TPTP Problem File: SLH0253^1.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_Calc2/0018_EPathHintikka/prob_00405_016492__13181416_1 [Des23]

% Status   : Theorem
% Rating   : ? v8.2.0
% Syntax   : Number of formulae    : 2010 ( 567 unt; 721 typ;   0 def)
%            Number of atoms       : 4470 ( 921 equ;  10 cnn)
%            Maximal formula atoms :   15 (   3 avg)
%            Number of connectives : 15356 ( 254   ~;  76   |; 544   &;12439   @)
%                                         (   0 <=>;2043  =>;   0  <=;   0 <~>)
%            Maximal formula depth :   22 (   8 avg)
%            Number of types       :  131 ( 130 usr)
%            Number of type conns  : 3105 (3105   >;   0   *;   0   +;   0  <<)
%            Number of symbols     :  595 ( 591 usr;  32 con; 0-4 aty)
%            Number of variables   : 4517 ( 676   ^;3522   !; 319   ?;4517   :)
% SPC      : TH0_THM_EQU_NAR

% Comments : This file was generated by Isabelle (most likely Sledgehammer)
%            2023-01-19 08:45:35.391
%------------------------------------------------------------------------------
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thf(ty_n_t__Set__Oset_It__Sum____Type__Osum_It__Product____Type__Oprod_It__List__Olist_It__SeCaV__Otm_J_Mt__List__Olist_It__SeCaV__Ofm_J_J_Mt__Nat__Onat_J_J,type,
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thf(ty_n_t__Set__Oset_It__Sum____Type__Osum_It__Nat__Onat_Mt__Product____Type__Oprod_It__List__Olist_It__SeCaV__Otm_J_Mt__List__Olist_It__SeCaV__Ofm_J_J_J_J,type,
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thf(ty_n_t__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(ty_n_t__Product____Type__Oprod_It__Product____Type__Oprod_It__List__Olist_It__SeCaV__Otm_J_Mt__List__Olist_It__SeCaV__Ofm_J_J_Mt__Nat__Onat_J,type,
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thf(ty_n_t__Set__Oset_It__Stream__Ostream_It__Product____Type__Oprod_It__List__Olist_It__SeCaV__Otm_J_Mt__List__Olist_It__SeCaV__Ofm_J_J_J_J,type,
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thf(ty_n_t__Set__Oset_It__Option__Ooption_It__Product____Type__Oprod_It__List__Olist_It__SeCaV__Otm_J_Mt__List__Olist_It__SeCaV__Ofm_J_J_J_J,type,
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thf(ty_n_t__Abstract____Completeness__Otree_It__Product____Type__Oprod_It__List__Olist_It__SeCaV__Otm_J_Mt__List__Olist_It__SeCaV__Ofm_J_J_J,type,
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thf(ty_n_t__Set__Oset_It__Set__Oset_It__Product____Type__Oprod_It__List__Olist_It__SeCaV__Otm_J_Mt__List__Olist_It__SeCaV__Ofm_J_J_J_J,type,
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thf(ty_n_t__Stream__Ostream_It__Product____Type__Oprod_It__List__Olist_It__SeCaV__Otm_J_Mt__List__Olist_It__SeCaV__Ofm_J_J_J,type,
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thf(ty_n_t__FSet__Ofset_It__List__Olist_It__SeCaV__Otm_J_J,type,
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thf(ty_n_t__Set__Oset_It__Set__Oset_It__Prover__Orule_J_J,type,
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thf(ty_n_t__Set__Oset_It__List__Olist_It__SeCaV__Ofm_J_J,type,
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thf(ty_n_t__Set__Oset_It__Set__Oset_It__SeCaV__Otm_J_J,type,
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thf(ty_n_t__Set__Oset_It__Set__Oset_It__Nat__Onat_J_J,type,
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thf(ty_n_t__Stream__Ostream_It__Prover__Orule_J,type,
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thf(ty_n_t__Stream__Ostream_It__SeCaV__Otm_J,type,
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thf(ty_n_t__Stream__Ostream_It__Nat__Onat_J,type,
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thf(ty_n_t__List__Olist_It__Prover__Orule_J,type,
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thf(ty_n_t__FSet__Ofset_It__Prover__Orule_J,type,
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thf(ty_n_t__Set__Oset_It__Prover__Orule_J,type,
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thf(ty_n_t__List__Olist_It__SeCaV__Otm_J,type,
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thf(ty_n_t__List__Olist_It__SeCaV__Ofm_J,type,
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thf(ty_n_t__FSet__Ofset_It__Nat__Onat_J,type,
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thf(ty_n_t__Set__Oset_It__SeCaV__Otm_J,type,
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thf(ty_n_t__Set__Oset_It__Nat__Onat_J,type,
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thf(ty_n_t__Prover__Orule,type,
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thf(ty_n_t__SeCaV__Otm,type,
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thf(ty_n_t__Nat__Onat,type,
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% Explicit typings (591)
thf(sy_c_Abstract__Completeness_OPersistentRuleSystem_001t__List__Olist_It__SeCaV__Ofm_J_001t__List__Olist_It__SeCaV__Otm_J,type,
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thf(sy_c_Abstract__Completeness_OPersistentRuleSystem_001t__Prover__Orule_001t__Prover__Orule,type,
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thf(sy_c_Abstract__Completeness_ORuleSystem_001t__List__Olist_It__SeCaV__Ofm_J_001t__List__Olist_It__SeCaV__Otm_J,type,
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thf(sy_c_Abstract__Completeness_ORuleSystem_001t__Nat__Onat_001t__Nat__Onat,type,
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thf(sy_c_Linear__Temporal__Logic__on__Streams_Oholds_001t__Prover__Orule,type,
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thf(sy_c_Linear__Temporal__Logic__on__Streams_Oholds_001t__SeCaV__Otm,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_Olist__all_001t__Prover__Orule,type,
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thf(sy_c_List_Olist_Oset_001t__SeCaV__Otm,type,
    set_tm2: list_tm > set_tm ).

thf(sy_c_Nat_OSuc,type,
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thf(sy_c_Nat_Ocompow_001_062_I_062_It__Nat__Onat_Mt__Nat__Onat_J_M_062_It__Nat__Onat_Mt__Nat__Onat_J_J,type,
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thf(sy_c_Nat_Ocompow_001_062_I_062_It__Stream__Ostream_It__Prover__Orule_J_M_Eo_J_M_062_It__Stream__Ostream_It__Prover__Orule_J_M_Eo_J_J,type,
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thf(sy_c_Nat_Ocompow_001_062_It__Nat__Onat_Mt__Nat__Onat_J,type,
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thf(sy_c_Nat__Bijection_Oprod__decode__aux,type,
    nat_prod_decode_aux: nat > nat > product_prod_nat_nat ).

thf(sy_c_Orderings_Oord__class_OLeast_001t__Nat__Onat,type,
    ord_Least_nat: ( nat > $o ) > nat ).

thf(sy_c_Orderings_Oord__class_OLeast_001t__Set__Oset_It__Prover__Orule_J,type,
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thf(sy_c_Orderings_Oord__class_OLeast_001t__Set__Oset_It__SeCaV__Otm_J,type,
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thf(sy_c_Orderings_Oord__class_Oless_001t__Nat__Onat,type,
    ord_less_nat: nat > nat > $o ).

thf(sy_c_Orderings_Oord__class_Oless__eq_001_062_It__List__Olist_It__SeCaV__Otm_J_M_062_It__List__Olist_It__SeCaV__Ofm_J_M_Eo_J_J,type,
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thf(sy_c_Orderings_Oord__class_Oless__eq_001_062_It__Nat__Onat_M_062_It__Nat__Onat_M_Eo_J_J,type,
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thf(sy_c_Orderings_Oord__class_Oless__eq_001_062_It__Nat__Onat_M_Eo_J,type,
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    ord_le2289920624603415501rule_o: ( produc6018962875968178549ist_fm > rule > $o ) > ( produc6018962875968178549ist_fm > rule > $o ) > $o ).

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thf(sy_c_Orderings_Oord__class_Oless__eq_001_062_It__Prover__Orule_M_Eo_J,type,
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thf(sy_c_Orderings_Oord__class_Oless__eq_001_062_It__SeCaV__Otm_M_Eo_J,type,
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thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Nat__Onat,type,
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thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_It__Nat__Onat_J,type,
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thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_It__Prover__Orule_J,type,
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thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_It__SeCaV__Otm_J,type,
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thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_It__Set__Oset_It__Prover__Orule_J_J,type,
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thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_It__Set__Oset_It__SeCaV__Otm_J_J,type,
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thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_It__Stream__Ostream_It__Prover__Orule_J_J,type,
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thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_It__Stream__Ostream_It__SeCaV__Otm_J_J,type,
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thf(sy_c_Orderings_Oordering__top_001t__Nat__Onat,type,
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thf(sy_c_Orderings_Otop__class_Otop_001_062_It__List__Olist_It__SeCaV__Otm_J_M_062_It__List__Olist_It__SeCaV__Ofm_J_M_Eo_J_J,type,
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thf(sy_c_Orderings_Otop__class_Otop_001_062_It__Nat__Onat_M_062_It__Nat__Onat_M_Eo_J_J,type,
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thf(sy_c_Orderings_Otop__class_Otop_001_062_It__Nat__Onat_M_Eo_J,type,
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thf(sy_c_Orderings_Otop__class_Otop_001_062_It__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_M_Eo_J,type,
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thf(sy_c_Orderings_Otop__class_Otop_001_062_It__Prover__Orule_M_Eo_J,type,
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thf(sy_c_Orderings_Otop__class_Otop_001t__Set__Oset_It__List__Olist_It__SeCaV__Ofm_J_J,type,
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thf(sy_c_Orderings_Otop__class_Otop_001t__Set__Oset_It__List__Olist_It__SeCaV__Otm_J_J,type,
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thf(sy_c_Orderings_Otop__class_Otop_001t__Set__Oset_It__Nat__Onat_J,type,
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thf(sy_c_Orderings_Otop__class_Otop_001t__Set__Oset_It__Option__Ooption_It__Nat__Onat_J_J,type,
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thf(sy_c_Orderings_Otop__class_Otop_001t__Set__Oset_It__SeCaV__Otm_J,type,
    top_top_set_tm: set_tm ).

thf(sy_c_Orderings_Otop__class_Otop_001t__Set__Oset_It__Set__Oset_It__Nat__Onat_J_J,type,
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thf(sy_c_Orderings_Otop__class_Otop_001t__Set__Oset_It__Stream__Ostream_It__Nat__Onat_J_J,type,
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thf(sy_c_member_001t__Stream__Ostream_It__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_J,type,
    member1992146023263842070at_nat: stream8372878641218411373at_nat > set_st6020995191862378445at_nat > $o ).

thf(sy_c_member_001t__Stream__Ostream_It__Prover__Orule_J,type,
    member_stream_rule: stream_rule > set_stream_rule > $o ).

thf(sy_c_member_001t__Stream__Ostream_It__SeCaV__Otm_J,type,
    member_stream_tm: stream_tm > set_stream_tm > $o ).

thf(sy_v_n____,type,
    n: nat ).

thf(sy_v_steps,type,
    steps: stream2709947120125613254m_rule ).

% Relevant facts (1275)
thf(fact_0_assms_I2_J,axiom,
    abstra6097777249025082867ist_fm @ eff @ rules @ steps ).

% assms(2)
thf(fact_1_Saturated__ev__rule,axiom,
    ! [Steps: stream2709947120125613254m_rule,R: rule,N: nat] :
      ( ( abstra6097777249025082867ist_fm @ eff @ rules @ Steps )
     => ( linear1816559245158234865m_rule
        @ ( linear7925859179493631106m_rule
          @ ^ [Step: produc340336539035504054m_rule] :
              ( ( produc6514711208036230658m_rule @ Step )
              = R ) )
        @ ( sdrop_8169176516188972301m_rule @ N @ Steps ) ) ) ).

% Saturated_ev_rule
thf(fact_2_ev__ev,axiom,
    ! [Phi: stream2709947120125613254m_rule > $o] :
      ( ( linear1816559245158234865m_rule @ ( linear1816559245158234865m_rule @ Phi ) )
      = ( linear1816559245158234865m_rule @ Phi ) ) ).

% ev_ev
thf(fact_3_ev__ev,axiom,
    ! [Phi: stream_rule > $o] :
      ( ( linear292482573288640205v_rule @ ( linear292482573288640205v_rule @ Phi ) )
      = ( linear292482573288640205v_rule @ Phi ) ) ).

% ev_ev
thf(fact_4_assms_I1_J,axiom,
    abstra6789711989322986974ist_fm @ eff @ rules @ steps ).

% assms(1)
thf(fact_5_ev__iff__sdrop,axiom,
    ( linear1816559245158234865m_rule
    = ( ^ [P: stream2709947120125613254m_rule > $o,Omega: stream2709947120125613254m_rule] :
        ? [M: nat] : ( P @ ( sdrop_8169176516188972301m_rule @ M @ Omega ) ) ) ) ).

% ev_iff_sdrop
thf(fact_6_ev__iff__sdrop,axiom,
    ( linear292482573288640205v_rule
    = ( ^ [P: stream_rule > $o,Omega: stream_rule] :
        ? [M: nat] : ( P @ ( sdrop_rule @ M @ Omega ) ) ) ) ).

% ev_iff_sdrop
thf(fact_7_holds__aand,axiom,
    ! [P2: produc340336539035504054m_rule > $o,Steps: stream2709947120125613254m_rule,Q: produc340336539035504054m_rule > $o] :
      ( ( ( linear7925859179493631106m_rule @ P2 @ Steps )
        & ( linear7925859179493631106m_rule @ Q @ Steps ) )
      = ( linear7925859179493631106m_rule
        @ ^ [Step: produc340336539035504054m_rule] :
            ( ( P2 @ Step )
            & ( Q @ Step ) )
        @ Steps ) ) ).

% holds_aand
thf(fact_8_holds__aand,axiom,
    ! [P2: rule > $o,Steps: stream_rule,Q: rule > $o] :
      ( ( ( linear7354821883049305950s_rule @ P2 @ Steps )
        & ( linear7354821883049305950s_rule @ Q @ Steps ) )
      = ( linear7354821883049305950s_rule
        @ ^ [Step: rule] :
            ( ( P2 @ Step )
            & ( Q @ Step ) )
        @ Steps ) ) ).

% holds_aand
thf(fact_9_ev__or,axiom,
    ! [Phi: stream2709947120125613254m_rule > $o,Psi: stream2709947120125613254m_rule > $o] :
      ( ( linear1816559245158234865m_rule
        @ ^ [Xs: stream2709947120125613254m_rule] :
            ( ( Phi @ Xs )
            | ( Psi @ Xs ) ) )
      = ( ^ [Xs: stream2709947120125613254m_rule] :
            ( ( linear1816559245158234865m_rule @ Phi @ Xs )
            | ( linear1816559245158234865m_rule @ Psi @ Xs ) ) ) ) ).

% ev_or
thf(fact_10_ev__or,axiom,
    ! [Phi: stream_rule > $o,Psi: stream_rule > $o] :
      ( ( linear292482573288640205v_rule
        @ ^ [Xs: stream_rule] :
            ( ( Phi @ Xs )
            | ( Psi @ Xs ) ) )
      = ( ^ [Xs: stream_rule] :
            ( ( linear292482573288640205v_rule @ Phi @ Xs )
            | ( linear292482573288640205v_rule @ Psi @ Xs ) ) ) ) ).

% ev_or
thf(fact_11_ev__False,axiom,
    ! [Omega2: stream2709947120125613254m_rule] :
      ~ ( linear1816559245158234865m_rule
        @ ^ [X: stream2709947120125613254m_rule] : $false
        @ Omega2 ) ).

% ev_False
thf(fact_12_ev__False,axiom,
    ! [Omega2: stream_rule] :
      ~ ( linear292482573288640205v_rule
        @ ^ [X: stream_rule] : $false
        @ Omega2 ) ).

% ev_False
thf(fact_13_holds__mono,axiom,
    ! [P2: produc340336539035504054m_rule > $o,Xs2: stream2709947120125613254m_rule,Q: produc340336539035504054m_rule > $o] :
      ( ( linear7925859179493631106m_rule @ P2 @ Xs2 )
     => ( ! [X2: produc340336539035504054m_rule] :
            ( ( P2 @ X2 )
           => ( Q @ X2 ) )
       => ( linear7925859179493631106m_rule @ Q @ Xs2 ) ) ) ).

% holds_mono
thf(fact_14_holds__mono,axiom,
    ! [P2: rule > $o,Xs2: stream_rule,Q: rule > $o] :
      ( ( linear7354821883049305950s_rule @ P2 @ Xs2 )
     => ( ! [X2: rule] :
            ( ( P2 @ X2 )
           => ( Q @ X2 ) )
       => ( linear7354821883049305950s_rule @ Q @ Xs2 ) ) ) ).

% holds_mono
thf(fact_15_ev_Obase,axiom,
    ! [Phi: stream2709947120125613254m_rule > $o,Xs2: stream2709947120125613254m_rule] :
      ( ( Phi @ Xs2 )
     => ( linear1816559245158234865m_rule @ Phi @ Xs2 ) ) ).

% ev.base
thf(fact_16_ev_Obase,axiom,
    ! [Phi: stream_rule > $o,Xs2: stream_rule] :
      ( ( Phi @ Xs2 )
     => ( linear292482573288640205v_rule @ Phi @ Xs2 ) ) ).

% ev.base
thf(fact_17_ev__mono,axiom,
    ! [Phi: stream2709947120125613254m_rule > $o,Xs2: stream2709947120125613254m_rule,Psi: stream2709947120125613254m_rule > $o] :
      ( ( linear1816559245158234865m_rule @ Phi @ Xs2 )
     => ( ! [Xs3: stream2709947120125613254m_rule] :
            ( ( Phi @ Xs3 )
           => ( Psi @ Xs3 ) )
       => ( linear1816559245158234865m_rule @ Psi @ Xs2 ) ) ) ).

% ev_mono
thf(fact_18_ev__mono,axiom,
    ! [Phi: stream_rule > $o,Xs2: stream_rule,Psi: stream_rule > $o] :
      ( ( linear292482573288640205v_rule @ Phi @ Xs2 )
     => ( ! [Xs3: stream_rule] :
            ( ( Phi @ Xs3 )
           => ( Psi @ Xs3 ) )
       => ( linear292482573288640205v_rule @ Psi @ Xs2 ) ) ) ).

% ev_mono
thf(fact_19_sdrop__szip,axiom,
    ! [N: nat,S1: stream2709947120125613254m_rule,S2: stream2709947120125613254m_rule] :
      ( ( sdrop_7192298464603511222m_rule @ N @ ( szip_P811719526838699976m_rule @ S1 @ S2 ) )
      = ( szip_P811719526838699976m_rule @ ( sdrop_8169176516188972301m_rule @ N @ S1 ) @ ( sdrop_8169176516188972301m_rule @ N @ S2 ) ) ) ).

% sdrop_szip
thf(fact_20_sdrop__szip,axiom,
    ! [N: nat,S1: stream2709947120125613254m_rule,S2: stream_rule] :
      ( ( sdrop_1938960342593938834e_rule @ N @ ( szip_P2044787555563963556e_rule @ S1 @ S2 ) )
      = ( szip_P2044787555563963556e_rule @ ( sdrop_8169176516188972301m_rule @ N @ S1 ) @ ( sdrop_rule @ N @ S2 ) ) ) ).

% sdrop_szip
thf(fact_21_sdrop__szip,axiom,
    ! [N: nat,S1: stream_rule,S2: stream2709947120125613254m_rule] :
      ( ( sdrop_746751008794826386m_rule @ N @ ( szip_r6533552703345879972m_rule @ S1 @ S2 ) )
      = ( szip_r6533552703345879972m_rule @ ( sdrop_rule @ N @ S1 ) @ ( sdrop_8169176516188972301m_rule @ N @ S2 ) ) ) ).

% sdrop_szip
thf(fact_22_sdrop__szip,axiom,
    ! [N: nat,S1: stream_rule,S2: stream_rule] :
      ( ( sdrop_9113879250048157294e_rule @ N @ ( szip_rule_rule @ S1 @ S2 ) )
      = ( szip_rule_rule @ ( sdrop_rule @ N @ S1 ) @ ( sdrop_rule @ N @ S2 ) ) ) ).

% sdrop_szip
thf(fact_23_sdrop__szip,axiom,
    ! [N: nat,S1: stream4408948924543953275ist_fm,S2: stream_rule] :
      ( ( sdrop_8169176516188972301m_rule @ N @ ( szip_P2924820683901490861m_rule @ S1 @ S2 ) )
      = ( szip_P2924820683901490861m_rule @ ( sdrop_9176333610110415838ist_fm @ N @ S1 ) @ ( sdrop_rule @ N @ S2 ) ) ) ).

% sdrop_szip
thf(fact_24_sdrop__wait,axiom,
    ! [Phi: stream2709947120125613254m_rule > $o,Xs2: stream2709947120125613254m_rule] :
      ( ( linear1816559245158234865m_rule @ Phi @ Xs2 )
     => ( Phi @ ( sdrop_8169176516188972301m_rule @ ( linear8417479883454578709m_rule @ Phi @ Xs2 ) @ Xs2 ) ) ) ).

% sdrop_wait
thf(fact_25_sdrop__wait,axiom,
    ! [Phi: stream_rule > $o,Xs2: stream_rule] :
      ( ( linear292482573288640205v_rule @ Phi @ Xs2 )
     => ( Phi @ ( sdrop_rule @ ( linear2710171048243638257t_rule @ Phi @ Xs2 ) @ Xs2 ) ) ) ).

% sdrop_wait
thf(fact_26_Saturated__sdrop,axiom,
    ! [Steps: stream2709947120125613254m_rule,N: nat] :
      ( ( abstra6097777249025082867ist_fm @ eff @ rules @ Steps )
     => ( abstra6097777249025082867ist_fm @ eff @ rules @ ( sdrop_8169176516188972301m_rule @ N @ Steps ) ) ) ).

% Saturated_sdrop
thf(fact_27_epath__sdrop,axiom,
    ! [Steps: stream2709947120125613254m_rule,N: nat] :
      ( ( abstra6789711989322986974ist_fm @ eff @ rules @ Steps )
     => ( abstra6789711989322986974ist_fm @ eff @ rules @ ( sdrop_8169176516188972301m_rule @ N @ Steps ) ) ) ).

% epath_sdrop
thf(fact_28_RuleSystem__Defs_OSaturated_Ocong,axiom,
    abstra6097777249025082867ist_fm = abstra6097777249025082867ist_fm ).

% RuleSystem_Defs.Saturated.cong
thf(fact_29_sdrop__wait__least,axiom,
    ! [Phi: stream2709947120125613254m_rule > $o,Xs2: stream2709947120125613254m_rule,N: nat] :
      ( ( linear1816559245158234865m_rule @ Phi @ Xs2 )
     => ( ( Phi @ ( sdrop_8169176516188972301m_rule @ N @ Xs2 ) )
       => ( ord_less_eq_nat @ ( linear8417479883454578709m_rule @ Phi @ Xs2 ) @ N ) ) ) ).

% sdrop_wait_least
thf(fact_30_sdrop__wait__least,axiom,
    ! [Phi: stream_rule > $o,Xs2: stream_rule,N: nat] :
      ( ( linear292482573288640205v_rule @ Phi @ Xs2 )
     => ( ( Phi @ ( sdrop_rule @ N @ Xs2 ) )
       => ( ord_less_eq_nat @ ( linear2710171048243638257t_rule @ Phi @ Xs2 ) @ N ) ) ) ).

% sdrop_wait_least
thf(fact_31_RuleSystem__Defs_Oepath_Ocong,axiom,
    abstra6789711989322986974ist_fm = abstra6789711989322986974ist_fm ).

% RuleSystem_Defs.epath.cong
thf(fact_32_sdrop__smap2,axiom,
    ! [N: nat,F: produc340336539035504054m_rule > produc340336539035504054m_rule > produc340336539035504054m_rule,S1: stream2709947120125613254m_rule,S2: stream2709947120125613254m_rule] :
      ( ( sdrop_8169176516188972301m_rule @ N @ ( smap2_6945574070536835298m_rule @ F @ S1 @ S2 ) )
      = ( smap2_6945574070536835298m_rule @ F @ ( sdrop_8169176516188972301m_rule @ N @ S1 ) @ ( sdrop_8169176516188972301m_rule @ N @ S2 ) ) ) ).

% sdrop_smap2
thf(fact_33_sdrop__smap2,axiom,
    ! [N: nat,F: produc340336539035504054m_rule > rule > produc340336539035504054m_rule,S1: stream2709947120125613254m_rule,S2: stream_rule] :
      ( ( sdrop_8169176516188972301m_rule @ N @ ( smap2_6366920367380388542m_rule @ F @ S1 @ S2 ) )
      = ( smap2_6366920367380388542m_rule @ F @ ( sdrop_8169176516188972301m_rule @ N @ S1 ) @ ( sdrop_rule @ N @ S2 ) ) ) ).

% sdrop_smap2
thf(fact_34_sdrop__smap2,axiom,
    ! [N: nat,F: rule > produc340336539035504054m_rule > produc340336539035504054m_rule,S1: stream_rule,S2: stream2709947120125613254m_rule] :
      ( ( sdrop_8169176516188972301m_rule @ N @ ( smap2_4579943528986984894m_rule @ F @ S1 @ S2 ) )
      = ( smap2_4579943528986984894m_rule @ F @ ( sdrop_rule @ N @ S1 ) @ ( sdrop_8169176516188972301m_rule @ N @ S2 ) ) ) ).

% sdrop_smap2
thf(fact_35_sdrop__smap2,axiom,
    ! [N: nat,F: rule > rule > produc340336539035504054m_rule,S1: stream_rule,S2: stream_rule] :
      ( ( sdrop_8169176516188972301m_rule @ N @ ( smap2_4319354692778897306m_rule @ F @ S1 @ S2 ) )
      = ( smap2_4319354692778897306m_rule @ F @ ( sdrop_rule @ N @ S1 ) @ ( sdrop_rule @ N @ S2 ) ) ) ).

% sdrop_smap2
thf(fact_36_sdrop__smap2,axiom,
    ! [N: nat,F: produc340336539035504054m_rule > produc340336539035504054m_rule > rule,S1: stream2709947120125613254m_rule,S2: stream2709947120125613254m_rule] :
      ( ( sdrop_rule @ N @ ( smap2_1878155219598472126e_rule @ F @ S1 @ S2 ) )
      = ( smap2_1878155219598472126e_rule @ F @ ( sdrop_8169176516188972301m_rule @ N @ S1 ) @ ( sdrop_8169176516188972301m_rule @ N @ S2 ) ) ) ).

% sdrop_smap2
thf(fact_37_sdrop__smap2,axiom,
    ! [N: nat,F: produc340336539035504054m_rule > rule > rule,S1: stream2709947120125613254m_rule,S2: stream_rule] :
      ( ( sdrop_rule @ N @ ( smap2_8971878701134242202e_rule @ F @ S1 @ S2 ) )
      = ( smap2_8971878701134242202e_rule @ F @ ( sdrop_8169176516188972301m_rule @ N @ S1 ) @ ( sdrop_rule @ N @ S2 ) ) ) ).

% sdrop_smap2
thf(fact_38_sdrop__smap2,axiom,
    ! [N: nat,F: rule > produc340336539035504054m_rule > rule,S1: stream_rule,S2: stream2709947120125613254m_rule] :
      ( ( sdrop_rule @ N @ ( smap2_9053961581851756698e_rule @ F @ S1 @ S2 ) )
      = ( smap2_9053961581851756698e_rule @ F @ ( sdrop_rule @ N @ S1 ) @ ( sdrop_8169176516188972301m_rule @ N @ S2 ) ) ) ).

% sdrop_smap2
thf(fact_39_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_40_HLD__def,axiom,
    ( linear5551378545506465448LD_nat
    = ( ^ [S: set_nat] :
          ( linear3087052677352696476ds_nat
          @ ^ [X: nat] : ( member_nat @ X @ S ) ) ) ) ).

% HLD_def
thf(fact_41_HLD__def,axiom,
    ( linear6120129154741419165ist_fm
    = ( ^ [S: set_Pr5202636777678657877ist_fm] :
          ( linear4742913937519564201ist_fm
          @ ^ [X: produc6018962875968178549ist_fm] : ( member4699826688122452638ist_fm @ X @ S ) ) ) ) ).

% HLD_def
thf(fact_42_HLD__def,axiom,
    ( linear6066394698532776847at_nat
    = ( ^ [S: set_Pr8693737435421807431at_nat] :
          ( linear2637753954401252507at_nat
          @ ^ [X: produc859450856879609959at_nat] : ( member8206827879206165904at_nat @ X @ S ) ) ) ) ).

% HLD_def
thf(fact_43_HLD__def,axiom,
    ( linear2885739836753817742m_rule
    = ( ^ [S: set_Pr1822751329126368876m_rule] :
          ( linear7925859179493631106m_rule
          @ ^ [X: produc340336539035504054m_rule] : ( member7231649785386036813m_rule @ X @ S ) ) ) ) ).

% HLD_def
thf(fact_44_HLD__def,axiom,
    ( linear5032321834864961386D_rule
    = ( ^ [S: set_rule] :
          ( linear7354821883049305950s_rule
          @ ^ [X: rule] : ( member_rule @ X @ S ) ) ) ) ).

% HLD_def
thf(fact_45_ev__holds__sset,axiom,
    ! [P2: produc340336539035504054m_rule > $o,Xs2: stream2709947120125613254m_rule] :
      ( ( linear1816559245158234865m_rule @ ( linear7925859179493631106m_rule @ P2 ) @ Xs2 )
      = ( ? [X: produc340336539035504054m_rule] :
            ( ( member7231649785386036813m_rule @ X @ ( sset_P4484857331586881186m_rule @ Xs2 ) )
            & ( P2 @ X ) ) ) ) ).

% ev_holds_sset
thf(fact_46_ev__holds__sset,axiom,
    ! [P2: rule > $o,Xs2: stream_rule] :
      ( ( linear292482573288640205v_rule @ ( linear7354821883049305950s_rule @ P2 ) @ Xs2 )
      = ( ? [X: rule] :
            ( ( member_rule @ X @ ( sset_rule @ Xs2 ) )
            & ( P2 @ X ) ) ) ) ).

% ev_holds_sset
thf(fact_47_ev__prefix__sdrop,axiom,
    ! [P2: produc340336539035504054m_rule > $o,Xs2: stream2709947120125613254m_rule] :
      ( ( linear1816559245158234865m_rule @ ( linear7925859179493631106m_rule @ P2 ) @ Xs2 )
     => ? [N2: nat] :
          ( ( list_a9038620553111681026m_rule
            @ ^ [Xs: produc340336539035504054m_rule] :
                ~ ( P2 @ Xs )
            @ ( stake_5421812949518764133m_rule @ N2 @ Xs2 ) )
          & ( linear7925859179493631106m_rule @ P2 @ ( sdrop_8169176516188972301m_rule @ N2 @ Xs2 ) ) ) ) ).

% ev_prefix_sdrop
thf(fact_48_ev__prefix__sdrop,axiom,
    ! [P2: rule > $o,Xs2: stream_rule] :
      ( ( linear292482573288640205v_rule @ ( linear7354821883049305950s_rule @ P2 ) @ Xs2 )
     => ? [N2: nat] :
          ( ( list_all_rule
            @ ^ [Xs: rule] :
                ~ ( P2 @ Xs )
            @ ( stake_rule @ N2 @ Xs2 ) )
          & ( linear7354821883049305950s_rule @ P2 @ ( sdrop_rule @ N2 @ Xs2 ) ) ) ) ).

% ev_prefix_sdrop
thf(fact_49_alw__ev__sdrop,axiom,
    ! [Phi: stream_rule > $o,M2: nat,Xs2: stream_rule] :
      ( ( linear1087404401031714582w_rule @ ( linear292482573288640205v_rule @ Phi ) @ ( sdrop_rule @ M2 @ Xs2 ) )
     => ( linear1087404401031714582w_rule @ ( linear292482573288640205v_rule @ Phi ) @ Xs2 ) ) ).

% alw_ev_sdrop
thf(fact_50_alw__ev__sdrop,axiom,
    ! [Phi: stream2709947120125613254m_rule > $o,M2: nat,Xs2: stream2709947120125613254m_rule] :
      ( ( linear4957294957385550906m_rule @ ( linear1816559245158234865m_rule @ Phi ) @ ( sdrop_8169176516188972301m_rule @ M2 @ Xs2 ) )
     => ( linear4957294957385550906m_rule @ ( linear1816559245158234865m_rule @ Phi ) @ Xs2 ) ) ).

% alw_ev_sdrop
thf(fact_51_until__must__release__ev,axiom,
    ! [Phi: stream2709947120125613254m_rule > $o,Psi: stream2709947120125613254m_rule > $o,Omega2: stream2709947120125613254m_rule] :
      ( ( linear8167355876404644844m_rule @ Phi @ Psi @ Omega2 )
     => ( ( linear1816559245158234865m_rule
          @ ^ [Xs: stream2709947120125613254m_rule] :
              ~ ( Phi @ Xs )
          @ Omega2 )
       => ( linear1816559245158234865m_rule @ Psi @ Omega2 ) ) ) ).

% until_must_release_ev
thf(fact_52_until__must__release__ev,axiom,
    ! [Phi: stream_rule > $o,Psi: stream_rule > $o,Omega2: stream_rule] :
      ( ( linear6513964619561280712L_rule @ Phi @ Psi @ Omega2 )
     => ( ( linear292482573288640205v_rule
          @ ^ [Xs: stream_rule] :
              ~ ( Phi @ Xs )
          @ Omega2 )
       => ( linear292482573288640205v_rule @ Psi @ Omega2 ) ) ) ).

% until_must_release_ev
thf(fact_53_ev__eq__suntil,axiom,
    ( linear1816559245158234865m_rule
    = ( ^ [P: stream2709947120125613254m_rule > $o] :
          ( linear5457325955354885675m_rule
          @ ^ [Xs: stream2709947120125613254m_rule] :
              ~ ( P @ Xs )
          @ P ) ) ) ).

% ev_eq_suntil
thf(fact_54_ev__eq__suntil,axiom,
    ( linear292482573288640205v_rule
    = ( ^ [P: stream_rule > $o] :
          ( linear1836410492716652551l_rule
          @ ^ [Xs: stream_rule] :
              ~ ( P @ Xs )
          @ P ) ) ) ).

% ev_eq_suntil
thf(fact_55_true__suntil,axiom,
    ! [P2: stream2709947120125613254m_rule > $o] :
      ( ( linear5457325955354885675m_rule
        @ ^ [Uu: stream2709947120125613254m_rule] : $true
        @ P2 )
      = ( linear1816559245158234865m_rule @ P2 ) ) ).

% true_suntil
thf(fact_56_true__suntil,axiom,
    ! [P2: stream_rule > $o] :
      ( ( linear1836410492716652551l_rule
        @ ^ [Uu: stream_rule] : $true
        @ P2 )
      = ( linear292482573288640205v_rule @ P2 ) ) ).

% true_suntil
thf(fact_57_ev__nxt,axiom,
    ( linear292482573288640205v_rule
    = ( ^ [Phi2: stream_rule > $o,Xs: stream_rule] :
          ( ( Phi2 @ Xs )
          | ( linear8837004798957005347rule_o @ ( linear292482573288640205v_rule @ Phi2 ) @ Xs ) ) ) ) ).

% ev_nxt
thf(fact_58_ev__nxt,axiom,
    ( linear1816559245158234865m_rule
    = ( ^ [Phi2: stream2709947120125613254m_rule > $o,Xs: stream2709947120125613254m_rule] :
          ( ( Phi2 @ Xs )
          | ( linear3460418925085718015rule_o @ ( linear1816559245158234865m_rule @ Phi2 ) @ Xs ) ) ) ) ).

% ev_nxt
thf(fact_59_alw__alw,axiom,
    ! [Phi: stream_rule > $o] :
      ( ( linear1087404401031714582w_rule @ ( linear1087404401031714582w_rule @ Phi ) )
      = ( linear1087404401031714582w_rule @ Phi ) ) ).

% alw_alw
thf(fact_60_alw__alw,axiom,
    ! [Phi: stream2709947120125613254m_rule > $o] :
      ( ( linear4957294957385550906m_rule @ ( linear4957294957385550906m_rule @ Phi ) )
      = ( linear4957294957385550906m_rule @ Phi ) ) ).

% alw_alw
thf(fact_61_not__ev__not,axiom,
    ! [Phi: stream_rule > $o] :
      ( ( ^ [Xs: stream_rule] :
            ~ ( linear292482573288640205v_rule
              @ ^ [Xt: stream_rule] :
                  ~ ( Phi @ Xt )
              @ Xs ) )
      = ( linear1087404401031714582w_rule @ Phi ) ) ).

% not_ev_not
thf(fact_62_not__ev__not,axiom,
    ! [Phi: stream2709947120125613254m_rule > $o] :
      ( ( ^ [Xs: stream2709947120125613254m_rule] :
            ~ ( linear1816559245158234865m_rule
              @ ^ [Xt: stream2709947120125613254m_rule] :
                  ~ ( Phi @ Xt )
              @ Xs ) )
      = ( linear4957294957385550906m_rule @ Phi ) ) ).

% not_ev_not
thf(fact_63_not__alw__not,axiom,
    ! [Phi: stream_rule > $o] :
      ( ( ^ [Xs: stream_rule] :
            ~ ( linear1087404401031714582w_rule
              @ ^ [Xt: stream_rule] :
                  ~ ( Phi @ Xt )
              @ Xs ) )
      = ( linear292482573288640205v_rule @ Phi ) ) ).

% not_alw_not
thf(fact_64_not__alw__not,axiom,
    ! [Phi: stream2709947120125613254m_rule > $o] :
      ( ( ^ [Xs: stream2709947120125613254m_rule] :
            ~ ( linear4957294957385550906m_rule
              @ ^ [Xt: stream2709947120125613254m_rule] :
                  ~ ( Phi @ Xt )
              @ Xs ) )
      = ( linear1816559245158234865m_rule @ Phi ) ) ).

% not_alw_not
thf(fact_65_alwD,axiom,
    ! [P2: stream_rule > $o,X3: stream_rule] :
      ( ( linear1087404401031714582w_rule @ P2 @ X3 )
     => ( P2 @ X3 ) ) ).

% alwD
thf(fact_66_alwD,axiom,
    ! [P2: stream2709947120125613254m_rule > $o,X3: stream2709947120125613254m_rule] :
      ( ( linear4957294957385550906m_rule @ P2 @ X3 )
     => ( P2 @ X3 ) ) ).

% alwD
thf(fact_67_alw__nxt,axiom,
    ( linear1087404401031714582w_rule
    = ( ^ [Phi2: stream_rule > $o,Xs: stream_rule] :
          ( ( Phi2 @ Xs )
          & ( linear8837004798957005347rule_o @ ( linear1087404401031714582w_rule @ Phi2 ) @ Xs ) ) ) ) ).

% alw_nxt
thf(fact_68_alw__nxt,axiom,
    ( linear4957294957385550906m_rule
    = ( ^ [Phi2: stream2709947120125613254m_rule > $o,Xs: stream2709947120125613254m_rule] :
          ( ( Phi2 @ Xs )
          & ( linear3460418925085718015rule_o @ ( linear4957294957385550906m_rule @ Phi2 ) @ Xs ) ) ) ) ).

% alw_nxt
thf(fact_69_alw__alwD,axiom,
    ! [P2: stream_rule > $o,Omega2: stream_rule] :
      ( ( linear1087404401031714582w_rule @ P2 @ Omega2 )
     => ( linear1087404401031714582w_rule @ ( linear1087404401031714582w_rule @ P2 ) @ Omega2 ) ) ).

% alw_alwD
thf(fact_70_alw__alwD,axiom,
    ! [P2: stream2709947120125613254m_rule > $o,Omega2: stream2709947120125613254m_rule] :
      ( ( linear4957294957385550906m_rule @ P2 @ Omega2 )
     => ( linear4957294957385550906m_rule @ ( linear4957294957385550906m_rule @ P2 ) @ Omega2 ) ) ).

% alw_alwD
thf(fact_71_alw__cong,axiom,
    ! [P2: stream_rule > $o,Omega2: stream_rule,Q1: stream_rule > $o,Q2: stream_rule > $o] :
      ( ( linear1087404401031714582w_rule @ P2 @ Omega2 )
     => ( ! [Omega3: stream_rule] :
            ( ( P2 @ Omega3 )
           => ( ( Q1 @ Omega3 )
              = ( Q2 @ Omega3 ) ) )
       => ( ( linear1087404401031714582w_rule @ Q1 @ Omega2 )
          = ( linear1087404401031714582w_rule @ Q2 @ Omega2 ) ) ) ) ).

% alw_cong
thf(fact_72_alw__cong,axiom,
    ! [P2: stream2709947120125613254m_rule > $o,Omega2: stream2709947120125613254m_rule,Q1: stream2709947120125613254m_rule > $o,Q2: stream2709947120125613254m_rule > $o] :
      ( ( linear4957294957385550906m_rule @ P2 @ Omega2 )
     => ( ! [Omega3: stream2709947120125613254m_rule] :
            ( ( P2 @ Omega3 )
           => ( ( Q1 @ Omega3 )
              = ( Q2 @ Omega3 ) ) )
       => ( ( linear4957294957385550906m_rule @ Q1 @ Omega2 )
          = ( linear4957294957385550906m_rule @ Q2 @ Omega2 ) ) ) ) ).

% alw_cong
thf(fact_73_alw__mono,axiom,
    ! [Phi: stream_rule > $o,Xs2: stream_rule,Psi: stream_rule > $o] :
      ( ( linear1087404401031714582w_rule @ Phi @ Xs2 )
     => ( ! [Xs3: stream_rule] :
            ( ( Phi @ Xs3 )
           => ( Psi @ Xs3 ) )
       => ( linear1087404401031714582w_rule @ Psi @ Xs2 ) ) ) ).

% alw_mono
thf(fact_74_alw__mono,axiom,
    ! [Phi: stream2709947120125613254m_rule > $o,Xs2: stream2709947120125613254m_rule,Psi: stream2709947120125613254m_rule > $o] :
      ( ( linear4957294957385550906m_rule @ Phi @ Xs2 )
     => ( ! [Xs3: stream2709947120125613254m_rule] :
            ( ( Phi @ Xs3 )
           => ( Psi @ Xs3 ) )
       => ( linear4957294957385550906m_rule @ Psi @ Xs2 ) ) ) ).

% alw_mono
thf(fact_75_nxt__mono,axiom,
    ! [Phi: stream2709947120125613254m_rule > $o,Xs2: stream2709947120125613254m_rule,Psi: stream2709947120125613254m_rule > $o] :
      ( ( linear3460418925085718015rule_o @ Phi @ Xs2 )
     => ( ! [Xs3: stream2709947120125613254m_rule] :
            ( ( Phi @ Xs3 )
           => ( Psi @ Xs3 ) )
       => ( linear3460418925085718015rule_o @ Psi @ Xs2 ) ) ) ).

% nxt_mono
thf(fact_76_alw__invar,axiom,
    ! [Phi: stream_rule > $o,Xs2: stream_rule] :
      ( ( Phi @ Xs2 )
     => ( ( linear1087404401031714582w_rule
          @ ^ [Xs: stream_rule] :
              ( ( Phi @ Xs )
             => ( linear8837004798957005347rule_o @ Phi @ Xs ) )
          @ Xs2 )
       => ( linear1087404401031714582w_rule @ Phi @ Xs2 ) ) ) ).

% alw_invar
thf(fact_77_alw__invar,axiom,
    ! [Phi: stream2709947120125613254m_rule > $o,Xs2: stream2709947120125613254m_rule] :
      ( ( Phi @ Xs2 )
     => ( ( linear4957294957385550906m_rule
          @ ^ [Xs: stream2709947120125613254m_rule] :
              ( ( Phi @ Xs )
             => ( linear3460418925085718015rule_o @ Phi @ Xs ) )
          @ Xs2 )
       => ( linear4957294957385550906m_rule @ Phi @ Xs2 ) ) ) ).

% alw_invar
thf(fact_78_not__suntil,axiom,
    ! [P3: stream_rule > $o,Q3: stream_rule > $o,Omega2: stream_rule] :
      ( ( ~ ( linear1836410492716652551l_rule @ P3 @ Q3 @ Omega2 ) )
      = ( ~ ( linear6513964619561280712L_rule @ P3 @ Q3 @ Omega2 )
        | ( linear1087404401031714582w_rule
          @ ^ [Xs: stream_rule] :
              ~ ( Q3 @ Xs )
          @ Omega2 ) ) ) ).

% not_suntil
thf(fact_79_not__suntil,axiom,
    ! [P3: stream2709947120125613254m_rule > $o,Q3: stream2709947120125613254m_rule > $o,Omega2: stream2709947120125613254m_rule] :
      ( ( ~ ( linear5457325955354885675m_rule @ P3 @ Q3 @ Omega2 ) )
      = ( ~ ( linear8167355876404644844m_rule @ P3 @ Q3 @ Omega2 )
        | ( linear4957294957385550906m_rule
          @ ^ [Xs: stream2709947120125613254m_rule] :
              ~ ( Q3 @ Xs )
          @ Omega2 ) ) ) ).

% not_suntil
thf(fact_80_all__imp__alw,axiom,
    ! [Phi: stream_rule > $o,Xs2: stream_rule] :
      ( ! [X_1: stream_rule] : ( Phi @ X_1 )
     => ( linear1087404401031714582w_rule @ Phi @ Xs2 ) ) ).

% all_imp_alw
thf(fact_81_all__imp__alw,axiom,
    ! [Phi: stream2709947120125613254m_rule > $o,Xs2: stream2709947120125613254m_rule] :
      ( ! [X_1: stream2709947120125613254m_rule] : ( Phi @ X_1 )
     => ( linear4957294957385550906m_rule @ Phi @ Xs2 ) ) ).

% all_imp_alw
thf(fact_82_suntil__cong,axiom,
    ! [P2: stream_rule > $o,Omega2: stream_rule,Q1: stream_rule > $o,Q2: stream_rule > $o,R1: stream_rule > $o,R2: stream_rule > $o] :
      ( ( linear1087404401031714582w_rule @ P2 @ Omega2 )
     => ( ! [Omega3: stream_rule] :
            ( ( P2 @ Omega3 )
           => ( ( Q1 @ Omega3 )
              = ( Q2 @ Omega3 ) ) )
       => ( ! [Omega3: stream_rule] :
              ( ( P2 @ Omega3 )
             => ( ( R1 @ Omega3 )
                = ( R2 @ Omega3 ) ) )
         => ( ( linear1836410492716652551l_rule @ Q1 @ R1 @ Omega2 )
            = ( linear1836410492716652551l_rule @ Q2 @ R2 @ Omega2 ) ) ) ) ) ).

% suntil_cong
thf(fact_83_suntil__cong,axiom,
    ! [P2: stream2709947120125613254m_rule > $o,Omega2: stream2709947120125613254m_rule,Q1: stream2709947120125613254m_rule > $o,Q2: stream2709947120125613254m_rule > $o,R1: stream2709947120125613254m_rule > $o,R2: stream2709947120125613254m_rule > $o] :
      ( ( linear4957294957385550906m_rule @ P2 @ Omega2 )
     => ( ! [Omega3: stream2709947120125613254m_rule] :
            ( ( P2 @ Omega3 )
           => ( ( Q1 @ Omega3 )
              = ( Q2 @ Omega3 ) ) )
       => ( ! [Omega3: stream2709947120125613254m_rule] :
              ( ( P2 @ Omega3 )
             => ( ( R1 @ Omega3 )
                = ( R2 @ Omega3 ) ) )
         => ( ( linear5457325955354885675m_rule @ Q1 @ R1 @ Omega2 )
            = ( linear5457325955354885675m_rule @ Q2 @ R2 @ Omega2 ) ) ) ) ) ).

% suntil_cong
thf(fact_84_suntil__mono,axiom,
    ! [P2: stream_rule > $o,Q1: stream_rule > $o,Q2: stream_rule > $o,R1: stream_rule > $o,R2: stream_rule > $o,Omega2: stream_rule] :
      ( ! [Omega3: stream_rule] :
          ( ( P2 @ Omega3 )
         => ( ( Q1 @ Omega3 )
           => ( Q2 @ Omega3 ) ) )
     => ( ! [Omega3: stream_rule] :
            ( ( P2 @ Omega3 )
           => ( ( R1 @ Omega3 )
             => ( R2 @ Omega3 ) ) )
       => ( ( linear1836410492716652551l_rule @ Q1 @ R1 @ Omega2 )
         => ( ( linear1087404401031714582w_rule @ P2 @ Omega2 )
           => ( linear1836410492716652551l_rule @ Q2 @ R2 @ Omega2 ) ) ) ) ) ).

% suntil_mono
thf(fact_85_suntil__mono,axiom,
    ! [P2: stream2709947120125613254m_rule > $o,Q1: stream2709947120125613254m_rule > $o,Q2: stream2709947120125613254m_rule > $o,R1: stream2709947120125613254m_rule > $o,R2: stream2709947120125613254m_rule > $o,Omega2: stream2709947120125613254m_rule] :
      ( ! [Omega3: stream2709947120125613254m_rule] :
          ( ( P2 @ Omega3 )
         => ( ( Q1 @ Omega3 )
           => ( Q2 @ Omega3 ) ) )
     => ( ! [Omega3: stream2709947120125613254m_rule] :
            ( ( P2 @ Omega3 )
           => ( ( R1 @ Omega3 )
             => ( R2 @ Omega3 ) ) )
       => ( ( linear5457325955354885675m_rule @ Q1 @ R1 @ Omega2 )
         => ( ( linear4957294957385550906m_rule @ P2 @ Omega2 )
           => ( linear5457325955354885675m_rule @ Q2 @ R2 @ Omega2 ) ) ) ) ) ).

% suntil_mono
thf(fact_86_until__false,axiom,
    ! [Phi: stream_rule > $o] :
      ( ( linear6513964619561280712L_rule @ Phi
        @ ^ [Xs: stream_rule] : $false )
      = ( linear1087404401031714582w_rule @ Phi ) ) ).

% until_false
thf(fact_87_until__false,axiom,
    ! [Phi: stream2709947120125613254m_rule > $o] :
      ( ( linear8167355876404644844m_rule @ Phi
        @ ^ [Xs: stream2709947120125613254m_rule] : $false )
      = ( linear4957294957385550906m_rule @ Phi ) ) ).

% until_false
thf(fact_88_mem__Collect__eq,axiom,
    ! [A: produc6018962875968178549ist_fm,P2: produc6018962875968178549ist_fm > $o] :
      ( ( member4699826688122452638ist_fm @ A @ ( collec5845330005437105888ist_fm @ P2 ) )
      = ( P2 @ A ) ) ).

% mem_Collect_eq
thf(fact_89_mem__Collect__eq,axiom,
    ! [A: rule,P2: rule > $o] :
      ( ( member_rule @ A @ ( collect_rule @ P2 ) )
      = ( P2 @ A ) ) ).

% mem_Collect_eq
thf(fact_90_mem__Collect__eq,axiom,
    ! [A: produc859450856879609959at_nat,P2: produc859450856879609959at_nat > $o] :
      ( ( member8206827879206165904at_nat @ A @ ( collec7088162979684241874at_nat @ P2 ) )
      = ( P2 @ A ) ) ).

% mem_Collect_eq
thf(fact_91_mem__Collect__eq,axiom,
    ! [A: nat,P2: nat > $o] :
      ( ( member_nat @ A @ ( collect_nat @ P2 ) )
      = ( P2 @ A ) ) ).

% mem_Collect_eq
thf(fact_92_Collect__mem__eq,axiom,
    ! [A2: set_Pr5202636777678657877ist_fm] :
      ( ( collec5845330005437105888ist_fm
        @ ^ [X: produc6018962875968178549ist_fm] : ( member4699826688122452638ist_fm @ X @ A2 ) )
      = A2 ) ).

% Collect_mem_eq
thf(fact_93_Collect__mem__eq,axiom,
    ! [A2: set_rule] :
      ( ( collect_rule
        @ ^ [X: rule] : ( member_rule @ X @ A2 ) )
      = A2 ) ).

% Collect_mem_eq
thf(fact_94_Collect__mem__eq,axiom,
    ! [A2: set_Pr8693737435421807431at_nat] :
      ( ( collec7088162979684241874at_nat
        @ ^ [X: produc859450856879609959at_nat] : ( member8206827879206165904at_nat @ X @ A2 ) )
      = A2 ) ).

% Collect_mem_eq
thf(fact_95_Collect__mem__eq,axiom,
    ! [A2: set_nat] :
      ( ( collect_nat
        @ ^ [X: nat] : ( member_nat @ X @ A2 ) )
      = A2 ) ).

% Collect_mem_eq
thf(fact_96_Collect__cong,axiom,
    ! [P2: nat > $o,Q: nat > $o] :
      ( ! [X2: nat] :
          ( ( P2 @ X2 )
          = ( Q @ X2 ) )
     => ( ( collect_nat @ P2 )
        = ( collect_nat @ Q ) ) ) ).

% Collect_cong
thf(fact_97_suntil__aand__nxt,axiom,
    ! [Phi: stream2709947120125613254m_rule > $o,Psi: stream2709947120125613254m_rule > $o,Omega2: stream2709947120125613254m_rule] :
      ( ( linear5457325955354885675m_rule @ Phi
        @ ^ [Xs: stream2709947120125613254m_rule] :
            ( ( Phi @ Xs )
            & ( linear3460418925085718015rule_o @ Psi @ Xs ) )
        @ Omega2 )
      = ( ( Phi @ Omega2 )
        & ( linear3460418925085718015rule_o @ ( linear5457325955354885675m_rule @ Phi @ Psi ) @ Omega2 ) ) ) ).

% suntil_aand_nxt
thf(fact_98_until__as__suntil,axiom,
    ( linear6513964619561280712L_rule
    = ( ^ [Phi2: stream_rule > $o,Psi2: stream_rule > $o,Omega: stream_rule] :
          ( ( linear1836410492716652551l_rule @ Phi2 @ Psi2 @ Omega )
          | ( linear1087404401031714582w_rule @ Phi2 @ Omega ) ) ) ) ).

% until_as_suntil
thf(fact_99_until__as__suntil,axiom,
    ( linear8167355876404644844m_rule
    = ( ^ [Phi2: stream2709947120125613254m_rule > $o,Psi2: stream2709947120125613254m_rule > $o,Omega: stream2709947120125613254m_rule] :
          ( ( linear5457325955354885675m_rule @ Phi2 @ Psi2 @ Omega )
          | ( linear4957294957385550906m_rule @ Phi2 @ Omega ) ) ) ) ).

% until_as_suntil
thf(fact_100_alw__implies__until,axiom,
    ! [Phi: stream_rule > $o,Omega2: stream_rule,Psi: stream_rule > $o] :
      ( ( linear1087404401031714582w_rule @ Phi @ Omega2 )
     => ( linear6513964619561280712L_rule @ Phi @ Psi @ Omega2 ) ) ).

% alw_implies_until
thf(fact_101_alw__implies__until,axiom,
    ! [Phi: stream2709947120125613254m_rule > $o,Omega2: stream2709947120125613254m_rule,Psi: stream2709947120125613254m_rule > $o] :
      ( ( linear4957294957385550906m_rule @ Phi @ Omega2 )
     => ( linear8167355876404644844m_rule @ Phi @ Psi @ Omega2 ) ) ).

% alw_implies_until
thf(fact_102_alw__False,axiom,
    ! [Omega2: stream_rule] :
      ~ ( linear1087404401031714582w_rule
        @ ^ [X: stream_rule] : $false
        @ Omega2 ) ).

% alw_False
thf(fact_103_alw__False,axiom,
    ! [Omega2: stream2709947120125613254m_rule] :
      ~ ( linear4957294957385550906m_rule
        @ ^ [X: stream2709947120125613254m_rule] : $false
        @ Omega2 ) ).

% alw_False
thf(fact_104_alw__aand,axiom,
    ! [Phi: stream_rule > $o,Psi: stream_rule > $o] :
      ( ( linear1087404401031714582w_rule
        @ ^ [Xs: stream_rule] :
            ( ( Phi @ Xs )
            & ( Psi @ Xs ) ) )
      = ( ^ [Xs: stream_rule] :
            ( ( linear1087404401031714582w_rule @ Phi @ Xs )
            & ( linear1087404401031714582w_rule @ Psi @ Xs ) ) ) ) ).

% alw_aand
thf(fact_105_alw__aand,axiom,
    ! [Phi: stream2709947120125613254m_rule > $o,Psi: stream2709947120125613254m_rule > $o] :
      ( ( linear4957294957385550906m_rule
        @ ^ [Xs: stream2709947120125613254m_rule] :
            ( ( Phi @ Xs )
            & ( Psi @ Xs ) ) )
      = ( ^ [Xs: stream2709947120125613254m_rule] :
            ( ( linear4957294957385550906m_rule @ Phi @ Xs )
            & ( linear4957294957385550906m_rule @ Psi @ Xs ) ) ) ) ).

% alw_aand
thf(fact_106_alw__mp,axiom,
    ! [Phi: stream_rule > $o,Xs2: stream_rule,Psi: stream_rule > $o] :
      ( ( linear1087404401031714582w_rule @ Phi @ Xs2 )
     => ( ( linear1087404401031714582w_rule
          @ ^ [Xs: stream_rule] :
              ( ( Phi @ Xs )
             => ( Psi @ Xs ) )
          @ Xs2 )
       => ( linear1087404401031714582w_rule @ Psi @ Xs2 ) ) ) ).

% alw_mp
thf(fact_107_alw__mp,axiom,
    ! [Phi: stream2709947120125613254m_rule > $o,Xs2: stream2709947120125613254m_rule,Psi: stream2709947120125613254m_rule > $o] :
      ( ( linear4957294957385550906m_rule @ Phi @ Xs2 )
     => ( ( linear4957294957385550906m_rule
          @ ^ [Xs: stream2709947120125613254m_rule] :
              ( ( Phi @ Xs )
             => ( Psi @ Xs ) )
          @ Xs2 )
       => ( linear4957294957385550906m_rule @ Psi @ Xs2 ) ) ) ).

% alw_mp
thf(fact_108_until__ev__suntil,axiom,
    ! [Phi: stream2709947120125613254m_rule > $o,Psi: stream2709947120125613254m_rule > $o,Omega2: stream2709947120125613254m_rule] :
      ( ( linear8167355876404644844m_rule @ Phi @ Psi @ Omega2 )
     => ( ( linear1816559245158234865m_rule @ Psi @ Omega2 )
       => ( linear5457325955354885675m_rule @ Phi @ Psi @ Omega2 ) ) ) ).

% until_ev_suntil
thf(fact_109_until__ev__suntil,axiom,
    ! [Phi: stream_rule > $o,Psi: stream_rule > $o,Omega2: stream_rule] :
      ( ( linear6513964619561280712L_rule @ Phi @ Psi @ Omega2 )
     => ( ( linear292482573288640205v_rule @ Psi @ Omega2 )
       => ( linear1836410492716652551l_rule @ Phi @ Psi @ Omega2 ) ) ) ).

% until_ev_suntil
thf(fact_110_suntil__as__until,axiom,
    ( linear5457325955354885675m_rule
    = ( ^ [Phi2: stream2709947120125613254m_rule > $o,Psi2: stream2709947120125613254m_rule > $o,Omega: stream2709947120125613254m_rule] :
          ( ( linear8167355876404644844m_rule @ Phi2 @ Psi2 @ Omega )
          & ( linear1816559245158234865m_rule @ Psi2 @ Omega ) ) ) ) ).

% suntil_as_until
thf(fact_111_suntil__as__until,axiom,
    ( linear1836410492716652551l_rule
    = ( ^ [Phi2: stream_rule > $o,Psi2: stream_rule > $o,Omega: stream_rule] :
          ( ( linear6513964619561280712L_rule @ Phi2 @ Psi2 @ Omega )
          & ( linear292482573288640205v_rule @ Psi2 @ Omega ) ) ) ) ).

% suntil_as_until
thf(fact_112_variance,axiom,
    ! [Phi: stream_rule > $o,Xs2: stream_rule,Psi: stream_rule > $o] :
      ( ( Phi @ Xs2 )
     => ( ( linear1087404401031714582w_rule
          @ ^ [Xs: stream_rule] :
              ( ( Phi @ Xs )
             => ( ( Psi @ Xs )
                | ( linear8837004798957005347rule_o @ Phi @ Xs ) ) )
          @ Xs2 )
       => ( ( linear1087404401031714582w_rule @ Phi @ Xs2 )
          | ( linear292482573288640205v_rule @ Psi @ Xs2 ) ) ) ) ).

% variance
thf(fact_113_variance,axiom,
    ! [Phi: stream2709947120125613254m_rule > $o,Xs2: stream2709947120125613254m_rule,Psi: stream2709947120125613254m_rule > $o] :
      ( ( Phi @ Xs2 )
     => ( ( linear4957294957385550906m_rule
          @ ^ [Xs: stream2709947120125613254m_rule] :
              ( ( Phi @ Xs )
             => ( ( Psi @ Xs )
                | ( linear3460418925085718015rule_o @ Phi @ Xs ) ) )
          @ Xs2 )
       => ( ( linear4957294957385550906m_rule @ Phi @ Xs2 )
          | ( linear1816559245158234865m_rule @ Psi @ Xs2 ) ) ) ) ).

% variance
thf(fact_114_ev__alw__imp__nxt,axiom,
    ! [Phi: stream_rule > $o,Xs2: stream_rule] :
      ( ( linear292482573288640205v_rule @ Phi @ Xs2 )
     => ( ( linear1087404401031714582w_rule
          @ ^ [Xs: stream_rule] :
              ( ( Phi @ Xs )
             => ( linear8837004798957005347rule_o @ Phi @ Xs ) )
          @ Xs2 )
       => ( linear292482573288640205v_rule @ ( linear1087404401031714582w_rule @ Phi ) @ Xs2 ) ) ) ).

% ev_alw_imp_nxt
thf(fact_115_ev__alw__imp__nxt,axiom,
    ! [Phi: stream2709947120125613254m_rule > $o,Xs2: stream2709947120125613254m_rule] :
      ( ( linear1816559245158234865m_rule @ Phi @ Xs2 )
     => ( ( linear4957294957385550906m_rule
          @ ^ [Xs: stream2709947120125613254m_rule] :
              ( ( Phi @ Xs )
             => ( linear3460418925085718015rule_o @ Phi @ Xs ) )
          @ Xs2 )
       => ( linear1816559245158234865m_rule @ ( linear4957294957385550906m_rule @ Phi ) @ Xs2 ) ) ) ).

% ev_alw_imp_nxt
thf(fact_116_ev__cong,axiom,
    ! [P2: stream_rule > $o,Omega2: stream_rule,Q1: stream_rule > $o,Q2: stream_rule > $o] :
      ( ( linear1087404401031714582w_rule @ P2 @ Omega2 )
     => ( ! [Omega3: stream_rule] :
            ( ( P2 @ Omega3 )
           => ( ( Q1 @ Omega3 )
              = ( Q2 @ Omega3 ) ) )
       => ( ( linear292482573288640205v_rule @ Q1 @ Omega2 )
          = ( linear292482573288640205v_rule @ Q2 @ Omega2 ) ) ) ) ).

% ev_cong
thf(fact_117_ev__cong,axiom,
    ! [P2: stream2709947120125613254m_rule > $o,Omega2: stream2709947120125613254m_rule,Q1: stream2709947120125613254m_rule > $o,Q2: stream2709947120125613254m_rule > $o] :
      ( ( linear4957294957385550906m_rule @ P2 @ Omega2 )
     => ( ! [Omega3: stream2709947120125613254m_rule] :
            ( ( P2 @ Omega3 )
           => ( ( Q1 @ Omega3 )
              = ( Q2 @ Omega3 ) ) )
       => ( ( linear1816559245158234865m_rule @ Q1 @ Omega2 )
          = ( linear1816559245158234865m_rule @ Q2 @ Omega2 ) ) ) ) ).

% ev_cong
thf(fact_118_ev__alw__imp__alw__ev,axiom,
    ! [Phi: stream_rule > $o,Xs2: stream_rule] :
      ( ( linear292482573288640205v_rule @ ( linear1087404401031714582w_rule @ Phi ) @ Xs2 )
     => ( linear1087404401031714582w_rule @ ( linear292482573288640205v_rule @ Phi ) @ Xs2 ) ) ).

% ev_alw_imp_alw_ev
thf(fact_119_ev__alw__imp__alw__ev,axiom,
    ! [Phi: stream2709947120125613254m_rule > $o,Xs2: stream2709947120125613254m_rule] :
      ( ( linear1816559245158234865m_rule @ ( linear4957294957385550906m_rule @ Phi ) @ Xs2 )
     => ( linear4957294957385550906m_rule @ ( linear1816559245158234865m_rule @ Phi ) @ Xs2 ) ) ).

% ev_alw_imp_alw_ev
thf(fact_120_alw__sdrop,axiom,
    ! [Phi: stream_rule > $o,Xs2: stream_rule,N: nat] :
      ( ( linear1087404401031714582w_rule @ Phi @ Xs2 )
     => ( linear1087404401031714582w_rule @ Phi @ ( sdrop_rule @ N @ Xs2 ) ) ) ).

% alw_sdrop
thf(fact_121_alw__sdrop,axiom,
    ! [Phi: stream2709947120125613254m_rule > $o,Xs2: stream2709947120125613254m_rule,N: nat] :
      ( ( linear4957294957385550906m_rule @ Phi @ Xs2 )
     => ( linear4957294957385550906m_rule @ Phi @ ( sdrop_8169176516188972301m_rule @ N @ Xs2 ) ) ) ).

% alw_sdrop
thf(fact_122_alw__iff__sdrop,axiom,
    ( linear1087404401031714582w_rule
    = ( ^ [P: stream_rule > $o,Omega: stream_rule] :
        ! [M: nat] : ( P @ ( sdrop_rule @ M @ Omega ) ) ) ) ).

% alw_iff_sdrop
thf(fact_123_alw__iff__sdrop,axiom,
    ( linear4957294957385550906m_rule
    = ( ^ [P: stream2709947120125613254m_rule > $o,Omega: stream2709947120125613254m_rule] :
        ! [M: nat] : ( P @ ( sdrop_8169176516188972301m_rule @ M @ Omega ) ) ) ) ).

% alw_iff_sdrop
thf(fact_124_ev__suntil,axiom,
    ! [Phi: stream2709947120125613254m_rule > $o,Psi: stream2709947120125613254m_rule > $o,Omega2: stream2709947120125613254m_rule] :
      ( ( linear5457325955354885675m_rule @ Phi @ Psi @ Omega2 )
     => ( linear1816559245158234865m_rule @ Psi @ Omega2 ) ) ).

% ev_suntil
thf(fact_125_ev__suntil,axiom,
    ! [Phi: stream_rule > $o,Psi: stream_rule > $o,Omega2: stream_rule] :
      ( ( linear1836410492716652551l_rule @ Phi @ Psi @ Omega2 )
     => ( linear292482573288640205v_rule @ Psi @ Omega2 ) ) ).

% ev_suntil
thf(fact_126_ev__suntil__iff,axiom,
    ! [P2: stream2709947120125613254m_rule > $o,Q: stream2709947120125613254m_rule > $o,Omega2: stream2709947120125613254m_rule] :
      ( ( linear1816559245158234865m_rule @ ( linear5457325955354885675m_rule @ P2 @ Q ) @ Omega2 )
      = ( linear1816559245158234865m_rule @ Q @ Omega2 ) ) ).

% ev_suntil_iff
thf(fact_127_ev__suntil__iff,axiom,
    ! [P2: stream_rule > $o,Q: stream_rule > $o,Omega2: stream_rule] :
      ( ( linear292482573288640205v_rule @ ( linear1836410492716652551l_rule @ P2 @ Q ) @ Omega2 )
      = ( linear292482573288640205v_rule @ Q @ Omega2 ) ) ).

% ev_suntil_iff
thf(fact_128_not__ev,axiom,
    ! [Phi: stream_rule > $o] :
      ( ( ^ [Xs: stream_rule] :
            ~ ( linear292482573288640205v_rule @ Phi @ Xs ) )
      = ( linear1087404401031714582w_rule
        @ ^ [Xs: stream_rule] :
            ~ ( Phi @ Xs ) ) ) ).

% not_ev
thf(fact_129_not__ev,axiom,
    ! [Phi: stream2709947120125613254m_rule > $o] :
      ( ( ^ [Xs: stream2709947120125613254m_rule] :
            ~ ( linear1816559245158234865m_rule @ Phi @ Xs ) )
      = ( linear4957294957385550906m_rule
        @ ^ [Xs: stream2709947120125613254m_rule] :
            ~ ( Phi @ Xs ) ) ) ).

% not_ev
thf(fact_130_not__alw,axiom,
    ! [Phi: stream_rule > $o] :
      ( ( ^ [Xs: stream_rule] :
            ~ ( linear1087404401031714582w_rule @ Phi @ Xs ) )
      = ( linear292482573288640205v_rule
        @ ^ [Xs: stream_rule] :
            ~ ( Phi @ Xs ) ) ) ).

% not_alw
thf(fact_131_not__alw,axiom,
    ! [Phi: stream2709947120125613254m_rule > $o] :
      ( ( ^ [Xs: stream2709947120125613254m_rule] :
            ~ ( linear4957294957385550906m_rule @ Phi @ Xs ) )
      = ( linear1816559245158234865m_rule
        @ ^ [Xs: stream2709947120125613254m_rule] :
            ~ ( Phi @ Xs ) ) ) ).

% not_alw
thf(fact_132_not__ev__iff,axiom,
    ! [P2: stream_rule > $o,Omega2: stream_rule] :
      ( ( ~ ( linear292482573288640205v_rule @ P2 @ Omega2 ) )
      = ( linear1087404401031714582w_rule
        @ ^ [Xs: stream_rule] :
            ~ ( P2 @ Xs )
        @ Omega2 ) ) ).

% not_ev_iff
thf(fact_133_not__ev__iff,axiom,
    ! [P2: stream2709947120125613254m_rule > $o,Omega2: stream2709947120125613254m_rule] :
      ( ( ~ ( linear1816559245158234865m_rule @ P2 @ Omega2 ) )
      = ( linear4957294957385550906m_rule
        @ ^ [Xs: stream2709947120125613254m_rule] :
            ~ ( P2 @ Xs )
        @ Omega2 ) ) ).

% not_ev_iff
thf(fact_134_ev__alw__aand,axiom,
    ! [Phi: stream_rule > $o,Xs2: stream_rule,Psi: stream_rule > $o] :
      ( ( linear292482573288640205v_rule @ ( linear1087404401031714582w_rule @ Phi ) @ Xs2 )
     => ( ( linear292482573288640205v_rule @ ( linear1087404401031714582w_rule @ Psi ) @ Xs2 )
       => ( linear292482573288640205v_rule
          @ ( linear1087404401031714582w_rule
            @ ^ [Xs: stream_rule] :
                ( ( Phi @ Xs )
                & ( Psi @ Xs ) ) )
          @ Xs2 ) ) ) ).

% ev_alw_aand
thf(fact_135_ev__alw__aand,axiom,
    ! [Phi: stream2709947120125613254m_rule > $o,Xs2: stream2709947120125613254m_rule,Psi: stream2709947120125613254m_rule > $o] :
      ( ( linear1816559245158234865m_rule @ ( linear4957294957385550906m_rule @ Phi ) @ Xs2 )
     => ( ( linear1816559245158234865m_rule @ ( linear4957294957385550906m_rule @ Psi ) @ Xs2 )
       => ( linear1816559245158234865m_rule
          @ ( linear4957294957385550906m_rule
            @ ^ [Xs: stream2709947120125613254m_rule] :
                ( ( Phi @ Xs )
                & ( Psi @ Xs ) ) )
          @ Xs2 ) ) ) ).

% ev_alw_aand
thf(fact_136_ev__alw__impl,axiom,
    ! [Phi: stream_rule > $o,Xs2: stream_rule,Psi: stream_rule > $o] :
      ( ( linear292482573288640205v_rule @ Phi @ Xs2 )
     => ( ( linear1087404401031714582w_rule
          @ ^ [Xs: stream_rule] :
              ( ( Phi @ Xs )
             => ( Psi @ Xs ) )
          @ Xs2 )
       => ( linear292482573288640205v_rule @ Psi @ Xs2 ) ) ) ).

% ev_alw_impl
thf(fact_137_ev__alw__impl,axiom,
    ! [Phi: stream2709947120125613254m_rule > $o,Xs2: stream2709947120125613254m_rule,Psi: stream2709947120125613254m_rule > $o] :
      ( ( linear1816559245158234865m_rule @ Phi @ Xs2 )
     => ( ( linear4957294957385550906m_rule
          @ ^ [Xs: stream2709947120125613254m_rule] :
              ( ( Phi @ Xs )
             => ( Psi @ Xs ) )
          @ Xs2 )
       => ( linear1816559245158234865m_rule @ Psi @ Xs2 ) ) ) ).

% ev_alw_impl
thf(fact_138_not__alw__iff,axiom,
    ! [P2: stream_rule > $o,Omega2: stream_rule] :
      ( ( ~ ( linear1087404401031714582w_rule @ P2 @ Omega2 ) )
      = ( linear292482573288640205v_rule
        @ ^ [Xs: stream_rule] :
            ~ ( P2 @ Xs )
        @ Omega2 ) ) ).

% not_alw_iff
thf(fact_139_not__alw__iff,axiom,
    ! [P2: stream2709947120125613254m_rule > $o,Omega2: stream2709947120125613254m_rule] :
      ( ( ~ ( linear4957294957385550906m_rule @ P2 @ Omega2 ) )
      = ( linear1816559245158234865m_rule
        @ ^ [Xs: stream2709947120125613254m_rule] :
            ~ ( P2 @ Xs )
        @ Omega2 ) ) ).

% not_alw_iff
thf(fact_140_ev__alw__impl__ev,axiom,
    ! [Phi: stream_rule > $o,Xs2: stream_rule,Psi: stream_rule > $o] :
      ( ( linear292482573288640205v_rule @ Phi @ Xs2 )
     => ( ( linear1087404401031714582w_rule
          @ ^ [Xs: stream_rule] :
              ( ( Phi @ Xs )
             => ( linear292482573288640205v_rule @ Psi @ Xs ) )
          @ Xs2 )
       => ( linear292482573288640205v_rule @ Psi @ Xs2 ) ) ) ).

% ev_alw_impl_ev
thf(fact_141_ev__alw__impl__ev,axiom,
    ! [Phi: stream2709947120125613254m_rule > $o,Xs2: stream2709947120125613254m_rule,Psi: stream2709947120125613254m_rule > $o] :
      ( ( linear1816559245158234865m_rule @ Phi @ Xs2 )
     => ( ( linear4957294957385550906m_rule
          @ ^ [Xs: stream2709947120125613254m_rule] :
              ( ( Phi @ Xs )
             => ( linear1816559245158234865m_rule @ Psi @ Xs ) )
          @ Xs2 )
       => ( linear1816559245158234865m_rule @ Psi @ Xs2 ) ) ) ).

% ev_alw_impl_ev
thf(fact_142_alw__alw__impl__ev,axiom,
    ! [Phi: stream_rule > $o,Psi: stream_rule > $o] :
      ( ( linear1087404401031714582w_rule
        @ ^ [Xs: stream_rule] :
            ( ( linear1087404401031714582w_rule @ Phi @ Xs )
           => ( linear292482573288640205v_rule @ Psi @ Xs ) ) )
      = ( ^ [Xs: stream_rule] :
            ( ( linear292482573288640205v_rule @ ( linear1087404401031714582w_rule @ Phi ) @ Xs )
           => ( linear1087404401031714582w_rule @ ( linear292482573288640205v_rule @ Psi ) @ Xs ) ) ) ) ).

% alw_alw_impl_ev
thf(fact_143_alw__alw__impl__ev,axiom,
    ! [Phi: stream2709947120125613254m_rule > $o,Psi: stream2709947120125613254m_rule > $o] :
      ( ( linear4957294957385550906m_rule
        @ ^ [Xs: stream2709947120125613254m_rule] :
            ( ( linear4957294957385550906m_rule @ Phi @ Xs )
           => ( linear1816559245158234865m_rule @ Psi @ Xs ) ) )
      = ( ^ [Xs: stream2709947120125613254m_rule] :
            ( ( linear1816559245158234865m_rule @ ( linear4957294957385550906m_rule @ Phi ) @ Xs )
           => ( linear4957294957385550906m_rule @ ( linear1816559245158234865m_rule @ Psi ) @ Xs ) ) ) ) ).

% alw_alw_impl_ev
thf(fact_144_alw__impl__ev__alw,axiom,
    ! [Phi: stream_rule > $o,Psi: stream_rule > $o,Xs2: stream_rule] :
      ( ( linear1087404401031714582w_rule
        @ ^ [Xs: stream_rule] :
            ( ( Phi @ Xs )
           => ( linear292482573288640205v_rule @ Psi @ Xs ) )
        @ Xs2 )
     => ( linear1087404401031714582w_rule
        @ ^ [Xs: stream_rule] :
            ( ( linear292482573288640205v_rule @ Phi @ Xs )
           => ( linear292482573288640205v_rule @ Psi @ Xs ) )
        @ Xs2 ) ) ).

% alw_impl_ev_alw
thf(fact_145_alw__impl__ev__alw,axiom,
    ! [Phi: stream2709947120125613254m_rule > $o,Psi: stream2709947120125613254m_rule > $o,Xs2: stream2709947120125613254m_rule] :
      ( ( linear4957294957385550906m_rule
        @ ^ [Xs: stream2709947120125613254m_rule] :
            ( ( Phi @ Xs )
           => ( linear1816559245158234865m_rule @ Psi @ Xs ) )
        @ Xs2 )
     => ( linear4957294957385550906m_rule
        @ ^ [Xs: stream2709947120125613254m_rule] :
            ( ( linear1816559245158234865m_rule @ Phi @ Xs )
           => ( linear1816559245158234865m_rule @ Psi @ Xs ) )
        @ Xs2 ) ) ).

% alw_impl_ev_alw
thf(fact_146_ev__alw__alw__impl,axiom,
    ! [Phi: stream_rule > $o,Xs2: stream_rule,Psi: stream_rule > $o] :
      ( ( linear292482573288640205v_rule @ ( linear1087404401031714582w_rule @ Phi ) @ Xs2 )
     => ( ( linear1087404401031714582w_rule
          @ ^ [Xs: stream_rule] :
              ( ( linear1087404401031714582w_rule @ Phi @ Xs )
             => ( linear292482573288640205v_rule @ Psi @ Xs ) )
          @ Xs2 )
       => ( linear292482573288640205v_rule @ Psi @ Xs2 ) ) ) ).

% ev_alw_alw_impl
thf(fact_147_ev__alw__alw__impl,axiom,
    ! [Phi: stream2709947120125613254m_rule > $o,Xs2: stream2709947120125613254m_rule,Psi: stream2709947120125613254m_rule > $o] :
      ( ( linear1816559245158234865m_rule @ ( linear4957294957385550906m_rule @ Phi ) @ Xs2 )
     => ( ( linear4957294957385550906m_rule
          @ ^ [Xs: stream2709947120125613254m_rule] :
              ( ( linear4957294957385550906m_rule @ Phi @ Xs )
             => ( linear1816559245158234865m_rule @ Psi @ Xs ) )
          @ Xs2 )
       => ( linear1816559245158234865m_rule @ Psi @ Xs2 ) ) ) ).

% ev_alw_alw_impl
thf(fact_148_alw__ev__imp__ev__alw,axiom,
    ! [P2: stream_rule > $o,Omega2: stream_rule] :
      ( ( linear1087404401031714582w_rule @ ( linear292482573288640205v_rule @ P2 ) @ Omega2 )
     => ( linear292482573288640205v_rule
        @ ^ [Xs: stream_rule] :
            ( ( P2 @ Xs )
            & ( linear1087404401031714582w_rule @ ( linear292482573288640205v_rule @ P2 ) @ Xs ) )
        @ Omega2 ) ) ).

% alw_ev_imp_ev_alw
thf(fact_149_alw__ev__imp__ev__alw,axiom,
    ! [P2: stream2709947120125613254m_rule > $o,Omega2: stream2709947120125613254m_rule] :
      ( ( linear4957294957385550906m_rule @ ( linear1816559245158234865m_rule @ P2 ) @ Omega2 )
     => ( linear1816559245158234865m_rule
        @ ^ [Xs: stream2709947120125613254m_rule] :
            ( ( P2 @ Xs )
            & ( linear4957294957385550906m_rule @ ( linear1816559245158234865m_rule @ P2 ) @ Xs ) )
        @ Omega2 ) ) ).

% alw_ev_imp_ev_alw
thf(fact_150_dual__order_Orefl,axiom,
    ! [A: nat] : ( ord_less_eq_nat @ A @ A ) ).

% dual_order.refl
thf(fact_151_dual__order_Orefl,axiom,
    ! [A: set_rule] : ( ord_less_eq_set_rule @ A @ A ) ).

% dual_order.refl
thf(fact_152_dual__order_Orefl,axiom,
    ! [A: set_tm] : ( ord_less_eq_set_tm @ A @ A ) ).

% dual_order.refl
thf(fact_153_order__refl,axiom,
    ! [X3: nat] : ( ord_less_eq_nat @ X3 @ X3 ) ).

% order_refl
thf(fact_154_order__refl,axiom,
    ! [X3: set_rule] : ( ord_less_eq_set_rule @ X3 @ X3 ) ).

% order_refl
thf(fact_155_order__refl,axiom,
    ! [X3: set_tm] : ( ord_less_eq_set_tm @ X3 @ X3 ) ).

% order_refl
thf(fact_156_holds__sfilter,axiom,
    ! [Q: produc340336539035504054m_rule > $o,S3: stream2709947120125613254m_rule,P2: produc340336539035504054m_rule > $o] :
      ( ( linear1816559245158234865m_rule @ ( linear7925859179493631106m_rule @ Q ) @ S3 )
     => ( ( linear7925859179493631106m_rule @ P2 @ ( sfilte3304650064355440278m_rule @ Q @ S3 ) )
        = ( linear5457325955354885675m_rule
          @ ^ [Xs: stream2709947120125613254m_rule] :
              ~ ( linear7925859179493631106m_rule @ Q @ Xs )
          @ ( linear7925859179493631106m_rule
            @ ^ [Xs: produc340336539035504054m_rule] :
                ( ( Q @ Xs )
                & ( P2 @ Xs ) ) )
          @ S3 ) ) ) ).

% holds_sfilter
thf(fact_157_holds__sfilter,axiom,
    ! [Q: rule > $o,S3: stream_rule,P2: rule > $o] :
      ( ( linear292482573288640205v_rule @ ( linear7354821883049305950s_rule @ Q ) @ S3 )
     => ( ( linear7354821883049305950s_rule @ P2 @ ( sfilter_rule @ Q @ S3 ) )
        = ( linear1836410492716652551l_rule
          @ ^ [Xs: stream_rule] :
              ~ ( linear7354821883049305950s_rule @ Q @ Xs )
          @ ( linear7354821883049305950s_rule
            @ ^ [Xs: rule] :
                ( ( Q @ Xs )
                & ( P2 @ Xs ) ) )
          @ S3 ) ) ) ).

% holds_sfilter
thf(fact_158_ev__sfilter,axiom,
    ! [P2: produc340336539035504054m_rule > $o,S3: stream2709947120125613254m_rule,Q: stream2709947120125613254m_rule > $o] :
      ( ( linear4957294957385550906m_rule @ ( linear1816559245158234865m_rule @ ( linear7925859179493631106m_rule @ P2 ) ) @ S3 )
     => ( ( linear1816559245158234865m_rule @ Q @ ( sfilte3304650064355440278m_rule @ P2 @ S3 ) )
        = ( linear1816559245158234865m_rule
          @ ^ [X: stream2709947120125613254m_rule] : ( Q @ ( sfilte3304650064355440278m_rule @ P2 @ X ) )
          @ S3 ) ) ) ).

% ev_sfilter
thf(fact_159_ev__sfilter,axiom,
    ! [P2: rule > $o,S3: stream_rule,Q: stream_rule > $o] :
      ( ( linear1087404401031714582w_rule @ ( linear292482573288640205v_rule @ ( linear7354821883049305950s_rule @ P2 ) ) @ S3 )
     => ( ( linear292482573288640205v_rule @ Q @ ( sfilter_rule @ P2 @ S3 ) )
        = ( linear292482573288640205v_rule
          @ ^ [X: stream_rule] : ( Q @ ( sfilter_rule @ P2 @ X ) )
          @ S3 ) ) ) ).

% ev_sfilter
thf(fact_160_alw__sfilter,axiom,
    ! [P2: produc340336539035504054m_rule > $o,S3: stream2709947120125613254m_rule,Q: stream2709947120125613254m_rule > $o] :
      ( ( linear4957294957385550906m_rule @ ( linear1816559245158234865m_rule @ ( linear7925859179493631106m_rule @ P2 ) ) @ S3 )
     => ( ( linear4957294957385550906m_rule @ Q @ ( sfilte3304650064355440278m_rule @ P2 @ S3 ) )
        = ( linear4957294957385550906m_rule
          @ ^ [X: stream2709947120125613254m_rule] : ( Q @ ( sfilte3304650064355440278m_rule @ P2 @ X ) )
          @ S3 ) ) ) ).

% alw_sfilter
thf(fact_161_alw__sfilter,axiom,
    ! [P2: rule > $o,S3: stream_rule,Q: stream_rule > $o] :
      ( ( linear1087404401031714582w_rule @ ( linear292482573288640205v_rule @ ( linear7354821883049305950s_rule @ P2 ) ) @ S3 )
     => ( ( linear1087404401031714582w_rule @ Q @ ( sfilter_rule @ P2 @ S3 ) )
        = ( linear1087404401031714582w_rule
          @ ^ [X: stream_rule] : ( Q @ ( sfilter_rule @ P2 @ X ) )
          @ S3 ) ) ) ).

% alw_sfilter
thf(fact_162_RuleSystem__Defs_OSaturated__def,axiom,
    ( abstra6097777249025082867ist_fm
    = ( ^ [Eff: rule > produc6018962875968178549ist_fm > fset_P8989946509869081563ist_fm > $o,Rules: stream_rule,Steps2: stream2709947120125613254m_rule] :
        ! [X: rule] :
          ( ( member_rule @ X @ ( sset_rule @ Rules ) )
         => ( abstra2533313685312581075ist_fm @ Eff @ X @ Steps2 ) ) ) ) ).

% RuleSystem_Defs.Saturated_def
thf(fact_163_ev__prefix,axiom,
    ! [P2: produc340336539035504054m_rule > $o,Xs2: stream2709947120125613254m_rule] :
      ( ( linear1816559245158234865m_rule @ ( linear7925859179493631106m_rule @ P2 ) @ Xs2 )
     => ? [Pre: list_P2774625669004309958m_rule,Suf: stream2709947120125613254m_rule] :
          ( ( list_a9038620553111681026m_rule
            @ ^ [Xs: produc340336539035504054m_rule] :
                ~ ( P2 @ Xs )
            @ Pre )
          & ( linear7925859179493631106m_rule @ P2 @ Suf )
          & ( Xs2
            = ( shift_2334844276748245581m_rule @ Pre @ Suf ) ) ) ) ).

% ev_prefix
thf(fact_164_ev__prefix,axiom,
    ! [P2: rule > $o,Xs2: stream_rule] :
      ( ( linear292482573288640205v_rule @ ( linear7354821883049305950s_rule @ P2 ) @ Xs2 )
     => ? [Pre: list_rule,Suf: stream_rule] :
          ( ( list_all_rule
            @ ^ [Xs: rule] :
                ~ ( P2 @ Xs )
            @ Pre )
          & ( linear7354821883049305950s_rule @ P2 @ Suf )
          & ( Xs2
            = ( shift_rule @ Pre @ Suf ) ) ) ) ).

% ev_prefix
thf(fact_165_nxt__wait__least,axiom,
    ! [Phi: stream_rule > $o,Xs2: stream_rule,N: nat] :
      ( ( linear292482573288640205v_rule @ Phi @ Xs2 )
     => ( ( compow7134168332496650290rule_o @ N @ linear8837004798957005347rule_o @ Phi @ Xs2 )
       => ( ord_less_eq_nat @ ( linear2710171048243638257t_rule @ Phi @ Xs2 ) @ N ) ) ) ).

% nxt_wait_least
thf(fact_166_nxt__wait__least,axiom,
    ! [Phi: stream2709947120125613254m_rule > $o,Xs2: stream2709947120125613254m_rule,N: nat] :
      ( ( linear1816559245158234865m_rule @ Phi @ Xs2 )
     => ( ( compow6896429539834504498rule_o @ N @ linear3460418925085718015rule_o @ Phi @ Xs2 )
       => ( ord_less_eq_nat @ ( linear8417479883454578709m_rule @ Phi @ Xs2 ) @ N ) ) ) ).

% nxt_wait_least
thf(fact_167_Stream_Osmember__def,axiom,
    ( smember_nat
    = ( ^ [X: nat,S: stream_nat] : ( member_nat @ X @ ( sset_nat @ S ) ) ) ) ).

% Stream.smember_def
thf(fact_168_Stream_Osmember__def,axiom,
    ( smembe2974859016285698739ist_fm
    = ( ^ [X: produc6018962875968178549ist_fm,S: stream4408948924543953275ist_fm] : ( member4699826688122452638ist_fm @ X @ ( sset_P5379989128061332361ist_fm @ S ) ) ) ) ).

% Stream.smember_def
thf(fact_169_Stream_Osmember__def,axiom,
    ( smembe1741262129464730021at_nat
    = ( ^ [X: produc859450856879609959at_nat,S: stream8372878641218411373at_nat] : ( member8206827879206165904at_nat @ X @ ( sset_P9164763103296375419at_nat @ S ) ) ) ) ).

% Stream.smember_def
thf(fact_170_Stream_Osmember__def,axiom,
    ( smember_rule
    = ( ^ [X: rule,S: stream_rule] : ( member_rule @ X @ ( sset_rule @ S ) ) ) ) ).

% Stream.smember_def
thf(fact_171_infinite__iff__alw__ev,axiom,
    ! [P2: stream_rule > $o,Omega2: stream_rule] :
      ( ( ~ ( finite_finite_nat
            @ ( collect_nat
              @ ^ [M: nat] : ( P2 @ ( sdrop_rule @ M @ Omega2 ) ) ) ) )
      = ( linear1087404401031714582w_rule @ ( linear292482573288640205v_rule @ P2 ) @ Omega2 ) ) ).

% infinite_iff_alw_ev
thf(fact_172_infinite__iff__alw__ev,axiom,
    ! [P2: stream2709947120125613254m_rule > $o,Omega2: stream2709947120125613254m_rule] :
      ( ( ~ ( finite_finite_nat
            @ ( collect_nat
              @ ^ [M: nat] : ( P2 @ ( sdrop_8169176516188972301m_rule @ M @ Omega2 ) ) ) ) )
      = ( linear4957294957385550906m_rule @ ( linear1816559245158234865m_rule @ P2 ) @ Omega2 ) ) ).

% infinite_iff_alw_ev
thf(fact_173_RuleSystem__Defs_Osaturated_Ocong,axiom,
    abstra2533313685312581075ist_fm = abstra2533313685312581075ist_fm ).

% RuleSystem_Defs.saturated.cong
thf(fact_174_alw__shift,axiom,
    ! [Phi: stream_rule > $o,Xl: list_rule,Xs2: stream_rule] :
      ( ( linear1087404401031714582w_rule @ Phi @ ( shift_rule @ Xl @ Xs2 ) )
     => ( linear1087404401031714582w_rule @ Phi @ Xs2 ) ) ).

% alw_shift
thf(fact_175_alw__shift,axiom,
    ! [Phi: stream2709947120125613254m_rule > $o,Xl: list_P2774625669004309958m_rule,Xs2: stream2709947120125613254m_rule] :
      ( ( linear4957294957385550906m_rule @ Phi @ ( shift_2334844276748245581m_rule @ Xl @ Xs2 ) )
     => ( linear4957294957385550906m_rule @ Phi @ Xs2 ) ) ).

% alw_shift
thf(fact_176_ev__imp__shift,axiom,
    ! [Phi: stream2709947120125613254m_rule > $o,Xs2: stream2709947120125613254m_rule] :
      ( ( linear1816559245158234865m_rule @ Phi @ Xs2 )
     => ? [Xl2: list_P2774625669004309958m_rule,Xs22: stream2709947120125613254m_rule] :
          ( ( Xs2
            = ( shift_2334844276748245581m_rule @ Xl2 @ Xs22 ) )
          & ( Phi @ Xs22 ) ) ) ).

% ev_imp_shift
thf(fact_177_ev__imp__shift,axiom,
    ! [Phi: stream_rule > $o,Xs2: stream_rule] :
      ( ( linear292482573288640205v_rule @ Phi @ Xs2 )
     => ? [Xl2: list_rule,Xs22: stream_rule] :
          ( ( Xs2
            = ( shift_rule @ Xl2 @ Xs22 ) )
          & ( Phi @ Xs22 ) ) ) ).

% ev_imp_shift
thf(fact_178_ev__shift,axiom,
    ! [Phi: stream2709947120125613254m_rule > $o,Xs2: stream2709947120125613254m_rule,Xl: list_P2774625669004309958m_rule] :
      ( ( linear1816559245158234865m_rule @ Phi @ Xs2 )
     => ( linear1816559245158234865m_rule @ Phi @ ( shift_2334844276748245581m_rule @ Xl @ Xs2 ) ) ) ).

% ev_shift
thf(fact_179_ev__shift,axiom,
    ! [Phi: stream_rule > $o,Xs2: stream_rule,Xl: list_rule] :
      ( ( linear292482573288640205v_rule @ Phi @ Xs2 )
     => ( linear292482573288640205v_rule @ Phi @ ( shift_rule @ Xl @ Xs2 ) ) ) ).

% ev_shift
thf(fact_180_nxt__ev,axiom,
    ! [N: nat,Phi: stream_rule > $o,Xs2: stream_rule] :
      ( ( compow7134168332496650290rule_o @ N @ linear8837004798957005347rule_o @ Phi @ Xs2 )
     => ( linear292482573288640205v_rule @ Phi @ Xs2 ) ) ).

% nxt_ev
thf(fact_181_nxt__ev,axiom,
    ! [N: nat,Phi: stream2709947120125613254m_rule > $o,Xs2: stream2709947120125613254m_rule] :
      ( ( compow6896429539834504498rule_o @ N @ linear3460418925085718015rule_o @ Phi @ Xs2 )
     => ( linear1816559245158234865m_rule @ Phi @ Xs2 ) ) ).

