TPTP Problem File: SLH0513^1.p
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%------------------------------------------------------------------------------
% 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_00388_015792__13170318_1 [Des23]
% Status : Theorem
% Rating : ? v8.2.0
% Syntax : Number of formulae : 1990 ( 596 unt; 687 typ; 0 def)
% Number of atoms : 3874 (1273 equ; 0 cnn)
% Maximal formula atoms : 15 ( 2 avg)
% Number of connectives : 15500 ( 396 ~; 75 |; 521 &;12879 @)
% ( 0 <=>;1629 =>; 0 <=; 0 <~>)
% Maximal formula depth : 20 ( 7 avg)
% Number of types : 107 ( 106 usr)
% Number of type conns : 2268 (2268 >; 0 *; 0 +; 0 <<)
% Number of symbols : 584 ( 581 usr; 25 con; 0-4 aty)
% Number of variables : 4475 ( 414 ^;3665 !; 396 ?;4475 :)
% SPC : TH0_THM_EQU_NAR
% Comments : This file was generated by Isabelle (most likely Sledgehammer)
% 2023-01-19 08:45:29.446
%------------------------------------------------------------------------------
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thf(ty_n_t__Nat__Onat,type,
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% Explicit typings (581)
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thf(sy_c_List_Ocan__select_001t__SeCaV__Ofm,type,
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thf(sy_c_List_Ocan__select_001t__SeCaV__Otm,type,
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thf(sy_c_List_Oconcat_001t__SeCaV__Ofm,type,
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thf(sy_c_List_Olist_OCons_001_062_It__Nat__Onat_Mt__Nat__Onat_J,type,
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thf(sy_c_List_Olist_OCons_001t__SeCaV__Ofm,type,
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thf(sy_c_List_Olist_OCons_001t__SeCaV__Otm,type,
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thf(sy_c_List_Olist_Ohd_001t__SeCaV__Ofm,type,
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thf(sy_c_List_Olist_Ohd_001t__SeCaV__Otm,type,
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thf(sy_c_List_Olist_Oset_001t__List__Olist_It__Prover__Orule_J,type,
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thf(sy_c_List_Olist_Oset_001t__SeCaV__Otm,type,
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thf(sy_c_List_Olist_Otl_001t__SeCaV__Ofm,type,
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thf(sy_c_List_Olist_Otl_001t__SeCaV__Otm,type,
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thf(sy_c_List_Olist__ex1_001t__SeCaV__Ofm,type,
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thf(sy_c_List_Olist__ex1_001t__SeCaV__Otm,type,
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thf(sy_c_List_Olist__update_001t__Prover__Orule,type,
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thf(sy_c_List_Olist__update_001t__SeCaV__Ofm,type,
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thf(sy_c_List_Olist__update_001t__SeCaV__Otm,type,
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thf(sy_c_List_Olistrel1_001t__Nat__Onat,type,
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thf(sy_c_List_Olistrel1_001t__SeCaV__Otm,type,
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thf(sy_c_List_Omeasures_001t__Nat__Onat,type,
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thf(sy_c_List_On__lists_001t__Prover__Orule,type,
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thf(sy_c_List_On__lists_001t__SeCaV__Ofm,type,
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thf(sy_c_List_On__lists_001t__SeCaV__Otm,type,
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thf(sy_c_List_Onth_001t__List__Olist_It__SeCaV__Ofm_J,type,
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thf(sy_c_List_Onth_001t__SeCaV__Ofm,type,
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thf(sy_c_List_Onth_001t__SeCaV__Otm,type,
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stl_fm: stream_fm > stream_fm ).
thf(sy_c_Stream_Ostream_Ostl_001t__SeCaV__Otm,type,
stl_tm: stream_tm > stream_tm ).
thf(sy_c_Stream_Ostream__all_001t__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,type,
stream2134157564947672044m_rule: ( produc340336539035504054m_rule > $o ) > stream2709947120125613254m_rule > $o ).
thf(sy_c_Stream_Ostream__all_001t__Prover__Orule,type,
stream_all_rule: ( rule > $o ) > stream_rule > $o ).
thf(sy_c_Stream_Oszip_001t__List__Olist_It__SeCaV__Otm_J_001t__List__Olist_It__SeCaV__Ofm_J,type,
szip_list_tm_list_fm: stream_list_tm > stream_list_fm > stream4408948924543953275ist_fm ).
thf(sy_c_Stream_Oszip_001t__Nat__Onat_001t__Nat__Onat,type,
szip_nat_nat: stream_nat > stream_nat > stream6724221391990029191at_nat ).
thf(sy_c_Stream_Oszip_001t__Product____Type__Oprod_It__List__Olist_It__SeCaV__Otm_J_Mt__List__Olist_It__SeCaV__Ofm_J_J_001t__Prover__Orule,type,
szip_P2924820683901490861m_rule: stream4408948924543953275ist_fm > stream_rule > stream2709947120125613254m_rule ).
thf(sy_c_Stream_Oszip_001t__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_001t__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,type,
szip_P811719526838699976m_rule: stream2709947120125613254m_rule > stream2709947120125613254m_rule > stream6210534828274662995m_rule ).
thf(sy_c_Stream_Oszip_001t__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_001t__Prover__Orule,type,
szip_P2044787555563963556e_rule: stream2709947120125613254m_rule > stream_rule > stream8099677779113257519e_rule ).
thf(sy_c_Stream_Oszip_001t__Prover__Orule_001t__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,type,
szip_r6533552703345879972m_rule: stream_rule > stream2709947120125613254m_rule > stream8953843411776101167m_rule ).
thf(sy_c_Stream_Oszip_001t__Prover__Orule_001t__Prover__Orule,type,
szip_rule_rule: stream_rule > stream_rule > stream4385846686851721995e_rule ).
thf(sy_c_Sublist_Oprefixes_001t__SeCaV__Otm,type,
prefixes_tm: list_tm > list_list_tm ).
thf(sy_c_member_001t__List__Olist_It__List__Olist_It__SeCaV__Ofm_J_J,type,
member_list_list_fm: list_list_fm > set_list_list_fm > $o ).
thf(sy_c_member_001t__List__Olist_It__Product____Type__Oprod_It__List__Olist_It__SeCaV__Otm_J_Mt__List__Olist_It__SeCaV__Ofm_J_J_J,type,
member4266112761388634404ist_fm: list_P2887561121880082555ist_fm > set_li7039099389128218331ist_fm > $o ).
thf(sy_c_member_001t__List__Olist_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,type,
member522667613741702877m_rule: list_P2774625669004309958m_rule > set_li5047378834958542076m_rule > $o ).
thf(sy_c_member_001t__List__Olist_It__Prover__Orule_J,type,
member_list_rule: list_rule > set_list_rule > $o ).
thf(sy_c_member_001t__List__Olist_It__SeCaV__Ofm_J,type,
member_list_fm: list_fm > set_list_fm > $o ).
thf(sy_c_member_001t__List__Olist_It__SeCaV__Otm_J,type,
member_list_tm: list_tm > set_list_tm > $o ).
thf(sy_c_member_001t__Nat__Onat,type,
member_nat: nat > set_nat > $o ).
thf(sy_c_member_001t__Product____Type__Oprod_It__List__Olist_It__List__Olist_It__SeCaV__Ofm_J_J_Mt__List__Olist_It__List__Olist_It__SeCaV__Ofm_J_J_J,type,
member5102842307574975760ist_fm: produc4386893285136772327ist_fm > set_Pr3202885091612042951ist_fm > $o ).
thf(sy_c_member_001t__Product____Type__Oprod_It__List__Olist_It__Nat__Onat_J_Mt__List__Olist_It__Nat__Onat_J_J,type,
member7340969449405702474st_nat: produc1828647624359046049st_nat > set_Pr3451248702717554689st_nat > $o ).
thf(sy_c_member_001t__Product____Type__Oprod_It__List__Olist_It__Product____Type__Oprod_It__List__Olist_It__SeCaV__Otm_J_Mt__List__Olist_It__SeCaV__Ofm_J_J_J_Mt__List__Olist_It__Product____Type__Oprod_It__List__Olist_It__SeCaV__Otm_J_Mt__List__Olist_It__SeCaV__Ofm_J_J_J_J,type,
member6238937682488541200ist_fm: produc8729865333125084903ist_fm > set_Pr8947555413577213383ist_fm > $o ).
thf(sy_c_member_001t__Product____Type__Oprod_It__List__Olist_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_Mt__List__Olist_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_J,type,
member3128744185454049558m_rule: produc7139673775237694317m_rule > set_Pr8447737590334104013m_rule > $o ).
thf(sy_c_member_001t__Product____Type__Oprod_It__List__Olist_It__Prover__Orule_J_Mt__List__Olist_It__Prover__Orule_J_J,type,
member695053538577043918t_rule: produc1530960993530998309t_rule > set_Pr5482552573296100997t_rule > $o ).
thf(sy_c_member_001t__Product____Type__Oprod_It__List__Olist_It__SeCaV__Ofm_J_Mt__List__Olist_It__SeCaV__Ofm_J_J,type,
member1926098302810316688ist_fm: produc3245234490656042599ist_fm > set_Pr1916137733696349511ist_fm > $o ).
thf(sy_c_member_001t__Product____Type__Oprod_It__List__Olist_It__SeCaV__Otm_J_Mt__List__Olist_It__SeCaV__Ofm_J_J,type,
member4699826688122452638ist_fm: produc6018962875968178549ist_fm > set_Pr5202636777678657877ist_fm > $o ).
thf(sy_c_member_001t__Product____Type__Oprod_It__List__Olist_It__SeCaV__Otm_J_Mt__List__Olist_It__SeCaV__Otm_J_J,type,
member4457312017796942864ist_tm: produc5776448205642668775ist_tm > set_Pr1507011332596240839ist_tm > $o ).
thf(sy_c_member_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J,type,
member8440522571783428010at_nat: product_prod_nat_nat > set_Pr1261947904930325089at_nat > $o ).
thf(sy_c_member_001t__Product____Type__Oprod_It__Product____Type__Oprod_It__List__Olist_It__SeCaV__Otm_J_Mt__List__Olist_It__SeCaV__Ofm_J_J_Mt__Product____Type__Oprod_It__List__Olist_It__SeCaV__Otm_J_Mt__List__Olist_It__SeCaV__Ofm_J_J_J,type,
member858341428514850704ist_fm: produc7850321100735229287ist_fm > set_Pr6724411253394807111ist_fm > $o ).
thf(sy_c_member_001t__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,type,
member7231649785386036813m_rule: produc340336539035504054m_rule > set_Pr1822751329126368876m_rule > $o ).
thf(sy_c_member_001t__Product____Type__Oprod_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_Mt__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,type,
member1411645983481675894m_rule: produc8828831911945107917m_rule > set_Pr4971326047967503661m_rule > $o ).
thf(sy_c_member_001t__Product____Type__Oprod_It__Prover__Orule_Mt__Prover__Orule_J,type,
member1444691403335511342e_rule: produc7694839378271647877e_rule > set_Pr7340090144965549541e_rule > $o ).
thf(sy_c_member_001t__Product____Type__Oprod_It__SeCaV__Ofm_Mt__SeCaV__Ofm_J,type,
member7780952600467998736_fm_fm: product_prod_fm_fm > set_Pr4706815898642364871_fm_fm > $o ).
thf(sy_c_member_001t__Product____Type__Oprod_It__SeCaV__Otm_Mt__SeCaV__Otm_J,type,
member3121616906494481296_tm_tm: product_prod_tm_tm > set_Pr2455929065695642951_tm_tm > $o ).
thf(sy_c_member_001t__Prover__Orule,type,
member_rule: rule > set_rule > $o ).
thf(sy_c_member_001t__SeCaV__Ofm,type,
member_fm: fm > set_fm > $o ).
thf(sy_c_member_001t__SeCaV__Otm,type,
member_tm: tm > set_tm > $o ).
thf(sy_v_n____,type,
n: nat ).
thf(sy_v_p____,type,
p: fm ).
thf(sy_v_pre____,type,
pre: list_P2774625669004309958m_rule ).
thf(sy_v_q____,type,
q: fm ).
thf(sy_v_steps,type,
steps: stream2709947120125613254m_rule ).
thf(sy_v_suf____,type,
suf: stream2709947120125613254m_rule ).
thf(sy_v_thesis____,type,
thesis: $o ).
% Relevant facts (1276)
thf(fact_0__092_060open_062epath_Asuf_092_060close_062,axiom,
abstra6789711989322986974ist_fm @ eff @ rules @ suf ).
% \<open>epath suf\<close>
thf(fact_1_epath__effect,axiom,
! [Steps: stream2709947120125613254m_rule,A: list_tm,Z: list_fm,R: rule] :
( ( abstra6789711989322986974ist_fm @ eff @ rules @ Steps )
=> ( ( ( shd_Pr4562317740776619530m_rule @ Steps )
= ( produc1733806532565653680m_rule @ ( produc1414352766439514085ist_fm @ A @ Z ) @ R ) )
=> ? [B: list_tm,Z2: list_fm,R2: rule] :
( ( fmembe3754813877001230652ist_fm @ ( produc1414352766439514085ist_fm @ B @ Z2 ) @ ( effect @ R @ ( produc1414352766439514085ist_fm @ A @ Z ) ) )
& ( ( shd_Pr4562317740776619530m_rule @ ( stl_Pr12655793849929990m_rule @ Steps ) )
= ( produc1733806532565653680m_rule @ ( produc1414352766439514085ist_fm @ B @ Z2 ) @ R2 ) )
& ( B
= ( remdups_tm @ ( append_tm @ A @ ( append_tm @ ( subterms @ Z ) @ ( subterms @ Z2 ) ) ) ) ) ) ) ) ).
% epath_effect
thf(fact_2_prod_Oinject,axiom,
! [X1: list_tm,X2: list_fm,Y1: list_tm,Y2: list_fm] :
( ( ( produc1414352766439514085ist_fm @ X1 @ X2 )
= ( produc1414352766439514085ist_fm @ Y1 @ Y2 ) )
= ( ( X1 = Y1 )
& ( X2 = Y2 ) ) ) ).
% prod.inject
thf(fact_3_prod_Oinject,axiom,
! [X1: produc6018962875968178549ist_fm,X2: rule,Y1: produc6018962875968178549ist_fm,Y2: rule] :
( ( ( produc1733806532565653680m_rule @ X1 @ X2 )
= ( produc1733806532565653680m_rule @ Y1 @ Y2 ) )
= ( ( X1 = Y1 )
& ( X2 = Y2 ) ) ) ).
% prod.inject
thf(fact_4_prod_Oinject,axiom,
! [X1: nat,X2: nat,Y1: nat,Y2: nat] :
( ( ( product_Pair_nat_nat @ X1 @ X2 )
= ( product_Pair_nat_nat @ Y1 @ Y2 ) )
= ( ( X1 = Y1 )
& ( X2 = Y2 ) ) ) ).
% prod.inject
thf(fact_5_old_Oprod_Oinject,axiom,
! [A2: list_tm,B2: list_fm,A3: list_tm,B3: list_fm] :
( ( ( produc1414352766439514085ist_fm @ A2 @ B2 )
= ( produc1414352766439514085ist_fm @ A3 @ B3 ) )
= ( ( A2 = A3 )
& ( B2 = B3 ) ) ) ).
% old.prod.inject
thf(fact_6_old_Oprod_Oinject,axiom,
! [A2: produc6018962875968178549ist_fm,B2: rule,A3: produc6018962875968178549ist_fm,B3: rule] :
( ( ( produc1733806532565653680m_rule @ A2 @ B2 )
= ( produc1733806532565653680m_rule @ A3 @ B3 ) )
= ( ( A2 = A3 )
& ( B2 = B3 ) ) ) ).
% old.prod.inject
thf(fact_7_old_Oprod_Oinject,axiom,
! [A2: nat,B2: nat,A3: nat,B3: nat] :
( ( ( product_Pair_nat_nat @ A2 @ B2 )
= ( product_Pair_nat_nat @ A3 @ B3 ) )
= ( ( A2 = A3 )
& ( B2 = B3 ) ) ) ).
% old.prod.inject
thf(fact_8_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_9_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_10_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_11_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_12_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_13_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_14_effect__tms,axiom,
! [B4: list_tm,Z3: list_fm,R: rule,A: list_tm,Z: list_fm] :
( ( fmembe3754813877001230652ist_fm @ ( produc1414352766439514085ist_fm @ B4 @ Z3 ) @ ( effect @ R @ ( produc1414352766439514085ist_fm @ A @ Z ) ) )
=> ( B4
= ( remdups_tm @ ( append_tm @ A @ ( append_tm @ ( subterms @ Z ) @ ( subterms @ Z3 ) ) ) ) ) ) ).
% effect_tms
thf(fact_15_epath__stl__ptms,axiom,
! [Steps: stream2709947120125613254m_rule] :
( ( abstra6789711989322986974ist_fm @ eff @ rules @ Steps )
=> ( ( ptms @ ( shd_Pr4562317740776619530m_rule @ ( stl_Pr12655793849929990m_rule @ Steps ) ) )
= ( remdups_tm @ ( append_tm @ ( ptms @ ( shd_Pr4562317740776619530m_rule @ Steps ) ) @ ( append_tm @ ( subterms @ ( pseq @ ( shd_Pr4562317740776619530m_rule @ Steps ) ) ) @ ( subterms @ ( pseq @ ( shd_Pr4562317740776619530m_rule @ ( stl_Pr12655793849929990m_rule @ Steps ) ) ) ) ) ) ) ) ) ).
% epath_stl_ptms
thf(fact_16_fset__eqI,axiom,
! [A: fset_P8989946509869081563ist_fm,B4: fset_P8989946509869081563ist_fm] :
( ! [X: produc6018962875968178549ist_fm] :
( ( fmembe3754813877001230652ist_fm @ X @ A )
= ( fmembe3754813877001230652ist_fm @ X @ B4 ) )
=> ( A = B4 ) ) ).
% fset_eqI
thf(fact_17_fset__eqI,axiom,
! [A: fset_A8849196716589201442m_rule,B4: fset_A8849196716589201442m_rule] :
( ! [X: abstra8520858721168826610m_rule] :
( ( fmembe1099044791951439339m_rule @ X @ A )
= ( fmembe1099044791951439339m_rule @ X @ B4 ) )
=> ( A = B4 ) ) ).
% fset_eqI
thf(fact_18_fequalityCE,axiom,
! [A: fset_P8989946509869081563ist_fm,B4: fset_P8989946509869081563ist_fm,C: produc6018962875968178549ist_fm] :
( ( A = B4 )
=> ( ( ( fmembe3754813877001230652ist_fm @ C @ A )
=> ~ ( fmembe3754813877001230652ist_fm @ C @ B4 ) )
=> ~ ( ~ ( fmembe3754813877001230652ist_fm @ C @ A )
=> ( fmembe3754813877001230652ist_fm @ C @ B4 ) ) ) ) ).
% fequalityCE
thf(fact_19_fequalityCE,axiom,
! [A: fset_A8849196716589201442m_rule,B4: fset_A8849196716589201442m_rule,C: abstra8520858721168826610m_rule] :
( ( A = B4 )
=> ( ( ( fmembe1099044791951439339m_rule @ C @ A )
=> ~ ( fmembe1099044791951439339m_rule @ C @ B4 ) )
=> ~ ( ~ ( fmembe1099044791951439339m_rule @ C @ A )
=> ( fmembe1099044791951439339m_rule @ C @ B4 ) ) ) ) ).
% fequalityCE
thf(fact_20_eq__fmem__trans,axiom,
! [A2: produc6018962875968178549ist_fm,B2: produc6018962875968178549ist_fm,A: fset_P8989946509869081563ist_fm] :
( ( A2 = B2 )
=> ( ( fmembe3754813877001230652ist_fm @ B2 @ A )
=> ( fmembe3754813877001230652ist_fm @ A2 @ A ) ) ) ).
% eq_fmem_trans
thf(fact_21_eq__fmem__trans,axiom,
! [A2: abstra8520858721168826610m_rule,B2: abstra8520858721168826610m_rule,A: fset_A8849196716589201442m_rule] :
( ( A2 = B2 )
=> ( ( fmembe1099044791951439339m_rule @ B2 @ A )
=> ( fmembe1099044791951439339m_rule @ A2 @ A ) ) ) ).
% eq_fmem_trans
thf(fact_22_eqfset__imp__iff,axiom,
! [A: fset_P8989946509869081563ist_fm,B4: fset_P8989946509869081563ist_fm,X3: produc6018962875968178549ist_fm] :
( ( A = B4 )
=> ( ( fmembe3754813877001230652ist_fm @ X3 @ A )
= ( fmembe3754813877001230652ist_fm @ X3 @ B4 ) ) ) ).
% eqfset_imp_iff
thf(fact_23_eqfset__imp__iff,axiom,
! [A: fset_A8849196716589201442m_rule,B4: fset_A8849196716589201442m_rule,X3: abstra8520858721168826610m_rule] :
( ( A = B4 )
=> ( ( fmembe1099044791951439339m_rule @ X3 @ A )
= ( fmembe1099044791951439339m_rule @ X3 @ B4 ) ) ) ).
% eqfset_imp_iff
thf(fact_24_assms_I1_J,axiom,
abstra6789711989322986974ist_fm @ eff @ rules @ steps ).
% assms(1)
thf(fact_25_Pair__inject,axiom,
! [A2: list_tm,B2: list_fm,A3: list_tm,B3: list_fm] :
( ( ( produc1414352766439514085ist_fm @ A2 @ B2 )
= ( produc1414352766439514085ist_fm @ A3 @ B3 ) )
=> ~ ( ( A2 = A3 )
=> ( B2 != B3 ) ) ) ).
% Pair_inject
thf(fact_26_Pair__inject,axiom,
! [A2: produc6018962875968178549ist_fm,B2: rule,A3: produc6018962875968178549ist_fm,B3: rule] :
( ( ( produc1733806532565653680m_rule @ A2 @ B2 )
= ( produc1733806532565653680m_rule @ A3 @ B3 ) )
=> ~ ( ( A2 = A3 )
=> ( B2 != B3 ) ) ) ).
% Pair_inject
thf(fact_27_Pair__inject,axiom,
! [A2: nat,B2: nat,A3: nat,B3: nat] :
( ( ( product_Pair_nat_nat @ A2 @ B2 )
= ( product_Pair_nat_nat @ A3 @ B3 ) )
=> ~ ( ( A2 = A3 )
=> ( B2 != B3 ) ) ) ).
% Pair_inject
thf(fact_28_prod__cases,axiom,
! [P: produc6018962875968178549ist_fm > $o,P2: produc6018962875968178549ist_fm] :
( ! [A4: list_tm,B5: list_fm] : ( P @ ( produc1414352766439514085ist_fm @ A4 @ B5 ) )
=> ( P @ P2 ) ) ).
% prod_cases
thf(fact_29_prod__cases,axiom,
! [P: produc340336539035504054m_rule > $o,P2: produc340336539035504054m_rule] :
( ! [A4: produc6018962875968178549ist_fm,B5: rule] : ( P @ ( produc1733806532565653680m_rule @ A4 @ B5 ) )
=> ( P @ P2 ) ) ).
% prod_cases
thf(fact_30_prod__cases,axiom,
! [P: product_prod_nat_nat > $o,P2: product_prod_nat_nat] :
( ! [A4: nat,B5: nat] : ( P @ ( product_Pair_nat_nat @ A4 @ B5 ) )
=> ( P @ P2 ) ) ).
% prod_cases
thf(fact_31_surj__pair,axiom,
! [P2: produc6018962875968178549ist_fm] :
? [X: list_tm,Y: list_fm] :
( P2
= ( produc1414352766439514085ist_fm @ X @ Y ) ) ).
% surj_pair
thf(fact_32_surj__pair,axiom,
! [P2: produc340336539035504054m_rule] :
? [X: produc6018962875968178549ist_fm,Y: rule] :
( P2
= ( produc1733806532565653680m_rule @ X @ Y ) ) ).
% surj_pair
thf(fact_33_surj__pair,axiom,
! [P2: product_prod_nat_nat] :
? [X: nat,Y: nat] :
( P2
= ( product_Pair_nat_nat @ X @ Y ) ) ).
% surj_pair
thf(fact_34_old_Oprod_Oexhaust,axiom,
! [Y3: produc6018962875968178549ist_fm] :
~ ! [A4: list_tm,B5: list_fm] :
( Y3
!= ( produc1414352766439514085ist_fm @ A4 @ B5 ) ) ).
% old.prod.exhaust
thf(fact_35_old_Oprod_Oexhaust,axiom,
! [Y3: produc340336539035504054m_rule] :
~ ! [A4: produc6018962875968178549ist_fm,B5: rule] :
( Y3
!= ( produc1733806532565653680m_rule @ A4 @ B5 ) ) ).
% old.prod.exhaust
thf(fact_36_old_Oprod_Oexhaust,axiom,
! [Y3: product_prod_nat_nat] :
~ ! [A4: nat,B5: nat] :
( Y3
!= ( product_Pair_nat_nat @ A4 @ B5 ) ) ).
% old.prod.exhaust
thf(fact_37_eqfelem__imp__iff,axiom,
! [X3: produc6018962875968178549ist_fm,Y3: produc6018962875968178549ist_fm,A: fset_P8989946509869081563ist_fm] :
( ( X3 = Y3 )
=> ( ( fmembe3754813877001230652ist_fm @ X3 @ A )
= ( fmembe3754813877001230652ist_fm @ Y3 @ A ) ) ) ).
% eqfelem_imp_iff
thf(fact_38_eqfelem__imp__iff,axiom,
! [X3: abstra8520858721168826610m_rule,Y3: abstra8520858721168826610m_rule,A: fset_A8849196716589201442m_rule] :
( ( X3 = Y3 )
=> ( ( fmembe1099044791951439339m_rule @ X3 @ A )
= ( fmembe1099044791951439339m_rule @ Y3 @ A ) ) ) ).
% eqfelem_imp_iff
thf(fact_39_if__split__fmem2,axiom,
! [A2: produc6018962875968178549ist_fm,Q: $o,X3: fset_P8989946509869081563ist_fm,Y3: fset_P8989946509869081563ist_fm] :
( ( fmembe3754813877001230652ist_fm @ A2 @ ( if_fse7999432387889793441ist_fm @ Q @ X3 @ Y3 ) )
= ( ( Q
=> ( fmembe3754813877001230652ist_fm @ A2 @ X3 ) )
& ( ~ Q
=> ( fmembe3754813877001230652ist_fm @ A2 @ Y3 ) ) ) ) ).
% if_split_fmem2
thf(fact_40_if__split__fmem2,axiom,
! [A2: abstra8520858721168826610m_rule,Q: $o,X3: fset_A8849196716589201442m_rule,Y3: fset_A8849196716589201442m_rule] :
( ( fmembe1099044791951439339m_rule @ A2 @ ( if_fse9105678332773368284m_rule @ Q @ X3 @ Y3 ) )
= ( ( Q
=> ( fmembe1099044791951439339m_rule @ A2 @ X3 ) )
& ( ~ Q
=> ( fmembe1099044791951439339m_rule @ A2 @ Y3 ) ) ) ) ).
% if_split_fmem2
thf(fact_41_if__split__fmem1,axiom,
! [Q: $o,X3: produc6018962875968178549ist_fm,Y3: produc6018962875968178549ist_fm,B2: fset_P8989946509869081563ist_fm] :
( ( fmembe3754813877001230652ist_fm @ ( if_Pro4760001780252510779ist_fm @ Q @ X3 @ Y3 ) @ B2 )
= ( ( Q
=> ( fmembe3754813877001230652ist_fm @ X3 @ B2 ) )
& ( ~ Q
=> ( fmembe3754813877001230652ist_fm @ Y3 @ B2 ) ) ) ) ).
% if_split_fmem1
thf(fact_42_if__split__fmem1,axiom,
! [Q: $o,X3: abstra8520858721168826610m_rule,Y3: abstra8520858721168826610m_rule,B2: fset_A8849196716589201442m_rule] :
( ( fmembe1099044791951439339m_rule @ ( if_Abs1293773526126661548m_rule @ Q @ X3 @ Y3 ) @ B2 )
= ( ( Q
=> ( fmembe1099044791951439339m_rule @ X3 @ B2 ) )
& ( ~ Q
=> ( fmembe1099044791951439339m_rule @ Y3 @ B2 ) ) ) ) ).
% if_split_fmem1
thf(fact_43_assms_I2_J,axiom,
abstra6097777249025082867ist_fm @ eff @ rules @ steps ).
% assms(2)
thf(fact_44_append_Oassoc,axiom,
! [A2: list_tm,B2: list_tm,C: list_tm] :
( ( append_tm @ ( append_tm @ A2 @ B2 ) @ C )
= ( append_tm @ A2 @ ( append_tm @ B2 @ C ) ) ) ).
% append.assoc
thf(fact_45_append__assoc,axiom,
! [Xs: list_tm,Ys: list_tm,Zs: list_tm] :
( ( append_tm @ ( append_tm @ Xs @ Ys ) @ Zs )
= ( append_tm @ Xs @ ( append_tm @ Ys @ Zs ) ) ) ).
% append_assoc
thf(fact_46_append__same__eq,axiom,
! [Ys: list_tm,Xs: list_tm,Zs: list_tm] :
( ( ( append_tm @ Ys @ Xs )
= ( append_tm @ Zs @ Xs ) )
= ( Ys = Zs ) ) ).
% append_same_eq
thf(fact_47_same__append__eq,axiom,
! [Xs: list_tm,Ys: list_tm,Zs: list_tm] :
( ( ( append_tm @ Xs @ Ys )
= ( append_tm @ Xs @ Zs ) )
= ( Ys = Zs ) ) ).
% same_append_eq
thf(fact_48_remdups__append2,axiom,
! [Xs: list_tm,Ys: list_tm] :
( ( remdups_tm @ ( append_tm @ Xs @ ( remdups_tm @ Ys ) ) )
= ( remdups_tm @ ( append_tm @ Xs @ Ys ) ) ) ).
% remdups_append2
thf(fact_49__092_060open_062epath_A_Ipre_A_064_N_Asuf_J_092_060close_062,axiom,
abstra6789711989322986974ist_fm @ eff @ rules @ ( shift_2334844276748245581m_rule @ pre @ suf ) ).
% \<open>epath (pre @- suf)\<close>
thf(fact_50_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_51_szip_Osimps_I1_J,axiom,
! [S1: stream_list_tm,S2: stream_list_fm] :
( ( shd_Pr3211216682057661985ist_fm @ ( szip_list_tm_list_fm @ S1 @ S2 ) )
= ( produc1414352766439514085ist_fm @ ( shd_list_tm @ S1 ) @ ( shd_list_fm @ S2 ) ) ) ).
% szip.simps(1)
thf(fact_52_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_53_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_54_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_55_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_56_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_57_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_58_snd__conv,axiom,
! [X1: nat,X2: nat] :
( ( product_snd_nat_nat @ ( product_Pair_nat_nat @ X1 @ X2 ) )
= X2 ) ).
% snd_conv
thf(fact_59_snd__conv,axiom,
! [X1: produc6018962875968178549ist_fm,X2: rule] :
( ( produc6514711208036230658m_rule @ ( produc1733806532565653680m_rule @ X1 @ X2 ) )
= X2 ) ).
% snd_conv
thf(fact_60_snd__conv,axiom,
! [X1: list_tm,X2: list_fm] :
( ( produc3242453064496107283ist_fm @ ( produc1414352766439514085ist_fm @ X1 @ X2 ) )
= X2 ) ).
% snd_conv
thf(fact_61_snd__eqD,axiom,
! [X3: nat,Y3: nat,A2: nat] :
( ( ( product_snd_nat_nat @ ( product_Pair_nat_nat @ X3 @ Y3 ) )
= A2 )
=> ( Y3 = A2 ) ) ).
% snd_eqD
thf(fact_62_snd__eqD,axiom,
! [X3: produc6018962875968178549ist_fm,Y3: rule,A2: rule] :
( ( ( produc6514711208036230658m_rule @ ( produc1733806532565653680m_rule @ X3 @ Y3 ) )
= A2 )
=> ( Y3 = A2 ) ) ).
% snd_eqD
thf(fact_63_snd__eqD,axiom,
! [X3: list_tm,Y3: list_fm,A2: list_fm] :
( ( ( produc3242453064496107283ist_fm @ ( produc1414352766439514085ist_fm @ X3 @ Y3 ) )
= A2 )
=> ( Y3 = A2 ) ) ).
% snd_eqD
thf(fact_64_mem__Collect__eq,axiom,
! [A2: rule,P: rule > $o] :
( ( member_rule @ A2 @ ( collect_rule @ P ) )
= ( P @ A2 ) ) ).
% mem_Collect_eq
thf(fact_65_mem__Collect__eq,axiom,
! [A2: fm,P: fm > $o] :
( ( member_fm @ A2 @ ( collect_fm @ P ) )
= ( P @ A2 ) ) ).
% mem_Collect_eq
thf(fact_66_mem__Collect__eq,axiom,
! [A2: produc6018962875968178549ist_fm,P: produc6018962875968178549ist_fm > $o] :
( ( member4699826688122452638ist_fm @ A2 @ ( collec5845330005437105888ist_fm @ P ) )
= ( P @ A2 ) ) ).
% mem_Collect_eq
thf(fact_67_mem__Collect__eq,axiom,
! [A2: produc340336539035504054m_rule,P: produc340336539035504054m_rule > $o] :
( ( member7231649785386036813m_rule @ A2 @ ( collec1009276759030335883m_rule @ P ) )
= ( P @ A2 ) ) ).
% mem_Collect_eq
thf(fact_68_mem__Collect__eq,axiom,
! [A2: list_fm,P: list_fm > $o] :
( ( member_list_fm @ A2 @ ( collect_list_fm @ P ) )
= ( P @ A2 ) ) ).
% mem_Collect_eq
thf(fact_69_Collect__mem__eq,axiom,
! [A: set_rule] :
( ( collect_rule
@ ^ [X4: rule] : ( member_rule @ X4 @ A ) )
= A ) ).
% Collect_mem_eq
thf(fact_70_Collect__mem__eq,axiom,
! [A: set_fm] :
( ( collect_fm
@ ^ [X4: fm] : ( member_fm @ X4 @ A ) )
= A ) ).
% Collect_mem_eq
thf(fact_71_Collect__mem__eq,axiom,
! [A: set_Pr5202636777678657877ist_fm] :
( ( collec5845330005437105888ist_fm
@ ^ [X4: produc6018962875968178549ist_fm] : ( member4699826688122452638ist_fm @ X4 @ A ) )
= A ) ).
% Collect_mem_eq
thf(fact_72_Collect__mem__eq,axiom,
! [A: set_Pr1822751329126368876m_rule] :
( ( collec1009276759030335883m_rule
@ ^ [X4: produc340336539035504054m_rule] : ( member7231649785386036813m_rule @ X4 @ A ) )
= A ) ).
% Collect_mem_eq
thf(fact_73_Collect__mem__eq,axiom,
! [A: set_list_fm] :
( ( collect_list_fm
@ ^ [X4: list_fm] : ( member_list_fm @ X4 @ A ) )
= A ) ).
% Collect_mem_eq
thf(fact_74_shift__left__inj,axiom,
! [Xs: list_P2774625669004309958m_rule,S1: stream2709947120125613254m_rule,S2: stream2709947120125613254m_rule] :
( ( ( shift_2334844276748245581m_rule @ Xs @ S1 )
= ( shift_2334844276748245581m_rule @ Xs @ S2 ) )
= ( S1 = S2 ) ) ).
% shift_left_inj
thf(fact_75_shift__append,axiom,
! [Xs: list_tm,Ys: list_tm,S: stream_tm] :
( ( shift_tm @ ( append_tm @ Xs @ Ys ) @ S )
= ( shift_tm @ Xs @ ( shift_tm @ Ys @ S ) ) ) ).
% shift_append
thf(fact_76_shift__append,axiom,
! [Xs: list_P2774625669004309958m_rule,Ys: list_P2774625669004309958m_rule,S: stream2709947120125613254m_rule] :
( ( shift_2334844276748245581m_rule @ ( append2714808343334685275m_rule @ Xs @ Ys ) @ S )
= ( shift_2334844276748245581m_rule @ Xs @ ( shift_2334844276748245581m_rule @ Ys @ S ) ) ) ).
% shift_append
thf(fact_77_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_78_prod_Ocollapse,axiom,
! [Prod: produc340336539035504054m_rule] :
( ( produc1733806532565653680m_rule @ ( produc993771546023551940m_rule @ Prod ) @ ( produc6514711208036230658m_rule @ Prod ) )
= Prod ) ).
% prod.collapse
thf(fact_79_prod_Ocollapse,axiom,
! [Prod: produc6018962875968178549ist_fm] :
( ( produc1414352766439514085ist_fm @ ( produc4874987732838623697ist_fm @ Prod ) @ ( produc3242453064496107283ist_fm @ Prod ) )
= Prod ) ).
% prod.collapse
thf(fact_80_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_81_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_82_prod__eqI,axiom,
! [P2: produc340336539035504054m_rule,Q2: produc340336539035504054m_rule] :
( ( ( produc993771546023551940m_rule @ P2 )
= ( produc993771546023551940m_rule @ Q2 ) )
=> ( ( ( produc6514711208036230658m_rule @ P2 )
= ( produc6514711208036230658m_rule @ Q2 ) )
=> ( P2 = Q2 ) ) ) ).
% prod_eqI
thf(fact_83_prod__eqI,axiom,
! [P2: produc6018962875968178549ist_fm,Q2: produc6018962875968178549ist_fm] :
( ( ( produc4874987732838623697ist_fm @ P2 )
= ( produc4874987732838623697ist_fm @ Q2 ) )
=> ( ( ( produc3242453064496107283ist_fm @ P2 )
= ( produc3242453064496107283ist_fm @ Q2 ) )
=> ( P2 = Q2 ) ) ) ).
% prod_eqI
thf(fact_84_prod__eq__iff,axiom,
( ( ^ [Y4: produc340336539035504054m_rule,Z4: produc340336539035504054m_rule] : ( Y4 = Z4 ) )
= ( ^ [S3: produc340336539035504054m_rule,T: produc340336539035504054m_rule] :
( ( ( produc993771546023551940m_rule @ S3 )
= ( produc993771546023551940m_rule @ T ) )
& ( ( produc6514711208036230658m_rule @ S3 )
= ( produc6514711208036230658m_rule @ T ) ) ) ) ) ).
% prod_eq_iff
thf(fact_85_prod__eq__iff,axiom,
( ( ^ [Y4: produc6018962875968178549ist_fm,Z4: produc6018962875968178549ist_fm] : ( Y4 = Z4 ) )
= ( ^ [S3: produc6018962875968178549ist_fm,T: produc6018962875968178549ist_fm] :
( ( ( produc4874987732838623697ist_fm @ S3 )
= ( produc4874987732838623697ist_fm @ T ) )
& ( ( produc3242453064496107283ist_fm @ S3 )
= ( produc3242453064496107283ist_fm @ T ) ) ) ) ) ).
% prod_eq_iff
thf(fact_86_pseq__def,axiom,
( pseq
= ( ^ [Z5: produc340336539035504054m_rule] : ( produc3242453064496107283ist_fm @ ( produc993771546023551940m_rule @ Z5 ) ) ) ) ).
% pseq_def
thf(fact_87_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_88_prod_Oexhaust__sel,axiom,
! [Prod: produc340336539035504054m_rule] :
( Prod
= ( produc1733806532565653680m_rule @ ( produc993771546023551940m_rule @ Prod ) @ ( produc6514711208036230658m_rule @ Prod ) ) ) ).
% prod.exhaust_sel
thf(fact_89_prod_Oexhaust__sel,axiom,
! [Prod: produc6018962875968178549ist_fm] :
( Prod
= ( produc1414352766439514085ist_fm @ ( produc4874987732838623697ist_fm @ Prod ) @ ( produc3242453064496107283ist_fm @ Prod ) ) ) ).
% prod.exhaust_sel
thf(fact_90_surjective__pairing,axiom,
! [T2: product_prod_nat_nat] :
( T2
= ( product_Pair_nat_nat @ ( product_fst_nat_nat @ T2 ) @ ( product_snd_nat_nat @ T2 ) ) ) ).
% surjective_pairing
thf(fact_91_surjective__pairing,axiom,
! [T2: produc340336539035504054m_rule] :
( T2
= ( produc1733806532565653680m_rule @ ( produc993771546023551940m_rule @ T2 ) @ ( produc6514711208036230658m_rule @ T2 ) ) ) ).
% surjective_pairing
thf(fact_92_surjective__pairing,axiom,
! [T2: produc6018962875968178549ist_fm] :
( T2
= ( produc1414352766439514085ist_fm @ ( produc4874987732838623697ist_fm @ T2 ) @ ( produc3242453064496107283ist_fm @ T2 ) ) ) ).
% surjective_pairing
thf(fact_93_fst__eqD,axiom,
! [X3: nat,Y3: nat,A2: nat] :
( ( ( product_fst_nat_nat @ ( product_Pair_nat_nat @ X3 @ Y3 ) )
= A2 )
=> ( X3 = A2 ) ) ).
% fst_eqD
thf(fact_94_fst__eqD,axiom,
! [X3: produc6018962875968178549ist_fm,Y3: rule,A2: produc6018962875968178549ist_fm] :
( ( ( produc993771546023551940m_rule @ ( produc1733806532565653680m_rule @ X3 @ Y3 ) )
= A2 )
=> ( X3 = A2 ) ) ).
% fst_eqD
thf(fact_95_fst__eqD,axiom,
! [X3: list_tm,Y3: list_fm,A2: list_tm] :
( ( ( produc4874987732838623697ist_fm @ ( produc1414352766439514085ist_fm @ X3 @ Y3 ) )
= A2 )
=> ( X3 = A2 ) ) ).
% fst_eqD
thf(fact_96_fst__conv,axiom,
! [X1: nat,X2: nat] :
( ( product_fst_nat_nat @ ( product_Pair_nat_nat @ X1 @ X2 ) )
= X1 ) ).
% fst_conv
thf(fact_97_fst__conv,axiom,
! [X1: produc6018962875968178549ist_fm,X2: rule] :
( ( produc993771546023551940m_rule @ ( produc1733806532565653680m_rule @ X1 @ X2 ) )
= X1 ) ).
% fst_conv
thf(fact_98_fst__conv,axiom,
! [X1: list_tm,X2: list_fm] :
( ( produc4874987732838623697ist_fm @ ( produc1414352766439514085ist_fm @ X1 @ X2 ) )
= X1 ) ).
% fst_conv
thf(fact_99_append__eq__append__conv2,axiom,
! [Xs: list_tm,Ys: list_tm,Zs: list_tm,Ts: list_tm] :
( ( ( append_tm @ Xs @ Ys )
= ( append_tm @ Zs @ Ts ) )
= ( ? [Us: list_tm] :
( ( ( Xs
= ( append_tm @ Zs @ Us ) )
& ( ( append_tm @ Us @ Ys )
= Ts ) )
| ( ( ( append_tm @ Xs @ Us )
= Zs )
& ( Ys
= ( append_tm @ Us @ Ts ) ) ) ) ) ) ).
% append_eq_append_conv2
thf(fact_100_append__eq__appendI,axiom,
! [Xs: list_tm,Xs1: list_tm,Zs: list_tm,Ys: list_tm,Us2: list_tm] :
( ( ( append_tm @ Xs @ Xs1 )
= Zs )
=> ( ( Ys
= ( append_tm @ Xs1 @ Us2 ) )
=> ( ( append_tm @ Xs @ Ys )
= ( append_tm @ Zs @ Us2 ) ) ) ) ).
% append_eq_appendI
thf(fact_101_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_102_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_103_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_104_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_105_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_106_remdups__remdups,axiom,
! [Xs: list_tm] :
( ( remdups_tm @ ( remdups_tm @ Xs ) )
= ( remdups_tm @ Xs ) ) ).
% remdups_remdups
thf(fact_107_ori,axiom,
( ( sdrop_8169176516188972301m_rule @ n @ steps )
= ( shift_2334844276748245581m_rule @ pre @ suf ) ) ).
% ori
thf(fact_108_exI__realizer,axiom,
! [P: nat > nat > $o,Y3: nat,X3: nat] :
( ( P @ Y3 @ X3 )
=> ( P @ ( product_snd_nat_nat @ ( product_Pair_nat_nat @ X3 @ Y3 ) ) @ ( product_fst_nat_nat @ ( product_Pair_nat_nat @ X3 @ Y3 ) ) ) ) ).
% exI_realizer
thf(fact_109_exI__realizer,axiom,
! [P: rule > produc6018962875968178549ist_fm > $o,Y3: rule,X3: produc6018962875968178549ist_fm] :
( ( P @ Y3 @ X3 )
=> ( P @ ( produc6514711208036230658m_rule @ ( produc1733806532565653680m_rule @ X3 @ Y3 ) ) @ ( produc993771546023551940m_rule @ ( produc1733806532565653680m_rule @ X3 @ Y3 ) ) ) ) ).
% exI_realizer
thf(fact_110_exI__realizer,axiom,
! [P: list_fm > list_tm > $o,Y3: list_fm,X3: list_tm] :
( ( P @ Y3 @ X3 )
=> ( P @ ( produc3242453064496107283ist_fm @ ( produc1414352766439514085ist_fm @ X3 @ Y3 ) ) @ ( produc4874987732838623697ist_fm @ ( produc1414352766439514085ist_fm @ X3 @ Y3 ) ) ) ) ).
% exI_realizer
thf(fact_111_conjI__realizer,axiom,
! [P: nat > $o,P2: nat,Q: nat > $o,Q2: nat] :
( ( P @ P2 )
=> ( ( Q @ Q2 )
=> ( ( P @ ( product_fst_nat_nat @ ( product_Pair_nat_nat @ P2 @ Q2 ) ) )
& ( Q @ ( product_snd_nat_nat @ ( product_Pair_nat_nat @ P2 @ Q2 ) ) ) ) ) ) ).
% conjI_realizer
thf(fact_112_conjI__realizer,axiom,
! [P: produc6018962875968178549ist_fm > $o,P2: produc6018962875968178549ist_fm,Q: rule > $o,Q2: rule] :
( ( P @ P2 )
=> ( ( Q @ Q2 )
=> ( ( P @ ( produc993771546023551940m_rule @ ( produc1733806532565653680m_rule @ P2 @ Q2 ) ) )
& ( Q @ ( produc6514711208036230658m_rule @ ( produc1733806532565653680m_rule @ P2 @ Q2 ) ) ) ) ) ) ).
% conjI_realizer
thf(fact_113_conjI__realizer,axiom,
! [P: list_tm > $o,P2: list_tm,Q: list_fm > $o,Q2: list_fm] :
( ( P @ P2 )
=> ( ( Q @ Q2 )
=> ( ( P @ ( produc4874987732838623697ist_fm @ ( produc1414352766439514085ist_fm @ P2 @ Q2 ) ) )
& ( Q @ ( produc3242453064496107283ist_fm @ ( produc1414352766439514085ist_fm @ P2 @ Q2 ) ) ) ) ) ) ).
% conjI_realizer
thf(fact_114_BNF__Greatest__Fixpoint_Osubst__Pair,axiom,
! [P: nat > nat > $o,X3: nat,Y3: nat,A2: product_prod_nat_nat] :
( ( P @ X3 @ Y3 )
=> ( ( A2
= ( product_Pair_nat_nat @ X3 @ Y3 ) )
=> ( P @ ( product_fst_nat_nat @ A2 ) @ ( product_snd_nat_nat @ A2 ) ) ) ) ).
% BNF_Greatest_Fixpoint.subst_Pair
thf(fact_115_BNF__Greatest__Fixpoint_Osubst__Pair,axiom,
! [P: produc6018962875968178549ist_fm > rule > $o,X3: produc6018962875968178549ist_fm,Y3: rule,A2: produc340336539035504054m_rule] :
( ( P @ X3 @ Y3 )
=> ( ( A2
= ( produc1733806532565653680m_rule @ X3 @ Y3 ) )
=> ( P @ ( produc993771546023551940m_rule @ A2 ) @ ( produc6514711208036230658m_rule @ A2 ) ) ) ) ).
% BNF_Greatest_Fixpoint.subst_Pair
thf(fact_116_BNF__Greatest__Fixpoint_Osubst__Pair,axiom,
! [P: list_tm > list_fm > $o,X3: list_tm,Y3: list_fm,A2: produc6018962875968178549ist_fm] :
( ( P @ X3 @ Y3 )
=> ( ( A2
= ( produc1414352766439514085ist_fm @ X3 @ Y3 ) )
=> ( P @ ( produc4874987732838623697ist_fm @ A2 ) @ ( produc3242453064496107283ist_fm @ A2 ) ) ) ) ).
% BNF_Greatest_Fixpoint.subst_Pair
thf(fact_117_exE__realizer_H,axiom,
! [P: rule > produc6018962875968178549ist_fm > $o,P2: produc340336539035504054m_rule] :
( ( P @ ( produc6514711208036230658m_rule @ P2 ) @ ( produc993771546023551940m_rule @ P2 ) )
=> ~ ! [X: produc6018962875968178549ist_fm,Y: rule] :
~ ( P @ Y @ X ) ) ).
% exE_realizer'
thf(fact_118_exE__realizer_H,axiom,
! [P: list_fm > list_tm > $o,P2: produc6018962875968178549ist_fm] :
( ( P @ ( produc3242453064496107283ist_fm @ P2 ) @ ( produc4874987732838623697ist_fm @ P2 ) )
=> ~ ! [X: list_tm,Y: list_fm] :
~ ( P @ Y @ X ) ) ).
% exE_realizer'
thf(fact_119_sndI,axiom,
! [X3: product_prod_nat_nat,Y3: nat,Z: nat] :
( ( X3
= ( product_Pair_nat_nat @ Y3 @ Z ) )
=> ( ( product_snd_nat_nat @ X3 )
= Z ) ) ).
% sndI
thf(fact_120_sndI,axiom,
! [X3: produc340336539035504054m_rule,Y3: produc6018962875968178549ist_fm,Z: rule] :
( ( X3
= ( produc1733806532565653680m_rule @ Y3 @ Z ) )
=> ( ( produc6514711208036230658m_rule @ X3 )
= Z ) ) ).
% sndI
thf(fact_121_sndI,axiom,
! [X3: produc6018962875968178549ist_fm,Y3: list_tm,Z: list_fm] :
( ( X3
= ( produc1414352766439514085ist_fm @ Y3 @ Z ) )
=> ( ( produc3242453064496107283ist_fm @ X3 )
= Z ) ) ).
% sndI
thf(fact_122_eq__snd__iff,axiom,
! [B2: nat,P2: product_prod_nat_nat] :
( ( B2
= ( product_snd_nat_nat @ P2 ) )
= ( ? [A5: nat] :
( P2
= ( product_Pair_nat_nat @ A5 @ B2 ) ) ) ) ).
% eq_snd_iff
thf(fact_123_eq__snd__iff,axiom,
! [B2: rule,P2: produc340336539035504054m_rule] :
( ( B2
= ( produc6514711208036230658m_rule @ P2 ) )
= ( ? [A5: produc6018962875968178549ist_fm] :
( P2
= ( produc1733806532565653680m_rule @ A5 @ B2 ) ) ) ) ).
% eq_snd_iff
thf(fact_124_eq__snd__iff,axiom,
! [B2: list_fm,P2: produc6018962875968178549ist_fm] :
( ( B2
= ( produc3242453064496107283ist_fm @ P2 ) )
= ( ? [A5: list_tm] :
( P2
= ( produc1414352766439514085ist_fm @ A5 @ B2 ) ) ) ) ).
% eq_snd_iff
thf(fact_125_fstI,axiom,
! [X3: product_prod_nat_nat,Y3: nat,Z: nat] :
( ( X3
= ( product_Pair_nat_nat @ Y3 @ Z ) )
=> ( ( product_fst_nat_nat @ X3 )
= Y3 ) ) ).
% fstI
thf(fact_126_fstI,axiom,
! [X3: produc340336539035504054m_rule,Y3: produc6018962875968178549ist_fm,Z: rule] :
( ( X3
= ( produc1733806532565653680m_rule @ Y3 @ Z ) )
=> ( ( produc993771546023551940m_rule @ X3 )
= Y3 ) ) ).
% fstI
thf(fact_127_fstI,axiom,
! [X3: produc6018962875968178549ist_fm,Y3: list_tm,Z: list_fm] :
( ( X3
= ( produc1414352766439514085ist_fm @ Y3 @ Z ) )
=> ( ( produc4874987732838623697ist_fm @ X3 )
= Y3 ) ) ).
% fstI
thf(fact_128_eq__fst__iff,axiom,
! [A2: nat,P2: product_prod_nat_nat] :
( ( A2
= ( product_fst_nat_nat @ P2 ) )
= ( ? [B6: nat] :
( P2
= ( product_Pair_nat_nat @ A2 @ B6 ) ) ) ) ).
% eq_fst_iff
thf(fact_129_eq__fst__iff,axiom,
! [A2: produc6018962875968178549ist_fm,P2: produc340336539035504054m_rule] :
( ( A2
= ( produc993771546023551940m_rule @ P2 ) )
= ( ? [B6: rule] :
( P2
= ( produc1733806532565653680m_rule @ A2 @ B6 ) ) ) ) ).
% eq_fst_iff
thf(fact_130_eq__fst__iff,axiom,
! [A2: list_tm,P2: produc6018962875968178549ist_fm] :
( ( A2
= ( produc4874987732838623697ist_fm @ P2 ) )
= ( ? [B6: list_fm] :
( P2
= ( produc1414352766439514085ist_fm @ A2 @ B6 ) ) ) ) ).
% eq_fst_iff
thf(fact_131_epath_Ocases,axiom,
! [A2: stream2709947120125613254m_rule] :
( ( abstra6789711989322986974ist_fm @ eff @ rules @ A2 )
=> ~ ( ( member_rule @ ( produc6514711208036230658m_rule @ ( shd_Pr4562317740776619530m_rule @ A2 ) ) @ ( sset_rule @ rules ) )
=> ! [Sl: fset_P8989946509869081563ist_fm] :
( ( fmembe3754813877001230652ist_fm @ ( produc993771546023551940m_rule @ ( shd_Pr4562317740776619530m_rule @ ( stl_Pr12655793849929990m_rule @ A2 ) ) ) @ Sl )
=> ( ( eff @ ( produc6514711208036230658m_rule @ ( shd_Pr4562317740776619530m_rule @ A2 ) ) @ ( produc993771546023551940m_rule @ ( shd_Pr4562317740776619530m_rule @ A2 ) ) @ Sl )
=> ~ ( abstra6789711989322986974ist_fm @ eff @ rules @ ( stl_Pr12655793849929990m_rule @ A2 ) ) ) ) ) ) ).
% epath.cases
thf(fact_132_epath_Osimps,axiom,
! [A2: stream2709947120125613254m_rule] :
( ( abstra6789711989322986974ist_fm @ eff @ rules @ A2 )
= ( ? [Steps2: stream2709947120125613254m_rule,Sl2: fset_P8989946509869081563ist_fm] :
( ( A2 = Steps2 )
& ( member_rule @ ( produc6514711208036230658m_rule @ ( shd_Pr4562317740776619530m_rule @ Steps2 ) ) @ ( sset_rule @ rules ) )
& ( fmembe3754813877001230652ist_fm @ ( produc993771546023551940m_rule @ ( shd_Pr4562317740776619530m_rule @ ( stl_Pr12655793849929990m_rule @ Steps2 ) ) ) @ Sl2 )
& ( eff @ ( produc6514711208036230658m_rule @ ( shd_Pr4562317740776619530m_rule @ Steps2 ) ) @ ( produc993771546023551940m_rule @ ( shd_Pr4562317740776619530m_rule @ Steps2 ) ) @ Sl2 )
& ( abstra6789711989322986974ist_fm @ eff @ rules @ ( stl_Pr12655793849929990m_rule @ Steps2 ) ) ) ) ) ).
% epath.simps
thf(fact_133_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_134_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_135_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_136_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_137_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_138_shd__sset,axiom,
! [A2: stream_fm] : ( member_fm @ ( shd_fm @ A2 ) @ ( sset_fm @ A2 ) ) ).
% shd_sset
thf(fact_139_shd__sset,axiom,
! [A2: stream4408948924543953275ist_fm] : ( member4699826688122452638ist_fm @ ( shd_Pr3211216682057661985ist_fm @ A2 ) @ ( sset_P5379989128061332361ist_fm @ A2 ) ) ).
% shd_sset
thf(fact_140_shd__sset,axiom,
! [A2: stream_list_fm] : ( member_list_fm @ ( shd_list_fm @ A2 ) @ ( sset_list_fm @ A2 ) ) ).
% shd_sset
thf(fact_141_shd__sset,axiom,
! [A2: stream_rule] : ( member_rule @ ( shd_rule @ A2 ) @ ( sset_rule @ A2 ) ) ).
% shd_sset
thf(fact_142_shd__sset,axiom,
! [A2: stream2709947120125613254m_rule] : ( member7231649785386036813m_rule @ ( shd_Pr4562317740776619530m_rule @ A2 ) @ ( sset_P4484857331586881186m_rule @ A2 ) ) ).
% shd_sset
thf(fact_143_stl__sset,axiom,
! [X3: fm,A2: stream_fm] :
( ( member_fm @ X3 @ ( sset_fm @ ( stl_fm @ A2 ) ) )
=> ( member_fm @ X3 @ ( sset_fm @ A2 ) ) ) ).
% stl_sset
thf(fact_144_stl__sset,axiom,
! [X3: produc6018962875968178549ist_fm,A2: stream4408948924543953275ist_fm] :
( ( member4699826688122452638ist_fm @ X3 @ ( sset_P5379989128061332361ist_fm @ ( stl_Pr1506262294867171877ist_fm @ A2 ) ) )
=> ( member4699826688122452638ist_fm @ X3 @ ( sset_P5379989128061332361ist_fm @ A2 ) ) ) ).
% stl_sset
thf(fact_145_stl__sset,axiom,
! [X3: list_fm,A2: stream_list_fm] :
( ( member_list_fm @ X3 @ ( sset_list_fm @ ( stl_list_fm @ A2 ) ) )
=> ( member_list_fm @ X3 @ ( sset_list_fm @ A2 ) ) ) ).
% stl_sset
thf(fact_146_stl__sset,axiom,
! [X3: rule,A2: stream_rule] :
( ( member_rule @ X3 @ ( sset_rule @ ( stl_rule @ A2 ) ) )
=> ( member_rule @ X3 @ ( sset_rule @ A2 ) ) ) ).
% stl_sset
thf(fact_147_stl__sset,axiom,
! [X3: produc340336539035504054m_rule,A2: stream2709947120125613254m_rule] :
( ( member7231649785386036813m_rule @ X3 @ ( sset_P4484857331586881186m_rule @ ( stl_Pr12655793849929990m_rule @ A2 ) ) )
=> ( member7231649785386036813m_rule @ X3 @ ( sset_P4484857331586881186m_rule @ A2 ) ) ) ).
% stl_sset
thf(fact_148_sdrop__stl,axiom,
! [N: nat,S: stream2709947120125613254m_rule] :
( ( sdrop_8169176516188972301m_rule @ N @ ( stl_Pr12655793849929990m_rule @ S ) )
= ( stl_Pr12655793849929990m_rule @ ( sdrop_8169176516188972301m_rule @ N @ S ) ) ) ).
% sdrop_stl
thf(fact_149_sdrop__stl,axiom,
! [N: nat,S: stream_rule] :
( ( sdrop_rule @ N @ ( stl_rule @ S ) )
= ( stl_rule @ ( sdrop_rule @ N @ S ) ) ) ).
% sdrop_stl
thf(fact_150_sset__induct,axiom,
! [Y3: fm,S: stream_fm,P: fm > stream_fm > $o] :
( ( member_fm @ Y3 @ ( sset_fm @ S ) )
=> ( ! [S4: stream_fm] : ( P @ ( shd_fm @ S4 ) @ S4 )
=> ( ! [S4: stream_fm,Y: fm] :
( ( member_fm @ Y @ ( sset_fm @ ( stl_fm @ S4 ) ) )
=> ( ( P @ Y @ ( stl_fm @ S4 ) )
=> ( P @ Y @ S4 ) ) )
=> ( P @ Y3 @ S ) ) ) ) ).
% sset_induct
thf(fact_151_sset__induct,axiom,
! [Y3: produc6018962875968178549ist_fm,S: stream4408948924543953275ist_fm,P: produc6018962875968178549ist_fm > stream4408948924543953275ist_fm > $o] :
( ( member4699826688122452638ist_fm @ Y3 @ ( sset_P5379989128061332361ist_fm @ S ) )
=> ( ! [S4: stream4408948924543953275ist_fm] : ( P @ ( shd_Pr3211216682057661985ist_fm @ S4 ) @ S4 )
=> ( ! [S4: stream4408948924543953275ist_fm,Y: produc6018962875968178549ist_fm] :
( ( member4699826688122452638ist_fm @ Y @ ( sset_P5379989128061332361ist_fm @ ( stl_Pr1506262294867171877ist_fm @ S4 ) ) )
=> ( ( P @ Y @ ( stl_Pr1506262294867171877ist_fm @ S4 ) )
=> ( P @ Y @ S4 ) ) )
=> ( P @ Y3 @ S ) ) ) ) ).
% sset_induct
thf(fact_152_sset__induct,axiom,
! [Y3: list_fm,S: stream_list_fm,P: list_fm > stream_list_fm > $o] :
( ( member_list_fm @ Y3 @ ( sset_list_fm @ S ) )
=> ( ! [S4: stream_list_fm] : ( P @ ( shd_list_fm @ S4 ) @ S4 )
=> ( ! [S4: stream_list_fm,Y: list_fm] :
( ( member_list_fm @ Y @ ( sset_list_fm @ ( stl_list_fm @ S4 ) ) )
=> ( ( P @ Y @ ( stl_list_fm @ S4 ) )
=> ( P @ Y @ S4 ) ) )
=> ( P @ Y3 @ S ) ) ) ) ).
% sset_induct
thf(fact_153_sset__induct,axiom,
! [Y3: rule,S: stream_rule,P: rule > stream_rule > $o] :
( ( member_rule @ Y3 @ ( sset_rule @ S ) )
=> ( ! [S4: stream_rule] : ( P @ ( shd_rule @ S4 ) @ S4 )
=> ( ! [S4: stream_rule,Y: rule] :
( ( member_rule @ Y @ ( sset_rule @ ( stl_rule @ S4 ) ) )
=> ( ( P @ Y @ ( stl_rule @ S4 ) )
=> ( P @ Y @ S4 ) ) )
=> ( P @ Y3 @ S ) ) ) ) ).
% sset_induct
thf(fact_154_sset__induct,axiom,
! [Y3: produc340336539035504054m_rule,S: stream2709947120125613254m_rule,P: produc340336539035504054m_rule > stream2709947120125613254m_rule > $o] :
( ( member7231649785386036813m_rule @ Y3 @ ( sset_P4484857331586881186m_rule @ S ) )
=> ( ! [S4: stream2709947120125613254m_rule] : ( P @ ( shd_Pr4562317740776619530m_rule @ S4 ) @ S4 )
=> ( ! [S4: stream2709947120125613254m_rule,Y: produc340336539035504054m_rule] :
( ( member7231649785386036813m_rule @ Y @ ( sset_P4484857331586881186m_rule @ ( stl_Pr12655793849929990m_rule @ S4 ) ) )
=> ( ( P @ Y @ ( stl_Pr12655793849929990m_rule @ S4 ) )
=> ( P @ Y @ S4 ) ) )
=> ( P @ Y3 @ S ) ) ) ) ).
% sset_induct
thf(fact_155_ptms__def,axiom,
( ptms
= ( ^ [Z5: produc340336539035504054m_rule] : ( produc4874987732838623697ist_fm @ ( produc993771546023551940m_rule @ Z5 ) ) ) ) ).
% ptms_def
thf(fact_156_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_157_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_158_epath_Ocoinduct,axiom,
! [X5: stream2709947120125613254m_rule > $o,X3: stream2709947120125613254m_rule] :
( ( X5 @ X3 )
=> ( ! [X: stream2709947120125613254m_rule] :
( ( X5 @ X )
=> ? [Steps3: stream2709947120125613254m_rule,Sl3: fset_P8989946509869081563ist_fm] :
( ( X = Steps3 )
& ( member_rule @ ( produc6514711208036230658m_rule @ ( shd_Pr4562317740776619530m_rule @ Steps3 ) ) @ ( sset_rule @ rules ) )
& ( fmembe3754813877001230652ist_fm @ ( produc993771546023551940m_rule @ ( shd_Pr4562317740776619530m_rule @ ( stl_Pr12655793849929990m_rule @ Steps3 ) ) ) @ Sl3 )
& ( eff @ ( produc6514711208036230658m_rule @ ( shd_Pr4562317740776619530m_rule @ Steps3 ) ) @ ( produc993771546023551940m_rule @ ( shd_Pr4562317740776619530m_rule @ Steps3 ) ) @ Sl3 )
& ( ( X5 @ ( stl_Pr12655793849929990m_rule @ Steps3 ) )
| ( abstra6789711989322986974ist_fm @ eff @ rules @ ( stl_Pr12655793849929990m_rule @ Steps3 ) ) ) ) )
=> ( abstra6789711989322986974ist_fm @ eff @ rules @ X3 ) ) ) ).
% epath.coinduct
thf(fact_159_epath_Ointros,axiom,
! [Steps: stream2709947120125613254m_rule,Sl4: 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 ) ) ) @ Sl4 )
=> ( ( eff @ ( produc6514711208036230658m_rule @ ( shd_Pr4562317740776619530m_rule @ Steps ) ) @ ( produc993771546023551940m_rule @ ( shd_Pr4562317740776619530m_rule @ Steps ) ) @ Sl4 )
=> ( ( abstra6789711989322986974ist_fm @ eff @ rules @ ( stl_Pr12655793849929990m_rule @ Steps ) )
=> ( abstra6789711989322986974ist_fm @ eff @ rules @ Steps ) ) ) ) ) ).
% epath.intros
thf(fact_160_RuleSystem__Defs_Oepath_Ocoinduct,axiom,
! [X5: stream4408948924543953275ist_fm > $o,X3: stream4408948924543953275ist_fm,Rules: stream_list_fm,Eff: list_fm > list_tm > fset_list_tm > $o] :
( ( X5 @ X3 )
=> ( ! [X: stream4408948924543953275ist_fm] :
( ( X5 @ X )
=> ? [Steps3: stream4408948924543953275ist_fm,Sl3: fset_list_tm] :
( ( X = Steps3 )
& ( member_list_fm @ ( produc3242453064496107283ist_fm @ ( shd_Pr3211216682057661985ist_fm @ Steps3 ) ) @ ( sset_list_fm @ Rules ) )
& ( fmember_list_tm @ ( produc4874987732838623697ist_fm @ ( shd_Pr3211216682057661985ist_fm @ ( stl_Pr1506262294867171877ist_fm @ Steps3 ) ) ) @ Sl3 )
& ( Eff @ ( produc3242453064496107283ist_fm @ ( shd_Pr3211216682057661985ist_fm @ Steps3 ) ) @ ( produc4874987732838623697ist_fm @ ( shd_Pr3211216682057661985ist_fm @ Steps3 ) ) @ Sl3 )
& ( ( X5 @ ( stl_Pr1506262294867171877ist_fm @ Steps3 ) )
| ( abstra302294732650053493ist_tm @ Eff @ Rules @ ( stl_Pr1506262294867171877ist_fm @ Steps3 ) ) ) ) )
=> ( abstra302294732650053493ist_tm @ Eff @ Rules @ X3 ) ) ) ).
% RuleSystem_Defs.epath.coinduct
thf(fact_161_RuleSystem__Defs_Oepath_Ocoinduct,axiom,
! [X5: stream7995241112960703083_fm_fm > $o,X3: stream7995241112960703083_fm_fm,Rules: stream_fm,Eff: fm > produc6018962875968178549ist_fm > fset_P8989946509869081563ist_fm > $o] :
( ( X5 @ X3 )
=> ( ! [X: stream7995241112960703083_fm_fm] :
( ( X5 @ X )
=> ? [Steps3: stream7995241112960703083_fm_fm,Sl3: fset_P8989946509869081563ist_fm] :
( ( X = Steps3 )
& ( member_fm @ ( produc5710729445452762371_fm_fm @ ( shd_Pr3833923502784695313_fm_fm @ Steps3 ) ) @ ( sset_fm @ Rules ) )
& ( fmembe3754813877001230652ist_fm @ ( produc8064570269430693313_fm_fm @ ( shd_Pr3833923502784695313_fm_fm @ ( stl_Pr2438619550280678421_fm_fm @ Steps3 ) ) ) @ Sl3 )
& ( Eff @ ( produc5710729445452762371_fm_fm @ ( shd_Pr3833923502784695313_fm_fm @ Steps3 ) ) @ ( produc8064570269430693313_fm_fm @ ( shd_Pr3833923502784695313_fm_fm @ Steps3 ) ) @ Sl3 )
& ( ( X5 @ ( stl_Pr2438619550280678421_fm_fm @ Steps3 ) )
| ( abstra3580388122067503493ist_fm @ Eff @ Rules @ ( stl_Pr2438619550280678421_fm_fm @ Steps3 ) ) ) ) )
=> ( abstra3580388122067503493ist_fm @ Eff @ Rules @ X3 ) ) ) ).
% RuleSystem_Defs.epath.coinduct
thf(fact_162_RuleSystem__Defs_Oepath_Ocoinduct,axiom,
! [X5: stream2709947120125613254m_rule > $o,X3: stream2709947120125613254m_rule,Rules: stream_rule,Eff: rule > produc6018962875968178549ist_fm > fset_P8989946509869081563ist_fm > $o] :
( ( X5 @ X3 )
=> ( ! [X: stream2709947120125613254m_rule] :
( ( X5 @ X )
=> ? [Steps3: stream2709947120125613254m_rule,Sl3: fset_P8989946509869081563ist_fm] :
( ( X = Steps3 )
& ( member_rule @ ( produc6514711208036230658m_rule @ ( shd_Pr4562317740776619530m_rule @ Steps3 ) ) @ ( sset_rule @ Rules ) )
& ( fmembe3754813877001230652ist_fm @ ( produc993771546023551940m_rule @ ( shd_Pr4562317740776619530m_rule @ ( stl_Pr12655793849929990m_rule @ Steps3 ) ) ) @ Sl3 )
& ( Eff @ ( produc6514711208036230658m_rule @ ( shd_Pr4562317740776619530m_rule @ Steps3 ) ) @ ( produc993771546023551940m_rule @ ( shd_Pr4562317740776619530m_rule @ Steps3 ) ) @ Sl3 )
& ( ( X5 @ ( stl_Pr12655793849929990m_rule @ Steps3 ) )
| ( abstra6789711989322986974ist_fm @ Eff @ Rules @ ( stl_Pr12655793849929990m_rule @ Steps3 ) ) ) ) )
=> ( abstra6789711989322986974ist_fm @ Eff @ Rules @ X3 ) ) ) ).
% RuleSystem_Defs.epath.coinduct
thf(fact_163_RuleSystem__Defs_Oepath_Ocoinduct,axiom,
! [X5: stream4656891395932198769ist_fm > $o,X3: stream4656891395932198769ist_fm,Rules: stream_list_fm,Eff: list_fm > produc6018962875968178549ist_fm > fset_P8989946509869081563ist_fm > $o] :
( ( X5 @ X3 )
=> ( ! [X: stream4656891395932198769ist_fm] :
( ( X5 @ X )
=> ? [Steps3: stream4656891395932198769ist_fm,Sl3: fset_P8989946509869081563ist_fm] :
( ( X = Steps3 )
& ( member_list_fm @ ( produc2977072740996481289ist_fm @ ( shd_Pr311069668031106967ist_fm @ Steps3 ) ) @ ( sset_list_fm @ Rules ) )
& ( fmembe3754813877001230652ist_fm @ ( produc5700618299769748167ist_fm @ ( shd_Pr311069668031106967ist_fm @ ( stl_Pr7978862689335602587ist_fm @ Steps3 ) ) ) @ Sl3 )
& ( Eff @ ( produc2977072740996481289ist_fm @ ( shd_Pr311069668031106967ist_fm @ Steps3 ) ) @ ( produc5700618299769748167ist_fm @ ( shd_Pr311069668031106967ist_fm @ Steps3 ) ) @ Sl3 )
& ( ( X5 @ ( stl_Pr7978862689335602587ist_fm @ Steps3 ) )
| ( abstra5075300408540101887ist_fm @ Eff @ Rules @ ( stl_Pr7978862689335602587ist_fm @ Steps3 ) ) ) ) )
=> ( abstra5075300408540101887ist_fm @ Eff @ Rules @ X3 ) ) ) ).
% RuleSystem_Defs.epath.coinduct
thf(fact_164_RuleSystem__Defs_Oepath_Ocoinduct,axiom,
! [X5: stream2858270464523782ule_fm > $o,X3: stream2858270464523782ule_fm,Rules: stream_fm,Eff: fm > abstra8520858721168826610m_rule > fset_A8849196716589201442m_rule > $o] :
( ( X5 @ X3 )
=> ( ! [X: stream2858270464523782ule_fm] :
( ( X5 @ X )
=> ? [Steps3: stream2858270464523782ule_fm,Sl3: fset_A8849196716589201442m_rule] :
( ( X = Steps3 )
& ( member_fm @ ( produc5983953905017532922ule_fm @ ( shd_Pr142054665586430922ule_fm @ Steps3 ) ) @ ( sset_fm @ Rules ) )
& ( fmembe1099044791951439339m_rule @ ( produc7765541624337483452ule_fm @ ( shd_Pr142054665586430922ule_fm @ ( stl_Pr7787596663505189062ule_fm @ Steps3 ) ) ) @ Sl3 )
& ( Eff @ ( produc5983953905017532922ule_fm @ ( shd_Pr142054665586430922ule_fm @ Steps3 ) ) @ ( produc7765541624337483452ule_fm @ ( shd_Pr142054665586430922ule_fm @ Steps3 ) ) @ Sl3 )
& ( ( X5 @ ( stl_Pr7787596663505189062ule_fm @ Steps3 ) )
| ( abstra6170950473048798434m_rule @ Eff @ Rules @ ( stl_Pr7787596663505189062ule_fm @ Steps3 ) ) ) ) )
=> ( abstra6170950473048798434m_rule @ Eff @ Rules @ X3 ) ) ) ).
% RuleSystem_Defs.epath.coinduct
thf(fact_165_RuleSystem__Defs_Oepath_Ocoinduct,axiom,
! [X5: stream3393403210778678635e_rule > $o,X3: stream3393403210778678635e_rule,Rules: stream_rule,Eff: rule > abstra8520858721168826610m_rule > fset_A8849196716589201442m_rule > $o] :
( ( X5 @ X3 )
=> ( ! [X: stream3393403210778678635e_rule] :
( ( X5 @ X )
=> ? [Steps3: stream3393403210778678635e_rule,Sl3: fset_A8849196716589201442m_rule] :
( ( X = Steps3 )
& ( member_rule @ ( produc9166542125499771467e_rule @ ( shd_Pr8663275782010802449e_rule @ Steps3 ) ) @ ( sset_rule @ Rules ) )
& ( fmembe1099044791951439339m_rule @ ( produc41289011304007433e_rule @ ( shd_Pr8663275782010802449e_rule @ ( stl_Pr654857084227610901e_rule @ Steps3 ) ) ) @ Sl3 )
& ( Eff @ ( produc9166542125499771467e_rule @ ( shd_Pr8663275782010802449e_rule @ Steps3 ) ) @ ( produc41289011304007433e_rule @ ( shd_Pr8663275782010802449e_rule @ Steps3 ) ) @ Sl3 )
& ( ( X5 @ ( stl_Pr654857084227610901e_rule @ Steps3 ) )
| ( abstra7067474524808760265m_rule @ Eff @ Rules @ ( stl_Pr654857084227610901e_rule @ Steps3 ) ) ) ) )
=> ( abstra7067474524808760265m_rule @ Eff @ Rules @ X3 ) ) ) ).
% RuleSystem_Defs.epath.coinduct
thf(fact_166_RuleSystem__Defs_Oepath_Ocoinduct,axiom,
! [X5: stream1747862497754140525ist_fm > $o,X3: stream1747862497754140525ist_fm,Rules: stream4408948924543953275ist_fm,Eff: produc6018962875968178549ist_fm > produc6018962875968178549ist_fm > fset_P8989946509869081563ist_fm > $o] :
( ( X5 @ X3 )
=> ( ! [X: stream1747862497754140525ist_fm] :
( ( X5 @ X )
=> ? [Steps3: stream1747862497754140525ist_fm,Sl3: fset_P8989946509869081563ist_fm] :
( ( X = Steps3 )
& ( member4699826688122452638ist_fm @ ( produc6364584182369725445ist_fm @ ( shd_Pr4098038590278241811ist_fm @ Steps3 ) ) @ ( sset_P5379989128061332361ist_fm @ Rules ) )
& ( fmembe3754813877001230652ist_fm @ ( produc8141473855448433859ist_fm @ ( shd_Pr4098038590278241811ist_fm @ ( stl_Pr1377318027900677655ist_fm @ Steps3 ) ) ) @ Sl3 )
& ( Eff @ ( produc6364584182369725445ist_fm @ ( shd_Pr4098038590278241811ist_fm @ Steps3 ) ) @ ( produc8141473855448433859ist_fm @ ( shd_Pr4098038590278241811ist_fm @ Steps3 ) ) @ Sl3 )
& ( ( X5 @ ( stl_Pr1377318027900677655ist_fm @ Steps3 ) )
| ( abstra6295994116336782083ist_fm @ Eff @ Rules @ ( stl_Pr1377318027900677655ist_fm @ Steps3 ) ) ) ) )
=> ( abstra6295994116336782083ist_fm @ Eff @ Rules @ X3 ) ) ) ).
% RuleSystem_Defs.epath.coinduct
thf(fact_167_RuleSystem__Defs_Oepath_Ocoinduct,axiom,
! [X5: stream5329691398384333196ist_fm > $o,X3: stream5329691398384333196ist_fm,Rules: stream_list_fm,Eff: list_fm > abstra8520858721168826610m_rule > fset_A8849196716589201442m_rule > $o] :
( ( X5 @ X3 )
=> ( ! [X: stream5329691398384333196ist_fm] :
( ( X5 @ X )
=> ? [Steps3: stream5329691398384333196ist_fm,Sl3: fset_A8849196716589201442m_rule] :
( ( X = Steps3 )
& ( member_list_fm @ ( produc6440468272734011008ist_fm @ ( shd_Pr2916036590756128464ist_fm @ Steps3 ) ) @ ( sset_list_fm @ Rules ) )
& ( fmembe1099044791951439339m_rule @ ( produc162143410373610050ist_fm @ ( shd_Pr2916036590756128464ist_fm @ ( stl_Pr213190007691795916ist_fm @ Steps3 ) ) ) @ Sl3 )
& ( Eff @ ( produc6440468272734011008ist_fm @ ( shd_Pr2916036590756128464ist_fm @ Steps3 ) ) @ ( produc162143410373610050ist_fm @ ( shd_Pr2916036590756128464ist_fm @ Steps3 ) ) @ Sl3 )
& ( ( X5 @ ( stl_Pr213190007691795916ist_fm @ Steps3 ) )
| ( abstra4310285689605054952m_rule @ Eff @ Rules @ ( stl_Pr213190007691795916ist_fm @ Steps3 ) ) ) ) )
=> ( abstra4310285689605054952m_rule @ Eff @ Rules @ X3 ) ) ) ).
% RuleSystem_Defs.epath.coinduct
thf(fact_168_RuleSystem__Defs_Oepath_Ocoinduct,axiom,
! [X5: stream6742152135616053226m_rule > $o,X3: stream6742152135616053226m_rule,Rules: stream2709947120125613254m_rule,Eff: produc340336539035504054m_rule > produc6018962875968178549ist_fm > fset_P8989946509869081563ist_fm > $o] :
( ( X5 @ X3 )
=> ( ! [X: stream6742152135616053226m_rule] :
( ( X5 @ X )
=> ? [Steps3: stream6742152135616053226m_rule,Sl3: fset_P8989946509869081563ist_fm] :
( ( X = Steps3 )
& ( member7231649785386036813m_rule @ ( produc1227651196481849126m_rule @ ( shd_Pr6124039728035424814m_rule @ Steps3 ) ) @ ( sset_P4484857331586881186m_rule @ Rules ) )
& ( fmembe3754813877001230652ist_fm @ ( produc6387030584228546280m_rule @ ( shd_Pr6124039728035424814m_rule @ ( stl_Pr7706092684847479082m_rule @ Steps3 ) ) ) @ Sl3 )
& ( Eff @ ( produc1227651196481849126m_rule @ ( shd_Pr6124039728035424814m_rule @ Steps3 ) ) @ ( produc6387030584228546280m_rule @ ( shd_Pr6124039728035424814m_rule @ Steps3 ) ) @ Sl3 )
& ( ( X5 @ ( stl_Pr7706092684847479082m_rule @ Steps3 ) )
| ( abstra5381393010826267066ist_fm @ Eff @ Rules @ ( stl_Pr7706092684847479082m_rule @ Steps3 ) ) ) ) )
=> ( abstra5381393010826267066ist_fm @ Eff @ Rules @ X3 ) ) ) ).
% RuleSystem_Defs.epath.coinduct
thf(fact_169_RuleSystem__Defs_Oepath_Ocoinduct,axiom,
! [X5: stream2118054573033641736ist_fm > $o,X3: stream2118054573033641736ist_fm,Rules: stream4408948924543953275ist_fm,Eff: produc6018962875968178549ist_fm > abstra8520858721168826610m_rule > fset_A8849196716589201442m_rule > $o] :
( ( X5 @ X3 )
=> ( ! [X: stream2118054573033641736ist_fm] :
( ( X5 @ X )
=> ? [Steps3: stream2118054573033641736ist_fm,Sl3: fset_A8849196716589201442m_rule] :
( ( X = Steps3 )
& ( member4699826688122452638ist_fm @ ( produc1196760618833626364ist_fm @ ( shd_Pr1348735749115486668ist_fm @ Steps3 ) ) @ ( sset_P5379989128061332361ist_fm @ Rules ) )
& ( fmembe1099044791951439339m_rule @ ( produc7096817939255439806ist_fm @ ( shd_Pr1348735749115486668ist_fm @ ( stl_Pr1735168262451467976ist_fm @ Steps3 ) ) ) @ Sl3 )
& ( Eff @ ( produc1196760618833626364ist_fm @ ( shd_Pr1348735749115486668ist_fm @ Steps3 ) ) @ ( produc7096817939255439806ist_fm @ ( shd_Pr1348735749115486668ist_fm @ Steps3 ) ) @ Sl3 )
& ( ( X5 @ ( stl_Pr1735168262451467976ist_fm @ Steps3 ) )
| ( abstra8533012240466880740m_rule @ Eff @ Rules @ ( stl_Pr1735168262451467976ist_fm @ Steps3 ) ) ) ) )
=> ( abstra8533012240466880740m_rule @ Eff @ Rules @ X3 ) ) ) ).
% RuleSystem_Defs.epath.coinduct
thf(fact_170_RuleSystem__Defs_Oepath_Ointros,axiom,
! [Steps: stream4408948924543953275ist_fm,Rules: stream_list_fm,Sl4: fset_list_tm,Eff: list_fm > list_tm > fset_list_tm > $o] :
( ( member_list_fm @ ( produc3242453064496107283ist_fm @ ( shd_Pr3211216682057661985ist_fm @ Steps ) ) @ ( sset_list_fm @ Rules ) )
=> ( ( fmember_list_tm @ ( produc4874987732838623697ist_fm @ ( shd_Pr3211216682057661985ist_fm @ ( stl_Pr1506262294867171877ist_fm @ Steps ) ) ) @ Sl4 )
=> ( ( Eff @ ( produc3242453064496107283ist_fm @ ( shd_Pr3211216682057661985ist_fm @ Steps ) ) @ ( produc4874987732838623697ist_fm @ ( shd_Pr3211216682057661985ist_fm @ Steps ) ) @ Sl4 )
=> ( ( abstra302294732650053493ist_tm @ Eff @ Rules @ ( stl_Pr1506262294867171877ist_fm @ Steps ) )
=> ( abstra302294732650053493ist_tm @ Eff @ Rules @ Steps ) ) ) ) ) ).
% RuleSystem_Defs.epath.intros
thf(fact_171_RuleSystem__Defs_Oepath_Ointros,axiom,
! [Steps: stream7995241112960703083_fm_fm,Rules: stream_fm,Sl4: fset_P8989946509869081563ist_fm,Eff: fm > produc6018962875968178549ist_fm > fset_P8989946509869081563ist_fm > $o] :
( ( member_fm @ ( produc5710729445452762371_fm_fm @ ( shd_Pr3833923502784695313_fm_fm @ Steps ) ) @ ( sset_fm @ Rules ) )
=> ( ( fmembe3754813877001230652ist_fm @ ( produc8064570269430693313_fm_fm @ ( shd_Pr3833923502784695313_fm_fm @ ( stl_Pr2438619550280678421_fm_fm @ Steps ) ) ) @ Sl4 )
=> ( ( Eff @ ( produc5710729445452762371_fm_fm @ ( shd_Pr3833923502784695313_fm_fm @ Steps ) ) @ ( produc8064570269430693313_fm_fm @ ( shd_Pr3833923502784695313_fm_fm @ Steps ) ) @ Sl4 )
=> ( ( abstra3580388122067503493ist_fm @ Eff @ Rules @ ( stl_Pr2438619550280678421_fm_fm @ Steps ) )
=> ( abstra3580388122067503493ist_fm @ Eff @ Rules @ Steps ) ) ) ) ) ).
