & ( ~ Q2
         => ( fmembe3381613331217039976ist_fm @ Y3 @ B ) ) ) ) ).

% if_split_fmem1
thf(fact_532_if__split__fmem1,axiom,
    ! [Q2: $o,X3: abstra4572470701423150929m_rule,Y3: abstra4572470701423150929m_rule,B: fset_A5396818216090053761m_rule] :
      ( ( fmembe1635983029874454474m_rule @ ( if_Abs4566652138351786251m_rule @ Q2 @ X3 @ Y3 ) @ B )
      = ( ( Q2
         => ( fmembe1635983029874454474m_rule @ X3 @ B ) )
        & ( ~ Q2
         => ( fmembe1635983029874454474m_rule @ Y3 @ B ) ) ) ) ).

% if_split_fmem1
thf(fact_533_eqfset__imp__iff,axiom,
    ! [A5: fset_P661503646757059847ist_fm,B5: fset_P661503646757059847ist_fm,X3: produc1996495991257130529ist_fm] :
      ( ( A5 = B5 )
     => ( ( fmembe3381613331217039976ist_fm @ X3 @ A5 )
        = ( fmembe3381613331217039976ist_fm @ X3 @ B5 ) ) ) ).

% eqfset_imp_iff
thf(fact_534_eqfset__imp__iff,axiom,
    ! [A5: fset_A5396818216090053761m_rule,B5: fset_A5396818216090053761m_rule,X3: abstra4572470701423150929m_rule] :
      ( ( A5 = B5 )
     => ( ( fmembe1635983029874454474m_rule @ X3 @ A5 )
        = ( fmembe1635983029874454474m_rule @ X3 @ B5 ) ) ) ).

% eqfset_imp_iff
thf(fact_535_eq__fmem__trans,axiom,
    ! [A: produc1996495991257130529ist_fm,B: produc1996495991257130529ist_fm,A5: fset_P661503646757059847ist_fm] :
      ( ( A = B )
     => ( ( fmembe3381613331217039976ist_fm @ B @ A5 )
       => ( fmembe3381613331217039976ist_fm @ A @ A5 ) ) ) ).

% eq_fmem_trans
thf(fact_536_eq__fmem__trans,axiom,
    ! [A: abstra4572470701423150929m_rule,B: abstra4572470701423150929m_rule,A5: fset_A5396818216090053761m_rule] :
      ( ( A = B )
     => ( ( fmembe1635983029874454474m_rule @ B @ A5 )
       => ( fmembe1635983029874454474m_rule @ A @ A5 ) ) ) ).

% eq_fmem_trans
thf(fact_537_fequalityCE,axiom,
    ! [A5: fset_P661503646757059847ist_fm,B5: fset_P661503646757059847ist_fm,C2: produc1996495991257130529ist_fm] :
      ( ( A5 = B5 )
     => ( ( ( fmembe3381613331217039976ist_fm @ C2 @ A5 )
         => ~ ( fmembe3381613331217039976ist_fm @ C2 @ B5 ) )
       => ~ ( ~ ( fmembe3381613331217039976ist_fm @ C2 @ A5 )
           => ( fmembe3381613331217039976ist_fm @ C2 @ B5 ) ) ) ) ).

% fequalityCE
thf(fact_538_fequalityCE,axiom,
    ! [A5: fset_A5396818216090053761m_rule,B5: fset_A5396818216090053761m_rule,C2: abstra4572470701423150929m_rule] :
      ( ( A5 = B5 )
     => ( ( ( fmembe1635983029874454474m_rule @ C2 @ A5 )
         => ~ ( fmembe1635983029874454474m_rule @ C2 @ B5 ) )
       => ~ ( ~ ( fmembe1635983029874454474m_rule @ C2 @ A5 )
           => ( fmembe1635983029874454474m_rule @ C2 @ B5 ) ) ) ) ).

% fequalityCE
thf(fact_539_fset__eqI,axiom,
    ! [A5: fset_P661503646757059847ist_fm,B5: fset_P661503646757059847ist_fm] :
      ( ! [X: produc1996495991257130529ist_fm] :
          ( ( fmembe3381613331217039976ist_fm @ X @ A5 )
          = ( fmembe3381613331217039976ist_fm @ X @ B5 ) )
     => ( A5 = B5 ) ) ).

% fset_eqI
thf(fact_540_fset__eqI,axiom,
    ! [A5: fset_A5396818216090053761m_rule,B5: fset_A5396818216090053761m_rule] :
      ( ! [X: abstra4572470701423150929m_rule] :
          ( ( fmembe1635983029874454474m_rule @ X @ A5 )
          = ( fmembe1635983029874454474m_rule @ X @ B5 ) )
     => ( A5 = B5 ) ) ).

% fset_eqI
thf(fact_541_fsubsetD,axiom,
    ! [A5: fset_P661503646757059847ist_fm,B5: fset_P661503646757059847ist_fm,C2: produc1996495991257130529ist_fm] :
      ( ( ord_le2064643713053750439ist_fm @ A5 @ B5 )
     => ( ( fmembe3381613331217039976ist_fm @ C2 @ A5 )
       => ( fmembe3381613331217039976ist_fm @ C2 @ B5 ) ) ) ).

% fsubsetD
thf(fact_542_fsubsetD,axiom,
    ! [A5: fset_A5396818216090053761m_rule,B5: fset_A5396818216090053761m_rule,C2: abstra4572470701423150929m_rule] :
      ( ( ord_le6951374404268211425m_rule @ A5 @ B5 )
     => ( ( fmembe1635983029874454474m_rule @ C2 @ A5 )
       => ( fmembe1635983029874454474m_rule @ C2 @ B5 ) ) ) ).

% fsubsetD
thf(fact_543_fin__mono,axiom,
    ! [A5: fset_P661503646757059847ist_fm,B5: fset_P661503646757059847ist_fm,X3: produc1996495991257130529ist_fm] :
      ( ( ord_le2064643713053750439ist_fm @ A5 @ B5 )
     => ( ( fmembe3381613331217039976ist_fm @ X3 @ A5 )
       => ( fmembe3381613331217039976ist_fm @ X3 @ B5 ) ) ) ).

% fin_mono
thf(fact_544_fin__mono,axiom,
    ! [A5: fset_A5396818216090053761m_rule,B5: fset_A5396818216090053761m_rule,X3: abstra4572470701423150929m_rule] :
      ( ( ord_le6951374404268211425m_rule @ A5 @ B5 )
     => ( ( fmembe1635983029874454474m_rule @ X3 @ A5 )
       => ( fmembe1635983029874454474m_rule @ X3 @ B5 ) ) ) ).

% fin_mono
thf(fact_545_RuleSystem__Defs_OSaturated__def,axiom,
    ( abstra8647779336736900908ist_fm
    = ( ^ [Eff: rule > produc1996495991257130529ist_fm > fset_P661503646757059847ist_fm > $o,Rules2: stream_rule,Steps3: stream727092118206550309m_rule] :
        ! [X6: rule] :
          ( ( member_rule @ X6 @ ( sset_rule @ Rules2 ) )
         => ( abstra7252176601000738572ist_fm @ Eff @ X6 @ Steps3 ) ) ) ) ).

% RuleSystem_Defs.Saturated_def
thf(fact_546_enabled__R,axiom,
    ! [S3: produc1996495991257130529ist_fm] :
      ( ( member8102475879199740618ist_fm @ S3 @ top_to728902761071950545ist_fm )
     => ? [X: rule] :
          ( ( member_rule @ X @ ( sset_rule @ rules ) )
          & ? [Sl4: fset_P661503646757059847ist_fm] :
              ( ( eff @ X @ S3 )
              = ( some_f2162114139285431020ist_fm @ Sl4 ) ) ) ) ).

% enabled_R
thf(fact_547_saturated__effG__uu__def,axiom,
    ( abstra7873564507807625473ist_fm
    = ( ^ [Eff4: rule > produc1996495991257130529ist_fm > option6029102822600979917ist_fm] : ( abstra7252176601000738572ist_fm @ ( abstra5459341389059251418ist_fm @ Eff4 ) ) ) ) ).

% saturated_effG_uu_def
thf(fact_548_Saturated__effG__uu__uu__def,axiom,
    ( abstra5023634976456092961ist_fm
    = ( ^ [Eff4: rule > produc1996495991257130529ist_fm > option6029102822600979917ist_fm] : ( abstra8647779336736900908ist_fm @ ( abstra5459341389059251418ist_fm @ Eff4 ) ) ) ) ).

% Saturated_effG_uu_uu_def
thf(fact_549_mkTree__effG__uu__def,axiom,
    ( abstra3917079402916513406ist_fm
    = ( ^ [Eff4: rule > produc1996495991257130529ist_fm > option6029102822600979917ist_fm] : ( abstra178918799166095863ist_fm @ ( abstra5459341389059251418ist_fm @ Eff4 ) ) ) ) ).

% mkTree_effG_uu_def
thf(fact_550_per,axiom,
    ! [R2: rule] :
      ( ( member_rule @ R2 @ ( sset_rule @ rules ) )
     => ( abstra4153889242452539894ist_fm
        @ ^ [R: rule,S: produc1996495991257130529ist_fm,Ss: fset_P661503646757059847ist_fm] :
            ( ( eff @ R @ S )
            = ( some_f2162114139285431020ist_fm @ Ss ) )
        @ rules
        @ top_to728902761071950545ist_fm
        @ R2 ) ) ).

% per
thf(fact_551_minWait__ex,axiom,
    ! [S3: produc1996495991257130529ist_fm,Rs: stream_rule] :
      ( ( member8102475879199740618ist_fm @ S3 @ top_to728902761071950545ist_fm )
     => ( ( abstra1114877437032970289r_rule @ rules @ Rs )
       => ? [N2: nat] :
            ( abstra3055540779341473792ist_fm
            @ ^ [R: rule,S: produc1996495991257130529ist_fm,Ss: fset_P661503646757059847ist_fm] :
                ( ( eff @ R @ S )
                = ( some_f2162114139285431020ist_fm @ Ss ) )
            @ ( shd_rule @ ( sdrop_rule @ N2 @ Rs ) )
            @ S3 ) ) ) ).

% minWait_ex
thf(fact_552_minWait__def,axiom,
    ! [Rs: stream_rule,S3: produc1996495991257130529ist_fm] :
      ( ( abstra738319790293016300ist_fm
        @ ^ [R: rule,S: produc1996495991257130529ist_fm,Ss: fset_P661503646757059847ist_fm] :
            ( ( eff @ R @ S )
            = ( some_f2162114139285431020ist_fm @ Ss ) )
        @ Rs
        @ S3 )
      = ( ord_Least_nat
        @ ^ [N3: nat] :
            ( abstra3055540779341473792ist_fm
            @ ^ [R: rule,S: produc1996495991257130529ist_fm,Ss: fset_P661503646757059847ist_fm] :
                ( ( eff @ R @ S )
                = ( some_f2162114139285431020ist_fm @ Ss ) )
            @ ( shd_rule @ ( sdrop_rule @ N3 @ Rs ) )
            @ S3 ) ) ) ).

% minWait_def
thf(fact_553_PersistentRuleSystem__axioms,axiom,
    ( abstra7887161495092839066ist_fm
    @ ^ [R: rule,S: produc1996495991257130529ist_fm,Ss: fset_P661503646757059847ist_fm] :
        ( ( eff @ R @ S )
        = ( some_f2162114139285431020ist_fm @ Ss ) )
    @ rules
    @ top_to728902761071950545ist_fm ) ).

% PersistentRuleSystem_axioms
thf(fact_554_per__rules,axiom,
    ! [R2: rule] :
      ( abstra4153889242452539894ist_fm
      @ ^ [R: rule,S: produc1996495991257130529ist_fm,Ss: fset_P661503646757059847ist_fm] :
          ( ( eff @ R @ S )
          = ( some_f2162114139285431020ist_fm @ Ss ) )
      @ rules
      @ top_to728902761071950545ist_fm
      @ R2 ) ).

% per_rules
thf(fact_555_RuleSystem__Defs_Ofair_Ocong,axiom,
    abstra1114877437032970289r_rule = abstra1114877437032970289r_rule ).

% RuleSystem_Defs.fair.cong
thf(fact_556_RuleSystem_Oper_Ocong,axiom,
    abstra4153889242452539894ist_fm = abstra4153889242452539894ist_fm ).

% RuleSystem.per.cong
thf(fact_557_LeastI2__ex,axiom,
    ! [P2: nat > $o,Q2: nat > $o] :
      ( ? [X_1: nat] : ( P2 @ X_1 )
     => ( ! [X: nat] :
            ( ( P2 @ X )
           => ( Q2 @ X ) )
       => ( Q2 @ ( ord_Least_nat @ P2 ) ) ) ) ).

% LeastI2_ex
thf(fact_558_LeastI__ex,axiom,
    ! [P2: nat > $o] :
      ( ? [X_1: nat] : ( P2 @ X_1 )
     => ( P2 @ ( ord_Least_nat @ P2 ) ) ) ).

% LeastI_ex
thf(fact_559_LeastI2,axiom,
    ! [P2: nat > $o,A: nat,Q2: nat > $o] :
      ( ( P2 @ A )
     => ( ! [X: nat] :
            ( ( P2 @ X )
           => ( Q2 @ X ) )
       => ( Q2 @ ( ord_Least_nat @ P2 ) ) ) ) ).

% LeastI2
thf(fact_560_LeastI,axiom,
    ! [P2: nat > $o,K2: nat] :
      ( ( P2 @ K2 )
     => ( P2 @ ( ord_Least_nat @ P2 ) ) ) ).

% LeastI
thf(fact_561_top__empty__eq,axiom,
    ( top_top_fm_o
    = ( ^ [X6: fm] : ( member_fm @ X6 @ top_top_set_fm ) ) ) ).

% top_empty_eq
thf(fact_562_top__empty__eq,axiom,
    ( top_to1313005431924226822_nat_o
    = ( ^ [X6: produc859450856879609959at_nat] : ( member8206827879206165904at_nat @ X6 @ top_to8454300755403694231at_nat ) ) ) ).

% top_empty_eq
thf(fact_563_top__empty__eq,axiom,
    ( top_to6798613745154499404t_fm_o
    = ( ^ [X6: produc1996495991257130529ist_fm] : ( member8102475879199740618ist_fm @ X6 @ top_to728902761071950545ist_fm ) ) ) ).

% top_empty_eq
thf(fact_564_top__empty__eq,axiom,
    ( top_top_rule_o
    = ( ^ [X6: rule] : ( member_rule @ X6 @ top_top_set_rule ) ) ) ).

% top_empty_eq
thf(fact_565_top__set__def,axiom,
    ( top_to728902761071950545ist_fm
    = ( collec7231575776666803724ist_fm @ top_to6798613745154499404t_fm_o ) ) ).

% top_set_def
thf(fact_566_top__set__def,axiom,
    ( top_top_set_rule
    = ( collect_rule @ top_top_rule_o ) ) ).

% top_set_def
thf(fact_567_PersistentRuleSystem_Oper,axiom,
    ! [Eff2: rule > produc1996495991257130529ist_fm > fset_P661503646757059847ist_fm > $o,Rules: stream_rule,S4: set_Pr7058068377845519745ist_fm,R2: rule] :
      ( ( abstra7887161495092839066ist_fm @ Eff2 @ Rules @ S4 )
     => ( ( member_rule @ R2 @ ( sset_rule @ Rules ) )
       => ( abstra4153889242452539894ist_fm @ Eff2 @ Rules @ S4 @ R2 ) ) ) ).

% PersistentRuleSystem.per
thf(fact_568_fair__stl,axiom,
    ! [Rs: stream_rule] :
      ( ( abstra1114877437032970289r_rule @ rules @ Rs )
     => ( abstra1114877437032970289r_rule @ rules @ ( stl_rule @ Rs ) ) ) ).

% fair_stl
thf(fact_569_Least1I,axiom,
    ! [P2: set_fm > $o] :
      ( ? [X8: set_fm] :
          ( ( P2 @ X8 )
          & ! [Y: set_fm] :
              ( ( P2 @ Y )
             => ( ord_less_eq_set_fm @ X8 @ Y ) )
          & ! [Y: set_fm] :
              ( ( ( P2 @ Y )
                & ! [Ya: set_fm] :
                    ( ( P2 @ Ya )
                   => ( ord_less_eq_set_fm @ Y @ Ya ) ) )
             => ( Y = X8 ) ) )
     => ( P2 @ ( ord_Least_set_fm @ P2 ) ) ) ).

% Least1I
thf(fact_570_Least1I,axiom,
    ! [P2: nat > $o] :
      ( ? [X8: nat] :
          ( ( P2 @ X8 )
          & ! [Y: nat] :
              ( ( P2 @ Y )
             => ( ord_less_eq_nat @ X8 @ Y ) )
          & ! [Y: nat] :
              ( ( ( P2 @ Y )
                & ! [Ya: nat] :
                    ( ( P2 @ Ya )
                   => ( ord_less_eq_nat @ Y @ Ya ) ) )
             => ( Y = X8 ) ) )
     => ( P2 @ ( ord_Least_nat @ P2 ) ) ) ).

% Least1I
thf(fact_571_Least1__le,axiom,
    ! [P2: set_fm > $o,Z2: set_fm] :
      ( ? [X8: set_fm] :
          ( ( P2 @ X8 )
          & ! [Y: set_fm] :
              ( ( P2 @ Y )
             => ( ord_less_eq_set_fm @ X8 @ Y ) )
          & ! [Y: set_fm] :
              ( ( ( P2 @ Y )
                & ! [Ya: set_fm] :
                    ( ( P2 @ Ya )
                   => ( ord_less_eq_set_fm @ Y @ Ya ) ) )
             => ( Y = X8 ) ) )
     => ( ( P2 @ Z2 )
       => ( ord_less_eq_set_fm @ ( ord_Least_set_fm @ P2 ) @ Z2 ) ) ) ).

% Least1_le
thf(fact_572_Least1__le,axiom,
    ! [P2: nat > $o,Z2: nat] :
      ( ? [X8: nat] :
          ( ( P2 @ X8 )
          & ! [Y: nat] :
              ( ( P2 @ Y )
             => ( ord_less_eq_nat @ X8 @ Y ) )
          & ! [Y: nat] :
              ( ( ( P2 @ Y )
                & ! [Ya: nat] :
                    ( ( P2 @ Ya )
                   => ( ord_less_eq_nat @ Y @ Ya ) ) )
             => ( Y = X8 ) ) )
     => ( ( P2 @ Z2 )
       => ( ord_less_eq_nat @ ( ord_Least_nat @ P2 ) @ Z2 ) ) ) ).

% Least1_le
thf(fact_573_LeastI2__order,axiom,
    ! [P2: set_fm > $o,X3: set_fm,Q2: set_fm > $o] :
      ( ( P2 @ X3 )
     => ( ! [Y: set_fm] :
            ( ( P2 @ Y )
           => ( ord_less_eq_set_fm @ X3 @ Y ) )
       => ( ! [X: set_fm] :
              ( ( P2 @ X )
             => ( ! [Y6: set_fm] :
                    ( ( P2 @ Y6 )
                   => ( ord_less_eq_set_fm @ X @ Y6 ) )
               => ( Q2 @ X ) ) )
         => ( Q2 @ ( ord_Least_set_fm @ P2 ) ) ) ) ) ).

% LeastI2_order
thf(fact_574_LeastI2__order,axiom,
    ! [P2: nat > $o,X3: nat,Q2: nat > $o] :
      ( ( P2 @ X3 )
     => ( ! [Y: nat] :
            ( ( P2 @ Y )
           => ( ord_less_eq_nat @ X3 @ Y ) )
       => ( ! [X: nat] :
              ( ( P2 @ X )
             => ( ! [Y6: nat] :
                    ( ( P2 @ Y6 )
                   => ( ord_less_eq_nat @ X @ Y6 ) )
               => ( Q2 @ X ) ) )
         => ( Q2 @ ( ord_Least_nat @ P2 ) ) ) ) ) ).

% LeastI2_order
thf(fact_575_Least__equality,axiom,
    ! [P2: set_fm > $o,X3: set_fm] :
      ( ( P2 @ X3 )
     => ( ! [Y: set_fm] :
            ( ( P2 @ Y )
           => ( ord_less_eq_set_fm @ X3 @ Y ) )
       => ( ( ord_Least_set_fm @ P2 )
          = X3 ) ) ) ).

% Least_equality
thf(fact_576_Least__equality,axiom,
    ! [P2: nat > $o,X3: nat] :
      ( ( P2 @ X3 )
     => ( ! [Y: nat] :
            ( ( P2 @ Y )
           => ( ord_less_eq_nat @ X3 @ Y ) )
       => ( ( ord_Least_nat @ P2 )
          = X3 ) ) ) ).

% Least_equality
thf(fact_577_LeastI2__wellorder,axiom,
    ! [P2: nat > $o,A: nat,Q2: nat > $o] :
      ( ( P2 @ A )
     => ( ! [A4: nat] :
            ( ( P2 @ A4 )
           => ( ! [B8: nat] :
                  ( ( P2 @ B8 )
                 => ( ord_less_eq_nat @ A4 @ B8 ) )
             => ( Q2 @ A4 ) ) )
       => ( Q2 @ ( ord_Least_nat @ P2 ) ) ) ) ).

% LeastI2_wellorder
thf(fact_578_LeastI2__wellorder__ex,axiom,
    ! [P2: nat > $o,Q2: nat > $o] :
      ( ? [X_1: nat] : ( P2 @ X_1 )
     => ( ! [A4: nat] :
            ( ( P2 @ A4 )
           => ( ! [B8: nat] :
                  ( ( P2 @ B8 )
                 => ( ord_less_eq_nat @ A4 @ B8 ) )
             => ( Q2 @ A4 ) ) )
       => ( Q2 @ ( ord_Least_nat @ P2 ) ) ) ) ).

% LeastI2_wellorder_ex
thf(fact_579_RuleSystem__Defs_Ofair__stl,axiom,
    ! [Rules: stream727092118206550309m_rule,Rs: stream727092118206550309m_rule] :
      ( ( abstra4997497885011821313m_rule @ Rules @ Rs )
     => ( abstra4997497885011821313m_rule @ Rules @ ( stl_Pr950425576149878629m_rule @ Rs ) ) ) ).

% RuleSystem_Defs.fair_stl
thf(fact_580_RuleSystem__Defs_Ofair__stl,axiom,
    ! [Rules: stream_rule,Rs: stream_rule] :
      ( ( abstra1114877437032970289r_rule @ Rules @ Rs )
     => ( abstra1114877437032970289r_rule @ Rules @ ( stl_rule @ Rs ) ) ) ).

% RuleSystem_Defs.fair_stl
thf(fact_581_i_Ofair__stl,axiom,
    ! [Rules: stream727092118206550309m_rule,Rs: stream727092118206550309m_rule] :
      ( ( abstra4997497885011821313m_rule @ Rules @ Rs )
     => ( abstra4997497885011821313m_rule @ Rules @ ( stl_Pr950425576149878629m_rule @ Rs ) ) ) ).

% i.fair_stl
thf(fact_582_i_Ofair__stl,axiom,
    ! [Rules: stream_rule,Rs: stream_rule] :
      ( ( abstra1114877437032970289r_rule @ Rules @ Rs )
     => ( abstra1114877437032970289r_rule @ Rules @ ( stl_rule @ Rs ) ) ) ).

% i.fair_stl
thf(fact_583_RuleSystem__Defs_Osdrop__fair,axiom,
    ! [Rules: stream727092118206550309m_rule,Rs: stream727092118206550309m_rule,M: nat] :
      ( ( abstra4997497885011821313m_rule @ Rules @ Rs )
     => ( abstra4997497885011821313m_rule @ Rules @ ( sdrop_7224736112439592940m_rule @ M @ Rs ) ) ) ).

% RuleSystem_Defs.sdrop_fair
thf(fact_584_RuleSystem__Defs_Osdrop__fair,axiom,
    ! [Rules: stream8299795917829157543ist_fm,Rs: stream8299795917829157543ist_fm,M: nat] :
      ( ( abstra7454370078233524789ist_fm @ Rules @ Rs )
     => ( abstra7454370078233524789ist_fm @ Rules @ ( sdrop_4442373711808556042ist_fm @ M @ Rs ) ) ) ).

% RuleSystem_Defs.sdrop_fair
thf(fact_585_RuleSystem__Defs_Osdrop__fair,axiom,
    ! [Rules: stream3588992565182678257e_rule,Rs: stream3588992565182678257e_rule,M: nat] :
      ( ( abstra4837122928373946751e_rule @ Rules @ Rs )
     => ( abstra4837122928373946751e_rule @ Rules @ ( sdrop_2199339672054592340e_rule @ M @ Rs ) ) ) ).

% RuleSystem_Defs.sdrop_fair
thf(fact_586_RuleSystem__Defs_Osdrop__fair,axiom,
    ! [Rules: stream1878305877988316353m_rule,Rs: stream1878305877988316353m_rule,M: nat] :
      ( ( abstra5219219999209536975m_rule @ Rules @ Rs )
     => ( abstra5219219999209536975m_rule @ Rules @ ( sdrop_1764812713384251940m_rule @ M @ Rs ) ) ) ).

% RuleSystem_Defs.sdrop_fair
thf(fact_587_RuleSystem__Defs_Osdrop__fair,axiom,
    ! [Rules: stream2312013117288958913e_rule,Rs: stream2312013117288958913e_rule,M: nat] :
      ( ( abstra623789749151397071e_rule @ Rules @ Rs )
     => ( abstra623789749151397071e_rule @ Rules @ ( sdrop_6392754500180887844e_rule @ M @ Rs ) ) ) ).

% RuleSystem_Defs.sdrop_fair
thf(fact_588_RuleSystem__Defs_Osdrop__fair,axiom,
    ! [Rules: stream6494289010434245521m_rule,Rs: stream6494289010434245521m_rule,M: nat] :
      ( ( abstra7961590535338393887m_rule @ Rules @ Rs )
     => ( abstra7961590535338393887m_rule @ Rules @ ( sdrop_7373388980841566196m_rule @ M @ Rs ) ) ) ).

% RuleSystem_Defs.sdrop_fair
thf(fact_589_RuleSystem__Defs_Osdrop__fair,axiom,
    ! [Rules: stream_rule,Rs: stream_rule,M: nat] :
      ( ( abstra1114877437032970289r_rule @ Rules @ Rs )
     => ( abstra1114877437032970289r_rule @ Rules @ ( sdrop_rule @ M @ Rs ) ) ) ).

% RuleSystem_Defs.sdrop_fair
thf(fact_590_i_Osdrop__fair,axiom,
    ! [Rules: stream727092118206550309m_rule,Rs: stream727092118206550309m_rule,M: nat] :
      ( ( abstra4997497885011821313m_rule @ Rules @ Rs )
     => ( abstra4997497885011821313m_rule @ Rules @ ( sdrop_7224736112439592940m_rule @ M @ Rs ) ) ) ).

% i.sdrop_fair
thf(fact_591_i_Osdrop__fair,axiom,
    ! [Rules: stream8299795917829157543ist_fm,Rs: stream8299795917829157543ist_fm,M: nat] :
      ( ( abstra7454370078233524789ist_fm @ Rules @ Rs )
     => ( abstra7454370078233524789ist_fm @ Rules @ ( sdrop_4442373711808556042ist_fm @ M @ Rs ) ) ) ).

% i.sdrop_fair
thf(fact_592_i_Osdrop__fair,axiom,
    ! [Rules: stream3588992565182678257e_rule,Rs: stream3588992565182678257e_rule,M: nat] :
      ( ( abstra4837122928373946751e_rule @ Rules @ Rs )
     => ( abstra4837122928373946751e_rule @ Rules @ ( sdrop_2199339672054592340e_rule @ M @ Rs ) ) ) ).

% i.sdrop_fair
thf(fact_593_i_Osdrop__fair,axiom,
    ! [Rules: stream1878305877988316353m_rule,Rs: stream1878305877988316353m_rule,M: nat] :
      ( ( abstra5219219999209536975m_rule @ Rules @ Rs )
     => ( abstra5219219999209536975m_rule @ Rules @ ( sdrop_1764812713384251940m_rule @ M @ Rs ) ) ) ).

% i.sdrop_fair
thf(fact_594_i_Osdrop__fair,axiom,
    ! [Rules: stream2312013117288958913e_rule,Rs: stream2312013117288958913e_rule,M: nat] :
      ( ( abstra623789749151397071e_rule @ Rules @ Rs )
     => ( abstra623789749151397071e_rule @ Rules @ ( sdrop_6392754500180887844e_rule @ M @ Rs ) ) ) ).

% i.sdrop_fair
thf(fact_595_i_Osdrop__fair,axiom,
    ! [Rules: stream6494289010434245521m_rule,Rs: stream6494289010434245521m_rule,M: nat] :
      ( ( abstra7961590535338393887m_rule @ Rules @ Rs )
     => ( abstra7961590535338393887m_rule @ Rules @ ( sdrop_7373388980841566196m_rule @ M @ Rs ) ) ) ).

% i.sdrop_fair
thf(fact_596_i_Osdrop__fair,axiom,
    ! [Rules: stream_rule,Rs: stream_rule,M: nat] :
      ( ( abstra1114877437032970289r_rule @ Rules @ Rs )
     => ( abstra1114877437032970289r_rule @ Rules @ ( sdrop_rule @ M @ Rs ) ) ) ).

% i.sdrop_fair
thf(fact_597_Least__le,axiom,
    ! [P2: nat > $o,K2: nat] :
      ( ( P2 @ K2 )
     => ( ord_less_eq_nat @ ( ord_Least_nat @ P2 ) @ K2 ) ) ).

% Least_le
thf(fact_598_top__empty__eq2,axiom,
    ( top_to1955928418919630078rule_o
    = ( ^ [X6: produc1996495991257130529ist_fm,Y5: rule] : ( member4220325220686508332m_rule @ ( produc491467635432902671m_rule @ X6 @ Y5 ) @ top_to1603951219661559419m_rule ) ) ) ).

% top_empty_eq2
thf(fact_599_top__empty__eq2,axiom,
    ( top_top_nat_nat_o
    = ( ^ [X6: nat,Y5: nat] : ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X6 @ Y5 ) @ top_to4669805908274784177at_nat ) ) ) ).

% top_empty_eq2
thf(fact_600_top__empty__eq2,axiom,
    ( top_to2659251575160290430_nat_o
    = ( ^ [X6: product_prod_nat_nat,Y5: product_prod_nat_nat] : ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ X6 @ Y5 ) @ top_to8454300755403694231at_nat ) ) ) ).

% top_empty_eq2
thf(fact_601_top__empty__eq2,axiom,
    ( top_to5144090402559349552t_fm_o
    = ( ^ [X6: rule,Y5: produc1996495991257130529ist_fm] : ( member8686995742288324264ist_fm @ ( produc7063755442767259331ist_fm @ X6 @ Y5 ) @ top_to2837172092223966711ist_fm ) ) ) ).

% top_empty_eq2
thf(fact_602_top__empty__eq2,axiom,
    ( top_to7326020031456794390t_fm_o
    = ( ^ [X6: list_fm,Y5: list_fm] : ( member8102475879199740618ist_fm @ ( produc381145313068854617ist_fm @ X6 @ Y5 ) @ top_to728902761071950545ist_fm ) ) ) ).

% top_empty_eq2
thf(fact_603_UNIV__rules,axiom,
    ( ( sset_rule @ rules )
    = top_top_set_rule ) ).

% UNIV_rules
thf(fact_604_sdrop__fair,axiom,
    ! [Rs: stream_rule,M: nat] :
      ( ( abstra1114877437032970289r_rule @ rules @ Rs )
     => ( abstra1114877437032970289r_rule @ rules @ ( sdrop_rule @ M @ Rs ) ) ) ).

% sdrop_fair
thf(fact_605_minWait__le__pos,axiom,
    ! [Rs: stream_rule,R2: rule,S3: produc1996495991257130529ist_fm] :
      ( ( abstra1114877437032970289r_rule @ rules @ Rs )
     => ( ( member_rule @ R2 @ ( sset_rule @ rules ) )
       => ( ( abstra3055540779341473792ist_fm
            @ ^ [R: rule,S: produc1996495991257130529ist_fm,Ss: fset_P661503646757059847ist_fm] :
                ( ( eff @ R @ S )
                = ( some_f2162114139285431020ist_fm @ Ss ) )
            @ R2
            @ S3 )
         => ( ord_less_eq_nat
            @ ( abstra738319790293016300ist_fm
              @ ^ [R: rule,S: produc1996495991257130529ist_fm,Ss: fset_P661503646757059847ist_fm] :
                  ( ( eff @ R @ S )
                  = ( some_f2162114139285431020ist_fm @ Ss ) )
              @ Rs
              @ S3 )
            @ ( abstract_pos_rule @ Rs @ R2 ) ) ) ) ) ).

% minWait_le_pos
thf(fact_606_ipath__mkTree__saturated,axiom,
    ! [S3: produc1996495991257130529ist_fm,Rs: stream_rule,Steps: stream727092118206550309m_rule,R2: rule] :
      ( ( member8102475879199740618ist_fm @ S3 @ top_to728902761071950545ist_fm )
     => ( ( abstra1114877437032970289r_rule @ rules @ Rs )
       => ( ( abstra2285467793620430502m_rule
            @ ( abstra178918799166095863ist_fm
              @ ^ [R: rule,S: produc1996495991257130529ist_fm,Ss: fset_P661503646757059847ist_fm] :
                  ( ( eff @ R @ S )
                  = ( some_f2162114139285431020ist_fm @ Ss ) )
              @ Rs
              @ S3 )
            @ Steps )
         => ( ( member_rule @ R2 @ ( sset_rule @ rules ) )
           => ( abstra7252176601000738572ist_fm
              @ ^ [R: rule,S: produc1996495991257130529ist_fm,Ss: fset_P661503646757059847ist_fm] :
                  ( ( eff @ R @ S )
                  = ( some_f2162114139285431020ist_fm @ Ss ) )
              @ R2
              @ Steps ) ) ) ) ) ).

% ipath_mkTree_saturated
thf(fact_607_ipath__mkTree__sdrop,axiom,
    ! [S3: produc1996495991257130529ist_fm,Rs: stream_rule,Steps: stream727092118206550309m_rule,M: nat] :
      ( ( member8102475879199740618ist_fm @ S3 @ top_to728902761071950545ist_fm )
     => ( ( abstra1114877437032970289r_rule @ rules @ Rs )
       => ( ( abstra2285467793620430502m_rule
            @ ( abstra178918799166095863ist_fm
              @ ^ [R: rule,S: produc1996495991257130529ist_fm,Ss: fset_P661503646757059847ist_fm] :
                  ( ( eff @ R @ S )
                  = ( some_f2162114139285431020ist_fm @ Ss ) )
              @ Rs
              @ S3 )
            @ Steps )
         => ? [N2: nat,S7: produc1996495991257130529ist_fm] :
              ( ( member8102475879199740618ist_fm @ S7 @ top_to728902761071950545ist_fm )
              & ( abstra2285467793620430502m_rule
                @ ( abstra178918799166095863ist_fm
                  @ ^ [R: rule,S: produc1996495991257130529ist_fm,Ss: fset_P661503646757059847ist_fm] :
                      ( ( eff @ R @ S )
                      = ( some_f2162114139285431020ist_fm @ Ss ) )
                  @ ( sdrop_rule @ N2 @ Rs )
                  @ S7 )
                @ ( sdrop_7224736112439592940m_rule @ M @ Steps ) ) ) ) ) ) ).

% ipath_mkTree_sdrop
thf(fact_608_ipath__mkTree__Saturated,axiom,
    ! [S3: produc1996495991257130529ist_fm,Rs: stream_rule,Steps: stream727092118206550309m_rule] :
      ( ( member8102475879199740618ist_fm @ S3 @ top_to728902761071950545ist_fm )
     => ( ( abstra1114877437032970289r_rule @ rules @ Rs )
       => ( ( abstra2285467793620430502m_rule
            @ ( abstra178918799166095863ist_fm
              @ ^ [R: rule,S: produc1996495991257130529ist_fm,Ss: fset_P661503646757059847ist_fm] :
                  ( ( eff @ R @ S )
                  = ( some_f2162114139285431020ist_fm @ Ss ) )
              @ Rs
              @ S3 )
            @ Steps )
         => ( abstra8647779336736900908ist_fm
            @ ^ [R: rule,S: produc1996495991257130529ist_fm,Ss: fset_P661503646757059847ist_fm] :
                ( ( eff @ R @ S )
                = ( some_f2162114139285431020ist_fm @ Ss ) )
            @ rules
            @ Steps ) ) ) ) ).

% ipath_mkTree_Saturated
thf(fact_609_pos,axiom,
    ! [Rs: stream_rule,R2: rule] :
      ( ( abstra1114877437032970289r_rule @ rules @ Rs )
     => ( ( member_rule @ R2 @ ( sset_rule @ rules ) )
       => ( ( shd_rule @ ( sdrop_rule @ ( abstract_pos_rule @ Rs @ R2 ) @ Rs ) )
          = R2 ) ) ) ).

% pos
thf(fact_610_trim__alt,axiom,
    ! [S3: produc1996495991257130529ist_fm,Rs: stream_rule] :
      ( ( member8102475879199740618ist_fm @ S3 @ top_to728902761071950545ist_fm )
     => ( ( abstra1114877437032970289r_rule @ rules @ Rs )
       => ( ( abstra2363873759530810705ist_fm
            @ ^ [R: rule,S: produc1996495991257130529ist_fm,Ss: fset_P661503646757059847ist_fm] :
                ( ( eff @ R @ S )
                = ( some_f2162114139285431020ist_fm @ Ss ) )
            @ Rs
            @ S3 )
          = ( sdrop_rule
            @ ( abstra738319790293016300ist_fm
              @ ^ [R: rule,S: produc1996495991257130529ist_fm,Ss: fset_P661503646757059847ist_fm] :
                  ( ( eff @ R @ S )
                  = ( some_f2162114139285431020ist_fm @ Ss ) )
              @ Rs
              @ S3 )
            @ Rs ) ) ) ) ).

% trim_alt
thf(fact_611_trim__enabled,axiom,
    ! [S3: produc1996495991257130529ist_fm,Rs: stream_rule] :
      ( ( member8102475879199740618ist_fm @ S3 @ top_to728902761071950545ist_fm )
     => ( ( abstra1114877437032970289r_rule @ rules @ Rs )
       => ( abstra3055540779341473792ist_fm
          @ ^ [R: rule,S: produc1996495991257130529ist_fm,Ss: fset_P661503646757059847ist_fm] :
              ( ( eff @ R @ S )
              = ( some_f2162114139285431020ist_fm @ Ss ) )
          @ ( shd_rule
            @ ( abstra2363873759530810705ist_fm
              @ ^ [R: rule,S: produc1996495991257130529ist_fm,Ss: fset_P661503646757059847ist_fm] :
                  ( ( eff @ R @ S )
                  = ( some_f2162114139285431020ist_fm @ Ss ) )
              @ Rs
              @ S3 ) )
          @ S3 ) ) ) ).

% trim_enabled
thf(fact_612_RuleSystem__Defs_Otrim_Ocong,axiom,
    abstra2363873759530810705ist_fm = abstra2363873759530810705ist_fm ).

% RuleSystem_Defs.trim.cong
thf(fact_613_pos__least,axiom,
    ! [N: nat,Rs: stream8299795917829157543ist_fm,R2: produc1996495991257130529ist_fm] :
      ( ( ( shd_Pr772355297128350925ist_fm @ ( sdrop_4442373711808556042ist_fm @ N @ Rs ) )
        = R2 )
     => ( ord_less_eq_nat @ ( abstra3513618393749879589ist_fm @ Rs @ R2 ) @ N ) ) ).

% pos_least
thf(fact_614_pos__least,axiom,
    ! [N: nat,Rs: stream3588992565182678257e_rule,R2: produc780299800278082923e_rule] :
      ( ( ( shd_Pr4573454438745646103e_rule @ ( sdrop_2199339672054592340e_rule @ N @ Rs ) )
        = R2 )
     => ( ord_less_eq_nat @ ( abstra883788664900704879e_rule @ Rs @ R2 ) @ N ) ) ).

% pos_least
thf(fact_615_pos__least,axiom,
    ! [N: nat,Rs: stream1878305877988316353m_rule,R2: produc1660549859252256699m_rule] :
      ( ( ( shd_Pr3564872132220007m_rule @ ( sdrop_1764812713384251940m_rule @ N @ Rs ) )
        = R2 )
     => ( ord_less_eq_nat @ ( abstra7431954940529686207m_rule @ Rs @ R2 ) @ N ) ) ).

% pos_least
thf(fact_616_pos__least,axiom,
    ! [N: nat,Rs: stream2312013117288958913e_rule,R2: produc6288491646048892603e_rule] :
      ( ( ( shd_Pr4631506658928855911e_rule @ ( sdrop_6392754500180887844e_rule @ N @ Rs ) )
        = R2 )
     => ( ord_less_eq_nat @ ( abstra2836524690471546303e_rule @ Rs @ R2 ) @ N ) ) ).

% pos_least
thf(fact_617_pos__least,axiom,
    ! [N: nat,Rs: stream6494289010434245521m_rule,R2: produc3793571666044074251m_rule] :
      ( ( ( shd_Pr2340755956080993207m_rule @ ( sdrop_7373388980841566196m_rule @ N @ Rs ) )
        = R2 )
     => ( ord_less_eq_nat @ ( abstra3193765552906301967m_rule @ Rs @ R2 ) @ N ) ) ).

% pos_least
thf(fact_618_pos__least,axiom,
    ! [N: nat,Rs: stream727092118206550309m_rule,R2: produc164195504107695125m_rule] :
      ( ( ( shd_Pr7235097944458474089m_rule @ ( sdrop_7224736112439592940m_rule @ N @ Rs ) )
        = R2 )
     => ( ord_less_eq_nat @ ( abstra1882453022147397137m_rule @ Rs @ R2 ) @ N ) ) ).

% pos_least
thf(fact_619_pos__least,axiom,
    ! [N: nat,Rs: stream_rule,R2: rule] :
      ( ( ( shd_rule @ ( sdrop_rule @ N @ Rs ) )
        = R2 )
     => ( ord_less_eq_nat @ ( abstract_pos_rule @ Rs @ R2 ) @ N ) ) ).

% pos_least
thf(fact_620_trim__fair,axiom,
    ! [S3: produc1996495991257130529ist_fm,Rs: stream_rule] :
      ( ( member8102475879199740618ist_fm @ S3 @ top_to728902761071950545ist_fm )
     => ( ( abstra1114877437032970289r_rule @ rules @ Rs )
       => ( abstra1114877437032970289r_rule @ rules
          @ ( abstra2363873759530810705ist_fm
            @ ^ [R: rule,S: produc1996495991257130529ist_fm,Ss: fset_P661503646757059847ist_fm] :
                ( ( eff @ R @ S )
                = ( some_f2162114139285431020ist_fm @ Ss ) )
            @ Rs
            @ S3 ) ) ) ) ).

% trim_fair
thf(fact_621_pos__def,axiom,
    ( abstra3513618393749879589ist_fm
    = ( ^ [Rs2: stream8299795917829157543ist_fm,R: produc1996495991257130529ist_fm] :
          ( ord_Least_nat
          @ ^ [N3: nat] :
              ( ( shd_Pr772355297128350925ist_fm @ ( sdrop_4442373711808556042ist_fm @ N3 @ Rs2 ) )
              = R ) ) ) ) ).

% pos_def
thf(fact_622_pos__def,axiom,
    ( abstra883788664900704879e_rule
    = ( ^ [Rs2: stream3588992565182678257e_rule,R: produc780299800278082923e_rule] :
          ( ord_Least_nat
          @ ^ [N3: nat] :
              ( ( shd_Pr4573454438745646103e_rule @ ( sdrop_2199339672054592340e_rule @ N3 @ Rs2 ) )
              = R ) ) ) ) ).

% pos_def
thf(fact_623_pos__def,axiom,
    ( abstra7431954940529686207m_rule
    = ( ^ [Rs2: stream1878305877988316353m_rule,R: produc1660549859252256699m_rule] :
          ( ord_Least_nat
          @ ^ [N3: nat] :
              ( ( shd_Pr3564872132220007m_rule @ ( sdrop_1764812713384251940m_rule @ N3 @ Rs2 ) )
              = R ) ) ) ) ).

% pos_def
thf(fact_624_pos__def,axiom,
    ( abstra2836524690471546303e_rule
    = ( ^ [Rs2: stream2312013117288958913e_rule,R: produc6288491646048892603e_rule] :
          ( ord_Least_nat
          @ ^ [N3: nat] :
              ( ( shd_Pr4631506658928855911e_rule @ ( sdrop_6392754500180887844e_rule @ N3 @ Rs2 ) )
              = R ) ) ) ) ).

% pos_def
thf(fact_625_pos__def,axiom,
    ( abstra3193765552906301967m_rule
    = ( ^ [Rs2: stream6494289010434245521m_rule,R: produc3793571666044074251m_rule] :
          ( ord_Least_nat
          @ ^ [N3: nat] :
              ( ( shd_Pr2340755956080993207m_rule @ ( sdrop_7373388980841566196m_rule @ N3 @ Rs2 ) )
              = R ) ) ) ) ).

% pos_def
thf(fact_626_pos__def,axiom,
    ( abstra1882453022147397137m_rule
    = ( ^ [Rs2: stream727092118206550309m_rule,R: produc164195504107695125m_rule] :
          ( ord_Least_nat
          @ ^ [N3: nat] :
              ( ( shd_Pr7235097944458474089m_rule @ ( sdrop_7224736112439592940m_rule @ N3 @ Rs2 ) )
              = R ) ) ) ) ).

% pos_def
thf(fact_627_pos__def,axiom,
    ( abstract_pos_rule
    = ( ^ [Rs2: stream_rule,R: rule] :
          ( ord_Least_nat
          @ ^ [N3: nat] :
              ( ( shd_rule @ ( sdrop_rule @ N3 @ Rs2 ) )
              = R ) ) ) ) ).

% pos_def
thf(fact_628_PersistentRuleSystem_Oipath__mkTree__Saturated,axiom,
    ! [Eff2: rule > fm > fset_fm > $o,Rules: stream_rule,S4: set_fm,S3: fm,Rs: stream_rule,Steps: stream8579757915198507644m_rule] :
      ( ( abstra9012132111595872407ule_fm @ Eff2 @ Rules @ S4 )
     => ( ( member_fm @ S3 @ S4 )
       => ( ( abstra1114877437032970289r_rule @ Rules @ Rs )
         => ( ( abstra7716921463347034149m_rule @ ( abstra3991228042706630138ule_fm @ Eff2 @ Rs @ S3 ) @ Steps )
           => ( abstra7168897066996117381ule_fm @ Eff2 @ Rules @ Steps ) ) ) ) ) ).

% PersistentRuleSystem.ipath_mkTree_Saturated
thf(fact_629_PersistentRuleSystem_Oipath__mkTree__Saturated,axiom,
    ! [Eff2: rule > rule > fset_rule > $o,Rules: stream_rule,S4: set_rule,S3: rule,Rs: stream_rule,Steps: stream3588992565182678257e_rule] :
      ( ( abstra8115150628584006924e_rule @ Eff2 @ Rules @ S4 )
     => ( ( member_rule @ S3 @ S4 )
       => ( ( abstra1114877437032970289r_rule @ Rules @ Rs )
         => ( ( abstra3969007714708337434e_rule @ ( abstra7443870075974203631e_rule @ Eff2 @ Rs @ S3 ) @ Steps )
           => ( abstra4228102177642093818e_rule @ Eff2 @ Rules @ Steps ) ) ) ) ) ).

% PersistentRuleSystem.ipath_mkTree_Saturated
thf(fact_630_PersistentRuleSystem_Oipath__mkTree__Saturated,axiom,
    ! [Eff2: rule > produc859450856879609959at_nat > fset_P3619101297660450253at_nat > $o,Rules: stream_rule,S4: set_Pr8693737435421807431at_nat,S3: produc859450856879609959at_nat,Rs: stream_rule,Steps: stream1370203160359889003t_rule] :
      ( ( abstra1526291686923892064at_nat @ Eff2 @ Rules @ S4 )
     => ( ( member8206827879206165904at_nat @ S3 @ S4 )
       => ( ( abstra1114877437032970289r_rule @ Rules @ Rs )
         => ( ( abstra4530403282039525100t_rule @ ( abstra193881638956283709at_nat @ Eff2 @ Rs @ S3 ) @ Steps )
           => ( abstra7052252741274667762at_nat @ Eff2 @ Rules @ Steps ) ) ) ) ) ).

% PersistentRuleSystem.ipath_mkTree_Saturated
thf(fact_631_PersistentRuleSystem_Oipath__mkTree__Saturated,axiom,
    ! [Eff2: rule > produc1996495991257130529ist_fm > fset_P661503646757059847ist_fm > $o,Rules: stream_rule,S4: set_Pr7058068377845519745ist_fm,S3: produc1996495991257130529ist_fm,Rs: stream_rule,Steps: stream727092118206550309m_rule] :
      ( ( abstra7887161495092839066ist_fm @ Eff2 @ Rules @ S4 )
     => ( ( member8102475879199740618ist_fm @ S3 @ S4 )
       => ( ( abstra1114877437032970289r_rule @ Rules @ Rs )
         => ( ( abstra2285467793620430502m_rule @ ( abstra178918799166095863ist_fm @ Eff2 @ Rs @ S3 ) @ Steps )
           => ( abstra8647779336736900908ist_fm @ Eff2 @ Rules @ Steps ) ) ) ) ) ).

% PersistentRuleSystem.ipath_mkTree_Saturated
thf(fact_632_PersistentRuleSystem_Oipath__mkTree__saturated,axiom,
    ! [Eff2: fm > fm > fset_fm > $o,Rules: stream_fm,S4: set_fm,S3: fm,Rs: stream_fm,Steps: stream1375025385734440967_fm_fm,R2: fm] :
      ( ( abstra1990789280842237858_fm_fm @ Eff2 @ Rules @ S4 )
     => ( ( member_fm @ S3 @ S4 )
       => ( ( abstra5119165595139683900air_fm @ Rules @ Rs )
         => ( ( abstra724614664783131952_fm_fm @ ( abstra5285091746773999237_fm_fm @ Eff2 @ Rs @ S3 ) @ Steps )
           => ( ( member_fm @ R2 @ ( sset_fm @ Rules ) )
             => ( abstra81079133870296496_fm_fm @ Eff2 @ R2 @ Steps ) ) ) ) ) ) ).

% PersistentRuleSystem.ipath_mkTree_saturated
thf(fact_633_PersistentRuleSystem_Oipath__mkTree__saturated,axiom,
    ! [Eff2: fm > rule > fset_rule > $o,Rules: stream_fm,S4: set_rule,S3: rule,Rs: stream_fm,Steps: stream1800043145262542076ule_fm,R2: fm] :
      ( ( abstra4039608574777915031m_rule @ Eff2 @ Rules @ S4 )
     => ( ( member_rule @ S3 @ S4 )
       => ( ( abstra5119165595139683900air_fm @ Rules @ Rs )
         => ( ( abstra8780395760585530021ule_fm @ ( abstra8242076542743448570m_rule @ Eff2 @ Rs @ S3 ) @ Steps )
           => ( ( member_fm @ R2 @ ( sset_fm @ Rules ) )
             => ( abstra5359274141611169189m_rule @ Eff2 @ R2 @ Steps ) ) ) ) ) ) ).

% PersistentRuleSystem.ipath_mkTree_saturated
thf(fact_634_PersistentRuleSystem_Oipath__mkTree__saturated,axiom,
    ! [Eff2: rule > fm > fset_fm > $o,Rules: stream_rule,S4: set_fm,S3: fm,Rs: stream_rule,Steps: stream8579757915198507644m_rule,R2: rule] :
      ( ( abstra9012132111595872407ule_fm @ Eff2 @ Rules @ S4 )
     => ( ( member_fm @ S3 @ S4 )
       => ( ( abstra1114877437032970289r_rule @ Rules @ Rs )
         => ( ( abstra7716921463347034149m_rule @ ( abstra3991228042706630138ule_fm @ Eff2 @ Rs @ S3 ) @ Steps )
           => ( ( member_rule @ R2 @ ( sset_rule @ Rules ) )
             => ( abstra1108425641574350757ule_fm @ Eff2 @ R2 @ Steps ) ) ) ) ) ) ).

% PersistentRuleSystem.ipath_mkTree_saturated
thf(fact_635_PersistentRuleSystem_Oipath__mkTree__saturated,axiom,
    ! [Eff2: rule > rule > fset_rule > $o,Rules: stream_rule,S4: set_rule,S3: rule,Rs: stream_rule,Steps: stream3588992565182678257e_rule,R2: rule] :
      ( ( abstra8115150628584006924e_rule @ Eff2 @ Rules @ S4 )
     => ( ( member_rule @ S3 @ S4 )
       => ( ( abstra1114877437032970289r_rule @ Rules @ Rs )
         => ( ( abstra3969007714708337434e_rule @ ( abstra7443870075974203631e_rule @ Eff2 @ Rs @ S3 ) @ Steps )
           => ( ( member_rule @ R2 @ ( sset_rule @ Rules ) )
             => ( abstra6372423936129899802e_rule @ Eff2 @ R2 @ Steps ) ) ) ) ) ) ).

% PersistentRuleSystem.ipath_mkTree_saturated
thf(fact_636_PersistentRuleSystem_Oipath__mkTree__saturated,axiom,
    ! [Eff2: produc1996495991257130529ist_fm > fm > fset_fm > $o,Rules: stream8299795917829157543ist_fm,S4: set_fm,S3: fm,Rs: stream8299795917829157543ist_fm,Steps: stream5068232101399153750ist_fm,R2: produc1996495991257130529ist_fm] :
      ( ( abstra1665324712063988849_fm_fm @ Eff2 @ Rules @ S4 )
     => ( ( member_fm @ S3 @ S4 )
       => ( ( abstra7454370078233524789ist_fm @ Rules @ Rs )
         => ( ( abstra8787874164724549719ist_fm @ ( abstra5537592759599600206_fm_fm @ Eff2 @ Rs @ S3 ) @ Steps )
           => ( ( member8102475879199740618ist_fm @ R2 @ ( sset_P3431374123360452149ist_fm @ Rules ) )
             => ( abstra230716855031908835_fm_fm @ Eff2 @ R2 @ Steps ) ) ) ) ) ) ).

% PersistentRuleSystem.ipath_mkTree_saturated
thf(fact_637_PersistentRuleSystem_Oipath__mkTree__saturated,axiom,
    ! [Eff2: produc1996495991257130529ist_fm > rule > fset_rule > $o,Rules: stream8299795917829157543ist_fm,S4: set_rule,S3: rule,Rs: stream8299795917829157543ist_fm,Steps: stream1960312990768957601ist_fm,R2: produc1996495991257130529ist_fm] :
      ( ( abstra1314873687758482406m_rule @ Eff2 @ Rules @ S4 )
     => ( ( member_rule @ S3 @ S4 )
       => ( ( abstra7454370078233524789ist_fm @ Rules @ Rs )
         => ( ( abstra6752138315222246434ist_fm @ ( abstra2830003028686515011m_rule @ Eff2 @ Rs @ S3 ) @ Steps )
           => ( ( member8102475879199740618ist_fm @ R2 @ ( sset_P3431374123360452149ist_fm @ Rules ) )
             => ( abstra679888793666381912m_rule @ Eff2 @ R2 @ Steps ) ) ) ) ) ) ).

% PersistentRuleSystem.ipath_mkTree_saturated
thf(fact_638_PersistentRuleSystem_Oipath__mkTree__saturated,axiom,
    ! [Eff2: fm > produc1996495991257130529ist_fm > fset_P661503646757059847ist_fm > $o,Rules: stream_fm,S4: set_Pr7058068377845519745ist_fm,S3: produc1996495991257130529ist_fm,Rs: stream_fm,Steps: stream5128679079952569136_fm_fm,R2: fm] :
      ( ( abstra6333699301575899727ist_fm @ Eff2 @ Rules @ S4 )
     => ( ( member8102475879199740618ist_fm @ S3 @ S4 )
       => ( ( abstra5119165595139683900air_fm @ Rules @ Rs )
         => ( ( abstra2316067621365718065_fm_fm @ ( abstra982595312256735276ist_fm @ Eff2 @ Rs @ S3 ) @ Steps )
           => ( ( member_fm @ R2 @ ( sset_fm @ Rules ) )
             => ( abstra4899091444543819713ist_fm @ Eff2 @ R2 @ Steps ) ) ) ) ) ) ).

% PersistentRuleSystem.ipath_mkTree_saturated
thf(fact_639_PersistentRuleSystem_Oipath__mkTree__saturated,axiom,
    ! [Eff2: rule > produc1996495991257130529ist_fm > fset_P661503646757059847ist_fm > $o,Rules: stream_rule,S4: set_Pr7058068377845519745ist_fm,S3: produc1996495991257130529ist_fm,Rs: stream_rule,Steps: stream727092118206550309m_rule,R2: rule] :
      ( ( abstra7887161495092839066ist_fm @ Eff2 @ Rules @ S4 )
     => ( ( member8102475879199740618ist_fm @ S3 @ S4 )
       => ( ( abstra1114877437032970289r_rule @ Rules @ Rs )
         => ( ( abstra2285467793620430502m_rule @ ( abstra178918799166095863ist_fm @ Eff2 @ Rs @ S3 ) @ Steps )
           => ( ( member_rule @ R2 @ ( sset_rule @ Rules ) )
             => ( abstra7252176601000738572ist_fm @ Eff2 @ R2 @ Steps ) ) ) ) ) ) ).

% PersistentRuleSystem.ipath_mkTree_saturated
thf(fact_640_PersistentRuleSystem_Oipath__mkTree__saturated,axiom,
    ! [Eff2: produc859450856879609959at_nat > fm > fset_fm > $o,Rules: stream8372878641218411373at_nat,S4: set_fm,S3: fm,Rs: stream8372878641218411373at_nat,Steps: stream8224063793758848028at_nat,R2: produc859450856879609959at_nat] :
      ( ( abstra8110488454511597367nat_fm @ Eff2 @ Rules @ S4 )
     => ( ( member_fm @ S3 @ S4 )
       => ( ( abstra5618069685180253819at_nat @ Rules @ Rs )
         => ( ( abstra8590151062464399645at_nat @ ( abstra8755028071118092180nat_fm @ Eff2 @ Rs @ S3 ) @ Steps )
           => ( ( member8206827879206165904at_nat @ R2 @ ( sset_P9164763103296375419at_nat @ Rules ) )
             => ( abstra7502804401140805033nat_fm @ Eff2 @ R2 @ Steps ) ) ) ) ) ) ).

% PersistentRuleSystem.ipath_mkTree_saturated
thf(fact_641_PersistentRuleSystem_Oipath__mkTree__saturated,axiom,
    ! [Eff2: produc859450856879609959at_nat > rule > fset_rule > $o,Rules: stream8372878641218411373at_nat,S4: set_rule,S3: rule,Rs: stream8372878641218411373at_nat,Steps: stream7497242749156405735at_nat,R2: produc859450856879609959at_nat] :
      ( ( abstra7673653355469228972t_rule @ Eff2 @ Rules @ S4 )
     => ( ( member_rule @ S3 @ S4 )
       => ( ( abstra5618069685180253819at_nat @ Rules @ Rs )
         => ( ( abstra3018042991430833768at_nat @ ( abstra6341243307501620617t_rule @ Eff2 @ Rs @ S3 ) @ Steps )
           => ( ( member8206827879206165904at_nat @ R2 @ ( sset_P9164763103296375419at_nat @ Rules ) )
             => ( abstra436705010757945630t_rule @ Eff2 @ R2 @ Steps ) ) ) ) ) ) ).

