TPTP Problem File: SLH0798^1.p
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%------------------------------------------------------------------------------
% File : SLH0000^1 : TPTP v8.2.0. Released v8.2.0.
% Domain : Archive of Formal Proofs
% Problem :
% Version : Especial.
% English :
% Refs : [Des23] Desharnais (2023), Email to Geoff Sutcliffe
% Source : [Des23]
% Names : IMP_Compiler_Reuse/0006_Compiler2/prob_00254_013742__6022660_1 [Des23]
% Status : Theorem
% Rating : ? v8.2.0
% Syntax : Number of formulae : 1463 ( 580 unt; 193 typ; 0 def)
% Number of atoms : 3712 (1439 equ; 0 cnn)
% Maximal formula atoms : 14 ( 2 avg)
% Number of connectives : 10914 ( 463 ~; 81 |; 321 &;8515 @)
% ( 0 <=>;1534 =>; 0 <=; 0 <~>)
% Maximal formula depth : 22 ( 6 avg)
% Number of types : 32 ( 31 usr)
% Number of type conns : 680 ( 680 >; 0 *; 0 +; 0 <<)
% Number of symbols : 165 ( 162 usr; 22 con; 0-3 aty)
% Number of variables : 3687 ( 322 ^;3104 !; 261 ?;3687 :)
% SPC : TH0_THM_EQU_NAR
% Comments : This file was generated by Isabelle (most likely Sledgehammer)
% 2023-01-18 16:49:00.811
%------------------------------------------------------------------------------
% Could-be-implicit typings (31)
thf(ty_n_t__Product____Type__Oprod_It__List__Olist_It__Compiler__Oinstr_J_Mt__List__Olist_It__Product____Type__Oprod_It__Int__Oint_Mt__Product____Type__Oprod_I_062_It__List__Olist_It__String__Ochar_J_Mt__Int__Oint_J_Mt__List__Olist_It__Int__Oint_J_J_J_J_J,type,
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% Explicit typings (162)
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thf(sy_c_List_Olist_Omap_001t__Product____Type__Oprod_It__Int__Oint_Mt__Product____Type__Oprod_I_062_It__List__Olist_It__String__Ochar_J_Mt__Int__Oint_J_Mt__List__Olist_It__Int__Oint_J_J_J_001t__List__Olist_It__String__Ochar_J,type,
map_Pr5338895242362685753t_char: ( produc6425607678544837394st_int > list_char ) > list_P419547167332244514st_int > list_list_char ).
thf(sy_c_List_Olist_Omap_001t__Product____Type__Oprod_It__Int__Oint_Mt__Product____Type__Oprod_I_062_It__List__Olist_It__String__Ochar_J_Mt__Int__Oint_J_Mt__List__Olist_It__Int__Oint_J_J_J_001t__Product____Type__Oprod_It__Int__Oint_Mt__Product____Type__Oprod_I_062_It__List__Olist_It__String__Ochar_J_Mt__Int__Oint_J_Mt__List__Olist_It__Int__Oint_J_J_J,type,
map_Pr8625093460839654122st_int: ( produc6425607678544837394st_int > produc6425607678544837394st_int ) > list_P419547167332244514st_int > list_P419547167332244514st_int ).
thf(sy_c_List_Olist_Oset_001t__Compiler__Oinstr,type,
set_instr2: list_instr > set_instr ).
thf(sy_c_List_Olist_Oset_001t__Int__Oint,type,
set_int2: list_int > set_int ).
thf(sy_c_List_Olist_Oset_001t__List__Olist_It__String__Ochar_J,type,
set_list_char2: list_list_char > set_list_char ).
thf(sy_c_List_Olist_Oset_001t__Nat__Onat,type,
set_nat2: list_nat > set_nat ).
thf(sy_c_List_Olist_Oset_001t__Product____Type__Oprod_It__Int__Oint_Mt__Product____Type__Oprod_I_062_It__List__Olist_It__String__Ochar_J_Mt__Int__Oint_J_Mt__List__Olist_It__Int__Oint_J_J_J,type,
set_Pr3303432077563008829st_int: list_P419547167332244514st_int > set_Pr2116125220558876360st_int ).
thf(sy_c_List_Olist__ex_001t__Compiler__Oinstr,type,
list_ex_instr: ( instr > $o ) > list_instr > $o ).
thf(sy_c_List_Olist__ex_001t__Int__Oint,type,
list_ex_int: ( int > $o ) > list_int > $o ).
thf(sy_c_List_Olist__ex_001t__List__Olist_It__String__Ochar_J,type,
list_ex_list_char: ( list_char > $o ) > list_list_char > $o ).
thf(sy_c_List_Olist__ex_001t__Product____Type__Oprod_It__Int__Oint_Mt__Product____Type__Oprod_I_062_It__List__Olist_It__String__Ochar_J_Mt__Int__Oint_J_Mt__List__Olist_It__Int__Oint_J_J_J,type,
list_e3437881803614826702st_int: ( produc6425607678544837394st_int > $o ) > list_P419547167332244514st_int > $o ).
thf(sy_c_List_Onth_001t__Compiler__Oinstr,type,
nth_instr: list_instr > nat > instr ).
thf(sy_c_List_Onth_001t__Int__Oint,type,
nth_int: list_int > nat > int ).
thf(sy_c_List_Onth_001t__List__Olist_It__String__Ochar_J,type,
nth_list_char: list_list_char > nat > list_char ).
thf(sy_c_List_Onth_001t__Product____Type__Oprod_It__Int__Oint_Mt__Product____Type__Oprod_I_062_It__List__Olist_It__String__Ochar_J_Mt__Int__Oint_J_Mt__List__Olist_It__Int__Oint_J_J_J,type,
nth_Pr8327173284404383691st_int: list_P419547167332244514st_int > nat > produc6425607678544837394st_int ).
thf(sy_c_List_Oupto,type,
upto: int > int > list_int ).
thf(sy_c_List_Oupto__aux,type,
upto_aux: int > int > list_int > list_int ).
thf(sy_c_List_Oupto__rel,type,
upto_rel: product_prod_int_int > product_prod_int_int > $o ).
thf(sy_c_Nat_OSuc,type,
suc: nat > nat ).
thf(sy_c_Nat_Osemiring__1__class_Oof__nat_001t__Int__Oint,type,
semiri1314217659103216013at_int: nat > int ).
thf(sy_c_Nat_Osemiring__1__class_Oof__nat_001t__Nat__Onat,type,
semiri1316708129612266289at_nat: nat > nat ).
thf(sy_c_Nat_Osize__class_Osize_001t__List__Olist_It__Compiler__Oinstr_J,type,
size_size_list_instr: list_instr > nat ).
thf(sy_c_Nat_Osize__class_Osize_001t__List__Olist_It__Int__Oint_J,type,
size_size_list_int: list_int > nat ).
thf(sy_c_Nat_Osize__class_Osize_001t__List__Olist_It__List__Olist_It__String__Ochar_J_J,type,
size_s356637359517785349t_char: list_list_char > nat ).
thf(sy_c_Nat_Osize__class_Osize_001t__List__Olist_It__Nat__Onat_J,type,
size_size_list_nat: list_nat > nat ).
thf(sy_c_Nat_Osize__class_Osize_001t__List__Olist_It__Product____Type__Oprod_It__Int__Oint_Mt__Product____Type__Oprod_I_062_It__List__Olist_It__String__Ochar_J_Mt__Int__Oint_J_Mt__List__Olist_It__Int__Oint_J_J_J_J,type,
size_s229347708945616566st_int: list_P419547167332244514st_int > nat ).
thf(sy_c_Num_Onumeral__class_Onumeral_001t__Int__Oint,type,
numeral_numeral_int: num > int ).
thf(sy_c_Orderings_Oord__class_Oless_001_062_It__Int__Oint_M_Eo_J,type,
ord_less_int_o: ( int > $o ) > ( int > $o ) > $o ).
thf(sy_c_Orderings_Oord__class_Oless_001_062_It__List__Olist_It__String__Ochar_J_M_Eo_J,type,
ord_less_list_char_o: ( list_char > $o ) > ( list_char > $o ) > $o ).
thf(sy_c_Orderings_Oord__class_Oless_001_062_It__Nat__Onat_M_Eo_J,type,
ord_less_nat_o: ( nat > $o ) > ( nat > $o ) > $o ).
thf(sy_c_Orderings_Oord__class_Oless_001t__Int__Oint,type,
ord_less_int: int > int > $o ).
thf(sy_c_Orderings_Oord__class_Oless_001t__Nat__Onat,type,
ord_less_nat: nat > nat > $o ).
thf(sy_c_Orderings_Oord__class_Oless_001t__Set__Oset_It__Int__Oint_J,type,
ord_less_set_int: set_int > set_int > $o ).
thf(sy_c_Orderings_Oord__class_Oless_001t__Set__Oset_It__List__Olist_It__String__Ochar_J_J,type,
ord_le5286566838938544235t_char: set_list_char > set_list_char > $o ).
thf(sy_c_Orderings_Oord__class_Oless_001t__Set__Oset_It__Nat__Onat_J,type,
ord_less_set_nat: set_nat > set_nat > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001_062_It__Int__Oint_M_Eo_J,type,
ord_less_eq_int_o: ( int > $o ) > ( int > $o ) > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001_062_It__List__Olist_It__String__Ochar_J_M_Eo_J,type,
ord_le4796328588573674190char_o: ( list_char > $o ) > ( list_char > $o ) > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001_062_It__Nat__Onat_M_Eo_J,type,
ord_less_eq_nat_o: ( nat > $o ) > ( nat > $o ) > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Int__Oint,type,
ord_less_eq_int: int > int > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Nat__Onat,type,
ord_less_eq_nat: nat > nat > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_It__Int__Oint_J,type,
ord_less_eq_set_int: set_int > set_int > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_It__List__Olist_It__String__Ochar_J_J,type,
ord_le1207710660481778807t_char: set_list_char > set_list_char > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_It__Nat__Onat_J,type,
ord_less_eq_set_nat: set_nat > set_nat > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_It__Product____Type__Oprod_It__Int__Oint_Mt__Product____Type__Oprod_I_062_It__List__Olist_It__String__Ochar_J_Mt__Int__Oint_J_Mt__List__Olist_It__Int__Oint_J_J_J_J,type,
ord_le8444298136753458984st_int: set_Pr2116125220558876360st_int > set_Pr2116125220558876360st_int > $o ).
thf(sy_c_Product__Type_OPair_001_062_It__List__Olist_It__String__Ochar_J_Mt__Int__Oint_J_001t__Int__Oint,type,
produc5790713362662368625nt_int: ( list_char > int ) > int > produc4435102495419491129nt_int ).
thf(sy_c_Product__Type_OPair_001_062_It__List__Olist_It__String__Ochar_J_Mt__Int__Oint_J_001t__List__Olist_It__Int__Oint_J,type,
produc8650753666468850689st_int: ( list_char > int ) > list_int > produc369741286924889929st_int ).
thf(sy_c_Product__Type_OPair_001t__Compiler__Oinstr_001t__Product____Type__Oprod_It__Int__Oint_Mt__Product____Type__Oprod_I_062_It__List__Olist_It__String__Ochar_J_Mt__Int__Oint_J_Mt__List__Olist_It__Int__Oint_J_J_J,type,
produc1484136438817787418st_int: instr > produc6425607678544837394st_int > produc5995290525303592096st_int ).
thf(sy_c_Product__Type_OPair_001t__Int__Oint_001t__Int__Oint,type,
product_Pair_int_int: int > int > product_prod_int_int ).
thf(sy_c_Product__Type_OPair_001t__Int__Oint_001t__Product____Type__Oprod_I_062_It__List__Olist_It__String__Ochar_J_Mt__Int__Oint_J_Mt__List__Olist_It__Int__Oint_J_J,type,
produc5086643055186798020st_int: int > produc369741286924889929st_int > produc6425607678544837394st_int ).
thf(sy_c_Product__Type_OPair_001t__List__Olist_It__Compiler__Oinstr_J_001t__List__Olist_It__Product____Type__Oprod_It__Int__Oint_Mt__Product____Type__Oprod_I_062_It__List__Olist_It__String__Ochar_J_Mt__Int__Oint_J_Mt__List__Olist_It__Int__Oint_J_J_J_J,type,
produc7764413618310988720st_int: list_instr > list_P419547167332244514st_int > produc4400946599758736694st_int ).
thf(sy_c_Product__Type_OPair_001t__List__Olist_It__List__Olist_It__String__Ochar_J_J_001t__Product____Type__Oprod_I_062_It__List__Olist_It__String__Ochar_J_Mt__Int__Oint_J_Mt__Int__Oint_J,type,
produc3965054194175396271nt_int: list_list_char > produc4435102495419491129nt_int > produc4189061985984568957nt_int ).
thf(sy_c_Product__Type_Oprod_Ofst_001t__Int__Oint_001t__Product____Type__Oprod_I_062_It__List__Olist_It__String__Ochar_J_Mt__Int__Oint_J_Mt__List__Olist_It__Int__Oint_J_J,type,
produc8941309049508465880st_int: produc6425607678544837394st_int > int ).
thf(sy_c_Set_OCollect_001t__Int__Oint,type,
collect_int: ( int > $o ) > set_int ).
thf(sy_c_Set_OCollect_001t__List__Olist_It__String__Ochar_J,type,
collect_list_char: ( list_char > $o ) > set_list_char ).
thf(sy_c_Set_OCollect_001t__Nat__Onat,type,
collect_nat: ( nat > $o ) > set_nat ).
thf(sy_c_Set__Interval_Oord__class_OatLeastLessThan_001t__Int__Oint,type,
set_or4662586982721622107an_int: int > int > set_int ).
thf(sy_c_Set__Interval_Oord__class_OatLeastLessThan_001t__Nat__Onat,type,
set_or4665077453230672383an_nat: nat > nat > set_nat ).
thf(sy_c_Set__Interval_Oord__class_OatLeastLessThan_001t__Set__Oset_It__Int__Oint_J,type,
set_or8585797421378605585et_int: set_int > set_int > set_set_int ).
thf(sy_c_Set__Interval_Oord__class_OatLeastLessThan_001t__Set__Oset_It__Nat__Onat_J,type,
set_or3540276404033026485et_nat: set_nat > set_nat > set_set_nat ).
thf(sy_c_Star_Ostar_001t__Product____Type__Oprod_It__Int__Oint_Mt__Product____Type__Oprod_I_062_It__List__Olist_It__String__Ochar_J_Mt__Int__Oint_J_Mt__List__Olist_It__Int__Oint_J_J_J,type,
star_P707599355569300323st_int: ( produc6425607678544837394st_int > produc6425607678544837394st_int > $o ) > produc6425607678544837394st_int > produc6425607678544837394st_int > $o ).
thf(sy_c_Wellfounded_Oaccp_001t__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J,type,
accp_P1096762738010456898nt_int: ( product_prod_int_int > product_prod_int_int > $o ) > product_prod_int_int > $o ).
thf(sy_c_Wellfounded_Oaccp_001t__Product____Type__Oprod_It__List__Olist_It__Compiler__Oinstr_J_Mt__List__Olist_It__Product____Type__Oprod_It__Int__Oint_Mt__Product____Type__Oprod_I_062_It__List__Olist_It__String__Ochar_J_Mt__Int__Oint_J_Mt__List__Olist_It__Int__Oint_J_J_J_J_J,type,
accp_P1482788143903795053st_int: ( produc4400946599758736694st_int > produc4400946599758736694st_int > $o ) > produc4400946599758736694st_int > $o ).
thf(sy_c_Wellfounded_Oaccp_001t__Product____Type__Oprod_It__List__Olist_It__List__Olist_It__String__Ochar_J_J_Mt__Product____Type__Oprod_I_062_It__List__Olist_It__String__Ochar_J_Mt__Int__Oint_J_Mt__Int__Oint_J_J,type,
accp_P4562858270263085236nt_int: ( produc4189061985984568957nt_int > produc4189061985984568957nt_int > $o ) > produc4189061985984568957nt_int > $o ).
thf(sy_c_member_001t__Compiler__Oinstr,type,
member_instr: instr > set_instr > $o ).
thf(sy_c_member_001t__Int__Oint,type,
member_int: int > set_int > $o ).
thf(sy_c_member_001t__List__Olist_It__String__Ochar_J,type,
member_list_char: list_char > set_list_char > $o ).
thf(sy_c_member_001t__Nat__Onat,type,
member_nat: nat > set_nat > $o ).
thf(sy_c_member_001t__Product____Type__Oprod_It__Int__Oint_Mt__Product____Type__Oprod_I_062_It__List__Olist_It__String__Ochar_J_Mt__Int__Oint_J_Mt__List__Olist_It__Int__Oint_J_J_J,type,
member9100552558438320297st_int: produc6425607678544837394st_int > set_Pr2116125220558876360st_int > $o ).
thf(sy_c_member_001t__Set__Oset_It__Int__Oint_J,type,
member_set_int: set_int > set_set_int > $o ).
thf(sy_c_member_001t__Set__Oset_It__Nat__Onat_J,type,
member_set_nat: set_nat > set_set_nat > $o ).
thf(sy_v_P,type,
p: list_instr ).
thf(sy_v_P_H,type,
p2: a > list_instr ).
thf(sy_v_P_H_H,type,
p3: list_instr ).
thf(sy_v_cfs,type,
cfs: list_P419547167332244514st_int ).
thf(sy_v_k,type,
k: nat ).
thf(sy_v_x,type,
x: a ).
% Relevant facts (1262)
thf(fact_0_B,axiom,
ord_less_nat @ k @ ( size_s229347708945616566st_int @ cfs ) ).
% B
thf(fact_1_C,axiom,
( ( produc8941309049508465880st_int @ ( nth_Pr8327173284404383691st_int @ cfs @ k ) )
= ( semiri1314217659103216013at_int @ ( size_size_list_instr @ p ) ) ) ).
% C
thf(fact_2_A,axiom,
execl_all @ ( append_instr @ p @ ( append_instr @ ( p2 @ x ) @ p3 ) ) @ cfs ).
% A
thf(fact_3_atLeastLessThan__iff,axiom,
! [I: set_int,L: set_int,U: set_int] :
( ( member_set_int @ I @ ( set_or8585797421378605585et_int @ L @ U ) )
= ( ( ord_less_eq_set_int @ L @ I )
& ( ord_less_set_int @ I @ U ) ) ) ).
% atLeastLessThan_iff
thf(fact_4_atLeastLessThan__iff,axiom,
! [I: set_nat,L: set_nat,U: set_nat] :
( ( member_set_nat @ I @ ( set_or3540276404033026485et_nat @ L @ U ) )
= ( ( ord_less_eq_set_nat @ L @ I )
& ( ord_less_set_nat @ I @ U ) ) ) ).
% atLeastLessThan_iff
thf(fact_5_atLeastLessThan__iff,axiom,
! [I: nat,L: nat,U: nat] :
( ( member_nat @ I @ ( set_or4665077453230672383an_nat @ L @ U ) )
= ( ( ord_less_eq_nat @ L @ I )
& ( ord_less_nat @ I @ U ) ) ) ).
% atLeastLessThan_iff
thf(fact_6_atLeastLessThan__iff,axiom,
! [I: int,L: int,U: int] :
( ( member_int @ I @ ( set_or4662586982721622107an_int @ L @ U ) )
= ( ( ord_less_eq_int @ L @ I )
& ( ord_less_int @ I @ U ) ) ) ).
% atLeastLessThan_iff
thf(fact_7_ivl__subset,axiom,
! [I: nat,J: nat,M: nat,N: nat] :
( ( ord_less_eq_set_nat @ ( set_or4665077453230672383an_nat @ I @ J ) @ ( set_or4665077453230672383an_nat @ M @ N ) )
= ( ( ord_less_eq_nat @ J @ I )
| ( ( ord_less_eq_nat @ M @ I )
& ( ord_less_eq_nat @ J @ N ) ) ) ) ).
% ivl_subset
thf(fact_8_ivl__subset,axiom,
! [I: int,J: int,M: int,N: int] :
( ( ord_less_eq_set_int @ ( set_or4662586982721622107an_int @ I @ J ) @ ( set_or4662586982721622107an_int @ M @ N ) )
= ( ( ord_less_eq_int @ J @ I )
| ( ( ord_less_eq_int @ M @ I )
& ( ord_less_eq_int @ J @ N ) ) ) ) ).
% ivl_subset
thf(fact_9_of__nat__less__iff,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ ( semiri1316708129612266289at_nat @ M ) @ ( semiri1316708129612266289at_nat @ N ) )
= ( ord_less_nat @ M @ N ) ) ).
% of_nat_less_iff
thf(fact_10_of__nat__less__iff,axiom,
! [M: nat,N: nat] :
( ( ord_less_int @ ( semiri1314217659103216013at_int @ M ) @ ( semiri1314217659103216013at_int @ N ) )
= ( ord_less_nat @ M @ N ) ) ).
% of_nat_less_iff
thf(fact_11_of__nat__le__iff,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq_int @ ( semiri1314217659103216013at_int @ M ) @ ( semiri1314217659103216013at_int @ N ) )
= ( ord_less_eq_nat @ M @ N ) ) ).
% of_nat_le_iff
thf(fact_12_of__nat__le__iff,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq_nat @ ( semiri1316708129612266289at_nat @ M ) @ ( semiri1316708129612266289at_nat @ N ) )
= ( ord_less_eq_nat @ M @ N ) ) ).
% of_nat_le_iff
thf(fact_13_of__nat__eq__iff,axiom,
! [M: nat,N: nat] :
( ( ( semiri1316708129612266289at_nat @ M )
= ( semiri1316708129612266289at_nat @ N ) )
= ( M = N ) ) ).
% of_nat_eq_iff
thf(fact_14_of__nat__eq__iff,axiom,
! [M: nat,N: nat] :
( ( ( semiri1314217659103216013at_int @ M )
= ( semiri1314217659103216013at_int @ N ) )
= ( M = N ) ) ).
% of_nat_eq_iff
thf(fact_15_order__refl,axiom,
! [X: set_int] : ( ord_less_eq_set_int @ X @ X ) ).
% order_refl
thf(fact_16_order__refl,axiom,
! [X: set_nat] : ( ord_less_eq_set_nat @ X @ X ) ).
% order_refl
thf(fact_17_order__refl,axiom,
! [X: int] : ( ord_less_eq_int @ X @ X ) ).
% order_refl
thf(fact_18_order__refl,axiom,
! [X: nat] : ( ord_less_eq_nat @ X @ X ) ).
% order_refl
thf(fact_19_dual__order_Orefl,axiom,
! [A: int] : ( ord_less_eq_int @ A @ A ) ).
% dual_order.refl
thf(fact_20_dual__order_Orefl,axiom,
! [A: nat] : ( ord_less_eq_nat @ A @ A ) ).
% dual_order.refl
thf(fact_21_dual__order_Orefl,axiom,
! [A: set_int] : ( ord_less_eq_set_int @ A @ A ) ).
% dual_order.refl
thf(fact_22_dual__order_Orefl,axiom,
! [A: set_nat] : ( ord_less_eq_set_nat @ A @ A ) ).
% dual_order.refl
thf(fact_23_int__diff__cases,axiom,
! [Z: int] :
~ ! [M2: nat,N2: nat] :
( Z
!= ( minus_minus_int @ ( semiri1314217659103216013at_int @ M2 ) @ ( semiri1314217659103216013at_int @ N2 ) ) ) ).
% int_diff_cases
thf(fact_24_atLeastLessThan__inj_I2_J,axiom,
! [A: nat,B: nat,C: nat,D: nat] :
( ( ( set_or4665077453230672383an_nat @ A @ B )
= ( set_or4665077453230672383an_nat @ C @ D ) )
=> ( ( ord_less_nat @ A @ B )
=> ( ( ord_less_nat @ C @ D )
=> ( B = D ) ) ) ) ).
% atLeastLessThan_inj(2)
thf(fact_25_atLeastLessThan__inj_I2_J,axiom,
! [A: int,B: int,C: int,D: int] :
( ( ( set_or4662586982721622107an_int @ A @ B )
= ( set_or4662586982721622107an_int @ C @ D ) )
=> ( ( ord_less_int @ A @ B )
=> ( ( ord_less_int @ C @ D )
=> ( B = D ) ) ) ) ).
% atLeastLessThan_inj(2)
thf(fact_26_atLeastLessThan__inj_I1_J,axiom,
! [A: nat,B: nat,C: nat,D: nat] :
( ( ( set_or4665077453230672383an_nat @ A @ B )
= ( set_or4665077453230672383an_nat @ C @ D ) )
=> ( ( ord_less_nat @ A @ B )
=> ( ( ord_less_nat @ C @ D )
=> ( A = C ) ) ) ) ).
% atLeastLessThan_inj(1)
thf(fact_27_atLeastLessThan__inj_I1_J,axiom,
! [A: int,B: int,C: int,D: int] :
( ( ( set_or4662586982721622107an_int @ A @ B )
= ( set_or4662586982721622107an_int @ C @ D ) )
=> ( ( ord_less_int @ A @ B )
=> ( ( ord_less_int @ C @ D )
=> ( A = C ) ) ) ) ).
% atLeastLessThan_inj(1)
thf(fact_28_Ico__eq__Ico,axiom,
! [L: nat,H: nat,L2: nat,H2: nat] :
( ( ( set_or4665077453230672383an_nat @ L @ H )
= ( set_or4665077453230672383an_nat @ L2 @ H2 ) )
= ( ( ( L = L2 )
& ( H = H2 ) )
| ( ~ ( ord_less_nat @ L @ H )
& ~ ( ord_less_nat @ L2 @ H2 ) ) ) ) ).
% Ico_eq_Ico
thf(fact_29_Ico__eq__Ico,axiom,
! [L: int,H: int,L2: int,H2: int] :
( ( ( set_or4662586982721622107an_int @ L @ H )
= ( set_or4662586982721622107an_int @ L2 @ H2 ) )
= ( ( ( L = L2 )
& ( H = H2 ) )
| ( ~ ( ord_less_int @ L @ H )
& ~ ( ord_less_int @ L2 @ H2 ) ) ) ) ).
% Ico_eq_Ico
thf(fact_30_diff__diff__cancel,axiom,
! [I: nat,N: nat] :
( ( ord_less_eq_nat @ I @ N )
=> ( ( minus_minus_nat @ N @ ( minus_minus_nat @ N @ I ) )
= I ) ) ).
% diff_diff_cancel
thf(fact_31_ivl__diff,axiom,
! [I: nat,N: nat,M: nat] :
( ( ord_less_eq_nat @ I @ N )
=> ( ( minus_minus_set_nat @ ( set_or4665077453230672383an_nat @ I @ M ) @ ( set_or4665077453230672383an_nat @ I @ N ) )
= ( set_or4665077453230672383an_nat @ N @ M ) ) ) ).
% ivl_diff
thf(fact_32_ivl__diff,axiom,
! [I: int,N: int,M: int] :
( ( ord_less_eq_int @ I @ N )
=> ( ( minus_minus_set_int @ ( set_or4662586982721622107an_int @ I @ M ) @ ( set_or4662586982721622107an_int @ I @ N ) )
= ( set_or4662586982721622107an_int @ N @ M ) ) ) ).
% ivl_diff
thf(fact_33_le__refl,axiom,
! [N: nat] : ( ord_less_eq_nat @ N @ N ) ).
% le_refl
thf(fact_34_le__trans,axiom,
! [I: nat,J: nat,K: nat] :
( ( ord_less_eq_nat @ I @ J )
=> ( ( ord_less_eq_nat @ J @ K )
=> ( ord_less_eq_nat @ I @ K ) ) ) ).
% le_trans
thf(fact_35_eq__imp__le,axiom,
! [M: nat,N: nat] :
( ( M = N )
=> ( ord_less_eq_nat @ M @ N ) ) ).
% eq_imp_le
thf(fact_36_le__antisym,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq_nat @ M @ N )
=> ( ( ord_less_eq_nat @ N @ M )
=> ( M = N ) ) ) ).
% le_antisym
thf(fact_37_eq__diff__iff,axiom,
! [K: nat,M: nat,N: nat] :
( ( ord_less_eq_nat @ K @ M )
=> ( ( ord_less_eq_nat @ K @ N )
=> ( ( ( minus_minus_nat @ M @ K )
= ( minus_minus_nat @ N @ K ) )
= ( M = N ) ) ) ) ).
% eq_diff_iff
thf(fact_38_le__diff__iff,axiom,
! [K: nat,M: nat,N: nat] :
( ( ord_less_eq_nat @ K @ M )
=> ( ( ord_less_eq_nat @ K @ N )
=> ( ( ord_less_eq_nat @ ( minus_minus_nat @ M @ K ) @ ( minus_minus_nat @ N @ K ) )
= ( ord_less_eq_nat @ M @ N ) ) ) ) ).
% le_diff_iff
thf(fact_39_nat__less__le,axiom,
( ord_less_nat
= ( ^ [M3: nat,N3: nat] :
( ( ord_less_eq_nat @ M3 @ N3 )
& ( M3 != N3 ) ) ) ) ).
% nat_less_le
thf(fact_40_nat__neq__iff,axiom,
! [M: nat,N: nat] :
( ( M != N )
= ( ( ord_less_nat @ M @ N )
| ( ord_less_nat @ N @ M ) ) ) ).
% nat_neq_iff
thf(fact_41_Nat_Odiff__diff__eq,axiom,
! [K: nat,M: nat,N: nat] :
( ( ord_less_eq_nat @ K @ M )
=> ( ( ord_less_eq_nat @ K @ N )
=> ( ( minus_minus_nat @ ( minus_minus_nat @ M @ K ) @ ( minus_minus_nat @ N @ K ) )
= ( minus_minus_nat @ M @ N ) ) ) ) ).
% Nat.diff_diff_eq
thf(fact_42_diff__le__mono,axiom,
! [M: nat,N: nat,L: nat] :
( ( ord_less_eq_nat @ M @ N )
=> ( ord_less_eq_nat @ ( minus_minus_nat @ M @ L ) @ ( minus_minus_nat @ N @ L ) ) ) ).
% diff_le_mono
thf(fact_43_diff__le__self,axiom,
! [M: nat,N: nat] : ( ord_less_eq_nat @ ( minus_minus_nat @ M @ N ) @ M ) ).
% diff_le_self
thf(fact_44_le__diff__iff_H,axiom,
! [A: nat,C: nat,B: nat] :
( ( ord_less_eq_nat @ A @ C )
=> ( ( ord_less_eq_nat @ B @ C )
=> ( ( ord_less_eq_nat @ ( minus_minus_nat @ C @ A ) @ ( minus_minus_nat @ C @ B ) )
= ( ord_less_eq_nat @ B @ A ) ) ) ) ).
% le_diff_iff'
thf(fact_45_diff__le__mono2,axiom,
! [M: nat,N: nat,L: nat] :
( ( ord_less_eq_nat @ M @ N )
=> ( ord_less_eq_nat @ ( minus_minus_nat @ L @ N ) @ ( minus_minus_nat @ L @ M ) ) ) ).
% diff_le_mono2
thf(fact_46_less__diff__iff,axiom,
! [K: nat,M: nat,N: nat] :
( ( ord_less_eq_nat @ K @ M )
=> ( ( ord_less_eq_nat @ K @ N )
=> ( ( ord_less_nat @ ( minus_minus_nat @ M @ K ) @ ( minus_minus_nat @ N @ K ) )
= ( ord_less_nat @ M @ N ) ) ) ) ).
% less_diff_iff
thf(fact_47_less__not__refl,axiom,
! [N: nat] :
~ ( ord_less_nat @ N @ N ) ).
% less_not_refl
thf(fact_48_nat__le__linear,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq_nat @ M @ N )
| ( ord_less_eq_nat @ N @ M ) ) ).
% nat_le_linear
thf(fact_49_diff__less__mono,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_nat @ A @ B )
=> ( ( ord_less_eq_nat @ C @ A )
=> ( ord_less_nat @ ( minus_minus_nat @ A @ C ) @ ( minus_minus_nat @ B @ C ) ) ) ) ).
% diff_less_mono
thf(fact_50_less__not__refl2,axiom,
! [N: nat,M: nat] :
( ( ord_less_nat @ N @ M )
=> ( M != N ) ) ).
% less_not_refl2
thf(fact_51_less__not__refl3,axiom,
! [S: nat,T: nat] :
( ( ord_less_nat @ S @ T )
=> ( S != T ) ) ).
% less_not_refl3
thf(fact_52_diff__less__mono2,axiom,
! [M: nat,N: nat,L: nat] :
( ( ord_less_nat @ M @ N )
=> ( ( ord_less_nat @ M @ L )
=> ( ord_less_nat @ ( minus_minus_nat @ L @ N ) @ ( minus_minus_nat @ L @ M ) ) ) ) ).
% diff_less_mono2
thf(fact_53_less__imp__le__nat,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ M @ N )
=> ( ord_less_eq_nat @ M @ N ) ) ).
% less_imp_le_nat
thf(fact_54_less__irrefl__nat,axiom,
! [N: nat] :
~ ( ord_less_nat @ N @ N ) ).
% less_irrefl_nat
thf(fact_55_nat__less__induct,axiom,
! [P: nat > $o,N: nat] :
( ! [N2: nat] :
( ! [M4: nat] :
( ( ord_less_nat @ M4 @ N2 )
=> ( P @ M4 ) )
=> ( P @ N2 ) )
=> ( P @ N ) ) ).
% nat_less_induct
thf(fact_56_infinite__descent,axiom,
! [P: nat > $o,N: nat] :
( ! [N2: nat] :
( ~ ( P @ N2 )
=> ? [M4: nat] :
( ( ord_less_nat @ M4 @ N2 )
& ~ ( P @ M4 ) ) )
=> ( P @ N ) ) ).
% infinite_descent
thf(fact_57_le__eq__less__or__eq,axiom,
( ord_less_eq_nat
= ( ^ [M3: nat,N3: nat] :
( ( ord_less_nat @ M3 @ N3 )
| ( M3 = N3 ) ) ) ) ).
% le_eq_less_or_eq
thf(fact_58_less__or__eq__imp__le,axiom,
! [M: nat,N: nat] :
( ( ( ord_less_nat @ M @ N )
| ( M = N ) )
=> ( ord_less_eq_nat @ M @ N ) ) ).
% less_or_eq_imp_le
thf(fact_59_linorder__neqE__nat,axiom,
! [X: nat,Y: nat] :
( ( X != Y )
=> ( ~ ( ord_less_nat @ X @ Y )
=> ( ord_less_nat @ Y @ X ) ) ) ).
% linorder_neqE_nat
thf(fact_60_less__imp__diff__less,axiom,
! [J: nat,K: nat,N: nat] :
( ( ord_less_nat @ J @ K )
=> ( ord_less_nat @ ( minus_minus_nat @ J @ N ) @ K ) ) ).
% less_imp_diff_less
thf(fact_61_mem__Collect__eq,axiom,
! [A: int,P: int > $o] :
( ( member_int @ A @ ( collect_int @ P ) )
= ( P @ A ) ) ).
% mem_Collect_eq
thf(fact_62_mem__Collect__eq,axiom,
! [A: list_char,P: list_char > $o] :
( ( member_list_char @ A @ ( collect_list_char @ P ) )
= ( P @ A ) ) ).
% mem_Collect_eq
thf(fact_63_mem__Collect__eq,axiom,
! [A: nat,P: nat > $o] :
( ( member_nat @ A @ ( collect_nat @ P ) )
= ( P @ A ) ) ).
% mem_Collect_eq
thf(fact_64_Collect__mem__eq,axiom,
! [A2: set_int] :
( ( collect_int
@ ^ [X2: int] : ( member_int @ X2 @ A2 ) )
= A2 ) ).
% Collect_mem_eq
thf(fact_65_Collect__mem__eq,axiom,
! [A2: set_list_char] :
( ( collect_list_char
@ ^ [X2: list_char] : ( member_list_char @ X2 @ A2 ) )
= A2 ) ).
% Collect_mem_eq
thf(fact_66_Collect__mem__eq,axiom,
! [A2: set_nat] :
( ( collect_nat
@ ^ [X2: nat] : ( member_nat @ X2 @ A2 ) )
= A2 ) ).
% Collect_mem_eq
thf(fact_67_Collect__cong,axiom,
! [P: nat > $o,Q: nat > $o] :
( ! [X3: nat] :
( ( P @ X3 )
= ( Q @ X3 ) )
=> ( ( collect_nat @ P )
= ( collect_nat @ Q ) ) ) ).
% Collect_cong
thf(fact_68_Nat_Oex__has__greatest__nat,axiom,
! [P: nat > $o,K: nat,B: nat] :
( ( P @ K )
=> ( ! [Y2: nat] :
( ( P @ Y2 )
=> ( ord_less_eq_nat @ Y2 @ B ) )
=> ? [X3: nat] :
( ( P @ X3 )
& ! [Y3: nat] :
( ( P @ Y3 )
=> ( ord_less_eq_nat @ Y3 @ X3 ) ) ) ) ) ).
% Nat.ex_has_greatest_nat
thf(fact_69_le__neq__implies__less,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq_nat @ M @ N )
=> ( ( M != N )
=> ( ord_less_nat @ M @ N ) ) ) ).
% le_neq_implies_less
thf(fact_70_less__mono__imp__le__mono,axiom,
! [F: nat > nat,I: nat,J: nat] :
( ! [I2: nat,J2: nat] :
( ( ord_less_nat @ I2 @ J2 )
=> ( ord_less_nat @ ( F @ I2 ) @ ( F @ J2 ) ) )
=> ( ( ord_less_eq_nat @ I @ J )
=> ( ord_less_eq_nat @ ( F @ I ) @ ( F @ J ) ) ) ) ).
% less_mono_imp_le_mono
thf(fact_71_bounded__Max__nat,axiom,
! [P: nat > $o,X: nat,M5: nat] :
( ( P @ X )
=> ( ! [X3: nat] :
( ( P @ X3 )
=> ( ord_less_eq_nat @ X3 @ M5 ) )
=> ~ ! [M2: nat] :
( ( P @ M2 )
=> ~ ! [X4: nat] :
( ( P @ X4 )
=> ( ord_less_eq_nat @ X4 @ M2 ) ) ) ) ) ).
% bounded_Max_nat
thf(fact_72_of__nat__diff,axiom,
! [N: nat,M: nat] :
( ( ord_less_eq_nat @ N @ M )
=> ( ( semiri1314217659103216013at_int @ ( minus_minus_nat @ M @ N ) )
= ( minus_minus_int @ ( semiri1314217659103216013at_int @ M ) @ ( semiri1314217659103216013at_int @ N ) ) ) ) ).
% of_nat_diff
thf(fact_73_of__nat__diff,axiom,
! [N: nat,M: nat] :
( ( ord_less_eq_nat @ N @ M )
=> ( ( semiri1316708129612266289at_nat @ ( minus_minus_nat @ M @ N ) )
= ( minus_minus_nat @ ( semiri1316708129612266289at_nat @ M ) @ ( semiri1316708129612266289at_nat @ N ) ) ) ) ).
% of_nat_diff
thf(fact_74_zle__int,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq_int @ ( semiri1314217659103216013at_int @ M ) @ ( semiri1314217659103216013at_int @ N ) )
= ( ord_less_eq_nat @ M @ N ) ) ).
% zle_int
thf(fact_75_order__antisym__conv,axiom,
! [Y: int,X: int] :
( ( ord_less_eq_int @ Y @ X )
=> ( ( ord_less_eq_int @ X @ Y )
= ( X = Y ) ) ) ).
% order_antisym_conv
thf(fact_76_order__antisym__conv,axiom,
! [Y: nat,X: nat] :
( ( ord_less_eq_nat @ Y @ X )
=> ( ( ord_less_eq_nat @ X @ Y )
= ( X = Y ) ) ) ).
% order_antisym_conv
thf(fact_77_order__antisym__conv,axiom,
! [Y: set_int,X: set_int] :
( ( ord_less_eq_set_int @ Y @ X )
=> ( ( ord_less_eq_set_int @ X @ Y )
= ( X = Y ) ) ) ).
% order_antisym_conv
thf(fact_78_order__antisym__conv,axiom,
! [Y: set_nat,X: set_nat] :
( ( ord_less_eq_set_nat @ Y @ X )
=> ( ( ord_less_eq_set_nat @ X @ Y )
= ( X = Y ) ) ) ).
% order_antisym_conv
thf(fact_79_linorder__le__cases,axiom,
! [X: int,Y: int] :
( ~ ( ord_less_eq_int @ X @ Y )
=> ( ord_less_eq_int @ Y @ X ) ) ).
% linorder_le_cases
thf(fact_80_linorder__le__cases,axiom,
! [X: nat,Y: nat] :
( ~ ( ord_less_eq_nat @ X @ Y )
=> ( ord_less_eq_nat @ Y @ X ) ) ).
% linorder_le_cases
thf(fact_81_ord__le__eq__subst,axiom,
! [A: int,B: int,F: int > int,C: int] :
( ( ord_less_eq_int @ A @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X3: int,Y2: int] :
( ( ord_less_eq_int @ X3 @ Y2 )
=> ( ord_less_eq_int @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_int @ ( F @ A ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_82_ord__le__eq__subst,axiom,
! [A: int,B: int,F: int > nat,C: nat] :
( ( ord_less_eq_int @ A @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X3: int,Y2: int] :
( ( ord_less_eq_int @ X3 @ Y2 )
=> ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_nat @ ( F @ A ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_83_ord__le__eq__subst,axiom,
! [A: nat,B: nat,F: nat > int,C: int] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X3: nat,Y2: nat] :
( ( ord_less_eq_nat @ X3 @ Y2 )
=> ( ord_less_eq_int @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_int @ ( F @ A ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_84_ord__le__eq__subst,axiom,
! [A: nat,B: nat,F: nat > nat,C: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X3: nat,Y2: nat] :
( ( ord_less_eq_nat @ X3 @ Y2 )
=> ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_nat @ ( F @ A ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_85_ord__le__eq__subst,axiom,
! [A: int,B: int,F: int > set_int,C: set_int] :
( ( ord_less_eq_int @ A @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X3: int,Y2: int] :
( ( ord_less_eq_int @ X3 @ Y2 )
=> ( ord_less_eq_set_int @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_set_int @ ( F @ A ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_86_ord__le__eq__subst,axiom,
! [A: int,B: int,F: int > set_nat,C: set_nat] :
( ( ord_less_eq_int @ A @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X3: int,Y2: int] :
( ( ord_less_eq_int @ X3 @ Y2 )
=> ( ord_less_eq_set_nat @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_set_nat @ ( F @ A ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_87_ord__le__eq__subst,axiom,
! [A: nat,B: nat,F: nat > set_int,C: set_int] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X3: nat,Y2: nat] :
( ( ord_less_eq_nat @ X3 @ Y2 )
=> ( ord_less_eq_set_int @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_set_int @ ( F @ A ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_88_ord__le__eq__subst,axiom,
! [A: nat,B: nat,F: nat > set_nat,C: set_nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X3: nat,Y2: nat] :
( ( ord_less_eq_nat @ X3 @ Y2 )
=> ( ord_less_eq_set_nat @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_set_nat @ ( F @ A ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_89_ord__le__eq__subst,axiom,
! [A: set_int,B: set_int,F: set_int > int,C: int] :
( ( ord_less_eq_set_int @ A @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X3: set_int,Y2: set_int] :
( ( ord_less_eq_set_int @ X3 @ Y2 )
=> ( ord_less_eq_int @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_int @ ( F @ A ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_90_ord__le__eq__subst,axiom,
! [A: set_int,B: set_int,F: set_int > nat,C: nat] :
( ( ord_less_eq_set_int @ A @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X3: set_int,Y2: set_int] :
( ( ord_less_eq_set_int @ X3 @ Y2 )
=> ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_nat @ ( F @ A ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_91_ord__eq__le__subst,axiom,
! [A: int,F: int > int,B: int,C: int] :
( ( A
= ( F @ B ) )
=> ( ( ord_less_eq_int @ B @ C )
=> ( ! [X3: int,Y2: int] :
( ( ord_less_eq_int @ X3 @ Y2 )
=> ( ord_less_eq_int @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_int @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_92_ord__eq__le__subst,axiom,
! [A: nat,F: int > nat,B: int,C: int] :
( ( A
= ( F @ B ) )
=> ( ( ord_less_eq_int @ B @ C )
=> ( ! [X3: int,Y2: int] :
( ( ord_less_eq_int @ X3 @ Y2 )
=> ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_nat @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_93_ord__eq__le__subst,axiom,
! [A: int,F: nat > int,B: nat,C: nat] :
( ( A
= ( F @ B ) )
=> ( ( ord_less_eq_nat @ B @ C )
=> ( ! [X3: nat,Y2: nat] :
( ( ord_less_eq_nat @ X3 @ Y2 )
=> ( ord_less_eq_int @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_int @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_94_ord__eq__le__subst,axiom,
! [A: nat,F: nat > nat,B: nat,C: nat] :
( ( A
= ( F @ B ) )
=> ( ( ord_less_eq_nat @ B @ C )
=> ( ! [X3: nat,Y2: nat] :
( ( ord_less_eq_nat @ X3 @ Y2 )
=> ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_nat @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_95_ord__eq__le__subst,axiom,
! [A: set_int,F: int > set_int,B: int,C: int] :
( ( A
= ( F @ B ) )
=> ( ( ord_less_eq_int @ B @ C )
=> ( ! [X3: int,Y2: int] :
( ( ord_less_eq_int @ X3 @ Y2 )
=> ( ord_less_eq_set_int @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_set_int @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_96_ord__eq__le__subst,axiom,
! [A: set_nat,F: int > set_nat,B: int,C: int] :
( ( A
= ( F @ B ) )
=> ( ( ord_less_eq_int @ B @ C )
=> ( ! [X3: int,Y2: int] :
( ( ord_less_eq_int @ X3 @ Y2 )
=> ( ord_less_eq_set_nat @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_set_nat @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_97_ord__eq__le__subst,axiom,
! [A: set_int,F: nat > set_int,B: nat,C: nat] :
( ( A
= ( F @ B ) )
=> ( ( ord_less_eq_nat @ B @ C )
=> ( ! [X3: nat,Y2: nat] :
( ( ord_less_eq_nat @ X3 @ Y2 )
=> ( ord_less_eq_set_int @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_set_int @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_98_ord__eq__le__subst,axiom,
! [A: set_nat,F: nat > set_nat,B: nat,C: nat] :
( ( A
= ( F @ B ) )
=> ( ( ord_less_eq_nat @ B @ C )
=> ( ! [X3: nat,Y2: nat] :
( ( ord_less_eq_nat @ X3 @ Y2 )
=> ( ord_less_eq_set_nat @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_set_nat @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_99_ord__eq__le__subst,axiom,
! [A: int,F: set_int > int,B: set_int,C: set_int] :
( ( A
= ( F @ B ) )
=> ( ( ord_less_eq_set_int @ B @ C )
=> ( ! [X3: set_int,Y2: set_int] :
( ( ord_less_eq_set_int @ X3 @ Y2 )
=> ( ord_less_eq_int @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_int @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_100_ord__eq__le__subst,axiom,
! [A: nat,F: set_int > nat,B: set_int,C: set_int] :
( ( A
= ( F @ B ) )
=> ( ( ord_less_eq_set_int @ B @ C )
=> ( ! [X3: set_int,Y2: set_int] :
( ( ord_less_eq_set_int @ X3 @ Y2 )
=> ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_nat @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_101_linorder__linear,axiom,
! [X: int,Y: int] :
( ( ord_less_eq_int @ X @ Y )
| ( ord_less_eq_int @ Y @ X ) ) ).
% linorder_linear
thf(fact_102_linorder__linear,axiom,
! [X: nat,Y: nat] :
( ( ord_less_eq_nat @ X @ Y )
| ( ord_less_eq_nat @ Y @ X ) ) ).
% linorder_linear
thf(fact_103_order__eq__refl,axiom,
! [X: int,Y: int] :
( ( X = Y )
=> ( ord_less_eq_int @ X @ Y ) ) ).
% order_eq_refl
thf(fact_104_order__eq__refl,axiom,
! [X: nat,Y: nat] :
( ( X = Y )
=> ( ord_less_eq_nat @ X @ Y ) ) ).
% order_eq_refl
thf(fact_105_order__eq__refl,axiom,
! [X: set_int,Y: set_int] :
( ( X = Y )
=> ( ord_less_eq_set_int @ X @ Y ) ) ).
% order_eq_refl
thf(fact_106_order__eq__refl,axiom,
! [X: set_nat,Y: set_nat] :
( ( X = Y )
=> ( ord_less_eq_set_nat @ X @ Y ) ) ).
% order_eq_refl
thf(fact_107_order__subst2,axiom,
! [A: int,B: int,F: int > int,C: int] :
( ( ord_less_eq_int @ A @ B )
=> ( ( ord_less_eq_int @ ( F @ B ) @ C )
=> ( ! [X3: int,Y2: int] :
( ( ord_less_eq_int @ X3 @ Y2 )
=> ( ord_less_eq_int @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_int @ ( F @ A ) @ C ) ) ) ) ).
% order_subst2
thf(fact_108_order__subst2,axiom,
! [A: int,B: int,F: int > nat,C: nat] :
( ( ord_less_eq_int @ A @ B )
=> ( ( ord_less_eq_nat @ ( F @ B ) @ C )
=> ( ! [X3: int,Y2: int] :
( ( ord_less_eq_int @ X3 @ Y2 )
=> ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_nat @ ( F @ A ) @ C ) ) ) ) ).
% order_subst2
thf(fact_109_order__subst2,axiom,
! [A: nat,B: nat,F: nat > int,C: int] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ord_less_eq_int @ ( F @ B ) @ C )
=> ( ! [X3: nat,Y2: nat] :
( ( ord_less_eq_nat @ X3 @ Y2 )
=> ( ord_less_eq_int @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_int @ ( F @ A ) @ C ) ) ) ) ).
% order_subst2
thf(fact_110_order__subst2,axiom,
! [A: nat,B: nat,F: nat > nat,C: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ord_less_eq_nat @ ( F @ B ) @ C )
=> ( ! [X3: nat,Y2: nat] :
( ( ord_less_eq_nat @ X3 @ Y2 )
=> ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_nat @ ( F @ A ) @ C ) ) ) ) ).
% order_subst2
thf(fact_111_order__subst2,axiom,
! [A: int,B: int,F: int > set_int,C: set_int] :
( ( ord_less_eq_int @ A @ B )
=> ( ( ord_less_eq_set_int @ ( F @ B ) @ C )
=> ( ! [X3: int,Y2: int] :
( ( ord_less_eq_int @ X3 @ Y2 )
=> ( ord_less_eq_set_int @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_set_int @ ( F @ A ) @ C ) ) ) ) ).
% order_subst2
thf(fact_112_order__subst2,axiom,
! [A: int,B: int,F: int > set_nat,C: set_nat] :
( ( ord_less_eq_int @ A @ B )
=> ( ( ord_less_eq_set_nat @ ( F @ B ) @ C )
=> ( ! [X3: int,Y2: int] :
( ( ord_less_eq_int @ X3 @ Y2 )
=> ( ord_less_eq_set_nat @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_set_nat @ ( F @ A ) @ C ) ) ) ) ).
% order_subst2
thf(fact_113_order__subst2,axiom,
! [A: nat,B: nat,F: nat > set_int,C: set_int] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ord_less_eq_set_int @ ( F @ B ) @ C )
=> ( ! [X3: nat,Y2: nat] :
( ( ord_less_eq_nat @ X3 @ Y2 )
=> ( ord_less_eq_set_int @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_set_int @ ( F @ A ) @ C ) ) ) ) ).
% order_subst2
thf(fact_114_order__subst2,axiom,
! [A: nat,B: nat,F: nat > set_nat,C: set_nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ord_less_eq_set_nat @ ( F @ B ) @ C )
=> ( ! [X3: nat,Y2: nat] :
( ( ord_less_eq_nat @ X3 @ Y2 )
=> ( ord_less_eq_set_nat @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_set_nat @ ( F @ A ) @ C ) ) ) ) ).
% order_subst2
thf(fact_115_order__subst2,axiom,
! [A: set_int,B: set_int,F: set_int > int,C: int] :
( ( ord_less_eq_set_int @ A @ B )
=> ( ( ord_less_eq_int @ ( F @ B ) @ C )
=> ( ! [X3: set_int,Y2: set_int] :
( ( ord_less_eq_set_int @ X3 @ Y2 )
=> ( ord_less_eq_int @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_int @ ( F @ A ) @ C ) ) ) ) ).
% order_subst2
thf(fact_116_order__subst2,axiom,
! [A: set_int,B: set_int,F: set_int > nat,C: nat] :
( ( ord_less_eq_set_int @ A @ B )
=> ( ( ord_less_eq_nat @ ( F @ B ) @ C )
=> ( ! [X3: set_int,Y2: set_int] :
( ( ord_less_eq_set_int @ X3 @ Y2 )
=> ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_nat @ ( F @ A ) @ C ) ) ) ) ).
% order_subst2
thf(fact_117_order__subst1,axiom,
! [A: int,F: int > int,B: int,C: int] :
( ( ord_less_eq_int @ A @ ( F @ B ) )
=> ( ( ord_less_eq_int @ B @ C )
=> ( ! [X3: int,Y2: int] :
( ( ord_less_eq_int @ X3 @ Y2 )
=> ( ord_less_eq_int @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_int @ A @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_118_order__subst1,axiom,
! [A: int,F: nat > int,B: nat,C: nat] :
( ( ord_less_eq_int @ A @ ( F @ B ) )
=> ( ( ord_less_eq_nat @ B @ C )
=> ( ! [X3: nat,Y2: nat] :
( ( ord_less_eq_nat @ X3 @ Y2 )
=> ( ord_less_eq_int @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_int @ A @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_119_order__subst1,axiom,
! [A: nat,F: int > nat,B: int,C: int] :
( ( ord_less_eq_nat @ A @ ( F @ B ) )
=> ( ( ord_less_eq_int @ B @ C )
=> ( ! [X3: int,Y2: int] :
( ( ord_less_eq_int @ X3 @ Y2 )
=> ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_nat @ A @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_120_order__subst1,axiom,
! [A: nat,F: nat > nat,B: nat,C: nat] :
( ( ord_less_eq_nat @ A @ ( F @ B ) )
=> ( ( ord_less_eq_nat @ B @ C )
=> ( ! [X3: nat,Y2: nat] :
( ( ord_less_eq_nat @ X3 @ Y2 )
=> ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_nat @ A @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_121_order__subst1,axiom,
! [A: int,F: set_int > int,B: set_int,C: set_int] :
( ( ord_less_eq_int @ A @ ( F @ B ) )
=> ( ( ord_less_eq_set_int @ B @ C )
=> ( ! [X3: set_int,Y2: set_int] :
( ( ord_less_eq_set_int @ X3 @ Y2 )
=> ( ord_less_eq_int @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_int @ A @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_122_order__subst1,axiom,
! [A: int,F: set_nat > int,B: set_nat,C: set_nat] :
( ( ord_less_eq_int @ A @ ( F @ B ) )
=> ( ( ord_less_eq_set_nat @ B @ C )
=> ( ! [X3: set_nat,Y2: set_nat] :
( ( ord_less_eq_set_nat @ X3 @ Y2 )
=> ( ord_less_eq_int @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_int @ A @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_123_order__subst1,axiom,
! [A: nat,F: set_int > nat,B: set_int,C: set_int] :
( ( ord_less_eq_nat @ A @ ( F @ B ) )
=> ( ( ord_less_eq_set_int @ B @ C )
=> ( ! [X3: set_int,Y2: set_int] :
( ( ord_less_eq_set_int @ X3 @ Y2 )
=> ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_nat @ A @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_124_order__subst1,axiom,
! [A: nat,F: set_nat > nat,B: set_nat,C: set_nat] :
( ( ord_less_eq_nat @ A @ ( F @ B ) )
=> ( ( ord_less_eq_set_nat @ B @ C )
=> ( ! [X3: set_nat,Y2: set_nat] :
( ( ord_less_eq_set_nat @ X3 @ Y2 )
=> ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_nat @ A @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_125_order__subst1,axiom,
! [A: set_int,F: int > set_int,B: int,C: int] :
( ( ord_less_eq_set_int @ A @ ( F @ B ) )
=> ( ( ord_less_eq_int @ B @ C )
=> ( ! [X3: int,Y2: int] :
( ( ord_less_eq_int @ X3 @ Y2 )
=> ( ord_less_eq_set_int @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_set_int @ A @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_126_order__subst1,axiom,
! [A: set_int,F: nat > set_int,B: nat,C: nat] :
( ( ord_less_eq_set_int @ A @ ( F @ B ) )
=> ( ( ord_less_eq_nat @ B @ C )
=> ( ! [X3: nat,Y2: nat] :
( ( ord_less_eq_nat @ X3 @ Y2 )
=> ( ord_less_eq_set_int @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_set_int @ A @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_127_Orderings_Oorder__eq__iff,axiom,
( ( ^ [Y4: int,Z2: int] : ( Y4 = Z2 ) )
= ( ^ [A3: int,B2: int] :
( ( ord_less_eq_int @ A3 @ B2 )
& ( ord_less_eq_int @ B2 @ A3 ) ) ) ) ).
% Orderings.order_eq_iff
thf(fact_128_Orderings_Oorder__eq__iff,axiom,
( ( ^ [Y4: nat,Z2: nat] : ( Y4 = Z2 ) )
= ( ^ [A3: nat,B2: nat] :
( ( ord_less_eq_nat @ A3 @ B2 )
& ( ord_less_eq_nat @ B2 @ A3 ) ) ) ) ).
