TPTP Problem File: ITP142^1.p
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%------------------------------------------------------------------------------
% File : ITP142^1 : TPTP v8.2.0. Released v7.5.0.
% Domain : Interactive Theorem Proving
% Problem : Sledgehammer PHoareTotal problem prob_304__3261994_1
% Version : Especial.
% English :
% Refs : [BH+15] Blanchette et al. (2015), Mining the Archive of Formal
% : [Des21] Desharnais (2021), Email to Geoff Sutcliffe
% Source : [Des21]
% Names : PHoareTotal/prob_304__3261994_1 [Des21]
% Status : Theorem
% Rating : 0.20 v8.2.0, 0.15 v8.1.0, 0.18 v7.5.0
% Syntax : Number of formulae : 522 ( 152 unt; 164 typ; 0 def)
% Number of atoms : 1042 ( 497 equ; 0 cnn)
% Maximal formula atoms : 7 ( 2 avg)
% Number of connectives : 5299 ( 222 ~; 42 |; 115 &;4342 @)
% ( 0 <=>; 578 =>; 0 <=; 0 <~>)
% Maximal formula depth : 23 ( 10 avg)
% Number of types : 41 ( 40 usr)
% Number of type conns : 389 ( 389 >; 0 *; 0 +; 0 <<)
% Number of symbols : 125 ( 124 usr; 13 con; 0-3 aty)
% Number of variables : 1572 ( 8 ^;1499 !; 65 ?;1572 :)
% SPC : TH0_THM_EQU_NAR
% Comments : This file was generated by Sledgehammer 2021-02-23 15:33:07.515
%------------------------------------------------------------------------------
% Could-be-implicit typings (40)
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thf(ty_n_t__PLang__Ocom,type,
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% Explicit typings (124)
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thf(sy_c_List_Oconcat_001t__PLang__Ocom,type,
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thf(sy_c_List_Oinsert_001t__PLang__Ocom,type,
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thf(sy_c_List_Olast_001t__PLang__Ocom,type,
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thf(sy_c_List_Olexord_001t__PLang__Ocom,type,
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thf(sy_c_List_Olexord_001t__Product____Type__Oprod_It__List__Olist_It__PLang__Ocom_J_Mt__PLang__Ostate_J,type,
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thf(sy_c_List_Olist_OCons_001t__List__Olist_It__PLang__Ocom_J,type,
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thf(sy_c_List_Olist_OCons_001t__PLang__Ocom,type,
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thf(sy_c_Transitive__Closure_Ortrancl_001t__Product____Type__Oprod_It__List__Olist_It__Product____Type__Oprod_It__List__Olist_It__PLang__Ocom_J_Mt__PLang__Ostate_J_J_Mt__List__Olist_It__Product____Type__Oprod_It__List__Olist_It__PLang__Ocom_J_Mt__PLang__Ostate_J_J_J,type,
transi1428749162_state: set_Pr1847866567_state > set_Pr1847866567_state ).
thf(sy_c_Transitive__Closure_Ortrancl_001t__Product____Type__Oprod_It__List__Olist_It__Product____Type__Oprod_It__Product____Type__Oprod_It__List__Olist_It__PLang__Ocom_J_Mt__PLang__Ostate_J_Mt__Product____Type__Oprod_It__List__Olist_It__PLang__Ocom_J_Mt__PLang__Ostate_J_J_J_Mt__List__Olist_It__Product____Type__Oprod_It__Product____Type__Oprod_It__List__Olist_It__PLang__Ocom_J_Mt__PLang__Ostate_J_Mt__Product____Type__Oprod_It__List__Olist_It__PLang__Ocom_J_Mt__PLang__Ostate_J_J_J_J,type,
transi445721450_state: set_Pr1696918727_state > set_Pr1696918727_state ).
thf(sy_c_Transitive__Closure_Ortrancl_001t__Product____Type__Oprod_It__Product____Type__Oprod_It__List__Olist_It__PLang__Ocom_J_Mt__PLang__Ostate_J_Mt__Product____Type__Oprod_It__List__Olist_It__PLang__Ocom_J_Mt__PLang__Ostate_J_J,type,
transi463013610_state: set_Pr2096893127_state > set_Pr2096893127_state ).
thf(sy_c_Transitive__Closure_Ortrancl_001t__Product____Type__Oprod_It__Product____Type__Oprod_It__Product____Type__Oprod_It__List__Olist_It__PLang__Ocom_J_Mt__PLang__Ostate_J_Mt__Product____Type__Oprod_It__List__Olist_It__PLang__Ocom_J_Mt__PLang__Ostate_J_J_Mt__Product____Type__Oprod_It__Product____Type__Oprod_It__List__Olist_It__PLang__Ocom_J_Mt__PLang__Ostate_J_Mt__Product____Type__Oprod_It__List__Olist_It__PLang__Ocom_J_Mt__PLang__Ostate_J_J_J,type,
transi152726506_state: set_Pr541394119_state > set_Pr541394119_state ).
thf(sy_c_member_001t__PLang__Ostate,type,
member_state: state > set_state > $o ).
thf(sy_c_member_001t__Product____Type__Oprod_It__List__Olist_It__List__Olist_It__PLang__Ocom_J_J_Mt__List__Olist_It__PLang__Ostate_J_J,type,
member1442949316_state: produc1393453339_state > set_Pr391631483_state > $o ).
thf(sy_c_member_001t__Product____Type__Oprod_It__List__Olist_It__PLang__Ocom_J_Mt__List__Olist_It__PLang__Ocom_J_J,type,
member32358020st_com: produc258871771st_com > set_Pr602326075st_com > $o ).
thf(sy_c_member_001t__Product____Type__Oprod_It__List__Olist_It__PLang__Ocom_J_Mt__PLang__Ostate_J,type,
member64234660_state: produc2041926651_state > set_Pr849496667_state > $o ).
thf(sy_c_member_001t__Product____Type__Oprod_It__List__Olist_It__Product____Type__Oprod_It__List__Olist_It__PLang__Ocom_J_Mt__PLang__Ostate_J_J_Mt__List__Olist_It__Product____Type__Oprod_It__List__Olist_It__PLang__Ocom_J_Mt__PLang__Ostate_J_J_J,type,
member1267359376_state: produc1225889383_state > set_Pr1142642759_state > $o ).
thf(sy_c_member_001t__Product____Type__Oprod_It__List__Olist_It__Product____Type__Oprod_It__Product____Type__Oprod_It__List__Olist_It__PLang__Ocom_J_Mt__PLang__Ostate_J_Mt__Product____Type__Oprod_It__List__Olist_It__PLang__Ocom_J_Mt__PLang__Ostate_J_J_J_Mt__List__Olist_It__Product____Type__Oprod_It__Product____Type__Oprod_It__List__Olist_It__PLang__Ocom_J_Mt__PLang__Ostate_J_Mt__Product____Type__Oprod_It__List__Olist_It__PLang__Ocom_J_Mt__PLang__Ostate_J_J_J_J,type,
member423306384_state: produc1586269031_state > set_Pr1325664839_state > $o ).
thf(sy_c_member_001t__Product____Type__Oprod_It__PLang__Ocom_Mt__PLang__Ocom_J,type,
member1333916004om_com: product_prod_com_com > set_Pr1822651163om_com > $o ).
thf(sy_c_member_001t__Product____Type__Oprod_It__Product____Type__Oprod_I_062_It__PLang__Ocom_M_062_It__PLang__Ocom_M_Eo_J_J_Mt__List__Olist_It__PLang__Ocom_J_J_Mt__Product____Type__Oprod_I_062_It__PLang__Ocom_M_062_It__PLang__Ocom_M_Eo_J_J_Mt__List__Olist_It__PLang__Ocom_J_J_J,type,
member89779770st_com: produc1440412305st_com > set_Pr135335153st_com > $o ).
thf(sy_c_member_001t__Product____Type__Oprod_It__Product____Type__Oprod_It__List__Olist_It__List__Olist_It__PLang__Ocom_J_J_Mt__List__Olist_It__PLang__Ostate_J_J_Mt__Product____Type__Oprod_It__List__Olist_It__List__Olist_It__PLang__Ocom_J_J_Mt__List__Olist_It__PLang__Ostate_J_J_J,type,
member1414265872_state: produc392316135_state > set_Pr1419004359_state > $o ).
thf(sy_c_member_001t__Product____Type__Oprod_It__Product____Type__Oprod_It__List__Olist_It__PLang__Ocom_J_Mt__List__Olist_It__PLang__Ocom_J_J_Mt__Product____Type__Oprod_It__List__Olist_It__PLang__Ocom_J_Mt__List__Olist_It__PLang__Ocom_J_J_J,type,
member919914512st_com: produc1762014951st_com > set_Pr2083163591st_com > $o ).
thf(sy_c_member_001t__Product____Type__Oprod_It__Product____Type__Oprod_It__List__Olist_It__PLang__Ocom_J_Mt__PLang__Ostate_J_Mt__Product____Type__Oprod_It__List__Olist_It__PLang__Ocom_J_Mt__PLang__Ostate_J_J,type,
member1069318160_state: produc410756839_state > set_Pr1165141447_state > $o ).
thf(sy_c_member_001t__Product____Type__Oprod_It__Product____Type__Oprod_It__List__Olist_It__Product____Type__Oprod_It__List__Olist_It__PLang__Ocom_J_Mt__PLang__Ostate_J_J_Mt__List__Olist_It__Product____Type__Oprod_It__List__Olist_It__PLang__Ocom_J_Mt__PLang__Ostate_J_J_J_Mt__Product____Type__Oprod_It__List__Olist_It__Product____Type__Oprod_It__List__Olist_It__PLang__Ocom_J_Mt__PLang__Ostate_J_J_Mt__List__Olist_It__Product____Type__Oprod_It__List__Olist_It__PLang__Ocom_J_Mt__PLang__Ostate_J_J_J_J,type,
member2092727056_state: produc1519776487_state > set_Pr1847866567_state > $o ).
thf(sy_c_member_001t__Product____Type__Oprod_It__Product____Type__Oprod_It__List__Olist_It__Product____Type__Oprod_It__Product____Type__Oprod_It__List__Olist_It__PLang__Ocom_J_Mt__PLang__Ostate_J_Mt__Product____Type__Oprod_It__List__Olist_It__PLang__Ocom_J_Mt__PLang__Ostate_J_J_J_Mt__List__Olist_It__Product____Type__Oprod_It__Product____Type__Oprod_It__List__Olist_It__PLang__Ocom_J_Mt__PLang__Ostate_J_Mt__Product____Type__Oprod_It__List__Olist_It__PLang__Ocom_J_Mt__PLang__Ostate_J_J_J_J_Mt__Product____Type__Oprod_It__List__Olist_It__Product____Type__Oprod_It__Product____Type__Oprod_It__List__Olist_It__PLang__Ocom_J_Mt__PLang__Ostate_J_Mt__Product____Type__Oprod_It__List__Olist_It__PLang__Ocom_J_Mt__PLang__Ostate_J_J_J_Mt__List__Olist_It__Product____Type__Oprod_It__Product____Type__Oprod_It__List__Olist_It__PLang__Ocom_J_Mt__PLang__Ostate_J_Mt__Product____Type__Oprod_It__List__Olist_It__PLang__Ocom_J_Mt__PLang__Ostate_J_J_J_J_J,type,
member1642814224_state: produc270545127_state > set_Pr1696918727_state > $o ).
thf(sy_c_member_001t__Product____Type__Oprod_It__Product____Type__Oprod_It__Product____Type__Oprod_It__List__Olist_It__PLang__Ocom_J_Mt__PLang__Ostate_J_Mt__Product____Type__Oprod_It__List__Olist_It__PLang__Ocom_J_Mt__PLang__Ostate_J_J_Mt__Product____Type__Oprod_It__Product____Type__Oprod_It__List__Olist_It__PLang__Ocom_J_Mt__PLang__Ostate_J_Mt__Product____Type__Oprod_It__List__Olist_It__PLang__Ocom_J_Mt__PLang__Ostate_J_J_J,type,
member512523024_state: produc186437863_state > set_Pr2096893127_state > $o ).
thf(sy_c_member_001t__Product____Type__Oprod_It__Product____Type__Oprod_It__Product____Type__Oprod_It__Product____Type__Oprod_It__List__Olist_It__PLang__Ocom_J_Mt__PLang__Ostate_J_Mt__Product____Type__Oprod_It__List__Olist_It__PLang__Ocom_J_Mt__PLang__Ostate_J_J_Mt__Product____Type__Oprod_It__Product____Type__Oprod_It__List__Olist_It__PLang__Ocom_J_Mt__PLang__Ostate_J_Mt__Product____Type__Oprod_It__List__Olist_It__PLang__Ocom_J_Mt__PLang__Ostate_J_J_J_Mt__Product____Type__Oprod_It__Product____Type__Oprod_It__Product____Type__Oprod_It__List__Olist_It__PLang__Ocom_J_Mt__PLang__Ostate_J_Mt__Product____Type__Oprod_It__List__Olist_It__PLang__Ocom_J_Mt__PLang__Ostate_J_J_Mt__Product____Type__Oprod_It__Product____Type__Oprod_It__List__Olist_It__PLang__Ocom_J_Mt__PLang__Ostate_J_Mt__Product____Type__Oprod_It__List__Olist_It__PLang__Ocom_J_Mt__PLang__Ostate_J_J_J_J,type,
member284400400_state: produc2046512871_state > set_Pr541394119_state > $o ).
thf(sy_v_c,type,
c: com ).
thf(sy_v_c_H,type,
c2: com ).
thf(sy_v_cs_H,type,
cs: list_com ).
thf(sy_v_s,type,
s: state ).
thf(sy_v_s_H,type,
s2: state ).
% Relevant facts (354)
thf(fact_0_exec1E_I1_J,axiom,
! [S: state,Cs: list_com,S2: state] :
~ ( member1069318160_state @ ( produc1909270103_state @ ( produc1204172211_state @ nil_com @ S ) @ ( produc1204172211_state @ Cs @ S2 ) ) @ pHoare259243666_exec1 ) ).
% exec1E(1)
thf(fact_1_termis_Osimps_I2_J,axiom,
! [C: com,Cs2: list_com,S: state] :
( ( pHoare806997636termis @ ( cons_com @ C @ Cs2 ) @ S )
= ( ( termi @ C @ S )
& ! [T: state] :
( ( exec @ S @ C @ T )
=> ( pHoare806997636termis @ Cs2 @ T ) ) ) ) ).
% termis.simps(2)
thf(fact_2_termis_Osimps_I1_J,axiom,
! [S: state] : ( pHoare806997636termis @ nil_com @ S ) ).
% termis.simps(1)
thf(fact_3_exec__impl__execs,axiom,
! [S: state,C: com,S2: state,Cs2: list_com] :
( ( exec @ S @ C @ S2 )
=> ( member1069318160_state @ ( produc1909270103_state @ ( produc1204172211_state @ ( cons_com @ C @ Cs2 ) @ S ) @ ( produc1204172211_state @ Cs2 @ S2 ) ) @ ( transi1302705790_state @ pHoare259243666_exec1 ) ) ) ).
% exec_impl_execs
thf(fact_4_exec1s__impl__exec,axiom,
! [C: com,S: state,T2: state] :
( ( member1069318160_state @ ( produc1909270103_state @ ( produc1204172211_state @ ( cons_com @ C @ nil_com ) @ S ) @ ( produc1204172211_state @ nil_com @ T2 ) ) @ ( transi1302705790_state @ pHoare259243666_exec1 ) )
=> ( exec @ S @ C @ T2 ) ) ).
% exec1s_impl_exec
thf(fact_5_exec1__pres__termis,axiom,
! [Cs2: list_com,S: state,Cs: list_com,S2: state] :
( ( member1069318160_state @ ( produc1909270103_state @ ( produc1204172211_state @ Cs2 @ S ) @ ( produc1204172211_state @ Cs @ S2 ) ) @ pHoare259243666_exec1 )
=> ( ( pHoare806997636termis @ Cs2 @ S )
=> ( pHoare806997636termis @ Cs @ S2 ) ) ) ).
% exec1_pres_termis
thf(fact_6_execs__pres__termis,axiom,
! [Cs2: list_com,S: state,Cs: list_com,S2: state] :
( ( member1069318160_state @ ( produc1909270103_state @ ( produc1204172211_state @ Cs2 @ S ) @ ( produc1204172211_state @ Cs @ S2 ) ) @ ( transi1302705790_state @ pHoare259243666_exec1 ) )
=> ( ( pHoare806997636termis @ Cs2 @ S )
=> ( pHoare806997636termis @ Cs @ S2 ) ) ) ).
% execs_pres_termis
thf(fact_7_rtrancl__induct2,axiom,
! [Ax: list_P890094829_state,Ay: list_P890094829_state,Bx: list_P890094829_state,By: list_P890094829_state,R: set_Pr1696918727_state,P: list_P890094829_state > list_P890094829_state > $o] :
( ( member1642814224_state @ ( produc1576663895_state @ ( produc1354769367_state @ Ax @ Ay ) @ ( produc1354769367_state @ Bx @ By ) ) @ ( transi445721450_state @ R ) )
=> ( ( P @ Ax @ Ay )
=> ( ! [A: list_P890094829_state,B: list_P890094829_state,Aa: list_P890094829_state,Ba: list_P890094829_state] :
( ( member1642814224_state @ ( produc1576663895_state @ ( produc1354769367_state @ Ax @ Ay ) @ ( produc1354769367_state @ A @ B ) ) @ ( transi445721450_state @ R ) )
=> ( ( member1642814224_state @ ( produc1576663895_state @ ( produc1354769367_state @ A @ B ) @ ( produc1354769367_state @ Aa @ Ba ) ) @ R )
=> ( ( P @ A @ B )
=> ( P @ Aa @ Ba ) ) ) )
=> ( P @ Bx @ By ) ) ) ) ).
% rtrancl_induct2
thf(fact_8_rtrancl__induct2,axiom,
! [Ax: list_P1040014977_state,Ay: list_P1040014977_state,Bx: list_P1040014977_state,By: list_P1040014977_state,R: set_Pr1847866567_state,P: list_P1040014977_state > list_P1040014977_state > $o] :
( ( member2092727056_state @ ( produc984060759_state @ ( produc1298457559_state @ Ax @ Ay ) @ ( produc1298457559_state @ Bx @ By ) ) @ ( transi1428749162_state @ R ) )
=> ( ( P @ Ax @ Ay )
=> ( ! [A: list_P1040014977_state,B: list_P1040014977_state,Aa: list_P1040014977_state,Ba: list_P1040014977_state] :
( ( member2092727056_state @ ( produc984060759_state @ ( produc1298457559_state @ Ax @ Ay ) @ ( produc1298457559_state @ A @ B ) ) @ ( transi1428749162_state @ R ) )
=> ( ( member2092727056_state @ ( produc984060759_state @ ( produc1298457559_state @ A @ B ) @ ( produc1298457559_state @ Aa @ Ba ) ) @ R )
=> ( ( P @ A @ B )
=> ( P @ Aa @ Ba ) ) ) )
=> ( P @ Bx @ By ) ) ) ) ).
% rtrancl_induct2
thf(fact_9_rtrancl__induct2,axiom,
! [Ax: list_com,Ay: list_com,Bx: list_com,By: list_com,R: set_Pr2083163591st_com,P: list_com > list_com > $o] :
( ( member919914512st_com @ ( produc313736791st_com @ ( produc1307105171st_com @ Ax @ Ay ) @ ( produc1307105171st_com @ Bx @ By ) ) @ ( transi1782245470st_com @ R ) )
=> ( ( P @ Ax @ Ay )
=> ( ! [A: list_com,B: list_com,Aa: list_com,Ba: list_com] :
( ( member919914512st_com @ ( produc313736791st_com @ ( produc1307105171st_com @ Ax @ Ay ) @ ( produc1307105171st_com @ A @ B ) ) @ ( transi1782245470st_com @ R ) )
=> ( ( member919914512st_com @ ( produc313736791st_com @ ( produc1307105171st_com @ A @ B ) @ ( produc1307105171st_com @ Aa @ Ba ) ) @ R )
=> ( ( P @ A @ B )
=> ( P @ Aa @ Ba ) ) ) )
=> ( P @ Bx @ By ) ) ) ) ).
% rtrancl_induct2
thf(fact_10_rtrancl__induct2,axiom,
! [Ax: list_list_com,Ay: list_state,Bx: list_list_com,By: list_state,R: set_Pr1419004359_state,P: list_list_com > list_state > $o] :
( ( member1414265872_state @ ( produc67381847_state @ ( produc1742567891_state @ Ax @ Ay ) @ ( produc1742567891_state @ Bx @ By ) ) @ ( transi89280670_state @ R ) )
=> ( ( P @ Ax @ Ay )
=> ( ! [A: list_list_com,B: list_state,Aa: list_list_com,Ba: list_state] :
( ( member1414265872_state @ ( produc67381847_state @ ( produc1742567891_state @ Ax @ Ay ) @ ( produc1742567891_state @ A @ B ) ) @ ( transi89280670_state @ R ) )
=> ( ( member1414265872_state @ ( produc67381847_state @ ( produc1742567891_state @ A @ B ) @ ( produc1742567891_state @ Aa @ Ba ) ) @ R )
=> ( ( P @ A @ B )
=> ( P @ Aa @ Ba ) ) ) )
=> ( P @ Bx @ By ) ) ) ) ).
% rtrancl_induct2
thf(fact_11_rtrancl__induct2,axiom,
! [Ax: com > com > $o,Ay: list_com,Bx: com > com > $o,By: list_com,R: set_Pr135335153st_com,P: ( com > com > $o ) > list_com > $o] :
( ( member89779770st_com @ ( produc67337417st_com @ ( produc1914379986st_com @ Ax @ Ay ) @ ( produc1914379986st_com @ Bx @ By ) ) @ ( transi1592315797st_com @ R ) )
=> ( ( P @ Ax @ Ay )
=> ( ! [A: com > com > $o,B: list_com,Aa: com > com > $o,Ba: list_com] :
( ( member89779770st_com @ ( produc67337417st_com @ ( produc1914379986st_com @ Ax @ Ay ) @ ( produc1914379986st_com @ A @ B ) ) @ ( transi1592315797st_com @ R ) )
=> ( ( member89779770st_com @ ( produc67337417st_com @ ( produc1914379986st_com @ A @ B ) @ ( produc1914379986st_com @ Aa @ Ba ) ) @ R )
=> ( ( P @ A @ B )
=> ( P @ Aa @ Ba ) ) ) )
=> ( P @ Bx @ By ) ) ) ) ).
% rtrancl_induct2
thf(fact_12_rtrancl__induct2,axiom,
! [Ax: produc410756839_state,Ay: produc410756839_state,Bx: produc410756839_state,By: produc410756839_state,R: set_Pr541394119_state,P: produc410756839_state > produc410756839_state > $o] :
( ( member284400400_state @ ( produc2049704791_state @ ( produc2065881943_state @ Ax @ Ay ) @ ( produc2065881943_state @ Bx @ By ) ) @ ( transi152726506_state @ R ) )
=> ( ( P @ Ax @ Ay )
=> ( ! [A: produc410756839_state,B: produc410756839_state,Aa: produc410756839_state,Ba: produc410756839_state] :
( ( member284400400_state @ ( produc2049704791_state @ ( produc2065881943_state @ Ax @ Ay ) @ ( produc2065881943_state @ A @ B ) ) @ ( transi152726506_state @ R ) )
=> ( ( member284400400_state @ ( produc2049704791_state @ ( produc2065881943_state @ A @ B ) @ ( produc2065881943_state @ Aa @ Ba ) ) @ R )
=> ( ( P @ A @ B )
=> ( P @ Aa @ Ba ) ) ) )
=> ( P @ Bx @ By ) ) ) ) ).
% rtrancl_induct2
thf(fact_13_rtrancl__induct2,axiom,
! [Ax: produc2041926651_state,Ay: produc2041926651_state,Bx: produc2041926651_state,By: produc2041926651_state,R: set_Pr2096893127_state,P: produc2041926651_state > produc2041926651_state > $o] :
( ( member512523024_state @ ( produc2065881943_state @ ( produc1909270103_state @ Ax @ Ay ) @ ( produc1909270103_state @ Bx @ By ) ) @ ( transi463013610_state @ R ) )
=> ( ( P @ Ax @ Ay )
=> ( ! [A: produc2041926651_state,B: produc2041926651_state,Aa: produc2041926651_state,Ba: produc2041926651_state] :
( ( member512523024_state @ ( produc2065881943_state @ ( produc1909270103_state @ Ax @ Ay ) @ ( produc1909270103_state @ A @ B ) ) @ ( transi463013610_state @ R ) )
=> ( ( member512523024_state @ ( produc2065881943_state @ ( produc1909270103_state @ A @ B ) @ ( produc1909270103_state @ Aa @ Ba ) ) @ R )
=> ( ( P @ A @ B )
=> ( P @ Aa @ Ba ) ) ) )
=> ( P @ Bx @ By ) ) ) ) ).
% rtrancl_induct2
thf(fact_14_rtrancl__induct2,axiom,
! [Ax: list_com,Ay: state,Bx: list_com,By: state,R: set_Pr1165141447_state,P: list_com > state > $o] :
( ( member1069318160_state @ ( produc1909270103_state @ ( produc1204172211_state @ Ax @ Ay ) @ ( produc1204172211_state @ Bx @ By ) ) @ ( transi1302705790_state @ R ) )
=> ( ( P @ Ax @ Ay )
=> ( ! [A: list_com,B: state,Aa: list_com,Ba: state] :
( ( member1069318160_state @ ( produc1909270103_state @ ( produc1204172211_state @ Ax @ Ay ) @ ( produc1204172211_state @ A @ B ) ) @ ( transi1302705790_state @ R ) )
=> ( ( member1069318160_state @ ( produc1909270103_state @ ( produc1204172211_state @ A @ B ) @ ( produc1204172211_state @ Aa @ Ba ) ) @ R )
=> ( ( P @ A @ B )
=> ( P @ Aa @ Ba ) ) ) )
=> ( P @ Bx @ By ) ) ) ) ).
