TPTP Problem File: SLH0802^1.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/0005_Compiler/prob_00441_022088__5879336_1 [Des23]
% Status : Theorem
% Rating : ? v8.2.0
% Syntax : Number of formulae : 1378 ( 639 unt; 104 typ; 0 def)
% Number of atoms : 3243 (1195 equ; 0 cnn)
% Maximal formula atoms : 12 ( 2 avg)
% Number of connectives : 9427 ( 306 ~; 78 |; 145 &;7617 @)
% ( 0 <=>;1281 =>; 0 <=; 0 <~>)
% Maximal formula depth : 18 ( 6 avg)
% Number of types : 15 ( 14 usr)
% Number of type conns : 318 ( 318 >; 0 *; 0 +; 0 <<)
% Number of symbols : 93 ( 90 usr; 10 con; 0-3 aty)
% Number of variables : 3039 ( 138 ^;2810 !; 91 ?;3039 :)
% SPC : TH0_THM_EQU_NAR
% Comments : This file was generated by Isabelle (most likely Sledgehammer)
% 2023-01-18 16:42:53.196
%------------------------------------------------------------------------------
% Could-be-implicit typings (14)
thf(ty_n_t__List__Olist_It__List__Olist_It__List__Olist_It__String__Ochar_J_J_J,type,
list_list_list_char: $tType ).
thf(ty_n_t__Set__Oset_It__List__Olist_It__List__Olist_It__String__Ochar_J_J_J,type,
set_list_list_char: $tType ).
thf(ty_n_t__List__Olist_It__List__Olist_It__Compiler__Oinstr_J_J,type,
list_list_instr: $tType ).
thf(ty_n_t__Set__Oset_It__List__Olist_It__Compiler__Oinstr_J_J,type,
set_list_instr: $tType ).
thf(ty_n_t__List__Olist_It__List__Olist_It__String__Ochar_J_J,type,
list_list_char: $tType ).
thf(ty_n_t__Set__Oset_It__List__Olist_It__String__Ochar_J_J,type,
set_list_char: $tType ).
thf(ty_n_t__List__Olist_It__Compiler__Oinstr_J,type,
list_instr: $tType ).
thf(ty_n_t__Set__Oset_It__Compiler__Oinstr_J,type,
set_instr: $tType ).
thf(ty_n_t__List__Olist_It__String__Ochar_J,type,
list_char: $tType ).
thf(ty_n_t__Set__Oset_It__String__Ochar_J,type,
set_char: $tType ).
thf(ty_n_t__Compiler__Oinstr,type,
instr: $tType ).
thf(ty_n_t__String__Ochar,type,
char: $tType ).
thf(ty_n_t__Nat__Onat,type,
nat: $tType ).
thf(ty_n_t__Int__Oint,type,
int: $tType ).
% Explicit typings (90)
thf(sy_c_Compiler_Oinstr_OADD,type,
add: instr ).
thf(sy_c_Compiler_Oinstr_OLOAD,type,
load: list_char > instr ).
thf(sy_c_Compiler_Oinstr_OSTORE,type,
store: list_char > instr ).
thf(sy_c_Compiler_Oinstr_Osize__instr,type,
size_instr: instr > nat ).
thf(sy_c_Compiler_Ointh_001t__Compiler__Oinstr,type,
inth_instr: list_instr > int > instr ).
thf(sy_c_Compiler_Ointh_001t__List__Olist_It__Compiler__Oinstr_J,type,
inth_list_instr: list_list_instr > int > list_instr ).
thf(sy_c_Compiler_Ointh_001t__List__Olist_It__List__Olist_It__String__Ochar_J_J,type,
inth_list_list_char: list_list_list_char > int > list_list_char ).
thf(sy_c_Compiler_Ointh_001t__List__Olist_It__String__Ochar_J,type,
inth_list_char: list_list_char > int > list_char ).
thf(sy_c_Compiler_Ointh_001t__String__Ochar,type,
inth_char: list_char > int > char ).
thf(sy_c_Compiler_Oto__m__state,type,
to_m_state: list_list_char > ( list_char > int ) > int > int ).
thf(sy_c_Compiler_Ovars,type,
vars: list_instr > list_list_char ).
thf(sy_c_Groups_Oabs__class_Oabs_001t__Int__Oint,type,
abs_abs_int: int > int ).
thf(sy_c_Groups_Ominus__class_Ominus_001t__Int__Oint,type,
minus_minus_int: int > int > int ).
thf(sy_c_Groups_Ominus__class_Ominus_001t__Nat__Onat,type,
minus_minus_nat: nat > nat > nat ).
thf(sy_c_Groups_Ominus__class_Ominus_001t__Set__Oset_It__Compiler__Oinstr_J,type,
minus_5481936877629709990_instr: set_instr > set_instr > set_instr ).
thf(sy_c_Groups_Ominus__class_Ominus_001t__Set__Oset_It__List__Olist_It__String__Ochar_J_J,type,
minus_269717477326218608t_char: set_list_char > set_list_char > set_list_char ).
thf(sy_c_Groups_Oone__class_Oone_001t__Int__Oint,type,
one_one_int: int ).
thf(sy_c_Groups_Oone__class_Oone_001t__Nat__Onat,type,
one_one_nat: nat ).
thf(sy_c_Groups_Oplus__class_Oplus_001t__Int__Oint,type,
plus_plus_int: int > int > int ).
thf(sy_c_Groups_Oplus__class_Oplus_001t__Nat__Onat,type,
plus_plus_nat: nat > nat > nat ).
thf(sy_c_Groups_Osgn__class_Osgn_001t__Int__Oint,type,
sgn_sgn_int: int > int ).
thf(sy_c_Groups_Otimes__class_Otimes_001t__Int__Oint,type,
times_times_int: int > int > int ).
thf(sy_c_Groups_Otimes__class_Otimes_001t__Nat__Onat,type,
times_times_nat: nat > nat > nat ).
thf(sy_c_Groups_Ouminus__class_Ouminus_001t__Int__Oint,type,
uminus_uminus_int: int > int ).
thf(sy_c_Groups_Ouminus__class_Ouminus_001t__Set__Oset_It__Compiler__Oinstr_J,type,
uminus1347503074550245174_instr: set_instr > set_instr ).
thf(sy_c_Groups_Ouminus__class_Ouminus_001t__Set__Oset_It__List__Olist_It__String__Ochar_J_J,type,
uminus5517088710498760928t_char: set_list_char > set_list_char ).
thf(sy_c_Groups_Ozero__class_Ozero_001t__Int__Oint,type,
zero_zero_int: int ).
thf(sy_c_Groups_Ozero__class_Ozero_001t__Nat__Onat,type,
zero_zero_nat: nat ).
thf(sy_c_If_001t__Int__Oint,type,
if_int: $o > int > int > int ).
thf(sy_c_If_001t__Nat__Onat,type,
if_nat: $o > nat > nat > nat ).
thf(sy_c_Int_Onat,type,
nat2: int > nat ).
thf(sy_c_Int_Oring__1__class_Oof__int_001t__Int__Oint,type,
ring_1_of_int_int: int > int ).
thf(sy_c_List_Oappend_001t__Compiler__Oinstr,type,
append_instr: list_instr > list_instr > list_instr ).
thf(sy_c_List_Oappend_001t__List__Olist_It__String__Ochar_J,type,
append_list_char: list_list_char > list_list_char > list_list_char ).
thf(sy_c_List_Ogen__length_001t__Compiler__Oinstr,type,
gen_length_instr: nat > list_instr > nat ).
thf(sy_c_List_Ogen__length_001t__List__Olist_It__String__Ochar_J,type,
gen_length_list_char: nat > list_list_char > nat ).
thf(sy_c_List_Olist_Omap_001t__Compiler__Oinstr_001t__Compiler__Oinstr,type,
map_instr_instr: ( instr > instr ) > list_instr > list_instr ).
thf(sy_c_List_Olist_Omap_001t__Compiler__Oinstr_001t__List__Olist_It__String__Ochar_J,type,
map_instr_list_char: ( instr > list_char ) > list_instr > list_list_char ).
thf(sy_c_List_Olist_Omap_001t__List__Olist_It__Compiler__Oinstr_J_001t__Compiler__Oinstr,type,
map_list_instr_instr: ( list_instr > instr ) > list_list_instr > list_instr ).
thf(sy_c_List_Olist_Omap_001t__List__Olist_It__Compiler__Oinstr_J_001t__List__Olist_It__String__Ochar_J,type,
map_li418958971439749762t_char: ( list_instr > list_char ) > list_list_instr > list_list_char ).
thf(sy_c_List_Olist_Omap_001t__List__Olist_It__String__Ochar_J_001t__Compiler__Oinstr,type,
map_list_char_instr: ( list_char > instr ) > list_list_char > list_instr ).
thf(sy_c_List_Olist_Omap_001t__List__Olist_It__String__Ochar_J_001t__List__Olist_It__String__Ochar_J,type,
map_li116305933131242120t_char: ( list_char > list_char ) > list_list_char > list_list_char ).
thf(sy_c_List_Olist_Oset_001t__Compiler__Oinstr,type,
set_instr2: list_instr > set_instr ).
thf(sy_c_List_Olist_Oset_001t__List__Olist_It__Compiler__Oinstr_J,type,
set_list_instr2: list_list_instr > set_list_instr ).
thf(sy_c_List_Olist_Oset_001t__List__Olist_It__List__Olist_It__String__Ochar_J_J,type,
set_list_list_char2: list_list_list_char > set_list_list_char ).
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__String__Ochar,type,
set_char2: list_char > set_char ).
thf(sy_c_List_On__lists_001t__Compiler__Oinstr,type,
n_lists_instr: nat > list_instr > list_list_instr ).
thf(sy_c_List_On__lists_001t__List__Olist_It__String__Ochar_J,type,
n_lists_list_char: nat > list_list_char > list_list_list_char ).
thf(sy_c_List_On__lists_001t__String__Ochar,type,
n_lists_char: nat > list_char > list_list_char ).
thf(sy_c_List_Oproduct__lists_001t__Compiler__Oinstr,type,
product_lists_instr: list_list_instr > list_list_instr ).
thf(sy_c_List_Oproduct__lists_001t__List__Olist_It__String__Ochar_J,type,
produc4928866681682397746t_char: list_list_list_char > list_list_list_char ).
thf(sy_c_List_Oproduct__lists_001t__String__Ochar,type,
product_lists_char: list_list_char > list_list_char ).
thf(sy_c_List_Orotate1_001t__Compiler__Oinstr,type,
rotate1_instr: list_instr > list_instr ).
thf(sy_c_List_Orotate1_001t__List__Olist_It__String__Ochar_J,type,
rotate1_list_char: list_list_char > list_list_char ).
thf(sy_c_List_Orotate_001t__Compiler__Oinstr,type,
rotate_instr: nat > list_instr > list_instr ).
thf(sy_c_List_Orotate_001t__List__Olist_It__String__Ochar_J,type,
rotate_list_char: nat > list_list_char > list_list_char ).
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__Compiler__Oinstr,type,
size_size_instr: instr > 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__List__Olist_It__Compiler__Oinstr_J_J,type,
size_s1289461246499836759_instr: list_list_instr > nat ).
thf(sy_c_Nat_Osize__class_Osize_001t__List__Olist_It__List__Olist_It__List__Olist_It__String__Ochar_J_J_J,type,
size_s2562795203299699605t_char: list_list_list_char > 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__String__Ochar_J,type,
size_size_list_char: list_char > nat ).
thf(sy_c_Nat_Osize__class_Osize_001t__String__Ochar,type,
size_size_char: char > nat ).
thf(sy_c_Num_Oneg__numeral__class_Odbl__dec_001t__Int__Oint,type,
neg_nu3811975205180677377ec_int: int > int ).
thf(sy_c_Num_Oneg__numeral__class_Odbl__inc_001t__Int__Oint,type,
neg_nu5851722552734809277nc_int: int > int ).
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__Compiler__Oinstr_J,type,
ord_less_set_instr: set_instr > set_instr > $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__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__Compiler__Oinstr_J,type,
ord_le4956778034350794079_instr: set_instr > set_instr > $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_Set_OCollect_001t__Compiler__Oinstr,type,
collect_instr: ( instr > $o ) > set_instr ).
thf(sy_c_Set_OCollect_001t__List__Olist_It__Compiler__Oinstr_J,type,
collect_list_instr: ( list_instr > $o ) > set_list_instr ).
thf(sy_c_Set_OCollect_001t__List__Olist_It__List__Olist_It__String__Ochar_J_J,type,
collec7404787413733948998t_char: ( list_list_char > $o ) > set_list_list_char ).
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_String_Ochar_Osize__char,type,
size_char: char > nat ).
thf(sy_c_member_001t__Compiler__Oinstr,type,
member_instr: instr > set_instr > $o ).
thf(sy_c_member_001t__List__Olist_It__Compiler__Oinstr_J,type,
member_list_instr: list_instr > set_list_instr > $o ).
thf(sy_c_member_001t__List__Olist_It__List__Olist_It__String__Ochar_J_J,type,
member4762036990631661320t_char: list_list_char > set_list_list_char > $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__String__Ochar,type,
member_char: char > set_char > $o ).
thf(sy_v_P,type,
p: list_instr ).
thf(sy_v_i,type,
i: int ).
thf(sy_v_x,type,
x: list_char ).
% Relevant facts (1265)
thf(fact_0_instr_Oinject_I3_J,axiom,
! [X4: list_char,Y4: list_char] :
( ( ( store @ X4 )
= ( store @ Y4 ) )
= ( X4 = Y4 ) ) ).
% instr.inject(3)
thf(fact_1_inth__set,axiom,
! [I: int,Xs: list_char] :
( ( ord_less_eq_int @ zero_zero_int @ I )
=> ( ( ord_less_int @ I @ ( semiri1314217659103216013at_int @ ( size_size_list_char @ Xs ) ) )
=> ( member_char @ ( inth_char @ Xs @ I ) @ ( set_char2 @ Xs ) ) ) ) ).
% inth_set
thf(fact_2_inth__set,axiom,
! [I: int,Xs: list_list_list_char] :
( ( ord_less_eq_int @ zero_zero_int @ I )
=> ( ( ord_less_int @ I @ ( semiri1314217659103216013at_int @ ( size_s2562795203299699605t_char @ Xs ) ) )
=> ( member4762036990631661320t_char @ ( inth_list_list_char @ Xs @ I ) @ ( set_list_list_char2 @ Xs ) ) ) ) ).
% inth_set
thf(fact_3_inth__set,axiom,
! [I: int,Xs: list_list_instr] :
( ( ord_less_eq_int @ zero_zero_int @ I )
=> ( ( ord_less_int @ I @ ( semiri1314217659103216013at_int @ ( size_s1289461246499836759_instr @ Xs ) ) )
=> ( member_list_instr @ ( inth_list_instr @ Xs @ I ) @ ( set_list_instr2 @ Xs ) ) ) ) ).
% inth_set
thf(fact_4_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_5_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_6_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_7_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_8_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_9_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_10_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_11_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_12_of__nat__0,axiom,
( ( semiri1316708129612266289at_nat @ zero_zero_nat )
= zero_zero_nat ) ).
% of_nat_0
thf(fact_13_of__nat__0,axiom,
( ( semiri1314217659103216013at_int @ zero_zero_nat )
= zero_zero_int ) ).
% of_nat_0
thf(fact_14_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_15_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_16_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_17_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_18_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_19_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_20_not__gr__zero,axiom,
! [N: nat] :
( ( ~ ( ord_less_nat @ zero_zero_nat @ N ) )
= ( N = zero_zero_nat ) ) ).
% not_gr_zero
thf(fact_21_le__zero__eq,axiom,
! [N: nat] :
( ( ord_less_eq_nat @ N @ zero_zero_nat )
= ( N = zero_zero_nat ) ) ).
% le_zero_eq
thf(fact_22_vars__load,axiom,
! [I: int,P: list_instr,X: list_char] :
( ( ord_less_eq_int @ zero_zero_int @ I )
=> ( ( ord_less_int @ I @ ( semiri1314217659103216013at_int @ ( size_size_list_instr @ P ) ) )
=> ( ( ( inth_instr @ P @ I )
= ( load @ X ) )
=> ( member_list_char @ X @ ( set_list_char2 @ ( vars @ P ) ) ) ) ) ) ).
% vars_load
thf(fact_23_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_24_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_25_of__nat__eq__iff,axiom,
! [M: nat,N: nat] :
( ( ( semiri1314217659103216013at_int @ M )
= ( semiri1314217659103216013at_int @ N ) )
= ( M = N ) ) ).
% of_nat_eq_iff
thf(fact_26_of__nat__eq__iff,axiom,
! [M: nat,N: nat] :
( ( ( semiri1316708129612266289at_nat @ M )
= ( semiri1316708129612266289at_nat @ N ) )
= ( M = N ) ) ).
% of_nat_eq_iff
thf(fact_27_instr_Oinject_I2_J,axiom,
! [X2: list_char,Y2: list_char] :
( ( ( load @ X2 )
= ( load @ Y2 ) )
= ( X2 = Y2 ) ) ).
% instr.inject(2)
thf(fact_28_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_29_neq0__conv,axiom,
! [N: nat] :
( ( N != zero_zero_nat )
= ( ord_less_nat @ zero_zero_nat @ N ) ) ).
% neq0_conv
thf(fact_30_less__nat__zero__code,axiom,
! [N: nat] :
~ ( ord_less_nat @ N @ zero_zero_nat ) ).
% less_nat_zero_code
thf(fact_31_bot__nat__0_Oextremum,axiom,
! [A: nat] : ( ord_less_eq_nat @ zero_zero_nat @ A ) ).
% bot_nat_0.extremum
thf(fact_32_le0,axiom,
! [N: nat] : ( ord_less_eq_nat @ zero_zero_nat @ N ) ).
% le0
thf(fact_33_less__eq__nat_Osimps_I1_J,axiom,
! [N: nat] : ( ord_less_eq_nat @ zero_zero_nat @ N ) ).
% less_eq_nat.simps(1)
thf(fact_34_bot__nat__0_Oextremum__strict,axiom,
! [A: nat] :
~ ( ord_less_nat @ A @ zero_zero_nat ) ).
% bot_nat_0.extremum_strict
thf(fact_35_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_36_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_37_gr0I,axiom,
! [N: nat] :
( ( N != zero_zero_nat )
=> ( ord_less_nat @ zero_zero_nat @ N ) ) ).
% gr0I
thf(fact_38_le__0__eq,axiom,
! [N: nat] :
( ( ord_less_eq_nat @ N @ zero_zero_nat )
= ( N = zero_zero_nat ) ) ).
% le_0_eq
thf(fact_39_le__refl,axiom,
! [N: nat] : ( ord_less_eq_nat @ N @ N ) ).
% le_refl
thf(fact_40_not__gr0,axiom,
! [N: nat] :
( ( ~ ( ord_less_nat @ zero_zero_nat @ N ) )
= ( N = zero_zero_nat ) ) ).
% not_gr0
thf(fact_41_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_42_eq__imp__le,axiom,
! [M: nat,N: nat] :
( ( M = N )
=> ( ord_less_eq_nat @ M @ N ) ) ).
% eq_imp_le
thf(fact_43_not__less0,axiom,
! [N: nat] :
~ ( ord_less_nat @ N @ zero_zero_nat ) ).
% not_less0
thf(fact_44_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_45_less__zeroE,axiom,
! [N: nat] :
~ ( ord_less_nat @ N @ zero_zero_nat ) ).
% less_zeroE
thf(fact_46_nat__less__le,axiom,
( ord_less_nat
= ( ^ [M2: nat,N3: nat] :
( ( ord_less_eq_nat @ M2 @ N3 )
& ( M2 != N3 ) ) ) ) ).
% nat_less_le
thf(fact_47_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_48_less__not__refl,axiom,
! [N: nat] :
~ ( ord_less_nat @ N @ N ) ).
% less_not_refl
thf(fact_49_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_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_ex__least__nat__le,axiom,
! [P: nat > $o,N: nat] :
( ( P @ N )
=> ( ~ ( P @ zero_zero_nat )
=> ? [K2: nat] :
( ( ord_less_eq_nat @ K2 @ N )
& ! [I2: nat] :
( ( ord_less_nat @ I2 @ K2 )
=> ~ ( P @ I2 ) )
& ( P @ K2 ) ) ) ) ).
% ex_least_nat_le
thf(fact_53_gr__implies__not0,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ M @ N )
=> ( N != zero_zero_nat ) ) ).
% gr_implies_not0
thf(fact_54_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_55_mem__Collect__eq,axiom,
! [A: list_list_char,P: list_list_char > $o] :
( ( member4762036990631661320t_char @ A @ ( collec7404787413733948998t_char @ P ) )
= ( P @ A ) ) ).
% mem_Collect_eq
thf(fact_56_mem__Collect__eq,axiom,
! [A: list_instr,P: list_instr > $o] :
( ( member_list_instr @ A @ ( collect_list_instr @ P ) )
= ( P @ A ) ) ).
% mem_Collect_eq
thf(fact_57_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_58_mem__Collect__eq,axiom,
! [A: instr,P: instr > $o] :
( ( member_instr @ A @ ( collect_instr @ P ) )
= ( P @ A ) ) ).
% mem_Collect_eq
thf(fact_59_Collect__mem__eq,axiom,
! [A2: set_list_list_char] :
( ( collec7404787413733948998t_char
@ ^ [X3: list_list_char] : ( member4762036990631661320t_char @ X3 @ A2 ) )
= A2 ) ).
% Collect_mem_eq
thf(fact_60_Collect__mem__eq,axiom,
! [A2: set_list_instr] :
( ( collect_list_instr
@ ^ [X3: list_instr] : ( member_list_instr @ X3 @ A2 ) )
= A2 ) ).
% Collect_mem_eq
thf(fact_61_Collect__mem__eq,axiom,
! [A2: set_list_char] :
( ( collect_list_char
@ ^ [X3: list_char] : ( member_list_char @ X3 @ A2 ) )
= A2 ) ).
% Collect_mem_eq
thf(fact_62_Collect__mem__eq,axiom,
! [A2: set_instr] :
( ( collect_instr
@ ^ [X3: instr] : ( member_instr @ X3 @ A2 ) )
= A2 ) ).
% Collect_mem_eq
thf(fact_63_Collect__cong,axiom,
! [P: instr > $o,Q: instr > $o] :
( ! [X5: instr] :
( ( P @ X5 )
= ( Q @ X5 ) )
=> ( ( collect_instr @ P )
= ( collect_instr @ Q ) ) ) ).
% Collect_cong
thf(fact_64_Collect__cong,axiom,
! [P: list_char > $o,Q: list_char > $o] :
( ! [X5: list_char] :
( ( P @ X5 )
= ( Q @ X5 ) )
=> ( ( collect_list_char @ P )
= ( collect_list_char @ Q ) ) ) ).
% Collect_cong
thf(fact_65_less__irrefl__nat,axiom,
! [N: nat] :
~ ( ord_less_nat @ N @ N ) ).
% less_irrefl_nat
thf(fact_66_nat__less__induct,axiom,
! [P: nat > $o,N: nat] :
( ! [N2: nat] :
( ! [M3: nat] :
( ( ord_less_nat @ M3 @ N2 )
=> ( P @ M3 ) )
=> ( P @ N2 ) )
=> ( P @ N ) ) ).
% nat_less_induct
thf(fact_67_infinite__descent,axiom,
! [P: nat > $o,N: nat] :
( ! [N2: nat] :
( ~ ( P @ N2 )
=> ? [M3: nat] :
( ( ord_less_nat @ M3 @ N2 )
& ~ ( P @ M3 ) ) )
=> ( P @ N ) ) ).
% infinite_descent
thf(fact_68_le__eq__less__or__eq,axiom,
( ord_less_eq_nat
= ( ^ [M2: nat,N3: nat] :
( ( ord_less_nat @ M2 @ N3 )
| ( M2 = N3 ) ) ) ) ).
% le_eq_less_or_eq
thf(fact_69_infinite__descent0,axiom,
! [P: nat > $o,N: nat] :
( ( P @ zero_zero_nat )
=> ( ! [N2: nat] :
( ( ord_less_nat @ zero_zero_nat @ N2 )
=> ( ~ ( P @ N2 )
=> ? [M3: nat] :
( ( ord_less_nat @ M3 @ N2 )
& ~ ( P @ M3 ) ) ) )
=> ( P @ N ) ) ) ).
% infinite_descent0
thf(fact_70_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_71_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_72_Nat_Oex__has__greatest__nat,axiom,
! [P: nat > $o,K: nat,B: nat] :
( ( P @ K )
=> ( ! [Y3: nat] :
( ( P @ Y3 )
=> ( ord_less_eq_nat @ Y3 @ B ) )
=> ? [X5: nat] :
( ( P @ X5 )
& ! [Y5: nat] :
( ( P @ Y5 )
=> ( ord_less_eq_nat @ Y5 @ X5 ) ) ) ) ) ).
% Nat.ex_has_greatest_nat
thf(fact_73_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_74_less__mono__imp__le__mono,axiom,
! [F: nat > nat,I: nat,J: nat] :
( ! [I3: nat,J2: nat] :
( ( ord_less_nat @ I3 @ J2 )
=> ( ord_less_nat @ ( F @ I3 ) @ ( F @ J2 ) ) )
=> ( ( ord_less_eq_nat @ I @ J )
=> ( ord_less_eq_nat @ ( F @ I ) @ ( F @ J ) ) ) ) ).
% less_mono_imp_le_mono
thf(fact_75_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_76_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_77_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_78_instr_Odistinct_I15_J,axiom,
! [X2: list_char,X4: list_char] :
( ( load @ X2 )
!= ( store @ X4 ) ) ).
% instr.distinct(15)
thf(fact_79_zero__reorient,axiom,
! [X: int] :
( ( zero_zero_int = X )
= ( X = zero_zero_int ) ) ).
% zero_reorient
thf(fact_80_zero__reorient,axiom,
! [X: nat] :
( ( zero_zero_nat = X )
= ( X = zero_zero_nat ) ) ).
% zero_reorient
thf(fact_81_size__neq__size__imp__neq,axiom,
! [X: list_char,Y: list_char] :
( ( ( size_size_list_char @ X )
!= ( size_size_list_char @ Y ) )
=> ( X != Y ) ) ).
% size_neq_size_imp_neq
thf(fact_82_size__neq__size__imp__neq,axiom,
! [X: list_list_list_char,Y: list_list_list_char] :
( ( ( size_s2562795203299699605t_char @ X )
!= ( size_s2562795203299699605t_char @ Y ) )
=> ( X != Y ) ) ).
% size_neq_size_imp_neq
thf(fact_83_size__neq__size__imp__neq,axiom,
! [X: list_list_instr,Y: list_list_instr] :
( ( ( size_s1289461246499836759_instr @ X )
!= ( size_s1289461246499836759_instr @ Y ) )
=> ( X != Y ) ) ).
% size_neq_size_imp_neq
thf(fact_84_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_85_size__neq__size__imp__neq,axiom,
! [X: instr,Y: instr] :
( ( ( size_size_instr @ X )
!= ( size_size_instr @ Y ) )
=> ( X != Y ) ) ).
% size_neq_size_imp_neq
thf(fact_86_size__neq__size__imp__neq,axiom,
! [X: char,Y: char] :
( ( ( size_size_char @ X )
!= ( size_size_char @ Y ) )
=> ( X != Y ) ) ).
% size_neq_size_imp_neq
thf(fact_87_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_88_int__int__eq,axiom,
! [M: nat,N: nat] :
( ( ( semiri1314217659103216013at_int @ M )
= ( semiri1314217659103216013at_int @ N ) )
= ( M = N ) ) ).
% int_int_eq
thf(fact_89_zero__le,axiom,
! [X: nat] : ( ord_less_eq_nat @ zero_zero_nat @ X ) ).
% zero_le
thf(fact_90_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_91_gr__implies__not__zero,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ M @ N )
=> ( N != zero_zero_nat ) ) ).
% gr_implies_not_zero
thf(fact_92_not__less__zero,axiom,
! [N: nat] :
~ ( ord_less_nat @ N @ zero_zero_nat ) ).
% not_less_zero
thf(fact_93_gr__zeroI,axiom,
! [N: nat] :
( ( N != zero_zero_nat )
=> ( ord_less_nat @ zero_zero_nat @ N ) ) ).
% gr_zeroI
thf(fact_94_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_95_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_96_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_97_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_98_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_99_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_100_less__eq__int__code_I1_J,axiom,
ord_less_eq_int @ zero_zero_int @ zero_zero_int ).
% less_eq_int_code(1)
thf(fact_101_less__int__code_I1_J,axiom,
~ ( ord_less_int @ zero_zero_int @ zero_zero_int ) ).
% less_int_code(1)
thf(fact_102_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_103_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_104_of__nat__less__0__iff,axiom,
! [M: nat] :
~ ( ord_less_int @ ( semiri1314217659103216013at_int @ M ) @ zero_zero_int ) ).
% of_nat_less_0_iff
thf(fact_105_of__nat__less__0__iff,axiom,
! [M: nat] :
~ ( ord_less_nat @ ( semiri1316708129612266289at_nat @ M ) @ zero_zero_nat ) ).
% of_nat_less_0_iff
thf(fact_106_length__pos__if__in__set,axiom,
! [X: char,Xs: list_char] :
( ( member_char @ X @ ( set_char2 @ Xs ) )
=> ( ord_less_nat @ zero_zero_nat @ ( size_size_list_char @ Xs ) ) ) ).
% length_pos_if_in_set
thf(fact_107_length__pos__if__in__set,axiom,
! [X: list_list_char,Xs: list_list_list_char] :
( ( member4762036990631661320t_char @ X @ ( set_list_list_char2 @ Xs ) )
=> ( ord_less_nat @ zero_zero_nat @ ( size_s2562795203299699605t_char @ Xs ) ) ) ).
% length_pos_if_in_set
thf(fact_108_length__pos__if__in__set,axiom,
! [X: list_instr,Xs: list_list_instr] :
( ( member_list_instr @ X @ ( set_list_instr2 @ Xs ) )
=> ( ord_less_nat @ zero_zero_nat @ ( size_s1289461246499836759_instr @ Xs ) ) ) ).
% length_pos_if_in_set
thf(fact_109_length__pos__if__in__set,axiom,
! [X: instr,Xs: list_instr] :
( ( member_instr @ X @ ( set_instr2 @ Xs ) )
=> ( ord_less_nat @ zero_zero_nat @ ( size_size_list_instr @ Xs ) ) ) ).
% length_pos_if_in_set
thf(fact_110_length__pos__if__in__set,axiom,
! [X: list_char,Xs: list_list_char] :
( ( member_list_char @ X @ ( set_list_char2 @ Xs ) )
=> ( ord_less_nat @ zero_zero_nat @ ( size_s356637359517785349t_char @ Xs ) ) ) ).
% length_pos_if_in_set
thf(fact_111_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_112_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_113_int__ops_I1_J,axiom,
( ( semiri1314217659103216013at_int @ zero_zero_nat )
= zero_zero_int ) ).
% int_ops(1)
thf(fact_114_dual__order_Orefl,axiom,
! [A: set_instr] : ( ord_le4956778034350794079_instr @ A @ A ) ).
% dual_order.refl
thf(fact_115_dual__order_Orefl,axiom,
! [A: set_list_char] : ( ord_le1207710660481778807t_char @ A @ A ) ).
% dual_order.refl
thf(fact_116_dual__order_Orefl,axiom,
! [A: int] : ( ord_less_eq_int @ A @ A ) ).
% dual_order.refl
thf(fact_117_dual__order_Orefl,axiom,
! [A: nat] : ( ord_less_eq_nat @ A @ A ) ).
% dual_order.refl
thf(fact_118_order__refl,axiom,
! [X: set_instr] : ( ord_le4956778034350794079_instr @ X @ X ) ).
% order_refl
thf(fact_119_order__refl,axiom,
! [X: set_list_char] : ( ord_le1207710660481778807t_char @ X @ X ) ).
% order_refl
thf(fact_120_order__refl,axiom,
! [X: int] : ( ord_less_eq_int @ X @ X ) ).
% order_refl
thf(fact_121_order__refl,axiom,
! [X: nat] : ( ord_less_eq_nat @ X @ X ) ).
% order_refl
thf(fact_122_conj__le__cong,axiom,
! [X: int,X6: int,P: $o,P2: $o] :
( ( X = X6 )
=> ( ( ( ord_less_eq_int @ zero_zero_int @ X6 )
=> ( P = P2 ) )
=> ( ( ( ord_less_eq_int @ zero_zero_int @ X )
& P )
= ( ( ord_less_eq_int @ zero_zero_int @ X6 )
& P2 ) ) ) ) ).
% conj_le_cong
thf(fact_123_imp__le__cong,axiom,
! [X: int,X6: int,P: $o,P2: $o] :
( ( X = X6 )
=> ( ( ( ord_less_eq_int @ zero_zero_int @ X6 )
=> ( P = P2 ) )
=> ( ( ( ord_less_eq_int @ zero_zero_int @ X )
=> P )
= ( ( ord_less_eq_int @ zero_zero_int @ X6 )
=> P2 ) ) ) ) ).
% imp_le_cong
thf(fact_124_length__induct,axiom,
! [P: list_char > $o,Xs: list_char] :
( ! [Xs2: list_char] :
( ! [Ys: list_char] :
( ( ord_less_nat @ ( size_size_list_char @ Ys ) @ ( size_size_list_char @ Xs2 ) )
=> ( P @ Ys ) )
=> ( P @ Xs2 ) )
=> ( P @ Xs ) ) ).
% length_induct
thf(fact_125_length__induct,axiom,
! [P: list_list_list_char > $o,Xs: list_list_list_char] :
( ! [Xs2: list_list_list_char] :
( ! [Ys: list_list_list_char] :
( ( ord_less_nat @ ( size_s2562795203299699605t_char @ Ys ) @ ( size_s2562795203299699605t_char @ Xs2 ) )
=> ( P @ Ys ) )
=> ( P @ Xs2 ) )
=> ( P @ Xs ) ) ).
% length_induct
thf(fact_126_length__induct,axiom,
! [P: list_list_instr > $o,Xs: list_list_instr] :
( ! [Xs2: list_list_instr] :
( ! [Ys: list_list_instr] :
( ( ord_less_nat @ ( size_s1289461246499836759_instr @ Ys ) @ ( size_s1289461246499836759_instr @ Xs2 ) )
=> ( P @ Ys ) )
=> ( P @ Xs2 ) )
=> ( P @ Xs ) ) ).
% length_induct
thf(fact_127_length__induct,axiom,
! [P: list_instr > $o,Xs: list_instr] :
( ! [Xs2: list_instr] :
( ! [Ys: list_instr] :
( ( ord_less_nat @ ( size_size_list_instr @ Ys ) @ ( size_size_list_instr @ Xs2 ) )
=> ( P @ Ys ) )
=> ( P @ Xs2 ) )
=> ( P @ Xs ) ) ).
% length_induct
thf(fact_128_length__induct,axiom,
! [P: list_list_char > $o,Xs: list_list_char] :
( ! [Xs2: list_list_char] :
( ! [Ys: list_list_char] :
( ( ord_less_nat @ ( size_s356637359517785349t_char @ Ys ) @ ( size_s356637359517785349t_char @ Xs2 ) )
=> ( P @ Ys ) )
=> ( P @ Xs2 ) )
=> ( P @ Xs ) ) ).
% length_induct
thf(fact_129_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 )
=> ( ! [I2: nat] :
( ( ord_less_nat @ K2 @ I2 )
=> ( P @ I2 ) )
=> ( P @ K2 ) ) )
=> ( P @ M ) ) ) ).
% nat_descend_induct
thf(fact_130_inth__map,axiom,
! [I: int,Xs: list_list_instr,F: list_instr > instr] :
( ( ord_less_eq_int @ zero_zero_int @ I )
=> ( ( ord_less_int @ I @ ( semiri1314217659103216013at_int @ ( size_s1289461246499836759_instr @ Xs ) ) )
=> ( ( inth_instr @ ( map_list_instr_instr @ F @ Xs ) @ I )
= ( F @ ( inth_list_instr @ Xs @ I ) ) ) ) ) ).
% inth_map
thf(fact_131_inth__map,axiom,
! [I: int,Xs: list_list_instr,F: list_instr > list_char] :
( ( ord_less_eq_int @ zero_zero_int @ I )
=> ( ( ord_less_int @ I @ ( semiri1314217659103216013at_int @ ( size_s1289461246499836759_instr @ Xs ) ) )
=> ( ( inth_list_char @ ( map_li418958971439749762t_char @ F @ Xs ) @ I )
= ( F @ ( inth_list_instr @ Xs @ I ) ) ) ) ) ).
% inth_map
thf(fact_132_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_133_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_134_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_135_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_136_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_137_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_138_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_139_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_140_list_Omap__ident__strong,axiom,
! [T: list_list_char,F: list_char > list_char] :
( ! [Z: list_char] :
( ( member_list_char @ Z @ ( set_list_char2 @ T ) )
=> ( ( F @ Z )
= Z ) )
=> ( ( map_li116305933131242120t_char @ F @ T )
= T ) ) ).
% list.map_ident_strong
thf(fact_141_list_Omap__ident__strong,axiom,
! [T: list_instr,F: instr > instr] :
( ! [Z: instr] :
( ( member_instr @ Z @ ( set_instr2 @ T ) )
=> ( ( F @ Z )
= Z ) )
=> ( ( map_instr_instr @ F @ T )
= T ) ) ).
% list.map_ident_strong
thf(fact_142_map__idI,axiom,
! [Xs: list_list_char,F: list_char > list_char] :
( ! [X5: list_char] :
( ( member_list_char @ X5 @ ( set_list_char2 @ Xs ) )
=> ( ( F @ X5 )
= X5 ) )
=> ( ( map_li116305933131242120t_char @ F @ Xs )
= Xs ) ) ).
% map_idI
thf(fact_143_map__idI,axiom,
! [Xs: list_instr,F: instr > instr] :
( ! [X5: instr] :
( ( member_instr @ X5 @ ( set_instr2 @ Xs ) )
=> ( ( F @ X5 )
= X5 ) )
=> ( ( map_instr_instr @ F @ Xs )
= Xs ) ) ).
% map_idI
thf(fact_144_verit__comp__simplify1_I2_J,axiom,
! [A: int] : ( ord_less_eq_int @ A @ A ) ).
% verit_comp_simplify1(2)
thf(fact_145_verit__comp__simplify1_I2_J,axiom,
! [A: nat] : ( ord_less_eq_nat @ A @ A ) ).
% verit_comp_simplify1(2)
thf(fact_146_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_147_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_148_le__cases3,axiom,
! [X: int,Y: int,Z2: int] :
( ( ( ord_less_eq_int @ X @ Y )
=> ~ ( ord_less_eq_int @ Y @ Z2 ) )
=> ( ( ( ord_less_eq_int @ Y @ X )
=> ~ ( ord_less_eq_int @ X @ Z2 ) )
=> ( ( ( ord_less_eq_int @ X @ Z2 )
=> ~ ( ord_less_eq_int @ Z2 @ Y ) )
=> ( ( ( ord_less_eq_int @ Z2 @ Y )
=> ~ ( ord_less_eq_int @ Y @ X ) )
=> ( ( ( ord_less_eq_int @ Y @ Z2 )
=> ~ ( ord_less_eq_int @ Z2 @ X ) )
=> ~ ( ( ord_less_eq_int @ Z2 @ X )
=> ~ ( ord_less_eq_int @ X @ Y ) ) ) ) ) ) ) ).
