TPTP Problem File: ITP142^2.p
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%------------------------------------------------------------------------------
% File : ITP142^2 : 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 : 0.00 v7.5.0
% Syntax : Number of formulae : 319 ( 123 unt; 54 typ; 0 def)
% Number of atoms : 656 ( 363 equ; 0 cnn)
% Maximal formula atoms : 7 ( 2 avg)
% Number of connectives : 7177 ( 151 ~; 21 |; 67 &;6586 @)
% ( 0 <=>; 352 =>; 0 <=; 0 <~>)
% Maximal formula depth : 28 ( 10 avg)
% Number of types : 4 ( 3 usr)
% Number of type conns : 247 ( 247 >; 0 *; 0 +; 0 <<)
% Number of symbols : 52 ( 51 usr; 6 con; 0-5 aty)
% Number of variables : 1364 ( 9 ^;1267 !; 46 ?;1364 :)
% ( 42 !>; 0 ?*; 0 @-; 0 @+)
% SPC : TH1_THM_EQU_NAR
% Comments : This file was generated by Sledgehammer 2021-02-23 16:18:28.966
%------------------------------------------------------------------------------
% Could-be-implicit typings (6)
thf(ty_t_Product__Type_Oprod,type,
product_prod: $tType > $tType > $tType ).
thf(ty_t_PLang_Ostate,type,
state: $tType ).
thf(ty_t_PLang_Ocom,type,
com: $tType ).
thf(ty_t_List_Olist,type,
list: $tType > $tType ).
thf(ty_t_Set_Oset,type,
set: $tType > $tType ).
thf(ty_t_Nat_Onat,type,
nat: $tType ).
% Explicit typings (48)
thf(sy_cl_HOL_Otype,type,
type:
!>[A: $tType] : $o ).
thf(sy_cl_Orderings_Oord,type,
ord:
!>[A: $tType] : $o ).
thf(sy_cl_Orderings_Olinorder,type,
linorder:
!>[A: $tType] : $o ).
thf(sy_c_List_Oappend,type,
append:
!>[A: $tType] : ( ( list @ A ) > ( list @ A ) > ( list @ A ) ) ).
thf(sy_c_List_Obind,type,
bind:
!>[A: $tType,B: $tType] : ( ( list @ A ) > ( A > ( list @ B ) ) > ( list @ B ) ) ).
thf(sy_c_List_Obutlast,type,
butlast:
!>[A: $tType] : ( ( list @ A ) > ( list @ A ) ) ).
thf(sy_c_List_Oconcat,type,
concat:
!>[A: $tType] : ( ( list @ ( list @ A ) ) > ( list @ A ) ) ).
thf(sy_c_List_Oinsert,type,
insert:
!>[A: $tType] : ( A > ( list @ A ) > ( list @ A ) ) ).
thf(sy_c_List_Olast,type,
last:
!>[A: $tType] : ( ( list @ A ) > A ) ).
thf(sy_c_List_Olenlex,type,
lenlex:
!>[A: $tType] : ( ( set @ ( product_prod @ A @ A ) ) > ( set @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) ) ) ).
thf(sy_c_List_Olex,type,
lex:
!>[A: $tType] : ( ( set @ ( product_prod @ A @ A ) ) > ( set @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) ) ) ).
thf(sy_c_List_Olexord,type,
lexord:
!>[A: $tType] : ( ( set @ ( product_prod @ A @ A ) ) > ( set @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) ) ) ).
thf(sy_c_List_Olist_OCons,type,
cons:
!>[A: $tType] : ( A > ( list @ A ) > ( list @ A ) ) ).
thf(sy_c_List_Olist_ONil,type,
nil:
!>[A: $tType] : ( list @ A ) ).
thf(sy_c_List_Olist__ex1,type,
list_ex1:
!>[A: $tType] : ( ( A > $o ) > ( list @ A ) > $o ) ).
thf(sy_c_List_Olist__update,type,
list_update:
!>[A: $tType] : ( ( list @ A ) > nat > A > ( list @ A ) ) ).
thf(sy_c_List_Olistrel,type,
listrel:
!>[A: $tType,B: $tType] : ( ( set @ ( product_prod @ A @ B ) ) > ( set @ ( product_prod @ ( list @ A ) @ ( list @ B ) ) ) ) ).
thf(sy_c_List_Olistrel1,type,
listrel1:
!>[A: $tType] : ( ( set @ ( product_prod @ A @ A ) ) > ( set @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) ) ) ).
thf(sy_c_List_Omaps,type,
maps:
!>[A: $tType,B: $tType] : ( ( A > ( list @ B ) ) > ( list @ A ) > ( list @ B ) ) ).
thf(sy_c_List_Omeasures,type,
measures:
!>[A: $tType] : ( ( list @ ( A > nat ) ) > ( set @ ( product_prod @ A @ A ) ) ) ).
thf(sy_c_List_Onth,type,
nth:
!>[A: $tType] : ( ( list @ A ) > nat > A ) ).
thf(sy_c_List_Oproduct__lists,type,
product_lists:
!>[A: $tType] : ( ( list @ ( list @ A ) ) > ( list @ ( list @ A ) ) ) ).
thf(sy_c_List_Orotate1,type,
rotate1:
!>[A: $tType] : ( ( list @ A ) > ( list @ A ) ) ).
thf(sy_c_List_Osubseqs,type,
subseqs:
!>[A: $tType] : ( ( list @ A ) > ( list @ ( list @ A ) ) ) ).
thf(sy_c_List_Otake,type,
take:
!>[A: $tType] : ( nat > ( list @ A ) > ( list @ A ) ) ).
thf(sy_c_Nat_Osize__class_Osize,type,
size_size:
!>[A: $tType] : ( A > nat ) ).
thf(sy_c_Orderings_Oord__class_Oless,type,
ord_less:
!>[A: $tType] : ( A > A > $o ) ).
thf(sy_c_PHoareTotal__Mirabelle__nfzgdgthbw_Oexec1,type,
pHoare1053570893_exec1: set @ ( product_prod @ ( product_prod @ ( list @ com ) @ state ) @ ( product_prod @ ( list @ com ) @ state ) ) ).
thf(sy_c_PHoareTotal__Mirabelle__nfzgdgthbw_Oexecs,type,
pHoare1053570959_execs: state > ( list @ com ) > state > $o ).
thf(sy_c_PHoareTotal__Mirabelle__nfzgdgthbw_Otermis,type,
pHoare1335526537termis: ( list @ com ) > state > $o ).
thf(sy_c_PLang_Ocom_OCond,type,
cond: ( state > $o ) > com > com > com ).
thf(sy_c_PLang_Ocom_ODo,type,
do: ( state > ( set @ state ) ) > com ).
thf(sy_c_PLang_Ocom_OSemi,type,
semi: com > com > com ).
thf(sy_c_PLang_Ocom_OWhile,type,
while: ( state > $o ) > com > com ).
thf(sy_c_PLang_Oexec,type,
exec: state > com > state > $o ).
thf(sy_c_PTermi_Otermi,type,
termi: com > state > $o ).
thf(sy_c_Product__Type_OPair,type,
product_Pair:
!>[A: $tType,B: $tType] : ( A > B > ( product_prod @ A @ B ) ) ).
thf(sy_c_Product__Type_Ointernal__case__prod,type,
produc2004651681e_prod:
!>[A: $tType,B: $tType,C: $tType] : ( ( A > B > C ) > ( product_prod @ A @ B ) > C ) ).
thf(sy_c_Product__Type_Oold_Oprod_Orec__prod,type,
product_rec_prod:
!>[A: $tType,B: $tType,T: $tType] : ( ( A > B > T ) > ( product_prod @ A @ B ) > T ) ).
thf(sy_c_Relation_Oirrefl,type,
irrefl:
!>[A: $tType] : ( ( set @ ( product_prod @ A @ A ) ) > $o ) ).
thf(sy_c_Set_OCollect,type,
collect:
!>[A: $tType] : ( ( A > $o ) > ( set @ A ) ) ).
thf(sy_c_Transitive__Closure_Ortrancl,type,
transitive_rtrancl:
!>[A: $tType] : ( ( set @ ( product_prod @ A @ A ) ) > ( set @ ( product_prod @ A @ A ) ) ) ).
thf(sy_c_member,type,
member:
!>[A: $tType] : ( A > ( set @ A ) > $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 (255)
thf(fact_0_exec1E_I1_J,axiom,
! [S: state,Cs: list @ com,S2: state] :
~ ( member @ ( product_prod @ ( product_prod @ ( list @ com ) @ state ) @ ( product_prod @ ( list @ com ) @ state ) ) @ ( product_Pair @ ( product_prod @ ( list @ com ) @ state ) @ ( product_prod @ ( list @ com ) @ state ) @ ( product_Pair @ ( list @ com ) @ state @ ( nil @ com ) @ S ) @ ( product_Pair @ ( list @ com ) @ state @ Cs @ S2 ) ) @ pHoare1053570893_exec1 ) ).
% exec1E(1)
thf(fact_1_termis_Osimps_I2_J,axiom,
! [C2: com,Cs2: list @ com,S: state] :
( ( pHoare1335526537termis @ ( cons @ com @ C2 @ Cs2 ) @ S )
= ( ( termi @ C2 @ S )
& ! [T2: state] :
( ( exec @ S @ C2 @ T2 )
=> ( pHoare1335526537termis @ Cs2 @ T2 ) ) ) ) ).
% termis.simps(2)
thf(fact_2_termis_Osimps_I1_J,axiom,
! [S: state] : ( pHoare1335526537termis @ ( nil @ com ) @ S ) ).
% termis.simps(1)
thf(fact_3_exec__impl__execs,axiom,
! [S: state,C2: com,S2: state,Cs2: list @ com] :
( ( exec @ S @ C2 @ S2 )
=> ( member @ ( product_prod @ ( product_prod @ ( list @ com ) @ state ) @ ( product_prod @ ( list @ com ) @ state ) ) @ ( product_Pair @ ( product_prod @ ( list @ com ) @ state ) @ ( product_prod @ ( list @ com ) @ state ) @ ( product_Pair @ ( list @ com ) @ state @ ( cons @ com @ C2 @ Cs2 ) @ S ) @ ( product_Pair @ ( list @ com ) @ state @ Cs2 @ S2 ) ) @ ( transitive_rtrancl @ ( product_prod @ ( list @ com ) @ state ) @ pHoare1053570893_exec1 ) ) ) ).
% exec_impl_execs
thf(fact_4_exec1s__impl__exec,axiom,
! [C2: com,S: state,T3: state] :
( ( member @ ( product_prod @ ( product_prod @ ( list @ com ) @ state ) @ ( product_prod @ ( list @ com ) @ state ) ) @ ( product_Pair @ ( product_prod @ ( list @ com ) @ state ) @ ( product_prod @ ( list @ com ) @ state ) @ ( product_Pair @ ( list @ com ) @ state @ ( cons @ com @ C2 @ ( nil @ com ) ) @ S ) @ ( product_Pair @ ( list @ com ) @ state @ ( nil @ com ) @ T3 ) ) @ ( transitive_rtrancl @ ( product_prod @ ( list @ com ) @ state ) @ pHoare1053570893_exec1 ) )
=> ( exec @ S @ C2 @ T3 ) ) ).
% exec1s_impl_exec
thf(fact_5_exec1__pres__termis,axiom,
! [Cs2: list @ com,S: state,Cs: list @ com,S2: state] :
( ( member @ ( product_prod @ ( product_prod @ ( list @ com ) @ state ) @ ( product_prod @ ( list @ com ) @ state ) ) @ ( product_Pair @ ( product_prod @ ( list @ com ) @ state ) @ ( product_prod @ ( list @ com ) @ state ) @ ( product_Pair @ ( list @ com ) @ state @ Cs2 @ S ) @ ( product_Pair @ ( list @ com ) @ state @ Cs @ S2 ) ) @ pHoare1053570893_exec1 )
=> ( ( pHoare1335526537termis @ Cs2 @ S )
=> ( pHoare1335526537termis @ Cs @ S2 ) ) ) ).
% exec1_pres_termis
thf(fact_6_execs__pres__termis,axiom,
! [Cs2: list @ com,S: state,Cs: list @ com,S2: state] :
( ( member @ ( product_prod @ ( product_prod @ ( list @ com ) @ state ) @ ( product_prod @ ( list @ com ) @ state ) ) @ ( product_Pair @ ( product_prod @ ( list @ com ) @ state ) @ ( product_prod @ ( list @ com ) @ state ) @ ( product_Pair @ ( list @ com ) @ state @ Cs2 @ S ) @ ( product_Pair @ ( list @ com ) @ state @ Cs @ S2 ) ) @ ( transitive_rtrancl @ ( product_prod @ ( list @ com ) @ state ) @ pHoare1053570893_exec1 ) )
=> ( ( pHoare1335526537termis @ Cs2 @ S )
=> ( pHoare1335526537termis @ Cs @ S2 ) ) ) ).
% execs_pres_termis
thf(fact_7_rtrancl__induct2,axiom,
! [A: $tType,B: $tType,Ax: A,Ay: B,Bx: A,By: B,R: set @ ( product_prod @ ( product_prod @ A @ B ) @ ( product_prod @ A @ B ) ),P: A > B > $o] :
( ( member @ ( product_prod @ ( product_prod @ A @ B ) @ ( product_prod @ A @ B ) ) @ ( product_Pair @ ( product_prod @ A @ B ) @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ Ax @ Ay ) @ ( product_Pair @ A @ B @ Bx @ By ) ) @ ( transitive_rtrancl @ ( product_prod @ A @ B ) @ R ) )
=> ( ( P @ Ax @ Ay )
=> ( ! [A2: A,B2: B,Aa: A,Ba: B] :
( ( member @ ( product_prod @ ( product_prod @ A @ B ) @ ( product_prod @ A @ B ) ) @ ( product_Pair @ ( product_prod @ A @ B ) @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ Ax @ Ay ) @ ( product_Pair @ A @ B @ A2 @ B2 ) ) @ ( transitive_rtrancl @ ( product_prod @ A @ B ) @ R ) )
=> ( ( member @ ( product_prod @ ( product_prod @ A @ B ) @ ( product_prod @ A @ B ) ) @ ( product_Pair @ ( product_prod @ A @ B ) @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ A2 @ B2 ) @ ( product_Pair @ A @ B @ Aa @ Ba ) ) @ R )
=> ( ( P @ A2 @ B2 )
=> ( P @ Aa @ Ba ) ) ) )
=> ( P @ Bx @ By ) ) ) ) ).
% rtrancl_induct2
thf(fact_8_converse__rtranclE2,axiom,
! [B: $tType,A: $tType,Xa: A,Xb: B,Za: A,Zb: B,R: set @ ( product_prod @ ( product_prod @ A @ B ) @ ( product_prod @ A @ B ) )] :
( ( member @ ( product_prod @ ( product_prod @ A @ B ) @ ( product_prod @ A @ B ) ) @ ( product_Pair @ ( product_prod @ A @ B ) @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ Xa @ Xb ) @ ( product_Pair @ A @ B @ Za @ Zb ) ) @ ( transitive_rtrancl @ ( product_prod @ A @ B ) @ R ) )
=> ( ( ( product_Pair @ A @ B @ Xa @ Xb )
!= ( product_Pair @ A @ B @ Za @ Zb ) )
=> ~ ! [A2: A,B2: B] :
( ( member @ ( product_prod @ ( product_prod @ A @ B ) @ ( product_prod @ A @ B ) ) @ ( product_Pair @ ( product_prod @ A @ B ) @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ Xa @ Xb ) @ ( product_Pair @ A @ B @ A2 @ B2 ) ) @ R )
=> ~ ( member @ ( product_prod @ ( product_prod @ A @ B ) @ ( product_prod @ A @ B ) ) @ ( product_Pair @ ( product_prod @ A @ B ) @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ A2 @ B2 ) @ ( product_Pair @ A @ B @ Za @ Zb ) ) @ ( transitive_rtrancl @ ( product_prod @ A @ B ) @ R ) ) ) ) ) ).
% converse_rtranclE2
thf(fact_9_converse__rtrancl__induct2,axiom,
! [A: $tType,B: $tType,Ax: A,Ay: B,Bx: A,By: B,R: set @ ( product_prod @ ( product_prod @ A @ B ) @ ( product_prod @ A @ B ) ),P: A > B > $o] :
( ( member @ ( product_prod @ ( product_prod @ A @ B ) @ ( product_prod @ A @ B ) ) @ ( product_Pair @ ( product_prod @ A @ B ) @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ Ax @ Ay ) @ ( product_Pair @ A @ B @ Bx @ By ) ) @ ( transitive_rtrancl @ ( product_prod @ A @ B ) @ R ) )
=> ( ( P @ Bx @ By )
=> ( ! [A2: A,B2: B,Aa: A,Ba: B] :
( ( member @ ( product_prod @ ( product_prod @ A @ B ) @ ( product_prod @ A @ B ) ) @ ( product_Pair @ ( product_prod @ A @ B ) @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ A2 @ B2 ) @ ( product_Pair @ A @ B @ Aa @ Ba ) ) @ R )
=> ( ( member @ ( product_prod @ ( product_prod @ A @ B ) @ ( product_prod @ A @ B ) ) @ ( product_Pair @ ( product_prod @ A @ B ) @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ Aa @ Ba ) @ ( product_Pair @ A @ B @ Bx @ By ) ) @ ( transitive_rtrancl @ ( product_prod @ A @ B ) @ R ) )
=> ( ( P @ Aa @ Ba )
=> ( P @ A2 @ B2 ) ) ) )
=> ( P @ Ax @ Ay ) ) ) ) ).