% Orderings.order_eq_iff
thf(fact_129_Orderings_Oorder__eq__iff,axiom,
( ( ^ [Y4: set_int,Z2: set_int] : ( Y4 = Z2 ) )
= ( ^ [A3: set_int,B2: set_int] :
( ( ord_less_eq_set_int @ A3 @ B2 )
& ( ord_less_eq_set_int @ B2 @ A3 ) ) ) ) ).
% Orderings.order_eq_iff
thf(fact_130_Orderings_Oorder__eq__iff,axiom,
( ( ^ [Y4: set_nat,Z2: set_nat] : ( Y4 = Z2 ) )
= ( ^ [A3: set_nat,B2: set_nat] :
( ( ord_less_eq_set_nat @ A3 @ B2 )
& ( ord_less_eq_set_nat @ B2 @ A3 ) ) ) ) ).
% Orderings.order_eq_iff
thf(fact_131_antisym,axiom,
! [A: int,B: int] :
( ( ord_less_eq_int @ A @ B )
=> ( ( ord_less_eq_int @ B @ A )
=> ( A = B ) ) ) ).
% antisym
thf(fact_132_antisym,axiom,
! [A: nat,B: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ord_less_eq_nat @ B @ A )
=> ( A = B ) ) ) ).
% antisym
thf(fact_133_antisym,axiom,
! [A: set_int,B: set_int] :
( ( ord_less_eq_set_int @ A @ B )
=> ( ( ord_less_eq_set_int @ B @ A )
=> ( A = B ) ) ) ).
% antisym
thf(fact_134_antisym,axiom,
! [A: set_nat,B: set_nat] :
( ( ord_less_eq_set_nat @ A @ B )
=> ( ( ord_less_eq_set_nat @ B @ A )
=> ( A = B ) ) ) ).
% antisym
thf(fact_135_dual__order_Otrans,axiom,
! [B: int,A: int,C: int] :
( ( ord_less_eq_int @ B @ A )
=> ( ( ord_less_eq_int @ C @ B )
=> ( ord_less_eq_int @ C @ A ) ) ) ).
% dual_order.trans
thf(fact_136_dual__order_Otrans,axiom,
! [B: nat,A: nat,C: nat] :
( ( ord_less_eq_nat @ B @ A )
=> ( ( ord_less_eq_nat @ C @ B )
=> ( ord_less_eq_nat @ C @ A ) ) ) ).
% dual_order.trans
thf(fact_137_dual__order_Otrans,axiom,
! [B: set_int,A: set_int,C: set_int] :
( ( ord_less_eq_set_int @ B @ A )
=> ( ( ord_less_eq_set_int @ C @ B )
=> ( ord_less_eq_set_int @ C @ A ) ) ) ).
% dual_order.trans
thf(fact_138_dual__order_Otrans,axiom,
! [B: set_nat,A: set_nat,C: set_nat] :
( ( ord_less_eq_set_nat @ B @ A )
=> ( ( ord_less_eq_set_nat @ C @ B )
=> ( ord_less_eq_set_nat @ C @ A ) ) ) ).
% dual_order.trans
thf(fact_139_dual__order_Oantisym,axiom,
! [B: int,A: int] :
( ( ord_less_eq_int @ B @ A )
=> ( ( ord_less_eq_int @ A @ B )
=> ( A = B ) ) ) ).
% dual_order.antisym
thf(fact_140_dual__order_Oantisym,axiom,
! [B: nat,A: nat] :
( ( ord_less_eq_nat @ B @ A )
=> ( ( ord_less_eq_nat @ A @ B )
=> ( A = B ) ) ) ).
% dual_order.antisym
thf(fact_141_dual__order_Oantisym,axiom,
! [B: set_int,A: set_int] :
( ( ord_less_eq_set_int @ B @ A )
=> ( ( ord_less_eq_set_int @ A @ B )
=> ( A = B ) ) ) ).
% dual_order.antisym
thf(fact_142_dual__order_Oantisym,axiom,
! [B: set_nat,A: set_nat] :
( ( ord_less_eq_set_nat @ B @ A )
=> ( ( ord_less_eq_set_nat @ A @ B )
=> ( A = B ) ) ) ).
% dual_order.antisym
thf(fact_143_dual__order_Oeq__iff,axiom,
( ( ^ [Y4: int,Z2: int] : ( Y4 = Z2 ) )
= ( ^ [A3: int,B2: int] :
( ( ord_less_eq_int @ B2 @ A3 )
& ( ord_less_eq_int @ A3 @ B2 ) ) ) ) ).
% dual_order.eq_iff
thf(fact_144_dual__order_Oeq__iff,axiom,
( ( ^ [Y4: nat,Z2: nat] : ( Y4 = Z2 ) )
= ( ^ [A3: nat,B2: nat] :
( ( ord_less_eq_nat @ B2 @ A3 )
& ( ord_less_eq_nat @ A3 @ B2 ) ) ) ) ).
% dual_order.eq_iff
thf(fact_145_dual__order_Oeq__iff,axiom,
( ( ^ [Y4: set_int,Z2: set_int] : ( Y4 = Z2 ) )
= ( ^ [A3: set_int,B2: set_int] :
( ( ord_less_eq_set_int @ B2 @ A3 )
& ( ord_less_eq_set_int @ A3 @ B2 ) ) ) ) ).
% dual_order.eq_iff
thf(fact_146_dual__order_Oeq__iff,axiom,
( ( ^ [Y4: set_nat,Z2: set_nat] : ( Y4 = Z2 ) )
= ( ^ [A3: set_nat,B2: set_nat] :
( ( ord_less_eq_set_nat @ B2 @ A3 )
& ( ord_less_eq_set_nat @ A3 @ B2 ) ) ) ) ).
% dual_order.eq_iff
thf(fact_147_linorder__wlog,axiom,
! [P: int > int > $o,A: int,B: int] :
( ! [A4: int,B3: int] :
( ( ord_less_eq_int @ A4 @ B3 )
=> ( P @ A4 @ B3 ) )
=> ( ! [A4: int,B3: int] :
( ( P @ B3 @ A4 )
=> ( P @ A4 @ B3 ) )
=> ( P @ A @ B ) ) ) ).
% linorder_wlog
thf(fact_148_linorder__wlog,axiom,
! [P: nat > nat > $o,A: nat,B: nat] :
( ! [A4: nat,B3: nat] :
( ( ord_less_eq_nat @ A4 @ B3 )
=> ( P @ A4 @ B3 ) )
=> ( ! [A4: nat,B3: nat] :
( ( P @ B3 @ A4 )
=> ( P @ A4 @ B3 ) )
=> ( P @ A @ B ) ) ) ).
% linorder_wlog
thf(fact_149_order__trans,axiom,
! [X: int,Y: int,Z: int] :
( ( ord_less_eq_int @ X @ Y )
=> ( ( ord_less_eq_int @ Y @ Z )
=> ( ord_less_eq_int @ X @ Z ) ) ) ).
% order_trans
thf(fact_150_order__trans,axiom,
! [X: nat,Y: nat,Z: nat] :
( ( ord_less_eq_nat @ X @ Y )
=> ( ( ord_less_eq_nat @ Y @ Z )
=> ( ord_less_eq_nat @ X @ Z ) ) ) ).
% order_trans
thf(fact_151_order__trans,axiom,
! [X: set_int,Y: set_int,Z: set_int] :
( ( ord_less_eq_set_int @ X @ Y )
=> ( ( ord_less_eq_set_int @ Y @ Z )
=> ( ord_less_eq_set_int @ X @ Z ) ) ) ).
% order_trans
thf(fact_152_order__trans,axiom,
! [X: set_nat,Y: set_nat,Z: set_nat] :
( ( ord_less_eq_set_nat @ X @ Y )
=> ( ( ord_less_eq_set_nat @ Y @ Z )
=> ( ord_less_eq_set_nat @ X @ Z ) ) ) ).
% order_trans
thf(fact_153_order_Otrans,axiom,
! [A: int,B: int,C: int] :
( ( ord_less_eq_int @ A @ B )
=> ( ( ord_less_eq_int @ B @ C )
=> ( ord_less_eq_int @ A @ C ) ) ) ).
% order.trans
thf(fact_154_order_Otrans,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ord_less_eq_nat @ B @ C )
=> ( ord_less_eq_nat @ A @ C ) ) ) ).
% order.trans
thf(fact_155_order_Otrans,axiom,
! [A: set_int,B: set_int,C: set_int] :
( ( ord_less_eq_set_int @ A @ B )
=> ( ( ord_less_eq_set_int @ B @ C )
=> ( ord_less_eq_set_int @ A @ C ) ) ) ).
% order.trans
thf(fact_156_order_Otrans,axiom,
! [A: set_nat,B: set_nat,C: set_nat] :
( ( ord_less_eq_set_nat @ A @ B )
=> ( ( ord_less_eq_set_nat @ B @ C )
=> ( ord_less_eq_set_nat @ A @ C ) ) ) ).
% order.trans
thf(fact_157_order__antisym,axiom,
! [X: int,Y: int] :
( ( ord_less_eq_int @ X @ Y )
=> ( ( ord_less_eq_int @ Y @ X )
=> ( X = Y ) ) ) ).
% order_antisym
thf(fact_158_order__antisym,axiom,
! [X: nat,Y: nat] :
( ( ord_less_eq_nat @ X @ Y )
=> ( ( ord_less_eq_nat @ Y @ X )
=> ( X = Y ) ) ) ).
% order_antisym
thf(fact_159_order__antisym,axiom,
! [X: set_int,Y: set_int] :
( ( ord_less_eq_set_int @ X @ Y )
=> ( ( ord_less_eq_set_int @ Y @ X )
=> ( X = Y ) ) ) ).
% order_antisym
thf(fact_160_order__antisym,axiom,
! [X: set_nat,Y: set_nat] :
( ( ord_less_eq_set_nat @ X @ Y )
=> ( ( ord_less_eq_set_nat @ Y @ X )
=> ( X = Y ) ) ) ).
% order_antisym
thf(fact_161_ord__le__eq__trans,axiom,
! [A: int,B: int,C: int] :
( ( ord_less_eq_int @ A @ B )
=> ( ( B = C )
=> ( ord_less_eq_int @ A @ C ) ) ) ).
% ord_le_eq_trans
thf(fact_162_ord__le__eq__trans,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( B = C )
=> ( ord_less_eq_nat @ A @ C ) ) ) ).
% ord_le_eq_trans
thf(fact_163_ord__le__eq__trans,axiom,
! [A: set_int,B: set_int,C: set_int] :
( ( ord_less_eq_set_int @ A @ B )
=> ( ( B = C )
=> ( ord_less_eq_set_int @ A @ C ) ) ) ).
% ord_le_eq_trans
thf(fact_164_ord__le__eq__trans,axiom,
! [A: set_nat,B: set_nat,C: set_nat] :
( ( ord_less_eq_set_nat @ A @ B )
=> ( ( B = C )
=> ( ord_less_eq_set_nat @ A @ C ) ) ) ).
% ord_le_eq_trans
thf(fact_165_ord__eq__le__trans,axiom,
! [A: int,B: int,C: int] :
( ( A = B )
=> ( ( ord_less_eq_int @ B @ C )
=> ( ord_less_eq_int @ A @ C ) ) ) ).
% ord_eq_le_trans
thf(fact_166_ord__eq__le__trans,axiom,
! [A: nat,B: nat,C: nat] :
( ( A = B )
=> ( ( ord_less_eq_nat @ B @ C )
=> ( ord_less_eq_nat @ A @ C ) ) ) ).
% ord_eq_le_trans
thf(fact_167_ord__eq__le__trans,axiom,
! [A: set_int,B: set_int,C: set_int] :
( ( A = B )
=> ( ( ord_less_eq_set_int @ B @ C )
=> ( ord_less_eq_set_int @ A @ C ) ) ) ).
% ord_eq_le_trans
thf(fact_168_ord__eq__le__trans,axiom,
! [A: set_nat,B: set_nat,C: set_nat] :
( ( A = B )
=> ( ( ord_less_eq_set_nat @ B @ C )
=> ( ord_less_eq_set_nat @ A @ C ) ) ) ).
% ord_eq_le_trans
thf(fact_169_order__class_Oorder__eq__iff,axiom,
( ( ^ [Y4: int,Z2: int] : ( Y4 = Z2 ) )
= ( ^ [X2: int,Y5: int] :
( ( ord_less_eq_int @ X2 @ Y5 )
& ( ord_less_eq_int @ Y5 @ X2 ) ) ) ) ).
% order_class.order_eq_iff
thf(fact_170_order__class_Oorder__eq__iff,axiom,
( ( ^ [Y4: nat,Z2: nat] : ( Y4 = Z2 ) )
= ( ^ [X2: nat,Y5: nat] :
( ( ord_less_eq_nat @ X2 @ Y5 )
& ( ord_less_eq_nat @ Y5 @ X2 ) ) ) ) ).
% order_class.order_eq_iff
thf(fact_171_order__class_Oorder__eq__iff,axiom,
( ( ^ [Y4: set_int,Z2: set_int] : ( Y4 = Z2 ) )
= ( ^ [X2: set_int,Y5: set_int] :
( ( ord_less_eq_set_int @ X2 @ Y5 )
& ( ord_less_eq_set_int @ Y5 @ X2 ) ) ) ) ).
% order_class.order_eq_iff
thf(fact_172_order__class_Oorder__eq__iff,axiom,
( ( ^ [Y4: set_nat,Z2: set_nat] : ( Y4 = Z2 ) )
= ( ^ [X2: set_nat,Y5: set_nat] :
( ( ord_less_eq_set_nat @ X2 @ Y5 )
& ( ord_less_eq_set_nat @ Y5 @ X2 ) ) ) ) ).
% order_class.order_eq_iff
thf(fact_173_le__cases3,axiom,
! [X: int,Y: int,Z: int] :
( ( ( ord_less_eq_int @ X @ Y )
=> ~ ( ord_less_eq_int @ Y @ Z ) )
=> ( ( ( ord_less_eq_int @ Y @ X )
=> ~ ( ord_less_eq_int @ X @ Z ) )
=> ( ( ( ord_less_eq_int @ X @ Z )
=> ~ ( ord_less_eq_int @ Z @ Y ) )
=> ( ( ( ord_less_eq_int @ Z @ Y )
=> ~ ( ord_less_eq_int @ Y @ X ) )
=> ( ( ( ord_less_eq_int @ Y @ Z )
=> ~ ( ord_less_eq_int @ Z @ X ) )
=> ~ ( ( ord_less_eq_int @ Z @ X )
=> ~ ( ord_less_eq_int @ X @ Y ) ) ) ) ) ) ) ).
% le_cases3
thf(fact_174_le__cases3,axiom,
! [X: nat,Y: nat,Z: nat] :
( ( ( ord_less_eq_nat @ X @ Y )
=> ~ ( ord_less_eq_nat @ Y @ Z ) )
=> ( ( ( ord_less_eq_nat @ Y @ X )
=> ~ ( ord_less_eq_nat @ X @ Z ) )
=> ( ( ( ord_less_eq_nat @ X @ Z )
=> ~ ( ord_less_eq_nat @ Z @ Y ) )
=> ( ( ( ord_less_eq_nat @ Z @ Y )
=> ~ ( ord_less_eq_nat @ Y @ X ) )
=> ( ( ( ord_less_eq_nat @ Y @ Z )
=> ~ ( ord_less_eq_nat @ Z @ X ) )
=> ~ ( ( ord_less_eq_nat @ Z @ X )
=> ~ ( ord_less_eq_nat @ X @ Y ) ) ) ) ) ) ) ).
% le_cases3
thf(fact_175_nle__le,axiom,
! [A: int,B: int] :
( ( ~ ( ord_less_eq_int @ A @ B ) )
= ( ( ord_less_eq_int @ B @ A )
& ( B != A ) ) ) ).
% nle_le
thf(fact_176_nle__le,axiom,
! [A: nat,B: nat] :
( ( ~ ( ord_less_eq_nat @ A @ B ) )
= ( ( ord_less_eq_nat @ B @ A )
& ( B != A ) ) ) ).
% nle_le
thf(fact_177_order__less__imp__not__less,axiom,
! [X: int,Y: int] :
( ( ord_less_int @ X @ Y )
=> ~ ( ord_less_int @ Y @ X ) ) ).
% order_less_imp_not_less
thf(fact_178_order__less__imp__not__less,axiom,
! [X: nat,Y: nat] :
( ( ord_less_nat @ X @ Y )
=> ~ ( ord_less_nat @ Y @ X ) ) ).
% order_less_imp_not_less
thf(fact_179_order__less__imp__not__eq2,axiom,
! [X: int,Y: int] :
( ( ord_less_int @ X @ Y )
=> ( Y != X ) ) ).
% order_less_imp_not_eq2
thf(fact_180_order__less__imp__not__eq2,axiom,
! [X: nat,Y: nat] :
( ( ord_less_nat @ X @ Y )
=> ( Y != X ) ) ).
% order_less_imp_not_eq2
thf(fact_181_order__less__imp__not__eq,axiom,
! [X: int,Y: int] :
( ( ord_less_int @ X @ Y )
=> ( X != Y ) ) ).
% order_less_imp_not_eq
thf(fact_182_order__less__imp__not__eq,axiom,
! [X: nat,Y: nat] :
( ( ord_less_nat @ X @ Y )
=> ( X != Y ) ) ).
% order_less_imp_not_eq
thf(fact_183_linorder__less__linear,axiom,
! [X: int,Y: int] :
( ( ord_less_int @ X @ Y )
| ( X = Y )
| ( ord_less_int @ Y @ X ) ) ).
% linorder_less_linear
thf(fact_184_linorder__less__linear,axiom,
! [X: nat,Y: nat] :
( ( ord_less_nat @ X @ Y )
| ( X = Y )
| ( ord_less_nat @ Y @ X ) ) ).
% linorder_less_linear
thf(fact_185_order__less__imp__triv,axiom,
! [X: int,Y: int,P: $o] :
( ( ord_less_int @ X @ Y )
=> ( ( ord_less_int @ Y @ X )
=> P ) ) ).
% order_less_imp_triv
thf(fact_186_order__less__imp__triv,axiom,
! [X: nat,Y: nat,P: $o] :
( ( ord_less_nat @ X @ Y )
=> ( ( ord_less_nat @ Y @ X )
=> P ) ) ).
% order_less_imp_triv
thf(fact_187_order__less__not__sym,axiom,
! [X: int,Y: int] :
( ( ord_less_int @ X @ Y )
=> ~ ( ord_less_int @ Y @ X ) ) ).
% order_less_not_sym
thf(fact_188_order__less__not__sym,axiom,
! [X: nat,Y: nat] :
( ( ord_less_nat @ X @ Y )
=> ~ ( ord_less_nat @ Y @ X ) ) ).
% order_less_not_sym
thf(fact_189_order__less__subst2,axiom,
! [A: int,B: int,F: int > int,C: int] :
( ( ord_less_int @ A @ B )
=> ( ( ord_less_int @ ( F @ B ) @ C )
=> ( ! [X3: int,Y2: int] :
( ( ord_less_int @ X3 @ Y2 )
=> ( ord_less_int @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_int @ ( F @ A ) @ C ) ) ) ) ).
% order_less_subst2
thf(fact_190_order__less__subst2,axiom,
! [A: int,B: int,F: int > nat,C: nat] :
( ( ord_less_int @ A @ B )
=> ( ( ord_less_nat @ ( F @ B ) @ C )
=> ( ! [X3: int,Y2: int] :
( ( ord_less_int @ X3 @ Y2 )
=> ( ord_less_nat @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_nat @ ( F @ A ) @ C ) ) ) ) ).
% order_less_subst2
thf(fact_191_order__less__subst2,axiom,
! [A: nat,B: nat,F: nat > int,C: int] :
( ( ord_less_nat @ A @ B )
=> ( ( ord_less_int @ ( F @ B ) @ C )
=> ( ! [X3: nat,Y2: nat] :
( ( ord_less_nat @ X3 @ Y2 )
=> ( ord_less_int @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_int @ ( F @ A ) @ C ) ) ) ) ).
% order_less_subst2
thf(fact_192_order__less__subst2,axiom,
! [A: nat,B: nat,F: nat > nat,C: nat] :
( ( ord_less_nat @ A @ B )
=> ( ( ord_less_nat @ ( F @ B ) @ C )
=> ( ! [X3: nat,Y2: nat] :
( ( ord_less_nat @ X3 @ Y2 )
=> ( ord_less_nat @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_nat @ ( F @ A ) @ C ) ) ) ) ).
% order_less_subst2
thf(fact_193_order__less__subst1,axiom,
! [A: int,F: int > int,B: int,C: int] :
( ( ord_less_int @ A @ ( F @ B ) )
=> ( ( ord_less_int @ B @ C )
=> ( ! [X3: int,Y2: int] :
( ( ord_less_int @ X3 @ Y2 )
=> ( ord_less_int @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_int @ A @ ( F @ C ) ) ) ) ) ).
% order_less_subst1
thf(fact_194_order__less__subst1,axiom,
! [A: int,F: nat > int,B: nat,C: nat] :
( ( ord_less_int @ A @ ( F @ B ) )
=> ( ( ord_less_nat @ B @ C )
=> ( ! [X3: nat,Y2: nat] :
( ( ord_less_nat @ X3 @ Y2 )
=> ( ord_less_int @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_int @ A @ ( F @ C ) ) ) ) ) ).
% order_less_subst1
thf(fact_195_order__less__subst1,axiom,
! [A: nat,F: int > nat,B: int,C: int] :
( ( ord_less_nat @ A @ ( F @ B ) )
=> ( ( ord_less_int @ B @ C )
=> ( ! [X3: int,Y2: int] :
( ( ord_less_int @ X3 @ Y2 )
=> ( ord_less_nat @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_nat @ A @ ( F @ C ) ) ) ) ) ).
% order_less_subst1
thf(fact_196_order__less__subst1,axiom,
! [A: nat,F: nat > nat,B: nat,C: nat] :
( ( ord_less_nat @ A @ ( F @ B ) )
=> ( ( ord_less_nat @ B @ C )
=> ( ! [X3: nat,Y2: nat] :
( ( ord_less_nat @ X3 @ Y2 )
=> ( ord_less_nat @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_nat @ A @ ( F @ C ) ) ) ) ) ).
% order_less_subst1
thf(fact_197_order__less__irrefl,axiom,
! [X: int] :
~ ( ord_less_int @ X @ X ) ).
% order_less_irrefl
thf(fact_198_order__less__irrefl,axiom,
! [X: nat] :
~ ( ord_less_nat @ X @ X ) ).
% order_less_irrefl
thf(fact_199_ord__less__eq__subst,axiom,
! [A: int,B: int,F: int > int,C: int] :
( ( ord_less_int @ A @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X3: int,Y2: int] :
( ( ord_less_int @ X3 @ Y2 )
=> ( ord_less_int @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_int @ ( F @ A ) @ C ) ) ) ) ).
% ord_less_eq_subst
thf(fact_200_ord__less__eq__subst,axiom,
! [A: int,B: int,F: int > nat,C: nat] :
( ( ord_less_int @ A @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X3: int,Y2: int] :
( ( ord_less_int @ X3 @ Y2 )
=> ( ord_less_nat @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_nat @ ( F @ A ) @ C ) ) ) ) ).
% ord_less_eq_subst
thf(fact_201_ord__less__eq__subst,axiom,
! [A: nat,B: nat,F: nat > int,C: int] :
( ( ord_less_nat @ A @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X3: nat,Y2: nat] :
( ( ord_less_nat @ X3 @ Y2 )
=> ( ord_less_int @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_int @ ( F @ A ) @ C ) ) ) ) ).
% ord_less_eq_subst
thf(fact_202_ord__less__eq__subst,axiom,
! [A: nat,B: nat,F: nat > nat,C: nat] :
( ( ord_less_nat @ A @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X3: nat,Y2: nat] :
( ( ord_less_nat @ X3 @ Y2 )
=> ( ord_less_nat @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_nat @ ( F @ A ) @ C ) ) ) ) ).
% ord_less_eq_subst
thf(fact_203_ord__eq__less__subst,axiom,
! [A: int,F: int > int,B: int,C: int] :
( ( A
= ( F @ B ) )
=> ( ( ord_less_int @ B @ C )
=> ( ! [X3: int,Y2: int] :
( ( ord_less_int @ X3 @ Y2 )
=> ( ord_less_int @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_int @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_less_subst
thf(fact_204_ord__eq__less__subst,axiom,
! [A: nat,F: int > nat,B: int,C: int] :
( ( A
= ( F @ B ) )
=> ( ( ord_less_int @ B @ C )
=> ( ! [X3: int,Y2: int] :
( ( ord_less_int @ X3 @ Y2 )
=> ( ord_less_nat @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_nat @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_less_subst
thf(fact_205_ord__eq__less__subst,axiom,
! [A: int,F: nat > int,B: nat,C: nat] :
( ( A
= ( F @ B ) )
=> ( ( ord_less_nat @ B @ C )
=> ( ! [X3: nat,Y2: nat] :
( ( ord_less_nat @ X3 @ Y2 )
=> ( ord_less_int @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_int @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_less_subst
thf(fact_206_ord__eq__less__subst,axiom,
! [A: nat,F: nat > nat,B: nat,C: nat] :
( ( A
= ( F @ B ) )
=> ( ( ord_less_nat @ B @ C )
=> ( ! [X3: nat,Y2: nat] :
( ( ord_less_nat @ X3 @ Y2 )
=> ( ord_less_nat @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_nat @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_less_subst
thf(fact_207_order__less__trans,axiom,
! [X: int,Y: int,Z: int] :
( ( ord_less_int @ X @ Y )
=> ( ( ord_less_int @ Y @ Z )
=> ( ord_less_int @ X @ Z ) ) ) ).
% order_less_trans
thf(fact_208_order__less__trans,axiom,
! [X: nat,Y: nat,Z: nat] :
( ( ord_less_nat @ X @ Y )
=> ( ( ord_less_nat @ Y @ Z )
=> ( ord_less_nat @ X @ Z ) ) ) ).
% order_less_trans
thf(fact_209_order__less__asym_H,axiom,
! [A: int,B: int] :
( ( ord_less_int @ A @ B )
=> ~ ( ord_less_int @ B @ A ) ) ).
% order_less_asym'
thf(fact_210_order__less__asym_H,axiom,
! [A: nat,B: nat] :
( ( ord_less_nat @ A @ B )
=> ~ ( ord_less_nat @ B @ A ) ) ).
% order_less_asym'
thf(fact_211_linorder__neq__iff,axiom,
! [X: int,Y: int] :
( ( X != Y )
= ( ( ord_less_int @ X @ Y )
| ( ord_less_int @ Y @ X ) ) ) ).
% linorder_neq_iff
thf(fact_212_linorder__neq__iff,axiom,
! [X: nat,Y: nat] :
( ( X != Y )
= ( ( ord_less_nat @ X @ Y )
| ( ord_less_nat @ Y @ X ) ) ) ).
% linorder_neq_iff
thf(fact_213_order__less__asym,axiom,
! [X: int,Y: int] :
( ( ord_less_int @ X @ Y )
=> ~ ( ord_less_int @ Y @ X ) ) ).
% order_less_asym
thf(fact_214_order__less__asym,axiom,
! [X: nat,Y: nat] :
( ( ord_less_nat @ X @ Y )
=> ~ ( ord_less_nat @ Y @ X ) ) ).
% order_less_asym
thf(fact_215_linorder__neqE,axiom,
! [X: int,Y: int] :
( ( X != Y )
=> ( ~ ( ord_less_int @ X @ Y )
=> ( ord_less_int @ Y @ X ) ) ) ).
% linorder_neqE
thf(fact_216_linorder__neqE,axiom,
! [X: nat,Y: nat] :
( ( X != Y )
=> ( ~ ( ord_less_nat @ X @ Y )
=> ( ord_less_nat @ Y @ X ) ) ) ).
% linorder_neqE
thf(fact_217_dual__order_Ostrict__implies__not__eq,axiom,
! [B: int,A: int] :
( ( ord_less_int @ B @ A )
=> ( A != B ) ) ).
% dual_order.strict_implies_not_eq
thf(fact_218_dual__order_Ostrict__implies__not__eq,axiom,
! [B: nat,A: nat] :
( ( ord_less_nat @ B @ A )
=> ( A != B ) ) ).
% dual_order.strict_implies_not_eq
thf(fact_219_order_Ostrict__implies__not__eq,axiom,
! [A: int,B: int] :
( ( ord_less_int @ A @ B )
=> ( A != B ) ) ).
% order.strict_implies_not_eq
thf(fact_220_order_Ostrict__implies__not__eq,axiom,
! [A: nat,B: nat] :
( ( ord_less_nat @ A @ B )
=> ( A != B ) ) ).
% order.strict_implies_not_eq
thf(fact_221_dual__order_Ostrict__trans,axiom,
! [B: int,A: int,C: int] :
( ( ord_less_int @ B @ A )
=> ( ( ord_less_int @ C @ B )
=> ( ord_less_int @ C @ A ) ) ) ).
% dual_order.strict_trans
thf(fact_222_dual__order_Ostrict__trans,axiom,
! [B: nat,A: nat,C: nat] :
( ( ord_less_nat @ B @ A )
=> ( ( ord_less_nat @ C @ B )
=> ( ord_less_nat @ C @ A ) ) ) ).
% dual_order.strict_trans
thf(fact_223_not__less__iff__gr__or__eq,axiom,
! [X: int,Y: int] :
( ( ~ ( ord_less_int @ X @ Y ) )
= ( ( ord_less_int @ Y @ X )
| ( X = Y ) ) ) ).
% not_less_iff_gr_or_eq
thf(fact_224_not__less__iff__gr__or__eq,axiom,
! [X: nat,Y: nat] :
( ( ~ ( ord_less_nat @ X @ Y ) )
= ( ( ord_less_nat @ Y @ X )
| ( X = Y ) ) ) ).
% not_less_iff_gr_or_eq
thf(fact_225_order_Ostrict__trans,axiom,
! [A: int,B: int,C: int] :
( ( ord_less_int @ A @ B )
=> ( ( ord_less_int @ B @ C )
=> ( ord_less_int @ A @ C ) ) ) ).
% order.strict_trans
thf(fact_226_order_Ostrict__trans,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_nat @ A @ B )
=> ( ( ord_less_nat @ B @ C )
=> ( ord_less_nat @ A @ C ) ) ) ).
% order.strict_trans
thf(fact_227_linorder__less__wlog,axiom,
! [P: int > int > $o,A: int,B: int] :
( ! [A4: int,B3: int] :
( ( ord_less_int @ A4 @ B3 )
=> ( P @ A4 @ B3 ) )
=> ( ! [A4: int] : ( P @ A4 @ A4 )
=> ( ! [A4: int,B3: int] :
( ( P @ B3 @ A4 )
=> ( P @ A4 @ B3 ) )
=> ( P @ A @ B ) ) ) ) ).
% linorder_less_wlog
thf(fact_228_linorder__less__wlog,axiom,
! [P: nat > nat > $o,A: nat,B: nat] :
( ! [A4: nat,B3: nat] :
( ( ord_less_nat @ A4 @ B3 )
=> ( P @ A4 @ B3 ) )
=> ( ! [A4: nat] : ( P @ A4 @ A4 )
=> ( ! [A4: nat,B3: nat] :
( ( P @ B3 @ A4 )
=> ( P @ A4 @ B3 ) )
=> ( P @ A @ B ) ) ) ) ).
% linorder_less_wlog
thf(fact_229_exists__least__iff,axiom,
( ( ^ [P2: nat > $o] :
? [X5: nat] : ( P2 @ X5 ) )
= ( ^ [P3: nat > $o] :
? [N3: nat] :
( ( P3 @ N3 )
& ! [M3: nat] :
( ( ord_less_nat @ M3 @ N3 )
=> ~ ( P3 @ M3 ) ) ) ) ) ).
% exists_least_iff
thf(fact_230_dual__order_Oirrefl,axiom,
! [A: int] :
~ ( ord_less_int @ A @ A ) ).
% dual_order.irrefl
thf(fact_231_dual__order_Oirrefl,axiom,
! [A: nat] :
~ ( ord_less_nat @ A @ A ) ).
% dual_order.irrefl
thf(fact_232_dual__order_Oasym,axiom,
! [B: int,A: int] :
( ( ord_less_int @ B @ A )
=> ~ ( ord_less_int @ A @ B ) ) ).
% dual_order.asym
thf(fact_233_dual__order_Oasym,axiom,
! [B: nat,A: nat] :
( ( ord_less_nat @ B @ A )
=> ~ ( ord_less_nat @ A @ B ) ) ).
% dual_order.asym
thf(fact_234_linorder__cases,axiom,
! [X: int,Y: int] :
( ~ ( ord_less_int @ X @ Y )
=> ( ( X != Y )
=> ( ord_less_int @ Y @ X ) ) ) ).
% linorder_cases
thf(fact_235_linorder__cases,axiom,
! [X: nat,Y: nat] :
( ~ ( ord_less_nat @ X @ Y )
=> ( ( X != Y )
=> ( ord_less_nat @ Y @ X ) ) ) ).
% linorder_cases
thf(fact_236_antisym__conv3,axiom,
! [Y: int,X: int] :
( ~ ( ord_less_int @ Y @ X )
=> ( ( ~ ( ord_less_int @ X @ Y ) )
= ( X = Y ) ) ) ).
% antisym_conv3
thf(fact_237_antisym__conv3,axiom,
! [Y: nat,X: nat] :
( ~ ( ord_less_nat @ Y @ X )
=> ( ( ~ ( ord_less_nat @ X @ Y ) )
= ( X = Y ) ) ) ).
% antisym_conv3
thf(fact_238_less__induct,axiom,
! [P: nat > $o,A: nat] :
( ! [X3: nat] :
( ! [Y3: nat] :
( ( ord_less_nat @ Y3 @ X3 )
=> ( P @ Y3 ) )
=> ( P @ X3 ) )
=> ( P @ A ) ) ).
% less_induct
thf(fact_239_ord__less__eq__trans,axiom,
! [A: int,B: int,C: int] :
( ( ord_less_int @ A @ B )
=> ( ( B = C )
=> ( ord_less_int @ A @ C ) ) ) ).
% ord_less_eq_trans
thf(fact_240_ord__less__eq__trans,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_nat @ A @ B )
=> ( ( B = C )
=> ( ord_less_nat @ A @ C ) ) ) ).
% ord_less_eq_trans
thf(fact_241_ord__eq__less__trans,axiom,
! [A: int,B: int,C: int] :
( ( A = B )
=> ( ( ord_less_int @ B @ C )
=> ( ord_less_int @ A @ C ) ) ) ).
% ord_eq_less_trans
thf(fact_242_ord__eq__less__trans,axiom,
! [A: nat,B: nat,C: nat] :
( ( A = B )
=> ( ( ord_less_nat @ B @ C )
=> ( ord_less_nat @ A @ C ) ) ) ).
% ord_eq_less_trans
thf(fact_243_order_Oasym,axiom,
! [A: int,B: int] :
( ( ord_less_int @ A @ B )
=> ~ ( ord_less_int @ B @ A ) ) ).
% order.asym
thf(fact_244_order_Oasym,axiom,
! [A: nat,B: nat] :
( ( ord_less_nat @ A @ B )
=> ~ ( ord_less_nat @ B @ A ) ) ).
% order.asym
thf(fact_245_less__imp__neq,axiom,
! [X: int,Y: int] :
( ( ord_less_int @ X @ Y )
=> ( X != Y ) ) ).
% less_imp_neq
thf(fact_246_less__imp__neq,axiom,
! [X: nat,Y: nat] :
( ( ord_less_nat @ X @ Y )
=> ( X != Y ) ) ).
% less_imp_neq
thf(fact_247_gt__ex,axiom,
! [X: int] :
? [X_1: int] : ( ord_less_int @ X @ X_1 ) ).
% gt_ex
thf(fact_248_gt__ex,axiom,
! [X: nat] :
? [X_1: nat] : ( ord_less_nat @ X @ X_1 ) ).
% gt_ex
thf(fact_249_lt__ex,axiom,
! [X: int] :
? [Y2: int] : ( ord_less_int @ Y2 @ X ) ).
% lt_ex
thf(fact_250_size__neq__size__imp__neq,axiom,
! [X: list_P419547167332244514st_int,Y: list_P419547167332244514st_int] :
( ( ( size_s229347708945616566st_int @ X )
!= ( size_s229347708945616566st_int @ Y ) )
=> ( X != Y ) ) ).
% size_neq_size_imp_neq
thf(fact_251_size__neq__size__imp__neq,axiom,
! [X: list_instr,Y: list_instr] :
( ( ( size_size_list_instr @ X )
!= ( size_size_list_instr @ Y ) )
=> ( X != Y ) ) ).
% size_neq_size_imp_neq
thf(fact_252_size__neq__size__imp__neq,axiom,
! [X: list_list_char,Y: list_list_char] :
( ( ( size_s356637359517785349t_char @ X )
!= ( size_s356637359517785349t_char @ Y ) )
=> ( X != Y ) ) ).
% size_neq_size_imp_neq
thf(fact_253_size__neq__size__imp__neq,axiom,
! [X: list_int,Y: list_int] :
( ( ( size_size_list_int @ X )
!= ( size_size_list_int @ Y ) )
=> ( X != Y ) ) ).
% size_neq_size_imp_neq
thf(fact_254_int__int__eq,axiom,
! [M: nat,N: nat] :
( ( ( semiri1314217659103216013at_int @ M )
= ( semiri1314217659103216013at_int @ N ) )
= ( M = N ) ) ).
% int_int_eq
thf(fact_255_order__le__imp__less__or__eq,axiom,
! [X: int,Y: int] :
( ( ord_less_eq_int @ X @ Y )
=> ( ( ord_less_int @ X @ Y )
| ( X = Y ) ) ) ).
% order_le_imp_less_or_eq
thf(fact_256_order__le__imp__less__or__eq,axiom,
! [X: nat,Y: nat] :
( ( ord_less_eq_nat @ X @ Y )
=> ( ( ord_less_nat @ X @ Y )
| ( X = Y ) ) ) ).
% order_le_imp_less_or_eq
thf(fact_257_order__le__imp__less__or__eq,axiom,
! [X: set_int,Y: set_int] :
( ( ord_less_eq_set_int @ X @ Y )
=> ( ( ord_less_set_int @ X @ Y )
| ( X = Y ) ) ) ).
% order_le_imp_less_or_eq
thf(fact_258_order__le__imp__less__or__eq,axiom,
! [X: set_nat,Y: set_nat] :
( ( ord_less_eq_set_nat @ X @ Y )
=> ( ( ord_less_set_nat @ X @ Y )
| ( X = Y ) ) ) ).
% order_le_imp_less_or_eq
thf(fact_259_linorder__le__less__linear,axiom,
! [X: int,Y: int] :
( ( ord_less_eq_int @ X @ Y )
| ( ord_less_int @ Y @ X ) ) ).
% linorder_le_less_linear
thf(fact_260_linorder__le__less__linear,axiom,
! [X: nat,Y: nat] :
( ( ord_less_eq_nat @ X @ Y )
| ( ord_less_nat @ Y @ X ) ) ).
% linorder_le_less_linear
thf(fact_261_order__less__le__subst2,axiom,
! [A: int,B: int,F: int > int,C: int] :
( ( ord_less_int @ A @ B )
=> ( ( ord_less_eq_int @ ( F @ B ) @ C )
=> ( ! [X3: int,Y2: int] :
( ( ord_less_int @ X3 @ Y2 )
=> ( ord_less_int @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_int @ ( F @ A ) @ C ) ) ) ) ).
% order_less_le_subst2
thf(fact_262_order__less__le__subst2,axiom,
! [A: nat,B: nat,F: nat > int,C: int] :
( ( ord_less_nat @ A @ B )
=> ( ( ord_less_eq_int @ ( F @ B ) @ C )
=> ( ! [X3: nat,Y2: nat] :
( ( ord_less_nat @ X3 @ Y2 )
=> ( ord_less_int @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_int @ ( F @ A ) @ C ) ) ) ) ).
% order_less_le_subst2
thf(fact_263_order__less__le__subst2,axiom,
! [A: int,B: int,F: int > nat,C: nat] :
( ( ord_less_int @ A @ B )
=> ( ( ord_less_eq_nat @ ( F @ B ) @ C )
=> ( ! [X3: int,Y2: int] :
( ( ord_less_int @ X3 @ Y2 )
=> ( ord_less_nat @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_nat @ ( F @ A ) @ C ) ) ) ) ).
% order_less_le_subst2
thf(fact_264_order__less__le__subst2,axiom,
! [A: nat,B: nat,F: nat > nat,C: nat] :
( ( ord_less_nat @ A @ B )
=> ( ( ord_less_eq_nat @ ( F @ B ) @ C )
=> ( ! [X3: nat,Y2: nat] :
( ( ord_less_nat @ X3 @ Y2 )
=> ( ord_less_nat @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_nat @ ( F @ A ) @ C ) ) ) ) ).
% order_less_le_subst2
thf(fact_265_order__less__le__subst2,axiom,
! [A: int,B: int,F: int > set_int,C: set_int] :
( ( ord_less_int @ A @ B )
=> ( ( ord_less_eq_set_int @ ( F @ B ) @ C )
=> ( ! [X3: int,Y2: int] :
( ( ord_less_int @ X3 @ Y2 )
=> ( ord_less_set_int @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_set_int @ ( F @ A ) @ C ) ) ) ) ).
% order_less_le_subst2
thf(fact_266_order__less__le__subst2,axiom,
! [A: nat,B: nat,F: nat > set_int,C: set_int] :
( ( ord_less_nat @ A @ B )
=> ( ( ord_less_eq_set_int @ ( F @ B ) @ C )
=> ( ! [X3: nat,Y2: nat] :
( ( ord_less_nat @ X3 @ Y2 )
=> ( ord_less_set_int @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_set_int @ ( F @ A ) @ C ) ) ) ) ).
% order_less_le_subst2
thf(fact_267_order__less__le__subst2,axiom,
! [A: int,B: int,F: int > set_nat,C: set_nat] :
( ( ord_less_int @ A @ B )
=> ( ( ord_less_eq_set_nat @ ( F @ B ) @ C )
=> ( ! [X3: int,Y2: int] :
( ( ord_less_int @ X3 @ Y2 )
=> ( ord_less_set_nat @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_set_nat @ ( F @ A ) @ C ) ) ) ) ).
% order_less_le_subst2
thf(fact_268_order__less__le__subst2,axiom,
! [A: nat,B: nat,F: nat > set_nat,C: set_nat] :
( ( ord_less_nat @ A @ B )
=> ( ( ord_less_eq_set_nat @ ( F @ B ) @ C )
=> ( ! [X3: nat,Y2: nat] :
( ( ord_less_nat @ X3 @ Y2 )
=> ( ord_less_set_nat @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_set_nat @ ( F @ A ) @ C ) ) ) ) ).
% order_less_le_subst2
thf(fact_269_order__less__le__subst1,axiom,
! [A: int,F: int > int,B: int,C: int] :
( ( ord_less_int @ A @ ( F @ B ) )
=> ( ( ord_less_eq_int @ B @ C )
=> ( ! [X3: int,Y2: int] :
( ( ord_less_eq_int @ X3 @ Y2 )
=> ( ord_less_eq_int @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_int @ A @ ( F @ C ) ) ) ) ) ).
% order_less_le_subst1
thf(fact_270_order__less__le__subst1,axiom,
! [A: nat,F: int > nat,B: int,C: int] :
( ( ord_less_nat @ A @ ( F @ B ) )
=> ( ( ord_less_eq_int @ B @ C )
=> ( ! [X3: int,Y2: int] :
( ( ord_less_eq_int @ X3 @ Y2 )
=> ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_nat @ A @ ( F @ C ) ) ) ) ) ).
% order_less_le_subst1
thf(fact_271_order__less__le__subst1,axiom,
! [A: int,F: nat > int,B: nat,C: nat] :
( ( ord_less_int @ A @ ( F @ B ) )
=> ( ( ord_less_eq_nat @ B @ C )
=> ( ! [X3: nat,Y2: nat] :
( ( ord_less_eq_nat @ X3 @ Y2 )
=> ( ord_less_eq_int @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_int @ A @ ( F @ C ) ) ) ) ) ).
% order_less_le_subst1
thf(fact_272_order__less__le__subst1,axiom,
! [A: nat,F: nat > nat,B: nat,C: nat] :
( ( ord_less_nat @ A @ ( F @ B ) )
=> ( ( ord_less_eq_nat @ B @ C )
=> ( ! [X3: nat,Y2: nat] :
( ( ord_less_eq_nat @ X3 @ Y2 )
=> ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_nat @ A @ ( F @ C ) ) ) ) ) ).
% order_less_le_subst1
thf(fact_273_order__less__le__subst1,axiom,
! [A: set_int,F: int > set_int,B: int,C: int] :
( ( ord_less_set_int @ A @ ( F @ B ) )
=> ( ( ord_less_eq_int @ B @ C )
=> ( ! [X3: int,Y2: int] :
( ( ord_less_eq_int @ X3 @ Y2 )
=> ( ord_less_eq_set_int @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_set_int @ A @ ( F @ C ) ) ) ) ) ).
% order_less_le_subst1
thf(fact_274_order__less__le__subst1,axiom,
! [A: set_nat,F: int > set_nat,B: int,C: int] :
( ( ord_less_set_nat @ A @ ( F @ B ) )
=> ( ( ord_less_eq_int @ B @ C )
=> ( ! [X3: int,Y2: int] :
( ( ord_less_eq_int @ X3 @ Y2 )
=> ( ord_less_eq_set_nat @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_set_nat @ A @ ( F @ C ) ) ) ) ) ).
% order_less_le_subst1
thf(fact_275_order__less__le__subst1,axiom,
! [A: set_int,F: nat > set_int,B: nat,C: nat] :
( ( ord_less_set_int @ A @ ( F @ B ) )
=> ( ( ord_less_eq_nat @ B @ C )
=> ( ! [X3: nat,Y2: nat] :
( ( ord_less_eq_nat @ X3 @ Y2 )
=> ( ord_less_eq_set_int @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_set_int @ A @ ( F @ C ) ) ) ) ) ).
% order_less_le_subst1
thf(fact_276_order__less__le__subst1,axiom,
! [A: set_nat,F: nat > set_nat,B: nat,C: nat] :
( ( ord_less_set_nat @ A @ ( F @ B ) )
=> ( ( ord_less_eq_nat @ B @ C )
=> ( ! [X3: nat,Y2: nat] :
( ( ord_less_eq_nat @ X3 @ Y2 )
=> ( ord_less_eq_set_nat @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_set_nat @ A @ ( F @ C ) ) ) ) ) ).
% order_less_le_subst1
thf(fact_277_order__less__le__subst1,axiom,
! [A: int,F: set_int > int,B: set_int,C: set_int] :
( ( ord_less_int @ A @ ( F @ B ) )
=> ( ( ord_less_eq_set_int @ B @ C )
=> ( ! [X3: set_int,Y2: set_int] :
( ( ord_less_eq_set_int @ X3 @ Y2 )
=> ( ord_less_eq_int @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_int @ A @ ( F @ C ) ) ) ) ) ).
% order_less_le_subst1
thf(fact_278_order__less__le__subst1,axiom,
! [A: nat,F: set_int > nat,B: set_int,C: set_int] :
( ( ord_less_nat @ A @ ( F @ B ) )
=> ( ( ord_less_eq_set_int @ B @ C )
=> ( ! [X3: set_int,Y2: set_int] :
( ( ord_less_eq_set_int @ X3 @ Y2 )
=> ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_nat @ A @ ( F @ C ) ) ) ) ) ).
% order_less_le_subst1
thf(fact_279_order__le__less__subst2,axiom,
! [A: int,B: int,F: int > int,C: int] :
( ( ord_less_eq_int @ A @ B )
=> ( ( ord_less_int @ ( F @ B ) @ C )
=> ( ! [X3: int,Y2: int] :
( ( ord_less_eq_int @ X3 @ Y2 )
=> ( ord_less_eq_int @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_int @ ( F @ A ) @ C ) ) ) ) ).
% order_le_less_subst2
thf(fact_280_order__le__less__subst2,axiom,
! [A: int,B: int,F: int > nat,C: nat] :
( ( ord_less_eq_int @ A @ B )
=> ( ( ord_less_nat @ ( F @ B ) @ C )
=> ( ! [X3: int,Y2: int] :
( ( ord_less_eq_int @ X3 @ Y2 )
=> ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_nat @ ( F @ A ) @ C ) ) ) ) ).
% order_le_less_subst2
thf(fact_281_order__le__less__subst2,axiom,
! [A: nat,B: nat,F: nat > int,C: int] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ord_less_int @ ( F @ B ) @ C )
=> ( ! [X3: nat,Y2: nat] :
( ( ord_less_eq_nat @ X3 @ Y2 )
=> ( ord_less_eq_int @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_int @ ( F @ A ) @ C ) ) ) ) ).
% order_le_less_subst2
thf(fact_282_order__le__less__subst2,axiom,
! [A: nat,B: nat,F: nat > nat,C: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ord_less_nat @ ( F @ B ) @ C )
=> ( ! [X3: nat,Y2: nat] :
( ( ord_less_eq_nat @ X3 @ Y2 )
=> ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_nat @ ( F @ A ) @ C ) ) ) ) ).
% order_le_less_subst2
thf(fact_283_order__le__less__subst2,axiom,
! [A: int,B: int,F: int > set_int,C: set_int] :
( ( ord_less_eq_int @ A @ B )
=> ( ( ord_less_set_int @ ( F @ B ) @ C )
=> ( ! [X3: int,Y2: int] :
( ( ord_less_eq_int @ X3 @ Y2 )
=> ( ord_less_eq_set_int @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_set_int @ ( F @ A ) @ C ) ) ) ) ).
% order_le_less_subst2
thf(fact_284_order__le__less__subst2,axiom,
! [A: int,B: int,F: int > set_nat,C: set_nat] :
( ( ord_less_eq_int @ A @ B )
=> ( ( ord_less_set_nat @ ( F @ B ) @ C )
=> ( ! [X3: int,Y2: int] :
( ( ord_less_eq_int @ X3 @ Y2 )
=> ( ord_less_eq_set_nat @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_set_nat @ ( F @ A ) @ C ) ) ) ) ).
% order_le_less_subst2
thf(fact_285_order__le__less__subst2,axiom,
! [A: nat,B: nat,F: nat > set_int,C: set_int] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ord_less_set_int @ ( F @ B ) @ C )
=> ( ! [X3: nat,Y2: nat] :
( ( ord_less_eq_nat @ X3 @ Y2 )
=> ( ord_less_eq_set_int @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_set_int @ ( F @ A ) @ C ) ) ) ) ).
% order_le_less_subst2
thf(fact_286_order__le__less__subst2,axiom,
! [A: nat,B: nat,F: nat > set_nat,C: set_nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ord_less_set_nat @ ( F @ B ) @ C )
=> ( ! [X3: nat,Y2: nat] :
( ( ord_less_eq_nat @ X3 @ Y2 )
=> ( ord_less_eq_set_nat @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_set_nat @ ( F @ A ) @ C ) ) ) ) ).
% order_le_less_subst2
thf(fact_287_order__le__less__subst2,axiom,
! [A: set_int,B: set_int,F: set_int > int,C: int] :
( ( ord_less_eq_set_int @ A @ B )
=> ( ( ord_less_int @ ( F @ B ) @ C )
=> ( ! [X3: set_int,Y2: set_int] :
( ( ord_less_eq_set_int @ X3 @ Y2 )
=> ( ord_less_eq_int @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_int @ ( F @ A ) @ C ) ) ) ) ).
% order_le_less_subst2
thf(fact_288_order__le__less__subst2,axiom,
! [A: set_int,B: set_int,F: set_int > nat,C: nat] :
( ( ord_less_eq_set_int @ A @ B )
=> ( ( ord_less_nat @ ( F @ B ) @ C )
=> ( ! [X3: set_int,Y2: set_int] :
( ( ord_less_eq_set_int @ X3 @ Y2 )
=> ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_nat @ ( F @ A ) @ C ) ) ) ) ).
% order_le_less_subst2
thf(fact_289_order__le__less__subst1,axiom,
! [A: int,F: int > int,B: int,C: int] :
( ( ord_less_eq_int @ A @ ( F @ B ) )
=> ( ( ord_less_int @ B @ C )
=> ( ! [X3: int,Y2: int] :
( ( ord_less_int @ X3 @ Y2 )
=> ( ord_less_int @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_int @ A @ ( F @ C ) ) ) ) ) ).
% order_le_less_subst1
thf(fact_290_order__le__less__subst1,axiom,
! [A: int,F: nat > int,B: nat,C: nat] :
( ( ord_less_eq_int @ A @ ( F @ B ) )
=> ( ( ord_less_nat @ B @ C )
=> ( ! [X3: nat,Y2: nat] :
( ( ord_less_nat @ X3 @ Y2 )
=> ( ord_less_int @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_int @ A @ ( F @ C ) ) ) ) ) ).
% order_le_less_subst1
thf(fact_291_order__le__less__subst1,axiom,
! [A: nat,F: int > nat,B: int,C: int] :
( ( ord_less_eq_nat @ A @ ( F @ B ) )
=> ( ( ord_less_int @ B @ C )
=> ( ! [X3: int,Y2: int] :
( ( ord_less_int @ X3 @ Y2 )
=> ( ord_less_nat @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_nat @ A @ ( F @ C ) ) ) ) ) ).
% order_le_less_subst1
thf(fact_292_order__le__less__subst1,axiom,
! [A: nat,F: nat > nat,B: nat,C: nat] :
( ( ord_less_eq_nat @ A @ ( F @ B ) )
=> ( ( ord_less_nat @ B @ C )
=> ( ! [X3: nat,Y2: nat] :
( ( ord_less_nat @ X3 @ Y2 )
=> ( ord_less_nat @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_nat @ A @ ( F @ C ) ) ) ) ) ).
% order_le_less_subst1
thf(fact_293_order__le__less__subst1,axiom,
! [A: set_int,F: int > set_int,B: int,C: int] :
( ( ord_less_eq_set_int @ A @ ( F @ B ) )
=> ( ( ord_less_int @ B @ C )
=> ( ! [X3: int,Y2: int] :
( ( ord_less_int @ X3 @ Y2 )
=> ( ord_less_set_int @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_set_int @ A @ ( F @ C ) ) ) ) ) ).
% order_le_less_subst1
thf(fact_294_order__le__less__subst1,axiom,
! [A: set_int,F: nat > set_int,B: nat,C: nat] :
( ( ord_less_eq_set_int @ A @ ( F @ B ) )
=> ( ( ord_less_nat @ B @ C )
=> ( ! [X3: nat,Y2: nat] :
( ( ord_less_nat @ X3 @ Y2 )
=> ( ord_less_set_int @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_set_int @ A @ ( F @ C ) ) ) ) ) ).
% order_le_less_subst1
thf(fact_295_order__le__less__subst1,axiom,
! [A: set_nat,F: int > set_nat,B: int,C: int] :
( ( ord_less_eq_set_nat @ A @ ( F @ B ) )
=> ( ( ord_less_int @ B @ C )
=> ( ! [X3: int,Y2: int] :
( ( ord_less_int @ X3 @ Y2 )
=> ( ord_less_set_nat @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_set_nat @ A @ ( F @ C ) ) ) ) ) ).
% order_le_less_subst1
thf(fact_296_order__le__less__subst1,axiom,
! [A: set_nat,F: nat > set_nat,B: nat,C: nat] :
( ( ord_less_eq_set_nat @ A @ ( F @ B ) )
=> ( ( ord_less_nat @ B @ C )
=> ( ! [X3: nat,Y2: nat] :
( ( ord_less_nat @ X3 @ Y2 )
=> ( ord_less_set_nat @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_set_nat @ A @ ( F @ C ) ) ) ) ) ).
% order_le_less_subst1
thf(fact_297_order__less__le__trans,axiom,
! [X: int,Y: int,Z: int] :
( ( ord_less_int @ X @ Y )
=> ( ( ord_less_eq_int @ Y @ Z )
=> ( ord_less_int @ X @ Z ) ) ) ).
% order_less_le_trans
thf(fact_298_order__less__le__trans,axiom,
! [X: nat,Y: nat,Z: nat] :
( ( ord_less_nat @ X @ Y )
=> ( ( ord_less_eq_nat @ Y @ Z )
=> ( ord_less_nat @ X @ Z ) ) ) ).
% order_less_le_trans
thf(fact_299_order__less__le__trans,axiom,
! [X: set_int,Y: set_int,Z: set_int] :
( ( ord_less_set_int @ X @ Y )
=> ( ( ord_less_eq_set_int @ Y @ Z )
=> ( ord_less_set_int @ X @ Z ) ) ) ).
% order_less_le_trans
thf(fact_300_order__less__le__trans,axiom,
! [X: set_nat,Y: set_nat,Z: set_nat] :
( ( ord_less_set_nat @ X @ Y )
=> ( ( ord_less_eq_set_nat @ Y @ Z )
=> ( ord_less_set_nat @ X @ Z ) ) ) ).
% order_less_le_trans
thf(fact_301_order__le__less__trans,axiom,
! [X: int,Y: int,Z: int] :
( ( ord_less_eq_int @ X @ Y )
=> ( ( ord_less_int @ Y @ Z )
=> ( ord_less_int @ X @ Z ) ) ) ).