% rtrancl_induct2
thf(fact_15_converse__rtranclE2,axiom,
! [Xa: list_P890094829_state,Xb: list_P890094829_state,Za: list_P890094829_state,Zb: list_P890094829_state,R: set_Pr1696918727_state] :
( ( member1642814224_state @ ( produc1576663895_state @ ( produc1354769367_state @ Xa @ Xb ) @ ( produc1354769367_state @ Za @ Zb ) ) @ ( transi445721450_state @ R ) )
=> ( ( ( produc1354769367_state @ Xa @ Xb )
!= ( produc1354769367_state @ Za @ Zb ) )
=> ~ ! [A: list_P890094829_state,B: list_P890094829_state] :
( ( member1642814224_state @ ( produc1576663895_state @ ( produc1354769367_state @ Xa @ Xb ) @ ( produc1354769367_state @ A @ B ) ) @ R )
=> ~ ( member1642814224_state @ ( produc1576663895_state @ ( produc1354769367_state @ A @ B ) @ ( produc1354769367_state @ Za @ Zb ) ) @ ( transi445721450_state @ R ) ) ) ) ) ).
% converse_rtranclE2
thf(fact_16_converse__rtranclE2,axiom,
! [Xa: list_P1040014977_state,Xb: list_P1040014977_state,Za: list_P1040014977_state,Zb: list_P1040014977_state,R: set_Pr1847866567_state] :
( ( member2092727056_state @ ( produc984060759_state @ ( produc1298457559_state @ Xa @ Xb ) @ ( produc1298457559_state @ Za @ Zb ) ) @ ( transi1428749162_state @ R ) )
=> ( ( ( produc1298457559_state @ Xa @ Xb )
!= ( produc1298457559_state @ Za @ Zb ) )
=> ~ ! [A: list_P1040014977_state,B: list_P1040014977_state] :
( ( member2092727056_state @ ( produc984060759_state @ ( produc1298457559_state @ Xa @ Xb ) @ ( produc1298457559_state @ A @ B ) ) @ R )
=> ~ ( member2092727056_state @ ( produc984060759_state @ ( produc1298457559_state @ A @ B ) @ ( produc1298457559_state @ Za @ Zb ) ) @ ( transi1428749162_state @ R ) ) ) ) ) ).
% converse_rtranclE2
thf(fact_17_converse__rtranclE2,axiom,
! [Xa: list_com,Xb: list_com,Za: list_com,Zb: list_com,R: set_Pr2083163591st_com] :
( ( member919914512st_com @ ( produc313736791st_com @ ( produc1307105171st_com @ Xa @ Xb ) @ ( produc1307105171st_com @ Za @ Zb ) ) @ ( transi1782245470st_com @ R ) )
=> ( ( ( produc1307105171st_com @ Xa @ Xb )
!= ( produc1307105171st_com @ Za @ Zb ) )
=> ~ ! [A: list_com,B: list_com] :
( ( member919914512st_com @ ( produc313736791st_com @ ( produc1307105171st_com @ Xa @ Xb ) @ ( produc1307105171st_com @ A @ B ) ) @ R )
=> ~ ( member919914512st_com @ ( produc313736791st_com @ ( produc1307105171st_com @ A @ B ) @ ( produc1307105171st_com @ Za @ Zb ) ) @ ( transi1782245470st_com @ R ) ) ) ) ) ).
% converse_rtranclE2
thf(fact_18_converse__rtranclE2,axiom,
! [Xa: list_list_com,Xb: list_state,Za: list_list_com,Zb: list_state,R: set_Pr1419004359_state] :
( ( member1414265872_state @ ( produc67381847_state @ ( produc1742567891_state @ Xa @ Xb ) @ ( produc1742567891_state @ Za @ Zb ) ) @ ( transi89280670_state @ R ) )
=> ( ( ( produc1742567891_state @ Xa @ Xb )
!= ( produc1742567891_state @ Za @ Zb ) )
=> ~ ! [A: list_list_com,B: list_state] :
( ( member1414265872_state @ ( produc67381847_state @ ( produc1742567891_state @ Xa @ Xb ) @ ( produc1742567891_state @ A @ B ) ) @ R )
=> ~ ( member1414265872_state @ ( produc67381847_state @ ( produc1742567891_state @ A @ B ) @ ( produc1742567891_state @ Za @ Zb ) ) @ ( transi89280670_state @ R ) ) ) ) ) ).
% converse_rtranclE2
thf(fact_19_converse__rtranclE2,axiom,
! [Xa: com > com > $o,Xb: list_com,Za: com > com > $o,Zb: list_com,R: set_Pr135335153st_com] :
( ( member89779770st_com @ ( produc67337417st_com @ ( produc1914379986st_com @ Xa @ Xb ) @ ( produc1914379986st_com @ Za @ Zb ) ) @ ( transi1592315797st_com @ R ) )
=> ( ( ( produc1914379986st_com @ Xa @ Xb )
!= ( produc1914379986st_com @ Za @ Zb ) )
=> ~ ! [A: com > com > $o,B: list_com] :
( ( member89779770st_com @ ( produc67337417st_com @ ( produc1914379986st_com @ Xa @ Xb ) @ ( produc1914379986st_com @ A @ B ) ) @ R )
=> ~ ( member89779770st_com @ ( produc67337417st_com @ ( produc1914379986st_com @ A @ B ) @ ( produc1914379986st_com @ Za @ Zb ) ) @ ( transi1592315797st_com @ R ) ) ) ) ) ).
% converse_rtranclE2
thf(fact_20_converse__rtranclE2,axiom,
! [Xa: produc410756839_state,Xb: produc410756839_state,Za: produc410756839_state,Zb: produc410756839_state,R: set_Pr541394119_state] :
( ( member284400400_state @ ( produc2049704791_state @ ( produc2065881943_state @ Xa @ Xb ) @ ( produc2065881943_state @ Za @ Zb ) ) @ ( transi152726506_state @ R ) )
=> ( ( ( produc2065881943_state @ Xa @ Xb )
!= ( produc2065881943_state @ Za @ Zb ) )
=> ~ ! [A: produc410756839_state,B: produc410756839_state] :
( ( member284400400_state @ ( produc2049704791_state @ ( produc2065881943_state @ Xa @ Xb ) @ ( produc2065881943_state @ A @ B ) ) @ R )
=> ~ ( member284400400_state @ ( produc2049704791_state @ ( produc2065881943_state @ A @ B ) @ ( produc2065881943_state @ Za @ Zb ) ) @ ( transi152726506_state @ R ) ) ) ) ) ).
% converse_rtranclE2
thf(fact_21_converse__rtranclE2,axiom,
! [Xa: produc2041926651_state,Xb: produc2041926651_state,Za: produc2041926651_state,Zb: produc2041926651_state,R: set_Pr2096893127_state] :
( ( member512523024_state @ ( produc2065881943_state @ ( produc1909270103_state @ Xa @ Xb ) @ ( produc1909270103_state @ Za @ Zb ) ) @ ( transi463013610_state @ R ) )
=> ( ( ( produc1909270103_state @ Xa @ Xb )
!= ( produc1909270103_state @ Za @ Zb ) )
=> ~ ! [A: produc2041926651_state,B: produc2041926651_state] :
( ( member512523024_state @ ( produc2065881943_state @ ( produc1909270103_state @ Xa @ Xb ) @ ( produc1909270103_state @ A @ B ) ) @ R )
=> ~ ( member512523024_state @ ( produc2065881943_state @ ( produc1909270103_state @ A @ B ) @ ( produc1909270103_state @ Za @ Zb ) ) @ ( transi463013610_state @ R ) ) ) ) ) ).
% converse_rtranclE2
thf(fact_22_converse__rtranclE2,axiom,
! [Xa: list_com,Xb: state,Za: list_com,Zb: state,R: set_Pr1165141447_state] :
( ( member1069318160_state @ ( produc1909270103_state @ ( produc1204172211_state @ Xa @ Xb ) @ ( produc1204172211_state @ Za @ Zb ) ) @ ( transi1302705790_state @ R ) )
=> ( ( ( produc1204172211_state @ Xa @ Xb )
!= ( produc1204172211_state @ Za @ Zb ) )
=> ~ ! [A: list_com,B: state] :
( ( member1069318160_state @ ( produc1909270103_state @ ( produc1204172211_state @ Xa @ Xb ) @ ( produc1204172211_state @ A @ B ) ) @ R )
=> ~ ( member1069318160_state @ ( produc1909270103_state @ ( produc1204172211_state @ A @ B ) @ ( produc1204172211_state @ Za @ Zb ) ) @ ( transi1302705790_state @ R ) ) ) ) ) ).
% converse_rtranclE2
thf(fact_23_converse__rtrancl__induct2,axiom,
! [Ax: list_P890094829_state,Ay: list_P890094829_state,Bx: list_P890094829_state,By: list_P890094829_state,R: set_Pr1696918727_state,P: list_P890094829_state > list_P890094829_state > $o] :
( ( member1642814224_state @ ( produc1576663895_state @ ( produc1354769367_state @ Ax @ Ay ) @ ( produc1354769367_state @ Bx @ By ) ) @ ( transi445721450_state @ R ) )
=> ( ( P @ Bx @ By )
=> ( ! [A: list_P890094829_state,B: list_P890094829_state,Aa: list_P890094829_state,Ba: list_P890094829_state] :
( ( member1642814224_state @ ( produc1576663895_state @ ( produc1354769367_state @ A @ B ) @ ( produc1354769367_state @ Aa @ Ba ) ) @ R )
=> ( ( member1642814224_state @ ( produc1576663895_state @ ( produc1354769367_state @ Aa @ Ba ) @ ( produc1354769367_state @ Bx @ By ) ) @ ( transi445721450_state @ R ) )
=> ( ( P @ Aa @ Ba )
=> ( P @ A @ B ) ) ) )
=> ( P @ Ax @ Ay ) ) ) ) ).
% converse_rtrancl_induct2
thf(fact_24_converse__rtrancl__induct2,axiom,
! [Ax: list_P1040014977_state,Ay: list_P1040014977_state,Bx: list_P1040014977_state,By: list_P1040014977_state,R: set_Pr1847866567_state,P: list_P1040014977_state > list_P1040014977_state > $o] :
( ( member2092727056_state @ ( produc984060759_state @ ( produc1298457559_state @ Ax @ Ay ) @ ( produc1298457559_state @ Bx @ By ) ) @ ( transi1428749162_state @ R ) )
=> ( ( P @ Bx @ By )
=> ( ! [A: list_P1040014977_state,B: list_P1040014977_state,Aa: list_P1040014977_state,Ba: list_P1040014977_state] :
( ( member2092727056_state @ ( produc984060759_state @ ( produc1298457559_state @ A @ B ) @ ( produc1298457559_state @ Aa @ Ba ) ) @ R )
=> ( ( member2092727056_state @ ( produc984060759_state @ ( produc1298457559_state @ Aa @ Ba ) @ ( produc1298457559_state @ Bx @ By ) ) @ ( transi1428749162_state @ R ) )
=> ( ( P @ Aa @ Ba )
=> ( P @ A @ B ) ) ) )
=> ( P @ Ax @ Ay ) ) ) ) ).
% converse_rtrancl_induct2
thf(fact_25_converse__rtrancl__induct2,axiom,
! [Ax: list_com,Ay: list_com,Bx: list_com,By: list_com,R: set_Pr2083163591st_com,P: list_com > list_com > $o] :
( ( member919914512st_com @ ( produc313736791st_com @ ( produc1307105171st_com @ Ax @ Ay ) @ ( produc1307105171st_com @ Bx @ By ) ) @ ( transi1782245470st_com @ R ) )
=> ( ( P @ Bx @ By )
=> ( ! [A: list_com,B: list_com,Aa: list_com,Ba: list_com] :
( ( member919914512st_com @ ( produc313736791st_com @ ( produc1307105171st_com @ A @ B ) @ ( produc1307105171st_com @ Aa @ Ba ) ) @ R )
=> ( ( member919914512st_com @ ( produc313736791st_com @ ( produc1307105171st_com @ Aa @ Ba ) @ ( produc1307105171st_com @ Bx @ By ) ) @ ( transi1782245470st_com @ R ) )
=> ( ( P @ Aa @ Ba )
=> ( P @ A @ B ) ) ) )
=> ( P @ Ax @ Ay ) ) ) ) ).
% converse_rtrancl_induct2
thf(fact_26_converse__rtrancl__induct2,axiom,
! [Ax: list_list_com,Ay: list_state,Bx: list_list_com,By: list_state,R: set_Pr1419004359_state,P: list_list_com > list_state > $o] :
( ( member1414265872_state @ ( produc67381847_state @ ( produc1742567891_state @ Ax @ Ay ) @ ( produc1742567891_state @ Bx @ By ) ) @ ( transi89280670_state @ R ) )
=> ( ( P @ Bx @ By )
=> ( ! [A: list_list_com,B: list_state,Aa: list_list_com,Ba: list_state] :
( ( member1414265872_state @ ( produc67381847_state @ ( produc1742567891_state @ A @ B ) @ ( produc1742567891_state @ Aa @ Ba ) ) @ R )
=> ( ( member1414265872_state @ ( produc67381847_state @ ( produc1742567891_state @ Aa @ Ba ) @ ( produc1742567891_state @ Bx @ By ) ) @ ( transi89280670_state @ R ) )
=> ( ( P @ Aa @ Ba )
=> ( P @ A @ B ) ) ) )
=> ( P @ Ax @ Ay ) ) ) ) ).
% converse_rtrancl_induct2
thf(fact_27_converse__rtrancl__induct2,axiom,
! [Ax: com > com > $o,Ay: list_com,Bx: com > com > $o,By: list_com,R: set_Pr135335153st_com,P: ( com > com > $o ) > list_com > $o] :
( ( member89779770st_com @ ( produc67337417st_com @ ( produc1914379986st_com @ Ax @ Ay ) @ ( produc1914379986st_com @ Bx @ By ) ) @ ( transi1592315797st_com @ R ) )
=> ( ( P @ Bx @ By )
=> ( ! [A: com > com > $o,B: list_com,Aa: com > com > $o,Ba: list_com] :
( ( member89779770st_com @ ( produc67337417st_com @ ( produc1914379986st_com @ A @ B ) @ ( produc1914379986st_com @ Aa @ Ba ) ) @ R )
=> ( ( member89779770st_com @ ( produc67337417st_com @ ( produc1914379986st_com @ Aa @ Ba ) @ ( produc1914379986st_com @ Bx @ By ) ) @ ( transi1592315797st_com @ R ) )
=> ( ( P @ Aa @ Ba )
=> ( P @ A @ B ) ) ) )
=> ( P @ Ax @ Ay ) ) ) ) ).
% converse_rtrancl_induct2
thf(fact_28_converse__rtrancl__induct2,axiom,
! [Ax: produc410756839_state,Ay: produc410756839_state,Bx: produc410756839_state,By: produc410756839_state,R: set_Pr541394119_state,P: produc410756839_state > produc410756839_state > $o] :
( ( member284400400_state @ ( produc2049704791_state @ ( produc2065881943_state @ Ax @ Ay ) @ ( produc2065881943_state @ Bx @ By ) ) @ ( transi152726506_state @ R ) )
=> ( ( P @ Bx @ By )
=> ( ! [A: produc410756839_state,B: produc410756839_state,Aa: produc410756839_state,Ba: produc410756839_state] :
( ( member284400400_state @ ( produc2049704791_state @ ( produc2065881943_state @ A @ B ) @ ( produc2065881943_state @ Aa @ Ba ) ) @ R )
=> ( ( member284400400_state @ ( produc2049704791_state @ ( produc2065881943_state @ Aa @ Ba ) @ ( produc2065881943_state @ Bx @ By ) ) @ ( transi152726506_state @ R ) )
=> ( ( P @ Aa @ Ba )
=> ( P @ A @ B ) ) ) )
=> ( P @ Ax @ Ay ) ) ) ) ).
% converse_rtrancl_induct2
thf(fact_29_converse__rtrancl__induct2,axiom,
! [Ax: produc2041926651_state,Ay: produc2041926651_state,Bx: produc2041926651_state,By: produc2041926651_state,R: set_Pr2096893127_state,P: produc2041926651_state > produc2041926651_state > $o] :
( ( member512523024_state @ ( produc2065881943_state @ ( produc1909270103_state @ Ax @ Ay ) @ ( produc1909270103_state @ Bx @ By ) ) @ ( transi463013610_state @ R ) )
=> ( ( P @ Bx @ By )
=> ( ! [A: produc2041926651_state,B: produc2041926651_state,Aa: produc2041926651_state,Ba: produc2041926651_state] :
( ( member512523024_state @ ( produc2065881943_state @ ( produc1909270103_state @ A @ B ) @ ( produc1909270103_state @ Aa @ Ba ) ) @ R )
=> ( ( member512523024_state @ ( produc2065881943_state @ ( produc1909270103_state @ Aa @ Ba ) @ ( produc1909270103_state @ Bx @ By ) ) @ ( transi463013610_state @ R ) )
=> ( ( P @ Aa @ Ba )
=> ( P @ A @ B ) ) ) )
=> ( P @ Ax @ Ay ) ) ) ) ).
% converse_rtrancl_induct2
thf(fact_30_converse__rtrancl__induct2,axiom,
! [Ax: list_com,Ay: state,Bx: list_com,By: state,R: set_Pr1165141447_state,P: list_com > state > $o] :
( ( member1069318160_state @ ( produc1909270103_state @ ( produc1204172211_state @ Ax @ Ay ) @ ( produc1204172211_state @ Bx @ By ) ) @ ( transi1302705790_state @ R ) )
=> ( ( P @ Bx @ By )
=> ( ! [A: list_com,B: state,Aa: list_com,Ba: state] :
( ( member1069318160_state @ ( produc1909270103_state @ ( produc1204172211_state @ A @ B ) @ ( produc1204172211_state @ Aa @ Ba ) ) @ R )
=> ( ( member1069318160_state @ ( produc1909270103_state @ ( produc1204172211_state @ Aa @ Ba ) @ ( produc1204172211_state @ Bx @ By ) ) @ ( transi1302705790_state @ R ) )
=> ( ( P @ Aa @ Ba )
=> ( P @ A @ B ) ) ) )
=> ( P @ Ax @ Ay ) ) ) ) ).
% converse_rtrancl_induct2
thf(fact_31_exec1s__impl__execs,axiom,
! [Cs2: list_com,S: state,T2: state] :
( ( member1069318160_state @ ( produc1909270103_state @ ( produc1204172211_state @ Cs2 @ S ) @ ( produc1204172211_state @ nil_com @ T2 ) ) @ ( transi1302705790_state @ pHoare259243666_exec1 ) )
=> ( pHoare259243732_execs @ S @ Cs2 @ T2 ) ) ).
% exec1s_impl_execs
thf(fact_32_rtrancl__idemp,axiom,
! [R: set_Pr541394119_state] :
( ( transi152726506_state @ ( transi152726506_state @ R ) )
= ( transi152726506_state @ R ) ) ).
% rtrancl_idemp
thf(fact_33_rtrancl__idemp,axiom,
! [R: set_Pr602326075st_com] :
( ( transi436345274st_com @ ( transi436345274st_com @ R ) )
= ( transi436345274st_com @ R ) ) ).
% rtrancl_idemp
thf(fact_34_rtrancl__idemp,axiom,
! [R: set_Pr1325664839_state] :
( ( transi1965148016_state @ ( transi1965148016_state @ R ) )
= ( transi1965148016_state @ R ) ) ).
% rtrancl_idemp
thf(fact_35_rtrancl__idemp,axiom,
! [R: set_Pr1142642759_state] :
( ( transi339813252_state @ ( transi339813252_state @ R ) )
= ( transi339813252_state @ R ) ) ).
% rtrancl_idemp
thf(fact_36_rtrancl__idemp,axiom,
! [R: set_Pr1822651163om_com] :
( ( transi863150634cl_com @ ( transi863150634cl_com @ R ) )
= ( transi863150634cl_com @ R ) ) ).
% rtrancl_idemp
thf(fact_37_rtrancl__idemp,axiom,
! [R: set_Pr1165141447_state] :
( ( transi1302705790_state @ ( transi1302705790_state @ R ) )
= ( transi1302705790_state @ R ) ) ).
% rtrancl_idemp
thf(fact_38_rtrancl__idemp,axiom,
! [R: set_Pr2096893127_state] :
( ( transi463013610_state @ ( transi463013610_state @ R ) )
= ( transi463013610_state @ R ) ) ).
% rtrancl_idemp
thf(fact_39_r__into__rtrancl,axiom,
! [P2: produc2046512871_state,R: set_Pr541394119_state] :
( ( member284400400_state @ P2 @ R )
=> ( member284400400_state @ P2 @ ( transi152726506_state @ R ) ) ) ).
% r_into_rtrancl
thf(fact_40_r__into__rtrancl,axiom,
! [P2: produc258871771st_com,R: set_Pr602326075st_com] :
( ( member32358020st_com @ P2 @ R )
=> ( member32358020st_com @ P2 @ ( transi436345274st_com @ R ) ) ) ).
% r_into_rtrancl
thf(fact_41_r__into__rtrancl,axiom,
! [P2: produc1586269031_state,R: set_Pr1325664839_state] :
( ( member423306384_state @ P2 @ R )
=> ( member423306384_state @ P2 @ ( transi1965148016_state @ R ) ) ) ).
% r_into_rtrancl
thf(fact_42_r__into__rtrancl,axiom,
! [P2: produc1225889383_state,R: set_Pr1142642759_state] :
( ( member1267359376_state @ P2 @ R )
=> ( member1267359376_state @ P2 @ ( transi339813252_state @ R ) ) ) ).
% r_into_rtrancl
thf(fact_43_r__into__rtrancl,axiom,
! [P2: product_prod_com_com,R: set_Pr1822651163om_com] :
( ( member1333916004om_com @ P2 @ R )
=> ( member1333916004om_com @ P2 @ ( transi863150634cl_com @ R ) ) ) ).
% r_into_rtrancl
thf(fact_44_r__into__rtrancl,axiom,
! [P2: produc410756839_state,R: set_Pr1165141447_state] :
( ( member1069318160_state @ P2 @ R )
=> ( member1069318160_state @ P2 @ ( transi1302705790_state @ R ) ) ) ).
% r_into_rtrancl
thf(fact_45_r__into__rtrancl,axiom,
! [P2: produc186437863_state,R: set_Pr2096893127_state] :
( ( member512523024_state @ P2 @ R )
=> ( member512523024_state @ P2 @ ( transi463013610_state @ R ) ) ) ).
% r_into_rtrancl
thf(fact_46_list_Oinject,axiom,
! [X21: produc2041926651_state,X22: list_P1040014977_state,Y21: produc2041926651_state,Y22: list_P1040014977_state] :
( ( ( cons_P465369265_state @ X21 @ X22 )
= ( cons_P465369265_state @ Y21 @ Y22 ) )
= ( ( X21 = Y21 )
& ( X22 = Y22 ) ) ) ).
% list.inject
thf(fact_47_list_Oinject,axiom,
! [X21: state,X22: list_state,Y21: state,Y22: list_state] :
( ( ( cons_state @ X21 @ X22 )
= ( cons_state @ Y21 @ Y22 ) )
= ( ( X21 = Y21 )
& ( X22 = Y22 ) ) ) ).
% list.inject
thf(fact_48_list_Oinject,axiom,
! [X21: list_com,X22: list_list_com,Y21: list_com,Y22: list_list_com] :
( ( ( cons_list_com @ X21 @ X22 )
= ( cons_list_com @ Y21 @ Y22 ) )
= ( ( X21 = Y21 )
& ( X22 = Y22 ) ) ) ).
% list.inject
thf(fact_49_list_Oinject,axiom,
! [X21: produc410756839_state > nat,X22: list_P1627643054te_nat,Y21: produc410756839_state > nat,Y22: list_P1627643054te_nat] :
( ( ( cons_P1631380062te_nat @ X21 @ X22 )
= ( cons_P1631380062te_nat @ Y21 @ Y22 ) )
= ( ( X21 = Y21 )
& ( X22 = Y22 ) ) ) ).
% list.inject
thf(fact_50_list_Oinject,axiom,
! [X21: produc2041926651_state > nat,X22: list_P1178430914te_nat,Y21: produc2041926651_state > nat,Y22: list_P1178430914te_nat] :
( ( ( cons_P1821701106te_nat @ X21 @ X22 )
= ( cons_P1821701106te_nat @ Y21 @ Y22 ) )
= ( ( X21 = Y21 )
& ( X22 = Y22 ) ) ) ).
% list.inject
thf(fact_51_list_Oinject,axiom,
! [X21: com,X22: list_com,Y21: com,Y22: list_com] :
( ( ( cons_com @ X21 @ X22 )
= ( cons_com @ Y21 @ Y22 ) )
= ( ( X21 = Y21 )
& ( X22 = Y22 ) ) ) ).
% list.inject
thf(fact_52_prod_Oinject,axiom,
! [X1: list_com,X2: list_com,Y1: list_com,Y2: list_com] :
( ( ( produc1307105171st_com @ X1 @ X2 )
= ( produc1307105171st_com @ Y1 @ Y2 ) )
= ( ( X1 = Y1 )
& ( X2 = Y2 ) ) ) ).
% prod.inject
thf(fact_53_prod_Oinject,axiom,
! [X1: list_list_com,X2: list_state,Y1: list_list_com,Y2: list_state] :
( ( ( produc1742567891_state @ X1 @ X2 )
= ( produc1742567891_state @ Y1 @ Y2 ) )
= ( ( X1 = Y1 )
& ( X2 = Y2 ) ) ) ).
% prod.inject
thf(fact_54_prod_Oinject,axiom,
! [X1: com > com > $o,X2: list_com,Y1: com > com > $o,Y2: list_com] :
( ( ( produc1914379986st_com @ X1 @ X2 )
= ( produc1914379986st_com @ Y1 @ Y2 ) )
= ( ( X1 = Y1 )
& ( X2 = Y2 ) ) ) ).
% prod.inject
thf(fact_55_prod_Oinject,axiom,
! [X1: produc2041926651_state,X2: produc2041926651_state,Y1: produc2041926651_state,Y2: produc2041926651_state] :
( ( ( produc1909270103_state @ X1 @ X2 )
= ( produc1909270103_state @ Y1 @ Y2 ) )
= ( ( X1 = Y1 )
& ( X2 = Y2 ) ) ) ).
% prod.inject
thf(fact_56_prod_Oinject,axiom,
! [X1: list_com,X2: state,Y1: list_com,Y2: state] :
( ( ( produc1204172211_state @ X1 @ X2 )
= ( produc1204172211_state @ Y1 @ Y2 ) )
= ( ( X1 = Y1 )
& ( X2 = Y2 ) ) ) ).
