TPTP Problem File: SLH0896^1.p
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- Solve Problem
%------------------------------------------------------------------------------
% 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 : Knights_Tour/0000_KnightsTour/prob_01716_067387__6039412_1 [Des23]
% Status : Theorem
% Rating : ? v8.2.0
% Syntax : Number of formulae : 1437 ( 998 unt; 169 typ; 0 def)
% Number of atoms : 2209 (1811 equ; 0 cnn)
% Maximal formula atoms : 12 ( 1 avg)
% Number of connectives : 9866 ( 388 ~; 54 |; 77 &;8873 @)
% ( 0 <=>; 474 =>; 0 <=; 0 <~>)
% Maximal formula depth : 34 ( 4 avg)
% Number of types : 34 ( 33 usr)
% Number of type conns : 314 ( 314 >; 0 *; 0 +; 0 <<)
% Number of symbols : 139 ( 136 usr; 14 con; 0-3 aty)
% Number of variables : 2774 ( 80 ^;2646 !; 48 ?;2774 :)
% SPC : TH0_THM_EQU_NAR
% Comments : This file was generated by Isabelle (most likely Sledgehammer)
% 2023-01-19 09:01:13.359
%------------------------------------------------------------------------------
% Could-be-implicit typings (33)
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% Explicit typings (136)
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thf(sy_c_Groups_Oplus__class_Oplus_001t__Int__Oint,type,
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thf(sy_c_Groups_Oplus__class_Oplus_001t__Nat__Onat,type,
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thf(sy_c_Groups_Ouminus__class_Ouminus_001t__Int__Oint,type,
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thf(sy_c_Groups_Ozero__class_Ozero_001t__Int__Oint,type,
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thf(sy_c_Groups_Ozero__class_Ozero_001t__Nat__Onat,type,
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thf(sy_c_If_001t__Nat__Onat,type,
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thf(sy_c_KnightsTour_Oboard,type,
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thf(sy_c_KnightsTour_Omirror1,type,
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thf(sy_c_KnightsTour_Omirror2,type,
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thf(sy_c_KnightsTour_Ostep__checker,type,
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thf(sy_c_KnightsTour_Ostep__in,type,
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thf(sy_c_KnightsTour_Oto__path,type,
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thf(sy_c_KnightsTour_Otrans__board,type,
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thf(sy_c_KnightsTour_Otrans__path,type,
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thf(sy_c_KnightsTour_Otranspose,type,
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thf(sy_c_KnightsTour_Ovalid__step,type,
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thf(sy_c_List_Oinsert_001t__List__Olist_It__Nat__Onat_J,type,
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product_lists_nat: list_list_nat > list_list_nat ).
thf(sy_c_List_Oproduct__lists_001t__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J,type,
produc5568053154996169768nt_int: list_l1670014477004246597nt_int > list_l1670014477004246597nt_int ).
thf(sy_c_List_Osubseqs_001t__List__Olist_It__List__Olist_It__Nat__Onat_J_J,type,
subseq6786254687447370203st_nat: list_list_list_nat > list_l5212752354702395664st_nat ).
thf(sy_c_List_Osubseqs_001t__List__Olist_It__Nat__Onat_J,type,
subseqs_list_nat: list_list_nat > list_list_list_nat ).
thf(sy_c_List_Osubseqs_001t__List__Olist_It__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J_J,type,
subseq2030970315493690260nt_int: list_l1670014477004246597nt_int > list_l2734933928136073931nt_int ).
thf(sy_c_List_Osubseqs_001t__Nat__Onat,type,
subseqs_nat: list_nat > list_list_nat ).
thf(sy_c_List_Osubseqs_001t__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J,type,
subseq1357044202310323342nt_int: list_P5707943133018811711nt_int > list_l1670014477004246597nt_int ).
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_Num_OBitM,type,
bitM: num > num ).
thf(sy_c_Num_Oinc,type,
inc: num > num ).
thf(sy_c_Num_Oneg__numeral__class_Odbl_001t__Int__Oint,type,
neg_numeral_dbl_int: int > int ).
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_Num_Oneg__numeral__class_Osub_001t__Int__Oint,type,
neg_numeral_sub_int: num > num > int ).
thf(sy_c_Num_Onum_OBit0,type,
bit0: num > num ).
thf(sy_c_Num_Onum_OBit1,type,
bit1: num > num ).
thf(sy_c_Num_Onum_OOne,type,
one: num ).
thf(sy_c_Num_Onum_Osize__num,type,
size_num: num > nat ).
thf(sy_c_Num_Onumeral__class_Onumeral_001t__Int__Oint,type,
numeral_numeral_int: num > int ).
thf(sy_c_Num_Onumeral__class_Onumeral_001t__Nat__Onat,type,
numeral_numeral_nat: num > nat ).
thf(sy_c_Num_Opow,type,
pow: num > num > num ).
thf(sy_c_Num_Opred__numeral,type,
pred_numeral: num > nat ).
thf(sy_c_Option_Ooption_Othe_001t__List__Olist_It__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J_J,type,
the_li7431803565598847443nt_int: option3674462004671404037nt_int > list_P5707943133018811711nt_int ).
thf(sy_c_Parity_Ounique__euclidean__semiring__with__nat__division__class_Odivides__aux_001t__Int__Oint,type,
unique5329631941980267465ux_int: product_prod_int_int > $o ).
thf(sy_c_Parity_Ounique__euclidean__semiring__with__nat__division__class_Odivides__aux_001t__Nat__Onat,type,
unique5332122412489317741ux_nat: product_prod_nat_nat > $o ).
thf(sy_c_Parity_Ounique__euclidean__semiring__with__nat__division__class_Odivmod_001t__Int__Oint,type,
unique5403075989570733136od_int: num > num > product_prod_int_int ).
thf(sy_c_Parity_Ounique__euclidean__semiring__with__nat__division__class_Odivmod_001t__Nat__Onat,type,
unique5405566460079783412od_nat: num > num > product_prod_nat_nat ).
thf(sy_c_Power_Opower__class_Opower_001t__Int__Oint,type,
power_power_int: int > nat > int ).
thf(sy_c_Power_Opower__class_Opower_001t__Nat__Onat,type,
power_power_nat: nat > nat > nat ).
thf(sy_c_Product__Type_OPair_001_062_It__List__Olist_It__List__Olist_It__Nat__Onat_J_J_M_062_It__List__Olist_It__List__Olist_It__Nat__Onat_J_J_M_Eo_J_J_001t__List__Olist_It__List__Olist_It__List__Olist_It__Nat__Onat_J_J_J,type,
produc842716015072215631st_nat: ( list_list_nat > list_list_nat > $o ) > list_list_list_nat > produc9085372331848283477st_nat ).
thf(sy_c_Product__Type_OPair_001_062_It__List__Olist_It__Nat__Onat_J_M_062_It__List__Olist_It__Nat__Onat_J_M_Eo_J_J_001t__List__Olist_It__List__Olist_It__Nat__Onat_J_J,type,
produc6344517630863869919st_nat: ( list_nat > list_nat > $o ) > list_list_nat > produc2068713617857708901st_nat ).
thf(sy_c_Product__Type_OPair_001_062_It__List__Olist_It__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J_J_M_062_It__List__Olist_It__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J_J_M_Eo_J_J_001t__List__Olist_It__List__Olist_It__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J_J_J,type,
produc7365201557946222510nt_int: ( list_P5707943133018811711nt_int > list_P5707943133018811711nt_int > $o ) > list_l1670014477004246597nt_int > produc5798082952504421822nt_int ).
thf(sy_c_Product__Type_OPair_001_062_It__Nat__Onat_M_062_It__Nat__Onat_M_Eo_J_J_001t__List__Olist_It__Nat__Onat_J,type,
produc4727192421694094319st_nat: ( nat > nat > $o ) > list_nat > produc254973753779126261st_nat ).
thf(sy_c_Product__Type_OPair_001_062_It__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J_M_062_It__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J_M_Eo_J_J_001t__List__Olist_It__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J_J,type,
produc3328129369365053992nt_int: ( product_prod_int_int > product_prod_int_int > $o ) > list_P5707943133018811711nt_int > produc1050408459402128056nt_int ).
thf(sy_c_Product__Type_OPair_001t__Int__Oint_001t__Int__Oint,type,
product_Pair_int_int: int > int > product_prod_int_int ).
thf(sy_c_Product__Type_OPair_001t__Int__Oint_001t__List__Olist_It__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J_J,type,
produc8814303788642274490nt_int: int > list_P5707943133018811711nt_int > produc661532565036771336nt_int ).
thf(sy_c_Product__Type_OPair_001t__List__Olist_It__List__Olist_It__List__Olist_It__Nat__Onat_J_J_J_001t__List__Olist_It__List__Olist_It__List__Olist_It__Nat__Onat_J_J_J,type,
produc7456843779855495193st_nat: list_list_list_nat > list_list_list_nat > produc9189680189847484897st_nat ).
thf(sy_c_Product__Type_OPair_001t__List__Olist_It__List__Olist_It__Nat__Onat_J_J_001t__List__Olist_It__List__Olist_It__Nat__Onat_J_J,type,
produc7129799990162260089st_nat: list_list_nat > list_list_nat > produc4326814125627636033st_nat ).
thf(sy_c_Product__Type_OPair_001t__List__Olist_It__List__Olist_It__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J_J_J_001t__List__Olist_It__List__Olist_It__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J_J_J,type,
produc4603928489606670295nt_int: list_l1670014477004246597nt_int > list_l1670014477004246597nt_int > produc9134732803953422695nt_int ).
thf(sy_c_Product__Type_OPair_001t__List__Olist_It__Nat__Onat_J_001t__List__Olist_It__Nat__Onat_J,type,
produc2694037385005941721st_nat: list_nat > list_nat > produc1828647624359046049st_nat ).
thf(sy_c_Product__Type_OPair_001t__List__Olist_It__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J_J_001t__List__Olist_It__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J_J,type,
produc1932183703851549015nt_int: list_P5707943133018811711nt_int > list_P5707943133018811711nt_int > produc1089560213143673063nt_int ).
thf(sy_c_Product__Type_OPair_001t__Nat__Onat_001t__List__Olist_It__List__Olist_It__Nat__Onat_J_J,type,
produc1673703249076330713st_nat: nat > list_list_nat > produc2814713032259027617st_nat ).
thf(sy_c_Product__Type_OPair_001t__Nat__Onat_001t__List__Olist_It__Nat__Onat_J,type,
produc8282810413953273033st_nat: nat > list_nat > produc4575160907756185873st_nat ).
thf(sy_c_Product__Type_OPair_001t__Nat__Onat_001t__Nat__Onat,type,
product_Pair_nat_nat: nat > nat > product_prod_nat_nat ).
thf(sy_c_Product__Type_OPair_001t__Nat__Onat_001t__Set__Oset_It__Int__Oint_J,type,
produc29655638201817675et_int: nat > set_int > produc9133624956312949779et_int ).
thf(sy_c_Product__Type_OPair_001t__Num__Onum_001t__Num__Onum,type,
product_Pair_num_num: num > num > product_prod_num_num ).
thf(sy_c_Product__Type_OPair_001t__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J_001t__List__Olist_It__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J_J,type,
produc8677244595851196253nt_int: product_prod_int_int > list_P5707943133018811711nt_int > produc4058024888802116461nt_int ).
thf(sy_c_Product__Type_OPair_001t__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J_001t__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J,type,
produc3646306378393792727nt_int: product_prod_int_int > product_prod_int_int > produc1219242969750017639nt_int ).
thf(sy_c_Product__Type_OPair_001t__Set__Oset_It__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J_J_001t__List__Olist_It__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J_J,type,
produc2261658324281137661nt_int: set_Pr958786334691620121nt_int > list_P5707943133018811711nt_int > produc2007852851243229709nt_int ).
thf(sy_c_Rings_Odivide__class_Odivide_001t__Int__Oint,type,
divide_divide_int: int > int > int ).
thf(sy_c_Rings_Odivide__class_Odivide_001t__Nat__Onat,type,
divide_divide_nat: nat > nat > nat ).
thf(sy_c_Rings_Odvd__class_Odvd_001t__Int__Oint,type,
dvd_dvd_int: int > int > $o ).
thf(sy_c_Rings_Odvd__class_Odvd_001t__Nat__Onat,type,
dvd_dvd_nat: nat > nat > $o ).
thf(sy_c_Set_OCollect_001t__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J,type,
collec213857154873943460nt_int: ( product_prod_int_int > $o ) > set_Pr958786334691620121nt_int ).
thf(sy_c_member_001t__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J,type,
member5262025264175285858nt_int: product_prod_int_int > set_Pr958786334691620121nt_int > $o ).
% Relevant facts (1264)
thf(fact_0_num__elems_Ocases,axiom,
! [X: list_list_nat] :
( ! [R: list_nat,Rs: list_list_nat] :
( X
!= ( cons_list_nat @ R @ Rs ) )
=> ( X = nil_list_nat ) ) ).
% num_elems.cases
thf(fact_1_numeral__eq__one__iff,axiom,
! [N: num] :
( ( ( numeral_numeral_nat @ N )
= one_one_nat )
= ( N = one ) ) ).
% numeral_eq_one_iff
thf(fact_2_numeral__eq__one__iff,axiom,
! [N: num] :
( ( ( numeral_numeral_int @ N )
= one_one_int )
= ( N = one ) ) ).
% numeral_eq_one_iff
thf(fact_3_one__eq__numeral__iff,axiom,
! [N: num] :
( ( one_one_nat
= ( numeral_numeral_nat @ N ) )
= ( one = N ) ) ).
% one_eq_numeral_iff
thf(fact_4_one__eq__numeral__iff,axiom,
! [N: num] :
( ( one_one_int
= ( numeral_numeral_int @ N ) )
= ( one = N ) ) ).
% one_eq_numeral_iff
thf(fact_5_semiring__norm_I86_J,axiom,
! [M: num] :
( ( bit1 @ M )
!= one ) ).
% semiring_norm(86)
thf(fact_6_semiring__norm_I84_J,axiom,
! [N: num] :
( one
!= ( bit1 @ N ) ) ).
% semiring_norm(84)
thf(fact_7_semiring__norm_I89_J,axiom,
! [M: num,N: num] :
( ( bit1 @ M )
!= ( bit0 @ N ) ) ).
% semiring_norm(89)
thf(fact_8_semiring__norm_I88_J,axiom,
! [M: num,N: num] :
( ( bit0 @ M )
!= ( bit1 @ N ) ) ).
% semiring_norm(88)
thf(fact_9_semiring__norm_I85_J,axiom,
! [M: num] :
( ( bit0 @ M )
!= one ) ).
% semiring_norm(85)
thf(fact_10_semiring__norm_I83_J,axiom,
! [N: num] :
( one
!= ( bit0 @ N ) ) ).
% semiring_norm(83)
thf(fact_11_List_Otranspose_Ocases,axiom,
! [X: list_l2734933928136073931nt_int] :
( ( X != nil_li8254795547437800779nt_int )
=> ( ! [Xss: list_l2734933928136073931nt_int] :
( X
!= ( cons_l4848925899132326139nt_int @ nil_li8670148097206105925nt_int @ Xss ) )
=> ~ ! [X2: list_P5707943133018811711nt_int,Xs: list_l1670014477004246597nt_int,Xss: list_l2734933928136073931nt_int] :
( X
!= ( cons_l4848925899132326139nt_int @ ( cons_l7309679040211256053nt_int @ X2 @ Xs ) @ Xss ) ) ) ) ).
% List.transpose.cases
thf(fact_12_List_Otranspose_Ocases,axiom,
! [X: list_l5212752354702395664st_nat] :
( ( X != nil_li5455921165481563386st_nat )
=> ( ! [Xss: list_l5212752354702395664st_nat] :
( X
!= ( cons_l7310388135179752778st_nat @ nil_list_list_nat @ Xss ) )
=> ~ ! [X2: list_list_nat,Xs: list_list_list_nat,Xss: list_l5212752354702395664st_nat] :
( X
!= ( cons_l7310388135179752778st_nat @ ( cons_list_list_nat @ X2 @ Xs ) @ Xss ) ) ) ) ).
% List.transpose.cases
thf(fact_13_List_Otranspose_Ocases,axiom,
! [X: list_list_list_nat] :
( ( X != nil_list_list_nat )
=> ( ! [Xss: list_list_list_nat] :
( X
!= ( cons_list_list_nat @ nil_list_nat @ Xss ) )
=> ~ ! [X2: list_nat,Xs: list_list_nat,Xss: list_list_list_nat] :
( X
!= ( cons_list_list_nat @ ( cons_list_nat @ X2 @ Xs ) @ Xss ) ) ) ) ).
% List.transpose.cases
thf(fact_14_List_Otranspose_Ocases,axiom,
! [X: list_l1670014477004246597nt_int] :
( ( X != nil_li8670148097206105925nt_int )
=> ( ! [Xss: list_l1670014477004246597nt_int] :
( X
!= ( cons_l7309679040211256053nt_int @ nil_Pr2300489316682597567nt_int @ Xss ) )
=> ~ ! [X2: product_prod_int_int,Xs: list_P5707943133018811711nt_int,Xss: list_l1670014477004246597nt_int] :
( X
!= ( cons_l7309679040211256053nt_int @ ( cons_P3334398858971670639nt_int @ X2 @ Xs ) @ Xss ) ) ) ) ).
% List.transpose.cases
thf(fact_15_List_Otranspose_Ocases,axiom,
! [X: list_list_nat] :
( ( X != nil_list_nat )
=> ( ! [Xss: list_list_nat] :
( X
!= ( cons_list_nat @ nil_nat @ Xss ) )
=> ~ ! [X2: nat,Xs: list_nat,Xss: list_list_nat] :
( X
!= ( cons_list_nat @ ( cons_nat @ X2 @ Xs ) @ Xss ) ) ) ) ).
% List.transpose.cases
thf(fact_16_last_Osimps,axiom,
! [Xs2: list_list_nat,X: list_nat] :
( ( ( Xs2 = nil_list_nat )
=> ( ( last_list_nat @ ( cons_list_nat @ X @ Xs2 ) )
= X ) )
& ( ( Xs2 != nil_list_nat )
=> ( ( last_list_nat @ ( cons_list_nat @ X @ Xs2 ) )
= ( last_list_nat @ Xs2 ) ) ) ) ).
% last.simps
thf(fact_17_last_Osimps,axiom,
! [Xs2: list_nat,X: nat] :
( ( ( Xs2 = nil_nat )
=> ( ( last_nat @ ( cons_nat @ X @ Xs2 ) )
= X ) )
& ( ( Xs2 != nil_nat )
=> ( ( last_nat @ ( cons_nat @ X @ Xs2 ) )
= ( last_nat @ Xs2 ) ) ) ) ).
% last.simps
thf(fact_18_last_Osimps,axiom,
! [Xs2: list_P5707943133018811711nt_int,X: product_prod_int_int] :
( ( ( Xs2 = nil_Pr2300489316682597567nt_int )
=> ( ( last_P3305686521732843992nt_int @ ( cons_P3334398858971670639nt_int @ X @ Xs2 ) )
= X ) )
& ( ( Xs2 != nil_Pr2300489316682597567nt_int )
=> ( ( last_P3305686521732843992nt_int @ ( cons_P3334398858971670639nt_int @ X @ Xs2 ) )
= ( last_P3305686521732843992nt_int @ Xs2 ) ) ) ) ).
% last.simps
thf(fact_19_last_Osimps,axiom,
! [Xs2: list_l1670014477004246597nt_int,X: list_P5707943133018811711nt_int] :
( ( ( Xs2 = nil_li8670148097206105925nt_int )
=> ( ( last_l5818330359162608606nt_int @ ( cons_l7309679040211256053nt_int @ X @ Xs2 ) )
= X ) )
& ( ( Xs2 != nil_li8670148097206105925nt_int )
=> ( ( last_l5818330359162608606nt_int @ ( cons_l7309679040211256053nt_int @ X @ Xs2 ) )
= ( last_l5818330359162608606nt_int @ Xs2 ) ) ) ) ).
% last.simps
thf(fact_20_last_Osimps,axiom,
! [Xs2: list_list_list_nat,X: list_list_nat] :
( ( ( Xs2 = nil_list_list_nat )
=> ( ( last_list_list_nat @ ( cons_list_list_nat @ X @ Xs2 ) )
= X ) )
& ( ( Xs2 != nil_list_list_nat )
=> ( ( last_list_list_nat @ ( cons_list_list_nat @ X @ Xs2 ) )
= ( last_list_list_nat @ Xs2 ) ) ) ) ).
% last.simps
thf(fact_21_last__ConsL,axiom,
! [Xs2: list_list_nat,X: list_nat] :
( ( Xs2 = nil_list_nat )
=> ( ( last_list_nat @ ( cons_list_nat @ X @ Xs2 ) )
= X ) ) ).
% last_ConsL
thf(fact_22_last__ConsL,axiom,
! [Xs2: list_nat,X: nat] :
( ( Xs2 = nil_nat )
=> ( ( last_nat @ ( cons_nat @ X @ Xs2 ) )
= X ) ) ).
% last_ConsL
thf(fact_23_last__ConsL,axiom,
! [Xs2: list_P5707943133018811711nt_int,X: product_prod_int_int] :
( ( Xs2 = nil_Pr2300489316682597567nt_int )
=> ( ( last_P3305686521732843992nt_int @ ( cons_P3334398858971670639nt_int @ X @ Xs2 ) )
= X ) ) ).
% last_ConsL
thf(fact_24_last__ConsL,axiom,
! [Xs2: list_l1670014477004246597nt_int,X: list_P5707943133018811711nt_int] :
( ( Xs2 = nil_li8670148097206105925nt_int )
=> ( ( last_l5818330359162608606nt_int @ ( cons_l7309679040211256053nt_int @ X @ Xs2 ) )
= X ) ) ).
% last_ConsL
thf(fact_25_last__ConsL,axiom,
! [Xs2: list_list_list_nat,X: list_list_nat] :
( ( Xs2 = nil_list_list_nat )
=> ( ( last_list_list_nat @ ( cons_list_list_nat @ X @ Xs2 ) )
= X ) ) ).
% last_ConsL
thf(fact_26_last__ConsR,axiom,
! [Xs2: list_list_nat,X: list_nat] :
( ( Xs2 != nil_list_nat )
=> ( ( last_list_nat @ ( cons_list_nat @ X @ Xs2 ) )
= ( last_list_nat @ Xs2 ) ) ) ).
% last_ConsR
thf(fact_27_last__ConsR,axiom,
! [Xs2: list_nat,X: nat] :
( ( Xs2 != nil_nat )
=> ( ( last_nat @ ( cons_nat @ X @ Xs2 ) )
= ( last_nat @ Xs2 ) ) ) ).
% last_ConsR
thf(fact_28_last__ConsR,axiom,
! [Xs2: list_P5707943133018811711nt_int,X: product_prod_int_int] :
( ( Xs2 != nil_Pr2300489316682597567nt_int )
=> ( ( last_P3305686521732843992nt_int @ ( cons_P3334398858971670639nt_int @ X @ Xs2 ) )
= ( last_P3305686521732843992nt_int @ Xs2 ) ) ) ).
% last_ConsR
thf(fact_29_last__ConsR,axiom,
! [Xs2: list_l1670014477004246597nt_int,X: list_P5707943133018811711nt_int] :
( ( Xs2 != nil_li8670148097206105925nt_int )
=> ( ( last_l5818330359162608606nt_int @ ( cons_l7309679040211256053nt_int @ X @ Xs2 ) )
= ( last_l5818330359162608606nt_int @ Xs2 ) ) ) ).
% last_ConsR
thf(fact_30_last__ConsR,axiom,
! [Xs2: list_list_list_nat,X: list_list_nat] :
( ( Xs2 != nil_list_list_nat )
=> ( ( last_list_list_nat @ ( cons_list_list_nat @ X @ Xs2 ) )
= ( last_list_list_nat @ Xs2 ) ) ) ).
% last_ConsR
thf(fact_31_numeral__eq__iff,axiom,
! [M: num,N: num] :
( ( ( numeral_numeral_nat @ M )
= ( numeral_numeral_nat @ N ) )
= ( M = N ) ) ).
% numeral_eq_iff
thf(fact_32_numeral__eq__iff,axiom,
! [M: num,N: num] :
( ( ( numeral_numeral_int @ M )
= ( numeral_numeral_int @ N ) )
= ( M = N ) ) ).
% numeral_eq_iff
thf(fact_33_semiring__norm_I87_J,axiom,
! [M: num,N: num] :
( ( ( bit0 @ M )
= ( bit0 @ N ) )
= ( M = N ) ) ).
% semiring_norm(87)
thf(fact_34_list_Oinject,axiom,
! [X21: list_nat,X22: list_list_nat,Y21: list_nat,Y22: list_list_nat] :
( ( ( cons_list_nat @ X21 @ X22 )
= ( cons_list_nat @ Y21 @ Y22 ) )
= ( ( X21 = Y21 )
& ( X22 = Y22 ) ) ) ).
% list.inject
thf(fact_35_list_Oinject,axiom,
! [X21: nat,X22: list_nat,Y21: nat,Y22: list_nat] :
( ( ( cons_nat @ X21 @ X22 )
= ( cons_nat @ Y21 @ Y22 ) )
= ( ( X21 = Y21 )
& ( X22 = Y22 ) ) ) ).
% list.inject
thf(fact_36_list_Oinject,axiom,
! [X21: product_prod_int_int,X22: list_P5707943133018811711nt_int,Y21: product_prod_int_int,Y22: list_P5707943133018811711nt_int] :
( ( ( cons_P3334398858971670639nt_int @ X21 @ X22 )
= ( cons_P3334398858971670639nt_int @ Y21 @ Y22 ) )
= ( ( X21 = Y21 )
& ( X22 = Y22 ) ) ) ).
% list.inject
thf(fact_37_list_Oinject,axiom,
! [X21: list_P5707943133018811711nt_int,X22: list_l1670014477004246597nt_int,Y21: list_P5707943133018811711nt_int,Y22: list_l1670014477004246597nt_int] :
( ( ( cons_l7309679040211256053nt_int @ X21 @ X22 )
= ( cons_l7309679040211256053nt_int @ Y21 @ Y22 ) )
= ( ( X21 = Y21 )
& ( X22 = Y22 ) ) ) ).
% list.inject
thf(fact_38_list_Oinject,axiom,
! [X21: list_list_nat,X22: list_list_list_nat,Y21: list_list_nat,Y22: list_list_list_nat] :
( ( ( cons_list_list_nat @ X21 @ X22 )
= ( cons_list_list_nat @ Y21 @ Y22 ) )
= ( ( X21 = Y21 )
& ( X22 = Y22 ) ) ) ).
% list.inject
thf(fact_39_semiring__norm_I90_J,axiom,
! [M: num,N: num] :
( ( ( bit1 @ M )
= ( bit1 @ N ) )
= ( M = N ) ) ).
% semiring_norm(90)
thf(fact_40_not__Cons__self2,axiom,
! [X: list_nat,Xs2: list_list_nat] :
( ( cons_list_nat @ X @ Xs2 )
!= Xs2 ) ).
% not_Cons_self2
thf(fact_41_not__Cons__self2,axiom,
! [X: nat,Xs2: list_nat] :
( ( cons_nat @ X @ Xs2 )
!= Xs2 ) ).
% not_Cons_self2
thf(fact_42_not__Cons__self2,axiom,
! [X: product_prod_int_int,Xs2: list_P5707943133018811711nt_int] :
( ( cons_P3334398858971670639nt_int @ X @ Xs2 )
!= Xs2 ) ).
% not_Cons_self2
thf(fact_43_not__Cons__self2,axiom,
! [X: list_P5707943133018811711nt_int,Xs2: list_l1670014477004246597nt_int] :
( ( cons_l7309679040211256053nt_int @ X @ Xs2 )
!= Xs2 ) ).
% not_Cons_self2
thf(fact_44_not__Cons__self2,axiom,
! [X: list_list_nat,Xs2: list_list_list_nat] :
( ( cons_list_list_nat @ X @ Xs2 )
!= Xs2 ) ).
% not_Cons_self2
thf(fact_45_list__nonempty__induct,axiom,
! [Xs2: list_list_nat,P: list_list_nat > $o] :
( ( Xs2 != nil_list_nat )
=> ( ! [X2: list_nat] : ( P @ ( cons_list_nat @ X2 @ nil_list_nat ) )
=> ( ! [X2: list_nat,Xs: list_list_nat] :
( ( Xs != nil_list_nat )
=> ( ( P @ Xs )
=> ( P @ ( cons_list_nat @ X2 @ Xs ) ) ) )
=> ( P @ Xs2 ) ) ) ) ).
% list_nonempty_induct
thf(fact_46_list__nonempty__induct,axiom,
! [Xs2: list_nat,P: list_nat > $o] :
( ( Xs2 != nil_nat )
=> ( ! [X2: nat] : ( P @ ( cons_nat @ X2 @ nil_nat ) )
=> ( ! [X2: nat,Xs: list_nat] :
( ( Xs != nil_nat )
=> ( ( P @ Xs )
=> ( P @ ( cons_nat @ X2 @ Xs ) ) ) )
=> ( P @ Xs2 ) ) ) ) ).
% list_nonempty_induct
thf(fact_47_list__nonempty__induct,axiom,
! [Xs2: list_P5707943133018811711nt_int,P: list_P5707943133018811711nt_int > $o] :
( ( Xs2 != nil_Pr2300489316682597567nt_int )
=> ( ! [X2: product_prod_int_int] : ( P @ ( cons_P3334398858971670639nt_int @ X2 @ nil_Pr2300489316682597567nt_int ) )
=> ( ! [X2: product_prod_int_int,Xs: list_P5707943133018811711nt_int] :
( ( Xs != nil_Pr2300489316682597567nt_int )
=> ( ( P @ Xs )
=> ( P @ ( cons_P3334398858971670639nt_int @ X2 @ Xs ) ) ) )
=> ( P @ Xs2 ) ) ) ) ).
% list_nonempty_induct
thf(fact_48_list__nonempty__induct,axiom,
! [Xs2: list_l1670014477004246597nt_int,P: list_l1670014477004246597nt_int > $o] :
( ( Xs2 != nil_li8670148097206105925nt_int )
=> ( ! [X2: list_P5707943133018811711nt_int] : ( P @ ( cons_l7309679040211256053nt_int @ X2 @ nil_li8670148097206105925nt_int ) )
=> ( ! [X2: list_P5707943133018811711nt_int,Xs: list_l1670014477004246597nt_int] :
( ( Xs != nil_li8670148097206105925nt_int )
=> ( ( P @ Xs )
=> ( P @ ( cons_l7309679040211256053nt_int @ X2 @ Xs ) ) ) )
=> ( P @ Xs2 ) ) ) ) ).
% list_nonempty_induct
thf(fact_49_list__nonempty__induct,axiom,
! [Xs2: list_list_list_nat,P: list_list_list_nat > $o] :
( ( Xs2 != nil_list_list_nat )
=> ( ! [X2: list_list_nat] : ( P @ ( cons_list_list_nat @ X2 @ nil_list_list_nat ) )
=> ( ! [X2: list_list_nat,Xs: list_list_list_nat] :
( ( Xs != nil_list_list_nat )
=> ( ( P @ Xs )
=> ( P @ ( cons_list_list_nat @ X2 @ Xs ) ) ) )
=> ( P @ Xs2 ) ) ) ) ).
% list_nonempty_induct
thf(fact_50_list__induct2_H,axiom,
! [P: list_nat > list_nat > $o,Xs2: list_nat,Ys: list_nat] :
( ( P @ nil_nat @ nil_nat )
=> ( ! [X2: nat,Xs: list_nat] : ( P @ ( cons_nat @ X2 @ Xs ) @ nil_nat )
=> ( ! [Y: nat,Ys2: list_nat] : ( P @ nil_nat @ ( cons_nat @ Y @ Ys2 ) )
=> ( ! [X2: nat,Xs: list_nat,Y: nat,Ys2: list_nat] :
( ( P @ Xs @ Ys2 )
=> ( P @ ( cons_nat @ X2 @ Xs ) @ ( cons_nat @ Y @ Ys2 ) ) )
=> ( P @ Xs2 @ Ys ) ) ) ) ) ).
% list_induct2'
thf(fact_51_list__induct2_H,axiom,
! [P: list_list_nat > list_nat > $o,Xs2: list_list_nat,Ys: list_nat] :
( ( P @ nil_list_nat @ nil_nat )
=> ( ! [X2: list_nat,Xs: list_list_nat] : ( P @ ( cons_list_nat @ X2 @ Xs ) @ nil_nat )
=> ( ! [Y: nat,Ys2: list_nat] : ( P @ nil_list_nat @ ( cons_nat @ Y @ Ys2 ) )
=> ( ! [X2: list_nat,Xs: list_list_nat,Y: nat,Ys2: list_nat] :
( ( P @ Xs @ Ys2 )
=> ( P @ ( cons_list_nat @ X2 @ Xs ) @ ( cons_nat @ Y @ Ys2 ) ) )
=> ( P @ Xs2 @ Ys ) ) ) ) ) ).
% list_induct2'
thf(fact_52_list__induct2_H,axiom,
! [P: list_nat > list_list_nat > $o,Xs2: list_nat,Ys: list_list_nat] :
( ( P @ nil_nat @ nil_list_nat )
=> ( ! [X2: nat,Xs: list_nat] : ( P @ ( cons_nat @ X2 @ Xs ) @ nil_list_nat )
=> ( ! [Y: list_nat,Ys2: list_list_nat] : ( P @ nil_nat @ ( cons_list_nat @ Y @ Ys2 ) )
=> ( ! [X2: nat,Xs: list_nat,Y: list_nat,Ys2: list_list_nat] :
( ( P @ Xs @ Ys2 )
=> ( P @ ( cons_nat @ X2 @ Xs ) @ ( cons_list_nat @ Y @ Ys2 ) ) )
=> ( P @ Xs2 @ Ys ) ) ) ) ) ).
% list_induct2'
thf(fact_53_list__induct2_H,axiom,
! [P: list_list_nat > list_list_nat > $o,Xs2: list_list_nat,Ys: list_list_nat] :
( ( P @ nil_list_nat @ nil_list_nat )
=> ( ! [X2: list_nat,Xs: list_list_nat] : ( P @ ( cons_list_nat @ X2 @ Xs ) @ nil_list_nat )
=> ( ! [Y: list_nat,Ys2: list_list_nat] : ( P @ nil_list_nat @ ( cons_list_nat @ Y @ Ys2 ) )
=> ( ! [X2: list_nat,Xs: list_list_nat,Y: list_nat,Ys2: list_list_nat] :
( ( P @ Xs @ Ys2 )
=> ( P @ ( cons_list_nat @ X2 @ Xs ) @ ( cons_list_nat @ Y @ Ys2 ) ) )
=> ( P @ Xs2 @ Ys ) ) ) ) ) ).
% list_induct2'
thf(fact_54_list__induct2_H,axiom,
! [P: list_nat > list_P5707943133018811711nt_int > $o,Xs2: list_nat,Ys: list_P5707943133018811711nt_int] :
( ( P @ nil_nat @ nil_Pr2300489316682597567nt_int )
=> ( ! [X2: nat,Xs: list_nat] : ( P @ ( cons_nat @ X2 @ Xs ) @ nil_Pr2300489316682597567nt_int )
=> ( ! [Y: product_prod_int_int,Ys2: list_P5707943133018811711nt_int] : ( P @ nil_nat @ ( cons_P3334398858971670639nt_int @ Y @ Ys2 ) )
=> ( ! [X2: nat,Xs: list_nat,Y: product_prod_int_int,Ys2: list_P5707943133018811711nt_int] :
( ( P @ Xs @ Ys2 )
=> ( P @ ( cons_nat @ X2 @ Xs ) @ ( cons_P3334398858971670639nt_int @ Y @ Ys2 ) ) )
=> ( P @ Xs2 @ Ys ) ) ) ) ) ).
% list_induct2'
thf(fact_55_list__induct2_H,axiom,
! [P: list_nat > list_list_list_nat > $o,Xs2: list_nat,Ys: list_list_list_nat] :
( ( P @ nil_nat @ nil_list_list_nat )
=> ( ! [X2: nat,Xs: list_nat] : ( P @ ( cons_nat @ X2 @ Xs ) @ nil_list_list_nat )
=> ( ! [Y: list_list_nat,Ys2: list_list_list_nat] : ( P @ nil_nat @ ( cons_list_list_nat @ Y @ Ys2 ) )
=> ( ! [X2: nat,Xs: list_nat,Y: list_list_nat,Ys2: list_list_list_nat] :
( ( P @ Xs @ Ys2 )
=> ( P @ ( cons_nat @ X2 @ Xs ) @ ( cons_list_list_nat @ Y @ Ys2 ) ) )
=> ( P @ Xs2 @ Ys ) ) ) ) ) ).
% list_induct2'
thf(fact_56_list__induct2_H,axiom,
! [P: list_P5707943133018811711nt_int > list_nat > $o,Xs2: list_P5707943133018811711nt_int,Ys: list_nat] :
( ( P @ nil_Pr2300489316682597567nt_int @ nil_nat )
=> ( ! [X2: product_prod_int_int,Xs: list_P5707943133018811711nt_int] : ( P @ ( cons_P3334398858971670639nt_int @ X2 @ Xs ) @ nil_nat )
=> ( ! [Y: nat,Ys2: list_nat] : ( P @ nil_Pr2300489316682597567nt_int @ ( cons_nat @ Y @ Ys2 ) )
=> ( ! [X2: product_prod_int_int,Xs: list_P5707943133018811711nt_int,Y: nat,Ys2: list_nat] :
( ( P @ Xs @ Ys2 )
=> ( P @ ( cons_P3334398858971670639nt_int @ X2 @ Xs ) @ ( cons_nat @ Y @ Ys2 ) ) )
=> ( P @ Xs2 @ Ys ) ) ) ) ) ).
% list_induct2'
thf(fact_57_list__induct2_H,axiom,
! [P: list_list_list_nat > list_nat > $o,Xs2: list_list_list_nat,Ys: list_nat] :
( ( P @ nil_list_list_nat @ nil_nat )
=> ( ! [X2: list_list_nat,Xs: list_list_list_nat] : ( P @ ( cons_list_list_nat @ X2 @ Xs ) @ nil_nat )
=> ( ! [Y: nat,Ys2: list_nat] : ( P @ nil_list_list_nat @ ( cons_nat @ Y @ Ys2 ) )
=> ( ! [X2: list_list_nat,Xs: list_list_list_nat,Y: nat,Ys2: list_nat] :
( ( P @ Xs @ Ys2 )
=> ( P @ ( cons_list_list_nat @ X2 @ Xs ) @ ( cons_nat @ Y @ Ys2 ) ) )
=> ( P @ Xs2 @ Ys ) ) ) ) ) ).
% list_induct2'
thf(fact_58_list__induct2_H,axiom,
! [P: list_list_nat > list_P5707943133018811711nt_int > $o,Xs2: list_list_nat,Ys: list_P5707943133018811711nt_int] :
( ( P @ nil_list_nat @ nil_Pr2300489316682597567nt_int )
=> ( ! [X2: list_nat,Xs: list_list_nat] : ( P @ ( cons_list_nat @ X2 @ Xs ) @ nil_Pr2300489316682597567nt_int )
=> ( ! [Y: product_prod_int_int,Ys2: list_P5707943133018811711nt_int] : ( P @ nil_list_nat @ ( cons_P3334398858971670639nt_int @ Y @ Ys2 ) )
=> ( ! [X2: list_nat,Xs: list_list_nat,Y: product_prod_int_int,Ys2: list_P5707943133018811711nt_int] :
( ( P @ Xs @ Ys2 )
=> ( P @ ( cons_list_nat @ X2 @ Xs ) @ ( cons_P3334398858971670639nt_int @ Y @ Ys2 ) ) )
=> ( P @ Xs2 @ Ys ) ) ) ) ) ).
% list_induct2'
thf(fact_59_list__induct2_H,axiom,
! [P: list_list_nat > list_list_list_nat > $o,Xs2: list_list_nat,Ys: list_list_list_nat] :
( ( P @ nil_list_nat @ nil_list_list_nat )
=> ( ! [X2: list_nat,Xs: list_list_nat] : ( P @ ( cons_list_nat @ X2 @ Xs ) @ nil_list_list_nat )
=> ( ! [Y: list_list_nat,Ys2: list_list_list_nat] : ( P @ nil_list_nat @ ( cons_list_list_nat @ Y @ Ys2 ) )
=> ( ! [X2: list_nat,Xs: list_list_nat,Y: list_list_nat,Ys2: list_list_list_nat] :
( ( P @ Xs @ Ys2 )
=> ( P @ ( cons_list_nat @ X2 @ Xs ) @ ( cons_list_list_nat @ Y @ Ys2 ) ) )
=> ( P @ Xs2 @ Ys ) ) ) ) ) ).
% list_induct2'
thf(fact_60_neq__Nil__conv,axiom,
! [Xs2: list_list_nat] :
( ( Xs2 != nil_list_nat )
= ( ? [Y2: list_nat,Ys3: list_list_nat] :
( Xs2
= ( cons_list_nat @ Y2 @ Ys3 ) ) ) ) ).
% neq_Nil_conv
thf(fact_61_neq__Nil__conv,axiom,
! [Xs2: list_nat] :
( ( Xs2 != nil_nat )
= ( ? [Y2: nat,Ys3: list_nat] :
( Xs2
= ( cons_nat @ Y2 @ Ys3 ) ) ) ) ).
% neq_Nil_conv
thf(fact_62_neq__Nil__conv,axiom,
! [Xs2: list_P5707943133018811711nt_int] :
( ( Xs2 != nil_Pr2300489316682597567nt_int )
= ( ? [Y2: product_prod_int_int,Ys3: list_P5707943133018811711nt_int] :
( Xs2
= ( cons_P3334398858971670639nt_int @ Y2 @ Ys3 ) ) ) ) ).
% neq_Nil_conv
thf(fact_63_neq__Nil__conv,axiom,
! [Xs2: list_l1670014477004246597nt_int] :
( ( Xs2 != nil_li8670148097206105925nt_int )
= ( ? [Y2: list_P5707943133018811711nt_int,Ys3: list_l1670014477004246597nt_int] :
( Xs2
= ( cons_l7309679040211256053nt_int @ Y2 @ Ys3 ) ) ) ) ).
% neq_Nil_conv
thf(fact_64_neq__Nil__conv,axiom,
! [Xs2: list_list_list_nat] :
( ( Xs2 != nil_list_list_nat )
= ( ? [Y2: list_list_nat,Ys3: list_list_list_nat] :
( Xs2
= ( cons_list_list_nat @ Y2 @ Ys3 ) ) ) ) ).
% neq_Nil_conv
thf(fact_65_remdups__adj_Ocases,axiom,
! [X: list_list_nat] :
( ( X != nil_list_nat )
=> ( ! [X2: list_nat] :
( X
!= ( cons_list_nat @ X2 @ nil_list_nat ) )
=> ~ ! [X2: list_nat,Y: list_nat,Xs: list_list_nat] :
( X
!= ( cons_list_nat @ X2 @ ( cons_list_nat @ Y @ Xs ) ) ) ) ) ).
% remdups_adj.cases
thf(fact_66_remdups__adj_Ocases,axiom,
! [X: list_nat] :
( ( X != nil_nat )
=> ( ! [X2: nat] :
( X
!= ( cons_nat @ X2 @ nil_nat ) )
=> ~ ! [X2: nat,Y: nat,Xs: list_nat] :
( X
!= ( cons_nat @ X2 @ ( cons_nat @ Y @ Xs ) ) ) ) ) ).
% remdups_adj.cases
thf(fact_67_remdups__adj_Ocases,axiom,
! [X: list_P5707943133018811711nt_int] :
( ( X != nil_Pr2300489316682597567nt_int )
=> ( ! [X2: product_prod_int_int] :
( X
!= ( cons_P3334398858971670639nt_int @ X2 @ nil_Pr2300489316682597567nt_int ) )
=> ~ ! [X2: product_prod_int_int,Y: product_prod_int_int,Xs: list_P5707943133018811711nt_int] :
( X
!= ( cons_P3334398858971670639nt_int @ X2 @ ( cons_P3334398858971670639nt_int @ Y @ Xs ) ) ) ) ) ).
% remdups_adj.cases
thf(fact_68_remdups__adj_Ocases,axiom,
! [X: list_l1670014477004246597nt_int] :
( ( X != nil_li8670148097206105925nt_int )
=> ( ! [X2: list_P5707943133018811711nt_int] :
( X
!= ( cons_l7309679040211256053nt_int @ X2 @ nil_li8670148097206105925nt_int ) )
=> ~ ! [X2: list_P5707943133018811711nt_int,Y: list_P5707943133018811711nt_int,Xs: list_l1670014477004246597nt_int] :
( X
!= ( cons_l7309679040211256053nt_int @ X2 @ ( cons_l7309679040211256053nt_int @ Y @ Xs ) ) ) ) ) ).
% remdups_adj.cases
thf(fact_69_remdups__adj_Ocases,axiom,
! [X: list_list_list_nat] :
( ( X != nil_list_list_nat )
=> ( ! [X2: list_list_nat] :
( X
!= ( cons_list_list_nat @ X2 @ nil_list_list_nat ) )
=> ~ ! [X2: list_list_nat,Y: list_list_nat,Xs: list_list_list_nat] :
( X
!= ( cons_list_list_nat @ X2 @ ( cons_list_list_nat @ Y @ Xs ) ) ) ) ) ).
% remdups_adj.cases
thf(fact_70_min__list_Ocases,axiom,
! [X: list_nat] :
( ! [X2: nat,Xs: list_nat] :
( X
!= ( cons_nat @ X2 @ Xs ) )
=> ( X = nil_nat ) ) ).
% min_list.cases
thf(fact_71_list_Oexhaust,axiom,
! [Y3: list_list_nat] :
( ( Y3 != nil_list_nat )
=> ~ ! [X212: list_nat,X222: list_list_nat] :
( Y3
!= ( cons_list_nat @ X212 @ X222 ) ) ) ).
% list.exhaust
thf(fact_72_list_Oexhaust,axiom,
! [Y3: list_nat] :
( ( Y3 != nil_nat )
=> ~ ! [X212: nat,X222: list_nat] :
( Y3
!= ( cons_nat @ X212 @ X222 ) ) ) ).
% list.exhaust
thf(fact_73_list_Oexhaust,axiom,
! [Y3: list_P5707943133018811711nt_int] :
( ( Y3 != nil_Pr2300489316682597567nt_int )
=> ~ ! [X212: product_prod_int_int,X222: list_P5707943133018811711nt_int] :
( Y3
!= ( cons_P3334398858971670639nt_int @ X212 @ X222 ) ) ) ).
% list.exhaust
thf(fact_74_list_Oexhaust,axiom,
! [Y3: list_l1670014477004246597nt_int] :
( ( Y3 != nil_li8670148097206105925nt_int )
=> ~ ! [X212: list_P5707943133018811711nt_int,X222: list_l1670014477004246597nt_int] :
( Y3
!= ( cons_l7309679040211256053nt_int @ X212 @ X222 ) ) ) ).
% list.exhaust
thf(fact_75_list_Oexhaust,axiom,
! [Y3: list_list_list_nat] :
( ( Y3 != nil_list_list_nat )
=> ~ ! [X212: list_list_nat,X222: list_list_list_nat] :
( Y3
!= ( cons_list_list_nat @ X212 @ X222 ) ) ) ).
% list.exhaust
thf(fact_76_list_OdiscI,axiom,
! [List: list_list_nat,X21: list_nat,X22: list_list_nat] :
( ( List
= ( cons_list_nat @ X21 @ X22 ) )
=> ( List != nil_list_nat ) ) ).
% list.discI
thf(fact_77_list_OdiscI,axiom,
! [List: list_nat,X21: nat,X22: list_nat] :
( ( List
= ( cons_nat @ X21 @ X22 ) )
=> ( List != nil_nat ) ) ).
% list.discI
thf(fact_78_list_OdiscI,axiom,
! [List: list_P5707943133018811711nt_int,X21: product_prod_int_int,X22: list_P5707943133018811711nt_int] :
( ( List
= ( cons_P3334398858971670639nt_int @ X21 @ X22 ) )
=> ( List != nil_Pr2300489316682597567nt_int ) ) ).
% list.discI
thf(fact_79_list_OdiscI,axiom,
! [List: list_l1670014477004246597nt_int,X21: list_P5707943133018811711nt_int,X22: list_l1670014477004246597nt_int] :
( ( List
= ( cons_l7309679040211256053nt_int @ X21 @ X22 ) )
=> ( List != nil_li8670148097206105925nt_int ) ) ).
% list.discI
thf(fact_80_list_OdiscI,axiom,
! [List: list_list_list_nat,X21: list_list_nat,X22: list_list_list_nat] :
( ( List
= ( cons_list_list_nat @ X21 @ X22 ) )
=> ( List != nil_list_list_nat ) ) ).
% list.discI
thf(fact_81_list_Odistinct_I1_J,axiom,
! [X21: list_nat,X22: list_list_nat] :
( nil_list_nat
!= ( cons_list_nat @ X21 @ X22 ) ) ).
% list.distinct(1)
thf(fact_82_list_Odistinct_I1_J,axiom,
! [X21: nat,X22: list_nat] :
( nil_nat
!= ( cons_nat @ X21 @ X22 ) ) ).
% list.distinct(1)
thf(fact_83_list_Odistinct_I1_J,axiom,
! [X21: product_prod_int_int,X22: list_P5707943133018811711nt_int] :
( nil_Pr2300489316682597567nt_int
!= ( cons_P3334398858971670639nt_int @ X21 @ X22 ) ) ).
% list.distinct(1)
thf(fact_84_list_Odistinct_I1_J,axiom,
! [X21: list_P5707943133018811711nt_int,X22: list_l1670014477004246597nt_int] :
( nil_li8670148097206105925nt_int
!= ( cons_l7309679040211256053nt_int @ X21 @ X22 ) ) ).
% list.distinct(1)
thf(fact_85_list_Odistinct_I1_J,axiom,
! [X21: list_list_nat,X22: list_list_list_nat] :
( nil_list_list_nat
!= ( cons_list_list_nat @ X21 @ X22 ) ) ).
% list.distinct(1)
thf(fact_86_numeral__One,axiom,
( ( numeral_numeral_nat @ one )
= one_one_nat ) ).
% numeral_One
thf(fact_87_numeral__One,axiom,
( ( numeral_numeral_int @ one )
= one_one_int ) ).
% numeral_One
thf(fact_88_num_Oexhaust,axiom,
! [Y3: num] :
( ( Y3 != one )
=> ( ! [X23: num] :
( Y3
!= ( bit0 @ X23 ) )
=> ~ ! [X3: num] :
( Y3
!= ( bit1 @ X3 ) ) ) ) ).
% num.exhaust
thf(fact_89_mem__Collect__eq,axiom,
! [A: product_prod_int_int,P: product_prod_int_int > $o] :
( ( member5262025264175285858nt_int @ A @ ( collec213857154873943460nt_int @ P ) )
= ( P @ A ) ) ).
% mem_Collect_eq
thf(fact_90_Collect__mem__eq,axiom,
! [A2: set_Pr958786334691620121nt_int] :
( ( collec213857154873943460nt_int
@ ^ [X4: product_prod_int_int] : ( member5262025264175285858nt_int @ X4 @ A2 ) )
= A2 ) ).
% Collect_mem_eq
thf(fact_91_numerals_I1_J,axiom,
( ( numeral_numeral_nat @ one )
= one_one_nat ) ).
% numerals(1)
thf(fact_92_dbl__inc__simps_I3_J,axiom,
( ( neg_nu5851722552734809277nc_int @ one_one_int )
= ( numeral_numeral_int @ ( bit1 @ one ) ) ) ).
% dbl_inc_simps(3)
thf(fact_93_dbl__simps_I3_J,axiom,
( ( neg_numeral_dbl_int @ one_one_int )
= ( numeral_numeral_int @ ( bit0 @ one ) ) ) ).
