TPTP Problem File: SLH0839^1.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 : Multiset_Ordering_NPC/0002_Multiset_Ordering_in_NP/prob_00931_041632__13972610_1 [Des23]
% Status : Theorem
% Rating : ? v8.2.0
% Syntax : Number of formulae : 1409 ( 821 unt; 134 typ; 0 def)
% Number of atoms : 2881 (1532 equ; 0 cnn)
% Maximal formula atoms : 9 ( 2 avg)
% Number of connectives : 9546 ( 423 ~; 61 |; 114 &;8183 @)
% ( 0 <=>; 765 =>; 0 <=; 0 <~>)
% Maximal formula depth : 17 ( 5 avg)
% Number of types : 28 ( 27 usr)
% Number of type conns : 259 ( 259 >; 0 *; 0 +; 0 <<)
% Number of symbols : 110 ( 107 usr; 20 con; 0-4 aty)
% Number of variables : 2685 ( 38 ^;2617 !; 30 ?;2685 :)
% SPC : TH0_THM_EQU_NAR
% Comments : This file was generated by Isabelle (most likely Sledgehammer)
% 2023-01-18 16:30:10.077
%------------------------------------------------------------------------------
% Could-be-implicit typings (27)
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% Explicit typings (107)
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thf(sy_c_Groups_Ominus__class_Ominus_001t__Int__Oint,type,
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thf(sy_c_Groups_Ominus__class_Ominus_001t__Nat__Onat,type,
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thf(sy_c_Groups_Oone__class_Oone_001t__Int__Oint,type,
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thf(sy_c_Groups_Oone__class_Oone_001t__Nat__Onat,type,
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thf(sy_c_Groups_Oplus__class_Oplus_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__Int__Oint,type,
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thf(sy_c_If_001t__Nat__Onat,type,
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thf(sy_c_List_Olist_OCons_001t__Product____Type__Oprod_It__Multiset____Ordering____in____NP__OPropVar_M_Eo_J,type,
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thf(sy_c_Multiset__Ordering__in__NP_Ocnf__encodeGammaCond,type,
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thf(sy_c_Num_Onum_OBit0,type,
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thf(sy_c_Num_Onum_OBit1,type,
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thf(sy_c_Num_Onum_OOne,type,
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thf(sy_c_Num_Opow,type,
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thf(sy_c_Num_Oring__1__class_Oiszero_001t__Int__Oint,type,
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thf(sy_c_Orderings_Oord__class_Oless_001t__Int__Oint,type,
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thf(sy_c_Orderings_Oord__class_Oless_001t__Nat__Onat,type,
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thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Int__Oint,type,
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thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Nat__Onat,type,
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thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Num__Onum,type,
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thf(sy_c_Parity_Oadjust__div,type,
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thf(sy_c_Parity_Ounique__euclidean__semiring__with__nat__division__class_Odivides__aux_001t__Int__Oint,type,
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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__Product____Type__Oprod_It__Multiset____Ordering____in____NP__OPropVar_M_Eo_J_J_J_M_062_It__List__Olist_It__List__Olist_It__Product____Type__Oprod_It__Multiset____Ordering____in____NP__OPropVar_M_Eo_J_J_J_M_Eo_J_J_001t__List__Olist_It__List__Olist_It__List__Olist_It__Product____Type__Oprod_It__Multiset____Ordering____in____NP__OPropVar_M_Eo_J_J_J_J,type,
produc8106139234514114142pVar_o: ( list_l1255154591254203759pVar_o > list_l1255154591254203759pVar_o > $o ) > list_l985434930288970485pVar_o > produc3853458377486834414pVar_o ).
thf(sy_c_Product__Type_OPair_001_062_It__List__Olist_It__Product____Type__Oprod_It__Multiset____Ordering____in____NP__OPropVar_M_Eo_J_J_M_062_It__List__Olist_It__Product____Type__Oprod_It__Multiset____Ordering____in____NP__OPropVar_M_Eo_J_J_M_Eo_J_J_001t__List__Olist_It__List__Olist_It__Product____Type__Oprod_It__Multiset____Ordering____in____NP__OPropVar_M_Eo_J_J_J,type,
produc1228089817268051288pVar_o: ( list_P1665994083459302249pVar_o > list_P1665994083459302249pVar_o > $o ) > list_l1255154591254203759pVar_o > produc3305181862331950184pVar_o ).
thf(sy_c_Product__Type_OPair_001_062_It__List__Olist_It__Product____Type__Oprod_It__Multiset____Ordering____in____NP__OPropVar_M_Eo_J_J_Mt__List__Olist_It__List__Olist_It__Product____Type__Oprod_It__Multiset____Ordering____in____NP__OPropVar_M_Eo_J_J_J_J_001t__Product____Type__Oprod_It__List__Olist_It__List__Olist_It__Product____Type__Oprod_It__Multiset____Ordering____in____NP__OPropVar_M_Eo_J_J_J_Mt__List__Olist_It__List__Olist_It__List__Olist_It__Product____Type__Oprod_It__Multiset____Ordering____in____NP__OPropVar_M_Eo_J_J_J_J_J,type,
produc1538423139514444329pVar_o: ( list_P1665994083459302249pVar_o > list_l1255154591254203759pVar_o ) > produc5436468477138497005pVar_o > produc6278577299168974967pVar_o ).
thf(sy_c_Product__Type_OPair_001_062_It__Product____Type__Oprod_It__Multiset____Ordering____in____NP__OPropVar_M_Eo_J_M_062_It__Product____Type__Oprod_It__Multiset____Ordering____in____NP__OPropVar_M_Eo_J_M_Eo_J_J_001t__List__Olist_It__Product____Type__Oprod_It__Multiset____Ordering____in____NP__OPropVar_M_Eo_J_J,type,
produc3277683845840367058pVar_o: ( produc8999643685545340131pVar_o > produc8999643685545340131pVar_o > $o ) > list_P1665994083459302249pVar_o > produc7264647458362488674pVar_o ).
thf(sy_c_Product__Type_OPair_001_062_It__Product____Type__Oprod_It__Multiset____Ordering____in____NP__OPropVar_Mt__Product____Type__Oprod_It__Multiset____Ordering____in____NP__OPropVar_Mt__Multiset____Ordering____in____NP__OPropVar_J_J_M_062_It__Product____Type__Oprod_It__Multiset____Ordering____in____NP__OPropVar_Mt__Product____Type__Oprod_It__Multiset____Ordering____in____NP__OPropVar_Mt__Multiset____Ordering____in____NP__OPropVar_J_J_M_Eo_J_J_001t__List__Olist_It__Product____Type__Oprod_It__Multiset____Ordering____in____NP__OPropVar_Mt__Product____Type__Oprod_It__Multiset____Ordering____in____NP__OPropVar_Mt__Multiset____Ordering____in____NP__OPropVar_J_J_J,type,
produc3887254912111605537ropVar: ( produc2642885279816024434ropVar > produc2642885279816024434ropVar > $o ) > list_P8791008543742759928ropVar > produc5533280449304468145ropVar ).
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__List__Olist_It__List__Olist_It__List__Olist_It__Product____Type__Oprod_It__Multiset____Ordering____in____NP__OPropVar_M_Eo_J_J_J_J_001t__List__Olist_It__List__Olist_It__List__Olist_It__Product____Type__Oprod_It__Multiset____Ordering____in____NP__OPropVar_M_Eo_J_J_J_J,type,
produc2080502776689584855pVar_o: list_l985434930288970485pVar_o > list_l985434930288970485pVar_o > produc1241035484025441639pVar_o ).
thf(sy_c_Product__Type_OPair_001t__List__Olist_It__List__Olist_It__List__Olist_It__Product____Type__Oprod_It__Multiset____Ordering____in____NP__OPropVar_M_Eo_J_J_J_J_001t__Product____Type__Oprod_It__Multiset____Ordering____in____NP__OPropVar_Mt__Multiset____Ordering____in____NP__OPropVar_J,type,
produc2520692699247754633ropVar: list_l985434930288970485pVar_o > produc7679832412844898215ropVar > produc9206049318796031513ropVar ).
thf(sy_c_Product__Type_OPair_001t__List__Olist_It__List__Olist_It__Product____Type__Oprod_It__Multiset____Ordering____in____NP__OPropVar_M_Eo_J_J_J_001t__List__Olist_It__List__Olist_It__List__Olist_It__Product____Type__Oprod_It__Multiset____Ordering____in____NP__OPropVar_M_Eo_J_J_J_J,type,
produc4064290870397114077pVar_o: list_l1255154591254203759pVar_o > list_l985434930288970485pVar_o > produc5436468477138497005pVar_o ).
thf(sy_c_Product__Type_OPair_001t__List__Olist_It__List__Olist_It__Product____Type__Oprod_It__Multiset____Ordering____in____NP__OPropVar_M_Eo_J_J_J_001t__List__Olist_It__List__Olist_It__Product____Type__Oprod_It__Multiset____Ordering____in____NP__OPropVar_M_Eo_J_J_J,type,
produc5036572322992794967pVar_o: list_l1255154591254203759pVar_o > list_l1255154591254203759pVar_o > produc2246139759411191271pVar_o ).
thf(sy_c_Product__Type_OPair_001t__List__Olist_It__Product____Type__Oprod_It__Multiset____Ordering____in____NP__OPropVar_M_Eo_J_J_001t__List__Olist_It__Product____Type__Oprod_It__Multiset____Ordering____in____NP__OPropVar_M_Eo_J_J,type,
produc9064796567173055959pVar_o: list_P1665994083459302249pVar_o > list_P1665994083459302249pVar_o > produc4416794199230284903pVar_o ).
thf(sy_c_Product__Type_OPair_001t__List__Olist_It__Product____Type__Oprod_It__Multiset____Ordering____in____NP__OPropVar_Mt__Product____Type__Oprod_It__Multiset____Ordering____in____NP__OPropVar_Mt__Multiset____Ordering____in____NP__OPropVar_J_J_J_001t__List__Olist_It__Product____Type__Oprod_It__Multiset____Ordering____in____NP__OPropVar_Mt__Product____Type__Oprod_It__Multiset____Ordering____in____NP__OPropVar_Mt__Multiset____Ordering____in____NP__OPropVar_J_J_J,type,
produc5333478016634530327ropVar: list_P8791008543742759928ropVar > list_P8791008543742759928ropVar > produc3638977808798937127ropVar ).
thf(sy_c_Product__Type_OPair_001t__Multiset____Ordering____in____NP__OPropVar_001_Eo,type,
produc2127521181850112851pVar_o: multis3193088007478089820ropVar > $o > produc8999643685545340131pVar_o ).
thf(sy_c_Product__Type_OPair_001t__Multiset____Ordering____in____NP__OPropVar_001t__Multiset____Ordering____in____NP__OPropVar,type,
produc6936306168532358295ropVar: multis3193088007478089820ropVar > multis3193088007478089820ropVar > produc7679832412844898215ropVar ).
thf(sy_c_Product__Type_OPair_001t__Multiset____Ordering____in____NP__OPropVar_001t__Product____Type__Oprod_It__Multiset____Ordering____in____NP__OPropVar_Mt__Multiset____Ordering____in____NP__OPropVar_J,type,
produc6194868611276206818ropVar: multis3193088007478089820ropVar > produc7679832412844898215ropVar > produc2642885279816024434ropVar ).
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__Num__Onum_001t__Num__Onum,type,
product_Pair_num_num: num > num > product_prod_num_num ).
thf(sy_c_Product__Type_Oprod_Ofst_001t__Int__Oint_001t__Int__Oint,type,
product_fst_int_int: product_prod_int_int > int ).
thf(sy_c_Product__Type_Oprod_Ofst_001t__List__Olist_It__List__Olist_It__List__Olist_It__Product____Type__Oprod_It__Multiset____Ordering____in____NP__OPropVar_M_Eo_J_J_J_J_001t__Product____Type__Oprod_It__Multiset____Ordering____in____NP__OPropVar_Mt__Multiset____Ordering____in____NP__OPropVar_J,type,
produc284539208842135413ropVar: produc9206049318796031513ropVar > list_l985434930288970485pVar_o ).
thf(sy_c_Product__Type_Oprod_Ofst_001t__List__Olist_It__List__Olist_It__Product____Type__Oprod_It__Multiset____Ordering____in____NP__OPropVar_M_Eo_J_J_J_001t__List__Olist_It__List__Olist_It__List__Olist_It__Product____Type__Oprod_It__Multiset____Ordering____in____NP__OPropVar_M_Eo_J_J_J_J,type,
produc1814762917671617225pVar_o: produc5436468477138497005pVar_o > list_l1255154591254203759pVar_o ).
thf(sy_c_Product__Type_Oprod_Ofst_001t__Multiset____Ordering____in____NP__OPropVar_001_Eo,type,
produc7654750731481044799pVar_o: produc8999643685545340131pVar_o > multis3193088007478089820ropVar ).
thf(sy_c_Product__Type_Oprod_Ofst_001t__Nat__Onat_001t__Nat__Onat,type,
product_fst_nat_nat: product_prod_nat_nat > nat ).
thf(sy_c_Product__Type_Oprod_Ofst_001t__Num__Onum_001t__Num__Onum,type,
product_fst_num_num: product_prod_num_num > num ).
thf(sy_c_Propositional__Formula_Oeval__cnf_001t__Multiset____Ordering____in____NP__OPropVar,type,
propos2885041177628046573ropVar: ( multis3193088007478089820ropVar > $o ) > list_l1255154591254203759pVar_o > $o ).
thf(sy_c_Propositional__Formula_Osize__cnf_001t__Multiset____Ordering____in____NP__OPropVar,type,
propos4736024705303644274ropVar: list_l1255154591254203759pVar_o > nat ).
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_Rings_Ozero__neq__one__class_Oof__bool_001t__Int__Oint,type,
zero_n2684676970156552555ol_int: $o > int ).
thf(sy_c_Rings_Ozero__neq__one__class_Oof__bool_001t__Nat__Onat,type,
zero_n2687167440665602831ol_nat: $o > nat ).
thf(sy_v_conds,type,
conds: list_l985434930288970485pVar_o ).
thf(sy_v_on_H____,type,
on: multis3193088007478089820ropVar ).
thf(sy_v_one,type,
one2: list_l1255154591254203759pVar_o ).
thf(sy_v_vars,type,
vars: list_P8791008543742759928ropVar ).
thf(sy_v_vs____,type,
vs: list_P8791008543742759928ropVar ).
thf(sy_v_x____,type,
x: multis3193088007478089820ropVar ).
thf(sy_v_ze_H____,type,
ze: multis3193088007478089820ropVar ).
% Relevant facts (1269)
thf(fact_0_semiring__norm_I85_J,axiom,
! [M: num] :
( ( bit0 @ M )
!= one ) ).
% semiring_norm(85)
thf(fact_1_semiring__norm_I83_J,axiom,
! [N: num] :
( one
!= ( bit0 @ N ) ) ).
% semiring_norm(83)
thf(fact_2_verit__eq__simplify_I8_J,axiom,
! [X2: num,Y2: num] :
( ( ( bit0 @ X2 )
= ( bit0 @ Y2 ) )
= ( X2 = Y2 ) ) ).
% verit_eq_simplify(8)
thf(fact_3_semiring__norm_I87_J,axiom,
! [M: num,N: num] :
( ( ( bit0 @ M )
= ( bit0 @ N ) )
= ( M = N ) ) ).
% semiring_norm(87)
thf(fact_4_numeral__eq__iff,axiom,
! [M: num,N: num] :
( ( ( numeral_numeral_nat @ M )
= ( numeral_numeral_nat @ N ) )
= ( M = N ) ) ).
% numeral_eq_iff
thf(fact_5_numeral__eq__iff,axiom,
! [M: num,N: num] :
( ( ( numeral_numeral_int @ M )
= ( numeral_numeral_int @ N ) )
= ( M = N ) ) ).
% numeral_eq_iff
thf(fact_6_verit__eq__simplify_I10_J,axiom,
! [X2: num] :
( one
!= ( bit0 @ X2 ) ) ).
% verit_eq_simplify(10)
thf(fact_7_assms,axiom,
( ( multis8929244183426903067ly_one @ vars )
= ( produc4064290870397114077pVar_o @ one2 @ conds ) ) ).
% assms
thf(fact_8_fact__2,axiom,
( ( semiri1406184849735516958ct_int @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
= ( numeral_numeral_int @ ( bit0 @ one ) ) ) ).
% fact_2
thf(fact_9_fact__2,axiom,
( ( semiri1408675320244567234ct_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
= ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ).
% fact_2
thf(fact_10_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_11_numeral__Bit0__div__2,axiom,
! [N: num] :
( ( divide_divide_int @ ( numeral_numeral_int @ ( bit0 @ N ) ) @ ( numeral_numeral_int @ ( bit0 @ one ) ) )
= ( numeral_numeral_int @ N ) ) ).
% numeral_Bit0_div_2
thf(fact_12_numeral__Bit0__div__2,axiom,
! [N: num] :
( ( divide_divide_nat @ ( numeral_numeral_nat @ ( bit0 @ N ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
= ( numeral_numeral_nat @ N ) ) ).
% numeral_Bit0_div_2
thf(fact_13_False,axiom,
vars != nil_Pr6119456312066593144ropVar ).
% False
thf(fact_14_zdiv__numeral__Bit0,axiom,
! [V: num,W: num] :
( ( divide_divide_int @ ( numeral_numeral_int @ ( bit0 @ V ) ) @ ( numeral_numeral_int @ ( bit0 @ W ) ) )
= ( divide_divide_int @ ( numeral_numeral_int @ V ) @ ( numeral_numeral_int @ W ) ) ) ).
% zdiv_numeral_Bit0
thf(fact_15_prod_Oinject,axiom,
! [X1: multis3193088007478089820ropVar,X2: $o,Y1: multis3193088007478089820ropVar,Y2: $o] :
( ( ( produc2127521181850112851pVar_o @ X1 @ X2 )
= ( produc2127521181850112851pVar_o @ Y1 @ Y2 ) )
= ( ( X1 = Y1 )
& ( X2 = Y2 ) ) ) ).
% prod.inject
thf(fact_16_prod_Oinject,axiom,
! [X1: list_l1255154591254203759pVar_o,X2: list_l985434930288970485pVar_o,Y1: list_l1255154591254203759pVar_o,Y2: list_l985434930288970485pVar_o] :
( ( ( produc4064290870397114077pVar_o @ X1 @ X2 )
= ( produc4064290870397114077pVar_o @ Y1 @ Y2 ) )
= ( ( X1 = Y1 )
& ( X2 = Y2 ) ) ) ).
% prod.inject
thf(fact_17_prod_Oinject,axiom,
! [X1: list_l985434930288970485pVar_o,X2: produc7679832412844898215ropVar,Y1: list_l985434930288970485pVar_o,Y2: produc7679832412844898215ropVar] :
( ( ( produc2520692699247754633ropVar @ X1 @ X2 )
= ( produc2520692699247754633ropVar @ Y1 @ Y2 ) )
= ( ( X1 = Y1 )
& ( X2 = Y2 ) ) ) ).
% prod.inject
thf(fact_18_prod_Oinject,axiom,
! [X1: num,X2: num,Y1: num,Y2: num] :
( ( ( product_Pair_num_num @ X1 @ X2 )
= ( product_Pair_num_num @ Y1 @ Y2 ) )
= ( ( X1 = Y1 )
& ( X2 = Y2 ) ) ) ).
% prod.inject
thf(fact_19_prod_Oinject,axiom,
! [X1: int,X2: int,Y1: int,Y2: int] :
( ( ( product_Pair_int_int @ X1 @ X2 )
= ( product_Pair_int_int @ Y1 @ Y2 ) )
= ( ( X1 = Y1 )
& ( X2 = Y2 ) ) ) ).
% prod.inject
thf(fact_20_old_Oprod_Oinject,axiom,
! [A: multis3193088007478089820ropVar,B: $o,A2: multis3193088007478089820ropVar,B2: $o] :
( ( ( produc2127521181850112851pVar_o @ A @ B )
= ( produc2127521181850112851pVar_o @ A2 @ B2 ) )
= ( ( A = A2 )
& ( B = B2 ) ) ) ).
% old.prod.inject
thf(fact_21_old_Oprod_Oinject,axiom,
! [A: list_l1255154591254203759pVar_o,B: list_l985434930288970485pVar_o,A2: list_l1255154591254203759pVar_o,B2: list_l985434930288970485pVar_o] :
( ( ( produc4064290870397114077pVar_o @ A @ B )
= ( produc4064290870397114077pVar_o @ A2 @ B2 ) )
= ( ( A = A2 )
& ( B = B2 ) ) ) ).
% old.prod.inject
thf(fact_22_old_Oprod_Oinject,axiom,
! [A: list_l985434930288970485pVar_o,B: produc7679832412844898215ropVar,A2: list_l985434930288970485pVar_o,B2: produc7679832412844898215ropVar] :
( ( ( produc2520692699247754633ropVar @ A @ B )
= ( produc2520692699247754633ropVar @ A2 @ B2 ) )
= ( ( A = A2 )
& ( B = B2 ) ) ) ).
% old.prod.inject
thf(fact_23_old_Oprod_Oinject,axiom,
! [A: num,B: num,A2: num,B2: num] :
( ( ( product_Pair_num_num @ A @ B )
= ( product_Pair_num_num @ A2 @ B2 ) )
= ( ( A = A2 )
& ( B = B2 ) ) ) ).
% old.prod.inject
thf(fact_24_old_Oprod_Oinject,axiom,
! [A: int,B: int,A2: int,B2: int] :
( ( ( product_Pair_int_int @ A @ B )
= ( product_Pair_int_int @ A2 @ B2 ) )
= ( ( A = A2 )
& ( B = B2 ) ) ) ).
% old.prod.inject
thf(fact_25_dbl__simps_I3_J,axiom,
( ( neg_numeral_dbl_int @ one_one_int )
= ( numeral_numeral_int @ ( bit0 @ one ) ) ) ).
% dbl_simps(3)
thf(fact_26_numeral__Bit1__div__2,axiom,
! [N: num] :
( ( divide_divide_int @ ( numeral_numeral_int @ ( bit1 @ N ) ) @ ( numeral_numeral_int @ ( bit0 @ one ) ) )
= ( numeral_numeral_int @ N ) ) ).
% numeral_Bit1_div_2
thf(fact_27_numeral__Bit1__div__2,axiom,
! [N: num] :
( ( divide_divide_nat @ ( numeral_numeral_nat @ ( bit1 @ N ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
= ( numeral_numeral_nat @ N ) ) ).
% numeral_Bit1_div_2
thf(fact_28_old_Oprod_Oexhaust,axiom,
! [Y: produc8999643685545340131pVar_o] :
~ ! [A3: multis3193088007478089820ropVar,B3: $o] :
( Y
!= ( produc2127521181850112851pVar_o @ A3 @ B3 ) ) ).
% old.prod.exhaust
thf(fact_29_old_Oprod_Oexhaust,axiom,
! [Y: produc5436468477138497005pVar_o] :
~ ! [A3: list_l1255154591254203759pVar_o,B3: list_l985434930288970485pVar_o] :
( Y
!= ( produc4064290870397114077pVar_o @ A3 @ B3 ) ) ).
% old.prod.exhaust
thf(fact_30_old_Oprod_Oexhaust,axiom,
! [Y: produc9206049318796031513ropVar] :
~ ! [A3: list_l985434930288970485pVar_o,B3: produc7679832412844898215ropVar] :
( Y
!= ( produc2520692699247754633ropVar @ A3 @ B3 ) ) ).
% old.prod.exhaust
thf(fact_31_old_Oprod_Oexhaust,axiom,
! [Y: product_prod_num_num] :
~ ! [A3: num,B3: num] :
( Y
!= ( product_Pair_num_num @ A3 @ B3 ) ) ).
% old.prod.exhaust
thf(fact_32_old_Oprod_Oexhaust,axiom,
! [Y: product_prod_int_int] :
~ ! [A3: int,B3: int] :
( Y
!= ( product_Pair_int_int @ A3 @ B3 ) ) ).
% old.prod.exhaust
thf(fact_33_surj__pair,axiom,
! [P: produc8999643685545340131pVar_o] :
? [X: multis3193088007478089820ropVar,Y3: $o] :
( P
= ( produc2127521181850112851pVar_o @ X @ Y3 ) ) ).
% surj_pair
thf(fact_34_surj__pair,axiom,
! [P: produc5436468477138497005pVar_o] :
? [X: list_l1255154591254203759pVar_o,Y3: list_l985434930288970485pVar_o] :
( P
= ( produc4064290870397114077pVar_o @ X @ Y3 ) ) ).
% surj_pair
thf(fact_35_surj__pair,axiom,
! [P: produc9206049318796031513ropVar] :
? [X: list_l985434930288970485pVar_o,Y3: produc7679832412844898215ropVar] :
( P
= ( produc2520692699247754633ropVar @ X @ Y3 ) ) ).
% surj_pair
thf(fact_36_surj__pair,axiom,
! [P: product_prod_num_num] :
? [X: num,Y3: num] :
( P
= ( product_Pair_num_num @ X @ Y3 ) ) ).
% surj_pair
thf(fact_37_surj__pair,axiom,
! [P: product_prod_int_int] :
? [X: int,Y3: int] :
( P
= ( product_Pair_int_int @ X @ Y3 ) ) ).
% surj_pair
thf(fact_38_prod__cases,axiom,
! [P2: produc8999643685545340131pVar_o > $o,P: produc8999643685545340131pVar_o] :
( ! [A3: multis3193088007478089820ropVar,B3: $o] : ( P2 @ ( produc2127521181850112851pVar_o @ A3 @ B3 ) )
=> ( P2 @ P ) ) ).
% prod_cases
thf(fact_39_prod__cases,axiom,
! [P2: produc5436468477138497005pVar_o > $o,P: produc5436468477138497005pVar_o] :
( ! [A3: list_l1255154591254203759pVar_o,B3: list_l985434930288970485pVar_o] : ( P2 @ ( produc4064290870397114077pVar_o @ A3 @ B3 ) )
=> ( P2 @ P ) ) ).
% prod_cases
thf(fact_40_prod__cases,axiom,
! [P2: produc9206049318796031513ropVar > $o,P: produc9206049318796031513ropVar] :
( ! [A3: list_l985434930288970485pVar_o,B3: produc7679832412844898215ropVar] : ( P2 @ ( produc2520692699247754633ropVar @ A3 @ B3 ) )
=> ( P2 @ P ) ) ).
% prod_cases
thf(fact_41_prod__cases,axiom,
! [P2: product_prod_num_num > $o,P: product_prod_num_num] :
( ! [A3: num,B3: num] : ( P2 @ ( product_Pair_num_num @ A3 @ B3 ) )
=> ( P2 @ P ) ) ).
% prod_cases
thf(fact_42_prod__cases,axiom,
! [P2: product_prod_int_int > $o,P: product_prod_int_int] :
( ! [A3: int,B3: int] : ( P2 @ ( product_Pair_int_int @ A3 @ B3 ) )
=> ( P2 @ P ) ) ).
% prod_cases
thf(fact_43_Pair__inject,axiom,
! [A: multis3193088007478089820ropVar,B: $o,A2: multis3193088007478089820ropVar,B2: $o] :
( ( ( produc2127521181850112851pVar_o @ A @ B )
= ( produc2127521181850112851pVar_o @ A2 @ B2 ) )
=> ~ ( ( A = A2 )
=> ( B = ~ B2 ) ) ) ).
% Pair_inject
thf(fact_44_Pair__inject,axiom,
! [A: list_l1255154591254203759pVar_o,B: list_l985434930288970485pVar_o,A2: list_l1255154591254203759pVar_o,B2: list_l985434930288970485pVar_o] :
( ( ( produc4064290870397114077pVar_o @ A @ B )
= ( produc4064290870397114077pVar_o @ A2 @ B2 ) )
=> ~ ( ( A = A2 )
=> ( B != B2 ) ) ) ).
% Pair_inject
thf(fact_45_Pair__inject,axiom,
! [A: list_l985434930288970485pVar_o,B: produc7679832412844898215ropVar,A2: list_l985434930288970485pVar_o,B2: produc7679832412844898215ropVar] :
( ( ( produc2520692699247754633ropVar @ A @ B )
= ( produc2520692699247754633ropVar @ A2 @ B2 ) )
=> ~ ( ( A = A2 )
=> ( B != B2 ) ) ) ).
% Pair_inject
thf(fact_46_Pair__inject,axiom,
! [A: num,B: num,A2: num,B2: num] :
( ( ( product_Pair_num_num @ A @ B )
= ( product_Pair_num_num @ A2 @ B2 ) )
=> ~ ( ( A = A2 )
=> ( B != B2 ) ) ) ).
% Pair_inject
thf(fact_47_Pair__inject,axiom,
! [A: int,B: int,A2: int,B2: int] :
( ( ( product_Pair_int_int @ A @ B )
= ( product_Pair_int_int @ A2 @ B2 ) )
=> ~ ( ( A = A2 )
=> ( B != B2 ) ) ) ).
% Pair_inject
thf(fact_48_prod__cases3,axiom,
! [Y: produc9206049318796031513ropVar] :
~ ! [A3: list_l985434930288970485pVar_o,B3: multis3193088007478089820ropVar,C: multis3193088007478089820ropVar] :
( Y
!= ( produc2520692699247754633ropVar @ A3 @ ( produc6936306168532358295ropVar @ B3 @ C ) ) ) ).
% prod_cases3
thf(fact_49_local_ONil,axiom,
vs = nil_Pr6119456312066593144ropVar ).
% local.Nil
thf(fact_50_verit__eq__simplify_I9_J,axiom,
! [X3: num,Y32: num] :
( ( ( bit1 @ X3 )
= ( bit1 @ Y32 ) )
= ( X3 = Y32 ) ) ).
% verit_eq_simplify(9)
thf(fact_51_semiring__norm_I90_J,axiom,
! [M: num,N: num] :
( ( ( bit1 @ M )
= ( bit1 @ N ) )
= ( M = N ) ) ).
% semiring_norm(90)
thf(fact_52_semiring__norm_I89_J,axiom,
! [M: num,N: num] :
( ( bit1 @ M )
!= ( bit0 @ N ) ) ).
% semiring_norm(89)
thf(fact_53_semiring__norm_I88_J,axiom,
! [M: num,N: num] :
( ( bit0 @ M )
!= ( bit1 @ N ) ) ).
% semiring_norm(88)
thf(fact_54_semiring__norm_I86_J,axiom,
! [M: num] :
( ( bit1 @ M )
!= one ) ).
% semiring_norm(86)
thf(fact_55_semiring__norm_I84_J,axiom,
! [N: num] :
( one
!= ( bit1 @ N ) ) ).
% semiring_norm(84)
thf(fact_56_fact__1,axiom,
( ( semiri1408675320244567234ct_nat @ one_one_nat )
= one_one_nat ) ).
% fact_1
thf(fact_57_fact__1,axiom,
( ( semiri1406184849735516958ct_int @ one_one_nat )
= one_one_int ) ).
% fact_1
thf(fact_58_numeral__eq__one__iff,axiom,
! [N: num] :
( ( ( numeral_numeral_nat @ N )
= one_one_nat )
= ( N = one ) ) ).
% numeral_eq_one_iff
thf(fact_59_numeral__eq__one__iff,axiom,
! [N: num] :
( ( ( numeral_numeral_int @ N )
= one_one_int )
= ( N = one ) ) ).
% numeral_eq_one_iff
thf(fact_60_one__eq__numeral__iff,axiom,
! [N: num] :
( ( one_one_nat
= ( numeral_numeral_nat @ N ) )
= ( one = N ) ) ).
% one_eq_numeral_iff
thf(fact_61_one__eq__numeral__iff,axiom,
! [N: num] :
( ( one_one_int
= ( numeral_numeral_int @ N ) )
= ( one = N ) ) ).
% one_eq_numeral_iff
thf(fact_62_zdiv__numeral__Bit1,axiom,
! [V: num,W: num] :
( ( divide_divide_int @ ( numeral_numeral_int @ ( bit1 @ V ) ) @ ( numeral_numeral_int @ ( bit0 @ W ) ) )
= ( divide_divide_int @ ( numeral_numeral_int @ V ) @ ( numeral_numeral_int @ W ) ) ) ).
% zdiv_numeral_Bit1
thf(fact_63_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_64_verit__eq__simplify_I14_J,axiom,
! [X2: num,X3: num] :
( ( bit0 @ X2 )
!= ( bit1 @ X3 ) ) ).
% verit_eq_simplify(14)
thf(fact_65_verit__eq__simplify_I12_J,axiom,
! [X3: num] :
( one
!= ( bit1 @ X3 ) ) ).
% verit_eq_simplify(12)
thf(fact_66_num_Oexhaust,axiom,
! [Y: num] :
( ( Y != one )
=> ( ! [X22: num] :
( Y
!= ( bit0 @ X22 ) )
=> ~ ! [X32: num] :
( Y
!= ( bit1 @ X32 ) ) ) ) ).
% num.exhaust
thf(fact_67_numeral__One,axiom,
( ( numeral_numeral_nat @ one )
= one_one_nat ) ).
% numeral_One
thf(fact_68_numeral__One,axiom,
( ( numeral_numeral_int @ one )
= one_one_int ) ).
% numeral_One
thf(fact_69_numerals_I1_J,axiom,
( ( numeral_numeral_nat @ one )
= one_one_nat ) ).
% numerals(1)
thf(fact_70_prod__induct3,axiom,
! [P2: produc9206049318796031513ropVar > $o,X4: produc9206049318796031513ropVar] :
( ! [A3: list_l985434930288970485pVar_o,B3: multis3193088007478089820ropVar,C: multis3193088007478089820ropVar] : ( P2 @ ( produc2520692699247754633ropVar @ A3 @ ( produc6936306168532358295ropVar @ B3 @ C ) ) )
=> ( P2 @ X4 ) ) ).
% prod_induct3
thf(fact_71_bits__div__by__1,axiom,
! [A: int] :
( ( divide_divide_int @ A @ one_one_int )
= A ) ).
% bits_div_by_1
thf(fact_72_bits__div__by__1,axiom,
! [A: nat] :
( ( divide_divide_nat @ A @ one_one_nat )
= A ) ).
% bits_div_by_1
thf(fact_73_div__by__1,axiom,
! [A: int] :
( ( divide_divide_int @ A @ one_one_int )
= A ) ).
% div_by_1
thf(fact_74_div__by__1,axiom,
! [A: nat] :
( ( divide_divide_nat @ A @ one_one_nat )
= A ) ).
% div_by_1
thf(fact_75_dbl__inc__simps_I3_J,axiom,
( ( neg_nu5851722552734809277nc_int @ one_one_int )
= ( numeral_numeral_int @ ( bit1 @ one ) ) ) ).