% prod.inject
thf(fact_57_prod_Oinject,axiom,
! [X1: produc410756839_state,X2: produc410756839_state,Y1: produc410756839_state,Y2: produc410756839_state] :
( ( ( produc2065881943_state @ X1 @ X2 )
= ( produc2065881943_state @ Y1 @ Y2 ) )
= ( ( X1 = Y1 )
& ( X2 = Y2 ) ) ) ).
% prod.inject
thf(fact_58_old_Oprod_Oinject,axiom,
! [A2: produc2041926651_state,B2: produc2041926651_state,A3: produc2041926651_state,B3: produc2041926651_state] :
( ( ( produc1909270103_state @ A2 @ B2 )
= ( produc1909270103_state @ A3 @ B3 ) )
= ( ( A2 = A3 )
& ( B2 = B3 ) ) ) ).
% old.prod.inject
thf(fact_59_old_Oprod_Oinject,axiom,
! [A2: list_com,B2: state,A3: list_com,B3: state] :
( ( ( produc1204172211_state @ A2 @ B2 )
= ( produc1204172211_state @ A3 @ B3 ) )
= ( ( A2 = A3 )
& ( B2 = B3 ) ) ) ).
% old.prod.inject
thf(fact_60_old_Oprod_Oinject,axiom,
! [A2: produc410756839_state,B2: produc410756839_state,A3: produc410756839_state,B3: produc410756839_state] :
( ( ( produc2065881943_state @ A2 @ B2 )
= ( produc2065881943_state @ A3 @ B3 ) )
= ( ( A2 = A3 )
& ( B2 = B3 ) ) ) ).
% old.prod.inject
thf(fact_61_prod__cases3,axiom,
! [Y: produc410756839_state] :
~ ! [A: produc2041926651_state,B: list_com,C2: state] :
( Y
!= ( produc1909270103_state @ A @ ( produc1204172211_state @ B @ C2 ) ) ) ).
% prod_cases3
thf(fact_62_prod__cases3,axiom,
! [Y: produc186437863_state] :
~ ! [A: produc410756839_state,B: produc2041926651_state,C2: produc2041926651_state] :
( Y
!= ( produc2065881943_state @ A @ ( produc1909270103_state @ B @ C2 ) ) ) ).
% prod_cases3
thf(fact_63_prod__cases4,axiom,
! [Y: produc186437863_state] :
~ ! [A: produc410756839_state,B: produc2041926651_state,C2: list_com,D: state] :
( Y
!= ( produc2065881943_state @ A @ ( produc1909270103_state @ B @ ( produc1204172211_state @ C2 @ D ) ) ) ) ).
% prod_cases4
thf(fact_64_successively_Ocases,axiom,
! [X: produc2058209240st_com] :
( ! [P3: com > com > $o] :
( X
!= ( produc1914379986st_com @ P3 @ nil_com ) )
=> ( ! [P3: com > com > $o,X3: com] :
( X
!= ( produc1914379986st_com @ P3 @ ( cons_com @ X3 @ nil_com ) ) )
=> ~ ! [P3: com > com > $o,X3: com,Y3: com,Xs: list_com] :
( X
!= ( produc1914379986st_com @ P3 @ ( cons_com @ X3 @ ( cons_com @ Y3 @ Xs ) ) ) ) ) ) ).
% successively.cases
thf(fact_65_sorted__wrt_Ocases,axiom,
! [X: produc2058209240st_com] :
( ! [P3: com > com > $o] :
( X
!= ( produc1914379986st_com @ P3 @ nil_com ) )
=> ~ ! [P3: com > com > $o,X3: com,Ys: list_com] :
( X
!= ( produc1914379986st_com @ P3 @ ( cons_com @ X3 @ Ys ) ) ) ) ).
% sorted_wrt.cases
thf(fact_66_shuffles_Ocases,axiom,
! [X: produc258871771st_com] :
( ! [Ys: list_com] :
( X
!= ( produc1307105171st_com @ nil_com @ Ys ) )
=> ( ! [Xs: list_com] :
( X
!= ( produc1307105171st_com @ Xs @ nil_com ) )
=> ~ ! [X3: com,Xs: list_com,Y3: com,Ys: list_com] :
( X
!= ( produc1307105171st_com @ ( cons_com @ X3 @ Xs ) @ ( cons_com @ Y3 @ Ys ) ) ) ) ) ).
% shuffles.cases
thf(fact_67_splice_Ocases,axiom,
! [X: produc258871771st_com] :
( ! [Ys: list_com] :
( X
!= ( produc1307105171st_com @ nil_com @ Ys ) )
=> ~ ! [X3: com,Xs: list_com,Ys: list_com] :
( X
!= ( produc1307105171st_com @ ( cons_com @ X3 @ Xs ) @ Ys ) ) ) ).
% splice.cases
thf(fact_68_execs_Ointros_I1_J,axiom,
! [S: state] : ( pHoare259243732_execs @ S @ nil_com @ S ) ).
% execs.intros(1)
thf(fact_69_old_Oprod_Oinducts,axiom,
! [P: produc410756839_state > $o,Prod: produc410756839_state] :
( ! [A: produc2041926651_state,B: produc2041926651_state] : ( P @ ( produc1909270103_state @ A @ B ) )
=> ( P @ Prod ) ) ).
% old.prod.inducts
thf(fact_70_old_Oprod_Oinducts,axiom,
! [P: produc2041926651_state > $o,Prod: produc2041926651_state] :
( ! [A: list_com,B: state] : ( P @ ( produc1204172211_state @ A @ B ) )
=> ( P @ Prod ) ) ).
% old.prod.inducts
thf(fact_71_old_Oprod_Oinducts,axiom,
! [P: produc186437863_state > $o,Prod: produc186437863_state] :
( ! [A: produc410756839_state,B: produc410756839_state] : ( P @ ( produc2065881943_state @ A @ B ) )
=> ( P @ Prod ) ) ).
% old.prod.inducts
thf(fact_72_old_Oprod_Oexhaust,axiom,
! [Y: produc410756839_state] :
~ ! [A: produc2041926651_state,B: produc2041926651_state] :
( Y
!= ( produc1909270103_state @ A @ B ) ) ).
% old.prod.exhaust
thf(fact_73_old_Oprod_Oexhaust,axiom,
! [Y: produc2041926651_state] :
~ ! [A: list_com,B: state] :
( Y
!= ( produc1204172211_state @ A @ B ) ) ).
% old.prod.exhaust
thf(fact_74_old_Oprod_Oexhaust,axiom,
! [Y: produc186437863_state] :
~ ! [A: produc410756839_state,B: produc410756839_state] :
( Y
!= ( produc2065881943_state @ A @ B ) ) ).
% old.prod.exhaust
thf(fact_75_Pair__inject,axiom,
! [A2: produc2041926651_state,B2: produc2041926651_state,A3: produc2041926651_state,B3: produc2041926651_state] :
( ( ( produc1909270103_state @ A2 @ B2 )
= ( produc1909270103_state @ A3 @ B3 ) )
=> ~ ( ( A2 = A3 )
=> ( B2 != B3 ) ) ) ).
% Pair_inject
thf(fact_76_Pair__inject,axiom,
! [A2: list_com,B2: state,A3: list_com,B3: state] :
( ( ( produc1204172211_state @ A2 @ B2 )
= ( produc1204172211_state @ A3 @ B3 ) )
=> ~ ( ( A2 = A3 )
=> ( B2 != B3 ) ) ) ).
% Pair_inject
thf(fact_77_Pair__inject,axiom,
! [A2: produc410756839_state,B2: produc410756839_state,A3: produc410756839_state,B3: produc410756839_state] :
( ( ( produc2065881943_state @ A2 @ B2 )
= ( produc2065881943_state @ A3 @ B3 ) )
=> ~ ( ( A2 = A3 )
=> ( B2 != B3 ) ) ) ).
% Pair_inject
thf(fact_78_prod__cases,axiom,
! [P: produc410756839_state > $o,P2: produc410756839_state] :
( ! [A: produc2041926651_state,B: produc2041926651_state] : ( P @ ( produc1909270103_state @ A @ B ) )
=> ( P @ P2 ) ) ).
% prod_cases
thf(fact_79_prod__cases,axiom,
! [P: produc2041926651_state > $o,P2: produc2041926651_state] :
( ! [A: list_com,B: state] : ( P @ ( produc1204172211_state @ A @ B ) )
=> ( P @ P2 ) ) ).
% prod_cases
thf(fact_80_prod__cases,axiom,
! [P: produc186437863_state > $o,P2: produc186437863_state] :
( ! [A: produc410756839_state,B: produc410756839_state] : ( P @ ( produc2065881943_state @ A @ B ) )
=> ( P @ P2 ) ) ).
% prod_cases
thf(fact_81_surj__pair,axiom,
! [P2: produc410756839_state] :
? [X3: produc2041926651_state,Y3: produc2041926651_state] :
( P2
= ( produc1909270103_state @ X3 @ Y3 ) ) ).
% surj_pair
thf(fact_82_surj__pair,axiom,
! [P2: produc2041926651_state] :
? [X3: list_com,Y3: state] :
( P2
= ( produc1204172211_state @ X3 @ Y3 ) ) ).
% surj_pair
thf(fact_83_surj__pair,axiom,
! [P2: produc186437863_state] :
? [X3: produc410756839_state,Y3: produc410756839_state] :
( P2
= ( produc2065881943_state @ X3 @ Y3 ) ) ).
% surj_pair
thf(fact_84_not__Cons__self2,axiom,
! [X: com,Xs2: list_com] :
( ( cons_com @ X @ Xs2 )
!= Xs2 ) ).
% not_Cons_self2
thf(fact_85_execs_Ointros_I2_J,axiom,
! [S: state,C: com,T2: state,Cs2: list_com,U: state] :
( ( exec @ S @ C @ T2 )
=> ( ( pHoare259243732_execs @ T2 @ Cs2 @ U )
=> ( pHoare259243732_execs @ S @ ( cons_com @ C @ Cs2 ) @ U ) ) ) ).
% execs.intros(2)
thf(fact_86_map__tailrec__rev_Oinduct,axiom,
! [P: ( com > com ) > list_com > list_com > $o,A0: com > com,A1: list_com,A22: list_com] :
( ! [F: com > com,X_1: list_com] : ( P @ F @ nil_com @ X_1 )
=> ( ! [F: com > com,A: com,As: list_com,Bs: list_com] :
( ( P @ F @ As @ ( cons_com @ ( F @ A ) @ Bs ) )
=> ( P @ F @ ( cons_com @ A @ As ) @ Bs ) )
=> ( P @ A0 @ A1 @ A22 ) ) ) ).
% map_tailrec_rev.induct
thf(fact_87_list__nonempty__induct,axiom,
! [Xs2: list_com,P: list_com > $o] :
( ( Xs2 != nil_com )
=> ( ! [X3: com] : ( P @ ( cons_com @ X3 @ nil_com ) )
=> ( ! [X3: com,Xs: list_com] :
( ( Xs != nil_com )
=> ( ( P @ Xs )
=> ( P @ ( cons_com @ X3 @ Xs ) ) ) )
=> ( P @ Xs2 ) ) ) ) ).
% list_nonempty_induct
thf(fact_88_successively_Oinduct,axiom,
! [P: ( com > com > $o ) > list_com > $o,A0: com > com > $o,A1: list_com] :
( ! [P3: com > com > $o] : ( P @ P3 @ nil_com )
=> ( ! [P3: com > com > $o,X3: com] : ( P @ P3 @ ( cons_com @ X3 @ nil_com ) )
=> ( ! [P3: com > com > $o,X3: com,Y3: com,Xs: list_com] :
( ( P @ P3 @ ( cons_com @ Y3 @ Xs ) )
=> ( P @ P3 @ ( cons_com @ X3 @ ( cons_com @ Y3 @ Xs ) ) ) )
=> ( P @ A0 @ A1 ) ) ) ) ).
% successively.induct
thf(fact_89_remdups__adj_Oinduct,axiom,
! [P: list_com > $o,A0: list_com] :
( ( P @ nil_com )
=> ( ! [X3: com] : ( P @ ( cons_com @ X3 @ nil_com ) )
=> ( ! [X3: com,Y3: com,Xs: list_com] :
( ( ( X3 = Y3 )
=> ( P @ ( cons_com @ X3 @ Xs ) ) )
=> ( ( ( X3 != Y3 )
=> ( P @ ( cons_com @ Y3 @ Xs ) ) )
=> ( P @ ( cons_com @ X3 @ ( cons_com @ Y3 @ Xs ) ) ) ) )
=> ( P @ A0 ) ) ) ) ).
% remdups_adj.induct
thf(fact_90_sorted__wrt_Oinduct,axiom,
! [P: ( com > com > $o ) > list_com > $o,A0: com > com > $o,A1: list_com] :
( ! [P3: com > com > $o] : ( P @ P3 @ nil_com )
=> ( ! [P3: com > com > $o,X3: com,Ys: list_com] :
( ( P @ P3 @ Ys )
=> ( P @ P3 @ ( cons_com @ X3 @ Ys ) ) )
=> ( P @ A0 @ A1 ) ) ) ).
% sorted_wrt.induct
thf(fact_91_remdups__adj_Ocases,axiom,
! [X: list_com] :
( ( X != nil_com )
=> ( ! [X3: com] :
( X
!= ( cons_com @ X3 @ nil_com ) )
=> ~ ! [X3: com,Y3: com,Xs: list_com] :
( X
!= ( cons_com @ X3 @ ( cons_com @ Y3 @ Xs ) ) ) ) ) ).
% remdups_adj.cases
thf(fact_92_transpose_Ocases,axiom,
! [X: list_list_com] :
( ( X != nil_list_com )
=> ( ! [Xss: list_list_com] :
( X
!= ( cons_list_com @ nil_com @ Xss ) )
=> ~ ! [X3: com,Xs: list_com,Xss: list_list_com] :
( X
!= ( cons_list_com @ ( cons_com @ X3 @ Xs ) @ Xss ) ) ) ) ).
% transpose.cases
thf(fact_93_shuffles_Oinduct,axiom,
! [P: list_com > list_com > $o,A0: list_com,A1: list_com] :
( ! [X_1: list_com] : ( P @ nil_com @ X_1 )
=> ( ! [Xs: list_com] : ( P @ Xs @ nil_com )
=> ( ! [X3: com,Xs: list_com,Y3: com,Ys: list_com] :
( ( P @ Xs @ ( cons_com @ Y3 @ Ys ) )
=> ( ( P @ ( cons_com @ X3 @ Xs ) @ Ys )
=> ( P @ ( cons_com @ X3 @ Xs ) @ ( cons_com @ Y3 @ Ys ) ) ) )
=> ( P @ A0 @ A1 ) ) ) ) ).
% shuffles.induct
thf(fact_94_mem__Collect__eq,axiom,
! [A2: produc410756839_state,P: produc410756839_state > $o] :
( ( member1069318160_state @ A2 @ ( collec300063826_state @ P ) )
= ( P @ A2 ) ) ).
% mem_Collect_eq
thf(fact_95_mem__Collect__eq,axiom,
! [A2: state,P: state > $o] :
( ( member_state @ A2 @ ( collect_state @ P ) )
= ( P @ A2 ) ) ).
% mem_Collect_eq
thf(fact_96_mem__Collect__eq,axiom,
! [A2: produc186437863_state,P: produc186437863_state > $o] :
( ( member512523024_state @ A2 @ ( collec489924946_state @ P ) )
= ( P @ A2 ) ) ).
% mem_Collect_eq
thf(fact_97_Collect__mem__eq,axiom,
! [A4: set_Pr1165141447_state] :
( ( collec300063826_state
@ ^ [X4: produc410756839_state] : ( member1069318160_state @ X4 @ A4 ) )
= A4 ) ).
% Collect_mem_eq
thf(fact_98_Collect__mem__eq,axiom,
! [A4: set_state] :
( ( collect_state
@ ^ [X4: state] : ( member_state @ X4 @ A4 ) )
= A4 ) ).
% Collect_mem_eq
thf(fact_99_Collect__mem__eq,axiom,
! [A4: set_Pr2096893127_state] :
( ( collec489924946_state
@ ^ [X4: produc186437863_state] : ( member512523024_state @ X4 @ A4 ) )
= A4 ) ).
% Collect_mem_eq
thf(fact_100_induct__list012,axiom,
! [P: list_com > $o,Xs2: list_com] :
( ( P @ nil_com )
=> ( ! [X3: com] : ( P @ ( cons_com @ X3 @ nil_com ) )
=> ( ! [X3: com,Y3: com,Zs: list_com] :
( ( P @ Zs )
=> ( ( P @ ( cons_com @ Y3 @ Zs ) )
=> ( P @ ( cons_com @ X3 @ ( cons_com @ Y3 @ Zs ) ) ) ) )
=> ( P @ Xs2 ) ) ) ) ).
% induct_list012
thf(fact_101_splice_Oinduct,axiom,
! [P: list_com > list_com > $o,A0: list_com,A1: list_com] :
( ! [X_1: list_com] : ( P @ nil_com @ X_1 )
=> ( ! [X3: com,Xs: list_com,Ys: list_com] :
( ( P @ Ys @ Xs )
=> ( P @ ( cons_com @ X3 @ Xs ) @ Ys ) )
=> ( P @ A0 @ A1 ) ) ) ).
% splice.induct
thf(fact_102_list__induct2_H,axiom,
! [P: list_com > list_com > $o,Xs2: list_com,Ys2: list_com] :
( ( P @ nil_com @ nil_com )
=> ( ! [X3: com,Xs: list_com] : ( P @ ( cons_com @ X3 @ Xs ) @ nil_com )
=> ( ! [Y3: com,Ys: list_com] : ( P @ nil_com @ ( cons_com @ Y3 @ Ys ) )
=> ( ! [X3: com,Xs: list_com,Y3: com,Ys: list_com] :
( ( P @ Xs @ Ys )
=> ( P @ ( cons_com @ X3 @ Xs ) @ ( cons_com @ Y3 @ Ys ) ) )
=> ( P @ Xs2 @ Ys2 ) ) ) ) ) ).
% list_induct2'
thf(fact_103_neq__Nil__conv,axiom,
! [Xs2: list_com] :
( ( Xs2 != nil_com )
= ( ? [Y4: com,Ys3: list_com] :
( Xs2
= ( cons_com @ Y4 @ Ys3 ) ) ) ) ).
% neq_Nil_conv
thf(fact_104_list_Oinducts,axiom,
! [P: list_com > $o,List: list_com] :
( ( P @ nil_com )
=> ( ! [X12: com,X23: list_com] :
( ( P @ X23 )
=> ( P @ ( cons_com @ X12 @ X23 ) ) )
=> ( P @ List ) ) ) ).
% list.inducts
thf(fact_105_list_Oexhaust,axiom,
! [Y: list_com] :
( ( Y != nil_com )
=> ~ ! [X212: com,X222: list_com] :
( Y
!= ( cons_com @ X212 @ X222 ) ) ) ).
% list.exhaust
thf(fact_106_list_OdiscI,axiom,
! [List: list_com,X21: com,X22: list_com] :
( ( List
= ( cons_com @ X21 @ X22 ) )
=> ( List != nil_com ) ) ).
% list.discI
thf(fact_107_list_Odistinct_I1_J,axiom,
! [X21: com,X22: list_com] :
( nil_com
!= ( cons_com @ X21 @ X22 ) ) ).
% list.distinct(1)
thf(fact_108_execs_Oinducts,axiom,
! [X1: state,X2: list_com,X32: state,P: state > list_com > state > $o] :
( ( pHoare259243732_execs @ X1 @ X2 @ X32 )
=> ( ! [S3: state] : ( P @ S3 @ nil_com @ S3 )
=> ( ! [S3: state,C2: com,T3: state,Cs3: list_com,U2: state] :
( ( exec @ S3 @ C2 @ T3 )
=> ( ( pHoare259243732_execs @ T3 @ Cs3 @ U2 )
=> ( ( P @ T3 @ Cs3 @ U2 )
=> ( P @ S3 @ ( cons_com @ C2 @ Cs3 ) @ U2 ) ) ) )
=> ( P @ X1 @ X2 @ X32 ) ) ) ) ).
% execs.inducts
thf(fact_109_execs_Osimps,axiom,
( pHoare259243732_execs
= ( ^ [A12: state,A23: list_com,A32: state] :
( ? [S4: state] :
( ( A12 = S4 )
& ( A23 = nil_com )
& ( A32 = S4 ) )
| ? [S4: state,C3: com,T: state,Cs4: list_com,U3: state] :
( ( A12 = S4 )
& ( A23
= ( cons_com @ C3 @ Cs4 ) )
& ( A32 = U3 )
& ( exec @ S4 @ C3 @ T )
& ( pHoare259243732_execs @ T @ Cs4 @ U3 ) ) ) ) ) ).
% execs.simps
thf(fact_110_execs_Ocases,axiom,
! [A1: state,A22: list_com,A33: state] :
( ( pHoare259243732_execs @ A1 @ A22 @ A33 )
=> ( ( ( A22 = nil_com )
=> ( A33 != A1 ) )
=> ~ ! [C2: com,T3: state,Cs3: list_com] :
( ( A22
= ( cons_com @ C2 @ Cs3 ) )
=> ( ( exec @ A1 @ C2 @ T3 )
=> ~ ( pHoare259243732_execs @ T3 @ Cs3 @ A33 ) ) ) ) ) ).
% execs.cases
thf(fact_111_converse__rtrancl__into__rtrancl,axiom,
! [A2: produc2041926651_state,B2: produc2041926651_state,R: set_Pr1165141447_state,C: produc2041926651_state] :
( ( member1069318160_state @ ( produc1909270103_state @ A2 @ B2 ) @ R )
=> ( ( member1069318160_state @ ( produc1909270103_state @ B2 @ C ) @ ( transi1302705790_state @ R ) )
=> ( member1069318160_state @ ( produc1909270103_state @ A2 @ C ) @ ( transi1302705790_state @ R ) ) ) ) ).
% converse_rtrancl_into_rtrancl
thf(fact_112_converse__rtrancl__into__rtrancl,axiom,
! [A2: produc410756839_state,B2: produc410756839_state,R: set_Pr2096893127_state,C: produc410756839_state] :
( ( member512523024_state @ ( produc2065881943_state @ A2 @ B2 ) @ R )
=> ( ( member512523024_state @ ( produc2065881943_state @ B2 @ C ) @ ( transi463013610_state @ R ) )
=> ( member512523024_state @ ( produc2065881943_state @ A2 @ C ) @ ( transi463013610_state @ R ) ) ) ) ).
% converse_rtrancl_into_rtrancl
thf(fact_113_rtrancl_Ortrancl__into__rtrancl,axiom,
! [A2: produc2041926651_state,B2: produc2041926651_state,R: set_Pr1165141447_state,C: produc2041926651_state] :
( ( member1069318160_state @ ( produc1909270103_state @ A2 @ B2 ) @ ( transi1302705790_state @ R ) )
=> ( ( member1069318160_state @ ( produc1909270103_state @ B2 @ C ) @ R )
=> ( member1069318160_state @ ( produc1909270103_state @ A2 @ C ) @ ( transi1302705790_state @ R ) ) ) ) ).
% rtrancl.rtrancl_into_rtrancl
thf(fact_114_rtrancl_Ortrancl__into__rtrancl,axiom,
! [A2: produc410756839_state,B2: produc410756839_state,R: set_Pr2096893127_state,C: produc410756839_state] :
( ( member512523024_state @ ( produc2065881943_state @ A2 @ B2 ) @ ( transi463013610_state @ R ) )
=> ( ( member512523024_state @ ( produc2065881943_state @ B2 @ C ) @ R )
=> ( member512523024_state @ ( produc2065881943_state @ A2 @ C ) @ ( transi463013610_state @ R ) ) ) ) ).
% rtrancl.rtrancl_into_rtrancl
thf(fact_115_converse__rtrancl__induct,axiom,
! [A2: produc2041926651_state,B2: produc2041926651_state,R: set_Pr1165141447_state,P: produc2041926651_state > $o] :
( ( member1069318160_state @ ( produc1909270103_state @ A2 @ B2 ) @ ( transi1302705790_state @ R ) )
=> ( ( P @ B2 )
=> ( ! [Y3: produc2041926651_state,Z: produc2041926651_state] :
( ( member1069318160_state @ ( produc1909270103_state @ Y3 @ Z ) @ R )
=> ( ( member1069318160_state @ ( produc1909270103_state @ Z @ B2 ) @ ( transi1302705790_state @ R ) )
=> ( ( P @ Z )
=> ( P @ Y3 ) ) ) )
=> ( P @ A2 ) ) ) ) ).
% converse_rtrancl_induct
thf(fact_116_converse__rtrancl__induct,axiom,
! [A2: produc410756839_state,B2: produc410756839_state,R: set_Pr2096893127_state,P: produc410756839_state > $o] :
( ( member512523024_state @ ( produc2065881943_state @ A2 @ B2 ) @ ( transi463013610_state @ R ) )
=> ( ( P @ B2 )
=> ( ! [Y3: produc410756839_state,Z: produc410756839_state] :
( ( member512523024_state @ ( produc2065881943_state @ Y3 @ Z ) @ R )
=> ( ( member512523024_state @ ( produc2065881943_state @ Z @ B2 ) @ ( transi463013610_state @ R ) )
=> ( ( P @ Z )
=> ( P @ Y3 ) ) ) )
=> ( P @ A2 ) ) ) ) ).
% converse_rtrancl_induct
thf(fact_117_rtrancl_Ortrancl__refl,axiom,
! [A2: produc2041926651_state,R: set_Pr1165141447_state] : ( member1069318160_state @ ( produc1909270103_state @ A2 @ A2 ) @ ( transi1302705790_state @ R ) ) ).
% rtrancl.rtrancl_refl
thf(fact_118_rtrancl_Ortrancl__refl,axiom,
! [A2: produc410756839_state,R: set_Pr2096893127_state] : ( member512523024_state @ ( produc2065881943_state @ A2 @ A2 ) @ ( transi463013610_state @ R ) ) ).
% rtrancl.rtrancl_refl
thf(fact_119_converse__rtranclE,axiom,
! [X: produc2041926651_state,Z2: produc2041926651_state,R: set_Pr1165141447_state] :
( ( member1069318160_state @ ( produc1909270103_state @ X @ Z2 ) @ ( transi1302705790_state @ R ) )
=> ( ( X != Z2 )
=> ~ ! [Y3: produc2041926651_state] :
( ( member1069318160_state @ ( produc1909270103_state @ X @ Y3 ) @ R )
=> ~ ( member1069318160_state @ ( produc1909270103_state @ Y3 @ Z2 ) @ ( transi1302705790_state @ R ) ) ) ) ) ).