% converse_rtrancl_induct2
thf(fact_10_exec1s__impl__execs,axiom,
! [Cs2: list @ com,S: state,T3: state] :
( ( member @ ( product_prod @ ( product_prod @ ( list @ com ) @ state ) @ ( product_prod @ ( list @ com ) @ state ) ) @ ( product_Pair @ ( product_prod @ ( list @ com ) @ state ) @ ( product_prod @ ( list @ com ) @ state ) @ ( product_Pair @ ( list @ com ) @ state @ Cs2 @ S ) @ ( product_Pair @ ( list @ com ) @ state @ ( nil @ com ) @ T3 ) ) @ ( transitive_rtrancl @ ( product_prod @ ( list @ com ) @ state ) @ pHoare1053570893_exec1 ) )
=> ( pHoare1053570959_execs @ S @ Cs2 @ T3 ) ) ).
% exec1s_impl_execs
thf(fact_11_rtrancl__idemp,axiom,
! [A: $tType,R: set @ ( product_prod @ A @ A )] :
( ( transitive_rtrancl @ A @ ( transitive_rtrancl @ A @ R ) )
= ( transitive_rtrancl @ A @ R ) ) ).
% rtrancl_idemp
thf(fact_12_r__into__rtrancl,axiom,
! [A: $tType,P2: product_prod @ A @ A,R: set @ ( product_prod @ A @ A )] :
( ( member @ ( product_prod @ A @ A ) @ P2 @ R )
=> ( member @ ( product_prod @ A @ A ) @ P2 @ ( transitive_rtrancl @ A @ R ) ) ) ).
% r_into_rtrancl
thf(fact_13_list_Oinject,axiom,
! [A: $tType,X21: A,X22: list @ A,Y21: A,Y22: list @ A] :
( ( ( cons @ A @ X21 @ X22 )
= ( cons @ A @ Y21 @ Y22 ) )
= ( ( X21 = Y21 )
& ( X22 = Y22 ) ) ) ).
% list.inject
thf(fact_14_prod_Oinject,axiom,
! [A: $tType,B: $tType,X1: A,X2: B,Y1: A,Y2: B] :
( ( ( product_Pair @ A @ B @ X1 @ X2 )
= ( product_Pair @ A @ B @ Y1 @ Y2 ) )
= ( ( X1 = Y1 )
& ( X2 = Y2 ) ) ) ).
% prod.inject
thf(fact_15_old_Oprod_Oinject,axiom,
! [A: $tType,B: $tType,A3: A,B3: B,A4: A,B4: B] :
( ( ( product_Pair @ A @ B @ A3 @ B3 )
= ( product_Pair @ A @ B @ A4 @ B4 ) )
= ( ( A3 = A4 )
& ( B3 = B4 ) ) ) ).
% old.prod.inject
thf(fact_16_prod__cases3,axiom,
! [A: $tType,B: $tType,C: $tType,Y: product_prod @ A @ ( product_prod @ B @ C )] :
~ ! [A2: A,B2: B,C3: C] :
( Y
!= ( product_Pair @ A @ ( product_prod @ B @ C ) @ A2 @ ( product_Pair @ B @ C @ B2 @ C3 ) ) ) ).
% prod_cases3
thf(fact_17_prod__cases4,axiom,
! [A: $tType,B: $tType,C: $tType,D: $tType,Y: product_prod @ A @ ( product_prod @ B @ ( product_prod @ C @ D ) )] :
~ ! [A2: A,B2: B,C3: C,D2: D] :
( Y
!= ( product_Pair @ A @ ( product_prod @ B @ ( product_prod @ C @ D ) ) @ A2 @ ( product_Pair @ B @ ( product_prod @ C @ D ) @ B2 @ ( product_Pair @ C @ D @ C3 @ D2 ) ) ) ) ).
% prod_cases4
thf(fact_18_prod__cases5,axiom,
! [A: $tType,B: $tType,C: $tType,D: $tType,E: $tType,Y: product_prod @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D @ E ) ) )] :
~ ! [A2: A,B2: B,C3: C,D2: D,E2: E] :
( Y
!= ( product_Pair @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D @ E ) ) ) @ A2 @ ( product_Pair @ B @ ( product_prod @ C @ ( product_prod @ D @ E ) ) @ B2 @ ( product_Pair @ C @ ( product_prod @ D @ E ) @ C3 @ ( product_Pair @ D @ E @ D2 @ E2 ) ) ) ) ) ).
% prod_cases5
thf(fact_19_map__tailrec__rev_Ocases,axiom,
! [A: $tType,B: $tType,X: product_prod @ ( A > B ) @ ( product_prod @ ( list @ A ) @ ( list @ B ) )] :
( ! [F: A > B,Bs: list @ B] :
( X
!= ( product_Pair @ ( A > B ) @ ( product_prod @ ( list @ A ) @ ( list @ B ) ) @ F @ ( product_Pair @ ( list @ A ) @ ( list @ B ) @ ( nil @ A ) @ Bs ) ) )
=> ~ ! [F: A > B,A2: A,As: list @ A,Bs: list @ B] :
( X
!= ( product_Pair @ ( A > B ) @ ( product_prod @ ( list @ A ) @ ( list @ B ) ) @ F @ ( product_Pair @ ( list @ A ) @ ( list @ B ) @ ( cons @ A @ A2 @ As ) @ Bs ) ) ) ) ).
% map_tailrec_rev.cases
thf(fact_20_arg__min__list_Ocases,axiom,
! [B: $tType,A: $tType] :
( ( linorder @ B )
=> ! [X: product_prod @ ( A > B ) @ ( list @ A )] :
( ! [F: A > B,X3: A] :
( X
!= ( product_Pair @ ( A > B ) @ ( list @ A ) @ F @ ( cons @ A @ X3 @ ( nil @ A ) ) ) )
=> ( ! [F: A > B,X3: A,Y3: A,Zs: list @ A] :
( X
!= ( product_Pair @ ( A > B ) @ ( list @ A ) @ F @ ( cons @ A @ X3 @ ( cons @ A @ Y3 @ Zs ) ) ) )
=> ~ ! [A2: A > B] :
( X
!= ( product_Pair @ ( A > B ) @ ( list @ A ) @ A2 @ ( nil @ A ) ) ) ) ) ) ).
% arg_min_list.cases
thf(fact_21_successively_Ocases,axiom,
! [A: $tType,X: product_prod @ ( A > A > $o ) @ ( list @ A )] :
( ! [P3: A > A > $o] :
( X
!= ( product_Pair @ ( A > A > $o ) @ ( list @ A ) @ P3 @ ( nil @ A ) ) )
=> ( ! [P3: A > A > $o,X3: A] :
( X
!= ( product_Pair @ ( A > A > $o ) @ ( list @ A ) @ P3 @ ( cons @ A @ X3 @ ( nil @ A ) ) ) )
=> ~ ! [P3: A > A > $o,X3: A,Y3: A,Xs: list @ A] :
( X
!= ( product_Pair @ ( A > A > $o ) @ ( list @ A ) @ P3 @ ( cons @ A @ X3 @ ( cons @ A @ Y3 @ Xs ) ) ) ) ) ) ).
% successively.cases
thf(fact_22_sorted__wrt_Ocases,axiom,
! [A: $tType,X: product_prod @ ( A > A > $o ) @ ( list @ A )] :
( ! [P3: A > A > $o] :
( X
!= ( product_Pair @ ( A > A > $o ) @ ( list @ A ) @ P3 @ ( nil @ A ) ) )
=> ~ ! [P3: A > A > $o,X3: A,Ys: list @ A] :
( X
!= ( product_Pair @ ( A > A > $o ) @ ( list @ A ) @ P3 @ ( cons @ A @ X3 @ Ys ) ) ) ) ).
% sorted_wrt.cases
thf(fact_23_shuffles_Ocases,axiom,
! [A: $tType,X: product_prod @ ( list @ A ) @ ( list @ A )] :
( ! [Ys: list @ A] :
( X
!= ( product_Pair @ ( list @ A ) @ ( list @ A ) @ ( nil @ A ) @ Ys ) )
=> ( ! [Xs: list @ A] :
( X
!= ( product_Pair @ ( list @ A ) @ ( list @ A ) @ Xs @ ( nil @ A ) ) )
=> ~ ! [X3: A,Xs: list @ A,Y3: A,Ys: list @ A] :
( X
!= ( product_Pair @ ( list @ A ) @ ( list @ A ) @ ( cons @ A @ X3 @ Xs ) @ ( cons @ A @ Y3 @ Ys ) ) ) ) ) ).
% shuffles.cases
thf(fact_24_splice_Ocases,axiom,
! [A: $tType,X: product_prod @ ( list @ A ) @ ( list @ A )] :
( ! [Ys: list @ A] :
( X
!= ( product_Pair @ ( list @ A ) @ ( list @ A ) @ ( nil @ A ) @ Ys ) )
=> ~ ! [X3: A,Xs: list @ A,Ys: list @ A] :
( X
!= ( product_Pair @ ( list @ A ) @ ( list @ A ) @ ( cons @ A @ X3 @ Xs ) @ Ys ) ) ) ).
% splice.cases
thf(fact_25_execs_Ointros_I1_J,axiom,
! [S: state] : ( pHoare1053570959_execs @ S @ ( nil @ com ) @ S ) ).
% execs.intros(1)
thf(fact_26_old_Oprod_Oinducts,axiom,
! [B: $tType,A: $tType,P: ( product_prod @ A @ B ) > $o,Prod: product_prod @ A @ B] :
( ! [A2: A,B2: B] : ( P @ ( product_Pair @ A @ B @ A2 @ B2 ) )
=> ( P @ Prod ) ) ).
% old.prod.inducts
thf(fact_27_old_Oprod_Oexhaust,axiom,
! [A: $tType,B: $tType,Y: product_prod @ A @ B] :
~ ! [A2: A,B2: B] :
( Y
!= ( product_Pair @ A @ B @ A2 @ B2 ) ) ).
% old.prod.exhaust
thf(fact_28_Pair__inject,axiom,
! [A: $tType,B: $tType,A3: A,B3: B,A4: A,B4: B] :
( ( ( product_Pair @ A @ B @ A3 @ B3 )
= ( product_Pair @ A @ B @ A4 @ B4 ) )
=> ~ ( ( A3 = A4 )
=> ( B3 != B4 ) ) ) ).
% Pair_inject
thf(fact_29_prod__cases,axiom,
! [B: $tType,A: $tType,P: ( product_prod @ A @ B ) > $o,P2: product_prod @ A @ B] :
( ! [A2: A,B2: B] : ( P @ ( product_Pair @ A @ B @ A2 @ B2 ) )
=> ( P @ P2 ) ) ).
% prod_cases
thf(fact_30_surj__pair,axiom,
! [A: $tType,B: $tType,P2: product_prod @ A @ B] :
? [X3: A,Y3: B] :
( P2
= ( product_Pair @ A @ B @ X3 @ Y3 ) ) ).
% surj_pair
thf(fact_31_not__Cons__self2,axiom,
! [A: $tType,X: A,Xs2: list @ A] :
( ( cons @ A @ X @ Xs2 )
!= Xs2 ) ).
% not_Cons_self2
thf(fact_32_execs_Ointros_I2_J,axiom,
! [S: state,C2: com,T3: state,Cs2: list @ com,U: state] :
( ( exec @ S @ C2 @ T3 )
=> ( ( pHoare1053570959_execs @ T3 @ Cs2 @ U )
=> ( pHoare1053570959_execs @ S @ ( cons @ com @ C2 @ Cs2 ) @ U ) ) ) ).
% execs.intros(2)
thf(fact_33_strict__sorted_Oinduct,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [P: ( list @ A ) > $o,A0: list @ A] :
( ( P @ ( nil @ A ) )
=> ( ! [X3: A,Ys: list @ A] :
( ( P @ Ys )
=> ( P @ ( cons @ A @ X3 @ Ys ) ) )
=> ( P @ A0 ) ) ) ) ).
% strict_sorted.induct
thf(fact_34_strict__sorted_Ocases,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [X: list @ A] :
( ( X
!= ( nil @ A ) )
=> ~ ! [X3: A,Ys: list @ A] :
( X
!= ( cons @ A @ X3 @ Ys ) ) ) ) ).
% strict_sorted.cases
thf(fact_35_map__tailrec__rev_Oinduct,axiom,
! [A: $tType,B: $tType,P: ( A > B ) > ( list @ A ) > ( list @ B ) > $o,A0: A > B,A1: list @ A,A22: list @ B] :
( ! [F: A > B,X_1: list @ B] : ( P @ F @ ( nil @ A ) @ X_1 )
=> ( ! [F: A > B,A2: A,As: list @ A,Bs: list @ B] :
( ( P @ F @ As @ ( cons @ B @ ( F @ A2 ) @ Bs ) )
=> ( P @ F @ ( cons @ A @ A2 @ As ) @ Bs ) )
=> ( P @ A0 @ A1 @ A22 ) ) ) ).
% map_tailrec_rev.induct
thf(fact_36_list__nonempty__induct,axiom,
! [A: $tType,Xs2: list @ A,P: ( list @ A ) > $o] :
( ( Xs2
!= ( nil @ A ) )
=> ( ! [X3: A] : ( P @ ( cons @ A @ X3 @ ( nil @ A ) ) )
=> ( ! [X3: A,Xs: list @ A] :
( ( Xs
!= ( nil @ A ) )
=> ( ( P @ Xs )
=> ( P @ ( cons @ A @ X3 @ Xs ) ) ) )
=> ( P @ Xs2 ) ) ) ) ).
% list_nonempty_induct
thf(fact_37_successively_Oinduct,axiom,
! [A: $tType,P: ( A > A > $o ) > ( list @ A ) > $o,A0: A > A > $o,A1: list @ A] :
( ! [P3: A > A > $o] : ( P @ P3 @ ( nil @ A ) )
=> ( ! [P3: A > A > $o,X3: A] : ( P @ P3 @ ( cons @ A @ X3 @ ( nil @ A ) ) )
=> ( ! [P3: A > A > $o,X3: A,Y3: A,Xs: list @ A] :
( ( P @ P3 @ ( cons @ A @ Y3 @ Xs ) )
=> ( P @ P3 @ ( cons @ A @ X3 @ ( cons @ A @ Y3 @ Xs ) ) ) )
=> ( P @ A0 @ A1 ) ) ) ) ).
% successively.induct
thf(fact_38_arg__min__list_Oinduct,axiom,
! [B: $tType,A: $tType] :
( ( linorder @ B )
=> ! [P: ( A > B ) > ( list @ A ) > $o,A0: A > B,A1: list @ A] :
( ! [F: A > B,X3: A] : ( P @ F @ ( cons @ A @ X3 @ ( nil @ A ) ) )
=> ( ! [F: A > B,X3: A,Y3: A,Zs: list @ A] :
( ( P @ F @ ( cons @ A @ Y3 @ Zs ) )
=> ( P @ F @ ( cons @ A @ X3 @ ( cons @ A @ Y3 @ Zs ) ) ) )
=> ( ! [A2: A > B] : ( P @ A2 @ ( nil @ A ) )
=> ( P @ A0 @ A1 ) ) ) ) ) ).
% arg_min_list.induct
thf(fact_39_remdups__adj_Oinduct,axiom,
! [A: $tType,P: ( list @ A ) > $o,A0: list @ A] :
( ( P @ ( nil @ A ) )
=> ( ! [X3: A] : ( P @ ( cons @ A @ X3 @ ( nil @ A ) ) )
=> ( ! [X3: A,Y3: A,Xs: list @ A] :
( ( ( X3 = Y3 )
=> ( P @ ( cons @ A @ X3 @ Xs ) ) )
=> ( ( ( X3 != Y3 )
=> ( P @ ( cons @ A @ Y3 @ Xs ) ) )
=> ( P @ ( cons @ A @ X3 @ ( cons @ A @ Y3 @ Xs ) ) ) ) )
=> ( P @ A0 ) ) ) ) ).
% remdups_adj.induct
thf(fact_40_sorted__wrt_Oinduct,axiom,
! [A: $tType,P: ( A > A > $o ) > ( list @ A ) > $o,A0: A > A > $o,A1: list @ A] :
( ! [P3: A > A > $o] : ( P @ P3 @ ( nil @ A ) )
=> ( ! [P3: A > A > $o,X3: A,Ys: list @ A] :
( ( P @ P3 @ Ys )
=> ( P @ P3 @ ( cons @ A @ X3 @ Ys ) ) )
=> ( P @ A0 @ A1 ) ) ) ).
% sorted_wrt.induct
thf(fact_41_remdups__adj_Ocases,axiom,
! [A: $tType,X: list @ A] :
( ( X
!= ( nil @ A ) )
=> ( ! [X3: A] :
( X
!= ( cons @ A @ X3 @ ( nil @ A ) ) )
=> ~ ! [X3: A,Y3: A,Xs: list @ A] :
( X
!= ( cons @ A @ X3 @ ( cons @ A @ Y3 @ Xs ) ) ) ) ) ).
% remdups_adj.cases
thf(fact_42_transpose_Ocases,axiom,
! [A: $tType,X: list @ ( list @ A )] :
( ( X
!= ( nil @ ( list @ A ) ) )
=> ( ! [Xss: list @ ( list @ A )] :
( X
!= ( cons @ ( list @ A ) @ ( nil @ A ) @ Xss ) )
=> ~ ! [X3: A,Xs: list @ A,Xss: list @ ( list @ A )] :
( X
!= ( cons @ ( list @ A ) @ ( cons @ A @ X3 @ Xs ) @ Xss ) ) ) ) ).