% nxt_ev
thf(fact_182_ev__ex__nxt,axiom,
    ! [Phi: stream_rule > $o,Xs2: stream_rule] :
      ( ( linear292482573288640205v_rule @ Phi @ Xs2 )
     => ? [N2: nat] : ( compow7134168332496650290rule_o @ N2 @ linear8837004798957005347rule_o @ Phi @ Xs2 ) ) ).

% ev_ex_nxt
thf(fact_183_ev__ex__nxt,axiom,
    ! [Phi: stream2709947120125613254m_rule > $o,Xs2: stream2709947120125613254m_rule] :
      ( ( linear1816559245158234865m_rule @ Phi @ Xs2 )
     => ? [N2: nat] : ( compow6896429539834504498rule_o @ N2 @ linear3460418925085718015rule_o @ Phi @ Xs2 ) ) ).

% ev_ex_nxt
thf(fact_184_nxt__sdrop,axiom,
    ! [N: nat,Phi: stream_rule > $o,Xs2: stream_rule] :
      ( ( compow7134168332496650290rule_o @ N @ linear8837004798957005347rule_o @ Phi @ Xs2 )
      = ( Phi @ ( sdrop_rule @ N @ Xs2 ) ) ) ).

% nxt_sdrop
thf(fact_185_nxt__sdrop,axiom,
    ! [N: nat,Phi: stream2709947120125613254m_rule > $o,Xs2: stream2709947120125613254m_rule] :
      ( ( compow6896429539834504498rule_o @ N @ linear3460418925085718015rule_o @ Phi @ Xs2 )
      = ( Phi @ ( sdrop_8169176516188972301m_rule @ N @ Xs2 ) ) ) ).

% nxt_sdrop
thf(fact_186_sset__sdrop,axiom,
    ! [N: nat,S3: stream2709947120125613254m_rule] : ( ord_le6390412330253371084m_rule @ ( sset_P4484857331586881186m_rule @ ( sdrop_8169176516188972301m_rule @ N @ S3 ) ) @ ( sset_P4484857331586881186m_rule @ S3 ) ) ).

% sset_sdrop
thf(fact_187_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_188_sset__sdrop,axiom,
    ! [N: nat,S3: stream_tm] : ( ord_less_eq_set_tm @ ( sset_tm @ ( sdrop_tm @ N @ S3 ) ) @ ( sset_tm @ S3 ) ) ).

% sset_sdrop
thf(fact_189_alw__ev__shift,axiom,
    ! [Phi: stream_rule > $o,Xs1: stream_rule,Xl: list_rule] :
      ( ( linear1087404401031714582w_rule @ Phi @ Xs1 )
     => ( linear292482573288640205v_rule @ ( linear1087404401031714582w_rule @ Phi ) @ ( shift_rule @ Xl @ Xs1 ) ) ) ).

% alw_ev_shift
thf(fact_190_alw__ev__shift,axiom,
    ! [Phi: stream2709947120125613254m_rule > $o,Xs1: stream2709947120125613254m_rule,Xl: list_P2774625669004309958m_rule] :
      ( ( linear4957294957385550906m_rule @ Phi @ Xs1 )
     => ( linear1816559245158234865m_rule @ ( linear4957294957385550906m_rule @ Phi ) @ ( shift_2334844276748245581m_rule @ Xl @ Xs1 ) ) ) ).

% alw_ev_shift
thf(fact_191_stake__sdrop,axiom,
    ! [N: nat,S3: stream2709947120125613254m_rule] :
      ( ( shift_2334844276748245581m_rule @ ( stake_5421812949518764133m_rule @ N @ S3 ) @ ( sdrop_8169176516188972301m_rule @ N @ S3 ) )
      = S3 ) ).

% stake_sdrop
thf(fact_192_stake__sdrop,axiom,
    ! [N: nat,S3: stream_rule] :
      ( ( shift_rule @ ( stake_rule @ N @ S3 ) @ ( sdrop_rule @ N @ S3 ) )
      = S3 ) ).

% stake_sdrop
thf(fact_193_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_194_le__cases3,axiom,
    ! [X3: nat,Y: nat,Z: nat] :
      ( ( ( ord_less_eq_nat @ X3 @ Y )
       => ~ ( ord_less_eq_nat @ Y @ Z ) )
     => ( ( ( ord_less_eq_nat @ Y @ X3 )
         => ~ ( ord_less_eq_nat @ X3 @ Z ) )
       => ( ( ( ord_less_eq_nat @ X3 @ Z )
           => ~ ( ord_less_eq_nat @ Z @ Y ) )
         => ( ( ( ord_less_eq_nat @ Z @ Y )
             => ~ ( ord_less_eq_nat @ Y @ X3 ) )
           => ( ( ( ord_less_eq_nat @ Y @ Z )
               => ~ ( ord_less_eq_nat @ Z @ X3 ) )
             => ~ ( ( ord_less_eq_nat @ Z @ X3 )
                 => ~ ( ord_less_eq_nat @ X3 @ Y ) ) ) ) ) ) ) ).

% le_cases3
thf(fact_195_order__class_Oorder__eq__iff,axiom,
    ( ( ^ [Y2: nat,Z2: nat] : ( Y2 = Z2 ) )
    = ( ^ [X: nat,Y3: nat] :
          ( ( ord_less_eq_nat @ X @ Y3 )
          & ( ord_less_eq_nat @ Y3 @ X ) ) ) ) ).

% order_class.order_eq_iff
thf(fact_196_order__class_Oorder__eq__iff,axiom,
    ( ( ^ [Y2: set_rule,Z2: set_rule] : ( Y2 = Z2 ) )
    = ( ^ [X: set_rule,Y3: set_rule] :
          ( ( ord_less_eq_set_rule @ X @ Y3 )
          & ( ord_less_eq_set_rule @ Y3 @ X ) ) ) ) ).

% order_class.order_eq_iff
thf(fact_197_order__class_Oorder__eq__iff,axiom,
    ( ( ^ [Y2: set_tm,Z2: set_tm] : ( Y2 = Z2 ) )
    = ( ^ [X: set_tm,Y3: set_tm] :
          ( ( ord_less_eq_set_tm @ X @ Y3 )
          & ( ord_less_eq_set_tm @ Y3 @ X ) ) ) ) ).

% order_class.order_eq_iff
thf(fact_198_ord__eq__le__trans,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( ( A = B )
     => ( ( ord_less_eq_nat @ B @ C )
       => ( ord_less_eq_nat @ A @ C ) ) ) ).

% ord_eq_le_trans
thf(fact_199_ord__eq__le__trans,axiom,
    ! [A: set_rule,B: set_rule,C: set_rule] :
      ( ( A = B )
     => ( ( ord_less_eq_set_rule @ B @ C )
       => ( ord_less_eq_set_rule @ A @ C ) ) ) ).

% ord_eq_le_trans
thf(fact_200_ord__eq__le__trans,axiom,
    ! [A: set_tm,B: set_tm,C: set_tm] :
      ( ( A = B )
     => ( ( ord_less_eq_set_tm @ B @ C )
       => ( ord_less_eq_set_tm @ A @ C ) ) ) ).

% ord_eq_le_trans
thf(fact_201_ord__le__eq__trans,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( ( ord_less_eq_nat @ A @ B )
     => ( ( B = C )
       => ( ord_less_eq_nat @ A @ C ) ) ) ).

% ord_le_eq_trans
thf(fact_202_ord__le__eq__trans,axiom,
    ! [A: set_rule,B: set_rule,C: set_rule] :
      ( ( ord_less_eq_set_rule @ A @ B )
     => ( ( B = C )
       => ( ord_less_eq_set_rule @ A @ C ) ) ) ).

% ord_le_eq_trans
thf(fact_203_ord__le__eq__trans,axiom,
    ! [A: set_tm,B: set_tm,C: set_tm] :
      ( ( ord_less_eq_set_tm @ A @ B )
     => ( ( B = C )
       => ( ord_less_eq_set_tm @ A @ C ) ) ) ).

% ord_le_eq_trans
thf(fact_204_order__antisym,axiom,
    ! [X3: nat,Y: nat] :
      ( ( ord_less_eq_nat @ X3 @ Y )
     => ( ( ord_less_eq_nat @ Y @ X3 )
       => ( X3 = Y ) ) ) ).

% order_antisym
thf(fact_205_order__antisym,axiom,
    ! [X3: set_rule,Y: set_rule] :
      ( ( ord_less_eq_set_rule @ X3 @ Y )
     => ( ( ord_less_eq_set_rule @ Y @ X3 )
       => ( X3 = Y ) ) ) ).

% order_antisym
thf(fact_206_order__antisym,axiom,
    ! [X3: set_tm,Y: set_tm] :
      ( ( ord_less_eq_set_tm @ X3 @ Y )
     => ( ( ord_less_eq_set_tm @ Y @ X3 )
       => ( X3 = Y ) ) ) ).

% order_antisym
thf(fact_207_order_Otrans,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( ( ord_less_eq_nat @ A @ B )
     => ( ( ord_less_eq_nat @ B @ C )
       => ( ord_less_eq_nat @ A @ C ) ) ) ).

% order.trans
thf(fact_208_order_Otrans,axiom,
    ! [A: set_rule,B: set_rule,C: set_rule] :
      ( ( ord_less_eq_set_rule @ A @ B )
     => ( ( ord_less_eq_set_rule @ B @ C )
       => ( ord_less_eq_set_rule @ A @ C ) ) ) ).

% order.trans
thf(fact_209_order_Otrans,axiom,
    ! [A: set_tm,B: set_tm,C: set_tm] :
      ( ( ord_less_eq_set_tm @ A @ B )
     => ( ( ord_less_eq_set_tm @ B @ C )
       => ( ord_less_eq_set_tm @ A @ C ) ) ) ).

% order.trans
thf(fact_210_order__trans,axiom,
    ! [X3: nat,Y: nat,Z: nat] :
      ( ( ord_less_eq_nat @ X3 @ Y )
     => ( ( ord_less_eq_nat @ Y @ Z )
       => ( ord_less_eq_nat @ X3 @ Z ) ) ) ).

% order_trans
thf(fact_211_order__trans,axiom,
    ! [X3: set_rule,Y: set_rule,Z: set_rule] :
      ( ( ord_less_eq_set_rule @ X3 @ Y )
     => ( ( ord_less_eq_set_rule @ Y @ Z )
       => ( ord_less_eq_set_rule @ X3 @ Z ) ) ) ).

% order_trans
thf(fact_212_order__trans,axiom,
    ! [X3: set_tm,Y: set_tm,Z: set_tm] :
      ( ( ord_less_eq_set_tm @ X3 @ Y )
     => ( ( ord_less_eq_set_tm @ Y @ Z )
       => ( ord_less_eq_set_tm @ X3 @ Z ) ) ) ).

% order_trans
thf(fact_213_linorder__wlog,axiom,
    ! [P2: nat > nat > $o,A: nat,B: nat] :
      ( ! [A3: nat,B2: nat] :
          ( ( ord_less_eq_nat @ A3 @ B2 )
         => ( P2 @ A3 @ B2 ) )
     => ( ! [A3: nat,B2: nat] :
            ( ( P2 @ B2 @ A3 )
           => ( P2 @ A3 @ B2 ) )
       => ( P2 @ A @ B ) ) ) ).

% linorder_wlog
thf(fact_214_dual__order_Oeq__iff,axiom,
    ( ( ^ [Y2: nat,Z2: nat] : ( Y2 = Z2 ) )
    = ( ^ [A4: nat,B3: nat] :
          ( ( ord_less_eq_nat @ B3 @ A4 )
          & ( ord_less_eq_nat @ A4 @ B3 ) ) ) ) ).

% dual_order.eq_iff
thf(fact_215_dual__order_Oeq__iff,axiom,
    ( ( ^ [Y2: set_rule,Z2: set_rule] : ( Y2 = Z2 ) )
    = ( ^ [A4: set_rule,B3: set_rule] :
          ( ( ord_less_eq_set_rule @ B3 @ A4 )
          & ( ord_less_eq_set_rule @ A4 @ B3 ) ) ) ) ).

% dual_order.eq_iff
thf(fact_216_dual__order_Oeq__iff,axiom,
    ( ( ^ [Y2: set_tm,Z2: set_tm] : ( Y2 = Z2 ) )
    = ( ^ [A4: set_tm,B3: set_tm] :
          ( ( ord_less_eq_set_tm @ B3 @ A4 )
          & ( ord_less_eq_set_tm @ A4 @ B3 ) ) ) ) ).

% dual_order.eq_iff
thf(fact_217_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_218_dual__order_Oantisym,axiom,
    ! [B: set_rule,A: set_rule] :
      ( ( ord_less_eq_set_rule @ B @ A )
     => ( ( ord_less_eq_set_rule @ A @ B )
       => ( A = B ) ) ) ).

% dual_order.antisym
thf(fact_219_dual__order_Oantisym,axiom,
    ! [B: set_tm,A: set_tm] :
      ( ( ord_less_eq_set_tm @ B @ A )
     => ( ( ord_less_eq_set_tm @ A @ B )
       => ( A = B ) ) ) ).

% dual_order.antisym
thf(fact_220_dual__order_Otrans,axiom,
    ! [B: nat,A: nat,C: nat] :
      ( ( ord_less_eq_nat @ B @ A )
     => ( ( ord_less_eq_nat @ C @ B )
       => ( ord_less_eq_nat @ C @ A ) ) ) ).

% dual_order.trans
thf(fact_221_dual__order_Otrans,axiom,
    ! [B: set_rule,A: set_rule,C: set_rule] :
      ( ( ord_less_eq_set_rule @ B @ A )
     => ( ( ord_less_eq_set_rule @ C @ B )
       => ( ord_less_eq_set_rule @ C @ A ) ) ) ).

% dual_order.trans
thf(fact_222_dual__order_Otrans,axiom,
    ! [B: set_tm,A: set_tm,C: set_tm] :
      ( ( ord_less_eq_set_tm @ B @ A )
     => ( ( ord_less_eq_set_tm @ C @ B )
       => ( ord_less_eq_set_tm @ C @ A ) ) ) ).

% dual_order.trans
thf(fact_223_antisym,axiom,
    ! [A: nat,B: nat] :
      ( ( ord_less_eq_nat @ A @ B )
     => ( ( ord_less_eq_nat @ B @ A )
       => ( A = B ) ) ) ).

% antisym
thf(fact_224_antisym,axiom,
    ! [A: set_rule,B: set_rule] :
      ( ( ord_less_eq_set_rule @ A @ B )
     => ( ( ord_less_eq_set_rule @ B @ A )
       => ( A = B ) ) ) ).

% antisym
thf(fact_225_antisym,axiom,
    ! [A: set_tm,B: set_tm] :
      ( ( ord_less_eq_set_tm @ A @ B )
     => ( ( ord_less_eq_set_tm @ B @ A )
       => ( A = B ) ) ) ).

% antisym
thf(fact_226_Orderings_Oorder__eq__iff,axiom,
    ( ( ^ [Y2: nat,Z2: nat] : ( Y2 = Z2 ) )
    = ( ^ [A4: nat,B3: nat] :
          ( ( ord_less_eq_nat @ A4 @ B3 )
          & ( ord_less_eq_nat @ B3 @ A4 ) ) ) ) ).

% Orderings.order_eq_iff
thf(fact_227_Orderings_Oorder__eq__iff,axiom,
    ( ( ^ [Y2: set_rule,Z2: set_rule] : ( Y2 = Z2 ) )
    = ( ^ [A4: set_rule,B3: set_rule] :
          ( ( ord_less_eq_set_rule @ A4 @ B3 )
          & ( ord_less_eq_set_rule @ B3 @ A4 ) ) ) ) ).

% Orderings.order_eq_iff
thf(fact_228_Orderings_Oorder__eq__iff,axiom,
    ( ( ^ [Y2: set_tm,Z2: set_tm] : ( Y2 = Z2 ) )
    = ( ^ [A4: set_tm,B3: set_tm] :
          ( ( ord_less_eq_set_tm @ A4 @ B3 )
          & ( ord_less_eq_set_tm @ B3 @ A4 ) ) ) ) ).

% Orderings.order_eq_iff
thf(fact_229_order__subst1,axiom,
    ! [A: nat,F: nat > nat,B: nat,C: nat] :
      ( ( ord_less_eq_nat @ A @ ( F @ B ) )
     => ( ( ord_less_eq_nat @ B @ C )
       => ( ! [X2: nat,Y4: nat] :
              ( ( ord_less_eq_nat @ X2 @ Y4 )
             => ( ord_less_eq_nat @ ( F @ X2 ) @ ( F @ Y4 ) ) )
         => ( ord_less_eq_nat @ A @ ( F @ C ) ) ) ) ) ).

% order_subst1
thf(fact_230_order__subst1,axiom,
    ! [A: nat,F: set_rule > nat,B: set_rule,C: set_rule] :
      ( ( ord_less_eq_nat @ A @ ( F @ B ) )
     => ( ( ord_less_eq_set_rule @ B @ C )
       => ( ! [X2: set_rule,Y4: set_rule] :
              ( ( ord_less_eq_set_rule @ X2 @ Y4 )
             => ( ord_less_eq_nat @ ( F @ X2 ) @ ( F @ Y4 ) ) )
         => ( ord_less_eq_nat @ A @ ( F @ C ) ) ) ) ) ).

% order_subst1
thf(fact_231_order__subst1,axiom,
    ! [A: nat,F: set_tm > nat,B: set_tm,C: set_tm] :
      ( ( ord_less_eq_nat @ A @ ( F @ B ) )
     => ( ( ord_less_eq_set_tm @ B @ C )
       => ( ! [X2: set_tm,Y4: set_tm] :
              ( ( ord_less_eq_set_tm @ X2 @ Y4 )
             => ( ord_less_eq_nat @ ( F @ X2 ) @ ( F @ Y4 ) ) )
         => ( ord_less_eq_nat @ A @ ( F @ C ) ) ) ) ) ).

% order_subst1
thf(fact_232_order__subst1,axiom,
    ! [A: set_rule,F: nat > set_rule,B: nat,C: nat] :
      ( ( ord_less_eq_set_rule @ A @ ( F @ B ) )
     => ( ( ord_less_eq_nat @ B @ C )
       => ( ! [X2: nat,Y4: nat] :
              ( ( ord_less_eq_nat @ X2 @ Y4 )
             => ( ord_less_eq_set_rule @ ( F @ X2 ) @ ( F @ Y4 ) ) )
         => ( ord_less_eq_set_rule @ A @ ( F @ C ) ) ) ) ) ).

% order_subst1
thf(fact_233_order__subst1,axiom,
    ! [A: set_rule,F: set_rule > set_rule,B: set_rule,C: set_rule] :
      ( ( ord_less_eq_set_rule @ A @ ( F @ B ) )
     => ( ( ord_less_eq_set_rule @ B @ C )
       => ( ! [X2: set_rule,Y4: set_rule] :
              ( ( ord_less_eq_set_rule @ X2 @ Y4 )
             => ( ord_less_eq_set_rule @ ( F @ X2 ) @ ( F @ Y4 ) ) )
         => ( ord_less_eq_set_rule @ A @ ( F @ C ) ) ) ) ) ).

% order_subst1
thf(fact_234_order__subst1,axiom,
    ! [A: set_rule,F: set_tm > set_rule,B: set_tm,C: set_tm] :
      ( ( ord_less_eq_set_rule @ A @ ( F @ B ) )
     => ( ( ord_less_eq_set_tm @ B @ C )
       => ( ! [X2: set_tm,Y4: set_tm] :
              ( ( ord_less_eq_set_tm @ X2 @ Y4 )
             => ( ord_less_eq_set_rule @ ( F @ X2 ) @ ( F @ Y4 ) ) )
         => ( ord_less_eq_set_rule @ A @ ( F @ C ) ) ) ) ) ).

% order_subst1
thf(fact_235_order__subst1,axiom,
    ! [A: set_tm,F: nat > set_tm,B: nat,C: nat] :
      ( ( ord_less_eq_set_tm @ A @ ( F @ B ) )
     => ( ( ord_less_eq_nat @ B @ C )
       => ( ! [X2: nat,Y4: nat] :
              ( ( ord_less_eq_nat @ X2 @ Y4 )
             => ( ord_less_eq_set_tm @ ( F @ X2 ) @ ( F @ Y4 ) ) )
         => ( ord_less_eq_set_tm @ A @ ( F @ C ) ) ) ) ) ).

% order_subst1
thf(fact_236_order__subst1,axiom,
    ! [A: set_tm,F: set_rule > set_tm,B: set_rule,C: set_rule] :
      ( ( ord_less_eq_set_tm @ A @ ( F @ B ) )
     => ( ( ord_less_eq_set_rule @ B @ C )
       => ( ! [X2: set_rule,Y4: set_rule] :
              ( ( ord_less_eq_set_rule @ X2 @ Y4 )
             => ( ord_less_eq_set_tm @ ( F @ X2 ) @ ( F @ Y4 ) ) )
         => ( ord_less_eq_set_tm @ A @ ( F @ C ) ) ) ) ) ).

% order_subst1
thf(fact_237_order__subst1,axiom,
    ! [A: set_tm,F: set_tm > set_tm,B: set_tm,C: set_tm] :
      ( ( ord_less_eq_set_tm @ A @ ( F @ B ) )
     => ( ( ord_less_eq_set_tm @ B @ C )
       => ( ! [X2: set_tm,Y4: set_tm] :
              ( ( ord_less_eq_set_tm @ X2 @ Y4 )
             => ( ord_less_eq_set_tm @ ( F @ X2 ) @ ( F @ Y4 ) ) )
         => ( ord_less_eq_set_tm @ A @ ( F @ C ) ) ) ) ) ).

% order_subst1
thf(fact_238_order__subst2,axiom,
    ! [A: nat,B: nat,F: nat > nat,C: nat] :
      ( ( ord_less_eq_nat @ A @ B )
     => ( ( ord_less_eq_nat @ ( F @ B ) @ C )
       => ( ! [X2: nat,Y4: nat] :
              ( ( ord_less_eq_nat @ X2 @ Y4 )
             => ( ord_less_eq_nat @ ( F @ X2 ) @ ( F @ Y4 ) ) )
         => ( ord_less_eq_nat @ ( F @ A ) @ C ) ) ) ) ).

% order_subst2
thf(fact_239_order__subst2,axiom,
    ! [A: nat,B: nat,F: nat > set_rule,C: set_rule] :
      ( ( ord_less_eq_nat @ A @ B )
     => ( ( ord_less_eq_set_rule @ ( F @ B ) @ C )
       => ( ! [X2: nat,Y4: nat] :
              ( ( ord_less_eq_nat @ X2 @ Y4 )
             => ( ord_less_eq_set_rule @ ( F @ X2 ) @ ( F @ Y4 ) ) )
         => ( ord_less_eq_set_rule @ ( F @ A ) @ C ) ) ) ) ).

% order_subst2
thf(fact_240_order__subst2,axiom,
    ! [A: nat,B: nat,F: nat > set_tm,C: set_tm] :
      ( ( ord_less_eq_nat @ A @ B )
     => ( ( ord_less_eq_set_tm @ ( F @ B ) @ C )
       => ( ! [X2: nat,Y4: nat] :
              ( ( ord_less_eq_nat @ X2 @ Y4 )
             => ( ord_less_eq_set_tm @ ( F @ X2 ) @ ( F @ Y4 ) ) )
         => ( ord_less_eq_set_tm @ ( F @ A ) @ C ) ) ) ) ).

% order_subst2
thf(fact_241_order__subst2,axiom,
    ! [A: set_rule,B: set_rule,F: set_rule > nat,C: nat] :
      ( ( ord_less_eq_set_rule @ A @ B )
     => ( ( ord_less_eq_nat @ ( F @ B ) @ C )
       => ( ! [X2: set_rule,Y4: set_rule] :
              ( ( ord_less_eq_set_rule @ X2 @ Y4 )
             => ( ord_less_eq_nat @ ( F @ X2 ) @ ( F @ Y4 ) ) )
         => ( ord_less_eq_nat @ ( F @ A ) @ C ) ) ) ) ).

% order_subst2
thf(fact_242_order__subst2,axiom,
    ! [A: set_rule,B: set_rule,F: set_rule > set_rule,C: set_rule] :
      ( ( ord_less_eq_set_rule @ A @ B )
     => ( ( ord_less_eq_set_rule @ ( F @ B ) @ C )
       => ( ! [X2: set_rule,Y4: set_rule] :
              ( ( ord_less_eq_set_rule @ X2 @ Y4 )
             => ( ord_less_eq_set_rule @ ( F @ X2 ) @ ( F @ Y4 ) ) )
         => ( ord_less_eq_set_rule @ ( F @ A ) @ C ) ) ) ) ).

% order_subst2
thf(fact_243_order__subst2,axiom,
    ! [A: set_rule,B: set_rule,F: set_rule > set_tm,C: set_tm] :
      ( ( ord_less_eq_set_rule @ A @ B )
     => ( ( ord_less_eq_set_tm @ ( F @ B ) @ C )
       => ( ! [X2: set_rule,Y4: set_rule] :
              ( ( ord_less_eq_set_rule @ X2 @ Y4 )
             => ( ord_less_eq_set_tm @ ( F @ X2 ) @ ( F @ Y4 ) ) )
         => ( ord_less_eq_set_tm @ ( F @ A ) @ C ) ) ) ) ).

% order_subst2
thf(fact_244_order__subst2,axiom,
    ! [A: set_tm,B: set_tm,F: set_tm > nat,C: nat] :
      ( ( ord_less_eq_set_tm @ A @ B )
     => ( ( ord_less_eq_nat @ ( F @ B ) @ C )
       => ( ! [X2: set_tm,Y4: set_tm] :
              ( ( ord_less_eq_set_tm @ X2 @ Y4 )
             => ( ord_less_eq_nat @ ( F @ X2 ) @ ( F @ Y4 ) ) )
         => ( ord_less_eq_nat @ ( F @ A ) @ C ) ) ) ) ).

% order_subst2
thf(fact_245_order__subst2,axiom,
    ! [A: set_tm,B: set_tm,F: set_tm > set_rule,C: set_rule] :
      ( ( ord_less_eq_set_tm @ A @ B )
     => ( ( ord_less_eq_set_rule @ ( F @ B ) @ C )
       => ( ! [X2: set_tm,Y4: set_tm] :
              ( ( ord_less_eq_set_tm @ X2 @ Y4 )
             => ( ord_less_eq_set_rule @ ( F @ X2 ) @ ( F @ Y4 ) ) )
         => ( ord_less_eq_set_rule @ ( F @ A ) @ C ) ) ) ) ).

% order_subst2
thf(fact_246_order__subst2,axiom,
    ! [A: set_tm,B: set_tm,F: set_tm > set_tm,C: set_tm] :
      ( ( ord_less_eq_set_tm @ A @ B )
     => ( ( ord_less_eq_set_tm @ ( F @ B ) @ C )
       => ( ! [X2: set_tm,Y4: set_tm] :
              ( ( ord_less_eq_set_tm @ X2 @ Y4 )
             => ( ord_less_eq_set_tm @ ( F @ X2 ) @ ( F @ Y4 ) ) )
         => ( ord_less_eq_set_tm @ ( F @ A ) @ C ) ) ) ) ).

% order_subst2
thf(fact_247_order__eq__refl,axiom,
    ! [X3: nat,Y: nat] :
      ( ( X3 = Y )
     => ( ord_less_eq_nat @ X3 @ Y ) ) ).

% order_eq_refl
thf(fact_248_order__eq__refl,axiom,
    ! [X3: set_rule,Y: set_rule] :
      ( ( X3 = Y )
     => ( ord_less_eq_set_rule @ X3 @ Y ) ) ).

% order_eq_refl
thf(fact_249_order__eq__refl,axiom,
    ! [X3: set_tm,Y: set_tm] :
      ( ( X3 = Y )
     => ( ord_less_eq_set_tm @ X3 @ Y ) ) ).

% order_eq_refl
thf(fact_250_linorder__linear,axiom,
    ! [X3: nat,Y: nat] :
      ( ( ord_less_eq_nat @ X3 @ Y )
      | ( ord_less_eq_nat @ Y @ X3 ) ) ).

% linorder_linear
thf(fact_251_ord__eq__le__subst,axiom,
    ! [A: nat,F: nat > nat,B: nat,C: nat] :
      ( ( A
        = ( F @ B ) )
     => ( ( ord_less_eq_nat @ B @ C )
       => ( ! [X2: nat,Y4: nat] :
              ( ( ord_less_eq_nat @ X2 @ Y4 )
             => ( ord_less_eq_nat @ ( F @ X2 ) @ ( F @ Y4 ) ) )
         => ( ord_less_eq_nat @ A @ ( F @ C ) ) ) ) ) ).

% ord_eq_le_subst
thf(fact_252_ord__eq__le__subst,axiom,
    ! [A: set_rule,F: nat > set_rule,B: nat,C: nat] :
      ( ( A
        = ( F @ B ) )
     => ( ( ord_less_eq_nat @ B @ C )
       => ( ! [X2: nat,Y4: nat] :
              ( ( ord_less_eq_nat @ X2 @ Y4 )
             => ( ord_less_eq_set_rule @ ( F @ X2 ) @ ( F @ Y4 ) ) )
         => ( ord_less_eq_set_rule @ A @ ( F @ C ) ) ) ) ) ).

% ord_eq_le_subst
thf(fact_253_ord__eq__le__subst,axiom,
    ! [A: set_tm,F: nat > set_tm,B: nat,C: nat] :
      ( ( A
        = ( F @ B ) )
     => ( ( ord_less_eq_nat @ B @ C )
       => ( ! [X2: nat,Y4: nat] :
              ( ( ord_less_eq_nat @ X2 @ Y4 )
             => ( ord_less_eq_set_tm @ ( F @ X2 ) @ ( F @ Y4 ) ) )
         => ( ord_less_eq_set_tm @ A @ ( F @ C ) ) ) ) ) ).

% ord_eq_le_subst
thf(fact_254_ord__eq__le__subst,axiom,
    ! [A: nat,F: set_rule > nat,B: set_rule,C: set_rule] :
      ( ( A
        = ( F @ B ) )
     => ( ( ord_less_eq_set_rule @ B @ C )
       => ( ! [X2: set_rule,Y4: set_rule] :
              ( ( ord_less_eq_set_rule @ X2 @ Y4 )
             => ( ord_less_eq_nat @ ( F @ X2 ) @ ( F @ Y4 ) ) )
         => ( ord_less_eq_nat @ A @ ( F @ C ) ) ) ) ) ).

% ord_eq_le_subst
thf(fact_255_ord__eq__le__subst,axiom,
    ! [A: set_rule,F: set_rule > set_rule,B: set_rule,C: set_rule] :
      ( ( A
        = ( F @ B ) )
     => ( ( ord_less_eq_set_rule @ B @ C )
       => ( ! [X2: set_rule,Y4: set_rule] :
              ( ( ord_less_eq_set_rule @ X2 @ Y4 )
             => ( ord_less_eq_set_rule @ ( F @ X2 ) @ ( F @ Y4 ) ) )
         => ( ord_less_eq_set_rule @ A @ ( F @ C ) ) ) ) ) ).

% ord_eq_le_subst
thf(fact_256_ord__eq__le__subst,axiom,
    ! [A: set_tm,F: set_rule > set_tm,B: set_rule,C: set_rule] :
      ( ( A
        = ( F @ B ) )
     => ( ( ord_less_eq_set_rule @ B @ C )
       => ( ! [X2: set_rule,Y4: set_rule] :
              ( ( ord_less_eq_set_rule @ X2 @ Y4 )
             => ( ord_less_eq_set_tm @ ( F @ X2 ) @ ( F @ Y4 ) ) )
         => ( ord_less_eq_set_tm @ A @ ( F @ C ) ) ) ) ) ).

% ord_eq_le_subst
thf(fact_257_ord__eq__le__subst,axiom,
    ! [A: nat,F: set_tm > nat,B: set_tm,C: set_tm] :
      ( ( A
        = ( F @ B ) )
     => ( ( ord_less_eq_set_tm @ B @ C )
       => ( ! [X2: set_tm,Y4: set_tm] :
              ( ( ord_less_eq_set_tm @ X2 @ Y4 )
             => ( ord_less_eq_nat @ ( F @ X2 ) @ ( F @ Y4 ) ) )
         => ( ord_less_eq_nat @ A @ ( F @ C ) ) ) ) ) ).

% ord_eq_le_subst
thf(fact_258_ord__eq__le__subst,axiom,
    ! [A: set_rule,F: set_tm > set_rule,B: set_tm,C: set_tm] :
      ( ( A
        = ( F @ B ) )
     => ( ( ord_less_eq_set_tm @ B @ C )
       => ( ! [X2: set_tm,Y4: set_tm] :
              ( ( ord_less_eq_set_tm @ X2 @ Y4 )
             => ( ord_less_eq_set_rule @ ( F @ X2 ) @ ( F @ Y4 ) ) )
         => ( ord_less_eq_set_rule @ A @ ( F @ C ) ) ) ) ) ).

% ord_eq_le_subst
thf(fact_259_ord__eq__le__subst,axiom,
    ! [A: set_tm,F: set_tm > set_tm,B: set_tm,C: set_tm] :
      ( ( A
        = ( F @ B ) )
     => ( ( ord_less_eq_set_tm @ B @ C )
       => ( ! [X2: set_tm,Y4: set_tm] :
              ( ( ord_less_eq_set_tm @ X2 @ Y4 )
             => ( ord_less_eq_set_tm @ ( F @ X2 ) @ ( F @ Y4 ) ) )
         => ( ord_less_eq_set_tm @ A @ ( F @ C ) ) ) ) ) ).

% ord_eq_le_subst
thf(fact_260_ord__le__eq__subst,axiom,
    ! [A: nat,B: nat,F: nat > nat,C: nat] :
      ( ( ord_less_eq_nat @ A @ B )
     => ( ( ( F @ B )
          = C )
       => ( ! [X2: nat,Y4: nat] :
              ( ( ord_less_eq_nat @ X2 @ Y4 )
             => ( ord_less_eq_nat @ ( F @ X2 ) @ ( F @ Y4 ) ) )
         => ( ord_less_eq_nat @ ( F @ A ) @ C ) ) ) ) ).

% ord_le_eq_subst
thf(fact_261_ord__le__eq__subst,axiom,
    ! [A: nat,B: nat,F: nat > set_rule,C: set_rule] :
      ( ( ord_less_eq_nat @ A @ B )
     => ( ( ( F @ B )
          = C )
       => ( ! [X2: nat,Y4: nat] :
              ( ( ord_less_eq_nat @ X2 @ Y4 )
             => ( ord_less_eq_set_rule @ ( F @ X2 ) @ ( F @ Y4 ) ) )
         => ( ord_less_eq_set_rule @ ( F @ A ) @ C ) ) ) ) ).

% ord_le_eq_subst
thf(fact_262_ord__le__eq__subst,axiom,
    ! [A: nat,B: nat,F: nat > set_tm,C: set_tm] :
      ( ( ord_less_eq_nat @ A @ B )
     => ( ( ( F @ B )
          = C )
       => ( ! [X2: nat,Y4: nat] :
              ( ( ord_less_eq_nat @ X2 @ Y4 )
             => ( ord_less_eq_set_tm @ ( F @ X2 ) @ ( F @ Y4 ) ) )
         => ( ord_less_eq_set_tm @ ( F @ A ) @ C ) ) ) ) ).

% ord_le_eq_subst
thf(fact_263_ord__le__eq__subst,axiom,
    ! [A: set_rule,B: set_rule,F: set_rule > nat,C: nat] :
      ( ( ord_less_eq_set_rule @ A @ B )
     => ( ( ( F @ B )
          = C )
       => ( ! [X2: set_rule,Y4: set_rule] :
              ( ( ord_less_eq_set_rule @ X2 @ Y4 )
             => ( ord_less_eq_nat @ ( F @ X2 ) @ ( F @ Y4 ) ) )
         => ( ord_less_eq_nat @ ( F @ A ) @ C ) ) ) ) ).

% ord_le_eq_subst
thf(fact_264_ord__le__eq__subst,axiom,
    ! [A: set_rule,B: set_rule,F: set_rule > set_rule,C: set_rule] :
      ( ( ord_less_eq_set_rule @ A @ B )
     => ( ( ( F @ B )
          = C )
       => ( ! [X2: set_rule,Y4: set_rule] :
              ( ( ord_less_eq_set_rule @ X2 @ Y4 )
             => ( ord_less_eq_set_rule @ ( F @ X2 ) @ ( F @ Y4 ) ) )
         => ( ord_less_eq_set_rule @ ( F @ A ) @ C ) ) ) ) ).

% ord_le_eq_subst
thf(fact_265_ord__le__eq__subst,axiom,
    ! [A: set_rule,B: set_rule,F: set_rule > set_tm,C: set_tm] :
      ( ( ord_less_eq_set_rule @ A @ B )
     => ( ( ( F @ B )
          = C )
       => ( ! [X2: set_rule,Y4: set_rule] :
              ( ( ord_less_eq_set_rule @ X2 @ Y4 )
             => ( ord_less_eq_set_tm @ ( F @ X2 ) @ ( F @ Y4 ) ) )
         => ( ord_less_eq_set_tm @ ( F @ A ) @ C ) ) ) ) ).

% ord_le_eq_subst
thf(fact_266_ord__le__eq__subst,axiom,
    ! [A: set_tm,B: set_tm,F: set_tm > nat,C: nat] :
      ( ( ord_less_eq_set_tm @ A @ B )
     => ( ( ( F @ B )
          = C )
       => ( ! [X2: set_tm,Y4: set_tm] :
              ( ( ord_less_eq_set_tm @ X2 @ Y4 )
             => ( ord_less_eq_nat @ ( F @ X2 ) @ ( F @ Y4 ) ) )
         => ( ord_less_eq_nat @ ( F @ A ) @ C ) ) ) ) ).

% ord_le_eq_subst
thf(fact_267_ord__le__eq__subst,axiom,
    ! [A: set_tm,B: set_tm,F: set_tm > set_rule,C: set_rule] :
      ( ( ord_less_eq_set_tm @ A @ B )
     => ( ( ( F @ B )
          = C )
       => ( ! [X2: set_tm,Y4: set_tm] :
              ( ( ord_less_eq_set_tm @ X2 @ Y4 )
             => ( ord_less_eq_set_rule @ ( F @ X2 ) @ ( F @ Y4 ) ) )
         => ( ord_less_eq_set_rule @ ( F @ A ) @ C ) ) ) ) ).

% ord_le_eq_subst
thf(fact_268_ord__le__eq__subst,axiom,
    ! [A: set_tm,B: set_tm,F: set_tm > set_tm,C: set_tm] :
      ( ( ord_less_eq_set_tm @ A @ B )
     => ( ( ( F @ B )
          = C )
       => ( ! [X2: set_tm,Y4: set_tm] :
              ( ( ord_less_eq_set_tm @ X2 @ Y4 )
             => ( ord_less_eq_set_tm @ ( F @ X2 ) @ ( F @ Y4 ) ) )
         => ( ord_less_eq_set_tm @ ( F @ A ) @ C ) ) ) ) ).

% ord_le_eq_subst
thf(fact_269_linorder__le__cases,axiom,
    ! [X3: nat,Y: nat] :
      ( ~ ( ord_less_eq_nat @ X3 @ Y )
     => ( ord_less_eq_nat @ Y @ X3 ) ) ).

% linorder_le_cases
thf(fact_270_order__antisym__conv,axiom,
    ! [Y: nat,X3: nat] :
      ( ( ord_less_eq_nat @ Y @ X3 )
     => ( ( ord_less_eq_nat @ X3 @ Y )
        = ( X3 = Y ) ) ) ).

% order_antisym_conv
thf(fact_271_order__antisym__conv,axiom,
    ! [Y: set_rule,X3: set_rule] :
      ( ( ord_less_eq_set_rule @ Y @ X3 )
     => ( ( ord_less_eq_set_rule @ X3 @ Y )
        = ( X3 = Y ) ) ) ).

% order_antisym_conv
thf(fact_272_order__antisym__conv,axiom,
    ! [Y: set_tm,X3: set_tm] :
      ( ( ord_less_eq_set_tm @ Y @ X3 )
     => ( ( ord_less_eq_set_tm @ X3 @ Y )
        = ( X3 = Y ) ) ) ).

% order_antisym_conv
thf(fact_273_nxt__wait,axiom,
    ! [Phi: stream_rule > $o,Xs2: stream_rule] :
      ( ( linear292482573288640205v_rule @ Phi @ Xs2 )
     => ( compow7134168332496650290rule_o @ ( linear2710171048243638257t_rule @ Phi @ Xs2 ) @ linear8837004798957005347rule_o @ Phi @ Xs2 ) ) ).

% nxt_wait
thf(fact_274_nxt__wait,axiom,
    ! [Phi: stream2709947120125613254m_rule > $o,Xs2: stream2709947120125613254m_rule] :
      ( ( linear1816559245158234865m_rule @ Phi @ Xs2 )
     => ( compow6896429539834504498rule_o @ ( linear8417479883454578709m_rule @ Phi @ Xs2 ) @ linear3460418925085718015rule_o @ Phi @ Xs2 ) ) ).

% nxt_wait
thf(fact_275_finite__Collect__le__nat,axiom,
    ! [K: nat] :
      ( finite_finite_nat
      @ ( collect_nat
        @ ^ [N3: nat] : ( ord_less_eq_nat @ N3 @ K ) ) ) ).

% finite_Collect_le_nat
thf(fact_276_Saturated__def,axiom,
    ! [Steps: stream2709947120125613254m_rule] :
      ( ( abstra6097777249025082867ist_fm @ eff @ rules @ Steps )
      = ( ! [X: rule] :
            ( ( member_rule @ X @ ( sset_rule @ rules ) )
           => ( abstra2533313685312581075ist_fm @ eff @ X @ Steps ) ) ) ) ).

% Saturated_def
thf(fact_277_finite__Collect__subsets,axiom,
    ! [A2: set_nat] :
      ( ( finite_finite_nat @ A2 )
     => ( finite1152437895449049373et_nat
        @ ( collect_set_nat
          @ ^ [B4: set_nat] : ( ord_less_eq_set_nat @ B4 @ A2 ) ) ) ) ).

% finite_Collect_subsets
thf(fact_278_finite__Collect__subsets,axiom,
    ! [A2: set_rule] :
      ( ( finite_finite_rule @ A2 )
     => ( finite6045212024966962911t_rule
        @ ( collect_set_rule
          @ ^ [B4: set_rule] : ( ord_less_eq_set_rule @ B4 @ A2 ) ) ) ) ).

% finite_Collect_subsets
thf(fact_279_finite__Collect__subsets,axiom,
    ! [A2: set_tm] :
      ( ( finite_finite_tm @ A2 )
     => ( finite_finite_set_tm
        @ ( collect_set_tm
          @ ^ [B4: set_tm] : ( ord_less_eq_set_tm @ B4 @ A2 ) ) ) ) ).

% finite_Collect_subsets
thf(fact_280_finite__Collect__conjI,axiom,
    ! [P2: nat > $o,Q: nat > $o] :
      ( ( ( finite_finite_nat @ ( collect_nat @ P2 ) )
        | ( finite_finite_nat @ ( collect_nat @ Q ) ) )
     => ( finite_finite_nat
        @ ( collect_nat
          @ ^ [X: nat] :
              ( ( P2 @ X )
              & ( Q @ X ) ) ) ) ) ).

% finite_Collect_conjI
thf(fact_281_finite__Collect__disjI,axiom,
    ! [P2: nat > $o,Q: nat > $o] :
      ( ( finite_finite_nat
        @ ( collect_nat
          @ ^ [X: nat] :
              ( ( P2 @ X )
              | ( Q @ X ) ) ) )
      = ( ( finite_finite_nat @ ( collect_nat @ P2 ) )
        & ( finite_finite_nat @ ( collect_nat @ Q ) ) ) ) ).

% finite_Collect_disjI
thf(fact_282_finite__less__ub,axiom,
    ! [F: nat > nat,U: nat] :
      ( ! [N2: nat] : ( ord_less_eq_nat @ N2 @ ( F @ N2 ) )
     => ( finite_finite_nat
        @ ( collect_nat
          @ ^ [N3: nat] : ( ord_less_eq_nat @ ( F @ N3 ) @ U ) ) ) ) ).

% finite_less_ub
thf(fact_283_infinite__nat__iff__unbounded__le,axiom,
    ! [S4: set_nat] :
      ( ( ~ ( finite_finite_nat @ S4 ) )
      = ( ! [M: nat] :
          ? [N3: nat] :
            ( ( ord_less_eq_nat @ M @ N3 )
            & ( member_nat @ N3 @ S4 ) ) ) ) ).

% infinite_nat_iff_unbounded_le
thf(fact_284_finite__nat__set__iff__bounded__le,axiom,
    ( finite_finite_nat
    = ( ^ [N4: set_nat] :
        ? [M: nat] :
        ! [X: nat] :
          ( ( member_nat @ X @ N4 )
         => ( ord_less_eq_nat @ X @ M ) ) ) ) ).

% finite_nat_set_iff_bounded_le
thf(fact_285_finite__subset,axiom,
    ! [A2: set_nat,B5: set_nat] :
      ( ( ord_less_eq_set_nat @ A2 @ B5 )
     => ( ( finite_finite_nat @ B5 )
       => ( finite_finite_nat @ A2 ) ) ) ).

% finite_subset
thf(fact_286_finite__subset,axiom,
    ! [A2: set_rule,B5: set_rule] :
      ( ( ord_less_eq_set_rule @ A2 @ B5 )
     => ( ( finite_finite_rule @ B5 )
       => ( finite_finite_rule @ A2 ) ) ) ).

% finite_subset
thf(fact_287_finite__subset,axiom,
    ! [A2: set_tm,B5: set_tm] :
      ( ( ord_less_eq_set_tm @ A2 @ B5 )
     => ( ( finite_finite_tm @ B5 )
       => ( finite_finite_tm @ A2 ) ) ) ).

% finite_subset
thf(fact_288_infinite__super,axiom,
    ! [S4: set_nat,T: set_nat] :
      ( ( ord_less_eq_set_nat @ S4 @ T )
     => ( ~ ( finite_finite_nat @ S4 )
       => ~ ( finite_finite_nat @ T ) ) ) ).

% infinite_super
thf(fact_289_infinite__super,axiom,
    ! [S4: set_rule,T: set_rule] :
      ( ( ord_less_eq_set_rule @ S4 @ T )
     => ( ~ ( finite_finite_rule @ S4 )
       => ~ ( finite_finite_rule @ T ) ) ) ).

% infinite_super
thf(fact_290_infinite__super,axiom,
    ! [S4: set_tm,T: set_tm] :
      ( ( ord_less_eq_set_tm @ S4 @ T )
     => ( ~ ( finite_finite_tm @ S4 )
       => ~ ( finite_finite_tm @ T ) ) ) ).

% infinite_super
thf(fact_291_rev__finite__subset,axiom,
    ! [B5: set_nat,A2: set_nat] :
      ( ( finite_finite_nat @ B5 )
     => ( ( ord_less_eq_set_nat @ A2 @ B5 )
       => ( finite_finite_nat @ A2 ) ) ) ).

% rev_finite_subset
thf(fact_292_rev__finite__subset,axiom,
    ! [B5: set_rule,A2: set_rule] :
      ( ( finite_finite_rule @ B5 )
     => ( ( ord_less_eq_set_rule @ A2 @ B5 )
       => ( finite_finite_rule @ A2 ) ) ) ).

% rev_finite_subset
thf(fact_293_rev__finite__subset,axiom,
    ! [B5: set_tm,A2: set_tm] :
      ( ( finite_finite_tm @ B5 )
     => ( ( ord_less_eq_set_tm @ A2 @ B5 )
       => ( finite_finite_tm @ A2 ) ) ) ).

% rev_finite_subset
thf(fact_294_bounded__Max__nat,axiom,
    ! [P2: nat > $o,X3: nat,M3: nat] :
      ( ( P2 @ X3 )
     => ( ! [X2: nat] :
            ( ( P2 @ X2 )
           => ( ord_less_eq_nat @ X2 @ M3 ) )
       => ~ ! [M4: nat] :
              ( ( P2 @ M4 )
             => ~ ! [X4: nat] :
                    ( ( P2 @ X4 )
                   => ( ord_less_eq_nat @ X4 @ M4 ) ) ) ) ) ).

% bounded_Max_nat
thf(fact_295_pigeonhole__infinite__rel,axiom,
    ! [A2: set_Pr5202636777678657877ist_fm,B5: set_nat,R3: produc6018962875968178549ist_fm > nat > $o] :
      ( ~ ( finite600313802051052958ist_fm @ A2 )
     => ( ( finite_finite_nat @ B5 )
       => ( ! [X2: produc6018962875968178549ist_fm] :
              ( ( member4699826688122452638ist_fm @ X2 @ A2 )
             => ? [Xa: nat] :
                  ( ( member_nat @ Xa @ B5 )
                  & ( R3 @ X2 @ Xa ) ) )
         => ? [X2: nat] :
              ( ( member_nat @ X2 @ B5 )
              & ~ ( finite600313802051052958ist_fm
                  @ ( collec5845330005437105888ist_fm
                    @ ^ [A4: produc6018962875968178549ist_fm] :
                        ( ( member4699826688122452638ist_fm @ A4 @ A2 )
                        & ( R3 @ A4 @ X2 ) ) ) ) ) ) ) ) ).

% pigeonhole_infinite_rel
thf(fact_296_pigeonhole__infinite__rel,axiom,
    ! [A2: set_rule,B5: set_nat,R3: rule > nat > $o] :
      ( ~ ( finite_finite_rule @ A2 )
     => ( ( finite_finite_nat @ B5 )
       => ( ! [X2: rule] :
              ( ( member_rule @ X2 @ A2 )
             => ? [Xa: nat] :
                  ( ( member_nat @ Xa @ B5 )
                  & ( R3 @ X2 @ Xa ) ) )
         => ? [X2: nat] :
              ( ( member_nat @ X2 @ B5 )
              & ~ ( finite_finite_rule
                  @ ( collect_rule
                    @ ^ [A4: rule] :
                        ( ( member_rule @ A4 @ A2 )
                        & ( R3 @ A4 @ X2 ) ) ) ) ) ) ) ) ).

% pigeonhole_infinite_rel
thf(fact_297_pigeonhole__infinite__rel,axiom,
    ! [A2: set_Pr8693737435421807431at_nat,B5: set_nat,R3: produc859450856879609959at_nat > nat > $o] :
      ( ~ ( finite4392333629123659920at_nat @ A2 )
     => ( ( finite_finite_nat @ B5 )
       => ( ! [X2: produc859450856879609959at_nat] :
              ( ( member8206827879206165904at_nat @ X2 @ A2 )
             => ? [Xa: nat] :
                  ( ( member_nat @ Xa @ B5 )
                  & ( R3 @ X2 @ Xa ) ) )
         => ? [X2: nat] :
              ( ( member_nat @ X2 @ B5 )
              & ~ ( finite4392333629123659920at_nat
                  @ ( collec7088162979684241874at_nat
                    @ ^ [A4: produc859450856879609959at_nat] :
                        ( ( member8206827879206165904at_nat @ A4 @ A2 )
                        & ( R3 @ A4 @ X2 ) ) ) ) ) ) ) ) ).

% pigeonhole_infinite_rel
thf(fact_298_pigeonhole__infinite__rel,axiom,
    ! [A2: set_nat,B5: set_nat,R3: nat > nat > $o] :
      ( ~ ( finite_finite_nat @ A2 )
     => ( ( finite_finite_nat @ B5 )
       => ( ! [X2: nat] :
              ( ( member_nat @ X2 @ A2 )
             => ? [Xa: nat] :
                  ( ( member_nat @ Xa @ B5 )
                  & ( R3 @ X2 @ Xa ) ) )
         => ? [X2: nat] :
              ( ( member_nat @ X2 @ B5 )
              & ~ ( finite_finite_nat
                  @ ( collect_nat
                    @ ^ [A4: nat] :
                        ( ( member_nat @ A4 @ A2 )
                        & ( R3 @ A4 @ X2 ) ) ) ) ) ) ) ) ).

% pigeonhole_infinite_rel
thf(fact_299_not__finite__existsD,axiom,
    ! [P2: nat > $o] :
      ( ~ ( finite_finite_nat @ ( collect_nat @ P2 ) )
     => ? [X_1: nat] : ( P2 @ X_1 ) ) ).

% not_finite_existsD
thf(fact_300_finite__has__minimal2,axiom,
    ! [A2: set_nat,A: nat] :
      ( ( finite_finite_nat @ A2 )
     => ( ( member_nat @ A @ A2 )
       => ? [X2: nat] :
            ( ( member_nat @ X2 @ A2 )
            & ( ord_less_eq_nat @ X2 @ A )
            & ! [Xa: nat] :
                ( ( member_nat @ Xa @ A2 )
               => ( ( ord_less_eq_nat @ Xa @ X2 )
                 => ( X2 = Xa ) ) ) ) ) ) ).

% finite_has_minimal2
thf(fact_301_finite__has__minimal2,axiom,
    ! [A2: set_set_rule,A: set_rule] :
      ( ( finite6045212024966962911t_rule @ A2 )
     => ( ( member_set_rule @ A @ A2 )
       => ? [X2: set_rule] :
            ( ( member_set_rule @ X2 @ A2 )
            & ( ord_less_eq_set_rule @ X2 @ A )
            & ! [Xa: set_rule] :
                ( ( member_set_rule @ Xa @ A2 )
               => ( ( ord_less_eq_set_rule @ Xa @ X2 )
                 => ( X2 = Xa ) ) ) ) ) ) ).

% finite_has_minimal2
thf(fact_302_finite__has__minimal2,axiom,
    ! [A2: set_set_tm,A: set_tm] :
      ( ( finite_finite_set_tm @ A2 )
     => ( ( member_set_tm @ A @ A2 )
       => ? [X2: set_tm] :
            ( ( member_set_tm @ X2 @ A2 )
            & ( ord_less_eq_set_tm @ X2 @ A )
            & ! [Xa: set_tm] :
                ( ( member_set_tm @ Xa @ A2 )
               => ( ( ord_less_eq_set_tm @ Xa @ X2 )
                 => ( X2 = Xa ) ) ) ) ) ) ).

% finite_has_minimal2
thf(fact_303_finite__has__maximal2,axiom,
    ! [A2: set_nat,A: nat] :
      ( ( finite_finite_nat @ A2 )
     => ( ( member_nat @ A @ A2 )
       => ? [X2: nat] :
            ( ( member_nat @ X2 @ A2 )
            & ( ord_less_eq_nat @ A @ X2 )
            & ! [Xa: nat] :
                ( ( member_nat @ Xa @ A2 )
               => ( ( ord_less_eq_nat @ X2 @ Xa )
                 => ( X2 = Xa ) ) ) ) ) ) ).

% finite_has_maximal2
thf(fact_304_finite__has__maximal2,axiom,
    ! [A2: set_set_rule,A: set_rule] :
      ( ( finite6045212024966962911t_rule @ A2 )
     => ( ( member_set_rule @ A @ A2 )
       => ? [X2: set_rule] :
            ( ( member_set_rule @ X2 @ A2 )
            & ( ord_less_eq_set_rule @ A @ X2 )
            & ! [Xa: set_rule] :
                ( ( member_set_rule @ Xa @ A2 )
               => ( ( ord_less_eq_set_rule @ X2 @ Xa )
                 => ( X2 = Xa ) ) ) ) ) ) ).

% finite_has_maximal2
thf(fact_305_finite__has__maximal2,axiom,
    ! [A2: set_set_tm,A: set_tm] :
      ( ( finite_finite_set_tm @ A2 )
     => ( ( member_set_tm @ A @ A2 )
       => ? [X2: set_tm] :
            ( ( member_set_tm @ X2 @ A2 )
            & ( ord_less_eq_set_tm @ A @ X2 )
            & ! [Xa: set_tm] :
                ( ( member_set_tm @ Xa @ A2 )
               => ( ( ord_less_eq_set_tm @ X2 @ Xa )
                 => ( X2 = Xa ) ) ) ) ) ) ).

% finite_has_maximal2
thf(fact_306_pred__subset__eq,axiom,
    ! [R3: set_nat,S4: set_nat] :
      ( ( ord_less_eq_nat_o
        @ ^ [X: nat] : ( member_nat @ X @ R3 )
        @ ^ [X: nat] : ( member_nat @ X @ S4 ) )
      = ( ord_less_eq_set_nat @ R3 @ S4 ) ) ).

% pred_subset_eq
thf(fact_307_pred__subset__eq,axiom,
    ! [R3: set_Pr5202636777678657877ist_fm,S4: set_Pr5202636777678657877ist_fm] :
      ( ( ord_le1264026566993285160t_fm_o
        @ ^ [X: produc6018962875968178549ist_fm] : ( member4699826688122452638ist_fm @ X @ R3 )
        @ ^ [X: produc6018962875968178549ist_fm] : ( member4699826688122452638ist_fm @ X @ S4 ) )
      = ( ord_le1771420097867575541ist_fm @ R3 @ S4 ) ) ).

% pred_subset_eq
thf(fact_308_pred__subset__eq,axiom,
    ! [R3: set_Pr8693737435421807431at_nat,S4: set_Pr8693737435421807431at_nat] :
      ( ( ord_le7858099551454983350_nat_o
        @ ^ [X: produc859450856879609959at_nat] : ( member8206827879206165904at_nat @ X @ R3 )
        @ ^ [X: produc859450856879609959at_nat] : ( member8206827879206165904at_nat @ X @ S4 ) )
      = ( ord_le3000389064537975527at_nat @ R3 @ S4 ) ) ).

% pred_subset_eq
thf(fact_309_pred__subset__eq,axiom,
    ! [R3: set_rule,S4: set_rule] :
      ( ( ord_less_eq_rule_o
        @ ^ [X: rule] : ( member_rule @ X @ R3 )
        @ ^ [X: rule] : ( member_rule @ X @ S4 ) )
      = ( ord_less_eq_set_rule @ R3 @ S4 ) ) ).

% pred_subset_eq
thf(fact_310_pred__subset__eq,axiom,
    ! [R3: set_tm,S4: set_tm] :
      ( ( ord_less_eq_tm_o
        @ ^ [X: tm] : ( member_tm @ X @ R3 )
        @ ^ [X: tm] : ( member_tm @ X @ S4 ) )
      = ( ord_less_eq_set_tm @ R3 @ S4 ) ) ).

% pred_subset_eq
thf(fact_311_less__eq__set__def,axiom,
    ( ord_less_eq_set_nat
    = ( ^ [A5: set_nat,B4: set_nat] :
          ( ord_less_eq_nat_o
          @ ^ [X: nat] : ( member_nat @ X @ A5 )
          @ ^ [X: nat] : ( member_nat @ X @ B4 ) ) ) ) ).

% less_eq_set_def
thf(fact_312_less__eq__set__def,axiom,
    ( ord_le1771420097867575541ist_fm
    = ( ^ [A5: set_Pr5202636777678657877ist_fm,B4: set_Pr5202636777678657877ist_fm] :
          ( ord_le1264026566993285160t_fm_o
          @ ^ [X: produc6018962875968178549ist_fm] : ( member4699826688122452638ist_fm @ X @ A5 )
          @ ^ [X: produc6018962875968178549ist_fm] : ( member4699826688122452638ist_fm @ X @ B4 ) ) ) ) ).

% less_eq_set_def
thf(fact_313_less__eq__set__def,axiom,
    ( ord_le3000389064537975527at_nat
    = ( ^ [A5: set_Pr8693737435421807431at_nat,B4: set_Pr8693737435421807431at_nat] :
          ( ord_le7858099551454983350_nat_o
          @ ^ [X: produc859450856879609959at_nat] : ( member8206827879206165904at_nat @ X @ A5 )
          @ ^ [X: produc859450856879609959at_nat] : ( member8206827879206165904at_nat @ X @ B4 ) ) ) ) ).

% less_eq_set_def
thf(fact_314_less__eq__set__def,axiom,
    ( ord_less_eq_set_rule
    = ( ^ [A5: set_rule,B4: set_rule] :
          ( ord_less_eq_rule_o
          @ ^ [X: rule] : ( member_rule @ X @ A5 )
          @ ^ [X: rule] : ( member_rule @ X @ B4 ) ) ) ) ).

% less_eq_set_def
thf(fact_315_less__eq__set__def,axiom,
    ( ord_less_eq_set_tm
    = ( ^ [A5: set_tm,B4: set_tm] :
          ( ord_less_eq_tm_o
          @ ^ [X: tm] : ( member_tm @ X @ A5 )
          @ ^ [X: tm] : ( member_tm @ X @ B4 ) ) ) ) ).

% less_eq_set_def
thf(fact_316_subset__antisym,axiom,
    ! [A2: set_rule,B5: set_rule] :
      ( ( ord_less_eq_set_rule @ A2 @ B5 )
     => ( ( ord_less_eq_set_rule @ B5 @ A2 )
       => ( A2 = B5 ) ) ) ).

% subset_antisym
thf(fact_317_subset__antisym,axiom,
    ! [A2: set_tm,B5: set_tm] :
      ( ( ord_less_eq_set_tm @ A2 @ B5 )
     => ( ( ord_less_eq_set_tm @ B5 @ A2 )
       => ( A2 = B5 ) ) ) ).

% subset_antisym
thf(fact_318_subsetI,axiom,
    ! [A2: set_nat,B5: set_nat] :
      ( ! [X2: nat] :
          ( ( member_nat @ X2 @ A2 )
         => ( member_nat @ X2 @ B5 ) )
     => ( ord_less_eq_set_nat @ A2 @ B5 ) ) ).

% subsetI
thf(fact_319_subsetI,axiom,
    ! [A2: set_Pr5202636777678657877ist_fm,B5: set_Pr5202636777678657877ist_fm] :
      ( ! [X2: produc6018962875968178549ist_fm] :
          ( ( member4699826688122452638ist_fm @ X2 @ A2 )
         => ( member4699826688122452638ist_fm @ X2 @ B5 ) )
     => ( ord_le1771420097867575541ist_fm @ A2 @ B5 ) ) ).

% subsetI
thf(fact_320_subsetI,axiom,
    ! [A2: set_Pr8693737435421807431at_nat,B5: set_Pr8693737435421807431at_nat] :
      ( ! [X2: produc859450856879609959at_nat] :
          ( ( member8206827879206165904at_nat @ X2 @ A2 )
         => ( member8206827879206165904at_nat @ X2 @ B5 ) )
     => ( ord_le3000389064537975527at_nat @ A2 @ B5 ) ) ).

% subsetI
thf(fact_321_subsetI,axiom,
    ! [A2: set_rule,B5: set_rule] :
      ( ! [X2: rule] :
          ( ( member_rule @ X2 @ A2 )
         => ( member_rule @ X2 @ B5 ) )
     => ( ord_less_eq_set_rule @ A2 @ B5 ) ) ).

% subsetI
thf(fact_322_subsetI,axiom,
    ! [A2: set_tm,B5: set_tm] :
      ( ! [X2: tm] :
          ( ( member_tm @ X2 @ A2 )
         => ( member_tm @ X2 @ B5 ) )
     => ( ord_less_eq_set_tm @ A2 @ B5 ) ) ).

% subsetI
thf(fact_323_i_Ofair__def,axiom,
    ( abstra6979500642109889529air_tm
    = ( ^ [Rules: stream_tm,Rs: stream_tm] :
          ( ( ord_less_eq_set_tm @ ( sset_tm @ Rs ) @ ( sset_tm @ Rules ) )
          & ! [X: tm] :
              ( ( member_tm @ X @ ( sset_tm @ Rules ) )
             => ( linear1925250572922474593alw_tm
                @ ( linear627519065254162282_ev_tm
                  @ ( linear3702449374085696025lds_tm
                    @ ( ^ [Y2: tm,Z2: tm] : ( Y2 = Z2 )
                      @ X ) ) )
                @ Rs ) ) ) ) ) ).

% i.fair_def
thf(fact_324_i_Ofair__def,axiom,
    ( abstra3665372904625986210m_rule
    = ( ^ [Rules: stream2709947120125613254m_rule,Rs: stream2709947120125613254m_rule] :
          ( ( ord_le6390412330253371084m_rule @ ( sset_P4484857331586881186m_rule @ Rs ) @ ( sset_P4484857331586881186m_rule @ Rules ) )
          & ! [X: produc340336539035504054m_rule] :
              ( ( member7231649785386036813m_rule @ X @ ( sset_P4484857331586881186m_rule @ Rules ) )
             => ( linear4957294957385550906m_rule
                @ ( linear1816559245158234865m_rule
                  @ ( linear7925859179493631106m_rule
                    @ ( ^ [Y2: produc340336539035504054m_rule,Z2: produc340336539035504054m_rule] : ( Y2 = Z2 )
                      @ X ) ) )
                @ Rs ) ) ) ) ) ).

% i.fair_def
thf(fact_325_i_Ofair__def,axiom,
    ( abstra3799686578551160190r_rule
    = ( ^ [Rules: stream_rule,Rs: stream_rule] :
          ( ( ord_less_eq_set_rule @ ( sset_rule @ Rs ) @ ( sset_rule @ Rules ) )
          & ! [X: rule] :
              ( ( member_rule @ X @ ( sset_rule @ Rules ) )
             => ( linear1087404401031714582w_rule
                @ ( linear292482573288640205v_rule
                  @ ( linear7354821883049305950s_rule
                    @ ( ^ [Y2: rule,Z2: rule] : ( Y2 = Z2 )
                      @ X ) ) )
                @ Rs ) ) ) ) ) ).

% i.fair_def
thf(fact_326_RuleSystem__Defs_Ofair__def,axiom,
    ( abstra6979500642109889529air_tm
    = ( ^ [Rules: stream_tm,Rs: stream_tm] :
          ( ( ord_less_eq_set_tm @ ( sset_tm @ Rs ) @ ( sset_tm @ Rules ) )
          & ! [X: tm] :
              ( ( member_tm @ X @ ( sset_tm @ Rules ) )
             => ( linear1925250572922474593alw_tm
                @ ( linear627519065254162282_ev_tm
                  @ ( linear3702449374085696025lds_tm
                    @ ( ^ [Y2: tm,Z2: tm] : ( Y2 = Z2 )
                      @ X ) ) )
                @ Rs ) ) ) ) ) ).

% RuleSystem_Defs.fair_def
thf(fact_327_RuleSystem__Defs_Ofair__def,axiom,
    ( abstra3665372904625986210m_rule
    = ( ^ [Rules: stream2709947120125613254m_rule,Rs: stream2709947120125613254m_rule] :
          ( ( ord_le6390412330253371084m_rule @ ( sset_P4484857331586881186m_rule @ Rs ) @ ( sset_P4484857331586881186m_rule @ Rules ) )
          & ! [X: produc340336539035504054m_rule] :
              ( ( member7231649785386036813m_rule @ X @ ( sset_P4484857331586881186m_rule @ Rules ) )
             => ( linear4957294957385550906m_rule
                @ ( linear1816559245158234865m_rule
                  @ ( linear7925859179493631106m_rule
                    @ ( ^ [Y2: produc340336539035504054m_rule,Z2: produc340336539035504054m_rule] : ( Y2 = Z2 )
                      @ X ) ) )
                @ Rs ) ) ) ) ) ).

% RuleSystem_Defs.fair_def
thf(fact_328_RuleSystem__Defs_Ofair__def,axiom,
    ( abstra3799686578551160190r_rule
    = ( ^ [Rules: stream_rule,Rs: stream_rule] :
          ( ( ord_less_eq_set_rule @ ( sset_rule @ Rs ) @ ( sset_rule @ Rules ) )
          & ! [X: rule] :
              ( ( member_rule @ X @ ( sset_rule @ Rules ) )
             => ( linear1087404401031714582w_rule
                @ ( linear292482573288640205v_rule
                  @ ( linear7354821883049305950s_rule
                    @ ( ^ [Y2: rule,Z2: rule] : ( Y2 = Z2 )
                      @ X ) ) )
                @ Rs ) ) ) ) ) ).

% RuleSystem_Defs.fair_def
thf(fact_329_sfilter__streams,axiom,
    ! [P2: produc6018962875968178549ist_fm > $o,Omega2: stream4408948924543953275ist_fm,A2: set_Pr5202636777678657877ist_fm] :
      ( ( linear2489540979234913265ist_fm @ ( linear2237513815067535098ist_fm @ ( linear4742913937519564201ist_fm @ P2 ) ) @ Omega2 )
     => ( ( member197948898507729188ist_fm @ Omega2 @ ( stream1217541351357117421ist_fm @ A2 ) )
       => ( member197948898507729188ist_fm @ ( sfilte223592874874493077ist_fm @ P2 @ Omega2 )
          @ ( stream1217541351357117421ist_fm
            @ ( collec5845330005437105888ist_fm
              @ ^ [X: produc6018962875968178549ist_fm] :
                  ( ( member4699826688122452638ist_fm @ X @ A2 )
                  & ( P2 @ X ) ) ) ) ) ) ) ).

% sfilter_streams
thf(fact_330_sfilter__streams,axiom,
    ! [P2: produc859450856879609959at_nat > $o,Omega2: stream8372878641218411373at_nat,A2: set_Pr8693737435421807431at_nat] :
      ( ( linear847162001329847011at_nat @ ( linear3409734863606872556at_nat @ ( linear2637753954401252507at_nat @ P2 ) ) @ Omega2 )
     => ( ( member1992146023263842070at_nat @ Omega2 @ ( stream5885727000947712735at_nat @ A2 ) )
       => ( member1992146023263842070at_nat @ ( sfilte509745310137608071at_nat @ P2 @ Omega2 )
          @ ( stream5885727000947712735at_nat
            @ ( collec7088162979684241874at_nat
              @ ^ [X: produc859450856879609959at_nat] :
                  ( ( member8206827879206165904at_nat @ X @ A2 )
                  & ( P2 @ X ) ) ) ) ) ) ) ).

% sfilter_streams
thf(fact_331_sfilter__streams,axiom,
    ! [P2: nat > $o,Omega2: stream_nat,A2: set_nat] :
      ( ( linear1448867875752120916lw_nat @ ( linear5502553777413005579ev_nat @ ( linear3087052677352696476ds_nat @ P2 ) ) @ Omega2 )
     => ( ( member_stream_nat @ Omega2 @ ( streams_nat @ A2 ) )
       => ( member_stream_nat @ ( sfilter_nat @ P2 @ Omega2 )
          @ ( streams_nat
            @ ( collect_nat
              @ ^ [X: nat] :
                  ( ( member_nat @ X @ A2 )
                  & ( P2 @ X ) ) ) ) ) ) ) ).

% sfilter_streams
thf(fact_332_sfilter__streams,axiom,
    ! [P2: produc340336539035504054m_rule > $o,Omega2: stream2709947120125613254m_rule,A2: set_Pr1822751329126368876m_rule] :
      ( ( linear4957294957385550906m_rule @ ( linear1816559245158234865m_rule @ ( linear7925859179493631106m_rule @ P2 ) ) @ Omega2 )
     => ( ( member5228318729319728861m_rule @ Omega2 @ ( stream3418479298939905342m_rule @ A2 ) )
       => ( member5228318729319728861m_rule @ ( sfilte3304650064355440278m_rule @ P2 @ Omega2 )
          @ ( stream3418479298939905342m_rule
            @ ( collec1009276759030335883m_rule
              @ ^ [X: produc340336539035504054m_rule] :
                  ( ( member7231649785386036813m_rule @ X @ A2 )
                  & ( P2 @ X ) ) ) ) ) ) ) ).

% sfilter_streams
thf(fact_333_sfilter__streams,axiom,
    ! [P2: rule > $o,Omega2: stream_rule,A2: set_rule] :
      ( ( linear1087404401031714582w_rule @ ( linear292482573288640205v_rule @ ( linear7354821883049305950s_rule @ P2 ) ) @ Omega2 )
     => ( ( member_stream_rule @ Omega2 @ ( streams_rule @ A2 ) )
       => ( member_stream_rule @ ( sfilter_rule @ P2 @ Omega2 )
          @ ( streams_rule
            @ ( collect_rule
              @ ^ [X: rule] :
                  ( ( member_rule @ X @ A2 )
                  & ( P2 @ X ) ) ) ) ) ) ) ).

% sfilter_streams
thf(fact_334_wait__def,axiom,
    ( linear8417479883454578709m_rule
    = ( ^ [Phi2: stream2709947120125613254m_rule > $o,Xs: stream2709947120125613254m_rule] :
          ( ord_Least_nat
          @ ^ [N3: nat] : ( compow6896429539834504498rule_o @ N3 @ linear3460418925085718015rule_o @ Phi2 @ Xs ) ) ) ) ).

% wait_def
thf(fact_335_subset__Collect__iff,axiom,
    ! [B5: set_Pr5202636777678657877ist_fm,A2: set_Pr5202636777678657877ist_fm,P2: produc6018962875968178549ist_fm > $o] :
      ( ( ord_le1771420097867575541ist_fm @ B5 @ A2 )
     => ( ( ord_le1771420097867575541ist_fm @ B5
          @ ( collec5845330005437105888ist_fm
            @ ^ [X: produc6018962875968178549ist_fm] :
                ( ( member4699826688122452638ist_fm @ X @ A2 )
                & ( P2 @ X ) ) ) )
        = ( ! [X: produc6018962875968178549ist_fm] :
              ( ( member4699826688122452638ist_fm @ X @ B5 )
             => ( P2 @ X ) ) ) ) ) ).

% subset_Collect_iff
thf(fact_336_subset__Collect__iff,axiom,
    ! [B5: set_Pr8693737435421807431at_nat,A2: set_Pr8693737435421807431at_nat,P2: produc859450856879609959at_nat > $o] :
      ( ( ord_le3000389064537975527at_nat @ B5 @ A2 )
     => ( ( ord_le3000389064537975527at_nat @ B5
          @ ( collec7088162979684241874at_nat
            @ ^ [X: produc859450856879609959at_nat] :
                ( ( member8206827879206165904at_nat @ X @ A2 )
                & ( P2 @ X ) ) ) )
        = ( ! [X: produc859450856879609959at_nat] :
              ( ( member8206827879206165904at_nat @ X @ B5 )
             => ( P2 @ X ) ) ) ) ) ).

% subset_Collect_iff
thf(fact_337_subset__Collect__iff,axiom,
    ! [B5: set_nat,A2: set_nat,P2: nat > $o] :
      ( ( ord_less_eq_set_nat @ B5 @ A2 )
     => ( ( ord_less_eq_set_nat @ B5
          @ ( collect_nat
            @ ^ [X: nat] :
                ( ( member_nat @ X @ A2 )
                & ( P2 @ X ) ) ) )
        = ( ! [X: nat] :
              ( ( member_nat @ X @ B5 )
             => ( P2 @ X ) ) ) ) ) ).

% subset_Collect_iff
thf(fact_338_subset__Collect__iff,axiom,
    ! [B5: set_rule,A2: set_rule,P2: rule > $o] :
      ( ( ord_less_eq_set_rule @ B5 @ A2 )
     => ( ( ord_less_eq_set_rule @ B5
          @ ( collect_rule
            @ ^ [X: rule] :
                ( ( member_rule @ X @ A2 )
                & ( P2 @ X ) ) ) )
        = ( ! [X: rule] :
              ( ( member_rule @ X @ B5 )
             => ( P2 @ X ) ) ) ) ) ).

% subset_Collect_iff
thf(fact_339_subset__Collect__iff,axiom,
    ! [B5: set_tm,A2: set_tm,P2: tm > $o] :
      ( ( ord_less_eq_set_tm @ B5 @ A2 )
     => ( ( ord_less_eq_set_tm @ B5
          @ ( collect_tm
            @ ^ [X: tm] :
                ( ( member_tm @ X @ A2 )
                & ( P2 @ X ) ) ) )
        = ( ! [X: tm] :
              ( ( member_tm @ X @ B5 )
             => ( P2 @ X ) ) ) ) ) ).

% subset_Collect_iff
thf(fact_340_RuleSystem__Defs_Ofair_Ocong,axiom,
    abstra3799686578551160190r_rule = abstra3799686578551160190r_rule ).

% RuleSystem_Defs.fair.cong
thf(fact_341_LeastI2__ex,axiom,
    ! [P2: nat > $o,Q: nat > $o] :
      ( ? [X_12: nat] : ( P2 @ X_12 )
     => ( ! [X2: nat] :
            ( ( P2 @ X2 )
           => ( Q @ X2 ) )
       => ( Q @ ( ord_Least_nat @ P2 ) ) ) ) ).

% LeastI2_ex
thf(fact_342_LeastI__ex,axiom,
    ! [P2: nat > $o] :
      ( ? [X_12: nat] : ( P2 @ X_12 )
     => ( P2 @ ( ord_Least_nat @ P2 ) ) ) ).

% LeastI_ex
thf(fact_343_LeastI2,axiom,
    ! [P2: nat > $o,A: nat,Q: nat > $o] :
      ( ( P2 @ A )
     => ( ! [X2: nat] :
            ( ( P2 @ X2 )
           => ( Q @ X2 ) )
       => ( Q @ ( ord_Least_nat @ P2 ) ) ) ) ).

% LeastI2
thf(fact_344_LeastI,axiom,
    ! [P2: nat > $o,K: nat] :
      ( ( P2 @ K )
     => ( P2 @ ( ord_Least_nat @ P2 ) ) ) ).

% LeastI
thf(fact_345_sdrop__fair,axiom,
    ! [Rs2: stream_rule,M2: nat] :
      ( ( abstra3799686578551160190r_rule @ rules @ Rs2 )
     => ( abstra3799686578551160190r_rule @ rules @ ( sdrop_rule @ M2 @ Rs2 ) ) ) ).

% sdrop_fair
thf(fact_346_streams__mono2,axiom,
    ! [S4: set_rule,T: set_rule] :
      ( ( ord_less_eq_set_rule @ S4 @ T )
     => ( ord_le1988440220176329784m_rule @ ( streams_rule @ S4 ) @ ( streams_rule @ T ) ) ) ).

% streams_mono2
thf(fact_347_streams__mono2,axiom,
    ! [S4: set_tm,T: set_tm] :
      ( ( ord_less_eq_set_tm @ S4 @ T )
     => ( ord_le2662260337392385771eam_tm @ ( streams_tm @ S4 ) @ ( streams_tm @ T ) ) ) ).

% streams_mono2
thf(fact_348_streams__mono,axiom,
    ! [S3: stream_rule,A2: set_rule,B5: set_rule] :
      ( ( member_stream_rule @ S3 @ ( streams_rule @ A2 ) )
     => ( ( ord_less_eq_set_rule @ A2 @ B5 )
       => ( member_stream_rule @ S3 @ ( streams_rule @ B5 ) ) ) ) ).

% streams_mono
thf(fact_349_streams__mono,axiom,
    ! [S3: stream_tm,A2: set_tm,B5: set_tm] :
      ( ( member_stream_tm @ S3 @ ( streams_tm @ A2 ) )
     => ( ( ord_less_eq_set_tm @ A2 @ B5 )
       => ( member_stream_tm @ S3 @ ( streams_tm @ B5 ) ) ) ) ).

% streams_mono
thf(fact_350_Least1I,axiom,
    ! [P2: nat > $o] :
      ( ? [X4: nat] :
          ( ( P2 @ X4 )
          & ! [Y4: nat] :
              ( ( P2 @ Y4 )
             => ( ord_less_eq_nat @ X4 @ Y4 ) )
          & ! [Y4: nat] :
              ( ( ( P2 @ Y4 )
                & ! [Ya: nat] :
                    ( ( P2 @ Ya )
                   => ( ord_less_eq_nat @ Y4 @ Ya ) ) )
             => ( Y4 = X4 ) ) )
     => ( P2 @ ( ord_Least_nat @ P2 ) ) ) ).

% Least1I
thf(fact_351_Least1I,axiom,
    ! [P2: set_rule > $o] :
      ( ? [X4: set_rule] :
          ( ( P2 @ X4 )
          & ! [Y4: set_rule] :
              ( ( P2 @ Y4 )
             => ( ord_less_eq_set_rule @ X4 @ Y4 ) )
          & ! [Y4: set_rule] :
              ( ( ( P2 @ Y4 )
                & ! [Ya: set_rule] :
                    ( ( P2 @ Ya )
                   => ( ord_less_eq_set_rule @ Y4 @ Ya ) ) )
             => ( Y4 = X4 ) ) )
     => ( P2 @ ( ord_Least_set_rule @ P2 ) ) ) ).

% Least1I
thf(fact_352_Least1I,axiom,
    ! [P2: set_tm > $o] :
      ( ? [X4: set_tm] :
          ( ( P2 @ X4 )
          & ! [Y4: set_tm] :
              ( ( P2 @ Y4 )
             => ( ord_less_eq_set_tm @ X4 @ Y4 ) )
          & ! [Y4: set_tm] :
              ( ( ( P2 @ Y4 )
                & ! [Ya: set_tm] :
                    ( ( P2 @ Ya )
                   => ( ord_less_eq_set_tm @ Y4 @ Ya ) ) )
             => ( Y4 = X4 ) ) )
     => ( P2 @ ( ord_Least_set_tm @ P2 ) ) ) ).

% Least1I
thf(fact_353_Least1__le,axiom,
    ! [P2: nat > $o,Z: nat] :
      ( ? [X4: nat] :
          ( ( P2 @ X4 )
          & ! [Y4: nat] :
              ( ( P2 @ Y4 )
             => ( ord_less_eq_nat @ X4 @ Y4 ) )
          & ! [Y4: nat] :
              ( ( ( P2 @ Y4 )
                & ! [Ya: nat] :
                    ( ( P2 @ Ya )
                   => ( ord_less_eq_nat @ Y4 @ Ya ) ) )
             => ( Y4 = X4 ) ) )
     => ( ( P2 @ Z )
       => ( ord_less_eq_nat @ ( ord_Least_nat @ P2 ) @ Z ) ) ) ).

% Least1_le
thf(fact_354_Least1__le,axiom,
    ! [P2: set_rule > $o,Z: set_rule] :
      ( ? [X4: set_rule] :
          ( ( P2 @ X4 )
          & ! [Y4: set_rule] :
              ( ( P2 @ Y4 )
             => ( ord_less_eq_set_rule @ X4 @ Y4 ) )
          & ! [Y4: set_rule] :
              ( ( ( P2 @ Y4 )
                & ! [Ya: set_rule] :
                    ( ( P2 @ Ya )
                   => ( ord_less_eq_set_rule @ Y4 @ Ya ) ) )
             => ( Y4 = X4 ) ) )
     => ( ( P2 @ Z )
       => ( ord_less_eq_set_rule @ ( ord_Least_set_rule @ P2 ) @ Z ) ) ) ).

% Least1_le
thf(fact_355_Least1__le,axiom,
    ! [P2: set_tm > $o,Z: set_tm] :
      ( ? [X4: set_tm] :
          ( ( P2 @ X4 )
          & ! [Y4: set_tm] :
              ( ( P2 @ Y4 )
             => ( ord_less_eq_set_tm @ X4 @ Y4 ) )
          & ! [Y4: set_tm] :
              ( ( ( P2 @ Y4 )
                & ! [Ya: set_tm] :
                    ( ( P2 @ Ya )
                   => ( ord_less_eq_set_tm @ Y4 @ Ya ) ) )
             => ( Y4 = X4 ) ) )
     => ( ( P2 @ Z )
       => ( ord_less_eq_set_tm @ ( ord_Least_set_tm @ P2 ) @ Z ) ) ) ).

% Least1_le
thf(fact_356_LeastI2__order,axiom,
    ! [P2: nat > $o,X3: nat,Q: nat > $o] :
      ( ( P2 @ X3 )
     => ( ! [Y4: nat] :
            ( ( P2 @ Y4 )
           => ( ord_less_eq_nat @ X3 @ Y4 ) )
       => ( ! [X2: nat] :
              ( ( P2 @ X2 )
             => ( ! [Y5: nat] :
                    ( ( P2 @ Y5 )
                   => ( ord_less_eq_nat @ X2 @ Y5 ) )
               => ( Q @ X2 ) ) )
         => ( Q @ ( ord_Least_nat @ P2 ) ) ) ) ) ).

% LeastI2_order
thf(fact_357_LeastI2__order,axiom,
    ! [P2: set_rule > $o,X3: set_rule,Q: set_rule > $o] :
      ( ( P2 @ X3 )
     => ( ! [Y4: set_rule] :
            ( ( P2 @ Y4 )
           => ( ord_less_eq_set_rule @ X3 @ Y4 ) )
       => ( ! [X2: set_rule] :
              ( ( P2 @ X2 )
             => ( ! [Y5: set_rule] :
                    ( ( P2 @ Y5 )
                   => ( ord_less_eq_set_rule @ X2 @ Y5 ) )
               => ( Q @ X2 ) ) )
         => ( Q @ ( ord_Least_set_rule @ P2 ) ) ) ) ) ).

% LeastI2_order
thf(fact_358_LeastI2__order,axiom,
    ! [P2: set_tm > $o,X3: set_tm,Q: set_tm > $o] :
      ( ( P2 @ X3 )
     => ( ! [Y4: set_tm] :
            ( ( P2 @ Y4 )
           => ( ord_less_eq_set_tm @ X3 @ Y4 ) )
       => ( ! [X2: set_tm] :
              ( ( P2 @ X2 )
             => ( ! [Y5: set_tm] :
                    ( ( P2 @ Y5 )
                   => ( ord_less_eq_set_tm @ X2 @ Y5 ) )
               => ( Q @ X2 ) ) )
         => ( Q @ ( ord_Least_set_tm @ P2 ) ) ) ) ) ).

% LeastI2_order
thf(fact_359_Least__equality,axiom,
    ! [P2: nat > $o,X3: nat] :
      ( ( P2 @ X3 )
     => ( ! [Y4: nat] :
            ( ( P2 @ Y4 )
           => ( ord_less_eq_nat @ X3 @ Y4 ) )
       => ( ( ord_Least_nat @ P2 )
          = X3 ) ) ) ).

% Least_equality
thf(fact_360_Least__equality,axiom,
    ! [P2: set_rule > $o,X3: set_rule] :
      ( ( P2 @ X3 )
     => ( ! [Y4: set_rule] :
            ( ( P2 @ Y4 )
           => ( ord_less_eq_set_rule @ X3 @ Y4 ) )
       => ( ( ord_Least_set_rule @ P2 )
          = X3 ) ) ) ).

% Least_equality
thf(fact_361_Least__equality,axiom,
    ! [P2: set_tm > $o,X3: set_tm] :
      ( ( P2 @ X3 )
     => ( ! [Y4: set_tm] :
            ( ( P2 @ Y4 )
           => ( ord_less_eq_set_tm @ X3 @ Y4 ) )
       => ( ( ord_Least_set_tm @ P2 )
          = X3 ) ) ) ).

% Least_equality
thf(fact_362_LeastI2__wellorder,axiom,
    ! [P2: nat > $o,A: nat,Q: nat > $o] :
      ( ( P2 @ A )
     => ( ! [A3: nat] :
            ( ( P2 @ A3 )
           => ( ! [B6: nat] :
                  ( ( P2 @ B6 )
                 => ( ord_less_eq_nat @ A3 @ B6 ) )
             => ( Q @ A3 ) ) )
       => ( Q @ ( ord_Least_nat @ P2 ) ) ) ) ).

% LeastI2_wellorder
thf(fact_363_LeastI2__wellorder__ex,axiom,
    ! [P2: nat > $o,Q: nat > $o] :
      ( ? [X_12: nat] : ( P2 @ X_12 )
     => ( ! [A3: nat] :
            ( ( P2 @ A3 )
           => ( ! [B6: nat] :
                  ( ( P2 @ B6 )
                 => ( ord_less_eq_nat @ A3 @ B6 ) )
             => ( Q @ A3 ) ) )
       => ( Q @ ( ord_Least_nat @ P2 ) ) ) ) ).

% LeastI2_wellorder_ex
thf(fact_364_RuleSystem__Defs_Osdrop__fair,axiom,
    ! [Rules2: stream2709947120125613254m_rule,Rs2: stream2709947120125613254m_rule,M2: nat] :
      ( ( abstra3665372904625986210m_rule @ Rules2 @ Rs2 )
     => ( abstra3665372904625986210m_rule @ Rules2 @ ( sdrop_8169176516188972301m_rule @ M2 @ Rs2 ) ) ) ).

% RuleSystem_Defs.sdrop_fair
thf(fact_365_RuleSystem__Defs_Osdrop__fair,axiom,
    ! [Rules2: stream_rule,Rs2: stream_rule,M2: nat] :
      ( ( abstra3799686578551160190r_rule @ Rules2 @ Rs2 )
     => ( abstra3799686578551160190r_rule @ Rules2 @ ( sdrop_rule @ M2 @ Rs2 ) ) ) ).

% RuleSystem_Defs.sdrop_fair
thf(fact_366_i_Osdrop__fair,axiom,
    ! [Rules2: stream2709947120125613254m_rule,Rs2: stream2709947120125613254m_rule,M2: nat] :
      ( ( abstra3665372904625986210m_rule @ Rules2 @ Rs2 )
     => ( abstra3665372904625986210m_rule @ Rules2 @ ( sdrop_8169176516188972301m_rule @ M2 @ Rs2 ) ) ) ).

% i.sdrop_fair
thf(fact_367_i_Osdrop__fair,axiom,
    ! [Rules2: stream_rule,Rs2: stream_rule,M2: nat] :
      ( ( abstra3799686578551160190r_rule @ Rules2 @ Rs2 )
     => ( abstra3799686578551160190r_rule @ Rules2 @ ( sdrop_rule @ M2 @ Rs2 ) ) ) ).

% i.sdrop_fair
thf(fact_368_Least__le,axiom,
    ! [P2: nat > $o,K: nat] :
      ( ( P2 @ K )
     => ( ord_less_eq_nat @ ( ord_Least_nat @ P2 ) @ K ) ) ).

% Least_le
thf(fact_369_streams__iff__sset,axiom,
    ! [S3: stream_rule,A2: set_rule] :
      ( ( member_stream_rule @ S3 @ ( streams_rule @ A2 ) )
      = ( ord_less_eq_set_rule @ ( sset_rule @ S3 ) @ A2 ) ) ).

% streams_iff_sset
thf(fact_370_streams__iff__sset,axiom,
    ! [S3: stream_tm,A2: set_tm] :
      ( ( member_stream_tm @ S3 @ ( streams_tm @ A2 ) )
      = ( ord_less_eq_set_tm @ ( sset_tm @ S3 ) @ A2 ) ) ).

% streams_iff_sset
thf(fact_371_streams__sset,axiom,
    ! [S3: stream_rule,A2: set_rule] :
      ( ( member_stream_rule @ S3 @ ( streams_rule @ A2 ) )
     => ( ord_less_eq_set_rule @ ( sset_rule @ S3 ) @ A2 ) ) ).

% streams_sset
thf(fact_372_streams__sset,axiom,
    ! [S3: stream_tm,A2: set_tm] :
      ( ( member_stream_tm @ S3 @ ( streams_tm @ A2 ) )
     => ( ord_less_eq_set_tm @ ( sset_tm @ S3 ) @ A2 ) ) ).

% streams_sset
thf(fact_373_sset__streams,axiom,
    ! [S3: stream_rule,A2: set_rule] :
      ( ( ord_less_eq_set_rule @ ( sset_rule @ S3 ) @ A2 )
     => ( member_stream_rule @ S3 @ ( streams_rule @ A2 ) ) ) ).

% sset_streams
thf(fact_374_sset__streams,axiom,
    ! [S3: stream_tm,A2: set_tm] :
      ( ( ord_less_eq_set_tm @ ( sset_tm @ S3 ) @ A2 )
     => ( member_stream_tm @ S3 @ ( streams_tm @ A2 ) ) ) ).

% sset_streams
thf(fact_375_fair__def,axiom,
    ! [Rs2: stream_rule] :
      ( ( abstra3799686578551160190r_rule @ rules @ Rs2 )
      = ( ( ord_less_eq_set_rule @ ( sset_rule @ Rs2 ) @ ( sset_rule @ rules ) )
        & ! [X: rule] :
            ( ( member_rule @ X @ ( sset_rule @ rules ) )
           => ( linear1087404401031714582w_rule
              @ ( linear292482573288640205v_rule
                @ ( linear7354821883049305950s_rule
                  @ ( ^ [Y2: rule,Z2: rule] : ( Y2 = Z2 )
                    @ X ) ) )
              @ Rs2 ) ) ) ) ).

% fair_def
thf(fact_376_alw__HLD__iff__streams,axiom,
    ! [X5: set_rule,Omega2: stream_rule] :
      ( ( linear1087404401031714582w_rule @ ( linear5032321834864961386D_rule @ X5 ) @ Omega2 )
      = ( member_stream_rule @ Omega2 @ ( streams_rule @ X5 ) ) ) ).

% alw_HLD_iff_streams
thf(fact_377_alw__HLD__iff__streams,axiom,
    ! [X5: set_Pr1822751329126368876m_rule,Omega2: stream2709947120125613254m_rule] :
      ( ( linear4957294957385550906m_rule @ ( linear2885739836753817742m_rule @ X5 ) @ Omega2 )
      = ( member5228318729319728861m_rule @ Omega2 @ ( stream3418479298939905342m_rule @ X5 ) ) ) ).

% alw_HLD_iff_streams
thf(fact_378_in__mono,axiom,
    ! [A2: set_nat,B5: set_nat,X3: nat] :
      ( ( ord_less_eq_set_nat @ A2 @ B5 )
     => ( ( member_nat @ X3 @ A2 )
       => ( member_nat @ X3 @ B5 ) ) ) ).

% in_mono
thf(fact_379_in__mono,axiom,
    ! [A2: set_Pr5202636777678657877ist_fm,B5: set_Pr5202636777678657877ist_fm,X3: produc6018962875968178549ist_fm] :
      ( ( ord_le1771420097867575541ist_fm @ A2 @ B5 )
     => ( ( member4699826688122452638ist_fm @ X3 @ A2 )
       => ( member4699826688122452638ist_fm @ X3 @ B5 ) ) ) ).

% in_mono
thf(fact_380_in__mono,axiom,
    ! [A2: set_Pr8693737435421807431at_nat,B5: set_Pr8693737435421807431at_nat,X3: produc859450856879609959at_nat] :
      ( ( ord_le3000389064537975527at_nat @ A2 @ B5 )
     => ( ( member8206827879206165904at_nat @ X3 @ A2 )
       => ( member8206827879206165904at_nat @ X3 @ B5 ) ) ) ).