% converse_rtranclE
thf(fact_120_converse__rtranclE,axiom,
! [X: produc410756839_state,Z2: produc410756839_state,R: set_Pr2096893127_state] :
( ( member512523024_state @ ( produc2065881943_state @ X @ Z2 ) @ ( transi463013610_state @ R ) )
=> ( ( X != Z2 )
=> ~ ! [Y3: produc410756839_state] :
( ( member512523024_state @ ( produc2065881943_state @ X @ Y3 ) @ R )
=> ~ ( member512523024_state @ ( produc2065881943_state @ Y3 @ Z2 ) @ ( transi463013610_state @ R ) ) ) ) ) ).
% converse_rtranclE
thf(fact_121_rtrancl_Oinducts,axiom,
! [X1: produc2041926651_state,X2: produc2041926651_state,R: set_Pr1165141447_state,P: produc2041926651_state > produc2041926651_state > $o] :
( ( member1069318160_state @ ( produc1909270103_state @ X1 @ X2 ) @ ( transi1302705790_state @ R ) )
=> ( ! [A: produc2041926651_state] : ( P @ A @ A )
=> ( ! [A: produc2041926651_state,B: produc2041926651_state,C2: produc2041926651_state] :
( ( member1069318160_state @ ( produc1909270103_state @ A @ B ) @ ( transi1302705790_state @ R ) )
=> ( ( P @ A @ B )
=> ( ( member1069318160_state @ ( produc1909270103_state @ B @ C2 ) @ R )
=> ( P @ A @ C2 ) ) ) )
=> ( P @ X1 @ X2 ) ) ) ) ).
% rtrancl.inducts
thf(fact_122_rtrancl_Oinducts,axiom,
! [X1: produc410756839_state,X2: produc410756839_state,R: set_Pr2096893127_state,P: produc410756839_state > produc410756839_state > $o] :
( ( member512523024_state @ ( produc2065881943_state @ X1 @ X2 ) @ ( transi463013610_state @ R ) )
=> ( ! [A: produc410756839_state] : ( P @ A @ A )
=> ( ! [A: produc410756839_state,B: produc410756839_state,C2: produc410756839_state] :
( ( member512523024_state @ ( produc2065881943_state @ A @ B ) @ ( transi463013610_state @ R ) )
=> ( ( P @ A @ B )
=> ( ( member512523024_state @ ( produc2065881943_state @ B @ C2 ) @ R )
=> ( P @ A @ C2 ) ) ) )
=> ( P @ X1 @ X2 ) ) ) ) ).
% rtrancl.inducts
thf(fact_123_rtrancl__induct,axiom,
! [A2: produc2041926651_state,B2: produc2041926651_state,R: set_Pr1165141447_state,P: produc2041926651_state > $o] :
( ( member1069318160_state @ ( produc1909270103_state @ A2 @ B2 ) @ ( transi1302705790_state @ R ) )
=> ( ( P @ A2 )
=> ( ! [Y3: produc2041926651_state,Z: produc2041926651_state] :
( ( member1069318160_state @ ( produc1909270103_state @ A2 @ Y3 ) @ ( transi1302705790_state @ R ) )
=> ( ( member1069318160_state @ ( produc1909270103_state @ Y3 @ Z ) @ R )
=> ( ( P @ Y3 )
=> ( P @ Z ) ) ) )
=> ( P @ B2 ) ) ) ) ).
% rtrancl_induct
thf(fact_124_rtrancl__induct,axiom,
! [A2: produc410756839_state,B2: produc410756839_state,R: set_Pr2096893127_state,P: produc410756839_state > $o] :
( ( member512523024_state @ ( produc2065881943_state @ A2 @ B2 ) @ ( transi463013610_state @ R ) )
=> ( ( P @ A2 )
=> ( ! [Y3: produc410756839_state,Z: produc410756839_state] :
( ( member512523024_state @ ( produc2065881943_state @ A2 @ Y3 ) @ ( transi463013610_state @ R ) )
=> ( ( member512523024_state @ ( produc2065881943_state @ Y3 @ Z ) @ R )
=> ( ( P @ Y3 )
=> ( P @ Z ) ) ) )
=> ( P @ B2 ) ) ) ) ).
% rtrancl_induct
thf(fact_125_rtrancl__trans,axiom,
! [X: produc2041926651_state,Y: produc2041926651_state,R: set_Pr1165141447_state,Z2: produc2041926651_state] :
( ( member1069318160_state @ ( produc1909270103_state @ X @ Y ) @ ( transi1302705790_state @ R ) )
=> ( ( member1069318160_state @ ( produc1909270103_state @ Y @ Z2 ) @ ( transi1302705790_state @ R ) )
=> ( member1069318160_state @ ( produc1909270103_state @ X @ Z2 ) @ ( transi1302705790_state @ R ) ) ) ) ).
% rtrancl_trans
thf(fact_126_rtrancl__trans,axiom,
! [X: produc410756839_state,Y: produc410756839_state,R: set_Pr2096893127_state,Z2: produc410756839_state] :
( ( member512523024_state @ ( produc2065881943_state @ X @ Y ) @ ( transi463013610_state @ R ) )
=> ( ( member512523024_state @ ( produc2065881943_state @ Y @ Z2 ) @ ( transi463013610_state @ R ) )
=> ( member512523024_state @ ( produc2065881943_state @ X @ Z2 ) @ ( transi463013610_state @ R ) ) ) ) ).
% rtrancl_trans
thf(fact_127_rtrancl_Osimps,axiom,
! [A1: produc2041926651_state,A22: produc2041926651_state,R: set_Pr1165141447_state] :
( ( member1069318160_state @ ( produc1909270103_state @ A1 @ A22 ) @ ( transi1302705790_state @ R ) )
= ( ? [A5: produc2041926651_state] :
( ( A1 = A5 )
& ( A22 = A5 ) )
| ? [A5: produc2041926651_state,B4: produc2041926651_state,C3: produc2041926651_state] :
( ( A1 = A5 )
& ( A22 = C3 )
& ( member1069318160_state @ ( produc1909270103_state @ A5 @ B4 ) @ ( transi1302705790_state @ R ) )
& ( member1069318160_state @ ( produc1909270103_state @ B4 @ C3 ) @ R ) ) ) ) ).
% rtrancl.simps
thf(fact_128_rtrancl_Osimps,axiom,
! [A1: produc410756839_state,A22: produc410756839_state,R: set_Pr2096893127_state] :
( ( member512523024_state @ ( produc2065881943_state @ A1 @ A22 ) @ ( transi463013610_state @ R ) )
= ( ? [A5: produc410756839_state] :
( ( A1 = A5 )
& ( A22 = A5 ) )
| ? [A5: produc410756839_state,B4: produc410756839_state,C3: produc410756839_state] :
( ( A1 = A5 )
& ( A22 = C3 )
& ( member512523024_state @ ( produc2065881943_state @ A5 @ B4 ) @ ( transi463013610_state @ R ) )
& ( member512523024_state @ ( produc2065881943_state @ B4 @ C3 ) @ R ) ) ) ) ).
% rtrancl.simps
thf(fact_129_rtrancl_Ocases,axiom,
! [A1: produc2041926651_state,A22: produc2041926651_state,R: set_Pr1165141447_state] :
( ( member1069318160_state @ ( produc1909270103_state @ A1 @ A22 ) @ ( transi1302705790_state @ R ) )
=> ( ( A22 != A1 )
=> ~ ! [B: produc2041926651_state] :
( ( member1069318160_state @ ( produc1909270103_state @ A1 @ B ) @ ( transi1302705790_state @ R ) )
=> ~ ( member1069318160_state @ ( produc1909270103_state @ B @ A22 ) @ R ) ) ) ) ).
% rtrancl.cases
thf(fact_130_rtrancl_Ocases,axiom,
! [A1: produc410756839_state,A22: produc410756839_state,R: set_Pr2096893127_state] :
( ( member512523024_state @ ( produc2065881943_state @ A1 @ A22 ) @ ( transi463013610_state @ R ) )
=> ( ( A22 != A1 )
=> ~ ! [B: produc410756839_state] :
( ( member512523024_state @ ( produc2065881943_state @ A1 @ B ) @ ( transi463013610_state @ R ) )
=> ~ ( member512523024_state @ ( produc2065881943_state @ B @ A22 ) @ R ) ) ) ) ).
% rtrancl.cases
thf(fact_131_rtranclE,axiom,
! [A2: produc2041926651_state,B2: produc2041926651_state,R: set_Pr1165141447_state] :
( ( member1069318160_state @ ( produc1909270103_state @ A2 @ B2 ) @ ( transi1302705790_state @ R ) )
=> ( ( A2 != B2 )
=> ~ ! [Y3: produc2041926651_state] :
( ( member1069318160_state @ ( produc1909270103_state @ A2 @ Y3 ) @ ( transi1302705790_state @ R ) )
=> ~ ( member1069318160_state @ ( produc1909270103_state @ Y3 @ B2 ) @ R ) ) ) ) ).
% rtranclE
thf(fact_132_rtranclE,axiom,
! [A2: produc410756839_state,B2: produc410756839_state,R: set_Pr2096893127_state] :
( ( member512523024_state @ ( produc2065881943_state @ A2 @ B2 ) @ ( transi463013610_state @ R ) )
=> ( ( A2 != B2 )
=> ~ ! [Y3: produc410756839_state] :
( ( member512523024_state @ ( produc2065881943_state @ A2 @ Y3 ) @ ( transi463013610_state @ R ) )
=> ~ ( member512523024_state @ ( produc2065881943_state @ Y3 @ B2 ) @ R ) ) ) ) ).
% rtranclE
thf(fact_133_prod__induct4,axiom,
! [P: produc186437863_state > $o,X: produc186437863_state] :
( ! [A: produc410756839_state,B: produc2041926651_state,C2: list_com,D: state] : ( P @ ( produc2065881943_state @ A @ ( produc1909270103_state @ B @ ( produc1204172211_state @ C2 @ D ) ) ) )
=> ( P @ X ) ) ).
% prod_induct4
thf(fact_134_prod__induct3,axiom,
! [P: produc410756839_state > $o,X: produc410756839_state] :
( ! [A: produc2041926651_state,B: list_com,C2: state] : ( P @ ( produc1909270103_state @ A @ ( produc1204172211_state @ B @ C2 ) ) )
=> ( P @ X ) ) ).
% prod_induct3
thf(fact_135_prod__induct3,axiom,
! [P: produc186437863_state > $o,X: produc186437863_state] :
( ! [A: produc410756839_state,B: produc2041926651_state,C2: produc2041926651_state] : ( P @ ( produc2065881943_state @ A @ ( produc1909270103_state @ B @ C2 ) ) )
=> ( P @ X ) ) ).
% prod_induct3
thf(fact_136_exec1_OSemi,axiom,
! [C1: com,C22: com,Cs2: list_com,S: state] : ( member1069318160_state @ ( produc1909270103_state @ ( produc1204172211_state @ ( cons_com @ ( semi @ C1 @ C22 ) @ Cs2 ) @ S ) @ ( produc1204172211_state @ ( cons_com @ C1 @ ( cons_com @ C22 @ Cs2 ) ) @ S ) ) @ pHoare259243666_exec1 ) ).
% exec1.Semi
thf(fact_137_exec1E_I3_J,axiom,
! [C1: com,C22: com,Cs2: list_com,S: state,Cs: list_com,S2: state] :
( ( member1069318160_state @ ( produc1909270103_state @ ( produc1204172211_state @ ( cons_com @ ( semi @ C1 @ C22 ) @ Cs2 ) @ S ) @ ( produc1204172211_state @ Cs @ S2 ) ) @ pHoare259243666_exec1 )
=> ~ ( ( Cs
= ( cons_com @ C1 @ ( cons_com @ C22 @ Cs2 ) ) )
=> ( S2 != S ) ) ) ).
% exec1E(3)
thf(fact_138_exec1_ODo,axiom,
! [T2: state,F2: state > set_state,S: state,Cs2: list_com] :
( ( member_state @ T2 @ ( F2 @ S ) )
=> ( member1069318160_state @ ( produc1909270103_state @ ( produc1204172211_state @ ( cons_com @ ( do @ F2 ) @ Cs2 ) @ S ) @ ( produc1204172211_state @ Cs2 @ T2 ) ) @ pHoare259243666_exec1 ) ) ).
% exec1.Do
thf(fact_139_exec1E_I2_J,axiom,
! [F2: state > set_state,Cs2: list_com,S: state,Cs: list_com,S2: state] :
( ( member1069318160_state @ ( produc1909270103_state @ ( produc1204172211_state @ ( cons_com @ ( do @ F2 ) @ Cs2 ) @ S ) @ ( produc1204172211_state @ Cs @ S2 ) ) @ pHoare259243666_exec1 )
=> ~ ( ( Cs = Cs2 )
=> ~ ( member_state @ S2 @ ( F2 @ S ) ) ) ) ).
% exec1E(2)
thf(fact_140_exec1_OIfFalse,axiom,
! [B2: state > $o,S: state,C1: com,C22: com,Cs2: list_com] :
( ~ ( B2 @ S )
=> ( member1069318160_state @ ( produc1909270103_state @ ( produc1204172211_state @ ( cons_com @ ( cond @ B2 @ C1 @ C22 ) @ Cs2 ) @ S ) @ ( produc1204172211_state @ ( cons_com @ C22 @ Cs2 ) @ S ) ) @ pHoare259243666_exec1 ) ) ).
% exec1.IfFalse
thf(fact_141_exec1_OIfTrue,axiom,
! [B2: state > $o,S: state,C1: com,C22: com,Cs2: list_com] :
( ( B2 @ S )
=> ( member1069318160_state @ ( produc1909270103_state @ ( produc1204172211_state @ ( cons_com @ ( cond @ B2 @ C1 @ C22 ) @ Cs2 ) @ S ) @ ( produc1204172211_state @ ( cons_com @ C1 @ Cs2 ) @ S ) ) @ pHoare259243666_exec1 ) ) ).
% exec1.IfTrue
thf(fact_142_exec1E_I4_J,axiom,
! [B2: state > $o,C1: com,C22: com,Cs2: list_com,S: state,Cs: list_com,S2: state] :
( ( member1069318160_state @ ( produc1909270103_state @ ( produc1204172211_state @ ( cons_com @ ( cond @ B2 @ C1 @ C22 ) @ Cs2 ) @ S ) @ ( produc1204172211_state @ Cs @ S2 ) ) @ pHoare259243666_exec1 )
=> ( ( ( Cs
= ( cons_com @ C1 @ Cs2 ) )
=> ( ( S2 = S )
=> ~ ( B2 @ S ) ) )
=> ~ ( ( Cs
= ( cons_com @ C22 @ Cs2 ) )
=> ( ( S2 = S )
=> ( B2 @ S ) ) ) ) ) ).
% exec1E(4)
thf(fact_143_exec1_OWhileFalse,axiom,
! [B2: state > $o,S: state,C: com,Cs2: list_com] :
( ~ ( B2 @ S )
=> ( member1069318160_state @ ( produc1909270103_state @ ( produc1204172211_state @ ( cons_com @ ( while @ B2 @ C ) @ Cs2 ) @ S ) @ ( produc1204172211_state @ Cs2 @ S ) ) @ pHoare259243666_exec1 ) ) ).
% exec1.WhileFalse
thf(fact_144_exec1_OWhileTrue,axiom,
! [B2: state > $o,S: state,C: com,Cs2: list_com] :
( ( B2 @ S )
=> ( member1069318160_state @ ( produc1909270103_state @ ( produc1204172211_state @ ( cons_com @ ( while @ B2 @ C ) @ Cs2 ) @ S ) @ ( produc1204172211_state @ ( cons_com @ C @ ( cons_com @ ( while @ B2 @ C ) @ Cs2 ) ) @ S ) ) @ pHoare259243666_exec1 ) ) ).
% exec1.WhileTrue
thf(fact_145_while__termiE,axiom,
! [B2: state > $o,C: com,S: state] :
( ( termi @ ( while @ B2 @ C ) @ S )
=> ( ( B2 @ S )
=> ( termi @ C @ S ) ) ) ).
% while_termiE
thf(fact_146_while__termiE2,axiom,
! [B2: state > $o,C: com,S: state,T2: state] :
( ( termi @ ( while @ B2 @ C ) @ S )
=> ( ( B2 @ S )
=> ( ( exec @ S @ C @ T2 )
=> ( termi @ ( while @ B2 @ C ) @ T2 ) ) ) ) ).
% while_termiE2
thf(fact_147_exec1E_I5_J,axiom,
! [B2: state > $o,C: com,Cs2: list_com,S: state,Cs: list_com,S2: state] :
( ( member1069318160_state @ ( produc1909270103_state @ ( produc1204172211_state @ ( cons_com @ ( while @ B2 @ C ) @ Cs2 ) @ S ) @ ( produc1204172211_state @ Cs @ S2 ) ) @ pHoare259243666_exec1 )
=> ( ( ( Cs = Cs2 )
=> ( ( S2 = S )
=> ( B2 @ S ) ) )
=> ~ ( ( Cs
= ( cons_com @ C @ ( cons_com @ ( while @ B2 @ C ) @ Cs2 ) ) )
=> ( ( S2 = S )
=> ~ ( B2 @ S ) ) ) ) ) ).
% exec1E(5)
thf(fact_148_termi_OSemi,axiom,
! [C1: com,S0: state,C22: com] :
( ( termi @ C1 @ S0 )
=> ( ! [S1: state] :
( ( exec @ S0 @ C1 @ S1 )
=> ( termi @ C22 @ S1 ) )
=> ( termi @ ( semi @ C1 @ C22 ) @ S0 ) ) ) ).
% termi.Semi
thf(fact_149_termi_OWhileTrue,axiom,
! [B2: state > $o,S: state,C: com] :
( ( B2 @ S )
=> ( ( termi @ C @ S )
=> ( ! [T3: state] :
( ( exec @ S @ C @ T3 )
=> ( termi @ ( while @ B2 @ C ) @ T3 ) )
=> ( termi @ ( while @ B2 @ C ) @ S ) ) ) ) ).
% termi.WhileTrue
thf(fact_150_com_Oinject_I2_J,axiom,
! [X21: com,X22: com,Y21: com,Y22: com] :
( ( ( semi @ X21 @ X22 )
= ( semi @ Y21 @ Y22 ) )
= ( ( X21 = Y21 )
& ( X22 = Y22 ) ) ) ).
% com.inject(2)
thf(fact_151_com_Oinject_I1_J,axiom,
! [X1: state > set_state,Y1: state > set_state] :
( ( ( do @ X1 )
= ( do @ Y1 ) )
= ( X1 = Y1 ) ) ).
% com.inject(1)
thf(fact_152_com_Oinject_I3_J,axiom,
! [X31: state > $o,X322: com,X33: com,Y31: state > $o,Y32: com,Y33: com] :
( ( ( cond @ X31 @ X322 @ X33 )
= ( cond @ Y31 @ Y32 @ Y33 ) )
= ( ( X31 = Y31 )
& ( X322 = Y32 )
& ( X33 = Y33 ) ) ) ).
% com.inject(3)
thf(fact_153_com_Oinject_I4_J,axiom,
! [X41: state > $o,X42: com,Y41: state > $o,Y42: com] :
( ( ( while @ X41 @ X42 )
= ( while @ Y41 @ Y42 ) )
= ( ( X41 = Y41 )
& ( X42 = Y42 ) ) ) ).
% com.inject(4)
thf(fact_154_exec_OWhileFalse,axiom,
! [B2: state > $o,S: state,C: com] :
( ~ ( B2 @ S )
=> ( exec @ S @ ( while @ B2 @ C ) @ S ) ) ).
% exec.WhileFalse
thf(fact_155_exec_OWhileTrue,axiom,
! [B2: state > $o,S: state,C: com,T2: state,U: state] :
( ( B2 @ S )
=> ( ( exec @ S @ C @ T2 )
=> ( ( exec @ T2 @ ( while @ B2 @ C ) @ U )
=> ( exec @ S @ ( while @ B2 @ C ) @ U ) ) ) ) ).
% exec.WhileTrue
thf(fact_156_com_Odistinct_I19_J,axiom,
! [X31: state > $o,X322: com,X33: com,X41: state > $o,X42: com] :
( ( cond @ X31 @ X322 @ X33 )
!= ( while @ X41 @ X42 ) ) ).
% com.distinct(19)
thf(fact_157_termi_OWhileFalse,axiom,
! [B2: state > $o,S: state,C: com] :
( ~ ( B2 @ S )
=> ( termi @ ( while @ B2 @ C ) @ S ) ) ).
% termi.WhileFalse
thf(fact_158_com_Odistinct_I5_J,axiom,
! [X1: state > set_state,X41: state > $o,X42: com] :
( ( do @ X1 )
!= ( while @ X41 @ X42 ) ) ).
% com.distinct(5)
thf(fact_159_exec_OIfFalse,axiom,
! [B2: state > $o,S: state,C22: com,T2: state,C1: com] :
( ~ ( B2 @ S )
=> ( ( exec @ S @ C22 @ T2 )
=> ( exec @ S @ ( cond @ B2 @ C1 @ C22 ) @ T2 ) ) ) ).
% exec.IfFalse
thf(fact_160_exec_OIfTrue,axiom,
! [B2: state > $o,S: state,C1: com,T2: state,C22: com] :
( ( B2 @ S )
=> ( ( exec @ S @ C1 @ T2 )
=> ( exec @ S @ ( cond @ B2 @ C1 @ C22 ) @ T2 ) ) ) ).
% exec.IfTrue
thf(fact_161_com_Odistinct_I13_J,axiom,
! [X21: com,X22: com,X41: state > $o,X42: com] :
( ( semi @ X21 @ X22 )
!= ( while @ X41 @ X42 ) ) ).
% com.distinct(13)
thf(fact_162_exec_ODo,axiom,
! [T2: state,F2: state > set_state,S: state] :
( ( member_state @ T2 @ ( F2 @ S ) )
=> ( exec @ S @ ( do @ F2 ) @ T2 ) ) ).
% exec.Do
thf(fact_163_exec_OSemi,axiom,
! [S0: state,C1: com,S12: state,C22: com,S22: state] :
( ( exec @ S0 @ C1 @ S12 )
=> ( ( exec @ S12 @ C22 @ S22 )
=> ( exec @ S0 @ ( semi @ C1 @ C22 ) @ S22 ) ) ) ).
% exec.Semi
thf(fact_164_termi_OIfFalse,axiom,
! [B2: state > $o,S: state,C22: com,C1: com] :
( ~ ( B2 @ S )
=> ( ( termi @ C22 @ S )
=> ( termi @ ( cond @ B2 @ C1 @ C22 ) @ S ) ) ) ).
% termi.IfFalse
thf(fact_165_termi_OIfTrue,axiom,
! [B2: state > $o,S: state,C1: com,C22: com] :
( ( B2 @ S )
=> ( ( termi @ C1 @ S )
=> ( termi @ ( cond @ B2 @ C1 @ C22 ) @ S ) ) ) ).
% termi.IfTrue
thf(fact_166_com_Odistinct_I3_J,axiom,
! [X1: state > set_state,X31: state > $o,X322: com,X33: com] :
( ( do @ X1 )
!= ( cond @ X31 @ X322 @ X33 ) ) ).
% com.distinct(3)
thf(fact_167_com_Odistinct_I11_J,axiom,
! [X21: com,X22: com,X31: state > $o,X322: com,X33: com] :
( ( semi @ X21 @ X22 )
!= ( cond @ X31 @ X322 @ X33 ) ) ).
% com.distinct(11)
thf(fact_168_com_Odistinct_I1_J,axiom,
! [X1: state > set_state,X21: com,X22: com] :
( ( do @ X1 )
!= ( semi @ X21 @ X22 ) ) ).
% com.distinct(1)
thf(fact_169_product__lists_Osimps_I1_J,axiom,
( ( product_lists_com @ nil_list_com )
= ( cons_list_com @ nil_com @ nil_list_com ) ) ).
% product_lists.simps(1)
thf(fact_170_subseqs_Osimps_I1_J,axiom,
( ( subseqs_com @ nil_com )
= ( cons_list_com @ nil_com @ nil_list_com ) ) ).
% subseqs.simps(1)
thf(fact_171_app__execs,axiom,
! [Cs2: list_com,S: state,Cs: list_com,S2: state,Cs22: list_com] :
( ( member1069318160_state @ ( produc1909270103_state @ ( produc1204172211_state @ Cs2 @ S ) @ ( produc1204172211_state @ Cs @ S2 ) ) @ ( transi1302705790_state @ pHoare259243666_exec1 ) )
=> ( member1069318160_state @ ( produc1909270103_state @ ( produc1204172211_state @ ( append_com @ Cs2 @ Cs22 ) @ S ) @ ( produc1204172211_state @ ( append_com @ Cs @ Cs22 ) @ S2 ) ) @ ( transi1302705790_state @ pHoare259243666_exec1 ) ) ) ).
% app_execs
thf(fact_172_insert__Nil,axiom,
! [X: com] :
( ( insert_com @ X @ nil_com )
= ( cons_com @ X @ nil_com ) ) ).
% insert_Nil
thf(fact_173_listrel_Oinducts,axiom,
! [X1: list_com,X2: list_com,R: set_Pr1822651163om_com,P: list_com > list_com > $o] :
( ( member32358020st_com @ ( produc1307105171st_com @ X1 @ X2 ) @ ( listrel_com_com @ R ) )
=> ( ( P @ nil_com @ nil_com )
=> ( ! [X3: com,Y3: com,Xs: list_com,Ys: list_com] :
( ( member1333916004om_com @ ( product_Pair_com_com @ X3 @ Y3 ) @ R )
=> ( ( member32358020st_com @ ( produc1307105171st_com @ Xs @ Ys ) @ ( listrel_com_com @ R ) )
=> ( ( P @ Xs @ Ys )
=> ( P @ ( cons_com @ X3 @ Xs ) @ ( cons_com @ Y3 @ Ys ) ) ) ) )
=> ( P @ X1 @ X2 ) ) ) ) ).