% le_cases3
thf(fact_149_le__cases3,axiom,
! [X: nat,Y: nat,Z2: nat] :
( ( ( ord_less_eq_nat @ X @ Y )
=> ~ ( ord_less_eq_nat @ Y @ Z2 ) )
=> ( ( ( ord_less_eq_nat @ Y @ X )
=> ~ ( ord_less_eq_nat @ X @ Z2 ) )
=> ( ( ( ord_less_eq_nat @ X @ Z2 )
=> ~ ( ord_less_eq_nat @ Z2 @ Y ) )
=> ( ( ( ord_less_eq_nat @ Z2 @ Y )
=> ~ ( ord_less_eq_nat @ Y @ X ) )
=> ( ( ( ord_less_eq_nat @ Y @ Z2 )
=> ~ ( ord_less_eq_nat @ Z2 @ X ) )
=> ~ ( ( ord_less_eq_nat @ Z2 @ X )
=> ~ ( ord_less_eq_nat @ X @ Y ) ) ) ) ) ) ) ).
% le_cases3
thf(fact_150_order__class_Oorder__eq__iff,axiom,
( ( ^ [Y6: int,Z3: int] : ( Y6 = Z3 ) )
= ( ^ [X3: int,Y7: int] :
( ( ord_less_eq_int @ X3 @ Y7 )
& ( ord_less_eq_int @ Y7 @ X3 ) ) ) ) ).
% order_class.order_eq_iff
thf(fact_151_order__class_Oorder__eq__iff,axiom,
( ( ^ [Y6: nat,Z3: nat] : ( Y6 = Z3 ) )
= ( ^ [X3: nat,Y7: nat] :
( ( ord_less_eq_nat @ X3 @ Y7 )
& ( ord_less_eq_nat @ Y7 @ X3 ) ) ) ) ).
% order_class.order_eq_iff
thf(fact_152_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_153_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_154_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_155_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_156_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_157_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_158_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_159_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_160_order__trans,axiom,
! [X: int,Y: int,Z2: int] :
( ( ord_less_eq_int @ X @ Y )
=> ( ( ord_less_eq_int @ Y @ Z2 )
=> ( ord_less_eq_int @ X @ Z2 ) ) ) ).
% order_trans
thf(fact_161_order__trans,axiom,
! [X: nat,Y: nat,Z2: nat] :
( ( ord_less_eq_nat @ X @ Y )
=> ( ( ord_less_eq_nat @ Y @ Z2 )
=> ( ord_less_eq_nat @ X @ Z2 ) ) ) ).
% order_trans
thf(fact_162_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_163_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_164_dual__order_Oeq__iff,axiom,
( ( ^ [Y6: int,Z3: int] : ( Y6 = Z3 ) )
= ( ^ [A3: int,B2: int] :
( ( ord_less_eq_int @ B2 @ A3 )
& ( ord_less_eq_int @ A3 @ B2 ) ) ) ) ).
% dual_order.eq_iff
thf(fact_165_dual__order_Oeq__iff,axiom,
( ( ^ [Y6: nat,Z3: nat] : ( Y6 = Z3 ) )
= ( ^ [A3: nat,B2: nat] :
( ( ord_less_eq_nat @ B2 @ A3 )
& ( ord_less_eq_nat @ A3 @ B2 ) ) ) ) ).
% dual_order.eq_iff
thf(fact_166_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_167_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_168_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_169_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_170_antisym,axiom,
! [A: int,B: int] :
( ( ord_less_eq_int @ A @ B )
=> ( ( ord_less_eq_int @ B @ A )
=> ( A = B ) ) ) ).
% antisym
thf(fact_171_antisym,axiom,
! [A: nat,B: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ord_less_eq_nat @ B @ A )
=> ( A = B ) ) ) ).
% antisym
thf(fact_172_Orderings_Oorder__eq__iff,axiom,
( ( ^ [Y6: int,Z3: int] : ( Y6 = Z3 ) )
= ( ^ [A3: int,B2: int] :
( ( ord_less_eq_int @ A3 @ B2 )
& ( ord_less_eq_int @ B2 @ A3 ) ) ) ) ).
% Orderings.order_eq_iff
thf(fact_173_Orderings_Oorder__eq__iff,axiom,
( ( ^ [Y6: nat,Z3: nat] : ( Y6 = Z3 ) )
= ( ^ [A3: nat,B2: nat] :
( ( ord_less_eq_nat @ A3 @ B2 )
& ( ord_less_eq_nat @ B2 @ A3 ) ) ) ) ).
% Orderings.order_eq_iff
thf(fact_174_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 )
=> ( ! [X5: int,Y3: int] :
( ( ord_less_eq_int @ X5 @ Y3 )
=> ( ord_less_eq_int @ ( F @ X5 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_int @ A @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_175_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 )
=> ( ! [X5: nat,Y3: nat] :
( ( ord_less_eq_nat @ X5 @ Y3 )
=> ( ord_less_eq_int @ ( F @ X5 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_int @ A @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_176_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 )
=> ( ! [X5: int,Y3: int] :
( ( ord_less_eq_int @ X5 @ Y3 )
=> ( ord_less_eq_nat @ ( F @ X5 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_nat @ A @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_177_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 )
=> ( ! [X5: nat,Y3: nat] :
( ( ord_less_eq_nat @ X5 @ Y3 )
=> ( ord_less_eq_nat @ ( F @ X5 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_nat @ A @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_178_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 )
=> ( ! [X5: int,Y3: int] :
( ( ord_less_eq_int @ X5 @ Y3 )
=> ( ord_less_eq_int @ ( F @ X5 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_int @ ( F @ A ) @ C ) ) ) ) ).
% order_subst2
thf(fact_179_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 )
=> ( ! [X5: int,Y3: int] :
( ( ord_less_eq_int @ X5 @ Y3 )
=> ( ord_less_eq_nat @ ( F @ X5 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_nat @ ( F @ A ) @ C ) ) ) ) ).
% order_subst2
thf(fact_180_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 )
=> ( ! [X5: nat,Y3: nat] :
( ( ord_less_eq_nat @ X5 @ Y3 )
=> ( ord_less_eq_int @ ( F @ X5 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_int @ ( F @ A ) @ C ) ) ) ) ).
% order_subst2
thf(fact_181_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 )
=> ( ! [X5: nat,Y3: nat] :
( ( ord_less_eq_nat @ X5 @ Y3 )
=> ( ord_less_eq_nat @ ( F @ X5 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_nat @ ( F @ A ) @ C ) ) ) ) ).
% order_subst2
thf(fact_182_order__eq__refl,axiom,
! [X: int,Y: int] :
( ( X = Y )
=> ( ord_less_eq_int @ X @ Y ) ) ).
% order_eq_refl
thf(fact_183_order__eq__refl,axiom,
! [X: nat,Y: nat] :
( ( X = Y )
=> ( ord_less_eq_nat @ X @ Y ) ) ).
% order_eq_refl
thf(fact_184_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_185_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_186_linorder__linear,axiom,
! [X: int,Y: int] :
( ( ord_less_eq_int @ X @ Y )
| ( ord_less_eq_int @ Y @ X ) ) ).
% linorder_linear
thf(fact_187_linorder__linear,axiom,
! [X: nat,Y: nat] :
( ( ord_less_eq_nat @ X @ Y )
| ( ord_less_eq_nat @ Y @ X ) ) ).
% linorder_linear
thf(fact_188_ord__eq__le__subst,axiom,
! [A: int,F: int > int,B: int,C: int] :
( ( A
= ( F @ B ) )
=> ( ( ord_less_eq_int @ B @ C )
=> ( ! [X5: int,Y3: int] :
( ( ord_less_eq_int @ X5 @ Y3 )
=> ( ord_less_eq_int @ ( F @ X5 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_int @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_189_ord__eq__le__subst,axiom,
! [A: nat,F: int > nat,B: int,C: int] :
( ( A
= ( F @ B ) )
=> ( ( ord_less_eq_int @ B @ C )
=> ( ! [X5: int,Y3: int] :
( ( ord_less_eq_int @ X5 @ Y3 )
=> ( ord_less_eq_nat @ ( F @ X5 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_nat @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_190_ord__eq__le__subst,axiom,
! [A: int,F: nat > int,B: nat,C: nat] :
( ( A
= ( F @ B ) )
=> ( ( ord_less_eq_nat @ B @ C )
=> ( ! [X5: nat,Y3: nat] :
( ( ord_less_eq_nat @ X5 @ Y3 )
=> ( ord_less_eq_int @ ( F @ X5 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_int @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_191_ord__eq__le__subst,axiom,
! [A: nat,F: nat > nat,B: nat,C: nat] :
( ( A
= ( F @ B ) )
=> ( ( ord_less_eq_nat @ B @ C )
=> ( ! [X5: nat,Y3: nat] :
( ( ord_less_eq_nat @ X5 @ Y3 )
=> ( ord_less_eq_nat @ ( F @ X5 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_nat @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_192_ord__le__eq__subst,axiom,
! [A: int,B: int,F: int > int,C: int] :
( ( ord_less_eq_int @ A @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X5: int,Y3: int] :
( ( ord_less_eq_int @ X5 @ Y3 )
=> ( ord_less_eq_int @ ( F @ X5 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_int @ ( F @ A ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_193_ord__le__eq__subst,axiom,
! [A: int,B: int,F: int > nat,C: nat] :
( ( ord_less_eq_int @ A @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X5: int,Y3: int] :
( ( ord_less_eq_int @ X5 @ Y3 )
=> ( ord_less_eq_nat @ ( F @ X5 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_nat @ ( F @ A ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_194_ord__le__eq__subst,axiom,
! [A: nat,B: nat,F: nat > int,C: int] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X5: nat,Y3: nat] :
( ( ord_less_eq_nat @ X5 @ Y3 )
=> ( ord_less_eq_int @ ( F @ X5 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_int @ ( F @ A ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_195_ord__le__eq__subst,axiom,
! [A: nat,B: nat,F: nat > nat,C: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X5: nat,Y3: nat] :
( ( ord_less_eq_nat @ X5 @ Y3 )
=> ( ord_less_eq_nat @ ( F @ X5 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_nat @ ( F @ A ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_196_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_197_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_198_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_199_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_200_minf_I7_J,axiom,
! [T: int] :
? [Z: int] :
! [X7: int] :
( ( ord_less_int @ X7 @ Z )
=> ~ ( ord_less_int @ T @ X7 ) ) ).
% minf(7)
thf(fact_201_minf_I7_J,axiom,
! [T: nat] :
? [Z: nat] :
! [X7: nat] :
( ( ord_less_nat @ X7 @ Z )
=> ~ ( ord_less_nat @ T @ X7 ) ) ).
% minf(7)
thf(fact_202_minf_I5_J,axiom,
! [T: int] :
? [Z: int] :
! [X7: int] :
( ( ord_less_int @ X7 @ Z )
=> ( ord_less_int @ X7 @ T ) ) ).
% minf(5)
thf(fact_203_minf_I5_J,axiom,
! [T: nat] :
? [Z: nat] :
! [X7: nat] :
( ( ord_less_nat @ X7 @ Z )
=> ( ord_less_nat @ X7 @ T ) ) ).
% minf(5)
thf(fact_204_minf_I4_J,axiom,
! [T: int] :
? [Z: int] :
! [X7: int] :
( ( ord_less_int @ X7 @ Z )
=> ( X7 != T ) ) ).
% minf(4)
thf(fact_205_minf_I4_J,axiom,
! [T: nat] :
? [Z: nat] :
! [X7: nat] :
( ( ord_less_nat @ X7 @ Z )
=> ( X7 != T ) ) ).
% minf(4)
thf(fact_206_minf_I3_J,axiom,
! [T: int] :
? [Z: int] :
! [X7: int] :
( ( ord_less_int @ X7 @ Z )
=> ( X7 != T ) ) ).
% minf(3)
thf(fact_207_minf_I3_J,axiom,
! [T: nat] :
? [Z: nat] :
! [X7: nat] :
( ( ord_less_nat @ X7 @ Z )
=> ( X7 != T ) ) ).
% minf(3)
thf(fact_208_minf_I2_J,axiom,
! [P: int > $o,P2: int > $o,Q: int > $o,Q2: int > $o] :
( ? [Z4: int] :
! [X5: int] :
( ( ord_less_int @ X5 @ Z4 )
=> ( ( P @ X5 )
= ( P2 @ X5 ) ) )
=> ( ? [Z4: int] :
! [X5: int] :
( ( ord_less_int @ X5 @ Z4 )
=> ( ( Q @ X5 )
= ( Q2 @ X5 ) ) )
=> ? [Z: int] :
! [X7: int] :
( ( ord_less_int @ X7 @ Z )
=> ( ( ( P @ X7 )
| ( Q @ X7 ) )
= ( ( P2 @ X7 )
| ( Q2 @ X7 ) ) ) ) ) ) ).
% minf(2)
thf(fact_209_minf_I2_J,axiom,
! [P: nat > $o,P2: nat > $o,Q: nat > $o,Q2: nat > $o] :
( ? [Z4: nat] :
! [X5: nat] :
( ( ord_less_nat @ X5 @ Z4 )
=> ( ( P @ X5 )
= ( P2 @ X5 ) ) )
=> ( ? [Z4: nat] :
! [X5: nat] :
( ( ord_less_nat @ X5 @ Z4 )
=> ( ( Q @ X5 )
= ( Q2 @ X5 ) ) )
=> ? [Z: nat] :
! [X7: nat] :
( ( ord_less_nat @ X7 @ Z )
=> ( ( ( P @ X7 )
| ( Q @ X7 ) )
= ( ( P2 @ X7 )
| ( Q2 @ X7 ) ) ) ) ) ) ).
% minf(2)
thf(fact_210_minf_I1_J,axiom,
! [P: int > $o,P2: int > $o,Q: int > $o,Q2: int > $o] :
( ? [Z4: int] :
! [X5: int] :
( ( ord_less_int @ X5 @ Z4 )
=> ( ( P @ X5 )
= ( P2 @ X5 ) ) )
=> ( ? [Z4: int] :
! [X5: int] :
( ( ord_less_int @ X5 @ Z4 )
=> ( ( Q @ X5 )
= ( Q2 @ X5 ) ) )
=> ? [Z: int] :
! [X7: int] :
( ( ord_less_int @ X7 @ Z )
=> ( ( ( P @ X7 )
& ( Q @ X7 ) )
= ( ( P2 @ X7 )
& ( Q2 @ X7 ) ) ) ) ) ) ).
% minf(1)
thf(fact_211_minf_I1_J,axiom,
! [P: nat > $o,P2: nat > $o,Q: nat > $o,Q2: nat > $o] :
( ? [Z4: nat] :
! [X5: nat] :
( ( ord_less_nat @ X5 @ Z4 )
=> ( ( P @ X5 )
= ( P2 @ X5 ) ) )
=> ( ? [Z4: nat] :
! [X5: nat] :
( ( ord_less_nat @ X5 @ Z4 )
=> ( ( Q @ X5 )
= ( Q2 @ X5 ) ) )
=> ? [Z: nat] :
! [X7: nat] :
( ( ord_less_nat @ X7 @ Z )
=> ( ( ( P @ X7 )
& ( Q @ X7 ) )
= ( ( P2 @ X7 )
& ( Q2 @ X7 ) ) ) ) ) ) ).
% minf(1)
thf(fact_212_pinf_I7_J,axiom,
! [T: int] :
? [Z: int] :
! [X7: int] :
( ( ord_less_int @ Z @ X7 )
=> ( ord_less_int @ T @ X7 ) ) ).
% pinf(7)
thf(fact_213_pinf_I7_J,axiom,
! [T: nat] :
? [Z: nat] :
! [X7: nat] :
( ( ord_less_nat @ Z @ X7 )
=> ( ord_less_nat @ T @ X7 ) ) ).
% pinf(7)
thf(fact_214_pinf_I5_J,axiom,
! [T: int] :
? [Z: int] :
! [X7: int] :
( ( ord_less_int @ Z @ X7 )
=> ~ ( ord_less_int @ X7 @ T ) ) ).
% pinf(5)
thf(fact_215_pinf_I5_J,axiom,
! [T: nat] :
? [Z: nat] :
! [X7: nat] :
( ( ord_less_nat @ Z @ X7 )
=> ~ ( ord_less_nat @ X7 @ T ) ) ).
% pinf(5)
thf(fact_216_pinf_I4_J,axiom,
! [T: int] :
? [Z: int] :
! [X7: int] :
( ( ord_less_int @ Z @ X7 )
=> ( X7 != T ) ) ).
% pinf(4)
thf(fact_217_pinf_I4_J,axiom,
! [T: nat] :
? [Z: nat] :
! [X7: nat] :
( ( ord_less_nat @ Z @ X7 )
=> ( X7 != T ) ) ).
% pinf(4)
thf(fact_218_pinf_I3_J,axiom,
! [T: int] :
? [Z: int] :
! [X7: int] :
( ( ord_less_int @ Z @ X7 )
=> ( X7 != T ) ) ).
% pinf(3)
thf(fact_219_pinf_I3_J,axiom,
! [T: nat] :
? [Z: nat] :
! [X7: nat] :
( ( ord_less_nat @ Z @ X7 )
=> ( X7 != T ) ) ).
% pinf(3)
thf(fact_220_pinf_I2_J,axiom,
! [P: int > $o,P2: int > $o,Q: int > $o,Q2: int > $o] :
( ? [Z4: int] :
! [X5: int] :
( ( ord_less_int @ Z4 @ X5 )
=> ( ( P @ X5 )
= ( P2 @ X5 ) ) )
=> ( ? [Z4: int] :
! [X5: int] :
( ( ord_less_int @ Z4 @ X5 )
=> ( ( Q @ X5 )
= ( Q2 @ X5 ) ) )
=> ? [Z: int] :
! [X7: int] :
( ( ord_less_int @ Z @ X7 )
=> ( ( ( P @ X7 )
| ( Q @ X7 ) )
= ( ( P2 @ X7 )
| ( Q2 @ X7 ) ) ) ) ) ) ).
% pinf(2)
thf(fact_221_pinf_I2_J,axiom,
! [P: nat > $o,P2: nat > $o,Q: nat > $o,Q2: nat > $o] :
( ? [Z4: nat] :
! [X5: nat] :
( ( ord_less_nat @ Z4 @ X5 )
=> ( ( P @ X5 )
= ( P2 @ X5 ) ) )
=> ( ? [Z4: nat] :
! [X5: nat] :
( ( ord_less_nat @ Z4 @ X5 )
=> ( ( Q @ X5 )
= ( Q2 @ X5 ) ) )
=> ? [Z: nat] :
! [X7: nat] :
( ( ord_less_nat @ Z @ X7 )
=> ( ( ( P @ X7 )
| ( Q @ X7 ) )
= ( ( P2 @ X7 )
| ( Q2 @ X7 ) ) ) ) ) ) ).
% pinf(2)
thf(fact_222_pinf_I1_J,axiom,
! [P: int > $o,P2: int > $o,Q: int > $o,Q2: int > $o] :
( ? [Z4: int] :
! [X5: int] :
( ( ord_less_int @ Z4 @ X5 )
=> ( ( P @ X5 )
= ( P2 @ X5 ) ) )
=> ( ? [Z4: int] :
! [X5: int] :
( ( ord_less_int @ Z4 @ X5 )
=> ( ( Q @ X5 )
= ( Q2 @ X5 ) ) )
=> ? [Z: int] :
! [X7: int] :
( ( ord_less_int @ Z @ X7 )
=> ( ( ( P @ X7 )
& ( Q @ X7 ) )
= ( ( P2 @ X7 )
& ( Q2 @ X7 ) ) ) ) ) ) ).
% pinf(1)
thf(fact_223_pinf_I1_J,axiom,
! [P: nat > $o,P2: nat > $o,Q: nat > $o,Q2: nat > $o] :
( ? [Z4: nat] :
! [X5: nat] :
( ( ord_less_nat @ Z4 @ X5 )
=> ( ( P @ X5 )
= ( P2 @ X5 ) ) )
=> ( ? [Z4: nat] :
! [X5: nat] :
( ( ord_less_nat @ Z4 @ X5 )
=> ( ( Q @ X5 )
= ( Q2 @ X5 ) ) )
=> ? [Z: nat] :
! [X7: nat] :
( ( ord_less_nat @ Z @ X7 )
=> ( ( ( P @ X7 )
& ( Q @ X7 ) )
= ( ( P2 @ X7 )
& ( Q2 @ X7 ) ) ) ) ) ) ).
% pinf(1)
thf(fact_224_verit__comp__simplify1_I1_J,axiom,
! [A: int] :
~ ( ord_less_int @ A @ A ) ).
% verit_comp_simplify1(1)
thf(fact_225_verit__comp__simplify1_I1_J,axiom,
! [A: nat] :
~ ( ord_less_nat @ A @ A ) ).
% verit_comp_simplify1(1)
thf(fact_226_lt__ex,axiom,
! [X: int] :
? [Y3: int] : ( ord_less_int @ Y3 @ X ) ).
% lt_ex
thf(fact_227_gt__ex,axiom,
! [X: int] :
? [X_1: int] : ( ord_less_int @ X @ X_1 ) ).
% gt_ex
thf(fact_228_gt__ex,axiom,
! [X: nat] :
? [X_1: nat] : ( ord_less_nat @ X @ X_1 ) ).
% gt_ex
thf(fact_229_less__imp__neq,axiom,
! [X: int,Y: int] :
( ( ord_less_int @ X @ Y )
=> ( X != Y ) ) ).
% less_imp_neq
thf(fact_230_less__imp__neq,axiom,
! [X: nat,Y: nat] :
( ( ord_less_nat @ X @ Y )
=> ( X != Y ) ) ).
% less_imp_neq
thf(fact_231_order_Oasym,axiom,
! [A: int,B: int] :
( ( ord_less_int @ A @ B )
=> ~ ( ord_less_int @ B @ A ) ) ).
% order.asym
thf(fact_232_order_Oasym,axiom,
! [A: nat,B: nat] :
( ( ord_less_nat @ A @ B )
=> ~ ( ord_less_nat @ B @ A ) ) ).
% order.asym
thf(fact_233_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_234_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_235_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_236_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_237_less__induct,axiom,
! [P: nat > $o,A: nat] :
( ! [X5: nat] :
( ! [Y5: nat] :
( ( ord_less_nat @ Y5 @ X5 )
=> ( P @ Y5 ) )
=> ( P @ X5 ) )
=> ( P @ A ) ) ).
% less_induct
thf(fact_238_antisym__conv3,axiom,
! [Y: int,X: int] :
( ~ ( ord_less_int @ Y @ X )
=> ( ( ~ ( ord_less_int @ X @ Y ) )
= ( X = Y ) ) ) ).
% antisym_conv3
thf(fact_239_antisym__conv3,axiom,
! [Y: nat,X: nat] :
( ~ ( ord_less_nat @ Y @ X )
=> ( ( ~ ( ord_less_nat @ X @ Y ) )
= ( X = Y ) ) ) ).
% antisym_conv3
thf(fact_240_linorder__cases,axiom,
! [X: int,Y: int] :
( ~ ( ord_less_int @ X @ Y )
=> ( ( X != Y )
=> ( ord_less_int @ Y @ X ) ) ) ).
% linorder_cases
thf(fact_241_linorder__cases,axiom,
! [X: nat,Y: nat] :
( ~ ( ord_less_nat @ X @ Y )
=> ( ( X != Y )
=> ( ord_less_nat @ Y @ X ) ) ) ).
% linorder_cases
thf(fact_242_dual__order_Oasym,axiom,
! [B: int,A: int] :
( ( ord_less_int @ B @ A )
=> ~ ( ord_less_int @ A @ B ) ) ).
% dual_order.asym
thf(fact_243_dual__order_Oasym,axiom,
! [B: nat,A: nat] :
( ( ord_less_nat @ B @ A )
=> ~ ( ord_less_nat @ A @ B ) ) ).
% dual_order.asym
thf(fact_244_dual__order_Oirrefl,axiom,
! [A: int] :
~ ( ord_less_int @ A @ A ) ).
% dual_order.irrefl
thf(fact_245_dual__order_Oirrefl,axiom,
! [A: nat] :
~ ( ord_less_nat @ A @ A ) ).
% dual_order.irrefl
thf(fact_246_exists__least__iff,axiom,
( ( ^ [P3: nat > $o] :
? [X8: nat] : ( P3 @ X8 ) )
= ( ^ [P4: nat > $o] :
? [N3: nat] :
( ( P4 @ N3 )
& ! [M2: nat] :
( ( ord_less_nat @ M2 @ N3 )
=> ~ ( P4 @ M2 ) ) ) ) ) ).
% exists_least_iff
thf(fact_247_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_248_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_249_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_250_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_251_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_252_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_253_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_254_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_255_order_Ostrict__implies__not__eq,axiom,
! [A: int,B: int] :
( ( ord_less_int @ A @ B )
=> ( A != B ) ) ).
% order.strict_implies_not_eq
thf(fact_256_order_Ostrict__implies__not__eq,axiom,
! [A: nat,B: nat] :
( ( ord_less_nat @ A @ B )
=> ( A != B ) ) ).
% order.strict_implies_not_eq
thf(fact_257_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_258_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_259_linorder__neqE,axiom,
! [X: int,Y: int] :
( ( X != Y )
=> ( ~ ( ord_less_int @ X @ Y )
=> ( ord_less_int @ Y @ X ) ) ) ).
% linorder_neqE
thf(fact_260_linorder__neqE,axiom,
! [X: nat,Y: nat] :
( ( X != Y )
=> ( ~ ( ord_less_nat @ X @ Y )
=> ( ord_less_nat @ Y @ X ) ) ) ).
% linorder_neqE
thf(fact_261_order__less__asym,axiom,
! [X: int,Y: int] :
( ( ord_less_int @ X @ Y )
=> ~ ( ord_less_int @ Y @ X ) ) ).
% order_less_asym
thf(fact_262_order__less__asym,axiom,
! [X: nat,Y: nat] :
( ( ord_less_nat @ X @ Y )
=> ~ ( ord_less_nat @ Y @ X ) ) ).
% order_less_asym
thf(fact_263_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_264_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_265_order__less__asym_H,axiom,
! [A: int,B: int] :
( ( ord_less_int @ A @ B )
=> ~ ( ord_less_int @ B @ A ) ) ).
% order_less_asym'
thf(fact_266_order__less__asym_H,axiom,
! [A: nat,B: nat] :
( ( ord_less_nat @ A @ B )
=> ~ ( ord_less_nat @ B @ A ) ) ).
% order_less_asym'
thf(fact_267_order__less__trans,axiom,
! [X: int,Y: int,Z2: int] :
( ( ord_less_int @ X @ Y )
=> ( ( ord_less_int @ Y @ Z2 )
=> ( ord_less_int @ X @ Z2 ) ) ) ).
% order_less_trans
thf(fact_268_order__less__trans,axiom,
! [X: nat,Y: nat,Z2: nat] :
( ( ord_less_nat @ X @ Y )
=> ( ( ord_less_nat @ Y @ Z2 )
=> ( ord_less_nat @ X @ Z2 ) ) ) ).
% order_less_trans
thf(fact_269_ord__eq__less__subst,axiom,
! [A: int,F: int > int,B: int,C: int] :
( ( A
= ( F @ B ) )
=> ( ( ord_less_int @ B @ C )
=> ( ! [X5: int,Y3: int] :
( ( ord_less_int @ X5 @ Y3 )
=> ( ord_less_int @ ( F @ X5 ) @ ( F @ Y3 ) ) )
=> ( ord_less_int @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_less_subst
thf(fact_270_ord__eq__less__subst,axiom,
! [A: nat,F: int > nat,B: int,C: int] :
( ( A
= ( F @ B ) )
=> ( ( ord_less_int @ B @ C )
=> ( ! [X5: int,Y3: int] :
( ( ord_less_int @ X5 @ Y3 )
=> ( ord_less_nat @ ( F @ X5 ) @ ( F @ Y3 ) ) )
=> ( ord_less_nat @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_less_subst
thf(fact_271_ord__eq__less__subst,axiom,
! [A: int,F: nat > int,B: nat,C: nat] :
( ( A
= ( F @ B ) )
=> ( ( ord_less_nat @ B @ C )
=> ( ! [X5: nat,Y3: nat] :
( ( ord_less_nat @ X5 @ Y3 )
=> ( ord_less_int @ ( F @ X5 ) @ ( F @ Y3 ) ) )
=> ( ord_less_int @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_less_subst
thf(fact_272_ord__eq__less__subst,axiom,
! [A: nat,F: nat > nat,B: nat,C: nat] :
( ( A
= ( F @ B ) )
=> ( ( ord_less_nat @ B @ C )
=> ( ! [X5: nat,Y3: nat] :
( ( ord_less_nat @ X5 @ Y3 )
=> ( ord_less_nat @ ( F @ X5 ) @ ( F @ Y3 ) ) )
=> ( ord_less_nat @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_less_subst
thf(fact_273_ord__less__eq__subst,axiom,
! [A: int,B: int,F: int > int,C: int] :
( ( ord_less_int @ A @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X5: int,Y3: int] :
( ( ord_less_int @ X5 @ Y3 )
=> ( ord_less_int @ ( F @ X5 ) @ ( F @ Y3 ) ) )
=> ( ord_less_int @ ( F @ A ) @ C ) ) ) ) ).
% ord_less_eq_subst
thf(fact_274_ord__less__eq__subst,axiom,
! [A: int,B: int,F: int > nat,C: nat] :
( ( ord_less_int @ A @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X5: int,Y3: int] :
( ( ord_less_int @ X5 @ Y3 )
=> ( ord_less_nat @ ( F @ X5 ) @ ( F @ Y3 ) ) )
=> ( ord_less_nat @ ( F @ A ) @ C ) ) ) ) ).
% ord_less_eq_subst
thf(fact_275_ord__less__eq__subst,axiom,
! [A: nat,B: nat,F: nat > int,C: int] :
( ( ord_less_nat @ A @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X5: nat,Y3: nat] :
( ( ord_less_nat @ X5 @ Y3 )
=> ( ord_less_int @ ( F @ X5 ) @ ( F @ Y3 ) ) )
=> ( ord_less_int @ ( F @ A ) @ C ) ) ) ) ).
% ord_less_eq_subst
thf(fact_276_ord__less__eq__subst,axiom,
! [A: nat,B: nat,F: nat > nat,C: nat] :
( ( ord_less_nat @ A @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X5: nat,Y3: nat] :
( ( ord_less_nat @ X5 @ Y3 )
=> ( ord_less_nat @ ( F @ X5 ) @ ( F @ Y3 ) ) )
=> ( ord_less_nat @ ( F @ A ) @ C ) ) ) ) ).
% ord_less_eq_subst
thf(fact_277_order__less__irrefl,axiom,
! [X: int] :
~ ( ord_less_int @ X @ X ) ).
% order_less_irrefl
thf(fact_278_order__less__irrefl,axiom,
! [X: nat] :
~ ( ord_less_nat @ X @ X ) ).
% order_less_irrefl
thf(fact_279_order__less__subst1,axiom,
! [A: int,F: int > int,B: int,C: int] :
( ( ord_less_int @ A @ ( F @ B ) )
=> ( ( ord_less_int @ B @ C )
=> ( ! [X5: int,Y3: int] :
( ( ord_less_int @ X5 @ Y3 )
=> ( ord_less_int @ ( F @ X5 ) @ ( F @ Y3 ) ) )
=> ( ord_less_int @ A @ ( F @ C ) ) ) ) ) ).
% order_less_subst1
thf(fact_280_order__less__subst1,axiom,
! [A: int,F: nat > int,B: nat,C: nat] :
( ( ord_less_int @ A @ ( F @ B ) )
=> ( ( ord_less_nat @ B @ C )
=> ( ! [X5: nat,Y3: nat] :
( ( ord_less_nat @ X5 @ Y3 )
=> ( ord_less_int @ ( F @ X5 ) @ ( F @ Y3 ) ) )
=> ( ord_less_int @ A @ ( F @ C ) ) ) ) ) ).
% order_less_subst1
thf(fact_281_order__less__subst1,axiom,
! [A: nat,F: int > nat,B: int,C: int] :
( ( ord_less_nat @ A @ ( F @ B ) )
=> ( ( ord_less_int @ B @ C )
=> ( ! [X5: int,Y3: int] :
( ( ord_less_int @ X5 @ Y3 )
=> ( ord_less_nat @ ( F @ X5 ) @ ( F @ Y3 ) ) )
=> ( ord_less_nat @ A @ ( F @ C ) ) ) ) ) ).
% order_less_subst1
thf(fact_282_order__less__subst1,axiom,
! [A: nat,F: nat > nat,B: nat,C: nat] :
( ( ord_less_nat @ A @ ( F @ B ) )
=> ( ( ord_less_nat @ B @ C )
=> ( ! [X5: nat,Y3: nat] :
( ( ord_less_nat @ X5 @ Y3 )
=> ( ord_less_nat @ ( F @ X5 ) @ ( F @ Y3 ) ) )
=> ( ord_less_nat @ A @ ( F @ C ) ) ) ) ) ).
% order_less_subst1
thf(fact_283_order__less__subst2,axiom,
! [A: int,B: int,F: int > int,C: int] :
( ( ord_less_int @ A @ B )
=> ( ( ord_less_int @ ( F @ B ) @ C )
=> ( ! [X5: int,Y3: int] :
( ( ord_less_int @ X5 @ Y3 )
=> ( ord_less_int @ ( F @ X5 ) @ ( F @ Y3 ) ) )
=> ( ord_less_int @ ( F @ A ) @ C ) ) ) ) ).
% order_less_subst2
thf(fact_284_order__less__subst2,axiom,
! [A: int,B: int,F: int > nat,C: nat] :
( ( ord_less_int @ A @ B )
=> ( ( ord_less_nat @ ( F @ B ) @ C )
=> ( ! [X5: int,Y3: int] :
( ( ord_less_int @ X5 @ Y3 )
=> ( ord_less_nat @ ( F @ X5 ) @ ( F @ Y3 ) ) )
=> ( ord_less_nat @ ( F @ A ) @ C ) ) ) ) ).
% order_less_subst2
thf(fact_285_order__less__subst2,axiom,
! [A: nat,B: nat,F: nat > int,C: int] :
( ( ord_less_nat @ A @ B )
=> ( ( ord_less_int @ ( F @ B ) @ C )
=> ( ! [X5: nat,Y3: nat] :
( ( ord_less_nat @ X5 @ Y3 )
=> ( ord_less_int @ ( F @ X5 ) @ ( F @ Y3 ) ) )
=> ( ord_less_int @ ( F @ A ) @ C ) ) ) ) ).
% order_less_subst2
thf(fact_286_order__less__subst2,axiom,
! [A: nat,B: nat,F: nat > nat,C: nat] :
( ( ord_less_nat @ A @ B )
=> ( ( ord_less_nat @ ( F @ B ) @ C )
=> ( ! [X5: nat,Y3: nat] :
( ( ord_less_nat @ X5 @ Y3 )
=> ( ord_less_nat @ ( F @ X5 ) @ ( F @ Y3 ) ) )
=> ( ord_less_nat @ ( F @ A ) @ C ) ) ) ) ).
% order_less_subst2
thf(fact_287_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_288_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_289_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_290_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_291_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_292_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_293_order__less__imp__not__eq,axiom,
! [X: int,Y: int] :
( ( ord_less_int @ X @ Y )
=> ( X != Y ) ) ).
% order_less_imp_not_eq
thf(fact_294_order__less__imp__not__eq,axiom,
! [X: nat,Y: nat] :
( ( ord_less_nat @ X @ Y )
=> ( X != Y ) ) ).
% order_less_imp_not_eq
thf(fact_295_order__less__imp__not__eq2,axiom,
! [X: int,Y: int] :
( ( ord_less_int @ X @ Y )
=> ( Y != X ) ) ).
% order_less_imp_not_eq2
thf(fact_296_order__less__imp__not__eq2,axiom,
! [X: nat,Y: nat] :
( ( ord_less_nat @ X @ Y )
=> ( Y != X ) ) ).
% order_less_imp_not_eq2
thf(fact_297_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_298_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_299_subset__code_I1_J,axiom,
! [Xs: list_list_char,B4: set_list_char] :
( ( ord_le1207710660481778807t_char @ ( set_list_char2 @ Xs ) @ B4 )
= ( ! [X3: list_char] :
( ( member_list_char @ X3 @ ( set_list_char2 @ Xs ) )
=> ( member_list_char @ X3 @ B4 ) ) ) ) ).
% subset_code(1)
thf(fact_300_subset__code_I1_J,axiom,
! [Xs: list_instr,B4: set_instr] :
( ( ord_le4956778034350794079_instr @ ( set_instr2 @ Xs ) @ B4 )
= ( ! [X3: instr] :
( ( member_instr @ X3 @ ( set_instr2 @ Xs ) )
=> ( member_instr @ X3 @ B4 ) ) ) ) ).
% subset_code(1)
thf(fact_301_Ex__list__of__length,axiom,
! [N: nat] :
? [Xs2: list_instr] :
( ( size_size_list_instr @ Xs2 )
= N ) ).
% Ex_list_of_length
thf(fact_302_Ex__list__of__length,axiom,
! [N: nat] :
? [Xs2: list_list_char] :
( ( size_s356637359517785349t_char @ Xs2 )
= N ) ).
% Ex_list_of_length
thf(fact_303_neq__if__length__neq,axiom,
! [Xs: list_instr,Ys2: list_instr] :
( ( ( size_size_list_instr @ Xs )
!= ( size_size_list_instr @ Ys2 ) )
=> ( Xs != Ys2 ) ) ).
% neq_if_length_neq
thf(fact_304_neq__if__length__neq,axiom,
! [Xs: list_list_char,Ys2: list_list_char] :
( ( ( size_s356637359517785349t_char @ Xs )
!= ( size_s356637359517785349t_char @ Ys2 ) )
=> ( Xs != Ys2 ) ) ).
% neq_if_length_neq
thf(fact_305_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_306_nat__int__comparison_I1_J,axiom,
( ( ^ [Y6: nat,Z3: nat] : ( Y6 = Z3 ) )
= ( ^ [A3: nat,B2: nat] :
( ( semiri1314217659103216013at_int @ A3 )
= ( semiri1314217659103216013at_int @ B2 ) ) ) ) ).
% nat_int_comparison(1)
thf(fact_307_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_308_instr_Osize_I9_J,axiom,
! [X2: list_char] :
( ( size_size_instr @ ( load @ X2 ) )
= zero_zero_nat ) ).
% instr.size(9)
thf(fact_309_instr_Osize_I11_J,axiom,
! [X4: list_char] :
( ( size_size_instr @ ( store @ X4 ) )
= zero_zero_nat ) ).