% PersistentRuleSystem.ipath_mkTree_saturated
thf(fact_642_trim__in__R,axiom,
    ! [S3: produc1996495991257130529ist_fm,Rs: stream_rule] :
      ( ( member8102475879199740618ist_fm @ S3 @ top_to728902761071950545ist_fm )
     => ( ( abstra1114877437032970289r_rule @ rules @ Rs )
       => ( member_rule
          @ ( shd_rule
            @ ( abstra2363873759530810705ist_fm
              @ ^ [R: rule,S: produc1996495991257130529ist_fm,Ss: fset_P661503646757059847ist_fm] :
                  ( ( eff @ R @ S )
                  = ( some_f2162114139285431020ist_fm @ Ss ) )
              @ Rs
              @ S3 ) )
          @ ( sset_rule @ rules ) ) ) ) ).

% trim_in_R
thf(fact_643_in__cont__mkTree,axiom,
    ! [S3: produc1996495991257130529ist_fm,Rs: stream_rule,T3: abstra4572470701423150929m_rule] :
      ( ( member8102475879199740618ist_fm @ S3 @ top_to728902761071950545ist_fm )
     => ( ( abstra1114877437032970289r_rule @ rules @ Rs )
       => ( ( fmembe1635983029874454474m_rule @ T3
            @ ( abstra4962253831105676628m_rule
              @ ( abstra178918799166095863ist_fm
                @ ^ [R: rule,S: produc1996495991257130529ist_fm,Ss: fset_P661503646757059847ist_fm] :
                    ( ( eff @ R @ S )
                    = ( some_f2162114139285431020ist_fm @ Ss ) )
                @ Rs
                @ S3 ) ) )
         => ? [Sl5: fset_P661503646757059847ist_fm,S7: produc1996495991257130529ist_fm] :
              ( ( member8102475879199740618ist_fm @ S7 @ top_to728902761071950545ist_fm )
              & ( ( eff
                  @ ( shd_rule
                    @ ( abstra2363873759530810705ist_fm
                      @ ^ [R: rule,S: produc1996495991257130529ist_fm,Ss: fset_P661503646757059847ist_fm] :
                          ( ( eff @ R @ S )
                          = ( some_f2162114139285431020ist_fm @ Ss ) )
                      @ Rs
                      @ S3 ) )
                  @ S3 )
                = ( some_f2162114139285431020ist_fm @ Sl5 ) )
              & ( fmembe3381613331217039976ist_fm @ S7 @ Sl5 )
              & ( T3
                = ( abstra178918799166095863ist_fm
                  @ ^ [R: rule,S: produc1996495991257130529ist_fm,Ss: fset_P661503646757059847ist_fm] :
                      ( ( eff @ R @ S )
                      = ( some_f2162114139285431020ist_fm @ Ss ) )
                  @ ( stl_rule
                    @ ( abstra2363873759530810705ist_fm
                      @ ^ [R: rule,S: produc1996495991257130529ist_fm,Ss: fset_P661503646757059847ist_fm] :
                          ( ( eff @ R @ S )
                          = ( some_f2162114139285431020ist_fm @ Ss ) )
                      @ Rs
                      @ S3 ) )
                  @ S7 ) ) ) ) ) ) ).

% in_cont_mkTree
thf(fact_644_fair__uu__def,axiom,
    abstra1913438439042302728u_rule = abstra1114877437032970289r_rule ).

% fair_uu_def
thf(fact_645_RuleSystem_OminWait__le__pos,axiom,
    ! [Eff2: rule > produc1996495991257130529ist_fm > fset_P661503646757059847ist_fm > $o,Rules: stream_rule,S4: set_Pr7058068377845519745ist_fm,Rs: stream_rule,R2: rule,S3: produc1996495991257130529ist_fm] :
      ( ( abstra6455275554694158129ist_fm @ Eff2 @ Rules @ S4 )
     => ( ( abstra1114877437032970289r_rule @ Rules @ Rs )
       => ( ( member_rule @ R2 @ ( sset_rule @ Rules ) )
         => ( ( abstra3055540779341473792ist_fm @ Eff2 @ R2 @ S3 )
           => ( ord_less_eq_nat @ ( abstra738319790293016300ist_fm @ Eff2 @ Rs @ S3 ) @ ( abstract_pos_rule @ Rs @ R2 ) ) ) ) ) ) ).

% RuleSystem.minWait_le_pos
thf(fact_646_wf__mkTree,axiom,
    ! [S3: produc1996495991257130529ist_fm,Rs: stream_rule] :
      ( ( member8102475879199740618ist_fm @ S3 @ top_to728902761071950545ist_fm )
     => ( ( abstra1114877437032970289r_rule @ rules @ Rs )
       => ( abstra4476735159789068068ist_fm
          @ ^ [R: rule,S: produc1996495991257130529ist_fm,Ss: fset_P661503646757059847ist_fm] :
              ( ( eff @ R @ S )
              = ( some_f2162114139285431020ist_fm @ Ss ) )
          @ rules
          @ ( abstra178918799166095863ist_fm
            @ ^ [R: rule,S: produc1996495991257130529ist_fm,Ss: fset_P661503646757059847ist_fm] :
                ( ( eff @ R @ S )
                = ( some_f2162114139285431020ist_fm @ Ss ) )
            @ Rs
            @ S3 ) ) ) ) ).

% wf_mkTree
thf(fact_647_RuleSystem_OminWait__def,axiom,
    ! [Eff2: rule > produc1996495991257130529ist_fm > fset_P661503646757059847ist_fm > $o,Rules: stream_rule,S4: set_Pr7058068377845519745ist_fm,Rs: stream_rule,S3: produc1996495991257130529ist_fm] :
      ( ( abstra6455275554694158129ist_fm @ Eff2 @ Rules @ S4 )
     => ( ( abstra738319790293016300ist_fm @ Eff2 @ Rs @ S3 )
        = ( ord_Least_nat
          @ ^ [N3: nat] : ( abstra3055540779341473792ist_fm @ Eff2 @ ( shd_rule @ ( sdrop_rule @ N3 @ Rs ) ) @ S3 ) ) ) ) ).

% RuleSystem.minWait_def
thf(fact_648_trim__def,axiom,
    ! [Rs: stream_rule,S3: produc1996495991257130529ist_fm] :
      ( ( abstra2363873759530810705ist_fm
        @ ^ [R: rule,S: produc1996495991257130529ist_fm,Ss: fset_P661503646757059847ist_fm] :
            ( ( eff @ R @ S )
            = ( some_f2162114139285431020ist_fm @ Ss ) )
        @ Rs
        @ S3 )
      = ( sdrop_while_rule
        @ ^ [R: rule] :
            ~ ( abstra3055540779341473792ist_fm
              @ ^ [S: rule,T: produc1996495991257130529ist_fm,Ss: fset_P661503646757059847ist_fm] :
                  ( ( eff @ S @ T )
                  = ( some_f2162114139285431020ist_fm @ Ss ) )
              @ R
              @ S3 )
        @ Rs ) ) ).

% trim_def
thf(fact_649_mkTree_Osimps_I1_J,axiom,
    ! [Rs: stream_rule,S3: produc1996495991257130529ist_fm] :
      ( ( abstra3503897793822247108m_rule
        @ ( abstra178918799166095863ist_fm
          @ ^ [R: rule,S: produc1996495991257130529ist_fm,Ss: fset_P661503646757059847ist_fm] :
              ( ( eff @ R @ S )
              = ( some_f2162114139285431020ist_fm @ Ss ) )
          @ Rs
          @ S3 ) )
      = ( produc491467635432902671m_rule @ S3
        @ ( shd_rule
          @ ( abstra2363873759530810705ist_fm
            @ ^ [R: rule,S: produc1996495991257130529ist_fm,Ss: fset_P661503646757059847ist_fm] :
                ( ( eff @ R @ S )
                = ( some_f2162114139285431020ist_fm @ Ss ) )
            @ Rs
            @ S3 ) ) ) ) ).

% mkTree.simps(1)
thf(fact_650_RuleSystem__Defs_Owf_Ocong,axiom,
    abstra4476735159789068068ist_fm = abstra4476735159789068068ist_fm ).

% RuleSystem_Defs.wf.cong
thf(fact_651_tree_Oexpand,axiom,
    ! [Tree: abstra4572470701423150929m_rule,Tree2: abstra4572470701423150929m_rule] :
      ( ( ( ( abstra3503897793822247108m_rule @ Tree )
          = ( abstra3503897793822247108m_rule @ Tree2 ) )
        & ( ( abstra4962253831105676628m_rule @ Tree )
          = ( abstra4962253831105676628m_rule @ Tree2 ) ) )
     => ( Tree = Tree2 ) ) ).

% tree.expand
thf(fact_652_RuleSystem_Owf__mkTree,axiom,
    ! [Eff2: rule > fm > fset_fm > $o,Rules: stream_rule,S4: set_fm,S3: fm,Rs: stream_rule] :
      ( ( abstra916514217863227968ule_fm @ Eff2 @ Rules @ S4 )
     => ( ( member_fm @ S3 @ S4 )
       => ( ( abstra1114877437032970289r_rule @ Rules @ Rs )
         => ( abstra2794488811626936717ule_fm @ Eff2 @ Rules @ ( abstra3991228042706630138ule_fm @ Eff2 @ Rs @ S3 ) ) ) ) ) ).

% RuleSystem.wf_mkTree
thf(fact_653_RuleSystem_Owf__mkTree,axiom,
    ! [Eff2: rule > rule > fset_rule > $o,Rules: stream_rule,S4: set_rule,S3: rule,Rs: stream_rule] :
      ( ( abstra2725590884229999157e_rule @ Eff2 @ Rules @ S4 )
     => ( ( member_rule @ S3 @ S4 )
       => ( ( abstra1114877437032970289r_rule @ Rules @ Rs )
         => ( abstra5602818997816140546e_rule @ Eff2 @ Rules @ ( abstra7443870075974203631e_rule @ Eff2 @ Rs @ S3 ) ) ) ) ) ).

% RuleSystem.wf_mkTree
thf(fact_654_RuleSystem_Owf__mkTree,axiom,
    ! [Eff2: rule > produc859450856879609959at_nat > fset_P3619101297660450253at_nat > $o,Rules: stream_rule,S4: set_Pr8693737435421807431at_nat,S3: produc859450856879609959at_nat,Rs: stream_rule] :
      ( ( abstra5655617845878133111at_nat @ Eff2 @ Rules @ S4 )
     => ( ( member8206827879206165904at_nat @ S3 @ S4 )
       => ( ( abstra1114877437032970289r_rule @ Rules @ Rs )
         => ( abstra1446413340331638506at_nat @ Eff2 @ Rules @ ( abstra193881638956283709at_nat @ Eff2 @ Rs @ S3 ) ) ) ) ) ).

% RuleSystem.wf_mkTree
thf(fact_655_RuleSystem_Owf__mkTree,axiom,
    ! [Eff2: rule > produc1996495991257130529ist_fm > fset_P661503646757059847ist_fm > $o,Rules: stream_rule,S4: set_Pr7058068377845519745ist_fm,S3: produc1996495991257130529ist_fm,Rs: stream_rule] :
      ( ( abstra6455275554694158129ist_fm @ Eff2 @ Rules @ S4 )
     => ( ( member8102475879199740618ist_fm @ S3 @ S4 )
       => ( ( abstra1114877437032970289r_rule @ Rules @ Rs )
         => ( abstra4476735159789068068ist_fm @ Eff2 @ Rules @ ( abstra178918799166095863ist_fm @ Eff2 @ Rs @ S3 ) ) ) ) ) ).

% RuleSystem.wf_mkTree
thf(fact_656_RuleSystem_Oenabled__R,axiom,
    ! [Eff2: rule > fm > fset_fm > $o,Rules: stream_rule,S4: set_fm,S3: fm] :
      ( ( abstra916514217863227968ule_fm @ Eff2 @ Rules @ S4 )
     => ( ( member_fm @ S3 @ S4 )
       => ? [X: rule] :
            ( ( member_rule @ X @ ( sset_rule @ Rules ) )
            & ? [X_12: fset_fm] : ( Eff2 @ X @ S3 @ X_12 ) ) ) ) ).

% RuleSystem.enabled_R
thf(fact_657_RuleSystem_Oenabled__R,axiom,
    ! [Eff2: rule > rule > fset_rule > $o,Rules: stream_rule,S4: set_rule,S3: rule] :
      ( ( abstra2725590884229999157e_rule @ Eff2 @ Rules @ S4 )
     => ( ( member_rule @ S3 @ S4 )
       => ? [X: rule] :
            ( ( member_rule @ X @ ( sset_rule @ Rules ) )
            & ? [X_12: fset_rule] : ( Eff2 @ X @ S3 @ X_12 ) ) ) ) ).

% RuleSystem.enabled_R
thf(fact_658_RuleSystem_Oenabled__R,axiom,
    ! [Eff2: rule > produc859450856879609959at_nat > fset_P3619101297660450253at_nat > $o,Rules: stream_rule,S4: set_Pr8693737435421807431at_nat,S3: produc859450856879609959at_nat] :
      ( ( abstra5655617845878133111at_nat @ Eff2 @ Rules @ S4 )
     => ( ( member8206827879206165904at_nat @ S3 @ S4 )
       => ? [X: rule] :
            ( ( member_rule @ X @ ( sset_rule @ Rules ) )
            & ? [X_12: fset_P3619101297660450253at_nat] : ( Eff2 @ X @ S3 @ X_12 ) ) ) ) ).

% RuleSystem.enabled_R
thf(fact_659_RuleSystem_Oenabled__R,axiom,
    ! [Eff2: produc164195504107695125m_rule > fm > fset_fm > $o,Rules: stream727092118206550309m_rule,S4: set_fm,S3: fm] :
      ( ( abstra3764363818050485264ule_fm @ Eff2 @ Rules @ S4 )
     => ( ( member_fm @ S3 @ S4 )
       => ? [X: produc164195504107695125m_rule] :
            ( ( member4220325220686508332m_rule @ X @ ( sset_P4979403222752902913m_rule @ Rules ) )
            & ? [X_12: fset_fm] : ( Eff2 @ X @ S3 @ X_12 ) ) ) ) ).

% RuleSystem.enabled_R
thf(fact_660_RuleSystem_Oenabled__R,axiom,
    ! [Eff2: produc164195504107695125m_rule > rule > fset_rule > $o,Rules: stream727092118206550309m_rule,S4: set_rule,S3: rule] :
      ( ( abstra3778134268586021893e_rule @ Eff2 @ Rules @ S4 )
     => ( ( member_rule @ S3 @ S4 )
       => ? [X: produc164195504107695125m_rule] :
            ( ( member4220325220686508332m_rule @ X @ ( sset_P4979403222752902913m_rule @ Rules ) )
            & ? [X_12: fset_rule] : ( Eff2 @ X @ S3 @ X_12 ) ) ) ) ).

% RuleSystem.enabled_R
thf(fact_661_RuleSystem_Oenabled__R,axiom,
    ! [Eff2: produc164195504107695125m_rule > produc1996495991257130529ist_fm > fset_P661503646757059847ist_fm > $o,Rules: stream727092118206550309m_rule,S4: set_Pr7058068377845519745ist_fm,S3: produc1996495991257130529ist_fm] :
      ( ( abstra3408323020314982753ist_fm @ Eff2 @ Rules @ S4 )
     => ( ( member8102475879199740618ist_fm @ S3 @ S4 )
       => ? [X: produc164195504107695125m_rule] :
            ( ( member4220325220686508332m_rule @ X @ ( sset_P4979403222752902913m_rule @ Rules ) )
            & ? [X_12: fset_P661503646757059847ist_fm] : ( Eff2 @ X @ S3 @ X_12 ) ) ) ) ).

% RuleSystem.enabled_R
thf(fact_662_RuleSystem_Oenabled__R,axiom,
    ! [Eff2: produc164195504107695125m_rule > produc859450856879609959at_nat > fset_P3619101297660450253at_nat > $o,Rules: stream727092118206550309m_rule,S4: set_Pr8693737435421807431at_nat,S3: produc859450856879609959at_nat] :
      ( ( abstra8123243527896255399at_nat @ Eff2 @ Rules @ S4 )
     => ( ( member8206827879206165904at_nat @ S3 @ S4 )
       => ? [X: produc164195504107695125m_rule] :
            ( ( member4220325220686508332m_rule @ X @ ( sset_P4979403222752902913m_rule @ Rules ) )
            & ? [X_12: fset_P3619101297660450253at_nat] : ( Eff2 @ X @ S3 @ X_12 ) ) ) ) ).

% RuleSystem.enabled_R
thf(fact_663_RuleSystem_Oenabled__R,axiom,
    ! [Eff2: rule > produc1996495991257130529ist_fm > fset_P661503646757059847ist_fm > $o,Rules: stream_rule,S4: set_Pr7058068377845519745ist_fm,S3: produc1996495991257130529ist_fm] :
      ( ( abstra6455275554694158129ist_fm @ Eff2 @ Rules @ S4 )
     => ( ( member8102475879199740618ist_fm @ S3 @ S4 )
       => ? [X: rule] :
            ( ( member_rule @ X @ ( sset_rule @ Rules ) )
            & ? [X_12: fset_P661503646757059847ist_fm] : ( Eff2 @ X @ S3 @ X_12 ) ) ) ) ).

% RuleSystem.enabled_R
thf(fact_664_PersistentRuleSystem_Oaxioms_I1_J,axiom,
    ! [Eff2: rule > produc1996495991257130529ist_fm > fset_P661503646757059847ist_fm > $o,Rules: stream_rule,S4: set_Pr7058068377845519745ist_fm] :
      ( ( abstra7887161495092839066ist_fm @ Eff2 @ Rules @ S4 )
     => ( abstra6455275554694158129ist_fm @ Eff2 @ Rules @ S4 ) ) ).

% PersistentRuleSystem.axioms(1)
thf(fact_665_ipath_Ocases,axiom,
    ! [A1: abstract_tree_rule,A22: stream_rule] :
      ( ( abstract_ipath_rule @ A1 @ A22 )
     => ~ ( ( ( abstract_root_rule @ A1 )
            = ( shd_rule @ A22 ) )
         => ! [T4: abstract_tree_rule] :
              ( ( fmembe7964111328986570234e_rule @ T4 @ ( abstract_cont_rule @ A1 ) )
             => ~ ( abstract_ipath_rule @ T4 @ ( stl_rule @ A22 ) ) ) ) ) ).

% ipath.cases
thf(fact_666_ipath_Ocases,axiom,
    ! [A1: abstra4572470701423150929m_rule,A22: stream727092118206550309m_rule] :
      ( ( abstra2285467793620430502m_rule @ A1 @ A22 )
     => ~ ( ( ( abstra3503897793822247108m_rule @ A1 )
            = ( shd_Pr7235097944458474089m_rule @ A22 ) )
         => ! [T4: abstra4572470701423150929m_rule] :
              ( ( fmembe1635983029874454474m_rule @ T4 @ ( abstra4962253831105676628m_rule @ A1 ) )
             => ~ ( abstra2285467793620430502m_rule @ T4 @ ( stl_Pr950425576149878629m_rule @ A22 ) ) ) ) ) ).

% ipath.cases
thf(fact_667_ipath_Osimps,axiom,
    ( abstract_ipath_rule
    = ( ^ [A12: abstract_tree_rule,A23: stream_rule] :
        ? [T: abstract_tree_rule,Steps3: stream_rule,T5: abstract_tree_rule] :
          ( ( A12 = T )
          & ( A23 = Steps3 )
          & ( ( abstract_root_rule @ T )
            = ( shd_rule @ Steps3 ) )
          & ( fmembe7964111328986570234e_rule @ T5 @ ( abstract_cont_rule @ T ) )
          & ( abstract_ipath_rule @ T5 @ ( stl_rule @ Steps3 ) ) ) ) ) ).

% ipath.simps
thf(fact_668_ipath_Osimps,axiom,
    ( abstra2285467793620430502m_rule
    = ( ^ [A12: abstra4572470701423150929m_rule,A23: stream727092118206550309m_rule] :
        ? [T: abstra4572470701423150929m_rule,Steps3: stream727092118206550309m_rule,T5: abstra4572470701423150929m_rule] :
          ( ( A12 = T )
          & ( A23 = Steps3 )
          & ( ( abstra3503897793822247108m_rule @ T )
            = ( shd_Pr7235097944458474089m_rule @ Steps3 ) )
          & ( fmembe1635983029874454474m_rule @ T5 @ ( abstra4962253831105676628m_rule @ T ) )
          & ( abstra2285467793620430502m_rule @ T5 @ ( stl_Pr950425576149878629m_rule @ Steps3 ) ) ) ) ) ).

% ipath.simps
thf(fact_669_ipath_Ointros,axiom,
    ! [T2: abstract_tree_rule,Steps: stream_rule,T3: abstract_tree_rule] :
      ( ( ( abstract_root_rule @ T2 )
        = ( shd_rule @ Steps ) )
     => ( ( fmembe7964111328986570234e_rule @ T3 @ ( abstract_cont_rule @ T2 ) )
       => ( ( abstract_ipath_rule @ T3 @ ( stl_rule @ Steps ) )
         => ( abstract_ipath_rule @ T2 @ Steps ) ) ) ) ).

% ipath.intros
thf(fact_670_ipath_Ointros,axiom,
    ! [T2: abstra4572470701423150929m_rule,Steps: stream727092118206550309m_rule,T3: abstra4572470701423150929m_rule] :
      ( ( ( abstra3503897793822247108m_rule @ T2 )
        = ( shd_Pr7235097944458474089m_rule @ Steps ) )
     => ( ( fmembe1635983029874454474m_rule @ T3 @ ( abstra4962253831105676628m_rule @ T2 ) )
       => ( ( abstra2285467793620430502m_rule @ T3 @ ( stl_Pr950425576149878629m_rule @ Steps ) )
         => ( abstra2285467793620430502m_rule @ T2 @ Steps ) ) ) ) ).

% ipath.intros
thf(fact_671_ipath_Ocoinduct,axiom,
    ! [X7: abstract_tree_rule > stream_rule > $o,X1: abstract_tree_rule,X2: stream_rule] :
      ( ( X7 @ X1 @ X2 )
     => ( ! [X12: abstract_tree_rule,X23: stream_rule] :
            ( ( X7 @ X12 @ X23 )
           => ? [T6: abstract_tree_rule,Steps2: stream_rule,T7: abstract_tree_rule] :
                ( ( X12 = T6 )
                & ( X23 = Steps2 )
                & ( ( abstract_root_rule @ T6 )
                  = ( shd_rule @ Steps2 ) )
                & ( fmembe7964111328986570234e_rule @ T7 @ ( abstract_cont_rule @ T6 ) )
                & ( ( X7 @ T7 @ ( stl_rule @ Steps2 ) )
                  | ( abstract_ipath_rule @ T7 @ ( stl_rule @ Steps2 ) ) ) ) )
       => ( abstract_ipath_rule @ X1 @ X2 ) ) ) ).

% ipath.coinduct
thf(fact_672_ipath_Ocoinduct,axiom,
    ! [X7: abstra4572470701423150929m_rule > stream727092118206550309m_rule > $o,X1: abstra4572470701423150929m_rule,X2: stream727092118206550309m_rule] :
      ( ( X7 @ X1 @ X2 )
     => ( ! [X12: abstra4572470701423150929m_rule,X23: stream727092118206550309m_rule] :
            ( ( X7 @ X12 @ X23 )
           => ? [T6: abstra4572470701423150929m_rule,Steps2: stream727092118206550309m_rule,T7: abstra4572470701423150929m_rule] :
                ( ( X12 = T6 )
                & ( X23 = Steps2 )
                & ( ( abstra3503897793822247108m_rule @ T6 )
                  = ( shd_Pr7235097944458474089m_rule @ Steps2 ) )
                & ( fmembe1635983029874454474m_rule @ T7 @ ( abstra4962253831105676628m_rule @ T6 ) )
                & ( ( X7 @ T7 @ ( stl_Pr950425576149878629m_rule @ Steps2 ) )
                  | ( abstra2285467793620430502m_rule @ T7 @ ( stl_Pr950425576149878629m_rule @ Steps2 ) ) ) ) )
       => ( abstra2285467793620430502m_rule @ X1 @ X2 ) ) ) ).

% ipath.coinduct
thf(fact_673_RuleSystem__def,axiom,
    ( abstra886934874521163211_fm_fm
    = ( ^ [Eff: fm > fm > fset_fm > $o,Rules2: stream_fm,S8: set_fm] :
          ( ! [S: fm] :
              ( ( member_fm @ S @ S8 )
             => ! [R: fm] :
                  ( ( member_fm @ R @ ( sset_fm @ Rules2 ) )
                 => ! [Sl: fset_fm] :
                      ( ( Eff @ R @ S @ Sl )
                     => ! [S9: fm] :
                          ( ( fmember_fm @ S9 @ Sl )
                         => ( member_fm @ S9 @ S8 ) ) ) ) )
          & ! [S: fm] :
              ( ( member_fm @ S @ S8 )
             => ? [X6: fm] :
                  ( ( member_fm @ X6 @ ( sset_fm @ Rules2 ) )
                  & ? [X5: fset_fm] : ( Eff @ X6 @ S @ X5 ) ) ) ) ) ) ).

% RuleSystem_def
thf(fact_674_RuleSystem__def,axiom,
    ( abstra5167362717900046400m_rule
    = ( ^ [Eff: fm > rule > fset_rule > $o,Rules2: stream_fm,S8: set_rule] :
          ( ! [S: rule] :
              ( ( member_rule @ S @ S8 )
             => ! [R: fm] :
                  ( ( member_fm @ R @ ( sset_fm @ Rules2 ) )
                 => ! [Sl: fset_rule] :
                      ( ( Eff @ R @ S @ Sl )
                     => ! [S9: rule] :
                          ( ( fmember_rule @ S9 @ Sl )
                         => ( member_rule @ S9 @ S8 ) ) ) ) )
          & ! [S: rule] :
              ( ( member_rule @ S @ S8 )
             => ? [X6: fm] :
                  ( ( member_fm @ X6 @ ( sset_fm @ Rules2 ) )
                  & ? [X5: fset_rule] : ( Eff @ X6 @ S @ X5 ) ) ) ) ) ) ).

% RuleSystem_def
thf(fact_675_RuleSystem__def,axiom,
    ( abstra916514217863227968ule_fm
    = ( ^ [Eff: rule > fm > fset_fm > $o,Rules2: stream_rule,S8: set_fm] :
          ( ! [S: fm] :
              ( ( member_fm @ S @ S8 )
             => ! [R: rule] :
                  ( ( member_rule @ R @ ( sset_rule @ Rules2 ) )
                 => ! [Sl: fset_fm] :
                      ( ( Eff @ R @ S @ Sl )
                     => ! [S9: fm] :
                          ( ( fmember_fm @ S9 @ Sl )
                         => ( member_fm @ S9 @ S8 ) ) ) ) )
          & ! [S: fm] :
              ( ( member_fm @ S @ S8 )
             => ? [X6: rule] :
                  ( ( member_rule @ X6 @ ( sset_rule @ Rules2 ) )
                  & ? [X5: fset_fm] : ( Eff @ X6 @ S @ X5 ) ) ) ) ) ) ).

% RuleSystem_def
thf(fact_676_RuleSystem__def,axiom,
    ( abstra2725590884229999157e_rule
    = ( ^ [Eff: rule > rule > fset_rule > $o,Rules2: stream_rule,S8: set_rule] :
          ( ! [S: rule] :
              ( ( member_rule @ S @ S8 )
             => ! [R: rule] :
                  ( ( member_rule @ R @ ( sset_rule @ Rules2 ) )
                 => ! [Sl: fset_rule] :
                      ( ( Eff @ R @ S @ Sl )
                     => ! [S9: rule] :
                          ( ( fmember_rule @ S9 @ Sl )
                         => ( member_rule @ S9 @ S8 ) ) ) ) )
          & ! [S: rule] :
              ( ( member_rule @ S @ S8 )
             => ? [X6: rule] :
                  ( ( member_rule @ X6 @ ( sset_rule @ Rules2 ) )
                  & ? [X5: fset_rule] : ( Eff @ X6 @ S @ X5 ) ) ) ) ) ) ).

% RuleSystem_def
thf(fact_677_RuleSystem__def,axiom,
    ( abstra2919136520687232136_fm_fm
    = ( ^ [Eff: produc1996495991257130529ist_fm > fm > fset_fm > $o,Rules2: stream8299795917829157543ist_fm,S8: set_fm] :
          ( ! [S: fm] :
              ( ( member_fm @ S @ S8 )
             => ! [R: produc1996495991257130529ist_fm] :
                  ( ( member8102475879199740618ist_fm @ R @ ( sset_P3431374123360452149ist_fm @ Rules2 ) )
                 => ! [Sl: fset_fm] :
                      ( ( Eff @ R @ S @ Sl )
                     => ! [S9: fm] :
                          ( ( fmember_fm @ S9 @ Sl )
                         => ( member_fm @ S9 @ S8 ) ) ) ) )
          & ! [S: fm] :
              ( ( member_fm @ S @ S8 )
             => ? [X6: produc1996495991257130529ist_fm] :
                  ( ( member8102475879199740618ist_fm @ X6 @ ( sset_P3431374123360452149ist_fm @ Rules2 ) )
                  & ? [X5: fset_fm] : ( Eff @ X6 @ S @ X5 ) ) ) ) ) ) ).

% RuleSystem_def
thf(fact_678_RuleSystem__def,axiom,
    ( abstra9106359784214577277m_rule
    = ( ^ [Eff: produc1996495991257130529ist_fm > rule > fset_rule > $o,Rules2: stream8299795917829157543ist_fm,S8: set_rule] :
          ( ! [S: rule] :
              ( ( member_rule @ S @ S8 )
             => ! [R: produc1996495991257130529ist_fm] :
                  ( ( member8102475879199740618ist_fm @ R @ ( sset_P3431374123360452149ist_fm @ Rules2 ) )
                 => ! [Sl: fset_rule] :
                      ( ( Eff @ R @ S @ Sl )
                     => ! [S9: rule] :
                          ( ( fmember_rule @ S9 @ Sl )
                         => ( member_rule @ S9 @ S8 ) ) ) ) )
          & ! [S: rule] :
              ( ( member_rule @ S @ S8 )
             => ? [X6: produc1996495991257130529ist_fm] :
                  ( ( member8102475879199740618ist_fm @ X6 @ ( sset_P3431374123360452149ist_fm @ Rules2 ) )
                  & ? [X5: fset_rule] : ( Eff @ X6 @ S @ X5 ) ) ) ) ) ) ).

% RuleSystem_def
thf(fact_679_RuleSystem__def,axiom,
    ( abstra7587511110199143014ist_fm
    = ( ^ [Eff: fm > produc1996495991257130529ist_fm > fset_P661503646757059847ist_fm > $o,Rules2: stream_fm,S8: set_Pr7058068377845519745ist_fm] :
          ( ! [S: produc1996495991257130529ist_fm] :
              ( ( member8102475879199740618ist_fm @ S @ S8 )
             => ! [R: fm] :
                  ( ( member_fm @ R @ ( sset_fm @ Rules2 ) )
                 => ! [Sl: fset_P661503646757059847ist_fm] :
                      ( ( Eff @ R @ S @ Sl )
                     => ! [S9: produc1996495991257130529ist_fm] :
                          ( ( fmembe3381613331217039976ist_fm @ S9 @ Sl )
                         => ( member8102475879199740618ist_fm @ S9 @ S8 ) ) ) ) )
          & ! [S: produc1996495991257130529ist_fm] :
              ( ( member8102475879199740618ist_fm @ S @ S8 )
             => ? [X6: fm] :
                  ( ( member_fm @ X6 @ ( sset_fm @ Rules2 ) )
                  & ? [X5: fset_P661503646757059847ist_fm] : ( Eff @ X6 @ S @ X5 ) ) ) ) ) ) ).

% RuleSystem_def
thf(fact_680_RuleSystem__def,axiom,
    ( abstra6455275554694158129ist_fm
    = ( ^ [Eff: rule > produc1996495991257130529ist_fm > fset_P661503646757059847ist_fm > $o,Rules2: stream_rule,S8: set_Pr7058068377845519745ist_fm] :
          ( ! [S: produc1996495991257130529ist_fm] :
              ( ( member8102475879199740618ist_fm @ S @ S8 )
             => ! [R: rule] :
                  ( ( member_rule @ R @ ( sset_rule @ Rules2 ) )
                 => ! [Sl: fset_P661503646757059847ist_fm] :
                      ( ( Eff @ R @ S @ Sl )
                     => ! [S9: produc1996495991257130529ist_fm] :
                          ( ( fmembe3381613331217039976ist_fm @ S9 @ Sl )
                         => ( member8102475879199740618ist_fm @ S9 @ S8 ) ) ) ) )
          & ! [S: produc1996495991257130529ist_fm] :
              ( ( member8102475879199740618ist_fm @ S @ S8 )
             => ? [X6: rule] :
                  ( ( member_rule @ X6 @ ( sset_rule @ Rules2 ) )
                  & ? [X5: fset_P661503646757059847ist_fm] : ( Eff @ X6 @ S @ X5 ) ) ) ) ) ) ).

% RuleSystem_def
thf(fact_681_RuleSystem__def,axiom,
    ( abstra849841822169861838nat_fm
    = ( ^ [Eff: produc859450856879609959at_nat > fm > fset_fm > $o,Rules2: stream8372878641218411373at_nat,S8: set_fm] :
          ( ! [S: fm] :
              ( ( member_fm @ S @ S8 )
             => ! [R: produc859450856879609959at_nat] :
                  ( ( member8206827879206165904at_nat @ R @ ( sset_P9164763103296375419at_nat @ Rules2 ) )
                 => ! [Sl: fset_fm] :
                      ( ( Eff @ R @ S @ Sl )
                     => ! [S9: fm] :
                          ( ( fmember_fm @ S9 @ Sl )
                         => ( member_fm @ S9 @ S8 ) ) ) ) )
          & ! [S: fm] :
              ( ( member_fm @ S @ S8 )
             => ? [X6: produc859450856879609959at_nat] :
                  ( ( member8206827879206165904at_nat @ X6 @ ( sset_P9164763103296375419at_nat @ Rules2 ) )
                  & ? [X5: fset_fm] : ( Eff @ X6 @ S @ X5 ) ) ) ) ) ) ).

% RuleSystem_def
thf(fact_682_RuleSystem__def,axiom,
    ( abstra2579607477568694211t_rule
    = ( ^ [Eff: produc859450856879609959at_nat > rule > fset_rule > $o,Rules2: stream8372878641218411373at_nat,S8: set_rule] :
          ( ! [S: rule] :
              ( ( member_rule @ S @ S8 )
             => ! [R: produc859450856879609959at_nat] :
                  ( ( member8206827879206165904at_nat @ R @ ( sset_P9164763103296375419at_nat @ Rules2 ) )
                 => ! [Sl: fset_rule] :
                      ( ( Eff @ R @ S @ Sl )
                     => ! [S9: rule] :
                          ( ( fmember_rule @ S9 @ Sl )
                         => ( member_rule @ S9 @ S8 ) ) ) ) )
          & ! [S: rule] :
              ( ( member_rule @ S @ S8 )
             => ? [X6: produc859450856879609959at_nat] :
                  ( ( member8206827879206165904at_nat @ X6 @ ( sset_P9164763103296375419at_nat @ Rules2 ) )
                  & ? [X5: fset_rule] : ( Eff @ X6 @ S @ X5 ) ) ) ) ) ) ).

% RuleSystem_def
thf(fact_683_RuleSystem_Ointro,axiom,
    ! [S4: set_fm,Rules: stream_fm,Eff2: fm > fm > fset_fm > $o] :
      ( ! [S6: fm] :
          ( ( member_fm @ S6 @ S4 )
         => ! [R5: fm] :
              ( ( member_fm @ R5 @ ( sset_fm @ Rules ) )
             => ! [Sl4: fset_fm] :
                  ( ( Eff2 @ R5 @ S6 @ Sl4 )
                 => ! [S7: fm] :
                      ( ( fmember_fm @ S7 @ Sl4 )
                     => ( member_fm @ S7 @ S4 ) ) ) ) )
     => ( ! [S6: fm] :
            ( ( member_fm @ S6 @ S4 )
           => ? [X8: fm] :
                ( ( member_fm @ X8 @ ( sset_fm @ Rules ) )
                & ? [X_1: fset_fm] : ( Eff2 @ X8 @ S6 @ X_1 ) ) )
       => ( abstra886934874521163211_fm_fm @ Eff2 @ Rules @ S4 ) ) ) ).

% RuleSystem.intro
thf(fact_684_RuleSystem_Ointro,axiom,
    ! [S4: set_rule,Rules: stream_fm,Eff2: fm > rule > fset_rule > $o] :
      ( ! [S6: rule] :
          ( ( member_rule @ S6 @ S4 )
         => ! [R5: fm] :
              ( ( member_fm @ R5 @ ( sset_fm @ Rules ) )
             => ! [Sl4: fset_rule] :
                  ( ( Eff2 @ R5 @ S6 @ Sl4 )
                 => ! [S7: rule] :
                      ( ( fmember_rule @ S7 @ Sl4 )
                     => ( member_rule @ S7 @ S4 ) ) ) ) )
     => ( ! [S6: rule] :
            ( ( member_rule @ S6 @ S4 )
           => ? [X8: fm] :
                ( ( member_fm @ X8 @ ( sset_fm @ Rules ) )
                & ? [X_1: fset_rule] : ( Eff2 @ X8 @ S6 @ X_1 ) ) )
       => ( abstra5167362717900046400m_rule @ Eff2 @ Rules @ S4 ) ) ) ).

% RuleSystem.intro
thf(fact_685_RuleSystem_Ointro,axiom,
    ! [S4: set_fm,Rules: stream_rule,Eff2: rule > fm > fset_fm > $o] :
      ( ! [S6: fm] :
          ( ( member_fm @ S6 @ S4 )
         => ! [R5: rule] :
              ( ( member_rule @ R5 @ ( sset_rule @ Rules ) )
             => ! [Sl4: fset_fm] :
                  ( ( Eff2 @ R5 @ S6 @ Sl4 )
                 => ! [S7: fm] :
                      ( ( fmember_fm @ S7 @ Sl4 )
                     => ( member_fm @ S7 @ S4 ) ) ) ) )
     => ( ! [S6: fm] :
            ( ( member_fm @ S6 @ S4 )
           => ? [X8: rule] :
                ( ( member_rule @ X8 @ ( sset_rule @ Rules ) )
                & ? [X_1: fset_fm] : ( Eff2 @ X8 @ S6 @ X_1 ) ) )
       => ( abstra916514217863227968ule_fm @ Eff2 @ Rules @ S4 ) ) ) ).

% RuleSystem.intro
thf(fact_686_RuleSystem_Ointro,axiom,
    ! [S4: set_rule,Rules: stream_rule,Eff2: rule > rule > fset_rule > $o] :
      ( ! [S6: rule] :
          ( ( member_rule @ S6 @ S4 )
         => ! [R5: rule] :
              ( ( member_rule @ R5 @ ( sset_rule @ Rules ) )
             => ! [Sl4: fset_rule] :
                  ( ( Eff2 @ R5 @ S6 @ Sl4 )
                 => ! [S7: rule] :
                      ( ( fmember_rule @ S7 @ Sl4 )
                     => ( member_rule @ S7 @ S4 ) ) ) ) )
     => ( ! [S6: rule] :
            ( ( member_rule @ S6 @ S4 )
           => ? [X8: rule] :
                ( ( member_rule @ X8 @ ( sset_rule @ Rules ) )
                & ? [X_1: fset_rule] : ( Eff2 @ X8 @ S6 @ X_1 ) ) )
       => ( abstra2725590884229999157e_rule @ Eff2 @ Rules @ S4 ) ) ) ).

% RuleSystem.intro
thf(fact_687_RuleSystem_Ointro,axiom,
    ! [S4: set_fm,Rules: stream8299795917829157543ist_fm,Eff2: produc1996495991257130529ist_fm > fm > fset_fm > $o] :
      ( ! [S6: fm] :
          ( ( member_fm @ S6 @ S4 )
         => ! [R5: produc1996495991257130529ist_fm] :
              ( ( member8102475879199740618ist_fm @ R5 @ ( sset_P3431374123360452149ist_fm @ Rules ) )
             => ! [Sl4: fset_fm] :
                  ( ( Eff2 @ R5 @ S6 @ Sl4 )
                 => ! [S7: fm] :
                      ( ( fmember_fm @ S7 @ Sl4 )
                     => ( member_fm @ S7 @ S4 ) ) ) ) )
     => ( ! [S6: fm] :
            ( ( member_fm @ S6 @ S4 )
           => ? [X8: produc1996495991257130529ist_fm] :
                ( ( member8102475879199740618ist_fm @ X8 @ ( sset_P3431374123360452149ist_fm @ Rules ) )
                & ? [X_1: fset_fm] : ( Eff2 @ X8 @ S6 @ X_1 ) ) )
       => ( abstra2919136520687232136_fm_fm @ Eff2 @ Rules @ S4 ) ) ) ).

% RuleSystem.intro
thf(fact_688_RuleSystem_Ointro,axiom,
    ! [S4: set_rule,Rules: stream8299795917829157543ist_fm,Eff2: produc1996495991257130529ist_fm > rule > fset_rule > $o] :
      ( ! [S6: rule] :
          ( ( member_rule @ S6 @ S4 )
         => ! [R5: produc1996495991257130529ist_fm] :
              ( ( member8102475879199740618ist_fm @ R5 @ ( sset_P3431374123360452149ist_fm @ Rules ) )
             => ! [Sl4: fset_rule] :
                  ( ( Eff2 @ R5 @ S6 @ Sl4 )
                 => ! [S7: rule] :
                      ( ( fmember_rule @ S7 @ Sl4 )
                     => ( member_rule @ S7 @ S4 ) ) ) ) )
     => ( ! [S6: rule] :
            ( ( member_rule @ S6 @ S4 )
           => ? [X8: produc1996495991257130529ist_fm] :
                ( ( member8102475879199740618ist_fm @ X8 @ ( sset_P3431374123360452149ist_fm @ Rules ) )
                & ? [X_1: fset_rule] : ( Eff2 @ X8 @ S6 @ X_1 ) ) )
       => ( abstra9106359784214577277m_rule @ Eff2 @ Rules @ S4 ) ) ) ).

% RuleSystem.intro
thf(fact_689_RuleSystem_Ointro,axiom,
    ! [S4: set_Pr7058068377845519745ist_fm,Rules: stream_fm,Eff2: fm > produc1996495991257130529ist_fm > fset_P661503646757059847ist_fm > $o] :
      ( ! [S6: produc1996495991257130529ist_fm] :
          ( ( member8102475879199740618ist_fm @ S6 @ S4 )
         => ! [R5: fm] :
              ( ( member_fm @ R5 @ ( sset_fm @ Rules ) )
             => ! [Sl4: fset_P661503646757059847ist_fm] :
                  ( ( Eff2 @ R5 @ S6 @ Sl4 )
                 => ! [S7: produc1996495991257130529ist_fm] :
                      ( ( fmembe3381613331217039976ist_fm @ S7 @ Sl4 )
                     => ( member8102475879199740618ist_fm @ S7 @ S4 ) ) ) ) )
     => ( ! [S6: produc1996495991257130529ist_fm] :
            ( ( member8102475879199740618ist_fm @ S6 @ S4 )
           => ? [X8: fm] :
                ( ( member_fm @ X8 @ ( sset_fm @ Rules ) )
                & ? [X_1: fset_P661503646757059847ist_fm] : ( Eff2 @ X8 @ S6 @ X_1 ) ) )
       => ( abstra7587511110199143014ist_fm @ Eff2 @ Rules @ S4 ) ) ) ).

% RuleSystem.intro
thf(fact_690_RuleSystem_Ointro,axiom,
    ! [S4: set_Pr7058068377845519745ist_fm,Rules: stream_rule,Eff2: rule > produc1996495991257130529ist_fm > fset_P661503646757059847ist_fm > $o] :
      ( ! [S6: produc1996495991257130529ist_fm] :
          ( ( member8102475879199740618ist_fm @ S6 @ S4 )
         => ! [R5: rule] :
              ( ( member_rule @ R5 @ ( sset_rule @ Rules ) )
             => ! [Sl4: fset_P661503646757059847ist_fm] :
                  ( ( Eff2 @ R5 @ S6 @ Sl4 )
                 => ! [S7: produc1996495991257130529ist_fm] :
                      ( ( fmembe3381613331217039976ist_fm @ S7 @ Sl4 )
                     => ( member8102475879199740618ist_fm @ S7 @ S4 ) ) ) ) )
     => ( ! [S6: produc1996495991257130529ist_fm] :
            ( ( member8102475879199740618ist_fm @ S6 @ S4 )
           => ? [X8: rule] :
                ( ( member_rule @ X8 @ ( sset_rule @ Rules ) )
                & ? [X_1: fset_P661503646757059847ist_fm] : ( Eff2 @ X8 @ S6 @ X_1 ) ) )
       => ( abstra6455275554694158129ist_fm @ Eff2 @ Rules @ S4 ) ) ) ).

% RuleSystem.intro
thf(fact_691_RuleSystem_Ointro,axiom,
    ! [S4: set_fm,Rules: stream8372878641218411373at_nat,Eff2: produc859450856879609959at_nat > fm > fset_fm > $o] :
      ( ! [S6: fm] :
          ( ( member_fm @ S6 @ S4 )
         => ! [R5: produc859450856879609959at_nat] :
              ( ( member8206827879206165904at_nat @ R5 @ ( sset_P9164763103296375419at_nat @ Rules ) )
             => ! [Sl4: fset_fm] :
                  ( ( Eff2 @ R5 @ S6 @ Sl4 )
                 => ! [S7: fm] :
                      ( ( fmember_fm @ S7 @ Sl4 )
                     => ( member_fm @ S7 @ S4 ) ) ) ) )
     => ( ! [S6: fm] :
            ( ( member_fm @ S6 @ S4 )
           => ? [X8: produc859450856879609959at_nat] :
                ( ( member8206827879206165904at_nat @ X8 @ ( sset_P9164763103296375419at_nat @ Rules ) )
                & ? [X_1: fset_fm] : ( Eff2 @ X8 @ S6 @ X_1 ) ) )
       => ( abstra849841822169861838nat_fm @ Eff2 @ Rules @ S4 ) ) ) ).

% RuleSystem.intro
thf(fact_692_RuleSystem_Ointro,axiom,
    ! [S4: set_rule,Rules: stream8372878641218411373at_nat,Eff2: produc859450856879609959at_nat > rule > fset_rule > $o] :
      ( ! [S6: rule] :
          ( ( member_rule @ S6 @ S4 )
         => ! [R5: produc859450856879609959at_nat] :
              ( ( member8206827879206165904at_nat @ R5 @ ( sset_P9164763103296375419at_nat @ Rules ) )
             => ! [Sl4: fset_rule] :
                  ( ( Eff2 @ R5 @ S6 @ Sl4 )
                 => ! [S7: rule] :
                      ( ( fmember_rule @ S7 @ Sl4 )
                     => ( member_rule @ S7 @ S4 ) ) ) ) )
     => ( ! [S6: rule] :
            ( ( member_rule @ S6 @ S4 )
           => ? [X8: produc859450856879609959at_nat] :
                ( ( member8206827879206165904at_nat @ X8 @ ( sset_P9164763103296375419at_nat @ Rules ) )
                & ? [X_1: fset_rule] : ( Eff2 @ X8 @ S6 @ X_1 ) ) )
       => ( abstra2579607477568694211t_rule @ Eff2 @ Rules @ S4 ) ) ) ).

% RuleSystem.intro
thf(fact_693_RuleSystem_Oeff__S,axiom,
    ! [Eff2: fm > fm > fset_fm > $o,Rules: stream_fm,S4: set_fm,S3: fm,R2: fm,Sl3: fset_fm,S5: fm] :
      ( ( abstra886934874521163211_fm_fm @ Eff2 @ Rules @ S4 )
     => ( ( member_fm @ S3 @ S4 )
       => ( ( member_fm @ R2 @ ( sset_fm @ Rules ) )
         => ( ( Eff2 @ R2 @ S3 @ Sl3 )
           => ( ( fmember_fm @ S5 @ Sl3 )
             => ( member_fm @ S5 @ S4 ) ) ) ) ) ) ).

% RuleSystem.eff_S
thf(fact_694_RuleSystem_Oeff__S,axiom,
    ! [Eff2: fm > rule > fset_rule > $o,Rules: stream_fm,S4: set_rule,S3: rule,R2: fm,Sl3: fset_rule,S5: rule] :
      ( ( abstra5167362717900046400m_rule @ Eff2 @ Rules @ S4 )
     => ( ( member_rule @ S3 @ S4 )
       => ( ( member_fm @ R2 @ ( sset_fm @ Rules ) )
         => ( ( Eff2 @ R2 @ S3 @ Sl3 )
           => ( ( fmember_rule @ S5 @ Sl3 )
             => ( member_rule @ S5 @ S4 ) ) ) ) ) ) ).

% RuleSystem.eff_S
thf(fact_695_RuleSystem_Oeff__S,axiom,
    ! [Eff2: rule > fm > fset_fm > $o,Rules: stream_rule,S4: set_fm,S3: fm,R2: rule,Sl3: fset_fm,S5: fm] :
      ( ( abstra916514217863227968ule_fm @ Eff2 @ Rules @ S4 )
     => ( ( member_fm @ S3 @ S4 )
       => ( ( member_rule @ R2 @ ( sset_rule @ Rules ) )
         => ( ( Eff2 @ R2 @ S3 @ Sl3 )
           => ( ( fmember_fm @ S5 @ Sl3 )
             => ( member_fm @ S5 @ S4 ) ) ) ) ) ) ).

% RuleSystem.eff_S
thf(fact_696_RuleSystem_Oeff__S,axiom,
    ! [Eff2: rule > rule > fset_rule > $o,Rules: stream_rule,S4: set_rule,S3: rule,R2: rule,Sl3: fset_rule,S5: rule] :
      ( ( abstra2725590884229999157e_rule @ Eff2 @ Rules @ S4 )
     => ( ( member_rule @ S3 @ S4 )
       => ( ( member_rule @ R2 @ ( sset_rule @ Rules ) )
         => ( ( Eff2 @ R2 @ S3 @ Sl3 )
           => ( ( fmember_rule @ S5 @ Sl3 )
             => ( member_rule @ S5 @ S4 ) ) ) ) ) ) ).

% RuleSystem.eff_S
thf(fact_697_RuleSystem_Oeff__S,axiom,
    ! [Eff2: produc1996495991257130529ist_fm > fm > fset_fm > $o,Rules: stream8299795917829157543ist_fm,S4: set_fm,S3: fm,R2: produc1996495991257130529ist_fm,Sl3: fset_fm,S5: fm] :
      ( ( abstra2919136520687232136_fm_fm @ Eff2 @ Rules @ S4 )
     => ( ( member_fm @ S3 @ S4 )
       => ( ( member8102475879199740618ist_fm @ R2 @ ( sset_P3431374123360452149ist_fm @ Rules ) )
         => ( ( Eff2 @ R2 @ S3 @ Sl3 )
           => ( ( fmember_fm @ S5 @ Sl3 )
             => ( member_fm @ S5 @ S4 ) ) ) ) ) ) ).

% RuleSystem.eff_S
thf(fact_698_RuleSystem_Oeff__S,axiom,
    ! [Eff2: produc1996495991257130529ist_fm > rule > fset_rule > $o,Rules: stream8299795917829157543ist_fm,S4: set_rule,S3: rule,R2: produc1996495991257130529ist_fm,Sl3: fset_rule,S5: rule] :
      ( ( abstra9106359784214577277m_rule @ Eff2 @ Rules @ S4 )
     => ( ( member_rule @ S3 @ S4 )
       => ( ( member8102475879199740618ist_fm @ R2 @ ( sset_P3431374123360452149ist_fm @ Rules ) )
         => ( ( Eff2 @ R2 @ S3 @ Sl3 )
           => ( ( fmember_rule @ S5 @ Sl3 )
             => ( member_rule @ S5 @ S4 ) ) ) ) ) ) ).

% RuleSystem.eff_S
thf(fact_699_RuleSystem_Oeff__S,axiom,
    ! [Eff2: fm > produc1996495991257130529ist_fm > fset_P661503646757059847ist_fm > $o,Rules: stream_fm,S4: set_Pr7058068377845519745ist_fm,S3: produc1996495991257130529ist_fm,R2: fm,Sl3: fset_P661503646757059847ist_fm,S5: produc1996495991257130529ist_fm] :
      ( ( abstra7587511110199143014ist_fm @ Eff2 @ Rules @ S4 )
     => ( ( member8102475879199740618ist_fm @ S3 @ S4 )
       => ( ( member_fm @ R2 @ ( sset_fm @ Rules ) )
         => ( ( Eff2 @ R2 @ S3 @ Sl3 )
           => ( ( fmembe3381613331217039976ist_fm @ S5 @ Sl3 )
             => ( member8102475879199740618ist_fm @ S5 @ S4 ) ) ) ) ) ) ).

% RuleSystem.eff_S
thf(fact_700_RuleSystem_Oeff__S,axiom,
    ! [Eff2: rule > produc1996495991257130529ist_fm > fset_P661503646757059847ist_fm > $o,Rules: stream_rule,S4: set_Pr7058068377845519745ist_fm,S3: produc1996495991257130529ist_fm,R2: rule,Sl3: fset_P661503646757059847ist_fm,S5: produc1996495991257130529ist_fm] :
      ( ( abstra6455275554694158129ist_fm @ Eff2 @ Rules @ S4 )
     => ( ( member8102475879199740618ist_fm @ S3 @ S4 )
       => ( ( member_rule @ R2 @ ( sset_rule @ Rules ) )
         => ( ( Eff2 @ R2 @ S3 @ Sl3 )
           => ( ( fmembe3381613331217039976ist_fm @ S5 @ Sl3 )
             => ( member8102475879199740618ist_fm @ S5 @ S4 ) ) ) ) ) ) ).

% RuleSystem.eff_S
thf(fact_701_RuleSystem_Oeff__S,axiom,
    ! [Eff2: produc859450856879609959at_nat > fm > fset_fm > $o,Rules: stream8372878641218411373at_nat,S4: set_fm,S3: fm,R2: produc859450856879609959at_nat,Sl3: fset_fm,S5: fm] :
      ( ( abstra849841822169861838nat_fm @ Eff2 @ Rules @ S4 )
     => ( ( member_fm @ S3 @ S4 )
       => ( ( member8206827879206165904at_nat @ R2 @ ( sset_P9164763103296375419at_nat @ Rules ) )
         => ( ( Eff2 @ R2 @ S3 @ Sl3 )
           => ( ( fmember_fm @ S5 @ Sl3 )
             => ( member_fm @ S5 @ S4 ) ) ) ) ) ) ).

% RuleSystem.eff_S
thf(fact_702_RuleSystem_Oeff__S,axiom,
    ! [Eff2: produc859450856879609959at_nat > rule > fset_rule > $o,Rules: stream8372878641218411373at_nat,S4: set_rule,S3: rule,R2: produc859450856879609959at_nat,Sl3: fset_rule,S5: rule] :
      ( ( abstra2579607477568694211t_rule @ Eff2 @ Rules @ S4 )
     => ( ( member_rule @ S3 @ S4 )
       => ( ( member8206827879206165904at_nat @ R2 @ ( sset_P9164763103296375419at_nat @ Rules ) )
         => ( ( Eff2 @ R2 @ S3 @ Sl3 )
           => ( ( fmember_rule @ S5 @ Sl3 )
             => ( member_rule @ S5 @ S4 ) ) ) ) ) ) ).

% RuleSystem.eff_S
thf(fact_703_RuleSystem_Otrim__fair,axiom,
    ! [Eff2: rule > fm > fset_fm > $o,Rules: stream_rule,S4: set_fm,S3: fm,Rs: stream_rule] :
      ( ( abstra916514217863227968ule_fm @ Eff2 @ Rules @ S4 )
     => ( ( member_fm @ S3 @ S4 )
       => ( ( abstra1114877437032970289r_rule @ Rules @ Rs )
         => ( abstra1114877437032970289r_rule @ Rules @ ( abstra4245643633226316832ule_fm @ Eff2 @ Rs @ S3 ) ) ) ) ) ).

% RuleSystem.trim_fair
thf(fact_704_RuleSystem_Otrim__fair,axiom,
    ! [Eff2: rule > rule > fset_rule > $o,Rules: stream_rule,S4: set_rule,S3: rule,Rs: stream_rule] :
      ( ( abstra2725590884229999157e_rule @ Eff2 @ Rules @ S4 )
     => ( ( member_rule @ S3 @ S4 )
       => ( ( abstra1114877437032970289r_rule @ Rules @ Rs )
         => ( abstra1114877437032970289r_rule @ Rules @ ( abstra7824469560916757525e_rule @ Eff2 @ Rs @ S3 ) ) ) ) ) ).

% RuleSystem.trim_fair
thf(fact_705_RuleSystem_Otrim__fair,axiom,
    ! [Eff2: rule > produc859450856879609959at_nat > fset_P3619101297660450253at_nat > $o,Rules: stream_rule,S4: set_Pr8693737435421807431at_nat,S3: produc859450856879609959at_nat,Rs: stream_rule] :
      ( ( abstra5655617845878133111at_nat @ Eff2 @ Rules @ S4 )
     => ( ( member8206827879206165904at_nat @ S3 @ S4 )
       => ( ( abstra1114877437032970289r_rule @ Rules @ Rs )
         => ( abstra1114877437032970289r_rule @ Rules @ ( abstra3437498576158148503at_nat @ Eff2 @ Rs @ S3 ) ) ) ) ) ).

% RuleSystem.trim_fair
thf(fact_706_RuleSystem_Otrim__fair,axiom,
    ! [Eff2: rule > produc1996495991257130529ist_fm > fset_P661503646757059847ist_fm > $o,Rules: stream_rule,S4: set_Pr7058068377845519745ist_fm,S3: produc1996495991257130529ist_fm,Rs: stream_rule] :
      ( ( abstra6455275554694158129ist_fm @ Eff2 @ Rules @ S4 )
     => ( ( member8102475879199740618ist_fm @ S3 @ S4 )
       => ( ( abstra1114877437032970289r_rule @ Rules @ Rs )
         => ( abstra1114877437032970289r_rule @ Rules @ ( abstra2363873759530810705ist_fm @ Eff2 @ Rs @ S3 ) ) ) ) ) ).

% RuleSystem.trim_fair
thf(fact_707_RuleSystem__Defs_Owf__ipath__epath,axiom,
    ! [Eff2: rule > produc1996495991257130529ist_fm > fset_P661503646757059847ist_fm > $o,Rules: stream_rule,T2: abstra4572470701423150929m_rule,Steps: stream727092118206550309m_rule] :
      ( ( abstra4476735159789068068ist_fm @ Eff2 @ Rules @ T2 )
     => ( ( abstra2285467793620430502m_rule @ T2 @ Steps )
       => ( abstra5415202973855822231ist_fm @ Eff2 @ Rules @ Steps ) ) ) ).

% RuleSystem_Defs.wf_ipath_epath
thf(fact_708_sdrop__while_Osimps,axiom,
    ( sdrop_5541336384294084785m_rule
    = ( ^ [P4: produc164195504107695125m_rule > $o,S: stream727092118206550309m_rule] : ( if_str8948254419368749791m_rule @ ( P4 @ ( shd_Pr7235097944458474089m_rule @ S ) ) @ ( sdrop_5541336384294084785m_rule @ P4 @ ( stl_Pr950425576149878629m_rule @ S ) ) @ S ) ) ) ).