% listrel.inducts
thf(fact_174_listrel_Oinducts,axiom,
! [X1: list_P1040014977_state,X2: list_P1040014977_state,R: set_Pr1165141447_state,P: list_P1040014977_state > list_P1040014977_state > $o] :
( ( member1267359376_state @ ( produc1298457559_state @ X1 @ X2 ) @ ( listre194873521_state @ R ) )
=> ( ( P @ nil_Pr1844458753_state @ nil_Pr1844458753_state )
=> ( ! [X3: produc2041926651_state,Y3: produc2041926651_state,Xs: list_P1040014977_state,Ys: list_P1040014977_state] :
( ( member1069318160_state @ ( produc1909270103_state @ X3 @ Y3 ) @ R )
=> ( ( member1267359376_state @ ( produc1298457559_state @ Xs @ Ys ) @ ( listre194873521_state @ R ) )
=> ( ( P @ Xs @ Ys )
=> ( P @ ( cons_P465369265_state @ X3 @ Xs ) @ ( cons_P465369265_state @ Y3 @ Ys ) ) ) ) )
=> ( P @ X1 @ X2 ) ) ) ) ).
% listrel.inducts
thf(fact_175_listrel_Oinducts,axiom,
! [X1: list_list_com,X2: list_state,R: set_Pr849496667_state,P: list_list_com > list_state > $o] :
( ( member1442949316_state @ ( produc1742567891_state @ X1 @ X2 ) @ ( listre1268245261_state @ R ) )
=> ( ( P @ nil_list_com @ nil_state )
=> ( ! [X3: list_com,Y3: state,Xs: list_list_com,Ys: list_state] :
( ( member64234660_state @ ( produc1204172211_state @ X3 @ Y3 ) @ R )
=> ( ( member1442949316_state @ ( produc1742567891_state @ Xs @ Ys ) @ ( listre1268245261_state @ R ) )
=> ( ( P @ Xs @ Ys )
=> ( P @ ( cons_list_com @ X3 @ Xs ) @ ( cons_state @ Y3 @ Ys ) ) ) ) )
=> ( P @ X1 @ X2 ) ) ) ) ).
% listrel.inducts
thf(fact_176_listrel_Oinducts,axiom,
! [X1: list_P890094829_state,X2: list_P890094829_state,R: set_Pr2096893127_state,P: list_P890094829_state > list_P890094829_state > $o] :
( ( member423306384_state @ ( produc1354769367_state @ X1 @ X2 ) @ ( listre1244407729_state @ R ) )
=> ( ( P @ nil_Pr1347322605_state @ nil_Pr1347322605_state )
=> ( ! [X3: produc410756839_state,Y3: produc410756839_state,Xs: list_P890094829_state,Ys: list_P890094829_state] :
( ( member512523024_state @ ( produc2065881943_state @ X3 @ Y3 ) @ R )
=> ( ( member423306384_state @ ( produc1354769367_state @ Xs @ Ys ) @ ( listre1244407729_state @ R ) )
=> ( ( P @ Xs @ Ys )
=> ( P @ ( cons_P438230685_state @ X3 @ Xs ) @ ( cons_P438230685_state @ Y3 @ Ys ) ) ) ) )
=> ( P @ X1 @ X2 ) ) ) ) ).
% listrel.inducts
thf(fact_177_append_Oassoc,axiom,
! [A2: list_com,B2: list_com,C: list_com] :
( ( append_com @ ( append_com @ A2 @ B2 ) @ C )
= ( append_com @ A2 @ ( append_com @ B2 @ C ) ) ) ).
% append.assoc
thf(fact_178_append__assoc,axiom,
! [Xs2: list_com,Ys2: list_com,Zs2: list_com] :
( ( append_com @ ( append_com @ Xs2 @ Ys2 ) @ Zs2 )
= ( append_com @ Xs2 @ ( append_com @ Ys2 @ Zs2 ) ) ) ).
% append_assoc
thf(fact_179_append__same__eq,axiom,
! [Ys2: list_com,Xs2: list_com,Zs2: list_com] :
( ( ( append_com @ Ys2 @ Xs2 )
= ( append_com @ Zs2 @ Xs2 ) )
= ( Ys2 = Zs2 ) ) ).
% append_same_eq
thf(fact_180_same__append__eq,axiom,
! [Xs2: list_com,Ys2: list_com,Zs2: list_com] :
( ( ( append_com @ Xs2 @ Ys2 )
= ( append_com @ Xs2 @ Zs2 ) )
= ( Ys2 = Zs2 ) ) ).
% same_append_eq
thf(fact_181_append__Nil2,axiom,
! [Xs2: list_com] :
( ( append_com @ Xs2 @ nil_com )
= Xs2 ) ).
% append_Nil2
thf(fact_182_append__self__conv,axiom,
! [Xs2: list_com,Ys2: list_com] :
( ( ( append_com @ Xs2 @ Ys2 )
= Xs2 )
= ( Ys2 = nil_com ) ) ).
% append_self_conv
thf(fact_183_self__append__conv,axiom,
! [Xs2: list_com,Ys2: list_com] :
( ( Xs2
= ( append_com @ Xs2 @ Ys2 ) )
= ( Ys2 = nil_com ) ) ).
% self_append_conv
thf(fact_184_append__self__conv2,axiom,
! [Xs2: list_com,Ys2: list_com] :
( ( ( append_com @ Xs2 @ Ys2 )
= Ys2 )
= ( Xs2 = nil_com ) ) ).
% append_self_conv2
thf(fact_185_self__append__conv2,axiom,
! [Ys2: list_com,Xs2: list_com] :
( ( Ys2
= ( append_com @ Xs2 @ Ys2 ) )
= ( Xs2 = nil_com ) ) ).
% self_append_conv2
thf(fact_186_Nil__is__append__conv,axiom,
! [Xs2: list_com,Ys2: list_com] :
( ( nil_com
= ( append_com @ Xs2 @ Ys2 ) )
= ( ( Xs2 = nil_com )
& ( Ys2 = nil_com ) ) ) ).
% Nil_is_append_conv
thf(fact_187_append__is__Nil__conv,axiom,
! [Xs2: list_com,Ys2: list_com] :
( ( ( append_com @ Xs2 @ Ys2 )
= nil_com )
= ( ( Xs2 = nil_com )
& ( Ys2 = nil_com ) ) ) ).
% append_is_Nil_conv
thf(fact_188_append_Oright__neutral,axiom,
! [A2: list_com] :
( ( append_com @ A2 @ nil_com )
= A2 ) ).
% append.right_neutral
thf(fact_189_append1__eq__conv,axiom,
! [Xs2: list_com,X: com,Ys2: list_com,Y: com] :
( ( ( append_com @ Xs2 @ ( cons_com @ X @ nil_com ) )
= ( append_com @ Ys2 @ ( cons_com @ Y @ nil_com ) ) )
= ( ( Xs2 = Ys2 )
& ( X = Y ) ) ) ).
% append1_eq_conv
thf(fact_190_listrel__rtrancl__refl,axiom,
! [Xs2: list_P1040014977_state,R: set_Pr1165141447_state] : ( member1267359376_state @ ( produc1298457559_state @ Xs2 @ Xs2 ) @ ( listre194873521_state @ ( transi1302705790_state @ R ) ) ) ).
% listrel_rtrancl_refl
thf(fact_191_listrel__rtrancl__refl,axiom,
! [Xs2: list_P890094829_state,R: set_Pr2096893127_state] : ( member423306384_state @ ( produc1354769367_state @ Xs2 @ Xs2 ) @ ( listre1244407729_state @ ( transi463013610_state @ R ) ) ) ).
% listrel_rtrancl_refl
thf(fact_192_append__Cons,axiom,
! [X: com,Xs2: list_com,Ys2: list_com] :
( ( append_com @ ( cons_com @ X @ Xs2 ) @ Ys2 )
= ( cons_com @ X @ ( append_com @ Xs2 @ Ys2 ) ) ) ).
% append_Cons
thf(fact_193_Cons__eq__appendI,axiom,
! [X: com,Xs1: list_com,Ys2: list_com,Xs2: list_com,Zs2: list_com] :
( ( ( cons_com @ X @ Xs1 )
= Ys2 )
=> ( ( Xs2
= ( append_com @ Xs1 @ Zs2 ) )
=> ( ( cons_com @ X @ Xs2 )
= ( append_com @ Ys2 @ Zs2 ) ) ) ) ).
% Cons_eq_appendI
thf(fact_194_append_Oleft__neutral,axiom,
! [A2: list_com] :
( ( append_com @ nil_com @ A2 )
= A2 ) ).
% append.left_neutral
thf(fact_195_append__Nil,axiom,
! [Ys2: list_com] :
( ( append_com @ nil_com @ Ys2 )
= Ys2 ) ).
% append_Nil
thf(fact_196_eq__Nil__appendI,axiom,
! [Xs2: list_com,Ys2: list_com] :
( ( Xs2 = Ys2 )
=> ( Xs2
= ( append_com @ nil_com @ Ys2 ) ) ) ).
% eq_Nil_appendI
thf(fact_197_append__eq__appendI,axiom,
! [Xs2: list_com,Xs1: list_com,Zs2: list_com,Ys2: list_com,Us: list_com] :
( ( ( append_com @ Xs2 @ Xs1 )
= Zs2 )
=> ( ( Ys2
= ( append_com @ Xs1 @ Us ) )
=> ( ( append_com @ Xs2 @ Ys2 )
= ( append_com @ Zs2 @ Us ) ) ) ) ).
% append_eq_appendI
thf(fact_198_append__eq__append__conv2,axiom,
! [Xs2: list_com,Ys2: list_com,Zs2: list_com,Ts: list_com] :
( ( ( append_com @ Xs2 @ Ys2 )
= ( append_com @ Zs2 @ Ts ) )
= ( ? [Us2: list_com] :
( ( ( Xs2
= ( append_com @ Zs2 @ Us2 ) )
& ( ( append_com @ Us2 @ Ys2 )
= Ts ) )
| ( ( ( append_com @ Xs2 @ Us2 )
= Zs2 )
& ( Ys2
= ( append_com @ Us2 @ Ts ) ) ) ) ) ) ).
% append_eq_append_conv2
thf(fact_199_rev__induct,axiom,
! [P: list_com > $o,Xs2: list_com] :
( ( P @ nil_com )
=> ( ! [X3: com,Xs: list_com] :
( ( P @ Xs )
=> ( P @ ( append_com @ Xs @ ( cons_com @ X3 @ nil_com ) ) ) )
=> ( P @ Xs2 ) ) ) ).
% rev_induct
thf(fact_200_rev__exhaust,axiom,
! [Xs2: list_com] :
( ( Xs2 != nil_com )
=> ~ ! [Ys: list_com,Y3: com] :
( Xs2
!= ( append_com @ Ys @ ( cons_com @ Y3 @ nil_com ) ) ) ) ).
% rev_exhaust
thf(fact_201_Cons__eq__append__conv,axiom,
! [X: com,Xs2: list_com,Ys2: list_com,Zs2: list_com] :
( ( ( cons_com @ X @ Xs2 )
= ( append_com @ Ys2 @ Zs2 ) )
= ( ( ( Ys2 = nil_com )
& ( ( cons_com @ X @ Xs2 )
= Zs2 ) )
| ? [Ys4: list_com] :
( ( ( cons_com @ X @ Ys4 )
= Ys2 )
& ( Xs2
= ( append_com @ Ys4 @ Zs2 ) ) ) ) ) ).
% Cons_eq_append_conv
thf(fact_202_append__eq__Cons__conv,axiom,
! [Ys2: list_com,Zs2: list_com,X: com,Xs2: list_com] :
( ( ( append_com @ Ys2 @ Zs2 )
= ( cons_com @ X @ Xs2 ) )
= ( ( ( Ys2 = nil_com )
& ( Zs2
= ( cons_com @ X @ Xs2 ) ) )
| ? [Ys4: list_com] :
( ( Ys2
= ( cons_com @ X @ Ys4 ) )
& ( ( append_com @ Ys4 @ Zs2 )
= Xs2 ) ) ) ) ).
% append_eq_Cons_conv
thf(fact_203_rev__nonempty__induct,axiom,
! [Xs2: list_com,P: list_com > $o] :
( ( Xs2 != nil_com )
=> ( ! [X3: com] : ( P @ ( cons_com @ X3 @ nil_com ) )
=> ( ! [X3: com,Xs: list_com] :
( ( Xs != nil_com )
=> ( ( P @ Xs )
=> ( P @ ( append_com @ Xs @ ( cons_com @ X3 @ nil_com ) ) ) ) )
=> ( P @ Xs2 ) ) ) ) ).
% rev_nonempty_induct
thf(fact_204_listrel__Nil2,axiom,
! [Xs2: list_com,R: set_Pr1822651163om_com] :
( ( member32358020st_com @ ( produc1307105171st_com @ Xs2 @ nil_com ) @ ( listrel_com_com @ R ) )
=> ( Xs2 = nil_com ) ) ).
% listrel_Nil2
thf(fact_205_listrel__Nil1,axiom,
! [Xs2: list_com,R: set_Pr1822651163om_com] :
( ( member32358020st_com @ ( produc1307105171st_com @ nil_com @ Xs2 ) @ ( listrel_com_com @ R ) )
=> ( Xs2 = nil_com ) ) ).
% listrel_Nil1
thf(fact_206_listrel_ONil,axiom,
! [R: set_Pr1822651163om_com] : ( member32358020st_com @ ( produc1307105171st_com @ nil_com @ nil_com ) @ ( listrel_com_com @ R ) ) ).
% listrel.Nil
thf(fact_207_listrel__rtrancl__trans,axiom,
! [Xs2: list_P1040014977_state,Ys2: list_P1040014977_state,R: set_Pr1165141447_state,Zs2: list_P1040014977_state] :
( ( member1267359376_state @ ( produc1298457559_state @ Xs2 @ Ys2 ) @ ( listre194873521_state @ ( transi1302705790_state @ R ) ) )
=> ( ( member1267359376_state @ ( produc1298457559_state @ Ys2 @ Zs2 ) @ ( listre194873521_state @ ( transi1302705790_state @ R ) ) )
=> ( member1267359376_state @ ( produc1298457559_state @ Xs2 @ Zs2 ) @ ( listre194873521_state @ ( transi1302705790_state @ R ) ) ) ) ) ).
% listrel_rtrancl_trans
thf(fact_208_listrel__rtrancl__trans,axiom,
! [Xs2: list_P890094829_state,Ys2: list_P890094829_state,R: set_Pr2096893127_state,Zs2: list_P890094829_state] :
( ( member423306384_state @ ( produc1354769367_state @ Xs2 @ Ys2 ) @ ( listre1244407729_state @ ( transi463013610_state @ R ) ) )
=> ( ( member423306384_state @ ( produc1354769367_state @ Ys2 @ Zs2 ) @ ( listre1244407729_state @ ( transi463013610_state @ R ) ) )
=> ( member423306384_state @ ( produc1354769367_state @ Xs2 @ Zs2 ) @ ( listre1244407729_state @ ( transi463013610_state @ R ) ) ) ) ) ).
% listrel_rtrancl_trans
thf(fact_209_listrel_OCons,axiom,
! [X: com,Y: com,R: set_Pr1822651163om_com,Xs2: list_com,Ys2: list_com] :
( ( member1333916004om_com @ ( product_Pair_com_com @ X @ Y ) @ R )
=> ( ( member32358020st_com @ ( produc1307105171st_com @ Xs2 @ Ys2 ) @ ( listrel_com_com @ R ) )
=> ( member32358020st_com @ ( produc1307105171st_com @ ( cons_com @ X @ Xs2 ) @ ( cons_com @ Y @ Ys2 ) ) @ ( listrel_com_com @ R ) ) ) ) ).
% listrel.Cons
thf(fact_210_listrel_OCons,axiom,
! [X: produc2041926651_state,Y: produc2041926651_state,R: set_Pr1165141447_state,Xs2: list_P1040014977_state,Ys2: list_P1040014977_state] :
( ( member1069318160_state @ ( produc1909270103_state @ X @ Y ) @ R )
=> ( ( member1267359376_state @ ( produc1298457559_state @ Xs2 @ Ys2 ) @ ( listre194873521_state @ R ) )
=> ( member1267359376_state @ ( produc1298457559_state @ ( cons_P465369265_state @ X @ Xs2 ) @ ( cons_P465369265_state @ Y @ Ys2 ) ) @ ( listre194873521_state @ R ) ) ) ) ).
% listrel.Cons
thf(fact_211_listrel_OCons,axiom,
! [X: list_com,Y: state,R: set_Pr849496667_state,Xs2: list_list_com,Ys2: list_state] :
( ( member64234660_state @ ( produc1204172211_state @ X @ Y ) @ R )
=> ( ( member1442949316_state @ ( produc1742567891_state @ Xs2 @ Ys2 ) @ ( listre1268245261_state @ R ) )
=> ( member1442949316_state @ ( produc1742567891_state @ ( cons_list_com @ X @ Xs2 ) @ ( cons_state @ Y @ Ys2 ) ) @ ( listre1268245261_state @ R ) ) ) ) ).
% listrel.Cons
thf(fact_212_listrel_OCons,axiom,
! [X: produc410756839_state,Y: produc410756839_state,R: set_Pr2096893127_state,Xs2: list_P890094829_state,Ys2: list_P890094829_state] :
( ( member512523024_state @ ( produc2065881943_state @ X @ Y ) @ R )
=> ( ( member423306384_state @ ( produc1354769367_state @ Xs2 @ Ys2 ) @ ( listre1244407729_state @ R ) )
=> ( member423306384_state @ ( produc1354769367_state @ ( cons_P438230685_state @ X @ Xs2 ) @ ( cons_P438230685_state @ Y @ Ys2 ) ) @ ( listre1244407729_state @ R ) ) ) ) ).
% listrel.Cons
thf(fact_213_listrel__Cons1,axiom,
! [Y: com,Ys2: list_com,Xs2: list_com,R: set_Pr1822651163om_com] :
( ( member32358020st_com @ ( produc1307105171st_com @ ( cons_com @ Y @ Ys2 ) @ Xs2 ) @ ( listrel_com_com @ R ) )
=> ~ ! [Y3: com,Ys: list_com] :
( ( Xs2
= ( cons_com @ Y3 @ Ys ) )
=> ( ( member1333916004om_com @ ( product_Pair_com_com @ Y @ Y3 ) @ R )
=> ~ ( member32358020st_com @ ( produc1307105171st_com @ Ys2 @ Ys ) @ ( listrel_com_com @ R ) ) ) ) ) ).
% listrel_Cons1
thf(fact_214_listrel__Cons1,axiom,
! [Y: produc2041926651_state,Ys2: list_P1040014977_state,Xs2: list_P1040014977_state,R: set_Pr1165141447_state] :
( ( member1267359376_state @ ( produc1298457559_state @ ( cons_P465369265_state @ Y @ Ys2 ) @ Xs2 ) @ ( listre194873521_state @ R ) )
=> ~ ! [Y3: produc2041926651_state,Ys: list_P1040014977_state] :
( ( Xs2
= ( cons_P465369265_state @ Y3 @ Ys ) )
=> ( ( member1069318160_state @ ( produc1909270103_state @ Y @ Y3 ) @ R )
=> ~ ( member1267359376_state @ ( produc1298457559_state @ Ys2 @ Ys ) @ ( listre194873521_state @ R ) ) ) ) ) ).
% listrel_Cons1
thf(fact_215_listrel__Cons1,axiom,
! [Y: list_com,Ys2: list_list_com,Xs2: list_state,R: set_Pr849496667_state] :
( ( member1442949316_state @ ( produc1742567891_state @ ( cons_list_com @ Y @ Ys2 ) @ Xs2 ) @ ( listre1268245261_state @ R ) )
=> ~ ! [Y3: state,Ys: list_state] :
( ( Xs2
= ( cons_state @ Y3 @ Ys ) )
=> ( ( member64234660_state @ ( produc1204172211_state @ Y @ Y3 ) @ R )
=> ~ ( member1442949316_state @ ( produc1742567891_state @ Ys2 @ Ys ) @ ( listre1268245261_state @ R ) ) ) ) ) ).
% listrel_Cons1
thf(fact_216_listrel__Cons1,axiom,
! [Y: produc410756839_state,Ys2: list_P890094829_state,Xs2: list_P890094829_state,R: set_Pr2096893127_state] :
( ( member423306384_state @ ( produc1354769367_state @ ( cons_P438230685_state @ Y @ Ys2 ) @ Xs2 ) @ ( listre1244407729_state @ R ) )
=> ~ ! [Y3: produc410756839_state,Ys: list_P890094829_state] :
( ( Xs2
= ( cons_P438230685_state @ Y3 @ Ys ) )
=> ( ( member512523024_state @ ( produc2065881943_state @ Y @ Y3 ) @ R )
=> ~ ( member423306384_state @ ( produc1354769367_state @ Ys2 @ Ys ) @ ( listre1244407729_state @ R ) ) ) ) ) ).
% listrel_Cons1
thf(fact_217_listrel__Cons2,axiom,
! [Xs2: list_com,Y: com,Ys2: list_com,R: set_Pr1822651163om_com] :
( ( member32358020st_com @ ( produc1307105171st_com @ Xs2 @ ( cons_com @ Y @ Ys2 ) ) @ ( listrel_com_com @ R ) )
=> ~ ! [X3: com,Xs: list_com] :
( ( Xs2
= ( cons_com @ X3 @ Xs ) )
=> ( ( member1333916004om_com @ ( product_Pair_com_com @ X3 @ Y ) @ R )
=> ~ ( member32358020st_com @ ( produc1307105171st_com @ Xs @ Ys2 ) @ ( listrel_com_com @ R ) ) ) ) ) ).
% listrel_Cons2
thf(fact_218_listrel__Cons2,axiom,
! [Xs2: list_P1040014977_state,Y: produc2041926651_state,Ys2: list_P1040014977_state,R: set_Pr1165141447_state] :
( ( member1267359376_state @ ( produc1298457559_state @ Xs2 @ ( cons_P465369265_state @ Y @ Ys2 ) ) @ ( listre194873521_state @ R ) )
=> ~ ! [X3: produc2041926651_state,Xs: list_P1040014977_state] :
( ( Xs2
= ( cons_P465369265_state @ X3 @ Xs ) )
=> ( ( member1069318160_state @ ( produc1909270103_state @ X3 @ Y ) @ R )
=> ~ ( member1267359376_state @ ( produc1298457559_state @ Xs @ Ys2 ) @ ( listre194873521_state @ R ) ) ) ) ) ).
% listrel_Cons2
thf(fact_219_listrel__Cons2,axiom,
! [Xs2: list_list_com,Y: state,Ys2: list_state,R: set_Pr849496667_state] :
( ( member1442949316_state @ ( produc1742567891_state @ Xs2 @ ( cons_state @ Y @ Ys2 ) ) @ ( listre1268245261_state @ R ) )
=> ~ ! [X3: list_com,Xs: list_list_com] :
( ( Xs2
= ( cons_list_com @ X3 @ Xs ) )
=> ( ( member64234660_state @ ( produc1204172211_state @ X3 @ Y ) @ R )
=> ~ ( member1442949316_state @ ( produc1742567891_state @ Xs @ Ys2 ) @ ( listre1268245261_state @ R ) ) ) ) ) ).
% listrel_Cons2
thf(fact_220_listrel__Cons2,axiom,
! [Xs2: list_P890094829_state,Y: produc410756839_state,Ys2: list_P890094829_state,R: set_Pr2096893127_state] :
( ( member423306384_state @ ( produc1354769367_state @ Xs2 @ ( cons_P438230685_state @ Y @ Ys2 ) ) @ ( listre1244407729_state @ R ) )
=> ~ ! [X3: produc410756839_state,Xs: list_P890094829_state] :
( ( Xs2
= ( cons_P438230685_state @ X3 @ Xs ) )
=> ( ( member512523024_state @ ( produc2065881943_state @ X3 @ Y ) @ R )
=> ~ ( member423306384_state @ ( produc1354769367_state @ Xs @ Ys2 ) @ ( listre1244407729_state @ R ) ) ) ) ) ).
% listrel_Cons2
thf(fact_221_app__exec,axiom,
! [Cs2: list_com,S: state,Cs: list_com,S2: state,Cs22: list_com] :
( ( member1069318160_state @ ( produc1909270103_state @ ( produc1204172211_state @ Cs2 @ S ) @ ( produc1204172211_state @ Cs @ S2 ) ) @ pHoare259243666_exec1 )
=> ( member1069318160_state @ ( produc1909270103_state @ ( produc1204172211_state @ ( append_com @ Cs2 @ Cs22 ) @ S ) @ ( produc1204172211_state @ ( append_com @ Cs @ Cs22 ) @ S2 ) ) @ pHoare259243666_exec1 ) ) ).
% app_exec
thf(fact_222_listrel_Ocases,axiom,
! [A1: list_com,A22: list_com,R: set_Pr1822651163om_com] :
( ( member32358020st_com @ ( produc1307105171st_com @ A1 @ A22 ) @ ( listrel_com_com @ R ) )
=> ( ( ( A1 = nil_com )
=> ( A22 != nil_com ) )
=> ~ ! [X3: com,Y3: com,Xs: list_com] :
( ( A1
= ( cons_com @ X3 @ Xs ) )
=> ! [Ys: list_com] :
( ( A22
= ( cons_com @ Y3 @ Ys ) )
=> ( ( member1333916004om_com @ ( product_Pair_com_com @ X3 @ Y3 ) @ R )
=> ~ ( member32358020st_com @ ( produc1307105171st_com @ Xs @ Ys ) @ ( listrel_com_com @ R ) ) ) ) ) ) ) ).
% listrel.cases
thf(fact_223_listrel_Ocases,axiom,
! [A1: list_P1040014977_state,A22: list_P1040014977_state,R: set_Pr1165141447_state] :
( ( member1267359376_state @ ( produc1298457559_state @ A1 @ A22 ) @ ( listre194873521_state @ R ) )
=> ( ( ( A1 = nil_Pr1844458753_state )
=> ( A22 != nil_Pr1844458753_state ) )
=> ~ ! [X3: produc2041926651_state,Y3: produc2041926651_state,Xs: list_P1040014977_state] :
( ( A1
= ( cons_P465369265_state @ X3 @ Xs ) )
=> ! [Ys: list_P1040014977_state] :
( ( A22
= ( cons_P465369265_state @ Y3 @ Ys ) )
=> ( ( member1069318160_state @ ( produc1909270103_state @ X3 @ Y3 ) @ R )
=> ~ ( member1267359376_state @ ( produc1298457559_state @ Xs @ Ys ) @ ( listre194873521_state @ R ) ) ) ) ) ) ) ).