% RuleSystem_Defs.epath.intros
thf(fact_172_RuleSystem__Defs_Oepath_Ointros,axiom,
! [Steps: stream2709947120125613254m_rule,Rules: stream_rule,Sl4: fset_P8989946509869081563ist_fm,Eff: rule > produc6018962875968178549ist_fm > fset_P8989946509869081563ist_fm > $o] :
( ( member_rule @ ( produc6514711208036230658m_rule @ ( shd_Pr4562317740776619530m_rule @ Steps ) ) @ ( sset_rule @ Rules ) )
=> ( ( fmembe3754813877001230652ist_fm @ ( produc993771546023551940m_rule @ ( shd_Pr4562317740776619530m_rule @ ( stl_Pr12655793849929990m_rule @ Steps ) ) ) @ Sl4 )
=> ( ( Eff @ ( produc6514711208036230658m_rule @ ( shd_Pr4562317740776619530m_rule @ Steps ) ) @ ( produc993771546023551940m_rule @ ( shd_Pr4562317740776619530m_rule @ Steps ) ) @ Sl4 )
=> ( ( abstra6789711989322986974ist_fm @ Eff @ Rules @ ( stl_Pr12655793849929990m_rule @ Steps ) )
=> ( abstra6789711989322986974ist_fm @ Eff @ Rules @ Steps ) ) ) ) ) ).
% RuleSystem_Defs.epath.intros
thf(fact_173_RuleSystem__Defs_Oepath_Ointros,axiom,
! [Steps: stream4656891395932198769ist_fm,Rules: stream_list_fm,Sl4: fset_P8989946509869081563ist_fm,Eff: list_fm > produc6018962875968178549ist_fm > fset_P8989946509869081563ist_fm > $o] :
( ( member_list_fm @ ( produc2977072740996481289ist_fm @ ( shd_Pr311069668031106967ist_fm @ Steps ) ) @ ( sset_list_fm @ Rules ) )
=> ( ( fmembe3754813877001230652ist_fm @ ( produc5700618299769748167ist_fm @ ( shd_Pr311069668031106967ist_fm @ ( stl_Pr7978862689335602587ist_fm @ Steps ) ) ) @ Sl4 )
=> ( ( Eff @ ( produc2977072740996481289ist_fm @ ( shd_Pr311069668031106967ist_fm @ Steps ) ) @ ( produc5700618299769748167ist_fm @ ( shd_Pr311069668031106967ist_fm @ Steps ) ) @ Sl4 )
=> ( ( abstra5075300408540101887ist_fm @ Eff @ Rules @ ( stl_Pr7978862689335602587ist_fm @ Steps ) )
=> ( abstra5075300408540101887ist_fm @ Eff @ Rules @ Steps ) ) ) ) ) ).
% RuleSystem_Defs.epath.intros
thf(fact_174_RuleSystem__Defs_Oepath_Ointros,axiom,
! [Steps: stream2858270464523782ule_fm,Rules: stream_fm,Sl4: fset_A8849196716589201442m_rule,Eff: fm > abstra8520858721168826610m_rule > fset_A8849196716589201442m_rule > $o] :
( ( member_fm @ ( produc5983953905017532922ule_fm @ ( shd_Pr142054665586430922ule_fm @ Steps ) ) @ ( sset_fm @ Rules ) )
=> ( ( fmembe1099044791951439339m_rule @ ( produc7765541624337483452ule_fm @ ( shd_Pr142054665586430922ule_fm @ ( stl_Pr7787596663505189062ule_fm @ Steps ) ) ) @ Sl4 )
=> ( ( Eff @ ( produc5983953905017532922ule_fm @ ( shd_Pr142054665586430922ule_fm @ Steps ) ) @ ( produc7765541624337483452ule_fm @ ( shd_Pr142054665586430922ule_fm @ Steps ) ) @ Sl4 )
=> ( ( abstra6170950473048798434m_rule @ Eff @ Rules @ ( stl_Pr7787596663505189062ule_fm @ Steps ) )
=> ( abstra6170950473048798434m_rule @ Eff @ Rules @ Steps ) ) ) ) ) ).
% RuleSystem_Defs.epath.intros
thf(fact_175_RuleSystem__Defs_Oepath_Ointros,axiom,
! [Steps: stream3393403210778678635e_rule,Rules: stream_rule,Sl4: fset_A8849196716589201442m_rule,Eff: rule > abstra8520858721168826610m_rule > fset_A8849196716589201442m_rule > $o] :
( ( member_rule @ ( produc9166542125499771467e_rule @ ( shd_Pr8663275782010802449e_rule @ Steps ) ) @ ( sset_rule @ Rules ) )
=> ( ( fmembe1099044791951439339m_rule @ ( produc41289011304007433e_rule @ ( shd_Pr8663275782010802449e_rule @ ( stl_Pr654857084227610901e_rule @ Steps ) ) ) @ Sl4 )
=> ( ( Eff @ ( produc9166542125499771467e_rule @ ( shd_Pr8663275782010802449e_rule @ Steps ) ) @ ( produc41289011304007433e_rule @ ( shd_Pr8663275782010802449e_rule @ Steps ) ) @ Sl4 )
=> ( ( abstra7067474524808760265m_rule @ Eff @ Rules @ ( stl_Pr654857084227610901e_rule @ Steps ) )
=> ( abstra7067474524808760265m_rule @ Eff @ Rules @ Steps ) ) ) ) ) ).
% RuleSystem_Defs.epath.intros
thf(fact_176_RuleSystem__Defs_Oepath_Ointros,axiom,
! [Steps: stream1747862497754140525ist_fm,Rules: stream4408948924543953275ist_fm,Sl4: fset_P8989946509869081563ist_fm,Eff: produc6018962875968178549ist_fm > produc6018962875968178549ist_fm > fset_P8989946509869081563ist_fm > $o] :
( ( member4699826688122452638ist_fm @ ( produc6364584182369725445ist_fm @ ( shd_Pr4098038590278241811ist_fm @ Steps ) ) @ ( sset_P5379989128061332361ist_fm @ Rules ) )
=> ( ( fmembe3754813877001230652ist_fm @ ( produc8141473855448433859ist_fm @ ( shd_Pr4098038590278241811ist_fm @ ( stl_Pr1377318027900677655ist_fm @ Steps ) ) ) @ Sl4 )
=> ( ( Eff @ ( produc6364584182369725445ist_fm @ ( shd_Pr4098038590278241811ist_fm @ Steps ) ) @ ( produc8141473855448433859ist_fm @ ( shd_Pr4098038590278241811ist_fm @ Steps ) ) @ Sl4 )
=> ( ( abstra6295994116336782083ist_fm @ Eff @ Rules @ ( stl_Pr1377318027900677655ist_fm @ Steps ) )
=> ( abstra6295994116336782083ist_fm @ Eff @ Rules @ Steps ) ) ) ) ) ).
% RuleSystem_Defs.epath.intros
thf(fact_177_RuleSystem__Defs_Oepath_Ointros,axiom,
! [Steps: stream5329691398384333196ist_fm,Rules: stream_list_fm,Sl4: fset_A8849196716589201442m_rule,Eff: list_fm > abstra8520858721168826610m_rule > fset_A8849196716589201442m_rule > $o] :
( ( member_list_fm @ ( produc6440468272734011008ist_fm @ ( shd_Pr2916036590756128464ist_fm @ Steps ) ) @ ( sset_list_fm @ Rules ) )
=> ( ( fmembe1099044791951439339m_rule @ ( produc162143410373610050ist_fm @ ( shd_Pr2916036590756128464ist_fm @ ( stl_Pr213190007691795916ist_fm @ Steps ) ) ) @ Sl4 )
=> ( ( Eff @ ( produc6440468272734011008ist_fm @ ( shd_Pr2916036590756128464ist_fm @ Steps ) ) @ ( produc162143410373610050ist_fm @ ( shd_Pr2916036590756128464ist_fm @ Steps ) ) @ Sl4 )
=> ( ( abstra4310285689605054952m_rule @ Eff @ Rules @ ( stl_Pr213190007691795916ist_fm @ Steps ) )
=> ( abstra4310285689605054952m_rule @ Eff @ Rules @ Steps ) ) ) ) ) ).
% RuleSystem_Defs.epath.intros
thf(fact_178_RuleSystem__Defs_Oepath_Ointros,axiom,
! [Steps: stream6742152135616053226m_rule,Rules: stream2709947120125613254m_rule,Sl4: fset_P8989946509869081563ist_fm,Eff: produc340336539035504054m_rule > produc6018962875968178549ist_fm > fset_P8989946509869081563ist_fm > $o] :
( ( member7231649785386036813m_rule @ ( produc1227651196481849126m_rule @ ( shd_Pr6124039728035424814m_rule @ Steps ) ) @ ( sset_P4484857331586881186m_rule @ Rules ) )
=> ( ( fmembe3754813877001230652ist_fm @ ( produc6387030584228546280m_rule @ ( shd_Pr6124039728035424814m_rule @ ( stl_Pr7706092684847479082m_rule @ Steps ) ) ) @ Sl4 )
=> ( ( Eff @ ( produc1227651196481849126m_rule @ ( shd_Pr6124039728035424814m_rule @ Steps ) ) @ ( produc6387030584228546280m_rule @ ( shd_Pr6124039728035424814m_rule @ Steps ) ) @ Sl4 )
=> ( ( abstra5381393010826267066ist_fm @ Eff @ Rules @ ( stl_Pr7706092684847479082m_rule @ Steps ) )
=> ( abstra5381393010826267066ist_fm @ Eff @ Rules @ Steps ) ) ) ) ) ).
% RuleSystem_Defs.epath.intros
thf(fact_179_RuleSystem__Defs_Oepath_Ointros,axiom,
! [Steps: stream2118054573033641736ist_fm,Rules: stream4408948924543953275ist_fm,Sl4: fset_A8849196716589201442m_rule,Eff: produc6018962875968178549ist_fm > abstra8520858721168826610m_rule > fset_A8849196716589201442m_rule > $o] :
( ( member4699826688122452638ist_fm @ ( produc1196760618833626364ist_fm @ ( shd_Pr1348735749115486668ist_fm @ Steps ) ) @ ( sset_P5379989128061332361ist_fm @ Rules ) )
=> ( ( fmembe1099044791951439339m_rule @ ( produc7096817939255439806ist_fm @ ( shd_Pr1348735749115486668ist_fm @ ( stl_Pr1735168262451467976ist_fm @ Steps ) ) ) @ Sl4 )
=> ( ( Eff @ ( produc1196760618833626364ist_fm @ ( shd_Pr1348735749115486668ist_fm @ Steps ) ) @ ( produc7096817939255439806ist_fm @ ( shd_Pr1348735749115486668ist_fm @ Steps ) ) @ Sl4 )
=> ( ( abstra8533012240466880740m_rule @ Eff @ Rules @ ( stl_Pr1735168262451467976ist_fm @ Steps ) )
=> ( abstra8533012240466880740m_rule @ Eff @ Rules @ Steps ) ) ) ) ) ).
% RuleSystem_Defs.epath.intros
thf(fact_180_RuleSystem__Defs_Oepath_Osimps,axiom,
( abstra302294732650053493ist_tm
= ( ^ [Eff2: list_fm > list_tm > fset_list_tm > $o,Rules2: stream_list_fm,A5: stream4408948924543953275ist_fm] :
? [Steps2: stream4408948924543953275ist_fm,Sl2: fset_list_tm] :
( ( A5 = 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 ) ) ) @ Sl2 )
& ( Eff2 @ ( produc3242453064496107283ist_fm @ ( shd_Pr3211216682057661985ist_fm @ Steps2 ) ) @ ( produc4874987732838623697ist_fm @ ( shd_Pr3211216682057661985ist_fm @ Steps2 ) ) @ Sl2 )
& ( abstra302294732650053493ist_tm @ Eff2 @ Rules2 @ ( stl_Pr1506262294867171877ist_fm @ Steps2 ) ) ) ) ) ).
% RuleSystem_Defs.epath.simps
thf(fact_181_RuleSystem__Defs_Oepath_Osimps,axiom,
( abstra3580388122067503493ist_fm
= ( ^ [Eff2: fm > produc6018962875968178549ist_fm > fset_P8989946509869081563ist_fm > $o,Rules2: stream_fm,A5: stream7995241112960703083_fm_fm] :
? [Steps2: stream7995241112960703083_fm_fm,Sl2: fset_P8989946509869081563ist_fm] :
( ( A5 = Steps2 )
& ( member_fm @ ( produc5710729445452762371_fm_fm @ ( shd_Pr3833923502784695313_fm_fm @ Steps2 ) ) @ ( sset_fm @ Rules2 ) )
& ( fmembe3754813877001230652ist_fm @ ( produc8064570269430693313_fm_fm @ ( shd_Pr3833923502784695313_fm_fm @ ( stl_Pr2438619550280678421_fm_fm @ Steps2 ) ) ) @ Sl2 )
& ( Eff2 @ ( produc5710729445452762371_fm_fm @ ( shd_Pr3833923502784695313_fm_fm @ Steps2 ) ) @ ( produc8064570269430693313_fm_fm @ ( shd_Pr3833923502784695313_fm_fm @ Steps2 ) ) @ Sl2 )
& ( abstra3580388122067503493ist_fm @ Eff2 @ Rules2 @ ( stl_Pr2438619550280678421_fm_fm @ Steps2 ) ) ) ) ) ).
% RuleSystem_Defs.epath.simps
thf(fact_182_RuleSystem__Defs_Oepath_Osimps,axiom,
( abstra6789711989322986974ist_fm
= ( ^ [Eff2: rule > produc6018962875968178549ist_fm > fset_P8989946509869081563ist_fm > $o,Rules2: stream_rule,A5: stream2709947120125613254m_rule] :
? [Steps2: stream2709947120125613254m_rule,Sl2: fset_P8989946509869081563ist_fm] :
( ( A5 = Steps2 )
& ( member_rule @ ( produc6514711208036230658m_rule @ ( shd_Pr4562317740776619530m_rule @ Steps2 ) ) @ ( sset_rule @ Rules2 ) )
& ( fmembe3754813877001230652ist_fm @ ( produc993771546023551940m_rule @ ( shd_Pr4562317740776619530m_rule @ ( stl_Pr12655793849929990m_rule @ Steps2 ) ) ) @ Sl2 )
& ( Eff2 @ ( produc6514711208036230658m_rule @ ( shd_Pr4562317740776619530m_rule @ Steps2 ) ) @ ( produc993771546023551940m_rule @ ( shd_Pr4562317740776619530m_rule @ Steps2 ) ) @ Sl2 )
& ( abstra6789711989322986974ist_fm @ Eff2 @ Rules2 @ ( stl_Pr12655793849929990m_rule @ Steps2 ) ) ) ) ) ).
% RuleSystem_Defs.epath.simps
thf(fact_183_RuleSystem__Defs_Oepath_Osimps,axiom,
( abstra5075300408540101887ist_fm
= ( ^ [Eff2: list_fm > produc6018962875968178549ist_fm > fset_P8989946509869081563ist_fm > $o,Rules2: stream_list_fm,A5: stream4656891395932198769ist_fm] :
? [Steps2: stream4656891395932198769ist_fm,Sl2: fset_P8989946509869081563ist_fm] :
( ( A5 = Steps2 )
& ( member_list_fm @ ( produc2977072740996481289ist_fm @ ( shd_Pr311069668031106967ist_fm @ Steps2 ) ) @ ( sset_list_fm @ Rules2 ) )
& ( fmembe3754813877001230652ist_fm @ ( produc5700618299769748167ist_fm @ ( shd_Pr311069668031106967ist_fm @ ( stl_Pr7978862689335602587ist_fm @ Steps2 ) ) ) @ Sl2 )
& ( Eff2 @ ( produc2977072740996481289ist_fm @ ( shd_Pr311069668031106967ist_fm @ Steps2 ) ) @ ( produc5700618299769748167ist_fm @ ( shd_Pr311069668031106967ist_fm @ Steps2 ) ) @ Sl2 )
& ( abstra5075300408540101887ist_fm @ Eff2 @ Rules2 @ ( stl_Pr7978862689335602587ist_fm @ Steps2 ) ) ) ) ) ).
% RuleSystem_Defs.epath.simps
thf(fact_184_RuleSystem__Defs_Oepath_Osimps,axiom,
( abstra6170950473048798434m_rule
= ( ^ [Eff2: fm > abstra8520858721168826610m_rule > fset_A8849196716589201442m_rule > $o,Rules2: stream_fm,A5: stream2858270464523782ule_fm] :
? [Steps2: stream2858270464523782ule_fm,Sl2: fset_A8849196716589201442m_rule] :
( ( A5 = Steps2 )
& ( member_fm @ ( produc5983953905017532922ule_fm @ ( shd_Pr142054665586430922ule_fm @ Steps2 ) ) @ ( sset_fm @ Rules2 ) )
& ( fmembe1099044791951439339m_rule @ ( produc7765541624337483452ule_fm @ ( shd_Pr142054665586430922ule_fm @ ( stl_Pr7787596663505189062ule_fm @ Steps2 ) ) ) @ Sl2 )
& ( Eff2 @ ( produc5983953905017532922ule_fm @ ( shd_Pr142054665586430922ule_fm @ Steps2 ) ) @ ( produc7765541624337483452ule_fm @ ( shd_Pr142054665586430922ule_fm @ Steps2 ) ) @ Sl2 )
& ( abstra6170950473048798434m_rule @ Eff2 @ Rules2 @ ( stl_Pr7787596663505189062ule_fm @ Steps2 ) ) ) ) ) ).
% RuleSystem_Defs.epath.simps
thf(fact_185_RuleSystem__Defs_Oepath_Osimps,axiom,
( abstra7067474524808760265m_rule
= ( ^ [Eff2: rule > abstra8520858721168826610m_rule > fset_A8849196716589201442m_rule > $o,Rules2: stream_rule,A5: stream3393403210778678635e_rule] :
? [Steps2: stream3393403210778678635e_rule,Sl2: fset_A8849196716589201442m_rule] :
( ( A5 = Steps2 )
& ( member_rule @ ( produc9166542125499771467e_rule @ ( shd_Pr8663275782010802449e_rule @ Steps2 ) ) @ ( sset_rule @ Rules2 ) )
& ( fmembe1099044791951439339m_rule @ ( produc41289011304007433e_rule @ ( shd_Pr8663275782010802449e_rule @ ( stl_Pr654857084227610901e_rule @ Steps2 ) ) ) @ Sl2 )
& ( Eff2 @ ( produc9166542125499771467e_rule @ ( shd_Pr8663275782010802449e_rule @ Steps2 ) ) @ ( produc41289011304007433e_rule @ ( shd_Pr8663275782010802449e_rule @ Steps2 ) ) @ Sl2 )
& ( abstra7067474524808760265m_rule @ Eff2 @ Rules2 @ ( stl_Pr654857084227610901e_rule @ Steps2 ) ) ) ) ) ).
% RuleSystem_Defs.epath.simps
thf(fact_186_RuleSystem__Defs_Oepath_Osimps,axiom,
( abstra6295994116336782083ist_fm
= ( ^ [Eff2: produc6018962875968178549ist_fm > produc6018962875968178549ist_fm > fset_P8989946509869081563ist_fm > $o,Rules2: stream4408948924543953275ist_fm,A5: stream1747862497754140525ist_fm] :
? [Steps2: stream1747862497754140525ist_fm,Sl2: fset_P8989946509869081563ist_fm] :
( ( A5 = Steps2 )
& ( member4699826688122452638ist_fm @ ( produc6364584182369725445ist_fm @ ( shd_Pr4098038590278241811ist_fm @ Steps2 ) ) @ ( sset_P5379989128061332361ist_fm @ Rules2 ) )
& ( fmembe3754813877001230652ist_fm @ ( produc8141473855448433859ist_fm @ ( shd_Pr4098038590278241811ist_fm @ ( stl_Pr1377318027900677655ist_fm @ Steps2 ) ) ) @ Sl2 )
& ( Eff2 @ ( produc6364584182369725445ist_fm @ ( shd_Pr4098038590278241811ist_fm @ Steps2 ) ) @ ( produc8141473855448433859ist_fm @ ( shd_Pr4098038590278241811ist_fm @ Steps2 ) ) @ Sl2 )
& ( abstra6295994116336782083ist_fm @ Eff2 @ Rules2 @ ( stl_Pr1377318027900677655ist_fm @ Steps2 ) ) ) ) ) ).
% RuleSystem_Defs.epath.simps
thf(fact_187_RuleSystem__Defs_Oepath_Osimps,axiom,
( abstra4310285689605054952m_rule
= ( ^ [Eff2: list_fm > abstra8520858721168826610m_rule > fset_A8849196716589201442m_rule > $o,Rules2: stream_list_fm,A5: stream5329691398384333196ist_fm] :
? [Steps2: stream5329691398384333196ist_fm,Sl2: fset_A8849196716589201442m_rule] :
( ( A5 = Steps2 )
& ( member_list_fm @ ( produc6440468272734011008ist_fm @ ( shd_Pr2916036590756128464ist_fm @ Steps2 ) ) @ ( sset_list_fm @ Rules2 ) )
& ( fmembe1099044791951439339m_rule @ ( produc162143410373610050ist_fm @ ( shd_Pr2916036590756128464ist_fm @ ( stl_Pr213190007691795916ist_fm @ Steps2 ) ) ) @ Sl2 )
& ( Eff2 @ ( produc6440468272734011008ist_fm @ ( shd_Pr2916036590756128464ist_fm @ Steps2 ) ) @ ( produc162143410373610050ist_fm @ ( shd_Pr2916036590756128464ist_fm @ Steps2 ) ) @ Sl2 )
& ( abstra4310285689605054952m_rule @ Eff2 @ Rules2 @ ( stl_Pr213190007691795916ist_fm @ Steps2 ) ) ) ) ) ).
% RuleSystem_Defs.epath.simps
thf(fact_188_RuleSystem__Defs_Oepath_Osimps,axiom,
( abstra5381393010826267066ist_fm
= ( ^ [Eff2: produc340336539035504054m_rule > produc6018962875968178549ist_fm > fset_P8989946509869081563ist_fm > $o,Rules2: stream2709947120125613254m_rule,A5: stream6742152135616053226m_rule] :
? [Steps2: stream6742152135616053226m_rule,Sl2: fset_P8989946509869081563ist_fm] :
( ( A5 = Steps2 )
& ( member7231649785386036813m_rule @ ( produc1227651196481849126m_rule @ ( shd_Pr6124039728035424814m_rule @ Steps2 ) ) @ ( sset_P4484857331586881186m_rule @ Rules2 ) )
& ( fmembe3754813877001230652ist_fm @ ( produc6387030584228546280m_rule @ ( shd_Pr6124039728035424814m_rule @ ( stl_Pr7706092684847479082m_rule @ Steps2 ) ) ) @ Sl2 )
& ( Eff2 @ ( produc1227651196481849126m_rule @ ( shd_Pr6124039728035424814m_rule @ Steps2 ) ) @ ( produc6387030584228546280m_rule @ ( shd_Pr6124039728035424814m_rule @ Steps2 ) ) @ Sl2 )
& ( abstra5381393010826267066ist_fm @ Eff2 @ Rules2 @ ( stl_Pr7706092684847479082m_rule @ Steps2 ) ) ) ) ) ).
% RuleSystem_Defs.epath.simps
thf(fact_189_RuleSystem__Defs_Oepath_Osimps,axiom,
( abstra8533012240466880740m_rule
= ( ^ [Eff2: produc6018962875968178549ist_fm > abstra8520858721168826610m_rule > fset_A8849196716589201442m_rule > $o,Rules2: stream4408948924543953275ist_fm,A5: stream2118054573033641736ist_fm] :
? [Steps2: stream2118054573033641736ist_fm,Sl2: fset_A8849196716589201442m_rule] :
( ( A5 = Steps2 )
& ( member4699826688122452638ist_fm @ ( produc1196760618833626364ist_fm @ ( shd_Pr1348735749115486668ist_fm @ Steps2 ) ) @ ( sset_P5379989128061332361ist_fm @ Rules2 ) )
& ( fmembe1099044791951439339m_rule @ ( produc7096817939255439806ist_fm @ ( shd_Pr1348735749115486668ist_fm @ ( stl_Pr1735168262451467976ist_fm @ Steps2 ) ) ) @ Sl2 )
& ( Eff2 @ ( produc1196760618833626364ist_fm @ ( shd_Pr1348735749115486668ist_fm @ Steps2 ) ) @ ( produc7096817939255439806ist_fm @ ( shd_Pr1348735749115486668ist_fm @ Steps2 ) ) @ Sl2 )
& ( abstra8533012240466880740m_rule @ Eff2 @ Rules2 @ ( stl_Pr1735168262451467976ist_fm @ Steps2 ) ) ) ) ) ).
% RuleSystem_Defs.epath.simps
thf(fact_190_RuleSystem__Defs_Oepath_Ocases,axiom,
! [Eff: list_fm > list_tm > fset_list_tm > $o,Rules: stream_list_fm,A2: stream4408948924543953275ist_fm] :
( ( abstra302294732650053493ist_tm @ Eff @ Rules @ A2 )
=> ~ ( ( member_list_fm @ ( produc3242453064496107283ist_fm @ ( shd_Pr3211216682057661985ist_fm @ A2 ) ) @ ( sset_list_fm @ Rules ) )
=> ! [Sl: fset_list_tm] :
( ( fmember_list_tm @ ( produc4874987732838623697ist_fm @ ( shd_Pr3211216682057661985ist_fm @ ( stl_Pr1506262294867171877ist_fm @ A2 ) ) ) @ Sl )
=> ( ( Eff @ ( produc3242453064496107283ist_fm @ ( shd_Pr3211216682057661985ist_fm @ A2 ) ) @ ( produc4874987732838623697ist_fm @ ( shd_Pr3211216682057661985ist_fm @ A2 ) ) @ Sl )
=> ~ ( abstra302294732650053493ist_tm @ Eff @ Rules @ ( stl_Pr1506262294867171877ist_fm @ A2 ) ) ) ) ) ) ).
% RuleSystem_Defs.epath.cases
thf(fact_191_RuleSystem__Defs_Oepath_Ocases,axiom,
! [Eff: fm > produc6018962875968178549ist_fm > fset_P8989946509869081563ist_fm > $o,Rules: stream_fm,A2: stream7995241112960703083_fm_fm] :
( ( abstra3580388122067503493ist_fm @ Eff @ Rules @ A2 )
=> ~ ( ( member_fm @ ( produc5710729445452762371_fm_fm @ ( shd_Pr3833923502784695313_fm_fm @ A2 ) ) @ ( sset_fm @ Rules ) )
=> ! [Sl: fset_P8989946509869081563ist_fm] :
( ( fmembe3754813877001230652ist_fm @ ( produc8064570269430693313_fm_fm @ ( shd_Pr3833923502784695313_fm_fm @ ( stl_Pr2438619550280678421_fm_fm @ A2 ) ) ) @ Sl )
=> ( ( Eff @ ( produc5710729445452762371_fm_fm @ ( shd_Pr3833923502784695313_fm_fm @ A2 ) ) @ ( produc8064570269430693313_fm_fm @ ( shd_Pr3833923502784695313_fm_fm @ A2 ) ) @ Sl )
=> ~ ( abstra3580388122067503493ist_fm @ Eff @ Rules @ ( stl_Pr2438619550280678421_fm_fm @ A2 ) ) ) ) ) ) ).
% RuleSystem_Defs.epath.cases
thf(fact_192_RuleSystem__Defs_Oepath_Ocases,axiom,
! [Eff: rule > produc6018962875968178549ist_fm > fset_P8989946509869081563ist_fm > $o,Rules: stream_rule,A2: stream2709947120125613254m_rule] :
( ( abstra6789711989322986974ist_fm @ Eff @ Rules @ A2 )
=> ~ ( ( member_rule @ ( produc6514711208036230658m_rule @ ( shd_Pr4562317740776619530m_rule @ A2 ) ) @ ( sset_rule @ Rules ) )
=> ! [Sl: fset_P8989946509869081563ist_fm] :
( ( fmembe3754813877001230652ist_fm @ ( produc993771546023551940m_rule @ ( shd_Pr4562317740776619530m_rule @ ( stl_Pr12655793849929990m_rule @ A2 ) ) ) @ Sl )
=> ( ( Eff @ ( produc6514711208036230658m_rule @ ( shd_Pr4562317740776619530m_rule @ A2 ) ) @ ( produc993771546023551940m_rule @ ( shd_Pr4562317740776619530m_rule @ A2 ) ) @ Sl )
=> ~ ( abstra6789711989322986974ist_fm @ Eff @ Rules @ ( stl_Pr12655793849929990m_rule @ A2 ) ) ) ) ) ) ).
% RuleSystem_Defs.epath.cases
thf(fact_193_RuleSystem__Defs_Oepath_Ocases,axiom,
! [Eff: list_fm > produc6018962875968178549ist_fm > fset_P8989946509869081563ist_fm > $o,Rules: stream_list_fm,A2: stream4656891395932198769ist_fm] :
( ( abstra5075300408540101887ist_fm @ Eff @ Rules @ A2 )
=> ~ ( ( member_list_fm @ ( produc2977072740996481289ist_fm @ ( shd_Pr311069668031106967ist_fm @ A2 ) ) @ ( sset_list_fm @ Rules ) )
=> ! [Sl: fset_P8989946509869081563ist_fm] :
( ( fmembe3754813877001230652ist_fm @ ( produc5700618299769748167ist_fm @ ( shd_Pr311069668031106967ist_fm @ ( stl_Pr7978862689335602587ist_fm @ A2 ) ) ) @ Sl )
=> ( ( Eff @ ( produc2977072740996481289ist_fm @ ( shd_Pr311069668031106967ist_fm @ A2 ) ) @ ( produc5700618299769748167ist_fm @ ( shd_Pr311069668031106967ist_fm @ A2 ) ) @ Sl )
=> ~ ( abstra5075300408540101887ist_fm @ Eff @ Rules @ ( stl_Pr7978862689335602587ist_fm @ A2 ) ) ) ) ) ) ).
% RuleSystem_Defs.epath.cases
thf(fact_194_RuleSystem__Defs_Oepath_Ocases,axiom,
! [Eff: fm > abstra8520858721168826610m_rule > fset_A8849196716589201442m_rule > $o,Rules: stream_fm,A2: stream2858270464523782ule_fm] :
( ( abstra6170950473048798434m_rule @ Eff @ Rules @ A2 )
=> ~ ( ( member_fm @ ( produc5983953905017532922ule_fm @ ( shd_Pr142054665586430922ule_fm @ A2 ) ) @ ( sset_fm @ Rules ) )
=> ! [Sl: fset_A8849196716589201442m_rule] :
( ( fmembe1099044791951439339m_rule @ ( produc7765541624337483452ule_fm @ ( shd_Pr142054665586430922ule_fm @ ( stl_Pr7787596663505189062ule_fm @ A2 ) ) ) @ Sl )
=> ( ( Eff @ ( produc5983953905017532922ule_fm @ ( shd_Pr142054665586430922ule_fm @ A2 ) ) @ ( produc7765541624337483452ule_fm @ ( shd_Pr142054665586430922ule_fm @ A2 ) ) @ Sl )
=> ~ ( abstra6170950473048798434m_rule @ Eff @ Rules @ ( stl_Pr7787596663505189062ule_fm @ A2 ) ) ) ) ) ) ).
% RuleSystem_Defs.epath.cases
thf(fact_195_RuleSystem__Defs_Oepath_Ocases,axiom,
! [Eff: rule > abstra8520858721168826610m_rule > fset_A8849196716589201442m_rule > $o,Rules: stream_rule,A2: stream3393403210778678635e_rule] :
( ( abstra7067474524808760265m_rule @ Eff @ Rules @ A2 )
=> ~ ( ( member_rule @ ( produc9166542125499771467e_rule @ ( shd_Pr8663275782010802449e_rule @ A2 ) ) @ ( sset_rule @ Rules ) )
=> ! [Sl: fset_A8849196716589201442m_rule] :
( ( fmembe1099044791951439339m_rule @ ( produc41289011304007433e_rule @ ( shd_Pr8663275782010802449e_rule @ ( stl_Pr654857084227610901e_rule @ A2 ) ) ) @ Sl )
=> ( ( Eff @ ( produc9166542125499771467e_rule @ ( shd_Pr8663275782010802449e_rule @ A2 ) ) @ ( produc41289011304007433e_rule @ ( shd_Pr8663275782010802449e_rule @ A2 ) ) @ Sl )
=> ~ ( abstra7067474524808760265m_rule @ Eff @ Rules @ ( stl_Pr654857084227610901e_rule @ A2 ) ) ) ) ) ) ).
% RuleSystem_Defs.epath.cases
thf(fact_196_RuleSystem__Defs_Oepath_Ocases,axiom,
! [Eff: produc6018962875968178549ist_fm > produc6018962875968178549ist_fm > fset_P8989946509869081563ist_fm > $o,Rules: stream4408948924543953275ist_fm,A2: stream1747862497754140525ist_fm] :
( ( abstra6295994116336782083ist_fm @ Eff @ Rules @ A2 )
=> ~ ( ( member4699826688122452638ist_fm @ ( produc6364584182369725445ist_fm @ ( shd_Pr4098038590278241811ist_fm @ A2 ) ) @ ( sset_P5379989128061332361ist_fm @ Rules ) )
=> ! [Sl: fset_P8989946509869081563ist_fm] :
( ( fmembe3754813877001230652ist_fm @ ( produc8141473855448433859ist_fm @ ( shd_Pr4098038590278241811ist_fm @ ( stl_Pr1377318027900677655ist_fm @ A2 ) ) ) @ Sl )
=> ( ( Eff @ ( produc6364584182369725445ist_fm @ ( shd_Pr4098038590278241811ist_fm @ A2 ) ) @ ( produc8141473855448433859ist_fm @ ( shd_Pr4098038590278241811ist_fm @ A2 ) ) @ Sl )
=> ~ ( abstra6295994116336782083ist_fm @ Eff @ Rules @ ( stl_Pr1377318027900677655ist_fm @ A2 ) ) ) ) ) ) ).
% RuleSystem_Defs.epath.cases
thf(fact_197_RuleSystem__Defs_Oepath_Ocases,axiom,
! [Eff: list_fm > abstra8520858721168826610m_rule > fset_A8849196716589201442m_rule > $o,Rules: stream_list_fm,A2: stream5329691398384333196ist_fm] :
( ( abstra4310285689605054952m_rule @ Eff @ Rules @ A2 )
=> ~ ( ( member_list_fm @ ( produc6440468272734011008ist_fm @ ( shd_Pr2916036590756128464ist_fm @ A2 ) ) @ ( sset_list_fm @ Rules ) )
=> ! [Sl: fset_A8849196716589201442m_rule] :
( ( fmembe1099044791951439339m_rule @ ( produc162143410373610050ist_fm @ ( shd_Pr2916036590756128464ist_fm @ ( stl_Pr213190007691795916ist_fm @ A2 ) ) ) @ Sl )
=> ( ( Eff @ ( produc6440468272734011008ist_fm @ ( shd_Pr2916036590756128464ist_fm @ A2 ) ) @ ( produc162143410373610050ist_fm @ ( shd_Pr2916036590756128464ist_fm @ A2 ) ) @ Sl )
=> ~ ( abstra4310285689605054952m_rule @ Eff @ Rules @ ( stl_Pr213190007691795916ist_fm @ A2 ) ) ) ) ) ) ).
% RuleSystem_Defs.epath.cases
thf(fact_198_RuleSystem__Defs_Oepath_Ocases,axiom,
! [Eff: produc340336539035504054m_rule > produc6018962875968178549ist_fm > fset_P8989946509869081563ist_fm > $o,Rules: stream2709947120125613254m_rule,A2: stream6742152135616053226m_rule] :
( ( abstra5381393010826267066ist_fm @ Eff @ Rules @ A2 )
=> ~ ( ( member7231649785386036813m_rule @ ( produc1227651196481849126m_rule @ ( shd_Pr6124039728035424814m_rule @ A2 ) ) @ ( sset_P4484857331586881186m_rule @ Rules ) )
=> ! [Sl: fset_P8989946509869081563ist_fm] :
( ( fmembe3754813877001230652ist_fm @ ( produc6387030584228546280m_rule @ ( shd_Pr6124039728035424814m_rule @ ( stl_Pr7706092684847479082m_rule @ A2 ) ) ) @ Sl )
=> ( ( Eff @ ( produc1227651196481849126m_rule @ ( shd_Pr6124039728035424814m_rule @ A2 ) ) @ ( produc6387030584228546280m_rule @ ( shd_Pr6124039728035424814m_rule @ A2 ) ) @ Sl )
=> ~ ( abstra5381393010826267066ist_fm @ Eff @ Rules @ ( stl_Pr7706092684847479082m_rule @ A2 ) ) ) ) ) ) ).
% RuleSystem_Defs.epath.cases
thf(fact_199_RuleSystem__Defs_Oepath_Ocases,axiom,
! [Eff: produc6018962875968178549ist_fm > abstra8520858721168826610m_rule > fset_A8849196716589201442m_rule > $o,Rules: stream4408948924543953275ist_fm,A2: stream2118054573033641736ist_fm] :
( ( abstra8533012240466880740m_rule @ Eff @ Rules @ A2 )
=> ~ ( ( member4699826688122452638ist_fm @ ( produc1196760618833626364ist_fm @ ( shd_Pr1348735749115486668ist_fm @ A2 ) ) @ ( sset_P5379989128061332361ist_fm @ Rules ) )
=> ! [Sl: fset_A8849196716589201442m_rule] :
( ( fmembe1099044791951439339m_rule @ ( produc7096817939255439806ist_fm @ ( shd_Pr1348735749115486668ist_fm @ ( stl_Pr1735168262451467976ist_fm @ A2 ) ) ) @ Sl )
=> ( ( Eff @ ( produc1196760618833626364ist_fm @ ( shd_Pr1348735749115486668ist_fm @ A2 ) ) @ ( produc7096817939255439806ist_fm @ ( shd_Pr1348735749115486668ist_fm @ A2 ) ) @ Sl )
=> ~ ( abstra8533012240466880740m_rule @ Eff @ Rules @ ( stl_Pr1735168262451467976ist_fm @ A2 ) ) ) ) ) ) ).
% RuleSystem_Defs.epath.cases
thf(fact_200_Saturated__def,axiom,
! [Steps: stream2709947120125613254m_rule] :
( ( abstra6097777249025082867ist_fm @ eff @ rules @ Steps )
= ( ! [X4: rule] :
( ( member_rule @ X4 @ ( sset_rule @ rules ) )
=> ( abstra2533313685312581075ist_fm @ eff @ X4 @ Steps ) ) ) ) ).
% Saturated_def
thf(fact_201_suf,axiom,
( linear7925859179493631106m_rule
@ ^ [Step: produc340336539035504054m_rule] :
( ( produc6514711208036230658m_rule @ Step )
= betaCon )
@ suf ) ).
% suf
thf(fact_202_i_Oepath_Ocoinduct,axiom,
! [X5: stream4408948924543953275ist_fm > $o,X3: stream4408948924543953275ist_fm,Rules: stream_list_fm,Eff3: list_fm > list_tm > option_fset_list_tm] :
( ( X5 @ X3 )
=> ( ! [X: stream4408948924543953275ist_fm] :
( ( X5 @ X )
=> ? [Steps3: stream4408948924543953275ist_fm,Sl3: fset_list_tm] :
( ( X = Steps3 )
& ( member_list_fm @ ( produc3242453064496107283ist_fm @ ( shd_Pr3211216682057661985ist_fm @ Steps3 ) ) @ ( sset_list_fm @ Rules ) )
& ( fmember_list_tm @ ( produc4874987732838623697ist_fm @ ( shd_Pr3211216682057661985ist_fm @ ( stl_Pr1506262294867171877ist_fm @ Steps3 ) ) ) @ Sl3 )
& ( abstra9141620886324064114ist_tm @ Eff3 @ ( produc3242453064496107283ist_fm @ ( shd_Pr3211216682057661985ist_fm @ Steps3 ) ) @ ( produc4874987732838623697ist_fm @ ( shd_Pr3211216682057661985ist_fm @ Steps3 ) ) @ Sl3 )
& ( ( X5 @ ( stl_Pr1506262294867171877ist_fm @ Steps3 ) )
| ( abstra302294732650053493ist_tm @ ( abstra9141620886324064114ist_tm @ Eff3 ) @ Rules @ ( stl_Pr1506262294867171877ist_fm @ Steps3 ) ) ) ) )
=> ( abstra302294732650053493ist_tm @ ( abstra9141620886324064114ist_tm @ Eff3 ) @ Rules @ X3 ) ) ) ).
% i.epath.coinduct
thf(fact_203_i_Oepath_Ocoinduct,axiom,
! [X5: stream7995241112960703083_fm_fm > $o,X3: stream7995241112960703083_fm_fm,Rules: stream_fm,Eff3: fm > produc6018962875968178549ist_fm > option6967287582980624417ist_fm] :
( ( X5 @ X3 )
=> ( ! [X: stream7995241112960703083_fm_fm] :
( ( X5 @ X )
=> ? [Steps3: stream7995241112960703083_fm_fm,Sl3: fset_P8989946509869081563ist_fm] :
( ( X = Steps3 )
& ( member_fm @ ( produc5710729445452762371_fm_fm @ ( shd_Pr3833923502784695313_fm_fm @ Steps3 ) ) @ ( sset_fm @ Rules ) )
& ( fmembe3754813877001230652ist_fm @ ( produc8064570269430693313_fm_fm @ ( shd_Pr3833923502784695313_fm_fm @ ( stl_Pr2438619550280678421_fm_fm @ Steps3 ) ) ) @ Sl3 )
& ( abstra7874326357579037314ist_fm @ Eff3 @ ( produc5710729445452762371_fm_fm @ ( shd_Pr3833923502784695313_fm_fm @ Steps3 ) ) @ ( produc8064570269430693313_fm_fm @ ( shd_Pr3833923502784695313_fm_fm @ Steps3 ) ) @ Sl3 )
& ( ( X5 @ ( stl_Pr2438619550280678421_fm_fm @ Steps3 ) )
| ( abstra3580388122067503493ist_fm @ ( abstra7874326357579037314ist_fm @ Eff3 ) @ Rules @ ( stl_Pr2438619550280678421_fm_fm @ Steps3 ) ) ) ) )
=> ( abstra3580388122067503493ist_fm @ ( abstra7874326357579037314ist_fm @ Eff3 ) @ Rules @ X3 ) ) ) ).
% i.epath.coinduct
thf(fact_204_i_Oepath_Ocoinduct,axiom,
! [X5: stream2709947120125613254m_rule > $o,X3: stream2709947120125613254m_rule,Rules: stream_rule,Eff3: rule > produc6018962875968178549ist_fm > option6967287582980624417ist_fm] :
( ( X5 @ X3 )
=> ( ! [X: stream2709947120125613254m_rule] :
( ( X5 @ X )
=> ? [Steps3: stream2709947120125613254m_rule,Sl3: fset_P8989946509869081563ist_fm] :
( ( X = Steps3 )
& ( member_rule @ ( produc6514711208036230658m_rule @ ( shd_Pr4562317740776619530m_rule @ Steps3 ) ) @ ( sset_rule @ Rules ) )
& ( fmembe3754813877001230652ist_fm @ ( produc993771546023551940m_rule @ ( shd_Pr4562317740776619530m_rule @ ( stl_Pr12655793849929990m_rule @ Steps3 ) ) ) @ Sl3 )
& ( abstra2682625350522704545ist_fm @ Eff3 @ ( produc6514711208036230658m_rule @ ( shd_Pr4562317740776619530m_rule @ Steps3 ) ) @ ( produc993771546023551940m_rule @ ( shd_Pr4562317740776619530m_rule @ Steps3 ) ) @ Sl3 )
& ( ( X5 @ ( stl_Pr12655793849929990m_rule @ Steps3 ) )
| ( abstra6789711989322986974ist_fm @ ( abstra2682625350522704545ist_fm @ Eff3 ) @ Rules @ ( stl_Pr12655793849929990m_rule @ Steps3 ) ) ) ) )
=> ( abstra6789711989322986974ist_fm @ ( abstra2682625350522704545ist_fm @ Eff3 ) @ Rules @ X3 ) ) ) ).
% i.epath.coinduct
thf(fact_205_i_Oepath_Ocoinduct,axiom,
! [X5: stream4656891395932198769ist_fm > $o,X3: stream4656891395932198769ist_fm,Rules: stream_list_fm,Eff3: list_fm > produc6018962875968178549ist_fm > option6967287582980624417ist_fm] :
( ( X5 @ X3 )
=> ( ! [X: stream4656891395932198769ist_fm] :
( ( X5 @ X )
=> ? [Steps3: stream4656891395932198769ist_fm,Sl3: fset_P8989946509869081563ist_fm] :
( ( X = Steps3 )
& ( member_list_fm @ ( produc2977072740996481289ist_fm @ ( shd_Pr311069668031106967ist_fm @ Steps3 ) ) @ ( sset_list_fm @ Rules ) )
& ( fmembe3754813877001230652ist_fm @ ( produc5700618299769748167ist_fm @ ( shd_Pr311069668031106967ist_fm @ ( stl_Pr7978862689335602587ist_fm @ Steps3 ) ) ) @ Sl3 )
& ( abstra1000122179166948220ist_fm @ Eff3 @ ( produc2977072740996481289ist_fm @ ( shd_Pr311069668031106967ist_fm @ Steps3 ) ) @ ( produc5700618299769748167ist_fm @ ( shd_Pr311069668031106967ist_fm @ Steps3 ) ) @ Sl3 )
& ( ( X5 @ ( stl_Pr7978862689335602587ist_fm @ Steps3 ) )
| ( abstra5075300408540101887ist_fm @ ( abstra1000122179166948220ist_fm @ Eff3 ) @ Rules @ ( stl_Pr7978862689335602587ist_fm @ Steps3 ) ) ) ) )
=> ( abstra5075300408540101887ist_fm @ ( abstra1000122179166948220ist_fm @ Eff3 ) @ Rules @ X3 ) ) ) ).
% i.epath.coinduct
thf(fact_206_i_Oepath_Ocoinduct,axiom,
! [X5: stream2858270464523782ule_fm > $o,X3: stream2858270464523782ule_fm,Rules: stream_fm,Eff3: fm > abstra8520858721168826610m_rule > option9127550908420139634m_rule] :
( ( X5 @ X3 )
=> ( ! [X: stream2858270464523782ule_fm] :
( ( X5 @ X )
=> ? [Steps3: stream2858270464523782ule_fm,Sl3: fset_A8849196716589201442m_rule] :
( ( X = Steps3 )
& ( member_fm @ ( produc5983953905017532922ule_fm @ ( shd_Pr142054665586430922ule_fm @ Steps3 ) ) @ ( sset_fm @ Rules ) )
& ( fmembe1099044791951439339m_rule @ ( produc7765541624337483452ule_fm @ ( shd_Pr142054665586430922ule_fm @ ( stl_Pr7787596663505189062ule_fm @ Steps3 ) ) ) @ Sl3 )
& ( abstra7618830265118336037m_rule @ Eff3 @ ( produc5983953905017532922ule_fm @ ( shd_Pr142054665586430922ule_fm @ Steps3 ) ) @ ( produc7765541624337483452ule_fm @ ( shd_Pr142054665586430922ule_fm @ Steps3 ) ) @ Sl3 )
& ( ( X5 @ ( stl_Pr7787596663505189062ule_fm @ Steps3 ) )
| ( abstra6170950473048798434m_rule @ ( abstra7618830265118336037m_rule @ Eff3 ) @ Rules @ ( stl_Pr7787596663505189062ule_fm @ Steps3 ) ) ) ) )
=> ( abstra6170950473048798434m_rule @ ( abstra7618830265118336037m_rule @ Eff3 ) @ Rules @ X3 ) ) ) ).
% i.epath.coinduct
thf(fact_207_i_Oepath_Ocoinduct,axiom,
! [X5: stream3393403210778678635e_rule > $o,X3: stream3393403210778678635e_rule,Rules: stream_rule,Eff3: rule > abstra8520858721168826610m_rule > option9127550908420139634m_rule] :
( ( X5 @ X3 )
=> ( ! [X: stream3393403210778678635e_rule] :
( ( X5 @ X )
=> ? [Steps3: stream3393403210778678635e_rule,Sl3: fset_A8849196716589201442m_rule] :
( ( X = Steps3 )
& ( member_rule @ ( produc9166542125499771467e_rule @ ( shd_Pr8663275782010802449e_rule @ Steps3 ) ) @ ( sset_rule @ Rules ) )
& ( fmembe1099044791951439339m_rule @ ( produc41289011304007433e_rule @ ( shd_Pr8663275782010802449e_rule @ ( stl_Pr654857084227610901e_rule @ Steps3 ) ) ) @ Sl3 )
& ( abstra3599228160297647814m_rule @ Eff3 @ ( produc9166542125499771467e_rule @ ( shd_Pr8663275782010802449e_rule @ Steps3 ) ) @ ( produc41289011304007433e_rule @ ( shd_Pr8663275782010802449e_rule @ Steps3 ) ) @ Sl3 )
& ( ( X5 @ ( stl_Pr654857084227610901e_rule @ Steps3 ) )
| ( abstra7067474524808760265m_rule @ ( abstra3599228160297647814m_rule @ Eff3 ) @ Rules @ ( stl_Pr654857084227610901e_rule @ Steps3 ) ) ) ) )
=> ( abstra7067474524808760265m_rule @ ( abstra3599228160297647814m_rule @ Eff3 ) @ Rules @ X3 ) ) ) ).
% i.epath.coinduct
thf(fact_208_i_Oepath_Ocoinduct,axiom,
! [X5: stream1747862497754140525ist_fm > $o,X3: stream1747862497754140525ist_fm,Rules: stream4408948924543953275ist_fm,Eff3: produc6018962875968178549ist_fm > produc6018962875968178549ist_fm > option6967287582980624417ist_fm] :
( ( X5 @ X3 )
=> ( ! [X: stream1747862497754140525ist_fm] :
( ( X5 @ X )
=> ? [Steps3: stream1747862497754140525ist_fm,Sl3: fset_P8989946509869081563ist_fm] :
( ( X = Steps3 )
& ( member4699826688122452638ist_fm @ ( produc6364584182369725445ist_fm @ ( shd_Pr4098038590278241811ist_fm @ Steps3 ) ) @ ( sset_P5379989128061332361ist_fm @ Rules ) )
& ( fmembe3754813877001230652ist_fm @ ( produc8141473855448433859ist_fm @ ( shd_Pr4098038590278241811ist_fm @ ( stl_Pr1377318027900677655ist_fm @ Steps3 ) ) ) @ Sl3 )
& ( abstra2084124586061969408ist_fm @ Eff3 @ ( produc6364584182369725445ist_fm @ ( shd_Pr4098038590278241811ist_fm @ Steps3 ) ) @ ( produc8141473855448433859ist_fm @ ( shd_Pr4098038590278241811ist_fm @ Steps3 ) ) @ Sl3 )
& ( ( X5 @ ( stl_Pr1377318027900677655ist_fm @ Steps3 ) )
| ( abstra6295994116336782083ist_fm @ ( abstra2084124586061969408ist_fm @ Eff3 ) @ Rules @ ( stl_Pr1377318027900677655ist_fm @ Steps3 ) ) ) ) )
=> ( abstra6295994116336782083ist_fm @ ( abstra2084124586061969408ist_fm @ Eff3 ) @ Rules @ X3 ) ) ) ).
% i.epath.coinduct
thf(fact_209_i_Oepath_Ocoinduct,axiom,
! [X5: stream5329691398384333196ist_fm > $o,X3: stream5329691398384333196ist_fm,Rules: stream_list_fm,Eff3: list_fm > abstra8520858721168826610m_rule > option9127550908420139634m_rule] :
( ( X5 @ X3 )
=> ( ! [X: stream5329691398384333196ist_fm] :
( ( X5 @ X )
=> ? [Steps3: stream5329691398384333196ist_fm,Sl3: fset_A8849196716589201442m_rule] :
( ( X = Steps3 )
& ( member_list_fm @ ( produc6440468272734011008ist_fm @ ( shd_Pr2916036590756128464ist_fm @ Steps3 ) ) @ ( sset_list_fm @ Rules ) )
& ( fmembe1099044791951439339m_rule @ ( produc162143410373610050ist_fm @ ( shd_Pr2916036590756128464ist_fm @ ( stl_Pr213190007691795916ist_fm @ Steps3 ) ) ) @ Sl3 )
& ( abstra5667912417467126699m_rule @ Eff3 @ ( produc6440468272734011008ist_fm @ ( shd_Pr2916036590756128464ist_fm @ Steps3 ) ) @ ( produc162143410373610050ist_fm @ ( shd_Pr2916036590756128464ist_fm @ Steps3 ) ) @ Sl3 )
& ( ( X5 @ ( stl_Pr213190007691795916ist_fm @ Steps3 ) )
| ( abstra4310285689605054952m_rule @ ( abstra5667912417467126699m_rule @ Eff3 ) @ Rules @ ( stl_Pr213190007691795916ist_fm @ Steps3 ) ) ) ) )
=> ( abstra4310285689605054952m_rule @ ( abstra5667912417467126699m_rule @ Eff3 ) @ Rules @ X3 ) ) ) ).
% i.epath.coinduct
thf(fact_210_i_Oepath_Ocoinduct,axiom,
! [X5: stream6742152135616053226m_rule > $o,X3: stream6742152135616053226m_rule,Rules: stream2709947120125613254m_rule,Eff3: produc340336539035504054m_rule > produc6018962875968178549ist_fm > option6967287582980624417ist_fm] :
( ( X5 @ X3 )
=> ( ! [X: stream6742152135616053226m_rule] :
( ( X5 @ X )
=> ? [Steps3: stream6742152135616053226m_rule,Sl3: fset_P8989946509869081563ist_fm] :
( ( X = Steps3 )
& ( member7231649785386036813m_rule @ ( produc1227651196481849126m_rule @ ( shd_Pr6124039728035424814m_rule @ Steps3 ) ) @ ( sset_P4484857331586881186m_rule @ Rules ) )
& ( fmembe3754813877001230652ist_fm @ ( produc6387030584228546280m_rule @ ( shd_Pr6124039728035424814m_rule @ ( stl_Pr7706092684847479082m_rule @ Steps3 ) ) ) @ Sl3 )
& ( abstra7153543546181413245ist_fm @ Eff3 @ ( produc1227651196481849126m_rule @ ( shd_Pr6124039728035424814m_rule @ Steps3 ) ) @ ( produc6387030584228546280m_rule @ ( shd_Pr6124039728035424814m_rule @ Steps3 ) ) @ Sl3 )
& ( ( X5 @ ( stl_Pr7706092684847479082m_rule @ Steps3 ) )
| ( abstra5381393010826267066ist_fm @ ( abstra7153543546181413245ist_fm @ Eff3 ) @ Rules @ ( stl_Pr7706092684847479082m_rule @ Steps3 ) ) ) ) )
=> ( abstra5381393010826267066ist_fm @ ( abstra7153543546181413245ist_fm @ Eff3 ) @ Rules @ X3 ) ) ) ).
% i.epath.coinduct
thf(fact_211_i_Oepath_Ocoinduct,axiom,
! [X5: stream2118054573033641736ist_fm > $o,X3: stream2118054573033641736ist_fm,Rules: stream4408948924543953275ist_fm,Eff3: produc6018962875968178549ist_fm > abstra8520858721168826610m_rule > option9127550908420139634m_rule] :
( ( X5 @ X3 )
=> ( ! [X: stream2118054573033641736ist_fm] :
( ( X5 @ X )
=> ? [Steps3: stream2118054573033641736ist_fm,Sl3: fset_A8849196716589201442m_rule] :
( ( X = Steps3 )
& ( member4699826688122452638ist_fm @ ( produc1196760618833626364ist_fm @ ( shd_Pr1348735749115486668ist_fm @ Steps3 ) ) @ ( sset_P5379989128061332361ist_fm @ Rules ) )
& ( fmembe1099044791951439339m_rule @ ( produc7096817939255439806ist_fm @ ( shd_Pr1348735749115486668ist_fm @ ( stl_Pr1735168262451467976ist_fm @ Steps3 ) ) ) @ Sl3 )
& ( abstra2762543317532761127m_rule @ Eff3 @ ( produc1196760618833626364ist_fm @ ( shd_Pr1348735749115486668ist_fm @ Steps3 ) ) @ ( produc7096817939255439806ist_fm @ ( shd_Pr1348735749115486668ist_fm @ Steps3 ) ) @ Sl3 )
& ( ( X5 @ ( stl_Pr1735168262451467976ist_fm @ Steps3 ) )
| ( abstra8533012240466880740m_rule @ ( abstra2762543317532761127m_rule @ Eff3 ) @ Rules @ ( stl_Pr1735168262451467976ist_fm @ Steps3 ) ) ) ) )
=> ( abstra8533012240466880740m_rule @ ( abstra2762543317532761127m_rule @ Eff3 ) @ Rules @ X3 ) ) ) ).
% i.epath.coinduct
thf(fact_212_i_Oepath_Ointros,axiom,
! [Steps: stream4408948924543953275ist_fm,Rules: stream_list_fm,Sl4: fset_list_tm,Eff3: list_fm > list_tm > option_fset_list_tm] :
( ( member_list_fm @ ( produc3242453064496107283ist_fm @ ( shd_Pr3211216682057661985ist_fm @ Steps ) ) @ ( sset_list_fm @ Rules ) )
=> ( ( fmember_list_tm @ ( produc4874987732838623697ist_fm @ ( shd_Pr3211216682057661985ist_fm @ ( stl_Pr1506262294867171877ist_fm @ Steps ) ) ) @ Sl4 )
=> ( ( abstra9141620886324064114ist_tm @ Eff3 @ ( produc3242453064496107283ist_fm @ ( shd_Pr3211216682057661985ist_fm @ Steps ) ) @ ( produc4874987732838623697ist_fm @ ( shd_Pr3211216682057661985ist_fm @ Steps ) ) @ Sl4 )
=> ( ( abstra302294732650053493ist_tm @ ( abstra9141620886324064114ist_tm @ Eff3 ) @ Rules @ ( stl_Pr1506262294867171877ist_fm @ Steps ) )
=> ( abstra302294732650053493ist_tm @ ( abstra9141620886324064114ist_tm @ Eff3 ) @ Rules @ Steps ) ) ) ) ) ).
% i.epath.intros
thf(fact_213_i_Oepath_Ointros,axiom,
! [Steps: stream7995241112960703083_fm_fm,Rules: stream_fm,Sl4: fset_P8989946509869081563ist_fm,Eff3: fm > produc6018962875968178549ist_fm > option6967287582980624417ist_fm] :
( ( member_fm @ ( produc5710729445452762371_fm_fm @ ( shd_Pr3833923502784695313_fm_fm @ Steps ) ) @ ( sset_fm @ Rules ) )
=> ( ( fmembe3754813877001230652ist_fm @ ( produc8064570269430693313_fm_fm @ ( shd_Pr3833923502784695313_fm_fm @ ( stl_Pr2438619550280678421_fm_fm @ Steps ) ) ) @ Sl4 )
=> ( ( abstra7874326357579037314ist_fm @ Eff3 @ ( produc5710729445452762371_fm_fm @ ( shd_Pr3833923502784695313_fm_fm @ Steps ) ) @ ( produc8064570269430693313_fm_fm @ ( shd_Pr3833923502784695313_fm_fm @ Steps ) ) @ Sl4 )
=> ( ( abstra3580388122067503493ist_fm @ ( abstra7874326357579037314ist_fm @ Eff3 ) @ Rules @ ( stl_Pr2438619550280678421_fm_fm @ Steps ) )
=> ( abstra3580388122067503493ist_fm @ ( abstra7874326357579037314ist_fm @ Eff3 ) @ Rules @ Steps ) ) ) ) ) ).
% i.epath.intros
thf(fact_214_i_Oepath_Ointros,axiom,
! [Steps: stream2709947120125613254m_rule,Rules: stream_rule,Sl4: fset_P8989946509869081563ist_fm,Eff3: rule > produc6018962875968178549ist_fm > option6967287582980624417ist_fm] :
( ( member_rule @ ( produc6514711208036230658m_rule @ ( shd_Pr4562317740776619530m_rule @ Steps ) ) @ ( sset_rule @ Rules ) )
=> ( ( fmembe3754813877001230652ist_fm @ ( produc993771546023551940m_rule @ ( shd_Pr4562317740776619530m_rule @ ( stl_Pr12655793849929990m_rule @ Steps ) ) ) @ Sl4 )
=> ( ( abstra2682625350522704545ist_fm @ Eff3 @ ( produc6514711208036230658m_rule @ ( shd_Pr4562317740776619530m_rule @ Steps ) ) @ ( produc993771546023551940m_rule @ ( shd_Pr4562317740776619530m_rule @ Steps ) ) @ Sl4 )
=> ( ( abstra6789711989322986974ist_fm @ ( abstra2682625350522704545ist_fm @ Eff3 ) @ Rules @ ( stl_Pr12655793849929990m_rule @ Steps ) )
=> ( abstra6789711989322986974ist_fm @ ( abstra2682625350522704545ist_fm @ Eff3 ) @ Rules @ Steps ) ) ) ) ) ).
% i.epath.intros
thf(fact_215_i_Oepath_Ointros,axiom,
! [Steps: stream4656891395932198769ist_fm,Rules: stream_list_fm,Sl4: fset_P8989946509869081563ist_fm,Eff3: list_fm > produc6018962875968178549ist_fm > option6967287582980624417ist_fm] :
( ( member_list_fm @ ( produc2977072740996481289ist_fm @ ( shd_Pr311069668031106967ist_fm @ Steps ) ) @ ( sset_list_fm @ Rules ) )
=> ( ( fmembe3754813877001230652ist_fm @ ( produc5700618299769748167ist_fm @ ( shd_Pr311069668031106967ist_fm @ ( stl_Pr7978862689335602587ist_fm @ Steps ) ) ) @ Sl4 )
=> ( ( abstra1000122179166948220ist_fm @ Eff3 @ ( produc2977072740996481289ist_fm @ ( shd_Pr311069668031106967ist_fm @ Steps ) ) @ ( produc5700618299769748167ist_fm @ ( shd_Pr311069668031106967ist_fm @ Steps ) ) @ Sl4 )
=> ( ( abstra5075300408540101887ist_fm @ ( abstra1000122179166948220ist_fm @ Eff3 ) @ Rules @ ( stl_Pr7978862689335602587ist_fm @ Steps ) )
=> ( abstra5075300408540101887ist_fm @ ( abstra1000122179166948220ist_fm @ Eff3 ) @ Rules @ Steps ) ) ) ) ) ).
% i.epath.intros
thf(fact_216_i_Oepath_Ointros,axiom,
! [Steps: stream2858270464523782ule_fm,Rules: stream_fm,Sl4: fset_A8849196716589201442m_rule,Eff3: fm > abstra8520858721168826610m_rule > option9127550908420139634m_rule] :
( ( member_fm @ ( produc5983953905017532922ule_fm @ ( shd_Pr142054665586430922ule_fm @ Steps ) ) @ ( sset_fm @ Rules ) )
=> ( ( fmembe1099044791951439339m_rule @ ( produc7765541624337483452ule_fm @ ( shd_Pr142054665586430922ule_fm @ ( stl_Pr7787596663505189062ule_fm @ Steps ) ) ) @ Sl4 )
=> ( ( abstra7618830265118336037m_rule @ Eff3 @ ( produc5983953905017532922ule_fm @ ( shd_Pr142054665586430922ule_fm @ Steps ) ) @ ( produc7765541624337483452ule_fm @ ( shd_Pr142054665586430922ule_fm @ Steps ) ) @ Sl4 )
=> ( ( abstra6170950473048798434m_rule @ ( abstra7618830265118336037m_rule @ Eff3 ) @ Rules @ ( stl_Pr7787596663505189062ule_fm @ Steps ) )
=> ( abstra6170950473048798434m_rule @ ( abstra7618830265118336037m_rule @ Eff3 ) @ Rules @ Steps ) ) ) ) ) ).
% i.epath.intros
thf(fact_217_i_Oepath_Ointros,axiom,
! [Steps: stream3393403210778678635e_rule,Rules: stream_rule,Sl4: fset_A8849196716589201442m_rule,Eff3: rule > abstra8520858721168826610m_rule > option9127550908420139634m_rule] :
( ( member_rule @ ( produc9166542125499771467e_rule @ ( shd_Pr8663275782010802449e_rule @ Steps ) ) @ ( sset_rule @ Rules ) )
=> ( ( fmembe1099044791951439339m_rule @ ( produc41289011304007433e_rule @ ( shd_Pr8663275782010802449e_rule @ ( stl_Pr654857084227610901e_rule @ Steps ) ) ) @ Sl4 )
=> ( ( abstra3599228160297647814m_rule @ Eff3 @ ( produc9166542125499771467e_rule @ ( shd_Pr8663275782010802449e_rule @ Steps ) ) @ ( produc41289011304007433e_rule @ ( shd_Pr8663275782010802449e_rule @ Steps ) ) @ Sl4 )
=> ( ( abstra7067474524808760265m_rule @ ( abstra3599228160297647814m_rule @ Eff3 ) @ Rules @ ( stl_Pr654857084227610901e_rule @ Steps ) )
=> ( abstra7067474524808760265m_rule @ ( abstra3599228160297647814m_rule @ Eff3 ) @ Rules @ Steps ) ) ) ) ) ).
% i.epath.intros
thf(fact_218_i_Oepath_Ointros,axiom,
! [Steps: stream1747862497754140525ist_fm,Rules: stream4408948924543953275ist_fm,Sl4: fset_P8989946509869081563ist_fm,Eff3: produc6018962875968178549ist_fm > produc6018962875968178549ist_fm > option6967287582980624417ist_fm] :
( ( member4699826688122452638ist_fm @ ( produc6364584182369725445ist_fm @ ( shd_Pr4098038590278241811ist_fm @ Steps ) ) @ ( sset_P5379989128061332361ist_fm @ Rules ) )
=> ( ( fmembe3754813877001230652ist_fm @ ( produc8141473855448433859ist_fm @ ( shd_Pr4098038590278241811ist_fm @ ( stl_Pr1377318027900677655ist_fm @ Steps ) ) ) @ Sl4 )
=> ( ( abstra2084124586061969408ist_fm @ Eff3 @ ( produc6364584182369725445ist_fm @ ( shd_Pr4098038590278241811ist_fm @ Steps ) ) @ ( produc8141473855448433859ist_fm @ ( shd_Pr4098038590278241811ist_fm @ Steps ) ) @ Sl4 )
=> ( ( abstra6295994116336782083ist_fm @ ( abstra2084124586061969408ist_fm @ Eff3 ) @ Rules @ ( stl_Pr1377318027900677655ist_fm @ Steps ) )
=> ( abstra6295994116336782083ist_fm @ ( abstra2084124586061969408ist_fm @ Eff3 ) @ Rules @ Steps ) ) ) ) ) ).
% i.epath.intros
thf(fact_219_i_Oepath_Ointros,axiom,
! [Steps: stream5329691398384333196ist_fm,Rules: stream_list_fm,Sl4: fset_A8849196716589201442m_rule,Eff3: list_fm > abstra8520858721168826610m_rule > option9127550908420139634m_rule] :
( ( member_list_fm @ ( produc6440468272734011008ist_fm @ ( shd_Pr2916036590756128464ist_fm @ Steps ) ) @ ( sset_list_fm @ Rules ) )
=> ( ( fmembe1099044791951439339m_rule @ ( produc162143410373610050ist_fm @ ( shd_Pr2916036590756128464ist_fm @ ( stl_Pr213190007691795916ist_fm @ Steps ) ) ) @ Sl4 )
=> ( ( abstra5667912417467126699m_rule @ Eff3 @ ( produc6440468272734011008ist_fm @ ( shd_Pr2916036590756128464ist_fm @ Steps ) ) @ ( produc162143410373610050ist_fm @ ( shd_Pr2916036590756128464ist_fm @ Steps ) ) @ Sl4 )
=> ( ( abstra4310285689605054952m_rule @ ( abstra5667912417467126699m_rule @ Eff3 ) @ Rules @ ( stl_Pr213190007691795916ist_fm @ Steps ) )
=> ( abstra4310285689605054952m_rule @ ( abstra5667912417467126699m_rule @ Eff3 ) @ Rules @ Steps ) ) ) ) ) ).
% i.epath.intros
thf(fact_220_i_Oepath_Ointros,axiom,
! [Steps: stream6742152135616053226m_rule,Rules: stream2709947120125613254m_rule,Sl4: fset_P8989946509869081563ist_fm,Eff3: produc340336539035504054m_rule > produc6018962875968178549ist_fm > option6967287582980624417ist_fm] :
( ( member7231649785386036813m_rule @ ( produc1227651196481849126m_rule @ ( shd_Pr6124039728035424814m_rule @ Steps ) ) @ ( sset_P4484857331586881186m_rule @ Rules ) )
=> ( ( fmembe3754813877001230652ist_fm @ ( produc6387030584228546280m_rule @ ( shd_Pr6124039728035424814m_rule @ ( stl_Pr7706092684847479082m_rule @ Steps ) ) ) @ Sl4 )
=> ( ( abstra7153543546181413245ist_fm @ Eff3 @ ( produc1227651196481849126m_rule @ ( shd_Pr6124039728035424814m_rule @ Steps ) ) @ ( produc6387030584228546280m_rule @ ( shd_Pr6124039728035424814m_rule @ Steps ) ) @ Sl4 )
=> ( ( abstra5381393010826267066ist_fm @ ( abstra7153543546181413245ist_fm @ Eff3 ) @ Rules @ ( stl_Pr7706092684847479082m_rule @ Steps ) )
=> ( abstra5381393010826267066ist_fm @ ( abstra7153543546181413245ist_fm @ Eff3 ) @ Rules @ Steps ) ) ) ) ) ).
% i.epath.intros
thf(fact_221_i_Oepath_Ointros,axiom,
! [Steps: stream2118054573033641736ist_fm,Rules: stream4408948924543953275ist_fm,Sl4: fset_A8849196716589201442m_rule,Eff3: produc6018962875968178549ist_fm > abstra8520858721168826610m_rule > option9127550908420139634m_rule] :
( ( member4699826688122452638ist_fm @ ( produc1196760618833626364ist_fm @ ( shd_Pr1348735749115486668ist_fm @ Steps ) ) @ ( sset_P5379989128061332361ist_fm @ Rules ) )
=> ( ( fmembe1099044791951439339m_rule @ ( produc7096817939255439806ist_fm @ ( shd_Pr1348735749115486668ist_fm @ ( stl_Pr1735168262451467976ist_fm @ Steps ) ) ) @ Sl4 )
=> ( ( abstra2762543317532761127m_rule @ Eff3 @ ( produc1196760618833626364ist_fm @ ( shd_Pr1348735749115486668ist_fm @ Steps ) ) @ ( produc7096817939255439806ist_fm @ ( shd_Pr1348735749115486668ist_fm @ Steps ) ) @ Sl4 )
=> ( ( abstra8533012240466880740m_rule @ ( abstra2762543317532761127m_rule @ Eff3 ) @ Rules @ ( stl_Pr1735168262451467976ist_fm @ Steps ) )
=> ( abstra8533012240466880740m_rule @ ( abstra2762543317532761127m_rule @ Eff3 ) @ Rules @ Steps ) ) ) ) ) ).
% i.epath.intros
thf(fact_222_i_Oepath_Osimps,axiom,
! [Eff3: list_fm > list_tm > option_fset_list_tm,Rules: stream_list_fm,A2: stream4408948924543953275ist_fm] :
( ( abstra302294732650053493ist_tm @ ( abstra9141620886324064114ist_tm @ Eff3 ) @ Rules @ A2 )
= ( ? [Steps2: stream4408948924543953275ist_fm,Sl2: fset_list_tm] :
( ( A2 = 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 ) ) ) @ Sl2 )
& ( abstra9141620886324064114ist_tm @ Eff3 @ ( produc3242453064496107283ist_fm @ ( shd_Pr3211216682057661985ist_fm @ Steps2 ) ) @ ( produc4874987732838623697ist_fm @ ( shd_Pr3211216682057661985ist_fm @ Steps2 ) ) @ Sl2 )
& ( abstra302294732650053493ist_tm @ ( abstra9141620886324064114ist_tm @ Eff3 ) @ Rules @ ( stl_Pr1506262294867171877ist_fm @ Steps2 ) ) ) ) ) ).
% i.epath.simps
thf(fact_223_i_Oepath_Osimps,axiom,
! [Eff3: fm > produc6018962875968178549ist_fm > option6967287582980624417ist_fm,Rules: stream_fm,A2: stream7995241112960703083_fm_fm] :
( ( abstra3580388122067503493ist_fm @ ( abstra7874326357579037314ist_fm @ Eff3 ) @ Rules @ A2 )
= ( ? [Steps2: stream7995241112960703083_fm_fm,Sl2: fset_P8989946509869081563ist_fm] :
( ( A2 = Steps2 )
& ( member_fm @ ( produc5710729445452762371_fm_fm @ ( shd_Pr3833923502784695313_fm_fm @ Steps2 ) ) @ ( sset_fm @ Rules ) )
& ( fmembe3754813877001230652ist_fm @ ( produc8064570269430693313_fm_fm @ ( shd_Pr3833923502784695313_fm_fm @ ( stl_Pr2438619550280678421_fm_fm @ Steps2 ) ) ) @ Sl2 )
& ( abstra7874326357579037314ist_fm @ Eff3 @ ( produc5710729445452762371_fm_fm @ ( shd_Pr3833923502784695313_fm_fm @ Steps2 ) ) @ ( produc8064570269430693313_fm_fm @ ( shd_Pr3833923502784695313_fm_fm @ Steps2 ) ) @ Sl2 )
& ( abstra3580388122067503493ist_fm @ ( abstra7874326357579037314ist_fm @ Eff3 ) @ Rules @ ( stl_Pr2438619550280678421_fm_fm @ Steps2 ) ) ) ) ) ).
% i.epath.simps
thf(fact_224_i_Oepath_Osimps,axiom,
! [Eff3: rule > produc6018962875968178549ist_fm > option6967287582980624417ist_fm,Rules: stream_rule,A2: stream2709947120125613254m_rule] :
( ( abstra6789711989322986974ist_fm @ ( abstra2682625350522704545ist_fm @ Eff3 ) @ Rules @ A2 )
= ( ? [Steps2: stream2709947120125613254m_rule,Sl2: fset_P8989946509869081563ist_fm] :
( ( A2 = Steps2 )
& ( member_rule @ ( produc6514711208036230658m_rule @ ( shd_Pr4562317740776619530m_rule @ Steps2 ) ) @ ( sset_rule @ Rules ) )
& ( fmembe3754813877001230652ist_fm @ ( produc993771546023551940m_rule @ ( shd_Pr4562317740776619530m_rule @ ( stl_Pr12655793849929990m_rule @ Steps2 ) ) ) @ Sl2 )
& ( abstra2682625350522704545ist_fm @ Eff3 @ ( produc6514711208036230658m_rule @ ( shd_Pr4562317740776619530m_rule @ Steps2 ) ) @ ( produc993771546023551940m_rule @ ( shd_Pr4562317740776619530m_rule @ Steps2 ) ) @ Sl2 )
& ( abstra6789711989322986974ist_fm @ ( abstra2682625350522704545ist_fm @ Eff3 ) @ Rules @ ( stl_Pr12655793849929990m_rule @ Steps2 ) ) ) ) ) ).
% i.epath.simps
thf(fact_225_i_Oepath_Osimps,axiom,
! [Eff3: list_fm > produc6018962875968178549ist_fm > option6967287582980624417ist_fm,Rules: stream_list_fm,A2: stream4656891395932198769ist_fm] :
( ( abstra5075300408540101887ist_fm @ ( abstra1000122179166948220ist_fm @ Eff3 ) @ Rules @ A2 )
= ( ? [Steps2: stream4656891395932198769ist_fm,Sl2: fset_P8989946509869081563ist_fm] :
( ( A2 = Steps2 )
& ( member_list_fm @ ( produc2977072740996481289ist_fm @ ( shd_Pr311069668031106967ist_fm @ Steps2 ) ) @ ( sset_list_fm @ Rules ) )
& ( fmembe3754813877001230652ist_fm @ ( produc5700618299769748167ist_fm @ ( shd_Pr311069668031106967ist_fm @ ( stl_Pr7978862689335602587ist_fm @ Steps2 ) ) ) @ Sl2 )
& ( abstra1000122179166948220ist_fm @ Eff3 @ ( produc2977072740996481289ist_fm @ ( shd_Pr311069668031106967ist_fm @ Steps2 ) ) @ ( produc5700618299769748167ist_fm @ ( shd_Pr311069668031106967ist_fm @ Steps2 ) ) @ Sl2 )
& ( abstra5075300408540101887ist_fm @ ( abstra1000122179166948220ist_fm @ Eff3 ) @ Rules @ ( stl_Pr7978862689335602587ist_fm @ Steps2 ) ) ) ) ) ).
% i.epath.simps
thf(fact_226_i_Oepath_Osimps,axiom,
! [Eff3: fm > abstra8520858721168826610m_rule > option9127550908420139634m_rule,Rules: stream_fm,A2: stream2858270464523782ule_fm] :
( ( abstra6170950473048798434m_rule @ ( abstra7618830265118336037m_rule @ Eff3 ) @ Rules @ A2 )
= ( ? [Steps2: stream2858270464523782ule_fm,Sl2: fset_A8849196716589201442m_rule] :
( ( A2 = Steps2 )
& ( member_fm @ ( produc5983953905017532922ule_fm @ ( shd_Pr142054665586430922ule_fm @ Steps2 ) ) @ ( sset_fm @ Rules ) )
& ( fmembe1099044791951439339m_rule @ ( produc7765541624337483452ule_fm @ ( shd_Pr142054665586430922ule_fm @ ( stl_Pr7787596663505189062ule_fm @ Steps2 ) ) ) @ Sl2 )
& ( abstra7618830265118336037m_rule @ Eff3 @ ( produc5983953905017532922ule_fm @ ( shd_Pr142054665586430922ule_fm @ Steps2 ) ) @ ( produc7765541624337483452ule_fm @ ( shd_Pr142054665586430922ule_fm @ Steps2 ) ) @ Sl2 )
& ( abstra6170950473048798434m_rule @ ( abstra7618830265118336037m_rule @ Eff3 ) @ Rules @ ( stl_Pr7787596663505189062ule_fm @ Steps2 ) ) ) ) ) ).
% i.epath.simps
thf(fact_227_i_Oepath_Osimps,axiom,
! [Eff3: rule > abstra8520858721168826610m_rule > option9127550908420139634m_rule,Rules: stream_rule,A2: stream3393403210778678635e_rule] :
( ( abstra7067474524808760265m_rule @ ( abstra3599228160297647814m_rule @ Eff3 ) @ Rules @ A2 )
= ( ? [Steps2: stream3393403210778678635e_rule,Sl2: fset_A8849196716589201442m_rule] :
( ( A2 = Steps2 )
& ( member_rule @ ( produc9166542125499771467e_rule @ ( shd_Pr8663275782010802449e_rule @ Steps2 ) ) @ ( sset_rule @ Rules ) )
& ( fmembe1099044791951439339m_rule @ ( produc41289011304007433e_rule @ ( shd_Pr8663275782010802449e_rule @ ( stl_Pr654857084227610901e_rule @ Steps2 ) ) ) @ Sl2 )
& ( abstra3599228160297647814m_rule @ Eff3 @ ( produc9166542125499771467e_rule @ ( shd_Pr8663275782010802449e_rule @ Steps2 ) ) @ ( produc41289011304007433e_rule @ ( shd_Pr8663275782010802449e_rule @ Steps2 ) ) @ Sl2 )
& ( abstra7067474524808760265m_rule @ ( abstra3599228160297647814m_rule @ Eff3 ) @ Rules @ ( stl_Pr654857084227610901e_rule @ Steps2 ) ) ) ) ) ).
% i.epath.simps
thf(fact_228_i_Oepath_Osimps,axiom,
! [Eff3: produc6018962875968178549ist_fm > produc6018962875968178549ist_fm > option6967287582980624417ist_fm,Rules: stream4408948924543953275ist_fm,A2: stream1747862497754140525ist_fm] :
( ( abstra6295994116336782083ist_fm @ ( abstra2084124586061969408ist_fm @ Eff3 ) @ Rules @ A2 )
= ( ? [Steps2: stream1747862497754140525ist_fm,Sl2: fset_P8989946509869081563ist_fm] :
( ( A2 = Steps2 )
& ( member4699826688122452638ist_fm @ ( produc6364584182369725445ist_fm @ ( shd_Pr4098038590278241811ist_fm @ Steps2 ) ) @ ( sset_P5379989128061332361ist_fm @ Rules ) )
& ( fmembe3754813877001230652ist_fm @ ( produc8141473855448433859ist_fm @ ( shd_Pr4098038590278241811ist_fm @ ( stl_Pr1377318027900677655ist_fm @ Steps2 ) ) ) @ Sl2 )
& ( abstra2084124586061969408ist_fm @ Eff3 @ ( produc6364584182369725445ist_fm @ ( shd_Pr4098038590278241811ist_fm @ Steps2 ) ) @ ( produc8141473855448433859ist_fm @ ( shd_Pr4098038590278241811ist_fm @ Steps2 ) ) @ Sl2 )
& ( abstra6295994116336782083ist_fm @ ( abstra2084124586061969408ist_fm @ Eff3 ) @ Rules @ ( stl_Pr1377318027900677655ist_fm @ Steps2 ) ) ) ) ) ).
% i.epath.simps
thf(fact_229_i_Oepath_Osimps,axiom,
! [Eff3: list_fm > abstra8520858721168826610m_rule > option9127550908420139634m_rule,Rules: stream_list_fm,A2: stream5329691398384333196ist_fm] :
( ( abstra4310285689605054952m_rule @ ( abstra5667912417467126699m_rule @ Eff3 ) @ Rules @ A2 )
= ( ? [Steps2: stream5329691398384333196ist_fm,Sl2: fset_A8849196716589201442m_rule] :
( ( A2 = Steps2 )
& ( member_list_fm @ ( produc6440468272734011008ist_fm @ ( shd_Pr2916036590756128464ist_fm @ Steps2 ) ) @ ( sset_list_fm @ Rules ) )
& ( fmembe1099044791951439339m_rule @ ( produc162143410373610050ist_fm @ ( shd_Pr2916036590756128464ist_fm @ ( stl_Pr213190007691795916ist_fm @ Steps2 ) ) ) @ Sl2 )
& ( abstra5667912417467126699m_rule @ Eff3 @ ( produc6440468272734011008ist_fm @ ( shd_Pr2916036590756128464ist_fm @ Steps2 ) ) @ ( produc162143410373610050ist_fm @ ( shd_Pr2916036590756128464ist_fm @ Steps2 ) ) @ Sl2 )
& ( abstra4310285689605054952m_rule @ ( abstra5667912417467126699m_rule @ Eff3 ) @ Rules @ ( stl_Pr213190007691795916ist_fm @ Steps2 ) ) ) ) ) ).
% i.epath.simps
thf(fact_230_i_Oepath_Osimps,axiom,
! [Eff3: produc340336539035504054m_rule > produc6018962875968178549ist_fm > option6967287582980624417ist_fm,Rules: stream2709947120125613254m_rule,A2: stream6742152135616053226m_rule] :
( ( abstra5381393010826267066ist_fm @ ( abstra7153543546181413245ist_fm @ Eff3 ) @ Rules @ A2 )
= ( ? [Steps2: stream6742152135616053226m_rule,Sl2: fset_P8989946509869081563ist_fm] :
( ( A2 = Steps2 )
& ( member7231649785386036813m_rule @ ( produc1227651196481849126m_rule @ ( shd_Pr6124039728035424814m_rule @ Steps2 ) ) @ ( sset_P4484857331586881186m_rule @ Rules ) )
& ( fmembe3754813877001230652ist_fm @ ( produc6387030584228546280m_rule @ ( shd_Pr6124039728035424814m_rule @ ( stl_Pr7706092684847479082m_rule @ Steps2 ) ) ) @ Sl2 )
& ( abstra7153543546181413245ist_fm @ Eff3 @ ( produc1227651196481849126m_rule @ ( shd_Pr6124039728035424814m_rule @ Steps2 ) ) @ ( produc6387030584228546280m_rule @ ( shd_Pr6124039728035424814m_rule @ Steps2 ) ) @ Sl2 )
& ( abstra5381393010826267066ist_fm @ ( abstra7153543546181413245ist_fm @ Eff3 ) @ Rules @ ( stl_Pr7706092684847479082m_rule @ Steps2 ) ) ) ) ) ).
% i.epath.simps
thf(fact_231_i_Oepath_Osimps,axiom,
! [Eff3: produc6018962875968178549ist_fm > abstra8520858721168826610m_rule > option9127550908420139634m_rule,Rules: stream4408948924543953275ist_fm,A2: stream2118054573033641736ist_fm] :
( ( abstra8533012240466880740m_rule @ ( abstra2762543317532761127m_rule @ Eff3 ) @ Rules @ A2 )
= ( ? [Steps2: stream2118054573033641736ist_fm,Sl2: fset_A8849196716589201442m_rule] :
( ( A2 = Steps2 )
& ( member4699826688122452638ist_fm @ ( produc1196760618833626364ist_fm @ ( shd_Pr1348735749115486668ist_fm @ Steps2 ) ) @ ( sset_P5379989128061332361ist_fm @ Rules ) )
& ( fmembe1099044791951439339m_rule @ ( produc7096817939255439806ist_fm @ ( shd_Pr1348735749115486668ist_fm @ ( stl_Pr1735168262451467976ist_fm @ Steps2 ) ) ) @ Sl2 )
& ( abstra2762543317532761127m_rule @ Eff3 @ ( produc1196760618833626364ist_fm @ ( shd_Pr1348735749115486668ist_fm @ Steps2 ) ) @ ( produc7096817939255439806ist_fm @ ( shd_Pr1348735749115486668ist_fm @ Steps2 ) ) @ Sl2 )
& ( abstra8533012240466880740m_rule @ ( abstra2762543317532761127m_rule @ Eff3 ) @ Rules @ ( stl_Pr1735168262451467976ist_fm @ Steps2 ) ) ) ) ) ).