% dbl_inc_simps(3)
thf(fact_76_xor__num_Ocases,axiom,
! [X4: product_prod_num_num] :
( ( X4
!= ( product_Pair_num_num @ one @ one ) )
=> ( ! [N2: num] :
( X4
!= ( product_Pair_num_num @ one @ ( bit0 @ N2 ) ) )
=> ( ! [N2: num] :
( X4
!= ( product_Pair_num_num @ one @ ( bit1 @ N2 ) ) )
=> ( ! [M2: num] :
( X4
!= ( product_Pair_num_num @ ( bit0 @ M2 ) @ one ) )
=> ( ! [M2: num,N2: num] :
( X4
!= ( product_Pair_num_num @ ( bit0 @ M2 ) @ ( bit0 @ N2 ) ) )
=> ( ! [M2: num,N2: num] :
( X4
!= ( product_Pair_num_num @ ( bit0 @ M2 ) @ ( bit1 @ N2 ) ) )
=> ( ! [M2: num] :
( X4
!= ( product_Pair_num_num @ ( bit1 @ M2 ) @ one ) )
=> ( ! [M2: num,N2: num] :
( X4
!= ( product_Pair_num_num @ ( bit1 @ M2 ) @ ( bit0 @ N2 ) ) )
=> ~ ! [M2: num,N2: num] :
( X4
!= ( product_Pair_num_num @ ( bit1 @ M2 ) @ ( bit1 @ N2 ) ) ) ) ) ) ) ) ) ) ) ).
% xor_num.cases
thf(fact_77__092_060open_062_092_060And_062thesis_O_A_I_092_060And_062x_Aze_H_Aon_H_Avs_O_Avars_A_061_A_Ix_M_Aze_H_M_Aon_H_J_A_D_Avs_A_092_060Longrightarrow_062_Athesis_J_A_092_060Longrightarrow_062_Athesis_092_060close_062,axiom,
~ ! [X: multis3193088007478089820ropVar,Ze: multis3193088007478089820ropVar,On: multis3193088007478089820ropVar,Vs: list_P8791008543742759928ropVar] :
( vars
!= ( cons_P1504208554336013608ropVar @ ( produc6194868611276206818ropVar @ X @ ( produc6936306168532358295ropVar @ Ze @ On ) ) @ Vs ) ) ).
% \<open>\<And>thesis. (\<And>x ze' on' vs. vars = (x, ze', on') # vs \<Longrightarrow> thesis) \<Longrightarrow> thesis\<close>
thf(fact_78_one__div__two__eq__zero,axiom,
( ( divide_divide_int @ one_one_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) )
= zero_zero_int ) ).
% one_div_two_eq_zero
thf(fact_79_one__div__two__eq__zero,axiom,
( ( divide_divide_nat @ one_one_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
= zero_zero_nat ) ).
% one_div_two_eq_zero
thf(fact_80_div__0,axiom,
! [A: int] :
( ( divide_divide_int @ zero_zero_int @ A )
= zero_zero_int ) ).
% div_0
thf(fact_81_div__0,axiom,
! [A: nat] :
( ( divide_divide_nat @ zero_zero_nat @ A )
= zero_zero_nat ) ).
% div_0
thf(fact_82_div__by__0,axiom,
! [A: int] :
( ( divide_divide_int @ A @ zero_zero_int )
= zero_zero_int ) ).
% div_by_0
thf(fact_83_div__by__0,axiom,
! [A: nat] :
( ( divide_divide_nat @ A @ zero_zero_nat )
= zero_zero_nat ) ).
% div_by_0
thf(fact_84_bits__div__0,axiom,
! [A: int] :
( ( divide_divide_int @ zero_zero_int @ A )
= zero_zero_int ) ).
% bits_div_0
thf(fact_85_bits__div__0,axiom,
! [A: nat] :
( ( divide_divide_nat @ zero_zero_nat @ A )
= zero_zero_nat ) ).
% bits_div_0
thf(fact_86_bits__div__by__0,axiom,
! [A: int] :
( ( divide_divide_int @ A @ zero_zero_int )
= zero_zero_int ) ).
% bits_div_by_0
thf(fact_87_bits__div__by__0,axiom,
! [A: nat] :
( ( divide_divide_nat @ A @ zero_zero_nat )
= zero_zero_nat ) ).
% bits_div_by_0
thf(fact_88_fact__0,axiom,
( ( semiri1408675320244567234ct_nat @ zero_zero_nat )
= one_one_nat ) ).
% fact_0
thf(fact_89_fact__0,axiom,
( ( semiri1406184849735516958ct_int @ zero_zero_nat )
= one_one_int ) ).
% fact_0
thf(fact_90_dbl__simps_I2_J,axiom,
( ( neg_numeral_dbl_int @ zero_zero_int )
= zero_zero_int ) ).
% dbl_simps(2)
thf(fact_91_dbl__dec__simps_I3_J,axiom,
( ( neg_nu3811975205180677377ec_int @ one_one_int )
= one_one_int ) ).
% dbl_dec_simps(3)
thf(fact_92_div__self,axiom,
! [A: int] :
( ( A != zero_zero_int )
=> ( ( divide_divide_int @ A @ A )
= one_one_int ) ) ).
% div_self
thf(fact_93_div__self,axiom,
! [A: nat] :
( ( A != zero_zero_nat )
=> ( ( divide_divide_nat @ A @ A )
= one_one_nat ) ) ).
% div_self
thf(fact_94_sub__num__simps_I1_J,axiom,
( ( neg_numeral_sub_int @ one @ one )
= zero_zero_int ) ).
% sub_num_simps(1)
thf(fact_95_vars,axiom,
( vars
= ( cons_P1504208554336013608ropVar @ ( produc6194868611276206818ropVar @ x @ ( produc6936306168532358295ropVar @ ze @ on ) ) @ vs ) ) ).
% vars
thf(fact_96_dbl__inc__simps_I2_J,axiom,
( ( neg_nu5851722552734809277nc_int @ zero_zero_int )
= one_one_int ) ).
% dbl_inc_simps(2)
thf(fact_97_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_98_bits__1__div__2,axiom,
( ( divide_divide_int @ one_one_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) )
= zero_zero_int ) ).
% bits_1_div_2
thf(fact_99_bits__1__div__2,axiom,
( ( divide_divide_nat @ one_one_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
= zero_zero_nat ) ).
% bits_1_div_2
thf(fact_100_zero__reorient,axiom,
! [X4: nat] :
( ( zero_zero_nat = X4 )
= ( X4 = zero_zero_nat ) ) ).
% zero_reorient
thf(fact_101_zero__reorient,axiom,
! [X4: int] :
( ( zero_zero_int = X4 )
= ( X4 = zero_zero_int ) ) ).
% zero_reorient
thf(fact_102_zero__neq__one,axiom,
zero_zero_nat != one_one_nat ).
% zero_neq_one
thf(fact_103_zero__neq__one,axiom,
zero_zero_int != one_one_int ).
% zero_neq_one
thf(fact_104_zero__neq__numeral,axiom,
! [N: num] :
( zero_zero_nat
!= ( numeral_numeral_nat @ N ) ) ).
% zero_neq_numeral
thf(fact_105_zero__neq__numeral,axiom,
! [N: num] :
( zero_zero_int
!= ( numeral_numeral_int @ N ) ) ).
% zero_neq_numeral
thf(fact_106_fact__nonzero,axiom,
! [N: nat] :
( ( semiri1408675320244567234ct_nat @ N )
!= zero_zero_nat ) ).
% fact_nonzero
thf(fact_107_fact__nonzero,axiom,
! [N: nat] :
( ( semiri1406184849735516958ct_int @ N )
!= zero_zero_int ) ).
% fact_nonzero
thf(fact_108_cnf__encoder_Oencode__sum__0__1__main_Ocases,axiom,
! [X4: list_P8791008543742759928ropVar] :
( ! [X: multis3193088007478089820ropVar,Zero: multis3193088007478089820ropVar,One: multis3193088007478089820ropVar] :
( X4
!= ( cons_P1504208554336013608ropVar @ ( produc6194868611276206818ropVar @ X @ ( produc6936306168532358295ropVar @ Zero @ One ) ) @ nil_Pr6119456312066593144ropVar ) )
=> ( ! [X: multis3193088007478089820ropVar,Zero: multis3193088007478089820ropVar,One: multis3193088007478089820ropVar,V2: produc2642885279816024434ropVar,Va: list_P8791008543742759928ropVar] :
( X4
!= ( cons_P1504208554336013608ropVar @ ( produc6194868611276206818ropVar @ X @ ( produc6936306168532358295ropVar @ Zero @ One ) ) @ ( cons_P1504208554336013608ropVar @ V2 @ Va ) ) )
=> ( X4 = nil_Pr6119456312066593144ropVar ) ) ) ).
% cnf_encoder.encode_sum_0_1_main.cases
thf(fact_109_one__reorient,axiom,
! [X4: nat] :
( ( one_one_nat = X4 )
= ( X4 = one_one_nat ) ) ).
% one_reorient
thf(fact_110_one__reorient,axiom,
! [X4: int] :
( ( one_one_int = X4 )
= ( X4 = one_one_int ) ) ).
% one_reorient
thf(fact_111_divides__aux__eq,axiom,
! [Q: nat,R: nat] :
( ( unique5332122412489317741ux_nat @ ( product_Pair_nat_nat @ Q @ R ) )
= ( R = zero_zero_nat ) ) ).
% divides_aux_eq
thf(fact_112_divides__aux__eq,axiom,
! [Q: int,R: int] :
( ( unique5329631941980267465ux_int @ ( product_Pair_int_int @ Q @ R ) )
= ( R = zero_zero_int ) ) ).
% divides_aux_eq
thf(fact_113_shuffles_Ocases,axiom,
! [X4: produc3638977808798937127ropVar] :
( ! [Ys: list_P8791008543742759928ropVar] :
( X4
!= ( produc5333478016634530327ropVar @ nil_Pr6119456312066593144ropVar @ Ys ) )
=> ( ! [Xs: list_P8791008543742759928ropVar] :
( X4
!= ( produc5333478016634530327ropVar @ Xs @ nil_Pr6119456312066593144ropVar ) )
=> ~ ! [X: produc2642885279816024434ropVar,Xs: list_P8791008543742759928ropVar,Y3: produc2642885279816024434ropVar,Ys: list_P8791008543742759928ropVar] :
( X4
!= ( produc5333478016634530327ropVar @ ( cons_P1504208554336013608ropVar @ X @ Xs ) @ ( cons_P1504208554336013608ropVar @ Y3 @ Ys ) ) ) ) ) ).
% shuffles.cases
thf(fact_114_shuffles_Ocases,axiom,
! [X4: produc1241035484025441639pVar_o] :
( ! [Ys: list_l985434930288970485pVar_o] :
( X4
!= ( produc2080502776689584855pVar_o @ nil_li590035233778055285pVar_o @ Ys ) )
=> ( ! [Xs: list_l985434930288970485pVar_o] :
( X4
!= ( produc2080502776689584855pVar_o @ Xs @ nil_li590035233778055285pVar_o ) )
=> ~ ! [X: list_l1255154591254203759pVar_o,Xs: list_l985434930288970485pVar_o,Y3: list_l1255154591254203759pVar_o,Ys: list_l985434930288970485pVar_o] :
( X4
!= ( produc2080502776689584855pVar_o @ ( cons_l7316790070876032037pVar_o @ X @ Xs ) @ ( cons_l7316790070876032037pVar_o @ Y3 @ Ys ) ) ) ) ) ).
% shuffles.cases
thf(fact_115_shuffles_Ocases,axiom,
! [X4: produc2246139759411191271pVar_o] :
( ! [Ys: list_l1255154591254203759pVar_o] :
( X4
!= ( produc5036572322992794967pVar_o @ nil_li1310946081133177199pVar_o @ Ys ) )
=> ( ! [Xs: list_l1255154591254203759pVar_o] :
( X4
!= ( produc5036572322992794967pVar_o @ Xs @ nil_li1310946081133177199pVar_o ) )
=> ~ ! [X: list_P1665994083459302249pVar_o,Xs: list_l1255154591254203759pVar_o,Y3: list_P1665994083459302249pVar_o,Ys: list_l1255154591254203759pVar_o] :
( X4
!= ( produc5036572322992794967pVar_o @ ( cons_l3452452106213086495pVar_o @ X @ Xs ) @ ( cons_l3452452106213086495pVar_o @ Y3 @ Ys ) ) ) ) ) ).
% shuffles.cases
thf(fact_116_shuffles_Ocases,axiom,
! [X4: produc4416794199230284903pVar_o] :
( ! [Ys: list_P1665994083459302249pVar_o] :
( X4
!= ( produc9064796567173055959pVar_o @ nil_Pr6413368191065476585pVar_o @ Ys ) )
=> ( ! [Xs: list_P1665994083459302249pVar_o] :
( X4
!= ( produc9064796567173055959pVar_o @ Xs @ nil_Pr6413368191065476585pVar_o ) )
=> ~ ! [X: produc8999643685545340131pVar_o,Xs: list_P1665994083459302249pVar_o,Y3: produc8999643685545340131pVar_o,Ys: list_P1665994083459302249pVar_o] :
( X4
!= ( produc9064796567173055959pVar_o @ ( cons_P6860203055867761049pVar_o @ X @ Xs ) @ ( cons_P6860203055867761049pVar_o @ Y3 @ Ys ) ) ) ) ) ).
% shuffles.cases
thf(fact_117_map__tailrec__rev_Ocases,axiom,
! [X4: produc6278577299168974967pVar_o] :
( ! [F: list_P1665994083459302249pVar_o > list_l1255154591254203759pVar_o,Bs: list_l985434930288970485pVar_o] :
( X4
!= ( produc1538423139514444329pVar_o @ F @ ( produc4064290870397114077pVar_o @ nil_li1310946081133177199pVar_o @ Bs ) ) )
=> ~ ! [F: list_P1665994083459302249pVar_o > list_l1255154591254203759pVar_o,A3: list_P1665994083459302249pVar_o,As: list_l1255154591254203759pVar_o,Bs: list_l985434930288970485pVar_o] :
( X4
!= ( produc1538423139514444329pVar_o @ F @ ( produc4064290870397114077pVar_o @ ( cons_l3452452106213086495pVar_o @ A3 @ As ) @ Bs ) ) ) ) ).
% map_tailrec_rev.cases
thf(fact_118_subset__eq__mset__impl_Ocases,axiom,
! [X4: produc3638977808798937127ropVar] :
( ! [Ys: list_P8791008543742759928ropVar] :
( X4
!= ( produc5333478016634530327ropVar @ nil_Pr6119456312066593144ropVar @ Ys ) )
=> ~ ! [X: produc2642885279816024434ropVar,Xs: list_P8791008543742759928ropVar,Ys: list_P8791008543742759928ropVar] :
( X4
!= ( produc5333478016634530327ropVar @ ( cons_P1504208554336013608ropVar @ X @ Xs ) @ Ys ) ) ) ).
% subset_eq_mset_impl.cases
thf(fact_119_subset__eq__mset__impl_Ocases,axiom,
! [X4: produc1241035484025441639pVar_o] :
( ! [Ys: list_l985434930288970485pVar_o] :
( X4
!= ( produc2080502776689584855pVar_o @ nil_li590035233778055285pVar_o @ Ys ) )
=> ~ ! [X: list_l1255154591254203759pVar_o,Xs: list_l985434930288970485pVar_o,Ys: list_l985434930288970485pVar_o] :
( X4
!= ( produc2080502776689584855pVar_o @ ( cons_l7316790070876032037pVar_o @ X @ Xs ) @ Ys ) ) ) ).
% subset_eq_mset_impl.cases
thf(fact_120_subset__eq__mset__impl_Ocases,axiom,
! [X4: produc2246139759411191271pVar_o] :
( ! [Ys: list_l1255154591254203759pVar_o] :
( X4
!= ( produc5036572322992794967pVar_o @ nil_li1310946081133177199pVar_o @ Ys ) )
=> ~ ! [X: list_P1665994083459302249pVar_o,Xs: list_l1255154591254203759pVar_o,Ys: list_l1255154591254203759pVar_o] :
( X4
!= ( produc5036572322992794967pVar_o @ ( cons_l3452452106213086495pVar_o @ X @ Xs ) @ Ys ) ) ) ).
% subset_eq_mset_impl.cases
thf(fact_121_subset__eq__mset__impl_Ocases,axiom,
! [X4: produc4416794199230284903pVar_o] :
( ! [Ys: list_P1665994083459302249pVar_o] :
( X4
!= ( produc9064796567173055959pVar_o @ nil_Pr6413368191065476585pVar_o @ Ys ) )
=> ~ ! [X: produc8999643685545340131pVar_o,Xs: list_P1665994083459302249pVar_o,Ys: list_P1665994083459302249pVar_o] :
( X4
!= ( produc9064796567173055959pVar_o @ ( cons_P6860203055867761049pVar_o @ X @ Xs ) @ Ys ) ) ) ).
% subset_eq_mset_impl.cases
thf(fact_122_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_123_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_124_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_125_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_126_list_Oinject,axiom,
! [X21: produc2642885279816024434ropVar,X222: list_P8791008543742759928ropVar,Y21: produc2642885279816024434ropVar,Y22: list_P8791008543742759928ropVar] :
( ( ( cons_P1504208554336013608ropVar @ X21 @ X222 )
= ( cons_P1504208554336013608ropVar @ Y21 @ Y22 ) )
= ( ( X21 = Y21 )
& ( X222 = Y22 ) ) ) ).
% list.inject
thf(fact_127_list_Oinject,axiom,
! [X21: list_l1255154591254203759pVar_o,X222: list_l985434930288970485pVar_o,Y21: list_l1255154591254203759pVar_o,Y22: list_l985434930288970485pVar_o] :
( ( ( cons_l7316790070876032037pVar_o @ X21 @ X222 )
= ( cons_l7316790070876032037pVar_o @ Y21 @ Y22 ) )
= ( ( X21 = Y21 )
& ( X222 = Y22 ) ) ) ).
% list.inject
thf(fact_128_list_Oinject,axiom,
! [X21: list_P1665994083459302249pVar_o,X222: list_l1255154591254203759pVar_o,Y21: list_P1665994083459302249pVar_o,Y22: list_l1255154591254203759pVar_o] :
( ( ( cons_l3452452106213086495pVar_o @ X21 @ X222 )
= ( cons_l3452452106213086495pVar_o @ Y21 @ Y22 ) )
= ( ( X21 = Y21 )
& ( X222 = Y22 ) ) ) ).
% list.inject
thf(fact_129_list_Oinject,axiom,
! [X21: produc8999643685545340131pVar_o,X222: list_P1665994083459302249pVar_o,Y21: produc8999643685545340131pVar_o,Y22: list_P1665994083459302249pVar_o] :
( ( ( cons_P6860203055867761049pVar_o @ X21 @ X222 )
= ( cons_P6860203055867761049pVar_o @ Y21 @ Y22 ) )
= ( ( X21 = Y21 )
& ( X222 = Y22 ) ) ) ).
% list.inject
thf(fact_130_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_131_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_132_sorted__wrt_Ocases,axiom,
! [X4: produc5533280449304468145ropVar] :
( ! [P3: produc2642885279816024434ropVar > produc2642885279816024434ropVar > $o] :
( X4
!= ( produc3887254912111605537ropVar @ P3 @ nil_Pr6119456312066593144ropVar ) )
=> ~ ! [P3: produc2642885279816024434ropVar > produc2642885279816024434ropVar > $o,X: produc2642885279816024434ropVar,Ys: list_P8791008543742759928ropVar] :
( X4
!= ( produc3887254912111605537ropVar @ P3 @ ( cons_P1504208554336013608ropVar @ X @ Ys ) ) ) ) ).
% sorted_wrt.cases
thf(fact_133_sorted__wrt_Ocases,axiom,
! [X4: produc3853458377486834414pVar_o] :
( ! [P3: list_l1255154591254203759pVar_o > list_l1255154591254203759pVar_o > $o] :
( X4
!= ( produc8106139234514114142pVar_o @ P3 @ nil_li590035233778055285pVar_o ) )
=> ~ ! [P3: list_l1255154591254203759pVar_o > list_l1255154591254203759pVar_o > $o,X: list_l1255154591254203759pVar_o,Ys: list_l985434930288970485pVar_o] :
( X4
!= ( produc8106139234514114142pVar_o @ P3 @ ( cons_l7316790070876032037pVar_o @ X @ Ys ) ) ) ) ).
% sorted_wrt.cases
thf(fact_134_sorted__wrt_Ocases,axiom,
! [X4: produc3305181862331950184pVar_o] :
( ! [P3: list_P1665994083459302249pVar_o > list_P1665994083459302249pVar_o > $o] :
( X4
!= ( produc1228089817268051288pVar_o @ P3 @ nil_li1310946081133177199pVar_o ) )
=> ~ ! [P3: list_P1665994083459302249pVar_o > list_P1665994083459302249pVar_o > $o,X: list_P1665994083459302249pVar_o,Ys: list_l1255154591254203759pVar_o] :
( X4
!= ( produc1228089817268051288pVar_o @ P3 @ ( cons_l3452452106213086495pVar_o @ X @ Ys ) ) ) ) ).
% sorted_wrt.cases
thf(fact_135_sorted__wrt_Ocases,axiom,
! [X4: produc7264647458362488674pVar_o] :
( ! [P3: produc8999643685545340131pVar_o > produc8999643685545340131pVar_o > $o] :
( X4
!= ( produc3277683845840367058pVar_o @ P3 @ nil_Pr6413368191065476585pVar_o ) )
=> ~ ! [P3: produc8999643685545340131pVar_o > produc8999643685545340131pVar_o > $o,X: produc8999643685545340131pVar_o,Ys: list_P1665994083459302249pVar_o] :
( X4
!= ( produc3277683845840367058pVar_o @ P3 @ ( cons_P6860203055867761049pVar_o @ X @ Ys ) ) ) ) ).
% sorted_wrt.cases
thf(fact_136_not__Cons__self2,axiom,
! [X4: produc2642885279816024434ropVar,Xs2: list_P8791008543742759928ropVar] :
( ( cons_P1504208554336013608ropVar @ X4 @ Xs2 )
!= Xs2 ) ).
% not_Cons_self2
thf(fact_137_not__Cons__self2,axiom,
! [X4: list_l1255154591254203759pVar_o,Xs2: list_l985434930288970485pVar_o] :
( ( cons_l7316790070876032037pVar_o @ X4 @ Xs2 )
!= Xs2 ) ).
% not_Cons_self2
thf(fact_138_not__Cons__self2,axiom,
! [X4: list_P1665994083459302249pVar_o,Xs2: list_l1255154591254203759pVar_o] :
( ( cons_l3452452106213086495pVar_o @ X4 @ Xs2 )
!= Xs2 ) ).
% not_Cons_self2
thf(fact_139_not__Cons__self2,axiom,
! [X4: produc8999643685545340131pVar_o,Xs2: list_P1665994083459302249pVar_o] :
( ( cons_P6860203055867761049pVar_o @ X4 @ Xs2 )
!= Xs2 ) ).
% not_Cons_self2
thf(fact_140_transpose_Ocases,axiom,
! [X4: list_l3415868495287753214ropVar] :
( ( X4 != nil_li2679230897895549182ropVar )
=> ( ! [Xss: list_l3415868495287753214ropVar] :
( X4
!= ( cons_l5280711761816980654ropVar @ nil_Pr6119456312066593144ropVar @ Xss ) )
=> ~ ! [X: produc2642885279816024434ropVar,Xs: list_P8791008543742759928ropVar,Xss: list_l3415868495287753214ropVar] :
( X4
!= ( cons_l5280711761816980654ropVar @ ( cons_P1504208554336013608ropVar @ X @ Xs ) @ Xss ) ) ) ) ).
% transpose.cases
thf(fact_141_transpose_Ocases,axiom,
! [X4: list_l6341520962544925691pVar_o] :
( ( X4 != nil_li5579170522155219707pVar_o )
=> ( ! [Xss: list_l6341520962544925691pVar_o] :
( X4
!= ( cons_l8096070148660426411pVar_o @ nil_li590035233778055285pVar_o @ Xss ) )
=> ~ ! [X: list_l1255154591254203759pVar_o,Xs: list_l985434930288970485pVar_o,Xss: list_l6341520962544925691pVar_o] :
( X4
!= ( cons_l8096070148660426411pVar_o @ ( cons_l7316790070876032037pVar_o @ X @ Xs ) @ Xss ) ) ) ) ).
% transpose.cases
thf(fact_142_transpose_Ocases,axiom,
! [X4: list_l985434930288970485pVar_o] :
( ( X4 != nil_li590035233778055285pVar_o )
=> ( ! [Xss: list_l985434930288970485pVar_o] :
( X4
!= ( cons_l7316790070876032037pVar_o @ nil_li1310946081133177199pVar_o @ Xss ) )
=> ~ ! [X: list_P1665994083459302249pVar_o,Xs: list_l1255154591254203759pVar_o,Xss: list_l985434930288970485pVar_o] :
( X4
!= ( cons_l7316790070876032037pVar_o @ ( cons_l3452452106213086495pVar_o @ X @ Xs ) @ Xss ) ) ) ) ).
% transpose.cases
thf(fact_143_transpose_Ocases,axiom,
! [X4: list_l1255154591254203759pVar_o] :
( ( X4 != nil_li1310946081133177199pVar_o )
=> ( ! [Xss: list_l1255154591254203759pVar_o] :
( X4
!= ( cons_l3452452106213086495pVar_o @ nil_Pr6413368191065476585pVar_o @ Xss ) )
=> ~ ! [X: produc8999643685545340131pVar_o,Xs: list_P1665994083459302249pVar_o,Xss: list_l1255154591254203759pVar_o] :
( X4
!= ( cons_l3452452106213086495pVar_o @ ( cons_P6860203055867761049pVar_o @ X @ Xs ) @ Xss ) ) ) ) ).
% transpose.cases
thf(fact_144_list_Odistinct_I1_J,axiom,
! [X21: produc2642885279816024434ropVar,X222: list_P8791008543742759928ropVar] :
( nil_Pr6119456312066593144ropVar
!= ( cons_P1504208554336013608ropVar @ X21 @ X222 ) ) ).
% list.distinct(1)
thf(fact_145_list_Odistinct_I1_J,axiom,
! [X21: list_l1255154591254203759pVar_o,X222: list_l985434930288970485pVar_o] :
( nil_li590035233778055285pVar_o
!= ( cons_l7316790070876032037pVar_o @ X21 @ X222 ) ) ).
% list.distinct(1)
thf(fact_146_list_Odistinct_I1_J,axiom,
! [X21: list_P1665994083459302249pVar_o,X222: list_l1255154591254203759pVar_o] :
( nil_li1310946081133177199pVar_o
!= ( cons_l3452452106213086495pVar_o @ X21 @ X222 ) ) ).
% list.distinct(1)
thf(fact_147_list_Odistinct_I1_J,axiom,
! [X21: produc8999643685545340131pVar_o,X222: list_P1665994083459302249pVar_o] :
( nil_Pr6413368191065476585pVar_o
!= ( cons_P6860203055867761049pVar_o @ X21 @ X222 ) ) ).
% list.distinct(1)
thf(fact_148_list_OdiscI,axiom,
! [List: list_P8791008543742759928ropVar,X21: produc2642885279816024434ropVar,X222: list_P8791008543742759928ropVar] :
( ( List
= ( cons_P1504208554336013608ropVar @ X21 @ X222 ) )
=> ( List != nil_Pr6119456312066593144ropVar ) ) ).
% list.discI
thf(fact_149_list_OdiscI,axiom,
! [List: list_l985434930288970485pVar_o,X21: list_l1255154591254203759pVar_o,X222: list_l985434930288970485pVar_o] :
( ( List
= ( cons_l7316790070876032037pVar_o @ X21 @ X222 ) )
=> ( List != nil_li590035233778055285pVar_o ) ) ).
% list.discI
thf(fact_150_list_OdiscI,axiom,
! [List: list_l1255154591254203759pVar_o,X21: list_P1665994083459302249pVar_o,X222: list_l1255154591254203759pVar_o] :
( ( List
= ( cons_l3452452106213086495pVar_o @ X21 @ X222 ) )
=> ( List != nil_li1310946081133177199pVar_o ) ) ).
% list.discI
thf(fact_151_list_OdiscI,axiom,
! [List: list_P1665994083459302249pVar_o,X21: produc8999643685545340131pVar_o,X222: list_P1665994083459302249pVar_o] :
( ( List
= ( cons_P6860203055867761049pVar_o @ X21 @ X222 ) )
=> ( List != nil_Pr6413368191065476585pVar_o ) ) ).
% list.discI
thf(fact_152_list_Oexhaust,axiom,
! [Y: list_P8791008543742759928ropVar] :
( ( Y != nil_Pr6119456312066593144ropVar )
=> ~ ! [X212: produc2642885279816024434ropVar,X223: list_P8791008543742759928ropVar] :
( Y
!= ( cons_P1504208554336013608ropVar @ X212 @ X223 ) ) ) ).
% list.exhaust
thf(fact_153_list_Oexhaust,axiom,
! [Y: list_l985434930288970485pVar_o] :
( ( Y != nil_li590035233778055285pVar_o )
=> ~ ! [X212: list_l1255154591254203759pVar_o,X223: list_l985434930288970485pVar_o] :
( Y
!= ( cons_l7316790070876032037pVar_o @ X212 @ X223 ) ) ) ).
% list.exhaust
thf(fact_154_list_Oexhaust,axiom,
! [Y: list_l1255154591254203759pVar_o] :
( ( Y != nil_li1310946081133177199pVar_o )
=> ~ ! [X212: list_P1665994083459302249pVar_o,X223: list_l1255154591254203759pVar_o] :
( Y
!= ( cons_l3452452106213086495pVar_o @ X212 @ X223 ) ) ) ).
% list.exhaust
thf(fact_155_list_Oexhaust,axiom,
! [Y: list_P1665994083459302249pVar_o] :
( ( Y != nil_Pr6413368191065476585pVar_o )
=> ~ ! [X212: produc8999643685545340131pVar_o,X223: list_P1665994083459302249pVar_o] :
( Y
!= ( cons_P6860203055867761049pVar_o @ X212 @ X223 ) ) ) ).
% list.exhaust
thf(fact_156_remdups__adj_Ocases,axiom,
! [X4: list_P8791008543742759928ropVar] :
( ( X4 != nil_Pr6119456312066593144ropVar )
=> ( ! [X: produc2642885279816024434ropVar] :
( X4
!= ( cons_P1504208554336013608ropVar @ X @ nil_Pr6119456312066593144ropVar ) )
=> ~ ! [X: produc2642885279816024434ropVar,Y3: produc2642885279816024434ropVar,Xs: list_P8791008543742759928ropVar] :
( X4
!= ( cons_P1504208554336013608ropVar @ X @ ( cons_P1504208554336013608ropVar @ Y3 @ Xs ) ) ) ) ) ).
% remdups_adj.cases
thf(fact_157_remdups__adj_Ocases,axiom,
! [X4: list_l985434930288970485pVar_o] :
( ( X4 != nil_li590035233778055285pVar_o )
=> ( ! [X: list_l1255154591254203759pVar_o] :
( X4
!= ( cons_l7316790070876032037pVar_o @ X @ nil_li590035233778055285pVar_o ) )
=> ~ ! [X: list_l1255154591254203759pVar_o,Y3: list_l1255154591254203759pVar_o,Xs: list_l985434930288970485pVar_o] :
( X4
!= ( cons_l7316790070876032037pVar_o @ X @ ( cons_l7316790070876032037pVar_o @ Y3 @ Xs ) ) ) ) ) ).
% remdups_adj.cases
thf(fact_158_remdups__adj_Ocases,axiom,
! [X4: list_l1255154591254203759pVar_o] :
( ( X4 != nil_li1310946081133177199pVar_o )
=> ( ! [X: list_P1665994083459302249pVar_o] :
( X4
!= ( cons_l3452452106213086495pVar_o @ X @ nil_li1310946081133177199pVar_o ) )
=> ~ ! [X: list_P1665994083459302249pVar_o,Y3: list_P1665994083459302249pVar_o,Xs: list_l1255154591254203759pVar_o] :
( X4
!= ( cons_l3452452106213086495pVar_o @ X @ ( cons_l3452452106213086495pVar_o @ Y3 @ Xs ) ) ) ) ) ).
% remdups_adj.cases
thf(fact_159_remdups__adj_Ocases,axiom,
! [X4: list_P1665994083459302249pVar_o] :
( ( X4 != nil_Pr6413368191065476585pVar_o )
=> ( ! [X: produc8999643685545340131pVar_o] :
( X4
!= ( cons_P6860203055867761049pVar_o @ X @ nil_Pr6413368191065476585pVar_o ) )
=> ~ ! [X: produc8999643685545340131pVar_o,Y3: produc8999643685545340131pVar_o,Xs: list_P1665994083459302249pVar_o] :
( X4
!= ( cons_P6860203055867761049pVar_o @ X @ ( cons_P6860203055867761049pVar_o @ Y3 @ Xs ) ) ) ) ) ).
% remdups_adj.cases
thf(fact_160_neq__Nil__conv,axiom,
! [Xs2: list_P8791008543742759928ropVar] :
( ( Xs2 != nil_Pr6119456312066593144ropVar )
= ( ? [Y4: produc2642885279816024434ropVar,Ys2: list_P8791008543742759928ropVar] :
( Xs2
= ( cons_P1504208554336013608ropVar @ Y4 @ Ys2 ) ) ) ) ).
% neq_Nil_conv
thf(fact_161_neq__Nil__conv,axiom,
! [Xs2: list_l985434930288970485pVar_o] :
( ( Xs2 != nil_li590035233778055285pVar_o )
= ( ? [Y4: list_l1255154591254203759pVar_o,Ys2: list_l985434930288970485pVar_o] :
( Xs2
= ( cons_l7316790070876032037pVar_o @ Y4 @ Ys2 ) ) ) ) ).
% neq_Nil_conv
thf(fact_162_neq__Nil__conv,axiom,
! [Xs2: list_l1255154591254203759pVar_o] :
( ( Xs2 != nil_li1310946081133177199pVar_o )
= ( ? [Y4: list_P1665994083459302249pVar_o,Ys2: list_l1255154591254203759pVar_o] :
( Xs2
= ( cons_l3452452106213086495pVar_o @ Y4 @ Ys2 ) ) ) ) ).
% neq_Nil_conv
thf(fact_163_neq__Nil__conv,axiom,
! [Xs2: list_P1665994083459302249pVar_o] :
( ( Xs2 != nil_Pr6413368191065476585pVar_o )
= ( ? [Y4: produc8999643685545340131pVar_o,Ys2: list_P1665994083459302249pVar_o] :
( Xs2
= ( cons_P6860203055867761049pVar_o @ Y4 @ Ys2 ) ) ) ) ).