% listrel.cases
thf(fact_224_listrel_Ocases,axiom,
! [A1: list_list_com,A22: list_state,R: set_Pr849496667_state] :
( ( member1442949316_state @ ( produc1742567891_state @ A1 @ A22 ) @ ( listre1268245261_state @ R ) )
=> ( ( ( A1 = nil_list_com )
=> ( A22 != nil_state ) )
=> ~ ! [X3: list_com,Y3: state,Xs: list_list_com] :
( ( A1
= ( cons_list_com @ X3 @ Xs ) )
=> ! [Ys: list_state] :
( ( A22
= ( cons_state @ Y3 @ Ys ) )
=> ( ( member64234660_state @ ( produc1204172211_state @ X3 @ Y3 ) @ R )
=> ~ ( member1442949316_state @ ( produc1742567891_state @ Xs @ Ys ) @ ( listre1268245261_state @ R ) ) ) ) ) ) ) ).
% listrel.cases
thf(fact_225_listrel_Ocases,axiom,
! [A1: list_P890094829_state,A22: list_P890094829_state,R: set_Pr2096893127_state] :
( ( member423306384_state @ ( produc1354769367_state @ A1 @ A22 ) @ ( listre1244407729_state @ R ) )
=> ( ( ( A1 = nil_Pr1347322605_state )
=> ( A22 != nil_Pr1347322605_state ) )
=> ~ ! [X3: produc410756839_state,Y3: produc410756839_state,Xs: list_P890094829_state] :
( ( A1
= ( cons_P438230685_state @ X3 @ Xs ) )
=> ! [Ys: list_P890094829_state] :
( ( A22
= ( cons_P438230685_state @ Y3 @ Ys ) )
=> ( ( member512523024_state @ ( produc2065881943_state @ X3 @ Y3 ) @ R )
=> ~ ( member423306384_state @ ( produc1354769367_state @ Xs @ Ys ) @ ( listre1244407729_state @ R ) ) ) ) ) ) ) ).
% listrel.cases
thf(fact_226_listrel_Osimps,axiom,
! [A1: list_com,A22: list_com,R: set_Pr1822651163om_com] :
( ( member32358020st_com @ ( produc1307105171st_com @ A1 @ A22 ) @ ( listrel_com_com @ R ) )
= ( ( ( A1 = nil_com )
& ( A22 = nil_com ) )
| ? [X4: com,Y4: com,Xs3: list_com,Ys3: list_com] :
( ( A1
= ( cons_com @ X4 @ Xs3 ) )
& ( A22
= ( cons_com @ Y4 @ Ys3 ) )
& ( member1333916004om_com @ ( product_Pair_com_com @ X4 @ Y4 ) @ R )
& ( member32358020st_com @ ( produc1307105171st_com @ Xs3 @ Ys3 ) @ ( listrel_com_com @ R ) ) ) ) ) ).
% listrel.simps
thf(fact_227_listrel_Osimps,axiom,
! [A1: list_P1040014977_state,A22: list_P1040014977_state,R: set_Pr1165141447_state] :
( ( member1267359376_state @ ( produc1298457559_state @ A1 @ A22 ) @ ( listre194873521_state @ R ) )
= ( ( ( A1 = nil_Pr1844458753_state )
& ( A22 = nil_Pr1844458753_state ) )
| ? [X4: produc2041926651_state,Y4: produc2041926651_state,Xs3: list_P1040014977_state,Ys3: list_P1040014977_state] :
( ( A1
= ( cons_P465369265_state @ X4 @ Xs3 ) )
& ( A22
= ( cons_P465369265_state @ Y4 @ Ys3 ) )
& ( member1069318160_state @ ( produc1909270103_state @ X4 @ Y4 ) @ R )
& ( member1267359376_state @ ( produc1298457559_state @ Xs3 @ Ys3 ) @ ( listre194873521_state @ R ) ) ) ) ) ).
% listrel.simps
thf(fact_228_listrel_Osimps,axiom,
! [A1: list_list_com,A22: list_state,R: set_Pr849496667_state] :
( ( member1442949316_state @ ( produc1742567891_state @ A1 @ A22 ) @ ( listre1268245261_state @ R ) )
= ( ( ( A1 = nil_list_com )
& ( A22 = nil_state ) )
| ? [X4: list_com,Y4: state,Xs3: list_list_com,Ys3: list_state] :
( ( A1
= ( cons_list_com @ X4 @ Xs3 ) )
& ( A22
= ( cons_state @ Y4 @ Ys3 ) )
& ( member64234660_state @ ( produc1204172211_state @ X4 @ Y4 ) @ R )
& ( member1442949316_state @ ( produc1742567891_state @ Xs3 @ Ys3 ) @ ( listre1268245261_state @ R ) ) ) ) ) ).
% listrel.simps
thf(fact_229_listrel_Osimps,axiom,
! [A1: list_P890094829_state,A22: list_P890094829_state,R: set_Pr2096893127_state] :
( ( member423306384_state @ ( produc1354769367_state @ A1 @ A22 ) @ ( listre1244407729_state @ R ) )
= ( ( ( A1 = nil_Pr1347322605_state )
& ( A22 = nil_Pr1347322605_state ) )
| ? [X4: produc410756839_state,Y4: produc410756839_state,Xs3: list_P890094829_state,Ys3: list_P890094829_state] :
( ( A1
= ( cons_P438230685_state @ X4 @ Xs3 ) )
& ( A22
= ( cons_P438230685_state @ Y4 @ Ys3 ) )
& ( member512523024_state @ ( produc2065881943_state @ X4 @ Y4 ) @ R )
& ( member423306384_state @ ( produc1354769367_state @ Xs3 @ Ys3 ) @ ( listre1244407729_state @ R ) ) ) ) ) ).
% listrel.simps
thf(fact_230_bind__simps_I2_J,axiom,
! [X: com,Xs2: list_com,F2: com > list_com] :
( ( bind_com_com @ ( cons_com @ X @ Xs2 ) @ F2 )
= ( append_com @ ( F2 @ X ) @ ( bind_com_com @ Xs2 @ F2 ) ) ) ).
% bind_simps(2)
thf(fact_231_snoc__listrel1__snoc__iff,axiom,
! [Xs2: list_com,X: com,Ys2: list_com,Y: com,R: set_Pr1822651163om_com] :
( ( member32358020st_com @ ( produc1307105171st_com @ ( append_com @ Xs2 @ ( cons_com @ X @ nil_com ) ) @ ( append_com @ Ys2 @ ( cons_com @ Y @ nil_com ) ) ) @ ( listrel1_com @ R ) )
= ( ( ( member32358020st_com @ ( produc1307105171st_com @ Xs2 @ Ys2 ) @ ( listrel1_com @ R ) )
& ( X = Y ) )
| ( ( Xs2 = Ys2 )
& ( member1333916004om_com @ ( product_Pair_com_com @ X @ Y ) @ R ) ) ) ) ).
% snoc_listrel1_snoc_iff
thf(fact_232_snoc__listrel1__snoc__iff,axiom,
! [Xs2: list_P1040014977_state,X: produc2041926651_state,Ys2: list_P1040014977_state,Y: produc2041926651_state,R: set_Pr1165141447_state] :
( ( member1267359376_state @ ( produc1298457559_state @ ( append982487830_state @ Xs2 @ ( cons_P465369265_state @ X @ nil_Pr1844458753_state ) ) @ ( append982487830_state @ Ys2 @ ( cons_P465369265_state @ Y @ nil_Pr1844458753_state ) ) ) @ ( listre1738592858_state @ R ) )
= ( ( ( member1267359376_state @ ( produc1298457559_state @ Xs2 @ Ys2 ) @ ( listre1738592858_state @ R ) )
& ( X = Y ) )
| ( ( Xs2 = Ys2 )
& ( member1069318160_state @ ( produc1909270103_state @ X @ Y ) @ R ) ) ) ) ).
% snoc_listrel1_snoc_iff
thf(fact_233_snoc__listrel1__snoc__iff,axiom,
! [Xs2: list_P890094829_state,X: produc410756839_state,Ys2: list_P890094829_state,Y: produc410756839_state,R: set_Pr2096893127_state] :
( ( member423306384_state @ ( produc1354769367_state @ ( append221843330_state @ Xs2 @ ( cons_P438230685_state @ X @ nil_Pr1347322605_state ) ) @ ( append221843330_state @ Ys2 @ ( cons_P438230685_state @ Y @ nil_Pr1347322605_state ) ) ) @ ( listre1725776070_state @ R ) )
= ( ( ( member423306384_state @ ( produc1354769367_state @ Xs2 @ Ys2 ) @ ( listre1725776070_state @ R ) )
& ( X = Y ) )
| ( ( Xs2 = Ys2 )
& ( member512523024_state @ ( produc2065881943_state @ X @ Y ) @ R ) ) ) ) ).
% snoc_listrel1_snoc_iff
thf(fact_234_maps__simps_I1_J,axiom,
! [F2: com > list_com,X: com,Xs2: list_com] :
( ( maps_com_com @ F2 @ ( cons_com @ X @ Xs2 ) )
= ( append_com @ ( F2 @ X ) @ ( maps_com_com @ F2 @ Xs2 ) ) ) ).
% maps_simps(1)
thf(fact_235_concat__eq__append__conv,axiom,
! [Xss2: list_list_com,Ys2: list_com,Zs2: list_com] :
( ( ( concat_com @ Xss2 )
= ( append_com @ Ys2 @ Zs2 ) )
= ( ( ( Xss2 = nil_list_com )
=> ( ( Ys2 = nil_com )
& ( Zs2 = nil_com ) ) )
& ( ( Xss2 != nil_list_com )
=> ? [Xss1: list_list_com,Xs3: list_com,Xs4: list_com,Xss22: list_list_com] :
( ( Xss2
= ( append_list_com @ Xss1 @ ( cons_list_com @ ( append_com @ Xs3 @ Xs4 ) @ Xss22 ) ) )
& ( Ys2
= ( append_com @ ( concat_com @ Xss1 ) @ Xs3 ) )
& ( Zs2
= ( append_com @ Xs4 @ ( concat_com @ Xss22 ) ) ) ) ) ) ) ).
% concat_eq_append_conv
thf(fact_236_bind__simps_I1_J,axiom,
! [F2: com > list_com] :
( ( bind_com_com @ nil_com @ F2 )
= nil_com ) ).
% bind_simps(1)
thf(fact_237_concat__append,axiom,
! [Xs2: list_list_com,Ys2: list_list_com] :
( ( concat_com @ ( append_list_com @ Xs2 @ Ys2 ) )
= ( append_com @ ( concat_com @ Xs2 ) @ ( concat_com @ Ys2 ) ) ) ).
% concat_append
thf(fact_238_Cons__listrel1__Cons,axiom,
! [X: com,Xs2: list_com,Y: com,Ys2: list_com,R: set_Pr1822651163om_com] :
( ( member32358020st_com @ ( produc1307105171st_com @ ( cons_com @ X @ Xs2 ) @ ( cons_com @ Y @ Ys2 ) ) @ ( listrel1_com @ R ) )
= ( ( ( member1333916004om_com @ ( product_Pair_com_com @ X @ Y ) @ R )
& ( Xs2 = Ys2 ) )
| ( ( X = Y )
& ( member32358020st_com @ ( produc1307105171st_com @ Xs2 @ Ys2 ) @ ( listrel1_com @ R ) ) ) ) ) ).
% Cons_listrel1_Cons
thf(fact_239_Cons__listrel1__Cons,axiom,
! [X: produc2041926651_state,Xs2: list_P1040014977_state,Y: produc2041926651_state,Ys2: list_P1040014977_state,R: set_Pr1165141447_state] :
( ( member1267359376_state @ ( produc1298457559_state @ ( cons_P465369265_state @ X @ Xs2 ) @ ( cons_P465369265_state @ Y @ Ys2 ) ) @ ( listre1738592858_state @ R ) )
= ( ( ( member1069318160_state @ ( produc1909270103_state @ X @ Y ) @ R )
& ( Xs2 = Ys2 ) )
| ( ( X = Y )
& ( member1267359376_state @ ( produc1298457559_state @ Xs2 @ Ys2 ) @ ( listre1738592858_state @ R ) ) ) ) ) ).
% Cons_listrel1_Cons
thf(fact_240_Cons__listrel1__Cons,axiom,
! [X: produc410756839_state,Xs2: list_P890094829_state,Y: produc410756839_state,Ys2: list_P890094829_state,R: set_Pr2096893127_state] :
( ( member423306384_state @ ( produc1354769367_state @ ( cons_P438230685_state @ X @ Xs2 ) @ ( cons_P438230685_state @ Y @ Ys2 ) ) @ ( listre1725776070_state @ R ) )
= ( ( ( member512523024_state @ ( produc2065881943_state @ X @ Y ) @ R )
& ( Xs2 = Ys2 ) )
| ( ( X = Y )
& ( member423306384_state @ ( produc1354769367_state @ Xs2 @ Ys2 ) @ ( listre1725776070_state @ R ) ) ) ) ) ).
% Cons_listrel1_Cons
thf(fact_241_listrel1I2,axiom,
! [Xs2: list_com,Ys2: list_com,R: set_Pr1822651163om_com,X: com] :
( ( member32358020st_com @ ( produc1307105171st_com @ Xs2 @ Ys2 ) @ ( listrel1_com @ R ) )
=> ( member32358020st_com @ ( produc1307105171st_com @ ( cons_com @ X @ Xs2 ) @ ( cons_com @ X @ Ys2 ) ) @ ( listrel1_com @ R ) ) ) ).
% listrel1I2
thf(fact_242_rtrancl__listrel1__ConsI1,axiom,
! [Xs2: list_com,Ys2: list_com,R: set_Pr1822651163om_com,X: com] :
( ( member32358020st_com @ ( produc1307105171st_com @ Xs2 @ Ys2 ) @ ( transi436345274st_com @ ( listrel1_com @ R ) ) )
=> ( member32358020st_com @ ( produc1307105171st_com @ ( cons_com @ X @ Xs2 ) @ ( cons_com @ X @ Ys2 ) ) @ ( transi436345274st_com @ ( listrel1_com @ R ) ) ) ) ).
% rtrancl_listrel1_ConsI1
thf(fact_243_not__Nil__listrel1,axiom,
! [Xs2: list_com,R: set_Pr1822651163om_com] :
~ ( member32358020st_com @ ( produc1307105171st_com @ nil_com @ Xs2 ) @ ( listrel1_com @ R ) ) ).
% not_Nil_listrel1
thf(fact_244_not__listrel1__Nil,axiom,
! [Xs2: list_com,R: set_Pr1822651163om_com] :
~ ( member32358020st_com @ ( produc1307105171st_com @ Xs2 @ nil_com ) @ ( listrel1_com @ R ) ) ).
% not_listrel1_Nil
thf(fact_245_append__listrel1I,axiom,
! [Xs2: list_com,Ys2: list_com,R: set_Pr1822651163om_com,Us: list_com,Vs: list_com] :
( ( ( ( member32358020st_com @ ( produc1307105171st_com @ Xs2 @ Ys2 ) @ ( listrel1_com @ R ) )
& ( Us = Vs ) )
| ( ( Xs2 = Ys2 )
& ( member32358020st_com @ ( produc1307105171st_com @ Us @ Vs ) @ ( listrel1_com @ R ) ) ) )
=> ( member32358020st_com @ ( produc1307105171st_com @ ( append_com @ Xs2 @ Us ) @ ( append_com @ Ys2 @ Vs ) ) @ ( listrel1_com @ R ) ) ) ).
% append_listrel1I
thf(fact_246_concat_Osimps_I1_J,axiom,
( ( concat_com @ nil_list_com )
= nil_com ) ).
% concat.simps(1)
thf(fact_247_concat_Osimps_I2_J,axiom,
! [X: list_com,Xs2: list_list_com] :
( ( concat_com @ ( cons_list_com @ X @ Xs2 ) )
= ( append_com @ X @ ( concat_com @ Xs2 ) ) ) ).
% concat.simps(2)
thf(fact_248_listrel__rtrancl__eq__rtrancl__listrel1,axiom,
! [R: set_Pr1165141447_state] :
( ( listre194873521_state @ ( transi1302705790_state @ R ) )
= ( transi339813252_state @ ( listre1738592858_state @ R ) ) ) ).
% listrel_rtrancl_eq_rtrancl_listrel1
thf(fact_249_listrel__rtrancl__eq__rtrancl__listrel1,axiom,
! [R: set_Pr2096893127_state] :
( ( listre1244407729_state @ ( transi463013610_state @ R ) )
= ( transi1965148016_state @ ( listre1725776070_state @ R ) ) ) ).
% listrel_rtrancl_eq_rtrancl_listrel1
thf(fact_250_maps__simps_I2_J,axiom,
! [F2: com > list_com] :
( ( maps_com_com @ F2 @ nil_com )
= nil_com ) ).
% maps_simps(2)
thf(fact_251_Cons__listrel1E2,axiom,
! [Xs2: list_com,Y: com,Ys2: list_com,R: set_Pr1822651163om_com] :
( ( member32358020st_com @ ( produc1307105171st_com @ Xs2 @ ( cons_com @ Y @ Ys2 ) ) @ ( listrel1_com @ R ) )
=> ( ! [X3: com] :
( ( Xs2
= ( cons_com @ X3 @ Ys2 ) )
=> ~ ( member1333916004om_com @ ( product_Pair_com_com @ X3 @ Y ) @ R ) )
=> ~ ! [Zs: list_com] :
( ( Xs2
= ( cons_com @ Y @ Zs ) )
=> ~ ( member32358020st_com @ ( produc1307105171st_com @ Zs @ Ys2 ) @ ( listrel1_com @ R ) ) ) ) ) ).
% Cons_listrel1E2
thf(fact_252_Cons__listrel1E2,axiom,
! [Xs2: list_P1040014977_state,Y: produc2041926651_state,Ys2: list_P1040014977_state,R: set_Pr1165141447_state] :
( ( member1267359376_state @ ( produc1298457559_state @ Xs2 @ ( cons_P465369265_state @ Y @ Ys2 ) ) @ ( listre1738592858_state @ R ) )
=> ( ! [X3: produc2041926651_state] :
( ( Xs2
= ( cons_P465369265_state @ X3 @ Ys2 ) )
=> ~ ( member1069318160_state @ ( produc1909270103_state @ X3 @ Y ) @ R ) )
=> ~ ! [Zs: list_P1040014977_state] :
( ( Xs2
= ( cons_P465369265_state @ Y @ Zs ) )
=> ~ ( member1267359376_state @ ( produc1298457559_state @ Zs @ Ys2 ) @ ( listre1738592858_state @ R ) ) ) ) ) ).
% Cons_listrel1E2
thf(fact_253_Cons__listrel1E2,axiom,
! [Xs2: list_P890094829_state,Y: produc410756839_state,Ys2: list_P890094829_state,R: set_Pr2096893127_state] :
( ( member423306384_state @ ( produc1354769367_state @ Xs2 @ ( cons_P438230685_state @ Y @ Ys2 ) ) @ ( listre1725776070_state @ R ) )
=> ( ! [X3: produc410756839_state] :
( ( Xs2
= ( cons_P438230685_state @ X3 @ Ys2 ) )
=> ~ ( member512523024_state @ ( produc2065881943_state @ X3 @ Y ) @ R ) )
=> ~ ! [Zs: list_P890094829_state] :
( ( Xs2
= ( cons_P438230685_state @ Y @ Zs ) )
=> ~ ( member423306384_state @ ( produc1354769367_state @ Zs @ Ys2 ) @ ( listre1725776070_state @ R ) ) ) ) ) ).
% Cons_listrel1E2
thf(fact_254_Cons__listrel1E1,axiom,
! [X: com,Xs2: list_com,Ys2: list_com,R: set_Pr1822651163om_com] :
( ( member32358020st_com @ ( produc1307105171st_com @ ( cons_com @ X @ Xs2 ) @ Ys2 ) @ ( listrel1_com @ R ) )
=> ( ! [Y3: com] :
( ( Ys2
= ( cons_com @ Y3 @ Xs2 ) )
=> ~ ( member1333916004om_com @ ( product_Pair_com_com @ X @ Y3 ) @ R ) )
=> ~ ! [Zs: list_com] :
( ( Ys2
= ( cons_com @ X @ Zs ) )
=> ~ ( member32358020st_com @ ( produc1307105171st_com @ Xs2 @ Zs ) @ ( listrel1_com @ R ) ) ) ) ) ).
% Cons_listrel1E1
thf(fact_255_Cons__listrel1E1,axiom,
! [X: produc2041926651_state,Xs2: list_P1040014977_state,Ys2: list_P1040014977_state,R: set_Pr1165141447_state] :
( ( member1267359376_state @ ( produc1298457559_state @ ( cons_P465369265_state @ X @ Xs2 ) @ Ys2 ) @ ( listre1738592858_state @ R ) )
=> ( ! [Y3: produc2041926651_state] :
( ( Ys2
= ( cons_P465369265_state @ Y3 @ Xs2 ) )
=> ~ ( member1069318160_state @ ( produc1909270103_state @ X @ Y3 ) @ R ) )
=> ~ ! [Zs: list_P1040014977_state] :
( ( Ys2
= ( cons_P465369265_state @ X @ Zs ) )
=> ~ ( member1267359376_state @ ( produc1298457559_state @ Xs2 @ Zs ) @ ( listre1738592858_state @ R ) ) ) ) ) ).
% Cons_listrel1E1
thf(fact_256_Cons__listrel1E1,axiom,
! [X: produc410756839_state,Xs2: list_P890094829_state,Ys2: list_P890094829_state,R: set_Pr2096893127_state] :
( ( member423306384_state @ ( produc1354769367_state @ ( cons_P438230685_state @ X @ Xs2 ) @ Ys2 ) @ ( listre1725776070_state @ R ) )
=> ( ! [Y3: produc410756839_state] :
( ( Ys2
= ( cons_P438230685_state @ Y3 @ Xs2 ) )
=> ~ ( member512523024_state @ ( produc2065881943_state @ X @ Y3 ) @ R ) )
=> ~ ! [Zs: list_P890094829_state] :
( ( Ys2
= ( cons_P438230685_state @ X @ Zs ) )
=> ~ ( member423306384_state @ ( produc1354769367_state @ Xs2 @ Zs ) @ ( listre1725776070_state @ R ) ) ) ) ) ).
% Cons_listrel1E1
thf(fact_257_listrel1I1,axiom,
! [X: com,Y: com,R: set_Pr1822651163om_com,Xs2: list_com] :
( ( member1333916004om_com @ ( product_Pair_com_com @ X @ Y ) @ R )
=> ( member32358020st_com @ ( produc1307105171st_com @ ( cons_com @ X @ Xs2 ) @ ( cons_com @ Y @ Xs2 ) ) @ ( listrel1_com @ R ) ) ) ).
% listrel1I1
thf(fact_258_listrel1I1,axiom,
! [X: produc2041926651_state,Y: produc2041926651_state,R: set_Pr1165141447_state,Xs2: list_P1040014977_state] :
( ( member1069318160_state @ ( produc1909270103_state @ X @ Y ) @ R )
=> ( member1267359376_state @ ( produc1298457559_state @ ( cons_P465369265_state @ X @ Xs2 ) @ ( cons_P465369265_state @ Y @ Xs2 ) ) @ ( listre1738592858_state @ R ) ) ) ).
% listrel1I1
thf(fact_259_listrel1I1,axiom,
! [X: produc410756839_state,Y: produc410756839_state,R: set_Pr2096893127_state,Xs2: list_P890094829_state] :
( ( member512523024_state @ ( produc2065881943_state @ X @ Y ) @ R )
=> ( member423306384_state @ ( produc1354769367_state @ ( cons_P438230685_state @ X @ Xs2 ) @ ( cons_P438230685_state @ Y @ Xs2 ) ) @ ( listre1725776070_state @ R ) ) ) ).
% listrel1I1
thf(fact_260_listrel__reflcl__if__listrel1,axiom,
! [Xs2: list_P1040014977_state,Ys2: list_P1040014977_state,R: set_Pr1165141447_state] :
( ( member1267359376_state @ ( produc1298457559_state @ Xs2 @ Ys2 ) @ ( listre1738592858_state @ R ) )
=> ( member1267359376_state @ ( produc1298457559_state @ Xs2 @ Ys2 ) @ ( listre194873521_state @ ( transi1302705790_state @ R ) ) ) ) ).
% listrel_reflcl_if_listrel1
thf(fact_261_listrel__reflcl__if__listrel1,axiom,
! [Xs2: list_P890094829_state,Ys2: list_P890094829_state,R: set_Pr2096893127_state] :
( ( member423306384_state @ ( produc1354769367_state @ Xs2 @ Ys2 ) @ ( listre1725776070_state @ R ) )
=> ( member423306384_state @ ( produc1354769367_state @ Xs2 @ Ys2 ) @ ( listre1244407729_state @ ( transi463013610_state @ R ) ) ) ) ).
% listrel_reflcl_if_listrel1
thf(fact_262_concat__eq__appendD,axiom,
! [Xss2: list_list_com,Ys2: list_com,Zs2: list_com] :
( ( ( concat_com @ Xss2 )
= ( append_com @ Ys2 @ Zs2 ) )
=> ( ( Xss2 != nil_list_com )
=> ? [Xss12: list_list_com,Xs: list_com,Xs5: list_com,Xss23: list_list_com] :
( ( Xss2
= ( append_list_com @ Xss12 @ ( cons_list_com @ ( append_com @ Xs @ Xs5 ) @ Xss23 ) ) )
& ( Ys2
= ( append_com @ ( concat_com @ Xss12 ) @ Xs ) )
& ( Zs2
= ( append_com @ Xs5 @ ( concat_com @ Xss23 ) ) ) ) ) ) ).
% concat_eq_appendD
thf(fact_263_listrel1I,axiom,
! [X: com,Y: com,R: set_Pr1822651163om_com,Xs2: list_com,Us: list_com,Vs: list_com,Ys2: list_com] :
( ( member1333916004om_com @ ( product_Pair_com_com @ X @ Y ) @ R )
=> ( ( Xs2
= ( append_com @ Us @ ( cons_com @ X @ Vs ) ) )
=> ( ( Ys2
= ( append_com @ Us @ ( cons_com @ Y @ Vs ) ) )
=> ( member32358020st_com @ ( produc1307105171st_com @ Xs2 @ Ys2 ) @ ( listrel1_com @ R ) ) ) ) ) ).