% i.epath.simps
thf(fact_232_i_Oepath_Ocases,axiom,
! [Eff3: list_fm > list_tm > option_fset_list_tm,Rules: stream_list_fm,A2: stream4408948924543953275ist_fm] :
( ( abstra302294732650053493ist_tm @ ( abstra9141620886324064114ist_tm @ Eff3 ) @ Rules @ A2 )
=> ~ ( ( member_list_fm @ ( produc3242453064496107283ist_fm @ ( shd_Pr3211216682057661985ist_fm @ A2 ) ) @ ( sset_list_fm @ Rules ) )
=> ! [Sl: fset_list_tm] :
( ( fmember_list_tm @ ( produc4874987732838623697ist_fm @ ( shd_Pr3211216682057661985ist_fm @ ( stl_Pr1506262294867171877ist_fm @ A2 ) ) ) @ Sl )
=> ( ( abstra9141620886324064114ist_tm @ Eff3 @ ( produc3242453064496107283ist_fm @ ( shd_Pr3211216682057661985ist_fm @ A2 ) ) @ ( produc4874987732838623697ist_fm @ ( shd_Pr3211216682057661985ist_fm @ A2 ) ) @ Sl )
=> ~ ( abstra302294732650053493ist_tm @ ( abstra9141620886324064114ist_tm @ Eff3 ) @ Rules @ ( stl_Pr1506262294867171877ist_fm @ A2 ) ) ) ) ) ) ).
% i.epath.cases
thf(fact_233_i_Oepath_Ocases,axiom,
! [Eff3: fm > produc6018962875968178549ist_fm > option6967287582980624417ist_fm,Rules: stream_fm,A2: stream7995241112960703083_fm_fm] :
( ( abstra3580388122067503493ist_fm @ ( abstra7874326357579037314ist_fm @ Eff3 ) @ Rules @ A2 )
=> ~ ( ( member_fm @ ( produc5710729445452762371_fm_fm @ ( shd_Pr3833923502784695313_fm_fm @ A2 ) ) @ ( sset_fm @ Rules ) )
=> ! [Sl: fset_P8989946509869081563ist_fm] :
( ( fmembe3754813877001230652ist_fm @ ( produc8064570269430693313_fm_fm @ ( shd_Pr3833923502784695313_fm_fm @ ( stl_Pr2438619550280678421_fm_fm @ A2 ) ) ) @ Sl )
=> ( ( abstra7874326357579037314ist_fm @ Eff3 @ ( produc5710729445452762371_fm_fm @ ( shd_Pr3833923502784695313_fm_fm @ A2 ) ) @ ( produc8064570269430693313_fm_fm @ ( shd_Pr3833923502784695313_fm_fm @ A2 ) ) @ Sl )
=> ~ ( abstra3580388122067503493ist_fm @ ( abstra7874326357579037314ist_fm @ Eff3 ) @ Rules @ ( stl_Pr2438619550280678421_fm_fm @ A2 ) ) ) ) ) ) ).
% i.epath.cases
thf(fact_234_i_Oepath_Ocases,axiom,
! [Eff3: rule > produc6018962875968178549ist_fm > option6967287582980624417ist_fm,Rules: stream_rule,A2: stream2709947120125613254m_rule] :
( ( abstra6789711989322986974ist_fm @ ( abstra2682625350522704545ist_fm @ Eff3 ) @ Rules @ A2 )
=> ~ ( ( member_rule @ ( produc6514711208036230658m_rule @ ( shd_Pr4562317740776619530m_rule @ A2 ) ) @ ( sset_rule @ Rules ) )
=> ! [Sl: fset_P8989946509869081563ist_fm] :
( ( fmembe3754813877001230652ist_fm @ ( produc993771546023551940m_rule @ ( shd_Pr4562317740776619530m_rule @ ( stl_Pr12655793849929990m_rule @ A2 ) ) ) @ Sl )
=> ( ( abstra2682625350522704545ist_fm @ Eff3 @ ( produc6514711208036230658m_rule @ ( shd_Pr4562317740776619530m_rule @ A2 ) ) @ ( produc993771546023551940m_rule @ ( shd_Pr4562317740776619530m_rule @ A2 ) ) @ Sl )
=> ~ ( abstra6789711989322986974ist_fm @ ( abstra2682625350522704545ist_fm @ Eff3 ) @ Rules @ ( stl_Pr12655793849929990m_rule @ A2 ) ) ) ) ) ) ).
% i.epath.cases
thf(fact_235_i_Oepath_Ocases,axiom,
! [Eff3: list_fm > produc6018962875968178549ist_fm > option6967287582980624417ist_fm,Rules: stream_list_fm,A2: stream4656891395932198769ist_fm] :
( ( abstra5075300408540101887ist_fm @ ( abstra1000122179166948220ist_fm @ Eff3 ) @ Rules @ A2 )
=> ~ ( ( member_list_fm @ ( produc2977072740996481289ist_fm @ ( shd_Pr311069668031106967ist_fm @ A2 ) ) @ ( sset_list_fm @ Rules ) )
=> ! [Sl: fset_P8989946509869081563ist_fm] :
( ( fmembe3754813877001230652ist_fm @ ( produc5700618299769748167ist_fm @ ( shd_Pr311069668031106967ist_fm @ ( stl_Pr7978862689335602587ist_fm @ A2 ) ) ) @ Sl )
=> ( ( abstra1000122179166948220ist_fm @ Eff3 @ ( produc2977072740996481289ist_fm @ ( shd_Pr311069668031106967ist_fm @ A2 ) ) @ ( produc5700618299769748167ist_fm @ ( shd_Pr311069668031106967ist_fm @ A2 ) ) @ Sl )
=> ~ ( abstra5075300408540101887ist_fm @ ( abstra1000122179166948220ist_fm @ Eff3 ) @ Rules @ ( stl_Pr7978862689335602587ist_fm @ A2 ) ) ) ) ) ) ).
% i.epath.cases
thf(fact_236_i_Oepath_Ocases,axiom,
! [Eff3: fm > abstra8520858721168826610m_rule > option9127550908420139634m_rule,Rules: stream_fm,A2: stream2858270464523782ule_fm] :
( ( abstra6170950473048798434m_rule @ ( abstra7618830265118336037m_rule @ Eff3 ) @ Rules @ A2 )
=> ~ ( ( member_fm @ ( produc5983953905017532922ule_fm @ ( shd_Pr142054665586430922ule_fm @ A2 ) ) @ ( sset_fm @ Rules ) )
=> ! [Sl: fset_A8849196716589201442m_rule] :
( ( fmembe1099044791951439339m_rule @ ( produc7765541624337483452ule_fm @ ( shd_Pr142054665586430922ule_fm @ ( stl_Pr7787596663505189062ule_fm @ A2 ) ) ) @ Sl )
=> ( ( abstra7618830265118336037m_rule @ Eff3 @ ( produc5983953905017532922ule_fm @ ( shd_Pr142054665586430922ule_fm @ A2 ) ) @ ( produc7765541624337483452ule_fm @ ( shd_Pr142054665586430922ule_fm @ A2 ) ) @ Sl )
=> ~ ( abstra6170950473048798434m_rule @ ( abstra7618830265118336037m_rule @ Eff3 ) @ Rules @ ( stl_Pr7787596663505189062ule_fm @ A2 ) ) ) ) ) ) ).
% i.epath.cases
thf(fact_237_i_Oepath_Ocases,axiom,
! [Eff3: rule > abstra8520858721168826610m_rule > option9127550908420139634m_rule,Rules: stream_rule,A2: stream3393403210778678635e_rule] :
( ( abstra7067474524808760265m_rule @ ( abstra3599228160297647814m_rule @ Eff3 ) @ Rules @ A2 )
=> ~ ( ( member_rule @ ( produc9166542125499771467e_rule @ ( shd_Pr8663275782010802449e_rule @ A2 ) ) @ ( sset_rule @ Rules ) )
=> ! [Sl: fset_A8849196716589201442m_rule] :
( ( fmembe1099044791951439339m_rule @ ( produc41289011304007433e_rule @ ( shd_Pr8663275782010802449e_rule @ ( stl_Pr654857084227610901e_rule @ A2 ) ) ) @ Sl )
=> ( ( abstra3599228160297647814m_rule @ Eff3 @ ( produc9166542125499771467e_rule @ ( shd_Pr8663275782010802449e_rule @ A2 ) ) @ ( produc41289011304007433e_rule @ ( shd_Pr8663275782010802449e_rule @ A2 ) ) @ Sl )
=> ~ ( abstra7067474524808760265m_rule @ ( abstra3599228160297647814m_rule @ Eff3 ) @ Rules @ ( stl_Pr654857084227610901e_rule @ A2 ) ) ) ) ) ) ).
% i.epath.cases
thf(fact_238_i_Oepath_Ocases,axiom,
! [Eff3: produc6018962875968178549ist_fm > produc6018962875968178549ist_fm > option6967287582980624417ist_fm,Rules: stream4408948924543953275ist_fm,A2: stream1747862497754140525ist_fm] :
( ( abstra6295994116336782083ist_fm @ ( abstra2084124586061969408ist_fm @ Eff3 ) @ Rules @ A2 )
=> ~ ( ( member4699826688122452638ist_fm @ ( produc6364584182369725445ist_fm @ ( shd_Pr4098038590278241811ist_fm @ A2 ) ) @ ( sset_P5379989128061332361ist_fm @ Rules ) )
=> ! [Sl: fset_P8989946509869081563ist_fm] :
( ( fmembe3754813877001230652ist_fm @ ( produc8141473855448433859ist_fm @ ( shd_Pr4098038590278241811ist_fm @ ( stl_Pr1377318027900677655ist_fm @ A2 ) ) ) @ Sl )
=> ( ( abstra2084124586061969408ist_fm @ Eff3 @ ( produc6364584182369725445ist_fm @ ( shd_Pr4098038590278241811ist_fm @ A2 ) ) @ ( produc8141473855448433859ist_fm @ ( shd_Pr4098038590278241811ist_fm @ A2 ) ) @ Sl )
=> ~ ( abstra6295994116336782083ist_fm @ ( abstra2084124586061969408ist_fm @ Eff3 ) @ Rules @ ( stl_Pr1377318027900677655ist_fm @ A2 ) ) ) ) ) ) ).
% i.epath.cases
thf(fact_239_i_Oepath_Ocases,axiom,
! [Eff3: list_fm > abstra8520858721168826610m_rule > option9127550908420139634m_rule,Rules: stream_list_fm,A2: stream5329691398384333196ist_fm] :
( ( abstra4310285689605054952m_rule @ ( abstra5667912417467126699m_rule @ Eff3 ) @ Rules @ A2 )
=> ~ ( ( member_list_fm @ ( produc6440468272734011008ist_fm @ ( shd_Pr2916036590756128464ist_fm @ A2 ) ) @ ( sset_list_fm @ Rules ) )
=> ! [Sl: fset_A8849196716589201442m_rule] :
( ( fmembe1099044791951439339m_rule @ ( produc162143410373610050ist_fm @ ( shd_Pr2916036590756128464ist_fm @ ( stl_Pr213190007691795916ist_fm @ A2 ) ) ) @ Sl )
=> ( ( abstra5667912417467126699m_rule @ Eff3 @ ( produc6440468272734011008ist_fm @ ( shd_Pr2916036590756128464ist_fm @ A2 ) ) @ ( produc162143410373610050ist_fm @ ( shd_Pr2916036590756128464ist_fm @ A2 ) ) @ Sl )
=> ~ ( abstra4310285689605054952m_rule @ ( abstra5667912417467126699m_rule @ Eff3 ) @ Rules @ ( stl_Pr213190007691795916ist_fm @ A2 ) ) ) ) ) ) ).
% i.epath.cases
thf(fact_240_i_Oepath_Ocases,axiom,
! [Eff3: produc340336539035504054m_rule > produc6018962875968178549ist_fm > option6967287582980624417ist_fm,Rules: stream2709947120125613254m_rule,A2: stream6742152135616053226m_rule] :
( ( abstra5381393010826267066ist_fm @ ( abstra7153543546181413245ist_fm @ Eff3 ) @ Rules @ A2 )
=> ~ ( ( member7231649785386036813m_rule @ ( produc1227651196481849126m_rule @ ( shd_Pr6124039728035424814m_rule @ A2 ) ) @ ( sset_P4484857331586881186m_rule @ Rules ) )
=> ! [Sl: fset_P8989946509869081563ist_fm] :
( ( fmembe3754813877001230652ist_fm @ ( produc6387030584228546280m_rule @ ( shd_Pr6124039728035424814m_rule @ ( stl_Pr7706092684847479082m_rule @ A2 ) ) ) @ Sl )
=> ( ( abstra7153543546181413245ist_fm @ Eff3 @ ( produc1227651196481849126m_rule @ ( shd_Pr6124039728035424814m_rule @ A2 ) ) @ ( produc6387030584228546280m_rule @ ( shd_Pr6124039728035424814m_rule @ A2 ) ) @ Sl )
=> ~ ( abstra5381393010826267066ist_fm @ ( abstra7153543546181413245ist_fm @ Eff3 ) @ Rules @ ( stl_Pr7706092684847479082m_rule @ A2 ) ) ) ) ) ) ).
% i.epath.cases
thf(fact_241_i_Oepath_Ocases,axiom,
! [Eff3: produc6018962875968178549ist_fm > abstra8520858721168826610m_rule > option9127550908420139634m_rule,Rules: stream4408948924543953275ist_fm,A2: stream2118054573033641736ist_fm] :
( ( abstra8533012240466880740m_rule @ ( abstra2762543317532761127m_rule @ Eff3 ) @ Rules @ A2 )
=> ~ ( ( member4699826688122452638ist_fm @ ( produc1196760618833626364ist_fm @ ( shd_Pr1348735749115486668ist_fm @ A2 ) ) @ ( sset_P5379989128061332361ist_fm @ Rules ) )
=> ! [Sl: fset_A8849196716589201442m_rule] :
( ( fmembe1099044791951439339m_rule @ ( produc7096817939255439806ist_fm @ ( shd_Pr1348735749115486668ist_fm @ ( stl_Pr1735168262451467976ist_fm @ A2 ) ) ) @ Sl )
=> ( ( abstra2762543317532761127m_rule @ Eff3 @ ( produc1196760618833626364ist_fm @ ( shd_Pr1348735749115486668ist_fm @ A2 ) ) @ ( produc7096817939255439806ist_fm @ ( shd_Pr1348735749115486668ist_fm @ A2 ) ) @ Sl )
=> ~ ( abstra8533012240466880740m_rule @ ( abstra2762543317532761127m_rule @ Eff3 ) @ Rules @ ( stl_Pr1735168262451467976ist_fm @ A2 ) ) ) ) ) ) ).
% i.epath.cases
thf(fact_242_pre,axiom,
( list_a9038620553111681026m_rule
@ ^ [Xs2: produc340336539035504054m_rule] :
( ( produc6514711208036230658m_rule @ Xs2 )
!= betaCon )
@ pre ) ).
% pre
thf(fact_243_RuleSystem__Defs_Osaturated_Ocong,axiom,
abstra2533313685312581075ist_fm = abstra2533313685312581075ist_fm ).
% RuleSystem_Defs.saturated.cong
thf(fact_244_i_OSaturated__def,axiom,
! [Eff3: rule > produc6018962875968178549ist_fm > option6967287582980624417ist_fm,Rules: stream_rule,Steps: stream2709947120125613254m_rule] :
( ( abstra6097777249025082867ist_fm @ ( abstra2682625350522704545ist_fm @ Eff3 ) @ Rules @ Steps )
= ( ! [X4: rule] :
( ( member_rule @ X4 @ ( sset_rule @ Rules ) )
=> ( abstra2533313685312581075ist_fm @ ( abstra2682625350522704545ist_fm @ Eff3 ) @ X4 @ Steps ) ) ) ) ).
% i.Saturated_def
thf(fact_245_eff__def,axiom,
( eff
= ( ^ [R4: rule,S3: produc6018962875968178549ist_fm] :
( ^ [Y4: fset_P8989946509869081563ist_fm,Z4: fset_P8989946509869081563ist_fm] : ( Y4 = Z4 )
@ ( effect @ R4 @ S3 ) ) ) ) ).
% eff_def
thf(fact_246_RuleSystem__Defs_OSaturated__def,axiom,
( abstra6097777249025082867ist_fm
= ( ^ [Eff2: rule > produc6018962875968178549ist_fm > fset_P8989946509869081563ist_fm > $o,Rules2: stream_rule,Steps2: stream2709947120125613254m_rule] :
! [X4: rule] :
( ( member_rule @ X4 @ ( sset_rule @ Rules2 ) )
=> ( abstra2533313685312581075ist_fm @ Eff2 @ X4 @ Steps2 ) ) ) ) ).
% RuleSystem_Defs.Saturated_def
thf(fact_247_RuleSystem__Defs_Oepath_Ocong,axiom,
abstra6789711989322986974ist_fm = abstra6789711989322986974ist_fm ).
% RuleSystem_Defs.epath.cong
thf(fact_248_RuleSystem__Defs_OSaturated_Ocong,axiom,
abstra6097777249025082867ist_fm = abstra6097777249025082867ist_fm ).
% RuleSystem_Defs.Saturated.cong
thf(fact_249__092_060open_062_092_060And_062thesis_O_A_I_092_060And_062pre_Asuf_O_A_092_060lbrakk_062list__all_A_I_092_060lambda_062xs_O_Asnd_Axs_A_092_060noteq_062_ABetaCon_J_Apre_059_Aholds_A_Iis__rule_ABetaCon_J_Asuf_059_Asdrop_An_Asteps_A_061_Apre_A_064_N_Asuf_092_060rbrakk_062_A_092_060Longrightarrow_062_Athesis_J_A_092_060Longrightarrow_062_Athesis_092_060close_062,axiom,
~ ! [Pre: list_P2774625669004309958m_rule] :
( ( list_a9038620553111681026m_rule
@ ^ [Xs2: produc340336539035504054m_rule] :
( ( produc6514711208036230658m_rule @ Xs2 )
!= betaCon )
@ Pre )
=> ! [Suf: stream2709947120125613254m_rule] :
( ( linear7925859179493631106m_rule
@ ^ [Step: produc340336539035504054m_rule] :
( ( produc6514711208036230658m_rule @ Step )
= betaCon )
@ Suf )
=> ( ( sdrop_8169176516188972301m_rule @ n @ steps )
!= ( shift_2334844276748245581m_rule @ Pre @ Suf ) ) ) ) ).
% \<open>\<And>thesis. (\<And>pre suf. \<lbrakk>list_all (\<lambda>xs. snd xs \<noteq> BetaCon) pre; holds (is_rule BetaCon) suf; sdrop n steps = pre @- suf\<rbrakk> \<Longrightarrow> thesis) \<Longrightarrow> thesis\<close>
thf(fact_250__092_060open_062ev_A_Iholds_A_Iis__rule_ABetaCon_J_J_A_Isdrop_An_Asteps_J_092_060close_062,axiom,
( linear1816559245158234865m_rule
@ ( linear7925859179493631106m_rule
@ ^ [Step: produc340336539035504054m_rule] :
( ( produc6514711208036230658m_rule @ Step )
= betaCon ) )
@ ( sdrop_8169176516188972301m_rule @ n @ steps ) ) ).
% \<open>ev (holds (is_rule BetaCon)) (sdrop n steps)\<close>
thf(fact_251_saturated__effG__uu__def,axiom,
( abstra8695313511658867272ist_fm
= ( ^ [Eff4: rule > produc6018962875968178549ist_fm > option6967287582980624417ist_fm] : ( abstra2533313685312581075ist_fm @ ( abstra2682625350522704545ist_fm @ Eff4 ) ) ) ) ).
% saturated_effG_uu_def
thf(fact_252_Saturated__effG__uu__uu__def,axiom,
( abstra1395361944017196648ist_fm
= ( ^ [Eff4: rule > produc6018962875968178549ist_fm > option6967287582980624417ist_fm] : ( abstra6097777249025082867ist_fm @ ( abstra2682625350522704545ist_fm @ Eff4 ) ) ) ) ).
% Saturated_effG_uu_uu_def
thf(fact_253_holds_Oelims_I3_J,axiom,
! [X3: produc340336539035504054m_rule > $o,Xa: stream2709947120125613254m_rule] :
( ~ ( linear7925859179493631106m_rule @ X3 @ Xa )
=> ~ ( X3 @ ( shd_Pr4562317740776619530m_rule @ Xa ) ) ) ).
% holds.elims(3)
thf(fact_254_holds_Oelims_I3_J,axiom,
! [X3: rule > $o,Xa: stream_rule] :
( ~ ( linear7354821883049305950s_rule @ X3 @ Xa )
=> ~ ( X3 @ ( shd_rule @ Xa ) ) ) ).
% holds.elims(3)
thf(fact_255_holds_Oelims_I2_J,axiom,
! [X3: produc340336539035504054m_rule > $o,Xa: stream2709947120125613254m_rule] :
( ( linear7925859179493631106m_rule @ X3 @ Xa )
=> ( X3 @ ( shd_Pr4562317740776619530m_rule @ Xa ) ) ) ).
% holds.elims(2)
thf(fact_256_holds_Oelims_I2_J,axiom,
! [X3: rule > $o,Xa: stream_rule] :
( ( linear7354821883049305950s_rule @ X3 @ Xa )
=> ( X3 @ ( shd_rule @ Xa ) ) ) ).
% holds.elims(2)
thf(fact_257_holds_Oelims_I1_J,axiom,
! [X3: produc340336539035504054m_rule > $o,Xa: stream2709947120125613254m_rule,Y3: $o] :
( ( ( linear7925859179493631106m_rule @ X3 @ Xa )
= Y3 )
=> ( Y3
= ( X3 @ ( shd_Pr4562317740776619530m_rule @ Xa ) ) ) ) ).
% holds.elims(1)
thf(fact_258_holds_Oelims_I1_J,axiom,
! [X3: rule > $o,Xa: stream_rule,Y3: $o] :
( ( ( linear7354821883049305950s_rule @ X3 @ Xa )
= Y3 )
=> ( Y3
= ( X3 @ ( shd_rule @ Xa ) ) ) ) ).
% holds.elims(1)
thf(fact_259_holds_Osimps,axiom,
( linear7925859179493631106m_rule
= ( ^ [P3: produc340336539035504054m_rule > $o,Xs2: stream2709947120125613254m_rule] : ( P3 @ ( shd_Pr4562317740776619530m_rule @ Xs2 ) ) ) ) ).
% holds.simps
thf(fact_260_holds_Osimps,axiom,
( linear7354821883049305950s_rule
= ( ^ [P3: rule > $o,Xs2: stream_rule] : ( P3 @ ( shd_rule @ Xs2 ) ) ) ) ).
% holds.simps
thf(fact_261_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_262_holds__aand,axiom,
! [P: produc340336539035504054m_rule > $o,Steps: stream2709947120125613254m_rule,Q: produc340336539035504054m_rule > $o] :
( ( ( linear7925859179493631106m_rule @ P @ Steps )
& ( linear7925859179493631106m_rule @ Q @ Steps ) )
= ( linear7925859179493631106m_rule
@ ^ [Step: produc340336539035504054m_rule] :
( ( P @ Step )
& ( Q @ Step ) )
@ Steps ) ) ).
% holds_aand
thf(fact_263_holds__aand,axiom,
! [P: rule > $o,Steps: stream_rule,Q: rule > $o] :
( ( ( linear7354821883049305950s_rule @ P @ Steps )
& ( linear7354821883049305950s_rule @ Q @ Steps ) )
= ( linear7354821883049305950s_rule
@ ^ [Step: rule] :
( ( P @ Step )
& ( Q @ Step ) )
@ Steps ) ) ).
% holds_aand
thf(fact_264_ev__ev,axiom,
! [Phi: stream2709947120125613254m_rule > $o] :
( ( linear1816559245158234865m_rule @ ( linear1816559245158234865m_rule @ Phi ) )
= ( linear1816559245158234865m_rule @ Phi ) ) ).
% ev_ev
thf(fact_265_ev__ev,axiom,
! [Phi: stream_rule > $o] :
( ( linear292482573288640205v_rule @ ( linear292482573288640205v_rule @ Phi ) )
= ( linear292482573288640205v_rule @ Phi ) ) ).
% ev_ev
thf(fact_266_list__all__append,axiom,
! [P: tm > $o,Xs: list_tm,Ys: list_tm] :
( ( list_all_tm @ P @ ( append_tm @ Xs @ Ys ) )
= ( ( list_all_tm @ P @ Xs )
& ( list_all_tm @ P @ Ys ) ) ) ).
% list_all_append
thf(fact_267_list__all__append,axiom,
! [P: produc340336539035504054m_rule > $o,Xs: list_P2774625669004309958m_rule,Ys: list_P2774625669004309958m_rule] :
( ( list_a9038620553111681026m_rule @ P @ ( append2714808343334685275m_rule @ Xs @ Ys ) )
= ( ( list_a9038620553111681026m_rule @ P @ Xs )
& ( list_a9038620553111681026m_rule @ P @ Ys ) ) ) ).
% list_all_append
thf(fact_268_ev__False,axiom,
! [Omega: stream2709947120125613254m_rule] :
~ ( linear1816559245158234865m_rule
@ ^ [X4: stream2709947120125613254m_rule] : $false
@ Omega ) ).
% ev_False
thf(fact_269_ev__False,axiom,
! [Omega: stream_rule] :
~ ( linear292482573288640205v_rule
@ ^ [X4: stream_rule] : $false
@ Omega ) ).
% ev_False
thf(fact_270_ev__or,axiom,
! [Phi: stream2709947120125613254m_rule > $o,Psi: stream2709947120125613254m_rule > $o] :
( ( linear1816559245158234865m_rule
@ ^ [Xs2: stream2709947120125613254m_rule] :
( ( Phi @ Xs2 )
| ( Psi @ Xs2 ) ) )
= ( ^ [Xs2: stream2709947120125613254m_rule] :
( ( linear1816559245158234865m_rule @ Phi @ Xs2 )
| ( linear1816559245158234865m_rule @ Psi @ Xs2 ) ) ) ) ).
% ev_or
thf(fact_271_ev__or,axiom,
! [Phi: stream_rule > $o,Psi: stream_rule > $o] :
( ( linear292482573288640205v_rule
@ ^ [Xs2: stream_rule] :
( ( Phi @ Xs2 )
| ( Psi @ Xs2 ) ) )
= ( ^ [Xs2: stream_rule] :
( ( linear292482573288640205v_rule @ Phi @ Xs2 )
| ( linear292482573288640205v_rule @ Psi @ Xs2 ) ) ) ) ).
% ev_or
thf(fact_272_ev_Obase,axiom,
! [Phi: stream2709947120125613254m_rule > $o,Xs: stream2709947120125613254m_rule] :
( ( Phi @ Xs )
=> ( linear1816559245158234865m_rule @ Phi @ Xs ) ) ).
% ev.base
thf(fact_273_ev_Obase,axiom,
! [Phi: stream_rule > $o,Xs: stream_rule] :
( ( Phi @ Xs )
=> ( linear292482573288640205v_rule @ Phi @ Xs ) ) ).
% ev.base
thf(fact_274_ev__mono,axiom,
! [Phi: stream2709947120125613254m_rule > $o,Xs: stream2709947120125613254m_rule,Psi: stream2709947120125613254m_rule > $o] :
( ( linear1816559245158234865m_rule @ Phi @ Xs )
=> ( ! [Xs3: stream2709947120125613254m_rule] :
( ( Phi @ Xs3 )
=> ( Psi @ Xs3 ) )
=> ( linear1816559245158234865m_rule @ Psi @ Xs ) ) ) ).
% ev_mono
thf(fact_275_ev__mono,axiom,
! [Phi: stream_rule > $o,Xs: stream_rule,Psi: stream_rule > $o] :
( ( linear292482573288640205v_rule @ Phi @ Xs )
=> ( ! [Xs3: stream_rule] :
( ( Phi @ Xs3 )
=> ( Psi @ Xs3 ) )
=> ( linear292482573288640205v_rule @ Psi @ Xs ) ) ) ).
% ev_mono
thf(fact_276_ev__induct__strong,axiom,
! [Phi: stream2709947120125613254m_rule > $o,X3: stream2709947120125613254m_rule,P: stream2709947120125613254m_rule > $o] :
( ( linear1816559245158234865m_rule @ Phi @ X3 )
=> ( ! [Xs3: stream2709947120125613254m_rule] :
( ( Phi @ Xs3 )
=> ( P @ Xs3 ) )
=> ( ! [Xs3: stream2709947120125613254m_rule] :
( ( linear1816559245158234865m_rule @ Phi @ ( stl_Pr12655793849929990m_rule @ Xs3 ) )
=> ( ~ ( Phi @ Xs3 )
=> ( ( P @ ( stl_Pr12655793849929990m_rule @ Xs3 ) )
=> ( P @ Xs3 ) ) ) )
=> ( P @ X3 ) ) ) ) ).
% ev_induct_strong
thf(fact_277_ev__induct__strong,axiom,
! [Phi: stream_rule > $o,X3: stream_rule,P: stream_rule > $o] :
( ( linear292482573288640205v_rule @ Phi @ X3 )
=> ( ! [Xs3: stream_rule] :
( ( Phi @ Xs3 )
=> ( P @ Xs3 ) )
=> ( ! [Xs3: stream_rule] :
( ( linear292482573288640205v_rule @ Phi @ ( stl_rule @ Xs3 ) )
=> ( ~ ( Phi @ Xs3 )
=> ( ( P @ ( stl_rule @ Xs3 ) )
=> ( P @ Xs3 ) ) ) )
=> ( P @ X3 ) ) ) ) ).
% ev_induct_strong
thf(fact_278_ev_Osimps,axiom,
( linear1816559245158234865m_rule
= ( ^ [Phi2: stream2709947120125613254m_rule > $o,A5: stream2709947120125613254m_rule] :
( ? [Xs2: stream2709947120125613254m_rule] :
( ( A5 = Xs2 )
& ( Phi2 @ Xs2 ) )
| ? [Xs2: stream2709947120125613254m_rule] :
( ( A5 = Xs2 )
& ( linear1816559245158234865m_rule @ Phi2 @ ( stl_Pr12655793849929990m_rule @ Xs2 ) ) ) ) ) ) ).
% ev.simps
thf(fact_279_ev_Osimps,axiom,
( linear292482573288640205v_rule
= ( ^ [Phi2: stream_rule > $o,A5: stream_rule] :
( ? [Xs2: stream_rule] :
( ( A5 = Xs2 )
& ( Phi2 @ Xs2 ) )
| ? [Xs2: stream_rule] :
( ( A5 = Xs2 )
& ( linear292482573288640205v_rule @ Phi2 @ ( stl_rule @ Xs2 ) ) ) ) ) ) ).
% ev.simps
thf(fact_280_ev_Ocases,axiom,
! [Phi: stream2709947120125613254m_rule > $o,A2: stream2709947120125613254m_rule] :
( ( linear1816559245158234865m_rule @ Phi @ A2 )
=> ( ~ ( Phi @ A2 )
=> ( linear1816559245158234865m_rule @ Phi @ ( stl_Pr12655793849929990m_rule @ A2 ) ) ) ) ).
% ev.cases
thf(fact_281_ev_Ocases,axiom,
! [Phi: stream_rule > $o,A2: stream_rule] :
( ( linear292482573288640205v_rule @ Phi @ A2 )
=> ( ~ ( Phi @ A2 )
=> ( linear292482573288640205v_rule @ Phi @ ( stl_rule @ A2 ) ) ) ) ).
% ev.cases
thf(fact_282_ev_Ostep,axiom,
! [Phi: stream2709947120125613254m_rule > $o,Xs: stream2709947120125613254m_rule] :
( ( linear1816559245158234865m_rule @ Phi @ ( stl_Pr12655793849929990m_rule @ Xs ) )
=> ( linear1816559245158234865m_rule @ Phi @ Xs ) ) ).
% ev.step
thf(fact_283_ev_Ostep,axiom,
! [Phi: stream_rule > $o,Xs: stream_rule] :
( ( linear292482573288640205v_rule @ Phi @ ( stl_rule @ Xs ) )
=> ( linear292482573288640205v_rule @ Phi @ Xs ) ) ).
% ev.step
thf(fact_284_ev__iff__sdrop,axiom,
( linear1816559245158234865m_rule
= ( ^ [P3: stream2709947120125613254m_rule > $o,Omega2: stream2709947120125613254m_rule] :
? [M: nat] : ( P3 @ ( sdrop_8169176516188972301m_rule @ M @ Omega2 ) ) ) ) ).
% ev_iff_sdrop
thf(fact_285_ev__iff__sdrop,axiom,
( linear292482573288640205v_rule
= ( ^ [P3: stream_rule > $o,Omega2: stream_rule] :
? [M: nat] : ( P3 @ ( sdrop_rule @ M @ Omega2 ) ) ) ) ).
% ev_iff_sdrop
thf(fact_286_ev__shift,axiom,
! [Phi: stream2709947120125613254m_rule > $o,Xs: stream2709947120125613254m_rule,Xl: list_P2774625669004309958m_rule] :
( ( linear1816559245158234865m_rule @ Phi @ Xs )
=> ( linear1816559245158234865m_rule @ Phi @ ( shift_2334844276748245581m_rule @ Xl @ Xs ) ) ) ).
% ev_shift
thf(fact_287_ev__shift,axiom,
! [Phi: stream_rule > $o,Xs: stream_rule,Xl: list_rule] :
( ( linear292482573288640205v_rule @ Phi @ Xs )
=> ( linear292482573288640205v_rule @ Phi @ ( shift_rule @ Xl @ Xs ) ) ) ).
% ev_shift
thf(fact_288_ev__imp__shift,axiom,
! [Phi: stream2709947120125613254m_rule > $o,Xs: stream2709947120125613254m_rule] :
( ( linear1816559245158234865m_rule @ Phi @ Xs )
=> ? [Xl2: list_P2774625669004309958m_rule,Xs22: stream2709947120125613254m_rule] :
( ( Xs
= ( shift_2334844276748245581m_rule @ Xl2 @ Xs22 ) )
& ( Phi @ Xs22 ) ) ) ).
% ev_imp_shift
thf(fact_289_ev__imp__shift,axiom,
! [Phi: stream_rule > $o,Xs: stream_rule] :
( ( linear292482573288640205v_rule @ Phi @ Xs )
=> ? [Xl2: list_rule,Xs22: stream_rule] :
( ( Xs
= ( shift_rule @ Xl2 @ Xs22 ) )
& ( Phi @ Xs22 ) ) ) ).
% ev_imp_shift
thf(fact_290_list_Opred__True,axiom,
( ( list_a9038620553111681026m_rule
@ ^ [Uu: produc340336539035504054m_rule] : $true )
= ( ^ [Uu: list_P2774625669004309958m_rule] : $true ) ) ).
% list.pred_True
thf(fact_291_ev__inv,axiom,
! [F: stream2709947120125613254m_rule > stream2709947120125613254m_rule,P: stream2709947120125613254m_rule > $o,S: stream2709947120125613254m_rule] :
( ! [S4: stream2709947120125613254m_rule] :
( ( F @ ( stl_Pr12655793849929990m_rule @ S4 ) )
= ( stl_Pr12655793849929990m_rule @ ( F @ S4 ) ) )
=> ( ( linear1816559245158234865m_rule @ P @ ( F @ S ) )
= ( linear1816559245158234865m_rule
@ ^ [X4: stream2709947120125613254m_rule] : ( P @ ( F @ X4 ) )
@ S ) ) ) ).
% ev_inv
thf(fact_292_ev__inv,axiom,
! [F: stream_rule > stream2709947120125613254m_rule,P: stream2709947120125613254m_rule > $o,S: stream_rule] :
( ! [S4: stream_rule] :
( ( F @ ( stl_rule @ S4 ) )
= ( stl_Pr12655793849929990m_rule @ ( F @ S4 ) ) )
=> ( ( linear1816559245158234865m_rule @ P @ ( F @ S ) )
= ( linear292482573288640205v_rule
@ ^ [X4: stream_rule] : ( P @ ( F @ X4 ) )
@ S ) ) ) ).
% ev_inv
thf(fact_293_ev__inv,axiom,
! [F: stream2709947120125613254m_rule > stream_rule,P: stream_rule > $o,S: stream2709947120125613254m_rule] :
( ! [S4: stream2709947120125613254m_rule] :
( ( F @ ( stl_Pr12655793849929990m_rule @ S4 ) )
= ( stl_rule @ ( F @ S4 ) ) )
=> ( ( linear292482573288640205v_rule @ P @ ( F @ S ) )
= ( linear1816559245158234865m_rule
@ ^ [X4: stream2709947120125613254m_rule] : ( P @ ( F @ X4 ) )
@ S ) ) ) ).
% ev_inv
thf(fact_294_ev__inv,axiom,
! [F: stream_rule > stream_rule,P: stream_rule > $o,S: stream_rule] :
( ! [S4: stream_rule] :
( ( F @ ( stl_rule @ S4 ) )
= ( stl_rule @ ( F @ S4 ) ) )
=> ( ( linear292482573288640205v_rule @ P @ ( F @ S ) )
= ( linear292482573288640205v_rule
@ ^ [X4: stream_rule] : ( P @ ( F @ X4 ) )
@ S ) ) ) ).
% ev_inv
thf(fact_295_ev__prefix,axiom,
! [P: produc340336539035504054m_rule > $o,Xs: stream2709947120125613254m_rule] :
( ( linear1816559245158234865m_rule @ ( linear7925859179493631106m_rule @ P ) @ Xs )
=> ? [Pre: list_P2774625669004309958m_rule,Suf: stream2709947120125613254m_rule] :
( ( list_a9038620553111681026m_rule
@ ^ [Xs2: produc340336539035504054m_rule] :
~ ( P @ Xs2 )
@ Pre )
& ( linear7925859179493631106m_rule @ P @ Suf )
& ( Xs
= ( shift_2334844276748245581m_rule @ Pre @ Suf ) ) ) ) ).
% ev_prefix
thf(fact_296_ev__prefix,axiom,
! [P: rule > $o,Xs: stream_rule] :
( ( linear292482573288640205v_rule @ ( linear7354821883049305950s_rule @ P ) @ Xs )
=> ? [Pre: list_rule,Suf: stream_rule] :
( ( list_all_rule
@ ^ [Xs2: rule] :
~ ( P @ Xs2 )
@ Pre )
& ( linear7354821883049305950s_rule @ P @ Suf )
& ( Xs
= ( shift_rule @ Pre @ Suf ) ) ) ) ).
% ev_prefix
thf(fact_297_ev__holds__sset,axiom,
! [P: produc340336539035504054m_rule > $o,Xs: stream2709947120125613254m_rule] :
( ( linear1816559245158234865m_rule @ ( linear7925859179493631106m_rule @ P ) @ Xs )
= ( ? [X4: produc340336539035504054m_rule] :
( ( member7231649785386036813m_rule @ X4 @ ( sset_P4484857331586881186m_rule @ Xs ) )
& ( P @ X4 ) ) ) ) ).
% ev_holds_sset
thf(fact_298_ev__holds__sset,axiom,
! [P: rule > $o,Xs: stream_rule] :
( ( linear292482573288640205v_rule @ ( linear7354821883049305950s_rule @ P ) @ Xs )
= ( ? [X4: rule] :
( ( member_rule @ X4 @ ( sset_rule @ Xs ) )
& ( P @ X4 ) ) ) ) ).
% ev_holds_sset
thf(fact_299_holds__mono,axiom,
! [P: produc340336539035504054m_rule > $o,Xs: stream2709947120125613254m_rule,Q: produc340336539035504054m_rule > $o] :
( ( linear7925859179493631106m_rule @ P @ Xs )
=> ( ! [X: produc340336539035504054m_rule] :
( ( P @ X )
=> ( Q @ X ) )
=> ( linear7925859179493631106m_rule @ Q @ Xs ) ) ) ).
% holds_mono
thf(fact_300_holds__mono,axiom,
! [P: rule > $o,Xs: stream_rule,Q: rule > $o] :
( ( linear7354821883049305950s_rule @ P @ Xs )
=> ( ! [X: rule] :
( ( P @ X )
=> ( Q @ X ) )
=> ( linear7354821883049305950s_rule @ Q @ Xs ) ) ) ).
% holds_mono
thf(fact_301_stream__all__shift,axiom,
! [P: produc340336539035504054m_rule > $o,Xs: list_P2774625669004309958m_rule,S: stream2709947120125613254m_rule] :
( ( stream2134157564947672044m_rule @ P @ ( shift_2334844276748245581m_rule @ Xs @ S ) )
= ( ( list_a9038620553111681026m_rule @ P @ Xs )
& ( stream2134157564947672044m_rule @ P @ S ) ) ) ).
% stream_all_shift
thf(fact_302_epath__preserves__unaffected,axiom,
! [P2: fm,Steps: stream2709947120125613254m_rule,Pre2: list_P2774625669004309958m_rule,Suf2: stream2709947120125613254m_rule] :
( ( member_fm @ P2 @ ( set_fm2 @ ( pseq @ ( shd_Pr4562317740776619530m_rule @ Steps ) ) ) )
=> ( ( abstra6789711989322986974ist_fm @ eff @ rules @ Steps )
=> ( ( Steps
= ( shift_2334844276748245581m_rule @ Pre2 @ Suf2 ) )
=> ( ( list_a9038620553111681026m_rule
@ ^ [Xs2: produc340336539035504054m_rule] :
~ ( affects @ ( produc6514711208036230658m_rule @ Xs2 ) @ P2 )
@ Pre2 )
=> ( member_fm @ P2 @ ( set_fm2 @ ( pseq @ ( shd_Pr4562317740776619530m_rule @ Suf2 ) ) ) ) ) ) ) ) ).
% epath_preserves_unaffected
thf(fact_303_ev__prefix__sdrop,axiom,
! [P: produc340336539035504054m_rule > $o,Xs: stream2709947120125613254m_rule] :
( ( linear1816559245158234865m_rule @ ( linear7925859179493631106m_rule @ P ) @ Xs )
=> ? [N2: nat] :
( ( list_a9038620553111681026m_rule
@ ^ [Xs2: produc340336539035504054m_rule] :
~ ( P @ Xs2 )
@ ( stake_5421812949518764133m_rule @ N2 @ Xs ) )
& ( linear7925859179493631106m_rule @ P @ ( sdrop_8169176516188972301m_rule @ N2 @ Xs ) ) ) ) ).
% ev_prefix_sdrop
thf(fact_304_ev__prefix__sdrop,axiom,
! [P: rule > $o,Xs: stream_rule] :
( ( linear292482573288640205v_rule @ ( linear7354821883049305950s_rule @ P ) @ Xs )
=> ? [N2: nat] :
( ( list_all_rule
@ ^ [Xs2: rule] :
~ ( P @ Xs2 )
@ ( stake_rule @ N2 @ Xs ) )
& ( linear7354821883049305950s_rule @ P @ ( sdrop_rule @ N2 @ Xs ) ) ) ) ).
% ev_prefix_sdrop
thf(fact_305_sdrop__wait,axiom,
! [Phi: stream2709947120125613254m_rule > $o,Xs: stream2709947120125613254m_rule] :
( ( linear1816559245158234865m_rule @ Phi @ Xs )
=> ( Phi @ ( sdrop_8169176516188972301m_rule @ ( linear8417479883454578709m_rule @ Phi @ Xs ) @ Xs ) ) ) ).
% sdrop_wait
thf(fact_306_sdrop__wait,axiom,
! [Phi: stream_rule > $o,Xs: stream_rule] :
( ( linear292482573288640205v_rule @ Phi @ Xs )
=> ( Phi @ ( sdrop_rule @ ( linear2710171048243638257t_rule @ Phi @ Xs ) @ Xs ) ) ) ).
% sdrop_wait
thf(fact_307_eff__S,axiom,
! [S: produc6018962875968178549ist_fm,R: rule,Sl4: fset_P8989946509869081563ist_fm,S5: produc6018962875968178549ist_fm] :
( ( member4699826688122452638ist_fm @ S @ top_to1730629564055774885ist_fm )
=> ( ( member_rule @ R @ ( sset_rule @ rules ) )
=> ( ( eff @ R @ S @ Sl4 )
=> ( ( fmembe3754813877001230652ist_fm @ S5 @ Sl4 )
=> ( member4699826688122452638ist_fm @ S5 @ top_to1730629564055774885ist_fm ) ) ) ) ) ).
% eff_S
thf(fact_308_prod_Oswap__def,axiom,
( produc3209917318656576745ist_tm
= ( ^ [P4: produc3002719820330532825ist_tm] : ( produc1414352766439514085ist_fm @ ( produc472676704269606519ist_tm @ P4 ) @ ( produc2105211372612122933ist_tm @ P4 ) ) ) ) ).
% prod.swap_def
thf(fact_309_prod_Oswap__def,axiom,
( produc5647469659010186090ist_fm
= ( ^ [P4: produc2353945479044264792ist_fm] : ( produc1733806532565653680m_rule @ ( produc9041034967185247836ist_fm @ P4 ) @ ( produc3520095305172569118ist_fm @ P4 ) ) ) ) ).
% prod.swap_def
thf(fact_310_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_311_prod_Oswap__def,axiom,
( produc3121145899861168912m_rule
= ( ^ [P4: produc340336539035504054m_rule] : ( produc4260130291714670858ist_fm @ ( produc6514711208036230658m_rule @ P4 ) @ ( produc993771546023551940m_rule @ P4 ) ) ) ) ).
% prod.swap_def
thf(fact_312_prod_Oswap__def,axiom,
( produc5979693678883077509ist_fm
= ( ^ [P4: produc6018962875968178549ist_fm] : ( produc7867948443067789129ist_tm @ ( produc3242453064496107283ist_fm @ P4 ) @ ( produc4874987732838623697ist_fm @ P4 ) ) ) ) ).
% prod.swap_def
thf(fact_313_pred__equals__eq2,axiom,
! [R3: set_Pr5202636777678657877ist_fm,S6: set_Pr5202636777678657877ist_fm] :
( ( ( ^ [X4: list_tm,Y5: list_fm] : ( member4699826688122452638ist_fm @ ( produc1414352766439514085ist_fm @ X4 @ Y5 ) @ R3 ) )
= ( ^ [X4: list_tm,Y5: list_fm] : ( member4699826688122452638ist_fm @ ( produc1414352766439514085ist_fm @ X4 @ Y5 ) @ S6 ) ) )
= ( R3 = S6 ) ) ).
% pred_equals_eq2
thf(fact_314_pred__equals__eq2,axiom,
! [R3: set_Pr1822751329126368876m_rule,S6: set_Pr1822751329126368876m_rule] :
( ( ( ^ [X4: produc6018962875968178549ist_fm,Y5: rule] : ( member7231649785386036813m_rule @ ( produc1733806532565653680m_rule @ X4 @ Y5 ) @ R3 ) )
= ( ^ [X4: produc6018962875968178549ist_fm,Y5: rule] : ( member7231649785386036813m_rule @ ( produc1733806532565653680m_rule @ X4 @ Y5 ) @ S6 ) ) )
= ( R3 = S6 ) ) ).
% pred_equals_eq2
thf(fact_315_pred__equals__eq2,axiom,
! [R3: set_Pr1261947904930325089at_nat,S6: set_Pr1261947904930325089at_nat] :
( ( ( ^ [X4: nat,Y5: nat] : ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X4 @ Y5 ) @ R3 ) )
= ( ^ [X4: nat,Y5: nat] : ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X4 @ Y5 ) @ S6 ) ) )
= ( R3 = S6 ) ) ).
% pred_equals_eq2
thf(fact_316_set__remdups,axiom,
! [Xs: list_fm] :
( ( set_fm2 @ ( remdups_fm @ Xs ) )
= ( set_fm2 @ Xs ) ) ).
% set_remdups
thf(fact_317_set__remdups,axiom,
! [Xs: list_tm] :
( ( set_tm2 @ ( remdups_tm @ Xs ) )
= ( set_tm2 @ Xs ) ) ).
% set_remdups
thf(fact_318_set__remdups,axiom,
! [Xs: list_list_fm] :
( ( set_list_fm2 @ ( remdups_list_fm @ Xs ) )
= ( set_list_fm2 @ Xs ) ) ).
% set_remdups
thf(fact_319_swap__simp,axiom,
! [X3: list_fm,Y3: list_tm] :
( ( produc3209917318656576745ist_tm @ ( produc7867948443067789129ist_tm @ X3 @ Y3 ) )
= ( produc1414352766439514085ist_fm @ Y3 @ X3 ) ) ).
% swap_simp
thf(fact_320_swap__simp,axiom,
! [X3: rule,Y3: produc6018962875968178549ist_fm] :
( ( produc5647469659010186090ist_fm @ ( produc4260130291714670858ist_fm @ X3 @ Y3 ) )
= ( produc1733806532565653680m_rule @ Y3 @ X3 ) ) ).
% swap_simp
thf(fact_321_swap__simp,axiom,
! [X3: list_tm,Y3: list_fm] :
( ( produc5979693678883077509ist_fm @ ( produc1414352766439514085ist_fm @ X3 @ Y3 ) )
= ( produc7867948443067789129ist_tm @ Y3 @ X3 ) ) ).
% swap_simp
thf(fact_322_swap__simp,axiom,
! [X3: produc6018962875968178549ist_fm,Y3: rule] :
( ( produc3121145899861168912m_rule @ ( produc1733806532565653680m_rule @ X3 @ Y3 ) )
= ( produc4260130291714670858ist_fm @ Y3 @ X3 ) ) ).
% swap_simp
thf(fact_323_swap__simp,axiom,
! [X3: nat,Y3: nat] :
( ( product_swap_nat_nat @ ( product_Pair_nat_nat @ X3 @ Y3 ) )
= ( product_Pair_nat_nat @ Y3 @ X3 ) ) ).
% swap_simp
thf(fact_324_fst__swap,axiom,
! [X3: produc2353945479044264792ist_fm] :
( ( produc993771546023551940m_rule @ ( produc5647469659010186090ist_fm @ X3 ) )
= ( produc9041034967185247836ist_fm @ X3 ) ) ).
% fst_swap
thf(fact_325_fst__swap,axiom,
! [X3: produc3002719820330532825ist_tm] :
( ( produc4874987732838623697ist_fm @ ( produc3209917318656576745ist_tm @ X3 ) )
= ( produc472676704269606519ist_tm @ X3 ) ) ).
% fst_swap
thf(fact_326_fst__swap,axiom,
! [X3: produc340336539035504054m_rule] :
( ( produc3520095305172569118ist_fm @ ( produc3121145899861168912m_rule @ X3 ) )
= ( produc6514711208036230658m_rule @ X3 ) ) ).
% fst_swap
thf(fact_327_fst__swap,axiom,
! [X3: produc6018962875968178549ist_fm] :
( ( produc2105211372612122933ist_tm @ ( produc5979693678883077509ist_fm @ X3 ) )
= ( produc3242453064496107283ist_fm @ X3 ) ) ).
% fst_swap
thf(fact_328_snd__swap,axiom,
! [X3: produc340336539035504054m_rule] :
( ( produc9041034967185247836ist_fm @ ( produc3121145899861168912m_rule @ X3 ) )
= ( produc993771546023551940m_rule @ X3 ) ) ).
% snd_swap
thf(fact_329_snd__swap,axiom,
! [X3: produc6018962875968178549ist_fm] :
( ( produc472676704269606519ist_tm @ ( produc5979693678883077509ist_fm @ X3 ) )
= ( produc4874987732838623697ist_fm @ X3 ) ) ).
% snd_swap
thf(fact_330_snd__swap,axiom,
! [X3: produc2353945479044264792ist_fm] :
( ( produc6514711208036230658m_rule @ ( produc5647469659010186090ist_fm @ X3 ) )
= ( produc3520095305172569118ist_fm @ X3 ) ) ).
% snd_swap
thf(fact_331_snd__swap,axiom,
! [X3: produc3002719820330532825ist_tm] :
( ( produc3242453064496107283ist_fm @ ( produc3209917318656576745ist_tm @ X3 ) )
= ( produc2105211372612122933ist_tm @ X3 ) ) ).
% snd_swap
thf(fact_332_list_Opred__mono__strong,axiom,
! [P: rule > $o,X3: list_rule,Pa: rule > $o] :
( ( list_all_rule @ P @ X3 )
=> ( ! [Z6: rule] :
( ( member_rule @ Z6 @ ( set_rule2 @ X3 ) )
=> ( ( P @ Z6 )
=> ( Pa @ Z6 ) ) )
=> ( list_all_rule @ Pa @ X3 ) ) ) ).
% list.pred_mono_strong
thf(fact_333_list_Opred__mono__strong,axiom,
! [P: produc6018962875968178549ist_fm > $o,X3: list_P2887561121880082555ist_fm,Pa: produc6018962875968178549ist_fm > $o] :
( ( list_a5314861712561513001ist_fm @ P @ X3 )
=> ( ! [Z6: produc6018962875968178549ist_fm] :
( ( member4699826688122452638ist_fm @ Z6 @ ( set_Pr5051287786238754058ist_fm @ X3 ) )
=> ( ( P @ Z6 )
=> ( Pa @ Z6 ) ) )
=> ( list_a5314861712561513001ist_fm @ Pa @ X3 ) ) ) ).
% list.pred_mono_strong
thf(fact_334_list_Opred__mono__strong,axiom,
! [P: fm > $o,X3: list_fm,Pa: fm > $o] :
( ( list_all_fm @ P @ X3 )
=> ( ! [Z6: fm] :
( ( member_fm @ Z6 @ ( set_fm2 @ X3 ) )
=> ( ( P @ Z6 )
=> ( Pa @ Z6 ) ) )
=> ( list_all_fm @ Pa @ X3 ) ) ) ).
% list.pred_mono_strong
thf(fact_335_list_Opred__mono__strong,axiom,
! [P: tm > $o,X3: list_tm,Pa: tm > $o] :
( ( list_all_tm @ P @ X3 )
=> ( ! [Z6: tm] :
( ( member_tm @ Z6 @ ( set_tm2 @ X3 ) )
=> ( ( P @ Z6 )
=> ( Pa @ Z6 ) ) )
=> ( list_all_tm @ Pa @ X3 ) ) ) ).
% list.pred_mono_strong
thf(fact_336_list_Opred__mono__strong,axiom,
! [P: list_fm > $o,X3: list_list_fm,Pa: list_fm > $o] :
( ( list_all_list_fm @ P @ X3 )
=> ( ! [Z6: list_fm] :
( ( member_list_fm @ Z6 @ ( set_list_fm2 @ X3 ) )
=> ( ( P @ Z6 )
=> ( Pa @ Z6 ) ) )
=> ( list_all_list_fm @ Pa @ X3 ) ) ) ).
% list.pred_mono_strong
thf(fact_337_list_Opred__mono__strong,axiom,
! [P: produc340336539035504054m_rule > $o,X3: list_P2774625669004309958m_rule,Pa: produc340336539035504054m_rule > $o] :
( ( list_a9038620553111681026m_rule @ P @ X3 )
=> ( ! [Z6: produc340336539035504054m_rule] :
( ( member7231649785386036813m_rule @ Z6 @ ( set_Pr4534715572506550497m_rule @ X3 ) )
=> ( ( P @ Z6 )
=> ( Pa @ Z6 ) ) )
=> ( list_a9038620553111681026m_rule @ Pa @ X3 ) ) ) ).
% list.pred_mono_strong
thf(fact_338_list__all__cong,axiom,
! [X3: list_rule,Ya: list_rule,P: rule > $o,Pa: rule > $o] :
( ( X3 = Ya )
=> ( ! [Z6: rule] :
( ( member_rule @ Z6 @ ( set_rule2 @ Ya ) )
=> ( ( P @ Z6 )
= ( Pa @ Z6 ) ) )
=> ( ( list_all_rule @ P @ X3 )
= ( list_all_rule @ Pa @ Ya ) ) ) ) ).
% list_all_cong
thf(fact_339_list__all__cong,axiom,
! [X3: list_P2887561121880082555ist_fm,Ya: list_P2887561121880082555ist_fm,P: produc6018962875968178549ist_fm > $o,Pa: produc6018962875968178549ist_fm > $o] :
( ( X3 = Ya )
=> ( ! [Z6: produc6018962875968178549ist_fm] :
( ( member4699826688122452638ist_fm @ Z6 @ ( set_Pr5051287786238754058ist_fm @ Ya ) )
=> ( ( P @ Z6 )
= ( Pa @ Z6 ) ) )
=> ( ( list_a5314861712561513001ist_fm @ P @ X3 )
= ( list_a5314861712561513001ist_fm @ Pa @ Ya ) ) ) ) ).
% list_all_cong
thf(fact_340_list__all__cong,axiom,
! [X3: list_fm,Ya: list_fm,P: fm > $o,Pa: fm > $o] :
( ( X3 = Ya )
=> ( ! [Z6: fm] :
( ( member_fm @ Z6 @ ( set_fm2 @ Ya ) )
=> ( ( P @ Z6 )
= ( Pa @ Z6 ) ) )
=> ( ( list_all_fm @ P @ X3 )
= ( list_all_fm @ Pa @ Ya ) ) ) ) ).
% list_all_cong
thf(fact_341_list__all__cong,axiom,
! [X3: list_tm,Ya: list_tm,P: tm > $o,Pa: tm > $o] :
( ( X3 = Ya )
=> ( ! [Z6: tm] :
( ( member_tm @ Z6 @ ( set_tm2 @ Ya ) )
=> ( ( P @ Z6 )
= ( Pa @ Z6 ) ) )
=> ( ( list_all_tm @ P @ X3 )
= ( list_all_tm @ Pa @ Ya ) ) ) ) ).
% list_all_cong
thf(fact_342_list__all__cong,axiom,
! [X3: list_list_fm,Ya: list_list_fm,P: list_fm > $o,Pa: list_fm > $o] :
( ( X3 = Ya )
=> ( ! [Z6: list_fm] :
( ( member_list_fm @ Z6 @ ( set_list_fm2 @ Ya ) )
=> ( ( P @ Z6 )
= ( Pa @ Z6 ) ) )
=> ( ( list_all_list_fm @ P @ X3 )
= ( list_all_list_fm @ Pa @ Ya ) ) ) ) ).
% list_all_cong
thf(fact_343_list__all__cong,axiom,
! [X3: list_P2774625669004309958m_rule,Ya: list_P2774625669004309958m_rule,P: produc340336539035504054m_rule > $o,Pa: produc340336539035504054m_rule > $o] :
( ( X3 = Ya )
=> ( ! [Z6: produc340336539035504054m_rule] :
( ( member7231649785386036813m_rule @ Z6 @ ( set_Pr4534715572506550497m_rule @ Ya ) )
=> ( ( P @ Z6 )
= ( Pa @ Z6 ) ) )
=> ( ( list_a9038620553111681026m_rule @ P @ X3 )
= ( list_a9038620553111681026m_rule @ Pa @ Ya ) ) ) ) ).
% list_all_cong
thf(fact_344_stake__sdrop,axiom,
! [N: nat,S: stream_rule] :
( ( shift_rule @ ( stake_rule @ N @ S ) @ ( sdrop_rule @ N @ S ) )
= S ) ).
% stake_sdrop
thf(fact_345_stake__sdrop,axiom,
! [N: nat,S: stream2709947120125613254m_rule] :
( ( shift_2334844276748245581m_rule @ ( stake_5421812949518764133m_rule @ N @ S ) @ ( sdrop_8169176516188972301m_rule @ N @ S ) )
= S ) ).
% stake_sdrop
thf(fact_346_enabled__R,axiom,
! [S: produc6018962875968178549ist_fm] :
( ( member4699826688122452638ist_fm @ S @ top_to1730629564055774885ist_fm )
=> ? [X: rule] :
( ( member_rule @ X @ ( sset_rule @ rules ) )
& ? [X_1: fset_P8989946509869081563ist_fm] : ( eff @ X @ S @ X_1 ) ) ) ).
% enabled_R
thf(fact_347_effect__preserves__unaffected,axiom,
! [P2: fm,Z: list_fm,R: rule,B4: list_tm,Z3: list_fm,A: list_tm] :
( ( member_fm @ P2 @ ( set_fm2 @ Z ) )
=> ( ~ ( affects @ R @ P2 )
=> ( ( fmembe3754813877001230652ist_fm @ ( produc1414352766439514085ist_fm @ B4 @ Z3 ) @ ( effect @ R @ ( produc1414352766439514085ist_fm @ A @ Z ) ) )
=> ( member_fm @ P2 @ ( set_fm2 @ Z3 ) ) ) ) ) ).
% effect_preserves_unaffected
thf(fact_348_n,axiom,
member_fm @ ( con @ p @ q ) @ ( set_fm2 @ ( pseq @ ( shd_Pr4562317740776619530m_rule @ ( sdrop_8169176516188972301m_rule @ n @ steps ) ) ) ) ).
% n
thf(fact_349_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_350_per,axiom,
! [R: rule] :
( ( member_rule @ R @ ( sset_rule @ rules ) )
=> ( abstra5255361903751151037ist_fm @ eff @ rules @ top_to1730629564055774885ist_fm @ R ) ) ).
% per
thf(fact_351_pseq__in__tree__fms,axiom,
! [X3: produc340336539035504054m_rule,Steps: stream2709947120125613254m_rule,P2: fm] :
( ( member7231649785386036813m_rule @ X3 @ ( sset_P4484857331586881186m_rule @ Steps ) )
=> ( ( member_fm @ P2 @ ( set_fm2 @ ( pseq @ X3 ) ) )
=> ( member_fm @ P2 @ ( tree_fms @ Steps ) ) ) ) ).
% pseq_in_tree_fms
thf(fact_352_PersistentRuleSystem__axioms,axiom,
abstra3967921542344661089ist_fm @ eff @ rules @ top_to1730629564055774885ist_fm ).
% PersistentRuleSystem_axioms
thf(fact_353_lexord__same__pref__iff,axiom,
! [Xs: list_fm,Ys: list_fm,Zs: list_fm,R: set_Pr4706815898642364871_fm_fm] :
( ( member1926098302810316688ist_fm @ ( produc7863996417982153943ist_fm @ ( append_fm @ Xs @ Ys ) @ ( append_fm @ Xs @ Zs ) ) @ ( lexord_fm @ R ) )
= ( ? [X4: fm] :
( ( member_fm @ X4 @ ( set_fm2 @ Xs ) )
& ( member7780952600467998736_fm_fm @ ( product_Pair_fm_fm @ X4 @ X4 ) @ R ) )
| ( member1926098302810316688ist_fm @ ( produc7863996417982153943ist_fm @ Ys @ Zs ) @ ( lexord_fm @ R ) ) ) ) ).
% lexord_same_pref_iff
thf(fact_354_lexord__same__pref__iff,axiom,
! [Xs: list_tm,Ys: list_tm,Zs: list_tm,R: set_Pr2455929065695642951_tm_tm] :
( ( member4457312017796942864ist_tm @ ( produc1418304791525149271ist_tm @ ( append_tm @ Xs @ Ys ) @ ( append_tm @ Xs @ Zs ) ) @ ( lexord_tm @ R ) )
= ( ? [X4: tm] :
( ( member_tm @ X4 @ ( set_tm2 @ Xs ) )
& ( member3121616906494481296_tm_tm @ ( product_Pair_tm_tm @ X4 @ X4 ) @ R ) )
| ( member4457312017796942864ist_tm @ ( produc1418304791525149271ist_tm @ Ys @ Zs ) @ ( lexord_tm @ R ) ) ) ) ).
% lexord_same_pref_iff
thf(fact_355_lexord__same__pref__iff,axiom,
! [Xs: list_list_fm,Ys: list_list_fm,Zs: list_list_fm,R: set_Pr1916137733696349511ist_fm] :
( ( member5102842307574975760ist_fm @ ( produc8321651870839017815ist_fm @ ( append_list_fm @ Xs @ Ys ) @ ( append_list_fm @ Xs @ Zs ) ) @ ( lexord_list_fm @ R ) )
= ( ? [X4: list_fm] :
( ( member_list_fm @ X4 @ ( set_list_fm2 @ Xs ) )
& ( member1926098302810316688ist_fm @ ( produc7863996417982153943ist_fm @ X4 @ X4 ) @ R ) )
| ( member5102842307574975760ist_fm @ ( produc8321651870839017815ist_fm @ Ys @ Zs ) @ ( lexord_list_fm @ R ) ) ) ) ).
% lexord_same_pref_iff
thf(fact_356_lexord__same__pref__iff,axiom,
! [Xs: list_nat,Ys: list_nat,Zs: list_nat,R: set_Pr1261947904930325089at_nat] :
( ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ ( append_nat @ Xs @ Ys ) @ ( append_nat @ Xs @ Zs ) ) @ ( lexord_nat @ R ) )
= ( ? [X4: nat] :
( ( member_nat @ X4 @ ( set_nat2 @ Xs ) )
& ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X4 @ X4 ) @ R ) )
| ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ Ys @ Zs ) @ ( lexord_nat @ R ) ) ) ) ).
% lexord_same_pref_iff
thf(fact_357_UNIV__def,axiom,
( top_to1730629564055774885ist_fm
= ( collec5845330005437105888ist_fm
@ ^ [X4: produc6018962875968178549ist_fm] : $true ) ) ).
% UNIV_def
thf(fact_358_szip_Ocode,axiom,
( szip_list_tm_list_fm
= ( ^ [S12: stream_list_tm,S22: stream_list_fm] : ( sCons_2518607988952603276ist_fm @ ( produc1414352766439514085ist_fm @ ( shd_list_tm @ S12 ) @ ( shd_list_fm @ S22 ) ) @ ( szip_list_tm_list_fm @ ( stl_list_tm @ S12 ) @ ( stl_list_fm @ S22 ) ) ) ) ) ).
% szip.code
thf(fact_359_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_360_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_361_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_362_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_363_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_364_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_365__092_060open_062Con_Ap_Aq_A_092_060in_062_Atree__fms_Asteps_092_060close_062,axiom,
member_fm @ ( con @ p @ q ) @ ( tree_fms @ steps ) ).
% \<open>Con p q \<in> tree_fms steps\<close>
thf(fact_366__092_060open_062_092_060And_062r_O_Aaffects_Ar_A_ICon_Ap_Aq_J_A_061_A_Ir_A_061_ABetaCon_J_092_060close_062,axiom,
! [R: rule] :
( ( affects @ R @ ( con @ p @ q ) )
= ( R = betaCon ) ) ).
% \<open>\<And>r. affects r (Con p q) = (r = BetaCon)\<close>
thf(fact_367_calculation,axiom,
member_fm @ ( con @ p @ q ) @ ( set_fm2 @ ( pseq @ ( shd_Pr4562317740776619530m_rule @ suf ) ) ) ).
% calculation
thf(fact_368_stream_Ocollapse,axiom,
! [Stream: stream2709947120125613254m_rule] :
( ( sCons_2955607467875998431m_rule @ ( shd_Pr4562317740776619530m_rule @ Stream ) @ ( stl_Pr12655793849929990m_rule @ Stream ) )
= Stream ) ).
% stream.collapse
thf(fact_369_stream_Ocollapse,axiom,
! [Stream: stream_rule] :
( ( sCons_rule @ ( shd_rule @ Stream ) @ ( stl_rule @ Stream ) )
= Stream ) ).
% stream.collapse
thf(fact_370__092_060open_062_092_060And_062thesis_O_A_I_092_060And_062n_O_ACon_Ap_Aq_A_092_060in_062_Aset_A_Ipseq_A_Ishd_A_Isdrop_An_Asteps_J_J_J_A_092_060Longrightarrow_062_Athesis_J_A_092_060Longrightarrow_062_Athesis_092_060close_062,axiom,
~ ! [N2: nat] :
~ ( member_fm @ ( con @ p @ q ) @ ( set_fm2 @ ( pseq @ ( shd_Pr4562317740776619530m_rule @ ( sdrop_8169176516188972301m_rule @ N2 @ steps ) ) ) ) ) ).
% \<open>\<And>thesis. (\<And>n. Con p q \<in> set (pseq (shd (sdrop n steps))) \<Longrightarrow> thesis) \<Longrightarrow> thesis\<close>
thf(fact_371__092_060open_062list__all_A_I_092_060lambda_062xs_O_A_092_060not_062_Aaffects_A_Isnd_Axs_J_A_ICon_Ap_Aq_J_J_Apre_092_060close_062,axiom,
( list_a9038620553111681026m_rule
@ ^ [Xs2: produc340336539035504054m_rule] :
~ ( affects @ ( produc6514711208036230658m_rule @ Xs2 ) @ ( con @ p @ q ) )
@ pre ) ).
% \<open>list_all (\<lambda>xs. \<not> affects (snd xs) (Con p q)) pre\<close>
thf(fact_372_RuleSystem_Oper_Ocong,axiom,
abstra5255361903751151037ist_fm = abstra5255361903751151037ist_fm ).
% RuleSystem.per.cong
thf(fact_373_szip__unfold,axiom,
! [A2: list_tm,S1: stream_list_tm,B2: list_fm,S2: stream_list_fm] :
( ( szip_list_tm_list_fm @ ( sCons_list_tm @ A2 @ S1 ) @ ( sCons_list_fm @ B2 @ S2 ) )
= ( sCons_2518607988952603276ist_fm @ ( produc1414352766439514085ist_fm @ A2 @ B2 ) @ ( szip_list_tm_list_fm @ S1 @ S2 ) ) ) ).
% szip_unfold
thf(fact_374_szip__unfold,axiom,
! [A2: produc6018962875968178549ist_fm,S1: stream4408948924543953275ist_fm,B2: rule,S2: stream_rule] :
( ( szip_P2924820683901490861m_rule @ ( sCons_2518607988952603276ist_fm @ A2 @ S1 ) @ ( sCons_rule @ B2 @ S2 ) )
= ( sCons_2955607467875998431m_rule @ ( produc1733806532565653680m_rule @ A2 @ B2 ) @ ( szip_P2924820683901490861m_rule @ S1 @ S2 ) ) ) ).
% szip_unfold
thf(fact_375_szip__unfold,axiom,
! [A2: nat,S1: stream_nat,B2: nat,S2: stream_nat] :
( ( szip_nat_nat @ ( sCons_nat @ A2 @ S1 ) @ ( sCons_nat @ B2 @ S2 ) )
= ( sCons_2147322719763279000at_nat @ ( product_Pair_nat_nat @ A2 @ B2 ) @ ( szip_nat_nat @ S1 @ S2 ) ) ) ).
% szip_unfold
thf(fact_376_PersistentRuleSystem_Oper,axiom,
! [Eff: rule > produc6018962875968178549ist_fm > fset_P8989946509869081563ist_fm > $o,Rules: stream_rule,S6: set_Pr5202636777678657877ist_fm,R: rule] :
( ( abstra3967921542344661089ist_fm @ Eff @ Rules @ S6 )
=> ( ( member_rule @ R @ ( sset_rule @ Rules ) )
=> ( abstra5255361903751151037ist_fm @ Eff @ Rules @ S6 @ R ) ) ) ).
% PersistentRuleSystem.per
thf(fact_377_stream_Osel_I1_J,axiom,
! [X1: produc340336539035504054m_rule,X2: stream2709947120125613254m_rule] :
( ( shd_Pr4562317740776619530m_rule @ ( sCons_2955607467875998431m_rule @ X1 @ X2 ) )
= X1 ) ).
% stream.sel(1)
thf(fact_378_stream_Osel_I1_J,axiom,
! [X1: rule,X2: stream_rule] :
( ( shd_rule @ ( sCons_rule @ X1 @ X2 ) )
= X1 ) ).
% stream.sel(1)
thf(fact_379_stream_Osel_I2_J,axiom,
! [X1: produc340336539035504054m_rule,X2: stream2709947120125613254m_rule] :
( ( stl_Pr12655793849929990m_rule @ ( sCons_2955607467875998431m_rule @ X1 @ X2 ) )
= X2 ) ).
% stream.sel(2)
thf(fact_380_stream_Osel_I2_J,axiom,
! [X1: rule,X2: stream_rule] :
( ( stl_rule @ ( sCons_rule @ X1 @ X2 ) )
= X2 ) ).
% stream.sel(2)
thf(fact_381_stream_Oset__induct,axiom,
! [X3: fm,A2: stream_fm,P: fm > stream_fm > $o] :
( ( member_fm @ X3 @ ( sset_fm @ A2 ) )
=> ( ! [Z1: fm,Z22: stream_fm] : ( P @ Z1 @ ( sCons_fm @ Z1 @ Z22 ) )
=> ( ! [Z1: fm,Z22: stream_fm,Xa2: fm] :
( ( member_fm @ Xa2 @ ( sset_fm @ Z22 ) )
=> ( ( P @ Xa2 @ Z22 )
=> ( P @ Xa2 @ ( sCons_fm @ Z1 @ Z22 ) ) ) )
=> ( P @ X3 @ A2 ) ) ) ) ).
% stream.set_induct
thf(fact_382_stream_Oset__induct,axiom,
! [X3: produc6018962875968178549ist_fm,A2: stream4408948924543953275ist_fm,P: produc6018962875968178549ist_fm > stream4408948924543953275ist_fm > $o] :
( ( member4699826688122452638ist_fm @ X3 @ ( sset_P5379989128061332361ist_fm @ A2 ) )
=> ( ! [Z1: produc6018962875968178549ist_fm,Z22: stream4408948924543953275ist_fm] : ( P @ Z1 @ ( sCons_2518607988952603276ist_fm @ Z1 @ Z22 ) )
=> ( ! [Z1: produc6018962875968178549ist_fm,Z22: stream4408948924543953275ist_fm,Xa2: produc6018962875968178549ist_fm] :
( ( member4699826688122452638ist_fm @ Xa2 @ ( sset_P5379989128061332361ist_fm @ Z22 ) )
=> ( ( P @ Xa2 @ Z22 )
=> ( P @ Xa2 @ ( sCons_2518607988952603276ist_fm @ Z1 @ Z22 ) ) ) )
=> ( P @ X3 @ A2 ) ) ) ) ).
% stream.set_induct
thf(fact_383_stream_Oset__induct,axiom,
! [X3: list_fm,A2: stream_list_fm,P: list_fm > stream_list_fm > $o] :
( ( member_list_fm @ X3 @ ( sset_list_fm @ A2 ) )
=> ( ! [Z1: list_fm,Z22: stream_list_fm] : ( P @ Z1 @ ( sCons_list_fm @ Z1 @ Z22 ) )
=> ( ! [Z1: list_fm,Z22: stream_list_fm,Xa2: list_fm] :
( ( member_list_fm @ Xa2 @ ( sset_list_fm @ Z22 ) )
=> ( ( P @ Xa2 @ Z22 )
=> ( P @ Xa2 @ ( sCons_list_fm @ Z1 @ Z22 ) ) ) )
=> ( P @ X3 @ A2 ) ) ) ) ).
% stream.set_induct
thf(fact_384_stream_Oset__induct,axiom,
! [X3: rule,A2: stream_rule,P: rule > stream_rule > $o] :
( ( member_rule @ X3 @ ( sset_rule @ A2 ) )
=> ( ! [Z1: rule,Z22: stream_rule] : ( P @ Z1 @ ( sCons_rule @ Z1 @ Z22 ) )
=> ( ! [Z1: rule,Z22: stream_rule,Xa2: rule] :
( ( member_rule @ Xa2 @ ( sset_rule @ Z22 ) )
=> ( ( P @ Xa2 @ Z22 )
=> ( P @ Xa2 @ ( sCons_rule @ Z1 @ Z22 ) ) ) )
=> ( P @ X3 @ A2 ) ) ) ) ).
% stream.set_induct
thf(fact_385_stream_Oset__induct,axiom,
! [X3: produc340336539035504054m_rule,A2: stream2709947120125613254m_rule,P: produc340336539035504054m_rule > stream2709947120125613254m_rule > $o] :
( ( member7231649785386036813m_rule @ X3 @ ( sset_P4484857331586881186m_rule @ A2 ) )
=> ( ! [Z1: produc340336539035504054m_rule,Z22: stream2709947120125613254m_rule] : ( P @ Z1 @ ( sCons_2955607467875998431m_rule @ Z1 @ Z22 ) )
=> ( ! [Z1: produc340336539035504054m_rule,Z22: stream2709947120125613254m_rule,Xa2: produc340336539035504054m_rule] :
( ( member7231649785386036813m_rule @ Xa2 @ ( sset_P4484857331586881186m_rule @ Z22 ) )
=> ( ( P @ Xa2 @ Z22 )
=> ( P @ Xa2 @ ( sCons_2955607467875998431m_rule @ Z1 @ Z22 ) ) ) )
=> ( P @ X3 @ A2 ) ) ) ) ).
% stream.set_induct
thf(fact_386_stream_Oset__cases,axiom,
! [E: fm,A2: stream_fm] :
( ( member_fm @ E @ ( sset_fm @ A2 ) )
=> ( ! [Z22: stream_fm] :
( A2
!= ( sCons_fm @ E @ Z22 ) )
=> ~ ! [Z1: fm,Z22: stream_fm] :
( ( A2
= ( sCons_fm @ Z1 @ Z22 ) )
=> ~ ( member_fm @ E @ ( sset_fm @ Z22 ) ) ) ) ) ).
% stream.set_cases
thf(fact_387_stream_Oset__cases,axiom,
! [E: produc6018962875968178549ist_fm,A2: stream4408948924543953275ist_fm] :
( ( member4699826688122452638ist_fm @ E @ ( sset_P5379989128061332361ist_fm @ A2 ) )
=> ( ! [Z22: stream4408948924543953275ist_fm] :
( A2
!= ( sCons_2518607988952603276ist_fm @ E @ Z22 ) )
=> ~ ! [Z1: produc6018962875968178549ist_fm,Z22: stream4408948924543953275ist_fm] :
( ( A2
= ( sCons_2518607988952603276ist_fm @ Z1 @ Z22 ) )
=> ~ ( member4699826688122452638ist_fm @ E @ ( sset_P5379989128061332361ist_fm @ Z22 ) ) ) ) ) ).
% stream.set_cases
thf(fact_388_stream_Oset__cases,axiom,
! [E: list_fm,A2: stream_list_fm] :
( ( member_list_fm @ E @ ( sset_list_fm @ A2 ) )
=> ( ! [Z22: stream_list_fm] :
( A2
!= ( sCons_list_fm @ E @ Z22 ) )
=> ~ ! [Z1: list_fm,Z22: stream_list_fm] :
( ( A2
= ( sCons_list_fm @ Z1 @ Z22 ) )
=> ~ ( member_list_fm @ E @ ( sset_list_fm @ Z22 ) ) ) ) ) ).
% stream.set_cases
thf(fact_389_stream_Oset__cases,axiom,
! [E: rule,A2: stream_rule] :
( ( member_rule @ E @ ( sset_rule @ A2 ) )
=> ( ! [Z22: stream_rule] :
( A2
!= ( sCons_rule @ E @ Z22 ) )
=> ~ ! [Z1: rule,Z22: stream_rule] :
( ( A2
= ( sCons_rule @ Z1 @ Z22 ) )
=> ~ ( member_rule @ E @ ( sset_rule @ Z22 ) ) ) ) ) ).
% stream.set_cases
thf(fact_390_stream_Oset__cases,axiom,
! [E: produc340336539035504054m_rule,A2: stream2709947120125613254m_rule] :
( ( member7231649785386036813m_rule @ E @ ( sset_P4484857331586881186m_rule @ A2 ) )
=> ( ! [Z22: stream2709947120125613254m_rule] :
( A2
!= ( sCons_2955607467875998431m_rule @ E @ Z22 ) )
=> ~ ! [Z1: produc340336539035504054m_rule,Z22: stream2709947120125613254m_rule] :
( ( A2
= ( sCons_2955607467875998431m_rule @ Z1 @ Z22 ) )
=> ~ ( member7231649785386036813m_rule @ E @ ( sset_P4484857331586881186m_rule @ Z22 ) ) ) ) ) ).
% stream.set_cases
thf(fact_391_stream_Oset__intros_I1_J,axiom,
! [X1: fm,X2: stream_fm] : ( member_fm @ X1 @ ( sset_fm @ ( sCons_fm @ X1 @ X2 ) ) ) ).
% stream.set_intros(1)
thf(fact_392_stream_Oset__intros_I1_J,axiom,
! [X1: produc6018962875968178549ist_fm,X2: stream4408948924543953275ist_fm] : ( member4699826688122452638ist_fm @ X1 @ ( sset_P5379989128061332361ist_fm @ ( sCons_2518607988952603276ist_fm @ X1 @ X2 ) ) ) ).
% stream.set_intros(1)
thf(fact_393_stream_Oset__intros_I1_J,axiom,
! [X1: list_fm,X2: stream_list_fm] : ( member_list_fm @ X1 @ ( sset_list_fm @ ( sCons_list_fm @ X1 @ X2 ) ) ) ).
% stream.set_intros(1)
thf(fact_394_stream_Oset__intros_I1_J,axiom,
! [X1: rule,X2: stream_rule] : ( member_rule @ X1 @ ( sset_rule @ ( sCons_rule @ X1 @ X2 ) ) ) ).
% stream.set_intros(1)
thf(fact_395_stream_Oset__intros_I1_J,axiom,
! [X1: produc340336539035504054m_rule,X2: stream2709947120125613254m_rule] : ( member7231649785386036813m_rule @ X1 @ ( sset_P4484857331586881186m_rule @ ( sCons_2955607467875998431m_rule @ X1 @ X2 ) ) ) ).
% stream.set_intros(1)
thf(fact_396_stream_Oset__intros_I2_J,axiom,
! [Y3: fm,X2: stream_fm,X1: fm] :
( ( member_fm @ Y3 @ ( sset_fm @ X2 ) )
=> ( member_fm @ Y3 @ ( sset_fm @ ( sCons_fm @ X1 @ X2 ) ) ) ) ).
% stream.set_intros(2)
thf(fact_397_stream_Oset__intros_I2_J,axiom,
! [Y3: produc6018962875968178549ist_fm,X2: stream4408948924543953275ist_fm,X1: produc6018962875968178549ist_fm] :
( ( member4699826688122452638ist_fm @ Y3 @ ( sset_P5379989128061332361ist_fm @ X2 ) )
=> ( member4699826688122452638ist_fm @ Y3 @ ( sset_P5379989128061332361ist_fm @ ( sCons_2518607988952603276ist_fm @ X1 @ X2 ) ) ) ) ).
% stream.set_intros(2)
thf(fact_398_stream_Oset__intros_I2_J,axiom,
! [Y3: list_fm,X2: stream_list_fm,X1: list_fm] :
( ( member_list_fm @ Y3 @ ( sset_list_fm @ X2 ) )
=> ( member_list_fm @ Y3 @ ( sset_list_fm @ ( sCons_list_fm @ X1 @ X2 ) ) ) ) ).
% stream.set_intros(2)
thf(fact_399_stream_Oset__intros_I2_J,axiom,
! [Y3: rule,X2: stream_rule,X1: rule] :
( ( member_rule @ Y3 @ ( sset_rule @ X2 ) )
=> ( member_rule @ Y3 @ ( sset_rule @ ( sCons_rule @ X1 @ X2 ) ) ) ) ).
% stream.set_intros(2)
thf(fact_400_stream_Oset__intros_I2_J,axiom,
! [Y3: produc340336539035504054m_rule,X2: stream2709947120125613254m_rule,X1: produc340336539035504054m_rule] :
( ( member7231649785386036813m_rule @ Y3 @ ( sset_P4484857331586881186m_rule @ X2 ) )
=> ( member7231649785386036813m_rule @ Y3 @ ( sset_P4484857331586881186m_rule @ ( sCons_2955607467875998431m_rule @ X1 @ X2 ) ) ) ) ).
% stream.set_intros(2)
thf(fact_401_ev__Stream,axiom,
! [P: stream2709947120125613254m_rule > $o,X3: produc340336539035504054m_rule,S: stream2709947120125613254m_rule] :
( ( linear1816559245158234865m_rule @ P @ ( sCons_2955607467875998431m_rule @ X3 @ S ) )
= ( ( P @ ( sCons_2955607467875998431m_rule @ X3 @ S ) )
| ( linear1816559245158234865m_rule @ P @ S ) ) ) ).
% ev_Stream
thf(fact_402_ev__Stream,axiom,
! [P: stream_rule > $o,X3: rule,S: stream_rule] :
( ( linear292482573288640205v_rule @ P @ ( sCons_rule @ X3 @ S ) )
= ( ( P @ ( sCons_rule @ X3 @ S ) )
| ( linear292482573288640205v_rule @ P @ S ) ) ) ).
% ev_Stream
thf(fact_403_holds__Stream,axiom,
! [P: produc340336539035504054m_rule > $o,X3: produc340336539035504054m_rule,S: stream2709947120125613254m_rule] :
( ( linear7925859179493631106m_rule @ P @ ( sCons_2955607467875998431m_rule @ X3 @ S ) )
= ( P @ X3 ) ) ).
% holds_Stream
thf(fact_404_holds__Stream,axiom,
! [P: rule > $o,X3: rule,S: stream_rule] :
( ( linear7354821883049305950s_rule @ P @ ( sCons_rule @ X3 @ S ) )
= ( P @ X3 ) ) ).
% holds_Stream
thf(fact_405_stream_Oexhaust__sel,axiom,
! [Stream: stream2709947120125613254m_rule] :
( Stream
= ( sCons_2955607467875998431m_rule @ ( shd_Pr4562317740776619530m_rule @ Stream ) @ ( stl_Pr12655793849929990m_rule @ Stream ) ) ) ).
% stream.exhaust_sel
thf(fact_406_stream_Oexhaust__sel,axiom,
! [Stream: stream_rule] :
( Stream
= ( sCons_rule @ ( shd_rule @ Stream ) @ ( stl_rule @ Stream ) ) ) ).