% neq_Nil_conv
thf(fact_164_list__induct2_H,axiom,
! [P2: list_P1665994083459302249pVar_o > list_P1665994083459302249pVar_o > $o,Xs2: list_P1665994083459302249pVar_o,Ys3: list_P1665994083459302249pVar_o] :
( ( P2 @ nil_Pr6413368191065476585pVar_o @ nil_Pr6413368191065476585pVar_o )
=> ( ! [X: produc8999643685545340131pVar_o,Xs: list_P1665994083459302249pVar_o] : ( P2 @ ( cons_P6860203055867761049pVar_o @ X @ Xs ) @ nil_Pr6413368191065476585pVar_o )
=> ( ! [Y3: produc8999643685545340131pVar_o,Ys: list_P1665994083459302249pVar_o] : ( P2 @ nil_Pr6413368191065476585pVar_o @ ( cons_P6860203055867761049pVar_o @ Y3 @ Ys ) )
=> ( ! [X: produc8999643685545340131pVar_o,Xs: list_P1665994083459302249pVar_o,Y3: produc8999643685545340131pVar_o,Ys: list_P1665994083459302249pVar_o] :
( ( P2 @ Xs @ Ys )
=> ( P2 @ ( cons_P6860203055867761049pVar_o @ X @ Xs ) @ ( cons_P6860203055867761049pVar_o @ Y3 @ Ys ) ) )
=> ( P2 @ Xs2 @ Ys3 ) ) ) ) ) ).
% list_induct2'
thf(fact_165_list__induct2_H,axiom,
! [P2: list_l1255154591254203759pVar_o > list_P1665994083459302249pVar_o > $o,Xs2: list_l1255154591254203759pVar_o,Ys3: list_P1665994083459302249pVar_o] :
( ( P2 @ nil_li1310946081133177199pVar_o @ nil_Pr6413368191065476585pVar_o )
=> ( ! [X: list_P1665994083459302249pVar_o,Xs: list_l1255154591254203759pVar_o] : ( P2 @ ( cons_l3452452106213086495pVar_o @ X @ Xs ) @ nil_Pr6413368191065476585pVar_o )
=> ( ! [Y3: produc8999643685545340131pVar_o,Ys: list_P1665994083459302249pVar_o] : ( P2 @ nil_li1310946081133177199pVar_o @ ( cons_P6860203055867761049pVar_o @ Y3 @ Ys ) )
=> ( ! [X: list_P1665994083459302249pVar_o,Xs: list_l1255154591254203759pVar_o,Y3: produc8999643685545340131pVar_o,Ys: list_P1665994083459302249pVar_o] :
( ( P2 @ Xs @ Ys )
=> ( P2 @ ( cons_l3452452106213086495pVar_o @ X @ Xs ) @ ( cons_P6860203055867761049pVar_o @ Y3 @ Ys ) ) )
=> ( P2 @ Xs2 @ Ys3 ) ) ) ) ) ).
% list_induct2'
thf(fact_166_list__induct2_H,axiom,
! [P2: list_P1665994083459302249pVar_o > list_l1255154591254203759pVar_o > $o,Xs2: list_P1665994083459302249pVar_o,Ys3: list_l1255154591254203759pVar_o] :
( ( P2 @ nil_Pr6413368191065476585pVar_o @ nil_li1310946081133177199pVar_o )
=> ( ! [X: produc8999643685545340131pVar_o,Xs: list_P1665994083459302249pVar_o] : ( P2 @ ( cons_P6860203055867761049pVar_o @ X @ Xs ) @ nil_li1310946081133177199pVar_o )
=> ( ! [Y3: list_P1665994083459302249pVar_o,Ys: list_l1255154591254203759pVar_o] : ( P2 @ nil_Pr6413368191065476585pVar_o @ ( cons_l3452452106213086495pVar_o @ Y3 @ Ys ) )
=> ( ! [X: produc8999643685545340131pVar_o,Xs: list_P1665994083459302249pVar_o,Y3: list_P1665994083459302249pVar_o,Ys: list_l1255154591254203759pVar_o] :
( ( P2 @ Xs @ Ys )
=> ( P2 @ ( cons_P6860203055867761049pVar_o @ X @ Xs ) @ ( cons_l3452452106213086495pVar_o @ Y3 @ Ys ) ) )
=> ( P2 @ Xs2 @ Ys3 ) ) ) ) ) ).
% list_induct2'
thf(fact_167_list__induct2_H,axiom,
! [P2: list_P8791008543742759928ropVar > list_P1665994083459302249pVar_o > $o,Xs2: list_P8791008543742759928ropVar,Ys3: list_P1665994083459302249pVar_o] :
( ( P2 @ nil_Pr6119456312066593144ropVar @ nil_Pr6413368191065476585pVar_o )
=> ( ! [X: produc2642885279816024434ropVar,Xs: list_P8791008543742759928ropVar] : ( P2 @ ( cons_P1504208554336013608ropVar @ X @ Xs ) @ nil_Pr6413368191065476585pVar_o )
=> ( ! [Y3: produc8999643685545340131pVar_o,Ys: list_P1665994083459302249pVar_o] : ( P2 @ nil_Pr6119456312066593144ropVar @ ( cons_P6860203055867761049pVar_o @ Y3 @ Ys ) )
=> ( ! [X: produc2642885279816024434ropVar,Xs: list_P8791008543742759928ropVar,Y3: produc8999643685545340131pVar_o,Ys: list_P1665994083459302249pVar_o] :
( ( P2 @ Xs @ Ys )
=> ( P2 @ ( cons_P1504208554336013608ropVar @ X @ Xs ) @ ( cons_P6860203055867761049pVar_o @ Y3 @ Ys ) ) )
=> ( P2 @ Xs2 @ Ys3 ) ) ) ) ) ).
% list_induct2'
thf(fact_168_list__induct2_H,axiom,
! [P2: list_l985434930288970485pVar_o > list_P1665994083459302249pVar_o > $o,Xs2: list_l985434930288970485pVar_o,Ys3: list_P1665994083459302249pVar_o] :
( ( P2 @ nil_li590035233778055285pVar_o @ nil_Pr6413368191065476585pVar_o )
=> ( ! [X: list_l1255154591254203759pVar_o,Xs: list_l985434930288970485pVar_o] : ( P2 @ ( cons_l7316790070876032037pVar_o @ X @ Xs ) @ nil_Pr6413368191065476585pVar_o )
=> ( ! [Y3: produc8999643685545340131pVar_o,Ys: list_P1665994083459302249pVar_o] : ( P2 @ nil_li590035233778055285pVar_o @ ( cons_P6860203055867761049pVar_o @ Y3 @ Ys ) )
=> ( ! [X: list_l1255154591254203759pVar_o,Xs: list_l985434930288970485pVar_o,Y3: produc8999643685545340131pVar_o,Ys: list_P1665994083459302249pVar_o] :
( ( P2 @ Xs @ Ys )
=> ( P2 @ ( cons_l7316790070876032037pVar_o @ X @ Xs ) @ ( cons_P6860203055867761049pVar_o @ Y3 @ Ys ) ) )
=> ( P2 @ Xs2 @ Ys3 ) ) ) ) ) ).
% list_induct2'
thf(fact_169_list__induct2_H,axiom,
! [P2: list_l1255154591254203759pVar_o > list_l1255154591254203759pVar_o > $o,Xs2: list_l1255154591254203759pVar_o,Ys3: list_l1255154591254203759pVar_o] :
( ( P2 @ nil_li1310946081133177199pVar_o @ nil_li1310946081133177199pVar_o )
=> ( ! [X: list_P1665994083459302249pVar_o,Xs: list_l1255154591254203759pVar_o] : ( P2 @ ( cons_l3452452106213086495pVar_o @ X @ Xs ) @ nil_li1310946081133177199pVar_o )
=> ( ! [Y3: list_P1665994083459302249pVar_o,Ys: list_l1255154591254203759pVar_o] : ( P2 @ nil_li1310946081133177199pVar_o @ ( cons_l3452452106213086495pVar_o @ Y3 @ Ys ) )
=> ( ! [X: list_P1665994083459302249pVar_o,Xs: list_l1255154591254203759pVar_o,Y3: list_P1665994083459302249pVar_o,Ys: list_l1255154591254203759pVar_o] :
( ( P2 @ Xs @ Ys )
=> ( P2 @ ( cons_l3452452106213086495pVar_o @ X @ Xs ) @ ( cons_l3452452106213086495pVar_o @ Y3 @ Ys ) ) )
=> ( P2 @ Xs2 @ Ys3 ) ) ) ) ) ).
% list_induct2'
thf(fact_170_list__induct2_H,axiom,
! [P2: list_P1665994083459302249pVar_o > list_P8791008543742759928ropVar > $o,Xs2: list_P1665994083459302249pVar_o,Ys3: list_P8791008543742759928ropVar] :
( ( P2 @ nil_Pr6413368191065476585pVar_o @ nil_Pr6119456312066593144ropVar )
=> ( ! [X: produc8999643685545340131pVar_o,Xs: list_P1665994083459302249pVar_o] : ( P2 @ ( cons_P6860203055867761049pVar_o @ X @ Xs ) @ nil_Pr6119456312066593144ropVar )
=> ( ! [Y3: produc2642885279816024434ropVar,Ys: list_P8791008543742759928ropVar] : ( P2 @ nil_Pr6413368191065476585pVar_o @ ( cons_P1504208554336013608ropVar @ Y3 @ Ys ) )
=> ( ! [X: produc8999643685545340131pVar_o,Xs: list_P1665994083459302249pVar_o,Y3: produc2642885279816024434ropVar,Ys: list_P8791008543742759928ropVar] :
( ( P2 @ Xs @ Ys )
=> ( P2 @ ( cons_P6860203055867761049pVar_o @ X @ Xs ) @ ( cons_P1504208554336013608ropVar @ Y3 @ Ys ) ) )
=> ( P2 @ Xs2 @ Ys3 ) ) ) ) ) ).
% list_induct2'
thf(fact_171_list__induct2_H,axiom,
! [P2: list_P1665994083459302249pVar_o > list_l985434930288970485pVar_o > $o,Xs2: list_P1665994083459302249pVar_o,Ys3: list_l985434930288970485pVar_o] :
( ( P2 @ nil_Pr6413368191065476585pVar_o @ nil_li590035233778055285pVar_o )
=> ( ! [X: produc8999643685545340131pVar_o,Xs: list_P1665994083459302249pVar_o] : ( P2 @ ( cons_P6860203055867761049pVar_o @ X @ Xs ) @ nil_li590035233778055285pVar_o )
=> ( ! [Y3: list_l1255154591254203759pVar_o,Ys: list_l985434930288970485pVar_o] : ( P2 @ nil_Pr6413368191065476585pVar_o @ ( cons_l7316790070876032037pVar_o @ Y3 @ Ys ) )
=> ( ! [X: produc8999643685545340131pVar_o,Xs: list_P1665994083459302249pVar_o,Y3: list_l1255154591254203759pVar_o,Ys: list_l985434930288970485pVar_o] :
( ( P2 @ Xs @ Ys )
=> ( P2 @ ( cons_P6860203055867761049pVar_o @ X @ Xs ) @ ( cons_l7316790070876032037pVar_o @ Y3 @ Ys ) ) )
=> ( P2 @ Xs2 @ Ys3 ) ) ) ) ) ).
% list_induct2'
thf(fact_172_list__induct2_H,axiom,
! [P2: list_P8791008543742759928ropVar > list_l1255154591254203759pVar_o > $o,Xs2: list_P8791008543742759928ropVar,Ys3: list_l1255154591254203759pVar_o] :
( ( P2 @ nil_Pr6119456312066593144ropVar @ nil_li1310946081133177199pVar_o )
=> ( ! [X: produc2642885279816024434ropVar,Xs: list_P8791008543742759928ropVar] : ( P2 @ ( cons_P1504208554336013608ropVar @ X @ Xs ) @ nil_li1310946081133177199pVar_o )
=> ( ! [Y3: list_P1665994083459302249pVar_o,Ys: list_l1255154591254203759pVar_o] : ( P2 @ nil_Pr6119456312066593144ropVar @ ( cons_l3452452106213086495pVar_o @ Y3 @ Ys ) )
=> ( ! [X: produc2642885279816024434ropVar,Xs: list_P8791008543742759928ropVar,Y3: list_P1665994083459302249pVar_o,Ys: list_l1255154591254203759pVar_o] :
( ( P2 @ Xs @ Ys )
=> ( P2 @ ( cons_P1504208554336013608ropVar @ X @ Xs ) @ ( cons_l3452452106213086495pVar_o @ Y3 @ Ys ) ) )
=> ( P2 @ Xs2 @ Ys3 ) ) ) ) ) ).
% list_induct2'
thf(fact_173_list__induct2_H,axiom,
! [P2: list_l985434930288970485pVar_o > list_l1255154591254203759pVar_o > $o,Xs2: list_l985434930288970485pVar_o,Ys3: list_l1255154591254203759pVar_o] :
( ( P2 @ nil_li590035233778055285pVar_o @ nil_li1310946081133177199pVar_o )
=> ( ! [X: list_l1255154591254203759pVar_o,Xs: list_l985434930288970485pVar_o] : ( P2 @ ( cons_l7316790070876032037pVar_o @ X @ Xs ) @ nil_li1310946081133177199pVar_o )
=> ( ! [Y3: list_P1665994083459302249pVar_o,Ys: list_l1255154591254203759pVar_o] : ( P2 @ nil_li590035233778055285pVar_o @ ( cons_l3452452106213086495pVar_o @ Y3 @ Ys ) )
=> ( ! [X: list_l1255154591254203759pVar_o,Xs: list_l985434930288970485pVar_o,Y3: list_P1665994083459302249pVar_o,Ys: list_l1255154591254203759pVar_o] :
( ( P2 @ Xs @ Ys )
=> ( P2 @ ( cons_l7316790070876032037pVar_o @ X @ Xs ) @ ( cons_l3452452106213086495pVar_o @ Y3 @ Ys ) ) )
=> ( P2 @ Xs2 @ Ys3 ) ) ) ) ) ).
% list_induct2'
thf(fact_174_list__nonempty__induct,axiom,
! [Xs2: list_P8791008543742759928ropVar,P2: list_P8791008543742759928ropVar > $o] :
( ( Xs2 != nil_Pr6119456312066593144ropVar )
=> ( ! [X: produc2642885279816024434ropVar] : ( P2 @ ( cons_P1504208554336013608ropVar @ X @ nil_Pr6119456312066593144ropVar ) )
=> ( ! [X: produc2642885279816024434ropVar,Xs: list_P8791008543742759928ropVar] :
( ( Xs != nil_Pr6119456312066593144ropVar )
=> ( ( P2 @ Xs )
=> ( P2 @ ( cons_P1504208554336013608ropVar @ X @ Xs ) ) ) )
=> ( P2 @ Xs2 ) ) ) ) ).
% list_nonempty_induct
thf(fact_175_list__nonempty__induct,axiom,
! [Xs2: list_l985434930288970485pVar_o,P2: list_l985434930288970485pVar_o > $o] :
( ( Xs2 != nil_li590035233778055285pVar_o )
=> ( ! [X: list_l1255154591254203759pVar_o] : ( P2 @ ( cons_l7316790070876032037pVar_o @ X @ nil_li590035233778055285pVar_o ) )
=> ( ! [X: list_l1255154591254203759pVar_o,Xs: list_l985434930288970485pVar_o] :
( ( Xs != nil_li590035233778055285pVar_o )
=> ( ( P2 @ Xs )
=> ( P2 @ ( cons_l7316790070876032037pVar_o @ X @ Xs ) ) ) )
=> ( P2 @ Xs2 ) ) ) ) ).
% list_nonempty_induct
thf(fact_176_list__nonempty__induct,axiom,
! [Xs2: list_l1255154591254203759pVar_o,P2: list_l1255154591254203759pVar_o > $o] :
( ( Xs2 != nil_li1310946081133177199pVar_o )
=> ( ! [X: list_P1665994083459302249pVar_o] : ( P2 @ ( cons_l3452452106213086495pVar_o @ X @ nil_li1310946081133177199pVar_o ) )
=> ( ! [X: list_P1665994083459302249pVar_o,Xs: list_l1255154591254203759pVar_o] :
( ( Xs != nil_li1310946081133177199pVar_o )
=> ( ( P2 @ Xs )
=> ( P2 @ ( cons_l3452452106213086495pVar_o @ X @ Xs ) ) ) )
=> ( P2 @ Xs2 ) ) ) ) ).
% list_nonempty_induct
thf(fact_177_list__nonempty__induct,axiom,
! [Xs2: list_P1665994083459302249pVar_o,P2: list_P1665994083459302249pVar_o > $o] :
( ( Xs2 != nil_Pr6413368191065476585pVar_o )
=> ( ! [X: produc8999643685545340131pVar_o] : ( P2 @ ( cons_P6860203055867761049pVar_o @ X @ nil_Pr6413368191065476585pVar_o ) )
=> ( ! [X: produc8999643685545340131pVar_o,Xs: list_P1665994083459302249pVar_o] :
( ( Xs != nil_Pr6413368191065476585pVar_o )
=> ( ( P2 @ Xs )
=> ( P2 @ ( cons_P6860203055867761049pVar_o @ X @ Xs ) ) ) )
=> ( P2 @ Xs2 ) ) ) ) ).
% list_nonempty_induct
thf(fact_178_successively_Ocases,axiom,
! [X4: produc5533280449304468145ropVar] :
( ! [P3: produc2642885279816024434ropVar > produc2642885279816024434ropVar > $o] :
( X4
!= ( produc3887254912111605537ropVar @ P3 @ nil_Pr6119456312066593144ropVar ) )
=> ( ! [P3: produc2642885279816024434ropVar > produc2642885279816024434ropVar > $o,X: produc2642885279816024434ropVar] :
( X4
!= ( produc3887254912111605537ropVar @ P3 @ ( cons_P1504208554336013608ropVar @ X @ nil_Pr6119456312066593144ropVar ) ) )
=> ~ ! [P3: produc2642885279816024434ropVar > produc2642885279816024434ropVar > $o,X: produc2642885279816024434ropVar,Y3: produc2642885279816024434ropVar,Xs: list_P8791008543742759928ropVar] :
( X4
!= ( produc3887254912111605537ropVar @ P3 @ ( cons_P1504208554336013608ropVar @ X @ ( cons_P1504208554336013608ropVar @ Y3 @ Xs ) ) ) ) ) ) ).
% successively.cases
thf(fact_179_successively_Ocases,axiom,
! [X4: produc3853458377486834414pVar_o] :
( ! [P3: list_l1255154591254203759pVar_o > list_l1255154591254203759pVar_o > $o] :
( X4
!= ( produc8106139234514114142pVar_o @ P3 @ nil_li590035233778055285pVar_o ) )
=> ( ! [P3: list_l1255154591254203759pVar_o > list_l1255154591254203759pVar_o > $o,X: list_l1255154591254203759pVar_o] :
( X4
!= ( produc8106139234514114142pVar_o @ P3 @ ( cons_l7316790070876032037pVar_o @ X @ nil_li590035233778055285pVar_o ) ) )
=> ~ ! [P3: list_l1255154591254203759pVar_o > list_l1255154591254203759pVar_o > $o,X: list_l1255154591254203759pVar_o,Y3: list_l1255154591254203759pVar_o,Xs: list_l985434930288970485pVar_o] :
( X4
!= ( produc8106139234514114142pVar_o @ P3 @ ( cons_l7316790070876032037pVar_o @ X @ ( cons_l7316790070876032037pVar_o @ Y3 @ Xs ) ) ) ) ) ) ).
% successively.cases
thf(fact_180_successively_Ocases,axiom,
! [X4: produc3305181862331950184pVar_o] :
( ! [P3: list_P1665994083459302249pVar_o > list_P1665994083459302249pVar_o > $o] :
( X4
!= ( produc1228089817268051288pVar_o @ P3 @ nil_li1310946081133177199pVar_o ) )
=> ( ! [P3: list_P1665994083459302249pVar_o > list_P1665994083459302249pVar_o > $o,X: list_P1665994083459302249pVar_o] :
( X4
!= ( produc1228089817268051288pVar_o @ P3 @ ( cons_l3452452106213086495pVar_o @ X @ nil_li1310946081133177199pVar_o ) ) )
=> ~ ! [P3: list_P1665994083459302249pVar_o > list_P1665994083459302249pVar_o > $o,X: list_P1665994083459302249pVar_o,Y3: list_P1665994083459302249pVar_o,Xs: list_l1255154591254203759pVar_o] :
( X4
!= ( produc1228089817268051288pVar_o @ P3 @ ( cons_l3452452106213086495pVar_o @ X @ ( cons_l3452452106213086495pVar_o @ Y3 @ Xs ) ) ) ) ) ) ).
% successively.cases
thf(fact_181_successively_Ocases,axiom,
! [X4: produc7264647458362488674pVar_o] :
( ! [P3: produc8999643685545340131pVar_o > produc8999643685545340131pVar_o > $o] :
( X4
!= ( produc3277683845840367058pVar_o @ P3 @ nil_Pr6413368191065476585pVar_o ) )
=> ( ! [P3: produc8999643685545340131pVar_o > produc8999643685545340131pVar_o > $o,X: produc8999643685545340131pVar_o] :
( X4
!= ( produc3277683845840367058pVar_o @ P3 @ ( cons_P6860203055867761049pVar_o @ X @ nil_Pr6413368191065476585pVar_o ) ) )
=> ~ ! [P3: produc8999643685545340131pVar_o > produc8999643685545340131pVar_o > $o,X: produc8999643685545340131pVar_o,Y3: produc8999643685545340131pVar_o,Xs: list_P1665994083459302249pVar_o] :
( X4
!= ( produc3277683845840367058pVar_o @ P3 @ ( cons_P6860203055867761049pVar_o @ X @ ( cons_P6860203055867761049pVar_o @ Y3 @ Xs ) ) ) ) ) ) ).
% successively.cases
thf(fact_182_cnf__encoder_Oencode__sum__0__1__main_Osimps_I1_J,axiom,
! [X4: multis3193088007478089820ropVar,Zero2: multis3193088007478089820ropVar,One2: multis3193088007478089820ropVar] :
( ( multis70479551085940736041_main @ ( cons_P1504208554336013608ropVar @ ( produc6194868611276206818ropVar @ X4 @ ( produc6936306168532358295ropVar @ Zero2 @ One2 ) ) @ nil_Pr6119456312066593144ropVar ) )
= ( produc2520692699247754633ropVar @ ( cons_l7316790070876032037pVar_o @ ( cons_l3452452106213086495pVar_o @ ( cons_P6860203055867761049pVar_o @ ( produc2127521181850112851pVar_o @ Zero2 @ $true ) @ ( cons_P6860203055867761049pVar_o @ ( produc2127521181850112851pVar_o @ X4 @ $true ) @ nil_Pr6413368191065476585pVar_o ) ) @ ( cons_l3452452106213086495pVar_o @ ( cons_P6860203055867761049pVar_o @ ( produc2127521181850112851pVar_o @ Zero2 @ $false ) @ ( cons_P6860203055867761049pVar_o @ ( produc2127521181850112851pVar_o @ X4 @ $false ) @ nil_Pr6413368191065476585pVar_o ) ) @ nil_li1310946081133177199pVar_o ) ) @ nil_li590035233778055285pVar_o ) @ ( produc6936306168532358295ropVar @ Zero2 @ X4 ) ) ) ).
% cnf_encoder.encode_sum_0_1_main.simps(1)
thf(fact_183_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_184_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_185_n__lists__Nil,axiom,
! [N: nat] :
( ( ( N = zero_zero_nat )
=> ( ( n_list6927042657299049856ropVar @ N @ nil_Pr6119456312066593144ropVar )
= ( cons_l5280711761816980654ropVar @ nil_Pr6119456312066593144ropVar @ nil_li2679230897895549182ropVar ) ) )
& ( ( N != zero_zero_nat )
=> ( ( n_list6927042657299049856ropVar @ N @ nil_Pr6119456312066593144ropVar )
= nil_li2679230897895549182ropVar ) ) ) ).
% n_lists_Nil
thf(fact_186_n__lists__Nil,axiom,
! [N: nat] :
( ( ( N = zero_zero_nat )
=> ( ( n_list2523503019017615485pVar_o @ N @ nil_li590035233778055285pVar_o )
= ( cons_l8096070148660426411pVar_o @ nil_li590035233778055285pVar_o @ nil_li5579170522155219707pVar_o ) ) )
& ( ( N != zero_zero_nat )
=> ( ( n_list2523503019017615485pVar_o @ N @ nil_li590035233778055285pVar_o )
= nil_li5579170522155219707pVar_o ) ) ) ).
% n_lists_Nil
thf(fact_187_n__lists__Nil,axiom,
! [N: nat] :
( ( ( N = zero_zero_nat )
=> ( ( n_list1372634133161135479pVar_o @ N @ nil_li1310946081133177199pVar_o )
= ( cons_l7316790070876032037pVar_o @ nil_li1310946081133177199pVar_o @ nil_li590035233778055285pVar_o ) ) )
& ( ( N != zero_zero_nat )
=> ( ( n_list1372634133161135479pVar_o @ N @ nil_li1310946081133177199pVar_o )
= nil_li590035233778055285pVar_o ) ) ) ).
% n_lists_Nil
thf(fact_188_n__lists__Nil,axiom,
! [N: nat] :
( ( ( N = zero_zero_nat )
=> ( ( n_list4068704761422681585pVar_o @ N @ nil_Pr6413368191065476585pVar_o )
= ( cons_l3452452106213086495pVar_o @ nil_Pr6413368191065476585pVar_o @ nil_li1310946081133177199pVar_o ) ) )
& ( ( N != zero_zero_nat )
=> ( ( n_list4068704761422681585pVar_o @ N @ nil_Pr6413368191065476585pVar_o )
= nil_li1310946081133177199pVar_o ) ) ) ).
% n_lists_Nil
thf(fact_189_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_190_minus__1__div__2__eq,axiom,
( ( divide_divide_int @ ( uminus_uminus_int @ one_one_int ) @ ( numeral_numeral_int @ ( bit0 @ one ) ) )
= ( uminus_uminus_int @ one_one_int ) ) ).
% minus_1_div_2_eq
thf(fact_191_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_192_one__div__numeral,axiom,
! [N: num] :
( ( divide_divide_nat @ one_one_nat @ ( numeral_numeral_nat @ N ) )
= ( product_fst_nat_nat @ ( unique5405566460079783412od_nat @ one @ N ) ) ) ).
% one_div_numeral
thf(fact_193_one__div__numeral,axiom,
! [N: num] :
( ( divide_divide_int @ one_one_int @ ( numeral_numeral_int @ N ) )
= ( product_fst_int_int @ ( unique5403075989570733136od_int @ one @ N ) ) ) ).
% one_div_numeral
thf(fact_194_eq__numeral__iff__iszero_I5_J,axiom,
! [X4: num] :
( ( ( numeral_numeral_int @ X4 )
= one_one_int )
= ( ring_1_iszero_int @ ( neg_numeral_sub_int @ X4 @ one ) ) ) ).
% eq_numeral_iff_iszero(5)
thf(fact_195_add_Oinverse__inverse,axiom,
! [A: int] :
( ( uminus_uminus_int @ ( uminus_uminus_int @ A ) )
= A ) ).
% add.inverse_inverse
thf(fact_196_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_197_verit__minus__simplify_I4_J,axiom,
! [B: int] :
( ( uminus_uminus_int @ ( uminus_uminus_int @ B ) )
= B ) ).
% verit_minus_simplify(4)
thf(fact_198_neg__equal__zero,axiom,
! [A: int] :
( ( ( uminus_uminus_int @ A )
= A )
= ( A = zero_zero_int ) ) ).
% neg_equal_zero
thf(fact_199_equal__neg__zero,axiom,
! [A: int] :
( ( A
= ( uminus_uminus_int @ A ) )
= ( A = zero_zero_int ) ) ).
% equal_neg_zero
thf(fact_200_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_201_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_202_add_Oinverse__neutral,axiom,
( ( uminus_uminus_int @ zero_zero_int )
= zero_zero_int ) ).
% add.inverse_neutral
thf(fact_203_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_204_div__minus__minus,axiom,
! [A: int,B: int] :
( ( divide_divide_int @ ( uminus_uminus_int @ A ) @ ( uminus_uminus_int @ B ) )
= ( divide_divide_int @ A @ B ) ) ).
% div_minus_minus
thf(fact_205_div__minus1__right,axiom,
! [A: int] :
( ( divide_divide_int @ A @ ( uminus_uminus_int @ one_one_int ) )
= ( uminus_uminus_int @ A ) ) ).
% div_minus1_right
thf(fact_206_iszero__neg__numeral,axiom,
! [W: num] :
( ( ring_1_iszero_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ W ) ) )
= ( ring_1_iszero_int @ ( numeral_numeral_int @ W ) ) ) ).
% iszero_neg_numeral
thf(fact_207_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_208_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_209_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_210_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_211_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_212_numeral__div__numeral,axiom,
! [K: num,L: num] :
( ( divide_divide_nat @ ( numeral_numeral_nat @ K ) @ ( numeral_numeral_nat @ L ) )
= ( product_fst_nat_nat @ ( unique5405566460079783412od_nat @ K @ L ) ) ) ).
% numeral_div_numeral
thf(fact_213_numeral__div__numeral,axiom,
! [K: num,L: num] :
( ( divide_divide_int @ ( numeral_numeral_int @ K ) @ ( numeral_numeral_int @ L ) )
= ( product_fst_int_int @ ( unique5403075989570733136od_int @ K @ L ) ) ) ).
% numeral_div_numeral
thf(fact_214_dbl__dec__simps_I2_J,axiom,
( ( neg_nu3811975205180677377ec_int @ zero_zero_int )
= ( uminus_uminus_int @ one_one_int ) ) ).
% dbl_dec_simps(2)
thf(fact_215_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_216_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_217_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_218_equation__minus__iff,axiom,
! [A: int,B: int] :
( ( A
= ( uminus_uminus_int @ B ) )
= ( B
= ( uminus_uminus_int @ A ) ) ) ).
% equation_minus_iff
thf(fact_219_minus__equation__iff,axiom,
! [A: int,B: int] :
( ( ( uminus_uminus_int @ A )
= B )
= ( ( uminus_uminus_int @ B )
= A ) ) ).
% minus_equation_iff
thf(fact_220_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_221_not__iszero__neg__1,axiom,
~ ( ring_1_iszero_int @ ( uminus_uminus_int @ one_one_int ) ) ).
% not_iszero_neg_1
thf(fact_222_fst__conv,axiom,
! [X1: multis3193088007478089820ropVar,X2: $o] :
( ( produc7654750731481044799pVar_o @ ( produc2127521181850112851pVar_o @ X1 @ X2 ) )
= X1 ) ).
% fst_conv
thf(fact_223_fst__conv,axiom,
! [X1: list_l1255154591254203759pVar_o,X2: list_l985434930288970485pVar_o] :
( ( produc1814762917671617225pVar_o @ ( produc4064290870397114077pVar_o @ X1 @ X2 ) )
= X1 ) ).
% fst_conv
thf(fact_224_fst__conv,axiom,
! [X1: list_l985434930288970485pVar_o,X2: produc7679832412844898215ropVar] :
( ( produc284539208842135413ropVar @ ( produc2520692699247754633ropVar @ X1 @ X2 ) )
= X1 ) ).
% fst_conv
thf(fact_225_fst__conv,axiom,
! [X1: num,X2: num] :
( ( product_fst_num_num @ ( product_Pair_num_num @ X1 @ X2 ) )
= X1 ) ).
% fst_conv
thf(fact_226_fst__conv,axiom,
! [X1: int,X2: int] :
( ( product_fst_int_int @ ( product_Pair_int_int @ X1 @ X2 ) )
= X1 ) ).
% fst_conv
thf(fact_227_fst__eqD,axiom,
! [X4: multis3193088007478089820ropVar,Y: $o,A: multis3193088007478089820ropVar] :
( ( ( produc7654750731481044799pVar_o @ ( produc2127521181850112851pVar_o @ X4 @ Y ) )
= A )
=> ( X4 = A ) ) ).
% fst_eqD
thf(fact_228_fst__eqD,axiom,
! [X4: list_l1255154591254203759pVar_o,Y: list_l985434930288970485pVar_o,A: list_l1255154591254203759pVar_o] :
( ( ( produc1814762917671617225pVar_o @ ( produc4064290870397114077pVar_o @ X4 @ Y ) )
= A )
=> ( X4 = A ) ) ).
% fst_eqD
thf(fact_229_fst__eqD,axiom,
! [X4: list_l985434930288970485pVar_o,Y: produc7679832412844898215ropVar,A: list_l985434930288970485pVar_o] :
( ( ( produc284539208842135413ropVar @ ( produc2520692699247754633ropVar @ X4 @ Y ) )
= A )
=> ( X4 = A ) ) ).
% fst_eqD
thf(fact_230_fst__eqD,axiom,
! [X4: num,Y: num,A: num] :
( ( ( product_fst_num_num @ ( product_Pair_num_num @ X4 @ Y ) )
= A )
=> ( X4 = A ) ) ).
% fst_eqD
thf(fact_231_fst__eqD,axiom,
! [X4: int,Y: int,A: int] :
( ( ( product_fst_int_int @ ( product_Pair_int_int @ X4 @ Y ) )
= A )
=> ( X4 = A ) ) ).
% fst_eqD
thf(fact_232_eq__numeral__iff__iszero_I11_J,axiom,
! [X4: num] :
( ( ( uminus_uminus_int @ ( numeral_numeral_int @ X4 ) )
= zero_zero_int )
= ( ring_1_iszero_int @ ( numeral_numeral_int @ X4 ) ) ) ).
% eq_numeral_iff_iszero(11)
thf(fact_233_eq__numeral__iff__iszero_I12_J,axiom,
! [Y: num] :
( ( zero_zero_int
= ( uminus_uminus_int @ ( numeral_numeral_int @ Y ) ) )
= ( ring_1_iszero_int @ ( numeral_numeral_int @ Y ) ) ) ).
% eq_numeral_iff_iszero(12)
thf(fact_234_not__iszero__neg__Numeral1,axiom,
~ ( ring_1_iszero_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ one ) ) ) ).
% not_iszero_neg_Numeral1
thf(fact_235_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_236_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_237_one__neq__neg__one,axiom,
( one_one_int
!= ( uminus_uminus_int @ one_one_int ) ) ).
% one_neq_neg_one
thf(fact_238_div__minus__right,axiom,
! [A: int,B: int] :
( ( divide_divide_int @ A @ ( uminus_uminus_int @ B ) )
= ( divide_divide_int @ ( uminus_uminus_int @ A ) @ B ) ) ).
% div_minus_right
thf(fact_239_eq__numeral__iff__iszero_I4_J,axiom,
! [X4: num,Y: num] :
( ( ( uminus_uminus_int @ ( numeral_numeral_int @ X4 ) )
= ( uminus_uminus_int @ ( numeral_numeral_int @ Y ) ) )
= ( ring_1_iszero_int @ ( neg_numeral_sub_int @ Y @ X4 ) ) ) ).