% transpose.cases
thf(fact_43_shuffles_Oinduct,axiom,
! [A: $tType,P: ( list @ A ) > ( list @ A ) > $o,A0: list @ A,A1: list @ A] :
( ! [X_1: list @ A] : ( P @ ( nil @ A ) @ X_1 )
=> ( ! [Xs: list @ A] : ( P @ Xs @ ( nil @ A ) )
=> ( ! [X3: A,Xs: list @ A,Y3: A,Ys: list @ A] :
( ( P @ Xs @ ( cons @ A @ Y3 @ Ys ) )
=> ( ( P @ ( cons @ A @ X3 @ Xs ) @ Ys )
=> ( P @ ( cons @ A @ X3 @ Xs ) @ ( cons @ A @ Y3 @ Ys ) ) ) )
=> ( P @ A0 @ A1 ) ) ) ) ).
% shuffles.induct
thf(fact_44_mem__Collect__eq,axiom,
! [A: $tType,A3: A,P: A > $o] :
( ( member @ A @ A3 @ ( collect @ A @ P ) )
= ( P @ A3 ) ) ).
% mem_Collect_eq
thf(fact_45_Collect__mem__eq,axiom,
! [A: $tType,A5: set @ A] :
( ( collect @ A
@ ^ [X4: A] : ( member @ A @ X4 @ A5 ) )
= A5 ) ).
% Collect_mem_eq
thf(fact_46_Collect__cong,axiom,
! [A: $tType,P: A > $o,Q: A > $o] :
( ! [X3: A] :
( ( P @ X3 )
= ( Q @ X3 ) )
=> ( ( collect @ A @ P )
= ( collect @ A @ Q ) ) ) ).
% Collect_cong
thf(fact_47_min__list_Oinduct,axiom,
! [A: $tType] :
( ( ord @ A )
=> ! [P: ( list @ A ) > $o,A0: list @ A] :
( ! [X3: A,Xs: list @ A] :
( ! [X212: A,X222: list @ A] :
( ( Xs
= ( cons @ A @ X212 @ X222 ) )
=> ( P @ Xs ) )
=> ( P @ ( cons @ A @ X3 @ Xs ) ) )
=> ( ( P @ ( nil @ A ) )
=> ( P @ A0 ) ) ) ) ).
% min_list.induct
thf(fact_48_min__list_Ocases,axiom,
! [A: $tType] :
( ( ord @ A )
=> ! [X: list @ A] :
( ! [X3: A,Xs: list @ A] :
( X
!= ( cons @ A @ X3 @ Xs ) )
=> ( X
= ( nil @ A ) ) ) ) ).
% min_list.cases
thf(fact_49_induct__list012,axiom,
! [A: $tType,P: ( list @ A ) > $o,Xs2: list @ A] :
( ( P @ ( nil @ A ) )
=> ( ! [X3: A] : ( P @ ( cons @ A @ X3 @ ( nil @ A ) ) )
=> ( ! [X3: A,Y3: A,Zs: list @ A] :
( ( P @ Zs )
=> ( ( P @ ( cons @ A @ Y3 @ Zs ) )
=> ( P @ ( cons @ A @ X3 @ ( cons @ A @ Y3 @ Zs ) ) ) ) )
=> ( P @ Xs2 ) ) ) ) ).
% induct_list012
thf(fact_50_splice_Oinduct,axiom,
! [A: $tType,P: ( list @ A ) > ( list @ A ) > $o,A0: list @ A,A1: list @ A] :
( ! [X_1: list @ A] : ( P @ ( nil @ A ) @ X_1 )
=> ( ! [X3: A,Xs: list @ A,Ys: list @ A] :
( ( P @ Ys @ Xs )
=> ( P @ ( cons @ A @ X3 @ Xs ) @ Ys ) )
=> ( P @ A0 @ A1 ) ) ) ).
% splice.induct
thf(fact_51_list__induct2_H,axiom,
! [A: $tType,B: $tType,P: ( list @ A ) > ( list @ B ) > $o,Xs2: list @ A,Ys2: list @ B] :
( ( P @ ( nil @ A ) @ ( nil @ B ) )
=> ( ! [X3: A,Xs: list @ A] : ( P @ ( cons @ A @ X3 @ Xs ) @ ( nil @ B ) )
=> ( ! [Y3: B,Ys: list @ B] : ( P @ ( nil @ A ) @ ( cons @ B @ Y3 @ Ys ) )
=> ( ! [X3: A,Xs: list @ A,Y3: B,Ys: list @ B] :
( ( P @ Xs @ Ys )
=> ( P @ ( cons @ A @ X3 @ Xs ) @ ( cons @ B @ Y3 @ Ys ) ) )
=> ( P @ Xs2 @ Ys2 ) ) ) ) ) ).
% list_induct2'
thf(fact_52_neq__Nil__conv,axiom,
! [A: $tType,Xs2: list @ A] :
( ( Xs2
!= ( nil @ A ) )
= ( ? [Y4: A,Ys3: list @ A] :
( Xs2
= ( cons @ A @ Y4 @ Ys3 ) ) ) ) ).
% neq_Nil_conv
thf(fact_53_list_Oinducts,axiom,
! [A: $tType,P: ( list @ A ) > $o,List: list @ A] :
( ( P @ ( nil @ A ) )
=> ( ! [X12: A,X23: list @ A] :
( ( P @ X23 )
=> ( P @ ( cons @ A @ X12 @ X23 ) ) )
=> ( P @ List ) ) ) ).
% list.inducts
thf(fact_54_list_Oexhaust,axiom,
! [A: $tType,Y: list @ A] :
( ( Y
!= ( nil @ A ) )
=> ~ ! [X213: A,X223: list @ A] :
( Y
!= ( cons @ A @ X213 @ X223 ) ) ) ).
% list.exhaust
thf(fact_55_list_OdiscI,axiom,
! [A: $tType,List: list @ A,X21: A,X22: list @ A] :
( ( List
= ( cons @ A @ X21 @ X22 ) )
=> ( List
!= ( nil @ A ) ) ) ).
% list.discI
thf(fact_56_list_Odistinct_I1_J,axiom,
! [A: $tType,X21: A,X22: list @ A] :
( ( nil @ A )
!= ( cons @ A @ X21 @ X22 ) ) ).
% list.distinct(1)
thf(fact_57_execs_Oinducts,axiom,
! [X1: state,X2: list @ com,X32: state,P: state > ( list @ com ) > state > $o] :
( ( pHoare1053570959_execs @ X1 @ X2 @ X32 )
=> ( ! [S3: state] : ( P @ S3 @ ( nil @ com ) @ S3 )
=> ( ! [S3: state,C3: com,T4: state,Cs3: list @ com,U2: state] :
( ( exec @ S3 @ C3 @ T4 )
=> ( ( pHoare1053570959_execs @ T4 @ Cs3 @ U2 )
=> ( ( P @ T4 @ Cs3 @ U2 )
=> ( P @ S3 @ ( cons @ com @ C3 @ Cs3 ) @ U2 ) ) ) )
=> ( P @ X1 @ X2 @ X32 ) ) ) ) ).
% execs.inducts
thf(fact_58_execs_Osimps,axiom,
( pHoare1053570959_execs
= ( ^ [A12: state,A23: list @ com,A32: state] :
( ? [S4: state] :
( ( A12 = S4 )
& ( A23
= ( nil @ com ) )
& ( A32 = S4 ) )
| ? [S4: state,C4: com,T2: state,Cs4: list @ com,U3: state] :
( ( A12 = S4 )
& ( A23
= ( cons @ com @ C4 @ Cs4 ) )
& ( A32 = U3 )
& ( exec @ S4 @ C4 @ T2 )
& ( pHoare1053570959_execs @ T2 @ Cs4 @ U3 ) ) ) ) ) ).
% execs.simps
thf(fact_59_execs_Ocases,axiom,
! [A1: state,A22: list @ com,A33: state] :
( ( pHoare1053570959_execs @ A1 @ A22 @ A33 )
=> ( ( ( A22
= ( nil @ com ) )
=> ( A33 != A1 ) )
=> ~ ! [C3: com,T4: state,Cs3: list @ com] :
( ( A22
= ( cons @ com @ C3 @ Cs3 ) )
=> ( ( exec @ A1 @ C3 @ T4 )
=> ~ ( pHoare1053570959_execs @ T4 @ Cs3 @ A33 ) ) ) ) ) ).
% execs.cases
thf(fact_60_converse__rtrancl__into__rtrancl,axiom,
! [A: $tType,A3: A,B3: A,R: set @ ( product_prod @ A @ A ),C2: A] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A3 @ B3 ) @ R )
=> ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ B3 @ C2 ) @ ( transitive_rtrancl @ A @ R ) )
=> ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A3 @ C2 ) @ ( transitive_rtrancl @ A @ R ) ) ) ) ).
% converse_rtrancl_into_rtrancl
thf(fact_61_rtrancl_Ortrancl__into__rtrancl,axiom,
! [A: $tType,A3: A,B3: A,R: set @ ( product_prod @ A @ A ),C2: A] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A3 @ B3 ) @ ( transitive_rtrancl @ A @ R ) )
=> ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ B3 @ C2 ) @ R )
=> ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A3 @ C2 ) @ ( transitive_rtrancl @ A @ R ) ) ) ) ).
% rtrancl.rtrancl_into_rtrancl
thf(fact_62_converse__rtrancl__induct,axiom,
! [A: $tType,A3: A,B3: A,R: set @ ( product_prod @ A @ A ),P: A > $o] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A3 @ B3 ) @ ( transitive_rtrancl @ A @ R ) )
=> ( ( P @ B3 )
=> ( ! [Y3: A,Z: A] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Y3 @ Z ) @ R )
=> ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Z @ B3 ) @ ( transitive_rtrancl @ A @ R ) )
=> ( ( P @ Z )
=> ( P @ Y3 ) ) ) )
=> ( P @ A3 ) ) ) ) ).
% converse_rtrancl_induct
thf(fact_63_rtrancl_Ortrancl__refl,axiom,
! [A: $tType,A3: A,R: set @ ( product_prod @ A @ A )] : ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A3 @ A3 ) @ ( transitive_rtrancl @ A @ R ) ) ).
% rtrancl.rtrancl_refl
thf(fact_64_converse__rtranclE,axiom,
! [A: $tType,X: A,Z2: A,R: set @ ( product_prod @ A @ A )] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X @ Z2 ) @ ( transitive_rtrancl @ A @ R ) )
=> ( ( X != Z2 )
=> ~ ! [Y3: A] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X @ Y3 ) @ R )
=> ~ ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Y3 @ Z2 ) @ ( transitive_rtrancl @ A @ R ) ) ) ) ) ).
% converse_rtranclE
thf(fact_65_rtrancl_Oinducts,axiom,
! [A: $tType,X1: A,X2: A,R: set @ ( product_prod @ A @ A ),P: A > A > $o] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X1 @ X2 ) @ ( transitive_rtrancl @ A @ R ) )
=> ( ! [A2: A] : ( P @ A2 @ A2 )
=> ( ! [A2: A,B2: A,C3: A] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A2 @ B2 ) @ ( transitive_rtrancl @ A @ R ) )
=> ( ( P @ A2 @ B2 )
=> ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ B2 @ C3 ) @ R )
=> ( P @ A2 @ C3 ) ) ) )
=> ( P @ X1 @ X2 ) ) ) ) ).
% rtrancl.inducts
thf(fact_66_rtrancl__induct,axiom,
! [A: $tType,A3: A,B3: A,R: set @ ( product_prod @ A @ A ),P: A > $o] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A3 @ B3 ) @ ( transitive_rtrancl @ A @ R ) )
=> ( ( P @ A3 )
=> ( ! [Y3: A,Z: A] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A3 @ Y3 ) @ ( transitive_rtrancl @ A @ R ) )
=> ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Y3 @ Z ) @ R )
=> ( ( P @ Y3 )
=> ( P @ Z ) ) ) )
=> ( P @ B3 ) ) ) ) ).
% rtrancl_induct
thf(fact_67_rtrancl__trans,axiom,
! [A: $tType,X: A,Y: A,R: set @ ( product_prod @ A @ A ),Z2: A] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X @ Y ) @ ( transitive_rtrancl @ A @ R ) )
=> ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Y @ Z2 ) @ ( transitive_rtrancl @ A @ R ) )
=> ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X @ Z2 ) @ ( transitive_rtrancl @ A @ R ) ) ) ) ).
% rtrancl_trans
thf(fact_68_rtrancl_Osimps,axiom,
! [A: $tType,A1: A,A22: A,R: set @ ( product_prod @ A @ A )] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A1 @ A22 ) @ ( transitive_rtrancl @ A @ R ) )
= ( ? [A6: A] :
( ( A1 = A6 )
& ( A22 = A6 ) )
| ? [A6: A,B5: A,C4: A] :
( ( A1 = A6 )
& ( A22 = C4 )
& ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A6 @ B5 ) @ ( transitive_rtrancl @ A @ R ) )
& ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ B5 @ C4 ) @ R ) ) ) ) ).
% rtrancl.simps
thf(fact_69_rtrancl_Ocases,axiom,
! [A: $tType,A1: A,A22: A,R: set @ ( product_prod @ A @ A )] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A1 @ A22 ) @ ( transitive_rtrancl @ A @ R ) )
=> ( ( A22 != A1 )
=> ~ ! [B2: A] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A1 @ B2 ) @ ( transitive_rtrancl @ A @ R ) )
=> ~ ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ B2 @ A22 ) @ R ) ) ) ) ).
% rtrancl.cases
thf(fact_70_rtranclE,axiom,
! [A: $tType,A3: A,B3: A,R: set @ ( product_prod @ A @ A )] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A3 @ B3 ) @ ( transitive_rtrancl @ A @ R ) )
=> ( ( A3 != B3 )
=> ~ ! [Y3: A] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A3 @ Y3 ) @ ( transitive_rtrancl @ A @ R ) )
=> ~ ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Y3 @ B3 ) @ R ) ) ) ) ).
% rtranclE
thf(fact_71_prod__induct7,axiom,
! [G: $tType,F2: $tType,E: $tType,D: $tType,C: $tType,B: $tType,A: $tType,P: ( product_prod @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D @ ( product_prod @ E @ ( product_prod @ F2 @ G ) ) ) ) ) ) > $o,X: product_prod @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D @ ( product_prod @ E @ ( product_prod @ F2 @ G ) ) ) ) )] :
( ! [A2: A,B2: B,C3: C,D2: D,E2: E,F: F2,G2: G] : ( P @ ( product_Pair @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D @ ( product_prod @ E @ ( product_prod @ F2 @ G ) ) ) ) ) @ A2 @ ( product_Pair @ B @ ( product_prod @ C @ ( product_prod @ D @ ( product_prod @ E @ ( product_prod @ F2 @ G ) ) ) ) @ B2 @ ( product_Pair @ C @ ( product_prod @ D @ ( product_prod @ E @ ( product_prod @ F2 @ G ) ) ) @ C3 @ ( product_Pair @ D @ ( product_prod @ E @ ( product_prod @ F2 @ G ) ) @ D2 @ ( product_Pair @ E @ ( product_prod @ F2 @ G ) @ E2 @ ( product_Pair @ F2 @ G @ F @ G2 ) ) ) ) ) ) )
=> ( P @ X ) ) ).
% prod_induct7
thf(fact_72_prod__induct6,axiom,
! [F2: $tType,E: $tType,D: $tType,C: $tType,B: $tType,A: $tType,P: ( product_prod @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D @ ( product_prod @ E @ F2 ) ) ) ) ) > $o,X: product_prod @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D @ ( product_prod @ E @ F2 ) ) ) )] :
( ! [A2: A,B2: B,C3: C,D2: D,E2: E,F: F2] : ( P @ ( product_Pair @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D @ ( product_prod @ E @ F2 ) ) ) ) @ A2 @ ( product_Pair @ B @ ( product_prod @ C @ ( product_prod @ D @ ( product_prod @ E @ F2 ) ) ) @ B2 @ ( product_Pair @ C @ ( product_prod @ D @ ( product_prod @ E @ F2 ) ) @ C3 @ ( product_Pair @ D @ ( product_prod @ E @ F2 ) @ D2 @ ( product_Pair @ E @ F2 @ E2 @ F ) ) ) ) ) )
=> ( P @ X ) ) ).
% prod_induct6
thf(fact_73_prod__induct5,axiom,
! [E: $tType,D: $tType,C: $tType,B: $tType,A: $tType,P: ( product_prod @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D @ E ) ) ) ) > $o,X: product_prod @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D @ E ) ) )] :
( ! [A2: A,B2: B,C3: C,D2: D,E2: E] : ( P @ ( product_Pair @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D @ E ) ) ) @ A2 @ ( product_Pair @ B @ ( product_prod @ C @ ( product_prod @ D @ E ) ) @ B2 @ ( product_Pair @ C @ ( product_prod @ D @ E ) @ C3 @ ( product_Pair @ D @ E @ D2 @ E2 ) ) ) ) )
=> ( P @ X ) ) ).
% prod_induct5
thf(fact_74_prod__induct4,axiom,
! [D: $tType,C: $tType,B: $tType,A: $tType,P: ( product_prod @ A @ ( product_prod @ B @ ( product_prod @ C @ D ) ) ) > $o,X: product_prod @ A @ ( product_prod @ B @ ( product_prod @ C @ D ) )] :
( ! [A2: A,B2: B,C3: C,D2: D] : ( P @ ( product_Pair @ A @ ( product_prod @ B @ ( product_prod @ C @ D ) ) @ A2 @ ( product_Pair @ B @ ( product_prod @ C @ D ) @ B2 @ ( product_Pair @ C @ D @ C3 @ D2 ) ) ) )
=> ( P @ X ) ) ).
% prod_induct4
thf(fact_75_prod__induct3,axiom,
! [C: $tType,B: $tType,A: $tType,P: ( product_prod @ A @ ( product_prod @ B @ C ) ) > $o,X: product_prod @ A @ ( product_prod @ B @ C )] :
( ! [A2: A,B2: B,C3: C] : ( P @ ( product_Pair @ A @ ( product_prod @ B @ C ) @ A2 @ ( product_Pair @ B @ C @ B2 @ C3 ) ) )
=> ( P @ X ) ) ).