% instr.size(11)
thf(fact_310_minf_I8_J,axiom,
! [T: int] :
? [Z: int] :
! [X7: int] :
( ( ord_less_int @ X7 @ Z )
=> ~ ( ord_less_eq_int @ T @ X7 ) ) ).
% minf(8)
thf(fact_311_minf_I8_J,axiom,
! [T: nat] :
? [Z: nat] :
! [X7: nat] :
( ( ord_less_nat @ X7 @ Z )
=> ~ ( ord_less_eq_nat @ T @ X7 ) ) ).
% minf(8)
thf(fact_312_minf_I6_J,axiom,
! [T: int] :
? [Z: int] :
! [X7: int] :
( ( ord_less_int @ X7 @ Z )
=> ( ord_less_eq_int @ X7 @ T ) ) ).
% minf(6)
thf(fact_313_minf_I6_J,axiom,
! [T: nat] :
? [Z: nat] :
! [X7: nat] :
( ( ord_less_nat @ X7 @ Z )
=> ( ord_less_eq_nat @ X7 @ T ) ) ).
% minf(6)
thf(fact_314_pinf_I8_J,axiom,
! [T: int] :
? [Z: int] :
! [X7: int] :
( ( ord_less_int @ Z @ X7 )
=> ( ord_less_eq_int @ T @ X7 ) ) ).
% pinf(8)
thf(fact_315_pinf_I8_J,axiom,
! [T: nat] :
? [Z: nat] :
! [X7: nat] :
( ( ord_less_nat @ Z @ X7 )
=> ( ord_less_eq_nat @ T @ X7 ) ) ).
% pinf(8)
thf(fact_316_pinf_I6_J,axiom,
! [T: int] :
? [Z: int] :
! [X7: int] :
( ( ord_less_int @ Z @ X7 )
=> ~ ( ord_less_eq_int @ X7 @ T ) ) ).
% pinf(6)
thf(fact_317_pinf_I6_J,axiom,
! [T: nat] :
? [Z: nat] :
! [X7: nat] :
( ( ord_less_nat @ Z @ X7 )
=> ~ ( ord_less_eq_nat @ X7 @ T ) ) ).
% pinf(6)
thf(fact_318_verit__comp__simplify1_I3_J,axiom,
! [B5: int,A5: int] :
( ( ~ ( ord_less_eq_int @ B5 @ A5 ) )
= ( ord_less_int @ A5 @ B5 ) ) ).
% verit_comp_simplify1(3)
thf(fact_319_verit__comp__simplify1_I3_J,axiom,
! [B5: nat,A5: nat] :
( ( ~ ( ord_less_eq_nat @ B5 @ A5 ) )
= ( ord_less_nat @ A5 @ B5 ) ) ).
% verit_comp_simplify1(3)
thf(fact_320_leD,axiom,
! [Y: int,X: int] :
( ( ord_less_eq_int @ Y @ X )
=> ~ ( ord_less_int @ X @ Y ) ) ).
% leD
thf(fact_321_leD,axiom,
! [Y: nat,X: nat] :
( ( ord_less_eq_nat @ Y @ X )
=> ~ ( ord_less_nat @ X @ Y ) ) ).
% leD
thf(fact_322_leI,axiom,
! [X: int,Y: int] :
( ~ ( ord_less_int @ X @ Y )
=> ( ord_less_eq_int @ Y @ X ) ) ).
% leI
thf(fact_323_leI,axiom,
! [X: nat,Y: nat] :
( ~ ( ord_less_nat @ X @ Y )
=> ( ord_less_eq_nat @ Y @ X ) ) ).
% leI
thf(fact_324_nless__le,axiom,
! [A: int,B: int] :
( ( ~ ( ord_less_int @ A @ B ) )
= ( ~ ( ord_less_eq_int @ A @ B )
| ( A = B ) ) ) ).
% nless_le
thf(fact_325_nless__le,axiom,
! [A: nat,B: nat] :
( ( ~ ( ord_less_nat @ A @ B ) )
= ( ~ ( ord_less_eq_nat @ A @ B )
| ( A = B ) ) ) ).
% nless_le
thf(fact_326_antisym__conv1,axiom,
! [X: int,Y: int] :
( ~ ( ord_less_int @ X @ Y )
=> ( ( ord_less_eq_int @ X @ Y )
= ( X = Y ) ) ) ).
% antisym_conv1
thf(fact_327_antisym__conv1,axiom,
! [X: nat,Y: nat] :
( ~ ( ord_less_nat @ X @ Y )
=> ( ( ord_less_eq_nat @ X @ Y )
= ( X = Y ) ) ) ).
% antisym_conv1
thf(fact_328_antisym__conv2,axiom,
! [X: int,Y: int] :
( ( ord_less_eq_int @ X @ Y )
=> ( ( ~ ( ord_less_int @ X @ Y ) )
= ( X = Y ) ) ) ).
% antisym_conv2
thf(fact_329_antisym__conv2,axiom,
! [X: nat,Y: nat] :
( ( ord_less_eq_nat @ X @ Y )
=> ( ( ~ ( ord_less_nat @ X @ Y ) )
= ( X = Y ) ) ) ).
% antisym_conv2
thf(fact_330_less__le__not__le,axiom,
( ord_less_int
= ( ^ [X3: int,Y7: int] :
( ( ord_less_eq_int @ X3 @ Y7 )
& ~ ( ord_less_eq_int @ Y7 @ X3 ) ) ) ) ).
% less_le_not_le
thf(fact_331_less__le__not__le,axiom,
( ord_less_nat
= ( ^ [X3: nat,Y7: nat] :
( ( ord_less_eq_nat @ X3 @ Y7 )
& ~ ( ord_less_eq_nat @ Y7 @ X3 ) ) ) ) ).
% less_le_not_le
thf(fact_332_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_333_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_334_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_335_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_336_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_337_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_338_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_339_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_340_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_341_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_342_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_343_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_344_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_345_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_346_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_347_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_348_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_349_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_350_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_351_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_352_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_353_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_354_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_355_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_356_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_357_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_358_order__le__less,axiom,
( ord_less_eq_int
= ( ^ [X3: int,Y7: int] :
( ( ord_less_int @ X3 @ Y7 )
| ( X3 = Y7 ) ) ) ) ).
% order_le_less
thf(fact_359_order__le__less,axiom,
( ord_less_eq_nat
= ( ^ [X3: nat,Y7: nat] :
( ( ord_less_nat @ X3 @ Y7 )
| ( X3 = Y7 ) ) ) ) ).
% order_le_less
thf(fact_360_order__less__le,axiom,
( ord_less_int
= ( ^ [X3: int,Y7: int] :
( ( ord_less_eq_int @ X3 @ Y7 )
& ( X3 != Y7 ) ) ) ) ).
% order_less_le
thf(fact_361_order__less__le,axiom,
( ord_less_nat
= ( ^ [X3: nat,Y7: nat] :
( ( ord_less_eq_nat @ X3 @ Y7 )
& ( X3 != Y7 ) ) ) ) ).
% order_less_le
thf(fact_362_linorder__not__le,axiom,
! [X: int,Y: int] :
( ( ~ ( ord_less_eq_int @ X @ Y ) )
= ( ord_less_int @ Y @ X ) ) ).
% linorder_not_le
thf(fact_363_linorder__not__le,axiom,
! [X: nat,Y: nat] :
( ( ~ ( ord_less_eq_nat @ X @ Y ) )
= ( ord_less_nat @ Y @ X ) ) ).
% linorder_not_le
thf(fact_364_linorder__not__less,axiom,
! [X: int,Y: int] :
( ( ~ ( ord_less_int @ X @ Y ) )
= ( ord_less_eq_int @ Y @ X ) ) ).
% linorder_not_less
thf(fact_365_linorder__not__less,axiom,
! [X: nat,Y: nat] :
( ( ~ ( ord_less_nat @ X @ Y ) )
= ( ord_less_eq_nat @ Y @ X ) ) ).
% linorder_not_less
thf(fact_366_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_367_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_368_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_369_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_370_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_371_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_372_order__le__less__trans,axiom,
! [X: int,Y: int,Z2: int] :
( ( ord_less_eq_int @ X @ Y )
=> ( ( ord_less_int @ Y @ Z2 )
=> ( ord_less_int @ X @ Z2 ) ) ) ).
% order_le_less_trans
thf(fact_373_order__le__less__trans,axiom,
! [X: nat,Y: nat,Z2: nat] :
( ( ord_less_eq_nat @ X @ Y )
=> ( ( ord_less_nat @ Y @ Z2 )
=> ( ord_less_nat @ X @ Z2 ) ) ) ).
% order_le_less_trans
thf(fact_374_order__less__le__trans,axiom,
! [X: int,Y: int,Z2: int] :
( ( ord_less_int @ X @ Y )
=> ( ( ord_less_eq_int @ Y @ Z2 )
=> ( ord_less_int @ X @ Z2 ) ) ) ).
% order_less_le_trans
thf(fact_375_order__less__le__trans,axiom,
! [X: nat,Y: nat,Z2: nat] :
( ( ord_less_nat @ X @ Y )
=> ( ( ord_less_eq_nat @ Y @ Z2 )
=> ( ord_less_nat @ X @ Z2 ) ) ) ).
% order_less_le_trans
thf(fact_376_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 )
=> ( ! [X5: int,Y3: int] :
( ( ord_less_int @ X5 @ Y3 )
=> ( ord_less_int @ ( F @ X5 ) @ ( F @ Y3 ) ) )
=> ( ord_less_int @ A @ ( F @ C ) ) ) ) ) ).
% order_le_less_subst1
thf(fact_377_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 )
=> ( ! [X5: nat,Y3: nat] :
( ( ord_less_nat @ X5 @ Y3 )
=> ( ord_less_int @ ( F @ X5 ) @ ( F @ Y3 ) ) )
=> ( ord_less_int @ A @ ( F @ C ) ) ) ) ) ).
% order_le_less_subst1
thf(fact_378_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 )
=> ( ! [X5: int,Y3: int] :
( ( ord_less_int @ X5 @ Y3 )
=> ( ord_less_nat @ ( F @ X5 ) @ ( F @ Y3 ) ) )
=> ( ord_less_nat @ A @ ( F @ C ) ) ) ) ) ).
% order_le_less_subst1
thf(fact_379_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 )
=> ( ! [X5: nat,Y3: nat] :
( ( ord_less_nat @ X5 @ Y3 )
=> ( ord_less_nat @ ( F @ X5 ) @ ( F @ Y3 ) ) )
=> ( ord_less_nat @ A @ ( F @ C ) ) ) ) ) ).
% order_le_less_subst1
thf(fact_380_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 )
=> ( ! [X5: int,Y3: int] :
( ( ord_less_eq_int @ X5 @ Y3 )
=> ( ord_less_eq_int @ ( F @ X5 ) @ ( F @ Y3 ) ) )
=> ( ord_less_int @ ( F @ A ) @ C ) ) ) ) ).
% order_le_less_subst2
thf(fact_381_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 )
=> ( ! [X5: int,Y3: int] :
( ( ord_less_eq_int @ X5 @ Y3 )
=> ( ord_less_eq_nat @ ( F @ X5 ) @ ( F @ Y3 ) ) )
=> ( ord_less_nat @ ( F @ A ) @ C ) ) ) ) ).
% order_le_less_subst2
thf(fact_382_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 )
=> ( ! [X5: nat,Y3: nat] :
( ( ord_less_eq_nat @ X5 @ Y3 )
=> ( ord_less_eq_int @ ( F @ X5 ) @ ( F @ Y3 ) ) )
=> ( ord_less_int @ ( F @ A ) @ C ) ) ) ) ).
% order_le_less_subst2
thf(fact_383_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 )
=> ( ! [X5: nat,Y3: nat] :
( ( ord_less_eq_nat @ X5 @ Y3 )
=> ( ord_less_eq_nat @ ( F @ X5 ) @ ( F @ Y3 ) ) )
=> ( ord_less_nat @ ( F @ A ) @ C ) ) ) ) ).
% order_le_less_subst2
thf(fact_384_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 )
=> ( ! [X5: int,Y3: int] :
( ( ord_less_eq_int @ X5 @ Y3 )
=> ( ord_less_eq_int @ ( F @ X5 ) @ ( F @ Y3 ) ) )
=> ( ord_less_int @ A @ ( F @ C ) ) ) ) ) ).
% order_less_le_subst1
thf(fact_385_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 )
=> ( ! [X5: int,Y3: int] :
( ( ord_less_eq_int @ X5 @ Y3 )
=> ( ord_less_eq_nat @ ( F @ X5 ) @ ( F @ Y3 ) ) )
=> ( ord_less_nat @ A @ ( F @ C ) ) ) ) ) ).
% order_less_le_subst1
thf(fact_386_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 )
=> ( ! [X5: nat,Y3: nat] :
( ( ord_less_eq_nat @ X5 @ Y3 )
=> ( ord_less_eq_int @ ( F @ X5 ) @ ( F @ Y3 ) ) )
=> ( ord_less_int @ A @ ( F @ C ) ) ) ) ) ).
% order_less_le_subst1
thf(fact_387_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 )
=> ( ! [X5: nat,Y3: nat] :
( ( ord_less_eq_nat @ X5 @ Y3 )
=> ( ord_less_eq_nat @ ( F @ X5 ) @ ( F @ Y3 ) ) )
=> ( ord_less_nat @ A @ ( F @ C ) ) ) ) ) ).
% order_less_le_subst1
thf(fact_388_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 )
=> ( ! [X5: int,Y3: int] :
( ( ord_less_int @ X5 @ Y3 )
=> ( ord_less_int @ ( F @ X5 ) @ ( F @ Y3 ) ) )
=> ( ord_less_int @ ( F @ A ) @ C ) ) ) ) ).
% order_less_le_subst2
thf(fact_389_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 )
=> ( ! [X5: nat,Y3: nat] :
( ( ord_less_nat @ X5 @ Y3 )
=> ( ord_less_int @ ( F @ X5 ) @ ( F @ Y3 ) ) )
=> ( ord_less_int @ ( F @ A ) @ C ) ) ) ) ).
% order_less_le_subst2
thf(fact_390_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 )
=> ( ! [X5: int,Y3: int] :
( ( ord_less_int @ X5 @ Y3 )
=> ( ord_less_nat @ ( F @ X5 ) @ ( F @ Y3 ) ) )
=> ( ord_less_nat @ ( F @ A ) @ C ) ) ) ) ).
% order_less_le_subst2
thf(fact_391_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 )
=> ( ! [X5: nat,Y3: nat] :
( ( ord_less_nat @ X5 @ Y3 )
=> ( ord_less_nat @ ( F @ X5 ) @ ( F @ Y3 ) ) )
=> ( ord_less_nat @ ( F @ A ) @ C ) ) ) ) ).
% order_less_le_subst2
thf(fact_392_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_393_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_394_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_395_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_396_size__char__eq__0,axiom,
( size_size_char
= ( ^ [C2: char] : zero_zero_nat ) ) ).
% size_char_eq_0
thf(fact_397_less__numeral__extra_I3_J,axiom,
~ ( ord_less_int @ zero_zero_int @ zero_zero_int ) ).
% less_numeral_extra(3)
thf(fact_398_less__numeral__extra_I3_J,axiom,
~ ( ord_less_nat @ zero_zero_nat @ zero_zero_nat ) ).
% less_numeral_extra(3)
thf(fact_399_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 )
& ! [X7: int] :
( ( ( ord_less_eq_int @ A @ X7 )
& ( ord_less_int @ X7 @ C3 ) )
=> ( P @ X7 ) )
& ! [D: int] :
( ! [X5: int] :
( ( ( ord_less_eq_int @ A @ X5 )
& ( ord_less_int @ X5 @ D ) )
=> ( P @ X5 ) )
=> ( ord_less_eq_int @ D @ C3 ) ) ) ) ) ) ).
% complete_interval
thf(fact_400_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 )
& ! [X7: nat] :
( ( ( ord_less_eq_nat @ A @ X7 )
& ( ord_less_nat @ X7 @ C3 ) )
=> ( P @ X7 ) )
& ! [D: nat] :
( ! [X5: nat] :
( ( ( ord_less_eq_nat @ A @ X5 )
& ( ord_less_nat @ X5 @ D ) )
=> ( P @ X5 ) )
=> ( ord_less_eq_nat @ D @ C3 ) ) ) ) ) ) ).
% complete_interval
thf(fact_401_le__numeral__extra_I3_J,axiom,
ord_less_eq_int @ zero_zero_int @ zero_zero_int ).
% le_numeral_extra(3)
thf(fact_402_le__numeral__extra_I3_J,axiom,
ord_less_eq_nat @ zero_zero_nat @ zero_zero_nat ).
% le_numeral_extra(3)
thf(fact_403_length__code,axiom,
( size_size_list_instr
= ( gen_length_instr @ zero_zero_nat ) ) ).
% length_code
thf(fact_404_length__code,axiom,
( size_s356637359517785349t_char
= ( gen_length_list_char @ zero_zero_nat ) ) ).
% length_code
thf(fact_405_neg__int__cases,axiom,
! [K: int] :
( ( ord_less_int @ K @ zero_zero_int )
=> ~ ! [N2: nat] :
( ( K
= ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ N2 ) ) )
=> ~ ( ord_less_nat @ zero_zero_nat @ N2 ) ) ) ).
% neg_int_cases
thf(fact_406_instr_Osize__gen_I4_J,axiom,
! [X4: list_char] :
( ( size_instr @ ( store @ X4 ) )
= zero_zero_nat ) ).
% instr.size_gen(4)
thf(fact_407_instr_Osize__gen_I2_J,axiom,
! [X2: list_char] :
( ( size_instr @ ( load @ X2 ) )
= zero_zero_nat ) ).
% instr.size_gen(2)
thf(fact_408_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_409_verit__minus__simplify_I4_J,axiom,
! [B: int] :
( ( uminus_uminus_int @ ( uminus_uminus_int @ B ) )
= B ) ).
% verit_minus_simplify(4)
thf(fact_410_neg__equal__iff__equal,axiom,
! [A: int,B: int] :
( ( ( uminus_uminus_int @ A )
= ( uminus_uminus_int @ B ) )
= ( A = B ) ) ).
% neg_equal_iff_equal
thf(fact_411_add_Oinverse__inverse,axiom,
! [A: int] :
( ( uminus_uminus_int @ ( uminus_uminus_int @ A ) )
= A ) ).
% add.inverse_inverse
thf(fact_412_neg__le__iff__le,axiom,
! [B: int,A: int] :
( ( ord_less_eq_int @ ( uminus_uminus_int @ B ) @ ( uminus_uminus_int @ A ) )
= ( ord_less_eq_int @ A @ B ) ) ).
% neg_le_iff_le
thf(fact_413_neg__equal__zero,axiom,
! [A: int] :
( ( ( uminus_uminus_int @ A )
= A )
= ( A = zero_zero_int ) ) ).
% neg_equal_zero
thf(fact_414_equal__neg__zero,axiom,
! [A: int] :
( ( A
= ( uminus_uminus_int @ A ) )
= ( A = zero_zero_int ) ) ).
% equal_neg_zero
thf(fact_415_neg__equal__0__iff__equal,axiom,
! [A: int] :
( ( ( uminus_uminus_int @ A )
= zero_zero_int )
= ( A = zero_zero_int ) ) ).
% neg_equal_0_iff_equal
thf(fact_416_neg__0__equal__iff__equal,axiom,
! [A: int] :
( ( zero_zero_int
= ( uminus_uminus_int @ A ) )
= ( zero_zero_int = A ) ) ).
% neg_0_equal_iff_equal
thf(fact_417_add_Oinverse__neutral,axiom,
( ( uminus_uminus_int @ zero_zero_int )
= zero_zero_int ) ).
% add.inverse_neutral
thf(fact_418_neg__less__iff__less,axiom,
! [B: int,A: int] :
( ( ord_less_int @ ( uminus_uminus_int @ B ) @ ( uminus_uminus_int @ A ) )
= ( ord_less_int @ A @ B ) ) ).
% neg_less_iff_less
thf(fact_419_nat__int,axiom,
! [N: nat] :
( ( nat2 @ ( semiri1314217659103216013at_int @ N ) )
= N ) ).
% nat_int
thf(fact_420_neg__less__eq__nonneg,axiom,
! [A: int] :
( ( ord_less_eq_int @ ( uminus_uminus_int @ A ) @ A )
= ( ord_less_eq_int @ zero_zero_int @ A ) ) ).
% neg_less_eq_nonneg
thf(fact_421_less__eq__neg__nonpos,axiom,
! [A: int] :
( ( ord_less_eq_int @ A @ ( uminus_uminus_int @ A ) )
= ( ord_less_eq_int @ A @ zero_zero_int ) ) ).
% less_eq_neg_nonpos
thf(fact_422_neg__le__0__iff__le,axiom,
! [A: int] :
( ( ord_less_eq_int @ ( uminus_uminus_int @ A ) @ zero_zero_int )
= ( ord_less_eq_int @ zero_zero_int @ A ) ) ).
% neg_le_0_iff_le
thf(fact_423_neg__0__le__iff__le,axiom,
! [A: int] :
( ( ord_less_eq_int @ zero_zero_int @ ( uminus_uminus_int @ A ) )
= ( ord_less_eq_int @ A @ zero_zero_int ) ) ).
% neg_0_le_iff_le
thf(fact_424_neg__less__0__iff__less,axiom,
! [A: int] :
( ( ord_less_int @ ( uminus_uminus_int @ A ) @ zero_zero_int )
= ( ord_less_int @ zero_zero_int @ A ) ) ).
% neg_less_0_iff_less
thf(fact_425_neg__0__less__iff__less,axiom,
! [A: int] :
( ( ord_less_int @ zero_zero_int @ ( uminus_uminus_int @ A ) )
= ( ord_less_int @ A @ zero_zero_int ) ) ).
% neg_0_less_iff_less
thf(fact_426_neg__less__pos,axiom,
! [A: int] :
( ( ord_less_int @ ( uminus_uminus_int @ A ) @ A )
= ( ord_less_int @ zero_zero_int @ A ) ) ).
% neg_less_pos
thf(fact_427_less__neg__neg,axiom,
! [A: int] :
( ( ord_less_int @ A @ ( uminus_uminus_int @ A ) )
= ( ord_less_int @ A @ zero_zero_int ) ) ).
% less_neg_neg
thf(fact_428_negative__eq__positive,axiom,
! [N: nat,M: nat] :
( ( ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ N ) )
= ( semiri1314217659103216013at_int @ M ) )
= ( ( N = zero_zero_nat )
& ( M = zero_zero_nat ) ) ) ).
% negative_eq_positive
thf(fact_429_negative__zle,axiom,
! [N: nat,M: nat] : ( ord_less_eq_int @ ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ N ) ) @ ( semiri1314217659103216013at_int @ M ) ) ).
% negative_zle
thf(fact_430_nat__0__iff,axiom,
! [I: int] :
( ( ( nat2 @ I )
= zero_zero_nat )
= ( ord_less_eq_int @ I @ zero_zero_int ) ) ).
% nat_0_iff
thf(fact_431_nat__le__0,axiom,
! [Z2: int] :
( ( ord_less_eq_int @ Z2 @ zero_zero_int )
=> ( ( nat2 @ Z2 )
= zero_zero_nat ) ) ).
% nat_le_0
thf(fact_432_zless__nat__conj,axiom,
! [W: int,Z2: int] :
( ( ord_less_nat @ ( nat2 @ W ) @ ( nat2 @ Z2 ) )
= ( ( ord_less_int @ zero_zero_int @ Z2 )
& ( ord_less_int @ W @ Z2 ) ) ) ).
% zless_nat_conj
thf(fact_433_nat__zminus__int,axiom,
! [N: nat] :
( ( nat2 @ ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ N ) ) )
= zero_zero_nat ) ).
% nat_zminus_int
thf(fact_434_int__nat__eq,axiom,
! [Z2: int] :
( ( ( ord_less_eq_int @ zero_zero_int @ Z2 )
=> ( ( semiri1314217659103216013at_int @ ( nat2 @ Z2 ) )
= Z2 ) )
& ( ~ ( ord_less_eq_int @ zero_zero_int @ Z2 )
=> ( ( semiri1314217659103216013at_int @ ( nat2 @ Z2 ) )
= zero_zero_int ) ) ) ).
% int_nat_eq
thf(fact_435_zero__less__nat__eq,axiom,
! [Z2: int] :
( ( ord_less_nat @ zero_zero_nat @ ( nat2 @ Z2 ) )
= ( ord_less_int @ zero_zero_int @ Z2 ) ) ).
% zero_less_nat_eq
thf(fact_436_verit__negate__coefficient_I3_J,axiom,
! [A: int,B: int] :
( ( A = B )
=> ( ( uminus_uminus_int @ A )
= ( uminus_uminus_int @ B ) ) ) ).
% verit_negate_coefficient(3)
thf(fact_437_minus__equation__iff,axiom,
! [A: int,B: int] :
( ( ( uminus_uminus_int @ A )
= B )
= ( ( uminus_uminus_int @ B )
= A ) ) ).
% minus_equation_iff
thf(fact_438_equation__minus__iff,axiom,
! [A: int,B: int] :
( ( A
= ( uminus_uminus_int @ B ) )
= ( B
= ( uminus_uminus_int @ A ) ) ) ).
% equation_minus_iff
thf(fact_439_le__imp__neg__le,axiom,
! [A: int,B: int] :
( ( ord_less_eq_int @ A @ B )
=> ( ord_less_eq_int @ ( uminus_uminus_int @ B ) @ ( uminus_uminus_int @ A ) ) ) ).
% le_imp_neg_le
thf(fact_440_minus__le__iff,axiom,
! [A: int,B: int] :
( ( ord_less_eq_int @ ( uminus_uminus_int @ A ) @ B )
= ( ord_less_eq_int @ ( uminus_uminus_int @ B ) @ A ) ) ).
% minus_le_iff
thf(fact_441_le__minus__iff,axiom,
! [A: int,B: int] :
( ( ord_less_eq_int @ A @ ( uminus_uminus_int @ B ) )
= ( ord_less_eq_int @ B @ ( uminus_uminus_int @ A ) ) ) ).
% le_minus_iff
thf(fact_442_minus__less__iff,axiom,
! [A: int,B: int] :
( ( ord_less_int @ ( uminus_uminus_int @ A ) @ B )
= ( ord_less_int @ ( uminus_uminus_int @ B ) @ A ) ) ).
% minus_less_iff
thf(fact_443_less__minus__iff,axiom,
! [A: int,B: int] :
( ( ord_less_int @ A @ ( uminus_uminus_int @ B ) )
= ( ord_less_int @ B @ ( uminus_uminus_int @ A ) ) ) ).
% less_minus_iff
thf(fact_444_verit__negate__coefficient_I2_J,axiom,
! [A: int,B: int] :
( ( ord_less_int @ A @ B )
=> ( ord_less_int @ ( uminus_uminus_int @ B ) @ ( uminus_uminus_int @ A ) ) ) ).
% verit_negate_coefficient(2)
thf(fact_445_uminus__int__code_I1_J,axiom,
( ( uminus_uminus_int @ zero_zero_int )
= zero_zero_int ) ).
% uminus_int_code(1)
thf(fact_446_int__cases2,axiom,
! [Z2: int] :
( ! [N2: nat] :
( Z2
!= ( semiri1314217659103216013at_int @ N2 ) )
=> ~ ! [N2: nat] :
( Z2
!= ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ N2 ) ) ) ) ).
% int_cases2
thf(fact_447_nat__zero__as__int,axiom,
( zero_zero_nat
= ( nat2 @ zero_zero_int ) ) ).
% nat_zero_as_int
thf(fact_448_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_449_eq__nat__nat__iff,axiom,
! [Z2: int,Z5: int] :
( ( ord_less_eq_int @ zero_zero_int @ Z2 )
=> ( ( ord_less_eq_int @ zero_zero_int @ Z5 )
=> ( ( ( nat2 @ Z2 )
= ( nat2 @ Z5 ) )
= ( Z2 = Z5 ) ) ) ) ).
% eq_nat_nat_iff
thf(fact_450_all__nat,axiom,
( ( ^ [P3: nat > $o] :
! [X8: nat] : ( P3 @ X8 ) )
= ( ^ [P4: nat > $o] :
! [X3: int] :
( ( ord_less_eq_int @ zero_zero_int @ X3 )
=> ( P4 @ ( nat2 @ X3 ) ) ) ) ) ).
% all_nat
thf(fact_451_ex__nat,axiom,
( ( ^ [P3: nat > $o] :
? [X8: nat] : ( P3 @ X8 ) )
= ( ^ [P4: nat > $o] :
? [X3: int] :
( ( ord_less_eq_int @ zero_zero_int @ X3 )
& ( P4 @ ( nat2 @ X3 ) ) ) ) ) ).
% ex_nat
thf(fact_452_not__int__zless__negative,axiom,
! [N: nat,M: nat] :
~ ( ord_less_int @ ( semiri1314217659103216013at_int @ N ) @ ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ M ) ) ) ).
% not_int_zless_negative
thf(fact_453_nat__mono__iff,axiom,
! [Z2: int,W: int] :
( ( ord_less_int @ zero_zero_int @ Z2 )
=> ( ( ord_less_nat @ ( nat2 @ W ) @ ( nat2 @ Z2 ) )
= ( ord_less_int @ W @ Z2 ) ) ) ).
% nat_mono_iff
thf(fact_454_zless__nat__eq__int__zless,axiom,
! [M: nat,Z2: int] :
( ( ord_less_nat @ M @ ( nat2 @ Z2 ) )
= ( ord_less_int @ ( semiri1314217659103216013at_int @ M ) @ Z2 ) ) ).
% zless_nat_eq_int_zless
thf(fact_455_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_456_int__eq__iff,axiom,
! [M: nat,Z2: int] :
( ( ( semiri1314217659103216013at_int @ M )
= Z2 )
= ( ( M
= ( nat2 @ Z2 ) )
& ( ord_less_eq_int @ zero_zero_int @ Z2 ) ) ) ).
% int_eq_iff
thf(fact_457_nat__0__le,axiom,
! [Z2: int] :
( ( ord_less_eq_int @ zero_zero_int @ Z2 )
=> ( ( semiri1314217659103216013at_int @ ( nat2 @ Z2 ) )
= Z2 ) ) ).
% nat_0_le
thf(fact_458_int__cases4,axiom,
! [M: int] :
( ! [N2: nat] :
( M
!= ( semiri1314217659103216013at_int @ N2 ) )
=> ~ ! [N2: nat] :
( ( ord_less_nat @ zero_zero_nat @ N2 )
=> ( M
!= ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ N2 ) ) ) ) ) ).
% int_cases4
thf(fact_459_int__zle__neg,axiom,
! [N: nat,M: nat] :
( ( ord_less_eq_int @ ( semiri1314217659103216013at_int @ N ) @ ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ M ) ) )
= ( ( N = zero_zero_nat )
& ( M = zero_zero_nat ) ) ) ).
% int_zle_neg
thf(fact_460_nonpos__int__cases,axiom,
! [K: int] :
( ( ord_less_eq_int @ K @ zero_zero_int )
=> ~ ! [N2: nat] :
( K
!= ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ N2 ) ) ) ) ).
% nonpos_int_cases
thf(fact_461_negative__zle__0,axiom,
! [N: nat] : ( ord_less_eq_int @ ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ N ) ) @ zero_zero_int ) ).
% negative_zle_0
thf(fact_462_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_463_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_464_nat__less__eq__zless,axiom,
! [W: int,Z2: int] :
( ( ord_less_eq_int @ zero_zero_int @ W )
=> ( ( ord_less_nat @ ( nat2 @ W ) @ ( nat2 @ Z2 ) )
= ( ord_less_int @ W @ Z2 ) ) ) ).
% nat_less_eq_zless
thf(fact_465_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_466_nat__le__eq__zle,axiom,
! [W: int,Z2: int] :
( ( ( ord_less_int @ zero_zero_int @ W )
| ( ord_less_eq_int @ zero_zero_int @ Z2 ) )
=> ( ( ord_less_eq_nat @ ( nat2 @ W ) @ ( nat2 @ Z2 ) )
= ( ord_less_eq_int @ W @ Z2 ) ) ) ).
% nat_le_eq_zle
thf(fact_467_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_468_int__cases3,axiom,
! [K: int] :
( ( K != zero_zero_int )
=> ( ! [N2: nat] :
( ( K
= ( semiri1314217659103216013at_int @ N2 ) )
=> ~ ( ord_less_nat @ zero_zero_nat @ N2 ) )
=> ~ ! [N2: nat] :
( ( K
= ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ N2 ) ) )
=> ~ ( ord_less_nat @ zero_zero_nat @ N2 ) ) ) ) ).
% int_cases3
thf(fact_469_size_H__char__eq__0,axiom,
( size_char
= ( ^ [C2: char] : zero_zero_nat ) ) ).
% size'_char_eq_0
thf(fact_470_subsetI,axiom,
! [A2: set_list_char,B4: set_list_char] :
( ! [X5: list_char] :
( ( member_list_char @ X5 @ A2 )
=> ( member_list_char @ X5 @ B4 ) )
=> ( ord_le1207710660481778807t_char @ A2 @ B4 ) ) ).
% subsetI
thf(fact_471_subsetI,axiom,
! [A2: set_instr,B4: set_instr] :
( ! [X5: instr] :
( ( member_instr @ X5 @ A2 )
=> ( member_instr @ X5 @ B4 ) )
=> ( ord_le4956778034350794079_instr @ A2 @ B4 ) ) ).
% subsetI
thf(fact_472_psubsetD,axiom,
! [A2: set_list_char,B4: set_list_char,C: list_char] :
( ( ord_le5286566838938544235t_char @ A2 @ B4 )
=> ( ( member_list_char @ C @ A2 )
=> ( member_list_char @ C @ B4 ) ) ) ).
% psubsetD
thf(fact_473_psubsetD,axiom,
! [A2: set_instr,B4: set_instr,C: instr] :
( ( ord_less_set_instr @ A2 @ B4 )
=> ( ( member_instr @ C @ A2 )
=> ( member_instr @ C @ B4 ) ) ) ).
% psubsetD
thf(fact_474_in__mono,axiom,
! [A2: set_list_char,B4: set_list_char,X: list_char] :
( ( ord_le1207710660481778807t_char @ A2 @ B4 )
=> ( ( member_list_char @ X @ A2 )
=> ( member_list_char @ X @ B4 ) ) ) ).
% in_mono
thf(fact_475_in__mono,axiom,
! [A2: set_instr,B4: set_instr,X: instr] :
( ( ord_le4956778034350794079_instr @ A2 @ B4 )
=> ( ( member_instr @ X @ A2 )
=> ( member_instr @ X @ B4 ) ) ) ).
% in_mono
thf(fact_476_subsetD,axiom,
! [A2: set_list_char,B4: set_list_char,C: list_char] :
( ( ord_le1207710660481778807t_char @ A2 @ B4 )
=> ( ( member_list_char @ C @ A2 )
=> ( member_list_char @ C @ B4 ) ) ) ).
% subsetD
thf(fact_477_subsetD,axiom,
! [A2: set_instr,B4: set_instr,C: instr] :
( ( ord_le4956778034350794079_instr @ A2 @ B4 )
=> ( ( member_instr @ C @ A2 )
=> ( member_instr @ C @ B4 ) ) ) ).
% subsetD
thf(fact_478_subset__eq,axiom,
( ord_le1207710660481778807t_char
= ( ^ [A6: set_list_char,B6: set_list_char] :
! [X3: list_char] :
( ( member_list_char @ X3 @ A6 )
=> ( member_list_char @ X3 @ B6 ) ) ) ) ).
% subset_eq
thf(fact_479_subset__eq,axiom,
( ord_le4956778034350794079_instr
= ( ^ [A6: set_instr,B6: set_instr] :
! [X3: instr] :
( ( member_instr @ X3 @ A6 )
=> ( member_instr @ X3 @ B6 ) ) ) ) ).
% subset_eq
thf(fact_480_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_481_subset__iff,axiom,
( ord_le4956778034350794079_instr
= ( ^ [A6: set_instr,B6: set_instr] :
! [T2: instr] :
( ( member_instr @ T2 @ A6 )
=> ( member_instr @ T2 @ B6 ) ) ) ) ).
% subset_iff
thf(fact_482_of__int__of__nat,axiom,
( ring_1_of_int_int
= ( ^ [K3: int] : ( if_int @ ( ord_less_int @ K3 @ zero_zero_int ) @ ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ ( nat2 @ ( uminus_uminus_int @ K3 ) ) ) ) @ ( semiri1314217659103216013at_int @ ( nat2 @ K3 ) ) ) ) ) ).
% of_int_of_nat
thf(fact_483_length__n__lists__elem,axiom,
! [Ys2: list_char,N: nat,Xs: list_char] :
( ( member_list_char @ Ys2 @ ( set_list_char2 @ ( n_lists_char @ N @ Xs ) ) )
=> ( ( size_size_list_char @ Ys2 )
= N ) ) ).
% length_n_lists_elem
thf(fact_484_length__n__lists__elem,axiom,
! [Ys2: list_instr,N: nat,Xs: list_instr] :
( ( member_list_instr @ Ys2 @ ( set_list_instr2 @ ( n_lists_instr @ N @ Xs ) ) )
=> ( ( size_size_list_instr @ Ys2 )
= N ) ) ).
% length_n_lists_elem
thf(fact_485_length__n__lists__elem,axiom,
! [Ys2: list_list_char,N: nat,Xs: list_list_char] :
( ( member4762036990631661320t_char @ Ys2 @ ( set_list_list_char2 @ ( n_lists_list_char @ N @ Xs ) ) )
=> ( ( size_s356637359517785349t_char @ Ys2 )
= N ) ) ).
% length_n_lists_elem
thf(fact_486_of__nat__nat,axiom,
! [Z2: int] :
( ( ord_less_eq_int @ zero_zero_int @ Z2 )
=> ( ( semiri1314217659103216013at_int @ ( nat2 @ Z2 ) )
= ( ring_1_of_int_int @ Z2 ) ) ) ).
% of_nat_nat
thf(fact_487_ComplI,axiom,
! [C: list_char,A2: set_list_char] :
( ~ ( member_list_char @ C @ A2 )
=> ( member_list_char @ C @ ( uminus5517088710498760928t_char @ A2 ) ) ) ).
% ComplI
thf(fact_488_ComplI,axiom,
! [C: instr,A2: set_instr] :
( ~ ( member_instr @ C @ A2 )
=> ( member_instr @ C @ ( uminus1347503074550245174_instr @ A2 ) ) ) ).
% ComplI
thf(fact_489_Compl__iff,axiom,
! [C: list_char,A2: set_list_char] :
( ( member_list_char @ C @ ( uminus5517088710498760928t_char @ A2 ) )
= ( ~ ( member_list_char @ C @ A2 ) ) ) ).
% Compl_iff
thf(fact_490_Compl__iff,axiom,
! [C: instr,A2: set_instr] :
( ( member_instr @ C @ ( uminus1347503074550245174_instr @ A2 ) )
= ( ~ ( member_instr @ C @ A2 ) ) ) ).