% eq_numeral_iff_iszero(4)
thf(fact_240_semiring__norm_I26_J,axiom,
( ( bitM @ one )
= one ) ).
% semiring_norm(26)
thf(fact_241_iszero__0,axiom,
ring_1_iszero_int @ zero_zero_int ).
% iszero_0
thf(fact_242_iszero__def,axiom,
( ring_1_iszero_int
= ( ^ [Z: int] : ( Z = zero_zero_int ) ) ) ).
% iszero_def
thf(fact_243_not__iszero__numeral,axiom,
! [W: num] :
~ ( ring_1_iszero_int @ ( numeral_numeral_int @ W ) ) ).
% not_iszero_numeral
thf(fact_244_not__iszero__1,axiom,
~ ( ring_1_iszero_int @ one_one_int ) ).
% not_iszero_1
thf(fact_245_zero__neq__neg__numeral,axiom,
! [N: num] :
( zero_zero_int
!= ( uminus_uminus_int @ ( numeral_numeral_int @ N ) ) ) ).
% zero_neq_neg_numeral
thf(fact_246_zero__neq__neg__one,axiom,
( zero_zero_int
!= ( uminus_uminus_int @ one_one_int ) ) ).
% zero_neq_neg_one
thf(fact_247_one__neq__neg__numeral,axiom,
! [N: num] :
( one_one_int
!= ( uminus_uminus_int @ ( numeral_numeral_int @ N ) ) ) ).
% one_neq_neg_numeral
thf(fact_248_numeral__neq__neg__one,axiom,
! [N: num] :
( ( numeral_numeral_int @ N )
!= ( uminus_uminus_int @ one_one_int ) ) ).
% numeral_neq_neg_one
thf(fact_249_semiring__norm_I27_J,axiom,
! [N: num] :
( ( bitM @ ( bit0 @ N ) )
= ( bit1 @ ( bitM @ N ) ) ) ).
% semiring_norm(27)
thf(fact_250_semiring__norm_I28_J,axiom,
! [N: num] :
( ( bitM @ ( bit1 @ N ) )
= ( bit1 @ ( bit0 @ N ) ) ) ).
% semiring_norm(28)
thf(fact_251_eq__numeral__iff__iszero_I9_J,axiom,
! [X4: num] :
( ( ( numeral_numeral_int @ X4 )
= zero_zero_int )
= ( ring_1_iszero_int @ ( numeral_numeral_int @ X4 ) ) ) ).
% eq_numeral_iff_iszero(9)
thf(fact_252_eq__numeral__iff__iszero_I10_J,axiom,
! [Y: num] :
( ( zero_zero_int
= ( numeral_numeral_int @ Y ) )
= ( ring_1_iszero_int @ ( numeral_numeral_int @ Y ) ) ) ).
% eq_numeral_iff_iszero(10)
thf(fact_253_not__iszero__Numeral1,axiom,
~ ( ring_1_iszero_int @ ( numeral_numeral_int @ one ) ) ).
% not_iszero_Numeral1
thf(fact_254_uminus__numeral__One,axiom,
( ( uminus_uminus_int @ ( numeral_numeral_int @ one ) )
= ( uminus_uminus_int @ one_one_int ) ) ).
% uminus_numeral_One
thf(fact_255_fst__divmod,axiom,
! [M: num,N: num] :
( ( product_fst_nat_nat @ ( unique5405566460079783412od_nat @ M @ N ) )
= ( divide_divide_nat @ ( numeral_numeral_nat @ M ) @ ( numeral_numeral_nat @ N ) ) ) ).
% fst_divmod
thf(fact_256_fst__divmod,axiom,
! [M: num,N: num] :
( ( product_fst_int_int @ ( unique5403075989570733136od_int @ M @ N ) )
= ( divide_divide_int @ ( numeral_numeral_int @ M ) @ ( numeral_numeral_int @ N ) ) ) ).
% fst_divmod
thf(fact_257_eq__numeral__iff__iszero_I1_J,axiom,
! [X4: num,Y: num] :
( ( ( numeral_numeral_int @ X4 )
= ( numeral_numeral_int @ Y ) )
= ( ring_1_iszero_int @ ( neg_numeral_sub_int @ X4 @ Y ) ) ) ).
% eq_numeral_iff_iszero(1)
thf(fact_258_n__lists_Osimps_I1_J,axiom,
! [Xs2: list_P8791008543742759928ropVar] :
( ( n_list6927042657299049856ropVar @ zero_zero_nat @ Xs2 )
= ( cons_l5280711761816980654ropVar @ nil_Pr6119456312066593144ropVar @ nil_li2679230897895549182ropVar ) ) ).
% n_lists.simps(1)
thf(fact_259_n__lists_Osimps_I1_J,axiom,
! [Xs2: list_l985434930288970485pVar_o] :
( ( n_list2523503019017615485pVar_o @ zero_zero_nat @ Xs2 )
= ( cons_l8096070148660426411pVar_o @ nil_li590035233778055285pVar_o @ nil_li5579170522155219707pVar_o ) ) ).
% n_lists.simps(1)
thf(fact_260_n__lists_Osimps_I1_J,axiom,
! [Xs2: list_l1255154591254203759pVar_o] :
( ( n_list1372634133161135479pVar_o @ zero_zero_nat @ Xs2 )
= ( cons_l7316790070876032037pVar_o @ nil_li1310946081133177199pVar_o @ nil_li590035233778055285pVar_o ) ) ).
% n_lists.simps(1)
thf(fact_261_n__lists_Osimps_I1_J,axiom,
! [Xs2: list_P1665994083459302249pVar_o] :
( ( n_list4068704761422681585pVar_o @ zero_zero_nat @ Xs2 )
= ( cons_l3452452106213086495pVar_o @ nil_Pr6413368191065476585pVar_o @ nil_li1310946081133177199pVar_o ) ) ).
% n_lists.simps(1)
thf(fact_262_eq__numeral__iff__iszero_I6_J,axiom,
! [Y: num] :
( ( one_one_int
= ( numeral_numeral_int @ Y ) )
= ( ring_1_iszero_int @ ( neg_numeral_sub_int @ one @ Y ) ) ) ).
% eq_numeral_iff_iszero(6)
thf(fact_263_minus__one__div__numeral,axiom,
! [N: num] :
( ( divide_divide_int @ ( uminus_uminus_int @ one_one_int ) @ ( numeral_numeral_int @ N ) )
= ( uminus_uminus_int @ ( adjust_div @ ( unique5403075989570733136od_int @ one @ N ) ) ) ) ).
% minus_one_div_numeral
thf(fact_264_one__div__minus__numeral,axiom,
! [N: num] :
( ( divide_divide_int @ one_one_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ N ) ) )
= ( uminus_uminus_int @ ( adjust_div @ ( unique5403075989570733136od_int @ one @ N ) ) ) ) ).
% one_div_minus_numeral
thf(fact_265_cnf__encoder_OencodeGammaCond_Oelims,axiom,
! [X4: multis3193088007478089820ropVar,Xa: multis3193088007478089820ropVar,Xb: $o,Xc: $o,Y: list_l1255154591254203759pVar_o] :
( ( ( multis334970729563149064maCond @ X4 @ Xa @ Xb @ Xc )
= Y )
=> ( ( Xb
=> ( Xc
=> ( Y != nil_li1310946081133177199pVar_o ) ) )
=> ( ( ~ Xb
=> ( ~ Xc
=> ( Y
!= ( cons_l3452452106213086495pVar_o @ ( cons_P6860203055867761049pVar_o @ ( produc2127521181850112851pVar_o @ X4 @ $false ) @ nil_Pr6413368191065476585pVar_o ) @ nil_li1310946081133177199pVar_o ) ) ) )
=> ( ( ~ Xb
=> ( Xc
=> ( Y
!= ( cons_l3452452106213086495pVar_o @ ( cons_P6860203055867761049pVar_o @ ( produc2127521181850112851pVar_o @ X4 @ $false ) @ ( cons_P6860203055867761049pVar_o @ ( produc2127521181850112851pVar_o @ Xa @ $true ) @ nil_Pr6413368191065476585pVar_o ) ) @ nil_li1310946081133177199pVar_o ) ) ) )
=> ~ ( Xb
=> ( ~ Xc
=> ( Y
!= ( cons_l3452452106213086495pVar_o @ ( cons_P6860203055867761049pVar_o @ ( produc2127521181850112851pVar_o @ X4 @ $false ) @ ( cons_P6860203055867761049pVar_o @ ( produc2127521181850112851pVar_o @ Xa @ $false ) @ nil_Pr6413368191065476585pVar_o ) ) @ nil_li1310946081133177199pVar_o ) ) ) ) ) ) ) ) ).
% cnf_encoder.encodeGammaCond.elims
thf(fact_266_cnf__encoder_OencodeGammaCond_Osimps_I2_J,axiom,
! [Gam: multis3193088007478089820ropVar,Eps: multis3193088007478089820ropVar] :
( ( multis334970729563149064maCond @ Gam @ Eps @ $false @ $false )
= ( cons_l3452452106213086495pVar_o @ ( cons_P6860203055867761049pVar_o @ ( produc2127521181850112851pVar_o @ Gam @ $false ) @ nil_Pr6413368191065476585pVar_o ) @ nil_li1310946081133177199pVar_o ) ) ).
% cnf_encoder.encodeGammaCond.simps(2)
thf(fact_267_cnf__encoder_OencodeGammaCond_Osimps_I3_J,axiom,
! [Gam: multis3193088007478089820ropVar,Eps: multis3193088007478089820ropVar] :
( ( multis334970729563149064maCond @ Gam @ Eps @ $false @ $true )
= ( cons_l3452452106213086495pVar_o @ ( cons_P6860203055867761049pVar_o @ ( produc2127521181850112851pVar_o @ Gam @ $false ) @ ( cons_P6860203055867761049pVar_o @ ( produc2127521181850112851pVar_o @ Eps @ $true ) @ nil_Pr6413368191065476585pVar_o ) ) @ nil_li1310946081133177199pVar_o ) ) ).
% cnf_encoder.encodeGammaCond.simps(3)
thf(fact_268_cnf__encoder_OencodeGammaCond_Osimps_I4_J,axiom,
! [Gam: multis3193088007478089820ropVar,Eps: multis3193088007478089820ropVar] :
( ( multis334970729563149064maCond @ Gam @ Eps @ $true @ $false )
= ( cons_l3452452106213086495pVar_o @ ( cons_P6860203055867761049pVar_o @ ( produc2127521181850112851pVar_o @ Gam @ $false ) @ ( cons_P6860203055867761049pVar_o @ ( produc2127521181850112851pVar_o @ Eps @ $false ) @ nil_Pr6413368191065476585pVar_o ) ) @ nil_li1310946081133177199pVar_o ) ) ).
% cnf_encoder.encodeGammaCond.simps(4)
thf(fact_269_minus__numeral__div__numeral,axiom,
! [M: num,N: num] :
( ( divide_divide_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ M ) ) @ ( numeral_numeral_int @ N ) )
= ( uminus_uminus_int @ ( adjust_div @ ( unique5403075989570733136od_int @ M @ N ) ) ) ) ).
% minus_numeral_div_numeral
thf(fact_270_numeral__div__minus__numeral,axiom,
! [M: num,N: num] :
( ( divide_divide_int @ ( numeral_numeral_int @ M ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ N ) ) )
= ( uminus_uminus_int @ ( adjust_div @ ( unique5403075989570733136od_int @ M @ N ) ) ) ) ).
% numeral_div_minus_numeral
thf(fact_271_eq__fst__iff,axiom,
! [A: multis3193088007478089820ropVar,P: produc8999643685545340131pVar_o] :
( ( A
= ( produc7654750731481044799pVar_o @ P ) )
= ( ? [B4: $o] :
( P
= ( produc2127521181850112851pVar_o @ A @ B4 ) ) ) ) ).
% eq_fst_iff
thf(fact_272_eq__fst__iff,axiom,
! [A: list_l1255154591254203759pVar_o,P: produc5436468477138497005pVar_o] :
( ( A
= ( produc1814762917671617225pVar_o @ P ) )
= ( ? [B4: list_l985434930288970485pVar_o] :
( P
= ( produc4064290870397114077pVar_o @ A @ B4 ) ) ) ) ).
% eq_fst_iff
thf(fact_273_eq__fst__iff,axiom,
! [A: list_l985434930288970485pVar_o,P: produc9206049318796031513ropVar] :
( ( A
= ( produc284539208842135413ropVar @ P ) )
= ( ? [B4: produc7679832412844898215ropVar] :
( P
= ( produc2520692699247754633ropVar @ A @ B4 ) ) ) ) ).
% eq_fst_iff
thf(fact_274_eq__fst__iff,axiom,
! [A: num,P: product_prod_num_num] :
( ( A
= ( product_fst_num_num @ P ) )
= ( ? [B4: num] :
( P
= ( product_Pair_num_num @ A @ B4 ) ) ) ) ).
% eq_fst_iff
thf(fact_275_eq__fst__iff,axiom,
! [A: int,P: product_prod_int_int] :
( ( A
= ( product_fst_int_int @ P ) )
= ( ? [B4: int] :
( P
= ( product_Pair_int_int @ A @ B4 ) ) ) ) ).
% eq_fst_iff
thf(fact_276_cnf__encoder_OencodeGammaCond_Osimps_I1_J,axiom,
! [Gam: multis3193088007478089820ropVar,Eps: multis3193088007478089820ropVar] :
( ( multis334970729563149064maCond @ Gam @ Eps @ $true @ $true )
= nil_li1310946081133177199pVar_o ) ).
% cnf_encoder.encodeGammaCond.simps(1)
thf(fact_277_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_278_fstI,axiom,
! [X4: produc8999643685545340131pVar_o,Y: multis3193088007478089820ropVar,Z2: $o] :
( ( X4
= ( produc2127521181850112851pVar_o @ Y @ Z2 ) )
=> ( ( produc7654750731481044799pVar_o @ X4 )
= Y ) ) ).
% fstI
thf(fact_279_fstI,axiom,
! [X4: produc5436468477138497005pVar_o,Y: list_l1255154591254203759pVar_o,Z2: list_l985434930288970485pVar_o] :
( ( X4
= ( produc4064290870397114077pVar_o @ Y @ Z2 ) )
=> ( ( produc1814762917671617225pVar_o @ X4 )
= Y ) ) ).
% fstI
thf(fact_280_fstI,axiom,
! [X4: produc9206049318796031513ropVar,Y: list_l985434930288970485pVar_o,Z2: produc7679832412844898215ropVar] :
( ( X4
= ( produc2520692699247754633ropVar @ Y @ Z2 ) )
=> ( ( produc284539208842135413ropVar @ X4 )
= Y ) ) ).
% fstI
thf(fact_281_fstI,axiom,
! [X4: product_prod_num_num,Y: num,Z2: num] :
( ( X4
= ( product_Pair_num_num @ Y @ Z2 ) )
=> ( ( product_fst_num_num @ X4 )
= Y ) ) ).
% fstI
thf(fact_282_fstI,axiom,
! [X4: product_prod_int_int,Y: int,Z2: int] :
( ( X4
= ( product_Pair_int_int @ Y @ Z2 ) )
=> ( ( product_fst_int_int @ X4 )
= Y ) ) ).
% fstI
thf(fact_283_cnf__encoder_Oenc__sound_I11_J,axiom,
! [Alpha: multis3193088007478089820ropVar > $o,X4: multis3193088007478089820ropVar,Y: multis3193088007478089820ropVar] :
( ( propos2885041177628046573ropVar @ Alpha @ ( cons_l3452452106213086495pVar_o @ ( cons_P6860203055867761049pVar_o @ ( produc2127521181850112851pVar_o @ X4 @ $false ) @ ( cons_P6860203055867761049pVar_o @ ( produc2127521181850112851pVar_o @ Y @ $false ) @ nil_Pr6413368191065476585pVar_o ) ) @ nil_li1310946081133177199pVar_o ) )
= ( ~ ( ( Alpha @ X4 )
& ( Alpha @ Y ) ) ) ) ).
% cnf_encoder.enc_sound(11)
thf(fact_284_cnf__encoder_Oenc__sound_I10_J,axiom,
! [Alpha: multis3193088007478089820ropVar > $o,X4: multis3193088007478089820ropVar,Y: multis3193088007478089820ropVar] :
( ( propos2885041177628046573ropVar @ Alpha @ ( cons_l3452452106213086495pVar_o @ ( cons_P6860203055867761049pVar_o @ ( produc2127521181850112851pVar_o @ X4 @ $false ) @ ( cons_P6860203055867761049pVar_o @ ( produc2127521181850112851pVar_o @ Y @ $true ) @ nil_Pr6413368191065476585pVar_o ) ) @ nil_li1310946081133177199pVar_o ) )
= ( ( Alpha @ X4 )
=> ( Alpha @ Y ) ) ) ).
% cnf_encoder.enc_sound(10)
thf(fact_285_cnf__encoder_Oenc__sound_I8_J,axiom,
! [Alpha: multis3193088007478089820ropVar > $o,X4: multis3193088007478089820ropVar,Z2: multis3193088007478089820ropVar,Y: multis3193088007478089820ropVar] :
( ( propos2885041177628046573ropVar @ Alpha @ ( cons_l3452452106213086495pVar_o @ ( cons_P6860203055867761049pVar_o @ ( produc2127521181850112851pVar_o @ X4 @ $true ) @ ( cons_P6860203055867761049pVar_o @ ( produc2127521181850112851pVar_o @ Z2 @ $true ) @ nil_Pr6413368191065476585pVar_o ) ) @ ( cons_l3452452106213086495pVar_o @ ( cons_P6860203055867761049pVar_o @ ( produc2127521181850112851pVar_o @ X4 @ $false ) @ ( cons_P6860203055867761049pVar_o @ ( produc2127521181850112851pVar_o @ Y @ $true ) @ nil_Pr6413368191065476585pVar_o ) ) @ nil_li1310946081133177199pVar_o ) ) )
= ( ( ( Alpha @ X4 )
=> ( Alpha @ Y ) )
& ( ~ ( Alpha @ X4 )
=> ( Alpha @ Z2 ) ) ) ) ).
% cnf_encoder.enc_sound(8)
thf(fact_286_cnf__encoder_Oenc__sound_I7_J,axiom,
! [Alpha: multis3193088007478089820ropVar > $o,X4: multis3193088007478089820ropVar,Y: multis3193088007478089820ropVar,U: multis3193088007478089820ropVar,Z2: multis3193088007478089820ropVar] :
( ( propos2885041177628046573ropVar @ Alpha @ ( cons_l3452452106213086495pVar_o @ ( cons_P6860203055867761049pVar_o @ ( produc2127521181850112851pVar_o @ X4 @ $true ) @ ( cons_P6860203055867761049pVar_o @ ( produc2127521181850112851pVar_o @ Y @ $true ) @ ( cons_P6860203055867761049pVar_o @ ( produc2127521181850112851pVar_o @ U @ $false ) @ nil_Pr6413368191065476585pVar_o ) ) ) @ ( cons_l3452452106213086495pVar_o @ ( cons_P6860203055867761049pVar_o @ ( produc2127521181850112851pVar_o @ X4 @ $true ) @ ( cons_P6860203055867761049pVar_o @ ( produc2127521181850112851pVar_o @ Y @ $false ) @ ( cons_P6860203055867761049pVar_o @ ( produc2127521181850112851pVar_o @ Z2 @ $false ) @ nil_Pr6413368191065476585pVar_o ) ) ) @ ( cons_l3452452106213086495pVar_o @ ( cons_P6860203055867761049pVar_o @ ( produc2127521181850112851pVar_o @ X4 @ $false ) @ ( cons_P6860203055867761049pVar_o @ ( produc2127521181850112851pVar_o @ Y @ $false ) @ ( cons_P6860203055867761049pVar_o @ ( produc2127521181850112851pVar_o @ Z2 @ $true ) @ nil_Pr6413368191065476585pVar_o ) ) ) @ ( cons_l3452452106213086495pVar_o @ ( cons_P6860203055867761049pVar_o @ ( produc2127521181850112851pVar_o @ X4 @ $false ) @ ( cons_P6860203055867761049pVar_o @ ( produc2127521181850112851pVar_o @ Y @ $true ) @ ( cons_P6860203055867761049pVar_o @ ( produc2127521181850112851pVar_o @ U @ $true ) @ nil_Pr6413368191065476585pVar_o ) ) ) @ nil_li1310946081133177199pVar_o ) ) ) ) )
= ( ( Alpha @ X4 )
= ( ( ( Alpha @ Y )
=> ( Alpha @ Z2 ) )
& ( ~ ( Alpha @ Y )
=> ( Alpha @ U ) ) ) ) ) ).
% cnf_encoder.enc_sound(7)
thf(fact_287_cnf__encoder_Oenc__sound_I6_J,axiom,
! [Alpha: multis3193088007478089820ropVar > $o,X4: multis3193088007478089820ropVar,Y: multis3193088007478089820ropVar,Z2: multis3193088007478089820ropVar] :
( ( propos2885041177628046573ropVar @ Alpha @ ( cons_l3452452106213086495pVar_o @ ( cons_P6860203055867761049pVar_o @ ( produc2127521181850112851pVar_o @ X4 @ $false ) @ ( cons_P6860203055867761049pVar_o @ ( produc2127521181850112851pVar_o @ Y @ $true ) @ nil_Pr6413368191065476585pVar_o ) ) @ ( cons_l3452452106213086495pVar_o @ ( cons_P6860203055867761049pVar_o @ ( produc2127521181850112851pVar_o @ X4 @ $false ) @ ( cons_P6860203055867761049pVar_o @ ( produc2127521181850112851pVar_o @ Z2 @ $false ) @ nil_Pr6413368191065476585pVar_o ) ) @ ( cons_l3452452106213086495pVar_o @ ( cons_P6860203055867761049pVar_o @ ( produc2127521181850112851pVar_o @ X4 @ $true ) @ ( cons_P6860203055867761049pVar_o @ ( produc2127521181850112851pVar_o @ Y @ $false ) @ ( cons_P6860203055867761049pVar_o @ ( produc2127521181850112851pVar_o @ Z2 @ $true ) @ nil_Pr6413368191065476585pVar_o ) ) ) @ nil_li1310946081133177199pVar_o ) ) ) )
= ( ( Alpha @ X4 )
= ( ( Alpha @ Y )
& ~ ( Alpha @ Z2 ) ) ) ) ).
% cnf_encoder.enc_sound(6)
thf(fact_288_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_289_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_290_diff__zero,axiom,
! [A: int] :
( ( minus_minus_int @ A @ zero_zero_int )
= A ) ).
% diff_zero
thf(fact_291_diff__zero,axiom,
! [A: nat] :
( ( minus_minus_nat @ A @ zero_zero_nat )
= A ) ).
% diff_zero
thf(fact_292_zero__diff,axiom,
! [A: nat] :
( ( minus_minus_nat @ zero_zero_nat @ A )
= zero_zero_nat ) ).
% zero_diff
thf(fact_293_diff__0__right,axiom,
! [A: int] :
( ( minus_minus_int @ A @ zero_zero_int )
= A ) ).
% diff_0_right
thf(fact_294_diff__self,axiom,
! [A: int] :
( ( minus_minus_int @ A @ A )
= zero_zero_int ) ).
% diff_self
thf(fact_295_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_296_cnf__encoder_Oenc__sound_I2_J,axiom,
! [Alpha: multis3193088007478089820ropVar > $o] : ( propos2885041177628046573ropVar @ Alpha @ nil_li1310946081133177199pVar_o ) ).
% cnf_encoder.enc_sound(2)
thf(fact_297_cnf__encoder_OencodeGammaCond,axiom,
! [Alpha: multis3193088007478089820ropVar > $o,Gam: multis3193088007478089820ropVar,Eps: multis3193088007478089820ropVar,S: $o,Ns: $o] :
( ( propos2885041177628046573ropVar @ Alpha @ ( multis334970729563149064maCond @ Gam @ Eps @ S @ Ns ) )
= ( ( Alpha @ Gam )
=> ( ( ( Alpha @ Eps )
=> Ns )
& ( ~ ( Alpha @ Eps )
=> S ) ) ) ) ).
% cnf_encoder.encodeGammaCond
thf(fact_298_diff__numeral__special_I9_J,axiom,
( ( minus_minus_int @ one_one_int @ one_one_int )
= zero_zero_int ) ).
% diff_numeral_special(9)
thf(fact_299_diff__0,axiom,
! [A: int] :
( ( minus_minus_int @ zero_zero_int @ A )
= ( uminus_uminus_int @ A ) ) ).
% diff_0
thf(fact_300_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_301_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_302_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_303_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_304_cnf__encoder_Oenc__sound_I1_J,axiom,
! [Alpha: multis3193088007478089820ropVar > $o] :
~ ( propos2885041177628046573ropVar @ Alpha @ ( cons_l3452452106213086495pVar_o @ nil_Pr6413368191065476585pVar_o @ nil_li1310946081133177199pVar_o ) ) ).
% cnf_encoder.enc_sound(1)
thf(fact_305_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_306_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_307_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_308_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_309_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_310_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_311_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_312_cnf__encoder_Oenc__sound_I3_J,axiom,
! [Alpha: multis3193088007478089820ropVar > $o,X4: multis3193088007478089820ropVar] :
( ( propos2885041177628046573ropVar @ Alpha @ ( cons_l3452452106213086495pVar_o @ ( cons_P6860203055867761049pVar_o @ ( produc2127521181850112851pVar_o @ X4 @ $true ) @ nil_Pr6413368191065476585pVar_o ) @ nil_li1310946081133177199pVar_o ) )
= ( Alpha @ X4 ) ) ).
% cnf_encoder.enc_sound(3)
thf(fact_313_cnf__encoder_Oenc__sound_I4_J,axiom,
! [Alpha: multis3193088007478089820ropVar > $o,X4: multis3193088007478089820ropVar] :
( ( propos2885041177628046573ropVar @ Alpha @ ( cons_l3452452106213086495pVar_o @ ( cons_P6860203055867761049pVar_o @ ( produc2127521181850112851pVar_o @ X4 @ $false ) @ nil_Pr6413368191065476585pVar_o ) @ nil_li1310946081133177199pVar_o ) )
= ( ~ ( Alpha @ X4 ) ) ) ).
% cnf_encoder.enc_sound(4)
thf(fact_314_cnf__encoder_Oenc__sound_I5_J,axiom,
! [Alpha: multis3193088007478089820ropVar > $o,X4: multis3193088007478089820ropVar,Y: multis3193088007478089820ropVar] :
( ( propos2885041177628046573ropVar @ Alpha @ ( cons_l3452452106213086495pVar_o @ ( cons_P6860203055867761049pVar_o @ ( produc2127521181850112851pVar_o @ X4 @ $true ) @ ( cons_P6860203055867761049pVar_o @ ( produc2127521181850112851pVar_o @ Y @ $true ) @ nil_Pr6413368191065476585pVar_o ) ) @ ( cons_l3452452106213086495pVar_o @ ( cons_P6860203055867761049pVar_o @ ( produc2127521181850112851pVar_o @ X4 @ $false ) @ ( cons_P6860203055867761049pVar_o @ ( produc2127521181850112851pVar_o @ Y @ $false ) @ nil_Pr6413368191065476585pVar_o ) ) @ nil_li1310946081133177199pVar_o ) ) )
= ( ( Alpha @ X4 )
= ( ~ ( Alpha @ Y ) ) ) ) ).
% cnf_encoder.enc_sound(5)
thf(fact_315_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_316_cancel__ab__semigroup__add__class_Odiff__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 ) ) ).
% cancel_ab_semigroup_add_class.diff_right_commute
thf(fact_317_cancel__ab__semigroup__add__class_Odiff__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 ) ) ).
% cancel_ab_semigroup_add_class.diff_right_commute
thf(fact_318_eq__iff__diff__eq__0,axiom,
( ( ^ [Y5: int,Z3: int] : ( Y5 = Z3 ) )
= ( ^ [A4: int,B4: int] :
( ( minus_minus_int @ A4 @ B4 )
= zero_zero_int ) ) ) ).
% eq_iff_diff_eq_0
thf(fact_319_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_320_eq__iff__iszero__diff,axiom,
( ( ^ [Y5: int,Z3: int] : ( Y5 = Z3 ) )
= ( ^ [X5: int,Y4: int] : ( ring_1_iszero_int @ ( minus_minus_int @ X5 @ Y4 ) ) ) ) ).
% eq_iff_iszero_diff
thf(fact_321_neg__numeral__class_Osub__def,axiom,
( neg_numeral_sub_int
= ( ^ [K2: num,L2: num] : ( minus_minus_int @ ( numeral_numeral_int @ K2 ) @ ( numeral_numeral_int @ L2 ) ) ) ) ).
% neg_numeral_class.sub_def
thf(fact_322_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_323_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_324_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_325_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_326_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_327_minus__1__div__exp__eq__int,axiom,
! [N: nat] :
( ( divide_divide_int @ ( uminus_uminus_int @ one_one_int ) @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) )
= ( uminus_uminus_int @ one_one_int ) ) ).
% minus_1_div_exp_eq_int
thf(fact_328_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_329_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_330_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_331_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_332_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_333_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_334_add_Oright__neutral,axiom,
! [A: nat] :
( ( plus_plus_nat @ A @ zero_zero_nat )
= A ) ).
% add.right_neutral
thf(fact_335_add_Oright__neutral,axiom,
! [A: int] :
( ( plus_plus_int @ A @ zero_zero_int )
= A ) ).
% add.right_neutral
thf(fact_336_double__zero__sym,axiom,
! [A: int] :
( ( zero_zero_int
= ( plus_plus_int @ A @ A ) )
= ( A = zero_zero_int ) ) ).
% double_zero_sym
thf(fact_337_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_338_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_339_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_340_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_341_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_342_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_343_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_344_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_345_add__eq__0__iff__both__eq__0,axiom,
! [X4: nat,Y: nat] :
( ( ( plus_plus_nat @ X4 @ Y )
= zero_zero_nat )
= ( ( X4 = zero_zero_nat )
& ( Y = zero_zero_nat ) ) ) ).
% add_eq_0_iff_both_eq_0
thf(fact_346_zero__eq__add__iff__both__eq__0,axiom,
! [X4: nat,Y: nat] :
( ( zero_zero_nat
= ( plus_plus_nat @ X4 @ Y ) )
= ( ( X4 = zero_zero_nat )
& ( Y = zero_zero_nat ) ) ) ).
% zero_eq_add_iff_both_eq_0
thf(fact_347_add__0,axiom,
! [A: nat] :
( ( plus_plus_nat @ zero_zero_nat @ A )
= A ) ).
% add_0
thf(fact_348_add__0,axiom,
! [A: int] :
( ( plus_plus_int @ zero_zero_int @ A )
= A ) ).
% add_0
thf(fact_349_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_350_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_351_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_352_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_353_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_354_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_355_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_356_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_357_diff__add__cancel,axiom,
! [A: int,B: int] :
( ( plus_plus_int @ ( minus_minus_int @ A @ B ) @ B )
= A ) ).
% diff_add_cancel
thf(fact_358_add__diff__cancel,axiom,
! [A: int,B: int] :
( ( minus_minus_int @ ( plus_plus_int @ A @ B ) @ B )
= A ) ).
% add_diff_cancel
thf(fact_359_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_360_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_361_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_362_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_363_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_364_ab__left__minus,axiom,
! [A: int] :
( ( plus_plus_int @ ( uminus_uminus_int @ A ) @ A )
= zero_zero_int ) ).
% ab_left_minus
thf(fact_365_add_Oright__inverse,axiom,
! [A: int] :
( ( plus_plus_int @ A @ ( uminus_uminus_int @ A ) )
= zero_zero_int ) ).
% add.right_inverse
thf(fact_366_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_367_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_368_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_369_add__self__div__2,axiom,
! [M: nat] :
( ( divide_divide_nat @ ( plus_plus_nat @ M @ M ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
= M ) ).
% add_self_div_2
thf(fact_370_add__numeral__left,axiom,
! [V: num,W: num,Z2: nat] :
( ( plus_plus_nat @ ( numeral_numeral_nat @ V ) @ ( plus_plus_nat @ ( numeral_numeral_nat @ W ) @ Z2 ) )
= ( plus_plus_nat @ ( numeral_numeral_nat @ ( plus_plus_num @ V @ W ) ) @ Z2 ) ) ).
% add_numeral_left
thf(fact_371_add__numeral__left,axiom,
! [V: num,W: num,Z2: int] :
( ( plus_plus_int @ ( numeral_numeral_int @ V ) @ ( plus_plus_int @ ( numeral_numeral_int @ W ) @ Z2 ) )
= ( plus_plus_int @ ( numeral_numeral_int @ ( plus_plus_num @ V @ W ) ) @ Z2 ) ) ).
% add_numeral_left
thf(fact_372_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_373_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_374_semiring__norm_I2_J,axiom,
( ( plus_plus_num @ one @ one )
= ( bit0 @ one ) ) ).
% semiring_norm(2)
thf(fact_375_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_376_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_377_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_378_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_379_semiring__norm_I167_J,axiom,
! [V: num,W: num,Y: int] :
( ( plus_plus_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ V ) ) @ ( plus_plus_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ W ) ) @ Y ) )
= ( plus_plus_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ ( plus_plus_num @ V @ W ) ) ) @ Y ) ) ).
% semiring_norm(167)
thf(fact_380_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_381_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_382_semiring__norm_I165_J,axiom,
! [V: num,W: num,Y: int] :
( ( plus_plus_int @ ( numeral_numeral_int @ V ) @ ( plus_plus_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ W ) ) @ Y ) )
= ( plus_plus_int @ ( neg_numeral_sub_int @ V @ W ) @ Y ) ) ).
% semiring_norm(165)
thf(fact_383_semiring__norm_I166_J,axiom,
! [V: num,W: num,Y: int] :
( ( plus_plus_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ V ) ) @ ( plus_plus_int @ ( numeral_numeral_int @ W ) @ Y ) )
= ( plus_plus_int @ ( neg_numeral_sub_int @ W @ V ) @ Y ) ) ).
% semiring_norm(166)
thf(fact_384_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_385_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_386_semiring__norm_I3_J,axiom,
! [N: num] :
( ( plus_plus_num @ one @ ( bit0 @ N ) )
= ( bit1 @ N ) ) ).
% semiring_norm(3)
thf(fact_387_semiring__norm_I4_J,axiom,
! [N: num] :
( ( plus_plus_num @ one @ ( bit1 @ N ) )
= ( bit0 @ ( plus_plus_num @ N @ one ) ) ) ).
% semiring_norm(4)
thf(fact_388_semiring__norm_I5_J,axiom,
! [M: num] :
( ( plus_plus_num @ ( bit0 @ M ) @ one )
= ( bit1 @ M ) ) ).