% stream.exhaust_sel
thf(fact_407_lexord__irreflexive,axiom,
! [R: set_Pr1261947904930325089at_nat,Xs: list_nat] :
( ! [X: nat] :
~ ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X @ X ) @ R )
=> ~ ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ Xs @ Xs ) @ ( lexord_nat @ R ) ) ) ).
% lexord_irreflexive
thf(fact_408_lexord__linear,axiom,
! [R: set_Pr1261947904930325089at_nat,X3: list_nat,Y3: list_nat] :
( ! [A4: nat,B5: nat] :
( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ A4 @ B5 ) @ R )
| ( A4 = B5 )
| ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ B5 @ A4 ) @ R ) )
=> ( ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ X3 @ Y3 ) @ ( lexord_nat @ R ) )
| ( X3 = Y3 )
| ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ Y3 @ X3 ) @ ( lexord_nat @ R ) ) ) ) ).
% lexord_linear
thf(fact_409_lexord__append__leftI,axiom,
! [U: list_tm,V: list_tm,R: set_Pr2455929065695642951_tm_tm,X3: list_tm] :
( ( member4457312017796942864ist_tm @ ( produc1418304791525149271ist_tm @ U @ V ) @ ( lexord_tm @ R ) )
=> ( member4457312017796942864ist_tm @ ( produc1418304791525149271ist_tm @ ( append_tm @ X3 @ U ) @ ( append_tm @ X3 @ V ) ) @ ( lexord_tm @ R ) ) ) ).
% lexord_append_leftI
thf(fact_410_lexord__partial__trans,axiom,
! [Xs: list_rule,R: set_Pr7340090144965549541e_rule,Ys: list_rule,Zs: list_rule] :
( ! [X: rule,Y: rule,Z6: rule] :
( ( member_rule @ X @ ( set_rule2 @ Xs ) )
=> ( ( member1444691403335511342e_rule @ ( produc5849431337705160893e_rule @ X @ Y ) @ R )
=> ( ( member1444691403335511342e_rule @ ( produc5849431337705160893e_rule @ Y @ Z6 ) @ R )
=> ( member1444691403335511342e_rule @ ( produc5849431337705160893e_rule @ X @ Z6 ) @ R ) ) ) )
=> ( ( member695053538577043918t_rule @ ( produc8979233614864344413t_rule @ Xs @ Ys ) @ ( lexord_rule @ R ) )
=> ( ( member695053538577043918t_rule @ ( produc8979233614864344413t_rule @ Ys @ Zs ) @ ( lexord_rule @ R ) )
=> ( member695053538577043918t_rule @ ( produc8979233614864344413t_rule @ Xs @ Zs ) @ ( lexord_rule @ R ) ) ) ) ) ).
% lexord_partial_trans
thf(fact_411_lexord__partial__trans,axiom,
! [Xs: list_P2887561121880082555ist_fm,R: set_Pr6724411253394807111ist_fm,Ys: list_P2887561121880082555ist_fm,Zs: list_P2887561121880082555ist_fm] :
( ! [X: produc6018962875968178549ist_fm,Y: produc6018962875968178549ist_fm,Z6: produc6018962875968178549ist_fm] :
( ( member4699826688122452638ist_fm @ X @ ( set_Pr5051287786238754058ist_fm @ Xs ) )
=> ( ( member858341428514850704ist_fm @ ( produc5488031402154692311ist_fm @ X @ Y ) @ R )
=> ( ( member858341428514850704ist_fm @ ( produc5488031402154692311ist_fm @ Y @ Z6 ) @ R )
=> ( member858341428514850704ist_fm @ ( produc5488031402154692311ist_fm @ X @ Z6 ) @ R ) ) ) )
=> ( ( member6238937682488541200ist_fm @ ( produc8740291170414093911ist_fm @ Xs @ Ys ) @ ( lexord4892917800681400680ist_fm @ R ) )
=> ( ( member6238937682488541200ist_fm @ ( produc8740291170414093911ist_fm @ Ys @ Zs ) @ ( lexord4892917800681400680ist_fm @ R ) )
=> ( member6238937682488541200ist_fm @ ( produc8740291170414093911ist_fm @ Xs @ Zs ) @ ( lexord4892917800681400680ist_fm @ R ) ) ) ) ) ).
% lexord_partial_trans
thf(fact_412_lexord__partial__trans,axiom,
! [Xs: list_P2774625669004309958m_rule,R: set_Pr4971326047967503661m_rule,Ys: list_P2774625669004309958m_rule,Zs: list_P2774625669004309958m_rule] :
( ! [X: produc340336539035504054m_rule,Y: produc340336539035504054m_rule,Z6: produc340336539035504054m_rule] :
( ( member7231649785386036813m_rule @ X @ ( set_Pr4534715572506550497m_rule @ Xs ) )
=> ( ( member1411645983481675894m_rule @ ( produc6261311607089640965m_rule @ X @ Y ) @ R )
=> ( ( member1411645983481675894m_rule @ ( produc6261311607089640965m_rule @ Y @ Z6 ) @ R )
=> ( member1411645983481675894m_rule @ ( produc6261311607089640965m_rule @ X @ Z6 ) @ R ) ) ) )
=> ( ( member3128744185454049558m_rule @ ( produc6692282267706994341m_rule @ Xs @ Ys ) @ ( lexord6196499238003582211m_rule @ R ) )
=> ( ( member3128744185454049558m_rule @ ( produc6692282267706994341m_rule @ Ys @ Zs ) @ ( lexord6196499238003582211m_rule @ R ) )
=> ( member3128744185454049558m_rule @ ( produc6692282267706994341m_rule @ Xs @ Zs ) @ ( lexord6196499238003582211m_rule @ R ) ) ) ) ) ).
% lexord_partial_trans
thf(fact_413_lexord__partial__trans,axiom,
! [Xs: list_fm,R: set_Pr4706815898642364871_fm_fm,Ys: list_fm,Zs: list_fm] :
( ! [X: fm,Y: fm,Z6: fm] :
( ( member_fm @ X @ ( set_fm2 @ Xs ) )
=> ( ( member7780952600467998736_fm_fm @ ( product_Pair_fm_fm @ X @ Y ) @ R )
=> ( ( member7780952600467998736_fm_fm @ ( product_Pair_fm_fm @ Y @ Z6 ) @ R )
=> ( member7780952600467998736_fm_fm @ ( product_Pair_fm_fm @ X @ Z6 ) @ R ) ) ) )
=> ( ( member1926098302810316688ist_fm @ ( produc7863996417982153943ist_fm @ Xs @ Ys ) @ ( lexord_fm @ R ) )
=> ( ( member1926098302810316688ist_fm @ ( produc7863996417982153943ist_fm @ Ys @ Zs ) @ ( lexord_fm @ R ) )
=> ( member1926098302810316688ist_fm @ ( produc7863996417982153943ist_fm @ Xs @ Zs ) @ ( lexord_fm @ R ) ) ) ) ) ).
% lexord_partial_trans
thf(fact_414_lexord__partial__trans,axiom,
! [Xs: list_tm,R: set_Pr2455929065695642951_tm_tm,Ys: list_tm,Zs: list_tm] :
( ! [X: tm,Y: tm,Z6: tm] :
( ( member_tm @ X @ ( set_tm2 @ Xs ) )
=> ( ( member3121616906494481296_tm_tm @ ( product_Pair_tm_tm @ X @ Y ) @ R )
=> ( ( member3121616906494481296_tm_tm @ ( product_Pair_tm_tm @ Y @ Z6 ) @ R )
=> ( member3121616906494481296_tm_tm @ ( product_Pair_tm_tm @ X @ Z6 ) @ R ) ) ) )
=> ( ( member4457312017796942864ist_tm @ ( produc1418304791525149271ist_tm @ Xs @ Ys ) @ ( lexord_tm @ R ) )
=> ( ( member4457312017796942864ist_tm @ ( produc1418304791525149271ist_tm @ Ys @ Zs ) @ ( lexord_tm @ R ) )
=> ( member4457312017796942864ist_tm @ ( produc1418304791525149271ist_tm @ Xs @ Zs ) @ ( lexord_tm @ R ) ) ) ) ) ).
% lexord_partial_trans
thf(fact_415_lexord__partial__trans,axiom,
! [Xs: list_list_fm,R: set_Pr1916137733696349511ist_fm,Ys: list_list_fm,Zs: list_list_fm] :
( ! [X: list_fm,Y: list_fm,Z6: list_fm] :
( ( member_list_fm @ X @ ( set_list_fm2 @ Xs ) )
=> ( ( member1926098302810316688ist_fm @ ( produc7863996417982153943ist_fm @ X @ Y ) @ R )
=> ( ( member1926098302810316688ist_fm @ ( produc7863996417982153943ist_fm @ Y @ Z6 ) @ R )
=> ( member1926098302810316688ist_fm @ ( produc7863996417982153943ist_fm @ X @ Z6 ) @ R ) ) ) )
=> ( ( member5102842307574975760ist_fm @ ( produc8321651870839017815ist_fm @ Xs @ Ys ) @ ( lexord_list_fm @ R ) )
=> ( ( member5102842307574975760ist_fm @ ( produc8321651870839017815ist_fm @ Ys @ Zs ) @ ( lexord_list_fm @ R ) )
=> ( member5102842307574975760ist_fm @ ( produc8321651870839017815ist_fm @ Xs @ Zs ) @ ( lexord_list_fm @ R ) ) ) ) ) ).
% lexord_partial_trans
thf(fact_416_lexord__partial__trans,axiom,
! [Xs: list_nat,R: set_Pr1261947904930325089at_nat,Ys: list_nat,Zs: list_nat] :
( ! [X: nat,Y: nat,Z6: nat] :
( ( member_nat @ X @ ( set_nat2 @ Xs ) )
=> ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X @ Y ) @ R )
=> ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ Y @ Z6 ) @ R )
=> ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X @ Z6 ) @ R ) ) ) )
=> ( ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ Xs @ Ys ) @ ( lexord_nat @ R ) )
=> ( ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ Ys @ Zs ) @ ( lexord_nat @ R ) )
=> ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ Xs @ Zs ) @ ( lexord_nat @ R ) ) ) ) ) ).
% lexord_partial_trans
thf(fact_417_lexord__append__leftD,axiom,
! [X3: list_tm,U: list_tm,V: list_tm,R: set_Pr2455929065695642951_tm_tm] :
( ( member4457312017796942864ist_tm @ ( produc1418304791525149271ist_tm @ ( append_tm @ X3 @ U ) @ ( append_tm @ X3 @ V ) ) @ ( lexord_tm @ R ) )
=> ( ! [A4: tm] :
~ ( member3121616906494481296_tm_tm @ ( product_Pair_tm_tm @ A4 @ A4 ) @ R )
=> ( member4457312017796942864ist_tm @ ( produc1418304791525149271ist_tm @ U @ V ) @ ( lexord_tm @ R ) ) ) ) ).
% lexord_append_leftD
thf(fact_418_lexord__append__leftD,axiom,
! [X3: list_nat,U: list_nat,V: list_nat,R: set_Pr1261947904930325089at_nat] :
( ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ ( append_nat @ X3 @ U ) @ ( append_nat @ X3 @ V ) ) @ ( lexord_nat @ R ) )
=> ( ! [A4: nat] :
~ ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ A4 @ A4 ) @ R )
=> ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ U @ V ) @ ( lexord_nat @ R ) ) ) ) ).
% lexord_append_leftD
thf(fact_419_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_420_lexord__same__pref__if__irrefl,axiom,
! [R: set_Pr2455929065695642951_tm_tm,Xs: list_tm,Ys: list_tm,Zs: list_tm] :
( ( irrefl_on_tm @ top_top_set_tm @ R )
=> ( ( member4457312017796942864ist_tm @ ( produc1418304791525149271ist_tm @ ( append_tm @ Xs @ Ys ) @ ( append_tm @ Xs @ Zs ) ) @ ( lexord_tm @ R ) )
= ( member4457312017796942864ist_tm @ ( produc1418304791525149271ist_tm @ Ys @ Zs ) @ ( lexord_tm @ R ) ) ) ) ).
% lexord_same_pref_if_irrefl
thf(fact_421_lexord__same__pref__if__irrefl,axiom,
! [R: set_Pr6724411253394807111ist_fm,Xs: list_P2887561121880082555ist_fm,Ys: list_P2887561121880082555ist_fm,Zs: list_P2887561121880082555ist_fm] :
( ( irrefl7757270364544296811ist_fm @ top_to1730629564055774885ist_fm @ R )
=> ( ( member6238937682488541200ist_fm @ ( produc8740291170414093911ist_fm @ ( append1476581234975291920ist_fm @ Xs @ Ys ) @ ( append1476581234975291920ist_fm @ Xs @ Zs ) ) @ ( lexord4892917800681400680ist_fm @ R ) )
= ( member6238937682488541200ist_fm @ ( produc8740291170414093911ist_fm @ Ys @ Zs ) @ ( lexord4892917800681400680ist_fm @ R ) ) ) ) ).
% lexord_same_pref_if_irrefl
thf(fact_422_PersistentRuleSystem__axioms_Ointro,axiom,
! [Rules: stream_rule,Eff: rule > produc6018962875968178549ist_fm > fset_P8989946509869081563ist_fm > $o,S6: set_Pr5202636777678657877ist_fm] :
( ! [R6: rule] :
( ( member_rule @ R6 @ ( sset_rule @ Rules ) )
=> ( abstra5255361903751151037ist_fm @ Eff @ Rules @ S6 @ R6 ) )
=> ( abstra9069655090277518462ist_fm @ Eff @ Rules @ S6 ) ) ).
% PersistentRuleSystem_axioms.intro
thf(fact_423_PersistentRuleSystem__axioms__def,axiom,
( abstra9069655090277518462ist_fm
= ( ^ [Eff2: rule > produc6018962875968178549ist_fm > fset_P8989946509869081563ist_fm > $o,Rules2: stream_rule,S7: set_Pr5202636777678657877ist_fm] :
! [R4: rule] :
( ( member_rule @ R4 @ ( sset_rule @ Rules2 ) )
=> ( abstra5255361903751151037ist_fm @ Eff2 @ Rules2 @ S7 @ R4 ) ) ) ) ).
% PersistentRuleSystem_axioms_def
thf(fact_424_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_425_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_426_Collect__subset,axiom,
! [A: set_Pr5202636777678657877ist_fm,P: produc6018962875968178549ist_fm > $o] :
( ord_le1771420097867575541ist_fm
@ ( collec5845330005437105888ist_fm
@ ^ [X4: produc6018962875968178549ist_fm] :
( ( member4699826688122452638ist_fm @ X4 @ A )
& ( P @ X4 ) ) )
@ A ) ).
% Collect_subset
thf(fact_427_Collect__subset,axiom,
! [A: set_Pr1822751329126368876m_rule,P: produc340336539035504054m_rule > $o] :
( ord_le6390412330253371084m_rule
@ ( collec1009276759030335883m_rule
@ ^ [X4: produc340336539035504054m_rule] :
( ( member7231649785386036813m_rule @ X4 @ A )
& ( P @ X4 ) ) )
@ A ) ).
% Collect_subset
thf(fact_428_Collect__subset,axiom,
! [A: set_list_fm,P: list_fm > $o] :
( ord_le7838213414353715577ist_fm
@ ( collect_list_fm
@ ^ [X4: list_fm] :
( ( member_list_fm @ X4 @ A )
& ( P @ X4 ) ) )
@ A ) ).
% Collect_subset
thf(fact_429_Collect__subset,axiom,
! [A: set_tm,P: tm > $o] :
( ord_less_eq_set_tm
@ ( collect_tm
@ ^ [X4: tm] :
( ( member_tm @ X4 @ A )
& ( P @ X4 ) ) )
@ A ) ).
% Collect_subset
thf(fact_430_Collect__subset,axiom,
! [A: set_fm,P: fm > $o] :
( ord_less_eq_set_fm
@ ( collect_fm
@ ^ [X4: fm] :
( ( member_fm @ X4 @ A )
& ( P @ X4 ) ) )
@ A ) ).
% Collect_subset
thf(fact_431_Collect__subset,axiom,
! [A: set_rule,P: rule > $o] :
( ord_less_eq_set_rule
@ ( collect_rule
@ ^ [X4: rule] :
( ( member_rule @ X4 @ A )
& ( P @ X4 ) ) )
@ A ) ).
% Collect_subset
thf(fact_432_subset__Collect__iff,axiom,
! [B4: set_Pr5202636777678657877ist_fm,A: set_Pr5202636777678657877ist_fm,P: produc6018962875968178549ist_fm > $o] :
( ( ord_le1771420097867575541ist_fm @ B4 @ A )
=> ( ( ord_le1771420097867575541ist_fm @ B4
@ ( collec5845330005437105888ist_fm
@ ^ [X4: produc6018962875968178549ist_fm] :
( ( member4699826688122452638ist_fm @ X4 @ A )
& ( P @ X4 ) ) ) )
= ( ! [X4: produc6018962875968178549ist_fm] :
( ( member4699826688122452638ist_fm @ X4 @ B4 )
=> ( P @ X4 ) ) ) ) ) ).
% subset_Collect_iff
thf(fact_433_subset__Collect__iff,axiom,
! [B4: set_Pr1822751329126368876m_rule,A: set_Pr1822751329126368876m_rule,P: produc340336539035504054m_rule > $o] :
( ( ord_le6390412330253371084m_rule @ B4 @ A )
=> ( ( ord_le6390412330253371084m_rule @ B4
@ ( collec1009276759030335883m_rule
@ ^ [X4: produc340336539035504054m_rule] :
( ( member7231649785386036813m_rule @ X4 @ A )
& ( P @ X4 ) ) ) )
= ( ! [X4: produc340336539035504054m_rule] :
( ( member7231649785386036813m_rule @ X4 @ B4 )
=> ( P @ X4 ) ) ) ) ) ).
% subset_Collect_iff
thf(fact_434_subset__Collect__iff,axiom,
! [B4: set_list_fm,A: set_list_fm,P: list_fm > $o] :
( ( ord_le7838213414353715577ist_fm @ B4 @ A )
=> ( ( ord_le7838213414353715577ist_fm @ B4
@ ( collect_list_fm
@ ^ [X4: list_fm] :
( ( member_list_fm @ X4 @ A )
& ( P @ X4 ) ) ) )
= ( ! [X4: list_fm] :
( ( member_list_fm @ X4 @ B4 )
=> ( P @ X4 ) ) ) ) ) ).
% subset_Collect_iff
thf(fact_435_subset__Collect__iff,axiom,
! [B4: set_tm,A: set_tm,P: tm > $o] :
( ( ord_less_eq_set_tm @ B4 @ A )
=> ( ( ord_less_eq_set_tm @ B4
@ ( collect_tm
@ ^ [X4: tm] :
( ( member_tm @ X4 @ A )
& ( P @ X4 ) ) ) )
= ( ! [X4: tm] :
( ( member_tm @ X4 @ B4 )
=> ( P @ X4 ) ) ) ) ) ).
% subset_Collect_iff
thf(fact_436_subset__Collect__iff,axiom,
! [B4: set_fm,A: set_fm,P: fm > $o] :
( ( ord_less_eq_set_fm @ B4 @ A )
=> ( ( ord_less_eq_set_fm @ B4
@ ( collect_fm
@ ^ [X4: fm] :
( ( member_fm @ X4 @ A )
& ( P @ X4 ) ) ) )
= ( ! [X4: fm] :
( ( member_fm @ X4 @ B4 )
=> ( P @ X4 ) ) ) ) ) ).
% subset_Collect_iff
thf(fact_437_subset__Collect__iff,axiom,
! [B4: set_rule,A: set_rule,P: rule > $o] :
( ( ord_less_eq_set_rule @ B4 @ A )
=> ( ( ord_less_eq_set_rule @ B4
@ ( collect_rule
@ ^ [X4: rule] :
( ( member_rule @ X4 @ A )
& ( P @ X4 ) ) ) )
= ( ! [X4: rule] :
( ( member_rule @ X4 @ B4 )
=> ( P @ X4 ) ) ) ) ) ).
% subset_Collect_iff
thf(fact_438_subset__CollectI,axiom,
! [B4: set_Pr5202636777678657877ist_fm,A: set_Pr5202636777678657877ist_fm,Q: produc6018962875968178549ist_fm > $o,P: produc6018962875968178549ist_fm > $o] :
( ( ord_le1771420097867575541ist_fm @ B4 @ A )
=> ( ! [X: produc6018962875968178549ist_fm] :
( ( member4699826688122452638ist_fm @ X @ B4 )
=> ( ( Q @ X )
=> ( P @ X ) ) )
=> ( ord_le1771420097867575541ist_fm
@ ( collec5845330005437105888ist_fm
@ ^ [X4: produc6018962875968178549ist_fm] :
( ( member4699826688122452638ist_fm @ X4 @ B4 )
& ( Q @ X4 ) ) )
@ ( collec5845330005437105888ist_fm
@ ^ [X4: produc6018962875968178549ist_fm] :
( ( member4699826688122452638ist_fm @ X4 @ A )
& ( P @ X4 ) ) ) ) ) ) ).
% subset_CollectI
thf(fact_439_subset__CollectI,axiom,
! [B4: set_Pr1822751329126368876m_rule,A: set_Pr1822751329126368876m_rule,Q: produc340336539035504054m_rule > $o,P: produc340336539035504054m_rule > $o] :
( ( ord_le6390412330253371084m_rule @ B4 @ A )
=> ( ! [X: produc340336539035504054m_rule] :
( ( member7231649785386036813m_rule @ X @ B4 )
=> ( ( Q @ X )
=> ( P @ X ) ) )
=> ( ord_le6390412330253371084m_rule
@ ( collec1009276759030335883m_rule
@ ^ [X4: produc340336539035504054m_rule] :
( ( member7231649785386036813m_rule @ X4 @ B4 )
& ( Q @ X4 ) ) )
@ ( collec1009276759030335883m_rule
@ ^ [X4: produc340336539035504054m_rule] :
( ( member7231649785386036813m_rule @ X4 @ A )
& ( P @ X4 ) ) ) ) ) ) ).
% subset_CollectI
thf(fact_440_subset__CollectI,axiom,
! [B4: set_list_fm,A: set_list_fm,Q: list_fm > $o,P: list_fm > $o] :
( ( ord_le7838213414353715577ist_fm @ B4 @ A )
=> ( ! [X: list_fm] :
( ( member_list_fm @ X @ B4 )
=> ( ( Q @ X )
=> ( P @ X ) ) )
=> ( ord_le7838213414353715577ist_fm
@ ( collect_list_fm
@ ^ [X4: list_fm] :
( ( member_list_fm @ X4 @ B4 )
& ( Q @ X4 ) ) )
@ ( collect_list_fm
@ ^ [X4: list_fm] :
( ( member_list_fm @ X4 @ A )
& ( P @ X4 ) ) ) ) ) ) ).
% subset_CollectI
thf(fact_441_subset__CollectI,axiom,
! [B4: set_tm,A: set_tm,Q: tm > $o,P: tm > $o] :
( ( ord_less_eq_set_tm @ B4 @ A )
=> ( ! [X: tm] :
( ( member_tm @ X @ B4 )
=> ( ( Q @ X )
=> ( P @ X ) ) )
=> ( ord_less_eq_set_tm
@ ( collect_tm
@ ^ [X4: tm] :
( ( member_tm @ X4 @ B4 )
& ( Q @ X4 ) ) )
@ ( collect_tm
@ ^ [X4: tm] :
( ( member_tm @ X4 @ A )
& ( P @ X4 ) ) ) ) ) ) ).
% subset_CollectI
thf(fact_442_subset__CollectI,axiom,
! [B4: set_fm,A: set_fm,Q: fm > $o,P: fm > $o] :
( ( ord_less_eq_set_fm @ B4 @ A )
=> ( ! [X: fm] :
( ( member_fm @ X @ B4 )
=> ( ( Q @ X )
=> ( P @ X ) ) )
=> ( ord_less_eq_set_fm
@ ( collect_fm
@ ^ [X4: fm] :
( ( member_fm @ X4 @ B4 )
& ( Q @ X4 ) ) )
@ ( collect_fm
@ ^ [X4: fm] :
( ( member_fm @ X4 @ A )
& ( P @ X4 ) ) ) ) ) ) ).
% subset_CollectI
thf(fact_443_subset__CollectI,axiom,
! [B4: set_rule,A: set_rule,Q: rule > $o,P: rule > $o] :
( ( ord_less_eq_set_rule @ B4 @ A )
=> ( ! [X: rule] :
( ( member_rule @ X @ B4 )
=> ( ( Q @ X )
=> ( P @ X ) ) )
=> ( ord_less_eq_set_rule
@ ( collect_rule
@ ^ [X4: rule] :
( ( member_rule @ X4 @ B4 )
& ( Q @ X4 ) ) )
@ ( collect_rule
@ ^ [X4: rule] :
( ( member_rule @ X4 @ A )
& ( P @ X4 ) ) ) ) ) ) ).
% subset_CollectI
thf(fact_444_subset__code_I1_J,axiom,
! [Xs: list_P2887561121880082555ist_fm,B4: set_Pr5202636777678657877ist_fm] :
( ( ord_le1771420097867575541ist_fm @ ( set_Pr5051287786238754058ist_fm @ Xs ) @ B4 )
= ( ! [X4: produc6018962875968178549ist_fm] :
( ( member4699826688122452638ist_fm @ X4 @ ( set_Pr5051287786238754058ist_fm @ Xs ) )
=> ( member4699826688122452638ist_fm @ X4 @ B4 ) ) ) ) ).
% subset_code(1)
thf(fact_445_subset__code_I1_J,axiom,
! [Xs: list_P2774625669004309958m_rule,B4: set_Pr1822751329126368876m_rule] :
( ( ord_le6390412330253371084m_rule @ ( set_Pr4534715572506550497m_rule @ Xs ) @ B4 )
= ( ! [X4: produc340336539035504054m_rule] :
( ( member7231649785386036813m_rule @ X4 @ ( set_Pr4534715572506550497m_rule @ Xs ) )
=> ( member7231649785386036813m_rule @ X4 @ B4 ) ) ) ) ).
% subset_code(1)
thf(fact_446_subset__code_I1_J,axiom,
! [Xs: list_list_fm,B4: set_list_fm] :
( ( ord_le7838213414353715577ist_fm @ ( set_list_fm2 @ Xs ) @ B4 )
= ( ! [X4: list_fm] :
( ( member_list_fm @ X4 @ ( set_list_fm2 @ Xs ) )
=> ( member_list_fm @ X4 @ B4 ) ) ) ) ).
% subset_code(1)
thf(fact_447_subset__code_I1_J,axiom,
! [Xs: list_tm,B4: set_tm] :
( ( ord_less_eq_set_tm @ ( set_tm2 @ Xs ) @ B4 )
= ( ! [X4: tm] :
( ( member_tm @ X4 @ ( set_tm2 @ Xs ) )
=> ( member_tm @ X4 @ B4 ) ) ) ) ).
% subset_code(1)
thf(fact_448_subset__code_I1_J,axiom,
! [Xs: list_fm,B4: set_fm] :
( ( ord_less_eq_set_fm @ ( set_fm2 @ Xs ) @ B4 )
= ( ! [X4: fm] :
( ( member_fm @ X4 @ ( set_fm2 @ Xs ) )
=> ( member_fm @ X4 @ B4 ) ) ) ) ).
% subset_code(1)
thf(fact_449_subset__code_I1_J,axiom,
! [Xs: list_rule,B4: set_rule] :
( ( ord_less_eq_set_rule @ ( set_rule2 @ Xs ) @ B4 )
= ( ! [X4: rule] :
( ( member_rule @ X4 @ ( set_rule2 @ Xs ) )
=> ( member_rule @ X4 @ B4 ) ) ) ) ).
% subset_code(1)
thf(fact_450_lexord__irrefl,axiom,
! [R3: set_Pr6724411253394807111ist_fm] :
( ( irrefl7757270364544296811ist_fm @ top_to1730629564055774885ist_fm @ R3 )
=> ( irrefl4218442000938873713ist_fm @ top_to7413187565660093483ist_fm @ ( lexord4892917800681400680ist_fm @ R3 ) ) ) ).
% lexord_irrefl
thf(fact_451_irrefl__onD,axiom,
! [A: set_rule,R: set_Pr7340090144965549541e_rule,A2: rule] :
( ( irrefl_on_rule @ A @ R )
=> ( ( member_rule @ A2 @ A )
=> ~ ( member1444691403335511342e_rule @ ( produc5849431337705160893e_rule @ A2 @ A2 ) @ R ) ) ) ).
% irrefl_onD
thf(fact_452_irrefl__onD,axiom,
! [A: set_fm,R: set_Pr4706815898642364871_fm_fm,A2: fm] :
( ( irrefl_on_fm @ A @ R )
=> ( ( member_fm @ A2 @ A )
=> ~ ( member7780952600467998736_fm_fm @ ( product_Pair_fm_fm @ A2 @ A2 ) @ R ) ) ) ).
% irrefl_onD
thf(fact_453_irrefl__onD,axiom,
! [A: set_Pr5202636777678657877ist_fm,R: set_Pr6724411253394807111ist_fm,A2: produc6018962875968178549ist_fm] :
( ( irrefl7757270364544296811ist_fm @ A @ R )
=> ( ( member4699826688122452638ist_fm @ A2 @ A )
=> ~ ( member858341428514850704ist_fm @ ( produc5488031402154692311ist_fm @ A2 @ A2 ) @ R ) ) ) ).
% irrefl_onD
thf(fact_454_irrefl__onD,axiom,
! [A: set_Pr1822751329126368876m_rule,R: set_Pr4971326047967503661m_rule,A2: produc340336539035504054m_rule] :
( ( irrefl2627423231736715840m_rule @ A @ R )
=> ( ( member7231649785386036813m_rule @ A2 @ A )
=> ~ ( member1411645983481675894m_rule @ ( produc6261311607089640965m_rule @ A2 @ A2 ) @ R ) ) ) ).
% irrefl_onD
thf(fact_455_irrefl__onD,axiom,
! [A: set_list_fm,R: set_Pr1916137733696349511ist_fm,A2: list_fm] :
( ( irrefl_on_list_fm @ A @ R )
=> ( ( member_list_fm @ A2 @ A )
=> ~ ( member1926098302810316688ist_fm @ ( produc7863996417982153943ist_fm @ A2 @ A2 ) @ R ) ) ) ).
% irrefl_onD
thf(fact_456_irrefl__onD,axiom,
! [A: set_nat,R: set_Pr1261947904930325089at_nat,A2: nat] :
( ( irrefl_on_nat @ A @ R )
=> ( ( member_nat @ A2 @ A )
=> ~ ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ A2 @ A2 ) @ R ) ) ) ).
% irrefl_onD
thf(fact_457_irrefl__onI,axiom,
! [A: set_rule,R: set_Pr7340090144965549541e_rule] :
( ! [A4: rule] :
( ( member_rule @ A4 @ A )
=> ~ ( member1444691403335511342e_rule @ ( produc5849431337705160893e_rule @ A4 @ A4 ) @ R ) )
=> ( irrefl_on_rule @ A @ R ) ) ).
% irrefl_onI
thf(fact_458_irrefl__onI,axiom,
! [A: set_fm,R: set_Pr4706815898642364871_fm_fm] :
( ! [A4: fm] :
( ( member_fm @ A4 @ A )
=> ~ ( member7780952600467998736_fm_fm @ ( product_Pair_fm_fm @ A4 @ A4 ) @ R ) )
=> ( irrefl_on_fm @ A @ R ) ) ).
% irrefl_onI
thf(fact_459_irrefl__onI,axiom,
! [A: set_Pr5202636777678657877ist_fm,R: set_Pr6724411253394807111ist_fm] :
( ! [A4: produc6018962875968178549ist_fm] :
( ( member4699826688122452638ist_fm @ A4 @ A )
=> ~ ( member858341428514850704ist_fm @ ( produc5488031402154692311ist_fm @ A4 @ A4 ) @ R ) )
=> ( irrefl7757270364544296811ist_fm @ A @ R ) ) ).
% irrefl_onI
thf(fact_460_irrefl__onI,axiom,
! [A: set_Pr1822751329126368876m_rule,R: set_Pr4971326047967503661m_rule] :
( ! [A4: produc340336539035504054m_rule] :
( ( member7231649785386036813m_rule @ A4 @ A )
=> ~ ( member1411645983481675894m_rule @ ( produc6261311607089640965m_rule @ A4 @ A4 ) @ R ) )
=> ( irrefl2627423231736715840m_rule @ A @ R ) ) ).
% irrefl_onI
thf(fact_461_irrefl__onI,axiom,
! [A: set_list_fm,R: set_Pr1916137733696349511ist_fm] :
( ! [A4: list_fm] :
( ( member_list_fm @ A4 @ A )
=> ~ ( member1926098302810316688ist_fm @ ( produc7863996417982153943ist_fm @ A4 @ A4 ) @ R ) )
=> ( irrefl_on_list_fm @ A @ R ) ) ).
% irrefl_onI
thf(fact_462_irrefl__onI,axiom,
! [A: set_nat,R: set_Pr1261947904930325089at_nat] :
( ! [A4: nat] :
( ( member_nat @ A4 @ A )
=> ~ ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ A4 @ A4 ) @ R ) )
=> ( irrefl_on_nat @ A @ R ) ) ).
% irrefl_onI
thf(fact_463_irrefl__on__def,axiom,
( irrefl_on_nat
= ( ^ [A6: set_nat,R4: set_Pr1261947904930325089at_nat] :
! [X4: nat] :
( ( member_nat @ X4 @ A6 )
=> ~ ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X4 @ X4 ) @ R4 ) ) ) ) ).
% irrefl_on_def
thf(fact_464_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_465_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_466_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_467_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_468_sset__sdrop,axiom,
! [N: nat,S: stream2709947120125613254m_rule] : ( ord_le6390412330253371084m_rule @ ( sset_P4484857331586881186m_rule @ ( sdrop_8169176516188972301m_rule @ N @ S ) ) @ ( sset_P4484857331586881186m_rule @ S ) ) ).
% sset_sdrop
thf(fact_469_sset__sdrop,axiom,
! [N: nat,S: stream_tm] : ( ord_less_eq_set_tm @ ( sset_tm @ ( sdrop_tm @ N @ S ) ) @ ( sset_tm @ S ) ) ).
% sset_sdrop
thf(fact_470_sset__sdrop,axiom,
! [N: nat,S: stream_fm] : ( ord_less_eq_set_fm @ ( sset_fm @ ( sdrop_fm @ N @ S ) ) @ ( sset_fm @ S ) ) ).
% sset_sdrop
thf(fact_471_sset__sdrop,axiom,
! [N: nat,S: stream_rule] : ( ord_less_eq_set_rule @ ( sset_rule @ ( sdrop_rule @ N @ S ) ) @ ( sset_rule @ S ) ) ).
% sset_sdrop
thf(fact_472_irreflI,axiom,
! [R: set_Pr1261947904930325089at_nat] :
( ! [A4: nat] :
~ ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ A4 @ A4 ) @ R )
=> ( irrefl_on_nat @ top_top_set_nat @ R ) ) ).
% irreflI
thf(fact_473_irreflI,axiom,
! [R: set_Pr6724411253394807111ist_fm] :
( ! [A4: produc6018962875968178549ist_fm] :
~ ( member858341428514850704ist_fm @ ( produc5488031402154692311ist_fm @ A4 @ A4 ) @ R )
=> ( irrefl7757270364544296811ist_fm @ top_to1730629564055774885ist_fm @ R ) ) ).
% irreflI
thf(fact_474_irreflD,axiom,
! [R: set_Pr1261947904930325089at_nat,X3: nat] :
( ( irrefl_on_nat @ top_top_set_nat @ R )
=> ~ ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X3 @ X3 ) @ R ) ) ).
% irreflD
thf(fact_475_irreflD,axiom,
! [R: set_Pr6724411253394807111ist_fm,X3: produc6018962875968178549ist_fm] :
( ( irrefl7757270364544296811ist_fm @ top_to1730629564055774885ist_fm @ R )
=> ~ ( member858341428514850704ist_fm @ ( produc5488031402154692311ist_fm @ X3 @ X3 ) @ R ) ) ).
% irreflD
thf(fact_476_stl__def,axiom,
( stl_Pr12655793849929990m_rule
= ( case_s7424745816878926122m_rule
@ ^ [X12: produc340336539035504054m_rule,X22: stream2709947120125613254m_rule] : X22 ) ) ).
% stl_def
thf(fact_477_stl__def,axiom,
( stl_rule
= ( case_s2939844242074606818m_rule
@ ^ [X12: rule,X22: stream_rule] : X22 ) ) ).
% stl_def
thf(fact_478_shd__def,axiom,
( shd_Pr4562317740776619530m_rule
= ( case_s9014419740221122586m_rule
@ ^ [X12: produc340336539035504054m_rule,X22: stream2709947120125613254m_rule] : X12 ) ) ).
% shd_def
thf(fact_479_shd__def,axiom,
( shd_rule
= ( case_s8060621433503779026e_rule
@ ^ [X12: rule,X22: stream_rule] : X12 ) ) ).
% shd_def
thf(fact_480_PersistentRuleSystem_Oaxioms_I2_J,axiom,
! [Eff: rule > produc6018962875968178549ist_fm > fset_P8989946509869081563ist_fm > $o,Rules: stream_rule,S6: set_Pr5202636777678657877ist_fm] :
( ( abstra3967921542344661089ist_fm @ Eff @ Rules @ S6 )
=> ( abstra9069655090277518462ist_fm @ Eff @ Rules @ S6 ) ) ).
% PersistentRuleSystem.axioms(2)
thf(fact_481_Stream_Osmember__def,axiom,
( smember_fm
= ( ^ [X4: fm,S3: stream_fm] : ( member_fm @ X4 @ ( sset_fm @ S3 ) ) ) ) ).
% Stream.smember_def
thf(fact_482_Stream_Osmember__def,axiom,
( smembe2974859016285698739ist_fm
= ( ^ [X4: produc6018962875968178549ist_fm,S3: stream4408948924543953275ist_fm] : ( member4699826688122452638ist_fm @ X4 @ ( sset_P5379989128061332361ist_fm @ S3 ) ) ) ) ).
% Stream.smember_def
thf(fact_483_Stream_Osmember__def,axiom,
( smember_list_fm
= ( ^ [X4: list_fm,S3: stream_list_fm] : ( member_list_fm @ X4 @ ( sset_list_fm @ S3 ) ) ) ) ).
% Stream.smember_def
thf(fact_484_Stream_Osmember__def,axiom,
( smember_rule
= ( ^ [X4: rule,S3: stream_rule] : ( member_rule @ X4 @ ( sset_rule @ S3 ) ) ) ) ).
% Stream.smember_def
thf(fact_485_Stream_Osmember__def,axiom,
( smembe4439892024482649336m_rule
= ( ^ [X4: produc340336539035504054m_rule,S3: stream2709947120125613254m_rule] : ( member7231649785386036813m_rule @ X4 @ ( sset_P4484857331586881186m_rule @ S3 ) ) ) ) ).
% Stream.smember_def
thf(fact_486_lenlex__append2,axiom,
! [R3: set_Pr2455929065695642951_tm_tm,Us2: list_tm,Xs: list_tm,Ys: list_tm] :
( ( irrefl_on_tm @ top_top_set_tm @ R3 )
=> ( ( member4457312017796942864ist_tm @ ( produc1418304791525149271ist_tm @ ( append_tm @ Us2 @ Xs ) @ ( append_tm @ Us2 @ Ys ) ) @ ( lenlex_tm @ R3 ) )
= ( member4457312017796942864ist_tm @ ( produc1418304791525149271ist_tm @ Xs @ Ys ) @ ( lenlex_tm @ R3 ) ) ) ) ).
% lenlex_append2
thf(fact_487_lenlex__append2,axiom,
! [R3: set_Pr6724411253394807111ist_fm,Us2: list_P2887561121880082555ist_fm,Xs: list_P2887561121880082555ist_fm,Ys: list_P2887561121880082555ist_fm] :
( ( irrefl7757270364544296811ist_fm @ top_to1730629564055774885ist_fm @ R3 )
=> ( ( member6238937682488541200ist_fm @ ( produc8740291170414093911ist_fm @ ( append1476581234975291920ist_fm @ Us2 @ Xs ) @ ( append1476581234975291920ist_fm @ Us2 @ Ys ) ) @ ( lenlex4019133117718051968ist_fm @ R3 ) )
= ( member6238937682488541200ist_fm @ ( produc8740291170414093911ist_fm @ Xs @ Ys ) @ ( lenlex4019133117718051968ist_fm @ R3 ) ) ) ) ).
% lenlex_append2
thf(fact_488_sfilter__P,axiom,
! [P: produc340336539035504054m_rule > $o,S: stream2709947120125613254m_rule] :
( ( P @ ( shd_Pr4562317740776619530m_rule @ S ) )
=> ( ( sfilte3304650064355440278m_rule @ P @ S )
= ( sCons_2955607467875998431m_rule @ ( shd_Pr4562317740776619530m_rule @ S ) @ ( sfilte3304650064355440278m_rule @ P @ ( stl_Pr12655793849929990m_rule @ S ) ) ) ) ) ).
% sfilter_P
thf(fact_489_sfilter__P,axiom,
! [P: rule > $o,S: stream_rule] :
( ( P @ ( shd_rule @ S ) )
=> ( ( sfilter_rule @ P @ S )
= ( sCons_rule @ ( shd_rule @ S ) @ ( sfilter_rule @ P @ ( stl_rule @ S ) ) ) ) ) ).
% sfilter_P
thf(fact_490_parts__in__effect,axiom,
! [P2: fm,Z: list_fm,B4: list_tm,Z3: list_fm,R: rule,A: list_tm] :
( ( member_fm @ P2 @ ( set_fm2 @ Z ) )
=> ( ( fmembe3754813877001230652ist_fm @ ( produc1414352766439514085ist_fm @ B4 @ Z3 ) @ ( effect @ R @ ( produc1414352766439514085ist_fm @ A @ Z ) ) )
=> ? [C2: list_tm,Xs3: list_fm] :
( ( ord_less_eq_set_tm @ ( set_tm2 @ A ) @ ( set_tm2 @ C2 ) )
& ( member_list_fm @ Xs3 @ ( set_list_fm2 @ ( parts @ C2 @ R @ P2 ) ) )
& ( ord_less_eq_set_fm @ ( set_fm2 @ Xs3 ) @ ( set_fm2 @ Z3 ) ) ) ) ) ).
% parts_in_effect
thf(fact_491_irreflp__irrefl__eq,axiom,
! [R: set_Pr1261947904930325089at_nat] :
( ( irreflp_on_nat @ top_top_set_nat
@ ^ [A5: nat,B6: nat] : ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ A5 @ B6 ) @ R ) )
= ( irrefl_on_nat @ top_top_set_nat @ R ) ) ).
% irreflp_irrefl_eq
thf(fact_492_irreflp__irrefl__eq,axiom,
! [R: set_Pr6724411253394807111ist_fm] :
( ( irrefl1015672864661668943ist_fm @ top_to1730629564055774885ist_fm
@ ^ [A5: produc6018962875968178549ist_fm,B6: produc6018962875968178549ist_fm] : ( member858341428514850704ist_fm @ ( produc5488031402154692311ist_fm @ A5 @ B6 ) @ R ) )
= ( irrefl7757270364544296811ist_fm @ top_to1730629564055774885ist_fm @ R ) ) ).
% irreflp_irrefl_eq
thf(fact_493_Collect__restrict,axiom,
! [X5: set_Pr5202636777678657877ist_fm,P: produc6018962875968178549ist_fm > $o] :
( ord_le1771420097867575541ist_fm
@ ( collec5845330005437105888ist_fm
@ ^ [X4: produc6018962875968178549ist_fm] :
( ( member4699826688122452638ist_fm @ X4 @ X5 )
& ( P @ X4 ) ) )
@ X5 ) ).
% Collect_restrict
thf(fact_494_Collect__restrict,axiom,
! [X5: set_Pr1822751329126368876m_rule,P: produc340336539035504054m_rule > $o] :
( ord_le6390412330253371084m_rule
@ ( collec1009276759030335883m_rule
@ ^ [X4: produc340336539035504054m_rule] :
( ( member7231649785386036813m_rule @ X4 @ X5 )
& ( P @ X4 ) ) )
@ X5 ) ).
% Collect_restrict
thf(fact_495_Collect__restrict,axiom,
! [X5: set_list_fm,P: list_fm > $o] :
( ord_le7838213414353715577ist_fm
@ ( collect_list_fm
@ ^ [X4: list_fm] :
( ( member_list_fm @ X4 @ X5 )
& ( P @ X4 ) ) )
@ X5 ) ).
% Collect_restrict
thf(fact_496_Collect__restrict,axiom,
! [X5: set_tm,P: tm > $o] :
( ord_less_eq_set_tm
@ ( collect_tm
@ ^ [X4: tm] :
( ( member_tm @ X4 @ X5 )
& ( P @ X4 ) ) )
@ X5 ) ).
% Collect_restrict
thf(fact_497_Collect__restrict,axiom,
! [X5: set_fm,P: fm > $o] :
( ord_less_eq_set_fm
@ ( collect_fm
@ ^ [X4: fm] :
( ( member_fm @ X4 @ X5 )
& ( P @ X4 ) ) )
@ X5 ) ).
% Collect_restrict
thf(fact_498_Collect__restrict,axiom,
! [X5: set_rule,P: rule > $o] :
( ord_less_eq_set_rule
@ ( collect_rule
@ ^ [X4: rule] :
( ( member_rule @ X4 @ X5 )
& ( P @ X4 ) ) )
@ X5 ) ).
% Collect_restrict
thf(fact_499_prop__restrict,axiom,
! [X3: produc6018962875968178549ist_fm,Z7: set_Pr5202636777678657877ist_fm,X5: set_Pr5202636777678657877ist_fm,P: produc6018962875968178549ist_fm > $o] :
( ( member4699826688122452638ist_fm @ X3 @ Z7 )
=> ( ( ord_le1771420097867575541ist_fm @ Z7
@ ( collec5845330005437105888ist_fm
@ ^ [X4: produc6018962875968178549ist_fm] :
( ( member4699826688122452638ist_fm @ X4 @ X5 )
& ( P @ X4 ) ) ) )
=> ( P @ X3 ) ) ) ).
% prop_restrict
thf(fact_500_prop__restrict,axiom,
! [X3: produc340336539035504054m_rule,Z7: set_Pr1822751329126368876m_rule,X5: set_Pr1822751329126368876m_rule,P: produc340336539035504054m_rule > $o] :
( ( member7231649785386036813m_rule @ X3 @ Z7 )
=> ( ( ord_le6390412330253371084m_rule @ Z7
@ ( collec1009276759030335883m_rule
@ ^ [X4: produc340336539035504054m_rule] :
( ( member7231649785386036813m_rule @ X4 @ X5 )
& ( P @ X4 ) ) ) )
=> ( P @ X3 ) ) ) ).
% prop_restrict
thf(fact_501_prop__restrict,axiom,
! [X3: list_fm,Z7: set_list_fm,X5: set_list_fm,P: list_fm > $o] :
( ( member_list_fm @ X3 @ Z7 )
=> ( ( ord_le7838213414353715577ist_fm @ Z7
@ ( collect_list_fm
@ ^ [X4: list_fm] :
( ( member_list_fm @ X4 @ X5 )
& ( P @ X4 ) ) ) )
=> ( P @ X3 ) ) ) ).
% prop_restrict
thf(fact_502_prop__restrict,axiom,
! [X3: tm,Z7: set_tm,X5: set_tm,P: tm > $o] :
( ( member_tm @ X3 @ Z7 )
=> ( ( ord_less_eq_set_tm @ Z7
@ ( collect_tm
@ ^ [X4: tm] :
( ( member_tm @ X4 @ X5 )
& ( P @ X4 ) ) ) )
=> ( P @ X3 ) ) ) ).
% prop_restrict
thf(fact_503_prop__restrict,axiom,
! [X3: fm,Z7: set_fm,X5: set_fm,P: fm > $o] :
( ( member_fm @ X3 @ Z7 )
=> ( ( ord_less_eq_set_fm @ Z7
@ ( collect_fm
@ ^ [X4: fm] :
( ( member_fm @ X4 @ X5 )
& ( P @ X4 ) ) ) )
=> ( P @ X3 ) ) ) ).
% prop_restrict
thf(fact_504_prop__restrict,axiom,
! [X3: rule,Z7: set_rule,X5: set_rule,P: rule > $o] :
( ( member_rule @ X3 @ Z7 )
=> ( ( ord_less_eq_set_rule @ Z7
@ ( collect_rule
@ ^ [X4: rule] :
( ( member_rule @ X4 @ X5 )
& ( P @ X4 ) ) ) )
=> ( P @ X3 ) ) ) ).
% prop_restrict
thf(fact_505_conj__subset__def,axiom,
! [A: set_tm,P: tm > $o,Q: tm > $o] :
( ( ord_less_eq_set_tm @ A
@ ( collect_tm
@ ^ [X4: tm] :
( ( P @ X4 )
& ( Q @ X4 ) ) ) )
= ( ( ord_less_eq_set_tm @ A @ ( collect_tm @ P ) )
& ( ord_less_eq_set_tm @ A @ ( collect_tm @ Q ) ) ) ) ).
% conj_subset_def
thf(fact_506_conj__subset__def,axiom,
! [A: set_fm,P: fm > $o,Q: fm > $o] :
( ( ord_less_eq_set_fm @ A
@ ( collect_fm
@ ^ [X4: fm] :
( ( P @ X4 )
& ( Q @ X4 ) ) ) )
= ( ( ord_less_eq_set_fm @ A @ ( collect_fm @ P ) )
& ( ord_less_eq_set_fm @ A @ ( collect_fm @ Q ) ) ) ) ).
% conj_subset_def
thf(fact_507_conj__subset__def,axiom,
! [A: set_rule,P: rule > $o,Q: rule > $o] :
( ( ord_less_eq_set_rule @ A
@ ( collect_rule
@ ^ [X4: rule] :
( ( P @ X4 )
& ( Q @ X4 ) ) ) )
= ( ( ord_less_eq_set_rule @ A @ ( collect_rule @ P ) )
& ( ord_less_eq_set_rule @ A @ ( collect_rule @ Q ) ) ) ) ).
% conj_subset_def
thf(fact_508_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_509_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_510_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_511_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_512_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_513_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_514_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_515_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_516_sfilter__not__P,axiom,
! [P: produc340336539035504054m_rule > $o,S: stream2709947120125613254m_rule] :
( ~ ( P @ ( shd_Pr4562317740776619530m_rule @ S ) )
=> ( ( sfilte3304650064355440278m_rule @ P @ S )
= ( sfilte3304650064355440278m_rule @ P @ ( stl_Pr12655793849929990m_rule @ S ) ) ) ) ).
% sfilter_not_P
thf(fact_517_sfilter__not__P,axiom,
! [P: rule > $o,S: stream_rule] :
( ~ ( P @ ( shd_rule @ S ) )
=> ( ( sfilter_rule @ P @ S )
= ( sfilter_rule @ P @ ( stl_rule @ S ) ) ) ) ).
% sfilter_not_P
thf(fact_518_subrelI,axiom,
! [R: set_Pr5202636777678657877ist_fm,S: set_Pr5202636777678657877ist_fm] :
( ! [X: list_tm,Y: list_fm] :
( ( member4699826688122452638ist_fm @ ( produc1414352766439514085ist_fm @ X @ Y ) @ R )
=> ( member4699826688122452638ist_fm @ ( produc1414352766439514085ist_fm @ X @ Y ) @ S ) )
=> ( ord_le1771420097867575541ist_fm @ R @ S ) ) ).
% subrelI
thf(fact_519_subrelI,axiom,
! [R: set_Pr1822751329126368876m_rule,S: set_Pr1822751329126368876m_rule] :
( ! [X: produc6018962875968178549ist_fm,Y: rule] :
( ( member7231649785386036813m_rule @ ( produc1733806532565653680m_rule @ X @ Y ) @ R )
=> ( member7231649785386036813m_rule @ ( produc1733806532565653680m_rule @ X @ Y ) @ S ) )
=> ( ord_le6390412330253371084m_rule @ R @ S ) ) ).
% subrelI
thf(fact_520_subrelI,axiom,
! [R: set_Pr1261947904930325089at_nat,S: set_Pr1261947904930325089at_nat] :
( ! [X: nat,Y: nat] :
( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X @ Y ) @ R )
=> ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X @ Y ) @ S ) )
=> ( ord_le3146513528884898305at_nat @ R @ S ) ) ).
% subrelI
thf(fact_521_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_522_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_523_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_524_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_525_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_526_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_527_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_528_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_529_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_530_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_531_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_532_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_533_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_534_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_535_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_536_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_537_lenlex__irreflexive,axiom,
! [R: set_Pr1261947904930325089at_nat,Xs: list_nat] :
( ! [X: nat] :
~ ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X @ X ) @ R )
=> ~ ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ Xs @ Xs ) @ ( lenlex_nat @ R ) ) ) ).
% lenlex_irreflexive
thf(fact_538_ssubst__Pair__rhs,axiom,
! [R: list_tm,S: list_fm,R3: set_Pr5202636777678657877ist_fm,S5: list_fm] :
( ( member4699826688122452638ist_fm @ ( produc1414352766439514085ist_fm @ R @ S ) @ R3 )
=> ( ( S5 = S )
=> ( member4699826688122452638ist_fm @ ( produc1414352766439514085ist_fm @ R @ S5 ) @ R3 ) ) ) ).
% ssubst_Pair_rhs
thf(fact_539_ssubst__Pair__rhs,axiom,
! [R: produc6018962875968178549ist_fm,S: rule,R3: set_Pr1822751329126368876m_rule,S5: rule] :
( ( member7231649785386036813m_rule @ ( produc1733806532565653680m_rule @ R @ S ) @ R3 )
=> ( ( S5 = S )
=> ( member7231649785386036813m_rule @ ( produc1733806532565653680m_rule @ R @ S5 ) @ R3 ) ) ) ).
% ssubst_Pair_rhs
thf(fact_540_ssubst__Pair__rhs,axiom,
! [R: nat,S: nat,R3: set_Pr1261947904930325089at_nat,S5: nat] :
( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ R @ S ) @ R3 )
=> ( ( S5 = S )
=> ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ R @ S5 ) @ R3 ) ) ) ).
% ssubst_Pair_rhs
thf(fact_541_irreflp__on__irrefl__on__eq,axiom,
! [A: set_nat,R: set_Pr1261947904930325089at_nat] :
( ( irreflp_on_nat @ A
@ ^ [A5: nat,B6: nat] : ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ A5 @ B6 ) @ R ) )
= ( irrefl_on_nat @ A @ R ) ) ).
% irreflp_on_irrefl_on_eq
thf(fact_542_irrefl__lex,axiom,
! [R: set_Pr6724411253394807111ist_fm] :
( ( irrefl7757270364544296811ist_fm @ top_to1730629564055774885ist_fm @ R )
=> ( irrefl4218442000938873713ist_fm @ top_to7413187565660093483ist_fm @ ( lex_Pr7037290254703045599ist_fm @ R ) ) ) ).
% irrefl_lex
thf(fact_543_tree__fms__in__pseq,axiom,
! [P2: fm,Steps: stream2709947120125613254m_rule] :
( ( member_fm @ P2 @ ( tree_fms @ Steps ) )
=> ? [N2: nat] : ( member_fm @ P2 @ ( set_fm2 @ ( pseq @ ( snth_P6679518042731451922m_rule @ Steps @ N2 ) ) ) ) ) ).
% tree_fms_in_pseq
thf(fact_544_lexl__not__refl,axiom,
! [R: set_Pr6724411253394807111ist_fm,X3: list_P2887561121880082555ist_fm] :
( ( irrefl7757270364544296811ist_fm @ top_to1730629564055774885ist_fm @ R )
=> ~ ( member6238937682488541200ist_fm @ ( produc8740291170414093911ist_fm @ X3 @ X3 ) @ ( lex_Pr7037290254703045599ist_fm @ R ) ) ) ).
% lexl_not_refl
thf(fact_545_lexord__append__left__rightI,axiom,
! [A2: tm,B2: tm,R: set_Pr2455929065695642951_tm_tm,U: list_tm,X3: list_tm,Y3: list_tm] :
( ( member3121616906494481296_tm_tm @ ( product_Pair_tm_tm @ A2 @ B2 ) @ R )
=> ( member4457312017796942864ist_tm @ ( produc1418304791525149271ist_tm @ ( append_tm @ U @ ( cons_tm @ A2 @ X3 ) ) @ ( append_tm @ U @ ( cons_tm @ B2 @ Y3 ) ) ) @ ( lexord_tm @ R ) ) ) ).
% lexord_append_left_rightI
thf(fact_546_lexord__append__left__rightI,axiom,
! [A2: nat,B2: nat,R: set_Pr1261947904930325089at_nat,U: list_nat,X3: list_nat,Y3: list_nat] :
( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ A2 @ B2 ) @ R )
=> ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ ( append_nat @ U @ ( cons_nat @ A2 @ X3 ) ) @ ( append_nat @ U @ ( cons_nat @ B2 @ Y3 ) ) ) @ ( lexord_nat @ R ) ) ) ).
% lexord_append_left_rightI
thf(fact_547_RuleSystem__axioms,axiom,
abstra5221733350967904376ist_fm @ eff @ rules @ top_to1730629564055774885ist_fm ).
% RuleSystem_axioms
thf(fact_548_holds__sfilter,axiom,
! [Q: produc340336539035504054m_rule > $o,S: stream2709947120125613254m_rule,P: produc340336539035504054m_rule > $o] :
( ( linear1816559245158234865m_rule @ ( linear7925859179493631106m_rule @ Q ) @ S )
=> ( ( linear7925859179493631106m_rule @ P @ ( sfilte3304650064355440278m_rule @ Q @ S ) )
= ( linear5457325955354885675m_rule
@ ^ [Xs2: stream2709947120125613254m_rule] :
~ ( linear7925859179493631106m_rule @ Q @ Xs2 )
@ ( linear7925859179493631106m_rule
@ ^ [Xs2: produc340336539035504054m_rule] :
( ( Q @ Xs2 )
& ( P @ Xs2 ) ) )
@ S ) ) ) ).
% holds_sfilter
thf(fact_549_holds__sfilter,axiom,
! [Q: rule > $o,S: stream_rule,P: rule > $o] :
( ( linear292482573288640205v_rule @ ( linear7354821883049305950s_rule @ Q ) @ S )
=> ( ( linear7354821883049305950s_rule @ P @ ( sfilter_rule @ Q @ S ) )
= ( linear1836410492716652551l_rule
@ ^ [Xs2: stream_rule] :
~ ( linear7354821883049305950s_rule @ Q @ Xs2 )
@ ( linear7354821883049305950s_rule
@ ^ [Xs2: rule] :
( ( Q @ Xs2 )
& ( P @ Xs2 ) ) )
@ S ) ) ) ).
% holds_sfilter
thf(fact_550_lexord__asymmetric,axiom,
! [R3: set_Pr6724411253394807111ist_fm,A2: list_P2887561121880082555ist_fm,B2: list_P2887561121880082555ist_fm] :
( ( asym_o6181534149136668839ist_fm @ top_to1730629564055774885ist_fm @ R3 )
=> ( ( member6238937682488541200ist_fm @ ( produc8740291170414093911ist_fm @ A2 @ B2 ) @ ( lexord4892917800681400680ist_fm @ R3 ) )
=> ~ ( member6238937682488541200ist_fm @ ( produc8740291170414093911ist_fm @ B2 @ A2 ) @ ( lexord4892917800681400680ist_fm @ R3 ) ) ) ) ).
% lexord_asymmetric
thf(fact_551_list__all__simps_I1_J,axiom,
! [P: produc340336539035504054m_rule > $o,X3: produc340336539035504054m_rule,Xs: list_P2774625669004309958m_rule] :
( ( list_a9038620553111681026m_rule @ P @ ( cons_P2818572562524668032m_rule @ X3 @ Xs ) )
= ( ( P @ X3 )
& ( list_a9038620553111681026m_rule @ P @ Xs ) ) ) ).
% list_all_simps(1)
thf(fact_552_list_Opred__inject_I2_J,axiom,
! [P: produc340336539035504054m_rule > $o,A2: produc340336539035504054m_rule,Aa: list_P2774625669004309958m_rule] :
( ( list_a9038620553111681026m_rule @ P @ ( cons_P2818572562524668032m_rule @ A2 @ Aa ) )
= ( ( P @ A2 )
& ( list_a9038620553111681026m_rule @ P @ Aa ) ) ) ).
% list.pred_inject(2)
thf(fact_553_asym__onI,axiom,
! [A: set_rule,R: set_Pr7340090144965549541e_rule] :
( ! [X: rule,Y: rule] :
( ( member_rule @ X @ A )
=> ( ( member_rule @ Y @ A )
=> ( ( member1444691403335511342e_rule @ ( produc5849431337705160893e_rule @ X @ Y ) @ R )
=> ~ ( member1444691403335511342e_rule @ ( produc5849431337705160893e_rule @ Y @ X ) @ R ) ) ) )
=> ( asym_on_rule @ A @ R ) ) ).
% asym_onI
thf(fact_554_asym__onI,axiom,
! [A: set_fm,R: set_Pr4706815898642364871_fm_fm] :
( ! [X: fm,Y: fm] :
( ( member_fm @ X @ A )
=> ( ( member_fm @ Y @ A )
=> ( ( member7780952600467998736_fm_fm @ ( product_Pair_fm_fm @ X @ Y ) @ R )
=> ~ ( member7780952600467998736_fm_fm @ ( product_Pair_fm_fm @ Y @ X ) @ R ) ) ) )
=> ( asym_on_fm @ A @ R ) ) ).
% asym_onI
thf(fact_555_asym__onI,axiom,
! [A: set_Pr5202636777678657877ist_fm,R: set_Pr6724411253394807111ist_fm] :
( ! [X: produc6018962875968178549ist_fm,Y: produc6018962875968178549ist_fm] :
( ( member4699826688122452638ist_fm @ X @ A )
=> ( ( member4699826688122452638ist_fm @ Y @ A )
=> ( ( member858341428514850704ist_fm @ ( produc5488031402154692311ist_fm @ X @ Y ) @ R )
=> ~ ( member858341428514850704ist_fm @ ( produc5488031402154692311ist_fm @ Y @ X ) @ R ) ) ) )
=> ( asym_o6181534149136668839ist_fm @ A @ R ) ) ).
% asym_onI
thf(fact_556_asym__onI,axiom,
! [A: set_Pr1822751329126368876m_rule,R: set_Pr4971326047967503661m_rule] :
( ! [X: produc340336539035504054m_rule,Y: produc340336539035504054m_rule] :
( ( member7231649785386036813m_rule @ X @ A )
=> ( ( member7231649785386036813m_rule @ Y @ A )
=> ( ( member1411645983481675894m_rule @ ( produc6261311607089640965m_rule @ X @ Y ) @ R )
=> ~ ( member1411645983481675894m_rule @ ( produc6261311607089640965m_rule @ Y @ X ) @ R ) ) ) )
=> ( asym_o3513717604774472196m_rule @ A @ R ) ) ).
% asym_onI
thf(fact_557_asym__onI,axiom,
! [A: set_list_fm,R: set_Pr1916137733696349511ist_fm] :
( ! [X: list_fm,Y: list_fm] :
( ( member_list_fm @ X @ A )
=> ( ( member_list_fm @ Y @ A )
=> ( ( member1926098302810316688ist_fm @ ( produc7863996417982153943ist_fm @ X @ Y ) @ R )
=> ~ ( member1926098302810316688ist_fm @ ( produc7863996417982153943ist_fm @ Y @ X ) @ R ) ) ) )
=> ( asym_on_list_fm @ A @ R ) ) ).
% asym_onI
thf(fact_558_asym__onI,axiom,
! [A: set_nat,R: set_Pr1261947904930325089at_nat] :
( ! [X: nat,Y: nat] :
( ( member_nat @ X @ A )
=> ( ( member_nat @ Y @ A )
=> ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X @ Y ) @ R )
=> ~ ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ Y @ X ) @ R ) ) ) )
=> ( asym_on_nat @ A @ R ) ) ).
% asym_onI
thf(fact_559_snth__smap2,axiom,
! [F: produc340336539035504054m_rule > produc340336539035504054m_rule > produc340336539035504054m_rule,S1: stream2709947120125613254m_rule,S2: stream2709947120125613254m_rule,N: nat] :
( ( snth_P6679518042731451922m_rule @ ( smap2_6945574070536835298m_rule @ F @ S1 @ S2 ) @ N )
= ( F @ ( snth_P6679518042731451922m_rule @ S1 @ N ) @ ( snth_P6679518042731451922m_rule @ S2 @ N ) ) ) ).
% snth_smap2
thf(fact_560_snth__smap2,axiom,
! [F: produc340336539035504054m_rule > rule > produc340336539035504054m_rule,S1: stream2709947120125613254m_rule,S2: stream_rule,N: nat] :
( ( snth_P6679518042731451922m_rule @ ( smap2_6366920367380388542m_rule @ F @ S1 @ S2 ) @ N )
= ( F @ ( snth_P6679518042731451922m_rule @ S1 @ N ) @ ( snth_rule @ S2 @ N ) ) ) ).
% snth_smap2
thf(fact_561_snth__smap2,axiom,
! [F: rule > produc340336539035504054m_rule > produc340336539035504054m_rule,S1: stream_rule,S2: stream2709947120125613254m_rule,N: nat] :
( ( snth_P6679518042731451922m_rule @ ( smap2_4579943528986984894m_rule @ F @ S1 @ S2 ) @ N )
= ( F @ ( snth_rule @ S1 @ N ) @ ( snth_P6679518042731451922m_rule @ S2 @ N ) ) ) ).
% snth_smap2
thf(fact_562_snth__smap2,axiom,
! [F: rule > rule > produc340336539035504054m_rule,S1: stream_rule,S2: stream_rule,N: nat] :
( ( snth_P6679518042731451922m_rule @ ( smap2_4319354692778897306m_rule @ F @ S1 @ S2 ) @ N )
= ( F @ ( snth_rule @ S1 @ N ) @ ( snth_rule @ S2 @ N ) ) ) ).
% snth_smap2
thf(fact_563_snth__smap2,axiom,
! [F: produc340336539035504054m_rule > produc340336539035504054m_rule > rule,S1: stream2709947120125613254m_rule,S2: stream2709947120125613254m_rule,N: nat] :
( ( snth_rule @ ( smap2_1878155219598472126e_rule @ F @ S1 @ S2 ) @ N )
= ( F @ ( snth_P6679518042731451922m_rule @ S1 @ N ) @ ( snth_P6679518042731451922m_rule @ S2 @ N ) ) ) ).
% snth_smap2
thf(fact_564_snth__smap2,axiom,
! [F: produc340336539035504054m_rule > rule > rule,S1: stream2709947120125613254m_rule,S2: stream_rule,N: nat] :
( ( snth_rule @ ( smap2_8971878701134242202e_rule @ F @ S1 @ S2 ) @ N )
= ( F @ ( snth_P6679518042731451922m_rule @ S1 @ N ) @ ( snth_rule @ S2 @ N ) ) ) ).
% snth_smap2
thf(fact_565_snth__smap2,axiom,
! [F: rule > produc340336539035504054m_rule > rule,S1: stream_rule,S2: stream2709947120125613254m_rule,N: nat] :
( ( snth_rule @ ( smap2_9053961581851756698e_rule @ F @ S1 @ S2 ) @ N )
= ( F @ ( snth_rule @ S1 @ N ) @ ( snth_P6679518042731451922m_rule @ S2 @ N ) ) ) ).
% snth_smap2
thf(fact_566_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_567_asymI,axiom,
! [R: set_Pr1261947904930325089at_nat] :
( ! [X: nat,Y: nat] :
( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X @ Y ) @ R )
=> ~ ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ Y @ X ) @ R ) )
=> ( asym_on_nat @ top_top_set_nat @ R ) ) ).
% asymI
thf(fact_568_asymI,axiom,
! [R: set_Pr6724411253394807111ist_fm] :
( ! [X: produc6018962875968178549ist_fm,Y: produc6018962875968178549ist_fm] :
( ( member858341428514850704ist_fm @ ( produc5488031402154692311ist_fm @ X @ Y ) @ R )
=> ~ ( member858341428514850704ist_fm @ ( produc5488031402154692311ist_fm @ Y @ X ) @ R ) )
=> ( asym_o6181534149136668839ist_fm @ top_to1730629564055774885ist_fm @ R ) ) ).
% asymI
thf(fact_569_sdrop__simps_I1_J,axiom,
! [N: nat,S: stream2709947120125613254m_rule] :
( ( shd_Pr4562317740776619530m_rule @ ( sdrop_8169176516188972301m_rule @ N @ S ) )
= ( snth_P6679518042731451922m_rule @ S @ N ) ) ).
% sdrop_simps(1)
thf(fact_570_sdrop__simps_I1_J,axiom,
! [N: nat,S: stream_rule] :
( ( shd_rule @ ( sdrop_rule @ N @ S ) )
= ( snth_rule @ S @ N ) ) ).
% sdrop_simps(1)
thf(fact_571_lexord__cons__cons,axiom,
! [A2: nat,X3: list_nat,B2: nat,Y3: list_nat,R: set_Pr1261947904930325089at_nat] :
( ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ ( cons_nat @ A2 @ X3 ) @ ( cons_nat @ B2 @ Y3 ) ) @ ( lexord_nat @ R ) )
= ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ A2 @ B2 ) @ R )
| ( ( A2 = B2 )
& ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ X3 @ Y3 ) @ ( lexord_nat @ R ) ) ) ) ) ).
% lexord_cons_cons
thf(fact_572_snth__szip,axiom,
! [S1: stream2709947120125613254m_rule,S2: stream2709947120125613254m_rule,N: nat] :
( ( snth_P8853764340393315953m_rule @ ( szip_P811719526838699976m_rule @ S1 @ S2 ) @ N )
= ( produc6261311607089640965m_rule @ ( snth_P6679518042731451922m_rule @ S1 @ N ) @ ( snth_P6679518042731451922m_rule @ S2 @ N ) ) ) ).
% snth_szip
thf(fact_573_snth__szip,axiom,
! [S1: stream2709947120125613254m_rule,S2: stream_rule,N: nat] :
( ( snth_P6395297581347168077e_rule @ ( szip_P2044787555563963556e_rule @ S1 @ S2 ) @ N )
= ( produc4831648765031708129e_rule @ ( snth_P6679518042731451922m_rule @ S1 @ N ) @ ( snth_rule @ S2 @ N ) ) ) ).
% snth_szip
thf(fact_574_snth__szip,axiom,
! [S1: stream_rule,S2: stream2709947120125613254m_rule,N: nat] :
( ( snth_P5203088247548055629m_rule @ ( szip_r6533552703345879972m_rule @ S1 @ S2 ) @ N )
= ( produc97041875958848737m_rule @ ( snth_rule @ S1 @ N ) @ ( snth_P6679518042731451922m_rule @ S2 @ N ) ) ) ).
% snth_szip
thf(fact_575_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_576_snth__szip,axiom,
! [S1: stream_list_tm,S2: stream_list_fm,N: nat] :
( ( snth_P7093566783922538521ist_fm @ ( szip_list_tm_list_fm @ S1 @ S2 ) @ N )
= ( produc1414352766439514085ist_fm @ ( snth_list_tm @ S1 @ N ) @ ( snth_list_fm @ S2 @ N ) ) ) ).
% snth_szip
thf(fact_577_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_578_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_579_asym__lex,axiom,
! [R3: set_Pr6724411253394807111ist_fm] :
( ( asym_o6181534149136668839ist_fm @ top_to1730629564055774885ist_fm @ R3 )
=> ( asym_o415074664785499821ist_fm @ top_to7413187565660093483ist_fm @ ( lex_Pr7037290254703045599ist_fm @ R3 ) ) ) ).
% asym_lex
thf(fact_580_list_Oset__intros_I2_J,axiom,
! [Y3: rule,X222: list_rule,X21: rule] :
( ( member_rule @ Y3 @ ( set_rule2 @ X222 ) )
=> ( member_rule @ Y3 @ ( set_rule2 @ ( cons_rule @ X21 @ X222 ) ) ) ) ).
% list.set_intros(2)
thf(fact_581_list_Oset__intros_I2_J,axiom,
! [Y3: produc6018962875968178549ist_fm,X222: list_P2887561121880082555ist_fm,X21: produc6018962875968178549ist_fm] :
( ( member4699826688122452638ist_fm @ Y3 @ ( set_Pr5051287786238754058ist_fm @ X222 ) )
=> ( member4699826688122452638ist_fm @ Y3 @ ( set_Pr5051287786238754058ist_fm @ ( cons_P6586734674047340843ist_fm @ X21 @ X222 ) ) ) ) ).
% list.set_intros(2)
thf(fact_582_list_Oset__intros_I2_J,axiom,
! [Y3: produc340336539035504054m_rule,X222: list_P2774625669004309958m_rule,X21: produc340336539035504054m_rule] :
( ( member7231649785386036813m_rule @ Y3 @ ( set_Pr4534715572506550497m_rule @ X222 ) )
=> ( member7231649785386036813m_rule @ Y3 @ ( set_Pr4534715572506550497m_rule @ ( cons_P2818572562524668032m_rule @ X21 @ X222 ) ) ) ) ).
% list.set_intros(2)
thf(fact_583_list_Oset__intros_I2_J,axiom,
! [Y3: fm,X222: list_fm,X21: fm] :
( ( member_fm @ Y3 @ ( set_fm2 @ X222 ) )
=> ( member_fm @ Y3 @ ( set_fm2 @ ( cons_fm @ X21 @ X222 ) ) ) ) ).
% list.set_intros(2)
thf(fact_584_list_Oset__intros_I2_J,axiom,
! [Y3: tm,X222: list_tm,X21: tm] :
( ( member_tm @ Y3 @ ( set_tm2 @ X222 ) )
=> ( member_tm @ Y3 @ ( set_tm2 @ ( cons_tm @ X21 @ X222 ) ) ) ) ).
% list.set_intros(2)
thf(fact_585_list_Oset__intros_I2_J,axiom,
! [Y3: list_fm,X222: list_list_fm,X21: list_fm] :
( ( member_list_fm @ Y3 @ ( set_list_fm2 @ X222 ) )
=> ( member_list_fm @ Y3 @ ( set_list_fm2 @ ( cons_list_fm @ X21 @ X222 ) ) ) ) ).
% list.set_intros(2)
thf(fact_586_list_Oset__intros_I1_J,axiom,
! [X21: rule,X222: list_rule] : ( member_rule @ X21 @ ( set_rule2 @ ( cons_rule @ X21 @ X222 ) ) ) ).
% list.set_intros(1)
thf(fact_587_list_Oset__intros_I1_J,axiom,
! [X21: produc6018962875968178549ist_fm,X222: list_P2887561121880082555ist_fm] : ( member4699826688122452638ist_fm @ X21 @ ( set_Pr5051287786238754058ist_fm @ ( cons_P6586734674047340843ist_fm @ X21 @ X222 ) ) ) ).
% list.set_intros(1)
thf(fact_588_list_Oset__intros_I1_J,axiom,
! [X21: produc340336539035504054m_rule,X222: list_P2774625669004309958m_rule] : ( member7231649785386036813m_rule @ X21 @ ( set_Pr4534715572506550497m_rule @ ( cons_P2818572562524668032m_rule @ X21 @ X222 ) ) ) ).
% list.set_intros(1)
thf(fact_589_list_Oset__intros_I1_J,axiom,
! [X21: fm,X222: list_fm] : ( member_fm @ X21 @ ( set_fm2 @ ( cons_fm @ X21 @ X222 ) ) ) ).
% list.set_intros(1)
thf(fact_590_list_Oset__intros_I1_J,axiom,
! [X21: tm,X222: list_tm] : ( member_tm @ X21 @ ( set_tm2 @ ( cons_tm @ X21 @ X222 ) ) ) ).
% list.set_intros(1)
thf(fact_591_list_Oset__intros_I1_J,axiom,
! [X21: list_fm,X222: list_list_fm] : ( member_list_fm @ X21 @ ( set_list_fm2 @ ( cons_list_fm @ X21 @ X222 ) ) ) ).
% list.set_intros(1)
thf(fact_592_list_Oset__cases,axiom,
! [E: rule,A2: list_rule] :
( ( member_rule @ E @ ( set_rule2 @ A2 ) )
=> ( ! [Z22: list_rule] :
( A2
!= ( cons_rule @ E @ Z22 ) )
=> ~ ! [Z1: rule,Z22: list_rule] :
( ( A2
= ( cons_rule @ Z1 @ Z22 ) )
=> ~ ( member_rule @ E @ ( set_rule2 @ Z22 ) ) ) ) ) ).
% list.set_cases
thf(fact_593_list_Oset__cases,axiom,
! [E: produc6018962875968178549ist_fm,A2: list_P2887561121880082555ist_fm] :
( ( member4699826688122452638ist_fm @ E @ ( set_Pr5051287786238754058ist_fm @ A2 ) )
=> ( ! [Z22: list_P2887561121880082555ist_fm] :
( A2
!= ( cons_P6586734674047340843ist_fm @ E @ Z22 ) )
=> ~ ! [Z1: produc6018962875968178549ist_fm,Z22: list_P2887561121880082555ist_fm] :
( ( A2
= ( cons_P6586734674047340843ist_fm @ Z1 @ Z22 ) )
=> ~ ( member4699826688122452638ist_fm @ E @ ( set_Pr5051287786238754058ist_fm @ Z22 ) ) ) ) ) ).
% list.set_cases
thf(fact_594_list_Oset__cases,axiom,
! [E: produc340336539035504054m_rule,A2: list_P2774625669004309958m_rule] :
( ( member7231649785386036813m_rule @ E @ ( set_Pr4534715572506550497m_rule @ A2 ) )
=> ( ! [Z22: list_P2774625669004309958m_rule] :
( A2
!= ( cons_P2818572562524668032m_rule @ E @ Z22 ) )
=> ~ ! [Z1: produc340336539035504054m_rule,Z22: list_P2774625669004309958m_rule] :
( ( A2
= ( cons_P2818572562524668032m_rule @ Z1 @ Z22 ) )
=> ~ ( member7231649785386036813m_rule @ E @ ( set_Pr4534715572506550497m_rule @ Z22 ) ) ) ) ) ).
% list.set_cases
thf(fact_595_list_Oset__cases,axiom,
! [E: fm,A2: list_fm] :
( ( member_fm @ E @ ( set_fm2 @ A2 ) )
=> ( ! [Z22: list_fm] :
( A2
!= ( cons_fm @ E @ Z22 ) )
=> ~ ! [Z1: fm,Z22: list_fm] :
( ( A2
= ( cons_fm @ Z1 @ Z22 ) )
=> ~ ( member_fm @ E @ ( set_fm2 @ Z22 ) ) ) ) ) ).
% list.set_cases
thf(fact_596_list_Oset__cases,axiom,
! [E: tm,A2: list_tm] :
( ( member_tm @ E @ ( set_tm2 @ A2 ) )
=> ( ! [Z22: list_tm] :
( A2
!= ( cons_tm @ E @ Z22 ) )
=> ~ ! [Z1: tm,Z22: list_tm] :
( ( A2
= ( cons_tm @ Z1 @ Z22 ) )
=> ~ ( member_tm @ E @ ( set_tm2 @ Z22 ) ) ) ) ) ).
% list.set_cases
thf(fact_597_list_Oset__cases,axiom,
! [E: list_fm,A2: list_list_fm] :
( ( member_list_fm @ E @ ( set_list_fm2 @ A2 ) )
=> ( ! [Z22: list_list_fm] :
( A2
!= ( cons_list_fm @ E @ Z22 ) )
=> ~ ! [Z1: list_fm,Z22: list_list_fm] :
( ( A2
= ( cons_list_fm @ Z1 @ Z22 ) )
=> ~ ( member_list_fm @ E @ ( set_list_fm2 @ Z22 ) ) ) ) ) ).
% list.set_cases
thf(fact_598_set__ConsD,axiom,
! [Y3: rule,X3: rule,Xs: list_rule] :
( ( member_rule @ Y3 @ ( set_rule2 @ ( cons_rule @ X3 @ Xs ) ) )
=> ( ( Y3 = X3 )
| ( member_rule @ Y3 @ ( set_rule2 @ Xs ) ) ) ) ).
% set_ConsD
thf(fact_599_set__ConsD,axiom,
! [Y3: produc6018962875968178549ist_fm,X3: produc6018962875968178549ist_fm,Xs: list_P2887561121880082555ist_fm] :
( ( member4699826688122452638ist_fm @ Y3 @ ( set_Pr5051287786238754058ist_fm @ ( cons_P6586734674047340843ist_fm @ X3 @ Xs ) ) )
=> ( ( Y3 = X3 )
| ( member4699826688122452638ist_fm @ Y3 @ ( set_Pr5051287786238754058ist_fm @ Xs ) ) ) ) ).
% set_ConsD
thf(fact_600_set__ConsD,axiom,
! [Y3: produc340336539035504054m_rule,X3: produc340336539035504054m_rule,Xs: list_P2774625669004309958m_rule] :
( ( member7231649785386036813m_rule @ Y3 @ ( set_Pr4534715572506550497m_rule @ ( cons_P2818572562524668032m_rule @ X3 @ Xs ) ) )
=> ( ( Y3 = X3 )
| ( member7231649785386036813m_rule @ Y3 @ ( set_Pr4534715572506550497m_rule @ Xs ) ) ) ) ).
% set_ConsD
thf(fact_601_set__ConsD,axiom,
! [Y3: fm,X3: fm,Xs: list_fm] :
( ( member_fm @ Y3 @ ( set_fm2 @ ( cons_fm @ X3 @ Xs ) ) )
=> ( ( Y3 = X3 )
| ( member_fm @ Y3 @ ( set_fm2 @ Xs ) ) ) ) ).
% set_ConsD
thf(fact_602_set__ConsD,axiom,
! [Y3: tm,X3: tm,Xs: list_tm] :
( ( member_tm @ Y3 @ ( set_tm2 @ ( cons_tm @ X3 @ Xs ) ) )
=> ( ( Y3 = X3 )
| ( member_tm @ Y3 @ ( set_tm2 @ Xs ) ) ) ) ).
% set_ConsD
thf(fact_603_set__ConsD,axiom,
! [Y3: list_fm,X3: list_fm,Xs: list_list_fm] :
( ( member_list_fm @ Y3 @ ( set_list_fm2 @ ( cons_list_fm @ X3 @ Xs ) ) )
=> ( ( Y3 = X3 )
| ( member_list_fm @ Y3 @ ( set_list_fm2 @ Xs ) ) ) ) ).
% set_ConsD
thf(fact_604_Cons__eq__appendI,axiom,
! [X3: tm,Xs1: list_tm,Ys: list_tm,Xs: list_tm,Zs: list_tm] :
( ( ( cons_tm @ X3 @ Xs1 )
= Ys )
=> ( ( Xs
= ( append_tm @ Xs1 @ Zs ) )
=> ( ( cons_tm @ X3 @ Xs )
= ( append_tm @ Ys @ Zs ) ) ) ) ).
% Cons_eq_appendI
thf(fact_605_append__Cons,axiom,
! [X3: tm,Xs: list_tm,Ys: list_tm] :
( ( append_tm @ ( cons_tm @ X3 @ Xs ) @ Ys )
= ( cons_tm @ X3 @ ( append_tm @ Xs @ Ys ) ) ) ).
% append_Cons
thf(fact_606_suntil_Ostep,axiom,
! [Phi: stream2709947120125613254m_rule > $o,Omega: stream2709947120125613254m_rule,Psi: stream2709947120125613254m_rule > $o] :
( ( Phi @ Omega )
=> ( ( linear5457325955354885675m_rule @ Phi @ Psi @ ( stl_Pr12655793849929990m_rule @ Omega ) )
=> ( linear5457325955354885675m_rule @ Phi @ Psi @ Omega ) ) ) ).
% suntil.step
thf(fact_607_suntil_Ostep,axiom,
! [Phi: stream_rule > $o,Omega: stream_rule,Psi: stream_rule > $o] :
( ( Phi @ Omega )
=> ( ( linear1836410492716652551l_rule @ Phi @ Psi @ ( stl_rule @ Omega ) )
=> ( linear1836410492716652551l_rule @ Phi @ Psi @ Omega ) ) ) ).
% suntil.step
thf(fact_608_suntil_Ocases,axiom,
! [Phi: stream2709947120125613254m_rule > $o,Psi: stream2709947120125613254m_rule > $o,A2: stream2709947120125613254m_rule] :
( ( linear5457325955354885675m_rule @ Phi @ Psi @ A2 )
=> ( ~ ( Psi @ A2 )
=> ~ ( ( Phi @ A2 )
=> ~ ( linear5457325955354885675m_rule @ Phi @ Psi @ ( stl_Pr12655793849929990m_rule @ A2 ) ) ) ) ) ).
% suntil.cases
thf(fact_609_suntil_Ocases,axiom,
! [Phi: stream_rule > $o,Psi: stream_rule > $o,A2: stream_rule] :
( ( linear1836410492716652551l_rule @ Phi @ Psi @ A2 )
=> ( ~ ( Psi @ A2 )
=> ~ ( ( Phi @ A2 )
=> ~ ( linear1836410492716652551l_rule @ Phi @ Psi @ ( stl_rule @ A2 ) ) ) ) ) ).
% suntil.cases
thf(fact_610_suntil_Osimps,axiom,
( linear5457325955354885675m_rule
= ( ^ [Phi2: stream2709947120125613254m_rule > $o,Psi2: stream2709947120125613254m_rule > $o,A5: stream2709947120125613254m_rule] :
( ? [Omega2: stream2709947120125613254m_rule] :
( ( A5 = Omega2 )
& ( Psi2 @ Omega2 ) )
| ? [Omega2: stream2709947120125613254m_rule] :
( ( A5 = Omega2 )
& ( Phi2 @ Omega2 )
& ( linear5457325955354885675m_rule @ Phi2 @ Psi2 @ ( stl_Pr12655793849929990m_rule @ Omega2 ) ) ) ) ) ) ).