% in_mono
thf(fact_381_in__mono,axiom,
    ! [A2: set_rule,B5: set_rule,X3: rule] :
      ( ( ord_less_eq_set_rule @ A2 @ B5 )
     => ( ( member_rule @ X3 @ A2 )
       => ( member_rule @ X3 @ B5 ) ) ) ).

% in_mono
thf(fact_382_in__mono,axiom,
    ! [A2: set_tm,B5: set_tm,X3: tm] :
      ( ( ord_less_eq_set_tm @ A2 @ B5 )
     => ( ( member_tm @ X3 @ A2 )
       => ( member_tm @ X3 @ B5 ) ) ) ).

% in_mono
thf(fact_383_subsetD,axiom,
    ! [A2: set_nat,B5: set_nat,C: nat] :
      ( ( ord_less_eq_set_nat @ A2 @ B5 )
     => ( ( member_nat @ C @ A2 )
       => ( member_nat @ C @ B5 ) ) ) ).

% subsetD
thf(fact_384_subsetD,axiom,
    ! [A2: set_Pr5202636777678657877ist_fm,B5: set_Pr5202636777678657877ist_fm,C: produc6018962875968178549ist_fm] :
      ( ( ord_le1771420097867575541ist_fm @ A2 @ B5 )
     => ( ( member4699826688122452638ist_fm @ C @ A2 )
       => ( member4699826688122452638ist_fm @ C @ B5 ) ) ) ).

% subsetD
thf(fact_385_subsetD,axiom,
    ! [A2: set_Pr8693737435421807431at_nat,B5: set_Pr8693737435421807431at_nat,C: produc859450856879609959at_nat] :
      ( ( ord_le3000389064537975527at_nat @ A2 @ B5 )
     => ( ( member8206827879206165904at_nat @ C @ A2 )
       => ( member8206827879206165904at_nat @ C @ B5 ) ) ) ).

% subsetD
thf(fact_386_subsetD,axiom,
    ! [A2: set_rule,B5: set_rule,C: rule] :
      ( ( ord_less_eq_set_rule @ A2 @ B5 )
     => ( ( member_rule @ C @ A2 )
       => ( member_rule @ C @ B5 ) ) ) ).

% subsetD
thf(fact_387_subsetD,axiom,
    ! [A2: set_tm,B5: set_tm,C: tm] :
      ( ( ord_less_eq_set_tm @ A2 @ B5 )
     => ( ( member_tm @ C @ A2 )
       => ( member_tm @ C @ B5 ) ) ) ).

% subsetD
thf(fact_388_equalityE,axiom,
    ! [A2: set_rule,B5: set_rule] :
      ( ( A2 = B5 )
     => ~ ( ( ord_less_eq_set_rule @ A2 @ B5 )
         => ~ ( ord_less_eq_set_rule @ B5 @ A2 ) ) ) ).

% equalityE
thf(fact_389_equalityE,axiom,
    ! [A2: set_tm,B5: set_tm] :
      ( ( A2 = B5 )
     => ~ ( ( ord_less_eq_set_tm @ A2 @ B5 )
         => ~ ( ord_less_eq_set_tm @ B5 @ A2 ) ) ) ).

% equalityE
thf(fact_390_subset__eq,axiom,
    ( ord_less_eq_set_nat
    = ( ^ [A5: set_nat,B4: set_nat] :
        ! [X: nat] :
          ( ( member_nat @ X @ A5 )
         => ( member_nat @ X @ B4 ) ) ) ) ).

% subset_eq
thf(fact_391_subset__eq,axiom,
    ( ord_le1771420097867575541ist_fm
    = ( ^ [A5: set_Pr5202636777678657877ist_fm,B4: set_Pr5202636777678657877ist_fm] :
        ! [X: produc6018962875968178549ist_fm] :
          ( ( member4699826688122452638ist_fm @ X @ A5 )
         => ( member4699826688122452638ist_fm @ X @ B4 ) ) ) ) ).

% subset_eq
thf(fact_392_subset__eq,axiom,
    ( ord_le3000389064537975527at_nat
    = ( ^ [A5: set_Pr8693737435421807431at_nat,B4: set_Pr8693737435421807431at_nat] :
        ! [X: produc859450856879609959at_nat] :
          ( ( member8206827879206165904at_nat @ X @ A5 )
         => ( member8206827879206165904at_nat @ X @ B4 ) ) ) ) ).

% subset_eq
thf(fact_393_subset__eq,axiom,
    ( ord_less_eq_set_rule
    = ( ^ [A5: set_rule,B4: set_rule] :
        ! [X: rule] :
          ( ( member_rule @ X @ A5 )
         => ( member_rule @ X @ B4 ) ) ) ) ).

% subset_eq
thf(fact_394_subset__eq,axiom,
    ( ord_less_eq_set_tm
    = ( ^ [A5: set_tm,B4: set_tm] :
        ! [X: tm] :
          ( ( member_tm @ X @ A5 )
         => ( member_tm @ X @ B4 ) ) ) ) ).

% subset_eq
thf(fact_395_equalityD1,axiom,
    ! [A2: set_rule,B5: set_rule] :
      ( ( A2 = B5 )
     => ( ord_less_eq_set_rule @ A2 @ B5 ) ) ).

% equalityD1
thf(fact_396_equalityD1,axiom,
    ! [A2: set_tm,B5: set_tm] :
      ( ( A2 = B5 )
     => ( ord_less_eq_set_tm @ A2 @ B5 ) ) ).

% equalityD1
thf(fact_397_equalityD2,axiom,
    ! [A2: set_rule,B5: set_rule] :
      ( ( A2 = B5 )
     => ( ord_less_eq_set_rule @ B5 @ A2 ) ) ).

% equalityD2
thf(fact_398_equalityD2,axiom,
    ! [A2: set_tm,B5: set_tm] :
      ( ( A2 = B5 )
     => ( ord_less_eq_set_tm @ B5 @ A2 ) ) ).

% equalityD2
thf(fact_399_subset__iff,axiom,
    ( ord_less_eq_set_nat
    = ( ^ [A5: set_nat,B4: set_nat] :
        ! [T2: nat] :
          ( ( member_nat @ T2 @ A5 )
         => ( member_nat @ T2 @ B4 ) ) ) ) ).

% subset_iff
thf(fact_400_subset__iff,axiom,
    ( ord_le1771420097867575541ist_fm
    = ( ^ [A5: set_Pr5202636777678657877ist_fm,B4: set_Pr5202636777678657877ist_fm] :
        ! [T2: produc6018962875968178549ist_fm] :
          ( ( member4699826688122452638ist_fm @ T2 @ A5 )
         => ( member4699826688122452638ist_fm @ T2 @ B4 ) ) ) ) ).

% subset_iff
thf(fact_401_subset__iff,axiom,
    ( ord_le3000389064537975527at_nat
    = ( ^ [A5: set_Pr8693737435421807431at_nat,B4: set_Pr8693737435421807431at_nat] :
        ! [T2: produc859450856879609959at_nat] :
          ( ( member8206827879206165904at_nat @ T2 @ A5 )
         => ( member8206827879206165904at_nat @ T2 @ B4 ) ) ) ) ).

% subset_iff
thf(fact_402_subset__iff,axiom,
    ( ord_less_eq_set_rule
    = ( ^ [A5: set_rule,B4: set_rule] :
        ! [T2: rule] :
          ( ( member_rule @ T2 @ A5 )
         => ( member_rule @ T2 @ B4 ) ) ) ) ).

% subset_iff
thf(fact_403_subset__iff,axiom,
    ( ord_less_eq_set_tm
    = ( ^ [A5: set_tm,B4: set_tm] :
        ! [T2: tm] :
          ( ( member_tm @ T2 @ A5 )
         => ( member_tm @ T2 @ B4 ) ) ) ) ).

% subset_iff
thf(fact_404_subset__refl,axiom,
    ! [A2: set_rule] : ( ord_less_eq_set_rule @ A2 @ A2 ) ).

% subset_refl
thf(fact_405_subset__refl,axiom,
    ! [A2: set_tm] : ( ord_less_eq_set_tm @ A2 @ A2 ) ).

% subset_refl
thf(fact_406_Collect__mono,axiom,
    ! [P2: nat > $o,Q: nat > $o] :
      ( ! [X2: nat] :
          ( ( P2 @ X2 )
         => ( Q @ X2 ) )
     => ( ord_less_eq_set_nat @ ( collect_nat @ P2 ) @ ( collect_nat @ Q ) ) ) ).

% Collect_mono
thf(fact_407_Collect__mono,axiom,
    ! [P2: rule > $o,Q: rule > $o] :
      ( ! [X2: rule] :
          ( ( P2 @ X2 )
         => ( Q @ X2 ) )
     => ( ord_less_eq_set_rule @ ( collect_rule @ P2 ) @ ( collect_rule @ Q ) ) ) ).

% Collect_mono
thf(fact_408_Collect__mono,axiom,
    ! [P2: tm > $o,Q: tm > $o] :
      ( ! [X2: tm] :
          ( ( P2 @ X2 )
         => ( Q @ X2 ) )
     => ( ord_less_eq_set_tm @ ( collect_tm @ P2 ) @ ( collect_tm @ Q ) ) ) ).

% Collect_mono
thf(fact_409_subset__trans,axiom,
    ! [A2: set_rule,B5: set_rule,C2: set_rule] :
      ( ( ord_less_eq_set_rule @ A2 @ B5 )
     => ( ( ord_less_eq_set_rule @ B5 @ C2 )
       => ( ord_less_eq_set_rule @ A2 @ C2 ) ) ) ).

% subset_trans
thf(fact_410_subset__trans,axiom,
    ! [A2: set_tm,B5: set_tm,C2: set_tm] :
      ( ( ord_less_eq_set_tm @ A2 @ B5 )
     => ( ( ord_less_eq_set_tm @ B5 @ C2 )
       => ( ord_less_eq_set_tm @ A2 @ C2 ) ) ) ).

% subset_trans
thf(fact_411_set__eq__subset,axiom,
    ( ( ^ [Y2: set_rule,Z2: set_rule] : ( Y2 = Z2 ) )
    = ( ^ [A5: set_rule,B4: set_rule] :
          ( ( ord_less_eq_set_rule @ A5 @ B4 )
          & ( ord_less_eq_set_rule @ B4 @ A5 ) ) ) ) ).

% set_eq_subset
thf(fact_412_set__eq__subset,axiom,
    ( ( ^ [Y2: set_tm,Z2: set_tm] : ( Y2 = Z2 ) )
    = ( ^ [A5: set_tm,B4: set_tm] :
          ( ( ord_less_eq_set_tm @ A5 @ B4 )
          & ( ord_less_eq_set_tm @ B4 @ A5 ) ) ) ) ).

% set_eq_subset
thf(fact_413_Collect__mono__iff,axiom,
    ! [P2: nat > $o,Q: nat > $o] :
      ( ( ord_less_eq_set_nat @ ( collect_nat @ P2 ) @ ( collect_nat @ Q ) )
      = ( ! [X: nat] :
            ( ( P2 @ X )
           => ( Q @ X ) ) ) ) ).

% Collect_mono_iff
thf(fact_414_Collect__mono__iff,axiom,
    ! [P2: rule > $o,Q: rule > $o] :
      ( ( ord_less_eq_set_rule @ ( collect_rule @ P2 ) @ ( collect_rule @ Q ) )
      = ( ! [X: rule] :
            ( ( P2 @ X )
           => ( Q @ X ) ) ) ) ).

% Collect_mono_iff
thf(fact_415_Collect__mono__iff,axiom,
    ! [P2: tm > $o,Q: tm > $o] :
      ( ( ord_less_eq_set_tm @ ( collect_tm @ P2 ) @ ( collect_tm @ Q ) )
      = ( ! [X: tm] :
            ( ( P2 @ X )
           => ( Q @ X ) ) ) ) ).

% Collect_mono_iff
thf(fact_416_Collect__subset,axiom,
    ! [A2: set_Pr5202636777678657877ist_fm,P2: produc6018962875968178549ist_fm > $o] :
      ( ord_le1771420097867575541ist_fm
      @ ( collec5845330005437105888ist_fm
        @ ^ [X: produc6018962875968178549ist_fm] :
            ( ( member4699826688122452638ist_fm @ X @ A2 )
            & ( P2 @ X ) ) )
      @ A2 ) ).

% Collect_subset
thf(fact_417_Collect__subset,axiom,
    ! [A2: set_Pr8693737435421807431at_nat,P2: produc859450856879609959at_nat > $o] :
      ( ord_le3000389064537975527at_nat
      @ ( collec7088162979684241874at_nat
        @ ^ [X: produc859450856879609959at_nat] :
            ( ( member8206827879206165904at_nat @ X @ A2 )
            & ( P2 @ X ) ) )
      @ A2 ) ).

% Collect_subset
thf(fact_418_Collect__subset,axiom,
    ! [A2: set_nat,P2: nat > $o] :
      ( ord_less_eq_set_nat
      @ ( collect_nat
        @ ^ [X: nat] :
            ( ( member_nat @ X @ A2 )
            & ( P2 @ X ) ) )
      @ A2 ) ).

% Collect_subset
thf(fact_419_Collect__subset,axiom,
    ! [A2: set_rule,P2: rule > $o] :
      ( ord_less_eq_set_rule
      @ ( collect_rule
        @ ^ [X: rule] :
            ( ( member_rule @ X @ A2 )
            & ( P2 @ X ) ) )
      @ A2 ) ).

% Collect_subset
thf(fact_420_Collect__subset,axiom,
    ! [A2: set_tm,P2: tm > $o] :
      ( ord_less_eq_set_tm
      @ ( collect_tm
        @ ^ [X: tm] :
            ( ( member_tm @ X @ A2 )
            & ( P2 @ X ) ) )
      @ A2 ) ).

% Collect_subset
thf(fact_421_subset__CollectI,axiom,
    ! [B5: set_Pr5202636777678657877ist_fm,A2: set_Pr5202636777678657877ist_fm,Q: produc6018962875968178549ist_fm > $o,P2: produc6018962875968178549ist_fm > $o] :
      ( ( ord_le1771420097867575541ist_fm @ B5 @ A2 )
     => ( ! [X2: produc6018962875968178549ist_fm] :
            ( ( member4699826688122452638ist_fm @ X2 @ B5 )
           => ( ( Q @ X2 )
             => ( P2 @ X2 ) ) )
       => ( ord_le1771420097867575541ist_fm
          @ ( collec5845330005437105888ist_fm
            @ ^ [X: produc6018962875968178549ist_fm] :
                ( ( member4699826688122452638ist_fm @ X @ B5 )
                & ( Q @ X ) ) )
          @ ( collec5845330005437105888ist_fm
            @ ^ [X: produc6018962875968178549ist_fm] :
                ( ( member4699826688122452638ist_fm @ X @ A2 )
                & ( P2 @ X ) ) ) ) ) ) ).

% subset_CollectI
thf(fact_422_subset__CollectI,axiom,
    ! [B5: set_Pr8693737435421807431at_nat,A2: set_Pr8693737435421807431at_nat,Q: produc859450856879609959at_nat > $o,P2: produc859450856879609959at_nat > $o] :
      ( ( ord_le3000389064537975527at_nat @ B5 @ A2 )
     => ( ! [X2: produc859450856879609959at_nat] :
            ( ( member8206827879206165904at_nat @ X2 @ B5 )
           => ( ( Q @ X2 )
             => ( P2 @ X2 ) ) )
       => ( ord_le3000389064537975527at_nat
          @ ( collec7088162979684241874at_nat
            @ ^ [X: produc859450856879609959at_nat] :
                ( ( member8206827879206165904at_nat @ X @ B5 )
                & ( Q @ X ) ) )
          @ ( collec7088162979684241874at_nat
            @ ^ [X: produc859450856879609959at_nat] :
                ( ( member8206827879206165904at_nat @ X @ A2 )
                & ( P2 @ X ) ) ) ) ) ) ).

% subset_CollectI
thf(fact_423_subset__CollectI,axiom,
    ! [B5: set_nat,A2: set_nat,Q: nat > $o,P2: nat > $o] :
      ( ( ord_less_eq_set_nat @ B5 @ A2 )
     => ( ! [X2: nat] :
            ( ( member_nat @ X2 @ B5 )
           => ( ( Q @ X2 )
             => ( P2 @ X2 ) ) )
       => ( ord_less_eq_set_nat
          @ ( collect_nat
            @ ^ [X: nat] :
                ( ( member_nat @ X @ B5 )
                & ( Q @ X ) ) )
          @ ( collect_nat
            @ ^ [X: nat] :
                ( ( member_nat @ X @ A2 )
                & ( P2 @ X ) ) ) ) ) ) ).

% subset_CollectI
thf(fact_424_subset__CollectI,axiom,
    ! [B5: set_rule,A2: set_rule,Q: rule > $o,P2: rule > $o] :
      ( ( ord_less_eq_set_rule @ B5 @ A2 )
     => ( ! [X2: rule] :
            ( ( member_rule @ X2 @ B5 )
           => ( ( Q @ X2 )
             => ( P2 @ X2 ) ) )
       => ( ord_less_eq_set_rule
          @ ( collect_rule
            @ ^ [X: rule] :
                ( ( member_rule @ X @ B5 )
                & ( Q @ X ) ) )
          @ ( collect_rule
            @ ^ [X: rule] :
                ( ( member_rule @ X @ A2 )
                & ( P2 @ X ) ) ) ) ) ) ).

% subset_CollectI
thf(fact_425_subset__CollectI,axiom,
    ! [B5: set_tm,A2: set_tm,Q: tm > $o,P2: tm > $o] :
      ( ( ord_less_eq_set_tm @ B5 @ A2 )
     => ( ! [X2: tm] :
            ( ( member_tm @ X2 @ B5 )
           => ( ( Q @ X2 )
             => ( P2 @ X2 ) ) )
       => ( ord_less_eq_set_tm
          @ ( collect_tm
            @ ^ [X: tm] :
                ( ( member_tm @ X @ B5 )
                & ( Q @ X ) ) )
          @ ( collect_tm
            @ ^ [X: tm] :
                ( ( member_tm @ X @ A2 )
                & ( P2 @ X ) ) ) ) ) ) ).

% subset_CollectI
thf(fact_426_fair__uu__def,axiom,
    abstra4598247580560492629u_rule = abstra3799686578551160190r_rule ).

% fair_uu_def
thf(fact_427_abort__Bleast__def,axiom,
    ( abort_Bleast_nat
    = ( ^ [S5: set_nat,P: nat > $o] :
          ( ord_Least_nat
          @ ^ [X: nat] :
              ( ( member_nat @ X @ S5 )
              & ( P @ X ) ) ) ) ) ).

% abort_Bleast_def
thf(fact_428_Bleast__def,axiom,
    ( bleast_nat
    = ( ^ [S5: set_nat,P: nat > $o] :
          ( ord_Least_nat
          @ ^ [X: nat] :
              ( ( member_nat @ X @ S5 )
              & ( P @ X ) ) ) ) ) ).

% Bleast_def
thf(fact_429_conj__subset__def,axiom,
    ! [A2: set_nat,P2: nat > $o,Q: nat > $o] :
      ( ( ord_less_eq_set_nat @ A2
        @ ( collect_nat
          @ ^ [X: nat] :
              ( ( P2 @ X )
              & ( Q @ X ) ) ) )
      = ( ( ord_less_eq_set_nat @ A2 @ ( collect_nat @ P2 ) )
        & ( ord_less_eq_set_nat @ A2 @ ( collect_nat @ Q ) ) ) ) ).

% conj_subset_def
thf(fact_430_conj__subset__def,axiom,
    ! [A2: set_rule,P2: rule > $o,Q: rule > $o] :
      ( ( ord_less_eq_set_rule @ A2
        @ ( collect_rule
          @ ^ [X: rule] :
              ( ( P2 @ X )
              & ( Q @ X ) ) ) )
      = ( ( ord_less_eq_set_rule @ A2 @ ( collect_rule @ P2 ) )
        & ( ord_less_eq_set_rule @ A2 @ ( collect_rule @ Q ) ) ) ) ).

% conj_subset_def
thf(fact_431_conj__subset__def,axiom,
    ! [A2: set_tm,P2: tm > $o,Q: tm > $o] :
      ( ( ord_less_eq_set_tm @ A2
        @ ( collect_tm
          @ ^ [X: tm] :
              ( ( P2 @ X )
              & ( Q @ X ) ) ) )
      = ( ( ord_less_eq_set_tm @ A2 @ ( collect_tm @ P2 ) )
        & ( ord_less_eq_set_tm @ A2 @ ( collect_tm @ Q ) ) ) ) ).

% conj_subset_def
thf(fact_432_prop__restrict,axiom,
    ! [X3: produc6018962875968178549ist_fm,Z3: set_Pr5202636777678657877ist_fm,X5: set_Pr5202636777678657877ist_fm,P2: produc6018962875968178549ist_fm > $o] :
      ( ( member4699826688122452638ist_fm @ X3 @ Z3 )
     => ( ( ord_le1771420097867575541ist_fm @ Z3
          @ ( collec5845330005437105888ist_fm
            @ ^ [X: produc6018962875968178549ist_fm] :
                ( ( member4699826688122452638ist_fm @ X @ X5 )
                & ( P2 @ X ) ) ) )
       => ( P2 @ X3 ) ) ) ).

% prop_restrict
thf(fact_433_prop__restrict,axiom,
    ! [X3: produc859450856879609959at_nat,Z3: set_Pr8693737435421807431at_nat,X5: set_Pr8693737435421807431at_nat,P2: produc859450856879609959at_nat > $o] :
      ( ( member8206827879206165904at_nat @ X3 @ Z3 )
     => ( ( ord_le3000389064537975527at_nat @ Z3
          @ ( collec7088162979684241874at_nat
            @ ^ [X: produc859450856879609959at_nat] :
                ( ( member8206827879206165904at_nat @ X @ X5 )
                & ( P2 @ X ) ) ) )
       => ( P2 @ X3 ) ) ) ).

% prop_restrict
thf(fact_434_prop__restrict,axiom,
    ! [X3: nat,Z3: set_nat,X5: set_nat,P2: nat > $o] :
      ( ( member_nat @ X3 @ Z3 )
     => ( ( ord_less_eq_set_nat @ Z3
          @ ( collect_nat
            @ ^ [X: nat] :
                ( ( member_nat @ X @ X5 )
                & ( P2 @ X ) ) ) )
       => ( P2 @ X3 ) ) ) ).

% prop_restrict
thf(fact_435_prop__restrict,axiom,
    ! [X3: rule,Z3: set_rule,X5: set_rule,P2: rule > $o] :
      ( ( member_rule @ X3 @ Z3 )
     => ( ( ord_less_eq_set_rule @ Z3
          @ ( collect_rule
            @ ^ [X: rule] :
                ( ( member_rule @ X @ X5 )
                & ( P2 @ X ) ) ) )
       => ( P2 @ X3 ) ) ) ).

% prop_restrict
thf(fact_436_prop__restrict,axiom,
    ! [X3: tm,Z3: set_tm,X5: set_tm,P2: tm > $o] :
      ( ( member_tm @ X3 @ Z3 )
     => ( ( ord_less_eq_set_tm @ Z3
          @ ( collect_tm
            @ ^ [X: tm] :
                ( ( member_tm @ X @ X5 )
                & ( P2 @ X ) ) ) )
       => ( P2 @ X3 ) ) ) ).

% prop_restrict
thf(fact_437_Collect__restrict,axiom,
    ! [X5: set_Pr5202636777678657877ist_fm,P2: produc6018962875968178549ist_fm > $o] :
      ( ord_le1771420097867575541ist_fm
      @ ( collec5845330005437105888ist_fm
        @ ^ [X: produc6018962875968178549ist_fm] :
            ( ( member4699826688122452638ist_fm @ X @ X5 )
            & ( P2 @ X ) ) )
      @ X5 ) ).

% Collect_restrict
thf(fact_438_Collect__restrict,axiom,
    ! [X5: set_Pr8693737435421807431at_nat,P2: produc859450856879609959at_nat > $o] :
      ( ord_le3000389064537975527at_nat
      @ ( collec7088162979684241874at_nat
        @ ^ [X: produc859450856879609959at_nat] :
            ( ( member8206827879206165904at_nat @ X @ X5 )
            & ( P2 @ X ) ) )
      @ X5 ) ).

% Collect_restrict
thf(fact_439_Collect__restrict,axiom,
    ! [X5: set_nat,P2: nat > $o] :
      ( ord_less_eq_set_nat
      @ ( collect_nat
        @ ^ [X: nat] :
            ( ( member_nat @ X @ X5 )
            & ( P2 @ X ) ) )
      @ X5 ) ).

% Collect_restrict
thf(fact_440_Collect__restrict,axiom,
    ! [X5: set_rule,P2: rule > $o] :
      ( ord_less_eq_set_rule
      @ ( collect_rule
        @ ^ [X: rule] :
            ( ( member_rule @ X @ X5 )
            & ( P2 @ X ) ) )
      @ X5 ) ).

% Collect_restrict
thf(fact_441_Collect__restrict,axiom,
    ! [X5: set_tm,P2: tm > $o] :
      ( ord_less_eq_set_tm
      @ ( collect_tm
        @ ^ [X: tm] :
            ( ( member_tm @ X @ X5 )
            & ( P2 @ X ) ) )
      @ X5 ) ).

% Collect_restrict
thf(fact_442_enumerate__mono__le__iff,axiom,
    ! [S4: set_nat,M2: nat,N: nat] :
      ( ~ ( finite_finite_nat @ S4 )
     => ( ( ord_less_eq_nat @ ( infini8530281810654367211te_nat @ S4 @ M2 ) @ ( infini8530281810654367211te_nat @ S4 @ N ) )
        = ( ord_less_eq_nat @ M2 @ N ) ) ) ).

% enumerate_mono_le_iff
thf(fact_443_Fpow__def,axiom,
    ( finite_Fpow_nat
    = ( ^ [A5: set_nat] :
          ( collect_set_nat
          @ ^ [X6: set_nat] :
              ( ( ord_less_eq_set_nat @ X6 @ A5 )
              & ( finite_finite_nat @ X6 ) ) ) ) ) ).

% Fpow_def
thf(fact_444_Fpow__def,axiom,
    ( finite_Fpow_rule
    = ( ^ [A5: set_rule] :
          ( collect_set_rule
          @ ^ [X6: set_rule] :
              ( ( ord_less_eq_set_rule @ X6 @ A5 )
              & ( finite_finite_rule @ X6 ) ) ) ) ) ).

% Fpow_def
thf(fact_445_Fpow__def,axiom,
    ( finite_Fpow_tm
    = ( ^ [A5: set_tm] :
          ( collect_set_tm
          @ ^ [X6: set_tm] :
              ( ( ord_less_eq_set_tm @ X6 @ A5 )
              & ( finite_finite_tm @ X6 ) ) ) ) ) ).

% Fpow_def
thf(fact_446_enumerate__in__set,axiom,
    ! [S4: set_nat,N: nat] :
      ( ~ ( finite_finite_nat @ S4 )
     => ( member_nat @ ( infini8530281810654367211te_nat @ S4 @ N ) @ S4 ) ) ).

% enumerate_in_set
thf(fact_447_enumerate__Ex,axiom,
    ! [S4: set_nat,S3: nat] :
      ( ~ ( finite_finite_nat @ S4 )
     => ( ( member_nat @ S3 @ S4 )
       => ? [N2: nat] :
            ( ( infini8530281810654367211te_nat @ S4 @ N2 )
            = S3 ) ) ) ).

% enumerate_Ex
thf(fact_448_Fpow__mono,axiom,
    ! [A2: set_rule,B5: set_rule] :
      ( ( ord_less_eq_set_rule @ A2 @ B5 )
     => ( ord_le4124860622784042334t_rule @ ( finite_Fpow_rule @ A2 ) @ ( finite_Fpow_rule @ B5 ) ) ) ).

% Fpow_mono
thf(fact_449_Fpow__mono,axiom,
    ! [A2: set_tm,B5: set_tm] :
      ( ( ord_less_eq_set_tm @ A2 @ B5 )
     => ( ord_le5601931644483074373set_tm @ ( finite_Fpow_tm @ A2 ) @ ( finite_Fpow_tm @ B5 ) ) ) ).

% Fpow_mono
thf(fact_450_le__enumerate,axiom,
    ! [S4: set_nat,N: nat] :
      ( ~ ( finite_finite_nat @ S4 )
     => ( ord_less_eq_nat @ N @ ( infini8530281810654367211te_nat @ S4 @ N ) ) ) ).

% le_enumerate
thf(fact_451_fair__rules,axiom,
    abstra3799686578551160190r_rule @ rules @ rules ).

% fair_rules
thf(fact_452_fair__fenum,axiom,
    abstra3799686578551160190r_rule @ rules @ ( abstra745658567949189203m_rule @ rules ) ).

% fair_fenum
thf(fact_453_sset__fenum,axiom,
    ( ( sset_rule @ ( abstra745658567949189203m_rule @ rules ) )
    = ( sset_rule @ rules ) ) ).

% sset_fenum
thf(fact_454_saturated__def,axiom,
    ! [R: rule] :
      ( ( abstra2533313685312581075ist_fm @ eff @ R )
      = ( linear4957294957385550906m_rule
        @ ^ [Xs: stream2709947120125613254m_rule] :
            ( ( linear7925859179493631106m_rule
              @ ^ [Step: produc340336539035504054m_rule] : ( abstra1707737187183579335ist_fm @ eff @ R @ ( produc993771546023551940m_rule @ Step ) )
              @ Xs )
           => ( linear1816559245158234865m_rule
              @ ( linear7925859179493631106m_rule
                @ ^ [Step: produc340336539035504054m_rule] :
                    ( ( produc6514711208036230658m_rule @ Step )
                    = R ) )
              @ Xs ) ) ) ) ).

% saturated_def
thf(fact_455_enabled__R,axiom,
    ! [S3: produc6018962875968178549ist_fm] :
      ( ( member4699826688122452638ist_fm @ S3 @ top_to1730629564055774885ist_fm )
     => ? [X2: rule] :
          ( ( member_rule @ X2 @ ( sset_rule @ rules ) )
          & ? [X_1: fset_P8989946509869081563ist_fm] : ( eff @ X2 @ S3 @ X_1 ) ) ) ).

% enabled_R
thf(fact_456_i_OSaturated__def,axiom,
    ! [Eff2: rule > produc6018962875968178549ist_fm > option6967287582980624417ist_fm,Rules2: stream_rule,Steps: stream2709947120125613254m_rule] :
      ( ( abstra6097777249025082867ist_fm @ ( abstra2682625350522704545ist_fm @ Eff2 ) @ Rules2 @ Steps )
      = ( ! [X: rule] :
            ( ( member_rule @ X @ ( sset_rule @ Rules2 ) )
           => ( abstra2533313685312581075ist_fm @ ( abstra2682625350522704545ist_fm @ Eff2 ) @ X @ Steps ) ) ) ) ).

% i.Saturated_def
thf(fact_457_UNIV__I,axiom,
    ! [X3: rule] : ( member_rule @ X3 @ top_top_set_rule ) ).

% UNIV_I
thf(fact_458_UNIV__I,axiom,
    ! [X3: produc859450856879609959at_nat] : ( member8206827879206165904at_nat @ X3 @ top_to8454300755403694231at_nat ) ).

% UNIV_I
thf(fact_459_UNIV__I,axiom,
    ! [X3: produc6018962875968178549ist_fm] : ( member4699826688122452638ist_fm @ X3 @ top_to1730629564055774885ist_fm ) ).

% UNIV_I
thf(fact_460_UNIV__I,axiom,
    ! [X3: nat] : ( member_nat @ X3 @ top_top_set_nat ) ).

% UNIV_I
thf(fact_461_finite__Plus__UNIV__iff,axiom,
    ( ( finite2960343109384769532ist_fm @ top_to3275842641920885323ist_fm )
    = ( ( finite600313802051052958ist_fm @ top_to1730629564055774885ist_fm )
      & ( finite600313802051052958ist_fm @ top_to1730629564055774885ist_fm ) ) ) ).

% finite_Plus_UNIV_iff
thf(fact_462_finite__Plus__UNIV__iff,axiom,
    ( ( finite1526080435621356319fm_nat @ top_to1087575569247603110fm_nat )
    = ( ( finite600313802051052958ist_fm @ top_to1730629564055774885ist_fm )
      & ( finite_finite_nat @ top_top_set_nat ) ) ) ).

% finite_Plus_UNIV_iff
thf(fact_463_finite__Plus__UNIV__iff,axiom,
    ( ( finite3904696927349524165ist_fm @ top_to1445627146280761932ist_fm )
    = ( ( finite_finite_nat @ top_top_set_nat )
      & ( finite600313802051052958ist_fm @ top_to1730629564055774885ist_fm ) ) ) ).

% finite_Plus_UNIV_iff
thf(fact_464_finite__Plus__UNIV__iff,axiom,
    ( ( finite6187706683773761046at_nat @ top_to6661820994512907621at_nat )
    = ( ( finite_finite_nat @ top_top_set_nat )
      & ( finite_finite_nat @ top_top_set_nat ) ) ) ).

% finite_Plus_UNIV_iff
thf(fact_465_streams__UNIV,axiom,
    ( ( stream1217541351357117421ist_fm @ top_to1730629564055774885ist_fm )
    = top_to7911878802019153963ist_fm ) ).

% streams_UNIV
thf(fact_466_streams__UNIV,axiom,
    ( ( streams_nat @ top_top_set_nat )
    = top_to7548458143485696966am_nat ) ).

% streams_UNIV
thf(fact_467_finite__Collect__not,axiom,
    ! [P2: produc6018962875968178549ist_fm > $o] :
      ( ( finite600313802051052958ist_fm @ ( collec5845330005437105888ist_fm @ P2 ) )
     => ( ( finite600313802051052958ist_fm
          @ ( collec5845330005437105888ist_fm
            @ ^ [X: produc6018962875968178549ist_fm] :
                ~ ( P2 @ X ) ) )
        = ( finite600313802051052958ist_fm @ top_to1730629564055774885ist_fm ) ) ) ).

% finite_Collect_not
thf(fact_468_finite__Collect__not,axiom,
    ! [P2: nat > $o] :
      ( ( finite_finite_nat @ ( collect_nat @ P2 ) )
     => ( ( finite_finite_nat
          @ ( collect_nat
            @ ^ [X: nat] :
                ~ ( P2 @ X ) ) )
        = ( finite_finite_nat @ top_top_set_nat ) ) ) ).

% finite_Collect_not
thf(fact_469_finite__prod,axiom,
    ( ( finite7504396045227389584ist_fm @ top_to8800349970593740439ist_fm )
    = ( ( finite600313802051052958ist_fm @ top_to1730629564055774885ist_fm )
      & ( finite600313802051052958ist_fm @ top_to1730629564055774885ist_fm ) ) ) ).

% finite_prod
thf(fact_470_finite__prod,axiom,
    ( ( finite6183226614883191691fm_nat @ top_to9116356840278544986fm_nat )
    = ( ( finite600313802051052958ist_fm @ top_to1730629564055774885ist_fm )
      & ( finite_finite_nat @ top_top_set_nat ) ) ) ).

% finite_prod
thf(fact_471_finite__prod,axiom,
    ( ( finite8561843106611359537ist_fm @ top_to251036380456928000ist_fm )
    = ( ( finite_finite_nat @ top_top_set_nat )
      & ( finite600313802051052958ist_fm @ top_to1730629564055774885ist_fm ) ) ) ).

% finite_prod
thf(fact_472_finite__prod,axiom,
    ( ( finite6177210948735845034at_nat @ top_to4669805908274784177at_nat )
    = ( ( finite_finite_nat @ top_top_set_nat )
      & ( finite_finite_nat @ top_top_set_nat ) ) ) ).

% finite_prod
thf(fact_473_finite__prod,axiom,
    ( ( finite600313802051052958ist_fm @ top_to1730629564055774885ist_fm )
    = ( ( finite837990543644588180ist_tm @ top_top_set_list_tm )
      & ( finite834038518558952994ist_fm @ top_top_set_list_fm ) ) ) ).

% finite_prod
thf(fact_474_finite__Prod__UNIV,axiom,
    ( ( finite837990543644588180ist_tm @ top_top_set_list_tm )
   => ( ( finite834038518558952994ist_fm @ top_top_set_list_fm )
     => ( finite600313802051052958ist_fm @ top_to1730629564055774885ist_fm ) ) ) ).

% finite_Prod_UNIV
thf(fact_475_finite__Prod__UNIV,axiom,
    ( ( finite600313802051052958ist_fm @ top_to1730629564055774885ist_fm )
   => ( ( finite600313802051052958ist_fm @ top_to1730629564055774885ist_fm )
     => ( finite7504396045227389584ist_fm @ top_to8800349970593740439ist_fm ) ) ) ).

% finite_Prod_UNIV
thf(fact_476_finite__Prod__UNIV,axiom,
    ( ( finite600313802051052958ist_fm @ top_to1730629564055774885ist_fm )
   => ( ( finite_finite_nat @ top_top_set_nat )
     => ( finite6183226614883191691fm_nat @ top_to9116356840278544986fm_nat ) ) ) ).

% finite_Prod_UNIV
thf(fact_477_finite__Prod__UNIV,axiom,
    ( ( finite_finite_nat @ top_top_set_nat )
   => ( ( finite600313802051052958ist_fm @ top_to1730629564055774885ist_fm )
     => ( finite8561843106611359537ist_fm @ top_to251036380456928000ist_fm ) ) ) ).

% finite_Prod_UNIV
thf(fact_478_finite__Prod__UNIV,axiom,
    ( ( finite_finite_nat @ top_top_set_nat )
   => ( ( finite_finite_nat @ top_top_set_nat )
     => ( finite6177210948735845034at_nat @ top_to4669805908274784177at_nat ) ) ) ).

% finite_Prod_UNIV
thf(fact_479_always__enabledAtStep,axiom,
    ! [R: rule,Xs2: produc340336539035504054m_rule] : ( abstra1707737187183579335ist_fm @ eff @ R @ ( produc993771546023551940m_rule @ Xs2 ) ) ).

% always_enabledAtStep
thf(fact_480_RuleSystem__Defs_Oenabled_Ocong,axiom,
    abstra1707737187183579335ist_fm = abstra1707737187183579335ist_fm ).

% RuleSystem_Defs.enabled.cong
thf(fact_481_RuleSystem__Defs_Oenabled__def,axiom,
    ( abstra1707737187183579335ist_fm
    = ( ^ [Eff: rule > produc6018962875968178549ist_fm > fset_P8989946509869081563ist_fm > $o,R4: rule,S: produc6018962875968178549ist_fm] :
        ? [X6: fset_P8989946509869081563ist_fm] : ( Eff @ R4 @ S @ X6 ) ) ) ).

% RuleSystem_Defs.enabled_def
thf(fact_482_i_Oenabled__def,axiom,
    ! [Eff2: rule > produc6018962875968178549ist_fm > option6967287582980624417ist_fm,R: rule,S3: produc6018962875968178549ist_fm] :
      ( ( abstra1707737187183579335ist_fm @ ( abstra2682625350522704545ist_fm @ Eff2 ) @ R @ S3 )
      = ( ? [X6: fset_P8989946509869081563ist_fm] : ( abstra2682625350522704545ist_fm @ Eff2 @ R @ S3 @ X6 ) ) ) ).

% i.enabled_def
thf(fact_483_RuleSystem__Defs_Ofenum_Ocong,axiom,
    abstra745658567949189203m_rule = abstra745658567949189203m_rule ).

% RuleSystem_Defs.fenum.cong
thf(fact_484_UNIV__witness,axiom,
    ? [X2: rule] : ( member_rule @ X2 @ top_top_set_rule ) ).

% UNIV_witness
thf(fact_485_UNIV__witness,axiom,
    ? [X2: produc859450856879609959at_nat] : ( member8206827879206165904at_nat @ X2 @ top_to8454300755403694231at_nat ) ).

% UNIV_witness
thf(fact_486_UNIV__witness,axiom,
    ? [X2: produc6018962875968178549ist_fm] : ( member4699826688122452638ist_fm @ X2 @ top_to1730629564055774885ist_fm ) ).

% UNIV_witness
thf(fact_487_UNIV__witness,axiom,
    ? [X2: nat] : ( member_nat @ X2 @ top_top_set_nat ) ).

% UNIV_witness
thf(fact_488_UNIV__eq__I,axiom,
    ! [A2: set_rule] :
      ( ! [X2: rule] : ( member_rule @ X2 @ A2 )
     => ( top_top_set_rule = A2 ) ) ).

% UNIV_eq_I
thf(fact_489_UNIV__eq__I,axiom,
    ! [A2: set_Pr8693737435421807431at_nat] :
      ( ! [X2: produc859450856879609959at_nat] : ( member8206827879206165904at_nat @ X2 @ A2 )
     => ( top_to8454300755403694231at_nat = A2 ) ) ).

% UNIV_eq_I
thf(fact_490_UNIV__eq__I,axiom,
    ! [A2: set_Pr5202636777678657877ist_fm] :
      ( ! [X2: produc6018962875968178549ist_fm] : ( member4699826688122452638ist_fm @ X2 @ A2 )
     => ( top_to1730629564055774885ist_fm = A2 ) ) ).

% UNIV_eq_I
thf(fact_491_UNIV__eq__I,axiom,
    ! [A2: set_nat] :
      ( ! [X2: nat] : ( member_nat @ X2 @ A2 )
     => ( top_top_set_nat = A2 ) ) ).

% UNIV_eq_I
thf(fact_492_UNIV__def,axiom,
    ( top_to1730629564055774885ist_fm
    = ( collec5845330005437105888ist_fm
      @ ^ [X: produc6018962875968178549ist_fm] : $true ) ) ).

% UNIV_def
thf(fact_493_UNIV__def,axiom,
    ( top_top_set_nat
    = ( collect_nat
      @ ^ [X: nat] : $true ) ) ).

% UNIV_def
thf(fact_494_top__greatest,axiom,
    ! [A: set_Pr5202636777678657877ist_fm] : ( ord_le1771420097867575541ist_fm @ A @ top_to1730629564055774885ist_fm ) ).

% top_greatest
thf(fact_495_top__greatest,axiom,
    ! [A: set_nat] : ( ord_less_eq_set_nat @ A @ top_top_set_nat ) ).

% top_greatest
thf(fact_496_top__greatest,axiom,
    ! [A: set_rule] : ( ord_less_eq_set_rule @ A @ top_top_set_rule ) ).

% top_greatest
thf(fact_497_top__greatest,axiom,
    ! [A: set_tm] : ( ord_less_eq_set_tm @ A @ top_top_set_tm ) ).

% top_greatest
thf(fact_498_top_Oextremum__unique,axiom,
    ! [A: set_Pr5202636777678657877ist_fm] :
      ( ( ord_le1771420097867575541ist_fm @ top_to1730629564055774885ist_fm @ A )
      = ( A = top_to1730629564055774885ist_fm ) ) ).

% top.extremum_unique
thf(fact_499_top_Oextremum__unique,axiom,
    ! [A: set_nat] :
      ( ( ord_less_eq_set_nat @ top_top_set_nat @ A )
      = ( A = top_top_set_nat ) ) ).

% top.extremum_unique
thf(fact_500_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_501_top_Oextremum__unique,axiom,
    ! [A: set_tm] :
      ( ( ord_less_eq_set_tm @ top_top_set_tm @ A )
      = ( A = top_top_set_tm ) ) ).

% top.extremum_unique
thf(fact_502_top_Oextremum__uniqueI,axiom,
    ! [A: set_Pr5202636777678657877ist_fm] :
      ( ( ord_le1771420097867575541ist_fm @ top_to1730629564055774885ist_fm @ A )
     => ( A = top_to1730629564055774885ist_fm ) ) ).

% top.extremum_uniqueI
thf(fact_503_top_Oextremum__uniqueI,axiom,
    ! [A: set_nat] :
      ( ( ord_less_eq_set_nat @ top_top_set_nat @ A )
     => ( A = top_top_set_nat ) ) ).

% top.extremum_uniqueI
thf(fact_504_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_505_top_Oextremum__uniqueI,axiom,
    ! [A: set_tm] :
      ( ( ord_less_eq_set_tm @ top_top_set_tm @ A )
     => ( A = top_top_set_tm ) ) ).

% top.extremum_uniqueI
thf(fact_506_Finite__Set_Ofinite__set,axiom,
    ( ( finite2992856023675031934ist_fm @ top_to4435265913292550789ist_fm )
    = ( finite600313802051052958ist_fm @ top_to1730629564055774885ist_fm ) ) ).

% Finite_Set.finite_set
thf(fact_507_Finite__Set_Ofinite__set,axiom,
    ( ( finite1152437895449049373et_nat @ top_top_set_set_nat )
    = ( finite_finite_nat @ top_top_set_nat ) ) ).

% Finite_Set.finite_set
thf(fact_508_ex__new__if__finite,axiom,
    ! [A2: set_rule] :
      ( ~ ( finite_finite_rule @ top_top_set_rule )
     => ( ( finite_finite_rule @ A2 )
       => ? [A3: rule] :
            ~ ( member_rule @ A3 @ A2 ) ) ) ).

% ex_new_if_finite
thf(fact_509_ex__new__if__finite,axiom,
    ! [A2: set_Pr8693737435421807431at_nat] :
      ( ~ ( finite4392333629123659920at_nat @ top_to8454300755403694231at_nat )
     => ( ( finite4392333629123659920at_nat @ A2 )
       => ? [A3: produc859450856879609959at_nat] :
            ~ ( member8206827879206165904at_nat @ A3 @ A2 ) ) ) ).

% ex_new_if_finite
thf(fact_510_ex__new__if__finite,axiom,
    ! [A2: set_Pr5202636777678657877ist_fm] :
      ( ~ ( finite600313802051052958ist_fm @ top_to1730629564055774885ist_fm )
     => ( ( finite600313802051052958ist_fm @ A2 )
       => ? [A3: produc6018962875968178549ist_fm] :
            ~ ( member4699826688122452638ist_fm @ A3 @ A2 ) ) ) ).

% ex_new_if_finite
thf(fact_511_ex__new__if__finite,axiom,
    ! [A2: set_nat] :
      ( ~ ( finite_finite_nat @ top_top_set_nat )
     => ( ( finite_finite_nat @ A2 )
       => ? [A3: nat] :
            ~ ( member_nat @ A3 @ A2 ) ) ) ).

% ex_new_if_finite
thf(fact_512_infinite__UNIV__char__0,axiom,
    ~ ( finite_finite_nat @ top_top_set_nat ) ).

% infinite_UNIV_char_0
thf(fact_513_subset__UNIV,axiom,
    ! [A2: set_Pr5202636777678657877ist_fm] : ( ord_le1771420097867575541ist_fm @ A2 @ top_to1730629564055774885ist_fm ) ).

% subset_UNIV
thf(fact_514_subset__UNIV,axiom,
    ! [A2: set_nat] : ( ord_less_eq_set_nat @ A2 @ top_top_set_nat ) ).

% subset_UNIV
thf(fact_515_subset__UNIV,axiom,
    ! [A2: set_rule] : ( ord_less_eq_set_rule @ A2 @ top_top_set_rule ) ).

% subset_UNIV
thf(fact_516_subset__UNIV,axiom,
    ! [A2: set_tm] : ( ord_less_eq_set_tm @ A2 @ top_top_set_tm ) ).

% subset_UNIV
thf(fact_517_enabled__def,axiom,
    ! [R: rule,S3: produc6018962875968178549ist_fm] :
      ( ( abstra1707737187183579335ist_fm @ eff @ R @ S3 )
      = ( ? [X6: fset_P8989946509869081563ist_fm] : ( eff @ R @ S3 @ X6 ) ) ) ).

% enabled_def
thf(fact_518_i_Osaturated__def,axiom,
    ! [Eff2: list_fm > list_tm > option_fset_list_tm,R: list_fm] :
      ( ( abstra5352578261360347328ist_tm @ ( abstra9141620886324064114ist_tm @ Eff2 ) @ R )
      = ( linear2489540979234913265ist_fm
        @ ^ [Xs: stream4408948924543953275ist_fm] :
            ( ( linear4742913937519564201ist_fm
              @ ^ [Step: produc6018962875968178549ist_fm] : ( abstra990286104146457292ist_tm @ ( abstra9141620886324064114ist_tm @ Eff2 ) @ R @ ( produc4874987732838623697ist_fm @ Step ) )
              @ Xs )
           => ( linear2237513815067535098ist_fm
              @ ( linear4742913937519564201ist_fm
                @ ^ [Step: produc6018962875968178549ist_fm] :
                    ( ( produc3242453064496107283ist_fm @ Step )
                    = R ) )
              @ Xs ) ) ) ) ).

% i.saturated_def
thf(fact_519_i_Osaturated__def,axiom,
    ! [Eff2: rule > produc6018962875968178549ist_fm > option6967287582980624417ist_fm,R: rule] :
      ( ( abstra2533313685312581075ist_fm @ ( abstra2682625350522704545ist_fm @ Eff2 ) @ R )
      = ( linear4957294957385550906m_rule
        @ ^ [Xs: stream2709947120125613254m_rule] :
            ( ( linear7925859179493631106m_rule
              @ ^ [Step: produc340336539035504054m_rule] : ( abstra1707737187183579335ist_fm @ ( abstra2682625350522704545ist_fm @ Eff2 ) @ R @ ( produc993771546023551940m_rule @ Step ) )
              @ Xs )
           => ( linear1816559245158234865m_rule
              @ ( linear7925859179493631106m_rule
                @ ^ [Step: produc340336539035504054m_rule] :
                    ( ( produc6514711208036230658m_rule @ Step )
                    = R ) )
              @ Xs ) ) ) ) ).

% i.saturated_def
thf(fact_520_i_Osset__fenum,axiom,
    ! [Rules2: stream_rule] :
      ( ( sset_rule @ ( abstra745658567949189203m_rule @ Rules2 ) )
      = ( sset_rule @ Rules2 ) ) ).

% i.sset_fenum
thf(fact_521_RuleSystem__Defs_Osset__fenum,axiom,
    ! [Rules2: stream_rule] :
      ( ( sset_rule @ ( abstra745658567949189203m_rule @ Rules2 ) )
      = ( sset_rule @ Rules2 ) ) ).

% RuleSystem_Defs.sset_fenum
thf(fact_522_RuleSystem__Defs_Ofair__fenum,axiom,
    ! [Rules2: stream_rule] : ( abstra3799686578551160190r_rule @ Rules2 @ ( abstra745658567949189203m_rule @ Rules2 ) ) ).

% RuleSystem_Defs.fair_fenum
thf(fact_523_i_Ofair__fenum,axiom,
    ! [Rules2: stream_rule] : ( abstra3799686578551160190r_rule @ Rules2 @ ( abstra745658567949189203m_rule @ Rules2 ) ) ).

% i.fair_fenum
thf(fact_524_RuleSystem__Defs_Osaturated__def,axiom,
    ( abstra5352578261360347328ist_tm
    = ( ^ [Eff: list_fm > list_tm > fset_list_tm > $o,R4: list_fm] :
          ( linear2489540979234913265ist_fm
          @ ^ [Xs: stream4408948924543953275ist_fm] :
              ( ( linear4742913937519564201ist_fm
                @ ^ [Step: produc6018962875968178549ist_fm] : ( abstra990286104146457292ist_tm @ Eff @ R4 @ ( produc4874987732838623697ist_fm @ Step ) )
                @ Xs )
             => ( linear2237513815067535098ist_fm
                @ ( linear4742913937519564201ist_fm
                  @ ^ [Step: produc6018962875968178549ist_fm] :
                      ( ( produc3242453064496107283ist_fm @ Step )
                      = R4 ) )
                @ Xs ) ) ) ) ) ).

% RuleSystem_Defs.saturated_def
thf(fact_525_RuleSystem__Defs_Osaturated__def,axiom,
    ( abstra2533313685312581075ist_fm
    = ( ^ [Eff: rule > produc6018962875968178549ist_fm > fset_P8989946509869081563ist_fm > $o,R4: rule] :
          ( linear4957294957385550906m_rule
          @ ^ [Xs: stream2709947120125613254m_rule] :
              ( ( linear7925859179493631106m_rule
                @ ^ [Step: produc340336539035504054m_rule] : ( abstra1707737187183579335ist_fm @ Eff @ R4 @ ( produc993771546023551940m_rule @ Step ) )
                @ Xs )
             => ( linear1816559245158234865m_rule
                @ ( linear7925859179493631106m_rule
                  @ ^ [Step: produc340336539035504054m_rule] :
                      ( ( produc6514711208036230658m_rule @ Step )
                      = R4 ) )
                @ Xs ) ) ) ) ) ).

% RuleSystem_Defs.saturated_def
thf(fact_526_finite__option__UNIV,axiom,
    ( ( finite5538963628675874020ist_fm @ top_to3677952576574767339ist_fm )
    = ( finite600313802051052958ist_fm @ top_to1730629564055774885ist_fm ) ) ).

% finite_option_UNIV
thf(fact_527_finite__option__UNIV,axiom,
    ( ( finite5523153139673422903on_nat @ top_to8920198386146353926on_nat )
    = ( finite_finite_nat @ top_top_set_nat ) ) ).

% finite_option_UNIV
thf(fact_528_saturated__effG__uu__def,axiom,
    ( abstra8695313511658867272ist_fm
    = ( ^ [Eff3: rule > produc6018962875968178549ist_fm > option6967287582980624417ist_fm] : ( abstra2533313685312581075ist_fm @ ( abstra2682625350522704545ist_fm @ Eff3 ) ) ) ) ).

% saturated_effG_uu_def
thf(fact_529_Saturated__effG__uu__uu__def,axiom,
    ( abstra1395361944017196648ist_fm
    = ( ^ [Eff3: rule > produc6018962875968178549ist_fm > option6967287582980624417ist_fm] : ( abstra6097777249025082867ist_fm @ ( abstra2682625350522704545ist_fm @ Eff3 ) ) ) ) ).

% Saturated_effG_uu_uu_def
thf(fact_530_fenum__uu__def,axiom,
    abstra1582897422107675196u_rule = abstra745658567949189203m_rule ).

% fenum_uu_def
thf(fact_531_ipath__mkTree__ev,axiom,
    ! [S3: produc6018962875968178549ist_fm,Rs2: stream_rule,Steps: stream2709947120125613254m_rule,R: rule] :
      ( ( member4699826688122452638ist_fm @ S3 @ top_to1730629564055774885ist_fm )
     => ( ( abstra3799686578551160190r_rule @ rules @ Rs2 )
       => ( ( abstra8836096083317999815m_rule @ ( abstra7840189589880272446ist_fm @ eff @ Rs2 @ S3 ) @ Steps )
         => ( ( member_rule @ R @ ( sset_rule @ rules ) )
           => ( ( linear4957294957385550906m_rule
                @ ( linear7925859179493631106m_rule
                  @ ^ [Step: produc340336539035504054m_rule] : ( abstra1707737187183579335ist_fm @ eff @ R @ ( produc993771546023551940m_rule @ Step ) ) )
                @ Steps )
             => ( linear1816559245158234865m_rule
                @ ( linear7925859179493631106m_rule
                  @ ^ [Step: produc340336539035504054m_rule] :
                      ( ( produc6514711208036230658m_rule @ Step )
                      = R ) )
                @ Steps ) ) ) ) ) ) ).

% ipath_mkTree_ev
thf(fact_532_per,axiom,
    ! [R: rule] :
      ( ( member_rule @ R @ ( sset_rule @ rules ) )
     => ( abstra5255361903751151037ist_fm @ eff @ rules @ top_to1730629564055774885ist_fm @ R ) ) ).

% per
thf(fact_533_RuleSystem__Defs_OmkTree_Ocong,axiom,
    abstra7840189589880272446ist_fm = abstra7840189589880272446ist_fm ).

% RuleSystem_Defs.mkTree.cong
thf(fact_534_RuleSystem_Oper_Ocong,axiom,
    abstra5255361903751151037ist_fm = abstra5255361903751151037ist_fm ).

% RuleSystem.per.cong
thf(fact_535_top__set__def,axiom,
    ( top_to1730629564055774885ist_fm
    = ( collec5845330005437105888ist_fm @ top_to4151594781713979000t_fm_o ) ) ).

% top_set_def
thf(fact_536_top__set__def,axiom,
    ( top_top_set_nat
    = ( collect_nat @ top_top_nat_o ) ) ).

% top_set_def
thf(fact_537_top__empty__eq,axiom,
    ( top_top_rule_o
    = ( ^ [X: rule] : ( member_rule @ X @ top_top_set_rule ) ) ) ).

% top_empty_eq
thf(fact_538_top__empty__eq,axiom,
    ( top_to1313005431924226822_nat_o
    = ( ^ [X: produc859450856879609959at_nat] : ( member8206827879206165904at_nat @ X @ top_to8454300755403694231at_nat ) ) ) ).

% top_empty_eq
thf(fact_539_top__empty__eq,axiom,
    ( top_to4151594781713979000t_fm_o
    = ( ^ [X: produc6018962875968178549ist_fm] : ( member4699826688122452638ist_fm @ X @ top_to1730629564055774885ist_fm ) ) ) ).

% top_empty_eq
thf(fact_540_top__empty__eq,axiom,
    ( top_top_nat_o
    = ( ^ [X: nat] : ( member_nat @ X @ top_top_set_nat ) ) ) ).

% top_empty_eq
thf(fact_541_infinite__UNIV__nat,axiom,
    ~ ( finite_finite_nat @ top_top_set_nat ) ).

% infinite_UNIV_nat
thf(fact_542_nat__not__finite,axiom,
    ~ ( finite_finite_nat @ top_top_set_nat ) ).

% nat_not_finite
thf(fact_543_ipath__mkTree__Saturated,axiom,
    ! [S3: produc6018962875968178549ist_fm,Rs2: stream_rule,Steps: stream2709947120125613254m_rule] :
      ( ( member4699826688122452638ist_fm @ S3 @ top_to1730629564055774885ist_fm )
     => ( ( abstra3799686578551160190r_rule @ rules @ Rs2 )
       => ( ( abstra8836096083317999815m_rule @ ( abstra7840189589880272446ist_fm @ eff @ Rs2 @ S3 ) @ Steps )
         => ( abstra6097777249025082867ist_fm @ eff @ rules @ Steps ) ) ) ) ).

% ipath_mkTree_Saturated
thf(fact_544_ipath__mkTree__sdrop,axiom,
    ! [S3: produc6018962875968178549ist_fm,Rs2: stream_rule,Steps: stream2709947120125613254m_rule,M2: nat] :
      ( ( member4699826688122452638ist_fm @ S3 @ top_to1730629564055774885ist_fm )
     => ( ( abstra3799686578551160190r_rule @ rules @ Rs2 )
       => ( ( abstra8836096083317999815m_rule @ ( abstra7840189589880272446ist_fm @ eff @ Rs2 @ S3 ) @ Steps )
         => ? [N2: nat,S6: produc6018962875968178549ist_fm] :
              ( ( member4699826688122452638ist_fm @ S6 @ top_to1730629564055774885ist_fm )
              & ( abstra8836096083317999815m_rule @ ( abstra7840189589880272446ist_fm @ eff @ ( sdrop_rule @ N2 @ Rs2 ) @ S6 ) @ ( sdrop_8169176516188972301m_rule @ M2 @ Steps ) ) ) ) ) ) ).

% ipath_mkTree_sdrop
thf(fact_545_ipath__mkTree__saturated,axiom,
    ! [S3: produc6018962875968178549ist_fm,Rs2: stream_rule,Steps: stream2709947120125613254m_rule,R: rule] :
      ( ( member4699826688122452638ist_fm @ S3 @ top_to1730629564055774885ist_fm )
     => ( ( abstra3799686578551160190r_rule @ rules @ Rs2 )
       => ( ( abstra8836096083317999815m_rule @ ( abstra7840189589880272446ist_fm @ eff @ Rs2 @ S3 ) @ Steps )
         => ( ( member_rule @ R @ ( sset_rule @ rules ) )
           => ( abstra2533313685312581075ist_fm @ eff @ R @ Steps ) ) ) ) ) ).

% ipath_mkTree_saturated
thf(fact_546_all__rules__persistent,axiom,
    ! [R5: rule] :
      ( ( member_rule @ R5 @ ( sset_rule @ rules ) )
     => ( abstra5255361903751151037ist_fm @ eff @ rules @ top_to1730629564055774885ist_fm @ R5 ) ) ).

% all_rules_persistent
thf(fact_547_secavProver__def,axiom,
    ( secavProver
    = ( abstra7840189589880272446ist_fm @ eff @ rules ) ) ).

% secavProver_def
thf(fact_548_RuleSystem_Oipath__mkTree__ev,axiom,
    ! [Eff4: nat > nat > fset_nat > $o,Rules2: stream_nat,S4: set_nat,S3: nat,Rs2: stream_nat,Steps: stream6724221391990029191at_nat,R: nat] :
      ( ( abstra5088622135990055243at_nat @ Eff4 @ Rules2 @ S4 )
     => ( ( member_nat @ S3 @ S4 )
       => ( ( abstra7018062486307922108ir_nat @ Rules2 @ Rs2 )
         => ( ( abstra5838625251101670576at_nat @ ( abstra1192804020863382533at_nat @ Eff4 @ Rs2 @ S3 ) @ Steps )
           => ( ( member_nat @ R @ ( sset_nat @ Rules2 ) )
             => ( ( linear1521848693801000317at_nat
                  @ ( linear2787706585215634229at_nat
                    @ ^ [Step: product_prod_nat_nat] : ( abstra2407503598237092412at_nat @ Eff4 @ R @ ( product_fst_nat_nat @ Step ) ) )
                  @ Steps )
               => ( linear869931615226441222at_nat
                  @ ( linear2787706585215634229at_nat
                    @ ^ [Step: product_prod_nat_nat] :
                        ( ( product_snd_nat_nat @ Step )
                        = R ) )
                  @ Steps ) ) ) ) ) ) ) ).

% RuleSystem.ipath_mkTree_ev
thf(fact_549_RuleSystem_Oipath__mkTree__ev,axiom,
    ! [Eff4: nat > rule > fset_rule > $o,Rules2: stream_nat,S4: set_rule,S3: rule,Rs2: stream_nat,Steps: stream8814882096039290697le_nat,R: nat] :
      ( ( abstra7777025902706499341t_rule @ Eff4 @ Rules2 @ S4 )
     => ( ( member_rule @ S3 @ S4 )
       => ( ( abstra7018062486307922108ir_nat @ Rules2 @ Rs2 )
         => ( ( abstra8430046938747295090le_nat @ ( abstra2951810738104559559t_rule @ Eff4 @ Rs2 @ S3 ) @ Steps )
           => ( ( member_nat @ R @ ( sset_nat @ Rules2 ) )
             => ( ( linear4049892813572868415le_nat
                  @ ( linear2020473048652005111le_nat
                    @ ^ [Step: produc1529498987567528131le_nat] : ( abstra3913749349918025470t_rule @ Eff4 @ R @ ( product_fst_rule_nat @ Step ) ) )
                  @ Steps )
               => ( linear6463589296130977992le_nat
                  @ ( linear2020473048652005111le_nat
                    @ ^ [Step: produc1529498987567528131le_nat] :
                        ( ( product_snd_rule_nat @ Step )
                        = R ) )
                  @ Steps ) ) ) ) ) ) ) ).

% RuleSystem.ipath_mkTree_ev
thf(fact_550_RuleSystem_Oipath__mkTree__ev,axiom,
    ! [Eff4: rule > nat > fset_nat > $o,Rules2: stream_rule,S4: set_nat,S3: nat,Rs2: stream_rule,Steps: stream8912087984624402761t_rule,R: rule] :
      ( ( abstra3263948797621512205le_nat @ Eff4 @ Rules2 @ S4 )
     => ( ( member_nat @ S3 @ S4 )
       => ( ( abstra3799686578551160190r_rule @ Rules2 @ Rs2 )
         => ( ( abstra5135089046934318962t_rule @ ( abstra7662105669874348231le_nat @ Eff4 @ Rs2 @ S3 ) @ Steps )
           => ( ( member_rule @ R @ ( sset_rule @ Rules2 ) )
             => ( ( linear754934921759892287t_rule
                  @ ( linear7948887193693804791t_rule
                    @ ^ [Step: produc7457913132609327811t_rule] : ( abstra8624044281687814142le_nat @ Eff4 @ R @ ( product_fst_nat_rule @ Step ) ) )
                  @ Steps )
               => ( linear3168631404318001864t_rule
                  @ ( linear7948887193693804791t_rule
                    @ ^ [Step: produc7457913132609327811t_rule] :
                        ( ( product_snd_nat_rule @ Step )
                        = R ) )
                  @ Steps ) ) ) ) ) ) ) ).

% RuleSystem.ipath_mkTree_ev
thf(fact_551_RuleSystem_Oipath__mkTree__ev,axiom,
    ! [Eff4: rule > rule > fset_rule > $o,Rules2: stream_rule,S4: set_rule,S3: rule,Rs2: stream_rule,Steps: stream4385846686851721995e_rule,R: rule] :
      ( ( abstra6805424310982398927e_rule @ Eff4 @ Rules2 @ S4 )
     => ( ( member_rule @ S3 @ S4 )
       => ( ( abstra3799686578551160190r_rule @ Rules2 @ Rs2 )
         => ( ( abstra1660175255847126580e_rule @ ( abstra2300331465871827593e_rule @ Eff4 @ Rs2 @ S3 ) @ Steps )
           => ( ( member_rule @ R @ ( sset_rule @ Rules2 ) )
             => ( ( linear4160411771378765057e_rule
                  @ ( linear5665384415496112825e_rule
                    @ ^ [Step: produc7694839378271647877e_rule] : ( abstra7234149737463204544e_rule @ Eff4 @ R @ ( produc2120989893493863593e_rule @ Step ) ) )
                  @ Steps )
               => ( linear4910413937529768842e_rule
                  @ ( linear5665384415496112825e_rule
                    @ ^ [Step: produc7694839378271647877e_rule] :
                        ( ( produc1581767551459125483e_rule @ Step )
                        = R ) )
                  @ Steps ) ) ) ) ) ) ) ).

% RuleSystem.ipath_mkTree_ev
thf(fact_552_RuleSystem_Oipath__mkTree__ev,axiom,
    ! [Eff4: list_fm > list_tm > fset_list_tm > $o,Rules2: stream_list_fm,S4: set_list_tm,S3: list_tm,Rs2: stream_list_fm,Steps: stream4408948924543953275ist_fm,R: list_fm] :
      ( ( abstra4205323827771120475ist_tm @ Eff4 @ Rules2 @ S4 )
     => ( ( member_list_tm @ S3 @ S4 )
       => ( ( abstra7078398738616236685ist_fm @ Rules2 @ Rs2 )
         => ( ( abstra2293663390402184868ist_fm @ ( abstra4388168957404043029ist_tm @ Eff4 @ Rs2 @ S3 ) @ Steps )
           => ( ( member_list_fm @ R @ ( sset_list_fm @ Rules2 ) )
             => ( ( linear2489540979234913265ist_fm
                  @ ( linear4742913937519564201ist_fm
                    @ ^ [Step: produc6018962875968178549ist_fm] : ( abstra990286104146457292ist_tm @ Eff4 @ R @ ( produc4874987732838623697ist_fm @ Step ) ) )
                  @ Steps )
               => ( linear2237513815067535098ist_fm
                  @ ( linear4742913937519564201ist_fm
                    @ ^ [Step: produc6018962875968178549ist_fm] :
                        ( ( produc3242453064496107283ist_fm @ Step )
                        = R ) )
                  @ Steps ) ) ) ) ) ) ) ).

% RuleSystem.ipath_mkTree_ev
thf(fact_553_RuleSystem_Oipath__mkTree__ev,axiom,
    ! [Eff4: produc6018962875968178549ist_fm > nat > fset_nat > $o,Rules2: stream4408948924543953275ist_fm,S4: set_nat,S3: nat,Rs2: stream4408948924543953275ist_fm,Steps: stream2110468692874468522ist_fm,R: produc6018962875968178549ist_fm] :
      ( ( abstra4534667231653962588fm_nat @ Eff4 @ Rules2 @ S4 )
     => ( ( member_nat @ S3 @ S4 )
       => ( ( abstra7670258426472572809ist_fm @ Rules2 @ Rs2 )
         => ( ( abstra2835824759249546923ist_fm @ ( abstra4727933336337454882fm_nat @ Eff4 @ Rs2 @ S3 ) @ Steps )
           => ( ( member4699826688122452638ist_fm @ R @ ( sset_P5379989128061332361ist_fm @ Rules2 ) )
             => ( ( linear8316169256737981726ist_fm
                  @ ( linear1342315010977244518ist_fm
                    @ ^ [Step: produc6317018572677857946ist_fm] : ( abstra337371019197618859fm_nat @ Eff4 @ R @ ( produc1587968209580003424ist_fm @ Step ) ) )
                  @ Steps )
               => ( linear9086108507079445205ist_fm
                  @ ( linear1342315010977244518ist_fm
                    @ ^ [Step: produc6317018572677857946ist_fm] :
                        ( ( produc3568175498200846494ist_fm @ Step )
                        = R ) )
                  @ Steps ) ) ) ) ) ) ) ).

% RuleSystem.ipath_mkTree_ev
thf(fact_554_RuleSystem_Oipath__mkTree__ev,axiom,
    ! [Eff4: nat > produc6018962875968178549ist_fm > fset_P8989946509869081563ist_fm > $o,Rules2: stream_nat,S4: set_Pr5202636777678657877ist_fm,S3: produc6018962875968178549ist_fm,Rs2: stream_nat,Steps: stream1752417115841309700fm_nat,R: nat] :
      ( ( abstra6010661502565574330ist_fm @ Eff4 @ Rules2 @ S4 )
     => ( ( member4699826688122452638ist_fm @ S3 @ S4 )
       => ( ( abstra7018062486307922108ir_nat @ Rules2 @ Rs2 )
         => ( ( abstra457208267521379077fm_nat @ ( abstra6203927607249066624ist_fm @ Eff4 @ Rs2 @ S3 ) @ Steps )
           => ( ( member_nat @ R @ ( sset_nat @ Rules2 ) )
             => ( ( linear5937552765009813880fm_nat
                  @ ( linear8187070556103852480fm_nat
                    @ ^ [Step: produc3938402080949690100fm_nat] : ( abstra1813365290109230601ist_fm @ Eff4 @ R @ ( produc111973938668391682fm_nat @ Step ) ) )
                  @ Steps )
               => ( linear6707492015351277359fm_nat
                  @ ( linear8187070556103852480fm_nat
                    @ ^ [Step: produc3938402080949690100fm_nat] :
                        ( ( produc2092181227289234752fm_nat @ Step )
                        = R ) )
                  @ Steps ) ) ) ) ) ) ) ).

% RuleSystem.ipath_mkTree_ev
thf(fact_555_RuleSystem_Oipath__mkTree__ev,axiom,
    ! [Eff4: produc6018962875968178549ist_fm > rule > fset_rule > $o,Rules2: stream4408948924543953275ist_fm,S4: set_rule,S3: rule,Rs2: stream4408948924543953275ist_fm,Steps: stream4490188412977367400ist_fm,R: produc6018962875968178549ist_fm] :
      ( ( abstra2695409591818887198m_rule @ Eff4 @ Rules2 @ S4 )
     => ( ( member_rule @ S3 @ S4 )
       => ( ( abstra7670258426472572809ist_fm @ Rules2 @ Rs2 )
         => ( ( abstra1626332986471984745ist_fm @ ( abstra5313865830731255268m_rule @ Eff4 @ Rs2 @ S3 ) @ Steps )
           => ( ( member4699826688122452638ist_fm @ R @ ( sset_P5379989128061332361ist_fm @ Rules2 ) )
             => ( ( linear6970903897394311644ist_fm
                  @ ( linear716096082647616036ist_fm
                    @ ^ [Step: produc2353945479044264792ist_fm] : ( abstra8404785464889337965m_rule @ Eff4 @ R @ ( produc3520095305172569118ist_fm @ Step ) ) )
                  @ Steps )
               => ( linear3830168185166995603ist_fm
                  @ ( linear716096082647616036ist_fm
                    @ ^ [Step: produc2353945479044264792ist_fm] :
                        ( ( produc9041034967185247836ist_fm @ Step )
                        = R ) )
                  @ Steps ) ) ) ) ) ) ) ).

% RuleSystem.ipath_mkTree_ev
thf(fact_556_RuleSystem_Oipath__mkTree__ev,axiom,
    ! [Eff4: rule > produc6018962875968178549ist_fm > fset_P8989946509869081563ist_fm > $o,Rules2: stream_rule,S4: set_Pr5202636777678657877ist_fm,S3: produc6018962875968178549ist_fm,Rs2: stream_rule,Steps: stream2709947120125613254m_rule,R: rule] :
      ( ( abstra5221733350967904376ist_fm @ Eff4 @ Rules2 @ S4 )
     => ( ( member4699826688122452638ist_fm @ S3 @ S4 )
       => ( ( abstra3799686578551160190r_rule @ Rules2 @ Rs2 )
         => ( ( abstra8836096083317999815m_rule @ ( abstra7840189589880272446ist_fm @ Eff4 @ Rs2 @ S3 ) @ Steps )
           => ( ( member_rule @ R @ ( sset_rule @ Rules2 ) )
             => ( ( linear4957294957385550906m_rule
                  @ ( linear7925859179493631106m_rule
                    @ ^ [Step: produc340336539035504054m_rule] : ( abstra1707737187183579335ist_fm @ Eff4 @ R @ ( produc993771546023551940m_rule @ Step ) ) )
                  @ Steps )
               => ( linear1816559245158234865m_rule
                  @ ( linear7925859179493631106m_rule
                    @ ^ [Step: produc340336539035504054m_rule] :
                        ( ( produc6514711208036230658m_rule @ Step )
                        = R ) )
                  @ Steps ) ) ) ) ) ) ) ).

% RuleSystem.ipath_mkTree_ev
thf(fact_557_RuleSystem_Oipath__mkTree__ev,axiom,
    ! [Eff4: produc859450856879609959at_nat > nat > fset_nat > $o,Rules2: stream8372878641218411373at_nat,S4: set_nat,S3: nat,Rs2: stream8372878641218411373at_nat,Steps: stream1775313132605703580at_nat,R: produc859450856879609959at_nat] :
      ( ( abstra1380253255819158094at_nat @ Eff4 @ Rules2 @ S4 )
     => ( ( member_nat @ S3 @ S4 )
       => ( ( abstra5618069685180253819at_nat @ Rules2 @ Rs2 )
         => ( ( abstra7405463511545700509at_nat @ ( abstra3915314607634222612at_nat @ Eff4 @ Rs2 @ S3 ) @ Steps )
           => ( ( member8206827879206165904at_nat @ R @ ( sset_P9164763103296375419at_nat @ Rules2 ) )
             => ( ( linear2645036804292154128at_nat
                  @ ( linear8939610272747787096at_nat
                    @ ^ [Step: produc2487518378626728076at_nat] : ( abstra5326277529176745373at_nat @ Eff4 @ R @ ( produc7178812058934418258at_nat @ Step ) ) )
                  @ Steps )
               => ( linear3718889564458697927at_nat
                  @ ( linear8939610272747787096at_nat
                    @ ^ [Step: produc2487518378626728076at_nat] :
                        ( ( produc3846061323874349968at_nat @ Step )
                        = R ) )
                  @ Steps ) ) ) ) ) ) ) ).

% RuleSystem.ipath_mkTree_ev
thf(fact_558_mkTree__effG__uu__def,axiom,
    ( abstra9183632535595798085ist_fm
    = ( ^ [Eff3: rule > produc6018962875968178549ist_fm > option6967287582980624417ist_fm] : ( abstra7840189589880272446ist_fm @ ( abstra2682625350522704545ist_fm @ Eff3 ) ) ) ) ).

% mkTree_effG_uu_def
thf(fact_559_per__alw,axiom,
    ! [R: rule,Steps: stream2709947120125613254m_rule] :
      ( ( abstra5255361903751151037ist_fm @ eff @ rules @ top_to1730629564055774885ist_fm @ R )
     => ( ( ( abstra6789711989322986974ist_fm @ eff @ rules @ Steps )
          & ( member4699826688122452638ist_fm @ ( produc993771546023551940m_rule @ ( shd_Pr4562317740776619530m_rule @ Steps ) ) @ top_to1730629564055774885ist_fm ) )
       => ( linear4957294957385550906m_rule
          @ ^ [Xs: stream2709947120125613254m_rule] :
              ( ( linear7925859179493631106m_rule
                @ ^ [Step: produc340336539035504054m_rule] : ( abstra1707737187183579335ist_fm @ eff @ R @ ( produc993771546023551940m_rule @ Step ) )
                @ Xs )
             => ( ( linear7925859179493631106m_rule
                  @ ^ [Step: produc340336539035504054m_rule] :
                      ( ( produc6514711208036230658m_rule @ Step )
                      = R )
                  @ Xs )
                | ( linear3460418925085718015rule_o
                  @ ( linear7925859179493631106m_rule
                    @ ^ [Step: produc340336539035504054m_rule] : ( abstra1707737187183579335ist_fm @ eff @ R @ ( produc993771546023551940m_rule @ Step ) ) )
                  @ Xs ) ) )
          @ Steps ) ) ) ).

% per_alw
thf(fact_560_PersistentRuleSystem_Oipath__mkTree__saturated,axiom,
    ! [Eff4: nat > nat > fset_nat > $o,Rules2: stream_nat,S4: set_nat,S3: nat,Rs2: stream_nat,Steps: stream6724221391990029191at_nat,R: nat] :
      ( ( abstra2374624847373867554at_nat @ Eff4 @ Rules2 @ S4 )
     => ( ( member_nat @ S3 @ S4 )
       => ( ( abstra7018062486307922108ir_nat @ Rules2 @ Rs2 )
         => ( ( abstra5838625251101670576at_nat @ ( abstra1192804020863382533at_nat @ Eff4 @ Rs2 @ S3 ) @ Steps )
           => ( ( member_nat @ R @ ( sset_nat @ Rules2 ) )
             => ( abstra5577456715138805296at_nat @ Eff4 @ R @ Steps ) ) ) ) ) ) ).

% PersistentRuleSystem.ipath_mkTree_saturated
thf(fact_561_PersistentRuleSystem_Oipath__mkTree__saturated,axiom,
    ! [Eff4: nat > rule > fset_rule > $o,Rules2: stream_nat,S4: set_rule,S3: rule,Rs2: stream_nat,Steps: stream8814882096039290697le_nat,R: nat] :
      ( ( abstra8880880309027573988t_rule @ Eff4 @ Rules2 @ S4 )
     => ( ( member_rule @ S3 @ S4 )
       => ( ( abstra7018062486307922108ir_nat @ Rules2 @ Rs2 )
         => ( ( abstra8430046938747295090le_nat @ ( abstra2951810738104559559t_rule @ Eff4 @ Rs2 @ S3 ) @ Steps )
           => ( ( member_nat @ R @ ( sset_nat @ Rules2 ) )
             => ( abstra6971170162055632626t_rule @ Eff4 @ R @ Steps ) ) ) ) ) ) ).

% PersistentRuleSystem.ipath_mkTree_saturated
thf(fact_562_PersistentRuleSystem_Oipath__mkTree__saturated,axiom,
    ! [Eff4: rule > nat > fset_nat > $o,Rules2: stream_rule,S4: set_nat,S3: nat,Rs2: stream_rule,Steps: stream8912087984624402761t_rule,R: rule] :
      ( ( abstra4367803203942586852le_nat @ Eff4 @ Rules2 @ S4 )
     => ( ( member_nat @ S3 @ S4 )
       => ( ( abstra3799686578551160190r_rule @ Rules2 @ Rs2 )
         => ( ( abstra5135089046934318962t_rule @ ( abstra7662105669874348231le_nat @ Eff4 @ Rs2 @ S3 ) @ Steps )
           => ( ( member_rule @ R @ ( sset_rule @ Rules2 ) )
             => ( abstra2458093056970645490le_nat @ Eff4 @ R @ Steps ) ) ) ) ) ) ).

% PersistentRuleSystem.ipath_mkTree_saturated
thf(fact_563_PersistentRuleSystem_Oipath__mkTree__saturated,axiom,
    ! [Eff4: rule > rule > fset_rule > $o,Rules2: stream_rule,S4: set_rule,S3: rule,Rs2: stream_rule,Steps: stream4385846686851721995e_rule,R: rule] :
      ( ( abstra2971612018481630886e_rule @ Eff4 @ Rules2 @ S4 )
     => ( ( member_rule @ S3 @ S4 )
       => ( ( abstra3799686578551160190r_rule @ Rules2 @ Rs2 )
         => ( ( abstra1660175255847126580e_rule @ ( abstra2300331465871827593e_rule @ Eff4 @ Rs2 @ S3 ) @ Steps )
           => ( ( member_rule @ R @ ( sset_rule @ Rules2 ) )
             => ( abstra1228885326027523764e_rule @ Eff4 @ R @ Steps ) ) ) ) ) ) ).

% PersistentRuleSystem.ipath_mkTree_saturated
thf(fact_564_PersistentRuleSystem_Oipath__mkTree__saturated,axiom,
    ! [Eff4: produc6018962875968178549ist_fm > nat > fset_nat > $o,Rules2: stream4408948924543953275ist_fm,S4: set_nat,S3: nat,Rs2: stream4408948924543953275ist_fm,Steps: stream2110468692874468522ist_fm,R: produc6018962875968178549ist_fm] :
      ( ( abstra7525679386635172933fm_nat @ Eff4 @ Rules2 @ S4 )
     => ( ( member_nat @ S3 @ S4 )
       => ( ( abstra7670258426472572809ist_fm @ Rules2 @ Rs2 )
         => ( ( abstra2835824759249546923ist_fm @ ( abstra4727933336337454882fm_nat @ Eff4 @ Rs2 @ S3 ) @ Steps )
           => ( ( member4699826688122452638ist_fm @ R @ ( sset_P5379989128061332361ist_fm @ Rules2 ) )
             => ( abstra7922471482953925559fm_nat @ Eff4 @ R @ Steps ) ) ) ) ) ) ).

% PersistentRuleSystem.ipath_mkTree_saturated
thf(fact_565_PersistentRuleSystem_Oipath__mkTree__saturated,axiom,
    ! [Eff4: nat > produc6018962875968178549ist_fm > fset_P8989946509869081563ist_fm > $o,Rules2: stream_nat,S4: set_Pr5202636777678657877ist_fm,S3: produc6018962875968178549ist_fm,Rs2: stream_nat,Steps: stream1752417115841309700fm_nat,R: nat] :
      ( ( abstra9001673657546784675ist_fm @ Eff4 @ Rules2 @ S4 )
     => ( ( member4699826688122452638ist_fm @ S3 @ S4 )
       => ( ( abstra7018062486307922108ir_nat @ Rules2 @ Rs2 )
         => ( ( abstra457208267521379077fm_nat @ ( abstra6203927607249066624ist_fm @ Eff4 @ Rs2 @ S3 ) @ Steps )
           => ( ( member_nat @ R @ ( sset_nat @ Rules2 ) )
             => ( abstra175093717010761493ist_fm @ Eff4 @ R @ Steps ) ) ) ) ) ) ).

% PersistentRuleSystem.ipath_mkTree_saturated
thf(fact_566_PersistentRuleSystem_Oipath__mkTree__saturated,axiom,
    ! [Eff4: produc6018962875968178549ist_fm > rule > fset_rule > $o,Rules2: stream4408948924543953275ist_fm,S4: set_rule,S3: rule,Rs2: stream4408948924543953275ist_fm,Steps: stream4490188412977367400ist_fm,R: produc6018962875968178549ist_fm] :
      ( ( abstra1441597783195643911m_rule @ Eff4 @ Rules2 @ S4 )
     => ( ( member_rule @ S3 @ S4 )
       => ( ( abstra7670258426472572809ist_fm @ Rules2 @ Rs2 )
         => ( ( abstra1626332986471984745ist_fm @ ( abstra5313865830731255268m_rule @ Eff4 @ Rs2 @ S3 ) @ Steps )
           => ( ( member4699826688122452638ist_fm @ R @ ( sset_P5379989128061332361ist_fm @ Rules2 ) )
             => ( abstra6989926163563897m_rule @ Eff4 @ R @ Steps ) ) ) ) ) ) ).

% PersistentRuleSystem.ipath_mkTree_saturated
thf(fact_567_PersistentRuleSystem_Oipath__mkTree__saturated,axiom,
    ! [Eff4: rule > produc6018962875968178549ist_fm > fset_P8989946509869081563ist_fm > $o,Rules2: stream_rule,S4: set_Pr5202636777678657877ist_fm,S3: produc6018962875968178549ist_fm,Rs2: stream_rule,Steps: stream2709947120125613254m_rule,R: rule] :
      ( ( abstra3967921542344661089ist_fm @ Eff4 @ Rules2 @ S4 )
     => ( ( member4699826688122452638ist_fm @ S3 @ S4 )
       => ( ( abstra3799686578551160190r_rule @ Rules2 @ Rs2 )
         => ( ( abstra8836096083317999815m_rule @ ( abstra7840189589880272446ist_fm @ Eff4 @ Rs2 @ S3 ) @ Steps )
           => ( ( member_rule @ R @ ( sset_rule @ Rules2 ) )
             => ( abstra2533313685312581075ist_fm @ Eff4 @ R @ Steps ) ) ) ) ) ) ).

% PersistentRuleSystem.ipath_mkTree_saturated
thf(fact_568_PersistentRuleSystem_Oipath__mkTree__saturated,axiom,
    ! [Eff4: produc859450856879609959at_nat > nat > fset_nat > $o,Rules2: stream8372878641218411373at_nat,S4: set_nat,S3: nat,Rs2: stream8372878641218411373at_nat,Steps: stream1775313132605703580at_nat,R: produc859450856879609959at_nat] :
      ( ( abstra8457097704303284023at_nat @ Eff4 @ Rules2 @ S4 )
     => ( ( member_nat @ S3 @ S4 )
       => ( ( abstra5618069685180253819at_nat @ Rules2 @ Rs2 )
         => ( ( abstra7405463511545700509at_nat @ ( abstra3915314607634222612at_nat @ Eff4 @ Rs2 @ S3 ) @ Steps )
           => ( ( member8206827879206165904at_nat @ R @ ( sset_P9164763103296375419at_nat @ Rules2 ) )
             => ( abstra1024696123990965929at_nat @ Eff4 @ R @ Steps ) ) ) ) ) ) ).

% PersistentRuleSystem.ipath_mkTree_saturated
thf(fact_569_PersistentRuleSystem_Oipath__mkTree__saturated,axiom,
    ! [Eff4: produc859450856879609959at_nat > rule > fset_rule > $o,Rules2: stream8372878641218411373at_nat,S4: set_rule,S3: rule,Rs2: stream8372878641218411373at_nat,Steps: stream1198919635141937754at_nat,R: produc859450856879609959at_nat] :
      ( ( abstra1135090460132643065t_rule @ Eff4 @ Rules2 @ S4 )
     => ( ( member_rule @ S3 @ S4 )
       => ( ( abstra5618069685180253819at_nat @ Rules2 @ Rs2 )
         => ( ( abstra649860893229217883at_nat @ ( abstra9026052449019810518t_rule @ Eff4 @ Rs2 @ S3 ) @ Steps )
           => ( ( member8206827879206165904at_nat @ R @ ( sset_P9164763103296375419at_nat @ Rules2 ) )
             => ( abstra3121514152276135531t_rule @ Eff4 @ R @ Steps ) ) ) ) ) ) ).

% PersistentRuleSystem.ipath_mkTree_saturated
thf(fact_570_PersistentRuleSystem_Oipath__mkTree__Saturated,axiom,
    ! [Eff4: rule > nat > fset_nat > $o,Rules2: stream_rule,S4: set_nat,S3: nat,Rs2: stream_rule,Steps: stream8912087984624402761t_rule] :
      ( ( abstra4367803203942586852le_nat @ Eff4 @ Rules2 @ S4 )
     => ( ( member_nat @ S3 @ S4 )
       => ( ( abstra3799686578551160190r_rule @ Rules2 @ Rs2 )
         => ( ( abstra5135089046934318962t_rule @ ( abstra7662105669874348231le_nat @ Eff4 @ Rs2 @ S3 ) @ Steps )
           => ( abstra3870052345063948242le_nat @ Eff4 @ Rules2 @ Steps ) ) ) ) ) ).

% PersistentRuleSystem.ipath_mkTree_Saturated
thf(fact_571_PersistentRuleSystem_Oipath__mkTree__Saturated,axiom,
    ! [Eff4: rule > rule > fset_rule > $o,Rules2: stream_rule,S4: set_rule,S3: rule,Rs2: stream_rule,Steps: stream4385846686851721995e_rule] :
      ( ( abstra2971612018481630886e_rule @ Eff4 @ Rules2 @ S4 )
     => ( ( member_rule @ S3 @ S4 )
       => ( ( abstra3799686578551160190r_rule @ Rules2 @ Rs2 )
         => ( ( abstra1660175255847126580e_rule @ ( abstra2300331465871827593e_rule @ Eff4 @ Rs2 @ S3 ) @ Steps )
           => ( abstra8307935604394493588e_rule @ Eff4 @ Rules2 @ Steps ) ) ) ) ) ).

% PersistentRuleSystem.ipath_mkTree_Saturated
thf(fact_572_PersistentRuleSystem_Oipath__mkTree__Saturated,axiom,
    ! [Eff4: rule > produc859450856879609959at_nat > fset_P3619101297660450253at_nat > $o,Rules2: stream_rule,S4: set_Pr8693737435421807431at_nat,S3: produc859450856879609959at_nat,Rs2: stream_rule,Steps: stream6749460841574940856t_rule] :
      ( ( abstra8381481625577051987at_nat @ Eff4 @ Rules2 @ S4 )
     => ( ( member8206827879206165904at_nat @ S3 @ S4 )
       => ( ( abstra3799686578551160190r_rule @ Rules2 @ Rs2 )
         => ( ( abstra826546537944104377t_rule @ ( abstra7049071577609443632at_nat @ Eff4 @ Rs2 @ S3 ) @ Steps )
           => ( abstra4684070643073051877at_nat @ Eff4 @ Rules2 @ Steps ) ) ) ) ) ).

% PersistentRuleSystem.ipath_mkTree_Saturated
thf(fact_573_PersistentRuleSystem_Oipath__mkTree__Saturated,axiom,
    ! [Eff4: rule > produc6018962875968178549ist_fm > fset_P8989946509869081563ist_fm > $o,Rules2: stream_rule,S4: set_Pr5202636777678657877ist_fm,S3: produc6018962875968178549ist_fm,Rs2: stream_rule,Steps: stream2709947120125613254m_rule] :
      ( ( abstra3967921542344661089ist_fm @ Eff4 @ Rules2 @ S4 )
     => ( ( member4699826688122452638ist_fm @ S3 @ S4 )
       => ( ( abstra3799686578551160190r_rule @ Rules2 @ Rs2 )
         => ( ( abstra8836096083317999815m_rule @ ( abstra7840189589880272446ist_fm @ Eff4 @ Rs2 @ S3 ) @ Steps )
           => ( abstra6097777249025082867ist_fm @ Eff4 @ Rules2 @ Steps ) ) ) ) ) ).

% PersistentRuleSystem.ipath_mkTree_Saturated
thf(fact_574_RuleSystem_Oipath__mkTree__sdrop,axiom,
    ! [Eff4: produc340336539035504054m_rule > nat > fset_nat > $o,Rules2: stream2709947120125613254m_rule,S4: set_nat,S3: nat,Rs2: stream2709947120125613254m_rule,Steps: stream4309476138983993197m_rule,M2: nat] :
      ( ( abstra2021474039004664625le_nat @ Eff4 @ Rules2 @ S4 )
     => ( ( member_nat @ S3 @ S4 )
       => ( ( abstra3665372904625986210m_rule @ Rules2 @ Rs2 )
         => ( ( abstra3711020939654513558m_rule @ ( abstra7903611026202203627le_nat @ Eff4 @ Rs2 @ S3 ) @ Steps )
           => ? [N2: nat,S6: nat] :
                ( ( member_nat @ S6 @ S4 )
                & ( abstra3711020939654513558m_rule @ ( abstra7903611026202203627le_nat @ Eff4 @ ( sdrop_8169176516188972301m_rule @ N2 @ Rs2 ) @ S6 ) @ ( sdrop_7346288004040472528m_rule @ M2 @ Steps ) ) ) ) ) ) ) ).

% RuleSystem.ipath_mkTree_sdrop
thf(fact_575_RuleSystem_Oipath__mkTree__sdrop,axiom,
    ! [Eff4: produc340336539035504054m_rule > produc6018962875968178549ist_fm > fset_P8989946509869081563ist_fm > $o,Rules2: stream2709947120125613254m_rule,S4: set_Pr5202636777678657877ist_fm,S3: produc6018962875968178549ist_fm,Rs2: stream2709947120125613254m_rule,Steps: stream6742152135616053226m_rule,M2: nat] :
      ( ( abstra406286406349174356ist_fm @ Eff4 @ Rules2 @ S4 )
     => ( ( member4699826688122452638ist_fm @ S3 @ S4 )
       => ( ( abstra3665372904625986210m_rule @ Rules2 @ Rs2 )
         => ( ( abstra7926801140857401835m_rule @ ( abstra7211189027691227674ist_fm @ Eff4 @ Rs2 @ S3 ) @ Steps )
           => ? [N2: nat,S6: produc6018962875968178549ist_fm] :
                ( ( member4699826688122452638ist_fm @ S6 @ S4 )
                & ( abstra7926801140857401835m_rule @ ( abstra7211189027691227674ist_fm @ Eff4 @ ( sdrop_8169176516188972301m_rule @ N2 @ Rs2 ) @ S6 ) @ ( sdrop_4865846802426565681m_rule @ M2 @ Steps ) ) ) ) ) ) ) ).

% RuleSystem.ipath_mkTree_sdrop
thf(fact_576_RuleSystem_Oipath__mkTree__sdrop,axiom,
    ! [Eff4: produc340336539035504054m_rule > rule > fset_rule > $o,Rules2: stream2709947120125613254m_rule,S4: set_rule,S3: rule,Rs2: stream2709947120125613254m_rule,Steps: stream8953843411776101167m_rule,M2: nat] :
      ( ( abstra7792261008698710771e_rule @ Eff4 @ Rules2 @ S4 )
     => ( ( member_rule @ S3 @ S4 )
       => ( ( abstra3665372904625986210m_rule @ Rules2 @ Rs2 )
         => ( ( abstra7237224312902573144m_rule @ ( abstra5235605640745886637e_rule @ Eff4 @ Rs2 @ S3 ) @ Steps )
           => ? [N2: nat,S6: rule] :
                ( ( member_rule @ S6 @ S4 )
                & ( abstra7237224312902573144m_rule @ ( abstra5235605640745886637e_rule @ Eff4 @ ( sdrop_8169176516188972301m_rule @ N2 @ Rs2 ) @ S6 ) @ ( sdrop_746751008794826386m_rule @ M2 @ Steps ) ) ) ) ) ) ) ).