% Compl_iff
thf(fact_491_of__int__eq__0__iff,axiom,
! [Z2: int] :
( ( ( ring_1_of_int_int @ Z2 )
= zero_zero_int )
= ( Z2 = zero_zero_int ) ) ).
% of_int_eq_0_iff
thf(fact_492_of__int__0__eq__iff,axiom,
! [Z2: int] :
( ( zero_zero_int
= ( ring_1_of_int_int @ Z2 ) )
= ( Z2 = zero_zero_int ) ) ).
% of_int_0_eq_iff
thf(fact_493_of__int__0,axiom,
( ( ring_1_of_int_int @ zero_zero_int )
= zero_zero_int ) ).
% of_int_0
thf(fact_494_of__int__le__iff,axiom,
! [W: int,Z2: int] :
( ( ord_less_eq_int @ ( ring_1_of_int_int @ W ) @ ( ring_1_of_int_int @ Z2 ) )
= ( ord_less_eq_int @ W @ Z2 ) ) ).
% of_int_le_iff
thf(fact_495_of__int__less__iff,axiom,
! [W: int,Z2: int] :
( ( ord_less_int @ ( ring_1_of_int_int @ W ) @ ( ring_1_of_int_int @ Z2 ) )
= ( ord_less_int @ W @ Z2 ) ) ).
% of_int_less_iff
thf(fact_496_of__int__minus,axiom,
! [Z2: int] :
( ( ring_1_of_int_int @ ( uminus_uminus_int @ Z2 ) )
= ( uminus_uminus_int @ ( ring_1_of_int_int @ Z2 ) ) ) ).
% of_int_minus
thf(fact_497_of__int__of__nat__eq,axiom,
! [N: nat] :
( ( ring_1_of_int_int @ ( semiri1314217659103216013at_int @ N ) )
= ( semiri1314217659103216013at_int @ N ) ) ).
% of_int_of_nat_eq
thf(fact_498_of__int__le__0__iff,axiom,
! [Z2: int] :
( ( ord_less_eq_int @ ( ring_1_of_int_int @ Z2 ) @ zero_zero_int )
= ( ord_less_eq_int @ Z2 @ zero_zero_int ) ) ).
% of_int_le_0_iff
thf(fact_499_of__int__0__le__iff,axiom,
! [Z2: int] :
( ( ord_less_eq_int @ zero_zero_int @ ( ring_1_of_int_int @ Z2 ) )
= ( ord_less_eq_int @ zero_zero_int @ Z2 ) ) ).
% of_int_0_le_iff
thf(fact_500_of__int__less__0__iff,axiom,
! [Z2: int] :
( ( ord_less_int @ ( ring_1_of_int_int @ Z2 ) @ zero_zero_int )
= ( ord_less_int @ Z2 @ zero_zero_int ) ) ).
% of_int_less_0_iff
thf(fact_501_of__int__0__less__iff,axiom,
! [Z2: int] :
( ( ord_less_int @ zero_zero_int @ ( ring_1_of_int_int @ Z2 ) )
= ( ord_less_int @ zero_zero_int @ Z2 ) ) ).
% of_int_0_less_iff
thf(fact_502_ComplD,axiom,
! [C: list_char,A2: set_list_char] :
( ( member_list_char @ C @ ( uminus5517088710498760928t_char @ A2 ) )
=> ~ ( member_list_char @ C @ A2 ) ) ).
% ComplD
thf(fact_503_ComplD,axiom,
! [C: instr,A2: set_instr] :
( ( member_instr @ C @ ( uminus1347503074550245174_instr @ A2 ) )
=> ~ ( member_instr @ C @ A2 ) ) ).
% ComplD
thf(fact_504_of__int__nonneg,axiom,
! [Z2: int] :
( ( ord_less_eq_int @ zero_zero_int @ Z2 )
=> ( ord_less_eq_int @ zero_zero_int @ ( ring_1_of_int_int @ Z2 ) ) ) ).
% of_int_nonneg
thf(fact_505_of__int__pos,axiom,
! [Z2: int] :
( ( ord_less_int @ zero_zero_int @ Z2 )
=> ( ord_less_int @ zero_zero_int @ ( ring_1_of_int_int @ Z2 ) ) ) ).
% of_int_pos
thf(fact_506_of__nat__less__of__int__iff,axiom,
! [N: nat,X: int] :
( ( ord_less_int @ ( semiri1314217659103216013at_int @ N ) @ ( ring_1_of_int_int @ X ) )
= ( ord_less_int @ ( semiri1314217659103216013at_int @ N ) @ X ) ) ).
% of_nat_less_of_int_iff
thf(fact_507_instr_Osize__gen_I3_J,axiom,
( ( size_instr @ add )
= zero_zero_nat ) ).
% instr.size_gen(3)
thf(fact_508_instr_Osize_I10_J,axiom,
( ( size_size_instr @ add )
= zero_zero_nat ) ).
% instr.size(10)
thf(fact_509_of__nat__1,axiom,
( ( semiri1314217659103216013at_int @ one_one_nat )
= one_one_int ) ).
% of_nat_1
thf(fact_510_of__nat__1,axiom,
( ( semiri1316708129612266289at_nat @ one_one_nat )
= one_one_nat ) ).
% of_nat_1
thf(fact_511_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_512_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_513_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_514_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_515_less__one,axiom,
! [N: nat] :
( ( ord_less_nat @ N @ one_one_nat )
= ( N = zero_zero_nat ) ) ).
% less_one
thf(fact_516_of__int__eq__1__iff,axiom,
! [Z2: int] :
( ( ( ring_1_of_int_int @ Z2 )
= one_one_int )
= ( Z2 = one_one_int ) ) ).
% of_int_eq_1_iff
thf(fact_517_of__int__1,axiom,
( ( ring_1_of_int_int @ one_one_int )
= one_one_int ) ).
% of_int_1
thf(fact_518_of__int__le__1__iff,axiom,
! [Z2: int] :
( ( ord_less_eq_int @ ( ring_1_of_int_int @ Z2 ) @ one_one_int )
= ( ord_less_eq_int @ Z2 @ one_one_int ) ) ).
% of_int_le_1_iff
thf(fact_519_of__int__1__le__iff,axiom,
! [Z2: int] :
( ( ord_less_eq_int @ one_one_int @ ( ring_1_of_int_int @ Z2 ) )
= ( ord_less_eq_int @ one_one_int @ Z2 ) ) ).
% of_int_1_le_iff
thf(fact_520_of__int__1__less__iff,axiom,
! [Z2: int] :
( ( ord_less_int @ one_one_int @ ( ring_1_of_int_int @ Z2 ) )
= ( ord_less_int @ one_one_int @ Z2 ) ) ).
% of_int_1_less_iff
thf(fact_521_of__int__less__1__iff,axiom,
! [Z2: int] :
( ( ord_less_int @ ( ring_1_of_int_int @ Z2 ) @ one_one_int )
= ( ord_less_int @ Z2 @ one_one_int ) ) ).
% of_int_less_1_iff
thf(fact_522_nat__one__as__int,axiom,
( one_one_nat
= ( nat2 @ one_one_int ) ) ).
% nat_one_as_int
thf(fact_523_int__ops_I2_J,axiom,
( ( semiri1314217659103216013at_int @ one_one_nat )
= one_one_int ) ).
% int_ops(2)
thf(fact_524_one__reorient,axiom,
! [X: nat] :
( ( one_one_nat = X )
= ( X = one_one_nat ) ) ).
% one_reorient
thf(fact_525_one__reorient,axiom,
! [X: int] :
( ( one_one_int = X )
= ( X = one_one_int ) ) ).
% one_reorient
thf(fact_526_le__numeral__extra_I4_J,axiom,
ord_less_eq_int @ one_one_int @ one_one_int ).
% le_numeral_extra(4)
thf(fact_527_le__numeral__extra_I4_J,axiom,
ord_less_eq_nat @ one_one_nat @ one_one_nat ).
% le_numeral_extra(4)
thf(fact_528_less__numeral__extra_I4_J,axiom,
~ ( ord_less_int @ one_one_int @ one_one_int ) ).
% less_numeral_extra(4)
thf(fact_529_less__numeral__extra_I4_J,axiom,
~ ( ord_less_nat @ one_one_nat @ one_one_nat ) ).
% less_numeral_extra(4)
thf(fact_530_one__neq__neg__one,axiom,
( one_one_int
!= ( uminus_uminus_int @ one_one_int ) ) ).
% one_neq_neg_one
thf(fact_531_less__numeral__extra_I1_J,axiom,
ord_less_int @ zero_zero_int @ one_one_int ).
% less_numeral_extra(1)
thf(fact_532_less__numeral__extra_I1_J,axiom,
ord_less_nat @ zero_zero_nat @ one_one_nat ).
% less_numeral_extra(1)
thf(fact_533_le__minus__one__simps_I4_J,axiom,
~ ( ord_less_eq_int @ one_one_int @ ( uminus_uminus_int @ one_one_int ) ) ).
% le_minus_one_simps(4)
thf(fact_534_le__minus__one__simps_I2_J,axiom,
ord_less_eq_int @ ( uminus_uminus_int @ one_one_int ) @ one_one_int ).
% le_minus_one_simps(2)
thf(fact_535_zero__neq__neg__one,axiom,
( zero_zero_int
!= ( uminus_uminus_int @ one_one_int ) ) ).
% zero_neq_neg_one
thf(fact_536_less__minus__one__simps_I4_J,axiom,
~ ( ord_less_int @ one_one_int @ ( uminus_uminus_int @ one_one_int ) ) ).
% less_minus_one_simps(4)
thf(fact_537_less__minus__one__simps_I2_J,axiom,
ord_less_int @ ( uminus_uminus_int @ one_one_int ) @ one_one_int ).
% less_minus_one_simps(2)
thf(fact_538_instr_Odistinct_I13_J,axiom,
! [X2: list_char] :
( ( load @ X2 )
!= add ) ).
% instr.distinct(13)
thf(fact_539_instr_Odistinct_I23_J,axiom,
! [X4: list_char] :
( add
!= ( store @ X4 ) ) ).
% instr.distinct(23)
thf(fact_540_le__minus__one__simps_I3_J,axiom,
~ ( ord_less_eq_int @ zero_zero_int @ ( uminus_uminus_int @ one_one_int ) ) ).
% le_minus_one_simps(3)
thf(fact_541_le__minus__one__simps_I1_J,axiom,
ord_less_eq_int @ ( uminus_uminus_int @ one_one_int ) @ zero_zero_int ).
% le_minus_one_simps(1)
thf(fact_542_less__minus__one__simps_I3_J,axiom,
~ ( ord_less_int @ zero_zero_int @ ( uminus_uminus_int @ one_one_int ) ) ).
% less_minus_one_simps(3)
thf(fact_543_less__minus__one__simps_I1_J,axiom,
ord_less_int @ ( uminus_uminus_int @ one_one_int ) @ zero_zero_int ).
% less_minus_one_simps(1)
thf(fact_544_int__one__le__iff__zero__less,axiom,
! [Z2: int] :
( ( ord_less_eq_int @ one_one_int @ Z2 )
= ( ord_less_int @ zero_zero_int @ Z2 ) ) ).
% int_one_le_iff_zero_less
thf(fact_545_zero__less__one,axiom,
ord_less_int @ zero_zero_int @ one_one_int ).
% zero_less_one
thf(fact_546_zero__less__one,axiom,
ord_less_nat @ zero_zero_nat @ one_one_nat ).
% zero_less_one
thf(fact_547_not__one__less__zero,axiom,
~ ( ord_less_int @ one_one_int @ zero_zero_int ) ).
% not_one_less_zero
thf(fact_548_not__one__less__zero,axiom,
~ ( ord_less_nat @ one_one_nat @ zero_zero_nat ) ).
% not_one_less_zero
thf(fact_549_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_550_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_551_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_552_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_553_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_554_zero__neq__one,axiom,
zero_zero_int != one_one_int ).
% zero_neq_one
thf(fact_555_zero__neq__one,axiom,
zero_zero_nat != one_one_nat ).
% zero_neq_one
thf(fact_556_not__one__le__zero,axiom,
~ ( ord_less_eq_int @ one_one_int @ zero_zero_int ) ).
% not_one_le_zero
thf(fact_557_not__one__le__zero,axiom,
~ ( ord_less_eq_nat @ one_one_nat @ zero_zero_nat ) ).
% not_one_le_zero
thf(fact_558_dbl__dec__simps_I2_J,axiom,
( ( neg_nu3811975205180677377ec_int @ zero_zero_int )
= ( uminus_uminus_int @ one_one_int ) ) ).
% dbl_dec_simps(2)
thf(fact_559_one__less__nat__eq,axiom,
! [Z2: int] :
( ( ord_less_nat @ ( suc @ zero_zero_nat ) @ ( nat2 @ Z2 ) )
= ( ord_less_int @ one_one_int @ Z2 ) ) ).
% one_less_nat_eq
thf(fact_560_dbl__inc__simps_I4_J,axiom,
( ( neg_nu5851722552734809277nc_int @ ( uminus_uminus_int @ one_one_int ) )
= ( uminus_uminus_int @ one_one_int ) ) ).
% dbl_inc_simps(4)
thf(fact_561_dbl__inc__simps_I2_J,axiom,
( ( neg_nu5851722552734809277nc_int @ zero_zero_int )
= one_one_int ) ).
% dbl_inc_simps(2)
thf(fact_562_rotate1__length01,axiom,
! [Xs: list_instr] :
( ( ord_less_eq_nat @ ( size_size_list_instr @ Xs ) @ one_one_nat )
=> ( ( rotate1_instr @ Xs )
= Xs ) ) ).
% rotate1_length01
thf(fact_563_rotate1__length01,axiom,
! [Xs: list_list_char] :
( ( ord_less_eq_nat @ ( size_s356637359517785349t_char @ Xs ) @ one_one_nat )
=> ( ( rotate1_list_char @ Xs )
= Xs ) ) ).
% rotate1_length01
thf(fact_564_nat_Oinject,axiom,
! [X2: nat,Y2: nat] :
( ( ( suc @ X2 )
= ( suc @ Y2 ) )
= ( X2 = Y2 ) ) ).
% nat.inject
thf(fact_565_old_Onat_Oinject,axiom,
! [Nat: nat,Nat2: nat] :
( ( ( suc @ Nat )
= ( suc @ Nat2 ) )
= ( Nat = Nat2 ) ) ).
% old.nat.inject
thf(fact_566_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_567_Suc__mono,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ M @ N )
=> ( ord_less_nat @ ( suc @ M ) @ ( suc @ N ) ) ) ).
% Suc_mono
thf(fact_568_lessI,axiom,
! [N: nat] : ( ord_less_nat @ N @ ( suc @ N ) ) ).
% lessI
thf(fact_569_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_570_set__rotate1,axiom,
! [Xs: list_list_char] :
( ( set_list_char2 @ ( rotate1_list_char @ Xs ) )
= ( set_list_char2 @ Xs ) ) ).
% set_rotate1
thf(fact_571_set__rotate1,axiom,
! [Xs: list_instr] :
( ( set_instr2 @ ( rotate1_instr @ Xs ) )
= ( set_instr2 @ Xs ) ) ).
% set_rotate1
thf(fact_572_length__rotate1,axiom,
! [Xs: list_instr] :
( ( size_size_list_instr @ ( rotate1_instr @ Xs ) )
= ( size_size_list_instr @ Xs ) ) ).
% length_rotate1
thf(fact_573_length__rotate1,axiom,
! [Xs: list_list_char] :
( ( size_s356637359517785349t_char @ ( rotate1_list_char @ Xs ) )
= ( size_s356637359517785349t_char @ Xs ) ) ).
% length_rotate1
thf(fact_574_dbl__dec__simps_I3_J,axiom,
( ( neg_nu3811975205180677377ec_int @ one_one_int )
= one_one_int ) ).
% dbl_dec_simps(3)
thf(fact_575_zero__less__Suc,axiom,
! [N: nat] : ( ord_less_nat @ zero_zero_nat @ ( suc @ N ) ) ).
% zero_less_Suc
thf(fact_576_less__Suc0,axiom,
! [N: nat] :
( ( ord_less_nat @ N @ ( suc @ zero_zero_nat ) )
= ( N = zero_zero_nat ) ) ).
% less_Suc0
thf(fact_577_nat__1,axiom,
( ( nat2 @ one_one_int )
= ( suc @ zero_zero_nat ) ) ).
% nat_1
thf(fact_578_negative__zless,axiom,
! [N: nat,M: nat] : ( ord_less_int @ ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ ( suc @ N ) ) ) @ ( semiri1314217659103216013at_int @ M ) ) ).
% negative_zless
thf(fact_579_nat_Odistinct_I1_J,axiom,
! [X2: nat] :
( zero_zero_nat
!= ( suc @ X2 ) ) ).
% nat.distinct(1)
thf(fact_580_old_Onat_Odistinct_I2_J,axiom,
! [Nat2: nat] :
( ( suc @ Nat2 )
!= zero_zero_nat ) ).
% old.nat.distinct(2)
thf(fact_581_old_Onat_Odistinct_I1_J,axiom,
! [Nat2: nat] :
( zero_zero_nat
!= ( suc @ Nat2 ) ) ).
% old.nat.distinct(1)
thf(fact_582_nat_OdiscI,axiom,
! [Nat: nat,X2: nat] :
( ( Nat
= ( suc @ X2 ) )
=> ( Nat != zero_zero_nat ) ) ).
% nat.discI
thf(fact_583_old_Onat_Oexhaust,axiom,
! [Y: nat] :
( ( Y != zero_zero_nat )
=> ~ ! [Nat3: nat] :
( Y
!= ( suc @ Nat3 ) ) ) ).
% old.nat.exhaust
thf(fact_584_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_585_diff__induct,axiom,
! [P: nat > nat > $o,M: nat,N: nat] :
( ! [X5: nat] : ( P @ X5 @ zero_zero_nat )
=> ( ! [Y3: nat] : ( P @ zero_zero_nat @ ( suc @ Y3 ) )
=> ( ! [X5: nat,Y3: nat] :
( ( P @ X5 @ Y3 )
=> ( P @ ( suc @ X5 ) @ ( suc @ Y3 ) ) )
=> ( P @ M @ N ) ) ) ) ).
% diff_induct
thf(fact_586_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_587_Suc__neq__Zero,axiom,
! [M: nat] :
( ( suc @ M )
!= zero_zero_nat ) ).
% Suc_neq_Zero
thf(fact_588_Zero__neq__Suc,axiom,
! [M: nat] :
( zero_zero_nat
!= ( suc @ M ) ) ).
% Zero_neq_Suc
thf(fact_589_Zero__not__Suc,axiom,
! [M: nat] :
( zero_zero_nat
!= ( suc @ M ) ) ).
% Zero_not_Suc
thf(fact_590_not0__implies__Suc,axiom,
! [N: nat] :
( ( N != zero_zero_nat )
=> ? [M4: nat] :
( N
= ( suc @ M4 ) ) ) ).
% not0_implies_Suc
thf(fact_591_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_592_Suc__lessD,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ ( suc @ M ) @ N )
=> ( ord_less_nat @ M @ N ) ) ).
% Suc_lessD
thf(fact_593_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_594_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_595_less__SucE,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ M @ ( suc @ N ) )
=> ( ~ ( ord_less_nat @ M @ N )
=> ( M = N ) ) ) ).
% less_SucE
thf(fact_596_less__SucI,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ M @ N )
=> ( ord_less_nat @ M @ ( suc @ N ) ) ) ).
% less_SucI
thf(fact_597_Ex__less__Suc,axiom,
! [N: nat,P: nat > $o] :
( ( ? [I4: nat] :
( ( ord_less_nat @ I4 @ ( suc @ N ) )
& ( P @ I4 ) ) )
= ( ( P @ N )
| ? [I4: nat] :
( ( ord_less_nat @ I4 @ N )
& ( P @ I4 ) ) ) ) ).
% Ex_less_Suc
thf(fact_598_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_599_not__less__eq,axiom,
! [M: nat,N: nat] :
( ( ~ ( ord_less_nat @ M @ N ) )
= ( ord_less_nat @ N @ ( suc @ M ) ) ) ).
% not_less_eq
thf(fact_600_All__less__Suc,axiom,
! [N: nat,P: nat > $o] :
( ( ! [I4: nat] :
( ( ord_less_nat @ I4 @ ( suc @ N ) )
=> ( P @ I4 ) ) )
= ( ( P @ N )
& ! [I4: nat] :
( ( ord_less_nat @ I4 @ N )
=> ( P @ I4 ) ) ) ) ).
% All_less_Suc
thf(fact_601_Suc__less__eq2,axiom,
! [N: nat,M: nat] :
( ( ord_less_nat @ ( suc @ N ) @ M )
= ( ? [M5: nat] :
( ( M
= ( suc @ M5 ) )
& ( ord_less_nat @ N @ M5 ) ) ) ) ).
% Suc_less_eq2
thf(fact_602_less__antisym,axiom,
! [N: nat,M: nat] :
( ~ ( ord_less_nat @ N @ M )
=> ( ( ord_less_nat @ N @ ( suc @ M ) )
=> ( M = N ) ) ) ).
% less_antisym
thf(fact_603_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_604_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_605_less__Suc__induct,axiom,
! [I: nat,J: nat,P: nat > nat > $o] :
( ( ord_less_nat @ I @ J )
=> ( ! [I3: nat] : ( P @ I3 @ ( suc @ I3 ) )
=> ( ! [I3: nat,J2: nat,K2: nat] :
( ( ord_less_nat @ I3 @ J2 )
=> ( ( ord_less_nat @ J2 @ K2 )
=> ( ( P @ I3 @ J2 )
=> ( ( P @ J2 @ K2 )
=> ( P @ I3 @ K2 ) ) ) ) )
=> ( P @ I @ J ) ) ) ) ).
% less_Suc_induct
thf(fact_606_strict__inc__induct,axiom,
! [I: nat,J: nat,P: nat > $o] :
( ( ord_less_nat @ I @ J )
=> ( ! [I3: nat] :
( ( J
= ( suc @ I3 ) )
=> ( P @ I3 ) )
=> ( ! [I3: nat] :
( ( ord_less_nat @ I3 @ J )
=> ( ( P @ ( suc @ I3 ) )
=> ( P @ I3 ) ) )
=> ( P @ I ) ) ) ) ).
% strict_inc_induct
thf(fact_607_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_608_Suc__leD,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq_nat @ ( suc @ M ) @ N )
=> ( ord_less_eq_nat @ M @ N ) ) ).
% Suc_leD
thf(fact_609_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_610_le__SucI,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq_nat @ M @ N )
=> ( ord_less_eq_nat @ M @ ( suc @ N ) ) ) ).
% le_SucI
thf(fact_611_Suc__le__D,axiom,
! [N: nat,M6: nat] :
( ( ord_less_eq_nat @ ( suc @ N ) @ M6 )
=> ? [M4: nat] :
( M6
= ( suc @ M4 ) ) ) ).
% Suc_le_D
thf(fact_612_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_613_Suc__n__not__le__n,axiom,
! [N: nat] :
~ ( ord_less_eq_nat @ ( suc @ N ) @ N ) ).
% Suc_n_not_le_n
thf(fact_614_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_615_full__nat__induct,axiom,
! [P: nat > $o,N: nat] :
( ! [N2: nat] :
( ! [M3: nat] :
( ( ord_less_eq_nat @ ( suc @ M3 ) @ N2 )
=> ( P @ M3 ) )
=> ( P @ N2 ) )
=> ( P @ N ) ) ).
% full_nat_induct
thf(fact_616_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_617_transitive__stepwise__le,axiom,
! [M: nat,N: nat,R: nat > nat > $o] :
( ( ord_less_eq_nat @ M @ N )
=> ( ! [X5: nat] : ( R @ X5 @ X5 )
=> ( ! [X5: nat,Y3: nat,Z: nat] :
( ( R @ X5 @ Y3 )
=> ( ( R @ Y3 @ Z )
=> ( R @ X5 @ Z ) ) )
=> ( ! [N2: nat] : ( R @ N2 @ ( suc @ N2 ) )
=> ( R @ M @ N ) ) ) ) ) ).
% transitive_stepwise_le
thf(fact_618_Suc__inject,axiom,
! [X: nat,Y: nat] :
( ( ( suc @ X )
= ( suc @ Y ) )
=> ( X = Y ) ) ).
% Suc_inject
thf(fact_619_n__not__Suc__n,axiom,
! [N: nat] :
( N
!= ( suc @ N ) ) ).
% n_not_Suc_n
thf(fact_620_lift__Suc__mono__le,axiom,
! [F: nat > int,N: nat,N4: nat] :
( ! [N2: nat] : ( ord_less_eq_int @ ( F @ N2 ) @ ( F @ ( suc @ N2 ) ) )
=> ( ( ord_less_eq_nat @ N @ N4 )
=> ( ord_less_eq_int @ ( F @ N ) @ ( F @ N4 ) ) ) ) ).
% lift_Suc_mono_le
thf(fact_621_lift__Suc__mono__le,axiom,
! [F: nat > nat,N: nat,N4: nat] :
( ! [N2: nat] : ( ord_less_eq_nat @ ( F @ N2 ) @ ( F @ ( suc @ N2 ) ) )
=> ( ( ord_less_eq_nat @ N @ N4 )
=> ( ord_less_eq_nat @ ( F @ N ) @ ( F @ N4 ) ) ) ) ).
% lift_Suc_mono_le
thf(fact_622_lift__Suc__antimono__le,axiom,
! [F: nat > int,N: nat,N4: nat] :
( ! [N2: nat] : ( ord_less_eq_int @ ( F @ ( suc @ N2 ) ) @ ( F @ N2 ) )
=> ( ( ord_less_eq_nat @ N @ N4 )
=> ( ord_less_eq_int @ ( F @ N4 ) @ ( F @ N ) ) ) ) ).
% lift_Suc_antimono_le
thf(fact_623_lift__Suc__antimono__le,axiom,
! [F: nat > nat,N: nat,N4: nat] :
( ! [N2: nat] : ( ord_less_eq_nat @ ( F @ ( suc @ N2 ) ) @ ( F @ N2 ) )
=> ( ( ord_less_eq_nat @ N @ N4 )
=> ( ord_less_eq_nat @ ( F @ N4 ) @ ( F @ N ) ) ) ) ).
% lift_Suc_antimono_le
thf(fact_624_lift__Suc__mono__less,axiom,
! [F: nat > int,N: nat,N4: nat] :
( ! [N2: nat] : ( ord_less_int @ ( F @ N2 ) @ ( F @ ( suc @ N2 ) ) )
=> ( ( ord_less_nat @ N @ N4 )
=> ( ord_less_int @ ( F @ N ) @ ( F @ N4 ) ) ) ) ).
% lift_Suc_mono_less
thf(fact_625_lift__Suc__mono__less,axiom,
! [F: nat > nat,N: nat,N4: nat] :
( ! [N2: nat] : ( ord_less_nat @ ( F @ N2 ) @ ( F @ ( suc @ N2 ) ) )
=> ( ( ord_less_nat @ N @ N4 )
=> ( ord_less_nat @ ( F @ N ) @ ( F @ N4 ) ) ) ) ).
% lift_Suc_mono_less
thf(fact_626_lift__Suc__mono__less__iff,axiom,
! [F: nat > int,N: nat,M: nat] :
( ! [N2: nat] : ( ord_less_int @ ( F @ N2 ) @ ( F @ ( suc @ N2 ) ) )
=> ( ( ord_less_int @ ( F @ N ) @ ( F @ M ) )
= ( ord_less_nat @ N @ M ) ) ) ).
% lift_Suc_mono_less_iff
thf(fact_627_lift__Suc__mono__less__iff,axiom,
! [F: nat > nat,N: nat,M: nat] :
( ! [N2: nat] : ( ord_less_nat @ ( F @ N2 ) @ ( F @ ( suc @ N2 ) ) )
=> ( ( ord_less_nat @ ( F @ N ) @ ( F @ M ) )
= ( ord_less_nat @ N @ M ) ) ) ).
% lift_Suc_mono_less_iff
thf(fact_628_of__nat__neq__0,axiom,
! [N: nat] :
( ( semiri1314217659103216013at_int @ ( suc @ N ) )
!= zero_zero_int ) ).
% of_nat_neq_0
thf(fact_629_of__nat__neq__0,axiom,
! [N: nat] :
( ( semiri1316708129612266289at_nat @ ( suc @ N ) )
!= zero_zero_nat ) ).
% of_nat_neq_0
thf(fact_630_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_631_gr0__implies__Suc,axiom,
! [N: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ? [M4: nat] :
( N
= ( suc @ M4 ) ) ) ).
% gr0_implies_Suc
thf(fact_632_All__less__Suc2,axiom,
! [N: nat,P: nat > $o] :
( ( ! [I4: nat] :
( ( ord_less_nat @ I4 @ ( suc @ N ) )
=> ( P @ I4 ) ) )
= ( ( P @ zero_zero_nat )
& ! [I4: nat] :
( ( ord_less_nat @ I4 @ N )
=> ( P @ ( suc @ I4 ) ) ) ) ) ).
% All_less_Suc2
thf(fact_633_gr0__conv__Suc,axiom,
! [N: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
= ( ? [M2: nat] :
( N
= ( suc @ M2 ) ) ) ) ).
% gr0_conv_Suc
thf(fact_634_Ex__less__Suc2,axiom,
! [N: nat,P: nat > $o] :
( ( ? [I4: nat] :
( ( ord_less_nat @ I4 @ ( suc @ N ) )
& ( P @ I4 ) ) )
= ( ( P @ zero_zero_nat )
| ? [I4: nat] :
( ( ord_less_nat @ I4 @ N )
& ( P @ ( suc @ I4 ) ) ) ) ) ).
% Ex_less_Suc2
thf(fact_635_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_636_less__eq__Suc__le,axiom,
( ord_less_nat
= ( ^ [N3: nat] : ( ord_less_eq_nat @ ( suc @ N3 ) ) ) ) ).
% less_eq_Suc_le
thf(fact_637_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_638_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_639_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_640_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_641_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_642_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_643_Suc__leI,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ M @ N )
=> ( ord_less_eq_nat @ ( suc @ M ) @ N ) ) ).
% Suc_leI
thf(fact_644_One__nat__def,axiom,
( one_one_nat
= ( suc @ zero_zero_nat ) ) ).
% One_nat_def
thf(fact_645_int__cases,axiom,
! [Z2: int] :
( ! [N2: nat] :
( Z2
!= ( semiri1314217659103216013at_int @ N2 ) )
=> ~ ! [N2: nat] :
( Z2
!= ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ ( suc @ N2 ) ) ) ) ) ).
% int_cases
thf(fact_646_int__of__nat__induct,axiom,
! [P: int > $o,Z2: int] :
( ! [N2: nat] : ( P @ ( semiri1314217659103216013at_int @ N2 ) )
=> ( ! [N2: nat] : ( P @ ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ ( suc @ N2 ) ) ) )
=> ( P @ Z2 ) ) ) ).
% int_of_nat_induct
thf(fact_647_ex__least__nat__less,axiom,
! [P: nat > $o,N: nat] :
( ( P @ N )
=> ( ~ ( P @ zero_zero_nat )
=> ? [K2: nat] :
( ( ord_less_nat @ K2 @ N )
& ! [I2: nat] :
( ( ord_less_eq_nat @ I2 @ K2 )
=> ~ ( P @ I2 ) )
& ( P @ ( suc @ K2 ) ) ) ) ) ).
% ex_least_nat_less
thf(fact_648_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_649_not__zle__0__negative,axiom,
! [N: nat] :
~ ( ord_less_eq_int @ zero_zero_int @ ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ ( suc @ N ) ) ) ) ).
% not_zle_0_negative
thf(fact_650_negD,axiom,
! [X: int] :
( ( ord_less_int @ X @ zero_zero_int )
=> ? [N2: nat] :
( X
= ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ ( suc @ N2 ) ) ) ) ) ).
% negD
thf(fact_651_negative__zless__0,axiom,
! [N: nat] : ( ord_less_int @ ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ ( suc @ N ) ) ) @ zero_zero_int ) ).
% negative_zless_0
thf(fact_652_Suc__nat__eq__nat__zadd1,axiom,
! [Z2: int] :
( ( ord_less_eq_int @ zero_zero_int @ Z2 )
=> ( ( suc @ ( nat2 @ Z2 ) )
= ( nat2 @ ( plus_plus_int @ one_one_int @ Z2 ) ) ) ) ).
% Suc_nat_eq_nat_zadd1
thf(fact_653_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_654_sgn__neg,axiom,
! [A: int] :
( ( ord_less_int @ A @ zero_zero_int )
=> ( ( sgn_sgn_int @ A )
= ( uminus_uminus_int @ one_one_int ) ) ) ).
% sgn_neg
thf(fact_655_rotate__length01,axiom,
! [Xs: list_instr,N: nat] :
( ( ord_less_eq_nat @ ( size_size_list_instr @ Xs ) @ one_one_nat )
=> ( ( rotate_instr @ N @ Xs )
= Xs ) ) ).
% rotate_length01
thf(fact_656_rotate__length01,axiom,
! [Xs: list_list_char,N: nat] :
( ( ord_less_eq_nat @ ( size_s356637359517785349t_char @ Xs ) @ one_one_nat )
=> ( ( rotate_list_char @ N @ Xs )
= Xs ) ) ).
% rotate_length01
thf(fact_657_add__right__cancel,axiom,
! [B: int,A: int,C: int] :
( ( ( plus_plus_int @ B @ A )
= ( plus_plus_int @ C @ A ) )
= ( B = C ) ) ).
% add_right_cancel
thf(fact_658_add__right__cancel,axiom,
! [B: nat,A: nat,C: nat] :
( ( ( plus_plus_nat @ B @ A )
= ( plus_plus_nat @ C @ A ) )
= ( B = C ) ) ).
% add_right_cancel
thf(fact_659_add__left__cancel,axiom,
! [A: int,B: int,C: int] :
( ( ( plus_plus_int @ A @ B )
= ( plus_plus_int @ A @ C ) )
= ( B = C ) ) ).
% add_left_cancel
thf(fact_660_add__left__cancel,axiom,
! [A: nat,B: nat,C: nat] :
( ( ( plus_plus_nat @ A @ B )
= ( plus_plus_nat @ A @ C ) )
= ( B = C ) ) ).
% add_left_cancel
thf(fact_661_add__le__cancel__right,axiom,
! [A: int,C: int,B: int] :
( ( ord_less_eq_int @ ( plus_plus_int @ A @ C ) @ ( plus_plus_int @ B @ C ) )
= ( ord_less_eq_int @ A @ B ) ) ).
% add_le_cancel_right
thf(fact_662_add__le__cancel__right,axiom,
! [A: nat,C: nat,B: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ A @ C ) @ ( plus_plus_nat @ B @ C ) )
= ( ord_less_eq_nat @ A @ B ) ) ).
% add_le_cancel_right
thf(fact_663_add__le__cancel__left,axiom,
! [C: int,A: int,B: int] :
( ( ord_less_eq_int @ ( plus_plus_int @ C @ A ) @ ( plus_plus_int @ C @ B ) )
= ( ord_less_eq_int @ A @ B ) ) ).
% add_le_cancel_left
thf(fact_664_add__le__cancel__left,axiom,
! [C: nat,A: nat,B: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ C @ A ) @ ( plus_plus_nat @ C @ B ) )
= ( ord_less_eq_nat @ A @ B ) ) ).
% add_le_cancel_left
thf(fact_665_double__eq__0__iff,axiom,
! [A: int] :
( ( ( plus_plus_int @ A @ A )
= zero_zero_int )
= ( A = zero_zero_int ) ) ).
% double_eq_0_iff
thf(fact_666_add__0,axiom,
! [A: int] :
( ( plus_plus_int @ zero_zero_int @ A )
= A ) ).
% add_0
thf(fact_667_add__0,axiom,
! [A: nat] :
( ( plus_plus_nat @ zero_zero_nat @ A )
= A ) ).
% add_0
thf(fact_668_zero__eq__add__iff__both__eq__0,axiom,
! [X: nat,Y: nat] :
( ( zero_zero_nat
= ( plus_plus_nat @ X @ Y ) )
= ( ( X = zero_zero_nat )
& ( Y = zero_zero_nat ) ) ) ).
% zero_eq_add_iff_both_eq_0
thf(fact_669_add__eq__0__iff__both__eq__0,axiom,
! [X: nat,Y: nat] :
( ( ( plus_plus_nat @ X @ Y )
= zero_zero_nat )
= ( ( X = zero_zero_nat )
& ( Y = zero_zero_nat ) ) ) ).
% add_eq_0_iff_both_eq_0
thf(fact_670_add__cancel__right__right,axiom,
! [A: int,B: int] :
( ( A
= ( plus_plus_int @ A @ B ) )
= ( B = zero_zero_int ) ) ).
% add_cancel_right_right
thf(fact_671_add__cancel__right__right,axiom,
! [A: nat,B: nat] :
( ( A
= ( plus_plus_nat @ A @ B ) )
= ( B = zero_zero_nat ) ) ).
% add_cancel_right_right
thf(fact_672_add__cancel__right__left,axiom,
! [A: int,B: int] :
( ( A
= ( plus_plus_int @ B @ A ) )
= ( B = zero_zero_int ) ) ).
% add_cancel_right_left
thf(fact_673_add__cancel__right__left,axiom,
! [A: nat,B: nat] :
( ( A
= ( plus_plus_nat @ B @ A ) )
= ( B = zero_zero_nat ) ) ).
% add_cancel_right_left
thf(fact_674_add__cancel__left__right,axiom,
! [A: int,B: int] :
( ( ( plus_plus_int @ A @ B )
= A )
= ( B = zero_zero_int ) ) ).
% add_cancel_left_right
thf(fact_675_add__cancel__left__right,axiom,
! [A: nat,B: nat] :
( ( ( plus_plus_nat @ A @ B )
= A )
= ( B = zero_zero_nat ) ) ).
% add_cancel_left_right
thf(fact_676_add__cancel__left__left,axiom,
! [B: int,A: int] :
( ( ( plus_plus_int @ B @ A )
= A )
= ( B = zero_zero_int ) ) ).
% add_cancel_left_left
thf(fact_677_add__cancel__left__left,axiom,
! [B: nat,A: nat] :
( ( ( plus_plus_nat @ B @ A )
= A )
= ( B = zero_zero_nat ) ) ).
% add_cancel_left_left
thf(fact_678_double__zero__sym,axiom,
! [A: int] :
( ( zero_zero_int
= ( plus_plus_int @ A @ A ) )
= ( A = zero_zero_int ) ) ).
% double_zero_sym
thf(fact_679_add_Oright__neutral,axiom,
! [A: int] :
( ( plus_plus_int @ A @ zero_zero_int )
= A ) ).
% add.right_neutral
thf(fact_680_add_Oright__neutral,axiom,
! [A: nat] :
( ( plus_plus_nat @ A @ zero_zero_nat )
= A ) ).
% add.right_neutral
thf(fact_681_add__less__cancel__right,axiom,
! [A: int,C: int,B: int] :
( ( ord_less_int @ ( plus_plus_int @ A @ C ) @ ( plus_plus_int @ B @ C ) )
= ( ord_less_int @ A @ B ) ) ).