% suntil.simps
thf(fact_611_suntil_Osimps,axiom,
( linear1836410492716652551l_rule
= ( ^ [Phi2: stream_rule > $o,Psi2: stream_rule > $o,A5: stream_rule] :
( ? [Omega2: stream_rule] :
( ( A5 = Omega2 )
& ( Psi2 @ Omega2 ) )
| ? [Omega2: stream_rule] :
( ( A5 = Omega2 )
& ( Phi2 @ Omega2 )
& ( linear1836410492716652551l_rule @ Phi2 @ Psi2 @ ( stl_rule @ Omega2 ) ) ) ) ) ) ).
% suntil.simps
thf(fact_612_suntil__induct__strong,axiom,
! [Phi: stream2709947120125613254m_rule > $o,Psi: stream2709947120125613254m_rule > $o,X3: stream2709947120125613254m_rule,P: stream2709947120125613254m_rule > $o] :
( ( linear5457325955354885675m_rule @ Phi @ Psi @ X3 )
=> ( ! [Omega3: stream2709947120125613254m_rule] :
( ( Psi @ Omega3 )
=> ( P @ Omega3 ) )
=> ( ! [Omega3: stream2709947120125613254m_rule] :
( ( Phi @ Omega3 )
=> ( ~ ( Psi @ Omega3 )
=> ( ( linear5457325955354885675m_rule @ Phi @ Psi @ ( stl_Pr12655793849929990m_rule @ Omega3 ) )
=> ( ( P @ ( stl_Pr12655793849929990m_rule @ Omega3 ) )
=> ( P @ Omega3 ) ) ) ) )
=> ( P @ X3 ) ) ) ) ).
% suntil_induct_strong
thf(fact_613_suntil__induct__strong,axiom,
! [Phi: stream_rule > $o,Psi: stream_rule > $o,X3: stream_rule,P: stream_rule > $o] :
( ( linear1836410492716652551l_rule @ Phi @ Psi @ X3 )
=> ( ! [Omega3: stream_rule] :
( ( Psi @ Omega3 )
=> ( P @ Omega3 ) )
=> ( ! [Omega3: stream_rule] :
( ( Phi @ Omega3 )
=> ( ~ ( Psi @ Omega3 )
=> ( ( linear1836410492716652551l_rule @ Phi @ Psi @ ( stl_rule @ Omega3 ) )
=> ( ( P @ ( stl_rule @ Omega3 ) )
=> ( P @ Omega3 ) ) ) ) )
=> ( P @ X3 ) ) ) ) ).
% suntil_induct_strong
thf(fact_614_snth__sset,axiom,
! [S: stream_fm,N: nat] : ( member_fm @ ( snth_fm @ S @ N ) @ ( sset_fm @ S ) ) ).
% snth_sset
thf(fact_615_snth__sset,axiom,
! [S: stream4408948924543953275ist_fm,N: nat] : ( member4699826688122452638ist_fm @ ( snth_P7093566783922538521ist_fm @ S @ N ) @ ( sset_P5379989128061332361ist_fm @ S ) ) ).
% snth_sset
thf(fact_616_snth__sset,axiom,
! [S: stream_list_fm,N: nat] : ( member_list_fm @ ( snth_list_fm @ S @ N ) @ ( sset_list_fm @ S ) ) ).
% snth_sset
thf(fact_617_snth__sset,axiom,
! [S: stream_rule,N: nat] : ( member_rule @ ( snth_rule @ S @ N ) @ ( sset_rule @ S ) ) ).
% snth_sset
thf(fact_618_snth__sset,axiom,
! [S: stream2709947120125613254m_rule,N: nat] : ( member7231649785386036813m_rule @ ( snth_P6679518042731451922m_rule @ S @ N ) @ ( sset_P4484857331586881186m_rule @ S ) ) ).
% snth_sset
thf(fact_619_ev__suntil__iff,axiom,
! [P: stream2709947120125613254m_rule > $o,Q: stream2709947120125613254m_rule > $o,Omega: stream2709947120125613254m_rule] :
( ( linear1816559245158234865m_rule @ ( linear5457325955354885675m_rule @ P @ Q ) @ Omega )
= ( linear1816559245158234865m_rule @ Q @ Omega ) ) ).
% ev_suntil_iff
thf(fact_620_ev__suntil__iff,axiom,
! [P: stream_rule > $o,Q: stream_rule > $o,Omega: stream_rule] :
( ( linear292482573288640205v_rule @ ( linear1836410492716652551l_rule @ P @ Q ) @ Omega )
= ( linear292482573288640205v_rule @ Q @ Omega ) ) ).
% ev_suntil_iff
thf(fact_621_ev__suntil,axiom,
! [Phi: stream2709947120125613254m_rule > $o,Psi: stream2709947120125613254m_rule > $o,Omega: stream2709947120125613254m_rule] :
( ( linear5457325955354885675m_rule @ Phi @ Psi @ Omega )
=> ( linear1816559245158234865m_rule @ Psi @ Omega ) ) ).
% ev_suntil
thf(fact_622_ev__suntil,axiom,
! [Phi: stream_rule > $o,Psi: stream_rule > $o,Omega: stream_rule] :
( ( linear1836410492716652551l_rule @ Phi @ Psi @ Omega )
=> ( linear292482573288640205v_rule @ Psi @ Omega ) ) ).
% ev_suntil
thf(fact_623_asym__onD,axiom,
! [A: set_rule,R: set_Pr7340090144965549541e_rule,X3: rule,Y3: rule] :
( ( asym_on_rule @ A @ R )
=> ( ( member_rule @ X3 @ A )
=> ( ( member_rule @ Y3 @ A )
=> ( ( member1444691403335511342e_rule @ ( produc5849431337705160893e_rule @ X3 @ Y3 ) @ R )
=> ~ ( member1444691403335511342e_rule @ ( produc5849431337705160893e_rule @ Y3 @ X3 ) @ R ) ) ) ) ) ).
% asym_onD
thf(fact_624_asym__onD,axiom,
! [A: set_fm,R: set_Pr4706815898642364871_fm_fm,X3: fm,Y3: fm] :
( ( asym_on_fm @ A @ R )
=> ( ( member_fm @ X3 @ A )
=> ( ( member_fm @ Y3 @ A )
=> ( ( member7780952600467998736_fm_fm @ ( product_Pair_fm_fm @ X3 @ Y3 ) @ R )
=> ~ ( member7780952600467998736_fm_fm @ ( product_Pair_fm_fm @ Y3 @ X3 ) @ R ) ) ) ) ) ).
% asym_onD
thf(fact_625_asym__onD,axiom,
! [A: set_Pr5202636777678657877ist_fm,R: set_Pr6724411253394807111ist_fm,X3: produc6018962875968178549ist_fm,Y3: produc6018962875968178549ist_fm] :
( ( asym_o6181534149136668839ist_fm @ A @ R )
=> ( ( member4699826688122452638ist_fm @ X3 @ A )
=> ( ( member4699826688122452638ist_fm @ Y3 @ A )
=> ( ( member858341428514850704ist_fm @ ( produc5488031402154692311ist_fm @ X3 @ Y3 ) @ R )
=> ~ ( member858341428514850704ist_fm @ ( produc5488031402154692311ist_fm @ Y3 @ X3 ) @ R ) ) ) ) ) ).
% asym_onD
thf(fact_626_asym__onD,axiom,
! [A: set_Pr1822751329126368876m_rule,R: set_Pr4971326047967503661m_rule,X3: produc340336539035504054m_rule,Y3: produc340336539035504054m_rule] :
( ( asym_o3513717604774472196m_rule @ A @ R )
=> ( ( member7231649785386036813m_rule @ X3 @ A )
=> ( ( member7231649785386036813m_rule @ Y3 @ A )
=> ( ( member1411645983481675894m_rule @ ( produc6261311607089640965m_rule @ X3 @ Y3 ) @ R )
=> ~ ( member1411645983481675894m_rule @ ( produc6261311607089640965m_rule @ Y3 @ X3 ) @ R ) ) ) ) ) ).
% asym_onD
thf(fact_627_asym__onD,axiom,
! [A: set_list_fm,R: set_Pr1916137733696349511ist_fm,X3: list_fm,Y3: list_fm] :
( ( asym_on_list_fm @ A @ R )
=> ( ( member_list_fm @ X3 @ A )
=> ( ( member_list_fm @ Y3 @ A )
=> ( ( member1926098302810316688ist_fm @ ( produc7863996417982153943ist_fm @ X3 @ Y3 ) @ R )
=> ~ ( member1926098302810316688ist_fm @ ( produc7863996417982153943ist_fm @ Y3 @ X3 ) @ R ) ) ) ) ) ).
% asym_onD
thf(fact_628_asym__onD,axiom,
! [A: set_nat,R: set_Pr1261947904930325089at_nat,X3: nat,Y3: nat] :
( ( asym_on_nat @ A @ R )
=> ( ( member_nat @ X3 @ A )
=> ( ( member_nat @ Y3 @ A )
=> ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X3 @ Y3 ) @ R )
=> ~ ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ Y3 @ X3 ) @ R ) ) ) ) ) ).
% asym_onD
thf(fact_629_asym__on__def,axiom,
( asym_on_nat
= ( ^ [A6: set_nat,R4: set_Pr1261947904930325089at_nat] :
! [X4: nat] :
( ( member_nat @ X4 @ A6 )
=> ! [Y5: nat] :
( ( member_nat @ Y5 @ A6 )
=> ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X4 @ Y5 ) @ R4 )
=> ~ ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ Y5 @ X4 ) @ R4 ) ) ) ) ) ) ).
% asym_on_def
thf(fact_630_RuleSystem_Oenabled__R,axiom,
! [Eff: rule > rule > fset_rule > $o,Rules: stream_rule,S6: set_rule,S: rule] :
( ( abstra6805424310982398927e_rule @ Eff @ Rules @ S6 )
=> ( ( member_rule @ S @ S6 )
=> ? [X: rule] :
( ( member_rule @ X @ ( sset_rule @ Rules ) )
& ? [X_1: fset_rule] : ( Eff @ X @ S @ X_1 ) ) ) ) ).
% RuleSystem.enabled_R
thf(fact_631_RuleSystem_Oenabled__R,axiom,
! [Eff: rule > fm > fset_fm > $o,Rules: stream_rule,S6: set_fm,S: fm] :
( ( abstra4909020524819817846ule_fm @ Eff @ Rules @ S6 )
=> ( ( member_fm @ S @ S6 )
=> ? [X: rule] :
( ( member_rule @ X @ ( sset_rule @ Rules ) )
& ? [X_1: fset_fm] : ( Eff @ X @ S @ X_1 ) ) ) ) ).
% RuleSystem.enabled_R
thf(fact_632_RuleSystem_Oenabled__R,axiom,
! [Eff: rule > produc340336539035504054m_rule > fset_P5628638355292684902m_rule > $o,Rules: stream_rule,S6: set_Pr1822751329126368876m_rule,S: produc340336539035504054m_rule] :
( ( abstra3057654119625851379m_rule @ Eff @ Rules @ S6 )
=> ( ( member7231649785386036813m_rule @ S @ S6 )
=> ? [X: rule] :
( ( member_rule @ X @ ( sset_rule @ Rules ) )
& ? [X_1: fset_P5628638355292684902m_rule] : ( Eff @ X @ S @ X_1 ) ) ) ) ).
% RuleSystem.enabled_R
thf(fact_633_RuleSystem_Oenabled__R,axiom,
! [Eff: rule > list_fm > fset_list_fm > $o,Rules: stream_rule,S6: set_list_fm,S: list_fm] :
( ( abstra9153614075763688444ist_fm @ Eff @ Rules @ S6 )
=> ( ( member_list_fm @ S @ S6 )
=> ? [X: rule] :
( ( member_rule @ X @ ( sset_rule @ Rules ) )
& ? [X_1: fset_list_fm] : ( Eff @ X @ S @ X_1 ) ) ) ) ).
% RuleSystem.enabled_R
thf(fact_634_RuleSystem_Oenabled__R,axiom,
! [Eff: produc340336539035504054m_rule > rule > fset_rule > $o,Rules: stream2709947120125613254m_rule,S6: set_rule,S: rule] :
( ( abstra7792261008698710771e_rule @ Eff @ Rules @ S6 )
=> ( ( member_rule @ S @ S6 )
=> ? [X: produc340336539035504054m_rule] :
( ( member7231649785386036813m_rule @ X @ ( sset_P4484857331586881186m_rule @ Rules ) )
& ? [X_1: fset_rule] : ( Eff @ X @ S @ X_1 ) ) ) ) ).
% RuleSystem.enabled_R
thf(fact_635_RuleSystem_Oenabled__R,axiom,
! [Eff: produc340336539035504054m_rule > fm > fset_fm > $o,Rules: stream2709947120125613254m_rule,S6: set_fm,S: fm] :
( ( abstra6746019703889232722ule_fm @ Eff @ Rules @ S6 )
=> ( ( member_fm @ S @ S6 )
=> ? [X: produc340336539035504054m_rule] :
( ( member7231649785386036813m_rule @ X @ ( sset_P4484857331586881186m_rule @ Rules ) )
& ? [X_1: fset_fm] : ( Eff @ X @ S @ X_1 ) ) ) ) ).
% RuleSystem.enabled_R
thf(fact_636_RuleSystem_Oenabled__R,axiom,
! [Eff: produc340336539035504054m_rule > produc6018962875968178549ist_fm > fset_P8989946509869081563ist_fm > $o,Rules: stream2709947120125613254m_rule,S6: set_Pr5202636777678657877ist_fm,S: produc6018962875968178549ist_fm] :
( ( abstra406286406349174356ist_fm @ Eff @ Rules @ S6 )
=> ( ( member4699826688122452638ist_fm @ S @ S6 )
=> ? [X: produc340336539035504054m_rule] :
( ( member7231649785386036813m_rule @ X @ ( sset_P4484857331586881186m_rule @ Rules ) )
& ? [X_1: fset_P8989946509869081563ist_fm] : ( Eff @ X @ S @ X_1 ) ) ) ) ).
% RuleSystem.enabled_R
thf(fact_637_RuleSystem_Oenabled__R,axiom,
! [Eff: produc340336539035504054m_rule > produc340336539035504054m_rule > fset_P5628638355292684902m_rule > $o,Rules: stream2709947120125613254m_rule,S6: set_Pr1822751329126368876m_rule,S: produc340336539035504054m_rule] :
( ( abstra7375212488576167191m_rule @ Eff @ Rules @ S6 )
=> ( ( member7231649785386036813m_rule @ S @ S6 )
=> ? [X: produc340336539035504054m_rule] :
( ( member7231649785386036813m_rule @ X @ ( sset_P4484857331586881186m_rule @ Rules ) )
& ? [X_1: fset_P5628638355292684902m_rule] : ( Eff @ X @ S @ X_1 ) ) ) ) ).
% RuleSystem.enabled_R
thf(fact_638_RuleSystem_Oenabled__R,axiom,
! [Eff: produc340336539035504054m_rule > list_fm > fset_list_fm > $o,Rules: stream2709947120125613254m_rule,S6: set_list_fm,S: list_fm] :
( ( abstra6353961794954871768ist_fm @ Eff @ Rules @ S6 )
=> ( ( member_list_fm @ S @ S6 )
=> ? [X: produc340336539035504054m_rule] :
( ( member7231649785386036813m_rule @ X @ ( sset_P4484857331586881186m_rule @ Rules ) )
& ? [X_1: fset_list_fm] : ( Eff @ X @ S @ X_1 ) ) ) ) ).
% RuleSystem.enabled_R
thf(fact_639_RuleSystem_Oenabled__R,axiom,
! [Eff: rule > produc6018962875968178549ist_fm > fset_P8989946509869081563ist_fm > $o,Rules: stream_rule,S6: set_Pr5202636777678657877ist_fm,S: produc6018962875968178549ist_fm] :
( ( abstra5221733350967904376ist_fm @ Eff @ Rules @ S6 )
=> ( ( member4699826688122452638ist_fm @ S @ S6 )
=> ? [X: rule] :
( ( member_rule @ X @ ( sset_rule @ Rules ) )
& ? [X_1: fset_P8989946509869081563ist_fm] : ( Eff @ X @ S @ X_1 ) ) ) ) ).
% RuleSystem.enabled_R
thf(fact_640_smap2__alt,axiom,
! [F: produc340336539035504054m_rule > produc340336539035504054m_rule > produc340336539035504054m_rule,S1: stream2709947120125613254m_rule,S2: stream2709947120125613254m_rule,S: stream2709947120125613254m_rule] :
( ( ( smap2_6945574070536835298m_rule @ F @ S1 @ S2 )
= S )
= ( ! [N3: nat] :
( ( F @ ( snth_P6679518042731451922m_rule @ S1 @ N3 ) @ ( snth_P6679518042731451922m_rule @ S2 @ N3 ) )
= ( snth_P6679518042731451922m_rule @ S @ N3 ) ) ) ) ).
% smap2_alt
thf(fact_641_smap2__alt,axiom,
! [F: produc340336539035504054m_rule > produc340336539035504054m_rule > rule,S1: stream2709947120125613254m_rule,S2: stream2709947120125613254m_rule,S: stream_rule] :
( ( ( smap2_1878155219598472126e_rule @ F @ S1 @ S2 )
= S )
= ( ! [N3: nat] :
( ( F @ ( snth_P6679518042731451922m_rule @ S1 @ N3 ) @ ( snth_P6679518042731451922m_rule @ S2 @ N3 ) )
= ( snth_rule @ S @ N3 ) ) ) ) ).
% smap2_alt
thf(fact_642_smap2__alt,axiom,
! [F: produc340336539035504054m_rule > rule > produc340336539035504054m_rule,S1: stream2709947120125613254m_rule,S2: stream_rule,S: stream2709947120125613254m_rule] :
( ( ( smap2_6366920367380388542m_rule @ F @ S1 @ S2 )
= S )
= ( ! [N3: nat] :
( ( F @ ( snth_P6679518042731451922m_rule @ S1 @ N3 ) @ ( snth_rule @ S2 @ N3 ) )
= ( snth_P6679518042731451922m_rule @ S @ N3 ) ) ) ) ).
% smap2_alt
thf(fact_643_smap2__alt,axiom,
! [F: produc340336539035504054m_rule > rule > rule,S1: stream2709947120125613254m_rule,S2: stream_rule,S: stream_rule] :
( ( ( smap2_8971878701134242202e_rule @ F @ S1 @ S2 )
= S )
= ( ! [N3: nat] :
( ( F @ ( snth_P6679518042731451922m_rule @ S1 @ N3 ) @ ( snth_rule @ S2 @ N3 ) )
= ( snth_rule @ S @ N3 ) ) ) ) ).
% smap2_alt
thf(fact_644_smap2__alt,axiom,
! [F: rule > produc340336539035504054m_rule > produc340336539035504054m_rule,S1: stream_rule,S2: stream2709947120125613254m_rule,S: stream2709947120125613254m_rule] :
( ( ( smap2_4579943528986984894m_rule @ F @ S1 @ S2 )
= S )
= ( ! [N3: nat] :
( ( F @ ( snth_rule @ S1 @ N3 ) @ ( snth_P6679518042731451922m_rule @ S2 @ N3 ) )
= ( snth_P6679518042731451922m_rule @ S @ N3 ) ) ) ) ).
% smap2_alt
thf(fact_645_smap2__alt,axiom,
! [F: rule > produc340336539035504054m_rule > rule,S1: stream_rule,S2: stream2709947120125613254m_rule,S: stream_rule] :
( ( ( smap2_9053961581851756698e_rule @ F @ S1 @ S2 )
= S )
= ( ! [N3: nat] :
( ( F @ ( snth_rule @ S1 @ N3 ) @ ( snth_P6679518042731451922m_rule @ S2 @ N3 ) )
= ( snth_rule @ S @ N3 ) ) ) ) ).
% smap2_alt
thf(fact_646_smap2__alt,axiom,
! [F: rule > rule > produc340336539035504054m_rule,S1: stream_rule,S2: stream_rule,S: stream2709947120125613254m_rule] :
( ( ( smap2_4319354692778897306m_rule @ F @ S1 @ S2 )
= S )
= ( ! [N3: nat] :
( ( F @ ( snth_rule @ S1 @ N3 ) @ ( snth_rule @ S2 @ N3 ) )
= ( snth_P6679518042731451922m_rule @ S @ N3 ) ) ) ) ).
% smap2_alt
thf(fact_647_smap2__alt,axiom,
! [F: rule > rule > rule,S1: stream_rule,S2: stream_rule,S: stream_rule] :
( ( ( smap2_rule_rule_rule @ F @ S1 @ S2 )
= S )
= ( ! [N3: nat] :
( ( F @ ( snth_rule @ S1 @ N3 ) @ ( snth_rule @ S2 @ N3 ) )
= ( snth_rule @ S @ N3 ) ) ) ) ).
% smap2_alt
thf(fact_648_suntil__inv,axiom,
! [F: stream2709947120125613254m_rule > stream2709947120125613254m_rule,P: stream2709947120125613254m_rule > $o,Q: stream2709947120125613254m_rule > $o,S: stream2709947120125613254m_rule] :
( ! [S4: stream2709947120125613254m_rule] :
( ( F @ ( stl_Pr12655793849929990m_rule @ S4 ) )
= ( stl_Pr12655793849929990m_rule @ ( F @ S4 ) ) )
=> ( ( linear5457325955354885675m_rule @ P @ Q @ ( F @ S ) )
= ( linear5457325955354885675m_rule
@ ^ [X4: stream2709947120125613254m_rule] : ( P @ ( F @ X4 ) )
@ ^ [X4: stream2709947120125613254m_rule] : ( Q @ ( F @ X4 ) )
@ S ) ) ) ).
% suntil_inv
thf(fact_649_suntil__inv,axiom,
! [F: stream2709947120125613254m_rule > stream_rule,P: stream_rule > $o,Q: stream_rule > $o,S: stream2709947120125613254m_rule] :
( ! [S4: stream2709947120125613254m_rule] :
( ( F @ ( stl_Pr12655793849929990m_rule @ S4 ) )
= ( stl_rule @ ( F @ S4 ) ) )
=> ( ( linear1836410492716652551l_rule @ P @ Q @ ( F @ S ) )
= ( linear5457325955354885675m_rule
@ ^ [X4: stream2709947120125613254m_rule] : ( P @ ( F @ X4 ) )
@ ^ [X4: stream2709947120125613254m_rule] : ( Q @ ( F @ X4 ) )
@ S ) ) ) ).
% suntil_inv
thf(fact_650_suntil__inv,axiom,
! [F: stream_rule > stream2709947120125613254m_rule,P: stream2709947120125613254m_rule > $o,Q: stream2709947120125613254m_rule > $o,S: stream_rule] :
( ! [S4: stream_rule] :
( ( F @ ( stl_rule @ S4 ) )
= ( stl_Pr12655793849929990m_rule @ ( F @ S4 ) ) )
=> ( ( linear5457325955354885675m_rule @ P @ Q @ ( F @ S ) )
= ( linear1836410492716652551l_rule
@ ^ [X4: stream_rule] : ( P @ ( F @ X4 ) )
@ ^ [X4: stream_rule] : ( Q @ ( F @ X4 ) )
@ S ) ) ) ).
% suntil_inv
thf(fact_651_suntil__inv,axiom,
! [F: stream_rule > stream_rule,P: stream_rule > $o,Q: stream_rule > $o,S: stream_rule] :
( ! [S4: stream_rule] :
( ( F @ ( stl_rule @ S4 ) )
= ( stl_rule @ ( F @ S4 ) ) )
=> ( ( linear1836410492716652551l_rule @ P @ Q @ ( F @ S ) )
= ( linear1836410492716652551l_rule
@ ^ [X4: stream_rule] : ( P @ ( F @ X4 ) )
@ ^ [X4: stream_rule] : ( Q @ ( F @ X4 ) )
@ S ) ) ) ).
% suntil_inv
thf(fact_652_true__suntil,axiom,
! [P: stream2709947120125613254m_rule > $o] :
( ( linear5457325955354885675m_rule
@ ^ [Uu: stream2709947120125613254m_rule] : $true
@ P )
= ( linear1816559245158234865m_rule @ P ) ) ).
% true_suntil
thf(fact_653_true__suntil,axiom,
! [P: stream_rule > $o] :
( ( linear1836410492716652551l_rule
@ ^ [Uu: stream_rule] : $true
@ P )
= ( linear292482573288640205v_rule @ P ) ) ).
% true_suntil
thf(fact_654_ev__eq__suntil,axiom,
( linear1816559245158234865m_rule
= ( ^ [P3: stream2709947120125613254m_rule > $o] :
( linear5457325955354885675m_rule
@ ^ [Xs2: stream2709947120125613254m_rule] :
~ ( P3 @ Xs2 )
@ P3 ) ) ) ).
% ev_eq_suntil
thf(fact_655_ev__eq__suntil,axiom,
( linear292482573288640205v_rule
= ( ^ [P3: stream_rule > $o] :
( linear1836410492716652551l_rule
@ ^ [Xs2: stream_rule] :
~ ( P3 @ Xs2 )
@ P3 ) ) ) ).
% ev_eq_suntil
thf(fact_656_stream__all__def,axiom,
( stream2134157564947672044m_rule
= ( ^ [P3: produc340336539035504054m_rule > $o,S3: stream2709947120125613254m_rule] :
! [P4: nat] : ( P3 @ ( snth_P6679518042731451922m_rule @ S3 @ P4 ) ) ) ) ).
% stream_all_def
thf(fact_657_stream__all__def,axiom,
( stream_all_rule
= ( ^ [P3: rule > $o,S3: stream_rule] :
! [P4: nat] : ( P3 @ ( snth_rule @ S3 @ P4 ) ) ) ) ).
% stream_all_def
thf(fact_658_PersistentRuleSystem_Oaxioms_I1_J,axiom,
! [Eff: rule > produc6018962875968178549ist_fm > fset_P8989946509869081563ist_fm > $o,Rules: stream_rule,S6: set_Pr5202636777678657877ist_fm] :
( ( abstra3967921542344661089ist_fm @ Eff @ Rules @ S6 )
=> ( abstra5221733350967904376ist_fm @ Eff @ Rules @ S6 ) ) ).
% PersistentRuleSystem.axioms(1)
thf(fact_659_lexord__asym,axiom,
! [R3: set_Pr6724411253394807111ist_fm] :
( ( asym_o6181534149136668839ist_fm @ top_to1730629564055774885ist_fm @ R3 )
=> ( asym_o415074664785499821ist_fm @ top_to7413187565660093483ist_fm @ ( lexord4892917800681400680ist_fm @ R3 ) ) ) ).
% lexord_asym
thf(fact_660_asym__lenlex,axiom,
! [R3: set_Pr6724411253394807111ist_fm] :
( ( asym_o6181534149136668839ist_fm @ top_to1730629564055774885ist_fm @ R3 )
=> ( asym_o415074664785499821ist_fm @ top_to7413187565660093483ist_fm @ ( lenlex4019133117718051968ist_fm @ R3 ) ) ) ).
% asym_lenlex
thf(fact_661_set__subset__Cons,axiom,
! [Xs: list_list_fm,X3: list_fm] : ( ord_le7838213414353715577ist_fm @ ( set_list_fm2 @ Xs ) @ ( set_list_fm2 @ ( cons_list_fm @ X3 @ Xs ) ) ) ).
% set_subset_Cons
thf(fact_662_set__subset__Cons,axiom,
! [Xs: list_tm,X3: tm] : ( ord_less_eq_set_tm @ ( set_tm2 @ Xs ) @ ( set_tm2 @ ( cons_tm @ X3 @ Xs ) ) ) ).
% set_subset_Cons
thf(fact_663_set__subset__Cons,axiom,
! [Xs: list_fm,X3: fm] : ( ord_less_eq_set_fm @ ( set_fm2 @ Xs ) @ ( set_fm2 @ ( cons_fm @ X3 @ Xs ) ) ) ).
% set_subset_Cons
thf(fact_664_set__subset__Cons,axiom,
! [Xs: list_rule,X3: rule] : ( ord_less_eq_set_rule @ ( set_rule2 @ Xs ) @ ( set_rule2 @ ( cons_rule @ X3 @ Xs ) ) ) ).
% set_subset_Cons
thf(fact_665_split__list__first__prop__iff,axiom,
! [Xs: list_fm,P: fm > $o] :
( ( ? [X4: fm] :
( ( member_fm @ X4 @ ( set_fm2 @ Xs ) )
& ( P @ X4 ) ) )
= ( ? [Ys2: list_fm,X4: fm] :
( ? [Zs2: list_fm] :
( Xs
= ( append_fm @ Ys2 @ ( cons_fm @ X4 @ Zs2 ) ) )
& ( P @ X4 )
& ! [Y5: fm] :
( ( member_fm @ Y5 @ ( set_fm2 @ Ys2 ) )
=> ~ ( P @ Y5 ) ) ) ) ) ).
% split_list_first_prop_iff
thf(fact_666_split__list__first__prop__iff,axiom,
! [Xs: list_tm,P: tm > $o] :
( ( ? [X4: tm] :
( ( member_tm @ X4 @ ( set_tm2 @ Xs ) )
& ( P @ X4 ) ) )
= ( ? [Ys2: list_tm,X4: tm] :
( ? [Zs2: list_tm] :
( Xs
= ( append_tm @ Ys2 @ ( cons_tm @ X4 @ Zs2 ) ) )
& ( P @ X4 )
& ! [Y5: tm] :
( ( member_tm @ Y5 @ ( set_tm2 @ Ys2 ) )
=> ~ ( P @ Y5 ) ) ) ) ) ).
% split_list_first_prop_iff
thf(fact_667_split__list__first__prop__iff,axiom,
! [Xs: list_list_fm,P: list_fm > $o] :
( ( ? [X4: list_fm] :
( ( member_list_fm @ X4 @ ( set_list_fm2 @ Xs ) )
& ( P @ X4 ) ) )
= ( ? [Ys2: list_list_fm,X4: list_fm] :
( ? [Zs2: list_list_fm] :
( Xs
= ( append_list_fm @ Ys2 @ ( cons_list_fm @ X4 @ Zs2 ) ) )
& ( P @ X4 )
& ! [Y5: list_fm] :
( ( member_list_fm @ Y5 @ ( set_list_fm2 @ Ys2 ) )
=> ~ ( P @ Y5 ) ) ) ) ) ).
% split_list_first_prop_iff
thf(fact_668_split__list__last__prop__iff,axiom,
! [Xs: list_fm,P: fm > $o] :
( ( ? [X4: fm] :
( ( member_fm @ X4 @ ( set_fm2 @ Xs ) )
& ( P @ X4 ) ) )
= ( ? [Ys2: list_fm,X4: fm,Zs2: list_fm] :
( ( Xs
= ( append_fm @ Ys2 @ ( cons_fm @ X4 @ Zs2 ) ) )
& ( P @ X4 )
& ! [Y5: fm] :
( ( member_fm @ Y5 @ ( set_fm2 @ Zs2 ) )
=> ~ ( P @ Y5 ) ) ) ) ) ).
% split_list_last_prop_iff
thf(fact_669_split__list__last__prop__iff,axiom,
! [Xs: list_tm,P: tm > $o] :
( ( ? [X4: tm] :
( ( member_tm @ X4 @ ( set_tm2 @ Xs ) )
& ( P @ X4 ) ) )
= ( ? [Ys2: list_tm,X4: tm,Zs2: list_tm] :
( ( Xs
= ( append_tm @ Ys2 @ ( cons_tm @ X4 @ Zs2 ) ) )
& ( P @ X4 )
& ! [Y5: tm] :
( ( member_tm @ Y5 @ ( set_tm2 @ Zs2 ) )
=> ~ ( P @ Y5 ) ) ) ) ) ).
% split_list_last_prop_iff
thf(fact_670_split__list__last__prop__iff,axiom,
! [Xs: list_list_fm,P: list_fm > $o] :
( ( ? [X4: list_fm] :
( ( member_list_fm @ X4 @ ( set_list_fm2 @ Xs ) )
& ( P @ X4 ) ) )
= ( ? [Ys2: list_list_fm,X4: list_fm,Zs2: list_list_fm] :
( ( Xs
= ( append_list_fm @ Ys2 @ ( cons_list_fm @ X4 @ Zs2 ) ) )
& ( P @ X4 )
& ! [Y5: list_fm] :
( ( member_list_fm @ Y5 @ ( set_list_fm2 @ Zs2 ) )
=> ~ ( P @ Y5 ) ) ) ) ) ).
% split_list_last_prop_iff
thf(fact_671_in__set__conv__decomp__first,axiom,
! [X3: rule,Xs: list_rule] :
( ( member_rule @ X3 @ ( set_rule2 @ Xs ) )
= ( ? [Ys2: list_rule,Zs2: list_rule] :
( ( Xs
= ( append_rule @ Ys2 @ ( cons_rule @ X3 @ Zs2 ) ) )
& ~ ( member_rule @ X3 @ ( set_rule2 @ Ys2 ) ) ) ) ) ).
% in_set_conv_decomp_first
thf(fact_672_in__set__conv__decomp__first,axiom,
! [X3: produc6018962875968178549ist_fm,Xs: list_P2887561121880082555ist_fm] :
( ( member4699826688122452638ist_fm @ X3 @ ( set_Pr5051287786238754058ist_fm @ Xs ) )
= ( ? [Ys2: list_P2887561121880082555ist_fm,Zs2: list_P2887561121880082555ist_fm] :
( ( Xs
= ( append1476581234975291920ist_fm @ Ys2 @ ( cons_P6586734674047340843ist_fm @ X3 @ Zs2 ) ) )
& ~ ( member4699826688122452638ist_fm @ X3 @ ( set_Pr5051287786238754058ist_fm @ Ys2 ) ) ) ) ) ).
% in_set_conv_decomp_first
thf(fact_673_in__set__conv__decomp__first,axiom,
! [X3: produc340336539035504054m_rule,Xs: list_P2774625669004309958m_rule] :
( ( member7231649785386036813m_rule @ X3 @ ( set_Pr4534715572506550497m_rule @ Xs ) )
= ( ? [Ys2: list_P2774625669004309958m_rule,Zs2: list_P2774625669004309958m_rule] :
( ( Xs
= ( append2714808343334685275m_rule @ Ys2 @ ( cons_P2818572562524668032m_rule @ X3 @ Zs2 ) ) )
& ~ ( member7231649785386036813m_rule @ X3 @ ( set_Pr4534715572506550497m_rule @ Ys2 ) ) ) ) ) ).
% in_set_conv_decomp_first
thf(fact_674_in__set__conv__decomp__first,axiom,
! [X3: fm,Xs: list_fm] :
( ( member_fm @ X3 @ ( set_fm2 @ Xs ) )
= ( ? [Ys2: list_fm,Zs2: list_fm] :
( ( Xs
= ( append_fm @ Ys2 @ ( cons_fm @ X3 @ Zs2 ) ) )
& ~ ( member_fm @ X3 @ ( set_fm2 @ Ys2 ) ) ) ) ) ).
% in_set_conv_decomp_first
thf(fact_675_in__set__conv__decomp__first,axiom,
! [X3: tm,Xs: list_tm] :
( ( member_tm @ X3 @ ( set_tm2 @ Xs ) )
= ( ? [Ys2: list_tm,Zs2: list_tm] :
( ( Xs
= ( append_tm @ Ys2 @ ( cons_tm @ X3 @ Zs2 ) ) )
& ~ ( member_tm @ X3 @ ( set_tm2 @ Ys2 ) ) ) ) ) ).
% in_set_conv_decomp_first
thf(fact_676_in__set__conv__decomp__first,axiom,
! [X3: list_fm,Xs: list_list_fm] :
( ( member_list_fm @ X3 @ ( set_list_fm2 @ Xs ) )
= ( ? [Ys2: list_list_fm,Zs2: list_list_fm] :
( ( Xs
= ( append_list_fm @ Ys2 @ ( cons_list_fm @ X3 @ Zs2 ) ) )
& ~ ( member_list_fm @ X3 @ ( set_list_fm2 @ Ys2 ) ) ) ) ) ).
% in_set_conv_decomp_first
thf(fact_677_in__set__conv__decomp__last,axiom,
! [X3: rule,Xs: list_rule] :
( ( member_rule @ X3 @ ( set_rule2 @ Xs ) )
= ( ? [Ys2: list_rule,Zs2: list_rule] :
( ( Xs
= ( append_rule @ Ys2 @ ( cons_rule @ X3 @ Zs2 ) ) )
& ~ ( member_rule @ X3 @ ( set_rule2 @ Zs2 ) ) ) ) ) ).
% in_set_conv_decomp_last
thf(fact_678_in__set__conv__decomp__last,axiom,
! [X3: produc6018962875968178549ist_fm,Xs: list_P2887561121880082555ist_fm] :
( ( member4699826688122452638ist_fm @ X3 @ ( set_Pr5051287786238754058ist_fm @ Xs ) )
= ( ? [Ys2: list_P2887561121880082555ist_fm,Zs2: list_P2887561121880082555ist_fm] :
( ( Xs
= ( append1476581234975291920ist_fm @ Ys2 @ ( cons_P6586734674047340843ist_fm @ X3 @ Zs2 ) ) )
& ~ ( member4699826688122452638ist_fm @ X3 @ ( set_Pr5051287786238754058ist_fm @ Zs2 ) ) ) ) ) ).
% in_set_conv_decomp_last
thf(fact_679_in__set__conv__decomp__last,axiom,
! [X3: produc340336539035504054m_rule,Xs: list_P2774625669004309958m_rule] :
( ( member7231649785386036813m_rule @ X3 @ ( set_Pr4534715572506550497m_rule @ Xs ) )
= ( ? [Ys2: list_P2774625669004309958m_rule,Zs2: list_P2774625669004309958m_rule] :
( ( Xs
= ( append2714808343334685275m_rule @ Ys2 @ ( cons_P2818572562524668032m_rule @ X3 @ Zs2 ) ) )
& ~ ( member7231649785386036813m_rule @ X3 @ ( set_Pr4534715572506550497m_rule @ Zs2 ) ) ) ) ) ).
% in_set_conv_decomp_last
thf(fact_680_in__set__conv__decomp__last,axiom,
! [X3: fm,Xs: list_fm] :
( ( member_fm @ X3 @ ( set_fm2 @ Xs ) )
= ( ? [Ys2: list_fm,Zs2: list_fm] :
( ( Xs
= ( append_fm @ Ys2 @ ( cons_fm @ X3 @ Zs2 ) ) )
& ~ ( member_fm @ X3 @ ( set_fm2 @ Zs2 ) ) ) ) ) ).
% in_set_conv_decomp_last
thf(fact_681_in__set__conv__decomp__last,axiom,
! [X3: tm,Xs: list_tm] :
( ( member_tm @ X3 @ ( set_tm2 @ Xs ) )
= ( ? [Ys2: list_tm,Zs2: list_tm] :
( ( Xs
= ( append_tm @ Ys2 @ ( cons_tm @ X3 @ Zs2 ) ) )
& ~ ( member_tm @ X3 @ ( set_tm2 @ Zs2 ) ) ) ) ) ).
% in_set_conv_decomp_last
thf(fact_682_in__set__conv__decomp__last,axiom,
! [X3: list_fm,Xs: list_list_fm] :
( ( member_list_fm @ X3 @ ( set_list_fm2 @ Xs ) )
= ( ? [Ys2: list_list_fm,Zs2: list_list_fm] :
( ( Xs
= ( append_list_fm @ Ys2 @ ( cons_list_fm @ X3 @ Zs2 ) ) )
& ~ ( member_list_fm @ X3 @ ( set_list_fm2 @ Zs2 ) ) ) ) ) ).
% in_set_conv_decomp_last
thf(fact_683_split__list__first__propE,axiom,
! [Xs: list_fm,P: fm > $o] :
( ? [X6: fm] :
( ( member_fm @ X6 @ ( set_fm2 @ Xs ) )
& ( P @ X6 ) )
=> ~ ! [Ys3: list_fm,X: fm] :
( ? [Zs3: list_fm] :
( Xs
= ( append_fm @ Ys3 @ ( cons_fm @ X @ Zs3 ) ) )
=> ( ( P @ X )
=> ~ ! [Xa3: fm] :
( ( member_fm @ Xa3 @ ( set_fm2 @ Ys3 ) )
=> ~ ( P @ Xa3 ) ) ) ) ) ).
% split_list_first_propE
thf(fact_684_split__list__first__propE,axiom,
! [Xs: list_tm,P: tm > $o] :
( ? [X6: tm] :
( ( member_tm @ X6 @ ( set_tm2 @ Xs ) )
& ( P @ X6 ) )
=> ~ ! [Ys3: list_tm,X: tm] :
( ? [Zs3: list_tm] :
( Xs
= ( append_tm @ Ys3 @ ( cons_tm @ X @ Zs3 ) ) )
=> ( ( P @ X )
=> ~ ! [Xa3: tm] :
( ( member_tm @ Xa3 @ ( set_tm2 @ Ys3 ) )
=> ~ ( P @ Xa3 ) ) ) ) ) ).
% split_list_first_propE
thf(fact_685_split__list__first__propE,axiom,
! [Xs: list_list_fm,P: list_fm > $o] :
( ? [X6: list_fm] :
( ( member_list_fm @ X6 @ ( set_list_fm2 @ Xs ) )
& ( P @ X6 ) )
=> ~ ! [Ys3: list_list_fm,X: list_fm] :
( ? [Zs3: list_list_fm] :
( Xs
= ( append_list_fm @ Ys3 @ ( cons_list_fm @ X @ Zs3 ) ) )
=> ( ( P @ X )
=> ~ ! [Xa3: list_fm] :
( ( member_list_fm @ Xa3 @ ( set_list_fm2 @ Ys3 ) )
=> ~ ( P @ Xa3 ) ) ) ) ) ).
% split_list_first_propE
thf(fact_686_split__list__last__propE,axiom,
! [Xs: list_fm,P: fm > $o] :
( ? [X6: fm] :
( ( member_fm @ X6 @ ( set_fm2 @ Xs ) )
& ( P @ X6 ) )
=> ~ ! [Ys3: list_fm,X: fm,Zs3: list_fm] :
( ( Xs
= ( append_fm @ Ys3 @ ( cons_fm @ X @ Zs3 ) ) )
=> ( ( P @ X )
=> ~ ! [Xa3: fm] :
( ( member_fm @ Xa3 @ ( set_fm2 @ Zs3 ) )
=> ~ ( P @ Xa3 ) ) ) ) ) ).
% split_list_last_propE
thf(fact_687_split__list__last__propE,axiom,
! [Xs: list_tm,P: tm > $o] :
( ? [X6: tm] :
( ( member_tm @ X6 @ ( set_tm2 @ Xs ) )
& ( P @ X6 ) )
=> ~ ! [Ys3: list_tm,X: tm,Zs3: list_tm] :
( ( Xs
= ( append_tm @ Ys3 @ ( cons_tm @ X @ Zs3 ) ) )
=> ( ( P @ X )
=> ~ ! [Xa3: tm] :
( ( member_tm @ Xa3 @ ( set_tm2 @ Zs3 ) )
=> ~ ( P @ Xa3 ) ) ) ) ) ).
% split_list_last_propE
thf(fact_688_split__list__last__propE,axiom,
! [Xs: list_list_fm,P: list_fm > $o] :
( ? [X6: list_fm] :
( ( member_list_fm @ X6 @ ( set_list_fm2 @ Xs ) )
& ( P @ X6 ) )
=> ~ ! [Ys3: list_list_fm,X: list_fm,Zs3: list_list_fm] :
( ( Xs
= ( append_list_fm @ Ys3 @ ( cons_list_fm @ X @ Zs3 ) ) )
=> ( ( P @ X )
=> ~ ! [Xa3: list_fm] :
( ( member_list_fm @ Xa3 @ ( set_list_fm2 @ Zs3 ) )
=> ~ ( P @ Xa3 ) ) ) ) ) ).
% split_list_last_propE
thf(fact_689_split__list__first__prop,axiom,
! [Xs: list_fm,P: fm > $o] :
( ? [X6: fm] :
( ( member_fm @ X6 @ ( set_fm2 @ Xs ) )
& ( P @ X6 ) )
=> ? [Ys3: list_fm,X: fm] :
( ? [Zs3: list_fm] :
( Xs
= ( append_fm @ Ys3 @ ( cons_fm @ X @ Zs3 ) ) )
& ( P @ X )
& ! [Xa3: fm] :
( ( member_fm @ Xa3 @ ( set_fm2 @ Ys3 ) )
=> ~ ( P @ Xa3 ) ) ) ) ).
% split_list_first_prop
thf(fact_690_split__list__first__prop,axiom,
! [Xs: list_tm,P: tm > $o] :
( ? [X6: tm] :
( ( member_tm @ X6 @ ( set_tm2 @ Xs ) )
& ( P @ X6 ) )
=> ? [Ys3: list_tm,X: tm] :
( ? [Zs3: list_tm] :
( Xs
= ( append_tm @ Ys3 @ ( cons_tm @ X @ Zs3 ) ) )
& ( P @ X )
& ! [Xa3: tm] :
( ( member_tm @ Xa3 @ ( set_tm2 @ Ys3 ) )
=> ~ ( P @ Xa3 ) ) ) ) ).
% split_list_first_prop
thf(fact_691_split__list__first__prop,axiom,
! [Xs: list_list_fm,P: list_fm > $o] :
( ? [X6: list_fm] :
( ( member_list_fm @ X6 @ ( set_list_fm2 @ Xs ) )
& ( P @ X6 ) )
=> ? [Ys3: list_list_fm,X: list_fm] :
( ? [Zs3: list_list_fm] :
( Xs
= ( append_list_fm @ Ys3 @ ( cons_list_fm @ X @ Zs3 ) ) )
& ( P @ X )
& ! [Xa3: list_fm] :
( ( member_list_fm @ Xa3 @ ( set_list_fm2 @ Ys3 ) )
=> ~ ( P @ Xa3 ) ) ) ) ).
% split_list_first_prop
thf(fact_692_split__list__last__prop,axiom,
! [Xs: list_fm,P: fm > $o] :
( ? [X6: fm] :
( ( member_fm @ X6 @ ( set_fm2 @ Xs ) )
& ( P @ X6 ) )
=> ? [Ys3: list_fm,X: fm,Zs3: list_fm] :
( ( Xs
= ( append_fm @ Ys3 @ ( cons_fm @ X @ Zs3 ) ) )
& ( P @ X )
& ! [Xa3: fm] :
( ( member_fm @ Xa3 @ ( set_fm2 @ Zs3 ) )
=> ~ ( P @ Xa3 ) ) ) ) ).
% split_list_last_prop
thf(fact_693_split__list__last__prop,axiom,
! [Xs: list_tm,P: tm > $o] :
( ? [X6: tm] :
( ( member_tm @ X6 @ ( set_tm2 @ Xs ) )
& ( P @ X6 ) )
=> ? [Ys3: list_tm,X: tm,Zs3: list_tm] :
( ( Xs
= ( append_tm @ Ys3 @ ( cons_tm @ X @ Zs3 ) ) )
& ( P @ X )
& ! [Xa3: tm] :
( ( member_tm @ Xa3 @ ( set_tm2 @ Zs3 ) )
=> ~ ( P @ Xa3 ) ) ) ) ).
% split_list_last_prop
thf(fact_694_split__list__last__prop,axiom,
! [Xs: list_list_fm,P: list_fm > $o] :
( ? [X6: list_fm] :
( ( member_list_fm @ X6 @ ( set_list_fm2 @ Xs ) )
& ( P @ X6 ) )
=> ? [Ys3: list_list_fm,X: list_fm,Zs3: list_list_fm] :
( ( Xs
= ( append_list_fm @ Ys3 @ ( cons_list_fm @ X @ Zs3 ) ) )
& ( P @ X )
& ! [Xa3: list_fm] :
( ( member_list_fm @ Xa3 @ ( set_list_fm2 @ Zs3 ) )
=> ~ ( P @ Xa3 ) ) ) ) ).
% split_list_last_prop
thf(fact_695_in__set__conv__decomp,axiom,
! [X3: rule,Xs: list_rule] :
( ( member_rule @ X3 @ ( set_rule2 @ Xs ) )
= ( ? [Ys2: list_rule,Zs2: list_rule] :
( Xs
= ( append_rule @ Ys2 @ ( cons_rule @ X3 @ Zs2 ) ) ) ) ) ).
% in_set_conv_decomp
thf(fact_696_in__set__conv__decomp,axiom,
! [X3: produc6018962875968178549ist_fm,Xs: list_P2887561121880082555ist_fm] :
( ( member4699826688122452638ist_fm @ X3 @ ( set_Pr5051287786238754058ist_fm @ Xs ) )
= ( ? [Ys2: list_P2887561121880082555ist_fm,Zs2: list_P2887561121880082555ist_fm] :
( Xs
= ( append1476581234975291920ist_fm @ Ys2 @ ( cons_P6586734674047340843ist_fm @ X3 @ Zs2 ) ) ) ) ) ).
% in_set_conv_decomp
thf(fact_697_in__set__conv__decomp,axiom,
! [X3: produc340336539035504054m_rule,Xs: list_P2774625669004309958m_rule] :
( ( member7231649785386036813m_rule @ X3 @ ( set_Pr4534715572506550497m_rule @ Xs ) )
= ( ? [Ys2: list_P2774625669004309958m_rule,Zs2: list_P2774625669004309958m_rule] :
( Xs
= ( append2714808343334685275m_rule @ Ys2 @ ( cons_P2818572562524668032m_rule @ X3 @ Zs2 ) ) ) ) ) ).
% in_set_conv_decomp
thf(fact_698_in__set__conv__decomp,axiom,
! [X3: fm,Xs: list_fm] :
( ( member_fm @ X3 @ ( set_fm2 @ Xs ) )
= ( ? [Ys2: list_fm,Zs2: list_fm] :
( Xs
= ( append_fm @ Ys2 @ ( cons_fm @ X3 @ Zs2 ) ) ) ) ) ).
% in_set_conv_decomp
thf(fact_699_in__set__conv__decomp,axiom,
! [X3: tm,Xs: list_tm] :
( ( member_tm @ X3 @ ( set_tm2 @ Xs ) )
= ( ? [Ys2: list_tm,Zs2: list_tm] :
( Xs
= ( append_tm @ Ys2 @ ( cons_tm @ X3 @ Zs2 ) ) ) ) ) ).
% in_set_conv_decomp
thf(fact_700_in__set__conv__decomp,axiom,
! [X3: list_fm,Xs: list_list_fm] :
( ( member_list_fm @ X3 @ ( set_list_fm2 @ Xs ) )
= ( ? [Ys2: list_list_fm,Zs2: list_list_fm] :
( Xs
= ( append_list_fm @ Ys2 @ ( cons_list_fm @ X3 @ Zs2 ) ) ) ) ) ).
% in_set_conv_decomp
thf(fact_701_append__Cons__eq__iff,axiom,
! [X3: rule,Xs: list_rule,Ys: list_rule,Xs4: list_rule,Ys4: list_rule] :
( ~ ( member_rule @ X3 @ ( set_rule2 @ Xs ) )
=> ( ~ ( member_rule @ X3 @ ( set_rule2 @ Ys ) )
=> ( ( ( append_rule @ Xs @ ( cons_rule @ X3 @ Ys ) )
= ( append_rule @ Xs4 @ ( cons_rule @ X3 @ Ys4 ) ) )
= ( ( Xs = Xs4 )
& ( Ys = Ys4 ) ) ) ) ) ).
% append_Cons_eq_iff
thf(fact_702_append__Cons__eq__iff,axiom,
! [X3: produc6018962875968178549ist_fm,Xs: list_P2887561121880082555ist_fm,Ys: list_P2887561121880082555ist_fm,Xs4: list_P2887561121880082555ist_fm,Ys4: list_P2887561121880082555ist_fm] :
( ~ ( member4699826688122452638ist_fm @ X3 @ ( set_Pr5051287786238754058ist_fm @ Xs ) )
=> ( ~ ( member4699826688122452638ist_fm @ X3 @ ( set_Pr5051287786238754058ist_fm @ Ys ) )
=> ( ( ( append1476581234975291920ist_fm @ Xs @ ( cons_P6586734674047340843ist_fm @ X3 @ Ys ) )
= ( append1476581234975291920ist_fm @ Xs4 @ ( cons_P6586734674047340843ist_fm @ X3 @ Ys4 ) ) )
= ( ( Xs = Xs4 )
& ( Ys = Ys4 ) ) ) ) ) ).
% append_Cons_eq_iff
thf(fact_703_append__Cons__eq__iff,axiom,
! [X3: produc340336539035504054m_rule,Xs: list_P2774625669004309958m_rule,Ys: list_P2774625669004309958m_rule,Xs4: list_P2774625669004309958m_rule,Ys4: list_P2774625669004309958m_rule] :
( ~ ( member7231649785386036813m_rule @ X3 @ ( set_Pr4534715572506550497m_rule @ Xs ) )
=> ( ~ ( member7231649785386036813m_rule @ X3 @ ( set_Pr4534715572506550497m_rule @ Ys ) )
=> ( ( ( append2714808343334685275m_rule @ Xs @ ( cons_P2818572562524668032m_rule @ X3 @ Ys ) )
= ( append2714808343334685275m_rule @ Xs4 @ ( cons_P2818572562524668032m_rule @ X3 @ Ys4 ) ) )
= ( ( Xs = Xs4 )
& ( Ys = Ys4 ) ) ) ) ) ).
% append_Cons_eq_iff
thf(fact_704_append__Cons__eq__iff,axiom,
! [X3: fm,Xs: list_fm,Ys: list_fm,Xs4: list_fm,Ys4: list_fm] :
( ~ ( member_fm @ X3 @ ( set_fm2 @ Xs ) )
=> ( ~ ( member_fm @ X3 @ ( set_fm2 @ Ys ) )
=> ( ( ( append_fm @ Xs @ ( cons_fm @ X3 @ Ys ) )
= ( append_fm @ Xs4 @ ( cons_fm @ X3 @ Ys4 ) ) )
= ( ( Xs = Xs4 )
& ( Ys = Ys4 ) ) ) ) ) ).
% append_Cons_eq_iff
thf(fact_705_append__Cons__eq__iff,axiom,
! [X3: tm,Xs: list_tm,Ys: list_tm,Xs4: list_tm,Ys4: list_tm] :
( ~ ( member_tm @ X3 @ ( set_tm2 @ Xs ) )
=> ( ~ ( member_tm @ X3 @ ( set_tm2 @ Ys ) )
=> ( ( ( append_tm @ Xs @ ( cons_tm @ X3 @ Ys ) )
= ( append_tm @ Xs4 @ ( cons_tm @ X3 @ Ys4 ) ) )
= ( ( Xs = Xs4 )
& ( Ys = Ys4 ) ) ) ) ) ).
% append_Cons_eq_iff
thf(fact_706_append__Cons__eq__iff,axiom,
! [X3: list_fm,Xs: list_list_fm,Ys: list_list_fm,Xs4: list_list_fm,Ys4: list_list_fm] :
( ~ ( member_list_fm @ X3 @ ( set_list_fm2 @ Xs ) )
=> ( ~ ( member_list_fm @ X3 @ ( set_list_fm2 @ Ys ) )
=> ( ( ( append_list_fm @ Xs @ ( cons_list_fm @ X3 @ Ys ) )
= ( append_list_fm @ Xs4 @ ( cons_list_fm @ X3 @ Ys4 ) ) )
= ( ( Xs = Xs4 )
& ( Ys = Ys4 ) ) ) ) ) ).
% append_Cons_eq_iff
thf(fact_707_split__list__propE,axiom,
! [Xs: list_fm,P: fm > $o] :
( ? [X6: fm] :
( ( member_fm @ X6 @ ( set_fm2 @ Xs ) )
& ( P @ X6 ) )
=> ~ ! [Ys3: list_fm,X: fm] :
( ? [Zs3: list_fm] :
( Xs
= ( append_fm @ Ys3 @ ( cons_fm @ X @ Zs3 ) ) )
=> ~ ( P @ X ) ) ) ).
% split_list_propE
thf(fact_708_split__list__propE,axiom,
! [Xs: list_tm,P: tm > $o] :
( ? [X6: tm] :
( ( member_tm @ X6 @ ( set_tm2 @ Xs ) )
& ( P @ X6 ) )
=> ~ ! [Ys3: list_tm,X: tm] :
( ? [Zs3: list_tm] :
( Xs
= ( append_tm @ Ys3 @ ( cons_tm @ X @ Zs3 ) ) )
=> ~ ( P @ X ) ) ) ).
% split_list_propE
thf(fact_709_split__list__propE,axiom,
! [Xs: list_list_fm,P: list_fm > $o] :
( ? [X6: list_fm] :
( ( member_list_fm @ X6 @ ( set_list_fm2 @ Xs ) )
& ( P @ X6 ) )
=> ~ ! [Ys3: list_list_fm,X: list_fm] :
( ? [Zs3: list_list_fm] :
( Xs
= ( append_list_fm @ Ys3 @ ( cons_list_fm @ X @ Zs3 ) ) )
=> ~ ( P @ X ) ) ) ).
% split_list_propE
thf(fact_710_split__list__first,axiom,
! [X3: rule,Xs: list_rule] :
( ( member_rule @ X3 @ ( set_rule2 @ Xs ) )
=> ? [Ys3: list_rule,Zs3: list_rule] :
( ( Xs
= ( append_rule @ Ys3 @ ( cons_rule @ X3 @ Zs3 ) ) )
& ~ ( member_rule @ X3 @ ( set_rule2 @ Ys3 ) ) ) ) ).
% split_list_first
thf(fact_711_split__list__first,axiom,
! [X3: produc6018962875968178549ist_fm,Xs: list_P2887561121880082555ist_fm] :
( ( member4699826688122452638ist_fm @ X3 @ ( set_Pr5051287786238754058ist_fm @ Xs ) )
=> ? [Ys3: list_P2887561121880082555ist_fm,Zs3: list_P2887561121880082555ist_fm] :
( ( Xs
= ( append1476581234975291920ist_fm @ Ys3 @ ( cons_P6586734674047340843ist_fm @ X3 @ Zs3 ) ) )
& ~ ( member4699826688122452638ist_fm @ X3 @ ( set_Pr5051287786238754058ist_fm @ Ys3 ) ) ) ) ).
% split_list_first
thf(fact_712_split__list__first,axiom,
! [X3: produc340336539035504054m_rule,Xs: list_P2774625669004309958m_rule] :
( ( member7231649785386036813m_rule @ X3 @ ( set_Pr4534715572506550497m_rule @ Xs ) )
=> ? [Ys3: list_P2774625669004309958m_rule,Zs3: list_P2774625669004309958m_rule] :
( ( Xs
= ( append2714808343334685275m_rule @ Ys3 @ ( cons_P2818572562524668032m_rule @ X3 @ Zs3 ) ) )
& ~ ( member7231649785386036813m_rule @ X3 @ ( set_Pr4534715572506550497m_rule @ Ys3 ) ) ) ) ).
% split_list_first
thf(fact_713_split__list__first,axiom,
! [X3: fm,Xs: list_fm] :
( ( member_fm @ X3 @ ( set_fm2 @ Xs ) )
=> ? [Ys3: list_fm,Zs3: list_fm] :
( ( Xs
= ( append_fm @ Ys3 @ ( cons_fm @ X3 @ Zs3 ) ) )
& ~ ( member_fm @ X3 @ ( set_fm2 @ Ys3 ) ) ) ) ).
% split_list_first
thf(fact_714_split__list__first,axiom,
! [X3: tm,Xs: list_tm] :
( ( member_tm @ X3 @ ( set_tm2 @ Xs ) )
=> ? [Ys3: list_tm,Zs3: list_tm] :
( ( Xs
= ( append_tm @ Ys3 @ ( cons_tm @ X3 @ Zs3 ) ) )
& ~ ( member_tm @ X3 @ ( set_tm2 @ Ys3 ) ) ) ) ).
% split_list_first
thf(fact_715_split__list__first,axiom,
! [X3: list_fm,Xs: list_list_fm] :
( ( member_list_fm @ X3 @ ( set_list_fm2 @ Xs ) )
=> ? [Ys3: list_list_fm,Zs3: list_list_fm] :
( ( Xs
= ( append_list_fm @ Ys3 @ ( cons_list_fm @ X3 @ Zs3 ) ) )
& ~ ( member_list_fm @ X3 @ ( set_list_fm2 @ Ys3 ) ) ) ) ).
% split_list_first
thf(fact_716_split__list__prop,axiom,
! [Xs: list_fm,P: fm > $o] :
( ? [X6: fm] :
( ( member_fm @ X6 @ ( set_fm2 @ Xs ) )
& ( P @ X6 ) )
=> ? [Ys3: list_fm,X: fm] :
( ? [Zs3: list_fm] :
( Xs
= ( append_fm @ Ys3 @ ( cons_fm @ X @ Zs3 ) ) )
& ( P @ X ) ) ) ).
% split_list_prop
thf(fact_717_split__list__prop,axiom,
! [Xs: list_tm,P: tm > $o] :
( ? [X6: tm] :
( ( member_tm @ X6 @ ( set_tm2 @ Xs ) )
& ( P @ X6 ) )
=> ? [Ys3: list_tm,X: tm] :
( ? [Zs3: list_tm] :
( Xs
= ( append_tm @ Ys3 @ ( cons_tm @ X @ Zs3 ) ) )
& ( P @ X ) ) ) ).
% split_list_prop
thf(fact_718_split__list__prop,axiom,
! [Xs: list_list_fm,P: list_fm > $o] :
( ? [X6: list_fm] :
( ( member_list_fm @ X6 @ ( set_list_fm2 @ Xs ) )
& ( P @ X6 ) )
=> ? [Ys3: list_list_fm,X: list_fm] :
( ? [Zs3: list_list_fm] :
( Xs
= ( append_list_fm @ Ys3 @ ( cons_list_fm @ X @ Zs3 ) ) )
& ( P @ X ) ) ) ).
% split_list_prop
thf(fact_719_split__list__last,axiom,
! [X3: rule,Xs: list_rule] :
( ( member_rule @ X3 @ ( set_rule2 @ Xs ) )
=> ? [Ys3: list_rule,Zs3: list_rule] :
( ( Xs
= ( append_rule @ Ys3 @ ( cons_rule @ X3 @ Zs3 ) ) )
& ~ ( member_rule @ X3 @ ( set_rule2 @ Zs3 ) ) ) ) ).
% split_list_last
thf(fact_720_split__list__last,axiom,
! [X3: produc6018962875968178549ist_fm,Xs: list_P2887561121880082555ist_fm] :
( ( member4699826688122452638ist_fm @ X3 @ ( set_Pr5051287786238754058ist_fm @ Xs ) )
=> ? [Ys3: list_P2887561121880082555ist_fm,Zs3: list_P2887561121880082555ist_fm] :
( ( Xs
= ( append1476581234975291920ist_fm @ Ys3 @ ( cons_P6586734674047340843ist_fm @ X3 @ Zs3 ) ) )
& ~ ( member4699826688122452638ist_fm @ X3 @ ( set_Pr5051287786238754058ist_fm @ Zs3 ) ) ) ) ).
% split_list_last
thf(fact_721_split__list__last,axiom,
! [X3: produc340336539035504054m_rule,Xs: list_P2774625669004309958m_rule] :
( ( member7231649785386036813m_rule @ X3 @ ( set_Pr4534715572506550497m_rule @ Xs ) )
=> ? [Ys3: list_P2774625669004309958m_rule,Zs3: list_P2774625669004309958m_rule] :
( ( Xs
= ( append2714808343334685275m_rule @ Ys3 @ ( cons_P2818572562524668032m_rule @ X3 @ Zs3 ) ) )
& ~ ( member7231649785386036813m_rule @ X3 @ ( set_Pr4534715572506550497m_rule @ Zs3 ) ) ) ) ).
% split_list_last
thf(fact_722_split__list__last,axiom,
! [X3: fm,Xs: list_fm] :
( ( member_fm @ X3 @ ( set_fm2 @ Xs ) )
=> ? [Ys3: list_fm,Zs3: list_fm] :
( ( Xs
= ( append_fm @ Ys3 @ ( cons_fm @ X3 @ Zs3 ) ) )
& ~ ( member_fm @ X3 @ ( set_fm2 @ Zs3 ) ) ) ) ).
% split_list_last
thf(fact_723_split__list__last,axiom,
! [X3: tm,Xs: list_tm] :
( ( member_tm @ X3 @ ( set_tm2 @ Xs ) )
=> ? [Ys3: list_tm,Zs3: list_tm] :
( ( Xs
= ( append_tm @ Ys3 @ ( cons_tm @ X3 @ Zs3 ) ) )
& ~ ( member_tm @ X3 @ ( set_tm2 @ Zs3 ) ) ) ) ).
% split_list_last
thf(fact_724_split__list__last,axiom,
! [X3: list_fm,Xs: list_list_fm] :
( ( member_list_fm @ X3 @ ( set_list_fm2 @ Xs ) )
=> ? [Ys3: list_list_fm,Zs3: list_list_fm] :
( ( Xs
= ( append_list_fm @ Ys3 @ ( cons_list_fm @ X3 @ Zs3 ) ) )
& ~ ( member_list_fm @ X3 @ ( set_list_fm2 @ Zs3 ) ) ) ) ).
% split_list_last
thf(fact_725_split__list,axiom,
! [X3: rule,Xs: list_rule] :
( ( member_rule @ X3 @ ( set_rule2 @ Xs ) )
=> ? [Ys3: list_rule,Zs3: list_rule] :
( Xs
= ( append_rule @ Ys3 @ ( cons_rule @ X3 @ Zs3 ) ) ) ) ).
% split_list
thf(fact_726_split__list,axiom,
! [X3: produc6018962875968178549ist_fm,Xs: list_P2887561121880082555ist_fm] :
( ( member4699826688122452638ist_fm @ X3 @ ( set_Pr5051287786238754058ist_fm @ Xs ) )
=> ? [Ys3: list_P2887561121880082555ist_fm,Zs3: list_P2887561121880082555ist_fm] :
( Xs
= ( append1476581234975291920ist_fm @ Ys3 @ ( cons_P6586734674047340843ist_fm @ X3 @ Zs3 ) ) ) ) ).
% split_list
thf(fact_727_split__list,axiom,
! [X3: produc340336539035504054m_rule,Xs: list_P2774625669004309958m_rule] :
( ( member7231649785386036813m_rule @ X3 @ ( set_Pr4534715572506550497m_rule @ Xs ) )
=> ? [Ys3: list_P2774625669004309958m_rule,Zs3: list_P2774625669004309958m_rule] :
( Xs
= ( append2714808343334685275m_rule @ Ys3 @ ( cons_P2818572562524668032m_rule @ X3 @ Zs3 ) ) ) ) ).
% split_list
thf(fact_728_split__list,axiom,
! [X3: fm,Xs: list_fm] :
( ( member_fm @ X3 @ ( set_fm2 @ Xs ) )
=> ? [Ys3: list_fm,Zs3: list_fm] :
( Xs
= ( append_fm @ Ys3 @ ( cons_fm @ X3 @ Zs3 ) ) ) ) ).
% split_list
thf(fact_729_split__list,axiom,
! [X3: tm,Xs: list_tm] :
( ( member_tm @ X3 @ ( set_tm2 @ Xs ) )
=> ? [Ys3: list_tm,Zs3: list_tm] :
( Xs
= ( append_tm @ Ys3 @ ( cons_tm @ X3 @ Zs3 ) ) ) ) ).
% split_list
thf(fact_730_split__list,axiom,
! [X3: list_fm,Xs: list_list_fm] :
( ( member_list_fm @ X3 @ ( set_list_fm2 @ Xs ) )
=> ? [Ys3: list_list_fm,Zs3: list_list_fm] :
( Xs
= ( append_list_fm @ Ys3 @ ( cons_list_fm @ X3 @ Zs3 ) ) ) ) ).
% split_list
thf(fact_731_remdups_Osimps_I2_J,axiom,
! [X3: rule,Xs: list_rule] :
( ( ( member_rule @ X3 @ ( set_rule2 @ Xs ) )
=> ( ( remdups_rule @ ( cons_rule @ X3 @ Xs ) )
= ( remdups_rule @ Xs ) ) )
& ( ~ ( member_rule @ X3 @ ( set_rule2 @ Xs ) )
=> ( ( remdups_rule @ ( cons_rule @ X3 @ Xs ) )
= ( cons_rule @ X3 @ ( remdups_rule @ Xs ) ) ) ) ) ).
% remdups.simps(2)
thf(fact_732_remdups_Osimps_I2_J,axiom,
! [X3: produc6018962875968178549ist_fm,Xs: list_P2887561121880082555ist_fm] :
( ( ( member4699826688122452638ist_fm @ X3 @ ( set_Pr5051287786238754058ist_fm @ Xs ) )
=> ( ( remdup8378870680097181200ist_fm @ ( cons_P6586734674047340843ist_fm @ X3 @ Xs ) )
= ( remdup8378870680097181200ist_fm @ Xs ) ) )
& ( ~ ( member4699826688122452638ist_fm @ X3 @ ( set_Pr5051287786238754058ist_fm @ Xs ) )
=> ( ( remdup8378870680097181200ist_fm @ ( cons_P6586734674047340843ist_fm @ X3 @ Xs ) )
= ( cons_P6586734674047340843ist_fm @ X3 @ ( remdup8378870680097181200ist_fm @ Xs ) ) ) ) ) ).
% remdups.simps(2)
thf(fact_733_remdups_Osimps_I2_J,axiom,
! [X3: produc340336539035504054m_rule,Xs: list_P2774625669004309958m_rule] :
( ( ( member7231649785386036813m_rule @ X3 @ ( set_Pr4534715572506550497m_rule @ Xs ) )
=> ( ( remdup7144978108267558491m_rule @ ( cons_P2818572562524668032m_rule @ X3 @ Xs ) )
= ( remdup7144978108267558491m_rule @ Xs ) ) )
& ( ~ ( member7231649785386036813m_rule @ X3 @ ( set_Pr4534715572506550497m_rule @ Xs ) )
=> ( ( remdup7144978108267558491m_rule @ ( cons_P2818572562524668032m_rule @ X3 @ Xs ) )
= ( cons_P2818572562524668032m_rule @ X3 @ ( remdup7144978108267558491m_rule @ Xs ) ) ) ) ) ).
% remdups.simps(2)
thf(fact_734_remdups_Osimps_I2_J,axiom,
! [X3: fm,Xs: list_fm] :
( ( ( member_fm @ X3 @ ( set_fm2 @ Xs ) )
=> ( ( remdups_fm @ ( cons_fm @ X3 @ Xs ) )
= ( remdups_fm @ Xs ) ) )
& ( ~ ( member_fm @ X3 @ ( set_fm2 @ Xs ) )
=> ( ( remdups_fm @ ( cons_fm @ X3 @ Xs ) )
= ( cons_fm @ X3 @ ( remdups_fm @ Xs ) ) ) ) ) ).
% remdups.simps(2)
thf(fact_735_remdups_Osimps_I2_J,axiom,
! [X3: tm,Xs: list_tm] :
( ( ( member_tm @ X3 @ ( set_tm2 @ Xs ) )
=> ( ( remdups_tm @ ( cons_tm @ X3 @ Xs ) )
= ( remdups_tm @ Xs ) ) )
& ( ~ ( member_tm @ X3 @ ( set_tm2 @ Xs ) )
=> ( ( remdups_tm @ ( cons_tm @ X3 @ Xs ) )
= ( cons_tm @ X3 @ ( remdups_tm @ Xs ) ) ) ) ) ).
% remdups.simps(2)
thf(fact_736_remdups_Osimps_I2_J,axiom,
! [X3: list_fm,Xs: list_list_fm] :
( ( ( member_list_fm @ X3 @ ( set_list_fm2 @ Xs ) )
=> ( ( remdups_list_fm @ ( cons_list_fm @ X3 @ Xs ) )
= ( remdups_list_fm @ Xs ) ) )
& ( ~ ( member_list_fm @ X3 @ ( set_list_fm2 @ Xs ) )
=> ( ( remdups_list_fm @ ( cons_list_fm @ X3 @ Xs ) )
= ( cons_list_fm @ X3 @ ( remdups_list_fm @ Xs ) ) ) ) ) ).
% remdups.simps(2)
thf(fact_737_shift_Osimps_I2_J,axiom,
! [X3: produc340336539035504054m_rule,Xs: list_P2774625669004309958m_rule,S: stream2709947120125613254m_rule] :
( ( shift_2334844276748245581m_rule @ ( cons_P2818572562524668032m_rule @ X3 @ Xs ) @ S )
= ( sCons_2955607467875998431m_rule @ X3 @ ( shift_2334844276748245581m_rule @ Xs @ S ) ) ) ).
% shift.simps(2)
thf(fact_738_asym__iff,axiom,
! [R: set_Pr1261947904930325089at_nat] :
( ( asym_on_nat @ top_top_set_nat @ R )
= ( ! [X4: nat,Y5: nat] :
( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X4 @ Y5 ) @ R )
=> ~ ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ Y5 @ X4 ) @ R ) ) ) ) ).
% asym_iff
thf(fact_739_asym__iff,axiom,
! [R: set_Pr6724411253394807111ist_fm] :
( ( asym_o6181534149136668839ist_fm @ top_to1730629564055774885ist_fm @ R )
= ( ! [X4: produc6018962875968178549ist_fm,Y5: produc6018962875968178549ist_fm] :
( ( member858341428514850704ist_fm @ ( produc5488031402154692311ist_fm @ X4 @ Y5 ) @ R )
=> ~ ( member858341428514850704ist_fm @ ( produc5488031402154692311ist_fm @ Y5 @ X4 ) @ R ) ) ) ) ).
% asym_iff
thf(fact_740_asymD,axiom,
! [R: set_Pr1261947904930325089at_nat,X3: nat,Y3: nat] :
( ( asym_on_nat @ top_top_set_nat @ R )
=> ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X3 @ Y3 ) @ R )
=> ~ ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ Y3 @ X3 ) @ R ) ) ) ).
% asymD
thf(fact_741_asymD,axiom,
! [R: set_Pr6724411253394807111ist_fm,X3: produc6018962875968178549ist_fm,Y3: produc6018962875968178549ist_fm] :
( ( asym_o6181534149136668839ist_fm @ top_to1730629564055774885ist_fm @ R )
=> ( ( member858341428514850704ist_fm @ ( produc5488031402154692311ist_fm @ X3 @ Y3 ) @ R )
=> ~ ( member858341428514850704ist_fm @ ( produc5488031402154692311ist_fm @ Y3 @ X3 ) @ R ) ) ) ).
% asymD
thf(fact_742_RuleSystem_Oeff__S,axiom,
! [Eff: fm > rule > fset_rule > $o,Rules: stream_fm,S6: set_rule,S: rule,R: fm,Sl4: fset_rule,S5: rule] :
( ( abstra7659285206637183772m_rule @ Eff @ Rules @ S6 )
=> ( ( member_rule @ S @ S6 )
=> ( ( member_fm @ R @ ( sset_fm @ Rules ) )
=> ( ( Eff @ R @ S @ Sl4 )
=> ( ( fmember_rule @ S5 @ Sl4 )
=> ( member_rule @ S5 @ S6 ) ) ) ) ) ) ).
% RuleSystem.eff_S
thf(fact_743_RuleSystem_Oeff__S,axiom,
! [Eff: fm > fm > fset_fm > $o,Rules: stream_fm,S6: set_fm,S: fm,R: fm,Sl4: fset_fm,S5: fm] :
( ( abstra966159493038840937_fm_fm @ Eff @ Rules @ S6 )
=> ( ( member_fm @ S @ S6 )
=> ( ( member_fm @ R @ ( sset_fm @ Rules ) )
=> ( ( Eff @ R @ S @ Sl4 )
=> ( ( fmember_fm @ S5 @ Sl4 )
=> ( member_fm @ S5 @ S6 ) ) ) ) ) ) ).
% RuleSystem.eff_S
thf(fact_744_RuleSystem_Oeff__S,axiom,
! [Eff: rule > rule > fset_rule > $o,Rules: stream_rule,S6: set_rule,S: rule,R: rule,Sl4: fset_rule,S5: rule] :
( ( abstra6805424310982398927e_rule @ Eff @ Rules @ S6 )
=> ( ( member_rule @ S @ S6 )
=> ( ( member_rule @ R @ ( sset_rule @ Rules ) )
=> ( ( Eff @ R @ S @ Sl4 )
=> ( ( fmember_rule @ S5 @ Sl4 )
=> ( member_rule @ S5 @ S6 ) ) ) ) ) ) ).
% RuleSystem.eff_S
thf(fact_745_RuleSystem_Oeff__S,axiom,
! [Eff: rule > fm > fset_fm > $o,Rules: stream_rule,S6: set_fm,S: fm,R: rule,Sl4: fset_fm,S5: fm] :
( ( abstra4909020524819817846ule_fm @ Eff @ Rules @ S6 )
=> ( ( member_fm @ S @ S6 )
=> ( ( member_rule @ R @ ( sset_rule @ Rules ) )
=> ( ( Eff @ R @ S @ Sl4 )
=> ( ( fmember_fm @ S5 @ Sl4 )
=> ( member_fm @ S5 @ S6 ) ) ) ) ) ) ).
% RuleSystem.eff_S
thf(fact_746_RuleSystem_Oeff__S,axiom,
! [Eff: list_fm > rule > fset_rule > $o,Rules: stream_list_fm,S6: set_rule,S: rule,R: list_fm,Sl4: fset_rule,S5: rule] :
( ( abstra6744069043672362530m_rule @ Eff @ Rules @ S6 )
=> ( ( member_rule @ S @ S6 )
=> ( ( member_list_fm @ R @ ( sset_list_fm @ Rules ) )
=> ( ( Eff @ R @ S @ Sl4 )
=> ( ( fmember_rule @ S5 @ Sl4 )
=> ( member_rule @ S5 @ S6 ) ) ) ) ) ) ).
% RuleSystem.eff_S
thf(fact_747_RuleSystem_Oeff__S,axiom,
! [Eff: list_fm > fm > fset_fm > $o,Rules: stream_list_fm,S6: set_fm,S: fm,R: list_fm,Sl4: fset_fm,S5: fm] :
( ( abstra8753327658175425763_fm_fm @ Eff @ Rules @ S6 )
=> ( ( member_fm @ S @ S6 )
=> ( ( member_list_fm @ R @ ( sset_list_fm @ Rules ) )
=> ( ( Eff @ R @ S @ Sl4 )
=> ( ( fmember_fm @ S5 @ Sl4 )
=> ( member_fm @ S5 @ S6 ) ) ) ) ) ) ).
% RuleSystem.eff_S
thf(fact_748_RuleSystem_Oeff__S,axiom,
! [Eff: fm > list_fm > fset_list_fm > $o,Rules: stream_fm,S6: set_list_fm,S: list_fm,R: fm,Sl4: fset_list_fm,S5: list_fm] :
( ( abstra1735665081823683951ist_fm @ Eff @ Rules @ S6 )
=> ( ( member_list_fm @ S @ S6 )
=> ( ( member_fm @ R @ ( sset_fm @ Rules ) )
=> ( ( Eff @ R @ S @ Sl4 )
=> ( ( fmember_list_fm @ S5 @ Sl4 )
=> ( member_list_fm @ S5 @ S6 ) ) ) ) ) ) ).
% RuleSystem.eff_S
thf(fact_749_RuleSystem_Oeff__S,axiom,
! [Eff: rule > list_fm > fset_list_fm > $o,Rules: stream_rule,S6: set_list_fm,S: list_fm,R: rule,Sl4: fset_list_fm,S5: list_fm] :
( ( abstra9153614075763688444ist_fm @ Eff @ Rules @ S6 )
=> ( ( member_list_fm @ S @ S6 )
=> ( ( member_rule @ R @ ( sset_rule @ Rules ) )
=> ( ( Eff @ R @ S @ Sl4 )
=> ( ( fmember_list_fm @ S5 @ Sl4 )
=> ( member_list_fm @ S5 @ S6 ) ) ) ) ) ) ).
% RuleSystem.eff_S
thf(fact_750_RuleSystem_Oeff__S,axiom,
! [Eff: list_fm > list_fm > fset_list_fm > $o,Rules: stream_list_fm,S6: set_list_fm,S: list_fm,R: list_fm,Sl4: fset_list_fm,S5: list_fm] :
( ( abstra4201371802685485289ist_fm @ Eff @ Rules @ S6 )
=> ( ( member_list_fm @ S @ S6 )
=> ( ( member_list_fm @ R @ ( sset_list_fm @ Rules ) )
=> ( ( Eff @ R @ S @ Sl4 )
=> ( ( fmember_list_fm @ S5 @ Sl4 )
=> ( member_list_fm @ S5 @ S6 ) ) ) ) ) ) ).
% RuleSystem.eff_S
thf(fact_751_RuleSystem_Oeff__S,axiom,
! [Eff: produc6018962875968178549ist_fm > rule > fset_rule > $o,Rules: stream4408948924543953275ist_fm,S6: set_rule,S: rule,R: produc6018962875968178549ist_fm,Sl4: fset_rule,S5: rule] :
( ( abstra2695409591818887198m_rule @ Eff @ Rules @ S6 )
=> ( ( member_rule @ S @ S6 )
=> ( ( member4699826688122452638ist_fm @ R @ ( sset_P5379989128061332361ist_fm @ Rules ) )
=> ( ( Eff @ R @ S @ Sl4 )
=> ( ( fmember_rule @ S5 @ Sl4 )
=> ( member_rule @ S5 @ S6 ) ) ) ) ) ) ).
% RuleSystem.eff_S
thf(fact_752_RuleSystem_Ointro,axiom,
! [S6: set_rule,Rules: stream_fm,Eff: fm > rule > fset_rule > $o] :
( ! [S4: rule] :
( ( member_rule @ S4 @ S6 )
=> ! [R6: fm] :
( ( member_fm @ R6 @ ( sset_fm @ Rules ) )
=> ! [Sl: fset_rule] :
( ( Eff @ R6 @ S4 @ Sl )
=> ! [S8: rule] :
( ( fmember_rule @ S8 @ Sl )
=> ( member_rule @ S8 @ S6 ) ) ) ) )
=> ( ! [S4: rule] :
( ( member_rule @ S4 @ S6 )
=> ? [X6: fm] :
( ( member_fm @ X6 @ ( sset_fm @ Rules ) )
& ? [X_12: fset_rule] : ( Eff @ X6 @ S4 @ X_12 ) ) )
=> ( abstra7659285206637183772m_rule @ Eff @ Rules @ S6 ) ) ) ).
% RuleSystem.intro
thf(fact_753_RuleSystem_Ointro,axiom,
! [S6: set_fm,Rules: stream_fm,Eff: fm > fm > fset_fm > $o] :
( ! [S4: fm] :
( ( member_fm @ S4 @ S6 )
=> ! [R6: fm] :
( ( member_fm @ R6 @ ( sset_fm @ Rules ) )
=> ! [Sl: fset_fm] :
( ( Eff @ R6 @ S4 @ Sl )
=> ! [S8: fm] :
( ( fmember_fm @ S8 @ Sl )
=> ( member_fm @ S8 @ S6 ) ) ) ) )
=> ( ! [S4: fm] :
( ( member_fm @ S4 @ S6 )
=> ? [X6: fm] :
( ( member_fm @ X6 @ ( sset_fm @ Rules ) )
& ? [X_12: fset_fm] : ( Eff @ X6 @ S4 @ X_12 ) ) )
=> ( abstra966159493038840937_fm_fm @ Eff @ Rules @ S6 ) ) ) ).
% RuleSystem.intro
thf(fact_754_RuleSystem_Ointro,axiom,
! [S6: set_rule,Rules: stream_rule,Eff: rule > rule > fset_rule > $o] :
( ! [S4: rule] :
( ( member_rule @ S4 @ S6 )
=> ! [R6: rule] :
( ( member_rule @ R6 @ ( sset_rule @ Rules ) )
=> ! [Sl: fset_rule] :
( ( Eff @ R6 @ S4 @ Sl )
=> ! [S8: rule] :
( ( fmember_rule @ S8 @ Sl )
=> ( member_rule @ S8 @ S6 ) ) ) ) )
=> ( ! [S4: rule] :
( ( member_rule @ S4 @ S6 )
=> ? [X6: rule] :
( ( member_rule @ X6 @ ( sset_rule @ Rules ) )
& ? [X_12: fset_rule] : ( Eff @ X6 @ S4 @ X_12 ) ) )
=> ( abstra6805424310982398927e_rule @ Eff @ Rules @ S6 ) ) ) ).
% RuleSystem.intro
thf(fact_755_RuleSystem_Ointro,axiom,
! [S6: set_fm,Rules: stream_rule,Eff: rule > fm > fset_fm > $o] :
( ! [S4: fm] :
( ( member_fm @ S4 @ S6 )
=> ! [R6: rule] :
( ( member_rule @ R6 @ ( sset_rule @ Rules ) )
=> ! [Sl: fset_fm] :
( ( Eff @ R6 @ S4 @ Sl )
=> ! [S8: fm] :
( ( fmember_fm @ S8 @ Sl )
=> ( member_fm @ S8 @ S6 ) ) ) ) )
=> ( ! [S4: fm] :
( ( member_fm @ S4 @ S6 )
=> ? [X6: rule] :
( ( member_rule @ X6 @ ( sset_rule @ Rules ) )
& ? [X_12: fset_fm] : ( Eff @ X6 @ S4 @ X_12 ) ) )
=> ( abstra4909020524819817846ule_fm @ Eff @ Rules @ S6 ) ) ) ).