% RuleSystem.ipath_mkTree_sdrop
thf(fact_577_RuleSystem_Oipath__mkTree__sdrop,axiom,
    ! [Eff4: produc340336539035504054m_rule > produc859450856879609959at_nat > fset_P3619101297660450253at_nat > $o,Rules2: stream2709947120125613254m_rule,S4: set_Pr8693737435421807431at_nat,S3: produc859450856879609959at_nat,Rs2: stream2709947120125613254m_rule,Steps: stream6015630362508254684m_rule,M2: nat] :
      ( ( abstra8713744819842732358at_nat @ Eff4 @ Rules2 @ S4 )
     => ( ( member8206827879206165904at_nat @ S3 @ S4 )
       => ( ( abstra3665372904625986210m_rule @ Rules2 @ Rs2 )
         => ( ( abstra8928944673429280989m_rule @ ( abstra2996675635236158732at_nat @ Eff4 @ Rs2 @ S3 ) @ Steps )
           => ? [N2: nat,S6: produc859450856879609959at_nat] :
                ( ( member8206827879206165904at_nat @ S6 @ S4 )
                & ( abstra8928944673429280989m_rule @ ( abstra2996675635236158732at_nat @ Eff4 @ ( sdrop_8169176516188972301m_rule @ N2 @ Rs2 ) @ S6 ) @ ( sdrop_5806731007647134499m_rule @ M2 @ Steps ) ) ) ) ) ) ) ).

% RuleSystem.ipath_mkTree_sdrop
thf(fact_578_RuleSystem_Oipath__mkTree__sdrop,axiom,
    ! [Eff4: rule > nat > fset_nat > $o,Rules2: stream_rule,S4: set_nat,S3: nat,Rs2: stream_rule,Steps: stream8912087984624402761t_rule,M2: nat] :
      ( ( abstra3263948797621512205le_nat @ Eff4 @ Rules2 @ S4 )
     => ( ( member_nat @ S3 @ S4 )
       => ( ( abstra3799686578551160190r_rule @ Rules2 @ Rs2 )
         => ( ( abstra5135089046934318962t_rule @ ( abstra7662105669874348231le_nat @ Eff4 @ Rs2 @ S3 ) @ Steps )
           => ? [N2: nat,S6: nat] :
                ( ( member_nat @ S6 @ S4 )
                & ( abstra5135089046934318962t_rule @ ( abstra7662105669874348231le_nat @ Eff4 @ ( sdrop_rule @ N2 @ Rs2 ) @ S6 ) @ ( sdrop_3122356783091591596t_rule @ M2 @ Steps ) ) ) ) ) ) ) ).

% RuleSystem.ipath_mkTree_sdrop
thf(fact_579_RuleSystem_Oipath__mkTree__sdrop,axiom,
    ! [Eff4: rule > rule > fset_rule > $o,Rules2: stream_rule,S4: set_rule,S3: rule,Rs2: stream_rule,Steps: stream4385846686851721995e_rule,M2: nat] :
      ( ( abstra6805424310982398927e_rule @ Eff4 @ Rules2 @ S4 )
     => ( ( member_rule @ S3 @ S4 )
       => ( ( abstra3799686578551160190r_rule @ Rules2 @ Rs2 )
         => ( ( abstra1660175255847126580e_rule @ ( abstra2300331465871827593e_rule @ Eff4 @ Rs2 @ S3 ) @ Steps )
           => ? [N2: nat,S6: rule] :
                ( ( member_rule @ S6 @ S4 )
                & ( abstra1660175255847126580e_rule @ ( abstra2300331465871827593e_rule @ Eff4 @ ( sdrop_rule @ N2 @ Rs2 ) @ S6 ) @ ( sdrop_9113879250048157294e_rule @ M2 @ Steps ) ) ) ) ) ) ) ).

% RuleSystem.ipath_mkTree_sdrop
thf(fact_580_RuleSystem_Oipath__mkTree__sdrop,axiom,
    ! [Eff4: rule > produc859450856879609959at_nat > fset_P3619101297660450253at_nat > $o,Rules2: stream_rule,S4: set_Pr8693737435421807431at_nat,S3: produc859450856879609959at_nat,Rs2: stream_rule,Steps: stream6749460841574940856t_rule,M2: nat] :
      ( ( abstra3287435747676517226at_nat @ Eff4 @ Rules2 @ S4 )
     => ( ( member8206827879206165904at_nat @ S3 @ S4 )
       => ( ( abstra3799686578551160190r_rule @ Rules2 @ Rs2 )
         => ( ( abstra826546537944104377t_rule @ ( abstra7049071577609443632at_nat @ Eff4 @ Rs2 @ S3 ) @ Steps )
           => ? [N2: nat,S6: produc859450856879609959at_nat] :
                ( ( member8206827879206165904at_nat @ S6 @ S4 )
                & ( abstra826546537944104377t_rule @ ( abstra7049071577609443632at_nat @ Eff4 @ ( sdrop_rule @ N2 @ Rs2 ) @ S6 ) @ ( sdrop_1568235664025375231t_rule @ M2 @ Steps ) ) ) ) ) ) ) ).

% RuleSystem.ipath_mkTree_sdrop
thf(fact_581_RuleSystem_Oipath__mkTree__sdrop,axiom,
    ! [Eff4: rule > produc6018962875968178549ist_fm > fset_P8989946509869081563ist_fm > $o,Rules2: stream_rule,S4: set_Pr5202636777678657877ist_fm,S3: produc6018962875968178549ist_fm,Rs2: stream_rule,Steps: stream2709947120125613254m_rule,M2: nat] :
      ( ( abstra5221733350967904376ist_fm @ Eff4 @ Rules2 @ S4 )
     => ( ( member4699826688122452638ist_fm @ S3 @ S4 )
       => ( ( abstra3799686578551160190r_rule @ Rules2 @ Rs2 )
         => ( ( abstra8836096083317999815m_rule @ ( abstra7840189589880272446ist_fm @ Eff4 @ Rs2 @ S3 ) @ Steps )
           => ? [N2: nat,S6: produc6018962875968178549ist_fm] :
                ( ( member4699826688122452638ist_fm @ S6 @ S4 )
                & ( abstra8836096083317999815m_rule @ ( abstra7840189589880272446ist_fm @ Eff4 @ ( sdrop_rule @ N2 @ Rs2 ) @ S6 ) @ ( sdrop_8169176516188972301m_rule @ M2 @ Steps ) ) ) ) ) ) ) ).

% RuleSystem.ipath_mkTree_sdrop
thf(fact_582_PersistentRuleSystem_Oaxioms_I1_J,axiom,
    ! [Eff4: rule > produc6018962875968178549ist_fm > fset_P8989946509869081563ist_fm > $o,Rules2: stream_rule,S4: set_Pr5202636777678657877ist_fm] :
      ( ( abstra3967921542344661089ist_fm @ Eff4 @ Rules2 @ S4 )
     => ( abstra5221733350967904376ist_fm @ Eff4 @ Rules2 @ S4 ) ) ).

% PersistentRuleSystem.axioms(1)
thf(fact_583_shd__sset,axiom,
    ! [A: stream_nat] : ( member_nat @ ( shd_nat @ A ) @ ( sset_nat @ A ) ) ).

% shd_sset
thf(fact_584_shd__sset,axiom,
    ! [A: stream4408948924543953275ist_fm] : ( member4699826688122452638ist_fm @ ( shd_Pr3211216682057661985ist_fm @ A ) @ ( sset_P5379989128061332361ist_fm @ A ) ) ).

% shd_sset
thf(fact_585_shd__sset,axiom,
    ! [A: stream8372878641218411373at_nat] : ( member8206827879206165904at_nat @ ( shd_Pr8412153233960533267at_nat @ A ) @ ( sset_P9164763103296375419at_nat @ A ) ) ).

% shd_sset
thf(fact_586_shd__sset,axiom,
    ! [A: stream2709947120125613254m_rule] : ( member7231649785386036813m_rule @ ( shd_Pr4562317740776619530m_rule @ A ) @ ( sset_P4484857331586881186m_rule @ A ) ) ).

% shd_sset
thf(fact_587_shd__sset,axiom,
    ! [A: stream_rule] : ( member_rule @ ( shd_rule @ A ) @ ( sset_rule @ A ) ) ).

% shd_sset
thf(fact_588_holds_Osimps,axiom,
    ( linear7925859179493631106m_rule
    = ( ^ [P: produc340336539035504054m_rule > $o,Xs: stream2709947120125613254m_rule] : ( P @ ( shd_Pr4562317740776619530m_rule @ Xs ) ) ) ) ).

% holds.simps
thf(fact_589_holds_Osimps,axiom,
    ( linear7354821883049305950s_rule
    = ( ^ [P: rule > $o,Xs: stream_rule] : ( P @ ( shd_rule @ Xs ) ) ) ) ).

% holds.simps
thf(fact_590_holds_Oelims_I1_J,axiom,
    ! [X3: produc340336539035504054m_rule > $o,Xa2: stream2709947120125613254m_rule,Y: $o] :
      ( ( ( linear7925859179493631106m_rule @ X3 @ Xa2 )
        = Y )
     => ( Y
        = ( X3 @ ( shd_Pr4562317740776619530m_rule @ Xa2 ) ) ) ) ).

% holds.elims(1)
thf(fact_591_holds_Oelims_I1_J,axiom,
    ! [X3: rule > $o,Xa2: stream_rule,Y: $o] :
      ( ( ( linear7354821883049305950s_rule @ X3 @ Xa2 )
        = Y )
     => ( Y
        = ( X3 @ ( shd_rule @ Xa2 ) ) ) ) ).

% holds.elims(1)
thf(fact_592_holds_Oelims_I2_J,axiom,
    ! [X3: produc340336539035504054m_rule > $o,Xa2: stream2709947120125613254m_rule] :
      ( ( linear7925859179493631106m_rule @ X3 @ Xa2 )
     => ( X3 @ ( shd_Pr4562317740776619530m_rule @ Xa2 ) ) ) ).

% holds.elims(2)
thf(fact_593_holds_Oelims_I2_J,axiom,
    ! [X3: rule > $o,Xa2: stream_rule] :
      ( ( linear7354821883049305950s_rule @ X3 @ Xa2 )
     => ( X3 @ ( shd_rule @ Xa2 ) ) ) ).

% holds.elims(2)
thf(fact_594_holds_Oelims_I3_J,axiom,
    ! [X3: produc340336539035504054m_rule > $o,Xa2: stream2709947120125613254m_rule] :
      ( ~ ( linear7925859179493631106m_rule @ X3 @ Xa2 )
     => ~ ( X3 @ ( shd_Pr4562317740776619530m_rule @ Xa2 ) ) ) ).

% holds.elims(3)
thf(fact_595_holds_Oelims_I3_J,axiom,
    ! [X3: rule > $o,Xa2: stream_rule] :
      ( ~ ( linear7354821883049305950s_rule @ X3 @ Xa2 )
     => ~ ( X3 @ ( shd_rule @ Xa2 ) ) ) ).

% holds.elims(3)
thf(fact_596_streams__shd,axiom,
    ! [S3: stream_nat,A2: set_nat] :
      ( ( member_stream_nat @ S3 @ ( streams_nat @ A2 ) )
     => ( member_nat @ ( shd_nat @ S3 ) @ A2 ) ) ).

% streams_shd
thf(fact_597_streams__shd,axiom,
    ! [S3: stream4408948924543953275ist_fm,A2: set_Pr5202636777678657877ist_fm] :
      ( ( member197948898507729188ist_fm @ S3 @ ( stream1217541351357117421ist_fm @ A2 ) )
     => ( member4699826688122452638ist_fm @ ( shd_Pr3211216682057661985ist_fm @ S3 ) @ A2 ) ) ).

% streams_shd
thf(fact_598_streams__shd,axiom,
    ! [S3: stream8372878641218411373at_nat,A2: set_Pr8693737435421807431at_nat] :
      ( ( member1992146023263842070at_nat @ S3 @ ( stream5885727000947712735at_nat @ A2 ) )
     => ( member8206827879206165904at_nat @ ( shd_Pr8412153233960533267at_nat @ S3 ) @ A2 ) ) ).

% streams_shd
thf(fact_599_streams__shd,axiom,
    ! [S3: stream2709947120125613254m_rule,A2: set_Pr1822751329126368876m_rule] :
      ( ( member5228318729319728861m_rule @ S3 @ ( stream3418479298939905342m_rule @ A2 ) )
     => ( member7231649785386036813m_rule @ ( shd_Pr4562317740776619530m_rule @ S3 ) @ A2 ) ) ).

% streams_shd
thf(fact_600_streams__shd,axiom,
    ! [S3: stream_rule,A2: set_rule] :
      ( ( member_stream_rule @ S3 @ ( streams_rule @ A2 ) )
     => ( member_rule @ ( shd_rule @ S3 ) @ A2 ) ) ).

% streams_shd
thf(fact_601_HLD__iff,axiom,
    ( linear5551378545506465448LD_nat
    = ( ^ [S: set_nat,Omega: stream_nat] : ( member_nat @ ( shd_nat @ Omega ) @ S ) ) ) ).

% HLD_iff
thf(fact_602_HLD__iff,axiom,
    ( linear6120129154741419165ist_fm
    = ( ^ [S: set_Pr5202636777678657877ist_fm,Omega: stream4408948924543953275ist_fm] : ( member4699826688122452638ist_fm @ ( shd_Pr3211216682057661985ist_fm @ Omega ) @ S ) ) ) ).

% HLD_iff
thf(fact_603_HLD__iff,axiom,
    ( linear6066394698532776847at_nat
    = ( ^ [S: set_Pr8693737435421807431at_nat,Omega: stream8372878641218411373at_nat] : ( member8206827879206165904at_nat @ ( shd_Pr8412153233960533267at_nat @ Omega ) @ S ) ) ) ).

% HLD_iff
thf(fact_604_HLD__iff,axiom,
    ( linear2885739836753817742m_rule
    = ( ^ [S: set_Pr1822751329126368876m_rule,Omega: stream2709947120125613254m_rule] : ( member7231649785386036813m_rule @ ( shd_Pr4562317740776619530m_rule @ Omega ) @ S ) ) ) ).

% HLD_iff
thf(fact_605_HLD__iff,axiom,
    ( linear5032321834864961386D_rule
    = ( ^ [S: set_rule,Omega: stream_rule] : ( member_rule @ ( shd_rule @ Omega ) @ S ) ) ) ).

% HLD_iff
thf(fact_606_smap2_Osimps_I1_J,axiom,
    ! [F: produc340336539035504054m_rule > produc340336539035504054m_rule > produc340336539035504054m_rule,S1: stream2709947120125613254m_rule,S2: stream2709947120125613254m_rule] :
      ( ( shd_Pr4562317740776619530m_rule @ ( smap2_6945574070536835298m_rule @ F @ S1 @ S2 ) )
      = ( F @ ( shd_Pr4562317740776619530m_rule @ S1 ) @ ( shd_Pr4562317740776619530m_rule @ S2 ) ) ) ).

% smap2.simps(1)
thf(fact_607_smap2_Osimps_I1_J,axiom,
    ! [F: produc340336539035504054m_rule > rule > produc340336539035504054m_rule,S1: stream2709947120125613254m_rule,S2: stream_rule] :
      ( ( shd_Pr4562317740776619530m_rule @ ( smap2_6366920367380388542m_rule @ F @ S1 @ S2 ) )
      = ( F @ ( shd_Pr4562317740776619530m_rule @ S1 ) @ ( shd_rule @ S2 ) ) ) ).

% smap2.simps(1)
thf(fact_608_smap2_Osimps_I1_J,axiom,
    ! [F: rule > produc340336539035504054m_rule > produc340336539035504054m_rule,S1: stream_rule,S2: stream2709947120125613254m_rule] :
      ( ( shd_Pr4562317740776619530m_rule @ ( smap2_4579943528986984894m_rule @ F @ S1 @ S2 ) )
      = ( F @ ( shd_rule @ S1 ) @ ( shd_Pr4562317740776619530m_rule @ S2 ) ) ) ).

% smap2.simps(1)
thf(fact_609_smap2_Osimps_I1_J,axiom,
    ! [F: rule > rule > produc340336539035504054m_rule,S1: stream_rule,S2: stream_rule] :
      ( ( shd_Pr4562317740776619530m_rule @ ( smap2_4319354692778897306m_rule @ F @ S1 @ S2 ) )
      = ( F @ ( shd_rule @ S1 ) @ ( shd_rule @ S2 ) ) ) ).

% smap2.simps(1)
thf(fact_610_smap2_Osimps_I1_J,axiom,
    ! [F: produc340336539035504054m_rule > produc340336539035504054m_rule > rule,S1: stream2709947120125613254m_rule,S2: stream2709947120125613254m_rule] :
      ( ( shd_rule @ ( smap2_1878155219598472126e_rule @ F @ S1 @ S2 ) )
      = ( F @ ( shd_Pr4562317740776619530m_rule @ S1 ) @ ( shd_Pr4562317740776619530m_rule @ S2 ) ) ) ).

% smap2.simps(1)
thf(fact_611_smap2_Osimps_I1_J,axiom,
    ! [F: produc340336539035504054m_rule > rule > rule,S1: stream2709947120125613254m_rule,S2: stream_rule] :
      ( ( shd_rule @ ( smap2_8971878701134242202e_rule @ F @ S1 @ S2 ) )
      = ( F @ ( shd_Pr4562317740776619530m_rule @ S1 ) @ ( shd_rule @ S2 ) ) ) ).

% smap2.simps(1)
thf(fact_612_smap2_Osimps_I1_J,axiom,
    ! [F: rule > produc340336539035504054m_rule > rule,S1: stream_rule,S2: stream2709947120125613254m_rule] :
      ( ( shd_rule @ ( smap2_9053961581851756698e_rule @ F @ S1 @ S2 ) )
      = ( F @ ( shd_rule @ S1 ) @ ( shd_Pr4562317740776619530m_rule @ S2 ) ) ) ).

% smap2.simps(1)
thf(fact_613_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_614_RuleSystem_Oenabled__R,axiom,
    ! [Eff4: rule > nat > fset_nat > $o,Rules2: stream_rule,S4: set_nat,S3: nat] :
      ( ( abstra3263948797621512205le_nat @ Eff4 @ Rules2 @ S4 )
     => ( ( member_nat @ S3 @ S4 )
       => ? [X2: rule] :
            ( ( member_rule @ X2 @ ( sset_rule @ Rules2 ) )
            & ? [X_1: fset_nat] : ( Eff4 @ X2 @ S3 @ X_1 ) ) ) ) ).

% RuleSystem.enabled_R
thf(fact_615_RuleSystem_Oenabled__R,axiom,
    ! [Eff4: rule > rule > fset_rule > $o,Rules2: stream_rule,S4: set_rule,S3: rule] :
      ( ( abstra6805424310982398927e_rule @ Eff4 @ Rules2 @ S4 )
     => ( ( member_rule @ S3 @ S4 )
       => ? [X2: rule] :
            ( ( member_rule @ X2 @ ( sset_rule @ Rules2 ) )
            & ? [X_1: fset_rule] : ( Eff4 @ X2 @ S3 @ X_1 ) ) ) ) ).

% RuleSystem.enabled_R
thf(fact_616_RuleSystem_Oenabled__R,axiom,
    ! [Eff4: rule > produc859450856879609959at_nat > fset_P3619101297660450253at_nat > $o,Rules2: stream_rule,S4: set_Pr8693737435421807431at_nat,S3: produc859450856879609959at_nat] :
      ( ( abstra3287435747676517226at_nat @ Eff4 @ Rules2 @ S4 )
     => ( ( member8206827879206165904at_nat @ S3 @ S4 )
       => ? [X2: rule] :
            ( ( member_rule @ X2 @ ( sset_rule @ Rules2 ) )
            & ? [X_1: fset_P3619101297660450253at_nat] : ( Eff4 @ X2 @ S3 @ X_1 ) ) ) ) ).

% RuleSystem.enabled_R
thf(fact_617_RuleSystem_Oenabled__R,axiom,
    ! [Eff4: rule > produc6018962875968178549ist_fm > fset_P8989946509869081563ist_fm > $o,Rules2: stream_rule,S4: set_Pr5202636777678657877ist_fm,S3: produc6018962875968178549ist_fm] :
      ( ( abstra5221733350967904376ist_fm @ Eff4 @ Rules2 @ S4 )
     => ( ( member4699826688122452638ist_fm @ S3 @ S4 )
       => ? [X2: rule] :
            ( ( member_rule @ X2 @ ( sset_rule @ Rules2 ) )
            & ? [X_1: fset_P8989946509869081563ist_fm] : ( Eff4 @ X2 @ S3 @ X_1 ) ) ) ) ).

% RuleSystem.enabled_R
thf(fact_618_RuleSystem_OminWait__ex,axiom,
    ! [Eff4: produc340336539035504054m_rule > nat > fset_nat > $o,Rules2: stream2709947120125613254m_rule,S4: set_nat,S3: nat,Rs2: stream2709947120125613254m_rule] :
      ( ( abstra2021474039004664625le_nat @ Eff4 @ Rules2 @ S4 )
     => ( ( member_nat @ S3 @ S4 )
       => ( ( abstra3665372904625986210m_rule @ Rules2 @ Rs2 )
         => ? [N2: nat] : ( abstra4513569833231518754le_nat @ Eff4 @ ( shd_Pr4562317740776619530m_rule @ ( sdrop_8169176516188972301m_rule @ N2 @ Rs2 ) ) @ S3 ) ) ) ) ).

% RuleSystem.minWait_ex
thf(fact_619_RuleSystem_OminWait__ex,axiom,
    ! [Eff4: produc340336539035504054m_rule > produc6018962875968178549ist_fm > fset_P8989946509869081563ist_fm > $o,Rules2: stream2709947120125613254m_rule,S4: set_Pr5202636777678657877ist_fm,S3: produc6018962875968178549ist_fm,Rs2: stream2709947120125613254m_rule] :
      ( ( abstra406286406349174356ist_fm @ Eff4 @ Rules2 @ S4 )
     => ( ( member4699826688122452638ist_fm @ S3 @ S4 )
       => ( ( abstra3665372904625986210m_rule @ Rules2 @ Rs2 )
         => ? [N2: nat] : ( abstra467781697624067491ist_fm @ Eff4 @ ( shd_Pr4562317740776619530m_rule @ ( sdrop_8169176516188972301m_rule @ N2 @ Rs2 ) ) @ S3 ) ) ) ) ).

% RuleSystem.minWait_ex
thf(fact_620_RuleSystem_OminWait__ex,axiom,
    ! [Eff4: produc340336539035504054m_rule > rule > fset_rule > $o,Rules2: stream2709947120125613254m_rule,S4: set_rule,S3: rule,Rs2: stream2709947120125613254m_rule] :
      ( ( abstra7792261008698710771e_rule @ Eff4 @ Rules2 @ S4 )
     => ( ( member_rule @ S3 @ S4 )
       => ( ( abstra3665372904625986210m_rule @ Rules2 @ Rs2 )
         => ? [N2: nat] : ( abstra1957041411669721828e_rule @ Eff4 @ ( shd_Pr4562317740776619530m_rule @ ( sdrop_8169176516188972301m_rule @ N2 @ Rs2 ) ) @ S3 ) ) ) ) ).

% RuleSystem.minWait_ex
thf(fact_621_RuleSystem_OminWait__ex,axiom,
    ! [Eff4: produc340336539035504054m_rule > produc859450856879609959at_nat > fset_P3619101297660450253at_nat > $o,Rules2: stream2709947120125613254m_rule,S4: set_Pr8693737435421807431at_nat,S3: produc859450856879609959at_nat,Rs2: stream2709947120125613254m_rule] :
      ( ( abstra8713744819842732358at_nat @ Eff4 @ Rules2 @ S4 )
     => ( ( member8206827879206165904at_nat @ S3 @ S4 )
       => ( ( abstra3665372904625986210m_rule @ Rules2 @ Rs2 )
         => ? [N2: nat] : ( abstra6415999825560908949at_nat @ Eff4 @ ( shd_Pr4562317740776619530m_rule @ ( sdrop_8169176516188972301m_rule @ N2 @ Rs2 ) ) @ S3 ) ) ) ) ).

% RuleSystem.minWait_ex
thf(fact_622_RuleSystem_OminWait__ex,axiom,
    ! [Eff4: rule > nat > fset_nat > $o,Rules2: stream_rule,S4: set_nat,S3: nat,Rs2: stream_rule] :
      ( ( abstra3263948797621512205le_nat @ Eff4 @ Rules2 @ S4 )
     => ( ( member_nat @ S3 @ S4 )
       => ( ( abstra3799686578551160190r_rule @ Rules2 @ Rs2 )
         => ? [N2: nat] : ( abstra8624044281687814142le_nat @ Eff4 @ ( shd_rule @ ( sdrop_rule @ N2 @ Rs2 ) ) @ S3 ) ) ) ) ).

% RuleSystem.minWait_ex
thf(fact_623_RuleSystem_OminWait__ex,axiom,
    ! [Eff4: rule > rule > fset_rule > $o,Rules2: stream_rule,S4: set_rule,S3: rule,Rs2: stream_rule] :
      ( ( abstra6805424310982398927e_rule @ Eff4 @ Rules2 @ S4 )
     => ( ( member_rule @ S3 @ S4 )
       => ( ( abstra3799686578551160190r_rule @ Rules2 @ Rs2 )
         => ? [N2: nat] : ( abstra7234149737463204544e_rule @ Eff4 @ ( shd_rule @ ( sdrop_rule @ N2 @ Rs2 ) ) @ S3 ) ) ) ) ).

% RuleSystem.minWait_ex
thf(fact_624_RuleSystem_OminWait__ex,axiom,
    ! [Eff4: rule > produc859450856879609959at_nat > fset_P3619101297660450253at_nat > $o,Rules2: stream_rule,S4: set_Pr8693737435421807431at_nat,S3: produc859450856879609959at_nat,Rs2: stream_rule] :
      ( ( abstra3287435747676517226at_nat @ Eff4 @ Rules2 @ S4 )
     => ( ( member8206827879206165904at_nat @ S3 @ S4 )
       => ( ( abstra3799686578551160190r_rule @ Rules2 @ Rs2 )
         => ? [N2: nat] : ( abstra7103363789936428985at_nat @ Eff4 @ ( shd_rule @ ( sdrop_rule @ N2 @ Rs2 ) ) @ S3 ) ) ) ) ).

% RuleSystem.minWait_ex
thf(fact_625_RuleSystem_OminWait__ex,axiom,
    ! [Eff4: rule > produc6018962875968178549ist_fm > fset_P8989946509869081563ist_fm > $o,Rules2: stream_rule,S4: set_Pr5202636777678657877ist_fm,S3: produc6018962875968178549ist_fm,Rs2: stream_rule] :
      ( ( abstra5221733350967904376ist_fm @ Eff4 @ Rules2 @ S4 )
     => ( ( member4699826688122452638ist_fm @ S3 @ S4 )
       => ( ( abstra3799686578551160190r_rule @ Rules2 @ Rs2 )
         => ? [N2: nat] : ( abstra1707737187183579335ist_fm @ Eff4 @ ( shd_rule @ ( sdrop_rule @ N2 @ Rs2 ) ) @ S3 ) ) ) ) ).

% RuleSystem.minWait_ex
thf(fact_626_RuleSystem__axioms,axiom,
    abstra5221733350967904376ist_fm @ eff @ rules @ top_to1730629564055774885ist_fm ).

% RuleSystem_axioms
thf(fact_627_RuleSystem_Oper__alw,axiom,
    ! [Eff4: list_fm > list_tm > fset_list_tm > $o,Rules2: stream_list_fm,S4: set_list_tm,R: list_fm,Steps: stream4408948924543953275ist_fm] :
      ( ( abstra4205323827771120475ist_tm @ Eff4 @ Rules2 @ S4 )
     => ( ( abstra8345049315893738326ist_tm @ Eff4 @ Rules2 @ S4 @ R )
       => ( ( ( abstra302294732650053493ist_tm @ Eff4 @ Rules2 @ Steps )
            & ( member_list_tm @ ( produc4874987732838623697ist_fm @ ( shd_Pr3211216682057661985ist_fm @ Steps ) ) @ S4 ) )
         => ( linear2489540979234913265ist_fm
            @ ^ [Xs: stream4408948924543953275ist_fm] :
                ( ( linear4742913937519564201ist_fm
                  @ ^ [Step: produc6018962875968178549ist_fm] : ( abstra990286104146457292ist_tm @ Eff4 @ R @ ( produc4874987732838623697ist_fm @ Step ) )
                  @ Xs )
               => ( ( linear4742913937519564201ist_fm
                    @ ^ [Step: produc6018962875968178549ist_fm] :
                        ( ( produc3242453064496107283ist_fm @ Step )
                        = R )
                    @ Xs )
                  | ( linear1228252369863192324t_fm_o
                    @ ( linear4742913937519564201ist_fm
                      @ ^ [Step: produc6018962875968178549ist_fm] : ( abstra990286104146457292ist_tm @ Eff4 @ R @ ( produc4874987732838623697ist_fm @ Step ) ) )
                    @ Xs ) ) )
            @ Steps ) ) ) ) ).

% RuleSystem.per_alw
thf(fact_628_RuleSystem_Oper__alw,axiom,
    ! [Eff4: rule > produc6018962875968178549ist_fm > fset_P8989946509869081563ist_fm > $o,Rules2: stream_rule,S4: set_Pr5202636777678657877ist_fm,R: rule,Steps: stream2709947120125613254m_rule] :
      ( ( abstra5221733350967904376ist_fm @ Eff4 @ Rules2 @ S4 )
     => ( ( abstra5255361903751151037ist_fm @ Eff4 @ Rules2 @ S4 @ R )
       => ( ( ( abstra6789711989322986974ist_fm @ Eff4 @ Rules2 @ Steps )
            & ( member4699826688122452638ist_fm @ ( produc993771546023551940m_rule @ ( shd_Pr4562317740776619530m_rule @ Steps ) ) @ S4 ) )
         => ( linear4957294957385550906m_rule
            @ ^ [Xs: stream2709947120125613254m_rule] :
                ( ( linear7925859179493631106m_rule
                  @ ^ [Step: produc340336539035504054m_rule] : ( abstra1707737187183579335ist_fm @ Eff4 @ R @ ( produc993771546023551940m_rule @ Step ) )
                  @ Xs )
               => ( ( linear7925859179493631106m_rule
                    @ ^ [Step: produc340336539035504054m_rule] :
                        ( ( produc6514711208036230658m_rule @ Step )
                        = R )
                    @ Xs )
                  | ( linear3460418925085718015rule_o
                    @ ( linear7925859179493631106m_rule
                      @ ^ [Step: produc340336539035504054m_rule] : ( abstra1707737187183579335ist_fm @ Eff4 @ R @ ( produc993771546023551940m_rule @ Step ) ) )
                    @ Xs ) ) )
            @ Steps ) ) ) ) ).

% RuleSystem.per_alw
thf(fact_629_PersistentRuleSystem_Oper,axiom,
    ! [Eff4: rule > produc6018962875968178549ist_fm > fset_P8989946509869081563ist_fm > $o,Rules2: stream_rule,S4: set_Pr5202636777678657877ist_fm,R: rule] :
      ( ( abstra3967921542344661089ist_fm @ Eff4 @ Rules2 @ S4 )
     => ( ( member_rule @ R @ ( sset_rule @ Rules2 ) )
       => ( abstra5255361903751151037ist_fm @ Eff4 @ Rules2 @ S4 @ R ) ) ) ).

% PersistentRuleSystem.per
thf(fact_630_PersistentRuleSystem__axioms,axiom,
    abstra3967921542344661089ist_fm @ eff @ rules @ top_to1730629564055774885ist_fm ).

% PersistentRuleSystem_axioms
thf(fact_631_RuleSystem_OminWait__def,axiom,
    ! [Eff4: rule > produc6018962875968178549ist_fm > fset_P8989946509869081563ist_fm > $o,Rules2: stream_rule,S4: set_Pr5202636777678657877ist_fm,Rs2: stream_rule,S3: produc6018962875968178549ist_fm] :
      ( ( abstra5221733350967904376ist_fm @ Eff4 @ Rules2 @ S4 )
     => ( ( abstra1963046427795717555ist_fm @ Eff4 @ Rs2 @ S3 )
        = ( ord_Least_nat
          @ ^ [N3: nat] : ( abstra1707737187183579335ist_fm @ Eff4 @ ( shd_rule @ ( sdrop_rule @ N3 @ Rs2 ) ) @ S3 ) ) ) ) ).

% RuleSystem.minWait_def
thf(fact_632_RuleSystem_Opos,axiom,
    ! [Eff4: rule > produc6018962875968178549ist_fm > fset_P8989946509869081563ist_fm > $o,Rules2: stream_rule,S4: set_Pr5202636777678657877ist_fm,Rs2: stream_rule,R: rule] :
      ( ( abstra5221733350967904376ist_fm @ Eff4 @ Rules2 @ S4 )
     => ( ( abstra3799686578551160190r_rule @ Rules2 @ Rs2 )
       => ( ( member_rule @ R @ ( sset_rule @ Rules2 ) )
         => ( ( shd_rule @ ( sdrop_rule @ ( abstract_pos_rule @ Rs2 @ R ) @ Rs2 ) )
            = R ) ) ) ) ).

% RuleSystem.pos
thf(fact_633_RuleSystem_OminWait__least,axiom,
    ! [Eff4: rule > produc6018962875968178549ist_fm > fset_P8989946509869081563ist_fm > $o,Rules2: stream_rule,S4: set_Pr5202636777678657877ist_fm,N: nat,Rs2: stream_rule,S3: produc6018962875968178549ist_fm] :
      ( ( abstra5221733350967904376ist_fm @ Eff4 @ Rules2 @ S4 )
     => ( ( abstra1707737187183579335ist_fm @ Eff4 @ ( shd_rule @ ( sdrop_rule @ N @ Rs2 ) ) @ S3 )
       => ( ord_less_eq_nat @ ( abstra1963046427795717555ist_fm @ Eff4 @ Rs2 @ S3 ) @ N ) ) ) ).

% RuleSystem.minWait_least
thf(fact_634_RuleSystem_Opos__def,axiom,
    ! [Eff4: rule > produc6018962875968178549ist_fm > fset_P8989946509869081563ist_fm > $o,Rules2: stream_rule,S4: set_Pr5202636777678657877ist_fm,Rs2: stream2709947120125613254m_rule,R: produc340336539035504054m_rule] :
      ( ( abstra5221733350967904376ist_fm @ Eff4 @ Rules2 @ S4 )
     => ( ( abstra4499547390127564210m_rule @ Rs2 @ R )
        = ( ord_Least_nat
          @ ^ [N3: nat] :
              ( ( shd_Pr4562317740776619530m_rule @ ( sdrop_8169176516188972301m_rule @ N3 @ Rs2 ) )
              = R ) ) ) ) ).

% RuleSystem.pos_def
thf(fact_635_RuleSystem_Opos__def,axiom,
    ! [Eff4: rule > produc6018962875968178549ist_fm > fset_P8989946509869081563ist_fm > $o,Rules2: stream_rule,S4: set_Pr5202636777678657877ist_fm,Rs2: stream_rule,R: rule] :
      ( ( abstra5221733350967904376ist_fm @ Eff4 @ Rules2 @ S4 )
     => ( ( abstract_pos_rule @ Rs2 @ R )
        = ( ord_Least_nat
          @ ^ [N3: nat] :
              ( ( shd_rule @ ( sdrop_rule @ N3 @ Rs2 ) )
              = R ) ) ) ) ).

% RuleSystem.pos_def
thf(fact_636_RuleSystem_Opos__least,axiom,
    ! [Eff4: rule > produc6018962875968178549ist_fm > fset_P8989946509869081563ist_fm > $o,Rules2: stream_rule,S4: set_Pr5202636777678657877ist_fm,N: nat,Rs2: stream2709947120125613254m_rule,R: produc340336539035504054m_rule] :
      ( ( abstra5221733350967904376ist_fm @ Eff4 @ Rules2 @ S4 )
     => ( ( ( shd_Pr4562317740776619530m_rule @ ( sdrop_8169176516188972301m_rule @ N @ Rs2 ) )
          = R )
       => ( ord_less_eq_nat @ ( abstra4499547390127564210m_rule @ Rs2 @ R ) @ N ) ) ) ).

% RuleSystem.pos_least
thf(fact_637_RuleSystem_Opos__least,axiom,
    ! [Eff4: rule > produc6018962875968178549ist_fm > fset_P8989946509869081563ist_fm > $o,Rules2: stream_rule,S4: set_Pr5202636777678657877ist_fm,N: nat,Rs2: stream_rule,R: rule] :
      ( ( abstra5221733350967904376ist_fm @ Eff4 @ Rules2 @ S4 )
     => ( ( ( shd_rule @ ( sdrop_rule @ N @ Rs2 ) )
          = R )
       => ( ord_less_eq_nat @ ( abstract_pos_rule @ Rs2 @ R ) @ N ) ) ) ).

% RuleSystem.pos_least
thf(fact_638_pos__def,axiom,
    ( abstra4499547390127564210m_rule
    = ( ^ [Rs: stream2709947120125613254m_rule,R4: produc340336539035504054m_rule] :
          ( ord_Least_nat
          @ ^ [N3: nat] :
              ( ( shd_Pr4562317740776619530m_rule @ ( sdrop_8169176516188972301m_rule @ N3 @ Rs ) )
              = R4 ) ) ) ) ).

% pos_def
thf(fact_639_pos__def,axiom,
    ( abstract_pos_rule
    = ( ^ [Rs: stream_rule,R4: rule] :
          ( ord_Least_nat
          @ ^ [N3: nat] :
              ( ( shd_rule @ ( sdrop_rule @ N3 @ Rs ) )
              = R4 ) ) ) ) ).

% pos_def
thf(fact_640_prod_Oexpand,axiom,
    ! [Prod: produc6018962875968178549ist_fm,Prod2: produc6018962875968178549ist_fm] :
      ( ( ( ( produc4874987732838623697ist_fm @ Prod )
          = ( produc4874987732838623697ist_fm @ Prod2 ) )
        & ( ( produc3242453064496107283ist_fm @ Prod )
          = ( produc3242453064496107283ist_fm @ Prod2 ) ) )
     => ( Prod = Prod2 ) ) ).

% prod.expand
thf(fact_641_prod_Oexpand,axiom,
    ! [Prod: produc340336539035504054m_rule,Prod2: produc340336539035504054m_rule] :
      ( ( ( ( produc993771546023551940m_rule @ Prod )
          = ( produc993771546023551940m_rule @ Prod2 ) )
        & ( ( produc6514711208036230658m_rule @ Prod )
          = ( produc6514711208036230658m_rule @ Prod2 ) ) )
     => ( Prod = Prod2 ) ) ).

% prod.expand
thf(fact_642_RuleSystem_OminWait_Ocong,axiom,
    abstra1963046427795717555ist_fm = abstra1963046427795717555ist_fm ).

% RuleSystem.minWait.cong
thf(fact_643_minWait__least,axiom,
    ! [N: nat,Rs2: stream_rule,S3: produc6018962875968178549ist_fm] :
      ( ( abstra1707737187183579335ist_fm @ eff @ ( shd_rule @ ( sdrop_rule @ N @ Rs2 ) ) @ S3 )
     => ( ord_less_eq_nat @ ( abstra1963046427795717555ist_fm @ eff @ Rs2 @ S3 ) @ N ) ) ).

% minWait_least
thf(fact_644_pos,axiom,
    ! [Rs2: stream_rule,R: rule] :
      ( ( abstra3799686578551160190r_rule @ rules @ Rs2 )
     => ( ( member_rule @ R @ ( sset_rule @ rules ) )
       => ( ( shd_rule @ ( sdrop_rule @ ( abstract_pos_rule @ Rs2 @ R ) @ Rs2 ) )
          = R ) ) ) ).

% pos
thf(fact_645_minWait__def,axiom,
    ! [Rs2: stream_rule,S3: produc6018962875968178549ist_fm] :
      ( ( abstra1963046427795717555ist_fm @ eff @ Rs2 @ S3 )
      = ( ord_Least_nat
        @ ^ [N3: nat] : ( abstra1707737187183579335ist_fm @ eff @ ( shd_rule @ ( sdrop_rule @ N3 @ Rs2 ) ) @ S3 ) ) ) ).

% minWait_def
thf(fact_646_RuleSystem_OminWait__le__pos,axiom,
    ! [Eff4: rule > produc6018962875968178549ist_fm > fset_P8989946509869081563ist_fm > $o,Rules2: stream_rule,S4: set_Pr5202636777678657877ist_fm,Rs2: stream_rule,R: rule,S3: produc6018962875968178549ist_fm] :
      ( ( abstra5221733350967904376ist_fm @ Eff4 @ Rules2 @ S4 )
     => ( ( abstra3799686578551160190r_rule @ Rules2 @ Rs2 )
       => ( ( member_rule @ R @ ( sset_rule @ Rules2 ) )
         => ( ( abstra1707737187183579335ist_fm @ Eff4 @ R @ S3 )
           => ( ord_less_eq_nat @ ( abstra1963046427795717555ist_fm @ Eff4 @ Rs2 @ S3 ) @ ( abstract_pos_rule @ Rs2 @ R ) ) ) ) ) ) ).

% RuleSystem.minWait_le_pos
thf(fact_647_minWait__le__pos,axiom,
    ! [Rs2: stream_rule,R: rule,S3: produc6018962875968178549ist_fm] :
      ( ( abstra3799686578551160190r_rule @ rules @ Rs2 )
     => ( ( member_rule @ R @ ( sset_rule @ rules ) )
       => ( ( abstra1707737187183579335ist_fm @ eff @ R @ S3 )
         => ( ord_less_eq_nat @ ( abstra1963046427795717555ist_fm @ eff @ Rs2 @ S3 ) @ ( abstract_pos_rule @ Rs2 @ R ) ) ) ) ) ).

% minWait_le_pos
thf(fact_648_pos__least,axiom,
    ! [N: nat,Rs2: stream2709947120125613254m_rule,R: produc340336539035504054m_rule] :
      ( ( ( shd_Pr4562317740776619530m_rule @ ( sdrop_8169176516188972301m_rule @ N @ Rs2 ) )
        = R )
     => ( ord_less_eq_nat @ ( abstra4499547390127564210m_rule @ Rs2 @ R ) @ N ) ) ).

% pos_least
thf(fact_649_pos__least,axiom,
    ! [N: nat,Rs2: stream_rule,R: rule] :
      ( ( ( shd_rule @ ( sdrop_rule @ N @ Rs2 ) )
        = R )
     => ( ord_less_eq_nat @ ( abstract_pos_rule @ Rs2 @ R ) @ N ) ) ).

% pos_least
thf(fact_650_minWait__ex,axiom,
    ! [S3: produc6018962875968178549ist_fm,Rs2: stream_rule] :
      ( ( member4699826688122452638ist_fm @ S3 @ top_to1730629564055774885ist_fm )
     => ( ( abstra3799686578551160190r_rule @ rules @ Rs2 )
       => ? [N2: nat] : ( abstra1707737187183579335ist_fm @ eff @ ( shd_rule @ ( sdrop_rule @ N2 @ Rs2 ) ) @ S3 ) ) ) ).

% minWait_ex
thf(fact_651_prod__eq__iff,axiom,
    ( ( ^ [Y2: produc6018962875968178549ist_fm,Z2: produc6018962875968178549ist_fm] : ( Y2 = Z2 ) )
    = ( ^ [S: produc6018962875968178549ist_fm,T2: produc6018962875968178549ist_fm] :
          ( ( ( produc4874987732838623697ist_fm @ S )
            = ( produc4874987732838623697ist_fm @ T2 ) )
          & ( ( produc3242453064496107283ist_fm @ S )
            = ( produc3242453064496107283ist_fm @ T2 ) ) ) ) ) ).

% prod_eq_iff
thf(fact_652_prod__eq__iff,axiom,
    ( ( ^ [Y2: produc340336539035504054m_rule,Z2: produc340336539035504054m_rule] : ( Y2 = Z2 ) )
    = ( ^ [S: produc340336539035504054m_rule,T2: produc340336539035504054m_rule] :
          ( ( ( produc993771546023551940m_rule @ S )
            = ( produc993771546023551940m_rule @ T2 ) )
          & ( ( produc6514711208036230658m_rule @ S )
            = ( produc6514711208036230658m_rule @ T2 ) ) ) ) ) ).

% prod_eq_iff
thf(fact_653_prod__eqI,axiom,
    ! [P3: produc6018962875968178549ist_fm,Q3: produc6018962875968178549ist_fm] :
      ( ( ( produc4874987732838623697ist_fm @ P3 )
        = ( produc4874987732838623697ist_fm @ Q3 ) )
     => ( ( ( produc3242453064496107283ist_fm @ P3 )
          = ( produc3242453064496107283ist_fm @ Q3 ) )
       => ( P3 = Q3 ) ) ) ).

% prod_eqI
thf(fact_654_prod__eqI,axiom,
    ! [P3: produc340336539035504054m_rule,Q3: produc340336539035504054m_rule] :
      ( ( ( produc993771546023551940m_rule @ P3 )
        = ( produc993771546023551940m_rule @ Q3 ) )
     => ( ( ( produc6514711208036230658m_rule @ P3 )
          = ( produc6514711208036230658m_rule @ Q3 ) )
       => ( P3 = Q3 ) ) ) ).

% prod_eqI
thf(fact_655_exE__realizer_H,axiom,
    ! [P2: list_fm > list_tm > $o,P3: produc6018962875968178549ist_fm] :
      ( ( P2 @ ( produc3242453064496107283ist_fm @ P3 ) @ ( produc4874987732838623697ist_fm @ P3 ) )
     => ~ ! [X2: list_tm,Y4: list_fm] :
            ~ ( P2 @ Y4 @ X2 ) ) ).

% exE_realizer'
thf(fact_656_exE__realizer_H,axiom,
    ! [P2: rule > produc6018962875968178549ist_fm > $o,P3: produc340336539035504054m_rule] :
      ( ( P2 @ ( produc6514711208036230658m_rule @ P3 ) @ ( produc993771546023551940m_rule @ P3 ) )
     => ~ ! [X2: produc6018962875968178549ist_fm,Y4: rule] :
            ~ ( P2 @ Y4 @ X2 ) ) ).

% exE_realizer'
thf(fact_657_wf__mkTree,axiom,
    ! [S3: produc6018962875968178549ist_fm,Rs2: stream_rule] :
      ( ( member4699826688122452638ist_fm @ S3 @ top_to1730629564055774885ist_fm )
     => ( ( abstra3799686578551160190r_rule @ rules @ Rs2 )
       => ( abstra8167393772827826667ist_fm @ eff @ rules @ ( abstra7840189589880272446ist_fm @ eff @ Rs2 @ S3 ) ) ) ) ).

% wf_mkTree
thf(fact_658_trim__alt,axiom,
    ! [S3: produc6018962875968178549ist_fm,Rs2: stream_rule] :
      ( ( member4699826688122452638ist_fm @ S3 @ top_to1730629564055774885ist_fm )
     => ( ( abstra3799686578551160190r_rule @ rules @ Rs2 )
       => ( ( abstra7806236140005899928ist_fm @ eff @ Rs2 @ S3 )
          = ( sdrop_rule @ ( abstra1963046427795717555ist_fm @ eff @ Rs2 @ S3 ) @ Rs2 ) ) ) ) ).

% trim_alt
thf(fact_659_trim__enabled,axiom,
    ! [S3: produc6018962875968178549ist_fm,Rs2: stream_rule] :
      ( ( member4699826688122452638ist_fm @ S3 @ top_to1730629564055774885ist_fm )
     => ( ( abstra3799686578551160190r_rule @ rules @ Rs2 )
       => ( abstra1707737187183579335ist_fm @ eff @ ( shd_rule @ ( abstra7806236140005899928ist_fm @ eff @ Rs2 @ S3 ) ) @ S3 ) ) ) ).

% trim_enabled
thf(fact_660_PersistentRuleSystem_Ointro,axiom,
    ! [Eff4: rule > produc6018962875968178549ist_fm > fset_P8989946509869081563ist_fm > $o,Rules2: stream_rule,S4: set_Pr5202636777678657877ist_fm] :
      ( ( abstra5221733350967904376ist_fm @ Eff4 @ Rules2 @ S4 )
     => ( ( abstra9069655090277518462ist_fm @ Eff4 @ Rules2 @ S4 )
       => ( abstra3967921542344661089ist_fm @ Eff4 @ Rules2 @ S4 ) ) ) ).

% PersistentRuleSystem.intro
thf(fact_661_PersistentRuleSystem__def,axiom,
    ( abstra3967921542344661089ist_fm
    = ( ^ [Eff: rule > produc6018962875968178549ist_fm > fset_P8989946509869081563ist_fm > $o,Rules: stream_rule,S5: set_Pr5202636777678657877ist_fm] :
          ( ( abstra5221733350967904376ist_fm @ Eff @ Rules @ S5 )
          & ( abstra9069655090277518462ist_fm @ Eff @ Rules @ S5 ) ) ) ) ).

% PersistentRuleSystem_def
thf(fact_662_RuleSystem__Defs_Otrim_Ocong,axiom,
    abstra7806236140005899928ist_fm = abstra7806236140005899928ist_fm ).

% RuleSystem_Defs.trim.cong
thf(fact_663_RuleSystem__Defs_Owf_Ocong,axiom,
    abstra8167393772827826667ist_fm = abstra8167393772827826667ist_fm ).

% RuleSystem_Defs.wf.cong
thf(fact_664_RuleSystem_Otrim__fair,axiom,
    ! [Eff4: rule > nat > fset_nat > $o,Rules2: stream_rule,S4: set_nat,S3: nat,Rs2: stream_rule] :
      ( ( abstra3263948797621512205le_nat @ Eff4 @ Rules2 @ S4 )
     => ( ( member_nat @ S3 @ S4 )
       => ( ( abstra3799686578551160190r_rule @ Rules2 @ Rs2 )
         => ( abstra3799686578551160190r_rule @ Rules2 @ ( abstra8319304012324366829le_nat @ Eff4 @ Rs2 @ S3 ) ) ) ) ) ).

% RuleSystem.trim_fair
thf(fact_665_RuleSystem_Otrim__fair,axiom,
    ! [Eff4: rule > rule > fset_rule > $o,Rules2: stream_rule,S4: set_rule,S3: rule,Rs2: stream_rule] :
      ( ( abstra6805424310982398927e_rule @ Eff4 @ Rules2 @ S4 )
     => ( ( member_rule @ S3 @ S4 )
       => ( ( abstra3799686578551160190r_rule @ Rules2 @ Rs2 )
         => ( abstra3799686578551160190r_rule @ Rules2 @ ( abstra2680930950814381487e_rule @ Eff4 @ Rs2 @ S3 ) ) ) ) ) ).

% RuleSystem.trim_fair
thf(fact_666_RuleSystem_Otrim__fair,axiom,
    ! [Eff4: rule > produc859450856879609959at_nat > fset_P3619101297660450253at_nat > $o,Rules2: stream_rule,S4: set_Pr8693737435421807431at_nat,S3: produc859450856879609959at_nat,Rs2: stream_rule] :
      ( ( abstra3287435747676517226at_nat @ Eff4 @ Rules2 @ S4 )
     => ( ( member8206827879206165904at_nat @ S3 @ S4 )
       => ( ( abstra3799686578551160190r_rule @ Rules2 @ Rs2 )
         => ( abstra3799686578551160190r_rule @ Rules2 @ ( abstra1069316477956532618at_nat @ Eff4 @ Rs2 @ S3 ) ) ) ) ) ).

% RuleSystem.trim_fair
thf(fact_667_RuleSystem_Otrim__fair,axiom,
    ! [Eff4: rule > produc6018962875968178549ist_fm > fset_P8989946509869081563ist_fm > $o,Rules2: stream_rule,S4: set_Pr5202636777678657877ist_fm,S3: produc6018962875968178549ist_fm,Rs2: stream_rule] :
      ( ( abstra5221733350967904376ist_fm @ Eff4 @ Rules2 @ S4 )
     => ( ( member4699826688122452638ist_fm @ S3 @ S4 )
       => ( ( abstra3799686578551160190r_rule @ Rules2 @ Rs2 )
         => ( abstra3799686578551160190r_rule @ Rules2 @ ( abstra7806236140005899928ist_fm @ Eff4 @ Rs2 @ S3 ) ) ) ) ) ).

% RuleSystem.trim_fair
thf(fact_668_RuleSystem__Defs_Owf__ipath__epath,axiom,
    ! [Eff4: rule > produc6018962875968178549ist_fm > fset_P8989946509869081563ist_fm > $o,Rules2: stream_rule,T3: abstra8520858721168826610m_rule,Steps: stream2709947120125613254m_rule] :
      ( ( abstra8167393772827826667ist_fm @ Eff4 @ Rules2 @ T3 )
     => ( ( abstra8836096083317999815m_rule @ T3 @ Steps )
       => ( abstra6789711989322986974ist_fm @ Eff4 @ Rules2 @ Steps ) ) ) ).

% RuleSystem_Defs.wf_ipath_epath
thf(fact_669_PersistentRuleSystem_Oaxioms_I2_J,axiom,
    ! [Eff4: rule > produc6018962875968178549ist_fm > fset_P8989946509869081563ist_fm > $o,Rules2: stream_rule,S4: set_Pr5202636777678657877ist_fm] :
      ( ( abstra3967921542344661089ist_fm @ Eff4 @ Rules2 @ S4 )
     => ( abstra9069655090277518462ist_fm @ Eff4 @ Rules2 @ S4 ) ) ).

% PersistentRuleSystem.axioms(2)
thf(fact_670_RuleSystem_Owf__mkTree,axiom,
    ! [Eff4: rule > nat > fset_nat > $o,Rules2: stream_rule,S4: set_nat,S3: nat,Rs2: stream_rule] :
      ( ( abstra3263948797621512205le_nat @ Eff4 @ Rules2 @ S4 )
     => ( ( member_nat @ S3 @ S4 )
       => ( ( abstra3799686578551160190r_rule @ Rules2 @ Rs2 )
         => ( abstra4678492489212612058le_nat @ Eff4 @ Rules2 @ ( abstra7662105669874348231le_nat @ Eff4 @ Rs2 @ S3 ) ) ) ) ) ).

% RuleSystem.wf_mkTree
thf(fact_671_RuleSystem_Owf__mkTree,axiom,
    ! [Eff4: rule > rule > fset_rule > $o,Rules2: stream_rule,S4: set_rule,S3: rule,Rs2: stream_rule] :
      ( ( abstra6805424310982398927e_rule @ Eff4 @ Rules2 @ S4 )
     => ( ( member_rule @ S3 @ S4 )
       => ( ( abstra3799686578551160190r_rule @ Rules2 @ Rs2 )
         => ( abstra459280387713764508e_rule @ Eff4 @ Rules2 @ ( abstra2300331465871827593e_rule @ Eff4 @ Rs2 @ S3 ) ) ) ) ) ).

% RuleSystem.wf_mkTree
thf(fact_672_RuleSystem_Owf__mkTree,axiom,
    ! [Eff4: rule > produc859450856879609959at_nat > fset_P3619101297660450253at_nat > $o,Rules2: stream_rule,S4: set_Pr8693737435421807431at_nat,S3: produc859450856879609959at_nat,Rs2: stream_rule] :
      ( ( abstra3287435747676517226at_nat @ Eff4 @ Rules2 @ S4 )
     => ( ( member8206827879206165904at_nat @ S3 @ S4 )
       => ( ( abstra3799686578551160190r_rule @ Rules2 @ Rs2 )
         => ( abstra8301603278984798429at_nat @ Eff4 @ Rules2 @ ( abstra7049071577609443632at_nat @ Eff4 @ Rs2 @ S3 ) ) ) ) ) ).

% RuleSystem.wf_mkTree
thf(fact_673_RuleSystem_Owf__mkTree,axiom,
    ! [Eff4: rule > produc6018962875968178549ist_fm > fset_P8989946509869081563ist_fm > $o,Rules2: stream_rule,S4: set_Pr5202636777678657877ist_fm,S3: produc6018962875968178549ist_fm,Rs2: stream_rule] :
      ( ( abstra5221733350967904376ist_fm @ Eff4 @ Rules2 @ S4 )
     => ( ( member4699826688122452638ist_fm @ S3 @ S4 )
       => ( ( abstra3799686578551160190r_rule @ Rules2 @ Rs2 )
         => ( abstra8167393772827826667ist_fm @ Eff4 @ Rules2 @ ( abstra7840189589880272446ist_fm @ Eff4 @ Rs2 @ S3 ) ) ) ) ) ).

% RuleSystem.wf_mkTree
thf(fact_674_i_Owf__ipath__epath,axiom,
    ! [Eff2: rule > produc6018962875968178549ist_fm > option6967287582980624417ist_fm,Rules2: stream_rule,T3: abstra8520858721168826610m_rule,Steps: stream2709947120125613254m_rule] :
      ( ( abstra8167393772827826667ist_fm @ ( abstra2682625350522704545ist_fm @ Eff2 ) @ Rules2 @ T3 )
     => ( ( abstra8836096083317999815m_rule @ T3 @ Steps )
       => ( abstra6789711989322986974ist_fm @ ( abstra2682625350522704545ist_fm @ Eff2 ) @ Rules2 @ Steps ) ) ) ).

% i.wf_ipath_epath
thf(fact_675_trim__fair,axiom,
    ! [S3: produc6018962875968178549ist_fm,Rs2: stream_rule] :
      ( ( member4699826688122452638ist_fm @ S3 @ top_to1730629564055774885ist_fm )
     => ( ( abstra3799686578551160190r_rule @ rules @ Rs2 )
       => ( abstra3799686578551160190r_rule @ rules @ ( abstra7806236140005899928ist_fm @ eff @ Rs2 @ S3 ) ) ) ) ).

% trim_fair
thf(fact_676_RuleSystem_Otrim__in__R,axiom,
    ! [Eff4: nat > nat > fset_nat > $o,Rules2: stream_nat,S4: set_nat,S3: nat,Rs2: stream_nat] :
      ( ( abstra5088622135990055243at_nat @ Eff4 @ Rules2 @ S4 )
     => ( ( member_nat @ S3 @ S4 )
       => ( ( abstra7018062486307922108ir_nat @ Rules2 @ Rs2 )
         => ( member_nat @ ( shd_nat @ ( abstra7573781713853518635at_nat @ Eff4 @ Rs2 @ S3 ) ) @ ( sset_nat @ Rules2 ) ) ) ) ) ).

% RuleSystem.trim_in_R
thf(fact_677_RuleSystem_Otrim__in__R,axiom,
    ! [Eff4: nat > rule > fset_rule > $o,Rules2: stream_nat,S4: set_rule,S3: rule,Rs2: stream_nat] :
      ( ( abstra7777025902706499341t_rule @ Eff4 @ Rules2 @ S4 )
     => ( ( member_rule @ S3 @ S4 )
       => ( ( abstra7018062486307922108ir_nat @ Rules2 @ Rs2 )
         => ( member_nat @ ( shd_nat @ ( abstra3609009080554578157t_rule @ Eff4 @ Rs2 @ S3 ) ) @ ( sset_nat @ Rules2 ) ) ) ) ) ).

% RuleSystem.trim_in_R
thf(fact_678_RuleSystem_Otrim__in__R,axiom,
    ! [Eff4: rule > nat > fset_nat > $o,Rules2: stream_rule,S4: set_nat,S3: nat,Rs2: stream_rule] :
      ( ( abstra3263948797621512205le_nat @ Eff4 @ Rules2 @ S4 )
     => ( ( member_nat @ S3 @ S4 )
       => ( ( abstra3799686578551160190r_rule @ Rules2 @ Rs2 )
         => ( member_rule @ ( shd_rule @ ( abstra8319304012324366829le_nat @ Eff4 @ Rs2 @ S3 ) ) @ ( sset_rule @ Rules2 ) ) ) ) ) ).

% RuleSystem.trim_in_R
thf(fact_679_RuleSystem_Otrim__in__R,axiom,
    ! [Eff4: rule > rule > fset_rule > $o,Rules2: stream_rule,S4: set_rule,S3: rule,Rs2: stream_rule] :
      ( ( abstra6805424310982398927e_rule @ Eff4 @ Rules2 @ S4 )
     => ( ( member_rule @ S3 @ S4 )
       => ( ( abstra3799686578551160190r_rule @ Rules2 @ Rs2 )
         => ( member_rule @ ( shd_rule @ ( abstra2680930950814381487e_rule @ Eff4 @ Rs2 @ S3 ) ) @ ( sset_rule @ Rules2 ) ) ) ) ) ).

% RuleSystem.trim_in_R
thf(fact_680_RuleSystem_Otrim__in__R,axiom,
    ! [Eff4: produc6018962875968178549ist_fm > nat > fset_nat > $o,Rules2: stream4408948924543953275ist_fm,S4: set_nat,S3: nat,Rs2: stream4408948924543953275ist_fm] :
      ( ( abstra4534667231653962588fm_nat @ Eff4 @ Rules2 @ S4 )
     => ( ( member_nat @ S3 @ S4 )
       => ( ( abstra7670258426472572809ist_fm @ Rules2 @ Rs2 )
         => ( member4699826688122452638ist_fm @ ( shd_Pr3211216682057661985ist_fm @ ( abstra1492044971799116156fm_nat @ Eff4 @ Rs2 @ S3 ) ) @ ( sset_P5379989128061332361ist_fm @ Rules2 ) ) ) ) ) ).

% RuleSystem.trim_in_R
thf(fact_681_RuleSystem_Otrim__in__R,axiom,
    ! [Eff4: nat > produc6018962875968178549ist_fm > fset_P8989946509869081563ist_fm > $o,Rules2: stream_nat,S4: set_Pr5202636777678657877ist_fm,S3: produc6018962875968178549ist_fm,Rs2: stream_nat] :
      ( ( abstra6010661502565574330ist_fm @ Eff4 @ Rules2 @ S4 )
     => ( ( member4699826688122452638ist_fm @ S3 @ S4 )
       => ( ( abstra7018062486307922108ir_nat @ Rules2 @ Rs2 )
         => ( member_nat @ ( shd_nat @ ( abstra2968039242710727898ist_fm @ Eff4 @ Rs2 @ S3 ) ) @ ( sset_nat @ Rules2 ) ) ) ) ) ).

% RuleSystem.trim_in_R
thf(fact_682_RuleSystem_Otrim__in__R,axiom,
    ! [Eff4: produc6018962875968178549ist_fm > rule > fset_rule > $o,Rules2: stream4408948924543953275ist_fm,S4: set_rule,S3: rule,Rs2: stream4408948924543953275ist_fm] :
      ( ( abstra2695409591818887198m_rule @ Eff4 @ Rules2 @ S4 )
     => ( ( member_rule @ S3 @ S4 )
       => ( ( abstra7670258426472572809ist_fm @ Rules2 @ Rs2 )
         => ( member4699826688122452638ist_fm @ ( shd_Pr3211216682057661985ist_fm @ ( abstra5279912380856882750m_rule @ Eff4 @ Rs2 @ S3 ) ) @ ( sset_P5379989128061332361ist_fm @ Rules2 ) ) ) ) ) ).

% RuleSystem.trim_in_R
thf(fact_683_RuleSystem_Otrim__in__R,axiom,
    ! [Eff4: rule > produc6018962875968178549ist_fm > fset_P8989946509869081563ist_fm > $o,Rules2: stream_rule,S4: set_Pr5202636777678657877ist_fm,S3: produc6018962875968178549ist_fm,Rs2: stream_rule] :
      ( ( abstra5221733350967904376ist_fm @ Eff4 @ Rules2 @ S4 )
     => ( ( member4699826688122452638ist_fm @ S3 @ S4 )
       => ( ( abstra3799686578551160190r_rule @ Rules2 @ Rs2 )
         => ( member_rule @ ( shd_rule @ ( abstra7806236140005899928ist_fm @ Eff4 @ Rs2 @ S3 ) ) @ ( sset_rule @ Rules2 ) ) ) ) ) ).

% RuleSystem.trim_in_R
thf(fact_684_RuleSystem_Otrim__in__R,axiom,
    ! [Eff4: produc859450856879609959at_nat > nat > fset_nat > $o,Rules2: stream8372878641218411373at_nat,S4: set_nat,S3: nat,Rs2: stream8372878641218411373at_nat] :
      ( ( abstra1380253255819158094at_nat @ Eff4 @ Rules2 @ S4 )
     => ( ( member_nat @ S3 @ S4 )
       => ( ( abstra5618069685180253819at_nat @ Rules2 @ Rs2 )
         => ( member8206827879206165904at_nat @ ( shd_Pr8412153233960533267at_nat @ ( abstra5497505521111644270at_nat @ Eff4 @ Rs2 @ S3 ) ) @ ( sset_P9164763103296375419at_nat @ Rules2 ) ) ) ) ) ).

% RuleSystem.trim_in_R
thf(fact_685_RuleSystem_Otrim__in__R,axiom,
    ! [Eff4: produc859450856879609959at_nat > rule > fset_rule > $o,Rules2: stream8372878641218411373at_nat,S4: set_rule,S3: rule,Rs2: stream8372878641218411373at_nat] :
      ( ( abstra5264416619086884112t_rule @ Eff4 @ Rules2 @ S4 )
     => ( ( member_rule @ S3 @ S4 )
       => ( ( abstra5618069685180253819at_nat @ Rules2 @ Rs2 )
         => ( member8206827879206165904at_nat @ ( shd_Pr8412153233960533267at_nat @ ( abstra3046297349366899504t_rule @ Eff4 @ Rs2 @ S3 ) ) @ ( sset_P9164763103296375419at_nat @ Rules2 ) ) ) ) ) ).

% RuleSystem.trim_in_R
thf(fact_686_wf__ipath__epath,axiom,
    ! [T3: abstra8520858721168826610m_rule,Steps: stream2709947120125613254m_rule] :
      ( ( abstra8167393772827826667ist_fm @ eff @ rules @ T3 )
     => ( ( abstra8836096083317999815m_rule @ T3 @ Steps )
       => ( abstra6789711989322986974ist_fm @ eff @ rules @ Steps ) ) ) ).

% wf_ipath_epath
thf(fact_687_RuleSystem_Otrim__enabled,axiom,
    ! [Eff4: produc340336539035504054m_rule > nat > fset_nat > $o,Rules2: stream2709947120125613254m_rule,S4: set_nat,S3: nat,Rs2: stream2709947120125613254m_rule] :
      ( ( abstra2021474039004664625le_nat @ Eff4 @ Rules2 @ S4 )
     => ( ( member_nat @ S3 @ S4 )
       => ( ( abstra3665372904625986210m_rule @ Rules2 @ Rs2 )
         => ( abstra4513569833231518754le_nat @ Eff4 @ ( shd_Pr4562317740776619530m_rule @ ( abstra7926412722278118673le_nat @ Eff4 @ Rs2 @ S3 ) ) @ S3 ) ) ) ) ).

% RuleSystem.trim_enabled
thf(fact_688_RuleSystem_Otrim__enabled,axiom,
    ! [Eff4: produc340336539035504054m_rule > produc6018962875968178549ist_fm > fset_P8989946509869081563ist_fm > $o,Rules2: stream2709947120125613254m_rule,S4: set_Pr5202636777678657877ist_fm,S3: produc6018962875968178549ist_fm,Rs2: stream2709947120125613254m_rule] :
      ( ( abstra406286406349174356ist_fm @ Eff4 @ Rules2 @ S4 )
     => ( ( member4699826688122452638ist_fm @ S3 @ S4 )
       => ( ( abstra3665372904625986210m_rule @ Rules2 @ Rs2 )
         => ( abstra467781697624067491ist_fm @ Eff4 @ ( shd_Pr4562317740776619530m_rule @ ( abstra1355475359686397044ist_fm @ Eff4 @ Rs2 @ S3 ) ) @ S3 ) ) ) ) ).

% RuleSystem.trim_enabled
thf(fact_689_RuleSystem_Otrim__enabled,axiom,
    ! [Eff4: produc340336539035504054m_rule > rule > fset_rule > $o,Rules2: stream2709947120125613254m_rule,S4: set_rule,S3: rule,Rs2: stream2709947120125613254m_rule] :
      ( ( abstra7792261008698710771e_rule @ Eff4 @ Rules2 @ S4 )
     => ( ( member_rule @ S3 @ S4 )
       => ( ( abstra3665372904625986210m_rule @ Rules2 @ Rs2 )
         => ( abstra1957041411669721828e_rule @ Eff4 @ ( shd_Pr4562317740776619530m_rule @ ( abstra6343601422196104403e_rule @ Eff4 @ Rs2 @ S3 ) ) @ S3 ) ) ) ) ).

% RuleSystem.trim_enabled
thf(fact_690_RuleSystem_Otrim__enabled,axiom,
    ! [Eff4: produc340336539035504054m_rule > produc859450856879609959at_nat > fset_P3619101297660450253at_nat > $o,Rules2: stream2709947120125613254m_rule,S4: set_Pr8693737435421807431at_nat,S3: produc859450856879609959at_nat,Rs2: stream2709947120125613254m_rule] :
      ( ( abstra8713744819842732358at_nat @ Eff4 @ Rules2 @ S4 )
     => ( ( member8206827879206165904at_nat @ S3 @ S4 )
       => ( ( abstra3665372904625986210m_rule @ Rules2 @ Rs2 )
         => ( abstra6415999825560908949at_nat @ Eff4 @ ( shd_Pr4562317740776619530m_rule @ ( abstra7961736532019816294at_nat @ Eff4 @ Rs2 @ S3 ) ) @ S3 ) ) ) ) ).

% RuleSystem.trim_enabled
thf(fact_691_RuleSystem_Otrim__enabled,axiom,
    ! [Eff4: rule > nat > fset_nat > $o,Rules2: stream_rule,S4: set_nat,S3: nat,Rs2: stream_rule] :
      ( ( abstra3263948797621512205le_nat @ Eff4 @ Rules2 @ S4 )
     => ( ( member_nat @ S3 @ S4 )
       => ( ( abstra3799686578551160190r_rule @ Rules2 @ Rs2 )
         => ( abstra8624044281687814142le_nat @ Eff4 @ ( shd_rule @ ( abstra8319304012324366829le_nat @ Eff4 @ Rs2 @ S3 ) ) @ S3 ) ) ) ) ).

% RuleSystem.trim_enabled
thf(fact_692_RuleSystem_Otrim__enabled,axiom,
    ! [Eff4: rule > rule > fset_rule > $o,Rules2: stream_rule,S4: set_rule,S3: rule,Rs2: stream_rule] :
      ( ( abstra6805424310982398927e_rule @ Eff4 @ Rules2 @ S4 )
     => ( ( member_rule @ S3 @ S4 )
       => ( ( abstra3799686578551160190r_rule @ Rules2 @ Rs2 )
         => ( abstra7234149737463204544e_rule @ Eff4 @ ( shd_rule @ ( abstra2680930950814381487e_rule @ Eff4 @ Rs2 @ S3 ) ) @ S3 ) ) ) ) ).

% RuleSystem.trim_enabled
thf(fact_693_RuleSystem_Otrim__enabled,axiom,
    ! [Eff4: rule > produc859450856879609959at_nat > fset_P3619101297660450253at_nat > $o,Rules2: stream_rule,S4: set_Pr8693737435421807431at_nat,S3: produc859450856879609959at_nat,Rs2: stream_rule] :
      ( ( abstra3287435747676517226at_nat @ Eff4 @ Rules2 @ S4 )
     => ( ( member8206827879206165904at_nat @ S3 @ S4 )
       => ( ( abstra3799686578551160190r_rule @ Rules2 @ Rs2 )
         => ( abstra7103363789936428985at_nat @ Eff4 @ ( shd_rule @ ( abstra1069316477956532618at_nat @ Eff4 @ Rs2 @ S3 ) ) @ S3 ) ) ) ) ).

% RuleSystem.trim_enabled
thf(fact_694_RuleSystem_Otrim__enabled,axiom,
    ! [Eff4: rule > produc6018962875968178549ist_fm > fset_P8989946509869081563ist_fm > $o,Rules2: stream_rule,S4: set_Pr5202636777678657877ist_fm,S3: produc6018962875968178549ist_fm,Rs2: stream_rule] :
      ( ( abstra5221733350967904376ist_fm @ Eff4 @ Rules2 @ S4 )
     => ( ( member4699826688122452638ist_fm @ S3 @ S4 )
       => ( ( abstra3799686578551160190r_rule @ Rules2 @ Rs2 )
         => ( abstra1707737187183579335ist_fm @ Eff4 @ ( shd_rule @ ( abstra7806236140005899928ist_fm @ Eff4 @ Rs2 @ S3 ) ) @ S3 ) ) ) ) ).

% RuleSystem.trim_enabled
thf(fact_695_PersistentRuleSystem__axioms_Ointro,axiom,
    ! [Rules2: stream_rule,Eff4: rule > produc6018962875968178549ist_fm > fset_P8989946509869081563ist_fm > $o,S4: set_Pr5202636777678657877ist_fm] :
      ( ! [R6: rule] :
          ( ( member_rule @ R6 @ ( sset_rule @ Rules2 ) )
         => ( abstra5255361903751151037ist_fm @ Eff4 @ Rules2 @ S4 @ R6 ) )
     => ( abstra9069655090277518462ist_fm @ Eff4 @ Rules2 @ S4 ) ) ).

% PersistentRuleSystem_axioms.intro
thf(fact_696_PersistentRuleSystem__axioms__def,axiom,
    ( abstra9069655090277518462ist_fm
    = ( ^ [Eff: rule > produc6018962875968178549ist_fm > fset_P8989946509869081563ist_fm > $o,Rules: stream_rule,S5: set_Pr5202636777678657877ist_fm] :
        ! [R4: rule] :
          ( ( member_rule @ R4 @ ( sset_rule @ Rules ) )
         => ( abstra5255361903751151037ist_fm @ Eff @ Rules @ S5 @ R4 ) ) ) ) ).

% PersistentRuleSystem_axioms_def
thf(fact_697_RuleSystem_Otrim__alt,axiom,
    ! [Eff4: produc340336539035504054m_rule > nat > fset_nat > $o,Rules2: stream2709947120125613254m_rule,S4: set_nat,S3: nat,Rs2: stream2709947120125613254m_rule] :
      ( ( abstra2021474039004664625le_nat @ Eff4 @ Rules2 @ S4 )
     => ( ( member_nat @ S3 @ S4 )
       => ( ( abstra3665372904625986210m_rule @ Rules2 @ Rs2 )
         => ( ( abstra7926412722278118673le_nat @ Eff4 @ Rs2 @ S3 )
            = ( sdrop_8169176516188972301m_rule @ ( abstra7995654914383480886le_nat @ Eff4 @ Rs2 @ S3 ) @ Rs2 ) ) ) ) ) ).

% RuleSystem.trim_alt
thf(fact_698_RuleSystem_Otrim__alt,axiom,
    ! [Eff4: produc340336539035504054m_rule > produc6018962875968178549ist_fm > fset_P8989946509869081563ist_fm > $o,Rules2: stream2709947120125613254m_rule,S4: set_Pr5202636777678657877ist_fm,S3: produc6018962875968178549ist_fm,Rs2: stream2709947120125613254m_rule] :
      ( ( abstra406286406349174356ist_fm @ Eff4 @ Rules2 @ S4 )
     => ( ( member4699826688122452638ist_fm @ S3 @ S4 )
       => ( ( abstra3665372904625986210m_rule @ Rules2 @ Rs2 )
         => ( ( abstra1355475359686397044ist_fm @ Eff4 @ Rs2 @ S3 )
            = ( sdrop_8169176516188972301m_rule @ ( abstra7932678668315420303ist_fm @ Eff4 @ Rs2 @ S3 ) @ Rs2 ) ) ) ) ) ).

% RuleSystem.trim_alt
thf(fact_699_RuleSystem_Otrim__alt,axiom,
    ! [Eff4: produc340336539035504054m_rule > rule > fset_rule > $o,Rules2: stream2709947120125613254m_rule,S4: set_rule,S3: rule,Rs2: stream2709947120125613254m_rule] :
      ( ( abstra7792261008698710771e_rule @ Eff4 @ Rules2 @ S4 )
     => ( ( member_rule @ S3 @ S4 )
       => ( ( abstra3665372904625986210m_rule @ Rules2 @ Rs2 )
         => ( ( abstra6343601422196104403e_rule @ Eff4 @ Rs2 @ S3 )
            = ( sdrop_8169176516188972301m_rule @ ( abstra898225168387205880e_rule @ Eff4 @ Rs2 @ S3 ) @ Rs2 ) ) ) ) ) ).

% RuleSystem.trim_alt
thf(fact_700_RuleSystem_Otrim__alt,axiom,
    ! [Eff4: produc340336539035504054m_rule > produc859450856879609959at_nat > fset_P3619101297660450253at_nat > $o,Rules2: stream2709947120125613254m_rule,S4: set_Pr8693737435421807431at_nat,S3: produc859450856879609959at_nat,Rs2: stream2709947120125613254m_rule] :
      ( ( abstra8713744819842732358at_nat @ Eff4 @ Rules2 @ S4 )
     => ( ( member8206827879206165904at_nat @ S3 @ S4 )
       => ( ( abstra3665372904625986210m_rule @ Rules2 @ Rs2 )
         => ( ( abstra7961736532019816294at_nat @ Eff4 @ Rs2 @ S3 )
            = ( sdrop_8169176516188972301m_rule @ ( abstra810980732209741185at_nat @ Eff4 @ Rs2 @ S3 ) @ Rs2 ) ) ) ) ) ).

% RuleSystem.trim_alt
thf(fact_701_RuleSystem_Otrim__alt,axiom,
    ! [Eff4: rule > nat > fset_nat > $o,Rules2: stream_rule,S4: set_nat,S3: nat,Rs2: stream_rule] :
      ( ( abstra3263948797621512205le_nat @ Eff4 @ Rules2 @ S4 )
     => ( ( member_nat @ S3 @ S4 )
       => ( ( abstra3799686578551160190r_rule @ Rules2 @ Rs2 )
         => ( ( abstra8319304012324366829le_nat @ Eff4 @ Rs2 @ S3 )
            = ( sdrop_rule @ ( abstra6265646195202037778le_nat @ Eff4 @ Rs2 @ S3 ) @ Rs2 ) ) ) ) ) ).

% RuleSystem.trim_alt
thf(fact_702_RuleSystem_Otrim__alt,axiom,
    ! [Eff4: rule > rule > fset_rule > $o,Rules2: stream_rule,S4: set_rule,S3: rule,Rs2: stream_rule] :
      ( ( abstra6805424310982398927e_rule @ Eff4 @ Rules2 @ S4 )
     => ( ( member_rule @ S3 @ S4 )
       => ( ( abstra3799686578551160190r_rule @ Rules2 @ Rs2 )
         => ( ( abstra2680930950814381487e_rule @ Eff4 @ Rs2 @ S3 )
            = ( sdrop_rule @ ( abstra538998172947062484e_rule @ Eff4 @ Rs2 @ S3 ) @ Rs2 ) ) ) ) ) ).

% RuleSystem.trim_alt
thf(fact_703_RuleSystem_Otrim__alt,axiom,
    ! [Eff4: rule > produc859450856879609959at_nat > fset_P3619101297660450253at_nat > $o,Rules2: stream_rule,S4: set_Pr8693737435421807431at_nat,S3: produc859450856879609959at_nat,Rs2: stream_rule] :
      ( ( abstra3287435747676517226at_nat @ Eff4 @ Rules2 @ S4 )
     => ( ( member8206827879206165904at_nat @ S3 @ S4 )
       => ( ( abstra3799686578551160190r_rule @ Rules2 @ Rs2 )
         => ( ( abstra1069316477956532618at_nat @ Eff4 @ Rs2 @ S3 )
            = ( sdrop_rule @ ( abstra1808058216634590373at_nat @ Eff4 @ Rs2 @ S3 ) @ Rs2 ) ) ) ) ) ).

% RuleSystem.trim_alt
thf(fact_704_RuleSystem_Otrim__alt,axiom,
    ! [Eff4: rule > produc6018962875968178549ist_fm > fset_P8989946509869081563ist_fm > $o,Rules2: stream_rule,S4: set_Pr5202636777678657877ist_fm,S3: produc6018962875968178549ist_fm,Rs2: stream_rule] :
      ( ( abstra5221733350967904376ist_fm @ Eff4 @ Rules2 @ S4 )
     => ( ( member4699826688122452638ist_fm @ S3 @ S4 )
       => ( ( abstra3799686578551160190r_rule @ Rules2 @ Rs2 )
         => ( ( abstra7806236140005899928ist_fm @ Eff4 @ Rs2 @ S3 )
            = ( sdrop_rule @ ( abstra1963046427795717555ist_fm @ Eff4 @ Rs2 @ S3 ) @ Rs2 ) ) ) ) ) ).

% RuleSystem.trim_alt
thf(fact_705_trim__in__R,axiom,
    ! [S3: produc6018962875968178549ist_fm,Rs2: stream_rule] :
      ( ( member4699826688122452638ist_fm @ S3 @ top_to1730629564055774885ist_fm )
     => ( ( abstra3799686578551160190r_rule @ rules @ Rs2 )
       => ( member_rule @ ( shd_rule @ ( abstra7806236140005899928ist_fm @ eff @ Rs2 @ S3 ) ) @ ( sset_rule @ rules ) ) ) ) ).

% trim_in_R
thf(fact_706_trim__def,axiom,
    ! [Rs2: stream_rule,S3: produc6018962875968178549ist_fm] :
      ( ( abstra7806236140005899928ist_fm @ eff @ Rs2 @ S3 )
      = ( sdrop_while_rule
        @ ^ [R4: rule] :
            ~ ( abstra1707737187183579335ist_fm @ eff @ R4 @ S3 )
        @ Rs2 ) ) ).

% trim_def
thf(fact_707_i_Otrim__def,axiom,
    ! [Eff2: rule > produc6018962875968178549ist_fm > option6967287582980624417ist_fm,Rs2: stream_rule,S3: produc6018962875968178549ist_fm] :
      ( ( abstra7806236140005899928ist_fm @ ( abstra2682625350522704545ist_fm @ Eff2 ) @ Rs2 @ S3 )
      = ( sdrop_while_rule
        @ ^ [R4: rule] :
            ~ ( abstra1707737187183579335ist_fm @ ( abstra2682625350522704545ist_fm @ Eff2 ) @ R4 @ S3 )
        @ Rs2 ) ) ).

% i.trim_def
thf(fact_708_epath__completeness__Saturated,axiom,
    ! [S3: produc6018962875968178549ist_fm] :
      ( ( member4699826688122452638ist_fm @ S3 @ top_to1730629564055774885ist_fm )
     => ( ? [T4: abstra8520858721168826610m_rule] :
            ( ( ( produc993771546023551940m_rule @ ( abstra5180277867158163941m_rule @ T4 ) )
              = S3 )
            & ( abstra8167393772827826667ist_fm @ eff @ rules @ T4 )
            & ( abstra3822255059661934556m_rule @ T4 ) )
        | ? [Steps3: stream2709947120125613254m_rule] :
            ( ( ( produc993771546023551940m_rule @ ( shd_Pr4562317740776619530m_rule @ Steps3 ) )
              = S3 )
            & ( abstra6789711989322986974ist_fm @ eff @ rules @ Steps3 )
            & ( abstra6097777249025082867ist_fm @ eff @ rules @ Steps3 ) ) ) ) ).

% epath_completeness_Saturated
thf(fact_709_eff__S,axiom,
    ! [S3: produc6018962875968178549ist_fm,R: rule,Sl: fset_P8989946509869081563ist_fm,S7: produc6018962875968178549ist_fm] :
      ( ( member4699826688122452638ist_fm @ S3 @ top_to1730629564055774885ist_fm )
     => ( ( member_rule @ R @ ( sset_rule @ rules ) )
       => ( ( eff @ R @ S3 @ Sl )
         => ( ( fmembe3754813877001230652ist_fm @ S7 @ Sl )
           => ( member4699826688122452638ist_fm @ S7 @ top_to1730629564055774885ist_fm ) ) ) ) ) ).

% eff_S
thf(fact_710_RuleSystem__Defs_Otrim__def,axiom,
    ( abstra7806236140005899928ist_fm
    = ( ^ [Eff: rule > produc6018962875968178549ist_fm > fset_P8989946509869081563ist_fm > $o,Rs: stream_rule,S: produc6018962875968178549ist_fm] :
          ( sdrop_while_rule
          @ ^ [R4: rule] :
              ~ ( abstra1707737187183579335ist_fm @ Eff @ R4 @ S )
          @ Rs ) ) ) ).

% RuleSystem_Defs.trim_def
thf(fact_711_RuleSystem_Oeff__S,axiom,
    ! [Eff4: nat > nat > fset_nat > $o,Rules2: stream_nat,S4: set_nat,S3: nat,R: nat,Sl: fset_nat,S7: nat] :
      ( ( abstra5088622135990055243at_nat @ Eff4 @ Rules2 @ S4 )
     => ( ( member_nat @ S3 @ S4 )
       => ( ( member_nat @ R @ ( sset_nat @ Rules2 ) )
         => ( ( Eff4 @ R @ S3 @ Sl )
           => ( ( fmember_nat @ S7 @ Sl )
             => ( member_nat @ S7 @ S4 ) ) ) ) ) ) ).

% RuleSystem.eff_S
thf(fact_712_RuleSystem_Oeff__S,axiom,
    ! [Eff4: nat > rule > fset_rule > $o,Rules2: stream_nat,S4: set_rule,S3: rule,R: nat,Sl: fset_rule,S7: rule] :
      ( ( abstra7777025902706499341t_rule @ Eff4 @ Rules2 @ S4 )
     => ( ( member_rule @ S3 @ S4 )
       => ( ( member_nat @ R @ ( sset_nat @ Rules2 ) )
         => ( ( Eff4 @ R @ S3 @ Sl )
           => ( ( fmember_rule @ S7 @ Sl )
             => ( member_rule @ S7 @ S4 ) ) ) ) ) ) ).

% RuleSystem.eff_S
thf(fact_713_RuleSystem_Oeff__S,axiom,
    ! [Eff4: rule > nat > fset_nat > $o,Rules2: stream_rule,S4: set_nat,S3: nat,R: rule,Sl: fset_nat,S7: nat] :
      ( ( abstra3263948797621512205le_nat @ Eff4 @ Rules2 @ S4 )
     => ( ( member_nat @ S3 @ S4 )
       => ( ( member_rule @ R @ ( sset_rule @ Rules2 ) )
         => ( ( Eff4 @ R @ S3 @ Sl )
           => ( ( fmember_nat @ S7 @ Sl )
             => ( member_nat @ S7 @ S4 ) ) ) ) ) ) ).

% RuleSystem.eff_S
thf(fact_714_RuleSystem_Oeff__S,axiom,
    ! [Eff4: rule > rule > fset_rule > $o,Rules2: stream_rule,S4: set_rule,S3: rule,R: rule,Sl: fset_rule,S7: rule] :
      ( ( abstra6805424310982398927e_rule @ Eff4 @ Rules2 @ S4 )
     => ( ( member_rule @ S3 @ S4 )
       => ( ( member_rule @ R @ ( sset_rule @ Rules2 ) )
         => ( ( Eff4 @ R @ S3 @ Sl )
           => ( ( fmember_rule @ S7 @ Sl )
             => ( member_rule @ S7 @ S4 ) ) ) ) ) ) ).

% RuleSystem.eff_S
thf(fact_715_RuleSystem_Oeff__S,axiom,
    ! [Eff4: produc6018962875968178549ist_fm > nat > fset_nat > $o,Rules2: stream4408948924543953275ist_fm,S4: set_nat,S3: nat,R: produc6018962875968178549ist_fm,Sl: fset_nat,S7: nat] :
      ( ( abstra4534667231653962588fm_nat @ Eff4 @ Rules2 @ S4 )
     => ( ( member_nat @ S3 @ S4 )
       => ( ( member4699826688122452638ist_fm @ R @ ( sset_P5379989128061332361ist_fm @ Rules2 ) )
         => ( ( Eff4 @ R @ S3 @ Sl )
           => ( ( fmember_nat @ S7 @ Sl )
             => ( member_nat @ S7 @ S4 ) ) ) ) ) ) ).

% RuleSystem.eff_S
thf(fact_716_RuleSystem_Oeff__S,axiom,
    ! [Eff4: produc6018962875968178549ist_fm > rule > fset_rule > $o,Rules2: stream4408948924543953275ist_fm,S4: set_rule,S3: rule,R: produc6018962875968178549ist_fm,Sl: fset_rule,S7: rule] :
      ( ( abstra2695409591818887198m_rule @ Eff4 @ Rules2 @ S4 )
     => ( ( member_rule @ S3 @ S4 )
       => ( ( member4699826688122452638ist_fm @ R @ ( sset_P5379989128061332361ist_fm @ Rules2 ) )
         => ( ( Eff4 @ R @ S3 @ Sl )
           => ( ( fmember_rule @ S7 @ Sl )
             => ( member_rule @ S7 @ S4 ) ) ) ) ) ) ).

% RuleSystem.eff_S
thf(fact_717_RuleSystem_Oeff__S,axiom,
    ! [Eff4: nat > produc6018962875968178549ist_fm > fset_P8989946509869081563ist_fm > $o,Rules2: stream_nat,S4: set_Pr5202636777678657877ist_fm,S3: produc6018962875968178549ist_fm,R: nat,Sl: fset_P8989946509869081563ist_fm,S7: produc6018962875968178549ist_fm] :
      ( ( abstra6010661502565574330ist_fm @ Eff4 @ Rules2 @ S4 )
     => ( ( member4699826688122452638ist_fm @ S3 @ S4 )
       => ( ( member_nat @ R @ ( sset_nat @ Rules2 ) )
         => ( ( Eff4 @ R @ S3 @ Sl )
           => ( ( fmembe3754813877001230652ist_fm @ S7 @ Sl )
             => ( member4699826688122452638ist_fm @ S7 @ S4 ) ) ) ) ) ) ).

% RuleSystem.eff_S
thf(fact_718_RuleSystem_Oeff__S,axiom,
    ! [Eff4: rule > produc6018962875968178549ist_fm > fset_P8989946509869081563ist_fm > $o,Rules2: stream_rule,S4: set_Pr5202636777678657877ist_fm,S3: produc6018962875968178549ist_fm,R: rule,Sl: fset_P8989946509869081563ist_fm,S7: produc6018962875968178549ist_fm] :
      ( ( abstra5221733350967904376ist_fm @ Eff4 @ Rules2 @ S4 )
     => ( ( member4699826688122452638ist_fm @ S3 @ S4 )
       => ( ( member_rule @ R @ ( sset_rule @ Rules2 ) )
         => ( ( Eff4 @ R @ S3 @ Sl )
           => ( ( fmembe3754813877001230652ist_fm @ S7 @ Sl )
             => ( member4699826688122452638ist_fm @ S7 @ S4 ) ) ) ) ) ) ).

% RuleSystem.eff_S
thf(fact_719_RuleSystem_Oeff__S,axiom,
    ! [Eff4: produc859450856879609959at_nat > nat > fset_nat > $o,Rules2: stream8372878641218411373at_nat,S4: set_nat,S3: nat,R: produc859450856879609959at_nat,Sl: fset_nat,S7: nat] :
      ( ( abstra1380253255819158094at_nat @ Eff4 @ Rules2 @ S4 )
     => ( ( member_nat @ S3 @ S4 )
       => ( ( member8206827879206165904at_nat @ R @ ( sset_P9164763103296375419at_nat @ Rules2 ) )
         => ( ( Eff4 @ R @ S3 @ Sl )
           => ( ( fmember_nat @ S7 @ Sl )
             => ( member_nat @ S7 @ S4 ) ) ) ) ) ) ).

% RuleSystem.eff_S
thf(fact_720_RuleSystem_Oeff__S,axiom,
    ! [Eff4: produc859450856879609959at_nat > rule > fset_rule > $o,Rules2: stream8372878641218411373at_nat,S4: set_rule,S3: rule,R: produc859450856879609959at_nat,Sl: fset_rule,S7: rule] :
      ( ( abstra5264416619086884112t_rule @ Eff4 @ Rules2 @ S4 )
     => ( ( member_rule @ S3 @ S4 )
       => ( ( member8206827879206165904at_nat @ R @ ( sset_P9164763103296375419at_nat @ Rules2 ) )
         => ( ( Eff4 @ R @ S3 @ Sl )
           => ( ( fmember_rule @ S7 @ Sl )
             => ( member_rule @ S7 @ S4 ) ) ) ) ) ) ).

% RuleSystem.eff_S
thf(fact_721_RuleSystem_Ointro,axiom,
    ! [S4: set_nat,Rules2: stream_nat,Eff4: nat > nat > fset_nat > $o] :
      ( ! [S8: nat] :
          ( ( member_nat @ S8 @ S4 )
         => ! [R6: nat] :
              ( ( member_nat @ R6 @ ( sset_nat @ Rules2 ) )
             => ! [Sl2: fset_nat] :
                  ( ( Eff4 @ R6 @ S8 @ Sl2 )
                 => ! [S6: nat] :
                      ( ( fmember_nat @ S6 @ Sl2 )
                     => ( member_nat @ S6 @ S4 ) ) ) ) )
     => ( ! [S8: nat] :
            ( ( member_nat @ S8 @ S4 )
           => ? [X4: nat] :
                ( ( member_nat @ X4 @ ( sset_nat @ Rules2 ) )
                & ? [X_12: fset_nat] : ( Eff4 @ X4 @ S8 @ X_12 ) ) )
       => ( abstra5088622135990055243at_nat @ Eff4 @ Rules2 @ S4 ) ) ) ).