% dbl_simps(3)
thf(fact_94_verit__eq__simplify_I9_J,axiom,
! [X32: num,Y32: num] :
( ( ( bit1 @ X32 )
= ( bit1 @ Y32 ) )
= ( X32 = Y32 ) ) ).
% verit_eq_simplify(9)
thf(fact_95_verit__eq__simplify_I8_J,axiom,
! [X24: num,Y23: num] :
( ( ( bit0 @ X24 )
= ( bit0 @ Y23 ) )
= ( X24 = Y23 ) ) ).
% verit_eq_simplify(8)
thf(fact_96_old_Oprod_Oinject,axiom,
! [A: nat,B: list_list_nat,A3: nat,B2: list_list_nat] :
( ( ( produc1673703249076330713st_nat @ A @ B )
= ( produc1673703249076330713st_nat @ A3 @ B2 ) )
= ( ( A = A3 )
& ( B = B2 ) ) ) ).
% old.prod.inject
thf(fact_97_old_Oprod_Oinject,axiom,
! [A: nat,B: list_nat,A3: nat,B2: list_nat] :
( ( ( produc8282810413953273033st_nat @ A @ B )
= ( produc8282810413953273033st_nat @ A3 @ B2 ) )
= ( ( A = A3 )
& ( B = B2 ) ) ) ).
% old.prod.inject
thf(fact_98_old_Oprod_Oinject,axiom,
! [A: nat,B: set_int,A3: nat,B2: set_int] :
( ( ( produc29655638201817675et_int @ A @ B )
= ( produc29655638201817675et_int @ A3 @ B2 ) )
= ( ( A = A3 )
& ( B = B2 ) ) ) ).
% old.prod.inject
thf(fact_99_old_Oprod_Oinject,axiom,
! [A: int,B: list_P5707943133018811711nt_int,A3: int,B2: list_P5707943133018811711nt_int] :
( ( ( produc8814303788642274490nt_int @ A @ B )
= ( produc8814303788642274490nt_int @ A3 @ B2 ) )
= ( ( A = A3 )
& ( B = B2 ) ) ) ).
% old.prod.inject
thf(fact_100_old_Oprod_Oinject,axiom,
! [A: int,B: int,A3: int,B2: int] :
( ( ( product_Pair_int_int @ A @ B )
= ( product_Pair_int_int @ A3 @ B2 ) )
= ( ( A = A3 )
& ( B = B2 ) ) ) ).
% old.prod.inject
thf(fact_101_prod_Oinject,axiom,
! [X1: nat,X24: list_list_nat,Y1: nat,Y23: list_list_nat] :
( ( ( produc1673703249076330713st_nat @ X1 @ X24 )
= ( produc1673703249076330713st_nat @ Y1 @ Y23 ) )
= ( ( X1 = Y1 )
& ( X24 = Y23 ) ) ) ).
% prod.inject
thf(fact_102_prod_Oinject,axiom,
! [X1: nat,X24: list_nat,Y1: nat,Y23: list_nat] :
( ( ( produc8282810413953273033st_nat @ X1 @ X24 )
= ( produc8282810413953273033st_nat @ Y1 @ Y23 ) )
= ( ( X1 = Y1 )
& ( X24 = Y23 ) ) ) ).
% prod.inject
thf(fact_103_prod_Oinject,axiom,
! [X1: nat,X24: set_int,Y1: nat,Y23: set_int] :
( ( ( produc29655638201817675et_int @ X1 @ X24 )
= ( produc29655638201817675et_int @ Y1 @ Y23 ) )
= ( ( X1 = Y1 )
& ( X24 = Y23 ) ) ) ).
% prod.inject
thf(fact_104_prod_Oinject,axiom,
! [X1: int,X24: list_P5707943133018811711nt_int,Y1: int,Y23: list_P5707943133018811711nt_int] :
( ( ( produc8814303788642274490nt_int @ X1 @ X24 )
= ( produc8814303788642274490nt_int @ Y1 @ Y23 ) )
= ( ( X1 = Y1 )
& ( X24 = Y23 ) ) ) ).
% prod.inject
thf(fact_105_prod_Oinject,axiom,
! [X1: int,X24: int,Y1: int,Y23: int] :
( ( ( product_Pair_int_int @ X1 @ X24 )
= ( product_Pair_int_int @ Y1 @ Y23 ) )
= ( ( X1 = Y1 )
& ( X24 = Y23 ) ) ) ).
% prod.inject
thf(fact_106_verit__eq__simplify_I12_J,axiom,
! [X32: num] :
( one
!= ( bit1 @ X32 ) ) ).
% verit_eq_simplify(12)
thf(fact_107_verit__eq__simplify_I14_J,axiom,
! [X24: num,X32: num] :
( ( bit0 @ X24 )
!= ( bit1 @ X32 ) ) ).
% verit_eq_simplify(14)
thf(fact_108_verit__eq__simplify_I10_J,axiom,
! [X24: num] :
( one
!= ( bit0 @ X24 ) ) ).
% verit_eq_simplify(10)
thf(fact_109_kp__5x5__ur,axiom,
knights_path @ ( board @ ( numeral_numeral_nat @ ( bit1 @ ( bit0 @ one ) ) ) @ ( numeral_numeral_nat @ ( bit1 @ ( bit0 @ one ) ) ) ) @ ( the_li7431803565598847443nt_int @ ( to_path @ ( cons_list_nat @ ( cons_nat @ ( numeral_numeral_nat @ ( bit1 @ ( bit1 @ one ) ) ) @ ( cons_nat @ ( numeral_numeral_nat @ ( bit0 @ ( bit0 @ ( bit1 @ one ) ) ) ) @ ( cons_nat @ ( numeral_numeral_nat @ ( bit1 @ ( bit1 @ ( bit1 @ one ) ) ) ) @ ( cons_nat @ ( numeral_numeral_nat @ ( bit0 @ ( bit0 @ ( bit1 @ ( bit0 @ one ) ) ) ) ) @ ( cons_nat @ ( numeral_numeral_nat @ ( bit1 @ ( bit0 @ one ) ) ) @ nil_nat ) ) ) ) ) @ ( cons_list_nat @ ( cons_nat @ ( numeral_numeral_nat @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ one ) ) ) ) ) @ ( cons_nat @ ( numeral_numeral_nat @ ( bit1 @ ( bit0 @ ( bit1 @ ( bit0 @ one ) ) ) ) ) @ ( cons_nat @ ( numeral_numeral_nat @ ( bit0 @ ( bit1 @ one ) ) ) @ ( cons_nat @ ( numeral_numeral_nat @ ( bit1 @ ( bit0 @ ( bit0 @ ( bit1 @ one ) ) ) ) ) @ ( cons_nat @ ( numeral_numeral_nat @ ( bit0 @ ( bit1 @ ( bit1 @ one ) ) ) ) @ nil_nat ) ) ) ) ) @ ( cons_list_nat @ ( cons_nat @ ( numeral_numeral_nat @ ( bit1 @ ( bit1 @ ( bit0 @ one ) ) ) ) @ ( cons_nat @ ( numeral_numeral_nat @ ( bit0 @ ( bit0 @ ( bit0 @ one ) ) ) ) @ ( cons_nat @ ( numeral_numeral_nat @ ( bit1 @ ( bit0 @ ( bit1 @ one ) ) ) ) @ ( cons_nat @ ( numeral_numeral_nat @ ( bit0 @ ( bit0 @ one ) ) ) @ ( cons_nat @ ( numeral_numeral_nat @ ( bit1 @ ( bit1 @ ( bit0 @ ( bit0 @ one ) ) ) ) ) @ nil_nat ) ) ) ) ) @ ( cons_list_nat @ ( cons_nat @ ( numeral_numeral_nat @ ( bit0 @ ( bit1 @ ( bit1 @ ( bit0 @ one ) ) ) ) ) @ ( cons_nat @ ( numeral_numeral_nat @ ( bit1 @ ( bit0 @ ( bit0 @ ( bit0 @ one ) ) ) ) ) @ ( cons_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( cons_nat @ ( numeral_numeral_nat @ ( bit1 @ ( bit0 @ ( bit0 @ one ) ) ) ) @ ( cons_nat @ ( numeral_numeral_nat @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit1 @ one ) ) ) ) ) @ nil_nat ) ) ) ) ) @ ( cons_list_nat @ ( cons_nat @ one_one_nat @ ( cons_nat @ ( numeral_numeral_nat @ ( bit0 @ ( bit1 @ ( bit0 @ one ) ) ) ) @ ( cons_nat @ ( numeral_numeral_nat @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit0 @ one ) ) ) ) ) @ ( cons_nat @ ( numeral_numeral_nat @ ( bit0 @ ( bit1 @ ( bit0 @ ( bit0 @ one ) ) ) ) ) @ ( cons_nat @ ( numeral_numeral_nat @ ( bit1 @ one ) ) @ nil_nat ) ) ) ) ) @ nil_list_nat ) ) ) ) ) ) ) ).
% kp_5x5_ur
thf(fact_110_dbl__simps_I5_J,axiom,
! [K: num] :
( ( neg_numeral_dbl_int @ ( numeral_numeral_int @ K ) )
= ( numeral_numeral_int @ ( bit0 @ K ) ) ) ).
% dbl_simps(5)
thf(fact_111_dbl__inc__simps_I5_J,axiom,
! [K: num] :
( ( neg_nu5851722552734809277nc_int @ ( numeral_numeral_int @ K ) )
= ( numeral_numeral_int @ ( bit1 @ K ) ) ) ).
% dbl_inc_simps(5)
thf(fact_112_test__path_Ocases,axiom,
! [X: list_P5707943133018811711nt_int] :
( ! [S_i: product_prod_int_int,S_j: product_prod_int_int,Xs: list_P5707943133018811711nt_int] :
( X
!= ( cons_P3334398858971670639nt_int @ S_i @ ( cons_P3334398858971670639nt_int @ S_j @ Xs ) ) )
=> ( ( X != nil_Pr2300489316682597567nt_int )
=> ~ ! [V: product_prod_int_int] :
( X
!= ( cons_P3334398858971670639nt_int @ V @ nil_Pr2300489316682597567nt_int ) ) ) ) ).
% test_path.cases
thf(fact_113_KnightsTour_Otranspose_Ocases,axiom,
! [X: list_P5707943133018811711nt_int] :
( ( X != nil_Pr2300489316682597567nt_int )
=> ~ ! [S_i: product_prod_int_int,Ps: list_P5707943133018811711nt_int] :
( X
!= ( cons_P3334398858971670639nt_int @ S_i @ Ps ) ) ) ).
% KnightsTour.transpose.cases
thf(fact_114_mirror2__aux_Ocases,axiom,
! [X: produc661532565036771336nt_int] :
( ! [M2: int] :
( X
!= ( produc8814303788642274490nt_int @ M2 @ nil_Pr2300489316682597567nt_int ) )
=> ~ ! [M2: int,S_i: product_prod_int_int,Ps: list_P5707943133018811711nt_int] :
( X
!= ( produc8814303788642274490nt_int @ M2 @ ( cons_P3334398858971670639nt_int @ S_i @ Ps ) ) ) ) ).
% mirror2_aux.cases
thf(fact_115_path__checker_Ocases,axiom,
! [X: produc2007852851243229709nt_int] :
( ! [B3: set_Pr958786334691620121nt_int] :
( X
!= ( produc2261658324281137661nt_int @ B3 @ nil_Pr2300489316682597567nt_int ) )
=> ( ! [B3: set_Pr958786334691620121nt_int,S_i: product_prod_int_int] :
( X
!= ( produc2261658324281137661nt_int @ B3 @ ( cons_P3334398858971670639nt_int @ S_i @ nil_Pr2300489316682597567nt_int ) ) )
=> ~ ! [B3: set_Pr958786334691620121nt_int,S_i: product_prod_int_int,S_j: product_prod_int_int,Ps: list_P5707943133018811711nt_int] :
( X
!= ( produc2261658324281137661nt_int @ B3 @ ( cons_P3334398858971670639nt_int @ S_i @ ( cons_P3334398858971670639nt_int @ S_j @ Ps ) ) ) ) ) ) ).
% path_checker.cases
thf(fact_116_knights__path__non__nil,axiom,
! [B: set_Pr958786334691620121nt_int,Ps2: list_P5707943133018811711nt_int] :
( ( knights_path @ B @ Ps2 )
=> ( Ps2 != nil_Pr2300489316682597567nt_int ) ) ).
% knights_path_non_nil
thf(fact_117_knights__path__board__unique,axiom,
! [B_1: set_Pr958786334691620121nt_int,Ps2: list_P5707943133018811711nt_int,B_2: set_Pr958786334691620121nt_int] :
( ( knights_path @ B_1 @ Ps2 )
=> ( ( knights_path @ B_2 @ Ps2 )
=> ( B_1 = B_2 ) ) ) ).
% knights_path_board_unique
thf(fact_118_trans__path_Ocases,axiom,
! [X: produc4058024888802116461nt_int] :
( ! [K_1: int,K_2: int] :
( X
!= ( produc8677244595851196253nt_int @ ( product_Pair_int_int @ K_1 @ K_2 ) @ nil_Pr2300489316682597567nt_int ) )
=> ~ ! [K_1: int,K_2: int,I: int,J: int,Xs: list_P5707943133018811711nt_int] :
( X
!= ( produc8677244595851196253nt_int @ ( product_Pair_int_int @ K_1 @ K_2 ) @ ( cons_P3334398858971670639nt_int @ ( product_Pair_int_int @ I @ J ) @ Xs ) ) ) ) ).
% trans_path.cases
thf(fact_119_step__checker_Ocases,axiom,
! [X: produc1219242969750017639nt_int] :
~ ! [I: int,J: int,I2: int,J2: int] :
( X
!= ( produc3646306378393792727nt_int @ ( product_Pair_int_int @ I @ J ) @ ( product_Pair_int_int @ I2 @ J2 ) ) ) ).
% step_checker.cases
thf(fact_120_sorted__wrt_Ocases,axiom,
! [X: produc2068713617857708901st_nat] :
( ! [P2: list_nat > list_nat > $o] :
( X
!= ( produc6344517630863869919st_nat @ P2 @ nil_list_nat ) )
=> ~ ! [P2: list_nat > list_nat > $o,X2: list_nat,Ys2: list_list_nat] :
( X
!= ( produc6344517630863869919st_nat @ P2 @ ( cons_list_nat @ X2 @ Ys2 ) ) ) ) ).
% sorted_wrt.cases
thf(fact_121_sorted__wrt_Ocases,axiom,
! [X: produc254973753779126261st_nat] :
( ! [P2: nat > nat > $o] :
( X
!= ( produc4727192421694094319st_nat @ P2 @ nil_nat ) )
=> ~ ! [P2: nat > nat > $o,X2: nat,Ys2: list_nat] :
( X
!= ( produc4727192421694094319st_nat @ P2 @ ( cons_nat @ X2 @ Ys2 ) ) ) ) ).
% sorted_wrt.cases
thf(fact_122_sorted__wrt_Ocases,axiom,
! [X: produc1050408459402128056nt_int] :
( ! [P2: product_prod_int_int > product_prod_int_int > $o] :
( X
!= ( produc3328129369365053992nt_int @ P2 @ nil_Pr2300489316682597567nt_int ) )
=> ~ ! [P2: product_prod_int_int > product_prod_int_int > $o,X2: product_prod_int_int,Ys2: list_P5707943133018811711nt_int] :
( X
!= ( produc3328129369365053992nt_int @ P2 @ ( cons_P3334398858971670639nt_int @ X2 @ Ys2 ) ) ) ) ).
% sorted_wrt.cases
thf(fact_123_sorted__wrt_Ocases,axiom,
! [X: produc5798082952504421822nt_int] :
( ! [P2: list_P5707943133018811711nt_int > list_P5707943133018811711nt_int > $o] :
( X
!= ( produc7365201557946222510nt_int @ P2 @ nil_li8670148097206105925nt_int ) )
=> ~ ! [P2: list_P5707943133018811711nt_int > list_P5707943133018811711nt_int > $o,X2: list_P5707943133018811711nt_int,Ys2: list_l1670014477004246597nt_int] :
( X
!= ( produc7365201557946222510nt_int @ P2 @ ( cons_l7309679040211256053nt_int @ X2 @ Ys2 ) ) ) ) ).
% sorted_wrt.cases
thf(fact_124_sorted__wrt_Ocases,axiom,
! [X: produc9085372331848283477st_nat] :
( ! [P2: list_list_nat > list_list_nat > $o] :
( X
!= ( produc842716015072215631st_nat @ P2 @ nil_list_list_nat ) )
=> ~ ! [P2: list_list_nat > list_list_nat > $o,X2: list_list_nat,Ys2: list_list_list_nat] :
( X
!= ( produc842716015072215631st_nat @ P2 @ ( cons_list_list_nat @ X2 @ Ys2 ) ) ) ) ).
% sorted_wrt.cases
thf(fact_125_successively_Ocases,axiom,
! [X: produc2068713617857708901st_nat] :
( ! [P2: list_nat > list_nat > $o] :
( X
!= ( produc6344517630863869919st_nat @ P2 @ nil_list_nat ) )
=> ( ! [P2: list_nat > list_nat > $o,X2: list_nat] :
( X
!= ( produc6344517630863869919st_nat @ P2 @ ( cons_list_nat @ X2 @ nil_list_nat ) ) )
=> ~ ! [P2: list_nat > list_nat > $o,X2: list_nat,Y: list_nat,Xs: list_list_nat] :
( X
!= ( produc6344517630863869919st_nat @ P2 @ ( cons_list_nat @ X2 @ ( cons_list_nat @ Y @ Xs ) ) ) ) ) ) ).
% successively.cases
thf(fact_126_successively_Ocases,axiom,
! [X: produc254973753779126261st_nat] :
( ! [P2: nat > nat > $o] :
( X
!= ( produc4727192421694094319st_nat @ P2 @ nil_nat ) )
=> ( ! [P2: nat > nat > $o,X2: nat] :
( X
!= ( produc4727192421694094319st_nat @ P2 @ ( cons_nat @ X2 @ nil_nat ) ) )
=> ~ ! [P2: nat > nat > $o,X2: nat,Y: nat,Xs: list_nat] :
( X
!= ( produc4727192421694094319st_nat @ P2 @ ( cons_nat @ X2 @ ( cons_nat @ Y @ Xs ) ) ) ) ) ) ).
% successively.cases
thf(fact_127_successively_Ocases,axiom,
! [X: produc1050408459402128056nt_int] :
( ! [P2: product_prod_int_int > product_prod_int_int > $o] :
( X
!= ( produc3328129369365053992nt_int @ P2 @ nil_Pr2300489316682597567nt_int ) )
=> ( ! [P2: product_prod_int_int > product_prod_int_int > $o,X2: product_prod_int_int] :
( X
!= ( produc3328129369365053992nt_int @ P2 @ ( cons_P3334398858971670639nt_int @ X2 @ nil_Pr2300489316682597567nt_int ) ) )
=> ~ ! [P2: product_prod_int_int > product_prod_int_int > $o,X2: product_prod_int_int,Y: product_prod_int_int,Xs: list_P5707943133018811711nt_int] :
( X
!= ( produc3328129369365053992nt_int @ P2 @ ( cons_P3334398858971670639nt_int @ X2 @ ( cons_P3334398858971670639nt_int @ Y @ Xs ) ) ) ) ) ) ).
% successively.cases
thf(fact_128_successively_Ocases,axiom,
! [X: produc5798082952504421822nt_int] :
( ! [P2: list_P5707943133018811711nt_int > list_P5707943133018811711nt_int > $o] :
( X
!= ( produc7365201557946222510nt_int @ P2 @ nil_li8670148097206105925nt_int ) )
=> ( ! [P2: list_P5707943133018811711nt_int > list_P5707943133018811711nt_int > $o,X2: list_P5707943133018811711nt_int] :
( X
!= ( produc7365201557946222510nt_int @ P2 @ ( cons_l7309679040211256053nt_int @ X2 @ nil_li8670148097206105925nt_int ) ) )
=> ~ ! [P2: list_P5707943133018811711nt_int > list_P5707943133018811711nt_int > $o,X2: list_P5707943133018811711nt_int,Y: list_P5707943133018811711nt_int,Xs: list_l1670014477004246597nt_int] :
( X
!= ( produc7365201557946222510nt_int @ P2 @ ( cons_l7309679040211256053nt_int @ X2 @ ( cons_l7309679040211256053nt_int @ Y @ Xs ) ) ) ) ) ) ).
% successively.cases
thf(fact_129_successively_Ocases,axiom,
! [X: produc9085372331848283477st_nat] :
( ! [P2: list_list_nat > list_list_nat > $o] :
( X
!= ( produc842716015072215631st_nat @ P2 @ nil_list_list_nat ) )
=> ( ! [P2: list_list_nat > list_list_nat > $o,X2: list_list_nat] :
( X
!= ( produc842716015072215631st_nat @ P2 @ ( cons_list_list_nat @ X2 @ nil_list_list_nat ) ) )
=> ~ ! [P2: list_list_nat > list_list_nat > $o,X2: list_list_nat,Y: list_list_nat,Xs: list_list_list_nat] :
( X
!= ( produc842716015072215631st_nat @ P2 @ ( cons_list_list_nat @ X2 @ ( cons_list_list_nat @ Y @ Xs ) ) ) ) ) ) ).
% successively.cases
thf(fact_130_xor__num_Ocases,axiom,
! [X: product_prod_num_num] :
( ( X
!= ( product_Pair_num_num @ one @ one ) )
=> ( ! [N2: num] :
( X
!= ( product_Pair_num_num @ one @ ( bit0 @ N2 ) ) )
=> ( ! [N2: num] :
( X
!= ( product_Pair_num_num @ one @ ( bit1 @ N2 ) ) )
=> ( ! [M2: num] :
( X
!= ( product_Pair_num_num @ ( bit0 @ M2 ) @ one ) )
=> ( ! [M2: num,N2: num] :
( X
!= ( product_Pair_num_num @ ( bit0 @ M2 ) @ ( bit0 @ N2 ) ) )
=> ( ! [M2: num,N2: num] :
( X
!= ( product_Pair_num_num @ ( bit0 @ M2 ) @ ( bit1 @ N2 ) ) )
=> ( ! [M2: num] :
( X
!= ( product_Pair_num_num @ ( bit1 @ M2 ) @ one ) )
=> ( ! [M2: num,N2: num] :
( X
!= ( product_Pair_num_num @ ( bit1 @ M2 ) @ ( bit0 @ N2 ) ) )
=> ~ ! [M2: num,N2: num] :
( X
!= ( product_Pair_num_num @ ( bit1 @ M2 ) @ ( bit1 @ N2 ) ) ) ) ) ) ) ) ) ) ) ).
% xor_num.cases
thf(fact_131_splice_Ocases,axiom,
! [X: produc4326814125627636033st_nat] :
( ! [Ys2: list_list_nat] :
( X
!= ( produc7129799990162260089st_nat @ nil_list_nat @ Ys2 ) )
=> ~ ! [X2: list_nat,Xs: list_list_nat,Ys2: list_list_nat] :
( X
!= ( produc7129799990162260089st_nat @ ( cons_list_nat @ X2 @ Xs ) @ Ys2 ) ) ) ).
% splice.cases
thf(fact_132_splice_Ocases,axiom,
! [X: produc1828647624359046049st_nat] :
( ! [Ys2: list_nat] :
( X
!= ( produc2694037385005941721st_nat @ nil_nat @ Ys2 ) )
=> ~ ! [X2: nat,Xs: list_nat,Ys2: list_nat] :
( X
!= ( produc2694037385005941721st_nat @ ( cons_nat @ X2 @ Xs ) @ Ys2 ) ) ) ).
% splice.cases
thf(fact_133_splice_Ocases,axiom,
! [X: produc1089560213143673063nt_int] :
( ! [Ys2: list_P5707943133018811711nt_int] :
( X
!= ( produc1932183703851549015nt_int @ nil_Pr2300489316682597567nt_int @ Ys2 ) )
=> ~ ! [X2: product_prod_int_int,Xs: list_P5707943133018811711nt_int,Ys2: list_P5707943133018811711nt_int] :
( X
!= ( produc1932183703851549015nt_int @ ( cons_P3334398858971670639nt_int @ X2 @ Xs ) @ Ys2 ) ) ) ).
% splice.cases
thf(fact_134_splice_Ocases,axiom,
! [X: produc9134732803953422695nt_int] :
( ! [Ys2: list_l1670014477004246597nt_int] :
( X
!= ( produc4603928489606670295nt_int @ nil_li8670148097206105925nt_int @ Ys2 ) )
=> ~ ! [X2: list_P5707943133018811711nt_int,Xs: list_l1670014477004246597nt_int,Ys2: list_l1670014477004246597nt_int] :
( X
!= ( produc4603928489606670295nt_int @ ( cons_l7309679040211256053nt_int @ X2 @ Xs ) @ Ys2 ) ) ) ).
% splice.cases
thf(fact_135_splice_Ocases,axiom,
! [X: produc9189680189847484897st_nat] :
( ! [Ys2: list_list_list_nat] :
( X
!= ( produc7456843779855495193st_nat @ nil_list_list_nat @ Ys2 ) )
=> ~ ! [X2: list_list_nat,Xs: list_list_list_nat,Ys2: list_list_list_nat] :
( X
!= ( produc7456843779855495193st_nat @ ( cons_list_list_nat @ X2 @ Xs ) @ Ys2 ) ) ) ).
% splice.cases
thf(fact_136_shuffles_Ocases,axiom,
! [X: produc4326814125627636033st_nat] :
( ! [Ys2: list_list_nat] :
( X
!= ( produc7129799990162260089st_nat @ nil_list_nat @ Ys2 ) )
=> ( ! [Xs: list_list_nat] :
( X
!= ( produc7129799990162260089st_nat @ Xs @ nil_list_nat ) )
=> ~ ! [X2: list_nat,Xs: list_list_nat,Y: list_nat,Ys2: list_list_nat] :
( X
!= ( produc7129799990162260089st_nat @ ( cons_list_nat @ X2 @ Xs ) @ ( cons_list_nat @ Y @ Ys2 ) ) ) ) ) ).
% shuffles.cases
thf(fact_137_shuffles_Ocases,axiom,
! [X: produc1828647624359046049st_nat] :
( ! [Ys2: list_nat] :
( X
!= ( produc2694037385005941721st_nat @ nil_nat @ Ys2 ) )
=> ( ! [Xs: list_nat] :
( X
!= ( produc2694037385005941721st_nat @ Xs @ nil_nat ) )
=> ~ ! [X2: nat,Xs: list_nat,Y: nat,Ys2: list_nat] :
( X
!= ( produc2694037385005941721st_nat @ ( cons_nat @ X2 @ Xs ) @ ( cons_nat @ Y @ Ys2 ) ) ) ) ) ).
% shuffles.cases
thf(fact_138_shuffles_Ocases,axiom,
! [X: produc1089560213143673063nt_int] :
( ! [Ys2: list_P5707943133018811711nt_int] :
( X
!= ( produc1932183703851549015nt_int @ nil_Pr2300489316682597567nt_int @ Ys2 ) )
=> ( ! [Xs: list_P5707943133018811711nt_int] :
( X
!= ( produc1932183703851549015nt_int @ Xs @ nil_Pr2300489316682597567nt_int ) )
=> ~ ! [X2: product_prod_int_int,Xs: list_P5707943133018811711nt_int,Y: product_prod_int_int,Ys2: list_P5707943133018811711nt_int] :
( X
!= ( produc1932183703851549015nt_int @ ( cons_P3334398858971670639nt_int @ X2 @ Xs ) @ ( cons_P3334398858971670639nt_int @ Y @ Ys2 ) ) ) ) ) ).
% shuffles.cases
thf(fact_139_shuffles_Ocases,axiom,
! [X: produc9134732803953422695nt_int] :
( ! [Ys2: list_l1670014477004246597nt_int] :
( X
!= ( produc4603928489606670295nt_int @ nil_li8670148097206105925nt_int @ Ys2 ) )
=> ( ! [Xs: list_l1670014477004246597nt_int] :
( X
!= ( produc4603928489606670295nt_int @ Xs @ nil_li8670148097206105925nt_int ) )
=> ~ ! [X2: list_P5707943133018811711nt_int,Xs: list_l1670014477004246597nt_int,Y: list_P5707943133018811711nt_int,Ys2: list_l1670014477004246597nt_int] :
( X
!= ( produc4603928489606670295nt_int @ ( cons_l7309679040211256053nt_int @ X2 @ Xs ) @ ( cons_l7309679040211256053nt_int @ Y @ Ys2 ) ) ) ) ) ).
% shuffles.cases
thf(fact_140_shuffles_Ocases,axiom,
! [X: produc9189680189847484897st_nat] :
( ! [Ys2: list_list_list_nat] :
( X
!= ( produc7456843779855495193st_nat @ nil_list_list_nat @ Ys2 ) )
=> ( ! [Xs: list_list_list_nat] :
( X
!= ( produc7456843779855495193st_nat @ Xs @ nil_list_list_nat ) )
=> ~ ! [X2: list_list_nat,Xs: list_list_list_nat,Y: list_list_nat,Ys2: list_list_list_nat] :
( X
!= ( produc7456843779855495193st_nat @ ( cons_list_list_nat @ X2 @ Xs ) @ ( cons_list_list_nat @ Y @ Ys2 ) ) ) ) ) ).
% shuffles.cases
thf(fact_141_find__k__in__col_Ocases,axiom,
! [X: produc4575160907756185873st_nat] :
( ! [K2: nat] :
( X
!= ( produc8282810413953273033st_nat @ K2 @ nil_nat ) )
=> ~ ! [K2: nat,C: nat,Cs: list_nat] :
( X
!= ( produc8282810413953273033st_nat @ K2 @ ( cons_nat @ C @ Cs ) ) ) ) ).
% find_k_in_col.cases
thf(fact_142_find__k__sqr_Ocases,axiom,
! [X: produc2814713032259027617st_nat] :
( ! [K2: nat] :
( X
!= ( produc1673703249076330713st_nat @ K2 @ nil_list_nat ) )
=> ~ ! [K2: nat,R: list_nat,Rs: list_list_nat] :
( X
!= ( produc1673703249076330713st_nat @ K2 @ ( cons_list_nat @ R @ Rs ) ) ) ) ).
% find_k_sqr.cases
thf(fact_143_old_Oprod_Oexhaust,axiom,
! [Y3: produc2814713032259027617st_nat] :
~ ! [A4: nat,B3: list_list_nat] :
( Y3
!= ( produc1673703249076330713st_nat @ A4 @ B3 ) ) ).
% old.prod.exhaust
thf(fact_144_old_Oprod_Oexhaust,axiom,
! [Y3: produc4575160907756185873st_nat] :
~ ! [A4: nat,B3: list_nat] :
( Y3
!= ( produc8282810413953273033st_nat @ A4 @ B3 ) ) ).
% old.prod.exhaust
thf(fact_145_old_Oprod_Oexhaust,axiom,
! [Y3: produc9133624956312949779et_int] :
~ ! [A4: nat,B3: set_int] :
( Y3
!= ( produc29655638201817675et_int @ A4 @ B3 ) ) ).
% old.prod.exhaust
thf(fact_146_old_Oprod_Oexhaust,axiom,
! [Y3: produc661532565036771336nt_int] :
~ ! [A4: int,B3: list_P5707943133018811711nt_int] :
( Y3
!= ( produc8814303788642274490nt_int @ A4 @ B3 ) ) ).
% old.prod.exhaust
thf(fact_147_old_Oprod_Oexhaust,axiom,
! [Y3: product_prod_int_int] :
~ ! [A4: int,B3: int] :
( Y3
!= ( product_Pair_int_int @ A4 @ B3 ) ) ).
% old.prod.exhaust
thf(fact_148_surj__pair,axiom,
! [P3: produc2814713032259027617st_nat] :
? [X2: nat,Y: list_list_nat] :
( P3
= ( produc1673703249076330713st_nat @ X2 @ Y ) ) ).
% surj_pair
thf(fact_149_surj__pair,axiom,
! [P3: produc4575160907756185873st_nat] :
? [X2: nat,Y: list_nat] :
( P3
= ( produc8282810413953273033st_nat @ X2 @ Y ) ) ).
% surj_pair
thf(fact_150_surj__pair,axiom,
! [P3: produc9133624956312949779et_int] :
? [X2: nat,Y: set_int] :
( P3
= ( produc29655638201817675et_int @ X2 @ Y ) ) ).
% surj_pair
thf(fact_151_surj__pair,axiom,
! [P3: produc661532565036771336nt_int] :
? [X2: int,Y: list_P5707943133018811711nt_int] :
( P3
= ( produc8814303788642274490nt_int @ X2 @ Y ) ) ).
% surj_pair
thf(fact_152_surj__pair,axiom,
! [P3: product_prod_int_int] :
? [X2: int,Y: int] :
( P3
= ( product_Pair_int_int @ X2 @ Y ) ) ).
% surj_pair
thf(fact_153_prod__cases,axiom,
! [P: produc2814713032259027617st_nat > $o,P3: produc2814713032259027617st_nat] :
( ! [A4: nat,B3: list_list_nat] : ( P @ ( produc1673703249076330713st_nat @ A4 @ B3 ) )
=> ( P @ P3 ) ) ).
% prod_cases
thf(fact_154_prod__cases,axiom,
! [P: produc4575160907756185873st_nat > $o,P3: produc4575160907756185873st_nat] :
( ! [A4: nat,B3: list_nat] : ( P @ ( produc8282810413953273033st_nat @ A4 @ B3 ) )
=> ( P @ P3 ) ) ).
% prod_cases
thf(fact_155_prod__cases,axiom,
! [P: produc9133624956312949779et_int > $o,P3: produc9133624956312949779et_int] :
( ! [A4: nat,B3: set_int] : ( P @ ( produc29655638201817675et_int @ A4 @ B3 ) )
=> ( P @ P3 ) ) ).
% prod_cases
thf(fact_156_prod__cases,axiom,
! [P: produc661532565036771336nt_int > $o,P3: produc661532565036771336nt_int] :
( ! [A4: int,B3: list_P5707943133018811711nt_int] : ( P @ ( produc8814303788642274490nt_int @ A4 @ B3 ) )
=> ( P @ P3 ) ) ).
% prod_cases
thf(fact_157_prod__cases,axiom,
! [P: product_prod_int_int > $o,P3: product_prod_int_int] :
( ! [A4: int,B3: int] : ( P @ ( product_Pair_int_int @ A4 @ B3 ) )
=> ( P @ P3 ) ) ).
% prod_cases
thf(fact_158_Pair__inject,axiom,
! [A: nat,B: list_list_nat,A3: nat,B2: list_list_nat] :
( ( ( produc1673703249076330713st_nat @ A @ B )
= ( produc1673703249076330713st_nat @ A3 @ B2 ) )
=> ~ ( ( A = A3 )
=> ( B != B2 ) ) ) ).
% Pair_inject
thf(fact_159_Pair__inject,axiom,
! [A: nat,B: list_nat,A3: nat,B2: list_nat] :
( ( ( produc8282810413953273033st_nat @ A @ B )
= ( produc8282810413953273033st_nat @ A3 @ B2 ) )
=> ~ ( ( A = A3 )
=> ( B != B2 ) ) ) ).
% Pair_inject
thf(fact_160_Pair__inject,axiom,
! [A: nat,B: set_int,A3: nat,B2: set_int] :
( ( ( produc29655638201817675et_int @ A @ B )
= ( produc29655638201817675et_int @ A3 @ B2 ) )
=> ~ ( ( A = A3 )
=> ( B != B2 ) ) ) ).
% Pair_inject
thf(fact_161_Pair__inject,axiom,
! [A: int,B: list_P5707943133018811711nt_int,A3: int,B2: list_P5707943133018811711nt_int] :
( ( ( produc8814303788642274490nt_int @ A @ B )
= ( produc8814303788642274490nt_int @ A3 @ B2 ) )
=> ~ ( ( A = A3 )
=> ( B != B2 ) ) ) ).
% Pair_inject
thf(fact_162_Pair__inject,axiom,
! [A: int,B: int,A3: int,B2: int] :
( ( ( product_Pair_int_int @ A @ B )
= ( product_Pair_int_int @ A3 @ B2 ) )
=> ~ ( ( A = A3 )
=> ( B != B2 ) ) ) ).
% Pair_inject
thf(fact_163_one__reorient,axiom,
! [X: nat] :
( ( one_one_nat = X )
= ( X = one_one_nat ) ) ).
% one_reorient
thf(fact_164_one__reorient,axiom,
! [X: int] :
( ( one_one_int = X )
= ( X = one_one_int ) ) ).
% one_reorient
thf(fact_165_kp__5x5__lr,axiom,
knights_path @ ( board @ ( numeral_numeral_nat @ ( bit1 @ ( bit0 @ one ) ) ) @ ( numeral_numeral_nat @ ( bit1 @ ( bit0 @ one ) ) ) ) @ ( the_li7431803565598847443nt_int @ ( to_path @ ( cons_list_nat @ ( cons_nat @ ( numeral_numeral_nat @ ( bit1 @ one ) ) @ ( cons_nat @ ( numeral_numeral_nat @ ( bit0 @ ( bit1 @ ( bit1 @ ( bit0 @ one ) ) ) ) ) @ ( cons_nat @ ( numeral_numeral_nat @ ( bit1 @ ( bit0 @ ( bit1 @ one ) ) ) ) @ ( cons_nat @ ( numeral_numeral_nat @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ one ) ) ) ) ) @ ( cons_nat @ ( numeral_numeral_nat @ ( bit1 @ ( bit0 @ one ) ) ) @ nil_nat ) ) ) ) ) @ ( cons_list_nat @ ( cons_nat @ ( numeral_numeral_nat @ ( bit0 @ ( bit0 @ ( bit1 @ one ) ) ) ) @ ( cons_nat @ ( numeral_numeral_nat @ ( bit1 @ ( bit0 @ ( bit0 @ ( bit0 @ one ) ) ) ) ) @ ( cons_nat @ ( numeral_numeral_nat @ ( bit0 @ ( bit0 @ one ) ) ) @ ( cons_nat @ ( numeral_numeral_nat @ ( bit1 @ ( bit0 @ ( bit1 @ ( bit0 @ one ) ) ) ) ) @ ( cons_nat @ ( numeral_numeral_nat @ ( bit0 @ ( bit1 @ ( bit1 @ one ) ) ) ) @ nil_nat ) ) ) ) ) @ ( cons_list_nat @ ( cons_nat @ ( numeral_numeral_nat @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit0 @ one ) ) ) ) ) @ ( cons_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( cons_nat @ ( numeral_numeral_nat @ ( bit1 @ ( bit1 @ ( bit1 @ one ) ) ) ) @ ( cons_nat @ ( numeral_numeral_nat @ ( bit0 @ ( bit1 @ one ) ) ) @ ( cons_nat @ ( numeral_numeral_nat @ ( bit1 @ ( bit0 @ ( bit0 @ one ) ) ) ) @ nil_nat ) ) ) ) ) @ ( cons_list_nat @ ( cons_nat @ ( numeral_numeral_nat @ ( bit0 @ ( bit1 @ ( bit0 @ ( bit0 @ one ) ) ) ) ) @ ( cons_nat @ ( numeral_numeral_nat @ ( bit1 @ ( bit1 @ ( bit0 @ one ) ) ) ) @ ( cons_nat @ ( numeral_numeral_nat @ ( bit0 @ ( bit0 @ ( bit0 @ one ) ) ) ) @ ( cons_nat @ ( numeral_numeral_nat @ ( bit1 @ ( bit0 @ ( bit0 @ ( bit1 @ one ) ) ) ) ) @ ( cons_nat @ ( numeral_numeral_nat @ ( bit0 @ ( bit0 @ ( bit1 @ ( bit0 @ one ) ) ) ) ) @ nil_nat ) ) ) ) ) @ ( cons_list_nat @ ( cons_nat @ one_one_nat @ ( cons_nat @ ( numeral_numeral_nat @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit1 @ one ) ) ) ) ) @ ( cons_nat @ ( numeral_numeral_nat @ ( bit1 @ ( bit1 @ ( bit0 @ ( bit0 @ one ) ) ) ) ) @ ( cons_nat @ ( numeral_numeral_nat @ ( bit0 @ ( bit1 @ ( bit0 @ one ) ) ) ) @ ( cons_nat @ ( numeral_numeral_nat @ ( bit1 @ ( bit1 @ one ) ) ) @ nil_nat ) ) ) ) ) @ nil_list_nat ) ) ) ) ) ) ) ).
% kp_5x5_lr
thf(fact_166_dbl__simps_I4_J,axiom,
( ( neg_numeral_dbl_int @ ( uminus_uminus_int @ one_one_int ) )
= ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ).
% dbl_simps(4)
thf(fact_167_product__lists_Osimps_I1_J,axiom,
( ( produc3398292703289546286nt_int @ nil_li8254795547437800779nt_int )
= ( cons_l4848925899132326139nt_int @ nil_li8670148097206105925nt_int @ nil_li8254795547437800779nt_int ) ) ).
% product_lists.simps(1)
thf(fact_168_product__lists_Osimps_I1_J,axiom,
( ( produc8156405240489450689st_nat @ nil_li5455921165481563386st_nat )
= ( cons_l7310388135179752778st_nat @ nil_list_list_nat @ nil_li5455921165481563386st_nat ) ) ).
% product_lists.simps(1)
thf(fact_169_product__lists_Osimps_I1_J,axiom,
( ( product_lists_nat @ nil_list_nat )
= ( cons_list_nat @ nil_nat @ nil_list_nat ) ) ).
% product_lists.simps(1)
thf(fact_170_product__lists_Osimps_I1_J,axiom,
( ( produc5568053154996169768nt_int @ nil_li8670148097206105925nt_int )
= ( cons_l7309679040211256053nt_int @ nil_Pr2300489316682597567nt_int @ nil_li8670148097206105925nt_int ) ) ).
% product_lists.simps(1)
thf(fact_171_product__lists_Osimps_I1_J,axiom,
( ( produc6783906451316923569st_nat @ nil_list_list_nat )
= ( cons_list_list_nat @ nil_list_nat @ nil_list_list_nat ) ) ).
% product_lists.simps(1)
thf(fact_172_trans__path__non__nil__last,axiom,
! [Ps2: list_P5707943133018811711nt_int,K_12: int,K_22: int,I3: int,J3: int] :
( ( Ps2 != nil_Pr2300489316682597567nt_int )
=> ( ( last_P3305686521732843992nt_int @ ( trans_path @ ( product_Pair_int_int @ K_12 @ K_22 ) @ Ps2 ) )
= ( last_P3305686521732843992nt_int @ ( trans_path @ ( product_Pair_int_int @ K_12 @ K_22 ) @ ( cons_P3334398858971670639nt_int @ ( product_Pair_int_int @ I3 @ J3 ) @ Ps2 ) ) ) ) ) ).
% trans_path_non_nil_last
thf(fact_173_sub__num__simps_I5_J,axiom,
! [K: num] :
( ( neg_numeral_sub_int @ ( bit1 @ K ) @ one )
= ( numeral_numeral_int @ ( bit0 @ K ) ) ) ).
% sub_num_simps(5)
thf(fact_174_subseqs_Osimps_I1_J,axiom,
( ( subseq2030970315493690260nt_int @ nil_li8670148097206105925nt_int )
= ( cons_l4848925899132326139nt_int @ nil_li8670148097206105925nt_int @ nil_li8254795547437800779nt_int ) ) ).
% subseqs.simps(1)
thf(fact_175_subseqs_Osimps_I1_J,axiom,
( ( subseq6786254687447370203st_nat @ nil_list_list_nat )
= ( cons_l7310388135179752778st_nat @ nil_list_list_nat @ nil_li5455921165481563386st_nat ) ) ).
% subseqs.simps(1)
thf(fact_176_subseqs_Osimps_I1_J,axiom,
( ( subseqs_nat @ nil_nat )
= ( cons_list_nat @ nil_nat @ nil_list_nat ) ) ).
% subseqs.simps(1)
thf(fact_177_subseqs_Osimps_I1_J,axiom,
( ( subseq1357044202310323342nt_int @ nil_Pr2300489316682597567nt_int )
= ( cons_l7309679040211256053nt_int @ nil_Pr2300489316682597567nt_int @ nil_li8670148097206105925nt_int ) ) ).
% subseqs.simps(1)
thf(fact_178_subseqs_Osimps_I1_J,axiom,
( ( subseqs_list_nat @ nil_list_nat )
= ( cons_list_list_nat @ nil_list_nat @ nil_list_list_nat ) ) ).
% subseqs.simps(1)
thf(fact_179_mirror2__knights__path,axiom,
! [N: nat,M: nat,Ps2: list_P5707943133018811711nt_int] :
( ( knights_path @ ( board @ N @ M ) @ Ps2 )
=> ( knights_path @ ( board @ N @ M ) @ ( mirror2 @ Ps2 ) ) ) ).
% mirror2_knights_path
thf(fact_180_Suc__1,axiom,
( ( suc @ one_one_nat )
= ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ).
% Suc_1
thf(fact_181_sub__num__simps_I8_J,axiom,
! [K: num,L: num] :
( ( neg_numeral_sub_int @ ( bit1 @ K ) @ ( bit0 @ L ) )
= ( neg_nu5851722552734809277nc_int @ ( neg_numeral_sub_int @ K @ L ) ) ) ).
% sub_num_simps(8)
thf(fact_182_insert__Nil,axiom,
! [X: list_nat] :
( ( insert_list_nat @ X @ nil_list_nat )
= ( cons_list_nat @ X @ nil_list_nat ) ) ).
% insert_Nil
thf(fact_183_insert__Nil,axiom,
! [X: nat] :
( ( insert_nat @ X @ nil_nat )
= ( cons_nat @ X @ nil_nat ) ) ).
% insert_Nil
thf(fact_184_insert__Nil,axiom,
! [X: product_prod_int_int] :
( ( insert5765537519290168021nt_int @ X @ nil_Pr2300489316682597567nt_int )
= ( cons_P3334398858971670639nt_int @ X @ nil_Pr2300489316682597567nt_int ) ) ).
% insert_Nil
thf(fact_185_insert__Nil,axiom,
! [X: list_P5707943133018811711nt_int] :
( ( insert2185137395918070363nt_int @ X @ nil_li8670148097206105925nt_int )
= ( cons_l7309679040211256053nt_int @ X @ nil_li8670148097206105925nt_int ) ) ).
% insert_Nil
thf(fact_186_insert__Nil,axiom,
! [X: list_list_nat] :
( ( insert_list_list_nat @ X @ nil_list_list_nat )
= ( cons_list_list_nat @ X @ nil_list_list_nat ) ) ).
% insert_Nil
thf(fact_187_xor__numerals_I8_J,axiom,
! [X: num] :
( ( bit_se6528837805403552850or_nat @ ( numeral_numeral_nat @ ( bit1 @ X ) ) @ one_one_nat )
= ( numeral_numeral_nat @ ( bit0 @ X ) ) ) ).
% xor_numerals(8)
thf(fact_188_xor__numerals_I8_J,axiom,
! [X: num] :
( ( bit_se6526347334894502574or_int @ ( numeral_numeral_int @ ( bit1 @ X ) ) @ one_one_int )
= ( numeral_numeral_int @ ( bit0 @ X ) ) ) ).
% xor_numerals(8)
thf(fact_189_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_190_add_Oinverse__inverse,axiom,
! [A: int] :
( ( uminus_uminus_int @ ( uminus_uminus_int @ A ) )
= A ) ).
% add.inverse_inverse
thf(fact_191_verit__minus__simplify_I4_J,axiom,
! [B: int] :
( ( uminus_uminus_int @ ( uminus_uminus_int @ B ) )
= B ) ).
% verit_minus_simplify(4)
thf(fact_192_bit_Oxor__left__self,axiom,
! [X: int,Y3: int] :
( ( bit_se6526347334894502574or_int @ X @ ( bit_se6526347334894502574or_int @ X @ Y3 ) )
= Y3 ) ).
% bit.xor_left_self
thf(fact_193_neg__numeral__eq__iff,axiom,
! [M: num,N: num] :
( ( ( uminus_uminus_int @ ( numeral_numeral_int @ M ) )
= ( uminus_uminus_int @ ( numeral_numeral_int @ N ) ) )
= ( M = N ) ) ).
% neg_numeral_eq_iff
thf(fact_194_dbl__simps_I1_J,axiom,
! [K: num] :
( ( neg_numeral_dbl_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ K ) ) )
= ( uminus_uminus_int @ ( neg_numeral_dbl_int @ ( numeral_numeral_int @ K ) ) ) ) ).
% dbl_simps(1)
thf(fact_195_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_196_sub__num__simps_I6_J,axiom,
! [K: num,L: num] :
( ( neg_numeral_sub_int @ ( bit0 @ K ) @ ( bit0 @ L ) )
= ( neg_numeral_dbl_int @ ( neg_numeral_sub_int @ K @ L ) ) ) ).
% sub_num_simps(6)
thf(fact_197_sub__num__simps_I9_J,axiom,
! [K: num,L: num] :
( ( neg_numeral_sub_int @ ( bit1 @ K ) @ ( bit1 @ L ) )
= ( neg_numeral_dbl_int @ ( neg_numeral_sub_int @ K @ L ) ) ) ).
% sub_num_simps(9)
thf(fact_198_numeral__eq__neg__one__iff,axiom,
! [N: num] :
( ( ( uminus_uminus_int @ ( numeral_numeral_int @ N ) )
= ( uminus_uminus_int @ one_one_int ) )
= ( N = one ) ) ).
% numeral_eq_neg_one_iff
thf(fact_199_neg__one__eq__numeral__iff,axiom,
! [N: num] :
( ( ( uminus_uminus_int @ one_one_int )
= ( uminus_uminus_int @ ( numeral_numeral_int @ N ) ) )
= ( N = one ) ) ).
% neg_one_eq_numeral_iff
thf(fact_200_xor__numerals_I1_J,axiom,
! [Y3: num] :
( ( bit_se6528837805403552850or_nat @ one_one_nat @ ( numeral_numeral_nat @ ( bit0 @ Y3 ) ) )
= ( numeral_numeral_nat @ ( bit1 @ Y3 ) ) ) ).
% xor_numerals(1)
thf(fact_201_xor__numerals_I1_J,axiom,
! [Y3: num] :
( ( bit_se6526347334894502574or_int @ one_one_int @ ( numeral_numeral_int @ ( bit0 @ Y3 ) ) )
= ( numeral_numeral_int @ ( bit1 @ Y3 ) ) ) ).
% xor_numerals(1)
thf(fact_202_xor__numerals_I2_J,axiom,
! [Y3: num] :
( ( bit_se6528837805403552850or_nat @ one_one_nat @ ( numeral_numeral_nat @ ( bit1 @ Y3 ) ) )
= ( numeral_numeral_nat @ ( bit0 @ Y3 ) ) ) ).
% xor_numerals(2)
thf(fact_203_xor__numerals_I2_J,axiom,
! [Y3: num] :
( ( bit_se6526347334894502574or_int @ one_one_int @ ( numeral_numeral_int @ ( bit1 @ Y3 ) ) )
= ( numeral_numeral_int @ ( bit0 @ Y3 ) ) ) ).
% xor_numerals(2)
thf(fact_204_xor__numerals_I5_J,axiom,
! [X: num] :
( ( bit_se6528837805403552850or_nat @ ( numeral_numeral_nat @ ( bit0 @ X ) ) @ one_one_nat )
= ( numeral_numeral_nat @ ( bit1 @ X ) ) ) ).
% xor_numerals(5)
thf(fact_205_xor__numerals_I5_J,axiom,
! [X: num] :
( ( bit_se6526347334894502574or_int @ ( numeral_numeral_int @ ( bit0 @ X ) ) @ one_one_int )
= ( numeral_numeral_int @ ( bit1 @ X ) ) ) ).
% xor_numerals(5)
thf(fact_206_sub__num__simps_I3_J,axiom,
! [L: num] :
( ( neg_numeral_sub_int @ one @ ( bit1 @ L ) )
= ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit0 @ L ) ) ) ) ).