% prod_induct3
thf(fact_76_prod__cases7,axiom,
! [A: $tType,B: $tType,C: $tType,D: $tType,E: $tType,F2: $tType,G: $tType,Y: product_prod @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D @ ( product_prod @ E @ ( product_prod @ F2 @ G ) ) ) ) )] :
~ ! [A2: A,B2: B,C3: C,D2: D,E2: E,F: F2,G2: G] :
( Y
!= ( product_Pair @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D @ ( product_prod @ E @ ( product_prod @ F2 @ G ) ) ) ) ) @ A2 @ ( product_Pair @ B @ ( product_prod @ C @ ( product_prod @ D @ ( product_prod @ E @ ( product_prod @ F2 @ G ) ) ) ) @ B2 @ ( product_Pair @ C @ ( product_prod @ D @ ( product_prod @ E @ ( product_prod @ F2 @ G ) ) ) @ C3 @ ( product_Pair @ D @ ( product_prod @ E @ ( product_prod @ F2 @ G ) ) @ D2 @ ( product_Pair @ E @ ( product_prod @ F2 @ G ) @ E2 @ ( product_Pair @ F2 @ G @ F @ G2 ) ) ) ) ) ) ) ).
% prod_cases7
thf(fact_77_prod__cases6,axiom,
! [A: $tType,B: $tType,C: $tType,D: $tType,E: $tType,F2: $tType,Y: product_prod @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D @ ( product_prod @ E @ F2 ) ) ) )] :
~ ! [A2: A,B2: B,C3: C,D2: D,E2: E,F: F2] :
( Y
!= ( product_Pair @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D @ ( product_prod @ E @ F2 ) ) ) ) @ A2 @ ( product_Pair @ B @ ( product_prod @ C @ ( product_prod @ D @ ( product_prod @ E @ F2 ) ) ) @ B2 @ ( product_Pair @ C @ ( product_prod @ D @ ( product_prod @ E @ F2 ) ) @ C3 @ ( product_Pair @ D @ ( product_prod @ E @ F2 ) @ D2 @ ( product_Pair @ E @ F2 @ E2 @ F ) ) ) ) ) ) ).
% prod_cases6
thf(fact_78_old_Oprod_Orec,axiom,
! [A: $tType,T: $tType,B: $tType,F1: A > B > T,A3: A,B3: B] :
( ( product_rec_prod @ A @ B @ T @ F1 @ ( product_Pair @ A @ B @ A3 @ B3 ) )
= ( F1 @ A3 @ B3 ) ) ).
% old.prod.rec
thf(fact_79_exec1_OSemi,axiom,
! [C1: com,C22: com,Cs2: list @ com,S: state] : ( member @ ( product_prod @ ( product_prod @ ( list @ com ) @ state ) @ ( product_prod @ ( list @ com ) @ state ) ) @ ( product_Pair @ ( product_prod @ ( list @ com ) @ state ) @ ( product_prod @ ( list @ com ) @ state ) @ ( product_Pair @ ( list @ com ) @ state @ ( cons @ com @ ( semi @ C1 @ C22 ) @ Cs2 ) @ S ) @ ( product_Pair @ ( list @ com ) @ state @ ( cons @ com @ C1 @ ( cons @ com @ C22 @ Cs2 ) ) @ S ) ) @ pHoare1053570893_exec1 ) ).
% exec1.Semi
thf(fact_80_exec1E_I3_J,axiom,
! [C1: com,C22: com,Cs2: list @ com,S: state,Cs: list @ com,S2: state] :
( ( member @ ( product_prod @ ( product_prod @ ( list @ com ) @ state ) @ ( product_prod @ ( list @ com ) @ state ) ) @ ( product_Pair @ ( product_prod @ ( list @ com ) @ state ) @ ( product_prod @ ( list @ com ) @ state ) @ ( product_Pair @ ( list @ com ) @ state @ ( cons @ com @ ( semi @ C1 @ C22 ) @ Cs2 ) @ S ) @ ( product_Pair @ ( list @ com ) @ state @ Cs @ S2 ) ) @ pHoare1053570893_exec1 )
=> ~ ( ( Cs
= ( cons @ com @ C1 @ ( cons @ com @ C22 @ Cs2 ) ) )
=> ( S2 != S ) ) ) ).
% exec1E(3)
thf(fact_81_exec1_ODo,axiom,
! [T3: state,F3: state > ( set @ state ),S: state,Cs2: list @ com] :
( ( member @ state @ T3 @ ( F3 @ S ) )
=> ( member @ ( product_prod @ ( product_prod @ ( list @ com ) @ state ) @ ( product_prod @ ( list @ com ) @ state ) ) @ ( product_Pair @ ( product_prod @ ( list @ com ) @ state ) @ ( product_prod @ ( list @ com ) @ state ) @ ( product_Pair @ ( list @ com ) @ state @ ( cons @ com @ ( do @ F3 ) @ Cs2 ) @ S ) @ ( product_Pair @ ( list @ com ) @ state @ Cs2 @ T3 ) ) @ pHoare1053570893_exec1 ) ) ).
% exec1.Do
thf(fact_82_exec1E_I2_J,axiom,
! [F3: state > ( set @ state ),Cs2: list @ com,S: state,Cs: list @ com,S2: state] :
( ( member @ ( product_prod @ ( product_prod @ ( list @ com ) @ state ) @ ( product_prod @ ( list @ com ) @ state ) ) @ ( product_Pair @ ( product_prod @ ( list @ com ) @ state ) @ ( product_prod @ ( list @ com ) @ state ) @ ( product_Pair @ ( list @ com ) @ state @ ( cons @ com @ ( do @ F3 ) @ Cs2 ) @ S ) @ ( product_Pair @ ( list @ com ) @ state @ Cs @ S2 ) ) @ pHoare1053570893_exec1 )
=> ~ ( ( Cs = Cs2 )
=> ~ ( member @ state @ S2 @ ( F3 @ S ) ) ) ) ).
% exec1E(2)
thf(fact_83_exec1_OIfFalse,axiom,
! [B3: state > $o,S: state,C1: com,C22: com,Cs2: list @ com] :
( ~ ( B3 @ S )
=> ( member @ ( product_prod @ ( product_prod @ ( list @ com ) @ state ) @ ( product_prod @ ( list @ com ) @ state ) ) @ ( product_Pair @ ( product_prod @ ( list @ com ) @ state ) @ ( product_prod @ ( list @ com ) @ state ) @ ( product_Pair @ ( list @ com ) @ state @ ( cons @ com @ ( cond @ B3 @ C1 @ C22 ) @ Cs2 ) @ S ) @ ( product_Pair @ ( list @ com ) @ state @ ( cons @ com @ C22 @ Cs2 ) @ S ) ) @ pHoare1053570893_exec1 ) ) ).
% exec1.IfFalse
thf(fact_84_exec1_OIfTrue,axiom,
! [B3: state > $o,S: state,C1: com,C22: com,Cs2: list @ com] :
( ( B3 @ S )
=> ( member @ ( product_prod @ ( product_prod @ ( list @ com ) @ state ) @ ( product_prod @ ( list @ com ) @ state ) ) @ ( product_Pair @ ( product_prod @ ( list @ com ) @ state ) @ ( product_prod @ ( list @ com ) @ state ) @ ( product_Pair @ ( list @ com ) @ state @ ( cons @ com @ ( cond @ B3 @ C1 @ C22 ) @ Cs2 ) @ S ) @ ( product_Pair @ ( list @ com ) @ state @ ( cons @ com @ C1 @ Cs2 ) @ S ) ) @ pHoare1053570893_exec1 ) ) ).
% exec1.IfTrue
thf(fact_85_exec1E_I4_J,axiom,
! [B3: state > $o,C1: com,C22: com,Cs2: list @ com,S: state,Cs: list @ com,S2: state] :
( ( member @ ( product_prod @ ( product_prod @ ( list @ com ) @ state ) @ ( product_prod @ ( list @ com ) @ state ) ) @ ( product_Pair @ ( product_prod @ ( list @ com ) @ state ) @ ( product_prod @ ( list @ com ) @ state ) @ ( product_Pair @ ( list @ com ) @ state @ ( cons @ com @ ( cond @ B3 @ C1 @ C22 ) @ Cs2 ) @ S ) @ ( product_Pair @ ( list @ com ) @ state @ Cs @ S2 ) ) @ pHoare1053570893_exec1 )
=> ( ( ( Cs
= ( cons @ com @ C1 @ Cs2 ) )
=> ( ( S2 = S )
=> ~ ( B3 @ S ) ) )
=> ~ ( ( Cs
= ( cons @ com @ C22 @ Cs2 ) )
=> ( ( S2 = S )
=> ( B3 @ S ) ) ) ) ) ).
% exec1E(4)
thf(fact_86_exec1_OWhileFalse,axiom,
! [B3: state > $o,S: state,C2: com,Cs2: list @ com] :
( ~ ( B3 @ S )
=> ( member @ ( product_prod @ ( product_prod @ ( list @ com ) @ state ) @ ( product_prod @ ( list @ com ) @ state ) ) @ ( product_Pair @ ( product_prod @ ( list @ com ) @ state ) @ ( product_prod @ ( list @ com ) @ state ) @ ( product_Pair @ ( list @ com ) @ state @ ( cons @ com @ ( while @ B3 @ C2 ) @ Cs2 ) @ S ) @ ( product_Pair @ ( list @ com ) @ state @ Cs2 @ S ) ) @ pHoare1053570893_exec1 ) ) ).
% exec1.WhileFalse
thf(fact_87_exec1_OWhileTrue,axiom,
! [B3: state > $o,S: state,C2: com,Cs2: list @ com] :
( ( B3 @ S )
=> ( member @ ( product_prod @ ( product_prod @ ( list @ com ) @ state ) @ ( product_prod @ ( list @ com ) @ state ) ) @ ( product_Pair @ ( product_prod @ ( list @ com ) @ state ) @ ( product_prod @ ( list @ com ) @ state ) @ ( product_Pair @ ( list @ com ) @ state @ ( cons @ com @ ( while @ B3 @ C2 ) @ Cs2 ) @ S ) @ ( product_Pair @ ( list @ com ) @ state @ ( cons @ com @ C2 @ ( cons @ com @ ( while @ B3 @ C2 ) @ Cs2 ) ) @ S ) ) @ pHoare1053570893_exec1 ) ) ).
% exec1.WhileTrue
thf(fact_88_while__termiE,axiom,
! [B3: state > $o,C2: com,S: state] :
( ( termi @ ( while @ B3 @ C2 ) @ S )
=> ( ( B3 @ S )
=> ( termi @ C2 @ S ) ) ) ).
% while_termiE
thf(fact_89_while__termiE2,axiom,
! [B3: state > $o,C2: com,S: state,T3: state] :
( ( termi @ ( while @ B3 @ C2 ) @ S )
=> ( ( B3 @ S )
=> ( ( exec @ S @ C2 @ T3 )
=> ( termi @ ( while @ B3 @ C2 ) @ T3 ) ) ) ) ).
% while_termiE2
thf(fact_90_exec1E_I5_J,axiom,
! [B3: state > $o,C2: com,Cs2: list @ com,S: state,Cs: list @ com,S2: state] :
( ( member @ ( product_prod @ ( product_prod @ ( list @ com ) @ state ) @ ( product_prod @ ( list @ com ) @ state ) ) @ ( product_Pair @ ( product_prod @ ( list @ com ) @ state ) @ ( product_prod @ ( list @ com ) @ state ) @ ( product_Pair @ ( list @ com ) @ state @ ( cons @ com @ ( while @ B3 @ C2 ) @ Cs2 ) @ S ) @ ( product_Pair @ ( list @ com ) @ state @ Cs @ S2 ) ) @ pHoare1053570893_exec1 )
=> ( ( ( Cs = Cs2 )
=> ( ( S2 = S )
=> ( B3 @ S ) ) )
=> ~ ( ( Cs
= ( cons @ com @ C2 @ ( cons @ com @ ( while @ B3 @ C2 ) @ Cs2 ) ) )
=> ( ( S2 = S )
=> ~ ( B3 @ S ) ) ) ) ) ).
% exec1E(5)
thf(fact_91_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_92_termi_OWhileTrue,axiom,
! [B3: state > $o,S: state,C2: com] :
( ( B3 @ S )
=> ( ( termi @ C2 @ S )
=> ( ! [T4: state] :
( ( exec @ S @ C2 @ T4 )
=> ( termi @ ( while @ B3 @ C2 ) @ T4 ) )
=> ( termi @ ( while @ B3 @ C2 ) @ S ) ) ) ) ).
% termi.WhileTrue
thf(fact_93_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_94_com_Oinject_I1_J,axiom,
! [X1: state > ( set @ state ),Y1: state > ( set @ state )] :
( ( ( do @ X1 )
= ( do @ Y1 ) )
= ( X1 = Y1 ) ) ).
% com.inject(1)
thf(fact_95_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_96_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_97_exec_OWhileFalse,axiom,
! [B3: state > $o,S: state,C2: com] :
( ~ ( B3 @ S )
=> ( exec @ S @ ( while @ B3 @ C2 ) @ S ) ) ).
% exec.WhileFalse
thf(fact_98_exec_OWhileTrue,axiom,
! [B3: state > $o,S: state,C2: com,T3: state,U: state] :
( ( B3 @ S )
=> ( ( exec @ S @ C2 @ T3 )
=> ( ( exec @ T3 @ ( while @ B3 @ C2 ) @ U )
=> ( exec @ S @ ( while @ B3 @ C2 ) @ U ) ) ) ) ).
% exec.WhileTrue
thf(fact_99_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_100_termi_OWhileFalse,axiom,
! [B3: state > $o,S: state,C2: com] :
( ~ ( B3 @ S )
=> ( termi @ ( while @ B3 @ C2 ) @ S ) ) ).
% termi.WhileFalse
thf(fact_101_com_Odistinct_I5_J,axiom,
! [X1: state > ( set @ state ),X41: state > $o,X42: com] :
( ( do @ X1 )
!= ( while @ X41 @ X42 ) ) ).
% com.distinct(5)
thf(fact_102_exec_OIfFalse,axiom,
! [B3: state > $o,S: state,C22: com,T3: state,C1: com] :
( ~ ( B3 @ S )
=> ( ( exec @ S @ C22 @ T3 )
=> ( exec @ S @ ( cond @ B3 @ C1 @ C22 ) @ T3 ) ) ) ).
% exec.IfFalse
thf(fact_103_exec_OIfTrue,axiom,
! [B3: state > $o,S: state,C1: com,T3: state,C22: com] :
( ( B3 @ S )
=> ( ( exec @ S @ C1 @ T3 )
=> ( exec @ S @ ( cond @ B3 @ C1 @ C22 ) @ T3 ) ) ) ).
% exec.IfTrue
thf(fact_104_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_105_exec_ODo,axiom,
! [T3: state,F3: state > ( set @ state ),S: state] :
( ( member @ state @ T3 @ ( F3 @ S ) )
=> ( exec @ S @ ( do @ F3 ) @ T3 ) ) ).
% exec.Do
thf(fact_106_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_107_termi_OIfFalse,axiom,
! [B3: state > $o,S: state,C22: com,C1: com] :
( ~ ( B3 @ S )
=> ( ( termi @ C22 @ S )
=> ( termi @ ( cond @ B3 @ C1 @ C22 ) @ S ) ) ) ).
% termi.IfFalse
thf(fact_108_termi_OIfTrue,axiom,
! [B3: state > $o,S: state,C1: com,C22: com] :
( ( B3 @ S )
=> ( ( termi @ C1 @ S )
=> ( termi @ ( cond @ B3 @ C1 @ C22 ) @ S ) ) ) ).
% termi.IfTrue
thf(fact_109_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_110_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_111_com_Odistinct_I1_J,axiom,
! [X1: state > ( set @ state ),X21: com,X22: com] :
( ( do @ X1 )
!= ( semi @ X21 @ X22 ) ) ).
% com.distinct(1)
thf(fact_112_product__lists_Osimps_I1_J,axiom,
! [A: $tType] :
( ( product_lists @ A @ ( nil @ ( list @ A ) ) )
= ( cons @ ( list @ A ) @ ( nil @ A ) @ ( nil @ ( list @ A ) ) ) ) ).
% product_lists.simps(1)
thf(fact_113_subseqs_Osimps_I1_J,axiom,
! [A: $tType] :
( ( subseqs @ A @ ( nil @ A ) )
= ( cons @ ( list @ A ) @ ( nil @ A ) @ ( nil @ ( list @ A ) ) ) ) ).
% subseqs.simps(1)
thf(fact_114_app__execs,axiom,
! [Cs2: list @ com,S: state,Cs: list @ com,S2: state,Cs22: list @ com] :
( ( member @ ( product_prod @ ( product_prod @ ( list @ com ) @ state ) @ ( product_prod @ ( list @ com ) @ state ) ) @ ( product_Pair @ ( product_prod @ ( list @ com ) @ state ) @ ( product_prod @ ( list @ com ) @ state ) @ ( product_Pair @ ( list @ com ) @ state @ Cs2 @ S ) @ ( product_Pair @ ( list @ com ) @ state @ Cs @ S2 ) ) @ ( transitive_rtrancl @ ( product_prod @ ( list @ com ) @ state ) @ pHoare1053570893_exec1 ) )
=> ( member @ ( product_prod @ ( product_prod @ ( list @ com ) @ state ) @ ( product_prod @ ( list @ com ) @ state ) ) @ ( product_Pair @ ( product_prod @ ( list @ com ) @ state ) @ ( product_prod @ ( list @ com ) @ state ) @ ( product_Pair @ ( list @ com ) @ state @ ( append @ com @ Cs2 @ Cs22 ) @ S ) @ ( product_Pair @ ( list @ com ) @ state @ ( append @ com @ Cs @ Cs22 ) @ S2 ) ) @ ( transitive_rtrancl @ ( product_prod @ ( list @ com ) @ state ) @ pHoare1053570893_exec1 ) ) ) ).