% RuleSystem.intro
thf(fact_722_RuleSystem_Ointro,axiom,
    ! [S4: set_rule,Rules2: stream_nat,Eff4: nat > rule > fset_rule > $o] :
      ( ! [S8: rule] :
          ( ( member_rule @ S8 @ S4 )
         => ! [R6: nat] :
              ( ( member_nat @ R6 @ ( sset_nat @ Rules2 ) )
             => ! [Sl2: fset_rule] :
                  ( ( Eff4 @ R6 @ S8 @ Sl2 )
                 => ! [S6: rule] :
                      ( ( fmember_rule @ S6 @ Sl2 )
                     => ( member_rule @ S6 @ S4 ) ) ) ) )
     => ( ! [S8: rule] :
            ( ( member_rule @ S8 @ S4 )
           => ? [X4: nat] :
                ( ( member_nat @ X4 @ ( sset_nat @ Rules2 ) )
                & ? [X_12: fset_rule] : ( Eff4 @ X4 @ S8 @ X_12 ) ) )
       => ( abstra7777025902706499341t_rule @ Eff4 @ Rules2 @ S4 ) ) ) ).

% RuleSystem.intro
thf(fact_723_RuleSystem_Ointro,axiom,
    ! [S4: set_nat,Rules2: stream_rule,Eff4: rule > nat > fset_nat > $o] :
      ( ! [S8: nat] :
          ( ( member_nat @ S8 @ S4 )
         => ! [R6: rule] :
              ( ( member_rule @ R6 @ ( sset_rule @ Rules2 ) )
             => ! [Sl2: fset_nat] :
                  ( ( Eff4 @ R6 @ S8 @ Sl2 )
                 => ! [S6: nat] :
                      ( ( fmember_nat @ S6 @ Sl2 )
                     => ( member_nat @ S6 @ S4 ) ) ) ) )
     => ( ! [S8: nat] :
            ( ( member_nat @ S8 @ S4 )
           => ? [X4: rule] :
                ( ( member_rule @ X4 @ ( sset_rule @ Rules2 ) )
                & ? [X_12: fset_nat] : ( Eff4 @ X4 @ S8 @ X_12 ) ) )
       => ( abstra3263948797621512205le_nat @ Eff4 @ Rules2 @ S4 ) ) ) ).

% RuleSystem.intro
thf(fact_724_RuleSystem_Ointro,axiom,
    ! [S4: set_rule,Rules2: stream_rule,Eff4: rule > rule > fset_rule > $o] :
      ( ! [S8: rule] :
          ( ( member_rule @ S8 @ S4 )
         => ! [R6: rule] :
              ( ( member_rule @ R6 @ ( sset_rule @ Rules2 ) )
             => ! [Sl2: fset_rule] :
                  ( ( Eff4 @ R6 @ S8 @ Sl2 )
                 => ! [S6: rule] :
                      ( ( fmember_rule @ S6 @ Sl2 )
                     => ( member_rule @ S6 @ S4 ) ) ) ) )
     => ( ! [S8: rule] :
            ( ( member_rule @ S8 @ S4 )
           => ? [X4: rule] :
                ( ( member_rule @ X4 @ ( sset_rule @ Rules2 ) )
                & ? [X_12: fset_rule] : ( Eff4 @ X4 @ S8 @ X_12 ) ) )
       => ( abstra6805424310982398927e_rule @ Eff4 @ Rules2 @ S4 ) ) ) ).

% RuleSystem.intro
thf(fact_725_RuleSystem_Ointro,axiom,
    ! [S4: set_nat,Rules2: stream4408948924543953275ist_fm,Eff4: produc6018962875968178549ist_fm > nat > fset_nat > $o] :
      ( ! [S8: nat] :
          ( ( member_nat @ S8 @ S4 )
         => ! [R6: produc6018962875968178549ist_fm] :
              ( ( member4699826688122452638ist_fm @ R6 @ ( sset_P5379989128061332361ist_fm @ Rules2 ) )
             => ! [Sl2: fset_nat] :
                  ( ( Eff4 @ R6 @ S8 @ Sl2 )
                 => ! [S6: nat] :
                      ( ( fmember_nat @ S6 @ Sl2 )
                     => ( member_nat @ S6 @ S4 ) ) ) ) )
     => ( ! [S8: nat] :
            ( ( member_nat @ S8 @ S4 )
           => ? [X4: produc6018962875968178549ist_fm] :
                ( ( member4699826688122452638ist_fm @ X4 @ ( sset_P5379989128061332361ist_fm @ Rules2 ) )
                & ? [X_12: fset_nat] : ( Eff4 @ X4 @ S8 @ X_12 ) ) )
       => ( abstra4534667231653962588fm_nat @ Eff4 @ Rules2 @ S4 ) ) ) ).

% RuleSystem.intro
thf(fact_726_RuleSystem_Ointro,axiom,
    ! [S4: set_rule,Rules2: stream4408948924543953275ist_fm,Eff4: produc6018962875968178549ist_fm > rule > fset_rule > $o] :
      ( ! [S8: rule] :
          ( ( member_rule @ S8 @ S4 )
         => ! [R6: produc6018962875968178549ist_fm] :
              ( ( member4699826688122452638ist_fm @ R6 @ ( sset_P5379989128061332361ist_fm @ Rules2 ) )
             => ! [Sl2: fset_rule] :
                  ( ( Eff4 @ R6 @ S8 @ Sl2 )
                 => ! [S6: rule] :
                      ( ( fmember_rule @ S6 @ Sl2 )
                     => ( member_rule @ S6 @ S4 ) ) ) ) )
     => ( ! [S8: rule] :
            ( ( member_rule @ S8 @ S4 )
           => ? [X4: produc6018962875968178549ist_fm] :
                ( ( member4699826688122452638ist_fm @ X4 @ ( sset_P5379989128061332361ist_fm @ Rules2 ) )
                & ? [X_12: fset_rule] : ( Eff4 @ X4 @ S8 @ X_12 ) ) )
       => ( abstra2695409591818887198m_rule @ Eff4 @ Rules2 @ S4 ) ) ) ).

% RuleSystem.intro
thf(fact_727_RuleSystem_Ointro,axiom,
    ! [S4: set_Pr5202636777678657877ist_fm,Rules2: stream_nat,Eff4: nat > produc6018962875968178549ist_fm > fset_P8989946509869081563ist_fm > $o] :
      ( ! [S8: produc6018962875968178549ist_fm] :
          ( ( member4699826688122452638ist_fm @ S8 @ S4 )
         => ! [R6: nat] :
              ( ( member_nat @ R6 @ ( sset_nat @ Rules2 ) )
             => ! [Sl2: fset_P8989946509869081563ist_fm] :
                  ( ( Eff4 @ R6 @ S8 @ Sl2 )
                 => ! [S6: produc6018962875968178549ist_fm] :
                      ( ( fmembe3754813877001230652ist_fm @ S6 @ Sl2 )
                     => ( member4699826688122452638ist_fm @ S6 @ S4 ) ) ) ) )
     => ( ! [S8: produc6018962875968178549ist_fm] :
            ( ( member4699826688122452638ist_fm @ S8 @ S4 )
           => ? [X4: nat] :
                ( ( member_nat @ X4 @ ( sset_nat @ Rules2 ) )
                & ? [X_12: fset_P8989946509869081563ist_fm] : ( Eff4 @ X4 @ S8 @ X_12 ) ) )
       => ( abstra6010661502565574330ist_fm @ Eff4 @ Rules2 @ S4 ) ) ) ).

% RuleSystem.intro
thf(fact_728_RuleSystem_Ointro,axiom,
    ! [S4: set_Pr5202636777678657877ist_fm,Rules2: stream_rule,Eff4: rule > produc6018962875968178549ist_fm > fset_P8989946509869081563ist_fm > $o] :
      ( ! [S8: produc6018962875968178549ist_fm] :
          ( ( member4699826688122452638ist_fm @ S8 @ S4 )
         => ! [R6: rule] :
              ( ( member_rule @ R6 @ ( sset_rule @ Rules2 ) )
             => ! [Sl2: fset_P8989946509869081563ist_fm] :
                  ( ( Eff4 @ R6 @ S8 @ Sl2 )
                 => ! [S6: produc6018962875968178549ist_fm] :
                      ( ( fmembe3754813877001230652ist_fm @ S6 @ Sl2 )
                     => ( member4699826688122452638ist_fm @ S6 @ S4 ) ) ) ) )
     => ( ! [S8: produc6018962875968178549ist_fm] :
            ( ( member4699826688122452638ist_fm @ S8 @ S4 )
           => ? [X4: rule] :
                ( ( member_rule @ X4 @ ( sset_rule @ Rules2 ) )
                & ? [X_12: fset_P8989946509869081563ist_fm] : ( Eff4 @ X4 @ S8 @ X_12 ) ) )
       => ( abstra5221733350967904376ist_fm @ Eff4 @ Rules2 @ S4 ) ) ) ).

% RuleSystem.intro
thf(fact_729_RuleSystem_Ointro,axiom,
    ! [S4: set_nat,Rules2: stream8372878641218411373at_nat,Eff4: produc859450856879609959at_nat > nat > fset_nat > $o] :
      ( ! [S8: nat] :
          ( ( member_nat @ S8 @ S4 )
         => ! [R6: produc859450856879609959at_nat] :
              ( ( member8206827879206165904at_nat @ R6 @ ( sset_P9164763103296375419at_nat @ Rules2 ) )
             => ! [Sl2: fset_nat] :
                  ( ( Eff4 @ R6 @ S8 @ Sl2 )
                 => ! [S6: nat] :
                      ( ( fmember_nat @ S6 @ Sl2 )
                     => ( member_nat @ S6 @ S4 ) ) ) ) )
     => ( ! [S8: nat] :
            ( ( member_nat @ S8 @ S4 )
           => ? [X4: produc859450856879609959at_nat] :
                ( ( member8206827879206165904at_nat @ X4 @ ( sset_P9164763103296375419at_nat @ Rules2 ) )
                & ? [X_12: fset_nat] : ( Eff4 @ X4 @ S8 @ X_12 ) ) )
       => ( abstra1380253255819158094at_nat @ Eff4 @ Rules2 @ S4 ) ) ) ).

% RuleSystem.intro
thf(fact_730_RuleSystem_Ointro,axiom,
    ! [S4: set_rule,Rules2: stream8372878641218411373at_nat,Eff4: produc859450856879609959at_nat > rule > fset_rule > $o] :
      ( ! [S8: rule] :
          ( ( member_rule @ S8 @ S4 )
         => ! [R6: produc859450856879609959at_nat] :
              ( ( member8206827879206165904at_nat @ R6 @ ( sset_P9164763103296375419at_nat @ Rules2 ) )
             => ! [Sl2: fset_rule] :
                  ( ( Eff4 @ R6 @ S8 @ Sl2 )
                 => ! [S6: rule] :
                      ( ( fmember_rule @ S6 @ Sl2 )
                     => ( member_rule @ S6 @ S4 ) ) ) ) )
     => ( ! [S8: rule] :
            ( ( member_rule @ S8 @ S4 )
           => ? [X4: produc859450856879609959at_nat] :
                ( ( member8206827879206165904at_nat @ X4 @ ( sset_P9164763103296375419at_nat @ Rules2 ) )
                & ? [X_12: fset_rule] : ( Eff4 @ X4 @ S8 @ X_12 ) ) )
       => ( abstra5264416619086884112t_rule @ Eff4 @ Rules2 @ S4 ) ) ) ).

% RuleSystem.intro
thf(fact_731_RuleSystem__def,axiom,
    ( abstra5088622135990055243at_nat
    = ( ^ [Eff: nat > nat > fset_nat > $o,Rules: stream_nat,S5: set_nat] :
          ( ! [S: nat] :
              ( ( member_nat @ S @ S5 )
             => ! [R4: nat] :
                  ( ( member_nat @ R4 @ ( sset_nat @ Rules ) )
                 => ! [Sl3: fset_nat] :
                      ( ( Eff @ R4 @ S @ Sl3 )
                     => ! [S9: nat] :
                          ( ( fmember_nat @ S9 @ Sl3 )
                         => ( member_nat @ S9 @ S5 ) ) ) ) )
          & ! [S: nat] :
              ( ( member_nat @ S @ S5 )
             => ? [X: nat] :
                  ( ( member_nat @ X @ ( sset_nat @ Rules ) )
                  & ? [X6: fset_nat] : ( Eff @ X @ S @ X6 ) ) ) ) ) ) ).

% RuleSystem_def
thf(fact_732_RuleSystem__def,axiom,
    ( abstra7777025902706499341t_rule
    = ( ^ [Eff: nat > rule > fset_rule > $o,Rules: stream_nat,S5: set_rule] :
          ( ! [S: rule] :
              ( ( member_rule @ S @ S5 )
             => ! [R4: nat] :
                  ( ( member_nat @ R4 @ ( sset_nat @ Rules ) )
                 => ! [Sl3: fset_rule] :
                      ( ( Eff @ R4 @ S @ Sl3 )
                     => ! [S9: rule] :
                          ( ( fmember_rule @ S9 @ Sl3 )
                         => ( member_rule @ S9 @ S5 ) ) ) ) )
          & ! [S: rule] :
              ( ( member_rule @ S @ S5 )
             => ? [X: nat] :
                  ( ( member_nat @ X @ ( sset_nat @ Rules ) )
                  & ? [X6: fset_rule] : ( Eff @ X @ S @ X6 ) ) ) ) ) ) ).

% RuleSystem_def
thf(fact_733_RuleSystem__def,axiom,
    ( abstra3263948797621512205le_nat
    = ( ^ [Eff: rule > nat > fset_nat > $o,Rules: stream_rule,S5: set_nat] :
          ( ! [S: nat] :
              ( ( member_nat @ S @ S5 )
             => ! [R4: rule] :
                  ( ( member_rule @ R4 @ ( sset_rule @ Rules ) )
                 => ! [Sl3: fset_nat] :
                      ( ( Eff @ R4 @ S @ Sl3 )
                     => ! [S9: nat] :
                          ( ( fmember_nat @ S9 @ Sl3 )
                         => ( member_nat @ S9 @ S5 ) ) ) ) )
          & ! [S: nat] :
              ( ( member_nat @ S @ S5 )
             => ? [X: rule] :
                  ( ( member_rule @ X @ ( sset_rule @ Rules ) )
                  & ? [X6: fset_nat] : ( Eff @ X @ S @ X6 ) ) ) ) ) ) ).

% RuleSystem_def
thf(fact_734_RuleSystem__def,axiom,
    ( abstra6805424310982398927e_rule
    = ( ^ [Eff: rule > rule > fset_rule > $o,Rules: stream_rule,S5: set_rule] :
          ( ! [S: rule] :
              ( ( member_rule @ S @ S5 )
             => ! [R4: rule] :
                  ( ( member_rule @ R4 @ ( sset_rule @ Rules ) )
                 => ! [Sl3: fset_rule] :
                      ( ( Eff @ R4 @ S @ Sl3 )
                     => ! [S9: rule] :
                          ( ( fmember_rule @ S9 @ Sl3 )
                         => ( member_rule @ S9 @ S5 ) ) ) ) )
          & ! [S: rule] :
              ( ( member_rule @ S @ S5 )
             => ? [X: rule] :
                  ( ( member_rule @ X @ ( sset_rule @ Rules ) )
                  & ? [X6: fset_rule] : ( Eff @ X @ S @ X6 ) ) ) ) ) ) ).

% RuleSystem_def
thf(fact_735_RuleSystem__def,axiom,
    ( abstra4534667231653962588fm_nat
    = ( ^ [Eff: produc6018962875968178549ist_fm > nat > fset_nat > $o,Rules: stream4408948924543953275ist_fm,S5: set_nat] :
          ( ! [S: nat] :
              ( ( member_nat @ S @ S5 )
             => ! [R4: produc6018962875968178549ist_fm] :
                  ( ( member4699826688122452638ist_fm @ R4 @ ( sset_P5379989128061332361ist_fm @ Rules ) )
                 => ! [Sl3: fset_nat] :
                      ( ( Eff @ R4 @ S @ Sl3 )
                     => ! [S9: nat] :
                          ( ( fmember_nat @ S9 @ Sl3 )
                         => ( member_nat @ S9 @ S5 ) ) ) ) )
          & ! [S: nat] :
              ( ( member_nat @ S @ S5 )
             => ? [X: produc6018962875968178549ist_fm] :
                  ( ( member4699826688122452638ist_fm @ X @ ( sset_P5379989128061332361ist_fm @ Rules ) )
                  & ? [X6: fset_nat] : ( Eff @ X @ S @ X6 ) ) ) ) ) ) ).

% RuleSystem_def
thf(fact_736_RuleSystem__def,axiom,
    ( abstra2695409591818887198m_rule
    = ( ^ [Eff: produc6018962875968178549ist_fm > rule > fset_rule > $o,Rules: stream4408948924543953275ist_fm,S5: set_rule] :
          ( ! [S: rule] :
              ( ( member_rule @ S @ S5 )
             => ! [R4: produc6018962875968178549ist_fm] :
                  ( ( member4699826688122452638ist_fm @ R4 @ ( sset_P5379989128061332361ist_fm @ Rules ) )
                 => ! [Sl3: fset_rule] :
                      ( ( Eff @ R4 @ S @ Sl3 )
                     => ! [S9: rule] :
                          ( ( fmember_rule @ S9 @ Sl3 )
                         => ( member_rule @ S9 @ S5 ) ) ) ) )
          & ! [S: rule] :
              ( ( member_rule @ S @ S5 )
             => ? [X: produc6018962875968178549ist_fm] :
                  ( ( member4699826688122452638ist_fm @ X @ ( sset_P5379989128061332361ist_fm @ Rules ) )
                  & ? [X6: fset_rule] : ( Eff @ X @ S @ X6 ) ) ) ) ) ) ).

% RuleSystem_def
thf(fact_737_RuleSystem__def,axiom,
    ( abstra6010661502565574330ist_fm
    = ( ^ [Eff: nat > produc6018962875968178549ist_fm > fset_P8989946509869081563ist_fm > $o,Rules: stream_nat,S5: set_Pr5202636777678657877ist_fm] :
          ( ! [S: produc6018962875968178549ist_fm] :
              ( ( member4699826688122452638ist_fm @ S @ S5 )
             => ! [R4: nat] :
                  ( ( member_nat @ R4 @ ( sset_nat @ Rules ) )
                 => ! [Sl3: fset_P8989946509869081563ist_fm] :
                      ( ( Eff @ R4 @ S @ Sl3 )
                     => ! [S9: produc6018962875968178549ist_fm] :
                          ( ( fmembe3754813877001230652ist_fm @ S9 @ Sl3 )
                         => ( member4699826688122452638ist_fm @ S9 @ S5 ) ) ) ) )
          & ! [S: produc6018962875968178549ist_fm] :
              ( ( member4699826688122452638ist_fm @ S @ S5 )
             => ? [X: nat] :
                  ( ( member_nat @ X @ ( sset_nat @ Rules ) )
                  & ? [X6: fset_P8989946509869081563ist_fm] : ( Eff @ X @ S @ X6 ) ) ) ) ) ) ).

% RuleSystem_def
thf(fact_738_RuleSystem__def,axiom,
    ( abstra5221733350967904376ist_fm
    = ( ^ [Eff: rule > produc6018962875968178549ist_fm > fset_P8989946509869081563ist_fm > $o,Rules: stream_rule,S5: set_Pr5202636777678657877ist_fm] :
          ( ! [S: produc6018962875968178549ist_fm] :
              ( ( member4699826688122452638ist_fm @ S @ S5 )
             => ! [R4: rule] :
                  ( ( member_rule @ R4 @ ( sset_rule @ Rules ) )
                 => ! [Sl3: fset_P8989946509869081563ist_fm] :
                      ( ( Eff @ R4 @ S @ Sl3 )
                     => ! [S9: produc6018962875968178549ist_fm] :
                          ( ( fmembe3754813877001230652ist_fm @ S9 @ Sl3 )
                         => ( member4699826688122452638ist_fm @ S9 @ S5 ) ) ) ) )
          & ! [S: produc6018962875968178549ist_fm] :
              ( ( member4699826688122452638ist_fm @ S @ S5 )
             => ? [X: rule] :
                  ( ( member_rule @ X @ ( sset_rule @ Rules ) )
                  & ? [X6: fset_P8989946509869081563ist_fm] : ( Eff @ X @ S @ X6 ) ) ) ) ) ) ).

% RuleSystem_def
thf(fact_739_RuleSystem__def,axiom,
    ( abstra1380253255819158094at_nat
    = ( ^ [Eff: produc859450856879609959at_nat > nat > fset_nat > $o,Rules: stream8372878641218411373at_nat,S5: set_nat] :
          ( ! [S: nat] :
              ( ( member_nat @ S @ S5 )
             => ! [R4: produc859450856879609959at_nat] :
                  ( ( member8206827879206165904at_nat @ R4 @ ( sset_P9164763103296375419at_nat @ Rules ) )
                 => ! [Sl3: fset_nat] :
                      ( ( Eff @ R4 @ S @ Sl3 )
                     => ! [S9: nat] :
                          ( ( fmember_nat @ S9 @ Sl3 )
                         => ( member_nat @ S9 @ S5 ) ) ) ) )
          & ! [S: nat] :
              ( ( member_nat @ S @ S5 )
             => ? [X: produc859450856879609959at_nat] :
                  ( ( member8206827879206165904at_nat @ X @ ( sset_P9164763103296375419at_nat @ Rules ) )
                  & ? [X6: fset_nat] : ( Eff @ X @ S @ X6 ) ) ) ) ) ) ).

% RuleSystem_def
thf(fact_740_RuleSystem__def,axiom,
    ( abstra5264416619086884112t_rule
    = ( ^ [Eff: produc859450856879609959at_nat > rule > fset_rule > $o,Rules: stream8372878641218411373at_nat,S5: set_rule] :
          ( ! [S: rule] :
              ( ( member_rule @ S @ S5 )
             => ! [R4: produc859450856879609959at_nat] :
                  ( ( member8206827879206165904at_nat @ R4 @ ( sset_P9164763103296375419at_nat @ Rules ) )
                 => ! [Sl3: fset_rule] :
                      ( ( Eff @ R4 @ S @ Sl3 )
                     => ! [S9: rule] :
                          ( ( fmember_rule @ S9 @ Sl3 )
                         => ( member_rule @ S9 @ S5 ) ) ) ) )
          & ! [S: rule] :
              ( ( member_rule @ S @ S5 )
             => ? [X: produc859450856879609959at_nat] :
                  ( ( member8206827879206165904at_nat @ X @ ( sset_P9164763103296375419at_nat @ Rules ) )
                  & ? [X6: fset_rule] : ( Eff @ X @ S @ X6 ) ) ) ) ) ) ).

% RuleSystem_def
thf(fact_741_ftree__no__ipath,axiom,
    ! [T3: abstra8520858721168826610m_rule,Steps: stream2709947120125613254m_rule] :
      ( ( abstra3822255059661934556m_rule @ T3 )
     => ~ ( abstra8836096083317999815m_rule @ T3 @ Steps ) ) ).

% ftree_no_ipath
thf(fact_742_PersistentRuleSystem_Oepath__completeness__Saturated,axiom,
    ! [Eff4: list_fm > list_tm > fset_list_tm > $o,Rules2: stream_list_fm,S4: set_list_tm,S3: list_tm] :
      ( ( abstra30754806901103026ist_tm @ Eff4 @ Rules2 @ S4 )
     => ( ( member_list_tm @ S3 @ S4 )
       => ( ? [T4: abstra4463927080622938191ist_fm] :
              ( ( ( produc4874987732838623697ist_fm @ ( abstra2656519403522841862ist_fm @ T4 ) )
                = S3 )
              & ( abstra7376083527646823080ist_tm @ Eff4 @ Rules2 @ T4 )
              & ( abstra1895643584677529807ist_fm @ T4 ) )
          | ? [Steps3: stream4408948924543953275ist_fm] :
              ( ( ( produc4874987732838623697ist_fm @ ( shd_Pr3211216682057661985ist_fm @ Steps3 ) )
                = S3 )
              & ( abstra302294732650053493ist_tm @ Eff4 @ Rules2 @ Steps3 )
              & ( abstra5290135422857967776ist_tm @ Eff4 @ Rules2 @ Steps3 ) ) ) ) ) ).

% PersistentRuleSystem.epath_completeness_Saturated
thf(fact_743_PersistentRuleSystem_Oepath__completeness__Saturated,axiom,
    ! [Eff4: rule > produc6018962875968178549ist_fm > fset_P8989946509869081563ist_fm > $o,Rules2: stream_rule,S4: set_Pr5202636777678657877ist_fm,S3: produc6018962875968178549ist_fm] :
      ( ( abstra3967921542344661089ist_fm @ Eff4 @ Rules2 @ S4 )
     => ( ( member4699826688122452638ist_fm @ S3 @ S4 )
       => ( ? [T4: abstra8520858721168826610m_rule] :
              ( ( ( produc993771546023551940m_rule @ ( abstra5180277867158163941m_rule @ T4 ) )
                = S3 )
              & ( abstra8167393772827826667ist_fm @ Eff4 @ Rules2 @ T4 )
              & ( abstra3822255059661934556m_rule @ T4 ) )
          | ? [Steps3: stream2709947120125613254m_rule] :
              ( ( ( produc993771546023551940m_rule @ ( shd_Pr4562317740776619530m_rule @ Steps3 ) )
                = S3 )
              & ( abstra6789711989322986974ist_fm @ Eff4 @ Rules2 @ Steps3 )
              & ( abstra6097777249025082867ist_fm @ Eff4 @ Rules2 @ Steps3 ) ) ) ) ) ).

% PersistentRuleSystem.epath_completeness_Saturated
thf(fact_744_epath_Ocoinduct,axiom,
    ! [X5: stream2709947120125613254m_rule > $o,X3: stream2709947120125613254m_rule] :
      ( ( X5 @ X3 )
     => ( ! [X2: stream2709947120125613254m_rule] :
            ( ( X5 @ X2 )
           => ? [Steps4: stream2709947120125613254m_rule,Sl4: fset_P8989946509869081563ist_fm] :
                ( ( X2 = Steps4 )
                & ( member_rule @ ( produc6514711208036230658m_rule @ ( shd_Pr4562317740776619530m_rule @ Steps4 ) ) @ ( sset_rule @ rules ) )
                & ( fmembe3754813877001230652ist_fm @ ( produc993771546023551940m_rule @ ( shd_Pr4562317740776619530m_rule @ ( stl_Pr12655793849929990m_rule @ Steps4 ) ) ) @ Sl4 )
                & ( eff @ ( produc6514711208036230658m_rule @ ( shd_Pr4562317740776619530m_rule @ Steps4 ) ) @ ( produc993771546023551940m_rule @ ( shd_Pr4562317740776619530m_rule @ Steps4 ) ) @ Sl4 )
                & ( ( X5 @ ( stl_Pr12655793849929990m_rule @ Steps4 ) )
                  | ( abstra6789711989322986974ist_fm @ eff @ rules @ ( stl_Pr12655793849929990m_rule @ Steps4 ) ) ) ) )
       => ( abstra6789711989322986974ist_fm @ eff @ rules @ X3 ) ) ) ).

% epath.coinduct
thf(fact_745_epath_Ointros,axiom,
    ! [Steps: stream2709947120125613254m_rule,Sl: fset_P8989946509869081563ist_fm] :
      ( ( member_rule @ ( produc6514711208036230658m_rule @ ( shd_Pr4562317740776619530m_rule @ Steps ) ) @ ( sset_rule @ rules ) )
     => ( ( fmembe3754813877001230652ist_fm @ ( produc993771546023551940m_rule @ ( shd_Pr4562317740776619530m_rule @ ( stl_Pr12655793849929990m_rule @ Steps ) ) ) @ Sl )
       => ( ( eff @ ( produc6514711208036230658m_rule @ ( shd_Pr4562317740776619530m_rule @ Steps ) ) @ ( produc993771546023551940m_rule @ ( shd_Pr4562317740776619530m_rule @ Steps ) ) @ Sl )
         => ( ( abstra6789711989322986974ist_fm @ eff @ rules @ ( stl_Pr12655793849929990m_rule @ Steps ) )
           => ( abstra6789711989322986974ist_fm @ eff @ rules @ Steps ) ) ) ) ) ).

% epath.intros
thf(fact_746_epath_Osimps,axiom,
    ! [A: stream2709947120125613254m_rule] :
      ( ( abstra6789711989322986974ist_fm @ eff @ rules @ A )
      = ( ? [Steps2: stream2709947120125613254m_rule,Sl3: fset_P8989946509869081563ist_fm] :
            ( ( A = Steps2 )
            & ( member_rule @ ( produc6514711208036230658m_rule @ ( shd_Pr4562317740776619530m_rule @ Steps2 ) ) @ ( sset_rule @ rules ) )
            & ( fmembe3754813877001230652ist_fm @ ( produc993771546023551940m_rule @ ( shd_Pr4562317740776619530m_rule @ ( stl_Pr12655793849929990m_rule @ Steps2 ) ) ) @ Sl3 )
            & ( eff @ ( produc6514711208036230658m_rule @ ( shd_Pr4562317740776619530m_rule @ Steps2 ) ) @ ( produc993771546023551940m_rule @ ( shd_Pr4562317740776619530m_rule @ Steps2 ) ) @ Sl3 )
            & ( abstra6789711989322986974ist_fm @ eff @ rules @ ( stl_Pr12655793849929990m_rule @ Steps2 ) ) ) ) ) ).

% epath.simps
thf(fact_747_epath_Ocases,axiom,
    ! [A: stream2709947120125613254m_rule] :
      ( ( abstra6789711989322986974ist_fm @ eff @ rules @ A )
     => ~ ( ( member_rule @ ( produc6514711208036230658m_rule @ ( shd_Pr4562317740776619530m_rule @ A ) ) @ ( sset_rule @ rules ) )
         => ! [Sl2: fset_P8989946509869081563ist_fm] :
              ( ( fmembe3754813877001230652ist_fm @ ( produc993771546023551940m_rule @ ( shd_Pr4562317740776619530m_rule @ ( stl_Pr12655793849929990m_rule @ A ) ) ) @ Sl2 )
             => ( ( eff @ ( produc6514711208036230658m_rule @ ( shd_Pr4562317740776619530m_rule @ A ) ) @ ( produc993771546023551940m_rule @ ( shd_Pr4562317740776619530m_rule @ A ) ) @ Sl2 )
               => ~ ( abstra6789711989322986974ist_fm @ eff @ rules @ ( stl_Pr12655793849929990m_rule @ A ) ) ) ) ) ) ).

% epath.cases
thf(fact_748_epath__eff,axiom,
    ! [Steps: stream2709947120125613254m_rule,Ss: fset_P8989946509869081563ist_fm] :
      ( ( abstra6789711989322986974ist_fm @ eff @ rules @ Steps )
     => ( ( eff @ ( produc6514711208036230658m_rule @ ( shd_Pr4562317740776619530m_rule @ Steps ) ) @ ( produc993771546023551940m_rule @ ( shd_Pr4562317740776619530m_rule @ Steps ) ) @ Ss )
       => ( fmembe3754813877001230652ist_fm @ ( produc993771546023551940m_rule @ ( shd_Pr4562317740776619530m_rule @ ( stl_Pr12655793849929990m_rule @ Steps ) ) ) @ Ss ) ) ) ).

% epath_eff
thf(fact_749_i_Oepath_Ocases,axiom,
    ! [Eff2: nat > produc6018962875968178549ist_fm > option6967287582980624417ist_fm,Rules2: stream_nat,A: stream1752417115841309700fm_nat] :
      ( ( abstra3661368698739669536ist_fm @ ( abstra6480510808789727715ist_fm @ Eff2 ) @ Rules2 @ A )
     => ~ ( ( member_nat @ ( produc2092181227289234752fm_nat @ ( shd_Pr4515903069162513224fm_nat @ A ) ) @ ( sset_nat @ Rules2 ) )
         => ! [Sl2: fset_P8989946509869081563ist_fm] :
              ( ( fmembe3754813877001230652ist_fm @ ( produc111973938668391682fm_nat @ ( shd_Pr4515903069162513224fm_nat @ ( stl_Pr1340940657110954564fm_nat @ A ) ) ) @ Sl2 )
             => ( ( abstra6480510808789727715ist_fm @ Eff2 @ ( produc2092181227289234752fm_nat @ ( shd_Pr4515903069162513224fm_nat @ A ) ) @ ( produc111973938668391682fm_nat @ ( shd_Pr4515903069162513224fm_nat @ A ) ) @ Sl2 )
               => ~ ( abstra3661368698739669536ist_fm @ ( abstra6480510808789727715ist_fm @ Eff2 ) @ Rules2 @ ( stl_Pr1340940657110954564fm_nat @ A ) ) ) ) ) ) ).

% i.epath.cases
thf(fact_750_i_Oepath_Ocases,axiom,
    ! [Eff2: produc6018962875968178549ist_fm > produc6018962875968178549ist_fm > option6967287582980624417ist_fm,Rules2: stream4408948924543953275ist_fm,A: stream1747862497754140525ist_fm] :
      ( ( abstra6295994116336782083ist_fm @ ( abstra2084124586061969408ist_fm @ Eff2 ) @ Rules2 @ A )
     => ~ ( ( member4699826688122452638ist_fm @ ( produc6364584182369725445ist_fm @ ( shd_Pr4098038590278241811ist_fm @ A ) ) @ ( sset_P5379989128061332361ist_fm @ Rules2 ) )
         => ! [Sl2: fset_P8989946509869081563ist_fm] :
              ( ( fmembe3754813877001230652ist_fm @ ( produc8141473855448433859ist_fm @ ( shd_Pr4098038590278241811ist_fm @ ( stl_Pr1377318027900677655ist_fm @ A ) ) ) @ Sl2 )
             => ( ( abstra2084124586061969408ist_fm @ Eff2 @ ( produc6364584182369725445ist_fm @ ( shd_Pr4098038590278241811ist_fm @ A ) ) @ ( produc8141473855448433859ist_fm @ ( shd_Pr4098038590278241811ist_fm @ A ) ) @ Sl2 )
               => ~ ( abstra6295994116336782083ist_fm @ ( abstra2084124586061969408ist_fm @ Eff2 ) @ Rules2 @ ( stl_Pr1377318027900677655ist_fm @ A ) ) ) ) ) ) ).

% i.epath.cases
thf(fact_751_i_Oepath_Ocases,axiom,
    ! [Eff2: produc859450856879609959at_nat > produc6018962875968178549ist_fm > option6967287582980624417ist_fm,Rules2: stream8372878641218411373at_nat,A: stream105034537780515935at_nat] :
      ( ( abstra409391092022950033ist_fm @ ( abstra331207336662862222ist_fm @ Eff2 ) @ Rules2 @ A )
     => ~ ( ( member8206827879206165904at_nat @ ( produc8336112126034639607at_nat @ ( shd_Pr7514066009280765957at_nat @ A ) ) @ ( sset_P9164763103296375419at_nat @ Rules2 ) )
         => ! [Sl2: fset_P8989946509869081563ist_fm] :
              ( ( fmembe3754813877001230652ist_fm @ ( produc6827676535622970293at_nat @ ( shd_Pr7514066009280765957at_nat @ ( stl_Pr7223821233291300873at_nat @ A ) ) ) @ Sl2 )
             => ( ( abstra331207336662862222ist_fm @ Eff2 @ ( produc8336112126034639607at_nat @ ( shd_Pr7514066009280765957at_nat @ A ) ) @ ( produc6827676535622970293at_nat @ ( shd_Pr7514066009280765957at_nat @ A ) ) @ Sl2 )
               => ~ ( abstra409391092022950033ist_fm @ ( abstra331207336662862222ist_fm @ Eff2 ) @ Rules2 @ ( stl_Pr7223821233291300873at_nat @ A ) ) ) ) ) ) ).

% i.epath.cases
thf(fact_752_i_Oepath_Ocases,axiom,
    ! [Eff2: nat > abstra8520858721168826610m_rule > option9127550908420139634m_rule,Rules2: stream_nat,A: stream2640339575054594473le_nat] :
      ( ( abstra2672558889254879751m_rule @ ( abstra1833466274487013892m_rule @ Eff2 ) @ Rules2 @ A )
     => ~ ( ( member_nat @ ( produc6965321437502945929le_nat @ ( shd_Pr4214957810428288335le_nat @ A ) ) @ ( sset_nat @ Rules2 ) )
         => ! [Sl2: fset_A8849196716589201442m_rule] :
              ( ( fmembe1099044791951439339m_rule @ ( produc1996884173541332295le_nat @ ( shd_Pr4214957810428288335le_nat @ ( stl_Pr8328254828799719763le_nat @ A ) ) ) @ Sl2 )
             => ( ( abstra1833466274487013892m_rule @ Eff2 @ ( produc6965321437502945929le_nat @ ( shd_Pr4214957810428288335le_nat @ A ) ) @ ( produc1996884173541332295le_nat @ ( shd_Pr4214957810428288335le_nat @ A ) ) @ Sl2 )
               => ~ ( abstra2672558889254879751m_rule @ ( abstra1833466274487013892m_rule @ Eff2 ) @ Rules2 @ ( stl_Pr8328254828799719763le_nat @ A ) ) ) ) ) ) ).

% i.epath.cases
thf(fact_753_i_Oepath_Ocases,axiom,
    ! [Eff2: produc6018962875968178549ist_fm > abstra8520858721168826610m_rule > option9127550908420139634m_rule,Rules2: stream4408948924543953275ist_fm,A: stream2118054573033641736ist_fm] :
      ( ( abstra8533012240466880740m_rule @ ( abstra2762543317532761127m_rule @ Eff2 ) @ Rules2 @ A )
     => ~ ( ( member4699826688122452638ist_fm @ ( produc1196760618833626364ist_fm @ ( shd_Pr1348735749115486668ist_fm @ A ) ) @ ( sset_P5379989128061332361ist_fm @ Rules2 ) )
         => ! [Sl2: fset_A8849196716589201442m_rule] :
              ( ( fmembe1099044791951439339m_rule @ ( produc7096817939255439806ist_fm @ ( shd_Pr1348735749115486668ist_fm @ ( stl_Pr1735168262451467976ist_fm @ A ) ) ) @ Sl2 )
             => ( ( abstra2762543317532761127m_rule @ Eff2 @ ( produc1196760618833626364ist_fm @ ( shd_Pr1348735749115486668ist_fm @ A ) ) @ ( produc7096817939255439806ist_fm @ ( shd_Pr1348735749115486668ist_fm @ A ) ) @ Sl2 )
               => ~ ( abstra8533012240466880740m_rule @ ( abstra2762543317532761127m_rule @ Eff2 ) @ Rules2 @ ( stl_Pr1735168262451467976ist_fm @ A ) ) ) ) ) ) ).

% i.epath.cases
thf(fact_754_i_Oepath_Ocases,axiom,
    ! [Eff2: produc859450856879609959at_nat > abstra8520858721168826610m_rule > option9127550908420139634m_rule,Rules2: stream8372878641218411373at_nat,A: stream7661288321842501626at_nat] :
      ( ( abstra7033530765613221078m_rule @ ( abstra56554827785262105m_rule @ Eff2 ) @ Rules2 @ A )
     => ~ ( ( member8206827879206165904at_nat @ ( produc740002071024140270at_nat @ ( shd_Pr7666802700379473854at_nat @ A ) ) @ ( sset_P9164763103296375419at_nat @ Rules2 ) )
         => ! [Sl2: fset_A8849196716589201442m_rule] :
              ( ( fmembe1099044791951439339m_rule @ ( produc1738731144906921136at_nat @ ( shd_Pr7666802700379473854at_nat @ ( stl_Pr6382724064935898298at_nat @ A ) ) ) @ Sl2 )
             => ( ( abstra56554827785262105m_rule @ Eff2 @ ( produc740002071024140270at_nat @ ( shd_Pr7666802700379473854at_nat @ A ) ) @ ( produc1738731144906921136at_nat @ ( shd_Pr7666802700379473854at_nat @ A ) ) @ Sl2 )
               => ~ ( abstra7033530765613221078m_rule @ ( abstra56554827785262105m_rule @ Eff2 ) @ Rules2 @ ( stl_Pr6382724064935898298at_nat @ A ) ) ) ) ) ) ).

% i.epath.cases
thf(fact_755_i_Oepath_Ocases,axiom,
    ! [Eff2: rule > abstra8520858721168826610m_rule > option9127550908420139634m_rule,Rules2: stream_rule,A: stream3393403210778678635e_rule] :
      ( ( abstra7067474524808760265m_rule @ ( abstra3599228160297647814m_rule @ Eff2 ) @ Rules2 @ A )
     => ~ ( ( member_rule @ ( produc9166542125499771467e_rule @ ( shd_Pr8663275782010802449e_rule @ A ) ) @ ( sset_rule @ Rules2 ) )
         => ! [Sl2: fset_A8849196716589201442m_rule] :
              ( ( fmembe1099044791951439339m_rule @ ( produc41289011304007433e_rule @ ( shd_Pr8663275782010802449e_rule @ ( stl_Pr654857084227610901e_rule @ A ) ) ) @ Sl2 )
             => ( ( abstra3599228160297647814m_rule @ Eff2 @ ( produc9166542125499771467e_rule @ ( shd_Pr8663275782010802449e_rule @ A ) ) @ ( produc41289011304007433e_rule @ ( shd_Pr8663275782010802449e_rule @ A ) ) @ Sl2 )
               => ~ ( abstra7067474524808760265m_rule @ ( abstra3599228160297647814m_rule @ Eff2 ) @ Rules2 @ ( stl_Pr654857084227610901e_rule @ A ) ) ) ) ) ) ).

% i.epath.cases
thf(fact_756_i_Oepath_Ocases,axiom,
    ! [Eff2: list_fm > list_tm > option_fset_list_tm,Rules2: stream_list_fm,A: stream4408948924543953275ist_fm] :
      ( ( abstra302294732650053493ist_tm @ ( abstra9141620886324064114ist_tm @ Eff2 ) @ Rules2 @ A )
     => ~ ( ( member_list_fm @ ( produc3242453064496107283ist_fm @ ( shd_Pr3211216682057661985ist_fm @ A ) ) @ ( sset_list_fm @ Rules2 ) )
         => ! [Sl2: fset_list_tm] :
              ( ( fmember_list_tm @ ( produc4874987732838623697ist_fm @ ( shd_Pr3211216682057661985ist_fm @ ( stl_Pr1506262294867171877ist_fm @ A ) ) ) @ Sl2 )
             => ( ( abstra9141620886324064114ist_tm @ Eff2 @ ( produc3242453064496107283ist_fm @ ( shd_Pr3211216682057661985ist_fm @ A ) ) @ ( produc4874987732838623697ist_fm @ ( shd_Pr3211216682057661985ist_fm @ A ) ) @ Sl2 )
               => ~ ( abstra302294732650053493ist_tm @ ( abstra9141620886324064114ist_tm @ Eff2 ) @ Rules2 @ ( stl_Pr1506262294867171877ist_fm @ A ) ) ) ) ) ) ).

% i.epath.cases
thf(fact_757_i_Oepath_Ocases,axiom,
    ! [Eff2: rule > produc6018962875968178549ist_fm > option6967287582980624417ist_fm,Rules2: stream_rule,A: stream2709947120125613254m_rule] :
      ( ( abstra6789711989322986974ist_fm @ ( abstra2682625350522704545ist_fm @ Eff2 ) @ Rules2 @ A )
     => ~ ( ( member_rule @ ( produc6514711208036230658m_rule @ ( shd_Pr4562317740776619530m_rule @ A ) ) @ ( sset_rule @ Rules2 ) )
         => ! [Sl2: fset_P8989946509869081563ist_fm] :
              ( ( fmembe3754813877001230652ist_fm @ ( produc993771546023551940m_rule @ ( shd_Pr4562317740776619530m_rule @ ( stl_Pr12655793849929990m_rule @ A ) ) ) @ Sl2 )
             => ( ( abstra2682625350522704545ist_fm @ Eff2 @ ( produc6514711208036230658m_rule @ ( shd_Pr4562317740776619530m_rule @ A ) ) @ ( produc993771546023551940m_rule @ ( shd_Pr4562317740776619530m_rule @ A ) ) @ Sl2 )
               => ~ ( abstra6789711989322986974ist_fm @ ( abstra2682625350522704545ist_fm @ Eff2 ) @ Rules2 @ ( stl_Pr12655793849929990m_rule @ A ) ) ) ) ) ) ).

% i.epath.cases
thf(fact_758_ev__alw__stl,axiom,
    ! [Phi: stream_rule > $o,X3: stream_rule] :
      ( ( linear292482573288640205v_rule @ ( linear1087404401031714582w_rule @ Phi ) @ ( stl_rule @ X3 ) )
      = ( linear292482573288640205v_rule @ ( linear1087404401031714582w_rule @ Phi ) @ X3 ) ) ).

% ev_alw_stl
thf(fact_759_ev__alw__stl,axiom,
    ! [Phi: stream2709947120125613254m_rule > $o,X3: stream2709947120125613254m_rule] :
      ( ( linear1816559245158234865m_rule @ ( linear4957294957385550906m_rule @ Phi ) @ ( stl_Pr12655793849929990m_rule @ X3 ) )
      = ( linear1816559245158234865m_rule @ ( linear4957294957385550906m_rule @ Phi ) @ X3 ) ) ).

% ev_alw_stl
thf(fact_760_sfilter__not__P,axiom,
    ! [P2: produc340336539035504054m_rule > $o,S3: stream2709947120125613254m_rule] :
      ( ~ ( P2 @ ( shd_Pr4562317740776619530m_rule @ S3 ) )
     => ( ( sfilte3304650064355440278m_rule @ P2 @ S3 )
        = ( sfilte3304650064355440278m_rule @ P2 @ ( stl_Pr12655793849929990m_rule @ S3 ) ) ) ) ).

% sfilter_not_P
thf(fact_761_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_762_szip_Osimps_I2_J,axiom,
    ! [S1: stream2709947120125613254m_rule,S2: stream2709947120125613254m_rule] :
      ( ( stl_Pr2679468901532635773m_rule @ ( szip_P811719526838699976m_rule @ S1 @ S2 ) )
      = ( szip_P811719526838699976m_rule @ ( stl_Pr12655793849929990m_rule @ S1 ) @ ( stl_Pr12655793849929990m_rule @ S2 ) ) ) ).

% szip.simps(2)
thf(fact_763_szip_Osimps_I2_J,axiom,
    ! [S1: stream2709947120125613254m_rule,S2: stream_rule] :
      ( ( stl_Pr497917230009538393e_rule @ ( szip_P2044787555563963556e_rule @ S1 @ S2 ) )
      = ( szip_P2044787555563963556e_rule @ ( stl_Pr12655793849929990m_rule @ S1 ) @ ( stl_rule @ S2 ) ) ) ).

% szip.simps(2)
thf(fact_764_szip_Osimps_I2_J,axiom,
    ! [S1: stream_rule,S2: stream2709947120125613254m_rule] :
      ( ( stl_Pr8529079933065201753m_rule @ ( szip_r6533552703345879972m_rule @ S1 @ S2 ) )
      = ( szip_r6533552703345879972m_rule @ ( stl_rule @ S1 ) @ ( stl_Pr12655793849929990m_rule @ S2 ) ) ) ).

% szip.simps(2)
thf(fact_765_szip_Osimps_I2_J,axiom,
    ! [S1: stream_rule,S2: stream_rule] :
      ( ( stl_Pr2595822214614905141e_rule @ ( szip_rule_rule @ S1 @ S2 ) )
      = ( szip_rule_rule @ ( stl_rule @ S1 ) @ ( stl_rule @ S2 ) ) ) ).

% szip.simps(2)
thf(fact_766_szip_Osimps_I2_J,axiom,
    ! [S1: stream4408948924543953275ist_fm,S2: stream_rule] :
      ( ( stl_Pr12655793849929990m_rule @ ( szip_P2924820683901490861m_rule @ S1 @ S2 ) )
      = ( szip_P2924820683901490861m_rule @ ( stl_Pr1506262294867171877ist_fm @ S1 ) @ ( stl_rule @ S2 ) ) ) ).

% szip.simps(2)
thf(fact_767_stream_Ocoinduct__strong,axiom,
    ! [R3: stream2709947120125613254m_rule > stream2709947120125613254m_rule > $o,Stream: stream2709947120125613254m_rule,Stream2: stream2709947120125613254m_rule] :
      ( ( R3 @ Stream @ Stream2 )
     => ( ! [Stream3: stream2709947120125613254m_rule,Stream4: stream2709947120125613254m_rule] :
            ( ( R3 @ Stream3 @ Stream4 )
           => ( ( ( shd_Pr4562317740776619530m_rule @ Stream3 )
                = ( shd_Pr4562317740776619530m_rule @ Stream4 ) )
              & ( ( R3 @ ( stl_Pr12655793849929990m_rule @ Stream3 ) @ ( stl_Pr12655793849929990m_rule @ Stream4 ) )
                | ( ( stl_Pr12655793849929990m_rule @ Stream3 )
                  = ( stl_Pr12655793849929990m_rule @ Stream4 ) ) ) ) )
       => ( Stream = Stream2 ) ) ) ).

% stream.coinduct_strong
thf(fact_768_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_769_stream_Ocoinduct,axiom,
    ! [R3: stream2709947120125613254m_rule > stream2709947120125613254m_rule > $o,Stream: stream2709947120125613254m_rule,Stream2: stream2709947120125613254m_rule] :
      ( ( R3 @ Stream @ Stream2 )
     => ( ! [Stream3: stream2709947120125613254m_rule,Stream4: stream2709947120125613254m_rule] :
            ( ( R3 @ Stream3 @ Stream4 )
           => ( ( ( shd_Pr4562317740776619530m_rule @ Stream3 )
                = ( shd_Pr4562317740776619530m_rule @ Stream4 ) )
              & ( R3 @ ( stl_Pr12655793849929990m_rule @ Stream3 ) @ ( stl_Pr12655793849929990m_rule @ Stream4 ) ) ) )
       => ( Stream = Stream2 ) ) ) ).

% stream.coinduct
thf(fact_770_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_771_stream_Oexpand,axiom,
    ! [Stream: stream2709947120125613254m_rule,Stream2: stream2709947120125613254m_rule] :
      ( ( ( ( shd_Pr4562317740776619530m_rule @ Stream )
          = ( shd_Pr4562317740776619530m_rule @ Stream2 ) )
        & ( ( stl_Pr12655793849929990m_rule @ Stream )
          = ( stl_Pr12655793849929990m_rule @ Stream2 ) ) )
     => ( Stream = Stream2 ) ) ).

% stream.expand
thf(fact_772_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_773_stl__sset,axiom,
    ! [X3: nat,A: stream_nat] :
      ( ( member_nat @ X3 @ ( sset_nat @ ( stl_nat @ A ) ) )
     => ( member_nat @ X3 @ ( sset_nat @ A ) ) ) ).

% stl_sset
thf(fact_774_stl__sset,axiom,
    ! [X3: produc6018962875968178549ist_fm,A: stream4408948924543953275ist_fm] :
      ( ( member4699826688122452638ist_fm @ X3 @ ( sset_P5379989128061332361ist_fm @ ( stl_Pr1506262294867171877ist_fm @ A ) ) )
     => ( member4699826688122452638ist_fm @ X3 @ ( sset_P5379989128061332361ist_fm @ A ) ) ) ).

% stl_sset
thf(fact_775_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_776_stl__sset,axiom,
    ! [X3: produc340336539035504054m_rule,A: stream2709947120125613254m_rule] :
      ( ( member7231649785386036813m_rule @ X3 @ ( sset_P4484857331586881186m_rule @ ( stl_Pr12655793849929990m_rule @ A ) ) )
     => ( member7231649785386036813m_rule @ X3 @ ( sset_P4484857331586881186m_rule @ A ) ) ) ).

% stl_sset
thf(fact_777_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_778_alw__coinduct,axiom,
    ! [X5: stream_rule > $o,X3: stream_rule,Phi: stream_rule > $o] :
      ( ( X5 @ X3 )
     => ( ! [X2: stream_rule] :
            ( ( X5 @ X2 )
           => ( Phi @ X2 ) )
       => ( ! [X2: stream_rule] :
              ( ( X5 @ X2 )
             => ( ~ ( linear1087404401031714582w_rule @ Phi @ ( stl_rule @ X2 ) )
               => ( X5 @ ( stl_rule @ X2 ) ) ) )
         => ( linear1087404401031714582w_rule @ Phi @ X3 ) ) ) ) ).

% alw_coinduct
thf(fact_779_alw__coinduct,axiom,
    ! [X5: stream2709947120125613254m_rule > $o,X3: stream2709947120125613254m_rule,Phi: stream2709947120125613254m_rule > $o] :
      ( ( X5 @ X3 )
     => ( ! [X2: stream2709947120125613254m_rule] :
            ( ( X5 @ X2 )
           => ( Phi @ X2 ) )
       => ( ! [X2: stream2709947120125613254m_rule] :
              ( ( X5 @ X2 )
             => ( ~ ( linear4957294957385550906m_rule @ Phi @ ( stl_Pr12655793849929990m_rule @ X2 ) )
               => ( X5 @ ( stl_Pr12655793849929990m_rule @ X2 ) ) ) )
         => ( linear4957294957385550906m_rule @ Phi @ X3 ) ) ) ) ).

% alw_coinduct
thf(fact_780_alw_Ocoinduct,axiom,
    ! [X5: stream_rule > $o,X3: stream_rule,Phi: stream_rule > $o] :
      ( ( X5 @ X3 )
     => ( ! [X2: stream_rule] :
            ( ( X5 @ X2 )
           => ? [Xs4: stream_rule] :
                ( ( X2 = Xs4 )
                & ( Phi @ Xs4 )
                & ( ( X5 @ ( stl_rule @ Xs4 ) )
                  | ( linear1087404401031714582w_rule @ Phi @ ( stl_rule @ Xs4 ) ) ) ) )
       => ( linear1087404401031714582w_rule @ Phi @ X3 ) ) ) ).

% alw.coinduct
thf(fact_781_alw_Ocoinduct,axiom,
    ! [X5: stream2709947120125613254m_rule > $o,X3: stream2709947120125613254m_rule,Phi: stream2709947120125613254m_rule > $o] :
      ( ( X5 @ X3 )
     => ( ! [X2: stream2709947120125613254m_rule] :
            ( ( X5 @ X2 )
           => ? [Xs4: stream2709947120125613254m_rule] :
                ( ( X2 = Xs4 )
                & ( Phi @ Xs4 )
                & ( ( X5 @ ( stl_Pr12655793849929990m_rule @ Xs4 ) )
                  | ( linear4957294957385550906m_rule @ Phi @ ( stl_Pr12655793849929990m_rule @ Xs4 ) ) ) ) )
       => ( linear4957294957385550906m_rule @ Phi @ X3 ) ) ) ).

% alw.coinduct
thf(fact_782_alw_Ointros,axiom,
    ! [Phi: stream_rule > $o,Xs2: stream_rule] :
      ( ( Phi @ Xs2 )
     => ( ( linear1087404401031714582w_rule @ Phi @ ( stl_rule @ Xs2 ) )
       => ( linear1087404401031714582w_rule @ Phi @ Xs2 ) ) ) ).

% alw.intros
thf(fact_783_alw_Ointros,axiom,
    ! [Phi: stream2709947120125613254m_rule > $o,Xs2: stream2709947120125613254m_rule] :
      ( ( Phi @ Xs2 )
     => ( ( linear4957294957385550906m_rule @ Phi @ ( stl_Pr12655793849929990m_rule @ Xs2 ) )
       => ( linear4957294957385550906m_rule @ Phi @ Xs2 ) ) ) ).

% alw.intros
thf(fact_784_alw_Osimps,axiom,
    ( linear1087404401031714582w_rule
    = ( ^ [Phi2: stream_rule > $o,A4: stream_rule] :
        ? [Xs: stream_rule] :
          ( ( A4 = Xs )
          & ( Phi2 @ Xs )
          & ( linear1087404401031714582w_rule @ Phi2 @ ( stl_rule @ Xs ) ) ) ) ) ).

% alw.simps
thf(fact_785_alw_Osimps,axiom,
    ( linear4957294957385550906m_rule
    = ( ^ [Phi2: stream2709947120125613254m_rule > $o,A4: stream2709947120125613254m_rule] :
        ? [Xs: stream2709947120125613254m_rule] :
          ( ( A4 = Xs )
          & ( Phi2 @ Xs )
          & ( linear4957294957385550906m_rule @ Phi2 @ ( stl_Pr12655793849929990m_rule @ Xs ) ) ) ) ) ).

% alw.simps
thf(fact_786_alw_Ocases,axiom,
    ! [Phi: stream_rule > $o,A: stream_rule] :
      ( ( linear1087404401031714582w_rule @ Phi @ A )
     => ~ ( ( Phi @ A )
         => ~ ( linear1087404401031714582w_rule @ Phi @ ( stl_rule @ A ) ) ) ) ).

% alw.cases
thf(fact_787_alw_Ocases,axiom,
    ! [Phi: stream2709947120125613254m_rule > $o,A: stream2709947120125613254m_rule] :
      ( ( linear4957294957385550906m_rule @ Phi @ A )
     => ~ ( ( Phi @ A )
         => ~ ( linear4957294957385550906m_rule @ Phi @ ( stl_Pr12655793849929990m_rule @ A ) ) ) ) ).

% alw.cases
thf(fact_788_ev__induct__strong,axiom,
    ! [Phi: stream2709947120125613254m_rule > $o,X3: stream2709947120125613254m_rule,P2: stream2709947120125613254m_rule > $o] :
      ( ( linear1816559245158234865m_rule @ Phi @ X3 )
     => ( ! [Xs3: stream2709947120125613254m_rule] :
            ( ( Phi @ Xs3 )
           => ( P2 @ Xs3 ) )
       => ( ! [Xs3: stream2709947120125613254m_rule] :
              ( ( linear1816559245158234865m_rule @ Phi @ ( stl_Pr12655793849929990m_rule @ Xs3 ) )
             => ( ~ ( Phi @ Xs3 )
               => ( ( P2 @ ( stl_Pr12655793849929990m_rule @ Xs3 ) )
                 => ( P2 @ Xs3 ) ) ) )
         => ( P2 @ X3 ) ) ) ) ).

% ev_induct_strong
thf(fact_789_ev__induct__strong,axiom,
    ! [Phi: stream_rule > $o,X3: stream_rule,P2: stream_rule > $o] :
      ( ( linear292482573288640205v_rule @ Phi @ X3 )
     => ( ! [Xs3: stream_rule] :
            ( ( Phi @ Xs3 )
           => ( P2 @ Xs3 ) )
       => ( ! [Xs3: stream_rule] :
              ( ( linear292482573288640205v_rule @ Phi @ ( stl_rule @ Xs3 ) )
             => ( ~ ( Phi @ Xs3 )
               => ( ( P2 @ ( stl_rule @ Xs3 ) )
                 => ( P2 @ Xs3 ) ) ) )
         => ( P2 @ X3 ) ) ) ) ).

% ev_induct_strong
thf(fact_790_ev_Osimps,axiom,
    ( linear1816559245158234865m_rule
    = ( ^ [Phi2: stream2709947120125613254m_rule > $o,A4: stream2709947120125613254m_rule] :
          ( ? [Xs: stream2709947120125613254m_rule] :
              ( ( A4 = Xs )
              & ( Phi2 @ Xs ) )
          | ? [Xs: stream2709947120125613254m_rule] :
              ( ( A4 = Xs )
              & ( linear1816559245158234865m_rule @ Phi2 @ ( stl_Pr12655793849929990m_rule @ Xs ) ) ) ) ) ) ).

% ev.simps
thf(fact_791_ev_Osimps,axiom,
    ( linear292482573288640205v_rule
    = ( ^ [Phi2: stream_rule > $o,A4: stream_rule] :
          ( ? [Xs: stream_rule] :
              ( ( A4 = Xs )
              & ( Phi2 @ Xs ) )
          | ? [Xs: stream_rule] :
              ( ( A4 = Xs )
              & ( linear292482573288640205v_rule @ Phi2 @ ( stl_rule @ Xs ) ) ) ) ) ) ).

% ev.simps
thf(fact_792_ev_Ocases,axiom,
    ! [Phi: stream2709947120125613254m_rule > $o,A: stream2709947120125613254m_rule] :
      ( ( linear1816559245158234865m_rule @ Phi @ A )
     => ( ~ ( Phi @ A )
       => ( linear1816559245158234865m_rule @ Phi @ ( stl_Pr12655793849929990m_rule @ A ) ) ) ) ).

% ev.cases
thf(fact_793_ev_Ocases,axiom,
    ! [Phi: stream_rule > $o,A: stream_rule] :
      ( ( linear292482573288640205v_rule @ Phi @ A )
     => ( ~ ( Phi @ A )
       => ( linear292482573288640205v_rule @ Phi @ ( stl_rule @ A ) ) ) ) ).

% ev.cases
thf(fact_794_ev_Ostep,axiom,
    ! [Phi: stream2709947120125613254m_rule > $o,Xs2: stream2709947120125613254m_rule] :
      ( ( linear1816559245158234865m_rule @ Phi @ ( stl_Pr12655793849929990m_rule @ Xs2 ) )
     => ( linear1816559245158234865m_rule @ Phi @ Xs2 ) ) ).

% ev.step
thf(fact_795_ev_Ostep,axiom,
    ! [Phi: stream_rule > $o,Xs2: stream_rule] :
      ( ( linear292482573288640205v_rule @ Phi @ ( stl_rule @ Xs2 ) )
     => ( linear292482573288640205v_rule @ Phi @ Xs2 ) ) ).

% ev.step
thf(fact_796_sdrop__stl,axiom,
    ! [N: nat,S3: stream2709947120125613254m_rule] :
      ( ( sdrop_8169176516188972301m_rule @ N @ ( stl_Pr12655793849929990m_rule @ S3 ) )
      = ( stl_Pr12655793849929990m_rule @ ( sdrop_8169176516188972301m_rule @ N @ S3 ) ) ) ).

% sdrop_stl
thf(fact_797_sdrop__stl,axiom,
    ! [N: nat,S3: stream_rule] :
      ( ( sdrop_rule @ N @ ( stl_rule @ S3 ) )
      = ( stl_rule @ ( sdrop_rule @ N @ S3 ) ) ) ).

% sdrop_stl
thf(fact_798_RuleSystem__Defs_Ofair__stl,axiom,
    ! [Rules2: stream2709947120125613254m_rule,Rs2: stream2709947120125613254m_rule] :
      ( ( abstra3665372904625986210m_rule @ Rules2 @ Rs2 )
     => ( abstra3665372904625986210m_rule @ Rules2 @ ( stl_Pr12655793849929990m_rule @ Rs2 ) ) ) ).

% RuleSystem_Defs.fair_stl
thf(fact_799_RuleSystem__Defs_Ofair__stl,axiom,
    ! [Rules2: stream_rule,Rs2: stream_rule] :
      ( ( abstra3799686578551160190r_rule @ Rules2 @ Rs2 )
     => ( abstra3799686578551160190r_rule @ Rules2 @ ( stl_rule @ Rs2 ) ) ) ).

% RuleSystem_Defs.fair_stl
thf(fact_800_i_Ofair__stl,axiom,
    ! [Rules2: stream2709947120125613254m_rule,Rs2: stream2709947120125613254m_rule] :
      ( ( abstra3665372904625986210m_rule @ Rules2 @ Rs2 )
     => ( abstra3665372904625986210m_rule @ Rules2 @ ( stl_Pr12655793849929990m_rule @ Rs2 ) ) ) ).

% i.fair_stl
thf(fact_801_i_Ofair__stl,axiom,
    ! [Rules2: stream_rule,Rs2: stream_rule] :
      ( ( abstra3799686578551160190r_rule @ Rules2 @ Rs2 )
     => ( abstra3799686578551160190r_rule @ Rules2 @ ( stl_rule @ Rs2 ) ) ) ).

% i.fair_stl
thf(fact_802_suntil__induct__strong,axiom,
    ! [Phi: stream2709947120125613254m_rule > $o,Psi: stream2709947120125613254m_rule > $o,X3: stream2709947120125613254m_rule,P2: stream2709947120125613254m_rule > $o] :
      ( ( linear5457325955354885675m_rule @ Phi @ Psi @ X3 )
     => ( ! [Omega3: stream2709947120125613254m_rule] :
            ( ( Psi @ Omega3 )
           => ( P2 @ Omega3 ) )
       => ( ! [Omega3: stream2709947120125613254m_rule] :
              ( ( Phi @ Omega3 )
             => ( ~ ( Psi @ Omega3 )
               => ( ( linear5457325955354885675m_rule @ Phi @ Psi @ ( stl_Pr12655793849929990m_rule @ Omega3 ) )
                 => ( ( P2 @ ( stl_Pr12655793849929990m_rule @ Omega3 ) )
                   => ( P2 @ Omega3 ) ) ) ) )
         => ( P2 @ X3 ) ) ) ) ).

% suntil_induct_strong
thf(fact_803_suntil__induct__strong,axiom,
    ! [Phi: stream_rule > $o,Psi: stream_rule > $o,X3: stream_rule,P2: stream_rule > $o] :
      ( ( linear1836410492716652551l_rule @ Phi @ Psi @ X3 )
     => ( ! [Omega3: stream_rule] :
            ( ( Psi @ Omega3 )
           => ( P2 @ Omega3 ) )
       => ( ! [Omega3: stream_rule] :
              ( ( Phi @ Omega3 )
             => ( ~ ( Psi @ Omega3 )
               => ( ( linear1836410492716652551l_rule @ Phi @ Psi @ ( stl_rule @ Omega3 ) )
                 => ( ( P2 @ ( stl_rule @ Omega3 ) )
                   => ( P2 @ Omega3 ) ) ) ) )
         => ( P2 @ X3 ) ) ) ) ).

% suntil_induct_strong
thf(fact_804_suntil_Osimps,axiom,
    ( linear5457325955354885675m_rule
    = ( ^ [Phi2: stream2709947120125613254m_rule > $o,Psi2: stream2709947120125613254m_rule > $o,A4: stream2709947120125613254m_rule] :
          ( ? [Omega: stream2709947120125613254m_rule] :
              ( ( A4 = Omega )
              & ( Psi2 @ Omega ) )
          | ? [Omega: stream2709947120125613254m_rule] :
              ( ( A4 = Omega )
              & ( Phi2 @ Omega )
              & ( linear5457325955354885675m_rule @ Phi2 @ Psi2 @ ( stl_Pr12655793849929990m_rule @ Omega ) ) ) ) ) ) ).

% suntil.simps
thf(fact_805_suntil_Osimps,axiom,
    ( linear1836410492716652551l_rule
    = ( ^ [Phi2: stream_rule > $o,Psi2: stream_rule > $o,A4: stream_rule] :
          ( ? [Omega: stream_rule] :
              ( ( A4 = Omega )
              & ( Psi2 @ Omega ) )
          | ? [Omega: stream_rule] :
              ( ( A4 = Omega )
              & ( Phi2 @ Omega )
              & ( linear1836410492716652551l_rule @ Phi2 @ Psi2 @ ( stl_rule @ Omega ) ) ) ) ) ) ).

% suntil.simps
thf(fact_806_suntil_Ocases,axiom,
    ! [Phi: stream2709947120125613254m_rule > $o,Psi: stream2709947120125613254m_rule > $o,A: stream2709947120125613254m_rule] :
      ( ( linear5457325955354885675m_rule @ Phi @ Psi @ A )
     => ( ~ ( Psi @ A )
       => ~ ( ( Phi @ A )
           => ~ ( linear5457325955354885675m_rule @ Phi @ Psi @ ( stl_Pr12655793849929990m_rule @ A ) ) ) ) ) ).

% suntil.cases
thf(fact_807_suntil_Ocases,axiom,
    ! [Phi: stream_rule > $o,Psi: stream_rule > $o,A: stream_rule] :
      ( ( linear1836410492716652551l_rule @ Phi @ Psi @ A )
     => ( ~ ( Psi @ A )
       => ~ ( ( Phi @ A )
           => ~ ( linear1836410492716652551l_rule @ Phi @ Psi @ ( stl_rule @ A ) ) ) ) ) ).

% suntil.cases
thf(fact_808_suntil_Ostep,axiom,
    ! [Phi: stream2709947120125613254m_rule > $o,Omega2: stream2709947120125613254m_rule,Psi: stream2709947120125613254m_rule > $o] :
      ( ( Phi @ Omega2 )
     => ( ( linear5457325955354885675m_rule @ Phi @ Psi @ ( stl_Pr12655793849929990m_rule @ Omega2 ) )
       => ( linear5457325955354885675m_rule @ Phi @ Psi @ Omega2 ) ) ) ).

% suntil.step
thf(fact_809_suntil_Ostep,axiom,
    ! [Phi: stream_rule > $o,Omega2: stream_rule,Psi: stream_rule > $o] :
      ( ( Phi @ Omega2 )
     => ( ( linear1836410492716652551l_rule @ Phi @ Psi @ ( stl_rule @ Omega2 ) )
       => ( linear1836410492716652551l_rule @ Phi @ Psi @ Omega2 ) ) ) ).

% suntil.step
thf(fact_810_streams__stl,axiom,
    ! [S3: stream2709947120125613254m_rule,A2: set_Pr1822751329126368876m_rule] :
      ( ( member5228318729319728861m_rule @ S3 @ ( stream3418479298939905342m_rule @ A2 ) )
     => ( member5228318729319728861m_rule @ ( stl_Pr12655793849929990m_rule @ S3 ) @ ( stream3418479298939905342m_rule @ A2 ) ) ) ).

% streams_stl
thf(fact_811_streams__stl,axiom,
    ! [S3: stream_rule,A2: set_rule] :
      ( ( member_stream_rule @ S3 @ ( streams_rule @ A2 ) )
     => ( member_stream_rule @ ( stl_rule @ S3 ) @ ( streams_rule @ A2 ) ) ) ).

% streams_stl
thf(fact_812_UNTIL_Ocoinduct,axiom,
    ! [X5: stream2709947120125613254m_rule > $o,X3: stream2709947120125613254m_rule,Psi: stream2709947120125613254m_rule > $o,Phi: stream2709947120125613254m_rule > $o] :
      ( ( X5 @ X3 )
     => ( ! [X2: stream2709947120125613254m_rule] :
            ( ( X5 @ X2 )
           => ( ? [Xs4: stream2709947120125613254m_rule] :
                  ( ( X2 = Xs4 )
                  & ( Psi @ Xs4 ) )
              | ? [Xs4: stream2709947120125613254m_rule] :
                  ( ( X2 = Xs4 )
                  & ( Phi @ Xs4 )
                  & ( ( X5 @ ( stl_Pr12655793849929990m_rule @ Xs4 ) )
                    | ( linear8167355876404644844m_rule @ Phi @ Psi @ ( stl_Pr12655793849929990m_rule @ Xs4 ) ) ) ) ) )
       => ( linear8167355876404644844m_rule @ Phi @ Psi @ X3 ) ) ) ).

% UNTIL.coinduct
thf(fact_813_UNTIL_Ocoinduct,axiom,
    ! [X5: stream_rule > $o,X3: stream_rule,Psi: stream_rule > $o,Phi: stream_rule > $o] :
      ( ( X5 @ X3 )
     => ( ! [X2: stream_rule] :
            ( ( X5 @ X2 )
           => ( ? [Xs4: stream_rule] :
                  ( ( X2 = Xs4 )
                  & ( Psi @ Xs4 ) )
              | ? [Xs4: stream_rule] :
                  ( ( X2 = Xs4 )
                  & ( Phi @ Xs4 )
                  & ( ( X5 @ ( stl_rule @ Xs4 ) )
                    | ( linear6513964619561280712L_rule @ Phi @ Psi @ ( stl_rule @ Xs4 ) ) ) ) ) )
       => ( linear6513964619561280712L_rule @ Phi @ Psi @ X3 ) ) ) ).

% UNTIL.coinduct
thf(fact_814_UNTIL_Osimps,axiom,
    ( linear8167355876404644844m_rule
    = ( ^ [Phi2: stream2709947120125613254m_rule > $o,Psi2: stream2709947120125613254m_rule > $o,A4: stream2709947120125613254m_rule] :
          ( ? [Xs: stream2709947120125613254m_rule] :
              ( ( A4 = Xs )
              & ( Psi2 @ Xs ) )
          | ? [Xs: stream2709947120125613254m_rule] :
              ( ( A4 = Xs )
              & ( Phi2 @ Xs )
              & ( linear8167355876404644844m_rule @ Phi2 @ Psi2 @ ( stl_Pr12655793849929990m_rule @ Xs ) ) ) ) ) ) ).

% UNTIL.simps
thf(fact_815_UNTIL_Osimps,axiom,
    ( linear6513964619561280712L_rule
    = ( ^ [Phi2: stream_rule > $o,Psi2: stream_rule > $o,A4: stream_rule] :
          ( ? [Xs: stream_rule] :
              ( ( A4 = Xs )
              & ( Psi2 @ Xs ) )
          | ? [Xs: stream_rule] :
              ( ( A4 = Xs )
              & ( Phi2 @ Xs )
              & ( linear6513964619561280712L_rule @ Phi2 @ Psi2 @ ( stl_rule @ Xs ) ) ) ) ) ) ).

% UNTIL.simps
thf(fact_816_UNTIL_Ocases,axiom,
    ! [Phi: stream2709947120125613254m_rule > $o,Psi: stream2709947120125613254m_rule > $o,A: stream2709947120125613254m_rule] :
      ( ( linear8167355876404644844m_rule @ Phi @ Psi @ A )
     => ( ~ ( Psi @ A )
       => ~ ( ( Phi @ A )
           => ~ ( linear8167355876404644844m_rule @ Phi @ Psi @ ( stl_Pr12655793849929990m_rule @ A ) ) ) ) ) ).

% UNTIL.cases
thf(fact_817_UNTIL_Ocases,axiom,
    ! [Phi: stream_rule > $o,Psi: stream_rule > $o,A: stream_rule] :
      ( ( linear6513964619561280712L_rule @ Phi @ Psi @ A )
     => ( ~ ( Psi @ A )
       => ~ ( ( Phi @ A )
           => ~ ( linear6513964619561280712L_rule @ Phi @ Psi @ ( stl_rule @ A ) ) ) ) ) ).

% UNTIL.cases
thf(fact_818_UNTIL_Ostep,axiom,
    ! [Phi: stream2709947120125613254m_rule > $o,Xs2: stream2709947120125613254m_rule,Psi: stream2709947120125613254m_rule > $o] :
      ( ( Phi @ Xs2 )
     => ( ( linear8167355876404644844m_rule @ Phi @ Psi @ ( stl_Pr12655793849929990m_rule @ Xs2 ) )
       => ( linear8167355876404644844m_rule @ Phi @ Psi @ Xs2 ) ) ) ).

% UNTIL.step
thf(fact_819_UNTIL_Ostep,axiom,
    ! [Phi: stream_rule > $o,Xs2: stream_rule,Psi: stream_rule > $o] :
      ( ( Phi @ Xs2 )
     => ( ( linear6513964619561280712L_rule @ Phi @ Psi @ ( stl_rule @ Xs2 ) )
       => ( linear6513964619561280712L_rule @ Phi @ Psi @ Xs2 ) ) ) ).

% UNTIL.step
thf(fact_820_nxt_Osimps,axiom,
    ( linear3460418925085718015rule_o
    = ( ^ [Phi2: stream2709947120125613254m_rule > $o,Xs: stream2709947120125613254m_rule] : ( Phi2 @ ( stl_Pr12655793849929990m_rule @ Xs ) ) ) ) ).

% nxt.simps
thf(fact_821_nxt_Oelims,axiom,
    ! [X3: stream2709947120125613254m_rule > $o,Xa2: stream2709947120125613254m_rule,Y: $o] :
      ( ( ( linear3460418925085718015rule_o @ X3 @ Xa2 )
        = Y )
     => ( Y
        = ( X3 @ ( stl_Pr12655793849929990m_rule @ Xa2 ) ) ) ) ).

% nxt.elims
thf(fact_822_smap2_Osimps_I2_J,axiom,
    ! [F: produc340336539035504054m_rule > produc340336539035504054m_rule > produc340336539035504054m_rule,S1: stream2709947120125613254m_rule,S2: stream2709947120125613254m_rule] :
      ( ( stl_Pr12655793849929990m_rule @ ( smap2_6945574070536835298m_rule @ F @ S1 @ S2 ) )
      = ( smap2_6945574070536835298m_rule @ F @ ( stl_Pr12655793849929990m_rule @ S1 ) @ ( stl_Pr12655793849929990m_rule @ S2 ) ) ) ).

% smap2.simps(2)
thf(fact_823_smap2_Osimps_I2_J,axiom,
    ! [F: produc340336539035504054m_rule > rule > produc340336539035504054m_rule,S1: stream2709947120125613254m_rule,S2: stream_rule] :
      ( ( stl_Pr12655793849929990m_rule @ ( smap2_6366920367380388542m_rule @ F @ S1 @ S2 ) )
      = ( smap2_6366920367380388542m_rule @ F @ ( stl_Pr12655793849929990m_rule @ S1 ) @ ( stl_rule @ S2 ) ) ) ).

% smap2.simps(2)
thf(fact_824_smap2_Osimps_I2_J,axiom,
    ! [F: rule > produc340336539035504054m_rule > produc340336539035504054m_rule,S1: stream_rule,S2: stream2709947120125613254m_rule] :
      ( ( stl_Pr12655793849929990m_rule @ ( smap2_4579943528986984894m_rule @ F @ S1 @ S2 ) )
      = ( smap2_4579943528986984894m_rule @ F @ ( stl_rule @ S1 ) @ ( stl_Pr12655793849929990m_rule @ S2 ) ) ) ).

% smap2.simps(2)
thf(fact_825_smap2_Osimps_I2_J,axiom,
    ! [F: rule > rule > produc340336539035504054m_rule,S1: stream_rule,S2: stream_rule] :
      ( ( stl_Pr12655793849929990m_rule @ ( smap2_4319354692778897306m_rule @ F @ S1 @ S2 ) )
      = ( smap2_4319354692778897306m_rule @ F @ ( stl_rule @ S1 ) @ ( stl_rule @ S2 ) ) ) ).

% smap2.simps(2)
thf(fact_826_smap2_Osimps_I2_J,axiom,
    ! [F: produc340336539035504054m_rule > produc340336539035504054m_rule > rule,S1: stream2709947120125613254m_rule,S2: stream2709947120125613254m_rule] :
      ( ( stl_rule @ ( smap2_1878155219598472126e_rule @ F @ S1 @ S2 ) )
      = ( smap2_1878155219598472126e_rule @ F @ ( stl_Pr12655793849929990m_rule @ S1 ) @ ( stl_Pr12655793849929990m_rule @ S2 ) ) ) ).

% smap2.simps(2)
thf(fact_827_smap2_Osimps_I2_J,axiom,
    ! [F: produc340336539035504054m_rule > rule > rule,S1: stream2709947120125613254m_rule,S2: stream_rule] :
      ( ( stl_rule @ ( smap2_8971878701134242202e_rule @ F @ S1 @ S2 ) )
      = ( smap2_8971878701134242202e_rule @ F @ ( stl_Pr12655793849929990m_rule @ S1 ) @ ( stl_rule @ S2 ) ) ) ).

% smap2.simps(2)
thf(fact_828_smap2_Osimps_I2_J,axiom,
    ! [F: rule > produc340336539035504054m_rule > rule,S1: stream_rule,S2: stream2709947120125613254m_rule] :
      ( ( stl_rule @ ( smap2_9053961581851756698e_rule @ F @ S1 @ S2 ) )
      = ( smap2_9053961581851756698e_rule @ F @ ( stl_rule @ S1 ) @ ( stl_Pr12655793849929990m_rule @ S2 ) ) ) ).

% smap2.simps(2)
thf(fact_829_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_830_alw__inv,axiom,
    ! [F: stream_rule > stream_rule,P2: stream_rule > $o,S3: stream_rule] :
      ( ! [S8: stream_rule] :
          ( ( F @ ( stl_rule @ S8 ) )
          = ( stl_rule @ ( F @ S8 ) ) )
     => ( ( linear1087404401031714582w_rule @ P2 @ ( F @ S3 ) )
        = ( linear1087404401031714582w_rule
          @ ^ [X: stream_rule] : ( P2 @ ( F @ X ) )
          @ S3 ) ) ) ).

% alw_inv
thf(fact_831_alw__inv,axiom,
    ! [F: stream2709947120125613254m_rule > stream_rule,P2: stream_rule > $o,S3: stream2709947120125613254m_rule] :
      ( ! [S8: stream2709947120125613254m_rule] :
          ( ( F @ ( stl_Pr12655793849929990m_rule @ S8 ) )
          = ( stl_rule @ ( F @ S8 ) ) )
     => ( ( linear1087404401031714582w_rule @ P2 @ ( F @ S3 ) )
        = ( linear4957294957385550906m_rule
          @ ^ [X: stream2709947120125613254m_rule] : ( P2 @ ( F @ X ) )
          @ S3 ) ) ) ).

% alw_inv
thf(fact_832_alw__inv,axiom,
    ! [F: stream_rule > stream2709947120125613254m_rule,P2: stream2709947120125613254m_rule > $o,S3: stream_rule] :
      ( ! [S8: stream_rule] :
          ( ( F @ ( stl_rule @ S8 ) )
          = ( stl_Pr12655793849929990m_rule @ ( F @ S8 ) ) )
     => ( ( linear4957294957385550906m_rule @ P2 @ ( F @ S3 ) )
        = ( linear1087404401031714582w_rule
          @ ^ [X: stream_rule] : ( P2 @ ( F @ X ) )
          @ S3 ) ) ) ).

% alw_inv
thf(fact_833_alw__inv,axiom,
    ! [F: stream2709947120125613254m_rule > stream2709947120125613254m_rule,P2: stream2709947120125613254m_rule > $o,S3: stream2709947120125613254m_rule] :
      ( ! [S8: stream2709947120125613254m_rule] :
          ( ( F @ ( stl_Pr12655793849929990m_rule @ S8 ) )
          = ( stl_Pr12655793849929990m_rule @ ( F @ S8 ) ) )
     => ( ( linear4957294957385550906m_rule @ P2 @ ( F @ S3 ) )
        = ( linear4957294957385550906m_rule
          @ ^ [X: stream2709947120125613254m_rule] : ( P2 @ ( F @ X ) )
          @ S3 ) ) ) ).

% alw_inv
thf(fact_834_ev__inv,axiom,
    ! [F: stream2709947120125613254m_rule > stream2709947120125613254m_rule,P2: stream2709947120125613254m_rule > $o,S3: stream2709947120125613254m_rule] :
      ( ! [S8: stream2709947120125613254m_rule] :
          ( ( F @ ( stl_Pr12655793849929990m_rule @ S8 ) )
          = ( stl_Pr12655793849929990m_rule @ ( F @ S8 ) ) )
     => ( ( linear1816559245158234865m_rule @ P2 @ ( F @ S3 ) )
        = ( linear1816559245158234865m_rule
          @ ^ [X: stream2709947120125613254m_rule] : ( P2 @ ( F @ X ) )
          @ S3 ) ) ) ).

% ev_inv
thf(fact_835_ev__inv,axiom,
    ! [F: stream_rule > stream2709947120125613254m_rule,P2: stream2709947120125613254m_rule > $o,S3: stream_rule] :
      ( ! [S8: stream_rule] :
          ( ( F @ ( stl_rule @ S8 ) )
          = ( stl_Pr12655793849929990m_rule @ ( F @ S8 ) ) )
     => ( ( linear1816559245158234865m_rule @ P2 @ ( F @ S3 ) )
        = ( linear292482573288640205v_rule
          @ ^ [X: stream_rule] : ( P2 @ ( F @ X ) )
          @ S3 ) ) ) ).

% ev_inv
thf(fact_836_ev__inv,axiom,
    ! [F: stream2709947120125613254m_rule > stream_rule,P2: stream_rule > $o,S3: stream2709947120125613254m_rule] :
      ( ! [S8: stream2709947120125613254m_rule] :
          ( ( F @ ( stl_Pr12655793849929990m_rule @ S8 ) )
          = ( stl_rule @ ( F @ S8 ) ) )
     => ( ( linear292482573288640205v_rule @ P2 @ ( F @ S3 ) )
        = ( linear1816559245158234865m_rule
          @ ^ [X: stream2709947120125613254m_rule] : ( P2 @ ( F @ X ) )
          @ S3 ) ) ) ).

% ev_inv
thf(fact_837_ev__inv,axiom,
    ! [F: stream_rule > stream_rule,P2: stream_rule > $o,S3: stream_rule] :
      ( ! [S8: stream_rule] :
          ( ( F @ ( stl_rule @ S8 ) )
          = ( stl_rule @ ( F @ S8 ) ) )
     => ( ( linear292482573288640205v_rule @ P2 @ ( F @ S3 ) )
        = ( linear292482573288640205v_rule
          @ ^ [X: stream_rule] : ( P2 @ ( F @ X ) )
          @ S3 ) ) ) ).

% ev_inv
thf(fact_838_suntil__inv,axiom,
    ! [F: stream2709947120125613254m_rule > stream2709947120125613254m_rule,P2: stream2709947120125613254m_rule > $o,Q: stream2709947120125613254m_rule > $o,S3: stream2709947120125613254m_rule] :
      ( ! [S8: stream2709947120125613254m_rule] :
          ( ( F @ ( stl_Pr12655793849929990m_rule @ S8 ) )
          = ( stl_Pr12655793849929990m_rule @ ( F @ S8 ) ) )
     => ( ( linear5457325955354885675m_rule @ P2 @ Q @ ( F @ S3 ) )
        = ( linear5457325955354885675m_rule
          @ ^ [X: stream2709947120125613254m_rule] : ( P2 @ ( F @ X ) )
          @ ^ [X: stream2709947120125613254m_rule] : ( Q @ ( F @ X ) )
          @ S3 ) ) ) ).

% suntil_inv
thf(fact_839_suntil__inv,axiom,
    ! [F: stream2709947120125613254m_rule > stream_rule,P2: stream_rule > $o,Q: stream_rule > $o,S3: stream2709947120125613254m_rule] :
      ( ! [S8: stream2709947120125613254m_rule] :
          ( ( F @ ( stl_Pr12655793849929990m_rule @ S8 ) )
          = ( stl_rule @ ( F @ S8 ) ) )
     => ( ( linear1836410492716652551l_rule @ P2 @ Q @ ( F @ S3 ) )
        = ( linear5457325955354885675m_rule
          @ ^ [X: stream2709947120125613254m_rule] : ( P2 @ ( F @ X ) )
          @ ^ [X: stream2709947120125613254m_rule] : ( Q @ ( F @ X ) )
          @ S3 ) ) ) ).

% suntil_inv
thf(fact_840_suntil__inv,axiom,
    ! [F: stream_rule > stream2709947120125613254m_rule,P2: stream2709947120125613254m_rule > $o,Q: stream2709947120125613254m_rule > $o,S3: stream_rule] :
      ( ! [S8: stream_rule] :
          ( ( F @ ( stl_rule @ S8 ) )
          = ( stl_Pr12655793849929990m_rule @ ( F @ S8 ) ) )
     => ( ( linear5457325955354885675m_rule @ P2 @ Q @ ( F @ S3 ) )
        = ( linear1836410492716652551l_rule
          @ ^ [X: stream_rule] : ( P2 @ ( F @ X ) )
          @ ^ [X: stream_rule] : ( Q @ ( F @ X ) )
          @ S3 ) ) ) ).

% suntil_inv
thf(fact_841_suntil__inv,axiom,
    ! [F: stream_rule > stream_rule,P2: stream_rule > $o,Q: stream_rule > $o,S3: stream_rule] :
      ( ! [S8: stream_rule] :
          ( ( F @ ( stl_rule @ S8 ) )
          = ( stl_rule @ ( F @ S8 ) ) )
     => ( ( linear1836410492716652551l_rule @ P2 @ Q @ ( F @ S3 ) )
        = ( linear1836410492716652551l_rule
          @ ^ [X: stream_rule] : ( P2 @ ( F @ X ) )
          @ ^ [X: stream_rule] : ( Q @ ( F @ X ) )
          @ S3 ) ) ) ).

% suntil_inv
thf(fact_842_sset__induct,axiom,
    ! [Y: nat,S3: stream_nat,P2: nat > stream_nat > $o] :
      ( ( member_nat @ Y @ ( sset_nat @ S3 ) )
     => ( ! [S8: stream_nat] : ( P2 @ ( shd_nat @ S8 ) @ S8 )
       => ( ! [S8: stream_nat,Y4: nat] :
              ( ( member_nat @ Y4 @ ( sset_nat @ ( stl_nat @ S8 ) ) )
             => ( ( P2 @ Y4 @ ( stl_nat @ S8 ) )
               => ( P2 @ Y4 @ S8 ) ) )
         => ( P2 @ Y @ S3 ) ) ) ) ).

% sset_induct
thf(fact_843_sset__induct,axiom,
    ! [Y: produc6018962875968178549ist_fm,S3: stream4408948924543953275ist_fm,P2: produc6018962875968178549ist_fm > stream4408948924543953275ist_fm > $o] :
      ( ( member4699826688122452638ist_fm @ Y @ ( sset_P5379989128061332361ist_fm @ S3 ) )
     => ( ! [S8: stream4408948924543953275ist_fm] : ( P2 @ ( shd_Pr3211216682057661985ist_fm @ S8 ) @ S8 )
       => ( ! [S8: stream4408948924543953275ist_fm,Y4: produc6018962875968178549ist_fm] :
              ( ( member4699826688122452638ist_fm @ Y4 @ ( sset_P5379989128061332361ist_fm @ ( stl_Pr1506262294867171877ist_fm @ S8 ) ) )
             => ( ( P2 @ Y4 @ ( stl_Pr1506262294867171877ist_fm @ S8 ) )
               => ( P2 @ Y4 @ S8 ) ) )
         => ( P2 @ Y @ S3 ) ) ) ) ).

% sset_induct
thf(fact_844_sset__induct,axiom,
    ! [Y: produc859450856879609959at_nat,S3: stream8372878641218411373at_nat,P2: produc859450856879609959at_nat > stream8372878641218411373at_nat > $o] :
      ( ( member8206827879206165904at_nat @ Y @ ( sset_P9164763103296375419at_nat @ S3 ) )
     => ( ! [S8: stream8372878641218411373at_nat] : ( P2 @ ( shd_Pr8412153233960533267at_nat @ S8 ) @ S8 )
       => ( ! [S8: stream8372878641218411373at_nat,Y4: produc859450856879609959at_nat] :
              ( ( member8206827879206165904at_nat @ Y4 @ ( sset_P9164763103296375419at_nat @ ( stl_Pr5010465336888994071at_nat @ S8 ) ) )
             => ( ( P2 @ Y4 @ ( stl_Pr5010465336888994071at_nat @ S8 ) )
               => ( P2 @ Y4 @ S8 ) ) )
         => ( P2 @ Y @ S3 ) ) ) ) ).

% sset_induct
thf(fact_845_sset__induct,axiom,
    ! [Y: produc340336539035504054m_rule,S3: stream2709947120125613254m_rule,P2: produc340336539035504054m_rule > stream2709947120125613254m_rule > $o] :
      ( ( member7231649785386036813m_rule @ Y @ ( sset_P4484857331586881186m_rule @ S3 ) )
     => ( ! [S8: stream2709947120125613254m_rule] : ( P2 @ ( shd_Pr4562317740776619530m_rule @ S8 ) @ S8 )
       => ( ! [S8: stream2709947120125613254m_rule,Y4: produc340336539035504054m_rule] :
              ( ( member7231649785386036813m_rule @ Y4 @ ( sset_P4484857331586881186m_rule @ ( stl_Pr12655793849929990m_rule @ S8 ) ) )
             => ( ( P2 @ Y4 @ ( stl_Pr12655793849929990m_rule @ S8 ) )
               => ( P2 @ Y4 @ S8 ) ) )
         => ( P2 @ Y @ S3 ) ) ) ) ).

% sset_induct
thf(fact_846_sset__induct,axiom,
    ! [Y: rule,S3: stream_rule,P2: rule > stream_rule > $o] :
      ( ( member_rule @ Y @ ( sset_rule @ S3 ) )
     => ( ! [S8: stream_rule] : ( P2 @ ( shd_rule @ S8 ) @ S8 )
       => ( ! [S8: stream_rule,Y4: rule] :
              ( ( member_rule @ Y4 @ ( sset_rule @ ( stl_rule @ S8 ) ) )
             => ( ( P2 @ Y4 @ ( stl_rule @ S8 ) )
               => ( P2 @ Y4 @ S8 ) ) )
         => ( P2 @ Y @ S3 ) ) ) ) ).

% sset_induct
thf(fact_847_alw__ev__stl,axiom,
    ! [P2: stream_rule > $o,Omega2: stream_rule] :
      ( ( linear1087404401031714582w_rule @ ( linear292482573288640205v_rule @ P2 ) @ ( stl_rule @ Omega2 ) )
      = ( linear1087404401031714582w_rule @ ( linear292482573288640205v_rule @ P2 ) @ Omega2 ) ) ).

% alw_ev_stl
thf(fact_848_alw__ev__stl,axiom,
    ! [P2: stream2709947120125613254m_rule > $o,Omega2: stream2709947120125613254m_rule] :
      ( ( linear4957294957385550906m_rule @ ( linear1816559245158234865m_rule @ P2 ) @ ( stl_Pr12655793849929990m_rule @ Omega2 ) )
      = ( linear4957294957385550906m_rule @ ( linear1816559245158234865m_rule @ P2 ) @ Omega2 ) ) ).

% alw_ev_stl
thf(fact_849_streamsE,axiom,
    ! [S3: stream_nat,A2: set_nat] :
      ( ( member_stream_nat @ S3 @ ( streams_nat @ A2 ) )
     => ~ ( ( member_nat @ ( shd_nat @ S3 ) @ A2 )
         => ~ ( member_stream_nat @ ( stl_nat @ S3 ) @ ( streams_nat @ A2 ) ) ) ) ).

% streamsE
thf(fact_850_streamsE,axiom,
    ! [S3: stream4408948924543953275ist_fm,A2: set_Pr5202636777678657877ist_fm] :
      ( ( member197948898507729188ist_fm @ S3 @ ( stream1217541351357117421ist_fm @ A2 ) )
     => ~ ( ( member4699826688122452638ist_fm @ ( shd_Pr3211216682057661985ist_fm @ S3 ) @ A2 )
         => ~ ( member197948898507729188ist_fm @ ( stl_Pr1506262294867171877ist_fm @ S3 ) @ ( stream1217541351357117421ist_fm @ A2 ) ) ) ) ).

% streamsE
thf(fact_851_streamsE,axiom,
    ! [S3: stream8372878641218411373at_nat,A2: set_Pr8693737435421807431at_nat] :
      ( ( member1992146023263842070at_nat @ S3 @ ( stream5885727000947712735at_nat @ A2 ) )
     => ~ ( ( member8206827879206165904at_nat @ ( shd_Pr8412153233960533267at_nat @ S3 ) @ A2 )
         => ~ ( member1992146023263842070at_nat @ ( stl_Pr5010465336888994071at_nat @ S3 ) @ ( stream5885727000947712735at_nat @ A2 ) ) ) ) ).