% order_le_less_trans
thf(fact_302_order__le__less__trans,axiom,
! [X: nat,Y: nat,Z: nat] :
( ( ord_less_eq_nat @ X @ Y )
=> ( ( ord_less_nat @ Y @ Z )
=> ( ord_less_nat @ X @ Z ) ) ) ).
% order_le_less_trans
thf(fact_303_order__le__less__trans,axiom,
! [X: set_int,Y: set_int,Z: set_int] :
( ( ord_less_eq_set_int @ X @ Y )
=> ( ( ord_less_set_int @ Y @ Z )
=> ( ord_less_set_int @ X @ Z ) ) ) ).
% order_le_less_trans
thf(fact_304_order__le__less__trans,axiom,
! [X: set_nat,Y: set_nat,Z: set_nat] :
( ( ord_less_eq_set_nat @ X @ Y )
=> ( ( ord_less_set_nat @ Y @ Z )
=> ( ord_less_set_nat @ X @ Z ) ) ) ).
% order_le_less_trans
thf(fact_305_order__neq__le__trans,axiom,
! [A: int,B: int] :
( ( A != B )
=> ( ( ord_less_eq_int @ A @ B )
=> ( ord_less_int @ A @ B ) ) ) ).
% order_neq_le_trans
thf(fact_306_order__neq__le__trans,axiom,
! [A: nat,B: nat] :
( ( A != B )
=> ( ( ord_less_eq_nat @ A @ B )
=> ( ord_less_nat @ A @ B ) ) ) ).
% order_neq_le_trans
thf(fact_307_order__neq__le__trans,axiom,
! [A: set_int,B: set_int] :
( ( A != B )
=> ( ( ord_less_eq_set_int @ A @ B )
=> ( ord_less_set_int @ A @ B ) ) ) ).
% order_neq_le_trans
thf(fact_308_order__neq__le__trans,axiom,
! [A: set_nat,B: set_nat] :
( ( A != B )
=> ( ( ord_less_eq_set_nat @ A @ B )
=> ( ord_less_set_nat @ A @ B ) ) ) ).
% order_neq_le_trans
thf(fact_309_order__le__neq__trans,axiom,
! [A: int,B: int] :
( ( ord_less_eq_int @ A @ B )
=> ( ( A != B )
=> ( ord_less_int @ A @ B ) ) ) ).
% order_le_neq_trans
thf(fact_310_order__le__neq__trans,axiom,
! [A: nat,B: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( A != B )
=> ( ord_less_nat @ A @ B ) ) ) ).
% order_le_neq_trans
thf(fact_311_order__le__neq__trans,axiom,
! [A: set_int,B: set_int] :
( ( ord_less_eq_set_int @ A @ B )
=> ( ( A != B )
=> ( ord_less_set_int @ A @ B ) ) ) ).
% order_le_neq_trans
thf(fact_312_order__le__neq__trans,axiom,
! [A: set_nat,B: set_nat] :
( ( ord_less_eq_set_nat @ A @ B )
=> ( ( A != B )
=> ( ord_less_set_nat @ A @ B ) ) ) ).
% order_le_neq_trans
thf(fact_313_order__less__imp__le,axiom,
! [X: int,Y: int] :
( ( ord_less_int @ X @ Y )
=> ( ord_less_eq_int @ X @ Y ) ) ).
% order_less_imp_le
thf(fact_314_order__less__imp__le,axiom,
! [X: nat,Y: nat] :
( ( ord_less_nat @ X @ Y )
=> ( ord_less_eq_nat @ X @ Y ) ) ).
% order_less_imp_le
thf(fact_315_order__less__imp__le,axiom,
! [X: set_int,Y: set_int] :
( ( ord_less_set_int @ X @ Y )
=> ( ord_less_eq_set_int @ X @ Y ) ) ).
% order_less_imp_le
thf(fact_316_order__less__imp__le,axiom,
! [X: set_nat,Y: set_nat] :
( ( ord_less_set_nat @ X @ Y )
=> ( ord_less_eq_set_nat @ X @ Y ) ) ).
% order_less_imp_le
thf(fact_317_linorder__not__less,axiom,
! [X: int,Y: int] :
( ( ~ ( ord_less_int @ X @ Y ) )
= ( ord_less_eq_int @ Y @ X ) ) ).
% linorder_not_less
thf(fact_318_linorder__not__less,axiom,
! [X: nat,Y: nat] :
( ( ~ ( ord_less_nat @ X @ Y ) )
= ( ord_less_eq_nat @ Y @ X ) ) ).
% linorder_not_less
thf(fact_319_linorder__not__le,axiom,
! [X: int,Y: int] :
( ( ~ ( ord_less_eq_int @ X @ Y ) )
= ( ord_less_int @ Y @ X ) ) ).
% linorder_not_le
thf(fact_320_linorder__not__le,axiom,
! [X: nat,Y: nat] :
( ( ~ ( ord_less_eq_nat @ X @ Y ) )
= ( ord_less_nat @ Y @ X ) ) ).
% linorder_not_le
thf(fact_321_order__less__le,axiom,
( ord_less_int
= ( ^ [X2: int,Y5: int] :
( ( ord_less_eq_int @ X2 @ Y5 )
& ( X2 != Y5 ) ) ) ) ).
% order_less_le
thf(fact_322_order__less__le,axiom,
( ord_less_nat
= ( ^ [X2: nat,Y5: nat] :
( ( ord_less_eq_nat @ X2 @ Y5 )
& ( X2 != Y5 ) ) ) ) ).
% order_less_le
thf(fact_323_order__less__le,axiom,
( ord_less_set_int
= ( ^ [X2: set_int,Y5: set_int] :
( ( ord_less_eq_set_int @ X2 @ Y5 )
& ( X2 != Y5 ) ) ) ) ).
% order_less_le
thf(fact_324_order__less__le,axiom,
( ord_less_set_nat
= ( ^ [X2: set_nat,Y5: set_nat] :
( ( ord_less_eq_set_nat @ X2 @ Y5 )
& ( X2 != Y5 ) ) ) ) ).
% order_less_le
thf(fact_325_order__le__less,axiom,
( ord_less_eq_int
= ( ^ [X2: int,Y5: int] :
( ( ord_less_int @ X2 @ Y5 )
| ( X2 = Y5 ) ) ) ) ).
% order_le_less
thf(fact_326_order__le__less,axiom,
( ord_less_eq_nat
= ( ^ [X2: nat,Y5: nat] :
( ( ord_less_nat @ X2 @ Y5 )
| ( X2 = Y5 ) ) ) ) ).
% order_le_less
thf(fact_327_order__le__less,axiom,
( ord_less_eq_set_int
= ( ^ [X2: set_int,Y5: set_int] :
( ( ord_less_set_int @ X2 @ Y5 )
| ( X2 = Y5 ) ) ) ) ).
% order_le_less
thf(fact_328_order__le__less,axiom,
( ord_less_eq_set_nat
= ( ^ [X2: set_nat,Y5: set_nat] :
( ( ord_less_set_nat @ X2 @ Y5 )
| ( X2 = Y5 ) ) ) ) ).
% order_le_less
thf(fact_329_dual__order_Ostrict__implies__order,axiom,
! [B: int,A: int] :
( ( ord_less_int @ B @ A )
=> ( ord_less_eq_int @ B @ A ) ) ).
% dual_order.strict_implies_order
thf(fact_330_dual__order_Ostrict__implies__order,axiom,
! [B: nat,A: nat] :
( ( ord_less_nat @ B @ A )
=> ( ord_less_eq_nat @ B @ A ) ) ).
% dual_order.strict_implies_order
thf(fact_331_dual__order_Ostrict__implies__order,axiom,
! [B: set_int,A: set_int] :
( ( ord_less_set_int @ B @ A )
=> ( ord_less_eq_set_int @ B @ A ) ) ).
% dual_order.strict_implies_order
thf(fact_332_dual__order_Ostrict__implies__order,axiom,
! [B: set_nat,A: set_nat] :
( ( ord_less_set_nat @ B @ A )
=> ( ord_less_eq_set_nat @ B @ A ) ) ).
% dual_order.strict_implies_order
thf(fact_333_order_Ostrict__implies__order,axiom,
! [A: int,B: int] :
( ( ord_less_int @ A @ B )
=> ( ord_less_eq_int @ A @ B ) ) ).
% order.strict_implies_order
thf(fact_334_order_Ostrict__implies__order,axiom,
! [A: nat,B: nat] :
( ( ord_less_nat @ A @ B )
=> ( ord_less_eq_nat @ A @ B ) ) ).
% order.strict_implies_order
thf(fact_335_order_Ostrict__implies__order,axiom,
! [A: set_int,B: set_int] :
( ( ord_less_set_int @ A @ B )
=> ( ord_less_eq_set_int @ A @ B ) ) ).
% order.strict_implies_order
thf(fact_336_order_Ostrict__implies__order,axiom,
! [A: set_nat,B: set_nat] :
( ( ord_less_set_nat @ A @ B )
=> ( ord_less_eq_set_nat @ A @ B ) ) ).
% order.strict_implies_order
thf(fact_337_dual__order_Ostrict__iff__not,axiom,
( ord_less_int
= ( ^ [B2: int,A3: int] :
( ( ord_less_eq_int @ B2 @ A3 )
& ~ ( ord_less_eq_int @ A3 @ B2 ) ) ) ) ).
% dual_order.strict_iff_not
thf(fact_338_dual__order_Ostrict__iff__not,axiom,
( ord_less_nat
= ( ^ [B2: nat,A3: nat] :
( ( ord_less_eq_nat @ B2 @ A3 )
& ~ ( ord_less_eq_nat @ A3 @ B2 ) ) ) ) ).
% dual_order.strict_iff_not
thf(fact_339_dual__order_Ostrict__iff__not,axiom,
( ord_less_set_int
= ( ^ [B2: set_int,A3: set_int] :
( ( ord_less_eq_set_int @ B2 @ A3 )
& ~ ( ord_less_eq_set_int @ A3 @ B2 ) ) ) ) ).
% dual_order.strict_iff_not
thf(fact_340_dual__order_Ostrict__iff__not,axiom,
( ord_less_set_nat
= ( ^ [B2: set_nat,A3: set_nat] :
( ( ord_less_eq_set_nat @ B2 @ A3 )
& ~ ( ord_less_eq_set_nat @ A3 @ B2 ) ) ) ) ).
% dual_order.strict_iff_not
thf(fact_341_dual__order_Ostrict__trans2,axiom,
! [B: int,A: int,C: int] :
( ( ord_less_int @ B @ A )
=> ( ( ord_less_eq_int @ C @ B )
=> ( ord_less_int @ C @ A ) ) ) ).
% dual_order.strict_trans2
thf(fact_342_dual__order_Ostrict__trans2,axiom,
! [B: nat,A: nat,C: nat] :
( ( ord_less_nat @ B @ A )
=> ( ( ord_less_eq_nat @ C @ B )
=> ( ord_less_nat @ C @ A ) ) ) ).
% dual_order.strict_trans2
thf(fact_343_dual__order_Ostrict__trans2,axiom,
! [B: set_int,A: set_int,C: set_int] :
( ( ord_less_set_int @ B @ A )
=> ( ( ord_less_eq_set_int @ C @ B )
=> ( ord_less_set_int @ C @ A ) ) ) ).
% dual_order.strict_trans2
thf(fact_344_dual__order_Ostrict__trans2,axiom,
! [B: set_nat,A: set_nat,C: set_nat] :
( ( ord_less_set_nat @ B @ A )
=> ( ( ord_less_eq_set_nat @ C @ B )
=> ( ord_less_set_nat @ C @ A ) ) ) ).
% dual_order.strict_trans2
thf(fact_345_dual__order_Ostrict__trans1,axiom,
! [B: int,A: int,C: int] :
( ( ord_less_eq_int @ B @ A )
=> ( ( ord_less_int @ C @ B )
=> ( ord_less_int @ C @ A ) ) ) ).
% dual_order.strict_trans1
thf(fact_346_dual__order_Ostrict__trans1,axiom,
! [B: nat,A: nat,C: nat] :
( ( ord_less_eq_nat @ B @ A )
=> ( ( ord_less_nat @ C @ B )
=> ( ord_less_nat @ C @ A ) ) ) ).
% dual_order.strict_trans1
thf(fact_347_dual__order_Ostrict__trans1,axiom,
! [B: set_int,A: set_int,C: set_int] :
( ( ord_less_eq_set_int @ B @ A )
=> ( ( ord_less_set_int @ C @ B )
=> ( ord_less_set_int @ C @ A ) ) ) ).
% dual_order.strict_trans1
thf(fact_348_dual__order_Ostrict__trans1,axiom,
! [B: set_nat,A: set_nat,C: set_nat] :
( ( ord_less_eq_set_nat @ B @ A )
=> ( ( ord_less_set_nat @ C @ B )
=> ( ord_less_set_nat @ C @ A ) ) ) ).
% dual_order.strict_trans1
thf(fact_349_dual__order_Ostrict__iff__order,axiom,
( ord_less_int
= ( ^ [B2: int,A3: int] :
( ( ord_less_eq_int @ B2 @ A3 )
& ( A3 != B2 ) ) ) ) ).
% dual_order.strict_iff_order
thf(fact_350_dual__order_Ostrict__iff__order,axiom,
( ord_less_nat
= ( ^ [B2: nat,A3: nat] :
( ( ord_less_eq_nat @ B2 @ A3 )
& ( A3 != B2 ) ) ) ) ).
% dual_order.strict_iff_order
thf(fact_351_dual__order_Ostrict__iff__order,axiom,
( ord_less_set_int
= ( ^ [B2: set_int,A3: set_int] :
( ( ord_less_eq_set_int @ B2 @ A3 )
& ( A3 != B2 ) ) ) ) ).
% dual_order.strict_iff_order
thf(fact_352_dual__order_Ostrict__iff__order,axiom,
( ord_less_set_nat
= ( ^ [B2: set_nat,A3: set_nat] :
( ( ord_less_eq_set_nat @ B2 @ A3 )
& ( A3 != B2 ) ) ) ) ).
% dual_order.strict_iff_order
thf(fact_353_dual__order_Oorder__iff__strict,axiom,
( ord_less_eq_int
= ( ^ [B2: int,A3: int] :
( ( ord_less_int @ B2 @ A3 )
| ( A3 = B2 ) ) ) ) ).
% dual_order.order_iff_strict
thf(fact_354_dual__order_Oorder__iff__strict,axiom,
( ord_less_eq_nat
= ( ^ [B2: nat,A3: nat] :
( ( ord_less_nat @ B2 @ A3 )
| ( A3 = B2 ) ) ) ) ).
% dual_order.order_iff_strict
thf(fact_355_dual__order_Oorder__iff__strict,axiom,
( ord_less_eq_set_int
= ( ^ [B2: set_int,A3: set_int] :
( ( ord_less_set_int @ B2 @ A3 )
| ( A3 = B2 ) ) ) ) ).
% dual_order.order_iff_strict
thf(fact_356_dual__order_Oorder__iff__strict,axiom,
( ord_less_eq_set_nat
= ( ^ [B2: set_nat,A3: set_nat] :
( ( ord_less_set_nat @ B2 @ A3 )
| ( A3 = B2 ) ) ) ) ).
% dual_order.order_iff_strict
thf(fact_357_order_Ostrict__iff__not,axiom,
( ord_less_int
= ( ^ [A3: int,B2: int] :
( ( ord_less_eq_int @ A3 @ B2 )
& ~ ( ord_less_eq_int @ B2 @ A3 ) ) ) ) ).
% order.strict_iff_not
thf(fact_358_order_Ostrict__iff__not,axiom,
( ord_less_nat
= ( ^ [A3: nat,B2: nat] :
( ( ord_less_eq_nat @ A3 @ B2 )
& ~ ( ord_less_eq_nat @ B2 @ A3 ) ) ) ) ).
% order.strict_iff_not
thf(fact_359_order_Ostrict__iff__not,axiom,
( ord_less_set_int
= ( ^ [A3: set_int,B2: set_int] :
( ( ord_less_eq_set_int @ A3 @ B2 )
& ~ ( ord_less_eq_set_int @ B2 @ A3 ) ) ) ) ).
% order.strict_iff_not
thf(fact_360_order_Ostrict__iff__not,axiom,
( ord_less_set_nat
= ( ^ [A3: set_nat,B2: set_nat] :
( ( ord_less_eq_set_nat @ A3 @ B2 )
& ~ ( ord_less_eq_set_nat @ B2 @ A3 ) ) ) ) ).
% order.strict_iff_not
thf(fact_361_order_Ostrict__trans2,axiom,
! [A: int,B: int,C: int] :
( ( ord_less_int @ A @ B )
=> ( ( ord_less_eq_int @ B @ C )
=> ( ord_less_int @ A @ C ) ) ) ).
% order.strict_trans2
thf(fact_362_order_Ostrict__trans2,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_nat @ A @ B )
=> ( ( ord_less_eq_nat @ B @ C )
=> ( ord_less_nat @ A @ C ) ) ) ).
% order.strict_trans2
thf(fact_363_order_Ostrict__trans2,axiom,
! [A: set_int,B: set_int,C: set_int] :
( ( ord_less_set_int @ A @ B )
=> ( ( ord_less_eq_set_int @ B @ C )
=> ( ord_less_set_int @ A @ C ) ) ) ).
% order.strict_trans2
thf(fact_364_order_Ostrict__trans2,axiom,
! [A: set_nat,B: set_nat,C: set_nat] :
( ( ord_less_set_nat @ A @ B )
=> ( ( ord_less_eq_set_nat @ B @ C )
=> ( ord_less_set_nat @ A @ C ) ) ) ).
% order.strict_trans2
thf(fact_365_order_Ostrict__trans1,axiom,
! [A: int,B: int,C: int] :
( ( ord_less_eq_int @ A @ B )
=> ( ( ord_less_int @ B @ C )
=> ( ord_less_int @ A @ C ) ) ) ).
% order.strict_trans1
thf(fact_366_order_Ostrict__trans1,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ord_less_nat @ B @ C )
=> ( ord_less_nat @ A @ C ) ) ) ).
% order.strict_trans1
thf(fact_367_order_Ostrict__trans1,axiom,
! [A: set_int,B: set_int,C: set_int] :
( ( ord_less_eq_set_int @ A @ B )
=> ( ( ord_less_set_int @ B @ C )
=> ( ord_less_set_int @ A @ C ) ) ) ).
% order.strict_trans1
thf(fact_368_order_Ostrict__trans1,axiom,
! [A: set_nat,B: set_nat,C: set_nat] :
( ( ord_less_eq_set_nat @ A @ B )
=> ( ( ord_less_set_nat @ B @ C )
=> ( ord_less_set_nat @ A @ C ) ) ) ).
% order.strict_trans1
thf(fact_369_order_Ostrict__iff__order,axiom,
( ord_less_int
= ( ^ [A3: int,B2: int] :
( ( ord_less_eq_int @ A3 @ B2 )
& ( A3 != B2 ) ) ) ) ).
% order.strict_iff_order
thf(fact_370_order_Ostrict__iff__order,axiom,
( ord_less_nat
= ( ^ [A3: nat,B2: nat] :
( ( ord_less_eq_nat @ A3 @ B2 )
& ( A3 != B2 ) ) ) ) ).
% order.strict_iff_order
thf(fact_371_order_Ostrict__iff__order,axiom,
( ord_less_set_int
= ( ^ [A3: set_int,B2: set_int] :
( ( ord_less_eq_set_int @ A3 @ B2 )
& ( A3 != B2 ) ) ) ) ).
% order.strict_iff_order
thf(fact_372_order_Ostrict__iff__order,axiom,
( ord_less_set_nat
= ( ^ [A3: set_nat,B2: set_nat] :
( ( ord_less_eq_set_nat @ A3 @ B2 )
& ( A3 != B2 ) ) ) ) ).
% order.strict_iff_order
thf(fact_373_order_Oorder__iff__strict,axiom,
( ord_less_eq_int
= ( ^ [A3: int,B2: int] :
( ( ord_less_int @ A3 @ B2 )
| ( A3 = B2 ) ) ) ) ).
% order.order_iff_strict
thf(fact_374_order_Oorder__iff__strict,axiom,
( ord_less_eq_nat
= ( ^ [A3: nat,B2: nat] :
( ( ord_less_nat @ A3 @ B2 )
| ( A3 = B2 ) ) ) ) ).
% order.order_iff_strict
thf(fact_375_order_Oorder__iff__strict,axiom,
( ord_less_eq_set_int
= ( ^ [A3: set_int,B2: set_int] :
( ( ord_less_set_int @ A3 @ B2 )
| ( A3 = B2 ) ) ) ) ).
% order.order_iff_strict
thf(fact_376_order_Oorder__iff__strict,axiom,
( ord_less_eq_set_nat
= ( ^ [A3: set_nat,B2: set_nat] :
( ( ord_less_set_nat @ A3 @ B2 )
| ( A3 = B2 ) ) ) ) ).
% order.order_iff_strict
thf(fact_377_not__le__imp__less,axiom,
! [Y: int,X: int] :
( ~ ( ord_less_eq_int @ Y @ X )
=> ( ord_less_int @ X @ Y ) ) ).
% not_le_imp_less
thf(fact_378_not__le__imp__less,axiom,
! [Y: nat,X: nat] :
( ~ ( ord_less_eq_nat @ Y @ X )
=> ( ord_less_nat @ X @ Y ) ) ).
% not_le_imp_less
thf(fact_379_less__le__not__le,axiom,
( ord_less_int
= ( ^ [X2: int,Y5: int] :
( ( ord_less_eq_int @ X2 @ Y5 )
& ~ ( ord_less_eq_int @ Y5 @ X2 ) ) ) ) ).
% less_le_not_le
thf(fact_380_less__le__not__le,axiom,
( ord_less_nat
= ( ^ [X2: nat,Y5: nat] :
( ( ord_less_eq_nat @ X2 @ Y5 )
& ~ ( ord_less_eq_nat @ Y5 @ X2 ) ) ) ) ).
% less_le_not_le
thf(fact_381_less__le__not__le,axiom,
( ord_less_set_int
= ( ^ [X2: set_int,Y5: set_int] :
( ( ord_less_eq_set_int @ X2 @ Y5 )
& ~ ( ord_less_eq_set_int @ Y5 @ X2 ) ) ) ) ).
% less_le_not_le
thf(fact_382_less__le__not__le,axiom,
( ord_less_set_nat
= ( ^ [X2: set_nat,Y5: set_nat] :
( ( ord_less_eq_set_nat @ X2 @ Y5 )
& ~ ( ord_less_eq_set_nat @ Y5 @ X2 ) ) ) ) ).
% less_le_not_le
thf(fact_383_antisym__conv2,axiom,
! [X: int,Y: int] :
( ( ord_less_eq_int @ X @ Y )
=> ( ( ~ ( ord_less_int @ X @ Y ) )
= ( X = Y ) ) ) ).
% antisym_conv2
thf(fact_384_antisym__conv2,axiom,
! [X: nat,Y: nat] :
( ( ord_less_eq_nat @ X @ Y )
=> ( ( ~ ( ord_less_nat @ X @ Y ) )
= ( X = Y ) ) ) ).
% antisym_conv2
thf(fact_385_antisym__conv2,axiom,
! [X: set_int,Y: set_int] :
( ( ord_less_eq_set_int @ X @ Y )
=> ( ( ~ ( ord_less_set_int @ X @ Y ) )
= ( X = Y ) ) ) ).
% antisym_conv2
thf(fact_386_antisym__conv2,axiom,
! [X: set_nat,Y: set_nat] :
( ( ord_less_eq_set_nat @ X @ Y )
=> ( ( ~ ( ord_less_set_nat @ X @ Y ) )
= ( X = Y ) ) ) ).
% antisym_conv2
thf(fact_387_antisym__conv1,axiom,
! [X: int,Y: int] :
( ~ ( ord_less_int @ X @ Y )
=> ( ( ord_less_eq_int @ X @ Y )
= ( X = Y ) ) ) ).
% antisym_conv1
thf(fact_388_antisym__conv1,axiom,
! [X: nat,Y: nat] :
( ~ ( ord_less_nat @ X @ Y )
=> ( ( ord_less_eq_nat @ X @ Y )
= ( X = Y ) ) ) ).
% antisym_conv1
thf(fact_389_antisym__conv1,axiom,
! [X: set_int,Y: set_int] :
( ~ ( ord_less_set_int @ X @ Y )
=> ( ( ord_less_eq_set_int @ X @ Y )
= ( X = Y ) ) ) ).
% antisym_conv1
thf(fact_390_antisym__conv1,axiom,
! [X: set_nat,Y: set_nat] :
( ~ ( ord_less_set_nat @ X @ Y )
=> ( ( ord_less_eq_set_nat @ X @ Y )
= ( X = Y ) ) ) ).
% antisym_conv1
thf(fact_391_nless__le,axiom,
! [A: int,B: int] :
( ( ~ ( ord_less_int @ A @ B ) )
= ( ~ ( ord_less_eq_int @ A @ B )
| ( A = B ) ) ) ).
% nless_le
thf(fact_392_nless__le,axiom,
! [A: nat,B: nat] :
( ( ~ ( ord_less_nat @ A @ B ) )
= ( ~ ( ord_less_eq_nat @ A @ B )
| ( A = B ) ) ) ).
% nless_le
thf(fact_393_nless__le,axiom,
! [A: set_int,B: set_int] :
( ( ~ ( ord_less_set_int @ A @ B ) )
= ( ~ ( ord_less_eq_set_int @ A @ B )
| ( A = B ) ) ) ).
% nless_le
thf(fact_394_nless__le,axiom,
! [A: set_nat,B: set_nat] :
( ( ~ ( ord_less_set_nat @ A @ B ) )
= ( ~ ( ord_less_eq_set_nat @ A @ B )
| ( A = B ) ) ) ).
% nless_le
thf(fact_395_leI,axiom,
! [X: int,Y: int] :
( ~ ( ord_less_int @ X @ Y )
=> ( ord_less_eq_int @ Y @ X ) ) ).
% leI
thf(fact_396_leI,axiom,
! [X: nat,Y: nat] :
( ~ ( ord_less_nat @ X @ Y )
=> ( ord_less_eq_nat @ Y @ X ) ) ).
% leI
thf(fact_397_leD,axiom,
! [Y: int,X: int] :
( ( ord_less_eq_int @ Y @ X )
=> ~ ( ord_less_int @ X @ Y ) ) ).
% leD
thf(fact_398_leD,axiom,
! [Y: nat,X: nat] :
( ( ord_less_eq_nat @ Y @ X )
=> ~ ( ord_less_nat @ X @ Y ) ) ).
% leD
thf(fact_399_leD,axiom,
! [Y: set_int,X: set_int] :
( ( ord_less_eq_set_int @ Y @ X )
=> ~ ( ord_less_set_int @ X @ Y ) ) ).
% leD
thf(fact_400_leD,axiom,
! [Y: set_nat,X: set_nat] :
( ( ord_less_eq_set_nat @ Y @ X )
=> ~ ( ord_less_set_nat @ X @ Y ) ) ).
% leD
thf(fact_401_of__nat__mono,axiom,
! [I: nat,J: nat] :
( ( ord_less_eq_nat @ I @ J )
=> ( ord_less_eq_int @ ( semiri1314217659103216013at_int @ I ) @ ( semiri1314217659103216013at_int @ J ) ) ) ).
% of_nat_mono
thf(fact_402_of__nat__mono,axiom,
! [I: nat,J: nat] :
( ( ord_less_eq_nat @ I @ J )
=> ( ord_less_eq_nat @ ( semiri1316708129612266289at_nat @ I ) @ ( semiri1316708129612266289at_nat @ J ) ) ) ).
% of_nat_mono
thf(fact_403_of__nat__less__imp__less,axiom,
! [M: nat,N: nat] :
( ( ord_less_int @ ( semiri1314217659103216013at_int @ M ) @ ( semiri1314217659103216013at_int @ N ) )
=> ( ord_less_nat @ M @ N ) ) ).
% of_nat_less_imp_less
thf(fact_404_of__nat__less__imp__less,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ ( semiri1316708129612266289at_nat @ M ) @ ( semiri1316708129612266289at_nat @ N ) )
=> ( ord_less_nat @ M @ N ) ) ).
% of_nat_less_imp_less
thf(fact_405_less__imp__of__nat__less,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ M @ N )
=> ( ord_less_int @ ( semiri1314217659103216013at_int @ M ) @ ( semiri1314217659103216013at_int @ N ) ) ) ).
% less_imp_of_nat_less
thf(fact_406_less__imp__of__nat__less,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ M @ N )
=> ( ord_less_nat @ ( semiri1316708129612266289at_nat @ M ) @ ( semiri1316708129612266289at_nat @ N ) ) ) ).
% less_imp_of_nat_less
thf(fact_407_atLeastLessThan__subset__iff,axiom,
! [A: nat,B: nat,C: nat,D: nat] :
( ( ord_less_eq_set_nat @ ( set_or4665077453230672383an_nat @ A @ B ) @ ( set_or4665077453230672383an_nat @ C @ D ) )
=> ( ( ord_less_eq_nat @ B @ A )
| ( ( ord_less_eq_nat @ C @ A )
& ( ord_less_eq_nat @ B @ D ) ) ) ) ).
% atLeastLessThan_subset_iff
thf(fact_408_atLeastLessThan__subset__iff,axiom,
! [A: int,B: int,C: int,D: int] :
( ( ord_less_eq_set_int @ ( set_or4662586982721622107an_int @ A @ B ) @ ( set_or4662586982721622107an_int @ C @ D ) )
=> ( ( ord_less_eq_int @ B @ A )
| ( ( ord_less_eq_int @ C @ A )
& ( ord_less_eq_int @ B @ D ) ) ) ) ).
% atLeastLessThan_subset_iff
thf(fact_409_atLeastLessThan__eq__iff,axiom,
! [A: nat,B: nat,C: nat,D: nat] :
( ( ord_less_nat @ A @ B )
=> ( ( ord_less_nat @ C @ D )
=> ( ( ( set_or4665077453230672383an_nat @ A @ B )
= ( set_or4665077453230672383an_nat @ C @ D ) )
= ( ( A = C )
& ( B = D ) ) ) ) ) ).
% atLeastLessThan_eq_iff
thf(fact_410_atLeastLessThan__eq__iff,axiom,
! [A: int,B: int,C: int,D: int] :
( ( ord_less_int @ A @ B )
=> ( ( ord_less_int @ C @ D )
=> ( ( ( set_or4662586982721622107an_int @ A @ B )
= ( set_or4662586982721622107an_int @ C @ D ) )
= ( ( A = C )
& ( B = D ) ) ) ) ) ).
% atLeastLessThan_eq_iff
thf(fact_411_append__eq__append__conv,axiom,
! [Xs: list_P419547167332244514st_int,Ys: list_P419547167332244514st_int,Us: list_P419547167332244514st_int,Vs: list_P419547167332244514st_int] :
( ( ( ( size_s229347708945616566st_int @ Xs )
= ( size_s229347708945616566st_int @ Ys ) )
| ( ( size_s229347708945616566st_int @ Us )
= ( size_s229347708945616566st_int @ Vs ) ) )
=> ( ( ( append5028292981072010935st_int @ Xs @ Us )
= ( append5028292981072010935st_int @ Ys @ Vs ) )
= ( ( Xs = Ys )
& ( Us = Vs ) ) ) ) ).
% append_eq_append_conv
thf(fact_412_append__eq__append__conv,axiom,
! [Xs: list_instr,Ys: list_instr,Us: list_instr,Vs: list_instr] :
( ( ( ( size_size_list_instr @ Xs )
= ( size_size_list_instr @ Ys ) )
| ( ( size_size_list_instr @ Us )
= ( size_size_list_instr @ Vs ) ) )
=> ( ( ( append_instr @ Xs @ Us )
= ( append_instr @ Ys @ Vs ) )
= ( ( Xs = Ys )
& ( Us = Vs ) ) ) ) ).
% append_eq_append_conv
thf(fact_413_append__eq__append__conv,axiom,
! [Xs: list_list_char,Ys: list_list_char,Us: list_list_char,Vs: list_list_char] :
( ( ( ( size_s356637359517785349t_char @ Xs )
= ( size_s356637359517785349t_char @ Ys ) )
| ( ( size_s356637359517785349t_char @ Us )
= ( size_s356637359517785349t_char @ Vs ) ) )
=> ( ( ( append_list_char @ Xs @ Us )
= ( append_list_char @ Ys @ Vs ) )
= ( ( Xs = Ys )
& ( Us = Vs ) ) ) ) ).
% append_eq_append_conv
thf(fact_414_append__eq__append__conv,axiom,
! [Xs: list_int,Ys: list_int,Us: list_int,Vs: list_int] :
( ( ( ( size_size_list_int @ Xs )
= ( size_size_list_int @ Ys ) )
| ( ( size_size_list_int @ Us )
= ( size_size_list_int @ Vs ) ) )
=> ( ( ( append_int @ Xs @ Us )
= ( append_int @ Ys @ Vs ) )
= ( ( Xs = Ys )
& ( Us = Vs ) ) ) ) ).
% append_eq_append_conv
thf(fact_415_nth__append,axiom,
! [N: nat,Xs: list_P419547167332244514st_int,Ys: list_P419547167332244514st_int] :
( ( ( ord_less_nat @ N @ ( size_s229347708945616566st_int @ Xs ) )
=> ( ( nth_Pr8327173284404383691st_int @ ( append5028292981072010935st_int @ Xs @ Ys ) @ N )
= ( nth_Pr8327173284404383691st_int @ Xs @ N ) ) )
& ( ~ ( ord_less_nat @ N @ ( size_s229347708945616566st_int @ Xs ) )
=> ( ( nth_Pr8327173284404383691st_int @ ( append5028292981072010935st_int @ Xs @ Ys ) @ N )
= ( nth_Pr8327173284404383691st_int @ Ys @ ( minus_minus_nat @ N @ ( size_s229347708945616566st_int @ Xs ) ) ) ) ) ) ).
% nth_append
thf(fact_416_nth__append,axiom,
! [N: nat,Xs: list_instr,Ys: list_instr] :
( ( ( ord_less_nat @ N @ ( size_size_list_instr @ Xs ) )
=> ( ( nth_instr @ ( append_instr @ Xs @ Ys ) @ N )
= ( nth_instr @ Xs @ N ) ) )
& ( ~ ( ord_less_nat @ N @ ( size_size_list_instr @ Xs ) )
=> ( ( nth_instr @ ( append_instr @ Xs @ Ys ) @ N )
= ( nth_instr @ Ys @ ( minus_minus_nat @ N @ ( size_size_list_instr @ Xs ) ) ) ) ) ) ).
% nth_append
thf(fact_417_nth__append,axiom,
! [N: nat,Xs: list_list_char,Ys: list_list_char] :
( ( ( ord_less_nat @ N @ ( size_s356637359517785349t_char @ Xs ) )
=> ( ( nth_list_char @ ( append_list_char @ Xs @ Ys ) @ N )
= ( nth_list_char @ Xs @ N ) ) )
& ( ~ ( ord_less_nat @ N @ ( size_s356637359517785349t_char @ Xs ) )
=> ( ( nth_list_char @ ( append_list_char @ Xs @ Ys ) @ N )
= ( nth_list_char @ Ys @ ( minus_minus_nat @ N @ ( size_s356637359517785349t_char @ Xs ) ) ) ) ) ) ).
% nth_append
thf(fact_418_nth__append,axiom,
! [N: nat,Xs: list_int,Ys: list_int] :
( ( ( ord_less_nat @ N @ ( size_size_list_int @ Xs ) )
=> ( ( nth_int @ ( append_int @ Xs @ Ys ) @ N )
= ( nth_int @ Xs @ N ) ) )
& ( ~ ( ord_less_nat @ N @ ( size_size_list_int @ Xs ) )
=> ( ( nth_int @ ( append_int @ Xs @ Ys ) @ N )
= ( nth_int @ Ys @ ( minus_minus_nat @ N @ ( size_size_list_int @ Xs ) ) ) ) ) ) ).
% nth_append
thf(fact_419_nat__leq__as__int,axiom,
( ord_less_eq_nat
= ( ^ [A3: nat,B2: nat] : ( ord_less_eq_int @ ( semiri1314217659103216013at_int @ A3 ) @ ( semiri1314217659103216013at_int @ B2 ) ) ) ) ).
% nat_leq_as_int
thf(fact_420_nat__less__as__int,axiom,
( ord_less_nat
= ( ^ [A3: nat,B2: nat] : ( ord_less_int @ ( semiri1314217659103216013at_int @ A3 ) @ ( semiri1314217659103216013at_int @ B2 ) ) ) ) ).
% nat_less_as_int
thf(fact_421_nat__int__comparison_I3_J,axiom,
( ord_less_eq_nat
= ( ^ [A3: nat,B2: nat] : ( ord_less_eq_int @ ( semiri1314217659103216013at_int @ A3 ) @ ( semiri1314217659103216013at_int @ B2 ) ) ) ) ).
% nat_int_comparison(3)
thf(fact_422_nat__int__comparison_I2_J,axiom,
( ord_less_nat
= ( ^ [A3: nat,B2: nat] : ( ord_less_int @ ( semiri1314217659103216013at_int @ A3 ) @ ( semiri1314217659103216013at_int @ B2 ) ) ) ) ).
% nat_int_comparison(2)
thf(fact_423_list__eq__iff__nth__eq,axiom,
( ( ^ [Y4: list_P419547167332244514st_int,Z2: list_P419547167332244514st_int] : ( Y4 = Z2 ) )
= ( ^ [Xs2: list_P419547167332244514st_int,Ys2: list_P419547167332244514st_int] :
( ( ( size_s229347708945616566st_int @ Xs2 )
= ( size_s229347708945616566st_int @ Ys2 ) )
& ! [I3: nat] :
( ( ord_less_nat @ I3 @ ( size_s229347708945616566st_int @ Xs2 ) )
=> ( ( nth_Pr8327173284404383691st_int @ Xs2 @ I3 )
= ( nth_Pr8327173284404383691st_int @ Ys2 @ I3 ) ) ) ) ) ) ).
% list_eq_iff_nth_eq
thf(fact_424_list__eq__iff__nth__eq,axiom,
( ( ^ [Y4: list_instr,Z2: list_instr] : ( Y4 = Z2 ) )
= ( ^ [Xs2: list_instr,Ys2: list_instr] :
( ( ( size_size_list_instr @ Xs2 )
= ( size_size_list_instr @ Ys2 ) )
& ! [I3: nat] :
( ( ord_less_nat @ I3 @ ( size_size_list_instr @ Xs2 ) )
=> ( ( nth_instr @ Xs2 @ I3 )
= ( nth_instr @ Ys2 @ I3 ) ) ) ) ) ) ).
% list_eq_iff_nth_eq
thf(fact_425_list__eq__iff__nth__eq,axiom,
( ( ^ [Y4: list_list_char,Z2: list_list_char] : ( Y4 = Z2 ) )
= ( ^ [Xs2: list_list_char,Ys2: list_list_char] :
( ( ( size_s356637359517785349t_char @ Xs2 )
= ( size_s356637359517785349t_char @ Ys2 ) )
& ! [I3: nat] :
( ( ord_less_nat @ I3 @ ( size_s356637359517785349t_char @ Xs2 ) )
=> ( ( nth_list_char @ Xs2 @ I3 )
= ( nth_list_char @ Ys2 @ I3 ) ) ) ) ) ) ).
% list_eq_iff_nth_eq
thf(fact_426_list__eq__iff__nth__eq,axiom,
( ( ^ [Y4: list_int,Z2: list_int] : ( Y4 = Z2 ) )
= ( ^ [Xs2: list_int,Ys2: list_int] :
( ( ( size_size_list_int @ Xs2 )
= ( size_size_list_int @ Ys2 ) )
& ! [I3: nat] :
( ( ord_less_nat @ I3 @ ( size_size_list_int @ Xs2 ) )
=> ( ( nth_int @ Xs2 @ I3 )
= ( nth_int @ Ys2 @ I3 ) ) ) ) ) ) ).
% list_eq_iff_nth_eq
thf(fact_427_Skolem__list__nth,axiom,
! [K: nat,P: nat > produc6425607678544837394st_int > $o] :
( ( ! [I3: nat] :
( ( ord_less_nat @ I3 @ K )
=> ? [X6: produc6425607678544837394st_int] : ( P @ I3 @ X6 ) ) )
= ( ? [Xs2: list_P419547167332244514st_int] :
( ( ( size_s229347708945616566st_int @ Xs2 )
= K )
& ! [I3: nat] :
( ( ord_less_nat @ I3 @ K )
=> ( P @ I3 @ ( nth_Pr8327173284404383691st_int @ Xs2 @ I3 ) ) ) ) ) ) ).
% Skolem_list_nth
thf(fact_428_Skolem__list__nth,axiom,
! [K: nat,P: nat > instr > $o] :
( ( ! [I3: nat] :
( ( ord_less_nat @ I3 @ K )
=> ? [X6: instr] : ( P @ I3 @ X6 ) ) )
= ( ? [Xs2: list_instr] :
( ( ( size_size_list_instr @ Xs2 )
= K )
& ! [I3: nat] :
( ( ord_less_nat @ I3 @ K )
=> ( P @ I3 @ ( nth_instr @ Xs2 @ I3 ) ) ) ) ) ) ).
% Skolem_list_nth
thf(fact_429_Skolem__list__nth,axiom,
! [K: nat,P: nat > list_char > $o] :
( ( ! [I3: nat] :
( ( ord_less_nat @ I3 @ K )
=> ? [X6: list_char] : ( P @ I3 @ X6 ) ) )
= ( ? [Xs2: list_list_char] :
( ( ( size_s356637359517785349t_char @ Xs2 )
= K )
& ! [I3: nat] :
( ( ord_less_nat @ I3 @ K )
=> ( P @ I3 @ ( nth_list_char @ Xs2 @ I3 ) ) ) ) ) ) ).
% Skolem_list_nth
thf(fact_430_Skolem__list__nth,axiom,
! [K: nat,P: nat > int > $o] :
( ( ! [I3: nat] :
( ( ord_less_nat @ I3 @ K )
=> ? [X6: int] : ( P @ I3 @ X6 ) ) )
= ( ? [Xs2: list_int] :
( ( ( size_size_list_int @ Xs2 )
= K )
& ! [I3: nat] :
( ( ord_less_nat @ I3 @ K )
=> ( P @ I3 @ ( nth_int @ Xs2 @ I3 ) ) ) ) ) ) ).
% Skolem_list_nth
thf(fact_431_nth__equalityI,axiom,
! [Xs: list_P419547167332244514st_int,Ys: list_P419547167332244514st_int] :
( ( ( size_s229347708945616566st_int @ Xs )
= ( size_s229347708945616566st_int @ Ys ) )
=> ( ! [I2: nat] :
( ( ord_less_nat @ I2 @ ( size_s229347708945616566st_int @ Xs ) )
=> ( ( nth_Pr8327173284404383691st_int @ Xs @ I2 )
= ( nth_Pr8327173284404383691st_int @ Ys @ I2 ) ) )
=> ( Xs = Ys ) ) ) ).
% nth_equalityI
thf(fact_432_nth__equalityI,axiom,
! [Xs: list_instr,Ys: list_instr] :
( ( ( size_size_list_instr @ Xs )
= ( size_size_list_instr @ Ys ) )
=> ( ! [I2: nat] :
( ( ord_less_nat @ I2 @ ( size_size_list_instr @ Xs ) )
=> ( ( nth_instr @ Xs @ I2 )
= ( nth_instr @ Ys @ I2 ) ) )
=> ( Xs = Ys ) ) ) ).
% nth_equalityI
thf(fact_433_nth__equalityI,axiom,
! [Xs: list_list_char,Ys: list_list_char] :
( ( ( size_s356637359517785349t_char @ Xs )
= ( size_s356637359517785349t_char @ Ys ) )
=> ( ! [I2: nat] :
( ( ord_less_nat @ I2 @ ( size_s356637359517785349t_char @ Xs ) )
=> ( ( nth_list_char @ Xs @ I2 )
= ( nth_list_char @ Ys @ I2 ) ) )
=> ( Xs = Ys ) ) ) ).
% nth_equalityI
thf(fact_434_nth__equalityI,axiom,
! [Xs: list_int,Ys: list_int] :
( ( ( size_size_list_int @ Xs )
= ( size_size_list_int @ Ys ) )
=> ( ! [I2: nat] :
( ( ord_less_nat @ I2 @ ( size_size_list_int @ Xs ) )
=> ( ( nth_int @ Xs @ I2 )
= ( nth_int @ Ys @ I2 ) ) )
=> ( Xs = Ys ) ) ) ).
% nth_equalityI
thf(fact_435_append_Oassoc,axiom,
! [A: list_instr,B: list_instr,C: list_instr] :
( ( append_instr @ ( append_instr @ A @ B ) @ C )
= ( append_instr @ A @ ( append_instr @ B @ C ) ) ) ).
% append.assoc
thf(fact_436_append_Oassoc,axiom,
! [A: list_int,B: list_int,C: list_int] :
( ( append_int @ ( append_int @ A @ B ) @ C )
= ( append_int @ A @ ( append_int @ B @ C ) ) ) ).
% append.assoc
thf(fact_437_append__assoc,axiom,
! [Xs: list_instr,Ys: list_instr,Zs: list_instr] :
( ( append_instr @ ( append_instr @ Xs @ Ys ) @ Zs )
= ( append_instr @ Xs @ ( append_instr @ Ys @ Zs ) ) ) ).
% append_assoc
thf(fact_438_append__assoc,axiom,
! [Xs: list_int,Ys: list_int,Zs: list_int] :
( ( append_int @ ( append_int @ Xs @ Ys ) @ Zs )
= ( append_int @ Xs @ ( append_int @ Ys @ Zs ) ) ) ).
% append_assoc
thf(fact_439_append__same__eq,axiom,
! [Ys: list_instr,Xs: list_instr,Zs: list_instr] :
( ( ( append_instr @ Ys @ Xs )
= ( append_instr @ Zs @ Xs ) )
= ( Ys = Zs ) ) ).
% append_same_eq
thf(fact_440_append__same__eq,axiom,
! [Ys: list_int,Xs: list_int,Zs: list_int] :
( ( ( append_int @ Ys @ Xs )
= ( append_int @ Zs @ Xs ) )
= ( Ys = Zs ) ) ).
% append_same_eq
thf(fact_441_same__append__eq,axiom,
! [Xs: list_instr,Ys: list_instr,Zs: list_instr] :
( ( ( append_instr @ Xs @ Ys )
= ( append_instr @ Xs @ Zs ) )
= ( Ys = Zs ) ) ).
% same_append_eq
thf(fact_442_same__append__eq,axiom,
! [Xs: list_int,Ys: list_int,Zs: list_int] :
( ( ( append_int @ Xs @ Ys )
= ( append_int @ Xs @ Zs ) )
= ( Ys = Zs ) ) ).
% same_append_eq
thf(fact_443_diff__commute,axiom,
! [I: nat,J: nat,K: nat] :
( ( minus_minus_nat @ ( minus_minus_nat @ I @ J ) @ K )
= ( minus_minus_nat @ ( minus_minus_nat @ I @ K ) @ J ) ) ).
% diff_commute
thf(fact_444_verit__comp__simplify1_I2_J,axiom,
! [A: int] : ( ord_less_eq_int @ A @ A ) ).
% verit_comp_simplify1(2)
thf(fact_445_verit__comp__simplify1_I2_J,axiom,
! [A: nat] : ( ord_less_eq_nat @ A @ A ) ).
% verit_comp_simplify1(2)
thf(fact_446_verit__comp__simplify1_I2_J,axiom,
! [A: set_int] : ( ord_less_eq_set_int @ A @ A ) ).
% verit_comp_simplify1(2)
thf(fact_447_verit__comp__simplify1_I2_J,axiom,
! [A: set_nat] : ( ord_less_eq_set_nat @ A @ A ) ).
% verit_comp_simplify1(2)
thf(fact_448_verit__la__disequality,axiom,
! [A: int,B: int] :
( ( A = B )
| ~ ( ord_less_eq_int @ A @ B )
| ~ ( ord_less_eq_int @ B @ A ) ) ).
% verit_la_disequality
thf(fact_449_verit__la__disequality,axiom,
! [A: nat,B: nat] :
( ( A = B )
| ~ ( ord_less_eq_nat @ A @ B )
| ~ ( ord_less_eq_nat @ B @ A ) ) ).
% verit_la_disequality
thf(fact_450_verit__comp__simplify1_I1_J,axiom,
! [A: int] :
~ ( ord_less_int @ A @ A ) ).
% verit_comp_simplify1(1)
thf(fact_451_verit__comp__simplify1_I1_J,axiom,
! [A: nat] :
~ ( ord_less_nat @ A @ A ) ).
% verit_comp_simplify1(1)
thf(fact_452_Ex__list__of__length,axiom,
! [N: nat] :
? [Xs3: list_P419547167332244514st_int] :
( ( size_s229347708945616566st_int @ Xs3 )
= N ) ).
% Ex_list_of_length
thf(fact_453_Ex__list__of__length,axiom,
! [N: nat] :
? [Xs3: list_instr] :
( ( size_size_list_instr @ Xs3 )
= N ) ).
% Ex_list_of_length
thf(fact_454_Ex__list__of__length,axiom,
! [N: nat] :
? [Xs3: list_list_char] :
( ( size_s356637359517785349t_char @ Xs3 )
= N ) ).
% Ex_list_of_length
thf(fact_455_Ex__list__of__length,axiom,
! [N: nat] :
? [Xs3: list_int] :
( ( size_size_list_int @ Xs3 )
= N ) ).
% Ex_list_of_length
thf(fact_456_neq__if__length__neq,axiom,
! [Xs: list_P419547167332244514st_int,Ys: list_P419547167332244514st_int] :
( ( ( size_s229347708945616566st_int @ Xs )
!= ( size_s229347708945616566st_int @ Ys ) )
=> ( Xs != Ys ) ) ).
% neq_if_length_neq
thf(fact_457_neq__if__length__neq,axiom,
! [Xs: list_instr,Ys: list_instr] :
( ( ( size_size_list_instr @ Xs )
!= ( size_size_list_instr @ Ys ) )
=> ( Xs != Ys ) ) ).
% neq_if_length_neq
thf(fact_458_neq__if__length__neq,axiom,
! [Xs: list_list_char,Ys: list_list_char] :
( ( ( size_s356637359517785349t_char @ Xs )
!= ( size_s356637359517785349t_char @ Ys ) )
=> ( Xs != Ys ) ) ).
% neq_if_length_neq
thf(fact_459_neq__if__length__neq,axiom,
! [Xs: list_int,Ys: list_int] :
( ( ( size_size_list_int @ Xs )
!= ( size_size_list_int @ Ys ) )
=> ( Xs != Ys ) ) ).
% neq_if_length_neq
thf(fact_460_append__eq__append__conv2,axiom,
! [Xs: list_instr,Ys: list_instr,Zs: list_instr,Ts: list_instr] :
( ( ( append_instr @ Xs @ Ys )
= ( append_instr @ Zs @ Ts ) )
= ( ? [Us2: list_instr] :
( ( ( Xs
= ( append_instr @ Zs @ Us2 ) )
& ( ( append_instr @ Us2 @ Ys )
= Ts ) )
| ( ( ( append_instr @ Xs @ Us2 )
= Zs )
& ( Ys
= ( append_instr @ Us2 @ Ts ) ) ) ) ) ) ).
% append_eq_append_conv2
thf(fact_461_append__eq__append__conv2,axiom,
! [Xs: list_int,Ys: list_int,Zs: list_int,Ts: list_int] :
( ( ( append_int @ Xs @ Ys )
= ( append_int @ Zs @ Ts ) )
= ( ? [Us2: list_int] :
( ( ( Xs
= ( append_int @ Zs @ Us2 ) )
& ( ( append_int @ Us2 @ Ys )
= Ts ) )
| ( ( ( append_int @ Xs @ Us2 )
= Zs )
& ( Ys
= ( append_int @ Us2 @ Ts ) ) ) ) ) ) ).
% append_eq_append_conv2
thf(fact_462_append__eq__appendI,axiom,
! [Xs: list_instr,Xs1: list_instr,Zs: list_instr,Ys: list_instr,Us: list_instr] :
( ( ( append_instr @ Xs @ Xs1 )
= Zs )
=> ( ( Ys
= ( append_instr @ Xs1 @ Us ) )
=> ( ( append_instr @ Xs @ Ys )
= ( append_instr @ Zs @ Us ) ) ) ) ).
% append_eq_appendI
thf(fact_463_append__eq__appendI,axiom,
! [Xs: list_int,Xs1: list_int,Zs: list_int,Ys: list_int,Us: list_int] :
( ( ( append_int @ Xs @ Xs1 )
= Zs )
=> ( ( Ys
= ( append_int @ Xs1 @ Us ) )
=> ( ( append_int @ Xs @ Ys )
= ( append_int @ Zs @ Us ) ) ) ) ).
% append_eq_appendI
thf(fact_464_verit__la__generic,axiom,
! [A: int,X: int] :
( ( ord_less_eq_int @ A @ X )
| ( A = X )
| ( ord_less_eq_int @ X @ A ) ) ).
% verit_la_generic
thf(fact_465_nat__int__comparison_I1_J,axiom,
( ( ^ [Y4: nat,Z2: nat] : ( Y4 = Z2 ) )
= ( ^ [A3: nat,B2: nat] :
( ( semiri1314217659103216013at_int @ A3 )
= ( semiri1314217659103216013at_int @ B2 ) ) ) ) ).
% nat_int_comparison(1)
thf(fact_466_int__if,axiom,
! [P: $o,A: nat,B: nat] :
( ( P
=> ( ( semiri1314217659103216013at_int @ ( if_nat @ P @ A @ B ) )
= ( semiri1314217659103216013at_int @ A ) ) )
& ( ~ P
=> ( ( semiri1314217659103216013at_int @ ( if_nat @ P @ A @ B ) )
= ( semiri1314217659103216013at_int @ B ) ) ) ) ).
% int_if
thf(fact_467_verit__comp__simplify1_I3_J,axiom,
! [B4: int,A5: int] :
( ( ~ ( ord_less_eq_int @ B4 @ A5 ) )
= ( ord_less_int @ A5 @ B4 ) ) ).
% verit_comp_simplify1(3)
thf(fact_468_verit__comp__simplify1_I3_J,axiom,
! [B4: nat,A5: nat] :
( ( ~ ( ord_less_eq_nat @ B4 @ A5 ) )
= ( ord_less_nat @ A5 @ B4 ) ) ).
% verit_comp_simplify1(3)
thf(fact_469_length__induct,axiom,
! [P: list_P419547167332244514st_int > $o,Xs: list_P419547167332244514st_int] :
( ! [Xs3: list_P419547167332244514st_int] :
( ! [Ys3: list_P419547167332244514st_int] :
( ( ord_less_nat @ ( size_s229347708945616566st_int @ Ys3 ) @ ( size_s229347708945616566st_int @ Xs3 ) )
=> ( P @ Ys3 ) )
=> ( P @ Xs3 ) )
=> ( P @ Xs ) ) ).
% length_induct
thf(fact_470_length__induct,axiom,
! [P: list_instr > $o,Xs: list_instr] :
( ! [Xs3: list_instr] :
( ! [Ys3: list_instr] :
( ( ord_less_nat @ ( size_size_list_instr @ Ys3 ) @ ( size_size_list_instr @ Xs3 ) )
=> ( P @ Ys3 ) )
=> ( P @ Xs3 ) )
=> ( P @ Xs ) ) ).
% length_induct
thf(fact_471_length__induct,axiom,
! [P: list_list_char > $o,Xs: list_list_char] :
( ! [Xs3: list_list_char] :
( ! [Ys3: list_list_char] :
( ( ord_less_nat @ ( size_s356637359517785349t_char @ Ys3 ) @ ( size_s356637359517785349t_char @ Xs3 ) )
=> ( P @ Ys3 ) )
=> ( P @ Xs3 ) )
=> ( P @ Xs ) ) ).
% length_induct
thf(fact_472_length__induct,axiom,
! [P: list_int > $o,Xs: list_int] :
( ! [Xs3: list_int] :
( ! [Ys3: list_int] :
( ( ord_less_nat @ ( size_size_list_int @ Ys3 ) @ ( size_size_list_int @ Xs3 ) )
=> ( P @ Ys3 ) )
=> ( P @ Xs3 ) )
=> ( P @ Xs ) ) ).
% length_induct
thf(fact_473_DiffI,axiom,
! [C: nat,A2: set_nat,B5: set_nat] :
( ( member_nat @ C @ A2 )
=> ( ~ ( member_nat @ C @ B5 )
=> ( member_nat @ C @ ( minus_minus_set_nat @ A2 @ B5 ) ) ) ) ).
% DiffI
thf(fact_474_DiffI,axiom,
! [C: int,A2: set_int,B5: set_int] :
( ( member_int @ C @ A2 )
=> ( ~ ( member_int @ C @ B5 )
=> ( member_int @ C @ ( minus_minus_set_int @ A2 @ B5 ) ) ) ) ).
% DiffI
thf(fact_475_DiffI,axiom,
! [C: list_char,A2: set_list_char,B5: set_list_char] :
( ( member_list_char @ C @ A2 )
=> ( ~ ( member_list_char @ C @ B5 )
=> ( member_list_char @ C @ ( minus_269717477326218608t_char @ A2 @ B5 ) ) ) ) ).
% DiffI
thf(fact_476_Diff__iff,axiom,
! [C: nat,A2: set_nat,B5: set_nat] :
( ( member_nat @ C @ ( minus_minus_set_nat @ A2 @ B5 ) )
= ( ( member_nat @ C @ A2 )
& ~ ( member_nat @ C @ B5 ) ) ) ).