% sdrop_while.simps
thf(fact_709_sdrop__while_Osimps,axiom,
    ( sdrop_while_rule
    = ( ^ [P4: rule > $o,S: stream_rule] : ( if_stream_rule @ ( P4 @ ( shd_rule @ S ) ) @ ( sdrop_while_rule @ P4 @ ( stl_rule @ S ) ) @ S ) ) ) ).

% sdrop_while.simps
thf(fact_710_RuleSystem_Oin__cont__mkTree,axiom,
    ! [Eff2: produc164195504107695125m_rule > fm > fset_fm > $o,Rules: stream727092118206550309m_rule,S4: set_fm,S3: fm,Rs: stream727092118206550309m_rule,T3: abstra4575282202256230560m_rule] :
      ( ( abstra3764363818050485264ule_fm @ Eff2 @ Rules @ S4 )
     => ( ( member_fm @ S3 @ S4 )
       => ( ( abstra4997497885011821313m_rule @ Rules @ Rs )
         => ( ( fmembe8691679578407899495m_rule @ T3 @ ( abstra2433431901973683783m_rule @ ( abstra1207708450097661130ule_fm @ Eff2 @ Rs @ S3 ) ) )
           => ? [Sl5: fset_fm,S7: fm] :
                ( ( member_fm @ S7 @ S4 )
                & ( Eff2 @ ( shd_Pr7235097944458474089m_rule @ ( abstra2315704231547878896ule_fm @ Eff2 @ Rs @ S3 ) ) @ S3 @ Sl5 )
                & ( fmember_fm @ S7 @ Sl5 )
                & ( T3
                  = ( abstra1207708450097661130ule_fm @ Eff2 @ ( stl_Pr950425576149878629m_rule @ ( abstra2315704231547878896ule_fm @ Eff2 @ Rs @ S3 ) ) @ S7 ) ) ) ) ) ) ) ).

% RuleSystem.in_cont_mkTree
thf(fact_711_RuleSystem_Oin__cont__mkTree,axiom,
    ! [Eff2: produc164195504107695125m_rule > rule > fset_rule > $o,Rules: stream727092118206550309m_rule,S4: set_rule,S3: rule,Rs: stream727092118206550309m_rule,T3: abstra4791331342562632597m_rule] :
      ( ( abstra3778134268586021893e_rule @ Eff2 @ Rules @ S4 )
     => ( ( member_rule @ S3 @ S4 )
       => ( ( abstra4997497885011821313m_rule @ Rules @ Rs )
         => ( ( fmembe1981849019698968796m_rule @ T3 @ ( abstra5403976286072314556m_rule @ ( abstra1231468025953361855e_rule @ Eff2 @ Rs @ S3 ) ) )
           => ? [Sl5: fset_rule,S7: rule] :
                ( ( member_rule @ S7 @ S4 )
                & ( Eff2 @ ( shd_Pr7235097944458474089m_rule @ ( abstra5289413328258868709e_rule @ Eff2 @ Rs @ S3 ) ) @ S3 @ Sl5 )
                & ( fmember_rule @ S7 @ Sl5 )
                & ( T3
                  = ( abstra1231468025953361855e_rule @ Eff2 @ ( stl_Pr950425576149878629m_rule @ ( abstra5289413328258868709e_rule @ Eff2 @ Rs @ S3 ) ) @ S7 ) ) ) ) ) ) ) ).

% RuleSystem.in_cont_mkTree
thf(fact_712_RuleSystem_Oin__cont__mkTree,axiom,
    ! [Eff2: produc164195504107695125m_rule > produc859450856879609959at_nat > fset_P3619101297660450253at_nat > $o,Rules: stream727092118206550309m_rule,S4: set_Pr8693737435421807431at_nat,S3: produc859450856879609959at_nat,Rs: stream727092118206550309m_rule,T3: abstra8173387307635926887m_rule] :
      ( ( abstra8123243527896255399at_nat @ Eff2 @ Rules @ S4 )
     => ( ( member8206827879206165904at_nat @ S3 @ S4 )
       => ( ( abstra4997497885011821313m_rule @ Rules @ Rs )
         => ( ( fmembe2908244859456314592m_rule @ T3 @ ( abstra8750048091207030378m_rule @ ( abstra8186850217458087021at_nat @ Eff2 @ Rs @ S3 ) ) )
           => ? [Sl5: fset_P3619101297660450253at_nat,S7: produc859450856879609959at_nat] :
                ( ( member8206827879206165904at_nat @ S7 @ S4 )
                & ( Eff2 @ ( shd_Pr7235097944458474089m_rule @ ( abstra276577985667043783at_nat @ Eff2 @ Rs @ S3 ) ) @ S3 @ Sl5 )
                & ( fmembe6431293532144391214at_nat @ S7 @ Sl5 )
                & ( T3
                  = ( abstra8186850217458087021at_nat @ Eff2 @ ( stl_Pr950425576149878629m_rule @ ( abstra276577985667043783at_nat @ Eff2 @ Rs @ S3 ) ) @ S7 ) ) ) ) ) ) ) ).

% RuleSystem.in_cont_mkTree
thf(fact_713_RuleSystem_Oin__cont__mkTree,axiom,
    ! [Eff2: produc164195504107695125m_rule > produc1996495991257130529ist_fm > fset_P661503646757059847ist_fm > $o,Rules: stream727092118206550309m_rule,S4: set_Pr7058068377845519745ist_fm,S3: produc1996495991257130529ist_fm,Rs: stream727092118206550309m_rule,T3: abstra4658698708444432161m_rule] :
      ( ( abstra3408323020314982753ist_fm @ Eff2 @ Rules @ S4 )
     => ( ( member8102475879199740618ist_fm @ S3 @ S4 )
       => ( ( abstra4997497885011821313m_rule @ Rules @ Rs )
         => ( ( fmembe2925761000581315610m_rule @ T3 @ ( abstra5924671108449608740m_rule @ ( abstra2161496378874750759ist_fm @ Eff2 @ Rs @ S3 ) ) )
           => ? [Sl5: fset_P661503646757059847ist_fm,S7: produc1996495991257130529ist_fm] :
                ( ( member8102475879199740618ist_fm @ S7 @ S4 )
                & ( Eff2 @ ( shd_Pr7235097944458474089m_rule @ ( abstra6201285654312316801ist_fm @ Eff2 @ Rs @ S3 ) ) @ S3 @ Sl5 )
                & ( fmembe3381613331217039976ist_fm @ S7 @ Sl5 )
                & ( T3
                  = ( abstra2161496378874750759ist_fm @ Eff2 @ ( stl_Pr950425576149878629m_rule @ ( abstra6201285654312316801ist_fm @ Eff2 @ Rs @ S3 ) ) @ S7 ) ) ) ) ) ) ) ).

% RuleSystem.in_cont_mkTree
thf(fact_714_RuleSystem_Oin__cont__mkTree,axiom,
    ! [Eff2: produc164195504107695125m_rule > abstra4572470701423150929m_rule > fset_A5396818216090053761m_rule > $o,Rules: stream727092118206550309m_rule,S4: set_Ab3744519486624674951m_rule,S3: abstra4572470701423150929m_rule,Rs: stream727092118206550309m_rule,T3: abstra5426939140630736801m_rule] :
      ( ( abstra6898608545798587409m_rule @ Eff2 @ Rules @ S4 )
     => ( ( member1590433669991072232m_rule @ S3 @ S4 )
       => ( ( abstra4997497885011821313m_rule @ Rules @ Rs )
         => ( ( fmembe8791921695509808616m_rule @ T3 @ ( abstra3071294317365906120m_rule @ ( abstra1631510761171731915m_rule @ Eff2 @ Rs @ S3 ) ) )
           => ? [Sl5: fset_A5396818216090053761m_rule,S7: abstra4572470701423150929m_rule] :
                ( ( member1590433669991072232m_rule @ S7 @ S4 )
                & ( Eff2 @ ( shd_Pr7235097944458474089m_rule @ ( abstra7609338345072424433m_rule @ Eff2 @ Rs @ S3 ) ) @ S3 @ Sl5 )
                & ( fmembe1635983029874454474m_rule @ S7 @ Sl5 )
                & ( T3
                  = ( abstra1631510761171731915m_rule @ Eff2 @ ( stl_Pr950425576149878629m_rule @ ( abstra7609338345072424433m_rule @ Eff2 @ Rs @ S3 ) ) @ S7 ) ) ) ) ) ) ) ).

% RuleSystem.in_cont_mkTree
thf(fact_715_RuleSystem_Oin__cont__mkTree,axiom,
    ! [Eff2: rule > fm > fset_fm > $o,Rules: stream_rule,S4: set_fm,S3: fm,Rs: stream_rule,T3: abstra4524330545568143696m_rule] :
      ( ( abstra916514217863227968ule_fm @ Eff2 @ Rules @ S4 )
     => ( ( member_fm @ S3 @ S4 )
       => ( ( abstra1114877437032970289r_rule @ Rules @ Rs )
         => ( ( fmembe2914889872492170903m_rule @ T3 @ ( abstra5134701107988495863m_rule @ ( abstra3991228042706630138ule_fm @ Eff2 @ Rs @ S3 ) ) )
           => ? [Sl5: fset_fm,S7: fm] :
                ( ( member_fm @ S7 @ S4 )
                & ( Eff2 @ ( shd_rule @ ( abstra4245643633226316832ule_fm @ Eff2 @ Rs @ S3 ) ) @ S3 @ Sl5 )
                & ( fmember_fm @ S7 @ Sl5 )
                & ( T3
                  = ( abstra3991228042706630138ule_fm @ Eff2 @ ( stl_rule @ ( abstra4245643633226316832ule_fm @ Eff2 @ Rs @ S3 ) ) @ S7 ) ) ) ) ) ) ) ).

% RuleSystem.in_cont_mkTree
thf(fact_716_RuleSystem_Oin__cont__mkTree,axiom,
    ! [Eff2: rule > rule > fset_rule > $o,Rules: stream_rule,S4: set_rule,S3: rule,Rs: stream_rule,T3: abstra1071693965251717573e_rule] :
      ( ( abstra2725590884229999157e_rule @ Eff2 @ Rules @ S4 )
     => ( ( member_rule @ S3 @ S4 )
       => ( ( abstra1114877437032970289r_rule @ Rules @ Rs )
         => ( ( fmembe5852619498312149900e_rule @ T3 @ ( abstra2651384284180525036e_rule @ ( abstra7443870075974203631e_rule @ Eff2 @ Rs @ S3 ) ) )
           => ? [Sl5: fset_rule,S7: rule] :
                ( ( member_rule @ S7 @ S4 )
                & ( Eff2 @ ( shd_rule @ ( abstra7824469560916757525e_rule @ Eff2 @ Rs @ S3 ) ) @ S3 @ Sl5 )
                & ( fmember_rule @ S7 @ Sl5 )
                & ( T3
                  = ( abstra7443870075974203631e_rule @ Eff2 @ ( stl_rule @ ( abstra7824469560916757525e_rule @ Eff2 @ Rs @ S3 ) ) @ S7 ) ) ) ) ) ) ) ).

% RuleSystem.in_cont_mkTree
thf(fact_717_RuleSystem_Oin__cont__mkTree,axiom,
    ! [Eff2: rule > produc859450856879609959at_nat > fset_P3619101297660450253at_nat > $o,Rules: stream_rule,S4: set_Pr8693737435421807431at_nat,S3: produc859450856879609959at_nat,Rs: stream_rule,T3: abstra5175790144536110231t_rule] :
      ( ( abstra5655617845878133111at_nat @ Eff2 @ Rules @ S4 )
     => ( ( member8206827879206165904at_nat @ S3 @ S4 )
       => ( ( abstra1114877437032970289r_rule @ Rules @ Rs )
         => ( ( fmembe4064145320595246480t_rule @ T3 @ ( abstra553533796049368218t_rule @ ( abstra193881638956283709at_nat @ Eff2 @ Rs @ S3 ) ) )
           => ? [Sl5: fset_P3619101297660450253at_nat,S7: produc859450856879609959at_nat] :
                ( ( member8206827879206165904at_nat @ S7 @ S4 )
                & ( Eff2 @ ( shd_rule @ ( abstra3437498576158148503at_nat @ Eff2 @ Rs @ S3 ) ) @ S3 @ Sl5 )
                & ( fmembe6431293532144391214at_nat @ S7 @ Sl5 )
                & ( T3
                  = ( abstra193881638956283709at_nat @ Eff2 @ ( stl_rule @ ( abstra3437498576158148503at_nat @ Eff2 @ Rs @ S3 ) ) @ S7 ) ) ) ) ) ) ) ).

% RuleSystem.in_cont_mkTree
thf(fact_718_RuleSystem_Oin__cont__mkTree,axiom,
    ! [Eff2: rule > abstra4572470701423150929m_rule > fset_A5396818216090053761m_rule > $o,Rules: stream_rule,S4: set_Ab3744519486624674951m_rule,S3: abstra4572470701423150929m_rule,Rs: stream_rule,T3: abstra4125161011840266193e_rule] :
      ( ( abstra6244190358723783233m_rule @ Eff2 @ Rules @ S4 )
     => ( ( member1590433669991072232m_rule @ S3 @ S4 )
       => ( ( abstra1114877437032970289r_rule @ Rules @ Rs )
         => ( ( fmembe8010993199532995224e_rule @ T3 @ ( abstra6162512231279892984e_rule @ ( abstra2861652164185304827m_rule @ Eff2 @ Rs @ S3 ) ) )
           => ? [Sl5: fset_A5396818216090053761m_rule,S7: abstra4572470701423150929m_rule] :
                ( ( member1590433669991072232m_rule @ S7 @ S4 )
                & ( Eff2 @ ( shd_rule @ ( abstra1865002795007470625m_rule @ Eff2 @ Rs @ S3 ) ) @ S3 @ Sl5 )
                & ( fmembe1635983029874454474m_rule @ S7 @ Sl5 )
                & ( T3
                  = ( abstra2861652164185304827m_rule @ Eff2 @ ( stl_rule @ ( abstra1865002795007470625m_rule @ Eff2 @ Rs @ S3 ) ) @ S7 ) ) ) ) ) ) ) ).

% RuleSystem.in_cont_mkTree
thf(fact_719_RuleSystem_Oin__cont__mkTree,axiom,
    ! [Eff2: rule > produc1996495991257130529ist_fm > fset_P661503646757059847ist_fm > $o,Rules: stream_rule,S4: set_Pr7058068377845519745ist_fm,S3: produc1996495991257130529ist_fm,Rs: stream_rule,T3: abstra4572470701423150929m_rule] :
      ( ( abstra6455275554694158129ist_fm @ Eff2 @ Rules @ S4 )
     => ( ( member8102475879199740618ist_fm @ S3 @ S4 )
       => ( ( abstra1114877437032970289r_rule @ Rules @ Rs )
         => ( ( fmembe1635983029874454474m_rule @ T3 @ ( abstra4962253831105676628m_rule @ ( abstra178918799166095863ist_fm @ Eff2 @ Rs @ S3 ) ) )
           => ? [Sl5: fset_P661503646757059847ist_fm,S7: produc1996495991257130529ist_fm] :
                ( ( member8102475879199740618ist_fm @ S7 @ S4 )
                & ( Eff2 @ ( shd_rule @ ( abstra2363873759530810705ist_fm @ Eff2 @ Rs @ S3 ) ) @ S3 @ Sl5 )
                & ( fmembe3381613331217039976ist_fm @ S7 @ Sl5 )
                & ( T3
                  = ( abstra178918799166095863ist_fm @ Eff2 @ ( stl_rule @ ( abstra2363873759530810705ist_fm @ Eff2 @ Rs @ S3 ) ) @ S7 ) ) ) ) ) ) ) ).

% RuleSystem.in_cont_mkTree
thf(fact_720_RuleSystem__Defs_Otrim__def,axiom,
    ( abstra2363873759530810705ist_fm
    = ( ^ [Eff: rule > produc1996495991257130529ist_fm > fset_P661503646757059847ist_fm > $o,Rs2: stream_rule,S: produc1996495991257130529ist_fm] :
          ( sdrop_while_rule
          @ ^ [R: rule] :
              ~ ( abstra3055540779341473792ist_fm @ Eff @ R @ S )
          @ Rs2 ) ) ) ).

% RuleSystem_Defs.trim_def
thf(fact_721_RuleSystem__Defs_OmkTree_Osimps_I1_J,axiom,
    ! [Eff2: list_fm > list_fm > fset_list_fm > $o,Rs: stream_list_fm,S3: list_fm] :
      ( ( abstra778364809706815794ist_fm @ ( abstra3846921640779263781ist_fm @ Eff2 @ Rs @ S3 ) )
      = ( produc381145313068854617ist_fm @ S3 @ ( shd_list_fm @ ( abstra2983386638146376267ist_fm @ Eff2 @ Rs @ S3 ) ) ) ) ).

% RuleSystem_Defs.mkTree.simps(1)
thf(fact_722_RuleSystem__Defs_OmkTree_Osimps_I1_J,axiom,
    ! [Eff2: nat > nat > fset_nat > $o,Rs: stream_nat,S3: nat] :
      ( ( abstra3259451721936639506at_nat @ ( abstra1192804020863382533at_nat @ Eff2 @ Rs @ S3 ) )
      = ( product_Pair_nat_nat @ S3 @ ( shd_nat @ ( abstra7573781713853518635at_nat @ Eff2 @ Rs @ S3 ) ) ) ) ).

% RuleSystem_Defs.mkTree.simps(1)
thf(fact_723_RuleSystem__Defs_OmkTree_Osimps_I1_J,axiom,
    ! [Eff2: product_prod_nat_nat > product_prod_nat_nat > fset_P5670320511379867111at_nat > $o,Rs: stream6724221391990029191at_nat,S3: product_prod_nat_nat] :
      ( ( abstra35009723297898488at_nat @ ( abstra8972947971889770147at_nat @ Eff2 @ Rs @ S3 ) )
      = ( produc6161850002892822231at_nat @ S3 @ ( shd_Pr4260400998323988397at_nat @ ( abstra4493250544927931593at_nat @ Eff2 @ Rs @ S3 ) ) ) ) ).

% RuleSystem_Defs.mkTree.simps(1)
thf(fact_724_RuleSystem__Defs_OmkTree_Osimps_I1_J,axiom,
    ! [Eff2: produc1996495991257130529ist_fm > rule > fset_rule > $o,Rs: stream8299795917829157543ist_fm,S3: rule] :
      ( ( abstra7970568315424063040ist_fm @ ( abstra2830003028686515011m_rule @ Eff2 @ Rs @ S3 ) )
      = ( produc7063755442767259331ist_fm @ S3 @ ( shd_Pr772355297128350925ist_fm @ ( abstra5014957989051229853m_rule @ Eff2 @ Rs @ S3 ) ) ) ) ).

% RuleSystem_Defs.mkTree.simps(1)
thf(fact_725_RuleSystem__Defs_OmkTree_Osimps_I1_J,axiom,
    ! [Eff2: rule > produc1996495991257130529ist_fm > fset_P661503646757059847ist_fm > $o,Rs: stream_rule,S3: produc1996495991257130529ist_fm] :
      ( ( abstra3503897793822247108m_rule @ ( abstra178918799166095863ist_fm @ Eff2 @ Rs @ S3 ) )
      = ( produc491467635432902671m_rule @ S3 @ ( shd_rule @ ( abstra2363873759530810705ist_fm @ Eff2 @ Rs @ S3 ) ) ) ) ).

% RuleSystem_Defs.mkTree.simps(1)
thf(fact_726_i_Owf__ipath__epath,axiom,
    ! [Eff3: rule > produc1996495991257130529ist_fm > option6029102822600979917ist_fm,Rules: stream_rule,T2: abstra4572470701423150929m_rule,Steps: stream727092118206550309m_rule] :
      ( ( abstra4476735159789068068ist_fm @ ( abstra5459341389059251418ist_fm @ Eff3 ) @ Rules @ T2 )
     => ( ( abstra2285467793620430502m_rule @ T2 @ Steps )
       => ( abstra5415202973855822231ist_fm @ ( abstra5459341389059251418ist_fm @ Eff3 ) @ Rules @ Steps ) ) ) ).

% i.wf_ipath_epath
thf(fact_727_i_OmkTree_Osimps_I1_J,axiom,
    ! [Eff3: list_fm > list_fm > option_fset_list_fm,Rs: stream_list_fm,S3: list_fm] :
      ( ( abstra778364809706815794ist_fm @ ( abstra3846921640779263781ist_fm @ ( abstra256862739949234434ist_fm @ Eff3 ) @ Rs @ S3 ) )
      = ( produc381145313068854617ist_fm @ S3 @ ( shd_list_fm @ ( abstra2983386638146376267ist_fm @ ( abstra256862739949234434ist_fm @ Eff3 ) @ Rs @ S3 ) ) ) ) ).

% i.mkTree.simps(1)
thf(fact_728_i_OmkTree_Osimps_I1_J,axiom,
    ! [Eff3: nat > nat > option_fset_nat,Rs: stream_nat,S3: nat] :
      ( ( abstra3259451721936639506at_nat @ ( abstra1192804020863382533at_nat @ ( abstra4671362528648695778at_nat @ Eff3 ) @ Rs @ S3 ) )
      = ( product_Pair_nat_nat @ S3 @ ( shd_nat @ ( abstra7573781713853518635at_nat @ ( abstra4671362528648695778at_nat @ Eff3 ) @ Rs @ S3 ) ) ) ) ).

% i.mkTree.simps(1)
thf(fact_729_i_OmkTree_Osimps_I1_J,axiom,
    ! [Eff3: product_prod_nat_nat > product_prod_nat_nat > option7922001270233832621at_nat,Rs: stream6724221391990029191at_nat,S3: product_prod_nat_nat] :
      ( ( abstra35009723297898488at_nat @ ( abstra8972947971889770147at_nat @ ( abstra426094224824445696at_nat @ Eff3 ) @ Rs @ S3 ) )
      = ( produc6161850002892822231at_nat @ S3 @ ( shd_Pr4260400998323988397at_nat @ ( abstra4493250544927931593at_nat @ ( abstra426094224824445696at_nat @ Eff3 ) @ Rs @ S3 ) ) ) ) ).

% i.mkTree.simps(1)
thf(fact_730_i_OmkTree_Osimps_I1_J,axiom,
    ! [Eff3: produc1996495991257130529ist_fm > rule > option_fset_rule,Rs: stream8299795917829157543ist_fm,S3: rule] :
      ( ( abstra7970568315424063040ist_fm @ ( abstra2830003028686515011m_rule @ ( abstra8110425618579670566m_rule @ Eff3 ) @ Rs @ S3 ) )
      = ( produc7063755442767259331ist_fm @ S3 @ ( shd_Pr772355297128350925ist_fm @ ( abstra5014957989051229853m_rule @ ( abstra8110425618579670566m_rule @ Eff3 ) @ Rs @ S3 ) ) ) ) ).

% i.mkTree.simps(1)
thf(fact_731_i_OmkTree_Osimps_I1_J,axiom,
    ! [Eff3: rule > produc1996495991257130529ist_fm > option6029102822600979917ist_fm,Rs: stream_rule,S3: produc1996495991257130529ist_fm] :
      ( ( abstra3503897793822247108m_rule @ ( abstra178918799166095863ist_fm @ ( abstra5459341389059251418ist_fm @ Eff3 ) @ Rs @ S3 ) )
      = ( produc491467635432902671m_rule @ S3 @ ( shd_rule @ ( abstra2363873759530810705ist_fm @ ( abstra5459341389059251418ist_fm @ Eff3 ) @ Rs @ S3 ) ) ) ) ).

% i.mkTree.simps(1)
thf(fact_732_RuleSystem_Otrim__in__R,axiom,
    ! [Eff2: fm > fm > fset_fm > $o,Rules: stream_fm,S4: set_fm,S3: fm,Rs: stream_fm] :
      ( ( abstra886934874521163211_fm_fm @ Eff2 @ Rules @ S4 )
     => ( ( member_fm @ S3 @ S4 )
       => ( ( abstra5119165595139683900air_fm @ Rules @ Rs )
         => ( member_fm @ ( shd_fm @ ( abstra5942290089224017835_fm_fm @ Eff2 @ Rs @ S3 ) ) @ ( sset_fm @ Rules ) ) ) ) ) ).

% RuleSystem.trim_in_R
thf(fact_733_RuleSystem_Otrim__in__R,axiom,
    ! [Eff2: fm > rule > fset_rule > $o,Rules: stream_fm,S4: set_rule,S3: rule,Rs: stream_fm] :
      ( ( abstra5167362717900046400m_rule @ Eff2 @ Rules @ S4 )
     => ( ( member_rule @ S3 @ S4 )
       => ( ( abstra5119165595139683900air_fm @ Rules @ Rs )
         => ( member_fm @ ( shd_fm @ ( abstra8496492133263135264m_rule @ Eff2 @ Rs @ S3 ) ) @ ( sset_fm @ Rules ) ) ) ) ) ).

% RuleSystem.trim_in_R
thf(fact_734_RuleSystem_Otrim__in__R,axiom,
    ! [Eff2: rule > fm > fset_fm > $o,Rules: stream_rule,S4: set_fm,S3: fm,Rs: stream_rule] :
      ( ( abstra916514217863227968ule_fm @ Eff2 @ Rules @ S4 )
     => ( ( member_fm @ S3 @ S4 )
       => ( ( abstra1114877437032970289r_rule @ Rules @ Rs )
         => ( member_rule @ ( shd_rule @ ( abstra4245643633226316832ule_fm @ Eff2 @ Rs @ S3 ) ) @ ( sset_rule @ Rules ) ) ) ) ) ).

% RuleSystem.trim_in_R
thf(fact_735_RuleSystem_Otrim__in__R,axiom,
    ! [Eff2: rule > rule > fset_rule > $o,Rules: stream_rule,S4: set_rule,S3: rule,Rs: stream_rule] :
      ( ( abstra2725590884229999157e_rule @ Eff2 @ Rules @ S4 )
     => ( ( member_rule @ S3 @ S4 )
       => ( ( abstra1114877437032970289r_rule @ Rules @ Rs )
         => ( member_rule @ ( shd_rule @ ( abstra7824469560916757525e_rule @ Eff2 @ Rs @ S3 ) ) @ ( sset_rule @ Rules ) ) ) ) ) ).

% RuleSystem.trim_in_R
thf(fact_736_RuleSystem_Otrim__in__R,axiom,
    ! [Eff2: produc1996495991257130529ist_fm > fm > fset_fm > $o,Rules: stream8299795917829157543ist_fm,S4: set_fm,S3: fm,Rs: stream8299795917829157543ist_fm] :
      ( ( abstra2919136520687232136_fm_fm @ Eff2 @ Rules @ S4 )
     => ( ( member_fm @ S3 @ S4 )
       => ( ( abstra7454370078233524789ist_fm @ Rules @ Rs )
         => ( member8102475879199740618ist_fm @ ( shd_Pr772355297128350925ist_fm @ ( abstra5503639309725227688_fm_fm @ Eff2 @ Rs @ S3 ) ) @ ( sset_P3431374123360452149ist_fm @ Rules ) ) ) ) ) ).

% RuleSystem.trim_in_R
thf(fact_737_RuleSystem_Otrim__in__R,axiom,
    ! [Eff2: produc1996495991257130529ist_fm > rule > fset_rule > $o,Rules: stream8299795917829157543ist_fm,S4: set_rule,S3: rule,Rs: stream8299795917829157543ist_fm] :
      ( ( abstra9106359784214577277m_rule @ Eff2 @ Rules @ S4 )
     => ( ( member_rule @ S3 @ S4 )
       => ( ( abstra7454370078233524789ist_fm @ Rules @ Rs )
         => ( member8102475879199740618ist_fm @ ( shd_Pr772355297128350925ist_fm @ ( abstra5014957989051229853m_rule @ Eff2 @ Rs @ S3 ) ) @ ( sset_P3431374123360452149ist_fm @ Rules ) ) ) ) ) ).

% RuleSystem.trim_in_R
thf(fact_738_RuleSystem_Otrim__in__R,axiom,
    ! [Eff2: fm > produc1996495991257130529ist_fm > fset_P661503646757059847ist_fm > $o,Rules: stream_fm,S4: set_Pr7058068377845519745ist_fm,S3: produc1996495991257130529ist_fm,Rs: stream_fm] :
      ( ( abstra7587511110199143014ist_fm @ Eff2 @ Rules @ S4 )
     => ( ( member8102475879199740618ist_fm @ S3 @ S4 )
       => ( ( abstra5119165595139683900air_fm @ Rules @ Rs )
         => ( member_fm @ ( shd_fm @ ( abstra948641862382362758ist_fm @ Eff2 @ Rs @ S3 ) ) @ ( sset_fm @ Rules ) ) ) ) ) ).

% RuleSystem.trim_in_R
thf(fact_739_RuleSystem_Otrim__in__R,axiom,
    ! [Eff2: rule > produc1996495991257130529ist_fm > fset_P661503646757059847ist_fm > $o,Rules: stream_rule,S4: set_Pr7058068377845519745ist_fm,S3: produc1996495991257130529ist_fm,Rs: stream_rule] :
      ( ( abstra6455275554694158129ist_fm @ Eff2 @ Rules @ S4 )
     => ( ( member8102475879199740618ist_fm @ S3 @ S4 )
       => ( ( abstra1114877437032970289r_rule @ Rules @ Rs )
         => ( member_rule @ ( shd_rule @ ( abstra2363873759530810705ist_fm @ Eff2 @ Rs @ S3 ) ) @ ( sset_rule @ Rules ) ) ) ) ) ).

% RuleSystem.trim_in_R
thf(fact_740_RuleSystem_Otrim__in__R,axiom,
    ! [Eff2: produc859450856879609959at_nat > fm > fset_fm > $o,Rules: stream8372878641218411373at_nat,S4: set_fm,S3: fm,Rs: stream8372878641218411373at_nat] :
      ( ( abstra849841822169861838nat_fm @ Eff2 @ Rules @ S4 )
     => ( ( member_fm @ S3 @ S4 )
       => ( ( abstra5618069685180253819at_nat @ Rules @ Rs )
         => ( member8206827879206165904at_nat @ ( shd_Pr8412153233960533267at_nat @ ( abstra2782398720704632046nat_fm @ Eff2 @ Rs @ S3 ) ) @ ( sset_P9164763103296375419at_nat @ Rules ) ) ) ) ) ).

% RuleSystem.trim_in_R
thf(fact_741_RuleSystem_Otrim__in__R,axiom,
    ! [Eff2: produc859450856879609959at_nat > rule > fset_rule > $o,Rules: stream8372878641218411373at_nat,S4: set_rule,S3: rule,Rs: stream8372878641218411373at_nat] :
      ( ( abstra2579607477568694211t_rule @ Eff2 @ Rules @ S4 )
     => ( ( member_rule @ S3 @ S4 )
       => ( ( abstra5618069685180253819at_nat @ Rules @ Rs )
         => ( member8206827879206165904at_nat @ ( shd_Pr8412153233960533267at_nat @ ( abstra361488207848709603t_rule @ Eff2 @ Rs @ S3 ) ) @ ( sset_P9164763103296375419at_nat @ Rules ) ) ) ) ) ).

% RuleSystem.trim_in_R
thf(fact_742_RuleSystem_OminWait__ex,axiom,
    ! [Eff2: rule > fm > fset_fm > $o,Rules: stream_rule,S4: set_fm,S3: fm,Rs: stream_rule] :
      ( ( abstra916514217863227968ule_fm @ Eff2 @ Rules @ S4 )
     => ( ( member_fm @ S3 @ S4 )
       => ( ( abstra1114877437032970289r_rule @ Rules @ Rs )
         => ? [N2: nat] : ( abstra7038331712872434097ule_fm @ Eff2 @ ( shd_rule @ ( sdrop_rule @ N2 @ Rs ) ) @ S3 ) ) ) ) ).

% RuleSystem.minWait_ex
thf(fact_743_RuleSystem_OminWait__ex,axiom,
    ! [Eff2: rule > rule > fset_rule > $o,Rules: stream_rule,S4: set_rule,S3: rule,Rs: stream_rule] :
      ( ( abstra2725590884229999157e_rule @ Eff2 @ Rules @ S4 )
     => ( ( member_rule @ S3 @ S4 )
       => ( ( abstra1114877437032970289r_rule @ Rules @ Rs )
         => ? [N2: nat] : ( abstra3154316310710804774e_rule @ Eff2 @ ( shd_rule @ ( sdrop_rule @ N2 @ Rs ) ) @ S3 ) ) ) ) ).

% RuleSystem.minWait_ex
thf(fact_744_RuleSystem_OminWait__ex,axiom,
    ! [Eff2: produc780299800278082923e_rule > fm > fset_fm > $o,Rules: stream3588992565182678257e_rule,S4: set_fm,S3: fm,Rs: stream3588992565182678257e_rule] :
      ( ( abstra6843014240698492626ule_fm @ Eff2 @ Rules @ S4 )
     => ( ( member_fm @ S3 @ S4 )
       => ( ( abstra4837122928373946751e_rule @ Rules @ Rs )
         => ? [N2: nat] : ( abstra1405456541278283553ule_fm @ Eff2 @ ( shd_Pr4573454438745646103e_rule @ ( sdrop_2199339672054592340e_rule @ N2 @ Rs ) ) @ S3 ) ) ) ) ).

% RuleSystem.minWait_ex
thf(fact_745_RuleSystem_OminWait__ex,axiom,
    ! [Eff2: produc780299800278082923e_rule > rule > fset_rule > $o,Rules: stream3588992565182678257e_rule,S4: set_rule,S3: rule,Rs: stream3588992565182678257e_rule] :
      ( ( abstra1742398642700549575e_rule @ Eff2 @ Rules @ S4 )
     => ( ( member_rule @ S3 @ S4 )
       => ( ( abstra4837122928373946751e_rule @ Rules @ Rs )
         => ? [N2: nat] : ( abstra5553215051456775830e_rule @ Eff2 @ ( shd_Pr4573454438745646103e_rule @ ( sdrop_2199339672054592340e_rule @ N2 @ Rs ) ) @ S3 ) ) ) ) ).

% RuleSystem.minWait_ex
thf(fact_746_RuleSystem_OminWait__ex,axiom,
    ! [Eff2: produc1996495991257130529ist_fm > fm > fset_fm > $o,Rules: stream8299795917829157543ist_fm,S4: set_fm,S3: fm,Rs: stream8299795917829157543ist_fm] :
      ( ( abstra2919136520687232136_fm_fm @ Eff2 @ Rules @ S4 )
     => ( ( member_fm @ S3 @ S4 )
       => ( ( abstra7454370078233524789ist_fm @ Rules @ Rs )
         => ? [N2: nat] : ( abstra8628512393757682903_fm_fm @ Eff2 @ ( shd_Pr772355297128350925ist_fm @ ( sdrop_4442373711808556042ist_fm @ N2 @ Rs ) ) @ S3 ) ) ) ) ).

% RuleSystem.minWait_ex
thf(fact_747_RuleSystem_OminWait__ex,axiom,
    ! [Eff2: produc1996495991257130529ist_fm > rule > fset_rule > $o,Rules: stream8299795917829157543ist_fm,S4: set_rule,S3: rule,Rs: stream8299795917829157543ist_fm] :
      ( ( abstra9106359784214577277m_rule @ Eff2 @ Rules @ S4 )
     => ( ( member_rule @ S3 @ S4 )
       => ( ( abstra7454370078233524789ist_fm @ Rules @ Rs )
         => ? [N2: nat] : ( abstra5706625008861892940m_rule @ Eff2 @ ( shd_Pr772355297128350925ist_fm @ ( sdrop_4442373711808556042ist_fm @ N2 @ Rs ) ) @ S3 ) ) ) ) ).

% RuleSystem.minWait_ex
thf(fact_748_RuleSystem_OminWait__ex,axiom,
    ! [Eff2: rule > produc1996495991257130529ist_fm > fset_P661503646757059847ist_fm > $o,Rules: stream_rule,S4: set_Pr7058068377845519745ist_fm,S3: produc1996495991257130529ist_fm,Rs: stream_rule] :
      ( ( abstra6455275554694158129ist_fm @ Eff2 @ Rules @ S4 )
     => ( ( member8102475879199740618ist_fm @ S3 @ S4 )
       => ( ( abstra1114877437032970289r_rule @ Rules @ Rs )
         => ? [N2: nat] : ( abstra3055540779341473792ist_fm @ Eff2 @ ( shd_rule @ ( sdrop_rule @ N2 @ Rs ) ) @ S3 ) ) ) ) ).

% RuleSystem.minWait_ex
thf(fact_749_RuleSystem_OminWait__ex,axiom,
    ! [Eff2: produc780299800278082923e_rule > produc1996495991257130529ist_fm > fset_P661503646757059847ist_fm > $o,Rules: stream3588992565182678257e_rule,S4: set_Pr7058068377845519745ist_fm,S3: produc1996495991257130529ist_fm,Rs: stream3588992565182678257e_rule] :
      ( ( abstra2302509236375889695ist_fm @ Eff2 @ Rules @ S4 )
     => ( ( member8102475879199740618ist_fm @ S3 @ S4 )
       => ( ( abstra4837122928373946751e_rule @ Rules @ Rs )
         => ? [N2: nat] : ( abstra1049073205711939216ist_fm @ Eff2 @ ( shd_Pr4573454438745646103e_rule @ ( sdrop_2199339672054592340e_rule @ N2 @ Rs ) ) @ S3 ) ) ) ) ).

% RuleSystem.minWait_ex
thf(fact_750_RuleSystem_OminWait__ex,axiom,
    ! [Eff2: produc164195504107695125m_rule > fm > fset_fm > $o,Rules: stream727092118206550309m_rule,S4: set_fm,S3: fm,Rs: stream727092118206550309m_rule] :
      ( ( abstra3764363818050485264ule_fm @ Eff2 @ Rules @ S4 )
     => ( ( member_fm @ S3 @ S4 )
       => ( ( abstra4997497885011821313m_rule @ Rules @ Rs )
         => ? [N2: nat] : ( abstra7152516257876272129ule_fm @ Eff2 @ ( shd_Pr7235097944458474089m_rule @ ( sdrop_7224736112439592940m_rule @ N2 @ Rs ) ) @ S3 ) ) ) ) ).

% RuleSystem.minWait_ex
thf(fact_751_RuleSystem_OminWait__ex,axiom,
    ! [Eff2: produc164195504107695125m_rule > rule > fset_rule > $o,Rules: stream727092118206550309m_rule,S4: set_rule,S3: rule,Rs: stream727092118206550309m_rule] :
      ( ( abstra3778134268586021893e_rule @ Eff2 @ Rules @ S4 )
     => ( ( member_rule @ S3 @ S4 )
       => ( ( abstra4997497885011821313m_rule @ Rules @ Rs )
         => ? [N2: nat] : ( abstra2524698237922071414e_rule @ Eff2 @ ( shd_Pr7235097944458474089m_rule @ ( sdrop_7224736112439592940m_rule @ N2 @ Rs ) ) @ S3 ) ) ) ) ).

% RuleSystem.minWait_ex
thf(fact_752_RuleSystem_Opos__least,axiom,
    ! [Eff2: rule > produc1996495991257130529ist_fm > fset_P661503646757059847ist_fm > $o,Rules: stream_rule,S4: set_Pr7058068377845519745ist_fm,N: nat,Rs: stream8299795917829157543ist_fm,R2: produc1996495991257130529ist_fm] :
      ( ( abstra6455275554694158129ist_fm @ Eff2 @ Rules @ S4 )
     => ( ( ( shd_Pr772355297128350925ist_fm @ ( sdrop_4442373711808556042ist_fm @ N @ Rs ) )
          = R2 )
       => ( ord_less_eq_nat @ ( abstra3513618393749879589ist_fm @ Rs @ R2 ) @ N ) ) ) ).

% RuleSystem.pos_least
thf(fact_753_RuleSystem_Opos__least,axiom,
    ! [Eff2: rule > produc1996495991257130529ist_fm > fset_P661503646757059847ist_fm > $o,Rules: stream_rule,S4: set_Pr7058068377845519745ist_fm,N: nat,Rs: stream3588992565182678257e_rule,R2: produc780299800278082923e_rule] :
      ( ( abstra6455275554694158129ist_fm @ Eff2 @ Rules @ S4 )
     => ( ( ( shd_Pr4573454438745646103e_rule @ ( sdrop_2199339672054592340e_rule @ N @ Rs ) )
          = R2 )
       => ( ord_less_eq_nat @ ( abstra883788664900704879e_rule @ Rs @ R2 ) @ N ) ) ) ).

% RuleSystem.pos_least
thf(fact_754_RuleSystem_Opos__least,axiom,
    ! [Eff2: rule > produc1996495991257130529ist_fm > fset_P661503646757059847ist_fm > $o,Rules: stream_rule,S4: set_Pr7058068377845519745ist_fm,N: nat,Rs: stream1878305877988316353m_rule,R2: produc1660549859252256699m_rule] :
      ( ( abstra6455275554694158129ist_fm @ Eff2 @ Rules @ S4 )
     => ( ( ( shd_Pr3564872132220007m_rule @ ( sdrop_1764812713384251940m_rule @ N @ Rs ) )
          = R2 )
       => ( ord_less_eq_nat @ ( abstra7431954940529686207m_rule @ Rs @ R2 ) @ N ) ) ) ).

% RuleSystem.pos_least
thf(fact_755_RuleSystem_Opos__least,axiom,
    ! [Eff2: rule > produc1996495991257130529ist_fm > fset_P661503646757059847ist_fm > $o,Rules: stream_rule,S4: set_Pr7058068377845519745ist_fm,N: nat,Rs: stream2312013117288958913e_rule,R2: produc6288491646048892603e_rule] :
      ( ( abstra6455275554694158129ist_fm @ Eff2 @ Rules @ S4 )
     => ( ( ( shd_Pr4631506658928855911e_rule @ ( sdrop_6392754500180887844e_rule @ N @ Rs ) )
          = R2 )
       => ( ord_less_eq_nat @ ( abstra2836524690471546303e_rule @ Rs @ R2 ) @ N ) ) ) ).

% RuleSystem.pos_least
thf(fact_756_RuleSystem_Opos__least,axiom,
    ! [Eff2: rule > produc1996495991257130529ist_fm > fset_P661503646757059847ist_fm > $o,Rules: stream_rule,S4: set_Pr7058068377845519745ist_fm,N: nat,Rs: stream6494289010434245521m_rule,R2: produc3793571666044074251m_rule] :
      ( ( abstra6455275554694158129ist_fm @ Eff2 @ Rules @ S4 )
     => ( ( ( shd_Pr2340755956080993207m_rule @ ( sdrop_7373388980841566196m_rule @ N @ Rs ) )
          = R2 )
       => ( ord_less_eq_nat @ ( abstra3193765552906301967m_rule @ Rs @ R2 ) @ N ) ) ) ).

% RuleSystem.pos_least
thf(fact_757_RuleSystem_Opos__least,axiom,
    ! [Eff2: rule > produc1996495991257130529ist_fm > fset_P661503646757059847ist_fm > $o,Rules: stream_rule,S4: set_Pr7058068377845519745ist_fm,N: nat,Rs: stream727092118206550309m_rule,R2: produc164195504107695125m_rule] :
      ( ( abstra6455275554694158129ist_fm @ Eff2 @ Rules @ S4 )
     => ( ( ( shd_Pr7235097944458474089m_rule @ ( sdrop_7224736112439592940m_rule @ N @ Rs ) )
          = R2 )
       => ( ord_less_eq_nat @ ( abstra1882453022147397137m_rule @ Rs @ R2 ) @ N ) ) ) ).

% RuleSystem.pos_least
thf(fact_758_RuleSystem_Opos__least,axiom,
    ! [Eff2: rule > produc1996495991257130529ist_fm > fset_P661503646757059847ist_fm > $o,Rules: stream_rule,S4: set_Pr7058068377845519745ist_fm,N: nat,Rs: stream_rule,R2: rule] :
      ( ( abstra6455275554694158129ist_fm @ Eff2 @ Rules @ S4 )
     => ( ( ( shd_rule @ ( sdrop_rule @ N @ Rs ) )
          = R2 )
       => ( ord_less_eq_nat @ ( abstract_pos_rule @ Rs @ R2 ) @ N ) ) ) ).

% RuleSystem.pos_least
thf(fact_759_RuleSystem_Otrim__enabled,axiom,
    ! [Eff2: produc164195504107695125m_rule > fm > fset_fm > $o,Rules: stream727092118206550309m_rule,S4: set_fm,S3: fm,Rs: stream727092118206550309m_rule] :
      ( ( abstra3764363818050485264ule_fm @ Eff2 @ Rules @ S4 )
     => ( ( member_fm @ S3 @ S4 )
       => ( ( abstra4997497885011821313m_rule @ Rules @ Rs )
         => ( abstra7152516257876272129ule_fm @ Eff2 @ ( shd_Pr7235097944458474089m_rule @ ( abstra2315704231547878896ule_fm @ Eff2 @ Rs @ S3 ) ) @ S3 ) ) ) ) ).

% RuleSystem.trim_enabled
thf(fact_760_RuleSystem_Otrim__enabled,axiom,
    ! [Eff2: produc164195504107695125m_rule > rule > fset_rule > $o,Rules: stream727092118206550309m_rule,S4: set_rule,S3: rule,Rs: stream727092118206550309m_rule] :
      ( ( abstra3778134268586021893e_rule @ Eff2 @ Rules @ S4 )
     => ( ( member_rule @ S3 @ S4 )
       => ( ( abstra4997497885011821313m_rule @ Rules @ Rs )
         => ( abstra2524698237922071414e_rule @ Eff2 @ ( shd_Pr7235097944458474089m_rule @ ( abstra5289413328258868709e_rule @ Eff2 @ Rs @ S3 ) ) @ S3 ) ) ) ) ).

% RuleSystem.trim_enabled
thf(fact_761_RuleSystem_Otrim__enabled,axiom,
    ! [Eff2: produc164195504107695125m_rule > produc1996495991257130529ist_fm > fset_P661503646757059847ist_fm > $o,Rules: stream727092118206550309m_rule,S4: set_Pr7058068377845519745ist_fm,S3: produc1996495991257130529ist_fm,Rs: stream727092118206550309m_rule] :
      ( ( abstra3408323020314982753ist_fm @ Eff2 @ Rules @ S4 )
     => ( ( member8102475879199740618ist_fm @ S3 @ S4 )
       => ( ( abstra4997497885011821313m_rule @ Rules @ Rs )
         => ( abstra7200325324992684464ist_fm @ Eff2 @ ( shd_Pr7235097944458474089m_rule @ ( abstra6201285654312316801ist_fm @ Eff2 @ Rs @ S3 ) ) @ S3 ) ) ) ) ).

% RuleSystem.trim_enabled
thf(fact_762_RuleSystem_Otrim__enabled,axiom,
    ! [Eff2: produc164195504107695125m_rule > produc859450856879609959at_nat > fset_P3619101297660450253at_nat > $o,Rules: stream727092118206550309m_rule,S4: set_Pr8693737435421807431at_nat,S3: produc859450856879609959at_nat,Rs: stream727092118206550309m_rule] :
      ( ( abstra8123243527896255399at_nat @ Eff2 @ Rules @ S4 )
     => ( ( member8206827879206165904at_nat @ S3 @ S4 )
       => ( ( abstra4997497885011821313m_rule @ Rules @ Rs )
         => ( abstra1800717694132746102at_nat @ Eff2 @ ( shd_Pr7235097944458474089m_rule @ ( abstra276577985667043783at_nat @ Eff2 @ Rs @ S3 ) ) @ S3 ) ) ) ) ).

% RuleSystem.trim_enabled
thf(fact_763_RuleSystem_Otrim__enabled,axiom,
    ! [Eff2: rule > fm > fset_fm > $o,Rules: stream_rule,S4: set_fm,S3: fm,Rs: stream_rule] :
      ( ( abstra916514217863227968ule_fm @ Eff2 @ Rules @ S4 )
     => ( ( member_fm @ S3 @ S4 )
       => ( ( abstra1114877437032970289r_rule @ Rules @ Rs )
         => ( abstra7038331712872434097ule_fm @ Eff2 @ ( shd_rule @ ( abstra4245643633226316832ule_fm @ Eff2 @ Rs @ S3 ) ) @ S3 ) ) ) ) ).

% RuleSystem.trim_enabled
thf(fact_764_RuleSystem_Otrim__enabled,axiom,
    ! [Eff2: rule > rule > fset_rule > $o,Rules: stream_rule,S4: set_rule,S3: rule,Rs: stream_rule] :
      ( ( abstra2725590884229999157e_rule @ Eff2 @ Rules @ S4 )
     => ( ( member_rule @ S3 @ S4 )
       => ( ( abstra1114877437032970289r_rule @ Rules @ Rs )
         => ( abstra3154316310710804774e_rule @ Eff2 @ ( shd_rule @ ( abstra7824469560916757525e_rule @ Eff2 @ Rs @ S3 ) ) @ S3 ) ) ) ) ).

% RuleSystem.trim_enabled
thf(fact_765_RuleSystem_Otrim__enabled,axiom,
    ! [Eff2: rule > produc859450856879609959at_nat > fset_P3619101297660450253at_nat > $o,Rules: stream_rule,S4: set_Pr8693737435421807431at_nat,S3: produc859450856879609959at_nat,Rs: stream_rule] :
      ( ( abstra5655617845878133111at_nat @ Eff2 @ Rules @ S4 )
     => ( ( member8206827879206165904at_nat @ S3 @ S4 )
       => ( ( abstra1114877437032970289r_rule @ Rules @ Rs )
         => ( abstra248173851283269062at_nat @ Eff2 @ ( shd_rule @ ( abstra3437498576158148503at_nat @ Eff2 @ Rs @ S3 ) ) @ S3 ) ) ) ) ).

% RuleSystem.trim_enabled
thf(fact_766_RuleSystem_Otrim__enabled,axiom,
    ! [Eff2: rule > produc1996495991257130529ist_fm > fset_P661503646757059847ist_fm > $o,Rules: stream_rule,S4: set_Pr7058068377845519745ist_fm,S3: produc1996495991257130529ist_fm,Rs: stream_rule] :
      ( ( abstra6455275554694158129ist_fm @ Eff2 @ Rules @ S4 )
     => ( ( member8102475879199740618ist_fm @ S3 @ S4 )
       => ( ( abstra1114877437032970289r_rule @ Rules @ Rs )
         => ( abstra3055540779341473792ist_fm @ Eff2 @ ( shd_rule @ ( abstra2363873759530810705ist_fm @ Eff2 @ Rs @ S3 ) ) @ S3 ) ) ) ) ).

% RuleSystem.trim_enabled
thf(fact_767_RuleSystem_Otrim__alt,axiom,
    ! [Eff2: rule > fm > fset_fm > $o,Rules: stream_rule,S4: set_fm,S3: fm,Rs: stream_rule] :
      ( ( abstra916514217863227968ule_fm @ Eff2 @ Rules @ S4 )
     => ( ( member_fm @ S3 @ S4 )
       => ( ( abstra1114877437032970289r_rule @ Rules @ Rs )
         => ( ( abstra4245643633226316832ule_fm @ Eff2 @ Rs @ S3 )
            = ( sdrop_rule @ ( abstra2294468009765627845ule_fm @ Eff2 @ Rs @ S3 ) @ Rs ) ) ) ) ) ).

% RuleSystem.trim_alt
thf(fact_768_RuleSystem_Otrim__alt,axiom,
    ! [Eff2: rule > rule > fset_rule > $o,Rules: stream_rule,S4: set_rule,S3: rule,Rs: stream_rule] :
      ( ( abstra2725590884229999157e_rule @ Eff2 @ Rules @ S4 )
     => ( ( member_rule @ S3 @ S4 )
       => ( ( abstra1114877437032970289r_rule @ Rules @ Rs )
         => ( ( abstra7824469560916757525e_rule @ Eff2 @ Rs @ S3 )
            = ( sdrop_rule @ ( abstra5682536783049438522e_rule @ Eff2 @ Rs @ S3 ) @ Rs ) ) ) ) ) ).

% RuleSystem.trim_alt
thf(fact_769_RuleSystem_Otrim__alt,axiom,
    ! [Eff2: produc780299800278082923e_rule > fm > fset_fm > $o,Rules: stream3588992565182678257e_rule,S4: set_fm,S3: fm,Rs: stream3588992565182678257e_rule] :
      ( ( abstra6843014240698492626ule_fm @ Eff2 @ Rules @ S4 )
     => ( ( member_fm @ S3 @ S4 )
       => ( ( abstra4837122928373946751e_rule @ Rules @ Rs )
         => ( ( abstra7283367714565168370ule_fm @ Eff2 @ Rs @ S3 )
            = ( sdrop_2199339672054592340e_rule @ ( abstra1933929482952324109ule_fm @ Eff2 @ Rs @ S3 ) @ Rs ) ) ) ) ) ).

% RuleSystem.trim_alt
thf(fact_770_RuleSystem_Otrim__alt,axiom,
    ! [Eff2: produc780299800278082923e_rule > rule > fset_rule > $o,Rules: stream3588992565182678257e_rule,S4: set_rule,S3: rule,Rs: stream3588992565182678257e_rule] :
      ( ( abstra1742398642700549575e_rule @ Eff2 @ Rules @ S4 )
     => ( ( member_rule @ S3 @ S4 )
       => ( ( abstra4837122928373946751e_rule @ Rules @ Rs )
         => ( ( abstra5027552996074955751e_rule @ Eff2 @ Rs @ S3 )
            = ( sdrop_2199339672054592340e_rule @ ( abstra5810752615369301378e_rule @ Eff2 @ Rs @ S3 ) @ Rs ) ) ) ) ) ).

% RuleSystem.trim_alt
thf(fact_771_RuleSystem_Otrim__alt,axiom,
    ! [Eff2: produc1996495991257130529ist_fm > fm > fset_fm > $o,Rules: stream8299795917829157543ist_fm,S4: set_fm,S3: fm,Rs: stream8299795917829157543ist_fm] :
      ( ( abstra2919136520687232136_fm_fm @ Eff2 @ Rules @ S4 )
     => ( ( member_fm @ S3 @ S4 )
       => ( ( abstra7454370078233524789ist_fm @ Rules @ Rs )
         => ( ( abstra5503639309725227688_fm_fm @ Eff2 @ Rs @ S3 )
            = ( sdrop_4442373711808556042ist_fm @ ( abstra8883821634369821123_fm_fm @ Eff2 @ Rs @ S3 ) @ Rs ) ) ) ) ) ).

% RuleSystem.trim_alt
thf(fact_772_RuleSystem_Otrim__alt,axiom,
    ! [Eff2: produc1996495991257130529ist_fm > rule > fset_rule > $o,Rules: stream8299795917829157543ist_fm,S4: set_rule,S3: rule,Rs: stream8299795917829157543ist_fm] :
      ( ( abstra9106359784214577277m_rule @ Eff2 @ Rules @ S4 )
     => ( ( member_rule @ S3 @ S4 )
       => ( ( abstra7454370078233524789ist_fm @ Rules @ Rs )
         => ( ( abstra5014957989051229853m_rule @ Eff2 @ Rs @ S3 )
            = ( sdrop_4442373711808556042ist_fm @ ( abstra3389404019813435448m_rule @ Eff2 @ Rs @ S3 ) @ Rs ) ) ) ) ) ).

% RuleSystem.trim_alt
thf(fact_773_RuleSystem_Otrim__alt,axiom,
    ! [Eff2: rule > produc1996495991257130529ist_fm > fset_P661503646757059847ist_fm > $o,Rules: stream_rule,S4: set_Pr7058068377845519745ist_fm,S3: produc1996495991257130529ist_fm,Rs: stream_rule] :
      ( ( abstra6455275554694158129ist_fm @ Eff2 @ Rules @ S4 )
     => ( ( member8102475879199740618ist_fm @ S3 @ S4 )
       => ( ( abstra1114877437032970289r_rule @ Rules @ Rs )
         => ( ( abstra2363873759530810705ist_fm @ Eff2 @ Rs @ S3 )
            = ( sdrop_rule @ ( abstra738319790293016300ist_fm @ Eff2 @ Rs @ S3 ) @ Rs ) ) ) ) ) ).

% RuleSystem.trim_alt
thf(fact_774_RuleSystem_Otrim__alt,axiom,
    ! [Eff2: produc164195504107695125m_rule > fm > fset_fm > $o,Rules: stream727092118206550309m_rule,S4: set_fm,S3: fm,Rs: stream727092118206550309m_rule] :
      ( ( abstra3764363818050485264ule_fm @ Eff2 @ Rules @ S4 )
     => ( ( member_fm @ S3 @ S4 )
       => ( ( abstra4997497885011821313m_rule @ Rules @ Rs )
         => ( ( abstra2315704231547878896ule_fm @ Eff2 @ Rs @ S3 )
            = ( sdrop_7224736112439592940m_rule @ ( abstra6093700014593756181ule_fm @ Eff2 @ Rs @ S3 ) @ Rs ) ) ) ) ) ).

% RuleSystem.trim_alt
thf(fact_775_RuleSystem_Otrim__alt,axiom,
    ! [Eff2: produc164195504107695125m_rule > rule > fset_rule > $o,Rules: stream727092118206550309m_rule,S4: set_rule,S3: rule,Rs: stream727092118206550309m_rule] :
      ( ( abstra3778134268586021893e_rule @ Eff2 @ Rules @ S4 )
     => ( ( member_rule @ S3 @ S4 )
       => ( ( abstra4997497885011821313m_rule @ Rules @ Rs )
         => ( ( abstra5289413328258868709e_rule @ Eff2 @ Rs @ S3 )
            = ( sdrop_7224736112439592940m_rule @ ( abstra6412047235177443722e_rule @ Eff2 @ Rs @ S3 ) @ Rs ) ) ) ) ) ).

% RuleSystem.trim_alt
thf(fact_776_RuleSystem_Otrim__alt,axiom,
    ! [Eff2: produc780299800278082923e_rule > produc1996495991257130529ist_fm > fset_P661503646757059847ist_fm > $o,Rules: stream3588992565182678257e_rule,S4: set_Pr7058068377845519745ist_fm,S3: produc1996495991257130529ist_fm,Rs: stream3588992565182678257e_rule] :
      ( ( abstra2302509236375889695ist_fm @ Eff2 @ Rules @ S4 )
     => ( ( member8102475879199740618ist_fm @ S3 @ S4 )
       => ( ( abstra4837122928373946751e_rule @ Rules @ Rs )
         => ( ( abstra3813788296048736511ist_fm @ Eff2 @ Rs @ S3 )
            = ( sdrop_2199339672054592340e_rule @ ( abstra4936422202967311524ist_fm @ Eff2 @ Rs @ S3 ) @ Rs ) ) ) ) ) ).

% RuleSystem.trim_alt
thf(fact_777_RuleSystem__axioms,axiom,
    ( abstra6455275554694158129ist_fm
    @ ^ [R: rule,S: produc1996495991257130529ist_fm,Ss: fset_P661503646757059847ist_fm] :
        ( ( eff @ R @ S )
        = ( some_f2162114139285431020ist_fm @ Ss ) )
    @ rules
    @ top_to728902761071950545ist_fm ) ).