% sub_num_simps(3)
thf(fact_207_xor_Oleft__commute,axiom,
! [B: nat,A: nat,C2: nat] :
( ( bit_se6528837805403552850or_nat @ B @ ( bit_se6528837805403552850or_nat @ A @ C2 ) )
= ( bit_se6528837805403552850or_nat @ A @ ( bit_se6528837805403552850or_nat @ B @ C2 ) ) ) ).
% xor.left_commute
thf(fact_208_xor_Oleft__commute,axiom,
! [B: int,A: int,C2: int] :
( ( bit_se6526347334894502574or_int @ B @ ( bit_se6526347334894502574or_int @ A @ C2 ) )
= ( bit_se6526347334894502574or_int @ A @ ( bit_se6526347334894502574or_int @ B @ C2 ) ) ) ).
% xor.left_commute
thf(fact_209_xor_Ocommute,axiom,
( bit_se6528837805403552850or_nat
= ( ^ [A5: nat,B4: nat] : ( bit_se6528837805403552850or_nat @ B4 @ A5 ) ) ) ).
% xor.commute
thf(fact_210_xor_Ocommute,axiom,
( bit_se6526347334894502574or_int
= ( ^ [A5: int,B4: int] : ( bit_se6526347334894502574or_int @ B4 @ A5 ) ) ) ).
% xor.commute
thf(fact_211_xor_Oassoc,axiom,
! [A: nat,B: nat,C2: nat] :
( ( bit_se6528837805403552850or_nat @ ( bit_se6528837805403552850or_nat @ A @ B ) @ C2 )
= ( bit_se6528837805403552850or_nat @ A @ ( bit_se6528837805403552850or_nat @ B @ C2 ) ) ) ).
% xor.assoc
thf(fact_212_xor_Oassoc,axiom,
! [A: int,B: int,C2: int] :
( ( bit_se6526347334894502574or_int @ ( bit_se6526347334894502574or_int @ A @ B ) @ C2 )
= ( bit_se6526347334894502574or_int @ A @ ( bit_se6526347334894502574or_int @ B @ C2 ) ) ) ).
% xor.assoc
thf(fact_213_minus__equation__iff,axiom,
! [A: int,B: int] :
( ( ( uminus_uminus_int @ A )
= B )
= ( ( uminus_uminus_int @ B )
= A ) ) ).
% minus_equation_iff
thf(fact_214_equation__minus__iff,axiom,
! [A: int,B: int] :
( ( A
= ( uminus_uminus_int @ B ) )
= ( B
= ( uminus_uminus_int @ A ) ) ) ).
% equation_minus_iff
thf(fact_215_circuit__checker_Ocases,axiom,
! [X: produc2007852851243229709nt_int] :
~ ! [B3: set_Pr958786334691620121nt_int,Ps: list_P5707943133018811711nt_int] :
( X
!= ( produc2261658324281137661nt_int @ B3 @ Ps ) ) ).
% circuit_checker.cases
thf(fact_216_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_217_numeral__neq__neg__numeral,axiom,
! [M: num,N: num] :
( ( numeral_numeral_int @ M )
!= ( uminus_uminus_int @ ( numeral_numeral_int @ N ) ) ) ).
% numeral_neq_neg_numeral
thf(fact_218_neg__numeral__neq__numeral,axiom,
! [M: num,N: num] :
( ( uminus_uminus_int @ ( numeral_numeral_int @ M ) )
!= ( numeral_numeral_int @ N ) ) ).
% neg_numeral_neq_numeral
thf(fact_219_one__neq__neg__one,axiom,
( one_one_int
!= ( uminus_uminus_int @ one_one_int ) ) ).
% one_neq_neg_one
thf(fact_220_one__neq__neg__numeral,axiom,
! [N: num] :
( one_one_int
!= ( uminus_uminus_int @ ( numeral_numeral_int @ N ) ) ) ).
% one_neq_neg_numeral
thf(fact_221_numeral__neq__neg__one,axiom,
! [N: num] :
( ( numeral_numeral_int @ N )
!= ( uminus_uminus_int @ one_one_int ) ) ).
% numeral_neq_neg_one
thf(fact_222_mirror2__nil,axiom,
! [Ps2: list_P5707943133018811711nt_int] :
( ( Ps2 = nil_Pr2300489316682597567nt_int )
= ( ( mirror2 @ Ps2 )
= nil_Pr2300489316682597567nt_int ) ) ).
% mirror2_nil
thf(fact_223_uminus__numeral__One,axiom,
( ( uminus_uminus_int @ ( numeral_numeral_int @ one ) )
= ( uminus_uminus_int @ one_one_int ) ) ).
% uminus_numeral_One
thf(fact_224_trans__path__non__nil,axiom,
! [Ps2: list_P5707943133018811711nt_int,K_12: int,K_22: int] :
( ( Ps2 != nil_Pr2300489316682597567nt_int )
=> ( ( trans_path @ ( product_Pair_int_int @ K_12 @ K_22 ) @ Ps2 )
!= nil_Pr2300489316682597567nt_int ) ) ).
% trans_path_non_nil
thf(fact_225_trans__path_Osimps_I1_J,axiom,
! [K_12: int,K_22: int] :
( ( trans_path @ ( product_Pair_int_int @ K_12 @ K_22 ) @ nil_Pr2300489316682597567nt_int )
= nil_Pr2300489316682597567nt_int ) ).
% trans_path.simps(1)
thf(fact_226_eval__nat__numeral_I3_J,axiom,
! [N: num] :
( ( numeral_numeral_nat @ ( bit1 @ N ) )
= ( suc @ ( numeral_numeral_nat @ ( bit0 @ N ) ) ) ) ).
% eval_nat_numeral(3)
thf(fact_227_dbl__dec__simps_I4_J,axiom,
( ( neg_nu3811975205180677377ec_int @ ( uminus_uminus_int @ one_one_int ) )
= ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit1 @ one ) ) ) ) ).
% dbl_dec_simps(4)
thf(fact_228_dbl__dec__simps_I1_J,axiom,
! [K: num] :
( ( neg_nu3811975205180677377ec_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ K ) ) )
= ( uminus_uminus_int @ ( neg_nu5851722552734809277nc_int @ ( numeral_numeral_int @ K ) ) ) ) ).
% dbl_dec_simps(1)
thf(fact_229_dbl__inc__simps_I1_J,axiom,
! [K: num] :
( ( neg_nu5851722552734809277nc_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ K ) ) )
= ( uminus_uminus_int @ ( neg_nu3811975205180677377ec_int @ ( numeral_numeral_int @ K ) ) ) ) ).
% dbl_inc_simps(1)
thf(fact_230_sub__num__simps_I7_J,axiom,
! [K: num,L: num] :
( ( neg_numeral_sub_int @ ( bit0 @ K ) @ ( bit1 @ L ) )
= ( neg_nu3811975205180677377ec_int @ ( neg_numeral_sub_int @ K @ L ) ) ) ).
% sub_num_simps(7)
thf(fact_231_nat_Oinject,axiom,
! [X24: nat,Y23: nat] :
( ( ( suc @ X24 )
= ( suc @ Y23 ) )
= ( X24 = Y23 ) ) ).
% nat.inject
thf(fact_232_old_Onat_Oinject,axiom,
! [Nat: nat,Nat2: nat] :
( ( ( suc @ Nat )
= ( suc @ Nat2 ) )
= ( Nat = Nat2 ) ) ).
% old.nat.inject
thf(fact_233_dbl__dec__simps_I3_J,axiom,
( ( neg_nu3811975205180677377ec_int @ one_one_int )
= one_one_int ) ).
% dbl_dec_simps(3)
thf(fact_234_n__not__Suc__n,axiom,
! [N: nat] :
( N
!= ( suc @ N ) ) ).
% n_not_Suc_n
thf(fact_235_Suc__inject,axiom,
! [X: nat,Y3: nat] :
( ( ( suc @ X )
= ( suc @ Y3 ) )
=> ( X = Y3 ) ) ).
% Suc_inject
thf(fact_236_sub__num__simps_I2_J,axiom,
! [L: num] :
( ( neg_numeral_sub_int @ one @ ( bit0 @ L ) )
= ( uminus_uminus_int @ ( numeral_numeral_int @ ( bitM @ L ) ) ) ) ).
% sub_num_simps(2)
thf(fact_237_trans__knights__path,axiom,
! [B: set_Pr958786334691620121nt_int,Ps2: list_P5707943133018811711nt_int,K_12: int,K_22: int] :
( ( knights_path @ B @ Ps2 )
=> ( knights_path @ ( trans_board @ ( product_Pair_int_int @ K_12 @ K_22 ) @ B ) @ ( trans_path @ ( product_Pair_int_int @ K_12 @ K_22 ) @ Ps2 ) ) ) ).
% trans_knights_path
thf(fact_238_xor__numerals_I7_J,axiom,
! [X: num,Y3: num] :
( ( bit_se6528837805403552850or_nat @ ( numeral_numeral_nat @ ( bit1 @ X ) ) @ ( numeral_numeral_nat @ ( bit1 @ Y3 ) ) )
= ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( bit_se6528837805403552850or_nat @ ( numeral_numeral_nat @ X ) @ ( numeral_numeral_nat @ Y3 ) ) ) ) ).
% xor_numerals(7)
thf(fact_239_xor__numerals_I7_J,axiom,
! [X: num,Y3: num] :
( ( bit_se6526347334894502574or_int @ ( numeral_numeral_int @ ( bit1 @ X ) ) @ ( numeral_numeral_int @ ( bit1 @ Y3 ) ) )
= ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( bit_se6526347334894502574or_int @ ( numeral_numeral_int @ X ) @ ( numeral_numeral_int @ Y3 ) ) ) ) ).
% xor_numerals(7)
thf(fact_240_add__neg__numeral__special_I4_J,axiom,
! [N: num] :
( ( plus_plus_int @ ( uminus_uminus_int @ one_one_int ) @ ( numeral_numeral_int @ N ) )
= ( neg_numeral_sub_int @ N @ one ) ) ).
% add_neg_numeral_special(4)
thf(fact_241_add__neg__numeral__special_I3_J,axiom,
! [M: num] :
( ( plus_plus_int @ ( numeral_numeral_int @ M ) @ ( uminus_uminus_int @ one_one_int ) )
= ( neg_numeral_sub_int @ M @ one ) ) ).
% add_neg_numeral_special(3)
thf(fact_242_add__neg__numeral__special_I2_J,axiom,
! [M: num] :
( ( plus_plus_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ M ) ) @ one_one_int )
= ( neg_numeral_sub_int @ one @ M ) ) ).
% add_neg_numeral_special(2)
thf(fact_243_add__neg__numeral__special_I1_J,axiom,
! [M: num] :
( ( plus_plus_int @ one_one_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ M ) ) )
= ( neg_numeral_sub_int @ one @ M ) ) ).
% add_neg_numeral_special(1)
thf(fact_244_sub__num__simps_I4_J,axiom,
! [K: num] :
( ( neg_numeral_sub_int @ ( bit0 @ K ) @ one )
= ( numeral_numeral_int @ ( bitM @ K ) ) ) ).
% sub_num_simps(4)
thf(fact_245_add__right__cancel,axiom,
! [B: nat,A: nat,C2: nat] :
( ( ( plus_plus_nat @ B @ A )
= ( plus_plus_nat @ C2 @ A ) )
= ( B = C2 ) ) ).
% add_right_cancel
thf(fact_246_add__right__cancel,axiom,
! [B: int,A: int,C2: int] :
( ( ( plus_plus_int @ B @ A )
= ( plus_plus_int @ C2 @ A ) )
= ( B = C2 ) ) ).
% add_right_cancel
thf(fact_247_add__left__cancel,axiom,
! [A: nat,B: nat,C2: nat] :
( ( ( plus_plus_nat @ A @ B )
= ( plus_plus_nat @ A @ C2 ) )
= ( B = C2 ) ) ).
% add_left_cancel
thf(fact_248_add__left__cancel,axiom,
! [A: int,B: int,C2: int] :
( ( ( plus_plus_int @ A @ B )
= ( plus_plus_int @ A @ C2 ) )
= ( B = C2 ) ) ).
% add_left_cancel
thf(fact_249_semiring__norm_I6_J,axiom,
! [M: num,N: num] :
( ( plus_plus_num @ ( bit0 @ M ) @ ( bit0 @ N ) )
= ( bit0 @ ( plus_plus_num @ M @ N ) ) ) ).
% semiring_norm(6)
thf(fact_250_semiring__norm_I13_J,axiom,
! [M: num,N: num] :
( ( times_times_num @ ( bit0 @ M ) @ ( bit0 @ N ) )
= ( bit0 @ ( bit0 @ ( times_times_num @ M @ N ) ) ) ) ).
% semiring_norm(13)
thf(fact_251_semiring__norm_I11_J,axiom,
! [M: num] :
( ( times_times_num @ M @ one )
= M ) ).
% semiring_norm(11)
thf(fact_252_semiring__norm_I12_J,axiom,
! [N: num] :
( ( times_times_num @ one @ N )
= N ) ).
% semiring_norm(12)
thf(fact_253_mult__Suc__right,axiom,
! [M: nat,N: nat] :
( ( times_times_nat @ M @ ( suc @ N ) )
= ( plus_plus_nat @ M @ ( times_times_nat @ M @ N ) ) ) ).
% mult_Suc_right
thf(fact_254_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_255_nat__mult__eq__1__iff,axiom,
! [M: nat,N: nat] :
( ( ( times_times_nat @ M @ N )
= one_one_nat )
= ( ( M = one_one_nat )
& ( N = one_one_nat ) ) ) ).
% nat_mult_eq_1_iff
thf(fact_256_nat__1__eq__mult__iff,axiom,
! [M: nat,N: nat] :
( ( one_one_nat
= ( times_times_nat @ M @ N ) )
= ( ( M = one_one_nat )
& ( N = one_one_nat ) ) ) ).
% nat_1_eq_mult_iff
thf(fact_257_mult__numeral__left__semiring__numeral,axiom,
! [V2: num,W: num,Z: nat] :
( ( times_times_nat @ ( numeral_numeral_nat @ V2 ) @ ( times_times_nat @ ( numeral_numeral_nat @ W ) @ Z ) )
= ( times_times_nat @ ( numeral_numeral_nat @ ( times_times_num @ V2 @ W ) ) @ Z ) ) ).
% mult_numeral_left_semiring_numeral
thf(fact_258_mult__numeral__left__semiring__numeral,axiom,
! [V2: num,W: num,Z: int] :
( ( times_times_int @ ( numeral_numeral_int @ V2 ) @ ( times_times_int @ ( numeral_numeral_int @ W ) @ Z ) )
= ( times_times_int @ ( numeral_numeral_int @ ( times_times_num @ V2 @ W ) ) @ Z ) ) ).
% mult_numeral_left_semiring_numeral
thf(fact_259_numeral__times__numeral,axiom,
! [M: num,N: num] :
( ( times_times_nat @ ( numeral_numeral_nat @ M ) @ ( numeral_numeral_nat @ N ) )
= ( numeral_numeral_nat @ ( times_times_num @ M @ N ) ) ) ).
% numeral_times_numeral
thf(fact_260_numeral__times__numeral,axiom,
! [M: num,N: num] :
( ( times_times_int @ ( numeral_numeral_int @ M ) @ ( numeral_numeral_int @ N ) )
= ( numeral_numeral_int @ ( times_times_num @ M @ N ) ) ) ).
% numeral_times_numeral
thf(fact_261_add__numeral__left,axiom,
! [V2: num,W: num,Z: nat] :
( ( plus_plus_nat @ ( numeral_numeral_nat @ V2 ) @ ( plus_plus_nat @ ( numeral_numeral_nat @ W ) @ Z ) )
= ( plus_plus_nat @ ( numeral_numeral_nat @ ( plus_plus_num @ V2 @ W ) ) @ Z ) ) ).
% add_numeral_left
thf(fact_262_add__numeral__left,axiom,
! [V2: num,W: num,Z: int] :
( ( plus_plus_int @ ( numeral_numeral_int @ V2 ) @ ( plus_plus_int @ ( numeral_numeral_int @ W ) @ Z ) )
= ( plus_plus_int @ ( numeral_numeral_int @ ( plus_plus_num @ V2 @ W ) ) @ Z ) ) ).
% add_numeral_left
thf(fact_263_numeral__plus__numeral,axiom,
! [M: num,N: num] :
( ( plus_plus_nat @ ( numeral_numeral_nat @ M ) @ ( numeral_numeral_nat @ N ) )
= ( numeral_numeral_nat @ ( plus_plus_num @ M @ N ) ) ) ).
% numeral_plus_numeral
thf(fact_264_numeral__plus__numeral,axiom,
! [M: num,N: num] :
( ( plus_plus_int @ ( numeral_numeral_int @ M ) @ ( numeral_numeral_int @ N ) )
= ( numeral_numeral_int @ ( plus_plus_num @ M @ N ) ) ) ).
% numeral_plus_numeral
thf(fact_265_mult__1,axiom,
! [A: nat] :
( ( times_times_nat @ one_one_nat @ A )
= A ) ).
% mult_1
thf(fact_266_mult__1,axiom,
! [A: int] :
( ( times_times_int @ one_one_int @ A )
= A ) ).
% mult_1
thf(fact_267_mult_Oright__neutral,axiom,
! [A: nat] :
( ( times_times_nat @ A @ one_one_nat )
= A ) ).
% mult.right_neutral
thf(fact_268_mult_Oright__neutral,axiom,
! [A: int] :
( ( times_times_int @ A @ one_one_int )
= A ) ).
% mult.right_neutral
thf(fact_269_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_270_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_271_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_272_semiring__norm_I2_J,axiom,
( ( plus_plus_num @ one @ one )
= ( bit0 @ one ) ) ).
% semiring_norm(2)
thf(fact_273_num__double,axiom,
! [N: num] :
( ( times_times_num @ ( bit0 @ one ) @ N )
= ( bit0 @ N ) ) ).
% num_double
thf(fact_274_semiring__norm_I16_J,axiom,
! [M: num,N: num] :
( ( times_times_num @ ( bit1 @ M ) @ ( bit1 @ N ) )
= ( bit1 @ ( plus_plus_num @ ( plus_plus_num @ M @ N ) @ ( bit0 @ ( times_times_num @ M @ N ) ) ) ) ) ).
% semiring_norm(16)
thf(fact_275_semiring__norm_I7_J,axiom,
! [M: num,N: num] :
( ( plus_plus_num @ ( bit0 @ M ) @ ( bit1 @ N ) )
= ( bit1 @ ( plus_plus_num @ M @ N ) ) ) ).
% semiring_norm(7)
thf(fact_276_semiring__norm_I9_J,axiom,
! [M: num,N: num] :
( ( plus_plus_num @ ( bit1 @ M ) @ ( bit0 @ N ) )
= ( bit1 @ ( plus_plus_num @ M @ N ) ) ) ).
% semiring_norm(9)
thf(fact_277_semiring__norm_I14_J,axiom,
! [M: num,N: num] :
( ( times_times_num @ ( bit0 @ M ) @ ( bit1 @ N ) )
= ( bit0 @ ( times_times_num @ M @ ( bit1 @ N ) ) ) ) ).
% semiring_norm(14)
thf(fact_278_semiring__norm_I15_J,axiom,
! [M: num,N: num] :
( ( times_times_num @ ( bit1 @ M ) @ ( bit0 @ N ) )
= ( bit0 @ ( times_times_num @ ( bit1 @ M ) @ N ) ) ) ).
% semiring_norm(15)
thf(fact_279_distrib__right__numeral,axiom,
! [A: nat,B: nat,V2: num] :
( ( times_times_nat @ ( plus_plus_nat @ A @ B ) @ ( numeral_numeral_nat @ V2 ) )
= ( plus_plus_nat @ ( times_times_nat @ A @ ( numeral_numeral_nat @ V2 ) ) @ ( times_times_nat @ B @ ( numeral_numeral_nat @ V2 ) ) ) ) ).
% distrib_right_numeral
thf(fact_280_distrib__right__numeral,axiom,
! [A: int,B: int,V2: num] :
( ( times_times_int @ ( plus_plus_int @ A @ B ) @ ( numeral_numeral_int @ V2 ) )
= ( plus_plus_int @ ( times_times_int @ A @ ( numeral_numeral_int @ V2 ) ) @ ( times_times_int @ B @ ( numeral_numeral_int @ V2 ) ) ) ) ).
% distrib_right_numeral
thf(fact_281_distrib__left__numeral,axiom,
! [V2: num,B: nat,C2: nat] :
( ( times_times_nat @ ( numeral_numeral_nat @ V2 ) @ ( plus_plus_nat @ B @ C2 ) )
= ( plus_plus_nat @ ( times_times_nat @ ( numeral_numeral_nat @ V2 ) @ B ) @ ( times_times_nat @ ( numeral_numeral_nat @ V2 ) @ C2 ) ) ) ).
% distrib_left_numeral
thf(fact_282_distrib__left__numeral,axiom,
! [V2: num,B: int,C2: int] :
( ( times_times_int @ ( numeral_numeral_int @ V2 ) @ ( plus_plus_int @ B @ C2 ) )
= ( plus_plus_int @ ( times_times_int @ ( numeral_numeral_int @ V2 ) @ B ) @ ( times_times_int @ ( numeral_numeral_int @ V2 ) @ C2 ) ) ) ).
% distrib_left_numeral
thf(fact_283_mult__neg__numeral__simps_I1_J,axiom,
! [M: num,N: num] :
( ( times_times_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ M ) ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ N ) ) )
= ( numeral_numeral_int @ ( times_times_num @ M @ N ) ) ) ).
% mult_neg_numeral_simps(1)
thf(fact_284_mult__neg__numeral__simps_I2_J,axiom,
! [M: num,N: num] :
( ( times_times_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ M ) ) @ ( numeral_numeral_int @ N ) )
= ( uminus_uminus_int @ ( numeral_numeral_int @ ( times_times_num @ M @ N ) ) ) ) ).
% mult_neg_numeral_simps(2)
thf(fact_285_mult__neg__numeral__simps_I3_J,axiom,
! [M: num,N: num] :
( ( times_times_int @ ( numeral_numeral_int @ M ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ N ) ) )
= ( uminus_uminus_int @ ( numeral_numeral_int @ ( times_times_num @ M @ N ) ) ) ) ).
% mult_neg_numeral_simps(3)
thf(fact_286_semiring__norm_I169_J,axiom,
! [V2: num,W: num,Y3: int] :
( ( times_times_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ V2 ) ) @ ( times_times_int @ ( numeral_numeral_int @ W ) @ Y3 ) )
= ( times_times_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ ( times_times_num @ V2 @ W ) ) ) @ Y3 ) ) ).
% semiring_norm(169)
thf(fact_287_semiring__norm_I170_J,axiom,
! [V2: num,W: num,Y3: int] :
( ( times_times_int @ ( numeral_numeral_int @ V2 ) @ ( times_times_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ W ) ) @ Y3 ) )
= ( times_times_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ ( times_times_num @ V2 @ W ) ) ) @ Y3 ) ) ).
% semiring_norm(170)
thf(fact_288_semiring__norm_I171_J,axiom,
! [V2: num,W: num,Y3: int] :
( ( times_times_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ V2 ) ) @ ( times_times_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ W ) ) @ Y3 ) )
= ( times_times_int @ ( numeral_numeral_int @ ( times_times_num @ V2 @ W ) ) @ Y3 ) ) ).
% semiring_norm(171)
thf(fact_289_semiring__norm_I167_J,axiom,
! [V2: num,W: num,Y3: int] :
( ( plus_plus_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ V2 ) ) @ ( plus_plus_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ W ) ) @ Y3 ) )
= ( plus_plus_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ ( plus_plus_num @ V2 @ W ) ) ) @ Y3 ) ) ).
% semiring_norm(167)
thf(fact_290_add__neg__numeral__simps_I3_J,axiom,
! [M: num,N: num] :
( ( plus_plus_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ M ) ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ N ) ) )
= ( uminus_uminus_int @ ( plus_plus_int @ ( numeral_numeral_int @ M ) @ ( numeral_numeral_int @ N ) ) ) ) ).
% add_neg_numeral_simps(3)
thf(fact_291_mult__minus1,axiom,
! [Z: int] :
( ( times_times_int @ ( uminus_uminus_int @ one_one_int ) @ Z )
= ( uminus_uminus_int @ Z ) ) ).
% mult_minus1
thf(fact_292_mult__minus1__right,axiom,
! [Z: int] :
( ( times_times_int @ Z @ ( uminus_uminus_int @ one_one_int ) )
= ( uminus_uminus_int @ Z ) ) ).
% mult_minus1_right
thf(fact_293_Suc__numeral,axiom,
! [N: num] :
( ( suc @ ( numeral_numeral_nat @ N ) )
= ( numeral_numeral_nat @ ( plus_plus_num @ N @ one ) ) ) ).
% Suc_numeral
thf(fact_294_semiring__norm_I3_J,axiom,
! [N: num] :
( ( plus_plus_num @ one @ ( bit0 @ N ) )
= ( bit1 @ N ) ) ).
% semiring_norm(3)
thf(fact_295_semiring__norm_I4_J,axiom,
! [N: num] :
( ( plus_plus_num @ one @ ( bit1 @ N ) )
= ( bit0 @ ( plus_plus_num @ N @ one ) ) ) ).
% semiring_norm(4)
thf(fact_296_semiring__norm_I5_J,axiom,
! [M: num] :
( ( plus_plus_num @ ( bit0 @ M ) @ one )
= ( bit1 @ M ) ) ).
% semiring_norm(5)
thf(fact_297_semiring__norm_I8_J,axiom,
! [M: num] :
( ( plus_plus_num @ ( bit1 @ M ) @ one )
= ( bit0 @ ( plus_plus_num @ M @ one ) ) ) ).
% semiring_norm(8)
thf(fact_298_semiring__norm_I10_J,axiom,
! [M: num,N: num] :
( ( plus_plus_num @ ( bit1 @ M ) @ ( bit1 @ N ) )
= ( bit0 @ ( plus_plus_num @ ( plus_plus_num @ M @ N ) @ one ) ) ) ).
% semiring_norm(10)
thf(fact_299_dbl__dec__simps_I5_J,axiom,
! [K: num] :
( ( neg_nu3811975205180677377ec_int @ ( numeral_numeral_int @ K ) )
= ( numeral_numeral_int @ ( bitM @ K ) ) ) ).
% dbl_dec_simps(5)
thf(fact_300_one__plus__numeral,axiom,
! [N: num] :
( ( plus_plus_nat @ one_one_nat @ ( numeral_numeral_nat @ N ) )
= ( numeral_numeral_nat @ ( plus_plus_num @ one @ N ) ) ) ).
% one_plus_numeral
thf(fact_301_one__plus__numeral,axiom,
! [N: num] :
( ( plus_plus_int @ one_one_int @ ( numeral_numeral_int @ N ) )
= ( numeral_numeral_int @ ( plus_plus_num @ one @ N ) ) ) ).
% one_plus_numeral
thf(fact_302_numeral__plus__one,axiom,
! [N: num] :
( ( plus_plus_nat @ ( numeral_numeral_nat @ N ) @ one_one_nat )
= ( numeral_numeral_nat @ ( plus_plus_num @ N @ one ) ) ) ).
% numeral_plus_one
thf(fact_303_numeral__plus__one,axiom,
! [N: num] :
( ( plus_plus_int @ ( numeral_numeral_int @ N ) @ one_one_int )
= ( numeral_numeral_int @ ( plus_plus_num @ N @ one ) ) ) ).
% numeral_plus_one
thf(fact_304_add__2__eq__Suc_H,axiom,
! [N: nat] :
( ( plus_plus_nat @ N @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
= ( suc @ ( suc @ N ) ) ) ).
% add_2_eq_Suc'
thf(fact_305_add__2__eq__Suc,axiom,
! [N: nat] :
( ( plus_plus_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
= ( suc @ ( suc @ N ) ) ) ).
% add_2_eq_Suc
thf(fact_306_add__neg__numeral__simps_I1_J,axiom,
! [M: num,N: num] :
( ( plus_plus_int @ ( numeral_numeral_int @ M ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ N ) ) )
= ( neg_numeral_sub_int @ M @ N ) ) ).
% add_neg_numeral_simps(1)
thf(fact_307_add__neg__numeral__simps_I2_J,axiom,
! [M: num,N: num] :
( ( plus_plus_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ M ) ) @ ( numeral_numeral_int @ N ) )
= ( neg_numeral_sub_int @ N @ M ) ) ).
% add_neg_numeral_simps(2)
thf(fact_308_semiring__norm_I165_J,axiom,
! [V2: num,W: num,Y3: int] :
( ( plus_plus_int @ ( numeral_numeral_int @ V2 ) @ ( plus_plus_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ W ) ) @ Y3 ) )
= ( plus_plus_int @ ( neg_numeral_sub_int @ V2 @ W ) @ Y3 ) ) ).
% semiring_norm(165)
thf(fact_309_semiring__norm_I166_J,axiom,
! [V2: num,W: num,Y3: int] :
( ( plus_plus_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ V2 ) ) @ ( plus_plus_int @ ( numeral_numeral_int @ W ) @ Y3 ) )
= ( plus_plus_int @ ( neg_numeral_sub_int @ W @ V2 ) @ Y3 ) ) ).
% semiring_norm(166)
thf(fact_310_one__add__one,axiom,
( ( plus_plus_nat @ one_one_nat @ one_one_nat )
= ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ).
% one_add_one
thf(fact_311_one__add__one,axiom,
( ( plus_plus_int @ one_one_int @ one_one_int )
= ( numeral_numeral_int @ ( bit0 @ one ) ) ) ).
% one_add_one
thf(fact_312_xor__numerals_I3_J,axiom,
! [X: num,Y3: num] :
( ( bit_se6528837805403552850or_nat @ ( numeral_numeral_nat @ ( bit0 @ X ) ) @ ( numeral_numeral_nat @ ( bit0 @ Y3 ) ) )
= ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( bit_se6528837805403552850or_nat @ ( numeral_numeral_nat @ X ) @ ( numeral_numeral_nat @ Y3 ) ) ) ) ).
% xor_numerals(3)
thf(fact_313_xor__numerals_I3_J,axiom,
! [X: num,Y3: num] :
( ( bit_se6526347334894502574or_int @ ( numeral_numeral_int @ ( bit0 @ X ) ) @ ( numeral_numeral_int @ ( bit0 @ Y3 ) ) )
= ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( bit_se6526347334894502574or_int @ ( numeral_numeral_int @ X ) @ ( numeral_numeral_int @ Y3 ) ) ) ) ).
% xor_numerals(3)
thf(fact_314_add__neg__numeral__special_I9_J,axiom,
( ( plus_plus_int @ ( uminus_uminus_int @ one_one_int ) @ ( uminus_uminus_int @ one_one_int ) )
= ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ).
% add_neg_numeral_special(9)
thf(fact_315_xor__numerals_I4_J,axiom,
! [X: num,Y3: num] :
( ( bit_se6528837805403552850or_nat @ ( numeral_numeral_nat @ ( bit0 @ X ) ) @ ( numeral_numeral_nat @ ( bit1 @ Y3 ) ) )
= ( plus_plus_nat @ one_one_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( bit_se6528837805403552850or_nat @ ( numeral_numeral_nat @ X ) @ ( numeral_numeral_nat @ Y3 ) ) ) ) ) ).
% xor_numerals(4)
thf(fact_316_xor__numerals_I4_J,axiom,
! [X: num,Y3: num] :
( ( bit_se6526347334894502574or_int @ ( numeral_numeral_int @ ( bit0 @ X ) ) @ ( numeral_numeral_int @ ( bit1 @ Y3 ) ) )
= ( plus_plus_int @ one_one_int @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( bit_se6526347334894502574or_int @ ( numeral_numeral_int @ X ) @ ( numeral_numeral_int @ Y3 ) ) ) ) ) ).
% xor_numerals(4)
thf(fact_317_xor__numerals_I6_J,axiom,
! [X: num,Y3: num] :
( ( bit_se6528837805403552850or_nat @ ( numeral_numeral_nat @ ( bit1 @ X ) ) @ ( numeral_numeral_nat @ ( bit0 @ Y3 ) ) )
= ( plus_plus_nat @ one_one_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( bit_se6528837805403552850or_nat @ ( numeral_numeral_nat @ X ) @ ( numeral_numeral_nat @ Y3 ) ) ) ) ) ).
% xor_numerals(6)
thf(fact_318_xor__numerals_I6_J,axiom,
! [X: num,Y3: num] :
( ( bit_se6526347334894502574or_int @ ( numeral_numeral_int @ ( bit1 @ X ) ) @ ( numeral_numeral_int @ ( bit0 @ Y3 ) ) )
= ( plus_plus_int @ one_one_int @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( bit_se6526347334894502574or_int @ ( numeral_numeral_int @ X ) @ ( numeral_numeral_int @ Y3 ) ) ) ) ) ).
% xor_numerals(6)
thf(fact_319_mult__Suc,axiom,
! [M: nat,N: nat] :
( ( times_times_nat @ ( suc @ M ) @ N )
= ( plus_plus_nat @ N @ ( times_times_nat @ M @ N ) ) ) ).
% mult_Suc
thf(fact_320_add__right__imp__eq,axiom,
! [B: nat,A: nat,C2: nat] :
( ( ( plus_plus_nat @ B @ A )
= ( plus_plus_nat @ C2 @ A ) )
=> ( B = C2 ) ) ).
% add_right_imp_eq
thf(fact_321_add__right__imp__eq,axiom,
! [B: int,A: int,C2: int] :
( ( ( plus_plus_int @ B @ A )
= ( plus_plus_int @ C2 @ A ) )
=> ( B = C2 ) ) ).
% add_right_imp_eq
thf(fact_322_add__left__imp__eq,axiom,
! [A: nat,B: nat,C2: nat] :
( ( ( plus_plus_nat @ A @ B )
= ( plus_plus_nat @ A @ C2 ) )
=> ( B = C2 ) ) ).
% add_left_imp_eq
thf(fact_323_add__left__imp__eq,axiom,
! [A: int,B: int,C2: int] :
( ( ( plus_plus_int @ A @ B )
= ( plus_plus_int @ A @ C2 ) )
=> ( B = C2 ) ) ).
% add_left_imp_eq
thf(fact_324_mult_Oleft__commute,axiom,
! [B: nat,A: nat,C2: nat] :
( ( times_times_nat @ B @ ( times_times_nat @ A @ C2 ) )
= ( times_times_nat @ A @ ( times_times_nat @ B @ C2 ) ) ) ).
% mult.left_commute
thf(fact_325_mult_Oleft__commute,axiom,
! [B: int,A: int,C2: int] :
( ( times_times_int @ B @ ( times_times_int @ A @ C2 ) )
= ( times_times_int @ A @ ( times_times_int @ B @ C2 ) ) ) ).
% mult.left_commute
thf(fact_326_add_Oleft__commute,axiom,
! [B: nat,A: nat,C2: nat] :
( ( plus_plus_nat @ B @ ( plus_plus_nat @ A @ C2 ) )
= ( plus_plus_nat @ A @ ( plus_plus_nat @ B @ C2 ) ) ) ).
% add.left_commute
thf(fact_327_add_Oleft__commute,axiom,
! [B: int,A: int,C2: int] :
( ( plus_plus_int @ B @ ( plus_plus_int @ A @ C2 ) )
= ( plus_plus_int @ A @ ( plus_plus_int @ B @ C2 ) ) ) ).
% add.left_commute
thf(fact_328_mult_Ocommute,axiom,
( times_times_nat
= ( ^ [A5: nat,B4: nat] : ( times_times_nat @ B4 @ A5 ) ) ) ).
% mult.commute
thf(fact_329_mult_Ocommute,axiom,
( times_times_int
= ( ^ [A5: int,B4: int] : ( times_times_int @ B4 @ A5 ) ) ) ).
% mult.commute
thf(fact_330_add_Ocommute,axiom,
( plus_plus_nat
= ( ^ [A5: nat,B4: nat] : ( plus_plus_nat @ B4 @ A5 ) ) ) ).
% add.commute
thf(fact_331_add_Ocommute,axiom,
( plus_plus_int
= ( ^ [A5: int,B4: int] : ( plus_plus_int @ B4 @ A5 ) ) ) ).
% add.commute
thf(fact_332_add_Oright__cancel,axiom,
! [B: int,A: int,C2: int] :
( ( ( plus_plus_int @ B @ A )
= ( plus_plus_int @ C2 @ A ) )
= ( B = C2 ) ) ).
% add.right_cancel
thf(fact_333_mult_Oassoc,axiom,
! [A: nat,B: nat,C2: nat] :
( ( times_times_nat @ ( times_times_nat @ A @ B ) @ C2 )
= ( times_times_nat @ A @ ( times_times_nat @ B @ C2 ) ) ) ).
% mult.assoc
thf(fact_334_mult_Oassoc,axiom,
! [A: int,B: int,C2: int] :
( ( times_times_int @ ( times_times_int @ A @ B ) @ C2 )
= ( times_times_int @ A @ ( times_times_int @ B @ C2 ) ) ) ).
% mult.assoc
thf(fact_335_add_Oleft__cancel,axiom,
! [A: int,B: int,C2: int] :
( ( ( plus_plus_int @ A @ B )
= ( plus_plus_int @ A @ C2 ) )
= ( B = C2 ) ) ).
% add.left_cancel
thf(fact_336_add_Oassoc,axiom,
! [A: nat,B: nat,C2: nat] :
( ( plus_plus_nat @ ( plus_plus_nat @ A @ B ) @ C2 )
= ( plus_plus_nat @ A @ ( plus_plus_nat @ B @ C2 ) ) ) ).
% add.assoc
thf(fact_337_add_Oassoc,axiom,
! [A: int,B: int,C2: int] :
( ( plus_plus_int @ ( plus_plus_int @ A @ B ) @ C2 )
= ( plus_plus_int @ A @ ( plus_plus_int @ B @ C2 ) ) ) ).
% add.assoc
thf(fact_338_group__cancel_Oadd2,axiom,
! [B5: nat,K: nat,B: nat,A: nat] :
( ( B5
= ( plus_plus_nat @ K @ B ) )
=> ( ( plus_plus_nat @ A @ B5 )
= ( plus_plus_nat @ K @ ( plus_plus_nat @ A @ B ) ) ) ) ).
% group_cancel.add2
thf(fact_339_group__cancel_Oadd2,axiom,
! [B5: int,K: int,B: int,A: int] :
( ( B5
= ( plus_plus_int @ K @ B ) )
=> ( ( plus_plus_int @ A @ B5 )
= ( plus_plus_int @ K @ ( plus_plus_int @ A @ B ) ) ) ) ).
% group_cancel.add2
thf(fact_340_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_341_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_342_add__mono__thms__linordered__semiring_I4_J,axiom,
! [I3: nat,J3: nat,K: nat,L: nat] :
( ( ( I3 = J3 )
& ( K = L ) )
=> ( ( plus_plus_nat @ I3 @ K )
= ( plus_plus_nat @ J3 @ L ) ) ) ).
% add_mono_thms_linordered_semiring(4)
thf(fact_343_add__mono__thms__linordered__semiring_I4_J,axiom,
! [I3: int,J3: int,K: int,L: int] :
( ( ( I3 = J3 )
& ( K = L ) )
=> ( ( plus_plus_int @ I3 @ K )
= ( plus_plus_int @ J3 @ L ) ) ) ).
% add_mono_thms_linordered_semiring(4)
thf(fact_344_ab__semigroup__mult__class_Omult__ac_I1_J,axiom,
! [A: nat,B: nat,C2: nat] :
( ( times_times_nat @ ( times_times_nat @ A @ B ) @ C2 )
= ( times_times_nat @ A @ ( times_times_nat @ B @ C2 ) ) ) ).
% ab_semigroup_mult_class.mult_ac(1)
thf(fact_345_ab__semigroup__mult__class_Omult__ac_I1_J,axiom,
! [A: int,B: int,C2: int] :
( ( times_times_int @ ( times_times_int @ A @ B ) @ C2 )
= ( times_times_int @ A @ ( times_times_int @ B @ C2 ) ) ) ).
% ab_semigroup_mult_class.mult_ac(1)
thf(fact_346_ab__semigroup__add__class_Oadd__ac_I1_J,axiom,
! [A: nat,B: nat,C2: nat] :
( ( plus_plus_nat @ ( plus_plus_nat @ A @ B ) @ C2 )
= ( plus_plus_nat @ A @ ( plus_plus_nat @ B @ C2 ) ) ) ).
% ab_semigroup_add_class.add_ac(1)
thf(fact_347_ab__semigroup__add__class_Oadd__ac_I1_J,axiom,
! [A: int,B: int,C2: int] :
( ( plus_plus_int @ ( plus_plus_int @ A @ B ) @ C2 )
= ( plus_plus_int @ A @ ( plus_plus_int @ B @ C2 ) ) ) ).
% ab_semigroup_add_class.add_ac(1)
thf(fact_348_is__num__normalize_I1_J,axiom,
! [A: int,B: int,C2: int] :
( ( plus_plus_int @ ( plus_plus_int @ A @ B ) @ C2 )
= ( plus_plus_int @ A @ ( plus_plus_int @ B @ C2 ) ) ) ).
% is_num_normalize(1)
thf(fact_349_trans__board__correct,axiom,
! [I3: int,J3: int,B: set_Pr958786334691620121nt_int,K_12: int,K_22: int] :
( ( member5262025264175285858nt_int @ ( product_Pair_int_int @ I3 @ J3 ) @ B )
= ( member5262025264175285858nt_int @ ( product_Pair_int_int @ ( plus_plus_int @ I3 @ K_12 ) @ ( plus_plus_int @ J3 @ K_22 ) ) @ ( trans_board @ ( product_Pair_int_int @ K_12 @ K_22 ) @ B ) ) ) ).
% trans_board_correct
thf(fact_350_one__plus__BitM,axiom,
! [N: num] :
( ( plus_plus_num @ one @ ( bitM @ N ) )
= ( bit0 @ N ) ) ).
% one_plus_BitM
thf(fact_351_BitM__plus__one,axiom,
! [N: num] :
( ( plus_plus_num @ ( bitM @ N ) @ one )
= ( bit0 @ N ) ) ).
% BitM_plus_one
thf(fact_352_add__One__commute,axiom,
! [N: num] :
( ( plus_plus_num @ one @ N )
= ( plus_plus_num @ N @ one ) ) ).
% add_One_commute
thf(fact_353_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_354_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_355_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_356_mult_Ocomm__neutral,axiom,
! [A: nat] :
( ( times_times_nat @ A @ one_one_nat )
= A ) ).
% mult.comm_neutral
thf(fact_357_mult_Ocomm__neutral,axiom,
! [A: int] :
( ( times_times_int @ A @ one_one_int )
= A ) ).
% mult.comm_neutral
thf(fact_358_comm__monoid__mult__class_Omult__1,axiom,
! [A: nat] :
( ( times_times_nat @ one_one_nat @ A )
= A ) ).
% comm_monoid_mult_class.mult_1
thf(fact_359_comm__monoid__mult__class_Omult__1,axiom,
! [A: int] :
( ( times_times_int @ one_one_int @ A )
= A ) ).
% comm_monoid_mult_class.mult_1
thf(fact_360_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_361_add__Suc,axiom,
! [M: nat,N: nat] :
( ( plus_plus_nat @ ( suc @ M ) @ N )
= ( suc @ ( plus_plus_nat @ M @ N ) ) ) ).
% add_Suc
thf(fact_362_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_363_Suc__mult__cancel1,axiom,
! [K: nat,M: nat,N: nat] :
( ( ( times_times_nat @ ( suc @ K ) @ M )
= ( times_times_nat @ ( suc @ K ) @ N ) )
= ( M = N ) ) ).
% Suc_mult_cancel1
thf(fact_364_nat__mult__1__right,axiom,
! [N: nat] :
( ( times_times_nat @ N @ one_one_nat )
= N ) ).
% nat_mult_1_right
thf(fact_365_nat__mult__1,axiom,
! [N: nat] :
( ( times_times_nat @ one_one_nat @ N )
= N ) ).
% nat_mult_1
thf(fact_366_semiring__norm_I26_J,axiom,
( ( bitM @ one )
= one ) ).
% semiring_norm(26)
thf(fact_367_left__add__twice,axiom,
! [A: nat,B: nat] :
( ( plus_plus_nat @ A @ ( plus_plus_nat @ A @ B ) )
= ( plus_plus_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A ) @ B ) ) ).
% left_add_twice
thf(fact_368_left__add__twice,axiom,
! [A: int,B: int] :
( ( plus_plus_int @ A @ ( plus_plus_int @ A @ B ) )
= ( plus_plus_int @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A ) @ B ) ) ).
% left_add_twice
thf(fact_369_mult__2__right,axiom,
! [Z: nat] :
( ( times_times_nat @ Z @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
= ( plus_plus_nat @ Z @ Z ) ) ).
% mult_2_right
thf(fact_370_mult__2__right,axiom,
! [Z: int] :
( ( times_times_int @ Z @ ( numeral_numeral_int @ ( bit0 @ one ) ) )
= ( plus_plus_int @ Z @ Z ) ) ).
% mult_2_right
thf(fact_371_mult__2,axiom,
! [Z: nat] :
( ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Z )
= ( plus_plus_nat @ Z @ Z ) ) ).
% mult_2
thf(fact_372_mult__2,axiom,
! [Z: int] :
( ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ Z )
= ( plus_plus_int @ Z @ Z ) ) ).
% mult_2
thf(fact_373_Suc__nat__number__of__add,axiom,
! [V2: num,N: nat] :
( ( suc @ ( plus_plus_nat @ ( numeral_numeral_nat @ V2 ) @ N ) )
= ( plus_plus_nat @ ( numeral_numeral_nat @ ( plus_plus_num @ V2 @ one ) ) @ N ) ) ).
% Suc_nat_number_of_add
thf(fact_374_dbl__def,axiom,
( neg_numeral_dbl_int
= ( ^ [X4: int] : ( plus_plus_int @ X4 @ X4 ) ) ) ).
% dbl_def
thf(fact_375_one__plus__numeral__commute,axiom,
! [X: num] :
( ( plus_plus_nat @ one_one_nat @ ( numeral_numeral_nat @ X ) )
= ( plus_plus_nat @ ( numeral_numeral_nat @ X ) @ one_one_nat ) ) ).
% one_plus_numeral_commute
thf(fact_376_one__plus__numeral__commute,axiom,
! [X: num] :
( ( plus_plus_int @ one_one_int @ ( numeral_numeral_int @ X ) )
= ( plus_plus_int @ ( numeral_numeral_int @ X ) @ one_one_int ) ) ).
% one_plus_numeral_commute
thf(fact_377_numeral__Bit0,axiom,
! [N: num] :
( ( numeral_numeral_nat @ ( bit0 @ N ) )
= ( plus_plus_nat @ ( numeral_numeral_nat @ N ) @ ( numeral_numeral_nat @ N ) ) ) ).
% numeral_Bit0
thf(fact_378_numeral__Bit0,axiom,
! [N: num] :
( ( numeral_numeral_int @ ( bit0 @ N ) )
= ( plus_plus_int @ ( numeral_numeral_int @ N ) @ ( numeral_numeral_int @ N ) ) ) ).
% numeral_Bit0
thf(fact_379_mult__numeral__1__right,axiom,
! [A: nat] :
( ( times_times_nat @ A @ ( numeral_numeral_nat @ one ) )
= A ) ).
% mult_numeral_1_right
thf(fact_380_mult__numeral__1__right,axiom,
! [A: int] :
( ( times_times_int @ A @ ( numeral_numeral_int @ one ) )
= A ) ).
% mult_numeral_1_right
thf(fact_381_mult__numeral__1,axiom,
! [A: nat] :
( ( times_times_nat @ ( numeral_numeral_nat @ one ) @ A )
= A ) ).
% mult_numeral_1
thf(fact_382_mult__numeral__1,axiom,
! [A: int] :
( ( times_times_int @ ( numeral_numeral_int @ one ) @ A )
= A ) ).
% mult_numeral_1
thf(fact_383_numeral__times__minus__swap,axiom,
! [W: num,X: int] :
( ( times_times_int @ ( numeral_numeral_int @ W ) @ ( uminus_uminus_int @ X ) )
= ( times_times_int @ X @ ( uminus_uminus_int @ ( numeral_numeral_int @ W ) ) ) ) ).
% numeral_times_minus_swap
thf(fact_384_Suc__eq__plus1__left,axiom,
( suc
= ( plus_plus_nat @ one_one_nat ) ) ).
% Suc_eq_plus1_left
thf(fact_385_plus__1__eq__Suc,axiom,
( ( plus_plus_nat @ one_one_nat )
= suc ) ).
% plus_1_eq_Suc
thf(fact_386_Suc__eq__plus1,axiom,
( suc
= ( ^ [N3: nat] : ( plus_plus_nat @ N3 @ one_one_nat ) ) ) ).
% Suc_eq_plus1
thf(fact_387_semiring__norm_I27_J,axiom,
! [N: num] :
( ( bitM @ ( bit0 @ N ) )
= ( bit1 @ ( bitM @ N ) ) ) ).
% semiring_norm(27)
thf(fact_388_semiring__norm_I28_J,axiom,
! [N: num] :
( ( bitM @ ( bit1 @ N ) )
= ( bit1 @ ( bit0 @ N ) ) ) ).
% semiring_norm(28)
thf(fact_389_dbl__inc__def,axiom,
( neg_nu5851722552734809277nc_int
= ( ^ [X4: int] : ( plus_plus_int @ ( plus_plus_int @ X4 @ X4 ) @ one_one_int ) ) ) ).
% dbl_inc_def
thf(fact_390_numeral__Bit1,axiom,
! [N: num] :
( ( numeral_numeral_nat @ ( bit1 @ N ) )
= ( plus_plus_nat @ ( plus_plus_nat @ ( numeral_numeral_nat @ N ) @ ( numeral_numeral_nat @ N ) ) @ one_one_nat ) ) ).
% numeral_Bit1
thf(fact_391_numeral__Bit1,axiom,
! [N: num] :
( ( numeral_numeral_int @ ( bit1 @ N ) )
= ( plus_plus_int @ ( plus_plus_int @ ( numeral_numeral_int @ N ) @ ( numeral_numeral_int @ N ) ) @ one_one_int ) ) ).
% numeral_Bit1
thf(fact_392_mult__1s__ring__1_I1_J,axiom,
! [B: int] :
( ( times_times_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ one ) ) @ B )
= ( uminus_uminus_int @ B ) ) ).
% mult_1s_ring_1(1)
thf(fact_393_mult__1s__ring__1_I2_J,axiom,
! [B: int] :
( ( times_times_int @ B @ ( uminus_uminus_int @ ( numeral_numeral_int @ one ) ) )
= ( uminus_uminus_int @ B ) ) ).
% mult_1s_ring_1(2)
thf(fact_394_trans__path_Osimps_I2_J,axiom,
! [K_12: int,K_22: int,I3: int,J3: int,Xs2: list_P5707943133018811711nt_int] :
( ( trans_path @ ( product_Pair_int_int @ K_12 @ K_22 ) @ ( cons_P3334398858971670639nt_int @ ( product_Pair_int_int @ I3 @ J3 ) @ Xs2 ) )
= ( cons_P3334398858971670639nt_int @ ( product_Pair_int_int @ ( plus_plus_int @ I3 @ K_12 ) @ ( plus_plus_int @ J3 @ K_22 ) ) @ ( trans_path @ ( product_Pair_int_int @ K_12 @ K_22 ) @ Xs2 ) ) ) ).
% trans_path.simps(2)
thf(fact_395_eval__nat__numeral_I2_J,axiom,
! [N: num] :
( ( numeral_numeral_nat @ ( bit0 @ N ) )
= ( suc @ ( numeral_numeral_nat @ ( bitM @ N ) ) ) ) ).