% Diff_iff
thf(fact_477_Diff__iff,axiom,
! [C: int,A2: set_int,B5: set_int] :
( ( member_int @ C @ ( minus_minus_set_int @ A2 @ B5 ) )
= ( ( member_int @ C @ A2 )
& ~ ( member_int @ C @ B5 ) ) ) ).
% Diff_iff
thf(fact_478_Diff__iff,axiom,
! [C: list_char,A2: set_list_char,B5: set_list_char] :
( ( member_list_char @ C @ ( minus_269717477326218608t_char @ A2 @ B5 ) )
= ( ( member_list_char @ C @ A2 )
& ~ ( member_list_char @ C @ B5 ) ) ) ).
% Diff_iff
thf(fact_479_subset__antisym,axiom,
! [A2: set_int,B5: set_int] :
( ( ord_less_eq_set_int @ A2 @ B5 )
=> ( ( ord_less_eq_set_int @ B5 @ A2 )
=> ( A2 = B5 ) ) ) ).
% subset_antisym
thf(fact_480_subset__antisym,axiom,
! [A2: set_nat,B5: set_nat] :
( ( ord_less_eq_set_nat @ A2 @ B5 )
=> ( ( ord_less_eq_set_nat @ B5 @ A2 )
=> ( A2 = B5 ) ) ) ).
% subset_antisym
thf(fact_481_psubsetI,axiom,
! [A2: set_int,B5: set_int] :
( ( ord_less_eq_set_int @ A2 @ B5 )
=> ( ( A2 != B5 )
=> ( ord_less_set_int @ A2 @ B5 ) ) ) ).
% psubsetI
thf(fact_482_psubsetI,axiom,
! [A2: set_nat,B5: set_nat] :
( ( ord_less_eq_set_nat @ A2 @ B5 )
=> ( ( A2 != B5 )
=> ( ord_less_set_nat @ A2 @ B5 ) ) ) ).
% psubsetI
thf(fact_483_subsetI,axiom,
! [A2: set_list_char,B5: set_list_char] :
( ! [X3: list_char] :
( ( member_list_char @ X3 @ A2 )
=> ( member_list_char @ X3 @ B5 ) )
=> ( ord_le1207710660481778807t_char @ A2 @ B5 ) ) ).
% subsetI
thf(fact_484_subsetI,axiom,
! [A2: set_int,B5: set_int] :
( ! [X3: int] :
( ( member_int @ X3 @ A2 )
=> ( member_int @ X3 @ B5 ) )
=> ( ord_less_eq_set_int @ A2 @ B5 ) ) ).
% subsetI
thf(fact_485_subsetI,axiom,
! [A2: set_nat,B5: set_nat] :
( ! [X3: nat] :
( ( member_nat @ X3 @ A2 )
=> ( member_nat @ X3 @ B5 ) )
=> ( ord_less_eq_set_nat @ A2 @ B5 ) ) ).
% subsetI
thf(fact_486_double__diff,axiom,
! [A2: set_int,B5: set_int,C2: set_int] :
( ( ord_less_eq_set_int @ A2 @ B5 )
=> ( ( ord_less_eq_set_int @ B5 @ C2 )
=> ( ( minus_minus_set_int @ B5 @ ( minus_minus_set_int @ C2 @ A2 ) )
= A2 ) ) ) ).
% double_diff
thf(fact_487_double__diff,axiom,
! [A2: set_nat,B5: set_nat,C2: set_nat] :
( ( ord_less_eq_set_nat @ A2 @ B5 )
=> ( ( ord_less_eq_set_nat @ B5 @ C2 )
=> ( ( minus_minus_set_nat @ B5 @ ( minus_minus_set_nat @ C2 @ A2 ) )
= A2 ) ) ) ).
% double_diff
thf(fact_488_Diff__subset,axiom,
! [A2: set_int,B5: set_int] : ( ord_less_eq_set_int @ ( minus_minus_set_int @ A2 @ B5 ) @ A2 ) ).
% Diff_subset
thf(fact_489_Diff__subset,axiom,
! [A2: set_nat,B5: set_nat] : ( ord_less_eq_set_nat @ ( minus_minus_set_nat @ A2 @ B5 ) @ A2 ) ).
% Diff_subset
thf(fact_490_less__set__def,axiom,
( ord_less_set_nat
= ( ^ [A6: set_nat,B6: set_nat] :
( ord_less_nat_o
@ ^ [X2: nat] : ( member_nat @ X2 @ A6 )
@ ^ [X2: nat] : ( member_nat @ X2 @ B6 ) ) ) ) ).
% less_set_def
thf(fact_491_less__set__def,axiom,
( ord_less_set_int
= ( ^ [A6: set_int,B6: set_int] :
( ord_less_int_o
@ ^ [X2: int] : ( member_int @ X2 @ A6 )
@ ^ [X2: int] : ( member_int @ X2 @ B6 ) ) ) ) ).
% less_set_def
thf(fact_492_less__set__def,axiom,
( ord_le5286566838938544235t_char
= ( ^ [A6: set_list_char,B6: set_list_char] :
( ord_less_list_char_o
@ ^ [X2: list_char] : ( member_list_char @ X2 @ A6 )
@ ^ [X2: list_char] : ( member_list_char @ X2 @ B6 ) ) ) ) ).
% less_set_def
thf(fact_493_psubsetD,axiom,
! [A2: set_nat,B5: set_nat,C: nat] :
( ( ord_less_set_nat @ A2 @ B5 )
=> ( ( member_nat @ C @ A2 )
=> ( member_nat @ C @ B5 ) ) ) ).
% psubsetD
thf(fact_494_psubsetD,axiom,
! [A2: set_int,B5: set_int,C: int] :
( ( ord_less_set_int @ A2 @ B5 )
=> ( ( member_int @ C @ A2 )
=> ( member_int @ C @ B5 ) ) ) ).
% psubsetD
thf(fact_495_psubsetD,axiom,
! [A2: set_list_char,B5: set_list_char,C: list_char] :
( ( ord_le5286566838938544235t_char @ A2 @ B5 )
=> ( ( member_list_char @ C @ A2 )
=> ( member_list_char @ C @ B5 ) ) ) ).
% psubsetD
thf(fact_496_in__mono,axiom,
! [A2: set_list_char,B5: set_list_char,X: list_char] :
( ( ord_le1207710660481778807t_char @ A2 @ B5 )
=> ( ( member_list_char @ X @ A2 )
=> ( member_list_char @ X @ B5 ) ) ) ).
% in_mono
thf(fact_497_in__mono,axiom,
! [A2: set_int,B5: set_int,X: int] :
( ( ord_less_eq_set_int @ A2 @ B5 )
=> ( ( member_int @ X @ A2 )
=> ( member_int @ X @ B5 ) ) ) ).
% in_mono
thf(fact_498_in__mono,axiom,
! [A2: set_nat,B5: set_nat,X: nat] :
( ( ord_less_eq_set_nat @ A2 @ B5 )
=> ( ( member_nat @ X @ A2 )
=> ( member_nat @ X @ B5 ) ) ) ).
% in_mono
thf(fact_499_subsetD,axiom,
! [A2: set_list_char,B5: set_list_char,C: list_char] :
( ( ord_le1207710660481778807t_char @ A2 @ B5 )
=> ( ( member_list_char @ C @ A2 )
=> ( member_list_char @ C @ B5 ) ) ) ).
% subsetD
thf(fact_500_subsetD,axiom,
! [A2: set_int,B5: set_int,C: int] :
( ( ord_less_eq_set_int @ A2 @ B5 )
=> ( ( member_int @ C @ A2 )
=> ( member_int @ C @ B5 ) ) ) ).
% subsetD
thf(fact_501_subsetD,axiom,
! [A2: set_nat,B5: set_nat,C: nat] :
( ( ord_less_eq_set_nat @ A2 @ B5 )
=> ( ( member_nat @ C @ A2 )
=> ( member_nat @ C @ B5 ) ) ) ).
% subsetD
thf(fact_502_psubsetE,axiom,
! [A2: set_int,B5: set_int] :
( ( ord_less_set_int @ A2 @ B5 )
=> ~ ( ( ord_less_eq_set_int @ A2 @ B5 )
=> ( ord_less_eq_set_int @ B5 @ A2 ) ) ) ).
% psubsetE
thf(fact_503_psubsetE,axiom,
! [A2: set_nat,B5: set_nat] :
( ( ord_less_set_nat @ A2 @ B5 )
=> ~ ( ( ord_less_eq_set_nat @ A2 @ B5 )
=> ( ord_less_eq_set_nat @ B5 @ A2 ) ) ) ).
% psubsetE
thf(fact_504_equalityE,axiom,
! [A2: set_int,B5: set_int] :
( ( A2 = B5 )
=> ~ ( ( ord_less_eq_set_int @ A2 @ B5 )
=> ~ ( ord_less_eq_set_int @ B5 @ A2 ) ) ) ).
% equalityE
thf(fact_505_equalityE,axiom,
! [A2: set_nat,B5: set_nat] :
( ( A2 = B5 )
=> ~ ( ( ord_less_eq_set_nat @ A2 @ B5 )
=> ~ ( ord_less_eq_set_nat @ B5 @ A2 ) ) ) ).
% equalityE
thf(fact_506_subset__eq,axiom,
( ord_le1207710660481778807t_char
= ( ^ [A6: set_list_char,B6: set_list_char] :
! [X2: list_char] :
( ( member_list_char @ X2 @ A6 )
=> ( member_list_char @ X2 @ B6 ) ) ) ) ).
% subset_eq
thf(fact_507_subset__eq,axiom,
( ord_less_eq_set_int
= ( ^ [A6: set_int,B6: set_int] :
! [X2: int] :
( ( member_int @ X2 @ A6 )
=> ( member_int @ X2 @ B6 ) ) ) ) ).
% subset_eq
thf(fact_508_subset__eq,axiom,
( ord_less_eq_set_nat
= ( ^ [A6: set_nat,B6: set_nat] :
! [X2: nat] :
( ( member_nat @ X2 @ A6 )
=> ( member_nat @ X2 @ B6 ) ) ) ) ).
% subset_eq
thf(fact_509_equalityD1,axiom,
! [A2: set_int,B5: set_int] :
( ( A2 = B5 )
=> ( ord_less_eq_set_int @ A2 @ B5 ) ) ).
% equalityD1
thf(fact_510_equalityD1,axiom,
! [A2: set_nat,B5: set_nat] :
( ( A2 = B5 )
=> ( ord_less_eq_set_nat @ A2 @ B5 ) ) ).
% equalityD1
thf(fact_511_equalityD2,axiom,
! [A2: set_int,B5: set_int] :
( ( A2 = B5 )
=> ( ord_less_eq_set_int @ B5 @ A2 ) ) ).
% equalityD2
thf(fact_512_equalityD2,axiom,
! [A2: set_nat,B5: set_nat] :
( ( A2 = B5 )
=> ( ord_less_eq_set_nat @ B5 @ A2 ) ) ).
% equalityD2
thf(fact_513_psubset__eq,axiom,
( ord_less_set_int
= ( ^ [A6: set_int,B6: set_int] :
( ( ord_less_eq_set_int @ A6 @ B6 )
& ( A6 != B6 ) ) ) ) ).
% psubset_eq
thf(fact_514_psubset__eq,axiom,
( ord_less_set_nat
= ( ^ [A6: set_nat,B6: set_nat] :
( ( ord_less_eq_set_nat @ A6 @ B6 )
& ( A6 != B6 ) ) ) ) ).
% psubset_eq
thf(fact_515_subset__iff,axiom,
( ord_le1207710660481778807t_char
= ( ^ [A6: set_list_char,B6: set_list_char] :
! [T2: list_char] :
( ( member_list_char @ T2 @ A6 )
=> ( member_list_char @ T2 @ B6 ) ) ) ) ).
% subset_iff
thf(fact_516_subset__iff,axiom,
( ord_less_eq_set_int
= ( ^ [A6: set_int,B6: set_int] :
! [T2: int] :
( ( member_int @ T2 @ A6 )
=> ( member_int @ T2 @ B6 ) ) ) ) ).
% subset_iff
thf(fact_517_subset__iff,axiom,
( ord_less_eq_set_nat
= ( ^ [A6: set_nat,B6: set_nat] :
! [T2: nat] :
( ( member_nat @ T2 @ A6 )
=> ( member_nat @ T2 @ B6 ) ) ) ) ).
% subset_iff
thf(fact_518_subset__refl,axiom,
! [A2: set_int] : ( ord_less_eq_set_int @ A2 @ A2 ) ).
% subset_refl
thf(fact_519_subset__refl,axiom,
! [A2: set_nat] : ( ord_less_eq_set_nat @ A2 @ A2 ) ).
% subset_refl
thf(fact_520_Collect__mono,axiom,
! [P: int > $o,Q: int > $o] :
( ! [X3: int] :
( ( P @ X3 )
=> ( Q @ X3 ) )
=> ( ord_less_eq_set_int @ ( collect_int @ P ) @ ( collect_int @ Q ) ) ) ).
% Collect_mono
thf(fact_521_Collect__mono,axiom,
! [P: nat > $o,Q: nat > $o] :
( ! [X3: nat] :
( ( P @ X3 )
=> ( Q @ X3 ) )
=> ( ord_less_eq_set_nat @ ( collect_nat @ P ) @ ( collect_nat @ Q ) ) ) ).
% Collect_mono
thf(fact_522_subset__trans,axiom,
! [A2: set_int,B5: set_int,C2: set_int] :
( ( ord_less_eq_set_int @ A2 @ B5 )
=> ( ( ord_less_eq_set_int @ B5 @ C2 )
=> ( ord_less_eq_set_int @ A2 @ C2 ) ) ) ).
% subset_trans
thf(fact_523_subset__trans,axiom,
! [A2: set_nat,B5: set_nat,C2: set_nat] :
( ( ord_less_eq_set_nat @ A2 @ B5 )
=> ( ( ord_less_eq_set_nat @ B5 @ C2 )
=> ( ord_less_eq_set_nat @ A2 @ C2 ) ) ) ).
% subset_trans
thf(fact_524_set__eq__subset,axiom,
( ( ^ [Y4: set_int,Z2: set_int] : ( Y4 = Z2 ) )
= ( ^ [A6: set_int,B6: set_int] :
( ( ord_less_eq_set_int @ A6 @ B6 )
& ( ord_less_eq_set_int @ B6 @ A6 ) ) ) ) ).
% set_eq_subset
thf(fact_525_set__eq__subset,axiom,
( ( ^ [Y4: set_nat,Z2: set_nat] : ( Y4 = Z2 ) )
= ( ^ [A6: set_nat,B6: set_nat] :
( ( ord_less_eq_set_nat @ A6 @ B6 )
& ( ord_less_eq_set_nat @ B6 @ A6 ) ) ) ) ).
% set_eq_subset
thf(fact_526_Collect__mono__iff,axiom,
! [P: int > $o,Q: int > $o] :
( ( ord_less_eq_set_int @ ( collect_int @ P ) @ ( collect_int @ Q ) )
= ( ! [X2: int] :
( ( P @ X2 )
=> ( Q @ X2 ) ) ) ) ).
% Collect_mono_iff
thf(fact_527_Collect__mono__iff,axiom,
! [P: nat > $o,Q: nat > $o] :
( ( ord_less_eq_set_nat @ ( collect_nat @ P ) @ ( collect_nat @ Q ) )
= ( ! [X2: nat] :
( ( P @ X2 )
=> ( Q @ X2 ) ) ) ) ).
% Collect_mono_iff
thf(fact_528_psubset__imp__subset,axiom,
! [A2: set_int,B5: set_int] :
( ( ord_less_set_int @ A2 @ B5 )
=> ( ord_less_eq_set_int @ A2 @ B5 ) ) ).
% psubset_imp_subset
thf(fact_529_psubset__imp__subset,axiom,
! [A2: set_nat,B5: set_nat] :
( ( ord_less_set_nat @ A2 @ B5 )
=> ( ord_less_eq_set_nat @ A2 @ B5 ) ) ).
% psubset_imp_subset
thf(fact_530_psubset__subset__trans,axiom,
! [A2: set_int,B5: set_int,C2: set_int] :
( ( ord_less_set_int @ A2 @ B5 )
=> ( ( ord_less_eq_set_int @ B5 @ C2 )
=> ( ord_less_set_int @ A2 @ C2 ) ) ) ).
% psubset_subset_trans
thf(fact_531_psubset__subset__trans,axiom,
! [A2: set_nat,B5: set_nat,C2: set_nat] :
( ( ord_less_set_nat @ A2 @ B5 )
=> ( ( ord_less_eq_set_nat @ B5 @ C2 )
=> ( ord_less_set_nat @ A2 @ C2 ) ) ) ).
% psubset_subset_trans
thf(fact_532_subset__not__subset__eq,axiom,
( ord_less_set_int
= ( ^ [A6: set_int,B6: set_int] :
( ( ord_less_eq_set_int @ A6 @ B6 )
& ~ ( ord_less_eq_set_int @ B6 @ A6 ) ) ) ) ).
% subset_not_subset_eq
thf(fact_533_subset__not__subset__eq,axiom,
( ord_less_set_nat
= ( ^ [A6: set_nat,B6: set_nat] :
( ( ord_less_eq_set_nat @ A6 @ B6 )
& ~ ( ord_less_eq_set_nat @ B6 @ A6 ) ) ) ) ).
% subset_not_subset_eq
thf(fact_534_subset__psubset__trans,axiom,
! [A2: set_int,B5: set_int,C2: set_int] :
( ( ord_less_eq_set_int @ A2 @ B5 )
=> ( ( ord_less_set_int @ B5 @ C2 )
=> ( ord_less_set_int @ A2 @ C2 ) ) ) ).
% subset_psubset_trans
thf(fact_535_subset__psubset__trans,axiom,
! [A2: set_nat,B5: set_nat,C2: set_nat] :
( ( ord_less_eq_set_nat @ A2 @ B5 )
=> ( ( ord_less_set_nat @ B5 @ C2 )
=> ( ord_less_set_nat @ A2 @ C2 ) ) ) ).
% subset_psubset_trans
thf(fact_536_subset__iff__psubset__eq,axiom,
( ord_less_eq_set_int
= ( ^ [A6: set_int,B6: set_int] :
( ( ord_less_set_int @ A6 @ B6 )
| ( A6 = B6 ) ) ) ) ).
% subset_iff_psubset_eq
thf(fact_537_subset__iff__psubset__eq,axiom,
( ord_less_eq_set_nat
= ( ^ [A6: set_nat,B6: set_nat] :
( ( ord_less_set_nat @ A6 @ B6 )
| ( A6 = B6 ) ) ) ) ).
% subset_iff_psubset_eq
thf(fact_538_psubset__imp__ex__mem,axiom,
! [A2: set_nat,B5: set_nat] :
( ( ord_less_set_nat @ A2 @ B5 )
=> ? [B3: nat] : ( member_nat @ B3 @ ( minus_minus_set_nat @ B5 @ A2 ) ) ) ).
% psubset_imp_ex_mem
thf(fact_539_psubset__imp__ex__mem,axiom,
! [A2: set_int,B5: set_int] :
( ( ord_less_set_int @ A2 @ B5 )
=> ? [B3: int] : ( member_int @ B3 @ ( minus_minus_set_int @ B5 @ A2 ) ) ) ).
% psubset_imp_ex_mem
thf(fact_540_psubset__imp__ex__mem,axiom,
! [A2: set_list_char,B5: set_list_char] :
( ( ord_le5286566838938544235t_char @ A2 @ B5 )
=> ? [B3: list_char] : ( member_list_char @ B3 @ ( minus_269717477326218608t_char @ B5 @ A2 ) ) ) ).
% psubset_imp_ex_mem
thf(fact_541_DiffD2,axiom,
! [C: nat,A2: set_nat,B5: set_nat] :
( ( member_nat @ C @ ( minus_minus_set_nat @ A2 @ B5 ) )
=> ~ ( member_nat @ C @ B5 ) ) ).
% DiffD2
thf(fact_542_DiffD2,axiom,
! [C: int,A2: set_int,B5: set_int] :
( ( member_int @ C @ ( minus_minus_set_int @ A2 @ B5 ) )
=> ~ ( member_int @ C @ B5 ) ) ).
% DiffD2
thf(fact_543_DiffD2,axiom,
! [C: list_char,A2: set_list_char,B5: set_list_char] :
( ( member_list_char @ C @ ( minus_269717477326218608t_char @ A2 @ B5 ) )
=> ~ ( member_list_char @ C @ B5 ) ) ).
% DiffD2
thf(fact_544_DiffD1,axiom,
! [C: nat,A2: set_nat,B5: set_nat] :
( ( member_nat @ C @ ( minus_minus_set_nat @ A2 @ B5 ) )
=> ( member_nat @ C @ A2 ) ) ).
% DiffD1
thf(fact_545_DiffD1,axiom,
! [C: int,A2: set_int,B5: set_int] :
( ( member_int @ C @ ( minus_minus_set_int @ A2 @ B5 ) )
=> ( member_int @ C @ A2 ) ) ).
% DiffD1
thf(fact_546_DiffD1,axiom,
! [C: list_char,A2: set_list_char,B5: set_list_char] :
( ( member_list_char @ C @ ( minus_269717477326218608t_char @ A2 @ B5 ) )
=> ( member_list_char @ C @ A2 ) ) ).
% DiffD1
thf(fact_547_DiffE,axiom,
! [C: nat,A2: set_nat,B5: set_nat] :
( ( member_nat @ C @ ( minus_minus_set_nat @ A2 @ B5 ) )
=> ~ ( ( member_nat @ C @ A2 )
=> ( member_nat @ C @ B5 ) ) ) ).
% DiffE
thf(fact_548_DiffE,axiom,
! [C: int,A2: set_int,B5: set_int] :
( ( member_int @ C @ ( minus_minus_set_int @ A2 @ B5 ) )
=> ~ ( ( member_int @ C @ A2 )
=> ( member_int @ C @ B5 ) ) ) ).
% DiffE
thf(fact_549_DiffE,axiom,
! [C: list_char,A2: set_list_char,B5: set_list_char] :
( ( member_list_char @ C @ ( minus_269717477326218608t_char @ A2 @ B5 ) )
=> ~ ( ( member_list_char @ C @ A2 )
=> ( member_list_char @ C @ B5 ) ) ) ).
% DiffE
thf(fact_550_Collect__subset,axiom,
! [A2: set_list_char,P: list_char > $o] :
( ord_le1207710660481778807t_char
@ ( collect_list_char
@ ^ [X2: list_char] :
( ( member_list_char @ X2 @ A2 )
& ( P @ X2 ) ) )
@ A2 ) ).
% Collect_subset
thf(fact_551_Collect__subset,axiom,
! [A2: set_int,P: int > $o] :
( ord_less_eq_set_int
@ ( collect_int
@ ^ [X2: int] :
( ( member_int @ X2 @ A2 )
& ( P @ X2 ) ) )
@ A2 ) ).
% Collect_subset
thf(fact_552_Collect__subset,axiom,
! [A2: set_nat,P: nat > $o] :
( ord_less_eq_set_nat
@ ( collect_nat
@ ^ [X2: nat] :
( ( member_nat @ X2 @ A2 )
& ( P @ X2 ) ) )
@ A2 ) ).
% Collect_subset
thf(fact_553_less__eq__set__def,axiom,
( ord_le1207710660481778807t_char
= ( ^ [A6: set_list_char,B6: set_list_char] :
( ord_le4796328588573674190char_o
@ ^ [X2: list_char] : ( member_list_char @ X2 @ A6 )
@ ^ [X2: list_char] : ( member_list_char @ X2 @ B6 ) ) ) ) ).
% less_eq_set_def
thf(fact_554_less__eq__set__def,axiom,
( ord_less_eq_set_int
= ( ^ [A6: set_int,B6: set_int] :
( ord_less_eq_int_o
@ ^ [X2: int] : ( member_int @ X2 @ A6 )
@ ^ [X2: int] : ( member_int @ X2 @ B6 ) ) ) ) ).
% less_eq_set_def
thf(fact_555_less__eq__set__def,axiom,
( ord_less_eq_set_nat
= ( ^ [A6: set_nat,B6: set_nat] :
( ord_less_eq_nat_o
@ ^ [X2: nat] : ( member_nat @ X2 @ A6 )
@ ^ [X2: nat] : ( member_nat @ X2 @ B6 ) ) ) ) ).
% less_eq_set_def
thf(fact_556_minus__set__def,axiom,
( minus_minus_set_int
= ( ^ [A6: set_int,B6: set_int] :
( collect_int
@ ( minus_minus_int_o
@ ^ [X2: int] : ( member_int @ X2 @ A6 )
@ ^ [X2: int] : ( member_int @ X2 @ B6 ) ) ) ) ) ).
% minus_set_def
thf(fact_557_minus__set__def,axiom,
( minus_269717477326218608t_char
= ( ^ [A6: set_list_char,B6: set_list_char] :
( collect_list_char
@ ( minus_6821098565838606101char_o
@ ^ [X2: list_char] : ( member_list_char @ X2 @ A6 )
@ ^ [X2: list_char] : ( member_list_char @ X2 @ B6 ) ) ) ) ) ).
% minus_set_def
thf(fact_558_minus__set__def,axiom,
( minus_minus_set_nat
= ( ^ [A6: set_nat,B6: set_nat] :
( collect_nat
@ ( minus_minus_nat_o
@ ^ [X2: nat] : ( member_nat @ X2 @ A6 )
@ ^ [X2: nat] : ( member_nat @ X2 @ B6 ) ) ) ) ) ).
% minus_set_def
thf(fact_559_set__diff__eq,axiom,
( minus_minus_set_int
= ( ^ [A6: set_int,B6: set_int] :
( collect_int
@ ^ [X2: int] :
( ( member_int @ X2 @ A6 )
& ~ ( member_int @ X2 @ B6 ) ) ) ) ) ).
% set_diff_eq
thf(fact_560_set__diff__eq,axiom,
( minus_269717477326218608t_char
= ( ^ [A6: set_list_char,B6: set_list_char] :
( collect_list_char
@ ^ [X2: list_char] :
( ( member_list_char @ X2 @ A6 )
& ~ ( member_list_char @ X2 @ B6 ) ) ) ) ) ).
% set_diff_eq
thf(fact_561_set__diff__eq,axiom,
( minus_minus_set_nat
= ( ^ [A6: set_nat,B6: set_nat] :
( collect_nat
@ ^ [X2: nat] :
( ( member_nat @ X2 @ A6 )
& ~ ( member_nat @ X2 @ B6 ) ) ) ) ) ).
% set_diff_eq
thf(fact_562_Diff__mono,axiom,
! [A2: set_int,C2: set_int,D2: set_int,B5: set_int] :
( ( ord_less_eq_set_int @ A2 @ C2 )
=> ( ( ord_less_eq_set_int @ D2 @ B5 )
=> ( ord_less_eq_set_int @ ( minus_minus_set_int @ A2 @ B5 ) @ ( minus_minus_set_int @ C2 @ D2 ) ) ) ) ).
% Diff_mono
thf(fact_563_Diff__mono,axiom,
! [A2: set_nat,C2: set_nat,D2: set_nat,B5: set_nat] :
( ( ord_less_eq_set_nat @ A2 @ C2 )
=> ( ( ord_less_eq_set_nat @ D2 @ B5 )
=> ( ord_less_eq_set_nat @ ( minus_minus_set_nat @ A2 @ B5 ) @ ( minus_minus_set_nat @ C2 @ D2 ) ) ) ) ).
% Diff_mono
thf(fact_564_nat__descend__induct,axiom,
! [N: nat,P: nat > $o,M: nat] :
( ! [K2: nat] :
( ( ord_less_nat @ N @ K2 )
=> ( P @ K2 ) )
=> ( ! [K2: nat] :
( ( ord_less_eq_nat @ K2 @ N )
=> ( ! [I4: nat] :
( ( ord_less_nat @ K2 @ I4 )
=> ( P @ I4 ) )
=> ( P @ K2 ) ) )
=> ( P @ M ) ) ) ).
% nat_descend_induct
thf(fact_565_diff__strict__right__mono,axiom,
! [A: int,B: int,C: int] :
( ( ord_less_int @ A @ B )
=> ( ord_less_int @ ( minus_minus_int @ A @ C ) @ ( minus_minus_int @ B @ C ) ) ) ).
% diff_strict_right_mono
thf(fact_566_diff__strict__left__mono,axiom,
! [B: int,A: int,C: int] :
( ( ord_less_int @ B @ A )
=> ( ord_less_int @ ( minus_minus_int @ C @ A ) @ ( minus_minus_int @ C @ B ) ) ) ).
% diff_strict_left_mono
thf(fact_567_diff__eq__diff__less,axiom,
! [A: int,B: int,C: int,D: int] :
( ( ( minus_minus_int @ A @ B )
= ( minus_minus_int @ C @ D ) )
=> ( ( ord_less_int @ A @ B )
= ( ord_less_int @ C @ D ) ) ) ).
% diff_eq_diff_less
thf(fact_568_diff__strict__mono,axiom,
! [A: int,B: int,D: int,C: int] :
( ( ord_less_int @ A @ B )
=> ( ( ord_less_int @ D @ C )
=> ( ord_less_int @ ( minus_minus_int @ A @ C ) @ ( minus_minus_int @ B @ D ) ) ) ) ).
% diff_strict_mono
thf(fact_569_diff__eq__diff__less__eq,axiom,
! [A: int,B: int,C: int,D: int] :
( ( ( minus_minus_int @ A @ B )
= ( minus_minus_int @ C @ D ) )
=> ( ( ord_less_eq_int @ A @ B )
= ( ord_less_eq_int @ C @ D ) ) ) ).
% diff_eq_diff_less_eq
thf(fact_570_diff__right__mono,axiom,
! [A: int,B: int,C: int] :
( ( ord_less_eq_int @ A @ B )
=> ( ord_less_eq_int @ ( minus_minus_int @ A @ C ) @ ( minus_minus_int @ B @ C ) ) ) ).
% diff_right_mono
thf(fact_571_diff__eq__diff__eq,axiom,
! [A: int,B: int,C: int,D: int] :
( ( ( minus_minus_int @ A @ B )
= ( minus_minus_int @ C @ D ) )
=> ( ( A = B )
= ( C = D ) ) ) ).
% diff_eq_diff_eq
thf(fact_572_diff__right__commute,axiom,
! [A: int,C: int,B: int] :
( ( minus_minus_int @ ( minus_minus_int @ A @ C ) @ B )
= ( minus_minus_int @ ( minus_minus_int @ A @ B ) @ C ) ) ).
% diff_right_commute
thf(fact_573_diff__right__commute,axiom,
! [A: nat,C: nat,B: nat] :
( ( minus_minus_nat @ ( minus_minus_nat @ A @ C ) @ B )
= ( minus_minus_nat @ ( minus_minus_nat @ A @ B ) @ C ) ) ).
% diff_right_commute
thf(fact_574_diff__mono,axiom,
! [A: int,B: int,D: int,C: int] :
( ( ord_less_eq_int @ A @ B )
=> ( ( ord_less_eq_int @ D @ C )
=> ( ord_less_eq_int @ ( minus_minus_int @ A @ C ) @ ( minus_minus_int @ B @ D ) ) ) ) ).
% diff_mono
thf(fact_575_diff__left__mono,axiom,
! [B: int,A: int,C: int] :
( ( ord_less_eq_int @ B @ A )
=> ( ord_less_eq_int @ ( minus_minus_int @ C @ A ) @ ( minus_minus_int @ C @ B ) ) ) ).
% diff_left_mono
thf(fact_576_pred__subset__eq,axiom,
! [R: set_list_char,S2: set_list_char] :
( ( ord_le4796328588573674190char_o
@ ^ [X2: list_char] : ( member_list_char @ X2 @ R )
@ ^ [X2: list_char] : ( member_list_char @ X2 @ S2 ) )
= ( ord_le1207710660481778807t_char @ R @ S2 ) ) ).
% pred_subset_eq
thf(fact_577_pred__subset__eq,axiom,
! [R: set_int,S2: set_int] :
( ( ord_less_eq_int_o
@ ^ [X2: int] : ( member_int @ X2 @ R )
@ ^ [X2: int] : ( member_int @ X2 @ S2 ) )
= ( ord_less_eq_set_int @ R @ S2 ) ) ).
% pred_subset_eq
thf(fact_578_pred__subset__eq,axiom,
! [R: set_nat,S2: set_nat] :
( ( ord_less_eq_nat_o
@ ^ [X2: nat] : ( member_nat @ X2 @ R )
@ ^ [X2: nat] : ( member_nat @ X2 @ S2 ) )
= ( ord_less_eq_set_nat @ R @ S2 ) ) ).
% pred_subset_eq
thf(fact_579_dual__Max,axiom,
( ( lattices_Max_int
@ ^ [X2: int,Y5: int] : ( ord_less_eq_int @ Y5 @ X2 ) )
= lattic8718645017227715691in_int ) ).
% dual_Max
thf(fact_580_dual__Max,axiom,
( ( lattices_Max_nat
@ ^ [X2: nat,Y5: nat] : ( ord_less_eq_nat @ Y5 @ X2 ) )
= lattic8721135487736765967in_nat ) ).
% dual_Max
thf(fact_581_minf_I8_J,axiom,
! [T: int] :
? [Z3: int] :
! [X4: int] :
( ( ord_less_int @ X4 @ Z3 )
=> ~ ( ord_less_eq_int @ T @ X4 ) ) ).
% minf(8)
thf(fact_582_minf_I8_J,axiom,
! [T: nat] :
? [Z3: nat] :
! [X4: nat] :
( ( ord_less_nat @ X4 @ Z3 )
=> ~ ( ord_less_eq_nat @ T @ X4 ) ) ).
% minf(8)
thf(fact_583_minf_I6_J,axiom,
! [T: int] :
? [Z3: int] :
! [X4: int] :
( ( ord_less_int @ X4 @ Z3 )
=> ( ord_less_eq_int @ X4 @ T ) ) ).
% minf(6)
thf(fact_584_minf_I6_J,axiom,
! [T: nat] :
? [Z3: nat] :
! [X4: nat] :
( ( ord_less_nat @ X4 @ Z3 )
=> ( ord_less_eq_nat @ X4 @ T ) ) ).
% minf(6)
thf(fact_585_pinf_I8_J,axiom,
! [T: int] :
? [Z3: int] :
! [X4: int] :
( ( ord_less_int @ Z3 @ X4 )
=> ( ord_less_eq_int @ T @ X4 ) ) ).
% pinf(8)
thf(fact_586_pinf_I8_J,axiom,
! [T: nat] :
? [Z3: nat] :
! [X4: nat] :
( ( ord_less_nat @ Z3 @ X4 )
=> ( ord_less_eq_nat @ T @ X4 ) ) ).
% pinf(8)
thf(fact_587_pinf_I6_J,axiom,
! [T: int] :
? [Z3: int] :
! [X4: int] :
( ( ord_less_int @ Z3 @ X4 )
=> ~ ( ord_less_eq_int @ X4 @ T ) ) ).
% pinf(6)
thf(fact_588_pinf_I6_J,axiom,
! [T: nat] :
? [Z3: nat] :
! [X4: nat] :
( ( ord_less_nat @ Z3 @ X4 )
=> ~ ( ord_less_eq_nat @ X4 @ T ) ) ).
% pinf(6)
thf(fact_589_complete__interval,axiom,
! [A: int,B: int,P: int > $o] :
( ( ord_less_int @ A @ B )
=> ( ( P @ A )
=> ( ~ ( P @ B )
=> ? [C3: int] :
( ( ord_less_eq_int @ A @ C3 )
& ( ord_less_eq_int @ C3 @ B )
& ! [X4: int] :
( ( ( ord_less_eq_int @ A @ X4 )
& ( ord_less_int @ X4 @ C3 ) )
=> ( P @ X4 ) )
& ! [D3: int] :
( ! [X3: int] :
( ( ( ord_less_eq_int @ A @ X3 )
& ( ord_less_int @ X3 @ D3 ) )
=> ( P @ X3 ) )
=> ( ord_less_eq_int @ D3 @ C3 ) ) ) ) ) ) ).
% complete_interval
thf(fact_590_complete__interval,axiom,
! [A: nat,B: nat,P: nat > $o] :
( ( ord_less_nat @ A @ B )
=> ( ( P @ A )
=> ( ~ ( P @ B )
=> ? [C3: nat] :
( ( ord_less_eq_nat @ A @ C3 )
& ( ord_less_eq_nat @ C3 @ B )
& ! [X4: nat] :
( ( ( ord_less_eq_nat @ A @ X4 )
& ( ord_less_nat @ X4 @ C3 ) )
=> ( P @ X4 ) )
& ! [D3: nat] :
( ! [X3: nat] :
( ( ( ord_less_eq_nat @ A @ X3 )
& ( ord_less_nat @ X3 @ D3 ) )
=> ( P @ X3 ) )
=> ( ord_less_eq_nat @ D3 @ C3 ) ) ) ) ) ) ).
% complete_interval
thf(fact_591_pinf_I1_J,axiom,
! [P: int > $o,P4: int > $o,Q: int > $o,Q2: int > $o] :
( ? [Z4: int] :
! [X3: int] :
( ( ord_less_int @ Z4 @ X3 )
=> ( ( P @ X3 )
= ( P4 @ X3 ) ) )
=> ( ? [Z4: int] :
! [X3: int] :
( ( ord_less_int @ Z4 @ X3 )
=> ( ( Q @ X3 )
= ( Q2 @ X3 ) ) )
=> ? [Z3: int] :
! [X4: int] :
( ( ord_less_int @ Z3 @ X4 )
=> ( ( ( P @ X4 )
& ( Q @ X4 ) )
= ( ( P4 @ X4 )
& ( Q2 @ X4 ) ) ) ) ) ) ).
% pinf(1)
thf(fact_592_pinf_I1_J,axiom,
! [P: nat > $o,P4: nat > $o,Q: nat > $o,Q2: nat > $o] :
( ? [Z4: nat] :
! [X3: nat] :
( ( ord_less_nat @ Z4 @ X3 )
=> ( ( P @ X3 )
= ( P4 @ X3 ) ) )
=> ( ? [Z4: nat] :
! [X3: nat] :
( ( ord_less_nat @ Z4 @ X3 )
=> ( ( Q @ X3 )
= ( Q2 @ X3 ) ) )
=> ? [Z3: nat] :
! [X4: nat] :
( ( ord_less_nat @ Z3 @ X4 )
=> ( ( ( P @ X4 )
& ( Q @ X4 ) )
= ( ( P4 @ X4 )
& ( Q2 @ X4 ) ) ) ) ) ) ).
% pinf(1)
thf(fact_593_pinf_I2_J,axiom,
! [P: int > $o,P4: int > $o,Q: int > $o,Q2: int > $o] :
( ? [Z4: int] :
! [X3: int] :
( ( ord_less_int @ Z4 @ X3 )
=> ( ( P @ X3 )
= ( P4 @ X3 ) ) )
=> ( ? [Z4: int] :
! [X3: int] :
( ( ord_less_int @ Z4 @ X3 )
=> ( ( Q @ X3 )
= ( Q2 @ X3 ) ) )
=> ? [Z3: int] :
! [X4: int] :
( ( ord_less_int @ Z3 @ X4 )
=> ( ( ( P @ X4 )
| ( Q @ X4 ) )
= ( ( P4 @ X4 )
| ( Q2 @ X4 ) ) ) ) ) ) ).
% pinf(2)
thf(fact_594_pinf_I2_J,axiom,
! [P: nat > $o,P4: nat > $o,Q: nat > $o,Q2: nat > $o] :
( ? [Z4: nat] :
! [X3: nat] :
( ( ord_less_nat @ Z4 @ X3 )
=> ( ( P @ X3 )
= ( P4 @ X3 ) ) )
=> ( ? [Z4: nat] :
! [X3: nat] :
( ( ord_less_nat @ Z4 @ X3 )
=> ( ( Q @ X3 )
= ( Q2 @ X3 ) ) )
=> ? [Z3: nat] :
! [X4: nat] :
( ( ord_less_nat @ Z3 @ X4 )
=> ( ( ( P @ X4 )
| ( Q @ X4 ) )
= ( ( P4 @ X4 )
| ( Q2 @ X4 ) ) ) ) ) ) ).
% pinf(2)
thf(fact_595_pinf_I3_J,axiom,
! [T: int] :
? [Z3: int] :
! [X4: int] :
( ( ord_less_int @ Z3 @ X4 )
=> ( X4 != T ) ) ).
% pinf(3)
thf(fact_596_pinf_I3_J,axiom,
! [T: nat] :
? [Z3: nat] :
! [X4: nat] :
( ( ord_less_nat @ Z3 @ X4 )
=> ( X4 != T ) ) ).
% pinf(3)
thf(fact_597_pinf_I4_J,axiom,
! [T: int] :
? [Z3: int] :
! [X4: int] :
( ( ord_less_int @ Z3 @ X4 )
=> ( X4 != T ) ) ).
% pinf(4)
thf(fact_598_pinf_I4_J,axiom,
! [T: nat] :
? [Z3: nat] :
! [X4: nat] :
( ( ord_less_nat @ Z3 @ X4 )
=> ( X4 != T ) ) ).
% pinf(4)
thf(fact_599_pinf_I5_J,axiom,
! [T: int] :
? [Z3: int] :
! [X4: int] :
( ( ord_less_int @ Z3 @ X4 )
=> ~ ( ord_less_int @ X4 @ T ) ) ).
% pinf(5)
thf(fact_600_pinf_I5_J,axiom,
! [T: nat] :
? [Z3: nat] :
! [X4: nat] :
( ( ord_less_nat @ Z3 @ X4 )
=> ~ ( ord_less_nat @ X4 @ T ) ) ).
% pinf(5)
thf(fact_601_pinf_I7_J,axiom,
! [T: int] :
? [Z3: int] :
! [X4: int] :
( ( ord_less_int @ Z3 @ X4 )
=> ( ord_less_int @ T @ X4 ) ) ).
% pinf(7)
thf(fact_602_pinf_I7_J,axiom,
! [T: nat] :
? [Z3: nat] :
! [X4: nat] :
( ( ord_less_nat @ Z3 @ X4 )
=> ( ord_less_nat @ T @ X4 ) ) ).
% pinf(7)
thf(fact_603_minf_I1_J,axiom,
! [P: int > $o,P4: int > $o,Q: int > $o,Q2: int > $o] :
( ? [Z4: int] :
! [X3: int] :
( ( ord_less_int @ X3 @ Z4 )
=> ( ( P @ X3 )
= ( P4 @ X3 ) ) )
=> ( ? [Z4: int] :
! [X3: int] :
( ( ord_less_int @ X3 @ Z4 )
=> ( ( Q @ X3 )
= ( Q2 @ X3 ) ) )
=> ? [Z3: int] :
! [X4: int] :
( ( ord_less_int @ X4 @ Z3 )
=> ( ( ( P @ X4 )
& ( Q @ X4 ) )
= ( ( P4 @ X4 )
& ( Q2 @ X4 ) ) ) ) ) ) ).
% minf(1)
thf(fact_604_minf_I1_J,axiom,
! [P: nat > $o,P4: nat > $o,Q: nat > $o,Q2: nat > $o] :
( ? [Z4: nat] :
! [X3: nat] :
( ( ord_less_nat @ X3 @ Z4 )
=> ( ( P @ X3 )
= ( P4 @ X3 ) ) )
=> ( ? [Z4: nat] :
! [X3: nat] :
( ( ord_less_nat @ X3 @ Z4 )
=> ( ( Q @ X3 )
= ( Q2 @ X3 ) ) )
=> ? [Z3: nat] :
! [X4: nat] :
( ( ord_less_nat @ X4 @ Z3 )
=> ( ( ( P @ X4 )
& ( Q @ X4 ) )
= ( ( P4 @ X4 )
& ( Q2 @ X4 ) ) ) ) ) ) ).
% minf(1)
thf(fact_605_minf_I2_J,axiom,
! [P: int > $o,P4: int > $o,Q: int > $o,Q2: int > $o] :
( ? [Z4: int] :
! [X3: int] :
( ( ord_less_int @ X3 @ Z4 )
=> ( ( P @ X3 )
= ( P4 @ X3 ) ) )
=> ( ? [Z4: int] :
! [X3: int] :
( ( ord_less_int @ X3 @ Z4 )
=> ( ( Q @ X3 )
= ( Q2 @ X3 ) ) )
=> ? [Z3: int] :
! [X4: int] :
( ( ord_less_int @ X4 @ Z3 )
=> ( ( ( P @ X4 )
| ( Q @ X4 ) )
= ( ( P4 @ X4 )
| ( Q2 @ X4 ) ) ) ) ) ) ).
% minf(2)
thf(fact_606_minf_I2_J,axiom,
! [P: nat > $o,P4: nat > $o,Q: nat > $o,Q2: nat > $o] :
( ? [Z4: nat] :
! [X3: nat] :
( ( ord_less_nat @ X3 @ Z4 )
=> ( ( P @ X3 )
= ( P4 @ X3 ) ) )
=> ( ? [Z4: nat] :
! [X3: nat] :
( ( ord_less_nat @ X3 @ Z4 )
=> ( ( Q @ X3 )
= ( Q2 @ X3 ) ) )
=> ? [Z3: nat] :
! [X4: nat] :
( ( ord_less_nat @ X4 @ Z3 )
=> ( ( ( P @ X4 )
| ( Q @ X4 ) )
= ( ( P4 @ X4 )
| ( Q2 @ X4 ) ) ) ) ) ) ).
% minf(2)
thf(fact_607_minf_I3_J,axiom,
! [T: int] :
? [Z3: int] :
! [X4: int] :
( ( ord_less_int @ X4 @ Z3 )
=> ( X4 != T ) ) ).
% minf(3)
thf(fact_608_minf_I3_J,axiom,
! [T: nat] :
? [Z3: nat] :
! [X4: nat] :
( ( ord_less_nat @ X4 @ Z3 )
=> ( X4 != T ) ) ).
% minf(3)
thf(fact_609_minf_I4_J,axiom,
! [T: int] :
? [Z3: int] :
! [X4: int] :
( ( ord_less_int @ X4 @ Z3 )
=> ( X4 != T ) ) ).
% minf(4)
thf(fact_610_minf_I4_J,axiom,
! [T: nat] :
? [Z3: nat] :
! [X4: nat] :
( ( ord_less_nat @ X4 @ Z3 )
=> ( X4 != T ) ) ).
% minf(4)
thf(fact_611_minf_I5_J,axiom,
! [T: int] :
? [Z3: int] :
! [X4: int] :
( ( ord_less_int @ X4 @ Z3 )
=> ( ord_less_int @ X4 @ T ) ) ).
% minf(5)
thf(fact_612_minf_I5_J,axiom,
! [T: nat] :
? [Z3: nat] :
! [X4: nat] :
( ( ord_less_nat @ X4 @ Z3 )
=> ( ord_less_nat @ X4 @ T ) ) ).
% minf(5)
thf(fact_613_minf_I7_J,axiom,
! [T: int] :
? [Z3: int] :
! [X4: int] :
( ( ord_less_int @ X4 @ Z3 )
=> ~ ( ord_less_int @ T @ X4 ) ) ).
% minf(7)
thf(fact_614_minf_I7_J,axiom,
! [T: nat] :
? [Z3: nat] :
! [X4: nat] :
( ( ord_less_nat @ X4 @ Z3 )
=> ~ ( ord_less_nat @ T @ X4 ) ) ).
% minf(7)
thf(fact_615_zdiff__int__split,axiom,
! [P: int > $o,X: nat,Y: nat] :
( ( P @ ( semiri1314217659103216013at_int @ ( minus_minus_nat @ X @ Y ) ) )
= ( ( ( ord_less_eq_nat @ Y @ X )
=> ( P @ ( minus_minus_int @ ( semiri1314217659103216013at_int @ X ) @ ( semiri1314217659103216013at_int @ Y ) ) ) )
& ( ( ord_less_nat @ X @ Y )
=> ( P @ zero_zero_int ) ) ) ) ).
% zdiff_int_split
thf(fact_616_list__ex__length,axiom,
( list_e3437881803614826702st_int
= ( ^ [P3: produc6425607678544837394st_int > $o,Xs2: list_P419547167332244514st_int] :
? [N3: nat] :
( ( ord_less_nat @ N3 @ ( size_s229347708945616566st_int @ Xs2 ) )
& ( P3 @ ( nth_Pr8327173284404383691st_int @ Xs2 @ N3 ) ) ) ) ) ).
% list_ex_length
thf(fact_617_list__ex__length,axiom,
( list_ex_instr
= ( ^ [P3: instr > $o,Xs2: list_instr] :
? [N3: nat] :
( ( ord_less_nat @ N3 @ ( size_size_list_instr @ Xs2 ) )
& ( P3 @ ( nth_instr @ Xs2 @ N3 ) ) ) ) ) ).
% list_ex_length
thf(fact_618_list__ex__length,axiom,
( list_ex_list_char
= ( ^ [P3: list_char > $o,Xs2: list_list_char] :
? [N3: nat] :
( ( ord_less_nat @ N3 @ ( size_s356637359517785349t_char @ Xs2 ) )
& ( P3 @ ( nth_list_char @ Xs2 @ N3 ) ) ) ) ) ).
% list_ex_length
thf(fact_619_list__ex__length,axiom,
( list_ex_int
= ( ^ [P3: int > $o,Xs2: list_int] :
? [N3: nat] :
( ( ord_less_nat @ N3 @ ( size_size_list_int @ Xs2 ) )
& ( P3 @ ( nth_int @ Xs2 @ N3 ) ) ) ) ) ).
% list_ex_length
thf(fact_620_conj__subset__def,axiom,
! [A2: set_int,P: int > $o,Q: int > $o] :
( ( ord_less_eq_set_int @ A2
@ ( collect_int
@ ^ [X2: int] :
( ( P @ X2 )
& ( Q @ X2 ) ) ) )
= ( ( ord_less_eq_set_int @ A2 @ ( collect_int @ P ) )
& ( ord_less_eq_set_int @ A2 @ ( collect_int @ Q ) ) ) ) ).
% conj_subset_def
thf(fact_621_conj__subset__def,axiom,
! [A2: set_nat,P: nat > $o,Q: nat > $o] :
( ( ord_less_eq_set_nat @ A2
@ ( collect_nat
@ ^ [X2: nat] :
( ( P @ X2 )
& ( Q @ X2 ) ) ) )
= ( ( ord_less_eq_set_nat @ A2 @ ( collect_nat @ P ) )
& ( ord_less_eq_set_nat @ A2 @ ( collect_nat @ Q ) ) ) ) ).
% conj_subset_def
thf(fact_622_prop__restrict,axiom,
! [X: list_char,Z5: set_list_char,X7: set_list_char,P: list_char > $o] :
( ( member_list_char @ X @ Z5 )
=> ( ( ord_le1207710660481778807t_char @ Z5
@ ( collect_list_char
@ ^ [X2: list_char] :
( ( member_list_char @ X2 @ X7 )
& ( P @ X2 ) ) ) )
=> ( P @ X ) ) ) ).
% prop_restrict
thf(fact_623_prop__restrict,axiom,
! [X: int,Z5: set_int,X7: set_int,P: int > $o] :
( ( member_int @ X @ Z5 )
=> ( ( ord_less_eq_set_int @ Z5
@ ( collect_int
@ ^ [X2: int] :
( ( member_int @ X2 @ X7 )
& ( P @ X2 ) ) ) )
=> ( P @ X ) ) ) ).
% prop_restrict
thf(fact_624_prop__restrict,axiom,
! [X: nat,Z5: set_nat,X7: set_nat,P: nat > $o] :
( ( member_nat @ X @ Z5 )
=> ( ( ord_less_eq_set_nat @ Z5
@ ( collect_nat
@ ^ [X2: nat] :
( ( member_nat @ X2 @ X7 )
& ( P @ X2 ) ) ) )
=> ( P @ X ) ) ) ).
% prop_restrict
thf(fact_625_Collect__restrict,axiom,
! [X7: set_list_char,P: list_char > $o] :
( ord_le1207710660481778807t_char
@ ( collect_list_char
@ ^ [X2: list_char] :
( ( member_list_char @ X2 @ X7 )
& ( P @ X2 ) ) )
@ X7 ) ).
% Collect_restrict
thf(fact_626_Collect__restrict,axiom,
! [X7: set_int,P: int > $o] :
( ord_less_eq_set_int
@ ( collect_int
@ ^ [X2: int] :
( ( member_int @ X2 @ X7 )
& ( P @ X2 ) ) )
@ X7 ) ).
% Collect_restrict
thf(fact_627_Collect__restrict,axiom,
! [X7: set_nat,P: nat > $o] :
( ord_less_eq_set_nat
@ ( collect_nat
@ ^ [X2: nat] :
( ( member_nat @ X2 @ X7 )
& ( P @ X2 ) ) )
@ X7 ) ).
% Collect_restrict
thf(fact_628_subset__CollectI,axiom,
! [B5: set_list_char,A2: set_list_char,Q: list_char > $o,P: list_char > $o] :
( ( ord_le1207710660481778807t_char @ B5 @ A2 )
=> ( ! [X3: list_char] :
( ( member_list_char @ X3 @ B5 )
=> ( ( Q @ X3 )
=> ( P @ X3 ) ) )
=> ( ord_le1207710660481778807t_char
@ ( collect_list_char
@ ^ [X2: list_char] :
( ( member_list_char @ X2 @ B5 )
& ( Q @ X2 ) ) )
@ ( collect_list_char
@ ^ [X2: list_char] :
( ( member_list_char @ X2 @ A2 )
& ( P @ X2 ) ) ) ) ) ) ).
% subset_CollectI
thf(fact_629_subset__CollectI,axiom,
! [B5: set_int,A2: set_int,Q: int > $o,P: int > $o] :
( ( ord_less_eq_set_int @ B5 @ A2 )
=> ( ! [X3: int] :
( ( member_int @ X3 @ B5 )
=> ( ( Q @ X3 )
=> ( P @ X3 ) ) )
=> ( ord_less_eq_set_int
@ ( collect_int
@ ^ [X2: int] :
( ( member_int @ X2 @ B5 )
& ( Q @ X2 ) ) )
@ ( collect_int
@ ^ [X2: int] :
( ( member_int @ X2 @ A2 )
& ( P @ X2 ) ) ) ) ) ) ).
% subset_CollectI
thf(fact_630_subset__CollectI,axiom,
! [B5: set_nat,A2: set_nat,Q: nat > $o,P: nat > $o] :
( ( ord_less_eq_set_nat @ B5 @ A2 )
=> ( ! [X3: nat] :
( ( member_nat @ X3 @ B5 )
=> ( ( Q @ X3 )
=> ( P @ X3 ) ) )
=> ( ord_less_eq_set_nat
@ ( collect_nat
@ ^ [X2: nat] :
( ( member_nat @ X2 @ B5 )
& ( Q @ X2 ) ) )
@ ( collect_nat
@ ^ [X2: nat] :
( ( member_nat @ X2 @ A2 )
& ( P @ X2 ) ) ) ) ) ) ).
% subset_CollectI
thf(fact_631_subset__Collect__iff,axiom,
! [B5: set_list_char,A2: set_list_char,P: list_char > $o] :
( ( ord_le1207710660481778807t_char @ B5 @ A2 )
=> ( ( ord_le1207710660481778807t_char @ B5
@ ( collect_list_char
@ ^ [X2: list_char] :
( ( member_list_char @ X2 @ A2 )
& ( P @ X2 ) ) ) )
= ( ! [X2: list_char] :
( ( member_list_char @ X2 @ B5 )
=> ( P @ X2 ) ) ) ) ) ).
% subset_Collect_iff
thf(fact_632_subset__Collect__iff,axiom,
! [B5: set_int,A2: set_int,P: int > $o] :
( ( ord_less_eq_set_int @ B5 @ A2 )
=> ( ( ord_less_eq_set_int @ B5
@ ( collect_int
@ ^ [X2: int] :
( ( member_int @ X2 @ A2 )
& ( P @ X2 ) ) ) )
= ( ! [X2: int] :
( ( member_int @ X2 @ B5 )
=> ( P @ X2 ) ) ) ) ) ).
% subset_Collect_iff
thf(fact_633_subset__Collect__iff,axiom,
! [B5: set_nat,A2: set_nat,P: nat > $o] :
( ( ord_less_eq_set_nat @ B5 @ A2 )
=> ( ( ord_less_eq_set_nat @ B5
@ ( collect_nat
@ ^ [X2: nat] :
( ( member_nat @ X2 @ A2 )
& ( P @ X2 ) ) ) )
= ( ! [X2: nat] :
( ( member_nat @ X2 @ B5 )
=> ( P @ X2 ) ) ) ) ) ).
% subset_Collect_iff
thf(fact_634_le__zero__eq,axiom,
! [N: nat] :
( ( ord_less_eq_nat @ N @ zero_zero_nat )
= ( N = zero_zero_nat ) ) ).
% le_zero_eq
thf(fact_635_not__gr__zero,axiom,
! [N: nat] :
( ( ~ ( ord_less_nat @ zero_zero_nat @ N ) )
= ( N = zero_zero_nat ) ) ).
% not_gr_zero
thf(fact_636_diff__self,axiom,
! [A: int] :
( ( minus_minus_int @ A @ A )
= zero_zero_int ) ).
% diff_self
thf(fact_637_diff__0__right,axiom,
! [A: int] :
( ( minus_minus_int @ A @ zero_zero_int )
= A ) ).