% RuleSystem_axioms
thf(fact_778_i_Otrim__def,axiom,
    ! [Eff3: rule > produc1996495991257130529ist_fm > option6029102822600979917ist_fm,Rs: stream_rule,S3: produc1996495991257130529ist_fm] :
      ( ( abstra2363873759530810705ist_fm @ ( abstra5459341389059251418ist_fm @ Eff3 ) @ Rs @ S3 )
      = ( sdrop_while_rule
        @ ^ [R: rule] :
            ~ ( abstra3055540779341473792ist_fm @ ( abstra5459341389059251418ist_fm @ Eff3 ) @ R @ S3 )
        @ Rs ) ) ).

% i.trim_def
thf(fact_779_RuleSystem_Opos__def,axiom,
    ! [Eff2: rule > produc1996495991257130529ist_fm > fset_P661503646757059847ist_fm > $o,Rules: stream_rule,S4: set_Pr7058068377845519745ist_fm,Rs: stream8299795917829157543ist_fm,R2: produc1996495991257130529ist_fm] :
      ( ( abstra6455275554694158129ist_fm @ Eff2 @ Rules @ S4 )
     => ( ( abstra3513618393749879589ist_fm @ Rs @ R2 )
        = ( ord_Least_nat
          @ ^ [N3: nat] :
              ( ( shd_Pr772355297128350925ist_fm @ ( sdrop_4442373711808556042ist_fm @ N3 @ Rs ) )
              = R2 ) ) ) ) ).

% RuleSystem.pos_def
thf(fact_780_RuleSystem_Opos__def,axiom,
    ! [Eff2: rule > produc1996495991257130529ist_fm > fset_P661503646757059847ist_fm > $o,Rules: stream_rule,S4: set_Pr7058068377845519745ist_fm,Rs: stream3588992565182678257e_rule,R2: produc780299800278082923e_rule] :
      ( ( abstra6455275554694158129ist_fm @ Eff2 @ Rules @ S4 )
     => ( ( abstra883788664900704879e_rule @ Rs @ R2 )
        = ( ord_Least_nat
          @ ^ [N3: nat] :
              ( ( shd_Pr4573454438745646103e_rule @ ( sdrop_2199339672054592340e_rule @ N3 @ Rs ) )
              = R2 ) ) ) ) ).

% RuleSystem.pos_def
thf(fact_781_RuleSystem_Opos__def,axiom,
    ! [Eff2: rule > produc1996495991257130529ist_fm > fset_P661503646757059847ist_fm > $o,Rules: stream_rule,S4: set_Pr7058068377845519745ist_fm,Rs: stream1878305877988316353m_rule,R2: produc1660549859252256699m_rule] :
      ( ( abstra6455275554694158129ist_fm @ Eff2 @ Rules @ S4 )
     => ( ( abstra7431954940529686207m_rule @ Rs @ R2 )
        = ( ord_Least_nat
          @ ^ [N3: nat] :
              ( ( shd_Pr3564872132220007m_rule @ ( sdrop_1764812713384251940m_rule @ N3 @ Rs ) )
              = R2 ) ) ) ) ).

% RuleSystem.pos_def
thf(fact_782_RuleSystem_Opos__def,axiom,
    ! [Eff2: rule > produc1996495991257130529ist_fm > fset_P661503646757059847ist_fm > $o,Rules: stream_rule,S4: set_Pr7058068377845519745ist_fm,Rs: stream2312013117288958913e_rule,R2: produc6288491646048892603e_rule] :
      ( ( abstra6455275554694158129ist_fm @ Eff2 @ Rules @ S4 )
     => ( ( abstra2836524690471546303e_rule @ Rs @ R2 )
        = ( ord_Least_nat
          @ ^ [N3: nat] :
              ( ( shd_Pr4631506658928855911e_rule @ ( sdrop_6392754500180887844e_rule @ N3 @ Rs ) )
              = R2 ) ) ) ) ).

% RuleSystem.pos_def
thf(fact_783_RuleSystem_Opos__def,axiom,
    ! [Eff2: rule > produc1996495991257130529ist_fm > fset_P661503646757059847ist_fm > $o,Rules: stream_rule,S4: set_Pr7058068377845519745ist_fm,Rs: stream6494289010434245521m_rule,R2: produc3793571666044074251m_rule] :
      ( ( abstra6455275554694158129ist_fm @ Eff2 @ Rules @ S4 )
     => ( ( abstra3193765552906301967m_rule @ Rs @ R2 )
        = ( ord_Least_nat
          @ ^ [N3: nat] :
              ( ( shd_Pr2340755956080993207m_rule @ ( sdrop_7373388980841566196m_rule @ N3 @ Rs ) )
              = R2 ) ) ) ) ).

% RuleSystem.pos_def
thf(fact_784_RuleSystem_Opos__def,axiom,
    ! [Eff2: rule > produc1996495991257130529ist_fm > fset_P661503646757059847ist_fm > $o,Rules: stream_rule,S4: set_Pr7058068377845519745ist_fm,Rs: stream727092118206550309m_rule,R2: produc164195504107695125m_rule] :
      ( ( abstra6455275554694158129ist_fm @ Eff2 @ Rules @ S4 )
     => ( ( abstra1882453022147397137m_rule @ Rs @ R2 )
        = ( ord_Least_nat
          @ ^ [N3: nat] :
              ( ( shd_Pr7235097944458474089m_rule @ ( sdrop_7224736112439592940m_rule @ N3 @ Rs ) )
              = R2 ) ) ) ) ).

% RuleSystem.pos_def
thf(fact_785_RuleSystem_Opos__def,axiom,
    ! [Eff2: rule > produc1996495991257130529ist_fm > fset_P661503646757059847ist_fm > $o,Rules: stream_rule,S4: set_Pr7058068377845519745ist_fm,Rs: stream_rule,R2: rule] :
      ( ( abstra6455275554694158129ist_fm @ Eff2 @ Rules @ S4 )
     => ( ( abstract_pos_rule @ Rs @ R2 )
        = ( ord_Least_nat
          @ ^ [N3: nat] :
              ( ( shd_rule @ ( sdrop_rule @ N3 @ Rs ) )
              = R2 ) ) ) ) ).

% RuleSystem.pos_def
thf(fact_786_RuleSystem_OminWait__least,axiom,
    ! [Eff2: rule > produc1996495991257130529ist_fm > fset_P661503646757059847ist_fm > $o,Rules: stream_rule,S4: set_Pr7058068377845519745ist_fm,N: nat,Rs: stream_rule,S3: produc1996495991257130529ist_fm] :
      ( ( abstra6455275554694158129ist_fm @ Eff2 @ Rules @ S4 )
     => ( ( abstra3055540779341473792ist_fm @ Eff2 @ ( shd_rule @ ( sdrop_rule @ N @ Rs ) ) @ S3 )
       => ( ord_less_eq_nat @ ( abstra738319790293016300ist_fm @ Eff2 @ Rs @ S3 ) @ N ) ) ) ).

% RuleSystem.minWait_least
thf(fact_787_RuleSystem_Opos,axiom,
    ! [Eff2: rule > produc1996495991257130529ist_fm > fset_P661503646757059847ist_fm > $o,Rules: stream_rule,S4: set_Pr7058068377845519745ist_fm,Rs: stream_rule,R2: rule] :
      ( ( abstra6455275554694158129ist_fm @ Eff2 @ Rules @ S4 )
     => ( ( abstra1114877437032970289r_rule @ Rules @ Rs )
       => ( ( member_rule @ R2 @ ( sset_rule @ Rules ) )
         => ( ( shd_rule @ ( sdrop_rule @ ( abstract_pos_rule @ Rs @ R2 ) @ Rs ) )
            = R2 ) ) ) ) ).

% RuleSystem.pos
thf(fact_788_wf__ipath__epath,axiom,
    ! [T2: abstra4572470701423150929m_rule,Steps: stream727092118206550309m_rule] :
      ( ( abstra4476735159789068068ist_fm
        @ ^ [R: rule,S: produc1996495991257130529ist_fm,Ss: fset_P661503646757059847ist_fm] :
            ( ( eff @ R @ S )
            = ( some_f2162114139285431020ist_fm @ Ss ) )
        @ rules
        @ T2 )
     => ( ( abstra2285467793620430502m_rule @ T2 @ Steps )
       => ( abstra5415202973855822231ist_fm
          @ ^ [R: rule,S: produc1996495991257130529ist_fm,Ss: fset_P661503646757059847ist_fm] :
              ( ( eff @ R @ S )
              = ( some_f2162114139285431020ist_fm @ Ss ) )
          @ rules
          @ Steps ) ) ) ).

% wf_ipath_epath
thf(fact_789_RuleSystem_Oipath__mkTree__sdrop,axiom,
    ! [Eff2: rule > fm > fset_fm > $o,Rules: stream_rule,S4: set_fm,S3: fm,Rs: stream_rule,Steps: stream8579757915198507644m_rule,M: nat] :
      ( ( abstra916514217863227968ule_fm @ Eff2 @ Rules @ S4 )
     => ( ( member_fm @ S3 @ S4 )
       => ( ( abstra1114877437032970289r_rule @ Rules @ Rs )
         => ( ( abstra7716921463347034149m_rule @ ( abstra3991228042706630138ule_fm @ Eff2 @ Rs @ S3 ) @ Steps )
           => ? [N2: nat,S7: fm] :
                ( ( member_fm @ S7 @ S4 )
                & ( abstra7716921463347034149m_rule @ ( abstra3991228042706630138ule_fm @ Eff2 @ ( sdrop_rule @ N2 @ Rs ) @ S7 ) @ ( sdrop_432990894411838815m_rule @ M @ Steps ) ) ) ) ) ) ) ).

% RuleSystem.ipath_mkTree_sdrop
thf(fact_790_RuleSystem_Oipath__mkTree__sdrop,axiom,
    ! [Eff2: rule > rule > fset_rule > $o,Rules: stream_rule,S4: set_rule,S3: rule,Rs: stream_rule,Steps: stream3588992565182678257e_rule,M: nat] :
      ( ( abstra2725590884229999157e_rule @ Eff2 @ Rules @ S4 )
     => ( ( member_rule @ S3 @ S4 )
       => ( ( abstra1114877437032970289r_rule @ Rules @ Rs )
         => ( ( abstra3969007714708337434e_rule @ ( abstra7443870075974203631e_rule @ Eff2 @ Rs @ S3 ) @ Steps )
           => ? [N2: nat,S7: rule] :
                ( ( member_rule @ S7 @ S4 )
                & ( abstra3969007714708337434e_rule @ ( abstra7443870075974203631e_rule @ Eff2 @ ( sdrop_rule @ N2 @ Rs ) @ S7 ) @ ( sdrop_2199339672054592340e_rule @ M @ Steps ) ) ) ) ) ) ) ).

% RuleSystem.ipath_mkTree_sdrop
thf(fact_791_RuleSystem_Oipath__mkTree__sdrop,axiom,
    ! [Eff2: list_fm > list_fm > fset_list_fm > $o,Rules: stream_list_fm,S4: set_list_fm,S3: list_fm,Rs: stream_list_fm,Steps: stream8299795917829157543ist_fm,M: nat] :
      ( ( abstra12045985430712427ist_fm @ Eff2 @ Rules @ S4 )
     => ( ( member_list_fm @ S3 @ S4 )
       => ( ( abstra274702546776099020ist_fm @ Rules @ Rs )
         => ( ( abstra5863983821914536912ist_fm @ ( abstra3846921640779263781ist_fm @ Eff2 @ Rs @ S3 ) @ Steps )
           => ? [N2: nat,S7: list_fm] :
                ( ( member_list_fm @ S7 @ S4 )
                & ( abstra5863983821914536912ist_fm @ ( abstra3846921640779263781ist_fm @ Eff2 @ ( sdrop_list_fm @ N2 @ Rs ) @ S7 ) @ ( sdrop_4442373711808556042ist_fm @ M @ Steps ) ) ) ) ) ) ) ).

% RuleSystem.ipath_mkTree_sdrop
thf(fact_792_RuleSystem_Oipath__mkTree__sdrop,axiom,
    ! [Eff2: produc780299800278082923e_rule > fm > fset_fm > $o,Rules: stream3588992565182678257e_rule,S4: set_fm,S3: fm,Rs: stream3588992565182678257e_rule,Steps: stream3505881084499982752e_rule,M: nat] :
      ( ( abstra6843014240698492626ule_fm @ Eff2 @ Rules @ S4 )
     => ( ( member_fm @ S3 @ S4 )
       => ( ( abstra4837122928373946751e_rule @ Rules @ Rs )
         => ( ( abstra8067268766995864225e_rule @ ( abstra5872912078930511000ule_fm @ Eff2 @ Rs @ S3 ) @ Steps )
           => ? [N2: nat,S7: fm] :
                ( ( member_fm @ S7 @ S4 )
                & ( abstra8067268766995864225e_rule @ ( abstra5872912078930511000ule_fm @ Eff2 @ ( sdrop_2199339672054592340e_rule @ N2 @ Rs ) @ S7 ) @ ( sdrop_3146478060095727335e_rule @ M @ Steps ) ) ) ) ) ) ) ).

% RuleSystem.ipath_mkTree_sdrop
thf(fact_793_RuleSystem_Oipath__mkTree__sdrop,axiom,
    ! [Eff2: produc780299800278082923e_rule > rule > fset_rule > $o,Rules: stream3588992565182678257e_rule,S4: set_rule,S3: rule,Rs: stream3588992565182678257e_rule,Steps: stream4003613590975974891e_rule,M: nat] :
      ( ( abstra1742398642700549575e_rule @ Eff2 @ Rules @ S4 )
     => ( ( member_rule @ S3 @ S4 )
       => ( ( abstra4837122928373946751e_rule @ Rules @ Rs )
         => ( ( abstra4067143171202666092e_rule @ ( abstra4338455777355667597e_rule @ Eff2 @ Rs @ S3 ) @ Steps )
           => ? [N2: nat,S7: rule] :
                ( ( member_rule @ S7 @ S4 )
                & ( abstra4067143171202666092e_rule @ ( abstra4338455777355667597e_rule @ Eff2 @ ( sdrop_2199339672054592340e_rule @ N2 @ Rs ) @ S7 ) @ ( sdrop_3185404920971892658e_rule @ M @ Steps ) ) ) ) ) ) ) ).

% RuleSystem.ipath_mkTree_sdrop
thf(fact_794_RuleSystem_Oipath__mkTree__sdrop,axiom,
    ! [Eff2: produc1996495991257130529ist_fm > fm > fset_fm > $o,Rules: stream8299795917829157543ist_fm,S4: set_fm,S3: fm,Rs: stream8299795917829157543ist_fm,Steps: stream5068232101399153750ist_fm,M: nat] :
      ( ( abstra2919136520687232136_fm_fm @ Eff2 @ Rules @ S4 )
     => ( ( member_fm @ S3 @ S4 )
       => ( ( abstra7454370078233524789ist_fm @ Rules @ Rs )
         => ( ( abstra8787874164724549719ist_fm @ ( abstra5537592759599600206_fm_fm @ Eff2 @ Rs @ S3 ) @ Steps )
           => ? [N2: nat,S7: fm] :
                ( ( member_fm @ S7 @ S4 )
                & ( abstra8787874164724549719ist_fm @ ( abstra5537592759599600206_fm_fm @ Eff2 @ ( sdrop_4442373711808556042ist_fm @ N2 @ Rs ) @ S7 ) @ ( sdrop_8120954597595522205ist_fm @ M @ Steps ) ) ) ) ) ) ) ).

% RuleSystem.ipath_mkTree_sdrop
thf(fact_795_RuleSystem_Oipath__mkTree__sdrop,axiom,
    ! [Eff2: produc1996495991257130529ist_fm > rule > fset_rule > $o,Rules: stream8299795917829157543ist_fm,S4: set_rule,S3: rule,Rs: stream8299795917829157543ist_fm,Steps: stream1960312990768957601ist_fm,M: nat] :
      ( ( abstra9106359784214577277m_rule @ Eff2 @ Rules @ S4 )
     => ( ( member_rule @ S3 @ S4 )
       => ( ( abstra7454370078233524789ist_fm @ Rules @ Rs )
         => ( ( abstra6752138315222246434ist_fm @ ( abstra2830003028686515011m_rule @ Eff2 @ Rs @ S3 ) @ Steps )
           => ? [N2: nat,S7: rule] :
                ( ( member_rule @ S7 @ S4 )
                & ( abstra6752138315222246434ist_fm @ ( abstra2830003028686515011m_rule @ Eff2 @ ( sdrop_4442373711808556042ist_fm @ N2 @ Rs ) @ S7 ) @ ( sdrop_2468034597186633064ist_fm @ M @ Steps ) ) ) ) ) ) ) ).

% RuleSystem.ipath_mkTree_sdrop
thf(fact_796_RuleSystem_Oipath__mkTree__sdrop,axiom,
    ! [Eff2: rule > produc1996495991257130529ist_fm > fset_P661503646757059847ist_fm > $o,Rules: stream_rule,S4: set_Pr7058068377845519745ist_fm,S3: produc1996495991257130529ist_fm,Rs: stream_rule,Steps: stream727092118206550309m_rule,M: nat] :
      ( ( abstra6455275554694158129ist_fm @ Eff2 @ Rules @ S4 )
     => ( ( member8102475879199740618ist_fm @ S3 @ S4 )
       => ( ( abstra1114877437032970289r_rule @ Rules @ Rs )
         => ( ( abstra2285467793620430502m_rule @ ( abstra178918799166095863ist_fm @ Eff2 @ Rs @ S3 ) @ Steps )
           => ? [N2: nat,S7: produc1996495991257130529ist_fm] :
                ( ( member8102475879199740618ist_fm @ S7 @ S4 )
                & ( abstra2285467793620430502m_rule @ ( abstra178918799166095863ist_fm @ Eff2 @ ( sdrop_rule @ N2 @ Rs ) @ S7 ) @ ( sdrop_7224736112439592940m_rule @ M @ Steps ) ) ) ) ) ) ) ).

% RuleSystem.ipath_mkTree_sdrop
thf(fact_797_RuleSystem_Oipath__mkTree__sdrop,axiom,
    ! [Eff2: produc164195504107695125m_rule > fm > fset_fm > $o,Rules: stream727092118206550309m_rule,S4: set_fm,S3: fm,Rs: stream727092118206550309m_rule,Steps: stream5001952251264143308m_rule,M: nat] :
      ( ( abstra3764363818050485264ule_fm @ Eff2 @ Rules @ S4 )
     => ( ( member_fm @ S3 @ S4 )
       => ( ( abstra4997497885011821313m_rule @ Rules @ Rs )
         => ( ( abstra3470856493162581877m_rule @ ( abstra1207708450097661130ule_fm @ Eff2 @ Rs @ S3 ) @ Steps )
           => ? [N2: nat,S7: fm] :
                ( ( member_fm @ S7 @ S4 )
                & ( abstra3470856493162581877m_rule @ ( abstra1207708450097661130ule_fm @ Eff2 @ ( sdrop_7224736112439592940m_rule @ N2 @ Rs ) @ S7 ) @ ( sdrop_6203755225909610927m_rule @ M @ Steps ) ) ) ) ) ) ) ).

% RuleSystem.ipath_mkTree_sdrop
thf(fact_798_RuleSystem_Oipath__mkTree__sdrop,axiom,
    ! [Eff2: produc164195504107695125m_rule > rule > fset_rule > $o,Rules: stream727092118206550309m_rule,S4: set_rule,S3: rule,Rs: stream727092118206550309m_rule,Steps: stream1878305877988316353m_rule,M: nat] :
      ( ( abstra3778134268586021893e_rule @ Eff2 @ Rules @ S4 )
     => ( ( member_rule @ S3 @ S4 )
       => ( ( abstra4997497885011821313m_rule @ Rules @ Rs )
         => ( ( abstra987325317039887594m_rule @ ( abstra1231468025953361855e_rule @ Eff2 @ Rs @ S3 ) @ Steps )
           => ? [N2: nat,S7: rule] :
                ( ( member_rule @ S7 @ S4 )
                & ( abstra987325317039887594m_rule @ ( abstra1231468025953361855e_rule @ Eff2 @ ( sdrop_7224736112439592940m_rule @ N2 @ Rs ) @ S7 ) @ ( sdrop_1764812713384251940m_rule @ M @ Steps ) ) ) ) ) ) ) ).

% RuleSystem.ipath_mkTree_sdrop
thf(fact_799_epath__completeness__Saturated,axiom,
    ! [S3: produc1996495991257130529ist_fm] :
      ( ( member8102475879199740618ist_fm @ S3 @ top_to728902761071950545ist_fm )
     => ( ? [T8: abstra4572470701423150929m_rule] :
            ( ( ( produc6879501374131015971m_rule @ ( abstra3503897793822247108m_rule @ T8 ) )
              = S3 )
            & ( abstra4476735159789068068ist_fm
              @ ^ [R: rule,S: produc1996495991257130529ist_fm,Ss: fset_P661503646757059847ist_fm] :
                  ( ( eff @ R @ S )
                  = ( some_f2162114139285431020ist_fm @ Ss ) )
              @ rules
              @ T8 )
            & ( abstra3507476289341504315m_rule @ T8 ) )
        | ? [Steps4: stream727092118206550309m_rule] :
            ( ( ( produc6879501374131015971m_rule @ ( shd_Pr7235097944458474089m_rule @ Steps4 ) )
              = S3 )
            & ( abstra5415202973855822231ist_fm
              @ ^ [R: rule,S: produc1996495991257130529ist_fm,Ss: fset_P661503646757059847ist_fm] :
                  ( ( eff @ R @ S )
                  = ( some_f2162114139285431020ist_fm @ Ss ) )
              @ rules
              @ Steps4 )
            & ( abstra8647779336736900908ist_fm
              @ ^ [R: rule,S: produc1996495991257130529ist_fm,Ss: fset_P661503646757059847ist_fm] :
                  ( ( eff @ R @ S )
                  = ( some_f2162114139285431020ist_fm @ Ss ) )
              @ rules
              @ Steps4 ) ) ) ) ).

% epath_completeness_Saturated
thf(fact_800_PersistentRuleSystem_Oepath__completeness__Saturated,axiom,
    ! [Eff2: list_fm > list_fm > fset_list_fm > $o,Rules: stream_list_fm,S4: set_list_fm,S3: list_fm] :
      ( ( abstra1616580364776158018ist_fm @ Eff2 @ Rules @ S4 )
     => ( ( member_list_fm @ S3 @ S4 )
       => ( ? [T8: abstra5042247676209264507ist_fm] :
              ( ( ( produc1501393135466168645ist_fm @ ( abstra778364809706815794ist_fm @ T8 ) )
                = S3 )
              & ( abstra8964572462466433336ist_fm @ Eff2 @ Rules @ T8 )
              & ( abstra5066959810000083067ist_fm @ T8 ) )
          | ? [Steps4: stream8299795917829157543ist_fm] :
              ( ( ( produc1501393135466168645ist_fm @ ( shd_Pr772355297128350925ist_fm @ Steps4 ) )
                = S3 )
              & ( abstra6579460235205289861ist_fm @ Eff2 @ Rules @ Steps4 )
              & ( abstra8946172763262761776ist_fm @ Eff2 @ Rules @ Steps4 ) ) ) ) ) ).

% PersistentRuleSystem.epath_completeness_Saturated
thf(fact_801_PersistentRuleSystem_Oepath__completeness__Saturated,axiom,
    ! [Eff2: rule > produc1996495991257130529ist_fm > fset_P661503646757059847ist_fm > $o,Rules: stream_rule,S4: set_Pr7058068377845519745ist_fm,S3: produc1996495991257130529ist_fm] :
      ( ( abstra7887161495092839066ist_fm @ Eff2 @ Rules @ S4 )
     => ( ( member8102475879199740618ist_fm @ S3 @ S4 )
       => ( ? [T8: abstra4572470701423150929m_rule] :
              ( ( ( produc6879501374131015971m_rule @ ( abstra3503897793822247108m_rule @ T8 ) )
                = S3 )
              & ( abstra4476735159789068068ist_fm @ Eff2 @ Rules @ T8 )
              & ( abstra3507476289341504315m_rule @ T8 ) )
          | ? [Steps4: stream727092118206550309m_rule] :
              ( ( ( produc6879501374131015971m_rule @ ( shd_Pr7235097944458474089m_rule @ Steps4 ) )
                = S3 )
              & ( abstra5415202973855822231ist_fm @ Eff2 @ Rules @ Steps4 )
              & ( abstra8647779336736900908ist_fm @ Eff2 @ Rules @ Steps4 ) ) ) ) ) ).

% PersistentRuleSystem.epath_completeness_Saturated
thf(fact_802_mkTree_Osimps_I2_J,axiom,
    ! [Rs: stream_rule,S3: produc1996495991257130529ist_fm] :
      ( ( abstra4962253831105676628m_rule
        @ ( abstra178918799166095863ist_fm
          @ ^ [R: rule,S: produc1996495991257130529ist_fm,Ss: fset_P661503646757059847ist_fm] :
              ( ( eff @ R @ S )
              = ( some_f2162114139285431020ist_fm @ Ss ) )
          @ Rs
          @ S3 ) )
      = ( fimage748043113761278389m_rule
        @ ( abstra178918799166095863ist_fm
          @ ^ [R: rule,S: produc1996495991257130529ist_fm,Ss: fset_P661503646757059847ist_fm] :
              ( ( eff @ R @ S )
              = ( some_f2162114139285431020ist_fm @ Ss ) )
          @ ( stl_rule
            @ ( abstra2363873759530810705ist_fm
              @ ^ [R: rule,S: produc1996495991257130529ist_fm,Ss: fset_P661503646757059847ist_fm] :
                  ( ( eff @ R @ S )
                  = ( some_f2162114139285431020ist_fm @ Ss ) )
              @ Rs
              @ S3 ) ) )
        @ ( abstra458490714785879965ist_fm
          @ ^ [R: rule,S: produc1996495991257130529ist_fm,Ss: fset_P661503646757059847ist_fm] :
              ( ( eff @ R @ S )
              = ( some_f2162114139285431020ist_fm @ Ss ) )
          @ ( shd_rule
            @ ( abstra2363873759530810705ist_fm
              @ ^ [R: rule,S: produc1996495991257130529ist_fm,Ss: fset_P661503646757059847ist_fm] :
                  ( ( eff @ R @ S )
                  = ( some_f2162114139285431020ist_fm @ Ss ) )
              @ Rs
              @ S3 ) )
          @ S3 ) ) ) ).

% mkTree.simps(2)
thf(fact_803_i_OmkTree_Osimps_I2_J,axiom,
    ! [Eff3: rule > produc1996495991257130529ist_fm > option6029102822600979917ist_fm,Rs: stream_rule,S3: produc1996495991257130529ist_fm] :
      ( ( abstra4962253831105676628m_rule @ ( abstra178918799166095863ist_fm @ ( abstra5459341389059251418ist_fm @ Eff3 ) @ Rs @ S3 ) )
      = ( fimage748043113761278389m_rule @ ( abstra178918799166095863ist_fm @ ( abstra5459341389059251418ist_fm @ Eff3 ) @ ( stl_rule @ ( abstra2363873759530810705ist_fm @ ( abstra5459341389059251418ist_fm @ Eff3 ) @ Rs @ S3 ) ) ) @ ( abstra458490714785879965ist_fm @ ( abstra5459341389059251418ist_fm @ Eff3 ) @ ( shd_rule @ ( abstra2363873759530810705ist_fm @ ( abstra5459341389059251418ist_fm @ Eff3 ) @ Rs @ S3 ) ) @ S3 ) ) ) ).

% i.mkTree.simps(2)
thf(fact_804_abort__Bleast__def,axiom,
    ( abort_Bleast_nat
    = ( ^ [S8: set_nat,P4: nat > $o] :
          ( ord_Least_nat
          @ ^ [X6: nat] :
              ( ( member_nat @ X6 @ S8 )
              & ( P4 @ X6 ) ) ) ) ) ).

% abort_Bleast_def
thf(fact_805_Bleast__def,axiom,
    ( bleast_nat
    = ( ^ [S8: set_nat,P4: nat > $o] :
          ( ord_Least_nat
          @ ^ [X6: nat] :
              ( ( member_nat @ X6 @ S8 )
              & ( P4 @ X6 ) ) ) ) ) ).

% Bleast_def
thf(fact_806_PersistentRuleSystem__def,axiom,
    ( abstra7887161495092839066ist_fm
    = ( ^ [Eff: rule > produc1996495991257130529ist_fm > fset_P661503646757059847ist_fm > $o,Rules2: stream_rule,S8: set_Pr7058068377845519745ist_fm] :
          ( ( abstra6455275554694158129ist_fm @ Eff @ Rules2 @ S8 )
          & ( abstra2778581409224112183ist_fm @ Eff @ Rules2 @ S8 ) ) ) ) ).

% PersistentRuleSystem_def
thf(fact_807_fimage__eqI,axiom,
    ! [B: produc1996495991257130529ist_fm,F: produc1996495991257130529ist_fm > produc1996495991257130529ist_fm,X3: produc1996495991257130529ist_fm,A5: fset_P661503646757059847ist_fm] :
      ( ( B
        = ( F @ X3 ) )
     => ( ( fmembe3381613331217039976ist_fm @ X3 @ A5 )
       => ( fmembe3381613331217039976ist_fm @ B @ ( fimage8674584286907012797ist_fm @ F @ A5 ) ) ) ) ).

% fimage_eqI
thf(fact_808_fimage__eqI,axiom,
    ! [B: abstra4572470701423150929m_rule,F: produc1996495991257130529ist_fm > abstra4572470701423150929m_rule,X3: produc1996495991257130529ist_fm,A5: fset_P661503646757059847ist_fm] :
      ( ( B
        = ( F @ X3 ) )
     => ( ( fmembe3381613331217039976ist_fm @ X3 @ A5 )
       => ( fmembe1635983029874454474m_rule @ B @ ( fimage748043113761278389m_rule @ F @ A5 ) ) ) ) ).

% fimage_eqI
thf(fact_809_fimage__eqI,axiom,
    ! [B: produc1996495991257130529ist_fm,F: abstra4572470701423150929m_rule > produc1996495991257130529ist_fm,X3: abstra4572470701423150929m_rule,A5: fset_A5396818216090053761m_rule] :
      ( ( B
        = ( F @ X3 ) )
     => ( ( fmembe1635983029874454474m_rule @ X3 @ A5 )
       => ( fmembe3381613331217039976ist_fm @ B @ ( fimage9164954772359936849ist_fm @ F @ A5 ) ) ) ) ).

% fimage_eqI
thf(fact_810_fimage__eqI,axiom,
    ! [B: abstra4572470701423150929m_rule,F: abstra4572470701423150929m_rule > abstra4572470701423150929m_rule,X3: abstra4572470701423150929m_rule,A5: fset_A5396818216090053761m_rule] :
      ( ( B
        = ( F @ X3 ) )
     => ( ( fmembe1635983029874454474m_rule @ X3 @ A5 )
       => ( fmembe1635983029874454474m_rule @ B @ ( fimage3349360726776475169m_rule @ F @ A5 ) ) ) ) ).

% fimage_eqI
thf(fact_811_fimage__fimage,axiom,
    ! [F: abstra4572470701423150929m_rule > abstra4572470701423150929m_rule,G: produc1996495991257130529ist_fm > abstra4572470701423150929m_rule,A5: fset_P661503646757059847ist_fm] :
      ( ( fimage3349360726776475169m_rule @ F @ ( fimage748043113761278389m_rule @ G @ A5 ) )
      = ( fimage748043113761278389m_rule
        @ ^ [X6: produc1996495991257130529ist_fm] : ( F @ ( G @ X6 ) )
        @ A5 ) ) ).

% fimage_fimage
thf(fact_812_fimage__fimage,axiom,
    ! [F: produc1996495991257130529ist_fm > produc1996495991257130529ist_fm,G: abstra4572470701423150929m_rule > produc1996495991257130529ist_fm,A5: fset_A5396818216090053761m_rule] :
      ( ( fimage8674584286907012797ist_fm @ F @ ( fimage9164954772359936849ist_fm @ G @ A5 ) )
      = ( fimage9164954772359936849ist_fm
        @ ^ [X6: abstra4572470701423150929m_rule] : ( F @ ( G @ X6 ) )
        @ A5 ) ) ).

% fimage_fimage
thf(fact_813_fimage__fimage,axiom,
    ! [F: produc1996495991257130529ist_fm > abstra4572470701423150929m_rule,G: produc1996495991257130529ist_fm > produc1996495991257130529ist_fm,A5: fset_P661503646757059847ist_fm] :
      ( ( fimage748043113761278389m_rule @ F @ ( fimage8674584286907012797ist_fm @ G @ A5 ) )
      = ( fimage748043113761278389m_rule
        @ ^ [X6: produc1996495991257130529ist_fm] : ( F @ ( G @ X6 ) )
        @ A5 ) ) ).

% fimage_fimage
thf(fact_814_fimage__fimage,axiom,
    ! [F: produc1996495991257130529ist_fm > abstra4572470701423150929m_rule,G: abstra4572470701423150929m_rule > produc1996495991257130529ist_fm,A5: fset_A5396818216090053761m_rule] :
      ( ( fimage748043113761278389m_rule @ F @ ( fimage9164954772359936849ist_fm @ G @ A5 ) )
      = ( fimage3349360726776475169m_rule
        @ ^ [X6: abstra4572470701423150929m_rule] : ( F @ ( G @ X6 ) )
        @ A5 ) ) ).

% fimage_fimage
thf(fact_815_fimage__fimage,axiom,
    ! [F: abstra4572470701423150929m_rule > produc1996495991257130529ist_fm,G: abstra4572470701423150929m_rule > abstra4572470701423150929m_rule,A5: fset_A5396818216090053761m_rule] :
      ( ( fimage9164954772359936849ist_fm @ F @ ( fimage3349360726776475169m_rule @ G @ A5 ) )
      = ( fimage9164954772359936849ist_fm
        @ ^ [X6: abstra4572470701423150929m_rule] : ( F @ ( G @ X6 ) )
        @ A5 ) ) ).

% fimage_fimage
thf(fact_816_fimage__fimage,axiom,
    ! [F: abstra4572470701423150929m_rule > produc1996495991257130529ist_fm,G: produc1996495991257130529ist_fm > abstra4572470701423150929m_rule,A5: fset_P661503646757059847ist_fm] :
      ( ( fimage9164954772359936849ist_fm @ F @ ( fimage748043113761278389m_rule @ G @ A5 ) )
      = ( fimage8674584286907012797ist_fm
        @ ^ [X6: produc1996495991257130529ist_fm] : ( F @ ( G @ X6 ) )
        @ A5 ) ) ).

% fimage_fimage
thf(fact_817_rev__fimage__eqI,axiom,
    ! [X3: produc1996495991257130529ist_fm,A5: fset_P661503646757059847ist_fm,B: produc1996495991257130529ist_fm,F: produc1996495991257130529ist_fm > produc1996495991257130529ist_fm] :
      ( ( fmembe3381613331217039976ist_fm @ X3 @ A5 )
     => ( ( B
          = ( F @ X3 ) )
       => ( fmembe3381613331217039976ist_fm @ B @ ( fimage8674584286907012797ist_fm @ F @ A5 ) ) ) ) ).

% rev_fimage_eqI
thf(fact_818_rev__fimage__eqI,axiom,
    ! [X3: produc1996495991257130529ist_fm,A5: fset_P661503646757059847ist_fm,B: abstra4572470701423150929m_rule,F: produc1996495991257130529ist_fm > abstra4572470701423150929m_rule] :
      ( ( fmembe3381613331217039976ist_fm @ X3 @ A5 )
     => ( ( B
          = ( F @ X3 ) )
       => ( fmembe1635983029874454474m_rule @ B @ ( fimage748043113761278389m_rule @ F @ A5 ) ) ) ) ).

% rev_fimage_eqI
thf(fact_819_rev__fimage__eqI,axiom,
    ! [X3: abstra4572470701423150929m_rule,A5: fset_A5396818216090053761m_rule,B: produc1996495991257130529ist_fm,F: abstra4572470701423150929m_rule > produc1996495991257130529ist_fm] :
      ( ( fmembe1635983029874454474m_rule @ X3 @ A5 )
     => ( ( B
          = ( F @ X3 ) )
       => ( fmembe3381613331217039976ist_fm @ B @ ( fimage9164954772359936849ist_fm @ F @ A5 ) ) ) ) ).

% rev_fimage_eqI
thf(fact_820_rev__fimage__eqI,axiom,
    ! [X3: abstra4572470701423150929m_rule,A5: fset_A5396818216090053761m_rule,B: abstra4572470701423150929m_rule,F: abstra4572470701423150929m_rule > abstra4572470701423150929m_rule] :
      ( ( fmembe1635983029874454474m_rule @ X3 @ A5 )
     => ( ( B
          = ( F @ X3 ) )
       => ( fmembe1635983029874454474m_rule @ B @ ( fimage3349360726776475169m_rule @ F @ A5 ) ) ) ) ).

% rev_fimage_eqI
thf(fact_821_fimage__cong,axiom,
    ! [M2: fset_P661503646757059847ist_fm,N4: fset_P661503646757059847ist_fm,F: produc1996495991257130529ist_fm > abstra4572470701423150929m_rule,G: produc1996495991257130529ist_fm > abstra4572470701423150929m_rule] :
      ( ( M2 = N4 )
     => ( ! [X: produc1996495991257130529ist_fm] :
            ( ( fmembe3381613331217039976ist_fm @ X @ N4 )
           => ( ( F @ X )
              = ( G @ X ) ) )
       => ( ( fimage748043113761278389m_rule @ F @ M2 )
          = ( fimage748043113761278389m_rule @ G @ N4 ) ) ) ) ).

% fimage_cong
thf(fact_822_fimage__cong,axiom,
    ! [M2: fset_A5396818216090053761m_rule,N4: fset_A5396818216090053761m_rule,F: abstra4572470701423150929m_rule > produc1996495991257130529ist_fm,G: abstra4572470701423150929m_rule > produc1996495991257130529ist_fm] :
      ( ( M2 = N4 )
     => ( ! [X: abstra4572470701423150929m_rule] :
            ( ( fmembe1635983029874454474m_rule @ X @ N4 )
           => ( ( F @ X )
              = ( G @ X ) ) )
       => ( ( fimage9164954772359936849ist_fm @ F @ M2 )
          = ( fimage9164954772359936849ist_fm @ G @ N4 ) ) ) ) ).

% fimage_cong
thf(fact_823_fimageI,axiom,
    ! [X3: produc1996495991257130529ist_fm,A5: fset_P661503646757059847ist_fm,F: produc1996495991257130529ist_fm > produc1996495991257130529ist_fm] :
      ( ( fmembe3381613331217039976ist_fm @ X3 @ A5 )
     => ( fmembe3381613331217039976ist_fm @ ( F @ X3 ) @ ( fimage8674584286907012797ist_fm @ F @ A5 ) ) ) ).

% fimageI
thf(fact_824_fimageI,axiom,
    ! [X3: produc1996495991257130529ist_fm,A5: fset_P661503646757059847ist_fm,F: produc1996495991257130529ist_fm > abstra4572470701423150929m_rule] :
      ( ( fmembe3381613331217039976ist_fm @ X3 @ A5 )
     => ( fmembe1635983029874454474m_rule @ ( F @ X3 ) @ ( fimage748043113761278389m_rule @ F @ A5 ) ) ) ).

% fimageI
thf(fact_825_fimageI,axiom,
    ! [X3: abstra4572470701423150929m_rule,A5: fset_A5396818216090053761m_rule,F: abstra4572470701423150929m_rule > produc1996495991257130529ist_fm] :
      ( ( fmembe1635983029874454474m_rule @ X3 @ A5 )
     => ( fmembe3381613331217039976ist_fm @ ( F @ X3 ) @ ( fimage9164954772359936849ist_fm @ F @ A5 ) ) ) ).

% fimageI
thf(fact_826_fimageI,axiom,
    ! [X3: abstra4572470701423150929m_rule,A5: fset_A5396818216090053761m_rule,F: abstra4572470701423150929m_rule > abstra4572470701423150929m_rule] :
      ( ( fmembe1635983029874454474m_rule @ X3 @ A5 )
     => ( fmembe1635983029874454474m_rule @ ( F @ X3 ) @ ( fimage3349360726776475169m_rule @ F @ A5 ) ) ) ).

% fimageI
thf(fact_827_fimageE,axiom,
    ! [B: produc1996495991257130529ist_fm,F: produc1996495991257130529ist_fm > produc1996495991257130529ist_fm,A5: fset_P661503646757059847ist_fm] :
      ( ( fmembe3381613331217039976ist_fm @ B @ ( fimage8674584286907012797ist_fm @ F @ A5 ) )
     => ~ ! [X: produc1996495991257130529ist_fm] :
            ( ( B
              = ( F @ X ) )
           => ~ ( fmembe3381613331217039976ist_fm @ X @ A5 ) ) ) ).

% fimageE
thf(fact_828_fimageE,axiom,
    ! [B: produc1996495991257130529ist_fm,F: abstra4572470701423150929m_rule > produc1996495991257130529ist_fm,A5: fset_A5396818216090053761m_rule] :
      ( ( fmembe3381613331217039976ist_fm @ B @ ( fimage9164954772359936849ist_fm @ F @ A5 ) )
     => ~ ! [X: abstra4572470701423150929m_rule] :
            ( ( B
              = ( F @ X ) )
           => ~ ( fmembe1635983029874454474m_rule @ X @ A5 ) ) ) ).

% fimageE
thf(fact_829_fimageE,axiom,
    ! [B: abstra4572470701423150929m_rule,F: produc1996495991257130529ist_fm > abstra4572470701423150929m_rule,A5: fset_P661503646757059847ist_fm] :
      ( ( fmembe1635983029874454474m_rule @ B @ ( fimage748043113761278389m_rule @ F @ A5 ) )
     => ~ ! [X: produc1996495991257130529ist_fm] :
            ( ( B
              = ( F @ X ) )
           => ~ ( fmembe3381613331217039976ist_fm @ X @ A5 ) ) ) ).

% fimageE
thf(fact_830_fimageE,axiom,
    ! [B: abstra4572470701423150929m_rule,F: abstra4572470701423150929m_rule > abstra4572470701423150929m_rule,A5: fset_A5396818216090053761m_rule] :
      ( ( fmembe1635983029874454474m_rule @ B @ ( fimage3349360726776475169m_rule @ F @ A5 ) )
     => ~ ! [X: abstra4572470701423150929m_rule] :
            ( ( B
              = ( F @ X ) )
           => ~ ( fmembe1635983029874454474m_rule @ X @ A5 ) ) ) ).

% fimageE
thf(fact_831_subset__fimage__iff,axiom,
    ! [B5: fset_A5396818216090053761m_rule,F: produc1996495991257130529ist_fm > abstra4572470701423150929m_rule,A5: fset_P661503646757059847ist_fm] :
      ( ( ord_le6951374404268211425m_rule @ B5 @ ( fimage748043113761278389m_rule @ F @ A5 ) )
      = ( ? [AA: fset_P661503646757059847ist_fm] :
            ( ( ord_le2064643713053750439ist_fm @ AA @ A5 )
            & ( B5
              = ( fimage748043113761278389m_rule @ F @ AA ) ) ) ) ) ).

% subset_fimage_iff
thf(fact_832_subset__fimage__iff,axiom,
    ! [B5: fset_P661503646757059847ist_fm,F: abstra4572470701423150929m_rule > produc1996495991257130529ist_fm,A5: fset_A5396818216090053761m_rule] :
      ( ( ord_le2064643713053750439ist_fm @ B5 @ ( fimage9164954772359936849ist_fm @ F @ A5 ) )
      = ( ? [AA: fset_A5396818216090053761m_rule] :
            ( ( ord_le6951374404268211425m_rule @ AA @ A5 )
            & ( B5
              = ( fimage9164954772359936849ist_fm @ F @ AA ) ) ) ) ) ).

% subset_fimage_iff
thf(fact_833_fimage__mono,axiom,
    ! [A5: fset_P661503646757059847ist_fm,B5: fset_P661503646757059847ist_fm,F: produc1996495991257130529ist_fm > abstra4572470701423150929m_rule] :
      ( ( ord_le2064643713053750439ist_fm @ A5 @ B5 )
     => ( ord_le6951374404268211425m_rule @ ( fimage748043113761278389m_rule @ F @ A5 ) @ ( fimage748043113761278389m_rule @ F @ B5 ) ) ) ).

% fimage_mono
thf(fact_834_fimage__mono,axiom,
    ! [A5: fset_A5396818216090053761m_rule,B5: fset_A5396818216090053761m_rule,F: abstra4572470701423150929m_rule > produc1996495991257130529ist_fm] :
      ( ( ord_le6951374404268211425m_rule @ A5 @ B5 )
     => ( ord_le2064643713053750439ist_fm @ ( fimage9164954772359936849ist_fm @ F @ A5 ) @ ( fimage9164954772359936849ist_fm @ F @ B5 ) ) ) ).

% fimage_mono
thf(fact_835_fimage__fsubsetI,axiom,
    ! [A5: fset_P661503646757059847ist_fm,F: produc1996495991257130529ist_fm > produc1996495991257130529ist_fm,B5: fset_P661503646757059847ist_fm] :
      ( ! [X: produc1996495991257130529ist_fm] :
          ( ( fmembe3381613331217039976ist_fm @ X @ A5 )
         => ( fmembe3381613331217039976ist_fm @ ( F @ X ) @ B5 ) )
     => ( ord_le2064643713053750439ist_fm @ ( fimage8674584286907012797ist_fm @ F @ A5 ) @ B5 ) ) ).

% fimage_fsubsetI
thf(fact_836_fimage__fsubsetI,axiom,
    ! [A5: fset_P661503646757059847ist_fm,F: produc1996495991257130529ist_fm > abstra4572470701423150929m_rule,B5: fset_A5396818216090053761m_rule] :
      ( ! [X: produc1996495991257130529ist_fm] :
          ( ( fmembe3381613331217039976ist_fm @ X @ A5 )
         => ( fmembe1635983029874454474m_rule @ ( F @ X ) @ B5 ) )
     => ( ord_le6951374404268211425m_rule @ ( fimage748043113761278389m_rule @ F @ A5 ) @ B5 ) ) ).

% fimage_fsubsetI
thf(fact_837_fimage__fsubsetI,axiom,
    ! [A5: fset_A5396818216090053761m_rule,F: abstra4572470701423150929m_rule > produc1996495991257130529ist_fm,B5: fset_P661503646757059847ist_fm] :
      ( ! [X: abstra4572470701423150929m_rule] :
          ( ( fmembe1635983029874454474m_rule @ X @ A5 )
         => ( fmembe3381613331217039976ist_fm @ ( F @ X ) @ B5 ) )
     => ( ord_le2064643713053750439ist_fm @ ( fimage9164954772359936849ist_fm @ F @ A5 ) @ B5 ) ) ).

% fimage_fsubsetI
thf(fact_838_fimage__fsubsetI,axiom,
    ! [A5: fset_A5396818216090053761m_rule,F: abstra4572470701423150929m_rule > abstra4572470701423150929m_rule,B5: fset_A5396818216090053761m_rule] :
      ( ! [X: abstra4572470701423150929m_rule] :
          ( ( fmembe1635983029874454474m_rule @ X @ A5 )
         => ( fmembe1635983029874454474m_rule @ ( F @ X ) @ B5 ) )
     => ( ord_le6951374404268211425m_rule @ ( fimage3349360726776475169m_rule @ F @ A5 ) @ B5 ) ) ).

% fimage_fsubsetI
thf(fact_839_ftree__no__ipath,axiom,
    ! [T2: abstra4572470701423150929m_rule,Steps: stream727092118206550309m_rule] :
      ( ( abstra3507476289341504315m_rule @ T2 )
     => ~ ( abstra2285467793620430502m_rule @ T2 @ Steps ) ) ).

% ftree_no_ipath
thf(fact_840_PersistentRuleSystem_Oaxioms_I2_J,axiom,
    ! [Eff2: rule > produc1996495991257130529ist_fm > fset_P661503646757059847ist_fm > $o,Rules: stream_rule,S4: set_Pr7058068377845519745ist_fm] :
      ( ( abstra7887161495092839066ist_fm @ Eff2 @ Rules @ S4 )
     => ( abstra2778581409224112183ist_fm @ Eff2 @ Rules @ S4 ) ) ).

% PersistentRuleSystem.axioms(2)
thf(fact_841_tfinite_Ocases,axiom,
    ! [A: abstra4572470701423150929m_rule] :
      ( ( abstra3507476289341504315m_rule @ A )
     => ! [T7: abstra4572470701423150929m_rule] :
          ( ( fmembe1635983029874454474m_rule @ T7 @ ( abstra4962253831105676628m_rule @ A ) )
         => ( abstra3507476289341504315m_rule @ T7 ) ) ) ).

% tfinite.cases
thf(fact_842_tfinite_Osimps,axiom,
    ( abstra3507476289341504315m_rule
    = ( ^ [A2: abstra4572470701423150929m_rule] :
        ? [T: abstra4572470701423150929m_rule] :
          ( ( A2 = T )
          & ! [X6: abstra4572470701423150929m_rule] :
              ( ( fmembe1635983029874454474m_rule @ X6 @ ( abstra4962253831105676628m_rule @ T ) )
             => ( abstra3507476289341504315m_rule @ X6 ) ) ) ) ) ).

% tfinite.simps
thf(fact_843_tfinite,axiom,
    ! [T2: abstra4572470701423150929m_rule] :
      ( ! [T4: abstra4572470701423150929m_rule] :
          ( ( fmembe1635983029874454474m_rule @ T4 @ ( abstra4962253831105676628m_rule @ T2 ) )
         => ( abstra3507476289341504315m_rule @ T4 ) )
     => ( abstra3507476289341504315m_rule @ T2 ) ) ).

% tfinite
thf(fact_844_PersistentRuleSystem__axioms_Ointro,axiom,
    ! [Rules: stream_rule,Eff2: rule > produc1996495991257130529ist_fm > fset_P661503646757059847ist_fm > $o,S4: set_Pr7058068377845519745ist_fm] :
      ( ! [R5: rule] :
          ( ( member_rule @ R5 @ ( sset_rule @ Rules ) )
         => ( abstra4153889242452539894ist_fm @ Eff2 @ Rules @ S4 @ R5 ) )
     => ( abstra2778581409224112183ist_fm @ Eff2 @ Rules @ S4 ) ) ).

% PersistentRuleSystem_axioms.intro
thf(fact_845_PersistentRuleSystem__axioms__def,axiom,
    ( abstra2778581409224112183ist_fm
    = ( ^ [Eff: rule > produc1996495991257130529ist_fm > fset_P661503646757059847ist_fm > $o,Rules2: stream_rule,S8: set_Pr7058068377845519745ist_fm] :
        ! [R: rule] :
          ( ( member_rule @ R @ ( sset_rule @ Rules2 ) )
         => ( abstra4153889242452539894ist_fm @ Eff @ Rules2 @ S8 @ R ) ) ) ) ).

% PersistentRuleSystem_axioms_def
thf(fact_846_RuleSystem__Defs_OmkTree_Osimps_I2_J,axiom,
    ! [Eff2: rule > produc1996495991257130529ist_fm > fset_P661503646757059847ist_fm > $o,Rs: stream_rule,S3: produc1996495991257130529ist_fm] :
      ( ( abstra4962253831105676628m_rule @ ( abstra178918799166095863ist_fm @ Eff2 @ Rs @ S3 ) )
      = ( fimage748043113761278389m_rule @ ( abstra178918799166095863ist_fm @ Eff2 @ ( stl_rule @ ( abstra2363873759530810705ist_fm @ Eff2 @ Rs @ S3 ) ) ) @ ( abstra458490714785879965ist_fm @ Eff2 @ ( shd_rule @ ( abstra2363873759530810705ist_fm @ Eff2 @ Rs @ S3 ) ) @ S3 ) ) ) ).

% RuleSystem_Defs.mkTree.simps(2)
thf(fact_847_PersistentRuleSystem_Ointro,axiom,
    ! [Eff2: rule > produc1996495991257130529ist_fm > fset_P661503646757059847ist_fm > $o,Rules: stream_rule,S4: set_Pr7058068377845519745ist_fm] :
      ( ( abstra6455275554694158129ist_fm @ Eff2 @ Rules @ S4 )
     => ( ( abstra2778581409224112183ist_fm @ Eff2 @ Rules @ S4 )
       => ( abstra7887161495092839066ist_fm @ Eff2 @ Rules @ S4 ) ) ) ).

% PersistentRuleSystem.intro
thf(fact_848_mkTree_Ocode,axiom,
    ! [Rs: stream_rule,S3: produc1996495991257130529ist_fm] :
      ( ( abstra178918799166095863ist_fm
        @ ^ [R: rule,S: produc1996495991257130529ist_fm,Ss: fset_P661503646757059847ist_fm] :
            ( ( eff @ R @ S )
            = ( some_f2162114139285431020ist_fm @ Ss ) )
        @ Rs
        @ S3 )
      = ( abstra7467780926338743076m_rule
        @ ( produc491467635432902671m_rule @ S3
          @ ( shd_rule
            @ ( abstra2363873759530810705ist_fm
              @ ^ [R: rule,S: produc1996495991257130529ist_fm,Ss: fset_P661503646757059847ist_fm] :
                  ( ( eff @ R @ S )
                  = ( some_f2162114139285431020ist_fm @ Ss ) )
              @ Rs
              @ S3 ) ) )
        @ ( fimage748043113761278389m_rule
          @ ( abstra178918799166095863ist_fm
            @ ^ [R: rule,S: produc1996495991257130529ist_fm,Ss: fset_P661503646757059847ist_fm] :
                ( ( eff @ R @ S )
                = ( some_f2162114139285431020ist_fm @ Ss ) )
            @ ( stl_rule
              @ ( abstra2363873759530810705ist_fm
                @ ^ [R: rule,S: produc1996495991257130529ist_fm,Ss: fset_P661503646757059847ist_fm] :
                    ( ( eff @ R @ S )
                    = ( some_f2162114139285431020ist_fm @ Ss ) )
                @ Rs
                @ S3 ) ) )
          @ ( abstra458490714785879965ist_fm
            @ ^ [R: rule,S: produc1996495991257130529ist_fm,Ss: fset_P661503646757059847ist_fm] :
                ( ( eff @ R @ S )
                = ( some_f2162114139285431020ist_fm @ Ss ) )
            @ ( shd_rule
              @ ( abstra2363873759530810705ist_fm
                @ ^ [R: rule,S: produc1996495991257130529ist_fm,Ss: fset_P661503646757059847ist_fm] :
                    ( ( eff @ R @ S )
                    = ( some_f2162114139285431020ist_fm @ Ss ) )
                @ Rs
                @ S3 ) )
            @ S3 ) ) ) ) ).

% mkTree.code
thf(fact_849_i_OmkTree_Ocode,axiom,
    ! [Eff3: list_fm > list_fm > option_fset_list_fm,Rs: stream_list_fm,S3: list_fm] :
      ( ( abstra3846921640779263781ist_fm @ ( abstra256862739949234434ist_fm @ Eff3 ) @ Rs @ S3 )
      = ( abstra6542830291470149330ist_fm @ ( produc381145313068854617ist_fm @ S3 @ ( shd_list_fm @ ( abstra2983386638146376267ist_fm @ ( abstra256862739949234434ist_fm @ Eff3 ) @ Rs @ S3 ) ) ) @ ( fimage27253841482530780ist_fm @ ( abstra3846921640779263781ist_fm @ ( abstra256862739949234434ist_fm @ Eff3 ) @ ( stl_list_fm @ ( abstra2983386638146376267ist_fm @ ( abstra256862739949234434ist_fm @ Eff3 ) @ Rs @ S3 ) ) ) @ ( abstra1097029126060538623ist_fm @ ( abstra256862739949234434ist_fm @ Eff3 ) @ ( shd_list_fm @ ( abstra2983386638146376267ist_fm @ ( abstra256862739949234434ist_fm @ Eff3 ) @ Rs @ S3 ) ) @ S3 ) ) ) ) ).

% i.mkTree.code
thf(fact_850_i_OmkTree_Ocode,axiom,
    ! [Eff3: nat > nat > option_fset_nat,Rs: stream_nat,S3: nat] :
      ( ( abstra1192804020863382533at_nat @ ( abstra4671362528648695778at_nat @ Eff3 ) @ Rs @ S3 )
      = ( abstra4999521449322675122at_nat @ ( product_Pair_nat_nat @ S3 @ ( shd_nat @ ( abstra7573781713853518635at_nat @ ( abstra4671362528648695778at_nat @ Eff3 ) @ Rs @ S3 ) ) ) @ ( fimage8970742173175350476at_nat @ ( abstra1192804020863382533at_nat @ ( abstra4671362528648695778at_nat @ Eff3 ) @ ( stl_nat @ ( abstra7573781713853518635at_nat @ ( abstra4671362528648695778at_nat @ Eff3 ) @ Rs @ S3 ) ) ) @ ( abstra969719415724737503at_nat @ ( abstra4671362528648695778at_nat @ Eff3 ) @ ( shd_nat @ ( abstra7573781713853518635at_nat @ ( abstra4671362528648695778at_nat @ Eff3 ) @ Rs @ S3 ) ) @ S3 ) ) ) ) ).

% i.mkTree.code
thf(fact_851_i_OmkTree_Ocode,axiom,
    ! [Eff3: product_prod_nat_nat > product_prod_nat_nat > option7922001270233832621at_nat,Rs: stream6724221391990029191at_nat,S3: product_prod_nat_nat] :
      ( ( abstra8972947971889770147at_nat @ ( abstra426094224824445696at_nat @ Eff3 ) @ Rs @ S3 )
      = ( abstra6467204794154587544at_nat @ ( produc6161850002892822231at_nat @ S3 @ ( shd_Pr4260400998323988397at_nat @ ( abstra4493250544927931593at_nat @ ( abstra426094224824445696at_nat @ Eff3 ) @ Rs @ S3 ) ) ) @ ( fimage9147729261254313981at_nat @ ( abstra8972947971889770147at_nat @ ( abstra426094224824445696at_nat @ Eff3 ) @ ( stl_Pr5027827701538482609at_nat @ ( abstra4493250544927931593at_nat @ ( abstra426094224824445696at_nat @ Eff3 ) @ Rs @ S3 ) ) ) @ ( abstra673899893360562045at_nat @ ( abstra426094224824445696at_nat @ Eff3 ) @ ( shd_Pr4260400998323988397at_nat @ ( abstra4493250544927931593at_nat @ ( abstra426094224824445696at_nat @ Eff3 ) @ Rs @ S3 ) ) @ S3 ) ) ) ) ).

% i.mkTree.code
thf(fact_852_i_OmkTree_Ocode,axiom,
    ! [Eff3: produc1996495991257130529ist_fm > rule > option_fset_rule,Rs: stream8299795917829157543ist_fm,S3: rule] :
      ( ( abstra2830003028686515011m_rule @ ( abstra8110425618579670566m_rule @ Eff3 ) @ Rs @ S3 )
      = ( abstra2711079411085783200ist_fm @ ( produc7063755442767259331ist_fm @ S3 @ ( shd_Pr772355297128350925ist_fm @ ( abstra5014957989051229853m_rule @ ( abstra8110425618579670566m_rule @ Eff3 ) @ Rs @ S3 ) ) ) @ ( fimage4184475965855427473ist_fm @ ( abstra2830003028686515011m_rule @ ( abstra8110425618579670566m_rule @ Eff3 ) @ ( stl_Pr5027778045650968273ist_fm @ ( abstra5014957989051229853m_rule @ ( abstra8110425618579670566m_rule @ Eff3 ) @ Rs @ S3 ) ) ) @ ( abstra3109574944306299113m_rule @ ( abstra8110425618579670566m_rule @ Eff3 ) @ ( shd_Pr772355297128350925ist_fm @ ( abstra5014957989051229853m_rule @ ( abstra8110425618579670566m_rule @ Eff3 ) @ Rs @ S3 ) ) @ S3 ) ) ) ) ).