% semiring_norm(5)
thf(fact_389_semiring__norm_I8_J,axiom,
! [M: num] :
( ( plus_plus_num @ ( bit1 @ M ) @ one )
= ( bit0 @ ( plus_plus_num @ M @ one ) ) ) ).
% semiring_norm(8)
thf(fact_390_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_391_one__add__one,axiom,
( ( plus_plus_nat @ one_one_nat @ one_one_nat )
= ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ).
% one_add_one
thf(fact_392_one__add__one,axiom,
( ( plus_plus_int @ one_one_int @ one_one_int )
= ( numeral_numeral_int @ ( bit0 @ one ) ) ) ).
% one_add_one
thf(fact_393_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_394_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_395_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_396_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_397_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_398_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_399_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_400_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_401_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_402_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_403_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_404_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_405_add_Ocommute,axiom,
( plus_plus_nat
= ( ^ [A4: nat,B4: nat] : ( plus_plus_nat @ B4 @ A4 ) ) ) ).
% add.commute
thf(fact_406_add_Ocommute,axiom,
( plus_plus_int
= ( ^ [A4: int,B4: int] : ( plus_plus_int @ B4 @ A4 ) ) ) ).
% add.commute
thf(fact_407_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_408_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_409_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_410_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_411_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_412_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_413_group__cancel_Oadd1,axiom,
! [A5: nat,K: nat,A: nat,B: nat] :
( ( A5
= ( plus_plus_nat @ K @ A ) )
=> ( ( plus_plus_nat @ A5 @ B )
= ( plus_plus_nat @ K @ ( plus_plus_nat @ A @ B ) ) ) ) ).
% group_cancel.add1
thf(fact_414_group__cancel_Oadd1,axiom,
! [A5: int,K: int,A: int,B: int] :
( ( A5
= ( plus_plus_int @ K @ A ) )
=> ( ( plus_plus_int @ A5 @ B )
= ( plus_plus_int @ K @ ( plus_plus_int @ A @ B ) ) ) ) ).
% group_cancel.add1
thf(fact_415_add__mono__thms__linordered__semiring_I4_J,axiom,
! [I: nat,J: nat,K: nat,L: nat] :
( ( ( I = J )
& ( K = L ) )
=> ( ( plus_plus_nat @ I @ K )
= ( plus_plus_nat @ J @ L ) ) ) ).
% add_mono_thms_linordered_semiring(4)
thf(fact_416_add__mono__thms__linordered__semiring_I4_J,axiom,
! [I: int,J: int,K: int,L: int] :
( ( ( I = J )
& ( K = L ) )
=> ( ( plus_plus_int @ I @ K )
= ( plus_plus_int @ J @ L ) ) ) ).
% add_mono_thms_linordered_semiring(4)
thf(fact_417_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_418_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_419_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_420_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_421_add_Ocomm__neutral,axiom,
! [A: nat] :
( ( plus_plus_nat @ A @ zero_zero_nat )
= A ) ).
% add.comm_neutral
thf(fact_422_add_Ocomm__neutral,axiom,
! [A: int] :
( ( plus_plus_int @ A @ zero_zero_int )
= A ) ).
% add.comm_neutral
thf(fact_423_add_Ogroup__left__neutral,axiom,
! [A: int] :
( ( plus_plus_int @ zero_zero_int @ A )
= A ) ).
% add.group_left_neutral
thf(fact_424_verit__sum__simplify,axiom,
! [A: nat] :
( ( plus_plus_nat @ A @ zero_zero_nat )
= A ) ).
% verit_sum_simplify
thf(fact_425_verit__sum__simplify,axiom,
! [A: int] :
( ( plus_plus_int @ A @ zero_zero_int )
= A ) ).
% verit_sum_simplify
thf(fact_426_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_427_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_428_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_429_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_430_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_431_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_432_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_433_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_434_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_435_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_436_group__cancel_Osub1,axiom,
! [A5: int,K: int,A: int,B: int] :
( ( A5
= ( plus_plus_int @ K @ A ) )
=> ( ( minus_minus_int @ A5 @ B )
= ( plus_plus_int @ K @ ( minus_minus_int @ A @ B ) ) ) ) ).
% group_cancel.sub1
thf(fact_437_add__One__commute,axiom,
! [N: num] :
( ( plus_plus_num @ one @ N )
= ( plus_plus_num @ N @ one ) ) ).
% add_One_commute
thf(fact_438_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_439_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_440_group__cancel_Oneg1,axiom,
! [A5: int,K: int,A: int] :
( ( A5
= ( plus_plus_int @ K @ A ) )
=> ( ( uminus_uminus_int @ A5 )
= ( plus_plus_int @ ( uminus_uminus_int @ K ) @ ( uminus_uminus_int @ A ) ) ) ) ).
% group_cancel.neg1
thf(fact_441_dbl__def,axiom,
( neg_numeral_dbl_int
= ( ^ [X5: int] : ( plus_plus_int @ X5 @ X5 ) ) ) ).
% dbl_def
thf(fact_442_exp__not__zero__imp__exp__diff__not__zero,axiom,
! [N: nat,M: nat] :
( ( ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
!= zero_zero_nat )
=> ( ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( minus_minus_nat @ N @ M ) )
!= zero_zero_nat ) ) ).
% exp_not_zero_imp_exp_diff_not_zero
thf(fact_443_exp__not__zero__imp__exp__diff__not__zero,axiom,
! [N: nat,M: nat] :
( ( ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N )
!= zero_zero_int )
=> ( ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( minus_minus_nat @ N @ M ) )
!= zero_zero_int ) ) ).
% exp_not_zero_imp_exp_diff_not_zero
thf(fact_444_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_445_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_446_exp__add__not__zero__imp__left,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 ) ) @ M )
!= zero_zero_nat ) ) ).
% exp_add_not_zero_imp_left
thf(fact_447_exp__add__not__zero__imp__left,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 ) ) @ M )
!= zero_zero_int ) ) ).
% exp_add_not_zero_imp_left
thf(fact_448_div__exp__eq,axiom,
! [A: int,M: nat,N: nat] :
( ( divide_divide_int @ ( divide_divide_int @ A @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ M ) ) @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) )
= ( divide_divide_int @ A @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( plus_plus_nat @ M @ N ) ) ) ) ).
% div_exp_eq
thf(fact_449_div__exp__eq,axiom,
! [A: nat,M: nat,N: nat] :
( ( divide_divide_nat @ ( divide_divide_nat @ A @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) )
= ( divide_divide_nat @ A @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( plus_plus_nat @ M @ N ) ) ) ) ).
% div_exp_eq
thf(fact_450_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_451_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_452_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_453_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_454_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_455_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_456_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_457_one__plus__numeral__commute,axiom,
! [X4: num] :
( ( plus_plus_nat @ one_one_nat @ ( numeral_numeral_nat @ X4 ) )
= ( plus_plus_nat @ ( numeral_numeral_nat @ X4 ) @ one_one_nat ) ) ).
% one_plus_numeral_commute
thf(fact_458_one__plus__numeral__commute,axiom,
! [X4: num] :
( ( plus_plus_int @ one_one_int @ ( numeral_numeral_int @ X4 ) )
= ( plus_plus_int @ ( numeral_numeral_int @ X4 ) @ one_one_int ) ) ).
% one_plus_numeral_commute
thf(fact_459_ab__group__add__class_Oab__diff__conv__add__uminus,axiom,
( minus_minus_int
= ( ^ [A4: int,B4: int] : ( plus_plus_int @ A4 @ ( uminus_uminus_int @ B4 ) ) ) ) ).
% ab_group_add_class.ab_diff_conv_add_uminus
thf(fact_460_diff__conv__add__uminus,axiom,
( minus_minus_int
= ( ^ [A4: int,B4: int] : ( plus_plus_int @ A4 @ ( uminus_uminus_int @ B4 ) ) ) ) ).
% diff_conv_add_uminus
thf(fact_461_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_462_dbl__inc__def,axiom,
( neg_nu5851722552734809277nc_int
= ( ^ [X5: int] : ( plus_plus_int @ ( plus_plus_int @ X5 @ X5 ) @ one_one_int ) ) ) ).
% dbl_inc_def
thf(fact_463_div__add__self2,axiom,
! [B: int,A: int] :
( ( B != zero_zero_int )
=> ( ( divide_divide_int @ ( plus_plus_int @ A @ B ) @ B )
= ( plus_plus_int @ ( divide_divide_int @ A @ B ) @ one_one_int ) ) ) ).
% div_add_self2
thf(fact_464_div__add__self2,axiom,
! [B: nat,A: nat] :
( ( B != zero_zero_nat )
=> ( ( divide_divide_nat @ ( plus_plus_nat @ A @ B ) @ B )
= ( plus_plus_nat @ ( divide_divide_nat @ A @ B ) @ one_one_nat ) ) ) ).
% div_add_self2
thf(fact_465_div__add__self1,axiom,
! [B: int,A: int] :
( ( B != zero_zero_int )
=> ( ( divide_divide_int @ ( plus_plus_int @ B @ A ) @ B )
= ( plus_plus_int @ ( divide_divide_int @ A @ B ) @ one_one_int ) ) ) ).
% div_add_self1
thf(fact_466_div__add__self1,axiom,
! [B: nat,A: nat] :
( ( B != zero_zero_nat )
=> ( ( divide_divide_nat @ ( plus_plus_nat @ B @ A ) @ B )
= ( plus_plus_nat @ ( divide_divide_nat @ A @ B ) @ one_one_nat ) ) ) ).
% div_add_self1
thf(fact_467_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_468_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_469_one__plus__BitM,axiom,
! [N: num] :
( ( plus_plus_num @ one @ ( bitM @ N ) )
= ( bit0 @ N ) ) ).
% one_plus_BitM
thf(fact_470_BitM__plus__one,axiom,
! [N: num] :
( ( plus_plus_num @ ( bitM @ N ) @ one )
= ( bit0 @ N ) ) ).
% BitM_plus_one
thf(fact_471_eq__numeral__iff__iszero_I3_J,axiom,
! [X4: num,Y: num] :
( ( ( uminus_uminus_int @ ( numeral_numeral_int @ X4 ) )
= ( numeral_numeral_int @ Y ) )
= ( ring_1_iszero_int @ ( numeral_numeral_int @ ( plus_plus_num @ X4 @ Y ) ) ) ) ).
% eq_numeral_iff_iszero(3)
thf(fact_472_eq__numeral__iff__iszero_I2_J,axiom,
! [X4: num,Y: num] :
( ( ( numeral_numeral_int @ X4 )
= ( uminus_uminus_int @ ( numeral_numeral_int @ Y ) ) )
= ( ring_1_iszero_int @ ( numeral_numeral_int @ ( plus_plus_num @ X4 @ Y ) ) ) ) ).
% eq_numeral_iff_iszero(2)
thf(fact_473_dbl__dec__def,axiom,
( neg_nu3811975205180677377ec_int
= ( ^ [X5: int] : ( minus_minus_int @ ( plus_plus_int @ X5 @ X5 ) @ one_one_int ) ) ) ).
% dbl_dec_def
thf(fact_474_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_475_nat__induct2,axiom,
! [P2: nat > $o,N: nat] :
( ( P2 @ zero_zero_nat )
=> ( ( P2 @ one_one_nat )
=> ( ! [N2: nat] :
( ( P2 @ N2 )
=> ( P2 @ ( plus_plus_nat @ N2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) )
=> ( P2 @ N ) ) ) ) ).
% nat_induct2
thf(fact_476_eq__numeral__iff__iszero_I8_J,axiom,
! [Y: num] :
( ( one_one_int
= ( uminus_uminus_int @ ( numeral_numeral_int @ Y ) ) )
= ( ring_1_iszero_int @ ( numeral_numeral_int @ ( plus_plus_num @ one @ Y ) ) ) ) ).
% eq_numeral_iff_iszero(8)
thf(fact_477_eq__numeral__iff__iszero_I7_J,axiom,
! [X4: num] :
( ( ( uminus_uminus_int @ ( numeral_numeral_int @ X4 ) )
= one_one_int )
= ( ring_1_iszero_int @ ( numeral_numeral_int @ ( plus_plus_num @ X4 @ one ) ) ) ) ).
% eq_numeral_iff_iszero(7)
thf(fact_478_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_479_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_480_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_481_power__zero__numeral,axiom,
! [K: num] :
( ( power_power_int @ zero_zero_int @ ( numeral_numeral_nat @ K ) )
= zero_zero_int ) ).
% power_zero_numeral
thf(fact_482_power__zero__numeral,axiom,
! [K: num] :
( ( power_power_nat @ zero_zero_nat @ ( numeral_numeral_nat @ K ) )
= zero_zero_nat ) ).
% power_zero_numeral
thf(fact_483_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_484_sum__power2__eq__zero__iff,axiom,
! [X4: int,Y: int] :
( ( ( plus_plus_int @ ( power_power_int @ X4 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_int @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
= zero_zero_int )
= ( ( X4 = zero_zero_int )
& ( Y = zero_zero_int ) ) ) ).
% sum_power2_eq_zero_iff
thf(fact_485_power__one__right,axiom,
! [A: int] :
( ( power_power_int @ A @ one_one_nat )
= A ) ).
% power_one_right
thf(fact_486_power__one__right,axiom,
! [A: nat] :
( ( power_power_nat @ A @ one_one_nat )
= A ) ).
% power_one_right
thf(fact_487_power2__eq__iff,axiom,
! [X4: int,Y: int] :
( ( ( power_power_int @ X4 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
= ( power_power_int @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
= ( ( X4 = Y )
| ( X4
= ( uminus_uminus_int @ Y ) ) ) ) ).
% power2_eq_iff
thf(fact_488_power__one,axiom,
! [N: nat] :
( ( power_power_int @ one_one_int @ N )
= one_one_int ) ).
% power_one
thf(fact_489_power__one,axiom,
! [N: nat] :
( ( power_power_nat @ one_one_nat @ N )
= one_one_nat ) ).
% power_one
thf(fact_490_power__not__zero,axiom,
! [A: int,N: nat] :
( ( A != zero_zero_int )
=> ( ( power_power_int @ A @ N )
!= zero_zero_int ) ) ).
% power_not_zero
thf(fact_491_power__not__zero,axiom,
! [A: nat,N: nat] :
( ( A != zero_zero_nat )
=> ( ( power_power_nat @ A @ N )
!= zero_zero_nat ) ) ).
% power_not_zero
thf(fact_492_power__0,axiom,
! [A: int] :
( ( power_power_int @ A @ zero_zero_nat )
= one_one_int ) ).
% power_0
thf(fact_493_power__0,axiom,
! [A: nat] :
( ( power_power_nat @ A @ zero_zero_nat )
= one_one_nat ) ).
% power_0
thf(fact_494_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_495_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_496_power__minus__Bit0,axiom,
! [X4: int,K: num] :
( ( power_power_int @ ( uminus_uminus_int @ X4 ) @ ( numeral_numeral_nat @ ( bit0 @ K ) ) )
= ( power_power_int @ X4 @ ( numeral_numeral_nat @ ( bit0 @ K ) ) ) ) ).
% power_minus_Bit0
thf(fact_497_power__minus__Bit1,axiom,
! [X4: int,K: num] :
( ( power_power_int @ ( uminus_uminus_int @ X4 ) @ ( numeral_numeral_nat @ ( bit1 @ K ) ) )
= ( uminus_uminus_int @ ( power_power_int @ X4 @ ( numeral_numeral_nat @ ( bit1 @ K ) ) ) ) ) ).
% power_minus_Bit1
thf(fact_498_zero__power2,axiom,
( ( power_power_int @ zero_zero_int @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
= zero_zero_int ) ).
% zero_power2
thf(fact_499_zero__power2,axiom,
( ( power_power_nat @ zero_zero_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
= zero_zero_nat ) ).
% zero_power2
thf(fact_500_power2__commute,axiom,
! [X4: int,Y: int] :
( ( power_power_int @ ( minus_minus_int @ X4 @ Y ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
= ( power_power_int @ ( minus_minus_int @ Y @ X4 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ).
% power2_commute
thf(fact_501_one__power2,axiom,
( ( power_power_int @ one_one_int @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
= one_one_int ) ).
% one_power2
thf(fact_502_one__power2,axiom,
( ( power_power_nat @ one_one_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
= one_one_nat ) ).
% one_power2
thf(fact_503_diff__0__eq__0,axiom,
! [N: nat] :
( ( minus_minus_nat @ zero_zero_nat @ N )
= zero_zero_nat ) ).
% diff_0_eq_0
thf(fact_504_diff__self__eq__0,axiom,
! [M: nat] :
( ( minus_minus_nat @ M @ M )
= zero_zero_nat ) ).
% diff_self_eq_0
thf(fact_505_Nat_Oadd__0__right,axiom,
! [M: nat] :
( ( plus_plus_nat @ M @ zero_zero_nat )
= M ) ).
% Nat.add_0_right
thf(fact_506_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_507_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_508_plus__nat_Oadd__0,axiom,
! [N: nat] :
( ( plus_plus_nat @ zero_zero_nat @ N )
= N ) ).
% plus_nat.add_0
thf(fact_509_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_510_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_511_minus__nat_Odiff__0,axiom,
! [M: nat] :
( ( minus_minus_nat @ M @ zero_zero_nat )
= M ) ).
% minus_nat.diff_0
thf(fact_512_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_513_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_514_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_515_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_516_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_517_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_518_bits__1__div__exp,axiom,
! [N: nat] :
( ( divide_divide_nat @ one_one_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) )
= ( zero_n2687167440665602831ol_nat @ ( N = zero_zero_nat ) ) ) ).
% bits_1_div_exp
thf(fact_519_bits__1__div__exp,axiom,
! [N: nat] :
( ( divide_divide_int @ one_one_int @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) )
= ( zero_n2684676970156552555ol_int @ ( N = zero_zero_nat ) ) ) ).
% bits_1_div_exp
thf(fact_520_one__div__2__pow__eq,axiom,
! [N: nat] :
( ( divide_divide_nat @ one_one_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) )
= ( zero_n2687167440665602831ol_nat @ ( N = zero_zero_nat ) ) ) ).
% one_div_2_pow_eq
thf(fact_521_one__div__2__pow__eq,axiom,
! [N: nat] :
( ( divide_divide_int @ one_one_int @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) )
= ( zero_n2684676970156552555ol_int @ ( N = zero_zero_nat ) ) ) ).
% one_div_2_pow_eq
thf(fact_522_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_523_dvd__0__left__iff,axiom,
! [A: nat] :
( ( dvd_dvd_nat @ zero_zero_nat @ A )
= ( A = zero_zero_nat ) ) ).
% dvd_0_left_iff
thf(fact_524_dvd__0__left__iff,axiom,
! [A: int] :
( ( dvd_dvd_int @ zero_zero_int @ A )
= ( A = zero_zero_int ) ) ).
% dvd_0_left_iff
thf(fact_525_dvd__0__right,axiom,
! [A: nat] : ( dvd_dvd_nat @ A @ zero_zero_nat ) ).
% dvd_0_right
thf(fact_526_dvd__0__right,axiom,
! [A: int] : ( dvd_dvd_int @ A @ zero_zero_int ) ).
% dvd_0_right
thf(fact_527_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_528_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_529_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_530_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_531_dvd__minus__iff,axiom,
! [X4: int,Y: int] :
( ( dvd_dvd_int @ X4 @ ( uminus_uminus_int @ Y ) )
= ( dvd_dvd_int @ X4 @ Y ) ) ).
% dvd_minus_iff
thf(fact_532_minus__dvd__iff,axiom,
! [X4: int,Y: int] :
( ( dvd_dvd_int @ ( uminus_uminus_int @ X4 ) @ Y )
= ( dvd_dvd_int @ X4 @ Y ) ) ).
% minus_dvd_iff
thf(fact_533_div__dvd__div,axiom,
! [A: int,B: int,C2: int] :
( ( dvd_dvd_int @ A @ B )
=> ( ( dvd_dvd_int @ A @ C2 )
=> ( ( dvd_dvd_int @ ( divide_divide_int @ B @ A ) @ ( divide_divide_int @ C2 @ A ) )
= ( dvd_dvd_int @ B @ C2 ) ) ) ) ).
% div_dvd_div
thf(fact_534_div__dvd__div,axiom,
! [A: nat,B: nat,C2: nat] :
( ( dvd_dvd_nat @ A @ B )
=> ( ( dvd_dvd_nat @ A @ C2 )
=> ( ( dvd_dvd_nat @ ( divide_divide_nat @ B @ A ) @ ( divide_divide_nat @ C2 @ A ) )
= ( dvd_dvd_nat @ B @ C2 ) ) ) ) ).
% div_dvd_div
thf(fact_535_of__bool__eq__0__iff,axiom,
! [P2: $o] :
( ( ( zero_n2687167440665602831ol_nat @ P2 )
= zero_zero_nat )
= ~ P2 ) ).
% of_bool_eq_0_iff
thf(fact_536_of__bool__eq__0__iff,axiom,
! [P2: $o] :
( ( ( zero_n2684676970156552555ol_int @ P2 )
= zero_zero_int )
= ~ P2 ) ).
% of_bool_eq_0_iff
thf(fact_537_of__bool__eq_I1_J,axiom,
( ( zero_n2687167440665602831ol_nat @ $false )
= zero_zero_nat ) ).
% of_bool_eq(1)
thf(fact_538_of__bool__eq_I1_J,axiom,
( ( zero_n2684676970156552555ol_int @ $false )
= zero_zero_int ) ).
% of_bool_eq(1)
thf(fact_539_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_540_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_541_push__bit__of__0,axiom,
! [N: nat] :
( ( bit_se547839408752420682it_nat @ N @ zero_zero_nat )
= zero_zero_nat ) ).
% push_bit_of_0
thf(fact_542_push__bit__of__0,axiom,
! [N: nat] :
( ( bit_se545348938243370406it_int @ N @ zero_zero_int )
= zero_zero_int ) ).
% push_bit_of_0
thf(fact_543_of__bool__eq__1__iff,axiom,
! [P2: $o] :
( ( ( zero_n2687167440665602831ol_nat @ P2 )
= one_one_nat )
= P2 ) ).
% of_bool_eq_1_iff
thf(fact_544_of__bool__eq__1__iff,axiom,
! [P2: $o] :
( ( ( zero_n2684676970156552555ol_int @ P2 )
= one_one_int )
= P2 ) ).
% of_bool_eq_1_iff
thf(fact_545_of__bool__eq_I2_J,axiom,
( ( zero_n2687167440665602831ol_nat @ $true )
= one_one_nat ) ).
% of_bool_eq(2)
thf(fact_546_of__bool__eq_I2_J,axiom,
( ( zero_n2684676970156552555ol_int @ $true )
= one_one_int ) ).
% of_bool_eq(2)
thf(fact_547_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_548_Parity_Oadjust__div__eq,axiom,
! [Q: int,R: int] :
( ( adjust_div @ ( product_Pair_int_int @ Q @ R ) )
= ( plus_plus_int @ Q @ ( zero_n2684676970156552555ol_int @ ( R != zero_zero_int ) ) ) ) ).
% Parity.adjust_div_eq
thf(fact_549_div__add,axiom,
! [C2: int,A: int,B: int] :
( ( dvd_dvd_int @ C2 @ A )
=> ( ( dvd_dvd_int @ C2 @ B )
=> ( ( divide_divide_int @ ( plus_plus_int @ A @ B ) @ C2 )
= ( plus_plus_int @ ( divide_divide_int @ A @ C2 ) @ ( divide_divide_int @ B @ C2 ) ) ) ) ) ).
% div_add
thf(fact_550_div__add,axiom,
! [C2: nat,A: nat,B: nat] :
( ( dvd_dvd_nat @ C2 @ A )
=> ( ( dvd_dvd_nat @ C2 @ B )
=> ( ( divide_divide_nat @ ( plus_plus_nat @ A @ B ) @ C2 )
= ( plus_plus_nat @ ( divide_divide_nat @ A @ C2 ) @ ( divide_divide_nat @ B @ C2 ) ) ) ) ) ).
% div_add
thf(fact_551_div__diff,axiom,
! [C2: int,A: int,B: int] :
( ( dvd_dvd_int @ C2 @ A )
=> ( ( dvd_dvd_int @ C2 @ B )
=> ( ( divide_divide_int @ ( minus_minus_int @ A @ B ) @ C2 )
= ( minus_minus_int @ ( divide_divide_int @ A @ C2 ) @ ( divide_divide_int @ B @ C2 ) ) ) ) ) ).
% div_diff
thf(fact_552_unit__div,axiom,
! [A: int,B: int] :
( ( dvd_dvd_int @ A @ one_one_int )
=> ( ( dvd_dvd_int @ B @ one_one_int )
=> ( dvd_dvd_int @ ( divide_divide_int @ A @ B ) @ one_one_int ) ) ) ).
% unit_div
thf(fact_553_unit__div,axiom,
! [A: nat,B: nat] :
( ( dvd_dvd_nat @ A @ one_one_nat )
=> ( ( dvd_dvd_nat @ B @ one_one_nat )
=> ( dvd_dvd_nat @ ( divide_divide_nat @ A @ B ) @ one_one_nat ) ) ) ).
% unit_div
thf(fact_554_unit__div__1__unit,axiom,
! [A: int] :
( ( dvd_dvd_int @ A @ one_one_int )
=> ( dvd_dvd_int @ ( divide_divide_int @ one_one_int @ A ) @ one_one_int ) ) ).
% unit_div_1_unit
thf(fact_555_unit__div__1__unit,axiom,
! [A: nat] :
( ( dvd_dvd_nat @ A @ one_one_nat )
=> ( dvd_dvd_nat @ ( divide_divide_nat @ one_one_nat @ A ) @ one_one_nat ) ) ).
% unit_div_1_unit
thf(fact_556_unit__div__1__div__1,axiom,
! [A: int] :
( ( dvd_dvd_int @ A @ one_one_int )
=> ( ( divide_divide_int @ one_one_int @ ( divide_divide_int @ one_one_int @ A ) )
= A ) ) ).
% unit_div_1_div_1
thf(fact_557_unit__div__1__div__1,axiom,
! [A: nat] :
( ( dvd_dvd_nat @ A @ one_one_nat )
=> ( ( divide_divide_nat @ one_one_nat @ ( divide_divide_nat @ one_one_nat @ A ) )
= A ) ) ).
% unit_div_1_div_1
thf(fact_558_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_559_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_560_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_561_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_562_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_563_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_564_odd__of__bool__self,axiom,
! [P: $o] :
( ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( zero_n2687167440665602831ol_nat @ P ) ) )
= P ) ).
% odd_of_bool_self
thf(fact_565_odd__of__bool__self,axiom,
! [P: $o] :
( ( ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( zero_n2684676970156552555ol_int @ P ) ) )
= P ) ).
% odd_of_bool_self
thf(fact_566_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_567_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_568_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_569_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_570_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_571_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_572_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_573_of__bool__half__eq__0,axiom,
! [B: $o] :
( ( divide_divide_nat @ ( zero_n2687167440665602831ol_nat @ B ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
= zero_zero_nat ) ).
% of_bool_half_eq_0
thf(fact_574_of__bool__half__eq__0,axiom,
! [B: $o] :
( ( divide_divide_int @ ( zero_n2684676970156552555ol_int @ B ) @ ( numeral_numeral_int @ ( bit0 @ one ) ) )
= zero_zero_int ) ).
% of_bool_half_eq_0
thf(fact_575_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_576_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_577_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_578_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_579_odd__succ__div__two,axiom,
! [A: int] :
( ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A )
=> ( ( divide_divide_int @ ( plus_plus_int @ A @ one_one_int ) @ ( numeral_numeral_int @ ( bit0 @ one ) ) )
= ( plus_plus_int @ ( divide_divide_int @ A @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) @ one_one_int ) ) ) ).
% odd_succ_div_two
thf(fact_580_odd__succ__div__two,axiom,
! [A: nat] :
( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A )
=> ( ( divide_divide_nat @ ( plus_plus_nat @ A @ one_one_nat ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
= ( plus_plus_nat @ ( divide_divide_nat @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ one_one_nat ) ) ) ).
% odd_succ_div_two
thf(fact_581_even__succ__div__two,axiom,
! [A: int] :
( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A )
=> ( ( divide_divide_int @ ( plus_plus_int @ A @ one_one_int ) @ ( numeral_numeral_int @ ( bit0 @ one ) ) )
= ( divide_divide_int @ A @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ) ).
% even_succ_div_two
thf(fact_582_even__succ__div__two,axiom,
! [A: nat] :
( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A )
=> ( ( divide_divide_nat @ ( plus_plus_nat @ A @ one_one_nat ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
= ( divide_divide_nat @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ).
% even_succ_div_two
thf(fact_583_even__succ__div__2,axiom,
! [A: int] :
( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A )
=> ( ( divide_divide_int @ ( plus_plus_int @ one_one_int @ A ) @ ( numeral_numeral_int @ ( bit0 @ one ) ) )
= ( divide_divide_int @ A @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ) ).
% even_succ_div_2
thf(fact_584_even__succ__div__2,axiom,
! [A: nat] :
( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A )
=> ( ( divide_divide_nat @ ( plus_plus_nat @ one_one_nat @ A ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
= ( divide_divide_nat @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ).
% even_succ_div_2
thf(fact_585_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_586_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_587_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_588_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_589_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_590_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_591_dvd__diff__commute,axiom,
! [A: int,C2: int,B: int] :
( ( dvd_dvd_int @ A @ ( minus_minus_int @ C2 @ B ) )
= ( dvd_dvd_int @ A @ ( minus_minus_int @ B @ C2 ) ) ) ).
% dvd_diff_commute
thf(fact_592_dvd__diff,axiom,
! [X4: int,Y: int,Z2: int] :
( ( dvd_dvd_int @ X4 @ Y )
=> ( ( dvd_dvd_int @ X4 @ Z2 )
=> ( dvd_dvd_int @ X4 @ ( minus_minus_int @ Y @ Z2 ) ) ) ) ).
% dvd_diff
thf(fact_593_one__dvd,axiom,
! [A: nat] : ( dvd_dvd_nat @ one_one_nat @ A ) ).
% one_dvd
thf(fact_594_one__dvd,axiom,
! [A: int] : ( dvd_dvd_int @ one_one_int @ A ) ).
% one_dvd
thf(fact_595_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_596_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_597_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_598_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_599_dvd__div__eq__iff,axiom,
! [C2: int,A: int,B: int] :
( ( dvd_dvd_int @ C2 @ A )
=> ( ( dvd_dvd_int @ C2 @ B )
=> ( ( ( divide_divide_int @ A @ C2 )
= ( divide_divide_int @ B @ C2 ) )
= ( A = B ) ) ) ) ).
% dvd_div_eq_iff
thf(fact_600_dvd__div__eq__iff,axiom,
! [C2: nat,A: nat,B: nat] :
( ( dvd_dvd_nat @ C2 @ A )
=> ( ( dvd_dvd_nat @ C2 @ B )
=> ( ( ( divide_divide_nat @ A @ C2 )
= ( divide_divide_nat @ B @ C2 ) )
= ( A = B ) ) ) ) ).
% dvd_div_eq_iff
thf(fact_601_dvd__div__eq__cancel,axiom,
! [A: int,C2: int,B: int] :
( ( ( divide_divide_int @ A @ C2 )
= ( divide_divide_int @ B @ C2 ) )
=> ( ( dvd_dvd_int @ C2 @ A )
=> ( ( dvd_dvd_int @ C2 @ B )
=> ( A = B ) ) ) ) ).
% dvd_div_eq_cancel
thf(fact_602_dvd__div__eq__cancel,axiom,
! [A: nat,C2: nat,B: nat] :
( ( ( divide_divide_nat @ A @ C2 )
= ( divide_divide_nat @ B @ C2 ) )
=> ( ( dvd_dvd_nat @ C2 @ A )
=> ( ( dvd_dvd_nat @ C2 @ B )
=> ( A = B ) ) ) ) ).
% dvd_div_eq_cancel
thf(fact_603_div__div__div__same,axiom,
! [D: int,B: int,A: int] :
( ( dvd_dvd_int @ D @ B )
=> ( ( dvd_dvd_int @ B @ A )
=> ( ( divide_divide_int @ ( divide_divide_int @ A @ D ) @ ( divide_divide_int @ B @ D ) )
= ( divide_divide_int @ A @ B ) ) ) ) ).
% div_div_div_same
thf(fact_604_div__div__div__same,axiom,
! [D: nat,B: nat,A: nat] :
( ( dvd_dvd_nat @ D @ B )
=> ( ( dvd_dvd_nat @ B @ A )
=> ( ( divide_divide_nat @ ( divide_divide_nat @ A @ D ) @ ( divide_divide_nat @ B @ D ) )
= ( divide_divide_nat @ A @ B ) ) ) ) ).
% div_div_div_same
thf(fact_605_dvd__refl,axiom,
! [A: nat] : ( dvd_dvd_nat @ A @ A ) ).
% dvd_refl
thf(fact_606_dvd__refl,axiom,
! [A: int] : ( dvd_dvd_int @ A @ A ) ).
% dvd_refl
thf(fact_607_dvd__trans,axiom,
! [A: nat,B: nat,C2: nat] :
( ( dvd_dvd_nat @ A @ B )
=> ( ( dvd_dvd_nat @ B @ C2 )
=> ( dvd_dvd_nat @ A @ C2 ) ) ) ).
% dvd_trans
thf(fact_608_dvd__trans,axiom,
! [A: int,B: int,C2: int] :
( ( dvd_dvd_int @ A @ B )
=> ( ( dvd_dvd_int @ B @ C2 )
=> ( dvd_dvd_int @ A @ C2 ) ) ) ).
% dvd_trans
thf(fact_609_of__bool__eq__iff,axiom,
! [P: $o,Q: $o] :
( ( ( zero_n2684676970156552555ol_int @ P )
= ( zero_n2684676970156552555ol_int @ Q ) )
= ( P = Q ) ) ).
% of_bool_eq_iff
thf(fact_610_dvd__0__left,axiom,
! [A: nat] :
( ( dvd_dvd_nat @ zero_zero_nat @ A )
=> ( A = zero_zero_nat ) ) ).
% dvd_0_left
thf(fact_611_dvd__0__left,axiom,
! [A: int] :
( ( dvd_dvd_int @ zero_zero_int @ A )
=> ( A = zero_zero_int ) ) ).
% dvd_0_left
thf(fact_612_push__bit__int__code_I1_J,axiom,
! [I: int] :
( ( bit_se545348938243370406it_int @ zero_zero_nat @ I )
= I ) ).
% push_bit_int_code(1)
thf(fact_613_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_614_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_615_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_616_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_617_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_618_of__bool__def,axiom,
( zero_n2687167440665602831ol_nat
= ( ^ [P4: $o] : ( if_nat @ P4 @ one_one_nat @ zero_zero_nat ) ) ) ).
% of_bool_def
thf(fact_619_of__bool__def,axiom,
( zero_n2684676970156552555ol_int
= ( ^ [P4: $o] : ( if_int @ P4 @ one_one_int @ zero_zero_int ) ) ) ).