% RuleSystem.intro
thf(fact_756_RuleSystem_Ointro,axiom,
! [S6: set_rule,Rules: stream_list_fm,Eff: list_fm > rule > fset_rule > $o] :
( ! [S4: rule] :
( ( member_rule @ S4 @ S6 )
=> ! [R6: list_fm] :
( ( member_list_fm @ R6 @ ( sset_list_fm @ Rules ) )
=> ! [Sl: fset_rule] :
( ( Eff @ R6 @ S4 @ Sl )
=> ! [S8: rule] :
( ( fmember_rule @ S8 @ Sl )
=> ( member_rule @ S8 @ S6 ) ) ) ) )
=> ( ! [S4: rule] :
( ( member_rule @ S4 @ S6 )
=> ? [X6: list_fm] :
( ( member_list_fm @ X6 @ ( sset_list_fm @ Rules ) )
& ? [X_12: fset_rule] : ( Eff @ X6 @ S4 @ X_12 ) ) )
=> ( abstra6744069043672362530m_rule @ Eff @ Rules @ S6 ) ) ) ).
% RuleSystem.intro
thf(fact_757_RuleSystem_Ointro,axiom,
! [S6: set_fm,Rules: stream_list_fm,Eff: list_fm > fm > fset_fm > $o] :
( ! [S4: fm] :
( ( member_fm @ S4 @ S6 )
=> ! [R6: list_fm] :
( ( member_list_fm @ R6 @ ( sset_list_fm @ Rules ) )
=> ! [Sl: fset_fm] :
( ( Eff @ R6 @ S4 @ Sl )
=> ! [S8: fm] :
( ( fmember_fm @ S8 @ Sl )
=> ( member_fm @ S8 @ S6 ) ) ) ) )
=> ( ! [S4: fm] :
( ( member_fm @ S4 @ S6 )
=> ? [X6: list_fm] :
( ( member_list_fm @ X6 @ ( sset_list_fm @ Rules ) )
& ? [X_12: fset_fm] : ( Eff @ X6 @ S4 @ X_12 ) ) )
=> ( abstra8753327658175425763_fm_fm @ Eff @ Rules @ S6 ) ) ) ).
% RuleSystem.intro
thf(fact_758_RuleSystem_Ointro,axiom,
! [S6: set_list_fm,Rules: stream_fm,Eff: fm > list_fm > fset_list_fm > $o] :
( ! [S4: list_fm] :
( ( member_list_fm @ S4 @ S6 )
=> ! [R6: fm] :
( ( member_fm @ R6 @ ( sset_fm @ Rules ) )
=> ! [Sl: fset_list_fm] :
( ( Eff @ R6 @ S4 @ Sl )
=> ! [S8: list_fm] :
( ( fmember_list_fm @ S8 @ Sl )
=> ( member_list_fm @ S8 @ S6 ) ) ) ) )
=> ( ! [S4: list_fm] :
( ( member_list_fm @ S4 @ S6 )
=> ? [X6: fm] :
( ( member_fm @ X6 @ ( sset_fm @ Rules ) )
& ? [X_12: fset_list_fm] : ( Eff @ X6 @ S4 @ X_12 ) ) )
=> ( abstra1735665081823683951ist_fm @ Eff @ Rules @ S6 ) ) ) ).
% RuleSystem.intro
thf(fact_759_RuleSystem_Ointro,axiom,
! [S6: set_list_fm,Rules: stream_rule,Eff: rule > list_fm > fset_list_fm > $o] :
( ! [S4: list_fm] :
( ( member_list_fm @ S4 @ S6 )
=> ! [R6: rule] :
( ( member_rule @ R6 @ ( sset_rule @ Rules ) )
=> ! [Sl: fset_list_fm] :
( ( Eff @ R6 @ S4 @ Sl )
=> ! [S8: list_fm] :
( ( fmember_list_fm @ S8 @ Sl )
=> ( member_list_fm @ S8 @ S6 ) ) ) ) )
=> ( ! [S4: list_fm] :
( ( member_list_fm @ S4 @ S6 )
=> ? [X6: rule] :
( ( member_rule @ X6 @ ( sset_rule @ Rules ) )
& ? [X_12: fset_list_fm] : ( Eff @ X6 @ S4 @ X_12 ) ) )
=> ( abstra9153614075763688444ist_fm @ Eff @ Rules @ S6 ) ) ) ).
% RuleSystem.intro
thf(fact_760_RuleSystem_Ointro,axiom,
! [S6: set_list_fm,Rules: stream_list_fm,Eff: list_fm > list_fm > fset_list_fm > $o] :
( ! [S4: list_fm] :
( ( member_list_fm @ S4 @ S6 )
=> ! [R6: list_fm] :
( ( member_list_fm @ R6 @ ( sset_list_fm @ Rules ) )
=> ! [Sl: fset_list_fm] :
( ( Eff @ R6 @ S4 @ Sl )
=> ! [S8: list_fm] :
( ( fmember_list_fm @ S8 @ Sl )
=> ( member_list_fm @ S8 @ S6 ) ) ) ) )
=> ( ! [S4: list_fm] :
( ( member_list_fm @ S4 @ S6 )
=> ? [X6: list_fm] :
( ( member_list_fm @ X6 @ ( sset_list_fm @ Rules ) )
& ? [X_12: fset_list_fm] : ( Eff @ X6 @ S4 @ X_12 ) ) )
=> ( abstra4201371802685485289ist_fm @ Eff @ Rules @ S6 ) ) ) ).
% RuleSystem.intro
thf(fact_761_RuleSystem_Ointro,axiom,
! [S6: set_rule,Rules: stream4408948924543953275ist_fm,Eff: produc6018962875968178549ist_fm > rule > fset_rule > $o] :
( ! [S4: rule] :
( ( member_rule @ S4 @ S6 )
=> ! [R6: produc6018962875968178549ist_fm] :
( ( member4699826688122452638ist_fm @ R6 @ ( sset_P5379989128061332361ist_fm @ Rules ) )
=> ! [Sl: fset_rule] :
( ( Eff @ R6 @ S4 @ Sl )
=> ! [S8: rule] :
( ( fmember_rule @ S8 @ Sl )
=> ( member_rule @ S8 @ S6 ) ) ) ) )
=> ( ! [S4: rule] :
( ( member_rule @ S4 @ S6 )
=> ? [X6: produc6018962875968178549ist_fm] :
( ( member4699826688122452638ist_fm @ X6 @ ( sset_P5379989128061332361ist_fm @ Rules ) )
& ? [X_12: fset_rule] : ( Eff @ X6 @ S4 @ X_12 ) ) )
=> ( abstra2695409591818887198m_rule @ Eff @ Rules @ S6 ) ) ) ).
% RuleSystem.intro
thf(fact_762_RuleSystem__def,axiom,
( abstra7659285206637183772m_rule
= ( ^ [Eff2: fm > rule > fset_rule > $o,Rules2: stream_fm,S7: set_rule] :
( ! [S3: rule] :
( ( member_rule @ S3 @ S7 )
=> ! [R4: fm] :
( ( member_fm @ R4 @ ( sset_fm @ Rules2 ) )
=> ! [Sl2: fset_rule] :
( ( Eff2 @ R4 @ S3 @ Sl2 )
=> ! [S9: rule] :
( ( fmember_rule @ S9 @ Sl2 )
=> ( member_rule @ S9 @ S7 ) ) ) ) )
& ! [S3: rule] :
( ( member_rule @ S3 @ S7 )
=> ? [X4: fm] :
( ( member_fm @ X4 @ ( sset_fm @ Rules2 ) )
& ? [X7: fset_rule] : ( Eff2 @ X4 @ S3 @ X7 ) ) ) ) ) ) ).
% RuleSystem_def
thf(fact_763_RuleSystem__def,axiom,
( abstra966159493038840937_fm_fm
= ( ^ [Eff2: fm > fm > fset_fm > $o,Rules2: stream_fm,S7: set_fm] :
( ! [S3: fm] :
( ( member_fm @ S3 @ S7 )
=> ! [R4: fm] :
( ( member_fm @ R4 @ ( sset_fm @ Rules2 ) )
=> ! [Sl2: fset_fm] :
( ( Eff2 @ R4 @ S3 @ Sl2 )
=> ! [S9: fm] :
( ( fmember_fm @ S9 @ Sl2 )
=> ( member_fm @ S9 @ S7 ) ) ) ) )
& ! [S3: fm] :
( ( member_fm @ S3 @ S7 )
=> ? [X4: fm] :
( ( member_fm @ X4 @ ( sset_fm @ Rules2 ) )
& ? [X7: fset_fm] : ( Eff2 @ X4 @ S3 @ X7 ) ) ) ) ) ) ).
% RuleSystem_def
thf(fact_764_RuleSystem__def,axiom,
( abstra6805424310982398927e_rule
= ( ^ [Eff2: rule > rule > fset_rule > $o,Rules2: stream_rule,S7: set_rule] :
( ! [S3: rule] :
( ( member_rule @ S3 @ S7 )
=> ! [R4: rule] :
( ( member_rule @ R4 @ ( sset_rule @ Rules2 ) )
=> ! [Sl2: fset_rule] :
( ( Eff2 @ R4 @ S3 @ Sl2 )
=> ! [S9: rule] :
( ( fmember_rule @ S9 @ Sl2 )
=> ( member_rule @ S9 @ S7 ) ) ) ) )
& ! [S3: rule] :
( ( member_rule @ S3 @ S7 )
=> ? [X4: rule] :
( ( member_rule @ X4 @ ( sset_rule @ Rules2 ) )
& ? [X7: fset_rule] : ( Eff2 @ X4 @ S3 @ X7 ) ) ) ) ) ) ).
% RuleSystem_def
thf(fact_765_RuleSystem__def,axiom,
( abstra4909020524819817846ule_fm
= ( ^ [Eff2: rule > fm > fset_fm > $o,Rules2: stream_rule,S7: set_fm] :
( ! [S3: fm] :
( ( member_fm @ S3 @ S7 )
=> ! [R4: rule] :
( ( member_rule @ R4 @ ( sset_rule @ Rules2 ) )
=> ! [Sl2: fset_fm] :
( ( Eff2 @ R4 @ S3 @ Sl2 )
=> ! [S9: fm] :
( ( fmember_fm @ S9 @ Sl2 )
=> ( member_fm @ S9 @ S7 ) ) ) ) )
& ! [S3: fm] :
( ( member_fm @ S3 @ S7 )
=> ? [X4: rule] :
( ( member_rule @ X4 @ ( sset_rule @ Rules2 ) )
& ? [X7: fset_fm] : ( Eff2 @ X4 @ S3 @ X7 ) ) ) ) ) ) ).
% RuleSystem_def
thf(fact_766_RuleSystem__def,axiom,
( abstra6744069043672362530m_rule
= ( ^ [Eff2: list_fm > rule > fset_rule > $o,Rules2: stream_list_fm,S7: set_rule] :
( ! [S3: rule] :
( ( member_rule @ S3 @ S7 )
=> ! [R4: list_fm] :
( ( member_list_fm @ R4 @ ( sset_list_fm @ Rules2 ) )
=> ! [Sl2: fset_rule] :
( ( Eff2 @ R4 @ S3 @ Sl2 )
=> ! [S9: rule] :
( ( fmember_rule @ S9 @ Sl2 )
=> ( member_rule @ S9 @ S7 ) ) ) ) )
& ! [S3: rule] :
( ( member_rule @ S3 @ S7 )
=> ? [X4: list_fm] :
( ( member_list_fm @ X4 @ ( sset_list_fm @ Rules2 ) )
& ? [X7: fset_rule] : ( Eff2 @ X4 @ S3 @ X7 ) ) ) ) ) ) ).
% RuleSystem_def
thf(fact_767_RuleSystem__def,axiom,
( abstra8753327658175425763_fm_fm
= ( ^ [Eff2: list_fm > fm > fset_fm > $o,Rules2: stream_list_fm,S7: set_fm] :
( ! [S3: fm] :
( ( member_fm @ S3 @ S7 )
=> ! [R4: list_fm] :
( ( member_list_fm @ R4 @ ( sset_list_fm @ Rules2 ) )
=> ! [Sl2: fset_fm] :
( ( Eff2 @ R4 @ S3 @ Sl2 )
=> ! [S9: fm] :
( ( fmember_fm @ S9 @ Sl2 )
=> ( member_fm @ S9 @ S7 ) ) ) ) )
& ! [S3: fm] :
( ( member_fm @ S3 @ S7 )
=> ? [X4: list_fm] :
( ( member_list_fm @ X4 @ ( sset_list_fm @ Rules2 ) )
& ? [X7: fset_fm] : ( Eff2 @ X4 @ S3 @ X7 ) ) ) ) ) ) ).
% RuleSystem_def
thf(fact_768_RuleSystem__def,axiom,
( abstra1735665081823683951ist_fm
= ( ^ [Eff2: fm > list_fm > fset_list_fm > $o,Rules2: stream_fm,S7: set_list_fm] :
( ! [S3: list_fm] :
( ( member_list_fm @ S3 @ S7 )
=> ! [R4: fm] :
( ( member_fm @ R4 @ ( sset_fm @ Rules2 ) )
=> ! [Sl2: fset_list_fm] :
( ( Eff2 @ R4 @ S3 @ Sl2 )
=> ! [S9: list_fm] :
( ( fmember_list_fm @ S9 @ Sl2 )
=> ( member_list_fm @ S9 @ S7 ) ) ) ) )
& ! [S3: list_fm] :
( ( member_list_fm @ S3 @ S7 )
=> ? [X4: fm] :
( ( member_fm @ X4 @ ( sset_fm @ Rules2 ) )
& ? [X7: fset_list_fm] : ( Eff2 @ X4 @ S3 @ X7 ) ) ) ) ) ) ).
% RuleSystem_def
thf(fact_769_RuleSystem__def,axiom,
( abstra9153614075763688444ist_fm
= ( ^ [Eff2: rule > list_fm > fset_list_fm > $o,Rules2: stream_rule,S7: set_list_fm] :
( ! [S3: list_fm] :
( ( member_list_fm @ S3 @ S7 )
=> ! [R4: rule] :
( ( member_rule @ R4 @ ( sset_rule @ Rules2 ) )
=> ! [Sl2: fset_list_fm] :
( ( Eff2 @ R4 @ S3 @ Sl2 )
=> ! [S9: list_fm] :
( ( fmember_list_fm @ S9 @ Sl2 )
=> ( member_list_fm @ S9 @ S7 ) ) ) ) )
& ! [S3: list_fm] :
( ( member_list_fm @ S3 @ S7 )
=> ? [X4: rule] :
( ( member_rule @ X4 @ ( sset_rule @ Rules2 ) )
& ? [X7: fset_list_fm] : ( Eff2 @ X4 @ S3 @ X7 ) ) ) ) ) ) ).
% RuleSystem_def
thf(fact_770_RuleSystem__def,axiom,
( abstra4201371802685485289ist_fm
= ( ^ [Eff2: list_fm > list_fm > fset_list_fm > $o,Rules2: stream_list_fm,S7: set_list_fm] :
( ! [S3: list_fm] :
( ( member_list_fm @ S3 @ S7 )
=> ! [R4: list_fm] :
( ( member_list_fm @ R4 @ ( sset_list_fm @ Rules2 ) )
=> ! [Sl2: fset_list_fm] :
( ( Eff2 @ R4 @ S3 @ Sl2 )
=> ! [S9: list_fm] :
( ( fmember_list_fm @ S9 @ Sl2 )
=> ( member_list_fm @ S9 @ S7 ) ) ) ) )
& ! [S3: list_fm] :
( ( member_list_fm @ S3 @ S7 )
=> ? [X4: list_fm] :
( ( member_list_fm @ X4 @ ( sset_list_fm @ Rules2 ) )
& ? [X7: fset_list_fm] : ( Eff2 @ X4 @ S3 @ X7 ) ) ) ) ) ) ).
% RuleSystem_def
thf(fact_771_RuleSystem__def,axiom,
( abstra2695409591818887198m_rule
= ( ^ [Eff2: produc6018962875968178549ist_fm > rule > fset_rule > $o,Rules2: stream4408948924543953275ist_fm,S7: set_rule] :
( ! [S3: rule] :
( ( member_rule @ S3 @ S7 )
=> ! [R4: produc6018962875968178549ist_fm] :
( ( member4699826688122452638ist_fm @ R4 @ ( sset_P5379989128061332361ist_fm @ Rules2 ) )
=> ! [Sl2: fset_rule] :
( ( Eff2 @ R4 @ S3 @ Sl2 )
=> ! [S9: rule] :
( ( fmember_rule @ S9 @ Sl2 )
=> ( member_rule @ S9 @ S7 ) ) ) ) )
& ! [S3: rule] :
( ( member_rule @ S3 @ S7 )
=> ? [X4: produc6018962875968178549ist_fm] :
( ( member4699826688122452638ist_fm @ X4 @ ( sset_P5379989128061332361ist_fm @ Rules2 ) )
& ? [X7: fset_rule] : ( Eff2 @ X4 @ S3 @ X7 ) ) ) ) ) ) ).
% RuleSystem_def
thf(fact_772_lex__append__leftI,axiom,
! [Ys: list_tm,Zs: list_tm,R: set_Pr2455929065695642951_tm_tm,Xs: list_tm] :
( ( member4457312017796942864ist_tm @ ( produc1418304791525149271ist_tm @ Ys @ Zs ) @ ( lex_tm @ R ) )
=> ( member4457312017796942864ist_tm @ ( produc1418304791525149271ist_tm @ ( append_tm @ Xs @ Ys ) @ ( append_tm @ Xs @ Zs ) ) @ ( lex_tm @ R ) ) ) ).
% lex_append_leftI
thf(fact_773_lexord__append__rightI,axiom,
! [Y3: list_tm,X3: list_tm,R: set_Pr2455929065695642951_tm_tm] :
( ? [B7: tm,Z8: list_tm] :
( Y3
= ( cons_tm @ B7 @ Z8 ) )
=> ( member4457312017796942864ist_tm @ ( produc1418304791525149271ist_tm @ X3 @ ( append_tm @ X3 @ Y3 ) ) @ ( lexord_tm @ R ) ) ) ).
% lexord_append_rightI
thf(fact_774_PersistentRuleSystem_Ointro,axiom,
! [Eff: rule > produc6018962875968178549ist_fm > fset_P8989946509869081563ist_fm > $o,Rules: stream_rule,S6: set_Pr5202636777678657877ist_fm] :
( ( abstra5221733350967904376ist_fm @ Eff @ Rules @ S6 )
=> ( ( abstra9069655090277518462ist_fm @ Eff @ Rules @ S6 )
=> ( abstra3967921542344661089ist_fm @ Eff @ Rules @ S6 ) ) ) ).
% PersistentRuleSystem.intro
thf(fact_775_PersistentRuleSystem__def,axiom,
( abstra3967921542344661089ist_fm
= ( ^ [Eff2: rule > produc6018962875968178549ist_fm > fset_P8989946509869081563ist_fm > $o,Rules2: stream_rule,S7: set_Pr5202636777678657877ist_fm] :
( ( abstra5221733350967904376ist_fm @ Eff2 @ Rules2 @ S7 )
& ( abstra9069655090277518462ist_fm @ Eff2 @ Rules2 @ S7 ) ) ) ) ).
% PersistentRuleSystem_def
thf(fact_776_lex__append__leftD,axiom,
! [R: set_Pr2455929065695642951_tm_tm,Xs: list_tm,Ys: list_tm,Zs: list_tm] :
( ! [X: tm] :
~ ( member3121616906494481296_tm_tm @ ( product_Pair_tm_tm @ X @ X ) @ R )
=> ( ( member4457312017796942864ist_tm @ ( produc1418304791525149271ist_tm @ ( append_tm @ Xs @ Ys ) @ ( append_tm @ Xs @ Zs ) ) @ ( lex_tm @ R ) )
=> ( member4457312017796942864ist_tm @ ( produc1418304791525149271ist_tm @ Ys @ Zs ) @ ( lex_tm @ R ) ) ) ) ).
% lex_append_leftD
thf(fact_777_lex__append__leftD,axiom,
! [R: set_Pr1261947904930325089at_nat,Xs: list_nat,Ys: list_nat,Zs: list_nat] :
( ! [X: nat] :
~ ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X @ X ) @ R )
=> ( ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ ( append_nat @ Xs @ Ys ) @ ( append_nat @ Xs @ Zs ) ) @ ( lex_nat @ R ) )
=> ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ Ys @ Zs ) @ ( lex_nat @ R ) ) ) ) ).
% lex_append_leftD
thf(fact_778_lex__append__left__iff,axiom,
! [R: set_Pr2455929065695642951_tm_tm,Xs: list_tm,Ys: list_tm,Zs: list_tm] :
( ! [X: tm] :
~ ( member3121616906494481296_tm_tm @ ( product_Pair_tm_tm @ X @ X ) @ R )
=> ( ( member4457312017796942864ist_tm @ ( produc1418304791525149271ist_tm @ ( append_tm @ Xs @ Ys ) @ ( append_tm @ Xs @ Zs ) ) @ ( lex_tm @ R ) )
= ( member4457312017796942864ist_tm @ ( produc1418304791525149271ist_tm @ Ys @ Zs ) @ ( lex_tm @ R ) ) ) ) ).
% lex_append_left_iff
thf(fact_779_lex__append__left__iff,axiom,
! [R: set_Pr1261947904930325089at_nat,Xs: list_nat,Ys: list_nat,Zs: list_nat] :
( ! [X: nat] :
~ ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X @ X ) @ R )
=> ( ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ ( append_nat @ Xs @ Ys ) @ ( append_nat @ Xs @ Zs ) ) @ ( lex_nat @ R ) )
= ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ Ys @ Zs ) @ ( lex_nat @ R ) ) ) ) ).
% lex_append_left_iff
thf(fact_780_list__ex1__simps_I2_J,axiom,
! [P: produc340336539035504054m_rule > $o,X3: produc340336539035504054m_rule,Xs: list_P2774625669004309958m_rule] :
( ( list_e4959145360368724013m_rule @ P @ ( cons_P2818572562524668032m_rule @ X3 @ Xs ) )
= ( ( ( P @ X3 )
=> ( list_a9038620553111681026m_rule
@ ^ [Y5: produc340336539035504054m_rule] :
( ~ ( P @ Y5 )
| ( X3 = Y5 ) )
@ Xs ) )
& ( ~ ( P @ X3 )
=> ( list_e4959145360368724013m_rule @ P @ Xs ) ) ) ) ).
% list_ex1_simps(2)
thf(fact_781_snth__sset__smerge,axiom,
! [Ss: stream_stream_fm,N: nat,M2: nat] : ( member_fm @ ( snth_fm @ ( snth_stream_fm @ Ss @ N ) @ M2 ) @ ( sset_fm @ ( smerge_fm @ Ss ) ) ) ).
% snth_sset_smerge
thf(fact_782_snth__sset__smerge,axiom,
! [Ss: stream5457498542113063425ist_fm,N: nat,M2: nat] : ( member4699826688122452638ist_fm @ ( snth_P7093566783922538521ist_fm @ ( snth_s3026855241675268639ist_fm @ Ss @ N ) @ M2 ) @ ( sset_P5379989128061332361ist_fm @ ( smerge6880477169801031811ist_fm @ Ss ) ) ) ).
% snth_sset_smerge
thf(fact_783_snth__sset__smerge,axiom,
! [Ss: stream6017534608192929797ist_fm,N: nat,M2: nat] : ( member_list_fm @ ( snth_list_fm @ ( snth_stream_list_fm @ Ss @ N ) @ M2 ) @ ( sset_list_fm @ ( smerge_list_fm @ Ss ) ) ) ).
% snth_sset_smerge
thf(fact_784_snth__sset__smerge,axiom,
! [Ss: stream_stream_rule,N: nat,M2: nat] : ( member_rule @ ( snth_rule @ ( snth_stream_rule @ Ss @ N ) @ M2 ) @ ( sset_rule @ ( smerge_rule @ Ss ) ) ) ).
% snth_sset_smerge
thf(fact_785_snth__sset__smerge,axiom,
! [Ss: stream3752074346242807894m_rule,N: nat,M2: nat] : ( member7231649785386036813m_rule @ ( snth_P6679518042731451922m_rule @ ( snth_s6182113952396108578m_rule @ Ss @ N ) @ M2 ) @ ( sset_P4484857331586881186m_rule @ ( smerge193809993764105000m_rule @ Ss ) ) ) ).
% snth_sset_smerge
thf(fact_786_id__stake__snth__sdrop,axiom,
! [S: stream_rule,I: nat] :
( S
= ( shift_rule @ ( stake_rule @ I @ S ) @ ( sCons_rule @ ( snth_rule @ S @ I ) @ ( sdrop_rule @ ( suc @ I ) @ S ) ) ) ) ).
% id_stake_snth_sdrop
thf(fact_787_id__stake__snth__sdrop,axiom,
! [S: stream2709947120125613254m_rule,I: nat] :
( S
= ( shift_2334844276748245581m_rule @ ( stake_5421812949518764133m_rule @ I @ S ) @ ( sCons_2955607467875998431m_rule @ ( snth_P6679518042731451922m_rule @ S @ I ) @ ( sdrop_8169176516188972301m_rule @ ( suc @ I ) @ S ) ) ) ) ).
% id_stake_snth_sdrop
thf(fact_788_Cons__in__lex,axiom,
! [X3: nat,Xs: list_nat,Y3: nat,Ys: list_nat,R: set_Pr1261947904930325089at_nat] :
( ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ ( cons_nat @ X3 @ Xs ) @ ( cons_nat @ Y3 @ Ys ) ) @ ( lex_nat @ R ) )
= ( ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X3 @ Y3 ) @ R )
& ( ( size_size_list_nat @ Xs )
= ( size_size_list_nat @ Ys ) ) )
| ( ( X3 = Y3 )
& ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ Xs @ Ys ) @ ( lex_nat @ R ) ) ) ) ) ).
% Cons_in_lex
thf(fact_789_asymp__asym__eq,axiom,
! [R: set_Pr1261947904930325089at_nat] :
( ( asymp_on_nat @ top_top_set_nat
@ ^ [X4: nat,Y5: nat] : ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X4 @ Y5 ) @ R ) )
= ( asym_on_nat @ top_top_set_nat @ R ) ) ).
% asymp_asym_eq
thf(fact_790_asymp__asym__eq,axiom,
! [R: set_Pr6724411253394807111ist_fm] :
( ( asymp_626095371146813971ist_fm @ top_to1730629564055774885ist_fm
@ ^ [X4: produc6018962875968178549ist_fm,Y5: produc6018962875968178549ist_fm] : ( member858341428514850704ist_fm @ ( produc5488031402154692311ist_fm @ X4 @ Y5 ) @ R ) )
= ( asym_o6181534149136668839ist_fm @ top_to1730629564055774885ist_fm @ R ) ) ).
% asymp_asym_eq
thf(fact_791_append__eq__append__conv,axiom,
! [Xs: list_tm,Ys: list_tm,Us2: list_tm,Vs: list_tm] :
( ( ( ( size_size_list_tm @ Xs )
= ( size_size_list_tm @ Ys ) )
| ( ( size_size_list_tm @ Us2 )
= ( size_size_list_tm @ Vs ) ) )
=> ( ( ( append_tm @ Xs @ Us2 )
= ( append_tm @ Ys @ Vs ) )
= ( ( Xs = Ys )
& ( Us2 = Vs ) ) ) ) ).
% append_eq_append_conv
thf(fact_792_length__remdups__eq,axiom,
! [Xs: list_tm] :
( ( ( size_size_list_tm @ ( remdups_tm @ Xs ) )
= ( size_size_list_tm @ Xs ) )
= ( ( remdups_tm @ Xs )
= Xs ) ) ).
% length_remdups_eq
thf(fact_793_sdrop__simps_I2_J,axiom,
! [N: nat,S: stream2709947120125613254m_rule] :
( ( stl_Pr12655793849929990m_rule @ ( sdrop_8169176516188972301m_rule @ N @ S ) )
= ( sdrop_8169176516188972301m_rule @ ( suc @ N ) @ S ) ) ).
% sdrop_simps(2)
thf(fact_794_sdrop__simps_I2_J,axiom,
! [N: nat,S: stream_rule] :
( ( stl_rule @ ( sdrop_rule @ N @ S ) )
= ( sdrop_rule @ ( suc @ N ) @ S ) ) ).
% sdrop_simps(2)
thf(fact_795_rules__repeat__sdrop,axiom,
! [K: nat,R: rule] :
? [N2: nat] :
( ( snth_rule @ ( sdrop_rule @ K @ rules ) @ N2 )
= R ) ).
% rules_repeat_sdrop
thf(fact_796_sdrop_Osimps_I2_J,axiom,
! [N: nat,S: stream2709947120125613254m_rule] :
( ( sdrop_8169176516188972301m_rule @ ( suc @ N ) @ S )
= ( sdrop_8169176516188972301m_rule @ N @ ( stl_Pr12655793849929990m_rule @ S ) ) ) ).
% sdrop.simps(2)
thf(fact_797_sdrop_Osimps_I2_J,axiom,
! [N: nat,S: stream_rule] :
( ( sdrop_rule @ ( suc @ N ) @ S )
= ( sdrop_rule @ N @ ( stl_rule @ S ) ) ) ).
% sdrop.simps(2)
thf(fact_798_snth_Osimps_I2_J,axiom,
! [S: stream2709947120125613254m_rule,N: nat] :
( ( snth_P6679518042731451922m_rule @ S @ ( suc @ N ) )
= ( snth_P6679518042731451922m_rule @ ( stl_Pr12655793849929990m_rule @ S ) @ N ) ) ).
% snth.simps(2)
thf(fact_799_snth_Osimps_I2_J,axiom,
! [S: stream_rule,N: nat] :
( ( snth_rule @ S @ ( suc @ N ) )
= ( snth_rule @ ( stl_rule @ S ) @ N ) ) ).
% snth.simps(2)
thf(fact_800_snth__Stream,axiom,
! [X3: produc340336539035504054m_rule,S: stream2709947120125613254m_rule,I: nat] :
( ( snth_P6679518042731451922m_rule @ ( sCons_2955607467875998431m_rule @ X3 @ S ) @ ( suc @ I ) )
= ( snth_P6679518042731451922m_rule @ S @ I ) ) ).
% snth_Stream
thf(fact_801_snth__Stream,axiom,
! [X3: rule,S: stream_rule,I: nat] :
( ( snth_rule @ ( sCons_rule @ X3 @ S ) @ ( suc @ I ) )
= ( snth_rule @ S @ I ) ) ).
% snth_Stream
thf(fact_802_list__ex1__iff,axiom,
( list_ex1_rule
= ( ^ [P3: rule > $o,Xs2: list_rule] :
? [X4: rule] :
( ( member_rule @ X4 @ ( set_rule2 @ Xs2 ) )
& ( P3 @ X4 )
& ! [Y5: rule] :
( ( ( member_rule @ Y5 @ ( set_rule2 @ Xs2 ) )
& ( P3 @ Y5 ) )
=> ( Y5 = X4 ) ) ) ) ) ).
% list_ex1_iff
thf(fact_803_list__ex1__iff,axiom,
( list_e8433478912307198142ist_fm
= ( ^ [P3: produc6018962875968178549ist_fm > $o,Xs2: list_P2887561121880082555ist_fm] :
? [X4: produc6018962875968178549ist_fm] :
( ( member4699826688122452638ist_fm @ X4 @ ( set_Pr5051287786238754058ist_fm @ Xs2 ) )
& ( P3 @ X4 )
& ! [Y5: produc6018962875968178549ist_fm] :
( ( ( member4699826688122452638ist_fm @ Y5 @ ( set_Pr5051287786238754058ist_fm @ Xs2 ) )
& ( P3 @ Y5 ) )
=> ( Y5 = X4 ) ) ) ) ) ).
% list_ex1_iff
thf(fact_804_list__ex1__iff,axiom,
( list_e4959145360368724013m_rule
= ( ^ [P3: produc340336539035504054m_rule > $o,Xs2: list_P2774625669004309958m_rule] :
? [X4: produc340336539035504054m_rule] :
( ( member7231649785386036813m_rule @ X4 @ ( set_Pr4534715572506550497m_rule @ Xs2 ) )
& ( P3 @ X4 )
& ! [Y5: produc340336539035504054m_rule] :
( ( ( member7231649785386036813m_rule @ Y5 @ ( set_Pr4534715572506550497m_rule @ Xs2 ) )
& ( P3 @ Y5 ) )
=> ( Y5 = X4 ) ) ) ) ) ).
% list_ex1_iff
thf(fact_805_list__ex1__iff,axiom,
( list_ex1_fm
= ( ^ [P3: fm > $o,Xs2: list_fm] :
? [X4: fm] :
( ( member_fm @ X4 @ ( set_fm2 @ Xs2 ) )
& ( P3 @ X4 )
& ! [Y5: fm] :
( ( ( member_fm @ Y5 @ ( set_fm2 @ Xs2 ) )
& ( P3 @ Y5 ) )
=> ( Y5 = X4 ) ) ) ) ) ).
% list_ex1_iff
thf(fact_806_list__ex1__iff,axiom,
( list_ex1_tm
= ( ^ [P3: tm > $o,Xs2: list_tm] :
? [X4: tm] :
( ( member_tm @ X4 @ ( set_tm2 @ Xs2 ) )
& ( P3 @ X4 )
& ! [Y5: tm] :
( ( ( member_tm @ Y5 @ ( set_tm2 @ Xs2 ) )
& ( P3 @ Y5 ) )
=> ( Y5 = X4 ) ) ) ) ) ).
% list_ex1_iff
thf(fact_807_list__ex1__iff,axiom,
( list_ex1_list_fm
= ( ^ [P3: list_fm > $o,Xs2: list_list_fm] :
? [X4: list_fm] :
( ( member_list_fm @ X4 @ ( set_list_fm2 @ Xs2 ) )
& ( P3 @ X4 )
& ! [Y5: list_fm] :
( ( ( member_list_fm @ Y5 @ ( set_list_fm2 @ Xs2 ) )
& ( P3 @ Y5 ) )
=> ( Y5 = X4 ) ) ) ) ) ).
% list_ex1_iff
thf(fact_808_lexord__sufE,axiom,
! [Xs: list_tm,Zs: list_tm,Ys: list_tm,Qs: list_tm,R: set_Pr2455929065695642951_tm_tm] :
( ( member4457312017796942864ist_tm @ ( produc1418304791525149271ist_tm @ ( append_tm @ Xs @ Zs ) @ ( append_tm @ Ys @ Qs ) ) @ ( lexord_tm @ R ) )
=> ( ( Xs != Ys )
=> ( ( ( size_size_list_tm @ Xs )
= ( size_size_list_tm @ Ys ) )
=> ( ( ( size_size_list_tm @ Zs )
= ( size_size_list_tm @ Qs ) )
=> ( member4457312017796942864ist_tm @ ( produc1418304791525149271ist_tm @ Xs @ Ys ) @ ( lexord_tm @ R ) ) ) ) ) ) ).
% lexord_sufE
thf(fact_809_lex__append__rightI,axiom,
! [Xs: list_tm,Ys: list_tm,R: set_Pr2455929065695642951_tm_tm,Vs: list_tm,Us2: list_tm] :
( ( member4457312017796942864ist_tm @ ( produc1418304791525149271ist_tm @ Xs @ Ys ) @ ( lex_tm @ R ) )
=> ( ( ( size_size_list_tm @ Vs )
= ( size_size_list_tm @ Us2 ) )
=> ( member4457312017796942864ist_tm @ ( produc1418304791525149271ist_tm @ ( append_tm @ Xs @ Us2 ) @ ( append_tm @ Ys @ Vs ) ) @ ( lex_tm @ R ) ) ) ) ).
% lex_append_rightI
thf(fact_810_lenlex__append1,axiom,
! [Us2: list_tm,Xs: list_tm,R3: set_Pr2455929065695642951_tm_tm,Vs: list_tm,Ys: list_tm] :
( ( member4457312017796942864ist_tm @ ( produc1418304791525149271ist_tm @ Us2 @ Xs ) @ ( lenlex_tm @ R3 ) )
=> ( ( ( size_size_list_tm @ Vs )
= ( size_size_list_tm @ Ys ) )
=> ( member4457312017796942864ist_tm @ ( produc1418304791525149271ist_tm @ ( append_tm @ Us2 @ Vs ) @ ( append_tm @ Xs @ Ys ) ) @ ( lenlex_tm @ R3 ) ) ) ) ).
% lenlex_append1
thf(fact_811_asymp__on__asym__on__eq,axiom,
! [A: set_nat,R: set_Pr1261947904930325089at_nat] :
( ( asymp_on_nat @ A
@ ^ [X4: nat,Y5: nat] : ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X4 @ Y5 ) @ R ) )
= ( asym_on_nat @ A @ R ) ) ).
% asymp_on_asym_on_eq
thf(fact_812_stake_Osimps_I2_J,axiom,
! [N: nat,S: stream2709947120125613254m_rule] :
( ( stake_5421812949518764133m_rule @ ( suc @ N ) @ S )
= ( cons_P2818572562524668032m_rule @ ( shd_Pr4562317740776619530m_rule @ S ) @ ( stake_5421812949518764133m_rule @ N @ ( stl_Pr12655793849929990m_rule @ S ) ) ) ) ).
% stake.simps(2)
thf(fact_813_stake_Osimps_I2_J,axiom,
! [N: nat,S: stream_rule] :
( ( stake_rule @ ( suc @ N ) @ S )
= ( cons_rule @ ( shd_rule @ S ) @ ( stake_rule @ N @ ( stl_rule @ S ) ) ) ) ).
% stake.simps(2)
thf(fact_814_can__select__set__list__ex1,axiom,
! [P: fm > $o,A: list_fm] :
( ( can_select_fm @ P @ ( set_fm2 @ A ) )
= ( list_ex1_fm @ P @ A ) ) ).
% can_select_set_list_ex1
thf(fact_815_can__select__set__list__ex1,axiom,
! [P: tm > $o,A: list_tm] :
( ( can_select_tm @ P @ ( set_tm2 @ A ) )
= ( list_ex1_tm @ P @ A ) ) ).
% can_select_set_list_ex1
thf(fact_816_can__select__set__list__ex1,axiom,
! [P: list_fm > $o,A: list_list_fm] :
( ( can_select_list_fm @ P @ ( set_list_fm2 @ A ) )
= ( list_ex1_list_fm @ P @ A ) ) ).
% can_select_set_list_ex1
thf(fact_817_set__n__lists,axiom,
! [N: nat,Xs: list_list_fm] :
( ( set_list_list_fm2 @ ( n_lists_list_fm @ N @ Xs ) )
= ( collect_list_list_fm
@ ^ [Ys2: list_list_fm] :
( ( ( size_s115229985653309035ist_fm @ Ys2 )
= N )
& ( ord_le7838213414353715577ist_fm @ ( set_list_fm2 @ Ys2 ) @ ( set_list_fm2 @ Xs ) ) ) ) ) ).
% set_n_lists
thf(fact_818_set__n__lists,axiom,
! [N: nat,Xs: list_tm] :
( ( set_list_tm2 @ ( n_lists_tm @ N @ Xs ) )
= ( collect_list_tm
@ ^ [Ys2: list_tm] :
( ( ( size_size_list_tm @ Ys2 )
= N )
& ( ord_less_eq_set_tm @ ( set_tm2 @ Ys2 ) @ ( set_tm2 @ Xs ) ) ) ) ) ).
% set_n_lists
thf(fact_819_set__n__lists,axiom,
! [N: nat,Xs: list_fm] :
( ( set_list_fm2 @ ( n_lists_fm @ N @ Xs ) )
= ( collect_list_fm
@ ^ [Ys2: list_fm] :
( ( ( size_size_list_fm @ Ys2 )
= N )
& ( ord_less_eq_set_fm @ ( set_fm2 @ Ys2 ) @ ( set_fm2 @ Xs ) ) ) ) ) ).
% set_n_lists
thf(fact_820_set__n__lists,axiom,
! [N: nat,Xs: list_rule] :
( ( set_list_rule2 @ ( n_lists_rule @ N @ Xs ) )
= ( collect_list_rule
@ ^ [Ys2: list_rule] :
( ( ( size_size_list_rule @ Ys2 )
= N )
& ( ord_less_eq_set_rule @ ( set_rule2 @ Ys2 ) @ ( set_rule2 @ Xs ) ) ) ) ) ).
% set_n_lists
thf(fact_821_stake__Suc,axiom,
! [N: nat,S: stream_tm] :
( ( stake_tm @ ( suc @ N ) @ S )
= ( append_tm @ ( stake_tm @ N @ S ) @ ( cons_tm @ ( snth_tm @ S @ N ) @ nil_tm ) ) ) ).
% stake_Suc
thf(fact_822_stake__Suc,axiom,
! [N: nat,S: stream2709947120125613254m_rule] :
( ( stake_5421812949518764133m_rule @ ( suc @ N ) @ S )
= ( append2714808343334685275m_rule @ ( stake_5421812949518764133m_rule @ N @ S ) @ ( cons_P2818572562524668032m_rule @ ( snth_P6679518042731451922m_rule @ S @ N ) @ nil_Pr2808214839852828720m_rule ) ) ) ).
% stake_Suc
thf(fact_823_stake__Suc,axiom,
! [N: nat,S: stream_rule] :
( ( stake_rule @ ( suc @ N ) @ S )
= ( append_rule @ ( stake_rule @ N @ S ) @ ( cons_rule @ ( snth_rule @ S @ N ) @ nil_rule ) ) ) ).
% stake_Suc
thf(fact_824_Cons__lenlex__iff,axiom,
! [M2: nat,Ms: list_nat,N: nat,Ns: list_nat,R: set_Pr1261947904930325089at_nat] :
( ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ ( cons_nat @ M2 @ Ms ) @ ( cons_nat @ N @ Ns ) ) @ ( lenlex_nat @ R ) )
= ( ( ord_less_nat @ ( size_size_list_nat @ Ms ) @ ( size_size_list_nat @ Ns ) )
| ( ( ( size_size_list_nat @ Ms )
= ( size_size_list_nat @ Ns ) )
& ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ M2 @ N ) @ R ) )
| ( ( M2 = N )
& ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ Ms @ Ns ) @ ( lenlex_nat @ R ) ) ) ) ) ).
% Cons_lenlex_iff
thf(fact_825_append__is__Nil__conv,axiom,
! [Xs: list_tm,Ys: list_tm] :
( ( ( append_tm @ Xs @ Ys )
= nil_tm )
= ( ( Xs = nil_tm )
& ( Ys = nil_tm ) ) ) ).
% append_is_Nil_conv
thf(fact_826_Nil__is__append__conv,axiom,
! [Xs: list_tm,Ys: list_tm] :
( ( nil_tm
= ( append_tm @ Xs @ Ys ) )
= ( ( Xs = nil_tm )
& ( Ys = nil_tm ) ) ) ).
% Nil_is_append_conv
thf(fact_827_self__append__conv2,axiom,
! [Y3: list_tm,Xs: list_tm] :
( ( Y3
= ( append_tm @ Xs @ Y3 ) )
= ( Xs = nil_tm ) ) ).
% self_append_conv2
thf(fact_828_append__self__conv2,axiom,
! [Xs: list_tm,Ys: list_tm] :
( ( ( append_tm @ Xs @ Ys )
= Ys )
= ( Xs = nil_tm ) ) ).
% append_self_conv2
thf(fact_829_self__append__conv,axiom,
! [Y3: list_tm,Ys: list_tm] :
( ( Y3
= ( append_tm @ Y3 @ Ys ) )
= ( Ys = nil_tm ) ) ).
% self_append_conv
thf(fact_830_append__self__conv,axiom,
! [Xs: list_tm,Ys: list_tm] :
( ( ( append_tm @ Xs @ Ys )
= Xs )
= ( Ys = nil_tm ) ) ).
% append_self_conv
thf(fact_831_append__Nil2,axiom,
! [Xs: list_tm] :
( ( append_tm @ Xs @ nil_tm )
= Xs ) ).
% append_Nil2
thf(fact_832_append_Oright__neutral,axiom,
! [A2: list_tm] :
( ( append_tm @ A2 @ nil_tm )
= A2 ) ).
% append.right_neutral
thf(fact_833_remdups__eq__nil__right__iff,axiom,
! [X3: list_tm] :
( ( nil_tm
= ( remdups_tm @ X3 ) )
= ( X3 = nil_tm ) ) ).
% remdups_eq_nil_right_iff
thf(fact_834_remdups__eq__nil__iff,axiom,
! [X3: list_tm] :
( ( ( remdups_tm @ X3 )
= nil_tm )
= ( X3 = nil_tm ) ) ).
% remdups_eq_nil_iff
thf(fact_835_list__all__simps_I2_J,axiom,
! [P: produc340336539035504054m_rule > $o] : ( list_a9038620553111681026m_rule @ P @ nil_Pr2808214839852828720m_rule ) ).
% list_all_simps(2)
thf(fact_836_append1__eq__conv,axiom,
! [Xs: list_tm,X3: tm,Ys: list_tm,Y3: tm] :
( ( ( append_tm @ Xs @ ( cons_tm @ X3 @ nil_tm ) )
= ( append_tm @ Ys @ ( cons_tm @ Y3 @ nil_tm ) ) )
= ( ( Xs = Ys )
& ( X3 = Y3 ) ) ) ).
% append1_eq_conv
thf(fact_837_can__select__def,axiom,
( can_select_rule
= ( ^ [P3: rule > $o,A6: set_rule] :
? [X4: rule] :
( ( member_rule @ X4 @ A6 )
& ( P3 @ X4 )
& ! [Y5: rule] :
( ( ( member_rule @ Y5 @ A6 )
& ( P3 @ Y5 ) )
=> ( Y5 = X4 ) ) ) ) ) ).
% can_select_def
thf(fact_838_can__select__def,axiom,
( can_select_fm
= ( ^ [P3: fm > $o,A6: set_fm] :
? [X4: fm] :
( ( member_fm @ X4 @ A6 )
& ( P3 @ X4 )
& ! [Y5: fm] :
( ( ( member_fm @ Y5 @ A6 )
& ( P3 @ Y5 ) )
=> ( Y5 = X4 ) ) ) ) ) ).
% can_select_def
thf(fact_839_can__select__def,axiom,
( can_se2863371813301722610ist_fm
= ( ^ [P3: produc6018962875968178549ist_fm > $o,A6: set_Pr5202636777678657877ist_fm] :
? [X4: produc6018962875968178549ist_fm] :
( ( member4699826688122452638ist_fm @ X4 @ A6 )
& ( P3 @ X4 )
& ! [Y5: produc6018962875968178549ist_fm] :
( ( ( member4699826688122452638ist_fm @ Y5 @ A6 )
& ( P3 @ Y5 ) )
=> ( Y5 = X4 ) ) ) ) ) ).
% can_select_def
thf(fact_840_can__select__def,axiom,
( can_se4506553828710032889m_rule
= ( ^ [P3: produc340336539035504054m_rule > $o,A6: set_Pr1822751329126368876m_rule] :
? [X4: produc340336539035504054m_rule] :
( ( member7231649785386036813m_rule @ X4 @ A6 )
& ( P3 @ X4 )
& ! [Y5: produc340336539035504054m_rule] :
( ( ( member7231649785386036813m_rule @ Y5 @ A6 )
& ( P3 @ Y5 ) )
=> ( Y5 = X4 ) ) ) ) ) ).
% can_select_def
thf(fact_841_can__select__def,axiom,
( can_select_list_fm
= ( ^ [P3: list_fm > $o,A6: set_list_fm] :
? [X4: list_fm] :
( ( member_list_fm @ X4 @ A6 )
& ( P3 @ X4 )
& ! [Y5: list_fm] :
( ( ( member_list_fm @ Y5 @ A6 )
& ( P3 @ Y5 ) )
=> ( Y5 = X4 ) ) ) ) ) ).
% can_select_def
thf(fact_842_eq__Nil__appendI,axiom,
! [Xs: list_tm,Ys: list_tm] :
( ( Xs = Ys )
=> ( Xs
= ( append_tm @ nil_tm @ Ys ) ) ) ).
% eq_Nil_appendI
thf(fact_843_append_Oleft__neutral,axiom,
! [A2: list_tm] :
( ( append_tm @ nil_tm @ A2 )
= A2 ) ).
% append.left_neutral
thf(fact_844_append__Nil,axiom,
! [Ys: list_tm] :
( ( append_tm @ nil_tm @ Ys )
= Ys ) ).
% append_Nil
thf(fact_845_rules__repeat,axiom,
! [M2: nat,R: rule] :
? [N2: nat] :
( ( ord_less_nat @ M2 @ N2 )
& ( ( snth_rule @ rules @ N2 )
= R ) ) ).
% rules_repeat
thf(fact_846_remdups_Osimps_I1_J,axiom,
( ( remdups_tm @ nil_tm )
= nil_tm ) ).
% remdups.simps(1)
thf(fact_847_shift_Osimps_I1_J,axiom,
! [S: stream2709947120125613254m_rule] :
( ( shift_2334844276748245581m_rule @ nil_Pr2808214839852828720m_rule @ S )
= S ) ).
% shift.simps(1)
thf(fact_848_list_Opred__inject_I1_J,axiom,
! [P: produc340336539035504054m_rule > $o] : ( list_a9038620553111681026m_rule @ P @ nil_Pr2808214839852828720m_rule ) ).
% list.pred_inject(1)
thf(fact_849_asymp__on__greater,axiom,
! [A: set_nat] :
( asymp_on_nat @ A
@ ^ [X4: nat,Y5: nat] : ( ord_less_nat @ Y5 @ X4 ) ) ).
% asymp_on_greater
thf(fact_850_irreflp__on__greater,axiom,
! [A: set_nat] :
( irreflp_on_nat @ A
@ ^ [X4: nat,Y5: nat] : ( ord_less_nat @ Y5 @ X4 ) ) ).
% irreflp_on_greater
thf(fact_851_rev__nonempty__induct,axiom,
! [Xs: list_tm,P: list_tm > $o] :
( ( Xs != nil_tm )
=> ( ! [X: tm] : ( P @ ( cons_tm @ X @ nil_tm ) )
=> ( ! [X: tm,Xs3: list_tm] :
( ( Xs3 != nil_tm )
=> ( ( P @ Xs3 )
=> ( P @ ( append_tm @ Xs3 @ ( cons_tm @ X @ nil_tm ) ) ) ) )
=> ( P @ Xs ) ) ) ) ).
% rev_nonempty_induct
thf(fact_852_append__eq__Cons__conv,axiom,
! [Ys: list_tm,Zs: list_tm,X3: tm,Xs: list_tm] :
( ( ( append_tm @ Ys @ Zs )
= ( cons_tm @ X3 @ Xs ) )
= ( ( ( Ys = nil_tm )
& ( Zs
= ( cons_tm @ X3 @ Xs ) ) )
| ? [Ys5: list_tm] :
( ( Ys
= ( cons_tm @ X3 @ Ys5 ) )
& ( ( append_tm @ Ys5 @ Zs )
= Xs ) ) ) ) ).
% append_eq_Cons_conv
thf(fact_853_Cons__eq__append__conv,axiom,
! [X3: tm,Xs: list_tm,Ys: list_tm,Zs: list_tm] :
( ( ( cons_tm @ X3 @ Xs )
= ( append_tm @ Ys @ Zs ) )
= ( ( ( Ys = nil_tm )
& ( ( cons_tm @ X3 @ Xs )
= Zs ) )
| ? [Ys5: list_tm] :
( ( ( cons_tm @ X3 @ Ys5 )
= Ys )
& ( Xs
= ( append_tm @ Ys5 @ Zs ) ) ) ) ) ).
% Cons_eq_append_conv
thf(fact_854_rev__exhaust,axiom,
! [Xs: list_tm] :
( ( Xs != nil_tm )
=> ~ ! [Ys3: list_tm,Y: tm] :
( Xs
!= ( append_tm @ Ys3 @ ( cons_tm @ Y @ nil_tm ) ) ) ) ).
% rev_exhaust
thf(fact_855_rev__induct,axiom,
! [P: list_tm > $o,Xs: list_tm] :
( ( P @ nil_tm )
=> ( ! [X: tm,Xs3: list_tm] :
( ( P @ Xs3 )
=> ( P @ ( append_tm @ Xs3 @ ( cons_tm @ X @ nil_tm ) ) ) )
=> ( P @ Xs ) ) ) ).
% rev_induct
thf(fact_856_map__tailrec__rev_Ocases,axiom,
! [X3: produc3792783349651969271ist_fm] :
( ! [F2: tm > fm,Bs: list_fm] :
( X3
!= ( produc4025633523336838569ist_fm @ F2 @ ( produc1414352766439514085ist_fm @ nil_tm @ Bs ) ) )
=> ~ ! [F2: tm > fm,A4: tm,As: list_tm,Bs: list_fm] :
( X3
!= ( produc4025633523336838569ist_fm @ F2 @ ( produc1414352766439514085ist_fm @ ( cons_tm @ A4 @ As ) @ Bs ) ) ) ) ).
% map_tailrec_rev.cases
thf(fact_857_sdrop__suntil,axiom,
! [Q2: stream2709947120125613254m_rule > $o,J: nat,Omega: stream2709947120125613254m_rule,P2: stream2709947120125613254m_rule > $o] :
( ( Q2 @ ( sdrop_8169176516188972301m_rule @ J @ Omega ) )
=> ( ! [K2: nat] :
( ( ord_less_nat @ K2 @ J )
=> ( P2 @ ( sdrop_8169176516188972301m_rule @ K2 @ Omega ) ) )
=> ( linear5457325955354885675m_rule @ P2 @ Q2 @ Omega ) ) ) ).
% sdrop_suntil
thf(fact_858_sdrop__suntil,axiom,
! [Q2: stream_rule > $o,J: nat,Omega: stream_rule,P2: stream_rule > $o] :
( ( Q2 @ ( sdrop_rule @ J @ Omega ) )
=> ( ! [K2: nat] :
( ( ord_less_nat @ K2 @ J )
=> ( P2 @ ( sdrop_rule @ K2 @ Omega ) ) )
=> ( linear1836410492716652551l_rule @ P2 @ Q2 @ Omega ) ) ) ).
% sdrop_suntil
thf(fact_859_sdrop__if__suntil,axiom,
! [P2: stream2709947120125613254m_rule > $o,Q2: stream2709947120125613254m_rule > $o,Omega: stream2709947120125613254m_rule] :
( ( linear5457325955354885675m_rule @ P2 @ Q2 @ Omega )
=> ? [J2: nat] :
( ( Q2 @ ( sdrop_8169176516188972301m_rule @ J2 @ Omega ) )
& ! [K3: nat] :
( ( ord_less_nat @ K3 @ J2 )
=> ( P2 @ ( sdrop_8169176516188972301m_rule @ K3 @ Omega ) ) ) ) ) ).
% sdrop_if_suntil
thf(fact_860_sdrop__if__suntil,axiom,
! [P2: stream_rule > $o,Q2: stream_rule > $o,Omega: stream_rule] :
( ( linear1836410492716652551l_rule @ P2 @ Q2 @ Omega )
=> ? [J2: nat] :
( ( Q2 @ ( sdrop_rule @ J2 @ Omega ) )
& ! [K3: nat] :
( ( ord_less_nat @ K3 @ J2 )
=> ( P2 @ ( sdrop_rule @ K3 @ Omega ) ) ) ) ) ).
% sdrop_if_suntil
thf(fact_861_suntil__iff__sdrop,axiom,
( linear5457325955354885675m_rule
= ( ^ [P4: stream2709947120125613254m_rule > $o,Q3: stream2709947120125613254m_rule > $o,Omega2: stream2709947120125613254m_rule] :
? [J3: nat] :
( ( Q3 @ ( sdrop_8169176516188972301m_rule @ J3 @ Omega2 ) )
& ! [K4: nat] :
( ( ord_less_nat @ K4 @ J3 )
=> ( P4 @ ( sdrop_8169176516188972301m_rule @ K4 @ Omega2 ) ) ) ) ) ) ).
% suntil_iff_sdrop
thf(fact_862_suntil__iff__sdrop,axiom,
( linear1836410492716652551l_rule
= ( ^ [P4: stream_rule > $o,Q3: stream_rule > $o,Omega2: stream_rule] :
? [J3: nat] :
( ( Q3 @ ( sdrop_rule @ J3 @ Omega2 ) )
& ! [K4: nat] :
( ( ord_less_nat @ K4 @ J3 )
=> ( P4 @ ( sdrop_rule @ K4 @ Omega2 ) ) ) ) ) ) ).
% suntil_iff_sdrop
thf(fact_863_same__length__different,axiom,
! [Xs: list_tm,Ys: list_tm] :
( ( Xs != Ys )
=> ( ( ( size_size_list_tm @ Xs )
= ( size_size_list_tm @ Ys ) )
=> ? [Pre: list_tm,X: tm,Xs5: list_tm,Y: tm,Ys6: list_tm] :
( ( X != Y )
& ( Xs
= ( append_tm @ Pre @ ( append_tm @ ( cons_tm @ X @ nil_tm ) @ Xs5 ) ) )
& ( Ys
= ( append_tm @ Pre @ ( append_tm @ ( cons_tm @ Y @ nil_tm ) @ Ys6 ) ) ) ) ) ) ).
% same_length_different
thf(fact_864_length__Suc__conv__rev,axiom,
! [Xs: list_tm,N: nat] :
( ( ( size_size_list_tm @ Xs )
= ( suc @ N ) )
= ( ? [Y5: tm,Ys2: list_tm] :
( ( Xs
= ( append_tm @ Ys2 @ ( cons_tm @ Y5 @ nil_tm ) ) )
& ( ( size_size_list_tm @ Ys2 )
= N ) ) ) ) ).
% length_Suc_conv_rev
thf(fact_865_length__append__singleton,axiom,
! [Xs: list_tm,X3: tm] :
( ( size_size_list_tm @ ( append_tm @ Xs @ ( cons_tm @ X3 @ nil_tm ) ) )
= ( suc @ ( size_size_list_tm @ Xs ) ) ) ).
% length_append_singleton
thf(fact_866_length__n__lists__elem,axiom,
! [Ys: list_fm,N: nat,Xs: list_fm] :
( ( member_list_fm @ Ys @ ( set_list_fm2 @ ( n_lists_fm @ N @ Xs ) ) )
=> ( ( size_size_list_fm @ Ys )
= N ) ) ).
% length_n_lists_elem
thf(fact_867_Succ__def,axiom,
( bNF_Greatest_Succ_fm
= ( ^ [Kl: set_list_fm,Kl2: list_fm] :
( collect_fm
@ ^ [K4: fm] : ( member_list_fm @ ( append_fm @ Kl2 @ ( cons_fm @ K4 @ nil_fm ) ) @ Kl ) ) ) ) ).
% Succ_def
thf(fact_868_Succ__def,axiom,
( bNF_Greatest_Succ_tm
= ( ^ [Kl: set_list_tm,Kl2: list_tm] :
( collect_tm
@ ^ [K4: tm] : ( member_list_tm @ ( append_tm @ Kl2 @ ( cons_tm @ K4 @ nil_tm ) ) @ Kl ) ) ) ) ).
% Succ_def
thf(fact_869_SuccD,axiom,
! [K: rule,Kl3: set_list_rule,Kl4: list_rule] :
( ( member_rule @ K @ ( bNF_Gr222962468618337463c_rule @ Kl3 @ Kl4 ) )
=> ( member_list_rule @ ( append_rule @ Kl4 @ ( cons_rule @ K @ nil_rule ) ) @ Kl3 ) ) ).
% SuccD
thf(fact_870_SuccD,axiom,
! [K: fm,Kl3: set_list_fm,Kl4: list_fm] :
( ( member_fm @ K @ ( bNF_Greatest_Succ_fm @ Kl3 @ Kl4 ) )
=> ( member_list_fm @ ( append_fm @ Kl4 @ ( cons_fm @ K @ nil_fm ) ) @ Kl3 ) ) ).
% SuccD
thf(fact_871_SuccD,axiom,
! [K: produc6018962875968178549ist_fm,Kl3: set_li7039099389128218331ist_fm,Kl4: list_P2887561121880082555ist_fm] :
( ( member4699826688122452638ist_fm @ K @ ( bNF_Gr8830538609425469840ist_fm @ Kl3 @ Kl4 ) )
=> ( member4266112761388634404ist_fm @ ( append1476581234975291920ist_fm @ Kl4 @ ( cons_P6586734674047340843ist_fm @ K @ nil_Pr2223031680201799035ist_fm ) ) @ Kl3 ) ) ).
% SuccD
thf(fact_872_SuccD,axiom,
! [K: produc340336539035504054m_rule,Kl3: set_li5047378834958542076m_rule,Kl4: list_P2774625669004309958m_rule] :
( ( member7231649785386036813m_rule @ K @ ( bNF_Gr6616380055184446171m_rule @ Kl3 @ Kl4 ) )
=> ( member522667613741702877m_rule @ ( append2714808343334685275m_rule @ Kl4 @ ( cons_P2818572562524668032m_rule @ K @ nil_Pr2808214839852828720m_rule ) ) @ Kl3 ) ) ).
% SuccD
thf(fact_873_SuccD,axiom,
! [K: list_fm,Kl3: set_list_list_fm,Kl4: list_list_fm] :
( ( member_list_fm @ K @ ( bNF_Gr8387611704671093012ist_fm @ Kl3 @ Kl4 ) )
=> ( member_list_list_fm @ ( append_list_fm @ Kl4 @ ( cons_list_fm @ K @ nil_list_fm ) ) @ Kl3 ) ) ).
% SuccD
thf(fact_874_SuccD,axiom,
! [K: tm,Kl3: set_list_tm,Kl4: list_tm] :
( ( member_tm @ K @ ( bNF_Greatest_Succ_tm @ Kl3 @ Kl4 ) )
=> ( member_list_tm @ ( append_tm @ Kl4 @ ( cons_tm @ K @ nil_tm ) ) @ Kl3 ) ) ).
% SuccD
thf(fact_875_SuccI,axiom,
! [Kl4: list_rule,K: rule,Kl3: set_list_rule] :
( ( member_list_rule @ ( append_rule @ Kl4 @ ( cons_rule @ K @ nil_rule ) ) @ Kl3 )
=> ( member_rule @ K @ ( bNF_Gr222962468618337463c_rule @ Kl3 @ Kl4 ) ) ) ).
% SuccI
thf(fact_876_SuccI,axiom,
! [Kl4: list_P2887561121880082555ist_fm,K: produc6018962875968178549ist_fm,Kl3: set_li7039099389128218331ist_fm] :
( ( member4266112761388634404ist_fm @ ( append1476581234975291920ist_fm @ Kl4 @ ( cons_P6586734674047340843ist_fm @ K @ nil_Pr2223031680201799035ist_fm ) ) @ Kl3 )
=> ( member4699826688122452638ist_fm @ K @ ( bNF_Gr8830538609425469840ist_fm @ Kl3 @ Kl4 ) ) ) ).
% SuccI
thf(fact_877_SuccI,axiom,
! [Kl4: list_P2774625669004309958m_rule,K: produc340336539035504054m_rule,Kl3: set_li5047378834958542076m_rule] :
( ( member522667613741702877m_rule @ ( append2714808343334685275m_rule @ Kl4 @ ( cons_P2818572562524668032m_rule @ K @ nil_Pr2808214839852828720m_rule ) ) @ Kl3 )
=> ( member7231649785386036813m_rule @ K @ ( bNF_Gr6616380055184446171m_rule @ Kl3 @ Kl4 ) ) ) ).
% SuccI
thf(fact_878_SuccI,axiom,
! [Kl4: list_list_fm,K: list_fm,Kl3: set_list_list_fm] :
( ( member_list_list_fm @ ( append_list_fm @ Kl4 @ ( cons_list_fm @ K @ nil_list_fm ) ) @ Kl3 )
=> ( member_list_fm @ K @ ( bNF_Gr8387611704671093012ist_fm @ Kl3 @ Kl4 ) ) ) ).
% SuccI
thf(fact_879_SuccI,axiom,
! [Kl4: list_fm,K: fm,Kl3: set_list_fm] :
( ( member_list_fm @ ( append_fm @ Kl4 @ ( cons_fm @ K @ nil_fm ) ) @ Kl3 )
=> ( member_fm @ K @ ( bNF_Greatest_Succ_fm @ Kl3 @ Kl4 ) ) ) ).
% SuccI
thf(fact_880_SuccI,axiom,
! [Kl4: list_tm,K: tm,Kl3: set_list_tm] :
( ( member_list_tm @ ( append_tm @ Kl4 @ ( cons_tm @ K @ nil_tm ) ) @ Kl3 )
=> ( member_tm @ K @ ( bNF_Greatest_Succ_tm @ Kl3 @ Kl4 ) ) ) ).
% SuccI
thf(fact_881_in__measures_I2_J,axiom,
! [X3: nat,Y3: nat,F: nat > nat,Fs: list_nat_nat] :
( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X3 @ Y3 ) @ ( measures_nat @ ( cons_nat_nat @ F @ Fs ) ) )
= ( ( ord_less_nat @ ( F @ X3 ) @ ( F @ Y3 ) )
| ( ( ( F @ X3 )
= ( F @ Y3 ) )
& ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X3 @ Y3 ) @ ( measures_nat @ Fs ) ) ) ) ) ).
% in_measures(2)
thf(fact_882_in__measures_I1_J,axiom,
! [X3: nat,Y3: nat] :
~ ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X3 @ Y3 ) @ ( measures_nat @ nil_nat_nat ) ) ).
% in_measures(1)
thf(fact_883_flat__Stream,axiom,
! [Xs: list_P2774625669004309958m_rule,Ws: stream2471014364565126742m_rule] :
( ( Xs != nil_Pr2808214839852828720m_rule )
=> ( ( flat_P7721466590633226428m_rule @ ( sCons_6947062437411232367m_rule @ Xs @ Ws ) )
= ( shift_2334844276748245581m_rule @ Xs @ ( flat_P7721466590633226428m_rule @ Ws ) ) ) ) ).
% flat_Stream
thf(fact_884_pfsubsetD,axiom,
! [A: fset_P8989946509869081563ist_fm,B4: fset_P8989946509869081563ist_fm,C: produc6018962875968178549ist_fm] :
( ( ord_le7716396445699002503ist_fm @ A @ B4 )
=> ( ( fmembe3754813877001230652ist_fm @ C @ A )
=> ( fmembe3754813877001230652ist_fm @ C @ B4 ) ) ) ).
% pfsubsetD
thf(fact_885_pfsubsetD,axiom,
! [A: fset_A8849196716589201442m_rule,B4: fset_A8849196716589201442m_rule,C: abstra8520858721168826610m_rule] :
( ( ord_le6302656846121046646m_rule @ A @ B4 )
=> ( ( fmembe1099044791951439339m_rule @ C @ A )
=> ( fmembe1099044791951439339m_rule @ C @ B4 ) ) ) ).
% pfsubsetD
thf(fact_886_flat__unfold,axiom,
! [Ws: stream2471014364565126742m_rule] :
( ( ( shd_li4676821617271663642m_rule @ Ws )
!= nil_Pr2808214839852828720m_rule )
=> ( ( flat_P7721466590633226428m_rule @ Ws )
= ( shift_2334844276748245581m_rule @ ( shd_li4676821617271663642m_rule @ Ws ) @ ( flat_P7721466590633226428m_rule @ ( stl_li6523153919213261078m_rule @ Ws ) ) ) ) ) ).
% flat_unfold
thf(fact_887_measures__less,axiom,
! [F: nat > nat,X3: nat,Y3: nat,Fs: list_nat_nat] :
( ( ord_less_nat @ ( F @ X3 ) @ ( F @ Y3 ) )
=> ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X3 @ Y3 ) @ ( measures_nat @ ( cons_nat_nat @ F @ Fs ) ) ) ) ).
% measures_less
thf(fact_888_prefixes__snoc,axiom,
! [Xs: list_tm,X3: tm] :
( ( prefixes_tm @ ( append_tm @ Xs @ ( cons_tm @ X3 @ nil_tm ) ) )
= ( append_list_tm @ ( prefixes_tm @ Xs ) @ ( cons_list_tm @ ( append_tm @ Xs @ ( cons_tm @ X3 @ nil_tm ) ) @ nil_list_tm ) ) ) ).
% prefixes_snoc
thf(fact_889_empty__Shift,axiom,
! [Kl3: set_list_rule,K: rule] :
( ( member_list_rule @ nil_rule @ Kl3 )
=> ( ( member_rule @ K @ ( bNF_Gr222962468618337463c_rule @ Kl3 @ nil_rule ) )
=> ( member_list_rule @ nil_rule @ ( bNF_Gr7799963440264979763t_rule @ Kl3 @ K ) ) ) ) ).
% empty_Shift
thf(fact_890_empty__Shift,axiom,
! [Kl3: set_li7039099389128218331ist_fm,K: produc6018962875968178549ist_fm] :
( ( member4266112761388634404ist_fm @ nil_Pr2223031680201799035ist_fm @ Kl3 )
=> ( ( member4699826688122452638ist_fm @ K @ ( bNF_Gr8830538609425469840ist_fm @ Kl3 @ nil_Pr2223031680201799035ist_fm ) )
=> ( member4266112761388634404ist_fm @ nil_Pr2223031680201799035ist_fm @ ( bNF_Gr6094329398807657492ist_fm @ Kl3 @ K ) ) ) ) ).
% empty_Shift
thf(fact_891_empty__Shift,axiom,
! [Kl3: set_li5047378834958542076m_rule,K: produc340336539035504054m_rule] :
( ( member522667613741702877m_rule @ nil_Pr2808214839852828720m_rule @ Kl3 )
=> ( ( member7231649785386036813m_rule @ K @ ( bNF_Gr6616380055184446171m_rule @ Kl3 @ nil_Pr2808214839852828720m_rule ) )
=> ( member522667613741702877m_rule @ nil_Pr2808214839852828720m_rule @ ( bNF_Gr2879600419463477079m_rule @ Kl3 @ K ) ) ) ) ).
% empty_Shift
thf(fact_892_empty__Shift,axiom,
! [Kl3: set_list_list_fm,K: list_fm] :
( ( member_list_list_fm @ nil_list_fm @ Kl3 )
=> ( ( member_list_fm @ K @ ( bNF_Gr8387611704671093012ist_fm @ Kl3 @ nil_list_fm ) )
=> ( member_list_list_fm @ nil_list_fm @ ( bNF_Gr4365904581682047384ist_fm @ Kl3 @ K ) ) ) ) ).
% empty_Shift
thf(fact_893_empty__Shift,axiom,
! [Kl3: set_list_fm,K: fm] :
( ( member_list_fm @ nil_fm @ Kl3 )
=> ( ( member_fm @ K @ ( bNF_Greatest_Succ_fm @ Kl3 @ nil_fm ) )
=> ( member_list_fm @ nil_fm @ ( bNF_Gr6607445516916672786ift_fm @ Kl3 @ K ) ) ) ) ).
% empty_Shift
thf(fact_894_prefixes__eq__snoc,axiom,
! [Ys: list_tm,Xs: list_list_tm,X3: list_tm] :
( ( ( prefixes_tm @ Ys )
= ( append_list_tm @ Xs @ ( cons_list_tm @ X3 @ nil_list_tm ) ) )
= ( ( ( ( Ys = nil_tm )
& ( Xs = nil_list_tm ) )
| ? [Z5: tm,Zs2: list_tm] :
( ( Ys
= ( append_tm @ Zs2 @ ( cons_tm @ Z5 @ nil_tm ) ) )
& ( Xs
= ( prefixes_tm @ Zs2 ) ) ) )
& ( X3 = Ys ) ) ) ).
% prefixes_eq_snoc
thf(fact_895_sdrop_Osimps_I1_J,axiom,
! [S: stream2709947120125613254m_rule] :
( ( sdrop_8169176516188972301m_rule @ zero_zero_nat @ S )
= S ) ).
% sdrop.simps(1)
thf(fact_896_sdrop_Osimps_I1_J,axiom,
! [S: stream_rule] :
( ( sdrop_rule @ zero_zero_nat @ S )
= S ) ).
% sdrop.simps(1)
thf(fact_897_snth_Osimps_I1_J,axiom,
! [S: stream2709947120125613254m_rule] :
( ( snth_P6679518042731451922m_rule @ S @ zero_zero_nat )
= ( shd_Pr4562317740776619530m_rule @ S ) ) ).
% snth.simps(1)
thf(fact_898_snth_Osimps_I1_J,axiom,
! [S: stream_rule] :
( ( snth_rule @ S @ zero_zero_nat )
= ( shd_rule @ S ) ) ).
% snth.simps(1)
thf(fact_899_ShiftD,axiom,
! [Kl4: list_fm,Kl3: set_list_fm,K: fm] :
( ( member_list_fm @ Kl4 @ ( bNF_Gr6607445516916672786ift_fm @ Kl3 @ K ) )
=> ( member_list_fm @ ( cons_fm @ K @ Kl4 ) @ Kl3 ) ) ).
% ShiftD
thf(fact_900_length__pos__if__in__set,axiom,
! [X3: rule,Xs: list_rule] :
( ( member_rule @ X3 @ ( set_rule2 @ Xs ) )
=> ( ord_less_nat @ zero_zero_nat @ ( size_size_list_rule @ Xs ) ) ) ).
% length_pos_if_in_set
thf(fact_901_length__pos__if__in__set,axiom,
! [X3: produc6018962875968178549ist_fm,Xs: list_P2887561121880082555ist_fm] :
( ( member4699826688122452638ist_fm @ X3 @ ( set_Pr5051287786238754058ist_fm @ Xs ) )
=> ( ord_less_nat @ zero_zero_nat @ ( size_s3138477486474831591ist_fm @ Xs ) ) ) ).
% length_pos_if_in_set
thf(fact_902_length__pos__if__in__set,axiom,
! [X3: produc340336539035504054m_rule,Xs: list_P2774625669004309958m_rule] :
( ( member7231649785386036813m_rule @ X3 @ ( set_Pr4534715572506550497m_rule @ Xs ) )
=> ( ord_less_nat @ zero_zero_nat @ ( size_s1575636608424004698m_rule @ Xs ) ) ) ).
% length_pos_if_in_set
thf(fact_903_length__pos__if__in__set,axiom,
! [X3: fm,Xs: list_fm] :
( ( member_fm @ X3 @ ( set_fm2 @ Xs ) )
=> ( ord_less_nat @ zero_zero_nat @ ( size_size_list_fm @ Xs ) ) ) ).
% length_pos_if_in_set
thf(fact_904_length__pos__if__in__set,axiom,
! [X3: tm,Xs: list_tm] :
( ( member_tm @ X3 @ ( set_tm2 @ Xs ) )
=> ( ord_less_nat @ zero_zero_nat @ ( size_size_list_tm @ Xs ) ) ) ).
% length_pos_if_in_set
thf(fact_905_length__pos__if__in__set,axiom,
! [X3: list_fm,Xs: list_list_fm] :
( ( member_list_fm @ X3 @ ( set_list_fm2 @ Xs ) )
=> ( ord_less_nat @ zero_zero_nat @ ( size_s115229985653309035ist_fm @ Xs ) ) ) ).
% length_pos_if_in_set
thf(fact_906_Shift__def,axiom,
( bNF_Gr6607445516916672786ift_fm
= ( ^ [Kl: set_list_fm,K4: fm] :
( collect_list_fm
@ ^ [Kl2: list_fm] : ( member_list_fm @ ( cons_fm @ K4 @ Kl2 ) @ Kl ) ) ) ) ).
% Shift_def
thf(fact_907_flat_Osimps_I2_J,axiom,
! [Ws: stream2471014364565126742m_rule] :
( ( stl_Pr12655793849929990m_rule @ ( flat_P7721466590633226428m_rule @ Ws ) )
= ( flat_P7721466590633226428m_rule
@ ( if_str2465195801493762064m_rule
@ ( ( tl_Pro2692115096043292423m_rule @ ( shd_li4676821617271663642m_rule @ Ws ) )
= nil_Pr2808214839852828720m_rule )
@ ( stl_li6523153919213261078m_rule @ Ws )
@ ( sCons_6947062437411232367m_rule @ ( tl_Pro2692115096043292423m_rule @ ( shd_li4676821617271663642m_rule @ Ws ) ) @ ( stl_li6523153919213261078m_rule @ Ws ) ) ) ) ) ).
% flat.simps(2)
thf(fact_908_flat_Osimps_I2_J,axiom,
! [Ws: stream_list_rule] :
( ( stl_rule @ ( flat_rule @ Ws ) )
= ( flat_rule
@ ( if_stream_list_rule
@ ( ( tl_rule @ ( shd_list_rule @ Ws ) )
= nil_rule )
@ ( stl_list_rule @ Ws )
@ ( sCons_list_rule @ ( tl_rule @ ( shd_list_rule @ Ws ) ) @ ( stl_list_rule @ Ws ) ) ) ) ) ).
% flat.simps(2)
thf(fact_909_nths__Cons,axiom,
! [X3: tm,L: list_tm,A: set_nat] :
( ( nths_tm @ ( cons_tm @ X3 @ L ) @ A )
= ( append_tm @ ( if_list_tm @ ( member_nat @ zero_zero_nat @ A ) @ ( cons_tm @ X3 @ nil_tm ) @ nil_tm )
@ ( nths_tm @ L
@ ( collect_nat
@ ^ [J3: nat] : ( member_nat @ ( suc @ J3 ) @ A ) ) ) ) ) ).
% nths_Cons
thf(fact_910_concat__eq__append__conv,axiom,
! [Xss: list_list_tm,Ys: list_tm,Zs: list_tm] :
( ( ( concat_tm @ Xss )
= ( append_tm @ Ys @ Zs ) )
= ( ( ( Xss = nil_list_tm )
=> ( ( Ys = nil_tm )
& ( Zs = nil_tm ) ) )
& ( ( Xss != nil_list_tm )
=> ? [Xss1: list_list_tm,Xs2: list_tm,Xs6: list_tm,Xss2: list_list_tm] :
( ( Xss
= ( append_list_tm @ Xss1 @ ( cons_list_tm @ ( append_tm @ Xs2 @ Xs6 ) @ Xss2 ) ) )
& ( Ys
= ( append_tm @ ( concat_tm @ Xss1 ) @ Xs2 ) )
& ( Zs
= ( append_tm @ Xs6 @ ( concat_tm @ Xss2 ) ) ) ) ) ) ) ).
% concat_eq_append_conv
thf(fact_911_snoc__listrel1__snoc__iff,axiom,
! [Xs: list_tm,X3: tm,Ys: list_tm,Y3: tm,R: set_Pr2455929065695642951_tm_tm] :
( ( member4457312017796942864ist_tm @ ( produc1418304791525149271ist_tm @ ( append_tm @ Xs @ ( cons_tm @ X3 @ nil_tm ) ) @ ( append_tm @ Ys @ ( cons_tm @ Y3 @ nil_tm ) ) ) @ ( listrel1_tm @ R ) )
= ( ( ( member4457312017796942864ist_tm @ ( produc1418304791525149271ist_tm @ Xs @ Ys ) @ ( listrel1_tm @ R ) )
& ( X3 = Y3 ) )
| ( ( Xs = Ys )
& ( member3121616906494481296_tm_tm @ ( product_Pair_tm_tm @ X3 @ Y3 ) @ R ) ) ) ) ).
% snoc_listrel1_snoc_iff
thf(fact_912_snoc__listrel1__snoc__iff,axiom,
! [Xs: list_nat,X3: nat,Ys: list_nat,Y3: nat,R: set_Pr1261947904930325089at_nat] :
( ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ ( append_nat @ Xs @ ( cons_nat @ X3 @ nil_nat ) ) @ ( append_nat @ Ys @ ( cons_nat @ Y3 @ nil_nat ) ) ) @ ( listrel1_nat @ R ) )
= ( ( ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ Xs @ Ys ) @ ( listrel1_nat @ R ) )
& ( X3 = Y3 ) )
| ( ( Xs = Ys )
& ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X3 @ Y3 ) @ R ) ) ) ) ).
% snoc_listrel1_snoc_iff
thf(fact_913_tl__append2,axiom,
! [Xs: list_tm,Ys: list_tm] :
( ( Xs != nil_tm )
=> ( ( tl_tm @ ( append_tm @ Xs @ Ys ) )
= ( append_tm @ ( tl_tm @ Xs ) @ Ys ) ) ) ).
% tl_append2
thf(fact_914_concat__eq__Nil__conv,axiom,
! [Xss: list_list_fm] :
( ( ( concat_fm @ Xss )
= nil_fm )
= ( ! [X4: list_fm] :
( ( member_list_fm @ X4 @ ( set_list_fm2 @ Xss ) )
=> ( X4 = nil_fm ) ) ) ) ).
% concat_eq_Nil_conv
thf(fact_915_Nil__eq__concat__conv,axiom,
! [Xss: list_list_fm] :
( ( nil_fm
= ( concat_fm @ Xss ) )
= ( ! [X4: list_fm] :
( ( member_list_fm @ X4 @ ( set_list_fm2 @ Xss ) )
=> ( X4 = nil_fm ) ) ) ) ).
% Nil_eq_concat_conv
thf(fact_916_concat__append,axiom,
! [Xs: list_list_tm,Ys: list_list_tm] :
( ( concat_tm @ ( append_list_tm @ Xs @ Ys ) )
= ( append_tm @ ( concat_tm @ Xs ) @ ( concat_tm @ Ys ) ) ) ).
% concat_append
thf(fact_917_Cons__listrel1__Cons,axiom,
! [X3: nat,Xs: list_nat,Y3: nat,Ys: list_nat,R: set_Pr1261947904930325089at_nat] :
( ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ ( cons_nat @ X3 @ Xs ) @ ( cons_nat @ Y3 @ Ys ) ) @ ( listrel1_nat @ R ) )
= ( ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X3 @ Y3 ) @ R )
& ( Xs = Ys ) )
| ( ( X3 = Y3 )
& ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ Xs @ Ys ) @ ( listrel1_nat @ R ) ) ) ) ) ).
% Cons_listrel1_Cons
thf(fact_918_shift__simps_I2_J,axiom,
! [Xs: list_P2774625669004309958m_rule,S: stream2709947120125613254m_rule] :
( ( ( Xs = nil_Pr2808214839852828720m_rule )
=> ( ( stl_Pr12655793849929990m_rule @ ( shift_2334844276748245581m_rule @ Xs @ S ) )
= ( stl_Pr12655793849929990m_rule @ S ) ) )
& ( ( Xs != nil_Pr2808214839852828720m_rule )
=> ( ( stl_Pr12655793849929990m_rule @ ( shift_2334844276748245581m_rule @ Xs @ S ) )
= ( shift_2334844276748245581m_rule @ ( tl_Pro2692115096043292423m_rule @ Xs ) @ S ) ) ) ) ).
% shift_simps(2)
thf(fact_919_shift__simps_I2_J,axiom,
! [Xs: list_rule,S: stream_rule] :
( ( ( Xs = nil_rule )
=> ( ( stl_rule @ ( shift_rule @ Xs @ S ) )
= ( stl_rule @ S ) ) )
& ( ( Xs != nil_rule )
=> ( ( stl_rule @ ( shift_rule @ Xs @ S ) )
= ( shift_rule @ ( tl_rule @ Xs ) @ S ) ) ) ) ).
% shift_simps(2)
thf(fact_920_notin__set__nthsI,axiom,
! [X3: rule,Xs: list_rule,I2: set_nat] :
( ~ ( member_rule @ X3 @ ( set_rule2 @ Xs ) )
=> ~ ( member_rule @ X3 @ ( set_rule2 @ ( nths_rule @ Xs @ I2 ) ) ) ) ).
% notin_set_nthsI
thf(fact_921_notin__set__nthsI,axiom,
! [X3: produc6018962875968178549ist_fm,Xs: list_P2887561121880082555ist_fm,I2: set_nat] :
( ~ ( member4699826688122452638ist_fm @ X3 @ ( set_Pr5051287786238754058ist_fm @ Xs ) )
=> ~ ( member4699826688122452638ist_fm @ X3 @ ( set_Pr5051287786238754058ist_fm @ ( nths_P7825489925148284537ist_fm @ Xs @ I2 ) ) ) ) ).
% notin_set_nthsI
thf(fact_922_notin__set__nthsI,axiom,
! [X3: produc340336539035504054m_rule,Xs: list_P2774625669004309958m_rule,I2: set_nat] :
( ~ ( member7231649785386036813m_rule @ X3 @ ( set_Pr4534715572506550497m_rule @ Xs ) )
=> ~ ( member7231649785386036813m_rule @ X3 @ ( set_Pr4534715572506550497m_rule @ ( nths_P1214564794791554994m_rule @ Xs @ I2 ) ) ) ) ).
% notin_set_nthsI
thf(fact_923_notin__set__nthsI,axiom,
! [X3: fm,Xs: list_fm,I2: set_nat] :
( ~ ( member_fm @ X3 @ ( set_fm2 @ Xs ) )
=> ~ ( member_fm @ X3 @ ( set_fm2 @ ( nths_fm @ Xs @ I2 ) ) ) ) ).
% notin_set_nthsI
thf(fact_924_notin__set__nthsI,axiom,
! [X3: tm,Xs: list_tm,I2: set_nat] :
( ~ ( member_tm @ X3 @ ( set_tm2 @ Xs ) )
=> ~ ( member_tm @ X3 @ ( set_tm2 @ ( nths_tm @ Xs @ I2 ) ) ) ) ).
% notin_set_nthsI
thf(fact_925_notin__set__nthsI,axiom,
! [X3: list_fm,Xs: list_list_fm,I2: set_nat] :
( ~ ( member_list_fm @ X3 @ ( set_list_fm2 @ Xs ) )
=> ~ ( member_list_fm @ X3 @ ( set_list_fm2 @ ( nths_list_fm @ Xs @ I2 ) ) ) ) ).
% notin_set_nthsI
thf(fact_926_in__set__nthsD,axiom,
! [X3: rule,Xs: list_rule,I2: set_nat] :
( ( member_rule @ X3 @ ( set_rule2 @ ( nths_rule @ Xs @ I2 ) ) )
=> ( member_rule @ X3 @ ( set_rule2 @ Xs ) ) ) ).