% i.mkTree.code
thf(fact_853_i_OmkTree_Ocode,axiom,
    ! [Eff3: rule > produc1996495991257130529ist_fm > option6029102822600979917ist_fm,Rs: stream_rule,S3: produc1996495991257130529ist_fm] :
      ( ( abstra178918799166095863ist_fm @ ( abstra5459341389059251418ist_fm @ Eff3 ) @ Rs @ S3 )
      = ( abstra7467780926338743076m_rule @ ( produc491467635432902671m_rule @ S3 @ ( shd_rule @ ( abstra2363873759530810705ist_fm @ ( abstra5459341389059251418ist_fm @ Eff3 ) @ Rs @ S3 ) ) ) @ ( fimage748043113761278389m_rule @ ( abstra178918799166095863ist_fm @ ( abstra5459341389059251418ist_fm @ Eff3 ) @ ( stl_rule @ ( abstra2363873759530810705ist_fm @ ( abstra5459341389059251418ist_fm @ Eff3 ) @ Rs @ S3 ) ) ) @ ( abstra458490714785879965ist_fm @ ( abstra5459341389059251418ist_fm @ Eff3 ) @ ( shd_rule @ ( abstra2363873759530810705ist_fm @ ( abstra5459341389059251418ist_fm @ Eff3 ) @ Rs @ S3 ) ) @ S3 ) ) ) ) ).

% i.mkTree.code
thf(fact_854_RuleSystem__Defs_OmkTree_Ocode,axiom,
    ( abstra3846921640779263781ist_fm
    = ( ^ [Eff: list_fm > list_fm > fset_list_fm > $o,Rs2: stream_list_fm,S: list_fm] : ( abstra6542830291470149330ist_fm @ ( produc381145313068854617ist_fm @ S @ ( shd_list_fm @ ( abstra2983386638146376267ist_fm @ Eff @ Rs2 @ S ) ) ) @ ( fimage27253841482530780ist_fm @ ( abstra3846921640779263781ist_fm @ Eff @ ( stl_list_fm @ ( abstra2983386638146376267ist_fm @ Eff @ Rs2 @ S ) ) ) @ ( abstra1097029126060538623ist_fm @ Eff @ ( shd_list_fm @ ( abstra2983386638146376267ist_fm @ Eff @ Rs2 @ S ) ) @ S ) ) ) ) ) ).

% RuleSystem_Defs.mkTree.code
thf(fact_855_RuleSystem__Defs_OmkTree_Ocode,axiom,
    ( abstra1192804020863382533at_nat
    = ( ^ [Eff: nat > nat > fset_nat > $o,Rs2: stream_nat,S: nat] : ( abstra4999521449322675122at_nat @ ( product_Pair_nat_nat @ S @ ( shd_nat @ ( abstra7573781713853518635at_nat @ Eff @ Rs2 @ S ) ) ) @ ( fimage8970742173175350476at_nat @ ( abstra1192804020863382533at_nat @ Eff @ ( stl_nat @ ( abstra7573781713853518635at_nat @ Eff @ Rs2 @ S ) ) ) @ ( abstra969719415724737503at_nat @ Eff @ ( shd_nat @ ( abstra7573781713853518635at_nat @ Eff @ Rs2 @ S ) ) @ S ) ) ) ) ) ).

% RuleSystem_Defs.mkTree.code
thf(fact_856_RuleSystem__Defs_OmkTree_Ocode,axiom,
    ( abstra8972947971889770147at_nat
    = ( ^ [Eff: product_prod_nat_nat > product_prod_nat_nat > fset_P5670320511379867111at_nat > $o,Rs2: stream6724221391990029191at_nat,S: product_prod_nat_nat] : ( abstra6467204794154587544at_nat @ ( produc6161850002892822231at_nat @ S @ ( shd_Pr4260400998323988397at_nat @ ( abstra4493250544927931593at_nat @ Eff @ Rs2 @ S ) ) ) @ ( fimage9147729261254313981at_nat @ ( abstra8972947971889770147at_nat @ Eff @ ( stl_Pr5027827701538482609at_nat @ ( abstra4493250544927931593at_nat @ Eff @ Rs2 @ S ) ) ) @ ( abstra673899893360562045at_nat @ Eff @ ( shd_Pr4260400998323988397at_nat @ ( abstra4493250544927931593at_nat @ Eff @ Rs2 @ S ) ) @ S ) ) ) ) ) ).

% RuleSystem_Defs.mkTree.code
thf(fact_857_RuleSystem__Defs_OmkTree_Ocode,axiom,
    ( abstra2830003028686515011m_rule
    = ( ^ [Eff: produc1996495991257130529ist_fm > rule > fset_rule > $o,Rs2: stream8299795917829157543ist_fm,S: rule] : ( abstra2711079411085783200ist_fm @ ( produc7063755442767259331ist_fm @ S @ ( shd_Pr772355297128350925ist_fm @ ( abstra5014957989051229853m_rule @ Eff @ Rs2 @ S ) ) ) @ ( fimage4184475965855427473ist_fm @ ( abstra2830003028686515011m_rule @ Eff @ ( stl_Pr5027778045650968273ist_fm @ ( abstra5014957989051229853m_rule @ Eff @ Rs2 @ S ) ) ) @ ( abstra3109574944306299113m_rule @ Eff @ ( shd_Pr772355297128350925ist_fm @ ( abstra5014957989051229853m_rule @ Eff @ Rs2 @ S ) ) @ S ) ) ) ) ) ).

% RuleSystem_Defs.mkTree.code
thf(fact_858_RuleSystem__Defs_OmkTree_Ocode,axiom,
    ( abstra178918799166095863ist_fm
    = ( ^ [Eff: rule > produc1996495991257130529ist_fm > fset_P661503646757059847ist_fm > $o,Rs2: stream_rule,S: produc1996495991257130529ist_fm] : ( abstra7467780926338743076m_rule @ ( produc491467635432902671m_rule @ S @ ( shd_rule @ ( abstra2363873759530810705ist_fm @ Eff @ Rs2 @ S ) ) ) @ ( fimage748043113761278389m_rule @ ( abstra178918799166095863ist_fm @ Eff @ ( stl_rule @ ( abstra2363873759530810705ist_fm @ Eff @ Rs2 @ S ) ) ) @ ( abstra458490714785879965ist_fm @ Eff @ ( shd_rule @ ( abstra2363873759530810705ist_fm @ Eff @ Rs2 @ S ) ) @ S ) ) ) ) ) ).

% RuleSystem_Defs.mkTree.code
thf(fact_859_mkTree__unfold,axiom,
    ! [Rs: stream_rule,S3: produc1996495991257130529ist_fm] :
      ( ( abstra178918799166095863ist_fm
        @ ^ [R: rule,S: produc1996495991257130529ist_fm,Ss: fset_P661503646757059847ist_fm] :
            ( ( eff @ R @ S )
            = ( some_f2162114139285431020ist_fm @ Ss ) )
        @ Rs
        @ S3 )
      = ( case_s2494923709252486212m_rule
        @ ^ [R: rule,S9: stream_rule] :
            ( abstra7467780926338743076m_rule @ ( produc491467635432902671m_rule @ S3 @ R )
            @ ( fimage748043113761278389m_rule
              @ ( abstra178918799166095863ist_fm
                @ ^ [S: rule,T: produc1996495991257130529ist_fm,Ss: fset_P661503646757059847ist_fm] :
                    ( ( eff @ S @ T )
                    = ( some_f2162114139285431020ist_fm @ Ss ) )
                @ S9 )
              @ ( abstra458490714785879965ist_fm
                @ ^ [S: rule,T: produc1996495991257130529ist_fm,Ss: fset_P661503646757059847ist_fm] :
                    ( ( eff @ S @ T )
                    = ( some_f2162114139285431020ist_fm @ Ss ) )
                @ R
                @ S3 ) ) )
        @ ( abstra2363873759530810705ist_fm
          @ ^ [R: rule,S: produc1996495991257130529ist_fm,Ss: fset_P661503646757059847ist_fm] :
              ( ( eff @ R @ S )
              = ( some_f2162114139285431020ist_fm @ Ss ) )
          @ Rs
          @ S3 ) ) ) ).

% mkTree_unfold
thf(fact_860_Konig,axiom,
    ! [T2: abstra4572470701423150929m_rule] :
      ( ~ ( abstra3507476289341504315m_rule @ T2 )
     => ( abstra2285467793620430502m_rule @ T2 @ ( abstra3079604473823497664m_rule @ T2 ) ) ) ).

% Konig
thf(fact_861_wf_Ocases,axiom,
    ! [A: abstra4572470701423150929m_rule] :
      ( ( abstra4476735159789068068ist_fm
        @ ^ [R: rule,S: produc1996495991257130529ist_fm,Ss: fset_P661503646757059847ist_fm] :
            ( ( eff @ R @ S )
            = ( some_f2162114139285431020ist_fm @ Ss ) )
        @ rules
        @ A )
     => ~ ( ( member_rule @ ( produc7165828336582415457m_rule @ ( abstra3503897793822247108m_rule @ A ) ) @ ( sset_rule @ rules ) )
         => ( ( ( eff @ ( produc7165828336582415457m_rule @ ( abstra3503897793822247108m_rule @ A ) ) @ ( produc6879501374131015971m_rule @ ( abstra3503897793822247108m_rule @ A ) ) )
              = ( some_f2162114139285431020ist_fm @ ( fimage9164954772359936849ist_fm @ ( comp_P4378417847848123130m_rule @ produc6879501374131015971m_rule @ abstra3503897793822247108m_rule ) @ ( abstra4962253831105676628m_rule @ A ) ) ) )
           => ~ ! [T7: abstra4572470701423150929m_rule] :
                  ( ( fmembe1635983029874454474m_rule @ T7 @ ( abstra4962253831105676628m_rule @ A ) )
                 => ( abstra4476735159789068068ist_fm
                    @ ^ [R: rule,S: produc1996495991257130529ist_fm,Ss: fset_P661503646757059847ist_fm] :
                        ( ( eff @ R @ S )
                        = ( some_f2162114139285431020ist_fm @ Ss ) )
                    @ rules
                    @ T7 ) ) ) ) ) ).

% wf.cases
thf(fact_862_tree_Oinject,axiom,
    ! [X1: produc164195504107695125m_rule,X2: fset_A5396818216090053761m_rule,Y1: produc164195504107695125m_rule,Y22: fset_A5396818216090053761m_rule] :
      ( ( ( abstra7467780926338743076m_rule @ X1 @ X2 )
        = ( abstra7467780926338743076m_rule @ Y1 @ Y22 ) )
      = ( ( X1 = Y1 )
        & ( X2 = Y22 ) ) ) ).

% tree.inject
thf(fact_863_tree_Ocollapse,axiom,
    ! [Tree: abstra4572470701423150929m_rule] :
      ( ( abstra7467780926338743076m_rule @ ( abstra3503897793822247108m_rule @ Tree ) @ ( abstra4962253831105676628m_rule @ Tree ) )
      = Tree ) ).

% tree.collapse
thf(fact_864_tree_Oexhaust,axiom,
    ! [Y3: abstra4572470701423150929m_rule] :
      ~ ! [X12: produc164195504107695125m_rule,X23: fset_A5396818216090053761m_rule] :
          ( Y3
         != ( abstra7467780926338743076m_rule @ X12 @ X23 ) ) ).

% tree.exhaust
thf(fact_865_fun_Omap__comp,axiom,
    ! [G: produc1996495991257130529ist_fm > produc1996495991257130529ist_fm,F: produc164195504107695125m_rule > produc1996495991257130529ist_fm,V: abstra4572470701423150929m_rule > produc164195504107695125m_rule] :
      ( ( comp_P2882693189979712244m_rule @ G @ ( comp_P4378417847848123130m_rule @ F @ V ) )
      = ( comp_P4378417847848123130m_rule @ ( comp_P8245093624798776376m_rule @ G @ F ) @ V ) ) ).

% fun.map_comp
thf(fact_866_fun_Omap__comp,axiom,
    ! [G: produc164195504107695125m_rule > produc1996495991257130529ist_fm,F: abstra4572470701423150929m_rule > produc164195504107695125m_rule,V: abstra4572470701423150929m_rule > abstra4572470701423150929m_rule] :
      ( ( comp_P4378417847848123130m_rule @ G @ ( comp_A7552847697395116572m_rule @ F @ V ) )
      = ( comp_A8152538925650202942m_rule @ ( comp_P4378417847848123130m_rule @ G @ F ) @ V ) ) ).

% fun.map_comp
thf(fact_867_fun_Omap__comp,axiom,
    ! [G: produc164195504107695125m_rule > produc1996495991257130529ist_fm,F: produc164195504107695125m_rule > produc164195504107695125m_rule,V: abstra4572470701423150929m_rule > produc164195504107695125m_rule] :
      ( ( comp_P4378417847848123130m_rule @ G @ ( comp_P8444364099071156576m_rule @ F @ V ) )
      = ( comp_P4378417847848123130m_rule @ ( comp_P7463969011078018366m_rule @ G @ F ) @ V ) ) ).

% fun.map_comp
thf(fact_868_comp__apply__eq,axiom,
    ! [F: produc164195504107695125m_rule > produc1996495991257130529ist_fm,G: abstra4572470701423150929m_rule > produc164195504107695125m_rule,X3: abstra4572470701423150929m_rule,H: produc164195504107695125m_rule > produc1996495991257130529ist_fm,K2: abstra4572470701423150929m_rule > produc164195504107695125m_rule] :
      ( ( ( F @ ( G @ X3 ) )
        = ( H @ ( K2 @ X3 ) ) )
     => ( ( comp_P4378417847848123130m_rule @ F @ G @ X3 )
        = ( comp_P4378417847848123130m_rule @ H @ K2 @ X3 ) ) ) ).

% comp_apply_eq
thf(fact_869_stream_Ocase__distrib,axiom,
    ! [H: abstra4572470701423150929m_rule > abstra4572470701423150929m_rule,F: rule > stream_rule > abstra4572470701423150929m_rule,Stream: stream_rule] :
      ( ( H @ ( case_s2494923709252486212m_rule @ F @ Stream ) )
      = ( case_s2494923709252486212m_rule
        @ ^ [X13: rule,X24: stream_rule] : ( H @ ( F @ X13 @ X24 ) )
        @ Stream ) ) ).

% stream.case_distrib
thf(fact_870_tree_Osel_I1_J,axiom,
    ! [X1: produc164195504107695125m_rule,X2: fset_A5396818216090053761m_rule] :
      ( ( abstra3503897793822247108m_rule @ ( abstra7467780926338743076m_rule @ X1 @ X2 ) )
      = X1 ) ).

% tree.sel(1)
thf(fact_871_tree_Osel_I2_J,axiom,
    ! [X1: produc164195504107695125m_rule,X2: fset_A5396818216090053761m_rule] :
      ( ( abstra4962253831105676628m_rule @ ( abstra7467780926338743076m_rule @ X1 @ X2 ) )
      = X2 ) ).

% tree.sel(2)
thf(fact_872_shd__def,axiom,
    ( shd_Pr7235097944458474089m_rule
    = ( case_s4599053733867249880m_rule
      @ ^ [X13: produc164195504107695125m_rule,X24: stream727092118206550309m_rule] : X13 ) ) ).

% shd_def
thf(fact_873_shd__def,axiom,
    ( shd_rule
    = ( case_s3980788006751379256e_rule
      @ ^ [X13: rule,X24: stream_rule] : X13 ) ) ).

% shd_def
thf(fact_874_tree_Oexhaust__sel,axiom,
    ! [Tree: abstra4572470701423150929m_rule] :
      ( Tree
      = ( abstra7467780926338743076m_rule @ ( abstra3503897793822247108m_rule @ Tree ) @ ( abstra4962253831105676628m_rule @ Tree ) ) ) ).

% tree.exhaust_sel
thf(fact_875_RuleSystem__Defs_Owf_Ocoinduct,axiom,
    ! [X7: abstra5042247676209264507ist_fm > $o,X3: abstra5042247676209264507ist_fm,Rules: stream_list_fm,Eff2: list_fm > list_fm > fset_list_fm > $o] :
      ( ( X7 @ X3 )
     => ( ! [X: abstra5042247676209264507ist_fm] :
            ( ( X7 @ X )
           => ? [T6: abstra5042247676209264507ist_fm] :
                ( ( X = T6 )
                & ( member_list_fm @ ( produc4588648349897876871ist_fm @ ( abstra778364809706815794ist_fm @ T6 ) ) @ ( sset_list_fm @ Rules ) )
                & ( Eff2 @ ( produc4588648349897876871ist_fm @ ( abstra778364809706815794ist_fm @ T6 ) ) @ ( produc1501393135466168645ist_fm @ ( abstra778364809706815794ist_fm @ T6 ) ) @ ( fimage9094613911859171998ist_fm @ ( comp_P8076707111405491611ist_fm @ produc1501393135466168645ist_fm @ abstra778364809706815794ist_fm ) @ ( abstra1905332785217431714ist_fm @ T6 ) ) )
                & ! [Xa: abstra5042247676209264507ist_fm] :
                    ( ( fmembe913231786053615938ist_fm @ Xa @ ( abstra1905332785217431714ist_fm @ T6 ) )
                   => ( ( X7 @ Xa )
                      | ( abstra8964572462466433336ist_fm @ Eff2 @ Rules @ Xa ) ) ) ) )
       => ( abstra8964572462466433336ist_fm @ Eff2 @ Rules @ X3 ) ) ) ).

% RuleSystem_Defs.wf.coinduct
thf(fact_876_RuleSystem__Defs_Owf_Ocoinduct,axiom,
    ! [X7: abstra4572470701423150929m_rule > $o,X3: abstra4572470701423150929m_rule,Rules: stream_rule,Eff2: rule > produc1996495991257130529ist_fm > fset_P661503646757059847ist_fm > $o] :
      ( ( X7 @ X3 )
     => ( ! [X: abstra4572470701423150929m_rule] :
            ( ( X7 @ X )
           => ? [T6: abstra4572470701423150929m_rule] :
                ( ( X = T6 )
                & ( member_rule @ ( produc7165828336582415457m_rule @ ( abstra3503897793822247108m_rule @ T6 ) ) @ ( sset_rule @ Rules ) )
                & ( Eff2 @ ( produc7165828336582415457m_rule @ ( abstra3503897793822247108m_rule @ T6 ) ) @ ( produc6879501374131015971m_rule @ ( abstra3503897793822247108m_rule @ T6 ) ) @ ( fimage9164954772359936849ist_fm @ ( comp_P4378417847848123130m_rule @ produc6879501374131015971m_rule @ abstra3503897793822247108m_rule ) @ ( abstra4962253831105676628m_rule @ T6 ) ) )
                & ! [Xa: abstra4572470701423150929m_rule] :
                    ( ( fmembe1635983029874454474m_rule @ Xa @ ( abstra4962253831105676628m_rule @ T6 ) )
                   => ( ( X7 @ Xa )
                      | ( abstra4476735159789068068ist_fm @ Eff2 @ Rules @ Xa ) ) ) ) )
       => ( abstra4476735159789068068ist_fm @ Eff2 @ Rules @ X3 ) ) ) ).

% RuleSystem_Defs.wf.coinduct
thf(fact_877_RuleSystem__Defs_Owf_Ointros,axiom,
    ! [T2: abstra5042247676209264507ist_fm,Rules: stream_list_fm,Eff2: list_fm > list_fm > fset_list_fm > $o] :
      ( ( member_list_fm @ ( produc4588648349897876871ist_fm @ ( abstra778364809706815794ist_fm @ T2 ) ) @ ( sset_list_fm @ Rules ) )
     => ( ( Eff2 @ ( produc4588648349897876871ist_fm @ ( abstra778364809706815794ist_fm @ T2 ) ) @ ( produc1501393135466168645ist_fm @ ( abstra778364809706815794ist_fm @ T2 ) ) @ ( fimage9094613911859171998ist_fm @ ( comp_P8076707111405491611ist_fm @ produc1501393135466168645ist_fm @ abstra778364809706815794ist_fm ) @ ( abstra1905332785217431714ist_fm @ T2 ) ) )
       => ( ! [T4: abstra5042247676209264507ist_fm] :
              ( ( fmembe913231786053615938ist_fm @ T4 @ ( abstra1905332785217431714ist_fm @ T2 ) )
             => ( abstra8964572462466433336ist_fm @ Eff2 @ Rules @ T4 ) )
         => ( abstra8964572462466433336ist_fm @ Eff2 @ Rules @ T2 ) ) ) ) ).

% RuleSystem_Defs.wf.intros
thf(fact_878_RuleSystem__Defs_Owf_Ointros,axiom,
    ! [T2: abstra4572470701423150929m_rule,Rules: stream_rule,Eff2: rule > produc1996495991257130529ist_fm > fset_P661503646757059847ist_fm > $o] :
      ( ( member_rule @ ( produc7165828336582415457m_rule @ ( abstra3503897793822247108m_rule @ T2 ) ) @ ( sset_rule @ Rules ) )
     => ( ( Eff2 @ ( produc7165828336582415457m_rule @ ( abstra3503897793822247108m_rule @ T2 ) ) @ ( produc6879501374131015971m_rule @ ( abstra3503897793822247108m_rule @ T2 ) ) @ ( fimage9164954772359936849ist_fm @ ( comp_P4378417847848123130m_rule @ produc6879501374131015971m_rule @ abstra3503897793822247108m_rule ) @ ( abstra4962253831105676628m_rule @ T2 ) ) )
       => ( ! [T4: abstra4572470701423150929m_rule] :
              ( ( fmembe1635983029874454474m_rule @ T4 @ ( abstra4962253831105676628m_rule @ T2 ) )
             => ( abstra4476735159789068068ist_fm @ Eff2 @ Rules @ T4 ) )
         => ( abstra4476735159789068068ist_fm @ Eff2 @ Rules @ T2 ) ) ) ) ).

% RuleSystem_Defs.wf.intros
thf(fact_879_RuleSystem__Defs_Owf_Osimps,axiom,
    ( abstra8964572462466433336ist_fm
    = ( ^ [Eff: list_fm > list_fm > fset_list_fm > $o,Rules2: stream_list_fm,A2: abstra5042247676209264507ist_fm] :
        ? [T: abstra5042247676209264507ist_fm] :
          ( ( A2 = T )
          & ( member_list_fm @ ( produc4588648349897876871ist_fm @ ( abstra778364809706815794ist_fm @ T ) ) @ ( sset_list_fm @ Rules2 ) )
          & ( Eff @ ( produc4588648349897876871ist_fm @ ( abstra778364809706815794ist_fm @ T ) ) @ ( produc1501393135466168645ist_fm @ ( abstra778364809706815794ist_fm @ T ) ) @ ( fimage9094613911859171998ist_fm @ ( comp_P8076707111405491611ist_fm @ produc1501393135466168645ist_fm @ abstra778364809706815794ist_fm ) @ ( abstra1905332785217431714ist_fm @ T ) ) )
          & ! [X6: abstra5042247676209264507ist_fm] :
              ( ( fmembe913231786053615938ist_fm @ X6 @ ( abstra1905332785217431714ist_fm @ T ) )
             => ( abstra8964572462466433336ist_fm @ Eff @ Rules2 @ X6 ) ) ) ) ) ).

% RuleSystem_Defs.wf.simps
thf(fact_880_RuleSystem__Defs_Owf_Osimps,axiom,
    ( abstra4476735159789068068ist_fm
    = ( ^ [Eff: rule > produc1996495991257130529ist_fm > fset_P661503646757059847ist_fm > $o,Rules2: stream_rule,A2: abstra4572470701423150929m_rule] :
        ? [T: abstra4572470701423150929m_rule] :
          ( ( A2 = T )
          & ( member_rule @ ( produc7165828336582415457m_rule @ ( abstra3503897793822247108m_rule @ T ) ) @ ( sset_rule @ Rules2 ) )
          & ( Eff @ ( produc7165828336582415457m_rule @ ( abstra3503897793822247108m_rule @ T ) ) @ ( produc6879501374131015971m_rule @ ( abstra3503897793822247108m_rule @ T ) ) @ ( fimage9164954772359936849ist_fm @ ( comp_P4378417847848123130m_rule @ produc6879501374131015971m_rule @ abstra3503897793822247108m_rule ) @ ( abstra4962253831105676628m_rule @ T ) ) )
          & ! [X6: abstra4572470701423150929m_rule] :
              ( ( fmembe1635983029874454474m_rule @ X6 @ ( abstra4962253831105676628m_rule @ T ) )
             => ( abstra4476735159789068068ist_fm @ Eff @ Rules2 @ X6 ) ) ) ) ) ).

% RuleSystem_Defs.wf.simps
thf(fact_881_RuleSystem__Defs_Owf_Ocases,axiom,
    ! [Eff2: list_fm > list_fm > fset_list_fm > $o,Rules: stream_list_fm,A: abstra5042247676209264507ist_fm] :
      ( ( abstra8964572462466433336ist_fm @ Eff2 @ Rules @ A )
     => ~ ( ( member_list_fm @ ( produc4588648349897876871ist_fm @ ( abstra778364809706815794ist_fm @ A ) ) @ ( sset_list_fm @ Rules ) )
         => ( ( Eff2 @ ( produc4588648349897876871ist_fm @ ( abstra778364809706815794ist_fm @ A ) ) @ ( produc1501393135466168645ist_fm @ ( abstra778364809706815794ist_fm @ A ) ) @ ( fimage9094613911859171998ist_fm @ ( comp_P8076707111405491611ist_fm @ produc1501393135466168645ist_fm @ abstra778364809706815794ist_fm ) @ ( abstra1905332785217431714ist_fm @ A ) ) )
           => ~ ! [T7: abstra5042247676209264507ist_fm] :
                  ( ( fmembe913231786053615938ist_fm @ T7 @ ( abstra1905332785217431714ist_fm @ A ) )
                 => ( abstra8964572462466433336ist_fm @ Eff2 @ Rules @ T7 ) ) ) ) ) ).

% RuleSystem_Defs.wf.cases
thf(fact_882_RuleSystem__Defs_Owf_Ocases,axiom,
    ! [Eff2: rule > produc1996495991257130529ist_fm > fset_P661503646757059847ist_fm > $o,Rules: stream_rule,A: abstra4572470701423150929m_rule] :
      ( ( abstra4476735159789068068ist_fm @ Eff2 @ Rules @ A )
     => ~ ( ( member_rule @ ( produc7165828336582415457m_rule @ ( abstra3503897793822247108m_rule @ A ) ) @ ( sset_rule @ Rules ) )
         => ( ( Eff2 @ ( produc7165828336582415457m_rule @ ( abstra3503897793822247108m_rule @ A ) ) @ ( produc6879501374131015971m_rule @ ( abstra3503897793822247108m_rule @ A ) ) @ ( fimage9164954772359936849ist_fm @ ( comp_P4378417847848123130m_rule @ produc6879501374131015971m_rule @ abstra3503897793822247108m_rule ) @ ( abstra4962253831105676628m_rule @ A ) ) )
           => ~ ! [T7: abstra4572470701423150929m_rule] :
                  ( ( fmembe1635983029874454474m_rule @ T7 @ ( abstra4962253831105676628m_rule @ A ) )
                 => ( abstra4476735159789068068ist_fm @ Eff2 @ Rules @ T7 ) ) ) ) ) ).

% RuleSystem_Defs.wf.cases
thf(fact_883_stream_Ocase__eq__if,axiom,
    ( case_s2494923709252486212m_rule
    = ( ^ [F2: rule > stream_rule > abstra4572470701423150929m_rule,Stream5: stream_rule] : ( F2 @ ( shd_rule @ Stream5 ) @ ( stl_rule @ Stream5 ) ) ) ) ).

% stream.case_eq_if
thf(fact_884_i_Owf_Ocases,axiom,
    ! [Eff3: list_fm > list_fm > option_fset_list_fm,Rules: stream_list_fm,A: abstra5042247676209264507ist_fm] :
      ( ( abstra8964572462466433336ist_fm @ ( abstra256862739949234434ist_fm @ Eff3 ) @ Rules @ A )
     => ~ ( ( member_list_fm @ ( produc4588648349897876871ist_fm @ ( abstra778364809706815794ist_fm @ A ) ) @ ( sset_list_fm @ Rules ) )
         => ( ( abstra256862739949234434ist_fm @ Eff3 @ ( produc4588648349897876871ist_fm @ ( abstra778364809706815794ist_fm @ A ) ) @ ( produc1501393135466168645ist_fm @ ( abstra778364809706815794ist_fm @ A ) ) @ ( fimage9094613911859171998ist_fm @ ( comp_P8076707111405491611ist_fm @ produc1501393135466168645ist_fm @ abstra778364809706815794ist_fm ) @ ( abstra1905332785217431714ist_fm @ A ) ) )
           => ~ ! [T7: abstra5042247676209264507ist_fm] :
                  ( ( fmembe913231786053615938ist_fm @ T7 @ ( abstra1905332785217431714ist_fm @ A ) )
                 => ( abstra8964572462466433336ist_fm @ ( abstra256862739949234434ist_fm @ Eff3 ) @ Rules @ T7 ) ) ) ) ) ).

% i.wf.cases
thf(fact_885_i_Owf_Ocases,axiom,
    ! [Eff3: rule > produc1996495991257130529ist_fm > option6029102822600979917ist_fm,Rules: stream_rule,A: abstra4572470701423150929m_rule] :
      ( ( abstra4476735159789068068ist_fm @ ( abstra5459341389059251418ist_fm @ Eff3 ) @ Rules @ A )
     => ~ ( ( member_rule @ ( produc7165828336582415457m_rule @ ( abstra3503897793822247108m_rule @ A ) ) @ ( sset_rule @ Rules ) )
         => ( ( abstra5459341389059251418ist_fm @ Eff3 @ ( produc7165828336582415457m_rule @ ( abstra3503897793822247108m_rule @ A ) ) @ ( produc6879501374131015971m_rule @ ( abstra3503897793822247108m_rule @ A ) ) @ ( fimage9164954772359936849ist_fm @ ( comp_P4378417847848123130m_rule @ produc6879501374131015971m_rule @ abstra3503897793822247108m_rule ) @ ( abstra4962253831105676628m_rule @ A ) ) )
           => ~ ! [T7: abstra4572470701423150929m_rule] :
                  ( ( fmembe1635983029874454474m_rule @ T7 @ ( abstra4962253831105676628m_rule @ A ) )
                 => ( abstra4476735159789068068ist_fm @ ( abstra5459341389059251418ist_fm @ Eff3 ) @ Rules @ T7 ) ) ) ) ) ).

% i.wf.cases
thf(fact_886_i_Owf_Osimps,axiom,
    ! [Eff3: list_fm > list_fm > option_fset_list_fm,Rules: stream_list_fm,A: abstra5042247676209264507ist_fm] :
      ( ( abstra8964572462466433336ist_fm @ ( abstra256862739949234434ist_fm @ Eff3 ) @ Rules @ A )
      = ( ? [T: abstra5042247676209264507ist_fm] :
            ( ( A = T )
            & ( member_list_fm @ ( produc4588648349897876871ist_fm @ ( abstra778364809706815794ist_fm @ T ) ) @ ( sset_list_fm @ Rules ) )
            & ( abstra256862739949234434ist_fm @ Eff3 @ ( produc4588648349897876871ist_fm @ ( abstra778364809706815794ist_fm @ T ) ) @ ( produc1501393135466168645ist_fm @ ( abstra778364809706815794ist_fm @ T ) ) @ ( fimage9094613911859171998ist_fm @ ( comp_P8076707111405491611ist_fm @ produc1501393135466168645ist_fm @ abstra778364809706815794ist_fm ) @ ( abstra1905332785217431714ist_fm @ T ) ) )
            & ! [X6: abstra5042247676209264507ist_fm] :
                ( ( fmembe913231786053615938ist_fm @ X6 @ ( abstra1905332785217431714ist_fm @ T ) )
               => ( abstra8964572462466433336ist_fm @ ( abstra256862739949234434ist_fm @ Eff3 ) @ Rules @ X6 ) ) ) ) ) ).

% i.wf.simps
thf(fact_887_i_Owf_Osimps,axiom,
    ! [Eff3: rule > produc1996495991257130529ist_fm > option6029102822600979917ist_fm,Rules: stream_rule,A: abstra4572470701423150929m_rule] :
      ( ( abstra4476735159789068068ist_fm @ ( abstra5459341389059251418ist_fm @ Eff3 ) @ Rules @ A )
      = ( ? [T: abstra4572470701423150929m_rule] :
            ( ( A = T )
            & ( member_rule @ ( produc7165828336582415457m_rule @ ( abstra3503897793822247108m_rule @ T ) ) @ ( sset_rule @ Rules ) )
            & ( abstra5459341389059251418ist_fm @ Eff3 @ ( produc7165828336582415457m_rule @ ( abstra3503897793822247108m_rule @ T ) ) @ ( produc6879501374131015971m_rule @ ( abstra3503897793822247108m_rule @ T ) ) @ ( fimage9164954772359936849ist_fm @ ( comp_P4378417847848123130m_rule @ produc6879501374131015971m_rule @ abstra3503897793822247108m_rule ) @ ( abstra4962253831105676628m_rule @ T ) ) )
            & ! [X6: abstra4572470701423150929m_rule] :
                ( ( fmembe1635983029874454474m_rule @ X6 @ ( abstra4962253831105676628m_rule @ T ) )
               => ( abstra4476735159789068068ist_fm @ ( abstra5459341389059251418ist_fm @ Eff3 ) @ Rules @ X6 ) ) ) ) ) ).

% i.wf.simps
thf(fact_888_i_Owf_Ointros,axiom,
    ! [T2: abstra5042247676209264507ist_fm,Rules: stream_list_fm,Eff3: list_fm > list_fm > option_fset_list_fm] :
      ( ( member_list_fm @ ( produc4588648349897876871ist_fm @ ( abstra778364809706815794ist_fm @ T2 ) ) @ ( sset_list_fm @ Rules ) )
     => ( ( abstra256862739949234434ist_fm @ Eff3 @ ( produc4588648349897876871ist_fm @ ( abstra778364809706815794ist_fm @ T2 ) ) @ ( produc1501393135466168645ist_fm @ ( abstra778364809706815794ist_fm @ T2 ) ) @ ( fimage9094613911859171998ist_fm @ ( comp_P8076707111405491611ist_fm @ produc1501393135466168645ist_fm @ abstra778364809706815794ist_fm ) @ ( abstra1905332785217431714ist_fm @ T2 ) ) )
       => ( ! [T4: abstra5042247676209264507ist_fm] :
              ( ( fmembe913231786053615938ist_fm @ T4 @ ( abstra1905332785217431714ist_fm @ T2 ) )
             => ( abstra8964572462466433336ist_fm @ ( abstra256862739949234434ist_fm @ Eff3 ) @ Rules @ T4 ) )
         => ( abstra8964572462466433336ist_fm @ ( abstra256862739949234434ist_fm @ Eff3 ) @ Rules @ T2 ) ) ) ) ).

% i.wf.intros
thf(fact_889_i_Owf_Ointros,axiom,
    ! [T2: abstra4572470701423150929m_rule,Rules: stream_rule,Eff3: rule > produc1996495991257130529ist_fm > option6029102822600979917ist_fm] :
      ( ( member_rule @ ( produc7165828336582415457m_rule @ ( abstra3503897793822247108m_rule @ T2 ) ) @ ( sset_rule @ Rules ) )
     => ( ( abstra5459341389059251418ist_fm @ Eff3 @ ( produc7165828336582415457m_rule @ ( abstra3503897793822247108m_rule @ T2 ) ) @ ( produc6879501374131015971m_rule @ ( abstra3503897793822247108m_rule @ T2 ) ) @ ( fimage9164954772359936849ist_fm @ ( comp_P4378417847848123130m_rule @ produc6879501374131015971m_rule @ abstra3503897793822247108m_rule ) @ ( abstra4962253831105676628m_rule @ T2 ) ) )
       => ( ! [T4: abstra4572470701423150929m_rule] :
              ( ( fmembe1635983029874454474m_rule @ T4 @ ( abstra4962253831105676628m_rule @ T2 ) )
             => ( abstra4476735159789068068ist_fm @ ( abstra5459341389059251418ist_fm @ Eff3 ) @ Rules @ T4 ) )
         => ( abstra4476735159789068068ist_fm @ ( abstra5459341389059251418ist_fm @ Eff3 ) @ Rules @ T2 ) ) ) ) ).

% i.wf.intros
thf(fact_890_i_Owf_Ocoinduct,axiom,
    ! [X7: abstra5042247676209264507ist_fm > $o,X3: abstra5042247676209264507ist_fm,Rules: stream_list_fm,Eff3: list_fm > list_fm > option_fset_list_fm] :
      ( ( X7 @ X3 )
     => ( ! [X: abstra5042247676209264507ist_fm] :
            ( ( X7 @ X )
           => ? [T6: abstra5042247676209264507ist_fm] :
                ( ( X = T6 )
                & ( member_list_fm @ ( produc4588648349897876871ist_fm @ ( abstra778364809706815794ist_fm @ T6 ) ) @ ( sset_list_fm @ Rules ) )
                & ( abstra256862739949234434ist_fm @ Eff3 @ ( produc4588648349897876871ist_fm @ ( abstra778364809706815794ist_fm @ T6 ) ) @ ( produc1501393135466168645ist_fm @ ( abstra778364809706815794ist_fm @ T6 ) ) @ ( fimage9094613911859171998ist_fm @ ( comp_P8076707111405491611ist_fm @ produc1501393135466168645ist_fm @ abstra778364809706815794ist_fm ) @ ( abstra1905332785217431714ist_fm @ T6 ) ) )
                & ! [Xa: abstra5042247676209264507ist_fm] :
                    ( ( fmembe913231786053615938ist_fm @ Xa @ ( abstra1905332785217431714ist_fm @ T6 ) )
                   => ( ( X7 @ Xa )
                      | ( abstra8964572462466433336ist_fm @ ( abstra256862739949234434ist_fm @ Eff3 ) @ Rules @ Xa ) ) ) ) )
       => ( abstra8964572462466433336ist_fm @ ( abstra256862739949234434ist_fm @ Eff3 ) @ Rules @ X3 ) ) ) ).

% i.wf.coinduct
thf(fact_891_i_Owf_Ocoinduct,axiom,
    ! [X7: abstra4572470701423150929m_rule > $o,X3: abstra4572470701423150929m_rule,Rules: stream_rule,Eff3: rule > produc1996495991257130529ist_fm > option6029102822600979917ist_fm] :
      ( ( X7 @ X3 )
     => ( ! [X: abstra4572470701423150929m_rule] :
            ( ( X7 @ X )
           => ? [T6: abstra4572470701423150929m_rule] :
                ( ( X = T6 )
                & ( member_rule @ ( produc7165828336582415457m_rule @ ( abstra3503897793822247108m_rule @ T6 ) ) @ ( sset_rule @ Rules ) )
                & ( abstra5459341389059251418ist_fm @ Eff3 @ ( produc7165828336582415457m_rule @ ( abstra3503897793822247108m_rule @ T6 ) ) @ ( produc6879501374131015971m_rule @ ( abstra3503897793822247108m_rule @ T6 ) ) @ ( fimage9164954772359936849ist_fm @ ( comp_P4378417847848123130m_rule @ produc6879501374131015971m_rule @ abstra3503897793822247108m_rule ) @ ( abstra4962253831105676628m_rule @ T6 ) ) )
                & ! [Xa: abstra4572470701423150929m_rule] :
                    ( ( fmembe1635983029874454474m_rule @ Xa @ ( abstra4962253831105676628m_rule @ T6 ) )
                   => ( ( X7 @ Xa )
                      | ( abstra4476735159789068068ist_fm @ ( abstra5459341389059251418ist_fm @ Eff3 ) @ Rules @ Xa ) ) ) ) )
       => ( abstra4476735159789068068ist_fm @ ( abstra5459341389059251418ist_fm @ Eff3 ) @ Rules @ X3 ) ) ) ).

% i.wf.coinduct
thf(fact_892_RuleSystem__Defs_OmkTree__unfold,axiom,
    ( abstra3846921640779263781ist_fm
    = ( ^ [Eff: list_fm > list_fm > fset_list_fm > $o,Rs2: stream_list_fm,S: list_fm] :
          ( case_s5690291548269216621ist_fm
          @ ^ [R: list_fm,S9: stream_list_fm] : ( abstra6542830291470149330ist_fm @ ( produc381145313068854617ist_fm @ S @ R ) @ ( fimage27253841482530780ist_fm @ ( abstra3846921640779263781ist_fm @ Eff @ S9 ) @ ( abstra1097029126060538623ist_fm @ Eff @ R @ S ) ) )
          @ ( abstra2983386638146376267ist_fm @ Eff @ Rs2 @ S ) ) ) ) ).

% RuleSystem_Defs.mkTree_unfold
thf(fact_893_RuleSystem__Defs_OmkTree__unfold,axiom,
    ( abstra1192804020863382533at_nat
    = ( ^ [Eff: nat > nat > fset_nat > $o,Rs2: stream_nat,S: nat] :
          ( case_s2280698752336340829at_nat
          @ ^ [R: nat,S9: stream_nat] : ( abstra4999521449322675122at_nat @ ( product_Pair_nat_nat @ S @ R ) @ ( fimage8970742173175350476at_nat @ ( abstra1192804020863382533at_nat @ Eff @ S9 ) @ ( abstra969719415724737503at_nat @ Eff @ R @ S ) ) )
          @ ( abstra7573781713853518635at_nat @ Eff @ Rs2 @ S ) ) ) ) ).

% RuleSystem_Defs.mkTree_unfold
thf(fact_894_RuleSystem__Defs_OmkTree__unfold,axiom,
    ( abstra8972947971889770147at_nat
    = ( ^ [Eff: product_prod_nat_nat > product_prod_nat_nat > fset_P5670320511379867111at_nat > $o,Rs2: stream6724221391990029191at_nat,S: product_prod_nat_nat] :
          ( case_s6993487711257930156at_nat
          @ ^ [R: product_prod_nat_nat,S9: stream6724221391990029191at_nat] : ( abstra6467204794154587544at_nat @ ( produc6161850002892822231at_nat @ S @ R ) @ ( fimage9147729261254313981at_nat @ ( abstra8972947971889770147at_nat @ Eff @ S9 ) @ ( abstra673899893360562045at_nat @ Eff @ R @ S ) ) )
          @ ( abstra4493250544927931593at_nat @ Eff @ Rs2 @ S ) ) ) ) ).

% RuleSystem_Defs.mkTree_unfold
thf(fact_895_RuleSystem__Defs_OmkTree__unfold,axiom,
    ( abstra2830003028686515011m_rule
    = ( ^ [Eff: produc1996495991257130529ist_fm > rule > fset_rule > $o,Rs2: stream8299795917829157543ist_fm,S: rule] :
          ( case_s482378023405642050ist_fm
          @ ^ [R: produc1996495991257130529ist_fm,S9: stream8299795917829157543ist_fm] : ( abstra2711079411085783200ist_fm @ ( produc7063755442767259331ist_fm @ S @ R ) @ ( fimage4184475965855427473ist_fm @ ( abstra2830003028686515011m_rule @ Eff @ S9 ) @ ( abstra3109574944306299113m_rule @ Eff @ R @ S ) ) )
          @ ( abstra5014957989051229853m_rule @ Eff @ Rs2 @ S ) ) ) ) ).

% RuleSystem_Defs.mkTree_unfold
thf(fact_896_RuleSystem__Defs_OmkTree__unfold,axiom,
    ( abstra178918799166095863ist_fm
    = ( ^ [Eff: rule > produc1996495991257130529ist_fm > fset_P661503646757059847ist_fm > $o,Rs2: stream_rule,S: produc1996495991257130529ist_fm] :
          ( case_s2494923709252486212m_rule
          @ ^ [R: rule,S9: stream_rule] : ( abstra7467780926338743076m_rule @ ( produc491467635432902671m_rule @ S @ R ) @ ( fimage748043113761278389m_rule @ ( abstra178918799166095863ist_fm @ Eff @ S9 ) @ ( abstra458490714785879965ist_fm @ Eff @ R @ S ) ) )
          @ ( abstra2363873759530810705ist_fm @ Eff @ Rs2 @ S ) ) ) ) ).

% RuleSystem_Defs.mkTree_unfold
thf(fact_897_i_OmkTree__unfold,axiom,
    ! [Eff3: list_fm > list_fm > option_fset_list_fm,Rs: stream_list_fm,S3: list_fm] :
      ( ( abstra3846921640779263781ist_fm @ ( abstra256862739949234434ist_fm @ Eff3 ) @ Rs @ S3 )
      = ( case_s5690291548269216621ist_fm
        @ ^ [R: list_fm,S9: stream_list_fm] : ( abstra6542830291470149330ist_fm @ ( produc381145313068854617ist_fm @ S3 @ R ) @ ( fimage27253841482530780ist_fm @ ( abstra3846921640779263781ist_fm @ ( abstra256862739949234434ist_fm @ Eff3 ) @ S9 ) @ ( abstra1097029126060538623ist_fm @ ( abstra256862739949234434ist_fm @ Eff3 ) @ R @ S3 ) ) )
        @ ( abstra2983386638146376267ist_fm @ ( abstra256862739949234434ist_fm @ Eff3 ) @ Rs @ S3 ) ) ) ).

% i.mkTree_unfold
thf(fact_898_i_OmkTree__unfold,axiom,
    ! [Eff3: nat > nat > option_fset_nat,Rs: stream_nat,S3: nat] :
      ( ( abstra1192804020863382533at_nat @ ( abstra4671362528648695778at_nat @ Eff3 ) @ Rs @ S3 )
      = ( case_s2280698752336340829at_nat
        @ ^ [R: nat,S9: stream_nat] : ( abstra4999521449322675122at_nat @ ( product_Pair_nat_nat @ S3 @ R ) @ ( fimage8970742173175350476at_nat @ ( abstra1192804020863382533at_nat @ ( abstra4671362528648695778at_nat @ Eff3 ) @ S9 ) @ ( abstra969719415724737503at_nat @ ( abstra4671362528648695778at_nat @ Eff3 ) @ R @ S3 ) ) )
        @ ( abstra7573781713853518635at_nat @ ( abstra4671362528648695778at_nat @ Eff3 ) @ Rs @ S3 ) ) ) ).

% i.mkTree_unfold
thf(fact_899_i_OmkTree__unfold,axiom,
    ! [Eff3: product_prod_nat_nat > product_prod_nat_nat > option7922001270233832621at_nat,Rs: stream6724221391990029191at_nat,S3: product_prod_nat_nat] :
      ( ( abstra8972947971889770147at_nat @ ( abstra426094224824445696at_nat @ Eff3 ) @ Rs @ S3 )
      = ( case_s6993487711257930156at_nat
        @ ^ [R: product_prod_nat_nat,S9: stream6724221391990029191at_nat] : ( abstra6467204794154587544at_nat @ ( produc6161850002892822231at_nat @ S3 @ R ) @ ( fimage9147729261254313981at_nat @ ( abstra8972947971889770147at_nat @ ( abstra426094224824445696at_nat @ Eff3 ) @ S9 ) @ ( abstra673899893360562045at_nat @ ( abstra426094224824445696at_nat @ Eff3 ) @ R @ S3 ) ) )
        @ ( abstra4493250544927931593at_nat @ ( abstra426094224824445696at_nat @ Eff3 ) @ Rs @ S3 ) ) ) ).

% i.mkTree_unfold
thf(fact_900_i_OmkTree__unfold,axiom,
    ! [Eff3: produc1996495991257130529ist_fm > rule > option_fset_rule,Rs: stream8299795917829157543ist_fm,S3: rule] :
      ( ( abstra2830003028686515011m_rule @ ( abstra8110425618579670566m_rule @ Eff3 ) @ Rs @ S3 )
      = ( case_s482378023405642050ist_fm
        @ ^ [R: produc1996495991257130529ist_fm,S9: stream8299795917829157543ist_fm] : ( abstra2711079411085783200ist_fm @ ( produc7063755442767259331ist_fm @ S3 @ R ) @ ( fimage4184475965855427473ist_fm @ ( abstra2830003028686515011m_rule @ ( abstra8110425618579670566m_rule @ Eff3 ) @ S9 ) @ ( abstra3109574944306299113m_rule @ ( abstra8110425618579670566m_rule @ Eff3 ) @ R @ S3 ) ) )
        @ ( abstra5014957989051229853m_rule @ ( abstra8110425618579670566m_rule @ Eff3 ) @ Rs @ S3 ) ) ) ).

% i.mkTree_unfold
thf(fact_901_i_OmkTree__unfold,axiom,
    ! [Eff3: rule > produc1996495991257130529ist_fm > option6029102822600979917ist_fm,Rs: stream_rule,S3: produc1996495991257130529ist_fm] :
      ( ( abstra178918799166095863ist_fm @ ( abstra5459341389059251418ist_fm @ Eff3 ) @ Rs @ S3 )
      = ( case_s2494923709252486212m_rule
        @ ^ [R: rule,S9: stream_rule] : ( abstra7467780926338743076m_rule @ ( produc491467635432902671m_rule @ S3 @ R ) @ ( fimage748043113761278389m_rule @ ( abstra178918799166095863ist_fm @ ( abstra5459341389059251418ist_fm @ Eff3 ) @ S9 ) @ ( abstra458490714785879965ist_fm @ ( abstra5459341389059251418ist_fm @ Eff3 ) @ R @ S3 ) ) )
        @ ( abstra2363873759530810705ist_fm @ ( abstra5459341389059251418ist_fm @ Eff3 ) @ Rs @ S3 ) ) ) ).

% i.mkTree_unfold
thf(fact_902_konig_Osimps_I1_J,axiom,
    ! [T2: abstract_tree_rule] :
      ( ( shd_rule @ ( abstract_konig_rule @ T2 ) )
      = ( abstract_root_rule @ T2 ) ) ).

% konig.simps(1)
thf(fact_903_konig_Osimps_I1_J,axiom,
    ! [T2: abstra4572470701423150929m_rule] :
      ( ( shd_Pr7235097944458474089m_rule @ ( abstra3079604473823497664m_rule @ T2 ) )
      = ( abstra3503897793822247108m_rule @ T2 ) ) ).

% konig.simps(1)
thf(fact_904_wf_Ocoinduct,axiom,
    ! [X7: abstra4572470701423150929m_rule > $o,X3: abstra4572470701423150929m_rule] :
      ( ( X7 @ X3 )
     => ( ! [X: abstra4572470701423150929m_rule] :
            ( ( X7 @ X )
           => ? [T6: abstra4572470701423150929m_rule] :
                ( ( X = T6 )
                & ( member_rule @ ( produc7165828336582415457m_rule @ ( abstra3503897793822247108m_rule @ T6 ) ) @ ( sset_rule @ rules ) )
                & ( ( eff @ ( produc7165828336582415457m_rule @ ( abstra3503897793822247108m_rule @ T6 ) ) @ ( produc6879501374131015971m_rule @ ( abstra3503897793822247108m_rule @ T6 ) ) )
                  = ( some_f2162114139285431020ist_fm @ ( fimage9164954772359936849ist_fm @ ( comp_P4378417847848123130m_rule @ produc6879501374131015971m_rule @ abstra3503897793822247108m_rule ) @ ( abstra4962253831105676628m_rule @ T6 ) ) ) )
                & ! [Xa: abstra4572470701423150929m_rule] :
                    ( ( fmembe1635983029874454474m_rule @ Xa @ ( abstra4962253831105676628m_rule @ T6 ) )
                   => ( ( X7 @ Xa )
                      | ( abstra4476735159789068068ist_fm
                        @ ^ [R: rule,S: produc1996495991257130529ist_fm,Ss: fset_P661503646757059847ist_fm] :
                            ( ( eff @ R @ S )
                            = ( some_f2162114139285431020ist_fm @ Ss ) )
                        @ rules
                        @ Xa ) ) ) ) )
       => ( abstra4476735159789068068ist_fm
          @ ^ [R: rule,S: produc1996495991257130529ist_fm,Ss: fset_P661503646757059847ist_fm] :
              ( ( eff @ R @ S )
              = ( some_f2162114139285431020ist_fm @ Ss ) )
          @ rules
          @ X3 ) ) ) ).

% wf.coinduct
thf(fact_905_wf_Ointros,axiom,
    ! [T2: abstra4572470701423150929m_rule] :
      ( ( member_rule @ ( produc7165828336582415457m_rule @ ( abstra3503897793822247108m_rule @ T2 ) ) @ ( sset_rule @ rules ) )
     => ( ( ( eff @ ( produc7165828336582415457m_rule @ ( abstra3503897793822247108m_rule @ T2 ) ) @ ( produc6879501374131015971m_rule @ ( abstra3503897793822247108m_rule @ T2 ) ) )
          = ( some_f2162114139285431020ist_fm @ ( fimage9164954772359936849ist_fm @ ( comp_P4378417847848123130m_rule @ produc6879501374131015971m_rule @ abstra3503897793822247108m_rule ) @ ( abstra4962253831105676628m_rule @ T2 ) ) ) )
       => ( ! [T4: abstra4572470701423150929m_rule] :
              ( ( fmembe1635983029874454474m_rule @ T4 @ ( abstra4962253831105676628m_rule @ T2 ) )
             => ( abstra4476735159789068068ist_fm
                @ ^ [R: rule,S: produc1996495991257130529ist_fm,Ss: fset_P661503646757059847ist_fm] :
                    ( ( eff @ R @ S )
                    = ( some_f2162114139285431020ist_fm @ Ss ) )
                @ rules
                @ T4 ) )
         => ( abstra4476735159789068068ist_fm
            @ ^ [R: rule,S: produc1996495991257130529ist_fm,Ss: fset_P661503646757059847ist_fm] :
                ( ( eff @ R @ S )
                = ( some_f2162114139285431020ist_fm @ Ss ) )
            @ rules
            @ T2 ) ) ) ) ).

% wf.intros
thf(fact_906_wf_Osimps,axiom,
    ! [A: abstra4572470701423150929m_rule] :
      ( ( abstra4476735159789068068ist_fm
        @ ^ [R: rule,S: produc1996495991257130529ist_fm,Ss: fset_P661503646757059847ist_fm] :
            ( ( eff @ R @ S )
            = ( some_f2162114139285431020ist_fm @ Ss ) )
        @ rules
        @ A )
      = ( ? [T: abstra4572470701423150929m_rule] :
            ( ( A = T )
            & ( member_rule @ ( produc7165828336582415457m_rule @ ( abstra3503897793822247108m_rule @ T ) ) @ ( sset_rule @ rules ) )
            & ( ( eff @ ( produc7165828336582415457m_rule @ ( abstra3503897793822247108m_rule @ T ) ) @ ( produc6879501374131015971m_rule @ ( abstra3503897793822247108m_rule @ T ) ) )
              = ( some_f2162114139285431020ist_fm @ ( fimage9164954772359936849ist_fm @ ( comp_P4378417847848123130m_rule @ produc6879501374131015971m_rule @ abstra3503897793822247108m_rule ) @ ( abstra4962253831105676628m_rule @ T ) ) ) )
            & ! [X6: abstra4572470701423150929m_rule] :
                ( ( fmembe1635983029874454474m_rule @ X6 @ ( abstra4962253831105676628m_rule @ T ) )
               => ( abstra4476735159789068068ist_fm
                  @ ^ [R: rule,S: produc1996495991257130529ist_fm,Ss: fset_P661503646757059847ist_fm] :
                      ( ( eff @ R @ S )
                      = ( some_f2162114139285431020ist_fm @ Ss ) )
                  @ rules
                  @ X6 ) ) ) ) ) ).

% wf.simps
thf(fact_907_konig_Osimps_I2_J,axiom,
    ! [T2: abstract_tree_rule] :
      ( ( stl_rule @ ( abstract_konig_rule @ T2 ) )
      = ( abstract_konig_rule
        @ ( fChoic8596484284512669613e_rule
          @ ^ [T5: abstract_tree_rule] :
              ( ( fmembe7964111328986570234e_rule @ T5 @ ( abstract_cont_rule @ T2 ) )
              & ~ ( abstra6797871900832723819e_rule @ T5 ) ) ) ) ) ).

% konig.simps(2)
thf(fact_908_konig_Osimps_I2_J,axiom,
    ! [T2: abstra4572470701423150929m_rule] :
      ( ( stl_Pr950425576149878629m_rule @ ( abstra3079604473823497664m_rule @ T2 ) )
      = ( abstra3079604473823497664m_rule
        @ ( fChoic7543385803022166781m_rule
          @ ^ [T5: abstra4572470701423150929m_rule] :
              ( ( fmembe1635983029874454474m_rule @ T5 @ ( abstra4962253831105676628m_rule @ T2 ) )
              & ~ ( abstra3507476289341504315m_rule @ T5 ) ) ) ) ) ).

% konig.simps(2)
thf(fact_909_fset__of__list__subset,axiom,
    ! [Xs: list_fm,Ys: list_fm] :
      ( ( ord_less_eq_set_fm @ ( set_fm2 @ Xs ) @ ( set_fm2 @ Ys ) )
     => ( ord_less_eq_fset_fm @ ( fset_of_list_fm @ Xs ) @ ( fset_of_list_fm @ Ys ) ) ) ).

% fset_of_list_subset
thf(fact_910_sdrop__while__sdrop__LEAST,axiom,
    ! [P2: produc164195504107695125m_rule > $o,S3: stream727092118206550309m_rule] :
      ( ? [N5: nat] : ( P2 @ ( snth_P1712579846118164081m_rule @ S3 @ N5 ) )
     => ( ( sdrop_5541336384294084785m_rule @ ( comp_o5517589812586031224m_rule @ (~) @ P2 ) @ S3 )
        = ( sdrop_7224736112439592940m_rule
          @ ( ord_Least_nat
            @ ^ [N3: nat] : ( P2 @ ( snth_P1712579846118164081m_rule @ S3 @ N3 ) ) )
          @ S3 ) ) ) ).

% sdrop_while_sdrop_LEAST
thf(fact_911_sdrop__while__sdrop__LEAST,axiom,
    ! [P2: produc1996495991257130529ist_fm > $o,S3: stream8299795917829157543ist_fm] :
      ( ? [N5: nat] : ( P2 @ ( snth_P2624749845746815685ist_fm @ S3 @ N5 ) )
     => ( ( sdrop_7013889347380152453ist_fm @ ( comp_o6353595986216377982ist_fm @ (~) @ P2 ) @ S3 )
        = ( sdrop_4442373711808556042ist_fm
          @ ( ord_Least_nat
            @ ^ [N3: nat] : ( P2 @ ( snth_P2624749845746815685ist_fm @ S3 @ N3 ) ) )
          @ S3 ) ) ) ).

% sdrop_while_sdrop_LEAST
thf(fact_912_sdrop__while__sdrop__LEAST,axiom,
    ! [P2: produc780299800278082923e_rule > $o,S3: stream3588992565182678257e_rule] :
      ( ? [N5: nat] : ( P2 @ ( snth_P8487267234472522255e_rule @ S3 @ N5 ) )
     => ( ( sdrop_4395986648051443663e_rule @ ( comp_o7949119091651446216e_rule @ (~) @ P2 ) @ S3 )
        = ( sdrop_2199339672054592340e_rule
          @ ( ord_Least_nat
            @ ^ [N3: nat] : ( P2 @ ( snth_P8487267234472522255e_rule @ S3 @ N3 ) ) )
          @ S3 ) ) ) ).