% of_bool_def
thf(fact_620_split__of__bool,axiom,
! [P2: nat > $o,P: $o] :
( ( P2 @ ( zero_n2687167440665602831ol_nat @ P ) )
= ( ( P
=> ( P2 @ one_one_nat ) )
& ( ~ P
=> ( P2 @ zero_zero_nat ) ) ) ) ).
% split_of_bool
thf(fact_621_split__of__bool,axiom,
! [P2: int > $o,P: $o] :
( ( P2 @ ( zero_n2684676970156552555ol_int @ P ) )
= ( ( P
=> ( P2 @ one_one_int ) )
& ( ~ P
=> ( P2 @ zero_zero_int ) ) ) ) ).
% split_of_bool
thf(fact_622_split__of__bool__asm,axiom,
! [P2: nat > $o,P: $o] :
( ( P2 @ ( zero_n2687167440665602831ol_nat @ P ) )
= ( ~ ( ( P
& ~ ( P2 @ one_one_nat ) )
| ( ~ P
& ~ ( P2 @ zero_zero_nat ) ) ) ) ) ).
% split_of_bool_asm
thf(fact_623_split__of__bool__asm,axiom,
! [P2: int > $o,P: $o] :
( ( P2 @ ( zero_n2684676970156552555ol_int @ P ) )
= ( ~ ( ( P
& ~ ( P2 @ one_one_int ) )
| ( ~ P
& ~ ( P2 @ zero_zero_int ) ) ) ) ) ).
% split_of_bool_asm
thf(fact_624_of__bool__not__iff,axiom,
! [P2: $o] :
( ( zero_n2684676970156552555ol_int @ ~ P2 )
= ( minus_minus_int @ one_one_int @ ( zero_n2684676970156552555ol_int @ P2 ) ) ) ).
% of_bool_not_iff
thf(fact_625_not__is__unit__0,axiom,
~ ( dvd_dvd_nat @ zero_zero_nat @ one_one_nat ) ).
% not_is_unit_0
thf(fact_626_not__is__unit__0,axiom,
~ ( dvd_dvd_int @ zero_zero_int @ one_one_int ) ).
% not_is_unit_0
thf(fact_627_dvd__div__eq__0__iff,axiom,
! [B: int,A: int] :
( ( dvd_dvd_int @ B @ A )
=> ( ( ( divide_divide_int @ A @ B )
= zero_zero_int )
= ( A = zero_zero_int ) ) ) ).
% dvd_div_eq_0_iff
thf(fact_628_dvd__div__eq__0__iff,axiom,
! [B: nat,A: nat] :
( ( dvd_dvd_nat @ B @ A )
=> ( ( ( divide_divide_nat @ A @ B )
= zero_zero_nat )
= ( A = zero_zero_nat ) ) ) ).
% dvd_div_eq_0_iff
thf(fact_629_div__plus__div__distrib__dvd__right,axiom,
! [C2: int,B: int,A: int] :
( ( dvd_dvd_int @ C2 @ B )
=> ( ( divide_divide_int @ ( plus_plus_int @ A @ B ) @ C2 )
= ( plus_plus_int @ ( divide_divide_int @ A @ C2 ) @ ( divide_divide_int @ B @ C2 ) ) ) ) ).
% div_plus_div_distrib_dvd_right
thf(fact_630_div__plus__div__distrib__dvd__right,axiom,
! [C2: nat,B: nat,A: nat] :
( ( dvd_dvd_nat @ C2 @ B )
=> ( ( divide_divide_nat @ ( plus_plus_nat @ A @ B ) @ C2 )
= ( plus_plus_nat @ ( divide_divide_nat @ A @ C2 ) @ ( divide_divide_nat @ B @ C2 ) ) ) ) ).
% div_plus_div_distrib_dvd_right
thf(fact_631_div__plus__div__distrib__dvd__left,axiom,
! [C2: int,A: int,B: int] :
( ( dvd_dvd_int @ C2 @ A )
=> ( ( divide_divide_int @ ( plus_plus_int @ A @ B ) @ C2 )
= ( plus_plus_int @ ( divide_divide_int @ A @ C2 ) @ ( divide_divide_int @ B @ C2 ) ) ) ) ).
% div_plus_div_distrib_dvd_left
thf(fact_632_div__plus__div__distrib__dvd__left,axiom,
! [C2: nat,A: nat,B: nat] :
( ( dvd_dvd_nat @ C2 @ A )
=> ( ( divide_divide_nat @ ( plus_plus_nat @ A @ B ) @ C2 )
= ( plus_plus_nat @ ( divide_divide_nat @ A @ C2 ) @ ( divide_divide_nat @ B @ C2 ) ) ) ) ).
% div_plus_div_distrib_dvd_left
thf(fact_633_unit__div__cancel,axiom,
! [A: int,B: int,C2: int] :
( ( dvd_dvd_int @ A @ one_one_int )
=> ( ( ( divide_divide_int @ B @ A )
= ( divide_divide_int @ C2 @ A ) )
= ( B = C2 ) ) ) ).
% unit_div_cancel
thf(fact_634_unit__div__cancel,axiom,
! [A: nat,B: nat,C2: nat] :
( ( dvd_dvd_nat @ A @ one_one_nat )
=> ( ( ( divide_divide_nat @ B @ A )
= ( divide_divide_nat @ C2 @ A ) )
= ( B = C2 ) ) ) ).
% unit_div_cancel
thf(fact_635_div__unit__dvd__iff,axiom,
! [B: int,A: int,C2: int] :
( ( dvd_dvd_int @ B @ one_one_int )
=> ( ( dvd_dvd_int @ ( divide_divide_int @ A @ B ) @ C2 )
= ( dvd_dvd_int @ A @ C2 ) ) ) ).
% div_unit_dvd_iff
thf(fact_636_div__unit__dvd__iff,axiom,
! [B: nat,A: nat,C2: nat] :
( ( dvd_dvd_nat @ B @ one_one_nat )
=> ( ( dvd_dvd_nat @ ( divide_divide_nat @ A @ B ) @ C2 )
= ( dvd_dvd_nat @ A @ C2 ) ) ) ).
% div_unit_dvd_iff
thf(fact_637_dvd__div__unit__iff,axiom,
! [B: int,A: int,C2: int] :
( ( dvd_dvd_int @ B @ one_one_int )
=> ( ( dvd_dvd_int @ A @ ( divide_divide_int @ C2 @ B ) )
= ( dvd_dvd_int @ A @ C2 ) ) ) ).
% dvd_div_unit_iff
thf(fact_638_dvd__div__unit__iff,axiom,
! [B: nat,A: nat,C2: nat] :
( ( dvd_dvd_nat @ B @ one_one_nat )
=> ( ( dvd_dvd_nat @ A @ ( divide_divide_nat @ C2 @ B ) )
= ( dvd_dvd_nat @ A @ C2 ) ) ) ).
% dvd_div_unit_iff
thf(fact_639_dvd__div__neg,axiom,
! [B: int,A: int] :
( ( dvd_dvd_int @ B @ A )
=> ( ( divide_divide_int @ A @ ( uminus_uminus_int @ B ) )
= ( uminus_uminus_int @ ( divide_divide_int @ A @ B ) ) ) ) ).
% dvd_div_neg
thf(fact_640_dvd__neg__div,axiom,
! [B: int,A: int] :
( ( dvd_dvd_int @ B @ A )
=> ( ( divide_divide_int @ ( uminus_uminus_int @ A ) @ B )
= ( uminus_uminus_int @ ( divide_divide_int @ A @ B ) ) ) ) ).
% dvd_neg_div
thf(fact_641_div__power,axiom,
! [B: int,A: int,N: nat] :
( ( dvd_dvd_int @ B @ A )
=> ( ( power_power_int @ ( divide_divide_int @ A @ B ) @ N )
= ( divide_divide_int @ ( power_power_int @ A @ N ) @ ( power_power_int @ B @ N ) ) ) ) ).
% div_power
thf(fact_642_div__power,axiom,
! [B: nat,A: nat,N: nat] :
( ( dvd_dvd_nat @ B @ A )
=> ( ( power_power_nat @ ( divide_divide_nat @ A @ B ) @ N )
= ( divide_divide_nat @ ( power_power_nat @ A @ N ) @ ( power_power_nat @ B @ N ) ) ) ) ).
% div_power
thf(fact_643_pow_Osimps_I1_J,axiom,
! [X4: num] :
( ( pow @ X4 @ one )
= X4 ) ).
% pow.simps(1)
thf(fact_644_unit__div__eq__0__iff,axiom,
! [B: int,A: int] :
( ( dvd_dvd_int @ B @ one_one_int )
=> ( ( ( divide_divide_int @ A @ B )
= zero_zero_int )
= ( A = zero_zero_int ) ) ) ).
% unit_div_eq_0_iff
thf(fact_645_unit__div__eq__0__iff,axiom,
! [B: nat,A: nat] :
( ( dvd_dvd_nat @ B @ one_one_nat )
=> ( ( ( divide_divide_nat @ A @ B )
= zero_zero_nat )
= ( A = zero_zero_nat ) ) ) ).
% unit_div_eq_0_iff
thf(fact_646_even__numeral,axiom,
! [N: num] : ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( numeral_numeral_nat @ ( bit0 @ N ) ) ) ).
% even_numeral
thf(fact_647_even__numeral,axiom,
! [N: num] : ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( numeral_numeral_int @ ( bit0 @ N ) ) ) ).
% even_numeral
thf(fact_648_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_649_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_650_even__zero,axiom,
dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ zero_zero_nat ).
% even_zero
thf(fact_651_even__zero,axiom,
dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ zero_zero_int ).
% even_zero
thf(fact_652_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_653_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_654_odd__one,axiom,
~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ one_one_nat ) ).
% odd_one
thf(fact_655_odd__one,axiom,
~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ one_one_int ) ).
% odd_one
thf(fact_656_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_657_bit__eq__rec,axiom,
( ( ^ [Y5: int,Z3: int] : ( Y5 = Z3 ) )
= ( ^ [A4: int,B4: int] :
( ( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A4 )
= ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ B4 ) )
& ( ( divide_divide_int @ A4 @ ( numeral_numeral_int @ ( bit0 @ one ) ) )
= ( divide_divide_int @ B4 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ) ) ) ).
% bit_eq_rec
thf(fact_658_bit__eq__rec,axiom,
( ( ^ [Y5: nat,Z3: nat] : ( Y5 = Z3 ) )
= ( ^ [A4: nat,B4: nat] :
( ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A4 )
= ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ B4 ) )
& ( ( divide_divide_nat @ A4 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
= ( divide_divide_nat @ B4 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ).
% bit_eq_rec
thf(fact_659_odd__numeral,axiom,
! [N: num] :
~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( numeral_numeral_nat @ ( bit1 @ N ) ) ) ).
% odd_numeral
thf(fact_660_odd__numeral,axiom,
! [N: num] :
~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( numeral_numeral_int @ ( bit1 @ N ) ) ) ).
% odd_numeral
thf(fact_661_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_662_odd__numeral__BitM,axiom,
! [W: num] :
~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( numeral_numeral_nat @ ( bitM @ W ) ) ) ).
% odd_numeral_BitM
thf(fact_663_odd__numeral__BitM,axiom,
! [W: num] :
~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( numeral_numeral_int @ ( bitM @ W ) ) ) ).
% odd_numeral_BitM
thf(fact_664_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_665_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_666_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_667_even__succ__div__exp,axiom,
! [A: int,N: nat] :
( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A )
=> ( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( divide_divide_int @ ( plus_plus_int @ one_one_int @ A ) @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) )
= ( divide_divide_int @ A @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) ) ) ) ) ).
% even_succ_div_exp
thf(fact_668_even__succ__div__exp,axiom,
! [A: nat,N: nat] :
( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A )
=> ( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( divide_divide_nat @ ( plus_plus_nat @ one_one_nat @ A ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) )
= ( divide_divide_nat @ A @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ) ) ) ).
% even_succ_div_exp
thf(fact_669_even__set__bit__iff,axiom,
! [M: nat,A: nat] :
( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( bit_se7882103937844011126it_nat @ M @ A ) )
= ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A )
& ( M != zero_zero_nat ) ) ) ).
% even_set_bit_iff
thf(fact_670_even__set__bit__iff,axiom,
! [M: nat,A: int] :
( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( bit_se7879613467334960850it_int @ M @ A ) )
= ( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A )
& ( M != zero_zero_nat ) ) ) ).
% even_set_bit_iff
thf(fact_671_even__flip__bit__iff,axiom,
! [M: nat,A: nat] :
( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( bit_se2161824704523386999it_nat @ M @ A ) )
= ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A )
!= ( M = zero_zero_nat ) ) ) ).
% even_flip_bit_iff
thf(fact_672_even__flip__bit__iff,axiom,
! [M: nat,A: int] :
( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( bit_se2159334234014336723it_int @ M @ A ) )
= ( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A )
!= ( M = zero_zero_nat ) ) ) ).
% even_flip_bit_iff
thf(fact_673_even__unset__bit__iff,axiom,
! [M: nat,A: nat] :
( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( bit_se4205575877204974255it_nat @ M @ A ) )
= ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A )
| ( M = zero_zero_nat ) ) ) ).
% even_unset_bit_iff
thf(fact_674_even__unset__bit__iff,axiom,
! [M: nat,A: int] :
( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( bit_se4203085406695923979it_int @ M @ A ) )
= ( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A )
| ( M = zero_zero_nat ) ) ) ).
% even_unset_bit_iff
thf(fact_675_even__mask__div__iff,axiom,
! [M: nat,N: nat] :
( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( divide_divide_int @ ( minus_minus_int @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ M ) @ one_one_int ) @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) ) )
= ( ( ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N )
= zero_zero_int )
| ( ord_less_eq_nat @ M @ N ) ) ) ).
% even_mask_div_iff
thf(fact_676_even__mask__div__iff,axiom,
! [M: nat,N: nat] :
( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( minus_minus_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M ) @ one_one_nat ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) )
= ( ( ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
= zero_zero_nat )
| ( ord_less_eq_nat @ M @ N ) ) ) ).
% even_mask_div_iff
thf(fact_677_le__zero__eq,axiom,
! [N: nat] :
( ( ord_less_eq_nat @ N @ zero_zero_nat )
= ( N = zero_zero_nat ) ) ).
% le_zero_eq
thf(fact_678_not__gr__zero,axiom,
! [N: nat] :
( ( ~ ( ord_less_nat @ zero_zero_nat @ N ) )
= ( N = zero_zero_nat ) ) ).
% not_gr_zero
thf(fact_679_add__le__cancel__right,axiom,
! [A: nat,C2: nat,B: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ A @ C2 ) @ ( plus_plus_nat @ B @ C2 ) )
= ( ord_less_eq_nat @ A @ B ) ) ).
% add_le_cancel_right
thf(fact_680_add__le__cancel__right,axiom,
! [A: int,C2: int,B: int] :
( ( ord_less_eq_int @ ( plus_plus_int @ A @ C2 ) @ ( plus_plus_int @ B @ C2 ) )
= ( ord_less_eq_int @ A @ B ) ) ).
% add_le_cancel_right
thf(fact_681_add__le__cancel__left,axiom,
! [C2: nat,A: nat,B: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ C2 @ A ) @ ( plus_plus_nat @ C2 @ B ) )
= ( ord_less_eq_nat @ A @ B ) ) ).
% add_le_cancel_left
thf(fact_682_add__le__cancel__left,axiom,
! [C2: int,A: int,B: int] :
( ( ord_less_eq_int @ ( plus_plus_int @ C2 @ A ) @ ( plus_plus_int @ C2 @ B ) )
= ( ord_less_eq_int @ A @ B ) ) ).
% add_le_cancel_left
thf(fact_683_numeral__le__iff,axiom,
! [M: num,N: num] :
( ( ord_less_eq_nat @ ( numeral_numeral_nat @ M ) @ ( numeral_numeral_nat @ N ) )
= ( ord_less_eq_num @ M @ N ) ) ).
% numeral_le_iff
thf(fact_684_numeral__le__iff,axiom,
! [M: num,N: num] :
( ( ord_less_eq_int @ ( numeral_numeral_int @ M ) @ ( numeral_numeral_int @ N ) )
= ( ord_less_eq_num @ M @ N ) ) ).
% numeral_le_iff
thf(fact_685_add__less__cancel__right,axiom,
! [A: nat,C2: nat,B: nat] :
( ( ord_less_nat @ ( plus_plus_nat @ A @ C2 ) @ ( plus_plus_nat @ B @ C2 ) )
= ( ord_less_nat @ A @ B ) ) ).
% add_less_cancel_right
thf(fact_686_add__less__cancel__right,axiom,
! [A: int,C2: int,B: int] :
( ( ord_less_int @ ( plus_plus_int @ A @ C2 ) @ ( plus_plus_int @ B @ C2 ) )
= ( ord_less_int @ A @ B ) ) ).
% add_less_cancel_right
thf(fact_687_add__less__cancel__left,axiom,
! [C2: nat,A: nat,B: nat] :
( ( ord_less_nat @ ( plus_plus_nat @ C2 @ A ) @ ( plus_plus_nat @ C2 @ B ) )
= ( ord_less_nat @ A @ B ) ) ).
% add_less_cancel_left
thf(fact_688_add__less__cancel__left,axiom,
! [C2: int,A: int,B: int] :
( ( ord_less_int @ ( plus_plus_int @ C2 @ A ) @ ( plus_plus_int @ C2 @ B ) )
= ( ord_less_int @ A @ B ) ) ).
% add_less_cancel_left
thf(fact_689_numeral__less__iff,axiom,
! [M: num,N: num] :
( ( ord_less_nat @ ( numeral_numeral_nat @ M ) @ ( numeral_numeral_nat @ N ) )
= ( ord_less_num @ M @ N ) ) ).
% numeral_less_iff
thf(fact_690_numeral__less__iff,axiom,
! [M: num,N: num] :
( ( ord_less_int @ ( numeral_numeral_int @ M ) @ ( numeral_numeral_int @ N ) )
= ( ord_less_num @ M @ N ) ) ).
% numeral_less_iff
thf(fact_691_neg__le__iff__le,axiom,
! [B: int,A: int] :
( ( ord_less_eq_int @ ( uminus_uminus_int @ B ) @ ( uminus_uminus_int @ A ) )
= ( ord_less_eq_int @ A @ B ) ) ).
% neg_le_iff_le
thf(fact_692_neg__less__iff__less,axiom,
! [B: int,A: int] :
( ( ord_less_int @ ( uminus_uminus_int @ B ) @ ( uminus_uminus_int @ A ) )
= ( ord_less_int @ A @ B ) ) ).
% neg_less_iff_less
thf(fact_693_bot__nat__0_Onot__eq__extremum,axiom,
! [A: nat] :
( ( A != zero_zero_nat )
= ( ord_less_nat @ zero_zero_nat @ A ) ) ).
% bot_nat_0.not_eq_extremum
thf(fact_694_neq0__conv,axiom,
! [N: nat] :
( ( N != zero_zero_nat )
= ( ord_less_nat @ zero_zero_nat @ N ) ) ).
% neq0_conv
thf(fact_695_less__nat__zero__code,axiom,
! [N: nat] :
~ ( ord_less_nat @ N @ zero_zero_nat ) ).
% less_nat_zero_code
thf(fact_696_nat__zero__less__power__iff,axiom,
! [X4: nat,N: nat] :
( ( ord_less_nat @ zero_zero_nat @ ( power_power_nat @ X4 @ N ) )
= ( ( ord_less_nat @ zero_zero_nat @ X4 )
| ( N = zero_zero_nat ) ) ) ).
% nat_zero_less_power_iff
thf(fact_697_bot__nat__0_Oextremum,axiom,
! [A: nat] : ( ord_less_eq_nat @ zero_zero_nat @ A ) ).
% bot_nat_0.extremum
thf(fact_698_le0,axiom,
! [N: nat] : ( ord_less_eq_nat @ zero_zero_nat @ N ) ).
% le0
thf(fact_699_of__bool__less__eq__iff,axiom,
! [P2: $o,Q2: $o] :
( ( ord_less_eq_nat @ ( zero_n2687167440665602831ol_nat @ P2 ) @ ( zero_n2687167440665602831ol_nat @ Q2 ) )
= ( P2
=> Q2 ) ) ).
% of_bool_less_eq_iff
thf(fact_700_of__bool__less__eq__iff,axiom,
! [P2: $o,Q2: $o] :
( ( ord_less_eq_int @ ( zero_n2684676970156552555ol_int @ P2 ) @ ( zero_n2684676970156552555ol_int @ Q2 ) )
= ( P2
=> Q2 ) ) ).
% of_bool_less_eq_iff
thf(fact_701_of__bool__less__iff,axiom,
! [P2: $o,Q2: $o] :
( ( ord_less_nat @ ( zero_n2687167440665602831ol_nat @ P2 ) @ ( zero_n2687167440665602831ol_nat @ Q2 ) )
= ( ~ P2
& Q2 ) ) ).
% of_bool_less_iff
thf(fact_702_of__bool__less__iff,axiom,
! [P2: $o,Q2: $o] :
( ( ord_less_int @ ( zero_n2684676970156552555ol_int @ P2 ) @ ( zero_n2684676970156552555ol_int @ Q2 ) )
= ( ~ P2
& Q2 ) ) ).
% of_bool_less_iff
thf(fact_703_zero__le__double__add__iff__zero__le__single__add,axiom,
! [A: int] :
( ( ord_less_eq_int @ zero_zero_int @ ( plus_plus_int @ A @ A ) )
= ( ord_less_eq_int @ zero_zero_int @ A ) ) ).
% zero_le_double_add_iff_zero_le_single_add
thf(fact_704_double__add__le__zero__iff__single__add__le__zero,axiom,
! [A: int] :
( ( ord_less_eq_int @ ( plus_plus_int @ A @ A ) @ zero_zero_int )
= ( ord_less_eq_int @ A @ zero_zero_int ) ) ).
% double_add_le_zero_iff_single_add_le_zero
thf(fact_705_le__add__same__cancel2,axiom,
! [A: nat,B: nat] :
( ( ord_less_eq_nat @ A @ ( plus_plus_nat @ B @ A ) )
= ( ord_less_eq_nat @ zero_zero_nat @ B ) ) ).
% le_add_same_cancel2
thf(fact_706_le__add__same__cancel2,axiom,
! [A: int,B: int] :
( ( ord_less_eq_int @ A @ ( plus_plus_int @ B @ A ) )
= ( ord_less_eq_int @ zero_zero_int @ B ) ) ).
% le_add_same_cancel2
thf(fact_707_le__add__same__cancel1,axiom,
! [A: nat,B: nat] :
( ( ord_less_eq_nat @ A @ ( plus_plus_nat @ A @ B ) )
= ( ord_less_eq_nat @ zero_zero_nat @ B ) ) ).
% le_add_same_cancel1
thf(fact_708_le__add__same__cancel1,axiom,
! [A: int,B: int] :
( ( ord_less_eq_int @ A @ ( plus_plus_int @ A @ B ) )
= ( ord_less_eq_int @ zero_zero_int @ B ) ) ).
% le_add_same_cancel1
thf(fact_709_add__le__same__cancel2,axiom,
! [A: nat,B: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ A @ B ) @ B )
= ( ord_less_eq_nat @ A @ zero_zero_nat ) ) ).
% add_le_same_cancel2
thf(fact_710_add__le__same__cancel2,axiom,
! [A: int,B: int] :
( ( ord_less_eq_int @ ( plus_plus_int @ A @ B ) @ B )
= ( ord_less_eq_int @ A @ zero_zero_int ) ) ).
% add_le_same_cancel2
thf(fact_711_add__le__same__cancel1,axiom,
! [B: nat,A: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ B @ A ) @ B )
= ( ord_less_eq_nat @ A @ zero_zero_nat ) ) ).
% add_le_same_cancel1
thf(fact_712_add__le__same__cancel1,axiom,
! [B: int,A: int] :
( ( ord_less_eq_int @ ( plus_plus_int @ B @ A ) @ B )
= ( ord_less_eq_int @ A @ zero_zero_int ) ) ).
% add_le_same_cancel1
thf(fact_713_diff__ge__0__iff__ge,axiom,
! [A: int,B: int] :
( ( ord_less_eq_int @ zero_zero_int @ ( minus_minus_int @ A @ B ) )
= ( ord_less_eq_int @ B @ A ) ) ).
% diff_ge_0_iff_ge
thf(fact_714_zero__less__double__add__iff__zero__less__single__add,axiom,
! [A: int] :
( ( ord_less_int @ zero_zero_int @ ( plus_plus_int @ A @ A ) )
= ( ord_less_int @ zero_zero_int @ A ) ) ).
% zero_less_double_add_iff_zero_less_single_add
thf(fact_715_double__add__less__zero__iff__single__add__less__zero,axiom,
! [A: int] :
( ( ord_less_int @ ( plus_plus_int @ A @ A ) @ zero_zero_int )
= ( ord_less_int @ A @ zero_zero_int ) ) ).
% double_add_less_zero_iff_single_add_less_zero
thf(fact_716_less__add__same__cancel2,axiom,
! [A: nat,B: nat] :
( ( ord_less_nat @ A @ ( plus_plus_nat @ B @ A ) )
= ( ord_less_nat @ zero_zero_nat @ B ) ) ).
% less_add_same_cancel2
thf(fact_717_less__add__same__cancel2,axiom,
! [A: int,B: int] :
( ( ord_less_int @ A @ ( plus_plus_int @ B @ A ) )
= ( ord_less_int @ zero_zero_int @ B ) ) ).
% less_add_same_cancel2
thf(fact_718_less__add__same__cancel1,axiom,
! [A: nat,B: nat] :
( ( ord_less_nat @ A @ ( plus_plus_nat @ A @ B ) )
= ( ord_less_nat @ zero_zero_nat @ B ) ) ).
% less_add_same_cancel1
thf(fact_719_less__add__same__cancel1,axiom,
! [A: int,B: int] :
( ( ord_less_int @ A @ ( plus_plus_int @ A @ B ) )
= ( ord_less_int @ zero_zero_int @ B ) ) ).
% less_add_same_cancel1
thf(fact_720_add__less__same__cancel2,axiom,
! [A: nat,B: nat] :
( ( ord_less_nat @ ( plus_plus_nat @ A @ B ) @ B )
= ( ord_less_nat @ A @ zero_zero_nat ) ) ).
% add_less_same_cancel2
thf(fact_721_add__less__same__cancel2,axiom,
! [A: int,B: int] :
( ( ord_less_int @ ( plus_plus_int @ A @ B ) @ B )
= ( ord_less_int @ A @ zero_zero_int ) ) ).
% add_less_same_cancel2
thf(fact_722_add__less__same__cancel1,axiom,
! [B: nat,A: nat] :
( ( ord_less_nat @ ( plus_plus_nat @ B @ A ) @ B )
= ( ord_less_nat @ A @ zero_zero_nat ) ) ).
% add_less_same_cancel1
thf(fact_723_add__less__same__cancel1,axiom,
! [B: int,A: int] :
( ( ord_less_int @ ( plus_plus_int @ B @ A ) @ B )
= ( ord_less_int @ A @ zero_zero_int ) ) ).
% add_less_same_cancel1
thf(fact_724_diff__gt__0__iff__gt,axiom,
! [A: int,B: int] :
( ( ord_less_int @ zero_zero_int @ ( minus_minus_int @ A @ B ) )
= ( ord_less_int @ B @ A ) ) ).
% diff_gt_0_iff_gt
thf(fact_725_neg__less__eq__nonneg,axiom,
! [A: int] :
( ( ord_less_eq_int @ ( uminus_uminus_int @ A ) @ A )
= ( ord_less_eq_int @ zero_zero_int @ A ) ) ).
% neg_less_eq_nonneg
thf(fact_726_less__eq__neg__nonpos,axiom,
! [A: int] :
( ( ord_less_eq_int @ A @ ( uminus_uminus_int @ A ) )
= ( ord_less_eq_int @ A @ zero_zero_int ) ) ).
% less_eq_neg_nonpos
thf(fact_727_neg__le__0__iff__le,axiom,
! [A: int] :
( ( ord_less_eq_int @ ( uminus_uminus_int @ A ) @ zero_zero_int )
= ( ord_less_eq_int @ zero_zero_int @ A ) ) ).
% neg_le_0_iff_le
thf(fact_728_neg__0__le__iff__le,axiom,
! [A: int] :
( ( ord_less_eq_int @ zero_zero_int @ ( uminus_uminus_int @ A ) )
= ( ord_less_eq_int @ A @ zero_zero_int ) ) ).
% neg_0_le_iff_le
thf(fact_729_le__add__diff__inverse,axiom,
! [B: nat,A: nat] :
( ( ord_less_eq_nat @ B @ A )
=> ( ( plus_plus_nat @ B @ ( minus_minus_nat @ A @ B ) )
= A ) ) ).
% le_add_diff_inverse
thf(fact_730_le__add__diff__inverse,axiom,
! [B: int,A: int] :
( ( ord_less_eq_int @ B @ A )
=> ( ( plus_plus_int @ B @ ( minus_minus_int @ A @ B ) )
= A ) ) ).
% le_add_diff_inverse
thf(fact_731_le__add__diff__inverse2,axiom,
! [B: nat,A: nat] :
( ( ord_less_eq_nat @ B @ A )
=> ( ( plus_plus_nat @ ( minus_minus_nat @ A @ B ) @ B )
= A ) ) ).
% le_add_diff_inverse2
thf(fact_732_le__add__diff__inverse2,axiom,
! [B: int,A: int] :
( ( ord_less_eq_int @ B @ A )
=> ( ( plus_plus_int @ ( minus_minus_int @ A @ B ) @ B )
= A ) ) ).
% le_add_diff_inverse2
thf(fact_733_neg__less__0__iff__less,axiom,
! [A: int] :
( ( ord_less_int @ ( uminus_uminus_int @ A ) @ zero_zero_int )
= ( ord_less_int @ zero_zero_int @ A ) ) ).
% neg_less_0_iff_less
thf(fact_734_neg__0__less__iff__less,axiom,
! [A: int] :
( ( ord_less_int @ zero_zero_int @ ( uminus_uminus_int @ A ) )
= ( ord_less_int @ A @ zero_zero_int ) ) ).
% neg_0_less_iff_less
thf(fact_735_neg__less__pos,axiom,
! [A: int] :
( ( ord_less_int @ ( uminus_uminus_int @ A ) @ A )
= ( ord_less_int @ zero_zero_int @ A ) ) ).
% neg_less_pos
thf(fact_736_less__neg__neg,axiom,
! [A: int] :
( ( ord_less_int @ A @ ( uminus_uminus_int @ A ) )
= ( ord_less_int @ A @ zero_zero_int ) ) ).
% less_neg_neg
thf(fact_737_neg__numeral__le__iff,axiom,
! [M: num,N: num] :
( ( ord_less_eq_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ M ) ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ N ) ) )
= ( ord_less_eq_num @ N @ M ) ) ).
% neg_numeral_le_iff
thf(fact_738_neg__numeral__less__iff,axiom,
! [M: num,N: num] :
( ( ord_less_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ M ) ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ N ) ) )
= ( ord_less_num @ N @ M ) ) ).
% neg_numeral_less_iff
thf(fact_739_power__inject__exp,axiom,
! [A: nat,M: nat,N: nat] :
( ( ord_less_nat @ one_one_nat @ A )
=> ( ( ( power_power_nat @ A @ M )
= ( power_power_nat @ A @ N ) )
= ( M = N ) ) ) ).
% power_inject_exp
thf(fact_740_power__inject__exp,axiom,
! [A: int,M: nat,N: nat] :
( ( ord_less_int @ one_one_int @ A )
=> ( ( ( power_power_int @ A @ M )
= ( power_power_int @ A @ N ) )
= ( M = N ) ) ) ).
% power_inject_exp
thf(fact_741_add__gr__0,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ zero_zero_nat @ ( plus_plus_nat @ M @ N ) )
= ( ( ord_less_nat @ zero_zero_nat @ M )
| ( ord_less_nat @ zero_zero_nat @ N ) ) ) ).
% add_gr_0
thf(fact_742_zero__less__of__bool__iff,axiom,
! [P2: $o] :
( ( ord_less_nat @ zero_zero_nat @ ( zero_n2687167440665602831ol_nat @ P2 ) )
= P2 ) ).
% zero_less_of_bool_iff
thf(fact_743_zero__less__of__bool__iff,axiom,
! [P2: $o] :
( ( ord_less_int @ zero_zero_int @ ( zero_n2684676970156552555ol_int @ P2 ) )
= P2 ) ).
% zero_less_of_bool_iff
thf(fact_744_zero__less__diff,axiom,
! [N: nat,M: nat] :
( ( ord_less_nat @ zero_zero_nat @ ( minus_minus_nat @ N @ M ) )
= ( ord_less_nat @ M @ N ) ) ).
% zero_less_diff
thf(fact_745_diff__is__0__eq,axiom,
! [M: nat,N: nat] :
( ( ( minus_minus_nat @ M @ N )
= zero_zero_nat )
= ( ord_less_eq_nat @ M @ N ) ) ).
% diff_is_0_eq
thf(fact_746_diff__is__0__eq_H,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq_nat @ M @ N )
=> ( ( minus_minus_nat @ M @ N )
= zero_zero_nat ) ) ).
% diff_is_0_eq'
thf(fact_747_of__bool__less__one__iff,axiom,
! [P2: $o] :
( ( ord_less_nat @ ( zero_n2687167440665602831ol_nat @ P2 ) @ one_one_nat )
= ~ P2 ) ).
% of_bool_less_one_iff
thf(fact_748_of__bool__less__one__iff,axiom,
! [P2: $o] :
( ( ord_less_int @ ( zero_n2684676970156552555ol_int @ P2 ) @ one_one_int )
= ~ P2 ) ).
% of_bool_less_one_iff
thf(fact_749_less__one,axiom,
! [N: nat] :
( ( ord_less_nat @ N @ one_one_nat )
= ( N = zero_zero_nat ) ) ).
% less_one
thf(fact_750_div__less,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ M @ N )
=> ( ( divide_divide_nat @ M @ N )
= zero_zero_nat ) ) ).
% div_less
thf(fact_751_numeral__le__one__iff,axiom,
! [N: num] :
( ( ord_less_eq_nat @ ( numeral_numeral_nat @ N ) @ one_one_nat )
= ( ord_less_eq_num @ N @ one ) ) ).
% numeral_le_one_iff
thf(fact_752_numeral__le__one__iff,axiom,
! [N: num] :
( ( ord_less_eq_int @ ( numeral_numeral_int @ N ) @ one_one_int )
= ( ord_less_eq_num @ N @ one ) ) ).
% numeral_le_one_iff
thf(fact_753_one__less__numeral__iff,axiom,
! [N: num] :
( ( ord_less_nat @ one_one_nat @ ( numeral_numeral_nat @ N ) )
= ( ord_less_num @ one @ N ) ) ).