% in_set_nthsD
thf(fact_927_in__set__nthsD,axiom,
! [X3: produc6018962875968178549ist_fm,Xs: list_P2887561121880082555ist_fm,I2: set_nat] :
( ( member4699826688122452638ist_fm @ X3 @ ( set_Pr5051287786238754058ist_fm @ ( nths_P7825489925148284537ist_fm @ Xs @ I2 ) ) )
=> ( member4699826688122452638ist_fm @ X3 @ ( set_Pr5051287786238754058ist_fm @ Xs ) ) ) ).
% in_set_nthsD
thf(fact_928_in__set__nthsD,axiom,
! [X3: produc340336539035504054m_rule,Xs: list_P2774625669004309958m_rule,I2: set_nat] :
( ( member7231649785386036813m_rule @ X3 @ ( set_Pr4534715572506550497m_rule @ ( nths_P1214564794791554994m_rule @ Xs @ I2 ) ) )
=> ( member7231649785386036813m_rule @ X3 @ ( set_Pr4534715572506550497m_rule @ Xs ) ) ) ).
% in_set_nthsD
thf(fact_929_in__set__nthsD,axiom,
! [X3: fm,Xs: list_fm,I2: set_nat] :
( ( member_fm @ X3 @ ( set_fm2 @ ( nths_fm @ Xs @ I2 ) ) )
=> ( member_fm @ X3 @ ( set_fm2 @ Xs ) ) ) ).
% in_set_nthsD
thf(fact_930_in__set__nthsD,axiom,
! [X3: tm,Xs: list_tm,I2: set_nat] :
( ( member_tm @ X3 @ ( set_tm2 @ ( nths_tm @ Xs @ I2 ) ) )
=> ( member_tm @ X3 @ ( set_tm2 @ Xs ) ) ) ).
% in_set_nthsD
thf(fact_931_in__set__nthsD,axiom,
! [X3: list_fm,Xs: list_list_fm,I2: set_nat] :
( ( member_list_fm @ X3 @ ( set_list_fm2 @ ( nths_list_fm @ Xs @ I2 ) ) )
=> ( member_list_fm @ X3 @ ( set_list_fm2 @ Xs ) ) ) ).
% in_set_nthsD
thf(fact_932_list_Oset__sel_I2_J,axiom,
! [A2: list_rule,X3: rule] :
( ( A2 != nil_rule )
=> ( ( member_rule @ X3 @ ( set_rule2 @ ( tl_rule @ A2 ) ) )
=> ( member_rule @ X3 @ ( set_rule2 @ A2 ) ) ) ) ).
% list.set_sel(2)
thf(fact_933_list_Oset__sel_I2_J,axiom,
! [A2: list_P2887561121880082555ist_fm,X3: produc6018962875968178549ist_fm] :
( ( A2 != nil_Pr2223031680201799035ist_fm )
=> ( ( member4699826688122452638ist_fm @ X3 @ ( set_Pr5051287786238754058ist_fm @ ( tl_Pro6900887359406299236ist_fm @ A2 ) ) )
=> ( member4699826688122452638ist_fm @ X3 @ ( set_Pr5051287786238754058ist_fm @ A2 ) ) ) ) ).
% list.set_sel(2)
thf(fact_934_list_Oset__sel_I2_J,axiom,
! [A2: list_P2774625669004309958m_rule,X3: produc340336539035504054m_rule] :
( ( A2 != nil_Pr2808214839852828720m_rule )
=> ( ( member7231649785386036813m_rule @ X3 @ ( set_Pr4534715572506550497m_rule @ ( tl_Pro2692115096043292423m_rule @ A2 ) ) )
=> ( member7231649785386036813m_rule @ X3 @ ( set_Pr4534715572506550497m_rule @ A2 ) ) ) ) ).
% list.set_sel(2)
thf(fact_935_list_Oset__sel_I2_J,axiom,
! [A2: list_fm,X3: fm] :
( ( A2 != nil_fm )
=> ( ( member_fm @ X3 @ ( set_fm2 @ ( tl_fm @ A2 ) ) )
=> ( member_fm @ X3 @ ( set_fm2 @ A2 ) ) ) ) ).
% list.set_sel(2)
thf(fact_936_list_Oset__sel_I2_J,axiom,
! [A2: list_tm,X3: tm] :
( ( A2 != nil_tm )
=> ( ( member_tm @ X3 @ ( set_tm2 @ ( tl_tm @ A2 ) ) )
=> ( member_tm @ X3 @ ( set_tm2 @ A2 ) ) ) ) ).
% list.set_sel(2)
thf(fact_937_list_Oset__sel_I2_J,axiom,
! [A2: list_list_fm,X3: list_fm] :
( ( A2 != nil_list_fm )
=> ( ( member_list_fm @ X3 @ ( set_list_fm2 @ ( tl_list_fm @ A2 ) ) )
=> ( member_list_fm @ X3 @ ( set_list_fm2 @ A2 ) ) ) ) ).
% list.set_sel(2)
thf(fact_938_tl__append__if,axiom,
! [Xs: list_tm,Ys: list_tm] :
( ( ( Xs = nil_tm )
=> ( ( tl_tm @ ( append_tm @ Xs @ Ys ) )
= ( tl_tm @ Ys ) ) )
& ( ( Xs != nil_tm )
=> ( ( tl_tm @ ( append_tm @ Xs @ Ys ) )
= ( append_tm @ ( tl_tm @ Xs ) @ Ys ) ) ) ) ).
% tl_append_if
thf(fact_939_concat_Osimps_I2_J,axiom,
! [X3: list_tm,Xs: list_list_tm] :
( ( concat_tm @ ( cons_list_tm @ X3 @ Xs ) )
= ( append_tm @ X3 @ ( concat_tm @ Xs ) ) ) ).
% concat.simps(2)
thf(fact_940_append__listrel1I,axiom,
! [Xs: list_tm,Ys: list_tm,R: set_Pr2455929065695642951_tm_tm,Us2: list_tm,Vs: list_tm] :
( ( ( ( member4457312017796942864ist_tm @ ( produc1418304791525149271ist_tm @ Xs @ Ys ) @ ( listrel1_tm @ R ) )
& ( Us2 = Vs ) )
| ( ( Xs = Ys )
& ( member4457312017796942864ist_tm @ ( produc1418304791525149271ist_tm @ Us2 @ Vs ) @ ( listrel1_tm @ R ) ) ) )
=> ( member4457312017796942864ist_tm @ ( produc1418304791525149271ist_tm @ ( append_tm @ Xs @ Us2 ) @ ( append_tm @ Ys @ Vs ) ) @ ( listrel1_tm @ R ) ) ) ).
% append_listrel1I
thf(fact_941_set__nths__subset,axiom,
! [Xs: list_list_fm,I2: set_nat] : ( ord_le7838213414353715577ist_fm @ ( set_list_fm2 @ ( nths_list_fm @ Xs @ I2 ) ) @ ( set_list_fm2 @ Xs ) ) ).
% set_nths_subset
thf(fact_942_set__nths__subset,axiom,
! [Xs: list_tm,I2: set_nat] : ( ord_less_eq_set_tm @ ( set_tm2 @ ( nths_tm @ Xs @ I2 ) ) @ ( set_tm2 @ Xs ) ) ).
% set_nths_subset
thf(fact_943_set__nths__subset,axiom,
! [Xs: list_fm,I2: set_nat] : ( ord_less_eq_set_fm @ ( set_fm2 @ ( nths_fm @ Xs @ I2 ) ) @ ( set_fm2 @ Xs ) ) ).
% set_nths_subset
thf(fact_944_set__nths__subset,axiom,
! [Xs: list_rule,I2: set_nat] : ( ord_less_eq_set_rule @ ( set_rule2 @ ( nths_rule @ Xs @ I2 ) ) @ ( set_rule2 @ Xs ) ) ).
% set_nths_subset
thf(fact_945_Cons__listrel1E2,axiom,
! [Xs: list_nat,Y3: nat,Ys: list_nat,R: set_Pr1261947904930325089at_nat] :
( ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ Xs @ ( cons_nat @ Y3 @ Ys ) ) @ ( listrel1_nat @ R ) )
=> ( ! [X: nat] :
( ( Xs
= ( cons_nat @ X @ Ys ) )
=> ~ ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X @ Y3 ) @ R ) )
=> ~ ! [Zs3: list_nat] :
( ( Xs
= ( cons_nat @ Y3 @ Zs3 ) )
=> ~ ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ Zs3 @ Ys ) @ ( listrel1_nat @ R ) ) ) ) ) ).
% Cons_listrel1E2
thf(fact_946_Cons__listrel1E1,axiom,
! [X3: nat,Xs: list_nat,Ys: list_nat,R: set_Pr1261947904930325089at_nat] :
( ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ ( cons_nat @ X3 @ Xs ) @ Ys ) @ ( listrel1_nat @ R ) )
=> ( ! [Y: nat] :
( ( Ys
= ( cons_nat @ Y @ Xs ) )
=> ~ ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X3 @ Y ) @ R ) )
=> ~ ! [Zs3: list_nat] :
( ( Ys
= ( cons_nat @ X3 @ Zs3 ) )
=> ~ ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ Xs @ Zs3 ) @ ( listrel1_nat @ R ) ) ) ) ) ).
% Cons_listrel1E1
thf(fact_947_listrel1I1,axiom,
! [X3: nat,Y3: nat,R: set_Pr1261947904930325089at_nat,Xs: list_nat] :
( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X3 @ Y3 ) @ R )
=> ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ ( cons_nat @ X3 @ Xs ) @ ( cons_nat @ Y3 @ Xs ) ) @ ( listrel1_nat @ R ) ) ) ).
% listrel1I1
thf(fact_948_in__set__product__lists__length,axiom,
! [Xs: list_fm,Xss: list_list_fm] :
( ( member_list_fm @ Xs @ ( set_list_fm2 @ ( product_lists_fm @ Xss ) ) )
=> ( ( size_size_list_fm @ Xs )
= ( size_s115229985653309035ist_fm @ Xss ) ) ) ).
% in_set_product_lists_length
thf(fact_949_listrel1I,axiom,
! [X3: tm,Y3: tm,R: set_Pr2455929065695642951_tm_tm,Xs: list_tm,Us2: list_tm,Vs: list_tm,Ys: list_tm] :
( ( member3121616906494481296_tm_tm @ ( product_Pair_tm_tm @ X3 @ Y3 ) @ R )
=> ( ( Xs
= ( append_tm @ Us2 @ ( cons_tm @ X3 @ Vs ) ) )
=> ( ( Ys
= ( append_tm @ Us2 @ ( cons_tm @ Y3 @ Vs ) ) )
=> ( member4457312017796942864ist_tm @ ( produc1418304791525149271ist_tm @ Xs @ Ys ) @ ( listrel1_tm @ R ) ) ) ) ) ).
% listrel1I
thf(fact_950_listrel1I,axiom,
! [X3: nat,Y3: nat,R: set_Pr1261947904930325089at_nat,Xs: list_nat,Us2: list_nat,Vs: list_nat,Ys: list_nat] :
( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X3 @ Y3 ) @ R )
=> ( ( Xs
= ( append_nat @ Us2 @ ( cons_nat @ X3 @ Vs ) ) )
=> ( ( Ys
= ( append_nat @ Us2 @ ( cons_nat @ Y3 @ Vs ) ) )
=> ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ Xs @ Ys ) @ ( listrel1_nat @ R ) ) ) ) ) ).
% listrel1I
thf(fact_951_listrel1E,axiom,
! [Xs: list_tm,Ys: list_tm,R: set_Pr2455929065695642951_tm_tm] :
( ( member4457312017796942864ist_tm @ ( produc1418304791525149271ist_tm @ Xs @ Ys ) @ ( listrel1_tm @ R ) )
=> ~ ! [X: tm,Y: tm] :
( ( member3121616906494481296_tm_tm @ ( product_Pair_tm_tm @ X @ Y ) @ R )
=> ! [Us3: list_tm,Vs2: list_tm] :
( ( Xs
= ( append_tm @ Us3 @ ( cons_tm @ X @ Vs2 ) ) )
=> ( Ys
!= ( append_tm @ Us3 @ ( cons_tm @ Y @ Vs2 ) ) ) ) ) ) ).
% listrel1E
thf(fact_952_listrel1E,axiom,
! [Xs: list_nat,Ys: list_nat,R: set_Pr1261947904930325089at_nat] :
( ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ Xs @ Ys ) @ ( listrel1_nat @ R ) )
=> ~ ! [X: nat,Y: nat] :
( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X @ Y ) @ R )
=> ! [Us3: list_nat,Vs2: list_nat] :
( ( Xs
= ( append_nat @ Us3 @ ( cons_nat @ X @ Vs2 ) ) )
=> ( Ys
!= ( append_nat @ Us3 @ ( cons_nat @ Y @ Vs2 ) ) ) ) ) ) ).
% listrel1E
thf(fact_953_concat__eq__appendD,axiom,
! [Xss: list_list_tm,Ys: list_tm,Zs: list_tm] :
( ( ( concat_tm @ Xss )
= ( append_tm @ Ys @ Zs ) )
=> ( ( Xss != nil_list_tm )
=> ? [Xss12: list_list_tm,Xs3: list_tm,Xs5: list_tm,Xss22: list_list_tm] :
( ( Xss
= ( append_list_tm @ Xss12 @ ( cons_list_tm @ ( append_tm @ Xs3 @ Xs5 ) @ Xss22 ) ) )
& ( Ys
= ( append_tm @ ( concat_tm @ Xss12 ) @ Xs3 ) )
& ( Zs
= ( append_tm @ Xs5 @ ( concat_tm @ Xss22 ) ) ) ) ) ) ).
% concat_eq_appendD
thf(fact_954_cycle__Cons,axiom,
! [X3: tm,Xs: list_tm] :
( ( cycle_tm @ ( cons_tm @ X3 @ Xs ) )
= ( sCons_tm @ X3 @ ( cycle_tm @ ( append_tm @ Xs @ ( cons_tm @ X3 @ nil_tm ) ) ) ) ) ).
% cycle_Cons
thf(fact_955_sdrop__cycle__eq,axiom,
! [U: list_P2774625669004309958m_rule] :
( ( U != nil_Pr2808214839852828720m_rule )
=> ( ( sdrop_8169176516188972301m_rule @ ( size_s1575636608424004698m_rule @ U ) @ ( cycle_5335117900641983313m_rule @ U ) )
= ( cycle_5335117900641983313m_rule @ U ) ) ) ).
% sdrop_cycle_eq
thf(fact_956_sdrop__cycle__eq,axiom,
! [U: list_rule] :
( ( U != nil_rule )
=> ( ( sdrop_rule @ ( size_size_list_rule @ U ) @ ( cycle_rule @ U ) )
= ( cycle_rule @ U ) ) ) ).
% sdrop_cycle_eq
thf(fact_957_case__prod__conv,axiom,
! [F: nat > nat > product_prod_nat_nat,A2: nat,B2: nat] :
( ( produc2626176000494625587at_nat @ F @ ( product_Pair_nat_nat @ A2 @ B2 ) )
= ( F @ A2 @ B2 ) ) ).
% case_prod_conv
thf(fact_958_case__prodI2,axiom,
! [P2: produc6018962875968178549ist_fm,C: list_tm > list_fm > $o] :
( ! [A4: list_tm,B5: list_fm] :
( ( P2
= ( produc1414352766439514085ist_fm @ A4 @ B5 ) )
=> ( C @ A4 @ B5 ) )
=> ( produc3862533337643388190t_fm_o @ C @ P2 ) ) ).
% case_prodI2
thf(fact_959_case__prodI2,axiom,
! [P2: produc340336539035504054m_rule,C: produc6018962875968178549ist_fm > rule > $o] :
( ! [A4: produc6018962875968178549ist_fm,B5: rule] :
( ( P2
= ( produc1733806532565653680m_rule @ A4 @ B5 ) )
=> ( C @ A4 @ B5 ) )
=> ( produc1247074552912856443rule_o @ C @ P2 ) ) ).
% case_prodI2
thf(fact_960_case__prodI2,axiom,
! [P2: product_prod_nat_nat,C: nat > nat > $o] :
( ! [A4: nat,B5: nat] :
( ( P2
= ( product_Pair_nat_nat @ A4 @ B5 ) )
=> ( C @ A4 @ B5 ) )
=> ( produc6081775807080527818_nat_o @ C @ P2 ) ) ).
% case_prodI2
thf(fact_961_case__prodI,axiom,
! [F: list_tm > list_fm > $o,A2: list_tm,B2: list_fm] :
( ( F @ A2 @ B2 )
=> ( produc3862533337643388190t_fm_o @ F @ ( produc1414352766439514085ist_fm @ A2 @ B2 ) ) ) ).
% case_prodI
thf(fact_962_case__prodI,axiom,
! [F: produc6018962875968178549ist_fm > rule > $o,A2: produc6018962875968178549ist_fm,B2: rule] :
( ( F @ A2 @ B2 )
=> ( produc1247074552912856443rule_o @ F @ ( produc1733806532565653680m_rule @ A2 @ B2 ) ) ) ).
% case_prodI
thf(fact_963_case__prodI,axiom,
! [F: nat > nat > $o,A2: nat,B2: nat] :
( ( F @ A2 @ B2 )
=> ( produc6081775807080527818_nat_o @ F @ ( product_Pair_nat_nat @ A2 @ B2 ) ) ) ).
% case_prodI
thf(fact_964_case__swap,axiom,
! [F: nat > nat > product_prod_nat_nat,P2: product_prod_nat_nat] :
( ( produc2626176000494625587at_nat
@ ^ [Y5: nat,X4: nat] : ( F @ X4 @ Y5 )
@ ( product_swap_nat_nat @ P2 ) )
= ( produc2626176000494625587at_nat @ F @ P2 ) ) ).
% case_swap
thf(fact_965_hd__append2,axiom,
! [Xs: list_tm,Ys: list_tm] :
( ( Xs != nil_tm )
=> ( ( hd_tm @ ( append_tm @ Xs @ Ys ) )
= ( hd_tm @ Xs ) ) ) ).
% hd_append2
thf(fact_966_sset__cycle,axiom,
! [Xs: list_fm] :
( ( Xs != nil_fm )
=> ( ( sset_fm @ ( cycle_fm @ Xs ) )
= ( set_fm2 @ Xs ) ) ) ).
% sset_cycle
thf(fact_967_sset__cycle,axiom,
! [Xs: list_tm] :
( ( Xs != nil_tm )
=> ( ( sset_tm @ ( cycle_tm @ Xs ) )
= ( set_tm2 @ Xs ) ) ) ).
% sset_cycle
thf(fact_968_sset__cycle,axiom,
! [Xs: list_list_fm] :
( ( Xs != nil_list_fm )
=> ( ( sset_list_fm @ ( cycle_list_fm @ Xs ) )
= ( set_list_fm2 @ Xs ) ) ) ).
% sset_cycle
thf(fact_969_sset__cycle,axiom,
! [Xs: list_rule] :
( ( Xs != nil_rule )
=> ( ( sset_rule @ ( cycle_rule @ Xs ) )
= ( set_rule2 @ Xs ) ) ) ).
% sset_cycle
thf(fact_970_sset__cycle,axiom,
! [Xs: list_P2774625669004309958m_rule] :
( ( Xs != nil_Pr2808214839852828720m_rule )
=> ( ( sset_P4484857331586881186m_rule @ ( cycle_5335117900641983313m_rule @ Xs ) )
= ( set_Pr4534715572506550497m_rule @ Xs ) ) ) ).
% sset_cycle
thf(fact_971_shift__simps_I1_J,axiom,
! [Xs: list_P2774625669004309958m_rule,S: stream2709947120125613254m_rule] :
( ( ( Xs = nil_Pr2808214839852828720m_rule )
=> ( ( shd_Pr4562317740776619530m_rule @ ( shift_2334844276748245581m_rule @ Xs @ S ) )
= ( shd_Pr4562317740776619530m_rule @ S ) ) )
& ( ( Xs != nil_Pr2808214839852828720m_rule )
=> ( ( shd_Pr4562317740776619530m_rule @ ( shift_2334844276748245581m_rule @ Xs @ S ) )
= ( hd_Pro7241777042969981963m_rule @ Xs ) ) ) ) ).
% shift_simps(1)
thf(fact_972_shift__simps_I1_J,axiom,
! [Xs: list_rule,S: stream_rule] :
( ( ( Xs = nil_rule )
=> ( ( shd_rule @ ( shift_rule @ Xs @ S ) )
= ( shd_rule @ S ) ) )
& ( ( Xs != nil_rule )
=> ( ( shd_rule @ ( shift_rule @ Xs @ S ) )
= ( hd_rule @ Xs ) ) ) ) ).
% shift_simps(1)
thf(fact_973_old_Oprod_Ocase,axiom,
! [F: nat > nat > product_prod_nat_nat,X1: nat,X2: nat] :
( ( produc2626176000494625587at_nat @ F @ ( product_Pair_nat_nat @ X1 @ X2 ) )
= ( F @ X1 @ X2 ) ) ).
% old.prod.case
thf(fact_974_prod_Ocase__distrib,axiom,
! [H: product_prod_nat_nat > product_prod_nat_nat,F: nat > nat > product_prod_nat_nat,Prod: product_prod_nat_nat] :
( ( H @ ( produc2626176000494625587at_nat @ F @ Prod ) )
= ( produc2626176000494625587at_nat
@ ^ [X12: nat,X22: nat] : ( H @ ( F @ X12 @ X22 ) )
@ Prod ) ) ).
% prod.case_distrib
thf(fact_975_cycle_Osimps_I1_J,axiom,
! [Xs: list_P2774625669004309958m_rule] :
( ( shd_Pr4562317740776619530m_rule @ ( cycle_5335117900641983313m_rule @ Xs ) )
= ( hd_Pro7241777042969981963m_rule @ Xs ) ) ).
% cycle.simps(1)
thf(fact_976_cycle_Osimps_I1_J,axiom,
! [Xs: list_rule] :
( ( shd_rule @ ( cycle_rule @ Xs ) )
= ( hd_rule @ Xs ) ) ).
% cycle.simps(1)
thf(fact_977_case__prodE,axiom,
! [C: list_tm > list_fm > $o,P2: produc6018962875968178549ist_fm] :
( ( produc3862533337643388190t_fm_o @ C @ P2 )
=> ~ ! [X: list_tm,Y: list_fm] :
( ( P2
= ( produc1414352766439514085ist_fm @ X @ Y ) )
=> ~ ( C @ X @ Y ) ) ) ).
% case_prodE
thf(fact_978_case__prodE,axiom,
! [C: produc6018962875968178549ist_fm > rule > $o,P2: produc340336539035504054m_rule] :
( ( produc1247074552912856443rule_o @ C @ P2 )
=> ~ ! [X: produc6018962875968178549ist_fm,Y: rule] :
( ( P2
= ( produc1733806532565653680m_rule @ X @ Y ) )
=> ~ ( C @ X @ Y ) ) ) ).
% case_prodE
thf(fact_979_case__prodE,axiom,
! [C: nat > nat > $o,P2: product_prod_nat_nat] :
( ( produc6081775807080527818_nat_o @ C @ P2 )
=> ~ ! [X: nat,Y: nat] :
( ( P2
= ( product_Pair_nat_nat @ X @ Y ) )
=> ~ ( C @ X @ Y ) ) ) ).
% case_prodE
thf(fact_980_case__prodD,axiom,
! [F: list_tm > list_fm > $o,A2: list_tm,B2: list_fm] :
( ( produc3862533337643388190t_fm_o @ F @ ( produc1414352766439514085ist_fm @ A2 @ B2 ) )
=> ( F @ A2 @ B2 ) ) ).
% case_prodD
thf(fact_981_case__prodD,axiom,
! [F: produc6018962875968178549ist_fm > rule > $o,A2: produc6018962875968178549ist_fm,B2: rule] :
( ( produc1247074552912856443rule_o @ F @ ( produc1733806532565653680m_rule @ A2 @ B2 ) )
=> ( F @ A2 @ B2 ) ) ).
% case_prodD
thf(fact_982_case__prodD,axiom,
! [F: nat > nat > $o,A2: nat,B2: nat] :
( ( produc6081775807080527818_nat_o @ F @ ( product_Pair_nat_nat @ A2 @ B2 ) )
=> ( F @ A2 @ B2 ) ) ).
% case_prodD
thf(fact_983_cond__case__prod__eta,axiom,
! [F: nat > nat > product_prod_nat_nat,G: product_prod_nat_nat > product_prod_nat_nat] :
( ! [X: nat,Y: nat] :
( ( F @ X @ Y )
= ( G @ ( product_Pair_nat_nat @ X @ Y ) ) )
=> ( ( produc2626176000494625587at_nat @ F )
= G ) ) ).
% cond_case_prod_eta
thf(fact_984_case__prod__eta,axiom,
! [F: product_prod_nat_nat > product_prod_nat_nat] :
( ( produc2626176000494625587at_nat
@ ^ [X4: nat,Y5: nat] : ( F @ ( product_Pair_nat_nat @ X4 @ Y5 ) ) )
= F ) ).
% case_prod_eta
thf(fact_985_case__prodE2,axiom,
! [Q: product_prod_nat_nat > $o,P: nat > nat > product_prod_nat_nat,Z: product_prod_nat_nat] :
( ( Q @ ( produc2626176000494625587at_nat @ P @ Z ) )
=> ~ ! [X: nat,Y: nat] :
( ( Z
= ( product_Pair_nat_nat @ X @ Y ) )
=> ~ ( Q @ ( P @ X @ Y ) ) ) ) ).
% case_prodE2
thf(fact_986_fst__def,axiom,
( produc993771546023551940m_rule
= ( produc3824424964374196184ist_fm
@ ^ [X12: produc6018962875968178549ist_fm,X22: rule] : X12 ) ) ).
% fst_def
thf(fact_987_fst__def,axiom,
( produc4874987732838623697ist_fm
= ( produc8091450810216765553ist_tm
@ ^ [X12: list_tm,X22: list_fm] : X12 ) ) ).
% fst_def
thf(fact_988_snd__def,axiom,
( produc6514711208036230658m_rule
= ( produc1291517305509475439e_rule
@ ^ [X12: produc6018962875968178549ist_fm,X22: rule] : X22 ) ) ).
% snd_def
thf(fact_989_snd__def,axiom,
( produc3242453064496107283ist_fm
= ( produc8087498785131130367ist_fm
@ ^ [X12: list_tm,X22: list_fm] : X22 ) ) ).
% snd_def
thf(fact_990_Product__Type_OCollect__case__prodD,axiom,
! [X3: produc340336539035504054m_rule,A: produc6018962875968178549ist_fm > rule > $o] :
( ( member7231649785386036813m_rule @ X3 @ ( collec1009276759030335883m_rule @ ( produc1247074552912856443rule_o @ A ) ) )
=> ( A @ ( produc993771546023551940m_rule @ X3 ) @ ( produc6514711208036230658m_rule @ X3 ) ) ) ).
% Product_Type.Collect_case_prodD
thf(fact_991_Product__Type_OCollect__case__prodD,axiom,
! [X3: produc6018962875968178549ist_fm,A: list_tm > list_fm > $o] :
( ( member4699826688122452638ist_fm @ X3 @ ( collec5845330005437105888ist_fm @ ( produc3862533337643388190t_fm_o @ A ) ) )
=> ( A @ ( produc4874987732838623697ist_fm @ X3 ) @ ( produc3242453064496107283ist_fm @ X3 ) ) ) ).
% Product_Type.Collect_case_prodD
thf(fact_992_list_Oset__sel_I1_J,axiom,
! [A2: list_rule] :
( ( A2 != nil_rule )
=> ( member_rule @ ( hd_rule @ A2 ) @ ( set_rule2 @ A2 ) ) ) ).
% list.set_sel(1)
thf(fact_993_list_Oset__sel_I1_J,axiom,
! [A2: list_P2887561121880082555ist_fm] :
( ( A2 != nil_Pr2223031680201799035ist_fm )
=> ( member4699826688122452638ist_fm @ ( hd_Pro8605841746596789344ist_fm @ A2 ) @ ( set_Pr5051287786238754058ist_fm @ A2 ) ) ) ).
% list.set_sel(1)
thf(fact_994_list_Oset__sel_I1_J,axiom,
! [A2: list_P2774625669004309958m_rule] :
( ( A2 != nil_Pr2808214839852828720m_rule )
=> ( member7231649785386036813m_rule @ ( hd_Pro7241777042969981963m_rule @ A2 ) @ ( set_Pr4534715572506550497m_rule @ A2 ) ) ) ).
% list.set_sel(1)
thf(fact_995_list_Oset__sel_I1_J,axiom,
! [A2: list_fm] :
( ( A2 != nil_fm )
=> ( member_fm @ ( hd_fm @ A2 ) @ ( set_fm2 @ A2 ) ) ) ).
% list.set_sel(1)
thf(fact_996_list_Oset__sel_I1_J,axiom,
! [A2: list_tm] :
( ( A2 != nil_tm )
=> ( member_tm @ ( hd_tm @ A2 ) @ ( set_tm2 @ A2 ) ) ) ).
% list.set_sel(1)
thf(fact_997_list_Oset__sel_I1_J,axiom,
! [A2: list_list_fm] :
( ( A2 != nil_list_fm )
=> ( member_list_fm @ ( hd_list_fm @ A2 ) @ ( set_list_fm2 @ A2 ) ) ) ).
% list.set_sel(1)
thf(fact_998_hd__in__set,axiom,
! [Xs: list_rule] :
( ( Xs != nil_rule )
=> ( member_rule @ ( hd_rule @ Xs ) @ ( set_rule2 @ Xs ) ) ) ).
% hd_in_set
thf(fact_999_hd__in__set,axiom,
! [Xs: list_P2887561121880082555ist_fm] :
( ( Xs != nil_Pr2223031680201799035ist_fm )
=> ( member4699826688122452638ist_fm @ ( hd_Pro8605841746596789344ist_fm @ Xs ) @ ( set_Pr5051287786238754058ist_fm @ Xs ) ) ) ).
% hd_in_set
thf(fact_1000_hd__in__set,axiom,
! [Xs: list_P2774625669004309958m_rule] :
( ( Xs != nil_Pr2808214839852828720m_rule )
=> ( member7231649785386036813m_rule @ ( hd_Pro7241777042969981963m_rule @ Xs ) @ ( set_Pr4534715572506550497m_rule @ Xs ) ) ) ).
% hd_in_set
thf(fact_1001_hd__in__set,axiom,
! [Xs: list_fm] :
( ( Xs != nil_fm )
=> ( member_fm @ ( hd_fm @ Xs ) @ ( set_fm2 @ Xs ) ) ) ).
% hd_in_set
thf(fact_1002_hd__in__set,axiom,
! [Xs: list_tm] :
( ( Xs != nil_tm )
=> ( member_tm @ ( hd_tm @ Xs ) @ ( set_tm2 @ Xs ) ) ) ).
% hd_in_set
thf(fact_1003_hd__in__set,axiom,
! [Xs: list_list_fm] :
( ( Xs != nil_list_fm )
=> ( member_list_fm @ ( hd_list_fm @ Xs ) @ ( set_list_fm2 @ Xs ) ) ) ).
% hd_in_set
thf(fact_1004_longest__common__prefix,axiom,
! [Xs: list_tm,Ys: list_tm] :
? [Ps: list_tm,Xs5: list_tm,Ys6: list_tm] :
( ( Xs
= ( append_tm @ Ps @ Xs5 ) )
& ( Ys
= ( append_tm @ Ps @ Ys6 ) )
& ( ( Xs5 = nil_tm )
| ( Ys6 = nil_tm )
| ( ( hd_tm @ Xs5 )
!= ( hd_tm @ Ys6 ) ) ) ) ).
% longest_common_prefix
thf(fact_1005_hd__append,axiom,
! [Xs: list_tm,Ys: list_tm] :
( ( ( Xs = nil_tm )
=> ( ( hd_tm @ ( append_tm @ Xs @ Ys ) )
= ( hd_tm @ Ys ) ) )
& ( ( Xs != nil_tm )
=> ( ( hd_tm @ ( append_tm @ Xs @ Ys ) )
= ( hd_tm @ Xs ) ) ) ) ).
% hd_append
thf(fact_1006_split__beta,axiom,
( produc2626176000494625587at_nat
= ( ^ [F3: nat > nat > product_prod_nat_nat,Prod3: product_prod_nat_nat] : ( F3 @ ( product_fst_nat_nat @ Prod3 ) @ ( product_snd_nat_nat @ Prod3 ) ) ) ) ).
% split_beta
thf(fact_1007_case__prod__beta,axiom,
( produc2626176000494625587at_nat
= ( ^ [F3: nat > nat > product_prod_nat_nat,P4: product_prod_nat_nat] : ( F3 @ ( product_fst_nat_nat @ P4 ) @ ( product_snd_nat_nat @ P4 ) ) ) ) ).
% case_prod_beta
thf(fact_1008_internal__case__prod__def,axiom,
produc7900996641596641245at_nat = produc2626176000494625587at_nat ).
% internal_case_prod_def
thf(fact_1009_case__prod__beta_H,axiom,
( produc2626176000494625587at_nat
= ( ^ [F3: nat > nat > product_prod_nat_nat,X4: product_prod_nat_nat] : ( F3 @ ( product_fst_nat_nat @ X4 ) @ ( product_snd_nat_nat @ X4 ) ) ) ) ).
% case_prod_beta'
thf(fact_1010_case__prod__unfold,axiom,
( produc2626176000494625587at_nat
= ( ^ [C3: nat > nat > product_prod_nat_nat,P4: product_prod_nat_nat] : ( C3 @ ( product_fst_nat_nat @ P4 ) @ ( product_snd_nat_nat @ P4 ) ) ) ) ).
% case_prod_unfold
thf(fact_1011_exE__realizer,axiom,
! [P: nat > nat > $o,P2: product_prod_nat_nat,Q: product_prod_nat_nat > $o,F: nat > nat > product_prod_nat_nat] :
( ( P @ ( product_snd_nat_nat @ P2 ) @ ( product_fst_nat_nat @ P2 ) )
=> ( ! [X: nat,Y: nat] :
( ( P @ Y @ X )
=> ( Q @ ( F @ X @ Y ) ) )
=> ( Q @ ( produc2626176000494625587at_nat @ F @ P2 ) ) ) ) ).
% exE_realizer
thf(fact_1012_cycle_Osimps_I2_J,axiom,
! [Xs: list_tm] :
( ( stl_tm @ ( cycle_tm @ Xs ) )
= ( cycle_tm @ ( append_tm @ ( tl_tm @ Xs ) @ ( cons_tm @ ( hd_tm @ Xs ) @ nil_tm ) ) ) ) ).
% cycle.simps(2)
thf(fact_1013_cycle_Osimps_I2_J,axiom,
! [Xs: list_P2774625669004309958m_rule] :
( ( stl_Pr12655793849929990m_rule @ ( cycle_5335117900641983313m_rule @ Xs ) )
= ( cycle_5335117900641983313m_rule @ ( append2714808343334685275m_rule @ ( tl_Pro2692115096043292423m_rule @ Xs ) @ ( cons_P2818572562524668032m_rule @ ( hd_Pro7241777042969981963m_rule @ Xs ) @ nil_Pr2808214839852828720m_rule ) ) ) ) ).
% cycle.simps(2)
thf(fact_1014_cycle_Osimps_I2_J,axiom,
! [Xs: list_rule] :
( ( stl_rule @ ( cycle_rule @ Xs ) )
= ( cycle_rule @ ( append_rule @ ( tl_rule @ Xs ) @ ( cons_rule @ ( hd_rule @ Xs ) @ nil_rule ) ) ) ) ).
% cycle.simps(2)
thf(fact_1015_cycle_Ocode,axiom,
( cycle_tm
= ( ^ [Xs2: list_tm] : ( sCons_tm @ ( hd_tm @ Xs2 ) @ ( cycle_tm @ ( append_tm @ ( tl_tm @ Xs2 ) @ ( cons_tm @ ( hd_tm @ Xs2 ) @ nil_tm ) ) ) ) ) ) ).
% cycle.code
thf(fact_1016_cycle__rotated,axiom,
! [V: list_tm,U: list_tm,S: stream_tm] :
( ( V != nil_tm )
=> ( ( ( cycle_tm @ U )
= ( shift_tm @ V @ S ) )
=> ( ( cycle_tm @ ( append_tm @ ( tl_tm @ U ) @ ( cons_tm @ ( hd_tm @ U ) @ nil_tm ) ) )
= ( shift_tm @ ( tl_tm @ V ) @ S ) ) ) ) ).
% cycle_rotated
thf(fact_1017_cycle__rotated,axiom,
! [V: list_P2774625669004309958m_rule,U: list_P2774625669004309958m_rule,S: stream2709947120125613254m_rule] :
( ( V != nil_Pr2808214839852828720m_rule )
=> ( ( ( cycle_5335117900641983313m_rule @ U )
= ( shift_2334844276748245581m_rule @ V @ S ) )
=> ( ( cycle_5335117900641983313m_rule @ ( append2714808343334685275m_rule @ ( tl_Pro2692115096043292423m_rule @ U ) @ ( cons_P2818572562524668032m_rule @ ( hd_Pro7241777042969981963m_rule @ U ) @ nil_Pr2808214839852828720m_rule ) ) )
= ( shift_2334844276748245581m_rule @ ( tl_Pro2692115096043292423m_rule @ V ) @ S ) ) ) ) ).
% cycle_rotated
thf(fact_1018_cycle__decomp,axiom,
! [U: list_P2774625669004309958m_rule] :
( ( U != nil_Pr2808214839852828720m_rule )
=> ( ( cycle_5335117900641983313m_rule @ U )
= ( shift_2334844276748245581m_rule @ U @ ( cycle_5335117900641983313m_rule @ U ) ) ) ) ).
% cycle_decomp
thf(fact_1019_prod_Osplit__sel,axiom,
! [P: product_prod_nat_nat > $o,F: nat > nat > product_prod_nat_nat,Prod: product_prod_nat_nat] :
( ( P @ ( produc2626176000494625587at_nat @ F @ Prod ) )
= ( ( Prod
= ( product_Pair_nat_nat @ ( product_fst_nat_nat @ Prod ) @ ( product_snd_nat_nat @ Prod ) ) )
=> ( P @ ( F @ ( product_fst_nat_nat @ Prod ) @ ( product_snd_nat_nat @ Prod ) ) ) ) ) ).
% prod.split_sel
thf(fact_1020_prod_Osplit__sel__asm,axiom,
! [P: product_prod_nat_nat > $o,F: nat > nat > product_prod_nat_nat,Prod: product_prod_nat_nat] :
( ( P @ ( produc2626176000494625587at_nat @ F @ Prod ) )
= ( ~ ( ( Prod
= ( product_Pair_nat_nat @ ( product_fst_nat_nat @ Prod ) @ ( product_snd_nat_nat @ Prod ) ) )
& ~ ( P @ ( F @ ( product_fst_nat_nat @ Prod ) @ ( product_snd_nat_nat @ Prod ) ) ) ) ) ) ).
% prod.split_sel_asm
thf(fact_1021_flat_Osimps_I1_J,axiom,
! [Ws: stream2471014364565126742m_rule] :
( ( shd_Pr4562317740776619530m_rule @ ( flat_P7721466590633226428m_rule @ Ws ) )
= ( hd_Pro7241777042969981963m_rule @ ( shd_li4676821617271663642m_rule @ Ws ) ) ) ).
% flat.simps(1)
thf(fact_1022_flat_Osimps_I1_J,axiom,
! [Ws: stream_list_rule] :
( ( shd_rule @ ( flat_rule @ Ws ) )
= ( hd_rule @ ( shd_list_rule @ Ws ) ) ) ).
% flat.simps(1)
thf(fact_1023_rotate1__hd__tl,axiom,
! [Xs: list_tm] :
( ( Xs != nil_tm )
=> ( ( rotate1_tm @ Xs )
= ( append_tm @ ( tl_tm @ Xs ) @ ( cons_tm @ ( hd_tm @ Xs ) @ nil_tm ) ) ) ) ).
% rotate1_hd_tl
thf(fact_1024_split__cong,axiom,
! [Q2: product_prod_nat_nat,F: nat > nat > product_prod_nat_nat,G: nat > nat > product_prod_nat_nat,P2: product_prod_nat_nat] :
( ! [X: nat,Y: nat] :
( ( ( product_Pair_nat_nat @ X @ Y )
= Q2 )
=> ( ( F @ X @ Y )
= ( G @ X @ Y ) ) )
=> ( ( P2 = Q2 )
=> ( ( produc2626176000494625587at_nat @ F @ P2 )
= ( produc2626176000494625587at_nat @ G @ Q2 ) ) ) ) ).
% split_cong
thf(fact_1025_mem__case__prodI,axiom,
! [Z: rule,C: nat > nat > set_rule,A2: nat,B2: nat] :
( ( member_rule @ Z @ ( C @ A2 @ B2 ) )
=> ( member_rule @ Z @ ( produc4888952194711710358t_rule @ C @ ( product_Pair_nat_nat @ A2 @ B2 ) ) ) ) ).
% mem_case_prodI
thf(fact_1026_mem__case__prodI,axiom,
! [Z: fm,C: nat > nat > set_fm,A2: nat,B2: nat] :
( ( member_fm @ Z @ ( C @ A2 @ B2 ) )
=> ( member_fm @ Z @ ( produc2762169475385161349set_fm @ C @ ( product_Pair_nat_nat @ A2 @ B2 ) ) ) ) ).
% mem_case_prodI
thf(fact_1027_mem__case__prodI,axiom,
! [Z: list_fm,C: nat > nat > set_list_fm,A2: nat,B2: nat] :
( ( member_list_fm @ Z @ ( C @ A2 @ B2 ) )
=> ( member_list_fm @ Z @ ( produc8363348377050837003ist_fm @ C @ ( product_Pair_nat_nat @ A2 @ B2 ) ) ) ) ).
% mem_case_prodI
thf(fact_1028_mem__case__prodI,axiom,
! [Z: rule,C: list_tm > list_fm > set_rule,A2: list_tm,B2: list_fm] :
( ( member_rule @ Z @ ( C @ A2 @ B2 ) )
=> ( member_rule @ Z @ ( produc1245229592201295426t_rule @ C @ ( produc1414352766439514085ist_fm @ A2 @ B2 ) ) ) ) ).
% mem_case_prodI
thf(fact_1029_mem__case__prodI,axiom,
! [Z: fm,C: list_tm > list_fm > set_fm,A2: list_tm,B2: list_fm] :
( ( member_fm @ Z @ ( C @ A2 @ B2 ) )
=> ( member_fm @ Z @ ( produc1673703176515561945set_fm @ C @ ( produc1414352766439514085ist_fm @ A2 @ B2 ) ) ) ) ).
% mem_case_prodI
thf(fact_1030_mem__case__prodI,axiom,
! [Z: list_fm,C: list_tm > list_fm > set_list_fm,A2: list_tm,B2: list_fm] :
( ( member_list_fm @ Z @ ( C @ A2 @ B2 ) )
=> ( member_list_fm @ Z @ ( produc4004295895882230111ist_fm @ C @ ( produc1414352766439514085ist_fm @ A2 @ B2 ) ) ) ) ).
% mem_case_prodI
thf(fact_1031_mem__case__prodI,axiom,
! [Z: rule,C: produc6018962875968178549ist_fm > rule > set_rule,A2: produc6018962875968178549ist_fm,B2: rule] :
( ( member_rule @ Z @ ( C @ A2 @ B2 ) )
=> ( member_rule @ Z @ ( produc2336968575880921125t_rule @ C @ ( produc1733806532565653680m_rule @ A2 @ B2 ) ) ) ) ).
% mem_case_prodI
thf(fact_1032_mem__case__prodI,axiom,
! [Z: fm,C: produc6018962875968178549ist_fm > rule > set_fm,A2: produc6018962875968178549ist_fm,B2: rule] :
( ( member_fm @ Z @ ( C @ A2 @ B2 ) )
=> ( member_fm @ Z @ ( produc271619708354958774set_fm @ C @ ( produc1733806532565653680m_rule @ A2 @ B2 ) ) ) ) ).
% mem_case_prodI
thf(fact_1033_mem__case__prodI,axiom,
! [Z: produc6018962875968178549ist_fm,C: nat > nat > set_Pr5202636777678657877ist_fm,A2: nat,B2: nat] :
( ( member4699826688122452638ist_fm @ Z @ ( C @ A2 @ B2 ) )
=> ( member4699826688122452638ist_fm @ Z @ ( produc8713957302166562951ist_fm @ C @ ( product_Pair_nat_nat @ A2 @ B2 ) ) ) ) ).
% mem_case_prodI
thf(fact_1034_mem__case__prodI,axiom,
! [Z: list_fm,C: produc6018962875968178549ist_fm > rule > set_list_fm,A2: produc6018962875968178549ist_fm,B2: rule] :
( ( member_list_fm @ Z @ ( C @ A2 @ B2 ) )
=> ( member_list_fm @ Z @ ( produc1230737422366950076ist_fm @ C @ ( produc1733806532565653680m_rule @ A2 @ B2 ) ) ) ) ).
% mem_case_prodI
thf(fact_1035_mem__case__prodI2,axiom,
! [P2: product_prod_nat_nat,Z: rule,C: nat > nat > set_rule] :
( ! [A4: nat,B5: nat] :
( ( P2
= ( product_Pair_nat_nat @ A4 @ B5 ) )
=> ( member_rule @ Z @ ( C @ A4 @ B5 ) ) )
=> ( member_rule @ Z @ ( produc4888952194711710358t_rule @ C @ P2 ) ) ) ).
% mem_case_prodI2
thf(fact_1036_mem__case__prodI2,axiom,
! [P2: product_prod_nat_nat,Z: fm,C: nat > nat > set_fm] :
( ! [A4: nat,B5: nat] :
( ( P2
= ( product_Pair_nat_nat @ A4 @ B5 ) )
=> ( member_fm @ Z @ ( C @ A4 @ B5 ) ) )
=> ( member_fm @ Z @ ( produc2762169475385161349set_fm @ C @ P2 ) ) ) ).
% mem_case_prodI2
thf(fact_1037_mem__case__prodI2,axiom,
! [P2: product_prod_nat_nat,Z: list_fm,C: nat > nat > set_list_fm] :
( ! [A4: nat,B5: nat] :
( ( P2
= ( product_Pair_nat_nat @ A4 @ B5 ) )
=> ( member_list_fm @ Z @ ( C @ A4 @ B5 ) ) )
=> ( member_list_fm @ Z @ ( produc8363348377050837003ist_fm @ C @ P2 ) ) ) ).
% mem_case_prodI2
thf(fact_1038_mem__case__prodI2,axiom,
! [P2: produc6018962875968178549ist_fm,Z: rule,C: list_tm > list_fm > set_rule] :
( ! [A4: list_tm,B5: list_fm] :
( ( P2
= ( produc1414352766439514085ist_fm @ A4 @ B5 ) )
=> ( member_rule @ Z @ ( C @ A4 @ B5 ) ) )
=> ( member_rule @ Z @ ( produc1245229592201295426t_rule @ C @ P2 ) ) ) ).
% mem_case_prodI2
thf(fact_1039_mem__case__prodI2,axiom,
! [P2: produc6018962875968178549ist_fm,Z: fm,C: list_tm > list_fm > set_fm] :
( ! [A4: list_tm,B5: list_fm] :
( ( P2
= ( produc1414352766439514085ist_fm @ A4 @ B5 ) )
=> ( member_fm @ Z @ ( C @ A4 @ B5 ) ) )
=> ( member_fm @ Z @ ( produc1673703176515561945set_fm @ C @ P2 ) ) ) ).
% mem_case_prodI2
thf(fact_1040_mem__case__prodI2,axiom,
! [P2: produc6018962875968178549ist_fm,Z: list_fm,C: list_tm > list_fm > set_list_fm] :
( ! [A4: list_tm,B5: list_fm] :
( ( P2
= ( produc1414352766439514085ist_fm @ A4 @ B5 ) )
=> ( member_list_fm @ Z @ ( C @ A4 @ B5 ) ) )
=> ( member_list_fm @ Z @ ( produc4004295895882230111ist_fm @ C @ P2 ) ) ) ).
% mem_case_prodI2
thf(fact_1041_mem__case__prodI2,axiom,
! [P2: produc340336539035504054m_rule,Z: rule,C: produc6018962875968178549ist_fm > rule > set_rule] :
( ! [A4: produc6018962875968178549ist_fm,B5: rule] :
( ( P2
= ( produc1733806532565653680m_rule @ A4 @ B5 ) )
=> ( member_rule @ Z @ ( C @ A4 @ B5 ) ) )
=> ( member_rule @ Z @ ( produc2336968575880921125t_rule @ C @ P2 ) ) ) ).
% mem_case_prodI2
thf(fact_1042_mem__case__prodI2,axiom,
! [P2: produc340336539035504054m_rule,Z: fm,C: produc6018962875968178549ist_fm > rule > set_fm] :
( ! [A4: produc6018962875968178549ist_fm,B5: rule] :
( ( P2
= ( produc1733806532565653680m_rule @ A4 @ B5 ) )
=> ( member_fm @ Z @ ( C @ A4 @ B5 ) ) )
=> ( member_fm @ Z @ ( produc271619708354958774set_fm @ C @ P2 ) ) ) ).
% mem_case_prodI2
thf(fact_1043_mem__case__prodI2,axiom,
! [P2: product_prod_nat_nat,Z: produc6018962875968178549ist_fm,C: nat > nat > set_Pr5202636777678657877ist_fm] :
( ! [A4: nat,B5: nat] :
( ( P2
= ( product_Pair_nat_nat @ A4 @ B5 ) )
=> ( member4699826688122452638ist_fm @ Z @ ( C @ A4 @ B5 ) ) )
=> ( member4699826688122452638ist_fm @ Z @ ( produc8713957302166562951ist_fm @ C @ P2 ) ) ) ).
% mem_case_prodI2
thf(fact_1044_mem__case__prodI2,axiom,
! [P2: produc340336539035504054m_rule,Z: list_fm,C: produc6018962875968178549ist_fm > rule > set_list_fm] :
( ! [A4: produc6018962875968178549ist_fm,B5: rule] :
( ( P2
= ( produc1733806532565653680m_rule @ A4 @ B5 ) )
=> ( member_list_fm @ Z @ ( C @ A4 @ B5 ) ) )
=> ( member_list_fm @ Z @ ( produc1230737422366950076ist_fm @ C @ P2 ) ) ) ).
% mem_case_prodI2
thf(fact_1045_set__rotate1,axiom,
! [Xs: list_fm] :
( ( set_fm2 @ ( rotate1_fm @ Xs ) )
= ( set_fm2 @ Xs ) ) ).
% set_rotate1
thf(fact_1046_set__rotate1,axiom,
! [Xs: list_tm] :
( ( set_tm2 @ ( rotate1_tm @ Xs ) )
= ( set_tm2 @ Xs ) ) ).
% set_rotate1
thf(fact_1047_set__rotate1,axiom,
! [Xs: list_list_fm] :
( ( set_list_fm2 @ ( rotate1_list_fm @ Xs ) )
= ( set_list_fm2 @ Xs ) ) ).
% set_rotate1
thf(fact_1048_mem__case__prodE,axiom,
! [Z: rule,C: nat > nat > set_rule,P2: product_prod_nat_nat] :
( ( member_rule @ Z @ ( produc4888952194711710358t_rule @ C @ P2 ) )
=> ~ ! [X: nat,Y: nat] :
( ( P2
= ( product_Pair_nat_nat @ X @ Y ) )
=> ~ ( member_rule @ Z @ ( C @ X @ Y ) ) ) ) ).
% mem_case_prodE
thf(fact_1049_mem__case__prodE,axiom,
! [Z: fm,C: nat > nat > set_fm,P2: product_prod_nat_nat] :
( ( member_fm @ Z @ ( produc2762169475385161349set_fm @ C @ P2 ) )
=> ~ ! [X: nat,Y: nat] :
( ( P2
= ( product_Pair_nat_nat @ X @ Y ) )
=> ~ ( member_fm @ Z @ ( C @ X @ Y ) ) ) ) ).
% mem_case_prodE
thf(fact_1050_mem__case__prodE,axiom,
! [Z: list_fm,C: nat > nat > set_list_fm,P2: product_prod_nat_nat] :
( ( member_list_fm @ Z @ ( produc8363348377050837003ist_fm @ C @ P2 ) )
=> ~ ! [X: nat,Y: nat] :
( ( P2
= ( product_Pair_nat_nat @ X @ Y ) )
=> ~ ( member_list_fm @ Z @ ( C @ X @ Y ) ) ) ) ).
% mem_case_prodE
thf(fact_1051_mem__case__prodE,axiom,
! [Z: rule,C: list_tm > list_fm > set_rule,P2: produc6018962875968178549ist_fm] :
( ( member_rule @ Z @ ( produc1245229592201295426t_rule @ C @ P2 ) )
=> ~ ! [X: list_tm,Y: list_fm] :
( ( P2
= ( produc1414352766439514085ist_fm @ X @ Y ) )
=> ~ ( member_rule @ Z @ ( C @ X @ Y ) ) ) ) ).
% mem_case_prodE
thf(fact_1052_mem__case__prodE,axiom,
! [Z: fm,C: list_tm > list_fm > set_fm,P2: produc6018962875968178549ist_fm] :
( ( member_fm @ Z @ ( produc1673703176515561945set_fm @ C @ P2 ) )
=> ~ ! [X: list_tm,Y: list_fm] :
( ( P2
= ( produc1414352766439514085ist_fm @ X @ Y ) )
=> ~ ( member_fm @ Z @ ( C @ X @ Y ) ) ) ) ).
% mem_case_prodE
thf(fact_1053_mem__case__prodE,axiom,
! [Z: list_fm,C: list_tm > list_fm > set_list_fm,P2: produc6018962875968178549ist_fm] :
( ( member_list_fm @ Z @ ( produc4004295895882230111ist_fm @ C @ P2 ) )
=> ~ ! [X: list_tm,Y: list_fm] :
( ( P2
= ( produc1414352766439514085ist_fm @ X @ Y ) )
=> ~ ( member_list_fm @ Z @ ( C @ X @ Y ) ) ) ) ).
% mem_case_prodE
thf(fact_1054_mem__case__prodE,axiom,
! [Z: rule,C: produc6018962875968178549ist_fm > rule > set_rule,P2: produc340336539035504054m_rule] :
( ( member_rule @ Z @ ( produc2336968575880921125t_rule @ C @ P2 ) )
=> ~ ! [X: produc6018962875968178549ist_fm,Y: rule] :
( ( P2
= ( produc1733806532565653680m_rule @ X @ Y ) )
=> ~ ( member_rule @ Z @ ( C @ X @ Y ) ) ) ) ).
% mem_case_prodE
thf(fact_1055_mem__case__prodE,axiom,
! [Z: fm,C: produc6018962875968178549ist_fm > rule > set_fm,P2: produc340336539035504054m_rule] :
( ( member_fm @ Z @ ( produc271619708354958774set_fm @ C @ P2 ) )
=> ~ ! [X: produc6018962875968178549ist_fm,Y: rule] :
( ( P2
= ( produc1733806532565653680m_rule @ X @ Y ) )
=> ~ ( member_fm @ Z @ ( C @ X @ Y ) ) ) ) ).
% mem_case_prodE
thf(fact_1056_mem__case__prodE,axiom,
! [Z: produc6018962875968178549ist_fm,C: nat > nat > set_Pr5202636777678657877ist_fm,P2: product_prod_nat_nat] :
( ( member4699826688122452638ist_fm @ Z @ ( produc8713957302166562951ist_fm @ C @ P2 ) )
=> ~ ! [X: nat,Y: nat] :
( ( P2
= ( product_Pair_nat_nat @ X @ Y ) )
=> ~ ( member4699826688122452638ist_fm @ Z @ ( C @ X @ Y ) ) ) ) ).
% mem_case_prodE
thf(fact_1057_mem__case__prodE,axiom,
! [Z: list_fm,C: produc6018962875968178549ist_fm > rule > set_list_fm,P2: produc340336539035504054m_rule] :
( ( member_list_fm @ Z @ ( produc1230737422366950076ist_fm @ C @ P2 ) )
=> ~ ! [X: produc6018962875968178549ist_fm,Y: rule] :
( ( P2
= ( produc1733806532565653680m_rule @ X @ Y ) )
=> ~ ( member_list_fm @ Z @ ( C @ X @ Y ) ) ) ) ).
% mem_case_prodE
thf(fact_1058_rotate1_Osimps_I2_J,axiom,
! [X3: tm,Xs: list_tm] :
( ( rotate1_tm @ ( cons_tm @ X3 @ Xs ) )
= ( append_tm @ Xs @ ( cons_tm @ X3 @ nil_tm ) ) ) ).
% rotate1.simps(2)
thf(fact_1059_sdrop__cycle__eq__mod__0,axiom,
! [U: list_P2774625669004309958m_rule,N: nat] :
( ( U != nil_Pr2808214839852828720m_rule )
=> ( ( ( modulo_modulo_nat @ N @ ( size_s1575636608424004698m_rule @ U ) )
= zero_zero_nat )
=> ( ( sdrop_8169176516188972301m_rule @ N @ ( cycle_5335117900641983313m_rule @ U ) )
= ( cycle_5335117900641983313m_rule @ U ) ) ) ) ).
% sdrop_cycle_eq_mod_0
thf(fact_1060_sdrop__cycle__eq__mod__0,axiom,
! [U: list_rule,N: nat] :
( ( U != nil_rule )
=> ( ( ( modulo_modulo_nat @ N @ ( size_size_list_rule @ U ) )
= zero_zero_nat )
=> ( ( sdrop_rule @ N @ ( cycle_rule @ U ) )
= ( cycle_rule @ U ) ) ) ) ).
% sdrop_cycle_eq_mod_0
thf(fact_1061_in__set__takeD,axiom,
! [X3: rule,N: nat,Xs: list_rule] :
( ( member_rule @ X3 @ ( set_rule2 @ ( take_rule @ N @ Xs ) ) )
=> ( member_rule @ X3 @ ( set_rule2 @ Xs ) ) ) ).
% in_set_takeD
thf(fact_1062_in__set__takeD,axiom,
! [X3: produc6018962875968178549ist_fm,N: nat,Xs: list_P2887561121880082555ist_fm] :
( ( member4699826688122452638ist_fm @ X3 @ ( set_Pr5051287786238754058ist_fm @ ( take_P3919814664433536355ist_fm @ N @ Xs ) ) )
=> ( member4699826688122452638ist_fm @ X3 @ ( set_Pr5051287786238754058ist_fm @ Xs ) ) ) ).
% in_set_takeD
thf(fact_1063_in__set__takeD,axiom,
! [X3: produc340336539035504054m_rule,N: nat,Xs: list_P2774625669004309958m_rule] :
( ( member7231649785386036813m_rule @ X3 @ ( set_Pr4534715572506550497m_rule @ ( take_P3067526267515409992m_rule @ N @ Xs ) ) )
=> ( member7231649785386036813m_rule @ X3 @ ( set_Pr4534715572506550497m_rule @ Xs ) ) ) ).
% in_set_takeD
thf(fact_1064_in__set__takeD,axiom,
! [X3: fm,N: nat,Xs: list_fm] :
( ( member_fm @ X3 @ ( set_fm2 @ ( take_fm @ N @ Xs ) ) )
=> ( member_fm @ X3 @ ( set_fm2 @ Xs ) ) ) ).
% in_set_takeD
thf(fact_1065_in__set__takeD,axiom,
! [X3: tm,N: nat,Xs: list_tm] :
( ( member_tm @ X3 @ ( set_tm2 @ ( take_tm @ N @ Xs ) ) )
=> ( member_tm @ X3 @ ( set_tm2 @ Xs ) ) ) ).
% in_set_takeD
thf(fact_1066_in__set__takeD,axiom,
! [X3: list_fm,N: nat,Xs: list_list_fm] :
( ( member_list_fm @ X3 @ ( set_list_fm2 @ ( take_list_fm @ N @ Xs ) ) )
=> ( member_list_fm @ X3 @ ( set_list_fm2 @ Xs ) ) ) ).
% in_set_takeD
thf(fact_1067_set__take__subset,axiom,
! [N: nat,Xs: list_list_fm] : ( ord_le7838213414353715577ist_fm @ ( set_list_fm2 @ ( take_list_fm @ N @ Xs ) ) @ ( set_list_fm2 @ Xs ) ) ).
% set_take_subset
thf(fact_1068_set__take__subset,axiom,
! [N: nat,Xs: list_tm] : ( ord_less_eq_set_tm @ ( set_tm2 @ ( take_tm @ N @ Xs ) ) @ ( set_tm2 @ Xs ) ) ).
% set_take_subset
thf(fact_1069_set__take__subset,axiom,
! [N: nat,Xs: list_fm] : ( ord_less_eq_set_fm @ ( set_fm2 @ ( take_fm @ N @ Xs ) ) @ ( set_fm2 @ Xs ) ) ).
% set_take_subset
thf(fact_1070_set__take__subset,axiom,
! [N: nat,Xs: list_rule] : ( ord_less_eq_set_rule @ ( set_rule2 @ ( take_rule @ N @ Xs ) ) @ ( set_rule2 @ Xs ) ) ).
% set_take_subset
thf(fact_1071_tl__append,axiom,
! [Xs: list_tm,Ys: list_tm] :
( ( tl_tm @ ( append_tm @ Xs @ Ys ) )
= ( case_list_list_tm_tm @ ( tl_tm @ Ys )
@ ^ [Z5: tm,Zs2: list_tm] : ( append_tm @ Zs2 @ Ys )
@ Xs ) ) ).
% tl_append
thf(fact_1072_take__hd__drop,axiom,
! [N: nat,Xs: list_tm] :
( ( ord_less_nat @ N @ ( size_size_list_tm @ Xs ) )
=> ( ( append_tm @ ( take_tm @ N @ Xs ) @ ( cons_tm @ ( hd_tm @ ( drop_tm @ N @ Xs ) ) @ nil_tm ) )
= ( take_tm @ ( suc @ N ) @ Xs ) ) ) ).
% take_hd_drop
thf(fact_1073_sdrop__cycle,axiom,
! [U: list_P2774625669004309958m_rule,N: nat] :
( ( U != nil_Pr2808214839852828720m_rule )
=> ( ( sdrop_8169176516188972301m_rule @ N @ ( cycle_5335117900641983313m_rule @ U ) )
= ( cycle_5335117900641983313m_rule @ ( rotate8779165051853931260m_rule @ ( modulo_modulo_nat @ N @ ( size_s1575636608424004698m_rule @ U ) ) @ U ) ) ) ) ).
% sdrop_cycle
thf(fact_1074_sdrop__cycle,axiom,
! [U: list_rule,N: nat] :
( ( U != nil_rule )
=> ( ( sdrop_rule @ N @ ( cycle_rule @ U ) )
= ( cycle_rule @ ( rotate_rule @ ( modulo_modulo_nat @ N @ ( size_size_list_rule @ U ) ) @ U ) ) ) ) ).
% sdrop_cycle
thf(fact_1075_take__Suc__conv__app__nth,axiom,
! [I: nat,Xs: list_tm] :
( ( ord_less_nat @ I @ ( size_size_list_tm @ Xs ) )
=> ( ( take_tm @ ( suc @ I ) @ Xs )
= ( append_tm @ ( take_tm @ I @ Xs ) @ ( cons_tm @ ( nth_tm @ Xs @ I ) @ nil_tm ) ) ) ) ).
% take_Suc_conv_app_nth
thf(fact_1076_set__rotate,axiom,
! [N: nat,Xs: list_fm] :
( ( set_fm2 @ ( rotate_fm @ N @ Xs ) )
= ( set_fm2 @ Xs ) ) ).
% set_rotate
thf(fact_1077_set__rotate,axiom,
! [N: nat,Xs: list_tm] :
( ( set_tm2 @ ( rotate_tm @ N @ Xs ) )
= ( set_tm2 @ Xs ) ) ).
% set_rotate
thf(fact_1078_set__rotate,axiom,
! [N: nat,Xs: list_list_fm] :
( ( set_list_fm2 @ ( rotate_list_fm @ N @ Xs ) )
= ( set_list_fm2 @ Xs ) ) ).
% set_rotate
thf(fact_1079_append__take__drop__id,axiom,
! [N: nat,Xs: list_tm] :
( ( append_tm @ ( take_tm @ N @ Xs ) @ ( drop_tm @ N @ Xs ) )
= Xs ) ).
% append_take_drop_id
thf(fact_1080_nth__append__length,axiom,
! [Xs: list_tm,X3: tm,Ys: list_tm] :
( ( nth_tm @ ( append_tm @ Xs @ ( cons_tm @ X3 @ Ys ) ) @ ( size_size_list_tm @ Xs ) )
= X3 ) ).
% nth_append_length
thf(fact_1081_stake__nth,axiom,
! [M2: nat,N: nat,S: stream2709947120125613254m_rule] :
( ( ord_less_nat @ M2 @ N )
=> ( ( nth_Pr3936752564452695919m_rule @ ( stake_5421812949518764133m_rule @ N @ S ) @ M2 )
= ( snth_P6679518042731451922m_rule @ S @ M2 ) ) ) ).
% stake_nth
thf(fact_1082_stake__nth,axiom,
! [M2: nat,N: nat,S: stream_rule] :
( ( ord_less_nat @ M2 @ N )
=> ( ( nth_rule @ ( stake_rule @ N @ S ) @ M2 )
= ( snth_rule @ S @ M2 ) ) ) ).
% stake_nth
thf(fact_1083_shift__snth__less,axiom,
! [P2: nat,Xs: list_rule,S: stream_rule] :
( ( ord_less_nat @ P2 @ ( size_size_list_rule @ Xs ) )
=> ( ( snth_rule @ ( shift_rule @ Xs @ S ) @ P2 )
= ( nth_rule @ Xs @ P2 ) ) ) ).
% shift_snth_less
thf(fact_1084_shift__snth__less,axiom,
! [P2: nat,Xs: list_P2774625669004309958m_rule,S: stream2709947120125613254m_rule] :
( ( ord_less_nat @ P2 @ ( size_s1575636608424004698m_rule @ Xs ) )
=> ( ( snth_P6679518042731451922m_rule @ ( shift_2334844276748245581m_rule @ Xs @ S ) @ P2 )
= ( nth_Pr3936752564452695919m_rule @ Xs @ P2 ) ) ) ).
% shift_snth_less
thf(fact_1085_in__set__dropD,axiom,
! [X3: rule,N: nat,Xs: list_rule] :
( ( member_rule @ X3 @ ( set_rule2 @ ( drop_rule @ N @ Xs ) ) )
=> ( member_rule @ X3 @ ( set_rule2 @ Xs ) ) ) ).
% in_set_dropD
thf(fact_1086_in__set__dropD,axiom,
! [X3: produc6018962875968178549ist_fm,N: nat,Xs: list_P2887561121880082555ist_fm] :
( ( member4699826688122452638ist_fm @ X3 @ ( set_Pr5051287786238754058ist_fm @ ( drop_P195108109085473467ist_fm @ N @ Xs ) ) )
=> ( member4699826688122452638ist_fm @ X3 @ ( set_Pr5051287786238754058ist_fm @ Xs ) ) ) ).
% in_set_dropD
thf(fact_1087_in__set__dropD,axiom,
! [X3: produc340336539035504054m_rule,N: nat,Xs: list_P2774625669004309958m_rule] :
( ( member7231649785386036813m_rule @ X3 @ ( set_Pr4534715572506550497m_rule @ ( drop_P5814889834185618160m_rule @ N @ Xs ) ) )
=> ( member7231649785386036813m_rule @ X3 @ ( set_Pr4534715572506550497m_rule @ Xs ) ) ) ).
% in_set_dropD
thf(fact_1088_in__set__dropD,axiom,
! [X3: fm,N: nat,Xs: list_fm] :
( ( member_fm @ X3 @ ( set_fm2 @ ( drop_fm @ N @ Xs ) ) )
=> ( member_fm @ X3 @ ( set_fm2 @ Xs ) ) ) ).
% in_set_dropD
thf(fact_1089_in__set__dropD,axiom,
! [X3: tm,N: nat,Xs: list_tm] :
( ( member_tm @ X3 @ ( set_tm2 @ ( drop_tm @ N @ Xs ) ) )
=> ( member_tm @ X3 @ ( set_tm2 @ Xs ) ) ) ).
% in_set_dropD
thf(fact_1090_in__set__dropD,axiom,
! [X3: list_fm,N: nat,Xs: list_list_fm] :
( ( member_list_fm @ X3 @ ( set_list_fm2 @ ( drop_list_fm @ N @ Xs ) ) )
=> ( member_list_fm @ X3 @ ( set_list_fm2 @ Xs ) ) ) ).
% in_set_dropD
thf(fact_1091_set__drop__subset,axiom,
! [N: nat,Xs: list_list_fm] : ( ord_le7838213414353715577ist_fm @ ( set_list_fm2 @ ( drop_list_fm @ N @ Xs ) ) @ ( set_list_fm2 @ Xs ) ) ).
% set_drop_subset
thf(fact_1092_set__drop__subset,axiom,
! [N: nat,Xs: list_tm] : ( ord_less_eq_set_tm @ ( set_tm2 @ ( drop_tm @ N @ Xs ) ) @ ( set_tm2 @ Xs ) ) ).
% set_drop_subset
thf(fact_1093_set__drop__subset,axiom,
! [N: nat,Xs: list_fm] : ( ord_less_eq_set_fm @ ( set_fm2 @ ( drop_fm @ N @ Xs ) ) @ ( set_fm2 @ Xs ) ) ).
% set_drop_subset
thf(fact_1094_set__drop__subset,axiom,
! [N: nat,Xs: list_rule] : ( ord_less_eq_set_rule @ ( set_rule2 @ ( drop_rule @ N @ Xs ) ) @ ( set_rule2 @ Xs ) ) ).
% set_drop_subset
thf(fact_1095_rotate__drop__take,axiom,
( rotate_tm
= ( ^ [N3: nat,Xs2: list_tm] : ( append_tm @ ( drop_tm @ ( modulo_modulo_nat @ N3 @ ( size_size_list_tm @ Xs2 ) ) @ Xs2 ) @ ( take_tm @ ( modulo_modulo_nat @ N3 @ ( size_size_list_tm @ Xs2 ) ) @ Xs2 ) ) ) ) ).
% rotate_drop_take
thf(fact_1096_rotate__append,axiom,
! [L: list_tm,Q2: list_tm] :
( ( rotate_tm @ ( size_size_list_tm @ L ) @ ( append_tm @ L @ Q2 ) )
= ( append_tm @ Q2 @ L ) ) ).
% rotate_append
thf(fact_1097_id__take__nth__drop,axiom,
! [I: nat,Xs: list_tm] :
( ( ord_less_nat @ I @ ( size_size_list_tm @ Xs ) )
=> ( Xs
= ( append_tm @ ( take_tm @ I @ Xs ) @ ( cons_tm @ ( nth_tm @ Xs @ I ) @ ( drop_tm @ ( suc @ I ) @ Xs ) ) ) ) ) ).
% id_take_nth_drop
thf(fact_1098_all__set__conv__all__nth,axiom,
! [Xs: list_fm,P: fm > $o] :
( ( ! [X4: fm] :
( ( member_fm @ X4 @ ( set_fm2 @ Xs ) )
=> ( P @ X4 ) ) )
= ( ! [I3: nat] :
( ( ord_less_nat @ I3 @ ( size_size_list_fm @ Xs ) )
=> ( P @ ( nth_fm @ Xs @ I3 ) ) ) ) ) ).
% all_set_conv_all_nth
thf(fact_1099_all__set__conv__all__nth,axiom,
! [Xs: list_tm,P: tm > $o] :
( ( ! [X4: tm] :
( ( member_tm @ X4 @ ( set_tm2 @ Xs ) )
=> ( P @ X4 ) ) )
= ( ! [I3: nat] :
( ( ord_less_nat @ I3 @ ( size_size_list_tm @ Xs ) )
=> ( P @ ( nth_tm @ Xs @ I3 ) ) ) ) ) ).
% all_set_conv_all_nth
thf(fact_1100_all__set__conv__all__nth,axiom,
! [Xs: list_list_fm,P: list_fm > $o] :
( ( ! [X4: list_fm] :
( ( member_list_fm @ X4 @ ( set_list_fm2 @ Xs ) )
=> ( P @ X4 ) ) )
= ( ! [I3: nat] :
( ( ord_less_nat @ I3 @ ( size_s115229985653309035ist_fm @ Xs ) )
=> ( P @ ( nth_list_fm @ Xs @ I3 ) ) ) ) ) ).
% all_set_conv_all_nth
thf(fact_1101_all__nth__imp__all__set,axiom,
! [Xs: list_rule,P: rule > $o,X3: rule] :
( ! [I4: nat] :
( ( ord_less_nat @ I4 @ ( size_size_list_rule @ Xs ) )
=> ( P @ ( nth_rule @ Xs @ I4 ) ) )
=> ( ( member_rule @ X3 @ ( set_rule2 @ Xs ) )
=> ( P @ X3 ) ) ) ).
% all_nth_imp_all_set
thf(fact_1102_all__nth__imp__all__set,axiom,
! [Xs: list_P2887561121880082555ist_fm,P: produc6018962875968178549ist_fm > $o,X3: produc6018962875968178549ist_fm] :
( ! [I4: nat] :
( ( ord_less_nat @ I4 @ ( size_s3138477486474831591ist_fm @ Xs ) )
=> ( P @ ( nth_Pr580027083122244092ist_fm @ Xs @ I4 ) ) )
=> ( ( member4699826688122452638ist_fm @ X3 @ ( set_Pr5051287786238754058ist_fm @ Xs ) )
=> ( P @ X3 ) ) ) ).
% all_nth_imp_all_set
thf(fact_1103_all__nth__imp__all__set,axiom,
! [Xs: list_P2774625669004309958m_rule,P: produc340336539035504054m_rule > $o,X3: produc340336539035504054m_rule] :
( ! [I4: nat] :
( ( ord_less_nat @ I4 @ ( size_s1575636608424004698m_rule @ Xs ) )
=> ( P @ ( nth_Pr3936752564452695919m_rule @ Xs @ I4 ) ) )
=> ( ( member7231649785386036813m_rule @ X3 @ ( set_Pr4534715572506550497m_rule @ Xs ) )
=> ( P @ X3 ) ) ) ).
% all_nth_imp_all_set
thf(fact_1104_all__nth__imp__all__set,axiom,
! [Xs: list_fm,P: fm > $o,X3: fm] :
( ! [I4: nat] :
( ( ord_less_nat @ I4 @ ( size_size_list_fm @ Xs ) )
=> ( P @ ( nth_fm @ Xs @ I4 ) ) )
=> ( ( member_fm @ X3 @ ( set_fm2 @ Xs ) )
=> ( P @ X3 ) ) ) ).
% all_nth_imp_all_set
thf(fact_1105_all__nth__imp__all__set,axiom,
! [Xs: list_tm,P: tm > $o,X3: tm] :
( ! [I4: nat] :
( ( ord_less_nat @ I4 @ ( size_size_list_tm @ Xs ) )
=> ( P @ ( nth_tm @ Xs @ I4 ) ) )
=> ( ( member_tm @ X3 @ ( set_tm2 @ Xs ) )
=> ( P @ X3 ) ) ) ).
% all_nth_imp_all_set
thf(fact_1106_all__nth__imp__all__set,axiom,
! [Xs: list_list_fm,P: list_fm > $o,X3: list_fm] :
( ! [I4: nat] :
( ( ord_less_nat @ I4 @ ( size_s115229985653309035ist_fm @ Xs ) )
=> ( P @ ( nth_list_fm @ Xs @ I4 ) ) )
=> ( ( member_list_fm @ X3 @ ( set_list_fm2 @ Xs ) )
=> ( P @ X3 ) ) ) ).
% all_nth_imp_all_set
thf(fact_1107_in__set__conv__nth,axiom,
! [X3: rule,Xs: list_rule] :
( ( member_rule @ X3 @ ( set_rule2 @ Xs ) )
= ( ? [I3: nat] :
( ( ord_less_nat @ I3 @ ( size_size_list_rule @ Xs ) )
& ( ( nth_rule @ Xs @ I3 )
= X3 ) ) ) ) ).
% in_set_conv_nth
thf(fact_1108_in__set__conv__nth,axiom,
! [X3: produc6018962875968178549ist_fm,Xs: list_P2887561121880082555ist_fm] :
( ( member4699826688122452638ist_fm @ X3 @ ( set_Pr5051287786238754058ist_fm @ Xs ) )
= ( ? [I3: nat] :
( ( ord_less_nat @ I3 @ ( size_s3138477486474831591ist_fm @ Xs ) )
& ( ( nth_Pr580027083122244092ist_fm @ Xs @ I3 )
= X3 ) ) ) ) ).
% in_set_conv_nth
thf(fact_1109_in__set__conv__nth,axiom,
! [X3: produc340336539035504054m_rule,Xs: list_P2774625669004309958m_rule] :
( ( member7231649785386036813m_rule @ X3 @ ( set_Pr4534715572506550497m_rule @ Xs ) )
= ( ? [I3: nat] :
( ( ord_less_nat @ I3 @ ( size_s1575636608424004698m_rule @ Xs ) )
& ( ( nth_Pr3936752564452695919m_rule @ Xs @ I3 )
= X3 ) ) ) ) ).
% in_set_conv_nth
thf(fact_1110_in__set__conv__nth,axiom,
! [X3: fm,Xs: list_fm] :
( ( member_fm @ X3 @ ( set_fm2 @ Xs ) )
= ( ? [I3: nat] :
( ( ord_less_nat @ I3 @ ( size_size_list_fm @ Xs ) )
& ( ( nth_fm @ Xs @ I3 )
= X3 ) ) ) ) ).
% in_set_conv_nth
thf(fact_1111_in__set__conv__nth,axiom,
! [X3: tm,Xs: list_tm] :
( ( member_tm @ X3 @ ( set_tm2 @ Xs ) )
= ( ? [I3: nat] :
( ( ord_less_nat @ I3 @ ( size_size_list_tm @ Xs ) )
& ( ( nth_tm @ Xs @ I3 )
= X3 ) ) ) ) ).
% in_set_conv_nth
thf(fact_1112_in__set__conv__nth,axiom,
! [X3: list_fm,Xs: list_list_fm] :
( ( member_list_fm @ X3 @ ( set_list_fm2 @ Xs ) )
= ( ? [I3: nat] :
( ( ord_less_nat @ I3 @ ( size_s115229985653309035ist_fm @ Xs ) )
& ( ( nth_list_fm @ Xs @ I3 )
= X3 ) ) ) ) ).
% in_set_conv_nth
thf(fact_1113_list__ball__nth,axiom,
! [N: nat,Xs: list_fm,P: fm > $o] :
( ( ord_less_nat @ N @ ( size_size_list_fm @ Xs ) )
=> ( ! [X: fm] :
( ( member_fm @ X @ ( set_fm2 @ Xs ) )
=> ( P @ X ) )
=> ( P @ ( nth_fm @ Xs @ N ) ) ) ) ).
% list_ball_nth
thf(fact_1114_list__ball__nth,axiom,
! [N: nat,Xs: list_tm,P: tm > $o] :
( ( ord_less_nat @ N @ ( size_size_list_tm @ Xs ) )
=> ( ! [X: tm] :
( ( member_tm @ X @ ( set_tm2 @ Xs ) )
=> ( P @ X ) )
=> ( P @ ( nth_tm @ Xs @ N ) ) ) ) ).
% list_ball_nth
thf(fact_1115_list__ball__nth,axiom,
! [N: nat,Xs: list_list_fm,P: list_fm > $o] :
( ( ord_less_nat @ N @ ( size_s115229985653309035ist_fm @ Xs ) )
=> ( ! [X: list_fm] :
( ( member_list_fm @ X @ ( set_list_fm2 @ Xs ) )
=> ( P @ X ) )
=> ( P @ ( nth_list_fm @ Xs @ N ) ) ) ) ).
% list_ball_nth
thf(fact_1116_nth__mem,axiom,
! [N: nat,Xs: list_rule] :
( ( ord_less_nat @ N @ ( size_size_list_rule @ Xs ) )
=> ( member_rule @ ( nth_rule @ Xs @ N ) @ ( set_rule2 @ Xs ) ) ) ).
% nth_mem
thf(fact_1117_nth__mem,axiom,
! [N: nat,Xs: list_P2887561121880082555ist_fm] :
( ( ord_less_nat @ N @ ( size_s3138477486474831591ist_fm @ Xs ) )
=> ( member4699826688122452638ist_fm @ ( nth_Pr580027083122244092ist_fm @ Xs @ N ) @ ( set_Pr5051287786238754058ist_fm @ Xs ) ) ) ).
% nth_mem
thf(fact_1118_nth__mem,axiom,
! [N: nat,Xs: list_P2774625669004309958m_rule] :
( ( ord_less_nat @ N @ ( size_s1575636608424004698m_rule @ Xs ) )
=> ( member7231649785386036813m_rule @ ( nth_Pr3936752564452695919m_rule @ Xs @ N ) @ ( set_Pr4534715572506550497m_rule @ Xs ) ) ) ).
% nth_mem
thf(fact_1119_nth__mem,axiom,
! [N: nat,Xs: list_fm] :
( ( ord_less_nat @ N @ ( size_size_list_fm @ Xs ) )
=> ( member_fm @ ( nth_fm @ Xs @ N ) @ ( set_fm2 @ Xs ) ) ) ).
% nth_mem
thf(fact_1120_nth__mem,axiom,
! [N: nat,Xs: list_tm] :
( ( ord_less_nat @ N @ ( size_size_list_tm @ Xs ) )
=> ( member_tm @ ( nth_tm @ Xs @ N ) @ ( set_tm2 @ Xs ) ) ) ).
% nth_mem
thf(fact_1121_nth__mem,axiom,
! [N: nat,Xs: list_list_fm] :
( ( ord_less_nat @ N @ ( size_s115229985653309035ist_fm @ Xs ) )
=> ( member_list_fm @ ( nth_list_fm @ Xs @ N ) @ ( set_list_fm2 @ Xs ) ) ) ).
% nth_mem
thf(fact_1122_append__eq__conv__conj,axiom,
! [Xs: list_tm,Ys: list_tm,Zs: list_tm] :
( ( ( append_tm @ Xs @ Ys )
= Zs )
= ( ( Xs
= ( take_tm @ ( size_size_list_tm @ Xs ) @ Zs ) )
& ( Ys
= ( drop_tm @ ( size_size_list_tm @ Xs ) @ Zs ) ) ) ) ).
% append_eq_conv_conj
thf(fact_1123_list__all__length,axiom,
( list_a9038620553111681026m_rule
= ( ^ [P3: produc340336539035504054m_rule > $o,Xs2: list_P2774625669004309958m_rule] :
! [N3: nat] :
( ( ord_less_nat @ N3 @ ( size_s1575636608424004698m_rule @ Xs2 ) )
=> ( P3 @ ( nth_Pr3936752564452695919m_rule @ Xs2 @ N3 ) ) ) ) ) ).
% list_all_length
thf(fact_1124_case__prod__Pair__iden,axiom,
! [P2: produc6018962875968178549ist_fm] :
( ( produc7510973280389026811ist_fm @ produc1414352766439514085ist_fm @ P2 )
= P2 ) ).
% case_prod_Pair_iden
thf(fact_1125_case__prod__Pair__iden,axiom,
! [P2: produc340336539035504054m_rule] :
( ( produc2469707509091222163m_rule @ produc1733806532565653680m_rule @ P2 )
= P2 ) ).
% case_prod_Pair_iden
thf(fact_1126_case__prod__Pair__iden,axiom,
! [P2: product_prod_nat_nat] :
( ( produc2626176000494625587at_nat @ product_Pair_nat_nat @ P2 )
= P2 ) ).
% case_prod_Pair_iden
thf(fact_1127_lex__take__index,axiom,
! [Xs: list_nat,Ys: list_nat,R: set_Pr1261947904930325089at_nat] :
( ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ Xs @ Ys ) @ ( lex_nat @ R ) )
=> ~ ! [I4: nat] :
( ( ord_less_nat @ I4 @ ( size_size_list_nat @ Xs ) )
=> ( ( ord_less_nat @ I4 @ ( size_size_list_nat @ Ys ) )
=> ( ( ( take_nat @ I4 @ Xs )
= ( take_nat @ I4 @ Ys ) )
=> ~ ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ ( nth_nat @ Xs @ I4 ) @ ( nth_nat @ Ys @ I4 ) ) @ R ) ) ) ) ) ).
% lex_take_index
thf(fact_1128_flat__snth,axiom,
! [S: stream2471014364565126742m_rule,N: nat] :
( ! [X: list_P2774625669004309958m_rule] :
( ( member522667613741702877m_rule @ X @ ( sset_l1945517436738345138m_rule @ S ) )
=> ( X != nil_Pr2808214839852828720m_rule ) )
=> ( ( ( ord_less_nat @ N @ ( size_s1575636608424004698m_rule @ ( shd_li4676821617271663642m_rule @ S ) ) )
=> ( ( snth_P6679518042731451922m_rule @ ( flat_P7721466590633226428m_rule @ S ) @ N )
= ( nth_Pr3936752564452695919m_rule @ ( shd_li4676821617271663642m_rule @ S ) @ N ) ) )
& ( ~ ( ord_less_nat @ N @ ( size_s1575636608424004698m_rule @ ( shd_li4676821617271663642m_rule @ S ) ) )
=> ( ( snth_P6679518042731451922m_rule @ ( flat_P7721466590633226428m_rule @ S ) @ N )
= ( snth_P6679518042731451922m_rule @ ( flat_P7721466590633226428m_rule @ ( stl_li6523153919213261078m_rule @ S ) ) @ ( minus_minus_nat @ N @ ( size_s1575636608424004698m_rule @ ( shd_li4676821617271663642m_rule @ S ) ) ) ) ) ) ) ) ).