% eval_nat_numeral(2)
thf(fact_396_Suc3__eq__add__3,axiom,
! [N: nat] :
( ( suc @ ( suc @ ( suc @ N ) ) )
= ( plus_plus_nat @ ( numeral_numeral_nat @ ( bit1 @ one ) ) @ N ) ) ).
% Suc3_eq_add_3
thf(fact_397_nat__1__add__1,axiom,
( ( plus_plus_nat @ one_one_nat @ one_one_nat )
= ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ).
% nat_1_add_1
thf(fact_398_trans__path_Oelims,axiom,
! [X: product_prod_int_int,Xa: list_P5707943133018811711nt_int,Y3: list_P5707943133018811711nt_int] :
( ( ( trans_path @ X @ Xa )
= Y3 )
=> ( ( ? [K_1: int,K_2: int] :
( X
= ( product_Pair_int_int @ K_1 @ K_2 ) )
=> ( ( Xa = nil_Pr2300489316682597567nt_int )
=> ( Y3 != nil_Pr2300489316682597567nt_int ) ) )
=> ~ ! [K_1: int,K_2: int] :
( ( X
= ( product_Pair_int_int @ K_1 @ K_2 ) )
=> ! [I: int,J: int,Xs: list_P5707943133018811711nt_int] :
( ( Xa
= ( cons_P3334398858971670639nt_int @ ( product_Pair_int_int @ I @ J ) @ Xs ) )
=> ( Y3
!= ( cons_P3334398858971670639nt_int @ ( product_Pair_int_int @ ( plus_plus_int @ I @ K_1 ) @ ( plus_plus_int @ J @ K_2 ) ) @ ( trans_path @ ( product_Pair_int_int @ K_1 @ K_2 ) @ Xs ) ) ) ) ) ) ) ).
% trans_path.elims
thf(fact_399_last__trans__path,axiom,
! [Ps2: list_P5707943133018811711nt_int,I3: int,J3: int,K_12: int,K_22: int] :
( ( Ps2 != nil_Pr2300489316682597567nt_int )
=> ( ( ( last_P3305686521732843992nt_int @ Ps2 )
= ( product_Pair_int_int @ I3 @ J3 ) )
=> ( ( last_P3305686521732843992nt_int @ ( trans_path @ ( product_Pair_int_int @ K_12 @ K_22 ) @ Ps2 ) )
= ( product_Pair_int_int @ ( plus_plus_int @ I3 @ K_12 ) @ ( plus_plus_int @ J3 @ K_22 ) ) ) ) ) ).
% last_trans_path
thf(fact_400_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_401_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_402_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_403_sub__BitM__One__eq,axiom,
! [N: num] :
( ( neg_numeral_sub_int @ ( bitM @ N ) @ one )
= ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( neg_numeral_sub_int @ N @ one ) ) ) ).
% sub_BitM_One_eq
thf(fact_404_double__not__eq__Suc__double,axiom,
! [M: nat,N: nat] :
( ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M )
!= ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ) ).
% double_not_eq_Suc_double
thf(fact_405_Suc__double__not__eq__double,axiom,
! [M: nat,N: nat] :
( ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M ) )
!= ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ).
% Suc_double_not_eq_double
thf(fact_406_square__eq__1__iff,axiom,
! [X: int] :
( ( ( times_times_int @ X @ X )
= one_one_int )
= ( ( X = one_one_int )
| ( X
= ( uminus_uminus_int @ one_one_int ) ) ) ) ).
% square_eq_1_iff
thf(fact_407_or__numerals_I4_J,axiom,
! [X: num,Y3: num] :
( ( bit_se1409905431419307370or_int @ ( numeral_numeral_int @ ( bit0 @ X ) ) @ ( numeral_numeral_int @ ( bit1 @ Y3 ) ) )
= ( plus_plus_int @ one_one_int @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( bit_se1409905431419307370or_int @ ( numeral_numeral_int @ X ) @ ( numeral_numeral_int @ Y3 ) ) ) ) ) ).
% or_numerals(4)
thf(fact_408_or__numerals_I4_J,axiom,
! [X: num,Y3: num] :
( ( bit_se1412395901928357646or_nat @ ( numeral_numeral_nat @ ( bit0 @ X ) ) @ ( numeral_numeral_nat @ ( bit1 @ Y3 ) ) )
= ( plus_plus_nat @ one_one_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( bit_se1412395901928357646or_nat @ ( numeral_numeral_nat @ X ) @ ( numeral_numeral_nat @ Y3 ) ) ) ) ) ).
% or_numerals(4)
thf(fact_409_or_Oidem,axiom,
! [A: int] :
( ( bit_se1409905431419307370or_int @ A @ A )
= A ) ).
% or.idem
thf(fact_410_or_Oidem,axiom,
! [A: nat] :
( ( bit_se1412395901928357646or_nat @ A @ A )
= A ) ).
% or.idem
thf(fact_411_or_Oleft__idem,axiom,
! [A: int,B: int] :
( ( bit_se1409905431419307370or_int @ A @ ( bit_se1409905431419307370or_int @ A @ B ) )
= ( bit_se1409905431419307370or_int @ A @ B ) ) ).
% or.left_idem
thf(fact_412_or_Oleft__idem,axiom,
! [A: nat,B: nat] :
( ( bit_se1412395901928357646or_nat @ A @ ( bit_se1412395901928357646or_nat @ A @ B ) )
= ( bit_se1412395901928357646or_nat @ A @ B ) ) ).
% or.left_idem
thf(fact_413_or_Oright__idem,axiom,
! [A: int,B: int] :
( ( bit_se1409905431419307370or_int @ ( bit_se1409905431419307370or_int @ A @ B ) @ B )
= ( bit_se1409905431419307370or_int @ A @ B ) ) ).
% or.right_idem
thf(fact_414_or_Oright__idem,axiom,
! [A: nat,B: nat] :
( ( bit_se1412395901928357646or_nat @ ( bit_se1412395901928357646or_nat @ A @ B ) @ B )
= ( bit_se1412395901928357646or_nat @ A @ B ) ) ).
% or.right_idem
thf(fact_415_or__minus__numerals_I6_J,axiom,
! [N: num] :
( ( bit_se1409905431419307370or_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit1 @ N ) ) ) @ one_one_int )
= ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit1 @ N ) ) ) ) ).
% or_minus_numerals(6)
thf(fact_416_or__minus__numerals_I2_J,axiom,
! [N: num] :
( ( bit_se1409905431419307370or_int @ one_one_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit1 @ N ) ) ) )
= ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit1 @ N ) ) ) ) ).
% or_minus_numerals(2)
thf(fact_417_bit_Odisj__one__left,axiom,
! [X: int] :
( ( bit_se1409905431419307370or_int @ ( uminus_uminus_int @ one_one_int ) @ X )
= ( uminus_uminus_int @ one_one_int ) ) ).
% bit.disj_one_left
thf(fact_418_bit_Odisj__one__right,axiom,
! [X: int] :
( ( bit_se1409905431419307370or_int @ X @ ( uminus_uminus_int @ one_one_int ) )
= ( uminus_uminus_int @ one_one_int ) ) ).
% bit.disj_one_right
thf(fact_419_or__numerals_I8_J,axiom,
! [X: num] :
( ( bit_se1409905431419307370or_int @ ( numeral_numeral_int @ ( bit1 @ X ) ) @ one_one_int )
= ( numeral_numeral_int @ ( bit1 @ X ) ) ) ).
% or_numerals(8)
thf(fact_420_or__numerals_I8_J,axiom,
! [X: num] :
( ( bit_se1412395901928357646or_nat @ ( numeral_numeral_nat @ ( bit1 @ X ) ) @ one_one_nat )
= ( numeral_numeral_nat @ ( bit1 @ X ) ) ) ).
% or_numerals(8)
thf(fact_421_or__numerals_I2_J,axiom,
! [Y3: num] :
( ( bit_se1409905431419307370or_int @ one_one_int @ ( numeral_numeral_int @ ( bit1 @ Y3 ) ) )
= ( numeral_numeral_int @ ( bit1 @ Y3 ) ) ) ).
% or_numerals(2)
thf(fact_422_or__numerals_I2_J,axiom,
! [Y3: num] :
( ( bit_se1412395901928357646or_nat @ one_one_nat @ ( numeral_numeral_nat @ ( bit1 @ Y3 ) ) )
= ( numeral_numeral_nat @ ( bit1 @ Y3 ) ) ) ).
% or_numerals(2)
thf(fact_423_or__numerals_I3_J,axiom,
! [X: num,Y3: num] :
( ( bit_se1409905431419307370or_int @ ( numeral_numeral_int @ ( bit0 @ X ) ) @ ( numeral_numeral_int @ ( bit0 @ Y3 ) ) )
= ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( bit_se1409905431419307370or_int @ ( numeral_numeral_int @ X ) @ ( numeral_numeral_int @ Y3 ) ) ) ) ).
% or_numerals(3)
thf(fact_424_or__numerals_I3_J,axiom,
! [X: num,Y3: num] :
( ( bit_se1412395901928357646or_nat @ ( numeral_numeral_nat @ ( bit0 @ X ) ) @ ( numeral_numeral_nat @ ( bit0 @ Y3 ) ) )
= ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( bit_se1412395901928357646or_nat @ ( numeral_numeral_nat @ X ) @ ( numeral_numeral_nat @ Y3 ) ) ) ) ).
% or_numerals(3)
thf(fact_425_or__numerals_I5_J,axiom,
! [X: num] :
( ( bit_se1409905431419307370or_int @ ( numeral_numeral_int @ ( bit0 @ X ) ) @ one_one_int )
= ( numeral_numeral_int @ ( bit1 @ X ) ) ) ).
% or_numerals(5)
thf(fact_426_or__numerals_I5_J,axiom,
! [X: num] :
( ( bit_se1412395901928357646or_nat @ ( numeral_numeral_nat @ ( bit0 @ X ) ) @ one_one_nat )
= ( numeral_numeral_nat @ ( bit1 @ X ) ) ) ).
% or_numerals(5)
thf(fact_427_or__numerals_I1_J,axiom,
! [Y3: num] :
( ( bit_se1409905431419307370or_int @ one_one_int @ ( numeral_numeral_int @ ( bit0 @ Y3 ) ) )
= ( numeral_numeral_int @ ( bit1 @ Y3 ) ) ) ).
% or_numerals(1)
thf(fact_428_or__numerals_I1_J,axiom,
! [Y3: num] :
( ( bit_se1412395901928357646or_nat @ one_one_nat @ ( numeral_numeral_nat @ ( bit0 @ Y3 ) ) )
= ( numeral_numeral_nat @ ( bit1 @ Y3 ) ) ) ).
% or_numerals(1)
thf(fact_429_or__numerals_I7_J,axiom,
! [X: num,Y3: num] :
( ( bit_se1409905431419307370or_int @ ( numeral_numeral_int @ ( bit1 @ X ) ) @ ( numeral_numeral_int @ ( bit1 @ Y3 ) ) )
= ( plus_plus_int @ one_one_int @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( bit_se1409905431419307370or_int @ ( numeral_numeral_int @ X ) @ ( numeral_numeral_int @ Y3 ) ) ) ) ) ).
% or_numerals(7)
thf(fact_430_or__numerals_I7_J,axiom,
! [X: num,Y3: num] :
( ( bit_se1412395901928357646or_nat @ ( numeral_numeral_nat @ ( bit1 @ X ) ) @ ( numeral_numeral_nat @ ( bit1 @ Y3 ) ) )
= ( plus_plus_nat @ one_one_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( bit_se1412395901928357646or_nat @ ( numeral_numeral_nat @ X ) @ ( numeral_numeral_nat @ Y3 ) ) ) ) ) ).
% or_numerals(7)
thf(fact_431_or__numerals_I6_J,axiom,
! [X: num,Y3: num] :
( ( bit_se1409905431419307370or_int @ ( numeral_numeral_int @ ( bit1 @ X ) ) @ ( numeral_numeral_int @ ( bit0 @ Y3 ) ) )
= ( plus_plus_int @ one_one_int @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( bit_se1409905431419307370or_int @ ( numeral_numeral_int @ X ) @ ( numeral_numeral_int @ Y3 ) ) ) ) ) ).
% or_numerals(6)
thf(fact_432_or__numerals_I6_J,axiom,
! [X: num,Y3: num] :
( ( bit_se1412395901928357646or_nat @ ( numeral_numeral_nat @ ( bit1 @ X ) ) @ ( numeral_numeral_nat @ ( bit0 @ Y3 ) ) )
= ( plus_plus_nat @ one_one_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( bit_se1412395901928357646or_nat @ ( numeral_numeral_nat @ X ) @ ( numeral_numeral_nat @ Y3 ) ) ) ) ) ).
% or_numerals(6)
thf(fact_433_add__mult__distrib2,axiom,
! [K: nat,M: nat,N: nat] :
( ( times_times_nat @ K @ ( plus_plus_nat @ M @ N ) )
= ( plus_plus_nat @ ( times_times_nat @ K @ M ) @ ( times_times_nat @ K @ N ) ) ) ).
% add_mult_distrib2
thf(fact_434_add__mult__distrib,axiom,
! [M: nat,N: nat,K: nat] :
( ( times_times_nat @ ( plus_plus_nat @ M @ N ) @ K )
= ( plus_plus_nat @ ( times_times_nat @ M @ K ) @ ( times_times_nat @ N @ K ) ) ) ).
% add_mult_distrib
thf(fact_435_int__distrib_I2_J,axiom,
! [W: int,Z1: int,Z2: int] :
( ( times_times_int @ W @ ( plus_plus_int @ Z1 @ Z2 ) )
= ( plus_plus_int @ ( times_times_int @ W @ Z1 ) @ ( times_times_int @ W @ Z2 ) ) ) ).
% int_distrib(2)
thf(fact_436_int__distrib_I1_J,axiom,
! [Z1: int,Z2: int,W: int] :
( ( times_times_int @ ( plus_plus_int @ Z1 @ Z2 ) @ W )
= ( plus_plus_int @ ( times_times_int @ Z1 @ W ) @ ( times_times_int @ Z2 @ W ) ) ) ).
% int_distrib(1)
thf(fact_437_left__add__mult__distrib,axiom,
! [I3: nat,U: nat,J3: nat,K: nat] :
( ( plus_plus_nat @ ( times_times_nat @ I3 @ U ) @ ( plus_plus_nat @ ( times_times_nat @ J3 @ U ) @ K ) )
= ( plus_plus_nat @ ( times_times_nat @ ( plus_plus_nat @ I3 @ J3 ) @ U ) @ K ) ) ).
% left_add_mult_distrib
thf(fact_438_zmult__eq__1__iff,axiom,
! [M: int,N: int] :
( ( ( times_times_int @ M @ N )
= one_one_int )
= ( ( ( M = one_one_int )
& ( N = one_one_int ) )
| ( ( M
= ( uminus_uminus_int @ one_one_int ) )
& ( N
= ( uminus_uminus_int @ one_one_int ) ) ) ) ) ).
% zmult_eq_1_iff
thf(fact_439_pos__zmult__eq__1__iff__lemma,axiom,
! [M: int,N: int] :
( ( ( times_times_int @ M @ N )
= one_one_int )
=> ( ( M = one_one_int )
| ( M
= ( uminus_uminus_int @ one_one_int ) ) ) ) ).
% pos_zmult_eq_1_iff_lemma
thf(fact_440_or_Oassoc,axiom,
! [A: int,B: int,C2: int] :
( ( bit_se1409905431419307370or_int @ ( bit_se1409905431419307370or_int @ A @ B ) @ C2 )
= ( bit_se1409905431419307370or_int @ A @ ( bit_se1409905431419307370or_int @ B @ C2 ) ) ) ).
% or.assoc
thf(fact_441_or_Oassoc,axiom,
! [A: nat,B: nat,C2: nat] :
( ( bit_se1412395901928357646or_nat @ ( bit_se1412395901928357646or_nat @ A @ B ) @ C2 )
= ( bit_se1412395901928357646or_nat @ A @ ( bit_se1412395901928357646or_nat @ B @ C2 ) ) ) ).
% or.assoc
thf(fact_442_or_Ocommute,axiom,
( bit_se1409905431419307370or_int
= ( ^ [A5: int,B4: int] : ( bit_se1409905431419307370or_int @ B4 @ A5 ) ) ) ).
% or.commute
thf(fact_443_or_Ocommute,axiom,
( bit_se1412395901928357646or_nat
= ( ^ [A5: nat,B4: nat] : ( bit_se1412395901928357646or_nat @ B4 @ A5 ) ) ) ).
% or.commute
thf(fact_444_or_Oleft__commute,axiom,
! [B: int,A: int,C2: int] :
( ( bit_se1409905431419307370or_int @ B @ ( bit_se1409905431419307370or_int @ A @ C2 ) )
= ( bit_se1409905431419307370or_int @ A @ ( bit_se1409905431419307370or_int @ B @ C2 ) ) ) ).
% or.left_commute
thf(fact_445_or_Oleft__commute,axiom,
! [B: nat,A: nat,C2: nat] :
( ( bit_se1412395901928357646or_nat @ B @ ( bit_se1412395901928357646or_nat @ A @ C2 ) )
= ( bit_se1412395901928357646or_nat @ A @ ( bit_se1412395901928357646or_nat @ B @ C2 ) ) ) ).
% or.left_commute
thf(fact_446_combine__common__factor,axiom,
! [A: nat,E: nat,B: nat,C2: nat] :
( ( plus_plus_nat @ ( times_times_nat @ A @ E ) @ ( plus_plus_nat @ ( times_times_nat @ B @ E ) @ C2 ) )
= ( plus_plus_nat @ ( times_times_nat @ ( plus_plus_nat @ A @ B ) @ E ) @ C2 ) ) ).
% combine_common_factor
thf(fact_447_combine__common__factor,axiom,
! [A: int,E: int,B: int,C2: int] :
( ( plus_plus_int @ ( times_times_int @ A @ E ) @ ( plus_plus_int @ ( times_times_int @ B @ E ) @ C2 ) )
= ( plus_plus_int @ ( times_times_int @ ( plus_plus_int @ A @ B ) @ E ) @ C2 ) ) ).
% combine_common_factor
thf(fact_448_distrib__right,axiom,
! [A: nat,B: nat,C2: nat] :
( ( times_times_nat @ ( plus_plus_nat @ A @ B ) @ C2 )
= ( plus_plus_nat @ ( times_times_nat @ A @ C2 ) @ ( times_times_nat @ B @ C2 ) ) ) ).
% distrib_right
thf(fact_449_distrib__right,axiom,
! [A: int,B: int,C2: int] :
( ( times_times_int @ ( plus_plus_int @ A @ B ) @ C2 )
= ( plus_plus_int @ ( times_times_int @ A @ C2 ) @ ( times_times_int @ B @ C2 ) ) ) ).
% distrib_right
thf(fact_450_distrib__left,axiom,
! [A: nat,B: nat,C2: nat] :
( ( times_times_nat @ A @ ( plus_plus_nat @ B @ C2 ) )
= ( plus_plus_nat @ ( times_times_nat @ A @ B ) @ ( times_times_nat @ A @ C2 ) ) ) ).
% distrib_left
thf(fact_451_distrib__left,axiom,
! [A: int,B: int,C2: int] :
( ( times_times_int @ A @ ( plus_plus_int @ B @ C2 ) )
= ( plus_plus_int @ ( times_times_int @ A @ B ) @ ( times_times_int @ A @ C2 ) ) ) ).
% distrib_left
thf(fact_452_comm__semiring__class_Odistrib,axiom,
! [A: nat,B: nat,C2: nat] :
( ( times_times_nat @ ( plus_plus_nat @ A @ B ) @ C2 )
= ( plus_plus_nat @ ( times_times_nat @ A @ C2 ) @ ( times_times_nat @ B @ C2 ) ) ) ).
% comm_semiring_class.distrib
thf(fact_453_comm__semiring__class_Odistrib,axiom,
! [A: int,B: int,C2: int] :
( ( times_times_int @ ( plus_plus_int @ A @ B ) @ C2 )
= ( plus_plus_int @ ( times_times_int @ A @ C2 ) @ ( times_times_int @ B @ C2 ) ) ) ).
% comm_semiring_class.distrib
thf(fact_454_ring__class_Oring__distribs_I1_J,axiom,
! [A: int,B: int,C2: int] :
( ( times_times_int @ A @ ( plus_plus_int @ B @ C2 ) )
= ( plus_plus_int @ ( times_times_int @ A @ B ) @ ( times_times_int @ A @ C2 ) ) ) ).
% ring_class.ring_distribs(1)
thf(fact_455_ring__class_Oring__distribs_I2_J,axiom,
! [A: int,B: int,C2: int] :
( ( times_times_int @ ( plus_plus_int @ A @ B ) @ C2 )
= ( plus_plus_int @ ( times_times_int @ A @ C2 ) @ ( times_times_int @ B @ C2 ) ) ) ).
% ring_class.ring_distribs(2)
thf(fact_456_square__eq__iff,axiom,
! [A: int,B: int] :
( ( ( times_times_int @ A @ A )
= ( times_times_int @ B @ B ) )
= ( ( A = B )
| ( A
= ( uminus_uminus_int @ B ) ) ) ) ).
% square_eq_iff
thf(fact_457_minus__mult__commute,axiom,
! [A: int,B: int] :
( ( times_times_int @ ( uminus_uminus_int @ A ) @ B )
= ( times_times_int @ A @ ( uminus_uminus_int @ B ) ) ) ).
% minus_mult_commute
thf(fact_458_mask__Suc__double,axiom,
! [N: nat] :
( ( bit_se2002935070580805687sk_nat @ ( suc @ N ) )
= ( bit_se1412395901928357646or_nat @ one_one_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( bit_se2002935070580805687sk_nat @ N ) ) ) ) ).
% mask_Suc_double
thf(fact_459_mask__Suc__double,axiom,
! [N: nat] :
( ( bit_se2000444600071755411sk_int @ ( suc @ N ) )
= ( bit_se1409905431419307370or_int @ one_one_int @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( bit_se2000444600071755411sk_int @ N ) ) ) ) ).
% mask_Suc_double
thf(fact_460_signed__take__bit__Suc__bit1,axiom,
! [N: nat,K: num] :
( ( bit_ri631733984087533419it_int @ ( suc @ N ) @ ( numeral_numeral_int @ ( bit1 @ K ) ) )
= ( plus_plus_int @ ( times_times_int @ ( bit_ri631733984087533419it_int @ N @ ( numeral_numeral_int @ K ) ) @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) @ one_one_int ) ) ).
% signed_take_bit_Suc_bit1
thf(fact_461_and__numerals_I7_J,axiom,
! [X: num,Y3: num] :
( ( bit_se725231765392027082nd_int @ ( numeral_numeral_int @ ( bit1 @ X ) ) @ ( numeral_numeral_int @ ( bit1 @ Y3 ) ) )
= ( plus_plus_int @ one_one_int @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( bit_se725231765392027082nd_int @ ( numeral_numeral_int @ X ) @ ( numeral_numeral_int @ Y3 ) ) ) ) ) ).
% and_numerals(7)
thf(fact_462_and__numerals_I7_J,axiom,
! [X: num,Y3: num] :
( ( bit_se727722235901077358nd_nat @ ( numeral_numeral_nat @ ( bit1 @ X ) ) @ ( numeral_numeral_nat @ ( bit1 @ Y3 ) ) )
= ( plus_plus_nat @ one_one_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( bit_se727722235901077358nd_nat @ ( numeral_numeral_nat @ X ) @ ( numeral_numeral_nat @ Y3 ) ) ) ) ) ).
% and_numerals(7)
thf(fact_463_take__bit__Suc__bit1,axiom,
! [N: nat,K: num] :
( ( bit_se2925701944663578781it_nat @ ( suc @ N ) @ ( numeral_numeral_nat @ ( bit1 @ K ) ) )
= ( plus_plus_nat @ ( times_times_nat @ ( bit_se2925701944663578781it_nat @ N @ ( numeral_numeral_nat @ K ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ one_one_nat ) ) ).
% take_bit_Suc_bit1
thf(fact_464_take__bit__Suc__bit1,axiom,
! [N: nat,K: num] :
( ( bit_se2923211474154528505it_int @ ( suc @ N ) @ ( numeral_numeral_int @ ( bit1 @ K ) ) )
= ( plus_plus_int @ ( times_times_int @ ( bit_se2923211474154528505it_int @ N @ ( numeral_numeral_int @ K ) ) @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) @ one_one_int ) ) ).
% take_bit_Suc_bit1
thf(fact_465_or__minus__numerals_I5_J,axiom,
! [N: num] :
( ( bit_se1409905431419307370or_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit0 @ N ) ) ) @ one_one_int )
= ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit_or_not_num_neg @ one @ ( bitM @ N ) ) ) ) ) ).
% or_minus_numerals(5)
thf(fact_466_or__minus__numerals_I1_J,axiom,
! [N: num] :
( ( bit_se1409905431419307370or_int @ one_one_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit0 @ N ) ) ) )
= ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit_or_not_num_neg @ one @ ( bitM @ N ) ) ) ) ) ).
% or_minus_numerals(1)
thf(fact_467_and_Oright__idem,axiom,
! [A: int,B: int] :
( ( bit_se725231765392027082nd_int @ ( bit_se725231765392027082nd_int @ A @ B ) @ B )
= ( bit_se725231765392027082nd_int @ A @ B ) ) ).
% and.right_idem
thf(fact_468_and_Oright__idem,axiom,
! [A: nat,B: nat] :
( ( bit_se727722235901077358nd_nat @ ( bit_se727722235901077358nd_nat @ A @ B ) @ B )
= ( bit_se727722235901077358nd_nat @ A @ B ) ) ).
% and.right_idem
thf(fact_469_and_Oleft__idem,axiom,
! [A: int,B: int] :
( ( bit_se725231765392027082nd_int @ A @ ( bit_se725231765392027082nd_int @ A @ B ) )
= ( bit_se725231765392027082nd_int @ A @ B ) ) ).
% and.left_idem
thf(fact_470_and_Oleft__idem,axiom,
! [A: nat,B: nat] :
( ( bit_se727722235901077358nd_nat @ A @ ( bit_se727722235901077358nd_nat @ A @ B ) )
= ( bit_se727722235901077358nd_nat @ A @ B ) ) ).
% and.left_idem
thf(fact_471_and_Oidem,axiom,
! [A: int] :
( ( bit_se725231765392027082nd_int @ A @ A )
= A ) ).
% and.idem
thf(fact_472_and_Oidem,axiom,
! [A: nat] :
( ( bit_se727722235901077358nd_nat @ A @ A )
= A ) ).
% and.idem
thf(fact_473_take__bit__and,axiom,
! [N: nat,A: nat,B: nat] :
( ( bit_se2925701944663578781it_nat @ N @ ( bit_se727722235901077358nd_nat @ A @ B ) )
= ( bit_se727722235901077358nd_nat @ ( bit_se2925701944663578781it_nat @ N @ A ) @ ( bit_se2925701944663578781it_nat @ N @ B ) ) ) ).
% take_bit_and
thf(fact_474_take__bit__and,axiom,
! [N: nat,A: int,B: int] :
( ( bit_se2923211474154528505it_int @ N @ ( bit_se725231765392027082nd_int @ A @ B ) )
= ( bit_se725231765392027082nd_int @ ( bit_se2923211474154528505it_int @ N @ A ) @ ( bit_se2923211474154528505it_int @ N @ B ) ) ) ).
% take_bit_and
thf(fact_475_take__bit__or,axiom,
! [N: nat,A: nat,B: nat] :
( ( bit_se2925701944663578781it_nat @ N @ ( bit_se1412395901928357646or_nat @ A @ B ) )
= ( bit_se1412395901928357646or_nat @ ( bit_se2925701944663578781it_nat @ N @ A ) @ ( bit_se2925701944663578781it_nat @ N @ B ) ) ) ).
% take_bit_or
thf(fact_476_take__bit__or,axiom,
! [N: nat,A: int,B: int] :
( ( bit_se2923211474154528505it_int @ N @ ( bit_se1409905431419307370or_int @ A @ B ) )
= ( bit_se1409905431419307370or_int @ ( bit_se2923211474154528505it_int @ N @ A ) @ ( bit_se2923211474154528505it_int @ N @ B ) ) ) ).
% take_bit_or
thf(fact_477_take__bit__xor,axiom,
! [N: nat,A: nat,B: nat] :
( ( bit_se2925701944663578781it_nat @ N @ ( bit_se6528837805403552850or_nat @ A @ B ) )
= ( bit_se6528837805403552850or_nat @ ( bit_se2925701944663578781it_nat @ N @ A ) @ ( bit_se2925701944663578781it_nat @ N @ B ) ) ) ).
% take_bit_xor
thf(fact_478_take__bit__xor,axiom,
! [N: nat,A: int,B: int] :
( ( bit_se2923211474154528505it_int @ N @ ( bit_se6526347334894502574or_int @ A @ B ) )
= ( bit_se6526347334894502574or_int @ ( bit_se2923211474154528505it_int @ N @ A ) @ ( bit_se2923211474154528505it_int @ N @ B ) ) ) ).
% take_bit_xor
thf(fact_479_bit_Oconj__one__right,axiom,
! [X: int] :
( ( bit_se725231765392027082nd_int @ X @ ( uminus_uminus_int @ one_one_int ) )
= X ) ).
% bit.conj_one_right
thf(fact_480_and_Oright__neutral,axiom,
! [A: int] :
( ( bit_se725231765392027082nd_int @ A @ ( uminus_uminus_int @ one_one_int ) )
= A ) ).
% and.right_neutral
thf(fact_481_and_Oleft__neutral,axiom,
! [A: int] :
( ( bit_se725231765392027082nd_int @ ( uminus_uminus_int @ one_one_int ) @ A )
= A ) ).
% and.left_neutral
thf(fact_482_take__bit__Suc__1,axiom,
! [N: nat] :
( ( bit_se2925701944663578781it_nat @ ( suc @ N ) @ one_one_nat )
= one_one_nat ) ).
% take_bit_Suc_1
thf(fact_483_take__bit__Suc__1,axiom,
! [N: nat] :
( ( bit_se2923211474154528505it_int @ ( suc @ N ) @ one_one_int )
= one_one_int ) ).
% take_bit_Suc_1
thf(fact_484_take__bit__numeral__1,axiom,
! [L: num] :
( ( bit_se2925701944663578781it_nat @ ( numeral_numeral_nat @ L ) @ one_one_nat )
= one_one_nat ) ).
% take_bit_numeral_1
thf(fact_485_take__bit__numeral__1,axiom,
! [L: num] :
( ( bit_se2923211474154528505it_int @ ( numeral_numeral_nat @ L ) @ one_one_int )
= one_one_int ) ).
% take_bit_numeral_1
thf(fact_486_signed__take__bit__of__minus__1,axiom,
! [N: nat] :
( ( bit_ri631733984087533419it_int @ N @ ( uminus_uminus_int @ one_one_int ) )
= ( uminus_uminus_int @ one_one_int ) ) ).
% signed_take_bit_of_minus_1
thf(fact_487_signed__take__bit__Suc__1,axiom,
! [N: nat] :
( ( bit_ri631733984087533419it_int @ ( suc @ N ) @ one_one_int )
= one_one_int ) ).
% signed_take_bit_Suc_1
thf(fact_488_signed__take__bit__numeral__of__1,axiom,
! [K: num] :
( ( bit_ri631733984087533419it_int @ ( numeral_numeral_nat @ K ) @ one_one_int )
= one_one_int ) ).
% signed_take_bit_numeral_of_1
thf(fact_489_and__numerals_I2_J,axiom,
! [Y3: num] :
( ( bit_se725231765392027082nd_int @ one_one_int @ ( numeral_numeral_int @ ( bit1 @ Y3 ) ) )
= one_one_int ) ).
% and_numerals(2)
thf(fact_490_and__numerals_I2_J,axiom,
! [Y3: num] :
( ( bit_se727722235901077358nd_nat @ one_one_nat @ ( numeral_numeral_nat @ ( bit1 @ Y3 ) ) )
= one_one_nat ) ).
% and_numerals(2)
thf(fact_491_and__numerals_I8_J,axiom,
! [X: num] :
( ( bit_se725231765392027082nd_int @ ( numeral_numeral_int @ ( bit1 @ X ) ) @ one_one_int )
= one_one_int ) ).
% and_numerals(8)
thf(fact_492_and__numerals_I8_J,axiom,
! [X: num] :
( ( bit_se727722235901077358nd_nat @ ( numeral_numeral_nat @ ( bit1 @ X ) ) @ one_one_nat )
= one_one_nat ) ).
% and_numerals(8)
thf(fact_493_and__minus__numerals_I2_J,axiom,
! [N: num] :
( ( bit_se725231765392027082nd_int @ one_one_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit1 @ N ) ) ) )
= one_one_int ) ).
% and_minus_numerals(2)
thf(fact_494_and__minus__numerals_I6_J,axiom,
! [N: num] :
( ( bit_se725231765392027082nd_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit1 @ N ) ) ) @ one_one_int )
= one_one_int ) ).
% and_minus_numerals(6)
thf(fact_495_take__bit__minus__one__eq__mask,axiom,
! [N: nat] :
( ( bit_se2923211474154528505it_int @ N @ ( uminus_uminus_int @ one_one_int ) )
= ( bit_se2000444600071755411sk_int @ N ) ) ).
% take_bit_minus_one_eq_mask
thf(fact_496_and__numerals_I3_J,axiom,
! [X: num,Y3: num] :
( ( bit_se725231765392027082nd_int @ ( numeral_numeral_int @ ( bit0 @ X ) ) @ ( numeral_numeral_int @ ( bit0 @ Y3 ) ) )
= ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( bit_se725231765392027082nd_int @ ( numeral_numeral_int @ X ) @ ( numeral_numeral_int @ Y3 ) ) ) ) ).
% and_numerals(3)
thf(fact_497_and__numerals_I3_J,axiom,
! [X: num,Y3: num] :
( ( bit_se727722235901077358nd_nat @ ( numeral_numeral_nat @ ( bit0 @ X ) ) @ ( numeral_numeral_nat @ ( bit0 @ Y3 ) ) )
= ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( bit_se727722235901077358nd_nat @ ( numeral_numeral_nat @ X ) @ ( numeral_numeral_nat @ Y3 ) ) ) ) ).
% and_numerals(3)
thf(fact_498_and__numerals_I4_J,axiom,
! [X: num,Y3: num] :
( ( bit_se725231765392027082nd_int @ ( numeral_numeral_int @ ( bit0 @ X ) ) @ ( numeral_numeral_int @ ( bit1 @ Y3 ) ) )
= ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( bit_se725231765392027082nd_int @ ( numeral_numeral_int @ X ) @ ( numeral_numeral_int @ Y3 ) ) ) ) ).
% and_numerals(4)
thf(fact_499_and__numerals_I4_J,axiom,
! [X: num,Y3: num] :
( ( bit_se727722235901077358nd_nat @ ( numeral_numeral_nat @ ( bit0 @ X ) ) @ ( numeral_numeral_nat @ ( bit1 @ Y3 ) ) )
= ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( bit_se727722235901077358nd_nat @ ( numeral_numeral_nat @ X ) @ ( numeral_numeral_nat @ Y3 ) ) ) ) ).
% and_numerals(4)
thf(fact_500_and__numerals_I6_J,axiom,
! [X: num,Y3: num] :
( ( bit_se725231765392027082nd_int @ ( numeral_numeral_int @ ( bit1 @ X ) ) @ ( numeral_numeral_int @ ( bit0 @ Y3 ) ) )
= ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( bit_se725231765392027082nd_int @ ( numeral_numeral_int @ X ) @ ( numeral_numeral_int @ Y3 ) ) ) ) ).
% and_numerals(6)
thf(fact_501_and__numerals_I6_J,axiom,
! [X: num,Y3: num] :
( ( bit_se727722235901077358nd_nat @ ( numeral_numeral_nat @ ( bit1 @ X ) ) @ ( numeral_numeral_nat @ ( bit0 @ Y3 ) ) )
= ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( bit_se727722235901077358nd_nat @ ( numeral_numeral_nat @ X ) @ ( numeral_numeral_nat @ Y3 ) ) ) ) ).
% and_numerals(6)
thf(fact_502_signed__take__bit__Suc__bit0,axiom,
! [N: nat,K: num] :
( ( bit_ri631733984087533419it_int @ ( suc @ N ) @ ( numeral_numeral_int @ ( bit0 @ K ) ) )
= ( times_times_int @ ( bit_ri631733984087533419it_int @ N @ ( numeral_numeral_int @ K ) ) @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ).
% signed_take_bit_Suc_bit0
thf(fact_503_or__minus__numerals_I4_J,axiom,
! [M: num,N: num] :
( ( bit_se1409905431419307370or_int @ ( numeral_numeral_int @ M ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit1 @ N ) ) ) )
= ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit_or_not_num_neg @ M @ ( bit0 @ N ) ) ) ) ) ).
% or_minus_numerals(4)
thf(fact_504_or__minus__numerals_I8_J,axiom,
! [N: num,M: num] :
( ( bit_se1409905431419307370or_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit1 @ N ) ) ) @ ( numeral_numeral_int @ M ) )
= ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit_or_not_num_neg @ M @ ( bit0 @ N ) ) ) ) ) ).
% or_minus_numerals(8)
thf(fact_505_or__minus__numerals_I3_J,axiom,
! [M: num,N: num] :
( ( bit_se1409905431419307370or_int @ ( numeral_numeral_int @ M ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit0 @ N ) ) ) )
= ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit_or_not_num_neg @ M @ ( bitM @ N ) ) ) ) ) ).
% or_minus_numerals(3)
thf(fact_506_or__minus__numerals_I7_J,axiom,
! [N: num,M: num] :
( ( bit_se1409905431419307370or_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit0 @ N ) ) ) @ ( numeral_numeral_int @ M ) )
= ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit_or_not_num_neg @ M @ ( bitM @ N ) ) ) ) ) ).
% or_minus_numerals(7)
thf(fact_507_signed__take__bit__Suc__minus__bit0,axiom,
! [N: nat,K: num] :
( ( bit_ri631733984087533419it_int @ ( suc @ N ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit0 @ K ) ) ) )
= ( times_times_int @ ( bit_ri631733984087533419it_int @ N @ ( uminus_uminus_int @ ( numeral_numeral_int @ K ) ) ) @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ).
% signed_take_bit_Suc_minus_bit0
thf(fact_508_take__bit__eq__mask,axiom,
( bit_se2925701944663578781it_nat
= ( ^ [N3: nat,A5: nat] : ( bit_se727722235901077358nd_nat @ A5 @ ( bit_se2002935070580805687sk_nat @ N3 ) ) ) ) ).
% take_bit_eq_mask
thf(fact_509_take__bit__eq__mask,axiom,
( bit_se2923211474154528505it_int
= ( ^ [N3: nat,A5: int] : ( bit_se725231765392027082nd_int @ A5 @ ( bit_se2000444600071755411sk_int @ N3 ) ) ) ) ).
% take_bit_eq_mask
thf(fact_510_and_Oleft__commute,axiom,
! [B: int,A: int,C2: int] :
( ( bit_se725231765392027082nd_int @ B @ ( bit_se725231765392027082nd_int @ A @ C2 ) )
= ( bit_se725231765392027082nd_int @ A @ ( bit_se725231765392027082nd_int @ B @ C2 ) ) ) ).
% and.left_commute
thf(fact_511_and_Oleft__commute,axiom,
! [B: nat,A: nat,C2: nat] :
( ( bit_se727722235901077358nd_nat @ B @ ( bit_se727722235901077358nd_nat @ A @ C2 ) )
= ( bit_se727722235901077358nd_nat @ A @ ( bit_se727722235901077358nd_nat @ B @ C2 ) ) ) ).
% and.left_commute
thf(fact_512_and_Ocommute,axiom,
( bit_se725231765392027082nd_int
= ( ^ [A5: int,B4: int] : ( bit_se725231765392027082nd_int @ B4 @ A5 ) ) ) ).
% and.commute
thf(fact_513_and_Ocommute,axiom,
( bit_se727722235901077358nd_nat
= ( ^ [A5: nat,B4: nat] : ( bit_se727722235901077358nd_nat @ B4 @ A5 ) ) ) ).
% and.commute
thf(fact_514_and_Oassoc,axiom,
! [A: int,B: int,C2: int] :
( ( bit_se725231765392027082nd_int @ ( bit_se725231765392027082nd_int @ A @ B ) @ C2 )
= ( bit_se725231765392027082nd_int @ A @ ( bit_se725231765392027082nd_int @ B @ C2 ) ) ) ).
% and.assoc
thf(fact_515_and_Oassoc,axiom,
! [A: nat,B: nat,C2: nat] :
( ( bit_se727722235901077358nd_nat @ ( bit_se727722235901077358nd_nat @ A @ B ) @ C2 )
= ( bit_se727722235901077358nd_nat @ A @ ( bit_se727722235901077358nd_nat @ B @ C2 ) ) ) ).
% and.assoc
thf(fact_516_signed__take__bit__eq__iff__take__bit__eq,axiom,
! [N: nat,A: int,B: int] :
( ( ( bit_ri631733984087533419it_int @ N @ A )
= ( bit_ri631733984087533419it_int @ N @ B ) )
= ( ( bit_se2923211474154528505it_int @ ( suc @ N ) @ A )
= ( bit_se2923211474154528505it_int @ ( suc @ N ) @ B ) ) ) ).
% signed_take_bit_eq_iff_take_bit_eq
thf(fact_517_signed__take__bit__minus,axiom,
! [N: nat,K: int] :
( ( bit_ri631733984087533419it_int @ N @ ( uminus_uminus_int @ ( bit_ri631733984087533419it_int @ N @ K ) ) )
= ( bit_ri631733984087533419it_int @ N @ ( uminus_uminus_int @ K ) ) ) ).
% signed_take_bit_minus
thf(fact_518_take__bit__minus,axiom,
! [N: nat,K: int] :
( ( bit_se2923211474154528505it_int @ N @ ( uminus_uminus_int @ ( bit_se2923211474154528505it_int @ N @ K ) ) )
= ( bit_se2923211474154528505it_int @ N @ ( uminus_uminus_int @ K ) ) ) ).
% take_bit_minus
thf(fact_519_take__bit__add,axiom,
! [N: nat,A: nat,B: nat] :
( ( bit_se2925701944663578781it_nat @ N @ ( plus_plus_nat @ ( bit_se2925701944663578781it_nat @ N @ A ) @ ( bit_se2925701944663578781it_nat @ N @ B ) ) )
= ( bit_se2925701944663578781it_nat @ N @ ( plus_plus_nat @ A @ B ) ) ) ).
% take_bit_add
thf(fact_520_take__bit__add,axiom,
! [N: nat,A: int,B: int] :
( ( bit_se2923211474154528505it_int @ N @ ( plus_plus_int @ ( bit_se2923211474154528505it_int @ N @ A ) @ ( bit_se2923211474154528505it_int @ N @ B ) ) )
= ( bit_se2923211474154528505it_int @ N @ ( plus_plus_int @ A @ B ) ) ) ).
% take_bit_add
thf(fact_521_bit_Odisj__conj__distrib2,axiom,
! [Y3: int,Z: int,X: int] :
( ( bit_se1409905431419307370or_int @ ( bit_se725231765392027082nd_int @ Y3 @ Z ) @ X )
= ( bit_se725231765392027082nd_int @ ( bit_se1409905431419307370or_int @ Y3 @ X ) @ ( bit_se1409905431419307370or_int @ Z @ X ) ) ) ).
% bit.disj_conj_distrib2
thf(fact_522_bit_Oconj__disj__distrib2,axiom,
! [Y3: int,Z: int,X: int] :
( ( bit_se725231765392027082nd_int @ ( bit_se1409905431419307370or_int @ Y3 @ Z ) @ X )
= ( bit_se1409905431419307370or_int @ ( bit_se725231765392027082nd_int @ Y3 @ X ) @ ( bit_se725231765392027082nd_int @ Z @ X ) ) ) ).
% bit.conj_disj_distrib2
thf(fact_523_bit_Odisj__conj__distrib,axiom,
! [X: int,Y3: int,Z: int] :
( ( bit_se1409905431419307370or_int @ X @ ( bit_se725231765392027082nd_int @ Y3 @ Z ) )
= ( bit_se725231765392027082nd_int @ ( bit_se1409905431419307370or_int @ X @ Y3 ) @ ( bit_se1409905431419307370or_int @ X @ Z ) ) ) ).
% bit.disj_conj_distrib
thf(fact_524_bit_Oconj__disj__distrib,axiom,
! [X: int,Y3: int,Z: int] :
( ( bit_se725231765392027082nd_int @ X @ ( bit_se1409905431419307370or_int @ Y3 @ Z ) )
= ( bit_se1409905431419307370or_int @ ( bit_se725231765392027082nd_int @ X @ Y3 ) @ ( bit_se725231765392027082nd_int @ X @ Z ) ) ) ).
% bit.conj_disj_distrib
thf(fact_525_plus__and__or,axiom,
! [X: int,Y3: int] :
( ( plus_plus_int @ ( bit_se725231765392027082nd_int @ X @ Y3 ) @ ( bit_se1409905431419307370or_int @ X @ Y3 ) )
= ( plus_plus_int @ X @ Y3 ) ) ).
% plus_and_or
thf(fact_526_bit_Oconj__xor__distrib2,axiom,
! [Y3: int,Z: int,X: int] :
( ( bit_se725231765392027082nd_int @ ( bit_se6526347334894502574or_int @ Y3 @ Z ) @ X )
= ( bit_se6526347334894502574or_int @ ( bit_se725231765392027082nd_int @ Y3 @ X ) @ ( bit_se725231765392027082nd_int @ Z @ X ) ) ) ).
% bit.conj_xor_distrib2
thf(fact_527_bit_Oconj__xor__distrib,axiom,
! [X: int,Y3: int,Z: int] :
( ( bit_se725231765392027082nd_int @ X @ ( bit_se6526347334894502574or_int @ Y3 @ Z ) )
= ( bit_se6526347334894502574or_int @ ( bit_se725231765392027082nd_int @ X @ Y3 ) @ ( bit_se725231765392027082nd_int @ X @ Z ) ) ) ).
% bit.conj_xor_distrib
thf(fact_528_take__bit__mult,axiom,
! [N: nat,K: int,L: int] :
( ( bit_se2923211474154528505it_int @ N @ ( times_times_int @ ( bit_se2923211474154528505it_int @ N @ K ) @ ( bit_se2923211474154528505it_int @ N @ L ) ) )
= ( bit_se2923211474154528505it_int @ N @ ( times_times_int @ K @ L ) ) ) ).
% take_bit_mult
thf(fact_529_signed__take__bit__add,axiom,
! [N: nat,K: int,L: int] :
( ( bit_ri631733984087533419it_int @ N @ ( plus_plus_int @ ( bit_ri631733984087533419it_int @ N @ K ) @ ( bit_ri631733984087533419it_int @ N @ L ) ) )
= ( bit_ri631733984087533419it_int @ N @ ( plus_plus_int @ K @ L ) ) ) ).
% signed_take_bit_add
thf(fact_530_signed__take__bit__mult,axiom,
! [N: nat,K: int,L: int] :
( ( bit_ri631733984087533419it_int @ N @ ( times_times_int @ ( bit_ri631733984087533419it_int @ N @ K ) @ ( bit_ri631733984087533419it_int @ N @ L ) ) )
= ( bit_ri631733984087533419it_int @ N @ ( times_times_int @ K @ L ) ) ) ).
% signed_take_bit_mult
thf(fact_531_or__not__num__neg_Osimps_I1_J,axiom,
( ( bit_or_not_num_neg @ one @ one )
= one ) ).
% or_not_num_neg.simps(1)
thf(fact_532_and__eq__minus__1__iff,axiom,
! [A: int,B: int] :
( ( ( bit_se725231765392027082nd_int @ A @ B )
= ( uminus_uminus_int @ one_one_int ) )
= ( ( A
= ( uminus_uminus_int @ one_one_int ) )
& ( B
= ( uminus_uminus_int @ one_one_int ) ) ) ) ).
% and_eq_minus_1_iff
thf(fact_533_or__not__num__neg_Osimps_I4_J,axiom,
! [N: num] :
( ( bit_or_not_num_neg @ ( bit0 @ N ) @ one )
= ( bit0 @ one ) ) ).
% or_not_num_neg.simps(4)
thf(fact_534_or__not__num__neg_Osimps_I6_J,axiom,
! [N: num,M: num] :
( ( bit_or_not_num_neg @ ( bit0 @ N ) @ ( bit1 @ M ) )
= ( bit0 @ ( bit_or_not_num_neg @ N @ M ) ) ) ).
% or_not_num_neg.simps(6)
thf(fact_535_or__not__num__neg_Osimps_I3_J,axiom,
! [M: num] :
( ( bit_or_not_num_neg @ one @ ( bit1 @ M ) )
= ( bit1 @ M ) ) ).
% or_not_num_neg.simps(3)
thf(fact_536_or__not__num__neg_Osimps_I7_J,axiom,
! [N: num] :
( ( bit_or_not_num_neg @ ( bit1 @ N ) @ one )
= one ) ).
% or_not_num_neg.simps(7)
thf(fact_537_or__not__num__neg_Osimps_I5_J,axiom,
! [N: num,M: num] :
( ( bit_or_not_num_neg @ ( bit0 @ N ) @ ( bit0 @ M ) )
= ( bitM @ ( bit_or_not_num_neg @ N @ M ) ) ) ).
% or_not_num_neg.simps(5)
thf(fact_538_or__not__num__neg_Osimps_I9_J,axiom,
! [N: num,M: num] :
( ( bit_or_not_num_neg @ ( bit1 @ N ) @ ( bit1 @ M ) )
= ( bitM @ ( bit_or_not_num_neg @ N @ M ) ) ) ).
% or_not_num_neg.simps(9)
thf(fact_539_or__not__num__neg_Osimps_I2_J,axiom,
! [M: num] :
( ( bit_or_not_num_neg @ one @ ( bit0 @ M ) )
= ( bit1 @ M ) ) ).
% or_not_num_neg.simps(2)
thf(fact_540_or__not__num__neg_Osimps_I8_J,axiom,
! [N: num,M: num] :
( ( bit_or_not_num_neg @ ( bit1 @ N ) @ ( bit0 @ M ) )
= ( bitM @ ( bit_or_not_num_neg @ N @ M ) ) ) ).
% or_not_num_neg.simps(8)
thf(fact_541_take__bit__Suc__bit0,axiom,
! [N: nat,K: num] :
( ( bit_se2925701944663578781it_nat @ ( suc @ N ) @ ( numeral_numeral_nat @ ( bit0 @ K ) ) )
= ( times_times_nat @ ( bit_se2925701944663578781it_nat @ N @ ( numeral_numeral_nat @ K ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ).
% take_bit_Suc_bit0
thf(fact_542_take__bit__Suc__bit0,axiom,
! [N: nat,K: num] :
( ( bit_se2923211474154528505it_int @ ( suc @ N ) @ ( numeral_numeral_int @ ( bit0 @ K ) ) )
= ( times_times_int @ ( bit_se2923211474154528505it_int @ N @ ( numeral_numeral_int @ K ) ) @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ).
% take_bit_Suc_bit0
thf(fact_543_take__bit__Suc__minus__bit0,axiom,
! [N: nat,K: num] :
( ( bit_se2923211474154528505it_int @ ( suc @ N ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit0 @ K ) ) ) )
= ( times_times_int @ ( bit_se2923211474154528505it_int @ N @ ( uminus_uminus_int @ ( numeral_numeral_int @ K ) ) ) @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ).
% take_bit_Suc_minus_bit0
thf(fact_544_signed__take__bit__Suc__minus__bit1,axiom,
! [N: nat,K: num] :
( ( bit_ri631733984087533419it_int @ ( suc @ N ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit1 @ K ) ) ) )
= ( plus_plus_int @ ( times_times_int @ ( bit_ri631733984087533419it_int @ N @ ( minus_minus_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ K ) ) @ one_one_int ) ) @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) @ one_one_int ) ) ).