% one_less_numeral_iff
thf(fact_754_one__less__numeral__iff,axiom,
! [N: num] :
( ( ord_less_int @ one_one_int @ ( numeral_numeral_int @ N ) )
= ( ord_less_num @ one @ N ) ) ).
% one_less_numeral_iff
thf(fact_755_power__eq__0__iff,axiom,
! [A: int,N: nat] :
( ( ( power_power_int @ A @ N )
= zero_zero_int )
= ( ( A = zero_zero_int )
& ( ord_less_nat @ zero_zero_nat @ N ) ) ) ).
% power_eq_0_iff
thf(fact_756_power__eq__0__iff,axiom,
! [A: nat,N: nat] :
( ( ( power_power_nat @ A @ N )
= zero_zero_nat )
= ( ( A = zero_zero_nat )
& ( ord_less_nat @ zero_zero_nat @ N ) ) ) ).
% power_eq_0_iff
thf(fact_757_power__strict__increasing__iff,axiom,
! [B: nat,X4: nat,Y: nat] :
( ( ord_less_nat @ one_one_nat @ B )
=> ( ( ord_less_nat @ ( power_power_nat @ B @ X4 ) @ ( power_power_nat @ B @ Y ) )
= ( ord_less_nat @ X4 @ Y ) ) ) ).
% power_strict_increasing_iff
thf(fact_758_power__strict__increasing__iff,axiom,
! [B: int,X4: nat,Y: nat] :
( ( ord_less_int @ one_one_int @ B )
=> ( ( ord_less_int @ ( power_power_int @ B @ X4 ) @ ( power_power_int @ B @ Y ) )
= ( ord_less_nat @ X4 @ Y ) ) ) ).
% power_strict_increasing_iff
thf(fact_759_not__neg__one__le__neg__numeral__iff,axiom,
! [M: num] :
( ( ~ ( ord_less_eq_int @ ( uminus_uminus_int @ one_one_int ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ M ) ) ) )
= ( M != one ) ) ).
% not_neg_one_le_neg_numeral_iff
thf(fact_760_neg__numeral__less__neg__one__iff,axiom,
! [M: num] :
( ( ord_less_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ M ) ) @ ( uminus_uminus_int @ one_one_int ) )
= ( M != one ) ) ).
% neg_numeral_less_neg_one_iff
thf(fact_761_power__mono__iff,axiom,
! [A: nat,B: nat,N: nat] :
( ( ord_less_eq_nat @ zero_zero_nat @ A )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ B )
=> ( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( ord_less_eq_nat @ ( power_power_nat @ A @ N ) @ ( power_power_nat @ B @ N ) )
= ( ord_less_eq_nat @ A @ B ) ) ) ) ) ).
% power_mono_iff
thf(fact_762_power__mono__iff,axiom,
! [A: int,B: int,N: nat] :
( ( ord_less_eq_int @ zero_zero_int @ A )
=> ( ( ord_less_eq_int @ zero_zero_int @ B )
=> ( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( ord_less_eq_int @ ( power_power_int @ A @ N ) @ ( power_power_int @ B @ N ) )
= ( ord_less_eq_int @ A @ B ) ) ) ) ) ).
% power_mono_iff
thf(fact_763_power__strict__decreasing__iff,axiom,
! [B: nat,M: nat,N: nat] :
( ( ord_less_nat @ zero_zero_nat @ B )
=> ( ( ord_less_nat @ B @ one_one_nat )
=> ( ( ord_less_nat @ ( power_power_nat @ B @ M ) @ ( power_power_nat @ B @ N ) )
= ( ord_less_nat @ N @ M ) ) ) ) ).
% power_strict_decreasing_iff
thf(fact_764_power__strict__decreasing__iff,axiom,
! [B: int,M: nat,N: nat] :
( ( ord_less_int @ zero_zero_int @ B )
=> ( ( ord_less_int @ B @ one_one_int )
=> ( ( ord_less_int @ ( power_power_int @ B @ M ) @ ( power_power_int @ B @ N ) )
= ( ord_less_nat @ N @ M ) ) ) ) ).
% power_strict_decreasing_iff
thf(fact_765_power__increasing__iff,axiom,
! [B: nat,X4: nat,Y: nat] :
( ( ord_less_nat @ one_one_nat @ B )
=> ( ( ord_less_eq_nat @ ( power_power_nat @ B @ X4 ) @ ( power_power_nat @ B @ Y ) )
= ( ord_less_eq_nat @ X4 @ Y ) ) ) ).
% power_increasing_iff
thf(fact_766_power__increasing__iff,axiom,
! [B: int,X4: nat,Y: nat] :
( ( ord_less_int @ one_one_int @ B )
=> ( ( ord_less_eq_int @ ( power_power_int @ B @ X4 ) @ ( power_power_int @ B @ Y ) )
= ( ord_less_eq_nat @ X4 @ Y ) ) ) ).
% power_increasing_iff
thf(fact_767_power__decreasing__iff,axiom,
! [B: nat,M: nat,N: nat] :
( ( ord_less_nat @ zero_zero_nat @ B )
=> ( ( ord_less_nat @ B @ one_one_nat )
=> ( ( ord_less_eq_nat @ ( power_power_nat @ B @ M ) @ ( power_power_nat @ B @ N ) )
= ( ord_less_eq_nat @ N @ M ) ) ) ) ).
% power_decreasing_iff
thf(fact_768_power__decreasing__iff,axiom,
! [B: int,M: nat,N: nat] :
( ( ord_less_int @ zero_zero_int @ B )
=> ( ( ord_less_int @ B @ one_one_int )
=> ( ( ord_less_eq_int @ ( power_power_int @ B @ M ) @ ( power_power_int @ B @ N ) )
= ( ord_less_eq_nat @ N @ M ) ) ) ) ).
% power_decreasing_iff
thf(fact_769_power2__eq__iff__nonneg,axiom,
! [X4: nat,Y: nat] :
( ( ord_less_eq_nat @ zero_zero_nat @ X4 )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ Y )
=> ( ( ( power_power_nat @ X4 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
= ( power_power_nat @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
= ( X4 = Y ) ) ) ) ).
% power2_eq_iff_nonneg
thf(fact_770_power2__eq__iff__nonneg,axiom,
! [X4: int,Y: int] :
( ( ord_less_eq_int @ zero_zero_int @ X4 )
=> ( ( ord_less_eq_int @ zero_zero_int @ Y )
=> ( ( ( power_power_int @ X4 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
= ( power_power_int @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
= ( X4 = Y ) ) ) ) ).
% power2_eq_iff_nonneg
thf(fact_771_power2__less__eq__zero__iff,axiom,
! [A: int] :
( ( ord_less_eq_int @ ( power_power_int @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ zero_zero_int )
= ( A = zero_zero_int ) ) ).
% power2_less_eq_zero_iff
thf(fact_772_zero__less__power2,axiom,
! [A: int] :
( ( ord_less_int @ zero_zero_int @ ( power_power_int @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
= ( A != zero_zero_int ) ) ).
% zero_less_power2
thf(fact_773_even__power,axiom,
! [A: nat,N: nat] :
( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( power_power_nat @ A @ N ) )
= ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A )
& ( ord_less_nat @ zero_zero_nat @ N ) ) ) ).
% even_power
thf(fact_774_even__power,axiom,
! [A: int,N: nat] :
( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( power_power_int @ A @ N ) )
= ( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A )
& ( ord_less_nat @ zero_zero_nat @ N ) ) ) ).
% even_power
thf(fact_775_zero__le__power__eq__numeral,axiom,
! [A: int,W: num] :
( ( ord_less_eq_int @ zero_zero_int @ ( power_power_int @ A @ ( numeral_numeral_nat @ W ) ) )
= ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( numeral_numeral_nat @ W ) )
| ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( numeral_numeral_nat @ W ) )
& ( ord_less_eq_int @ zero_zero_int @ A ) ) ) ) ).
% zero_le_power_eq_numeral
thf(fact_776_power__less__zero__eq__numeral,axiom,
! [A: int,W: num] :
( ( ord_less_int @ ( power_power_int @ A @ ( numeral_numeral_nat @ W ) ) @ zero_zero_int )
= ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( numeral_numeral_nat @ W ) )
& ( ord_less_int @ A @ zero_zero_int ) ) ) ).
% power_less_zero_eq_numeral
thf(fact_777_power__less__zero__eq,axiom,
! [A: int,N: nat] :
( ( ord_less_int @ ( power_power_int @ A @ N ) @ zero_zero_int )
= ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
& ( ord_less_int @ A @ zero_zero_int ) ) ) ).
% power_less_zero_eq
thf(fact_778_even__diff__nat,axiom,
! [M: nat,N: nat] :
( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( minus_minus_nat @ M @ N ) )
= ( ( ord_less_nat @ M @ N )
| ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( plus_plus_nat @ M @ N ) ) ) ) ).
% even_diff_nat
thf(fact_779_zero__less__power__eq__numeral,axiom,
! [A: int,W: num] :
( ( ord_less_int @ zero_zero_int @ ( power_power_int @ A @ ( numeral_numeral_nat @ W ) ) )
= ( ( ( numeral_numeral_nat @ W )
= zero_zero_nat )
| ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( numeral_numeral_nat @ W ) )
& ( A != zero_zero_int ) )
| ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( numeral_numeral_nat @ W ) )
& ( ord_less_int @ zero_zero_int @ A ) ) ) ) ).
% zero_less_power_eq_numeral
thf(fact_780_power__le__zero__eq__numeral,axiom,
! [A: int,W: num] :
( ( ord_less_eq_int @ ( power_power_int @ A @ ( numeral_numeral_nat @ W ) ) @ zero_zero_int )
= ( ( ord_less_nat @ zero_zero_nat @ ( numeral_numeral_nat @ W ) )
& ( ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( numeral_numeral_nat @ W ) )
& ( ord_less_eq_int @ A @ zero_zero_int ) )
| ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( numeral_numeral_nat @ W ) )
& ( A = zero_zero_int ) ) ) ) ) ).
% power_le_zero_eq_numeral
thf(fact_781_dvd__imp__le,axiom,
! [K: nat,N: nat] :
( ( dvd_dvd_nat @ K @ N )
=> ( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ord_less_eq_nat @ K @ N ) ) ) ).
% dvd_imp_le
thf(fact_782_power__dvd__imp__le,axiom,
! [I: nat,M: nat,N: nat] :
( ( dvd_dvd_nat @ ( power_power_nat @ I @ M ) @ ( power_power_nat @ I @ N ) )
=> ( ( ord_less_nat @ one_one_nat @ I )
=> ( ord_less_eq_nat @ M @ N ) ) ) ).
% power_dvd_imp_le
thf(fact_783_add__mono__thms__linordered__field_I5_J,axiom,
! [I: nat,J: nat,K: nat,L: nat] :
( ( ( ord_less_nat @ I @ J )
& ( ord_less_nat @ K @ L ) )
=> ( ord_less_nat @ ( plus_plus_nat @ I @ K ) @ ( plus_plus_nat @ J @ L ) ) ) ).
% add_mono_thms_linordered_field(5)
thf(fact_784_add__mono__thms__linordered__field_I5_J,axiom,
! [I: int,J: int,K: int,L: int] :
( ( ( ord_less_int @ I @ J )
& ( ord_less_int @ K @ L ) )
=> ( ord_less_int @ ( plus_plus_int @ I @ K ) @ ( plus_plus_int @ J @ L ) ) ) ).
% add_mono_thms_linordered_field(5)
thf(fact_785_add__mono__thms__linordered__field_I2_J,axiom,
! [I: nat,J: nat,K: nat,L: nat] :
( ( ( I = J )
& ( ord_less_nat @ K @ L ) )
=> ( ord_less_nat @ ( plus_plus_nat @ I @ K ) @ ( plus_plus_nat @ J @ L ) ) ) ).
% add_mono_thms_linordered_field(2)
thf(fact_786_add__mono__thms__linordered__field_I2_J,axiom,
! [I: int,J: int,K: int,L: int] :
( ( ( I = J )
& ( ord_less_int @ K @ L ) )
=> ( ord_less_int @ ( plus_plus_int @ I @ K ) @ ( plus_plus_int @ J @ L ) ) ) ).
% add_mono_thms_linordered_field(2)
thf(fact_787_add__mono__thms__linordered__field_I1_J,axiom,
! [I: nat,J: nat,K: nat,L: nat] :
( ( ( ord_less_nat @ I @ J )
& ( K = L ) )
=> ( ord_less_nat @ ( plus_plus_nat @ I @ K ) @ ( plus_plus_nat @ J @ L ) ) ) ).
% add_mono_thms_linordered_field(1)
thf(fact_788_add__mono__thms__linordered__field_I1_J,axiom,
! [I: int,J: int,K: int,L: int] :
( ( ( ord_less_int @ I @ J )
& ( K = L ) )
=> ( ord_less_int @ ( plus_plus_int @ I @ K ) @ ( plus_plus_int @ J @ L ) ) ) ).
% add_mono_thms_linordered_field(1)
thf(fact_789_add__strict__mono,axiom,
! [A: nat,B: nat,C2: nat,D: nat] :
( ( ord_less_nat @ A @ B )
=> ( ( ord_less_nat @ C2 @ D )
=> ( ord_less_nat @ ( plus_plus_nat @ A @ C2 ) @ ( plus_plus_nat @ B @ D ) ) ) ) ).
% add_strict_mono
thf(fact_790_add__strict__mono,axiom,
! [A: int,B: int,C2: int,D: int] :
( ( ord_less_int @ A @ B )
=> ( ( ord_less_int @ C2 @ D )
=> ( ord_less_int @ ( plus_plus_int @ A @ C2 ) @ ( plus_plus_int @ B @ D ) ) ) ) ).
% add_strict_mono
thf(fact_791_add__strict__left__mono,axiom,
! [A: nat,B: nat,C2: nat] :
( ( ord_less_nat @ A @ B )
=> ( ord_less_nat @ ( plus_plus_nat @ C2 @ A ) @ ( plus_plus_nat @ C2 @ B ) ) ) ).
% add_strict_left_mono
thf(fact_792_add__strict__left__mono,axiom,
! [A: int,B: int,C2: int] :
( ( ord_less_int @ A @ B )
=> ( ord_less_int @ ( plus_plus_int @ C2 @ A ) @ ( plus_plus_int @ C2 @ B ) ) ) ).
% add_strict_left_mono
thf(fact_793_add__strict__right__mono,axiom,
! [A: nat,B: nat,C2: nat] :
( ( ord_less_nat @ A @ B )
=> ( ord_less_nat @ ( plus_plus_nat @ A @ C2 ) @ ( plus_plus_nat @ B @ C2 ) ) ) ).
% add_strict_right_mono
thf(fact_794_add__strict__right__mono,axiom,
! [A: int,B: int,C2: int] :
( ( ord_less_int @ A @ B )
=> ( ord_less_int @ ( plus_plus_int @ A @ C2 ) @ ( plus_plus_int @ B @ C2 ) ) ) ).
% add_strict_right_mono
thf(fact_795_add__less__imp__less__left,axiom,
! [C2: nat,A: nat,B: nat] :
( ( ord_less_nat @ ( plus_plus_nat @ C2 @ A ) @ ( plus_plus_nat @ C2 @ B ) )
=> ( ord_less_nat @ A @ B ) ) ).
% add_less_imp_less_left
thf(fact_796_add__less__imp__less__left,axiom,
! [C2: int,A: int,B: int] :
( ( ord_less_int @ ( plus_plus_int @ C2 @ A ) @ ( plus_plus_int @ C2 @ B ) )
=> ( ord_less_int @ A @ B ) ) ).
% add_less_imp_less_left
thf(fact_797_add__less__imp__less__right,axiom,
! [A: nat,C2: nat,B: nat] :
( ( ord_less_nat @ ( plus_plus_nat @ A @ C2 ) @ ( plus_plus_nat @ B @ C2 ) )
=> ( ord_less_nat @ A @ B ) ) ).
% add_less_imp_less_right
thf(fact_798_add__less__imp__less__right,axiom,
! [A: int,C2: int,B: int] :
( ( ord_less_int @ ( plus_plus_int @ A @ C2 ) @ ( plus_plus_int @ B @ C2 ) )
=> ( ord_less_int @ A @ B ) ) ).
% add_less_imp_less_right
thf(fact_799_le__numeral__extra_I3_J,axiom,
ord_less_eq_nat @ zero_zero_nat @ zero_zero_nat ).
% le_numeral_extra(3)
thf(fact_800_le__numeral__extra_I3_J,axiom,
ord_less_eq_int @ zero_zero_int @ zero_zero_int ).
% le_numeral_extra(3)
thf(fact_801_zero__le,axiom,
! [X4: nat] : ( ord_less_eq_nat @ zero_zero_nat @ X4 ) ).
% zero_le
thf(fact_802_diff__strict__mono,axiom,
! [A: int,B: int,D: int,C2: int] :
( ( ord_less_int @ A @ B )
=> ( ( ord_less_int @ D @ C2 )
=> ( ord_less_int @ ( minus_minus_int @ A @ C2 ) @ ( minus_minus_int @ B @ D ) ) ) ) ).
% diff_strict_mono
thf(fact_803_diff__eq__diff__less,axiom,
! [A: int,B: int,C2: int,D: int] :
( ( ( minus_minus_int @ A @ B )
= ( minus_minus_int @ C2 @ D ) )
=> ( ( ord_less_int @ A @ B )
= ( ord_less_int @ C2 @ D ) ) ) ).
% diff_eq_diff_less
thf(fact_804_diff__strict__left__mono,axiom,
! [B: int,A: int,C2: int] :
( ( ord_less_int @ B @ A )
=> ( ord_less_int @ ( minus_minus_int @ C2 @ A ) @ ( minus_minus_int @ C2 @ B ) ) ) ).
% diff_strict_left_mono
thf(fact_805_diff__strict__right__mono,axiom,
! [A: int,B: int,C2: int] :
( ( ord_less_int @ A @ B )
=> ( ord_less_int @ ( minus_minus_int @ A @ C2 ) @ ( minus_minus_int @ B @ C2 ) ) ) ).
% diff_strict_right_mono
thf(fact_806_less__numeral__extra_I4_J,axiom,
~ ( ord_less_nat @ one_one_nat @ one_one_nat ) ).
% less_numeral_extra(4)
thf(fact_807_less__numeral__extra_I4_J,axiom,
~ ( ord_less_int @ one_one_int @ one_one_int ) ).
% less_numeral_extra(4)
thf(fact_808_add__mono__thms__linordered__semiring_I3_J,axiom,
! [I: nat,J: nat,K: nat,L: nat] :
( ( ( ord_less_eq_nat @ I @ J )
& ( K = L ) )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ I @ K ) @ ( plus_plus_nat @ J @ L ) ) ) ).
% add_mono_thms_linordered_semiring(3)
thf(fact_809_add__mono__thms__linordered__semiring_I3_J,axiom,
! [I: int,J: int,K: int,L: int] :
( ( ( ord_less_eq_int @ I @ J )
& ( K = L ) )
=> ( ord_less_eq_int @ ( plus_plus_int @ I @ K ) @ ( plus_plus_int @ J @ L ) ) ) ).
% add_mono_thms_linordered_semiring(3)
thf(fact_810_add__mono__thms__linordered__semiring_I2_J,axiom,
! [I: nat,J: nat,K: nat,L: nat] :
( ( ( I = J )
& ( ord_less_eq_nat @ K @ L ) )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ I @ K ) @ ( plus_plus_nat @ J @ L ) ) ) ).
% add_mono_thms_linordered_semiring(2)
thf(fact_811_add__mono__thms__linordered__semiring_I2_J,axiom,
! [I: int,J: int,K: int,L: int] :
( ( ( I = J )
& ( ord_less_eq_int @ K @ L ) )
=> ( ord_less_eq_int @ ( plus_plus_int @ I @ K ) @ ( plus_plus_int @ J @ L ) ) ) ).
% add_mono_thms_linordered_semiring(2)
thf(fact_812_add__mono__thms__linordered__semiring_I1_J,axiom,
! [I: nat,J: nat,K: nat,L: nat] :
( ( ( ord_less_eq_nat @ I @ J )
& ( ord_less_eq_nat @ K @ L ) )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ I @ K ) @ ( plus_plus_nat @ J @ L ) ) ) ).
% add_mono_thms_linordered_semiring(1)
thf(fact_813_add__mono__thms__linordered__semiring_I1_J,axiom,
! [I: int,J: int,K: int,L: int] :
( ( ( ord_less_eq_int @ I @ J )
& ( ord_less_eq_int @ K @ L ) )
=> ( ord_less_eq_int @ ( plus_plus_int @ I @ K ) @ ( plus_plus_int @ J @ L ) ) ) ).
% add_mono_thms_linordered_semiring(1)
thf(fact_814_add__mono,axiom,
! [A: nat,B: nat,C2: nat,D: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ord_less_eq_nat @ C2 @ D )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ A @ C2 ) @ ( plus_plus_nat @ B @ D ) ) ) ) ).
% add_mono
thf(fact_815_add__mono,axiom,
! [A: int,B: int,C2: int,D: int] :
( ( ord_less_eq_int @ A @ B )
=> ( ( ord_less_eq_int @ C2 @ D )
=> ( ord_less_eq_int @ ( plus_plus_int @ A @ C2 ) @ ( plus_plus_int @ B @ D ) ) ) ) ).
% add_mono
thf(fact_816_add__left__mono,axiom,
! [A: nat,B: nat,C2: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ C2 @ A ) @ ( plus_plus_nat @ C2 @ B ) ) ) ).
% add_left_mono
thf(fact_817_add__left__mono,axiom,
! [A: int,B: int,C2: int] :
( ( ord_less_eq_int @ A @ B )
=> ( ord_less_eq_int @ ( plus_plus_int @ C2 @ A ) @ ( plus_plus_int @ C2 @ B ) ) ) ).
% add_left_mono
thf(fact_818_less__eqE,axiom,
! [A: nat,B: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ~ ! [C: nat] :
( B
!= ( plus_plus_nat @ A @ C ) ) ) ).
% less_eqE
thf(fact_819_add__right__mono,axiom,
! [A: nat,B: nat,C2: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ A @ C2 ) @ ( plus_plus_nat @ B @ C2 ) ) ) ).
% add_right_mono
thf(fact_820_add__right__mono,axiom,
! [A: int,B: int,C2: int] :
( ( ord_less_eq_int @ A @ B )
=> ( ord_less_eq_int @ ( plus_plus_int @ A @ C2 ) @ ( plus_plus_int @ B @ C2 ) ) ) ).
% add_right_mono
thf(fact_821_le__iff__add,axiom,
( ord_less_eq_nat
= ( ^ [A4: nat,B4: nat] :
? [C3: nat] :
( B4
= ( plus_plus_nat @ A4 @ C3 ) ) ) ) ).
% le_iff_add
thf(fact_822_add__le__imp__le__left,axiom,
! [C2: nat,A: nat,B: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ C2 @ A ) @ ( plus_plus_nat @ C2 @ B ) )
=> ( ord_less_eq_nat @ A @ B ) ) ).
% add_le_imp_le_left
thf(fact_823_add__le__imp__le__left,axiom,
! [C2: int,A: int,B: int] :
( ( ord_less_eq_int @ ( plus_plus_int @ C2 @ A ) @ ( plus_plus_int @ C2 @ B ) )
=> ( ord_less_eq_int @ A @ B ) ) ).
% add_le_imp_le_left
thf(fact_824_add__le__imp__le__right,axiom,
! [A: nat,C2: nat,B: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ A @ C2 ) @ ( plus_plus_nat @ B @ C2 ) )
=> ( ord_less_eq_nat @ A @ B ) ) ).
% add_le_imp_le_right
thf(fact_825_add__le__imp__le__right,axiom,
! [A: int,C2: int,B: int] :
( ( ord_less_eq_int @ ( plus_plus_int @ A @ C2 ) @ ( plus_plus_int @ B @ C2 ) )
=> ( ord_less_eq_int @ A @ B ) ) ).
% add_le_imp_le_right
thf(fact_826_less__minus__iff,axiom,
! [A: int,B: int] :
( ( ord_less_int @ A @ ( uminus_uminus_int @ B ) )
= ( ord_less_int @ B @ ( uminus_uminus_int @ A ) ) ) ).
% less_minus_iff
thf(fact_827_minus__less__iff,axiom,
! [A: int,B: int] :
( ( ord_less_int @ ( uminus_uminus_int @ A ) @ B )
= ( ord_less_int @ ( uminus_uminus_int @ B ) @ A ) ) ).
% minus_less_iff
thf(fact_828_verit__negate__coefficient_I2_J,axiom,
! [A: int,B: int] :
( ( ord_less_int @ A @ B )
=> ( ord_less_int @ ( uminus_uminus_int @ B ) @ ( uminus_uminus_int @ A ) ) ) ).
% verit_negate_coefficient(2)
thf(fact_829_power__strict__mono,axiom,
! [A: nat,B: nat,N: nat] :
( ( ord_less_nat @ A @ B )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ A )
=> ( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ord_less_nat @ ( power_power_nat @ A @ N ) @ ( power_power_nat @ B @ N ) ) ) ) ) ).
% power_strict_mono
thf(fact_830_power__strict__mono,axiom,
! [A: int,B: int,N: nat] :
( ( ord_less_int @ A @ B )
=> ( ( ord_less_eq_int @ zero_zero_int @ A )
=> ( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ord_less_int @ ( power_power_int @ A @ N ) @ ( power_power_int @ B @ N ) ) ) ) ) ).
% power_strict_mono
thf(fact_831_power__less__imp__less__base,axiom,
! [A: nat,N: nat,B: nat] :
( ( ord_less_nat @ ( power_power_nat @ A @ N ) @ ( power_power_nat @ B @ N ) )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ B )
=> ( ord_less_nat @ A @ B ) ) ) ).
% power_less_imp_less_base
thf(fact_832_power__less__imp__less__base,axiom,
! [A: int,N: nat,B: int] :
( ( ord_less_int @ ( power_power_int @ A @ N ) @ ( power_power_int @ B @ N ) )
=> ( ( ord_less_eq_int @ zero_zero_int @ B )
=> ( ord_less_int @ A @ B ) ) ) ).
% power_less_imp_less_base
thf(fact_833_add__less__le__mono,axiom,
! [A: nat,B: nat,C2: nat,D: nat] :
( ( ord_less_nat @ A @ B )
=> ( ( ord_less_eq_nat @ C2 @ D )
=> ( ord_less_nat @ ( plus_plus_nat @ A @ C2 ) @ ( plus_plus_nat @ B @ D ) ) ) ) ).
% add_less_le_mono
thf(fact_834_add__less__le__mono,axiom,
! [A: int,B: int,C2: int,D: int] :
( ( ord_less_int @ A @ B )
=> ( ( ord_less_eq_int @ C2 @ D )
=> ( ord_less_int @ ( plus_plus_int @ A @ C2 ) @ ( plus_plus_int @ B @ D ) ) ) ) ).
% add_less_le_mono
thf(fact_835_add__le__less__mono,axiom,
! [A: nat,B: nat,C2: nat,D: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ord_less_nat @ C2 @ D )
=> ( ord_less_nat @ ( plus_plus_nat @ A @ C2 ) @ ( plus_plus_nat @ B @ D ) ) ) ) ).
% add_le_less_mono
thf(fact_836_add__le__less__mono,axiom,
! [A: int,B: int,C2: int,D: int] :
( ( ord_less_eq_int @ A @ B )
=> ( ( ord_less_int @ C2 @ D )
=> ( ord_less_int @ ( plus_plus_int @ A @ C2 ) @ ( plus_plus_int @ B @ D ) ) ) ) ).
% add_le_less_mono
thf(fact_837_add__mono__thms__linordered__field_I3_J,axiom,
! [I: nat,J: nat,K: nat,L: nat] :
( ( ( ord_less_nat @ I @ J )
& ( ord_less_eq_nat @ K @ L ) )
=> ( ord_less_nat @ ( plus_plus_nat @ I @ K ) @ ( plus_plus_nat @ J @ L ) ) ) ).
% add_mono_thms_linordered_field(3)
thf(fact_838_add__mono__thms__linordered__field_I3_J,axiom,
! [I: int,J: int,K: int,L: int] :
( ( ( ord_less_int @ I @ J )
& ( ord_less_eq_int @ K @ L ) )
=> ( ord_less_int @ ( plus_plus_int @ I @ K ) @ ( plus_plus_int @ J @ L ) ) ) ).
% add_mono_thms_linordered_field(3)
thf(fact_839_add__mono__thms__linordered__field_I4_J,axiom,
! [I: nat,J: nat,K: nat,L: nat] :
( ( ( ord_less_eq_nat @ I @ J )
& ( ord_less_nat @ K @ L ) )
=> ( ord_less_nat @ ( plus_plus_nat @ I @ K ) @ ( plus_plus_nat @ J @ L ) ) ) ).
% add_mono_thms_linordered_field(4)
thf(fact_840_add__mono__thms__linordered__field_I4_J,axiom,
! [I: int,J: int,K: int,L: int] :
( ( ( ord_less_eq_int @ I @ J )
& ( ord_less_int @ K @ L ) )
=> ( ord_less_int @ ( plus_plus_int @ I @ K ) @ ( plus_plus_int @ J @ L ) ) ) ).
% add_mono_thms_linordered_field(4)
thf(fact_841_power__strict__increasing,axiom,
! [N: nat,N3: nat,A: nat] :
( ( ord_less_nat @ N @ N3 )
=> ( ( ord_less_nat @ one_one_nat @ A )
=> ( ord_less_nat @ ( power_power_nat @ A @ N ) @ ( power_power_nat @ A @ N3 ) ) ) ) ).
% power_strict_increasing
thf(fact_842_power__strict__increasing,axiom,
! [N: nat,N3: nat,A: int] :
( ( ord_less_nat @ N @ N3 )
=> ( ( ord_less_int @ one_one_int @ A )
=> ( ord_less_int @ ( power_power_int @ A @ N ) @ ( power_power_int @ A @ N3 ) ) ) ) ).
% power_strict_increasing
thf(fact_843_power__less__imp__less__exp,axiom,
! [A: nat,M: nat,N: nat] :
( ( ord_less_nat @ one_one_nat @ A )
=> ( ( ord_less_nat @ ( power_power_nat @ A @ M ) @ ( power_power_nat @ A @ N ) )
=> ( ord_less_nat @ M @ N ) ) ) ).
% power_less_imp_less_exp
thf(fact_844_power__less__imp__less__exp,axiom,
! [A: int,M: nat,N: nat] :
( ( ord_less_int @ one_one_int @ A )
=> ( ( ord_less_int @ ( power_power_int @ A @ M ) @ ( power_power_int @ A @ N ) )
=> ( ord_less_nat @ M @ N ) ) ) ).
% power_less_imp_less_exp
thf(fact_845_power__le__imp__le__exp,axiom,
! [A: nat,M: nat,N: nat] :
( ( ord_less_nat @ one_one_nat @ A )
=> ( ( ord_less_eq_nat @ ( power_power_nat @ A @ M ) @ ( power_power_nat @ A @ N ) )
=> ( ord_less_eq_nat @ M @ N ) ) ) ).
% power_le_imp_le_exp
thf(fact_846_power__le__imp__le__exp,axiom,
! [A: int,M: nat,N: nat] :
( ( ord_less_int @ one_one_int @ A )
=> ( ( ord_less_eq_int @ ( power_power_int @ A @ M ) @ ( power_power_int @ A @ N ) )
=> ( ord_less_eq_nat @ M @ N ) ) ) ).
% power_le_imp_le_exp
thf(fact_847_power__increasing,axiom,
! [N: nat,N3: nat,A: nat] :
( ( ord_less_eq_nat @ N @ N3 )
=> ( ( ord_less_eq_nat @ one_one_nat @ A )
=> ( ord_less_eq_nat @ ( power_power_nat @ A @ N ) @ ( power_power_nat @ A @ N3 ) ) ) ) ).
% power_increasing
thf(fact_848_power__increasing,axiom,
! [N: nat,N3: nat,A: int] :
( ( ord_less_eq_nat @ N @ N3 )
=> ( ( ord_less_eq_int @ one_one_int @ A )
=> ( ord_less_eq_int @ ( power_power_int @ A @ N ) @ ( power_power_int @ A @ N3 ) ) ) ) ).
% power_increasing
thf(fact_849_div__greater__zero__iff,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ zero_zero_nat @ ( divide_divide_nat @ M @ N ) )
= ( ( ord_less_eq_nat @ N @ M )
& ( ord_less_nat @ zero_zero_nat @ N ) ) ) ).
% div_greater_zero_iff
thf(fact_850_div__le__mono2,axiom,
! [M: nat,N: nat,K: nat] :
( ( ord_less_nat @ zero_zero_nat @ M )
=> ( ( ord_less_eq_nat @ M @ N )
=> ( ord_less_eq_nat @ ( divide_divide_nat @ K @ N ) @ ( divide_divide_nat @ K @ M ) ) ) ) ).
% div_le_mono2
thf(fact_851_add__strict__increasing2,axiom,
! [A: nat,B: nat,C2: nat] :
( ( ord_less_eq_nat @ zero_zero_nat @ A )
=> ( ( ord_less_nat @ B @ C2 )
=> ( ord_less_nat @ B @ ( plus_plus_nat @ A @ C2 ) ) ) ) ).
% add_strict_increasing2
thf(fact_852_add__strict__increasing2,axiom,
! [A: int,B: int,C2: int] :
( ( ord_less_eq_int @ zero_zero_int @ A )
=> ( ( ord_less_int @ B @ C2 )
=> ( ord_less_int @ B @ ( plus_plus_int @ A @ C2 ) ) ) ) ).
% add_strict_increasing2
thf(fact_853_add__strict__increasing,axiom,
! [A: nat,B: nat,C2: nat] :
( ( ord_less_nat @ zero_zero_nat @ A )
=> ( ( ord_less_eq_nat @ B @ C2 )
=> ( ord_less_nat @ B @ ( plus_plus_nat @ A @ C2 ) ) ) ) ).
% add_strict_increasing
thf(fact_854_add__strict__increasing,axiom,
! [A: int,B: int,C2: int] :
( ( ord_less_int @ zero_zero_int @ A )
=> ( ( ord_less_eq_int @ B @ C2 )
=> ( ord_less_int @ B @ ( plus_plus_int @ A @ C2 ) ) ) ) ).
% add_strict_increasing
thf(fact_855_add__pos__nonneg,axiom,
! [A: nat,B: nat] :
( ( ord_less_nat @ zero_zero_nat @ A )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ B )
=> ( ord_less_nat @ zero_zero_nat @ ( plus_plus_nat @ A @ B ) ) ) ) ).
% add_pos_nonneg
thf(fact_856_add__pos__nonneg,axiom,
! [A: int,B: int] :
( ( ord_less_int @ zero_zero_int @ A )
=> ( ( ord_less_eq_int @ zero_zero_int @ B )
=> ( ord_less_int @ zero_zero_int @ ( plus_plus_int @ A @ B ) ) ) ) ).