% app_execs
thf(fact_115_insert__Nil,axiom,
! [A: $tType,X: A] :
( ( insert @ A @ X @ ( nil @ A ) )
= ( cons @ A @ X @ ( nil @ A ) ) ) ).
% insert_Nil
thf(fact_116_listrel_Oinducts,axiom,
! [A: $tType,B: $tType,X1: list @ A,X2: list @ B,R: set @ ( product_prod @ A @ B ),P: ( list @ A ) > ( list @ B ) > $o] :
( ( member @ ( product_prod @ ( list @ A ) @ ( list @ B ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ B ) @ X1 @ X2 ) @ ( listrel @ A @ B @ R ) )
=> ( ( P @ ( nil @ A ) @ ( nil @ B ) )
=> ( ! [X3: A,Y3: B,Xs: list @ A,Ys: list @ B] :
( ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X3 @ Y3 ) @ R )
=> ( ( member @ ( product_prod @ ( list @ A ) @ ( list @ B ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ B ) @ Xs @ Ys ) @ ( listrel @ A @ B @ R ) )
=> ( ( P @ Xs @ Ys )
=> ( P @ ( cons @ A @ X3 @ Xs ) @ ( cons @ B @ Y3 @ Ys ) ) ) ) )
=> ( P @ X1 @ X2 ) ) ) ) ).
% listrel.inducts
thf(fact_117_append_Oassoc,axiom,
! [A: $tType,A3: list @ A,B3: list @ A,C2: list @ A] :
( ( append @ A @ ( append @ A @ A3 @ B3 ) @ C2 )
= ( append @ A @ A3 @ ( append @ A @ B3 @ C2 ) ) ) ).
% append.assoc
thf(fact_118_append__assoc,axiom,
! [A: $tType,Xs2: list @ A,Ys2: list @ A,Zs2: list @ A] :
( ( append @ A @ ( append @ A @ Xs2 @ Ys2 ) @ Zs2 )
= ( append @ A @ Xs2 @ ( append @ A @ Ys2 @ Zs2 ) ) ) ).
% append_assoc
thf(fact_119_append__same__eq,axiom,
! [A: $tType,Ys2: list @ A,Xs2: list @ A,Zs2: list @ A] :
( ( ( append @ A @ Ys2 @ Xs2 )
= ( append @ A @ Zs2 @ Xs2 ) )
= ( Ys2 = Zs2 ) ) ).
% append_same_eq
thf(fact_120_same__append__eq,axiom,
! [A: $tType,Xs2: list @ A,Ys2: list @ A,Zs2: list @ A] :
( ( ( append @ A @ Xs2 @ Ys2 )
= ( append @ A @ Xs2 @ Zs2 ) )
= ( Ys2 = Zs2 ) ) ).
% same_append_eq
thf(fact_121_append__Nil2,axiom,
! [A: $tType,Xs2: list @ A] :
( ( append @ A @ Xs2 @ ( nil @ A ) )
= Xs2 ) ).
% append_Nil2
thf(fact_122_append__self__conv,axiom,
! [A: $tType,Xs2: list @ A,Ys2: list @ A] :
( ( ( append @ A @ Xs2 @ Ys2 )
= Xs2 )
= ( Ys2
= ( nil @ A ) ) ) ).
% append_self_conv
thf(fact_123_self__append__conv,axiom,
! [A: $tType,Xs2: list @ A,Ys2: list @ A] :
( ( Xs2
= ( append @ A @ Xs2 @ Ys2 ) )
= ( Ys2
= ( nil @ A ) ) ) ).
% self_append_conv
thf(fact_124_append__self__conv2,axiom,
! [A: $tType,Xs2: list @ A,Ys2: list @ A] :
( ( ( append @ A @ Xs2 @ Ys2 )
= Ys2 )
= ( Xs2
= ( nil @ A ) ) ) ).
% append_self_conv2
thf(fact_125_self__append__conv2,axiom,
! [A: $tType,Ys2: list @ A,Xs2: list @ A] :
( ( Ys2
= ( append @ A @ Xs2 @ Ys2 ) )
= ( Xs2
= ( nil @ A ) ) ) ).
% self_append_conv2
thf(fact_126_Nil__is__append__conv,axiom,
! [A: $tType,Xs2: list @ A,Ys2: list @ A] :
( ( ( nil @ A )
= ( append @ A @ Xs2 @ Ys2 ) )
= ( ( Xs2
= ( nil @ A ) )
& ( Ys2
= ( nil @ A ) ) ) ) ).
% Nil_is_append_conv
thf(fact_127_append__is__Nil__conv,axiom,
! [A: $tType,Xs2: list @ A,Ys2: list @ A] :
( ( ( append @ A @ Xs2 @ Ys2 )
= ( nil @ A ) )
= ( ( Xs2
= ( nil @ A ) )
& ( Ys2
= ( nil @ A ) ) ) ) ).
% append_is_Nil_conv
thf(fact_128_append_Oright__neutral,axiom,
! [A: $tType,A3: list @ A] :
( ( append @ A @ A3 @ ( nil @ A ) )
= A3 ) ).
% append.right_neutral
thf(fact_129_append1__eq__conv,axiom,
! [A: $tType,Xs2: list @ A,X: A,Ys2: list @ A,Y: A] :
( ( ( append @ A @ Xs2 @ ( cons @ A @ X @ ( nil @ A ) ) )
= ( append @ A @ Ys2 @ ( cons @ A @ Y @ ( nil @ A ) ) ) )
= ( ( Xs2 = Ys2 )
& ( X = Y ) ) ) ).
% append1_eq_conv
thf(fact_130_listrel__rtrancl__refl,axiom,
! [A: $tType,Xs2: list @ A,R: set @ ( product_prod @ A @ A )] : ( member @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ Xs2 @ Xs2 ) @ ( listrel @ A @ A @ ( transitive_rtrancl @ A @ R ) ) ) ).
% listrel_rtrancl_refl
thf(fact_131_append__Cons,axiom,
! [A: $tType,X: A,Xs2: list @ A,Ys2: list @ A] :
( ( append @ A @ ( cons @ A @ X @ Xs2 ) @ Ys2 )
= ( cons @ A @ X @ ( append @ A @ Xs2 @ Ys2 ) ) ) ).
% append_Cons
thf(fact_132_Cons__eq__appendI,axiom,
! [A: $tType,X: A,Xs1: list @ A,Ys2: list @ A,Xs2: list @ A,Zs2: list @ A] :
( ( ( cons @ A @ X @ Xs1 )
= Ys2 )
=> ( ( Xs2
= ( append @ A @ Xs1 @ Zs2 ) )
=> ( ( cons @ A @ X @ Xs2 )
= ( append @ A @ Ys2 @ Zs2 ) ) ) ) ).
% Cons_eq_appendI
thf(fact_133_append_Oleft__neutral,axiom,
! [A: $tType,A3: list @ A] :
( ( append @ A @ ( nil @ A ) @ A3 )
= A3 ) ).
% append.left_neutral
thf(fact_134_append__Nil,axiom,
! [A: $tType,Ys2: list @ A] :
( ( append @ A @ ( nil @ A ) @ Ys2 )
= Ys2 ) ).
% append_Nil
thf(fact_135_eq__Nil__appendI,axiom,
! [A: $tType,Xs2: list @ A,Ys2: list @ A] :
( ( Xs2 = Ys2 )
=> ( Xs2
= ( append @ A @ ( nil @ A ) @ Ys2 ) ) ) ).
% eq_Nil_appendI
thf(fact_136_append__eq__appendI,axiom,
! [A: $tType,Xs2: list @ A,Xs1: list @ A,Zs2: list @ A,Ys2: list @ A,Us: list @ A] :
( ( ( append @ A @ Xs2 @ Xs1 )
= Zs2 )
=> ( ( Ys2
= ( append @ A @ Xs1 @ Us ) )
=> ( ( append @ A @ Xs2 @ Ys2 )
= ( append @ A @ Zs2 @ Us ) ) ) ) ).
% append_eq_appendI
thf(fact_137_append__eq__append__conv2,axiom,
! [A: $tType,Xs2: list @ A,Ys2: list @ A,Zs2: list @ A,Ts: list @ A] :
( ( ( append @ A @ Xs2 @ Ys2 )
= ( append @ A @ Zs2 @ Ts ) )
= ( ? [Us2: list @ A] :
( ( ( Xs2
= ( append @ A @ Zs2 @ Us2 ) )
& ( ( append @ A @ Us2 @ Ys2 )
= Ts ) )
| ( ( ( append @ A @ Xs2 @ Us2 )
= Zs2 )
& ( Ys2
= ( append @ A @ Us2 @ Ts ) ) ) ) ) ) ).
% append_eq_append_conv2
thf(fact_138_rev__induct,axiom,
! [A: $tType,P: ( list @ A ) > $o,Xs2: list @ A] :
( ( P @ ( nil @ A ) )
=> ( ! [X3: A,Xs: list @ A] :
( ( P @ Xs )
=> ( P @ ( append @ A @ Xs @ ( cons @ A @ X3 @ ( nil @ A ) ) ) ) )
=> ( P @ Xs2 ) ) ) ).
% rev_induct
thf(fact_139_rev__exhaust,axiom,
! [A: $tType,Xs2: list @ A] :
( ( Xs2
!= ( nil @ A ) )
=> ~ ! [Ys: list @ A,Y3: A] :
( Xs2
!= ( append @ A @ Ys @ ( cons @ A @ Y3 @ ( nil @ A ) ) ) ) ) ).
% rev_exhaust
thf(fact_140_Cons__eq__append__conv,axiom,
! [A: $tType,X: A,Xs2: list @ A,Ys2: list @ A,Zs2: list @ A] :
( ( ( cons @ A @ X @ Xs2 )
= ( append @ A @ Ys2 @ Zs2 ) )
= ( ( ( Ys2
= ( nil @ A ) )
& ( ( cons @ A @ X @ Xs2 )
= Zs2 ) )
| ? [Ys4: list @ A] :
( ( ( cons @ A @ X @ Ys4 )
= Ys2 )
& ( Xs2
= ( append @ A @ Ys4 @ Zs2 ) ) ) ) ) ).
% Cons_eq_append_conv
thf(fact_141_append__eq__Cons__conv,axiom,
! [A: $tType,Ys2: list @ A,Zs2: list @ A,X: A,Xs2: list @ A] :
( ( ( append @ A @ Ys2 @ Zs2 )
= ( cons @ A @ X @ Xs2 ) )
= ( ( ( Ys2
= ( nil @ A ) )
& ( Zs2
= ( cons @ A @ X @ Xs2 ) ) )
| ? [Ys4: list @ A] :
( ( Ys2
= ( cons @ A @ X @ Ys4 ) )
& ( ( append @ A @ Ys4 @ Zs2 )
= Xs2 ) ) ) ) ).
% append_eq_Cons_conv
thf(fact_142_rev__nonempty__induct,axiom,
! [A: $tType,Xs2: list @ A,P: ( list @ A ) > $o] :
( ( Xs2
!= ( nil @ A ) )
=> ( ! [X3: A] : ( P @ ( cons @ A @ X3 @ ( nil @ A ) ) )
=> ( ! [X3: A,Xs: list @ A] :
( ( Xs
!= ( nil @ A ) )
=> ( ( P @ Xs )
=> ( P @ ( append @ A @ Xs @ ( cons @ A @ X3 @ ( nil @ A ) ) ) ) ) )
=> ( P @ Xs2 ) ) ) ) ).
% rev_nonempty_induct
thf(fact_143_listrel__Nil2,axiom,
! [B: $tType,A: $tType,Xs2: list @ A,R: set @ ( product_prod @ A @ B )] :
( ( member @ ( product_prod @ ( list @ A ) @ ( list @ B ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ B ) @ Xs2 @ ( nil @ B ) ) @ ( listrel @ A @ B @ R ) )
=> ( Xs2
= ( nil @ A ) ) ) ).
% listrel_Nil2
thf(fact_144_listrel__Nil1,axiom,
! [A: $tType,B: $tType,Xs2: list @ B,R: set @ ( product_prod @ A @ B )] :
( ( member @ ( product_prod @ ( list @ A ) @ ( list @ B ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ B ) @ ( nil @ A ) @ Xs2 ) @ ( listrel @ A @ B @ R ) )
=> ( Xs2
= ( nil @ B ) ) ) ).
% listrel_Nil1
thf(fact_145_listrel_ONil,axiom,
! [B: $tType,A: $tType,R: set @ ( product_prod @ A @ B )] : ( member @ ( product_prod @ ( list @ A ) @ ( list @ B ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ B ) @ ( nil @ A ) @ ( nil @ B ) ) @ ( listrel @ A @ B @ R ) ) ).
% listrel.Nil
thf(fact_146_listrel__rtrancl__trans,axiom,
! [A: $tType,Xs2: list @ A,Ys2: list @ A,R: set @ ( product_prod @ A @ A ),Zs2: list @ A] :
( ( member @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ Xs2 @ Ys2 ) @ ( listrel @ A @ A @ ( transitive_rtrancl @ A @ R ) ) )
=> ( ( member @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ Ys2 @ Zs2 ) @ ( listrel @ A @ A @ ( transitive_rtrancl @ A @ R ) ) )
=> ( member @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ Xs2 @ Zs2 ) @ ( listrel @ A @ A @ ( transitive_rtrancl @ A @ R ) ) ) ) ) ).
% listrel_rtrancl_trans
thf(fact_147_listrel_OCons,axiom,
! [B: $tType,A: $tType,X: A,Y: B,R: set @ ( product_prod @ A @ B ),Xs2: list @ A,Ys2: list @ B] :
( ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X @ Y ) @ R )
=> ( ( member @ ( product_prod @ ( list @ A ) @ ( list @ B ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ B ) @ Xs2 @ Ys2 ) @ ( listrel @ A @ B @ R ) )
=> ( member @ ( product_prod @ ( list @ A ) @ ( list @ B ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ B ) @ ( cons @ A @ X @ Xs2 ) @ ( cons @ B @ Y @ Ys2 ) ) @ ( listrel @ A @ B @ R ) ) ) ) ).
% listrel.Cons
thf(fact_148_listrel__Cons1,axiom,
! [B: $tType,A: $tType,Y: A,Ys2: list @ A,Xs2: list @ B,R: set @ ( product_prod @ A @ B )] :
( ( member @ ( product_prod @ ( list @ A ) @ ( list @ B ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ B ) @ ( cons @ A @ Y @ Ys2 ) @ Xs2 ) @ ( listrel @ A @ B @ R ) )
=> ~ ! [Y3: B,Ys: list @ B] :
( ( Xs2
= ( cons @ B @ Y3 @ Ys ) )
=> ( ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ Y @ Y3 ) @ R )
=> ~ ( member @ ( product_prod @ ( list @ A ) @ ( list @ B ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ B ) @ Ys2 @ Ys ) @ ( listrel @ A @ B @ R ) ) ) ) ) ).
% listrel_Cons1
thf(fact_149_listrel__Cons2,axiom,
! [B: $tType,A: $tType,Xs2: list @ A,Y: B,Ys2: list @ B,R: set @ ( product_prod @ A @ B )] :
( ( member @ ( product_prod @ ( list @ A ) @ ( list @ B ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ B ) @ Xs2 @ ( cons @ B @ Y @ Ys2 ) ) @ ( listrel @ A @ B @ R ) )
=> ~ ! [X3: A,Xs: list @ A] :
( ( Xs2
= ( cons @ A @ X3 @ Xs ) )
=> ( ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X3 @ Y ) @ R )
=> ~ ( member @ ( product_prod @ ( list @ A ) @ ( list @ B ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ B ) @ Xs @ Ys2 ) @ ( listrel @ A @ B @ R ) ) ) ) ) ).
% listrel_Cons2
thf(fact_150_app__exec,axiom,
! [Cs2: list @ com,S: state,Cs: list @ com,S2: state,Cs22: list @ com] :
( ( member @ ( product_prod @ ( product_prod @ ( list @ com ) @ state ) @ ( product_prod @ ( list @ com ) @ state ) ) @ ( product_Pair @ ( product_prod @ ( list @ com ) @ state ) @ ( product_prod @ ( list @ com ) @ state ) @ ( product_Pair @ ( list @ com ) @ state @ Cs2 @ S ) @ ( product_Pair @ ( list @ com ) @ state @ Cs @ S2 ) ) @ pHoare1053570893_exec1 )
=> ( member @ ( product_prod @ ( product_prod @ ( list @ com ) @ state ) @ ( product_prod @ ( list @ com ) @ state ) ) @ ( product_Pair @ ( product_prod @ ( list @ com ) @ state ) @ ( product_prod @ ( list @ com ) @ state ) @ ( product_Pair @ ( list @ com ) @ state @ ( append @ com @ Cs2 @ Cs22 ) @ S ) @ ( product_Pair @ ( list @ com ) @ state @ ( append @ com @ Cs @ Cs22 ) @ S2 ) ) @ pHoare1053570893_exec1 ) ) ).
% app_exec
thf(fact_151_listrel_Ocases,axiom,
! [B: $tType,A: $tType,A1: list @ A,A22: list @ B,R: set @ ( product_prod @ A @ B )] :
( ( member @ ( product_prod @ ( list @ A ) @ ( list @ B ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ B ) @ A1 @ A22 ) @ ( listrel @ A @ B @ R ) )
=> ( ( ( A1
= ( nil @ A ) )
=> ( A22
!= ( nil @ B ) ) )
=> ~ ! [X3: A,Y3: B,Xs: list @ A] :
( ( A1
= ( cons @ A @ X3 @ Xs ) )
=> ! [Ys: list @ B] :
( ( A22
= ( cons @ B @ Y3 @ Ys ) )
=> ( ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X3 @ Y3 ) @ R )
=> ~ ( member @ ( product_prod @ ( list @ A ) @ ( list @ B ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ B ) @ Xs @ Ys ) @ ( listrel @ A @ B @ R ) ) ) ) ) ) ) ).