% signed_take_bit_Suc_minus_bit1
thf(fact_545_signed__take__bit__numeral__bit1,axiom,
! [L: num,K: num] :
( ( bit_ri631733984087533419it_int @ ( numeral_numeral_nat @ L ) @ ( numeral_numeral_int @ ( bit1 @ K ) ) )
= ( plus_plus_int @ ( times_times_int @ ( bit_ri631733984087533419it_int @ ( pred_numeral @ L ) @ ( numeral_numeral_int @ K ) ) @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) @ one_one_int ) ) ).
% signed_take_bit_numeral_bit1
thf(fact_546_take__bit__Suc__minus__bit1,axiom,
! [N: nat,K: num] :
( ( bit_se2923211474154528505it_int @ ( suc @ N ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit1 @ K ) ) ) )
= ( plus_plus_int @ ( times_times_int @ ( bit_se2923211474154528505it_int @ N @ ( uminus_uminus_int @ ( numeral_numeral_int @ ( inc @ K ) ) ) ) @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) @ one_one_int ) ) ).
% take_bit_Suc_minus_bit1
thf(fact_547_take__bit__numeral__bit1,axiom,
! [L: num,K: num] :
( ( bit_se2925701944663578781it_nat @ ( numeral_numeral_nat @ L ) @ ( numeral_numeral_nat @ ( bit1 @ K ) ) )
= ( plus_plus_nat @ ( times_times_nat @ ( bit_se2925701944663578781it_nat @ ( pred_numeral @ L ) @ ( numeral_numeral_nat @ K ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ one_one_nat ) ) ).
% take_bit_numeral_bit1
thf(fact_548_take__bit__numeral__bit1,axiom,
! [L: num,K: num] :
( ( bit_se2923211474154528505it_int @ ( numeral_numeral_nat @ L ) @ ( numeral_numeral_int @ ( bit1 @ K ) ) )
= ( plus_plus_int @ ( times_times_int @ ( bit_se2923211474154528505it_int @ ( pred_numeral @ L ) @ ( numeral_numeral_int @ K ) ) @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) @ one_one_int ) ) ).
% take_bit_numeral_bit1
thf(fact_549_signed__take__bit__numeral__minus__bit0,axiom,
! [L: num,K: num] :
( ( bit_ri631733984087533419it_int @ ( numeral_numeral_nat @ L ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit0 @ K ) ) ) )
= ( times_times_int @ ( bit_ri631733984087533419it_int @ ( pred_numeral @ L ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ K ) ) ) @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ).
% signed_take_bit_numeral_minus_bit0
thf(fact_550_mask__numeral,axiom,
! [N: num] :
( ( bit_se2002935070580805687sk_nat @ ( numeral_numeral_nat @ N ) )
= ( plus_plus_nat @ one_one_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( bit_se2002935070580805687sk_nat @ ( pred_numeral @ N ) ) ) ) ) ).
% mask_numeral
thf(fact_551_mask__numeral,axiom,
! [N: num] :
( ( bit_se2000444600071755411sk_int @ ( numeral_numeral_nat @ N ) )
= ( plus_plus_int @ one_one_int @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( bit_se2000444600071755411sk_int @ ( pred_numeral @ N ) ) ) ) ) ).
% mask_numeral
thf(fact_552_add__diff__cancel,axiom,
! [A: int,B: int] :
( ( minus_minus_int @ ( plus_plus_int @ A @ B ) @ B )
= A ) ).
% add_diff_cancel
thf(fact_553_diff__add__cancel,axiom,
! [A: int,B: int] :
( ( plus_plus_int @ ( minus_minus_int @ A @ B ) @ B )
= A ) ).
% diff_add_cancel
thf(fact_554_add__diff__cancel__left,axiom,
! [C2: int,A: int,B: int] :
( ( minus_minus_int @ ( plus_plus_int @ C2 @ A ) @ ( plus_plus_int @ C2 @ B ) )
= ( minus_minus_int @ A @ B ) ) ).
% add_diff_cancel_left
thf(fact_555_add__diff__cancel__left,axiom,
! [C2: nat,A: nat,B: nat] :
( ( minus_minus_nat @ ( plus_plus_nat @ C2 @ A ) @ ( plus_plus_nat @ C2 @ B ) )
= ( minus_minus_nat @ A @ B ) ) ).
% add_diff_cancel_left
thf(fact_556_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_557_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_558_add__diff__cancel__right,axiom,
! [A: int,C2: int,B: int] :
( ( minus_minus_int @ ( plus_plus_int @ A @ C2 ) @ ( plus_plus_int @ B @ C2 ) )
= ( minus_minus_int @ A @ B ) ) ).
% add_diff_cancel_right
thf(fact_559_add__diff__cancel__right,axiom,
! [A: nat,C2: nat,B: nat] :
( ( minus_minus_nat @ ( plus_plus_nat @ A @ C2 ) @ ( plus_plus_nat @ B @ C2 ) )
= ( minus_minus_nat @ A @ B ) ) ).
% add_diff_cancel_right
thf(fact_560_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_561_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_562_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_563_left__diff__distrib__numeral,axiom,
! [A: int,B: int,V2: num] :
( ( times_times_int @ ( minus_minus_int @ A @ B ) @ ( numeral_numeral_int @ V2 ) )
= ( minus_minus_int @ ( times_times_int @ A @ ( numeral_numeral_int @ V2 ) ) @ ( times_times_int @ B @ ( numeral_numeral_int @ V2 ) ) ) ) ).
% left_diff_distrib_numeral
thf(fact_564_right__diff__distrib__numeral,axiom,
! [V2: num,B: int,C2: int] :
( ( times_times_int @ ( numeral_numeral_int @ V2 ) @ ( minus_minus_int @ B @ C2 ) )
= ( minus_minus_int @ ( times_times_int @ ( numeral_numeral_int @ V2 ) @ B ) @ ( times_times_int @ ( numeral_numeral_int @ V2 ) @ C2 ) ) ) ).
% right_diff_distrib_numeral
thf(fact_565_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_566_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_567_diff__numeral__simps_I1_J,axiom,
! [M: num,N: num] :
( ( minus_minus_int @ ( numeral_numeral_int @ M ) @ ( numeral_numeral_int @ N ) )
= ( neg_numeral_sub_int @ M @ N ) ) ).
% diff_numeral_simps(1)
thf(fact_568_eq__numeral__Suc,axiom,
! [K: num,N: nat] :
( ( ( numeral_numeral_nat @ K )
= ( suc @ N ) )
= ( ( pred_numeral @ K )
= N ) ) ).
% eq_numeral_Suc
thf(fact_569_Suc__eq__numeral,axiom,
! [N: nat,K: num] :
( ( ( suc @ N )
= ( numeral_numeral_nat @ K ) )
= ( N
= ( pred_numeral @ K ) ) ) ).
% Suc_eq_numeral
thf(fact_570_pred__numeral__inc,axiom,
! [K: num] :
( ( pred_numeral @ ( inc @ K ) )
= ( numeral_numeral_nat @ K ) ) ).
% pred_numeral_inc
thf(fact_571_diff__numeral__simps_I3_J,axiom,
! [M: num,N: num] :
( ( minus_minus_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ M ) ) @ ( numeral_numeral_int @ N ) )
= ( uminus_uminus_int @ ( numeral_numeral_int @ ( plus_plus_num @ M @ N ) ) ) ) ).
% diff_numeral_simps(3)
thf(fact_572_diff__numeral__simps_I2_J,axiom,
! [M: num,N: num] :
( ( minus_minus_int @ ( numeral_numeral_int @ M ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ N ) ) )
= ( numeral_numeral_int @ ( plus_plus_num @ M @ N ) ) ) ).
% diff_numeral_simps(2)
thf(fact_573_diff__numeral__simps_I4_J,axiom,
! [M: num,N: num] :
( ( minus_minus_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ M ) ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ N ) ) )
= ( neg_numeral_sub_int @ N @ M ) ) ).
% diff_numeral_simps(4)
thf(fact_574_pred__numeral__simps_I3_J,axiom,
! [K: num] :
( ( pred_numeral @ ( bit1 @ K ) )
= ( numeral_numeral_nat @ ( bit0 @ K ) ) ) ).
% pred_numeral_simps(3)
thf(fact_575_pred__numeral__simps_I2_J,axiom,
! [K: num] :
( ( pred_numeral @ ( bit0 @ K ) )
= ( numeral_numeral_nat @ ( bitM @ K ) ) ) ).
% pred_numeral_simps(2)
thf(fact_576_diff__numeral__special_I2_J,axiom,
! [M: num] :
( ( minus_minus_int @ ( numeral_numeral_int @ M ) @ one_one_int )
= ( neg_numeral_sub_int @ M @ one ) ) ).
% diff_numeral_special(2)
thf(fact_577_diff__numeral__special_I1_J,axiom,
! [N: num] :
( ( minus_minus_int @ one_one_int @ ( numeral_numeral_int @ N ) )
= ( neg_numeral_sub_int @ one @ N ) ) ).
% diff_numeral_special(1)
thf(fact_578_add__neg__numeral__special_I6_J,axiom,
! [M: num] :
( ( plus_plus_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ M ) ) @ ( uminus_uminus_int @ one_one_int ) )
= ( uminus_uminus_int @ ( numeral_numeral_int @ ( inc @ M ) ) ) ) ).
% add_neg_numeral_special(6)
thf(fact_579_add__neg__numeral__special_I5_J,axiom,
! [N: num] :
( ( plus_plus_int @ ( uminus_uminus_int @ one_one_int ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ N ) ) )
= ( uminus_uminus_int @ ( numeral_numeral_int @ ( inc @ N ) ) ) ) ).
% add_neg_numeral_special(5)
thf(fact_580_diff__numeral__special_I6_J,axiom,
! [M: num] :
( ( minus_minus_int @ ( numeral_numeral_int @ M ) @ ( uminus_uminus_int @ one_one_int ) )
= ( numeral_numeral_int @ ( inc @ M ) ) ) ).
% diff_numeral_special(6)
thf(fact_581_diff__numeral__special_I5_J,axiom,
! [N: num] :
( ( minus_minus_int @ ( uminus_uminus_int @ one_one_int ) @ ( numeral_numeral_int @ N ) )
= ( uminus_uminus_int @ ( numeral_numeral_int @ ( inc @ N ) ) ) ) ).
% diff_numeral_special(5)
thf(fact_582_diff__numeral__special_I11_J,axiom,
( ( minus_minus_int @ one_one_int @ ( uminus_uminus_int @ one_one_int ) )
= ( numeral_numeral_int @ ( bit0 @ one ) ) ) ).
% diff_numeral_special(11)
thf(fact_583_diff__numeral__special_I10_J,axiom,
( ( minus_minus_int @ ( uminus_uminus_int @ one_one_int ) @ one_one_int )
= ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ).
% diff_numeral_special(10)
thf(fact_584_diff__numeral__special_I4_J,axiom,
! [M: num] :
( ( minus_minus_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ M ) ) @ one_one_int )
= ( uminus_uminus_int @ ( numeral_numeral_int @ ( plus_plus_num @ M @ one ) ) ) ) ).
% diff_numeral_special(4)
thf(fact_585_diff__numeral__special_I3_J,axiom,
! [N: num] :
( ( minus_minus_int @ one_one_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ N ) ) )
= ( numeral_numeral_int @ ( plus_plus_num @ one @ N ) ) ) ).
% diff_numeral_special(3)
thf(fact_586_diff__numeral__special_I8_J,axiom,
! [M: num] :
( ( minus_minus_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ M ) ) @ ( uminus_uminus_int @ one_one_int ) )
= ( neg_numeral_sub_int @ one @ M ) ) ).
% diff_numeral_special(8)
thf(fact_587_diff__numeral__special_I7_J,axiom,
! [N: num] :
( ( minus_minus_int @ ( uminus_uminus_int @ one_one_int ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ N ) ) )
= ( neg_numeral_sub_int @ N @ one ) ) ).
% diff_numeral_special(7)
thf(fact_588_minus__sub__one__diff__one,axiom,
! [M: num] :
( ( minus_minus_int @ ( uminus_uminus_int @ ( neg_numeral_sub_int @ M @ one ) ) @ one_one_int )
= ( uminus_uminus_int @ ( numeral_numeral_int @ M ) ) ) ).
% minus_sub_one_diff_one
thf(fact_589_signed__take__bit__numeral__bit0,axiom,
! [L: num,K: num] :
( ( bit_ri631733984087533419it_int @ ( numeral_numeral_nat @ L ) @ ( numeral_numeral_int @ ( bit0 @ K ) ) )
= ( times_times_int @ ( bit_ri631733984087533419it_int @ ( pred_numeral @ L ) @ ( numeral_numeral_int @ K ) ) @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ).
% signed_take_bit_numeral_bit0
thf(fact_590_signed__take__bit__numeral__minus__bit1,axiom,
! [L: num,K: num] :
( ( bit_ri631733984087533419it_int @ ( numeral_numeral_nat @ L ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit1 @ K ) ) ) )
= ( plus_plus_int @ ( times_times_int @ ( bit_ri631733984087533419it_int @ ( pred_numeral @ L ) @ ( minus_minus_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ K ) ) @ one_one_int ) ) @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) @ one_one_int ) ) ).
% signed_take_bit_numeral_minus_bit1
thf(fact_591_take__bit__diff,axiom,
! [N: nat,K: int,L: int] :
( ( bit_se2923211474154528505it_int @ N @ ( minus_minus_int @ ( bit_se2923211474154528505it_int @ N @ K ) @ ( bit_se2923211474154528505it_int @ N @ L ) ) )
= ( bit_se2923211474154528505it_int @ N @ ( minus_minus_int @ K @ L ) ) ) ).
% take_bit_diff
thf(fact_592_diff__eq__diff__eq,axiom,
! [A: int,B: int,C2: int,D: int] :
( ( ( minus_minus_int @ A @ B )
= ( minus_minus_int @ C2 @ D ) )
=> ( ( A = B )
= ( C2 = D ) ) ) ).
% diff_eq_diff_eq
thf(fact_593_diff__right__commute,axiom,
! [A: int,C2: int,B: int] :
( ( minus_minus_int @ ( minus_minus_int @ A @ C2 ) @ B )
= ( minus_minus_int @ ( minus_minus_int @ A @ B ) @ C2 ) ) ).
% diff_right_commute
thf(fact_594_diff__right__commute,axiom,
! [A: nat,C2: nat,B: nat] :
( ( minus_minus_nat @ ( minus_minus_nat @ A @ C2 ) @ B )
= ( minus_minus_nat @ ( minus_minus_nat @ A @ B ) @ C2 ) ) ).
% diff_right_commute
thf(fact_595_left__diff__distrib,axiom,
! [A: int,B: int,C2: int] :
( ( times_times_int @ ( minus_minus_int @ A @ B ) @ C2 )
= ( minus_minus_int @ ( times_times_int @ A @ C2 ) @ ( times_times_int @ B @ C2 ) ) ) ).
% left_diff_distrib
thf(fact_596_right__diff__distrib,axiom,
! [A: int,B: int,C2: int] :
( ( times_times_int @ A @ ( minus_minus_int @ B @ C2 ) )
= ( minus_minus_int @ ( times_times_int @ A @ B ) @ ( times_times_int @ A @ C2 ) ) ) ).
% right_diff_distrib
thf(fact_597_left__diff__distrib_H,axiom,
! [B: nat,C2: nat,A: nat] :
( ( times_times_nat @ ( minus_minus_nat @ B @ C2 ) @ A )
= ( minus_minus_nat @ ( times_times_nat @ B @ A ) @ ( times_times_nat @ C2 @ A ) ) ) ).
% left_diff_distrib'
thf(fact_598_left__diff__distrib_H,axiom,
! [B: int,C2: int,A: int] :
( ( times_times_int @ ( minus_minus_int @ B @ C2 ) @ A )
= ( minus_minus_int @ ( times_times_int @ B @ A ) @ ( times_times_int @ C2 @ A ) ) ) ).
% left_diff_distrib'
thf(fact_599_right__diff__distrib_H,axiom,
! [A: nat,B: nat,C2: nat] :
( ( times_times_nat @ A @ ( minus_minus_nat @ B @ C2 ) )
= ( minus_minus_nat @ ( times_times_nat @ A @ B ) @ ( times_times_nat @ A @ C2 ) ) ) ).
% right_diff_distrib'
thf(fact_600_right__diff__distrib_H,axiom,
! [A: int,B: int,C2: int] :
( ( times_times_int @ A @ ( minus_minus_int @ B @ C2 ) )
= ( minus_minus_int @ ( times_times_int @ A @ B ) @ ( times_times_int @ A @ C2 ) ) ) ).
% right_diff_distrib'
thf(fact_601_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_602_diff__eq__eq,axiom,
! [A: int,B: int,C2: int] :
( ( ( minus_minus_int @ A @ B )
= C2 )
= ( A
= ( plus_plus_int @ C2 @ B ) ) ) ).
% diff_eq_eq
thf(fact_603_eq__diff__eq,axiom,
! [A: int,C2: int,B: int] :
( ( A
= ( minus_minus_int @ C2 @ B ) )
= ( ( plus_plus_int @ A @ B )
= C2 ) ) ).
% eq_diff_eq
thf(fact_604_add__diff__eq,axiom,
! [A: int,B: int,C2: int] :
( ( plus_plus_int @ A @ ( minus_minus_int @ B @ C2 ) )
= ( minus_minus_int @ ( plus_plus_int @ A @ B ) @ C2 ) ) ).
% add_diff_eq
thf(fact_605_diff__diff__eq2,axiom,
! [A: int,B: int,C2: int] :
( ( minus_minus_int @ A @ ( minus_minus_int @ B @ C2 ) )
= ( minus_minus_int @ ( plus_plus_int @ A @ C2 ) @ B ) ) ).
% diff_diff_eq2
thf(fact_606_diff__add__eq,axiom,
! [A: int,B: int,C2: int] :
( ( plus_plus_int @ ( minus_minus_int @ A @ B ) @ C2 )
= ( minus_minus_int @ ( plus_plus_int @ A @ C2 ) @ B ) ) ).
% diff_add_eq
thf(fact_607_diff__add__eq__diff__diff__swap,axiom,
! [A: int,B: int,C2: int] :
( ( minus_minus_int @ A @ ( plus_plus_int @ B @ C2 ) )
= ( minus_minus_int @ ( minus_minus_int @ A @ C2 ) @ B ) ) ).
% diff_add_eq_diff_diff_swap
thf(fact_608_add__implies__diff,axiom,
! [C2: int,B: int,A: int] :
( ( ( plus_plus_int @ C2 @ B )
= A )
=> ( C2
= ( minus_minus_int @ A @ B ) ) ) ).
% add_implies_diff
thf(fact_609_add__implies__diff,axiom,
! [C2: nat,B: nat,A: nat] :
( ( ( plus_plus_nat @ C2 @ B )
= A )
=> ( C2
= ( minus_minus_nat @ A @ B ) ) ) ).
% add_implies_diff
thf(fact_610_diff__diff__eq,axiom,
! [A: int,B: int,C2: int] :
( ( minus_minus_int @ ( minus_minus_int @ A @ B ) @ C2 )
= ( minus_minus_int @ A @ ( plus_plus_int @ B @ C2 ) ) ) ).
% diff_diff_eq
thf(fact_611_diff__diff__eq,axiom,
! [A: nat,B: nat,C2: nat] :
( ( minus_minus_nat @ ( minus_minus_nat @ A @ B ) @ C2 )
= ( minus_minus_nat @ A @ ( plus_plus_nat @ B @ C2 ) ) ) ).
% diff_diff_eq
thf(fact_612_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_613_int__distrib_I4_J,axiom,
! [W: int,Z1: int,Z2: int] :
( ( times_times_int @ W @ ( minus_minus_int @ Z1 @ Z2 ) )
= ( minus_minus_int @ ( times_times_int @ W @ Z1 ) @ ( times_times_int @ W @ Z2 ) ) ) ).
% int_distrib(4)
thf(fact_614_int__distrib_I3_J,axiom,
! [Z1: int,Z2: int,W: int] :
( ( times_times_int @ ( minus_minus_int @ Z1 @ Z2 ) @ W )
= ( minus_minus_int @ ( times_times_int @ Z1 @ W ) @ ( times_times_int @ Z2 @ W ) ) ) ).
% int_distrib(3)
thf(fact_615_signed__take__bit__diff,axiom,
! [N: nat,K: int,L: int] :
( ( bit_ri631733984087533419it_int @ N @ ( minus_minus_int @ ( bit_ri631733984087533419it_int @ N @ K ) @ ( bit_ri631733984087533419it_int @ N @ L ) ) )
= ( bit_ri631733984087533419it_int @ N @ ( minus_minus_int @ K @ L ) ) ) ).
% signed_take_bit_diff
thf(fact_616_num__induct,axiom,
! [P: num > $o,X: num] :
( ( P @ one )
=> ( ! [X2: num] :
( ( P @ X2 )
=> ( P @ ( inc @ X2 ) ) )
=> ( P @ X ) ) ) ).
% num_induct
thf(fact_617_add__inc,axiom,
! [X: num,Y3: num] :
( ( plus_plus_num @ X @ ( inc @ Y3 ) )
= ( inc @ ( plus_plus_num @ X @ Y3 ) ) ) ).
% add_inc
thf(fact_618_square__diff__square__factored,axiom,
! [X: int,Y3: int] :
( ( minus_minus_int @ ( times_times_int @ X @ X ) @ ( times_times_int @ Y3 @ Y3 ) )
= ( times_times_int @ ( plus_plus_int @ X @ Y3 ) @ ( minus_minus_int @ X @ Y3 ) ) ) ).
% square_diff_square_factored
thf(fact_619_eq__add__iff2,axiom,
! [A: int,E: int,C2: int,B: int,D: int] :
( ( ( plus_plus_int @ ( times_times_int @ A @ E ) @ C2 )
= ( plus_plus_int @ ( times_times_int @ B @ E ) @ D ) )
= ( C2
= ( plus_plus_int @ ( times_times_int @ ( minus_minus_int @ B @ A ) @ E ) @ D ) ) ) ).
% eq_add_iff2
thf(fact_620_eq__add__iff1,axiom,
! [A: int,E: int,C2: int,B: int,D: int] :
( ( ( plus_plus_int @ ( times_times_int @ A @ E ) @ C2 )
= ( plus_plus_int @ ( times_times_int @ B @ E ) @ D ) )
= ( ( plus_plus_int @ ( times_times_int @ ( minus_minus_int @ A @ B ) @ E ) @ C2 )
= D ) ) ).
% eq_add_iff1
thf(fact_621_ab__group__add__class_Oab__diff__conv__add__uminus,axiom,
( minus_minus_int
= ( ^ [A5: int,B4: int] : ( plus_plus_int @ A5 @ ( uminus_uminus_int @ B4 ) ) ) ) ).
% ab_group_add_class.ab_diff_conv_add_uminus
thf(fact_622_diff__conv__add__uminus,axiom,
( minus_minus_int
= ( ^ [A5: int,B4: int] : ( plus_plus_int @ A5 @ ( uminus_uminus_int @ B4 ) ) ) ) ).
% diff_conv_add_uminus
thf(fact_623_group__cancel_Osub2,axiom,
! [B5: int,K: int,B: int,A: int] :
( ( B5
= ( plus_plus_int @ K @ B ) )
=> ( ( minus_minus_int @ A @ B5 )
= ( plus_plus_int @ ( uminus_uminus_int @ K ) @ ( minus_minus_int @ A @ B ) ) ) ) ).
% group_cancel.sub2
thf(fact_624_neg__numeral__class_Osub__def,axiom,
( neg_numeral_sub_int
= ( ^ [K3: num,L2: num] : ( minus_minus_int @ ( numeral_numeral_int @ K3 ) @ ( numeral_numeral_int @ L2 ) ) ) ) ).
% neg_numeral_class.sub_def
thf(fact_625_numeral__eq__Suc,axiom,
( numeral_numeral_nat
= ( ^ [K3: num] : ( suc @ ( pred_numeral @ K3 ) ) ) ) ).
% numeral_eq_Suc
thf(fact_626_inc_Osimps_I1_J,axiom,
( ( inc @ one )
= ( bit0 @ one ) ) ).
% inc.simps(1)
thf(fact_627_inc_Osimps_I3_J,axiom,
! [X: num] :
( ( inc @ ( bit1 @ X ) )
= ( bit0 @ ( inc @ X ) ) ) ).
% inc.simps(3)
thf(fact_628_inc_Osimps_I2_J,axiom,
! [X: num] :
( ( inc @ ( bit0 @ X ) )
= ( bit1 @ X ) ) ).
% inc.simps(2)
thf(fact_629_add__One,axiom,
! [X: num] :
( ( plus_plus_num @ X @ one )
= ( inc @ X ) ) ).
% add_One
thf(fact_630_inc__BitM__eq,axiom,
! [N: num] :
( ( inc @ ( bitM @ N ) )
= ( bit0 @ N ) ) ).
% inc_BitM_eq
thf(fact_631_BitM__inc__eq,axiom,
! [N: num] :
( ( bitM @ ( inc @ N ) )
= ( bit1 @ N ) ) ).
% BitM_inc_eq
thf(fact_632_mult__inc,axiom,
! [X: num,Y3: num] :
( ( times_times_num @ X @ ( inc @ Y3 ) )
= ( plus_plus_num @ ( times_times_num @ X @ Y3 ) @ X ) ) ).
% mult_inc
thf(fact_633_square__diff__one__factored,axiom,
! [X: int] :
( ( minus_minus_int @ ( times_times_int @ X @ X ) @ one_one_int )
= ( times_times_int @ ( plus_plus_int @ X @ one_one_int ) @ ( minus_minus_int @ X @ one_one_int ) ) ) ).
% square_diff_one_factored
thf(fact_634_dbl__dec__def,axiom,
( neg_nu3811975205180677377ec_int
= ( ^ [X4: int] : ( minus_minus_int @ ( plus_plus_int @ X4 @ X4 ) @ one_one_int ) ) ) ).
% dbl_dec_def
thf(fact_635_numeral__inc,axiom,
! [X: num] :
( ( numeral_numeral_nat @ ( inc @ X ) )
= ( plus_plus_nat @ ( numeral_numeral_nat @ X ) @ one_one_nat ) ) ).
% numeral_inc
thf(fact_636_numeral__inc,axiom,
! [X: num] :
( ( numeral_numeral_int @ ( inc @ X ) )
= ( plus_plus_int @ ( numeral_numeral_int @ X ) @ one_one_int ) ) ).
% numeral_inc
thf(fact_637_sub__inc__One__eq,axiom,
! [N: num] :
( ( neg_numeral_sub_int @ ( inc @ N ) @ one )
= ( numeral_numeral_int @ N ) ) ).
% sub_inc_One_eq
thf(fact_638_numeral__BitM,axiom,
! [N: num] :
( ( numeral_numeral_int @ ( bitM @ N ) )
= ( minus_minus_int @ ( numeral_numeral_int @ ( bit0 @ N ) ) @ one_one_int ) ) ).
% numeral_BitM
thf(fact_639_take__bit__numeral__minus__bit1,axiom,
! [L: num,K: num] :
( ( bit_se2923211474154528505it_int @ ( numeral_numeral_nat @ L ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit1 @ K ) ) ) )
= ( plus_plus_int @ ( times_times_int @ ( bit_se2923211474154528505it_int @ ( pred_numeral @ L ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ ( inc @ K ) ) ) ) @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) @ one_one_int ) ) ).
% take_bit_numeral_minus_bit1
thf(fact_640_take__bit__numeral__bit0,axiom,
! [L: num,K: num] :
( ( bit_se2925701944663578781it_nat @ ( numeral_numeral_nat @ L ) @ ( numeral_numeral_nat @ ( bit0 @ K ) ) )
= ( times_times_nat @ ( bit_se2925701944663578781it_nat @ ( pred_numeral @ L ) @ ( numeral_numeral_nat @ K ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ).
% take_bit_numeral_bit0
thf(fact_641_take__bit__numeral__bit0,axiom,
! [L: num,K: num] :
( ( bit_se2923211474154528505it_int @ ( numeral_numeral_nat @ L ) @ ( numeral_numeral_int @ ( bit0 @ K ) ) )
= ( times_times_int @ ( bit_se2923211474154528505it_int @ ( pred_numeral @ L ) @ ( numeral_numeral_int @ K ) ) @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ).
% take_bit_numeral_bit0
thf(fact_642_take__bit__numeral__minus__bit0,axiom,
! [L: num,K: num] :
( ( bit_se2923211474154528505it_int @ ( numeral_numeral_nat @ L ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit0 @ K ) ) ) )
= ( times_times_int @ ( bit_se2923211474154528505it_int @ ( pred_numeral @ L ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ K ) ) ) @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ).
% take_bit_numeral_minus_bit0
thf(fact_643_divmod__BitM__2__eq,axiom,
! [M: num] :
( ( unique5403075989570733136od_int @ ( bitM @ M ) @ ( bit0 @ one ) )
= ( product_Pair_int_int @ ( minus_minus_int @ ( numeral_numeral_int @ M ) @ one_one_int ) @ one_one_int ) ) ).
% divmod_BitM_2_eq
thf(fact_644_step__checker_Osimps,axiom,
! [I3: int,J3: int,I4: int,J4: int] :
( ( step_checker @ ( product_Pair_int_int @ I3 @ J3 ) @ ( product_Pair_int_int @ I4 @ J4 ) )
= ( ( ( product_Pair_int_int @ ( plus_plus_int @ I3 @ one_one_int ) @ ( plus_plus_int @ J3 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) )
= ( product_Pair_int_int @ I4 @ J4 ) )
| ( ( product_Pair_int_int @ ( minus_minus_int @ I3 @ one_one_int ) @ ( plus_plus_int @ J3 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) )
= ( product_Pair_int_int @ I4 @ J4 ) )
| ( ( product_Pair_int_int @ ( plus_plus_int @ I3 @ one_one_int ) @ ( minus_minus_int @ J3 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) )
= ( product_Pair_int_int @ I4 @ J4 ) )
| ( ( product_Pair_int_int @ ( minus_minus_int @ I3 @ one_one_int ) @ ( minus_minus_int @ J3 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) )
= ( product_Pair_int_int @ I4 @ J4 ) )
| ( ( product_Pair_int_int @ ( plus_plus_int @ I3 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) @ ( plus_plus_int @ J3 @ one_one_int ) )
= ( product_Pair_int_int @ I4 @ J4 ) )
| ( ( product_Pair_int_int @ ( minus_minus_int @ I3 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) @ ( plus_plus_int @ J3 @ one_one_int ) )
= ( product_Pair_int_int @ I4 @ J4 ) )
| ( ( product_Pair_int_int @ ( plus_plus_int @ I3 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) @ ( minus_minus_int @ J3 @ one_one_int ) )
= ( product_Pair_int_int @ I4 @ J4 ) )
| ( ( product_Pair_int_int @ ( minus_minus_int @ I3 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) @ ( minus_minus_int @ J3 @ one_one_int ) )
= ( product_Pair_int_int @ I4 @ J4 ) ) ) ) ).
% step_checker.simps
thf(fact_645_last__mirror2,axiom,
! [N: nat,M: nat,Ps2: list_P5707943133018811711nt_int,I3: int,J3: int] :
( ( knights_path @ ( board @ N @ M ) @ Ps2 )
=> ( ( ( last_P3305686521732843992nt_int @ Ps2 )
= ( product_Pair_int_int @ I3 @ J3 ) )
=> ( ( last_P3305686521732843992nt_int @ ( mirror2 @ Ps2 ) )
= ( product_Pair_int_int @ I3 @ ( minus_minus_int @ ( plus_plus_int @ ( semiri1314217659103216013at_int @ M ) @ one_one_int ) @ J3 ) ) ) ) ) ).
% last_mirror2
thf(fact_646_signed__take__bit__eq__take__bit__shift,axiom,
( bit_ri631733984087533419it_int
= ( ^ [N3: nat,K3: int] : ( minus_minus_int @ ( bit_se2923211474154528505it_int @ ( suc @ N3 ) @ ( plus_plus_int @ K3 @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N3 ) ) ) @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N3 ) ) ) ) ).
% signed_take_bit_eq_take_bit_shift
thf(fact_647_take__bit__numeral__minus__1__eq,axiom,
! [K: num] :
( ( bit_se2923211474154528505it_int @ ( numeral_numeral_nat @ K ) @ ( uminus_uminus_int @ one_one_int ) )
= ( minus_minus_int @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( numeral_numeral_nat @ K ) ) @ one_one_int ) ) ).
% take_bit_numeral_minus_1_eq
thf(fact_648_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_649_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_650_of__nat__eq__iff,axiom,
! [M: nat,N: nat] :
( ( ( semiri1314217659103216013at_int @ M )
= ( semiri1314217659103216013at_int @ N ) )
= ( M = N ) ) ).
% of_nat_eq_iff
thf(fact_651_diff__diff__left,axiom,
! [I3: nat,J3: nat,K: nat] :
( ( minus_minus_nat @ ( minus_minus_nat @ I3 @ J3 ) @ K )
= ( minus_minus_nat @ I3 @ ( plus_plus_nat @ J3 @ K ) ) ) ).
% diff_diff_left
thf(fact_652_diff__Suc__1,axiom,
! [N: nat] :
( ( minus_minus_nat @ ( suc @ N ) @ one_one_nat )
= N ) ).
% diff_Suc_1
thf(fact_653_of__nat__numeral,axiom,
! [N: num] :
( ( semiri1316708129612266289at_nat @ ( numeral_numeral_nat @ N ) )
= ( numeral_numeral_nat @ N ) ) ).
% of_nat_numeral
thf(fact_654_of__nat__numeral,axiom,
! [N: num] :
( ( semiri1314217659103216013at_int @ ( numeral_numeral_nat @ N ) )
= ( numeral_numeral_int @ N ) ) ).
% of_nat_numeral
thf(fact_655_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_656_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_657_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_658_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_659_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_660_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_661_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_662_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_663_of__nat__1,axiom,
( ( semiri1316708129612266289at_nat @ one_one_nat )
= one_one_nat ) ).
% of_nat_1
thf(fact_664_of__nat__1,axiom,
( ( semiri1314217659103216013at_int @ one_one_nat )
= one_one_int ) ).
% of_nat_1
thf(fact_665_int__eq__iff__numeral,axiom,
! [M: nat,V2: num] :
( ( ( semiri1314217659103216013at_int @ M )
= ( numeral_numeral_int @ V2 ) )
= ( M
= ( numeral_numeral_nat @ V2 ) ) ) ).
% int_eq_iff_numeral
thf(fact_666_diff__numeral__Suc,axiom,
! [K: num,N: nat] :
( ( minus_minus_nat @ ( numeral_numeral_nat @ K ) @ ( suc @ N ) )
= ( minus_minus_nat @ ( pred_numeral @ K ) @ N ) ) ).
% diff_numeral_Suc
thf(fact_667_diff__Suc__numeral,axiom,
! [N: nat,K: num] :
( ( minus_minus_nat @ ( suc @ N ) @ ( numeral_numeral_nat @ K ) )
= ( minus_minus_nat @ N @ ( pred_numeral @ K ) ) ) ).
% diff_Suc_numeral
thf(fact_668_of__nat__Suc,axiom,
! [M: nat] :
( ( semiri1316708129612266289at_nat @ ( suc @ M ) )
= ( plus_plus_nat @ one_one_nat @ ( semiri1316708129612266289at_nat @ M ) ) ) ).
% of_nat_Suc
thf(fact_669_of__nat__Suc,axiom,
! [M: nat] :
( ( semiri1314217659103216013at_int @ ( suc @ M ) )
= ( plus_plus_int @ one_one_int @ ( semiri1314217659103216013at_int @ M ) ) ) ).
% of_nat_Suc
thf(fact_670_mult__of__nat__commute,axiom,
! [X: nat,Y3: nat] :
( ( times_times_nat @ ( semiri1316708129612266289at_nat @ X ) @ Y3 )
= ( times_times_nat @ Y3 @ ( semiri1316708129612266289at_nat @ X ) ) ) ).
% mult_of_nat_commute
thf(fact_671_mult__of__nat__commute,axiom,
! [X: nat,Y3: int] :
( ( times_times_int @ ( semiri1314217659103216013at_int @ X ) @ Y3 )
= ( times_times_int @ Y3 @ ( semiri1314217659103216013at_int @ X ) ) ) ).
% mult_of_nat_commute
thf(fact_672_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_673_nat__int__comparison_I1_J,axiom,
( ( ^ [Y4: nat,Z3: nat] : ( Y4 = Z3 ) )
= ( ^ [A5: nat,B4: nat] :
( ( semiri1314217659103216013at_int @ A5 )
= ( semiri1314217659103216013at_int @ B4 ) ) ) ) ).
% nat_int_comparison(1)
thf(fact_674_zero__induct__lemma,axiom,
! [P: nat > $o,K: nat,I3: nat] :
( ( P @ K )
=> ( ! [N2: nat] :
( ( P @ ( suc @ N2 ) )
=> ( P @ N2 ) )
=> ( P @ ( minus_minus_nat @ K @ I3 ) ) ) ) ).
% zero_induct_lemma
thf(fact_675_take__bit__of__nat,axiom,
! [N: nat,M: nat] :
( ( bit_se2923211474154528505it_int @ N @ ( semiri1314217659103216013at_int @ M ) )
= ( semiri1314217659103216013at_int @ ( bit_se2925701944663578781it_nat @ N @ M ) ) ) ).
% take_bit_of_nat
thf(fact_676_of__nat__and__eq,axiom,
! [M: nat,N: nat] :
( ( semiri1314217659103216013at_int @ ( bit_se727722235901077358nd_nat @ M @ N ) )
= ( bit_se725231765392027082nd_int @ ( semiri1314217659103216013at_int @ M ) @ ( semiri1314217659103216013at_int @ N ) ) ) ).
% of_nat_and_eq
thf(fact_677_of__nat__and__eq,axiom,
! [M: nat,N: nat] :
( ( semiri1316708129612266289at_nat @ ( bit_se727722235901077358nd_nat @ M @ N ) )
= ( bit_se727722235901077358nd_nat @ ( semiri1316708129612266289at_nat @ M ) @ ( semiri1316708129612266289at_nat @ N ) ) ) ).
% of_nat_and_eq
thf(fact_678_of__nat__or__eq,axiom,
! [M: nat,N: nat] :
( ( semiri1314217659103216013at_int @ ( bit_se1412395901928357646or_nat @ M @ N ) )
= ( bit_se1409905431419307370or_int @ ( semiri1314217659103216013at_int @ M ) @ ( semiri1314217659103216013at_int @ N ) ) ) ).
% of_nat_or_eq
thf(fact_679_of__nat__or__eq,axiom,
! [M: nat,N: nat] :
( ( semiri1316708129612266289at_nat @ ( bit_se1412395901928357646or_nat @ M @ N ) )
= ( bit_se1412395901928357646or_nat @ ( semiri1316708129612266289at_nat @ M ) @ ( semiri1316708129612266289at_nat @ N ) ) ) ).
% of_nat_or_eq
thf(fact_680_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_681_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_682_diff__add__inverse,axiom,
! [N: nat,M: nat] :
( ( minus_minus_nat @ ( plus_plus_nat @ N @ M ) @ N )
= M ) ).
% diff_add_inverse
thf(fact_683_diff__add__inverse2,axiom,
! [M: nat,N: nat] :
( ( minus_minus_nat @ ( plus_plus_nat @ M @ N ) @ N )
= M ) ).
% diff_add_inverse2
thf(fact_684_diff__mult__distrib,axiom,
! [M: nat,N: nat,K: nat] :
( ( times_times_nat @ ( minus_minus_nat @ M @ N ) @ K )
= ( minus_minus_nat @ ( times_times_nat @ M @ K ) @ ( times_times_nat @ N @ K ) ) ) ).
% diff_mult_distrib
thf(fact_685_diff__mult__distrib2,axiom,
! [K: nat,M: nat,N: nat] :
( ( times_times_nat @ K @ ( minus_minus_nat @ M @ N ) )
= ( minus_minus_nat @ ( times_times_nat @ K @ M ) @ ( times_times_nat @ K @ N ) ) ) ).
% diff_mult_distrib2
thf(fact_686_of__nat__xor__eq,axiom,
! [M: nat,N: nat] :
( ( semiri1316708129612266289at_nat @ ( bit_se6528837805403552850or_nat @ M @ N ) )
= ( bit_se6528837805403552850or_nat @ ( semiri1316708129612266289at_nat @ M ) @ ( semiri1316708129612266289at_nat @ N ) ) ) ).
% of_nat_xor_eq
thf(fact_687_of__nat__xor__eq,axiom,
! [M: nat,N: nat] :
( ( semiri1314217659103216013at_int @ ( bit_se6528837805403552850or_nat @ M @ N ) )
= ( bit_se6526347334894502574or_int @ ( semiri1314217659103216013at_int @ M ) @ ( semiri1314217659103216013at_int @ N ) ) ) ).
% of_nat_xor_eq
thf(fact_688_int__cases2,axiom,
! [Z: int] :
( ! [N2: nat] :
( Z
!= ( semiri1314217659103216013at_int @ N2 ) )
=> ~ ! [N2: nat] :
( Z
!= ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ N2 ) ) ) ) ).
% int_cases2
thf(fact_689_of__nat__mask__eq,axiom,
! [N: nat] :
( ( semiri1316708129612266289at_nat @ ( bit_se2002935070580805687sk_nat @ N ) )
= ( bit_se2002935070580805687sk_nat @ N ) ) ).
% of_nat_mask_eq
thf(fact_690_of__nat__mask__eq,axiom,
! [N: nat] :
( ( semiri1314217659103216013at_int @ ( bit_se2002935070580805687sk_nat @ N ) )
= ( bit_se2000444600071755411sk_int @ N ) ) ).
% of_nat_mask_eq
thf(fact_691_step__checker__rev,axiom,
! [I3: int,J3: int,I4: int,J4: int] :
( ( step_checker @ ( product_Pair_int_int @ I3 @ J3 ) @ ( product_Pair_int_int @ I4 @ J4 ) )
=> ( step_checker @ ( product_Pair_int_int @ I4 @ J4 ) @ ( product_Pair_int_int @ I3 @ J3 ) ) ) ).
% step_checker_rev
thf(fact_692_mask__nat__def,axiom,
( bit_se2002935070580805687sk_nat
= ( ^ [N3: nat] : ( minus_minus_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N3 ) @ one_one_nat ) ) ) ).
% mask_nat_def
thf(fact_693_int__ops_I3_J,axiom,
! [N: num] :
( ( semiri1314217659103216013at_int @ ( numeral_numeral_nat @ N ) )
= ( numeral_numeral_int @ N ) ) ).
% int_ops(3)
thf(fact_694_int__cases,axiom,
! [Z: int] :
( ! [N2: nat] :
( Z
!= ( semiri1314217659103216013at_int @ N2 ) )
=> ~ ! [N2: nat] :
( Z
!= ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ ( suc @ N2 ) ) ) ) ) ).
% int_cases
thf(fact_695_int__of__nat__induct,axiom,
! [P: int > $o,Z: int] :
( ! [N2: nat] : ( P @ ( semiri1314217659103216013at_int @ N2 ) )
=> ( ! [N2: nat] : ( P @ ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ ( suc @ N2 ) ) ) )
=> ( P @ Z ) ) ) ).
% int_of_nat_induct
thf(fact_696_zadd__int__left,axiom,
! [M: nat,N: nat,Z: int] :
( ( plus_plus_int @ ( semiri1314217659103216013at_int @ M ) @ ( plus_plus_int @ ( semiri1314217659103216013at_int @ N ) @ Z ) )
= ( plus_plus_int @ ( semiri1314217659103216013at_int @ ( plus_plus_nat @ M @ N ) ) @ Z ) ) ).
% zadd_int_left
thf(fact_697_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_698_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_699_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_700_int__ops_I2_J,axiom,
( ( semiri1314217659103216013at_int @ one_one_nat )
= one_one_int ) ).
% int_ops(2)
thf(fact_701_int__ops_I7_J,axiom,
! [A: nat,B: nat] :
( ( semiri1314217659103216013at_int @ ( times_times_nat @ A @ B ) )
= ( times_times_int @ ( semiri1314217659103216013at_int @ A ) @ ( semiri1314217659103216013at_int @ B ) ) ) ).
% int_ops(7)
thf(fact_702_int__Suc,axiom,
! [N: nat] :
( ( semiri1314217659103216013at_int @ ( suc @ N ) )
= ( plus_plus_int @ ( semiri1314217659103216013at_int @ N ) @ one_one_int ) ) ).
% int_Suc
thf(fact_703_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_704_pred__numeral__def,axiom,
( pred_numeral
= ( ^ [K3: num] : ( minus_minus_nat @ ( numeral_numeral_nat @ K3 ) @ one_one_nat ) ) ) ).
% pred_numeral_def
thf(fact_705_Suc__mask__eq__exp,axiom,
! [N: nat] :
( ( suc @ ( bit_se2002935070580805687sk_nat @ N ) )
= ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ).
% Suc_mask_eq_exp
thf(fact_706_mask__int__def,axiom,
( bit_se2000444600071755411sk_int
= ( ^ [N3: nat] : ( minus_minus_int @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N3 ) @ one_one_int ) ) ) ).
% mask_int_def
thf(fact_707_mask__eq__exp__minus__1,axiom,
( bit_se2002935070580805687sk_nat
= ( ^ [N3: nat] : ( minus_minus_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N3 ) @ one_one_nat ) ) ) ).
% mask_eq_exp_minus_1
thf(fact_708_mask__eq__exp__minus__1,axiom,
( bit_se2000444600071755411sk_int
= ( ^ [N3: nat] : ( minus_minus_int @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N3 ) @ one_one_int ) ) ) ).
% mask_eq_exp_minus_1
thf(fact_709_mask__Suc__exp,axiom,
! [N: nat] :
( ( bit_se2002935070580805687sk_nat @ ( suc @ N ) )
= ( bit_se1412395901928357646or_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) @ ( bit_se2002935070580805687sk_nat @ N ) ) ) ).
% mask_Suc_exp
thf(fact_710_mask__Suc__exp,axiom,
! [N: nat] :
( ( bit_se2000444600071755411sk_int @ ( suc @ N ) )
= ( bit_se1409905431419307370or_int @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) @ ( bit_se2000444600071755411sk_int @ N ) ) ) ).
% mask_Suc_exp
thf(fact_711_take__bit__incr__eq,axiom,
! [N: nat,K: int] :
( ( ( bit_se2923211474154528505it_int @ N @ K )
!= ( minus_minus_int @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) @ one_one_int ) )
=> ( ( bit_se2923211474154528505it_int @ N @ ( plus_plus_int @ K @ one_one_int ) )
= ( plus_plus_int @ one_one_int @ ( bit_se2923211474154528505it_int @ N @ K ) ) ) ) ).
% take_bit_incr_eq
thf(fact_712_take__bit__Suc__minus__1__eq,axiom,
! [N: nat] :
( ( bit_se2923211474154528505it_int @ ( suc @ N ) @ ( uminus_uminus_int @ one_one_int ) )
= ( minus_minus_int @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( suc @ N ) ) @ one_one_int ) ) ).
% take_bit_Suc_minus_1_eq
thf(fact_713_power__minus1__even,axiom,
! [N: nat] :
( ( power_power_int @ ( uminus_uminus_int @ one_one_int ) @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) )
= one_one_int ) ).
% power_minus1_even
thf(fact_714_Power_Oring__1__class_Opower__minus__even,axiom,
! [A: int,N: nat] :
( ( power_power_int @ ( uminus_uminus_int @ A ) @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) )
= ( power_power_int @ A @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ) ).
% Power.ring_1_class.power_minus_even
thf(fact_715_power2__minus,axiom,
! [A: int] :
( ( power_power_int @ ( uminus_uminus_int @ A ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
= ( power_power_int @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ).
% power2_minus
thf(fact_716_power__add__numeral2,axiom,
! [A: nat,M: num,N: num,B: nat] :
( ( times_times_nat @ ( power_power_nat @ A @ ( numeral_numeral_nat @ M ) ) @ ( times_times_nat @ ( power_power_nat @ A @ ( numeral_numeral_nat @ N ) ) @ B ) )
= ( times_times_nat @ ( power_power_nat @ A @ ( numeral_numeral_nat @ ( plus_plus_num @ M @ N ) ) ) @ B ) ) ).
% power_add_numeral2
thf(fact_717_power__add__numeral2,axiom,
! [A: int,M: num,N: num,B: int] :
( ( times_times_int @ ( power_power_int @ A @ ( numeral_numeral_nat @ M ) ) @ ( times_times_int @ ( power_power_int @ A @ ( numeral_numeral_nat @ N ) ) @ B ) )
= ( times_times_int @ ( power_power_int @ A @ ( numeral_numeral_nat @ ( plus_plus_num @ M @ N ) ) ) @ B ) ) ).
% power_add_numeral2
thf(fact_718_power__one,axiom,
! [N: nat] :
( ( power_power_int @ one_one_int @ N )
= one_one_int ) ).
% power_one
thf(fact_719_power__one,axiom,
! [N: nat] :
( ( power_power_nat @ one_one_nat @ N )
= one_one_nat ) ).
% power_one
thf(fact_720_power__one__right,axiom,
! [A: int] :
( ( power_power_int @ A @ one_one_nat )
= A ) ).
% power_one_right
thf(fact_721_power__one__right,axiom,
! [A: nat] :
( ( power_power_nat @ A @ one_one_nat )
= A ) ).
% power_one_right
thf(fact_722_power__mult__numeral,axiom,
! [A: int,M: num,N: num] :
( ( power_power_int @ ( power_power_int @ A @ ( numeral_numeral_nat @ M ) ) @ ( numeral_numeral_nat @ N ) )
= ( power_power_int @ A @ ( numeral_numeral_nat @ ( times_times_num @ M @ N ) ) ) ) ).
% power_mult_numeral
thf(fact_723_power__mult__numeral,axiom,
! [A: nat,M: num,N: num] :
( ( power_power_nat @ ( power_power_nat @ A @ ( numeral_numeral_nat @ M ) ) @ ( numeral_numeral_nat @ N ) )
= ( power_power_nat @ A @ ( numeral_numeral_nat @ ( times_times_num @ M @ N ) ) ) ) ).
% power_mult_numeral
thf(fact_724_minus__one__mult__self,axiom,
! [N: nat] :
( ( times_times_int @ ( power_power_int @ ( uminus_uminus_int @ one_one_int ) @ N ) @ ( power_power_int @ ( uminus_uminus_int @ one_one_int ) @ N ) )
= one_one_int ) ).
% minus_one_mult_self
thf(fact_725_left__minus__one__mult__self,axiom,
! [N: nat,A: int] :
( ( times_times_int @ ( power_power_int @ ( uminus_uminus_int @ one_one_int ) @ N ) @ ( times_times_int @ ( power_power_int @ ( uminus_uminus_int @ one_one_int ) @ N ) @ A ) )
= A ) ).
% left_minus_one_mult_self
thf(fact_726_power__add__numeral,axiom,
! [A: nat,M: num,N: num] :
( ( times_times_nat @ ( power_power_nat @ A @ ( numeral_numeral_nat @ M ) ) @ ( power_power_nat @ A @ ( numeral_numeral_nat @ N ) ) )
= ( power_power_nat @ A @ ( numeral_numeral_nat @ ( plus_plus_num @ M @ N ) ) ) ) ).
% power_add_numeral
thf(fact_727_power__add__numeral,axiom,
! [A: int,M: num,N: num] :
( ( times_times_int @ ( power_power_int @ A @ ( numeral_numeral_nat @ M ) ) @ ( power_power_int @ A @ ( numeral_numeral_nat @ N ) ) )
= ( power_power_int @ A @ ( numeral_numeral_nat @ ( plus_plus_num @ M @ N ) ) ) ) ).
% power_add_numeral
thf(fact_728_real__of__nat__eq__numeral__power__cancel__iff,axiom,
! [Y3: nat,X: num,N: nat] :
( ( ( semiri1316708129612266289at_nat @ Y3 )
= ( power_power_nat @ ( numeral_numeral_nat @ X ) @ N ) )
= ( Y3
= ( power_power_nat @ ( numeral_numeral_nat @ X ) @ N ) ) ) ).