% listrel1I
thf(fact_264_listrel1I,axiom,
! [X: produc2041926651_state,Y: produc2041926651_state,R: set_Pr1165141447_state,Xs2: list_P1040014977_state,Us: list_P1040014977_state,Vs: list_P1040014977_state,Ys2: list_P1040014977_state] :
( ( member1069318160_state @ ( produc1909270103_state @ X @ Y ) @ R )
=> ( ( Xs2
= ( append982487830_state @ Us @ ( cons_P465369265_state @ X @ Vs ) ) )
=> ( ( Ys2
= ( append982487830_state @ Us @ ( cons_P465369265_state @ Y @ Vs ) ) )
=> ( member1267359376_state @ ( produc1298457559_state @ Xs2 @ Ys2 ) @ ( listre1738592858_state @ R ) ) ) ) ) ).
% listrel1I
thf(fact_265_listrel1I,axiom,
! [X: produc410756839_state,Y: produc410756839_state,R: set_Pr2096893127_state,Xs2: list_P890094829_state,Us: list_P890094829_state,Vs: list_P890094829_state,Ys2: list_P890094829_state] :
( ( member512523024_state @ ( produc2065881943_state @ X @ Y ) @ R )
=> ( ( Xs2
= ( append221843330_state @ Us @ ( cons_P438230685_state @ X @ Vs ) ) )
=> ( ( Ys2
= ( append221843330_state @ Us @ ( cons_P438230685_state @ Y @ Vs ) ) )
=> ( member423306384_state @ ( produc1354769367_state @ Xs2 @ Ys2 ) @ ( listre1725776070_state @ R ) ) ) ) ) ).
% listrel1I
thf(fact_266_listrel1E,axiom,
! [Xs2: list_com,Ys2: list_com,R: set_Pr1822651163om_com] :
( ( member32358020st_com @ ( produc1307105171st_com @ Xs2 @ Ys2 ) @ ( listrel1_com @ R ) )
=> ~ ! [X3: com,Y3: com] :
( ( member1333916004om_com @ ( product_Pair_com_com @ X3 @ Y3 ) @ R )
=> ! [Us3: list_com,Vs2: list_com] :
( ( Xs2
= ( append_com @ Us3 @ ( cons_com @ X3 @ Vs2 ) ) )
=> ( Ys2
!= ( append_com @ Us3 @ ( cons_com @ Y3 @ Vs2 ) ) ) ) ) ) ).
% listrel1E
thf(fact_267_listrel1E,axiom,
! [Xs2: list_P1040014977_state,Ys2: list_P1040014977_state,R: set_Pr1165141447_state] :
( ( member1267359376_state @ ( produc1298457559_state @ Xs2 @ Ys2 ) @ ( listre1738592858_state @ R ) )
=> ~ ! [X3: produc2041926651_state,Y3: produc2041926651_state] :
( ( member1069318160_state @ ( produc1909270103_state @ X3 @ Y3 ) @ R )
=> ! [Us3: list_P1040014977_state,Vs2: list_P1040014977_state] :
( ( Xs2
= ( append982487830_state @ Us3 @ ( cons_P465369265_state @ X3 @ Vs2 ) ) )
=> ( Ys2
!= ( append982487830_state @ Us3 @ ( cons_P465369265_state @ Y3 @ Vs2 ) ) ) ) ) ) ).
% listrel1E
thf(fact_268_listrel1E,axiom,
! [Xs2: list_P890094829_state,Ys2: list_P890094829_state,R: set_Pr2096893127_state] :
( ( member423306384_state @ ( produc1354769367_state @ Xs2 @ Ys2 ) @ ( listre1725776070_state @ R ) )
=> ~ ! [X3: produc410756839_state,Y3: produc410756839_state] :
( ( member512523024_state @ ( produc2065881943_state @ X3 @ Y3 ) @ R )
=> ! [Us3: list_P890094829_state,Vs2: list_P890094829_state] :
( ( Xs2
= ( append221843330_state @ Us3 @ ( cons_P438230685_state @ X3 @ Vs2 ) ) )
=> ( Ys2
!= ( append221843330_state @ Us3 @ ( cons_P438230685_state @ Y3 @ Vs2 ) ) ) ) ) ) ).
% listrel1E
thf(fact_269_rtrancl__listrel1__ConsI2,axiom,
! [X: com,Y: com,R: set_Pr1822651163om_com,Xs2: list_com,Ys2: list_com] :
( ( member1333916004om_com @ ( product_Pair_com_com @ X @ Y ) @ ( transi863150634cl_com @ R ) )
=> ( ( member32358020st_com @ ( produc1307105171st_com @ Xs2 @ Ys2 ) @ ( transi436345274st_com @ ( listrel1_com @ R ) ) )
=> ( member32358020st_com @ ( produc1307105171st_com @ ( cons_com @ X @ Xs2 ) @ ( cons_com @ Y @ Ys2 ) ) @ ( transi436345274st_com @ ( listrel1_com @ R ) ) ) ) ) ).
% rtrancl_listrel1_ConsI2
thf(fact_270_rtrancl__listrel1__ConsI2,axiom,
! [X: produc2041926651_state,Y: produc2041926651_state,R: set_Pr1165141447_state,Xs2: list_P1040014977_state,Ys2: list_P1040014977_state] :
( ( member1069318160_state @ ( produc1909270103_state @ X @ Y ) @ ( transi1302705790_state @ R ) )
=> ( ( member1267359376_state @ ( produc1298457559_state @ Xs2 @ Ys2 ) @ ( transi339813252_state @ ( listre1738592858_state @ R ) ) )
=> ( member1267359376_state @ ( produc1298457559_state @ ( cons_P465369265_state @ X @ Xs2 ) @ ( cons_P465369265_state @ Y @ Ys2 ) ) @ ( transi339813252_state @ ( listre1738592858_state @ R ) ) ) ) ) ).
% rtrancl_listrel1_ConsI2
thf(fact_271_rtrancl__listrel1__ConsI2,axiom,
! [X: produc410756839_state,Y: produc410756839_state,R: set_Pr2096893127_state,Xs2: list_P890094829_state,Ys2: list_P890094829_state] :
( ( member512523024_state @ ( produc2065881943_state @ X @ Y ) @ ( transi463013610_state @ R ) )
=> ( ( member423306384_state @ ( produc1354769367_state @ Xs2 @ Ys2 ) @ ( transi1965148016_state @ ( listre1725776070_state @ R ) ) )
=> ( member423306384_state @ ( produc1354769367_state @ ( cons_P438230685_state @ X @ Xs2 ) @ ( cons_P438230685_state @ Y @ Ys2 ) ) @ ( transi1965148016_state @ ( listre1725776070_state @ R ) ) ) ) ) ).
% rtrancl_listrel1_ConsI2
thf(fact_272_lexord__append__left__rightI,axiom,
! [A2: com,B2: com,R: set_Pr1822651163om_com,U: list_com,X: list_com,Y: list_com] :
( ( member1333916004om_com @ ( product_Pair_com_com @ A2 @ B2 ) @ R )
=> ( member32358020st_com @ ( produc1307105171st_com @ ( append_com @ U @ ( cons_com @ A2 @ X ) ) @ ( append_com @ U @ ( cons_com @ B2 @ Y ) ) ) @ ( lexord_com @ R ) ) ) ).
% lexord_append_left_rightI
thf(fact_273_lexord__append__left__rightI,axiom,
! [A2: produc2041926651_state,B2: produc2041926651_state,R: set_Pr1165141447_state,U: list_P1040014977_state,X: list_P1040014977_state,Y: list_P1040014977_state] :
( ( member1069318160_state @ ( produc1909270103_state @ A2 @ B2 ) @ R )
=> ( member1267359376_state @ ( produc1298457559_state @ ( append982487830_state @ U @ ( cons_P465369265_state @ A2 @ X ) ) @ ( append982487830_state @ U @ ( cons_P465369265_state @ B2 @ Y ) ) ) @ ( lexord1062073966_state @ R ) ) ) ).
% lexord_append_left_rightI
thf(fact_274_lexord__append__left__rightI,axiom,
! [A2: produc410756839_state,B2: produc410756839_state,R: set_Pr2096893127_state,U: list_P890094829_state,X: list_P890094829_state,Y: list_P890094829_state] :
( ( member512523024_state @ ( produc2065881943_state @ A2 @ B2 ) @ R )
=> ( member423306384_state @ ( produc1354769367_state @ ( append221843330_state @ U @ ( cons_P438230685_state @ A2 @ X ) ) @ ( append221843330_state @ U @ ( cons_P438230685_state @ B2 @ Y ) ) ) @ ( lexord1893203674_state @ R ) ) ) ).
% lexord_append_left_rightI
thf(fact_275_rotate1_Osimps_I2_J,axiom,
! [X: com,Xs2: list_com] :
( ( rotate1_com @ ( cons_com @ X @ Xs2 ) )
= ( append_com @ Xs2 @ ( cons_com @ X @ nil_com ) ) ) ).
% rotate1.simps(2)
thf(fact_276_butlast__snoc,axiom,
! [Xs2: list_com,X: com] :
( ( butlast_com @ ( append_com @ Xs2 @ ( cons_com @ X @ nil_com ) ) )
= Xs2 ) ).
% butlast_snoc
thf(fact_277_rotate1__is__Nil__conv,axiom,
! [Xs2: list_com] :
( ( ( rotate1_com @ Xs2 )
= nil_com )
= ( Xs2 = nil_com ) ) ).
% rotate1_is_Nil_conv
thf(fact_278_lexord__cons__cons,axiom,
! [A2: com,X: list_com,B2: com,Y: list_com,R: set_Pr1822651163om_com] :
( ( member32358020st_com @ ( produc1307105171st_com @ ( cons_com @ A2 @ X ) @ ( cons_com @ B2 @ Y ) ) @ ( lexord_com @ R ) )
= ( ( member1333916004om_com @ ( product_Pair_com_com @ A2 @ B2 ) @ R )
| ( ( A2 = B2 )
& ( member32358020st_com @ ( produc1307105171st_com @ X @ Y ) @ ( lexord_com @ R ) ) ) ) ) ).
% lexord_cons_cons
thf(fact_279_lexord__cons__cons,axiom,
! [A2: produc2041926651_state,X: list_P1040014977_state,B2: produc2041926651_state,Y: list_P1040014977_state,R: set_Pr1165141447_state] :
( ( member1267359376_state @ ( produc1298457559_state @ ( cons_P465369265_state @ A2 @ X ) @ ( cons_P465369265_state @ B2 @ Y ) ) @ ( lexord1062073966_state @ R ) )
= ( ( member1069318160_state @ ( produc1909270103_state @ A2 @ B2 ) @ R )
| ( ( A2 = B2 )
& ( member1267359376_state @ ( produc1298457559_state @ X @ Y ) @ ( lexord1062073966_state @ R ) ) ) ) ) ).
% lexord_cons_cons
thf(fact_280_lexord__cons__cons,axiom,
! [A2: produc410756839_state,X: list_P890094829_state,B2: produc410756839_state,Y: list_P890094829_state,R: set_Pr2096893127_state] :
( ( member423306384_state @ ( produc1354769367_state @ ( cons_P438230685_state @ A2 @ X ) @ ( cons_P438230685_state @ B2 @ Y ) ) @ ( lexord1893203674_state @ R ) )
= ( ( member512523024_state @ ( produc2065881943_state @ A2 @ B2 ) @ R )
| ( ( A2 = B2 )
& ( member423306384_state @ ( produc1354769367_state @ X @ Y ) @ ( lexord1893203674_state @ R ) ) ) ) ) ).
% lexord_cons_cons
thf(fact_281_lexord__Nil__left,axiom,
! [Y: list_com,R: set_Pr1822651163om_com] :
( ( member32358020st_com @ ( produc1307105171st_com @ nil_com @ Y ) @ ( lexord_com @ R ) )
= ( ? [A5: com,X4: list_com] :
( Y
= ( cons_com @ A5 @ X4 ) ) ) ) ).
% lexord_Nil_left
thf(fact_282_butlast_Osimps_I1_J,axiom,
( ( butlast_com @ nil_com )
= nil_com ) ).
% butlast.simps(1)
thf(fact_283_rotate1_Osimps_I1_J,axiom,
( ( rotate1_com @ nil_com )
= nil_com ) ).
% rotate1.simps(1)
thf(fact_284_lexord__linear,axiom,
! [R: set_Pr1165141447_state,X: list_P1040014977_state,Y: list_P1040014977_state] :
( ! [A: produc2041926651_state,B: produc2041926651_state] :
( ( member1069318160_state @ ( produc1909270103_state @ A @ B ) @ R )
| ( A = B )
| ( member1069318160_state @ ( produc1909270103_state @ B @ A ) @ R ) )
=> ( ( member1267359376_state @ ( produc1298457559_state @ X @ Y ) @ ( lexord1062073966_state @ R ) )
| ( X = Y )
| ( member1267359376_state @ ( produc1298457559_state @ Y @ X ) @ ( lexord1062073966_state @ R ) ) ) ) ).
% lexord_linear
thf(fact_285_lexord__linear,axiom,
! [R: set_Pr2096893127_state,X: list_P890094829_state,Y: list_P890094829_state] :
( ! [A: produc410756839_state,B: produc410756839_state] :
( ( member512523024_state @ ( produc2065881943_state @ A @ B ) @ R )
| ( A = B )
| ( member512523024_state @ ( produc2065881943_state @ B @ A ) @ R ) )
=> ( ( member423306384_state @ ( produc1354769367_state @ X @ Y ) @ ( lexord1893203674_state @ R ) )
| ( X = Y )
| ( member423306384_state @ ( produc1354769367_state @ Y @ X ) @ ( lexord1893203674_state @ R ) ) ) ) ).
% lexord_linear
thf(fact_286_lexord__irreflexive,axiom,
! [R: set_Pr1165141447_state,Xs2: list_P1040014977_state] :
( ! [X3: produc2041926651_state] :
~ ( member1069318160_state @ ( produc1909270103_state @ X3 @ X3 ) @ R )
=> ~ ( member1267359376_state @ ( produc1298457559_state @ Xs2 @ Xs2 ) @ ( lexord1062073966_state @ R ) ) ) ).
% lexord_irreflexive
thf(fact_287_lexord__irreflexive,axiom,
! [R: set_Pr2096893127_state,Xs2: list_P890094829_state] :
( ! [X3: produc410756839_state] :
~ ( member512523024_state @ ( produc2065881943_state @ X3 @ X3 ) @ R )
=> ~ ( member423306384_state @ ( produc1354769367_state @ Xs2 @ Xs2 ) @ ( lexord1893203674_state @ R ) ) ) ).
% lexord_irreflexive
thf(fact_288_butlast_Osimps_I2_J,axiom,
! [Xs2: list_com,X: com] :
( ( ( Xs2 = nil_com )
=> ( ( butlast_com @ ( cons_com @ X @ Xs2 ) )
= nil_com ) )
& ( ( Xs2 != nil_com )
=> ( ( butlast_com @ ( cons_com @ X @ Xs2 ) )
= ( cons_com @ X @ ( butlast_com @ Xs2 ) ) ) ) ) ).
% butlast.simps(2)
thf(fact_289_lexord__Nil__right,axiom,
! [X: list_com,R: set_Pr1822651163om_com] :
~ ( member32358020st_com @ ( produc1307105171st_com @ X @ nil_com ) @ ( lexord_com @ R ) ) ).
% lexord_Nil_right
thf(fact_290_butlast__append,axiom,
! [Ys2: list_com,Xs2: list_com] :
( ( ( Ys2 = nil_com )
=> ( ( butlast_com @ ( append_com @ Xs2 @ Ys2 ) )
= ( butlast_com @ Xs2 ) ) )
& ( ( Ys2 != nil_com )
=> ( ( butlast_com @ ( append_com @ Xs2 @ Ys2 ) )
= ( append_com @ Xs2 @ ( butlast_com @ Ys2 ) ) ) ) ) ).
% butlast_append
thf(fact_291_lexord__append__leftI,axiom,
! [U: list_com,V: list_com,R: set_Pr1822651163om_com,X: list_com] :
( ( member32358020st_com @ ( produc1307105171st_com @ U @ V ) @ ( lexord_com @ R ) )
=> ( member32358020st_com @ ( produc1307105171st_com @ ( append_com @ X @ U ) @ ( append_com @ X @ V ) ) @ ( lexord_com @ R ) ) ) ).
% lexord_append_leftI
thf(fact_292_lexord__append__leftD,axiom,
! [X: list_com,U: list_com,V: list_com,R: set_Pr1822651163om_com] :
( ( member32358020st_com @ ( produc1307105171st_com @ ( append_com @ X @ U ) @ ( append_com @ X @ V ) ) @ ( lexord_com @ R ) )
=> ( ! [A: com] :
~ ( member1333916004om_com @ ( product_Pair_com_com @ A @ A ) @ R )
=> ( member32358020st_com @ ( produc1307105171st_com @ U @ V ) @ ( lexord_com @ R ) ) ) ) ).
% lexord_append_leftD
thf(fact_293_lexord__append__leftD,axiom,
! [X: list_P1040014977_state,U: list_P1040014977_state,V: list_P1040014977_state,R: set_Pr1165141447_state] :
( ( member1267359376_state @ ( produc1298457559_state @ ( append982487830_state @ X @ U ) @ ( append982487830_state @ X @ V ) ) @ ( lexord1062073966_state @ R ) )
=> ( ! [A: produc2041926651_state] :
~ ( member1069318160_state @ ( produc1909270103_state @ A @ A ) @ R )
=> ( member1267359376_state @ ( produc1298457559_state @ U @ V ) @ ( lexord1062073966_state @ R ) ) ) ) ).
% lexord_append_leftD
thf(fact_294_lexord__append__leftD,axiom,
! [X: list_P890094829_state,U: list_P890094829_state,V: list_P890094829_state,R: set_Pr2096893127_state] :
( ( member423306384_state @ ( produc1354769367_state @ ( append221843330_state @ X @ U ) @ ( append221843330_state @ X @ V ) ) @ ( lexord1893203674_state @ R ) )
=> ( ! [A: produc410756839_state] :
~ ( member512523024_state @ ( produc2065881943_state @ A @ A ) @ R )
=> ( member423306384_state @ ( produc1354769367_state @ U @ V ) @ ( lexord1893203674_state @ R ) ) ) ) ).
% lexord_append_leftD
thf(fact_295_lexord__append__rightI,axiom,
! [Y: list_com,X: list_com,R: set_Pr1822651163om_com] :
( ? [B5: com,Z3: list_com] :
( Y
= ( cons_com @ B5 @ Z3 ) )
=> ( member32358020st_com @ ( produc1307105171st_com @ X @ ( append_com @ X @ Y ) ) @ ( lexord_com @ R ) ) ) ).
% lexord_append_rightI
thf(fact_296_lexord__same__pref__if__irrefl,axiom,
! [R: set_Pr1822651163om_com,Xs2: list_com,Ys2: list_com,Zs2: list_com] :
( ( irrefl_com @ R )
=> ( ( member32358020st_com @ ( produc1307105171st_com @ ( append_com @ Xs2 @ Ys2 ) @ ( append_com @ Xs2 @ Zs2 ) ) @ ( lexord_com @ R ) )
= ( member32358020st_com @ ( produc1307105171st_com @ Ys2 @ Zs2 ) @ ( lexord_com @ R ) ) ) ) ).
% lexord_same_pref_if_irrefl
thf(fact_297_append__butlast__last__id,axiom,
! [Xs2: list_com] :
( ( Xs2 != nil_com )
=> ( ( append_com @ ( butlast_com @ Xs2 ) @ ( cons_com @ ( last_com @ Xs2 ) @ nil_com ) )
= Xs2 ) ) ).
% append_butlast_last_id
thf(fact_298_snoc__eq__iff__butlast,axiom,
! [Xs2: list_com,X: com,Ys2: list_com] :
( ( ( append_com @ Xs2 @ ( cons_com @ X @ nil_com ) )
= Ys2 )
= ( ( Ys2 != nil_com )
& ( ( butlast_com @ Ys2 )
= Xs2 )
& ( ( last_com @ Ys2 )
= X ) ) ) ).
% snoc_eq_iff_butlast
thf(fact_299_last__appendR,axiom,
! [Ys2: list_com,Xs2: list_com] :
( ( Ys2 != nil_com )
=> ( ( last_com @ ( append_com @ Xs2 @ Ys2 ) )
= ( last_com @ Ys2 ) ) ) ).
% last_appendR
thf(fact_300_last__appendL,axiom,
! [Ys2: list_com,Xs2: list_com] :
( ( Ys2 = nil_com )
=> ( ( last_com @ ( append_com @ Xs2 @ Ys2 ) )
= ( last_com @ Xs2 ) ) ) ).
% last_appendL
thf(fact_301_last__snoc,axiom,
! [Xs2: list_com,X: com] :
( ( last_com @ ( append_com @ Xs2 @ ( cons_com @ X @ nil_com ) ) )
= X ) ).
% last_snoc
thf(fact_302_last__ConsR,axiom,
! [Xs2: list_com,X: com] :
( ( Xs2 != nil_com )
=> ( ( last_com @ ( cons_com @ X @ Xs2 ) )
= ( last_com @ Xs2 ) ) ) ).
% last_ConsR
thf(fact_303_last__ConsL,axiom,
! [Xs2: list_com,X: com] :
( ( Xs2 = nil_com )
=> ( ( last_com @ ( cons_com @ X @ Xs2 ) )
= X ) ) ).
% last_ConsL
thf(fact_304_last_Osimps,axiom,
! [Xs2: list_com,X: com] :
( ( ( Xs2 = nil_com )
=> ( ( last_com @ ( cons_com @ X @ Xs2 ) )
= X ) )
& ( ( Xs2 != nil_com )
=> ( ( last_com @ ( cons_com @ X @ Xs2 ) )
= ( last_com @ Xs2 ) ) ) ) ).
% last.simps
thf(fact_305_longest__common__suffix,axiom,
! [Xs2: list_com,Ys2: list_com] :
? [Ss: list_com,Xs5: list_com,Ys5: list_com] :
( ( Xs2
= ( append_com @ Xs5 @ Ss ) )
& ( Ys2
= ( append_com @ Ys5 @ Ss ) )
& ( ( Xs5 = nil_com )
| ( Ys5 = nil_com )
| ( ( last_com @ Xs5 )
!= ( last_com @ Ys5 ) ) ) ) ).
% longest_common_suffix
thf(fact_306_last__append,axiom,
! [Ys2: list_com,Xs2: list_com] :
( ( ( Ys2 = nil_com )
=> ( ( last_com @ ( append_com @ Xs2 @ Ys2 ) )
= ( last_com @ Xs2 ) ) )
& ( ( Ys2 != nil_com )
=> ( ( last_com @ ( append_com @ Xs2 @ Ys2 ) )
= ( last_com @ Ys2 ) ) ) ) ).
% last_append
thf(fact_307_irreflI,axiom,
! [R2: set_Pr1165141447_state] :
( ! [A: produc2041926651_state] :
~ ( member1069318160_state @ ( produc1909270103_state @ A @ A ) @ R2 )
=> ( irrefl654089520_state @ R2 ) ) ).
% irreflI
thf(fact_308_irreflI,axiom,
! [R2: set_Pr2096893127_state] :
( ! [A: produc410756839_state] :
~ ( member512523024_state @ ( produc2065881943_state @ A @ A ) @ R2 )
=> ( irrefl1358048924_state @ R2 ) ) ).
% irreflI
thf(fact_309_irrefl__def,axiom,
( irrefl654089520_state
= ( ^ [R3: set_Pr1165141447_state] :
! [A5: produc2041926651_state] :
~ ( member1069318160_state @ ( produc1909270103_state @ A5 @ A5 ) @ R3 ) ) ) ).
% irrefl_def
thf(fact_310_irrefl__def,axiom,
( irrefl1358048924_state
= ( ^ [R3: set_Pr2096893127_state] :
! [A5: produc410756839_state] :
~ ( member512523024_state @ ( produc2065881943_state @ A5 @ A5 ) @ R3 ) ) ) ).
% irrefl_def
thf(fact_311_lenlex__append2,axiom,
! [R2: set_Pr1822651163om_com,Us: list_com,Xs2: list_com,Ys2: list_com] :
( ( irrefl_com @ R2 )
=> ( ( member32358020st_com @ ( produc1307105171st_com @ ( append_com @ Us @ Xs2 ) @ ( append_com @ Us @ Ys2 ) ) @ ( lenlex_com @ R2 ) )
= ( member32358020st_com @ ( produc1307105171st_com @ Xs2 @ Ys2 ) @ ( lenlex_com @ R2 ) ) ) ) ).
% lenlex_append2
thf(fact_312_Nil__lenlex__iff1,axiom,
! [Ns: list_com,R: set_Pr1822651163om_com] :
( ( member32358020st_com @ ( produc1307105171st_com @ nil_com @ Ns ) @ ( lenlex_com @ R ) )
= ( Ns != nil_com ) ) ).
% Nil_lenlex_iff1
thf(fact_313_Nil__notin__lex,axiom,
! [Ys2: list_com,R: set_Pr1822651163om_com] :
~ ( member32358020st_com @ ( produc1307105171st_com @ nil_com @ Ys2 ) @ ( lex_com @ R ) ) ).
% Nil_notin_lex
thf(fact_314_Nil2__notin__lex,axiom,
! [Xs2: list_com,R: set_Pr1822651163om_com] :
~ ( member32358020st_com @ ( produc1307105171st_com @ Xs2 @ nil_com ) @ ( lex_com @ R ) ) ).
% Nil2_notin_lex
thf(fact_315_lex__append__leftI,axiom,
! [Ys2: list_com,Zs2: list_com,R: set_Pr1822651163om_com,Xs2: list_com] :
( ( member32358020st_com @ ( produc1307105171st_com @ Ys2 @ Zs2 ) @ ( lex_com @ R ) )
=> ( member32358020st_com @ ( produc1307105171st_com @ ( append_com @ Xs2 @ Ys2 ) @ ( append_com @ Xs2 @ Zs2 ) ) @ ( lex_com @ R ) ) ) ).
% lex_append_leftI
thf(fact_316_lenlex__irreflexive,axiom,
! [R: set_Pr1165141447_state,Xs2: list_P1040014977_state] :
( ! [X3: produc2041926651_state] :
~ ( member1069318160_state @ ( produc1909270103_state @ X3 @ X3 ) @ R )
=> ~ ( member1267359376_state @ ( produc1298457559_state @ Xs2 @ Xs2 ) @ ( lenlex295247238_state @ R ) ) ) ).