% listrel.cases
thf(fact_152_listrel_Osimps,axiom,
! [B: $tType,A: $tType,A1: list @ A,A22: list @ B,R: set @ ( product_prod @ A @ B )] :
( ( member @ ( product_prod @ ( list @ A ) @ ( list @ B ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ B ) @ A1 @ A22 ) @ ( listrel @ A @ B @ R ) )
= ( ( ( A1
= ( nil @ A ) )
& ( A22
= ( nil @ B ) ) )
| ? [X4: A,Y4: B,Xs3: list @ A,Ys3: list @ B] :
( ( A1
= ( cons @ A @ X4 @ Xs3 ) )
& ( A22
= ( cons @ B @ Y4 @ Ys3 ) )
& ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X4 @ Y4 ) @ R )
& ( member @ ( product_prod @ ( list @ A ) @ ( list @ B ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ B ) @ Xs3 @ Ys3 ) @ ( listrel @ A @ B @ R ) ) ) ) ) ).
% listrel.simps
thf(fact_153_bind__simps_I2_J,axiom,
! [A: $tType,B: $tType,X: B,Xs2: list @ B,F3: B > ( list @ A )] :
( ( bind @ B @ A @ ( cons @ B @ X @ Xs2 ) @ F3 )
= ( append @ A @ ( F3 @ X ) @ ( bind @ B @ A @ Xs2 @ F3 ) ) ) ).
% bind_simps(2)
thf(fact_154_snoc__listrel1__snoc__iff,axiom,
! [A: $tType,Xs2: list @ A,X: A,Ys2: list @ A,Y: A,R: set @ ( product_prod @ A @ A )] :
( ( member @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ ( append @ A @ Xs2 @ ( cons @ A @ X @ ( nil @ A ) ) ) @ ( append @ A @ Ys2 @ ( cons @ A @ Y @ ( nil @ A ) ) ) ) @ ( listrel1 @ A @ R ) )
= ( ( ( member @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ Xs2 @ Ys2 ) @ ( listrel1 @ A @ R ) )
& ( X = Y ) )
| ( ( Xs2 = Ys2 )
& ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X @ Y ) @ R ) ) ) ) ).
% snoc_listrel1_snoc_iff
thf(fact_155_maps__simps_I1_J,axiom,
! [A: $tType,B: $tType,F3: B > ( list @ A ),X: B,Xs2: list @ B] :
( ( maps @ B @ A @ F3 @ ( cons @ B @ X @ Xs2 ) )
= ( append @ A @ ( F3 @ X ) @ ( maps @ B @ A @ F3 @ Xs2 ) ) ) ).
% maps_simps(1)
thf(fact_156_concat__eq__append__conv,axiom,
! [A: $tType,Xss2: list @ ( list @ A ),Ys2: list @ A,Zs2: list @ A] :
( ( ( concat @ A @ Xss2 )
= ( append @ A @ Ys2 @ Zs2 ) )
= ( ( ( Xss2
= ( nil @ ( list @ A ) ) )
=> ( ( Ys2
= ( nil @ A ) )
& ( Zs2
= ( nil @ A ) ) ) )
& ( ( Xss2
!= ( nil @ ( list @ A ) ) )
=> ? [Xss1: list @ ( list @ A ),Xs3: list @ A,Xs4: list @ A,Xss22: list @ ( list @ A )] :
( ( Xss2
= ( append @ ( list @ A ) @ Xss1 @ ( cons @ ( list @ A ) @ ( append @ A @ Xs3 @ Xs4 ) @ Xss22 ) ) )
& ( Ys2
= ( append @ A @ ( concat @ A @ Xss1 ) @ Xs3 ) )
& ( Zs2
= ( append @ A @ Xs4 @ ( concat @ A @ Xss22 ) ) ) ) ) ) ) ).
% concat_eq_append_conv
thf(fact_157_bind__simps_I1_J,axiom,
! [B: $tType,A: $tType,F3: B > ( list @ A )] :
( ( bind @ B @ A @ ( nil @ B ) @ F3 )
= ( nil @ A ) ) ).
% bind_simps(1)
thf(fact_158_concat__append,axiom,
! [A: $tType,Xs2: list @ ( list @ A ),Ys2: list @ ( list @ A )] :
( ( concat @ A @ ( append @ ( list @ A ) @ Xs2 @ Ys2 ) )
= ( append @ A @ ( concat @ A @ Xs2 ) @ ( concat @ A @ Ys2 ) ) ) ).
% concat_append
thf(fact_159_Cons__listrel1__Cons,axiom,
! [A: $tType,X: A,Xs2: list @ A,Y: A,Ys2: list @ A,R: set @ ( product_prod @ A @ A )] :
( ( member @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ ( cons @ A @ X @ Xs2 ) @ ( cons @ A @ Y @ Ys2 ) ) @ ( listrel1 @ A @ R ) )
= ( ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X @ Y ) @ R )
& ( Xs2 = Ys2 ) )
| ( ( X = Y )
& ( member @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ Xs2 @ Ys2 ) @ ( listrel1 @ A @ R ) ) ) ) ) ).
% Cons_listrel1_Cons
thf(fact_160_listrel1I2,axiom,
! [A: $tType,Xs2: list @ A,Ys2: list @ A,R: set @ ( product_prod @ A @ A ),X: A] :
( ( member @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ Xs2 @ Ys2 ) @ ( listrel1 @ A @ R ) )
=> ( member @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ ( cons @ A @ X @ Xs2 ) @ ( cons @ A @ X @ Ys2 ) ) @ ( listrel1 @ A @ R ) ) ) ).
% listrel1I2
thf(fact_161_rtrancl__listrel1__ConsI1,axiom,
! [A: $tType,Xs2: list @ A,Ys2: list @ A,R: set @ ( product_prod @ A @ A ),X: A] :
( ( member @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ Xs2 @ Ys2 ) @ ( transitive_rtrancl @ ( list @ A ) @ ( listrel1 @ A @ R ) ) )
=> ( member @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ ( cons @ A @ X @ Xs2 ) @ ( cons @ A @ X @ Ys2 ) ) @ ( transitive_rtrancl @ ( list @ A ) @ ( listrel1 @ A @ R ) ) ) ) ).
% rtrancl_listrel1_ConsI1
thf(fact_162_not__Nil__listrel1,axiom,
! [A: $tType,Xs2: list @ A,R: set @ ( product_prod @ A @ A )] :
~ ( member @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ ( nil @ A ) @ Xs2 ) @ ( listrel1 @ A @ R ) ) ).
% not_Nil_listrel1
thf(fact_163_not__listrel1__Nil,axiom,
! [A: $tType,Xs2: list @ A,R: set @ ( product_prod @ A @ A )] :
~ ( member @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ Xs2 @ ( nil @ A ) ) @ ( listrel1 @ A @ R ) ) ).
% not_listrel1_Nil
thf(fact_164_append__listrel1I,axiom,
! [A: $tType,Xs2: list @ A,Ys2: list @ A,R: set @ ( product_prod @ A @ A ),Us: list @ A,Vs: list @ A] :
( ( ( ( member @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ Xs2 @ Ys2 ) @ ( listrel1 @ A @ R ) )
& ( Us = Vs ) )
| ( ( Xs2 = Ys2 )
& ( member @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ Us @ Vs ) @ ( listrel1 @ A @ R ) ) ) )
=> ( member @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ ( append @ A @ Xs2 @ Us ) @ ( append @ A @ Ys2 @ Vs ) ) @ ( listrel1 @ A @ R ) ) ) ).
% append_listrel1I
thf(fact_165_concat_Osimps_I1_J,axiom,
! [A: $tType] :
( ( concat @ A @ ( nil @ ( list @ A ) ) )
= ( nil @ A ) ) ).
% concat.simps(1)
thf(fact_166_concat_Osimps_I2_J,axiom,
! [A: $tType,X: list @ A,Xs2: list @ ( list @ A )] :
( ( concat @ A @ ( cons @ ( list @ A ) @ X @ Xs2 ) )
= ( append @ A @ X @ ( concat @ A @ Xs2 ) ) ) ).
% concat.simps(2)
thf(fact_167_listrel__rtrancl__eq__rtrancl__listrel1,axiom,
! [A: $tType,R: set @ ( product_prod @ A @ A )] :
( ( listrel @ A @ A @ ( transitive_rtrancl @ A @ R ) )
= ( transitive_rtrancl @ ( list @ A ) @ ( listrel1 @ A @ R ) ) ) ).
% listrel_rtrancl_eq_rtrancl_listrel1
thf(fact_168_rtrancl__listrel1__if__listrel,axiom,
! [A: $tType,Xs2: list @ A,Ys2: list @ A,R: set @ ( product_prod @ A @ A )] :
( ( member @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ Xs2 @ Ys2 ) @ ( listrel @ A @ A @ R ) )
=> ( member @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ Xs2 @ Ys2 ) @ ( transitive_rtrancl @ ( list @ A ) @ ( listrel1 @ A @ R ) ) ) ) ).
% rtrancl_listrel1_if_listrel
thf(fact_169_maps__simps_I2_J,axiom,
! [B: $tType,A: $tType,F3: B > ( list @ A )] :
( ( maps @ B @ A @ F3 @ ( nil @ B ) )
= ( nil @ A ) ) ).
% maps_simps(2)
thf(fact_170_Cons__listrel1E2,axiom,
! [A: $tType,Xs2: list @ A,Y: A,Ys2: list @ A,R: set @ ( product_prod @ A @ A )] :
( ( member @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ Xs2 @ ( cons @ A @ Y @ Ys2 ) ) @ ( listrel1 @ A @ R ) )
=> ( ! [X3: A] :
( ( Xs2
= ( cons @ A @ X3 @ Ys2 ) )
=> ~ ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X3 @ Y ) @ R ) )
=> ~ ! [Zs: list @ A] :
( ( Xs2
= ( cons @ A @ Y @ Zs ) )
=> ~ ( member @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ Zs @ Ys2 ) @ ( listrel1 @ A @ R ) ) ) ) ) ).
% Cons_listrel1E2
thf(fact_171_Cons__listrel1E1,axiom,
! [A: $tType,X: A,Xs2: list @ A,Ys2: list @ A,R: set @ ( product_prod @ A @ A )] :
( ( member @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ ( cons @ A @ X @ Xs2 ) @ Ys2 ) @ ( listrel1 @ A @ R ) )
=> ( ! [Y3: A] :
( ( Ys2
= ( cons @ A @ Y3 @ Xs2 ) )
=> ~ ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X @ Y3 ) @ R ) )
=> ~ ! [Zs: list @ A] :
( ( Ys2
= ( cons @ A @ X @ Zs ) )
=> ~ ( member @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ Xs2 @ Zs ) @ ( listrel1 @ A @ R ) ) ) ) ) ).
% Cons_listrel1E1
thf(fact_172_listrel1I1,axiom,
! [A: $tType,X: A,Y: A,R: set @ ( product_prod @ A @ A ),Xs2: list @ A] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X @ Y ) @ R )
=> ( member @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ ( cons @ A @ X @ Xs2 ) @ ( cons @ A @ Y @ Xs2 ) ) @ ( listrel1 @ A @ R ) ) ) ).
% listrel1I1
thf(fact_173_listrel__reflcl__if__listrel1,axiom,
! [A: $tType,Xs2: list @ A,Ys2: list @ A,R: set @ ( product_prod @ A @ A )] :
( ( member @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ Xs2 @ Ys2 ) @ ( listrel1 @ A @ R ) )
=> ( member @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ Xs2 @ Ys2 ) @ ( listrel @ A @ A @ ( transitive_rtrancl @ A @ R ) ) ) ) ).
% listrel_reflcl_if_listrel1
thf(fact_174_concat__eq__appendD,axiom,
! [A: $tType,Xss2: list @ ( list @ A ),Ys2: list @ A,Zs2: list @ A] :
( ( ( concat @ A @ Xss2 )
= ( append @ A @ Ys2 @ Zs2 ) )
=> ( ( Xss2
!= ( nil @ ( list @ A ) ) )
=> ? [Xss12: list @ ( list @ A ),Xs: list @ A,Xs5: list @ A,Xss23: list @ ( list @ A )] :
( ( Xss2
= ( append @ ( list @ A ) @ Xss12 @ ( cons @ ( list @ A ) @ ( append @ A @ Xs @ Xs5 ) @ Xss23 ) ) )
& ( Ys2
= ( append @ A @ ( concat @ A @ Xss12 ) @ Xs ) )
& ( Zs2
= ( append @ A @ Xs5 @ ( concat @ A @ Xss23 ) ) ) ) ) ) ).
% concat_eq_appendD
thf(fact_175_listrel1I,axiom,
! [A: $tType,X: A,Y: A,R: set @ ( product_prod @ A @ A ),Xs2: list @ A,Us: list @ A,Vs: list @ A,Ys2: list @ A] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X @ Y ) @ R )
=> ( ( Xs2
= ( append @ A @ Us @ ( cons @ A @ X @ Vs ) ) )
=> ( ( Ys2
= ( append @ A @ Us @ ( cons @ A @ Y @ Vs ) ) )
=> ( member @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ Xs2 @ Ys2 ) @ ( listrel1 @ A @ R ) ) ) ) ) ).
% listrel1I
thf(fact_176_listrel1E,axiom,
! [A: $tType,Xs2: list @ A,Ys2: list @ A,R: set @ ( product_prod @ A @ A )] :
( ( member @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ Xs2 @ Ys2 ) @ ( listrel1 @ A @ R ) )
=> ~ ! [X3: A,Y3: A] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X3 @ Y3 ) @ R )
=> ! [Us3: list @ A,Vs2: list @ A] :
( ( Xs2
= ( append @ A @ Us3 @ ( cons @ A @ X3 @ Vs2 ) ) )
=> ( Ys2
!= ( append @ A @ Us3 @ ( cons @ A @ Y3 @ Vs2 ) ) ) ) ) ) ).
% listrel1E
thf(fact_177_rtrancl__listrel1__ConsI2,axiom,
! [A: $tType,X: A,Y: A,R: set @ ( product_prod @ A @ A ),Xs2: list @ A,Ys2: list @ A] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X @ Y ) @ ( transitive_rtrancl @ A @ R ) )
=> ( ( member @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ Xs2 @ Ys2 ) @ ( transitive_rtrancl @ ( list @ A ) @ ( listrel1 @ A @ R ) ) )
=> ( member @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ ( cons @ A @ X @ Xs2 ) @ ( cons @ A @ Y @ Ys2 ) ) @ ( transitive_rtrancl @ ( list @ A ) @ ( listrel1 @ A @ R ) ) ) ) ) ).
% rtrancl_listrel1_ConsI2
thf(fact_178_lexord__append__left__rightI,axiom,
! [A: $tType,A3: A,B3: A,R: set @ ( product_prod @ A @ A ),U: list @ A,X: list @ A,Y: list @ A] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A3 @ B3 ) @ R )
=> ( member @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ ( append @ A @ U @ ( cons @ A @ A3 @ X ) ) @ ( append @ A @ U @ ( cons @ A @ B3 @ Y ) ) ) @ ( lexord @ A @ R ) ) ) ).
% lexord_append_left_rightI
thf(fact_179_rotate1_Osimps_I2_J,axiom,
! [A: $tType,X: A,Xs2: list @ A] :
( ( rotate1 @ A @ ( cons @ A @ X @ Xs2 ) )
= ( append @ A @ Xs2 @ ( cons @ A @ X @ ( nil @ A ) ) ) ) ).
% rotate1.simps(2)
thf(fact_180_butlast__snoc,axiom,
! [A: $tType,Xs2: list @ A,X: A] :
( ( butlast @ A @ ( append @ A @ Xs2 @ ( cons @ A @ X @ ( nil @ A ) ) ) )
= Xs2 ) ).
% butlast_snoc
thf(fact_181_rotate1__is__Nil__conv,axiom,
! [A: $tType,Xs2: list @ A] :
( ( ( rotate1 @ A @ Xs2 )
= ( nil @ A ) )
= ( Xs2
= ( nil @ A ) ) ) ).
% rotate1_is_Nil_conv
thf(fact_182_lexord__cons__cons,axiom,
! [A: $tType,A3: A,X: list @ A,B3: A,Y: list @ A,R: set @ ( product_prod @ A @ A )] :
( ( member @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ ( cons @ A @ A3 @ X ) @ ( cons @ A @ B3 @ Y ) ) @ ( lexord @ A @ R ) )
= ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A3 @ B3 ) @ R )
| ( ( A3 = B3 )
& ( member @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ X @ Y ) @ ( lexord @ A @ R ) ) ) ) ) ).
% lexord_cons_cons
thf(fact_183_lexord__Nil__left,axiom,
! [A: $tType,Y: list @ A,R: set @ ( product_prod @ A @ A )] :
( ( member @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ ( nil @ A ) @ Y ) @ ( lexord @ A @ R ) )
= ( ? [A6: A,X4: list @ A] :
( Y
= ( cons @ A @ A6 @ X4 ) ) ) ) ).
% lexord_Nil_left
thf(fact_184_butlast_Osimps_I1_J,axiom,
! [A: $tType] :
( ( butlast @ A @ ( nil @ A ) )
= ( nil @ A ) ) ).
% butlast.simps(1)
thf(fact_185_rotate1_Osimps_I1_J,axiom,
! [A: $tType] :
( ( rotate1 @ A @ ( nil @ A ) )
= ( nil @ A ) ) ).