% streamsE
thf(fact_852_streamsE,axiom,
    ! [S3: stream2709947120125613254m_rule,A2: set_Pr1822751329126368876m_rule] :
      ( ( member5228318729319728861m_rule @ S3 @ ( stream3418479298939905342m_rule @ A2 ) )
     => ~ ( ( member7231649785386036813m_rule @ ( shd_Pr4562317740776619530m_rule @ S3 ) @ A2 )
         => ~ ( member5228318729319728861m_rule @ ( stl_Pr12655793849929990m_rule @ S3 ) @ ( stream3418479298939905342m_rule @ A2 ) ) ) ) ).

% streamsE
thf(fact_853_streamsE,axiom,
    ! [S3: stream_rule,A2: set_rule] :
      ( ( member_stream_rule @ S3 @ ( streams_rule @ A2 ) )
     => ~ ( ( member_rule @ ( shd_rule @ S3 ) @ A2 )
         => ~ ( member_stream_rule @ ( stl_rule @ S3 ) @ ( streams_rule @ A2 ) ) ) ) ).

% streamsE
thf(fact_854_in__streams,axiom,
    ! [S3: stream_nat,S4: set_nat] :
      ( ( member_stream_nat @ ( stl_nat @ S3 ) @ ( streams_nat @ S4 ) )
     => ( ( member_nat @ ( shd_nat @ S3 ) @ S4 )
       => ( member_stream_nat @ S3 @ ( streams_nat @ S4 ) ) ) ) ).

% in_streams
thf(fact_855_in__streams,axiom,
    ! [S3: stream4408948924543953275ist_fm,S4: set_Pr5202636777678657877ist_fm] :
      ( ( member197948898507729188ist_fm @ ( stl_Pr1506262294867171877ist_fm @ S3 ) @ ( stream1217541351357117421ist_fm @ S4 ) )
     => ( ( member4699826688122452638ist_fm @ ( shd_Pr3211216682057661985ist_fm @ S3 ) @ S4 )
       => ( member197948898507729188ist_fm @ S3 @ ( stream1217541351357117421ist_fm @ S4 ) ) ) ) ).

% in_streams
thf(fact_856_in__streams,axiom,
    ! [S3: stream8372878641218411373at_nat,S4: set_Pr8693737435421807431at_nat] :
      ( ( member1992146023263842070at_nat @ ( stl_Pr5010465336888994071at_nat @ S3 ) @ ( stream5885727000947712735at_nat @ S4 ) )
     => ( ( member8206827879206165904at_nat @ ( shd_Pr8412153233960533267at_nat @ S3 ) @ S4 )
       => ( member1992146023263842070at_nat @ S3 @ ( stream5885727000947712735at_nat @ S4 ) ) ) ) ).

% in_streams
thf(fact_857_in__streams,axiom,
    ! [S3: stream2709947120125613254m_rule,S4: set_Pr1822751329126368876m_rule] :
      ( ( member5228318729319728861m_rule @ ( stl_Pr12655793849929990m_rule @ S3 ) @ ( stream3418479298939905342m_rule @ S4 ) )
     => ( ( member7231649785386036813m_rule @ ( shd_Pr4562317740776619530m_rule @ S3 ) @ S4 )
       => ( member5228318729319728861m_rule @ S3 @ ( stream3418479298939905342m_rule @ S4 ) ) ) ) ).

% in_streams
thf(fact_858_in__streams,axiom,
    ! [S3: stream_rule,S4: set_rule] :
      ( ( member_stream_rule @ ( stl_rule @ S3 ) @ ( streams_rule @ S4 ) )
     => ( ( member_rule @ ( shd_rule @ S3 ) @ S4 )
       => ( member_stream_rule @ S3 @ ( streams_rule @ S4 ) ) ) ) ).

% in_streams
thf(fact_859_sdrop__while_Osimps,axiom,
    ( sdrop_8181906248673349714m_rule
    = ( ^ [P: produc340336539035504054m_rule > $o,S: stream2709947120125613254m_rule] : ( if_str2295835923641592704m_rule @ ( P @ ( shd_Pr4562317740776619530m_rule @ S ) ) @ ( sdrop_8181906248673349714m_rule @ P @ ( stl_Pr12655793849929990m_rule @ S ) ) @ S ) ) ) ).

% sdrop_while.simps
thf(fact_860_sdrop__while_Osimps,axiom,
    ( sdrop_while_rule
    = ( ^ [P: rule > $o,S: stream_rule] : ( if_stream_rule @ ( P @ ( shd_rule @ S ) ) @ ( sdrop_while_rule @ P @ ( stl_rule @ S ) ) @ S ) ) ) ).

% sdrop_while.simps
thf(fact_861_alw__holds2,axiom,
    ! [P2: produc340336539035504054m_rule > $o,Ss: stream2709947120125613254m_rule] :
      ( ( linear4957294957385550906m_rule @ ( linear7925859179493631106m_rule @ P2 ) @ Ss )
      = ( ( P2 @ ( shd_Pr4562317740776619530m_rule @ Ss ) )
        & ( linear4957294957385550906m_rule @ ( linear7925859179493631106m_rule @ P2 ) @ ( stl_Pr12655793849929990m_rule @ Ss ) ) ) ) ).

% alw_holds2
thf(fact_862_alw__holds2,axiom,
    ! [P2: rule > $o,Ss: stream_rule] :
      ( ( linear1087404401031714582w_rule @ ( linear7354821883049305950s_rule @ P2 ) @ Ss )
      = ( ( P2 @ ( shd_rule @ Ss ) )
        & ( linear1087404401031714582w_rule @ ( linear7354821883049305950s_rule @ P2 ) @ ( stl_rule @ Ss ) ) ) ) ).

% alw_holds2
thf(fact_863_RuleSystem__Defs_Oepath_Ocases,axiom,
    ! [Eff4: nat > produc6018962875968178549ist_fm > fset_P8989946509869081563ist_fm > $o,Rules2: stream_nat,A: stream1752417115841309700fm_nat] :
      ( ( abstra3661368698739669536ist_fm @ Eff4 @ Rules2 @ A )
     => ~ ( ( member_nat @ ( produc2092181227289234752fm_nat @ ( shd_Pr4515903069162513224fm_nat @ A ) ) @ ( sset_nat @ Rules2 ) )
         => ! [Sl2: fset_P8989946509869081563ist_fm] :
              ( ( fmembe3754813877001230652ist_fm @ ( produc111973938668391682fm_nat @ ( shd_Pr4515903069162513224fm_nat @ ( stl_Pr1340940657110954564fm_nat @ A ) ) ) @ Sl2 )
             => ( ( Eff4 @ ( produc2092181227289234752fm_nat @ ( shd_Pr4515903069162513224fm_nat @ A ) ) @ ( produc111973938668391682fm_nat @ ( shd_Pr4515903069162513224fm_nat @ A ) ) @ Sl2 )
               => ~ ( abstra3661368698739669536ist_fm @ Eff4 @ Rules2 @ ( stl_Pr1340940657110954564fm_nat @ A ) ) ) ) ) ) ).

% RuleSystem_Defs.epath.cases
thf(fact_864_RuleSystem__Defs_Oepath_Ocases,axiom,
    ! [Eff4: produc6018962875968178549ist_fm > produc6018962875968178549ist_fm > fset_P8989946509869081563ist_fm > $o,Rules2: stream4408948924543953275ist_fm,A: stream1747862497754140525ist_fm] :
      ( ( abstra6295994116336782083ist_fm @ Eff4 @ Rules2 @ A )
     => ~ ( ( member4699826688122452638ist_fm @ ( produc6364584182369725445ist_fm @ ( shd_Pr4098038590278241811ist_fm @ A ) ) @ ( sset_P5379989128061332361ist_fm @ Rules2 ) )
         => ! [Sl2: fset_P8989946509869081563ist_fm] :
              ( ( fmembe3754813877001230652ist_fm @ ( produc8141473855448433859ist_fm @ ( shd_Pr4098038590278241811ist_fm @ ( stl_Pr1377318027900677655ist_fm @ A ) ) ) @ Sl2 )
             => ( ( Eff4 @ ( produc6364584182369725445ist_fm @ ( shd_Pr4098038590278241811ist_fm @ A ) ) @ ( produc8141473855448433859ist_fm @ ( shd_Pr4098038590278241811ist_fm @ A ) ) @ Sl2 )
               => ~ ( abstra6295994116336782083ist_fm @ Eff4 @ Rules2 @ ( stl_Pr1377318027900677655ist_fm @ A ) ) ) ) ) ) ).

% RuleSystem_Defs.epath.cases
thf(fact_865_RuleSystem__Defs_Oepath_Ocases,axiom,
    ! [Eff4: produc859450856879609959at_nat > produc6018962875968178549ist_fm > fset_P8989946509869081563ist_fm > $o,Rules2: stream8372878641218411373at_nat,A: stream105034537780515935at_nat] :
      ( ( abstra409391092022950033ist_fm @ Eff4 @ Rules2 @ A )
     => ~ ( ( member8206827879206165904at_nat @ ( produc8336112126034639607at_nat @ ( shd_Pr7514066009280765957at_nat @ A ) ) @ ( sset_P9164763103296375419at_nat @ Rules2 ) )
         => ! [Sl2: fset_P8989946509869081563ist_fm] :
              ( ( fmembe3754813877001230652ist_fm @ ( produc6827676535622970293at_nat @ ( shd_Pr7514066009280765957at_nat @ ( stl_Pr7223821233291300873at_nat @ A ) ) ) @ Sl2 )
             => ( ( Eff4 @ ( produc8336112126034639607at_nat @ ( shd_Pr7514066009280765957at_nat @ A ) ) @ ( produc6827676535622970293at_nat @ ( shd_Pr7514066009280765957at_nat @ A ) ) @ Sl2 )
               => ~ ( abstra409391092022950033ist_fm @ Eff4 @ Rules2 @ ( stl_Pr7223821233291300873at_nat @ A ) ) ) ) ) ) ).

% RuleSystem_Defs.epath.cases
thf(fact_866_RuleSystem__Defs_Oepath_Ocases,axiom,
    ! [Eff4: nat > abstra8520858721168826610m_rule > fset_A8849196716589201442m_rule > $o,Rules2: stream_nat,A: stream2640339575054594473le_nat] :
      ( ( abstra2672558889254879751m_rule @ Eff4 @ Rules2 @ A )
     => ~ ( ( member_nat @ ( produc6965321437502945929le_nat @ ( shd_Pr4214957810428288335le_nat @ A ) ) @ ( sset_nat @ Rules2 ) )
         => ! [Sl2: fset_A8849196716589201442m_rule] :
              ( ( fmembe1099044791951439339m_rule @ ( produc1996884173541332295le_nat @ ( shd_Pr4214957810428288335le_nat @ ( stl_Pr8328254828799719763le_nat @ A ) ) ) @ Sl2 )
             => ( ( Eff4 @ ( produc6965321437502945929le_nat @ ( shd_Pr4214957810428288335le_nat @ A ) ) @ ( produc1996884173541332295le_nat @ ( shd_Pr4214957810428288335le_nat @ A ) ) @ Sl2 )
               => ~ ( abstra2672558889254879751m_rule @ Eff4 @ Rules2 @ ( stl_Pr8328254828799719763le_nat @ A ) ) ) ) ) ) ).

% RuleSystem_Defs.epath.cases
thf(fact_867_RuleSystem__Defs_Oepath_Ocases,axiom,
    ! [Eff4: produc6018962875968178549ist_fm > abstra8520858721168826610m_rule > fset_A8849196716589201442m_rule > $o,Rules2: stream4408948924543953275ist_fm,A: stream2118054573033641736ist_fm] :
      ( ( abstra8533012240466880740m_rule @ Eff4 @ Rules2 @ A )
     => ~ ( ( member4699826688122452638ist_fm @ ( produc1196760618833626364ist_fm @ ( shd_Pr1348735749115486668ist_fm @ A ) ) @ ( sset_P5379989128061332361ist_fm @ Rules2 ) )
         => ! [Sl2: fset_A8849196716589201442m_rule] :
              ( ( fmembe1099044791951439339m_rule @ ( produc7096817939255439806ist_fm @ ( shd_Pr1348735749115486668ist_fm @ ( stl_Pr1735168262451467976ist_fm @ A ) ) ) @ Sl2 )
             => ( ( Eff4 @ ( produc1196760618833626364ist_fm @ ( shd_Pr1348735749115486668ist_fm @ A ) ) @ ( produc7096817939255439806ist_fm @ ( shd_Pr1348735749115486668ist_fm @ A ) ) @ Sl2 )
               => ~ ( abstra8533012240466880740m_rule @ Eff4 @ Rules2 @ ( stl_Pr1735168262451467976ist_fm @ A ) ) ) ) ) ) ).

% RuleSystem_Defs.epath.cases
thf(fact_868_RuleSystem__Defs_Oepath_Ocases,axiom,
    ! [Eff4: produc859450856879609959at_nat > abstra8520858721168826610m_rule > fset_A8849196716589201442m_rule > $o,Rules2: stream8372878641218411373at_nat,A: stream7661288321842501626at_nat] :
      ( ( abstra7033530765613221078m_rule @ Eff4 @ Rules2 @ A )
     => ~ ( ( member8206827879206165904at_nat @ ( produc740002071024140270at_nat @ ( shd_Pr7666802700379473854at_nat @ A ) ) @ ( sset_P9164763103296375419at_nat @ Rules2 ) )
         => ! [Sl2: fset_A8849196716589201442m_rule] :
              ( ( fmembe1099044791951439339m_rule @ ( produc1738731144906921136at_nat @ ( shd_Pr7666802700379473854at_nat @ ( stl_Pr6382724064935898298at_nat @ A ) ) ) @ Sl2 )
             => ( ( Eff4 @ ( produc740002071024140270at_nat @ ( shd_Pr7666802700379473854at_nat @ A ) ) @ ( produc1738731144906921136at_nat @ ( shd_Pr7666802700379473854at_nat @ A ) ) @ Sl2 )
               => ~ ( abstra7033530765613221078m_rule @ Eff4 @ Rules2 @ ( stl_Pr6382724064935898298at_nat @ A ) ) ) ) ) ) ).

% RuleSystem_Defs.epath.cases
thf(fact_869_RuleSystem__Defs_Oepath_Ocases,axiom,
    ! [Eff4: rule > abstra8520858721168826610m_rule > fset_A8849196716589201442m_rule > $o,Rules2: stream_rule,A: stream3393403210778678635e_rule] :
      ( ( abstra7067474524808760265m_rule @ Eff4 @ Rules2 @ A )
     => ~ ( ( member_rule @ ( produc9166542125499771467e_rule @ ( shd_Pr8663275782010802449e_rule @ A ) ) @ ( sset_rule @ Rules2 ) )
         => ! [Sl2: fset_A8849196716589201442m_rule] :
              ( ( fmembe1099044791951439339m_rule @ ( produc41289011304007433e_rule @ ( shd_Pr8663275782010802449e_rule @ ( stl_Pr654857084227610901e_rule @ A ) ) ) @ Sl2 )
             => ( ( Eff4 @ ( produc9166542125499771467e_rule @ ( shd_Pr8663275782010802449e_rule @ A ) ) @ ( produc41289011304007433e_rule @ ( shd_Pr8663275782010802449e_rule @ A ) ) @ Sl2 )
               => ~ ( abstra7067474524808760265m_rule @ Eff4 @ Rules2 @ ( stl_Pr654857084227610901e_rule @ A ) ) ) ) ) ) ).

% RuleSystem_Defs.epath.cases
thf(fact_870_RuleSystem__Defs_Oepath_Ocases,axiom,
    ! [Eff4: list_fm > list_tm > fset_list_tm > $o,Rules2: stream_list_fm,A: stream4408948924543953275ist_fm] :
      ( ( abstra302294732650053493ist_tm @ Eff4 @ Rules2 @ A )
     => ~ ( ( member_list_fm @ ( produc3242453064496107283ist_fm @ ( shd_Pr3211216682057661985ist_fm @ A ) ) @ ( sset_list_fm @ Rules2 ) )
         => ! [Sl2: fset_list_tm] :
              ( ( fmember_list_tm @ ( produc4874987732838623697ist_fm @ ( shd_Pr3211216682057661985ist_fm @ ( stl_Pr1506262294867171877ist_fm @ A ) ) ) @ Sl2 )
             => ( ( Eff4 @ ( produc3242453064496107283ist_fm @ ( shd_Pr3211216682057661985ist_fm @ A ) ) @ ( produc4874987732838623697ist_fm @ ( shd_Pr3211216682057661985ist_fm @ A ) ) @ Sl2 )
               => ~ ( abstra302294732650053493ist_tm @ Eff4 @ Rules2 @ ( stl_Pr1506262294867171877ist_fm @ A ) ) ) ) ) ) ).

% RuleSystem_Defs.epath.cases
thf(fact_871_RuleSystem__Defs_Oepath_Ocases,axiom,
    ! [Eff4: rule > produc6018962875968178549ist_fm > fset_P8989946509869081563ist_fm > $o,Rules2: stream_rule,A: stream2709947120125613254m_rule] :
      ( ( abstra6789711989322986974ist_fm @ Eff4 @ Rules2 @ A )
     => ~ ( ( member_rule @ ( produc6514711208036230658m_rule @ ( shd_Pr4562317740776619530m_rule @ A ) ) @ ( sset_rule @ Rules2 ) )
         => ! [Sl2: fset_P8989946509869081563ist_fm] :
              ( ( fmembe3754813877001230652ist_fm @ ( produc993771546023551940m_rule @ ( shd_Pr4562317740776619530m_rule @ ( stl_Pr12655793849929990m_rule @ A ) ) ) @ Sl2 )
             => ( ( Eff4 @ ( produc6514711208036230658m_rule @ ( shd_Pr4562317740776619530m_rule @ A ) ) @ ( produc993771546023551940m_rule @ ( shd_Pr4562317740776619530m_rule @ A ) ) @ Sl2 )
               => ~ ( abstra6789711989322986974ist_fm @ Eff4 @ Rules2 @ ( stl_Pr12655793849929990m_rule @ A ) ) ) ) ) ) ).

% RuleSystem_Defs.epath.cases
thf(fact_872_RuleSystem__Defs_Oepath_Osimps,axiom,
    ( abstra3661368698739669536ist_fm
    = ( ^ [Eff: nat > produc6018962875968178549ist_fm > fset_P8989946509869081563ist_fm > $o,Rules: stream_nat,A4: stream1752417115841309700fm_nat] :
        ? [Steps2: stream1752417115841309700fm_nat,Sl3: fset_P8989946509869081563ist_fm] :
          ( ( A4 = Steps2 )
          & ( member_nat @ ( produc2092181227289234752fm_nat @ ( shd_Pr4515903069162513224fm_nat @ Steps2 ) ) @ ( sset_nat @ Rules ) )
          & ( fmembe3754813877001230652ist_fm @ ( produc111973938668391682fm_nat @ ( shd_Pr4515903069162513224fm_nat @ ( stl_Pr1340940657110954564fm_nat @ Steps2 ) ) ) @ Sl3 )
          & ( Eff @ ( produc2092181227289234752fm_nat @ ( shd_Pr4515903069162513224fm_nat @ Steps2 ) ) @ ( produc111973938668391682fm_nat @ ( shd_Pr4515903069162513224fm_nat @ Steps2 ) ) @ Sl3 )
          & ( abstra3661368698739669536ist_fm @ Eff @ Rules @ ( stl_Pr1340940657110954564fm_nat @ Steps2 ) ) ) ) ) ).

% RuleSystem_Defs.epath.simps
thf(fact_873_RuleSystem__Defs_Oepath_Osimps,axiom,
    ( abstra6295994116336782083ist_fm
    = ( ^ [Eff: produc6018962875968178549ist_fm > produc6018962875968178549ist_fm > fset_P8989946509869081563ist_fm > $o,Rules: stream4408948924543953275ist_fm,A4: stream1747862497754140525ist_fm] :
        ? [Steps2: stream1747862497754140525ist_fm,Sl3: fset_P8989946509869081563ist_fm] :
          ( ( A4 = Steps2 )
          & ( member4699826688122452638ist_fm @ ( produc6364584182369725445ist_fm @ ( shd_Pr4098038590278241811ist_fm @ Steps2 ) ) @ ( sset_P5379989128061332361ist_fm @ Rules ) )
          & ( fmembe3754813877001230652ist_fm @ ( produc8141473855448433859ist_fm @ ( shd_Pr4098038590278241811ist_fm @ ( stl_Pr1377318027900677655ist_fm @ Steps2 ) ) ) @ Sl3 )
          & ( Eff @ ( produc6364584182369725445ist_fm @ ( shd_Pr4098038590278241811ist_fm @ Steps2 ) ) @ ( produc8141473855448433859ist_fm @ ( shd_Pr4098038590278241811ist_fm @ Steps2 ) ) @ Sl3 )
          & ( abstra6295994116336782083ist_fm @ Eff @ Rules @ ( stl_Pr1377318027900677655ist_fm @ Steps2 ) ) ) ) ) ).

% RuleSystem_Defs.epath.simps
thf(fact_874_RuleSystem__Defs_Oepath_Osimps,axiom,
    ( abstra409391092022950033ist_fm
    = ( ^ [Eff: produc859450856879609959at_nat > produc6018962875968178549ist_fm > fset_P8989946509869081563ist_fm > $o,Rules: stream8372878641218411373at_nat,A4: stream105034537780515935at_nat] :
        ? [Steps2: stream105034537780515935at_nat,Sl3: fset_P8989946509869081563ist_fm] :
          ( ( A4 = Steps2 )
          & ( member8206827879206165904at_nat @ ( produc8336112126034639607at_nat @ ( shd_Pr7514066009280765957at_nat @ Steps2 ) ) @ ( sset_P9164763103296375419at_nat @ Rules ) )
          & ( fmembe3754813877001230652ist_fm @ ( produc6827676535622970293at_nat @ ( shd_Pr7514066009280765957at_nat @ ( stl_Pr7223821233291300873at_nat @ Steps2 ) ) ) @ Sl3 )
          & ( Eff @ ( produc8336112126034639607at_nat @ ( shd_Pr7514066009280765957at_nat @ Steps2 ) ) @ ( produc6827676535622970293at_nat @ ( shd_Pr7514066009280765957at_nat @ Steps2 ) ) @ Sl3 )
          & ( abstra409391092022950033ist_fm @ Eff @ Rules @ ( stl_Pr7223821233291300873at_nat @ Steps2 ) ) ) ) ) ).

% RuleSystem_Defs.epath.simps
thf(fact_875_RuleSystem__Defs_Oepath_Osimps,axiom,
    ( abstra2672558889254879751m_rule
    = ( ^ [Eff: nat > abstra8520858721168826610m_rule > fset_A8849196716589201442m_rule > $o,Rules: stream_nat,A4: stream2640339575054594473le_nat] :
        ? [Steps2: stream2640339575054594473le_nat,Sl3: fset_A8849196716589201442m_rule] :
          ( ( A4 = Steps2 )
          & ( member_nat @ ( produc6965321437502945929le_nat @ ( shd_Pr4214957810428288335le_nat @ Steps2 ) ) @ ( sset_nat @ Rules ) )
          & ( fmembe1099044791951439339m_rule @ ( produc1996884173541332295le_nat @ ( shd_Pr4214957810428288335le_nat @ ( stl_Pr8328254828799719763le_nat @ Steps2 ) ) ) @ Sl3 )
          & ( Eff @ ( produc6965321437502945929le_nat @ ( shd_Pr4214957810428288335le_nat @ Steps2 ) ) @ ( produc1996884173541332295le_nat @ ( shd_Pr4214957810428288335le_nat @ Steps2 ) ) @ Sl3 )
          & ( abstra2672558889254879751m_rule @ Eff @ Rules @ ( stl_Pr8328254828799719763le_nat @ Steps2 ) ) ) ) ) ).

% RuleSystem_Defs.epath.simps
thf(fact_876_RuleSystem__Defs_Oepath_Osimps,axiom,
    ( abstra8533012240466880740m_rule
    = ( ^ [Eff: produc6018962875968178549ist_fm > abstra8520858721168826610m_rule > fset_A8849196716589201442m_rule > $o,Rules: stream4408948924543953275ist_fm,A4: stream2118054573033641736ist_fm] :
        ? [Steps2: stream2118054573033641736ist_fm,Sl3: fset_A8849196716589201442m_rule] :
          ( ( A4 = Steps2 )
          & ( member4699826688122452638ist_fm @ ( produc1196760618833626364ist_fm @ ( shd_Pr1348735749115486668ist_fm @ Steps2 ) ) @ ( sset_P5379989128061332361ist_fm @ Rules ) )
          & ( fmembe1099044791951439339m_rule @ ( produc7096817939255439806ist_fm @ ( shd_Pr1348735749115486668ist_fm @ ( stl_Pr1735168262451467976ist_fm @ Steps2 ) ) ) @ Sl3 )
          & ( Eff @ ( produc1196760618833626364ist_fm @ ( shd_Pr1348735749115486668ist_fm @ Steps2 ) ) @ ( produc7096817939255439806ist_fm @ ( shd_Pr1348735749115486668ist_fm @ Steps2 ) ) @ Sl3 )
          & ( abstra8533012240466880740m_rule @ Eff @ Rules @ ( stl_Pr1735168262451467976ist_fm @ Steps2 ) ) ) ) ) ).

% RuleSystem_Defs.epath.simps
thf(fact_877_RuleSystem__Defs_Oepath_Osimps,axiom,
    ( abstra7033530765613221078m_rule
    = ( ^ [Eff: produc859450856879609959at_nat > abstra8520858721168826610m_rule > fset_A8849196716589201442m_rule > $o,Rules: stream8372878641218411373at_nat,A4: stream7661288321842501626at_nat] :
        ? [Steps2: stream7661288321842501626at_nat,Sl3: fset_A8849196716589201442m_rule] :
          ( ( A4 = Steps2 )
          & ( member8206827879206165904at_nat @ ( produc740002071024140270at_nat @ ( shd_Pr7666802700379473854at_nat @ Steps2 ) ) @ ( sset_P9164763103296375419at_nat @ Rules ) )
          & ( fmembe1099044791951439339m_rule @ ( produc1738731144906921136at_nat @ ( shd_Pr7666802700379473854at_nat @ ( stl_Pr6382724064935898298at_nat @ Steps2 ) ) ) @ Sl3 )
          & ( Eff @ ( produc740002071024140270at_nat @ ( shd_Pr7666802700379473854at_nat @ Steps2 ) ) @ ( produc1738731144906921136at_nat @ ( shd_Pr7666802700379473854at_nat @ Steps2 ) ) @ Sl3 )
          & ( abstra7033530765613221078m_rule @ Eff @ Rules @ ( stl_Pr6382724064935898298at_nat @ Steps2 ) ) ) ) ) ).

% RuleSystem_Defs.epath.simps
thf(fact_878_RuleSystem__Defs_Oepath_Osimps,axiom,
    ( abstra7067474524808760265m_rule
    = ( ^ [Eff: rule > abstra8520858721168826610m_rule > fset_A8849196716589201442m_rule > $o,Rules: stream_rule,A4: stream3393403210778678635e_rule] :
        ? [Steps2: stream3393403210778678635e_rule,Sl3: fset_A8849196716589201442m_rule] :
          ( ( A4 = Steps2 )
          & ( member_rule @ ( produc9166542125499771467e_rule @ ( shd_Pr8663275782010802449e_rule @ Steps2 ) ) @ ( sset_rule @ Rules ) )
          & ( fmembe1099044791951439339m_rule @ ( produc41289011304007433e_rule @ ( shd_Pr8663275782010802449e_rule @ ( stl_Pr654857084227610901e_rule @ Steps2 ) ) ) @ Sl3 )
          & ( Eff @ ( produc9166542125499771467e_rule @ ( shd_Pr8663275782010802449e_rule @ Steps2 ) ) @ ( produc41289011304007433e_rule @ ( shd_Pr8663275782010802449e_rule @ Steps2 ) ) @ Sl3 )
          & ( abstra7067474524808760265m_rule @ Eff @ Rules @ ( stl_Pr654857084227610901e_rule @ Steps2 ) ) ) ) ) ).

% RuleSystem_Defs.epath.simps
thf(fact_879_RuleSystem__Defs_Oepath_Osimps,axiom,
    ( abstra302294732650053493ist_tm
    = ( ^ [Eff: list_fm > list_tm > fset_list_tm > $o,Rules: stream_list_fm,A4: stream4408948924543953275ist_fm] :
        ? [Steps2: stream4408948924543953275ist_fm,Sl3: fset_list_tm] :
          ( ( A4 = Steps2 )
          & ( member_list_fm @ ( produc3242453064496107283ist_fm @ ( shd_Pr3211216682057661985ist_fm @ Steps2 ) ) @ ( sset_list_fm @ Rules ) )
          & ( fmember_list_tm @ ( produc4874987732838623697ist_fm @ ( shd_Pr3211216682057661985ist_fm @ ( stl_Pr1506262294867171877ist_fm @ Steps2 ) ) ) @ Sl3 )
          & ( Eff @ ( produc3242453064496107283ist_fm @ ( shd_Pr3211216682057661985ist_fm @ Steps2 ) ) @ ( produc4874987732838623697ist_fm @ ( shd_Pr3211216682057661985ist_fm @ Steps2 ) ) @ Sl3 )
          & ( abstra302294732650053493ist_tm @ Eff @ Rules @ ( stl_Pr1506262294867171877ist_fm @ Steps2 ) ) ) ) ) ).

% RuleSystem_Defs.epath.simps
thf(fact_880_RuleSystem__Defs_Oepath_Osimps,axiom,
    ( abstra6789711989322986974ist_fm
    = ( ^ [Eff: rule > produc6018962875968178549ist_fm > fset_P8989946509869081563ist_fm > $o,Rules: stream_rule,A4: stream2709947120125613254m_rule] :
        ? [Steps2: stream2709947120125613254m_rule,Sl3: fset_P8989946509869081563ist_fm] :
          ( ( A4 = Steps2 )
          & ( member_rule @ ( produc6514711208036230658m_rule @ ( shd_Pr4562317740776619530m_rule @ Steps2 ) ) @ ( sset_rule @ Rules ) )
          & ( fmembe3754813877001230652ist_fm @ ( produc993771546023551940m_rule @ ( shd_Pr4562317740776619530m_rule @ ( stl_Pr12655793849929990m_rule @ Steps2 ) ) ) @ Sl3 )
          & ( Eff @ ( produc6514711208036230658m_rule @ ( shd_Pr4562317740776619530m_rule @ Steps2 ) ) @ ( produc993771546023551940m_rule @ ( shd_Pr4562317740776619530m_rule @ Steps2 ) ) @ Sl3 )
          & ( abstra6789711989322986974ist_fm @ Eff @ Rules @ ( stl_Pr12655793849929990m_rule @ Steps2 ) ) ) ) ) ).

% RuleSystem_Defs.epath.simps
thf(fact_881_RuleSystem__Defs_Oepath_Ointros,axiom,
    ! [Steps: stream1752417115841309700fm_nat,Rules2: stream_nat,Sl: fset_P8989946509869081563ist_fm,Eff4: nat > produc6018962875968178549ist_fm > fset_P8989946509869081563ist_fm > $o] :
      ( ( member_nat @ ( produc2092181227289234752fm_nat @ ( shd_Pr4515903069162513224fm_nat @ Steps ) ) @ ( sset_nat @ Rules2 ) )
     => ( ( fmembe3754813877001230652ist_fm @ ( produc111973938668391682fm_nat @ ( shd_Pr4515903069162513224fm_nat @ ( stl_Pr1340940657110954564fm_nat @ Steps ) ) ) @ Sl )
       => ( ( Eff4 @ ( produc2092181227289234752fm_nat @ ( shd_Pr4515903069162513224fm_nat @ Steps ) ) @ ( produc111973938668391682fm_nat @ ( shd_Pr4515903069162513224fm_nat @ Steps ) ) @ Sl )
         => ( ( abstra3661368698739669536ist_fm @ Eff4 @ Rules2 @ ( stl_Pr1340940657110954564fm_nat @ Steps ) )
           => ( abstra3661368698739669536ist_fm @ Eff4 @ Rules2 @ Steps ) ) ) ) ) ).

% RuleSystem_Defs.epath.intros
thf(fact_882_RuleSystem__Defs_Oepath_Ointros,axiom,
    ! [Steps: stream1747862497754140525ist_fm,Rules2: stream4408948924543953275ist_fm,Sl: fset_P8989946509869081563ist_fm,Eff4: produc6018962875968178549ist_fm > produc6018962875968178549ist_fm > fset_P8989946509869081563ist_fm > $o] :
      ( ( member4699826688122452638ist_fm @ ( produc6364584182369725445ist_fm @ ( shd_Pr4098038590278241811ist_fm @ Steps ) ) @ ( sset_P5379989128061332361ist_fm @ Rules2 ) )
     => ( ( fmembe3754813877001230652ist_fm @ ( produc8141473855448433859ist_fm @ ( shd_Pr4098038590278241811ist_fm @ ( stl_Pr1377318027900677655ist_fm @ Steps ) ) ) @ Sl )
       => ( ( Eff4 @ ( produc6364584182369725445ist_fm @ ( shd_Pr4098038590278241811ist_fm @ Steps ) ) @ ( produc8141473855448433859ist_fm @ ( shd_Pr4098038590278241811ist_fm @ Steps ) ) @ Sl )
         => ( ( abstra6295994116336782083ist_fm @ Eff4 @ Rules2 @ ( stl_Pr1377318027900677655ist_fm @ Steps ) )
           => ( abstra6295994116336782083ist_fm @ Eff4 @ Rules2 @ Steps ) ) ) ) ) ).

% RuleSystem_Defs.epath.intros
thf(fact_883_RuleSystem__Defs_Oepath_Ointros,axiom,
    ! [Steps: stream105034537780515935at_nat,Rules2: stream8372878641218411373at_nat,Sl: fset_P8989946509869081563ist_fm,Eff4: produc859450856879609959at_nat > produc6018962875968178549ist_fm > fset_P8989946509869081563ist_fm > $o] :
      ( ( member8206827879206165904at_nat @ ( produc8336112126034639607at_nat @ ( shd_Pr7514066009280765957at_nat @ Steps ) ) @ ( sset_P9164763103296375419at_nat @ Rules2 ) )
     => ( ( fmembe3754813877001230652ist_fm @ ( produc6827676535622970293at_nat @ ( shd_Pr7514066009280765957at_nat @ ( stl_Pr7223821233291300873at_nat @ Steps ) ) ) @ Sl )
       => ( ( Eff4 @ ( produc8336112126034639607at_nat @ ( shd_Pr7514066009280765957at_nat @ Steps ) ) @ ( produc6827676535622970293at_nat @ ( shd_Pr7514066009280765957at_nat @ Steps ) ) @ Sl )
         => ( ( abstra409391092022950033ist_fm @ Eff4 @ Rules2 @ ( stl_Pr7223821233291300873at_nat @ Steps ) )
           => ( abstra409391092022950033ist_fm @ Eff4 @ Rules2 @ Steps ) ) ) ) ) ).

% RuleSystem_Defs.epath.intros
thf(fact_884_RuleSystem__Defs_Oepath_Ointros,axiom,
    ! [Steps: stream2640339575054594473le_nat,Rules2: stream_nat,Sl: fset_A8849196716589201442m_rule,Eff4: nat > abstra8520858721168826610m_rule > fset_A8849196716589201442m_rule > $o] :
      ( ( member_nat @ ( produc6965321437502945929le_nat @ ( shd_Pr4214957810428288335le_nat @ Steps ) ) @ ( sset_nat @ Rules2 ) )
     => ( ( fmembe1099044791951439339m_rule @ ( produc1996884173541332295le_nat @ ( shd_Pr4214957810428288335le_nat @ ( stl_Pr8328254828799719763le_nat @ Steps ) ) ) @ Sl )
       => ( ( Eff4 @ ( produc6965321437502945929le_nat @ ( shd_Pr4214957810428288335le_nat @ Steps ) ) @ ( produc1996884173541332295le_nat @ ( shd_Pr4214957810428288335le_nat @ Steps ) ) @ Sl )
         => ( ( abstra2672558889254879751m_rule @ Eff4 @ Rules2 @ ( stl_Pr8328254828799719763le_nat @ Steps ) )
           => ( abstra2672558889254879751m_rule @ Eff4 @ Rules2 @ Steps ) ) ) ) ) ).

% RuleSystem_Defs.epath.intros
thf(fact_885_RuleSystem__Defs_Oepath_Ointros,axiom,
    ! [Steps: stream2118054573033641736ist_fm,Rules2: stream4408948924543953275ist_fm,Sl: fset_A8849196716589201442m_rule,Eff4: produc6018962875968178549ist_fm > abstra8520858721168826610m_rule > fset_A8849196716589201442m_rule > $o] :
      ( ( member4699826688122452638ist_fm @ ( produc1196760618833626364ist_fm @ ( shd_Pr1348735749115486668ist_fm @ Steps ) ) @ ( sset_P5379989128061332361ist_fm @ Rules2 ) )
     => ( ( fmembe1099044791951439339m_rule @ ( produc7096817939255439806ist_fm @ ( shd_Pr1348735749115486668ist_fm @ ( stl_Pr1735168262451467976ist_fm @ Steps ) ) ) @ Sl )
       => ( ( Eff4 @ ( produc1196760618833626364ist_fm @ ( shd_Pr1348735749115486668ist_fm @ Steps ) ) @ ( produc7096817939255439806ist_fm @ ( shd_Pr1348735749115486668ist_fm @ Steps ) ) @ Sl )
         => ( ( abstra8533012240466880740m_rule @ Eff4 @ Rules2 @ ( stl_Pr1735168262451467976ist_fm @ Steps ) )
           => ( abstra8533012240466880740m_rule @ Eff4 @ Rules2 @ Steps ) ) ) ) ) ).

% RuleSystem_Defs.epath.intros
thf(fact_886_RuleSystem__Defs_Oepath_Ointros,axiom,
    ! [Steps: stream7661288321842501626at_nat,Rules2: stream8372878641218411373at_nat,Sl: fset_A8849196716589201442m_rule,Eff4: produc859450856879609959at_nat > abstra8520858721168826610m_rule > fset_A8849196716589201442m_rule > $o] :
      ( ( member8206827879206165904at_nat @ ( produc740002071024140270at_nat @ ( shd_Pr7666802700379473854at_nat @ Steps ) ) @ ( sset_P9164763103296375419at_nat @ Rules2 ) )
     => ( ( fmembe1099044791951439339m_rule @ ( produc1738731144906921136at_nat @ ( shd_Pr7666802700379473854at_nat @ ( stl_Pr6382724064935898298at_nat @ Steps ) ) ) @ Sl )
       => ( ( Eff4 @ ( produc740002071024140270at_nat @ ( shd_Pr7666802700379473854at_nat @ Steps ) ) @ ( produc1738731144906921136at_nat @ ( shd_Pr7666802700379473854at_nat @ Steps ) ) @ Sl )
         => ( ( abstra7033530765613221078m_rule @ Eff4 @ Rules2 @ ( stl_Pr6382724064935898298at_nat @ Steps ) )
           => ( abstra7033530765613221078m_rule @ Eff4 @ Rules2 @ Steps ) ) ) ) ) ).

% RuleSystem_Defs.epath.intros
thf(fact_887_RuleSystem__Defs_Oepath_Ointros,axiom,
    ! [Steps: stream3393403210778678635e_rule,Rules2: stream_rule,Sl: fset_A8849196716589201442m_rule,Eff4: rule > abstra8520858721168826610m_rule > fset_A8849196716589201442m_rule > $o] :
      ( ( member_rule @ ( produc9166542125499771467e_rule @ ( shd_Pr8663275782010802449e_rule @ Steps ) ) @ ( sset_rule @ Rules2 ) )
     => ( ( fmembe1099044791951439339m_rule @ ( produc41289011304007433e_rule @ ( shd_Pr8663275782010802449e_rule @ ( stl_Pr654857084227610901e_rule @ Steps ) ) ) @ Sl )
       => ( ( Eff4 @ ( produc9166542125499771467e_rule @ ( shd_Pr8663275782010802449e_rule @ Steps ) ) @ ( produc41289011304007433e_rule @ ( shd_Pr8663275782010802449e_rule @ Steps ) ) @ Sl )
         => ( ( abstra7067474524808760265m_rule @ Eff4 @ Rules2 @ ( stl_Pr654857084227610901e_rule @ Steps ) )
           => ( abstra7067474524808760265m_rule @ Eff4 @ Rules2 @ Steps ) ) ) ) ) ).

% RuleSystem_Defs.epath.intros
thf(fact_888_RuleSystem__Defs_Oepath_Ointros,axiom,
    ! [Steps: stream4408948924543953275ist_fm,Rules2: stream_list_fm,Sl: fset_list_tm,Eff4: list_fm > list_tm > fset_list_tm > $o] :
      ( ( member_list_fm @ ( produc3242453064496107283ist_fm @ ( shd_Pr3211216682057661985ist_fm @ Steps ) ) @ ( sset_list_fm @ Rules2 ) )
     => ( ( fmember_list_tm @ ( produc4874987732838623697ist_fm @ ( shd_Pr3211216682057661985ist_fm @ ( stl_Pr1506262294867171877ist_fm @ Steps ) ) ) @ Sl )
       => ( ( Eff4 @ ( produc3242453064496107283ist_fm @ ( shd_Pr3211216682057661985ist_fm @ Steps ) ) @ ( produc4874987732838623697ist_fm @ ( shd_Pr3211216682057661985ist_fm @ Steps ) ) @ Sl )
         => ( ( abstra302294732650053493ist_tm @ Eff4 @ Rules2 @ ( stl_Pr1506262294867171877ist_fm @ Steps ) )
           => ( abstra302294732650053493ist_tm @ Eff4 @ Rules2 @ Steps ) ) ) ) ) ).

% RuleSystem_Defs.epath.intros
thf(fact_889_RuleSystem__Defs_Oepath_Ointros,axiom,
    ! [Steps: stream2709947120125613254m_rule,Rules2: stream_rule,Sl: fset_P8989946509869081563ist_fm,Eff4: rule > produc6018962875968178549ist_fm > fset_P8989946509869081563ist_fm > $o] :
      ( ( member_rule @ ( produc6514711208036230658m_rule @ ( shd_Pr4562317740776619530m_rule @ Steps ) ) @ ( sset_rule @ Rules2 ) )
     => ( ( fmembe3754813877001230652ist_fm @ ( produc993771546023551940m_rule @ ( shd_Pr4562317740776619530m_rule @ ( stl_Pr12655793849929990m_rule @ Steps ) ) ) @ Sl )
       => ( ( Eff4 @ ( produc6514711208036230658m_rule @ ( shd_Pr4562317740776619530m_rule @ Steps ) ) @ ( produc993771546023551940m_rule @ ( shd_Pr4562317740776619530m_rule @ Steps ) ) @ Sl )
         => ( ( abstra6789711989322986974ist_fm @ Eff4 @ Rules2 @ ( stl_Pr12655793849929990m_rule @ Steps ) )
           => ( abstra6789711989322986974ist_fm @ Eff4 @ Rules2 @ Steps ) ) ) ) ) ).

% RuleSystem_Defs.epath.intros
thf(fact_890_RuleSystem__Defs_Oepath_Ocoinduct,axiom,
    ! [X5: stream1752417115841309700fm_nat > $o,X3: stream1752417115841309700fm_nat,Rules2: stream_nat,Eff4: nat > produc6018962875968178549ist_fm > fset_P8989946509869081563ist_fm > $o] :
      ( ( X5 @ X3 )
     => ( ! [X2: stream1752417115841309700fm_nat] :
            ( ( X5 @ X2 )
           => ? [Steps4: stream1752417115841309700fm_nat,Sl4: fset_P8989946509869081563ist_fm] :
                ( ( X2 = Steps4 )
                & ( member_nat @ ( produc2092181227289234752fm_nat @ ( shd_Pr4515903069162513224fm_nat @ Steps4 ) ) @ ( sset_nat @ Rules2 ) )
                & ( fmembe3754813877001230652ist_fm @ ( produc111973938668391682fm_nat @ ( shd_Pr4515903069162513224fm_nat @ ( stl_Pr1340940657110954564fm_nat @ Steps4 ) ) ) @ Sl4 )
                & ( Eff4 @ ( produc2092181227289234752fm_nat @ ( shd_Pr4515903069162513224fm_nat @ Steps4 ) ) @ ( produc111973938668391682fm_nat @ ( shd_Pr4515903069162513224fm_nat @ Steps4 ) ) @ Sl4 )
                & ( ( X5 @ ( stl_Pr1340940657110954564fm_nat @ Steps4 ) )
                  | ( abstra3661368698739669536ist_fm @ Eff4 @ Rules2 @ ( stl_Pr1340940657110954564fm_nat @ Steps4 ) ) ) ) )
       => ( abstra3661368698739669536ist_fm @ Eff4 @ Rules2 @ X3 ) ) ) ).

% RuleSystem_Defs.epath.coinduct
thf(fact_891_RuleSystem__Defs_Oepath_Ocoinduct,axiom,
    ! [X5: stream1747862497754140525ist_fm > $o,X3: stream1747862497754140525ist_fm,Rules2: stream4408948924543953275ist_fm,Eff4: produc6018962875968178549ist_fm > produc6018962875968178549ist_fm > fset_P8989946509869081563ist_fm > $o] :
      ( ( X5 @ X3 )
     => ( ! [X2: stream1747862497754140525ist_fm] :
            ( ( X5 @ X2 )
           => ? [Steps4: stream1747862497754140525ist_fm,Sl4: fset_P8989946509869081563ist_fm] :
                ( ( X2 = Steps4 )
                & ( member4699826688122452638ist_fm @ ( produc6364584182369725445ist_fm @ ( shd_Pr4098038590278241811ist_fm @ Steps4 ) ) @ ( sset_P5379989128061332361ist_fm @ Rules2 ) )
                & ( fmembe3754813877001230652ist_fm @ ( produc8141473855448433859ist_fm @ ( shd_Pr4098038590278241811ist_fm @ ( stl_Pr1377318027900677655ist_fm @ Steps4 ) ) ) @ Sl4 )
                & ( Eff4 @ ( produc6364584182369725445ist_fm @ ( shd_Pr4098038590278241811ist_fm @ Steps4 ) ) @ ( produc8141473855448433859ist_fm @ ( shd_Pr4098038590278241811ist_fm @ Steps4 ) ) @ Sl4 )
                & ( ( X5 @ ( stl_Pr1377318027900677655ist_fm @ Steps4 ) )
                  | ( abstra6295994116336782083ist_fm @ Eff4 @ Rules2 @ ( stl_Pr1377318027900677655ist_fm @ Steps4 ) ) ) ) )
       => ( abstra6295994116336782083ist_fm @ Eff4 @ Rules2 @ X3 ) ) ) ).

% RuleSystem_Defs.epath.coinduct
thf(fact_892_RuleSystem__Defs_Oepath_Ocoinduct,axiom,
    ! [X5: stream105034537780515935at_nat > $o,X3: stream105034537780515935at_nat,Rules2: stream8372878641218411373at_nat,Eff4: produc859450856879609959at_nat > produc6018962875968178549ist_fm > fset_P8989946509869081563ist_fm > $o] :
      ( ( X5 @ X3 )
     => ( ! [X2: stream105034537780515935at_nat] :
            ( ( X5 @ X2 )
           => ? [Steps4: stream105034537780515935at_nat,Sl4: fset_P8989946509869081563ist_fm] :
                ( ( X2 = Steps4 )
                & ( member8206827879206165904at_nat @ ( produc8336112126034639607at_nat @ ( shd_Pr7514066009280765957at_nat @ Steps4 ) ) @ ( sset_P9164763103296375419at_nat @ Rules2 ) )
                & ( fmembe3754813877001230652ist_fm @ ( produc6827676535622970293at_nat @ ( shd_Pr7514066009280765957at_nat @ ( stl_Pr7223821233291300873at_nat @ Steps4 ) ) ) @ Sl4 )
                & ( Eff4 @ ( produc8336112126034639607at_nat @ ( shd_Pr7514066009280765957at_nat @ Steps4 ) ) @ ( produc6827676535622970293at_nat @ ( shd_Pr7514066009280765957at_nat @ Steps4 ) ) @ Sl4 )
                & ( ( X5 @ ( stl_Pr7223821233291300873at_nat @ Steps4 ) )
                  | ( abstra409391092022950033ist_fm @ Eff4 @ Rules2 @ ( stl_Pr7223821233291300873at_nat @ Steps4 ) ) ) ) )
       => ( abstra409391092022950033ist_fm @ Eff4 @ Rules2 @ X3 ) ) ) ).

% RuleSystem_Defs.epath.coinduct
thf(fact_893_RuleSystem__Defs_Oepath_Ocoinduct,axiom,
    ! [X5: stream2640339575054594473le_nat > $o,X3: stream2640339575054594473le_nat,Rules2: stream_nat,Eff4: nat > abstra8520858721168826610m_rule > fset_A8849196716589201442m_rule > $o] :
      ( ( X5 @ X3 )
     => ( ! [X2: stream2640339575054594473le_nat] :
            ( ( X5 @ X2 )
           => ? [Steps4: stream2640339575054594473le_nat,Sl4: fset_A8849196716589201442m_rule] :
                ( ( X2 = Steps4 )
                & ( member_nat @ ( produc6965321437502945929le_nat @ ( shd_Pr4214957810428288335le_nat @ Steps4 ) ) @ ( sset_nat @ Rules2 ) )
                & ( fmembe1099044791951439339m_rule @ ( produc1996884173541332295le_nat @ ( shd_Pr4214957810428288335le_nat @ ( stl_Pr8328254828799719763le_nat @ Steps4 ) ) ) @ Sl4 )
                & ( Eff4 @ ( produc6965321437502945929le_nat @ ( shd_Pr4214957810428288335le_nat @ Steps4 ) ) @ ( produc1996884173541332295le_nat @ ( shd_Pr4214957810428288335le_nat @ Steps4 ) ) @ Sl4 )
                & ( ( X5 @ ( stl_Pr8328254828799719763le_nat @ Steps4 ) )
                  | ( abstra2672558889254879751m_rule @ Eff4 @ Rules2 @ ( stl_Pr8328254828799719763le_nat @ Steps4 ) ) ) ) )
       => ( abstra2672558889254879751m_rule @ Eff4 @ Rules2 @ X3 ) ) ) ).

% RuleSystem_Defs.epath.coinduct
thf(fact_894_RuleSystem__Defs_Oepath_Ocoinduct,axiom,
    ! [X5: stream2118054573033641736ist_fm > $o,X3: stream2118054573033641736ist_fm,Rules2: stream4408948924543953275ist_fm,Eff4: produc6018962875968178549ist_fm > abstra8520858721168826610m_rule > fset_A8849196716589201442m_rule > $o] :
      ( ( X5 @ X3 )
     => ( ! [X2: stream2118054573033641736ist_fm] :
            ( ( X5 @ X2 )
           => ? [Steps4: stream2118054573033641736ist_fm,Sl4: fset_A8849196716589201442m_rule] :
                ( ( X2 = Steps4 )
                & ( member4699826688122452638ist_fm @ ( produc1196760618833626364ist_fm @ ( shd_Pr1348735749115486668ist_fm @ Steps4 ) ) @ ( sset_P5379989128061332361ist_fm @ Rules2 ) )
                & ( fmembe1099044791951439339m_rule @ ( produc7096817939255439806ist_fm @ ( shd_Pr1348735749115486668ist_fm @ ( stl_Pr1735168262451467976ist_fm @ Steps4 ) ) ) @ Sl4 )
                & ( Eff4 @ ( produc1196760618833626364ist_fm @ ( shd_Pr1348735749115486668ist_fm @ Steps4 ) ) @ ( produc7096817939255439806ist_fm @ ( shd_Pr1348735749115486668ist_fm @ Steps4 ) ) @ Sl4 )
                & ( ( X5 @ ( stl_Pr1735168262451467976ist_fm @ Steps4 ) )
                  | ( abstra8533012240466880740m_rule @ Eff4 @ Rules2 @ ( stl_Pr1735168262451467976ist_fm @ Steps4 ) ) ) ) )
       => ( abstra8533012240466880740m_rule @ Eff4 @ Rules2 @ X3 ) ) ) ).

% RuleSystem_Defs.epath.coinduct
thf(fact_895_RuleSystem__Defs_Oepath_Ocoinduct,axiom,
    ! [X5: stream7661288321842501626at_nat > $o,X3: stream7661288321842501626at_nat,Rules2: stream8372878641218411373at_nat,Eff4: produc859450856879609959at_nat > abstra8520858721168826610m_rule > fset_A8849196716589201442m_rule > $o] :
      ( ( X5 @ X3 )
     => ( ! [X2: stream7661288321842501626at_nat] :
            ( ( X5 @ X2 )
           => ? [Steps4: stream7661288321842501626at_nat,Sl4: fset_A8849196716589201442m_rule] :
                ( ( X2 = Steps4 )
                & ( member8206827879206165904at_nat @ ( produc740002071024140270at_nat @ ( shd_Pr7666802700379473854at_nat @ Steps4 ) ) @ ( sset_P9164763103296375419at_nat @ Rules2 ) )
                & ( fmembe1099044791951439339m_rule @ ( produc1738731144906921136at_nat @ ( shd_Pr7666802700379473854at_nat @ ( stl_Pr6382724064935898298at_nat @ Steps4 ) ) ) @ Sl4 )
                & ( Eff4 @ ( produc740002071024140270at_nat @ ( shd_Pr7666802700379473854at_nat @ Steps4 ) ) @ ( produc1738731144906921136at_nat @ ( shd_Pr7666802700379473854at_nat @ Steps4 ) ) @ Sl4 )
                & ( ( X5 @ ( stl_Pr6382724064935898298at_nat @ Steps4 ) )
                  | ( abstra7033530765613221078m_rule @ Eff4 @ Rules2 @ ( stl_Pr6382724064935898298at_nat @ Steps4 ) ) ) ) )
       => ( abstra7033530765613221078m_rule @ Eff4 @ Rules2 @ X3 ) ) ) ).

% RuleSystem_Defs.epath.coinduct
thf(fact_896_RuleSystem__Defs_Oepath_Ocoinduct,axiom,
    ! [X5: stream3393403210778678635e_rule > $o,X3: stream3393403210778678635e_rule,Rules2: stream_rule,Eff4: rule > abstra8520858721168826610m_rule > fset_A8849196716589201442m_rule > $o] :
      ( ( X5 @ X3 )
     => ( ! [X2: stream3393403210778678635e_rule] :
            ( ( X5 @ X2 )
           => ? [Steps4: stream3393403210778678635e_rule,Sl4: fset_A8849196716589201442m_rule] :
                ( ( X2 = Steps4 )
                & ( member_rule @ ( produc9166542125499771467e_rule @ ( shd_Pr8663275782010802449e_rule @ Steps4 ) ) @ ( sset_rule @ Rules2 ) )
                & ( fmembe1099044791951439339m_rule @ ( produc41289011304007433e_rule @ ( shd_Pr8663275782010802449e_rule @ ( stl_Pr654857084227610901e_rule @ Steps4 ) ) ) @ Sl4 )
                & ( Eff4 @ ( produc9166542125499771467e_rule @ ( shd_Pr8663275782010802449e_rule @ Steps4 ) ) @ ( produc41289011304007433e_rule @ ( shd_Pr8663275782010802449e_rule @ Steps4 ) ) @ Sl4 )
                & ( ( X5 @ ( stl_Pr654857084227610901e_rule @ Steps4 ) )
                  | ( abstra7067474524808760265m_rule @ Eff4 @ Rules2 @ ( stl_Pr654857084227610901e_rule @ Steps4 ) ) ) ) )
       => ( abstra7067474524808760265m_rule @ Eff4 @ Rules2 @ X3 ) ) ) ).

% RuleSystem_Defs.epath.coinduct
thf(fact_897_RuleSystem__Defs_Oepath_Ocoinduct,axiom,
    ! [X5: stream4408948924543953275ist_fm > $o,X3: stream4408948924543953275ist_fm,Rules2: stream_list_fm,Eff4: list_fm > list_tm > fset_list_tm > $o] :
      ( ( X5 @ X3 )
     => ( ! [X2: stream4408948924543953275ist_fm] :
            ( ( X5 @ X2 )
           => ? [Steps4: stream4408948924543953275ist_fm,Sl4: fset_list_tm] :
                ( ( X2 = Steps4 )
                & ( member_list_fm @ ( produc3242453064496107283ist_fm @ ( shd_Pr3211216682057661985ist_fm @ Steps4 ) ) @ ( sset_list_fm @ Rules2 ) )
                & ( fmember_list_tm @ ( produc4874987732838623697ist_fm @ ( shd_Pr3211216682057661985ist_fm @ ( stl_Pr1506262294867171877ist_fm @ Steps4 ) ) ) @ Sl4 )
                & ( Eff4 @ ( produc3242453064496107283ist_fm @ ( shd_Pr3211216682057661985ist_fm @ Steps4 ) ) @ ( produc4874987732838623697ist_fm @ ( shd_Pr3211216682057661985ist_fm @ Steps4 ) ) @ Sl4 )
                & ( ( X5 @ ( stl_Pr1506262294867171877ist_fm @ Steps4 ) )
                  | ( abstra302294732650053493ist_tm @ Eff4 @ Rules2 @ ( stl_Pr1506262294867171877ist_fm @ Steps4 ) ) ) ) )
       => ( abstra302294732650053493ist_tm @ Eff4 @ Rules2 @ X3 ) ) ) ).

% RuleSystem_Defs.epath.coinduct
thf(fact_898_RuleSystem__Defs_Oepath_Ocoinduct,axiom,
    ! [X5: stream2709947120125613254m_rule > $o,X3: stream2709947120125613254m_rule,Rules2: stream_rule,Eff4: rule > produc6018962875968178549ist_fm > fset_P8989946509869081563ist_fm > $o] :
      ( ( X5 @ X3 )
     => ( ! [X2: stream2709947120125613254m_rule] :
            ( ( X5 @ X2 )
           => ? [Steps4: stream2709947120125613254m_rule,Sl4: fset_P8989946509869081563ist_fm] :
                ( ( X2 = Steps4 )
                & ( member_rule @ ( produc6514711208036230658m_rule @ ( shd_Pr4562317740776619530m_rule @ Steps4 ) ) @ ( sset_rule @ Rules2 ) )
                & ( fmembe3754813877001230652ist_fm @ ( produc993771546023551940m_rule @ ( shd_Pr4562317740776619530m_rule @ ( stl_Pr12655793849929990m_rule @ Steps4 ) ) ) @ Sl4 )
                & ( Eff4 @ ( produc6514711208036230658m_rule @ ( shd_Pr4562317740776619530m_rule @ Steps4 ) ) @ ( produc993771546023551940m_rule @ ( shd_Pr4562317740776619530m_rule @ Steps4 ) ) @ Sl4 )
                & ( ( X5 @ ( stl_Pr12655793849929990m_rule @ Steps4 ) )
                  | ( abstra6789711989322986974ist_fm @ Eff4 @ Rules2 @ ( stl_Pr12655793849929990m_rule @ Steps4 ) ) ) ) )
       => ( abstra6789711989322986974ist_fm @ Eff4 @ Rules2 @ X3 ) ) ) ).

% RuleSystem_Defs.epath.coinduct
thf(fact_899_i_Oepath_Ocoinduct,axiom,
    ! [X5: stream1752417115841309700fm_nat > $o,X3: stream1752417115841309700fm_nat,Rules2: stream_nat,Eff2: nat > produc6018962875968178549ist_fm > option6967287582980624417ist_fm] :
      ( ( X5 @ X3 )
     => ( ! [X2: stream1752417115841309700fm_nat] :
            ( ( X5 @ X2 )
           => ? [Steps4: stream1752417115841309700fm_nat,Sl4: fset_P8989946509869081563ist_fm] :
                ( ( X2 = Steps4 )
                & ( member_nat @ ( produc2092181227289234752fm_nat @ ( shd_Pr4515903069162513224fm_nat @ Steps4 ) ) @ ( sset_nat @ Rules2 ) )
                & ( fmembe3754813877001230652ist_fm @ ( produc111973938668391682fm_nat @ ( shd_Pr4515903069162513224fm_nat @ ( stl_Pr1340940657110954564fm_nat @ Steps4 ) ) ) @ Sl4 )
                & ( abstra6480510808789727715ist_fm @ Eff2 @ ( produc2092181227289234752fm_nat @ ( shd_Pr4515903069162513224fm_nat @ Steps4 ) ) @ ( produc111973938668391682fm_nat @ ( shd_Pr4515903069162513224fm_nat @ Steps4 ) ) @ Sl4 )
                & ( ( X5 @ ( stl_Pr1340940657110954564fm_nat @ Steps4 ) )
                  | ( abstra3661368698739669536ist_fm @ ( abstra6480510808789727715ist_fm @ Eff2 ) @ Rules2 @ ( stl_Pr1340940657110954564fm_nat @ Steps4 ) ) ) ) )
       => ( abstra3661368698739669536ist_fm @ ( abstra6480510808789727715ist_fm @ Eff2 ) @ Rules2 @ X3 ) ) ) ).

% i.epath.coinduct
thf(fact_900_i_Oepath_Ocoinduct,axiom,
    ! [X5: stream1747862497754140525ist_fm > $o,X3: stream1747862497754140525ist_fm,Rules2: stream4408948924543953275ist_fm,Eff2: produc6018962875968178549ist_fm > produc6018962875968178549ist_fm > option6967287582980624417ist_fm] :
      ( ( X5 @ X3 )
     => ( ! [X2: stream1747862497754140525ist_fm] :
            ( ( X5 @ X2 )
           => ? [Steps4: stream1747862497754140525ist_fm,Sl4: fset_P8989946509869081563ist_fm] :
                ( ( X2 = Steps4 )
                & ( member4699826688122452638ist_fm @ ( produc6364584182369725445ist_fm @ ( shd_Pr4098038590278241811ist_fm @ Steps4 ) ) @ ( sset_P5379989128061332361ist_fm @ Rules2 ) )
                & ( fmembe3754813877001230652ist_fm @ ( produc8141473855448433859ist_fm @ ( shd_Pr4098038590278241811ist_fm @ ( stl_Pr1377318027900677655ist_fm @ Steps4 ) ) ) @ Sl4 )
                & ( abstra2084124586061969408ist_fm @ Eff2 @ ( produc6364584182369725445ist_fm @ ( shd_Pr4098038590278241811ist_fm @ Steps4 ) ) @ ( produc8141473855448433859ist_fm @ ( shd_Pr4098038590278241811ist_fm @ Steps4 ) ) @ Sl4 )
                & ( ( X5 @ ( stl_Pr1377318027900677655ist_fm @ Steps4 ) )
                  | ( abstra6295994116336782083ist_fm @ ( abstra2084124586061969408ist_fm @ Eff2 ) @ Rules2 @ ( stl_Pr1377318027900677655ist_fm @ Steps4 ) ) ) ) )
       => ( abstra6295994116336782083ist_fm @ ( abstra2084124586061969408ist_fm @ Eff2 ) @ Rules2 @ X3 ) ) ) ).

% i.epath.coinduct
thf(fact_901_i_Oepath_Ocoinduct,axiom,
    ! [X5: stream105034537780515935at_nat > $o,X3: stream105034537780515935at_nat,Rules2: stream8372878641218411373at_nat,Eff2: produc859450856879609959at_nat > produc6018962875968178549ist_fm > option6967287582980624417ist_fm] :
      ( ( X5 @ X3 )
     => ( ! [X2: stream105034537780515935at_nat] :
            ( ( X5 @ X2 )
           => ? [Steps4: stream105034537780515935at_nat,Sl4: fset_P8989946509869081563ist_fm] :
                ( ( X2 = Steps4 )
                & ( member8206827879206165904at_nat @ ( produc8336112126034639607at_nat @ ( shd_Pr7514066009280765957at_nat @ Steps4 ) ) @ ( sset_P9164763103296375419at_nat @ Rules2 ) )
                & ( fmembe3754813877001230652ist_fm @ ( produc6827676535622970293at_nat @ ( shd_Pr7514066009280765957at_nat @ ( stl_Pr7223821233291300873at_nat @ Steps4 ) ) ) @ Sl4 )
                & ( abstra331207336662862222ist_fm @ Eff2 @ ( produc8336112126034639607at_nat @ ( shd_Pr7514066009280765957at_nat @ Steps4 ) ) @ ( produc6827676535622970293at_nat @ ( shd_Pr7514066009280765957at_nat @ Steps4 ) ) @ Sl4 )
                & ( ( X5 @ ( stl_Pr7223821233291300873at_nat @ Steps4 ) )
                  | ( abstra409391092022950033ist_fm @ ( abstra331207336662862222ist_fm @ Eff2 ) @ Rules2 @ ( stl_Pr7223821233291300873at_nat @ Steps4 ) ) ) ) )
       => ( abstra409391092022950033ist_fm @ ( abstra331207336662862222ist_fm @ Eff2 ) @ Rules2 @ X3 ) ) ) ).

% i.epath.coinduct
thf(fact_902_i_Oepath_Ocoinduct,axiom,
    ! [X5: stream2640339575054594473le_nat > $o,X3: stream2640339575054594473le_nat,Rules2: stream_nat,Eff2: nat > abstra8520858721168826610m_rule > option9127550908420139634m_rule] :
      ( ( X5 @ X3 )
     => ( ! [X2: stream2640339575054594473le_nat] :
            ( ( X5 @ X2 )
           => ? [Steps4: stream2640339575054594473le_nat,Sl4: fset_A8849196716589201442m_rule] :
                ( ( X2 = Steps4 )
                & ( member_nat @ ( produc6965321437502945929le_nat @ ( shd_Pr4214957810428288335le_nat @ Steps4 ) ) @ ( sset_nat @ Rules2 ) )
                & ( fmembe1099044791951439339m_rule @ ( produc1996884173541332295le_nat @ ( shd_Pr4214957810428288335le_nat @ ( stl_Pr8328254828799719763le_nat @ Steps4 ) ) ) @ Sl4 )
                & ( abstra1833466274487013892m_rule @ Eff2 @ ( produc6965321437502945929le_nat @ ( shd_Pr4214957810428288335le_nat @ Steps4 ) ) @ ( produc1996884173541332295le_nat @ ( shd_Pr4214957810428288335le_nat @ Steps4 ) ) @ Sl4 )
                & ( ( X5 @ ( stl_Pr8328254828799719763le_nat @ Steps4 ) )
                  | ( abstra2672558889254879751m_rule @ ( abstra1833466274487013892m_rule @ Eff2 ) @ Rules2 @ ( stl_Pr8328254828799719763le_nat @ Steps4 ) ) ) ) )
       => ( abstra2672558889254879751m_rule @ ( abstra1833466274487013892m_rule @ Eff2 ) @ Rules2 @ X3 ) ) ) ).

% i.epath.coinduct
thf(fact_903_i_Oepath_Ocoinduct,axiom,
    ! [X5: stream2118054573033641736ist_fm > $o,X3: stream2118054573033641736ist_fm,Rules2: stream4408948924543953275ist_fm,Eff2: produc6018962875968178549ist_fm > abstra8520858721168826610m_rule > option9127550908420139634m_rule] :
      ( ( X5 @ X3 )
     => ( ! [X2: stream2118054573033641736ist_fm] :
            ( ( X5 @ X2 )
           => ? [Steps4: stream2118054573033641736ist_fm,Sl4: fset_A8849196716589201442m_rule] :
                ( ( X2 = Steps4 )
                & ( member4699826688122452638ist_fm @ ( produc1196760618833626364ist_fm @ ( shd_Pr1348735749115486668ist_fm @ Steps4 ) ) @ ( sset_P5379989128061332361ist_fm @ Rules2 ) )
                & ( fmembe1099044791951439339m_rule @ ( produc7096817939255439806ist_fm @ ( shd_Pr1348735749115486668ist_fm @ ( stl_Pr1735168262451467976ist_fm @ Steps4 ) ) ) @ Sl4 )
                & ( abstra2762543317532761127m_rule @ Eff2 @ ( produc1196760618833626364ist_fm @ ( shd_Pr1348735749115486668ist_fm @ Steps4 ) ) @ ( produc7096817939255439806ist_fm @ ( shd_Pr1348735749115486668ist_fm @ Steps4 ) ) @ Sl4 )
                & ( ( X5 @ ( stl_Pr1735168262451467976ist_fm @ Steps4 ) )
                  | ( abstra8533012240466880740m_rule @ ( abstra2762543317532761127m_rule @ Eff2 ) @ Rules2 @ ( stl_Pr1735168262451467976ist_fm @ Steps4 ) ) ) ) )
       => ( abstra8533012240466880740m_rule @ ( abstra2762543317532761127m_rule @ Eff2 ) @ Rules2 @ X3 ) ) ) ).

% i.epath.coinduct
thf(fact_904_i_Oepath_Ocoinduct,axiom,
    ! [X5: stream7661288321842501626at_nat > $o,X3: stream7661288321842501626at_nat,Rules2: stream8372878641218411373at_nat,Eff2: produc859450856879609959at_nat > abstra8520858721168826610m_rule > option9127550908420139634m_rule] :
      ( ( X5 @ X3 )
     => ( ! [X2: stream7661288321842501626at_nat] :
            ( ( X5 @ X2 )
           => ? [Steps4: stream7661288321842501626at_nat,Sl4: fset_A8849196716589201442m_rule] :
                ( ( X2 = Steps4 )
                & ( member8206827879206165904at_nat @ ( produc740002071024140270at_nat @ ( shd_Pr7666802700379473854at_nat @ Steps4 ) ) @ ( sset_P9164763103296375419at_nat @ Rules2 ) )
                & ( fmembe1099044791951439339m_rule @ ( produc1738731144906921136at_nat @ ( shd_Pr7666802700379473854at_nat @ ( stl_Pr6382724064935898298at_nat @ Steps4 ) ) ) @ Sl4 )
                & ( abstra56554827785262105m_rule @ Eff2 @ ( produc740002071024140270at_nat @ ( shd_Pr7666802700379473854at_nat @ Steps4 ) ) @ ( produc1738731144906921136at_nat @ ( shd_Pr7666802700379473854at_nat @ Steps4 ) ) @ Sl4 )
                & ( ( X5 @ ( stl_Pr6382724064935898298at_nat @ Steps4 ) )
                  | ( abstra7033530765613221078m_rule @ ( abstra56554827785262105m_rule @ Eff2 ) @ Rules2 @ ( stl_Pr6382724064935898298at_nat @ Steps4 ) ) ) ) )
       => ( abstra7033530765613221078m_rule @ ( abstra56554827785262105m_rule @ Eff2 ) @ Rules2 @ X3 ) ) ) ).

% i.epath.coinduct
thf(fact_905_i_Oepath_Ocoinduct,axiom,
    ! [X5: stream3393403210778678635e_rule > $o,X3: stream3393403210778678635e_rule,Rules2: stream_rule,Eff2: rule > abstra8520858721168826610m_rule > option9127550908420139634m_rule] :
      ( ( X5 @ X3 )
     => ( ! [X2: stream3393403210778678635e_rule] :
            ( ( X5 @ X2 )
           => ? [Steps4: stream3393403210778678635e_rule,Sl4: fset_A8849196716589201442m_rule] :
                ( ( X2 = Steps4 )
                & ( member_rule @ ( produc9166542125499771467e_rule @ ( shd_Pr8663275782010802449e_rule @ Steps4 ) ) @ ( sset_rule @ Rules2 ) )
                & ( fmembe1099044791951439339m_rule @ ( produc41289011304007433e_rule @ ( shd_Pr8663275782010802449e_rule @ ( stl_Pr654857084227610901e_rule @ Steps4 ) ) ) @ Sl4 )
                & ( abstra3599228160297647814m_rule @ Eff2 @ ( produc9166542125499771467e_rule @ ( shd_Pr8663275782010802449e_rule @ Steps4 ) ) @ ( produc41289011304007433e_rule @ ( shd_Pr8663275782010802449e_rule @ Steps4 ) ) @ Sl4 )
                & ( ( X5 @ ( stl_Pr654857084227610901e_rule @ Steps4 ) )
                  | ( abstra7067474524808760265m_rule @ ( abstra3599228160297647814m_rule @ Eff2 ) @ Rules2 @ ( stl_Pr654857084227610901e_rule @ Steps4 ) ) ) ) )
       => ( abstra7067474524808760265m_rule @ ( abstra3599228160297647814m_rule @ Eff2 ) @ Rules2 @ X3 ) ) ) ).

% i.epath.coinduct
thf(fact_906_i_Oepath_Ocoinduct,axiom,
    ! [X5: stream4408948924543953275ist_fm > $o,X3: stream4408948924543953275ist_fm,Rules2: stream_list_fm,Eff2: list_fm > list_tm > option_fset_list_tm] :
      ( ( X5 @ X3 )
     => ( ! [X2: stream4408948924543953275ist_fm] :
            ( ( X5 @ X2 )
           => ? [Steps4: stream4408948924543953275ist_fm,Sl4: fset_list_tm] :
                ( ( X2 = Steps4 )
                & ( member_list_fm @ ( produc3242453064496107283ist_fm @ ( shd_Pr3211216682057661985ist_fm @ Steps4 ) ) @ ( sset_list_fm @ Rules2 ) )
                & ( fmember_list_tm @ ( produc4874987732838623697ist_fm @ ( shd_Pr3211216682057661985ist_fm @ ( stl_Pr1506262294867171877ist_fm @ Steps4 ) ) ) @ Sl4 )
                & ( abstra9141620886324064114ist_tm @ Eff2 @ ( produc3242453064496107283ist_fm @ ( shd_Pr3211216682057661985ist_fm @ Steps4 ) ) @ ( produc4874987732838623697ist_fm @ ( shd_Pr3211216682057661985ist_fm @ Steps4 ) ) @ Sl4 )
                & ( ( X5 @ ( stl_Pr1506262294867171877ist_fm @ Steps4 ) )
                  | ( abstra302294732650053493ist_tm @ ( abstra9141620886324064114ist_tm @ Eff2 ) @ Rules2 @ ( stl_Pr1506262294867171877ist_fm @ Steps4 ) ) ) ) )
       => ( abstra302294732650053493ist_tm @ ( abstra9141620886324064114ist_tm @ Eff2 ) @ Rules2 @ X3 ) ) ) ).

% i.epath.coinduct
thf(fact_907_i_Oepath_Ocoinduct,axiom,
    ! [X5: stream2709947120125613254m_rule > $o,X3: stream2709947120125613254m_rule,Rules2: stream_rule,Eff2: rule > produc6018962875968178549ist_fm > option6967287582980624417ist_fm] :
      ( ( X5 @ X3 )
     => ( ! [X2: stream2709947120125613254m_rule] :
            ( ( X5 @ X2 )
           => ? [Steps4: stream2709947120125613254m_rule,Sl4: fset_P8989946509869081563ist_fm] :
                ( ( X2 = Steps4 )
                & ( member_rule @ ( produc6514711208036230658m_rule @ ( shd_Pr4562317740776619530m_rule @ Steps4 ) ) @ ( sset_rule @ Rules2 ) )
                & ( fmembe3754813877001230652ist_fm @ ( produc993771546023551940m_rule @ ( shd_Pr4562317740776619530m_rule @ ( stl_Pr12655793849929990m_rule @ Steps4 ) ) ) @ Sl4 )
                & ( abstra2682625350522704545ist_fm @ Eff2 @ ( produc6514711208036230658m_rule @ ( shd_Pr4562317740776619530m_rule @ Steps4 ) ) @ ( produc993771546023551940m_rule @ ( shd_Pr4562317740776619530m_rule @ Steps4 ) ) @ Sl4 )
                & ( ( X5 @ ( stl_Pr12655793849929990m_rule @ Steps4 ) )
                  | ( abstra6789711989322986974ist_fm @ ( abstra2682625350522704545ist_fm @ Eff2 ) @ Rules2 @ ( stl_Pr12655793849929990m_rule @ Steps4 ) ) ) ) )
       => ( abstra6789711989322986974ist_fm @ ( abstra2682625350522704545ist_fm @ Eff2 ) @ Rules2 @ X3 ) ) ) ).

% i.epath.coinduct
thf(fact_908_i_Oepath_Ointros,axiom,
    ! [Steps: stream1752417115841309700fm_nat,Rules2: stream_nat,Sl: fset_P8989946509869081563ist_fm,Eff2: nat > produc6018962875968178549ist_fm > option6967287582980624417ist_fm] :
      ( ( member_nat @ ( produc2092181227289234752fm_nat @ ( shd_Pr4515903069162513224fm_nat @ Steps ) ) @ ( sset_nat @ Rules2 ) )
     => ( ( fmembe3754813877001230652ist_fm @ ( produc111973938668391682fm_nat @ ( shd_Pr4515903069162513224fm_nat @ ( stl_Pr1340940657110954564fm_nat @ Steps ) ) ) @ Sl )
       => ( ( abstra6480510808789727715ist_fm @ Eff2 @ ( produc2092181227289234752fm_nat @ ( shd_Pr4515903069162513224fm_nat @ Steps ) ) @ ( produc111973938668391682fm_nat @ ( shd_Pr4515903069162513224fm_nat @ Steps ) ) @ Sl )
         => ( ( abstra3661368698739669536ist_fm @ ( abstra6480510808789727715ist_fm @ Eff2 ) @ Rules2 @ ( stl_Pr1340940657110954564fm_nat @ Steps ) )
           => ( abstra3661368698739669536ist_fm @ ( abstra6480510808789727715ist_fm @ Eff2 ) @ Rules2 @ Steps ) ) ) ) ) ).

% i.epath.intros
thf(fact_909_i_Oepath_Ointros,axiom,
    ! [Steps: stream1747862497754140525ist_fm,Rules2: stream4408948924543953275ist_fm,Sl: fset_P8989946509869081563ist_fm,Eff2: produc6018962875968178549ist_fm > produc6018962875968178549ist_fm > option6967287582980624417ist_fm] :
      ( ( member4699826688122452638ist_fm @ ( produc6364584182369725445ist_fm @ ( shd_Pr4098038590278241811ist_fm @ Steps ) ) @ ( sset_P5379989128061332361ist_fm @ Rules2 ) )
     => ( ( fmembe3754813877001230652ist_fm @ ( produc8141473855448433859ist_fm @ ( shd_Pr4098038590278241811ist_fm @ ( stl_Pr1377318027900677655ist_fm @ Steps ) ) ) @ Sl )
       => ( ( abstra2084124586061969408ist_fm @ Eff2 @ ( produc6364584182369725445ist_fm @ ( shd_Pr4098038590278241811ist_fm @ Steps ) ) @ ( produc8141473855448433859ist_fm @ ( shd_Pr4098038590278241811ist_fm @ Steps ) ) @ Sl )
         => ( ( abstra6295994116336782083ist_fm @ ( abstra2084124586061969408ist_fm @ Eff2 ) @ Rules2 @ ( stl_Pr1377318027900677655ist_fm @ Steps ) )
           => ( abstra6295994116336782083ist_fm @ ( abstra2084124586061969408ist_fm @ Eff2 ) @ Rules2 @ Steps ) ) ) ) ) ).

% i.epath.intros
thf(fact_910_i_Oepath_Ointros,axiom,
    ! [Steps: stream105034537780515935at_nat,Rules2: stream8372878641218411373at_nat,Sl: fset_P8989946509869081563ist_fm,Eff2: produc859450856879609959at_nat > produc6018962875968178549ist_fm > option6967287582980624417ist_fm] :
      ( ( member8206827879206165904at_nat @ ( produc8336112126034639607at_nat @ ( shd_Pr7514066009280765957at_nat @ Steps ) ) @ ( sset_P9164763103296375419at_nat @ Rules2 ) )
     => ( ( fmembe3754813877001230652ist_fm @ ( produc6827676535622970293at_nat @ ( shd_Pr7514066009280765957at_nat @ ( stl_Pr7223821233291300873at_nat @ Steps ) ) ) @ Sl )
       => ( ( abstra331207336662862222ist_fm @ Eff2 @ ( produc8336112126034639607at_nat @ ( shd_Pr7514066009280765957at_nat @ Steps ) ) @ ( produc6827676535622970293at_nat @ ( shd_Pr7514066009280765957at_nat @ Steps ) ) @ Sl )
         => ( ( abstra409391092022950033ist_fm @ ( abstra331207336662862222ist_fm @ Eff2 ) @ Rules2 @ ( stl_Pr7223821233291300873at_nat @ Steps ) )
           => ( abstra409391092022950033ist_fm @ ( abstra331207336662862222ist_fm @ Eff2 ) @ Rules2 @ Steps ) ) ) ) ) ).

% i.epath.intros
thf(fact_911_i_Oepath_Ointros,axiom,
    ! [Steps: stream2640339575054594473le_nat,Rules2: stream_nat,Sl: fset_A8849196716589201442m_rule,Eff2: nat > abstra8520858721168826610m_rule > option9127550908420139634m_rule] :
      ( ( member_nat @ ( produc6965321437502945929le_nat @ ( shd_Pr4214957810428288335le_nat @ Steps ) ) @ ( sset_nat @ Rules2 ) )
     => ( ( fmembe1099044791951439339m_rule @ ( produc1996884173541332295le_nat @ ( shd_Pr4214957810428288335le_nat @ ( stl_Pr8328254828799719763le_nat @ Steps ) ) ) @ Sl )
       => ( ( abstra1833466274487013892m_rule @ Eff2 @ ( produc6965321437502945929le_nat @ ( shd_Pr4214957810428288335le_nat @ Steps ) ) @ ( produc1996884173541332295le_nat @ ( shd_Pr4214957810428288335le_nat @ Steps ) ) @ Sl )
         => ( ( abstra2672558889254879751m_rule @ ( abstra1833466274487013892m_rule @ Eff2 ) @ Rules2 @ ( stl_Pr8328254828799719763le_nat @ Steps ) )
           => ( abstra2672558889254879751m_rule @ ( abstra1833466274487013892m_rule @ Eff2 ) @ Rules2 @ Steps ) ) ) ) ) ).

% i.epath.intros
thf(fact_912_i_Oepath_Ointros,axiom,
    ! [Steps: stream2118054573033641736ist_fm,Rules2: stream4408948924543953275ist_fm,Sl: fset_A8849196716589201442m_rule,Eff2: produc6018962875968178549ist_fm > abstra8520858721168826610m_rule > option9127550908420139634m_rule] :
      ( ( member4699826688122452638ist_fm @ ( produc1196760618833626364ist_fm @ ( shd_Pr1348735749115486668ist_fm @ Steps ) ) @ ( sset_P5379989128061332361ist_fm @ Rules2 ) )
     => ( ( fmembe1099044791951439339m_rule @ ( produc7096817939255439806ist_fm @ ( shd_Pr1348735749115486668ist_fm @ ( stl_Pr1735168262451467976ist_fm @ Steps ) ) ) @ Sl )
       => ( ( abstra2762543317532761127m_rule @ Eff2 @ ( produc1196760618833626364ist_fm @ ( shd_Pr1348735749115486668ist_fm @ Steps ) ) @ ( produc7096817939255439806ist_fm @ ( shd_Pr1348735749115486668ist_fm @ Steps ) ) @ Sl )
         => ( ( abstra8533012240466880740m_rule @ ( abstra2762543317532761127m_rule @ Eff2 ) @ Rules2 @ ( stl_Pr1735168262451467976ist_fm @ Steps ) )
           => ( abstra8533012240466880740m_rule @ ( abstra2762543317532761127m_rule @ Eff2 ) @ Rules2 @ Steps ) ) ) ) ) ).

% i.epath.intros
thf(fact_913_i_Oepath_Ointros,axiom,
    ! [Steps: stream7661288321842501626at_nat,Rules2: stream8372878641218411373at_nat,Sl: fset_A8849196716589201442m_rule,Eff2: produc859450856879609959at_nat > abstra8520858721168826610m_rule > option9127550908420139634m_rule] :
      ( ( member8206827879206165904at_nat @ ( produc740002071024140270at_nat @ ( shd_Pr7666802700379473854at_nat @ Steps ) ) @ ( sset_P9164763103296375419at_nat @ Rules2 ) )
     => ( ( fmembe1099044791951439339m_rule @ ( produc1738731144906921136at_nat @ ( shd_Pr7666802700379473854at_nat @ ( stl_Pr6382724064935898298at_nat @ Steps ) ) ) @ Sl )
       => ( ( abstra56554827785262105m_rule @ Eff2 @ ( produc740002071024140270at_nat @ ( shd_Pr7666802700379473854at_nat @ Steps ) ) @ ( produc1738731144906921136at_nat @ ( shd_Pr7666802700379473854at_nat @ Steps ) ) @ Sl )
         => ( ( abstra7033530765613221078m_rule @ ( abstra56554827785262105m_rule @ Eff2 ) @ Rules2 @ ( stl_Pr6382724064935898298at_nat @ Steps ) )
           => ( abstra7033530765613221078m_rule @ ( abstra56554827785262105m_rule @ Eff2 ) @ Rules2 @ Steps ) ) ) ) ) ).

% i.epath.intros
thf(fact_914_i_Oepath_Ointros,axiom,
    ! [Steps: stream3393403210778678635e_rule,Rules2: stream_rule,Sl: fset_A8849196716589201442m_rule,Eff2: rule > abstra8520858721168826610m_rule > option9127550908420139634m_rule] :
      ( ( member_rule @ ( produc9166542125499771467e_rule @ ( shd_Pr8663275782010802449e_rule @ Steps ) ) @ ( sset_rule @ Rules2 ) )
     => ( ( fmembe1099044791951439339m_rule @ ( produc41289011304007433e_rule @ ( shd_Pr8663275782010802449e_rule @ ( stl_Pr654857084227610901e_rule @ Steps ) ) ) @ Sl )
       => ( ( abstra3599228160297647814m_rule @ Eff2 @ ( produc9166542125499771467e_rule @ ( shd_Pr8663275782010802449e_rule @ Steps ) ) @ ( produc41289011304007433e_rule @ ( shd_Pr8663275782010802449e_rule @ Steps ) ) @ Sl )
         => ( ( abstra7067474524808760265m_rule @ ( abstra3599228160297647814m_rule @ Eff2 ) @ Rules2 @ ( stl_Pr654857084227610901e_rule @ Steps ) )
           => ( abstra7067474524808760265m_rule @ ( abstra3599228160297647814m_rule @ Eff2 ) @ Rules2 @ Steps ) ) ) ) ) ).

% i.epath.intros
thf(fact_915_i_Oepath_Ointros,axiom,
    ! [Steps: stream4408948924543953275ist_fm,Rules2: stream_list_fm,Sl: fset_list_tm,Eff2: list_fm > list_tm > option_fset_list_tm] :
      ( ( member_list_fm @ ( produc3242453064496107283ist_fm @ ( shd_Pr3211216682057661985ist_fm @ Steps ) ) @ ( sset_list_fm @ Rules2 ) )
     => ( ( fmember_list_tm @ ( produc4874987732838623697ist_fm @ ( shd_Pr3211216682057661985ist_fm @ ( stl_Pr1506262294867171877ist_fm @ Steps ) ) ) @ Sl )
       => ( ( abstra9141620886324064114ist_tm @ Eff2 @ ( produc3242453064496107283ist_fm @ ( shd_Pr3211216682057661985ist_fm @ Steps ) ) @ ( produc4874987732838623697ist_fm @ ( shd_Pr3211216682057661985ist_fm @ Steps ) ) @ Sl )
         => ( ( abstra302294732650053493ist_tm @ ( abstra9141620886324064114ist_tm @ Eff2 ) @ Rules2 @ ( stl_Pr1506262294867171877ist_fm @ Steps ) )
           => ( abstra302294732650053493ist_tm @ ( abstra9141620886324064114ist_tm @ Eff2 ) @ Rules2 @ Steps ) ) ) ) ) ).

% i.epath.intros
thf(fact_916_i_Oepath_Ointros,axiom,
    ! [Steps: stream2709947120125613254m_rule,Rules2: stream_rule,Sl: fset_P8989946509869081563ist_fm,Eff2: rule > produc6018962875968178549ist_fm > option6967287582980624417ist_fm] :
      ( ( member_rule @ ( produc6514711208036230658m_rule @ ( shd_Pr4562317740776619530m_rule @ Steps ) ) @ ( sset_rule @ Rules2 ) )
     => ( ( fmembe3754813877001230652ist_fm @ ( produc993771546023551940m_rule @ ( shd_Pr4562317740776619530m_rule @ ( stl_Pr12655793849929990m_rule @ Steps ) ) ) @ Sl )
       => ( ( abstra2682625350522704545ist_fm @ Eff2 @ ( produc6514711208036230658m_rule @ ( shd_Pr4562317740776619530m_rule @ Steps ) ) @ ( produc993771546023551940m_rule @ ( shd_Pr4562317740776619530m_rule @ Steps ) ) @ Sl )
         => ( ( abstra6789711989322986974ist_fm @ ( abstra2682625350522704545ist_fm @ Eff2 ) @ Rules2 @ ( stl_Pr12655793849929990m_rule @ Steps ) )
           => ( abstra6789711989322986974ist_fm @ ( abstra2682625350522704545ist_fm @ Eff2 ) @ Rules2 @ Steps ) ) ) ) ) ).

% i.epath.intros
thf(fact_917_i_Oepath_Osimps,axiom,
    ! [Eff2: nat > produc6018962875968178549ist_fm > option6967287582980624417ist_fm,Rules2: stream_nat,A: stream1752417115841309700fm_nat] :
      ( ( abstra3661368698739669536ist_fm @ ( abstra6480510808789727715ist_fm @ Eff2 ) @ Rules2 @ A )
      = ( ? [Steps2: stream1752417115841309700fm_nat,Sl3: fset_P8989946509869081563ist_fm] :
            ( ( A = Steps2 )
            & ( member_nat @ ( produc2092181227289234752fm_nat @ ( shd_Pr4515903069162513224fm_nat @ Steps2 ) ) @ ( sset_nat @ Rules2 ) )
            & ( fmembe3754813877001230652ist_fm @ ( produc111973938668391682fm_nat @ ( shd_Pr4515903069162513224fm_nat @ ( stl_Pr1340940657110954564fm_nat @ Steps2 ) ) ) @ Sl3 )
            & ( abstra6480510808789727715ist_fm @ Eff2 @ ( produc2092181227289234752fm_nat @ ( shd_Pr4515903069162513224fm_nat @ Steps2 ) ) @ ( produc111973938668391682fm_nat @ ( shd_Pr4515903069162513224fm_nat @ Steps2 ) ) @ Sl3 )
            & ( abstra3661368698739669536ist_fm @ ( abstra6480510808789727715ist_fm @ Eff2 ) @ Rules2 @ ( stl_Pr1340940657110954564fm_nat @ Steps2 ) ) ) ) ) ).

% i.epath.simps
thf(fact_918_i_Oepath_Osimps,axiom,
    ! [Eff2: produc6018962875968178549ist_fm > produc6018962875968178549ist_fm > option6967287582980624417ist_fm,Rules2: stream4408948924543953275ist_fm,A: stream1747862497754140525ist_fm] :
      ( ( abstra6295994116336782083ist_fm @ ( abstra2084124586061969408ist_fm @ Eff2 ) @ Rules2 @ A )
      = ( ? [Steps2: stream1747862497754140525ist_fm,Sl3: fset_P8989946509869081563ist_fm] :
            ( ( A = Steps2 )
            & ( member4699826688122452638ist_fm @ ( produc6364584182369725445ist_fm @ ( shd_Pr4098038590278241811ist_fm @ Steps2 ) ) @ ( sset_P5379989128061332361ist_fm @ Rules2 ) )
            & ( fmembe3754813877001230652ist_fm @ ( produc8141473855448433859ist_fm @ ( shd_Pr4098038590278241811ist_fm @ ( stl_Pr1377318027900677655ist_fm @ Steps2 ) ) ) @ Sl3 )
            & ( abstra2084124586061969408ist_fm @ Eff2 @ ( produc6364584182369725445ist_fm @ ( shd_Pr4098038590278241811ist_fm @ Steps2 ) ) @ ( produc8141473855448433859ist_fm @ ( shd_Pr4098038590278241811ist_fm @ Steps2 ) ) @ Sl3 )
            & ( abstra6295994116336782083ist_fm @ ( abstra2084124586061969408ist_fm @ Eff2 ) @ Rules2 @ ( stl_Pr1377318027900677655ist_fm @ Steps2 ) ) ) ) ) ).

% i.epath.simps
thf(fact_919_i_Oepath_Osimps,axiom,
    ! [Eff2: produc859450856879609959at_nat > produc6018962875968178549ist_fm > option6967287582980624417ist_fm,Rules2: stream8372878641218411373at_nat,A: stream105034537780515935at_nat] :
      ( ( abstra409391092022950033ist_fm @ ( abstra331207336662862222ist_fm @ Eff2 ) @ Rules2 @ A )
      = ( ? [Steps2: stream105034537780515935at_nat,Sl3: fset_P8989946509869081563ist_fm] :
            ( ( A = Steps2 )
            & ( member8206827879206165904at_nat @ ( produc8336112126034639607at_nat @ ( shd_Pr7514066009280765957at_nat @ Steps2 ) ) @ ( sset_P9164763103296375419at_nat @ Rules2 ) )
            & ( fmembe3754813877001230652ist_fm @ ( produc6827676535622970293at_nat @ ( shd_Pr7514066009280765957at_nat @ ( stl_Pr7223821233291300873at_nat @ Steps2 ) ) ) @ Sl3 )
            & ( abstra331207336662862222ist_fm @ Eff2 @ ( produc8336112126034639607at_nat @ ( shd_Pr7514066009280765957at_nat @ Steps2 ) ) @ ( produc6827676535622970293at_nat @ ( shd_Pr7514066009280765957at_nat @ Steps2 ) ) @ Sl3 )
            & ( abstra409391092022950033ist_fm @ ( abstra331207336662862222ist_fm @ Eff2 ) @ Rules2 @ ( stl_Pr7223821233291300873at_nat @ Steps2 ) ) ) ) ) ).