% real_of_nat_eq_numeral_power_cancel_iff
thf(fact_729_real__of__nat__eq__numeral__power__cancel__iff,axiom,
! [Y3: nat,X: num,N: nat] :
( ( ( semiri1314217659103216013at_int @ Y3 )
= ( power_power_int @ ( numeral_numeral_int @ X ) @ N ) )
= ( Y3
= ( power_power_nat @ ( numeral_numeral_nat @ X ) @ N ) ) ) ).
% real_of_nat_eq_numeral_power_cancel_iff
thf(fact_730_numeral__power__eq__of__nat__cancel__iff,axiom,
! [X: num,N: nat,Y3: nat] :
( ( ( power_power_nat @ ( numeral_numeral_nat @ X ) @ N )
= ( semiri1316708129612266289at_nat @ Y3 ) )
= ( ( power_power_nat @ ( numeral_numeral_nat @ X ) @ N )
= Y3 ) ) ).
% numeral_power_eq_of_nat_cancel_iff
thf(fact_731_numeral__power__eq__of__nat__cancel__iff,axiom,
! [X: num,N: nat,Y3: nat] :
( ( ( power_power_int @ ( numeral_numeral_int @ X ) @ N )
= ( semiri1314217659103216013at_int @ Y3 ) )
= ( ( power_power_nat @ ( numeral_numeral_nat @ X ) @ N )
= Y3 ) ) ).
% numeral_power_eq_of_nat_cancel_iff
thf(fact_732_diff__commute,axiom,
! [I3: nat,J3: nat,K: nat] :
( ( minus_minus_nat @ ( minus_minus_nat @ I3 @ J3 ) @ K )
= ( minus_minus_nat @ ( minus_minus_nat @ I3 @ K ) @ J3 ) ) ).
% diff_commute
thf(fact_733_power__commuting__commutes,axiom,
! [X: nat,Y3: nat,N: nat] :
( ( ( times_times_nat @ X @ Y3 )
= ( times_times_nat @ Y3 @ X ) )
=> ( ( times_times_nat @ ( power_power_nat @ X @ N ) @ Y3 )
= ( times_times_nat @ Y3 @ ( power_power_nat @ X @ N ) ) ) ) ).
% power_commuting_commutes
thf(fact_734_power__commuting__commutes,axiom,
! [X: int,Y3: int,N: nat] :
( ( ( times_times_int @ X @ Y3 )
= ( times_times_int @ Y3 @ X ) )
=> ( ( times_times_int @ ( power_power_int @ X @ N ) @ Y3 )
= ( times_times_int @ Y3 @ ( power_power_int @ X @ N ) ) ) ) ).
% power_commuting_commutes
thf(fact_735_power__mult__distrib,axiom,
! [A: nat,B: nat,N: nat] :
( ( power_power_nat @ ( times_times_nat @ A @ B ) @ N )
= ( times_times_nat @ ( power_power_nat @ A @ N ) @ ( power_power_nat @ B @ N ) ) ) ).
% power_mult_distrib
thf(fact_736_power__mult__distrib,axiom,
! [A: int,B: int,N: nat] :
( ( power_power_int @ ( times_times_int @ A @ B ) @ N )
= ( times_times_int @ ( power_power_int @ A @ N ) @ ( power_power_int @ B @ N ) ) ) ).
% power_mult_distrib
thf(fact_737_power__commutes,axiom,
! [A: nat,N: nat] :
( ( times_times_nat @ ( power_power_nat @ A @ N ) @ A )
= ( times_times_nat @ A @ ( power_power_nat @ A @ N ) ) ) ).
% power_commutes
thf(fact_738_power__commutes,axiom,
! [A: int,N: nat] :
( ( times_times_int @ ( power_power_int @ A @ N ) @ A )
= ( times_times_int @ A @ ( power_power_int @ A @ N ) ) ) ).
% power_commutes
thf(fact_739_power__mult,axiom,
! [A: int,M: nat,N: nat] :
( ( power_power_int @ A @ ( times_times_nat @ M @ N ) )
= ( power_power_int @ ( power_power_int @ A @ M ) @ N ) ) ).
% power_mult
thf(fact_740_power__mult,axiom,
! [A: nat,M: nat,N: nat] :
( ( power_power_nat @ A @ ( times_times_nat @ M @ N ) )
= ( power_power_nat @ ( power_power_nat @ A @ M ) @ N ) ) ).
% power_mult
thf(fact_741_left__right__inverse__power,axiom,
! [X: nat,Y3: nat,N: nat] :
( ( ( times_times_nat @ X @ Y3 )
= one_one_nat )
=> ( ( times_times_nat @ ( power_power_nat @ X @ N ) @ ( power_power_nat @ Y3 @ N ) )
= one_one_nat ) ) ).
% left_right_inverse_power
thf(fact_742_left__right__inverse__power,axiom,
! [X: int,Y3: int,N: nat] :
( ( ( times_times_int @ X @ Y3 )
= one_one_int )
=> ( ( times_times_int @ ( power_power_int @ X @ N ) @ ( power_power_int @ Y3 @ N ) )
= one_one_int ) ) ).
% left_right_inverse_power
thf(fact_743_power__Suc,axiom,
! [A: nat,N: nat] :
( ( power_power_nat @ A @ ( suc @ N ) )
= ( times_times_nat @ A @ ( power_power_nat @ A @ N ) ) ) ).
% power_Suc
thf(fact_744_power__Suc,axiom,
! [A: int,N: nat] :
( ( power_power_int @ A @ ( suc @ N ) )
= ( times_times_int @ A @ ( power_power_int @ A @ N ) ) ) ).
% power_Suc
thf(fact_745_power__Suc2,axiom,
! [A: nat,N: nat] :
( ( power_power_nat @ A @ ( suc @ N ) )
= ( times_times_nat @ ( power_power_nat @ A @ N ) @ A ) ) ).
% power_Suc2
thf(fact_746_power__Suc2,axiom,
! [A: int,N: nat] :
( ( power_power_int @ A @ ( suc @ N ) )
= ( times_times_int @ ( power_power_int @ A @ N ) @ A ) ) ).
% power_Suc2
thf(fact_747_power__add,axiom,
! [A: nat,M: nat,N: nat] :
( ( power_power_nat @ A @ ( plus_plus_nat @ M @ N ) )
= ( times_times_nat @ ( power_power_nat @ A @ M ) @ ( power_power_nat @ A @ N ) ) ) ).
% power_add
thf(fact_748_power__add,axiom,
! [A: int,M: nat,N: nat] :
( ( power_power_int @ A @ ( plus_plus_nat @ M @ N ) )
= ( times_times_int @ ( power_power_int @ A @ M ) @ ( power_power_int @ A @ N ) ) ) ).
% power_add
thf(fact_749_power__minus,axiom,
! [A: int,N: nat] :
( ( power_power_int @ ( uminus_uminus_int @ A ) @ N )
= ( times_times_int @ ( power_power_int @ ( uminus_uminus_int @ one_one_int ) @ N ) @ ( power_power_int @ A @ N ) ) ) ).
% power_minus
thf(fact_750_power__minus__Bit0,axiom,
! [X: int,K: num] :
( ( power_power_int @ ( uminus_uminus_int @ X ) @ ( numeral_numeral_nat @ ( bit0 @ K ) ) )
= ( power_power_int @ X @ ( numeral_numeral_nat @ ( bit0 @ K ) ) ) ) ).
% power_minus_Bit0
thf(fact_751_power__minus__Bit1,axiom,
! [X: int,K: num] :
( ( power_power_int @ ( uminus_uminus_int @ X ) @ ( numeral_numeral_nat @ ( bit1 @ K ) ) )
= ( uminus_uminus_int @ ( power_power_int @ X @ ( numeral_numeral_nat @ ( bit1 @ K ) ) ) ) ) ).
% power_minus_Bit1
thf(fact_752_power4__eq__xxxx,axiom,
! [X: nat] :
( ( power_power_nat @ X @ ( numeral_numeral_nat @ ( bit0 @ ( bit0 @ one ) ) ) )
= ( times_times_nat @ ( times_times_nat @ ( times_times_nat @ X @ X ) @ X ) @ X ) ) ).
% power4_eq_xxxx
thf(fact_753_power4__eq__xxxx,axiom,
! [X: int] :
( ( power_power_int @ X @ ( numeral_numeral_nat @ ( bit0 @ ( bit0 @ one ) ) ) )
= ( times_times_int @ ( times_times_int @ ( times_times_int @ X @ X ) @ X ) @ X ) ) ).
% power4_eq_xxxx
thf(fact_754_power2__eq__square,axiom,
! [A: nat] :
( ( power_power_nat @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
= ( times_times_nat @ A @ A ) ) ).
% power2_eq_square
thf(fact_755_power2__eq__square,axiom,
! [A: int] :
( ( power_power_int @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
= ( times_times_int @ A @ A ) ) ).
% power2_eq_square
thf(fact_756_one__power2,axiom,
( ( power_power_int @ one_one_int @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
= one_one_int ) ).
% one_power2
thf(fact_757_one__power2,axiom,
( ( power_power_nat @ one_one_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
= one_one_nat ) ).
% one_power2
thf(fact_758_power2__commute,axiom,
! [X: int,Y3: int] :
( ( power_power_int @ ( minus_minus_int @ X @ Y3 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
= ( power_power_int @ ( minus_minus_int @ Y3 @ X ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ).
% power2_commute
thf(fact_759_power2__eq__iff,axiom,
! [X: int,Y3: int] :
( ( ( power_power_int @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
= ( power_power_int @ Y3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
= ( ( X = Y3 )
| ( X
= ( uminus_uminus_int @ Y3 ) ) ) ) ).
% power2_eq_iff
thf(fact_760_power3__eq__cube,axiom,
! [A: nat] :
( ( power_power_nat @ A @ ( numeral_numeral_nat @ ( bit1 @ one ) ) )
= ( times_times_nat @ ( times_times_nat @ A @ A ) @ A ) ) ).
% power3_eq_cube
thf(fact_761_power3__eq__cube,axiom,
! [A: int] :
( ( power_power_int @ A @ ( numeral_numeral_nat @ ( bit1 @ one ) ) )
= ( times_times_int @ ( times_times_int @ A @ A ) @ A ) ) ).
% power3_eq_cube
thf(fact_762_power__even__eq,axiom,
! [A: int,N: nat] :
( ( power_power_int @ A @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) )
= ( power_power_int @ ( power_power_int @ A @ N ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ).
% power_even_eq
thf(fact_763_power__even__eq,axiom,
! [A: nat,N: nat] :
( ( power_power_nat @ A @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) )
= ( power_power_nat @ ( power_power_nat @ A @ N ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ).
% power_even_eq
thf(fact_764_power2__eq__1__iff,axiom,
! [A: int] :
( ( ( power_power_int @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
= one_one_int )
= ( ( A = one_one_int )
| ( A
= ( uminus_uminus_int @ one_one_int ) ) ) ) ).
% power2_eq_1_iff
thf(fact_765_power2__sum,axiom,
! [X: nat,Y3: nat] :
( ( power_power_nat @ ( plus_plus_nat @ X @ Y3 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
= ( plus_plus_nat @ ( plus_plus_nat @ ( power_power_nat @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_nat @ Y3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( times_times_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ X ) @ Y3 ) ) ) ).
% power2_sum
thf(fact_766_power2__sum,axiom,
! [X: int,Y3: int] :
( ( power_power_int @ ( plus_plus_int @ X @ Y3 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
= ( plus_plus_int @ ( plus_plus_int @ ( power_power_int @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_int @ Y3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( times_times_int @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ X ) @ Y3 ) ) ) ).
% power2_sum
thf(fact_767_minus__power__mult__self,axiom,
! [A: int,N: nat] :
( ( times_times_int @ ( power_power_int @ ( uminus_uminus_int @ A ) @ N ) @ ( power_power_int @ ( uminus_uminus_int @ A ) @ N ) )
= ( power_power_int @ A @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ) ).
% minus_power_mult_self
thf(fact_768_power__odd__eq,axiom,
! [A: nat,N: nat] :
( ( power_power_nat @ A @ ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) )
= ( times_times_nat @ A @ ( power_power_nat @ ( power_power_nat @ A @ N ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ).
% power_odd_eq
thf(fact_769_power__odd__eq,axiom,
! [A: int,N: nat] :
( ( power_power_int @ A @ ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) )
= ( times_times_int @ A @ ( power_power_int @ ( power_power_int @ A @ N ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ).
% power_odd_eq
thf(fact_770_power2__diff,axiom,
! [X: int,Y3: int] :
( ( power_power_int @ ( minus_minus_int @ X @ Y3 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
= ( minus_minus_int @ ( plus_plus_int @ ( power_power_int @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_int @ Y3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( times_times_int @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ X ) @ Y3 ) ) ) ).
% power2_diff
thf(fact_771_power__minus1__odd,axiom,
! [N: nat] :
( ( power_power_int @ ( uminus_uminus_int @ one_one_int ) @ ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) )
= ( uminus_uminus_int @ one_one_int ) ) ).
% power_minus1_odd
thf(fact_772_last__mirror1,axiom,
! [N: nat,M: nat,Ps2: list_P5707943133018811711nt_int,I3: int,J3: int] :
( ( knights_path @ ( board @ N @ M ) @ Ps2 )
=> ( ( ( last_P3305686521732843992nt_int @ Ps2 )
= ( product_Pair_int_int @ I3 @ J3 ) )
=> ( ( last_P3305686521732843992nt_int @ ( mirror1 @ Ps2 ) )
= ( product_Pair_int_int @ ( minus_minus_int @ ( plus_plus_int @ ( semiri1314217659103216013at_int @ N ) @ one_one_int ) @ I3 ) @ J3 ) ) ) ) ).
% last_mirror1
thf(fact_773_power__numeral,axiom,
! [K: num,L: num] :
( ( power_power_nat @ ( numeral_numeral_nat @ K ) @ ( numeral_numeral_nat @ L ) )
= ( numeral_numeral_nat @ ( pow @ K @ L ) ) ) ).
% power_numeral
thf(fact_774_power__numeral,axiom,
! [K: num,L: num] :
( ( power_power_int @ ( numeral_numeral_int @ K ) @ ( numeral_numeral_nat @ L ) )
= ( numeral_numeral_int @ ( pow @ K @ L ) ) ) ).
% power_numeral
thf(fact_775_mirror2__board__id,axiom,
! [M: nat,N: nat] :
( ( mirror2_board @ ( plus_plus_int @ ( semiri1314217659103216013at_int @ M ) @ one_one_int ) @ ( board @ N @ M ) )
= ( board @ N @ M ) ) ).
% mirror2_board_id
thf(fact_776_mirror1__board__id,axiom,
! [N: nat,M: nat] :
( ( mirror1_board @ ( plus_plus_int @ ( semiri1314217659103216013at_int @ N ) @ one_one_int ) @ ( board @ N @ M ) )
= ( board @ N @ M ) ) ).
% mirror1_board_id
thf(fact_777_mirror1__nil,axiom,
! [Ps2: list_P5707943133018811711nt_int] :
( ( Ps2 = nil_Pr2300489316682597567nt_int )
= ( ( mirror1 @ Ps2 )
= nil_Pr2300489316682597567nt_int ) ) ).
% mirror1_nil
thf(fact_778_pow_Osimps_I1_J,axiom,
! [X: num] :
( ( pow @ X @ one )
= X ) ).
% pow.simps(1)
thf(fact_779_mirror1__knights__path,axiom,
! [N: nat,M: nat,Ps2: list_P5707943133018811711nt_int] :
( ( knights_path @ ( board @ N @ M ) @ Ps2 )
=> ( knights_path @ ( board @ N @ M ) @ ( mirror1 @ Ps2 ) ) ) ).
% mirror1_knights_path
thf(fact_780_last__rot90__knights__path,axiom,
! [N: nat,M: nat,Ps2: list_P5707943133018811711nt_int,I3: int,J3: int] :
( ( knights_path @ ( board @ N @ M ) @ Ps2 )
=> ( ( ( last_P3305686521732843992nt_int @ Ps2 )
= ( product_Pair_int_int @ I3 @ J3 ) )
=> ( ( last_P3305686521732843992nt_int @ ( mirror1 @ ( transpose @ Ps2 ) ) )
= ( product_Pair_int_int @ ( minus_minus_int @ ( plus_plus_int @ ( semiri1314217659103216013at_int @ M ) @ one_one_int ) @ J3 ) @ I3 ) ) ) ) ).
% last_rot90_knights_path
thf(fact_781_push__bit__numeral__minus__1,axiom,
! [N: num] :
( ( bit_se545348938243370406it_int @ ( numeral_numeral_nat @ N ) @ ( uminus_uminus_int @ one_one_int ) )
= ( uminus_uminus_int @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( numeral_numeral_nat @ N ) ) ) ) ).
% push_bit_numeral_minus_1
thf(fact_782_take__bit__eq__mask__iff__exp__dvd,axiom,
! [N: nat,K: int] :
( ( ( bit_se2923211474154528505it_int @ N @ K )
= ( bit_se2000444600071755411sk_int @ N ) )
= ( dvd_dvd_int @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) @ ( plus_plus_int @ K @ one_one_int ) ) ) ).
% take_bit_eq_mask_iff_exp_dvd
thf(fact_783_hd__mirror2,axiom,
! [N: nat,M: nat,Ps2: list_P5707943133018811711nt_int,I3: int,J3: int] :
( ( knights_path @ ( board @ N @ M ) @ Ps2 )
=> ( ( ( hd_Pro282112905867057956nt_int @ Ps2 )
= ( product_Pair_int_int @ I3 @ J3 ) )
=> ( ( hd_Pro282112905867057956nt_int @ ( mirror2 @ Ps2 ) )
= ( product_Pair_int_int @ I3 @ ( minus_minus_int @ ( plus_plus_int @ ( semiri1314217659103216013at_int @ M ) @ one_one_int ) @ J3 ) ) ) ) ) ).
% hd_mirror2
thf(fact_784_dvd__add__triv__left__iff,axiom,
! [A: nat,B: nat] :
( ( dvd_dvd_nat @ A @ ( plus_plus_nat @ A @ B ) )
= ( dvd_dvd_nat @ A @ B ) ) ).
% dvd_add_triv_left_iff
thf(fact_785_dvd__add__triv__left__iff,axiom,
! [A: int,B: int] :
( ( dvd_dvd_int @ A @ ( plus_plus_int @ A @ B ) )
= ( dvd_dvd_int @ A @ B ) ) ).
% dvd_add_triv_left_iff
thf(fact_786_dvd__add__triv__right__iff,axiom,
! [A: nat,B: nat] :
( ( dvd_dvd_nat @ A @ ( plus_plus_nat @ B @ A ) )
= ( dvd_dvd_nat @ A @ B ) ) ).
% dvd_add_triv_right_iff
thf(fact_787_dvd__add__triv__right__iff,axiom,
! [A: int,B: int] :
( ( dvd_dvd_int @ A @ ( plus_plus_int @ B @ A ) )
= ( dvd_dvd_int @ A @ B ) ) ).
% dvd_add_triv_right_iff
thf(fact_788_dvd__minus__iff,axiom,
! [X: int,Y3: int] :
( ( dvd_dvd_int @ X @ ( uminus_uminus_int @ Y3 ) )
= ( dvd_dvd_int @ X @ Y3 ) ) ).
% dvd_minus_iff
thf(fact_789_minus__dvd__iff,axiom,
! [X: int,Y3: int] :
( ( dvd_dvd_int @ ( uminus_uminus_int @ X ) @ Y3 )
= ( dvd_dvd_int @ X @ Y3 ) ) ).
% minus_dvd_iff
thf(fact_790_push__bit__push__bit,axiom,
! [M: nat,N: nat,A: int] :
( ( bit_se545348938243370406it_int @ M @ ( bit_se545348938243370406it_int @ N @ A ) )
= ( bit_se545348938243370406it_int @ ( plus_plus_nat @ M @ N ) @ A ) ) ).
% push_bit_push_bit
thf(fact_791_push__bit__push__bit,axiom,
! [M: nat,N: nat,A: nat] :
( ( bit_se547839408752420682it_nat @ M @ ( bit_se547839408752420682it_nat @ N @ A ) )
= ( bit_se547839408752420682it_nat @ ( plus_plus_nat @ M @ N ) @ A ) ) ).
% push_bit_push_bit
thf(fact_792_push__bit__and,axiom,
! [N: nat,A: int,B: int] :
( ( bit_se545348938243370406it_int @ N @ ( bit_se725231765392027082nd_int @ A @ B ) )
= ( bit_se725231765392027082nd_int @ ( bit_se545348938243370406it_int @ N @ A ) @ ( bit_se545348938243370406it_int @ N @ B ) ) ) ).
% push_bit_and
thf(fact_793_push__bit__and,axiom,
! [N: nat,A: nat,B: nat] :
( ( bit_se547839408752420682it_nat @ N @ ( bit_se727722235901077358nd_nat @ A @ B ) )
= ( bit_se727722235901077358nd_nat @ ( bit_se547839408752420682it_nat @ N @ A ) @ ( bit_se547839408752420682it_nat @ N @ B ) ) ) ).
% push_bit_and
thf(fact_794_push__bit__or,axiom,
! [N: nat,A: int,B: int] :
( ( bit_se545348938243370406it_int @ N @ ( bit_se1409905431419307370or_int @ A @ B ) )
= ( bit_se1409905431419307370or_int @ ( bit_se545348938243370406it_int @ N @ A ) @ ( bit_se545348938243370406it_int @ N @ B ) ) ) ).
% push_bit_or
thf(fact_795_push__bit__or,axiom,
! [N: nat,A: nat,B: nat] :
( ( bit_se547839408752420682it_nat @ N @ ( bit_se1412395901928357646or_nat @ A @ B ) )
= ( bit_se1412395901928357646or_nat @ ( bit_se547839408752420682it_nat @ N @ A ) @ ( bit_se547839408752420682it_nat @ N @ B ) ) ) ).
% push_bit_or
thf(fact_796_push__bit__xor,axiom,
! [N: nat,A: int,B: int] :
( ( bit_se545348938243370406it_int @ N @ ( bit_se6526347334894502574or_int @ A @ B ) )
= ( bit_se6526347334894502574or_int @ ( bit_se545348938243370406it_int @ N @ A ) @ ( bit_se545348938243370406it_int @ N @ B ) ) ) ).
% push_bit_xor
thf(fact_797_push__bit__xor,axiom,
! [N: nat,A: nat,B: nat] :
( ( bit_se547839408752420682it_nat @ N @ ( bit_se6528837805403552850or_nat @ A @ B ) )
= ( bit_se6528837805403552850or_nat @ ( bit_se547839408752420682it_nat @ N @ A ) @ ( bit_se547839408752420682it_nat @ N @ B ) ) ) ).
% push_bit_xor
thf(fact_798_dvd__add__times__triv__right__iff,axiom,
! [A: nat,B: nat,C2: nat] :
( ( dvd_dvd_nat @ A @ ( plus_plus_nat @ B @ ( times_times_nat @ C2 @ A ) ) )
= ( dvd_dvd_nat @ A @ B ) ) ).
% dvd_add_times_triv_right_iff
thf(fact_799_dvd__add__times__triv__right__iff,axiom,
! [A: int,B: int,C2: int] :
( ( dvd_dvd_int @ A @ ( plus_plus_int @ B @ ( times_times_int @ C2 @ A ) ) )
= ( dvd_dvd_int @ A @ B ) ) ).
% dvd_add_times_triv_right_iff
thf(fact_800_dvd__add__times__triv__left__iff,axiom,
! [A: nat,C2: nat,B: nat] :
( ( dvd_dvd_nat @ A @ ( plus_plus_nat @ ( times_times_nat @ C2 @ A ) @ B ) )
= ( dvd_dvd_nat @ A @ B ) ) ).
% dvd_add_times_triv_left_iff
thf(fact_801_dvd__add__times__triv__left__iff,axiom,
! [A: int,C2: int,B: int] :
( ( dvd_dvd_int @ A @ ( plus_plus_int @ ( times_times_int @ C2 @ A ) @ B ) )
= ( dvd_dvd_int @ A @ B ) ) ).
% dvd_add_times_triv_left_iff
thf(fact_802_unit__prod,axiom,
! [A: nat,B: nat] :
( ( dvd_dvd_nat @ A @ one_one_nat )
=> ( ( dvd_dvd_nat @ B @ one_one_nat )
=> ( dvd_dvd_nat @ ( times_times_nat @ A @ B ) @ one_one_nat ) ) ) ).
% unit_prod
thf(fact_803_unit__prod,axiom,
! [A: int,B: int] :
( ( dvd_dvd_int @ A @ one_one_int )
=> ( ( dvd_dvd_int @ B @ one_one_int )
=> ( dvd_dvd_int @ ( times_times_int @ A @ B ) @ one_one_int ) ) ) ).
% unit_prod
thf(fact_804_push__bit__Suc__numeral,axiom,
! [N: nat,K: num] :
( ( bit_se545348938243370406it_int @ ( suc @ N ) @ ( numeral_numeral_int @ K ) )
= ( bit_se545348938243370406it_int @ N @ ( numeral_numeral_int @ ( bit0 @ K ) ) ) ) ).
% push_bit_Suc_numeral
thf(fact_805_push__bit__Suc__numeral,axiom,
! [N: nat,K: num] :
( ( bit_se547839408752420682it_nat @ ( suc @ N ) @ ( numeral_numeral_nat @ K ) )
= ( bit_se547839408752420682it_nat @ N @ ( numeral_numeral_nat @ ( bit0 @ K ) ) ) ) ).
% push_bit_Suc_numeral
thf(fact_806_even__mult__iff,axiom,
! [A: nat,B: nat] :
( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( times_times_nat @ A @ B ) )
= ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A )
| ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ B ) ) ) ).
% even_mult_iff
thf(fact_807_even__mult__iff,axiom,
! [A: int,B: int] :
( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( times_times_int @ A @ B ) )
= ( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A )
| ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ B ) ) ) ).
% even_mult_iff
thf(fact_808_odd__add,axiom,
! [A: nat,B: nat] :
( ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( plus_plus_nat @ A @ B ) ) )
= ( ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A ) )
!= ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ B ) ) ) ) ).
% odd_add
thf(fact_809_odd__add,axiom,
! [A: int,B: int] :
( ( ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( plus_plus_int @ A @ B ) ) )
= ( ( ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A ) )
!= ( ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ B ) ) ) ) ).
% odd_add
thf(fact_810_even__add,axiom,
! [A: nat,B: nat] :
( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( plus_plus_nat @ A @ B ) )
= ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A )
= ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ B ) ) ) ).
% even_add
thf(fact_811_even__add,axiom,
! [A: int,B: int] :
( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( plus_plus_int @ A @ B ) )
= ( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A )
= ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ B ) ) ) ).
% even_add
thf(fact_812_push__bit__Suc__minus__numeral,axiom,
! [N: nat,K: num] :
( ( bit_se545348938243370406it_int @ ( suc @ N ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ K ) ) )
= ( bit_se545348938243370406it_int @ N @ ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit0 @ K ) ) ) ) ) ).
% push_bit_Suc_minus_numeral
thf(fact_813_push__bit__numeral,axiom,
! [L: num,K: num] :
( ( bit_se545348938243370406it_int @ ( numeral_numeral_nat @ L ) @ ( numeral_numeral_int @ K ) )
= ( bit_se545348938243370406it_int @ ( pred_numeral @ L ) @ ( numeral_numeral_int @ ( bit0 @ K ) ) ) ) ).
% push_bit_numeral
thf(fact_814_push__bit__numeral,axiom,
! [L: num,K: num] :
( ( bit_se547839408752420682it_nat @ ( numeral_numeral_nat @ L ) @ ( numeral_numeral_nat @ K ) )
= ( bit_se547839408752420682it_nat @ ( pred_numeral @ L ) @ ( numeral_numeral_nat @ ( bit0 @ K ) ) ) ) ).
% push_bit_numeral
thf(fact_815_even__plus__one__iff,axiom,
! [A: nat] :
( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( plus_plus_nat @ A @ one_one_nat ) )
= ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A ) ) ) ).
% even_plus_one_iff
thf(fact_816_even__plus__one__iff,axiom,
! [A: int] :
( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( plus_plus_int @ A @ one_one_int ) )
= ( ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A ) ) ) ).
% even_plus_one_iff
thf(fact_817_even__diff,axiom,
! [A: int,B: int] :
( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( minus_minus_int @ A @ B ) )
= ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( plus_plus_int @ A @ B ) ) ) ).
% even_diff
thf(fact_818_even__of__nat,axiom,
! [N: nat] :
( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( semiri1316708129612266289at_nat @ N ) )
= ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ).
% even_of_nat
thf(fact_819_even__of__nat,axiom,
! [N: nat] :
( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( semiri1314217659103216013at_int @ N ) )
= ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ).
% even_of_nat
thf(fact_820_push__bit__Suc,axiom,
! [N: nat,A: int] :
( ( bit_se545348938243370406it_int @ ( suc @ N ) @ A )
= ( bit_se545348938243370406it_int @ N @ ( times_times_int @ A @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ) ).
% push_bit_Suc
thf(fact_821_push__bit__Suc,axiom,
! [N: nat,A: nat] :
( ( bit_se547839408752420682it_nat @ ( suc @ N ) @ A )
= ( bit_se547839408752420682it_nat @ N @ ( times_times_nat @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ).
% push_bit_Suc
thf(fact_822_push__bit__of__1,axiom,
! [N: nat] :
( ( bit_se545348938243370406it_int @ N @ one_one_int )
= ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) ) ).
% push_bit_of_1
thf(fact_823_push__bit__of__1,axiom,
! [N: nat] :
( ( bit_se547839408752420682it_nat @ N @ one_one_nat )
= ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ).
% push_bit_of_1
thf(fact_824_push__bit__minus__numeral,axiom,
! [L: num,K: num] :
( ( bit_se545348938243370406it_int @ ( numeral_numeral_nat @ L ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ K ) ) )
= ( bit_se545348938243370406it_int @ ( pred_numeral @ L ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit0 @ K ) ) ) ) ) ).
% push_bit_minus_numeral
thf(fact_825_list_Osel_I1_J,axiom,
! [X21: list_nat,X22: list_list_nat] :
( ( hd_list_nat @ ( cons_list_nat @ X21 @ X22 ) )
= X21 ) ).
% list.sel(1)
thf(fact_826_list_Osel_I1_J,axiom,
! [X21: nat,X22: list_nat] :
( ( hd_nat @ ( cons_nat @ X21 @ X22 ) )
= X21 ) ).
% list.sel(1)
thf(fact_827_list_Osel_I1_J,axiom,
! [X21: product_prod_int_int,X22: list_P5707943133018811711nt_int] :
( ( hd_Pro282112905867057956nt_int @ ( cons_P3334398858971670639nt_int @ X21 @ X22 ) )
= X21 ) ).
% list.sel(1)
thf(fact_828_list_Osel_I1_J,axiom,
! [X21: list_P5707943133018811711nt_int,X22: list_l1670014477004246597nt_int] :
( ( hd_lis8550591025403828010nt_int @ ( cons_l7309679040211256053nt_int @ X21 @ X22 ) )
= X21 ) ).
% list.sel(1)
thf(fact_829_list_Osel_I1_J,axiom,
! [X21: list_list_nat,X22: list_list_list_nat] :
( ( hd_list_list_nat @ ( cons_list_list_nat @ X21 @ X22 ) )
= X21 ) ).
% list.sel(1)
thf(fact_830_push__bit__add,axiom,
! [N: nat,A: int,B: int] :
( ( bit_se545348938243370406it_int @ N @ ( plus_plus_int @ A @ B ) )
= ( plus_plus_int @ ( bit_se545348938243370406it_int @ N @ A ) @ ( bit_se545348938243370406it_int @ N @ B ) ) ) ).
% push_bit_add
thf(fact_831_push__bit__add,axiom,
! [N: nat,A: nat,B: nat] :
( ( bit_se547839408752420682it_nat @ N @ ( plus_plus_nat @ A @ B ) )
= ( plus_plus_nat @ ( bit_se547839408752420682it_nat @ N @ A ) @ ( bit_se547839408752420682it_nat @ N @ B ) ) ) ).
% push_bit_add
thf(fact_832_push__bit__minus,axiom,
! [N: nat,A: int] :
( ( bit_se545348938243370406it_int @ N @ ( uminus_uminus_int @ A ) )
= ( uminus_uminus_int @ ( bit_se545348938243370406it_int @ N @ A ) ) ) ).
% push_bit_minus
thf(fact_833_one__dvd,axiom,
! [A: nat] : ( dvd_dvd_nat @ one_one_nat @ A ) ).
% one_dvd
thf(fact_834_one__dvd,axiom,
! [A: int] : ( dvd_dvd_int @ one_one_int @ A ) ).
% one_dvd
thf(fact_835_unit__imp__dvd,axiom,
! [B: nat,A: nat] :
( ( dvd_dvd_nat @ B @ one_one_nat )
=> ( dvd_dvd_nat @ B @ A ) ) ).
% unit_imp_dvd
thf(fact_836_unit__imp__dvd,axiom,
! [B: int,A: int] :
( ( dvd_dvd_int @ B @ one_one_int )
=> ( dvd_dvd_int @ B @ A ) ) ).
% unit_imp_dvd
thf(fact_837_dvd__unit__imp__unit,axiom,
! [A: nat,B: nat] :
( ( dvd_dvd_nat @ A @ B )
=> ( ( dvd_dvd_nat @ B @ one_one_nat )
=> ( dvd_dvd_nat @ A @ one_one_nat ) ) ) ).
% dvd_unit_imp_unit
thf(fact_838_dvd__unit__imp__unit,axiom,
! [A: int,B: int] :
( ( dvd_dvd_int @ A @ B )
=> ( ( dvd_dvd_int @ B @ one_one_int )
=> ( dvd_dvd_int @ A @ one_one_int ) ) ) ).
% dvd_unit_imp_unit
thf(fact_839_dvd__add__right__iff,axiom,
! [A: nat,B: nat,C2: nat] :
( ( dvd_dvd_nat @ A @ B )
=> ( ( dvd_dvd_nat @ A @ ( plus_plus_nat @ B @ C2 ) )
= ( dvd_dvd_nat @ A @ C2 ) ) ) ).
% dvd_add_right_iff
thf(fact_840_dvd__add__right__iff,axiom,
! [A: int,B: int,C2: int] :
( ( dvd_dvd_int @ A @ B )
=> ( ( dvd_dvd_int @ A @ ( plus_plus_int @ B @ C2 ) )
= ( dvd_dvd_int @ A @ C2 ) ) ) ).
% dvd_add_right_iff
thf(fact_841_dvd__add__left__iff,axiom,
! [A: nat,C2: nat,B: nat] :
( ( dvd_dvd_nat @ A @ C2 )
=> ( ( dvd_dvd_nat @ A @ ( plus_plus_nat @ B @ C2 ) )
= ( dvd_dvd_nat @ A @ B ) ) ) ).
% dvd_add_left_iff
thf(fact_842_dvd__add__left__iff,axiom,
! [A: int,C2: int,B: int] :
( ( dvd_dvd_int @ A @ C2 )
=> ( ( dvd_dvd_int @ A @ ( plus_plus_int @ B @ C2 ) )
= ( dvd_dvd_int @ A @ B ) ) ) ).
% dvd_add_left_iff
thf(fact_843_dvd__add,axiom,
! [A: nat,B: nat,C2: nat] :
( ( dvd_dvd_nat @ A @ B )
=> ( ( dvd_dvd_nat @ A @ C2 )
=> ( dvd_dvd_nat @ A @ ( plus_plus_nat @ B @ C2 ) ) ) ) ).
% dvd_add
thf(fact_844_dvd__add,axiom,
! [A: int,B: int,C2: int] :
( ( dvd_dvd_int @ A @ B )
=> ( ( dvd_dvd_int @ A @ C2 )
=> ( dvd_dvd_int @ A @ ( plus_plus_int @ B @ C2 ) ) ) ) ).
% dvd_add
thf(fact_845_dvd__triv__right,axiom,
! [A: nat,B: nat] : ( dvd_dvd_nat @ A @ ( times_times_nat @ B @ A ) ) ).
% dvd_triv_right
thf(fact_846_dvd__triv__right,axiom,
! [A: int,B: int] : ( dvd_dvd_int @ A @ ( times_times_int @ B @ A ) ) ).
% dvd_triv_right
thf(fact_847_dvd__mult__right,axiom,
! [A: nat,B: nat,C2: nat] :
( ( dvd_dvd_nat @ ( times_times_nat @ A @ B ) @ C2 )
=> ( dvd_dvd_nat @ B @ C2 ) ) ).
% dvd_mult_right
thf(fact_848_dvd__mult__right,axiom,
! [A: int,B: int,C2: int] :
( ( dvd_dvd_int @ ( times_times_int @ A @ B ) @ C2 )
=> ( dvd_dvd_int @ B @ C2 ) ) ).
% dvd_mult_right
thf(fact_849_mult__dvd__mono,axiom,
! [A: nat,B: nat,C2: nat,D: nat] :
( ( dvd_dvd_nat @ A @ B )
=> ( ( dvd_dvd_nat @ C2 @ D )
=> ( dvd_dvd_nat @ ( times_times_nat @ A @ C2 ) @ ( times_times_nat @ B @ D ) ) ) ) ).
% mult_dvd_mono
thf(fact_850_mult__dvd__mono,axiom,
! [A: int,B: int,C2: int,D: int] :
( ( dvd_dvd_int @ A @ B )
=> ( ( dvd_dvd_int @ C2 @ D )
=> ( dvd_dvd_int @ ( times_times_int @ A @ C2 ) @ ( times_times_int @ B @ D ) ) ) ) ).
% mult_dvd_mono
thf(fact_851_dvd__triv__left,axiom,
! [A: nat,B: nat] : ( dvd_dvd_nat @ A @ ( times_times_nat @ A @ B ) ) ).
% dvd_triv_left
thf(fact_852_dvd__triv__left,axiom,
! [A: int,B: int] : ( dvd_dvd_int @ A @ ( times_times_int @ A @ B ) ) ).
% dvd_triv_left
thf(fact_853_dvd__mult__left,axiom,
! [A: nat,B: nat,C2: nat] :
( ( dvd_dvd_nat @ ( times_times_nat @ A @ B ) @ C2 )
=> ( dvd_dvd_nat @ A @ C2 ) ) ).
% dvd_mult_left
thf(fact_854_dvd__mult__left,axiom,
! [A: int,B: int,C2: int] :
( ( dvd_dvd_int @ ( times_times_int @ A @ B ) @ C2 )
=> ( dvd_dvd_int @ A @ C2 ) ) ).
% dvd_mult_left
thf(fact_855_dvd__mult2,axiom,
! [A: nat,B: nat,C2: nat] :
( ( dvd_dvd_nat @ A @ B )
=> ( dvd_dvd_nat @ A @ ( times_times_nat @ B @ C2 ) ) ) ).
% dvd_mult2
thf(fact_856_dvd__mult2,axiom,
! [A: int,B: int,C2: int] :
( ( dvd_dvd_int @ A @ B )
=> ( dvd_dvd_int @ A @ ( times_times_int @ B @ C2 ) ) ) ).
% dvd_mult2
thf(fact_857_dvd__mult,axiom,
! [A: nat,C2: nat,B: nat] :
( ( dvd_dvd_nat @ A @ C2 )
=> ( dvd_dvd_nat @ A @ ( times_times_nat @ B @ C2 ) ) ) ).
% dvd_mult
thf(fact_858_dvd__mult,axiom,
! [A: int,C2: int,B: int] :
( ( dvd_dvd_int @ A @ C2 )
=> ( dvd_dvd_int @ A @ ( times_times_int @ B @ C2 ) ) ) ).
% dvd_mult
thf(fact_859_dvd__def,axiom,
( dvd_dvd_nat
= ( ^ [B4: nat,A5: nat] :
? [K3: nat] :
( A5
= ( times_times_nat @ B4 @ K3 ) ) ) ) ).
% dvd_def
thf(fact_860_dvd__def,axiom,
( dvd_dvd_int
= ( ^ [B4: int,A5: int] :
? [K3: int] :
( A5
= ( times_times_int @ B4 @ K3 ) ) ) ) ).
% dvd_def
thf(fact_861_dvdI,axiom,
! [A: nat,B: nat,K: nat] :
( ( A
= ( times_times_nat @ B @ K ) )
=> ( dvd_dvd_nat @ B @ A ) ) ).
% dvdI
thf(fact_862_dvdI,axiom,
! [A: int,B: int,K: int] :
( ( A
= ( times_times_int @ B @ K ) )
=> ( dvd_dvd_int @ B @ A ) ) ).
% dvdI
thf(fact_863_dvdE,axiom,
! [B: nat,A: nat] :
( ( dvd_dvd_nat @ B @ A )
=> ~ ! [K2: nat] :
( A
!= ( times_times_nat @ B @ K2 ) ) ) ).
% dvdE
thf(fact_864_dvdE,axiom,
! [B: int,A: int] :
( ( dvd_dvd_int @ B @ A )
=> ~ ! [K2: int] :
( A
!= ( times_times_int @ B @ K2 ) ) ) ).
% dvdE
thf(fact_865_of__nat__push__bit,axiom,
! [M: nat,N: nat] :
( ( semiri1314217659103216013at_int @ ( bit_se547839408752420682it_nat @ M @ N ) )
= ( bit_se545348938243370406it_int @ M @ ( semiri1314217659103216013at_int @ N ) ) ) ).
% of_nat_push_bit
thf(fact_866_of__nat__push__bit,axiom,
! [M: nat,N: nat] :
( ( semiri1316708129612266289at_nat @ ( bit_se547839408752420682it_nat @ M @ N ) )
= ( bit_se547839408752420682it_nat @ M @ ( semiri1316708129612266289at_nat @ N ) ) ) ).
% of_nat_push_bit
thf(fact_867_push__bit__of__nat,axiom,
! [N: nat,M: nat] :
( ( bit_se545348938243370406it_int @ N @ ( semiri1314217659103216013at_int @ M ) )
= ( semiri1314217659103216013at_int @ ( bit_se547839408752420682it_nat @ N @ M ) ) ) ).
% push_bit_of_nat
thf(fact_868_push__bit__of__nat,axiom,
! [N: nat,M: nat] :
( ( bit_se547839408752420682it_nat @ N @ ( semiri1316708129612266289at_nat @ M ) )
= ( semiri1316708129612266289at_nat @ ( bit_se547839408752420682it_nat @ N @ M ) ) ) ).
% push_bit_of_nat
thf(fact_869_KnightsTour_Otranspose_Osimps_I1_J,axiom,
( ( transpose @ nil_Pr2300489316682597567nt_int )
= nil_Pr2300489316682597567nt_int ) ).
% KnightsTour.transpose.simps(1)
thf(fact_870_transpose__nil,axiom,
! [Ps2: list_P5707943133018811711nt_int] :
( ( Ps2 = nil_Pr2300489316682597567nt_int )
= ( ( transpose @ Ps2 )
= nil_Pr2300489316682597567nt_int ) ) ).
% transpose_nil
thf(fact_871_exp__dvdE,axiom,
! [N: nat,A: int] :
( ( dvd_dvd_int @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) @ A )
=> ~ ! [B3: int] :
( A
!= ( bit_se545348938243370406it_int @ N @ B3 ) ) ) ).
% exp_dvdE
thf(fact_872_exp__dvdE,axiom,
! [N: nat,A: nat] :
( ( dvd_dvd_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) @ A )
=> ~ ! [B3: nat] :
( A
!= ( bit_se547839408752420682it_nat @ N @ B3 ) ) ) ).
% exp_dvdE
thf(fact_873_is__unit__mult__iff,axiom,
! [A: nat,B: nat] :
( ( dvd_dvd_nat @ ( times_times_nat @ A @ B ) @ one_one_nat )
= ( ( dvd_dvd_nat @ A @ one_one_nat )
& ( dvd_dvd_nat @ B @ one_one_nat ) ) ) ).
% is_unit_mult_iff
thf(fact_874_is__unit__mult__iff,axiom,
! [A: int,B: int] :
( ( dvd_dvd_int @ ( times_times_int @ A @ B ) @ one_one_int )
= ( ( dvd_dvd_int @ A @ one_one_int )
& ( dvd_dvd_int @ B @ one_one_int ) ) ) ).
% is_unit_mult_iff
thf(fact_875_dvd__mult__unit__iff,axiom,
! [B: nat,A: nat,C2: nat] :
( ( dvd_dvd_nat @ B @ one_one_nat )
=> ( ( dvd_dvd_nat @ A @ ( times_times_nat @ C2 @ B ) )
= ( dvd_dvd_nat @ A @ C2 ) ) ) ).
% dvd_mult_unit_iff
thf(fact_876_dvd__mult__unit__iff,axiom,
! [B: int,A: int,C2: int] :
( ( dvd_dvd_int @ B @ one_one_int )
=> ( ( dvd_dvd_int @ A @ ( times_times_int @ C2 @ B ) )
= ( dvd_dvd_int @ A @ C2 ) ) ) ).
% dvd_mult_unit_iff
thf(fact_877_mult__unit__dvd__iff,axiom,
! [B: nat,A: nat,C2: nat] :
( ( dvd_dvd_nat @ B @ one_one_nat )
=> ( ( dvd_dvd_nat @ ( times_times_nat @ A @ B ) @ C2 )
= ( dvd_dvd_nat @ A @ C2 ) ) ) ).
% mult_unit_dvd_iff
thf(fact_878_mult__unit__dvd__iff,axiom,
! [B: int,A: int,C2: int] :
( ( dvd_dvd_int @ B @ one_one_int )
=> ( ( dvd_dvd_int @ ( times_times_int @ A @ B ) @ C2 )
= ( dvd_dvd_int @ A @ C2 ) ) ) ).
% mult_unit_dvd_iff
thf(fact_879_dvd__mult__unit__iff_H,axiom,
! [B: nat,A: nat,C2: nat] :
( ( dvd_dvd_nat @ B @ one_one_nat )
=> ( ( dvd_dvd_nat @ A @ ( times_times_nat @ B @ C2 ) )
= ( dvd_dvd_nat @ A @ C2 ) ) ) ).
% dvd_mult_unit_iff'
thf(fact_880_dvd__mult__unit__iff_H,axiom,
! [B: int,A: int,C2: int] :
( ( dvd_dvd_int @ B @ one_one_int )
=> ( ( dvd_dvd_int @ A @ ( times_times_int @ B @ C2 ) )
= ( dvd_dvd_int @ A @ C2 ) ) ) ).
% dvd_mult_unit_iff'
thf(fact_881_mult__unit__dvd__iff_H,axiom,
! [A: nat,B: nat,C2: nat] :
( ( dvd_dvd_nat @ A @ one_one_nat )
=> ( ( dvd_dvd_nat @ ( times_times_nat @ A @ B ) @ C2 )
= ( dvd_dvd_nat @ B @ C2 ) ) ) ).
% mult_unit_dvd_iff'
thf(fact_882_mult__unit__dvd__iff_H,axiom,
! [A: int,B: int,C2: int] :
( ( dvd_dvd_int @ A @ one_one_int )
=> ( ( dvd_dvd_int @ ( times_times_int @ A @ B ) @ C2 )
= ( dvd_dvd_int @ B @ C2 ) ) ) ).
% mult_unit_dvd_iff'
thf(fact_883_unit__mult__left__cancel,axiom,
! [A: nat,B: nat,C2: nat] :
( ( dvd_dvd_nat @ A @ one_one_nat )
=> ( ( ( times_times_nat @ A @ B )
= ( times_times_nat @ A @ C2 ) )
= ( B = C2 ) ) ) ).
% unit_mult_left_cancel
thf(fact_884_unit__mult__left__cancel,axiom,
! [A: int,B: int,C2: int] :
( ( dvd_dvd_int @ A @ one_one_int )
=> ( ( ( times_times_int @ A @ B )
= ( times_times_int @ A @ C2 ) )
= ( B = C2 ) ) ) ).
% unit_mult_left_cancel
thf(fact_885_unit__mult__right__cancel,axiom,
! [A: nat,B: nat,C2: nat] :
( ( dvd_dvd_nat @ A @ one_one_nat )
=> ( ( ( times_times_nat @ B @ A )
= ( times_times_nat @ C2 @ A ) )
= ( B = C2 ) ) ) ).
% unit_mult_right_cancel
thf(fact_886_unit__mult__right__cancel,axiom,
! [A: int,B: int,C2: int] :
( ( dvd_dvd_int @ A @ one_one_int )
=> ( ( ( times_times_int @ B @ A )
= ( times_times_int @ C2 @ A ) )
= ( B = C2 ) ) ) ).
% unit_mult_right_cancel
thf(fact_887_zdvd__period,axiom,
! [A: int,D: int,X: int,T: int,C2: int] :
( ( dvd_dvd_int @ A @ D )
=> ( ( dvd_dvd_int @ A @ ( plus_plus_int @ X @ T ) )
= ( dvd_dvd_int @ A @ ( plus_plus_int @ ( plus_plus_int @ X @ ( times_times_int @ C2 @ D ) ) @ T ) ) ) ) ).
% zdvd_period
thf(fact_888_zdvd__reduce,axiom,
! [K: int,N: int,M: int] :
( ( dvd_dvd_int @ K @ ( plus_plus_int @ N @ ( times_times_int @ K @ M ) ) )
= ( dvd_dvd_int @ K @ N ) ) ).
% zdvd_reduce
thf(fact_889_hd__Nil__eq__last,axiom,
( ( hd_nat @ nil_nat )
= ( last_nat @ nil_nat ) ) ).
% hd_Nil_eq_last
thf(fact_890_hd__Nil__eq__last,axiom,
( ( hd_list_nat @ nil_list_nat )
= ( last_list_nat @ nil_list_nat ) ) ).
% hd_Nil_eq_last
thf(fact_891_hd__Nil__eq__last,axiom,
( ( hd_Pro282112905867057956nt_int @ nil_Pr2300489316682597567nt_int )
= ( last_P3305686521732843992nt_int @ nil_Pr2300489316682597567nt_int ) ) ).
% hd_Nil_eq_last
thf(fact_892_hd__Nil__eq__last,axiom,
( ( hd_lis8550591025403828010nt_int @ nil_li8670148097206105925nt_int )
= ( last_l5818330359162608606nt_int @ nil_li8670148097206105925nt_int ) ) ).
% hd_Nil_eq_last
thf(fact_893_hd__Nil__eq__last,axiom,
( ( hd_list_list_nat @ nil_list_list_nat )
= ( last_list_list_nat @ nil_list_list_nat ) ) ).
% hd_Nil_eq_last
thf(fact_894_push__bit__take__bit,axiom,
! [M: nat,N: nat,A: nat] :
( ( bit_se547839408752420682it_nat @ M @ ( bit_se2925701944663578781it_nat @ N @ A ) )
= ( bit_se2925701944663578781it_nat @ ( plus_plus_nat @ M @ N ) @ ( bit_se547839408752420682it_nat @ M @ A ) ) ) ).
% push_bit_take_bit
thf(fact_895_push__bit__take__bit,axiom,
! [M: nat,N: nat,A: int] :
( ( bit_se545348938243370406it_int @ M @ ( bit_se2923211474154528505it_int @ N @ A ) )
= ( bit_se2923211474154528505it_int @ ( plus_plus_nat @ M @ N ) @ ( bit_se545348938243370406it_int @ M @ A ) ) ) ).
% push_bit_take_bit
thf(fact_896_take__bit__push__bit,axiom,
! [M: nat,N: nat,A: nat] :
( ( bit_se2925701944663578781it_nat @ M @ ( bit_se547839408752420682it_nat @ N @ A ) )
= ( bit_se547839408752420682it_nat @ N @ ( bit_se2925701944663578781it_nat @ ( minus_minus_nat @ M @ N ) @ A ) ) ) ).