% rotate1.simps(1)
thf(fact_186_lexord__linear,axiom,
! [A: $tType,R: set @ ( product_prod @ A @ A ),X: list @ A,Y: list @ A] :
( ! [A2: A,B2: A] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A2 @ B2 ) @ R )
| ( A2 = B2 )
| ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ B2 @ A2 ) @ R ) )
=> ( ( member @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ X @ Y ) @ ( lexord @ A @ R ) )
| ( X = Y )
| ( member @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ Y @ X ) @ ( lexord @ A @ R ) ) ) ) ).
% lexord_linear
thf(fact_187_lexord__irreflexive,axiom,
! [A: $tType,R: set @ ( product_prod @ A @ A ),Xs2: list @ A] :
( ! [X3: A] :
~ ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X3 @ X3 ) @ R )
=> ~ ( member @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ Xs2 @ Xs2 ) @ ( lexord @ A @ R ) ) ) ).
% lexord_irreflexive
thf(fact_188_butlast_Osimps_I2_J,axiom,
! [A: $tType,Xs2: list @ A,X: A] :
( ( ( Xs2
= ( nil @ A ) )
=> ( ( butlast @ A @ ( cons @ A @ X @ Xs2 ) )
= ( nil @ A ) ) )
& ( ( Xs2
!= ( nil @ A ) )
=> ( ( butlast @ A @ ( cons @ A @ X @ Xs2 ) )
= ( cons @ A @ X @ ( butlast @ A @ Xs2 ) ) ) ) ) ).
% butlast.simps(2)
thf(fact_189_lexord__Nil__right,axiom,
! [A: $tType,X: list @ A,R: set @ ( product_prod @ A @ A )] :
~ ( member @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ X @ ( nil @ A ) ) @ ( lexord @ A @ R ) ) ).
% lexord_Nil_right
thf(fact_190_butlast__append,axiom,
! [A: $tType,Ys2: list @ A,Xs2: list @ A] :
( ( ( Ys2
= ( nil @ A ) )
=> ( ( butlast @ A @ ( append @ A @ Xs2 @ Ys2 ) )
= ( butlast @ A @ Xs2 ) ) )
& ( ( Ys2
!= ( nil @ A ) )
=> ( ( butlast @ A @ ( append @ A @ Xs2 @ Ys2 ) )
= ( append @ A @ Xs2 @ ( butlast @ A @ Ys2 ) ) ) ) ) ).
% butlast_append
thf(fact_191_lexord__append__leftI,axiom,
! [A: $tType,U: list @ A,V: list @ A,R: set @ ( product_prod @ A @ A ),X: list @ A] :
( ( member @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ U @ V ) @ ( lexord @ A @ R ) )
=> ( member @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ ( append @ A @ X @ U ) @ ( append @ A @ X @ V ) ) @ ( lexord @ A @ R ) ) ) ).
% lexord_append_leftI
thf(fact_192_lexord__append__leftD,axiom,
! [A: $tType,X: list @ A,U: list @ A,V: list @ A,R: set @ ( product_prod @ A @ A )] :
( ( member @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ ( append @ A @ X @ U ) @ ( append @ A @ X @ V ) ) @ ( lexord @ A @ R ) )
=> ( ! [A2: A] :
~ ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A2 @ A2 ) @ R )
=> ( member @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ U @ V ) @ ( lexord @ A @ R ) ) ) ) ).
% lexord_append_leftD
thf(fact_193_lexord__append__rightI,axiom,
! [A: $tType,Y: list @ A,X: list @ A,R: set @ ( product_prod @ A @ A )] :
( ? [B6: A,Z3: list @ A] :
( Y
= ( cons @ A @ B6 @ Z3 ) )
=> ( member @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ X @ ( append @ A @ X @ Y ) ) @ ( lexord @ A @ R ) ) ) ).
% lexord_append_rightI
thf(fact_194_lexord__same__pref__if__irrefl,axiom,
! [A: $tType,R: set @ ( product_prod @ A @ A ),Xs2: list @ A,Ys2: list @ A,Zs2: list @ A] :
( ( irrefl @ A @ R )
=> ( ( member @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ ( append @ A @ Xs2 @ Ys2 ) @ ( append @ A @ Xs2 @ Zs2 ) ) @ ( lexord @ A @ R ) )
= ( member @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ Ys2 @ Zs2 ) @ ( lexord @ A @ R ) ) ) ) ).
% lexord_same_pref_if_irrefl
thf(fact_195_append__butlast__last__id,axiom,
! [A: $tType,Xs2: list @ A] :
( ( Xs2
!= ( nil @ A ) )
=> ( ( append @ A @ ( butlast @ A @ Xs2 ) @ ( cons @ A @ ( last @ A @ Xs2 ) @ ( nil @ A ) ) )
= Xs2 ) ) ).
% append_butlast_last_id
thf(fact_196_snoc__eq__iff__butlast,axiom,
! [A: $tType,Xs2: list @ A,X: A,Ys2: list @ A] :
( ( ( append @ A @ Xs2 @ ( cons @ A @ X @ ( nil @ A ) ) )
= Ys2 )
= ( ( Ys2
!= ( nil @ A ) )
& ( ( butlast @ A @ Ys2 )
= Xs2 )
& ( ( last @ A @ Ys2 )
= X ) ) ) ).
% snoc_eq_iff_butlast
thf(fact_197_last__appendR,axiom,
! [A: $tType,Ys2: list @ A,Xs2: list @ A] :
( ( Ys2
!= ( nil @ A ) )
=> ( ( last @ A @ ( append @ A @ Xs2 @ Ys2 ) )
= ( last @ A @ Ys2 ) ) ) ).
% last_appendR
thf(fact_198_last__appendL,axiom,
! [A: $tType,Ys2: list @ A,Xs2: list @ A] :
( ( Ys2
= ( nil @ A ) )
=> ( ( last @ A @ ( append @ A @ Xs2 @ Ys2 ) )
= ( last @ A @ Xs2 ) ) ) ).
% last_appendL
thf(fact_199_last__snoc,axiom,
! [A: $tType,Xs2: list @ A,X: A] :
( ( last @ A @ ( append @ A @ Xs2 @ ( cons @ A @ X @ ( nil @ A ) ) ) )
= X ) ).
% last_snoc
thf(fact_200_lexord__irrefl,axiom,
! [A: $tType,R2: set @ ( product_prod @ A @ A )] :
( ( irrefl @ A @ R2 )
=> ( irrefl @ ( list @ A ) @ ( lexord @ A @ R2 ) ) ) ).
% lexord_irrefl
thf(fact_201_last__ConsR,axiom,
! [A: $tType,Xs2: list @ A,X: A] :
( ( Xs2
!= ( nil @ A ) )
=> ( ( last @ A @ ( cons @ A @ X @ Xs2 ) )
= ( last @ A @ Xs2 ) ) ) ).
% last_ConsR
thf(fact_202_last__ConsL,axiom,
! [A: $tType,Xs2: list @ A,X: A] :
( ( Xs2
= ( nil @ A ) )
=> ( ( last @ A @ ( cons @ A @ X @ Xs2 ) )
= X ) ) ).
% last_ConsL
thf(fact_203_last_Osimps,axiom,
! [A: $tType,Xs2: list @ A,X: A] :
( ( ( Xs2
= ( nil @ A ) )
=> ( ( last @ A @ ( cons @ A @ X @ Xs2 ) )
= X ) )
& ( ( Xs2
!= ( nil @ A ) )
=> ( ( last @ A @ ( cons @ A @ X @ Xs2 ) )
= ( last @ A @ Xs2 ) ) ) ) ).
% last.simps
thf(fact_204_longest__common__suffix,axiom,
! [A: $tType,Xs2: list @ A,Ys2: list @ A] :
? [Ss: list @ A,Xs5: list @ A,Ys5: list @ A] :
( ( Xs2
= ( append @ A @ Xs5 @ Ss ) )
& ( Ys2
= ( append @ A @ Ys5 @ Ss ) )
& ( ( Xs5
= ( nil @ A ) )
| ( Ys5
= ( nil @ A ) )
| ( ( last @ A @ Xs5 )
!= ( last @ A @ Ys5 ) ) ) ) ).
% longest_common_suffix
thf(fact_205_last__append,axiom,
! [A: $tType,Ys2: list @ A,Xs2: list @ A] :
( ( ( Ys2
= ( nil @ A ) )
=> ( ( last @ A @ ( append @ A @ Xs2 @ Ys2 ) )
= ( last @ A @ Xs2 ) ) )
& ( ( Ys2
!= ( nil @ A ) )
=> ( ( last @ A @ ( append @ A @ Xs2 @ Ys2 ) )
= ( last @ A @ Ys2 ) ) ) ) ).
% last_append
thf(fact_206_irreflI,axiom,
! [A: $tType,R2: set @ ( product_prod @ A @ A )] :
( ! [A2: A] :
~ ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A2 @ A2 ) @ R2 )
=> ( irrefl @ A @ R2 ) ) ).
% irreflI
thf(fact_207_irrefl__def,axiom,
! [A: $tType] :
( ( irrefl @ A )
= ( ^ [R3: set @ ( product_prod @ A @ A )] :
! [A6: A] :
~ ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A6 @ A6 ) @ R3 ) ) ) ).
% irrefl_def
thf(fact_208_lenlex__append2,axiom,
! [A: $tType,R2: set @ ( product_prod @ A @ A ),Us: list @ A,Xs2: list @ A,Ys2: list @ A] :
( ( irrefl @ A @ R2 )
=> ( ( member @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ ( append @ A @ Us @ Xs2 ) @ ( append @ A @ Us @ Ys2 ) ) @ ( lenlex @ A @ R2 ) )
= ( member @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ Xs2 @ Ys2 ) @ ( lenlex @ A @ R2 ) ) ) ) ).
% lenlex_append2
thf(fact_209_irrefl__lex,axiom,
! [A: $tType,R: set @ ( product_prod @ A @ A )] :
( ( irrefl @ A @ R )
=> ( irrefl @ ( list @ A ) @ ( lex @ A @ R ) ) ) ).
% irrefl_lex
thf(fact_210_Nil__lenlex__iff1,axiom,
! [A: $tType,Ns: list @ A,R: set @ ( product_prod @ A @ A )] :
( ( member @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ ( nil @ A ) @ Ns ) @ ( lenlex @ A @ R ) )
= ( Ns
!= ( nil @ A ) ) ) ).
% Nil_lenlex_iff1
thf(fact_211_Nil__notin__lex,axiom,
! [A: $tType,Ys2: list @ A,R: set @ ( product_prod @ A @ A )] :
~ ( member @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ ( nil @ A ) @ Ys2 ) @ ( lex @ A @ R ) ) ).
% Nil_notin_lex
thf(fact_212_Nil2__notin__lex,axiom,
! [A: $tType,Xs2: list @ A,R: set @ ( product_prod @ A @ A )] :
~ ( member @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ Xs2 @ ( nil @ A ) ) @ ( lex @ A @ R ) ) ).
% Nil2_notin_lex
thf(fact_213_lex__append__leftI,axiom,
! [A: $tType,Ys2: list @ A,Zs2: list @ A,R: set @ ( product_prod @ A @ A ),Xs2: list @ A] :
( ( member @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ Ys2 @ Zs2 ) @ ( lex @ A @ R ) )
=> ( member @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ ( append @ A @ Xs2 @ Ys2 ) @ ( append @ A @ Xs2 @ Zs2 ) ) @ ( lex @ A @ R ) ) ) ).
% lex_append_leftI
thf(fact_214_lenlex__irreflexive,axiom,
! [A: $tType,R: set @ ( product_prod @ A @ A ),Xs2: list @ A] :
( ! [X3: A] :
~ ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X3 @ X3 ) @ R )
=> ~ ( member @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ Xs2 @ Xs2 ) @ ( lenlex @ A @ R ) ) ) ).
% lenlex_irreflexive
thf(fact_215_Nil__lenlex__iff2,axiom,
! [A: $tType,Ns: list @ A,R: set @ ( product_prod @ A @ A )] :
~ ( member @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ Ns @ ( nil @ A ) ) @ ( lenlex @ A @ R ) ) ).
% Nil_lenlex_iff2
thf(fact_216_lexl__not__refl,axiom,
! [A: $tType,R: set @ ( product_prod @ A @ A ),X: list @ A] :
( ( irrefl @ A @ R )
=> ~ ( member @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ X @ X ) @ ( lex @ A @ R ) ) ) ).
% lexl_not_refl
thf(fact_217_lex__append__leftD,axiom,
! [A: $tType,R: set @ ( product_prod @ A @ A ),Xs2: list @ A,Ys2: list @ A,Zs2: list @ A] :
( ! [X3: A] :
~ ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X3 @ X3 ) @ R )
=> ( ( member @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ ( append @ A @ Xs2 @ Ys2 ) @ ( append @ A @ Xs2 @ Zs2 ) ) @ ( lex @ A @ R ) )
=> ( member @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ Ys2 @ Zs2 ) @ ( lex @ A @ R ) ) ) ) ).
% lex_append_leftD
thf(fact_218_lex__append__left__iff,axiom,
! [A: $tType,R: set @ ( product_prod @ A @ A ),Xs2: list @ A,Ys2: list @ A,Zs2: list @ A] :
( ! [X3: A] :
~ ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X3 @ X3 ) @ R )
=> ( ( member @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ ( append @ A @ Xs2 @ Ys2 ) @ ( append @ A @ Xs2 @ Zs2 ) ) @ ( lex @ A @ R ) )
= ( member @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ Ys2 @ Zs2 ) @ ( lex @ A @ R ) ) ) ) ).
% lex_append_left_iff
thf(fact_219_Cons__in__lex,axiom,
! [A: $tType,X: A,Xs2: list @ A,Y: A,Ys2: list @ A,R: set @ ( product_prod @ A @ A )] :
( ( member @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ ( cons @ A @ X @ Xs2 ) @ ( cons @ A @ Y @ Ys2 ) ) @ ( lex @ A @ R ) )
= ( ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X @ Y ) @ R )
& ( ( size_size @ ( list @ A ) @ Xs2 )
= ( size_size @ ( list @ A ) @ Ys2 ) ) )
| ( ( X = Y )
& ( member @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ Xs2 @ Ys2 ) @ ( lex @ A @ R ) ) ) ) ) ).
% Cons_in_lex
thf(fact_220_internal__case__prod__conv,axiom,
! [B: $tType,A: $tType,C: $tType,C2: B > C > A,A3: B,B3: C] :
( ( produc2004651681e_prod @ B @ C @ A @ C2 @ ( product_Pair @ B @ C @ A3 @ B3 ) )
= ( C2 @ A3 @ B3 ) ) ).
% internal_case_prod_conv
thf(fact_221_append__eq__append__conv,axiom,
! [A: $tType,Xs2: list @ A,Ys2: list @ A,Us: list @ A,Vs: list @ A] :
( ( ( ( size_size @ ( list @ A ) @ Xs2 )
= ( size_size @ ( list @ A ) @ Ys2 ) )
| ( ( size_size @ ( list @ A ) @ Us )
= ( size_size @ ( list @ A ) @ Vs ) ) )
=> ( ( ( append @ A @ Xs2 @ Us )
= ( append @ A @ Ys2 @ Vs ) )
= ( ( Xs2 = Ys2 )
& ( Us = Vs ) ) ) ) ).
% append_eq_append_conv
thf(fact_222_length__rotate1,axiom,
! [A: $tType,Xs2: list @ A] :
( ( size_size @ ( list @ A ) @ ( rotate1 @ A @ Xs2 ) )
= ( size_size @ ( list @ A ) @ Xs2 ) ) ).
% length_rotate1
thf(fact_223_listrel1__eq__len,axiom,
! [A: $tType,Xs2: list @ A,Ys2: list @ A,R: set @ ( product_prod @ A @ A )] :
( ( member @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ Xs2 @ Ys2 ) @ ( listrel1 @ A @ R ) )
=> ( ( size_size @ ( list @ A ) @ Xs2 )
= ( size_size @ ( list @ A ) @ Ys2 ) ) ) ).
% listrel1_eq_len
thf(fact_224_listrel__eq__len,axiom,
! [A: $tType,B: $tType,Xs2: list @ A,Ys2: list @ B,R: set @ ( product_prod @ A @ B )] :
( ( member @ ( product_prod @ ( list @ A ) @ ( list @ B ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ B ) @ Xs2 @ Ys2 ) @ ( listrel @ A @ B @ R ) )
=> ( ( size_size @ ( list @ A ) @ Xs2 )
= ( size_size @ ( list @ B ) @ Ys2 ) ) ) ).
% listrel_eq_len
thf(fact_225_Ex__list__of__length,axiom,
! [A: $tType,N: nat] :
? [Xs: list @ A] :
( ( size_size @ ( list @ A ) @ Xs )
= N ) ).
% Ex_list_of_length
thf(fact_226_neq__if__length__neq,axiom,
! [A: $tType,Xs2: list @ A,Ys2: list @ A] :
( ( ( size_size @ ( list @ A ) @ Xs2 )
!= ( size_size @ ( list @ A ) @ Ys2 ) )
=> ( Xs2 != Ys2 ) ) ).
% neq_if_length_neq
thf(fact_227_list__induct3,axiom,
! [B: $tType,A: $tType,C: $tType,Xs2: list @ A,Ys2: list @ B,Zs2: list @ C,P: ( list @ A ) > ( list @ B ) > ( list @ C ) > $o] :
( ( ( size_size @ ( list @ A ) @ Xs2 )
= ( size_size @ ( list @ B ) @ Ys2 ) )
=> ( ( ( size_size @ ( list @ B ) @ Ys2 )
= ( size_size @ ( list @ C ) @ Zs2 ) )
=> ( ( P @ ( nil @ A ) @ ( nil @ B ) @ ( nil @ C ) )
=> ( ! [X3: A,Xs: list @ A,Y3: B,Ys: list @ B,Z: C,Zs: list @ C] :
( ( ( size_size @ ( list @ A ) @ Xs )
= ( size_size @ ( list @ B ) @ Ys ) )
=> ( ( ( size_size @ ( list @ B ) @ Ys )
= ( size_size @ ( list @ C ) @ Zs ) )
=> ( ( P @ Xs @ Ys @ Zs )
=> ( P @ ( cons @ A @ X3 @ Xs ) @ ( cons @ B @ Y3 @ Ys ) @ ( cons @ C @ Z @ Zs ) ) ) ) )
=> ( P @ Xs2 @ Ys2 @ Zs2 ) ) ) ) ) ).