% diff_0_right
thf(fact_638_zero__diff,axiom,
! [A: nat] :
( ( minus_minus_nat @ zero_zero_nat @ A )
= zero_zero_nat ) ).
% zero_diff
thf(fact_639_diff__zero,axiom,
! [A: int] :
( ( minus_minus_int @ A @ zero_zero_int )
= A ) ).
% diff_zero
thf(fact_640_diff__zero,axiom,
! [A: nat] :
( ( minus_minus_nat @ A @ zero_zero_nat )
= A ) ).
% diff_zero
thf(fact_641_cancel__comm__monoid__add__class_Odiff__cancel,axiom,
! [A: int] :
( ( minus_minus_int @ A @ A )
= zero_zero_int ) ).
% cancel_comm_monoid_add_class.diff_cancel
thf(fact_642_cancel__comm__monoid__add__class_Odiff__cancel,axiom,
! [A: nat] :
( ( minus_minus_nat @ A @ A )
= zero_zero_nat ) ).
% cancel_comm_monoid_add_class.diff_cancel
thf(fact_643_of__nat__0,axiom,
( ( semiri1314217659103216013at_int @ zero_zero_nat )
= zero_zero_int ) ).
% of_nat_0
thf(fact_644_of__nat__0,axiom,
( ( semiri1316708129612266289at_nat @ zero_zero_nat )
= zero_zero_nat ) ).
% of_nat_0
thf(fact_645_of__nat__0__eq__iff,axiom,
! [N: nat] :
( ( zero_zero_int
= ( semiri1314217659103216013at_int @ N ) )
= ( zero_zero_nat = N ) ) ).
% of_nat_0_eq_iff
thf(fact_646_of__nat__0__eq__iff,axiom,
! [N: nat] :
( ( zero_zero_nat
= ( semiri1316708129612266289at_nat @ N ) )
= ( zero_zero_nat = N ) ) ).
% of_nat_0_eq_iff
thf(fact_647_of__nat__eq__0__iff,axiom,
! [M: nat] :
( ( ( semiri1314217659103216013at_int @ M )
= zero_zero_int )
= ( M = zero_zero_nat ) ) ).
% of_nat_eq_0_iff
thf(fact_648_of__nat__eq__0__iff,axiom,
! [M: nat] :
( ( ( semiri1316708129612266289at_nat @ M )
= zero_zero_nat )
= ( M = zero_zero_nat ) ) ).
% of_nat_eq_0_iff
thf(fact_649_list__ex__append,axiom,
! [P: instr > $o,Xs: list_instr,Ys: list_instr] :
( ( list_ex_instr @ P @ ( append_instr @ Xs @ Ys ) )
= ( ( list_ex_instr @ P @ Xs )
| ( list_ex_instr @ P @ Ys ) ) ) ).
% list_ex_append
thf(fact_650_list__ex__append,axiom,
! [P: int > $o,Xs: list_int,Ys: list_int] :
( ( list_ex_int @ P @ ( append_int @ Xs @ Ys ) )
= ( ( list_ex_int @ P @ Xs )
| ( list_ex_int @ P @ Ys ) ) ) ).
% list_ex_append
thf(fact_651_diff__ge__0__iff__ge,axiom,
! [A: int,B: int] :
( ( ord_less_eq_int @ zero_zero_int @ ( minus_minus_int @ A @ B ) )
= ( ord_less_eq_int @ B @ A ) ) ).
% diff_ge_0_iff_ge
thf(fact_652_diff__gt__0__iff__gt,axiom,
! [A: int,B: int] :
( ( ord_less_int @ zero_zero_int @ ( minus_minus_int @ A @ B ) )
= ( ord_less_int @ B @ A ) ) ).
% diff_gt_0_iff_gt
thf(fact_653_of__nat__le__0__iff,axiom,
! [M: nat] :
( ( ord_less_eq_int @ ( semiri1314217659103216013at_int @ M ) @ zero_zero_int )
= ( M = zero_zero_nat ) ) ).
% of_nat_le_0_iff
thf(fact_654_of__nat__le__0__iff,axiom,
! [M: nat] :
( ( ord_less_eq_nat @ ( semiri1316708129612266289at_nat @ M ) @ zero_zero_nat )
= ( M = zero_zero_nat ) ) ).
% of_nat_le_0_iff
thf(fact_655_of__nat__0__less__iff,axiom,
! [N: nat] :
( ( ord_less_int @ zero_zero_int @ ( semiri1314217659103216013at_int @ N ) )
= ( ord_less_nat @ zero_zero_nat @ N ) ) ).
% of_nat_0_less_iff
thf(fact_656_of__nat__0__less__iff,axiom,
! [N: nat] :
( ( ord_less_nat @ zero_zero_nat @ ( semiri1316708129612266289at_nat @ N ) )
= ( ord_less_nat @ zero_zero_nat @ N ) ) ).
% of_nat_0_less_iff
thf(fact_657_zero__reorient,axiom,
! [X: int] :
( ( zero_zero_int = X )
= ( X = zero_zero_int ) ) ).
% zero_reorient
thf(fact_658_zero__reorient,axiom,
! [X: nat] :
( ( zero_zero_nat = X )
= ( X = zero_zero_nat ) ) ).
% zero_reorient
thf(fact_659_zero__le,axiom,
! [X: nat] : ( ord_less_eq_nat @ zero_zero_nat @ X ) ).
% zero_le
thf(fact_660_gr__zeroI,axiom,
! [N: nat] :
( ( N != zero_zero_nat )
=> ( ord_less_nat @ zero_zero_nat @ N ) ) ).
% gr_zeroI
thf(fact_661_not__less__zero,axiom,
! [N: nat] :
~ ( ord_less_nat @ N @ zero_zero_nat ) ).
% not_less_zero
thf(fact_662_gr__implies__not__zero,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ M @ N )
=> ( N != zero_zero_nat ) ) ).
% gr_implies_not_zero
thf(fact_663_zero__less__iff__neq__zero,axiom,
! [N: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
= ( N != zero_zero_nat ) ) ).
% zero_less_iff_neq_zero
thf(fact_664_eq__iff__diff__eq__0,axiom,
( ( ^ [Y4: int,Z2: int] : ( Y4 = Z2 ) )
= ( ^ [A3: int,B2: int] :
( ( minus_minus_int @ A3 @ B2 )
= zero_zero_int ) ) ) ).
% eq_iff_diff_eq_0
thf(fact_665_less__eq__int__code_I1_J,axiom,
ord_less_eq_int @ zero_zero_int @ zero_zero_int ).
% less_eq_int_code(1)
thf(fact_666_imp__le__cong,axiom,
! [X: int,X8: int,P: $o,P4: $o] :
( ( X = X8 )
=> ( ( ( ord_less_eq_int @ zero_zero_int @ X8 )
=> ( P = P4 ) )
=> ( ( ( ord_less_eq_int @ zero_zero_int @ X )
=> P )
= ( ( ord_less_eq_int @ zero_zero_int @ X8 )
=> P4 ) ) ) ) ).
% imp_le_cong
thf(fact_667_conj__le__cong,axiom,
! [X: int,X8: int,P: $o,P4: $o] :
( ( X = X8 )
=> ( ( ( ord_less_eq_int @ zero_zero_int @ X8 )
=> ( P = P4 ) )
=> ( ( ( ord_less_eq_int @ zero_zero_int @ X )
& P )
= ( ( ord_less_eq_int @ zero_zero_int @ X8 )
& P4 ) ) ) ) ).
% conj_le_cong
thf(fact_668_less__int__code_I1_J,axiom,
~ ( ord_less_int @ zero_zero_int @ zero_zero_int ) ).
% less_int_code(1)
thf(fact_669_minus__int__code_I1_J,axiom,
! [K: int] :
( ( minus_minus_int @ K @ zero_zero_int )
= K ) ).
% minus_int_code(1)
thf(fact_670_le__iff__diff__le__0,axiom,
( ord_less_eq_int
= ( ^ [A3: int,B2: int] : ( ord_less_eq_int @ ( minus_minus_int @ A3 @ B2 ) @ zero_zero_int ) ) ) ).
% le_iff_diff_le_0
thf(fact_671_less__iff__diff__less__0,axiom,
( ord_less_int
= ( ^ [A3: int,B2: int] : ( ord_less_int @ ( minus_minus_int @ A3 @ B2 ) @ zero_zero_int ) ) ) ).
% less_iff_diff_less_0
thf(fact_672_of__nat__0__le__iff,axiom,
! [N: nat] : ( ord_less_eq_int @ zero_zero_int @ ( semiri1314217659103216013at_int @ N ) ) ).
% of_nat_0_le_iff
thf(fact_673_of__nat__0__le__iff,axiom,
! [N: nat] : ( ord_less_eq_nat @ zero_zero_nat @ ( semiri1316708129612266289at_nat @ N ) ) ).
% of_nat_0_le_iff
thf(fact_674_of__nat__less__0__iff,axiom,
! [M: nat] :
~ ( ord_less_int @ ( semiri1314217659103216013at_int @ M ) @ zero_zero_int ) ).
% of_nat_less_0_iff
thf(fact_675_of__nat__less__0__iff,axiom,
! [M: nat] :
~ ( ord_less_nat @ ( semiri1316708129612266289at_nat @ M ) @ zero_zero_nat ) ).
% of_nat_less_0_iff
thf(fact_676_nonneg__int__cases,axiom,
! [K: int] :
( ( ord_less_eq_int @ zero_zero_int @ K )
=> ~ ! [N2: nat] :
( K
!= ( semiri1314217659103216013at_int @ N2 ) ) ) ).
% nonneg_int_cases
thf(fact_677_zero__le__imp__eq__int,axiom,
! [K: int] :
( ( ord_less_eq_int @ zero_zero_int @ K )
=> ? [N2: nat] :
( K
= ( semiri1314217659103216013at_int @ N2 ) ) ) ).
% zero_le_imp_eq_int
thf(fact_678_int__ops_I6_J,axiom,
! [A: nat,B: nat] :
( ( ( ord_less_int @ ( semiri1314217659103216013at_int @ A ) @ ( semiri1314217659103216013at_int @ B ) )
=> ( ( semiri1314217659103216013at_int @ ( minus_minus_nat @ A @ B ) )
= zero_zero_int ) )
& ( ~ ( ord_less_int @ ( semiri1314217659103216013at_int @ A ) @ ( semiri1314217659103216013at_int @ B ) )
=> ( ( semiri1314217659103216013at_int @ ( minus_minus_nat @ A @ B ) )
= ( minus_minus_int @ ( semiri1314217659103216013at_int @ A ) @ ( semiri1314217659103216013at_int @ B ) ) ) ) ) ).
% int_ops(6)
thf(fact_679_inth__append,axiom,
! [I: int,Xs: list_P419547167332244514st_int,Ys: list_P419547167332244514st_int] :
( ( ord_less_eq_int @ zero_zero_int @ I )
=> ( ( ( ord_less_int @ I @ ( semiri1314217659103216013at_int @ ( size_s229347708945616566st_int @ Xs ) ) )
=> ( ( inth_P8039235741264645271st_int @ ( append5028292981072010935st_int @ Xs @ Ys ) @ I )
= ( inth_P8039235741264645271st_int @ Xs @ I ) ) )
& ( ~ ( ord_less_int @ I @ ( semiri1314217659103216013at_int @ ( size_s229347708945616566st_int @ Xs ) ) )
=> ( ( inth_P8039235741264645271st_int @ ( append5028292981072010935st_int @ Xs @ Ys ) @ I )
= ( inth_P8039235741264645271st_int @ Ys @ ( minus_minus_int @ I @ ( semiri1314217659103216013at_int @ ( size_s229347708945616566st_int @ Xs ) ) ) ) ) ) ) ) ).
% inth_append
thf(fact_680_inth__append,axiom,
! [I: int,Xs: list_instr,Ys: list_instr] :
( ( ord_less_eq_int @ zero_zero_int @ I )
=> ( ( ( ord_less_int @ I @ ( semiri1314217659103216013at_int @ ( size_size_list_instr @ Xs ) ) )
=> ( ( inth_instr @ ( append_instr @ Xs @ Ys ) @ I )
= ( inth_instr @ Xs @ I ) ) )
& ( ~ ( ord_less_int @ I @ ( semiri1314217659103216013at_int @ ( size_size_list_instr @ Xs ) ) )
=> ( ( inth_instr @ ( append_instr @ Xs @ Ys ) @ I )
= ( inth_instr @ Ys @ ( minus_minus_int @ I @ ( semiri1314217659103216013at_int @ ( size_size_list_instr @ Xs ) ) ) ) ) ) ) ) ).
% inth_append
thf(fact_681_inth__append,axiom,
! [I: int,Xs: list_list_char,Ys: list_list_char] :
( ( ord_less_eq_int @ zero_zero_int @ I )
=> ( ( ( ord_less_int @ I @ ( semiri1314217659103216013at_int @ ( size_s356637359517785349t_char @ Xs ) ) )
=> ( ( inth_list_char @ ( append_list_char @ Xs @ Ys ) @ I )
= ( inth_list_char @ Xs @ I ) ) )
& ( ~ ( ord_less_int @ I @ ( semiri1314217659103216013at_int @ ( size_s356637359517785349t_char @ Xs ) ) )
=> ( ( inth_list_char @ ( append_list_char @ Xs @ Ys ) @ I )
= ( inth_list_char @ Ys @ ( minus_minus_int @ I @ ( semiri1314217659103216013at_int @ ( size_s356637359517785349t_char @ Xs ) ) ) ) ) ) ) ) ).
% inth_append
thf(fact_682_inth__append,axiom,
! [I: int,Xs: list_int,Ys: list_int] :
( ( ord_less_eq_int @ zero_zero_int @ I )
=> ( ( ( ord_less_int @ I @ ( semiri1314217659103216013at_int @ ( size_size_list_int @ Xs ) ) )
=> ( ( inth_int @ ( append_int @ Xs @ Ys ) @ I )
= ( inth_int @ Xs @ I ) ) )
& ( ~ ( ord_less_int @ I @ ( semiri1314217659103216013at_int @ ( size_size_list_int @ Xs ) ) )
=> ( ( inth_int @ ( append_int @ Xs @ Ys ) @ I )
= ( inth_int @ Ys @ ( minus_minus_int @ I @ ( semiri1314217659103216013at_int @ ( size_size_list_int @ Xs ) ) ) ) ) ) ) ) ).
% inth_append
thf(fact_683_less__numeral__extra_I3_J,axiom,
~ ( ord_less_int @ zero_zero_int @ zero_zero_int ) ).
% less_numeral_extra(3)
thf(fact_684_less__numeral__extra_I3_J,axiom,
~ ( ord_less_nat @ zero_zero_nat @ zero_zero_nat ) ).
% less_numeral_extra(3)
thf(fact_685_le__numeral__extra_I3_J,axiom,
ord_less_eq_int @ zero_zero_int @ zero_zero_int ).
% le_numeral_extra(3)
thf(fact_686_le__numeral__extra_I3_J,axiom,
ord_less_eq_nat @ zero_zero_nat @ zero_zero_nat ).
% le_numeral_extra(3)
thf(fact_687_execl__last,axiom,
! [P: list_instr,Cfs: list_P419547167332244514st_int,K: nat] :
( ( execl @ P @ Cfs )
=> ( ( ord_less_nat @ K @ ( size_s229347708945616566st_int @ Cfs ) )
=> ( ~ ( member_int @ ( produc8941309049508465880st_int @ ( nth_Pr8327173284404383691st_int @ Cfs @ K ) ) @ ( set_or4662586982721622107an_int @ zero_zero_int @ ( semiri1314217659103216013at_int @ ( size_size_list_instr @ P ) ) ) )
=> ( ( minus_minus_nat @ ( size_s229347708945616566st_int @ Cfs ) @ one_one_nat )
= K ) ) ) ) ).
% execl_last
thf(fact_688_nat__less__iff,axiom,
! [W: int,M: nat] :
( ( ord_less_eq_int @ zero_zero_int @ W )
=> ( ( ord_less_nat @ ( nat2 @ W ) @ M )
= ( ord_less_int @ W @ ( semiri1314217659103216013at_int @ M ) ) ) ) ).
% nat_less_iff
thf(fact_689_pos__int__cases,axiom,
! [K: int] :
( ( ord_less_int @ zero_zero_int @ K )
=> ~ ! [N2: nat] :
( ( K
= ( semiri1314217659103216013at_int @ N2 ) )
=> ~ ( ord_less_nat @ zero_zero_nat @ N2 ) ) ) ).
% pos_int_cases
thf(fact_690_bot__nat__0_Onot__eq__extremum,axiom,
! [A: nat] :
( ( A != zero_zero_nat )
= ( ord_less_nat @ zero_zero_nat @ A ) ) ).
% bot_nat_0.not_eq_extremum
thf(fact_691_neq0__conv,axiom,
! [N: nat] :
( ( N != zero_zero_nat )
= ( ord_less_nat @ zero_zero_nat @ N ) ) ).
% neq0_conv
thf(fact_692_less__nat__zero__code,axiom,
! [N: nat] :
~ ( ord_less_nat @ N @ zero_zero_nat ) ).
% less_nat_zero_code
thf(fact_693_bot__nat__0_Oextremum,axiom,
! [A: nat] : ( ord_less_eq_nat @ zero_zero_nat @ A ) ).
% bot_nat_0.extremum
thf(fact_694_le0,axiom,
! [N: nat] : ( ord_less_eq_nat @ zero_zero_nat @ N ) ).
% le0
thf(fact_695_diff__self__eq__0,axiom,
! [M: nat] :
( ( minus_minus_nat @ M @ M )
= zero_zero_nat ) ).
% diff_self_eq_0
thf(fact_696_diff__0__eq__0,axiom,
! [N: nat] :
( ( minus_minus_nat @ zero_zero_nat @ N )
= zero_zero_nat ) ).
% diff_0_eq_0
thf(fact_697_nat__int,axiom,
! [N: nat] :
( ( nat2 @ ( semiri1314217659103216013at_int @ N ) )
= N ) ).
% nat_int
thf(fact_698_diff__numeral__special_I9_J,axiom,
( ( minus_minus_int @ one_one_int @ one_one_int )
= zero_zero_int ) ).
% diff_numeral_special(9)
thf(fact_699_zero__less__diff,axiom,
! [N: nat,M: nat] :
( ( ord_less_nat @ zero_zero_nat @ ( minus_minus_nat @ N @ M ) )
= ( ord_less_nat @ M @ N ) ) ).
% zero_less_diff
thf(fact_700_diff__is__0__eq,axiom,
! [M: nat,N: nat] :
( ( ( minus_minus_nat @ M @ N )
= zero_zero_nat )
= ( ord_less_eq_nat @ M @ N ) ) ).
% diff_is_0_eq
thf(fact_701_diff__is__0__eq_H,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq_nat @ M @ N )
=> ( ( minus_minus_nat @ M @ N )
= zero_zero_nat ) ) ).
% diff_is_0_eq'
thf(fact_702_of__nat__eq__1__iff,axiom,
! [N: nat] :
( ( ( semiri1314217659103216013at_int @ N )
= one_one_int )
= ( N = one_one_nat ) ) ).
% of_nat_eq_1_iff
thf(fact_703_of__nat__eq__1__iff,axiom,
! [N: nat] :
( ( ( semiri1316708129612266289at_nat @ N )
= one_one_nat )
= ( N = one_one_nat ) ) ).
% of_nat_eq_1_iff
thf(fact_704_of__nat__1__eq__iff,axiom,
! [N: nat] :
( ( one_one_int
= ( semiri1314217659103216013at_int @ N ) )
= ( N = one_one_nat ) ) ).
% of_nat_1_eq_iff
thf(fact_705_of__nat__1__eq__iff,axiom,
! [N: nat] :
( ( one_one_nat
= ( semiri1316708129612266289at_nat @ N ) )
= ( N = one_one_nat ) ) ).
% of_nat_1_eq_iff
thf(fact_706_of__nat__1,axiom,
( ( semiri1314217659103216013at_int @ one_one_nat )
= one_one_int ) ).
% of_nat_1
thf(fact_707_of__nat__1,axiom,
( ( semiri1316708129612266289at_nat @ one_one_nat )
= one_one_nat ) ).
% of_nat_1
thf(fact_708_less__one,axiom,
! [N: nat] :
( ( ord_less_nat @ N @ one_one_nat )
= ( N = zero_zero_nat ) ) ).
% less_one
thf(fact_709_nat__0__iff,axiom,
! [I: int] :
( ( ( nat2 @ I )
= zero_zero_nat )
= ( ord_less_eq_int @ I @ zero_zero_int ) ) ).
% nat_0_iff
thf(fact_710_nat__le__0,axiom,
! [Z: int] :
( ( ord_less_eq_int @ Z @ zero_zero_int )
=> ( ( nat2 @ Z )
= zero_zero_nat ) ) ).
% nat_le_0
thf(fact_711_zless__nat__conj,axiom,
! [W: int,Z: int] :
( ( ord_less_nat @ ( nat2 @ W ) @ ( nat2 @ Z ) )
= ( ( ord_less_int @ zero_zero_int @ Z )
& ( ord_less_int @ W @ Z ) ) ) ).
% zless_nat_conj
thf(fact_712_int__nat__eq,axiom,
! [Z: int] :
( ( ( ord_less_eq_int @ zero_zero_int @ Z )
=> ( ( semiri1314217659103216013at_int @ ( nat2 @ Z ) )
= Z ) )
& ( ~ ( ord_less_eq_int @ zero_zero_int @ Z )
=> ( ( semiri1314217659103216013at_int @ ( nat2 @ Z ) )
= zero_zero_int ) ) ) ).
% int_nat_eq
thf(fact_713_zero__less__nat__eq,axiom,
! [Z: int] :
( ( ord_less_nat @ zero_zero_nat @ ( nat2 @ Z ) )
= ( ord_less_int @ zero_zero_int @ Z ) ) ).
% zero_less_nat_eq
thf(fact_714_nat__zero__as__int,axiom,
( zero_zero_nat
= ( nat2 @ zero_zero_int ) ) ).
% nat_zero_as_int
thf(fact_715_less__numeral__extra_I4_J,axiom,
~ ( ord_less_int @ one_one_int @ one_one_int ) ).
% less_numeral_extra(4)
thf(fact_716_less__numeral__extra_I4_J,axiom,
~ ( ord_less_nat @ one_one_nat @ one_one_nat ) ).
% less_numeral_extra(4)
thf(fact_717_one__reorient,axiom,
! [X: nat] :
( ( one_one_nat = X )
= ( X = one_one_nat ) ) ).
% one_reorient
thf(fact_718_one__reorient,axiom,
! [X: int] :
( ( one_one_int = X )
= ( X = one_one_int ) ) ).
% one_reorient
thf(fact_719_le__numeral__extra_I4_J,axiom,
ord_less_eq_int @ one_one_int @ one_one_int ).
% le_numeral_extra(4)
thf(fact_720_le__numeral__extra_I4_J,axiom,
ord_less_eq_nat @ one_one_nat @ one_one_nat ).
% le_numeral_extra(4)
thf(fact_721_bot__nat__0_Oextremum__strict,axiom,
! [A: nat] :
~ ( ord_less_nat @ A @ zero_zero_nat ) ).
% bot_nat_0.extremum_strict
thf(fact_722_gr0I,axiom,
! [N: nat] :
( ( N != zero_zero_nat )
=> ( ord_less_nat @ zero_zero_nat @ N ) ) ).
% gr0I
thf(fact_723_not__gr0,axiom,
! [N: nat] :
( ( ~ ( ord_less_nat @ zero_zero_nat @ N ) )
= ( N = zero_zero_nat ) ) ).
% not_gr0
thf(fact_724_not__less0,axiom,
! [N: nat] :
~ ( ord_less_nat @ N @ zero_zero_nat ) ).
% not_less0
thf(fact_725_less__zeroE,axiom,
! [N: nat] :
~ ( ord_less_nat @ N @ zero_zero_nat ) ).
% less_zeroE
thf(fact_726_gr__implies__not0,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ M @ N )
=> ( N != zero_zero_nat ) ) ).
% gr_implies_not0
thf(fact_727_infinite__descent0,axiom,
! [P: nat > $o,N: nat] :
( ( P @ zero_zero_nat )
=> ( ! [N2: nat] :
( ( ord_less_nat @ zero_zero_nat @ N2 )
=> ( ~ ( P @ N2 )
=> ? [M4: nat] :
( ( ord_less_nat @ M4 @ N2 )
& ~ ( P @ M4 ) ) ) )
=> ( P @ N ) ) ) ).
% infinite_descent0
thf(fact_728_less__eq__nat_Osimps_I1_J,axiom,
! [N: nat] : ( ord_less_eq_nat @ zero_zero_nat @ N ) ).
% less_eq_nat.simps(1)
thf(fact_729_bot__nat__0_Oextremum__unique,axiom,
! [A: nat] :
( ( ord_less_eq_nat @ A @ zero_zero_nat )
= ( A = zero_zero_nat ) ) ).
% bot_nat_0.extremum_unique
thf(fact_730_bot__nat__0_Oextremum__uniqueI,axiom,
! [A: nat] :
( ( ord_less_eq_nat @ A @ zero_zero_nat )
=> ( A = zero_zero_nat ) ) ).
% bot_nat_0.extremum_uniqueI
thf(fact_731_le__0__eq,axiom,
! [N: nat] :
( ( ord_less_eq_nat @ N @ zero_zero_nat )
= ( N = zero_zero_nat ) ) ).
% le_0_eq
thf(fact_732_diffs0__imp__equal,axiom,
! [M: nat,N: nat] :
( ( ( minus_minus_nat @ M @ N )
= zero_zero_nat )
=> ( ( ( minus_minus_nat @ N @ M )
= zero_zero_nat )
=> ( M = N ) ) ) ).
% diffs0_imp_equal
thf(fact_733_minus__nat_Odiff__0,axiom,
! [M: nat] :
( ( minus_minus_nat @ M @ zero_zero_nat )
= M ) ).
% minus_nat.diff_0
thf(fact_734_less__numeral__extra_I1_J,axiom,
ord_less_int @ zero_zero_int @ one_one_int ).
% less_numeral_extra(1)
thf(fact_735_less__numeral__extra_I1_J,axiom,
ord_less_nat @ zero_zero_nat @ one_one_nat ).
% less_numeral_extra(1)
thf(fact_736_nat__eq__iff2,axiom,
! [M: nat,W: int] :
( ( M
= ( nat2 @ W ) )
= ( ( ( ord_less_eq_int @ zero_zero_int @ W )
=> ( W
= ( semiri1314217659103216013at_int @ M ) ) )
& ( ~ ( ord_less_eq_int @ zero_zero_int @ W )
=> ( M = zero_zero_nat ) ) ) ) ).
% nat_eq_iff2
thf(fact_737_nat__eq__iff,axiom,
! [W: int,M: nat] :
( ( ( nat2 @ W )
= M )
= ( ( ( ord_less_eq_int @ zero_zero_int @ W )
=> ( W
= ( semiri1314217659103216013at_int @ M ) ) )
& ( ~ ( ord_less_eq_int @ zero_zero_int @ W )
=> ( M = zero_zero_nat ) ) ) ) ).
% nat_eq_iff
thf(fact_738_split__nat,axiom,
! [P: nat > $o,I: int] :
( ( P @ ( nat2 @ I ) )
= ( ! [N3: nat] :
( ( I
= ( semiri1314217659103216013at_int @ N3 ) )
=> ( P @ N3 ) )
& ( ( ord_less_int @ I @ zero_zero_int )
=> ( P @ zero_zero_nat ) ) ) ) ).
% split_nat
thf(fact_739_nat__mono,axiom,
! [X: int,Y: int] :
( ( ord_less_eq_int @ X @ Y )
=> ( ord_less_eq_nat @ ( nat2 @ X ) @ ( nat2 @ Y ) ) ) ).
% nat_mono
thf(fact_740_ex__nat,axiom,
( ( ^ [P2: nat > $o] :
? [X5: nat] : ( P2 @ X5 ) )
= ( ^ [P3: nat > $o] :
? [X2: int] :
( ( ord_less_eq_int @ zero_zero_int @ X2 )
& ( P3 @ ( nat2 @ X2 ) ) ) ) ) ).
% ex_nat
thf(fact_741_all__nat,axiom,
( ( ^ [P2: nat > $o] :
! [X5: nat] : ( P2 @ X5 ) )
= ( ^ [P3: nat > $o] :
! [X2: int] :
( ( ord_less_eq_int @ zero_zero_int @ X2 )
=> ( P3 @ ( nat2 @ X2 ) ) ) ) ) ).
% all_nat
thf(fact_742_eq__nat__nat__iff,axiom,
! [Z: int,Z6: int] :
( ( ord_less_eq_int @ zero_zero_int @ Z )
=> ( ( ord_less_eq_int @ zero_zero_int @ Z6 )
=> ( ( ( nat2 @ Z )
= ( nat2 @ Z6 ) )
= ( Z = Z6 ) ) ) ) ).
% eq_nat_nat_iff
thf(fact_743_diff__nat__eq__if,axiom,
! [Z6: int,Z: int] :
( ( ( ord_less_int @ Z6 @ zero_zero_int )
=> ( ( minus_minus_nat @ ( nat2 @ Z ) @ ( nat2 @ Z6 ) )
= ( nat2 @ Z ) ) )
& ( ~ ( ord_less_int @ Z6 @ zero_zero_int )
=> ( ( minus_minus_nat @ ( nat2 @ Z ) @ ( nat2 @ Z6 ) )
= ( if_nat @ ( ord_less_int @ ( minus_minus_int @ Z @ Z6 ) @ zero_zero_int ) @ zero_zero_nat @ ( nat2 @ ( minus_minus_int @ Z @ Z6 ) ) ) ) ) ) ).
% diff_nat_eq_if
thf(fact_744_ex__least__nat__le,axiom,
! [P: nat > $o,N: nat] :
( ( P @ N )
=> ( ~ ( P @ zero_zero_nat )
=> ? [K2: nat] :
( ( ord_less_eq_nat @ K2 @ N )
& ! [I4: nat] :
( ( ord_less_nat @ I4 @ K2 )
=> ~ ( P @ I4 ) )
& ( P @ K2 ) ) ) ) ).
% ex_least_nat_le
thf(fact_745_diff__less,axiom,
! [N: nat,M: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( ord_less_nat @ zero_zero_nat @ M )
=> ( ord_less_nat @ ( minus_minus_nat @ M @ N ) @ M ) ) ) ).
% diff_less
thf(fact_746_int__ops_I1_J,axiom,
( ( semiri1314217659103216013at_int @ zero_zero_nat )
= zero_zero_int ) ).
% int_ops(1)
thf(fact_747_all__nat__less__eq,axiom,
! [N: nat,P: nat > $o] :
( ( ! [M3: nat] :
( ( ord_less_nat @ M3 @ N )
=> ( P @ M3 ) ) )
= ( ! [X2: nat] :
( ( member_nat @ X2 @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ N ) )
=> ( P @ X2 ) ) ) ) ).
% all_nat_less_eq
thf(fact_748_ex__nat__less__eq,axiom,
! [N: nat,P: nat > $o] :
( ( ? [M3: nat] :
( ( ord_less_nat @ M3 @ N )
& ( P @ M3 ) ) )
= ( ? [X2: nat] :
( ( member_nat @ X2 @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ N ) )
& ( P @ X2 ) ) ) ) ).
% ex_nat_less_eq
thf(fact_749_nat__mono__iff,axiom,
! [Z: int,W: int] :
( ( ord_less_int @ zero_zero_int @ Z )
=> ( ( ord_less_nat @ ( nat2 @ W ) @ ( nat2 @ Z ) )
= ( ord_less_int @ W @ Z ) ) ) ).
% nat_mono_iff
thf(fact_750_zless__nat__eq__int__zless,axiom,
! [M: nat,Z: int] :
( ( ord_less_nat @ M @ ( nat2 @ Z ) )
= ( ord_less_int @ ( semiri1314217659103216013at_int @ M ) @ Z ) ) ).
% zless_nat_eq_int_zless
thf(fact_751_nat__le__iff,axiom,
! [X: int,N: nat] :
( ( ord_less_eq_nat @ ( nat2 @ X ) @ N )
= ( ord_less_eq_int @ X @ ( semiri1314217659103216013at_int @ N ) ) ) ).
% nat_le_iff
thf(fact_752_int__eq__iff,axiom,
! [M: nat,Z: int] :
( ( ( semiri1314217659103216013at_int @ M )
= Z )
= ( ( M
= ( nat2 @ Z ) )
& ( ord_less_eq_int @ zero_zero_int @ Z ) ) ) ).
% int_eq_iff
thf(fact_753_nat__0__le,axiom,
! [Z: int] :
( ( ord_less_eq_int @ zero_zero_int @ Z )
=> ( ( semiri1314217659103216013at_int @ ( nat2 @ Z ) )
= Z ) ) ).
% nat_0_le
thf(fact_754_int__minus,axiom,
! [N: nat,M: nat] :
( ( semiri1314217659103216013at_int @ ( minus_minus_nat @ N @ M ) )
= ( semiri1314217659103216013at_int @ ( nat2 @ ( minus_minus_int @ ( semiri1314217659103216013at_int @ N ) @ ( semiri1314217659103216013at_int @ M ) ) ) ) ) ).
% int_minus
thf(fact_755_nat__minus__as__int,axiom,
( minus_minus_nat
= ( ^ [A3: nat,B2: nat] : ( nat2 @ ( minus_minus_int @ ( semiri1314217659103216013at_int @ A3 ) @ ( semiri1314217659103216013at_int @ B2 ) ) ) ) ) ).
% nat_minus_as_int
thf(fact_756_nat__less__eq__zless,axiom,
! [W: int,Z: int] :
( ( ord_less_eq_int @ zero_zero_int @ W )
=> ( ( ord_less_nat @ ( nat2 @ W ) @ ( nat2 @ Z ) )
= ( ord_less_int @ W @ Z ) ) ) ).
% nat_less_eq_zless
thf(fact_757_le__nat__iff,axiom,
! [K: int,N: nat] :
( ( ord_less_eq_int @ zero_zero_int @ K )
=> ( ( ord_less_eq_nat @ N @ ( nat2 @ K ) )
= ( ord_less_eq_int @ ( semiri1314217659103216013at_int @ N ) @ K ) ) ) ).
% le_nat_iff
thf(fact_758_nat__le__eq__zle,axiom,
! [W: int,Z: int] :
( ( ( ord_less_int @ zero_zero_int @ W )
| ( ord_less_eq_int @ zero_zero_int @ Z ) )
=> ( ( ord_less_eq_nat @ ( nat2 @ W ) @ ( nat2 @ Z ) )
= ( ord_less_eq_int @ W @ Z ) ) ) ).
% nat_le_eq_zle
thf(fact_759_nat__diff__distrib_H,axiom,
! [X: int,Y: int] :
( ( ord_less_eq_int @ zero_zero_int @ X )
=> ( ( ord_less_eq_int @ zero_zero_int @ Y )
=> ( ( nat2 @ ( minus_minus_int @ X @ Y ) )
= ( minus_minus_nat @ ( nat2 @ X ) @ ( nat2 @ Y ) ) ) ) ) ).
% nat_diff_distrib'
thf(fact_760_nat__diff__distrib,axiom,
! [Z6: int,Z: int] :
( ( ord_less_eq_int @ zero_zero_int @ Z6 )
=> ( ( ord_less_eq_int @ Z6 @ Z )
=> ( ( nat2 @ ( minus_minus_int @ Z @ Z6 ) )
= ( minus_minus_nat @ ( nat2 @ Z ) @ ( nat2 @ Z6 ) ) ) ) ) ).
% nat_diff_distrib
thf(fact_761_zero__less__imp__eq__int,axiom,
! [K: int] :
( ( ord_less_int @ zero_zero_int @ K )
=> ? [N2: nat] :
( ( ord_less_nat @ zero_zero_nat @ N2 )
& ( K
= ( semiri1314217659103216013at_int @ N2 ) ) ) ) ).
% zero_less_imp_eq_int
thf(fact_762_execl__all__def,axiom,
( execl_all
= ( ^ [P3: list_instr,Cfs2: list_P419547167332244514st_int] :
( ( execl @ P3 @ Cfs2 )
& ( Cfs2 != nil_Pr749310521696113548st_int )
& ( ( produc8941309049508465880st_int @ ( nth_Pr8327173284404383691st_int @ Cfs2 @ zero_zero_nat ) )
= zero_zero_int )
& ~ ( member_int @ ( produc8941309049508465880st_int @ ( nth_Pr8327173284404383691st_int @ Cfs2 @ ( minus_minus_nat @ ( size_s229347708945616566st_int @ Cfs2 ) @ one_one_nat ) ) ) @ ( set_or4662586982721622107an_int @ zero_zero_int @ ( semiri1314217659103216013at_int @ ( size_size_list_instr @ P3 ) ) ) ) ) ) ) ).
% execl_all_def
thf(fact_763_to__m__state_Osimps,axiom,
( to_m_state
= ( ^ [Xs2: list_list_char,S3: list_char > int,A3: int] :
( if_int
@ ( ( ord_less_int @ zero_zero_int @ A3 )
& ( ord_less_eq_int @ A3 @ ( semiri1314217659103216013at_int @ ( size_s356637359517785349t_char @ Xs2 ) ) ) )
@ ( S3 @ ( inth_list_char @ Xs2 @ ( minus_minus_int @ ( semiri1314217659103216013at_int @ ( size_s356637359517785349t_char @ Xs2 ) ) @ A3 ) ) )
@ zero_zero_int ) ) ) ).
% to_m_state.simps
thf(fact_764_to__m__state_Oelims,axiom,
! [X: list_list_char,Xa: list_char > int,Xb: int,Y: int] :
( ( ( to_m_state @ X @ Xa @ Xb )
= Y )
=> ( ( ( ( ord_less_int @ zero_zero_int @ Xb )
& ( ord_less_eq_int @ Xb @ ( semiri1314217659103216013at_int @ ( size_s356637359517785349t_char @ X ) ) ) )
=> ( Y
= ( Xa @ ( inth_list_char @ X @ ( minus_minus_int @ ( semiri1314217659103216013at_int @ ( size_s356637359517785349t_char @ X ) ) @ Xb ) ) ) ) )
& ( ~ ( ( ord_less_int @ zero_zero_int @ Xb )
& ( ord_less_eq_int @ Xb @ ( semiri1314217659103216013at_int @ ( size_s356637359517785349t_char @ X ) ) ) )
=> ( Y = zero_zero_int ) ) ) ) ).
% to_m_state.elims
thf(fact_765_not__one__less__zero,axiom,
~ ( ord_less_int @ one_one_int @ zero_zero_int ) ).
% not_one_less_zero
thf(fact_766_not__one__less__zero,axiom,
~ ( ord_less_nat @ one_one_nat @ zero_zero_nat ) ).
% not_one_less_zero
thf(fact_767_zero__less__one,axiom,
ord_less_int @ zero_zero_int @ one_one_int ).
% zero_less_one
thf(fact_768_zero__less__one,axiom,
ord_less_nat @ zero_zero_nat @ one_one_nat ).
% zero_less_one
thf(fact_769_not__one__le__zero,axiom,
~ ( ord_less_eq_int @ one_one_int @ zero_zero_int ) ).
% not_one_le_zero
thf(fact_770_not__one__le__zero,axiom,
~ ( ord_less_eq_nat @ one_one_nat @ zero_zero_nat ) ).
% not_one_le_zero
thf(fact_771_append__is__Nil__conv,axiom,
! [Xs: list_P419547167332244514st_int,Ys: list_P419547167332244514st_int] :
( ( ( append5028292981072010935st_int @ Xs @ Ys )
= nil_Pr749310521696113548st_int )
= ( ( Xs = nil_Pr749310521696113548st_int )
& ( Ys = nil_Pr749310521696113548st_int ) ) ) ).
% append_is_Nil_conv
thf(fact_772_append__is__Nil__conv,axiom,
! [Xs: list_instr,Ys: list_instr] :
( ( ( append_instr @ Xs @ Ys )
= nil_instr )
= ( ( Xs = nil_instr )
& ( Ys = nil_instr ) ) ) ).
% append_is_Nil_conv
thf(fact_773_append__is__Nil__conv,axiom,
! [Xs: list_int,Ys: list_int] :
( ( ( append_int @ Xs @ Ys )
= nil_int )
= ( ( Xs = nil_int )
& ( Ys = nil_int ) ) ) ).
% append_is_Nil_conv
thf(fact_774_Nil__is__append__conv,axiom,
! [Xs: list_P419547167332244514st_int,Ys: list_P419547167332244514st_int] :
( ( nil_Pr749310521696113548st_int
= ( append5028292981072010935st_int @ Xs @ Ys ) )
= ( ( Xs = nil_Pr749310521696113548st_int )
& ( Ys = nil_Pr749310521696113548st_int ) ) ) ).
% Nil_is_append_conv
thf(fact_775_Nil__is__append__conv,axiom,
! [Xs: list_instr,Ys: list_instr] :
( ( nil_instr
= ( append_instr @ Xs @ Ys ) )
= ( ( Xs = nil_instr )
& ( Ys = nil_instr ) ) ) ).
% Nil_is_append_conv
thf(fact_776_Nil__is__append__conv,axiom,
! [Xs: list_int,Ys: list_int] :
( ( nil_int
= ( append_int @ Xs @ Ys ) )
= ( ( Xs = nil_int )
& ( Ys = nil_int ) ) ) ).
% Nil_is_append_conv
thf(fact_777_self__append__conv2,axiom,
! [Y: list_P419547167332244514st_int,Xs: list_P419547167332244514st_int] :
( ( Y
= ( append5028292981072010935st_int @ Xs @ Y ) )
= ( Xs = nil_Pr749310521696113548st_int ) ) ).
% self_append_conv2
thf(fact_778_self__append__conv2,axiom,
! [Y: list_instr,Xs: list_instr] :
( ( Y
= ( append_instr @ Xs @ Y ) )
= ( Xs = nil_instr ) ) ).
% self_append_conv2
thf(fact_779_self__append__conv2,axiom,
! [Y: list_int,Xs: list_int] :
( ( Y
= ( append_int @ Xs @ Y ) )
= ( Xs = nil_int ) ) ).
% self_append_conv2
thf(fact_780_append__self__conv2,axiom,
! [Xs: list_P419547167332244514st_int,Ys: list_P419547167332244514st_int] :
( ( ( append5028292981072010935st_int @ Xs @ Ys )
= Ys )
= ( Xs = nil_Pr749310521696113548st_int ) ) ).
% append_self_conv2
thf(fact_781_append__self__conv2,axiom,
! [Xs: list_instr,Ys: list_instr] :
( ( ( append_instr @ Xs @ Ys )
= Ys )
= ( Xs = nil_instr ) ) ).
% append_self_conv2
thf(fact_782_append__self__conv2,axiom,
! [Xs: list_int,Ys: list_int] :
( ( ( append_int @ Xs @ Ys )
= Ys )
= ( Xs = nil_int ) ) ).
% append_self_conv2
thf(fact_783_self__append__conv,axiom,
! [Y: list_P419547167332244514st_int,Ys: list_P419547167332244514st_int] :
( ( Y
= ( append5028292981072010935st_int @ Y @ Ys ) )
= ( Ys = nil_Pr749310521696113548st_int ) ) ).
% self_append_conv
thf(fact_784_self__append__conv,axiom,
! [Y: list_instr,Ys: list_instr] :
( ( Y
= ( append_instr @ Y @ Ys ) )
= ( Ys = nil_instr ) ) ).
% self_append_conv
thf(fact_785_self__append__conv,axiom,
! [Y: list_int,Ys: list_int] :
( ( Y
= ( append_int @ Y @ Ys ) )
= ( Ys = nil_int ) ) ).
% self_append_conv
thf(fact_786_append__self__conv,axiom,
! [Xs: list_P419547167332244514st_int,Ys: list_P419547167332244514st_int] :
( ( ( append5028292981072010935st_int @ Xs @ Ys )
= Xs )
= ( Ys = nil_Pr749310521696113548st_int ) ) ).
% append_self_conv
thf(fact_787_append__self__conv,axiom,
! [Xs: list_instr,Ys: list_instr] :
( ( ( append_instr @ Xs @ Ys )
= Xs )
= ( Ys = nil_instr ) ) ).
% append_self_conv
thf(fact_788_append__self__conv,axiom,
! [Xs: list_int,Ys: list_int] :
( ( ( append_int @ Xs @ Ys )
= Xs )
= ( Ys = nil_int ) ) ).
% append_self_conv
thf(fact_789_append__Nil2,axiom,
! [Xs: list_P419547167332244514st_int] :
( ( append5028292981072010935st_int @ Xs @ nil_Pr749310521696113548st_int )
= Xs ) ).
% append_Nil2
thf(fact_790_append__Nil2,axiom,
! [Xs: list_instr] :
( ( append_instr @ Xs @ nil_instr )
= Xs ) ).
% append_Nil2
thf(fact_791_append__Nil2,axiom,
! [Xs: list_int] :
( ( append_int @ Xs @ nil_int )
= Xs ) ).
% append_Nil2
thf(fact_792_append_Oright__neutral,axiom,
! [A: list_P419547167332244514st_int] :
( ( append5028292981072010935st_int @ A @ nil_Pr749310521696113548st_int )
= A ) ).
% append.right_neutral
thf(fact_793_append_Oright__neutral,axiom,
! [A: list_instr] :
( ( append_instr @ A @ nil_instr )
= A ) ).
% append.right_neutral
thf(fact_794_append_Oright__neutral,axiom,
! [A: list_int] :
( ( append_int @ A @ nil_int )
= A ) ).
% append.right_neutral
thf(fact_795_list__ex__simps_I2_J,axiom,
! [P: produc6425607678544837394st_int > $o] :
~ ( list_e3437881803614826702st_int @ P @ nil_Pr749310521696113548st_int ) ).
% list_ex_simps(2)
thf(fact_796_list__ex__simps_I2_J,axiom,
! [P: instr > $o] :
~ ( list_ex_instr @ P @ nil_instr ) ).
% list_ex_simps(2)
thf(fact_797_list__ex__simps_I2_J,axiom,
! [P: int > $o] :
~ ( list_ex_int @ P @ nil_int ) ).
% list_ex_simps(2)
thf(fact_798_length__0__conv,axiom,
! [Xs: list_P419547167332244514st_int] :
( ( ( size_s229347708945616566st_int @ Xs )
= zero_zero_nat )
= ( Xs = nil_Pr749310521696113548st_int ) ) ).
% length_0_conv
thf(fact_799_length__0__conv,axiom,
! [Xs: list_instr] :
( ( ( size_size_list_instr @ Xs )
= zero_zero_nat )
= ( Xs = nil_instr ) ) ).
% length_0_conv
thf(fact_800_length__0__conv,axiom,
! [Xs: list_list_char] :
( ( ( size_s356637359517785349t_char @ Xs )
= zero_zero_nat )
= ( Xs = nil_list_char ) ) ).
% length_0_conv
thf(fact_801_length__0__conv,axiom,
! [Xs: list_int] :
( ( ( size_size_list_int @ Xs )
= zero_zero_nat )
= ( Xs = nil_int ) ) ).
% length_0_conv
thf(fact_802_zle__diff1__eq,axiom,
! [W: int,Z: int] :
( ( ord_less_eq_int @ W @ ( minus_minus_int @ Z @ one_one_int ) )
= ( ord_less_int @ W @ Z ) ) ).
% zle_diff1_eq
thf(fact_803_length__greater__0__conv,axiom,
! [Xs: list_P419547167332244514st_int] :
( ( ord_less_nat @ zero_zero_nat @ ( size_s229347708945616566st_int @ Xs ) )
= ( Xs != nil_Pr749310521696113548st_int ) ) ).
% length_greater_0_conv
thf(fact_804_length__greater__0__conv,axiom,
! [Xs: list_instr] :
( ( ord_less_nat @ zero_zero_nat @ ( size_size_list_instr @ Xs ) )
= ( Xs != nil_instr ) ) ).
% length_greater_0_conv
thf(fact_805_length__greater__0__conv,axiom,
! [Xs: list_list_char] :
( ( ord_less_nat @ zero_zero_nat @ ( size_s356637359517785349t_char @ Xs ) )
= ( Xs != nil_list_char ) ) ).
% length_greater_0_conv
thf(fact_806_length__greater__0__conv,axiom,
! [Xs: list_int] :
( ( ord_less_nat @ zero_zero_nat @ ( size_size_list_int @ Xs ) )
= ( Xs != nil_int ) ) ).
% length_greater_0_conv
thf(fact_807_eq__Nil__appendI,axiom,
! [Xs: list_P419547167332244514st_int,Ys: list_P419547167332244514st_int] :
( ( Xs = Ys )
=> ( Xs
= ( append5028292981072010935st_int @ nil_Pr749310521696113548st_int @ Ys ) ) ) ).
% eq_Nil_appendI
thf(fact_808_eq__Nil__appendI,axiom,
! [Xs: list_instr,Ys: list_instr] :
( ( Xs = Ys )
=> ( Xs
= ( append_instr @ nil_instr @ Ys ) ) ) ).
% eq_Nil_appendI
thf(fact_809_eq__Nil__appendI,axiom,
! [Xs: list_int,Ys: list_int] :
( ( Xs = Ys )
=> ( Xs
= ( append_int @ nil_int @ Ys ) ) ) ).
% eq_Nil_appendI
thf(fact_810_append_Oleft__neutral,axiom,
! [A: list_P419547167332244514st_int] :
( ( append5028292981072010935st_int @ nil_Pr749310521696113548st_int @ A )
= A ) ).
% append.left_neutral
thf(fact_811_append_Oleft__neutral,axiom,
! [A: list_instr] :
( ( append_instr @ nil_instr @ A )
= A ) ).
% append.left_neutral
thf(fact_812_append_Oleft__neutral,axiom,
! [A: list_int] :
( ( append_int @ nil_int @ A )
= A ) ).
% append.left_neutral
thf(fact_813_append__Nil,axiom,
! [Ys: list_P419547167332244514st_int] :
( ( append5028292981072010935st_int @ nil_Pr749310521696113548st_int @ Ys )
= Ys ) ).
% append_Nil
thf(fact_814_append__Nil,axiom,
! [Ys: list_instr] :
( ( append_instr @ nil_instr @ Ys )
= Ys ) ).
% append_Nil
thf(fact_815_append__Nil,axiom,
! [Ys: list_int] :
( ( append_int @ nil_int @ Ys )
= Ys ) ).
% append_Nil
thf(fact_816_execl_Osimps_I2_J,axiom,
! [P: list_instr] : ( execl @ P @ nil_Pr749310521696113548st_int ) ).
% execl.simps(2)
thf(fact_817_list_Osize_I3_J,axiom,
( ( size_s229347708945616566st_int @ nil_Pr749310521696113548st_int )
= zero_zero_nat ) ).
% list.size(3)
thf(fact_818_list_Osize_I3_J,axiom,
( ( size_size_list_instr @ nil_instr )
= zero_zero_nat ) ).
% list.size(3)
thf(fact_819_list_Osize_I3_J,axiom,
( ( size_s356637359517785349t_char @ nil_list_char )
= zero_zero_nat ) ).
% list.size(3)
thf(fact_820_list_Osize_I3_J,axiom,
( ( size_size_list_int @ nil_int )
= zero_zero_nat ) ).
% list.size(3)
thf(fact_821_int__ops_I2_J,axiom,
( ( semiri1314217659103216013at_int @ one_one_nat )
= one_one_int ) ).
% int_ops(2)
thf(fact_822_int__le__induct,axiom,
! [I: int,K: int,P: int > $o] :
( ( ord_less_eq_int @ I @ K )
=> ( ( P @ K )
=> ( ! [I2: int] :
( ( ord_less_eq_int @ I2 @ K )
=> ( ( P @ I2 )
=> ( P @ ( minus_minus_int @ I2 @ one_one_int ) ) ) )
=> ( P @ I ) ) ) ) ).
% int_le_induct
thf(fact_823_int__less__induct,axiom,
! [I: int,K: int,P: int > $o] :
( ( ord_less_int @ I @ K )
=> ( ( P @ ( minus_minus_int @ K @ one_one_int ) )
=> ( ! [I2: int] :
( ( ord_less_int @ I2 @ K )
=> ( ( P @ I2 )
=> ( P @ ( minus_minus_int @ I2 @ one_one_int ) ) ) )
=> ( P @ I ) ) ) ) ).
% int_less_induct
thf(fact_824_nat__one__as__int,axiom,
( one_one_nat
= ( nat2 @ one_one_int ) ) ).
% nat_one_as_int
thf(fact_825_linorder__neqE__linordered__idom,axiom,
! [X: int,Y: int] :
( ( X != Y )
=> ( ~ ( ord_less_int @ X @ Y )
=> ( ord_less_int @ Y @ X ) ) ) ).
% linorder_neqE_linordered_idom
thf(fact_826_int__one__le__iff__zero__less,axiom,
! [Z: int] :
( ( ord_less_eq_int @ one_one_int @ Z )
= ( ord_less_int @ zero_zero_int @ Z ) ) ).
% int_one_le_iff_zero_less
thf(fact_827_zero__less__one__class_Ozero__le__one,axiom,
ord_less_eq_int @ zero_zero_int @ one_one_int ).
% zero_less_one_class.zero_le_one
thf(fact_828_zero__less__one__class_Ozero__le__one,axiom,
ord_less_eq_nat @ zero_zero_nat @ one_one_nat ).
% zero_less_one_class.zero_le_one
thf(fact_829_linordered__nonzero__semiring__class_Ozero__le__one,axiom,
ord_less_eq_int @ zero_zero_int @ one_one_int ).
% linordered_nonzero_semiring_class.zero_le_one
thf(fact_830_linordered__nonzero__semiring__class_Ozero__le__one,axiom,
ord_less_eq_nat @ zero_zero_nat @ one_one_nat ).
% linordered_nonzero_semiring_class.zero_le_one
thf(fact_831_inth__map,axiom,
! [I: int,Xs: list_P419547167332244514st_int,F: produc6425607678544837394st_int > list_char] :
( ( ord_less_eq_int @ zero_zero_int @ I )
=> ( ( ord_less_int @ I @ ( semiri1314217659103216013at_int @ ( size_s229347708945616566st_int @ Xs ) ) )
=> ( ( inth_list_char @ ( map_Pr5338895242362685753t_char @ F @ Xs ) @ I )
= ( F @ ( inth_P8039235741264645271st_int @ Xs @ I ) ) ) ) ) ).
% inth_map
thf(fact_832_inth__map,axiom,
! [I: int,Xs: list_P419547167332244514st_int,F: produc6425607678544837394st_int > instr] :
( ( ord_less_eq_int @ zero_zero_int @ I )
=> ( ( ord_less_int @ I @ ( semiri1314217659103216013at_int @ ( size_s229347708945616566st_int @ Xs ) ) )
=> ( ( inth_instr @ ( map_Pr7514435936640407687_instr @ F @ Xs ) @ I )
= ( F @ ( inth_P8039235741264645271st_int @ Xs @ I ) ) ) ) ) ).
% inth_map
thf(fact_833_inth__map,axiom,
! [I: int,Xs: list_instr,F: instr > list_char] :
( ( ord_less_eq_int @ zero_zero_int @ I )
=> ( ( ord_less_int @ I @ ( semiri1314217659103216013at_int @ ( size_size_list_instr @ Xs ) ) )
=> ( ( inth_list_char @ ( map_instr_list_char @ F @ Xs ) @ I )
= ( F @ ( inth_instr @ Xs @ I ) ) ) ) ) ).
% inth_map
thf(fact_834_inth__map,axiom,
! [I: int,Xs: list_instr,F: instr > instr] :
( ( ord_less_eq_int @ zero_zero_int @ I )
=> ( ( ord_less_int @ I @ ( semiri1314217659103216013at_int @ ( size_size_list_instr @ Xs ) ) )
=> ( ( inth_instr @ ( map_instr_instr @ F @ Xs ) @ I )
= ( F @ ( inth_instr @ Xs @ I ) ) ) ) ) ).
% inth_map
thf(fact_835_inth__map,axiom,
! [I: int,Xs: list_list_char,F: list_char > list_char] :
( ( ord_less_eq_int @ zero_zero_int @ I )
=> ( ( ord_less_int @ I @ ( semiri1314217659103216013at_int @ ( size_s356637359517785349t_char @ Xs ) ) )
=> ( ( inth_list_char @ ( map_li116305933131242120t_char @ F @ Xs ) @ I )
= ( F @ ( inth_list_char @ Xs @ I ) ) ) ) ) ).
% inth_map
thf(fact_836_inth__map,axiom,
! [I: int,Xs: list_list_char,F: list_char > instr] :
( ( ord_less_eq_int @ zero_zero_int @ I )
=> ( ( ord_less_int @ I @ ( semiri1314217659103216013at_int @ ( size_s356637359517785349t_char @ Xs ) ) )
=> ( ( inth_instr @ ( map_list_char_instr @ F @ Xs ) @ I )
= ( F @ ( inth_list_char @ Xs @ I ) ) ) ) ) ).
% inth_map
thf(fact_837_inth__map,axiom,
! [I: int,Xs: list_int,F: int > list_char] :
( ( ord_less_eq_int @ zero_zero_int @ I )
=> ( ( ord_less_int @ I @ ( semiri1314217659103216013at_int @ ( size_size_list_int @ Xs ) ) )
=> ( ( inth_list_char @ ( map_int_list_char @ F @ Xs ) @ I )
= ( F @ ( inth_int @ Xs @ I ) ) ) ) ) ).