% sdrop_while_sdrop_LEAST
thf(fact_913_sdrop__while__sdrop__LEAST,axiom,
    ! [P2: produc1660549859252256699m_rule > $o,S3: stream1878305877988316353m_rule] :
      ( ? [N5: nat] : ( P2 @ ( snth_P7967347330282200671m_rule @ S3 @ N5 ) )
     => ( ( sdrop_6423752018218906655m_rule @ ( comp_o6304513743058443928m_rule @ (~) @ P2 ) @ S3 )
        = ( sdrop_1764812713384251940m_rule
          @ ( ord_Least_nat
            @ ^ [N3: nat] : ( P2 @ ( snth_P7967347330282200671m_rule @ S3 @ N3 ) ) )
          @ S3 ) ) ) ).

% sdrop_while_sdrop_LEAST
thf(fact_914_sdrop__while__sdrop__LEAST,axiom,
    ! [P2: produc6288491646048892603e_rule > $o,S3: stream2312013117288958913e_rule] :
      ( ? [N5: nat] : ( P2 @ ( snth_P3371917080224060767e_rule @ S3 @ N5 ) )
     => ( ( sdrop_1828321768160766751e_rule @ ( comp_o1709083493000304024e_rule @ (~) @ P2 ) @ S3 )
        = ( sdrop_6392754500180887844e_rule
          @ ( ord_Least_nat
            @ ^ [N3: nat] : ( P2 @ ( snth_P3371917080224060767e_rule @ S3 @ N3 ) ) )
          @ S3 ) ) ) ).

% sdrop_while_sdrop_LEAST
thf(fact_915_sdrop__while__sdrop__LEAST,axiom,
    ! [P2: produc3793571666044074251m_rule > $o,S3: stream6494289010434245521m_rule] :
      ( ? [N5: nat] : ( P2 @ ( snth_P8950227884416379311m_rule @ S3 @ N5 ) )
     => ( ( sdrop_7771878303739699055m_rule @ ( comp_o8292784708554675816m_rule @ (~) @ P2 ) @ S3 )
        = ( sdrop_7373388980841566196m_rule
          @ ( ord_Least_nat
            @ ^ [N3: nat] : ( P2 @ ( snth_P8950227884416379311m_rule @ S3 @ N3 ) ) )
          @ S3 ) ) ) ).

% sdrop_while_sdrop_LEAST
thf(fact_916_sdrop__while__sdrop__LEAST,axiom,
    ! [P2: rule > $o,S3: stream_rule] :
      ( ? [N5: nat] : ( P2 @ ( snth_rule @ S3 @ N5 ) )
     => ( ( sdrop_while_rule @ ( comp_o_o_rule @ (~) @ P2 ) @ S3 )
        = ( sdrop_rule
          @ ( ord_Least_nat
            @ ^ [N3: nat] : ( P2 @ ( snth_rule @ S3 @ N3 ) ) )
          @ S3 ) ) ) ).

% sdrop_while_sdrop_LEAST
thf(fact_917_fst__diag__fst,axiom,
    ( ( comp_P1085356462900927218m_rule @ produc5894184038225292483ist_fm
      @ ( comp_P2017078995931194366m_rule
        @ ^ [X6: produc1996495991257130529ist_fm] : ( produc2592314732767339223ist_fm @ X6 @ X6 )
        @ produc6879501374131015971m_rule ) )
    = produc6879501374131015971m_rule ) ).

% fst_diag_fst
thf(fact_918_fst__diag__fst,axiom,
    ( ( comp_P1426762558556845121ist_fm @ produc1501393135466168645ist_fm
      @ ( comp_l2016392514137073923ist_fm
        @ ^ [X6: list_fm] : ( produc381145313068854617ist_fm @ X6 @ X6 )
        @ produc1501393135466168645ist_fm ) )
    = produc1501393135466168645ist_fm ) ).

% fst_diag_fst
thf(fact_919_snd__diag__snd,axiom,
    ( ( comp_P2548450393677710116m_rule @ produc6725306161561501521e_rule
      @ ( comp_r8929483813444710872m_rule
        @ ^ [X6: rule] : ( produc1769597910952761123e_rule @ X6 @ X6 )
        @ produc7165828336582415457m_rule ) )
    = produc7165828336582415457m_rule ) ).

% snd_diag_snd
thf(fact_920_snd__diag__snd,axiom,
    ( ( comp_P1426762558556845121ist_fm @ produc4588648349897876871ist_fm
      @ ( comp_l2016392514137073923ist_fm
        @ ^ [X6: list_fm] : ( produc381145313068854617ist_fm @ X6 @ X6 )
        @ produc4588648349897876871ist_fm ) )
    = produc4588648349897876871ist_fm ) ).

% snd_diag_snd
thf(fact_921_snd__diag__fst,axiom,
    ( ( comp_P1085356462900927218m_rule @ produc1639249265332130309ist_fm
      @ ( comp_P2017078995931194366m_rule
        @ ^ [X6: produc1996495991257130529ist_fm] : ( produc2592314732767339223ist_fm @ X6 @ X6 )
        @ produc6879501374131015971m_rule ) )
    = produc6879501374131015971m_rule ) ).

% snd_diag_fst
thf(fact_922_snd__diag__fst,axiom,
    ( ( comp_P1426762558556845121ist_fm @ produc4588648349897876871ist_fm
      @ ( comp_l2016392514137073923ist_fm
        @ ^ [X6: list_fm] : ( produc381145313068854617ist_fm @ X6 @ X6 )
        @ produc1501393135466168645ist_fm ) )
    = produc1501393135466168645ist_fm ) ).

% snd_diag_fst
thf(fact_923_fst__diag__snd,axiom,
    ( ( comp_P2548450393677710116m_rule @ produc7264528503596239631e_rule
      @ ( comp_r8929483813444710872m_rule
        @ ^ [X6: rule] : ( produc1769597910952761123e_rule @ X6 @ X6 )
        @ produc7165828336582415457m_rule ) )
    = produc7165828336582415457m_rule ) ).

% fst_diag_snd
thf(fact_924_fst__diag__snd,axiom,
    ( ( comp_P1426762558556845121ist_fm @ produc1501393135466168645ist_fm
      @ ( comp_l2016392514137073923ist_fm
        @ ^ [X6: list_fm] : ( produc381145313068854617ist_fm @ X6 @ X6 )
        @ produc4588648349897876871ist_fm ) )
    = produc4588648349897876871ist_fm ) ).

% fst_diag_snd
thf(fact_925_fst__fstOp,axiom,
    ! [P2: produc1996495991257130529ist_fm > rule > $o,Q2: rule > rule > $o] :
      ( produc6879501374131015971m_rule
      = ( comp_P7463969011078018366m_rule @ produc6879501374131015971m_rule @ ( bNF_fs2143400664727635963e_rule @ P2 @ Q2 ) ) ) ).

% fst_fstOp
thf(fact_926_fst__fstOp,axiom,
    ! [P2: list_fm > list_fm > $o,Q2: list_fm > list_fm > $o] :
      ( produc1501393135466168645ist_fm
      = ( comp_P1426762558556845121ist_fm @ produc1501393135466168645ist_fm @ ( bNF_fs6361025107161210196ist_fm @ P2 @ Q2 ) ) ) ).

% fst_fstOp
thf(fact_927_snd__sndOp,axiom,
    ! [P2: produc1996495991257130529ist_fm > produc1996495991257130529ist_fm > $o,Q2: produc1996495991257130529ist_fm > rule > $o] :
      ( produc7165828336582415457m_rule
      = ( comp_P422132551275928404m_rule @ produc7165828336582415457m_rule @ ( bNF_sn6617447237779055481m_rule @ P2 @ Q2 ) ) ) ).

% snd_sndOp
thf(fact_928_snd__sndOp,axiom,
    ! [P2: list_fm > list_fm > $o,Q2: list_fm > list_fm > $o] :
      ( produc4588648349897876871ist_fm
      = ( comp_P1426762558556845121ist_fm @ produc4588648349897876871ist_fm @ ( bNF_sn5292590578203848342ist_fm @ P2 @ Q2 ) ) ) ).

% snd_sndOp
thf(fact_929_stl__def,axiom,
    ( stl_Pr950425576149878629m_rule
    = ( case_s474603282864825704m_rule
      @ ^ [X13: produc164195504107695125m_rule,X24: stream727092118206550309m_rule] : X24 ) ) ).

% stl_def
thf(fact_930_stl__def,axiom,
    ( stl_rule
    = ( case_s5295311805034529096m_rule
      @ ^ [X13: rule,X24: stream_rule] : X24 ) ) ).

% stl_def
thf(fact_931_cont__def,axiom,
    ( abstra4962253831105676628m_rule
    = ( abstra1216198984563783442m_rule
      @ ^ [X13: produc164195504107695125m_rule,X24: fset_A5396818216090053761m_rule] : X24 ) ) ).

% cont_def
thf(fact_932_fset__of__list__elem,axiom,
    ! [X3: rule,Xs: list_rule] :
      ( ( fmember_rule @ X3 @ ( fset_of_list_rule @ Xs ) )
      = ( member_rule @ X3 @ ( set_rule2 @ Xs ) ) ) ).

% fset_of_list_elem
thf(fact_933_fset__of__list__elem,axiom,
    ! [X3: produc859450856879609959at_nat,Xs: list_P8469869581646625389at_nat] :
      ( ( fmembe6431293532144391214at_nat @ X3 @ ( fset_o2072219007702160685at_nat @ Xs ) )
      = ( member8206827879206165904at_nat @ X3 @ ( set_Pr5518436109238095868at_nat @ Xs ) ) ) ).

% fset_of_list_elem
thf(fact_934_fset__of__list__elem,axiom,
    ! [X3: produc1996495991257130529ist_fm,Xs: list_P5616295576739893671ist_fm] :
      ( ( fmembe3381613331217039976ist_fm @ X3 @ ( fset_o3706400737857578983ist_fm @ Xs ) )
      = ( member8102475879199740618ist_fm @ X3 @ ( set_Pr8767716839810916150ist_fm @ Xs ) ) ) ).

% fset_of_list_elem
thf(fact_935_fset__of__list__elem,axiom,
    ! [X3: abstra4572470701423150929m_rule,Xs: list_A1849579841884561377m_rule] :
      ( ( fmembe1635983029874454474m_rule @ X3 @ ( fset_o8294491781805274379m_rule @ Xs ) )
      = ( member1590433669991072232m_rule @ X3 @ ( set_Ab5312821370972875900m_rule @ Xs ) ) ) ).

% fset_of_list_elem
thf(fact_936_fset__of__list__elem,axiom,
    ! [X3: fm,Xs: list_fm] :
      ( ( fmember_fm @ X3 @ ( fset_of_list_fm @ Xs ) )
      = ( member_fm @ X3 @ ( set_fm2 @ Xs ) ) ) ).

% fset_of_list_elem
thf(fact_937_tree_Oroot__def,axiom,
    ( abstra3503897793822247108m_rule
    = ( abstra6581154509029876902m_rule
      @ ^ [X13: produc164195504107695125m_rule,X24: fset_A5396818216090053761m_rule] : X13 ) ) ).

% tree.root_def
thf(fact_938_some__sym__eq__trivial,axiom,
    ! [X3: fset_P661503646757059847ist_fm] :
      ( ( fChoic3892542562059901019ist_fm
        @ ( ^ [Y2: fset_P661503646757059847ist_fm,Z: fset_P661503646757059847ist_fm] : ( Y2 = Z )
          @ X3 ) )
      = X3 ) ).

% some_sym_eq_trivial
thf(fact_939_some__eq__trivial,axiom,
    ! [X3: fset_P661503646757059847ist_fm] :
      ( ( fChoic3892542562059901019ist_fm
        @ ^ [Y5: fset_P661503646757059847ist_fm] : ( Y5 = X3 ) )
      = X3 ) ).

% some_eq_trivial
thf(fact_940_some__equality,axiom,
    ! [P2: fset_P661503646757059847ist_fm > $o,A: fset_P661503646757059847ist_fm] :
      ( ( P2 @ A )
     => ( ! [X: fset_P661503646757059847ist_fm] :
            ( ( P2 @ X )
           => ( X = A ) )
       => ( ( fChoic3892542562059901019ist_fm @ P2 )
          = A ) ) ) ).

% some_equality
thf(fact_941_konig_Ocode,axiom,
    ( abstra3079604473823497664m_rule
    = ( ^ [T: abstra4572470701423150929m_rule] :
          ( sCons_5731567480799343806m_rule @ ( abstra3503897793822247108m_rule @ T )
          @ ( abstra3079604473823497664m_rule
            @ ( fChoic7543385803022166781m_rule
              @ ^ [T5: abstra4572470701423150929m_rule] :
                  ( ( fmembe1635983029874454474m_rule @ T5 @ ( abstra4962253831105676628m_rule @ T ) )
                  & ~ ( abstra3507476289341504315m_rule @ T5 ) ) ) ) ) ) ) ).

% konig.code
thf(fact_942_stream_Ocollapse,axiom,
    ! [Stream: stream727092118206550309m_rule] :
      ( ( sCons_5731567480799343806m_rule @ ( shd_Pr7235097944458474089m_rule @ Stream ) @ ( stl_Pr950425576149878629m_rule @ Stream ) )
      = Stream ) ).

% stream.collapse
thf(fact_943_stream_Ocollapse,axiom,
    ! [Stream: stream_rule] :
      ( ( sCons_rule @ ( shd_rule @ Stream ) @ ( stl_rule @ Stream ) )
      = Stream ) ).

% stream.collapse
thf(fact_944_szip__unfold,axiom,
    ! [A: rule,S1: stream_rule,B: rule,S2: stream_rule] :
      ( ( szip_rule_rule @ ( sCons_rule @ A @ S1 ) @ ( sCons_rule @ B @ S2 ) )
      = ( sCons_7940479310257918722e_rule @ ( produc1769597910952761123e_rule @ A @ B ) @ ( szip_rule_rule @ S1 @ S2 ) ) ) ).

% szip_unfold
thf(fact_945_szip__unfold,axiom,
    ! [A: rule,S1: stream_rule,B: produc164195504107695125m_rule,S2: stream727092118206550309m_rule] :
      ( ( szip_r2254114822203693110m_rule @ ( sCons_rule @ A @ S1 ) @ ( sCons_5731567480799343806m_rule @ B @ S2 ) )
      = ( sCons_2207919253591517394m_rule @ ( produc6084188410630241011m_rule @ A @ B ) @ ( szip_r2254114822203693110m_rule @ S1 @ S2 ) ) ) ).

% szip_unfold
thf(fact_946_szip__unfold,axiom,
    ! [A: produc164195504107695125m_rule,S1: stream727092118206550309m_rule,B: rule,S2: stream_rule] :
      ( ( szip_P553291425819358518e_rule @ ( sCons_5731567480799343806m_rule @ A @ S1 ) @ ( sCons_rule @ B @ S2 ) )
      = ( sCons_6835861040388153298e_rule @ ( produc4383365014245906419e_rule @ A @ B ) @ ( szip_P553291425819358518e_rule @ S1 @ S2 ) ) ) ).

% szip_unfold
thf(fact_947_szip__unfold,axiom,
    ! [A: produc164195504107695125m_rule,S1: stream727092118206550309m_rule,B: produc164195504107695125m_rule,S2: stream727092118206550309m_rule] :
      ( ( szip_P2499414959592755846m_rule @ ( sCons_5731567480799343806m_rule @ A @ S1 ) @ ( sCons_5731567480799343806m_rule @ B @ S2 ) )
      = ( sCons_4186928284053167522m_rule @ ( produc5927390650430071747m_rule @ A @ B ) @ ( szip_P2499414959592755846m_rule @ S1 @ S2 ) ) ) ).

% szip_unfold
thf(fact_948_szip__unfold,axiom,
    ! [A: produc1996495991257130529ist_fm,S1: stream8299795917829157543ist_fm,B: rule,S2: stream_rule] :
      ( ( szip_P1977448745965526924m_rule @ ( sCons_307821682964077496ist_fm @ A @ S1 ) @ ( sCons_rule @ B @ S2 ) )
      = ( sCons_5731567480799343806m_rule @ ( produc491467635432902671m_rule @ A @ B ) @ ( szip_P1977448745965526924m_rule @ S1 @ S2 ) ) ) ).

% szip_unfold
thf(fact_949_szip__unfold,axiom,
    ! [A: list_fm,S1: stream_list_fm,B: list_fm,S2: stream_list_fm] :
      ( ( szip_list_fm_list_fm @ ( sCons_list_fm @ A @ S1 ) @ ( sCons_list_fm @ B @ S2 ) )
      = ( sCons_307821682964077496ist_fm @ ( produc381145313068854617ist_fm @ A @ B ) @ ( szip_list_fm_list_fm @ S1 @ S2 ) ) ) ).

% szip_unfold
thf(fact_950_szip__unfold,axiom,
    ! [A: nat,S1: stream_nat,B: nat,S2: stream_nat] :
      ( ( szip_nat_nat @ ( sCons_nat @ A @ S1 ) @ ( sCons_nat @ B @ S2 ) )
      = ( sCons_2147322719763279000at_nat @ ( product_Pair_nat_nat @ A @ B ) @ ( szip_nat_nat @ S1 @ S2 ) ) ) ).

% szip_unfold
thf(fact_951_szip__unfold,axiom,
    ! [A: product_prod_nat_nat,S1: stream6724221391990029191at_nat,B: product_prod_nat_nat,S2: stream6724221391990029191at_nat] :
      ( ( szip_P4314639285670189082at_nat @ ( sCons_2147322719763279000at_nat @ A @ S1 ) @ ( sCons_2147322719763279000at_nat @ B @ S2 ) )
      = ( sCons_5297226365640217982at_nat @ ( produc6161850002892822231at_nat @ A @ B ) @ ( szip_P4314639285670189082at_nat @ S1 @ S2 ) ) ) ).

% szip_unfold
thf(fact_952_szip__unfold,axiom,
    ! [A: rule,S1: stream_rule,B: produc1996495991257130529ist_fm,S2: stream8299795917829157543ist_fm] :
      ( ( szip_r8549736553299883584ist_fm @ ( sCons_rule @ A @ S1 ) @ ( sCons_307821682964077496ist_fm @ B @ S2 ) )
      = ( sCons_974865965546383930ist_fm @ ( produc7063755442767259331ist_fm @ A @ B ) @ ( szip_r8549736553299883584ist_fm @ S1 @ S2 ) ) ) ).

% szip_unfold
thf(fact_953_stream_Osel_I1_J,axiom,
    ! [X1: produc164195504107695125m_rule,X2: stream727092118206550309m_rule] :
      ( ( shd_Pr7235097944458474089m_rule @ ( sCons_5731567480799343806m_rule @ X1 @ X2 ) )
      = X1 ) ).

% stream.sel(1)
thf(fact_954_stream_Osel_I1_J,axiom,
    ! [X1: rule,X2: stream_rule] :
      ( ( shd_rule @ ( sCons_rule @ X1 @ X2 ) )
      = X1 ) ).

% stream.sel(1)
thf(fact_955_stream_Osel_I2_J,axiom,
    ! [X1: produc164195504107695125m_rule,X2: stream727092118206550309m_rule] :
      ( ( stl_Pr950425576149878629m_rule @ ( sCons_5731567480799343806m_rule @ X1 @ X2 ) )
      = X2 ) ).

% stream.sel(2)
thf(fact_956_stream_Osel_I2_J,axiom,
    ! [X1: rule,X2: stream_rule] :
      ( ( stl_rule @ ( sCons_rule @ X1 @ X2 ) )
      = X2 ) ).

% stream.sel(2)
thf(fact_957_stream_Oset__induct,axiom,
    ! [X3: fm,A: stream_fm,P2: fm > stream_fm > $o] :
      ( ( member_fm @ X3 @ ( sset_fm @ A ) )
     => ( ! [Z1: fm,Z22: stream_fm] : ( P2 @ Z1 @ ( sCons_fm @ Z1 @ Z22 ) )
       => ( ! [Z1: fm,Z22: stream_fm,Xa: fm] :
              ( ( member_fm @ Xa @ ( sset_fm @ Z22 ) )
             => ( ( P2 @ Xa @ Z22 )
               => ( P2 @ Xa @ ( sCons_fm @ Z1 @ Z22 ) ) ) )
         => ( P2 @ X3 @ A ) ) ) ) ).

% stream.set_induct
thf(fact_958_stream_Oset__induct,axiom,
    ! [X3: produc1996495991257130529ist_fm,A: stream8299795917829157543ist_fm,P2: produc1996495991257130529ist_fm > stream8299795917829157543ist_fm > $o] :
      ( ( member8102475879199740618ist_fm @ X3 @ ( sset_P3431374123360452149ist_fm @ A ) )
     => ( ! [Z1: produc1996495991257130529ist_fm,Z22: stream8299795917829157543ist_fm] : ( P2 @ Z1 @ ( sCons_307821682964077496ist_fm @ Z1 @ Z22 ) )
       => ( ! [Z1: produc1996495991257130529ist_fm,Z22: stream8299795917829157543ist_fm,Xa: produc1996495991257130529ist_fm] :
              ( ( member8102475879199740618ist_fm @ Xa @ ( sset_P3431374123360452149ist_fm @ Z22 ) )
             => ( ( P2 @ Xa @ Z22 )
               => ( P2 @ Xa @ ( sCons_307821682964077496ist_fm @ Z1 @ Z22 ) ) ) )
         => ( P2 @ X3 @ A ) ) ) ) ).

% stream.set_induct
thf(fact_959_stream_Oset__induct,axiom,
    ! [X3: produc859450856879609959at_nat,A: stream8372878641218411373at_nat,P2: produc859450856879609959at_nat > stream8372878641218411373at_nat > $o] :
      ( ( member8206827879206165904at_nat @ X3 @ ( sset_P9164763103296375419at_nat @ A ) )
     => ( ! [Z1: produc859450856879609959at_nat,Z22: stream8372878641218411373at_nat] : ( P2 @ Z1 @ ( sCons_5297226365640217982at_nat @ Z1 @ Z22 ) )
       => ( ! [Z1: produc859450856879609959at_nat,Z22: stream8372878641218411373at_nat,Xa: produc859450856879609959at_nat] :
              ( ( member8206827879206165904at_nat @ Xa @ ( sset_P9164763103296375419at_nat @ Z22 ) )
             => ( ( P2 @ Xa @ Z22 )
               => ( P2 @ Xa @ ( sCons_5297226365640217982at_nat @ Z1 @ Z22 ) ) ) )
         => ( P2 @ X3 @ A ) ) ) ) ).

% stream.set_induct
thf(fact_960_stream_Oset__induct,axiom,
    ! [X3: rule,A: stream_rule,P2: rule > stream_rule > $o] :
      ( ( member_rule @ X3 @ ( sset_rule @ A ) )
     => ( ! [Z1: rule,Z22: stream_rule] : ( P2 @ Z1 @ ( sCons_rule @ Z1 @ Z22 ) )
       => ( ! [Z1: rule,Z22: stream_rule,Xa: rule] :
              ( ( member_rule @ Xa @ ( sset_rule @ Z22 ) )
             => ( ( P2 @ Xa @ Z22 )
               => ( P2 @ Xa @ ( sCons_rule @ Z1 @ Z22 ) ) ) )
         => ( P2 @ X3 @ A ) ) ) ) ).

% stream.set_induct
thf(fact_961_stream_Oset__induct,axiom,
    ! [X3: produc164195504107695125m_rule,A: stream727092118206550309m_rule,P2: produc164195504107695125m_rule > stream727092118206550309m_rule > $o] :
      ( ( member4220325220686508332m_rule @ X3 @ ( sset_P4979403222752902913m_rule @ A ) )
     => ( ! [Z1: produc164195504107695125m_rule,Z22: stream727092118206550309m_rule] : ( P2 @ Z1 @ ( sCons_5731567480799343806m_rule @ Z1 @ Z22 ) )
       => ( ! [Z1: produc164195504107695125m_rule,Z22: stream727092118206550309m_rule,Xa: produc164195504107695125m_rule] :
              ( ( member4220325220686508332m_rule @ Xa @ ( sset_P4979403222752902913m_rule @ Z22 ) )
             => ( ( P2 @ Xa @ Z22 )
               => ( P2 @ Xa @ ( sCons_5731567480799343806m_rule @ Z1 @ Z22 ) ) ) )
         => ( P2 @ X3 @ A ) ) ) ) ).

% stream.set_induct
thf(fact_962_stream_Oset__cases,axiom,
    ! [E: fm,A: stream_fm] :
      ( ( member_fm @ E @ ( sset_fm @ A ) )
     => ( ! [Z22: stream_fm] :
            ( A
           != ( sCons_fm @ E @ Z22 ) )
       => ~ ! [Z1: fm,Z22: stream_fm] :
              ( ( A
                = ( sCons_fm @ Z1 @ Z22 ) )
             => ~ ( member_fm @ E @ ( sset_fm @ Z22 ) ) ) ) ) ).

% stream.set_cases
thf(fact_963_stream_Oset__cases,axiom,
    ! [E: produc1996495991257130529ist_fm,A: stream8299795917829157543ist_fm] :
      ( ( member8102475879199740618ist_fm @ E @ ( sset_P3431374123360452149ist_fm @ A ) )
     => ( ! [Z22: stream8299795917829157543ist_fm] :
            ( A
           != ( sCons_307821682964077496ist_fm @ E @ Z22 ) )
       => ~ ! [Z1: produc1996495991257130529ist_fm,Z22: stream8299795917829157543ist_fm] :
              ( ( A
                = ( sCons_307821682964077496ist_fm @ Z1 @ Z22 ) )
             => ~ ( member8102475879199740618ist_fm @ E @ ( sset_P3431374123360452149ist_fm @ Z22 ) ) ) ) ) ).

% stream.set_cases
thf(fact_964_stream_Oset__cases,axiom,
    ! [E: produc859450856879609959at_nat,A: stream8372878641218411373at_nat] :
      ( ( member8206827879206165904at_nat @ E @ ( sset_P9164763103296375419at_nat @ A ) )
     => ( ! [Z22: stream8372878641218411373at_nat] :
            ( A
           != ( sCons_5297226365640217982at_nat @ E @ Z22 ) )
       => ~ ! [Z1: produc859450856879609959at_nat,Z22: stream8372878641218411373at_nat] :
              ( ( A
                = ( sCons_5297226365640217982at_nat @ Z1 @ Z22 ) )
             => ~ ( member8206827879206165904at_nat @ E @ ( sset_P9164763103296375419at_nat @ Z22 ) ) ) ) ) ).

% stream.set_cases
thf(fact_965_stream_Oset__cases,axiom,
    ! [E: rule,A: stream_rule] :
      ( ( member_rule @ E @ ( sset_rule @ A ) )
     => ( ! [Z22: stream_rule] :
            ( A
           != ( sCons_rule @ E @ Z22 ) )
       => ~ ! [Z1: rule,Z22: stream_rule] :
              ( ( A
                = ( sCons_rule @ Z1 @ Z22 ) )
             => ~ ( member_rule @ E @ ( sset_rule @ Z22 ) ) ) ) ) ).

% stream.set_cases
thf(fact_966_stream_Oset__cases,axiom,
    ! [E: produc164195504107695125m_rule,A: stream727092118206550309m_rule] :
      ( ( member4220325220686508332m_rule @ E @ ( sset_P4979403222752902913m_rule @ A ) )
     => ( ! [Z22: stream727092118206550309m_rule] :
            ( A
           != ( sCons_5731567480799343806m_rule @ E @ Z22 ) )
       => ~ ! [Z1: produc164195504107695125m_rule,Z22: stream727092118206550309m_rule] :
              ( ( A
                = ( sCons_5731567480799343806m_rule @ Z1 @ Z22 ) )
             => ~ ( member4220325220686508332m_rule @ E @ ( sset_P4979403222752902913m_rule @ Z22 ) ) ) ) ) ).

% stream.set_cases
thf(fact_967_stream_Oset__intros_I1_J,axiom,
    ! [X1: fm,X2: stream_fm] : ( member_fm @ X1 @ ( sset_fm @ ( sCons_fm @ X1 @ X2 ) ) ) ).

% stream.set_intros(1)
thf(fact_968_stream_Oset__intros_I1_J,axiom,
    ! [X1: produc1996495991257130529ist_fm,X2: stream8299795917829157543ist_fm] : ( member8102475879199740618ist_fm @ X1 @ ( sset_P3431374123360452149ist_fm @ ( sCons_307821682964077496ist_fm @ X1 @ X2 ) ) ) ).

% stream.set_intros(1)
thf(fact_969_stream_Oset__intros_I1_J,axiom,
    ! [X1: produc859450856879609959at_nat,X2: stream8372878641218411373at_nat] : ( member8206827879206165904at_nat @ X1 @ ( sset_P9164763103296375419at_nat @ ( sCons_5297226365640217982at_nat @ X1 @ X2 ) ) ) ).

% stream.set_intros(1)
thf(fact_970_stream_Oset__intros_I1_J,axiom,
    ! [X1: rule,X2: stream_rule] : ( member_rule @ X1 @ ( sset_rule @ ( sCons_rule @ X1 @ X2 ) ) ) ).

% stream.set_intros(1)
thf(fact_971_stream_Oset__intros_I1_J,axiom,
    ! [X1: produc164195504107695125m_rule,X2: stream727092118206550309m_rule] : ( member4220325220686508332m_rule @ X1 @ ( sset_P4979403222752902913m_rule @ ( sCons_5731567480799343806m_rule @ X1 @ X2 ) ) ) ).

% stream.set_intros(1)
thf(fact_972_stream_Oset__intros_I2_J,axiom,
    ! [Y3: fm,X2: stream_fm,X1: fm] :
      ( ( member_fm @ Y3 @ ( sset_fm @ X2 ) )
     => ( member_fm @ Y3 @ ( sset_fm @ ( sCons_fm @ X1 @ X2 ) ) ) ) ).

% stream.set_intros(2)
thf(fact_973_stream_Oset__intros_I2_J,axiom,
    ! [Y3: produc1996495991257130529ist_fm,X2: stream8299795917829157543ist_fm,X1: produc1996495991257130529ist_fm] :
      ( ( member8102475879199740618ist_fm @ Y3 @ ( sset_P3431374123360452149ist_fm @ X2 ) )
     => ( member8102475879199740618ist_fm @ Y3 @ ( sset_P3431374123360452149ist_fm @ ( sCons_307821682964077496ist_fm @ X1 @ X2 ) ) ) ) ).

% stream.set_intros(2)
thf(fact_974_stream_Oset__intros_I2_J,axiom,
    ! [Y3: produc859450856879609959at_nat,X2: stream8372878641218411373at_nat,X1: produc859450856879609959at_nat] :
      ( ( member8206827879206165904at_nat @ Y3 @ ( sset_P9164763103296375419at_nat @ X2 ) )
     => ( member8206827879206165904at_nat @ Y3 @ ( sset_P9164763103296375419at_nat @ ( sCons_5297226365640217982at_nat @ X1 @ X2 ) ) ) ) ).

% stream.set_intros(2)
thf(fact_975_stream_Oset__intros_I2_J,axiom,
    ! [Y3: rule,X2: stream_rule,X1: rule] :
      ( ( member_rule @ Y3 @ ( sset_rule @ X2 ) )
     => ( member_rule @ Y3 @ ( sset_rule @ ( sCons_rule @ X1 @ X2 ) ) ) ) ).

% stream.set_intros(2)
thf(fact_976_stream_Oset__intros_I2_J,axiom,
    ! [Y3: produc164195504107695125m_rule,X2: stream727092118206550309m_rule,X1: produc164195504107695125m_rule] :
      ( ( member4220325220686508332m_rule @ Y3 @ ( sset_P4979403222752902913m_rule @ X2 ) )
     => ( member4220325220686508332m_rule @ Y3 @ ( sset_P4979403222752902913m_rule @ ( sCons_5731567480799343806m_rule @ X1 @ X2 ) ) ) ) ).

% stream.set_intros(2)
thf(fact_977_stream_Ocase,axiom,
    ! [F: rule > stream_rule > abstra4572470701423150929m_rule,X1: rule,X2: stream_rule] :
      ( ( case_s2494923709252486212m_rule @ F @ ( sCons_rule @ X1 @ X2 ) )
      = ( F @ X1 @ X2 ) ) ).

% stream.case
thf(fact_978_sdrop__while__SCons,axiom,
    ! [P2: rule > $o,A: rule,S3: stream_rule] :
      ( ( ( P2 @ A )
       => ( ( sdrop_while_rule @ P2 @ ( sCons_rule @ A @ S3 ) )
          = ( sdrop_while_rule @ P2 @ S3 ) ) )
      & ( ~ ( P2 @ A )
       => ( ( sdrop_while_rule @ P2 @ ( sCons_rule @ A @ S3 ) )
          = ( sCons_rule @ A @ S3 ) ) ) ) ).

% sdrop_while_SCons
thf(fact_979_stream_Oexhaust__sel,axiom,
    ! [Stream: stream727092118206550309m_rule] :
      ( Stream
      = ( sCons_5731567480799343806m_rule @ ( shd_Pr7235097944458474089m_rule @ Stream ) @ ( stl_Pr950425576149878629m_rule @ Stream ) ) ) ).

% stream.exhaust_sel
thf(fact_980_stream_Oexhaust__sel,axiom,
    ! [Stream: stream_rule] :
      ( Stream
      = ( sCons_rule @ ( shd_rule @ Stream ) @ ( stl_rule @ Stream ) ) ) ).

% stream.exhaust_sel
thf(fact_981_stream_Osplit__sel,axiom,
    ! [P2: abstra4572470701423150929m_rule > $o,F: rule > stream_rule > abstra4572470701423150929m_rule,Stream: stream_rule] :
      ( ( P2 @ ( case_s2494923709252486212m_rule @ F @ Stream ) )
      = ( ( Stream
          = ( sCons_rule @ ( shd_rule @ Stream ) @ ( stl_rule @ Stream ) ) )
       => ( P2 @ ( F @ ( shd_rule @ Stream ) @ ( stl_rule @ Stream ) ) ) ) ) ).

% stream.split_sel
thf(fact_982_stream_Osplit__sel__asm,axiom,
    ! [P2: abstra4572470701423150929m_rule > $o,F: rule > stream_rule > abstra4572470701423150929m_rule,Stream: stream_rule] :
      ( ( P2 @ ( case_s2494923709252486212m_rule @ F @ Stream ) )
      = ( ~ ( ( Stream
              = ( sCons_rule @ ( shd_rule @ Stream ) @ ( stl_rule @ Stream ) ) )
            & ~ ( P2 @ ( F @ ( shd_rule @ Stream ) @ ( stl_rule @ Stream ) ) ) ) ) ) ).

% stream.split_sel_asm
thf(fact_983_szip_Ocode,axiom,
    ( szip_list_fm_list_fm
    = ( ^ [S12: stream_list_fm,S22: stream_list_fm] : ( sCons_307821682964077496ist_fm @ ( produc381145313068854617ist_fm @ ( shd_list_fm @ S12 ) @ ( shd_list_fm @ S22 ) ) @ ( szip_list_fm_list_fm @ ( stl_list_fm @ S12 ) @ ( stl_list_fm @ S22 ) ) ) ) ) ).

% szip.code
thf(fact_984_szip_Ocode,axiom,
    ( szip_nat_nat
    = ( ^ [S12: stream_nat,S22: stream_nat] : ( sCons_2147322719763279000at_nat @ ( product_Pair_nat_nat @ ( shd_nat @ S12 ) @ ( shd_nat @ S22 ) ) @ ( szip_nat_nat @ ( stl_nat @ S12 ) @ ( stl_nat @ S22 ) ) ) ) ) ).

% szip.code
thf(fact_985_szip_Ocode,axiom,
    ( szip_P4314639285670189082at_nat
    = ( ^ [S12: stream6724221391990029191at_nat,S22: stream6724221391990029191at_nat] : ( sCons_5297226365640217982at_nat @ ( produc6161850002892822231at_nat @ ( shd_Pr4260400998323988397at_nat @ S12 ) @ ( shd_Pr4260400998323988397at_nat @ S22 ) ) @ ( szip_P4314639285670189082at_nat @ ( stl_Pr5027827701538482609at_nat @ S12 ) @ ( stl_Pr5027827701538482609at_nat @ S22 ) ) ) ) ) ).

% szip.code
thf(fact_986_szip_Ocode,axiom,
    ( szip_P1977448745965526924m_rule
    = ( ^ [S12: stream8299795917829157543ist_fm,S22: stream_rule] : ( sCons_5731567480799343806m_rule @ ( produc491467635432902671m_rule @ ( shd_Pr772355297128350925ist_fm @ S12 ) @ ( shd_rule @ S22 ) ) @ ( szip_P1977448745965526924m_rule @ ( stl_Pr5027778045650968273ist_fm @ S12 ) @ ( stl_rule @ S22 ) ) ) ) ) ).

% szip.code
thf(fact_987_szip_Ocode,axiom,
    ( szip_P2499414959592755846m_rule
    = ( ^ [S12: stream727092118206550309m_rule,S22: stream727092118206550309m_rule] : ( sCons_4186928284053167522m_rule @ ( produc5927390650430071747m_rule @ ( shd_Pr7235097944458474089m_rule @ S12 ) @ ( shd_Pr7235097944458474089m_rule @ S22 ) ) @ ( szip_P2499414959592755846m_rule @ ( stl_Pr950425576149878629m_rule @ S12 ) @ ( stl_Pr950425576149878629m_rule @ S22 ) ) ) ) ) ).

% szip.code
thf(fact_988_szip_Ocode,axiom,
    ( szip_P553291425819358518e_rule
    = ( ^ [S12: stream727092118206550309m_rule,S22: stream_rule] : ( sCons_6835861040388153298e_rule @ ( produc4383365014245906419e_rule @ ( shd_Pr7235097944458474089m_rule @ S12 ) @ ( shd_rule @ S22 ) ) @ ( szip_P553291425819358518e_rule @ ( stl_Pr950425576149878629m_rule @ S12 ) @ ( stl_rule @ S22 ) ) ) ) ) ).

% szip.code
thf(fact_989_szip_Ocode,axiom,
    ( szip_r8549736553299883584ist_fm
    = ( ^ [S12: stream_rule,S22: stream8299795917829157543ist_fm] : ( sCons_974865965546383930ist_fm @ ( produc7063755442767259331ist_fm @ ( shd_rule @ S12 ) @ ( shd_Pr772355297128350925ist_fm @ S22 ) ) @ ( szip_r8549736553299883584ist_fm @ ( stl_rule @ S12 ) @ ( stl_Pr5027778045650968273ist_fm @ S22 ) ) ) ) ) ).

% szip.code
thf(fact_990_szip_Ocode,axiom,
    ( szip_r2254114822203693110m_rule
    = ( ^ [S12: stream_rule,S22: stream727092118206550309m_rule] : ( sCons_2207919253591517394m_rule @ ( produc6084188410630241011m_rule @ ( shd_rule @ S12 ) @ ( shd_Pr7235097944458474089m_rule @ S22 ) ) @ ( szip_r2254114822203693110m_rule @ ( stl_rule @ S12 ) @ ( stl_Pr950425576149878629m_rule @ S22 ) ) ) ) ) ).

% szip.code
thf(fact_991_szip_Ocode,axiom,
    ( szip_rule_rule
    = ( ^ [S12: stream_rule,S22: stream_rule] : ( sCons_7940479310257918722e_rule @ ( produc1769597910952761123e_rule @ ( shd_rule @ S12 ) @ ( shd_rule @ S22 ) ) @ ( szip_rule_rule @ ( stl_rule @ S12 ) @ ( stl_rule @ S22 ) ) ) ) ) ).

% szip.code
thf(fact_992_conj__subset__def,axiom,
    ! [A5: set_fm,P2: fm > $o,Q2: fm > $o] :
      ( ( ord_less_eq_set_fm @ A5
        @ ( collect_fm
          @ ^ [X6: fm] :
              ( ( P2 @ X6 )
              & ( Q2 @ X6 ) ) ) )
      = ( ( ord_less_eq_set_fm @ A5 @ ( collect_fm @ P2 ) )
        & ( ord_less_eq_set_fm @ A5 @ ( collect_fm @ Q2 ) ) ) ) ).

% conj_subset_def
thf(fact_993_someI2,axiom,
    ! [P2: fset_P661503646757059847ist_fm > $o,A: fset_P661503646757059847ist_fm,Q2: fset_P661503646757059847ist_fm > $o] :
      ( ( P2 @ A )
     => ( ! [X: fset_P661503646757059847ist_fm] :
            ( ( P2 @ X )
           => ( Q2 @ X ) )
       => ( Q2 @ ( fChoic3892542562059901019ist_fm @ P2 ) ) ) ) ).

% someI2
thf(fact_994_someI__ex,axiom,
    ! [P2: fset_P661503646757059847ist_fm > $o] :
      ( ? [X_1: fset_P661503646757059847ist_fm] : ( P2 @ X_1 )
     => ( P2 @ ( fChoic3892542562059901019ist_fm @ P2 ) ) ) ).

% someI_ex
thf(fact_995_someI2__ex,axiom,
    ! [P2: fset_P661503646757059847ist_fm > $o,Q2: fset_P661503646757059847ist_fm > $o] :
      ( ? [X_1: fset_P661503646757059847ist_fm] : ( P2 @ X_1 )
     => ( ! [X: fset_P661503646757059847ist_fm] :
            ( ( P2 @ X )
           => ( Q2 @ X ) )
       => ( Q2 @ ( fChoic3892542562059901019ist_fm @ P2 ) ) ) ) ).

% someI2_ex
thf(fact_996_someI2__bex,axiom,
    ! [A5: set_fm,P2: fm > $o,Q2: fm > $o] :
      ( ? [X8: fm] :
          ( ( member_fm @ X8 @ A5 )
          & ( P2 @ X8 ) )
     => ( ! [X: fm] :
            ( ( ( member_fm @ X @ A5 )
              & ( P2 @ X ) )
           => ( Q2 @ X ) )
       => ( Q2
          @ ( fChoice_fm
            @ ^ [X6: fm] :
                ( ( member_fm @ X6 @ A5 )
                & ( P2 @ X6 ) ) ) ) ) ) ).

% someI2_bex
thf(fact_997_someI2__bex,axiom,
    ! [A5: set_rule,P2: rule > $o,Q2: rule > $o] :
      ( ? [X8: rule] :
          ( ( member_rule @ X8 @ A5 )
          & ( P2 @ X8 ) )
     => ( ! [X: rule] :
            ( ( ( member_rule @ X @ A5 )
              & ( P2 @ X ) )
           => ( Q2 @ X ) )
       => ( Q2
          @ ( fChoice_rule
            @ ^ [X6: rule] :
                ( ( member_rule @ X6 @ A5 )
                & ( P2 @ X6 ) ) ) ) ) ) ).

% someI2_bex
thf(fact_998_someI2__bex,axiom,
    ! [A5: set_Pr7058068377845519745ist_fm,P2: produc1996495991257130529ist_fm > $o,Q2: produc1996495991257130529ist_fm > $o] :
      ( ? [X8: produc1996495991257130529ist_fm] :
          ( ( member8102475879199740618ist_fm @ X8 @ A5 )
          & ( P2 @ X8 ) )
     => ( ! [X: produc1996495991257130529ist_fm] :
            ( ( ( member8102475879199740618ist_fm @ X @ A5 )
              & ( P2 @ X ) )
           => ( Q2 @ X ) )
       => ( Q2
          @ ( fChoic1980167584695909237ist_fm
            @ ^ [X6: produc1996495991257130529ist_fm] :
                ( ( member8102475879199740618ist_fm @ X6 @ A5 )
                & ( P2 @ X6 ) ) ) ) ) ) ).

% someI2_bex
thf(fact_999_someI2__bex,axiom,
    ! [A5: set_Pr8693737435421807431at_nat,P2: produc859450856879609959at_nat > $o,Q2: produc859450856879609959at_nat > $o] :
      ( ? [X8: produc859450856879609959at_nat] :
          ( ( member8206827879206165904at_nat @ X8 @ A5 )
          & ( P2 @ X8 ) )
     => ( ! [X: produc859450856879609959at_nat] :
            ( ( ( member8206827879206165904at_nat @ X @ A5 )
              & ( P2 @ X ) )
           => ( Q2 @ X ) )
       => ( Q2
          @ ( fChoic6673916560228149179at_nat
            @ ^ [X6: produc859450856879609959at_nat] :
                ( ( member8206827879206165904at_nat @ X6 @ A5 )
                & ( P2 @ X6 ) ) ) ) ) ) ).

% someI2_bex
thf(fact_1000_someI2__bex,axiom,
    ! [A5: set_fs7534060674568567527ist_fm,P2: fset_P661503646757059847ist_fm > $o,Q2: fset_P661503646757059847ist_fm > $o] :
      ( ? [X8: fset_P661503646757059847ist_fm] :
          ( ( member3493611713653719344ist_fm @ X8 @ A5 )
          & ( P2 @ X8 ) )
     => ( ! [X: fset_P661503646757059847ist_fm] :
            ( ( ( member3493611713653719344ist_fm @ X @ A5 )
              & ( P2 @ X ) )
           => ( Q2 @ X ) )
       => ( Q2
          @ ( fChoic3892542562059901019ist_fm
            @ ^ [X6: fset_P661503646757059847ist_fm] :
                ( ( member3493611713653719344ist_fm @ X6 @ A5 )
                & ( P2 @ X6 ) ) ) ) ) ) ).

% someI2_bex
thf(fact_1001_some__eq__ex,axiom,
    ! [P2: fset_P661503646757059847ist_fm > $o] :
      ( ( P2 @ ( fChoic3892542562059901019ist_fm @ P2 ) )
      = ( ? [X5: fset_P661503646757059847ist_fm] : ( P2 @ X5 ) ) ) ).

% some_eq_ex
thf(fact_1002_some1__equality,axiom,
    ! [P2: fset_P661503646757059847ist_fm > $o,A: fset_P661503646757059847ist_fm] :
      ( ? [X8: fset_P661503646757059847ist_fm] :
          ( ( P2 @ X8 )
          & ! [Y: fset_P661503646757059847ist_fm] :
              ( ( P2 @ Y )
             => ( Y = X8 ) ) )
     => ( ( P2 @ A )
       => ( ( fChoic3892542562059901019ist_fm @ P2 )
          = A ) ) ) ).

% some1_equality
thf(fact_1003_sfilter_Ocode,axiom,
    ( sfilte7579662470208054517m_rule
    = ( ^ [P4: produc164195504107695125m_rule > $o,S: stream727092118206550309m_rule] : ( sCons_5731567480799343806m_rule @ ( shd_Pr7235097944458474089m_rule @ ( sdrop_5541336384294084785m_rule @ ( comp_o5517589812586031224m_rule @ (~) @ P4 ) @ S ) ) @ ( sfilte7579662470208054517m_rule @ P4 @ ( stl_Pr950425576149878629m_rule @ ( sdrop_5541336384294084785m_rule @ ( comp_o5517589812586031224m_rule @ (~) @ P4 ) @ S ) ) ) ) ) ) ).

% sfilter.code
thf(fact_1004_sfilter_Ocode,axiom,
    ( sfilter_rule
    = ( ^ [P4: rule > $o,S: stream_rule] : ( sCons_rule @ ( shd_rule @ ( sdrop_while_rule @ ( comp_o_o_rule @ (~) @ P4 ) @ S ) ) @ ( sfilter_rule @ P4 @ ( stl_rule @ ( sdrop_while_rule @ ( comp_o_o_rule @ (~) @ P4 ) @ S ) ) ) ) ) ) ).

% sfilter.code
thf(fact_1005_snd__comp__apsnd,axiom,
    ! [F: rule > rule] :
      ( ( comp_P422132551275928404m_rule @ produc7165828336582415457m_rule @ ( produc3063135074001183255ist_fm @ F ) )
      = ( comp_r641804198332660996m_rule @ F @ produc7165828336582415457m_rule ) ) ).

% snd_comp_apsnd
thf(fact_1006_snd__comp__apsnd,axiom,
    ! [F: list_fm > list_fm] :
      ( ( comp_P1426762558556845121ist_fm @ produc4588648349897876871ist_fm @ ( produc9047611296996456864ist_fm @ F ) )
      = ( comp_l1868218281415653180ist_fm @ F @ produc4588648349897876871ist_fm ) ) ).

% snd_comp_apsnd
thf(fact_1007_sinterleave_Ocode,axiom,
    ( sinter8084399408526207096m_rule
    = ( ^ [S12: stream727092118206550309m_rule,S22: stream727092118206550309m_rule] : ( sCons_5731567480799343806m_rule @ ( shd_Pr7235097944458474089m_rule @ S12 ) @ ( sinter8084399408526207096m_rule @ S22 @ ( stl_Pr950425576149878629m_rule @ S12 ) ) ) ) ) ).

% sinterleave.code
thf(fact_1008_sinterleave_Ocode,axiom,
    ( sinterleave_rule
    = ( ^ [S12: stream_rule,S22: stream_rule] : ( sCons_rule @ ( shd_rule @ S12 ) @ ( sinterleave_rule @ S22 @ ( stl_rule @ S12 ) ) ) ) ) ).

% sinterleave.code
thf(fact_1009_fst__comp__apfst,axiom,
    ! [F: produc1996495991257130529ist_fm > produc1996495991257130529ist_fm] :
      ( ( comp_P7463969011078018366m_rule @ produc6879501374131015971m_rule @ ( produc3731403639160201281m_rule @ F ) )
      = ( comp_P8245093624798776376m_rule @ F @ produc6879501374131015971m_rule ) ) ).

% fst_comp_apfst
thf(fact_1010_fst__comp__apfst,axiom,
    ! [F: list_fm > list_fm] :
      ( ( comp_P1426762558556845121ist_fm @ produc1501393135466168645ist_fm @ ( produc1130293176790411102ist_fm @ F ) )
      = ( comp_l1868218281415653180ist_fm @ F @ produc1501393135466168645ist_fm ) ) ).

% fst_comp_apfst
thf(fact_1011_apfst__conv,axiom,
    ! [F: produc1996495991257130529ist_fm > produc1996495991257130529ist_fm,X3: produc1996495991257130529ist_fm,Y3: rule] :
      ( ( produc3731403639160201281m_rule @ F @ ( produc491467635432902671m_rule @ X3 @ Y3 ) )
      = ( produc491467635432902671m_rule @ ( F @ X3 ) @ Y3 ) ) ).

% apfst_conv
thf(fact_1012_apfst__conv,axiom,
    ! [F: list_fm > list_fm,X3: list_fm,Y3: list_fm] :
      ( ( produc1130293176790411102ist_fm @ F @ ( produc381145313068854617ist_fm @ X3 @ Y3 ) )
      = ( produc381145313068854617ist_fm @ ( F @ X3 ) @ Y3 ) ) ).

% apfst_conv
thf(fact_1013_apfst__conv,axiom,
    ! [F: nat > nat,X3: nat,Y3: nat] :
      ( ( produc986720760941809198at_nat @ F @ ( product_Pair_nat_nat @ X3 @ Y3 ) )
      = ( product_Pair_nat_nat @ ( F @ X3 ) @ Y3 ) ) ).

% apfst_conv
thf(fact_1014_apfst__conv,axiom,
    ! [F: product_prod_nat_nat > product_prod_nat_nat,X3: product_prod_nat_nat,Y3: product_prod_nat_nat] :
      ( ( produc3057951030792887557at_nat @ F @ ( produc6161850002892822231at_nat @ X3 @ Y3 ) )
      = ( produc6161850002892822231at_nat @ ( F @ X3 ) @ Y3 ) ) ).

% apfst_conv
thf(fact_1015_apfst__conv,axiom,
    ! [F: rule > rule,X3: rule,Y3: produc1996495991257130529ist_fm] :
      ( ( produc5534506254562563289ist_fm @ F @ ( produc7063755442767259331ist_fm @ X3 @ Y3 ) )
      = ( produc7063755442767259331ist_fm @ ( F @ X3 ) @ Y3 ) ) ).

% apfst_conv
thf(fact_1016_apsnd__conv,axiom,
    ! [F: rule > rule,X3: produc1996495991257130529ist_fm,Y3: rule] :
      ( ( produc3063135074001183255ist_fm @ F @ ( produc491467635432902671m_rule @ X3 @ Y3 ) )
      = ( produc491467635432902671m_rule @ X3 @ ( F @ Y3 ) ) ) ).

% apsnd_conv
thf(fact_1017_apsnd__conv,axiom,
    ! [F: list_fm > list_fm,X3: list_fm,Y3: list_fm] :
      ( ( produc9047611296996456864ist_fm @ F @ ( produc381145313068854617ist_fm @ X3 @ Y3 ) )
      = ( produc381145313068854617ist_fm @ X3 @ ( F @ Y3 ) ) ) ).

% apsnd_conv
thf(fact_1018_apsnd__conv,axiom,
    ! [F: nat > nat,X3: nat,Y3: nat] :
      ( ( produc3094765310956027504at_nat @ F @ ( product_Pair_nat_nat @ X3 @ Y3 ) )
      = ( product_Pair_nat_nat @ X3 @ ( F @ Y3 ) ) ) ).

% apsnd_conv
thf(fact_1019_apsnd__conv,axiom,
    ! [F: product_prod_nat_nat > product_prod_nat_nat,X3: product_prod_nat_nat,Y3: product_prod_nat_nat] :
      ( ( produc8987972918798826819at_nat @ F @ ( produc6161850002892822231at_nat @ X3 @ Y3 ) )
      = ( produc6161850002892822231at_nat @ X3 @ ( F @ Y3 ) ) ) ).

% apsnd_conv
thf(fact_1020_apsnd__conv,axiom,
    ! [F: produc1996495991257130529ist_fm > produc1996495991257130529ist_fm,X3: rule,Y3: produc1996495991257130529ist_fm] :
      ( ( produc4332487942798490499m_rule @ F @ ( produc7063755442767259331ist_fm @ X3 @ Y3 ) )
      = ( produc7063755442767259331ist_fm @ X3 @ ( F @ Y3 ) ) ) ).

% apsnd_conv
thf(fact_1021_fst__apfst,axiom,
    ! [F: produc1996495991257130529ist_fm > produc1996495991257130529ist_fm,X3: produc164195504107695125m_rule] :
      ( ( produc6879501374131015971m_rule @ ( produc3731403639160201281m_rule @ F @ X3 ) )
      = ( F @ ( produc6879501374131015971m_rule @ X3 ) ) ) ).

% fst_apfst
thf(fact_1022_fst__apfst,axiom,
    ! [F: list_fm > list_fm,X3: produc1996495991257130529ist_fm] :
      ( ( produc1501393135466168645ist_fm @ ( produc1130293176790411102ist_fm @ F @ X3 ) )
      = ( F @ ( produc1501393135466168645ist_fm @ X3 ) ) ) ).

% fst_apfst
thf(fact_1023_snd__apfst,axiom,
    ! [F: produc1996495991257130529ist_fm > produc1996495991257130529ist_fm,X3: produc164195504107695125m_rule] :
      ( ( produc7165828336582415457m_rule @ ( produc3731403639160201281m_rule @ F @ X3 ) )
      = ( produc7165828336582415457m_rule @ X3 ) ) ).

% snd_apfst
thf(fact_1024_snd__apfst,axiom,
    ! [F: list_fm > list_fm,X3: produc1996495991257130529ist_fm] :
      ( ( produc4588648349897876871ist_fm @ ( produc1130293176790411102ist_fm @ F @ X3 ) )
      = ( produc4588648349897876871ist_fm @ X3 ) ) ).

% snd_apfst
thf(fact_1025_fst__apsnd,axiom,
    ! [F: rule > rule,X3: produc164195504107695125m_rule] :
      ( ( produc6879501374131015971m_rule @ ( produc3063135074001183255ist_fm @ F @ X3 ) )
      = ( produc6879501374131015971m_rule @ X3 ) ) ).

% fst_apsnd
thf(fact_1026_fst__apsnd,axiom,
    ! [F: list_fm > list_fm,X3: produc1996495991257130529ist_fm] :
      ( ( produc1501393135466168645ist_fm @ ( produc9047611296996456864ist_fm @ F @ X3 ) )
      = ( produc1501393135466168645ist_fm @ X3 ) ) ).

% fst_apsnd
thf(fact_1027_snd__apsnd,axiom,
    ! [F: rule > rule,X3: produc164195504107695125m_rule] :
      ( ( produc7165828336582415457m_rule @ ( produc3063135074001183255ist_fm @ F @ X3 ) )
      = ( F @ ( produc7165828336582415457m_rule @ X3 ) ) ) ).

% snd_apsnd
thf(fact_1028_snd__apsnd,axiom,
    ! [F: list_fm > list_fm,X3: produc1996495991257130529ist_fm] :
      ( ( produc4588648349897876871ist_fm @ ( produc9047611296996456864ist_fm @ F @ X3 ) )
      = ( F @ ( produc4588648349897876871ist_fm @ X3 ) ) ) ).

% snd_apsnd
thf(fact_1029_snd__comp__apfst,axiom,
    ! [F: produc1996495991257130529ist_fm > produc1996495991257130529ist_fm] :
      ( ( comp_P422132551275928404m_rule @ produc7165828336582415457m_rule @ ( produc3731403639160201281m_rule @ F ) )
      = produc7165828336582415457m_rule ) ).

% snd_comp_apfst
thf(fact_1030_snd__comp__apfst,axiom,
    ! [F: list_fm > list_fm] :
      ( ( comp_P1426762558556845121ist_fm @ produc4588648349897876871ist_fm @ ( produc1130293176790411102ist_fm @ F ) )
      = produc4588648349897876871ist_fm ) ).

% snd_comp_apfst
thf(fact_1031_fst__comp__apsnd,axiom,
    ! [F: rule > rule] :
      ( ( comp_P7463969011078018366m_rule @ produc6879501374131015971m_rule @ ( produc3063135074001183255ist_fm @ F ) )
      = produc6879501374131015971m_rule ) ).

% fst_comp_apsnd
thf(fact_1032_fst__comp__apsnd,axiom,
    ! [F: list_fm > list_fm] :
      ( ( comp_P1426762558556845121ist_fm @ produc1501393135466168645ist_fm @ ( produc9047611296996456864ist_fm @ F ) )
      = produc1501393135466168645ist_fm ) ).

% fst_comp_apsnd
thf(fact_1033_apsnd__apfst,axiom,
    ! [F: rule > rule,G: produc1996495991257130529ist_fm > produc1996495991257130529ist_fm,X3: produc164195504107695125m_rule] :
      ( ( produc3063135074001183255ist_fm @ F @ ( produc3731403639160201281m_rule @ G @ X3 ) )
      = ( produc491467635432902671m_rule @ ( G @ ( produc6879501374131015971m_rule @ X3 ) ) @ ( F @ ( produc7165828336582415457m_rule @ X3 ) ) ) ) ).

% apsnd_apfst
thf(fact_1034_apsnd__apfst,axiom,
    ! [F: rule > list_fm,G: produc1996495991257130529ist_fm > list_fm,X3: produc164195504107695125m_rule] :
      ( ( produc7464737497110386437ist_fm @ F @ ( produc7276395001879883020m_rule @ G @ X3 ) )
      = ( produc381145313068854617ist_fm @ ( G @ ( produc6879501374131015971m_rule @ X3 ) ) @ ( F @ ( produc7165828336582415457m_rule @ X3 ) ) ) ) ).

% apsnd_apfst
thf(fact_1035_apsnd__apfst,axiom,
    ! [F: rule > nat,G: produc1996495991257130529ist_fm > nat,X3: produc164195504107695125m_rule] :
      ( ( produc7007998740204313573at_nat @ F @ ( produc6002204213176801020t_rule @ G @ X3 ) )
      = ( product_Pair_nat_nat @ ( G @ ( produc6879501374131015971m_rule @ X3 ) ) @ ( F @ ( produc7165828336582415457m_rule @ X3 ) ) ) ) ).