% add_less_cancel_right
thf(fact_682_add__less__cancel__right,axiom,
! [A: nat,C: nat,B: nat] :
( ( ord_less_nat @ ( plus_plus_nat @ A @ C ) @ ( plus_plus_nat @ B @ C ) )
= ( ord_less_nat @ A @ B ) ) ).
% add_less_cancel_right
thf(fact_683_add__less__cancel__left,axiom,
! [C: int,A: int,B: int] :
( ( ord_less_int @ ( plus_plus_int @ C @ A ) @ ( plus_plus_int @ C @ B ) )
= ( ord_less_int @ A @ B ) ) ).
% add_less_cancel_left
thf(fact_684_add__less__cancel__left,axiom,
! [C: nat,A: nat,B: nat] :
( ( ord_less_nat @ ( plus_plus_nat @ C @ A ) @ ( plus_plus_nat @ C @ B ) )
= ( ord_less_nat @ A @ B ) ) ).
% add_less_cancel_left
thf(fact_685_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_686_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_687_diff__zero,axiom,
! [A: nat] :
( ( minus_minus_nat @ A @ zero_zero_nat )
= A ) ).
% diff_zero
thf(fact_688_diff__zero,axiom,
! [A: int] :
( ( minus_minus_int @ A @ zero_zero_int )
= A ) ).
% diff_zero
thf(fact_689_zero__diff,axiom,
! [A: nat] :
( ( minus_minus_nat @ zero_zero_nat @ A )
= zero_zero_nat ) ).
% zero_diff
thf(fact_690_diff__0__right,axiom,
! [A: int] :
( ( minus_minus_int @ A @ zero_zero_int )
= A ) ).
% diff_0_right
thf(fact_691_diff__self,axiom,
! [A: int] :
( ( minus_minus_int @ A @ A )
= zero_zero_int ) ).
% diff_self
thf(fact_692_add__diff__cancel__right_H,axiom,
! [A: nat,B: nat] :
( ( minus_minus_nat @ ( plus_plus_nat @ A @ B ) @ B )
= A ) ).
% add_diff_cancel_right'
thf(fact_693_add__diff__cancel__right_H,axiom,
! [A: int,B: int] :
( ( minus_minus_int @ ( plus_plus_int @ A @ B ) @ B )
= A ) ).
% add_diff_cancel_right'
thf(fact_694_add__diff__cancel__right,axiom,
! [A: nat,C: nat,B: nat] :
( ( minus_minus_nat @ ( plus_plus_nat @ A @ C ) @ ( plus_plus_nat @ B @ C ) )
= ( minus_minus_nat @ A @ B ) ) ).
% add_diff_cancel_right
thf(fact_695_add__diff__cancel__right,axiom,
! [A: int,C: int,B: int] :
( ( minus_minus_int @ ( plus_plus_int @ A @ C ) @ ( plus_plus_int @ B @ C ) )
= ( minus_minus_int @ A @ B ) ) ).
% add_diff_cancel_right
thf(fact_696_add__diff__cancel__left_H,axiom,
! [A: nat,B: nat] :
( ( minus_minus_nat @ ( plus_plus_nat @ A @ B ) @ A )
= B ) ).
% add_diff_cancel_left'
thf(fact_697_add__diff__cancel__left_H,axiom,
! [A: int,B: int] :
( ( minus_minus_int @ ( plus_plus_int @ A @ B ) @ A )
= B ) ).
% add_diff_cancel_left'
thf(fact_698_add__diff__cancel__left,axiom,
! [C: nat,A: nat,B: nat] :
( ( minus_minus_nat @ ( plus_plus_nat @ C @ A ) @ ( plus_plus_nat @ C @ B ) )
= ( minus_minus_nat @ A @ B ) ) ).
% add_diff_cancel_left
thf(fact_699_add__diff__cancel__left,axiom,
! [C: int,A: int,B: int] :
( ( minus_minus_int @ ( plus_plus_int @ C @ A ) @ ( plus_plus_int @ C @ B ) )
= ( minus_minus_int @ A @ B ) ) ).
% add_diff_cancel_left
thf(fact_700_diff__add__cancel,axiom,
! [A: int,B: int] :
( ( plus_plus_int @ ( minus_minus_int @ A @ B ) @ B )
= A ) ).
% diff_add_cancel
thf(fact_701_add__diff__cancel,axiom,
! [A: int,B: int] :
( ( minus_minus_int @ ( plus_plus_int @ A @ B ) @ B )
= A ) ).
% add_diff_cancel
thf(fact_702_minus__add__distrib,axiom,
! [A: int,B: int] :
( ( uminus_uminus_int @ ( plus_plus_int @ A @ B ) )
= ( plus_plus_int @ ( uminus_uminus_int @ A ) @ ( uminus_uminus_int @ B ) ) ) ).
% minus_add_distrib
thf(fact_703_minus__add__cancel,axiom,
! [A: int,B: int] :
( ( plus_plus_int @ ( uminus_uminus_int @ A ) @ ( plus_plus_int @ A @ B ) )
= B ) ).
% minus_add_cancel
thf(fact_704_add__minus__cancel,axiom,
! [A: int,B: int] :
( ( plus_plus_int @ A @ ( plus_plus_int @ ( uminus_uminus_int @ A ) @ B ) )
= B ) ).
% add_minus_cancel
thf(fact_705_minus__diff__eq,axiom,
! [A: int,B: int] :
( ( uminus_uminus_int @ ( minus_minus_int @ A @ B ) )
= ( minus_minus_int @ B @ A ) ) ).
% minus_diff_eq
thf(fact_706_of__nat__add,axiom,
! [M: nat,N: nat] :
( ( semiri1314217659103216013at_int @ ( plus_plus_nat @ M @ N ) )
= ( plus_plus_int @ ( semiri1314217659103216013at_int @ M ) @ ( semiri1314217659103216013at_int @ N ) ) ) ).
% of_nat_add
thf(fact_707_of__nat__add,axiom,
! [M: nat,N: nat] :
( ( semiri1316708129612266289at_nat @ ( plus_plus_nat @ M @ N ) )
= ( plus_plus_nat @ ( semiri1316708129612266289at_nat @ M ) @ ( semiri1316708129612266289at_nat @ N ) ) ) ).
% of_nat_add
thf(fact_708_diff__0__eq__0,axiom,
! [N: nat] :
( ( minus_minus_nat @ zero_zero_nat @ N )
= zero_zero_nat ) ).
% diff_0_eq_0
thf(fact_709_diff__self__eq__0,axiom,
! [M: nat] :
( ( minus_minus_nat @ M @ M )
= zero_zero_nat ) ).
% diff_self_eq_0
thf(fact_710_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_711_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_712_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_713_sgn__0,axiom,
( ( sgn_sgn_int @ zero_zero_int )
= zero_zero_int ) ).
% sgn_0
thf(fact_714_of__int__diff,axiom,
! [W: int,Z2: int] :
( ( ring_1_of_int_int @ ( minus_minus_int @ W @ Z2 ) )
= ( minus_minus_int @ ( ring_1_of_int_int @ W ) @ ( ring_1_of_int_int @ Z2 ) ) ) ).
% of_int_diff
thf(fact_715_sgn__minus,axiom,
! [A: int] :
( ( sgn_sgn_int @ ( uminus_uminus_int @ A ) )
= ( uminus_uminus_int @ ( sgn_sgn_int @ A ) ) ) ).
% sgn_minus
thf(fact_716_set__rotate,axiom,
! [N: nat,Xs: list_list_char] :
( ( set_list_char2 @ ( rotate_list_char @ N @ Xs ) )
= ( set_list_char2 @ Xs ) ) ).
% set_rotate
thf(fact_717_set__rotate,axiom,
! [N: nat,Xs: list_instr] :
( ( set_instr2 @ ( rotate_instr @ N @ Xs ) )
= ( set_instr2 @ Xs ) ) ).
% set_rotate
thf(fact_718_length__rotate,axiom,
! [N: nat,Xs: list_instr] :
( ( size_size_list_instr @ ( rotate_instr @ N @ Xs ) )
= ( size_size_list_instr @ Xs ) ) ).
% length_rotate
thf(fact_719_length__rotate,axiom,
! [N: nat,Xs: list_list_char] :
( ( size_s356637359517785349t_char @ ( rotate_list_char @ N @ Xs ) )
= ( size_s356637359517785349t_char @ Xs ) ) ).
% length_rotate
thf(fact_720_zero__le__double__add__iff__zero__le__single__add,axiom,
! [A: int] :
( ( ord_less_eq_int @ zero_zero_int @ ( plus_plus_int @ A @ A ) )
= ( ord_less_eq_int @ zero_zero_int @ A ) ) ).
% zero_le_double_add_iff_zero_le_single_add
thf(fact_721_double__add__le__zero__iff__single__add__le__zero,axiom,
! [A: int] :
( ( ord_less_eq_int @ ( plus_plus_int @ A @ A ) @ zero_zero_int )
= ( ord_less_eq_int @ A @ zero_zero_int ) ) ).
% double_add_le_zero_iff_single_add_le_zero
thf(fact_722_le__add__same__cancel2,axiom,
! [A: int,B: int] :
( ( ord_less_eq_int @ A @ ( plus_plus_int @ B @ A ) )
= ( ord_less_eq_int @ zero_zero_int @ B ) ) ).
% le_add_same_cancel2
thf(fact_723_le__add__same__cancel2,axiom,
! [A: nat,B: nat] :
( ( ord_less_eq_nat @ A @ ( plus_plus_nat @ B @ A ) )
= ( ord_less_eq_nat @ zero_zero_nat @ B ) ) ).
% le_add_same_cancel2
thf(fact_724_le__add__same__cancel1,axiom,
! [A: int,B: int] :
( ( ord_less_eq_int @ A @ ( plus_plus_int @ A @ B ) )
= ( ord_less_eq_int @ zero_zero_int @ B ) ) ).
% le_add_same_cancel1
thf(fact_725_le__add__same__cancel1,axiom,
! [A: nat,B: nat] :
( ( ord_less_eq_nat @ A @ ( plus_plus_nat @ A @ B ) )
= ( ord_less_eq_nat @ zero_zero_nat @ B ) ) ).
% le_add_same_cancel1
thf(fact_726_add__le__same__cancel2,axiom,
! [A: int,B: int] :
( ( ord_less_eq_int @ ( plus_plus_int @ A @ B ) @ B )
= ( ord_less_eq_int @ A @ zero_zero_int ) ) ).
% add_le_same_cancel2
thf(fact_727_add__le__same__cancel2,axiom,
! [A: nat,B: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ A @ B ) @ B )
= ( ord_less_eq_nat @ A @ zero_zero_nat ) ) ).
% add_le_same_cancel2
thf(fact_728_add__le__same__cancel1,axiom,
! [B: int,A: int] :
( ( ord_less_eq_int @ ( plus_plus_int @ B @ A ) @ B )
= ( ord_less_eq_int @ A @ zero_zero_int ) ) ).
% add_le_same_cancel1
thf(fact_729_add__le__same__cancel1,axiom,
! [B: nat,A: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ B @ A ) @ B )
= ( ord_less_eq_nat @ A @ zero_zero_nat ) ) ).
% add_le_same_cancel1
thf(fact_730_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_731_zero__less__double__add__iff__zero__less__single__add,axiom,
! [A: int] :
( ( ord_less_int @ zero_zero_int @ ( plus_plus_int @ A @ A ) )
= ( ord_less_int @ zero_zero_int @ A ) ) ).
% zero_less_double_add_iff_zero_less_single_add
thf(fact_732_double__add__less__zero__iff__single__add__less__zero,axiom,
! [A: int] :
( ( ord_less_int @ ( plus_plus_int @ A @ A ) @ zero_zero_int )
= ( ord_less_int @ A @ zero_zero_int ) ) ).
% double_add_less_zero_iff_single_add_less_zero
thf(fact_733_less__add__same__cancel2,axiom,
! [A: int,B: int] :
( ( ord_less_int @ A @ ( plus_plus_int @ B @ A ) )
= ( ord_less_int @ zero_zero_int @ B ) ) ).
% less_add_same_cancel2
thf(fact_734_less__add__same__cancel2,axiom,
! [A: nat,B: nat] :
( ( ord_less_nat @ A @ ( plus_plus_nat @ B @ A ) )
= ( ord_less_nat @ zero_zero_nat @ B ) ) ).
% less_add_same_cancel2
thf(fact_735_less__add__same__cancel1,axiom,
! [A: int,B: int] :
( ( ord_less_int @ A @ ( plus_plus_int @ A @ B ) )
= ( ord_less_int @ zero_zero_int @ B ) ) ).
% less_add_same_cancel1
thf(fact_736_less__add__same__cancel1,axiom,
! [A: nat,B: nat] :
( ( ord_less_nat @ A @ ( plus_plus_nat @ A @ B ) )
= ( ord_less_nat @ zero_zero_nat @ B ) ) ).
% less_add_same_cancel1
thf(fact_737_add__less__same__cancel2,axiom,
! [A: int,B: int] :
( ( ord_less_int @ ( plus_plus_int @ A @ B ) @ B )
= ( ord_less_int @ A @ zero_zero_int ) ) ).
% add_less_same_cancel2
thf(fact_738_add__less__same__cancel2,axiom,
! [A: nat,B: nat] :
( ( ord_less_nat @ ( plus_plus_nat @ A @ B ) @ B )
= ( ord_less_nat @ A @ zero_zero_nat ) ) ).
% add_less_same_cancel2
thf(fact_739_add__less__same__cancel1,axiom,
! [B: int,A: int] :
( ( ord_less_int @ ( plus_plus_int @ B @ A ) @ B )
= ( ord_less_int @ A @ zero_zero_int ) ) ).
% add_less_same_cancel1
thf(fact_740_add__less__same__cancel1,axiom,
! [B: nat,A: nat] :
( ( ord_less_nat @ ( plus_plus_nat @ B @ A ) @ B )
= ( ord_less_nat @ A @ zero_zero_nat ) ) ).
% add_less_same_cancel1
thf(fact_741_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_742_le__add__diff__inverse2,axiom,
! [B: int,A: int] :
( ( ord_less_eq_int @ B @ A )
=> ( ( plus_plus_int @ ( minus_minus_int @ A @ B ) @ B )
= A ) ) ).
% le_add_diff_inverse2
thf(fact_743_le__add__diff__inverse2,axiom,
! [B: nat,A: nat] :
( ( ord_less_eq_nat @ B @ A )
=> ( ( plus_plus_nat @ ( minus_minus_nat @ A @ B ) @ B )
= A ) ) ).
% le_add_diff_inverse2
thf(fact_744_le__add__diff__inverse,axiom,
! [B: int,A: int] :
( ( ord_less_eq_int @ B @ A )
=> ( ( plus_plus_int @ B @ ( minus_minus_int @ A @ B ) )
= A ) ) ).
% le_add_diff_inverse
thf(fact_745_le__add__diff__inverse,axiom,
! [B: nat,A: nat] :
( ( ord_less_eq_nat @ B @ A )
=> ( ( plus_plus_nat @ B @ ( minus_minus_nat @ A @ B ) )
= A ) ) ).
% le_add_diff_inverse
thf(fact_746_diff__add__zero,axiom,
! [A: nat,B: nat] :
( ( minus_minus_nat @ A @ ( plus_plus_nat @ A @ B ) )
= zero_zero_nat ) ).
% diff_add_zero
thf(fact_747_diff__numeral__special_I9_J,axiom,
( ( minus_minus_int @ one_one_int @ one_one_int )
= zero_zero_int ) ).
% diff_numeral_special(9)
thf(fact_748_add_Oright__inverse,axiom,
! [A: int] :
( ( plus_plus_int @ A @ ( uminus_uminus_int @ A ) )
= zero_zero_int ) ).
% add.right_inverse
thf(fact_749_ab__left__minus,axiom,
! [A: int] :
( ( plus_plus_int @ ( uminus_uminus_int @ A ) @ A )
= zero_zero_int ) ).
% ab_left_minus
thf(fact_750_verit__minus__simplify_I3_J,axiom,
! [B: int] :
( ( minus_minus_int @ zero_zero_int @ B )
= ( uminus_uminus_int @ B ) ) ).
% verit_minus_simplify(3)
thf(fact_751_diff__0,axiom,
! [A: int] :
( ( minus_minus_int @ zero_zero_int @ A )
= ( uminus_uminus_int @ A ) ) ).
% diff_0
thf(fact_752_diff__minus__eq__add,axiom,
! [A: int,B: int] :
( ( minus_minus_int @ A @ ( uminus_uminus_int @ B ) )
= ( plus_plus_int @ A @ B ) ) ).
% diff_minus_eq_add
thf(fact_753_uminus__add__conv__diff,axiom,
! [A: int,B: int] :
( ( plus_plus_int @ ( uminus_uminus_int @ A ) @ B )
= ( minus_minus_int @ B @ A ) ) ).
% uminus_add_conv_diff
thf(fact_754_sgn__greater,axiom,
! [A: int] :
( ( ord_less_int @ zero_zero_int @ ( sgn_sgn_int @ A ) )
= ( ord_less_int @ zero_zero_int @ A ) ) ).
% sgn_greater
thf(fact_755_sgn__less,axiom,
! [A: int] :
( ( ord_less_int @ ( sgn_sgn_int @ A ) @ zero_zero_int )
= ( ord_less_int @ A @ zero_zero_int ) ) ).
% sgn_less
thf(fact_756_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_757_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_758_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_759_diff__Suc__1,axiom,
! [N: nat] :
( ( minus_minus_nat @ ( suc @ N ) @ one_one_nat )
= N ) ).
% diff_Suc_1
thf(fact_760_of__int__add,axiom,
! [W: int,Z2: int] :
( ( ring_1_of_int_int @ ( plus_plus_int @ W @ Z2 ) )
= ( plus_plus_int @ ( ring_1_of_int_int @ W ) @ ( ring_1_of_int_int @ Z2 ) ) ) ).
% of_int_add
thf(fact_761_add__neg__numeral__special_I7_J,axiom,
( ( plus_plus_int @ one_one_int @ ( uminus_uminus_int @ one_one_int ) )
= zero_zero_int ) ).
% add_neg_numeral_special(7)
thf(fact_762_add__neg__numeral__special_I8_J,axiom,
( ( plus_plus_int @ ( uminus_uminus_int @ one_one_int ) @ one_one_int )
= zero_zero_int ) ).
% add_neg_numeral_special(8)
thf(fact_763_diff__numeral__special_I12_J,axiom,
( ( minus_minus_int @ ( uminus_uminus_int @ one_one_int ) @ ( uminus_uminus_int @ one_one_int ) )
= zero_zero_int ) ).
% diff_numeral_special(12)
thf(fact_764_of__nat__Suc,axiom,
! [M: nat] :
( ( semiri1314217659103216013at_int @ ( suc @ M ) )
= ( plus_plus_int @ one_one_int @ ( semiri1314217659103216013at_int @ M ) ) ) ).
% of_nat_Suc
thf(fact_765_of__nat__Suc,axiom,
! [M: nat] :
( ( semiri1316708129612266289at_nat @ ( suc @ M ) )
= ( plus_plus_nat @ one_one_nat @ ( semiri1316708129612266289at_nat @ M ) ) ) ).
% of_nat_Suc
thf(fact_766_sgn__pos,axiom,
! [A: int] :
( ( ord_less_int @ zero_zero_int @ A )
=> ( ( sgn_sgn_int @ A )
= one_one_int ) ) ).
% sgn_pos
thf(fact_767_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_768_zle__add1__eq__le,axiom,
! [W: int,Z2: int] :
( ( ord_less_int @ W @ ( plus_plus_int @ Z2 @ one_one_int ) )
= ( ord_less_eq_int @ W @ Z2 ) ) ).
% zle_add1_eq_le
thf(fact_769_dbl__dec__def,axiom,
( neg_nu3811975205180677377ec_int
= ( ^ [X3: int] : ( minus_minus_int @ ( plus_plus_int @ X3 @ X3 ) @ one_one_int ) ) ) ).
% dbl_dec_def
thf(fact_770_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_771_diff__mono,axiom,
! [A: int,B: int,D2: int,C: int] :
( ( ord_less_eq_int @ A @ B )
=> ( ( ord_less_eq_int @ D2 @ C )
=> ( ord_less_eq_int @ ( minus_minus_int @ A @ C ) @ ( minus_minus_int @ B @ D2 ) ) ) ) ).
% diff_mono
thf(fact_772_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_773_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_774_diff__eq__diff__less__eq,axiom,
! [A: int,B: int,C: int,D2: int] :
( ( ( minus_minus_int @ A @ B )
= ( minus_minus_int @ C @ D2 ) )
=> ( ( ord_less_eq_int @ A @ B )
= ( ord_less_eq_int @ C @ D2 ) ) ) ).
% diff_eq_diff_less_eq
thf(fact_775_eq__iff__diff__eq__0,axiom,
( ( ^ [Y6: int,Z3: int] : ( Y6 = Z3 ) )
= ( ^ [A3: int,B2: int] :
( ( minus_minus_int @ A3 @ B2 )
= zero_zero_int ) ) ) ).
% eq_iff_diff_eq_0
thf(fact_776_diff__strict__mono,axiom,
! [A: int,B: int,D2: int,C: int] :
( ( ord_less_int @ A @ B )
=> ( ( ord_less_int @ D2 @ C )
=> ( ord_less_int @ ( minus_minus_int @ A @ C ) @ ( minus_minus_int @ B @ D2 ) ) ) ) ).
% diff_strict_mono
thf(fact_777_diff__eq__diff__less,axiom,
! [A: int,B: int,C: int,D2: int] :
( ( ( minus_minus_int @ A @ B )
= ( minus_minus_int @ C @ D2 ) )
=> ( ( ord_less_int @ A @ B )
= ( ord_less_int @ C @ D2 ) ) ) ).
% diff_eq_diff_less
thf(fact_778_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_779_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_780_add__mono__thms__linordered__semiring_I3_J,axiom,
! [I: int,J: int,K: int,L: int] :
( ( ( ord_less_eq_int @ I @ J )
& ( K = L ) )
=> ( ord_less_eq_int @ ( plus_plus_int @ I @ K ) @ ( plus_plus_int @ J @ L ) ) ) ).
% add_mono_thms_linordered_semiring(3)
thf(fact_781_add__mono__thms__linordered__semiring_I3_J,axiom,
! [I: nat,J: nat,K: nat,L: nat] :
( ( ( ord_less_eq_nat @ I @ J )
& ( K = L ) )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ I @ K ) @ ( plus_plus_nat @ J @ L ) ) ) ).
% add_mono_thms_linordered_semiring(3)
thf(fact_782_add__mono__thms__linordered__semiring_I2_J,axiom,
! [I: int,J: int,K: int,L: int] :
( ( ( I = J )
& ( ord_less_eq_int @ K @ L ) )
=> ( ord_less_eq_int @ ( plus_plus_int @ I @ K ) @ ( plus_plus_int @ J @ L ) ) ) ).
% add_mono_thms_linordered_semiring(2)
thf(fact_783_add__mono__thms__linordered__semiring_I2_J,axiom,
! [I: nat,J: nat,K: nat,L: nat] :
( ( ( I = J )
& ( ord_less_eq_nat @ K @ L ) )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ I @ K ) @ ( plus_plus_nat @ J @ L ) ) ) ).
% add_mono_thms_linordered_semiring(2)
thf(fact_784_add__mono__thms__linordered__semiring_I1_J,axiom,
! [I: int,J: int,K: int,L: int] :
( ( ( ord_less_eq_int @ I @ J )
& ( ord_less_eq_int @ K @ L ) )
=> ( ord_less_eq_int @ ( plus_plus_int @ I @ K ) @ ( plus_plus_int @ J @ L ) ) ) ).
% add_mono_thms_linordered_semiring(1)
thf(fact_785_add__mono__thms__linordered__semiring_I1_J,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_mono_thms_linordered_semiring(1)
thf(fact_786_add__mono,axiom,
! [A: int,B: int,C: int,D2: int] :
( ( ord_less_eq_int @ A @ B )
=> ( ( ord_less_eq_int @ C @ D2 )
=> ( ord_less_eq_int @ ( plus_plus_int @ A @ C ) @ ( plus_plus_int @ B @ D2 ) ) ) ) ).
% add_mono
thf(fact_787_add__mono,axiom,
! [A: nat,B: nat,C: nat,D2: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ord_less_eq_nat @ C @ D2 )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ A @ C ) @ ( plus_plus_nat @ B @ D2 ) ) ) ) ).
% add_mono
thf(fact_788_add__left__mono,axiom,
! [A: int,B: int,C: int] :
( ( ord_less_eq_int @ A @ B )
=> ( ord_less_eq_int @ ( plus_plus_int @ C @ A ) @ ( plus_plus_int @ C @ B ) ) ) ).
% add_left_mono
thf(fact_789_add__left__mono,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ C @ A ) @ ( plus_plus_nat @ C @ B ) ) ) ).
% add_left_mono
thf(fact_790_less__eqE,axiom,
! [A: nat,B: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ~ ! [C3: nat] :
( B
!= ( plus_plus_nat @ A @ C3 ) ) ) ).
% less_eqE
thf(fact_791_add__right__mono,axiom,
! [A: int,B: int,C: int] :
( ( ord_less_eq_int @ A @ B )
=> ( ord_less_eq_int @ ( plus_plus_int @ A @ C ) @ ( plus_plus_int @ B @ C ) ) ) ).
% add_right_mono
thf(fact_792_add__right__mono,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ A @ C ) @ ( plus_plus_nat @ B @ C ) ) ) ).
% add_right_mono
thf(fact_793_le__iff__add,axiom,
( ord_less_eq_nat
= ( ^ [A3: nat,B2: nat] :
? [C2: nat] :
( B2
= ( plus_plus_nat @ A3 @ C2 ) ) ) ) ).
% le_iff_add
thf(fact_794_add__le__imp__le__left,axiom,
! [C: int,A: int,B: int] :
( ( ord_less_eq_int @ ( plus_plus_int @ C @ A ) @ ( plus_plus_int @ C @ B ) )
=> ( ord_less_eq_int @ A @ B ) ) ).
% add_le_imp_le_left
thf(fact_795_add__le__imp__le__left,axiom,
! [C: nat,A: nat,B: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ C @ A ) @ ( plus_plus_nat @ C @ B ) )
=> ( ord_less_eq_nat @ A @ B ) ) ).
% add_le_imp_le_left
thf(fact_796_add__le__imp__le__right,axiom,
! [A: int,C: int,B: int] :
( ( ord_less_eq_int @ ( plus_plus_int @ A @ C ) @ ( plus_plus_int @ B @ C ) )
=> ( ord_less_eq_int @ A @ B ) ) ).
% add_le_imp_le_right
thf(fact_797_add__le__imp__le__right,axiom,
! [A: nat,C: nat,B: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ A @ C ) @ ( plus_plus_nat @ B @ C ) )
=> ( ord_less_eq_nat @ A @ B ) ) ).
% add_le_imp_le_right
thf(fact_798_minus__diff__commute,axiom,
! [B: int,A: int] :
( ( minus_minus_int @ ( uminus_uminus_int @ B ) @ A )
= ( minus_minus_int @ ( uminus_uminus_int @ A ) @ B ) ) ).
% minus_diff_commute
thf(fact_799_diff__less__eq,axiom,
! [A: int,B: int,C: int] :
( ( ord_less_int @ ( minus_minus_int @ A @ B ) @ C )
= ( ord_less_int @ A @ ( plus_plus_int @ C @ B ) ) ) ).
% diff_less_eq
thf(fact_800_less__diff__eq,axiom,
! [A: int,C: int,B: int] :
( ( ord_less_int @ A @ ( minus_minus_int @ C @ B ) )
= ( ord_less_int @ ( plus_plus_int @ A @ B ) @ C ) ) ).
% less_diff_eq
thf(fact_801_diff__diff__eq,axiom,
! [A: nat,B: nat,C: nat] :
( ( minus_minus_nat @ ( minus_minus_nat @ A @ B ) @ C )
= ( minus_minus_nat @ A @ ( plus_plus_nat @ B @ C ) ) ) ).
% diff_diff_eq
thf(fact_802_diff__diff__eq,axiom,
! [A: int,B: int,C: int] :
( ( minus_minus_int @ ( minus_minus_int @ A @ B ) @ C )
= ( minus_minus_int @ A @ ( plus_plus_int @ B @ C ) ) ) ).
% diff_diff_eq
thf(fact_803_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_804_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_805_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_806_add__implies__diff,axiom,
! [C: nat,B: nat,A: nat] :
( ( ( plus_plus_nat @ C @ B )
= A )
=> ( C
= ( minus_minus_nat @ A @ B ) ) ) ).
% add_implies_diff
thf(fact_807_add__implies__diff,axiom,
! [C: int,B: int,A: int] :
( ( ( plus_plus_int @ C @ B )
= A )
=> ( C
= ( minus_minus_int @ A @ B ) ) ) ).
% add_implies_diff
thf(fact_808_add__right__imp__eq,axiom,
! [B: int,A: int,C: int] :
( ( ( plus_plus_int @ B @ A )
= ( plus_plus_int @ C @ A ) )
=> ( B = C ) ) ).
% add_right_imp_eq
thf(fact_809_add__right__imp__eq,axiom,
! [B: nat,A: nat,C: nat] :
( ( ( plus_plus_nat @ B @ A )
= ( plus_plus_nat @ C @ A ) )
=> ( B = C ) ) ).
% add_right_imp_eq
thf(fact_810_diff__add__eq__diff__diff__swap,axiom,
! [A: int,B: int,C: int] :
( ( minus_minus_int @ A @ ( plus_plus_int @ B @ C ) )
= ( minus_minus_int @ ( minus_minus_int @ A @ C ) @ B ) ) ).
% diff_add_eq_diff_diff_swap
thf(fact_811_add__left__imp__eq,axiom,
! [A: int,B: int,C: int] :
( ( ( plus_plus_int @ A @ B )
= ( plus_plus_int @ A @ C ) )
=> ( B = C ) ) ).
% add_left_imp_eq
thf(fact_812_add__left__imp__eq,axiom,
! [A: nat,B: nat,C: nat] :
( ( ( plus_plus_nat @ A @ B )
= ( plus_plus_nat @ A @ C ) )
=> ( B = C ) ) ).
% add_left_imp_eq
thf(fact_813_add_Oleft__commute,axiom,
! [B: int,A: int,C: int] :
( ( plus_plus_int @ B @ ( plus_plus_int @ A @ C ) )
= ( plus_plus_int @ A @ ( plus_plus_int @ B @ C ) ) ) ).
% add.left_commute
thf(fact_814_add_Oleft__commute,axiom,
! [B: nat,A: nat,C: nat] :
( ( plus_plus_nat @ B @ ( plus_plus_nat @ A @ C ) )
= ( plus_plus_nat @ A @ ( plus_plus_nat @ B @ C ) ) ) ).
% add.left_commute
thf(fact_815_add_Ocommute,axiom,
( plus_plus_int
= ( ^ [A3: int,B2: int] : ( plus_plus_int @ B2 @ A3 ) ) ) ).
% add.commute
thf(fact_816_add_Ocommute,axiom,
( plus_plus_nat
= ( ^ [A3: nat,B2: nat] : ( plus_plus_nat @ B2 @ A3 ) ) ) ).
% add.commute
thf(fact_817_add_Oright__cancel,axiom,
! [B: int,A: int,C: int] :
( ( ( plus_plus_int @ B @ A )
= ( plus_plus_int @ C @ A ) )
= ( B = C ) ) ).
% add.right_cancel
thf(fact_818_diff__eq__diff__eq,axiom,
! [A: int,B: int,C: int,D2: int] :
( ( ( minus_minus_int @ A @ B )
= ( minus_minus_int @ C @ D2 ) )
=> ( ( A = B )
= ( C = D2 ) ) ) ).
% diff_eq_diff_eq
thf(fact_819_add_Oleft__cancel,axiom,
! [A: int,B: int,C: int] :
( ( ( plus_plus_int @ A @ B )
= ( plus_plus_int @ A @ C ) )
= ( B = C ) ) ).
% add.left_cancel
thf(fact_820_diff__add__eq,axiom,
! [A: int,B: int,C: int] :
( ( plus_plus_int @ ( minus_minus_int @ A @ B ) @ C )
= ( minus_minus_int @ ( plus_plus_int @ A @ C ) @ B ) ) ).
% diff_add_eq
thf(fact_821_add_Oassoc,axiom,
! [A: int,B: int,C: int] :
( ( plus_plus_int @ ( plus_plus_int @ A @ B ) @ C )
= ( plus_plus_int @ A @ ( plus_plus_int @ B @ C ) ) ) ).
% add.assoc
thf(fact_822_add_Oassoc,axiom,
! [A: nat,B: nat,C: nat] :
( ( plus_plus_nat @ ( plus_plus_nat @ A @ B ) @ C )
= ( plus_plus_nat @ A @ ( plus_plus_nat @ B @ C ) ) ) ).
% add.assoc
thf(fact_823_diff__diff__eq2,axiom,
! [A: int,B: int,C: int] :
( ( minus_minus_int @ A @ ( minus_minus_int @ B @ C ) )
= ( minus_minus_int @ ( plus_plus_int @ A @ C ) @ B ) ) ).
% diff_diff_eq2
thf(fact_824_add__diff__eq,axiom,
! [A: int,B: int,C: int] :
( ( plus_plus_int @ A @ ( minus_minus_int @ B @ C ) )
= ( minus_minus_int @ ( plus_plus_int @ A @ B ) @ C ) ) ).
% add_diff_eq
thf(fact_825_eq__diff__eq,axiom,
! [A: int,C: int,B: int] :
( ( A
= ( minus_minus_int @ C @ B ) )
= ( ( plus_plus_int @ A @ B )
= C ) ) ).
% eq_diff_eq
thf(fact_826_diff__eq__eq,axiom,
! [A: int,B: int,C: int] :
( ( ( minus_minus_int @ A @ B )
= C )
= ( A
= ( plus_plus_int @ C @ B ) ) ) ).
% diff_eq_eq
thf(fact_827_group__cancel_Osub1,axiom,
! [A2: int,K: int,A: int,B: int] :
( ( A2
= ( plus_plus_int @ K @ A ) )
=> ( ( minus_minus_int @ A2 @ B )
= ( plus_plus_int @ K @ ( minus_minus_int @ A @ B ) ) ) ) ).
% group_cancel.sub1
thf(fact_828_group__cancel_Oadd2,axiom,
! [B4: int,K: int,B: int,A: int] :
( ( B4
= ( plus_plus_int @ K @ B ) )
=> ( ( plus_plus_int @ A @ B4 )
= ( plus_plus_int @ K @ ( plus_plus_int @ A @ B ) ) ) ) ).
% group_cancel.add2
thf(fact_829_group__cancel_Oadd2,axiom,
! [B4: nat,K: nat,B: nat,A: nat] :
( ( B4
= ( plus_plus_nat @ K @ B ) )
=> ( ( plus_plus_nat @ A @ B4 )
= ( plus_plus_nat @ K @ ( plus_plus_nat @ A @ B ) ) ) ) ).
% group_cancel.add2
thf(fact_830_group__cancel_Oadd1,axiom,
! [A2: int,K: int,A: int,B: int] :
( ( A2
= ( plus_plus_int @ K @ A ) )
=> ( ( plus_plus_int @ A2 @ B )
= ( plus_plus_int @ K @ ( plus_plus_int @ A @ B ) ) ) ) ).
% group_cancel.add1
thf(fact_831_group__cancel_Oadd1,axiom,
! [A2: nat,K: nat,A: nat,B: nat] :
( ( A2
= ( plus_plus_nat @ K @ A ) )
=> ( ( plus_plus_nat @ A2 @ B )
= ( plus_plus_nat @ K @ ( plus_plus_nat @ A @ B ) ) ) ) ).
% group_cancel.add1
thf(fact_832_add__mono__thms__linordered__semiring_I4_J,axiom,
! [I: int,J: int,K: int,L: int] :
( ( ( I = J )
& ( K = L ) )
=> ( ( plus_plus_int @ I @ K )
= ( plus_plus_int @ J @ L ) ) ) ).
% add_mono_thms_linordered_semiring(4)
thf(fact_833_add__mono__thms__linordered__semiring_I4_J,axiom,
! [I: nat,J: nat,K: nat,L: nat] :
( ( ( I = J )
& ( K = L ) )
=> ( ( plus_plus_nat @ I @ K )
= ( plus_plus_nat @ J @ L ) ) ) ).
% add_mono_thms_linordered_semiring(4)
thf(fact_834_ab__semigroup__add__class_Oadd__ac_I1_J,axiom,
! [A: int,B: int,C: int] :
( ( plus_plus_int @ ( plus_plus_int @ A @ B ) @ C )
= ( plus_plus_int @ A @ ( plus_plus_int @ B @ C ) ) ) ).
% ab_semigroup_add_class.add_ac(1)
thf(fact_835_ab__semigroup__add__class_Oadd__ac_I1_J,axiom,
! [A: nat,B: nat,C: nat] :
( ( plus_plus_nat @ ( plus_plus_nat @ A @ B ) @ C )
= ( plus_plus_nat @ A @ ( plus_plus_nat @ B @ C ) ) ) ).
% ab_semigroup_add_class.add_ac(1)
thf(fact_836_is__num__normalize_I1_J,axiom,
! [A: int,B: int,C: int] :
( ( plus_plus_int @ ( plus_plus_int @ A @ B ) @ C )
= ( plus_plus_int @ A @ ( plus_plus_int @ B @ C ) ) ) ).
% is_num_normalize(1)
thf(fact_837_diff__le__eq,axiom,
! [A: int,B: int,C: int] :
( ( ord_less_eq_int @ ( minus_minus_int @ A @ B ) @ C )
= ( ord_less_eq_int @ A @ ( plus_plus_int @ C @ B ) ) ) ).
% diff_le_eq
thf(fact_838_le__diff__eq,axiom,
! [A: int,C: int,B: int] :
( ( ord_less_eq_int @ A @ ( minus_minus_int @ C @ B ) )
= ( ord_less_eq_int @ ( plus_plus_int @ A @ B ) @ C ) ) ).
% le_diff_eq
thf(fact_839_diff__add,axiom,
! [A: nat,B: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( plus_plus_nat @ ( minus_minus_nat @ B @ A ) @ A )
= B ) ) ).
% diff_add
thf(fact_840_le__add__diff,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ord_less_eq_nat @ C @ ( minus_minus_nat @ ( plus_plus_nat @ B @ C ) @ A ) ) ) ).
% le_add_diff
thf(fact_841_ordered__cancel__comm__monoid__diff__class_Ole__diff__conv2,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ord_less_eq_nat @ C @ ( minus_minus_nat @ B @ A ) )
= ( ord_less_eq_nat @ ( plus_plus_nat @ C @ A ) @ B ) ) ) ).
% ordered_cancel_comm_monoid_diff_class.le_diff_conv2
thf(fact_842_ordered__cancel__comm__monoid__diff__class_Oadd__diff__assoc,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( plus_plus_nat @ C @ ( minus_minus_nat @ B @ A ) )
= ( minus_minus_nat @ ( plus_plus_nat @ C @ B ) @ A ) ) ) ).