% inth_map
thf(fact_838_inth__map,axiom,
! [I: int,Xs: list_int,F: int > instr] :
( ( ord_less_eq_int @ zero_zero_int @ I )
=> ( ( ord_less_int @ I @ ( semiri1314217659103216013at_int @ ( size_size_list_int @ Xs ) ) )
=> ( ( inth_instr @ ( map_int_instr @ F @ Xs ) @ I )
= ( F @ ( inth_int @ Xs @ I ) ) ) ) ) ).
% inth_map
thf(fact_839_inth__set,axiom,
! [I: int,Xs: list_nat] :
( ( ord_less_eq_int @ zero_zero_int @ I )
=> ( ( ord_less_int @ I @ ( semiri1314217659103216013at_int @ ( size_size_list_nat @ Xs ) ) )
=> ( member_nat @ ( inth_nat @ Xs @ I ) @ ( set_nat2 @ Xs ) ) ) ) ).
% inth_set
thf(fact_840_inth__set,axiom,
! [I: int,Xs: list_P419547167332244514st_int] :
( ( ord_less_eq_int @ zero_zero_int @ I )
=> ( ( ord_less_int @ I @ ( semiri1314217659103216013at_int @ ( size_s229347708945616566st_int @ Xs ) ) )
=> ( member9100552558438320297st_int @ ( inth_P8039235741264645271st_int @ Xs @ I ) @ ( set_Pr3303432077563008829st_int @ Xs ) ) ) ) ).
% inth_set
thf(fact_841_inth__set,axiom,
! [I: int,Xs: list_instr] :
( ( ord_less_eq_int @ zero_zero_int @ I )
=> ( ( ord_less_int @ I @ ( semiri1314217659103216013at_int @ ( size_size_list_instr @ Xs ) ) )
=> ( member_instr @ ( inth_instr @ Xs @ I ) @ ( set_instr2 @ Xs ) ) ) ) ).
% inth_set
thf(fact_842_inth__set,axiom,
! [I: int,Xs: list_list_char] :
( ( ord_less_eq_int @ zero_zero_int @ I )
=> ( ( ord_less_int @ I @ ( semiri1314217659103216013at_int @ ( size_s356637359517785349t_char @ Xs ) ) )
=> ( member_list_char @ ( inth_list_char @ Xs @ I ) @ ( set_list_char2 @ Xs ) ) ) ) ).
% inth_set
thf(fact_843_inth__set,axiom,
! [I: int,Xs: list_int] :
( ( ord_less_eq_int @ zero_zero_int @ I )
=> ( ( ord_less_int @ I @ ( semiri1314217659103216013at_int @ ( size_size_list_int @ Xs ) ) )
=> ( member_int @ ( inth_int @ Xs @ I ) @ ( set_int2 @ Xs ) ) ) ) ).
% inth_set
thf(fact_844_nth__Cons__pos,axiom,
! [N: nat,X: produc6425607678544837394st_int,Xs: list_P419547167332244514st_int] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( nth_Pr8327173284404383691st_int @ ( cons_P4151473878978182108st_int @ X @ Xs ) @ N )
= ( nth_Pr8327173284404383691st_int @ Xs @ ( minus_minus_nat @ N @ one_one_nat ) ) ) ) ).
% nth_Cons_pos
thf(fact_845_nth__Cons__pos,axiom,
! [N: nat,X: int,Xs: list_int] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( nth_int @ ( cons_int @ X @ Xs ) @ N )
= ( nth_int @ Xs @ ( minus_minus_nat @ N @ one_one_nat ) ) ) ) ).
% nth_Cons_pos
thf(fact_846_nth__Cons__pos,axiom,
! [N: nat,X: list_char,Xs: list_list_char] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( nth_list_char @ ( cons_list_char @ X @ Xs ) @ N )
= ( nth_list_char @ Xs @ ( minus_minus_nat @ N @ one_one_nat ) ) ) ) ).
% nth_Cons_pos
thf(fact_847_last__conv__nth,axiom,
! [Xs: list_P419547167332244514st_int] :
( ( Xs != nil_Pr749310521696113548st_int )
=> ( ( last_P5391958344443371187st_int @ Xs )
= ( nth_Pr8327173284404383691st_int @ Xs @ ( minus_minus_nat @ ( size_s229347708945616566st_int @ Xs ) @ one_one_nat ) ) ) ) ).
% last_conv_nth
thf(fact_848_last__conv__nth,axiom,
! [Xs: list_instr] :
( ( Xs != nil_instr )
=> ( ( last_instr @ Xs )
= ( nth_instr @ Xs @ ( minus_minus_nat @ ( size_size_list_instr @ Xs ) @ one_one_nat ) ) ) ) ).
% last_conv_nth
thf(fact_849_last__conv__nth,axiom,
! [Xs: list_list_char] :
( ( Xs != nil_list_char )
=> ( ( last_list_char @ Xs )
= ( nth_list_char @ Xs @ ( minus_minus_nat @ ( size_s356637359517785349t_char @ Xs ) @ one_one_nat ) ) ) ) ).
% last_conv_nth
thf(fact_850_last__conv__nth,axiom,
! [Xs: list_int] :
( ( Xs != nil_int )
=> ( ( last_int @ Xs )
= ( nth_int @ Xs @ ( minus_minus_nat @ ( size_size_list_int @ Xs ) @ one_one_nat ) ) ) ) ).
% last_conv_nth
thf(fact_851_list_Oinject,axiom,
! [X21: produc6425607678544837394st_int,X22: list_P419547167332244514st_int,Y21: produc6425607678544837394st_int,Y22: list_P419547167332244514st_int] :
( ( ( cons_P4151473878978182108st_int @ X21 @ X22 )
= ( cons_P4151473878978182108st_int @ Y21 @ Y22 ) )
= ( ( X21 = Y21 )
& ( X22 = Y22 ) ) ) ).
% list.inject
thf(fact_852_list_Oinject,axiom,
! [X21: int,X22: list_int,Y21: int,Y22: list_int] :
( ( ( cons_int @ X21 @ X22 )
= ( cons_int @ Y21 @ Y22 ) )
= ( ( X21 = Y21 )
& ( X22 = Y22 ) ) ) ).
% list.inject
thf(fact_853_list_Oinject,axiom,
! [X21: list_char,X22: list_list_char,Y21: list_char,Y22: list_list_char] :
( ( ( cons_list_char @ X21 @ X22 )
= ( cons_list_char @ Y21 @ Y22 ) )
= ( ( X21 = Y21 )
& ( X22 = Y22 ) ) ) ).
% list.inject
thf(fact_854_map__is__Nil__conv,axiom,
! [F: produc6425607678544837394st_int > produc6425607678544837394st_int,Xs: list_P419547167332244514st_int] :
( ( ( map_Pr8625093460839654122st_int @ F @ Xs )
= nil_Pr749310521696113548st_int )
= ( Xs = nil_Pr749310521696113548st_int ) ) ).
% map_is_Nil_conv
thf(fact_855_map__is__Nil__conv,axiom,
! [F: instr > produc6425607678544837394st_int,Xs: list_instr] :
( ( ( map_in7259965606695656109st_int @ F @ Xs )
= nil_Pr749310521696113548st_int )
= ( Xs = nil_instr ) ) ).
% map_is_Nil_conv
thf(fact_856_map__is__Nil__conv,axiom,
! [F: int > produc6425607678544837394st_int,Xs: list_int] :
( ( ( map_in8512336359594627332st_int @ F @ Xs )
= nil_Pr749310521696113548st_int )
= ( Xs = nil_int ) ) ).
% map_is_Nil_conv
thf(fact_857_map__is__Nil__conv,axiom,
! [F: produc6425607678544837394st_int > instr,Xs: list_P419547167332244514st_int] :
( ( ( map_Pr7514435936640407687_instr @ F @ Xs )
= nil_instr )
= ( Xs = nil_Pr749310521696113548st_int ) ) ).
% map_is_Nil_conv
thf(fact_858_map__is__Nil__conv,axiom,
! [F: instr > instr,Xs: list_instr] :
( ( ( map_instr_instr @ F @ Xs )
= nil_instr )
= ( Xs = nil_instr ) ) ).
% map_is_Nil_conv
thf(fact_859_map__is__Nil__conv,axiom,
! [F: int > instr,Xs: list_int] :
( ( ( map_int_instr @ F @ Xs )
= nil_instr )
= ( Xs = nil_int ) ) ).
% map_is_Nil_conv
thf(fact_860_map__is__Nil__conv,axiom,
! [F: produc6425607678544837394st_int > int,Xs: list_P419547167332244514st_int] :
( ( ( map_Pr3224476576807795204nt_int @ F @ Xs )
= nil_int )
= ( Xs = nil_Pr749310521696113548st_int ) ) ).
% map_is_Nil_conv
thf(fact_861_map__is__Nil__conv,axiom,
! [F: instr > int,Xs: list_instr] :
( ( ( map_instr_int @ F @ Xs )
= nil_int )
= ( Xs = nil_instr ) ) ).
% map_is_Nil_conv
thf(fact_862_map__is__Nil__conv,axiom,
! [F: int > int,Xs: list_int] :
( ( ( map_int_int @ F @ Xs )
= nil_int )
= ( Xs = nil_int ) ) ).
% map_is_Nil_conv
thf(fact_863_Nil__is__map__conv,axiom,
! [F: produc6425607678544837394st_int > produc6425607678544837394st_int,Xs: list_P419547167332244514st_int] :
( ( nil_Pr749310521696113548st_int
= ( map_Pr8625093460839654122st_int @ F @ Xs ) )
= ( Xs = nil_Pr749310521696113548st_int ) ) ).
% Nil_is_map_conv
thf(fact_864_Nil__is__map__conv,axiom,
! [F: instr > produc6425607678544837394st_int,Xs: list_instr] :
( ( nil_Pr749310521696113548st_int
= ( map_in7259965606695656109st_int @ F @ Xs ) )
= ( Xs = nil_instr ) ) ).
% Nil_is_map_conv
thf(fact_865_Nil__is__map__conv,axiom,
! [F: int > produc6425607678544837394st_int,Xs: list_int] :
( ( nil_Pr749310521696113548st_int
= ( map_in8512336359594627332st_int @ F @ Xs ) )
= ( Xs = nil_int ) ) ).
% Nil_is_map_conv
thf(fact_866_Nil__is__map__conv,axiom,
! [F: produc6425607678544837394st_int > instr,Xs: list_P419547167332244514st_int] :
( ( nil_instr
= ( map_Pr7514435936640407687_instr @ F @ Xs ) )
= ( Xs = nil_Pr749310521696113548st_int ) ) ).
% Nil_is_map_conv
thf(fact_867_Nil__is__map__conv,axiom,
! [F: instr > instr,Xs: list_instr] :
( ( nil_instr
= ( map_instr_instr @ F @ Xs ) )
= ( Xs = nil_instr ) ) ).
% Nil_is_map_conv
thf(fact_868_Nil__is__map__conv,axiom,
! [F: int > instr,Xs: list_int] :
( ( nil_instr
= ( map_int_instr @ F @ Xs ) )
= ( Xs = nil_int ) ) ).
% Nil_is_map_conv
thf(fact_869_Nil__is__map__conv,axiom,
! [F: produc6425607678544837394st_int > int,Xs: list_P419547167332244514st_int] :
( ( nil_int
= ( map_Pr3224476576807795204nt_int @ F @ Xs ) )
= ( Xs = nil_Pr749310521696113548st_int ) ) ).
% Nil_is_map_conv
thf(fact_870_Nil__is__map__conv,axiom,
! [F: instr > int,Xs: list_instr] :
( ( nil_int
= ( map_instr_int @ F @ Xs ) )
= ( Xs = nil_instr ) ) ).
% Nil_is_map_conv
thf(fact_871_Nil__is__map__conv,axiom,
! [F: int > int,Xs: list_int] :
( ( nil_int
= ( map_int_int @ F @ Xs ) )
= ( Xs = nil_int ) ) ).
% Nil_is_map_conv
thf(fact_872_list_Omap__disc__iff,axiom,
! [F: produc6425607678544837394st_int > produc6425607678544837394st_int,A: list_P419547167332244514st_int] :
( ( ( map_Pr8625093460839654122st_int @ F @ A )
= nil_Pr749310521696113548st_int )
= ( A = nil_Pr749310521696113548st_int ) ) ).
% list.map_disc_iff
thf(fact_873_list_Omap__disc__iff,axiom,
! [F: instr > produc6425607678544837394st_int,A: list_instr] :
( ( ( map_in7259965606695656109st_int @ F @ A )
= nil_Pr749310521696113548st_int )
= ( A = nil_instr ) ) ).
% list.map_disc_iff
thf(fact_874_list_Omap__disc__iff,axiom,
! [F: int > produc6425607678544837394st_int,A: list_int] :
( ( ( map_in8512336359594627332st_int @ F @ A )
= nil_Pr749310521696113548st_int )
= ( A = nil_int ) ) ).
% list.map_disc_iff
thf(fact_875_list_Omap__disc__iff,axiom,
! [F: produc6425607678544837394st_int > instr,A: list_P419547167332244514st_int] :
( ( ( map_Pr7514435936640407687_instr @ F @ A )
= nil_instr )
= ( A = nil_Pr749310521696113548st_int ) ) ).
% list.map_disc_iff
thf(fact_876_list_Omap__disc__iff,axiom,
! [F: instr > instr,A: list_instr] :
( ( ( map_instr_instr @ F @ A )
= nil_instr )
= ( A = nil_instr ) ) ).
% list.map_disc_iff
thf(fact_877_list_Omap__disc__iff,axiom,
! [F: int > instr,A: list_int] :
( ( ( map_int_instr @ F @ A )
= nil_instr )
= ( A = nil_int ) ) ).
% list.map_disc_iff
thf(fact_878_list_Omap__disc__iff,axiom,
! [F: produc6425607678544837394st_int > int,A: list_P419547167332244514st_int] :
( ( ( map_Pr3224476576807795204nt_int @ F @ A )
= nil_int )
= ( A = nil_Pr749310521696113548st_int ) ) ).
% list.map_disc_iff
thf(fact_879_list_Omap__disc__iff,axiom,
! [F: instr > int,A: list_instr] :
( ( ( map_instr_int @ F @ A )
= nil_int )
= ( A = nil_instr ) ) ).
% list.map_disc_iff
thf(fact_880_list_Omap__disc__iff,axiom,
! [F: int > int,A: list_int] :
( ( ( map_int_int @ F @ A )
= nil_int )
= ( A = nil_int ) ) ).
% list.map_disc_iff
thf(fact_881_length__map,axiom,
! [F: instr > instr,Xs: list_instr] :
( ( size_size_list_instr @ ( map_instr_instr @ F @ Xs ) )
= ( size_size_list_instr @ Xs ) ) ).
% length_map
thf(fact_882_length__map,axiom,
! [F: int > instr,Xs: list_int] :
( ( size_size_list_instr @ ( map_int_instr @ F @ Xs ) )
= ( size_size_list_int @ Xs ) ) ).
% length_map
thf(fact_883_length__map,axiom,
! [F: instr > int,Xs: list_instr] :
( ( size_size_list_int @ ( map_instr_int @ F @ Xs ) )
= ( size_size_list_instr @ Xs ) ) ).
% length_map
thf(fact_884_length__map,axiom,
! [F: int > int,Xs: list_int] :
( ( size_size_list_int @ ( map_int_int @ F @ Xs ) )
= ( size_size_list_int @ Xs ) ) ).
% length_map
thf(fact_885_length__map,axiom,
! [F: list_char > instr,Xs: list_list_char] :
( ( size_size_list_instr @ ( map_list_char_instr @ F @ Xs ) )
= ( size_s356637359517785349t_char @ Xs ) ) ).
% length_map
thf(fact_886_length__map,axiom,
! [F: instr > list_char,Xs: list_instr] :
( ( size_s356637359517785349t_char @ ( map_instr_list_char @ F @ Xs ) )
= ( size_size_list_instr @ Xs ) ) ).
% length_map
thf(fact_887_length__map,axiom,
! [F: int > list_char,Xs: list_int] :
( ( size_s356637359517785349t_char @ ( map_int_list_char @ F @ Xs ) )
= ( size_size_list_int @ Xs ) ) ).
% length_map
thf(fact_888_length__map,axiom,
! [F: list_char > int,Xs: list_list_char] :
( ( size_size_list_int @ ( map_list_char_int @ F @ Xs ) )
= ( size_s356637359517785349t_char @ Xs ) ) ).
% length_map
thf(fact_889_length__map,axiom,
! [F: list_char > list_char,Xs: list_list_char] :
( ( size_s356637359517785349t_char @ ( map_li116305933131242120t_char @ F @ Xs ) )
= ( size_s356637359517785349t_char @ Xs ) ) ).
% length_map
thf(fact_890_length__map,axiom,
! [F: instr > produc6425607678544837394st_int,Xs: list_instr] :
( ( size_s229347708945616566st_int @ ( map_in7259965606695656109st_int @ F @ Xs ) )
= ( size_size_list_instr @ Xs ) ) ).
% length_map
thf(fact_891_map__append,axiom,
! [F: instr > instr,Xs: list_instr,Ys: list_instr] :
( ( map_instr_instr @ F @ ( append_instr @ Xs @ Ys ) )
= ( append_instr @ ( map_instr_instr @ F @ Xs ) @ ( map_instr_instr @ F @ Ys ) ) ) ).
% map_append
thf(fact_892_map__append,axiom,
! [F: instr > int,Xs: list_instr,Ys: list_instr] :
( ( map_instr_int @ F @ ( append_instr @ Xs @ Ys ) )
= ( append_int @ ( map_instr_int @ F @ Xs ) @ ( map_instr_int @ F @ Ys ) ) ) ).
% map_append
thf(fact_893_map__append,axiom,
! [F: int > instr,Xs: list_int,Ys: list_int] :
( ( map_int_instr @ F @ ( append_int @ Xs @ Ys ) )
= ( append_instr @ ( map_int_instr @ F @ Xs ) @ ( map_int_instr @ F @ Ys ) ) ) ).
% map_append
thf(fact_894_map__append,axiom,
! [F: int > int,Xs: list_int,Ys: list_int] :
( ( map_int_int @ F @ ( append_int @ Xs @ Ys ) )
= ( append_int @ ( map_int_int @ F @ Xs ) @ ( map_int_int @ F @ Ys ) ) ) ).
% map_append
thf(fact_895_list__ex__simps_I1_J,axiom,
! [P: produc6425607678544837394st_int > $o,X: produc6425607678544837394st_int,Xs: list_P419547167332244514st_int] :
( ( list_e3437881803614826702st_int @ P @ ( cons_P4151473878978182108st_int @ X @ Xs ) )
= ( ( P @ X )
| ( list_e3437881803614826702st_int @ P @ Xs ) ) ) ).
% list_ex_simps(1)
thf(fact_896_list__ex__simps_I1_J,axiom,
! [P: int > $o,X: int,Xs: list_int] :
( ( list_ex_int @ P @ ( cons_int @ X @ Xs ) )
= ( ( P @ X )
| ( list_ex_int @ P @ Xs ) ) ) ).
% list_ex_simps(1)
thf(fact_897_list__ex__simps_I1_J,axiom,
! [P: list_char > $o,X: list_char,Xs: list_list_char] :
( ( list_ex_list_char @ P @ ( cons_list_char @ X @ Xs ) )
= ( ( P @ X )
| ( list_ex_list_char @ P @ Xs ) ) ) ).
% list_ex_simps(1)
thf(fact_898_append1__eq__conv,axiom,
! [Xs: list_instr,X: instr,Ys: list_instr,Y: instr] :
( ( ( append_instr @ Xs @ ( cons_instr @ X @ nil_instr ) )
= ( append_instr @ Ys @ ( cons_instr @ Y @ nil_instr ) ) )
= ( ( Xs = Ys )
& ( X = Y ) ) ) ).
% append1_eq_conv
thf(fact_899_append1__eq__conv,axiom,
! [Xs: list_P419547167332244514st_int,X: produc6425607678544837394st_int,Ys: list_P419547167332244514st_int,Y: produc6425607678544837394st_int] :
( ( ( append5028292981072010935st_int @ Xs @ ( cons_P4151473878978182108st_int @ X @ nil_Pr749310521696113548st_int ) )
= ( append5028292981072010935st_int @ Ys @ ( cons_P4151473878978182108st_int @ Y @ nil_Pr749310521696113548st_int ) ) )
= ( ( Xs = Ys )
& ( X = Y ) ) ) ).
% append1_eq_conv
thf(fact_900_append1__eq__conv,axiom,
! [Xs: list_int,X: int,Ys: list_int,Y: int] :
( ( ( append_int @ Xs @ ( cons_int @ X @ nil_int ) )
= ( append_int @ Ys @ ( cons_int @ Y @ nil_int ) ) )
= ( ( Xs = Ys )
& ( X = Y ) ) ) ).
% append1_eq_conv
thf(fact_901_append1__eq__conv,axiom,
! [Xs: list_list_char,X: list_char,Ys: list_list_char,Y: list_char] :
( ( ( append_list_char @ Xs @ ( cons_list_char @ X @ nil_list_char ) )
= ( append_list_char @ Ys @ ( cons_list_char @ Y @ nil_list_char ) ) )
= ( ( Xs = Ys )
& ( X = Y ) ) ) ).
% append1_eq_conv
thf(fact_902_nth__Cons__0,axiom,
! [X: produc6425607678544837394st_int,Xs: list_P419547167332244514st_int] :
( ( nth_Pr8327173284404383691st_int @ ( cons_P4151473878978182108st_int @ X @ Xs ) @ zero_zero_nat )
= X ) ).
% nth_Cons_0
thf(fact_903_nth__Cons__0,axiom,
! [X: int,Xs: list_int] :
( ( nth_int @ ( cons_int @ X @ Xs ) @ zero_zero_nat )
= X ) ).
% nth_Cons_0
thf(fact_904_nth__Cons__0,axiom,
! [X: list_char,Xs: list_list_char] :
( ( nth_list_char @ ( cons_list_char @ X @ Xs ) @ zero_zero_nat )
= X ) ).
% nth_Cons_0
thf(fact_905_last__appendL,axiom,
! [Ys: list_P419547167332244514st_int,Xs: list_P419547167332244514st_int] :
( ( Ys = nil_Pr749310521696113548st_int )
=> ( ( last_P5391958344443371187st_int @ ( append5028292981072010935st_int @ Xs @ Ys ) )
= ( last_P5391958344443371187st_int @ Xs ) ) ) ).
% last_appendL
thf(fact_906_last__appendL,axiom,
! [Ys: list_instr,Xs: list_instr] :
( ( Ys = nil_instr )
=> ( ( last_instr @ ( append_instr @ Xs @ Ys ) )
= ( last_instr @ Xs ) ) ) ).
% last_appendL
thf(fact_907_last__appendL,axiom,
! [Ys: list_int,Xs: list_int] :
( ( Ys = nil_int )
=> ( ( last_int @ ( append_int @ Xs @ Ys ) )
= ( last_int @ Xs ) ) ) ).
% last_appendL
thf(fact_908_last__appendR,axiom,
! [Ys: list_P419547167332244514st_int,Xs: list_P419547167332244514st_int] :
( ( Ys != nil_Pr749310521696113548st_int )
=> ( ( last_P5391958344443371187st_int @ ( append5028292981072010935st_int @ Xs @ Ys ) )
= ( last_P5391958344443371187st_int @ Ys ) ) ) ).
% last_appendR
thf(fact_909_last__appendR,axiom,
! [Ys: list_instr,Xs: list_instr] :
( ( Ys != nil_instr )
=> ( ( last_instr @ ( append_instr @ Xs @ Ys ) )
= ( last_instr @ Ys ) ) ) ).
% last_appendR
thf(fact_910_last__appendR,axiom,
! [Ys: list_int,Xs: list_int] :
( ( Ys != nil_int )
=> ( ( last_int @ ( append_int @ Xs @ Ys ) )
= ( last_int @ Ys ) ) ) ).
% last_appendR
thf(fact_911_nth__append__length,axiom,
! [Xs: list_P419547167332244514st_int,X: produc6425607678544837394st_int,Ys: list_P419547167332244514st_int] :
( ( nth_Pr8327173284404383691st_int @ ( append5028292981072010935st_int @ Xs @ ( cons_P4151473878978182108st_int @ X @ Ys ) ) @ ( size_s229347708945616566st_int @ Xs ) )
= X ) ).
% nth_append_length
thf(fact_912_nth__append__length,axiom,
! [Xs: list_instr,X: instr,Ys: list_instr] :
( ( nth_instr @ ( append_instr @ Xs @ ( cons_instr @ X @ Ys ) ) @ ( size_size_list_instr @ Xs ) )
= X ) ).
% nth_append_length
thf(fact_913_nth__append__length,axiom,
! [Xs: list_list_char,X: list_char,Ys: list_list_char] :
( ( nth_list_char @ ( append_list_char @ Xs @ ( cons_list_char @ X @ Ys ) ) @ ( size_s356637359517785349t_char @ Xs ) )
= X ) ).
% nth_append_length
thf(fact_914_nth__append__length,axiom,
! [Xs: list_int,X: int,Ys: list_int] :
( ( nth_int @ ( append_int @ Xs @ ( cons_int @ X @ Ys ) ) @ ( size_size_list_int @ Xs ) )
= X ) ).
% nth_append_length
thf(fact_915_nth__map,axiom,
! [N: nat,Xs: list_P419547167332244514st_int,F: produc6425607678544837394st_int > produc6425607678544837394st_int] :
( ( ord_less_nat @ N @ ( size_s229347708945616566st_int @ Xs ) )
=> ( ( nth_Pr8327173284404383691st_int @ ( map_Pr8625093460839654122st_int @ F @ Xs ) @ N )
= ( F @ ( nth_Pr8327173284404383691st_int @ Xs @ N ) ) ) ) ).
% nth_map
thf(fact_916_nth__map,axiom,
! [N: nat,Xs: list_P419547167332244514st_int,F: produc6425607678544837394st_int > int] :
( ( ord_less_nat @ N @ ( size_s229347708945616566st_int @ Xs ) )
=> ( ( nth_int @ ( map_Pr3224476576807795204nt_int @ F @ Xs ) @ N )
= ( F @ ( nth_Pr8327173284404383691st_int @ Xs @ N ) ) ) ) ).
% nth_map
thf(fact_917_nth__map,axiom,
! [N: nat,Xs: list_instr,F: instr > produc6425607678544837394st_int] :
( ( ord_less_nat @ N @ ( size_size_list_instr @ Xs ) )
=> ( ( nth_Pr8327173284404383691st_int @ ( map_in7259965606695656109st_int @ F @ Xs ) @ N )
= ( F @ ( nth_instr @ Xs @ N ) ) ) ) ).
% nth_map
thf(fact_918_nth__map,axiom,
! [N: nat,Xs: list_instr,F: instr > int] :
( ( ord_less_nat @ N @ ( size_size_list_instr @ Xs ) )
=> ( ( nth_int @ ( map_instr_int @ F @ Xs ) @ N )
= ( F @ ( nth_instr @ Xs @ N ) ) ) ) ).
% nth_map
thf(fact_919_nth__map,axiom,
! [N: nat,Xs: list_list_char,F: list_char > produc6425607678544837394st_int] :
( ( ord_less_nat @ N @ ( size_s356637359517785349t_char @ Xs ) )
=> ( ( nth_Pr8327173284404383691st_int @ ( map_li6440205260635360185st_int @ F @ Xs ) @ N )
= ( F @ ( nth_list_char @ Xs @ N ) ) ) ) ).
% nth_map
thf(fact_920_nth__map,axiom,
! [N: nat,Xs: list_list_char,F: list_char > int] :
( ( ord_less_nat @ N @ ( size_s356637359517785349t_char @ Xs ) )
=> ( ( nth_int @ ( map_list_char_int @ F @ Xs ) @ N )
= ( F @ ( nth_list_char @ Xs @ N ) ) ) ) ).
% nth_map
thf(fact_921_nth__map,axiom,
! [N: nat,Xs: list_int,F: int > produc6425607678544837394st_int] :
( ( ord_less_nat @ N @ ( size_size_list_int @ Xs ) )
=> ( ( nth_Pr8327173284404383691st_int @ ( map_in8512336359594627332st_int @ F @ Xs ) @ N )
= ( F @ ( nth_int @ Xs @ N ) ) ) ) ).
% nth_map
thf(fact_922_nth__map,axiom,
! [N: nat,Xs: list_int,F: int > int] :
( ( ord_less_nat @ N @ ( size_size_list_int @ Xs ) )
=> ( ( nth_int @ ( map_int_int @ F @ Xs ) @ N )
= ( F @ ( nth_int @ Xs @ N ) ) ) ) ).
% nth_map
thf(fact_923_last__snoc,axiom,
! [Xs: list_instr,X: instr] :
( ( last_instr @ ( append_instr @ Xs @ ( cons_instr @ X @ nil_instr ) ) )
= X ) ).
% last_snoc
thf(fact_924_last__snoc,axiom,
! [Xs: list_P419547167332244514st_int,X: produc6425607678544837394st_int] :
( ( last_P5391958344443371187st_int @ ( append5028292981072010935st_int @ Xs @ ( cons_P4151473878978182108st_int @ X @ nil_Pr749310521696113548st_int ) ) )
= X ) ).
% last_snoc
thf(fact_925_last__snoc,axiom,
! [Xs: list_int,X: int] :
( ( last_int @ ( append_int @ Xs @ ( cons_int @ X @ nil_int ) ) )
= X ) ).
% last_snoc
thf(fact_926_last__snoc,axiom,
! [Xs: list_list_char,X: list_char] :
( ( last_list_char @ ( append_list_char @ Xs @ ( cons_list_char @ X @ nil_list_char ) ) )
= X ) ).
% last_snoc
thf(fact_927_list_Osimps_I8_J,axiom,
! [F: produc6425607678544837394st_int > produc6425607678544837394st_int] :
( ( map_Pr8625093460839654122st_int @ F @ nil_Pr749310521696113548st_int )
= nil_Pr749310521696113548st_int ) ).
% list.simps(8)
thf(fact_928_list_Osimps_I8_J,axiom,
! [F: produc6425607678544837394st_int > instr] :
( ( map_Pr7514435936640407687_instr @ F @ nil_Pr749310521696113548st_int )
= nil_instr ) ).
% list.simps(8)
thf(fact_929_list_Osimps_I8_J,axiom,
! [F: produc6425607678544837394st_int > int] :
( ( map_Pr3224476576807795204nt_int @ F @ nil_Pr749310521696113548st_int )
= nil_int ) ).
% list.simps(8)
thf(fact_930_list_Osimps_I8_J,axiom,
! [F: instr > produc6425607678544837394st_int] :
( ( map_in7259965606695656109st_int @ F @ nil_instr )
= nil_Pr749310521696113548st_int ) ).
% list.simps(8)
thf(fact_931_list_Osimps_I8_J,axiom,
! [F: instr > instr] :
( ( map_instr_instr @ F @ nil_instr )
= nil_instr ) ).
% list.simps(8)
thf(fact_932_list_Osimps_I8_J,axiom,
! [F: instr > int] :
( ( map_instr_int @ F @ nil_instr )
= nil_int ) ).
% list.simps(8)
thf(fact_933_list_Osimps_I8_J,axiom,
! [F: int > produc6425607678544837394st_int] :
( ( map_in8512336359594627332st_int @ F @ nil_int )
= nil_Pr749310521696113548st_int ) ).
% list.simps(8)
thf(fact_934_list_Osimps_I8_J,axiom,
! [F: int > instr] :
( ( map_int_instr @ F @ nil_int )
= nil_instr ) ).
% list.simps(8)
thf(fact_935_list_Osimps_I8_J,axiom,
! [F: int > int] :
( ( map_int_int @ F @ nil_int )
= nil_int ) ).
% list.simps(8)
thf(fact_936_last_Osimps,axiom,
! [Xs: list_instr,X: instr] :
( ( ( Xs = nil_instr )
=> ( ( last_instr @ ( cons_instr @ X @ Xs ) )
= X ) )
& ( ( Xs != nil_instr )
=> ( ( last_instr @ ( cons_instr @ X @ Xs ) )
= ( last_instr @ Xs ) ) ) ) ).
% last.simps
thf(fact_937_last_Osimps,axiom,
! [Xs: list_P419547167332244514st_int,X: produc6425607678544837394st_int] :
( ( ( Xs = nil_Pr749310521696113548st_int )
=> ( ( last_P5391958344443371187st_int @ ( cons_P4151473878978182108st_int @ X @ Xs ) )
= X ) )
& ( ( Xs != nil_Pr749310521696113548st_int )
=> ( ( last_P5391958344443371187st_int @ ( cons_P4151473878978182108st_int @ X @ Xs ) )
= ( last_P5391958344443371187st_int @ Xs ) ) ) ) ).
% last.simps
thf(fact_938_last_Osimps,axiom,
! [Xs: list_int,X: int] :
( ( ( Xs = nil_int )
=> ( ( last_int @ ( cons_int @ X @ Xs ) )
= X ) )
& ( ( Xs != nil_int )
=> ( ( last_int @ ( cons_int @ X @ Xs ) )
= ( last_int @ Xs ) ) ) ) ).
% last.simps
thf(fact_939_last_Osimps,axiom,
! [Xs: list_list_char,X: list_char] :
( ( ( Xs = nil_list_char )
=> ( ( last_list_char @ ( cons_list_char @ X @ Xs ) )
= X ) )
& ( ( Xs != nil_list_char )
=> ( ( last_list_char @ ( cons_list_char @ X @ Xs ) )
= ( last_list_char @ Xs ) ) ) ) ).
% last.simps
thf(fact_940_transpose_Ocases,axiom,
! [X: list_list_instr] :
( ( X != nil_list_instr )
=> ( ! [Xss: list_list_instr] :
( X
!= ( cons_list_instr @ nil_instr @ Xss ) )
=> ~ ! [X3: instr,Xs3: list_instr,Xss: list_list_instr] :
( X
!= ( cons_list_instr @ ( cons_instr @ X3 @ Xs3 ) @ Xss ) ) ) ) ).
% transpose.cases
thf(fact_941_transpose_Ocases,axiom,
! [X: list_l6422821129637969586st_int] :
( ( X != nil_li4597314861645524892st_int )
=> ( ! [Xss: list_l6422821129637969586st_int] :
( X
!= ( cons_l8648290059330449388st_int @ nil_Pr749310521696113548st_int @ Xss ) )
=> ~ ! [X3: produc6425607678544837394st_int,Xs3: list_P419547167332244514st_int,Xss: list_l6422821129637969586st_int] :
( X
!= ( cons_l8648290059330449388st_int @ ( cons_P4151473878978182108st_int @ X3 @ Xs3 ) @ Xss ) ) ) ) ).
% transpose.cases
thf(fact_942_transpose_Ocases,axiom,
! [X: list_list_int] :
( ( X != nil_list_int )
=> ( ! [Xss: list_list_int] :
( X
!= ( cons_list_int @ nil_int @ Xss ) )
=> ~ ! [X3: int,Xs3: list_int,Xss: list_list_int] :
( X
!= ( cons_list_int @ ( cons_int @ X3 @ Xs3 ) @ Xss ) ) ) ) ).
% transpose.cases
thf(fact_943_transpose_Ocases,axiom,
! [X: list_list_list_char] :
( ( X != nil_list_list_char )
=> ( ! [Xss: list_list_list_char] :
( X
!= ( cons_list_list_char @ nil_list_char @ Xss ) )
=> ~ ! [X3: list_char,Xs3: list_list_char,Xss: list_list_list_char] :
( X
!= ( cons_list_list_char @ ( cons_list_char @ X3 @ Xs3 ) @ Xss ) ) ) ) ).
% transpose.cases
thf(fact_944_transpose_Ocases,axiom,
! [X: list_list_char] :
( ( X != nil_list_char )
=> ( ! [Xss: list_list_char] :
( X
!= ( cons_list_char @ nil_char @ Xss ) )
=> ~ ! [X3: char,Xs3: list_char,Xss: list_list_char] :
( X
!= ( cons_list_char @ ( cons_char @ X3 @ Xs3 ) @ Xss ) ) ) ) ).
% transpose.cases
thf(fact_945_last__ConsL,axiom,
! [Xs: list_instr,X: instr] :
( ( Xs = nil_instr )
=> ( ( last_instr @ ( cons_instr @ X @ Xs ) )
= X ) ) ).
% last_ConsL
thf(fact_946_last__ConsL,axiom,
! [Xs: list_P419547167332244514st_int,X: produc6425607678544837394st_int] :
( ( Xs = nil_Pr749310521696113548st_int )
=> ( ( last_P5391958344443371187st_int @ ( cons_P4151473878978182108st_int @ X @ Xs ) )
= X ) ) ).
% last_ConsL
thf(fact_947_last__ConsL,axiom,
! [Xs: list_int,X: int] :
( ( Xs = nil_int )
=> ( ( last_int @ ( cons_int @ X @ Xs ) )
= X ) ) ).
% last_ConsL
thf(fact_948_last__ConsL,axiom,
! [Xs: list_list_char,X: list_char] :
( ( Xs = nil_list_char )
=> ( ( last_list_char @ ( cons_list_char @ X @ Xs ) )
= X ) ) ).
% last_ConsL
thf(fact_949_last__ConsR,axiom,
! [Xs: list_instr,X: instr] :
( ( Xs != nil_instr )
=> ( ( last_instr @ ( cons_instr @ X @ Xs ) )
= ( last_instr @ Xs ) ) ) ).
% last_ConsR
thf(fact_950_last__ConsR,axiom,
! [Xs: list_P419547167332244514st_int,X: produc6425607678544837394st_int] :
( ( Xs != nil_Pr749310521696113548st_int )
=> ( ( last_P5391958344443371187st_int @ ( cons_P4151473878978182108st_int @ X @ Xs ) )
= ( last_P5391958344443371187st_int @ Xs ) ) ) ).
% last_ConsR
thf(fact_951_last__ConsR,axiom,
! [Xs: list_int,X: int] :
( ( Xs != nil_int )
=> ( ( last_int @ ( cons_int @ X @ Xs ) )
= ( last_int @ Xs ) ) ) ).
% last_ConsR
thf(fact_952_last__ConsR,axiom,
! [Xs: list_list_char,X: list_char] :
( ( Xs != nil_list_char )
=> ( ( last_list_char @ ( cons_list_char @ X @ Xs ) )
= ( last_list_char @ Xs ) ) ) ).
% last_ConsR
thf(fact_953_last__in__set,axiom,
! [As: list_nat] :
( ( As != nil_nat )
=> ( member_nat @ ( last_nat @ As ) @ ( set_nat2 @ As ) ) ) ).
% last_in_set
thf(fact_954_last__in__set,axiom,
! [As: list_P419547167332244514st_int] :
( ( As != nil_Pr749310521696113548st_int )
=> ( member9100552558438320297st_int @ ( last_P5391958344443371187st_int @ As ) @ ( set_Pr3303432077563008829st_int @ As ) ) ) ).
% last_in_set
thf(fact_955_last__in__set,axiom,
! [As: list_instr] :
( ( As != nil_instr )
=> ( member_instr @ ( last_instr @ As ) @ ( set_instr2 @ As ) ) ) ).
% last_in_set
thf(fact_956_last__in__set,axiom,
! [As: list_list_char] :
( ( As != nil_list_char )
=> ( member_list_char @ ( last_list_char @ As ) @ ( set_list_char2 @ As ) ) ) ).
% last_in_set
thf(fact_957_last__in__set,axiom,
! [As: list_int] :
( ( As != nil_int )
=> ( member_int @ ( last_int @ As ) @ ( set_int2 @ As ) ) ) ).
% last_in_set
thf(fact_958_map__eq__append__conv,axiom,
! [F: instr > instr,Xs: list_instr,Ys: list_instr,Zs: list_instr] :
( ( ( map_instr_instr @ F @ Xs )
= ( append_instr @ Ys @ Zs ) )
= ( ? [Us2: list_instr,Vs2: list_instr] :
( ( Xs
= ( append_instr @ Us2 @ Vs2 ) )
& ( Ys
= ( map_instr_instr @ F @ Us2 ) )
& ( Zs
= ( map_instr_instr @ F @ Vs2 ) ) ) ) ) ).
% map_eq_append_conv
thf(fact_959_map__eq__append__conv,axiom,
! [F: int > instr,Xs: list_int,Ys: list_instr,Zs: list_instr] :
( ( ( map_int_instr @ F @ Xs )
= ( append_instr @ Ys @ Zs ) )
= ( ? [Us2: list_int,Vs2: list_int] :
( ( Xs
= ( append_int @ Us2 @ Vs2 ) )
& ( Ys
= ( map_int_instr @ F @ Us2 ) )
& ( Zs
= ( map_int_instr @ F @ Vs2 ) ) ) ) ) ).
% map_eq_append_conv
thf(fact_960_map__eq__append__conv,axiom,
! [F: instr > int,Xs: list_instr,Ys: list_int,Zs: list_int] :
( ( ( map_instr_int @ F @ Xs )
= ( append_int @ Ys @ Zs ) )
= ( ? [Us2: list_instr,Vs2: list_instr] :
( ( Xs
= ( append_instr @ Us2 @ Vs2 ) )
& ( Ys
= ( map_instr_int @ F @ Us2 ) )
& ( Zs
= ( map_instr_int @ F @ Vs2 ) ) ) ) ) ).
% map_eq_append_conv
thf(fact_961_map__eq__append__conv,axiom,
! [F: int > int,Xs: list_int,Ys: list_int,Zs: list_int] :
( ( ( map_int_int @ F @ Xs )
= ( append_int @ Ys @ Zs ) )
= ( ? [Us2: list_int,Vs2: list_int] :
( ( Xs
= ( append_int @ Us2 @ Vs2 ) )
& ( Ys
= ( map_int_int @ F @ Us2 ) )
& ( Zs
= ( map_int_int @ F @ Vs2 ) ) ) ) ) ).
% map_eq_append_conv
thf(fact_962_append__eq__map__conv,axiom,
! [Ys: list_instr,Zs: list_instr,F: instr > instr,Xs: list_instr] :
( ( ( append_instr @ Ys @ Zs )
= ( map_instr_instr @ F @ Xs ) )
= ( ? [Us2: list_instr,Vs2: list_instr] :
( ( Xs
= ( append_instr @ Us2 @ Vs2 ) )
& ( Ys
= ( map_instr_instr @ F @ Us2 ) )
& ( Zs
= ( map_instr_instr @ F @ Vs2 ) ) ) ) ) ).
% append_eq_map_conv
thf(fact_963_append__eq__map__conv,axiom,
! [Ys: list_instr,Zs: list_instr,F: int > instr,Xs: list_int] :
( ( ( append_instr @ Ys @ Zs )
= ( map_int_instr @ F @ Xs ) )
= ( ? [Us2: list_int,Vs2: list_int] :
( ( Xs
= ( append_int @ Us2 @ Vs2 ) )
& ( Ys
= ( map_int_instr @ F @ Us2 ) )
& ( Zs
= ( map_int_instr @ F @ Vs2 ) ) ) ) ) ).
% append_eq_map_conv
thf(fact_964_append__eq__map__conv,axiom,
! [Ys: list_int,Zs: list_int,F: instr > int,Xs: list_instr] :
( ( ( append_int @ Ys @ Zs )
= ( map_instr_int @ F @ Xs ) )
= ( ? [Us2: list_instr,Vs2: list_instr] :
( ( Xs
= ( append_instr @ Us2 @ Vs2 ) )
& ( Ys
= ( map_instr_int @ F @ Us2 ) )
& ( Zs
= ( map_instr_int @ F @ Vs2 ) ) ) ) ) ).
% append_eq_map_conv
thf(fact_965_append__eq__map__conv,axiom,
! [Ys: list_int,Zs: list_int,F: int > int,Xs: list_int] :
( ( ( append_int @ Ys @ Zs )
= ( map_int_int @ F @ Xs ) )
= ( ? [Us2: list_int,Vs2: list_int] :
( ( Xs
= ( append_int @ Us2 @ Vs2 ) )
& ( Ys
= ( map_int_int @ F @ Us2 ) )
& ( Zs
= ( map_int_int @ F @ Vs2 ) ) ) ) ) ).
% append_eq_map_conv
thf(fact_966_subset__code_I1_J,axiom,
! [Xs: list_list_char,B5: set_list_char] :
( ( ord_le1207710660481778807t_char @ ( set_list_char2 @ Xs ) @ B5 )
= ( ! [X2: list_char] :
( ( member_list_char @ X2 @ ( set_list_char2 @ Xs ) )
=> ( member_list_char @ X2 @ B5 ) ) ) ) ).
% subset_code(1)
thf(fact_967_subset__code_I1_J,axiom,
! [Xs: list_int,B5: set_int] :
( ( ord_less_eq_set_int @ ( set_int2 @ Xs ) @ B5 )
= ( ! [X2: int] :
( ( member_int @ X2 @ ( set_int2 @ Xs ) )
=> ( member_int @ X2 @ B5 ) ) ) ) ).
% subset_code(1)
thf(fact_968_subset__code_I1_J,axiom,
! [Xs: list_nat,B5: set_nat] :
( ( ord_less_eq_set_nat @ ( set_nat2 @ Xs ) @ B5 )
= ( ! [X2: nat] :
( ( member_nat @ X2 @ ( set_nat2 @ Xs ) )
=> ( member_nat @ X2 @ B5 ) ) ) ) ).
% subset_code(1)
thf(fact_969_set__subset__Cons,axiom,
! [Xs: list_P419547167332244514st_int,X: produc6425607678544837394st_int] : ( ord_le8444298136753458984st_int @ ( set_Pr3303432077563008829st_int @ Xs ) @ ( set_Pr3303432077563008829st_int @ ( cons_P4151473878978182108st_int @ X @ Xs ) ) ) ).
% set_subset_Cons
thf(fact_970_set__subset__Cons,axiom,
! [Xs: list_list_char,X: list_char] : ( ord_le1207710660481778807t_char @ ( set_list_char2 @ Xs ) @ ( set_list_char2 @ ( cons_list_char @ X @ Xs ) ) ) ).
% set_subset_Cons
thf(fact_971_set__subset__Cons,axiom,
! [Xs: list_int,X: int] : ( ord_less_eq_set_int @ ( set_int2 @ Xs ) @ ( set_int2 @ ( cons_int @ X @ Xs ) ) ) ).
% set_subset_Cons
thf(fact_972_set__subset__Cons,axiom,
! [Xs: list_nat,X: nat] : ( ord_less_eq_set_nat @ ( set_nat2 @ Xs ) @ ( set_nat2 @ ( cons_nat @ X @ Xs ) ) ) ).
% set_subset_Cons
thf(fact_973_split__list,axiom,
! [X: nat,Xs: list_nat] :
( ( member_nat @ X @ ( set_nat2 @ Xs ) )
=> ? [Ys4: list_nat,Zs2: list_nat] :
( Xs
= ( append_nat @ Ys4 @ ( cons_nat @ X @ Zs2 ) ) ) ) ).
% split_list
thf(fact_974_split__list,axiom,
! [X: instr,Xs: list_instr] :
( ( member_instr @ X @ ( set_instr2 @ Xs ) )
=> ? [Ys4: list_instr,Zs2: list_instr] :
( Xs
= ( append_instr @ Ys4 @ ( cons_instr @ X @ Zs2 ) ) ) ) ).
% split_list
thf(fact_975_split__list,axiom,
! [X: produc6425607678544837394st_int,Xs: list_P419547167332244514st_int] :
( ( member9100552558438320297st_int @ X @ ( set_Pr3303432077563008829st_int @ Xs ) )
=> ? [Ys4: list_P419547167332244514st_int,Zs2: list_P419547167332244514st_int] :
( Xs
= ( append5028292981072010935st_int @ Ys4 @ ( cons_P4151473878978182108st_int @ X @ Zs2 ) ) ) ) ).
% split_list
thf(fact_976_split__list,axiom,
! [X: int,Xs: list_int] :
( ( member_int @ X @ ( set_int2 @ Xs ) )
=> ? [Ys4: list_int,Zs2: list_int] :
( Xs
= ( append_int @ Ys4 @ ( cons_int @ X @ Zs2 ) ) ) ) ).
% split_list
thf(fact_977_split__list,axiom,
! [X: list_char,Xs: list_list_char] :
( ( member_list_char @ X @ ( set_list_char2 @ Xs ) )
=> ? [Ys4: list_list_char,Zs2: list_list_char] :
( Xs
= ( append_list_char @ Ys4 @ ( cons_list_char @ X @ Zs2 ) ) ) ) ).
% split_list
thf(fact_978_split__list__last,axiom,
! [X: nat,Xs: list_nat] :
( ( member_nat @ X @ ( set_nat2 @ Xs ) )
=> ? [Ys4: list_nat,Zs2: list_nat] :
( ( Xs
= ( append_nat @ Ys4 @ ( cons_nat @ X @ Zs2 ) ) )
& ~ ( member_nat @ X @ ( set_nat2 @ Zs2 ) ) ) ) ).
% split_list_last
thf(fact_979_split__list__last,axiom,
! [X: instr,Xs: list_instr] :
( ( member_instr @ X @ ( set_instr2 @ Xs ) )
=> ? [Ys4: list_instr,Zs2: list_instr] :
( ( Xs
= ( append_instr @ Ys4 @ ( cons_instr @ X @ Zs2 ) ) )
& ~ ( member_instr @ X @ ( set_instr2 @ Zs2 ) ) ) ) ).
% split_list_last
thf(fact_980_split__list__last,axiom,
! [X: produc6425607678544837394st_int,Xs: list_P419547167332244514st_int] :
( ( member9100552558438320297st_int @ X @ ( set_Pr3303432077563008829st_int @ Xs ) )
=> ? [Ys4: list_P419547167332244514st_int,Zs2: list_P419547167332244514st_int] :
( ( Xs
= ( append5028292981072010935st_int @ Ys4 @ ( cons_P4151473878978182108st_int @ X @ Zs2 ) ) )
& ~ ( member9100552558438320297st_int @ X @ ( set_Pr3303432077563008829st_int @ Zs2 ) ) ) ) ).
% split_list_last
thf(fact_981_split__list__last,axiom,
! [X: int,Xs: list_int] :
( ( member_int @ X @ ( set_int2 @ Xs ) )
=> ? [Ys4: list_int,Zs2: list_int] :
( ( Xs
= ( append_int @ Ys4 @ ( cons_int @ X @ Zs2 ) ) )
& ~ ( member_int @ X @ ( set_int2 @ Zs2 ) ) ) ) ).
% split_list_last
thf(fact_982_split__list__last,axiom,
! [X: list_char,Xs: list_list_char] :
( ( member_list_char @ X @ ( set_list_char2 @ Xs ) )
=> ? [Ys4: list_list_char,Zs2: list_list_char] :
( ( Xs
= ( append_list_char @ Ys4 @ ( cons_list_char @ X @ Zs2 ) ) )
& ~ ( member_list_char @ X @ ( set_list_char2 @ Zs2 ) ) ) ) ).
% split_list_last
thf(fact_983_split__list__prop,axiom,
! [Xs: list_instr,P: instr > $o] :
( ? [X4: instr] :
( ( member_instr @ X4 @ ( set_instr2 @ Xs ) )
& ( P @ X4 ) )
=> ? [Ys4: list_instr,X3: instr] :
( ? [Zs2: list_instr] :
( Xs
= ( append_instr @ Ys4 @ ( cons_instr @ X3 @ Zs2 ) ) )
& ( P @ X3 ) ) ) ).
% split_list_prop
thf(fact_984_split__list__prop,axiom,
! [Xs: list_P419547167332244514st_int,P: produc6425607678544837394st_int > $o] :
( ? [X4: produc6425607678544837394st_int] :
( ( member9100552558438320297st_int @ X4 @ ( set_Pr3303432077563008829st_int @ Xs ) )
& ( P @ X4 ) )
=> ? [Ys4: list_P419547167332244514st_int,X3: produc6425607678544837394st_int] :
( ? [Zs2: list_P419547167332244514st_int] :
( Xs
= ( append5028292981072010935st_int @ Ys4 @ ( cons_P4151473878978182108st_int @ X3 @ Zs2 ) ) )
& ( P @ X3 ) ) ) ).
% split_list_prop
thf(fact_985_split__list__prop,axiom,
! [Xs: list_int,P: int > $o] :
( ? [X4: int] :
( ( member_int @ X4 @ ( set_int2 @ Xs ) )
& ( P @ X4 ) )
=> ? [Ys4: list_int,X3: int] :
( ? [Zs2: list_int] :
( Xs
= ( append_int @ Ys4 @ ( cons_int @ X3 @ Zs2 ) ) )
& ( P @ X3 ) ) ) ).
% split_list_prop
thf(fact_986_split__list__prop,axiom,
! [Xs: list_list_char,P: list_char > $o] :
( ? [X4: list_char] :
( ( member_list_char @ X4 @ ( set_list_char2 @ Xs ) )
& ( P @ X4 ) )
=> ? [Ys4: list_list_char,X3: list_char] :
( ? [Zs2: list_list_char] :
( Xs
= ( append_list_char @ Ys4 @ ( cons_list_char @ X3 @ Zs2 ) ) )
& ( P @ X3 ) ) ) ).
% split_list_prop
thf(fact_987_split__list__first,axiom,
! [X: nat,Xs: list_nat] :
( ( member_nat @ X @ ( set_nat2 @ Xs ) )
=> ? [Ys4: list_nat,Zs2: list_nat] :
( ( Xs
= ( append_nat @ Ys4 @ ( cons_nat @ X @ Zs2 ) ) )
& ~ ( member_nat @ X @ ( set_nat2 @ Ys4 ) ) ) ) ).
% split_list_first
thf(fact_988_split__list__first,axiom,
! [X: instr,Xs: list_instr] :
( ( member_instr @ X @ ( set_instr2 @ Xs ) )
=> ? [Ys4: list_instr,Zs2: list_instr] :
( ( Xs
= ( append_instr @ Ys4 @ ( cons_instr @ X @ Zs2 ) ) )
& ~ ( member_instr @ X @ ( set_instr2 @ Ys4 ) ) ) ) ).
% split_list_first
thf(fact_989_split__list__first,axiom,
! [X: produc6425607678544837394st_int,Xs: list_P419547167332244514st_int] :
( ( member9100552558438320297st_int @ X @ ( set_Pr3303432077563008829st_int @ Xs ) )
=> ? [Ys4: list_P419547167332244514st_int,Zs2: list_P419547167332244514st_int] :
( ( Xs
= ( append5028292981072010935st_int @ Ys4 @ ( cons_P4151473878978182108st_int @ X @ Zs2 ) ) )
& ~ ( member9100552558438320297st_int @ X @ ( set_Pr3303432077563008829st_int @ Ys4 ) ) ) ) ).
% split_list_first
thf(fact_990_split__list__first,axiom,
! [X: int,Xs: list_int] :
( ( member_int @ X @ ( set_int2 @ Xs ) )
=> ? [Ys4: list_int,Zs2: list_int] :
( ( Xs
= ( append_int @ Ys4 @ ( cons_int @ X @ Zs2 ) ) )
& ~ ( member_int @ X @ ( set_int2 @ Ys4 ) ) ) ) ).
% split_list_first
thf(fact_991_split__list__first,axiom,
! [X: list_char,Xs: list_list_char] :
( ( member_list_char @ X @ ( set_list_char2 @ Xs ) )
=> ? [Ys4: list_list_char,Zs2: list_list_char] :
( ( Xs
= ( append_list_char @ Ys4 @ ( cons_list_char @ X @ Zs2 ) ) )
& ~ ( member_list_char @ X @ ( set_list_char2 @ Ys4 ) ) ) ) ).
% split_list_first
thf(fact_992_split__list__propE,axiom,
! [Xs: list_instr,P: instr > $o] :
( ? [X4: instr] :
( ( member_instr @ X4 @ ( set_instr2 @ Xs ) )
& ( P @ X4 ) )
=> ~ ! [Ys4: list_instr,X3: instr] :
( ? [Zs2: list_instr] :
( Xs
= ( append_instr @ Ys4 @ ( cons_instr @ X3 @ Zs2 ) ) )
=> ~ ( P @ X3 ) ) ) ).
% split_list_propE
thf(fact_993_split__list__propE,axiom,
! [Xs: list_P419547167332244514st_int,P: produc6425607678544837394st_int > $o] :
( ? [X4: produc6425607678544837394st_int] :
( ( member9100552558438320297st_int @ X4 @ ( set_Pr3303432077563008829st_int @ Xs ) )
& ( P @ X4 ) )
=> ~ ! [Ys4: list_P419547167332244514st_int,X3: produc6425607678544837394st_int] :
( ? [Zs2: list_P419547167332244514st_int] :
( Xs
= ( append5028292981072010935st_int @ Ys4 @ ( cons_P4151473878978182108st_int @ X3 @ Zs2 ) ) )
=> ~ ( P @ X3 ) ) ) ).
% split_list_propE
thf(fact_994_split__list__propE,axiom,
! [Xs: list_int,P: int > $o] :
( ? [X4: int] :
( ( member_int @ X4 @ ( set_int2 @ Xs ) )
& ( P @ X4 ) )
=> ~ ! [Ys4: list_int,X3: int] :
( ? [Zs2: list_int] :
( Xs
= ( append_int @ Ys4 @ ( cons_int @ X3 @ Zs2 ) ) )
=> ~ ( P @ X3 ) ) ) ).