% lenlex_irreflexive
thf(fact_317_lenlex__irreflexive,axiom,
! [R: set_Pr2096893127_state,Xs2: list_P890094829_state] :
( ! [X3: produc410756839_state] :
~ ( member512523024_state @ ( produc2065881943_state @ X3 @ X3 ) @ R )
=> ~ ( member423306384_state @ ( produc1354769367_state @ Xs2 @ Xs2 ) @ ( lenlex1401983474_state @ R ) ) ) ).
% lenlex_irreflexive
thf(fact_318_Nil__lenlex__iff2,axiom,
! [Ns: list_com,R: set_Pr1822651163om_com] :
~ ( member32358020st_com @ ( produc1307105171st_com @ Ns @ nil_com ) @ ( lenlex_com @ R ) ) ).
% Nil_lenlex_iff2
thf(fact_319_lex__append__leftD,axiom,
! [R: set_Pr1822651163om_com,Xs2: list_com,Ys2: list_com,Zs2: list_com] :
( ! [X3: com] :
~ ( member1333916004om_com @ ( product_Pair_com_com @ X3 @ X3 ) @ R )
=> ( ( member32358020st_com @ ( produc1307105171st_com @ ( append_com @ Xs2 @ Ys2 ) @ ( append_com @ Xs2 @ Zs2 ) ) @ ( lex_com @ R ) )
=> ( member32358020st_com @ ( produc1307105171st_com @ Ys2 @ Zs2 ) @ ( lex_com @ R ) ) ) ) ).
% lex_append_leftD
thf(fact_320_lex__append__leftD,axiom,
! [R: set_Pr1165141447_state,Xs2: list_P1040014977_state,Ys2: list_P1040014977_state,Zs2: list_P1040014977_state] :
( ! [X3: produc2041926651_state] :
~ ( member1069318160_state @ ( produc1909270103_state @ X3 @ X3 ) @ R )
=> ( ( member1267359376_state @ ( produc1298457559_state @ ( append982487830_state @ Xs2 @ Ys2 ) @ ( append982487830_state @ Xs2 @ Zs2 ) ) @ ( lex_Pr570247525_state @ R ) )
=> ( member1267359376_state @ ( produc1298457559_state @ Ys2 @ Zs2 ) @ ( lex_Pr570247525_state @ R ) ) ) ) ).
% lex_append_leftD
thf(fact_321_lex__append__leftD,axiom,
! [R: set_Pr2096893127_state,Xs2: list_P890094829_state,Ys2: list_P890094829_state,Zs2: list_P890094829_state] :
( ! [X3: produc410756839_state] :
~ ( member512523024_state @ ( produc2065881943_state @ X3 @ X3 ) @ R )
=> ( ( member423306384_state @ ( produc1354769367_state @ ( append221843330_state @ Xs2 @ Ys2 ) @ ( append221843330_state @ Xs2 @ Zs2 ) ) @ ( lex_Pr1762629969_state @ R ) )
=> ( member423306384_state @ ( produc1354769367_state @ Ys2 @ Zs2 ) @ ( lex_Pr1762629969_state @ R ) ) ) ) ).
% lex_append_leftD
thf(fact_322_lex__append__left__iff,axiom,
! [R: set_Pr1822651163om_com,Xs2: list_com,Ys2: list_com,Zs2: list_com] :
( ! [X3: com] :
~ ( member1333916004om_com @ ( product_Pair_com_com @ X3 @ X3 ) @ R )
=> ( ( member32358020st_com @ ( produc1307105171st_com @ ( append_com @ Xs2 @ Ys2 ) @ ( append_com @ Xs2 @ Zs2 ) ) @ ( lex_com @ R ) )
= ( member32358020st_com @ ( produc1307105171st_com @ Ys2 @ Zs2 ) @ ( lex_com @ R ) ) ) ) ).
% lex_append_left_iff
thf(fact_323_lex__append__left__iff,axiom,
! [R: set_Pr1165141447_state,Xs2: list_P1040014977_state,Ys2: list_P1040014977_state,Zs2: list_P1040014977_state] :
( ! [X3: produc2041926651_state] :
~ ( member1069318160_state @ ( produc1909270103_state @ X3 @ X3 ) @ R )
=> ( ( member1267359376_state @ ( produc1298457559_state @ ( append982487830_state @ Xs2 @ Ys2 ) @ ( append982487830_state @ Xs2 @ Zs2 ) ) @ ( lex_Pr570247525_state @ R ) )
= ( member1267359376_state @ ( produc1298457559_state @ Ys2 @ Zs2 ) @ ( lex_Pr570247525_state @ R ) ) ) ) ).
% lex_append_left_iff
thf(fact_324_lex__append__left__iff,axiom,
! [R: set_Pr2096893127_state,Xs2: list_P890094829_state,Ys2: list_P890094829_state,Zs2: list_P890094829_state] :
( ! [X3: produc410756839_state] :
~ ( member512523024_state @ ( produc2065881943_state @ X3 @ X3 ) @ R )
=> ( ( member423306384_state @ ( produc1354769367_state @ ( append221843330_state @ Xs2 @ Ys2 ) @ ( append221843330_state @ Xs2 @ Zs2 ) ) @ ( lex_Pr1762629969_state @ R ) )
= ( member423306384_state @ ( produc1354769367_state @ Ys2 @ Zs2 ) @ ( lex_Pr1762629969_state @ R ) ) ) ) ).
% lex_append_left_iff
thf(fact_325_Cons__in__lex,axiom,
! [X: com,Xs2: list_com,Y: com,Ys2: list_com,R: set_Pr1822651163om_com] :
( ( member32358020st_com @ ( produc1307105171st_com @ ( cons_com @ X @ Xs2 ) @ ( cons_com @ Y @ Ys2 ) ) @ ( lex_com @ R ) )
= ( ( ( member1333916004om_com @ ( product_Pair_com_com @ X @ Y ) @ R )
& ( ( size_size_list_com @ Xs2 )
= ( size_size_list_com @ Ys2 ) ) )
| ( ( X = Y )
& ( member32358020st_com @ ( produc1307105171st_com @ Xs2 @ Ys2 ) @ ( lex_com @ R ) ) ) ) ) ).
% Cons_in_lex
thf(fact_326_Cons__in__lex,axiom,
! [X: produc2041926651_state,Xs2: list_P1040014977_state,Y: produc2041926651_state,Ys2: list_P1040014977_state,R: set_Pr1165141447_state] :
( ( member1267359376_state @ ( produc1298457559_state @ ( cons_P465369265_state @ X @ Xs2 ) @ ( cons_P465369265_state @ Y @ Ys2 ) ) @ ( lex_Pr570247525_state @ R ) )
= ( ( ( member1069318160_state @ ( produc1909270103_state @ X @ Y ) @ R )
& ( ( size_s213202669_state @ Xs2 )
= ( size_s213202669_state @ Ys2 ) ) )
| ( ( X = Y )
& ( member1267359376_state @ ( produc1298457559_state @ Xs2 @ Ys2 ) @ ( lex_Pr570247525_state @ R ) ) ) ) ) ).
% Cons_in_lex
thf(fact_327_Cons__in__lex,axiom,
! [X: produc410756839_state,Xs2: list_P890094829_state,Y: produc410756839_state,Ys2: list_P890094829_state,R: set_Pr2096893127_state] :
( ( member423306384_state @ ( produc1354769367_state @ ( cons_P438230685_state @ X @ Xs2 ) @ ( cons_P438230685_state @ Y @ Ys2 ) ) @ ( lex_Pr1762629969_state @ R ) )
= ( ( ( member512523024_state @ ( produc2065881943_state @ X @ Y ) @ R )
& ( ( size_s837829465_state @ Xs2 )
= ( size_s837829465_state @ Ys2 ) ) )
| ( ( X = Y )
& ( member423306384_state @ ( produc1354769367_state @ Xs2 @ Ys2 ) @ ( lex_Pr1762629969_state @ R ) ) ) ) ) ).
% Cons_in_lex
thf(fact_328_append__eq__append__conv,axiom,
! [Xs2: list_com,Ys2: list_com,Us: list_com,Vs: list_com] :
( ( ( ( size_size_list_com @ Xs2 )
= ( size_size_list_com @ Ys2 ) )
| ( ( size_size_list_com @ Us )
= ( size_size_list_com @ Vs ) ) )
=> ( ( ( append_com @ Xs2 @ Us )
= ( append_com @ Ys2 @ Vs ) )
= ( ( Xs2 = Ys2 )
& ( Us = Vs ) ) ) ) ).
% append_eq_append_conv
thf(fact_329_list__induct3,axiom,
! [Xs2: list_com,Ys2: list_com,Zs2: list_com,P: list_com > list_com > list_com > $o] :
( ( ( size_size_list_com @ Xs2 )
= ( size_size_list_com @ Ys2 ) )
=> ( ( ( size_size_list_com @ Ys2 )
= ( size_size_list_com @ Zs2 ) )
=> ( ( P @ nil_com @ nil_com @ nil_com )
=> ( ! [X3: com,Xs: list_com,Y3: com,Ys: list_com,Z: com,Zs: list_com] :
( ( ( size_size_list_com @ Xs )
= ( size_size_list_com @ Ys ) )
=> ( ( ( size_size_list_com @ Ys )
= ( size_size_list_com @ Zs ) )
=> ( ( P @ Xs @ Ys @ Zs )
=> ( P @ ( cons_com @ X3 @ Xs ) @ ( cons_com @ Y3 @ Ys ) @ ( cons_com @ Z @ Zs ) ) ) ) )
=> ( P @ Xs2 @ Ys2 @ Zs2 ) ) ) ) ) ).
% list_induct3
thf(fact_330_list__induct2,axiom,
! [Xs2: list_com,Ys2: list_com,P: list_com > list_com > $o] :
( ( ( size_size_list_com @ Xs2 )
= ( size_size_list_com @ Ys2 ) )
=> ( ( P @ nil_com @ nil_com )
=> ( ! [X3: com,Xs: list_com,Y3: com,Ys: list_com] :
( ( ( size_size_list_com @ Xs )
= ( size_size_list_com @ Ys ) )
=> ( ( P @ Xs @ Ys )
=> ( P @ ( cons_com @ X3 @ Xs ) @ ( cons_com @ Y3 @ Ys ) ) ) )
=> ( P @ Xs2 @ Ys2 ) ) ) ) ).
% list_induct2
thf(fact_331_same__length__different,axiom,
! [Xs2: list_com,Ys2: list_com] :
( ( Xs2 != Ys2 )
=> ( ( ( size_size_list_com @ Xs2 )
= ( size_size_list_com @ Ys2 ) )
=> ? [Pre: list_com,X3: com,Xs5: list_com,Y3: com,Ys5: list_com] :
( ( X3 != Y3 )
& ( Xs2
= ( append_com @ Pre @ ( append_com @ ( cons_com @ X3 @ nil_com ) @ Xs5 ) ) )
& ( Ys2
= ( append_com @ Pre @ ( append_com @ ( cons_com @ Y3 @ nil_com ) @ Ys5 ) ) ) ) ) ) ).
% same_length_different
thf(fact_332_lexord__sufE,axiom,
! [Xs2: list_com,Zs2: list_com,Ys2: list_com,Qs: list_com,R: set_Pr1822651163om_com] :
( ( member32358020st_com @ ( produc1307105171st_com @ ( append_com @ Xs2 @ Zs2 ) @ ( append_com @ Ys2 @ Qs ) ) @ ( lexord_com @ R ) )
=> ( ( Xs2 != Ys2 )
=> ( ( ( size_size_list_com @ Xs2 )
= ( size_size_list_com @ Ys2 ) )
=> ( ( ( size_size_list_com @ Zs2 )
= ( size_size_list_com @ Qs ) )
=> ( member32358020st_com @ ( produc1307105171st_com @ Xs2 @ Ys2 ) @ ( lexord_com @ R ) ) ) ) ) ) ).
% lexord_sufE
thf(fact_333_lex__append__rightI,axiom,
! [Xs2: list_com,Ys2: list_com,R: set_Pr1822651163om_com,Vs: list_com,Us: list_com] :
( ( member32358020st_com @ ( produc1307105171st_com @ Xs2 @ Ys2 ) @ ( lex_com @ R ) )
=> ( ( ( size_size_list_com @ Vs )
= ( size_size_list_com @ Us ) )
=> ( member32358020st_com @ ( produc1307105171st_com @ ( append_com @ Xs2 @ Us ) @ ( append_com @ Ys2 @ Vs ) ) @ ( lex_com @ R ) ) ) ) ).
% lex_append_rightI
thf(fact_334_lenlex__append1,axiom,
! [Us: list_com,Xs2: list_com,R2: set_Pr1822651163om_com,Vs: list_com,Ys2: list_com] :
( ( member32358020st_com @ ( produc1307105171st_com @ Us @ Xs2 ) @ ( lenlex_com @ R2 ) )
=> ( ( ( size_size_list_com @ Vs )
= ( size_size_list_com @ Ys2 ) )
=> ( member32358020st_com @ ( produc1307105171st_com @ ( append_com @ Us @ Vs ) @ ( append_com @ Xs2 @ Ys2 ) ) @ ( lenlex_com @ R2 ) ) ) ) ).
% lenlex_append1
thf(fact_335_Cons__lenlex__iff,axiom,
! [M: com,Ms: list_com,N: com,Ns: list_com,R: set_Pr1822651163om_com] :
( ( member32358020st_com @ ( produc1307105171st_com @ ( cons_com @ M @ Ms ) @ ( cons_com @ N @ Ns ) ) @ ( lenlex_com @ R ) )
= ( ( ord_less_nat @ ( size_size_list_com @ Ms ) @ ( size_size_list_com @ Ns ) )
| ( ( ( size_size_list_com @ Ms )
= ( size_size_list_com @ Ns ) )
& ( member1333916004om_com @ ( product_Pair_com_com @ M @ N ) @ R ) )
| ( ( M = N )
& ( member32358020st_com @ ( produc1307105171st_com @ Ms @ Ns ) @ ( lenlex_com @ R ) ) ) ) ) ).
% Cons_lenlex_iff
thf(fact_336_Cons__lenlex__iff,axiom,
! [M: produc2041926651_state,Ms: list_P1040014977_state,N: produc2041926651_state,Ns: list_P1040014977_state,R: set_Pr1165141447_state] :
( ( member1267359376_state @ ( produc1298457559_state @ ( cons_P465369265_state @ M @ Ms ) @ ( cons_P465369265_state @ N @ Ns ) ) @ ( lenlex295247238_state @ R ) )
= ( ( ord_less_nat @ ( size_s213202669_state @ Ms ) @ ( size_s213202669_state @ Ns ) )
| ( ( ( size_s213202669_state @ Ms )
= ( size_s213202669_state @ Ns ) )
& ( member1069318160_state @ ( produc1909270103_state @ M @ N ) @ R ) )
| ( ( M = N )
& ( member1267359376_state @ ( produc1298457559_state @ Ms @ Ns ) @ ( lenlex295247238_state @ R ) ) ) ) ) ).
% Cons_lenlex_iff
thf(fact_337_Cons__lenlex__iff,axiom,
! [M: produc410756839_state,Ms: list_P890094829_state,N: produc410756839_state,Ns: list_P890094829_state,R: set_Pr2096893127_state] :
( ( member423306384_state @ ( produc1354769367_state @ ( cons_P438230685_state @ M @ Ms ) @ ( cons_P438230685_state @ N @ Ns ) ) @ ( lenlex1401983474_state @ R ) )
= ( ( ord_less_nat @ ( size_s837829465_state @ Ms ) @ ( size_s837829465_state @ Ns ) )
| ( ( ( size_s837829465_state @ Ms )
= ( size_s837829465_state @ Ns ) )
& ( member512523024_state @ ( produc2065881943_state @ M @ N ) @ R ) )
| ( ( M = N )
& ( member423306384_state @ ( produc1354769367_state @ Ms @ Ns ) @ ( lenlex1401983474_state @ R ) ) ) ) ) ).
% Cons_lenlex_iff
thf(fact_338_list__ex1__simps_I1_J,axiom,
! [P: com > $o] :
~ ( list_ex1_com @ P @ nil_com ) ).
% list_ex1_simps(1)
thf(fact_339_in__measures_I2_J,axiom,
! [X: produc2041926651_state,Y: produc2041926651_state,F2: produc2041926651_state > nat,Fs: list_P1178430914te_nat] :
( ( member1069318160_state @ ( produc1909270103_state @ X @ Y ) @ ( measur1071343867_state @ ( cons_P1821701106te_nat @ F2 @ Fs ) ) )
= ( ( ord_less_nat @ ( F2 @ X ) @ ( F2 @ Y ) )
| ( ( ( F2 @ X )
= ( F2 @ Y ) )
& ( member1069318160_state @ ( produc1909270103_state @ X @ Y ) @ ( measur1071343867_state @ Fs ) ) ) ) ) ).
% in_measures(2)
thf(fact_340_in__measures_I2_J,axiom,
! [X: produc410756839_state,Y: produc410756839_state,F2: produc410756839_state > nat,Fs: list_P1627643054te_nat] :
( ( member512523024_state @ ( produc2065881943_state @ X @ Y ) @ ( measur768621543_state @ ( cons_P1631380062te_nat @ F2 @ Fs ) ) )
= ( ( ord_less_nat @ ( F2 @ X ) @ ( F2 @ Y ) )
| ( ( ( F2 @ X )
= ( F2 @ Y ) )
& ( member512523024_state @ ( produc2065881943_state @ X @ Y ) @ ( measur768621543_state @ Fs ) ) ) ) ) ).
% in_measures(2)
thf(fact_341_listrel__iff__nth,axiom,
! [Xs2: list_P1040014977_state,Ys2: list_P1040014977_state,R: set_Pr1165141447_state] :
( ( member1267359376_state @ ( produc1298457559_state @ Xs2 @ Ys2 ) @ ( listre194873521_state @ R ) )
= ( ( ( size_s213202669_state @ Xs2 )
= ( size_s213202669_state @ Ys2 ) )
& ! [N2: nat] :
( ( ord_less_nat @ N2 @ ( size_s213202669_state @ Xs2 ) )
=> ( member1069318160_state @ ( produc1909270103_state @ ( nth_Pr553241858_state @ Xs2 @ N2 ) @ ( nth_Pr553241858_state @ Ys2 @ N2 ) ) @ R ) ) ) ) ).
% listrel_iff_nth
thf(fact_342_listrel__iff__nth,axiom,
! [Xs2: list_list_com,Ys2: list_state,R: set_Pr849496667_state] :
( ( member1442949316_state @ ( produc1742567891_state @ Xs2 @ Ys2 ) @ ( listre1268245261_state @ R ) )
= ( ( ( size_s318434721st_com @ Xs2 )
= ( size_size_list_state @ Ys2 ) )
& ! [N2: nat] :
( ( ord_less_nat @ N2 @ ( size_s318434721st_com @ Xs2 ) )
=> ( member64234660_state @ ( produc1204172211_state @ ( nth_list_com @ Xs2 @ N2 ) @ ( nth_state @ Ys2 @ N2 ) ) @ R ) ) ) ) ).
% listrel_iff_nth
thf(fact_343_listrel__iff__nth,axiom,
! [Xs2: list_P890094829_state,Ys2: list_P890094829_state,R: set_Pr2096893127_state] :
( ( member423306384_state @ ( produc1354769367_state @ Xs2 @ Ys2 ) @ ( listre1244407729_state @ R ) )
= ( ( ( size_s837829465_state @ Xs2 )
= ( size_s837829465_state @ Ys2 ) )
& ! [N2: nat] :
( ( ord_less_nat @ N2 @ ( size_s837829465_state @ Xs2 ) )
=> ( member512523024_state @ ( produc2065881943_state @ ( nth_Pr526167918_state @ Xs2 @ N2 ) @ ( nth_Pr526167918_state @ Ys2 @ N2 ) ) @ R ) ) ) ) ).
% listrel_iff_nth
thf(fact_344_in__measures_I1_J,axiom,
! [X: produc2041926651_state,Y: produc2041926651_state] :
~ ( member1069318160_state @ ( produc1909270103_state @ X @ Y ) @ ( measur1071343867_state @ nil_Pr1876916290te_nat ) ) ).
% in_measures(1)
thf(fact_345_in__measures_I1_J,axiom,
! [X: produc410756839_state,Y: produc410756839_state] :
~ ( member512523024_state @ ( produc2065881943_state @ X @ Y ) @ ( measur768621543_state @ nil_Pr164089518te_nat ) ) ).
% in_measures(1)
thf(fact_346_nth__append__length,axiom,
! [Xs2: list_com,X: com,Ys2: list_com] :
( ( nth_com @ ( append_com @ Xs2 @ ( cons_com @ X @ Ys2 ) ) @ ( size_size_list_com @ Xs2 ) )
= X ) ).
% nth_append_length
thf(fact_347_measures__less,axiom,
! [F2: produc2041926651_state > nat,X: produc2041926651_state,Y: produc2041926651_state,Fs: list_P1178430914te_nat] :
( ( ord_less_nat @ ( F2 @ X ) @ ( F2 @ Y ) )
=> ( member1069318160_state @ ( produc1909270103_state @ X @ Y ) @ ( measur1071343867_state @ ( cons_P1821701106te_nat @ F2 @ Fs ) ) ) ) ).
% measures_less
thf(fact_348_measures__less,axiom,
! [F2: produc410756839_state > nat,X: produc410756839_state,Y: produc410756839_state,Fs: list_P1627643054te_nat] :
( ( ord_less_nat @ ( F2 @ X ) @ ( F2 @ Y ) )
=> ( member512523024_state @ ( produc2065881943_state @ X @ Y ) @ ( measur768621543_state @ ( cons_P1631380062te_nat @ F2 @ Fs ) ) ) ) ).
% measures_less
thf(fact_349_listrel1__iff__update,axiom,
! [Xs2: list_P1040014977_state,Ys2: list_P1040014977_state,R: set_Pr1165141447_state] :
( ( member1267359376_state @ ( produc1298457559_state @ Xs2 @ Ys2 ) @ ( listre1738592858_state @ R ) )
= ( ? [Y4: produc2041926651_state,N2: nat] :
( ( member1069318160_state @ ( produc1909270103_state @ ( nth_Pr553241858_state @ Xs2 @ N2 ) @ Y4 ) @ R )
& ( ord_less_nat @ N2 @ ( size_s213202669_state @ Xs2 ) )
& ( Ys2
= ( list_u737668969_state @ Xs2 @ N2 @ Y4 ) ) ) ) ) ).
% listrel1_iff_update
thf(fact_350_listrel1__iff__update,axiom,
! [Xs2: list_P890094829_state,Ys2: list_P890094829_state,R: set_Pr2096893127_state] :
( ( member423306384_state @ ( produc1354769367_state @ Xs2 @ Ys2 ) @ ( listre1725776070_state @ R ) )
= ( ? [Y4: produc410756839_state,N2: nat] :
( ( member512523024_state @ ( produc2065881943_state @ ( nth_Pr526167918_state @ Xs2 @ N2 ) @ Y4 ) @ R )
& ( ord_less_nat @ N2 @ ( size_s837829465_state @ Xs2 ) )
& ( Ys2
= ( list_u364351829_state @ Xs2 @ N2 @ Y4 ) ) ) ) ) ).
% listrel1_iff_update
thf(fact_351_lex__take__index,axiom,
! [Xs2: list_P1040014977_state,Ys2: list_P1040014977_state,R: set_Pr1165141447_state] :
( ( member1267359376_state @ ( produc1298457559_state @ Xs2 @ Ys2 ) @ ( lex_Pr570247525_state @ R ) )
=> ~ ! [I: nat] :
( ( ord_less_nat @ I @ ( size_s213202669_state @ Xs2 ) )
=> ( ( ord_less_nat @ I @ ( size_s213202669_state @ Ys2 ) )
=> ( ( ( take_P558217449_state @ I @ Xs2 )
= ( take_P558217449_state @ I @ Ys2 ) )
=> ~ ( member1069318160_state @ ( produc1909270103_state @ ( nth_Pr553241858_state @ Xs2 @ I ) @ ( nth_Pr553241858_state @ Ys2 @ I ) ) @ R ) ) ) ) ) ).
% lex_take_index
thf(fact_352_lex__take__index,axiom,
! [Xs2: list_P890094829_state,Ys2: list_P890094829_state,R: set_Pr2096893127_state] :
( ( member423306384_state @ ( produc1354769367_state @ Xs2 @ Ys2 ) @ ( lex_Pr1762629969_state @ R ) )
=> ~ ! [I: nat] :
( ( ord_less_nat @ I @ ( size_s837829465_state @ Xs2 ) )
=> ( ( ord_less_nat @ I @ ( size_s837829465_state @ Ys2 ) )
=> ( ( ( take_P1852316373_state @ I @ Xs2 )
= ( take_P1852316373_state @ I @ Ys2 ) )
=> ~ ( member512523024_state @ ( produc2065881943_state @ ( nth_Pr526167918_state @ Xs2 @ I ) @ ( nth_Pr526167918_state @ Ys2 @ I ) ) @ R ) ) ) ) ) ).
% lex_take_index
thf(fact_353_list__update__nonempty,axiom,
! [Xs2: list_com,K: nat,X: com] :
( ( ( list_update_com @ Xs2 @ K @ X )
= nil_com )
= ( Xs2 = nil_com ) ) ).
% list_update_nonempty
% Conjectures (4)
thf(conj_0,hypothesis,
member1069318160_state @ ( produc1909270103_state @ ( produc1204172211_state @ ( cons_com @ c @ nil_com ) @ s ) @ ( produc1204172211_state @ ( cons_com @ c2 @ cs ) @ s2 ) ) @ ( transi1302705790_state @ pHoare259243666_exec1 ) ).
thf(conj_1,hypothesis,
termi @ c @ s ).
thf(conj_2,hypothesis,
! [S5: state,S6: state] :
( ( member1069318160_state @ ( produc1909270103_state @ ( produc1204172211_state @ ( cons_com @ c @ nil_com ) @ S5 ) @ ( produc1204172211_state @ ( cons_com @ c2 @ cs ) @ S6 ) ) @ ( transi1302705790_state @ pHoare259243666_exec1 ) )
=> ( ( termi @ c @ S5 )
=> ( ( termi @ c2 @ S6 )
& ! [T4: state] :
( ( exec @ S6 @ c2 @ T4 )
=> ( pHoare806997636termis @ cs @ T4 ) ) ) ) ) ).
thf(conj_3,conjecture,
termi @ c2 @ s2 ).
%------------------------------------------------------------------------------