% ordered_cancel_comm_monoid_diff_class.add_diff_assoc
thf(fact_843_ordered__cancel__comm__monoid__diff__class_Odiff__add__assoc,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( minus_minus_nat @ ( plus_plus_nat @ C @ B ) @ A )
= ( plus_plus_nat @ C @ ( minus_minus_nat @ B @ A ) ) ) ) ).
% ordered_cancel_comm_monoid_diff_class.diff_add_assoc
thf(fact_844_ordered__cancel__comm__monoid__diff__class_Oadd__diff__assoc2,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( plus_plus_nat @ ( minus_minus_nat @ B @ A ) @ C )
= ( minus_minus_nat @ ( plus_plus_nat @ B @ C ) @ A ) ) ) ).
% ordered_cancel_comm_monoid_diff_class.add_diff_assoc2
thf(fact_845_ordered__cancel__comm__monoid__diff__class_Odiff__add__assoc2,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( minus_minus_nat @ ( plus_plus_nat @ B @ C ) @ A )
= ( plus_plus_nat @ ( minus_minus_nat @ B @ A ) @ C ) ) ) ).
% ordered_cancel_comm_monoid_diff_class.diff_add_assoc2
thf(fact_846_ordered__cancel__comm__monoid__diff__class_Odiff__diff__right,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( minus_minus_nat @ C @ ( minus_minus_nat @ B @ A ) )
= ( minus_minus_nat @ ( plus_plus_nat @ C @ A ) @ B ) ) ) ).
% ordered_cancel_comm_monoid_diff_class.diff_diff_right
thf(fact_847_ordered__cancel__comm__monoid__diff__class_Oadd__diff__inverse,axiom,
! [A: nat,B: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( plus_plus_nat @ A @ ( minus_minus_nat @ B @ A ) )
= B ) ) ).
% ordered_cancel_comm_monoid_diff_class.add_diff_inverse
thf(fact_848_ordered__cancel__comm__monoid__diff__class_Ole__imp__diff__is__add,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ord_less_eq_nat @ A @ B )
=> ( ( ( minus_minus_nat @ B @ A )
= C )
= ( B
= ( plus_plus_nat @ C @ A ) ) ) ) ) ).
% ordered_cancel_comm_monoid_diff_class.le_imp_diff_is_add
thf(fact_849_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_850_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_851_group__cancel_Osub2,axiom,
! [B4: int,K: int,B: int,A: int] :
( ( B4
= ( plus_plus_int @ K @ B ) )
=> ( ( minus_minus_int @ A @ B4 )
= ( plus_plus_int @ ( uminus_uminus_int @ K ) @ ( minus_minus_int @ A @ B ) ) ) ) ).
% group_cancel.sub2
thf(fact_852_diff__conv__add__uminus,axiom,
( minus_minus_int
= ( ^ [A3: int,B2: int] : ( plus_plus_int @ A3 @ ( uminus_uminus_int @ B2 ) ) ) ) ).
% diff_conv_add_uminus
thf(fact_853_ab__group__add__class_Oab__diff__conv__add__uminus,axiom,
( minus_minus_int
= ( ^ [A3: int,B2: int] : ( plus_plus_int @ A3 @ ( uminus_uminus_int @ B2 ) ) ) ) ).
% ab_group_add_class.ab_diff_conv_add_uminus
thf(fact_854_plus__int__code_I2_J,axiom,
! [L: int] :
( ( plus_plus_int @ zero_zero_int @ L )
= L ) ).
% plus_int_code(2)
thf(fact_855_plus__int__code_I1_J,axiom,
! [K: int] :
( ( plus_plus_int @ K @ zero_zero_int )
= K ) ).
% plus_int_code(1)
thf(fact_856_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_857_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_858_diff__le__self,axiom,
! [M: nat,N: nat] : ( ord_less_eq_nat @ ( minus_minus_nat @ M @ N ) @ M ) ).
% diff_le_self
thf(fact_859_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_860_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_861_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_862_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_863_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_864_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_865_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_866_minus__nat_Odiff__0,axiom,
! [M: nat] :
( ( minus_minus_nat @ M @ zero_zero_nat )
= M ) ).
% minus_nat.diff_0
thf(fact_867_group__cancel_Oneg1,axiom,
! [A2: int,K: int,A: int] :
( ( A2
= ( plus_plus_int @ K @ A ) )
=> ( ( uminus_uminus_int @ A2 )
= ( plus_plus_int @ ( uminus_uminus_int @ K ) @ ( uminus_uminus_int @ A ) ) ) ) ).
% group_cancel.neg1
thf(fact_868_add_Oinverse__distrib__swap,axiom,
! [A: int,B: int] :
( ( uminus_uminus_int @ ( plus_plus_int @ A @ B ) )
= ( plus_plus_int @ ( uminus_uminus_int @ B ) @ ( uminus_uminus_int @ A ) ) ) ).
% add.inverse_distrib_swap
thf(fact_869_is__num__normalize_I8_J,axiom,
! [A: int,B: int] :
( ( uminus_uminus_int @ ( plus_plus_int @ A @ B ) )
= ( plus_plus_int @ ( uminus_uminus_int @ B ) @ ( uminus_uminus_int @ A ) ) ) ).
% is_num_normalize(8)
thf(fact_870_add__mono__thms__linordered__field_I5_J,axiom,
! [I: int,J: int,K: int,L: int] :
( ( ( ord_less_int @ I @ J )
& ( ord_less_int @ K @ L ) )
=> ( ord_less_int @ ( plus_plus_int @ I @ K ) @ ( plus_plus_int @ J @ L ) ) ) ).
% add_mono_thms_linordered_field(5)
thf(fact_871_add__mono__thms__linordered__field_I5_J,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_mono_thms_linordered_field(5)
thf(fact_872_add__mono__thms__linordered__field_I2_J,axiom,
! [I: int,J: int,K: int,L: int] :
( ( ( I = J )
& ( ord_less_int @ K @ L ) )
=> ( ord_less_int @ ( plus_plus_int @ I @ K ) @ ( plus_plus_int @ J @ L ) ) ) ).
% add_mono_thms_linordered_field(2)
thf(fact_873_add__mono__thms__linordered__field_I2_J,axiom,
! [I: nat,J: nat,K: nat,L: nat] :
( ( ( I = J )
& ( ord_less_nat @ K @ L ) )
=> ( ord_less_nat @ ( plus_plus_nat @ I @ K ) @ ( plus_plus_nat @ J @ L ) ) ) ).
% add_mono_thms_linordered_field(2)
thf(fact_874_add__mono__thms__linordered__field_I1_J,axiom,
! [I: int,J: int,K: int,L: int] :
( ( ( ord_less_int @ I @ J )
& ( K = L ) )
=> ( ord_less_int @ ( plus_plus_int @ I @ K ) @ ( plus_plus_int @ J @ L ) ) ) ).
% add_mono_thms_linordered_field(1)
thf(fact_875_add__mono__thms__linordered__field_I1_J,axiom,
! [I: nat,J: nat,K: nat,L: nat] :
( ( ( ord_less_nat @ I @ J )
& ( K = L ) )
=> ( ord_less_nat @ ( plus_plus_nat @ I @ K ) @ ( plus_plus_nat @ J @ L ) ) ) ).
% add_mono_thms_linordered_field(1)
thf(fact_876_add__strict__mono,axiom,
! [A: int,B: int,C: int,D2: int] :
( ( ord_less_int @ A @ B )
=> ( ( ord_less_int @ C @ D2 )
=> ( ord_less_int @ ( plus_plus_int @ A @ C ) @ ( plus_plus_int @ B @ D2 ) ) ) ) ).
% add_strict_mono
thf(fact_877_add__strict__mono,axiom,
! [A: nat,B: nat,C: nat,D2: nat] :
( ( ord_less_nat @ A @ B )
=> ( ( ord_less_nat @ C @ D2 )
=> ( ord_less_nat @ ( plus_plus_nat @ A @ C ) @ ( plus_plus_nat @ B @ D2 ) ) ) ) ).
% add_strict_mono
thf(fact_878_add__strict__left__mono,axiom,
! [A: int,B: int,C: int] :
( ( ord_less_int @ A @ B )
=> ( ord_less_int @ ( plus_plus_int @ C @ A ) @ ( plus_plus_int @ C @ B ) ) ) ).
% add_strict_left_mono
thf(fact_879_add__strict__left__mono,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_nat @ A @ B )
=> ( ord_less_nat @ ( plus_plus_nat @ C @ A ) @ ( plus_plus_nat @ C @ B ) ) ) ).
% add_strict_left_mono
thf(fact_880_add__strict__right__mono,axiom,
! [A: int,B: int,C: int] :
( ( ord_less_int @ A @ B )
=> ( ord_less_int @ ( plus_plus_int @ A @ C ) @ ( plus_plus_int @ B @ C ) ) ) ).
% add_strict_right_mono
thf(fact_881_add__strict__right__mono,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_nat @ A @ B )
=> ( ord_less_nat @ ( plus_plus_nat @ A @ C ) @ ( plus_plus_nat @ B @ C ) ) ) ).
% add_strict_right_mono
thf(fact_882_add__less__imp__less__left,axiom,
! [C: int,A: int,B: int] :
( ( ord_less_int @ ( plus_plus_int @ C @ A ) @ ( plus_plus_int @ C @ B ) )
=> ( ord_less_int @ A @ B ) ) ).
% add_less_imp_less_left
thf(fact_883_add__less__imp__less__left,axiom,
! [C: nat,A: nat,B: nat] :
( ( ord_less_nat @ ( plus_plus_nat @ C @ A ) @ ( plus_plus_nat @ C @ B ) )
=> ( ord_less_nat @ A @ B ) ) ).
% add_less_imp_less_left
thf(fact_884_add__less__imp__less__right,axiom,
! [A: int,C: int,B: int] :
( ( ord_less_int @ ( plus_plus_int @ A @ C ) @ ( plus_plus_int @ B @ C ) )
=> ( ord_less_int @ A @ B ) ) ).
% add_less_imp_less_right
thf(fact_885_add__less__imp__less__right,axiom,
! [A: nat,C: nat,B: nat] :
( ( ord_less_nat @ ( plus_plus_nat @ A @ C ) @ ( plus_plus_nat @ B @ C ) )
=> ( ord_less_nat @ A @ B ) ) ).
% add_less_imp_less_right
thf(fact_886_verit__sum__simplify,axiom,
! [A: int] :
( ( plus_plus_int @ A @ zero_zero_int )
= A ) ).
% verit_sum_simplify
thf(fact_887_verit__sum__simplify,axiom,
! [A: nat] :
( ( plus_plus_nat @ A @ zero_zero_nat )
= A ) ).
% verit_sum_simplify
thf(fact_888_comm__monoid__add__class_Oadd__0,axiom,
! [A: int] :
( ( plus_plus_int @ zero_zero_int @ A )
= A ) ).
% comm_monoid_add_class.add_0
thf(fact_889_comm__monoid__add__class_Oadd__0,axiom,
! [A: nat] :
( ( plus_plus_nat @ zero_zero_nat @ A )
= A ) ).
% comm_monoid_add_class.add_0
thf(fact_890_add_Ocomm__neutral,axiom,
! [A: int] :
( ( plus_plus_int @ A @ zero_zero_int )
= A ) ).
% add.comm_neutral
thf(fact_891_add_Ocomm__neutral,axiom,
! [A: nat] :
( ( plus_plus_nat @ A @ zero_zero_nat )
= A ) ).
% add.comm_neutral
thf(fact_892_add_Ogroup__left__neutral,axiom,
! [A: int] :
( ( plus_plus_int @ zero_zero_int @ A )
= A ) ).
% add.group_left_neutral
thf(fact_893_add__le__add__imp__diff__le,axiom,
! [I: int,K: int,N: int,J: int] :
( ( ord_less_eq_int @ ( plus_plus_int @ I @ K ) @ N )
=> ( ( ord_less_eq_int @ N @ ( plus_plus_int @ J @ K ) )
=> ( ( ord_less_eq_int @ ( plus_plus_int @ I @ K ) @ N )
=> ( ( ord_less_eq_int @ N @ ( plus_plus_int @ J @ K ) )
=> ( ord_less_eq_int @ ( minus_minus_int @ N @ K ) @ J ) ) ) ) ) ).
% add_le_add_imp_diff_le
thf(fact_894_add__le__add__imp__diff__le,axiom,
! [I: nat,K: nat,N: nat,J: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ I @ K ) @ N )
=> ( ( ord_less_eq_nat @ N @ ( plus_plus_nat @ J @ K ) )
=> ( ( ord_less_eq_nat @ ( plus_plus_nat @ I @ K ) @ N )
=> ( ( ord_less_eq_nat @ N @ ( plus_plus_nat @ J @ K ) )
=> ( ord_less_eq_nat @ ( minus_minus_nat @ N @ K ) @ J ) ) ) ) ) ).
% add_le_add_imp_diff_le
thf(fact_895_add__le__imp__le__diff,axiom,
! [I: int,K: int,N: int] :
( ( ord_less_eq_int @ ( plus_plus_int @ I @ K ) @ N )
=> ( ord_less_eq_int @ I @ ( minus_minus_int @ N @ K ) ) ) ).
% add_le_imp_le_diff
thf(fact_896_add__le__imp__le__diff,axiom,
! [I: nat,K: nat,N: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ I @ K ) @ N )
=> ( ord_less_eq_nat @ I @ ( minus_minus_nat @ N @ K ) ) ) ).
% add_le_imp_le_diff
thf(fact_897_sgn__eq__0__iff,axiom,
! [A: int] :
( ( ( sgn_sgn_int @ A )
= zero_zero_int )
= ( A = zero_zero_int ) ) ).
% sgn_eq_0_iff
thf(fact_898_sgn__0__0,axiom,
! [A: int] :
( ( ( sgn_sgn_int @ A )
= zero_zero_int )
= ( A = zero_zero_int ) ) ).
% sgn_0_0
thf(fact_899_linordered__semidom__class_Oadd__diff__inverse,axiom,
! [A: int,B: int] :
( ~ ( ord_less_int @ A @ B )
=> ( ( plus_plus_int @ B @ ( minus_minus_int @ A @ B ) )
= A ) ) ).
% linordered_semidom_class.add_diff_inverse
thf(fact_900_linordered__semidom__class_Oadd__diff__inverse,axiom,
! [A: nat,B: nat] :
( ~ ( ord_less_nat @ A @ B )
=> ( ( plus_plus_nat @ B @ ( minus_minus_nat @ A @ B ) )
= A ) ) ).
% linordered_semidom_class.add_diff_inverse
thf(fact_901_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_902_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_903_add__nonpos__eq__0__iff,axiom,
! [X: int,Y: int] :
( ( ord_less_eq_int @ X @ zero_zero_int )
=> ( ( ord_less_eq_int @ Y @ zero_zero_int )
=> ( ( ( plus_plus_int @ X @ Y )
= zero_zero_int )
= ( ( X = zero_zero_int )
& ( Y = zero_zero_int ) ) ) ) ) ).
% add_nonpos_eq_0_iff
thf(fact_904_add__nonpos__eq__0__iff,axiom,
! [X: nat,Y: nat] :
( ( ord_less_eq_nat @ X @ zero_zero_nat )
=> ( ( ord_less_eq_nat @ Y @ zero_zero_nat )
=> ( ( ( plus_plus_nat @ X @ Y )
= zero_zero_nat )
= ( ( X = zero_zero_nat )
& ( Y = zero_zero_nat ) ) ) ) ) ).
% add_nonpos_eq_0_iff
thf(fact_905_add__nonneg__eq__0__iff,axiom,
! [X: int,Y: int] :
( ( ord_less_eq_int @ zero_zero_int @ X )
=> ( ( ord_less_eq_int @ zero_zero_int @ Y )
=> ( ( ( plus_plus_int @ X @ Y )
= zero_zero_int )
= ( ( X = zero_zero_int )
& ( Y = zero_zero_int ) ) ) ) ) ).
% add_nonneg_eq_0_iff
thf(fact_906_add__nonneg__eq__0__iff,axiom,
! [X: nat,Y: nat] :
( ( ord_less_eq_nat @ zero_zero_nat @ X )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ Y )
=> ( ( ( plus_plus_nat @ X @ Y )
= zero_zero_nat )
= ( ( X = zero_zero_nat )
& ( Y = zero_zero_nat ) ) ) ) ) ).
% add_nonneg_eq_0_iff
thf(fact_907_add__nonpos__nonpos,axiom,
! [A: int,B: int] :
( ( ord_less_eq_int @ A @ zero_zero_int )
=> ( ( ord_less_eq_int @ B @ zero_zero_int )
=> ( ord_less_eq_int @ ( plus_plus_int @ A @ B ) @ zero_zero_int ) ) ) ).
% add_nonpos_nonpos
thf(fact_908_add__nonpos__nonpos,axiom,
! [A: nat,B: nat] :
( ( ord_less_eq_nat @ A @ zero_zero_nat )
=> ( ( ord_less_eq_nat @ B @ zero_zero_nat )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ A @ B ) @ zero_zero_nat ) ) ) ).
% add_nonpos_nonpos
thf(fact_909_add__nonneg__nonneg,axiom,
! [A: int,B: int] :
( ( ord_less_eq_int @ zero_zero_int @ A )
=> ( ( ord_less_eq_int @ zero_zero_int @ B )
=> ( ord_less_eq_int @ zero_zero_int @ ( plus_plus_int @ A @ B ) ) ) ) ).
% add_nonneg_nonneg
thf(fact_910_add__nonneg__nonneg,axiom,
! [A: nat,B: nat] :
( ( ord_less_eq_nat @ zero_zero_nat @ A )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ B )
=> ( ord_less_eq_nat @ zero_zero_nat @ ( plus_plus_nat @ A @ B ) ) ) ) ).
% add_nonneg_nonneg
thf(fact_911_add__increasing2,axiom,
! [C: int,B: int,A: int] :
( ( ord_less_eq_int @ zero_zero_int @ C )
=> ( ( ord_less_eq_int @ B @ A )
=> ( ord_less_eq_int @ B @ ( plus_plus_int @ A @ C ) ) ) ) ).
% add_increasing2
thf(fact_912_add__increasing2,axiom,
! [C: nat,B: nat,A: nat] :
( ( ord_less_eq_nat @ zero_zero_nat @ C )
=> ( ( ord_less_eq_nat @ B @ A )
=> ( ord_less_eq_nat @ B @ ( plus_plus_nat @ A @ C ) ) ) ) ).
% add_increasing2
thf(fact_913_add__decreasing2,axiom,
! [C: int,A: int,B: int] :
( ( ord_less_eq_int @ C @ zero_zero_int )
=> ( ( ord_less_eq_int @ A @ B )
=> ( ord_less_eq_int @ ( plus_plus_int @ A @ C ) @ B ) ) ) ).
% add_decreasing2
thf(fact_914_add__decreasing2,axiom,
! [C: nat,A: nat,B: nat] :
( ( ord_less_eq_nat @ C @ zero_zero_nat )
=> ( ( ord_less_eq_nat @ A @ B )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ A @ C ) @ B ) ) ) ).
% add_decreasing2
thf(fact_915_add__increasing,axiom,
! [A: int,B: int,C: int] :
( ( ord_less_eq_int @ zero_zero_int @ A )
=> ( ( ord_less_eq_int @ B @ C )
=> ( ord_less_eq_int @ B @ ( plus_plus_int @ A @ C ) ) ) ) ).
% add_increasing
thf(fact_916_add__increasing,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_eq_nat @ zero_zero_nat @ A )
=> ( ( ord_less_eq_nat @ B @ C )
=> ( ord_less_eq_nat @ B @ ( plus_plus_nat @ A @ C ) ) ) ) ).
% add_increasing
thf(fact_917_add__decreasing,axiom,
! [A: int,C: int,B: int] :
( ( ord_less_eq_int @ A @ zero_zero_int )
=> ( ( ord_less_eq_int @ C @ B )
=> ( ord_less_eq_int @ ( plus_plus_int @ A @ C ) @ B ) ) ) ).
% add_decreasing
thf(fact_918_add__decreasing,axiom,
! [A: nat,C: nat,B: nat] :
( ( ord_less_eq_nat @ A @ zero_zero_nat )
=> ( ( ord_less_eq_nat @ C @ B )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ A @ C ) @ B ) ) ) ).
% add_decreasing
thf(fact_919_add__less__le__mono,axiom,
! [A: int,B: int,C: int,D2: int] :
( ( ord_less_int @ A @ B )
=> ( ( ord_less_eq_int @ C @ D2 )
=> ( ord_less_int @ ( plus_plus_int @ A @ C ) @ ( plus_plus_int @ B @ D2 ) ) ) ) ).
% add_less_le_mono
thf(fact_920_add__less__le__mono,axiom,
! [A: nat,B: nat,C: nat,D2: nat] :
( ( ord_less_nat @ A @ B )
=> ( ( ord_less_eq_nat @ C @ D2 )
=> ( ord_less_nat @ ( plus_plus_nat @ A @ C ) @ ( plus_plus_nat @ B @ D2 ) ) ) ) ).
% add_less_le_mono
thf(fact_921_add__le__less__mono,axiom,
! [A: int,B: int,C: int,D2: int] :
( ( ord_less_eq_int @ A @ B )
=> ( ( ord_less_int @ C @ D2 )
=> ( ord_less_int @ ( plus_plus_int @ A @ C ) @ ( plus_plus_int @ B @ D2 ) ) ) ) ).
% add_le_less_mono
thf(fact_922_add__le__less__mono,axiom,
! [A: nat,B: nat,C: nat,D2: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ord_less_nat @ C @ D2 )
=> ( ord_less_nat @ ( plus_plus_nat @ A @ C ) @ ( plus_plus_nat @ B @ D2 ) ) ) ) ).
% add_le_less_mono
thf(fact_923_add__mono__thms__linordered__field_I3_J,axiom,
! [I: int,J: int,K: int,L: int] :
( ( ( ord_less_int @ I @ J )
& ( ord_less_eq_int @ K @ L ) )
=> ( ord_less_int @ ( plus_plus_int @ I @ K ) @ ( plus_plus_int @ J @ L ) ) ) ).
% add_mono_thms_linordered_field(3)
thf(fact_924_add__mono__thms__linordered__field_I3_J,axiom,
! [I: nat,J: nat,K: nat,L: nat] :
( ( ( ord_less_nat @ I @ J )
& ( ord_less_eq_nat @ K @ L ) )
=> ( ord_less_nat @ ( plus_plus_nat @ I @ K ) @ ( plus_plus_nat @ J @ L ) ) ) ).
% add_mono_thms_linordered_field(3)
thf(fact_925_add__mono__thms__linordered__field_I4_J,axiom,
! [I: int,J: int,K: int,L: int] :
( ( ( ord_less_eq_int @ I @ J )
& ( ord_less_int @ K @ L ) )
=> ( ord_less_int @ ( plus_plus_int @ I @ K ) @ ( plus_plus_int @ J @ L ) ) ) ).
% add_mono_thms_linordered_field(4)
thf(fact_926_add__mono__thms__linordered__field_I4_J,axiom,
! [I: nat,J: nat,K: nat,L: nat] :
( ( ( ord_less_eq_nat @ I @ J )
& ( ord_less_nat @ K @ L ) )
=> ( ord_less_nat @ ( plus_plus_nat @ I @ K ) @ ( plus_plus_nat @ J @ L ) ) ) ).
% add_mono_thms_linordered_field(4)
thf(fact_927_add__less__zeroD,axiom,
! [X: int,Y: int] :
( ( ord_less_int @ ( plus_plus_int @ X @ Y ) @ zero_zero_int )
=> ( ( ord_less_int @ X @ zero_zero_int )
| ( ord_less_int @ Y @ zero_zero_int ) ) ) ).
% add_less_zeroD
thf(fact_928_pos__add__strict,axiom,
! [A: int,B: int,C: int] :
( ( ord_less_int @ zero_zero_int @ A )
=> ( ( ord_less_int @ B @ C )
=> ( ord_less_int @ B @ ( plus_plus_int @ A @ C ) ) ) ) ).
% pos_add_strict
thf(fact_929_pos__add__strict,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_nat @ zero_zero_nat @ A )
=> ( ( ord_less_nat @ B @ C )
=> ( ord_less_nat @ B @ ( plus_plus_nat @ A @ C ) ) ) ) ).
% pos_add_strict
thf(fact_930_canonically__ordered__monoid__add__class_OlessE,axiom,
! [A: nat,B: nat] :
( ( ord_less_nat @ A @ B )
=> ~ ! [C3: nat] :
( ( B
= ( plus_plus_nat @ A @ C3 ) )
=> ( C3 = zero_zero_nat ) ) ) ).
% canonically_ordered_monoid_add_class.lessE
thf(fact_931_add__pos__pos,axiom,
! [A: int,B: int] :
( ( ord_less_int @ zero_zero_int @ A )
=> ( ( ord_less_int @ zero_zero_int @ B )
=> ( ord_less_int @ zero_zero_int @ ( plus_plus_int @ A @ B ) ) ) ) ).
% add_pos_pos
thf(fact_932_add__pos__pos,axiom,
! [A: nat,B: nat] :
( ( ord_less_nat @ zero_zero_nat @ A )
=> ( ( ord_less_nat @ zero_zero_nat @ B )
=> ( ord_less_nat @ zero_zero_nat @ ( plus_plus_nat @ A @ B ) ) ) ) ).
% add_pos_pos
thf(fact_933_add__neg__neg,axiom,
! [A: int,B: int] :
( ( ord_less_int @ A @ zero_zero_int )
=> ( ( ord_less_int @ B @ zero_zero_int )
=> ( ord_less_int @ ( plus_plus_int @ A @ B ) @ zero_zero_int ) ) ) ).
% add_neg_neg
thf(fact_934_add__neg__neg,axiom,
! [A: nat,B: nat] :
( ( ord_less_nat @ A @ zero_zero_nat )
=> ( ( ord_less_nat @ B @ zero_zero_nat )
=> ( ord_less_nat @ ( plus_plus_nat @ A @ B ) @ zero_zero_nat ) ) ) ).
% add_neg_neg
thf(fact_935_less__add__one,axiom,
! [A: int] : ( ord_less_int @ A @ ( plus_plus_int @ A @ one_one_int ) ) ).
% less_add_one
thf(fact_936_less__add__one,axiom,
! [A: nat] : ( ord_less_nat @ A @ ( plus_plus_nat @ A @ one_one_nat ) ) ).
% less_add_one
thf(fact_937_add__mono1,axiom,
! [A: int,B: int] :
( ( ord_less_int @ A @ B )
=> ( ord_less_int @ ( plus_plus_int @ A @ one_one_int ) @ ( plus_plus_int @ B @ one_one_int ) ) ) ).
% add_mono1
thf(fact_938_add__mono1,axiom,
! [A: nat,B: nat] :
( ( ord_less_nat @ A @ B )
=> ( ord_less_nat @ ( plus_plus_nat @ A @ one_one_nat ) @ ( plus_plus_nat @ B @ one_one_nat ) ) ) ).
% add_mono1
thf(fact_939_neg__eq__iff__add__eq__0,axiom,
! [A: int,B: int] :
( ( ( uminus_uminus_int @ A )
= B )
= ( ( plus_plus_int @ A @ B )
= zero_zero_int ) ) ).
% neg_eq_iff_add_eq_0
thf(fact_940_eq__neg__iff__add__eq__0,axiom,
! [A: int,B: int] :
( ( A
= ( uminus_uminus_int @ B ) )
= ( ( plus_plus_int @ A @ B )
= zero_zero_int ) ) ).
% eq_neg_iff_add_eq_0
thf(fact_941_add_Oinverse__unique,axiom,
! [A: int,B: int] :
( ( ( plus_plus_int @ A @ B )
= zero_zero_int )
=> ( ( uminus_uminus_int @ A )
= B ) ) ).
% add.inverse_unique
thf(fact_942_ab__group__add__class_Oab__left__minus,axiom,
! [A: int] :
( ( plus_plus_int @ ( uminus_uminus_int @ A ) @ A )
= zero_zero_int ) ).
% ab_group_add_class.ab_left_minus
thf(fact_943_add__eq__0__iff,axiom,
! [A: int,B: int] :
( ( ( plus_plus_int @ A @ B )
= zero_zero_int )
= ( B
= ( uminus_uminus_int @ A ) ) ) ).
% add_eq_0_iff
thf(fact_944_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_945_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_946_diff__less__Suc,axiom,
! [M: nat,N: nat] : ( ord_less_nat @ ( minus_minus_nat @ M @ N ) @ ( suc @ M ) ) ).
% diff_less_Suc
thf(fact_947_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_948_sgn__not__eq__imp,axiom,
! [B: int,A: int] :
( ( ( sgn_sgn_int @ B )
!= ( sgn_sgn_int @ A ) )
=> ( ( ( sgn_sgn_int @ A )
!= zero_zero_int )
=> ( ( ( sgn_sgn_int @ B )
!= zero_zero_int )
=> ( ( sgn_sgn_int @ A )
= ( uminus_uminus_int @ ( sgn_sgn_int @ B ) ) ) ) ) ) ).
% sgn_not_eq_imp
thf(fact_949_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_950_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_951_sgn__minus__1,axiom,
( ( sgn_sgn_int @ ( uminus_uminus_int @ one_one_int ) )
= ( uminus_uminus_int @ one_one_int ) ) ).
% sgn_minus_1
thf(fact_952_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_953_odd__nonzero,axiom,
! [Z2: int] :
( ( plus_plus_int @ ( plus_plus_int @ one_one_int @ Z2 ) @ Z2 )
!= zero_zero_int ) ).
% odd_nonzero
thf(fact_954_int__ge__induct,axiom,
! [K: int,I: int,P: int > $o] :
( ( ord_less_eq_int @ K @ I )
=> ( ( P @ K )
=> ( ! [I3: int] :
( ( ord_less_eq_int @ K @ I3 )
=> ( ( P @ I3 )
=> ( P @ ( plus_plus_int @ I3 @ one_one_int ) ) ) )
=> ( P @ I ) ) ) ) ).
% int_ge_induct
thf(fact_955_int__gr__induct,axiom,
! [K: int,I: int,P: int > $o] :
( ( ord_less_int @ K @ I )
=> ( ( P @ ( plus_plus_int @ K @ one_one_int ) )
=> ( ! [I3: int] :
( ( ord_less_int @ K @ I3 )
=> ( ( P @ I3 )
=> ( P @ ( plus_plus_int @ I3 @ one_one_int ) ) ) )
=> ( P @ I ) ) ) ) ).
% int_gr_induct
thf(fact_956_zless__add1__eq,axiom,
! [W: int,Z2: int] :
( ( ord_less_int @ W @ ( plus_plus_int @ Z2 @ one_one_int ) )
= ( ( ord_less_int @ W @ Z2 )
| ( W = Z2 ) ) ) ).
% zless_add1_eq
thf(fact_957_zle__iff__zadd,axiom,
( ord_less_eq_int
= ( ^ [W2: int,Z6: int] :
? [N3: nat] :
( Z6
= ( plus_plus_int @ W2 @ ( semiri1314217659103216013at_int @ N3 ) ) ) ) ) ).
% zle_iff_zadd
thf(fact_958_dbl__inc__def,axiom,
( neg_nu5851722552734809277nc_int
= ( ^ [X3: int] : ( plus_plus_int @ ( plus_plus_int @ X3 @ X3 ) @ one_one_int ) ) ) ).
% dbl_inc_def
thf(fact_959_add__strict__increasing2,axiom,
! [A: int,B: int,C: int] :
( ( ord_less_eq_int @ zero_zero_int @ A )
=> ( ( ord_less_int @ B @ C )
=> ( ord_less_int @ B @ ( plus_plus_int @ A @ C ) ) ) ) ).
% add_strict_increasing2
thf(fact_960_add__strict__increasing2,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_eq_nat @ zero_zero_nat @ A )
=> ( ( ord_less_nat @ B @ C )
=> ( ord_less_nat @ B @ ( plus_plus_nat @ A @ C ) ) ) ) ).
% add_strict_increasing2
thf(fact_961_add__strict__increasing,axiom,
! [A: int,B: int,C: int] :
( ( ord_less_int @ zero_zero_int @ A )
=> ( ( ord_less_eq_int @ B @ C )
=> ( ord_less_int @ B @ ( plus_plus_int @ A @ C ) ) ) ) ).
% add_strict_increasing
thf(fact_962_add__strict__increasing,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_nat @ zero_zero_nat @ A )
=> ( ( ord_less_eq_nat @ B @ C )
=> ( ord_less_nat @ B @ ( plus_plus_nat @ A @ C ) ) ) ) ).
% add_strict_increasing
thf(fact_963_add__pos__nonneg,axiom,
! [A: int,B: int] :
( ( ord_less_int @ zero_zero_int @ A )
=> ( ( ord_less_eq_int @ zero_zero_int @ B )
=> ( ord_less_int @ zero_zero_int @ ( plus_plus_int @ A @ B ) ) ) ) ).
% add_pos_nonneg
thf(fact_964_add__pos__nonneg,axiom,
! [A: nat,B: nat] :
( ( ord_less_nat @ zero_zero_nat @ A )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ B )
=> ( ord_less_nat @ zero_zero_nat @ ( plus_plus_nat @ A @ B ) ) ) ) ).
% add_pos_nonneg
thf(fact_965_add__nonpos__neg,axiom,
! [A: int,B: int] :
( ( ord_less_eq_int @ A @ zero_zero_int )
=> ( ( ord_less_int @ B @ zero_zero_int )
=> ( ord_less_int @ ( plus_plus_int @ A @ B ) @ zero_zero_int ) ) ) ).
% add_nonpos_neg
thf(fact_966_add__nonpos__neg,axiom,
! [A: nat,B: nat] :
( ( ord_less_eq_nat @ A @ zero_zero_nat )
=> ( ( ord_less_nat @ B @ zero_zero_nat )
=> ( ord_less_nat @ ( plus_plus_nat @ A @ B ) @ zero_zero_nat ) ) ) ).
% add_nonpos_neg
thf(fact_967_add__nonneg__pos,axiom,
! [A: int,B: int] :
( ( ord_less_eq_int @ zero_zero_int @ A )
=> ( ( ord_less_int @ zero_zero_int @ B )
=> ( ord_less_int @ zero_zero_int @ ( plus_plus_int @ A @ B ) ) ) ) ).
% add_nonneg_pos
thf(fact_968_add__nonneg__pos,axiom,
! [A: nat,B: nat] :
( ( ord_less_eq_nat @ zero_zero_nat @ A )
=> ( ( ord_less_nat @ zero_zero_nat @ B )
=> ( ord_less_nat @ zero_zero_nat @ ( plus_plus_nat @ A @ B ) ) ) ) ).
% add_nonneg_pos
thf(fact_969_add__neg__nonpos,axiom,
! [A: int,B: int] :
( ( ord_less_int @ A @ zero_zero_int )
=> ( ( ord_less_eq_int @ B @ zero_zero_int )
=> ( ord_less_int @ ( plus_plus_int @ A @ B ) @ zero_zero_int ) ) ) ).
% add_neg_nonpos
thf(fact_970_add__neg__nonpos,axiom,
! [A: nat,B: nat] :
( ( ord_less_nat @ A @ zero_zero_nat )
=> ( ( ord_less_eq_nat @ B @ zero_zero_nat )
=> ( ord_less_nat @ ( plus_plus_nat @ A @ B ) @ zero_zero_nat ) ) ) ).
% add_neg_nonpos
thf(fact_971_zero__less__two,axiom,
ord_less_int @ zero_zero_int @ ( plus_plus_int @ one_one_int @ one_one_int ) ).
% zero_less_two
thf(fact_972_zero__less__two,axiom,
ord_less_nat @ zero_zero_nat @ ( plus_plus_nat @ one_one_nat @ one_one_nat ) ).
% zero_less_two
thf(fact_973_sgn__1__pos,axiom,
! [A: int] :
( ( ( sgn_sgn_int @ A )
= one_one_int )
= ( ord_less_int @ zero_zero_int @ A ) ) ).
% sgn_1_pos
thf(fact_974_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_975_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_976_int__Suc,axiom,
! [N: nat] :
( ( semiri1314217659103216013at_int @ ( suc @ N ) )
= ( plus_plus_int @ ( semiri1314217659103216013at_int @ N ) @ one_one_int ) ) ).
% int_Suc
thf(fact_977_zless__iff__Suc__zadd,axiom,
( ord_less_int
= ( ^ [W2: int,Z6: int] :
? [N3: nat] :
( Z6
= ( plus_plus_int @ W2 @ ( semiri1314217659103216013at_int @ ( suc @ N3 ) ) ) ) ) ) ).
% zless_iff_Suc_zadd
thf(fact_978_odd__less__0__iff,axiom,
! [Z2: int] :
( ( ord_less_int @ ( plus_plus_int @ ( plus_plus_int @ one_one_int @ Z2 ) @ Z2 ) @ zero_zero_int )
= ( ord_less_int @ Z2 @ zero_zero_int ) ) ).
% odd_less_0_iff
thf(fact_979_zless__imp__add1__zle,axiom,
! [W: int,Z2: int] :
( ( ord_less_int @ W @ Z2 )
=> ( ord_less_eq_int @ ( plus_plus_int @ W @ one_one_int ) @ Z2 ) ) ).
% zless_imp_add1_zle
thf(fact_980_add1__zle__eq,axiom,
! [W: int,Z2: int] :
( ( ord_less_eq_int @ ( plus_plus_int @ W @ one_one_int ) @ Z2 )
= ( ord_less_int @ W @ Z2 ) ) ).
% add1_zle_eq
thf(fact_981_zsgn__def,axiom,
( sgn_sgn_int
= ( ^ [I4: int] : ( if_int @ ( I4 = zero_zero_int ) @ zero_zero_int @ ( if_int @ ( ord_less_int @ zero_zero_int @ I4 ) @ one_one_int @ ( uminus_uminus_int @ one_one_int ) ) ) ) ) ).
% zsgn_def
thf(fact_982_sgn__if,axiom,
( sgn_sgn_int
= ( ^ [X3: int] : ( if_int @ ( X3 = zero_zero_int ) @ zero_zero_int @ ( if_int @ ( ord_less_int @ zero_zero_int @ X3 ) @ one_one_int @ ( uminus_uminus_int @ one_one_int ) ) ) ) ) ).
% sgn_if
thf(fact_983_sgn__1__neg,axiom,
! [A: int] :
( ( ( sgn_sgn_int @ A )
= ( uminus_uminus_int @ one_one_int ) )
= ( ord_less_int @ A @ zero_zero_int ) ) ).
% sgn_1_neg
thf(fact_984_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_985_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_986_le__imp__0__less,axiom,
! [Z2: int] :
( ( ord_less_eq_int @ zero_zero_int @ Z2 )
=> ( ord_less_int @ zero_zero_int @ ( plus_plus_int @ one_one_int @ Z2 ) ) ) ).
% le_imp_0_less
thf(fact_987_Suc__as__int,axiom,
( suc
= ( ^ [A3: nat] : ( nat2 @ ( plus_plus_int @ ( semiri1314217659103216013at_int @ A3 ) @ one_one_int ) ) ) ) ).