% split_list_propE
thf(fact_995_split__list__propE,axiom,
! [Xs: list_list_char,P: list_char > $o] :
( ? [X4: list_char] :
( ( member_list_char @ X4 @ ( set_list_char2 @ Xs ) )
& ( P @ X4 ) )
=> ~ ! [Ys4: list_list_char,X3: list_char] :
( ? [Zs2: list_list_char] :
( Xs
= ( append_list_char @ Ys4 @ ( cons_list_char @ X3 @ Zs2 ) ) )
=> ~ ( P @ X3 ) ) ) ).
% split_list_propE
thf(fact_996_append__Cons__eq__iff,axiom,
! [X: nat,Xs: list_nat,Ys: list_nat,Xs4: list_nat,Ys5: list_nat] :
( ~ ( member_nat @ X @ ( set_nat2 @ Xs ) )
=> ( ~ ( member_nat @ X @ ( set_nat2 @ Ys ) )
=> ( ( ( append_nat @ Xs @ ( cons_nat @ X @ Ys ) )
= ( append_nat @ Xs4 @ ( cons_nat @ X @ Ys5 ) ) )
= ( ( Xs = Xs4 )
& ( Ys = Ys5 ) ) ) ) ) ).
% append_Cons_eq_iff
thf(fact_997_append__Cons__eq__iff,axiom,
! [X: instr,Xs: list_instr,Ys: list_instr,Xs4: list_instr,Ys5: list_instr] :
( ~ ( member_instr @ X @ ( set_instr2 @ Xs ) )
=> ( ~ ( member_instr @ X @ ( set_instr2 @ Ys ) )
=> ( ( ( append_instr @ Xs @ ( cons_instr @ X @ Ys ) )
= ( append_instr @ Xs4 @ ( cons_instr @ X @ Ys5 ) ) )
= ( ( Xs = Xs4 )
& ( Ys = Ys5 ) ) ) ) ) ).
% append_Cons_eq_iff
thf(fact_998_append__Cons__eq__iff,axiom,
! [X: produc6425607678544837394st_int,Xs: list_P419547167332244514st_int,Ys: list_P419547167332244514st_int,Xs4: list_P419547167332244514st_int,Ys5: list_P419547167332244514st_int] :
( ~ ( member9100552558438320297st_int @ X @ ( set_Pr3303432077563008829st_int @ Xs ) )
=> ( ~ ( member9100552558438320297st_int @ X @ ( set_Pr3303432077563008829st_int @ Ys ) )
=> ( ( ( append5028292981072010935st_int @ Xs @ ( cons_P4151473878978182108st_int @ X @ Ys ) )
= ( append5028292981072010935st_int @ Xs4 @ ( cons_P4151473878978182108st_int @ X @ Ys5 ) ) )
= ( ( Xs = Xs4 )
& ( Ys = Ys5 ) ) ) ) ) ).
% append_Cons_eq_iff
thf(fact_999_append__Cons__eq__iff,axiom,
! [X: int,Xs: list_int,Ys: list_int,Xs4: list_int,Ys5: list_int] :
( ~ ( member_int @ X @ ( set_int2 @ Xs ) )
=> ( ~ ( member_int @ X @ ( set_int2 @ Ys ) )
=> ( ( ( append_int @ Xs @ ( cons_int @ X @ Ys ) )
= ( append_int @ Xs4 @ ( cons_int @ X @ Ys5 ) ) )
= ( ( Xs = Xs4 )
& ( Ys = Ys5 ) ) ) ) ) ).
% append_Cons_eq_iff
thf(fact_1000_append__Cons__eq__iff,axiom,
! [X: list_char,Xs: list_list_char,Ys: list_list_char,Xs4: list_list_char,Ys5: list_list_char] :
( ~ ( member_list_char @ X @ ( set_list_char2 @ Xs ) )
=> ( ~ ( member_list_char @ X @ ( set_list_char2 @ Ys ) )
=> ( ( ( append_list_char @ Xs @ ( cons_list_char @ X @ Ys ) )
= ( append_list_char @ Xs4 @ ( cons_list_char @ X @ Ys5 ) ) )
= ( ( Xs = Xs4 )
& ( Ys = Ys5 ) ) ) ) ) ).
% append_Cons_eq_iff
thf(fact_1001_in__set__conv__decomp,axiom,
! [X: nat,Xs: list_nat] :
( ( member_nat @ X @ ( set_nat2 @ Xs ) )
= ( ? [Ys2: list_nat,Zs3: list_nat] :
( Xs
= ( append_nat @ Ys2 @ ( cons_nat @ X @ Zs3 ) ) ) ) ) ).
% in_set_conv_decomp
thf(fact_1002_in__set__conv__decomp,axiom,
! [X: instr,Xs: list_instr] :
( ( member_instr @ X @ ( set_instr2 @ Xs ) )
= ( ? [Ys2: list_instr,Zs3: list_instr] :
( Xs
= ( append_instr @ Ys2 @ ( cons_instr @ X @ Zs3 ) ) ) ) ) ).
% in_set_conv_decomp
thf(fact_1003_in__set__conv__decomp,axiom,
! [X: produc6425607678544837394st_int,Xs: list_P419547167332244514st_int] :
( ( member9100552558438320297st_int @ X @ ( set_Pr3303432077563008829st_int @ Xs ) )
= ( ? [Ys2: list_P419547167332244514st_int,Zs3: list_P419547167332244514st_int] :
( Xs
= ( append5028292981072010935st_int @ Ys2 @ ( cons_P4151473878978182108st_int @ X @ Zs3 ) ) ) ) ) ).
% in_set_conv_decomp
thf(fact_1004_in__set__conv__decomp,axiom,
! [X: int,Xs: list_int] :
( ( member_int @ X @ ( set_int2 @ Xs ) )
= ( ? [Ys2: list_int,Zs3: list_int] :
( Xs
= ( append_int @ Ys2 @ ( cons_int @ X @ Zs3 ) ) ) ) ) ).
% in_set_conv_decomp
thf(fact_1005_in__set__conv__decomp,axiom,
! [X: list_char,Xs: list_list_char] :
( ( member_list_char @ X @ ( set_list_char2 @ Xs ) )
= ( ? [Ys2: list_list_char,Zs3: list_list_char] :
( Xs
= ( append_list_char @ Ys2 @ ( cons_list_char @ X @ Zs3 ) ) ) ) ) ).
% in_set_conv_decomp
thf(fact_1006_split__list__last__prop,axiom,
! [Xs: list_instr,P: instr > $o] :
( ? [X4: instr] :
( ( member_instr @ X4 @ ( set_instr2 @ Xs ) )
& ( P @ X4 ) )
=> ? [Ys4: list_instr,X3: instr,Zs2: list_instr] :
( ( Xs
= ( append_instr @ Ys4 @ ( cons_instr @ X3 @ Zs2 ) ) )
& ( P @ X3 )
& ! [Xa2: instr] :
( ( member_instr @ Xa2 @ ( set_instr2 @ Zs2 ) )
=> ~ ( P @ Xa2 ) ) ) ) ).
% split_list_last_prop
thf(fact_1007_split__list__last__prop,axiom,
! [Xs: list_P419547167332244514st_int,P: produc6425607678544837394st_int > $o] :
( ? [X4: produc6425607678544837394st_int] :
( ( member9100552558438320297st_int @ X4 @ ( set_Pr3303432077563008829st_int @ Xs ) )
& ( P @ X4 ) )
=> ? [Ys4: list_P419547167332244514st_int,X3: produc6425607678544837394st_int,Zs2: list_P419547167332244514st_int] :
( ( Xs
= ( append5028292981072010935st_int @ Ys4 @ ( cons_P4151473878978182108st_int @ X3 @ Zs2 ) ) )
& ( P @ X3 )
& ! [Xa2: produc6425607678544837394st_int] :
( ( member9100552558438320297st_int @ Xa2 @ ( set_Pr3303432077563008829st_int @ Zs2 ) )
=> ~ ( P @ Xa2 ) ) ) ) ).
% split_list_last_prop
thf(fact_1008_split__list__last__prop,axiom,
! [Xs: list_int,P: int > $o] :
( ? [X4: int] :
( ( member_int @ X4 @ ( set_int2 @ Xs ) )
& ( P @ X4 ) )
=> ? [Ys4: list_int,X3: int,Zs2: list_int] :
( ( Xs
= ( append_int @ Ys4 @ ( cons_int @ X3 @ Zs2 ) ) )
& ( P @ X3 )
& ! [Xa2: int] :
( ( member_int @ Xa2 @ ( set_int2 @ Zs2 ) )
=> ~ ( P @ Xa2 ) ) ) ) ).
% split_list_last_prop
thf(fact_1009_split__list__last__prop,axiom,
! [Xs: list_list_char,P: list_char > $o] :
( ? [X4: list_char] :
( ( member_list_char @ X4 @ ( set_list_char2 @ Xs ) )
& ( P @ X4 ) )
=> ? [Ys4: list_list_char,X3: list_char,Zs2: list_list_char] :
( ( Xs
= ( append_list_char @ Ys4 @ ( cons_list_char @ X3 @ Zs2 ) ) )
& ( P @ X3 )
& ! [Xa2: list_char] :
( ( member_list_char @ Xa2 @ ( set_list_char2 @ Zs2 ) )
=> ~ ( P @ Xa2 ) ) ) ) ).
% split_list_last_prop
thf(fact_1010_split__list__first__prop,axiom,
! [Xs: list_instr,P: instr > $o] :
( ? [X4: instr] :
( ( member_instr @ X4 @ ( set_instr2 @ Xs ) )
& ( P @ X4 ) )
=> ? [Ys4: list_instr,X3: instr] :
( ? [Zs2: list_instr] :
( Xs
= ( append_instr @ Ys4 @ ( cons_instr @ X3 @ Zs2 ) ) )
& ( P @ X3 )
& ! [Xa2: instr] :
( ( member_instr @ Xa2 @ ( set_instr2 @ Ys4 ) )
=> ~ ( P @ Xa2 ) ) ) ) ).
% split_list_first_prop
thf(fact_1011_split__list__first__prop,axiom,
! [Xs: list_P419547167332244514st_int,P: produc6425607678544837394st_int > $o] :
( ? [X4: produc6425607678544837394st_int] :
( ( member9100552558438320297st_int @ X4 @ ( set_Pr3303432077563008829st_int @ Xs ) )
& ( P @ X4 ) )
=> ? [Ys4: list_P419547167332244514st_int,X3: produc6425607678544837394st_int] :
( ? [Zs2: list_P419547167332244514st_int] :
( Xs
= ( append5028292981072010935st_int @ Ys4 @ ( cons_P4151473878978182108st_int @ X3 @ Zs2 ) ) )
& ( P @ X3 )
& ! [Xa2: produc6425607678544837394st_int] :
( ( member9100552558438320297st_int @ Xa2 @ ( set_Pr3303432077563008829st_int @ Ys4 ) )
=> ~ ( P @ Xa2 ) ) ) ) ).
% split_list_first_prop
thf(fact_1012_split__list__first__prop,axiom,
! [Xs: list_int,P: int > $o] :
( ? [X4: int] :
( ( member_int @ X4 @ ( set_int2 @ Xs ) )
& ( P @ X4 ) )
=> ? [Ys4: list_int,X3: int] :
( ? [Zs2: list_int] :
( Xs
= ( append_int @ Ys4 @ ( cons_int @ X3 @ Zs2 ) ) )
& ( P @ X3 )
& ! [Xa2: int] :
( ( member_int @ Xa2 @ ( set_int2 @ Ys4 ) )
=> ~ ( P @ Xa2 ) ) ) ) ).
% split_list_first_prop
thf(fact_1013_split__list__first__prop,axiom,
! [Xs: list_list_char,P: list_char > $o] :
( ? [X4: list_char] :
( ( member_list_char @ X4 @ ( set_list_char2 @ Xs ) )
& ( P @ X4 ) )
=> ? [Ys4: list_list_char,X3: list_char] :
( ? [Zs2: list_list_char] :
( Xs
= ( append_list_char @ Ys4 @ ( cons_list_char @ X3 @ Zs2 ) ) )
& ( P @ X3 )
& ! [Xa2: list_char] :
( ( member_list_char @ Xa2 @ ( set_list_char2 @ Ys4 ) )
=> ~ ( P @ Xa2 ) ) ) ) ).
% split_list_first_prop
thf(fact_1014_split__list__last__propE,axiom,
! [Xs: list_instr,P: instr > $o] :
( ? [X4: instr] :
( ( member_instr @ X4 @ ( set_instr2 @ Xs ) )
& ( P @ X4 ) )
=> ~ ! [Ys4: list_instr,X3: instr,Zs2: list_instr] :
( ( Xs
= ( append_instr @ Ys4 @ ( cons_instr @ X3 @ Zs2 ) ) )
=> ( ( P @ X3 )
=> ~ ! [Xa2: instr] :
( ( member_instr @ Xa2 @ ( set_instr2 @ Zs2 ) )
=> ~ ( P @ Xa2 ) ) ) ) ) ).
% split_list_last_propE
thf(fact_1015_split__list__last__propE,axiom,
! [Xs: list_P419547167332244514st_int,P: produc6425607678544837394st_int > $o] :
( ? [X4: produc6425607678544837394st_int] :
( ( member9100552558438320297st_int @ X4 @ ( set_Pr3303432077563008829st_int @ Xs ) )
& ( P @ X4 ) )
=> ~ ! [Ys4: list_P419547167332244514st_int,X3: produc6425607678544837394st_int,Zs2: list_P419547167332244514st_int] :
( ( Xs
= ( append5028292981072010935st_int @ Ys4 @ ( cons_P4151473878978182108st_int @ X3 @ Zs2 ) ) )
=> ( ( P @ X3 )
=> ~ ! [Xa2: produc6425607678544837394st_int] :
( ( member9100552558438320297st_int @ Xa2 @ ( set_Pr3303432077563008829st_int @ Zs2 ) )
=> ~ ( P @ Xa2 ) ) ) ) ) ).
% split_list_last_propE
thf(fact_1016_split__list__last__propE,axiom,
! [Xs: list_int,P: int > $o] :
( ? [X4: int] :
( ( member_int @ X4 @ ( set_int2 @ Xs ) )
& ( P @ X4 ) )
=> ~ ! [Ys4: list_int,X3: int,Zs2: list_int] :
( ( Xs
= ( append_int @ Ys4 @ ( cons_int @ X3 @ Zs2 ) ) )
=> ( ( P @ X3 )
=> ~ ! [Xa2: int] :
( ( member_int @ Xa2 @ ( set_int2 @ Zs2 ) )
=> ~ ( P @ Xa2 ) ) ) ) ) ).
% split_list_last_propE
thf(fact_1017_split__list__last__propE,axiom,
! [Xs: list_list_char,P: list_char > $o] :
( ? [X4: list_char] :
( ( member_list_char @ X4 @ ( set_list_char2 @ Xs ) )
& ( P @ X4 ) )
=> ~ ! [Ys4: list_list_char,X3: list_char,Zs2: list_list_char] :
( ( Xs
= ( append_list_char @ Ys4 @ ( cons_list_char @ X3 @ Zs2 ) ) )
=> ( ( P @ X3 )
=> ~ ! [Xa2: list_char] :
( ( member_list_char @ Xa2 @ ( set_list_char2 @ Zs2 ) )
=> ~ ( P @ Xa2 ) ) ) ) ) ).
% split_list_last_propE
thf(fact_1018_split__list__first__propE,axiom,
! [Xs: list_instr,P: instr > $o] :
( ? [X4: instr] :
( ( member_instr @ X4 @ ( set_instr2 @ Xs ) )
& ( P @ X4 ) )
=> ~ ! [Ys4: list_instr,X3: instr] :
( ? [Zs2: list_instr] :
( Xs
= ( append_instr @ Ys4 @ ( cons_instr @ X3 @ Zs2 ) ) )
=> ( ( P @ X3 )
=> ~ ! [Xa2: instr] :
( ( member_instr @ Xa2 @ ( set_instr2 @ Ys4 ) )
=> ~ ( P @ Xa2 ) ) ) ) ) ).
% split_list_first_propE
thf(fact_1019_split__list__first__propE,axiom,
! [Xs: list_P419547167332244514st_int,P: produc6425607678544837394st_int > $o] :
( ? [X4: produc6425607678544837394st_int] :
( ( member9100552558438320297st_int @ X4 @ ( set_Pr3303432077563008829st_int @ Xs ) )
& ( P @ X4 ) )
=> ~ ! [Ys4: list_P419547167332244514st_int,X3: produc6425607678544837394st_int] :
( ? [Zs2: list_P419547167332244514st_int] :
( Xs
= ( append5028292981072010935st_int @ Ys4 @ ( cons_P4151473878978182108st_int @ X3 @ Zs2 ) ) )
=> ( ( P @ X3 )
=> ~ ! [Xa2: produc6425607678544837394st_int] :
( ( member9100552558438320297st_int @ Xa2 @ ( set_Pr3303432077563008829st_int @ Ys4 ) )
=> ~ ( P @ Xa2 ) ) ) ) ) ).
% split_list_first_propE
thf(fact_1020_split__list__first__propE,axiom,
! [Xs: list_int,P: int > $o] :
( ? [X4: int] :
( ( member_int @ X4 @ ( set_int2 @ Xs ) )
& ( P @ X4 ) )
=> ~ ! [Ys4: list_int,X3: int] :
( ? [Zs2: list_int] :
( Xs
= ( append_int @ Ys4 @ ( cons_int @ X3 @ Zs2 ) ) )
=> ( ( P @ X3 )
=> ~ ! [Xa2: int] :
( ( member_int @ Xa2 @ ( set_int2 @ Ys4 ) )
=> ~ ( P @ Xa2 ) ) ) ) ) ).
% split_list_first_propE
thf(fact_1021_split__list__first__propE,axiom,
! [Xs: list_list_char,P: list_char > $o] :
( ? [X4: list_char] :
( ( member_list_char @ X4 @ ( set_list_char2 @ Xs ) )
& ( P @ X4 ) )
=> ~ ! [Ys4: list_list_char,X3: list_char] :
( ? [Zs2: list_list_char] :
( Xs
= ( append_list_char @ Ys4 @ ( cons_list_char @ X3 @ Zs2 ) ) )
=> ( ( P @ X3 )
=> ~ ! [Xa2: list_char] :
( ( member_list_char @ Xa2 @ ( set_list_char2 @ Ys4 ) )
=> ~ ( P @ Xa2 ) ) ) ) ) ).
% split_list_first_propE
thf(fact_1022_in__set__conv__decomp__last,axiom,
! [X: nat,Xs: list_nat] :
( ( member_nat @ X @ ( set_nat2 @ Xs ) )
= ( ? [Ys2: list_nat,Zs3: list_nat] :
( ( Xs
= ( append_nat @ Ys2 @ ( cons_nat @ X @ Zs3 ) ) )
& ~ ( member_nat @ X @ ( set_nat2 @ Zs3 ) ) ) ) ) ).
% in_set_conv_decomp_last
thf(fact_1023_in__set__conv__decomp__last,axiom,
! [X: instr,Xs: list_instr] :
( ( member_instr @ X @ ( set_instr2 @ Xs ) )
= ( ? [Ys2: list_instr,Zs3: list_instr] :
( ( Xs
= ( append_instr @ Ys2 @ ( cons_instr @ X @ Zs3 ) ) )
& ~ ( member_instr @ X @ ( set_instr2 @ Zs3 ) ) ) ) ) ).
% in_set_conv_decomp_last
thf(fact_1024_in__set__conv__decomp__last,axiom,
! [X: produc6425607678544837394st_int,Xs: list_P419547167332244514st_int] :
( ( member9100552558438320297st_int @ X @ ( set_Pr3303432077563008829st_int @ Xs ) )
= ( ? [Ys2: list_P419547167332244514st_int,Zs3: list_P419547167332244514st_int] :
( ( Xs
= ( append5028292981072010935st_int @ Ys2 @ ( cons_P4151473878978182108st_int @ X @ Zs3 ) ) )
& ~ ( member9100552558438320297st_int @ X @ ( set_Pr3303432077563008829st_int @ Zs3 ) ) ) ) ) ).
% in_set_conv_decomp_last
thf(fact_1025_in__set__conv__decomp__last,axiom,
! [X: int,Xs: list_int] :
( ( member_int @ X @ ( set_int2 @ Xs ) )
= ( ? [Ys2: list_int,Zs3: list_int] :
( ( Xs
= ( append_int @ Ys2 @ ( cons_int @ X @ Zs3 ) ) )
& ~ ( member_int @ X @ ( set_int2 @ Zs3 ) ) ) ) ) ).
% in_set_conv_decomp_last
thf(fact_1026_in__set__conv__decomp__last,axiom,
! [X: list_char,Xs: list_list_char] :
( ( member_list_char @ X @ ( set_list_char2 @ Xs ) )
= ( ? [Ys2: list_list_char,Zs3: list_list_char] :
( ( Xs
= ( append_list_char @ Ys2 @ ( cons_list_char @ X @ Zs3 ) ) )
& ~ ( member_list_char @ X @ ( set_list_char2 @ Zs3 ) ) ) ) ) ).
% in_set_conv_decomp_last
thf(fact_1027_in__set__conv__decomp__first,axiom,
! [X: nat,Xs: list_nat] :
( ( member_nat @ X @ ( set_nat2 @ Xs ) )
= ( ? [Ys2: list_nat,Zs3: list_nat] :
( ( Xs
= ( append_nat @ Ys2 @ ( cons_nat @ X @ Zs3 ) ) )
& ~ ( member_nat @ X @ ( set_nat2 @ Ys2 ) ) ) ) ) ).
% in_set_conv_decomp_first
thf(fact_1028_in__set__conv__decomp__first,axiom,
! [X: instr,Xs: list_instr] :
( ( member_instr @ X @ ( set_instr2 @ Xs ) )
= ( ? [Ys2: list_instr,Zs3: list_instr] :
( ( Xs
= ( append_instr @ Ys2 @ ( cons_instr @ X @ Zs3 ) ) )
& ~ ( member_instr @ X @ ( set_instr2 @ Ys2 ) ) ) ) ) ).
% in_set_conv_decomp_first
thf(fact_1029_in__set__conv__decomp__first,axiom,
! [X: produc6425607678544837394st_int,Xs: list_P419547167332244514st_int] :
( ( member9100552558438320297st_int @ X @ ( set_Pr3303432077563008829st_int @ Xs ) )
= ( ? [Ys2: list_P419547167332244514st_int,Zs3: list_P419547167332244514st_int] :
( ( Xs
= ( append5028292981072010935st_int @ Ys2 @ ( cons_P4151473878978182108st_int @ X @ Zs3 ) ) )
& ~ ( member9100552558438320297st_int @ X @ ( set_Pr3303432077563008829st_int @ Ys2 ) ) ) ) ) ).
% in_set_conv_decomp_first
thf(fact_1030_in__set__conv__decomp__first,axiom,
! [X: int,Xs: list_int] :
( ( member_int @ X @ ( set_int2 @ Xs ) )
= ( ? [Ys2: list_int,Zs3: list_int] :
( ( Xs
= ( append_int @ Ys2 @ ( cons_int @ X @ Zs3 ) ) )
& ~ ( member_int @ X @ ( set_int2 @ Ys2 ) ) ) ) ) ).
% in_set_conv_decomp_first
thf(fact_1031_in__set__conv__decomp__first,axiom,
! [X: list_char,Xs: list_list_char] :
( ( member_list_char @ X @ ( set_list_char2 @ Xs ) )
= ( ? [Ys2: list_list_char,Zs3: list_list_char] :
( ( Xs
= ( append_list_char @ Ys2 @ ( cons_list_char @ X @ Zs3 ) ) )
& ~ ( member_list_char @ X @ ( set_list_char2 @ Ys2 ) ) ) ) ) ).
% in_set_conv_decomp_first
thf(fact_1032_split__list__last__prop__iff,axiom,
! [Xs: list_int,P: int > $o] :
( ( ? [X2: int] :
( ( member_int @ X2 @ ( set_int2 @ Xs ) )
& ( P @ X2 ) ) )
= ( ? [Ys2: list_int,X2: int,Zs3: list_int] :
( ( Xs
= ( append_int @ Ys2 @ ( cons_int @ X2 @ Zs3 ) ) )
& ( P @ X2 )
& ! [Y5: int] :
( ( member_int @ Y5 @ ( set_int2 @ Zs3 ) )
=> ~ ( P @ Y5 ) ) ) ) ) ).
% split_list_last_prop_iff
thf(fact_1033_split__list__last__prop__iff,axiom,
! [Xs: list_list_char,P: list_char > $o] :
( ( ? [X2: list_char] :
( ( member_list_char @ X2 @ ( set_list_char2 @ Xs ) )
& ( P @ X2 ) ) )
= ( ? [Ys2: list_list_char,X2: list_char,Zs3: list_list_char] :
( ( Xs
= ( append_list_char @ Ys2 @ ( cons_list_char @ X2 @ Zs3 ) ) )
& ( P @ X2 )
& ! [Y5: list_char] :
( ( member_list_char @ Y5 @ ( set_list_char2 @ Zs3 ) )
=> ~ ( P @ Y5 ) ) ) ) ) ).
% split_list_last_prop_iff
thf(fact_1034_execl_Osimps_I3_J,axiom,
! [P: list_instr,V: produc6425607678544837394st_int] : ( execl @ P @ ( cons_P4151473878978182108st_int @ V @ nil_Pr749310521696113548st_int ) ) ).
% execl.simps(3)
thf(fact_1035_upto__aux__rec,axiom,
( upto_aux
= ( ^ [I3: int,J3: int,Js: list_int] : ( if_list_int @ ( ord_less_int @ J3 @ I3 ) @ Js @ ( upto_aux @ I3 @ ( minus_minus_int @ J3 @ one_one_int ) @ ( cons_int @ J3 @ Js ) ) ) ) ) ).
% upto_aux_rec
thf(fact_1036_addr__of__correct,axiom,
! [X: list_char,Xs: list_list_char] :
( ( member_list_char @ X @ ( set_list_char2 @ Xs ) )
=> ( ( inth_list_char @ Xs @ ( minus_minus_int @ ( semiri1314217659103216013at_int @ ( size_s356637359517785349t_char @ Xs ) ) @ ( addr_of @ Xs @ X ) ) )
= X ) ) ).
% addr_of_correct
thf(fact_1037_addr__of__nneg,axiom,
! [Xs: list_list_char,X: list_char] : ( ord_less_eq_int @ zero_zero_int @ ( addr_of @ Xs @ X ) ) ).
% addr_of_nneg
thf(fact_1038_addr__of__set,axiom,
! [X: list_char,Xs: list_list_char] :
( ( member_list_char @ X @ ( set_list_char2 @ Xs ) )
=> ( ord_less_int @ zero_zero_int @ ( addr_of @ Xs @ X ) ) ) ).
% addr_of_set
thf(fact_1039_addr__of__max,axiom,
! [Xs: list_list_char,X: list_char] : ( ord_less_eq_int @ ( addr_of @ Xs @ X ) @ ( semiri1314217659103216013at_int @ ( size_s356637359517785349t_char @ Xs ) ) ) ).
% addr_of_max
thf(fact_1040_addr__of__unique,axiom,
! [Xs: list_list_char,A: int] :
( ( distinct_list_char @ Xs )
=> ( ( ord_less_int @ zero_zero_int @ A )
=> ( ( ord_less_eq_int @ A @ ( semiri1314217659103216013at_int @ ( size_s356637359517785349t_char @ Xs ) ) )
=> ( ( addr_of @ Xs @ ( inth_list_char @ Xs @ ( minus_minus_int @ ( semiri1314217659103216013at_int @ ( size_s356637359517785349t_char @ Xs ) ) @ A ) ) )
= A ) ) ) ) ).
% addr_of_unique
thf(fact_1041_one__less__nat__eq,axiom,
! [Z: int] :
( ( ord_less_nat @ ( suc @ zero_zero_nat ) @ ( nat2 @ Z ) )
= ( ord_less_int @ one_one_int @ Z ) ) ).
% one_less_nat_eq
thf(fact_1042_old_Onat_Oinject,axiom,
! [Nat: nat,Nat2: nat] :
( ( ( suc @ Nat )
= ( suc @ Nat2 ) )
= ( Nat = Nat2 ) ) ).
% old.nat.inject
thf(fact_1043_nat_Oinject,axiom,
! [X23: nat,Y23: nat] :
( ( ( suc @ X23 )
= ( suc @ Y23 ) )
= ( X23 = Y23 ) ) ).
% nat.inject
thf(fact_1044_Suc__less__eq,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ ( suc @ M ) @ ( suc @ N ) )
= ( ord_less_nat @ M @ N ) ) ).
% Suc_less_eq
thf(fact_1045_Suc__mono,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ M @ N )
=> ( ord_less_nat @ ( suc @ M ) @ ( suc @ N ) ) ) ).
% Suc_mono
thf(fact_1046_lessI,axiom,
! [N: nat] : ( ord_less_nat @ N @ ( suc @ N ) ) ).
% lessI
thf(fact_1047_Suc__le__mono,axiom,
! [N: nat,M: nat] :
( ( ord_less_eq_nat @ ( suc @ N ) @ ( suc @ M ) )
= ( ord_less_eq_nat @ N @ M ) ) ).
% Suc_le_mono
thf(fact_1048_diff__Suc__Suc,axiom,
! [M: nat,N: nat] :
( ( minus_minus_nat @ ( suc @ M ) @ ( suc @ N ) )
= ( minus_minus_nat @ M @ N ) ) ).
% diff_Suc_Suc
thf(fact_1049_Suc__diff__diff,axiom,
! [M: nat,N: nat,K: nat] :
( ( minus_minus_nat @ ( minus_minus_nat @ ( suc @ M ) @ N ) @ ( suc @ K ) )
= ( minus_minus_nat @ ( minus_minus_nat @ M @ N ) @ K ) ) ).
% Suc_diff_diff
thf(fact_1050_less__Suc0,axiom,
! [N: nat] :
( ( ord_less_nat @ N @ ( suc @ zero_zero_nat ) )
= ( N = zero_zero_nat ) ) ).
% less_Suc0
thf(fact_1051_zero__less__Suc,axiom,
! [N: nat] : ( ord_less_nat @ zero_zero_nat @ ( suc @ N ) ) ).
% zero_less_Suc
thf(fact_1052_diff__Suc__1,axiom,
! [N: nat] :
( ( minus_minus_nat @ ( suc @ N ) @ one_one_nat )
= N ) ).
% diff_Suc_1
thf(fact_1053_Suc__pred,axiom,
! [N: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( suc @ ( minus_minus_nat @ N @ ( suc @ zero_zero_nat ) ) )
= N ) ) ).
% Suc_pred
thf(fact_1054_Suc__lessI2,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ M @ N )
=> ( ( M
!= ( minus_minus_nat @ N @ one_one_nat ) )
=> ( ord_less_nat @ ( suc @ M ) @ N ) ) ) ).
% Suc_lessI2
thf(fact_1055_nat__1,axiom,
( ( nat2 @ one_one_int )
= ( suc @ zero_zero_nat ) ) ).
% nat_1
thf(fact_1056_Suc__diff__1,axiom,
! [N: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( suc @ ( minus_minus_nat @ N @ one_one_nat ) )
= N ) ) ).
% Suc_diff_1
thf(fact_1057_n__not__Suc__n,axiom,
! [N: nat] :
( N
!= ( suc @ N ) ) ).
% n_not_Suc_n
thf(fact_1058_Suc__inject,axiom,
! [X: nat,Y: nat] :
( ( ( suc @ X )
= ( suc @ Y ) )
=> ( X = Y ) ) ).
% Suc_inject
thf(fact_1059_zero__induct__lemma,axiom,
! [P: nat > $o,K: nat,I: nat] :
( ( P @ K )
=> ( ! [N2: nat] :
( ( P @ ( suc @ N2 ) )
=> ( P @ N2 ) )
=> ( P @ ( minus_minus_nat @ K @ I ) ) ) ) ).
% zero_induct_lemma
thf(fact_1060_transitive__stepwise__le,axiom,
! [M: nat,N: nat,R: nat > nat > $o] :
( ( ord_less_eq_nat @ M @ N )
=> ( ! [X3: nat] : ( R @ X3 @ X3 )
=> ( ! [X3: nat,Y2: nat,Z3: nat] :
( ( R @ X3 @ Y2 )
=> ( ( R @ Y2 @ Z3 )
=> ( R @ X3 @ Z3 ) ) )
=> ( ! [N2: nat] : ( R @ N2 @ ( suc @ N2 ) )
=> ( R @ M @ N ) ) ) ) ) ).
% transitive_stepwise_le
thf(fact_1061_nat__induct__at__least,axiom,
! [M: nat,N: nat,P: nat > $o] :
( ( ord_less_eq_nat @ M @ N )
=> ( ( P @ M )
=> ( ! [N2: nat] :
( ( ord_less_eq_nat @ M @ N2 )
=> ( ( P @ N2 )
=> ( P @ ( suc @ N2 ) ) ) )
=> ( P @ N ) ) ) ) ).
% nat_induct_at_least
thf(fact_1062_full__nat__induct,axiom,
! [P: nat > $o,N: nat] :
( ! [N2: nat] :
( ! [M4: nat] :
( ( ord_less_eq_nat @ ( suc @ M4 ) @ N2 )
=> ( P @ M4 ) )
=> ( P @ N2 ) )
=> ( P @ N ) ) ).
% full_nat_induct
thf(fact_1063_not__less__eq__eq,axiom,
! [M: nat,N: nat] :
( ( ~ ( ord_less_eq_nat @ M @ N ) )
= ( ord_less_eq_nat @ ( suc @ N ) @ M ) ) ).
% not_less_eq_eq
thf(fact_1064_Suc__n__not__le__n,axiom,
! [N: nat] :
~ ( ord_less_eq_nat @ ( suc @ N ) @ N ) ).
% Suc_n_not_le_n
thf(fact_1065_le__Suc__eq,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq_nat @ M @ ( suc @ N ) )
= ( ( ord_less_eq_nat @ M @ N )
| ( M
= ( suc @ N ) ) ) ) ).
% le_Suc_eq
thf(fact_1066_Suc__le__D,axiom,
! [N: nat,M6: nat] :
( ( ord_less_eq_nat @ ( suc @ N ) @ M6 )
=> ? [M2: nat] :
( M6
= ( suc @ M2 ) ) ) ).
% Suc_le_D
thf(fact_1067_le__SucI,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq_nat @ M @ N )
=> ( ord_less_eq_nat @ M @ ( suc @ N ) ) ) ).
% le_SucI
thf(fact_1068_le__SucE,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq_nat @ M @ ( suc @ N ) )
=> ( ~ ( ord_less_eq_nat @ M @ N )
=> ( M
= ( suc @ N ) ) ) ) ).
% le_SucE
thf(fact_1069_Suc__leD,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq_nat @ ( suc @ M ) @ N )
=> ( ord_less_eq_nat @ M @ N ) ) ).
% Suc_leD
thf(fact_1070_not__less__less__Suc__eq,axiom,
! [N: nat,M: nat] :
( ~ ( ord_less_nat @ N @ M )
=> ( ( ord_less_nat @ N @ ( suc @ M ) )
= ( N = M ) ) ) ).
% not_less_less_Suc_eq
thf(fact_1071_strict__inc__induct,axiom,
! [I: nat,J: nat,P: nat > $o] :
( ( ord_less_nat @ I @ J )
=> ( ! [I2: nat] :
( ( J
= ( suc @ I2 ) )
=> ( P @ I2 ) )
=> ( ! [I2: nat] :
( ( ord_less_nat @ I2 @ J )
=> ( ( P @ ( suc @ I2 ) )
=> ( P @ I2 ) ) )
=> ( P @ I ) ) ) ) ).
% strict_inc_induct
thf(fact_1072_less__Suc__induct,axiom,
! [I: nat,J: nat,P: nat > nat > $o] :
( ( ord_less_nat @ I @ J )
=> ( ! [I2: nat] : ( P @ I2 @ ( suc @ I2 ) )
=> ( ! [I2: nat,J2: nat,K2: nat] :
( ( ord_less_nat @ I2 @ J2 )
=> ( ( ord_less_nat @ J2 @ K2 )
=> ( ( P @ I2 @ J2 )
=> ( ( P @ J2 @ K2 )
=> ( P @ I2 @ K2 ) ) ) ) )
=> ( P @ I @ J ) ) ) ) ).
% less_Suc_induct
thf(fact_1073_less__trans__Suc,axiom,
! [I: nat,J: nat,K: nat] :
( ( ord_less_nat @ I @ J )
=> ( ( ord_less_nat @ J @ K )
=> ( ord_less_nat @ ( suc @ I ) @ K ) ) ) ).
% less_trans_Suc
thf(fact_1074_Suc__less__SucD,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ ( suc @ M ) @ ( suc @ N ) )
=> ( ord_less_nat @ M @ N ) ) ).
% Suc_less_SucD
thf(fact_1075_less__antisym,axiom,
! [N: nat,M: nat] :
( ~ ( ord_less_nat @ N @ M )
=> ( ( ord_less_nat @ N @ ( suc @ M ) )
=> ( M = N ) ) ) ).
% less_antisym
thf(fact_1076_Suc__less__eq2,axiom,
! [N: nat,M: nat] :
( ( ord_less_nat @ ( suc @ N ) @ M )
= ( ? [M7: nat] :
( ( M
= ( suc @ M7 ) )
& ( ord_less_nat @ N @ M7 ) ) ) ) ).
% Suc_less_eq2
thf(fact_1077_All__less__Suc,axiom,
! [N: nat,P: nat > $o] :
( ( ! [I3: nat] :
( ( ord_less_nat @ I3 @ ( suc @ N ) )
=> ( P @ I3 ) ) )
= ( ( P @ N )
& ! [I3: nat] :
( ( ord_less_nat @ I3 @ N )
=> ( P @ I3 ) ) ) ) ).
% All_less_Suc
thf(fact_1078_not__less__eq,axiom,
! [M: nat,N: nat] :
( ( ~ ( ord_less_nat @ M @ N ) )
= ( ord_less_nat @ N @ ( suc @ M ) ) ) ).
% not_less_eq
thf(fact_1079_less__Suc__eq,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ M @ ( suc @ N ) )
= ( ( ord_less_nat @ M @ N )
| ( M = N ) ) ) ).
% less_Suc_eq
thf(fact_1080_Ex__less__Suc,axiom,
! [N: nat,P: nat > $o] :
( ( ? [I3: nat] :
( ( ord_less_nat @ I3 @ ( suc @ N ) )
& ( P @ I3 ) ) )
= ( ( P @ N )
| ? [I3: nat] :
( ( ord_less_nat @ I3 @ N )
& ( P @ I3 ) ) ) ) ).
% Ex_less_Suc
thf(fact_1081_less__SucI,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ M @ N )
=> ( ord_less_nat @ M @ ( suc @ N ) ) ) ).
% less_SucI
thf(fact_1082_less__SucE,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ M @ ( suc @ N ) )
=> ( ~ ( ord_less_nat @ M @ N )
=> ( M = N ) ) ) ).
% less_SucE
thf(fact_1083_Suc__lessI,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ M @ N )
=> ( ( ( suc @ M )
!= N )
=> ( ord_less_nat @ ( suc @ M ) @ N ) ) ) ).
% Suc_lessI
thf(fact_1084_Suc__lessE,axiom,
! [I: nat,K: nat] :
( ( ord_less_nat @ ( suc @ I ) @ K )
=> ~ ! [J2: nat] :
( ( ord_less_nat @ I @ J2 )
=> ( K
!= ( suc @ J2 ) ) ) ) ).
% Suc_lessE
thf(fact_1085_Suc__lessD,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ ( suc @ M ) @ N )
=> ( ord_less_nat @ M @ N ) ) ).
% Suc_lessD
thf(fact_1086_Nat_OlessE,axiom,
! [I: nat,K: nat] :
( ( ord_less_nat @ I @ K )
=> ( ( K
!= ( suc @ I ) )
=> ~ ! [J2: nat] :
( ( ord_less_nat @ I @ J2 )
=> ( K
!= ( suc @ J2 ) ) ) ) ) ).
% Nat.lessE
thf(fact_1087_not0__implies__Suc,axiom,
! [N: nat] :
( ( N != zero_zero_nat )
=> ? [M2: nat] :
( N
= ( suc @ M2 ) ) ) ).
% not0_implies_Suc
thf(fact_1088_Zero__not__Suc,axiom,
! [M: nat] :
( zero_zero_nat
!= ( suc @ M ) ) ).
% Zero_not_Suc
thf(fact_1089_Zero__neq__Suc,axiom,
! [M: nat] :
( zero_zero_nat
!= ( suc @ M ) ) ).
% Zero_neq_Suc
thf(fact_1090_Suc__neq__Zero,axiom,
! [M: nat] :
( ( suc @ M )
!= zero_zero_nat ) ).
% Suc_neq_Zero
thf(fact_1091_zero__induct,axiom,
! [P: nat > $o,K: nat] :
( ( P @ K )
=> ( ! [N2: nat] :
( ( P @ ( suc @ N2 ) )
=> ( P @ N2 ) )
=> ( P @ zero_zero_nat ) ) ) ).
% zero_induct
thf(fact_1092_diff__induct,axiom,
! [P: nat > nat > $o,M: nat,N: nat] :
( ! [X3: nat] : ( P @ X3 @ zero_zero_nat )
=> ( ! [Y2: nat] : ( P @ zero_zero_nat @ ( suc @ Y2 ) )
=> ( ! [X3: nat,Y2: nat] :
( ( P @ X3 @ Y2 )
=> ( P @ ( suc @ X3 ) @ ( suc @ Y2 ) ) )
=> ( P @ M @ N ) ) ) ) ).
% diff_induct
thf(fact_1093_nat__induct,axiom,
! [P: nat > $o,N: nat] :
( ( P @ zero_zero_nat )
=> ( ! [N2: nat] :
( ( P @ N2 )
=> ( P @ ( suc @ N2 ) ) )
=> ( P @ N ) ) ) ).
% nat_induct
thf(fact_1094_old_Onat_Oexhaust,axiom,
! [Y: nat] :
( ( Y != zero_zero_nat )
=> ~ ! [Nat3: nat] :
( Y
!= ( suc @ Nat3 ) ) ) ).
% old.nat.exhaust
thf(fact_1095_nat_OdiscI,axiom,
! [Nat: nat,X23: nat] :
( ( Nat
= ( suc @ X23 ) )
=> ( Nat != zero_zero_nat ) ) ).
% nat.discI
thf(fact_1096_old_Onat_Odistinct_I1_J,axiom,
! [Nat2: nat] :
( zero_zero_nat
!= ( suc @ Nat2 ) ) ).
% old.nat.distinct(1)
thf(fact_1097_old_Onat_Odistinct_I2_J,axiom,
! [Nat2: nat] :
( ( suc @ Nat2 )
!= zero_zero_nat ) ).
% old.nat.distinct(2)
thf(fact_1098_nat_Odistinct_I1_J,axiom,
! [X23: nat] :
( zero_zero_nat
!= ( suc @ X23 ) ) ).
% nat.distinct(1)
thf(fact_1099_Ex__less__Suc2,axiom,
! [N: nat,P: nat > $o] :
( ( ? [I3: nat] :
( ( ord_less_nat @ I3 @ ( suc @ N ) )
& ( P @ I3 ) ) )
= ( ( P @ zero_zero_nat )
| ? [I3: nat] :
( ( ord_less_nat @ I3 @ N )
& ( P @ ( suc @ I3 ) ) ) ) ) ).
% Ex_less_Suc2
thf(fact_1100_gr0__conv__Suc,axiom,
! [N: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
= ( ? [M3: nat] :
( N
= ( suc @ M3 ) ) ) ) ).
% gr0_conv_Suc
thf(fact_1101_All__less__Suc2,axiom,
! [N: nat,P: nat > $o] :
( ( ! [I3: nat] :
( ( ord_less_nat @ I3 @ ( suc @ N ) )
=> ( P @ I3 ) ) )
= ( ( P @ zero_zero_nat )
& ! [I3: nat] :
( ( ord_less_nat @ I3 @ N )
=> ( P @ ( suc @ I3 ) ) ) ) ) ).
% All_less_Suc2
thf(fact_1102_gr0__implies__Suc,axiom,
! [N: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ? [M2: nat] :
( N
= ( suc @ M2 ) ) ) ).
% gr0_implies_Suc
thf(fact_1103_less__Suc__eq__0__disj,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ M @ ( suc @ N ) )
= ( ( M = zero_zero_nat )
| ? [J3: nat] :
( ( M
= ( suc @ J3 ) )
& ( ord_less_nat @ J3 @ N ) ) ) ) ).
% less_Suc_eq_0_disj
thf(fact_1104_Suc__leI,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ M @ N )
=> ( ord_less_eq_nat @ ( suc @ M ) @ N ) ) ).
% Suc_leI
thf(fact_1105_Suc__le__eq,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq_nat @ ( suc @ M ) @ N )
= ( ord_less_nat @ M @ N ) ) ).
% Suc_le_eq
thf(fact_1106_dec__induct,axiom,
! [I: nat,J: nat,P: nat > $o] :
( ( ord_less_eq_nat @ I @ J )
=> ( ( P @ I )
=> ( ! [N2: nat] :
( ( ord_less_eq_nat @ I @ N2 )
=> ( ( ord_less_nat @ N2 @ J )
=> ( ( P @ N2 )
=> ( P @ ( suc @ N2 ) ) ) ) )
=> ( P @ J ) ) ) ) ).
% dec_induct
thf(fact_1107_inc__induct,axiom,
! [I: nat,J: nat,P: nat > $o] :
( ( ord_less_eq_nat @ I @ J )
=> ( ( P @ J )
=> ( ! [N2: nat] :
( ( ord_less_eq_nat @ I @ N2 )
=> ( ( ord_less_nat @ N2 @ J )
=> ( ( P @ ( suc @ N2 ) )
=> ( P @ N2 ) ) ) )
=> ( P @ I ) ) ) ) ).
% inc_induct
thf(fact_1108_Suc__le__lessD,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq_nat @ ( suc @ M ) @ N )
=> ( ord_less_nat @ M @ N ) ) ).
% Suc_le_lessD
thf(fact_1109_le__less__Suc__eq,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq_nat @ M @ N )
=> ( ( ord_less_nat @ N @ ( suc @ M ) )
= ( N = M ) ) ) ).
% le_less_Suc_eq
thf(fact_1110_less__Suc__eq__le,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ M @ ( suc @ N ) )
= ( ord_less_eq_nat @ M @ N ) ) ).
% less_Suc_eq_le
thf(fact_1111_less__eq__Suc__le,axiom,
( ord_less_nat
= ( ^ [N3: nat] : ( ord_less_eq_nat @ ( suc @ N3 ) ) ) ) ).
% less_eq_Suc_le
thf(fact_1112_le__imp__less__Suc,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq_nat @ M @ N )
=> ( ord_less_nat @ M @ ( suc @ N ) ) ) ).
% le_imp_less_Suc
thf(fact_1113_Suc__diff__Suc,axiom,
! [N: nat,M: nat] :
( ( ord_less_nat @ N @ M )
=> ( ( suc @ ( minus_minus_nat @ M @ ( suc @ N ) ) )
= ( minus_minus_nat @ M @ N ) ) ) ).
% Suc_diff_Suc
thf(fact_1114_diff__less__Suc,axiom,
! [M: nat,N: nat] : ( ord_less_nat @ ( minus_minus_nat @ M @ N ) @ ( suc @ M ) ) ).
% diff_less_Suc
thf(fact_1115_Suc__diff__le,axiom,
! [N: nat,M: nat] :
( ( ord_less_eq_nat @ N @ M )
=> ( ( minus_minus_nat @ ( suc @ M ) @ N )
= ( suc @ ( minus_minus_nat @ M @ N ) ) ) ) ).
% Suc_diff_le
thf(fact_1116_One__nat__def,axiom,
( one_one_nat
= ( suc @ zero_zero_nat ) ) ).
% One_nat_def
thf(fact_1117_diff__Suc__eq__diff__pred,axiom,
! [M: nat,N: nat] :
( ( minus_minus_nat @ M @ ( suc @ N ) )
= ( minus_minus_nat @ ( minus_minus_nat @ M @ one_one_nat ) @ N ) ) ).
% diff_Suc_eq_diff_pred
thf(fact_1118_ex__least__nat__less,axiom,
! [P: nat > $o,N: nat] :
( ( P @ N )
=> ( ~ ( P @ zero_zero_nat )
=> ? [K2: nat] :
( ( ord_less_nat @ K2 @ N )
& ! [I4: nat] :
( ( ord_less_eq_nat @ I4 @ K2 )
=> ~ ( P @ I4 ) )
& ( P @ ( suc @ K2 ) ) ) ) ) ).
% ex_least_nat_less
thf(fact_1119_diff__Suc__less,axiom,
! [N: nat,I: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ord_less_nat @ ( minus_minus_nat @ N @ ( suc @ I ) ) @ N ) ) ).
% diff_Suc_less
thf(fact_1120_nat__induct__non__zero,axiom,
! [N: nat,P: nat > $o] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( P @ one_one_nat )
=> ( ! [N2: nat] :
( ( ord_less_nat @ zero_zero_nat @ N2 )
=> ( ( P @ N2 )
=> ( P @ ( suc @ N2 ) ) ) )
=> ( P @ N ) ) ) ) ).
% nat_induct_non_zero
thf(fact_1121_Suc__diff__eq__diff__pred,axiom,
! [N: nat,M: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( minus_minus_nat @ ( suc @ M ) @ N )
= ( minus_minus_nat @ M @ ( minus_minus_nat @ N @ one_one_nat ) ) ) ) ).
% Suc_diff_eq_diff_pred
thf(fact_1122_Suc__pred_H,axiom,
! [N: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( N
= ( suc @ ( minus_minus_nat @ N @ one_one_nat ) ) ) ) ).
% Suc_pred'
thf(fact_1123_execl__next,axiom,
! [P: list_instr,Cfs: list_P419547167332244514st_int,K: nat] :
( ( execl @ P @ Cfs )
=> ( ( ord_less_nat @ K @ ( size_s229347708945616566st_int @ Cfs ) )
=> ( ( K
!= ( minus_minus_nat @ ( size_s229347708945616566st_int @ Cfs ) @ one_one_nat ) )
=> ( ( iexec_pred @ ( produc1484136438817787418st_int @ ( inth_instr @ P @ ( produc8941309049508465880st_int @ ( nth_Pr8327173284404383691st_int @ Cfs @ K ) ) ) @ ( nth_Pr8327173284404383691st_int @ Cfs @ K ) ) @ ( nth_Pr8327173284404383691st_int @ Cfs @ ( suc @ K ) ) )
& ( ord_less_eq_int @ zero_zero_int @ ( produc8941309049508465880st_int @ ( nth_Pr8327173284404383691st_int @ Cfs @ K ) ) )
& ( ord_less_int @ ( produc8941309049508465880st_int @ ( nth_Pr8327173284404383691st_int @ Cfs @ K ) ) @ ( semiri1314217659103216013at_int @ ( size_size_list_instr @ P ) ) ) ) ) ) ) ).
% execl_next
thf(fact_1124_execl__all__SKIP,axiom,
! [Cfs: list_P419547167332244514st_int] :
( ( execl_all @ nil_instr @ Cfs )
=> ( ( cpred @ skip @ ( nth_Pr8327173284404383691st_int @ Cfs @ zero_zero_nat ) @ ( nth_Pr8327173284404383691st_int @ Cfs @ ( minus_minus_nat @ ( size_s229347708945616566st_int @ Cfs ) @ ( suc @ zero_zero_nat ) ) ) )
& ( ( size_s229347708945616566st_int @ Cfs )
= ( suc @ zero_zero_nat ) ) ) ) ).
% execl_all_SKIP
thf(fact_1125_nat__ivt__aux,axiom,
! [N: nat,F: nat > int,K: int] :
( ! [I2: nat] :
( ( ord_less_nat @ I2 @ N )
=> ( ord_less_eq_int @ ( abs_abs_int @ ( minus_minus_int @ ( F @ ( suc @ I2 ) ) @ ( F @ I2 ) ) ) @ one_one_int ) )
=> ( ( ord_less_eq_int @ ( F @ zero_zero_nat ) @ K )
=> ( ( ord_less_eq_int @ K @ ( F @ N ) )
=> ? [I2: nat] :
( ( ord_less_eq_nat @ I2 @ N )
& ( ( F @ I2 )
= K ) ) ) ) ) ).
% nat_ivt_aux
thf(fact_1126_zabs__less__one__iff,axiom,
! [Z: int] :
( ( ord_less_int @ ( abs_abs_int @ Z ) @ one_one_int )
= ( Z = zero_zero_int ) ) ).
% zabs_less_one_iff
thf(fact_1127_nat__abs__int__diff,axiom,
! [A: nat,B: nat] :
( ( ( ord_less_eq_nat @ A @ B )
=> ( ( nat2 @ ( abs_abs_int @ ( minus_minus_int @ ( semiri1314217659103216013at_int @ A ) @ ( semiri1314217659103216013at_int @ B ) ) ) )
= ( minus_minus_nat @ B @ A ) ) )
& ( ~ ( ord_less_eq_nat @ A @ B )
=> ( ( nat2 @ ( abs_abs_int @ ( minus_minus_int @ ( semiri1314217659103216013at_int @ A ) @ ( semiri1314217659103216013at_int @ B ) ) ) )
= ( minus_minus_nat @ A @ B ) ) ) ) ).
% nat_abs_int_diff
thf(fact_1128_nat__intermed__int__val,axiom,
! [M: nat,N: nat,F: nat > int,K: int] :
( ! [I2: nat] :
( ( ( ord_less_eq_nat @ M @ I2 )
& ( ord_less_nat @ I2 @ N ) )
=> ( ord_less_eq_int @ ( abs_abs_int @ ( minus_minus_int @ ( F @ ( suc @ I2 ) ) @ ( F @ I2 ) ) ) @ one_one_int ) )
=> ( ( ord_less_eq_nat @ M @ N )
=> ( ( ord_less_eq_int @ ( F @ M ) @ K )
=> ( ( ord_less_eq_int @ K @ ( F @ N ) )
=> ? [I2: nat] :
( ( ord_less_eq_nat @ M @ I2 )
& ( ord_less_eq_nat @ I2 @ N )
& ( ( F @ I2 )
= K ) ) ) ) ) ) ).
% nat_intermed_int_val
thf(fact_1129_exec1__def,axiom,
( exec1
= ( ^ [P3: list_instr,Cf: produc6425607678544837394st_int,Cf2: produc6425607678544837394st_int] :
( ( iexec_pred @ ( produc1484136438817787418st_int @ ( inth_instr @ P3 @ ( produc8941309049508465880st_int @ Cf ) ) @ Cf ) @ Cf2 )
& ( ord_less_eq_int @ zero_zero_int @ ( produc8941309049508465880st_int @ Cf ) )
& ( ord_less_int @ ( produc8941309049508465880st_int @ Cf ) @ ( semiri1314217659103216013at_int @ ( size_size_list_instr @ P3 ) ) ) ) ) ) ).
% exec1_def
thf(fact_1130_nat0__intermed__int__val,axiom,
! [N: nat,F: nat > int,K: int] :
( ! [I2: nat] :
( ( ord_less_nat @ I2 @ N )
=> ( ord_less_eq_int @ ( abs_abs_int @ ( minus_minus_int @ ( F @ ( plus_plus_nat @ I2 @ one_one_nat ) ) @ ( F @ I2 ) ) ) @ one_one_int ) )
=> ( ( ord_less_eq_int @ ( F @ zero_zero_nat ) @ K )
=> ( ( ord_less_eq_int @ K @ ( F @ N ) )
=> ? [I2: nat] :
( ( ord_less_eq_nat @ I2 @ N )
& ( ( F @ I2 )
= K ) ) ) ) ) ).
% nat0_intermed_int_val
thf(fact_1131_Nat_Oadd__0__right,axiom,
! [M: nat] :
( ( plus_plus_nat @ M @ zero_zero_nat )
= M ) ).
% Nat.add_0_right
thf(fact_1132_add__is__0,axiom,
! [M: nat,N: nat] :
( ( ( plus_plus_nat @ M @ N )
= zero_zero_nat )
= ( ( M = zero_zero_nat )
& ( N = zero_zero_nat ) ) ) ).
% add_is_0
thf(fact_1133_add__Suc__right,axiom,
! [M: nat,N: nat] :
( ( plus_plus_nat @ M @ ( suc @ N ) )
= ( suc @ ( plus_plus_nat @ M @ N ) ) ) ).
% add_Suc_right
thf(fact_1134_nat__add__left__cancel__less,axiom,
! [K: nat,M: nat,N: nat] :
( ( ord_less_nat @ ( plus_plus_nat @ K @ M ) @ ( plus_plus_nat @ K @ N ) )
= ( ord_less_nat @ M @ N ) ) ).
% nat_add_left_cancel_less
thf(fact_1135_nat__add__left__cancel__le,axiom,
! [K: nat,M: nat,N: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ K @ M ) @ ( plus_plus_nat @ K @ N ) )
= ( ord_less_eq_nat @ M @ N ) ) ).
% nat_add_left_cancel_le
thf(fact_1136_diff__diff__left,axiom,
! [I: nat,J: nat,K: nat] :
( ( minus_minus_nat @ ( minus_minus_nat @ I @ J ) @ K )
= ( minus_minus_nat @ I @ ( plus_plus_nat @ J @ K ) ) ) ).
% diff_diff_left
thf(fact_1137_add__gr__0,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ zero_zero_nat @ ( plus_plus_nat @ M @ N ) )
= ( ( ord_less_nat @ zero_zero_nat @ M )
| ( ord_less_nat @ zero_zero_nat @ N ) ) ) ).