% take_bit_push_bit
thf(fact_897_take__bit__push__bit,axiom,
! [M: nat,N: nat,A: int] :
( ( bit_se2923211474154528505it_int @ M @ ( bit_se545348938243370406it_int @ N @ A ) )
= ( bit_se545348938243370406it_int @ N @ ( bit_se2923211474154528505it_int @ ( minus_minus_nat @ M @ N ) @ A ) ) ) ).
% take_bit_push_bit
thf(fact_898_even__numeral,axiom,
! [N: num] : ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( numeral_numeral_nat @ ( bit0 @ N ) ) ) ).
% even_numeral
thf(fact_899_even__numeral,axiom,
! [N: num] : ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( numeral_numeral_int @ ( bit0 @ N ) ) ) ).
% even_numeral
thf(fact_900_transpose__knights__path,axiom,
! [N: nat,M: nat,Ps2: list_P5707943133018811711nt_int] :
( ( knights_path @ ( board @ N @ M ) @ Ps2 )
=> ( knights_path @ ( board @ M @ N ) @ ( transpose @ Ps2 ) ) ) ).
% transpose_knights_path
thf(fact_901_evenE,axiom,
! [A: nat] :
( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A )
=> ~ ! [B3: nat] :
( A
!= ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ B3 ) ) ) ).
% evenE
thf(fact_902_evenE,axiom,
! [A: int] :
( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A )
=> ~ ! [B3: int] :
( A
!= ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ B3 ) ) ) ).
% evenE
thf(fact_903_odd__even__add,axiom,
! [A: nat,B: nat] :
( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A )
=> ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ B )
=> ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( plus_plus_nat @ A @ B ) ) ) ) ).
% odd_even_add
thf(fact_904_odd__even__add,axiom,
! [A: int,B: int] :
( ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A )
=> ( ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ B )
=> ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( plus_plus_int @ A @ B ) ) ) ) ).
% odd_even_add
thf(fact_905_odd__one,axiom,
~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ one_one_nat ) ).
% odd_one
thf(fact_906_odd__one,axiom,
~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ one_one_int ) ).
% odd_one
thf(fact_907_even__minus,axiom,
! [A: int] :
( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( uminus_uminus_int @ A ) )
= ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A ) ) ).
% even_minus
thf(fact_908_odd__numeral,axiom,
! [N: num] :
~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( numeral_numeral_nat @ ( bit1 @ N ) ) ) ).
% odd_numeral
thf(fact_909_odd__numeral,axiom,
! [N: num] :
~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( numeral_numeral_int @ ( bit1 @ N ) ) ) ).
% odd_numeral
thf(fact_910_even__and__iff,axiom,
! [A: int,B: int] :
( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( bit_se725231765392027082nd_int @ A @ B ) )
= ( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A )
| ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ B ) ) ) ).
% even_and_iff
thf(fact_911_even__and__iff,axiom,
! [A: nat,B: nat] :
( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( bit_se727722235901077358nd_nat @ A @ B ) )
= ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A )
| ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ B ) ) ) ).
% even_and_iff
thf(fact_912_even__or__iff,axiom,
! [A: int,B: int] :
( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( bit_se1409905431419307370or_int @ A @ B ) )
= ( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A )
& ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ B ) ) ) ).
% even_or_iff
thf(fact_913_even__or__iff,axiom,
! [A: nat,B: nat] :
( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( bit_se1412395901928357646or_nat @ A @ B ) )
= ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A )
& ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ B ) ) ) ).
% even_or_iff
thf(fact_914_even__xor__iff,axiom,
! [A: nat,B: nat] :
( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( bit_se6528837805403552850or_nat @ A @ B ) )
= ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A )
= ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ B ) ) ) ).
% even_xor_iff
thf(fact_915_even__xor__iff,axiom,
! [A: int,B: int] :
( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( bit_se6526347334894502574or_int @ A @ B ) )
= ( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A )
= ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ B ) ) ) ).
% even_xor_iff
thf(fact_916_hd__trans__path,axiom,
! [Ps2: list_P5707943133018811711nt_int,I3: int,J3: int,K_12: int,K_22: int] :
( ( Ps2 != nil_Pr2300489316682597567nt_int )
=> ( ( ( hd_Pro282112905867057956nt_int @ Ps2 )
= ( product_Pair_int_int @ I3 @ J3 ) )
=> ( ( hd_Pro282112905867057956nt_int @ ( trans_path @ ( product_Pair_int_int @ K_12 @ K_22 ) @ Ps2 ) )
= ( product_Pair_int_int @ ( plus_plus_int @ I3 @ K_12 ) @ ( plus_plus_int @ J3 @ K_22 ) ) ) ) ) ).
% hd_trans_path
thf(fact_917_even__signed__take__bit__iff,axiom,
! [M: nat,A: int] :
( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( bit_ri631733984087533419it_int @ M @ A ) )
= ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A ) ) ).
% even_signed_take_bit_iff
thf(fact_918_push__bit__double,axiom,
! [N: nat,A: int] :
( ( bit_se545348938243370406it_int @ N @ ( times_times_int @ A @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) )
= ( times_times_int @ ( bit_se545348938243370406it_int @ N @ A ) @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ).
% push_bit_double
thf(fact_919_push__bit__double,axiom,
! [N: nat,A: nat] :
( ( bit_se547839408752420682it_nat @ N @ ( times_times_nat @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
= ( times_times_nat @ ( bit_se547839408752420682it_nat @ N @ A ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ).
% push_bit_double
thf(fact_920_odd__numeral__BitM,axiom,
! [W: num] :
~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( numeral_numeral_nat @ ( bitM @ W ) ) ) ).
% odd_numeral_BitM
thf(fact_921_odd__numeral__BitM,axiom,
! [W: num] :
~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( numeral_numeral_int @ ( bitM @ W ) ) ) ).
% odd_numeral_BitM
thf(fact_922_even__and__iff__int,axiom,
! [K: int,L: int] :
( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( bit_se725231765392027082nd_int @ K @ L ) )
= ( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ K )
| ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ L ) ) ) ).
% even_and_iff_int
thf(fact_923_hd__rot90__knights__path,axiom,
! [N: nat,M: nat,Ps2: list_P5707943133018811711nt_int,I3: int,J3: int] :
( ( knights_path @ ( board @ N @ M ) @ Ps2 )
=> ( ( ( hd_Pro282112905867057956nt_int @ Ps2 )
= ( product_Pair_int_int @ I3 @ J3 ) )
=> ( ( hd_Pro282112905867057956nt_int @ ( mirror1 @ ( transpose @ Ps2 ) ) )
= ( product_Pair_int_int @ ( minus_minus_int @ ( plus_plus_int @ ( semiri1314217659103216013at_int @ M ) @ one_one_int ) @ J3 ) @ I3 ) ) ) ) ).
% hd_rot90_knights_path
thf(fact_924_push__bit__int__def,axiom,
( bit_se545348938243370406it_int
= ( ^ [N3: nat,K3: int] : ( times_times_int @ K3 @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N3 ) ) ) ) ).
% push_bit_int_def
thf(fact_925_push__bit__nat__def,axiom,
( bit_se547839408752420682it_nat
= ( ^ [N3: nat,M3: nat] : ( times_times_nat @ M3 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N3 ) ) ) ) ).
% push_bit_nat_def
thf(fact_926_rot90__knights__path,axiom,
! [N: nat,M: nat,Ps2: list_P5707943133018811711nt_int] :
( ( knights_path @ ( board @ N @ M ) @ Ps2 )
=> ( knights_path @ ( board @ M @ N ) @ ( mirror1 @ ( transpose @ Ps2 ) ) ) ) ).
% rot90_knights_path
thf(fact_927_even__diff__iff,axiom,
! [K: int,L: int] :
( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( minus_minus_int @ K @ L ) )
= ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( plus_plus_int @ K @ L ) ) ) ).
% even_diff_iff
thf(fact_928_push__bit__eq__mult,axiom,
( bit_se545348938243370406it_int
= ( ^ [N3: nat,A5: int] : ( times_times_int @ A5 @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N3 ) ) ) ) ).
% push_bit_eq_mult
thf(fact_929_push__bit__eq__mult,axiom,
( bit_se547839408752420682it_nat
= ( ^ [N3: nat,A5: nat] : ( times_times_nat @ A5 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N3 ) ) ) ) ).
% push_bit_eq_mult
thf(fact_930_oddE,axiom,
! [A: nat] :
( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A )
=> ~ ! [B3: nat] :
( A
!= ( plus_plus_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ B3 ) @ one_one_nat ) ) ) ).
% oddE
thf(fact_931_oddE,axiom,
! [A: int] :
( ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A )
=> ~ ! [B3: int] :
( A
!= ( plus_plus_int @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ B3 ) @ one_one_int ) ) ) ).
% oddE
thf(fact_932_push__bit__minus__one,axiom,
! [N: nat] :
( ( bit_se545348938243370406it_int @ N @ ( uminus_uminus_int @ one_one_int ) )
= ( uminus_uminus_int @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) ) ) ).
% push_bit_minus_one
thf(fact_933_hd__mirror1,axiom,
! [N: nat,M: nat,Ps2: list_P5707943133018811711nt_int,I3: int,J3: int] :
( ( knights_path @ ( board @ N @ M ) @ Ps2 )
=> ( ( ( hd_Pro282112905867057956nt_int @ Ps2 )
= ( product_Pair_int_int @ I3 @ J3 ) )
=> ( ( hd_Pro282112905867057956nt_int @ ( mirror1 @ Ps2 ) )
= ( product_Pair_int_int @ ( minus_minus_int @ ( plus_plus_int @ ( semiri1314217659103216013at_int @ N ) @ one_one_int ) @ I3 ) @ J3 ) ) ) ) ).
% hd_mirror1
thf(fact_934_dvd__numeral__simp,axiom,
! [M: num,N: num] :
( ( dvd_dvd_nat @ ( numeral_numeral_nat @ M ) @ ( numeral_numeral_nat @ N ) )
= ( unique5332122412489317741ux_nat @ ( unique5405566460079783412od_nat @ N @ M ) ) ) ).
% dvd_numeral_simp
thf(fact_935_dvd__numeral__simp,axiom,
! [M: num,N: num] :
( ( dvd_dvd_int @ ( numeral_numeral_int @ M ) @ ( numeral_numeral_int @ N ) )
= ( unique5329631941980267465ux_int @ ( unique5403075989570733136od_int @ N @ M ) ) ) ).
% dvd_numeral_simp
thf(fact_936_inf__period_I3_J,axiom,
! [D: int,D2: int,T: int] :
( ( dvd_dvd_int @ D @ D2 )
=> ! [X5: int,K4: int] :
( ( dvd_dvd_int @ D @ ( plus_plus_int @ X5 @ T ) )
= ( dvd_dvd_int @ D @ ( plus_plus_int @ ( minus_minus_int @ X5 @ ( times_times_int @ K4 @ D2 ) ) @ T ) ) ) ) ).
% inf_period(3)
thf(fact_937_inf__period_I4_J,axiom,
! [D: int,D2: int,T: int] :
( ( dvd_dvd_int @ D @ D2 )
=> ! [X5: int,K4: int] :
( ( ~ ( dvd_dvd_int @ D @ ( plus_plus_int @ X5 @ T ) ) )
= ( ~ ( dvd_dvd_int @ D @ ( plus_plus_int @ ( minus_minus_int @ X5 @ ( times_times_int @ K4 @ D2 ) ) @ T ) ) ) ) ) ).
% inf_period(4)
thf(fact_938_nat__dvd__1__iff__1,axiom,
! [M: nat] :
( ( dvd_dvd_nat @ M @ one_one_nat )
= ( M = one_one_nat ) ) ).
% nat_dvd_1_iff_1
thf(fact_939_even__Suc__Suc__iff,axiom,
! [N: nat] :
( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( suc @ ( suc @ N ) ) )
= ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ).
% even_Suc_Suc_iff
thf(fact_940_even__Suc,axiom,
! [N: nat] :
( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( suc @ N ) )
= ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ) ).
% even_Suc
thf(fact_941_Parity_Oring__1__class_Opower__minus__even,axiom,
! [N: nat,A: int] :
( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
=> ( ( power_power_int @ ( uminus_uminus_int @ A ) @ N )
= ( power_power_int @ A @ N ) ) ) ).
% Parity.ring_1_class.power_minus_even
thf(fact_942_power__minus__odd,axiom,
! [N: nat,A: int] :
( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
=> ( ( power_power_int @ ( uminus_uminus_int @ A ) @ N )
= ( uminus_uminus_int @ ( power_power_int @ A @ N ) ) ) ) ).
% power_minus_odd
thf(fact_943_neg__one__even__power,axiom,
! [N: nat] :
( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
=> ( ( power_power_int @ ( uminus_uminus_int @ one_one_int ) @ N )
= one_one_int ) ) ).
% neg_one_even_power
thf(fact_944_neg__one__odd__power,axiom,
! [N: nat] :
( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
=> ( ( power_power_int @ ( uminus_uminus_int @ one_one_int ) @ N )
= ( uminus_uminus_int @ one_one_int ) ) ) ).
% neg_one_odd_power
thf(fact_945_dvd__antisym,axiom,
! [M: nat,N: nat] :
( ( dvd_dvd_nat @ M @ N )
=> ( ( dvd_dvd_nat @ N @ M )
=> ( M = N ) ) ) ).
% dvd_antisym
thf(fact_946_dvd__diff__nat,axiom,
! [K: nat,M: nat,N: nat] :
( ( dvd_dvd_nat @ K @ M )
=> ( ( dvd_dvd_nat @ K @ N )
=> ( dvd_dvd_nat @ K @ ( minus_minus_nat @ M @ N ) ) ) ) ).
% dvd_diff_nat
thf(fact_947_uminus__power__if,axiom,
! [N: nat,A: int] :
( ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
=> ( ( power_power_int @ ( uminus_uminus_int @ A ) @ N )
= ( power_power_int @ A @ N ) ) )
& ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
=> ( ( power_power_int @ ( uminus_uminus_int @ A ) @ N )
= ( uminus_uminus_int @ ( power_power_int @ A @ N ) ) ) ) ) ).
% uminus_power_if
thf(fact_948_minus__one__power__iff,axiom,
! [N: nat] :
( ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
=> ( ( power_power_int @ ( uminus_uminus_int @ one_one_int ) @ N )
= one_one_int ) )
& ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
=> ( ( power_power_int @ ( uminus_uminus_int @ one_one_int ) @ N )
= ( uminus_uminus_int @ one_one_int ) ) ) ) ).
% minus_one_power_iff
thf(fact_949_inf__period_I2_J,axiom,
! [P: int > $o,D2: int,Q: int > $o] :
( ! [X2: int,K2: int] :
( ( P @ X2 )
= ( P @ ( minus_minus_int @ X2 @ ( times_times_int @ K2 @ D2 ) ) ) )
=> ( ! [X2: int,K2: int] :
( ( Q @ X2 )
= ( Q @ ( minus_minus_int @ X2 @ ( times_times_int @ K2 @ D2 ) ) ) )
=> ! [X5: int,K4: int] :
( ( ( P @ X5 )
| ( Q @ X5 ) )
= ( ( P @ ( minus_minus_int @ X5 @ ( times_times_int @ K4 @ D2 ) ) )
| ( Q @ ( minus_minus_int @ X5 @ ( times_times_int @ K4 @ D2 ) ) ) ) ) ) ) ).
% inf_period(2)
thf(fact_950_inf__period_I1_J,axiom,
! [P: int > $o,D2: int,Q: int > $o] :
( ! [X2: int,K2: int] :
( ( P @ X2 )
= ( P @ ( minus_minus_int @ X2 @ ( times_times_int @ K2 @ D2 ) ) ) )
=> ( ! [X2: int,K2: int] :
( ( Q @ X2 )
= ( Q @ ( minus_minus_int @ X2 @ ( times_times_int @ K2 @ D2 ) ) ) )
=> ! [X5: int,K4: int] :
( ( ( P @ X5 )
& ( Q @ X5 ) )
= ( ( P @ ( minus_minus_int @ X5 @ ( times_times_int @ K4 @ D2 ) ) )
& ( Q @ ( minus_minus_int @ X5 @ ( times_times_int @ K4 @ D2 ) ) ) ) ) ) ) ).
% inf_period(1)
thf(fact_951_uminus__dvd__conv_I1_J,axiom,
( dvd_dvd_int
= ( ^ [D3: int] : ( dvd_dvd_int @ ( uminus_uminus_int @ D3 ) ) ) ) ).
% uminus_dvd_conv(1)
thf(fact_952_uminus__dvd__conv_I2_J,axiom,
( dvd_dvd_int
= ( ^ [D3: int,T2: int] : ( dvd_dvd_int @ D3 @ ( uminus_uminus_int @ T2 ) ) ) ) ).
% uminus_dvd_conv(2)
thf(fact_953_bezout1__nat,axiom,
! [A: nat,B: nat] :
? [D4: nat,X2: nat,Y: nat] :
( ( dvd_dvd_nat @ D4 @ A )
& ( dvd_dvd_nat @ D4 @ B )
& ( ( ( minus_minus_nat @ ( times_times_nat @ A @ X2 ) @ ( times_times_nat @ B @ Y ) )
= D4 )
| ( ( minus_minus_nat @ ( times_times_nat @ B @ X2 ) @ ( times_times_nat @ A @ Y ) )
= D4 ) ) ) ).
% bezout1_nat
thf(fact_954_bezout__lemma__nat,axiom,
! [D: nat,A: nat,B: nat,X: nat,Y3: nat] :
( ( dvd_dvd_nat @ D @ A )
=> ( ( dvd_dvd_nat @ D @ B )
=> ( ( ( ( times_times_nat @ A @ X )
= ( plus_plus_nat @ ( times_times_nat @ B @ Y3 ) @ D ) )
| ( ( times_times_nat @ B @ X )
= ( plus_plus_nat @ ( times_times_nat @ A @ Y3 ) @ D ) ) )
=> ? [X2: nat,Y: nat] :
( ( dvd_dvd_nat @ D @ A )
& ( dvd_dvd_nat @ D @ ( plus_plus_nat @ A @ B ) )
& ( ( ( times_times_nat @ A @ X2 )
= ( plus_plus_nat @ ( times_times_nat @ ( plus_plus_nat @ A @ B ) @ Y ) @ D ) )
| ( ( times_times_nat @ ( plus_plus_nat @ A @ B ) @ X2 )
= ( plus_plus_nat @ ( times_times_nat @ A @ Y ) @ D ) ) ) ) ) ) ) ).
% bezout_lemma_nat
thf(fact_955_bezout__add__nat,axiom,
! [A: nat,B: nat] :
? [D4: nat,X2: nat,Y: nat] :
( ( dvd_dvd_nat @ D4 @ A )
& ( dvd_dvd_nat @ D4 @ B )
& ( ( ( times_times_nat @ A @ X2 )
= ( plus_plus_nat @ ( times_times_nat @ B @ Y ) @ D4 ) )
| ( ( times_times_nat @ B @ X2 )
= ( plus_plus_nat @ ( times_times_nat @ A @ Y ) @ D4 ) ) ) ) ).
% bezout_add_nat
thf(fact_956_division__decomp,axiom,
! [A: nat,B: nat,C2: nat] :
( ( dvd_dvd_nat @ A @ ( times_times_nat @ B @ C2 ) )
=> ? [B6: nat,C3: nat] :
( ( A
= ( times_times_nat @ B6 @ C3 ) )
& ( dvd_dvd_nat @ B6 @ B )
& ( dvd_dvd_nat @ C3 @ C2 ) ) ) ).
% division_decomp
thf(fact_957_division__decomp,axiom,
! [A: int,B: int,C2: int] :
( ( dvd_dvd_int @ A @ ( times_times_int @ B @ C2 ) )
=> ? [B6: int,C3: int] :
( ( A
= ( times_times_int @ B6 @ C3 ) )
& ( dvd_dvd_int @ B6 @ B )
& ( dvd_dvd_int @ C3 @ C2 ) ) ) ).
% division_decomp
thf(fact_958_dvd__productE,axiom,
! [P3: nat,A: nat,B: nat] :
( ( dvd_dvd_nat @ P3 @ ( times_times_nat @ A @ B ) )
=> ~ ! [X2: nat,Y: nat] :
( ( P3
= ( times_times_nat @ X2 @ Y ) )
=> ( ( dvd_dvd_nat @ X2 @ A )
=> ~ ( dvd_dvd_nat @ Y @ B ) ) ) ) ).
% dvd_productE
thf(fact_959_dvd__productE,axiom,
! [P3: int,A: int,B: int] :
( ( dvd_dvd_int @ P3 @ ( times_times_int @ A @ B ) )
=> ~ ! [X2: int,Y: int] :
( ( P3
= ( times_times_int @ X2 @ Y ) )
=> ( ( dvd_dvd_int @ X2 @ A )
=> ~ ( dvd_dvd_int @ Y @ B ) ) ) ) ).
% dvd_productE
thf(fact_960_semiring__parity__class_Oeven__mask__iff,axiom,
! [N: nat] :
( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( minus_minus_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) @ one_one_nat ) )
= ( N = zero_zero_nat ) ) ).
% semiring_parity_class.even_mask_iff
thf(fact_961_semiring__parity__class_Oeven__mask__iff,axiom,
! [N: nat] :
( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( minus_minus_int @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) @ one_one_int ) )
= ( N = zero_zero_nat ) ) ).
% semiring_parity_class.even_mask_iff
thf(fact_962_knights__circuit__lr__concat,axiom,
! [N: nat,M_1: nat,Ps_1: list_P5707943133018811711nt_int,M_2: nat,Ps_2: list_P5707943133018811711nt_int] :
( ( knights_circuit @ ( board @ N @ M_1 ) @ Ps_1 )
=> ( ( knights_circuit @ ( board @ N @ M_2 ) @ Ps_2 )
=> ( ( step_in @ Ps_1 @ ( product_Pair_int_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( minus_minus_int @ ( semiri1314217659103216013at_int @ M_1 ) @ one_one_int ) ) @ ( product_Pair_int_int @ ( numeral_numeral_int @ ( bit0 @ ( bit0 @ one ) ) ) @ ( semiri1314217659103216013at_int @ M_1 ) ) )
=> ( ( ( hd_Pro282112905867057956nt_int @ Ps_2 )
= ( product_Pair_int_int @ one_one_int @ one_one_int ) )
=> ( ( ( last_P3305686521732843992nt_int @ Ps_2 )
= ( product_Pair_int_int @ ( numeral_numeral_int @ ( bit1 @ one ) ) @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) )
=> ( ( step_in @ Ps_2 @ ( product_Pair_int_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( minus_minus_int @ ( semiri1314217659103216013at_int @ M_2 ) @ one_one_int ) ) @ ( product_Pair_int_int @ ( numeral_numeral_int @ ( bit0 @ ( bit0 @ one ) ) ) @ ( semiri1314217659103216013at_int @ M_2 ) ) )
=> ? [Ps: list_P5707943133018811711nt_int] :
( ( knights_circuit @ ( board @ N @ ( plus_plus_nat @ M_1 @ M_2 ) ) @ Ps )
& ( step_in @ Ps @ ( product_Pair_int_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( minus_minus_int @ ( semiri1314217659103216013at_int @ ( plus_plus_nat @ M_1 @ M_2 ) ) @ one_one_int ) ) @ ( product_Pair_int_int @ ( numeral_numeral_int @ ( bit0 @ ( bit0 @ one ) ) ) @ ( semiri1314217659103216013at_int @ ( plus_plus_nat @ M_1 @ M_2 ) ) ) ) ) ) ) ) ) ) ) ).
% knights_circuit_lr_concat
thf(fact_963_push__bit__of__Suc__0,axiom,
! [N: nat] :
( ( bit_se547839408752420682it_nat @ N @ ( suc @ zero_zero_nat ) )
= ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ).
% push_bit_of_Suc_0
thf(fact_964_mult__cancel__right,axiom,
! [A: nat,C2: nat,B: nat] :
( ( ( times_times_nat @ A @ C2 )
= ( times_times_nat @ B @ C2 ) )
= ( ( C2 = zero_zero_nat )
| ( A = B ) ) ) ).
% mult_cancel_right
thf(fact_965_mult__cancel__right,axiom,
! [A: int,C2: int,B: int] :
( ( ( times_times_int @ A @ C2 )
= ( times_times_int @ B @ C2 ) )
= ( ( C2 = zero_zero_int )
| ( A = B ) ) ) ).
% mult_cancel_right
thf(fact_966_mult__cancel__left,axiom,
! [C2: nat,A: nat,B: nat] :
( ( ( times_times_nat @ C2 @ A )
= ( times_times_nat @ C2 @ B ) )
= ( ( C2 = zero_zero_nat )
| ( A = B ) ) ) ).
% mult_cancel_left
thf(fact_967_mult__cancel__left,axiom,
! [C2: int,A: int,B: int] :
( ( ( times_times_int @ C2 @ A )
= ( times_times_int @ C2 @ B ) )
= ( ( C2 = zero_zero_int )
| ( A = B ) ) ) ).
% mult_cancel_left
thf(fact_968_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_969_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_970_mult__zero__right,axiom,
! [A: nat] :
( ( times_times_nat @ A @ zero_zero_nat )
= zero_zero_nat ) ).
% mult_zero_right
thf(fact_971_mult__zero__right,axiom,
! [A: int] :
( ( times_times_int @ A @ zero_zero_int )
= zero_zero_int ) ).
% mult_zero_right
thf(fact_972_mult__zero__left,axiom,
! [A: nat] :
( ( times_times_nat @ zero_zero_nat @ A )
= zero_zero_nat ) ).
% mult_zero_left
thf(fact_973_mult__zero__left,axiom,
! [A: int] :
( ( times_times_int @ zero_zero_int @ A )
= zero_zero_int ) ).
% mult_zero_left
thf(fact_974_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_975_add_Oright__neutral,axiom,
! [A: nat] :
( ( plus_plus_nat @ A @ zero_zero_nat )
= A ) ).
% add.right_neutral
thf(fact_976_add_Oright__neutral,axiom,
! [A: int] :
( ( plus_plus_int @ A @ zero_zero_int )
= A ) ).
% add.right_neutral
thf(fact_977_double__zero__sym,axiom,
! [A: int] :
( ( zero_zero_int
= ( plus_plus_int @ A @ A ) )
= ( A = zero_zero_int ) ) ).
% double_zero_sym
thf(fact_978_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_979_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_980_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_981_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_982_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_983_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_984_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_985_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_986_add__eq__0__iff__both__eq__0,axiom,
! [X: nat,Y3: nat] :
( ( ( plus_plus_nat @ X @ Y3 )
= zero_zero_nat )
= ( ( X = zero_zero_nat )
& ( Y3 = zero_zero_nat ) ) ) ).
% add_eq_0_iff_both_eq_0
thf(fact_987_zero__eq__add__iff__both__eq__0,axiom,
! [X: nat,Y3: nat] :
( ( zero_zero_nat
= ( plus_plus_nat @ X @ Y3 ) )
= ( ( X = zero_zero_nat )
& ( Y3 = zero_zero_nat ) ) ) ).
% zero_eq_add_iff_both_eq_0
thf(fact_988_add__0,axiom,
! [A: nat] :
( ( plus_plus_nat @ zero_zero_nat @ A )
= A ) ).
% add_0
thf(fact_989_add__0,axiom,
! [A: int] :
( ( plus_plus_int @ zero_zero_int @ A )
= A ) ).
% add_0
thf(fact_990_diff__self,axiom,
! [A: int] :
( ( minus_minus_int @ A @ A )
= zero_zero_int ) ).
% diff_self
thf(fact_991_diff__0__right,axiom,
! [A: int] :
( ( minus_minus_int @ A @ zero_zero_int )
= A ) ).
% diff_0_right
thf(fact_992_zero__diff,axiom,
! [A: nat] :
( ( minus_minus_nat @ zero_zero_nat @ A )
= zero_zero_nat ) ).
% zero_diff
thf(fact_993_diff__zero,axiom,
! [A: int] :
( ( minus_minus_int @ A @ zero_zero_int )
= A ) ).
% diff_zero
thf(fact_994_diff__zero,axiom,
! [A: nat] :
( ( minus_minus_nat @ A @ zero_zero_nat )
= A ) ).
% diff_zero
thf(fact_995_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_996_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_997_add_Oinverse__neutral,axiom,
( ( uminus_uminus_int @ zero_zero_int )
= zero_zero_int ) ).
% add.inverse_neutral
thf(fact_998_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_999_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_1000_equal__neg__zero,axiom,
! [A: int] :
( ( A
= ( uminus_uminus_int @ A ) )
= ( A = zero_zero_int ) ) ).
% equal_neg_zero
thf(fact_1001_neg__equal__zero,axiom,
! [A: int] :
( ( ( uminus_uminus_int @ A )
= A )
= ( A = zero_zero_int ) ) ).
% neg_equal_zero
thf(fact_1002_Nat_Oadd__0__right,axiom,
! [M: nat] :
( ( plus_plus_nat @ M @ zero_zero_nat )
= M ) ).
% Nat.add_0_right
thf(fact_1003_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_1004_diff__0__eq__0,axiom,
! [N: nat] :
( ( minus_minus_nat @ zero_zero_nat @ N )
= zero_zero_nat ) ).
% diff_0_eq_0
thf(fact_1005_diff__self__eq__0,axiom,
! [M: nat] :
( ( minus_minus_nat @ M @ M )
= zero_zero_nat ) ).
% diff_self_eq_0
thf(fact_1006_take__bit__of__0,axiom,
! [N: nat] :
( ( bit_se2925701944663578781it_nat @ N @ zero_zero_nat )
= zero_zero_nat ) ).
% take_bit_of_0
thf(fact_1007_take__bit__of__0,axiom,
! [N: nat] :
( ( bit_se2923211474154528505it_int @ N @ zero_zero_int )
= zero_zero_int ) ).
% take_bit_of_0
thf(fact_1008_and__zero__eq,axiom,
! [A: int] :
( ( bit_se725231765392027082nd_int @ A @ zero_zero_int )
= zero_zero_int ) ).
% and_zero_eq
thf(fact_1009_and__zero__eq,axiom,
! [A: nat] :
( ( bit_se727722235901077358nd_nat @ A @ zero_zero_nat )
= zero_zero_nat ) ).
% and_zero_eq
thf(fact_1010_zero__and__eq,axiom,
! [A: int] :
( ( bit_se725231765392027082nd_int @ zero_zero_int @ A )
= zero_zero_int ) ).
% zero_and_eq
thf(fact_1011_zero__and__eq,axiom,
! [A: nat] :
( ( bit_se727722235901077358nd_nat @ zero_zero_nat @ A )
= zero_zero_nat ) ).
% zero_and_eq
thf(fact_1012_bit_Oconj__zero__left,axiom,
! [X: int] :
( ( bit_se725231765392027082nd_int @ zero_zero_int @ X )
= zero_zero_int ) ).
% bit.conj_zero_left
thf(fact_1013_bit_Oconj__zero__right,axiom,
! [X: int] :
( ( bit_se725231765392027082nd_int @ X @ zero_zero_int )
= zero_zero_int ) ).
% bit.conj_zero_right
thf(fact_1014_mult__cancel2,axiom,
! [M: nat,K: nat,N: nat] :
( ( ( times_times_nat @ M @ K )
= ( times_times_nat @ N @ K ) )
= ( ( M = N )
| ( K = zero_zero_nat ) ) ) ).
% mult_cancel2
thf(fact_1015_mult__cancel1,axiom,
! [K: nat,M: nat,N: nat] :
( ( ( times_times_nat @ K @ M )
= ( times_times_nat @ K @ N ) )
= ( ( M = N )
| ( K = zero_zero_nat ) ) ) ).
% mult_cancel1
thf(fact_1016_mult__0__right,axiom,
! [M: nat] :
( ( times_times_nat @ M @ zero_zero_nat )
= zero_zero_nat ) ).
% mult_0_right
thf(fact_1017_mult__is__0,axiom,
! [M: nat,N: nat] :
( ( ( times_times_nat @ M @ N )
= zero_zero_nat )
= ( ( M = zero_zero_nat )
| ( N = zero_zero_nat ) ) ) ).
% mult_is_0
thf(fact_1018_or_Oleft__neutral,axiom,
! [A: int] :
( ( bit_se1409905431419307370or_int @ zero_zero_int @ A )
= A ) ).
% or.left_neutral
thf(fact_1019_or_Oleft__neutral,axiom,
! [A: nat] :
( ( bit_se1412395901928357646or_nat @ zero_zero_nat @ A )
= A ) ).
% or.left_neutral
thf(fact_1020_or_Oright__neutral,axiom,
! [A: int] :
( ( bit_se1409905431419307370or_int @ A @ zero_zero_int )
= A ) ).
% or.right_neutral
thf(fact_1021_or_Oright__neutral,axiom,
! [A: nat] :
( ( bit_se1412395901928357646or_nat @ A @ zero_zero_nat )
= A ) ).
% or.right_neutral
thf(fact_1022_bit_Oxor__self,axiom,
! [X: int] :
( ( bit_se6526347334894502574or_int @ X @ X )
= zero_zero_int ) ).
% bit.xor_self
thf(fact_1023_xor__self__eq,axiom,
! [A: nat] :
( ( bit_se6528837805403552850or_nat @ A @ A )
= zero_zero_nat ) ).
% xor_self_eq
thf(fact_1024_xor__self__eq,axiom,
! [A: int] :
( ( bit_se6526347334894502574or_int @ A @ A )
= zero_zero_int ) ).
% xor_self_eq
thf(fact_1025_xor_Oleft__neutral,axiom,
! [A: nat] :
( ( bit_se6528837805403552850or_nat @ zero_zero_nat @ A )
= A ) ).
% xor.left_neutral
thf(fact_1026_xor_Oleft__neutral,axiom,
! [A: int] :
( ( bit_se6526347334894502574or_int @ zero_zero_int @ A )
= A ) ).
% xor.left_neutral
thf(fact_1027_xor_Oright__neutral,axiom,
! [A: nat] :
( ( bit_se6528837805403552850or_nat @ A @ zero_zero_nat )
= A ) ).
% xor.right_neutral
thf(fact_1028_xor_Oright__neutral,axiom,
! [A: int] :
( ( bit_se6526347334894502574or_int @ A @ zero_zero_int )
= A ) ).
% xor.right_neutral
thf(fact_1029_push__bit__of__0,axiom,
! [N: nat] :
( ( bit_se545348938243370406it_int @ N @ zero_zero_int )
= zero_zero_int ) ).
% push_bit_of_0
thf(fact_1030_push__bit__of__0,axiom,
! [N: nat] :
( ( bit_se547839408752420682it_nat @ N @ zero_zero_nat )
= zero_zero_nat ) ).
% push_bit_of_0
thf(fact_1031_push__bit__eq__0__iff,axiom,
! [N: nat,A: int] :
( ( ( bit_se545348938243370406it_int @ N @ A )
= zero_zero_int )
= ( A = zero_zero_int ) ) ).
% push_bit_eq_0_iff
thf(fact_1032_push__bit__eq__0__iff,axiom,
! [N: nat,A: nat] :
( ( ( bit_se547839408752420682it_nat @ N @ A )
= zero_zero_nat )
= ( A = zero_zero_nat ) ) ).
% push_bit_eq_0_iff
thf(fact_1033_signed__take__bit__of__0,axiom,
! [N: nat] :
( ( bit_ri631733984087533419it_int @ N @ zero_zero_int )
= zero_zero_int ) ).
% signed_take_bit_of_0
thf(fact_1034_dbl__simps_I2_J,axiom,
( ( neg_numeral_dbl_int @ zero_zero_int )
= zero_zero_int ) ).
% dbl_simps(2)
thf(fact_1035_mult__cancel__right2,axiom,
! [A: int,C2: int] :
( ( ( times_times_int @ A @ C2 )
= C2 )
= ( ( C2 = zero_zero_int )
| ( A = one_one_int ) ) ) ).
% mult_cancel_right2
thf(fact_1036_mult__cancel__right1,axiom,
! [C2: int,B: int] :
( ( C2
= ( times_times_int @ B @ C2 ) )
= ( ( C2 = zero_zero_int )
| ( B = one_one_int ) ) ) ).
% mult_cancel_right1
thf(fact_1037_mult__cancel__left2,axiom,
! [C2: int,A: int] :
( ( ( times_times_int @ C2 @ A )
= C2 )
= ( ( C2 = zero_zero_int )
| ( A = one_one_int ) ) ) ).
% mult_cancel_left2
thf(fact_1038_mult__cancel__left1,axiom,
! [C2: int,B: int] :
( ( C2
= ( times_times_int @ C2 @ B ) )
= ( ( C2 = zero_zero_int )
| ( B = one_one_int ) ) ) ).
% mult_cancel_left1
thf(fact_1039_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_1040_diff__numeral__special_I9_J,axiom,
( ( minus_minus_int @ one_one_int @ one_one_int )
= zero_zero_int ) ).
% diff_numeral_special(9)
thf(fact_1041_add_Oright__inverse,axiom,
! [A: int] :
( ( plus_plus_int @ A @ ( uminus_uminus_int @ A ) )
= zero_zero_int ) ).
% add.right_inverse
thf(fact_1042_ab__left__minus,axiom,
! [A: int] :
( ( plus_plus_int @ ( uminus_uminus_int @ A ) @ A )
= zero_zero_int ) ).
% ab_left_minus
thf(fact_1043_dvd__mult__cancel__left,axiom,
! [C2: int,A: int,B: int] :
( ( dvd_dvd_int @ ( times_times_int @ C2 @ A ) @ ( times_times_int @ C2 @ B ) )
= ( ( C2 = zero_zero_int )
| ( dvd_dvd_int @ A @ B ) ) ) ).
% dvd_mult_cancel_left
thf(fact_1044_dvd__mult__cancel__right,axiom,
! [A: int,C2: int,B: int] :
( ( dvd_dvd_int @ ( times_times_int @ A @ C2 ) @ ( times_times_int @ B @ C2 ) )
= ( ( C2 = zero_zero_int )
| ( dvd_dvd_int @ A @ B ) ) ) ).
% dvd_mult_cancel_right
thf(fact_1045_dvd__times__left__cancel__iff,axiom,
! [A: nat,B: nat,C2: nat] :
( ( A != zero_zero_nat )
=> ( ( dvd_dvd_nat @ ( times_times_nat @ A @ B ) @ ( times_times_nat @ A @ C2 ) )
= ( dvd_dvd_nat @ B @ C2 ) ) ) ).
% dvd_times_left_cancel_iff
thf(fact_1046_dvd__times__left__cancel__iff,axiom,
! [A: int,B: int,C2: int] :
( ( A != zero_zero_int )
=> ( ( dvd_dvd_int @ ( times_times_int @ A @ B ) @ ( times_times_int @ A @ C2 ) )
= ( dvd_dvd_int @ B @ C2 ) ) ) ).
% dvd_times_left_cancel_iff
thf(fact_1047_dvd__times__right__cancel__iff,axiom,
! [A: nat,B: nat,C2: nat] :
( ( A != zero_zero_nat )
=> ( ( dvd_dvd_nat @ ( times_times_nat @ B @ A ) @ ( times_times_nat @ C2 @ A ) )
= ( dvd_dvd_nat @ B @ C2 ) ) ) ).
% dvd_times_right_cancel_iff
thf(fact_1048_dvd__times__right__cancel__iff,axiom,
! [A: int,B: int,C2: int] :
( ( A != zero_zero_int )
=> ( ( dvd_dvd_int @ ( times_times_int @ B @ A ) @ ( times_times_int @ C2 @ A ) )
= ( dvd_dvd_int @ B @ C2 ) ) ) ).
% dvd_times_right_cancel_iff
thf(fact_1049_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_1050_diff__0,axiom,
! [A: int] :
( ( minus_minus_int @ zero_zero_int @ A )
= ( uminus_uminus_int @ A ) ) ).
% diff_0
thf(fact_1051_power__0__Suc,axiom,
! [N: nat] :
( ( power_power_int @ zero_zero_int @ ( suc @ N ) )
= zero_zero_int ) ).
% power_0_Suc
thf(fact_1052_power__0__Suc,axiom,
! [N: nat] :
( ( power_power_nat @ zero_zero_nat @ ( suc @ N ) )
= zero_zero_nat ) ).
% power_0_Suc
thf(fact_1053_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_1054_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_1055_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_1056_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_1057_of__nat__0,axiom,
( ( semiri1316708129612266289at_nat @ zero_zero_nat )
= zero_zero_nat ) ).
% of_nat_0
thf(fact_1058_of__nat__0,axiom,
( ( semiri1314217659103216013at_int @ zero_zero_nat )
= zero_zero_int ) ).
% of_nat_0
thf(fact_1059_power__zero__numeral,axiom,
! [K: num] :
( ( power_power_int @ zero_zero_int @ ( numeral_numeral_nat @ K ) )
= zero_zero_int ) ).
% power_zero_numeral
thf(fact_1060_power__zero__numeral,axiom,
! [K: num] :
( ( power_power_nat @ zero_zero_nat @ ( numeral_numeral_nat @ K ) )
= zero_zero_nat ) ).
% power_zero_numeral
thf(fact_1061_power__Suc0__right,axiom,
! [A: int] :
( ( power_power_int @ A @ ( suc @ zero_zero_nat ) )
= A ) ).
% power_Suc0_right
thf(fact_1062_power__Suc0__right,axiom,
! [A: nat] :
( ( power_power_nat @ A @ ( suc @ zero_zero_nat ) )
= A ) ).
% power_Suc0_right
thf(fact_1063_take__bit__0,axiom,
! [A: nat] :
( ( bit_se2925701944663578781it_nat @ zero_zero_nat @ A )
= zero_zero_nat ) ).
% take_bit_0
thf(fact_1064_take__bit__0,axiom,
! [A: int] :
( ( bit_se2923211474154528505it_int @ zero_zero_nat @ A )
= zero_zero_int ) ).
% take_bit_0
thf(fact_1065_mult__eq__1__iff,axiom,
! [M: nat,N: nat] :
( ( ( times_times_nat @ M @ N )
= ( suc @ zero_zero_nat ) )
= ( ( M
= ( suc @ zero_zero_nat ) )
& ( N
= ( suc @ zero_zero_nat ) ) ) ) ).
% mult_eq_1_iff
thf(fact_1066_one__eq__mult__iff,axiom,
! [M: nat,N: nat] :
( ( ( suc @ zero_zero_nat )
= ( times_times_nat @ M @ N ) )
= ( ( M
= ( suc @ zero_zero_nat ) )
& ( N
= ( suc @ zero_zero_nat ) ) ) ) ).
% one_eq_mult_iff
thf(fact_1067_dvd__1__iff__1,axiom,
! [M: nat] :
( ( dvd_dvd_nat @ M @ ( suc @ zero_zero_nat ) )
= ( M
= ( suc @ zero_zero_nat ) ) ) ).
% dvd_1_iff_1
thf(fact_1068_dvd__1__left,axiom,
! [K: nat] : ( dvd_dvd_nat @ ( suc @ zero_zero_nat ) @ K ) ).
% dvd_1_left
thf(fact_1069_nat__mult__dvd__cancel__disj,axiom,
! [K: nat,M: nat,N: nat] :
( ( dvd_dvd_nat @ ( times_times_nat @ K @ M ) @ ( times_times_nat @ K @ N ) )
= ( ( K = zero_zero_nat )
| ( dvd_dvd_nat @ M @ N ) ) ) ).
% nat_mult_dvd_cancel_disj
thf(fact_1070_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_1071_mask__eq__0__iff,axiom,
! [N: nat] :
( ( ( bit_se2002935070580805687sk_nat @ N )
= zero_zero_nat )
= ( N = zero_zero_nat ) ) ).
% mask_eq_0_iff
thf(fact_1072_mask__eq__0__iff,axiom,
! [N: nat] :
( ( ( bit_se2000444600071755411sk_int @ N )
= zero_zero_int )
= ( N = zero_zero_nat ) ) ).
% mask_eq_0_iff
thf(fact_1073_mask__0,axiom,
( ( bit_se2002935070580805687sk_nat @ zero_zero_nat )
= zero_zero_nat ) ).
% mask_0
thf(fact_1074_mask__0,axiom,
( ( bit_se2000444600071755411sk_int @ zero_zero_nat )
= zero_zero_int ) ).
% mask_0
thf(fact_1075_sub__num__simps_I1_J,axiom,
( ( neg_numeral_sub_int @ one @ one )
= zero_zero_int ) ).
% sub_num_simps(1)
thf(fact_1076_power__Suc__0,axiom,
! [N: nat] :
( ( power_power_nat @ ( suc @ zero_zero_nat ) @ N )
= ( suc @ zero_zero_nat ) ) ).
% power_Suc_0
thf(fact_1077_nat__power__eq__Suc__0__iff,axiom,
! [X: nat,M: nat] :
( ( ( power_power_nat @ X @ M )
= ( suc @ zero_zero_nat ) )
= ( ( M = zero_zero_nat )
| ( X
= ( suc @ zero_zero_nat ) ) ) ) ).
% nat_power_eq_Suc_0_iff
thf(fact_1078_pred__numeral__simps_I1_J,axiom,
( ( pred_numeral @ one )
= zero_zero_nat ) ).
% pred_numeral_simps(1)
thf(fact_1079_dbl__inc__simps_I2_J,axiom,
( ( neg_nu5851722552734809277nc_int @ zero_zero_int )
= one_one_int ) ).
% dbl_inc_simps(2)
thf(fact_1080_divides__aux__eq,axiom,
! [Q2: nat,R2: nat] :
( ( unique5332122412489317741ux_nat @ ( product_Pair_nat_nat @ Q2 @ R2 ) )
= ( R2 = zero_zero_nat ) ) ).
% divides_aux_eq
thf(fact_1081_divides__aux__eq,axiom,
! [Q2: int,R2: int] :
( ( unique5329631941980267465ux_int @ ( product_Pair_int_int @ Q2 @ R2 ) )
= ( R2 = zero_zero_int ) ) ).
% divides_aux_eq
thf(fact_1082_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_1083_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_1084_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_1085_take__bit__of__1__eq__0__iff,axiom,
! [N: nat] :
( ( ( bit_se2925701944663578781it_nat @ N @ one_one_nat )
= zero_zero_nat )
= ( N = zero_zero_nat ) ) ).
% take_bit_of_1_eq_0_iff
thf(fact_1086_take__bit__of__1__eq__0__iff,axiom,
! [N: nat] :
( ( ( bit_se2923211474154528505it_int @ N @ one_one_int )
= zero_zero_int )
= ( N = zero_zero_nat ) ) ).
% take_bit_of_1_eq_0_iff
thf(fact_1087_mask__Suc__0,axiom,
( ( bit_se2002935070580805687sk_nat @ ( suc @ zero_zero_nat ) )
= one_one_nat ) ).
% mask_Suc_0
thf(fact_1088_mask__Suc__0,axiom,
( ( bit_se2000444600071755411sk_int @ ( suc @ zero_zero_nat ) )
= one_one_int ) ).
% mask_Suc_0
thf(fact_1089_dbl__dec__simps_I2_J,axiom,
( ( neg_nu3811975205180677377ec_int @ zero_zero_int )
= ( uminus_uminus_int @ one_one_int ) ) ).
% dbl_dec_simps(2)
thf(fact_1090_zero__eq__power2,axiom,
! [A: int] :
( ( ( power_power_int @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
= zero_zero_int )
= ( A = zero_zero_int ) ) ).
% zero_eq_power2
thf(fact_1091_zero__eq__power2,axiom,
! [A: nat] :
( ( ( power_power_nat @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
= zero_zero_nat )
= ( A = zero_zero_nat ) ) ).
% zero_eq_power2
thf(fact_1092_and__numerals_I1_J,axiom,
! [Y3: num] :
( ( bit_se725231765392027082nd_int @ one_one_int @ ( numeral_numeral_int @ ( bit0 @ Y3 ) ) )
= zero_zero_int ) ).
% and_numerals(1)
thf(fact_1093_and__numerals_I1_J,axiom,
! [Y3: num] :
( ( bit_se727722235901077358nd_nat @ one_one_nat @ ( numeral_numeral_nat @ ( bit0 @ Y3 ) ) )
= zero_zero_nat ) ).
% and_numerals(1)
thf(fact_1094_and__numerals_I5_J,axiom,
! [X: num] :
( ( bit_se725231765392027082nd_int @ ( numeral_numeral_int @ ( bit0 @ X ) ) @ one_one_int )
= zero_zero_int ) ).
% and_numerals(5)
thf(fact_1095_and__numerals_I5_J,axiom,
! [X: num] :
( ( bit_se727722235901077358nd_nat @ ( numeral_numeral_nat @ ( bit0 @ X ) ) @ one_one_nat )
= zero_zero_nat ) ).
% and_numerals(5)
thf(fact_1096_divmod__algorithm__code_I1_J,axiom,
! [M: num] :
( ( unique5405566460079783412od_nat @ M @ one )
= ( product_Pair_nat_nat @ ( numeral_numeral_nat @ M ) @ zero_zero_nat ) ) ).
% divmod_algorithm_code(1)
thf(fact_1097_divmod__algorithm__code_I1_J,axiom,
! [M: num] :
( ( unique5403075989570733136od_int @ M @ one )
= ( product_Pair_int_int @ ( numeral_numeral_int @ M ) @ zero_zero_int ) ) ).
% divmod_algorithm_code(1)
thf(fact_1098_and__nat__numerals_I1_J,axiom,
! [Y3: num] :
( ( bit_se727722235901077358nd_nat @ ( suc @ zero_zero_nat ) @ ( numeral_numeral_nat @ ( bit0 @ Y3 ) ) )
= zero_zero_nat ) ).
% and_nat_numerals(1)
thf(fact_1099_and__nat__numerals_I3_J,axiom,
! [X: num] :
( ( bit_se727722235901077358nd_nat @ ( numeral_numeral_nat @ ( bit0 @ X ) ) @ ( suc @ zero_zero_nat ) )
= zero_zero_nat ) ).
% and_nat_numerals(3)
thf(fact_1100_or__nat__numerals_I2_J,axiom,
! [Y3: num] :
( ( bit_se1412395901928357646or_nat @ ( suc @ zero_zero_nat ) @ ( numeral_numeral_nat @ ( bit1 @ Y3 ) ) )
= ( numeral_numeral_nat @ ( bit1 @ Y3 ) ) ) ).
% or_nat_numerals(2)
thf(fact_1101_or__nat__numerals_I4_J,axiom,
! [X: num] :
( ( bit_se1412395901928357646or_nat @ ( numeral_numeral_nat @ ( bit1 @ X ) ) @ ( suc @ zero_zero_nat ) )
= ( numeral_numeral_nat @ ( bit1 @ X ) ) ) ).
% or_nat_numerals(4)
thf(fact_1102_even__take__bit__eq,axiom,
! [N: nat,A: nat] :
( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( bit_se2925701944663578781it_nat @ N @ A ) )
= ( ( N = zero_zero_nat )
| ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A ) ) ) ).
% even_take_bit_eq
thf(fact_1103_even__take__bit__eq,axiom,
! [N: nat,A: int] :
( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( bit_se2923211474154528505it_int @ N @ A ) )
= ( ( N = zero_zero_nat )
| ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A ) ) ) ).
% even_take_bit_eq
thf(fact_1104_even__push__bit__iff,axiom,
! [N: nat,A: int] :
( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( bit_se545348938243370406it_int @ N @ A ) )
= ( ( N != zero_zero_nat )
| ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A ) ) ) ).
% even_push_bit_iff
thf(fact_1105_even__push__bit__iff,axiom,
! [N: nat,A: nat] :
( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( bit_se547839408752420682it_nat @ N @ A ) )
= ( ( N != zero_zero_nat )
| ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A ) ) ) ).
% even_push_bit_iff
thf(fact_1106_divmod__algorithm__code_I2_J,axiom,
! [N: num] :
( ( unique5405566460079783412od_nat @ one @ ( bit0 @ N ) )
= ( product_Pair_nat_nat @ zero_zero_nat @ ( numeral_numeral_nat @ one ) ) ) ).