% apsnd_apfst
thf(fact_1036_apsnd__apfst,axiom,
    ! [F: rule > product_prod_nat_nat,G: produc1996495991257130529ist_fm > product_prod_nat_nat,X3: produc164195504107695125m_rule] :
      ( ( produc5248576904221492867at_nat @ F @ ( produc7833360833687662369t_rule @ G @ X3 ) )
      = ( produc6161850002892822231at_nat @ ( G @ ( produc6879501374131015971m_rule @ X3 ) ) @ ( F @ ( produc7165828336582415457m_rule @ X3 ) ) ) ) ).

% apsnd_apfst
thf(fact_1037_apsnd__apfst,axiom,
    ! [F: rule > produc1996495991257130529ist_fm,G: produc1996495991257130529ist_fm > rule,X3: produc164195504107695125m_rule] :
      ( ( produc5714219303521602403m_rule @ F @ ( produc1151914582733876081e_rule @ G @ X3 ) )
      = ( produc7063755442767259331ist_fm @ ( G @ ( produc6879501374131015971m_rule @ X3 ) ) @ ( F @ ( produc7165828336582415457m_rule @ X3 ) ) ) ) ).

% apsnd_apfst
thf(fact_1038_apsnd__apfst,axiom,
    ! [F: list_fm > rule,G: list_fm > produc1996495991257130529ist_fm,X3: produc1996495991257130529ist_fm] :
      ( ( produc4802975340153367740ist_fm @ F @ ( produc147150953317395813ist_fm @ G @ X3 ) )
      = ( produc491467635432902671m_rule @ ( G @ ( produc1501393135466168645ist_fm @ X3 ) ) @ ( F @ ( produc4588648349897876871ist_fm @ X3 ) ) ) ) ).

% apsnd_apfst
thf(fact_1039_apsnd__apfst,axiom,
    ! [F: list_fm > list_fm,G: list_fm > list_fm,X3: produc1996495991257130529ist_fm] :
      ( ( produc9047611296996456864ist_fm @ F @ ( produc1130293176790411102ist_fm @ G @ X3 ) )
      = ( produc381145313068854617ist_fm @ ( G @ ( produc1501393135466168645ist_fm @ X3 ) ) @ ( F @ ( produc4588648349897876871ist_fm @ X3 ) ) ) ) ).

% apsnd_apfst
thf(fact_1040_apsnd__apfst,axiom,
    ! [F: list_fm > nat,G: list_fm > nat,X3: produc1996495991257130529ist_fm] :
      ( ( produc8916614678339685504at_nat @ F @ ( produc4198509902575289422ist_fm @ G @ X3 ) )
      = ( product_Pair_nat_nat @ ( G @ ( produc1501393135466168645ist_fm @ X3 ) ) @ ( F @ ( produc4588648349897876871ist_fm @ X3 ) ) ) ) ).

% apsnd_apfst
thf(fact_1041_apsnd__apfst,axiom,
    ! [F: list_fm > product_prod_nat_nat,G: list_fm > product_prod_nat_nat,X3: produc1996495991257130529ist_fm] :
      ( ( produc1584647392062276254at_nat @ F @ ( produc8169395892699911749ist_fm @ G @ X3 ) )
      = ( produc6161850002892822231at_nat @ ( G @ ( produc1501393135466168645ist_fm @ X3 ) ) @ ( F @ ( produc4588648349897876871ist_fm @ X3 ) ) ) ) ).

% apsnd_apfst
thf(fact_1042_apsnd__apfst,axiom,
    ! [F: list_fm > produc1996495991257130529ist_fm,G: list_fm > rule,X3: produc1996495991257130529ist_fm] :
      ( ( produc7454059569673786888m_rule @ F @ ( produc8579845201498058179ist_fm @ G @ X3 ) )
      = ( produc7063755442767259331ist_fm @ ( G @ ( produc1501393135466168645ist_fm @ X3 ) ) @ ( F @ ( produc4588648349897876871ist_fm @ X3 ) ) ) ) ).

% apsnd_apfst
thf(fact_1043_apfst__apsnd,axiom,
    ! [F: produc1996495991257130529ist_fm > produc1996495991257130529ist_fm,G: rule > rule,X3: produc164195504107695125m_rule] :
      ( ( produc3731403639160201281m_rule @ F @ ( produc3063135074001183255ist_fm @ G @ X3 ) )
      = ( produc491467635432902671m_rule @ ( F @ ( produc6879501374131015971m_rule @ X3 ) ) @ ( G @ ( produc7165828336582415457m_rule @ X3 ) ) ) ) ).

% apfst_apsnd
thf(fact_1044_apfst__apsnd,axiom,
    ! [F: produc1996495991257130529ist_fm > list_fm,G: rule > list_fm,X3: produc164195504107695125m_rule] :
      ( ( produc6153989603652719015ist_fm @ F @ ( produc2413407552729308604ist_fm @ G @ X3 ) )
      = ( produc381145313068854617ist_fm @ ( F @ ( produc6879501374131015971m_rule @ X3 ) ) @ ( G @ ( produc7165828336582415457m_rule @ X3 ) ) ) ) ).

% apfst_apsnd
thf(fact_1045_apfst__apsnd,axiom,
    ! [F: produc1996495991257130529ist_fm > nat,G: rule > nat,X3: produc164195504107695125m_rule] :
      ( ( produc8027278558657866375at_nat @ F @ ( produc5400006205771932108ist_fm @ G @ X3 ) )
      = ( product_Pair_nat_nat @ ( F @ ( produc6879501374131015971m_rule @ X3 ) ) @ ( G @ ( produc7165828336582415457m_rule @ X3 ) ) ) ) ).

% apfst_apsnd
thf(fact_1046_apfst__apsnd,axiom,
    ! [F: produc1996495991257130529ist_fm > product_prod_nat_nat,G: rule > product_prod_nat_nat,X3: produc164195504107695125m_rule] :
      ( ( produc3405531377997666853at_nat @ F @ ( produc5369481475936028579ist_fm @ G @ X3 ) )
      = ( produc6161850002892822231at_nat @ ( F @ ( produc6879501374131015971m_rule @ X3 ) ) @ ( G @ ( produc7165828336582415457m_rule @ X3 ) ) ) ) ).

% apfst_apsnd
thf(fact_1047_apfst__apsnd,axiom,
    ! [F: produc1996495991257130529ist_fm > rule,G: rule > produc1996495991257130529ist_fm,X3: produc164195504107695125m_rule] :
      ( ( produc1080319409639782133ist_fm @ F @ ( produc4729144337346781827ist_fm @ G @ X3 ) )
      = ( produc7063755442767259331ist_fm @ ( F @ ( produc6879501374131015971m_rule @ X3 ) ) @ ( G @ ( produc7165828336582415457m_rule @ X3 ) ) ) ) ).

% apfst_apsnd
thf(fact_1048_apfst__apsnd,axiom,
    ! [F: list_fm > produc1996495991257130529ist_fm,G: list_fm > rule,X3: produc1996495991257130529ist_fm] :
      ( ( produc7806619372436685002m_rule @ F @ ( produc4688919482198968325ist_fm @ G @ X3 ) )
      = ( produc491467635432902671m_rule @ ( F @ ( produc1501393135466168645ist_fm @ X3 ) ) @ ( G @ ( produc4588648349897876871ist_fm @ X3 ) ) ) ) ).

% apfst_apsnd
thf(fact_1049_apfst__apsnd,axiom,
    ! [F: list_fm > list_fm,G: list_fm > list_fm,X3: produc1996495991257130529ist_fm] :
      ( ( produc1130293176790411102ist_fm @ F @ ( produc9047611296996456864ist_fm @ G @ X3 ) )
      = ( produc381145313068854617ist_fm @ ( F @ ( produc1501393135466168645ist_fm @ X3 ) ) @ ( G @ ( produc4588648349897876871ist_fm @ X3 ) ) ) ) ).

% apfst_apsnd
thf(fact_1050_apfst__apsnd,axiom,
    ! [F: list_fm > nat,G: list_fm > nat,X3: produc1996495991257130529ist_fm] :
      ( ( produc4433738330514286142at_nat @ F @ ( produc5009473360894124176ist_fm @ G @ X3 ) )
      = ( product_Pair_nat_nat @ ( F @ ( produc1501393135466168645ist_fm @ X3 ) ) @ ( G @ ( produc4588648349897876871ist_fm @ X3 ) ) ) ) ).

% apfst_apsnd
thf(fact_1051_apfst__apsnd,axiom,
    ! [F: list_fm > product_prod_nat_nat,G: list_fm > product_prod_nat_nat,X3: produc1996495991257130529ist_fm] :
      ( ( produc2364714475711006556at_nat @ F @ ( produc9169599327578890115ist_fm @ G @ X3 ) )
      = ( produc6161850002892822231at_nat @ ( F @ ( produc1501393135466168645ist_fm @ X3 ) ) @ ( G @ ( produc4588648349897876871ist_fm @ X3 ) ) ) ) ).

% apfst_apsnd
thf(fact_1052_apfst__apsnd,axiom,
    ! [F: list_fm > rule,G: list_fm > produc1996495991257130529ist_fm,X3: produc1996495991257130529ist_fm] :
      ( ( produc5155535142916265854ist_fm @ F @ ( produc2501461190424677027ist_fm @ G @ X3 ) )
      = ( produc7063755442767259331ist_fm @ ( F @ ( produc1501393135466168645ist_fm @ X3 ) ) @ ( G @ ( produc4588648349897876871ist_fm @ X3 ) ) ) ) ).

% apfst_apsnd
thf(fact_1053_sinterleave_Osimps_I1_J,axiom,
    ! [S1: stream727092118206550309m_rule,S2: stream727092118206550309m_rule] :
      ( ( shd_Pr7235097944458474089m_rule @ ( sinter8084399408526207096m_rule @ S1 @ S2 ) )
      = ( shd_Pr7235097944458474089m_rule @ S1 ) ) ).

% sinterleave.simps(1)
thf(fact_1054_sinterleave_Osimps_I1_J,axiom,
    ! [S1: stream_rule,S2: stream_rule] :
      ( ( shd_rule @ ( sinterleave_rule @ S1 @ S2 ) )
      = ( shd_rule @ S1 ) ) ).

% sinterleave.simps(1)
thf(fact_1055_sinterleave_Osimps_I2_J,axiom,
    ! [S1: stream727092118206550309m_rule,S2: stream727092118206550309m_rule] :
      ( ( stl_Pr950425576149878629m_rule @ ( sinter8084399408526207096m_rule @ S1 @ S2 ) )
      = ( sinter8084399408526207096m_rule @ S2 @ ( stl_Pr950425576149878629m_rule @ S1 ) ) ) ).

% sinterleave.simps(2)
thf(fact_1056_sinterleave_Osimps_I2_J,axiom,
    ! [S1: stream_rule,S2: stream_rule] :
      ( ( stl_rule @ ( sinterleave_rule @ S1 @ S2 ) )
      = ( sinterleave_rule @ S2 @ ( stl_rule @ S1 ) ) ) ).

% sinterleave.simps(2)
thf(fact_1057_Stream_Osmember__def,axiom,
    ( smember_fm
    = ( ^ [X6: fm,S: stream_fm] : ( member_fm @ X6 @ ( sset_fm @ S ) ) ) ) ).

% Stream.smember_def
thf(fact_1058_Stream_Osmember__def,axiom,
    ( smembe2449974647026456671ist_fm
    = ( ^ [X6: produc1996495991257130529ist_fm,S: stream8299795917829157543ist_fm] : ( member8102475879199740618ist_fm @ X6 @ ( sset_P3431374123360452149ist_fm @ S ) ) ) ) ).

% Stream.smember_def
thf(fact_1059_Stream_Osmember__def,axiom,
    ( smembe1741262129464730021at_nat
    = ( ^ [X6: produc859450856879609959at_nat,S: stream8372878641218411373at_nat] : ( member8206827879206165904at_nat @ X6 @ ( sset_P9164763103296375419at_nat @ S ) ) ) ) ).

% Stream.smember_def
thf(fact_1060_Stream_Osmember__def,axiom,
    ( smember_rule
    = ( ^ [X6: rule,S: stream_rule] : ( member_rule @ X6 @ ( sset_rule @ S ) ) ) ) ).

% Stream.smember_def
thf(fact_1061_Stream_Osmember__def,axiom,
    ( smembe6335822699099550295m_rule
    = ( ^ [X6: produc164195504107695125m_rule,S: stream727092118206550309m_rule] : ( member4220325220686508332m_rule @ X6 @ ( sset_P4979403222752902913m_rule @ S ) ) ) ) ).

% Stream.smember_def
thf(fact_1062_sfilter_Osimps_I1_J,axiom,
    ! [P2: produc164195504107695125m_rule > $o,S3: stream727092118206550309m_rule] :
      ( ( shd_Pr7235097944458474089m_rule @ ( sfilte7579662470208054517m_rule @ P2 @ S3 ) )
      = ( shd_Pr7235097944458474089m_rule @ ( sdrop_5541336384294084785m_rule @ ( comp_o5517589812586031224m_rule @ (~) @ P2 ) @ S3 ) ) ) ).

% sfilter.simps(1)
thf(fact_1063_sfilter_Osimps_I1_J,axiom,
    ! [P2: rule > $o,S3: stream_rule] :
      ( ( shd_rule @ ( sfilter_rule @ P2 @ S3 ) )
      = ( shd_rule @ ( sdrop_while_rule @ ( comp_o_o_rule @ (~) @ P2 ) @ S3 ) ) ) ).

% sfilter.simps(1)
thf(fact_1064_sfilter_Osimps_I2_J,axiom,
    ! [P2: produc164195504107695125m_rule > $o,S3: stream727092118206550309m_rule] :
      ( ( stl_Pr950425576149878629m_rule @ ( sfilte7579662470208054517m_rule @ P2 @ S3 ) )
      = ( sfilte7579662470208054517m_rule @ P2 @ ( stl_Pr950425576149878629m_rule @ ( sdrop_5541336384294084785m_rule @ ( comp_o5517589812586031224m_rule @ (~) @ P2 ) @ S3 ) ) ) ) ).

% sfilter.simps(2)
thf(fact_1065_sfilter_Osimps_I2_J,axiom,
    ! [P2: rule > $o,S3: stream_rule] :
      ( ( stl_rule @ ( sfilter_rule @ P2 @ S3 ) )
      = ( sfilter_rule @ P2 @ ( stl_rule @ ( sdrop_while_rule @ ( comp_o_o_rule @ (~) @ P2 ) @ S3 ) ) ) ) ).

% sfilter.simps(2)
thf(fact_1066_sfilter__P,axiom,
    ! [P2: produc164195504107695125m_rule > $o,S3: stream727092118206550309m_rule] :
      ( ( P2 @ ( shd_Pr7235097944458474089m_rule @ S3 ) )
     => ( ( sfilte7579662470208054517m_rule @ P2 @ S3 )
        = ( sCons_5731567480799343806m_rule @ ( shd_Pr7235097944458474089m_rule @ S3 ) @ ( sfilte7579662470208054517m_rule @ P2 @ ( stl_Pr950425576149878629m_rule @ S3 ) ) ) ) ) ).

% sfilter_P
thf(fact_1067_sfilter__P,axiom,
    ! [P2: rule > $o,S3: stream_rule] :
      ( ( P2 @ ( shd_rule @ S3 ) )
     => ( ( sfilter_rule @ P2 @ S3 )
        = ( sCons_rule @ ( shd_rule @ S3 ) @ ( sfilter_rule @ P2 @ ( stl_rule @ S3 ) ) ) ) ) ).

% sfilter_P
thf(fact_1068_sfilter__not__P,axiom,
    ! [P2: produc164195504107695125m_rule > $o,S3: stream727092118206550309m_rule] :
      ( ~ ( P2 @ ( shd_Pr7235097944458474089m_rule @ S3 ) )
     => ( ( sfilte7579662470208054517m_rule @ P2 @ S3 )
        = ( sfilte7579662470208054517m_rule @ P2 @ ( stl_Pr950425576149878629m_rule @ S3 ) ) ) ) ).

% sfilter_not_P
thf(fact_1069_sfilter__not__P,axiom,
    ! [P2: rule > $o,S3: stream_rule] :
      ( ~ ( P2 @ ( shd_rule @ S3 ) )
     => ( ( sfilter_rule @ P2 @ S3 )
        = ( sfilter_rule @ P2 @ ( stl_rule @ S3 ) ) ) ) ).

% sfilter_not_P
thf(fact_1070_apfst__convE,axiom,
    ! [Q: produc164195504107695125m_rule,F: produc1996495991257130529ist_fm > produc1996495991257130529ist_fm,P: produc164195504107695125m_rule] :
      ( ( Q
        = ( produc3731403639160201281m_rule @ F @ P ) )
     => ~ ! [X: produc1996495991257130529ist_fm,Y: rule] :
            ( ( P
              = ( produc491467635432902671m_rule @ X @ Y ) )
           => ( Q
             != ( produc491467635432902671m_rule @ ( F @ X ) @ Y ) ) ) ) ).

% apfst_convE
thf(fact_1071_apfst__convE,axiom,
    ! [Q: produc1996495991257130529ist_fm,F: list_fm > list_fm,P: produc1996495991257130529ist_fm] :
      ( ( Q
        = ( produc1130293176790411102ist_fm @ F @ P ) )
     => ~ ! [X: list_fm,Y: list_fm] :
            ( ( P
              = ( produc381145313068854617ist_fm @ X @ Y ) )
           => ( Q
             != ( produc381145313068854617ist_fm @ ( F @ X ) @ Y ) ) ) ) ).

% apfst_convE
thf(fact_1072_apfst__convE,axiom,
    ! [Q: product_prod_nat_nat,F: nat > nat,P: product_prod_nat_nat] :
      ( ( Q
        = ( produc986720760941809198at_nat @ F @ P ) )
     => ~ ! [X: nat,Y: nat] :
            ( ( P
              = ( product_Pair_nat_nat @ X @ Y ) )
           => ( Q
             != ( product_Pair_nat_nat @ ( F @ X ) @ Y ) ) ) ) ).

% apfst_convE
thf(fact_1073_apfst__convE,axiom,
    ! [Q: produc859450856879609959at_nat,F: product_prod_nat_nat > product_prod_nat_nat,P: produc859450856879609959at_nat] :
      ( ( Q
        = ( produc3057951030792887557at_nat @ F @ P ) )
     => ~ ! [X: product_prod_nat_nat,Y: product_prod_nat_nat] :
            ( ( P
              = ( produc6161850002892822231at_nat @ X @ Y ) )
           => ( Q
             != ( produc6161850002892822231at_nat @ ( F @ X ) @ Y ) ) ) ) ).

% apfst_convE
thf(fact_1074_apfst__convE,axiom,
    ! [Q: produc4630866025709511057ist_fm,F: rule > rule,P: produc4630866025709511057ist_fm] :
      ( ( Q
        = ( produc5534506254562563289ist_fm @ F @ P ) )
     => ~ ! [X: rule,Y: produc1996495991257130529ist_fm] :
            ( ( P
              = ( produc7063755442767259331ist_fm @ X @ Y ) )
           => ( Q
             != ( produc7063755442767259331ist_fm @ ( F @ X ) @ Y ) ) ) ) ).

% apfst_convE
thf(fact_1075_verit__sko__forall__indirect2,axiom,
    ! [X3: fset_P661503646757059847ist_fm,P2: fset_P661503646757059847ist_fm > $o,P5: fset_P661503646757059847ist_fm > $o] :
      ( ( X3
        = ( fChoic3892542562059901019ist_fm
          @ ^ [X6: fset_P661503646757059847ist_fm] :
              ~ ( P2 @ X6 ) ) )
     => ( ! [X: fset_P661503646757059847ist_fm] :
            ( ( P2 @ X )
            = ( P5 @ X ) )
       => ( ( ! [X5: fset_P661503646757059847ist_fm] : ( P5 @ X5 ) )
          = ( P2 @ X3 ) ) ) ) ).

% verit_sko_forall_indirect2
thf(fact_1076_verit__sko__forall__indirect,axiom,
    ! [X3: fset_P661503646757059847ist_fm,P2: fset_P661503646757059847ist_fm > $o] :
      ( ( X3
        = ( fChoic3892542562059901019ist_fm
          @ ^ [X6: fset_P661503646757059847ist_fm] :
              ~ ( P2 @ X6 ) ) )
     => ( ( ! [X5: fset_P661503646757059847ist_fm] : ( P2 @ X5 ) )
        = ( P2 @ X3 ) ) ) ).

% verit_sko_forall_indirect
thf(fact_1077_verit__comp__simplify1_I2_J,axiom,
    ! [A: set_fm] : ( ord_less_eq_set_fm @ A @ A ) ).

% verit_comp_simplify1(2)
thf(fact_1078_verit__comp__simplify1_I2_J,axiom,
    ! [A: nat] : ( ord_less_eq_nat @ A @ A ) ).

% verit_comp_simplify1(2)
thf(fact_1079_verit__la__disequality,axiom,
    ! [A: nat,B: nat] :
      ( ( A = B )
      | ~ ( ord_less_eq_nat @ A @ B )
      | ~ ( ord_less_eq_nat @ B @ A ) ) ).

% verit_la_disequality
thf(fact_1080_verit__sko__ex_H,axiom,
    ! [P2: fset_P661503646757059847ist_fm > $o,A5: $o] :
      ( ( ( P2 @ ( fChoic3892542562059901019ist_fm @ P2 ) )
        = A5 )
     => ( ( ? [X5: fset_P661503646757059847ist_fm] : ( P2 @ X5 ) )
        = A5 ) ) ).

% verit_sko_ex'
thf(fact_1081_verit__sko__forall,axiom,
    ( ( ^ [P6: fset_P661503646757059847ist_fm > $o] :
        ! [X9: fset_P661503646757059847ist_fm] : ( P6 @ X9 ) )
    = ( ^ [P4: fset_P661503646757059847ist_fm > $o] :
          ( P4
          @ ( fChoic3892542562059901019ist_fm
            @ ^ [X6: fset_P661503646757059847ist_fm] :
                ~ ( P4 @ X6 ) ) ) ) ) ).

% verit_sko_forall
thf(fact_1082_verit__sko__forall_H,axiom,
    ! [P2: fset_P661503646757059847ist_fm > $o,A5: $o] :
      ( ( ( P2
          @ ( fChoic3892542562059901019ist_fm
            @ ^ [X6: fset_P661503646757059847ist_fm] :
                ~ ( P2 @ X6 ) ) )
        = A5 )
     => ( ( ! [X5: fset_P661503646757059847ist_fm] : ( P2 @ X5 ) )
        = A5 ) ) ).

% verit_sko_forall'
thf(fact_1083_verit__sko__forall_H_H,axiom,
    ! [B5: fset_P661503646757059847ist_fm,A5: fset_P661503646757059847ist_fm,P2: fset_P661503646757059847ist_fm > $o] :
      ( ( B5 = A5 )
     => ( ( ( fChoic3892542562059901019ist_fm @ P2 )
          = A5 )
        = ( ( fChoic3892542562059901019ist_fm @ P2 )
          = B5 ) ) ) ).

% verit_sko_forall''
thf(fact_1084_verit__sko__ex__indirect,axiom,
    ! [X3: fset_P661503646757059847ist_fm,P2: fset_P661503646757059847ist_fm > $o] :
      ( ( X3
        = ( fChoic3892542562059901019ist_fm @ P2 ) )
     => ( ( ? [X5: fset_P661503646757059847ist_fm] : ( P2 @ X5 ) )
        = ( P2 @ X3 ) ) ) ).

% verit_sko_ex_indirect
thf(fact_1085_verit__sko__ex__indirect2,axiom,
    ! [X3: fset_P661503646757059847ist_fm,P2: fset_P661503646757059847ist_fm > $o,P5: fset_P661503646757059847ist_fm > $o] :
      ( ( X3
        = ( fChoic3892542562059901019ist_fm @ P2 ) )
     => ( ! [X: fset_P661503646757059847ist_fm] :
            ( ( P2 @ X )
            = ( P5 @ X ) )
       => ( ( ? [X5: fset_P661503646757059847ist_fm] : ( P5 @ X5 ) )
          = ( P2 @ X3 ) ) ) ) ).

% verit_sko_ex_indirect2
thf(fact_1086_Fair__Stream_Ofair__def,axiom,
    ( fair_fair_rule
    = ( ^ [S: stream_rule] :
        ! [X6: rule] :
          ( ( member_rule @ X6 @ ( sset_rule @ S ) )
         => ! [M3: nat] :
            ? [N3: nat] :
              ( ( ord_less_eq_nat @ M3 @ N3 )
              & ( ( snth_rule @ S @ N3 )
                = X6 ) ) ) ) ) ).

% Fair_Stream.fair_def
thf(fact_1087_Fair__Stream_Ofair__def,axiom,
    ( fair_f6205644465234492970m_rule
    = ( ^ [S: stream727092118206550309m_rule] :
        ! [X6: produc164195504107695125m_rule] :
          ( ( member4220325220686508332m_rule @ X6 @ ( sset_P4979403222752902913m_rule @ S ) )
         => ! [M3: nat] :
            ? [N3: nat] :
              ( ( ord_less_eq_nat @ M3 @ N3 )
              & ( ( snth_P1712579846118164081m_rule @ S @ N3 )
                = X6 ) ) ) ) ) ).

% Fair_Stream.fair_def
thf(fact_1088_convol__expand__snd_H,axiom,
    ! [F: abstra4572470701423150929m_rule > produc164195504107695125m_rule,G: abstra4572470701423150929m_rule > produc1996495991257130529ist_fm,H: abstra4572470701423150929m_rule > rule] :
      ( ( ( comp_P4378417847848123130m_rule @ produc6879501374131015971m_rule @ F )
        = G )
     => ( ( H
          = ( comp_P1643750715206316048m_rule @ produc7165828336582415457m_rule @ F ) )
        = ( ( bNF_co2617238127527879852m_rule @ G @ H )
          = F ) ) ) ).

% convol_expand_snd'
thf(fact_1089_convol__expand__snd,axiom,
    ! [F: abstra4572470701423150929m_rule > produc164195504107695125m_rule,G: abstra4572470701423150929m_rule > produc1996495991257130529ist_fm] :
      ( ( ( comp_P4378417847848123130m_rule @ produc6879501374131015971m_rule @ F )
        = G )
     => ( ( bNF_co2617238127527879852m_rule @ G @ ( comp_P1643750715206316048m_rule @ produc7165828336582415457m_rule @ F ) )
        = F ) ) ).

% convol_expand_snd
thf(fact_1090_sdrop__smap2,axiom,
    ! [N: nat,F: rule > rule > rule,S1: stream_rule,S2: stream_rule] :
      ( ( sdrop_rule @ N @ ( smap2_rule_rule_rule @ F @ S1 @ S2 ) )
      = ( smap2_rule_rule_rule @ F @ ( sdrop_rule @ N @ S1 ) @ ( sdrop_rule @ N @ S2 ) ) ) ).

% sdrop_smap2
thf(fact_1091_sdrop__smap2,axiom,
    ! [N: nat,F: rule > produc780299800278082923e_rule > rule,S1: stream_rule,S2: stream3588992565182678257e_rule] :
      ( ( sdrop_rule @ N @ ( smap2_4638655784531377133e_rule @ F @ S1 @ S2 ) )
      = ( smap2_4638655784531377133e_rule @ F @ ( sdrop_rule @ N @ S1 ) @ ( sdrop_2199339672054592340e_rule @ N @ S2 ) ) ) ).

% sdrop_smap2
thf(fact_1092_sdrop__smap2,axiom,
    ! [N: nat,F: produc780299800278082923e_rule > rule > rule,S1: stream3588992565182678257e_rule,S2: stream_rule] :
      ( ( sdrop_rule @ N @ ( smap2_4273976090187245369e_rule @ F @ S1 @ S2 ) )
      = ( smap2_4273976090187245369e_rule @ F @ ( sdrop_2199339672054592340e_rule @ N @ S1 ) @ ( sdrop_rule @ N @ S2 ) ) ) ).

% sdrop_smap2
thf(fact_1093_sdrop__smap2,axiom,
    ! [N: nat,F: rule > rule > produc780299800278082923e_rule,S1: stream_rule,S2: stream_rule] :
      ( ( sdrop_2199339672054592340e_rule @ N @ ( smap2_2301042431756194977e_rule @ F @ S1 @ S2 ) )
      = ( smap2_2301042431756194977e_rule @ F @ ( sdrop_rule @ N @ S1 ) @ ( sdrop_rule @ N @ S2 ) ) ) ).

% sdrop_smap2
thf(fact_1094_sdrop__smap2,axiom,
    ! [N: nat,F: rule > produc1996495991257130529ist_fm > rule,S1: stream_rule,S2: stream8299795917829157543ist_fm] :
      ( ( sdrop_rule @ N @ ( smap2_3943848181979139747m_rule @ F @ S1 @ S2 ) )
      = ( smap2_3943848181979139747m_rule @ F @ ( sdrop_rule @ N @ S1 ) @ ( sdrop_4442373711808556042ist_fm @ N @ S2 ) ) ) ).

% sdrop_smap2
thf(fact_1095_sdrop__smap2,axiom,
    ! [N: nat,F: produc1996495991257130529ist_fm > rule > rule,S1: stream8299795917829157543ist_fm,S2: stream_rule] :
      ( ( sdrop_rule @ N @ ( smap2_6133544317484809199e_rule @ F @ S1 @ S2 ) )
      = ( smap2_6133544317484809199e_rule @ F @ ( sdrop_4442373711808556042ist_fm @ N @ S1 ) @ ( sdrop_rule @ N @ S2 ) ) ) ).

% sdrop_smap2
thf(fact_1096_sdrop__smap2,axiom,
    ! [N: nat,F: produc780299800278082923e_rule > produc780299800278082923e_rule > rule,S1: stream3588992565182678257e_rule,S2: stream3588992565182678257e_rule] :
      ( ( sdrop_rule @ N @ ( smap2_6314367768709263299e_rule @ F @ S1 @ S2 ) )
      = ( smap2_6314367768709263299e_rule @ F @ ( sdrop_2199339672054592340e_rule @ N @ S1 ) @ ( sdrop_2199339672054592340e_rule @ N @ S2 ) ) ) ).

% sdrop_smap2
thf(fact_1097_sdrop__smap2,axiom,
    ! [N: nat,F: rule > rule > produc1996495991257130529ist_fm,S1: stream_rule,S2: stream_rule] :
      ( ( sdrop_4442373711808556042ist_fm @ N @ ( smap2_1292763952458720599ist_fm @ F @ S1 @ S2 ) )
      = ( smap2_1292763952458720599ist_fm @ F @ ( sdrop_rule @ N @ S1 ) @ ( sdrop_rule @ N @ S2 ) ) ) ).

% sdrop_smap2
thf(fact_1098_sdrop__smap2,axiom,
    ! [N: nat,F: rule > produc780299800278082923e_rule > produc780299800278082923e_rule,S1: stream_rule,S2: stream3588992565182678257e_rule] :
      ( ( sdrop_2199339672054592340e_rule @ N @ ( smap2_644143445760767683e_rule @ F @ S1 @ S2 ) )
      = ( smap2_644143445760767683e_rule @ F @ ( sdrop_rule @ N @ S1 ) @ ( sdrop_2199339672054592340e_rule @ N @ S2 ) ) ) ).

% sdrop_smap2
thf(fact_1099_sdrop__smap2,axiom,
    ! [N: nat,F: produc780299800278082923e_rule > rule > produc780299800278082923e_rule,S1: stream3588992565182678257e_rule,S2: stream_rule] :
      ( ( sdrop_2199339672054592340e_rule @ N @ ( smap2_3976754415934081143e_rule @ F @ S1 @ S2 ) )
      = ( smap2_3976754415934081143e_rule @ F @ ( sdrop_2199339672054592340e_rule @ N @ S1 ) @ ( sdrop_rule @ N @ S2 ) ) ) ).

% sdrop_smap2
thf(fact_1100_snth__smap2,axiom,
    ! [F: rule > rule > rule,S1: stream_rule,S2: stream_rule,N: nat] :
      ( ( snth_rule @ ( smap2_rule_rule_rule @ F @ S1 @ S2 ) @ N )
      = ( F @ ( snth_rule @ S1 @ N ) @ ( snth_rule @ S2 @ N ) ) ) ).

% snth_smap2
thf(fact_1101_snth__smap2,axiom,
    ! [F: rule > produc164195504107695125m_rule > rule,S1: stream_rule,S2: stream727092118206550309m_rule,N: nat] :
      ( ( snth_rule @ ( smap2_2737362159257330783e_rule @ F @ S1 @ S2 ) @ N )
      = ( F @ ( snth_rule @ S1 @ N ) @ ( snth_P1712579846118164081m_rule @ S2 @ N ) ) ) ).

% snth_smap2
thf(fact_1102_snth__smap2,axiom,
    ! [F: produc164195504107695125m_rule > rule > rule,S1: stream727092118206550309m_rule,S2: stream_rule,N: nat] :
      ( ( snth_rule @ ( smap2_190827247425732959e_rule @ F @ S1 @ S2 ) @ N )
      = ( F @ ( snth_P1712579846118164081m_rule @ S1 @ N ) @ ( snth_rule @ S2 @ N ) ) ) ).

% snth_smap2
thf(fact_1103_snth__smap2,axiom,
    ! [F: produc164195504107695125m_rule > produc164195504107695125m_rule > rule,S1: stream727092118206550309m_rule,S2: stream727092118206550309m_rule,N: nat] :
      ( ( snth_rule @ ( smap2_1391242002117124143e_rule @ F @ S1 @ S2 ) @ N )
      = ( F @ ( snth_P1712579846118164081m_rule @ S1 @ N ) @ ( snth_P1712579846118164081m_rule @ S2 @ N ) ) ) ).

% snth_smap2
thf(fact_1104_snth__smap2,axiom,
    ! [F: rule > rule > produc164195504107695125m_rule,S1: stream_rule,S2: stream_rule,N: nat] :
      ( ( snth_P1712579846118164081m_rule @ ( smap2_4438185555641665375m_rule @ F @ S1 @ S2 ) @ N )
      = ( F @ ( snth_rule @ S1 @ N ) @ ( snth_rule @ S2 @ N ) ) ) ).

% snth_smap2
thf(fact_1105_snth__smap2,axiom,
    ! [F: rule > produc164195504107695125m_rule > produc164195504107695125m_rule,S1: stream_rule,S2: stream727092118206550309m_rule,N: nat] :
      ( ( snth_P1712579846118164081m_rule @ ( smap2_7722561687528711727m_rule @ F @ S1 @ S2 ) @ N )
      = ( F @ ( snth_rule @ S1 @ N ) @ ( snth_P1712579846118164081m_rule @ S2 @ N ) ) ) ).

% snth_smap2
thf(fact_1106_snth__smap2,axiom,
    ! [F: produc164195504107695125m_rule > rule > produc164195504107695125m_rule,S1: stream727092118206550309m_rule,S2: stream_rule,N: nat] :
      ( ( snth_P1712579846118164081m_rule @ ( smap2_3092065398501458735m_rule @ F @ S1 @ S2 ) @ N )
      = ( F @ ( snth_P1712579846118164081m_rule @ S1 @ N ) @ ( snth_rule @ S2 @ N ) ) ) ).

% snth_smap2
thf(fact_1107_snth__smap2,axiom,
    ! [F: produc164195504107695125m_rule > produc164195504107695125m_rule > produc164195504107695125m_rule,S1: stream727092118206550309m_rule,S2: stream727092118206550309m_rule,N: nat] :
      ( ( snth_P1712579846118164081m_rule @ ( smap2_3865418324013074943m_rule @ F @ S1 @ S2 ) @ N )
      = ( F @ ( snth_P1712579846118164081m_rule @ S1 @ N ) @ ( snth_P1712579846118164081m_rule @ S2 @ N ) ) ) ).

% snth_smap2
thf(fact_1108_smap2_Osimps_I1_J,axiom,
    ! [F: produc164195504107695125m_rule > produc164195504107695125m_rule > produc164195504107695125m_rule,S1: stream727092118206550309m_rule,S2: stream727092118206550309m_rule] :
      ( ( shd_Pr7235097944458474089m_rule @ ( smap2_3865418324013074943m_rule @ F @ S1 @ S2 ) )
      = ( F @ ( shd_Pr7235097944458474089m_rule @ S1 ) @ ( shd_Pr7235097944458474089m_rule @ S2 ) ) ) ).

% smap2.simps(1)
thf(fact_1109_smap2_Osimps_I1_J,axiom,
    ! [F: produc164195504107695125m_rule > rule > produc164195504107695125m_rule,S1: stream727092118206550309m_rule,S2: stream_rule] :
      ( ( shd_Pr7235097944458474089m_rule @ ( smap2_3092065398501458735m_rule @ F @ S1 @ S2 ) )
      = ( F @ ( shd_Pr7235097944458474089m_rule @ S1 ) @ ( shd_rule @ S2 ) ) ) ).

% smap2.simps(1)
thf(fact_1110_smap2_Osimps_I1_J,axiom,
    ! [F: rule > produc164195504107695125m_rule > produc164195504107695125m_rule,S1: stream_rule,S2: stream727092118206550309m_rule] :
      ( ( shd_Pr7235097944458474089m_rule @ ( smap2_7722561687528711727m_rule @ F @ S1 @ S2 ) )
      = ( F @ ( shd_rule @ S1 ) @ ( shd_Pr7235097944458474089m_rule @ S2 ) ) ) ).

% smap2.simps(1)
thf(fact_1111_smap2_Osimps_I1_J,axiom,
    ! [F: rule > rule > produc164195504107695125m_rule,S1: stream_rule,S2: stream_rule] :
      ( ( shd_Pr7235097944458474089m_rule @ ( smap2_4438185555641665375m_rule @ F @ S1 @ S2 ) )
      = ( F @ ( shd_rule @ S1 ) @ ( shd_rule @ S2 ) ) ) ).

% smap2.simps(1)
thf(fact_1112_smap2_Osimps_I1_J,axiom,
    ! [F: produc164195504107695125m_rule > produc164195504107695125m_rule > rule,S1: stream727092118206550309m_rule,S2: stream727092118206550309m_rule] :
      ( ( shd_rule @ ( smap2_1391242002117124143e_rule @ F @ S1 @ S2 ) )
      = ( F @ ( shd_Pr7235097944458474089m_rule @ S1 ) @ ( shd_Pr7235097944458474089m_rule @ S2 ) ) ) ).

% smap2.simps(1)
thf(fact_1113_smap2_Osimps_I1_J,axiom,
    ! [F: produc164195504107695125m_rule > rule > rule,S1: stream727092118206550309m_rule,S2: stream_rule] :
      ( ( shd_rule @ ( smap2_190827247425732959e_rule @ F @ S1 @ S2 ) )
      = ( F @ ( shd_Pr7235097944458474089m_rule @ S1 ) @ ( shd_rule @ S2 ) ) ) ).

% smap2.simps(1)
thf(fact_1114_smap2_Osimps_I1_J,axiom,
    ! [F: rule > produc164195504107695125m_rule > rule,S1: stream_rule,S2: stream727092118206550309m_rule] :
      ( ( shd_rule @ ( smap2_2737362159257330783e_rule @ F @ S1 @ S2 ) )
      = ( F @ ( shd_rule @ S1 ) @ ( shd_Pr7235097944458474089m_rule @ S2 ) ) ) ).

% smap2.simps(1)
thf(fact_1115_smap2_Osimps_I1_J,axiom,
    ! [F: rule > rule > rule,S1: stream_rule,S2: stream_rule] :
      ( ( shd_rule @ ( smap2_rule_rule_rule @ F @ S1 @ S2 ) )
      = ( F @ ( shd_rule @ S1 ) @ ( shd_rule @ S2 ) ) ) ).

% smap2.simps(1)
thf(fact_1116_smap2_Osimps_I2_J,axiom,
    ! [F: produc164195504107695125m_rule > produc164195504107695125m_rule > produc164195504107695125m_rule,S1: stream727092118206550309m_rule,S2: stream727092118206550309m_rule] :
      ( ( stl_Pr950425576149878629m_rule @ ( smap2_3865418324013074943m_rule @ F @ S1 @ S2 ) )
      = ( smap2_3865418324013074943m_rule @ F @ ( stl_Pr950425576149878629m_rule @ S1 ) @ ( stl_Pr950425576149878629m_rule @ S2 ) ) ) ).

% smap2.simps(2)
thf(fact_1117_smap2_Osimps_I2_J,axiom,
    ! [F: produc164195504107695125m_rule > rule > produc164195504107695125m_rule,S1: stream727092118206550309m_rule,S2: stream_rule] :
      ( ( stl_Pr950425576149878629m_rule @ ( smap2_3092065398501458735m_rule @ F @ S1 @ S2 ) )
      = ( smap2_3092065398501458735m_rule @ F @ ( stl_Pr950425576149878629m_rule @ S1 ) @ ( stl_rule @ S2 ) ) ) ).

% smap2.simps(2)
thf(fact_1118_smap2_Osimps_I2_J,axiom,
    ! [F: rule > produc164195504107695125m_rule > produc164195504107695125m_rule,S1: stream_rule,S2: stream727092118206550309m_rule] :
      ( ( stl_Pr950425576149878629m_rule @ ( smap2_7722561687528711727m_rule @ F @ S1 @ S2 ) )
      = ( smap2_7722561687528711727m_rule @ F @ ( stl_rule @ S1 ) @ ( stl_Pr950425576149878629m_rule @ S2 ) ) ) ).

% smap2.simps(2)
thf(fact_1119_smap2_Osimps_I2_J,axiom,
    ! [F: rule > rule > produc164195504107695125m_rule,S1: stream_rule,S2: stream_rule] :
      ( ( stl_Pr950425576149878629m_rule @ ( smap2_4438185555641665375m_rule @ F @ S1 @ S2 ) )
      = ( smap2_4438185555641665375m_rule @ F @ ( stl_rule @ S1 ) @ ( stl_rule @ S2 ) ) ) ).

% smap2.simps(2)
thf(fact_1120_smap2_Osimps_I2_J,axiom,
    ! [F: produc164195504107695125m_rule > produc164195504107695125m_rule > rule,S1: stream727092118206550309m_rule,S2: stream727092118206550309m_rule] :
      ( ( stl_rule @ ( smap2_1391242002117124143e_rule @ F @ S1 @ S2 ) )
      = ( smap2_1391242002117124143e_rule @ F @ ( stl_Pr950425576149878629m_rule @ S1 ) @ ( stl_Pr950425576149878629m_rule @ S2 ) ) ) ).

% smap2.simps(2)
thf(fact_1121_smap2_Osimps_I2_J,axiom,
    ! [F: produc164195504107695125m_rule > rule > rule,S1: stream727092118206550309m_rule,S2: stream_rule] :
      ( ( stl_rule @ ( smap2_190827247425732959e_rule @ F @ S1 @ S2 ) )
      = ( smap2_190827247425732959e_rule @ F @ ( stl_Pr950425576149878629m_rule @ S1 ) @ ( stl_rule @ S2 ) ) ) ).

% smap2.simps(2)
thf(fact_1122_smap2_Osimps_I2_J,axiom,
    ! [F: rule > produc164195504107695125m_rule > rule,S1: stream_rule,S2: stream727092118206550309m_rule] :
      ( ( stl_rule @ ( smap2_2737362159257330783e_rule @ F @ S1 @ S2 ) )
      = ( smap2_2737362159257330783e_rule @ F @ ( stl_rule @ S1 ) @ ( stl_Pr950425576149878629m_rule @ S2 ) ) ) ).

% smap2.simps(2)
thf(fact_1123_smap2_Osimps_I2_J,axiom,
    ! [F: rule > rule > rule,S1: stream_rule,S2: stream_rule] :
      ( ( stl_rule @ ( smap2_rule_rule_rule @ F @ S1 @ S2 ) )
      = ( smap2_rule_rule_rule @ F @ ( stl_rule @ S1 ) @ ( stl_rule @ S2 ) ) ) ).

% smap2.simps(2)
thf(fact_1124_smap2__alt,axiom,
    ! [F: rule > rule > rule,S1: stream_rule,S2: stream_rule,S3: stream_rule] :
      ( ( ( smap2_rule_rule_rule @ F @ S1 @ S2 )
        = S3 )
      = ( ! [N3: nat] :
            ( ( F @ ( snth_rule @ S1 @ N3 ) @ ( snth_rule @ S2 @ N3 ) )
            = ( snth_rule @ S3 @ N3 ) ) ) ) ).

% smap2_alt
thf(fact_1125_smap2__alt,axiom,
    ! [F: rule > rule > produc164195504107695125m_rule,S1: stream_rule,S2: stream_rule,S3: stream727092118206550309m_rule] :
      ( ( ( smap2_4438185555641665375m_rule @ F @ S1 @ S2 )
        = S3 )
      = ( ! [N3: nat] :
            ( ( F @ ( snth_rule @ S1 @ N3 ) @ ( snth_rule @ S2 @ N3 ) )
            = ( snth_P1712579846118164081m_rule @ S3 @ N3 ) ) ) ) ).

% smap2_alt
thf(fact_1126_smap2__alt,axiom,
    ! [F: rule > produc164195504107695125m_rule > rule,S1: stream_rule,S2: stream727092118206550309m_rule,S3: stream_rule] :
      ( ( ( smap2_2737362159257330783e_rule @ F @ S1 @ S2 )
        = S3 )
      = ( ! [N3: nat] :
            ( ( F @ ( snth_rule @ S1 @ N3 ) @ ( snth_P1712579846118164081m_rule @ S2 @ N3 ) )
            = ( snth_rule @ S3 @ N3 ) ) ) ) ).

% smap2_alt
thf(fact_1127_smap2__alt,axiom,
    ! [F: rule > produc164195504107695125m_rule > produc164195504107695125m_rule,S1: stream_rule,S2: stream727092118206550309m_rule,S3: stream727092118206550309m_rule] :
      ( ( ( smap2_7722561687528711727m_rule @ F @ S1 @ S2 )
        = S3 )
      = ( ! [N3: nat] :
            ( ( F @ ( snth_rule @ S1 @ N3 ) @ ( snth_P1712579846118164081m_rule @ S2 @ N3 ) )
            = ( snth_P1712579846118164081m_rule @ S3 @ N3 ) ) ) ) ).

% smap2_alt
thf(fact_1128_smap2__alt,axiom,
    ! [F: produc164195504107695125m_rule > rule > rule,S1: stream727092118206550309m_rule,S2: stream_rule,S3: stream_rule] :
      ( ( ( smap2_190827247425732959e_rule @ F @ S1 @ S2 )
        = S3 )
      = ( ! [N3: nat] :
            ( ( F @ ( snth_P1712579846118164081m_rule @ S1 @ N3 ) @ ( snth_rule @ S2 @ N3 ) )
            = ( snth_rule @ S3 @ N3 ) ) ) ) ).

% smap2_alt
thf(fact_1129_smap2__alt,axiom,
    ! [F: produc164195504107695125m_rule > rule > produc164195504107695125m_rule,S1: stream727092118206550309m_rule,S2: stream_rule,S3: stream727092118206550309m_rule] :
      ( ( ( smap2_3092065398501458735m_rule @ F @ S1 @ S2 )
        = S3 )
      = ( ! [N3: nat] :
            ( ( F @ ( snth_P1712579846118164081m_rule @ S1 @ N3 ) @ ( snth_rule @ S2 @ N3 ) )
            = ( snth_P1712579846118164081m_rule @ S3 @ N3 ) ) ) ) ).

% smap2_alt
thf(fact_1130_smap2__alt,axiom,
    ! [F: produc164195504107695125m_rule > produc164195504107695125m_rule > rule,S1: stream727092118206550309m_rule,S2: stream727092118206550309m_rule,S3: stream_rule] :
      ( ( ( smap2_1391242002117124143e_rule @ F @ S1 @ S2 )
        = S3 )
      = ( ! [N3: nat] :
            ( ( F @ ( snth_P1712579846118164081m_rule @ S1 @ N3 ) @ ( snth_P1712579846118164081m_rule @ S2 @ N3 ) )
            = ( snth_rule @ S3 @ N3 ) ) ) ) ).

% smap2_alt
thf(fact_1131_smap2__alt,axiom,
    ! [F: produc164195504107695125m_rule > produc164195504107695125m_rule > produc164195504107695125m_rule,S1: stream727092118206550309m_rule,S2: stream727092118206550309m_rule,S3: stream727092118206550309m_rule] :
      ( ( ( smap2_3865418324013074943m_rule @ F @ S1 @ S2 )
        = S3 )
      = ( ! [N3: nat] :
            ( ( F @ ( snth_P1712579846118164081m_rule @ S1 @ N3 ) @ ( snth_P1712579846118164081m_rule @ S2 @ N3 ) )
            = ( snth_P1712579846118164081m_rule @ S3 @ N3 ) ) ) ) ).

% smap2_alt
thf(fact_1132_fst__convol,axiom,
    ! [F: abstra4572470701423150929m_rule > produc1996495991257130529ist_fm,G: abstra4572470701423150929m_rule > rule] :
      ( ( comp_P4378417847848123130m_rule @ produc6879501374131015971m_rule @ ( bNF_co2617238127527879852m_rule @ F @ G ) )
      = F ) ).

% fst_convol
thf(fact_1133_K__record__comp,axiom,
    ! [C2: produc1996495991257130529ist_fm,F: abstra4572470701423150929m_rule > produc164195504107695125m_rule] :
      ( ( comp_P4378417847848123130m_rule
        @ ^ [X6: produc164195504107695125m_rule] : C2
        @ F )
      = ( ^ [X6: abstra4572470701423150929m_rule] : C2 ) ) ).

% K_record_comp
thf(fact_1134_snd__fst__flip,axiom,
    ( product_snd_nat_nat
    = ( comp_P1567445206330693457at_nat @ product_fst_nat_nat
      @ ( produc2626176000494625587at_nat
        @ ^ [X6: nat,Y5: nat] : ( product_Pair_nat_nat @ Y5 @ X6 ) ) ) ) ).

% snd_fst_flip
thf(fact_1135_snd__fst__flip,axiom,
    ( produc6408287024330202629at_nat
    = ( comp_P4814319827393324394at_nat @ produc3213797794245857475at_nat
      @ ( produc6744312701629110395at_nat
        @ ^ [X6: product_prod_nat_nat,Y5: product_prod_nat_nat] : ( produc6161850002892822231at_nat @ Y5 @ X6 ) ) ) ) ).

% snd_fst_flip
thf(fact_1136_snd__fst__flip,axiom,
    ( produc4514744107061996309ist_fm
    = ( comp_P2707267495825058490ist_fm @ produc6879501374131015971m_rule
      @ ( produc2185035559847904517m_rule
        @ ^ [X6: rule,Y5: produc1996495991257130529ist_fm] : ( produc491467635432902671m_rule @ Y5 @ X6 ) ) ) ) ).

% snd_fst_flip
thf(fact_1137_snd__fst__flip,axiom,
    ( produc7165828336582415457m_rule
    = ( comp_P3413117018928051920m_rule @ produc4228417144610596823ist_fm
      @ ( produc1775668667622460877ist_fm
        @ ^ [X6: produc1996495991257130529ist_fm,Y5: rule] : ( produc7063755442767259331ist_fm @ Y5 @ X6 ) ) ) ) ).

% snd_fst_flip
thf(fact_1138_snd__fst__flip,axiom,
    ( produc4588648349897876871ist_fm
    = ( comp_P1426762558556845121ist_fm @ produc1501393135466168645ist_fm
      @ ( produc4544959197559615795ist_fm
        @ ^ [X6: list_fm,Y5: list_fm] : ( produc381145313068854617ist_fm @ Y5 @ X6 ) ) ) ) ).

% snd_fst_flip
thf(fact_1139_fst__snd__flip,axiom,
    ( product_fst_nat_nat
    = ( comp_P1567445206330693457at_nat @ product_snd_nat_nat
      @ ( produc2626176000494625587at_nat
        @ ^ [X6: nat,Y5: nat] : ( product_Pair_nat_nat @ Y5 @ X6 ) ) ) ) ).

% fst_snd_flip
thf(fact_1140_fst__snd__flip,axiom,
    ( produc3213797794245857475at_nat
    = ( comp_P4814319827393324394at_nat @ produc6408287024330202629at_nat
      @ ( produc6744312701629110395at_nat
        @ ^ [X6: product_prod_nat_nat,Y5: product_prod_nat_nat] : ( produc6161850002892822231at_nat @ Y5 @ X6 ) ) ) ) ).

% fst_snd_flip
thf(fact_1141_fst__snd__flip,axiom,
    ( produc6879501374131015971m_rule
    = ( comp_P6646430376872094402m_rule @ produc4514744107061996309ist_fm
      @ ( produc1775668667622460877ist_fm
        @ ^ [X6: produc1996495991257130529ist_fm,Y5: rule] : ( produc7063755442767259331ist_fm @ Y5 @ X6 ) ) ) ) ).

% fst_snd_flip
thf(fact_1142_fst__snd__flip,axiom,
    ( produc4228417144610596823ist_fm
    = ( comp_P4888803072877744336ist_fm @ produc7165828336582415457m_rule
      @ ( produc2185035559847904517m_rule
        @ ^ [X6: rule,Y5: produc1996495991257130529ist_fm] : ( produc491467635432902671m_rule @ Y5 @ X6 ) ) ) ) ).

% fst_snd_flip
thf(fact_1143_fst__snd__flip,axiom,
    ( produc1501393135466168645ist_fm
    = ( comp_P1426762558556845121ist_fm @ produc4588648349897876871ist_fm
      @ ( produc4544959197559615795ist_fm
        @ ^ [X6: list_fm,Y5: list_fm] : ( produc381145313068854617ist_fm @ Y5 @ X6 ) ) ) ) ).

% fst_snd_flip
thf(fact_1144_stake__pos__minWait,axiom,
    ! [Rs: stream_rule,S3: produc1996495991257130529ist_fm,R2: rule] :
      ( ( abstra1114877437032970289r_rule @ rules @ Rs )
     => ( ( ord_less_nat
          @ ( abstra738319790293016300ist_fm
            @ ^ [R: rule,S: produc1996495991257130529ist_fm,Ss: fset_P661503646757059847ist_fm] :
                ( ( eff @ R @ S )
                = ( some_f2162114139285431020ist_fm @ Ss ) )
            @ Rs
            @ S3 )
          @ ( abstract_pos_rule @ Rs @ R2 ) )
       => ( ( member_rule @ R2 @ ( sset_rule @ rules ) )
         => ( ( member8102475879199740618ist_fm @ S3 @ top_to728902761071950545ist_fm )
           => ( ( abstract_pos_rule
                @ ( stl_rule
                  @ ( abstra2363873759530810705ist_fm
                    @ ^ [R: rule,S: produc1996495991257130529ist_fm,Ss: fset_P661503646757059847ist_fm] :
                        ( ( eff @ R @ S )
                        = ( some_f2162114139285431020ist_fm @ Ss ) )
                    @ Rs
                    @ S3 ) )
                @ R2 )
              = ( minus_minus_nat @ ( abstract_pos_rule @ Rs @ R2 )
                @ ( suc
                  @ ( abstra738319790293016300ist_fm
                    @ ^ [R: rule,S: produc1996495991257130529ist_fm,Ss: fset_P661503646757059847ist_fm] :
                        ( ( eff @ R @ S )
                        = ( some_f2162114139285431020ist_fm @ Ss ) )
                    @ Rs
                    @ S3 ) ) ) ) ) ) ) ) ).

% stake_pos_minWait
thf(fact_1145_sdrop__simps_I2_J,axiom,
    ! [N: nat,S3: stream8299795917829157543ist_fm] :
      ( ( stl_Pr5027778045650968273ist_fm @ ( sdrop_4442373711808556042ist_fm @ N @ S3 ) )
      = ( sdrop_4442373711808556042ist_fm @ ( suc @ N ) @ S3 ) ) ).

% sdrop_simps(2)
thf(fact_1146_sdrop__simps_I2_J,axiom,
    ! [N: nat,S3: stream3588992565182678257e_rule] :
      ( ( stl_Pr4904654673476115995e_rule @ ( sdrop_2199339672054592340e_rule @ N @ S3 ) )
      = ( sdrop_2199339672054592340e_rule @ ( suc @ N ) @ S3 ) ) ).

% sdrop_simps(2)
thf(fact_1147_sdrop__simps_I2_J,axiom,
    ! [N: nat,S3: stream1878305877988316353m_rule] :
      ( ( stl_Pr5425948690297933931m_rule @ ( sdrop_1764812713384251940m_rule @ N @ S3 ) )
      = ( sdrop_1764812713384251940m_rule @ ( suc @ N ) @ S3 ) ) ).

% sdrop_simps(2)
thf(fact_1148_sdrop__simps_I2_J,axiom,
    ! [N: nat,S3: stream2312013117288958913e_rule] :
      ( ( stl_Pr830518440239794027e_rule @ ( sdrop_6392754500180887844e_rule @ N @ S3 ) )
      = ( sdrop_6392754500180887844e_rule @ ( suc @ N ) @ S3 ) ) ).

% sdrop_simps(2)
thf(fact_1149_sdrop__simps_I2_J,axiom,
    ! [N: nat,S3: stream6494289010434245521m_rule] :
      ( ( stl_Pr2200982372225425851m_rule @ ( sdrop_7373388980841566196m_rule @ N @ S3 ) )
      = ( sdrop_7373388980841566196m_rule @ ( suc @ N ) @ S3 ) ) ).

% sdrop_simps(2)
thf(fact_1150_sdrop__simps_I2_J,axiom,
    ! [N: nat,S3: stream727092118206550309m_rule] :
      ( ( stl_Pr950425576149878629m_rule @ ( sdrop_7224736112439592940m_rule @ N @ S3 ) )
      = ( sdrop_7224736112439592940m_rule @ ( suc @ N ) @ S3 ) ) ).

% sdrop_simps(2)
thf(fact_1151_sdrop__simps_I2_J,axiom,
    ! [N: nat,S3: stream_rule] :
      ( ( stl_rule @ ( sdrop_rule @ N @ S3 ) )
      = ( sdrop_rule @ ( suc @ N ) @ S3 ) ) ).

% sdrop_simps(2)
thf(fact_1152_order__le__imp__less__or__eq,axiom,
    ! [X3: set_fm,Y3: set_fm] :
      ( ( ord_less_eq_set_fm @ X3 @ Y3 )
     => ( ( ord_less_set_fm @ X3 @ Y3 )
        | ( X3 = Y3 ) ) ) ).

% order_le_imp_less_or_eq
thf(fact_1153_order__le__imp__less__or__eq,axiom,
    ! [X3: nat,Y3: nat] :
      ( ( ord_less_eq_nat @ X3 @ Y3 )
     => ( ( ord_less_nat @ X3 @ Y3 )
        | ( X3 = Y3 ) ) ) ).

% order_le_imp_less_or_eq
thf(fact_1154_linorder__le__less__linear,axiom,
    ! [X3: nat,Y3: nat] :
      ( ( ord_less_eq_nat @ X3 @ Y3 )
      | ( ord_less_nat @ Y3 @ X3 ) ) ).

% linorder_le_less_linear
thf(fact_1155_order__less__le__subst2,axiom,
    ! [A: nat,B: nat,F: nat > set_fm,C2: set_fm] :
      ( ( ord_less_nat @ A @ B )
     => ( ( ord_less_eq_set_fm @ ( F @ B ) @ C2 )
       => ( ! [X: nat,Y: nat] :
              ( ( ord_less_nat @ X @ Y )
             => ( ord_less_set_fm @ ( F @ X ) @ ( F @ Y ) ) )
         => ( ord_less_set_fm @ ( F @ A ) @ C2 ) ) ) ) ).