% list_induct3
thf(fact_228_list__induct2,axiom,
! [A: $tType,B: $tType,Xs2: list @ A,Ys2: list @ B,P: ( list @ A ) > ( list @ B ) > $o] :
( ( ( size_size @ ( list @ A ) @ Xs2 )
= ( size_size @ ( list @ B ) @ Ys2 ) )
=> ( ( P @ ( nil @ A ) @ ( nil @ B ) )
=> ( ! [X3: A,Xs: list @ A,Y3: B,Ys: list @ B] :
( ( ( size_size @ ( list @ A ) @ Xs )
= ( size_size @ ( list @ B ) @ Ys ) )
=> ( ( P @ Xs @ Ys )
=> ( P @ ( cons @ A @ X3 @ Xs ) @ ( cons @ B @ Y3 @ Ys ) ) ) )
=> ( P @ Xs2 @ Ys2 ) ) ) ) ).
% list_induct2
thf(fact_229_same__length__different,axiom,
! [A: $tType,Xs2: list @ A,Ys2: list @ A] :
( ( Xs2 != Ys2 )
=> ( ( ( size_size @ ( list @ A ) @ Xs2 )
= ( size_size @ ( list @ A ) @ Ys2 ) )
=> ? [Pre: list @ A,X3: A,Xs5: list @ A,Y3: A,Ys5: list @ A] :
( ( X3 != Y3 )
& ( Xs2
= ( append @ A @ Pre @ ( append @ A @ ( cons @ A @ X3 @ ( nil @ A ) ) @ Xs5 ) ) )
& ( Ys2
= ( append @ A @ Pre @ ( append @ A @ ( cons @ A @ Y3 @ ( nil @ A ) ) @ Ys5 ) ) ) ) ) ) ).
% same_length_different
thf(fact_230_lexord__sufE,axiom,
! [A: $tType,Xs2: list @ A,Zs2: list @ A,Ys2: list @ A,Qs: list @ A,R: set @ ( product_prod @ A @ A )] :
( ( member @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ ( append @ A @ Xs2 @ Zs2 ) @ ( append @ A @ Ys2 @ Qs ) ) @ ( lexord @ A @ R ) )
=> ( ( Xs2 != Ys2 )
=> ( ( ( size_size @ ( list @ A ) @ Xs2 )
= ( size_size @ ( list @ A ) @ Ys2 ) )
=> ( ( ( size_size @ ( list @ A ) @ Zs2 )
= ( size_size @ ( list @ A ) @ Qs ) )
=> ( member @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ Xs2 @ Ys2 ) @ ( lexord @ A @ R ) ) ) ) ) ) ).
% lexord_sufE
thf(fact_231_lex__append__rightI,axiom,
! [A: $tType,Xs2: list @ A,Ys2: list @ A,R: set @ ( product_prod @ A @ A ),Vs: list @ A,Us: list @ A] :
( ( member @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ Xs2 @ Ys2 ) @ ( lex @ A @ R ) )
=> ( ( ( size_size @ ( list @ A ) @ Vs )
= ( size_size @ ( list @ A ) @ Us ) )
=> ( member @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ ( append @ A @ Xs2 @ Us ) @ ( append @ A @ Ys2 @ Vs ) ) @ ( lex @ A @ R ) ) ) ) ).
% lex_append_rightI
thf(fact_232_lenlex__append1,axiom,
! [A: $tType,Us: list @ A,Xs2: list @ A,R2: set @ ( product_prod @ A @ A ),Vs: list @ A,Ys2: list @ A] :
( ( member @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ Us @ Xs2 ) @ ( lenlex @ A @ R2 ) )
=> ( ( ( size_size @ ( list @ A ) @ Vs )
= ( size_size @ ( list @ A ) @ Ys2 ) )
=> ( member @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ ( append @ A @ Us @ Vs ) @ ( append @ A @ Xs2 @ Ys2 ) ) @ ( lenlex @ A @ R2 ) ) ) ) ).
% lenlex_append1
thf(fact_233_lexord__lex,axiom,
! [A: $tType,X: list @ A,Y: list @ A,R: set @ ( product_prod @ A @ A )] :
( ( member @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ X @ Y ) @ ( lex @ A @ R ) )
= ( ( member @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ X @ Y ) @ ( lexord @ A @ R ) )
& ( ( size_size @ ( list @ A ) @ X )
= ( size_size @ ( list @ A ) @ Y ) ) ) ) ).
% lexord_lex
thf(fact_234_rtrancl__listrel1__eq__len,axiom,
! [A: $tType,X: list @ A,Y: list @ A,R: set @ ( product_prod @ A @ A )] :
( ( member @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ X @ Y ) @ ( transitive_rtrancl @ ( list @ A ) @ ( listrel1 @ A @ R ) ) )
=> ( ( size_size @ ( list @ A ) @ X )
= ( size_size @ ( list @ A ) @ Y ) ) ) ).
% rtrancl_listrel1_eq_len
thf(fact_235_Cons__lenlex__iff,axiom,
! [A: $tType,M: A,Ms: list @ A,N: A,Ns: list @ A,R: set @ ( product_prod @ A @ A )] :
( ( member @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ ( cons @ A @ M @ Ms ) @ ( cons @ A @ N @ Ns ) ) @ ( lenlex @ A @ R ) )
= ( ( ord_less @ nat @ ( size_size @ ( list @ A ) @ Ms ) @ ( size_size @ ( list @ A ) @ Ns ) )
| ( ( ( size_size @ ( list @ A ) @ Ms )
= ( size_size @ ( list @ A ) @ Ns ) )
& ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ M @ N ) @ R ) )
| ( ( M = N )
& ( member @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ Ms @ Ns ) @ ( lenlex @ A @ R ) ) ) ) ) ).
% Cons_lenlex_iff
thf(fact_236_list__ex1__simps_I1_J,axiom,
! [A: $tType,P: A > $o] :
~ ( list_ex1 @ A @ P @ ( nil @ A ) ) ).
% list_ex1_simps(1)
thf(fact_237_length__induct,axiom,
! [A: $tType,P: ( list @ A ) > $o,Xs2: list @ A] :
( ! [Xs: list @ A] :
( ! [Ys6: list @ A] :
( ( ord_less @ nat @ ( size_size @ ( list @ A ) @ Ys6 ) @ ( size_size @ ( list @ A ) @ Xs ) )
=> ( P @ Ys6 ) )
=> ( P @ Xs ) )
=> ( P @ Xs2 ) ) ).
% length_induct
thf(fact_238_in__measures_I2_J,axiom,
! [A: $tType,X: A,Y: A,F3: A > nat,Fs: list @ ( A > nat )] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X @ Y ) @ ( measures @ A @ ( cons @ ( A > nat ) @ F3 @ Fs ) ) )
= ( ( ord_less @ nat @ ( F3 @ X ) @ ( F3 @ Y ) )
| ( ( ( F3 @ X )
= ( F3 @ Y ) )
& ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X @ Y ) @ ( measures @ A @ Fs ) ) ) ) ) ).
% in_measures(2)
thf(fact_239_listrel__iff__nth,axiom,
! [A: $tType,B: $tType,Xs2: list @ A,Ys2: list @ B,R: set @ ( product_prod @ A @ B )] :
( ( member @ ( product_prod @ ( list @ A ) @ ( list @ B ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ B ) @ Xs2 @ Ys2 ) @ ( listrel @ A @ B @ R ) )
= ( ( ( size_size @ ( list @ A ) @ Xs2 )
= ( size_size @ ( list @ B ) @ Ys2 ) )
& ! [N2: nat] :
( ( ord_less @ nat @ N2 @ ( size_size @ ( list @ A ) @ Xs2 ) )
=> ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ ( nth @ A @ Xs2 @ N2 ) @ ( nth @ B @ Ys2 @ N2 ) ) @ R ) ) ) ) ).
% listrel_iff_nth
thf(fact_240_in__measures_I1_J,axiom,
! [A: $tType,X: A,Y: A] :
~ ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X @ Y ) @ ( measures @ A @ ( nil @ ( A > nat ) ) ) ) ).
% in_measures(1)
thf(fact_241_nth__append__length,axiom,
! [A: $tType,Xs2: list @ A,X: A,Ys2: list @ A] :
( ( nth @ A @ ( append @ A @ Xs2 @ ( cons @ A @ X @ Ys2 ) ) @ ( size_size @ ( list @ A ) @ Xs2 ) )
= X ) ).
% nth_append_length
thf(fact_242_nth__equalityI,axiom,
! [A: $tType,Xs2: list @ A,Ys2: list @ A] :
( ( ( size_size @ ( list @ A ) @ Xs2 )
= ( size_size @ ( list @ A ) @ Ys2 ) )
=> ( ! [I: nat] :
( ( ord_less @ nat @ I @ ( size_size @ ( list @ A ) @ Xs2 ) )
=> ( ( nth @ A @ Xs2 @ I )
= ( nth @ A @ Ys2 @ I ) ) )
=> ( Xs2 = Ys2 ) ) ) ).
% nth_equalityI
thf(fact_243_Skolem__list__nth,axiom,
! [A: $tType,K: nat,P: nat > A > $o] :
( ( ! [I2: nat] :
( ( ord_less @ nat @ I2 @ K )
=> ? [X5: A] : ( P @ I2 @ X5 ) ) )
= ( ? [Xs3: list @ A] :
( ( ( size_size @ ( list @ A ) @ Xs3 )
= K )
& ! [I2: nat] :
( ( ord_less @ nat @ I2 @ K )
=> ( P @ I2 @ ( nth @ A @ Xs3 @ I2 ) ) ) ) ) ) ).
% Skolem_list_nth
thf(fact_244_list__eq__iff__nth__eq,axiom,
! [A: $tType] :
( ( ^ [Y5: list @ A,Z4: list @ A] : Y5 = Z4 )
= ( ^ [Xs3: list @ A,Ys3: list @ A] :
( ( ( size_size @ ( list @ A ) @ Xs3 )
= ( size_size @ ( list @ A ) @ Ys3 ) )
& ! [I2: nat] :
( ( ord_less @ nat @ I2 @ ( size_size @ ( list @ A ) @ Xs3 ) )
=> ( ( nth @ A @ Xs3 @ I2 )
= ( nth @ A @ Ys3 @ I2 ) ) ) ) ) ) ).
% list_eq_iff_nth_eq
thf(fact_245_nth__butlast,axiom,
! [A: $tType,N: nat,Xs2: list @ A] :
( ( ord_less @ nat @ N @ ( size_size @ ( list @ A ) @ ( butlast @ A @ Xs2 ) ) )
=> ( ( nth @ A @ ( butlast @ A @ Xs2 ) @ N )
= ( nth @ A @ Xs2 @ N ) ) ) ).
% nth_butlast
thf(fact_246_measures__less,axiom,
! [A: $tType,F3: A > nat,X: A,Y: A,Fs: list @ ( A > nat )] :
( ( ord_less @ nat @ ( F3 @ X ) @ ( F3 @ Y ) )
=> ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X @ Y ) @ ( measures @ A @ ( cons @ ( A > nat ) @ F3 @ Fs ) ) ) ) ).
% measures_less
thf(fact_247_listrel1__iff__update,axiom,
! [A: $tType,Xs2: list @ A,Ys2: list @ A,R: set @ ( product_prod @ A @ A )] :
( ( member @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ Xs2 @ Ys2 ) @ ( listrel1 @ A @ R ) )
= ( ? [Y4: A,N2: nat] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ ( nth @ A @ Xs2 @ N2 ) @ Y4 ) @ R )
& ( ord_less @ nat @ N2 @ ( size_size @ ( list @ A ) @ Xs2 ) )
& ( Ys2
= ( list_update @ A @ Xs2 @ N2 @ Y4 ) ) ) ) ) ).
% listrel1_iff_update
thf(fact_248_lex__take__index,axiom,
! [A: $tType,Xs2: list @ A,Ys2: list @ A,R: set @ ( product_prod @ A @ A )] :
( ( member @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ Xs2 @ Ys2 ) @ ( lex @ A @ R ) )
=> ~ ! [I: nat] :
( ( ord_less @ nat @ I @ ( size_size @ ( list @ A ) @ Xs2 ) )
=> ( ( ord_less @ nat @ I @ ( size_size @ ( list @ A ) @ Ys2 ) )
=> ( ( ( take @ A @ I @ Xs2 )
= ( take @ A @ I @ Ys2 ) )
=> ~ ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ ( nth @ A @ Xs2 @ I ) @ ( nth @ A @ Ys2 @ I ) ) @ R ) ) ) ) ) ).
% lex_take_index
thf(fact_249_list__update__overwrite,axiom,
! [A: $tType,Xs2: list @ A,I3: nat,X: A,Y: A] :
( ( list_update @ A @ ( list_update @ A @ Xs2 @ I3 @ X ) @ I3 @ Y )
= ( list_update @ A @ Xs2 @ I3 @ Y ) ) ).
% list_update_overwrite
thf(fact_250_list__update__nonempty,axiom,
! [A: $tType,Xs2: list @ A,K: nat,X: A] :
( ( ( list_update @ A @ Xs2 @ K @ X )
= ( nil @ A ) )
= ( Xs2
= ( nil @ A ) ) ) ).
% list_update_nonempty
thf(fact_251_length__list__update,axiom,
! [A: $tType,Xs2: list @ A,I3: nat,X: A] :
( ( size_size @ ( list @ A ) @ ( list_update @ A @ Xs2 @ I3 @ X ) )
= ( size_size @ ( list @ A ) @ Xs2 ) ) ).
% length_list_update
thf(fact_252_list__update__id,axiom,
! [A: $tType,Xs2: list @ A,I3: nat] :
( ( list_update @ A @ Xs2 @ I3 @ ( nth @ A @ Xs2 @ I3 ) )
= Xs2 ) ).
% list_update_id
thf(fact_253_nth__list__update__neq,axiom,
! [A: $tType,I3: nat,J: nat,Xs2: list @ A,X: A] :
( ( I3 != J )
=> ( ( nth @ A @ ( list_update @ A @ Xs2 @ I3 @ X ) @ J )
= ( nth @ A @ Xs2 @ J ) ) ) ).
% nth_list_update_neq
thf(fact_254_nth__take,axiom,
! [A: $tType,I3: nat,N: nat,Xs2: list @ A] :
( ( ord_less @ nat @ I3 @ N )
=> ( ( nth @ A @ ( take @ A @ N @ Xs2 ) @ I3 )
= ( nth @ A @ Xs2 @ I3 ) ) ) ).
% nth_take
% Type constructors (6)
thf(tcon_fun___Orderings_Oord,axiom,
! [A7: $tType,A8: $tType] :
( ( ord @ A8 )
=> ( ord @ ( A7 > A8 ) ) ) ).
thf(tcon_Nat_Onat___Orderings_Olinorder,axiom,
linorder @ nat ).
thf(tcon_Nat_Onat___Orderings_Oord_1,axiom,
ord @ nat ).
thf(tcon_Set_Oset___Orderings_Oord_2,axiom,
! [A7: $tType] : ( ord @ ( set @ A7 ) ) ).
thf(tcon_HOL_Obool___Orderings_Olinorder_3,axiom,
linorder @ $o ).
thf(tcon_HOL_Obool___Orderings_Oord_4,axiom,
ord @ $o ).
% Conjectures (4)
thf(conj_0,hypothesis,
member @ ( product_prod @ ( product_prod @ ( list @ com ) @ state ) @ ( product_prod @ ( list @ com ) @ state ) ) @ ( product_Pair @ ( product_prod @ ( list @ com ) @ state ) @ ( product_prod @ ( list @ com ) @ state ) @ ( product_Pair @ ( list @ com ) @ state @ ( cons @ com @ c @ ( nil @ com ) ) @ s ) @ ( product_Pair @ ( list @ com ) @ state @ ( cons @ com @ c2 @ cs ) @ s2 ) ) @ ( transitive_rtrancl @ ( product_prod @ ( list @ com ) @ state ) @ pHoare1053570893_exec1 ) ).
thf(conj_1,hypothesis,
termi @ c @ s ).
thf(conj_2,hypothesis,
! [S5: state,S6: state] :
( ( member @ ( product_prod @ ( product_prod @ ( list @ com ) @ state ) @ ( product_prod @ ( list @ com ) @ state ) ) @ ( product_Pair @ ( product_prod @ ( list @ com ) @ state ) @ ( product_prod @ ( list @ com ) @ state ) @ ( product_Pair @ ( list @ com ) @ state @ ( cons @ com @ c @ ( nil @ com ) ) @ S5 ) @ ( product_Pair @ ( list @ com ) @ state @ ( cons @ com @ c2 @ cs ) @ S6 ) ) @ ( transitive_rtrancl @ ( product_prod @ ( list @ com ) @ state ) @ pHoare1053570893_exec1 ) )
=> ( ( termi @ c @ S5 )
=> ( ( termi @ c2 @ S6 )
& ! [T5: state] :
( ( exec @ S6 @ c2 @ T5 )
=> ( pHoare1335526537termis @ cs @ T5 ) ) ) ) ) ).
thf(conj_3,conjecture,
termi @ c2 @ s2 ).
%------------------------------------------------------------------------------