% Suc_as_int
thf(fact_988_add__0__iff,axiom,
! [B: int,A: int] :
( ( B
= ( plus_plus_int @ B @ A ) )
= ( A = zero_zero_int ) ) ).
% add_0_iff
thf(fact_989_add__0__iff,axiom,
! [B: nat,A: nat] :
( ( B
= ( plus_plus_nat @ B @ A ) )
= ( A = zero_zero_nat ) ) ).
% add_0_iff
thf(fact_990_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_991_Nat_Oadd__0__right,axiom,
! [M: nat] :
( ( plus_plus_nat @ M @ zero_zero_nat )
= M ) ).
% Nat.add_0_right
thf(fact_992_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_993_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_994_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_995_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_996_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_997_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_998_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_999_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_1000_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_1001_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_1002_zle__diff1__eq,axiom,
! [W: int,Z2: int] :
( ( ord_less_eq_int @ W @ ( minus_minus_int @ Z2 @ one_one_int ) )
= ( ord_less_int @ W @ Z2 ) ) ).
% zle_diff1_eq
thf(fact_1003_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_1004_add__Suc,axiom,
! [M: nat,N: nat] :
( ( plus_plus_nat @ ( suc @ M ) @ N )
= ( suc @ ( plus_plus_nat @ M @ N ) ) ) ).
% add_Suc
thf(fact_1005_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_1006_diff__add__inverse2,axiom,
! [M: nat,N: nat] :
( ( minus_minus_nat @ ( plus_plus_nat @ M @ N ) @ N )
= M ) ).
% diff_add_inverse2
thf(fact_1007_diff__add__inverse,axiom,
! [N: nat,M: nat] :
( ( minus_minus_nat @ ( plus_plus_nat @ N @ M ) @ N )
= M ) ).
% diff_add_inverse
thf(fact_1008_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_1009_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_1010_psubset__imp__ex__mem,axiom,
! [A2: set_list_char,B4: set_list_char] :
( ( ord_le5286566838938544235t_char @ A2 @ B4 )
=> ? [B3: list_char] : ( member_list_char @ B3 @ ( minus_269717477326218608t_char @ B4 @ A2 ) ) ) ).
% psubset_imp_ex_mem
thf(fact_1011_psubset__imp__ex__mem,axiom,
! [A2: set_instr,B4: set_instr] :
( ( ord_less_set_instr @ A2 @ B4 )
=> ? [B3: instr] : ( member_instr @ B3 @ ( minus_5481936877629709990_instr @ B4 @ A2 ) ) ) ).
% psubset_imp_ex_mem
thf(fact_1012_nat__le__iff__add,axiom,
( ord_less_eq_nat
= ( ^ [M2: nat,N3: nat] :
? [K3: nat] :
( N3
= ( plus_plus_nat @ M2 @ K3 ) ) ) ) ).
% nat_le_iff_add
thf(fact_1013_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_1014_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_1015_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_1016_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_1017_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_1018_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_1019_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_1020_le__add2,axiom,
! [N: nat,M: nat] : ( ord_less_eq_nat @ N @ ( plus_plus_nat @ M @ N ) ) ).
% le_add2
thf(fact_1021_le__add1,axiom,
! [N: nat,M: nat] : ( ord_less_eq_nat @ N @ ( plus_plus_nat @ N @ M ) ) ).
% le_add1
thf(fact_1022_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_1023_int__diff__cases,axiom,
! [Z2: int] :
~ ! [M4: nat,N2: nat] :
( Z2
!= ( minus_minus_int @ ( semiri1314217659103216013at_int @ M4 ) @ ( semiri1314217659103216013at_int @ N2 ) ) ) ).
% int_diff_cases
thf(fact_1024_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_1025_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_1026_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_1027_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_1028_not__add__less2,axiom,
! [J: nat,I: nat] :
~ ( ord_less_nat @ ( plus_plus_nat @ J @ I ) @ I ) ).
% not_add_less2
thf(fact_1029_not__add__less1,axiom,
! [I: nat,J: nat] :
~ ( ord_less_nat @ ( plus_plus_nat @ I @ J ) @ I ) ).
% not_add_less1
thf(fact_1030_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_1031_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_1032_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_1033_plus__nat_Oadd__0,axiom,
! [N: nat] :
( ( plus_plus_nat @ zero_zero_nat @ N )
= N ) ).
% plus_nat.add_0
thf(fact_1034_minus__int__code_I1_J,axiom,
! [K: int] :
( ( minus_minus_int @ K @ zero_zero_int )
= K ) ).
% minus_int_code(1)
thf(fact_1035_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_1036_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_1037_minus__int__code_I2_J,axiom,
! [L: int] :
( ( minus_minus_int @ zero_zero_int @ L )
= ( uminus_uminus_int @ L ) ) ).
% minus_int_code(2)
thf(fact_1038_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_1039_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_1040_less__iff__Suc__add,axiom,
( ord_less_nat
= ( ^ [M2: nat,N3: nat] :
? [K3: nat] :
( N3
= ( suc @ ( plus_plus_nat @ M2 @ K3 ) ) ) ) ) ).
% less_iff_Suc_add
thf(fact_1041_less__add__Suc2,axiom,
! [I: nat,M: nat] : ( ord_less_nat @ I @ ( suc @ ( plus_plus_nat @ M @ I ) ) ) ).
% less_add_Suc2
thf(fact_1042_less__add__Suc1,axiom,
! [I: nat,M: nat] : ( ord_less_nat @ I @ ( suc @ ( plus_plus_nat @ I @ M ) ) ) ).
% less_add_Suc1
thf(fact_1043_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_1044_int__le__induct,axiom,
! [I: int,K: int,P: int > $o] :
( ( ord_less_eq_int @ I @ K )
=> ( ( P @ K )
=> ( ! [I3: int] :
( ( ord_less_eq_int @ I3 @ K )
=> ( ( P @ I3 )
=> ( P @ ( minus_minus_int @ I3 @ one_one_int ) ) ) )
=> ( P @ I ) ) ) ) ).
% int_le_induct
thf(fact_1045_int__less__induct,axiom,
! [I: int,K: int,P: int > $o] :
( ( ord_less_int @ I @ K )
=> ( ( P @ ( minus_minus_int @ K @ one_one_int ) )
=> ( ! [I3: int] :
( ( ord_less_int @ I3 @ K )
=> ( ( P @ I3 )
=> ( P @ ( minus_minus_int @ I3 @ one_one_int ) ) ) )
=> ( P @ I ) ) ) ) ).
% int_less_induct
thf(fact_1046_mono__nat__linear__lb,axiom,
! [F: nat > nat,M: nat,K: nat] :
( ! [M4: nat,N2: nat] :
( ( ord_less_nat @ M4 @ N2 )
=> ( ord_less_nat @ ( F @ M4 ) @ ( F @ N2 ) ) )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ ( F @ M ) @ K ) @ ( F @ ( plus_plus_nat @ M @ K ) ) ) ) ).
% mono_nat_linear_lb
thf(fact_1047_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_1048_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_1049_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_1050_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_1051_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_1052_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_1053_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_1054_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_1055_Suc__eq__plus1,axiom,
( suc
= ( ^ [N3: nat] : ( plus_plus_nat @ N3 @ one_one_nat ) ) ) ).
% Suc_eq_plus1
thf(fact_1056_plus__1__eq__Suc,axiom,
( ( plus_plus_nat @ one_one_nat )
= suc ) ).
% plus_1_eq_Suc
thf(fact_1057_Suc__eq__plus1__left,axiom,
( suc
= ( plus_plus_nat @ one_one_nat ) ) ).
% Suc_eq_plus1_left
thf(fact_1058_zadd__int__left,axiom,
! [M: nat,N: nat,Z2: int] :
( ( plus_plus_int @ ( semiri1314217659103216013at_int @ M ) @ ( plus_plus_int @ ( semiri1314217659103216013at_int @ N ) @ Z2 ) )
= ( plus_plus_int @ ( semiri1314217659103216013at_int @ ( plus_plus_nat @ M @ N ) ) @ Z2 ) ) ).
% zadd_int_left
thf(fact_1059_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_1060_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_1061_gen__length__def,axiom,
( gen_length_instr
= ( ^ [N3: nat,Xs3: list_instr] : ( plus_plus_nat @ N3 @ ( size_size_list_instr @ Xs3 ) ) ) ) ).
% gen_length_def
thf(fact_1062_gen__length__def,axiom,
( gen_length_list_char
= ( ^ [N3: nat,Xs3: list_list_char] : ( plus_plus_nat @ N3 @ ( size_s356637359517785349t_char @ Xs3 ) ) ) ) ).
% gen_length_def
thf(fact_1063_int__induct,axiom,
! [P: int > $o,K: int,I: int] :
( ( P @ K )
=> ( ! [I3: int] :
( ( ord_less_eq_int @ K @ I3 )
=> ( ( P @ I3 )
=> ( P @ ( plus_plus_int @ I3 @ one_one_int ) ) ) )
=> ( ! [I3: int] :
( ( ord_less_eq_int @ I3 @ K )
=> ( ( P @ I3 )
=> ( P @ ( minus_minus_int @ I3 @ one_one_int ) ) ) )
=> ( P @ I ) ) ) ) ).
% int_induct
thf(fact_1064_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 ) )
& ! [D3: nat] :
( ( A
= ( plus_plus_nat @ B @ D3 ) )
=> ( P @ D3 ) ) ) ) ).
% nat_diff_split
thf(fact_1065_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 ) )
| ? [D3: nat] :
( ( A
= ( plus_plus_nat @ B @ D3 ) )
& ~ ( P @ D3 ) ) ) ) ) ).
% nat_diff_split_asm
thf(fact_1066_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_1067_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_1068_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_1069_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_1070_nat__diff__distrib,axiom,
! [Z5: int,Z2: int] :
( ( ord_less_eq_int @ zero_zero_int @ Z5 )
=> ( ( ord_less_eq_int @ Z5 @ Z2 )
=> ( ( nat2 @ ( minus_minus_int @ Z2 @ Z5 ) )
= ( minus_minus_nat @ ( nat2 @ Z2 ) @ ( nat2 @ Z5 ) ) ) ) ) ).
% nat_diff_distrib
thf(fact_1071_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_1072_add__eq__if,axiom,
( plus_plus_nat
= ( ^ [M2: nat,N3: nat] : ( if_nat @ ( M2 = zero_zero_nat ) @ N3 @ ( suc @ ( plus_plus_nat @ ( minus_minus_nat @ M2 @ one_one_nat ) @ N3 ) ) ) ) ) ).
% add_eq_if
thf(fact_1073_nat__add__distrib,axiom,
! [Z2: int,Z5: int] :
( ( ord_less_eq_int @ zero_zero_int @ Z2 )
=> ( ( ord_less_eq_int @ zero_zero_int @ Z5 )
=> ( ( nat2 @ ( plus_plus_int @ Z2 @ Z5 ) )
= ( plus_plus_nat @ ( nat2 @ Z2 ) @ ( nat2 @ Z5 ) ) ) ) ) ).
% nat_add_distrib
thf(fact_1074_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_1075_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_1076_to__m__state_Osimps,axiom,
( to_m_state
= ( ^ [Xs3: list_list_char,S2: list_char > int,A3: int] :
( if_int
@ ( ( ord_less_int @ zero_zero_int @ A3 )
& ( ord_less_eq_int @ A3 @ ( semiri1314217659103216013at_int @ ( size_s356637359517785349t_char @ Xs3 ) ) ) )
@ ( S2 @ ( inth_list_char @ Xs3 @ ( minus_minus_int @ ( semiri1314217659103216013at_int @ ( size_s356637359517785349t_char @ Xs3 ) ) @ A3 ) ) )
@ zero_zero_int ) ) ) ).
% to_m_state.simps
thf(fact_1077_Diff__iff,axiom,
! [C: list_char,A2: set_list_char,B4: set_list_char] :
( ( member_list_char @ C @ ( minus_269717477326218608t_char @ A2 @ B4 ) )
= ( ( member_list_char @ C @ A2 )
& ~ ( member_list_char @ C @ B4 ) ) ) ).
% Diff_iff
thf(fact_1078_Diff__iff,axiom,
! [C: instr,A2: set_instr,B4: set_instr] :
( ( member_instr @ C @ ( minus_5481936877629709990_instr @ A2 @ B4 ) )
= ( ( member_instr @ C @ A2 )
& ~ ( member_instr @ C @ B4 ) ) ) ).
% Diff_iff
thf(fact_1079_DiffI,axiom,
! [C: list_char,A2: set_list_char,B4: set_list_char] :
( ( member_list_char @ C @ A2 )
=> ( ~ ( member_list_char @ C @ B4 )
=> ( member_list_char @ C @ ( minus_269717477326218608t_char @ A2 @ B4 ) ) ) ) ).
% DiffI
thf(fact_1080_DiffI,axiom,
! [C: instr,A2: set_instr,B4: set_instr] :
( ( member_instr @ C @ A2 )
=> ( ~ ( member_instr @ C @ B4 )
=> ( member_instr @ C @ ( minus_5481936877629709990_instr @ A2 @ B4 ) ) ) ) ).
% DiffI
thf(fact_1081_DiffD2,axiom,
! [C: list_char,A2: set_list_char,B4: set_list_char] :
( ( member_list_char @ C @ ( minus_269717477326218608t_char @ A2 @ B4 ) )
=> ~ ( member_list_char @ C @ B4 ) ) ).
% DiffD2
thf(fact_1082_DiffD2,axiom,
! [C: instr,A2: set_instr,B4: set_instr] :
( ( member_instr @ C @ ( minus_5481936877629709990_instr @ A2 @ B4 ) )
=> ~ ( member_instr @ C @ B4 ) ) ).
% DiffD2
thf(fact_1083_DiffD1,axiom,
! [C: list_char,A2: set_list_char,B4: set_list_char] :
( ( member_list_char @ C @ ( minus_269717477326218608t_char @ A2 @ B4 ) )
=> ( member_list_char @ C @ A2 ) ) ).
% DiffD1
thf(fact_1084_DiffD1,axiom,
! [C: instr,A2: set_instr,B4: set_instr] :
( ( member_instr @ C @ ( minus_5481936877629709990_instr @ A2 @ B4 ) )
=> ( member_instr @ C @ A2 ) ) ).
% DiffD1
thf(fact_1085_DiffE,axiom,
! [C: list_char,A2: set_list_char,B4: set_list_char] :
( ( member_list_char @ C @ ( minus_269717477326218608t_char @ A2 @ B4 ) )
=> ~ ( ( member_list_char @ C @ A2 )
=> ( member_list_char @ C @ B4 ) ) ) ).
% DiffE
thf(fact_1086_DiffE,axiom,
! [C: instr,A2: set_instr,B4: set_instr] :
( ( member_instr @ C @ ( minus_5481936877629709990_instr @ A2 @ B4 ) )
=> ~ ( ( member_instr @ C @ A2 )
=> ( member_instr @ C @ B4 ) ) ) ).
% DiffE
thf(fact_1087_bounded__Max__nat,axiom,
! [P: nat > $o,X: nat,M7: nat] :
( ( P @ X )
=> ( ! [X5: nat] :
( ( P @ X5 )
=> ( ord_less_eq_nat @ X5 @ M7 ) )
=> ~ ! [M4: nat] :
( ( P @ M4 )
=> ~ ! [X7: nat] :
( ( P @ X7 )
=> ( ord_less_eq_nat @ X7 @ M4 ) ) ) ) ) ).
% bounded_Max_nat
thf(fact_1088_inth__append,axiom,
! [I: int,Xs: list_instr,Ys2: 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 @ Ys2 ) @ I )
= ( inth_instr @ Xs @ I ) ) )
& ( ~ ( ord_less_int @ I @ ( semiri1314217659103216013at_int @ ( size_size_list_instr @ Xs ) ) )
=> ( ( inth_instr @ ( append_instr @ Xs @ Ys2 ) @ I )
= ( inth_instr @ Ys2 @ ( minus_minus_int @ I @ ( semiri1314217659103216013at_int @ ( size_size_list_instr @ Xs ) ) ) ) ) ) ) ) ).
% inth_append
thf(fact_1089_inth__append,axiom,
! [I: int,Xs: list_list_char,Ys2: 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 @ Ys2 ) @ I )
= ( inth_list_char @ Xs @ I ) ) )
& ( ~ ( ord_less_int @ I @ ( semiri1314217659103216013at_int @ ( size_s356637359517785349t_char @ Xs ) ) )
=> ( ( inth_list_char @ ( append_list_char @ Xs @ Ys2 ) @ I )
= ( inth_list_char @ Ys2 @ ( minus_minus_int @ I @ ( semiri1314217659103216013at_int @ ( size_s356637359517785349t_char @ Xs ) ) ) ) ) ) ) ) ).
% inth_append
thf(fact_1090_nat0__intermed__int__val,axiom,
! [N: nat,F: nat > int,K: int] :
( ! [I3: nat] :
( ( ord_less_nat @ I3 @ N )
=> ( ord_less_eq_int @ ( abs_abs_int @ ( minus_minus_int @ ( F @ ( plus_plus_nat @ I3 @ one_one_nat ) ) @ ( F @ I3 ) ) ) @ one_one_int ) )
=> ( ( ord_less_eq_int @ ( F @ zero_zero_nat ) @ K )
=> ( ( ord_less_eq_int @ K @ ( F @ N ) )
=> ? [I3: nat] :
( ( ord_less_eq_nat @ I3 @ N )
& ( ( F @ I3 )
= K ) ) ) ) ) ).
% nat0_intermed_int_val
thf(fact_1091_abs__idempotent,axiom,
! [A: int] :
( ( abs_abs_int @ ( abs_abs_int @ A ) )
= ( abs_abs_int @ A ) ) ).
% abs_idempotent
thf(fact_1092_abs__0__eq,axiom,
! [A: int] :
( ( zero_zero_int
= ( abs_abs_int @ A ) )
= ( A = zero_zero_int ) ) ).
% abs_0_eq
thf(fact_1093_abs__eq__0,axiom,
! [A: int] :
( ( ( abs_abs_int @ A )
= zero_zero_int )
= ( A = zero_zero_int ) ) ).
% abs_eq_0
thf(fact_1094_abs__zero,axiom,
( ( abs_abs_int @ zero_zero_int )
= zero_zero_int ) ).
% abs_zero
thf(fact_1095_abs__0,axiom,
( ( abs_abs_int @ zero_zero_int )
= zero_zero_int ) ).
% abs_0
thf(fact_1096_abs__add__abs,axiom,
! [A: int,B: int] :
( ( abs_abs_int @ ( plus_plus_int @ ( abs_abs_int @ A ) @ ( abs_abs_int @ B ) ) )
= ( plus_plus_int @ ( abs_abs_int @ A ) @ ( abs_abs_int @ B ) ) ) ).
% abs_add_abs
thf(fact_1097_abs__minus__cancel,axiom,
! [A: int] :
( ( abs_abs_int @ ( uminus_uminus_int @ A ) )
= ( abs_abs_int @ A ) ) ).
% abs_minus_cancel
thf(fact_1098_abs__minus,axiom,
! [A: int] :
( ( abs_abs_int @ ( uminus_uminus_int @ A ) )
= ( abs_abs_int @ A ) ) ).
% abs_minus
thf(fact_1099_append__eq__append__conv,axiom,
! [Xs: list_instr,Ys2: list_instr,Us: list_instr,Vs: list_instr] :
( ( ( ( size_size_list_instr @ Xs )
= ( size_size_list_instr @ Ys2 ) )
| ( ( size_size_list_instr @ Us )
= ( size_size_list_instr @ Vs ) ) )
=> ( ( ( append_instr @ Xs @ Us )
= ( append_instr @ Ys2 @ Vs ) )
= ( ( Xs = Ys2 )
& ( Us = Vs ) ) ) ) ).
% append_eq_append_conv
thf(fact_1100_append__eq__append__conv,axiom,
! [Xs: list_list_char,Ys2: list_list_char,Us: list_list_char,Vs: list_list_char] :
( ( ( ( size_s356637359517785349t_char @ Xs )
= ( size_s356637359517785349t_char @ Ys2 ) )
| ( ( size_s356637359517785349t_char @ Us )
= ( size_s356637359517785349t_char @ Vs ) ) )
=> ( ( ( append_list_char @ Xs @ Us )
= ( append_list_char @ Ys2 @ Vs ) )
= ( ( Xs = Ys2 )
& ( Us = Vs ) ) ) ) ).
% append_eq_append_conv
thf(fact_1101_abs__of__nat,axiom,
! [N: nat] :
( ( abs_abs_int @ ( semiri1314217659103216013at_int @ N ) )
= ( semiri1314217659103216013at_int @ N ) ) ).
% abs_of_nat
thf(fact_1102_abs__of__nonneg,axiom,
! [A: int] :
( ( ord_less_eq_int @ zero_zero_int @ A )
=> ( ( abs_abs_int @ A )
= A ) ) ).
% abs_of_nonneg
thf(fact_1103_abs__le__self__iff,axiom,
! [A: int] :
( ( ord_less_eq_int @ ( abs_abs_int @ A ) @ A )
= ( ord_less_eq_int @ zero_zero_int @ A ) ) ).
% abs_le_self_iff
thf(fact_1104_abs__le__zero__iff,axiom,
! [A: int] :
( ( ord_less_eq_int @ ( abs_abs_int @ A ) @ zero_zero_int )
= ( A = zero_zero_int ) ) ).
% abs_le_zero_iff
thf(fact_1105_zero__less__abs__iff,axiom,
! [A: int] :
( ( ord_less_int @ zero_zero_int @ ( abs_abs_int @ A ) )
= ( A != zero_zero_int ) ) ).
% zero_less_abs_iff
thf(fact_1106_abs__neg__one,axiom,
( ( abs_abs_int @ ( uminus_uminus_int @ one_one_int ) )
= one_one_int ) ).
% abs_neg_one
thf(fact_1107_length__append,axiom,
! [Xs: list_instr,Ys2: list_instr] :
( ( size_size_list_instr @ ( append_instr @ Xs @ Ys2 ) )
= ( plus_plus_nat @ ( size_size_list_instr @ Xs ) @ ( size_size_list_instr @ Ys2 ) ) ) ).
% length_append
thf(fact_1108_length__append,axiom,
! [Xs: list_list_char,Ys2: list_list_char] :
( ( size_s356637359517785349t_char @ ( append_list_char @ Xs @ Ys2 ) )
= ( plus_plus_nat @ ( size_s356637359517785349t_char @ Xs ) @ ( size_s356637359517785349t_char @ Ys2 ) ) ) ).
% length_append
thf(fact_1109_of__int__abs,axiom,
! [X: int] :
( ( ring_1_of_int_int @ ( abs_abs_int @ X ) )
= ( abs_abs_int @ ( ring_1_of_int_int @ X ) ) ) ).
% of_int_abs
thf(fact_1110_abs__of__nonpos,axiom,
! [A: int] :
( ( ord_less_eq_int @ A @ zero_zero_int )
=> ( ( abs_abs_int @ A )
= ( uminus_uminus_int @ A ) ) ) ).
% abs_of_nonpos
thf(fact_1111_abs__sgn__eq__1,axiom,
! [A: int] :
( ( A != zero_zero_int )
=> ( ( abs_abs_int @ ( sgn_sgn_int @ A ) )
= one_one_int ) ) ).
% abs_sgn_eq_1
thf(fact_1112_zabs__less__one__iff,axiom,
! [Z2: int] :
( ( ord_less_int @ ( abs_abs_int @ Z2 ) @ one_one_int )
= ( Z2 = zero_zero_int ) ) ).
% zabs_less_one_iff
thf(fact_1113_abs__ge__zero,axiom,
! [A: int] : ( ord_less_eq_int @ zero_zero_int @ ( abs_abs_int @ A ) ) ).
% abs_ge_zero
thf(fact_1114_abs__of__pos,axiom,
! [A: int] :
( ( ord_less_int @ zero_zero_int @ A )
=> ( ( abs_abs_int @ A )
= A ) ) ).
% abs_of_pos
thf(fact_1115_abs__not__less__zero,axiom,
! [A: int] :
~ ( ord_less_int @ ( abs_abs_int @ A ) @ zero_zero_int ) ).
% abs_not_less_zero
thf(fact_1116_abs__triangle__ineq,axiom,
! [A: int,B: int] : ( ord_less_eq_int @ ( abs_abs_int @ ( plus_plus_int @ A @ B ) ) @ ( plus_plus_int @ ( abs_abs_int @ A ) @ ( abs_abs_int @ B ) ) ) ).
% abs_triangle_ineq
thf(fact_1117_abs__leI,axiom,
! [A: int,B: int] :
( ( ord_less_eq_int @ A @ B )
=> ( ( ord_less_eq_int @ ( uminus_uminus_int @ A ) @ B )
=> ( ord_less_eq_int @ ( abs_abs_int @ A ) @ B ) ) ) ).
% abs_leI
thf(fact_1118_abs__le__D2,axiom,
! [A: int,B: int] :
( ( ord_less_eq_int @ ( abs_abs_int @ A ) @ B )
=> ( ord_less_eq_int @ ( uminus_uminus_int @ A ) @ B ) ) ).
% abs_le_D2
thf(fact_1119_abs__le__iff,axiom,
! [A: int,B: int] :
( ( ord_less_eq_int @ ( abs_abs_int @ A ) @ B )
= ( ( ord_less_eq_int @ A @ B )
& ( ord_less_eq_int @ ( uminus_uminus_int @ A ) @ B ) ) ) ).
% abs_le_iff
thf(fact_1120_abs__ge__minus__self,axiom,
! [A: int] : ( ord_less_eq_int @ ( uminus_uminus_int @ A ) @ ( abs_abs_int @ A ) ) ).
% abs_ge_minus_self
thf(fact_1121_abs__less__iff,axiom,
! [A: int,B: int] :
( ( ord_less_int @ ( abs_abs_int @ A ) @ B )
= ( ( ord_less_int @ A @ B )
& ( ord_less_int @ ( uminus_uminus_int @ A ) @ B ) ) ) ).
% abs_less_iff
thf(fact_1122_abs__le__D1,axiom,
! [A: int,B: int] :
( ( ord_less_eq_int @ ( abs_abs_int @ A ) @ B )
=> ( ord_less_eq_int @ A @ B ) ) ).
% abs_le_D1
thf(fact_1123_abs__ge__self,axiom,
! [A: int] : ( ord_less_eq_int @ A @ ( abs_abs_int @ A ) ) ).
% abs_ge_self
thf(fact_1124_abs__eq__iff,axiom,
! [X: int,Y: int] :
( ( ( abs_abs_int @ X )
= ( abs_abs_int @ Y ) )
= ( ( X = Y )
| ( X
= ( uminus_uminus_int @ Y ) ) ) ) ).
% abs_eq_iff
thf(fact_1125_abs__eq__0__iff,axiom,
! [A: int] :
( ( ( abs_abs_int @ A )
= zero_zero_int )
= ( A = zero_zero_int ) ) ).
% abs_eq_0_iff
thf(fact_1126_rotate__append,axiom,
! [L: list_instr,Q4: list_instr] :
( ( rotate_instr @ ( size_size_list_instr @ L ) @ ( append_instr @ L @ Q4 ) )
= ( append_instr @ Q4 @ L ) ) ).
% rotate_append
thf(fact_1127_rotate__append,axiom,
! [L: list_list_char,Q4: list_list_char] :
( ( rotate_list_char @ ( size_s356637359517785349t_char @ L ) @ ( append_list_char @ L @ Q4 ) )
= ( append_list_char @ Q4 @ L ) ) ).
% rotate_append
thf(fact_1128_abs__triangle__ineq2,axiom,
! [A: int,B: int] : ( ord_less_eq_int @ ( minus_minus_int @ ( abs_abs_int @ A ) @ ( abs_abs_int @ B ) ) @ ( abs_abs_int @ ( minus_minus_int @ A @ B ) ) ) ).
% abs_triangle_ineq2
thf(fact_1129_abs__triangle__ineq3,axiom,
! [A: int,B: int] : ( ord_less_eq_int @ ( abs_abs_int @ ( minus_minus_int @ ( abs_abs_int @ A ) @ ( abs_abs_int @ B ) ) ) @ ( abs_abs_int @ ( minus_minus_int @ A @ B ) ) ) ).
% abs_triangle_ineq3
thf(fact_1130_abs__triangle__ineq2__sym,axiom,
! [A: int,B: int] : ( ord_less_eq_int @ ( minus_minus_int @ ( abs_abs_int @ A ) @ ( abs_abs_int @ B ) ) @ ( abs_abs_int @ ( minus_minus_int @ B @ A ) ) ) ).
% abs_triangle_ineq2_sym
thf(fact_1131_abs__minus__commute,axiom,
! [A: int,B: int] :
( ( abs_abs_int @ ( minus_minus_int @ A @ B ) )
= ( abs_abs_int @ ( minus_minus_int @ B @ A ) ) ) ).
% abs_minus_commute
thf(fact_1132_eq__abs__iff_H,axiom,
! [A: int,B: int] :
( ( A
= ( abs_abs_int @ B ) )
= ( ( ord_less_eq_int @ zero_zero_int @ A )
& ( ( B = A )
| ( B
= ( uminus_uminus_int @ A ) ) ) ) ) ).
% eq_abs_iff'
thf(fact_1133_abs__eq__iff_H,axiom,
! [A: int,B: int] :
( ( ( abs_abs_int @ A )
= B )
= ( ( ord_less_eq_int @ zero_zero_int @ B )
& ( ( A = B )
| ( A
= ( uminus_uminus_int @ B ) ) ) ) ) ).
% abs_eq_iff'
thf(fact_1134_abs__minus__le__zero,axiom,
! [A: int] : ( ord_less_eq_int @ ( uminus_uminus_int @ ( abs_abs_int @ A ) ) @ zero_zero_int ) ).
% abs_minus_le_zero
thf(fact_1135_abs__triangle__ineq4,axiom,
! [A: int,B: int] : ( ord_less_eq_int @ ( abs_abs_int @ ( minus_minus_int @ A @ B ) ) @ ( plus_plus_int @ ( abs_abs_int @ A ) @ ( abs_abs_int @ B ) ) ) ).
% abs_triangle_ineq4
thf(fact_1136_abs__diff__triangle__ineq,axiom,
! [A: int,B: int,C: int,D2: int] : ( ord_less_eq_int @ ( abs_abs_int @ ( minus_minus_int @ ( plus_plus_int @ A @ B ) @ ( plus_plus_int @ C @ D2 ) ) ) @ ( plus_plus_int @ ( abs_abs_int @ ( minus_minus_int @ A @ C ) ) @ ( abs_abs_int @ ( minus_minus_int @ B @ D2 ) ) ) ) ).
% abs_diff_triangle_ineq
thf(fact_1137_abs__diff__le__iff,axiom,
! [X: int,A: int,R2: int] :
( ( ord_less_eq_int @ ( abs_abs_int @ ( minus_minus_int @ X @ A ) ) @ R2 )
= ( ( ord_less_eq_int @ ( minus_minus_int @ A @ R2 ) @ X )
& ( ord_less_eq_int @ X @ ( plus_plus_int @ A @ R2 ) ) ) ) ).
% abs_diff_le_iff
thf(fact_1138_abs__if,axiom,
( abs_abs_int
= ( ^ [A3: int] : ( if_int @ ( ord_less_int @ A3 @ zero_zero_int ) @ ( uminus_uminus_int @ A3 ) @ A3 ) ) ) ).
% abs_if
thf(fact_1139_abs__of__neg,axiom,
! [A: int] :
( ( ord_less_int @ A @ zero_zero_int )
=> ( ( abs_abs_int @ A )
= ( uminus_uminus_int @ A ) ) ) ).
% abs_of_neg
thf(fact_1140_abs__if__raw,axiom,
( abs_abs_int
= ( ^ [A3: int] : ( if_int @ ( ord_less_int @ A3 @ zero_zero_int ) @ ( uminus_uminus_int @ A3 ) @ A3 ) ) ) ).
% abs_if_raw
thf(fact_1141_abs__diff__less__iff,axiom,
! [X: int,A: int,R2: int] :
( ( ord_less_int @ ( abs_abs_int @ ( minus_minus_int @ X @ A ) ) @ R2 )
= ( ( ord_less_int @ ( minus_minus_int @ A @ R2 ) @ X )
& ( ord_less_int @ X @ ( plus_plus_int @ A @ R2 ) ) ) ) ).
% abs_diff_less_iff
thf(fact_1142_abs__sgn__eq,axiom,
! [A: int] :
( ( ( A = zero_zero_int )
=> ( ( abs_abs_int @ ( sgn_sgn_int @ A ) )
= zero_zero_int ) )
& ( ( A != zero_zero_int )
=> ( ( abs_abs_int @ ( sgn_sgn_int @ A ) )
= one_one_int ) ) ) ).
% abs_sgn_eq
thf(fact_1143_zabs__def,axiom,
( abs_abs_int
= ( ^ [I4: int] : ( if_int @ ( ord_less_int @ I4 @ zero_zero_int ) @ ( uminus_uminus_int @ I4 ) @ I4 ) ) ) ).
% zabs_def
thf(fact_1144_abs__add__one__gt__zero,axiom,
! [X: int] : ( ord_less_int @ zero_zero_int @ ( plus_plus_int @ one_one_int @ ( abs_abs_int @ X ) ) ) ).
% abs_add_one_gt_zero
thf(fact_1145_of__int__leD,axiom,
! [N: int,X: int] :
( ( ord_less_eq_int @ ( abs_abs_int @ ( ring_1_of_int_int @ N ) ) @ X )
=> ( ( N = zero_zero_int )
| ( ord_less_eq_int @ one_one_int @ X ) ) ) ).
% of_int_leD
thf(fact_1146_of__int__lessD,axiom,
! [N: int,X: int] :
( ( ord_less_int @ ( abs_abs_int @ ( ring_1_of_int_int @ N ) ) @ X )
=> ( ( N = zero_zero_int )
| ( ord_less_int @ one_one_int @ X ) ) ) ).
% of_int_lessD
thf(fact_1147_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_1148_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_1149_nat__intermed__int__val,axiom,
! [M: nat,N: nat,F: nat > int,K: int] :
( ! [I3: nat] :
( ( ( ord_less_eq_nat @ M @ I3 )
& ( ord_less_nat @ I3 @ N ) )
=> ( ord_less_eq_int @ ( abs_abs_int @ ( minus_minus_int @ ( F @ ( suc @ I3 ) ) @ ( F @ I3 ) ) ) @ one_one_int ) )
=> ( ( ord_less_eq_nat @ M @ N )
=> ( ( ord_less_eq_int @ ( F @ M ) @ K )
=> ( ( ord_less_eq_int @ K @ ( F @ N ) )
=> ? [I3: nat] :
( ( ord_less_eq_nat @ M @ I3 )
& ( ord_less_eq_nat @ I3 @ N )
& ( ( F @ I3 )
= K ) ) ) ) ) ) ).
% nat_intermed_int_val
thf(fact_1150_nat__ivt__aux,axiom,
! [N: nat,F: nat > int,K: int] :
( ! [I3: nat] :
( ( ord_less_nat @ I3 @ N )
=> ( ord_less_eq_int @ ( abs_abs_int @ ( minus_minus_int @ ( F @ ( suc @ I3 ) ) @ ( F @ I3 ) ) ) @ one_one_int ) )
=> ( ( ord_less_eq_int @ ( F @ zero_zero_nat ) @ K )
=> ( ( ord_less_eq_int @ K @ ( F @ N ) )
=> ? [I3: nat] :
( ( ord_less_eq_nat @ I3 @ N )
& ( ( F @ I3 )
= K ) ) ) ) ) ).
% nat_ivt_aux
thf(fact_1151_Euclid__induct,axiom,
! [P: nat > nat > $o,A: nat,B: nat] :
( ! [A4: nat,B3: nat] :
( ( P @ A4 @ B3 )
= ( P @ B3 @ A4 ) )
=> ( ! [A4: nat] : ( P @ A4 @ zero_zero_nat )
=> ( ! [A4: nat,B3: nat] :
( ( P @ A4 @ B3 )
=> ( P @ A4 @ ( plus_plus_nat @ A4 @ B3 ) ) )
=> ( P @ A @ B ) ) ) ) ).
% Euclid_induct
thf(fact_1152_in__set__product__lists__length,axiom,
! [Xs: list_char,Xss: list_list_char] :
( ( member_list_char @ Xs @ ( set_list_char2 @ ( product_lists_char @ Xss ) ) )
=> ( ( size_size_list_char @ Xs )
= ( size_s356637359517785349t_char @ Xss ) ) ) ).
% in_set_product_lists_length
thf(fact_1153_in__set__product__lists__length,axiom,
! [Xs: list_instr,Xss: list_list_instr] :
( ( member_list_instr @ Xs @ ( set_list_instr2 @ ( product_lists_instr @ Xss ) ) )
=> ( ( size_size_list_instr @ Xs )
= ( size_s1289461246499836759_instr @ Xss ) ) ) ).
% in_set_product_lists_length
thf(fact_1154_in__set__product__lists__length,axiom,
! [Xs: list_list_char,Xss: list_list_list_char] :
( ( member4762036990631661320t_char @ Xs @ ( set_list_list_char2 @ ( produc4928866681682397746t_char @ Xss ) ) )
=> ( ( size_s356637359517785349t_char @ Xs )
= ( size_s2562795203299699605t_char @ Xss ) ) ) ).
% in_set_product_lists_length
thf(fact_1155_decr__lemma,axiom,
! [D2: int,X: int,Z2: int] :
( ( ord_less_int @ zero_zero_int @ D2 )
=> ( ord_less_int @ ( minus_minus_int @ X @ ( times_times_int @ ( plus_plus_int @ ( abs_abs_int @ ( minus_minus_int @ X @ Z2 ) ) @ one_one_int ) @ D2 ) ) @ Z2 ) ) ).
% decr_lemma
thf(fact_1156_incr__lemma,axiom,
! [D2: int,Z2: int,X: int] :
( ( ord_less_int @ zero_zero_int @ D2 )
=> ( ord_less_int @ Z2 @ ( plus_plus_int @ X @ ( times_times_int @ ( plus_plus_int @ ( abs_abs_int @ ( minus_minus_int @ X @ Z2 ) ) @ one_one_int ) @ D2 ) ) ) ) ).
% incr_lemma
thf(fact_1157_mult__cancel__right,axiom,
! [A: nat,C: nat,B: nat] :
( ( ( times_times_nat @ A @ C )
= ( times_times_nat @ B @ C ) )
= ( ( C = zero_zero_nat )
| ( A = B ) ) ) ).
% mult_cancel_right
thf(fact_1158_mult__cancel__right,axiom,
! [A: int,C: int,B: int] :
( ( ( times_times_int @ A @ C )
= ( times_times_int @ B @ C ) )
= ( ( C = zero_zero_int )
| ( A = B ) ) ) ).
% mult_cancel_right
thf(fact_1159_mult__cancel__left,axiom,
! [C: nat,A: nat,B: nat] :
( ( ( times_times_nat @ C @ A )
= ( times_times_nat @ C @ B ) )
= ( ( C = zero_zero_nat )
| ( A = B ) ) ) ).