% i.epath.simps
thf(fact_920_i_Oepath_Osimps,axiom,
    ! [Eff2: nat > abstra8520858721168826610m_rule > option9127550908420139634m_rule,Rules2: stream_nat,A: stream2640339575054594473le_nat] :
      ( ( abstra2672558889254879751m_rule @ ( abstra1833466274487013892m_rule @ Eff2 ) @ Rules2 @ A )
      = ( ? [Steps2: stream2640339575054594473le_nat,Sl3: fset_A8849196716589201442m_rule] :
            ( ( A = Steps2 )
            & ( member_nat @ ( produc6965321437502945929le_nat @ ( shd_Pr4214957810428288335le_nat @ Steps2 ) ) @ ( sset_nat @ Rules2 ) )
            & ( fmembe1099044791951439339m_rule @ ( produc1996884173541332295le_nat @ ( shd_Pr4214957810428288335le_nat @ ( stl_Pr8328254828799719763le_nat @ Steps2 ) ) ) @ Sl3 )
            & ( abstra1833466274487013892m_rule @ Eff2 @ ( produc6965321437502945929le_nat @ ( shd_Pr4214957810428288335le_nat @ Steps2 ) ) @ ( produc1996884173541332295le_nat @ ( shd_Pr4214957810428288335le_nat @ Steps2 ) ) @ Sl3 )
            & ( abstra2672558889254879751m_rule @ ( abstra1833466274487013892m_rule @ Eff2 ) @ Rules2 @ ( stl_Pr8328254828799719763le_nat @ Steps2 ) ) ) ) ) ).

% i.epath.simps
thf(fact_921_i_Oepath_Osimps,axiom,
    ! [Eff2: produc6018962875968178549ist_fm > abstra8520858721168826610m_rule > option9127550908420139634m_rule,Rules2: stream4408948924543953275ist_fm,A: stream2118054573033641736ist_fm] :
      ( ( abstra8533012240466880740m_rule @ ( abstra2762543317532761127m_rule @ Eff2 ) @ Rules2 @ A )
      = ( ? [Steps2: stream2118054573033641736ist_fm,Sl3: fset_A8849196716589201442m_rule] :
            ( ( A = Steps2 )
            & ( member4699826688122452638ist_fm @ ( produc1196760618833626364ist_fm @ ( shd_Pr1348735749115486668ist_fm @ Steps2 ) ) @ ( sset_P5379989128061332361ist_fm @ Rules2 ) )
            & ( fmembe1099044791951439339m_rule @ ( produc7096817939255439806ist_fm @ ( shd_Pr1348735749115486668ist_fm @ ( stl_Pr1735168262451467976ist_fm @ Steps2 ) ) ) @ Sl3 )
            & ( abstra2762543317532761127m_rule @ Eff2 @ ( produc1196760618833626364ist_fm @ ( shd_Pr1348735749115486668ist_fm @ Steps2 ) ) @ ( produc7096817939255439806ist_fm @ ( shd_Pr1348735749115486668ist_fm @ Steps2 ) ) @ Sl3 )
            & ( abstra8533012240466880740m_rule @ ( abstra2762543317532761127m_rule @ Eff2 ) @ Rules2 @ ( stl_Pr1735168262451467976ist_fm @ Steps2 ) ) ) ) ) ).

% i.epath.simps
thf(fact_922_i_Oepath_Osimps,axiom,
    ! [Eff2: produc859450856879609959at_nat > abstra8520858721168826610m_rule > option9127550908420139634m_rule,Rules2: stream8372878641218411373at_nat,A: stream7661288321842501626at_nat] :
      ( ( abstra7033530765613221078m_rule @ ( abstra56554827785262105m_rule @ Eff2 ) @ Rules2 @ A )
      = ( ? [Steps2: stream7661288321842501626at_nat,Sl3: fset_A8849196716589201442m_rule] :
            ( ( A = Steps2 )
            & ( member8206827879206165904at_nat @ ( produc740002071024140270at_nat @ ( shd_Pr7666802700379473854at_nat @ Steps2 ) ) @ ( sset_P9164763103296375419at_nat @ Rules2 ) )
            & ( fmembe1099044791951439339m_rule @ ( produc1738731144906921136at_nat @ ( shd_Pr7666802700379473854at_nat @ ( stl_Pr6382724064935898298at_nat @ Steps2 ) ) ) @ Sl3 )
            & ( abstra56554827785262105m_rule @ Eff2 @ ( produc740002071024140270at_nat @ ( shd_Pr7666802700379473854at_nat @ Steps2 ) ) @ ( produc1738731144906921136at_nat @ ( shd_Pr7666802700379473854at_nat @ Steps2 ) ) @ Sl3 )
            & ( abstra7033530765613221078m_rule @ ( abstra56554827785262105m_rule @ Eff2 ) @ Rules2 @ ( stl_Pr6382724064935898298at_nat @ Steps2 ) ) ) ) ) ).

% i.epath.simps
thf(fact_923_i_Oepath_Osimps,axiom,
    ! [Eff2: rule > abstra8520858721168826610m_rule > option9127550908420139634m_rule,Rules2: stream_rule,A: stream3393403210778678635e_rule] :
      ( ( abstra7067474524808760265m_rule @ ( abstra3599228160297647814m_rule @ Eff2 ) @ Rules2 @ A )
      = ( ? [Steps2: stream3393403210778678635e_rule,Sl3: fset_A8849196716589201442m_rule] :
            ( ( A = Steps2 )
            & ( member_rule @ ( produc9166542125499771467e_rule @ ( shd_Pr8663275782010802449e_rule @ Steps2 ) ) @ ( sset_rule @ Rules2 ) )
            & ( fmembe1099044791951439339m_rule @ ( produc41289011304007433e_rule @ ( shd_Pr8663275782010802449e_rule @ ( stl_Pr654857084227610901e_rule @ Steps2 ) ) ) @ Sl3 )
            & ( abstra3599228160297647814m_rule @ Eff2 @ ( produc9166542125499771467e_rule @ ( shd_Pr8663275782010802449e_rule @ Steps2 ) ) @ ( produc41289011304007433e_rule @ ( shd_Pr8663275782010802449e_rule @ Steps2 ) ) @ Sl3 )
            & ( abstra7067474524808760265m_rule @ ( abstra3599228160297647814m_rule @ Eff2 ) @ Rules2 @ ( stl_Pr654857084227610901e_rule @ Steps2 ) ) ) ) ) ).

% i.epath.simps
thf(fact_924_i_Oepath_Osimps,axiom,
    ! [Eff2: list_fm > list_tm > option_fset_list_tm,Rules2: stream_list_fm,A: stream4408948924543953275ist_fm] :
      ( ( abstra302294732650053493ist_tm @ ( abstra9141620886324064114ist_tm @ Eff2 ) @ Rules2 @ A )
      = ( ? [Steps2: stream4408948924543953275ist_fm,Sl3: fset_list_tm] :
            ( ( A = Steps2 )
            & ( member_list_fm @ ( produc3242453064496107283ist_fm @ ( shd_Pr3211216682057661985ist_fm @ Steps2 ) ) @ ( sset_list_fm @ Rules2 ) )
            & ( fmember_list_tm @ ( produc4874987732838623697ist_fm @ ( shd_Pr3211216682057661985ist_fm @ ( stl_Pr1506262294867171877ist_fm @ Steps2 ) ) ) @ Sl3 )
            & ( abstra9141620886324064114ist_tm @ Eff2 @ ( produc3242453064496107283ist_fm @ ( shd_Pr3211216682057661985ist_fm @ Steps2 ) ) @ ( produc4874987732838623697ist_fm @ ( shd_Pr3211216682057661985ist_fm @ Steps2 ) ) @ Sl3 )
            & ( abstra302294732650053493ist_tm @ ( abstra9141620886324064114ist_tm @ Eff2 ) @ Rules2 @ ( stl_Pr1506262294867171877ist_fm @ Steps2 ) ) ) ) ) ).

% i.epath.simps
thf(fact_925_i_Oepath_Osimps,axiom,
    ! [Eff2: rule > produc6018962875968178549ist_fm > option6967287582980624417ist_fm,Rules2: stream_rule,A: stream2709947120125613254m_rule] :
      ( ( abstra6789711989322986974ist_fm @ ( abstra2682625350522704545ist_fm @ Eff2 ) @ Rules2 @ A )
      = ( ? [Steps2: stream2709947120125613254m_rule,Sl3: fset_P8989946509869081563ist_fm] :
            ( ( A = Steps2 )
            & ( member_rule @ ( produc6514711208036230658m_rule @ ( shd_Pr4562317740776619530m_rule @ Steps2 ) ) @ ( sset_rule @ Rules2 ) )
            & ( fmembe3754813877001230652ist_fm @ ( produc993771546023551940m_rule @ ( shd_Pr4562317740776619530m_rule @ ( stl_Pr12655793849929990m_rule @ Steps2 ) ) ) @ Sl3 )
            & ( abstra2682625350522704545ist_fm @ Eff2 @ ( produc6514711208036230658m_rule @ ( shd_Pr4562317740776619530m_rule @ Steps2 ) ) @ ( produc993771546023551940m_rule @ ( shd_Pr4562317740776619530m_rule @ Steps2 ) ) @ Sl3 )
            & ( abstra6789711989322986974ist_fm @ ( abstra2682625350522704545ist_fm @ Eff2 ) @ Rules2 @ ( stl_Pr12655793849929990m_rule @ Steps2 ) ) ) ) ) ).

% i.epath.simps
thf(fact_926_RuleSystem_Oin__cont__mkTree,axiom,
    ! [Eff4: produc340336539035504054m_rule > nat > fset_nat > $o,Rules2: stream2709947120125613254m_rule,S4: set_nat,S3: nat,Rs2: stream2709947120125613254m_rule,T5: abstra280919757892140097m_rule] :
      ( ( abstra2021474039004664625le_nat @ Eff4 @ Rules2 @ S4 )
     => ( ( member_nat @ S3 @ S4 )
       => ( ( abstra3665372904625986210m_rule @ Rules2 @ Rs2 )
         => ( ( fmembe1513992640624634888m_rule @ T5 @ ( abstra3621411569802224744m_rule @ ( abstra7903611026202203627le_nat @ Eff4 @ Rs2 @ S3 ) ) )
           => ? [Sl5: fset_nat,S6: nat] :
                ( ( member_nat @ S6 @ S4 )
                & ( Eff4 @ ( shd_Pr4562317740776619530m_rule @ ( abstra7926412722278118673le_nat @ Eff4 @ Rs2 @ S3 ) ) @ S3 @ Sl5 )
                & ( fmember_nat @ S6 @ Sl5 )
                & ( T5
                  = ( abstra7903611026202203627le_nat @ Eff4 @ ( stl_Pr12655793849929990m_rule @ ( abstra7926412722278118673le_nat @ Eff4 @ Rs2 @ S3 ) ) @ S6 ) ) ) ) ) ) ) ).

% RuleSystem.in_cont_mkTree
thf(fact_927_RuleSystem_Oin__cont__mkTree,axiom,
    ! [Eff4: produc340336539035504054m_rule > rule > fset_rule > $o,Rules2: stream2709947120125613254m_rule,S4: set_rule,S3: rule,Rs2: stream2709947120125613254m_rule,T5: abstra8527173362768188419m_rule] :
      ( ( abstra7792261008698710771e_rule @ Eff4 @ Rules2 @ S4 )
     => ( ( member_rule @ S3 @ S4 )
       => ( ( abstra3665372904625986210m_rule @ Rules2 @ Rs2 )
         => ( ( fmembe3420198702065081546m_rule @ T5 @ ( abstra6199799721713675050m_rule @ ( abstra5235605640745886637e_rule @ Eff4 @ Rs2 @ S3 ) ) )
           => ? [Sl5: fset_rule,S6: rule] :
                ( ( member_rule @ S6 @ S4 )
                & ( Eff4 @ ( shd_Pr4562317740776619530m_rule @ ( abstra6343601422196104403e_rule @ Eff4 @ Rs2 @ S3 ) ) @ S3 @ Sl5 )
                & ( fmember_rule @ S6 @ Sl5 )
                & ( T5
                  = ( abstra5235605640745886637e_rule @ Eff4 @ ( stl_Pr12655793849929990m_rule @ ( abstra6343601422196104403e_rule @ Eff4 @ Rs2 @ S3 ) ) @ S6 ) ) ) ) ) ) ) ).

% RuleSystem.in_cont_mkTree
thf(fact_928_RuleSystem_Oin__cont__mkTree,axiom,
    ! [Eff4: produc340336539035504054m_rule > produc859450856879609959at_nat > fset_P3619101297660450253at_nat > $o,Rules2: stream2709947120125613254m_rule,S4: set_Pr8693737435421807431at_nat,S3: produc859450856879609959at_nat,Rs2: stream2709947120125613254m_rule,T5: abstra3108524480970524168m_rule] :
      ( ( abstra8713744819842732358at_nat @ Eff4 @ Rules2 @ S4 )
     => ( ( member8206827879206165904at_nat @ S3 @ S4 )
       => ( ( abstra3665372904625986210m_rule @ Rules2 @ Rs2 )
         => ( ( fmembe5016087753231329281m_rule @ T5 @ ( abstra2671592689533873291m_rule @ ( abstra2996675635236158732at_nat @ Eff4 @ Rs2 @ S3 ) ) )
           => ? [Sl5: fset_P3619101297660450253at_nat,S6: produc859450856879609959at_nat] :
                ( ( member8206827879206165904at_nat @ S6 @ S4 )
                & ( Eff4 @ ( shd_Pr4562317740776619530m_rule @ ( abstra7961736532019816294at_nat @ Eff4 @ Rs2 @ S3 ) ) @ S3 @ Sl5 )
                & ( fmembe6431293532144391214at_nat @ S6 @ Sl5 )
                & ( T5
                  = ( abstra2996675635236158732at_nat @ Eff4 @ ( stl_Pr12655793849929990m_rule @ ( abstra7961736532019816294at_nat @ Eff4 @ Rs2 @ S3 ) ) @ S6 ) ) ) ) ) ) ) ).

% RuleSystem.in_cont_mkTree
thf(fact_929_RuleSystem_Oin__cont__mkTree,axiom,
    ! [Eff4: produc340336539035504054m_rule > produc6018962875968178549ist_fm > fset_P8989946509869081563ist_fm > $o,Rules2: stream2709947120125613254m_rule,S4: set_Pr5202636777678657877ist_fm,S3: produc6018962875968178549ist_fm,Rs2: stream2709947120125613254m_rule,T5: abstra5346343377569576982m_rule] :
      ( ( abstra406286406349174356ist_fm @ Eff4 @ Rules2 @ S4 )
     => ( ( member4699826688122452638ist_fm @ S3 @ S4 )
       => ( ( abstra3665372904625986210m_rule @ Rules2 @ Rs2 )
         => ( ( fmembe3383399666644223503m_rule @ T5 @ ( abstra564457536381300121m_rule @ ( abstra7211189027691227674ist_fm @ Eff4 @ Rs2 @ S3 ) ) )
           => ? [Sl5: fset_P8989946509869081563ist_fm,S6: produc6018962875968178549ist_fm] :
                ( ( member4699826688122452638ist_fm @ S6 @ S4 )
                & ( Eff4 @ ( shd_Pr4562317740776619530m_rule @ ( abstra1355475359686397044ist_fm @ Eff4 @ Rs2 @ S3 ) ) @ S3 @ Sl5 )
                & ( fmembe3754813877001230652ist_fm @ S6 @ Sl5 )
                & ( T5
                  = ( abstra7211189027691227674ist_fm @ Eff4 @ ( stl_Pr12655793849929990m_rule @ ( abstra1355475359686397044ist_fm @ Eff4 @ Rs2 @ S3 ) ) @ S6 ) ) ) ) ) ) ) ).

% RuleSystem.in_cont_mkTree
thf(fact_930_RuleSystem_Oin__cont__mkTree,axiom,
    ! [Eff4: produc340336539035504054m_rule > abstra8520858721168826610m_rule > fset_A8849196716589201442m_rule > $o,Rules2: stream2709947120125613254m_rule,S4: set_Ab3010718383674909480m_rule,S3: abstra8520858721168826610m_rule,Rs2: stream2709947120125613254m_rule,T5: abstra4980590702429328483m_rule] :
      ( ( abstra2126852716705863379m_rule @ Eff4 @ Rules2 @ S4 )
     => ( ( member5160799505749407497m_rule @ S3 @ S4 )
       => ( ( abstra3665372904625986210m_rule @ Rules2 @ Rs2 )
         => ( ( fmembe4831968196563866026m_rule @ T5 @ ( abstra1660610492346368650m_rule @ ( abstra2438427115914498701m_rule @ Eff4 @ Rs2 @ S3 ) ) )
           => ? [Sl5: fset_A8849196716589201442m_rule,S6: abstra8520858721168826610m_rule] :
                ( ( member5160799505749407497m_rule @ S6 @ S4 )
                & ( Eff4 @ ( shd_Pr4562317740776619530m_rule @ ( abstra5270706338319706291m_rule @ Eff4 @ Rs2 @ S3 ) ) @ S3 @ Sl5 )
                & ( fmembe1099044791951439339m_rule @ S6 @ Sl5 )
                & ( T5
                  = ( abstra2438427115914498701m_rule @ Eff4 @ ( stl_Pr12655793849929990m_rule @ ( abstra5270706338319706291m_rule @ Eff4 @ Rs2 @ S3 ) ) @ S6 ) ) ) ) ) ) ) ).

% RuleSystem.in_cont_mkTree
thf(fact_931_RuleSystem_Oin__cont__mkTree,axiom,
    ! [Eff4: rule > nat > fset_nat > $o,Rules2: stream_rule,S4: set_nat,S3: nat,Rs2: stream_rule,T5: abstra6807461288773466653t_rule] :
      ( ( abstra3263948797621512205le_nat @ Eff4 @ Rules2 @ S4 )
     => ( ( member_nat @ S3 @ S4 )
       => ( ( abstra3799686578551160190r_rule @ Rules2 @ Rs2 )
         => ( ( fmembe5167765794425430756t_rule @ T5 @ ( abstra6965617314692955716t_rule @ ( abstra7662105669874348231le_nat @ Eff4 @ Rs2 @ S3 ) ) )
           => ? [Sl5: fset_nat,S6: nat] :
                ( ( member_nat @ S6 @ S4 )
                & ( Eff4 @ ( shd_rule @ ( abstra8319304012324366829le_nat @ Eff4 @ Rs2 @ S3 ) ) @ S3 @ Sl5 )
                & ( fmember_nat @ S6 @ Sl5 )
                & ( T5
                  = ( abstra7662105669874348231le_nat @ Eff4 @ ( stl_rule @ ( abstra8319304012324366829le_nat @ Eff4 @ Rs2 @ S3 ) ) @ S6 ) ) ) ) ) ) ) ).

% RuleSystem.in_cont_mkTree
thf(fact_932_RuleSystem_Oin__cont__mkTree,axiom,
    ! [Eff4: rule > rule > fset_rule > $o,Rules2: stream_rule,S4: set_rule,S3: rule,Rs2: stream_rule,T5: abstra1868548086920761311e_rule] :
      ( ( abstra6805424310982398927e_rule @ Eff4 @ Rules2 @ S4 )
     => ( ( member_rule @ S3 @ S4 )
       => ( ( abstra3799686578551160190r_rule @ Rules2 @ Rs2 )
         => ( ( fmembe6649473619981193638e_rule @ T5 @ ( abstra342551825319314182e_rule @ ( abstra2300331465871827593e_rule @ Eff4 @ Rs2 @ S3 ) ) )
           => ? [Sl5: fset_rule,S6: rule] :
                ( ( member_rule @ S6 @ S4 )
                & ( Eff4 @ ( shd_rule @ ( abstra2680930950814381487e_rule @ Eff4 @ Rs2 @ S3 ) ) @ S3 @ Sl5 )
                & ( fmember_rule @ S6 @ Sl5 )
                & ( T5
                  = ( abstra2300331465871827593e_rule @ Eff4 @ ( stl_rule @ ( abstra2680930950814381487e_rule @ Eff4 @ Rs2 @ S3 ) ) @ S6 ) ) ) ) ) ) ) ).

% RuleSystem.in_cont_mkTree
thf(fact_933_RuleSystem_Oin__cont__mkTree,axiom,
    ! [Eff4: rule > produc859450856879609959at_nat > fset_P3619101297660450253at_nat > $o,Rules2: stream_rule,S4: set_Pr8693737435421807431at_nat,S3: produc859450856879609959at_nat,Rs2: stream_rule,T5: abstra1331675788896386276t_rule] :
      ( ( abstra3287435747676517226at_nat @ Eff4 @ Rules2 @ S4 )
     => ( ( member8206827879206165904at_nat @ S3 @ S4 )
       => ( ( abstra3799686578551160190r_rule @ Rules2 @ Rs2 )
         => ( ( fmembe220030964955522525t_rule @ T5 @ ( abstra6073049088808723303t_rule @ ( abstra7049071577609443632at_nat @ Eff4 @ Rs2 @ S3 ) ) )
           => ? [Sl5: fset_P3619101297660450253at_nat,S6: produc859450856879609959at_nat] :
                ( ( member8206827879206165904at_nat @ S6 @ S4 )
                & ( Eff4 @ ( shd_rule @ ( abstra1069316477956532618at_nat @ Eff4 @ Rs2 @ S3 ) ) @ S3 @ Sl5 )
                & ( fmembe6431293532144391214at_nat @ S6 @ Sl5 )
                & ( T5
                  = ( abstra7049071577609443632at_nat @ Eff4 @ ( stl_rule @ ( abstra1069316477956532618at_nat @ Eff4 @ Rs2 @ S3 ) ) @ S6 ) ) ) ) ) ) ) ).

% RuleSystem.in_cont_mkTree
thf(fact_934_RuleSystem_Oin__cont__mkTree,axiom,
    ! [Eff4: rule > abstra8520858721168826610m_rule > fset_A8849196716589201442m_rule > $o,Rules2: stream_rule,S4: set_Ab3010718383674909480m_rule,S3: abstra8520858721168826610m_rule,Rs2: stream_rule,T5: abstra8987725909520199743e_rule] :
      ( ( abstra8554342375581516207m_rule @ Eff4 @ Rules2 @ S4 )
     => ( ( member5160799505749407497m_rule @ S3 @ S4 )
       => ( ( abstra3799686578551160190r_rule @ Rules2 @ Rs2 )
         => ( ( fmembe5599296573807765126e_rule @ T5 @ ( abstra2862371701884369510e_rule @ ( abstra7342323255560930153m_rule @ Eff4 @ Rs2 @ S3 ) ) )
           => ? [Sl5: fset_A8849196716589201442m_rule,S6: abstra8520858721168826610m_rule] :
                ( ( member5160799505749407497m_rule @ S6 @ S4 )
                & ( Eff4 @ ( shd_rule @ ( abstra2567749246238847887m_rule @ Eff4 @ Rs2 @ S3 ) ) @ S3 @ Sl5 )
                & ( fmembe1099044791951439339m_rule @ S6 @ Sl5 )
                & ( T5
                  = ( abstra7342323255560930153m_rule @ Eff4 @ ( stl_rule @ ( abstra2567749246238847887m_rule @ Eff4 @ Rs2 @ S3 ) ) @ S6 ) ) ) ) ) ) ) ).

% RuleSystem.in_cont_mkTree
thf(fact_935_RuleSystem_Oin__cont__mkTree,axiom,
    ! [Eff4: rule > produc6018962875968178549ist_fm > fset_P8989946509869081563ist_fm > $o,Rules2: stream_rule,S4: set_Pr5202636777678657877ist_fm,S3: produc6018962875968178549ist_fm,Rs2: stream_rule,T5: abstra8520858721168826610m_rule] :
      ( ( abstra5221733350967904376ist_fm @ Eff4 @ Rules2 @ S4 )
     => ( ( member4699826688122452638ist_fm @ S3 @ S4 )
       => ( ( abstra3799686578551160190r_rule @ Rules2 @ Rs2 )
         => ( ( fmembe1099044791951439339m_rule @ T5 @ ( abstra2475333618431710837m_rule @ ( abstra7840189589880272446ist_fm @ Eff4 @ Rs2 @ S3 ) ) )
           => ? [Sl5: fset_P8989946509869081563ist_fm,S6: produc6018962875968178549ist_fm] :
                ( ( member4699826688122452638ist_fm @ S6 @ S4 )
                & ( Eff4 @ ( shd_rule @ ( abstra7806236140005899928ist_fm @ Eff4 @ Rs2 @ S3 ) ) @ S3 @ Sl5 )
                & ( fmembe3754813877001230652ist_fm @ S6 @ Sl5 )
                & ( T5
                  = ( abstra7840189589880272446ist_fm @ Eff4 @ ( stl_rule @ ( abstra7806236140005899928ist_fm @ Eff4 @ Rs2 @ S3 ) ) @ S6 ) ) ) ) ) ) ) ).

% RuleSystem.in_cont_mkTree
thf(fact_936_mkTree_Osimps_I1_J,axiom,
    ! [Rs2: stream_rule,S3: produc6018962875968178549ist_fm] :
      ( ( abstra5180277867158163941m_rule @ ( abstra7840189589880272446ist_fm @ eff @ Rs2 @ S3 ) )
      = ( produc1733806532565653680m_rule @ S3 @ ( shd_rule @ ( abstra7806236140005899928ist_fm @ eff @ Rs2 @ S3 ) ) ) ) ).

% mkTree.simps(1)
thf(fact_937_Konig,axiom,
    ! [T3: abstra8520858721168826610m_rule] :
      ( ~ ( abstra3822255059661934556m_rule @ T3 )
     => ( abstra8836096083317999815m_rule @ T3 @ ( abstra9098142006323500257m_rule @ T3 ) ) ) ).

% Konig
thf(fact_938_i_OmkTree_Osimps_I1_J,axiom,
    ! [Eff2: nat > nat > option_fset_nat,Rs2: stream_nat,S3: nat] :
      ( ( abstra3259451721936639506at_nat @ ( abstra1192804020863382533at_nat @ ( abstra4671362528648695778at_nat @ Eff2 ) @ Rs2 @ S3 ) )
      = ( product_Pair_nat_nat @ S3 @ ( shd_nat @ ( abstra7573781713853518635at_nat @ ( abstra4671362528648695778at_nat @ Eff2 ) @ Rs2 @ S3 ) ) ) ) ).

% i.mkTree.simps(1)
thf(fact_939_i_OmkTree_Osimps_I1_J,axiom,
    ! [Eff2: product_prod_nat_nat > product_prod_nat_nat > option7922001270233832621at_nat,Rs2: stream6724221391990029191at_nat,S3: product_prod_nat_nat] :
      ( ( abstra35009723297898488at_nat @ ( abstra8972947971889770147at_nat @ ( abstra426094224824445696at_nat @ Eff2 ) @ Rs2 @ S3 ) )
      = ( produc6161850002892822231at_nat @ S3 @ ( shd_Pr4260400998323988397at_nat @ ( abstra4493250544927931593at_nat @ ( abstra426094224824445696at_nat @ Eff2 ) @ Rs2 @ S3 ) ) ) ) ).

% i.mkTree.simps(1)
thf(fact_940_i_OmkTree_Osimps_I1_J,axiom,
    ! [Eff2: rule > produc6018962875968178549ist_fm > option6967287582980624417ist_fm,Rs2: stream_rule,S3: produc6018962875968178549ist_fm] :
      ( ( abstra5180277867158163941m_rule @ ( abstra7840189589880272446ist_fm @ ( abstra2682625350522704545ist_fm @ Eff2 ) @ Rs2 @ S3 ) )
      = ( produc1733806532565653680m_rule @ S3 @ ( shd_rule @ ( abstra7806236140005899928ist_fm @ ( abstra2682625350522704545ist_fm @ Eff2 ) @ Rs2 @ S3 ) ) ) ) ).

% i.mkTree.simps(1)
thf(fact_941_ipath_Ocoinduct,axiom,
    ! [X5: abstract_tree_rule > stream_rule > $o,X1: abstract_tree_rule,X22: stream_rule] :
      ( ( X5 @ X1 @ X22 )
     => ( ! [X12: abstract_tree_rule,X23: stream_rule] :
            ( ( X5 @ X12 @ X23 )
           => ? [T6: abstract_tree_rule,Steps4: stream_rule,T7: abstract_tree_rule] :
                ( ( X12 = T6 )
                & ( X23 = Steps4 )
                & ( ( abstract_root_rule @ T6 )
                  = ( shd_rule @ Steps4 ) )
                & ( fmembe4260254584891149511e_rule @ T7 @ ( abstract_cont_rule @ T6 ) )
                & ( ( X5 @ T7 @ ( stl_rule @ Steps4 ) )
                  | ( abstract_ipath_rule @ T7 @ ( stl_rule @ Steps4 ) ) ) ) )
       => ( abstract_ipath_rule @ X1 @ X22 ) ) ) ).

% ipath.coinduct
thf(fact_942_ipath_Ocoinduct,axiom,
    ! [X5: abstra8520858721168826610m_rule > stream2709947120125613254m_rule > $o,X1: abstra8520858721168826610m_rule,X22: stream2709947120125613254m_rule] :
      ( ( X5 @ X1 @ X22 )
     => ( ! [X12: abstra8520858721168826610m_rule,X23: stream2709947120125613254m_rule] :
            ( ( X5 @ X12 @ X23 )
           => ? [T6: abstra8520858721168826610m_rule,Steps4: stream2709947120125613254m_rule,T7: abstra8520858721168826610m_rule] :
                ( ( X12 = T6 )
                & ( X23 = Steps4 )
                & ( ( abstra5180277867158163941m_rule @ T6 )
                  = ( shd_Pr4562317740776619530m_rule @ Steps4 ) )
                & ( fmembe1099044791951439339m_rule @ T7 @ ( abstra2475333618431710837m_rule @ T6 ) )
                & ( ( X5 @ T7 @ ( stl_Pr12655793849929990m_rule @ Steps4 ) )
                  | ( abstra8836096083317999815m_rule @ T7 @ ( stl_Pr12655793849929990m_rule @ Steps4 ) ) ) ) )
       => ( abstra8836096083317999815m_rule @ X1 @ X22 ) ) ) ).

% ipath.coinduct
thf(fact_943_ipath_Ointros,axiom,
    ! [T3: abstract_tree_rule,Steps: stream_rule,T5: abstract_tree_rule] :
      ( ( ( abstract_root_rule @ T3 )
        = ( shd_rule @ Steps ) )
     => ( ( fmembe4260254584891149511e_rule @ T5 @ ( abstract_cont_rule @ T3 ) )
       => ( ( abstract_ipath_rule @ T5 @ ( stl_rule @ Steps ) )
         => ( abstract_ipath_rule @ T3 @ Steps ) ) ) ) ).

% ipath.intros
thf(fact_944_ipath_Ointros,axiom,
    ! [T3: abstra8520858721168826610m_rule,Steps: stream2709947120125613254m_rule,T5: abstra8520858721168826610m_rule] :
      ( ( ( abstra5180277867158163941m_rule @ T3 )
        = ( shd_Pr4562317740776619530m_rule @ Steps ) )
     => ( ( fmembe1099044791951439339m_rule @ T5 @ ( abstra2475333618431710837m_rule @ T3 ) )
       => ( ( abstra8836096083317999815m_rule @ T5 @ ( stl_Pr12655793849929990m_rule @ Steps ) )
         => ( abstra8836096083317999815m_rule @ T3 @ Steps ) ) ) ) ).

% ipath.intros
thf(fact_945_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_946_prod_Ocollapse,axiom,
    ! [Prod: produc859450856879609959at_nat] :
      ( ( produc6161850002892822231at_nat @ ( produc3213797794245857475at_nat @ Prod ) @ ( produc6408287024330202629at_nat @ Prod ) )
      = Prod ) ).

% prod.collapse
thf(fact_947_prod_Ocollapse,axiom,
    ! [Prod: produc6018962875968178549ist_fm] :
      ( ( produc1414352766439514085ist_fm @ ( produc4874987732838623697ist_fm @ Prod ) @ ( produc3242453064496107283ist_fm @ Prod ) )
      = Prod ) ).

% prod.collapse
thf(fact_948_prod_Ocollapse,axiom,
    ! [Prod: produc340336539035504054m_rule] :
      ( ( produc1733806532565653680m_rule @ ( produc993771546023551940m_rule @ Prod ) @ ( produc6514711208036230658m_rule @ Prod ) )
      = Prod ) ).

% prod.collapse
thf(fact_949_subrelI,axiom,
    ! [R: set_Pr5202636777678657877ist_fm,S3: set_Pr5202636777678657877ist_fm] :
      ( ! [X2: list_tm,Y4: list_fm] :
          ( ( member4699826688122452638ist_fm @ ( produc1414352766439514085ist_fm @ X2 @ Y4 ) @ R )
         => ( member4699826688122452638ist_fm @ ( produc1414352766439514085ist_fm @ X2 @ Y4 ) @ S3 ) )
     => ( ord_le1771420097867575541ist_fm @ R @ S3 ) ) ).

% subrelI
thf(fact_950_subrelI,axiom,
    ! [R: set_Pr1822751329126368876m_rule,S3: set_Pr1822751329126368876m_rule] :
      ( ! [X2: produc6018962875968178549ist_fm,Y4: rule] :
          ( ( member7231649785386036813m_rule @ ( produc1733806532565653680m_rule @ X2 @ Y4 ) @ R )
         => ( member7231649785386036813m_rule @ ( produc1733806532565653680m_rule @ X2 @ Y4 ) @ S3 ) )
     => ( ord_le6390412330253371084m_rule @ R @ S3 ) ) ).

% subrelI
thf(fact_951_subrelI,axiom,
    ! [R: set_Pr1261947904930325089at_nat,S3: set_Pr1261947904930325089at_nat] :
      ( ! [X2: nat,Y4: nat] :
          ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X2 @ Y4 ) @ R )
         => ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X2 @ Y4 ) @ S3 ) )
     => ( ord_le3146513528884898305at_nat @ R @ S3 ) ) ).

% subrelI
thf(fact_952_subrelI,axiom,
    ! [R: set_Pr8693737435421807431at_nat,S3: set_Pr8693737435421807431at_nat] :
      ( ! [X2: product_prod_nat_nat,Y4: product_prod_nat_nat] :
          ( ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ X2 @ Y4 ) @ R )
         => ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ X2 @ Y4 ) @ S3 ) )
     => ( ord_le3000389064537975527at_nat @ R @ S3 ) ) ).

% subrelI
thf(fact_953_pred__equals__eq2,axiom,
    ! [R3: set_Pr5202636777678657877ist_fm,S4: set_Pr5202636777678657877ist_fm] :
      ( ( ( ^ [X: list_tm,Y3: list_fm] : ( member4699826688122452638ist_fm @ ( produc1414352766439514085ist_fm @ X @ Y3 ) @ R3 ) )
        = ( ^ [X: list_tm,Y3: list_fm] : ( member4699826688122452638ist_fm @ ( produc1414352766439514085ist_fm @ X @ Y3 ) @ S4 ) ) )
      = ( R3 = S4 ) ) ).

% pred_equals_eq2
thf(fact_954_pred__equals__eq2,axiom,
    ! [R3: set_Pr1822751329126368876m_rule,S4: set_Pr1822751329126368876m_rule] :
      ( ( ( ^ [X: produc6018962875968178549ist_fm,Y3: rule] : ( member7231649785386036813m_rule @ ( produc1733806532565653680m_rule @ X @ Y3 ) @ R3 ) )
        = ( ^ [X: produc6018962875968178549ist_fm,Y3: rule] : ( member7231649785386036813m_rule @ ( produc1733806532565653680m_rule @ X @ Y3 ) @ S4 ) ) )
      = ( R3 = S4 ) ) ).

% pred_equals_eq2
thf(fact_955_pred__equals__eq2,axiom,
    ! [R3: set_Pr1261947904930325089at_nat,S4: set_Pr1261947904930325089at_nat] :
      ( ( ( ^ [X: nat,Y3: nat] : ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X @ Y3 ) @ R3 ) )
        = ( ^ [X: nat,Y3: nat] : ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X @ Y3 ) @ S4 ) ) )
      = ( R3 = S4 ) ) ).

% pred_equals_eq2
thf(fact_956_pred__equals__eq2,axiom,
    ! [R3: set_Pr8693737435421807431at_nat,S4: set_Pr8693737435421807431at_nat] :
      ( ( ( ^ [X: product_prod_nat_nat,Y3: product_prod_nat_nat] : ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ X @ Y3 ) @ R3 ) )
        = ( ^ [X: product_prod_nat_nat,Y3: product_prod_nat_nat] : ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ X @ Y3 ) @ S4 ) ) )
      = ( R3 = S4 ) ) ).

% pred_equals_eq2
thf(fact_957_pred__subset__eq2,axiom,
    ! [R3: set_Pr5202636777678657877ist_fm,S4: set_Pr5202636777678657877ist_fm] :
      ( ( ord_le7793033639944360256t_fm_o
        @ ^ [X: list_tm,Y3: list_fm] : ( member4699826688122452638ist_fm @ ( produc1414352766439514085ist_fm @ X @ Y3 ) @ R3 )
        @ ^ [X: list_tm,Y3: list_fm] : ( member4699826688122452638ist_fm @ ( produc1414352766439514085ist_fm @ X @ Y3 ) @ S4 ) )
      = ( ord_le1771420097867575541ist_fm @ R3 @ S4 ) ) ).

% pred_subset_eq2
thf(fact_958_pred__subset__eq2,axiom,
    ! [R3: set_Pr1822751329126368876m_rule,S4: set_Pr1822751329126368876m_rule] :
      ( ( ord_le2289920624603415501rule_o
        @ ^ [X: produc6018962875968178549ist_fm,Y3: rule] : ( member7231649785386036813m_rule @ ( produc1733806532565653680m_rule @ X @ Y3 ) @ R3 )
        @ ^ [X: produc6018962875968178549ist_fm,Y3: rule] : ( member7231649785386036813m_rule @ ( produc1733806532565653680m_rule @ X @ Y3 ) @ S4 ) )
      = ( ord_le6390412330253371084m_rule @ R3 @ S4 ) ) ).

% pred_subset_eq2
thf(fact_959_pred__subset__eq2,axiom,
    ! [R3: set_Pr1261947904930325089at_nat,S4: set_Pr1261947904930325089at_nat] :
      ( ( ord_le2646555220125990790_nat_o
        @ ^ [X: nat,Y3: nat] : ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X @ Y3 ) @ R3 )
        @ ^ [X: nat,Y3: nat] : ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X @ Y3 ) @ S4 ) )
      = ( ord_le3146513528884898305at_nat @ R3 @ S4 ) ) ).

% pred_subset_eq2
thf(fact_960_pred__subset__eq2,axiom,
    ! [R3: set_Pr8693737435421807431at_nat,S4: set_Pr8693737435421807431at_nat] :
      ( ( ord_le5604493270027003598_nat_o
        @ ^ [X: product_prod_nat_nat,Y3: product_prod_nat_nat] : ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ X @ Y3 ) @ R3 )
        @ ^ [X: product_prod_nat_nat,Y3: product_prod_nat_nat] : ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ X @ Y3 ) @ S4 ) )
      = ( ord_le3000389064537975527at_nat @ R3 @ S4 ) ) ).

% pred_subset_eq2
thf(fact_961_snd__eqD,axiom,
    ! [X3: nat,Y: nat,A: nat] :
      ( ( ( product_snd_nat_nat @ ( product_Pair_nat_nat @ X3 @ Y ) )
        = A )
     => ( Y = A ) ) ).

% snd_eqD
thf(fact_962_snd__eqD,axiom,
    ! [X3: product_prod_nat_nat,Y: product_prod_nat_nat,A: product_prod_nat_nat] :
      ( ( ( produc6408287024330202629at_nat @ ( produc6161850002892822231at_nat @ X3 @ Y ) )
        = A )
     => ( Y = A ) ) ).

% snd_eqD
thf(fact_963_snd__eqD,axiom,
    ! [X3: produc6018962875968178549ist_fm,Y: rule,A: rule] :
      ( ( ( produc6514711208036230658m_rule @ ( produc1733806532565653680m_rule @ X3 @ Y ) )
        = A )
     => ( Y = A ) ) ).

% snd_eqD
thf(fact_964_snd__conv,axiom,
    ! [X1: nat,X22: nat] :
      ( ( product_snd_nat_nat @ ( product_Pair_nat_nat @ X1 @ X22 ) )
      = X22 ) ).

% snd_conv
thf(fact_965_snd__conv,axiom,
    ! [X1: product_prod_nat_nat,X22: product_prod_nat_nat] :
      ( ( produc6408287024330202629at_nat @ ( produc6161850002892822231at_nat @ X1 @ X22 ) )
      = X22 ) ).

% snd_conv
thf(fact_966_snd__conv,axiom,
    ! [X1: produc6018962875968178549ist_fm,X22: rule] :
      ( ( produc6514711208036230658m_rule @ ( produc1733806532565653680m_rule @ X1 @ X22 ) )
      = X22 ) ).

% snd_conv
thf(fact_967_tree_Oexpand,axiom,
    ! [Tree: abstra8520858721168826610m_rule,Tree2: abstra8520858721168826610m_rule] :
      ( ( ( ( abstra5180277867158163941m_rule @ Tree )
          = ( abstra5180277867158163941m_rule @ Tree2 ) )
        & ( ( abstra2475333618431710837m_rule @ Tree )
          = ( abstra2475333618431710837m_rule @ Tree2 ) ) )
     => ( Tree = Tree2 ) ) ).

% tree.expand
thf(fact_968_top__empty__eq2,axiom,
    ( top_to5948207668388069917rule_o
    = ( ^ [X: produc6018962875968178549ist_fm,Y3: rule] : ( member7231649785386036813m_rule @ ( produc1733806532565653680m_rule @ X @ Y3 ) @ top_to8442024021002298652m_rule ) ) ) ).

% top_empty_eq2
thf(fact_969_top__empty__eq2,axiom,
    ( top_top_nat_nat_o
    = ( ^ [X: nat,Y3: nat] : ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X @ Y3 ) @ top_to4669805908274784177at_nat ) ) ) ).

% top_empty_eq2
thf(fact_970_top__empty__eq2,axiom,
    ( top_to2659251575160290430_nat_o
    = ( ^ [X: product_prod_nat_nat,Y3: product_prod_nat_nat] : ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ X @ Y3 ) @ top_to8454300755403694231at_nat ) ) ) ).

% top_empty_eq2
thf(fact_971_top__empty__eq2,axiom,
    ( top_to4753533911259588336t_fm_o
    = ( ^ [X: list_tm,Y3: list_fm] : ( member4699826688122452638ist_fm @ ( produc1414352766439514085ist_fm @ X @ Y3 ) @ top_to1730629564055774885ist_fm ) ) ) ).

% top_empty_eq2
thf(fact_972_tfinite,axiom,
    ! [T3: abstra8520858721168826610m_rule] :
      ( ! [T8: abstra8520858721168826610m_rule] :
          ( ( fmembe1099044791951439339m_rule @ T8 @ ( abstra2475333618431710837m_rule @ T3 ) )
         => ( abstra3822255059661934556m_rule @ T8 ) )
     => ( abstra3822255059661934556m_rule @ T3 ) ) ).

% tfinite
thf(fact_973_tfinite_Osimps,axiom,
    ( abstra3822255059661934556m_rule
    = ( ^ [A4: abstra8520858721168826610m_rule] :
        ? [T2: abstra8520858721168826610m_rule] :
          ( ( A4 = T2 )
          & ! [X: abstra8520858721168826610m_rule] :
              ( ( fmembe1099044791951439339m_rule @ X @ ( abstra2475333618431710837m_rule @ T2 ) )
             => ( abstra3822255059661934556m_rule @ X ) ) ) ) ) ).

% tfinite.simps
thf(fact_974_tfinite_Ocases,axiom,
    ! [A: abstra8520858721168826610m_rule] :
      ( ( abstra3822255059661934556m_rule @ A )
     => ! [T7: abstra8520858721168826610m_rule] :
          ( ( fmembe1099044791951439339m_rule @ T7 @ ( abstra2475333618431710837m_rule @ A ) )
         => ( abstra3822255059661934556m_rule @ T7 ) ) ) ).

% tfinite.cases
thf(fact_975_surjective__pairing,axiom,
    ! [T3: product_prod_nat_nat] :
      ( T3
      = ( product_Pair_nat_nat @ ( product_fst_nat_nat @ T3 ) @ ( product_snd_nat_nat @ T3 ) ) ) ).

% surjective_pairing
thf(fact_976_surjective__pairing,axiom,
    ! [T3: produc859450856879609959at_nat] :
      ( T3
      = ( produc6161850002892822231at_nat @ ( produc3213797794245857475at_nat @ T3 ) @ ( produc6408287024330202629at_nat @ T3 ) ) ) ).

% surjective_pairing
thf(fact_977_surjective__pairing,axiom,
    ! [T3: produc6018962875968178549ist_fm] :
      ( T3
      = ( produc1414352766439514085ist_fm @ ( produc4874987732838623697ist_fm @ T3 ) @ ( produc3242453064496107283ist_fm @ T3 ) ) ) ).

% surjective_pairing
thf(fact_978_surjective__pairing,axiom,
    ! [T3: produc340336539035504054m_rule] :
      ( T3
      = ( produc1733806532565653680m_rule @ ( produc993771546023551940m_rule @ T3 ) @ ( produc6514711208036230658m_rule @ T3 ) ) ) ).

% surjective_pairing
thf(fact_979_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_980_prod_Oexhaust__sel,axiom,
    ! [Prod: produc859450856879609959at_nat] :
      ( Prod
      = ( produc6161850002892822231at_nat @ ( produc3213797794245857475at_nat @ Prod ) @ ( produc6408287024330202629at_nat @ Prod ) ) ) ).

% prod.exhaust_sel
thf(fact_981_prod_Oexhaust__sel,axiom,
    ! [Prod: produc6018962875968178549ist_fm] :
      ( Prod
      = ( produc1414352766439514085ist_fm @ ( produc4874987732838623697ist_fm @ Prod ) @ ( produc3242453064496107283ist_fm @ Prod ) ) ) ).

% prod.exhaust_sel
thf(fact_982_prod_Oexhaust__sel,axiom,
    ! [Prod: produc340336539035504054m_rule] :
      ( Prod
      = ( produc1733806532565653680m_rule @ ( produc993771546023551940m_rule @ Prod ) @ ( produc6514711208036230658m_rule @ Prod ) ) ) ).

% prod.exhaust_sel
thf(fact_983_BNF__Greatest__Fixpoint_Osubst__Pair,axiom,
    ! [P2: nat > nat > $o,X3: nat,Y: nat,A: product_prod_nat_nat] :
      ( ( P2 @ X3 @ Y )
     => ( ( A
          = ( product_Pair_nat_nat @ X3 @ Y ) )
       => ( P2 @ ( product_fst_nat_nat @ A ) @ ( product_snd_nat_nat @ A ) ) ) ) ).

% BNF_Greatest_Fixpoint.subst_Pair
thf(fact_984_BNF__Greatest__Fixpoint_Osubst__Pair,axiom,
    ! [P2: product_prod_nat_nat > product_prod_nat_nat > $o,X3: product_prod_nat_nat,Y: product_prod_nat_nat,A: produc859450856879609959at_nat] :
      ( ( P2 @ X3 @ Y )
     => ( ( A
          = ( produc6161850002892822231at_nat @ X3 @ Y ) )
       => ( P2 @ ( produc3213797794245857475at_nat @ A ) @ ( produc6408287024330202629at_nat @ A ) ) ) ) ).

% BNF_Greatest_Fixpoint.subst_Pair
thf(fact_985_BNF__Greatest__Fixpoint_Osubst__Pair,axiom,
    ! [P2: list_tm > list_fm > $o,X3: list_tm,Y: list_fm,A: produc6018962875968178549ist_fm] :
      ( ( P2 @ X3 @ Y )
     => ( ( A
          = ( produc1414352766439514085ist_fm @ X3 @ Y ) )
       => ( P2 @ ( produc4874987732838623697ist_fm @ A ) @ ( produc3242453064496107283ist_fm @ A ) ) ) ) ).

% BNF_Greatest_Fixpoint.subst_Pair
thf(fact_986_BNF__Greatest__Fixpoint_Osubst__Pair,axiom,
    ! [P2: produc6018962875968178549ist_fm > rule > $o,X3: produc6018962875968178549ist_fm,Y: rule,A: produc340336539035504054m_rule] :
      ( ( P2 @ X3 @ Y )
     => ( ( A
          = ( produc1733806532565653680m_rule @ X3 @ Y ) )
       => ( P2 @ ( produc993771546023551940m_rule @ A ) @ ( produc6514711208036230658m_rule @ A ) ) ) ) ).

% BNF_Greatest_Fixpoint.subst_Pair
thf(fact_987_conjI__realizer,axiom,
    ! [P2: nat > $o,P3: nat,Q: nat > $o,Q3: nat] :
      ( ( P2 @ P3 )
     => ( ( Q @ Q3 )
       => ( ( P2 @ ( product_fst_nat_nat @ ( product_Pair_nat_nat @ P3 @ Q3 ) ) )
          & ( Q @ ( product_snd_nat_nat @ ( product_Pair_nat_nat @ P3 @ Q3 ) ) ) ) ) ) ).

% conjI_realizer
thf(fact_988_conjI__realizer,axiom,
    ! [P2: product_prod_nat_nat > $o,P3: product_prod_nat_nat,Q: product_prod_nat_nat > $o,Q3: product_prod_nat_nat] :
      ( ( P2 @ P3 )
     => ( ( Q @ Q3 )
       => ( ( P2 @ ( produc3213797794245857475at_nat @ ( produc6161850002892822231at_nat @ P3 @ Q3 ) ) )
          & ( Q @ ( produc6408287024330202629at_nat @ ( produc6161850002892822231at_nat @ P3 @ Q3 ) ) ) ) ) ) ).

% conjI_realizer
thf(fact_989_conjI__realizer,axiom,
    ! [P2: list_tm > $o,P3: list_tm,Q: list_fm > $o,Q3: list_fm] :
      ( ( P2 @ P3 )
     => ( ( Q @ Q3 )
       => ( ( P2 @ ( produc4874987732838623697ist_fm @ ( produc1414352766439514085ist_fm @ P3 @ Q3 ) ) )
          & ( Q @ ( produc3242453064496107283ist_fm @ ( produc1414352766439514085ist_fm @ P3 @ Q3 ) ) ) ) ) ) ).

% conjI_realizer
thf(fact_990_conjI__realizer,axiom,
    ! [P2: produc6018962875968178549ist_fm > $o,P3: produc6018962875968178549ist_fm,Q: rule > $o,Q3: rule] :
      ( ( P2 @ P3 )
     => ( ( Q @ Q3 )
       => ( ( P2 @ ( produc993771546023551940m_rule @ ( produc1733806532565653680m_rule @ P3 @ Q3 ) ) )
          & ( Q @ ( produc6514711208036230658m_rule @ ( produc1733806532565653680m_rule @ P3 @ Q3 ) ) ) ) ) ) ).

% conjI_realizer
thf(fact_991_exI__realizer,axiom,
    ! [P2: nat > nat > $o,Y: nat,X3: nat] :
      ( ( P2 @ Y @ X3 )
     => ( P2 @ ( product_snd_nat_nat @ ( product_Pair_nat_nat @ X3 @ Y ) ) @ ( product_fst_nat_nat @ ( product_Pair_nat_nat @ X3 @ Y ) ) ) ) ).

% exI_realizer
thf(fact_992_exI__realizer,axiom,
    ! [P2: product_prod_nat_nat > product_prod_nat_nat > $o,Y: product_prod_nat_nat,X3: product_prod_nat_nat] :
      ( ( P2 @ Y @ X3 )
     => ( P2 @ ( produc6408287024330202629at_nat @ ( produc6161850002892822231at_nat @ X3 @ Y ) ) @ ( produc3213797794245857475at_nat @ ( produc6161850002892822231at_nat @ X3 @ Y ) ) ) ) ).

% exI_realizer
thf(fact_993_exI__realizer,axiom,
    ! [P2: list_fm > list_tm > $o,Y: list_fm,X3: list_tm] :
      ( ( P2 @ Y @ X3 )
     => ( P2 @ ( produc3242453064496107283ist_fm @ ( produc1414352766439514085ist_fm @ X3 @ Y ) ) @ ( produc4874987732838623697ist_fm @ ( produc1414352766439514085ist_fm @ X3 @ Y ) ) ) ) ).

% exI_realizer
thf(fact_994_exI__realizer,axiom,
    ! [P2: rule > produc6018962875968178549ist_fm > $o,Y: rule,X3: produc6018962875968178549ist_fm] :
      ( ( P2 @ Y @ X3 )
     => ( P2 @ ( produc6514711208036230658m_rule @ ( produc1733806532565653680m_rule @ X3 @ Y ) ) @ ( produc993771546023551940m_rule @ ( produc1733806532565653680m_rule @ X3 @ Y ) ) ) ) ).

% exI_realizer
thf(fact_995_fair__stl,axiom,
    ! [Rs2: stream_rule] :
      ( ( abstra3799686578551160190r_rule @ rules @ Rs2 )
     => ( abstra3799686578551160190r_rule @ rules @ ( stl_rule @ Rs2 ) ) ) ).

% fair_stl
thf(fact_996_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_997_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_998_szip_Osimps_I1_J,axiom,
    ! [S1: stream2709947120125613254m_rule,S2: stream2709947120125613254m_rule] :
      ( ( shd_Pr4461660664618831993m_rule @ ( szip_P811719526838699976m_rule @ S1 @ S2 ) )
      = ( produc6261311607089640965m_rule @ ( shd_Pr4562317740776619530m_rule @ S1 ) @ ( shd_Pr4562317740776619530m_rule @ S2 ) ) ) ).

% szip.simps(1)
thf(fact_999_szip_Osimps_I1_J,axiom,
    ! [S1: stream2709947120125613254m_rule,S2: stream_rule] :
      ( ( shd_Pr2400962586966563157e_rule @ ( szip_P2044787555563963556e_rule @ S1 @ S2 ) )
      = ( produc4831648765031708129e_rule @ ( shd_Pr4562317740776619530m_rule @ S1 ) @ ( shd_rule @ S2 ) ) ) ).

% szip.simps(1)
thf(fact_1000_szip_Osimps_I1_J,axiom,
    ! [S1: stream_rule,S2: stream2709947120125613254m_rule] :
      ( ( shd_Pr1208753253167450709m_rule @ ( szip_r6533552703345879972m_rule @ S1 @ S2 ) )
      = ( produc97041875958848737m_rule @ ( shd_rule @ S1 ) @ ( shd_Pr4562317740776619530m_rule @ S2 ) ) ) ).

% szip.simps(1)
thf(fact_1001_szip_Osimps_I1_J,axiom,
    ! [S1: stream_rule,S2: stream_rule] :
      ( ( shd_Pr2264621979884435249e_rule @ ( szip_rule_rule @ S1 @ S2 ) )
      = ( produc5849431337705160893e_rule @ ( shd_rule @ S1 ) @ ( shd_rule @ S2 ) ) ) ).

% szip.simps(1)
thf(fact_1002_szip_Osimps_I1_J,axiom,
    ! [S1: stream4408948924543953275ist_fm,S2: stream_rule] :
      ( ( shd_Pr4562317740776619530m_rule @ ( szip_P2924820683901490861m_rule @ S1 @ S2 ) )
      = ( produc1733806532565653680m_rule @ ( shd_Pr3211216682057661985ist_fm @ S1 ) @ ( shd_rule @ S2 ) ) ) ).

% szip.simps(1)
thf(fact_1003_RuleSystem__Defs_OmkTree_Osimps_I1_J,axiom,
    ! [Eff4: nat > nat > fset_nat > $o,Rs2: stream_nat,S3: nat] :
      ( ( abstra3259451721936639506at_nat @ ( abstra1192804020863382533at_nat @ Eff4 @ Rs2 @ S3 ) )
      = ( product_Pair_nat_nat @ S3 @ ( shd_nat @ ( abstra7573781713853518635at_nat @ Eff4 @ Rs2 @ S3 ) ) ) ) ).

% RuleSystem_Defs.mkTree.simps(1)
thf(fact_1004_RuleSystem__Defs_OmkTree_Osimps_I1_J,axiom,
    ! [Eff4: product_prod_nat_nat > product_prod_nat_nat > fset_P5670320511379867111at_nat > $o,Rs2: stream6724221391990029191at_nat,S3: product_prod_nat_nat] :
      ( ( abstra35009723297898488at_nat @ ( abstra8972947971889770147at_nat @ Eff4 @ Rs2 @ S3 ) )
      = ( produc6161850002892822231at_nat @ S3 @ ( shd_Pr4260400998323988397at_nat @ ( abstra4493250544927931593at_nat @ Eff4 @ Rs2 @ S3 ) ) ) ) ).

% RuleSystem_Defs.mkTree.simps(1)
thf(fact_1005_RuleSystem__Defs_OmkTree_Osimps_I1_J,axiom,
    ! [Eff4: rule > produc6018962875968178549ist_fm > fset_P8989946509869081563ist_fm > $o,Rs2: stream_rule,S3: produc6018962875968178549ist_fm] :
      ( ( abstra5180277867158163941m_rule @ ( abstra7840189589880272446ist_fm @ Eff4 @ Rs2 @ S3 ) )
      = ( produc1733806532565653680m_rule @ S3 @ ( shd_rule @ ( abstra7806236140005899928ist_fm @ Eff4 @ Rs2 @ S3 ) ) ) ) ).

% RuleSystem_Defs.mkTree.simps(1)
thf(fact_1006_in__cont__mkTree,axiom,
    ! [S3: produc6018962875968178549ist_fm,Rs2: stream_rule,T5: abstra8520858721168826610m_rule] :
      ( ( member4699826688122452638ist_fm @ S3 @ top_to1730629564055774885ist_fm )
     => ( ( abstra3799686578551160190r_rule @ rules @ Rs2 )
       => ( ( fmembe1099044791951439339m_rule @ T5 @ ( abstra2475333618431710837m_rule @ ( abstra7840189589880272446ist_fm @ eff @ Rs2 @ S3 ) ) )
         => ? [Sl5: fset_P8989946509869081563ist_fm,S6: produc6018962875968178549ist_fm] :
              ( ( member4699826688122452638ist_fm @ S6 @ top_to1730629564055774885ist_fm )
              & ( eff @ ( shd_rule @ ( abstra7806236140005899928ist_fm @ eff @ Rs2 @ S3 ) ) @ S3 @ Sl5 )
              & ( fmembe3754813877001230652ist_fm @ S6 @ Sl5 )
              & ( T5
                = ( abstra7840189589880272446ist_fm @ eff @ ( stl_rule @ ( abstra7806236140005899928ist_fm @ eff @ Rs2 @ S3 ) ) @ S6 ) ) ) ) ) ) ).

% in_cont_mkTree
thf(fact_1007_konig_Osimps_I1_J,axiom,
    ! [T3: abstract_tree_rule] :
      ( ( shd_rule @ ( abstract_konig_rule @ T3 ) )
      = ( abstract_root_rule @ T3 ) ) ).

% konig.simps(1)
thf(fact_1008_konig_Osimps_I1_J,axiom,
    ! [T3: abstra8520858721168826610m_rule] :
      ( ( shd_Pr4562317740776619530m_rule @ ( abstra9098142006323500257m_rule @ T3 ) )
      = ( abstra5180277867158163941m_rule @ T3 ) ) ).

% konig.simps(1)
thf(fact_1009_ipath_Ocases,axiom,
    ! [A1: abstract_tree_rule,A22: stream_rule] :
      ( ( abstract_ipath_rule @ A1 @ A22 )
     => ~ ( ( ( abstract_root_rule @ A1 )
            = ( shd_rule @ A22 ) )
         => ! [T8: abstract_tree_rule] :
              ( ( fmembe4260254584891149511e_rule @ T8 @ ( abstract_cont_rule @ A1 ) )
             => ~ ( abstract_ipath_rule @ T8 @ ( stl_rule @ A22 ) ) ) ) ) ).

% ipath.cases
thf(fact_1010_ipath_Ocases,axiom,
    ! [A1: abstra8520858721168826610m_rule,A22: stream2709947120125613254m_rule] :
      ( ( abstra8836096083317999815m_rule @ A1 @ A22 )
     => ~ ( ( ( abstra5180277867158163941m_rule @ A1 )
            = ( shd_Pr4562317740776619530m_rule @ A22 ) )
         => ! [T8: abstra8520858721168826610m_rule] :
              ( ( fmembe1099044791951439339m_rule @ T8 @ ( abstra2475333618431710837m_rule @ A1 ) )
             => ~ ( abstra8836096083317999815m_rule @ T8 @ ( stl_Pr12655793849929990m_rule @ A22 ) ) ) ) ) ).

% ipath.cases
thf(fact_1011_ipath_Osimps,axiom,
    ( abstract_ipath_rule
    = ( ^ [A12: abstract_tree_rule,A23: stream_rule] :
        ? [T2: abstract_tree_rule,Steps2: stream_rule,T9: abstract_tree_rule] :
          ( ( A12 = T2 )
          & ( A23 = Steps2 )
          & ( ( abstract_root_rule @ T2 )
            = ( shd_rule @ Steps2 ) )
          & ( fmembe4260254584891149511e_rule @ T9 @ ( abstract_cont_rule @ T2 ) )
          & ( abstract_ipath_rule @ T9 @ ( stl_rule @ Steps2 ) ) ) ) ) ).

% ipath.simps
thf(fact_1012_ipath_Osimps,axiom,
    ( abstra8836096083317999815m_rule
    = ( ^ [A12: abstra8520858721168826610m_rule,A23: stream2709947120125613254m_rule] :
        ? [T2: abstra8520858721168826610m_rule,Steps2: stream2709947120125613254m_rule,T9: abstra8520858721168826610m_rule] :
          ( ( A12 = T2 )
          & ( A23 = Steps2 )
          & ( ( abstra5180277867158163941m_rule @ T2 )
            = ( shd_Pr4562317740776619530m_rule @ Steps2 ) )
          & ( fmembe1099044791951439339m_rule @ T9 @ ( abstra2475333618431710837m_rule @ T2 ) )
          & ( abstra8836096083317999815m_rule @ T9 @ ( stl_Pr12655793849929990m_rule @ Steps2 ) ) ) ) ) ).

% ipath.simps
thf(fact_1013_sndI,axiom,
    ! [X3: product_prod_nat_nat,Y: nat,Z: nat] :
      ( ( X3
        = ( product_Pair_nat_nat @ Y @ Z ) )
     => ( ( product_snd_nat_nat @ X3 )
        = Z ) ) ).

% sndI
thf(fact_1014_sndI,axiom,
    ! [X3: produc859450856879609959at_nat,Y: product_prod_nat_nat,Z: product_prod_nat_nat] :
      ( ( X3
        = ( produc6161850002892822231at_nat @ Y @ Z ) )
     => ( ( produc6408287024330202629at_nat @ X3 )
        = Z ) ) ).

% sndI
thf(fact_1015_sndI,axiom,
    ! [X3: produc340336539035504054m_rule,Y: produc6018962875968178549ist_fm,Z: rule] :
      ( ( X3
        = ( produc1733806532565653680m_rule @ Y @ Z ) )
     => ( ( produc6514711208036230658m_rule @ X3 )
        = Z ) ) ).

% sndI
thf(fact_1016_eq__snd__iff,axiom,
    ! [B: nat,P3: product_prod_nat_nat] :
      ( ( B
        = ( product_snd_nat_nat @ P3 ) )
      = ( ? [A4: nat] :
            ( P3
            = ( product_Pair_nat_nat @ A4 @ B ) ) ) ) ).

% eq_snd_iff
thf(fact_1017_eq__snd__iff,axiom,
    ! [B: product_prod_nat_nat,P3: produc859450856879609959at_nat] :
      ( ( B
        = ( produc6408287024330202629at_nat @ P3 ) )
      = ( ? [A4: product_prod_nat_nat] :
            ( P3
            = ( produc6161850002892822231at_nat @ A4 @ B ) ) ) ) ).

% eq_snd_iff
thf(fact_1018_eq__snd__iff,axiom,
    ! [B: rule,P3: produc340336539035504054m_rule] :
      ( ( B
        = ( produc6514711208036230658m_rule @ P3 ) )
      = ( ? [A4: produc6018962875968178549ist_fm] :
            ( P3
            = ( produc1733806532565653680m_rule @ A4 @ B ) ) ) ) ).

% eq_snd_iff
thf(fact_1019_konig_Osimps_I2_J,axiom,
    ! [T3: abstract_tree_rule] :
      ( ( stl_rule @ ( abstract_konig_rule @ T3 ) )
      = ( abstract_konig_rule
        @ ( fChoic4892627540417248890e_rule
          @ ^ [T9: abstract_tree_rule] :
              ( ( fmembe4260254584891149511e_rule @ T9 @ ( abstract_cont_rule @ T3 ) )
              & ~ ( abstra259309005496137912e_rule @ T9 ) ) ) ) ) ).

% konig.simps(2)
thf(fact_1020_konig_Osimps_I2_J,axiom,
    ! [T3: abstra8520858721168826610m_rule] :
      ( ( stl_Pr12655793849929990m_rule @ ( abstra9098142006323500257m_rule @ T3 ) )
      = ( abstra9098142006323500257m_rule
        @ ( fChoic3358775076371552926m_rule
          @ ^ [T9: abstra8520858721168826610m_rule] :
              ( ( fmembe1099044791951439339m_rule @ T9 @ ( abstra2475333618431710837m_rule @ T3 ) )
              & ~ ( abstra3822255059661934556m_rule @ T9 ) ) ) ) ) ).

% konig.simps(2)
thf(fact_1021_prod_Oswap__def,axiom,
    ( produc5647469659010186090ist_fm
    = ( ^ [P4: produc2353945479044264792ist_fm] : ( produc1733806532565653680m_rule @ ( produc9041034967185247836ist_fm @ P4 ) @ ( produc3520095305172569118ist_fm @ P4 ) ) ) ) ).

% prod.swap_def
thf(fact_1022_prod_Oswap__def,axiom,
    ( product_swap_nat_nat
    = ( ^ [P4: product_prod_nat_nat] : ( product_Pair_nat_nat @ ( product_snd_nat_nat @ P4 ) @ ( product_fst_nat_nat @ P4 ) ) ) ) ).

% prod.swap_def
thf(fact_1023_prod_Oswap__def,axiom,
    ( produc7225117575323628663at_nat
    = ( ^ [P4: produc859450856879609959at_nat] : ( produc6161850002892822231at_nat @ ( produc6408287024330202629at_nat @ P4 ) @ ( produc3213797794245857475at_nat @ P4 ) ) ) ) ).

% prod.swap_def
thf(fact_1024_prod_Oswap__def,axiom,
    ( produc5979693678883077509ist_fm
    = ( ^ [P4: produc6018962875968178549ist_fm] : ( produc7867948443067789129ist_tm @ ( produc3242453064496107283ist_fm @ P4 ) @ ( produc4874987732838623697ist_fm @ P4 ) ) ) ) ).

% prod.swap_def
thf(fact_1025_prod_Oswap__def,axiom,
    ( produc3121145899861168912m_rule
    = ( ^ [P4: produc340336539035504054m_rule] : ( produc4260130291714670858ist_fm @ ( produc6514711208036230658m_rule @ P4 ) @ ( produc993771546023551940m_rule @ P4 ) ) ) ) ).

% prod.swap_def
thf(fact_1026_snd__swap,axiom,
    ! [X3: produc340336539035504054m_rule] :
      ( ( produc9041034967185247836ist_fm @ ( produc3121145899861168912m_rule @ X3 ) )
      = ( produc993771546023551940m_rule @ X3 ) ) ).

% snd_swap
thf(fact_1027_snd__swap,axiom,
    ! [X3: produc6018962875968178549ist_fm] :
      ( ( produc472676704269606519ist_tm @ ( produc5979693678883077509ist_fm @ X3 ) )
      = ( produc4874987732838623697ist_fm @ X3 ) ) ).

% snd_swap
thf(fact_1028_snd__swap,axiom,
    ! [X3: produc2353945479044264792ist_fm] :
      ( ( produc6514711208036230658m_rule @ ( produc5647469659010186090ist_fm @ X3 ) )
      = ( produc3520095305172569118ist_fm @ X3 ) ) ).

% snd_swap
thf(fact_1029_fst__swap,axiom,
    ! [X3: produc2353945479044264792ist_fm] :
      ( ( produc993771546023551940m_rule @ ( produc5647469659010186090ist_fm @ X3 ) )
      = ( produc9041034967185247836ist_fm @ X3 ) ) ).

% fst_swap
thf(fact_1030_fst__swap,axiom,
    ! [X3: produc3002719820330532825ist_tm] :
      ( ( produc4874987732838623697ist_fm @ ( produc3209917318656576745ist_tm @ X3 ) )
      = ( produc472676704269606519ist_tm @ X3 ) ) ).

% fst_swap
thf(fact_1031_fst__swap,axiom,
    ! [X3: produc340336539035504054m_rule] :
      ( ( produc3520095305172569118ist_fm @ ( produc3121145899861168912m_rule @ X3 ) )
      = ( produc6514711208036230658m_rule @ X3 ) ) ).

% fst_swap
thf(fact_1032_konig_Ocode,axiom,
    ( abstra9098142006323500257m_rule
    = ( ^ [T2: abstra8520858721168826610m_rule] :
          ( sCons_2955607467875998431m_rule @ ( abstra5180277867158163941m_rule @ T2 )
          @ ( abstra9098142006323500257m_rule
            @ ( fChoic3358775076371552926m_rule
              @ ^ [T9: abstra8520858721168826610m_rule] :
                  ( ( fmembe1099044791951439339m_rule @ T9 @ ( abstra2475333618431710837m_rule @ T2 ) )
                  & ~ ( abstra3822255059661934556m_rule @ T9 ) ) ) ) ) ) ) ).

% konig.code
thf(fact_1033_HLD__Stream,axiom,
    ! [X5: set_nat,X3: nat,Omega2: stream_nat] :
      ( ( linear5551378545506465448LD_nat @ X5 @ ( sCons_nat @ X3 @ Omega2 ) )
      = ( member_nat @ X3 @ X5 ) ) ).

% HLD_Stream
thf(fact_1034_HLD__Stream,axiom,
    ! [X5: set_Pr5202636777678657877ist_fm,X3: produc6018962875968178549ist_fm,Omega2: stream4408948924543953275ist_fm] :
      ( ( linear6120129154741419165ist_fm @ X5 @ ( sCons_2518607988952603276ist_fm @ X3 @ Omega2 ) )
      = ( member4699826688122452638ist_fm @ X3 @ X5 ) ) ).

% HLD_Stream
thf(fact_1035_HLD__Stream,axiom,
    ! [X5: set_rule,X3: rule,Omega2: stream_rule] :
      ( ( linear5032321834864961386D_rule @ X5 @ ( sCons_rule @ X3 @ Omega2 ) )
      = ( member_rule @ X3 @ X5 ) ) ).

% HLD_Stream
thf(fact_1036_HLD__Stream,axiom,
    ! [X5: set_Pr8693737435421807431at_nat,X3: produc859450856879609959at_nat,Omega2: stream8372878641218411373at_nat] :
      ( ( linear6066394698532776847at_nat @ X5 @ ( sCons_5297226365640217982at_nat @ X3 @ Omega2 ) )
      = ( member8206827879206165904at_nat @ X3 @ X5 ) ) ).

% HLD_Stream
thf(fact_1037_stream_Ocollapse,axiom,
    ! [Stream: stream2709947120125613254m_rule] :
      ( ( sCons_2955607467875998431m_rule @ ( shd_Pr4562317740776619530m_rule @ Stream ) @ ( stl_Pr12655793849929990m_rule @ Stream ) )
      = Stream ) ).

% stream.collapse
thf(fact_1038_stream_Ocollapse,axiom,
    ! [Stream: stream_rule] :
      ( ( sCons_rule @ ( shd_rule @ Stream ) @ ( stl_rule @ Stream ) )
      = Stream ) ).

% stream.collapse
thf(fact_1039_sfilter__P,axiom,
    ! [P2: produc340336539035504054m_rule > $o,S3: stream2709947120125613254m_rule] :
      ( ( P2 @ ( shd_Pr4562317740776619530m_rule @ S3 ) )
     => ( ( sfilte3304650064355440278m_rule @ P2 @ S3 )
        = ( sCons_2955607467875998431m_rule @ ( shd_Pr4562317740776619530m_rule @ S3 ) @ ( sfilte3304650064355440278m_rule @ P2 @ ( stl_Pr12655793849929990m_rule @ S3 ) ) ) ) ) ).

% sfilter_P
thf(fact_1040_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_1041_stream_Osel_I1_J,axiom,
    ! [X1: produc340336539035504054m_rule,X22: stream2709947120125613254m_rule] :
      ( ( shd_Pr4562317740776619530m_rule @ ( sCons_2955607467875998431m_rule @ X1 @ X22 ) )
      = X1 ) ).

% stream.sel(1)
thf(fact_1042_stream_Osel_I1_J,axiom,
    ! [X1: rule,X22: stream_rule] :
      ( ( shd_rule @ ( sCons_rule @ X1 @ X22 ) )
      = X1 ) ).

% stream.sel(1)
thf(fact_1043_stream_Osel_I2_J,axiom,
    ! [X1: produc340336539035504054m_rule,X22: stream2709947120125613254m_rule] :
      ( ( stl_Pr12655793849929990m_rule @ ( sCons_2955607467875998431m_rule @ X1 @ X22 ) )
      = X22 ) ).

% stream.sel(2)
thf(fact_1044_stream_Osel_I2_J,axiom,
    ! [X1: rule,X22: stream_rule] :
      ( ( stl_rule @ ( sCons_rule @ X1 @ X22 ) )
      = X22 ) ).

% stream.sel(2)
thf(fact_1045_stream_Oset__induct,axiom,
    ! [X3: nat,A: stream_nat,P2: nat > stream_nat > $o] :
      ( ( member_nat @ X3 @ ( sset_nat @ A ) )
     => ( ! [Z1: nat,Z22: stream_nat] : ( P2 @ Z1 @ ( sCons_nat @ Z1 @ Z22 ) )
       => ( ! [Z1: nat,Z22: stream_nat,Xa3: nat] :
              ( ( member_nat @ Xa3 @ ( sset_nat @ Z22 ) )
             => ( ( P2 @ Xa3 @ Z22 )
               => ( P2 @ Xa3 @ ( sCons_nat @ Z1 @ Z22 ) ) ) )
         => ( P2 @ X3 @ A ) ) ) ) ).

% stream.set_induct
thf(fact_1046_stream_Oset__induct,axiom,
    ! [X3: produc6018962875968178549ist_fm,A: stream4408948924543953275ist_fm,P2: produc6018962875968178549ist_fm > stream4408948924543953275ist_fm > $o] :
      ( ( member4699826688122452638ist_fm @ X3 @ ( sset_P5379989128061332361ist_fm @ A ) )
     => ( ! [Z1: produc6018962875968178549ist_fm,Z22: stream4408948924543953275ist_fm] : ( P2 @ Z1 @ ( sCons_2518607988952603276ist_fm @ Z1 @ Z22 ) )
       => ( ! [Z1: produc6018962875968178549ist_fm,Z22: stream4408948924543953275ist_fm,Xa3: produc6018962875968178549ist_fm] :
              ( ( member4699826688122452638ist_fm @ Xa3 @ ( sset_P5379989128061332361ist_fm @ Z22 ) )
             => ( ( P2 @ Xa3 @ Z22 )
               => ( P2 @ Xa3 @ ( sCons_2518607988952603276ist_fm @ Z1 @ Z22 ) ) ) )
         => ( P2 @ X3 @ A ) ) ) ) ).

% stream.set_induct
thf(fact_1047_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,Xa3: produc859450856879609959at_nat] :
              ( ( member8206827879206165904at_nat @ Xa3 @ ( sset_P9164763103296375419at_nat @ Z22 ) )
             => ( ( P2 @ Xa3 @ Z22 )
               => ( P2 @ Xa3 @ ( sCons_5297226365640217982at_nat @ Z1 @ Z22 ) ) ) )
         => ( P2 @ X3 @ A ) ) ) ) ).

% stream.set_induct
thf(fact_1048_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,Xa3: rule] :
              ( ( member_rule @ Xa3 @ ( sset_rule @ Z22 ) )
             => ( ( P2 @ Xa3 @ Z22 )
               => ( P2 @ Xa3 @ ( sCons_rule @ Z1 @ Z22 ) ) ) )
         => ( P2 @ X3 @ A ) ) ) ) ).

% stream.set_induct
thf(fact_1049_stream_Oset__cases,axiom,
    ! [E: nat,A: stream_nat] :
      ( ( member_nat @ E @ ( sset_nat @ A ) )
     => ( ! [Z22: stream_nat] :
            ( A
           != ( sCons_nat @ E @ Z22 ) )
       => ~ ! [Z1: nat,Z22: stream_nat] :
              ( ( A
                = ( sCons_nat @ Z1 @ Z22 ) )
             => ~ ( member_nat @ E @ ( sset_nat @ Z22 ) ) ) ) ) ).

% stream.set_cases
thf(fact_1050_stream_Oset__cases,axiom,
    ! [E: produc6018962875968178549ist_fm,A: stream4408948924543953275ist_fm] :
      ( ( member4699826688122452638ist_fm @ E @ ( sset_P5379989128061332361ist_fm @ A ) )
     => ( ! [Z22: stream4408948924543953275ist_fm] :
            ( A
           != ( sCons_2518607988952603276ist_fm @ E @ Z22 ) )
       => ~ ! [Z1: produc6018962875968178549ist_fm,Z22: stream4408948924543953275ist_fm] :
              ( ( A
                = ( sCons_2518607988952603276ist_fm @ Z1 @ Z22 ) )
             => ~ ( member4699826688122452638ist_fm @ E @ ( sset_P5379989128061332361ist_fm @ Z22 ) ) ) ) ) ).

% stream.set_cases
thf(fact_1051_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_1052_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_1053_stream_Oset__intros_I1_J,axiom,
    ! [X1: nat,X22: stream_nat] : ( member_nat @ X1 @ ( sset_nat @ ( sCons_nat @ X1 @ X22 ) ) ) ).

% stream.set_intros(1)
thf(fact_1054_stream_Oset__intros_I1_J,axiom,
    ! [X1: produc6018962875968178549ist_fm,X22: stream4408948924543953275ist_fm] : ( member4699826688122452638ist_fm @ X1 @ ( sset_P5379989128061332361ist_fm @ ( sCons_2518607988952603276ist_fm @ X1 @ X22 ) ) ) ).

% stream.set_intros(1)
thf(fact_1055_stream_Oset__intros_I1_J,axiom,
    ! [X1: produc859450856879609959at_nat,X22: stream8372878641218411373at_nat] : ( member8206827879206165904at_nat @ X1 @ ( sset_P9164763103296375419at_nat @ ( sCons_5297226365640217982at_nat @ X1 @ X22 ) ) ) ).

% stream.set_intros(1)
thf(fact_1056_stream_Oset__intros_I1_J,axiom,
    ! [X1: rule,X22: stream_rule] : ( member_rule @ X1 @ ( sset_rule @ ( sCons_rule @ X1 @ X22 ) ) ) ).

% stream.set_intros(1)
thf(fact_1057_stream_Oset__intros_I2_J,axiom,
    ! [Y: nat,X22: stream_nat,X1: nat] :
      ( ( member_nat @ Y @ ( sset_nat @ X22 ) )
     => ( member_nat @ Y @ ( sset_nat @ ( sCons_nat @ X1 @ X22 ) ) ) ) ).

% stream.set_intros(2)
thf(fact_1058_stream_Oset__intros_I2_J,axiom,
    ! [Y: produc6018962875968178549ist_fm,X22: stream4408948924543953275ist_fm,X1: produc6018962875968178549ist_fm] :
      ( ( member4699826688122452638ist_fm @ Y @ ( sset_P5379989128061332361ist_fm @ X22 ) )
     => ( member4699826688122452638ist_fm @ Y @ ( sset_P5379989128061332361ist_fm @ ( sCons_2518607988952603276ist_fm @ X1 @ X22 ) ) ) ) ).

% stream.set_intros(2)
thf(fact_1059_stream_Oset__intros_I2_J,axiom,
    ! [Y: produc859450856879609959at_nat,X22: stream8372878641218411373at_nat,X1: produc859450856879609959at_nat] :
      ( ( member8206827879206165904at_nat @ Y @ ( sset_P9164763103296375419at_nat @ X22 ) )
     => ( member8206827879206165904at_nat @ Y @ ( sset_P9164763103296375419at_nat @ ( sCons_5297226365640217982at_nat @ X1 @ X22 ) ) ) ) ).

% stream.set_intros(2)
thf(fact_1060_stream_Oset__intros_I2_J,axiom,
    ! [Y: rule,X22: stream_rule,X1: rule] :
      ( ( member_rule @ Y @ ( sset_rule @ X22 ) )
     => ( member_rule @ Y @ ( sset_rule @ ( sCons_rule @ X1 @ X22 ) ) ) ) ).

% stream.set_intros(2)
thf(fact_1061_ev__Stream,axiom,
    ! [P2: stream2709947120125613254m_rule > $o,X3: produc340336539035504054m_rule,S3: stream2709947120125613254m_rule] :
      ( ( linear1816559245158234865m_rule @ P2 @ ( sCons_2955607467875998431m_rule @ X3 @ S3 ) )
      = ( ( P2 @ ( sCons_2955607467875998431m_rule @ X3 @ S3 ) )
        | ( linear1816559245158234865m_rule @ P2 @ S3 ) ) ) ).

% ev_Stream
thf(fact_1062_ev__Stream,axiom,
    ! [P2: stream_rule > $o,X3: rule,S3: stream_rule] :
      ( ( linear292482573288640205v_rule @ P2 @ ( sCons_rule @ X3 @ S3 ) )
      = ( ( P2 @ ( sCons_rule @ X3 @ S3 ) )
        | ( linear292482573288640205v_rule @ P2 @ S3 ) ) ) ).

% ev_Stream
thf(fact_1063_szip__unfold,axiom,
    ! [A: produc6018962875968178549ist_fm,S1: stream4408948924543953275ist_fm,B: rule,S2: stream_rule] :
      ( ( szip_P2924820683901490861m_rule @ ( sCons_2518607988952603276ist_fm @ A @ S1 ) @ ( sCons_rule @ B @ S2 ) )
      = ( sCons_2955607467875998431m_rule @ ( produc1733806532565653680m_rule @ A @ B ) @ ( szip_P2924820683901490861m_rule @ S1 @ S2 ) ) ) ).

% szip_unfold
thf(fact_1064_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_1065_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_1066_holds__Stream,axiom,
    ! [P2: produc340336539035504054m_rule > $o,X3: produc340336539035504054m_rule,S3: stream2709947120125613254m_rule] :
      ( ( linear7925859179493631106m_rule @ P2 @ ( sCons_2955607467875998431m_rule @ X3 @ S3 ) )
      = ( P2 @ X3 ) ) ).

% holds_Stream
thf(fact_1067_holds__Stream,axiom,
    ! [P2: rule > $o,X3: rule,S3: stream_rule] :
      ( ( linear7354821883049305950s_rule @ P2 @ ( sCons_rule @ X3 @ S3 ) )
      = ( P2 @ X3 ) ) ).

% holds_Stream
thf(fact_1068_streams_Ocases,axiom,
    ! [A: stream_nat,A2: set_nat] :
      ( ( member_stream_nat @ A @ ( streams_nat @ A2 ) )
     => ~ ! [A3: nat,S8: stream_nat] :
            ( ( A
              = ( sCons_nat @ A3 @ S8 ) )
           => ( ( member_nat @ A3 @ A2 )
             => ~ ( member_stream_nat @ S8 @ ( streams_nat @ A2 ) ) ) ) ) ).

% streams.cases
thf(fact_1069_streams_Ocases,axiom,
    ! [A: stream4408948924543953275ist_fm,A2: set_Pr5202636777678657877ist_fm] :
      ( ( member197948898507729188ist_fm @ A @ ( stream1217541351357117421ist_fm @ A2 ) )
     => ~ ! [A3: produc6018962875968178549ist_fm,S8: stream4408948924543953275ist_fm] :
            ( ( A
              = ( sCons_2518607988952603276ist_fm @ A3 @ S8 ) )
           => ( ( member4699826688122452638ist_fm @ A3 @ A2 )
             => ~ ( member197948898507729188ist_fm @ S8 @ ( stream1217541351357117421ist_fm @ A2 ) ) ) ) ) ).

% streams.cases
thf(fact_1070_streams_Ocases,axiom,
    ! [A: stream_rule,A2: set_rule] :
      ( ( member_stream_rule @ A @ ( streams_rule @ A2 ) )
     => ~ ! [A3: rule,S8: stream_rule] :
            ( ( A
              = ( sCons_rule @ A3 @ S8 ) )
           => ( ( member_rule @ A3 @ A2 )
             => ~ ( member_stream_rule @ S8 @ ( streams_rule @ A2 ) ) ) ) ) ).

% streams.cases
thf(fact_1071_streams_Ocases,axiom,
    ! [A: stream8372878641218411373at_nat,A2: set_Pr8693737435421807431at_nat] :
      ( ( member1992146023263842070at_nat @ A @ ( stream5885727000947712735at_nat @ A2 ) )
     => ~ ! [A3: produc859450856879609959at_nat,S8: stream8372878641218411373at_nat] :
            ( ( A
              = ( sCons_5297226365640217982at_nat @ A3 @ S8 ) )
           => ( ( member8206827879206165904at_nat @ A3 @ A2 )
             => ~ ( member1992146023263842070at_nat @ S8 @ ( stream5885727000947712735at_nat @ A2 ) ) ) ) ) ).

% streams.cases
thf(fact_1072_streams_Osimps,axiom,
    ! [A: stream_nat,A2: set_nat] :
      ( ( member_stream_nat @ A @ ( streams_nat @ A2 ) )
      = ( ? [A4: nat,S: stream_nat] :
            ( ( A
              = ( sCons_nat @ A4 @ S ) )
            & ( member_nat @ A4 @ A2 )
            & ( member_stream_nat @ S @ ( streams_nat @ A2 ) ) ) ) ) ).

% streams.simps
thf(fact_1073_streams_Osimps,axiom,
    ! [A: stream4408948924543953275ist_fm,A2: set_Pr5202636777678657877ist_fm] :
      ( ( member197948898507729188ist_fm @ A @ ( stream1217541351357117421ist_fm @ A2 ) )
      = ( ? [A4: produc6018962875968178549ist_fm,S: stream4408948924543953275ist_fm] :
            ( ( A
              = ( sCons_2518607988952603276ist_fm @ A4 @ S ) )
            & ( member4699826688122452638ist_fm @ A4 @ A2 )
            & ( member197948898507729188ist_fm @ S @ ( stream1217541351357117421ist_fm @ A2 ) ) ) ) ) ).

% streams.simps
thf(fact_1074_streams_Osimps,axiom,
    ! [A: stream_rule,A2: set_rule] :
      ( ( member_stream_rule @ A @ ( streams_rule @ A2 ) )
      = ( ? [A4: rule,S: stream_rule] :
            ( ( A
              = ( sCons_rule @ A4 @ S ) )
            & ( member_rule @ A4 @ A2 )
            & ( member_stream_rule @ S @ ( streams_rule @ A2 ) ) ) ) ) ).

% streams.simps
thf(fact_1075_streams_Osimps,axiom,
    ! [A: stream8372878641218411373at_nat,A2: set_Pr8693737435421807431at_nat] :
      ( ( member1992146023263842070at_nat @ A @ ( stream5885727000947712735at_nat @ A2 ) )
      = ( ? [A4: produc859450856879609959at_nat,S: stream8372878641218411373at_nat] :
            ( ( A
              = ( sCons_5297226365640217982at_nat @ A4 @ S ) )
            & ( member8206827879206165904at_nat @ A4 @ A2 )
            & ( member1992146023263842070at_nat @ S @ ( stream5885727000947712735at_nat @ A2 ) ) ) ) ) ).

% streams.simps
thf(fact_1076_streams_Ocoinduct,axiom,
    ! [X5: stream_nat > $o,X3: stream_nat,A2: set_nat] :
      ( ( X5 @ X3 )
     => ( ! [X2: stream_nat] :
            ( ( X5 @ X2 )
           => ? [A6: nat,S10: stream_nat] :
                ( ( X2
                  = ( sCons_nat @ A6 @ S10 ) )
                & ( member_nat @ A6 @ A2 )
                & ( ( X5 @ S10 )
                  | ( member_stream_nat @ S10 @ ( streams_nat @ A2 ) ) ) ) )
       => ( member_stream_nat @ X3 @ ( streams_nat @ A2 ) ) ) ) ).

% streams.coinduct
thf(fact_1077_streams_Ocoinduct,axiom,
    ! [X5: stream4408948924543953275ist_fm > $o,X3: stream4408948924543953275ist_fm,A2: set_Pr5202636777678657877ist_fm] :
      ( ( X5 @ X3 )
     => ( ! [X2: stream4408948924543953275ist_fm] :
            ( ( X5 @ X2 )
           => ? [A6: produc6018962875968178549ist_fm,S10: stream4408948924543953275ist_fm] :
                ( ( X2
                  = ( sCons_2518607988952603276ist_fm @ A6 @ S10 ) )
                & ( member4699826688122452638ist_fm @ A6 @ A2 )
                & ( ( X5 @ S10 )
                  | ( member197948898507729188ist_fm @ S10 @ ( stream1217541351357117421ist_fm @ A2 ) ) ) ) )
       => ( member197948898507729188ist_fm @ X3 @ ( stream1217541351357117421ist_fm @ A2 ) ) ) ) ).

% streams.coinduct
thf(fact_1078_streams_Ocoinduct,axiom,
    ! [X5: stream_rule > $o,X3: stream_rule,A2: set_rule] :
      ( ( X5 @ X3 )
     => ( ! [X2: stream_rule] :
            ( ( X5 @ X2 )
           => ? [A6: rule,S10: stream_rule] :
                ( ( X2
                  = ( sCons_rule @ A6 @ S10 ) )
                & ( member_rule @ A6 @ A2 )
                & ( ( X5 @ S10 )
                  | ( member_stream_rule @ S10 @ ( streams_rule @ A2 ) ) ) ) )
       => ( member_stream_rule @ X3 @ ( streams_rule @ A2 ) ) ) ) ).

% streams.coinduct
thf(fact_1079_streams_Ocoinduct,axiom,
    ! [X5: stream8372878641218411373at_nat > $o,X3: stream8372878641218411373at_nat,A2: set_Pr8693737435421807431at_nat] :
      ( ( X5 @ X3 )
     => ( ! [X2: stream8372878641218411373at_nat] :
            ( ( X5 @ X2 )
           => ? [A6: produc859450856879609959at_nat,S10: stream8372878641218411373at_nat] :
                ( ( X2
                  = ( sCons_5297226365640217982at_nat @ A6 @ S10 ) )
                & ( member8206827879206165904at_nat @ A6 @ A2 )
                & ( ( X5 @ S10 )
                  | ( member1992146023263842070at_nat @ S10 @ ( stream5885727000947712735at_nat @ A2 ) ) ) ) )
       => ( member1992146023263842070at_nat @ X3 @ ( stream5885727000947712735at_nat @ A2 ) ) ) ) ).

% streams.coinduct
thf(fact_1080_streams__Stream,axiom,
    ! [X3: nat,S3: stream_nat,A2: set_nat] :
      ( ( member_stream_nat @ ( sCons_nat @ X3 @ S3 ) @ ( streams_nat @ A2 ) )
      = ( ( member_nat @ X3 @ A2 )
        & ( member_stream_nat @ S3 @ ( streams_nat @ A2 ) ) ) ) ).

% streams_Stream
thf(fact_1081_streams__Stream,axiom,
    ! [X3: produc6018962875968178549ist_fm,S3: stream4408948924543953275ist_fm,A2: set_Pr5202636777678657877ist_fm] :
      ( ( member197948898507729188ist_fm @ ( sCons_2518607988952603276ist_fm @ X3 @ S3 ) @ ( stream1217541351357117421ist_fm @ A2 ) )
      = ( ( member4699826688122452638ist_fm @ X3 @ A2 )
        & ( member197948898507729188ist_fm @ S3 @ ( stream1217541351357117421ist_fm @ A2 ) ) ) ) ).

% streams_Stream
thf(fact_1082_streams__Stream,axiom,
    ! [X3: rule,S3: stream_rule,A2: set_rule] :
      ( ( member_stream_rule @ ( sCons_rule @ X3 @ S3 ) @ ( streams_rule @ A2 ) )
      = ( ( member_rule @ X3 @ A2 )
        & ( member_stream_rule @ S3 @ ( streams_rule @ A2 ) ) ) ) ).

% streams_Stream
thf(fact_1083_streams__Stream,axiom,
    ! [X3: produc859450856879609959at_nat,S3: stream8372878641218411373at_nat,A2: set_Pr8693737435421807431at_nat] :
      ( ( member1992146023263842070at_nat @ ( sCons_5297226365640217982at_nat @ X3 @ S3 ) @ ( stream5885727000947712735at_nat @ A2 ) )
      = ( ( member8206827879206165904at_nat @ X3 @ A2 )
        & ( member1992146023263842070at_nat @ S3 @ ( stream5885727000947712735at_nat @ A2 ) ) ) ) ).

% streams_Stream
thf(fact_1084_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_1085_stream_Oexhaust__sel,axiom,
    ! [Stream: stream2709947120125613254m_rule] :
      ( Stream
      = ( sCons_2955607467875998431m_rule @ ( shd_Pr4562317740776619530m_rule @ Stream ) @ ( stl_Pr12655793849929990m_rule @ Stream ) ) ) ).

% stream.exhaust_sel
thf(fact_1086_stream_Oexhaust__sel,axiom,
    ! [Stream: stream_rule] :
      ( Stream
      = ( sCons_rule @ ( shd_rule @ Stream ) @ ( stl_rule @ Stream ) ) ) ).

% stream.exhaust_sel
thf(fact_1087_alw__holds,axiom,
    ! [P2: produc340336539035504054m_rule > $o,H: produc340336539035504054m_rule,T3: stream2709947120125613254m_rule] :
      ( ( linear4957294957385550906m_rule @ ( linear7925859179493631106m_rule @ P2 ) @ ( sCons_2955607467875998431m_rule @ H @ T3 ) )
      = ( ( P2 @ H )
        & ( linear4957294957385550906m_rule @ ( linear7925859179493631106m_rule @ P2 ) @ T3 ) ) ) ).

% alw_holds
thf(fact_1088_alw__holds,axiom,
    ! [P2: rule > $o,H: rule,T3: stream_rule] :
      ( ( linear1087404401031714582w_rule @ ( linear7354821883049305950s_rule @ P2 ) @ ( sCons_rule @ H @ T3 ) )
      = ( ( P2 @ H )
        & ( linear1087404401031714582w_rule @ ( linear7354821883049305950s_rule @ P2 ) @ T3 ) ) ) ).

% alw_holds
thf(fact_1089_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_1090_verit__comp__simplify1_I2_J,axiom,
    ! [A: nat] : ( ord_less_eq_nat @ A @ A ) ).