% divmod_algorithm_code(2)
thf(fact_1107_divmod__algorithm__code_I2_J,axiom,
! [N: num] :
( ( unique5403075989570733136od_int @ one @ ( bit0 @ N ) )
= ( product_Pair_int_int @ zero_zero_int @ ( numeral_numeral_int @ one ) ) ) ).
% divmod_algorithm_code(2)
thf(fact_1108_divmod__algorithm__code_I3_J,axiom,
! [N: num] :
( ( unique5405566460079783412od_nat @ one @ ( bit1 @ N ) )
= ( product_Pair_nat_nat @ zero_zero_nat @ ( numeral_numeral_nat @ one ) ) ) ).
% divmod_algorithm_code(3)
thf(fact_1109_divmod__algorithm__code_I3_J,axiom,
! [N: num] :
( ( unique5403075989570733136od_int @ one @ ( bit1 @ N ) )
= ( product_Pair_int_int @ zero_zero_int @ ( numeral_numeral_int @ one ) ) ) ).
% divmod_algorithm_code(3)
thf(fact_1110_xor__nat__numerals_I1_J,axiom,
! [Y3: num] :
( ( bit_se6528837805403552850or_nat @ ( suc @ zero_zero_nat ) @ ( numeral_numeral_nat @ ( bit0 @ Y3 ) ) )
= ( numeral_numeral_nat @ ( bit1 @ Y3 ) ) ) ).
% xor_nat_numerals(1)
thf(fact_1111_xor__nat__numerals_I2_J,axiom,
! [Y3: num] :
( ( bit_se6528837805403552850or_nat @ ( suc @ zero_zero_nat ) @ ( numeral_numeral_nat @ ( bit1 @ Y3 ) ) )
= ( numeral_numeral_nat @ ( bit0 @ Y3 ) ) ) ).
% xor_nat_numerals(2)
thf(fact_1112_xor__nat__numerals_I3_J,axiom,
! [X: num] :
( ( bit_se6528837805403552850or_nat @ ( numeral_numeral_nat @ ( bit0 @ X ) ) @ ( suc @ zero_zero_nat ) )
= ( numeral_numeral_nat @ ( bit1 @ X ) ) ) ).
% xor_nat_numerals(3)
thf(fact_1113_xor__nat__numerals_I4_J,axiom,
! [X: num] :
( ( bit_se6528837805403552850or_nat @ ( numeral_numeral_nat @ ( bit1 @ X ) ) @ ( suc @ zero_zero_nat ) )
= ( numeral_numeral_nat @ ( bit0 @ X ) ) ) ).
% xor_nat_numerals(4)
thf(fact_1114_or__nat__numerals_I1_J,axiom,
! [Y3: num] :
( ( bit_se1412395901928357646or_nat @ ( suc @ zero_zero_nat ) @ ( numeral_numeral_nat @ ( bit0 @ Y3 ) ) )
= ( numeral_numeral_nat @ ( bit1 @ Y3 ) ) ) ).
% or_nat_numerals(1)
thf(fact_1115_or__nat__numerals_I3_J,axiom,
! [X: num] :
( ( bit_se1412395901928357646or_nat @ ( numeral_numeral_nat @ ( bit0 @ X ) ) @ ( suc @ zero_zero_nat ) )
= ( numeral_numeral_nat @ ( bit1 @ X ) ) ) ).
% or_nat_numerals(3)
thf(fact_1116_and__nat__numerals_I2_J,axiom,
! [Y3: num] :
( ( bit_se727722235901077358nd_nat @ ( suc @ zero_zero_nat ) @ ( numeral_numeral_nat @ ( bit1 @ Y3 ) ) )
= one_one_nat ) ).
% and_nat_numerals(2)
thf(fact_1117_and__nat__numerals_I4_J,axiom,
! [X: num] :
( ( bit_se727722235901077358nd_nat @ ( numeral_numeral_nat @ ( bit1 @ X ) ) @ ( suc @ zero_zero_nat ) )
= one_one_nat ) ).
% and_nat_numerals(4)
thf(fact_1118_odd__Suc__minus__one,axiom,
! [N: nat] :
( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
=> ( ( suc @ ( minus_minus_nat @ N @ ( suc @ zero_zero_nat ) ) )
= N ) ) ).
% odd_Suc_minus_one
thf(fact_1119_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_1120_minus__nat_Odiff__0,axiom,
! [M: nat] :
( ( minus_minus_nat @ M @ zero_zero_nat )
= M ) ).
% minus_nat.diff_0
thf(fact_1121_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_1122_power__0__left,axiom,
! [N: nat] :
( ( ( N = zero_zero_nat )
=> ( ( power_power_int @ zero_zero_int @ N )
= one_one_int ) )
& ( ( N != zero_zero_nat )
=> ( ( power_power_int @ zero_zero_int @ N )
= zero_zero_int ) ) ) ).
% power_0_left
thf(fact_1123_power__0__left,axiom,
! [N: nat] :
( ( ( N = zero_zero_nat )
=> ( ( power_power_nat @ zero_zero_nat @ N )
= one_one_nat ) )
& ( ( N != zero_zero_nat )
=> ( ( power_power_nat @ zero_zero_nat @ N )
= zero_zero_nat ) ) ) ).
% power_0_left
thf(fact_1124_eq__iff__diff__eq__0,axiom,
( ( ^ [Y4: int,Z3: int] : ( Y4 = Z3 ) )
= ( ^ [A5: int,B4: int] :
( ( minus_minus_int @ A5 @ B4 )
= zero_zero_int ) ) ) ).
% eq_iff_diff_eq_0
thf(fact_1125_bit_Odisj__zero__right,axiom,
! [X: int] :
( ( bit_se1409905431419307370or_int @ X @ zero_zero_int )
= X ) ).
% bit.disj_zero_right
thf(fact_1126_or__eq__0__iff,axiom,
! [A: int,B: int] :
( ( ( bit_se1409905431419307370or_int @ A @ B )
= zero_zero_int )
= ( ( A = zero_zero_int )
& ( B = zero_zero_int ) ) ) ).
% or_eq_0_iff
thf(fact_1127_or__eq__0__iff,axiom,
! [A: nat,B: nat] :
( ( ( bit_se1412395901928357646or_nat @ A @ B )
= zero_zero_nat )
= ( ( A = zero_zero_nat )
& ( B = zero_zero_nat ) ) ) ).
% or_eq_0_iff
thf(fact_1128_zero__reorient,axiom,
! [X: nat] :
( ( zero_zero_nat = X )
= ( X = zero_zero_nat ) ) ).
% zero_reorient
thf(fact_1129_zero__reorient,axiom,
! [X: int] :
( ( zero_zero_int = X )
= ( X = zero_zero_int ) ) ).
% zero_reorient
thf(fact_1130_mult__0,axiom,
! [N: nat] :
( ( times_times_nat @ zero_zero_nat @ N )
= zero_zero_nat ) ).
% mult_0
thf(fact_1131_nat__mult__eq__cancel__disj,axiom,
! [K: nat,M: nat,N: nat] :
( ( ( times_times_nat @ K @ M )
= ( times_times_nat @ K @ N ) )
= ( ( K = zero_zero_nat )
| ( M = N ) ) ) ).
% nat_mult_eq_cancel_disj
thf(fact_1132_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_1133_plus__nat_Oadd__0,axiom,
! [N: nat] :
( ( plus_plus_nat @ zero_zero_nat @ N )
= N ) ).
% plus_nat.add_0
thf(fact_1134_not0__implies__Suc,axiom,
! [N: nat] :
( ( N != zero_zero_nat )
=> ? [M2: nat] :
( N
= ( suc @ M2 ) ) ) ).
% not0_implies_Suc
thf(fact_1135_Zero__not__Suc,axiom,
! [M: nat] :
( zero_zero_nat
!= ( suc @ M ) ) ).
% Zero_not_Suc
thf(fact_1136_Zero__neq__Suc,axiom,
! [M: nat] :
( zero_zero_nat
!= ( suc @ M ) ) ).
% Zero_neq_Suc
thf(fact_1137_Suc__neq__Zero,axiom,
! [M: nat] :
( ( suc @ M )
!= zero_zero_nat ) ).
% Suc_neq_Zero
thf(fact_1138_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_1139_diff__induct,axiom,
! [P: nat > nat > $o,M: nat,N: nat] :
( ! [X2: nat] : ( P @ X2 @ zero_zero_nat )
=> ( ! [Y: nat] : ( P @ zero_zero_nat @ ( suc @ Y ) )
=> ( ! [X2: nat,Y: nat] :
( ( P @ X2 @ Y )
=> ( P @ ( suc @ X2 ) @ ( suc @ Y ) ) )
=> ( P @ M @ N ) ) ) ) ).
% diff_induct
thf(fact_1140_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_1141_old_Onat_Oexhaust,axiom,
! [Y3: nat] :
( ( Y3 != zero_zero_nat )
=> ~ ! [Nat3: nat] :
( Y3
!= ( suc @ Nat3 ) ) ) ).
% old.nat.exhaust
thf(fact_1142_nat_OdiscI,axiom,
! [Nat: nat,X24: nat] :
( ( Nat
= ( suc @ X24 ) )
=> ( Nat != zero_zero_nat ) ) ).
% nat.discI
thf(fact_1143_old_Onat_Odistinct_I1_J,axiom,
! [Nat2: nat] :
( zero_zero_nat
!= ( suc @ Nat2 ) ) ).
% old.nat.distinct(1)
thf(fact_1144_old_Onat_Odistinct_I2_J,axiom,
! [Nat2: nat] :
( ( suc @ Nat2 )
!= zero_zero_nat ) ).
% old.nat.distinct(2)
thf(fact_1145_nat_Odistinct_I1_J,axiom,
! [X24: nat] :
( zero_zero_nat
!= ( suc @ X24 ) ) ).
% nat.distinct(1)
thf(fact_1146_row__exec_Ocases,axiom,
! [X: nat] :
( ( X != zero_zero_nat )
=> ~ ! [V: nat] :
( X
!= ( suc @ V ) ) ) ).
% row_exec.cases
thf(fact_1147_board__exec__aux_Ocases,axiom,
! [X: produc9133624956312949779et_int] :
( ! [M4: set_int] :
( X
!= ( produc29655638201817675et_int @ zero_zero_nat @ M4 ) )
=> ~ ! [V: nat,M4: set_int] :
( X
!= ( produc29655638201817675et_int @ ( suc @ V ) @ M4 ) ) ) ).
% board_exec_aux.cases
thf(fact_1148_zero__neq__one,axiom,
zero_zero_nat != one_one_nat ).
% zero_neq_one
thf(fact_1149_zero__neq__one,axiom,
zero_zero_int != one_one_int ).
% zero_neq_one
thf(fact_1150_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_1151_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_1152_add_Ocomm__neutral,axiom,
! [A: nat] :
( ( plus_plus_nat @ A @ zero_zero_nat )
= A ) ).
% add.comm_neutral
thf(fact_1153_add_Ocomm__neutral,axiom,
! [A: int] :
( ( plus_plus_int @ A @ zero_zero_int )
= A ) ).
% add.comm_neutral
thf(fact_1154_add_Ogroup__left__neutral,axiom,
! [A: int] :
( ( plus_plus_int @ zero_zero_int @ A )
= A ) ).
% add.group_left_neutral
thf(fact_1155_verit__sum__simplify,axiom,
! [A: nat] :
( ( plus_plus_nat @ A @ zero_zero_nat )
= A ) ).
% verit_sum_simplify
thf(fact_1156_verit__sum__simplify,axiom,
! [A: int] :
( ( plus_plus_int @ A @ zero_zero_int )
= A ) ).
% verit_sum_simplify
thf(fact_1157_mult__not__zero,axiom,
! [A: nat,B: nat] :
( ( ( times_times_nat @ A @ B )
!= zero_zero_nat )
=> ( ( A != zero_zero_nat )
& ( B != zero_zero_nat ) ) ) ).
% mult_not_zero
thf(fact_1158_mult__not__zero,axiom,
! [A: int,B: int] :
( ( ( times_times_int @ A @ B )
!= zero_zero_int )
=> ( ( A != zero_zero_int )
& ( B != zero_zero_int ) ) ) ).
% mult_not_zero
thf(fact_1159_divisors__zero,axiom,
! [A: nat,B: nat] :
( ( ( times_times_nat @ A @ B )
= zero_zero_nat )
=> ( ( A = zero_zero_nat )
| ( B = zero_zero_nat ) ) ) ).
% divisors_zero
thf(fact_1160_divisors__zero,axiom,
! [A: int,B: int] :
( ( ( times_times_int @ A @ B )
= zero_zero_int )
=> ( ( A = zero_zero_int )
| ( B = zero_zero_int ) ) ) ).
% divisors_zero
thf(fact_1161_no__zero__divisors,axiom,
! [A: nat,B: nat] :
( ( A != zero_zero_nat )
=> ( ( B != zero_zero_nat )
=> ( ( times_times_nat @ A @ B )
!= zero_zero_nat ) ) ) ).
% no_zero_divisors
thf(fact_1162_no__zero__divisors,axiom,
! [A: int,B: int] :
( ( A != zero_zero_int )
=> ( ( B != zero_zero_int )
=> ( ( times_times_int @ A @ B )
!= zero_zero_int ) ) ) ).
% no_zero_divisors
thf(fact_1163_mult__left__cancel,axiom,
! [C2: nat,A: nat,B: nat] :
( ( C2 != zero_zero_nat )
=> ( ( ( times_times_nat @ C2 @ A )
= ( times_times_nat @ C2 @ B ) )
= ( A = B ) ) ) ).
% mult_left_cancel
thf(fact_1164_mult__left__cancel,axiom,
! [C2: int,A: int,B: int] :
( ( C2 != zero_zero_int )
=> ( ( ( times_times_int @ C2 @ A )
= ( times_times_int @ C2 @ B ) )
= ( A = B ) ) ) ).
% mult_left_cancel
thf(fact_1165_mult__right__cancel,axiom,
! [C2: nat,A: nat,B: nat] :
( ( C2 != zero_zero_nat )
=> ( ( ( times_times_nat @ A @ C2 )
= ( times_times_nat @ B @ C2 ) )
= ( A = B ) ) ) ).
% mult_right_cancel
thf(fact_1166_mult__right__cancel,axiom,
! [C2: int,A: int,B: int] :
( ( C2 != zero_zero_int )
=> ( ( ( times_times_int @ A @ C2 )
= ( times_times_int @ B @ C2 ) )
= ( A = B ) ) ) ).
% mult_right_cancel
thf(fact_1167_zero__neq__numeral,axiom,
! [N: num] :
( zero_zero_nat
!= ( numeral_numeral_nat @ N ) ) ).
% zero_neq_numeral
thf(fact_1168_zero__neq__numeral,axiom,
! [N: num] :
( zero_zero_int
!= ( numeral_numeral_int @ N ) ) ).
% zero_neq_numeral
thf(fact_1169_step__in__Cons,axiom,
! [Ps2: list_P5707943133018811711nt_int,S_i2: product_prod_int_int,S_j2: product_prod_int_int,S_k: product_prod_int_int] :
( ( step_in @ Ps2 @ S_i2 @ S_j2 )
=> ( step_in @ ( cons_P3334398858971670639nt_int @ S_k @ Ps2 ) @ S_i2 @ S_j2 ) ) ).
% step_in_Cons
thf(fact_1170_push__bit__int__code_I1_J,axiom,
! [I3: int] :
( ( bit_se545348938243370406it_int @ zero_zero_nat @ I3 )
= I3 ) ).
% push_bit_int_code(1)
thf(fact_1171_sum__squares__eq__zero__iff,axiom,
! [X: int,Y3: int] :
( ( ( plus_plus_int @ ( times_times_int @ X @ X ) @ ( times_times_int @ Y3 @ Y3 ) )
= zero_zero_int )
= ( ( X = zero_zero_int )
& ( Y3 = zero_zero_int ) ) ) ).
% sum_squares_eq_zero_iff
thf(fact_1172_zero__neq__neg__numeral,axiom,
! [N: num] :
( zero_zero_int
!= ( uminus_uminus_int @ ( numeral_numeral_int @ N ) ) ) ).
% zero_neq_neg_numeral
thf(fact_1173_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_1174_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_1175_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_1176_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_1177_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_1178_zero__neq__neg__one,axiom,
( zero_zero_int
!= ( uminus_uminus_int @ one_one_int ) ) ).
% zero_neq_neg_one
thf(fact_1179_not__is__unit__0,axiom,
~ ( dvd_dvd_nat @ zero_zero_nat @ one_one_nat ) ).
% not_is_unit_0
thf(fact_1180_not__is__unit__0,axiom,
~ ( dvd_dvd_int @ zero_zero_int @ one_one_int ) ).
% not_is_unit_0
thf(fact_1181_of__nat__neq__0,axiom,
! [N: nat] :
( ( semiri1316708129612266289at_nat @ ( suc @ N ) )
!= zero_zero_nat ) ).
% of_nat_neq_0
thf(fact_1182_of__nat__neq__0,axiom,
! [N: nat] :
( ( semiri1314217659103216013at_int @ ( suc @ N ) )
!= zero_zero_int ) ).
% of_nat_neq_0
thf(fact_1183_power__0,axiom,
! [A: int] :
( ( power_power_int @ A @ zero_zero_nat )
= one_one_int ) ).
% power_0
thf(fact_1184_power__0,axiom,
! [A: nat] :
( ( power_power_nat @ A @ zero_zero_nat )
= one_one_nat ) ).
% power_0
thf(fact_1185_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_1186_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_1187_One__nat__def,axiom,
( one_one_nat
= ( suc @ zero_zero_nat ) ) ).
% One_nat_def
thf(fact_1188_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_1189_mult__eq__self__implies__10,axiom,
! [M: nat,N: nat] :
( ( M
= ( times_times_nat @ M @ N ) )
=> ( ( N = one_one_nat )
| ( M = zero_zero_nat ) ) ) ).
% mult_eq_self_implies_10
thf(fact_1190_to__sqrs_Ocases,axiom,
! [X: produc2814713032259027617st_nat] :
( ! [Rs: list_list_nat] :
( X
!= ( produc1673703249076330713st_nat @ zero_zero_nat @ Rs ) )
=> ~ ! [V: nat,Rs: list_list_nat] :
( X
!= ( produc1673703249076330713st_nat @ ( suc @ V ) @ Rs ) ) ) ).
% to_sqrs.cases
thf(fact_1191_unity__coeff__ex,axiom,
! [P: nat > $o,L: nat] :
( ( ? [X4: nat] : ( P @ ( times_times_nat @ L @ X4 ) ) )
= ( ? [X4: nat] :
( ( dvd_dvd_nat @ L @ ( plus_plus_nat @ X4 @ zero_zero_nat ) )
& ( P @ X4 ) ) ) ) ).
% unity_coeff_ex
thf(fact_1192_unity__coeff__ex,axiom,
! [P: int > $o,L: int] :
( ( ? [X4: int] : ( P @ ( times_times_int @ L @ X4 ) ) )
= ( ? [X4: int] :
( ( dvd_dvd_int @ L @ ( plus_plus_int @ X4 @ zero_zero_int ) )
& ( P @ X4 ) ) ) ) ).
% unity_coeff_ex
thf(fact_1193_unit__dvdE,axiom,
! [A: nat,B: nat] :
( ( dvd_dvd_nat @ A @ one_one_nat )
=> ~ ( ( A != zero_zero_nat )
=> ! [C: nat] :
( B
!= ( times_times_nat @ A @ C ) ) ) ) ).
% unit_dvdE
thf(fact_1194_unit__dvdE,axiom,
! [A: int,B: int] :
( ( dvd_dvd_int @ A @ one_one_int )
=> ~ ( ( A != zero_zero_int )
=> ! [C: int] :
( B
!= ( times_times_int @ A @ C ) ) ) ) ).
% unit_dvdE
thf(fact_1195_is__unit__power__iff,axiom,
! [A: int,N: nat] :
( ( dvd_dvd_int @ ( power_power_int @ A @ N ) @ one_one_int )
= ( ( dvd_dvd_int @ A @ one_one_int )
| ( N = zero_zero_nat ) ) ) ).
% is_unit_power_iff
thf(fact_1196_is__unit__power__iff,axiom,
! [A: nat,N: nat] :
( ( dvd_dvd_nat @ ( power_power_nat @ A @ N ) @ one_one_nat )
= ( ( dvd_dvd_nat @ A @ one_one_nat )
| ( N = zero_zero_nat ) ) ) ).
% is_unit_power_iff
thf(fact_1197_numeral__1__eq__Suc__0,axiom,
( ( numeral_numeral_nat @ one )
= ( suc @ zero_zero_nat ) ) ).
% numeral_1_eq_Suc_0
thf(fact_1198_bezout__add__strong__nat,axiom,
! [A: nat,B: nat] :
( ( A != zero_zero_nat )
=> ? [D4: nat,X2: nat,Y: nat] :
( ( dvd_dvd_nat @ D4 @ A )
& ( dvd_dvd_nat @ D4 @ B )
& ( ( times_times_nat @ A @ X2 )
= ( plus_plus_nat @ ( times_times_nat @ B @ Y ) @ D4 ) ) ) ) ).
% bezout_add_strong_nat
thf(fact_1199_transpose__knights__circuit,axiom,
! [N: nat,M: nat,Ps2: list_P5707943133018811711nt_int] :
( ( knights_circuit @ ( board @ N @ M ) @ Ps2 )
=> ( knights_circuit @ ( board @ M @ N ) @ ( transpose @ Ps2 ) ) ) ).
% transpose_knights_circuit
thf(fact_1200_even__zero,axiom,
dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ zero_zero_nat ).
% even_zero
thf(fact_1201_even__zero,axiom,
dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ zero_zero_int ).
% even_zero
thf(fact_1202_zero__power2,axiom,
( ( power_power_int @ zero_zero_int @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
= zero_zero_int ) ).
% zero_power2
thf(fact_1203_zero__power2,axiom,
( ( power_power_nat @ zero_zero_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
= zero_zero_nat ) ).
% zero_power2
thf(fact_1204_numeral__2__eq__2,axiom,
( ( numeral_numeral_nat @ ( bit0 @ one ) )
= ( suc @ ( suc @ zero_zero_nat ) ) ) ).
% numeral_2_eq_2
thf(fact_1205_numeral__3__eq__3,axiom,
( ( numeral_numeral_nat @ ( bit1 @ one ) )
= ( suc @ ( suc @ ( suc @ zero_zero_nat ) ) ) ) ).
% numeral_3_eq_3
thf(fact_1206_bit_Ocomplement__unique,axiom,
! [A: int,X: int,Y3: int] :
( ( ( bit_se725231765392027082nd_int @ A @ X )
= zero_zero_int )
=> ( ( ( bit_se1409905431419307370or_int @ A @ X )
= ( uminus_uminus_int @ one_one_int ) )
=> ( ( ( bit_se725231765392027082nd_int @ A @ Y3 )
= zero_zero_int )
=> ( ( ( bit_se1409905431419307370or_int @ A @ Y3 )
= ( uminus_uminus_int @ one_one_int ) )
=> ( X = Y3 ) ) ) ) ) ).
% bit.complement_unique
thf(fact_1207_add__eq__if,axiom,
( plus_plus_nat
= ( ^ [M3: nat,N3: nat] : ( if_nat @ ( M3 = zero_zero_nat ) @ N3 @ ( suc @ ( plus_plus_nat @ ( minus_minus_nat @ M3 @ one_one_nat ) @ N3 ) ) ) ) ) ).
% add_eq_if
thf(fact_1208_mult__eq__if,axiom,
( times_times_nat
= ( ^ [M3: nat,N3: nat] : ( if_nat @ ( M3 = zero_zero_nat ) @ zero_zero_nat @ ( plus_plus_nat @ N3 @ ( times_times_nat @ ( minus_minus_nat @ M3 @ one_one_nat ) @ N3 ) ) ) ) ) ).
% mult_eq_if
thf(fact_1209_trans__step__in,axiom,
! [Ps2: list_P5707943133018811711nt_int,I3: int,J3: int,I4: int,J4: int,K_12: int,K_22: int] :
( ( step_in @ Ps2 @ ( product_Pair_int_int @ I3 @ J3 ) @ ( product_Pair_int_int @ I4 @ J4 ) )
=> ( step_in @ ( trans_path @ ( product_Pair_int_int @ K_12 @ K_22 ) @ Ps2 ) @ ( product_Pair_int_int @ ( plus_plus_int @ I3 @ K_12 ) @ ( plus_plus_int @ J3 @ K_22 ) ) @ ( product_Pair_int_int @ ( plus_plus_int @ I4 @ K_12 ) @ ( plus_plus_int @ J4 @ K_22 ) ) ) ) ).
% trans_step_in
thf(fact_1210_exp__add__not__zero__imp__right,axiom,
! [M: nat,N: nat] :
( ( ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( plus_plus_nat @ M @ N ) )
!= zero_zero_nat )
=> ( ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
!= zero_zero_nat ) ) ).
% exp_add_not_zero_imp_right
thf(fact_1211_exp__add__not__zero__imp__right,axiom,
! [M: nat,N: nat] :
( ( ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( plus_plus_nat @ M @ N ) )
!= zero_zero_int )
=> ( ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N )
!= zero_zero_int ) ) ).
% exp_add_not_zero_imp_right
thf(fact_1212_nat__induct2,axiom,
! [P: nat > $o,N: nat] :
( ( P @ zero_zero_nat )
=> ( ( P @ one_one_nat )
=> ( ! [N2: nat] :
( ( P @ N2 )
=> ( P @ ( plus_plus_nat @ N2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) )
=> ( P @ N ) ) ) ) ).
% nat_induct2
thf(fact_1213_knights__path__split__concat__si__prev,axiom,
! [N: nat,M_1: nat,Ps_1: list_P5707943133018811711nt_int,M_2: nat,Ps_2: list_P5707943133018811711nt_int,S_i2: product_prod_int_int,S_j2: product_prod_int_int,I_h: int,J_h: int,I_l: int,J_l: int,I3: int,J3: int,I4: int,J4: int] :
( ( knights_path @ ( board @ N @ M_1 ) @ Ps_1 )
=> ( ( knights_path @ ( board @ N @ M_2 ) @ Ps_2 )
=> ( ( step_in @ Ps_1 @ S_i2 @ S_j2 )
=> ( ( ( hd_Pro282112905867057956nt_int @ Ps_2 )
= ( product_Pair_int_int @ I_h @ J_h ) )
=> ( ( ( last_P3305686521732843992nt_int @ Ps_2 )
= ( product_Pair_int_int @ I_l @ J_l ) )
=> ( ( step_in @ Ps_2 @ ( product_Pair_int_int @ I3 @ J3 ) @ ( product_Pair_int_int @ I4 @ J4 ) )
=> ( ( valid_step @ S_i2 @ ( product_Pair_int_int @ I_h @ ( plus_plus_int @ ( semiri1314217659103216013at_int @ M_1 ) @ J_h ) ) )
=> ( ( valid_step @ ( product_Pair_int_int @ I_l @ ( plus_plus_int @ ( semiri1314217659103216013at_int @ M_1 ) @ J_l ) ) @ S_j2 )
=> ? [Ps: list_P5707943133018811711nt_int] :
( ( knights_path @ ( board @ N @ ( plus_plus_nat @ M_1 @ M_2 ) ) @ Ps )
& ( ( hd_Pro282112905867057956nt_int @ Ps )
= ( hd_Pro282112905867057956nt_int @ Ps_1 ) )
& ( ( last_P3305686521732843992nt_int @ Ps )
= ( last_P3305686521732843992nt_int @ Ps_1 ) )
& ( step_in @ Ps @ ( product_Pair_int_int @ I3 @ ( plus_plus_int @ ( semiri1314217659103216013at_int @ M_1 ) @ J3 ) ) @ ( product_Pair_int_int @ I4 @ ( plus_plus_int @ ( semiri1314217659103216013at_int @ M_1 ) @ J4 ) ) ) ) ) ) ) ) ) ) ) ) ).
% knights_path_split_concat_si_prev
thf(fact_1214_knights__path__split__concatT,axiom,
! [N_1: nat,M: nat,Ps_1: list_P5707943133018811711nt_int,N_2: nat,Ps_2: list_P5707943133018811711nt_int,S_i2: product_prod_int_int,S_j2: product_prod_int_int,I_h: int,J_h: int,I_l: int,J_l: int] :
( ( knights_path @ ( board @ N_1 @ M ) @ Ps_1 )
=> ( ( knights_path @ ( board @ N_2 @ M ) @ Ps_2 )
=> ( ( step_in @ Ps_1 @ S_i2 @ S_j2 )
=> ( ( ( hd_Pro282112905867057956nt_int @ Ps_2 )
= ( product_Pair_int_int @ I_h @ J_h ) )
=> ( ( ( last_P3305686521732843992nt_int @ Ps_2 )
= ( product_Pair_int_int @ I_l @ J_l ) )
=> ( ( valid_step @ S_i2 @ ( product_Pair_int_int @ ( plus_plus_int @ ( semiri1314217659103216013at_int @ N_1 ) @ I_h ) @ J_h ) )
=> ( ( valid_step @ ( product_Pair_int_int @ ( plus_plus_int @ ( semiri1314217659103216013at_int @ N_1 ) @ I_l ) @ J_l ) @ S_j2 )
=> ? [Ps: list_P5707943133018811711nt_int] :
( ( knights_path @ ( board @ ( plus_plus_nat @ N_1 @ N_2 ) @ M ) @ Ps )
& ( ( hd_Pro282112905867057956nt_int @ Ps )
= ( hd_Pro282112905867057956nt_int @ Ps_1 ) )
& ( ( last_P3305686521732843992nt_int @ Ps )
= ( last_P3305686521732843992nt_int @ Ps_1 ) ) ) ) ) ) ) ) ) ) ).
% knights_path_split_concatT
thf(fact_1215_and__minus__numerals_I1_J,axiom,
! [N: num] :
( ( bit_se725231765392027082nd_int @ one_one_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit0 @ N ) ) ) )
= zero_zero_int ) ).
% and_minus_numerals(1)
thf(fact_1216_and__minus__numerals_I5_J,axiom,
! [N: num] :
( ( bit_se725231765392027082nd_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit0 @ N ) ) ) @ one_one_int )
= zero_zero_int ) ).
% and_minus_numerals(5)
thf(fact_1217_and__int__code_I1_J,axiom,
! [J3: int] :
( ( bit_se725231765392027082nd_int @ zero_zero_int @ J3 )
= zero_zero_int ) ).
% and_int_code(1)
thf(fact_1218_and__int__code_I2_J,axiom,
! [I3: int] :
( ( bit_se725231765392027082nd_int @ I3 @ zero_zero_int )
= zero_zero_int ) ).
% and_int_code(2)
thf(fact_1219_or__int__code_I1_J,axiom,
! [J3: int] :
( ( bit_se1409905431419307370or_int @ zero_zero_int @ J3 )
= J3 ) ).
% or_int_code(1)
thf(fact_1220_or__int__code_I2_J,axiom,
! [I3: int] :
( ( bit_se1409905431419307370or_int @ I3 @ zero_zero_int )
= I3 ) ).
% or_int_code(2)
thf(fact_1221_times__int__code_I2_J,axiom,
! [L: int] :
( ( times_times_int @ zero_zero_int @ L )
= zero_zero_int ) ).
% times_int_code(2)
thf(fact_1222_times__int__code_I1_J,axiom,
! [K: int] :
( ( times_times_int @ K @ zero_zero_int )
= zero_zero_int ) ).
% times_int_code(1)
thf(fact_1223_uminus__int__code_I1_J,axiom,
( ( uminus_uminus_int @ zero_zero_int )
= zero_zero_int ) ).
% uminus_int_code(1)
thf(fact_1224_plus__int__code_I2_J,axiom,
! [L: int] :
( ( plus_plus_int @ zero_zero_int @ L )
= L ) ).
% plus_int_code(2)
thf(fact_1225_plus__int__code_I1_J,axiom,
! [K: int] :
( ( plus_plus_int @ K @ zero_zero_int )
= K ) ).
% plus_int_code(1)
thf(fact_1226_step__checker__correct,axiom,
step_checker = valid_step ).
% step_checker_correct
thf(fact_1227_valid__step__non__transitive,axiom,
! [S_i2: product_prod_int_int,S_j2: product_prod_int_int,S_k: product_prod_int_int] :
( ( valid_step @ S_i2 @ S_j2 )
=> ( ( valid_step @ S_j2 @ S_k )
=> ~ ( valid_step @ S_i2 @ S_k ) ) ) ).
% valid_step_non_transitive
thf(fact_1228_valid__step__rev,axiom,
! [S_i2: product_prod_int_int,S_j2: product_prod_int_int] :
( ( valid_step @ S_i2 @ S_j2 )
=> ( valid_step @ S_j2 @ S_i2 ) ) ).
% valid_step_rev
thf(fact_1229_valid__step__neq,axiom,
! [S_i2: product_prod_int_int,S_j2: product_prod_int_int] :
( ( valid_step @ S_i2 @ S_j2 )
=> ( S_i2 != S_j2 ) ) ).
% valid_step_neq
thf(fact_1230_int__ops_I1_J,axiom,
( ( semiri1314217659103216013at_int @ zero_zero_nat )
= zero_zero_int ) ).
% int_ops(1)
thf(fact_1231_odd__nonzero,axiom,
! [Z: int] :
( ( plus_plus_int @ ( plus_plus_int @ one_one_int @ Z ) @ Z )
!= zero_zero_int ) ).
% odd_nonzero
thf(fact_1232_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_1233_zdvd__mult__cancel,axiom,
! [K: int,M: int,N: int] :
( ( dvd_dvd_int @ ( times_times_int @ K @ M ) @ ( times_times_int @ K @ N ) )
=> ( ( K != zero_zero_int )
=> ( dvd_dvd_int @ M @ N ) ) ) ).
% zdvd_mult_cancel
thf(fact_1234_zdvd__mono,axiom,
! [K: int,M: int,T: int] :
( ( K != zero_zero_int )
=> ( ( dvd_dvd_int @ M @ T )
= ( dvd_dvd_int @ ( times_times_int @ K @ M ) @ ( times_times_int @ K @ T ) ) ) ) ).
% zdvd_mono
thf(fact_1235_trans__valid__step,axiom,
! [I3: int,J3: int,I4: int,J4: int,K_12: int,K_22: int] :
( ( valid_step @ ( product_Pair_int_int @ I3 @ J3 ) @ ( product_Pair_int_int @ I4 @ J4 ) )
=> ( valid_step @ ( product_Pair_int_int @ ( plus_plus_int @ I3 @ K_12 ) @ ( plus_plus_int @ J3 @ K_22 ) ) @ ( product_Pair_int_int @ ( plus_plus_int @ I4 @ K_12 ) @ ( plus_plus_int @ J4 @ K_22 ) ) ) ) ).
% trans_valid_step
thf(fact_1236_step__in__valid__step,axiom,
! [B: set_Pr958786334691620121nt_int,Ps2: list_P5707943133018811711nt_int,S_i2: product_prod_int_int,S_j2: product_prod_int_int] :
( ( knights_path @ B @ Ps2 )
=> ( ( step_in @ Ps2 @ S_i2 @ S_j2 )
=> ( valid_step @ S_i2 @ S_j2 ) ) ) ).
% step_in_valid_step
thf(fact_1237_take__bit__decr__eq,axiom,
! [N: nat,K: int] :
( ( ( bit_se2923211474154528505it_int @ N @ K )
!= zero_zero_int )
=> ( ( bit_se2923211474154528505it_int @ N @ ( minus_minus_int @ K @ one_one_int ) )
= ( minus_minus_int @ ( bit_se2923211474154528505it_int @ N @ K ) @ one_one_int ) ) ) ).
% take_bit_decr_eq
thf(fact_1238_take__bit__eq__mask__iff,axiom,
! [N: nat,K: int] :
( ( ( bit_se2923211474154528505it_int @ N @ K )
= ( bit_se2000444600071755411sk_int @ N ) )
= ( ( bit_se2923211474154528505it_int @ N @ ( plus_plus_int @ K @ one_one_int ) )
= zero_zero_int ) ) ).
% take_bit_eq_mask_iff
thf(fact_1239_knights__circuit__def,axiom,
( knights_circuit
= ( ^ [B4: set_Pr958786334691620121nt_int,Ps3: list_P5707943133018811711nt_int] :
( ( knights_path @ B4 @ Ps3 )
& ( valid_step @ ( last_P3305686521732843992nt_int @ Ps3 ) @ ( hd_Pro282112905867057956nt_int @ Ps3 ) ) ) ) ) ).
% knights_circuit_def
thf(fact_1240_int__bit__induct,axiom,
! [P: int > $o,K: int] :
( ( P @ zero_zero_int )
=> ( ( P @ ( uminus_uminus_int @ one_one_int ) )
=> ( ! [K2: int] :
( ( P @ K2 )
=> ( ( K2 != zero_zero_int )
=> ( P @ ( times_times_int @ K2 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ) )
=> ( ! [K2: int] :
( ( P @ K2 )
=> ( ( K2
!= ( uminus_uminus_int @ one_one_int ) )
=> ( P @ ( plus_plus_int @ one_one_int @ ( times_times_int @ K2 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ) ) )
=> ( P @ K ) ) ) ) ) ).
% int_bit_induct
thf(fact_1241_knights__path__split__concat,axiom,
! [N: nat,M_1: nat,Ps_1: list_P5707943133018811711nt_int,M_2: nat,Ps_2: list_P5707943133018811711nt_int,S_i2: product_prod_int_int,S_j2: product_prod_int_int,I_h: int,J_h: int,I_l: int,J_l: int] :
( ( knights_path @ ( board @ N @ M_1 ) @ Ps_1 )
=> ( ( knights_path @ ( board @ N @ M_2 ) @ Ps_2 )
=> ( ( step_in @ Ps_1 @ S_i2 @ S_j2 )
=> ( ( ( hd_Pro282112905867057956nt_int @ Ps_2 )
= ( product_Pair_int_int @ I_h @ J_h ) )
=> ( ( ( last_P3305686521732843992nt_int @ Ps_2 )
= ( product_Pair_int_int @ I_l @ J_l ) )
=> ( ( valid_step @ S_i2 @ ( product_Pair_int_int @ I_h @ ( plus_plus_int @ ( semiri1314217659103216013at_int @ M_1 ) @ J_h ) ) )
=> ( ( valid_step @ ( product_Pair_int_int @ I_l @ ( plus_plus_int @ ( semiri1314217659103216013at_int @ M_1 ) @ J_l ) ) @ S_j2 )
=> ? [Ps: list_P5707943133018811711nt_int] :
( ( knights_path @ ( board @ N @ ( plus_plus_nat @ M_1 @ M_2 ) ) @ Ps )
& ( ( hd_Pro282112905867057956nt_int @ Ps )
= ( hd_Pro282112905867057956nt_int @ Ps_1 ) )
& ( ( last_P3305686521732843992nt_int @ Ps )
= ( last_P3305686521732843992nt_int @ Ps_1 ) ) ) ) ) ) ) ) ) ) ).
% knights_path_split_concat
thf(fact_1242_knights__path__lr__concat,axiom,
! [N: nat,M_1: nat,Ps_1: list_P5707943133018811711nt_int,M_2: nat,Ps_2: list_P5707943133018811711nt_int] :
( ( knights_path @ ( board @ N @ M_1 ) @ Ps_1 )
=> ( ( knights_path @ ( board @ N @ M_2 ) @ Ps_2 )
=> ( ( ( last_P3305686521732843992nt_int @ Ps_1 )
= ( product_Pair_int_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( minus_minus_int @ ( semiri1314217659103216013at_int @ M_1 ) @ one_one_int ) ) )
=> ( ( ( hd_Pro282112905867057956nt_int @ Ps_2 )
= ( product_Pair_int_int @ one_one_int @ one_one_int ) )
=> ( knights_path @ ( board @ N @ ( plus_plus_nat @ M_1 @ M_2 ) ) @ ( append7030698103840186580nt_int @ Ps_1 @ ( trans_path @ ( product_Pair_int_int @ zero_zero_int @ ( semiri1314217659103216013at_int @ M_1 ) ) @ Ps_2 ) ) ) ) ) ) ) ).
% knights_path_lr_concat
thf(fact_1243_knights__path__concat,axiom,
! [N: nat,M_1: nat,Ps_1: list_P5707943133018811711nt_int,M_2: nat,Ps_2: list_P5707943133018811711nt_int,I_h: int,J_h: int] :
( ( knights_path @ ( board @ N @ M_1 ) @ Ps_1 )
=> ( ( knights_path @ ( board @ N @ M_2 ) @ Ps_2 )
=> ( ( ( hd_Pro282112905867057956nt_int @ Ps_2 )
= ( product_Pair_int_int @ I_h @ J_h ) )
=> ( ( valid_step @ ( last_P3305686521732843992nt_int @ Ps_1 ) @ ( product_Pair_int_int @ I_h @ ( plus_plus_int @ ( semiri1314217659103216013at_int @ M_1 ) @ J_h ) ) )
=> ( knights_path @ ( board @ N @ ( plus_plus_nat @ M_1 @ M_2 ) ) @ ( append7030698103840186580nt_int @ Ps_1 @ ( trans_path @ ( product_Pair_int_int @ zero_zero_int @ ( semiri1314217659103216013at_int @ M_1 ) ) @ Ps_2 ) ) ) ) ) ) ) ).
% knights_path_concat
thf(fact_1244_xor__int__code_I1_J,axiom,
! [J3: int] :
( ( bit_se6526347334894502574or_int @ zero_zero_int @ J3 )
= J3 ) ).
% xor_int_code(1)
thf(fact_1245_xor__int__code_I2_J,axiom,
! [I3: int] :
( ( bit_se6526347334894502574or_int @ I3 @ zero_zero_int )
= I3 ) ).
% xor_int_code(2)
thf(fact_1246_step__in__prepend,axiom,
! [Ps2: list_P5707943133018811711nt_int,S_i2: product_prod_int_int,S_j2: product_prod_int_int,Ps4: list_P5707943133018811711nt_int] :
( ( step_in @ Ps2 @ S_i2 @ S_j2 )
=> ( step_in @ ( append7030698103840186580nt_int @ Ps4 @ Ps2 ) @ S_i2 @ S_j2 ) ) ).
% step_in_prepend
thf(fact_1247_step__in__append,axiom,
! [Ps2: list_P5707943133018811711nt_int,S_i2: product_prod_int_int,S_j2: product_prod_int_int,Ps4: list_P5707943133018811711nt_int] :
( ( step_in @ Ps2 @ S_i2 @ S_j2 )
=> ( step_in @ ( append7030698103840186580nt_int @ Ps2 @ Ps4 ) @ S_i2 @ S_j2 ) ) ).
% step_in_append
thf(fact_1248_knights__circuit__rotate1,axiom,
! [B: set_Pr958786334691620121nt_int,S_i2: product_prod_int_int,Ps2: list_P5707943133018811711nt_int] :
( ( knights_circuit @ B @ ( cons_P3334398858971670639nt_int @ S_i2 @ Ps2 ) )
=> ( knights_circuit @ B @ ( append7030698103840186580nt_int @ Ps2 @ ( cons_P3334398858971670639nt_int @ S_i2 @ nil_Pr2300489316682597567nt_int ) ) ) ) ).
% knights_circuit_rotate1
thf(fact_1249_num_Osize__gen_I3_J,axiom,
! [X32: num] :
( ( size_num @ ( bit1 @ X32 ) )
= ( plus_plus_nat @ ( size_num @ X32 ) @ ( suc @ zero_zero_nat ) ) ) ).
% num.size_gen(3)
thf(fact_1250_num_Osize__gen_I1_J,axiom,
( ( size_num @ one )
= zero_zero_nat ) ).
% num.size_gen(1)
thf(fact_1251_num_Osize__gen_I2_J,axiom,
! [X24: num] :
( ( size_num @ ( bit0 @ X24 ) )
= ( plus_plus_nat @ ( size_num @ X24 ) @ ( suc @ zero_zero_nat ) ) ) ).
% num.size_gen(2)
thf(fact_1252_flip__bit__int__def,axiom,
( bit_se2159334234014336723it_int
= ( ^ [N3: nat,K3: int] : ( bit_se6526347334894502574or_int @ K3 @ ( bit_se545348938243370406it_int @ N3 @ one_one_int ) ) ) ) ).
% flip_bit_int_def
thf(fact_1253_flip__bit__nat__def,axiom,
( bit_se2161824704523386999it_nat
= ( ^ [M3: nat,N3: nat] : ( bit_se6528837805403552850or_nat @ N3 @ ( bit_se547839408752420682it_nat @ M3 @ one_one_nat ) ) ) ) ).
% flip_bit_nat_def
thf(fact_1254_Suc__0__div__numeral_I2_J,axiom,
! [N: num] :
( ( divide_divide_nat @ ( suc @ zero_zero_nat ) @ ( numeral_numeral_nat @ ( bit0 @ N ) ) )
= zero_zero_nat ) ).
% Suc_0_div_numeral(2)
thf(fact_1255_Suc__0__div__numeral_I3_J,axiom,
! [N: num] :
( ( divide_divide_nat @ ( suc @ zero_zero_nat ) @ ( numeral_numeral_nat @ ( bit1 @ N ) ) )
= zero_zero_nat ) ).
% Suc_0_div_numeral(3)
thf(fact_1256_odd__Suc__div__two,axiom,
! [N: nat] :
( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
=> ( ( divide_divide_nat @ ( suc @ N ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
= ( suc @ ( divide_divide_nat @ N @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ).
% odd_Suc_div_two
thf(fact_1257_even__Suc__div__two,axiom,
! [N: nat] :
( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
=> ( ( divide_divide_nat @ ( suc @ N ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
= ( divide_divide_nat @ N @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ).
% even_Suc_div_two
thf(fact_1258_Suc__0__div__numeral_I1_J,axiom,
( ( divide_divide_nat @ ( suc @ zero_zero_nat ) @ ( numeral_numeral_nat @ one ) )
= one_one_nat ) ).
% Suc_0_div_numeral(1)
thf(fact_1259_odd__two__times__div__two__nat,axiom,
! [N: nat] :
( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
=> ( ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ N @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
= ( minus_minus_nat @ N @ one_one_nat ) ) ) ).
% odd_two_times_div_two_nat
thf(fact_1260_nat__mult__div__cancel__disj,axiom,
! [K: nat,M: nat,N: nat] :
( ( ( K = zero_zero_nat )
=> ( ( divide_divide_nat @ ( times_times_nat @ K @ M ) @ ( times_times_nat @ K @ N ) )
= zero_zero_nat ) )
& ( ( K != zero_zero_nat )
=> ( ( divide_divide_nat @ ( times_times_nat @ K @ M ) @ ( times_times_nat @ K @ N ) )
= ( divide_divide_nat @ M @ N ) ) ) ) ).
% nat_mult_div_cancel_disj
thf(fact_1261_int__ops_I8_J,axiom,
! [A: nat,B: nat] :
( ( semiri1314217659103216013at_int @ ( divide_divide_nat @ A @ B ) )
= ( divide_divide_int @ ( semiri1314217659103216013at_int @ A ) @ ( semiri1314217659103216013at_int @ B ) ) ) ).
% int_ops(8)
thf(fact_1262_set__bit__int__def,axiom,
( bit_se7879613467334960850it_int
= ( ^ [N3: nat,K3: int] : ( bit_se1409905431419307370or_int @ K3 @ ( bit_se545348938243370406it_int @ N3 @ one_one_int ) ) ) ) ).
% set_bit_int_def
thf(fact_1263_set__bit__nat__def,axiom,
( bit_se7882103937844011126it_nat
= ( ^ [M3: nat,N3: nat] : ( bit_se1412395901928357646or_nat @ N3 @ ( bit_se547839408752420682it_nat @ M3 @ one_one_nat ) ) ) ) ).
% set_bit_nat_def
% Helper facts (3)
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,Y3: nat] :
( ( if_nat @ $false @ X @ Y3 )
= Y3 ) ).
thf(help_If_1_1_If_001t__Nat__Onat_T,axiom,
! [X: nat,Y3: nat] :
( ( if_nat @ $true @ X @ Y3 )
= X ) ).
% Conjectures (1)
thf(conj_0,conjecture,
( ( last_P3305686521732843992nt_int @ ( the_li7431803565598847443nt_int @ ( to_path @ ( cons_list_nat @ ( cons_nat @ ( numeral_numeral_nat @ ( bit1 @ ( bit1 @ one ) ) ) @ ( cons_nat @ ( numeral_numeral_nat @ ( bit0 @ ( bit0 @ ( bit1 @ one ) ) ) ) @ ( cons_nat @ ( numeral_numeral_nat @ ( bit1 @ ( bit1 @ ( bit1 @ one ) ) ) ) @ ( cons_nat @ ( numeral_numeral_nat @ ( bit0 @ ( bit0 @ ( bit1 @ ( bit0 @ one ) ) ) ) ) @ ( cons_nat @ ( numeral_numeral_nat @ ( bit1 @ ( bit0 @ one ) ) ) @ nil_nat ) ) ) ) ) @ ( cons_list_nat @ ( cons_nat @ ( numeral_numeral_nat @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ one ) ) ) ) ) @ ( cons_nat @ ( numeral_numeral_nat @ ( bit1 @ ( bit0 @ ( bit1 @ ( bit0 @ one ) ) ) ) ) @ ( cons_nat @ ( numeral_numeral_nat @ ( bit0 @ ( bit1 @ one ) ) ) @ ( cons_nat @ ( numeral_numeral_nat @ ( bit1 @ ( bit0 @ ( bit0 @ ( bit1 @ one ) ) ) ) ) @ ( cons_nat @ ( numeral_numeral_nat @ ( bit0 @ ( bit1 @ ( bit1 @ one ) ) ) ) @ nil_nat ) ) ) ) ) @ ( cons_list_nat @ ( cons_nat @ ( numeral_numeral_nat @ ( bit1 @ ( bit1 @ ( bit0 @ one ) ) ) ) @ ( cons_nat @ ( numeral_numeral_nat @ ( bit0 @ ( bit0 @ ( bit0 @ one ) ) ) ) @ ( cons_nat @ ( numeral_numeral_nat @ ( bit1 @ ( bit0 @ ( bit1 @ one ) ) ) ) @ ( cons_nat @ ( numeral_numeral_nat @ ( bit0 @ ( bit0 @ one ) ) ) @ ( cons_nat @ ( numeral_numeral_nat @ ( bit1 @ ( bit1 @ ( bit0 @ ( bit0 @ one ) ) ) ) ) @ nil_nat ) ) ) ) ) @ ( cons_list_nat @ ( cons_nat @ ( numeral_numeral_nat @ ( bit0 @ ( bit1 @ ( bit1 @ ( bit0 @ one ) ) ) ) ) @ ( cons_nat @ ( numeral_numeral_nat @ ( bit1 @ ( bit0 @ ( bit0 @ ( bit0 @ one ) ) ) ) ) @ ( cons_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( cons_nat @ ( numeral_numeral_nat @ ( bit1 @ ( bit0 @ ( bit0 @ one ) ) ) ) @ ( cons_nat @ ( numeral_numeral_nat @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit1 @ one ) ) ) ) ) @ nil_nat ) ) ) ) ) @ ( cons_list_nat @ ( cons_nat @ one_one_nat @ ( cons_nat @ ( numeral_numeral_nat @ ( bit0 @ ( bit1 @ ( bit0 @ one ) ) ) ) @ ( cons_nat @ ( numeral_numeral_nat @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit0 @ one ) ) ) ) ) @ ( cons_nat @ ( numeral_numeral_nat @ ( bit0 @ ( bit1 @ ( bit0 @ ( bit0 @ one ) ) ) ) ) @ ( cons_nat @ ( numeral_numeral_nat @ ( bit1 @ one ) ) @ nil_nat ) ) ) ) ) @ nil_list_nat ) ) ) ) ) ) ) )
= ( product_Pair_int_int @ ( numeral_numeral_int @ ( bit0 @ ( bit0 @ one ) ) ) @ ( numeral_numeral_int @ ( bit0 @ ( bit0 @ one ) ) ) ) ) ).
%------------------------------------------------------------------------------