% mult_cancel_left
thf(fact_1160_mult__cancel__left,axiom,
! [C: int,A: int,B: int] :
( ( ( times_times_int @ C @ A )
= ( times_times_int @ C @ B ) )
= ( ( C = zero_zero_int )
| ( A = B ) ) ) ).
% mult_cancel_left
thf(fact_1161_mult__eq__0__iff,axiom,
! [A: nat,B: nat] :
( ( ( times_times_nat @ A @ B )
= zero_zero_nat )
= ( ( A = zero_zero_nat )
| ( B = zero_zero_nat ) ) ) ).
% mult_eq_0_iff
thf(fact_1162_mult__eq__0__iff,axiom,
! [A: int,B: int] :
( ( ( times_times_int @ A @ B )
= zero_zero_int )
= ( ( A = zero_zero_int )
| ( B = zero_zero_int ) ) ) ).
% mult_eq_0_iff
thf(fact_1163_mult__zero__right,axiom,
! [A: nat] :
( ( times_times_nat @ A @ zero_zero_nat )
= zero_zero_nat ) ).
% mult_zero_right
thf(fact_1164_mult__zero__right,axiom,
! [A: int] :
( ( times_times_int @ A @ zero_zero_int )
= zero_zero_int ) ).
% mult_zero_right
thf(fact_1165_mult__zero__left,axiom,
! [A: nat] :
( ( times_times_nat @ zero_zero_nat @ A )
= zero_zero_nat ) ).
% mult_zero_left
thf(fact_1166_mult__zero__left,axiom,
! [A: int] :
( ( times_times_int @ zero_zero_int @ A )
= zero_zero_int ) ).
% mult_zero_left
thf(fact_1167_mult_Oright__neutral,axiom,
! [A: nat] :
( ( times_times_nat @ A @ one_one_nat )
= A ) ).
% mult.right_neutral
thf(fact_1168_mult_Oright__neutral,axiom,
! [A: int] :
( ( times_times_int @ A @ one_one_int )
= A ) ).
% mult.right_neutral
thf(fact_1169_mult__1,axiom,
! [A: nat] :
( ( times_times_nat @ one_one_nat @ A )
= A ) ).
% mult_1
thf(fact_1170_mult__1,axiom,
! [A: int] :
( ( times_times_int @ one_one_int @ A )
= A ) ).
% mult_1
thf(fact_1171_mult__minus__left,axiom,
! [A: int,B: int] :
( ( times_times_int @ ( uminus_uminus_int @ A ) @ B )
= ( uminus_uminus_int @ ( times_times_int @ A @ B ) ) ) ).
% mult_minus_left
thf(fact_1172_minus__mult__minus,axiom,
! [A: int,B: int] :
( ( times_times_int @ ( uminus_uminus_int @ A ) @ ( uminus_uminus_int @ B ) )
= ( times_times_int @ A @ B ) ) ).
% minus_mult_minus
thf(fact_1173_mult__minus__right,axiom,
! [A: int,B: int] :
( ( times_times_int @ A @ ( uminus_uminus_int @ B ) )
= ( uminus_uminus_int @ ( times_times_int @ A @ B ) ) ) ).
% mult_minus_right
thf(fact_1174_of__nat__mult,axiom,
! [M: nat,N: nat] :
( ( semiri1314217659103216013at_int @ ( times_times_nat @ M @ N ) )
= ( times_times_int @ ( semiri1314217659103216013at_int @ M ) @ ( semiri1314217659103216013at_int @ N ) ) ) ).
% of_nat_mult
thf(fact_1175_of__nat__mult,axiom,
! [M: nat,N: nat] :
( ( semiri1316708129612266289at_nat @ ( times_times_nat @ M @ N ) )
= ( times_times_nat @ ( semiri1316708129612266289at_nat @ M ) @ ( semiri1316708129612266289at_nat @ N ) ) ) ).
% of_nat_mult
thf(fact_1176_mult__cancel__left1,axiom,
! [C: int,B: int] :
( ( C
= ( times_times_int @ C @ B ) )
= ( ( C = zero_zero_int )
| ( B = one_one_int ) ) ) ).
% mult_cancel_left1
thf(fact_1177_mult__cancel__left2,axiom,
! [C: int,A: int] :
( ( ( times_times_int @ C @ A )
= C )
= ( ( C = zero_zero_int )
| ( A = one_one_int ) ) ) ).
% mult_cancel_left2
thf(fact_1178_mult__cancel__right1,axiom,
! [C: int,B: int] :
( ( C
= ( times_times_int @ B @ C ) )
= ( ( C = zero_zero_int )
| ( B = one_one_int ) ) ) ).
% mult_cancel_right1
thf(fact_1179_mult__cancel__right2,axiom,
! [A: int,C: int] :
( ( ( times_times_int @ A @ C )
= C )
= ( ( C = zero_zero_int )
| ( A = one_one_int ) ) ) ).
% mult_cancel_right2
thf(fact_1180_mult__minus1,axiom,
! [Z2: int] :
( ( times_times_int @ ( uminus_uminus_int @ one_one_int ) @ Z2 )
= ( uminus_uminus_int @ Z2 ) ) ).
% mult_minus1
thf(fact_1181_mult__minus1__right,axiom,
! [Z2: int] :
( ( times_times_int @ Z2 @ ( uminus_uminus_int @ one_one_int ) )
= ( uminus_uminus_int @ Z2 ) ) ).
% mult_minus1_right
thf(fact_1182_of__int__mult,axiom,
! [W: int,Z2: int] :
( ( ring_1_of_int_int @ ( times_times_int @ W @ Z2 ) )
= ( times_times_int @ ( ring_1_of_int_int @ W ) @ ( ring_1_of_int_int @ Z2 ) ) ) ).
% of_int_mult
thf(fact_1183_abs__mult__less,axiom,
! [A: int,C: int,B: int,D2: int] :
( ( ord_less_int @ ( abs_abs_int @ A ) @ C )
=> ( ( ord_less_int @ ( abs_abs_int @ B ) @ D2 )
=> ( ord_less_int @ ( times_times_int @ ( abs_abs_int @ A ) @ ( abs_abs_int @ B ) ) @ ( times_times_int @ C @ D2 ) ) ) ) ).
% abs_mult_less
thf(fact_1184_abs__zmult__eq__1,axiom,
! [M: int,N: int] :
( ( ( abs_abs_int @ ( times_times_int @ M @ N ) )
= one_one_int )
=> ( ( abs_abs_int @ M )
= one_one_int ) ) ).
% abs_zmult_eq_1
thf(fact_1185_mult__le__cancel__left,axiom,
! [C: int,A: int,B: int] :
( ( ord_less_eq_int @ ( times_times_int @ C @ A ) @ ( times_times_int @ C @ B ) )
= ( ( ( ord_less_int @ zero_zero_int @ C )
=> ( ord_less_eq_int @ A @ B ) )
& ( ( ord_less_int @ C @ zero_zero_int )
=> ( ord_less_eq_int @ B @ A ) ) ) ) ).
% mult_le_cancel_left
thf(fact_1186_mult__le__cancel__right,axiom,
! [A: int,C: int,B: int] :
( ( ord_less_eq_int @ ( times_times_int @ A @ C ) @ ( times_times_int @ B @ C ) )
= ( ( ( ord_less_int @ zero_zero_int @ C )
=> ( ord_less_eq_int @ A @ B ) )
& ( ( ord_less_int @ C @ zero_zero_int )
=> ( ord_less_eq_int @ B @ A ) ) ) ) ).
% mult_le_cancel_right
thf(fact_1187_mult__left__less__imp__less,axiom,
! [C: int,A: int,B: int] :
( ( ord_less_int @ ( times_times_int @ C @ A ) @ ( times_times_int @ C @ B ) )
=> ( ( ord_less_eq_int @ zero_zero_int @ C )
=> ( ord_less_int @ A @ B ) ) ) ).
% mult_left_less_imp_less
thf(fact_1188_mult__left__less__imp__less,axiom,
! [C: nat,A: nat,B: nat] :
( ( ord_less_nat @ ( times_times_nat @ C @ A ) @ ( times_times_nat @ C @ B ) )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ C )
=> ( ord_less_nat @ A @ B ) ) ) ).
% mult_left_less_imp_less
thf(fact_1189_mult__strict__mono,axiom,
! [A: int,B: int,C: int,D2: int] :
( ( ord_less_int @ A @ B )
=> ( ( ord_less_int @ C @ D2 )
=> ( ( ord_less_int @ zero_zero_int @ B )
=> ( ( ord_less_eq_int @ zero_zero_int @ C )
=> ( ord_less_int @ ( times_times_int @ A @ C ) @ ( times_times_int @ B @ D2 ) ) ) ) ) ) ).
% mult_strict_mono
thf(fact_1190_mult__strict__mono,axiom,
! [A: nat,B: nat,C: nat,D2: nat] :
( ( ord_less_nat @ A @ B )
=> ( ( ord_less_nat @ C @ D2 )
=> ( ( ord_less_nat @ zero_zero_nat @ B )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ C )
=> ( ord_less_nat @ ( times_times_nat @ A @ C ) @ ( times_times_nat @ B @ D2 ) ) ) ) ) ) ).
% mult_strict_mono
thf(fact_1191_mult__less__cancel__left,axiom,
! [C: int,A: int,B: int] :
( ( ord_less_int @ ( times_times_int @ C @ A ) @ ( times_times_int @ C @ B ) )
= ( ( ( ord_less_eq_int @ zero_zero_int @ C )
=> ( ord_less_int @ A @ B ) )
& ( ( ord_less_eq_int @ C @ zero_zero_int )
=> ( ord_less_int @ B @ A ) ) ) ) ).
% mult_less_cancel_left
thf(fact_1192_mult__right__less__imp__less,axiom,
! [A: int,C: int,B: int] :
( ( ord_less_int @ ( times_times_int @ A @ C ) @ ( times_times_int @ B @ C ) )
=> ( ( ord_less_eq_int @ zero_zero_int @ C )
=> ( ord_less_int @ A @ B ) ) ) ).
% mult_right_less_imp_less
thf(fact_1193_mult__right__less__imp__less,axiom,
! [A: nat,C: nat,B: nat] :
( ( ord_less_nat @ ( times_times_nat @ A @ C ) @ ( times_times_nat @ B @ C ) )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ C )
=> ( ord_less_nat @ A @ B ) ) ) ).
% mult_right_less_imp_less
thf(fact_1194_mult__strict__mono_H,axiom,
! [A: int,B: int,C: int,D2: int] :
( ( ord_less_int @ A @ B )
=> ( ( ord_less_int @ C @ D2 )
=> ( ( ord_less_eq_int @ zero_zero_int @ A )
=> ( ( ord_less_eq_int @ zero_zero_int @ C )
=> ( ord_less_int @ ( times_times_int @ A @ C ) @ ( times_times_int @ B @ D2 ) ) ) ) ) ) ).
% mult_strict_mono'
thf(fact_1195_mult__strict__mono_H,axiom,
! [A: nat,B: nat,C: nat,D2: nat] :
( ( ord_less_nat @ A @ B )
=> ( ( ord_less_nat @ C @ D2 )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ A )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ C )
=> ( ord_less_nat @ ( times_times_nat @ A @ C ) @ ( times_times_nat @ B @ D2 ) ) ) ) ) ) ).
% mult_strict_mono'
thf(fact_1196_mult__less__cancel__right,axiom,
! [A: int,C: int,B: int] :
( ( ord_less_int @ ( times_times_int @ A @ C ) @ ( times_times_int @ B @ C ) )
= ( ( ( ord_less_eq_int @ zero_zero_int @ C )
=> ( ord_less_int @ A @ B ) )
& ( ( ord_less_eq_int @ C @ zero_zero_int )
=> ( ord_less_int @ B @ A ) ) ) ) ).
% mult_less_cancel_right
thf(fact_1197_mult__le__cancel__left__neg,axiom,
! [C: int,A: int,B: int] :
( ( ord_less_int @ C @ zero_zero_int )
=> ( ( ord_less_eq_int @ ( times_times_int @ C @ A ) @ ( times_times_int @ C @ B ) )
= ( ord_less_eq_int @ B @ A ) ) ) ).
% mult_le_cancel_left_neg
thf(fact_1198_mult__le__cancel__left__pos,axiom,
! [C: int,A: int,B: int] :
( ( ord_less_int @ zero_zero_int @ C )
=> ( ( ord_less_eq_int @ ( times_times_int @ C @ A ) @ ( times_times_int @ C @ B ) )
= ( ord_less_eq_int @ A @ B ) ) ) ).
% mult_le_cancel_left_pos
thf(fact_1199_mult__left__le__imp__le,axiom,
! [C: int,A: int,B: int] :
( ( ord_less_eq_int @ ( times_times_int @ C @ A ) @ ( times_times_int @ C @ B ) )
=> ( ( ord_less_int @ zero_zero_int @ C )
=> ( ord_less_eq_int @ A @ B ) ) ) ).
% mult_left_le_imp_le
thf(fact_1200_mult__left__le__imp__le,axiom,
! [C: nat,A: nat,B: nat] :
( ( ord_less_eq_nat @ ( times_times_nat @ C @ A ) @ ( times_times_nat @ C @ B ) )
=> ( ( ord_less_nat @ zero_zero_nat @ C )
=> ( ord_less_eq_nat @ A @ B ) ) ) ).
% mult_left_le_imp_le
thf(fact_1201_mult__right__le__imp__le,axiom,
! [A: int,C: int,B: int] :
( ( ord_less_eq_int @ ( times_times_int @ A @ C ) @ ( times_times_int @ B @ C ) )
=> ( ( ord_less_int @ zero_zero_int @ C )
=> ( ord_less_eq_int @ A @ B ) ) ) ).
% mult_right_le_imp_le
thf(fact_1202_mult__right__le__imp__le,axiom,
! [A: nat,C: nat,B: nat] :
( ( ord_less_eq_nat @ ( times_times_nat @ A @ C ) @ ( times_times_nat @ B @ C ) )
=> ( ( ord_less_nat @ zero_zero_nat @ C )
=> ( ord_less_eq_nat @ A @ B ) ) ) ).
% mult_right_le_imp_le
thf(fact_1203_mult__le__less__imp__less,axiom,
! [A: int,B: int,C: int,D2: int] :
( ( ord_less_eq_int @ A @ B )
=> ( ( ord_less_int @ C @ D2 )
=> ( ( ord_less_int @ zero_zero_int @ A )
=> ( ( ord_less_eq_int @ zero_zero_int @ C )
=> ( ord_less_int @ ( times_times_int @ A @ C ) @ ( times_times_int @ B @ D2 ) ) ) ) ) ) ).
% mult_le_less_imp_less
thf(fact_1204_mult__le__less__imp__less,axiom,
! [A: nat,B: nat,C: nat,D2: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ord_less_nat @ C @ D2 )
=> ( ( ord_less_nat @ zero_zero_nat @ A )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ C )
=> ( ord_less_nat @ ( times_times_nat @ A @ C ) @ ( times_times_nat @ B @ D2 ) ) ) ) ) ) ).
% mult_le_less_imp_less
thf(fact_1205_mult__less__le__imp__less,axiom,
! [A: int,B: int,C: int,D2: int] :
( ( ord_less_int @ A @ B )
=> ( ( ord_less_eq_int @ C @ D2 )
=> ( ( ord_less_eq_int @ zero_zero_int @ A )
=> ( ( ord_less_int @ zero_zero_int @ C )
=> ( ord_less_int @ ( times_times_int @ A @ C ) @ ( times_times_int @ B @ D2 ) ) ) ) ) ) ).
% mult_less_le_imp_less
thf(fact_1206_mult__less__le__imp__less,axiom,
! [A: nat,B: nat,C: nat,D2: nat] :
( ( ord_less_nat @ A @ B )
=> ( ( ord_less_eq_nat @ C @ D2 )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ A )
=> ( ( ord_less_nat @ zero_zero_nat @ C )
=> ( ord_less_nat @ ( times_times_nat @ A @ C ) @ ( times_times_nat @ B @ D2 ) ) ) ) ) ) ).
% mult_less_le_imp_less
thf(fact_1207_sum__squares__ge__zero,axiom,
! [X: int,Y: int] : ( ord_less_eq_int @ zero_zero_int @ ( plus_plus_int @ ( times_times_int @ X @ X ) @ ( times_times_int @ Y @ Y ) ) ) ).
% sum_squares_ge_zero
thf(fact_1208_mult__left__le,axiom,
! [C: int,A: int] :
( ( ord_less_eq_int @ C @ one_one_int )
=> ( ( ord_less_eq_int @ zero_zero_int @ A )
=> ( ord_less_eq_int @ ( times_times_int @ A @ C ) @ A ) ) ) ).
% mult_left_le
thf(fact_1209_mult__left__le,axiom,
! [C: nat,A: nat] :
( ( ord_less_eq_nat @ C @ one_one_nat )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ A )
=> ( ord_less_eq_nat @ ( times_times_nat @ A @ C ) @ A ) ) ) ).
% mult_left_le
thf(fact_1210_mult__le__one,axiom,
! [A: int,B: int] :
( ( ord_less_eq_int @ A @ one_one_int )
=> ( ( ord_less_eq_int @ zero_zero_int @ B )
=> ( ( ord_less_eq_int @ B @ one_one_int )
=> ( ord_less_eq_int @ ( times_times_int @ A @ B ) @ one_one_int ) ) ) ) ).
% mult_le_one
thf(fact_1211_mult__le__one,axiom,
! [A: nat,B: nat] :
( ( ord_less_eq_nat @ A @ one_one_nat )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ B )
=> ( ( ord_less_eq_nat @ B @ one_one_nat )
=> ( ord_less_eq_nat @ ( times_times_nat @ A @ B ) @ one_one_nat ) ) ) ) ).
% mult_le_one
thf(fact_1212_mult__right__le__one__le,axiom,
! [X: int,Y: int] :
( ( ord_less_eq_int @ zero_zero_int @ X )
=> ( ( ord_less_eq_int @ zero_zero_int @ Y )
=> ( ( ord_less_eq_int @ Y @ one_one_int )
=> ( ord_less_eq_int @ ( times_times_int @ X @ Y ) @ X ) ) ) ) ).
% mult_right_le_one_le
thf(fact_1213_mult__left__le__one__le,axiom,
! [X: int,Y: int] :
( ( ord_less_eq_int @ zero_zero_int @ X )
=> ( ( ord_less_eq_int @ zero_zero_int @ Y )
=> ( ( ord_less_eq_int @ Y @ one_one_int )
=> ( ord_less_eq_int @ ( times_times_int @ Y @ X ) @ X ) ) ) ) ).
% mult_left_le_one_le
thf(fact_1214_not__sum__squares__lt__zero,axiom,
! [X: int,Y: int] :
~ ( ord_less_int @ ( plus_plus_int @ ( times_times_int @ X @ X ) @ ( times_times_int @ Y @ Y ) ) @ zero_zero_int ) ).
% not_sum_squares_lt_zero
thf(fact_1215_le__add__iff1,axiom,
! [A: int,E: int,C: int,B: int,D2: int] :
( ( ord_less_eq_int @ ( plus_plus_int @ ( times_times_int @ A @ E ) @ C ) @ ( plus_plus_int @ ( times_times_int @ B @ E ) @ D2 ) )
= ( ord_less_eq_int @ ( plus_plus_int @ ( times_times_int @ ( minus_minus_int @ A @ B ) @ E ) @ C ) @ D2 ) ) ).
% le_add_iff1
thf(fact_1216_le__add__iff2,axiom,
! [A: int,E: int,C: int,B: int,D2: int] :
( ( ord_less_eq_int @ ( plus_plus_int @ ( times_times_int @ A @ E ) @ C ) @ ( plus_plus_int @ ( times_times_int @ B @ E ) @ D2 ) )
= ( ord_less_eq_int @ C @ ( plus_plus_int @ ( times_times_int @ ( minus_minus_int @ B @ A ) @ E ) @ D2 ) ) ) ).
% le_add_iff2
thf(fact_1217_less__add__iff1,axiom,
! [A: int,E: int,C: int,B: int,D2: int] :
( ( ord_less_int @ ( plus_plus_int @ ( times_times_int @ A @ E ) @ C ) @ ( plus_plus_int @ ( times_times_int @ B @ E ) @ D2 ) )
= ( ord_less_int @ ( plus_plus_int @ ( times_times_int @ ( minus_minus_int @ A @ B ) @ E ) @ C ) @ D2 ) ) ).
% less_add_iff1
thf(fact_1218_less__add__iff2,axiom,
! [A: int,E: int,C: int,B: int,D2: int] :
( ( ord_less_int @ ( plus_plus_int @ ( times_times_int @ A @ E ) @ C ) @ ( plus_plus_int @ ( times_times_int @ B @ E ) @ D2 ) )
= ( ord_less_int @ C @ ( plus_plus_int @ ( times_times_int @ ( minus_minus_int @ B @ A ) @ E ) @ D2 ) ) ) ).
% less_add_iff2
thf(fact_1219_abs__mult__pos,axiom,
! [X: int,Y: int] :
( ( ord_less_eq_int @ zero_zero_int @ X )
=> ( ( times_times_int @ ( abs_abs_int @ Y ) @ X )
= ( abs_abs_int @ ( times_times_int @ Y @ X ) ) ) ) ).
% abs_mult_pos
thf(fact_1220_abs__eq__mult,axiom,
! [A: int,B: int] :
( ( ( ( ord_less_eq_int @ zero_zero_int @ A )
| ( ord_less_eq_int @ A @ zero_zero_int ) )
& ( ( ord_less_eq_int @ zero_zero_int @ B )
| ( ord_less_eq_int @ B @ zero_zero_int ) ) )
=> ( ( abs_abs_int @ ( times_times_int @ A @ B ) )
= ( times_times_int @ ( abs_abs_int @ A ) @ ( abs_abs_int @ B ) ) ) ) ).
% abs_eq_mult
thf(fact_1221_abs__mult__pos_H,axiom,
! [X: int,Y: int] :
( ( ord_less_eq_int @ zero_zero_int @ X )
=> ( ( times_times_int @ X @ ( abs_abs_int @ Y ) )
= ( abs_abs_int @ ( times_times_int @ X @ Y ) ) ) ) ).
% abs_mult_pos'
thf(fact_1222_int__sgnE,axiom,
! [K: int] :
~ ! [N2: nat,L2: int] :
( K
!= ( times_times_int @ ( sgn_sgn_int @ L2 ) @ ( semiri1314217659103216013at_int @ N2 ) ) ) ).
% int_sgnE
thf(fact_1223_add__scale__eq__noteq,axiom,
! [R2: nat,A: nat,B: nat,C: nat,D2: nat] :
( ( R2 != zero_zero_nat )
=> ( ( ( A = B )
& ( C != D2 ) )
=> ( ( plus_plus_nat @ A @ ( times_times_nat @ R2 @ C ) )
!= ( plus_plus_nat @ B @ ( times_times_nat @ R2 @ D2 ) ) ) ) ) ).
% add_scale_eq_noteq
thf(fact_1224_add__scale__eq__noteq,axiom,
! [R2: int,A: int,B: int,C: int,D2: int] :
( ( R2 != zero_zero_int )
=> ( ( ( A = B )
& ( C != D2 ) )
=> ( ( plus_plus_int @ A @ ( times_times_int @ R2 @ C ) )
!= ( plus_plus_int @ B @ ( times_times_int @ R2 @ D2 ) ) ) ) ) ).
% add_scale_eq_noteq
thf(fact_1225_int__distrib_I1_J,axiom,
! [Z1: int,Z22: int,W: int] :
( ( times_times_int @ ( plus_plus_int @ Z1 @ Z22 ) @ W )
= ( plus_plus_int @ ( times_times_int @ Z1 @ W ) @ ( times_times_int @ Z22 @ W ) ) ) ).
% int_distrib(1)
thf(fact_1226_int__distrib_I2_J,axiom,
! [W: int,Z1: int,Z22: int] :
( ( times_times_int @ W @ ( plus_plus_int @ Z1 @ Z22 ) )
= ( plus_plus_int @ ( times_times_int @ W @ Z1 ) @ ( times_times_int @ W @ Z22 ) ) ) ).
% int_distrib(2)
thf(fact_1227_mult__mono,axiom,
! [A: int,B: int,C: int,D2: int] :
( ( ord_less_eq_int @ A @ B )
=> ( ( ord_less_eq_int @ C @ D2 )
=> ( ( ord_less_eq_int @ zero_zero_int @ B )
=> ( ( ord_less_eq_int @ zero_zero_int @ C )
=> ( ord_less_eq_int @ ( times_times_int @ A @ C ) @ ( times_times_int @ B @ D2 ) ) ) ) ) ) ).
% mult_mono
thf(fact_1228_mult__mono,axiom,
! [A: nat,B: nat,C: nat,D2: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ord_less_eq_nat @ C @ D2 )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ B )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ C )
=> ( ord_less_eq_nat @ ( times_times_nat @ A @ C ) @ ( times_times_nat @ B @ D2 ) ) ) ) ) ) ).
% mult_mono
thf(fact_1229_mult__mono_H,axiom,
! [A: int,B: int,C: int,D2: int] :
( ( ord_less_eq_int @ A @ B )
=> ( ( ord_less_eq_int @ C @ D2 )
=> ( ( ord_less_eq_int @ zero_zero_int @ A )
=> ( ( ord_less_eq_int @ zero_zero_int @ C )
=> ( ord_less_eq_int @ ( times_times_int @ A @ C ) @ ( times_times_int @ B @ D2 ) ) ) ) ) ) ).
% mult_mono'
thf(fact_1230_mult__mono_H,axiom,
! [A: nat,B: nat,C: nat,D2: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ord_less_eq_nat @ C @ D2 )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ A )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ C )
=> ( ord_less_eq_nat @ ( times_times_nat @ A @ C ) @ ( times_times_nat @ B @ D2 ) ) ) ) ) ) ).
% mult_mono'
thf(fact_1231_zero__le__square,axiom,
! [A: int] : ( ord_less_eq_int @ zero_zero_int @ ( times_times_int @ A @ A ) ) ).
% zero_le_square
thf(fact_1232_split__mult__pos__le,axiom,
! [A: int,B: int] :
( ( ( ( ord_less_eq_int @ zero_zero_int @ A )
& ( ord_less_eq_int @ zero_zero_int @ B ) )
| ( ( ord_less_eq_int @ A @ zero_zero_int )
& ( ord_less_eq_int @ B @ zero_zero_int ) ) )
=> ( ord_less_eq_int @ zero_zero_int @ ( times_times_int @ A @ B ) ) ) ).
% split_mult_pos_le
thf(fact_1233_mult__left__mono__neg,axiom,
! [B: int,A: int,C: int] :
( ( ord_less_eq_int @ B @ A )
=> ( ( ord_less_eq_int @ C @ zero_zero_int )
=> ( ord_less_eq_int @ ( times_times_int @ C @ A ) @ ( times_times_int @ C @ B ) ) ) ) ).
% mult_left_mono_neg
thf(fact_1234_mult__nonpos__nonpos,axiom,
! [A: int,B: int] :
( ( ord_less_eq_int @ A @ zero_zero_int )
=> ( ( ord_less_eq_int @ B @ zero_zero_int )
=> ( ord_less_eq_int @ zero_zero_int @ ( times_times_int @ A @ B ) ) ) ) ).
% mult_nonpos_nonpos
thf(fact_1235_mult__left__mono,axiom,
! [A: int,B: int,C: int] :
( ( ord_less_eq_int @ A @ B )
=> ( ( ord_less_eq_int @ zero_zero_int @ C )
=> ( ord_less_eq_int @ ( times_times_int @ C @ A ) @ ( times_times_int @ C @ B ) ) ) ) ).
% mult_left_mono
thf(fact_1236_mult__left__mono,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ C )
=> ( ord_less_eq_nat @ ( times_times_nat @ C @ A ) @ ( times_times_nat @ C @ B ) ) ) ) ).
% mult_left_mono
thf(fact_1237_mult__right__mono__neg,axiom,
! [B: int,A: int,C: int] :
( ( ord_less_eq_int @ B @ A )
=> ( ( ord_less_eq_int @ C @ zero_zero_int )
=> ( ord_less_eq_int @ ( times_times_int @ A @ C ) @ ( times_times_int @ B @ C ) ) ) ) ).
% mult_right_mono_neg
thf(fact_1238_mult__right__mono,axiom,
! [A: int,B: int,C: int] :
( ( ord_less_eq_int @ A @ B )
=> ( ( ord_less_eq_int @ zero_zero_int @ C )
=> ( ord_less_eq_int @ ( times_times_int @ A @ C ) @ ( times_times_int @ B @ C ) ) ) ) ).
% mult_right_mono
thf(fact_1239_mult__right__mono,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ C )
=> ( ord_less_eq_nat @ ( times_times_nat @ A @ C ) @ ( times_times_nat @ B @ C ) ) ) ) ).
% mult_right_mono
thf(fact_1240_mult__le__0__iff,axiom,
! [A: int,B: int] :
( ( ord_less_eq_int @ ( times_times_int @ A @ B ) @ zero_zero_int )
= ( ( ( ord_less_eq_int @ zero_zero_int @ A )
& ( ord_less_eq_int @ B @ zero_zero_int ) )
| ( ( ord_less_eq_int @ A @ zero_zero_int )
& ( ord_less_eq_int @ zero_zero_int @ B ) ) ) ) ).
% mult_le_0_iff
thf(fact_1241_split__mult__neg__le,axiom,
! [A: int,B: int] :
( ( ( ( ord_less_eq_int @ zero_zero_int @ A )
& ( ord_less_eq_int @ B @ zero_zero_int ) )
| ( ( ord_less_eq_int @ A @ zero_zero_int )
& ( ord_less_eq_int @ zero_zero_int @ B ) ) )
=> ( ord_less_eq_int @ ( times_times_int @ A @ B ) @ zero_zero_int ) ) ).
% split_mult_neg_le
thf(fact_1242_split__mult__neg__le,axiom,
! [A: nat,B: nat] :
( ( ( ( ord_less_eq_nat @ zero_zero_nat @ A )
& ( ord_less_eq_nat @ B @ zero_zero_nat ) )
| ( ( ord_less_eq_nat @ A @ zero_zero_nat )
& ( ord_less_eq_nat @ zero_zero_nat @ B ) ) )
=> ( ord_less_eq_nat @ ( times_times_nat @ A @ B ) @ zero_zero_nat ) ) ).
% split_mult_neg_le
thf(fact_1243_mult__nonneg__nonneg,axiom,
! [A: int,B: int] :
( ( ord_less_eq_int @ zero_zero_int @ A )
=> ( ( ord_less_eq_int @ zero_zero_int @ B )
=> ( ord_less_eq_int @ zero_zero_int @ ( times_times_int @ A @ B ) ) ) ) ).
% mult_nonneg_nonneg
thf(fact_1244_mult__nonneg__nonneg,axiom,
! [A: nat,B: nat] :
( ( ord_less_eq_nat @ zero_zero_nat @ A )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ B )
=> ( ord_less_eq_nat @ zero_zero_nat @ ( times_times_nat @ A @ B ) ) ) ) ).
% mult_nonneg_nonneg
thf(fact_1245_mult__nonneg__nonpos,axiom,
! [A: int,B: int] :
( ( ord_less_eq_int @ zero_zero_int @ A )
=> ( ( ord_less_eq_int @ B @ zero_zero_int )
=> ( ord_less_eq_int @ ( times_times_int @ A @ B ) @ zero_zero_int ) ) ) ).
% mult_nonneg_nonpos
thf(fact_1246_mult__nonneg__nonpos,axiom,
! [A: nat,B: nat] :
( ( ord_less_eq_nat @ zero_zero_nat @ A )
=> ( ( ord_less_eq_nat @ B @ zero_zero_nat )
=> ( ord_less_eq_nat @ ( times_times_nat @ A @ B ) @ zero_zero_nat ) ) ) ).
% mult_nonneg_nonpos
thf(fact_1247_mult__nonpos__nonneg,axiom,
! [A: int,B: int] :
( ( ord_less_eq_int @ A @ zero_zero_int )
=> ( ( ord_less_eq_int @ zero_zero_int @ B )
=> ( ord_less_eq_int @ ( times_times_int @ A @ B ) @ zero_zero_int ) ) ) ).
% mult_nonpos_nonneg
thf(fact_1248_mult__nonpos__nonneg,axiom,
! [A: nat,B: nat] :
( ( ord_less_eq_nat @ A @ zero_zero_nat )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ B )
=> ( ord_less_eq_nat @ ( times_times_nat @ A @ B ) @ zero_zero_nat ) ) ) ).
% mult_nonpos_nonneg
thf(fact_1249_mult__nonneg__nonpos2,axiom,
! [A: int,B: int] :
( ( ord_less_eq_int @ zero_zero_int @ A )
=> ( ( ord_less_eq_int @ B @ zero_zero_int )
=> ( ord_less_eq_int @ ( times_times_int @ B @ A ) @ zero_zero_int ) ) ) ).
% mult_nonneg_nonpos2
thf(fact_1250_mult__nonneg__nonpos2,axiom,
! [A: nat,B: nat] :
( ( ord_less_eq_nat @ zero_zero_nat @ A )
=> ( ( ord_less_eq_nat @ B @ zero_zero_nat )
=> ( ord_less_eq_nat @ ( times_times_nat @ B @ A ) @ zero_zero_nat ) ) ) ).
% mult_nonneg_nonpos2
thf(fact_1251_zero__le__mult__iff,axiom,
! [A: int,B: int] :
( ( ord_less_eq_int @ zero_zero_int @ ( times_times_int @ A @ B ) )
= ( ( ( ord_less_eq_int @ zero_zero_int @ A )
& ( ord_less_eq_int @ zero_zero_int @ B ) )
| ( ( ord_less_eq_int @ A @ zero_zero_int )
& ( ord_less_eq_int @ B @ zero_zero_int ) ) ) ) ).
% zero_le_mult_iff
thf(fact_1252_ordered__comm__semiring__class_Ocomm__mult__left__mono,axiom,
! [A: int,B: int,C: int] :
( ( ord_less_eq_int @ A @ B )
=> ( ( ord_less_eq_int @ zero_zero_int @ C )
=> ( ord_less_eq_int @ ( times_times_int @ C @ A ) @ ( times_times_int @ C @ B ) ) ) ) ).
% ordered_comm_semiring_class.comm_mult_left_mono
thf(fact_1253_ordered__comm__semiring__class_Ocomm__mult__left__mono,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ C )
=> ( ord_less_eq_nat @ ( times_times_nat @ C @ A ) @ ( times_times_nat @ C @ B ) ) ) ) ).
% ordered_comm_semiring_class.comm_mult_left_mono
thf(fact_1254_mult__neg__neg,axiom,
! [A: int,B: int] :
( ( ord_less_int @ A @ zero_zero_int )
=> ( ( ord_less_int @ B @ zero_zero_int )
=> ( ord_less_int @ zero_zero_int @ ( times_times_int @ A @ B ) ) ) ) ).
% mult_neg_neg
thf(fact_1255_not__square__less__zero,axiom,
! [A: int] :
~ ( ord_less_int @ ( times_times_int @ A @ A ) @ zero_zero_int ) ).
% not_square_less_zero
thf(fact_1256_mult__less__0__iff,axiom,
! [A: int,B: int] :
( ( ord_less_int @ ( times_times_int @ A @ B ) @ zero_zero_int )
= ( ( ( ord_less_int @ zero_zero_int @ A )
& ( ord_less_int @ B @ zero_zero_int ) )
| ( ( ord_less_int @ A @ zero_zero_int )
& ( ord_less_int @ zero_zero_int @ B ) ) ) ) ).
% mult_less_0_iff
thf(fact_1257_mult__neg__pos,axiom,
! [A: int,B: int] :
( ( ord_less_int @ A @ zero_zero_int )
=> ( ( ord_less_int @ zero_zero_int @ B )
=> ( ord_less_int @ ( times_times_int @ A @ B ) @ zero_zero_int ) ) ) ).
% mult_neg_pos
thf(fact_1258_mult__neg__pos,axiom,
! [A: nat,B: nat] :
( ( ord_less_nat @ A @ zero_zero_nat )
=> ( ( ord_less_nat @ zero_zero_nat @ B )
=> ( ord_less_nat @ ( times_times_nat @ A @ B ) @ zero_zero_nat ) ) ) ).
% mult_neg_pos
thf(fact_1259_mult__pos__neg,axiom,
! [A: int,B: int] :
( ( ord_less_int @ zero_zero_int @ A )
=> ( ( ord_less_int @ B @ zero_zero_int )
=> ( ord_less_int @ ( times_times_int @ A @ B ) @ zero_zero_int ) ) ) ).
% mult_pos_neg
thf(fact_1260_mult__pos__neg,axiom,
! [A: nat,B: nat] :
( ( ord_less_nat @ zero_zero_nat @ A )
=> ( ( ord_less_nat @ B @ zero_zero_nat )
=> ( ord_less_nat @ ( times_times_nat @ A @ B ) @ zero_zero_nat ) ) ) ).
% mult_pos_neg
thf(fact_1261_mult__pos__pos,axiom,
! [A: int,B: int] :
( ( ord_less_int @ zero_zero_int @ A )
=> ( ( ord_less_int @ zero_zero_int @ B )
=> ( ord_less_int @ zero_zero_int @ ( times_times_int @ A @ B ) ) ) ) ).
% mult_pos_pos
thf(fact_1262_mult__pos__pos,axiom,
! [A: nat,B: nat] :
( ( ord_less_nat @ zero_zero_nat @ A )
=> ( ( ord_less_nat @ zero_zero_nat @ B )
=> ( ord_less_nat @ zero_zero_nat @ ( times_times_nat @ A @ B ) ) ) ) ).
% mult_pos_pos
thf(fact_1263_mult__pos__neg2,axiom,
! [A: int,B: int] :
( ( ord_less_int @ zero_zero_int @ A )
=> ( ( ord_less_int @ B @ zero_zero_int )
=> ( ord_less_int @ ( times_times_int @ B @ A ) @ zero_zero_int ) ) ) ).
% mult_pos_neg2
thf(fact_1264_mult__pos__neg2,axiom,
! [A: nat,B: nat] :
( ( ord_less_nat @ zero_zero_nat @ A )
=> ( ( ord_less_nat @ B @ zero_zero_nat )
=> ( ord_less_nat @ ( times_times_nat @ B @ A ) @ zero_zero_nat ) ) ) ).
% mult_pos_neg2
% Helper facts (5)
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_3_1_If_001t__Nat__Onat_T,axiom,
! [P: $o] :
( ( P = $true )
| ( P = $false ) ) ).
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 ) ).
% Conjectures (4)
thf(conj_0,hypothesis,
ord_less_eq_int @ zero_zero_int @ i ).
thf(conj_1,hypothesis,
ord_less_int @ i @ ( semiri1314217659103216013at_int @ ( size_size_list_instr @ p ) ) ).
thf(conj_2,hypothesis,
( ( inth_instr @ p @ i )
= ( store @ x ) ) ).
thf(conj_3,conjecture,
member_list_char @ x @ ( set_list_char2 @ ( vars @ p ) ) ).
%------------------------------------------------------------------------------