% verit_comp_simplify1(2)
thf(fact_1091_verit__comp__simplify1_I2_J,axiom,
    ! [A: set_rule] : ( ord_less_eq_set_rule @ A @ A ) ).

% verit_comp_simplify1(2)
thf(fact_1092_verit__comp__simplify1_I2_J,axiom,
    ! [A: set_tm] : ( ord_less_eq_set_tm @ A @ A ) ).

% verit_comp_simplify1(2)
thf(fact_1093_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_1094_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_1095_szip_Ocode,axiom,
    ( szip_P2924820683901490861m_rule
    = ( ^ [S12: stream4408948924543953275ist_fm,S22: stream_rule] : ( sCons_2955607467875998431m_rule @ ( produc1733806532565653680m_rule @ ( shd_Pr3211216682057661985ist_fm @ S12 ) @ ( shd_rule @ S22 ) ) @ ( szip_P2924820683901490861m_rule @ ( stl_Pr1506262294867171877ist_fm @ S12 ) @ ( stl_rule @ S22 ) ) ) ) ) ).

% szip.code
thf(fact_1096_szip_Ocode,axiom,
    ( szip_P811719526838699976m_rule
    = ( ^ [S12: stream2709947120125613254m_rule,S22: stream2709947120125613254m_rule] : ( sCons_8419621085341357412m_rule @ ( produc6261311607089640965m_rule @ ( shd_Pr4562317740776619530m_rule @ S12 ) @ ( shd_Pr4562317740776619530m_rule @ S22 ) ) @ ( szip_P811719526838699976m_rule @ ( stl_Pr12655793849929990m_rule @ S12 ) @ ( stl_Pr12655793849929990m_rule @ S22 ) ) ) ) ) ).

% szip.code
thf(fact_1097_szip_Ocode,axiom,
    ( szip_P2044787555563963556e_rule
    = ( ^ [S12: stream2709947120125613254m_rule,S22: stream_rule] : ( sCons_2189745227781010240e_rule @ ( produc4831648765031708129e_rule @ ( shd_Pr4562317740776619530m_rule @ S12 ) @ ( shd_rule @ S22 ) ) @ ( szip_P2044787555563963556e_rule @ ( stl_Pr12655793849929990m_rule @ S12 ) @ ( stl_rule @ S22 ) ) ) ) ) ).

% szip.code
thf(fact_1098_szip_Ocode,axiom,
    ( szip_r6533552703345879972m_rule
    = ( ^ [S12: stream_rule,S22: stream2709947120125613254m_rule] : ( sCons_997535893981897792m_rule @ ( produc97041875958848737m_rule @ ( shd_rule @ S12 ) @ ( shd_Pr4562317740776619530m_rule @ S22 ) ) @ ( szip_r6533552703345879972m_rule @ ( stl_rule @ S12 ) @ ( stl_Pr12655793849929990m_rule @ S22 ) ) ) ) ) ).

% szip.code
thf(fact_1099_szip_Ocode,axiom,
    ( szip_rule_rule
    = ( ^ [S12: stream_rule,S22: stream_rule] : ( sCons_5631646851396707868e_rule @ ( produc5849431337705160893e_rule @ ( shd_rule @ S12 ) @ ( shd_rule @ S22 ) ) @ ( szip_rule_rule @ ( stl_rule @ S12 ) @ ( stl_rule @ S22 ) ) ) ) ) ).

% szip.code
thf(fact_1100_someI2__bex,axiom,
    ! [A2: set_nat,P2: nat > $o,Q: nat > $o] :
      ( ? [X4: nat] :
          ( ( member_nat @ X4 @ A2 )
          & ( P2 @ X4 ) )
     => ( ! [X2: nat] :
            ( ( ( member_nat @ X2 @ A2 )
              & ( P2 @ X2 ) )
           => ( Q @ X2 ) )
       => ( Q
          @ ( fChoice_nat
            @ ^ [X: nat] :
                ( ( member_nat @ X @ A2 )
                & ( P2 @ X ) ) ) ) ) ) ).

% someI2_bex
thf(fact_1101_someI2__bex,axiom,
    ! [A2: set_Pr5202636777678657877ist_fm,P2: produc6018962875968178549ist_fm > $o,Q: produc6018962875968178549ist_fm > $o] :
      ( ? [X4: produc6018962875968178549ist_fm] :
          ( ( member4699826688122452638ist_fm @ X4 @ A2 )
          & ( P2 @ X4 ) )
     => ( ! [X2: produc6018962875968178549ist_fm] :
            ( ( ( member4699826688122452638ist_fm @ X2 @ A2 )
              & ( P2 @ X2 ) )
           => ( Q @ X2 ) )
       => ( Q
          @ ( fChoic652867151139883209ist_fm
            @ ^ [X: produc6018962875968178549ist_fm] :
                ( ( member4699826688122452638ist_fm @ X @ A2 )
                & ( P2 @ X ) ) ) ) ) ) ).

% someI2_bex
thf(fact_1102_someI2__bex,axiom,
    ! [A2: set_rule,P2: rule > $o,Q: rule > $o] :
      ( ? [X4: rule] :
          ( ( member_rule @ X4 @ A2 )
          & ( P2 @ X4 ) )
     => ( ! [X2: rule] :
            ( ( ( member_rule @ X2 @ A2 )
              & ( P2 @ X2 ) )
           => ( Q @ X2 ) )
       => ( Q
          @ ( fChoice_rule
            @ ^ [X: rule] :
                ( ( member_rule @ X @ A2 )
                & ( P2 @ X ) ) ) ) ) ) ).

% someI2_bex
thf(fact_1103_someI2__bex,axiom,
    ! [A2: set_Pr8693737435421807431at_nat,P2: produc859450856879609959at_nat > $o,Q: produc859450856879609959at_nat > $o] :
      ( ? [X4: produc859450856879609959at_nat] :
          ( ( member8206827879206165904at_nat @ X4 @ A2 )
          & ( P2 @ X4 ) )
     => ( ! [X2: produc859450856879609959at_nat] :
            ( ( ( member8206827879206165904at_nat @ X2 @ A2 )
              & ( P2 @ X2 ) )
           => ( Q @ X2 ) )
       => ( Q
          @ ( fChoic6673916560228149179at_nat
            @ ^ [X: produc859450856879609959at_nat] :
                ( ( member8206827879206165904at_nat @ X @ A2 )
                & ( P2 @ X ) ) ) ) ) ) ).

% someI2_bex
thf(fact_1104_sinterleave_Ocode,axiom,
    ( sinter2808979888547679129m_rule
    = ( ^ [S12: stream2709947120125613254m_rule,S22: stream2709947120125613254m_rule] : ( sCons_2955607467875998431m_rule @ ( shd_Pr4562317740776619530m_rule @ S12 ) @ ( sinter2808979888547679129m_rule @ S22 @ ( stl_Pr12655793849929990m_rule @ S12 ) ) ) ) ) ).

% sinterleave.code
thf(fact_1105_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_1106_stream_Osplit__sel__asm,axiom,
    ! [P2: abstra8520858721168826610m_rule > $o,F: rule > stream_rule > abstra8520858721168826610m_rule,Stream: stream_rule] :
      ( ( P2 @ ( case_s7672397449011031346m_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_1107_stream_Osplit__sel,axiom,
    ! [P2: abstra8520858721168826610m_rule > $o,F: rule > stream_rule > abstra8520858721168826610m_rule,Stream: stream_rule] :
      ( ( P2 @ ( case_s7672397449011031346m_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_1108_sfilter_Ocode,axiom,
    ( sfilte3304650064355440278m_rule
    = ( ^ [P: produc340336539035504054m_rule > $o,S: stream2709947120125613254m_rule] : ( sCons_2955607467875998431m_rule @ ( shd_Pr4562317740776619530m_rule @ ( sdrop_8181906248673349714m_rule @ ( comp_o1746713580868586393m_rule @ (~) @ P ) @ S ) ) @ ( sfilte3304650064355440278m_rule @ P @ ( stl_Pr12655793849929990m_rule @ ( sdrop_8181906248673349714m_rule @ ( comp_o1746713580868586393m_rule @ (~) @ P ) @ S ) ) ) ) ) ) ).

% sfilter.code
thf(fact_1109_sfilter_Ocode,axiom,
    ( sfilter_rule
    = ( ^ [P: rule > $o,S: stream_rule] : ( sCons_rule @ ( shd_rule @ ( sdrop_while_rule @ ( comp_o_o_rule @ (~) @ P ) @ S ) ) @ ( sfilter_rule @ P @ ( stl_rule @ ( sdrop_while_rule @ ( comp_o_o_rule @ (~) @ P ) @ S ) ) ) ) ) ) ).

% sfilter.code
thf(fact_1110_wf_Ocoinduct,axiom,
    ! [X5: abstra8520858721168826610m_rule > $o,X3: abstra8520858721168826610m_rule] :
      ( ( X5 @ X3 )
     => ( ! [X2: abstra8520858721168826610m_rule] :
            ( ( X5 @ X2 )
           => ? [T6: abstra8520858721168826610m_rule] :
                ( ( X2 = T6 )
                & ( member_rule @ ( produc6514711208036230658m_rule @ ( abstra5180277867158163941m_rule @ T6 ) ) @ ( sset_rule @ rules ) )
                & ( eff @ ( produc6514711208036230658m_rule @ ( abstra5180277867158163941m_rule @ T6 ) ) @ ( produc993771546023551940m_rule @ ( abstra5180277867158163941m_rule @ T6 ) ) @ ( fimage3741683731962187332ist_fm @ ( comp_P5969275118686715022m_rule @ produc993771546023551940m_rule @ abstra5180277867158163941m_rule ) @ ( abstra2475333618431710837m_rule @ T6 ) ) )
                & ! [Xa3: abstra8520858721168826610m_rule] :
                    ( ( fmembe1099044791951439339m_rule @ Xa3 @ ( abstra2475333618431710837m_rule @ T6 ) )
                   => ( ( X5 @ Xa3 )
                      | ( abstra8167393772827826667ist_fm @ eff @ rules @ Xa3 ) ) ) ) )
       => ( abstra8167393772827826667ist_fm @ eff @ rules @ X3 ) ) ) ).

% wf.coinduct
thf(fact_1111_stream_Ocase__distrib,axiom,
    ! [H: abstra8520858721168826610m_rule > abstra8520858721168826610m_rule,F: rule > stream_rule > abstra8520858721168826610m_rule,Stream: stream_rule] :
      ( ( H @ ( case_s7672397449011031346m_rule @ F @ Stream ) )
      = ( case_s7672397449011031346m_rule
        @ ^ [X13: rule,X24: stream_rule] : ( H @ ( F @ X13 @ X24 ) )
        @ Stream ) ) ).

% stream.case_distrib
thf(fact_1112_sinterleave_Osimps_I1_J,axiom,
    ! [S1: stream2709947120125613254m_rule,S2: stream2709947120125613254m_rule] :
      ( ( shd_Pr4562317740776619530m_rule @ ( sinter2808979888547679129m_rule @ S1 @ S2 ) )
      = ( shd_Pr4562317740776619530m_rule @ S1 ) ) ).

% sinterleave.simps(1)
thf(fact_1113_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_1114_sinterleave_Osimps_I2_J,axiom,
    ! [S1: stream2709947120125613254m_rule,S2: stream2709947120125613254m_rule] :
      ( ( stl_Pr12655793849929990m_rule @ ( sinter2808979888547679129m_rule @ S1 @ S2 ) )
      = ( sinter2808979888547679129m_rule @ S2 @ ( stl_Pr12655793849929990m_rule @ S1 ) ) ) ).

% sinterleave.simps(2)
thf(fact_1115_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_1116_stream_Ocase,axiom,
    ! [F: rule > stream_rule > abstra8520858721168826610m_rule,X1: rule,X22: stream_rule] :
      ( ( case_s7672397449011031346m_rule @ F @ ( sCons_rule @ X1 @ X22 ) )
      = ( F @ X1 @ X22 ) ) ).

% stream.case
thf(fact_1117_stl__def,axiom,
    ( stl_Pr12655793849929990m_rule
    = ( case_s7424745816878926122m_rule
      @ ^ [X13: produc340336539035504054m_rule,X24: stream2709947120125613254m_rule] : X24 ) ) ).

% stl_def
thf(fact_1118_stl__def,axiom,
    ( stl_rule
    = ( case_s2939844242074606818m_rule
      @ ^ [X13: rule,X24: stream_rule] : X24 ) ) ).

% stl_def
thf(fact_1119_shd__def,axiom,
    ( shd_Pr4562317740776619530m_rule
    = ( case_s9014419740221122586m_rule
      @ ^ [X13: produc340336539035504054m_rule,X24: stream2709947120125613254m_rule] : X13 ) ) ).

% shd_def
thf(fact_1120_shd__def,axiom,
    ( shd_rule
    = ( case_s8060621433503779026e_rule
      @ ^ [X13: rule,X24: stream_rule] : X13 ) ) ).

% shd_def
thf(fact_1121_sfilter_Osimps_I1_J,axiom,
    ! [P2: produc340336539035504054m_rule > $o,S3: stream2709947120125613254m_rule] :
      ( ( shd_Pr4562317740776619530m_rule @ ( sfilte3304650064355440278m_rule @ P2 @ S3 ) )
      = ( shd_Pr4562317740776619530m_rule @ ( sdrop_8181906248673349714m_rule @ ( comp_o1746713580868586393m_rule @ (~) @ P2 ) @ S3 ) ) ) ).

% sfilter.simps(1)
thf(fact_1122_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_1123_sfilter_Osimps_I2_J,axiom,
    ! [P2: produc340336539035504054m_rule > $o,S3: stream2709947120125613254m_rule] :
      ( ( stl_Pr12655793849929990m_rule @ ( sfilte3304650064355440278m_rule @ P2 @ S3 ) )
      = ( sfilte3304650064355440278m_rule @ P2 @ ( stl_Pr12655793849929990m_rule @ ( sdrop_8181906248673349714m_rule @ ( comp_o1746713580868586393m_rule @ (~) @ P2 ) @ S3 ) ) ) ) ).

% sfilter.simps(2)
thf(fact_1124_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_1125_RuleSystem__Defs_Owf_Ocoinduct,axiom,
    ! [X5: abstra4463927080622938191ist_fm > $o,X3: abstra4463927080622938191ist_fm,Rules2: stream_list_fm,Eff4: list_fm > list_tm > fset_list_tm > $o] :
      ( ( X5 @ X3 )
     => ( ! [X2: abstra4463927080622938191ist_fm] :
            ( ( X5 @ X2 )
           => ? [T6: abstra4463927080622938191ist_fm] :
                ( ( X2 = T6 )
                & ( member_list_fm @ ( produc3242453064496107283ist_fm @ ( abstra2656519403522841862ist_fm @ T6 ) ) @ ( sset_list_fm @ Rules2 ) )
                & ( Eff4 @ ( produc3242453064496107283ist_fm @ ( abstra2656519403522841862ist_fm @ T6 ) ) @ ( produc4874987732838623697ist_fm @ ( abstra2656519403522841862ist_fm @ T6 ) ) @ ( fimage3333425779666891737ist_tm @ ( comp_P8595014820038962806ist_fm @ produc4874987732838623697ist_fm @ abstra2656519403522841862ist_fm ) @ ( abstra417404627459193974ist_fm @ T6 ) ) )
                & ! [Xa3: abstra4463927080622938191ist_fm] :
                    ( ( fmembe7982432732373538198ist_fm @ Xa3 @ ( abstra417404627459193974ist_fm @ T6 ) )
                   => ( ( X5 @ Xa3 )
                      | ( abstra7376083527646823080ist_tm @ Eff4 @ Rules2 @ Xa3 ) ) ) ) )
       => ( abstra7376083527646823080ist_tm @ Eff4 @ Rules2 @ X3 ) ) ) ).

% RuleSystem_Defs.wf.coinduct
thf(fact_1126_RuleSystem__Defs_Owf_Ocoinduct,axiom,
    ! [X5: abstra8520858721168826610m_rule > $o,X3: abstra8520858721168826610m_rule,Rules2: stream_rule,Eff4: rule > produc6018962875968178549ist_fm > fset_P8989946509869081563ist_fm > $o] :
      ( ( X5 @ X3 )
     => ( ! [X2: abstra8520858721168826610m_rule] :
            ( ( X5 @ X2 )
           => ? [T6: abstra8520858721168826610m_rule] :
                ( ( X2 = T6 )
                & ( member_rule @ ( produc6514711208036230658m_rule @ ( abstra5180277867158163941m_rule @ T6 ) ) @ ( sset_rule @ Rules2 ) )
                & ( Eff4 @ ( produc6514711208036230658m_rule @ ( abstra5180277867158163941m_rule @ T6 ) ) @ ( produc993771546023551940m_rule @ ( abstra5180277867158163941m_rule @ T6 ) ) @ ( fimage3741683731962187332ist_fm @ ( comp_P5969275118686715022m_rule @ produc993771546023551940m_rule @ abstra5180277867158163941m_rule ) @ ( abstra2475333618431710837m_rule @ T6 ) ) )
                & ! [Xa3: abstra8520858721168826610m_rule] :
                    ( ( fmembe1099044791951439339m_rule @ Xa3 @ ( abstra2475333618431710837m_rule @ T6 ) )
                   => ( ( X5 @ Xa3 )
                      | ( abstra8167393772827826667ist_fm @ Eff4 @ Rules2 @ Xa3 ) ) ) ) )
       => ( abstra8167393772827826667ist_fm @ Eff4 @ Rules2 @ X3 ) ) ) ).

% RuleSystem_Defs.wf.coinduct
thf(fact_1127_RuleSystem__Defs_Owf_Ointros,axiom,
    ! [T3: abstra4463927080622938191ist_fm,Rules2: stream_list_fm,Eff4: list_fm > list_tm > fset_list_tm > $o] :
      ( ( member_list_fm @ ( produc3242453064496107283ist_fm @ ( abstra2656519403522841862ist_fm @ T3 ) ) @ ( sset_list_fm @ Rules2 ) )
     => ( ( Eff4 @ ( produc3242453064496107283ist_fm @ ( abstra2656519403522841862ist_fm @ T3 ) ) @ ( produc4874987732838623697ist_fm @ ( abstra2656519403522841862ist_fm @ T3 ) ) @ ( fimage3333425779666891737ist_tm @ ( comp_P8595014820038962806ist_fm @ produc4874987732838623697ist_fm @ abstra2656519403522841862ist_fm ) @ ( abstra417404627459193974ist_fm @ T3 ) ) )
       => ( ! [T8: abstra4463927080622938191ist_fm] :
              ( ( fmembe7982432732373538198ist_fm @ T8 @ ( abstra417404627459193974ist_fm @ T3 ) )
             => ( abstra7376083527646823080ist_tm @ Eff4 @ Rules2 @ T8 ) )
         => ( abstra7376083527646823080ist_tm @ Eff4 @ Rules2 @ T3 ) ) ) ) ).

% RuleSystem_Defs.wf.intros
thf(fact_1128_RuleSystem__Defs_Owf_Ointros,axiom,
    ! [T3: abstra8520858721168826610m_rule,Rules2: stream_rule,Eff4: rule > produc6018962875968178549ist_fm > fset_P8989946509869081563ist_fm > $o] :
      ( ( member_rule @ ( produc6514711208036230658m_rule @ ( abstra5180277867158163941m_rule @ T3 ) ) @ ( sset_rule @ Rules2 ) )
     => ( ( Eff4 @ ( produc6514711208036230658m_rule @ ( abstra5180277867158163941m_rule @ T3 ) ) @ ( produc993771546023551940m_rule @ ( abstra5180277867158163941m_rule @ T3 ) ) @ ( fimage3741683731962187332ist_fm @ ( comp_P5969275118686715022m_rule @ produc993771546023551940m_rule @ abstra5180277867158163941m_rule ) @ ( abstra2475333618431710837m_rule @ T3 ) ) )
       => ( ! [T8: abstra8520858721168826610m_rule] :
              ( ( fmembe1099044791951439339m_rule @ T8 @ ( abstra2475333618431710837m_rule @ T3 ) )
             => ( abstra8167393772827826667ist_fm @ Eff4 @ Rules2 @ T8 ) )
         => ( abstra8167393772827826667ist_fm @ Eff4 @ Rules2 @ T3 ) ) ) ) ).

% RuleSystem_Defs.wf.intros
thf(fact_1129_RuleSystem__Defs_Owf_Osimps,axiom,
    ( abstra7376083527646823080ist_tm
    = ( ^ [Eff: list_fm > list_tm > fset_list_tm > $o,Rules: stream_list_fm,A4: abstra4463927080622938191ist_fm] :
        ? [T2: abstra4463927080622938191ist_fm] :
          ( ( A4 = T2 )
          & ( member_list_fm @ ( produc3242453064496107283ist_fm @ ( abstra2656519403522841862ist_fm @ T2 ) ) @ ( sset_list_fm @ Rules ) )
          & ( Eff @ ( produc3242453064496107283ist_fm @ ( abstra2656519403522841862ist_fm @ T2 ) ) @ ( produc4874987732838623697ist_fm @ ( abstra2656519403522841862ist_fm @ T2 ) ) @ ( fimage3333425779666891737ist_tm @ ( comp_P8595014820038962806ist_fm @ produc4874987732838623697ist_fm @ abstra2656519403522841862ist_fm ) @ ( abstra417404627459193974ist_fm @ T2 ) ) )
          & ! [X: abstra4463927080622938191ist_fm] :
              ( ( fmembe7982432732373538198ist_fm @ X @ ( abstra417404627459193974ist_fm @ T2 ) )
             => ( abstra7376083527646823080ist_tm @ Eff @ Rules @ X ) ) ) ) ) ).

% RuleSystem_Defs.wf.simps
thf(fact_1130_RuleSystem__Defs_Owf_Osimps,axiom,
    ( abstra8167393772827826667ist_fm
    = ( ^ [Eff: rule > produc6018962875968178549ist_fm > fset_P8989946509869081563ist_fm > $o,Rules: stream_rule,A4: abstra8520858721168826610m_rule] :
        ? [T2: abstra8520858721168826610m_rule] :
          ( ( A4 = T2 )
          & ( member_rule @ ( produc6514711208036230658m_rule @ ( abstra5180277867158163941m_rule @ T2 ) ) @ ( sset_rule @ Rules ) )
          & ( Eff @ ( produc6514711208036230658m_rule @ ( abstra5180277867158163941m_rule @ T2 ) ) @ ( produc993771546023551940m_rule @ ( abstra5180277867158163941m_rule @ T2 ) ) @ ( fimage3741683731962187332ist_fm @ ( comp_P5969275118686715022m_rule @ produc993771546023551940m_rule @ abstra5180277867158163941m_rule ) @ ( abstra2475333618431710837m_rule @ T2 ) ) )
          & ! [X: abstra8520858721168826610m_rule] :
              ( ( fmembe1099044791951439339m_rule @ X @ ( abstra2475333618431710837m_rule @ T2 ) )
             => ( abstra8167393772827826667ist_fm @ Eff @ Rules @ X ) ) ) ) ) ).

% RuleSystem_Defs.wf.simps
thf(fact_1131_RuleSystem__Defs_Owf_Ocases,axiom,
    ! [Eff4: list_fm > list_tm > fset_list_tm > $o,Rules2: stream_list_fm,A: abstra4463927080622938191ist_fm] :
      ( ( abstra7376083527646823080ist_tm @ Eff4 @ Rules2 @ A )
     => ~ ( ( member_list_fm @ ( produc3242453064496107283ist_fm @ ( abstra2656519403522841862ist_fm @ A ) ) @ ( sset_list_fm @ Rules2 ) )
         => ( ( Eff4 @ ( produc3242453064496107283ist_fm @ ( abstra2656519403522841862ist_fm @ A ) ) @ ( produc4874987732838623697ist_fm @ ( abstra2656519403522841862ist_fm @ A ) ) @ ( fimage3333425779666891737ist_tm @ ( comp_P8595014820038962806ist_fm @ produc4874987732838623697ist_fm @ abstra2656519403522841862ist_fm ) @ ( abstra417404627459193974ist_fm @ A ) ) )
           => ~ ! [T7: abstra4463927080622938191ist_fm] :
                  ( ( fmembe7982432732373538198ist_fm @ T7 @ ( abstra417404627459193974ist_fm @ A ) )
                 => ( abstra7376083527646823080ist_tm @ Eff4 @ Rules2 @ T7 ) ) ) ) ) ).

% RuleSystem_Defs.wf.cases
thf(fact_1132_RuleSystem__Defs_Owf_Ocases,axiom,
    ! [Eff4: rule > produc6018962875968178549ist_fm > fset_P8989946509869081563ist_fm > $o,Rules2: stream_rule,A: abstra8520858721168826610m_rule] :
      ( ( abstra8167393772827826667ist_fm @ Eff4 @ Rules2 @ A )
     => ~ ( ( member_rule @ ( produc6514711208036230658m_rule @ ( abstra5180277867158163941m_rule @ A ) ) @ ( sset_rule @ Rules2 ) )
         => ( ( Eff4 @ ( produc6514711208036230658m_rule @ ( abstra5180277867158163941m_rule @ A ) ) @ ( produc993771546023551940m_rule @ ( abstra5180277867158163941m_rule @ A ) ) @ ( fimage3741683731962187332ist_fm @ ( comp_P5969275118686715022m_rule @ produc993771546023551940m_rule @ abstra5180277867158163941m_rule ) @ ( abstra2475333618431710837m_rule @ A ) ) )
           => ~ ! [T7: abstra8520858721168826610m_rule] :
                  ( ( fmembe1099044791951439339m_rule @ T7 @ ( abstra2475333618431710837m_rule @ A ) )
                 => ( abstra8167393772827826667ist_fm @ Eff4 @ Rules2 @ T7 ) ) ) ) ) ).

% RuleSystem_Defs.wf.cases
thf(fact_1133_stream_Ocase__eq__if,axiom,
    ( case_s7672397449011031346m_rule
    = ( ^ [F2: rule > stream_rule > abstra8520858721168826610m_rule,Stream5: stream_rule] : ( F2 @ ( shd_rule @ Stream5 ) @ ( stl_rule @ Stream5 ) ) ) ) ).

% stream.case_eq_if
thf(fact_1134_i_Owf_Ocoinduct,axiom,
    ! [X5: abstra4463927080622938191ist_fm > $o,X3: abstra4463927080622938191ist_fm,Rules2: stream_list_fm,Eff2: list_fm > list_tm > option_fset_list_tm] :
      ( ( X5 @ X3 )
     => ( ! [X2: abstra4463927080622938191ist_fm] :
            ( ( X5 @ X2 )
           => ? [T6: abstra4463927080622938191ist_fm] :
                ( ( X2 = T6 )
                & ( member_list_fm @ ( produc3242453064496107283ist_fm @ ( abstra2656519403522841862ist_fm @ T6 ) ) @ ( sset_list_fm @ Rules2 ) )
                & ( abstra9141620886324064114ist_tm @ Eff2 @ ( produc3242453064496107283ist_fm @ ( abstra2656519403522841862ist_fm @ T6 ) ) @ ( produc4874987732838623697ist_fm @ ( abstra2656519403522841862ist_fm @ T6 ) ) @ ( fimage3333425779666891737ist_tm @ ( comp_P8595014820038962806ist_fm @ produc4874987732838623697ist_fm @ abstra2656519403522841862ist_fm ) @ ( abstra417404627459193974ist_fm @ T6 ) ) )
                & ! [Xa3: abstra4463927080622938191ist_fm] :
                    ( ( fmembe7982432732373538198ist_fm @ Xa3 @ ( abstra417404627459193974ist_fm @ T6 ) )
                   => ( ( X5 @ Xa3 )
                      | ( abstra7376083527646823080ist_tm @ ( abstra9141620886324064114ist_tm @ Eff2 ) @ Rules2 @ Xa3 ) ) ) ) )
       => ( abstra7376083527646823080ist_tm @ ( abstra9141620886324064114ist_tm @ Eff2 ) @ Rules2 @ X3 ) ) ) ).

% i.wf.coinduct
thf(fact_1135_i_Owf_Ocoinduct,axiom,
    ! [X5: abstra8520858721168826610m_rule > $o,X3: abstra8520858721168826610m_rule,Rules2: stream_rule,Eff2: rule > produc6018962875968178549ist_fm > option6967287582980624417ist_fm] :
      ( ( X5 @ X3 )
     => ( ! [X2: abstra8520858721168826610m_rule] :
            ( ( X5 @ X2 )
           => ? [T6: abstra8520858721168826610m_rule] :
                ( ( X2 = T6 )
                & ( member_rule @ ( produc6514711208036230658m_rule @ ( abstra5180277867158163941m_rule @ T6 ) ) @ ( sset_rule @ Rules2 ) )
                & ( abstra2682625350522704545ist_fm @ Eff2 @ ( produc6514711208036230658m_rule @ ( abstra5180277867158163941m_rule @ T6 ) ) @ ( produc993771546023551940m_rule @ ( abstra5180277867158163941m_rule @ T6 ) ) @ ( fimage3741683731962187332ist_fm @ ( comp_P5969275118686715022m_rule @ produc993771546023551940m_rule @ abstra5180277867158163941m_rule ) @ ( abstra2475333618431710837m_rule @ T6 ) ) )
                & ! [Xa3: abstra8520858721168826610m_rule] :
                    ( ( fmembe1099044791951439339m_rule @ Xa3 @ ( abstra2475333618431710837m_rule @ T6 ) )
                   => ( ( X5 @ Xa3 )
                      | ( abstra8167393772827826667ist_fm @ ( abstra2682625350522704545ist_fm @ Eff2 ) @ Rules2 @ Xa3 ) ) ) ) )
       => ( abstra8167393772827826667ist_fm @ ( abstra2682625350522704545ist_fm @ Eff2 ) @ Rules2 @ X3 ) ) ) ).

% i.wf.coinduct
thf(fact_1136_i_Owf_Ointros,axiom,
    ! [T3: abstra4463927080622938191ist_fm,Rules2: stream_list_fm,Eff2: list_fm > list_tm > option_fset_list_tm] :
      ( ( member_list_fm @ ( produc3242453064496107283ist_fm @ ( abstra2656519403522841862ist_fm @ T3 ) ) @ ( sset_list_fm @ Rules2 ) )
     => ( ( abstra9141620886324064114ist_tm @ Eff2 @ ( produc3242453064496107283ist_fm @ ( abstra2656519403522841862ist_fm @ T3 ) ) @ ( produc4874987732838623697ist_fm @ ( abstra2656519403522841862ist_fm @ T3 ) ) @ ( fimage3333425779666891737ist_tm @ ( comp_P8595014820038962806ist_fm @ produc4874987732838623697ist_fm @ abstra2656519403522841862ist_fm ) @ ( abstra417404627459193974ist_fm @ T3 ) ) )
       => ( ! [T8: abstra4463927080622938191ist_fm] :
              ( ( fmembe7982432732373538198ist_fm @ T8 @ ( abstra417404627459193974ist_fm @ T3 ) )
             => ( abstra7376083527646823080ist_tm @ ( abstra9141620886324064114ist_tm @ Eff2 ) @ Rules2 @ T8 ) )
         => ( abstra7376083527646823080ist_tm @ ( abstra9141620886324064114ist_tm @ Eff2 ) @ Rules2 @ T3 ) ) ) ) ).

% i.wf.intros
thf(fact_1137_i_Owf_Ointros,axiom,
    ! [T3: abstra8520858721168826610m_rule,Rules2: stream_rule,Eff2: rule > produc6018962875968178549ist_fm > option6967287582980624417ist_fm] :
      ( ( member_rule @ ( produc6514711208036230658m_rule @ ( abstra5180277867158163941m_rule @ T3 ) ) @ ( sset_rule @ Rules2 ) )
     => ( ( abstra2682625350522704545ist_fm @ Eff2 @ ( produc6514711208036230658m_rule @ ( abstra5180277867158163941m_rule @ T3 ) ) @ ( produc993771546023551940m_rule @ ( abstra5180277867158163941m_rule @ T3 ) ) @ ( fimage3741683731962187332ist_fm @ ( comp_P5969275118686715022m_rule @ produc993771546023551940m_rule @ abstra5180277867158163941m_rule ) @ ( abstra2475333618431710837m_rule @ T3 ) ) )
       => ( ! [T8: abstra8520858721168826610m_rule] :
              ( ( fmembe1099044791951439339m_rule @ T8 @ ( abstra2475333618431710837m_rule @ T3 ) )
             => ( abstra8167393772827826667ist_fm @ ( abstra2682625350522704545ist_fm @ Eff2 ) @ Rules2 @ T8 ) )
         => ( abstra8167393772827826667ist_fm @ ( abstra2682625350522704545ist_fm @ Eff2 ) @ Rules2 @ T3 ) ) ) ) ).

% i.wf.intros
thf(fact_1138_i_Owf_Osimps,axiom,
    ! [Eff2: list_fm > list_tm > option_fset_list_tm,Rules2: stream_list_fm,A: abstra4463927080622938191ist_fm] :
      ( ( abstra7376083527646823080ist_tm @ ( abstra9141620886324064114ist_tm @ Eff2 ) @ Rules2 @ A )
      = ( ? [T2: abstra4463927080622938191ist_fm] :
            ( ( A = T2 )
            & ( member_list_fm @ ( produc3242453064496107283ist_fm @ ( abstra2656519403522841862ist_fm @ T2 ) ) @ ( sset_list_fm @ Rules2 ) )
            & ( abstra9141620886324064114ist_tm @ Eff2 @ ( produc3242453064496107283ist_fm @ ( abstra2656519403522841862ist_fm @ T2 ) ) @ ( produc4874987732838623697ist_fm @ ( abstra2656519403522841862ist_fm @ T2 ) ) @ ( fimage3333425779666891737ist_tm @ ( comp_P8595014820038962806ist_fm @ produc4874987732838623697ist_fm @ abstra2656519403522841862ist_fm ) @ ( abstra417404627459193974ist_fm @ T2 ) ) )
            & ! [X: abstra4463927080622938191ist_fm] :
                ( ( fmembe7982432732373538198ist_fm @ X @ ( abstra417404627459193974ist_fm @ T2 ) )
               => ( abstra7376083527646823080ist_tm @ ( abstra9141620886324064114ist_tm @ Eff2 ) @ Rules2 @ X ) ) ) ) ) ).

% i.wf.simps
thf(fact_1139_i_Owf_Osimps,axiom,
    ! [Eff2: rule > produc6018962875968178549ist_fm > option6967287582980624417ist_fm,Rules2: stream_rule,A: abstra8520858721168826610m_rule] :
      ( ( abstra8167393772827826667ist_fm @ ( abstra2682625350522704545ist_fm @ Eff2 ) @ Rules2 @ A )
      = ( ? [T2: abstra8520858721168826610m_rule] :
            ( ( A = T2 )
            & ( member_rule @ ( produc6514711208036230658m_rule @ ( abstra5180277867158163941m_rule @ T2 ) ) @ ( sset_rule @ Rules2 ) )
            & ( abstra2682625350522704545ist_fm @ Eff2 @ ( produc6514711208036230658m_rule @ ( abstra5180277867158163941m_rule @ T2 ) ) @ ( produc993771546023551940m_rule @ ( abstra5180277867158163941m_rule @ T2 ) ) @ ( fimage3741683731962187332ist_fm @ ( comp_P5969275118686715022m_rule @ produc993771546023551940m_rule @ abstra5180277867158163941m_rule ) @ ( abstra2475333618431710837m_rule @ T2 ) ) )
            & ! [X: abstra8520858721168826610m_rule] :
                ( ( fmembe1099044791951439339m_rule @ X @ ( abstra2475333618431710837m_rule @ T2 ) )
               => ( abstra8167393772827826667ist_fm @ ( abstra2682625350522704545ist_fm @ Eff2 ) @ Rules2 @ X ) ) ) ) ) ).

% i.wf.simps
thf(fact_1140_i_Owf_Ocases,axiom,
    ! [Eff2: list_fm > list_tm > option_fset_list_tm,Rules2: stream_list_fm,A: abstra4463927080622938191ist_fm] :
      ( ( abstra7376083527646823080ist_tm @ ( abstra9141620886324064114ist_tm @ Eff2 ) @ Rules2 @ A )
     => ~ ( ( member_list_fm @ ( produc3242453064496107283ist_fm @ ( abstra2656519403522841862ist_fm @ A ) ) @ ( sset_list_fm @ Rules2 ) )
         => ( ( abstra9141620886324064114ist_tm @ Eff2 @ ( produc3242453064496107283ist_fm @ ( abstra2656519403522841862ist_fm @ A ) ) @ ( produc4874987732838623697ist_fm @ ( abstra2656519403522841862ist_fm @ A ) ) @ ( fimage3333425779666891737ist_tm @ ( comp_P8595014820038962806ist_fm @ produc4874987732838623697ist_fm @ abstra2656519403522841862ist_fm ) @ ( abstra417404627459193974ist_fm @ A ) ) )
           => ~ ! [T7: abstra4463927080622938191ist_fm] :
                  ( ( fmembe7982432732373538198ist_fm @ T7 @ ( abstra417404627459193974ist_fm @ A ) )
                 => ( abstra7376083527646823080ist_tm @ ( abstra9141620886324064114ist_tm @ Eff2 ) @ Rules2 @ T7 ) ) ) ) ) ).

% i.wf.cases
thf(fact_1141_i_Owf_Ocases,axiom,
    ! [Eff2: rule > produc6018962875968178549ist_fm > option6967287582980624417ist_fm,Rules2: stream_rule,A: abstra8520858721168826610m_rule] :
      ( ( abstra8167393772827826667ist_fm @ ( abstra2682625350522704545ist_fm @ Eff2 ) @ Rules2 @ A )
     => ~ ( ( member_rule @ ( produc6514711208036230658m_rule @ ( abstra5180277867158163941m_rule @ A ) ) @ ( sset_rule @ Rules2 ) )
         => ( ( abstra2682625350522704545ist_fm @ Eff2 @ ( produc6514711208036230658m_rule @ ( abstra5180277867158163941m_rule @ A ) ) @ ( produc993771546023551940m_rule @ ( abstra5180277867158163941m_rule @ A ) ) @ ( fimage3741683731962187332ist_fm @ ( comp_P5969275118686715022m_rule @ produc993771546023551940m_rule @ abstra5180277867158163941m_rule ) @ ( abstra2475333618431710837m_rule @ A ) ) )
           => ~ ! [T7: abstra8520858721168826610m_rule] :
                  ( ( fmembe1099044791951439339m_rule @ T7 @ ( abstra2475333618431710837m_rule @ A ) )
                 => ( abstra8167393772827826667ist_fm @ ( abstra2682625350522704545ist_fm @ Eff2 ) @ Rules2 @ T7 ) ) ) ) ) ).

% i.wf.cases
thf(fact_1142_wf_Ocases,axiom,
    ! [A: abstra8520858721168826610m_rule] :
      ( ( abstra8167393772827826667ist_fm @ eff @ rules @ A )
     => ~ ( ( member_rule @ ( produc6514711208036230658m_rule @ ( abstra5180277867158163941m_rule @ A ) ) @ ( sset_rule @ rules ) )
         => ( ( eff @ ( produc6514711208036230658m_rule @ ( abstra5180277867158163941m_rule @ A ) ) @ ( produc993771546023551940m_rule @ ( abstra5180277867158163941m_rule @ A ) ) @ ( fimage3741683731962187332ist_fm @ ( comp_P5969275118686715022m_rule @ produc993771546023551940m_rule @ abstra5180277867158163941m_rule ) @ ( abstra2475333618431710837m_rule @ A ) ) )
           => ~ ! [T7: abstra8520858721168826610m_rule] :
                  ( ( fmembe1099044791951439339m_rule @ T7 @ ( abstra2475333618431710837m_rule @ A ) )
                 => ( abstra8167393772827826667ist_fm @ eff @ rules @ T7 ) ) ) ) ) ).

% wf.cases
thf(fact_1143_wf_Osimps,axiom,
    ! [A: abstra8520858721168826610m_rule] :
      ( ( abstra8167393772827826667ist_fm @ eff @ rules @ A )
      = ( ? [T2: abstra8520858721168826610m_rule] :
            ( ( A = T2 )
            & ( member_rule @ ( produc6514711208036230658m_rule @ ( abstra5180277867158163941m_rule @ T2 ) ) @ ( sset_rule @ rules ) )
            & ( eff @ ( produc6514711208036230658m_rule @ ( abstra5180277867158163941m_rule @ T2 ) ) @ ( produc993771546023551940m_rule @ ( abstra5180277867158163941m_rule @ T2 ) ) @ ( fimage3741683731962187332ist_fm @ ( comp_P5969275118686715022m_rule @ produc993771546023551940m_rule @ abstra5180277867158163941m_rule ) @ ( abstra2475333618431710837m_rule @ T2 ) ) )
            & ! [X: abstra8520858721168826610m_rule] :
                ( ( fmembe1099044791951439339m_rule @ X @ ( abstra2475333618431710837m_rule @ T2 ) )
               => ( abstra8167393772827826667ist_fm @ eff @ rules @ X ) ) ) ) ) ).

% wf.simps
thf(fact_1144_wf_Ointros,axiom,
    ! [T3: abstra8520858721168826610m_rule] :
      ( ( member_rule @ ( produc6514711208036230658m_rule @ ( abstra5180277867158163941m_rule @ T3 ) ) @ ( sset_rule @ rules ) )
     => ( ( eff @ ( produc6514711208036230658m_rule @ ( abstra5180277867158163941m_rule @ T3 ) ) @ ( produc993771546023551940m_rule @ ( abstra5180277867158163941m_rule @ T3 ) ) @ ( fimage3741683731962187332ist_fm @ ( comp_P5969275118686715022m_rule @ produc993771546023551940m_rule @ abstra5180277867158163941m_rule ) @ ( abstra2475333618431710837m_rule @ T3 ) ) )
       => ( ! [T8: abstra8520858721168826610m_rule] :
              ( ( fmembe1099044791951439339m_rule @ T8 @ ( abstra2475333618431710837m_rule @ T3 ) )
             => ( abstra8167393772827826667ist_fm @ eff @ rules @ T8 ) )
         => ( abstra8167393772827826667ist_fm @ eff @ rules @ T3 ) ) ) ) ).

% wf.intros
thf(fact_1145_comp__funpow,axiom,
    ! [N: nat,F: nat > nat] :
      ( ( compow4371056112333882794at_nat @ N @ ( comp_nat_nat_nat @ F ) )
      = ( comp_nat_nat_nat @ ( compow_nat_nat @ N @ F ) ) ) ).

% comp_funpow
thf(fact_1146_sdrop__while__sdrop__LEAST,axiom,
    ! [P2: produc340336539035504054m_rule > $o,S3: stream2709947120125613254m_rule] :
      ( ? [N5: nat] : ( P2 @ ( snth_P6679518042731451922m_rule @ S3 @ N5 ) )
     => ( ( sdrop_8181906248673349714m_rule @ ( comp_o1746713580868586393m_rule @ (~) @ P2 ) @ S3 )
        = ( sdrop_8169176516188972301m_rule
          @ ( ord_Least_nat
            @ ^ [N3: nat] : ( P2 @ ( snth_P6679518042731451922m_rule @ S3 @ N3 ) ) )
          @ S3 ) ) ) ).

% sdrop_while_sdrop_LEAST
thf(fact_1147_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_1148_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_1149_sdrop__simps_I1_J,axiom,
    ! [N: nat,S3: stream2709947120125613254m_rule] :
      ( ( shd_Pr4562317740776619530m_rule @ ( sdrop_8169176516188972301m_rule @ N @ S3 ) )
      = ( snth_P6679518042731451922m_rule @ S3 @ N ) ) ).

% sdrop_simps(1)
thf(fact_1150_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_1151_snth__szip,axiom,
    ! [S1: stream_rule,S2: stream_rule,N: nat] :
      ( ( snth_P6178434775611311401e_rule @ ( szip_rule_rule @ S1 @ S2 ) @ N )
      = ( produc5849431337705160893e_rule @ ( snth_rule @ S1 @ N ) @ ( snth_rule @ S2 @ N ) ) ) ).

% snth_szip
thf(fact_1152_snth__szip,axiom,
    ! [S1: stream4408948924543953275ist_fm,S2: stream_rule,N: nat] :
      ( ( snth_P6679518042731451922m_rule @ ( szip_P2924820683901490861m_rule @ S1 @ S2 ) @ N )
      = ( produc1733806532565653680m_rule @ ( snth_P7093566783922538521ist_fm @ S1 @ N ) @ ( snth_rule @ S2 @ N ) ) ) ).

% snth_szip
thf(fact_1153_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_1154_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_1155_fst__diag__fst,axiom,
    ( ( comp_P7261744286926158311m_rule @ produc8141473855448433859ist_fm
      @ ( comp_P8526740770942824267m_rule
        @ ^ [X: produc6018962875968178549ist_fm] : ( produc5488031402154692311ist_fm @ X @ X )
        @ produc993771546023551940m_rule ) )
    = produc993771546023551940m_rule ) ).

% fst_diag_fst
thf(fact_1156_fst__diag__fst,axiom,
    ( ( comp_P8084197151785673102ist_fm @ produc4878939757924258883ist_tm
      @ ( comp_l1296997340443160214ist_fm
        @ ^ [X: list_tm] : ( produc1418304791525149271ist_tm @ X @ X )
        @ produc4874987732838623697ist_fm ) )
    = produc4874987732838623697ist_fm ) ).

% fst_diag_fst
thf(fact_1157_snd__diag__snd,axiom,
    ( ( comp_P4471343452172326572m_rule @ produc1581767551459125483e_rule
      @ ( comp_r2696336692066588806m_rule
        @ ^ [X: rule] : ( produc5849431337705160893e_rule @ X @ X )
        @ produc6514711208036230658m_rule ) )
    = produc6514711208036230658m_rule ) ).

% snd_diag_snd
thf(fact_1158_snth__sset,axiom,
    ! [S3: stream_nat,N: nat] : ( member_nat @ ( snth_nat @ S3 @ N ) @ ( sset_nat @ S3 ) ) ).

% snth_sset
thf(fact_1159_snth__sset,axiom,
    ! [S3: stream4408948924543953275ist_fm,N: nat] : ( member4699826688122452638ist_fm @ ( snth_P7093566783922538521ist_fm @ S3 @ N ) @ ( sset_P5379989128061332361ist_fm @ S3 ) ) ).

% snth_sset
thf(fact_1160_snth__sset,axiom,
    ! [S3: stream8372878641218411373at_nat,N: nat] : ( member8206827879206165904at_nat @ ( snth_P9098630209796499211at_nat @ S3 @ N ) @ ( sset_P9164763103296375419at_nat @ S3 ) ) ).

% snth_sset
thf(fact_1161_snth__sset,axiom,
    ! [S3: stream_rule,N: nat] : ( member_rule @ ( snth_rule @ S3 @ N ) @ ( sset_rule @ S3 ) ) ).

% snth_sset
thf(fact_1162_snth__in,axiom,
    ! [S3: stream_nat,X5: set_nat,N: nat] :
      ( ( member_stream_nat @ S3 @ ( streams_nat @ X5 ) )
     => ( member_nat @ ( snth_nat @ S3 @ N ) @ X5 ) ) ).

% snth_in
thf(fact_1163_snth__in,axiom,
    ! [S3: stream4408948924543953275ist_fm,X5: set_Pr5202636777678657877ist_fm,N: nat] :
      ( ( member197948898507729188ist_fm @ S3 @ ( stream1217541351357117421ist_fm @ X5 ) )
     => ( member4699826688122452638ist_fm @ ( snth_P7093566783922538521ist_fm @ S3 @ N ) @ X5 ) ) ).

% snth_in
thf(fact_1164_snth__in,axiom,
    ! [S3: stream8372878641218411373at_nat,X5: set_Pr8693737435421807431at_nat,N: nat] :
      ( ( member1992146023263842070at_nat @ S3 @ ( stream5885727000947712735at_nat @ X5 ) )
     => ( member8206827879206165904at_nat @ ( snth_P9098630209796499211at_nat @ S3 @ N ) @ X5 ) ) ).

% snth_in
thf(fact_1165_snth__in,axiom,
    ! [S3: stream_rule,X5: set_rule,N: nat] :
      ( ( member_stream_rule @ S3 @ ( streams_rule @ X5 ) )
     => ( member_rule @ ( snth_rule @ S3 @ N ) @ X5 ) ) ).

% snth_in
thf(fact_1166_streams__iff__snth,axiom,
    ! [S3: stream_nat,X5: set_nat] :
      ( ( member_stream_nat @ S3 @ ( streams_nat @ X5 ) )
      = ( ! [N3: nat] : ( member_nat @ ( snth_nat @ S3 @ N3 ) @ X5 ) ) ) ).

% streams_iff_snth
thf(fact_1167_streams__iff__snth,axiom,
    ! [S3: stream4408948924543953275ist_fm,X5: set_Pr5202636777678657877ist_fm] :
      ( ( member197948898507729188ist_fm @ S3 @ ( stream1217541351357117421ist_fm @ X5 ) )
      = ( ! [N3: nat] : ( member4699826688122452638ist_fm @ ( snth_P7093566783922538521ist_fm @ S3 @ N3 ) @ X5 ) ) ) ).

% streams_iff_snth
thf(fact_1168_streams__iff__snth,axiom,
    ! [S3: stream8372878641218411373at_nat,X5: set_Pr8693737435421807431at_nat] :
      ( ( member1992146023263842070at_nat @ S3 @ ( stream5885727000947712735at_nat @ X5 ) )
      = ( ! [N3: nat] : ( member8206827879206165904at_nat @ ( snth_P9098630209796499211at_nat @ S3 @ N3 ) @ X5 ) ) ) ).

% streams_iff_snth
thf(fact_1169_streams__iff__snth,axiom,
    ! [S3: stream_rule,X5: set_rule] :
      ( ( member_stream_rule @ S3 @ ( streams_rule @ X5 ) )
      = ( ! [N3: nat] : ( member_rule @ ( snth_rule @ S3 @ N3 ) @ X5 ) ) ) ).

% streams_iff_snth
thf(fact_1170_snd__diag__fst,axiom,
    ( ( comp_P7261744286926158311m_rule @ produc6364584182369725445ist_fm
      @ ( comp_P8526740770942824267m_rule
        @ ^ [X: produc6018962875968178549ist_fm] : ( produc5488031402154692311ist_fm @ X @ X )
        @ produc993771546023551940m_rule ) )
    = produc993771546023551940m_rule ) ).

% snd_diag_fst
thf(fact_1171_snd__diag__fst,axiom,
    ( ( comp_P8084197151785673102ist_fm @ produc3246405089581742469ist_tm
      @ ( comp_l1296997340443160214ist_fm
        @ ^ [X: list_tm] : ( produc1418304791525149271ist_tm @ X @ X )
        @ produc4874987732838623697ist_fm ) )
    = produc4874987732838623697ist_fm ) ).

% snd_diag_fst
thf(fact_1172_fst__diag__snd,axiom,
    ( ( comp_P4471343452172326572m_rule @ produc2120989893493863593e_rule
      @ ( comp_r2696336692066588806m_rule
        @ ^ [X: rule] : ( produc5849431337705160893e_rule @ X @ X )
        @ produc6514711208036230658m_rule ) )
    = produc6514711208036230658m_rule ) ).

% fst_diag_snd
thf(fact_1173_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_1174_stream__all__def,axiom,
    ( stream_all_rule
    = ( ^ [P: rule > $o,S: stream_rule] :
        ! [P4: nat] : ( P @ ( snth_rule @ S @ P4 ) ) ) ) ).

% stream_all_def
thf(fact_1175_rules__repeat__sdrop,axiom,
    ! [K: nat,R: rule] :
    ? [N2: nat] :
      ( ( snth_rule @ ( sdrop_rule @ K @ rules ) @ N2 )
      = R ) ).

% rules_repeat_sdrop
thf(fact_1176_Nat_Oex__has__greatest__nat,axiom,
    ! [P2: nat > $o,K: nat,B: nat] :
      ( ( P2 @ K )
     => ( ! [Y4: nat] :
            ( ( P2 @ Y4 )
           => ( ord_less_eq_nat @ Y4 @ B ) )
       => ? [X2: nat] :
            ( ( P2 @ X2 )
            & ! [Y5: nat] :
                ( ( P2 @ Y5 )
               => ( ord_less_eq_nat @ Y5 @ X2 ) ) ) ) ) ).

% Nat.ex_has_greatest_nat
thf(fact_1177_nat__le__linear,axiom,
    ! [M2: nat,N: nat] :
      ( ( ord_less_eq_nat @ M2 @ N )
      | ( ord_less_eq_nat @ N @ M2 ) ) ).

% nat_le_linear
thf(fact_1178_le__antisym,axiom,
    ! [M2: nat,N: nat] :
      ( ( ord_less_eq_nat @ M2 @ N )
     => ( ( ord_less_eq_nat @ N @ M2 )
       => ( M2 = N ) ) ) ).

% le_antisym
thf(fact_1179_eq__imp__le,axiom,
    ! [M2: nat,N: nat] :
      ( ( M2 = N )
     => ( ord_less_eq_nat @ M2 @ N ) ) ).

% eq_imp_le
thf(fact_1180_le__trans,axiom,
    ! [I: nat,J: nat,K: nat] :
      ( ( ord_less_eq_nat @ I @ J )
     => ( ( ord_less_eq_nat @ J @ K )
       => ( ord_less_eq_nat @ I @ K ) ) ) ).

% le_trans
thf(fact_1181_le__refl,axiom,
    ! [N: nat] : ( ord_less_eq_nat @ N @ N ) ).

% le_refl
thf(fact_1182_fimage__fimage,axiom,
    ! [F: produc6018962875968178549ist_fm > produc6018962875968178549ist_fm,G: abstra8520858721168826610m_rule > produc6018962875968178549ist_fm,A2: fset_A8849196716589201442m_rule] :
      ( ( fimage4669962711938197693ist_fm @ F @ ( fimage3741683731962187332ist_fm @ G @ A2 ) )
      = ( fimage3741683731962187332ist_fm
        @ ^ [X: abstra8520858721168826610m_rule] : ( F @ ( G @ X ) )
        @ A2 ) ) ).

% fimage_fimage
thf(fact_1183_fimage__fimage,axiom,
    ! [F: abstra8520858721168826610m_rule > abstra8520858721168826610m_rule,G: produc6018962875968178549ist_fm > abstra8520858721168826610m_rule,A2: fset_P8989946509869081563ist_fm] :
      ( ( fimage7018686944362031843m_rule @ F @ ( fimage2815761261218872234m_rule @ G @ A2 ) )
      = ( fimage2815761261218872234m_rule
        @ ^ [X: produc6018962875968178549ist_fm] : ( F @ ( G @ X ) )
        @ A2 ) ) ).

% fimage_fimage
thf(fact_1184_fimage__fimage,axiom,
    ! [F: abstra8520858721168826610m_rule > produc6018962875968178549ist_fm,G: abstra8520858721168826610m_rule > abstra8520858721168826610m_rule,A2: fset_A8849196716589201442m_rule] :
      ( ( fimage3741683731962187332ist_fm @ F @ ( fimage7018686944362031843m_rule @ G @ A2 ) )
      = ( fimage3741683731962187332ist_fm
        @ ^ [X: abstra8520858721168826610m_rule] : ( F @ ( G @ X ) )
        @ A2 ) ) ).

% fimage_fimage
thf(fact_1185_fimage__fimage,axiom,
    ! [F: abstra8520858721168826610m_rule > produc6018962875968178549ist_fm,G: produc6018962875968178549ist_fm > abstra8520858721168826610m_rule,A2: fset_P8989946509869081563ist_fm] :
      ( ( fimage3741683731962187332ist_fm @ F @ ( fimage2815761261218872234m_rule @ G @ A2 ) )
      = ( fimage4669962711938197693ist_fm
        @ ^ [X: produc6018962875968178549ist_fm] : ( F @ ( G @ X ) )
        @ A2 ) ) ).

% fimage_fimage
thf(fact_1186_fimage__fimage,axiom,
    ! [F: produc6018962875968178549ist_fm > abstra8520858721168826610m_rule,G: produc6018962875968178549ist_fm > produc6018962875968178549ist_fm,A2: fset_P8989946509869081563ist_fm] :
      ( ( fimage2815761261218872234m_rule @ F @ ( fimage4669962711938197693ist_fm @ G @ A2 ) )
      = ( fimage2815761261218872234m_rule
        @ ^ [X: produc6018962875968178549ist_fm] : ( F @ ( G @ X ) )
        @ A2 ) ) ).

% fimage_fimage
thf(fact_1187_fimage__fimage,axiom,
    ! [F: produc6018962875968178549ist_fm > abstra8520858721168826610m_rule,G: abstra8520858721168826610m_rule > produc6018962875968178549ist_fm,A2: fset_A8849196716589201442m_rule] :
      ( ( fimage2815761261218872234m_rule @ F @ ( fimage3741683731962187332ist_fm @ G @ A2 ) )
      = ( fimage7018686944362031843m_rule
        @ ^ [X: abstra8520858721168826610m_rule] : ( F @ ( G @ X ) )
        @ A2 ) ) ).

% fimage_fimage
thf(fact_1188_mkTree_Osimps_I2_J,axiom,
    ! [Rs2: stream_rule,S3: produc6018962875968178549ist_fm] :
      ( ( abstra2475333618431710837m_rule @ ( abstra7840189589880272446ist_fm @ eff @ Rs2 @ S3 ) )
      = ( fimage2815761261218872234m_rule @ ( abstra7840189589880272446ist_fm @ eff @ ( stl_rule @ ( abstra7806236140005899928ist_fm @ eff @ Rs2 @ S3 ) ) ) @ ( abstra3309574903381802724ist_fm @ eff @ ( shd_rule @ ( abstra7806236140005899928ist_fm @ eff @ Rs2 @ S3 ) ) @ S3 ) ) ) ).

% mkTree.simps(2)
thf(fact_1189_i_OmkTree_Osimps_I2_J,axiom,
    ! [Eff2: rule > produc6018962875968178549ist_fm > option6967287582980624417ist_fm,Rs2: stream_rule,S3: produc6018962875968178549ist_fm] :
      ( ( abstra2475333618431710837m_rule @ ( abstra7840189589880272446ist_fm @ ( abstra2682625350522704545ist_fm @ Eff2 ) @ Rs2 @ S3 ) )
      = ( fimage2815761261218872234m_rule @ ( abstra7840189589880272446ist_fm @ ( abstra2682625350522704545ist_fm @ Eff2 ) @ ( stl_rule @ ( abstra7806236140005899928ist_fm @ ( abstra2682625350522704545ist_fm @ Eff2 ) @ Rs2 @ S3 ) ) ) @ ( abstra3309574903381802724ist_fm @ ( abstra2682625350522704545ist_fm @ Eff2 ) @ ( shd_rule @ ( abstra7806236140005899928ist_fm @ ( abstra2682625350522704545ist_fm @ Eff2 ) @ Rs2 @ S3 ) ) @ S3 ) ) ) ).

% i.mkTree.simps(2)
thf(fact_1190_RuleSystem__Defs_OmkTree_Osimps_I2_J,axiom,
    ! [Eff4: rule > produc6018962875968178549ist_fm > fset_P8989946509869081563ist_fm > $o,Rs2: stream_rule,S3: produc6018962875968178549ist_fm] :
      ( ( abstra2475333618431710837m_rule @ ( abstra7840189589880272446ist_fm @ Eff4 @ Rs2 @ S3 ) )
      = ( fimage2815761261218872234m_rule @ ( abstra7840189589880272446ist_fm @ Eff4 @ ( stl_rule @ ( abstra7806236140005899928ist_fm @ Eff4 @ Rs2 @ S3 ) ) ) @ ( abstra3309574903381802724ist_fm @ Eff4 @ ( shd_rule @ ( abstra7806236140005899928ist_fm @ Eff4 @ Rs2 @ S3 ) ) @ S3 ) ) ) ).

% RuleSystem_Defs.mkTree.simps(2)
thf(fact_1191_RuleSystem__Defs_OpickEff_Ocong,axiom,
    abstra3309574903381802724ist_fm = abstra3309574903381802724ist_fm ).

% RuleSystem_Defs.pickEff.cong
thf(fact_1192_RuleSystem__Defs_OpickEff,axiom,
    ! [Eff4: rule > produc6018962875968178549ist_fm > fset_P8989946509869081563ist_fm > $o,R: rule,S3: produc6018962875968178549ist_fm] :
      ( ( abstra1707737187183579335ist_fm @ Eff4 @ R @ S3 )
     => ( Eff4 @ R @ S3 @ ( abstra3309574903381802724ist_fm @ Eff4 @ R @ S3 ) ) ) ).

% RuleSystem_Defs.pickEff
thf(fact_1193_pickEff,axiom,
    ! [R: rule,S3: produc6018962875968178549ist_fm] :
      ( ( abstra1707737187183579335ist_fm @ eff @ R @ S3 )
     => ( eff @ R @ S3 @ ( abstra3309574903381802724ist_fm @ eff @ R @ S3 ) ) ) ).

% pickEff
thf(fact_1194_mkTree_Ocode,axiom,
    ! [Rs2: stream_rule,S3: produc6018962875968178549ist_fm] :
      ( ( abstra7840189589880272446ist_fm @ eff @ Rs2 @ S3 )
      = ( abstra5937941257295885381m_rule @ ( produc1733806532565653680m_rule @ S3 @ ( shd_rule @ ( abstra7806236140005899928ist_fm @ eff @ Rs2 @ S3 ) ) ) @ ( fimage2815761261218872234m_rule @ ( abstra7840189589880272446ist_fm @ eff @ ( stl_rule @ ( abstra7806236140005899928ist_fm @ eff @ Rs2 @ S3 ) ) ) @ ( abstra3309574903381802724ist_fm @ eff @ ( shd_rule @ ( abstra7806236140005899928ist_fm @ eff @ Rs2 @ S3 ) ) @ S3 ) ) ) ) ).

% mkTree.code
thf(fact_1195_mkTree__unfold,axiom,
    ! [Rs2: stream_rule,S3: produc6018962875968178549ist_fm] :
      ( ( abstra7840189589880272446ist_fm @ eff @ Rs2 @ S3 )
      = ( case_s7672397449011031346m_rule
        @ ^ [R4: rule,S9: stream_rule] : ( abstra5937941257295885381m_rule @ ( produc1733806532565653680m_rule @ S3 @ R4 ) @ ( fimage2815761261218872234m_rule @ ( abstra7840189589880272446ist_fm @ eff @ S9 ) @ ( abstra3309574903381802724ist_fm @ eff @ R4 @ S3 ) ) )
        @ ( abstra7806236140005899928ist_fm @ eff @ Rs2 @ S3 ) ) ) ).

% mkTree_unfold
thf(fact_1196_Suc__le__mono,axiom,
    ! [N: nat,M2: nat] :
      ( ( ord_less_eq_nat @ ( suc @ N ) @ ( suc @ M2 ) )
      = ( ord_less_eq_nat @ N @ M2 ) ) ).

% Suc_le_mono
thf(fact_1197_card_Ocomp__fun__commute__on,axiom,
    ( ( comp_nat_nat_nat @ suc @ suc )
    = ( comp_nat_nat_nat @ suc @ suc ) ) ).

% card.comp_fun_commute_on
thf(fact_1198_Suc__leD,axiom,
    ! [M2: nat,N: nat] :
      ( ( ord_less_eq_nat @ ( suc @ M2 ) @ N )
     => ( ord_less_eq_nat @ M2 @ N ) ) ).

% Suc_leD
thf(fact_1199_le__SucE,axiom,
    ! [M2: nat,N: nat] :
      ( ( ord_less_eq_nat @ M2 @ ( suc @ N ) )
     => ( ~ ( ord_less_eq_nat @ M2 @ N )
       => ( M2
          = ( suc @ N ) ) ) ) ).

% le_SucE
thf(fact_1200_le__SucI,axiom,
    ! [M2: nat,N: nat] :
      ( ( ord_less_eq_nat @ M2 @ N )
     => ( ord_less_eq_nat @ M2 @ ( suc @ N ) ) ) ).

% le_SucI
thf(fact_1201_Suc__le__D,axiom,
    ! [N: nat,M5: nat] :
      ( ( ord_less_eq_nat @ ( suc @ N ) @ M5 )
     => ? [M4: nat] :
          ( M5
          = ( suc @ M4 ) ) ) ).

% Suc_le_D
thf(fact_1202_le__Suc__eq,axiom,
    ! [M2: nat,N: nat] :
      ( ( ord_less_eq_nat @ M2 @ ( suc @ N ) )
      = ( ( ord_less_eq_nat @ M2 @ N )
        | ( M2
          = ( suc @ N ) ) ) ) ).

% le_Suc_eq
thf(fact_1203_Suc__n__not__le__n,axiom,
    ! [N: nat] :
      ~ ( ord_less_eq_nat @ ( suc @ N ) @ N ) ).

% Suc_n_not_le_n
thf(fact_1204_not__less__eq__eq,axiom,
    ! [M2: nat,N: nat] :
      ( ( ~ ( ord_less_eq_nat @ M2 @ N ) )
      = ( ord_less_eq_nat @ ( suc @ N ) @ M2 ) ) ).

% not_less_eq_eq
thf(fact_1205_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_1206_nat__induct__at__least,axiom,
    ! [M2: nat,N: nat,P2: nat > $o] :
      ( ( ord_less_eq_nat @ M2 @ N )
     => ( ( P2 @ M2 )
       => ( ! [N2: nat] :
              ( ( ord_less_eq_nat @ M2 @ N2 )
             => ( ( P2 @ N2 )
               => ( P2 @ ( suc @ N2 ) ) ) )
         => ( P2 @ N ) ) ) ) ).

% nat_induct_at_least
thf(fact_1207_transitive__stepwise__le,axiom,
    ! [M2: nat,N: nat,R3: nat > nat > $o] :
      ( ( ord_less_eq_nat @ M2 @ N )
     => ( ! [X2: nat] : ( R3 @ X2 @ X2 )
       => ( ! [X2: nat,Y4: nat,Z4: nat] :
              ( ( R3 @ X2 @ Y4 )
             => ( ( R3 @ Y4 @ Z4 )
               => ( R3 @ X2 @ Z4 ) ) )
         => ( ! [N2: nat] : ( R3 @ N2 @ ( suc @ N2 ) )
           => ( R3 @ M2 @ N ) ) ) ) ) ).

% transitive_stepwise_le
thf(fact_1208_stake__pos__minWait,axiom,
    ! [Rs2: stream_rule,S3: produc6018962875968178549ist_fm,R: rule] :
      ( ( abstra3799686578551160190r_rule @ rules @ Rs2 )
     => ( ( ord_less_nat @ ( abstra1963046427795717555ist_fm @ eff @ Rs2 @ S3 ) @ ( abstract_pos_rule @ Rs2 @ R ) )
       => ( ( member_rule @ R @ ( sset_rule @ rules ) )
         => ( ( member4699826688122452638ist_fm @ S3 @ top_to1730629564055774885ist_fm )
           => ( ( abstract_pos_rule @ ( stl_rule @ ( abstra7806236140005899928ist_fm @ eff @ Rs2 @ S3 ) ) @ R )
              = ( minus_minus_nat @ ( abstract_pos_rule @ Rs2 @ R ) @ ( suc @ ( abstra1963046427795717555ist_fm @ eff @ Rs2 @ S3 ) ) ) ) ) ) ) ) ).

% stake_pos_minWait
thf(fact_1209_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_1210_finite__Collect__less__nat,axiom,
    ! [K: nat] :
      ( finite_finite_nat
      @ ( collect_nat
        @ ^ [N3: nat] : ( ord_less_nat @ N3 @ K ) ) ) ).

% finite_Collect_less_nat
thf(fact_1211_diff__less__mono,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( ( ord_less_nat @ A @ B )
     => ( ( ord_less_eq_nat @ C @ A )
       => ( ord_less_nat @ ( minus_minus_nat @ A @ C ) @ ( minus_minus_nat @ B @ C ) ) ) ) ).

% diff_less_mono
thf(fact_1212_less__diff__iff,axiom,
    ! [K: nat,M2: nat,N: nat] :
      ( ( ord_less_eq_nat @ K @ M2 )
     => ( ( ord_less_eq_nat @ K @ N )
       => ( ( ord_less_nat @ ( minus_minus_nat @ M2 @ K ) @ ( minus_minus_nat @ N @ K ) )
          = ( ord_less_nat @ M2 @ N ) ) ) ) ).

% less_diff_iff
thf(fact_1213_diff__le__mono2,axiom,
    ! [M2: nat,N: nat,L: nat] :
      ( ( ord_less_eq_nat @ M2 @ N )
     => ( ord_less_eq_nat @ ( minus_minus_nat @ L @ N ) @ ( minus_minus_nat @ L @ M2 ) ) ) ).

% diff_le_mono2
thf(fact_1214_le__diff__iff_H,axiom,
    ! [A: nat,C: nat,B: nat] :
      ( ( ord_less_eq_nat @ A @ C )
     => ( ( ord_less_eq_nat @ B @ C )
       => ( ( ord_less_eq_nat @ ( minus_minus_nat @ C @ A ) @ ( minus_minus_nat @ C @ B ) )
          = ( ord_less_eq_nat @ B @ A ) ) ) ) ).

% le_diff_iff'
thf(fact_1215_diff__le__self,axiom,
    ! [M2: nat,N: nat] : ( ord_less_eq_nat @ ( minus_minus_nat @ M2 @ N ) @ M2 ) ).

% diff_le_self
thf(fact_1216_diff__le__mono,axiom,
    ! [M2: nat,N: nat,L: nat] :
      ( ( ord_less_eq_nat @ M2 @ N )
     => ( ord_less_eq_nat @ ( minus_minus_nat @ M2 @ L ) @ ( minus_minus_nat @ N @ L ) ) ) ).

% diff_le_mono
thf(fact_1217_Nat_Odiff__diff__eq,axiom,
    ! [K: nat,M2: nat,N: nat] :
      ( ( ord_less_eq_nat @ K @ M2 )
     => ( ( ord_less_eq_nat @ K @ N )
       => ( ( minus_minus_nat @ ( minus_minus_nat @ M2 @ K ) @ ( minus_minus_nat @ N @ K ) )
          = ( minus_minus_nat @ M2 @ N ) ) ) ) ).

% Nat.diff_diff_eq
thf(fact_1218_le__diff__iff,axiom,
    ! [K: nat,M2: nat,N: nat] :
      ( ( ord_less_eq_nat @ K @ M2 )
     => ( ( ord_less_eq_nat @ K @ N )
       => ( ( ord_less_eq_nat @ ( minus_minus_nat @ M2 @ K ) @ ( minus_minus_nat @ N @ K ) )
          = ( ord_less_eq_nat @ M2 @ N ) ) ) ) ).

% le_diff_iff
thf(fact_1219_eq__diff__iff,axiom,
    ! [K: nat,M2: nat,N: nat] :
      ( ( ord_less_eq_nat @ K @ M2 )
     => ( ( ord_less_eq_nat @ K @ N )
       => ( ( ( minus_minus_nat @ M2 @ K )
            = ( minus_minus_nat @ N @ K ) )
          = ( M2 = N ) ) ) ) ).

% eq_diff_iff
thf(fact_1220_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_1221_le__neq__implies__less,axiom,
    ! [M2: nat,N: nat] :
      ( ( ord_less_eq_nat @ M2 @ N )
     => ( ( M2 != N )
       => ( ord_less_nat @ M2 @ N ) ) ) ).

% le_neq_implies_less
thf(fact_1222_less__or__eq__imp__le,axiom,
    ! [M2: nat,N: nat] :
      ( ( ( ord_less_nat @ M2 @ N )
        | ( M2 = N ) )
     => ( ord_less_eq_nat @ M2 @ N ) ) ).

% less_or_eq_imp_le
thf(fact_1223_le__eq__less__or__eq,axiom,
    ( ord_less_eq_nat
    = ( ^ [M: nat,N3: nat] :
          ( ( ord_less_nat @ M @ N3 )
          | ( M = N3 ) ) ) ) ).

% le_eq_less_or_eq
thf(fact_1224_less__imp__le__nat,axiom,
    ! [M2: nat,N: nat] :
      ( ( ord_less_nat @ M2 @ N )
     => ( ord_less_eq_nat @ M2 @ N ) ) ).

% less_imp_le_nat
thf(fact_1225_nat__less__le,axiom,
    ( ord_less_nat
    = ( ^ [M: nat,N3: nat] :
          ( ( ord_less_eq_nat @ M @ N3 )
          & ( M != N3 ) ) ) ) ).

% nat_less_le
thf(fact_1226_finite__nat__set__iff__bounded,axiom,
    ( finite_finite_nat
    = ( ^ [N4: set_nat] :
        ? [M: nat] :
        ! [X: nat] :
          ( ( member_nat @ X @ N4 )
         => ( ord_less_nat @ X @ M ) ) ) ) ).

% finite_nat_set_iff_bounded
thf(fact_1227_infinite__nat__iff__unbounded,axiom,
    ! [S4: set_nat] :
      ( ( ~ ( finite_finite_nat @ S4 ) )
      = ( ! [M: nat] :
          ? [N3: nat] :
            ( ( ord_less_nat @ M @ N3 )
            & ( member_nat @ N3 @ S4 ) ) ) ) ).

% infinite_nat_iff_unbounded
thf(fact_1228_bounded__nat__set__is__finite,axiom,
    ! [N6: set_nat,N: nat] :
      ( ! [X2: nat] :
          ( ( member_nat @ X2 @ N6 )
         => ( ord_less_nat @ X2 @ N ) )
     => ( finite_finite_nat @ N6 ) ) ).

% bounded_nat_set_is_finite
thf(fact_1229_unbounded__k__infinite,axiom,
    ! [K: nat,S4: set_nat] :
      ( ! [M4: nat] :
          ( ( ord_less_nat @ K @ M4 )
         => ? [N5: nat] :
              ( ( ord_less_nat @ M4 @ N5 )
              & ( member_nat @ N5 @ S4 ) ) )
     => ~ ( finite_finite_nat @ S4 ) ) ).

% unbounded_k_infinite
thf(fact_1230_finite__M__bounded__by__nat,axiom,
    ! [P2: nat > $o,I: nat] :
      ( finite_finite_nat
      @ ( collect_nat
        @ ^ [K2: nat] :
            ( ( P2 @ K2 )
            & ( ord_less_nat @ K2 @ I ) ) ) ) ).

% finite_M_bounded_by_nat
thf(fact_1231_Suc__diff__le,axiom,
    ! [N: nat,M2: nat] :
      ( ( ord_less_eq_nat @ N @ M2 )
     => ( ( minus_minus_nat @ ( suc @ M2 ) @ N )
        = ( suc @ ( minus_minus_nat @ M2 @ N ) ) ) ) ).

% Suc_diff_le
thf(fact_1232_le__imp__less__Suc,axiom,
    ! [M2: nat,N: nat] :
      ( ( ord_less_eq_nat @ M2 @ N )
     => ( ord_less_nat @ M2 @ ( suc @ N ) ) ) ).

% le_imp_less_Suc
thf(fact_1233_less__eq__Suc__le,axiom,
    ( ord_less_nat
    = ( ^ [N3: nat] : ( ord_less_eq_nat @ ( suc @ N3 ) ) ) ) ).

% less_eq_Suc_le
thf(fact_1234_less__Suc__eq__le,axiom,
    ! [M2: nat,N: nat] :
      ( ( ord_less_nat @ M2 @ ( suc @ N ) )
      = ( ord_less_eq_nat @ M2 @ N ) ) ).

% less_Suc_eq_le
thf(fact_1235_le__less__Suc__eq,axiom,
    ! [M2: nat,N: nat] :
      ( ( ord_less_eq_nat @ M2 @ N )
     => ( ( ord_less_nat @ N @ ( suc @ M2 ) )
        = ( N = M2 ) ) ) ).

% le_less_Suc_eq
thf(fact_1236_Suc__le__lessD,axiom,
    ! [M2: nat,N: nat] :
      ( ( ord_less_eq_nat @ ( suc @ M2 ) @ N )
     => ( ord_less_nat @ M2 @ N ) ) ).

% Suc_le_lessD
thf(fact_1237_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_1238_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_1239_Suc__le__eq,axiom,
    ! [M2: nat,N: nat] :
      ( ( ord_less_eq_nat @ ( suc @ M2 ) @ N )
      = ( ord_less_nat @ M2 @ N ) ) ).

% Suc_le_eq
thf(fact_1240_Suc__leI,axiom,
    ! [M2: nat,N: nat] :
      ( ( ord_less_nat @ M2 @ N )
     => ( ord_less_eq_nat @ ( suc @ M2 ) @ N ) ) ).

% Suc_leI
thf(fact_1241_rules__repeat,axiom,
    ! [M2: nat,R: rule] :
    ? [N2: nat] :
      ( ( ord_less_nat @ M2 @ N2 )
      & ( ( snth_rule @ rules @ N2 )
        = R ) ) ).

% rules_repeat
thf(fact_1242_nat__descend__induct,axiom,
    ! [N: nat,P2: nat > $o,M2: nat] :
      ( ! [K3: nat] :
          ( ( ord_less_nat @ N @ K3 )
         => ( P2 @ K3 ) )
     => ( ! [K3: nat] :
            ( ( ord_less_eq_nat @ K3 @ N )
           => ( ! [I3: nat] :
                  ( ( ord_less_nat @ K3 @ I3 )
                 => ( P2 @ I3 ) )
             => ( P2 @ K3 ) ) )
       => ( P2 @ M2 ) ) ) ).

% nat_descend_induct
thf(fact_1243_card__Collect__less__nat,axiom,
    ! [N: nat] :
      ( ( finite_card_nat
        @ ( collect_nat
          @ ^ [I4: nat] : ( ord_less_nat @ I4 @ N ) ) )
      = N ) ).

% card_Collect_less_nat
thf(fact_1244_card__Collect__le__nat,axiom,
    ! [N: nat] :
      ( ( finite_card_nat
        @ ( collect_nat
          @ ^ [I4: nat] : ( ord_less_eq_nat @ I4 @ N ) ) )
      = ( suc @ N ) ) ).

% card_Collect_le_nat
thf(fact_1245_finite__le__enumerate,axiom,
    ! [S4: set_nat,N: nat] :
      ( ( finite_finite_nat @ S4 )
     => ( ( ord_less_nat @ N @ ( finite_card_nat @ S4 ) )
       => ( ord_less_eq_nat @ N @ ( infini8530281810654367211te_nat @ S4 @ N ) ) ) ) ).

% finite_le_enumerate
thf(fact_1246_prod__decode__aux_Oelims,axiom,
    ! [X3: nat,Xa2: nat,Y: product_prod_nat_nat] :
      ( ( ( nat_prod_decode_aux @ X3 @ Xa2 )
        = Y )
     => ( ( ( ord_less_eq_nat @ Xa2 @ X3 )
         => ( Y
            = ( product_Pair_nat_nat @ Xa2 @ ( minus_minus_nat @ X3 @ Xa2 ) ) ) )
        & ( ~ ( ord_less_eq_nat @ Xa2 @ X3 )
         => ( Y
            = ( nat_prod_decode_aux @ ( suc @ X3 ) @ ( minus_minus_nat @ Xa2 @ ( suc @ X3 ) ) ) ) ) ) ) ).

% prod_decode_aux.elims
thf(fact_1247_bot__nat__0_Oextremum,axiom,
    ! [A: nat] : ( ord_less_eq_nat @ zero_zero_nat @ A ) ).

% bot_nat_0.extremum
thf(fact_1248_le0,axiom,
    ! [N: nat] : ( ord_less_eq_nat @ zero_zero_nat @ N ) ).

% le0
thf(fact_1249_Least__eq__0,axiom,
    ! [P2: nat > $o] :
      ( ( P2 @ zero_zero_nat )
     => ( ( ord_Least_nat @ P2 )
        = zero_zero_nat ) ) ).

% Least_eq_0
thf(fact_1250_diff__is__0__eq,axiom,
    ! [M2: nat,N: nat] :
      ( ( ( minus_minus_nat @ M2 @ N )
        = zero_zero_nat )
      = ( ord_less_eq_nat @ M2 @ N ) ) ).

% diff_is_0_eq
thf(fact_1251_diff__is__0__eq_H,axiom,
    ! [M2: nat,N: nat] :
      ( ( ord_less_eq_nat @ M2 @ N )
     => ( ( minus_minus_nat @ M2 @ N )
        = zero_zero_nat ) ) ).

% diff_is_0_eq'
thf(fact_1252_less__eq__nat_Osimps_I1_J,axiom,
    ! [N: nat] : ( ord_less_eq_nat @ zero_zero_nat @ N ) ).

% less_eq_nat.simps(1)
thf(fact_1253_bot__nat__0_Oextremum__unique,axiom,
    ! [A: nat] :
      ( ( ord_less_eq_nat @ A @ zero_zero_nat )
      = ( A = zero_zero_nat ) ) ).

% bot_nat_0.extremum_unique
thf(fact_1254_bot__nat__0_Oextremum__uniqueI,axiom,
    ! [A: nat] :
      ( ( ord_less_eq_nat @ A @ zero_zero_nat )
     => ( A = zero_zero_nat ) ) ).

% bot_nat_0.extremum_uniqueI
thf(fact_1255_le__0__eq,axiom,
    ! [N: nat] :
      ( ( ord_less_eq_nat @ N @ zero_zero_nat )
      = ( N = zero_zero_nat ) ) ).

% le_0_eq
thf(fact_1256_ex__least__nat__le,axiom,
    ! [P2: nat > $o,N: nat] :
      ( ( P2 @ N )
     => ( ~ ( P2 @ zero_zero_nat )
       => ? [K3: nat] :
            ( ( ord_less_eq_nat @ K3 @ N )
            & ! [I3: nat] :
                ( ( ord_less_nat @ I3 @ K3 )
               => ~ ( P2 @ I3 ) )
            & ( P2 @ K3 ) ) ) ) ).

% ex_least_nat_le
thf(fact_1257_Least__Suc2,axiom,
    ! [P2: nat > $o,N: nat,Q: nat > $o,M2: nat] :
      ( ( P2 @ N )
     => ( ( Q @ M2 )
       => ( ~ ( P2 @ zero_zero_nat )
         => ( ! [K3: nat] :
                ( ( P2 @ ( suc @ K3 ) )
                = ( Q @ K3 ) )
           => ( ( ord_Least_nat @ P2 )
              = ( suc @ ( ord_Least_nat @ Q ) ) ) ) ) ) ) ).

% Least_Suc2
thf(fact_1258_card__less,axiom,
    ! [M3: set_nat,I: nat] :
      ( ( member_nat @ zero_zero_nat @ M3 )
     => ( ( finite_card_nat
          @ ( collect_nat
            @ ^ [K2: nat] :
                ( ( member_nat @ K2 @ M3 )
                & ( ord_less_nat @ K2 @ ( suc @ I ) ) ) ) )
       != zero_zero_nat ) ) ).

% card_less
thf(fact_1259_card__less__Suc,axiom,
    ! [M3: set_nat,I: nat] :
      ( ( member_nat @ zero_zero_nat @ M3 )
     => ( ( suc
          @ ( finite_card_nat
            @ ( collect_nat
              @ ^ [K2: nat] :
                  ( ( member_nat @ ( suc @ K2 ) @ M3 )
                  & ( ord_less_nat @ K2 @ I ) ) ) ) )
        = ( finite_card_nat
          @ ( collect_nat
            @ ^ [K2: nat] :
                ( ( member_nat @ K2 @ M3 )
                & ( ord_less_nat @ K2 @ ( suc @ I ) ) ) ) ) ) ) ).

% card_less_Suc
thf(fact_1260_card__less__Suc2,axiom,
    ! [M3: set_nat,I: nat] :
      ( ~ ( member_nat @ zero_zero_nat @ M3 )
     => ( ( finite_card_nat
          @ ( collect_nat
            @ ^ [K2: nat] :
                ( ( member_nat @ ( suc @ K2 ) @ M3 )
                & ( ord_less_nat @ K2 @ I ) ) ) )
        = ( finite_card_nat
          @ ( collect_nat
            @ ^ [K2: nat] :
                ( ( member_nat @ K2 @ M3 )
                & ( ord_less_nat @ K2 @ ( suc @ I ) ) ) ) ) ) ) ).

% card_less_Suc2
thf(fact_1261_Least__Suc,axiom,
    ! [P2: nat > $o,N: nat] :
      ( ( P2 @ N )
     => ( ~ ( P2 @ zero_zero_nat )
       => ( ( ord_Least_nat @ P2 )
          = ( suc
            @ ( ord_Least_nat
              @ ^ [M: nat] : ( P2 @ ( suc @ M ) ) ) ) ) ) ) ).

% Least_Suc
thf(fact_1262_ex__least__nat__less,axiom,
    ! [P2: nat > $o,N: nat] :
      ( ( P2 @ N )
     => ( ~ ( P2 @ zero_zero_nat )
       => ? [K3: nat] :
            ( ( ord_less_nat @ K3 @ N )
            & ! [I3: nat] :
                ( ( ord_less_eq_nat @ I3 @ K3 )
               => ~ ( P2 @ I3 ) )
            & ( P2 @ ( suc @ K3 ) ) ) ) ) ).

% ex_least_nat_less
thf(fact_1263_prod__decode__aux_Osimps,axiom,
    ( nat_prod_decode_aux
    = ( ^ [K2: nat,M: nat] : ( if_Pro6206227464963214023at_nat @ ( ord_less_eq_nat @ M @ K2 ) @ ( product_Pair_nat_nat @ M @ ( minus_minus_nat @ K2 @ M ) ) @ ( nat_prod_decode_aux @ ( suc @ K2 ) @ ( minus_minus_nat @ M @ ( suc @ K2 ) ) ) ) ) ) ).

% prod_decode_aux.simps
thf(fact_1264_pair__lessI2,axiom,
    ! [A: nat,B: nat,S3: nat,T3: nat] :
      ( ( ord_less_eq_nat @ A @ B )
     => ( ( ord_less_nat @ S3 @ T3 )
       => ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ ( product_Pair_nat_nat @ A @ S3 ) @ ( product_Pair_nat_nat @ B @ T3 ) ) @ fun_pair_less ) ) ) ).

% pair_lessI2
thf(fact_1265_pair__leqI2,axiom,
    ! [A: nat,B: nat,S3: nat,T3: nat] :
      ( ( ord_less_eq_nat @ A @ B )
     => ( ( ord_less_eq_nat @ S3 @ T3 )
       => ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ ( product_Pair_nat_nat @ A @ S3 ) @ ( product_Pair_nat_nat @ B @ T3 ) ) @ fun_pair_leq ) ) ) ).

% pair_leqI2
thf(fact_1266_bot__nat__0_Oordering__top__axioms,axiom,
    ( ordering_top_nat
    @ ^ [X: nat,Y3: nat] : ( ord_less_eq_nat @ Y3 @ X )
    @ ^ [X: nat,Y3: nat] : ( ord_less_nat @ Y3 @ X )
    @ zero_zero_nat ) ).

% bot_nat_0.ordering_top_axioms
thf(fact_1267_zero__notin__Suc__image,axiom,
    ! [A2: set_nat] :
      ~ ( member_nat @ zero_zero_nat @ ( image_nat_nat @ suc @ A2 ) ) ).

% zero_notin_Suc_image
thf(fact_1268_range__enumerate,axiom,
    ! [S4: set_nat] :
      ( ~ ( finite_finite_nat @ S4 )
     => ( ( image_nat_nat @ ( infini8530281810654367211te_nat @ S4 ) @ top_top_set_nat )
        = S4 ) ) ).

% range_enumerate
thf(fact_1269_epath__sdrop__ptms,axiom,
    ! [Steps: stream2709947120125613254m_rule,N: nat] :
      ( ( abstra6789711989322986974ist_fm @ eff @ rules @ Steps )
     => ( ord_less_eq_set_tm @ ( set_tm2 @ ( ptms @ ( shd_Pr4562317740776619530m_rule @ Steps ) ) ) @ ( set_tm2 @ ( ptms @ ( shd_Pr4562317740776619530m_rule @ ( sdrop_8169176516188972301m_rule @ N @ Steps ) ) ) ) ) ) ).

% epath_sdrop_ptms
thf(fact_1270_ptms__def,axiom,
    ( ptms
    = ( ^ [Z5: produc340336539035504054m_rule] : ( produc4874987732838623697ist_fm @ ( produc993771546023551940m_rule @ Z5 ) ) ) ) ).

% ptms_def
thf(fact_1271_mono__Suc,axiom,
    monotone_on_nat_nat @ top_top_set_nat @ ord_less_eq_nat @ ord_less_eq_nat @ suc ).

% mono_Suc
thf(fact_1272_strict__mono__imp__increasing,axiom,
    ! [F: nat > nat,N: nat] :
      ( ( monotone_on_nat_nat @ top_top_set_nat @ ord_less_nat @ ord_less_nat @ F )
     => ( ord_less_eq_nat @ N @ ( F @ N ) ) ) ).

% strict_mono_imp_increasing
thf(fact_1273_infinite__enumerate,axiom,
    ! [S4: set_nat] :
      ( ~ ( finite_finite_nat @ S4 )
     => ? [R6: nat > nat] :
          ( ( monotone_on_nat_nat @ top_top_set_nat @ ord_less_nat @ ord_less_nat @ R6 )
          & ! [N5: nat] : ( member_nat @ ( R6 @ N5 ) @ S4 ) ) ) ).

% infinite_enumerate
thf(fact_1274_strict__mono__enumerate,axiom,
    ! [S4: set_nat] :
      ( ~ ( finite_finite_nat @ S4 )
     => ( monotone_on_nat_nat @ top_top_set_nat @ ord_less_nat @ ord_less_nat @ ( infini8530281810654367211te_nat @ S4 ) ) ) ).

% strict_mono_enumerate

% Helper facts (13)
thf(help_fChoice_1_1_fChoice_001t__Nat__Onat_T,axiom,
    ! [P2: nat > $o] :
      ( ( P2 @ ( fChoice_nat @ P2 ) )
      = ( ? [X6: nat] : ( P2 @ X6 ) ) ) ).

thf(help_fChoice_1_1_fChoice_001t__Prover__Orule_T,axiom,
    ! [P2: rule > $o] :
      ( ( P2 @ ( fChoice_rule @ P2 ) )
      = ( ? [X6: rule] : ( P2 @ X6 ) ) ) ).

thf(help_If_2_1_If_001t__Stream__Ostream_It__Prover__Orule_J_T,axiom,
    ! [X3: stream_rule,Y: stream_rule] :
      ( ( if_stream_rule @ $false @ X3 @ Y )
      = Y ) ).

thf(help_If_1_1_If_001t__Stream__Ostream_It__Prover__Orule_J_T,axiom,
    ! [X3: stream_rule,Y: stream_rule] :
      ( ( if_stream_rule @ $true @ X3 @ Y )
      = 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,Y: product_prod_nat_nat] :
      ( ( if_Pro6206227464963214023at_nat @ $false @ X3 @ Y )
      = Y ) ).

thf(help_If_1_1_If_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_T,axiom,
    ! [X3: product_prod_nat_nat,Y: product_prod_nat_nat] :
      ( ( if_Pro6206227464963214023at_nat @ $true @ X3 @ Y )
      = X3 ) ).

thf(help_fChoice_1_1_fChoice_001t__Abstract____Completeness__Otree_It__Prover__Orule_J_T,axiom,
    ! [P2: abstract_tree_rule > $o] :
      ( ( P2 @ ( fChoic4892627540417248890e_rule @ P2 ) )
      = ( ? [X6: abstract_tree_rule] : ( P2 @ X6 ) ) ) ).

thf(help_fChoice_1_1_fChoice_001t__Product____Type__Oprod_It__List__Olist_It__SeCaV__Otm_J_Mt__List__Olist_It__SeCaV__Ofm_J_J_T,axiom,
    ! [P2: produc6018962875968178549ist_fm > $o] :
      ( ( P2 @ ( fChoic652867151139883209ist_fm @ P2 ) )
      = ( ? [X6: produc6018962875968178549ist_fm] : ( P2 @ X6 ) ) ) ).

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 ) )
      = ( ? [X6: produc859450856879609959at_nat] : ( P2 @ X6 ) ) ) ).

thf(help_If_3_1_If_001t__Stream__Ostream_It__Product____Type__Oprod_It__Product____Type__Oprod_It__List__Olist_It__SeCaV__Otm_J_Mt__List__Olist_It__SeCaV__Ofm_J_J_Mt__Prover__Orule_J_J_T,axiom,
    ! [P2: $o] :
      ( ( P2 = $true )
      | ( P2 = $false ) ) ).

thf(help_If_2_1_If_001t__Stream__Ostream_It__Product____Type__Oprod_It__Product____Type__Oprod_It__List__Olist_It__SeCaV__Otm_J_Mt__List__Olist_It__SeCaV__Ofm_J_J_Mt__Prover__Orule_J_J_T,axiom,
    ! [X3: stream2709947120125613254m_rule,Y: stream2709947120125613254m_rule] :
      ( ( if_str2295835923641592704m_rule @ $false @ X3 @ Y )
      = Y ) ).

thf(help_If_1_1_If_001t__Stream__Ostream_It__Product____Type__Oprod_It__Product____Type__Oprod_It__List__Olist_It__SeCaV__Otm_J_Mt__List__Olist_It__SeCaV__Ofm_J_J_Mt__Prover__Orule_J_J_T,axiom,
    ! [X3: stream2709947120125613254m_rule,Y: stream2709947120125613254m_rule] :
      ( ( if_str2295835923641592704m_rule @ $true @ X3 @ Y )
      = X3 ) ).

thf(help_fChoice_1_1_fChoice_001t__Abstract____Completeness__Otree_It__Product____Type__Oprod_It__Product____Type__Oprod_It__List__Olist_It__SeCaV__Otm_J_Mt__List__Olist_It__SeCaV__Ofm_J_J_Mt__Prover__Orule_J_J_T,axiom,
    ! [P2: abstra8520858721168826610m_rule > $o] :
      ( ( P2 @ ( fChoic3358775076371552926m_rule @ P2 ) )
      = ( ? [X6: abstra8520858721168826610m_rule] : ( P2 @ X6 ) ) ) ).

% Conjectures (1)
thf(conj_0,conjecture,
    ( linear1816559245158234865m_rule
    @ ( linear7925859179493631106m_rule
      @ ^ [Step: produc340336539035504054m_rule] :
          ( ( produc6514711208036230658m_rule @ Step )
          = betaImp ) )
    @ ( sdrop_8169176516188972301m_rule @ n @ steps ) ) ).

%------------------------------------------------------------------------------