% order_less_le_subst2
thf(fact_1156_order__less__le__subst2,axiom,
    ! [A: nat,B: nat,F: nat > nat,C2: nat] :
      ( ( ord_less_nat @ A @ B )
     => ( ( ord_less_eq_nat @ ( F @ B ) @ C2 )
       => ( ! [X: nat,Y: nat] :
              ( ( ord_less_nat @ X @ Y )
             => ( ord_less_nat @ ( F @ X ) @ ( F @ Y ) ) )
         => ( ord_less_nat @ ( F @ A ) @ C2 ) ) ) ) ).

% order_less_le_subst2
thf(fact_1157_order__less__le__subst1,axiom,
    ! [A: set_fm,F: set_fm > set_fm,B: set_fm,C2: set_fm] :
      ( ( ord_less_set_fm @ A @ ( F @ B ) )
     => ( ( ord_less_eq_set_fm @ B @ C2 )
       => ( ! [X: set_fm,Y: set_fm] :
              ( ( ord_less_eq_set_fm @ X @ Y )
             => ( ord_less_eq_set_fm @ ( F @ X ) @ ( F @ Y ) ) )
         => ( ord_less_set_fm @ A @ ( F @ C2 ) ) ) ) ) ).

% order_less_le_subst1
thf(fact_1158_order__less__le__subst1,axiom,
    ! [A: nat,F: set_fm > nat,B: set_fm,C2: set_fm] :
      ( ( ord_less_nat @ A @ ( F @ B ) )
     => ( ( ord_less_eq_set_fm @ B @ C2 )
       => ( ! [X: set_fm,Y: set_fm] :
              ( ( ord_less_eq_set_fm @ X @ Y )
             => ( ord_less_eq_nat @ ( F @ X ) @ ( F @ Y ) ) )
         => ( ord_less_nat @ A @ ( F @ C2 ) ) ) ) ) ).

% order_less_le_subst1
thf(fact_1159_order__less__le__subst1,axiom,
    ! [A: set_fm,F: nat > set_fm,B: nat,C2: nat] :
      ( ( ord_less_set_fm @ A @ ( F @ B ) )
     => ( ( ord_less_eq_nat @ B @ C2 )
       => ( ! [X: nat,Y: nat] :
              ( ( ord_less_eq_nat @ X @ Y )
             => ( ord_less_eq_set_fm @ ( F @ X ) @ ( F @ Y ) ) )
         => ( ord_less_set_fm @ A @ ( F @ C2 ) ) ) ) ) ).

% order_less_le_subst1
thf(fact_1160_order__less__le__subst1,axiom,
    ! [A: nat,F: nat > nat,B: nat,C2: nat] :
      ( ( ord_less_nat @ A @ ( F @ B ) )
     => ( ( ord_less_eq_nat @ B @ C2 )
       => ( ! [X: nat,Y: nat] :
              ( ( ord_less_eq_nat @ X @ Y )
             => ( ord_less_eq_nat @ ( F @ X ) @ ( F @ Y ) ) )
         => ( ord_less_nat @ A @ ( F @ C2 ) ) ) ) ) ).

% order_less_le_subst1
thf(fact_1161_order__le__less__subst2,axiom,
    ! [A: set_fm,B: set_fm,F: set_fm > set_fm,C2: set_fm] :
      ( ( ord_less_eq_set_fm @ A @ B )
     => ( ( ord_less_set_fm @ ( F @ B ) @ C2 )
       => ( ! [X: set_fm,Y: set_fm] :
              ( ( ord_less_eq_set_fm @ X @ Y )
             => ( ord_less_eq_set_fm @ ( F @ X ) @ ( F @ Y ) ) )
         => ( ord_less_set_fm @ ( F @ A ) @ C2 ) ) ) ) ).

% order_le_less_subst2
thf(fact_1162_order__le__less__subst2,axiom,
    ! [A: set_fm,B: set_fm,F: set_fm > nat,C2: nat] :
      ( ( ord_less_eq_set_fm @ A @ B )
     => ( ( ord_less_nat @ ( F @ B ) @ C2 )
       => ( ! [X: set_fm,Y: set_fm] :
              ( ( ord_less_eq_set_fm @ X @ Y )
             => ( ord_less_eq_nat @ ( F @ X ) @ ( F @ Y ) ) )
         => ( ord_less_nat @ ( F @ A ) @ C2 ) ) ) ) ).

% order_le_less_subst2
thf(fact_1163_order__le__less__subst2,axiom,
    ! [A: nat,B: nat,F: nat > set_fm,C2: set_fm] :
      ( ( ord_less_eq_nat @ A @ B )
     => ( ( ord_less_set_fm @ ( F @ B ) @ C2 )
       => ( ! [X: nat,Y: nat] :
              ( ( ord_less_eq_nat @ X @ Y )
             => ( ord_less_eq_set_fm @ ( F @ X ) @ ( F @ Y ) ) )
         => ( ord_less_set_fm @ ( F @ A ) @ C2 ) ) ) ) ).

% order_le_less_subst2
thf(fact_1164_order__le__less__subst2,axiom,
    ! [A: nat,B: nat,F: nat > nat,C2: nat] :
      ( ( ord_less_eq_nat @ A @ B )
     => ( ( ord_less_nat @ ( F @ B ) @ C2 )
       => ( ! [X: nat,Y: nat] :
              ( ( ord_less_eq_nat @ X @ Y )
             => ( ord_less_eq_nat @ ( F @ X ) @ ( F @ Y ) ) )
         => ( ord_less_nat @ ( F @ A ) @ C2 ) ) ) ) ).

% order_le_less_subst2
thf(fact_1165_order__le__less__subst1,axiom,
    ! [A: set_fm,F: nat > set_fm,B: nat,C2: nat] :
      ( ( ord_less_eq_set_fm @ A @ ( F @ B ) )
     => ( ( ord_less_nat @ B @ C2 )
       => ( ! [X: nat,Y: nat] :
              ( ( ord_less_nat @ X @ Y )
             => ( ord_less_set_fm @ ( F @ X ) @ ( F @ Y ) ) )
         => ( ord_less_set_fm @ A @ ( F @ C2 ) ) ) ) ) ).

% order_le_less_subst1
thf(fact_1166_order__le__less__subst1,axiom,
    ! [A: nat,F: nat > nat,B: nat,C2: nat] :
      ( ( ord_less_eq_nat @ A @ ( F @ B ) )
     => ( ( ord_less_nat @ B @ C2 )
       => ( ! [X: nat,Y: nat] :
              ( ( ord_less_nat @ X @ Y )
             => ( ord_less_nat @ ( F @ X ) @ ( F @ Y ) ) )
         => ( ord_less_nat @ A @ ( F @ C2 ) ) ) ) ) ).

% order_le_less_subst1
thf(fact_1167_order__less__le__trans,axiom,
    ! [X3: set_fm,Y3: set_fm,Z2: set_fm] :
      ( ( ord_less_set_fm @ X3 @ Y3 )
     => ( ( ord_less_eq_set_fm @ Y3 @ Z2 )
       => ( ord_less_set_fm @ X3 @ Z2 ) ) ) ).

% order_less_le_trans
thf(fact_1168_order__less__le__trans,axiom,
    ! [X3: nat,Y3: nat,Z2: nat] :
      ( ( ord_less_nat @ X3 @ Y3 )
     => ( ( ord_less_eq_nat @ Y3 @ Z2 )
       => ( ord_less_nat @ X3 @ Z2 ) ) ) ).

% order_less_le_trans
thf(fact_1169_order__le__less__trans,axiom,
    ! [X3: set_fm,Y3: set_fm,Z2: set_fm] :
      ( ( ord_less_eq_set_fm @ X3 @ Y3 )
     => ( ( ord_less_set_fm @ Y3 @ Z2 )
       => ( ord_less_set_fm @ X3 @ Z2 ) ) ) ).

% order_le_less_trans
thf(fact_1170_order__le__less__trans,axiom,
    ! [X3: nat,Y3: nat,Z2: nat] :
      ( ( ord_less_eq_nat @ X3 @ Y3 )
     => ( ( ord_less_nat @ Y3 @ Z2 )
       => ( ord_less_nat @ X3 @ Z2 ) ) ) ).

% order_le_less_trans
thf(fact_1171_order__neq__le__trans,axiom,
    ! [A: set_fm,B: set_fm] :
      ( ( A != B )
     => ( ( ord_less_eq_set_fm @ A @ B )
       => ( ord_less_set_fm @ A @ B ) ) ) ).

% order_neq_le_trans
thf(fact_1172_order__neq__le__trans,axiom,
    ! [A: nat,B: nat] :
      ( ( A != B )
     => ( ( ord_less_eq_nat @ A @ B )
       => ( ord_less_nat @ A @ B ) ) ) ).

% order_neq_le_trans
thf(fact_1173_order__le__neq__trans,axiom,
    ! [A: set_fm,B: set_fm] :
      ( ( ord_less_eq_set_fm @ A @ B )
     => ( ( A != B )
       => ( ord_less_set_fm @ A @ B ) ) ) ).

% order_le_neq_trans
thf(fact_1174_order__le__neq__trans,axiom,
    ! [A: nat,B: nat] :
      ( ( ord_less_eq_nat @ A @ B )
     => ( ( A != B )
       => ( ord_less_nat @ A @ B ) ) ) ).

% order_le_neq_trans
thf(fact_1175_order__less__imp__le,axiom,
    ! [X3: set_fm,Y3: set_fm] :
      ( ( ord_less_set_fm @ X3 @ Y3 )
     => ( ord_less_eq_set_fm @ X3 @ Y3 ) ) ).

% order_less_imp_le
thf(fact_1176_order__less__imp__le,axiom,
    ! [X3: nat,Y3: nat] :
      ( ( ord_less_nat @ X3 @ Y3 )
     => ( ord_less_eq_nat @ X3 @ Y3 ) ) ).

% order_less_imp_le
thf(fact_1177_linorder__not__less,axiom,
    ! [X3: nat,Y3: nat] :
      ( ( ~ ( ord_less_nat @ X3 @ Y3 ) )
      = ( ord_less_eq_nat @ Y3 @ X3 ) ) ).

% linorder_not_less
thf(fact_1178_linorder__not__le,axiom,
    ! [X3: nat,Y3: nat] :
      ( ( ~ ( ord_less_eq_nat @ X3 @ Y3 ) )
      = ( ord_less_nat @ Y3 @ X3 ) ) ).

% linorder_not_le
thf(fact_1179_order__less__le,axiom,
    ( ord_less_set_fm
    = ( ^ [X6: set_fm,Y5: set_fm] :
          ( ( ord_less_eq_set_fm @ X6 @ Y5 )
          & ( X6 != Y5 ) ) ) ) ).

% order_less_le
thf(fact_1180_order__less__le,axiom,
    ( ord_less_nat
    = ( ^ [X6: nat,Y5: nat] :
          ( ( ord_less_eq_nat @ X6 @ Y5 )
          & ( X6 != Y5 ) ) ) ) ).

% order_less_le
thf(fact_1181_order__le__less,axiom,
    ( ord_less_eq_set_fm
    = ( ^ [X6: set_fm,Y5: set_fm] :
          ( ( ord_less_set_fm @ X6 @ Y5 )
          | ( X6 = Y5 ) ) ) ) ).

% order_le_less
thf(fact_1182_order__le__less,axiom,
    ( ord_less_eq_nat
    = ( ^ [X6: nat,Y5: nat] :
          ( ( ord_less_nat @ X6 @ Y5 )
          | ( X6 = Y5 ) ) ) ) ).

% order_le_less
thf(fact_1183_dual__order_Ostrict__implies__order,axiom,
    ! [B: set_fm,A: set_fm] :
      ( ( ord_less_set_fm @ B @ A )
     => ( ord_less_eq_set_fm @ B @ A ) ) ).

% dual_order.strict_implies_order
thf(fact_1184_dual__order_Ostrict__implies__order,axiom,
    ! [B: nat,A: nat] :
      ( ( ord_less_nat @ B @ A )
     => ( ord_less_eq_nat @ B @ A ) ) ).

% dual_order.strict_implies_order
thf(fact_1185_order_Ostrict__implies__order,axiom,
    ! [A: set_fm,B: set_fm] :
      ( ( ord_less_set_fm @ A @ B )
     => ( ord_less_eq_set_fm @ A @ B ) ) ).

% order.strict_implies_order
thf(fact_1186_order_Ostrict__implies__order,axiom,
    ! [A: nat,B: nat] :
      ( ( ord_less_nat @ A @ B )
     => ( ord_less_eq_nat @ A @ B ) ) ).

% order.strict_implies_order
thf(fact_1187_dual__order_Ostrict__iff__not,axiom,
    ( ord_less_set_fm
    = ( ^ [B4: set_fm,A2: set_fm] :
          ( ( ord_less_eq_set_fm @ B4 @ A2 )
          & ~ ( ord_less_eq_set_fm @ A2 @ B4 ) ) ) ) ).

% dual_order.strict_iff_not
thf(fact_1188_dual__order_Ostrict__iff__not,axiom,
    ( ord_less_nat
    = ( ^ [B4: nat,A2: nat] :
          ( ( ord_less_eq_nat @ B4 @ A2 )
          & ~ ( ord_less_eq_nat @ A2 @ B4 ) ) ) ) ).

% dual_order.strict_iff_not
thf(fact_1189_dual__order_Ostrict__trans2,axiom,
    ! [B: set_fm,A: set_fm,C2: set_fm] :
      ( ( ord_less_set_fm @ B @ A )
     => ( ( ord_less_eq_set_fm @ C2 @ B )
       => ( ord_less_set_fm @ C2 @ A ) ) ) ).

% dual_order.strict_trans2
thf(fact_1190_dual__order_Ostrict__trans2,axiom,
    ! [B: nat,A: nat,C2: nat] :
      ( ( ord_less_nat @ B @ A )
     => ( ( ord_less_eq_nat @ C2 @ B )
       => ( ord_less_nat @ C2 @ A ) ) ) ).

% dual_order.strict_trans2
thf(fact_1191_dual__order_Ostrict__trans1,axiom,
    ! [B: set_fm,A: set_fm,C2: set_fm] :
      ( ( ord_less_eq_set_fm @ B @ A )
     => ( ( ord_less_set_fm @ C2 @ B )
       => ( ord_less_set_fm @ C2 @ A ) ) ) ).

% dual_order.strict_trans1
thf(fact_1192_dual__order_Ostrict__trans1,axiom,
    ! [B: nat,A: nat,C2: nat] :
      ( ( ord_less_eq_nat @ B @ A )
     => ( ( ord_less_nat @ C2 @ B )
       => ( ord_less_nat @ C2 @ A ) ) ) ).

% dual_order.strict_trans1
thf(fact_1193_dual__order_Ostrict__iff__order,axiom,
    ( ord_less_set_fm
    = ( ^ [B4: set_fm,A2: set_fm] :
          ( ( ord_less_eq_set_fm @ B4 @ A2 )
          & ( A2 != B4 ) ) ) ) ).

% dual_order.strict_iff_order
thf(fact_1194_dual__order_Ostrict__iff__order,axiom,
    ( ord_less_nat
    = ( ^ [B4: nat,A2: nat] :
          ( ( ord_less_eq_nat @ B4 @ A2 )
          & ( A2 != B4 ) ) ) ) ).

% dual_order.strict_iff_order
thf(fact_1195_dual__order_Oorder__iff__strict,axiom,
    ( ord_less_eq_set_fm
    = ( ^ [B4: set_fm,A2: set_fm] :
          ( ( ord_less_set_fm @ B4 @ A2 )
          | ( A2 = B4 ) ) ) ) ).

% dual_order.order_iff_strict
thf(fact_1196_dual__order_Oorder__iff__strict,axiom,
    ( ord_less_eq_nat
    = ( ^ [B4: nat,A2: nat] :
          ( ( ord_less_nat @ B4 @ A2 )
          | ( A2 = B4 ) ) ) ) ).

% dual_order.order_iff_strict
thf(fact_1197_order_Ostrict__iff__not,axiom,
    ( ord_less_set_fm
    = ( ^ [A2: set_fm,B4: set_fm] :
          ( ( ord_less_eq_set_fm @ A2 @ B4 )
          & ~ ( ord_less_eq_set_fm @ B4 @ A2 ) ) ) ) ).

% order.strict_iff_not
thf(fact_1198_order_Ostrict__iff__not,axiom,
    ( ord_less_nat
    = ( ^ [A2: nat,B4: nat] :
          ( ( ord_less_eq_nat @ A2 @ B4 )
          & ~ ( ord_less_eq_nat @ B4 @ A2 ) ) ) ) ).

% order.strict_iff_not
thf(fact_1199_order_Ostrict__trans2,axiom,
    ! [A: nat,B: nat,C2: nat] :
      ( ( ord_less_nat @ A @ B )
     => ( ( ord_less_eq_nat @ B @ C2 )
       => ( ord_less_nat @ A @ C2 ) ) ) ).

% order.strict_trans2
thf(fact_1200_diff__diff__cancel,axiom,
    ! [I: nat,N: nat] :
      ( ( ord_less_eq_nat @ I @ N )
     => ( ( minus_minus_nat @ N @ ( minus_minus_nat @ N @ I ) )
        = I ) ) ).

% diff_diff_cancel
thf(fact_1201_Suc__le__mono,axiom,
    ! [N: nat,M: nat] :
      ( ( ord_less_eq_nat @ ( suc @ N ) @ ( suc @ M ) )
      = ( ord_less_eq_nat @ N @ M ) ) ).

% Suc_le_mono
thf(fact_1202_diff__less__mono,axiom,
    ! [A: nat,B: nat,C2: nat] :
      ( ( ord_less_nat @ A @ B )
     => ( ( ord_less_eq_nat @ C2 @ A )
       => ( ord_less_nat @ ( minus_minus_nat @ A @ C2 ) @ ( minus_minus_nat @ B @ C2 ) ) ) ) ).

% diff_less_mono
thf(fact_1203_less__diff__iff,axiom,
    ! [K2: nat,M: nat,N: nat] :
      ( ( ord_less_eq_nat @ K2 @ M )
     => ( ( ord_less_eq_nat @ K2 @ N )
       => ( ( ord_less_nat @ ( minus_minus_nat @ M @ K2 ) @ ( minus_minus_nat @ N @ K2 ) )
          = ( ord_less_nat @ M @ N ) ) ) ) ).

% less_diff_iff
thf(fact_1204_Nat_Oex__has__greatest__nat,axiom,
    ! [P2: nat > $o,K2: nat,B: nat] :
      ( ( P2 @ K2 )
     => ( ! [Y: nat] :
            ( ( P2 @ Y )
           => ( ord_less_eq_nat @ Y @ B ) )
       => ? [X: nat] :
            ( ( P2 @ X )
            & ! [Y6: nat] :
                ( ( P2 @ Y6 )
               => ( ord_less_eq_nat @ Y6 @ X ) ) ) ) ) ).

% Nat.ex_has_greatest_nat
thf(fact_1205_nat__le__linear,axiom,
    ! [M: nat,N: nat] :
      ( ( ord_less_eq_nat @ M @ N )
      | ( ord_less_eq_nat @ N @ M ) ) ).

% nat_le_linear
thf(fact_1206_le__antisym,axiom,
    ! [M: nat,N: nat] :
      ( ( ord_less_eq_nat @ M @ N )
     => ( ( ord_less_eq_nat @ N @ M )
       => ( M = N ) ) ) ).

% le_antisym
thf(fact_1207_eq__imp__le,axiom,
    ! [M: nat,N: nat] :
      ( ( M = N )
     => ( ord_less_eq_nat @ M @ N ) ) ).

% eq_imp_le
thf(fact_1208_le__trans,axiom,
    ! [I: nat,J: nat,K2: nat] :
      ( ( ord_less_eq_nat @ I @ J )
     => ( ( ord_less_eq_nat @ J @ K2 )
       => ( ord_less_eq_nat @ I @ K2 ) ) ) ).

% le_trans
thf(fact_1209_le__refl,axiom,
    ! [N: nat] : ( ord_less_eq_nat @ N @ N ) ).

% le_refl
thf(fact_1210_Suc__leD,axiom,
    ! [M: nat,N: nat] :
      ( ( ord_less_eq_nat @ ( suc @ M ) @ N )
     => ( ord_less_eq_nat @ M @ N ) ) ).

% Suc_leD
thf(fact_1211_le__SucE,axiom,
    ! [M: nat,N: nat] :
      ( ( ord_less_eq_nat @ M @ ( suc @ N ) )
     => ( ~ ( ord_less_eq_nat @ M @ N )
       => ( M
          = ( suc @ N ) ) ) ) ).

% le_SucE
thf(fact_1212_le__SucI,axiom,
    ! [M: nat,N: nat] :
      ( ( ord_less_eq_nat @ M @ N )
     => ( ord_less_eq_nat @ M @ ( suc @ N ) ) ) ).

% le_SucI
thf(fact_1213_Suc__le__D,axiom,
    ! [N: nat,M4: nat] :
      ( ( ord_less_eq_nat @ ( suc @ N ) @ M4 )
     => ? [M5: nat] :
          ( M4
          = ( suc @ M5 ) ) ) ).

% Suc_le_D
thf(fact_1214_le__Suc__eq,axiom,
    ! [M: nat,N: nat] :
      ( ( ord_less_eq_nat @ M @ ( suc @ N ) )
      = ( ( ord_less_eq_nat @ M @ N )
        | ( M
          = ( suc @ N ) ) ) ) ).

% le_Suc_eq
thf(fact_1215_Suc__n__not__le__n,axiom,
    ! [N: nat] :
      ~ ( ord_less_eq_nat @ ( suc @ N ) @ N ) ).

% Suc_n_not_le_n
thf(fact_1216_not__less__eq__eq,axiom,
    ! [M: nat,N: nat] :
      ( ( ~ ( ord_less_eq_nat @ M @ N ) )
      = ( ord_less_eq_nat @ ( suc @ N ) @ M ) ) ).

% not_less_eq_eq
thf(fact_1217_full__nat__induct,axiom,
    ! [P2: nat > $o,N: nat] :
      ( ! [N2: nat] :
          ( ! [M6: nat] :
              ( ( ord_less_eq_nat @ ( suc @ M6 ) @ N2 )
             => ( P2 @ M6 ) )
         => ( P2 @ N2 ) )
     => ( P2 @ N ) ) ).

% full_nat_induct
thf(fact_1218_nat__induct__at__least,axiom,
    ! [M: nat,N: nat,P2: nat > $o] :
      ( ( ord_less_eq_nat @ M @ N )
     => ( ( P2 @ M )
       => ( ! [N2: nat] :
              ( ( ord_less_eq_nat @ M @ N2 )
             => ( ( P2 @ N2 )
               => ( P2 @ ( suc @ N2 ) ) ) )
         => ( P2 @ N ) ) ) ) ).

% nat_induct_at_least
thf(fact_1219_transitive__stepwise__le,axiom,
    ! [M: nat,N: nat,R3: nat > nat > $o] :
      ( ( ord_less_eq_nat @ M @ N )
     => ( ! [X: nat] : ( R3 @ X @ X )
       => ( ! [X: nat,Y: nat,Z4: nat] :
              ( ( R3 @ X @ Y )
             => ( ( R3 @ Y @ Z4 )
               => ( R3 @ X @ Z4 ) ) )
         => ( ! [N2: nat] : ( R3 @ N2 @ ( suc @ N2 ) )
           => ( R3 @ M @ N ) ) ) ) ) ).

% transitive_stepwise_le
thf(fact_1220_nat__less__le,axiom,
    ( ord_less_nat
    = ( ^ [M3: nat,N3: nat] :
          ( ( ord_less_eq_nat @ M3 @ N3 )
          & ( M3 != N3 ) ) ) ) ).

% nat_less_le
thf(fact_1221_less__imp__le__nat,axiom,
    ! [M: nat,N: nat] :
      ( ( ord_less_nat @ M @ N )
     => ( ord_less_eq_nat @ M @ N ) ) ).

% less_imp_le_nat
thf(fact_1222_le__eq__less__or__eq,axiom,
    ( ord_less_eq_nat
    = ( ^ [M3: nat,N3: nat] :
          ( ( ord_less_nat @ M3 @ N3 )
          | ( M3 = N3 ) ) ) ) ).

% le_eq_less_or_eq
thf(fact_1223_less__or__eq__imp__le,axiom,
    ! [M: nat,N: nat] :
      ( ( ( ord_less_nat @ M @ N )
        | ( M = N ) )
     => ( ord_less_eq_nat @ M @ N ) ) ).

% less_or_eq_imp_le
thf(fact_1224_le__neq__implies__less,axiom,
    ! [M: nat,N: nat] :
      ( ( ord_less_eq_nat @ M @ N )
     => ( ( M != N )
       => ( ord_less_nat @ M @ N ) ) ) ).

% le_neq_implies_less
thf(fact_1225_less__mono__imp__le__mono,axiom,
    ! [F: nat > nat,I: nat,J: nat] :
      ( ! [I2: nat,J2: nat] :
          ( ( ord_less_nat @ I2 @ J2 )
         => ( ord_less_nat @ ( F @ I2 ) @ ( F @ J2 ) ) )
     => ( ( ord_less_eq_nat @ I @ J )
       => ( ord_less_eq_nat @ ( F @ I ) @ ( F @ J ) ) ) ) ).

% less_mono_imp_le_mono
thf(fact_1226_eq__diff__iff,axiom,
    ! [K2: nat,M: nat,N: nat] :
      ( ( ord_less_eq_nat @ K2 @ M )
     => ( ( ord_less_eq_nat @ K2 @ N )
       => ( ( ( minus_minus_nat @ M @ K2 )
            = ( minus_minus_nat @ N @ K2 ) )
          = ( M = N ) ) ) ) ).

% eq_diff_iff
thf(fact_1227_le__diff__iff,axiom,
    ! [K2: nat,M: nat,N: nat] :
      ( ( ord_less_eq_nat @ K2 @ M )
     => ( ( ord_less_eq_nat @ K2 @ N )
       => ( ( ord_less_eq_nat @ ( minus_minus_nat @ M @ K2 ) @ ( minus_minus_nat @ N @ K2 ) )
          = ( ord_less_eq_nat @ M @ N ) ) ) ) ).

% le_diff_iff
thf(fact_1228_Nat_Odiff__diff__eq,axiom,
    ! [K2: nat,M: nat,N: nat] :
      ( ( ord_less_eq_nat @ K2 @ M )
     => ( ( ord_less_eq_nat @ K2 @ N )
       => ( ( minus_minus_nat @ ( minus_minus_nat @ M @ K2 ) @ ( minus_minus_nat @ N @ K2 ) )
          = ( minus_minus_nat @ M @ N ) ) ) ) ).

% Nat.diff_diff_eq
thf(fact_1229_diff__le__mono,axiom,
    ! [M: nat,N: nat,L: nat] :
      ( ( ord_less_eq_nat @ M @ N )
     => ( ord_less_eq_nat @ ( minus_minus_nat @ M @ L ) @ ( minus_minus_nat @ N @ L ) ) ) ).

% diff_le_mono
thf(fact_1230_diff__le__self,axiom,
    ! [M: nat,N: nat] : ( ord_less_eq_nat @ ( minus_minus_nat @ M @ N ) @ M ) ).

% diff_le_self
thf(fact_1231_le__diff__iff_H,axiom,
    ! [A: nat,C2: nat,B: nat] :
      ( ( ord_less_eq_nat @ A @ C2 )
     => ( ( ord_less_eq_nat @ B @ C2 )
       => ( ( ord_less_eq_nat @ ( minus_minus_nat @ C2 @ A ) @ ( minus_minus_nat @ C2 @ B ) )
          = ( ord_less_eq_nat @ B @ A ) ) ) ) ).

% le_diff_iff'
thf(fact_1232_diff__le__mono2,axiom,
    ! [M: nat,N: nat,L: nat] :
      ( ( ord_less_eq_nat @ M @ N )
     => ( ord_less_eq_nat @ ( minus_minus_nat @ L @ N ) @ ( minus_minus_nat @ L @ M ) ) ) ).

% diff_le_mono2
thf(fact_1233_Suc__leI,axiom,
    ! [M: nat,N: nat] :
      ( ( ord_less_nat @ M @ N )
     => ( ord_less_eq_nat @ ( suc @ M ) @ N ) ) ).

% Suc_leI
thf(fact_1234_Suc__le__eq,axiom,
    ! [M: nat,N: nat] :
      ( ( ord_less_eq_nat @ ( suc @ M ) @ N )
      = ( ord_less_nat @ M @ N ) ) ).

% Suc_le_eq
thf(fact_1235_dec__induct,axiom,
    ! [I: nat,J: nat,P2: nat > $o] :
      ( ( ord_less_eq_nat @ I @ J )
     => ( ( P2 @ I )
       => ( ! [N2: nat] :
              ( ( ord_less_eq_nat @ I @ N2 )
             => ( ( ord_less_nat @ N2 @ J )
               => ( ( P2 @ N2 )
                 => ( P2 @ ( suc @ N2 ) ) ) ) )
         => ( P2 @ J ) ) ) ) ).

% dec_induct
thf(fact_1236_inc__induct,axiom,
    ! [I: nat,J: nat,P2: nat > $o] :
      ( ( ord_less_eq_nat @ I @ J )
     => ( ( P2 @ J )
       => ( ! [N2: nat] :
              ( ( ord_less_eq_nat @ I @ N2 )
             => ( ( ord_less_nat @ N2 @ J )
               => ( ( P2 @ ( suc @ N2 ) )
                 => ( P2 @ N2 ) ) ) )
         => ( P2 @ I ) ) ) ) ).

% inc_induct
thf(fact_1237_Suc__le__lessD,axiom,
    ! [M: nat,N: nat] :
      ( ( ord_less_eq_nat @ ( suc @ M ) @ N )
     => ( ord_less_nat @ M @ N ) ) ).

% Suc_le_lessD
thf(fact_1238_le__less__Suc__eq,axiom,
    ! [M: nat,N: nat] :
      ( ( ord_less_eq_nat @ M @ N )
     => ( ( ord_less_nat @ N @ ( suc @ M ) )
        = ( N = M ) ) ) ).

% le_less_Suc_eq
thf(fact_1239_less__Suc__eq__le,axiom,
    ! [M: nat,N: nat] :
      ( ( ord_less_nat @ M @ ( suc @ N ) )
      = ( ord_less_eq_nat @ M @ N ) ) ).

% less_Suc_eq_le
thf(fact_1240_less__eq__Suc__le,axiom,
    ( ord_less_nat
    = ( ^ [N3: nat] : ( ord_less_eq_nat @ ( suc @ N3 ) ) ) ) ).

% less_eq_Suc_le
thf(fact_1241_le__imp__less__Suc,axiom,
    ! [M: nat,N: nat] :
      ( ( ord_less_eq_nat @ M @ N )
     => ( ord_less_nat @ M @ ( suc @ N ) ) ) ).

% le_imp_less_Suc
thf(fact_1242_Suc__diff__le,axiom,
    ! [N: nat,M: nat] :
      ( ( ord_less_eq_nat @ N @ M )
     => ( ( minus_minus_nat @ ( suc @ M ) @ N )
        = ( suc @ ( minus_minus_nat @ M @ N ) ) ) ) ).

% Suc_diff_le
thf(fact_1243_nat__descend__induct,axiom,
    ! [N: nat,P2: nat > $o,M: nat] :
      ( ! [K: nat] :
          ( ( ord_less_nat @ N @ K )
         => ( P2 @ K ) )
     => ( ! [K: nat] :
            ( ( ord_less_eq_nat @ K @ N )
           => ( ! [I3: nat] :
                  ( ( ord_less_nat @ K @ I3 )
                 => ( P2 @ I3 ) )
             => ( P2 @ K ) ) )
       => ( P2 @ M ) ) ) ).

% nat_descend_induct
thf(fact_1244_prod__decode__aux_Osimps,axiom,
    ( nat_prod_decode_aux
    = ( ^ [K3: nat,M3: nat] : ( if_Pro6206227464963214023at_nat @ ( ord_less_eq_nat @ M3 @ K3 ) @ ( product_Pair_nat_nat @ M3 @ ( minus_minus_nat @ K3 @ M3 ) ) @ ( nat_prod_decode_aux @ ( suc @ K3 ) @ ( minus_minus_nat @ M3 @ ( suc @ K3 ) ) ) ) ) ) ).

% prod_decode_aux.simps
thf(fact_1245_prod__decode__aux_Oelims,axiom,
    ! [X3: nat,Xa2: nat,Y3: product_prod_nat_nat] :
      ( ( ( nat_prod_decode_aux @ X3 @ Xa2 )
        = Y3 )
     => ( ( ( ord_less_eq_nat @ Xa2 @ X3 )
         => ( Y3
            = ( product_Pair_nat_nat @ Xa2 @ ( minus_minus_nat @ X3 @ Xa2 ) ) ) )
        & ( ~ ( ord_less_eq_nat @ Xa2 @ X3 )
         => ( Y3
            = ( nat_prod_decode_aux @ ( suc @ X3 ) @ ( minus_minus_nat @ Xa2 @ ( suc @ X3 ) ) ) ) ) ) ) ).

% prod_decode_aux.elims
thf(fact_1246_pair__lessI2,axiom,
    ! [A: nat,B: nat,S3: nat,T2: nat] :
      ( ( ord_less_eq_nat @ A @ B )
     => ( ( ord_less_nat @ S3 @ T2 )
       => ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ ( product_Pair_nat_nat @ A @ S3 ) @ ( product_Pair_nat_nat @ B @ T2 ) ) @ fun_pair_less ) ) ) ).

% pair_lessI2
thf(fact_1247_per__def,axiom,
    ! [R2: rule] :
      ( ( abstra4153889242452539894ist_fm
        @ ^ [R: rule,S: produc1996495991257130529ist_fm,Ss: fset_P661503646757059847ist_fm] :
            ( ( eff @ R @ S )
            = ( some_f2162114139285431020ist_fm @ Ss ) )
        @ rules
        @ top_to728902761071950545ist_fm
        @ R2 )
      = ( ! [S: produc1996495991257130529ist_fm,R1: rule,Sl6: fset_P661503646757059847ist_fm,S9: produc1996495991257130529ist_fm] :
            ( ( ( member8102475879199740618ist_fm @ S @ top_to728902761071950545ist_fm )
              & ( abstra3055540779341473792ist_fm
                @ ^ [R: rule,T: produc1996495991257130529ist_fm,Ss: fset_P661503646757059847ist_fm] :
                    ( ( eff @ R @ T )
                    = ( some_f2162114139285431020ist_fm @ Ss ) )
                @ R2
                @ S )
              & ( member_rule @ R1 @ ( minus_minus_set_rule @ ( sset_rule @ rules ) @ ( insert_rule @ R2 @ bot_bot_set_rule ) ) )
              & ( ( eff @ R1 @ S )
                = ( some_f2162114139285431020ist_fm @ Sl6 ) )
              & ( fmembe3381613331217039976ist_fm @ S9 @ Sl6 ) )
           => ( abstra3055540779341473792ist_fm
              @ ^ [R: rule,T: produc1996495991257130529ist_fm,Ss: fset_P661503646757059847ist_fm] :
                  ( ( eff @ R @ T )
                  = ( some_f2162114139285431020ist_fm @ Ss ) )
              @ R2
              @ S9 ) ) ) ) ).

% per_def
thf(fact_1248_NE__R,axiom,
    ( ( sset_rule @ rules )
   != bot_bot_set_rule ) ).

% NE_R
thf(fact_1249_pair__leqI2,axiom,
    ! [A: nat,B: nat,S3: nat,T2: nat] :
      ( ( ord_less_eq_nat @ A @ B )
     => ( ( ord_less_eq_nat @ S3 @ T2 )
       => ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ ( product_Pair_nat_nat @ A @ S3 ) @ ( product_Pair_nat_nat @ B @ T2 ) ) @ fun_pair_leq ) ) ) ).

% pair_leqI2
thf(fact_1250_eff_Osimps_I2_J,axiom,
    ! [P2: nat,Ts: list_tm,A5: list_fm,B5: list_fm] :
      ( ( ( ( member_fm @ ( pre @ P2 @ Ts ) @ ( set_fm2 @ A5 ) )
          & ( member_fm @ ( pre @ P2 @ Ts ) @ ( set_fm2 @ B5 ) ) )
       => ( ( eff @ ( axiom @ P2 @ Ts ) @ ( produc381145313068854617ist_fm @ A5 @ B5 ) )
          = ( some_f2162114139285431020ist_fm @ bot_bo2367426573206113139ist_fm ) ) )
      & ( ~ ( ( member_fm @ ( pre @ P2 @ Ts ) @ ( set_fm2 @ A5 ) )
            & ( member_fm @ ( pre @ P2 @ Ts ) @ ( set_fm2 @ B5 ) ) )
       => ( ( eff @ ( axiom @ P2 @ Ts ) @ ( produc381145313068854617ist_fm @ A5 @ B5 ) )
          = none_f2701453242279119720ist_fm ) ) ) ).

% eff.simps(2)
thf(fact_1251_pickEff__def,axiom,
    ! [R2: rule,S3: produc1996495991257130529ist_fm] :
      ( ( ( abstra3055540779341473792ist_fm
          @ ^ [R: rule,S: produc1996495991257130529ist_fm,Ss: fset_P661503646757059847ist_fm] :
              ( ( eff @ R @ S )
              = ( some_f2162114139285431020ist_fm @ Ss ) )
          @ R2
          @ S3 )
       => ( ( abstra458490714785879965ist_fm
            @ ^ [R: rule,S: produc1996495991257130529ist_fm,Ss: fset_P661503646757059847ist_fm] :
                ( ( eff @ R @ S )
                = ( some_f2162114139285431020ist_fm @ Ss ) )
            @ R2
            @ S3 )
          = ( fChoic3892542562059901019ist_fm
            @ ^ [Sl: fset_P661503646757059847ist_fm] :
                ( ( eff @ R2 @ S3 )
                = ( some_f2162114139285431020ist_fm @ Sl ) ) ) ) )
      & ( ~ ( abstra3055540779341473792ist_fm
            @ ^ [R: rule,S: produc1996495991257130529ist_fm,Ss: fset_P661503646757059847ist_fm] :
                ( ( eff @ R @ S )
                = ( some_f2162114139285431020ist_fm @ Ss ) )
            @ R2
            @ S3 )
       => ( ( abstra458490714785879965ist_fm
            @ ^ [R: rule,S: produc1996495991257130529ist_fm,Ss: fset_P661503646757059847ist_fm] :
                ( ( eff @ R @ S )
                = ( some_f2162114139285431020ist_fm @ Ss ) )
            @ R2
            @ S3 )
          = ( the_fs400605564276472219ist_fm @ none_f2701453242279119720ist_fm ) ) ) ) ).

% pickEff_def
thf(fact_1252_rule_Oinject_I1_J,axiom,
    ! [X21: nat,X22: list_tm,Y21: nat,Y222: list_tm] :
      ( ( ( axiom @ X21 @ X22 )
        = ( axiom @ Y21 @ Y222 ) )
      = ( ( X21 = Y21 )
        & ( X22 = Y222 ) ) ) ).

% rule.inject(1)
thf(fact_1253_rule_Odistinct_I23_J,axiom,
    ! [X21: nat,X22: list_tm,X71: tm,X72: fm] :
      ( ( axiom @ X21 @ X22 )
     != ( uniL @ X71 @ X72 ) ) ).

% rule.distinct(23)
thf(fact_1254_eff_Osimps_I1_J,axiom,
    ! [A5: list_fm,B5: list_fm] :
      ( ( eff @ idle @ ( produc381145313068854617ist_fm @ A5 @ B5 ) )
      = ( some_f2162114139285431020ist_fm @ ( finser3446675674286072169ist_fm @ ( produc381145313068854617ist_fm @ A5 @ B5 ) @ bot_bo2367426573206113139ist_fm ) ) ) ).

% eff.simps(1)
thf(fact_1255_rule_Odistinct_I1_J,axiom,
    ! [X21: nat,X22: list_tm] :
      ( idle
     != ( axiom @ X21 @ X22 ) ) ).

% rule.distinct(1)
thf(fact_1256_rule_Odistinct_I11_J,axiom,
    ! [X71: tm,X72: fm] :
      ( idle
     != ( uniL @ X71 @ X72 ) ) ).

% rule.distinct(11)
thf(fact_1257_eff_Osimps_I4_J,axiom,
    ! [B5: list_fm,A5: list_fm] :
      ( ( ( member_fm @ falsity @ ( set_fm2 @ B5 ) )
       => ( ( eff @ flsR @ ( produc381145313068854617ist_fm @ A5 @ B5 ) )
          = ( some_f2162114139285431020ist_fm @ ( finser3446675674286072169ist_fm @ ( produc381145313068854617ist_fm @ A5 @ ( removeAll_fm @ falsity @ B5 ) ) @ bot_bo2367426573206113139ist_fm ) ) ) )
      & ( ~ ( member_fm @ falsity @ ( set_fm2 @ B5 ) )
       => ( ( eff @ flsR @ ( produc381145313068854617ist_fm @ A5 @ B5 ) )
          = none_f2701453242279119720ist_fm ) ) ) ).

% eff.simps(4)
thf(fact_1258_rule_Odistinct_I17_J,axiom,
    ! [X21: nat,X22: list_tm] :
      ( ( axiom @ X21 @ X22 )
     != flsR ) ).

% rule.distinct(17)
thf(fact_1259_fm_Odistinct_I1_J,axiom,
    ! [X21: nat,X22: list_tm] :
      ( falsity
     != ( pre @ X21 @ X22 ) ) ).

% fm.distinct(1)
thf(fact_1260_fm_Odistinct_I5_J,axiom,
    ! [X4: fm] :
      ( falsity
     != ( uni @ X4 ) ) ).

% fm.distinct(5)
thf(fact_1261_rule_Odistinct_I41_J,axiom,
    ! [X71: tm,X72: fm] :
      ( flsR
     != ( uniL @ X71 @ X72 ) ) ).

% rule.distinct(41)
thf(fact_1262_rule_Odistinct_I5_J,axiom,
    idle != flsR ).

% rule.distinct(5)
thf(fact_1263_eff_Osimps_I3_J,axiom,
    ! [A5: list_fm,B5: list_fm] :
      ( ( ( member_fm @ falsity @ ( set_fm2 @ A5 ) )
       => ( ( eff @ flsL @ ( produc381145313068854617ist_fm @ A5 @ B5 ) )
          = ( some_f2162114139285431020ist_fm @ bot_bo2367426573206113139ist_fm ) ) )
      & ( ~ ( member_fm @ falsity @ ( set_fm2 @ A5 ) )
       => ( ( eff @ flsL @ ( produc381145313068854617ist_fm @ A5 @ B5 ) )
          = none_f2701453242279119720ist_fm ) ) ) ).

% eff.simps(3)
thf(fact_1264_eff_Opsimps_I4_J,axiom,
    ! [A5: list_fm,B5: list_fm] :
      ( ( accp_P2380921852949180872ist_fm @ eff_rel @ ( produc7063755442767259331ist_fm @ flsR @ ( produc381145313068854617ist_fm @ A5 @ B5 ) ) )
     => ( ( ( member_fm @ falsity @ ( set_fm2 @ B5 ) )
         => ( ( eff @ flsR @ ( produc381145313068854617ist_fm @ A5 @ B5 ) )
            = ( some_f2162114139285431020ist_fm @ ( finser3446675674286072169ist_fm @ ( produc381145313068854617ist_fm @ A5 @ ( removeAll_fm @ falsity @ B5 ) ) @ bot_bo2367426573206113139ist_fm ) ) ) )
        & ( ~ ( member_fm @ falsity @ ( set_fm2 @ B5 ) )
         => ( ( eff @ flsR @ ( produc381145313068854617ist_fm @ A5 @ B5 ) )
            = none_f2701453242279119720ist_fm ) ) ) ) ).

% eff.psimps(4)
thf(fact_1265_rule_Odistinct_I27_J,axiom,
    flsL != flsR ).

% rule.distinct(27)
thf(fact_1266_rule_Odistinct_I3_J,axiom,
    idle != flsL ).

% rule.distinct(3)
thf(fact_1267_rule_Odistinct_I33_J,axiom,
    ! [X71: tm,X72: fm] :
      ( flsL
     != ( uniL @ X71 @ X72 ) ) ).

% rule.distinct(33)
thf(fact_1268_rule_Odistinct_I15_J,axiom,
    ! [X21: nat,X22: list_tm] :
      ( ( axiom @ X21 @ X22 )
     != flsL ) ).

% rule.distinct(15)
thf(fact_1269_eff_Opsimps_I3_J,axiom,
    ! [A5: list_fm,B5: list_fm] :
      ( ( accp_P2380921852949180872ist_fm @ eff_rel @ ( produc7063755442767259331ist_fm @ flsL @ ( produc381145313068854617ist_fm @ A5 @ B5 ) ) )
     => ( ( ( member_fm @ falsity @ ( set_fm2 @ A5 ) )
         => ( ( eff @ flsL @ ( produc381145313068854617ist_fm @ A5 @ B5 ) )
            = ( some_f2162114139285431020ist_fm @ bot_bo2367426573206113139ist_fm ) ) )
        & ( ~ ( member_fm @ falsity @ ( set_fm2 @ A5 ) )
         => ( ( eff @ flsL @ ( produc381145313068854617ist_fm @ A5 @ B5 ) )
            = none_f2701453242279119720ist_fm ) ) ) ) ).

% eff.psimps(3)
thf(fact_1270_eff_Opsimps_I1_J,axiom,
    ! [A5: list_fm,B5: list_fm] :
      ( ( accp_P2380921852949180872ist_fm @ eff_rel @ ( produc7063755442767259331ist_fm @ idle @ ( produc381145313068854617ist_fm @ A5 @ B5 ) ) )
     => ( ( eff @ idle @ ( produc381145313068854617ist_fm @ A5 @ B5 ) )
        = ( some_f2162114139285431020ist_fm @ ( finser3446675674286072169ist_fm @ ( produc381145313068854617ist_fm @ A5 @ B5 ) @ bot_bo2367426573206113139ist_fm ) ) ) ) ).

% eff.psimps(1)
thf(fact_1271_eff_Opsimps_I2_J,axiom,
    ! [P2: nat,Ts: list_tm,A5: list_fm,B5: list_fm] :
      ( ( accp_P2380921852949180872ist_fm @ eff_rel @ ( produc7063755442767259331ist_fm @ ( axiom @ P2 @ Ts ) @ ( produc381145313068854617ist_fm @ A5 @ B5 ) ) )
     => ( ( ( ( member_fm @ ( pre @ P2 @ Ts ) @ ( set_fm2 @ A5 ) )
            & ( member_fm @ ( pre @ P2 @ Ts ) @ ( set_fm2 @ B5 ) ) )
         => ( ( eff @ ( axiom @ P2 @ Ts ) @ ( produc381145313068854617ist_fm @ A5 @ B5 ) )
            = ( some_f2162114139285431020ist_fm @ bot_bo2367426573206113139ist_fm ) ) )
        & ( ~ ( ( member_fm @ ( pre @ P2 @ Ts ) @ ( set_fm2 @ A5 ) )
              & ( member_fm @ ( pre @ P2 @ Ts ) @ ( set_fm2 @ B5 ) ) )
         => ( ( eff @ ( axiom @ P2 @ Ts ) @ ( produc381145313068854617ist_fm @ A5 @ B5 ) )
            = none_f2701453242279119720ist_fm ) ) ) ) ).

% eff.psimps(2)
thf(fact_1272_prod__decode__aux_Opelims,axiom,
    ! [X3: nat,Xa2: nat,Y3: product_prod_nat_nat] :
      ( ( ( nat_prod_decode_aux @ X3 @ Xa2 )
        = Y3 )
     => ( ( accp_P4275260045618599050at_nat @ nat_pr5047031295181774490ux_rel @ ( product_Pair_nat_nat @ X3 @ Xa2 ) )
       => ~ ( ( ( ( ord_less_eq_nat @ Xa2 @ X3 )
               => ( Y3
                  = ( product_Pair_nat_nat @ Xa2 @ ( minus_minus_nat @ X3 @ Xa2 ) ) ) )
              & ( ~ ( ord_less_eq_nat @ Xa2 @ X3 )
               => ( Y3
                  = ( nat_prod_decode_aux @ ( suc @ X3 ) @ ( minus_minus_nat @ Xa2 @ ( suc @ X3 ) ) ) ) ) )
           => ~ ( accp_P4275260045618599050at_nat @ nat_pr5047031295181774490ux_rel @ ( product_Pair_nat_nat @ X3 @ Xa2 ) ) ) ) ) ).

% prod_decode_aux.pelims
thf(fact_1273_treeB__def,axiom,
    ( treeB
    = ( ^ [Steps3: stream727092118206550309m_rule] :
          ( comple7677669044296858969set_fm
          @ ( image_5698427363735666338set_fm
            @ ^ [S: produc164195504107695125m_rule] : ( set_fm2 @ ( produc4588648349897876871ist_fm @ ( produc6879501374131015971m_rule @ S ) ) )
            @ ( sset_P4979403222752902913m_rule @ Steps3 ) ) ) ) ) ).

% treeB_def
thf(fact_1274_treeA__def,axiom,
    ( treeA
    = ( ^ [Steps3: stream727092118206550309m_rule] :
          ( comple7677669044296858969set_fm
          @ ( image_5698427363735666338set_fm
            @ ^ [S: produc164195504107695125m_rule] : ( set_fm2 @ ( produc1501393135466168645ist_fm @ ( produc6879501374131015971m_rule @ S ) ) )
            @ ( sset_P4979403222752902913m_rule @ Steps3 ) ) ) ) ) ).

% treeA_def

% Helper facts (22)
thf(help_fChoice_1_1_fChoice_001t__Syntax__Ofm_T,axiom,
    ! [P2: fm > $o] :
      ( ( P2 @ ( fChoice_fm @ P2 ) )
      = ( ? [X5: fm] : ( P2 @ X5 ) ) ) ).

thf(help_fChoice_1_1_fChoice_001t__Syntax__Orule_T,axiom,
    ! [P2: rule > $o] :
      ( ( P2 @ ( fChoice_rule @ P2 ) )
      = ( ? [X5: rule] : ( P2 @ X5 ) ) ) ).

thf(help_If_2_1_If_001t__Stream__Ostream_It__Syntax__Orule_J_T,axiom,
    ! [X3: stream_rule,Y3: stream_rule] :
      ( ( if_stream_rule @ $false @ X3 @ Y3 )
      = Y3 ) ).

thf(help_If_1_1_If_001t__Stream__Ostream_It__Syntax__Orule_J_T,axiom,
    ! [X3: stream_rule,Y3: stream_rule] :
      ( ( if_stream_rule @ $true @ X3 @ Y3 )
      = X3 ) ).

thf(help_If_2_1_If_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_T,axiom,
    ! [X3: product_prod_nat_nat,Y3: product_prod_nat_nat] :
      ( ( if_Pro6206227464963214023at_nat @ $false @ X3 @ Y3 )
      = Y3 ) ).

thf(help_If_1_1_If_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_T,axiom,
    ! [X3: product_prod_nat_nat,Y3: product_prod_nat_nat] :
      ( ( if_Pro6206227464963214023at_nat @ $true @ X3 @ Y3 )
      = X3 ) ).

thf(help_fChoice_1_1_fChoice_001t__Abstract____Completeness__Otree_It__Syntax__Orule_J_T,axiom,
    ! [P2: abstract_tree_rule > $o] :
      ( ( P2 @ ( fChoic8596484284512669613e_rule @ P2 ) )
      = ( ? [X5: abstract_tree_rule] : ( P2 @ X5 ) ) ) ).

thf(help_If_2_1_If_001t__Product____Type__Oprod_It__List__Olist_It__Syntax__Ofm_J_Mt__List__Olist_It__Syntax__Ofm_J_J_T,axiom,
    ! [X3: produc1996495991257130529ist_fm,Y3: produc1996495991257130529ist_fm] :
      ( ( if_Pro3930376587665744871ist_fm @ $false @ X3 @ Y3 )
      = Y3 ) ).

thf(help_If_1_1_If_001t__Product____Type__Oprod_It__List__Olist_It__Syntax__Ofm_J_Mt__List__Olist_It__Syntax__Ofm_J_J_T,axiom,
    ! [X3: produc1996495991257130529ist_fm,Y3: produc1996495991257130529ist_fm] :
      ( ( if_Pro3930376587665744871ist_fm @ $true @ X3 @ Y3 )
      = X3 ) ).

thf(help_fChoice_1_1_fChoice_001t__Product____Type__Oprod_It__List__Olist_It__Syntax__Ofm_J_Mt__List__Olist_It__Syntax__Ofm_J_J_T,axiom,
    ! [P2: produc1996495991257130529ist_fm > $o] :
      ( ( P2 @ ( fChoic1980167584695909237ist_fm @ P2 ) )
      = ( ? [X5: produc1996495991257130529ist_fm] : ( P2 @ X5 ) ) ) ).

thf(help_If_2_1_If_001t__FSet__Ofset_It__Product____Type__Oprod_It__List__Olist_It__Syntax__Ofm_J_Mt__List__Olist_It__Syntax__Ofm_J_J_J_T,axiom,
    ! [X3: fset_P661503646757059847ist_fm,Y3: fset_P661503646757059847ist_fm] :
      ( ( if_fse3714977293194272717ist_fm @ $false @ X3 @ Y3 )
      = Y3 ) ).

thf(help_If_1_1_If_001t__FSet__Ofset_It__Product____Type__Oprod_It__List__Olist_It__Syntax__Ofm_J_Mt__List__Olist_It__Syntax__Ofm_J_J_J_T,axiom,
    ! [X3: fset_P661503646757059847ist_fm,Y3: fset_P661503646757059847ist_fm] :
      ( ( if_fse3714977293194272717ist_fm @ $true @ X3 @ Y3 )
      = X3 ) ).

thf(help_fChoice_1_1_fChoice_001t__FSet__Ofset_It__Product____Type__Oprod_It__List__Olist_It__Syntax__Ofm_J_Mt__List__Olist_It__Syntax__Ofm_J_J_J_T,axiom,
    ! [P2: fset_P661503646757059847ist_fm > $o] :
      ( ( P2 @ ( fChoic3892542562059901019ist_fm @ P2 ) )
      = ( ? [X5: fset_P661503646757059847ist_fm] : ( P2 @ X5 ) ) ) ).

thf(help_fChoice_1_1_fChoice_001t__Product____Type__Oprod_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_Mt__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J_T,axiom,
    ! [P2: produc859450856879609959at_nat > $o] :
      ( ( P2 @ ( fChoic6673916560228149179at_nat @ P2 ) )
      = ( ? [X5: produc859450856879609959at_nat] : ( P2 @ X5 ) ) ) ).

thf(help_If_2_1_If_001t__Stream__Ostream_It__Product____Type__Oprod_It__Product____Type__Oprod_It__List__Olist_It__Syntax__Ofm_J_Mt__List__Olist_It__Syntax__Ofm_J_J_Mt__Syntax__Orule_J_J_T,axiom,
    ! [X3: stream727092118206550309m_rule,Y3: stream727092118206550309m_rule] :
      ( ( if_str8948254419368749791m_rule @ $false @ X3 @ Y3 )
      = Y3 ) ).

thf(help_If_1_1_If_001t__Stream__Ostream_It__Product____Type__Oprod_It__Product____Type__Oprod_It__List__Olist_It__Syntax__Ofm_J_Mt__List__Olist_It__Syntax__Ofm_J_J_Mt__Syntax__Orule_J_J_T,axiom,
    ! [X3: stream727092118206550309m_rule,Y3: stream727092118206550309m_rule] :
      ( ( if_str8948254419368749791m_rule @ $true @ X3 @ Y3 )
      = X3 ) ).

thf(help_If_2_1_If_001t__Abstract____Completeness__Otree_It__Product____Type__Oprod_It__Product____Type__Oprod_It__List__Olist_It__Syntax__Ofm_J_Mt__List__Olist_It__Syntax__Ofm_J_J_Mt__Syntax__Orule_J_J_T,axiom,
    ! [X3: abstra4572470701423150929m_rule,Y3: abstra4572470701423150929m_rule] :
      ( ( if_Abs4566652138351786251m_rule @ $false @ X3 @ Y3 )
      = Y3 ) ).

thf(help_If_1_1_If_001t__Abstract____Completeness__Otree_It__Product____Type__Oprod_It__Product____Type__Oprod_It__List__Olist_It__Syntax__Ofm_J_Mt__List__Olist_It__Syntax__Ofm_J_J_Mt__Syntax__Orule_J_J_T,axiom,
    ! [X3: abstra4572470701423150929m_rule,Y3: abstra4572470701423150929m_rule] :
      ( ( if_Abs4566652138351786251m_rule @ $true @ X3 @ Y3 )
      = X3 ) ).

thf(help_fChoice_1_1_fChoice_001t__Abstract____Completeness__Otree_It__Product____Type__Oprod_It__Product____Type__Oprod_It__List__Olist_It__Syntax__Ofm_J_Mt__List__Olist_It__Syntax__Ofm_J_J_Mt__Syntax__Orule_J_J_T,axiom,
    ! [P2: abstra4572470701423150929m_rule > $o] :
      ( ( P2 @ ( fChoic7543385803022166781m_rule @ P2 ) )
      = ( ? [X5: abstra4572470701423150929m_rule] : ( P2 @ X5 ) ) ) ).

thf(help_If_3_1_If_001t__FSet__Ofset_It__Abstract____Completeness__Otree_It__Product____Type__Oprod_It__Product____Type__Oprod_It__List__Olist_It__Syntax__Ofm_J_Mt__List__Olist_It__Syntax__Ofm_J_J_Mt__Syntax__Orule_J_J_J_T,axiom,
    ! [P2: $o] :
      ( ( P2 = $true )
      | ( P2 = $false ) ) ).

thf(help_If_2_1_If_001t__FSet__Ofset_It__Abstract____Completeness__Otree_It__Product____Type__Oprod_It__Product____Type__Oprod_It__List__Olist_It__Syntax__Ofm_J_Mt__List__Olist_It__Syntax__Ofm_J_J_Mt__Syntax__Orule_J_J_J_T,axiom,
    ! [X3: fset_A5396818216090053761m_rule,Y3: fset_A5396818216090053761m_rule] :
      ( ( if_fse3370020817956878651m_rule @ $false @ X3 @ Y3 )
      = Y3 ) ).

thf(help_If_1_1_If_001t__FSet__Ofset_It__Abstract____Completeness__Otree_It__Product____Type__Oprod_It__Product____Type__Oprod_It__List__Olist_It__Syntax__Ofm_J_Mt__List__Olist_It__Syntax__Ofm_J_J_Mt__Syntax__Orule_J_J_J_T,axiom,
    ! [X3: fset_A5396818216090053761m_rule,Y3: fset_A5396818216090053761m_rule] :
      ( ( if_fse3370020817956878651m_rule @ $true @ X3 @ Y3 )
      = X3 ) ).

% Conjectures (2)
thf(conj_0,hypothesis,
    ! [J3: nat] :
      ( ( ( produc7165828336582415457m_rule @ ( shd_Pr7235097944458474089m_rule @ ( sdrop_7224736112439592940m_rule @ J3 @ steps ) ) )
        = ( uniL @ t @ p ) )
     => thesis ) ).

thf(conj_1,conjecture,
    thesis ).

%------------------------------------------------------------------------------