% add_gr_0
thf(fact_1138_Nat_Oadd__diff__assoc,axiom,
! [K: nat,J: nat,I: nat] :
( ( ord_less_eq_nat @ K @ J )
=> ( ( plus_plus_nat @ I @ ( minus_minus_nat @ J @ K ) )
= ( minus_minus_nat @ ( plus_plus_nat @ I @ J ) @ K ) ) ) ).
% Nat.add_diff_assoc
thf(fact_1139_Nat_Oadd__diff__assoc2,axiom,
! [K: nat,J: nat,I: nat] :
( ( ord_less_eq_nat @ K @ J )
=> ( ( plus_plus_nat @ ( minus_minus_nat @ J @ K ) @ I )
= ( minus_minus_nat @ ( plus_plus_nat @ J @ I ) @ K ) ) ) ).
% Nat.add_diff_assoc2
thf(fact_1140_Nat_Odiff__diff__right,axiom,
! [K: nat,J: nat,I: nat] :
( ( ord_less_eq_nat @ K @ J )
=> ( ( minus_minus_nat @ I @ ( minus_minus_nat @ J @ K ) )
= ( minus_minus_nat @ ( plus_plus_nat @ I @ K ) @ J ) ) ) ).
% Nat.diff_diff_right
thf(fact_1141_iexec__offset,axiom,
! [Ins: instr,Pc: int,S: list_char > int,Stk: list_int,Pc2: int,S4: list_char > int,Stk2: list_int,I: int] :
( ( iexec_pred @ ( produc1484136438817787418st_int @ Ins @ ( produc5086643055186798020st_int @ Pc @ ( produc8650753666468850689st_int @ S @ Stk ) ) ) @ ( produc5086643055186798020st_int @ Pc2 @ ( produc8650753666468850689st_int @ S4 @ Stk2 ) ) )
=> ( iexec_pred @ ( produc1484136438817787418st_int @ Ins @ ( produc5086643055186798020st_int @ ( minus_minus_int @ Pc @ I ) @ ( produc8650753666468850689st_int @ S @ Stk ) ) ) @ ( produc5086643055186798020st_int @ ( minus_minus_int @ Pc2 @ I ) @ ( produc8650753666468850689st_int @ S4 @ Stk2 ) ) ) ) ).
% iexec_offset
thf(fact_1142_diff__Suc__diff__eq2,axiom,
! [K: nat,J: nat,I: nat] :
( ( ord_less_eq_nat @ K @ J )
=> ( ( minus_minus_nat @ ( suc @ ( minus_minus_nat @ J @ K ) ) @ I )
= ( minus_minus_nat @ ( suc @ J ) @ ( plus_plus_nat @ K @ I ) ) ) ) ).
% diff_Suc_diff_eq2
thf(fact_1143_diff__Suc__diff__eq1,axiom,
! [K: nat,J: nat,I: nat] :
( ( ord_less_eq_nat @ K @ J )
=> ( ( minus_minus_nat @ I @ ( suc @ ( minus_minus_nat @ J @ K ) ) )
= ( minus_minus_nat @ ( plus_plus_nat @ I @ K ) @ ( suc @ J ) ) ) ) ).
% diff_Suc_diff_eq1
thf(fact_1144_exec1__appendR,axiom,
! [P: list_instr,C: produc6425607678544837394st_int,C4: produc6425607678544837394st_int,P4: list_instr] :
( ( exec1 @ P @ C @ C4 )
=> ( exec1 @ ( append_instr @ P @ P4 ) @ C @ C4 ) ) ).
% exec1_appendR
thf(fact_1145_Nat_Odiff__cancel,axiom,
! [K: nat,M: nat,N: nat] :
( ( minus_minus_nat @ ( plus_plus_nat @ K @ M ) @ ( plus_plus_nat @ K @ N ) )
= ( minus_minus_nat @ M @ N ) ) ).
% Nat.diff_cancel
thf(fact_1146_diff__cancel2,axiom,
! [M: nat,K: nat,N: nat] :
( ( minus_minus_nat @ ( plus_plus_nat @ M @ K ) @ ( plus_plus_nat @ N @ K ) )
= ( minus_minus_nat @ M @ N ) ) ).
% diff_cancel2
thf(fact_1147_diff__add__inverse,axiom,
! [N: nat,M: nat] :
( ( minus_minus_nat @ ( plus_plus_nat @ N @ M ) @ N )
= M ) ).
% diff_add_inverse
thf(fact_1148_diff__add__inverse2,axiom,
! [M: nat,N: nat] :
( ( minus_minus_nat @ ( plus_plus_nat @ M @ N ) @ N )
= M ) ).
% diff_add_inverse2
thf(fact_1149_nat__le__iff__add,axiom,
( ord_less_eq_nat
= ( ^ [M3: nat,N3: nat] :
? [K3: nat] :
( N3
= ( plus_plus_nat @ M3 @ K3 ) ) ) ) ).
% nat_le_iff_add
thf(fact_1150_trans__le__add2,axiom,
! [I: nat,J: nat,M: nat] :
( ( ord_less_eq_nat @ I @ J )
=> ( ord_less_eq_nat @ I @ ( plus_plus_nat @ M @ J ) ) ) ).
% trans_le_add2
thf(fact_1151_trans__le__add1,axiom,
! [I: nat,J: nat,M: nat] :
( ( ord_less_eq_nat @ I @ J )
=> ( ord_less_eq_nat @ I @ ( plus_plus_nat @ J @ M ) ) ) ).
% trans_le_add1
thf(fact_1152_add__le__mono1,axiom,
! [I: nat,J: nat,K: nat] :
( ( ord_less_eq_nat @ I @ J )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ I @ K ) @ ( plus_plus_nat @ J @ K ) ) ) ).
% add_le_mono1
thf(fact_1153_add__le__mono,axiom,
! [I: nat,J: nat,K: nat,L: nat] :
( ( ord_less_eq_nat @ I @ J )
=> ( ( ord_less_eq_nat @ K @ L )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ I @ K ) @ ( plus_plus_nat @ J @ L ) ) ) ) ).
% add_le_mono
thf(fact_1154_le__Suc__ex,axiom,
! [K: nat,L: nat] :
( ( ord_less_eq_nat @ K @ L )
=> ? [N2: nat] :
( L
= ( plus_plus_nat @ K @ N2 ) ) ) ).
% le_Suc_ex
thf(fact_1155_add__leD2,axiom,
! [M: nat,K: nat,N: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ M @ K ) @ N )
=> ( ord_less_eq_nat @ K @ N ) ) ).
% add_leD2
thf(fact_1156_add__leD1,axiom,
! [M: nat,K: nat,N: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ M @ K ) @ N )
=> ( ord_less_eq_nat @ M @ N ) ) ).
% add_leD1
thf(fact_1157_le__add2,axiom,
! [N: nat,M: nat] : ( ord_less_eq_nat @ N @ ( plus_plus_nat @ M @ N ) ) ).
% le_add2
thf(fact_1158_le__add1,axiom,
! [N: nat,M: nat] : ( ord_less_eq_nat @ N @ ( plus_plus_nat @ N @ M ) ) ).
% le_add1
thf(fact_1159_add__leE,axiom,
! [M: nat,K: nat,N: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ M @ K ) @ N )
=> ~ ( ( ord_less_eq_nat @ M @ N )
=> ~ ( ord_less_eq_nat @ K @ N ) ) ) ).
% add_leE
thf(fact_1160_less__add__eq__less,axiom,
! [K: nat,L: nat,M: nat,N: nat] :
( ( ord_less_nat @ K @ L )
=> ( ( ( plus_plus_nat @ M @ L )
= ( plus_plus_nat @ K @ N ) )
=> ( ord_less_nat @ M @ N ) ) ) ).
% less_add_eq_less
thf(fact_1161_trans__less__add2,axiom,
! [I: nat,J: nat,M: nat] :
( ( ord_less_nat @ I @ J )
=> ( ord_less_nat @ I @ ( plus_plus_nat @ M @ J ) ) ) ).
% trans_less_add2
thf(fact_1162_trans__less__add1,axiom,
! [I: nat,J: nat,M: nat] :
( ( ord_less_nat @ I @ J )
=> ( ord_less_nat @ I @ ( plus_plus_nat @ J @ M ) ) ) ).
% trans_less_add1
thf(fact_1163_add__less__mono1,axiom,
! [I: nat,J: nat,K: nat] :
( ( ord_less_nat @ I @ J )
=> ( ord_less_nat @ ( plus_plus_nat @ I @ K ) @ ( plus_plus_nat @ J @ K ) ) ) ).
% add_less_mono1
thf(fact_1164_not__add__less2,axiom,
! [J: nat,I: nat] :
~ ( ord_less_nat @ ( plus_plus_nat @ J @ I ) @ I ) ).
% not_add_less2
thf(fact_1165_not__add__less1,axiom,
! [I: nat,J: nat] :
~ ( ord_less_nat @ ( plus_plus_nat @ I @ J ) @ I ) ).
% not_add_less1
thf(fact_1166_add__less__mono,axiom,
! [I: nat,J: nat,K: nat,L: nat] :
( ( ord_less_nat @ I @ J )
=> ( ( ord_less_nat @ K @ L )
=> ( ord_less_nat @ ( plus_plus_nat @ I @ K ) @ ( plus_plus_nat @ J @ L ) ) ) ) ).
% add_less_mono
thf(fact_1167_add__lessD1,axiom,
! [I: nat,J: nat,K: nat] :
( ( ord_less_nat @ ( plus_plus_nat @ I @ J ) @ K )
=> ( ord_less_nat @ I @ K ) ) ).
% add_lessD1
thf(fact_1168_add__eq__self__zero,axiom,
! [M: nat,N: nat] :
( ( ( plus_plus_nat @ M @ N )
= M )
=> ( N = zero_zero_nat ) ) ).
% add_eq_self_zero
thf(fact_1169_plus__nat_Oadd__0,axiom,
! [N: nat] :
( ( plus_plus_nat @ zero_zero_nat @ N )
= N ) ).
% plus_nat.add_0
thf(fact_1170_nat__arith_Osuc1,axiom,
! [A2: nat,K: nat,A: nat] :
( ( A2
= ( plus_plus_nat @ K @ A ) )
=> ( ( suc @ A2 )
= ( plus_plus_nat @ K @ ( suc @ A ) ) ) ) ).
% nat_arith.suc1
thf(fact_1171_add__Suc,axiom,
! [M: nat,N: nat] :
( ( plus_plus_nat @ ( suc @ M ) @ N )
= ( suc @ ( plus_plus_nat @ M @ N ) ) ) ).
% add_Suc
thf(fact_1172_add__Suc__shift,axiom,
! [M: nat,N: nat] :
( ( plus_plus_nat @ ( suc @ M ) @ N )
= ( plus_plus_nat @ M @ ( suc @ N ) ) ) ).
% add_Suc_shift
thf(fact_1173_one__is__add,axiom,
! [M: nat,N: nat] :
( ( ( suc @ zero_zero_nat )
= ( plus_plus_nat @ M @ N ) )
= ( ( ( M
= ( suc @ zero_zero_nat ) )
& ( N = zero_zero_nat ) )
| ( ( M = zero_zero_nat )
& ( N
= ( suc @ zero_zero_nat ) ) ) ) ) ).
% one_is_add
thf(fact_1174_add__is__1,axiom,
! [M: nat,N: nat] :
( ( ( plus_plus_nat @ M @ N )
= ( suc @ zero_zero_nat ) )
= ( ( ( M
= ( suc @ zero_zero_nat ) )
& ( N = zero_zero_nat ) )
| ( ( M = zero_zero_nat )
& ( N
= ( suc @ zero_zero_nat ) ) ) ) ) ).
% add_is_1
thf(fact_1175_less__imp__add__positive,axiom,
! [I: nat,J: nat] :
( ( ord_less_nat @ I @ J )
=> ? [K2: nat] :
( ( ord_less_nat @ zero_zero_nat @ K2 )
& ( ( plus_plus_nat @ I @ K2 )
= J ) ) ) ).
% less_imp_add_positive
thf(fact_1176_less__imp__Suc__add,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ M @ N )
=> ? [K2: nat] :
( N
= ( suc @ ( plus_plus_nat @ M @ K2 ) ) ) ) ).
% less_imp_Suc_add
thf(fact_1177_less__iff__Suc__add,axiom,
( ord_less_nat
= ( ^ [M3: nat,N3: nat] :
? [K3: nat] :
( N3
= ( suc @ ( plus_plus_nat @ M3 @ K3 ) ) ) ) ) ).
% less_iff_Suc_add
thf(fact_1178_less__add__Suc2,axiom,
! [I: nat,M: nat] : ( ord_less_nat @ I @ ( suc @ ( plus_plus_nat @ M @ I ) ) ) ).
% less_add_Suc2
thf(fact_1179_less__add__Suc1,axiom,
! [I: nat,M: nat] : ( ord_less_nat @ I @ ( suc @ ( plus_plus_nat @ I @ M ) ) ) ).
% less_add_Suc1
thf(fact_1180_less__natE,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ M @ N )
=> ~ ! [Q3: nat] :
( N
!= ( suc @ ( plus_plus_nat @ M @ Q3 ) ) ) ) ).
% less_natE
thf(fact_1181_mono__nat__linear__lb,axiom,
! [F: nat > nat,M: nat,K: nat] :
( ! [M2: nat,N2: nat] :
( ( ord_less_nat @ M2 @ N2 )
=> ( ord_less_nat @ ( F @ M2 ) @ ( F @ N2 ) ) )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ ( F @ M ) @ K ) @ ( F @ ( plus_plus_nat @ M @ K ) ) ) ) ).
% mono_nat_linear_lb
thf(fact_1182_diff__add__0,axiom,
! [N: nat,M: nat] :
( ( minus_minus_nat @ N @ ( plus_plus_nat @ N @ M ) )
= zero_zero_nat ) ).
% diff_add_0
thf(fact_1183_less__diff__conv,axiom,
! [I: nat,J: nat,K: nat] :
( ( ord_less_nat @ I @ ( minus_minus_nat @ J @ K ) )
= ( ord_less_nat @ ( plus_plus_nat @ I @ K ) @ J ) ) ).
% less_diff_conv
thf(fact_1184_add__diff__inverse__nat,axiom,
! [M: nat,N: nat] :
( ~ ( ord_less_nat @ M @ N )
=> ( ( plus_plus_nat @ N @ ( minus_minus_nat @ M @ N ) )
= M ) ) ).
% add_diff_inverse_nat
thf(fact_1185_le__diff__conv,axiom,
! [J: nat,K: nat,I: nat] :
( ( ord_less_eq_nat @ ( minus_minus_nat @ J @ K ) @ I )
= ( ord_less_eq_nat @ J @ ( plus_plus_nat @ I @ K ) ) ) ).
% le_diff_conv
thf(fact_1186_Nat_Ole__diff__conv2,axiom,
! [K: nat,J: nat,I: nat] :
( ( ord_less_eq_nat @ K @ J )
=> ( ( ord_less_eq_nat @ I @ ( minus_minus_nat @ J @ K ) )
= ( ord_less_eq_nat @ ( plus_plus_nat @ I @ K ) @ J ) ) ) ).
% Nat.le_diff_conv2
thf(fact_1187_Nat_Odiff__add__assoc,axiom,
! [K: nat,J: nat,I: nat] :
( ( ord_less_eq_nat @ K @ J )
=> ( ( minus_minus_nat @ ( plus_plus_nat @ I @ J ) @ K )
= ( plus_plus_nat @ I @ ( minus_minus_nat @ J @ K ) ) ) ) ).
% Nat.diff_add_assoc
thf(fact_1188_Nat_Odiff__add__assoc2,axiom,
! [K: nat,J: nat,I: nat] :
( ( ord_less_eq_nat @ K @ J )
=> ( ( minus_minus_nat @ ( plus_plus_nat @ J @ I ) @ K )
= ( plus_plus_nat @ ( minus_minus_nat @ J @ K ) @ I ) ) ) ).
% Nat.diff_add_assoc2
thf(fact_1189_Nat_Ole__imp__diff__is__add,axiom,
! [I: nat,J: nat,K: nat] :
( ( ord_less_eq_nat @ I @ J )
=> ( ( ( minus_minus_nat @ J @ I )
= K )
= ( J
= ( plus_plus_nat @ K @ I ) ) ) ) ).
% Nat.le_imp_diff_is_add
thf(fact_1190_Suc__eq__plus1,axiom,
( suc
= ( ^ [N3: nat] : ( plus_plus_nat @ N3 @ one_one_nat ) ) ) ).
% Suc_eq_plus1
thf(fact_1191_plus__1__eq__Suc,axiom,
( ( plus_plus_nat @ one_one_nat )
= suc ) ).
% plus_1_eq_Suc
thf(fact_1192_Suc__eq__plus1__left,axiom,
( suc
= ( plus_plus_nat @ one_one_nat ) ) ).
% Suc_eq_plus1_left
thf(fact_1193_execl_Ocases,axiom,
! [X: produc4400946599758736694st_int] :
( ! [P5: list_instr,Cf3: produc6425607678544837394st_int,Cf4: produc6425607678544837394st_int,Cfs3: list_P419547167332244514st_int] :
( X
!= ( produc7764413618310988720st_int @ P5 @ ( cons_P4151473878978182108st_int @ Cf3 @ ( cons_P4151473878978182108st_int @ Cf4 @ Cfs3 ) ) ) )
=> ( ! [P5: list_instr] :
( X
!= ( produc7764413618310988720st_int @ P5 @ nil_Pr749310521696113548st_int ) )
=> ~ ! [P5: list_instr,V2: produc6425607678544837394st_int] :
( X
!= ( produc7764413618310988720st_int @ P5 @ ( cons_P4151473878978182108st_int @ V2 @ nil_Pr749310521696113548st_int ) ) ) ) ) ).
% execl.cases
thf(fact_1194_execl_Osimps_I1_J,axiom,
! [P: list_instr,Cf5: produc6425607678544837394st_int,Cf6: produc6425607678544837394st_int,Cfs: list_P419547167332244514st_int] :
( ( execl @ P @ ( cons_P4151473878978182108st_int @ Cf5 @ ( cons_P4151473878978182108st_int @ Cf6 @ Cfs ) ) )
= ( ( exec1 @ P @ Cf5 @ Cf6 )
& ( execl @ P @ ( cons_P4151473878978182108st_int @ Cf6 @ Cfs ) ) ) ) ).
% execl.simps(1)
thf(fact_1195_execl_Oelims_I3_J,axiom,
! [X: list_instr,Xa: list_P419547167332244514st_int] :
( ~ ( execl @ X @ Xa )
=> ~ ! [Cf3: produc6425607678544837394st_int,Cf4: produc6425607678544837394st_int,Cfs3: list_P419547167332244514st_int] :
( ( Xa
= ( cons_P4151473878978182108st_int @ Cf3 @ ( cons_P4151473878978182108st_int @ Cf4 @ Cfs3 ) ) )
=> ( ( exec1 @ X @ Cf3 @ Cf4 )
& ( execl @ X @ ( cons_P4151473878978182108st_int @ Cf4 @ Cfs3 ) ) ) ) ) ).
% execl.elims(3)
thf(fact_1196_nat__diff__split,axiom,
! [P: nat > $o,A: nat,B: nat] :
( ( P @ ( minus_minus_nat @ A @ B ) )
= ( ( ( ord_less_nat @ A @ B )
=> ( P @ zero_zero_nat ) )
& ! [D4: nat] :
( ( A
= ( plus_plus_nat @ B @ D4 ) )
=> ( P @ D4 ) ) ) ) ).
% nat_diff_split
thf(fact_1197_nat__diff__split__asm,axiom,
! [P: nat > $o,A: nat,B: nat] :
( ( P @ ( minus_minus_nat @ A @ B ) )
= ( ~ ( ( ( ord_less_nat @ A @ B )
& ~ ( P @ zero_zero_nat ) )
| ? [D4: nat] :
( ( A
= ( plus_plus_nat @ B @ D4 ) )
& ~ ( P @ D4 ) ) ) ) ) ).
% nat_diff_split_asm
thf(fact_1198_less__diff__conv2,axiom,
! [K: nat,J: nat,I: nat] :
( ( ord_less_eq_nat @ K @ J )
=> ( ( ord_less_nat @ ( minus_minus_nat @ J @ K ) @ I )
= ( ord_less_nat @ J @ ( plus_plus_nat @ I @ K ) ) ) ) ).
% less_diff_conv2
thf(fact_1199_add__eq__if,axiom,
( plus_plus_nat
= ( ^ [M3: nat,N3: nat] : ( if_nat @ ( M3 = zero_zero_nat ) @ N3 @ ( suc @ ( plus_plus_nat @ ( minus_minus_nat @ M3 @ one_one_nat ) @ N3 ) ) ) ) ) ).
% add_eq_if
thf(fact_1200_execl_Oelims_I1_J,axiom,
! [X: list_instr,Xa: list_P419547167332244514st_int,Y: $o] :
( ( ( execl @ X @ Xa )
= Y )
=> ( ! [Cf3: produc6425607678544837394st_int,Cf4: produc6425607678544837394st_int,Cfs3: list_P419547167332244514st_int] :
( ( Xa
= ( cons_P4151473878978182108st_int @ Cf3 @ ( cons_P4151473878978182108st_int @ Cf4 @ Cfs3 ) ) )
=> ( Y
= ( ~ ( ( exec1 @ X @ Cf3 @ Cf4 )
& ( execl @ X @ ( cons_P4151473878978182108st_int @ Cf4 @ Cfs3 ) ) ) ) ) )
=> ( ( ( Xa = nil_Pr749310521696113548st_int )
=> ~ Y )
=> ~ ( ? [V2: produc6425607678544837394st_int] :
( Xa
= ( cons_P4151473878978182108st_int @ V2 @ nil_Pr749310521696113548st_int ) )
=> ~ Y ) ) ) ) ).
% execl.elims(1)
thf(fact_1201_execl_Oelims_I2_J,axiom,
! [X: list_instr,Xa: list_P419547167332244514st_int] :
( ( execl @ X @ Xa )
=> ( ! [Cf3: produc6425607678544837394st_int,Cf4: produc6425607678544837394st_int,Cfs3: list_P419547167332244514st_int] :
( ( Xa
= ( cons_P4151473878978182108st_int @ Cf3 @ ( cons_P4151473878978182108st_int @ Cf4 @ Cfs3 ) ) )
=> ~ ( ( exec1 @ X @ Cf3 @ Cf4 )
& ( execl @ X @ ( cons_P4151473878978182108st_int @ Cf4 @ Cfs3 ) ) ) )
=> ( ( Xa != nil_Pr749310521696113548st_int )
=> ~ ! [V2: produc6425607678544837394st_int] :
( Xa
!= ( cons_P4151473878978182108st_int @ V2 @ nil_Pr749310521696113548st_int ) ) ) ) ) ).
% execl.elims(2)
thf(fact_1202_exec1I,axiom,
! [C4: produc6425607678544837394st_int,P: list_instr,I: int,S: list_char > int,Stk: list_int] :
( ( C4
= ( iexec @ ( inth_instr @ P @ I ) @ ( produc5086643055186798020st_int @ I @ ( produc8650753666468850689st_int @ S @ Stk ) ) ) )
=> ( ( ord_less_eq_int @ zero_zero_int @ I )
=> ( ( ord_less_int @ I @ ( semiri1314217659103216013at_int @ ( size_size_list_instr @ P ) ) )
=> ( exec1 @ P @ ( produc5086643055186798020st_int @ I @ ( produc8650753666468850689st_int @ S @ Stk ) ) @ C4 ) ) ) ) ).
% exec1I
thf(fact_1203_zle__add1__eq__le,axiom,
! [W: int,Z: int] :
( ( ord_less_int @ W @ ( plus_plus_int @ Z @ one_one_int ) )
= ( ord_less_eq_int @ W @ Z ) ) ).
% zle_add1_eq_le
thf(fact_1204_plus__int__code_I2_J,axiom,
! [L: int] :
( ( plus_plus_int @ zero_zero_int @ L )
= L ) ).
% plus_int_code(2)
thf(fact_1205_plus__int__code_I1_J,axiom,
! [K: int] :
( ( plus_plus_int @ K @ zero_zero_int )
= K ) ).
% plus_int_code(1)
thf(fact_1206_odd__nonzero,axiom,
! [Z: int] :
( ( plus_plus_int @ ( plus_plus_int @ one_one_int @ Z ) @ Z )
!= zero_zero_int ) ).
% odd_nonzero
thf(fact_1207_zadd__int__left,axiom,
! [M: nat,N: nat,Z: int] :
( ( plus_plus_int @ ( semiri1314217659103216013at_int @ M ) @ ( plus_plus_int @ ( semiri1314217659103216013at_int @ N ) @ Z ) )
= ( plus_plus_int @ ( semiri1314217659103216013at_int @ ( plus_plus_nat @ M @ N ) ) @ Z ) ) ).
% zadd_int_left
thf(fact_1208_int__plus,axiom,
! [N: nat,M: nat] :
( ( semiri1314217659103216013at_int @ ( plus_plus_nat @ N @ M ) )
= ( plus_plus_int @ ( semiri1314217659103216013at_int @ N ) @ ( semiri1314217659103216013at_int @ M ) ) ) ).
% int_plus
thf(fact_1209_int__ops_I5_J,axiom,
! [A: nat,B: nat] :
( ( semiri1314217659103216013at_int @ ( plus_plus_nat @ A @ B ) )
= ( plus_plus_int @ ( semiri1314217659103216013at_int @ A ) @ ( semiri1314217659103216013at_int @ B ) ) ) ).
% int_ops(5)
thf(fact_1210_int__ge__induct,axiom,
! [K: int,I: int,P: int > $o] :
( ( ord_less_eq_int @ K @ I )
=> ( ( P @ K )
=> ( ! [I2: int] :
( ( ord_less_eq_int @ K @ I2 )
=> ( ( P @ I2 )
=> ( P @ ( plus_plus_int @ I2 @ one_one_int ) ) ) )
=> ( P @ I ) ) ) ) ).
% int_ge_induct
thf(fact_1211_zle__iff__zadd,axiom,
( ord_less_eq_int
= ( ^ [W2: int,Z7: int] :
? [N3: nat] :
( Z7
= ( plus_plus_int @ W2 @ ( semiri1314217659103216013at_int @ N3 ) ) ) ) ) ).
% zle_iff_zadd
thf(fact_1212_zless__add1__eq,axiom,
! [W: int,Z: int] :
( ( ord_less_int @ W @ ( plus_plus_int @ Z @ one_one_int ) )
= ( ( ord_less_int @ W @ Z )
| ( W = Z ) ) ) ).
% zless_add1_eq
thf(fact_1213_int__gr__induct,axiom,
! [K: int,I: int,P: int > $o] :
( ( ord_less_int @ K @ I )
=> ( ( P @ ( plus_plus_int @ K @ one_one_int ) )
=> ( ! [I2: int] :
( ( ord_less_int @ K @ I2 )
=> ( ( P @ I2 )
=> ( P @ ( plus_plus_int @ I2 @ one_one_int ) ) ) )
=> ( P @ I ) ) ) ) ).
% int_gr_induct
thf(fact_1214_int__ops_I4_J,axiom,
! [A: nat] :
( ( semiri1314217659103216013at_int @ ( suc @ A ) )
= ( plus_plus_int @ ( semiri1314217659103216013at_int @ A ) @ one_one_int ) ) ).
% int_ops(4)
thf(fact_1215_int__Suc,axiom,
! [N: nat] :
( ( semiri1314217659103216013at_int @ ( suc @ N ) )
= ( plus_plus_int @ ( semiri1314217659103216013at_int @ N ) @ one_one_int ) ) ).
% int_Suc
thf(fact_1216_zless__iff__Suc__zadd,axiom,
( ord_less_int
= ( ^ [W2: int,Z7: int] :
? [N3: nat] :
( Z7
= ( plus_plus_int @ W2 @ ( semiri1314217659103216013at_int @ ( suc @ N3 ) ) ) ) ) ) ).
% zless_iff_Suc_zadd
thf(fact_1217_odd__less__0__iff,axiom,
! [Z: int] :
( ( ord_less_int @ ( plus_plus_int @ ( plus_plus_int @ one_one_int @ Z ) @ Z ) @ zero_zero_int )
= ( ord_less_int @ Z @ zero_zero_int ) ) ).
% odd_less_0_iff
thf(fact_1218_add1__zle__eq,axiom,
! [W: int,Z: int] :
( ( ord_less_eq_int @ ( plus_plus_int @ W @ one_one_int ) @ Z )
= ( ord_less_int @ W @ Z ) ) ).
% add1_zle_eq
thf(fact_1219_zless__imp__add1__zle,axiom,
! [W: int,Z: int] :
( ( ord_less_int @ W @ Z )
=> ( ord_less_eq_int @ ( plus_plus_int @ W @ one_one_int ) @ Z ) ) ).
% zless_imp_add1_zle
thf(fact_1220_nat__int__add,axiom,
! [A: nat,B: nat] :
( ( nat2 @ ( plus_plus_int @ ( semiri1314217659103216013at_int @ A ) @ ( semiri1314217659103216013at_int @ B ) ) )
= ( plus_plus_nat @ A @ B ) ) ).
% nat_int_add
thf(fact_1221_int__induct,axiom,
! [P: int > $o,K: int,I: int] :
( ( P @ K )
=> ( ! [I2: int] :
( ( ord_less_eq_int @ K @ I2 )
=> ( ( P @ I2 )
=> ( P @ ( plus_plus_int @ I2 @ one_one_int ) ) ) )
=> ( ! [I2: int] :
( ( ord_less_eq_int @ I2 @ K )
=> ( ( P @ I2 )
=> ( P @ ( minus_minus_int @ I2 @ one_one_int ) ) ) )
=> ( P @ I ) ) ) ) ).
% int_induct
thf(fact_1222_nat__plus__as__int,axiom,
( plus_plus_nat
= ( ^ [A3: nat,B2: nat] : ( nat2 @ ( plus_plus_int @ ( semiri1314217659103216013at_int @ A3 ) @ ( semiri1314217659103216013at_int @ B2 ) ) ) ) ) ).
% nat_plus_as_int
thf(fact_1223_le__imp__0__less,axiom,
! [Z: int] :
( ( ord_less_eq_int @ zero_zero_int @ Z )
=> ( ord_less_int @ zero_zero_int @ ( plus_plus_int @ one_one_int @ Z ) ) ) ).
% le_imp_0_less
thf(fact_1224_nat__add__distrib,axiom,
! [Z: int,Z6: int] :
( ( ord_less_eq_int @ zero_zero_int @ Z )
=> ( ( ord_less_eq_int @ zero_zero_int @ Z6 )
=> ( ( nat2 @ ( plus_plus_int @ Z @ Z6 ) )
= ( plus_plus_nat @ ( nat2 @ Z ) @ ( nat2 @ Z6 ) ) ) ) ) ).
% nat_add_distrib
thf(fact_1225_Suc__as__int,axiom,
( suc
= ( ^ [A3: nat] : ( nat2 @ ( plus_plus_int @ ( semiri1314217659103216013at_int @ A3 ) @ one_one_int ) ) ) ) ).
% Suc_as_int
thf(fact_1226_nat__abs__triangle__ineq,axiom,
! [K: int,L: int] : ( ord_less_eq_nat @ ( nat2 @ ( abs_abs_int @ ( plus_plus_int @ K @ L ) ) ) @ ( plus_plus_nat @ ( nat2 @ ( abs_abs_int @ K ) ) @ ( nat2 @ ( abs_abs_int @ L ) ) ) ) ).
% nat_abs_triangle_ineq
thf(fact_1227_addr__of__neq,axiom,
! [Xs: list_list_char,X: list_char] :
( ( plus_plus_int @ one_one_int @ ( semiri1314217659103216013at_int @ ( size_s356637359517785349t_char @ Xs ) ) )
!= ( addr_of @ Xs @ X ) ) ).
% addr_of_neq
thf(fact_1228_addr__of_Osimps_I2_J,axiom,
! [X: list_char,Y: list_char,Xs: list_list_char] :
( ( ( X = Y )
=> ( ( addr_of @ ( cons_list_char @ X @ Xs ) @ Y )
= ( plus_plus_int @ ( semiri1314217659103216013at_int @ ( size_s356637359517785349t_char @ Xs ) ) @ one_one_int ) ) )
& ( ( X != Y )
=> ( ( addr_of @ ( cons_list_char @ X @ Xs ) @ Y )
= ( addr_of @ Xs @ Y ) ) ) ) ).
% addr_of.simps(2)
thf(fact_1229_exec1__appendL,axiom,
! [P: list_instr,I: int,S: list_char > int,Stk: list_int,I5: int,S4: list_char > int,Stk2: list_int,P4: list_instr] :
( ( exec1 @ P @ ( produc5086643055186798020st_int @ I @ ( produc8650753666468850689st_int @ S @ Stk ) ) @ ( produc5086643055186798020st_int @ I5 @ ( produc8650753666468850689st_int @ S4 @ Stk2 ) ) )
=> ( exec1 @ ( append_instr @ P4 @ P ) @ ( produc5086643055186798020st_int @ ( plus_plus_int @ ( semiri1314217659103216013at_int @ ( size_size_list_instr @ P4 ) ) @ I ) @ ( produc8650753666468850689st_int @ S @ Stk ) ) @ ( produc5086643055186798020st_int @ ( plus_plus_int @ ( semiri1314217659103216013at_int @ ( size_size_list_instr @ P4 ) ) @ I5 ) @ ( produc8650753666468850689st_int @ S4 @ Stk2 ) ) ) ) ).
% exec1_appendL
thf(fact_1230_Suc__nat__eq__nat__zadd1,axiom,
! [Z: int] :
( ( ord_less_eq_int @ zero_zero_int @ Z )
=> ( ( suc @ ( nat2 @ Z ) )
= ( nat2 @ ( plus_plus_int @ one_one_int @ Z ) ) ) ) ).
% Suc_nat_eq_nat_zadd1
thf(fact_1231_exec__append__trans,axiom,
! [P: list_instr,S: list_char > int,Stk: list_int,I5: int,S4: list_char > int,Stk2: list_int,P4: list_instr,I6: int,S5: list_char > int,Stk3: list_int,J4: int] :
( ( star_P707599355569300323st_int @ ( exec1 @ P ) @ ( produc5086643055186798020st_int @ zero_zero_int @ ( produc8650753666468850689st_int @ S @ Stk ) ) @ ( produc5086643055186798020st_int @ I5 @ ( produc8650753666468850689st_int @ S4 @ Stk2 ) ) )
=> ( ( ord_less_eq_int @ ( semiri1314217659103216013at_int @ ( size_size_list_instr @ P ) ) @ I5 )
=> ( ( star_P707599355569300323st_int @ ( exec1 @ P4 ) @ ( produc5086643055186798020st_int @ ( minus_minus_int @ I5 @ ( semiri1314217659103216013at_int @ ( size_size_list_instr @ P ) ) ) @ ( produc8650753666468850689st_int @ S4 @ Stk2 ) ) @ ( produc5086643055186798020st_int @ I6 @ ( produc8650753666468850689st_int @ S5 @ Stk3 ) ) )
=> ( ( J4
= ( plus_plus_int @ ( semiri1314217659103216013at_int @ ( size_size_list_instr @ P ) ) @ I6 ) )
=> ( star_P707599355569300323st_int @ ( exec1 @ ( append_instr @ P @ P4 ) ) @ ( produc5086643055186798020st_int @ zero_zero_int @ ( produc8650753666468850689st_int @ S @ Stk ) ) @ ( produc5086643055186798020st_int @ J4 @ ( produc8650753666468850689st_int @ S5 @ Stk3 ) ) ) ) ) ) ) ).
% exec_append_trans
thf(fact_1232_exec__appendL__if,axiom,
! [P4: list_instr,I: int,P: list_instr,S: list_char > int,Stk: list_int,J: int,S4: list_char > int,Stk2: list_int,I5: int] :
( ( ord_less_eq_int @ ( semiri1314217659103216013at_int @ ( size_size_list_instr @ P4 ) ) @ I )
=> ( ( star_P707599355569300323st_int @ ( exec1 @ P ) @ ( produc5086643055186798020st_int @ ( minus_minus_int @ I @ ( semiri1314217659103216013at_int @ ( size_size_list_instr @ P4 ) ) ) @ ( produc8650753666468850689st_int @ S @ Stk ) ) @ ( produc5086643055186798020st_int @ J @ ( produc8650753666468850689st_int @ S4 @ Stk2 ) ) )
=> ( ( I5
= ( plus_plus_int @ ( semiri1314217659103216013at_int @ ( size_size_list_instr @ P4 ) ) @ J ) )
=> ( star_P707599355569300323st_int @ ( exec1 @ ( append_instr @ P4 @ P ) ) @ ( produc5086643055186798020st_int @ I @ ( produc8650753666468850689st_int @ S @ Stk ) ) @ ( produc5086643055186798020st_int @ I5 @ ( produc8650753666468850689st_int @ S4 @ Stk2 ) ) ) ) ) ) ).
% exec_appendL_if
thf(fact_1233_exec__appendR,axiom,
! [P: list_instr,C: produc6425607678544837394st_int,C4: produc6425607678544837394st_int,P4: list_instr] :
( ( star_P707599355569300323st_int @ ( exec1 @ P ) @ C @ C4 )
=> ( star_P707599355569300323st_int @ ( exec1 @ ( append_instr @ P @ P4 ) ) @ C @ C4 ) ) ).
% exec_appendR
thf(fact_1234_exec__appendL,axiom,
! [P: list_instr,I: int,S: list_char > int,Stk: list_int,I5: int,S4: list_char > int,Stk2: list_int,P4: list_instr] :
( ( star_P707599355569300323st_int @ ( exec1 @ P ) @ ( produc5086643055186798020st_int @ I @ ( produc8650753666468850689st_int @ S @ Stk ) ) @ ( produc5086643055186798020st_int @ I5 @ ( produc8650753666468850689st_int @ S4 @ Stk2 ) ) )
=> ( star_P707599355569300323st_int @ ( exec1 @ ( append_instr @ P4 @ P ) ) @ ( produc5086643055186798020st_int @ ( plus_plus_int @ ( semiri1314217659103216013at_int @ ( size_size_list_instr @ P4 ) ) @ I ) @ ( produc8650753666468850689st_int @ S @ Stk ) ) @ ( produc5086643055186798020st_int @ ( plus_plus_int @ ( semiri1314217659103216013at_int @ ( size_size_list_instr @ P4 ) ) @ I5 ) @ ( produc8650753666468850689st_int @ S4 @ Stk2 ) ) ) ) ).
% exec_appendL
thf(fact_1235_to__m__state_Opelims,axiom,
! [X: list_list_char,Xa: list_char > int,Xb: int,Y: int] :
( ( ( to_m_state @ X @ Xa @ Xb )
= Y )
=> ( ( accp_P4562858270263085236nt_int @ to_m_state_rel @ ( produc3965054194175396271nt_int @ X @ ( produc5790713362662368625nt_int @ Xa @ Xb ) ) )
=> ~ ( ( ( ( ( ord_less_int @ zero_zero_int @ Xb )
& ( ord_less_eq_int @ Xb @ ( semiri1314217659103216013at_int @ ( size_s356637359517785349t_char @ X ) ) ) )
=> ( Y
= ( Xa @ ( inth_list_char @ X @ ( minus_minus_int @ ( semiri1314217659103216013at_int @ ( size_s356637359517785349t_char @ X ) ) @ Xb ) ) ) ) )
& ( ~ ( ( ord_less_int @ zero_zero_int @ Xb )
& ( ord_less_eq_int @ Xb @ ( semiri1314217659103216013at_int @ ( size_s356637359517785349t_char @ X ) ) ) )
=> ( Y = zero_zero_int ) ) )
=> ~ ( accp_P4562858270263085236nt_int @ to_m_state_rel @ ( produc3965054194175396271nt_int @ X @ ( produc5790713362662368625nt_int @ Xa @ Xb ) ) ) ) ) ) ).
% to_m_state.pelims
thf(fact_1236_execl_Opelims_I2_J,axiom,
! [X: list_instr,Xa: list_P419547167332244514st_int] :
( ( execl @ X @ Xa )
=> ( ( accp_P1482788143903795053st_int @ execl_rel @ ( produc7764413618310988720st_int @ X @ Xa ) )
=> ( ! [Cf3: produc6425607678544837394st_int,Cf4: produc6425607678544837394st_int,Cfs3: list_P419547167332244514st_int] :
( ( Xa
= ( cons_P4151473878978182108st_int @ Cf3 @ ( cons_P4151473878978182108st_int @ Cf4 @ Cfs3 ) ) )
=> ( ( accp_P1482788143903795053st_int @ execl_rel @ ( produc7764413618310988720st_int @ X @ ( cons_P4151473878978182108st_int @ Cf3 @ ( cons_P4151473878978182108st_int @ Cf4 @ Cfs3 ) ) ) )
=> ~ ( ( exec1 @ X @ Cf3 @ Cf4 )
& ( execl @ X @ ( cons_P4151473878978182108st_int @ Cf4 @ Cfs3 ) ) ) ) )
=> ( ( ( Xa = nil_Pr749310521696113548st_int )
=> ~ ( accp_P1482788143903795053st_int @ execl_rel @ ( produc7764413618310988720st_int @ X @ nil_Pr749310521696113548st_int ) ) )
=> ~ ! [V2: produc6425607678544837394st_int] :
( ( Xa
= ( cons_P4151473878978182108st_int @ V2 @ nil_Pr749310521696113548st_int ) )
=> ~ ( accp_P1482788143903795053st_int @ execl_rel @ ( produc7764413618310988720st_int @ X @ ( cons_P4151473878978182108st_int @ V2 @ nil_Pr749310521696113548st_int ) ) ) ) ) ) ) ) ).
% execl.pelims(2)
thf(fact_1237_execl_Opelims_I3_J,axiom,
! [X: list_instr,Xa: list_P419547167332244514st_int] :
( ~ ( execl @ X @ Xa )
=> ( ( accp_P1482788143903795053st_int @ execl_rel @ ( produc7764413618310988720st_int @ X @ Xa ) )
=> ~ ! [Cf3: produc6425607678544837394st_int,Cf4: produc6425607678544837394st_int,Cfs3: list_P419547167332244514st_int] :
( ( Xa
= ( cons_P4151473878978182108st_int @ Cf3 @ ( cons_P4151473878978182108st_int @ Cf4 @ Cfs3 ) ) )
=> ( ( accp_P1482788143903795053st_int @ execl_rel @ ( produc7764413618310988720st_int @ X @ ( cons_P4151473878978182108st_int @ Cf3 @ ( cons_P4151473878978182108st_int @ Cf4 @ Cfs3 ) ) ) )
=> ( ( exec1 @ X @ Cf3 @ Cf4 )
& ( execl @ X @ ( cons_P4151473878978182108st_int @ Cf4 @ Cfs3 ) ) ) ) ) ) ) ).
% execl.pelims(3)
thf(fact_1238_execl_Opelims_I1_J,axiom,
! [X: list_instr,Xa: list_P419547167332244514st_int,Y: $o] :
( ( ( execl @ X @ Xa )
= Y )
=> ( ( accp_P1482788143903795053st_int @ execl_rel @ ( produc7764413618310988720st_int @ X @ Xa ) )
=> ( ! [Cf3: produc6425607678544837394st_int,Cf4: produc6425607678544837394st_int,Cfs3: list_P419547167332244514st_int] :
( ( Xa
= ( cons_P4151473878978182108st_int @ Cf3 @ ( cons_P4151473878978182108st_int @ Cf4 @ Cfs3 ) ) )
=> ( ( Y
= ( ( exec1 @ X @ Cf3 @ Cf4 )
& ( execl @ X @ ( cons_P4151473878978182108st_int @ Cf4 @ Cfs3 ) ) ) )
=> ~ ( accp_P1482788143903795053st_int @ execl_rel @ ( produc7764413618310988720st_int @ X @ ( cons_P4151473878978182108st_int @ Cf3 @ ( cons_P4151473878978182108st_int @ Cf4 @ Cfs3 ) ) ) ) ) )
=> ( ( ( Xa = nil_Pr749310521696113548st_int )
=> ( Y
=> ~ ( accp_P1482788143903795053st_int @ execl_rel @ ( produc7764413618310988720st_int @ X @ nil_Pr749310521696113548st_int ) ) ) )
=> ~ ! [V2: produc6425607678544837394st_int] :
( ( Xa
= ( cons_P4151473878978182108st_int @ V2 @ nil_Pr749310521696113548st_int ) )
=> ( Y
=> ~ ( accp_P1482788143903795053st_int @ execl_rel @ ( produc7764413618310988720st_int @ X @ ( cons_P4151473878978182108st_int @ V2 @ nil_Pr749310521696113548st_int ) ) ) ) ) ) ) ) ) ).
% execl.pelims(1)
thf(fact_1239_upto_Opinduct,axiom,
! [A0: int,A1: int,P: int > int > $o] :
( ( accp_P1096762738010456898nt_int @ upto_rel @ ( product_Pair_int_int @ A0 @ A1 ) )
=> ( ! [I2: int,J2: int] :
( ( accp_P1096762738010456898nt_int @ upto_rel @ ( product_Pair_int_int @ I2 @ J2 ) )
=> ( ( ( ord_less_eq_int @ I2 @ J2 )
=> ( P @ ( plus_plus_int @ I2 @ one_one_int ) @ J2 ) )
=> ( P @ I2 @ J2 ) ) )
=> ( P @ A0 @ A1 ) ) ) ).
% upto.pinduct
thf(fact_1240_upto_Opsimps,axiom,
! [I: int,J: int] :
( ( accp_P1096762738010456898nt_int @ upto_rel @ ( product_Pair_int_int @ I @ J ) )
=> ( ( ( ord_less_eq_int @ I @ J )
=> ( ( upto @ I @ J )
= ( cons_int @ I @ ( upto @ ( plus_plus_int @ I @ one_one_int ) @ J ) ) ) )
& ( ~ ( ord_less_eq_int @ I @ J )
=> ( ( upto @ I @ J )
= nil_int ) ) ) ) ).
% upto.psimps
thf(fact_1241_upto_Opelims,axiom,
! [X: int,Xa: int,Y: list_int] :
( ( ( upto @ X @ Xa )
= Y )
=> ( ( accp_P1096762738010456898nt_int @ upto_rel @ ( product_Pair_int_int @ X @ Xa ) )
=> ~ ( ( ( ( ord_less_eq_int @ X @ Xa )
=> ( Y
= ( cons_int @ X @ ( upto @ ( plus_plus_int @ X @ one_one_int ) @ Xa ) ) ) )
& ( ~ ( ord_less_eq_int @ X @ Xa )
=> ( Y = nil_int ) ) )
=> ~ ( accp_P1096762738010456898nt_int @ upto_rel @ ( product_Pair_int_int @ X @ Xa ) ) ) ) ) ).
% upto.pelims
thf(fact_1242_upto__empty,axiom,
! [J: int,I: int] :
( ( ord_less_int @ J @ I )
=> ( ( upto @ I @ J )
= nil_int ) ) ).
% upto_empty
thf(fact_1243_upto__Nil2,axiom,
! [I: int,J: int] :
( ( nil_int
= ( upto @ I @ J ) )
= ( ord_less_int @ J @ I ) ) ).
% upto_Nil2
thf(fact_1244_upto__Nil,axiom,
! [I: int,J: int] :
( ( ( upto @ I @ J )
= nil_int )
= ( ord_less_int @ J @ I ) ) ).
% upto_Nil
thf(fact_1245_upto__single,axiom,
! [I: int] :
( ( upto @ I @ I )
= ( cons_int @ I @ nil_int ) ) ).
% upto_single
thf(fact_1246_nth__upto,axiom,
! [I: int,K: nat,J: int] :
( ( ord_less_eq_int @ ( plus_plus_int @ I @ ( semiri1314217659103216013at_int @ K ) ) @ J )
=> ( ( nth_int @ ( upto @ I @ J ) @ K )
= ( plus_plus_int @ I @ ( semiri1314217659103216013at_int @ K ) ) ) ) ).
% nth_upto
thf(fact_1247_length__upto,axiom,
! [I: int,J: int] :
( ( size_size_list_int @ ( upto @ I @ J ) )
= ( nat2 @ ( plus_plus_int @ ( minus_minus_int @ J @ I ) @ one_one_int ) ) ) ).
% length_upto
thf(fact_1248_upto__code,axiom,
( upto
= ( ^ [I3: int,J3: int] : ( upto_aux @ I3 @ J3 @ nil_int ) ) ) ).
% upto_code
thf(fact_1249_upto__aux__def,axiom,
( upto_aux
= ( ^ [I3: int,J3: int] : ( append_int @ ( upto @ I3 @ J3 ) ) ) ) ).
% upto_aux_def
thf(fact_1250_distinct__upto,axiom,
! [I: int,J: int] : ( distinct_int @ ( upto @ I @ J ) ) ).
% distinct_upto
thf(fact_1251_upto__split2,axiom,
! [I: int,J: int,K: int] :
( ( ord_less_eq_int @ I @ J )
=> ( ( ord_less_eq_int @ J @ K )
=> ( ( upto @ I @ K )
= ( append_int @ ( upto @ I @ J ) @ ( upto @ ( plus_plus_int @ J @ one_one_int ) @ K ) ) ) ) ) ).
% upto_split2
thf(fact_1252_upto__split1,axiom,
! [I: int,J: int,K: int] :
( ( ord_less_eq_int @ I @ J )
=> ( ( ord_less_eq_int @ J @ K )
=> ( ( upto @ I @ K )
= ( append_int @ ( upto @ I @ ( minus_minus_int @ J @ one_one_int ) ) @ ( upto @ J @ K ) ) ) ) ) ).
% upto_split1
thf(fact_1253_atLeastLessThan__upto,axiom,
( set_or4662586982721622107an_int
= ( ^ [I3: int,J3: int] : ( set_int2 @ ( upto @ I3 @ ( minus_minus_int @ J3 @ one_one_int ) ) ) ) ) ).
% atLeastLessThan_upto
thf(fact_1254_upto_Oelims,axiom,
! [X: int,Xa: int,Y: list_int] :
( ( ( upto @ X @ Xa )
= Y )
=> ( ( ( ord_less_eq_int @ X @ Xa )
=> ( Y
= ( cons_int @ X @ ( upto @ ( plus_plus_int @ X @ one_one_int ) @ Xa ) ) ) )
& ( ~ ( ord_less_eq_int @ X @ Xa )
=> ( Y = nil_int ) ) ) ) ).
% upto.elims
thf(fact_1255_upto_Osimps,axiom,
( upto
= ( ^ [I3: int,J3: int] : ( if_list_int @ ( ord_less_eq_int @ I3 @ J3 ) @ ( cons_int @ I3 @ ( upto @ ( plus_plus_int @ I3 @ one_one_int ) @ J3 ) ) @ nil_int ) ) ) ).
% upto.simps
thf(fact_1256_upto__rec1,axiom,
! [I: int,J: int] :
( ( ord_less_eq_int @ I @ J )
=> ( ( upto @ I @ J )
= ( cons_int @ I @ ( upto @ ( plus_plus_int @ I @ one_one_int ) @ J ) ) ) ) ).
% upto_rec1
thf(fact_1257_upto__rec2,axiom,
! [I: int,J: int] :
( ( ord_less_eq_int @ I @ J )
=> ( ( upto @ I @ J )
= ( append_int @ ( upto @ I @ ( minus_minus_int @ J @ one_one_int ) ) @ ( cons_int @ J @ nil_int ) ) ) ) ).
% upto_rec2
thf(fact_1258_upto__split3,axiom,
! [I: int,J: int,K: int] :
( ( ord_less_eq_int @ I @ J )
=> ( ( ord_less_eq_int @ J @ K )
=> ( ( upto @ I @ K )
= ( append_int @ ( upto @ I @ ( minus_minus_int @ J @ one_one_int ) ) @ ( cons_int @ J @ ( upto @ ( plus_plus_int @ J @ one_one_int ) @ K ) ) ) ) ) ) ).
% upto_split3
thf(fact_1259_upto__rec__numeral_I1_J,axiom,
! [M: num,N: num] :
( ( ( ord_less_eq_int @ ( numeral_numeral_int @ M ) @ ( numeral_numeral_int @ N ) )
=> ( ( upto @ ( numeral_numeral_int @ M ) @ ( numeral_numeral_int @ N ) )
= ( cons_int @ ( numeral_numeral_int @ M ) @ ( upto @ ( plus_plus_int @ ( numeral_numeral_int @ M ) @ one_one_int ) @ ( numeral_numeral_int @ N ) ) ) ) )
& ( ~ ( ord_less_eq_int @ ( numeral_numeral_int @ M ) @ ( numeral_numeral_int @ N ) )
=> ( ( upto @ ( numeral_numeral_int @ M ) @ ( numeral_numeral_int @ N ) )
= nil_int ) ) ) ).
% upto_rec_numeral(1)
thf(fact_1260_upto__rec__numeral_I4_J,axiom,
! [M: num,N: num] :
( ( ( ord_less_eq_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ M ) ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ N ) ) )
=> ( ( upto @ ( uminus_uminus_int @ ( numeral_numeral_int @ M ) ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ N ) ) )
= ( cons_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ M ) ) @ ( upto @ ( plus_plus_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ M ) ) @ one_one_int ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ N ) ) ) ) ) )
& ( ~ ( ord_less_eq_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ M ) ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ N ) ) )
=> ( ( upto @ ( uminus_uminus_int @ ( numeral_numeral_int @ M ) ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ N ) ) )
= nil_int ) ) ) ).
% upto_rec_numeral(4)
thf(fact_1261_upto__rec__numeral_I3_J,axiom,
! [M: num,N: num] :
( ( ( ord_less_eq_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ M ) ) @ ( numeral_numeral_int @ N ) )
=> ( ( upto @ ( uminus_uminus_int @ ( numeral_numeral_int @ M ) ) @ ( numeral_numeral_int @ N ) )
= ( cons_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ M ) ) @ ( upto @ ( plus_plus_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ M ) ) @ one_one_int ) @ ( numeral_numeral_int @ N ) ) ) ) )
& ( ~ ( ord_less_eq_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ M ) ) @ ( numeral_numeral_int @ N ) )
=> ( ( upto @ ( uminus_uminus_int @ ( numeral_numeral_int @ M ) ) @ ( numeral_numeral_int @ N ) )
= nil_int ) ) ) ).
% upto_rec_numeral(3)
% Helper facts (7)
thf(help_If_2_1_If_001t__Int__Oint_T,axiom,
! [X: int,Y: int] :
( ( if_int @ $false @ X @ Y )
= Y ) ).
thf(help_If_1_1_If_001t__Int__Oint_T,axiom,
! [X: int,Y: int] :
( ( if_int @ $true @ X @ Y )
= X ) ).
thf(help_If_2_1_If_001t__Nat__Onat_T,axiom,
! [X: nat,Y: nat] :
( ( if_nat @ $false @ X @ Y )
= Y ) ).
thf(help_If_1_1_If_001t__Nat__Onat_T,axiom,
! [X: nat,Y: nat] :
( ( if_nat @ $true @ X @ Y )
= X ) ).
thf(help_If_3_1_If_001t__List__Olist_It__Int__Oint_J_T,axiom,
! [P: $o] :
( ( P = $true )
| ( P = $false ) ) ).
thf(help_If_2_1_If_001t__List__Olist_It__Int__Oint_J_T,axiom,
! [X: list_int,Y: list_int] :
( ( if_list_int @ $false @ X @ Y )
= Y ) ).
thf(help_If_1_1_If_001t__List__Olist_It__Int__Oint_J_T,axiom,
! [X: list_int,Y: list_int] :
( ( if_list_int @ $true @ X @ Y )
= X ) ).
% Conjectures (1)
thf(conj_0,conjecture,
( member_nat
@ ( lattic8721135487736765967in_nat
@ ( collect_nat
@ ^ [K4: nat] :
( ( member_nat @ K4 @ ( set_or4665077453230672383an_nat @ k @ ( size_s229347708945616566st_int @ cfs ) ) )
& ~ ( ( ord_less_eq_int @ ( semiri1314217659103216013at_int @ ( size_size_list_instr @ p ) ) @ ( produc8941309049508465880st_int @ ( nth_Pr8327173284404383691st_int @ cfs @ K4 ) ) )
& ( ord_less_int @ ( minus_minus_int @ ( produc8941309049508465880st_int @ ( nth_Pr8327173284404383691st_int @ cfs @ K4 ) ) @ ( semiri1314217659103216013at_int @ ( size_size_list_instr @ p ) ) ) @ ( semiri1314217659103216013at_int @ ( size_size_list_instr @ ( p2 @ x ) ) ) ) ) ) ) )
@ ( collect_nat
@ ^ [K4: nat] :
( ( member_nat @ K4 @ ( set_or4665077453230672383an_nat @ k @ ( size_s229347708945616566st_int @ cfs ) ) )
& ~ ( ( ord_less_eq_int @ ( semiri1314217659103216013at_int @ ( size_size_list_instr @ p ) ) @ ( produc8941309049508465880st_int @ ( nth_Pr8327173284404383691st_int @ cfs @ K4 ) ) )
& ( ord_less_int @ ( minus_minus_int @ ( produc8941309049508465880st_int @ ( nth_Pr8327173284404383691st_int @ cfs @ K4 ) ) @ ( semiri1314217659103216013at_int @ ( size_size_list_instr @ p ) ) ) @ ( semiri1314217659103216013at_int @ ( size_size_list_instr @ ( p2 @ x ) ) ) ) ) ) ) ) ).
%------------------------------------------------------------------------------