% flat_snth
thf(fact_1129_flat__snth,axiom,
! [S: stream_list_rule,N: nat] :
( ! [X: list_rule] :
( ( member_list_rule @ X @ ( sset_list_rule @ S ) )
=> ( X != nil_rule ) )
=> ( ( ( ord_less_nat @ N @ ( size_size_list_rule @ ( shd_list_rule @ S ) ) )
=> ( ( snth_rule @ ( flat_rule @ S ) @ N )
= ( nth_rule @ ( shd_list_rule @ S ) @ N ) ) )
& ( ~ ( ord_less_nat @ N @ ( size_size_list_rule @ ( shd_list_rule @ S ) ) )
=> ( ( snth_rule @ ( flat_rule @ S ) @ N )
= ( snth_rule @ ( flat_rule @ ( stl_list_rule @ S ) ) @ ( minus_minus_nat @ N @ ( size_size_list_rule @ ( shd_list_rule @ S ) ) ) ) ) ) ) ) ).
% flat_snth
thf(fact_1130_upd__conv__take__nth__drop,axiom,
! [I: nat,Xs: list_tm,A2: tm] :
( ( ord_less_nat @ I @ ( size_size_list_tm @ Xs ) )
=> ( ( list_update_tm @ Xs @ I @ A2 )
= ( append_tm @ ( take_tm @ I @ Xs ) @ ( cons_tm @ A2 @ ( drop_tm @ ( suc @ I ) @ Xs ) ) ) ) ) ).
% upd_conv_take_nth_drop
thf(fact_1131_lexord__take__index__conv,axiom,
! [X3: list_nat,Y3: list_nat,R: set_Pr1261947904930325089at_nat] :
( ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ X3 @ Y3 ) @ ( lexord_nat @ R ) )
= ( ( ( ord_less_nat @ ( size_size_list_nat @ X3 ) @ ( size_size_list_nat @ Y3 ) )
& ( ( take_nat @ ( size_size_list_nat @ X3 ) @ Y3 )
= X3 ) )
| ? [I3: nat] :
( ( ord_less_nat @ I3 @ ( ord_min_nat @ ( size_size_list_nat @ X3 ) @ ( size_size_list_nat @ Y3 ) ) )
& ( ( take_nat @ I3 @ X3 )
= ( take_nat @ I3 @ Y3 ) )
& ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ ( nth_nat @ X3 @ I3 ) @ ( nth_nat @ Y3 @ I3 ) ) @ R ) ) ) ) ).
% lexord_take_index_conv
thf(fact_1132_listrel1__iff__update,axiom,
! [Xs: list_nat,Ys: list_nat,R: set_Pr1261947904930325089at_nat] :
( ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ Xs @ Ys ) @ ( listrel1_nat @ R ) )
= ( ? [Y5: nat,N3: nat] :
( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ ( nth_nat @ Xs @ N3 ) @ Y5 ) @ R )
& ( ord_less_nat @ N3 @ ( size_size_list_nat @ Xs ) )
& ( Ys
= ( list_update_nat @ Xs @ N3 @ Y5 ) ) ) ) ) ).
% listrel1_iff_update
thf(fact_1133_list__update__length,axiom,
! [Xs: list_tm,X3: tm,Ys: list_tm,Y3: tm] :
( ( list_update_tm @ ( append_tm @ Xs @ ( cons_tm @ X3 @ Ys ) ) @ ( size_size_list_tm @ Xs ) @ Y3 )
= ( append_tm @ Xs @ ( cons_tm @ Y3 @ Ys ) ) ) ).
% list_update_length
thf(fact_1134_take__append,axiom,
! [N: nat,Xs: list_tm,Ys: list_tm] :
( ( take_tm @ N @ ( append_tm @ Xs @ Ys ) )
= ( append_tm @ ( take_tm @ N @ Xs ) @ ( take_tm @ ( minus_minus_nat @ N @ ( size_size_list_tm @ Xs ) ) @ Ys ) ) ) ).
% take_append
thf(fact_1135_drop__append,axiom,
! [N: nat,Xs: list_tm,Ys: list_tm] :
( ( drop_tm @ N @ ( append_tm @ Xs @ Ys ) )
= ( append_tm @ ( drop_tm @ N @ Xs ) @ ( drop_tm @ ( minus_minus_nat @ N @ ( size_size_list_tm @ Xs ) ) @ Ys ) ) ) ).
% drop_append
thf(fact_1136_set__swap,axiom,
! [I: nat,Xs: list_fm,J: nat] :
( ( ord_less_nat @ I @ ( size_size_list_fm @ Xs ) )
=> ( ( ord_less_nat @ J @ ( size_size_list_fm @ Xs ) )
=> ( ( set_fm2 @ ( list_update_fm @ ( list_update_fm @ Xs @ I @ ( nth_fm @ Xs @ J ) ) @ J @ ( nth_fm @ Xs @ I ) ) )
= ( set_fm2 @ Xs ) ) ) ) ).
% set_swap
thf(fact_1137_set__swap,axiom,
! [I: nat,Xs: list_tm,J: nat] :
( ( ord_less_nat @ I @ ( size_size_list_tm @ Xs ) )
=> ( ( ord_less_nat @ J @ ( size_size_list_tm @ Xs ) )
=> ( ( set_tm2 @ ( list_update_tm @ ( list_update_tm @ Xs @ I @ ( nth_tm @ Xs @ J ) ) @ J @ ( nth_tm @ Xs @ I ) ) )
= ( set_tm2 @ Xs ) ) ) ) ).
% set_swap
thf(fact_1138_set__swap,axiom,
! [I: nat,Xs: list_list_fm,J: nat] :
( ( ord_less_nat @ I @ ( size_s115229985653309035ist_fm @ Xs ) )
=> ( ( ord_less_nat @ J @ ( size_s115229985653309035ist_fm @ Xs ) )
=> ( ( set_list_fm2 @ ( list_update_list_fm @ ( list_update_list_fm @ Xs @ I @ ( nth_list_fm @ Xs @ J ) ) @ J @ ( nth_list_fm @ Xs @ I ) ) )
= ( set_list_fm2 @ Xs ) ) ) ) ).
% set_swap
thf(fact_1139_list__update__append,axiom,
! [N: nat,Xs: list_tm,Ys: list_tm,X3: tm] :
( ( ( ord_less_nat @ N @ ( size_size_list_tm @ Xs ) )
=> ( ( list_update_tm @ ( append_tm @ Xs @ Ys ) @ N @ X3 )
= ( append_tm @ ( list_update_tm @ Xs @ N @ X3 ) @ Ys ) ) )
& ( ~ ( ord_less_nat @ N @ ( size_size_list_tm @ Xs ) )
=> ( ( list_update_tm @ ( append_tm @ Xs @ Ys ) @ N @ X3 )
= ( append_tm @ Xs @ ( list_update_tm @ Ys @ ( minus_minus_nat @ N @ ( size_size_list_tm @ Xs ) ) @ X3 ) ) ) ) ) ).
% list_update_append
thf(fact_1140_set__update__subsetI,axiom,
! [Xs: list_P2887561121880082555ist_fm,A: set_Pr5202636777678657877ist_fm,X3: produc6018962875968178549ist_fm,I: nat] :
( ( ord_le1771420097867575541ist_fm @ ( set_Pr5051287786238754058ist_fm @ Xs ) @ A )
=> ( ( member4699826688122452638ist_fm @ X3 @ A )
=> ( ord_le1771420097867575541ist_fm @ ( set_Pr5051287786238754058ist_fm @ ( list_u2187756355100148707ist_fm @ Xs @ I @ X3 ) ) @ A ) ) ) ).
% set_update_subsetI
thf(fact_1141_set__update__subsetI,axiom,
! [Xs: list_P2774625669004309958m_rule,A: set_Pr1822751329126368876m_rule,X3: produc340336539035504054m_rule,I: nat] :
( ( ord_le6390412330253371084m_rule @ ( set_Pr4534715572506550497m_rule @ Xs ) @ A )
=> ( ( member7231649785386036813m_rule @ X3 @ A )
=> ( ord_le6390412330253371084m_rule @ ( set_Pr4534715572506550497m_rule @ ( list_u1216070999260037064m_rule @ Xs @ I @ X3 ) ) @ A ) ) ) ).
% set_update_subsetI
thf(fact_1142_set__update__subsetI,axiom,
! [Xs: list_list_fm,A: set_list_fm,X3: list_fm,I: nat] :
( ( ord_le7838213414353715577ist_fm @ ( set_list_fm2 @ Xs ) @ A )
=> ( ( member_list_fm @ X3 @ A )
=> ( ord_le7838213414353715577ist_fm @ ( set_list_fm2 @ ( list_update_list_fm @ Xs @ I @ X3 ) ) @ A ) ) ) ).
% set_update_subsetI
thf(fact_1143_set__update__subsetI,axiom,
! [Xs: list_tm,A: set_tm,X3: tm,I: nat] :
( ( ord_less_eq_set_tm @ ( set_tm2 @ Xs ) @ A )
=> ( ( member_tm @ X3 @ A )
=> ( ord_less_eq_set_tm @ ( set_tm2 @ ( list_update_tm @ Xs @ I @ X3 ) ) @ A ) ) ) ).
% set_update_subsetI
thf(fact_1144_set__update__subsetI,axiom,
! [Xs: list_fm,A: set_fm,X3: fm,I: nat] :
( ( ord_less_eq_set_fm @ ( set_fm2 @ Xs ) @ A )
=> ( ( member_fm @ X3 @ A )
=> ( ord_less_eq_set_fm @ ( set_fm2 @ ( list_update_fm @ Xs @ I @ X3 ) ) @ A ) ) ) ).
% set_update_subsetI
thf(fact_1145_set__update__subsetI,axiom,
! [Xs: list_rule,A: set_rule,X3: rule,I: nat] :
( ( ord_less_eq_set_rule @ ( set_rule2 @ Xs ) @ A )
=> ( ( member_rule @ X3 @ A )
=> ( ord_less_eq_set_rule @ ( set_rule2 @ ( list_update_rule @ Xs @ I @ X3 ) ) @ A ) ) ) ).
% set_update_subsetI
thf(fact_1146_stake__append,axiom,
! [N: nat,U: list_tm,S: stream_tm] :
( ( stake_tm @ N @ ( shift_tm @ U @ S ) )
= ( append_tm @ ( take_tm @ ( ord_min_nat @ ( size_size_list_tm @ U ) @ N ) @ U ) @ ( stake_tm @ ( minus_minus_nat @ N @ ( size_size_list_tm @ U ) ) @ S ) ) ) ).
% stake_append
thf(fact_1147_stake__append,axiom,
! [N: nat,U: list_P2774625669004309958m_rule,S: stream2709947120125613254m_rule] :
( ( stake_5421812949518764133m_rule @ N @ ( shift_2334844276748245581m_rule @ U @ S ) )
= ( append2714808343334685275m_rule @ ( take_P3067526267515409992m_rule @ ( ord_min_nat @ ( size_s1575636608424004698m_rule @ U ) @ N ) @ U ) @ ( stake_5421812949518764133m_rule @ ( minus_minus_nat @ N @ ( size_s1575636608424004698m_rule @ U ) ) @ S ) ) ) ).
% stake_append
thf(fact_1148_set__update__memI,axiom,
! [N: nat,Xs: list_rule,X3: rule] :
( ( ord_less_nat @ N @ ( size_size_list_rule @ Xs ) )
=> ( member_rule @ X3 @ ( set_rule2 @ ( list_update_rule @ Xs @ N @ X3 ) ) ) ) ).
% set_update_memI
thf(fact_1149_set__update__memI,axiom,
! [N: nat,Xs: list_P2887561121880082555ist_fm,X3: produc6018962875968178549ist_fm] :
( ( ord_less_nat @ N @ ( size_s3138477486474831591ist_fm @ Xs ) )
=> ( member4699826688122452638ist_fm @ X3 @ ( set_Pr5051287786238754058ist_fm @ ( list_u2187756355100148707ist_fm @ Xs @ N @ X3 ) ) ) ) ).
% set_update_memI
thf(fact_1150_set__update__memI,axiom,
! [N: nat,Xs: list_P2774625669004309958m_rule,X3: produc340336539035504054m_rule] :
( ( ord_less_nat @ N @ ( size_s1575636608424004698m_rule @ Xs ) )
=> ( member7231649785386036813m_rule @ X3 @ ( set_Pr4534715572506550497m_rule @ ( list_u1216070999260037064m_rule @ Xs @ N @ X3 ) ) ) ) ).
% set_update_memI
thf(fact_1151_set__update__memI,axiom,
! [N: nat,Xs: list_fm,X3: fm] :
( ( ord_less_nat @ N @ ( size_size_list_fm @ Xs ) )
=> ( member_fm @ X3 @ ( set_fm2 @ ( list_update_fm @ Xs @ N @ X3 ) ) ) ) ).
% set_update_memI
thf(fact_1152_set__update__memI,axiom,
! [N: nat,Xs: list_tm,X3: tm] :
( ( ord_less_nat @ N @ ( size_size_list_tm @ Xs ) )
=> ( member_tm @ X3 @ ( set_tm2 @ ( list_update_tm @ Xs @ N @ X3 ) ) ) ) ).
% set_update_memI
thf(fact_1153_set__update__memI,axiom,
! [N: nat,Xs: list_list_fm,X3: list_fm] :
( ( ord_less_nat @ N @ ( size_s115229985653309035ist_fm @ Xs ) )
=> ( member_list_fm @ X3 @ ( set_list_fm2 @ ( list_update_list_fm @ Xs @ N @ X3 ) ) ) ) ).
% set_update_memI
thf(fact_1154_list__update__append1,axiom,
! [I: nat,Xs: list_tm,Ys: list_tm,X3: tm] :
( ( ord_less_nat @ I @ ( size_size_list_tm @ Xs ) )
=> ( ( list_update_tm @ ( append_tm @ Xs @ Ys ) @ I @ X3 )
= ( append_tm @ ( list_update_tm @ Xs @ I @ X3 ) @ Ys ) ) ) ).
% list_update_append1
thf(fact_1155_drop__stake,axiom,
! [N: nat,M2: nat,S: stream2709947120125613254m_rule] :
( ( drop_P5814889834185618160m_rule @ N @ ( stake_5421812949518764133m_rule @ M2 @ S ) )
= ( stake_5421812949518764133m_rule @ ( minus_minus_nat @ M2 @ N ) @ ( sdrop_8169176516188972301m_rule @ N @ S ) ) ) ).
% drop_stake
thf(fact_1156_drop__stake,axiom,
! [N: nat,M2: nat,S: stream_rule] :
( ( drop_rule @ N @ ( stake_rule @ M2 @ S ) )
= ( stake_rule @ ( minus_minus_nat @ M2 @ N ) @ ( sdrop_rule @ N @ S ) ) ) ).
% drop_stake
thf(fact_1157_nth__append,axiom,
! [N: nat,Xs: list_tm,Ys: list_tm] :
( ( ( ord_less_nat @ N @ ( size_size_list_tm @ Xs ) )
=> ( ( nth_tm @ ( append_tm @ Xs @ Ys ) @ N )
= ( nth_tm @ Xs @ N ) ) )
& ( ~ ( ord_less_nat @ N @ ( size_size_list_tm @ Xs ) )
=> ( ( nth_tm @ ( append_tm @ Xs @ Ys ) @ N )
= ( nth_tm @ Ys @ ( minus_minus_nat @ N @ ( size_size_list_tm @ Xs ) ) ) ) ) ) ).
% nth_append
thf(fact_1158_sdrop__shift,axiom,
! [I: nat,W: list_rule,S: stream_rule] :
( ( sdrop_rule @ I @ ( shift_rule @ W @ S ) )
= ( shift_rule @ ( drop_rule @ I @ W ) @ ( sdrop_rule @ ( minus_minus_nat @ I @ ( size_size_list_rule @ W ) ) @ S ) ) ) ).
% sdrop_shift
thf(fact_1159_sdrop__shift,axiom,
! [I: nat,W: list_P2774625669004309958m_rule,S: stream2709947120125613254m_rule] :
( ( sdrop_8169176516188972301m_rule @ I @ ( shift_2334844276748245581m_rule @ W @ S ) )
= ( shift_2334844276748245581m_rule @ ( drop_P5814889834185618160m_rule @ I @ W ) @ ( sdrop_8169176516188972301m_rule @ ( minus_minus_nat @ I @ ( size_s1575636608424004698m_rule @ W ) ) @ S ) ) ) ).
% sdrop_shift
thf(fact_1160_shift__snth,axiom,
! [N: nat,Xs: list_rule,S: stream_rule] :
( ( ( ord_less_nat @ N @ ( size_size_list_rule @ Xs ) )
=> ( ( snth_rule @ ( shift_rule @ Xs @ S ) @ N )
= ( nth_rule @ Xs @ N ) ) )
& ( ~ ( ord_less_nat @ N @ ( size_size_list_rule @ Xs ) )
=> ( ( snth_rule @ ( shift_rule @ Xs @ S ) @ N )
= ( snth_rule @ S @ ( minus_minus_nat @ N @ ( size_size_list_rule @ Xs ) ) ) ) ) ) ).
% shift_snth
thf(fact_1161_shift__snth,axiom,
! [N: nat,Xs: list_P2774625669004309958m_rule,S: stream2709947120125613254m_rule] :
( ( ( ord_less_nat @ N @ ( size_s1575636608424004698m_rule @ Xs ) )
=> ( ( snth_P6679518042731451922m_rule @ ( shift_2334844276748245581m_rule @ Xs @ S ) @ N )
= ( nth_Pr3936752564452695919m_rule @ Xs @ N ) ) )
& ( ~ ( ord_less_nat @ N @ ( size_s1575636608424004698m_rule @ Xs ) )
=> ( ( snth_P6679518042731451922m_rule @ ( shift_2334844276748245581m_rule @ Xs @ S ) @ N )
= ( snth_P6679518042731451922m_rule @ S @ ( minus_minus_nat @ N @ ( size_s1575636608424004698m_rule @ Xs ) ) ) ) ) ) ).
% shift_snth
thf(fact_1162_stake__shift,axiom,
! [I: nat,W: list_tm,S: stream_tm] :
( ( stake_tm @ I @ ( shift_tm @ W @ S ) )
= ( append_tm @ ( take_tm @ I @ W ) @ ( stake_tm @ ( minus_minus_nat @ I @ ( size_size_list_tm @ W ) ) @ S ) ) ) ).
% stake_shift
thf(fact_1163_stake__shift,axiom,
! [I: nat,W: list_P2774625669004309958m_rule,S: stream2709947120125613254m_rule] :
( ( stake_5421812949518764133m_rule @ I @ ( shift_2334844276748245581m_rule @ W @ S ) )
= ( append2714808343334685275m_rule @ ( take_P3067526267515409992m_rule @ I @ W ) @ ( stake_5421812949518764133m_rule @ ( minus_minus_nat @ I @ ( size_s1575636608424004698m_rule @ W ) ) @ S ) ) ) ).
% stake_shift
thf(fact_1164_min__def__raw,axiom,
( ord_min_set_tm
= ( ^ [A5: set_tm,B6: set_tm] : ( if_set_tm @ ( ord_less_eq_set_tm @ A5 @ B6 ) @ A5 @ B6 ) ) ) ).
% min_def_raw
thf(fact_1165_min__def__raw,axiom,
( ord_min_set_fm
= ( ^ [A5: set_fm,B6: set_fm] : ( if_set_fm @ ( ord_less_eq_set_fm @ A5 @ B6 ) @ A5 @ B6 ) ) ) ).
% min_def_raw
thf(fact_1166_min__def__raw,axiom,
( ord_min_nat
= ( ^ [A5: nat,B6: nat] : ( if_nat @ ( ord_less_eq_nat @ A5 @ B6 ) @ A5 @ B6 ) ) ) ).
% min_def_raw
thf(fact_1167_min__def__raw,axiom,
( ord_min_set_rule
= ( ^ [A5: set_rule,B6: set_rule] : ( if_set_rule @ ( ord_less_eq_set_rule @ A5 @ B6 ) @ A5 @ B6 ) ) ) ).
% min_def_raw
thf(fact_1168_fsubsetI,axiom,
! [A: fset_P8989946509869081563ist_fm,B4: fset_P8989946509869081563ist_fm] :
( ! [X: produc6018962875968178549ist_fm] :
( ( fmembe3754813877001230652ist_fm @ X @ A )
=> ( fmembe3754813877001230652ist_fm @ X @ B4 ) )
=> ( ord_le3986950534092794747ist_fm @ A @ B4 ) ) ).
% fsubsetI
thf(fact_1169_fsubsetI,axiom,
! [A: fset_A8849196716589201442m_rule,B4: fset_A8849196716589201442m_rule] :
( ! [X: abstra8520858721168826610m_rule] :
( ( fmembe1099044791951439339m_rule @ X @ A )
=> ( fmembe1099044791951439339m_rule @ X @ B4 ) )
=> ( ord_le59936386403465346m_rule @ A @ B4 ) ) ).
% fsubsetI
thf(fact_1170_length__remdups__leq,axiom,
! [Xs: list_tm] : ( ord_less_eq_nat @ ( size_size_list_tm @ ( remdups_tm @ Xs ) ) @ ( size_size_list_tm @ Xs ) ) ).
% length_remdups_leq
thf(fact_1171_shift__snth__ge,axiom,
! [Xs: list_rule,P2: nat,S: stream_rule] :
( ( ord_less_eq_nat @ ( size_size_list_rule @ Xs ) @ P2 )
=> ( ( snth_rule @ ( shift_rule @ Xs @ S ) @ P2 )
= ( snth_rule @ S @ ( minus_minus_nat @ P2 @ ( size_size_list_rule @ Xs ) ) ) ) ) ).
% shift_snth_ge
thf(fact_1172_shift__snth__ge,axiom,
! [Xs: list_P2774625669004309958m_rule,P2: nat,S: stream2709947120125613254m_rule] :
( ( ord_less_eq_nat @ ( size_s1575636608424004698m_rule @ Xs ) @ P2 )
=> ( ( snth_P6679518042731451922m_rule @ ( shift_2334844276748245581m_rule @ Xs @ S ) @ P2 )
= ( snth_P6679518042731451922m_rule @ S @ ( minus_minus_nat @ P2 @ ( size_s1575636608424004698m_rule @ Xs ) ) ) ) ) ).
% shift_snth_ge
thf(fact_1173_fsubsetD,axiom,
! [A: fset_P8989946509869081563ist_fm,B4: fset_P8989946509869081563ist_fm,C: produc6018962875968178549ist_fm] :
( ( ord_le3986950534092794747ist_fm @ A @ B4 )
=> ( ( fmembe3754813877001230652ist_fm @ C @ A )
=> ( fmembe3754813877001230652ist_fm @ C @ B4 ) ) ) ).
% fsubsetD
thf(fact_1174_fsubsetD,axiom,
! [A: fset_A8849196716589201442m_rule,B4: fset_A8849196716589201442m_rule,C: abstra8520858721168826610m_rule] :
( ( ord_le59936386403465346m_rule @ A @ B4 )
=> ( ( fmembe1099044791951439339m_rule @ C @ A )
=> ( fmembe1099044791951439339m_rule @ C @ B4 ) ) ) ).
% fsubsetD
thf(fact_1175_fin__mono,axiom,
! [A: fset_P8989946509869081563ist_fm,B4: fset_P8989946509869081563ist_fm,X3: produc6018962875968178549ist_fm] :
( ( ord_le3986950534092794747ist_fm @ A @ B4 )
=> ( ( fmembe3754813877001230652ist_fm @ X3 @ A )
=> ( fmembe3754813877001230652ist_fm @ X3 @ B4 ) ) ) ).
% fin_mono
thf(fact_1176_fin__mono,axiom,
! [A: fset_A8849196716589201442m_rule,B4: fset_A8849196716589201442m_rule,X3: abstra8520858721168826610m_rule] :
( ( ord_le59936386403465346m_rule @ A @ B4 )
=> ( ( fmembe1099044791951439339m_rule @ X3 @ A )
=> ( fmembe1099044791951439339m_rule @ X3 @ B4 ) ) ) ).
% fin_mono
thf(fact_1177_list_Opred__mono,axiom,
! [P: produc340336539035504054m_rule > $o,Pa: produc340336539035504054m_rule > $o] :
( ( ord_le2781455385998958777rule_o @ P @ Pa )
=> ( ord_le8241708723349629993rule_o @ ( list_a9038620553111681026m_rule @ P ) @ ( list_a9038620553111681026m_rule @ Pa ) ) ) ).
% list.pred_mono
thf(fact_1178_less__eq__set__def,axiom,
( ord_le1771420097867575541ist_fm
= ( ^ [A6: set_Pr5202636777678657877ist_fm,B8: set_Pr5202636777678657877ist_fm] :
( ord_le1264026566993285160t_fm_o
@ ^ [X4: produc6018962875968178549ist_fm] : ( member4699826688122452638ist_fm @ X4 @ A6 )
@ ^ [X4: produc6018962875968178549ist_fm] : ( member4699826688122452638ist_fm @ X4 @ B8 ) ) ) ) ).
% less_eq_set_def
thf(fact_1179_less__eq__set__def,axiom,
( ord_le6390412330253371084m_rule
= ( ^ [A6: set_Pr1822751329126368876m_rule,B8: set_Pr1822751329126368876m_rule] :
( ord_le2781455385998958777rule_o
@ ^ [X4: produc340336539035504054m_rule] : ( member7231649785386036813m_rule @ X4 @ A6 )
@ ^ [X4: produc340336539035504054m_rule] : ( member7231649785386036813m_rule @ X4 @ B8 ) ) ) ) ).
% less_eq_set_def
thf(fact_1180_less__eq__set__def,axiom,
( ord_le7838213414353715577ist_fm
= ( ^ [A6: set_list_fm,B8: set_list_fm] :
( ord_le6518561683347902116t_fm_o
@ ^ [X4: list_fm] : ( member_list_fm @ X4 @ A6 )
@ ^ [X4: list_fm] : ( member_list_fm @ X4 @ B8 ) ) ) ) ).
% less_eq_set_def
thf(fact_1181_less__eq__set__def,axiom,
( ord_less_eq_set_tm
= ( ^ [A6: set_tm,B8: set_tm] :
( ord_less_eq_tm_o
@ ^ [X4: tm] : ( member_tm @ X4 @ A6 )
@ ^ [X4: tm] : ( member_tm @ X4 @ B8 ) ) ) ) ).
% less_eq_set_def
thf(fact_1182_less__eq__set__def,axiom,
( ord_less_eq_set_fm
= ( ^ [A6: set_fm,B8: set_fm] :
( ord_less_eq_fm_o
@ ^ [X4: fm] : ( member_fm @ X4 @ A6 )
@ ^ [X4: fm] : ( member_fm @ X4 @ B8 ) ) ) ) ).
% less_eq_set_def
thf(fact_1183_less__eq__set__def,axiom,
( ord_less_eq_set_rule
= ( ^ [A6: set_rule,B8: set_rule] :
( ord_less_eq_rule_o
@ ^ [X4: rule] : ( member_rule @ X4 @ A6 )
@ ^ [X4: rule] : ( member_rule @ X4 @ B8 ) ) ) ) ).
% less_eq_set_def
thf(fact_1184_pred__subset__eq,axiom,
! [R3: set_Pr5202636777678657877ist_fm,S6: set_Pr5202636777678657877ist_fm] :
( ( ord_le1264026566993285160t_fm_o
@ ^ [X4: produc6018962875968178549ist_fm] : ( member4699826688122452638ist_fm @ X4 @ R3 )
@ ^ [X4: produc6018962875968178549ist_fm] : ( member4699826688122452638ist_fm @ X4 @ S6 ) )
= ( ord_le1771420097867575541ist_fm @ R3 @ S6 ) ) ).
% pred_subset_eq
thf(fact_1185_pred__subset__eq,axiom,
! [R3: set_Pr1822751329126368876m_rule,S6: set_Pr1822751329126368876m_rule] :
( ( ord_le2781455385998958777rule_o
@ ^ [X4: produc340336539035504054m_rule] : ( member7231649785386036813m_rule @ X4 @ R3 )
@ ^ [X4: produc340336539035504054m_rule] : ( member7231649785386036813m_rule @ X4 @ S6 ) )
= ( ord_le6390412330253371084m_rule @ R3 @ S6 ) ) ).
% pred_subset_eq
thf(fact_1186_pred__subset__eq,axiom,
! [R3: set_list_fm,S6: set_list_fm] :
( ( ord_le6518561683347902116t_fm_o
@ ^ [X4: list_fm] : ( member_list_fm @ X4 @ R3 )
@ ^ [X4: list_fm] : ( member_list_fm @ X4 @ S6 ) )
= ( ord_le7838213414353715577ist_fm @ R3 @ S6 ) ) ).
% pred_subset_eq
thf(fact_1187_pred__subset__eq,axiom,
! [R3: set_tm,S6: set_tm] :
( ( ord_less_eq_tm_o
@ ^ [X4: tm] : ( member_tm @ X4 @ R3 )
@ ^ [X4: tm] : ( member_tm @ X4 @ S6 ) )
= ( ord_less_eq_set_tm @ R3 @ S6 ) ) ).
% pred_subset_eq
thf(fact_1188_pred__subset__eq,axiom,
! [R3: set_fm,S6: set_fm] :
( ( ord_less_eq_fm_o
@ ^ [X4: fm] : ( member_fm @ X4 @ R3 )
@ ^ [X4: fm] : ( member_fm @ X4 @ S6 ) )
= ( ord_less_eq_set_fm @ R3 @ S6 ) ) ).
% pred_subset_eq
thf(fact_1189_pred__subset__eq,axiom,
! [R3: set_rule,S6: set_rule] :
( ( ord_less_eq_rule_o
@ ^ [X4: rule] : ( member_rule @ X4 @ R3 )
@ ^ [X4: rule] : ( member_rule @ X4 @ S6 ) )
= ( ord_less_eq_set_rule @ R3 @ S6 ) ) ).
% pred_subset_eq
thf(fact_1190_pred__subset__eq2,axiom,
! [R3: set_Pr5202636777678657877ist_fm,S6: set_Pr5202636777678657877ist_fm] :
( ( ord_le7793033639944360256t_fm_o
@ ^ [X4: list_tm,Y5: list_fm] : ( member4699826688122452638ist_fm @ ( produc1414352766439514085ist_fm @ X4 @ Y5 ) @ R3 )
@ ^ [X4: list_tm,Y5: list_fm] : ( member4699826688122452638ist_fm @ ( produc1414352766439514085ist_fm @ X4 @ Y5 ) @ S6 ) )
= ( ord_le1771420097867575541ist_fm @ R3 @ S6 ) ) ).
% pred_subset_eq2
thf(fact_1191_pred__subset__eq2,axiom,
! [R3: set_Pr1822751329126368876m_rule,S6: set_Pr1822751329126368876m_rule] :
( ( ord_le2289920624603415501rule_o
@ ^ [X4: produc6018962875968178549ist_fm,Y5: rule] : ( member7231649785386036813m_rule @ ( produc1733806532565653680m_rule @ X4 @ Y5 ) @ R3 )
@ ^ [X4: produc6018962875968178549ist_fm,Y5: rule] : ( member7231649785386036813m_rule @ ( produc1733806532565653680m_rule @ X4 @ Y5 ) @ S6 ) )
= ( ord_le6390412330253371084m_rule @ R3 @ S6 ) ) ).
% pred_subset_eq2
thf(fact_1192_pred__subset__eq2,axiom,
! [R3: set_Pr1261947904930325089at_nat,S6: set_Pr1261947904930325089at_nat] :
( ( ord_le2646555220125990790_nat_o
@ ^ [X4: nat,Y5: nat] : ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X4 @ Y5 ) @ R3 )
@ ^ [X4: nat,Y5: nat] : ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X4 @ Y5 ) @ S6 ) )
= ( ord_le3146513528884898305at_nat @ R3 @ S6 ) ) ).
% pred_subset_eq2
thf(fact_1193_set__take__subset__set__take,axiom,
! [M2: nat,N: nat,Xs: list_list_fm] :
( ( ord_less_eq_nat @ M2 @ N )
=> ( ord_le7838213414353715577ist_fm @ ( set_list_fm2 @ ( take_list_fm @ M2 @ Xs ) ) @ ( set_list_fm2 @ ( take_list_fm @ N @ Xs ) ) ) ) ).
% set_take_subset_set_take
thf(fact_1194_set__take__subset__set__take,axiom,
! [M2: nat,N: nat,Xs: list_tm] :
( ( ord_less_eq_nat @ M2 @ N )
=> ( ord_less_eq_set_tm @ ( set_tm2 @ ( take_tm @ M2 @ Xs ) ) @ ( set_tm2 @ ( take_tm @ N @ Xs ) ) ) ) ).
% set_take_subset_set_take
thf(fact_1195_set__take__subset__set__take,axiom,
! [M2: nat,N: nat,Xs: list_fm] :
( ( ord_less_eq_nat @ M2 @ N )
=> ( ord_less_eq_set_fm @ ( set_fm2 @ ( take_fm @ M2 @ Xs ) ) @ ( set_fm2 @ ( take_fm @ N @ Xs ) ) ) ) ).
% set_take_subset_set_take
thf(fact_1196_set__take__subset__set__take,axiom,
! [M2: nat,N: nat,Xs: list_rule] :
( ( ord_less_eq_nat @ M2 @ N )
=> ( ord_less_eq_set_rule @ ( set_rule2 @ ( take_rule @ M2 @ Xs ) ) @ ( set_rule2 @ ( take_rule @ N @ Xs ) ) ) ) ).
% set_take_subset_set_take
thf(fact_1197_set__drop__subset__set__drop,axiom,
! [N: nat,M2: nat,Xs: list_list_fm] :
( ( ord_less_eq_nat @ N @ M2 )
=> ( ord_le7838213414353715577ist_fm @ ( set_list_fm2 @ ( drop_list_fm @ M2 @ Xs ) ) @ ( set_list_fm2 @ ( drop_list_fm @ N @ Xs ) ) ) ) ).
% set_drop_subset_set_drop
thf(fact_1198_set__drop__subset__set__drop,axiom,
! [N: nat,M2: nat,Xs: list_tm] :
( ( ord_less_eq_nat @ N @ M2 )
=> ( ord_less_eq_set_tm @ ( set_tm2 @ ( drop_tm @ M2 @ Xs ) ) @ ( set_tm2 @ ( drop_tm @ N @ Xs ) ) ) ) ).
% set_drop_subset_set_drop
thf(fact_1199_set__drop__subset__set__drop,axiom,
! [N: nat,M2: nat,Xs: list_fm] :
( ( ord_less_eq_nat @ N @ M2 )
=> ( ord_less_eq_set_fm @ ( set_fm2 @ ( drop_fm @ M2 @ Xs ) ) @ ( set_fm2 @ ( drop_fm @ N @ Xs ) ) ) ) ).
% set_drop_subset_set_drop
thf(fact_1200_set__drop__subset__set__drop,axiom,
! [N: nat,M2: nat,Xs: list_rule] :
( ( ord_less_eq_nat @ N @ M2 )
=> ( ord_less_eq_set_rule @ ( set_rule2 @ ( drop_rule @ M2 @ Xs ) ) @ ( set_rule2 @ ( drop_rule @ N @ Xs ) ) ) ) ).
% set_drop_subset_set_drop
thf(fact_1201_sdrop__wait__least,axiom,
! [Phi: stream2709947120125613254m_rule > $o,Xs: stream2709947120125613254m_rule,N: nat] :
( ( linear1816559245158234865m_rule @ Phi @ Xs )
=> ( ( Phi @ ( sdrop_8169176516188972301m_rule @ N @ Xs ) )
=> ( ord_less_eq_nat @ ( linear8417479883454578709m_rule @ Phi @ Xs ) @ N ) ) ) ).
% sdrop_wait_least
thf(fact_1202_sdrop__wait__least,axiom,
! [Phi: stream_rule > $o,Xs: stream_rule,N: nat] :
( ( linear292482573288640205v_rule @ Phi @ Xs )
=> ( ( Phi @ ( sdrop_rule @ N @ Xs ) )
=> ( ord_less_eq_nat @ ( linear2710171048243638257t_rule @ Phi @ Xs ) @ N ) ) ) ).
% sdrop_wait_least
thf(fact_1203_append__eq__append__conv__if,axiom,
! [Xs_1: list_tm,Xs_2: list_tm,Ys_1: list_tm,Ys_2: list_tm] :
( ( ( append_tm @ Xs_1 @ Xs_2 )
= ( append_tm @ Ys_1 @ Ys_2 ) )
= ( ( ( ord_less_eq_nat @ ( size_size_list_tm @ Xs_1 ) @ ( size_size_list_tm @ Ys_1 ) )
=> ( ( Xs_1
= ( take_tm @ ( size_size_list_tm @ Xs_1 ) @ Ys_1 ) )
& ( Xs_2
= ( append_tm @ ( drop_tm @ ( size_size_list_tm @ Xs_1 ) @ Ys_1 ) @ Ys_2 ) ) ) )
& ( ~ ( ord_less_eq_nat @ ( size_size_list_tm @ Xs_1 ) @ ( size_size_list_tm @ Ys_1 ) )
=> ( ( ( take_tm @ ( size_size_list_tm @ Ys_1 ) @ Xs_1 )
= Ys_1 )
& ( ( append_tm @ ( drop_tm @ ( size_size_list_tm @ Ys_1 ) @ Xs_1 ) @ Xs_2 )
= Ys_2 ) ) ) ) ) ).
% append_eq_append_conv_if
thf(fact_1204_lexord__sufI,axiom,
! [U: list_tm,W: list_tm,R: set_Pr2455929065695642951_tm_tm,V: list_tm,Z: list_tm] :
( ( member4457312017796942864ist_tm @ ( produc1418304791525149271ist_tm @ U @ W ) @ ( lexord_tm @ R ) )
=> ( ( ord_less_eq_nat @ ( size_size_list_tm @ W ) @ ( size_size_list_tm @ U ) )
=> ( member4457312017796942864ist_tm @ ( produc1418304791525149271ist_tm @ ( append_tm @ U @ V ) @ ( append_tm @ W @ Z ) ) @ ( lexord_tm @ R ) ) ) ) ).
% lexord_sufI
thf(fact_1205_measures__lesseq,axiom,
! [F: nat > nat,X3: nat,Y3: nat,Fs: list_nat_nat] :
( ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y3 ) )
=> ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X3 @ Y3 ) @ ( measures_nat @ Fs ) )
=> ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X3 @ Y3 ) @ ( measures_nat @ ( cons_nat_nat @ F @ Fs ) ) ) ) ) ).
% measures_lesseq
thf(fact_1206_nth__equal__first__eq,axiom,
! [X3: rule,Xs: list_rule,N: nat] :
( ~ ( member_rule @ X3 @ ( set_rule2 @ Xs ) )
=> ( ( ord_less_eq_nat @ N @ ( size_size_list_rule @ Xs ) )
=> ( ( ( nth_rule @ ( cons_rule @ X3 @ Xs ) @ N )
= X3 )
= ( N = zero_zero_nat ) ) ) ) ).
% nth_equal_first_eq
thf(fact_1207_nth__equal__first__eq,axiom,
! [X3: produc6018962875968178549ist_fm,Xs: list_P2887561121880082555ist_fm,N: nat] :
( ~ ( member4699826688122452638ist_fm @ X3 @ ( set_Pr5051287786238754058ist_fm @ Xs ) )
=> ( ( ord_less_eq_nat @ N @ ( size_s3138477486474831591ist_fm @ Xs ) )
=> ( ( ( nth_Pr580027083122244092ist_fm @ ( cons_P6586734674047340843ist_fm @ X3 @ Xs ) @ N )
= X3 )
= ( N = zero_zero_nat ) ) ) ) ).
% nth_equal_first_eq
thf(fact_1208_nth__equal__first__eq,axiom,
! [X3: produc340336539035504054m_rule,Xs: list_P2774625669004309958m_rule,N: nat] :
( ~ ( member7231649785386036813m_rule @ X3 @ ( set_Pr4534715572506550497m_rule @ Xs ) )
=> ( ( ord_less_eq_nat @ N @ ( size_s1575636608424004698m_rule @ Xs ) )
=> ( ( ( nth_Pr3936752564452695919m_rule @ ( cons_P2818572562524668032m_rule @ X3 @ Xs ) @ N )
= X3 )
= ( N = zero_zero_nat ) ) ) ) ).
% nth_equal_first_eq
thf(fact_1209_nth__equal__first__eq,axiom,
! [X3: fm,Xs: list_fm,N: nat] :
( ~ ( member_fm @ X3 @ ( set_fm2 @ Xs ) )
=> ( ( ord_less_eq_nat @ N @ ( size_size_list_fm @ Xs ) )
=> ( ( ( nth_fm @ ( cons_fm @ X3 @ Xs ) @ N )
= X3 )
= ( N = zero_zero_nat ) ) ) ) ).
% nth_equal_first_eq
thf(fact_1210_nth__equal__first__eq,axiom,
! [X3: tm,Xs: list_tm,N: nat] :
( ~ ( member_tm @ X3 @ ( set_tm2 @ Xs ) )
=> ( ( ord_less_eq_nat @ N @ ( size_size_list_tm @ Xs ) )
=> ( ( ( nth_tm @ ( cons_tm @ X3 @ Xs ) @ N )
= X3 )
= ( N = zero_zero_nat ) ) ) ) ).
% nth_equal_first_eq
thf(fact_1211_nth__equal__first__eq,axiom,
! [X3: list_fm,Xs: list_list_fm,N: nat] :
( ~ ( member_list_fm @ X3 @ ( set_list_fm2 @ Xs ) )
=> ( ( ord_less_eq_nat @ N @ ( size_s115229985653309035ist_fm @ Xs ) )
=> ( ( ( nth_list_fm @ ( cons_list_fm @ X3 @ Xs ) @ N )
= X3 )
= ( N = zero_zero_nat ) ) ) ) ).
% nth_equal_first_eq
thf(fact_1212_RuleSystem_Opos__least,axiom,
! [Eff: rule > produc6018962875968178549ist_fm > fset_P8989946509869081563ist_fm > $o,Rules: stream_rule,S6: set_Pr5202636777678657877ist_fm,N: nat,Rs: stream2709947120125613254m_rule,R: produc340336539035504054m_rule] :
( ( abstra5221733350967904376ist_fm @ Eff @ Rules @ S6 )
=> ( ( ( shd_Pr4562317740776619530m_rule @ ( sdrop_8169176516188972301m_rule @ N @ Rs ) )
= R )
=> ( ord_less_eq_nat @ ( abstra4499547390127564210m_rule @ Rs @ R ) @ N ) ) ) ).
% RuleSystem.pos_least
thf(fact_1213_RuleSystem_Opos__least,axiom,
! [Eff: rule > produc6018962875968178549ist_fm > fset_P8989946509869081563ist_fm > $o,Rules: stream_rule,S6: set_Pr5202636777678657877ist_fm,N: nat,Rs: stream_rule,R: rule] :
( ( abstra5221733350967904376ist_fm @ Eff @ Rules @ S6 )
=> ( ( ( shd_rule @ ( sdrop_rule @ N @ Rs ) )
= R )
=> ( ord_less_eq_nat @ ( abstract_pos_rule @ Rs @ R ) @ N ) ) ) ).
% RuleSystem.pos_least
thf(fact_1214_fMin_Osemilattice__order__fset__axioms,axiom,
semila6541227223090250811et_nat @ ord_min_nat @ ord_less_eq_nat @ ord_less_nat ).
% fMin.semilattice_order_fset_axioms
thf(fact_1215_pos__least,axiom,
! [N: nat,Rs: stream2709947120125613254m_rule,R: produc340336539035504054m_rule] :
( ( ( shd_Pr4562317740776619530m_rule @ ( sdrop_8169176516188972301m_rule @ N @ Rs ) )
= R )
=> ( ord_less_eq_nat @ ( abstra4499547390127564210m_rule @ Rs @ R ) @ N ) ) ).
% pos_least
thf(fact_1216_pos__least,axiom,
! [N: nat,Rs: stream_rule,R: rule] :
( ( ( shd_rule @ ( sdrop_rule @ N @ Rs ) )
= R )
=> ( ord_less_eq_nat @ ( abstract_pos_rule @ Rs @ R ) @ N ) ) ).
% pos_least
thf(fact_1217_RuleSystem_Opos,axiom,
! [Eff: rule > produc6018962875968178549ist_fm > fset_P8989946509869081563ist_fm > $o,Rules: stream_rule,S6: set_Pr5202636777678657877ist_fm,Rs: stream_rule,R: rule] :
( ( abstra5221733350967904376ist_fm @ Eff @ Rules @ S6 )
=> ( ( abstra3799686578551160190r_rule @ Rules @ Rs )
=> ( ( member_rule @ R @ ( sset_rule @ Rules ) )
=> ( ( shd_rule @ ( sdrop_rule @ ( abstract_pos_rule @ Rs @ R ) @ Rs ) )
= R ) ) ) ) ).
% RuleSystem.pos
thf(fact_1218_butlast__snoc,axiom,
! [Xs: list_tm,X3: tm] :
( ( butlast_tm @ ( append_tm @ Xs @ ( cons_tm @ X3 @ nil_tm ) ) )
= Xs ) ).
% butlast_snoc
thf(fact_1219_fair__stl,axiom,
! [Rs: stream_rule] :
( ( abstra3799686578551160190r_rule @ rules @ Rs )
=> ( abstra3799686578551160190r_rule @ rules @ ( stl_rule @ Rs ) ) ) ).
% fair_stl
thf(fact_1220_fair__rules,axiom,
abstra3799686578551160190r_rule @ rules @ rules ).
% fair_rules
thf(fact_1221_in__set__butlastD,axiom,
! [X3: rule,Xs: list_rule] :
( ( member_rule @ X3 @ ( set_rule2 @ ( butlast_rule @ Xs ) ) )
=> ( member_rule @ X3 @ ( set_rule2 @ Xs ) ) ) ).
% in_set_butlastD
thf(fact_1222_in__set__butlastD,axiom,
! [X3: produc6018962875968178549ist_fm,Xs: list_P2887561121880082555ist_fm] :
( ( member4699826688122452638ist_fm @ X3 @ ( set_Pr5051287786238754058ist_fm @ ( butlas6635528968590783175ist_fm @ Xs ) ) )
=> ( member4699826688122452638ist_fm @ X3 @ ( set_Pr5051287786238754058ist_fm @ Xs ) ) ) ).
% in_set_butlastD
thf(fact_1223_in__set__butlastD,axiom,
! [X3: produc340336539035504054m_rule,Xs: list_P2774625669004309958m_rule] :
( ( member7231649785386036813m_rule @ X3 @ ( set_Pr4534715572506550497m_rule @ ( butlas8786434687291705316m_rule @ Xs ) ) )
=> ( member7231649785386036813m_rule @ X3 @ ( set_Pr4534715572506550497m_rule @ Xs ) ) ) ).
% in_set_butlastD
thf(fact_1224_in__set__butlastD,axiom,
! [X3: fm,Xs: list_fm] :
( ( member_fm @ X3 @ ( set_fm2 @ ( butlast_fm @ Xs ) ) )
=> ( member_fm @ X3 @ ( set_fm2 @ Xs ) ) ) ).
% in_set_butlastD
thf(fact_1225_in__set__butlastD,axiom,
! [X3: tm,Xs: list_tm] :
( ( member_tm @ X3 @ ( set_tm2 @ ( butlast_tm @ Xs ) ) )
=> ( member_tm @ X3 @ ( set_tm2 @ Xs ) ) ) ).
% in_set_butlastD
thf(fact_1226_in__set__butlastD,axiom,
! [X3: list_fm,Xs: list_list_fm] :
( ( member_list_fm @ X3 @ ( set_list_fm2 @ ( butlast_list_fm @ Xs ) ) )
=> ( member_list_fm @ X3 @ ( set_list_fm2 @ Xs ) ) ) ).
% in_set_butlastD
thf(fact_1227_RuleSystem__Defs_Ofair_Ocong,axiom,
abstra3799686578551160190r_rule = abstra3799686578551160190r_rule ).
% RuleSystem_Defs.fair.cong
thf(fact_1228_i_Ofair__stl,axiom,
! [Rules: stream2709947120125613254m_rule,Rs: stream2709947120125613254m_rule] :
( ( abstra3665372904625986210m_rule @ Rules @ Rs )
=> ( abstra3665372904625986210m_rule @ Rules @ ( stl_Pr12655793849929990m_rule @ Rs ) ) ) ).
% i.fair_stl
thf(fact_1229_i_Ofair__stl,axiom,
! [Rules: stream_rule,Rs: stream_rule] :
( ( abstra3799686578551160190r_rule @ Rules @ Rs )
=> ( abstra3799686578551160190r_rule @ Rules @ ( stl_rule @ Rs ) ) ) ).
% i.fair_stl
thf(fact_1230_RuleSystem__Defs_Ofair__stl,axiom,
! [Rules: stream2709947120125613254m_rule,Rs: stream2709947120125613254m_rule] :
( ( abstra3665372904625986210m_rule @ Rules @ Rs )
=> ( abstra3665372904625986210m_rule @ Rules @ ( stl_Pr12655793849929990m_rule @ Rs ) ) ) ).
% RuleSystem_Defs.fair_stl
thf(fact_1231_RuleSystem__Defs_Ofair__stl,axiom,
! [Rules: stream_rule,Rs: stream_rule] :
( ( abstra3799686578551160190r_rule @ Rules @ Rs )
=> ( abstra3799686578551160190r_rule @ Rules @ ( stl_rule @ Rs ) ) ) ).
% RuleSystem_Defs.fair_stl
thf(fact_1232_RuleSystem__Defs_Osdrop__fair,axiom,
! [Rules: stream2709947120125613254m_rule,Rs: stream2709947120125613254m_rule,M2: nat] :
( ( abstra3665372904625986210m_rule @ Rules @ Rs )
=> ( abstra3665372904625986210m_rule @ Rules @ ( sdrop_8169176516188972301m_rule @ M2 @ Rs ) ) ) ).
% RuleSystem_Defs.sdrop_fair
thf(fact_1233_RuleSystem__Defs_Osdrop__fair,axiom,
! [Rules: stream_rule,Rs: stream_rule,M2: nat] :
( ( abstra3799686578551160190r_rule @ Rules @ Rs )
=> ( abstra3799686578551160190r_rule @ Rules @ ( sdrop_rule @ M2 @ Rs ) ) ) ).
% RuleSystem_Defs.sdrop_fair
thf(fact_1234_i_Osdrop__fair,axiom,
! [Rules: stream2709947120125613254m_rule,Rs: stream2709947120125613254m_rule,M2: nat] :
( ( abstra3665372904625986210m_rule @ Rules @ Rs )
=> ( abstra3665372904625986210m_rule @ Rules @ ( sdrop_8169176516188972301m_rule @ M2 @ Rs ) ) ) ).
% i.sdrop_fair
thf(fact_1235_i_Osdrop__fair,axiom,
! [Rules: stream_rule,Rs: stream_rule,M2: nat] :
( ( abstra3799686578551160190r_rule @ Rules @ Rs )
=> ( abstra3799686578551160190r_rule @ Rules @ ( sdrop_rule @ M2 @ Rs ) ) ) ).
% i.sdrop_fair
thf(fact_1236_butlast__append,axiom,
! [Ys: list_tm,Xs: list_tm] :
( ( ( Ys = nil_tm )
=> ( ( butlast_tm @ ( append_tm @ Xs @ Ys ) )
= ( butlast_tm @ Xs ) ) )
& ( ( Ys != nil_tm )
=> ( ( butlast_tm @ ( append_tm @ Xs @ Ys ) )
= ( append_tm @ Xs @ ( butlast_tm @ Ys ) ) ) ) ) ).
% butlast_append
thf(fact_1237_in__set__butlast__appendI,axiom,
! [X3: rule,Xs: list_rule,Ys: list_rule] :
( ( ( member_rule @ X3 @ ( set_rule2 @ ( butlast_rule @ Xs ) ) )
| ( member_rule @ X3 @ ( set_rule2 @ ( butlast_rule @ Ys ) ) ) )
=> ( member_rule @ X3 @ ( set_rule2 @ ( butlast_rule @ ( append_rule @ Xs @ Ys ) ) ) ) ) ).
% in_set_butlast_appendI
thf(fact_1238_in__set__butlast__appendI,axiom,
! [X3: produc6018962875968178549ist_fm,Xs: list_P2887561121880082555ist_fm,Ys: list_P2887561121880082555ist_fm] :
( ( ( member4699826688122452638ist_fm @ X3 @ ( set_Pr5051287786238754058ist_fm @ ( butlas6635528968590783175ist_fm @ Xs ) ) )
| ( member4699826688122452638ist_fm @ X3 @ ( set_Pr5051287786238754058ist_fm @ ( butlas6635528968590783175ist_fm @ Ys ) ) ) )
=> ( member4699826688122452638ist_fm @ X3 @ ( set_Pr5051287786238754058ist_fm @ ( butlas6635528968590783175ist_fm @ ( append1476581234975291920ist_fm @ Xs @ Ys ) ) ) ) ) ).
% in_set_butlast_appendI
thf(fact_1239_in__set__butlast__appendI,axiom,
! [X3: produc340336539035504054m_rule,Xs: list_P2774625669004309958m_rule,Ys: list_P2774625669004309958m_rule] :
( ( ( member7231649785386036813m_rule @ X3 @ ( set_Pr4534715572506550497m_rule @ ( butlas8786434687291705316m_rule @ Xs ) ) )
| ( member7231649785386036813m_rule @ X3 @ ( set_Pr4534715572506550497m_rule @ ( butlas8786434687291705316m_rule @ Ys ) ) ) )
=> ( member7231649785386036813m_rule @ X3 @ ( set_Pr4534715572506550497m_rule @ ( butlas8786434687291705316m_rule @ ( append2714808343334685275m_rule @ Xs @ Ys ) ) ) ) ) ).
% in_set_butlast_appendI
thf(fact_1240_in__set__butlast__appendI,axiom,
! [X3: fm,Xs: list_fm,Ys: list_fm] :
( ( ( member_fm @ X3 @ ( set_fm2 @ ( butlast_fm @ Xs ) ) )
| ( member_fm @ X3 @ ( set_fm2 @ ( butlast_fm @ Ys ) ) ) )
=> ( member_fm @ X3 @ ( set_fm2 @ ( butlast_fm @ ( append_fm @ Xs @ Ys ) ) ) ) ) ).
% in_set_butlast_appendI
thf(fact_1241_in__set__butlast__appendI,axiom,
! [X3: tm,Xs: list_tm,Ys: list_tm] :
( ( ( member_tm @ X3 @ ( set_tm2 @ ( butlast_tm @ Xs ) ) )
| ( member_tm @ X3 @ ( set_tm2 @ ( butlast_tm @ Ys ) ) ) )
=> ( member_tm @ X3 @ ( set_tm2 @ ( butlast_tm @ ( append_tm @ Xs @ Ys ) ) ) ) ) ).
% in_set_butlast_appendI
thf(fact_1242_in__set__butlast__appendI,axiom,
! [X3: list_fm,Xs: list_list_fm,Ys: list_list_fm] :
( ( ( member_list_fm @ X3 @ ( set_list_fm2 @ ( butlast_list_fm @ Xs ) ) )
| ( member_list_fm @ X3 @ ( set_list_fm2 @ ( butlast_list_fm @ Ys ) ) ) )
=> ( member_list_fm @ X3 @ ( set_list_fm2 @ ( butlast_list_fm @ ( append_list_fm @ Xs @ Ys ) ) ) ) ) ).
% in_set_butlast_appendI
thf(fact_1243_sdrop__fair,axiom,
! [Rs: stream_rule,M2: nat] :
( ( abstra3799686578551160190r_rule @ rules @ Rs )
=> ( abstra3799686578551160190r_rule @ rules @ ( sdrop_rule @ M2 @ Rs ) ) ) ).
% sdrop_fair
thf(fact_1244_pos,axiom,
! [Rs: stream_rule,R: rule] :
( ( abstra3799686578551160190r_rule @ rules @ Rs )
=> ( ( member_rule @ R @ ( sset_rule @ rules ) )
=> ( ( shd_rule @ ( sdrop_rule @ ( abstract_pos_rule @ Rs @ R ) @ Rs ) )
= R ) ) ) ).
% pos
thf(fact_1245_fair__uu__def,axiom,
abstra4598247580560492629u_rule = abstra3799686578551160190r_rule ).
% fair_uu_def
thf(fact_1246_RuleSystem_Ostake__pos__minWait,axiom,
! [Eff: rule > fm > fset_fm > $o,Rules: stream_rule,S6: set_fm,Rs: stream_rule,S: fm,R: rule] :
( ( abstra4909020524819817846ule_fm @ Eff @ Rules @ S6 )
=> ( ( abstra3799686578551160190r_rule @ Rules @ Rs )
=> ( ( ord_less_nat @ ( abstra3486989597108156081ule_fm @ Eff @ Rs @ S ) @ ( abstract_pos_rule @ Rs @ R ) )
=> ( ( member_rule @ R @ ( sset_rule @ Rules ) )
=> ( ( member_fm @ S @ S6 )
=> ( ( abstract_pos_rule @ ( stl_rule @ ( abstra4814164703916120982ule_fm @ Eff @ Rs @ S ) ) @ R )
= ( minus_minus_nat @ ( abstract_pos_rule @ Rs @ R ) @ ( suc @ ( abstra3486989597108156081ule_fm @ Eff @ Rs @ S ) ) ) ) ) ) ) ) ) ).
% RuleSystem.stake_pos_minWait
thf(fact_1247_RuleSystem_Ostake__pos__minWait,axiom,
! [Eff: rule > produc340336539035504054m_rule > fset_P5628638355292684902m_rule > $o,Rules: stream_rule,S6: set_Pr1822751329126368876m_rule,Rs: stream_rule,S: produc340336539035504054m_rule,R: rule] :
( ( abstra3057654119625851379m_rule @ Eff @ Rules @ S6 )
=> ( ( abstra3799686578551160190r_rule @ Rules @ Rs )
=> ( ( ord_less_nat @ ( abstra5386990316169122296m_rule @ Eff @ Rs @ S ) @ ( abstract_pos_rule @ Rs @ R ) )
=> ( ( member_rule @ R @ ( sset_rule @ Rules ) )
=> ( ( member7231649785386036813m_rule @ S @ S6 )
=> ( ( abstract_pos_rule @ ( stl_rule @ ( abstra1608994533123245011m_rule @ Eff @ Rs @ S ) ) @ R )
= ( minus_minus_nat @ ( abstract_pos_rule @ Rs @ R ) @ ( suc @ ( abstra5386990316169122296m_rule @ Eff @ Rs @ S ) ) ) ) ) ) ) ) ) ).
% RuleSystem.stake_pos_minWait
thf(fact_1248_RuleSystem_Ostake__pos__minWait,axiom,
! [Eff: rule > list_fm > fset_list_fm > $o,Rules: stream_rule,S6: set_list_fm,Rs: stream_rule,S: list_fm,R: rule] :
( ( abstra9153614075763688444ist_fm @ Eff @ Rules @ S6 )
=> ( ( abstra3799686578551160190r_rule @ Rules @ Rs )
=> ( ( ord_less_nat @ ( abstra101201851149073335ist_fm @ Eff @ Rs @ S ) @ ( abstract_pos_rule @ Rs @ R ) )
=> ( ( member_rule @ R @ ( sset_rule @ Rules ) )
=> ( ( member_list_fm @ S @ S6 )
=> ( ( abstract_pos_rule @ ( stl_rule @ ( abstra593219384675117596ist_fm @ Eff @ Rs @ S ) ) @ R )
= ( minus_minus_nat @ ( abstract_pos_rule @ Rs @ R ) @ ( suc @ ( abstra101201851149073335ist_fm @ Eff @ Rs @ S ) ) ) ) ) ) ) ) ) ).
% RuleSystem.stake_pos_minWait
thf(fact_1249_RuleSystem_Ostake__pos__minWait,axiom,
! [Eff: rule > produc6018962875968178549ist_fm > fset_P8989946509869081563ist_fm > $o,Rules: stream_rule,S6: set_Pr5202636777678657877ist_fm,Rs: stream_rule,S: produc6018962875968178549ist_fm,R: rule] :
( ( abstra5221733350967904376ist_fm @ Eff @ Rules @ S6 )
=> ( ( abstra3799686578551160190r_rule @ Rules @ Rs )
=> ( ( ord_less_nat @ ( abstra1963046427795717555ist_fm @ Eff @ Rs @ S ) @ ( abstract_pos_rule @ Rs @ R ) )
=> ( ( member_rule @ R @ ( sset_rule @ Rules ) )
=> ( ( member4699826688122452638ist_fm @ S @ S6 )
=> ( ( abstract_pos_rule @ ( stl_rule @ ( abstra7806236140005899928ist_fm @ Eff @ Rs @ S ) ) @ R )
= ( minus_minus_nat @ ( abstract_pos_rule @ Rs @ R ) @ ( suc @ ( abstra1963046427795717555ist_fm @ Eff @ Rs @ S ) ) ) ) ) ) ) ) ) ).
% RuleSystem.stake_pos_minWait
thf(fact_1250_trim__alt,axiom,
! [S: produc6018962875968178549ist_fm,Rs: stream_rule] :
( ( member4699826688122452638ist_fm @ S @ top_to1730629564055774885ist_fm )
=> ( ( abstra3799686578551160190r_rule @ rules @ Rs )
=> ( ( abstra7806236140005899928ist_fm @ eff @ Rs @ S )
= ( sdrop_rule @ ( abstra1963046427795717555ist_fm @ eff @ Rs @ S ) @ Rs ) ) ) ) ).
% trim_alt
thf(fact_1251_trim__fair,axiom,
! [S: produc6018962875968178549ist_fm,Rs: stream_rule] :
( ( member4699826688122452638ist_fm @ S @ top_to1730629564055774885ist_fm )
=> ( ( abstra3799686578551160190r_rule @ rules @ Rs )
=> ( abstra3799686578551160190r_rule @ rules @ ( abstra7806236140005899928ist_fm @ eff @ Rs @ S ) ) ) ) ).
% trim_fair
thf(fact_1252_stake__pos__minWait,axiom,
! [Rs: stream_rule,S: produc6018962875968178549ist_fm,R: rule] :
( ( abstra3799686578551160190r_rule @ rules @ Rs )
=> ( ( ord_less_nat @ ( abstra1963046427795717555ist_fm @ eff @ Rs @ S ) @ ( abstract_pos_rule @ Rs @ R ) )
=> ( ( member_rule @ R @ ( sset_rule @ rules ) )
=> ( ( member4699826688122452638ist_fm @ S @ top_to1730629564055774885ist_fm )
=> ( ( abstract_pos_rule @ ( stl_rule @ ( abstra7806236140005899928ist_fm @ eff @ Rs @ S ) ) @ R )
= ( minus_minus_nat @ ( abstract_pos_rule @ Rs @ R ) @ ( suc @ ( abstra1963046427795717555ist_fm @ eff @ Rs @ S ) ) ) ) ) ) ) ) ).
% stake_pos_minWait
thf(fact_1253_trim__in__R,axiom,
! [S: produc6018962875968178549ist_fm,Rs: stream_rule] :
( ( member4699826688122452638ist_fm @ S @ top_to1730629564055774885ist_fm )
=> ( ( abstra3799686578551160190r_rule @ rules @ Rs )
=> ( member_rule @ ( shd_rule @ ( abstra7806236140005899928ist_fm @ eff @ Rs @ S ) ) @ ( sset_rule @ rules ) ) ) ) ).
% trim_in_R
thf(fact_1254_minWait__ex,axiom,
! [S: produc6018962875968178549ist_fm,Rs: stream_rule] :
( ( member4699826688122452638ist_fm @ S @ top_to1730629564055774885ist_fm )
=> ( ( abstra3799686578551160190r_rule @ rules @ Rs )
=> ? [N2: nat] : ( abstra1707737187183579335ist_fm @ eff @ ( shd_rule @ ( sdrop_rule @ N2 @ Rs ) ) @ S ) ) ) ).
% minWait_ex
thf(fact_1255_enabled__def,axiom,
! [R: rule,S: produc6018962875968178549ist_fm] :
( ( abstra1707737187183579335ist_fm @ eff @ R @ S )
= ( ? [X7: fset_P8989946509869081563ist_fm] : ( eff @ R @ S @ X7 ) ) ) ).
% enabled_def
thf(fact_1256_minWait__least,axiom,
! [N: nat,Rs: stream_rule,S: produc6018962875968178549ist_fm] :
( ( abstra1707737187183579335ist_fm @ eff @ ( shd_rule @ ( sdrop_rule @ N @ Rs ) ) @ S )
=> ( ord_less_eq_nat @ ( abstra1963046427795717555ist_fm @ eff @ Rs @ S ) @ N ) ) ).
% minWait_least
thf(fact_1257_trim__enabled,axiom,
! [S: produc6018962875968178549ist_fm,Rs: stream_rule] :
( ( member4699826688122452638ist_fm @ S @ top_to1730629564055774885ist_fm )
=> ( ( abstra3799686578551160190r_rule @ rules @ Rs )
=> ( abstra1707737187183579335ist_fm @ eff @ ( shd_rule @ ( abstra7806236140005899928ist_fm @ eff @ Rs @ S ) ) @ S ) ) ) ).
% trim_enabled
thf(fact_1258_minWait__le__pos,axiom,
! [Rs: stream_rule,R: rule,S: produc6018962875968178549ist_fm] :
( ( abstra3799686578551160190r_rule @ rules @ Rs )
=> ( ( member_rule @ R @ ( sset_rule @ rules ) )
=> ( ( abstra1707737187183579335ist_fm @ eff @ R @ S )
=> ( ord_less_eq_nat @ ( abstra1963046427795717555ist_fm @ eff @ Rs @ S ) @ ( abstract_pos_rule @ Rs @ R ) ) ) ) ) ).
% minWait_le_pos
thf(fact_1259_saturated__def,axiom,
! [R: rule] :
( ( abstra2533313685312581075ist_fm @ eff @ R )
= ( linear4957294957385550906m_rule
@ ^ [Xs2: stream2709947120125613254m_rule] :
( ( linear7925859179493631106m_rule
@ ^ [Step: produc340336539035504054m_rule] : ( abstra1707737187183579335ist_fm @ eff @ R @ ( produc993771546023551940m_rule @ Step ) )
@ Xs2 )
=> ( linear1816559245158234865m_rule
@ ( linear7925859179493631106m_rule
@ ^ [Step: produc340336539035504054m_rule] :
( ( produc6514711208036230658m_rule @ Step )
= R ) )
@ Xs2 ) ) ) ) ).
% saturated_def
thf(fact_1260_minWait__def,axiom,
! [Rs: stream_rule,S: produc6018962875968178549ist_fm] :
( ( abstra1963046427795717555ist_fm @ eff @ Rs @ S )
= ( ord_Least_nat
@ ^ [N3: nat] : ( abstra1707737187183579335ist_fm @ eff @ ( shd_rule @ ( sdrop_rule @ N3 @ Rs ) ) @ S ) ) ) ).
% minWait_def
thf(fact_1261_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
@ ^ [Xs2: stream2709947120125613254m_rule] :
( ( linear7925859179493631106m_rule
@ ^ [Step: produc340336539035504054m_rule] : ( abstra1707737187183579335ist_fm @ eff @ R @ ( produc993771546023551940m_rule @ Step ) )
@ Xs2 )
=> ( ( linear7925859179493631106m_rule
@ ^ [Step: produc340336539035504054m_rule] :
( ( produc6514711208036230658m_rule @ Step )
= R )
@ Xs2 )
| ( linear3460418925085718015rule_o
@ ( linear7925859179493631106m_rule
@ ^ [Step: produc340336539035504054m_rule] : ( abstra1707737187183579335ist_fm @ eff @ R @ ( produc993771546023551940m_rule @ Step ) ) )
@ Xs2 ) ) )
@ Steps ) ) ) ).
% per_alw
thf(fact_1262_fair__def,axiom,
! [Rs: stream_rule] :
( ( abstra3799686578551160190r_rule @ rules @ Rs )
= ( ( ord_less_eq_set_rule @ ( sset_rule @ Rs ) @ ( sset_rule @ rules ) )
& ! [X4: rule] :
( ( member_rule @ X4 @ ( sset_rule @ rules ) )
=> ( linear1087404401031714582w_rule
@ ( linear292482573288640205v_rule
@ ( linear7354821883049305950s_rule
@ ( ^ [Y4: rule,Z4: rule] : ( Y4 = Z4 )
@ X4 ) ) )
@ Rs ) ) ) ) ).
% fair_def
thf(fact_1263_Least__Suc,axiom,
! [P: nat > $o,N: nat] :
( ( P @ N )
=> ( ~ ( P @ zero_zero_nat )
=> ( ( ord_Least_nat @ P )
= ( suc
@ ( ord_Least_nat
@ ^ [M: nat] : ( P @ ( suc @ M ) ) ) ) ) ) ) ).
% Least_Suc
thf(fact_1264_ipath__mkTree__ev,axiom,
! [S: produc6018962875968178549ist_fm,Rs: stream_rule,Steps: stream2709947120125613254m_rule,R: rule] :
( ( member4699826688122452638ist_fm @ S @ top_to1730629564055774885ist_fm )
=> ( ( abstra3799686578551160190r_rule @ rules @ Rs )
=> ( ( abstra8836096083317999815m_rule @ ( abstra7840189589880272446ist_fm @ eff @ Rs @ S ) @ 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_1265_ipath__mkTree__Saturated,axiom,
! [S: produc6018962875968178549ist_fm,Rs: stream_rule,Steps: stream2709947120125613254m_rule] :
( ( member4699826688122452638ist_fm @ S @ top_to1730629564055774885ist_fm )
=> ( ( abstra3799686578551160190r_rule @ rules @ Rs )
=> ( ( abstra8836096083317999815m_rule @ ( abstra7840189589880272446ist_fm @ eff @ Rs @ S ) @ Steps )
=> ( abstra6097777249025082867ist_fm @ eff @ rules @ Steps ) ) ) ) ).
% ipath_mkTree_Saturated
thf(fact_1266_ipath__mkTree__sdrop,axiom,
! [S: produc6018962875968178549ist_fm,Rs: stream_rule,Steps: stream2709947120125613254m_rule,M2: nat] :
( ( member4699826688122452638ist_fm @ S @ top_to1730629564055774885ist_fm )
=> ( ( abstra3799686578551160190r_rule @ rules @ Rs )
=> ( ( abstra8836096083317999815m_rule @ ( abstra7840189589880272446ist_fm @ eff @ Rs @ S ) @ Steps )
=> ? [N2: nat,S8: produc6018962875968178549ist_fm] :
( ( member4699826688122452638ist_fm @ S8 @ top_to1730629564055774885ist_fm )
& ( abstra8836096083317999815m_rule @ ( abstra7840189589880272446ist_fm @ eff @ ( sdrop_rule @ N2 @ Rs ) @ S8 ) @ ( sdrop_8169176516188972301m_rule @ M2 @ Steps ) ) ) ) ) ) ).
% ipath_mkTree_sdrop
thf(fact_1267_ipath__mkTree__saturated,axiom,
! [S: produc6018962875968178549ist_fm,Rs: stream_rule,Steps: stream2709947120125613254m_rule,R: rule] :
( ( member4699826688122452638ist_fm @ S @ top_to1730629564055774885ist_fm )
=> ( ( abstra3799686578551160190r_rule @ rules @ Rs )
=> ( ( abstra8836096083317999815m_rule @ ( abstra7840189589880272446ist_fm @ eff @ Rs @ S ) @ Steps )
=> ( ( member_rule @ R @ ( sset_rule @ rules ) )
=> ( abstra2533313685312581075ist_fm @ eff @ R @ Steps ) ) ) ) ) ).
% ipath_mkTree_saturated
thf(fact_1268_secavProver__def,axiom,
( secavProver
= ( abstra7840189589880272446ist_fm @ eff @ rules ) ) ).
% secavProver_def
thf(fact_1269_in__cont__mkTree,axiom,
! [S: produc6018962875968178549ist_fm,Rs: stream_rule,T3: abstra8520858721168826610m_rule] :
( ( member4699826688122452638ist_fm @ S @ top_to1730629564055774885ist_fm )
=> ( ( abstra3799686578551160190r_rule @ rules @ Rs )
=> ( ( fmembe1099044791951439339m_rule @ T3 @ ( abstra2475333618431710837m_rule @ ( abstra7840189589880272446ist_fm @ eff @ Rs @ S ) ) )
=> ? [Sl5: fset_P8989946509869081563ist_fm,S8: produc6018962875968178549ist_fm] :
( ( member4699826688122452638ist_fm @ S8 @ top_to1730629564055774885ist_fm )
& ( eff @ ( shd_rule @ ( abstra7806236140005899928ist_fm @ eff @ Rs @ S ) ) @ S @ Sl5 )
& ( fmembe3754813877001230652ist_fm @ S8 @ Sl5 )
& ( T3
= ( abstra7840189589880272446ist_fm @ eff @ ( stl_rule @ ( abstra7806236140005899928ist_fm @ eff @ Rs @ S ) ) @ S8 ) ) ) ) ) ) ).
% in_cont_mkTree
thf(fact_1270_mkTree_Osimps_I1_J,axiom,
! [Rs: stream_rule,S: produc6018962875968178549ist_fm] :
( ( abstra5180277867158163941m_rule @ ( abstra7840189589880272446ist_fm @ eff @ Rs @ S ) )
= ( produc1733806532565653680m_rule @ S @ ( shd_rule @ ( abstra7806236140005899928ist_fm @ eff @ Rs @ S ) ) ) ) ).
% mkTree.simps(1)
thf(fact_1271_wf__mkTree,axiom,
! [S: produc6018962875968178549ist_fm,Rs: stream_rule] :
( ( member4699826688122452638ist_fm @ S @ top_to1730629564055774885ist_fm )
=> ( ( abstra3799686578551160190r_rule @ rules @ Rs )
=> ( abstra8167393772827826667ist_fm @ eff @ rules @ ( abstra7840189589880272446ist_fm @ eff @ Rs @ S ) ) ) ) ).
% wf_mkTree
thf(fact_1272_wf__ipath__epath,axiom,
! [T2: abstra8520858721168826610m_rule,Steps: stream2709947120125613254m_rule] :
( ( abstra8167393772827826667ist_fm @ eff @ rules @ T2 )
=> ( ( abstra8836096083317999815m_rule @ T2 @ Steps )
=> ( abstra6789711989322986974ist_fm @ eff @ rules @ Steps ) ) ) ).
% wf_ipath_epath
thf(fact_1273_epath__completeness__Saturated,axiom,
! [S: produc6018962875968178549ist_fm] :
( ( member4699826688122452638ist_fm @ S @ top_to1730629564055774885ist_fm )
=> ( ? [T4: abstra8520858721168826610m_rule] :
( ( ( produc993771546023551940m_rule @ ( abstra5180277867158163941m_rule @ T4 ) )
= S )
& ( abstra8167393772827826667ist_fm @ eff @ rules @ T4 )
& ( abstra3822255059661934556m_rule @ T4 ) )
| ? [Steps4: stream2709947120125613254m_rule] :
( ( ( produc993771546023551940m_rule @ ( shd_Pr4562317740776619530m_rule @ Steps4 ) )
= S )
& ( abstra6789711989322986974ist_fm @ eff @ rules @ Steps4 )
& ( abstra6097777249025082867ist_fm @ eff @ rules @ Steps4 ) ) ) ) ).
% epath_completeness_Saturated
thf(fact_1274_trim__def,axiom,
! [Rs: stream_rule,S: produc6018962875968178549ist_fm] :
( ( abstra7806236140005899928ist_fm @ eff @ Rs @ S )
= ( sdrop_while_rule
@ ^ [R4: rule] :
~ ( abstra1707737187183579335ist_fm @ eff @ R4 @ S )
@ Rs ) ) ).
% trim_def
thf(fact_1275_Euclidean__Division_Odivmod__nat__if,axiom,
( euclidean_divmod_nat
= ( ^ [M: nat,N3: nat] :
( if_Pro6206227464963214023at_nat
@ ( ( N3 = zero_zero_nat )
| ( ord_less_nat @ M @ N3 ) )
@ ( product_Pair_nat_nat @ zero_zero_nat @ M )
@ ( produc2626176000494625587at_nat
@ ^ [Q3: nat] : ( product_Pair_nat_nat @ ( suc @ Q3 ) )
@ ( euclidean_divmod_nat @ ( minus_minus_nat @ M @ N3 ) @ N3 ) ) ) ) ) ).
% Euclidean_Division.divmod_nat_if
% Helper facts (25)
thf(help_If_2_1_If_001t__Nat__Onat_T,axiom,
! [X3: nat,Y3: nat] :
( ( if_nat @ $false @ X3 @ Y3 )
= Y3 ) ).
thf(help_If_1_1_If_001t__Nat__Onat_T,axiom,
! [X3: nat,Y3: nat] :
( ( if_nat @ $true @ X3 @ Y3 )
= X3 ) ).
thf(help_If_2_1_If_001t__Set__Oset_It__SeCaV__Ofm_J_T,axiom,
! [X3: set_fm,Y3: set_fm] :
( ( if_set_fm @ $false @ X3 @ Y3 )
= Y3 ) ).
thf(help_If_1_1_If_001t__Set__Oset_It__SeCaV__Ofm_J_T,axiom,
! [X3: set_fm,Y3: set_fm] :
( ( if_set_fm @ $true @ X3 @ Y3 )
= X3 ) ).
thf(help_If_2_1_If_001t__Set__Oset_It__SeCaV__Otm_J_T,axiom,
! [X3: set_tm,Y3: set_tm] :
( ( if_set_tm @ $false @ X3 @ Y3 )
= Y3 ) ).
thf(help_If_1_1_If_001t__Set__Oset_It__SeCaV__Otm_J_T,axiom,
! [X3: set_tm,Y3: set_tm] :
( ( if_set_tm @ $true @ X3 @ Y3 )
= X3 ) ).
thf(help_If_2_1_If_001t__List__Olist_It__SeCaV__Otm_J_T,axiom,
! [X3: list_tm,Y3: list_tm] :
( ( if_list_tm @ $false @ X3 @ Y3 )
= Y3 ) ).
thf(help_If_1_1_If_001t__List__Olist_It__SeCaV__Otm_J_T,axiom,
! [X3: list_tm,Y3: list_tm] :
( ( if_list_tm @ $true @ X3 @ Y3 )
= X3 ) ).
thf(help_If_2_1_If_001t__Set__Oset_It__Prover__Orule_J_T,axiom,
! [X3: set_rule,Y3: set_rule] :
( ( if_set_rule @ $false @ X3 @ Y3 )
= Y3 ) ).
thf(help_If_1_1_If_001t__Set__Oset_It__Prover__Orule_J_T,axiom,
! [X3: set_rule,Y3: set_rule] :
( ( if_set_rule @ $true @ X3 @ Y3 )
= X3 ) ).
thf(help_If_2_1_If_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_T,axiom,
! [X3: product_prod_nat_nat,Y3: product_prod_nat_nat] :
( ( if_Pro6206227464963214023at_nat @ $false @ X3 @ Y3 )
= Y3 ) ).
thf(help_If_1_1_If_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_T,axiom,
! [X3: product_prod_nat_nat,Y3: product_prod_nat_nat] :
( ( if_Pro6206227464963214023at_nat @ $true @ X3 @ Y3 )
= X3 ) ).
thf(help_If_2_1_If_001t__Stream__Ostream_It__List__Olist_It__Prover__Orule_J_J_T,axiom,
! [X3: stream_list_rule,Y3: stream_list_rule] :
( ( if_stream_list_rule @ $false @ X3 @ Y3 )
= Y3 ) ).
thf(help_If_1_1_If_001t__Stream__Ostream_It__List__Olist_It__Prover__Orule_J_J_T,axiom,
! [X3: stream_list_rule,Y3: stream_list_rule] :
( ( if_stream_list_rule @ $true @ X3 @ Y3 )
= X3 ) ).
thf(help_If_2_1_If_001t__Product____Type__Oprod_It__List__Olist_It__SeCaV__Otm_J_Mt__List__Olist_It__SeCaV__Ofm_J_J_T,axiom,
! [X3: produc6018962875968178549ist_fm,Y3: produc6018962875968178549ist_fm] :
( ( if_Pro4760001780252510779ist_fm @ $false @ X3 @ Y3 )
= Y3 ) ).
thf(help_If_1_1_If_001t__Product____Type__Oprod_It__List__Olist_It__SeCaV__Otm_J_Mt__List__Olist_It__SeCaV__Ofm_J_J_T,axiom,
! [X3: produc6018962875968178549ist_fm,Y3: produc6018962875968178549ist_fm] :
( ( if_Pro4760001780252510779ist_fm @ $true @ X3 @ Y3 )
= X3 ) ).
thf(help_If_2_1_If_001t__FSet__Ofset_It__Product____Type__Oprod_It__List__Olist_It__SeCaV__Otm_J_Mt__List__Olist_It__SeCaV__Ofm_J_J_J_T,axiom,
! [X3: fset_P8989946509869081563ist_fm,Y3: fset_P8989946509869081563ist_fm] :
( ( if_fse7999432387889793441ist_fm @ $false @ X3 @ Y3 )
= Y3 ) ).
thf(help_If_1_1_If_001t__FSet__Ofset_It__Product____Type__Oprod_It__List__Olist_It__SeCaV__Otm_J_Mt__List__Olist_It__SeCaV__Ofm_J_J_J_T,axiom,
! [X3: fset_P8989946509869081563ist_fm,Y3: fset_P8989946509869081563ist_fm] :
( ( if_fse7999432387889793441ist_fm @ $true @ X3 @ Y3 )
= X3 ) ).
thf(help_If_2_1_If_001t__Abstract____Completeness__Otree_It__Product____Type__Oprod_It__Product____Type__Oprod_It__List__Olist_It__SeCaV__Otm_J_Mt__List__Olist_It__SeCaV__Ofm_J_J_Mt__Prover__Orule_J_J_T,axiom,
! [X3: abstra8520858721168826610m_rule,Y3: abstra8520858721168826610m_rule] :
( ( if_Abs1293773526126661548m_rule @ $false @ X3 @ Y3 )
= Y3 ) ).
thf(help_If_1_1_If_001t__Abstract____Completeness__Otree_It__Product____Type__Oprod_It__Product____Type__Oprod_It__List__Olist_It__SeCaV__Otm_J_Mt__List__Olist_It__SeCaV__Ofm_J_J_Mt__Prover__Orule_J_J_T,axiom,
! [X3: abstra8520858721168826610m_rule,Y3: abstra8520858721168826610m_rule] :
( ( if_Abs1293773526126661548m_rule @ $true @ X3 @ Y3 )
= X3 ) ).
thf(help_If_2_1_If_001t__Stream__Ostream_It__List__Olist_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_J_T,axiom,
! [X3: stream2471014364565126742m_rule,Y3: stream2471014364565126742m_rule] :
( ( if_str2465195801493762064m_rule @ $false @ X3 @ Y3 )
= Y3 ) ).
thf(help_If_1_1_If_001t__Stream__Ostream_It__List__Olist_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_J_T,axiom,
! [X3: stream2471014364565126742m_rule,Y3: stream2471014364565126742m_rule] :
( ( if_str2465195801493762064m_rule @ $true @ X3 @ Y3 )
= X3 ) ).
thf(help_If_3_1_If_001t__FSet__Ofset_It__Abstract____Completeness__Otree_It__Product____Type__Oprod_It__Product____Type__Oprod_It__List__Olist_It__SeCaV__Otm_J_Mt__List__Olist_It__SeCaV__Ofm_J_J_Mt__Prover__Orule_J_J_J_T,axiom,
! [P: $o] :
( ( P = $true )
| ( P = $false ) ) ).
thf(help_If_2_1_If_001t__FSet__Ofset_It__Abstract____Completeness__Otree_It__Product____Type__Oprod_It__Product____Type__Oprod_It__List__Olist_It__SeCaV__Otm_J_Mt__List__Olist_It__SeCaV__Ofm_J_J_Mt__Prover__Orule_J_J_J_T,axiom,
! [X3: fset_A8849196716589201442m_rule,Y3: fset_A8849196716589201442m_rule] :
( ( if_fse9105678332773368284m_rule @ $false @ X3 @ Y3 )
= Y3 ) ).
thf(help_If_1_1_If_001t__FSet__Ofset_It__Abstract____Completeness__Otree_It__Product____Type__Oprod_It__Product____Type__Oprod_It__List__Olist_It__SeCaV__Otm_J_Mt__List__Olist_It__SeCaV__Ofm_J_J_Mt__Prover__Orule_J_J_J_T,axiom,
! [X3: fset_A8849196716589201442m_rule,Y3: fset_A8849196716589201442m_rule] :
( ( if_fse9105678332773368284m_rule @ $true @ X3 @ Y3 )
= X3 ) ).
% Conjectures (2)
thf(conj_0,hypothesis,
! [B9: list_tm,Z9: list_fm,R7: rule] :
( ( fmembe3754813877001230652ist_fm @ ( produc1414352766439514085ist_fm @ B9 @ Z9 ) @ ( effect @ betaCon @ ( produc1414352766439514085ist_fm @ ( ptms @ ( shd_Pr4562317740776619530m_rule @ suf ) ) @ ( pseq @ ( shd_Pr4562317740776619530m_rule @ suf ) ) ) ) )
=> ( ( ( shd_Pr4562317740776619530m_rule @ ( stl_Pr12655793849929990m_rule @ suf ) )
= ( produc1733806532565653680m_rule @ ( produc1414352766439514085ist_fm @ B9 @ Z9 ) @ R7 ) )
=> thesis ) ) ).
thf(conj_1,conjecture,
thesis ).
%------------------------------------------------------------------------------