% add_pos_nonneg
thf(fact_857_add__nonpos__neg,axiom,
! [A: nat,B: nat] :
( ( ord_less_eq_nat @ A @ zero_zero_nat )
=> ( ( ord_less_nat @ B @ zero_zero_nat )
=> ( ord_less_nat @ ( plus_plus_nat @ A @ B ) @ zero_zero_nat ) ) ) ).
% add_nonpos_neg
thf(fact_858_add__nonpos__neg,axiom,
! [A: int,B: int] :
( ( ord_less_eq_int @ A @ zero_zero_int )
=> ( ( ord_less_int @ B @ zero_zero_int )
=> ( ord_less_int @ ( plus_plus_int @ A @ B ) @ zero_zero_int ) ) ) ).
% add_nonpos_neg
thf(fact_859_add__nonneg__pos,axiom,
! [A: nat,B: nat] :
( ( ord_less_eq_nat @ zero_zero_nat @ A )
=> ( ( ord_less_nat @ zero_zero_nat @ B )
=> ( ord_less_nat @ zero_zero_nat @ ( plus_plus_nat @ A @ B ) ) ) ) ).
% add_nonneg_pos
thf(fact_860_add__nonneg__pos,axiom,
! [A: int,B: int] :
( ( ord_less_eq_int @ zero_zero_int @ A )
=> ( ( ord_less_int @ zero_zero_int @ B )
=> ( ord_less_int @ zero_zero_int @ ( plus_plus_int @ A @ B ) ) ) ) ).
% add_nonneg_pos
thf(fact_861_add__neg__nonpos,axiom,
! [A: nat,B: nat] :
( ( ord_less_nat @ A @ zero_zero_nat )
=> ( ( ord_less_eq_nat @ B @ zero_zero_nat )
=> ( ord_less_nat @ ( plus_plus_nat @ A @ B ) @ zero_zero_nat ) ) ) ).
% add_neg_nonpos
thf(fact_862_add__neg__nonpos,axiom,
! [A: int,B: int] :
( ( ord_less_int @ A @ zero_zero_int )
=> ( ( ord_less_eq_int @ B @ zero_zero_int )
=> ( ord_less_int @ ( plus_plus_int @ A @ B ) @ zero_zero_int ) ) ) ).
% add_neg_nonpos
thf(fact_863_sub__positive,axiom,
! [N: num,M: num] :
( ( ord_less_int @ zero_zero_int @ ( neg_numeral_sub_int @ N @ M ) )
= ( ord_less_num @ M @ N ) ) ).
% sub_positive
thf(fact_864_sub__negative,axiom,
! [N: num,M: num] :
( ( ord_less_int @ ( neg_numeral_sub_int @ N @ M ) @ zero_zero_int )
= ( ord_less_num @ N @ M ) ) ).
% sub_negative
thf(fact_865_fact__less__mono,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ zero_zero_nat @ M )
=> ( ( ord_less_nat @ M @ N )
=> ( ord_less_nat @ ( semiri1408675320244567234ct_nat @ M ) @ ( semiri1408675320244567234ct_nat @ N ) ) ) ) ).
% fact_less_mono
thf(fact_866_fact__less__mono,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ zero_zero_nat @ M )
=> ( ( ord_less_nat @ M @ N )
=> ( ord_less_int @ ( semiri1406184849735516958ct_int @ M ) @ ( semiri1406184849735516958ct_int @ N ) ) ) ) ).
% fact_less_mono
thf(fact_867_sub__non__positive,axiom,
! [N: num,M: num] :
( ( ord_less_eq_int @ ( neg_numeral_sub_int @ N @ M ) @ zero_zero_int )
= ( ord_less_eq_num @ N @ M ) ) ).
% sub_non_positive
thf(fact_868_sub__non__negative,axiom,
! [N: num,M: num] :
( ( ord_less_eq_int @ zero_zero_int @ ( neg_numeral_sub_int @ N @ M ) )
= ( ord_less_eq_num @ M @ N ) ) ).
% sub_non_negative
thf(fact_869_fact__mono,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq_nat @ M @ N )
=> ( ord_less_eq_nat @ ( semiri1408675320244567234ct_nat @ M ) @ ( semiri1408675320244567234ct_nat @ N ) ) ) ).
% fact_mono
thf(fact_870_fact__mono,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq_nat @ M @ N )
=> ( ord_less_eq_int @ ( semiri1406184849735516958ct_int @ M ) @ ( semiri1406184849735516958ct_int @ N ) ) ) ).
% fact_mono
thf(fact_871_verit__la__disequality,axiom,
! [A: nat,B: nat] :
( ( A = B )
| ~ ( ord_less_eq_nat @ A @ B )
| ~ ( ord_less_eq_nat @ B @ A ) ) ).
% verit_la_disequality
thf(fact_872_verit__la__disequality,axiom,
! [A: int,B: int] :
( ( A = B )
| ~ ( ord_less_eq_int @ A @ B )
| ~ ( ord_less_eq_int @ B @ A ) ) ).
% verit_la_disequality
thf(fact_873_verit__la__disequality,axiom,
! [A: num,B: num] :
( ( A = B )
| ~ ( ord_less_eq_num @ A @ B )
| ~ ( ord_less_eq_num @ B @ A ) ) ).
% verit_la_disequality
thf(fact_874_fact__mono__nat,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq_nat @ M @ N )
=> ( ord_less_eq_nat @ ( semiri1408675320244567234ct_nat @ M ) @ ( semiri1408675320244567234ct_nat @ N ) ) ) ).
% fact_mono_nat
thf(fact_875_fact__ge__self,axiom,
! [N: nat] : ( ord_less_eq_nat @ N @ ( semiri1408675320244567234ct_nat @ N ) ) ).
% fact_ge_self
thf(fact_876_verit__comp__simplify1_I1_J,axiom,
! [A: nat] :
~ ( ord_less_nat @ A @ A ) ).
% verit_comp_simplify1(1)
thf(fact_877_verit__comp__simplify1_I1_J,axiom,
! [A: int] :
~ ( ord_less_int @ A @ A ) ).
% verit_comp_simplify1(1)
thf(fact_878_verit__comp__simplify1_I1_J,axiom,
! [A: num] :
~ ( ord_less_num @ A @ A ) ).
% verit_comp_simplify1(1)
thf(fact_879_verit__comp__simplify1_I2_J,axiom,
! [A: nat] : ( ord_less_eq_nat @ A @ A ) ).
% verit_comp_simplify1(2)
thf(fact_880_verit__comp__simplify1_I2_J,axiom,
! [A: int] : ( ord_less_eq_int @ A @ A ) ).
% verit_comp_simplify1(2)
thf(fact_881_verit__comp__simplify1_I2_J,axiom,
! [A: num] : ( ord_less_eq_num @ A @ A ) ).
% verit_comp_simplify1(2)
thf(fact_882_verit__comp__simplify1_I3_J,axiom,
! [B2: nat,A2: nat] :
( ( ~ ( ord_less_eq_nat @ B2 @ A2 ) )
= ( ord_less_nat @ A2 @ B2 ) ) ).
% verit_comp_simplify1(3)
thf(fact_883_verit__comp__simplify1_I3_J,axiom,
! [B2: int,A2: int] :
( ( ~ ( ord_less_eq_int @ B2 @ A2 ) )
= ( ord_less_int @ A2 @ B2 ) ) ).
% verit_comp_simplify1(3)
thf(fact_884_verit__comp__simplify1_I3_J,axiom,
! [B2: num,A2: num] :
( ( ~ ( ord_less_eq_num @ B2 @ A2 ) )
= ( ord_less_num @ A2 @ B2 ) ) ).
% verit_comp_simplify1(3)
thf(fact_885_linorder__neqE__linordered__idom,axiom,
! [X4: int,Y: int] :
( ( X4 != Y )
=> ( ~ ( ord_less_int @ X4 @ Y )
=> ( ord_less_int @ Y @ X4 ) ) ) ).
% linorder_neqE_linordered_idom
thf(fact_886_less__numeral__extra_I3_J,axiom,
~ ( ord_less_nat @ zero_zero_nat @ zero_zero_nat ) ).
% less_numeral_extra(3)
thf(fact_887_less__numeral__extra_I3_J,axiom,
~ ( ord_less_int @ zero_zero_int @ zero_zero_int ) ).
% less_numeral_extra(3)
thf(fact_888_zero__less__iff__neq__zero,axiom,
! [N: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
= ( N != zero_zero_nat ) ) ).
% zero_less_iff_neq_zero
thf(fact_889_gr__implies__not__zero,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ M @ N )
=> ( N != zero_zero_nat ) ) ).
% gr_implies_not_zero
thf(fact_890_not__less__zero,axiom,
! [N: nat] :
~ ( ord_less_nat @ N @ zero_zero_nat ) ).
% not_less_zero
thf(fact_891_gr__zeroI,axiom,
! [N: nat] :
( ( N != zero_zero_nat )
=> ( ord_less_nat @ zero_zero_nat @ N ) ) ).
% gr_zeroI
thf(fact_892_ex__least__nat__le,axiom,
! [P2: nat > $o,N: nat] :
( ( P2 @ N )
=> ( ~ ( P2 @ zero_zero_nat )
=> ? [K3: nat] :
( ( ord_less_eq_nat @ K3 @ N )
& ! [I2: nat] :
( ( ord_less_nat @ I2 @ K3 )
=> ~ ( P2 @ I2 ) )
& ( P2 @ K3 ) ) ) ) ).
% ex_least_nat_le
thf(fact_893_diff__eq__diff__less__eq,axiom,
! [A: int,B: int,C2: int,D: int] :
( ( ( minus_minus_int @ A @ B )
= ( minus_minus_int @ C2 @ D ) )
=> ( ( ord_less_eq_int @ A @ B )
= ( ord_less_eq_int @ C2 @ D ) ) ) ).
% diff_eq_diff_less_eq
thf(fact_894_diff__right__mono,axiom,
! [A: int,B: int,C2: int] :
( ( ord_less_eq_int @ A @ B )
=> ( ord_less_eq_int @ ( minus_minus_int @ A @ C2 ) @ ( minus_minus_int @ B @ C2 ) ) ) ).
% diff_right_mono
thf(fact_895_diff__left__mono,axiom,
! [B: int,A: int,C2: int] :
( ( ord_less_eq_int @ B @ A )
=> ( ord_less_eq_int @ ( minus_minus_int @ C2 @ A ) @ ( minus_minus_int @ C2 @ B ) ) ) ).
% diff_left_mono
thf(fact_896_diff__mono,axiom,
! [A: int,B: int,D: int,C2: int] :
( ( ord_less_eq_int @ A @ B )
=> ( ( ord_less_eq_int @ D @ C2 )
=> ( ord_less_eq_int @ ( minus_minus_int @ A @ C2 ) @ ( minus_minus_int @ B @ D ) ) ) ) ).
% diff_mono
thf(fact_897_le__numeral__extra_I4_J,axiom,
ord_less_eq_nat @ one_one_nat @ one_one_nat ).
% le_numeral_extra(4)
thf(fact_898_le__numeral__extra_I4_J,axiom,
ord_less_eq_int @ one_one_int @ one_one_int ).
% le_numeral_extra(4)
thf(fact_899_bot__nat__0_Oextremum__strict,axiom,
! [A: nat] :
~ ( ord_less_nat @ A @ zero_zero_nat ) ).
% bot_nat_0.extremum_strict
thf(fact_900_gr0I,axiom,
! [N: nat] :
( ( N != zero_zero_nat )
=> ( ord_less_nat @ zero_zero_nat @ N ) ) ).
% gr0I
thf(fact_901_not__gr0,axiom,
! [N: nat] :
( ( ~ ( ord_less_nat @ zero_zero_nat @ N ) )
= ( N = zero_zero_nat ) ) ).
% not_gr0
thf(fact_902_not__less0,axiom,
! [N: nat] :
~ ( ord_less_nat @ N @ zero_zero_nat ) ).
% not_less0
thf(fact_903_less__zeroE,axiom,
! [N: nat] :
~ ( ord_less_nat @ N @ zero_zero_nat ) ).
% less_zeroE
thf(fact_904_gr__implies__not0,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ M @ N )
=> ( N != zero_zero_nat ) ) ).
% gr_implies_not0
thf(fact_905_infinite__descent0,axiom,
! [P2: nat > $o,N: nat] :
( ( P2 @ zero_zero_nat )
=> ( ! [N2: nat] :
( ( ord_less_nat @ zero_zero_nat @ N2 )
=> ( ~ ( P2 @ N2 )
=> ? [M3: nat] :
( ( ord_less_nat @ M3 @ N2 )
& ~ ( P2 @ M3 ) ) ) )
=> ( P2 @ N ) ) ) ).
% infinite_descent0
thf(fact_906_fact__less__mono__nat,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ zero_zero_nat @ M )
=> ( ( ord_less_nat @ M @ N )
=> ( ord_less_nat @ ( semiri1408675320244567234ct_nat @ M ) @ ( semiri1408675320244567234ct_nat @ N ) ) ) ) ).
% fact_less_mono_nat
thf(fact_907_nat__power__less__imp__less,axiom,
! [I: nat,M: nat,N: nat] :
( ( ord_less_nat @ zero_zero_nat @ I )
=> ( ( ord_less_nat @ ( power_power_nat @ I @ M ) @ ( power_power_nat @ I @ N ) )
=> ( ord_less_nat @ M @ N ) ) ) ).
% nat_power_less_imp_less
thf(fact_908_nat__dvd__not__less,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ zero_zero_nat @ M )
=> ( ( ord_less_nat @ M @ N )
=> ~ ( dvd_dvd_nat @ N @ M ) ) ) ).
% nat_dvd_not_less
thf(fact_909_le__imp__neg__le,axiom,
! [A: int,B: int] :
( ( ord_less_eq_int @ A @ B )
=> ( ord_less_eq_int @ ( uminus_uminus_int @ B ) @ ( uminus_uminus_int @ A ) ) ) ).
% le_imp_neg_le
thf(fact_910_minus__le__iff,axiom,
! [A: int,B: int] :
( ( ord_less_eq_int @ ( uminus_uminus_int @ A ) @ B )
= ( ord_less_eq_int @ ( uminus_uminus_int @ B ) @ A ) ) ).
% minus_le_iff
thf(fact_911_le__minus__iff,axiom,
! [A: int,B: int] :
( ( ord_less_eq_int @ A @ ( uminus_uminus_int @ B ) )
= ( ord_less_eq_int @ B @ ( uminus_uminus_int @ A ) ) ) ).
% le_minus_iff
thf(fact_912_less__eq__nat_Osimps_I1_J,axiom,
! [N: nat] : ( ord_less_eq_nat @ zero_zero_nat @ N ) ).
% less_eq_nat.simps(1)
thf(fact_913_bot__nat__0_Oextremum__unique,axiom,
! [A: nat] :
( ( ord_less_eq_nat @ A @ zero_zero_nat )
= ( A = zero_zero_nat ) ) ).
% bot_nat_0.extremum_unique
thf(fact_914_bot__nat__0_Oextremum__uniqueI,axiom,
! [A: nat] :
( ( ord_less_eq_nat @ A @ zero_zero_nat )
=> ( A = zero_zero_nat ) ) ).
% bot_nat_0.extremum_uniqueI
thf(fact_915_le__0__eq,axiom,
! [N: nat] :
( ( ord_less_eq_nat @ N @ zero_zero_nat )
= ( N = zero_zero_nat ) ) ).
% le_0_eq
thf(fact_916_dvd__fact,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq_nat @ one_one_nat @ M )
=> ( ( ord_less_eq_nat @ M @ N )
=> ( dvd_dvd_nat @ M @ ( semiri1408675320244567234ct_nat @ N ) ) ) ) ).
% dvd_fact
thf(fact_917_div__le__mono,axiom,
! [M: nat,N: nat,K: nat] :
( ( ord_less_eq_nat @ M @ N )
=> ( ord_less_eq_nat @ ( divide_divide_nat @ M @ K ) @ ( divide_divide_nat @ N @ K ) ) ) ).
% div_le_mono
thf(fact_918_div__le__dividend,axiom,
! [M: nat,N: nat] : ( ord_less_eq_nat @ ( divide_divide_nat @ M @ N ) @ M ) ).
% div_le_dividend
thf(fact_919_power__strict__decreasing,axiom,
! [N: nat,N3: nat,A: nat] :
( ( ord_less_nat @ N @ N3 )
=> ( ( ord_less_nat @ zero_zero_nat @ A )
=> ( ( ord_less_nat @ A @ one_one_nat )
=> ( ord_less_nat @ ( power_power_nat @ A @ N3 ) @ ( power_power_nat @ A @ N ) ) ) ) ) ).
% power_strict_decreasing
thf(fact_920_power__strict__decreasing,axiom,
! [N: nat,N3: nat,A: int] :
( ( ord_less_nat @ N @ N3 )
=> ( ( ord_less_int @ zero_zero_int @ A )
=> ( ( ord_less_int @ A @ one_one_int )
=> ( ord_less_int @ ( power_power_int @ A @ N3 ) @ ( power_power_int @ A @ N ) ) ) ) ) ).
% power_strict_decreasing
thf(fact_921_power__eq__imp__eq__base,axiom,
! [A: nat,N: nat,B: nat] :
( ( ( power_power_nat @ A @ N )
= ( power_power_nat @ B @ N ) )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ A )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ B )
=> ( ( ord_less_nat @ zero_zero_nat @ N )
=> ( A = B ) ) ) ) ) ).
% power_eq_imp_eq_base
thf(fact_922_power__eq__imp__eq__base,axiom,
! [A: int,N: nat,B: int] :
( ( ( power_power_int @ A @ N )
= ( power_power_int @ B @ N ) )
=> ( ( ord_less_eq_int @ zero_zero_int @ A )
=> ( ( ord_less_eq_int @ zero_zero_int @ B )
=> ( ( ord_less_nat @ zero_zero_nat @ N )
=> ( A = B ) ) ) ) ) ).
% power_eq_imp_eq_base
thf(fact_923_power__eq__iff__eq__base,axiom,
! [N: nat,A: nat,B: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ A )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ B )
=> ( ( ( power_power_nat @ A @ N )
= ( power_power_nat @ B @ N ) )
= ( A = B ) ) ) ) ) ).
% power_eq_iff_eq_base
thf(fact_924_power__eq__iff__eq__base,axiom,
! [N: nat,A: int,B: int] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( ord_less_eq_int @ zero_zero_int @ A )
=> ( ( ord_less_eq_int @ zero_zero_int @ B )
=> ( ( ( power_power_int @ A @ N )
= ( power_power_int @ B @ N ) )
= ( A = B ) ) ) ) ) ).
% power_eq_iff_eq_base
thf(fact_925_power__decreasing,axiom,
! [N: nat,N3: nat,A: nat] :
( ( ord_less_eq_nat @ N @ N3 )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ A )
=> ( ( ord_less_eq_nat @ A @ one_one_nat )
=> ( ord_less_eq_nat @ ( power_power_nat @ A @ N3 ) @ ( power_power_nat @ A @ N ) ) ) ) ) ).
% power_decreasing
thf(fact_926_power__decreasing,axiom,
! [N: nat,N3: nat,A: int] :
( ( ord_less_eq_nat @ N @ N3 )
=> ( ( ord_less_eq_int @ zero_zero_int @ A )
=> ( ( ord_less_eq_int @ A @ one_one_int )
=> ( ord_less_eq_int @ ( power_power_int @ A @ N3 ) @ ( power_power_int @ A @ N ) ) ) ) ) ).
% power_decreasing
thf(fact_927_one__less__power,axiom,
! [A: nat,N: nat] :
( ( ord_less_nat @ one_one_nat @ A )
=> ( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ord_less_nat @ one_one_nat @ ( power_power_nat @ A @ N ) ) ) ) ).
% one_less_power
thf(fact_928_one__less__power,axiom,
! [A: int,N: nat] :
( ( ord_less_int @ one_one_int @ A )
=> ( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ord_less_int @ one_one_int @ ( power_power_int @ A @ N ) ) ) ) ).
% one_less_power
thf(fact_929_self__le__power,axiom,
! [A: nat,N: nat] :
( ( ord_less_eq_nat @ one_one_nat @ A )
=> ( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ord_less_eq_nat @ A @ ( power_power_nat @ A @ N ) ) ) ) ).
% self_le_power
thf(fact_930_self__le__power,axiom,
! [A: int,N: nat] :
( ( ord_less_eq_int @ one_one_int @ A )
=> ( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ord_less_eq_int @ A @ ( power_power_int @ A @ N ) ) ) ) ).
% self_le_power
thf(fact_931_add__less__zeroD,axiom,
! [X4: int,Y: int] :
( ( ord_less_int @ ( plus_plus_int @ X4 @ Y ) @ zero_zero_int )
=> ( ( ord_less_int @ X4 @ zero_zero_int )
| ( ord_less_int @ Y @ zero_zero_int ) ) ) ).
% add_less_zeroD
thf(fact_932_add__neg__neg,axiom,
! [A: nat,B: nat] :
( ( ord_less_nat @ A @ zero_zero_nat )
=> ( ( ord_less_nat @ B @ zero_zero_nat )
=> ( ord_less_nat @ ( plus_plus_nat @ A @ B ) @ zero_zero_nat ) ) ) ).
% add_neg_neg
thf(fact_933_add__neg__neg,axiom,
! [A: int,B: int] :
( ( ord_less_int @ A @ zero_zero_int )
=> ( ( ord_less_int @ B @ zero_zero_int )
=> ( ord_less_int @ ( plus_plus_int @ A @ B ) @ zero_zero_int ) ) ) ).
% add_neg_neg
thf(fact_934_add__pos__pos,axiom,
! [A: nat,B: nat] :
( ( ord_less_nat @ zero_zero_nat @ A )
=> ( ( ord_less_nat @ zero_zero_nat @ B )
=> ( ord_less_nat @ zero_zero_nat @ ( plus_plus_nat @ A @ B ) ) ) ) ).
% add_pos_pos
thf(fact_935_add__pos__pos,axiom,
! [A: int,B: int] :
( ( ord_less_int @ zero_zero_int @ A )
=> ( ( ord_less_int @ zero_zero_int @ B )
=> ( ord_less_int @ zero_zero_int @ ( plus_plus_int @ A @ B ) ) ) ) ).
% add_pos_pos
thf(fact_936_canonically__ordered__monoid__add__class_OlessE,axiom,
! [A: nat,B: nat] :
( ( ord_less_nat @ A @ B )
=> ~ ! [C: nat] :
( ( B
= ( plus_plus_nat @ A @ C ) )
=> ( C = zero_zero_nat ) ) ) ).
% canonically_ordered_monoid_add_class.lessE
thf(fact_937_pos__add__strict,axiom,
! [A: nat,B: nat,C2: nat] :
( ( ord_less_nat @ zero_zero_nat @ A )
=> ( ( ord_less_nat @ B @ C2 )
=> ( ord_less_nat @ B @ ( plus_plus_nat @ A @ C2 ) ) ) ) ).
% pos_add_strict
thf(fact_938_pos__add__strict,axiom,
! [A: int,B: int,C2: int] :
( ( ord_less_int @ zero_zero_int @ A )
=> ( ( ord_less_int @ B @ C2 )
=> ( ord_less_int @ B @ ( plus_plus_int @ A @ C2 ) ) ) ) ).
% pos_add_strict
thf(fact_939_zero__less__numeral,axiom,
! [N: num] : ( ord_less_nat @ zero_zero_nat @ ( numeral_numeral_nat @ N ) ) ).
% zero_less_numeral
thf(fact_940_zero__less__numeral,axiom,
! [N: num] : ( ord_less_int @ zero_zero_int @ ( numeral_numeral_int @ N ) ) ).
% zero_less_numeral
thf(fact_941_not__numeral__less__zero,axiom,
! [N: num] :
~ ( ord_less_nat @ ( numeral_numeral_nat @ N ) @ zero_zero_nat ) ).
% not_numeral_less_zero
thf(fact_942_not__numeral__less__zero,axiom,
! [N: num] :
~ ( ord_less_int @ ( numeral_numeral_int @ N ) @ zero_zero_int ) ).
% not_numeral_less_zero
thf(fact_943_less__iff__diff__less__0,axiom,
( ord_less_int
= ( ^ [A4: int,B4: int] : ( ord_less_int @ ( minus_minus_int @ A4 @ B4 ) @ zero_zero_int ) ) ) ).
% less_iff_diff_less_0
thf(fact_944_not__one__less__zero,axiom,
~ ( ord_less_nat @ one_one_nat @ zero_zero_nat ) ).
% not_one_less_zero
thf(fact_945_not__one__less__zero,axiom,
~ ( ord_less_int @ one_one_int @ zero_zero_int ) ).
% not_one_less_zero
thf(fact_946_zero__less__one,axiom,
ord_less_nat @ zero_zero_nat @ one_one_nat ).
% zero_less_one
thf(fact_947_zero__less__one,axiom,
ord_less_int @ zero_zero_int @ one_one_int ).
% zero_less_one
thf(fact_948_less__numeral__extra_I1_J,axiom,
ord_less_nat @ zero_zero_nat @ one_one_nat ).
% less_numeral_extra(1)
thf(fact_949_less__numeral__extra_I1_J,axiom,
ord_less_int @ zero_zero_int @ one_one_int ).
% less_numeral_extra(1)
thf(fact_950_diff__less__eq,axiom,
! [A: int,B: int,C2: int] :
( ( ord_less_int @ ( minus_minus_int @ A @ B ) @ C2 )
= ( ord_less_int @ A @ ( plus_plus_int @ C2 @ B ) ) ) ).
% diff_less_eq
thf(fact_951_less__diff__eq,axiom,
! [A: int,C2: int,B: int] :
( ( ord_less_int @ A @ ( minus_minus_int @ C2 @ B ) )
= ( ord_less_int @ ( plus_plus_int @ A @ B ) @ C2 ) ) ).
% less_diff_eq
thf(fact_952_linordered__semidom__class_Oadd__diff__inverse,axiom,
! [A: nat,B: nat] :
( ~ ( ord_less_nat @ A @ B )
=> ( ( plus_plus_nat @ B @ ( minus_minus_nat @ A @ B ) )
= A ) ) ).
% linordered_semidom_class.add_diff_inverse
thf(fact_953_linordered__semidom__class_Oadd__diff__inverse,axiom,
! [A: int,B: int] :
( ~ ( ord_less_int @ A @ B )
=> ( ( plus_plus_int @ B @ ( minus_minus_int @ A @ B ) )
= A ) ) ).
% linordered_semidom_class.add_diff_inverse
thf(fact_954_add__mono1,axiom,
! [A: nat,B: nat] :
( ( ord_less_nat @ A @ B )
=> ( ord_less_nat @ ( plus_plus_nat @ A @ one_one_nat ) @ ( plus_plus_nat @ B @ one_one_nat ) ) ) ).
% add_mono1
thf(fact_955_add__mono1,axiom,
! [A: int,B: int] :
( ( ord_less_int @ A @ B )
=> ( ord_less_int @ ( plus_plus_int @ A @ one_one_int ) @ ( plus_plus_int @ B @ one_one_int ) ) ) ).
% add_mono1
thf(fact_956_less__add__one,axiom,
! [A: nat] : ( ord_less_nat @ A @ ( plus_plus_nat @ A @ one_one_nat ) ) ).
% less_add_one
thf(fact_957_less__add__one,axiom,
! [A: int] : ( ord_less_int @ A @ ( plus_plus_int @ A @ one_one_int ) ) ).
% less_add_one
thf(fact_958_not__numeral__less__one,axiom,
! [N: num] :
~ ( ord_less_nat @ ( numeral_numeral_nat @ N ) @ one_one_nat ) ).
% not_numeral_less_one
thf(fact_959_not__numeral__less__one,axiom,
! [N: num] :
~ ( ord_less_int @ ( numeral_numeral_int @ N ) @ one_one_int ) ).
% not_numeral_less_one
thf(fact_960_add__decreasing,axiom,
! [A: nat,C2: nat,B: nat] :
( ( ord_less_eq_nat @ A @ zero_zero_nat )
=> ( ( ord_less_eq_nat @ C2 @ B )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ A @ C2 ) @ B ) ) ) ).
% add_decreasing
thf(fact_961_add__decreasing,axiom,
! [A: int,C2: int,B: int] :
( ( ord_less_eq_int @ A @ zero_zero_int )
=> ( ( ord_less_eq_int @ C2 @ B )
=> ( ord_less_eq_int @ ( plus_plus_int @ A @ C2 ) @ B ) ) ) ).
% add_decreasing
thf(fact_962_add__increasing,axiom,
! [A: nat,B: nat,C2: nat] :
( ( ord_less_eq_nat @ zero_zero_nat @ A )
=> ( ( ord_less_eq_nat @ B @ C2 )
=> ( ord_less_eq_nat @ B @ ( plus_plus_nat @ A @ C2 ) ) ) ) ).
% add_increasing
thf(fact_963_add__increasing,axiom,
! [A: int,B: int,C2: int] :
( ( ord_less_eq_int @ zero_zero_int @ A )
=> ( ( ord_less_eq_int @ B @ C2 )
=> ( ord_less_eq_int @ B @ ( plus_plus_int @ A @ C2 ) ) ) ) ).
% add_increasing
thf(fact_964_add__decreasing2,axiom,
! [C2: nat,A: nat,B: nat] :
( ( ord_less_eq_nat @ C2 @ zero_zero_nat )
=> ( ( ord_less_eq_nat @ A @ B )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ A @ C2 ) @ B ) ) ) ).
% add_decreasing2
thf(fact_965_add__decreasing2,axiom,
! [C2: int,A: int,B: int] :
( ( ord_less_eq_int @ C2 @ zero_zero_int )
=> ( ( ord_less_eq_int @ A @ B )
=> ( ord_less_eq_int @ ( plus_plus_int @ A @ C2 ) @ B ) ) ) ).
% add_decreasing2
thf(fact_966_add__increasing2,axiom,
! [C2: nat,B: nat,A: nat] :
( ( ord_less_eq_nat @ zero_zero_nat @ C2 )
=> ( ( ord_less_eq_nat @ B @ A )
=> ( ord_less_eq_nat @ B @ ( plus_plus_nat @ A @ C2 ) ) ) ) ).
% add_increasing2
thf(fact_967_add__increasing2,axiom,
! [C2: int,B: int,A: int] :
( ( ord_less_eq_int @ zero_zero_int @ C2 )
=> ( ( ord_less_eq_int @ B @ A )
=> ( ord_less_eq_int @ B @ ( plus_plus_int @ A @ C2 ) ) ) ) ).
% add_increasing2
thf(fact_968_add__nonneg__nonneg,axiom,
! [A: nat,B: nat] :
( ( ord_less_eq_nat @ zero_zero_nat @ A )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ B )
=> ( ord_less_eq_nat @ zero_zero_nat @ ( plus_plus_nat @ A @ B ) ) ) ) ).
% add_nonneg_nonneg
thf(fact_969_add__nonneg__nonneg,axiom,
! [A: int,B: int] :
( ( ord_less_eq_int @ zero_zero_int @ A )
=> ( ( ord_less_eq_int @ zero_zero_int @ B )
=> ( ord_less_eq_int @ zero_zero_int @ ( plus_plus_int @ A @ B ) ) ) ) ).
% add_nonneg_nonneg
thf(fact_970_add__nonpos__nonpos,axiom,
! [A: nat,B: nat] :
( ( ord_less_eq_nat @ A @ zero_zero_nat )
=> ( ( ord_less_eq_nat @ B @ zero_zero_nat )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ A @ B ) @ zero_zero_nat ) ) ) ).
% add_nonpos_nonpos
thf(fact_971_add__nonpos__nonpos,axiom,
! [A: int,B: int] :
( ( ord_less_eq_int @ A @ zero_zero_int )
=> ( ( ord_less_eq_int @ B @ zero_zero_int )
=> ( ord_less_eq_int @ ( plus_plus_int @ A @ B ) @ zero_zero_int ) ) ) ).
% add_nonpos_nonpos
thf(fact_972_add__nonneg__eq__0__iff,axiom,
! [X4: nat,Y: nat] :
( ( ord_less_eq_nat @ zero_zero_nat @ X4 )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ Y )
=> ( ( ( plus_plus_nat @ X4 @ Y )
= zero_zero_nat )
= ( ( X4 = zero_zero_nat )
& ( Y = zero_zero_nat ) ) ) ) ) ).
% add_nonneg_eq_0_iff
thf(fact_973_add__nonneg__eq__0__iff,axiom,
! [X4: int,Y: int] :
( ( ord_less_eq_int @ zero_zero_int @ X4 )
=> ( ( ord_less_eq_int @ zero_zero_int @ Y )
=> ( ( ( plus_plus_int @ X4 @ Y )
= zero_zero_int )
= ( ( X4 = zero_zero_int )
& ( Y = zero_zero_int ) ) ) ) ) ).
% add_nonneg_eq_0_iff
thf(fact_974_add__nonpos__eq__0__iff,axiom,
! [X4: nat,Y: nat] :
( ( ord_less_eq_nat @ X4 @ zero_zero_nat )
=> ( ( ord_less_eq_nat @ Y @ zero_zero_nat )
=> ( ( ( plus_plus_nat @ X4 @ Y )
= zero_zero_nat )
= ( ( X4 = zero_zero_nat )
& ( Y = zero_zero_nat ) ) ) ) ) ).
% add_nonpos_eq_0_iff
thf(fact_975_add__nonpos__eq__0__iff,axiom,
! [X4: int,Y: int] :
( ( ord_less_eq_int @ X4 @ zero_zero_int )
=> ( ( ord_less_eq_int @ Y @ zero_zero_int )
=> ( ( ( plus_plus_int @ X4 @ Y )
= zero_zero_int )
= ( ( X4 = zero_zero_int )
& ( Y = zero_zero_int ) ) ) ) ) ).
% add_nonpos_eq_0_iff
thf(fact_976_zero__le__numeral,axiom,
! [N: num] : ( ord_less_eq_nat @ zero_zero_nat @ ( numeral_numeral_nat @ N ) ) ).
% zero_le_numeral
thf(fact_977_zero__le__numeral,axiom,
! [N: num] : ( ord_less_eq_int @ zero_zero_int @ ( numeral_numeral_int @ N ) ) ).
% zero_le_numeral
thf(fact_978_not__numeral__le__zero,axiom,
! [N: num] :
~ ( ord_less_eq_nat @ ( numeral_numeral_nat @ N ) @ zero_zero_nat ) ).
% not_numeral_le_zero
thf(fact_979_not__numeral__le__zero,axiom,
! [N: num] :
~ ( ord_less_eq_int @ ( numeral_numeral_int @ N ) @ zero_zero_int ) ).
% not_numeral_le_zero
thf(fact_980_le__iff__diff__le__0,axiom,
( ord_less_eq_int
= ( ^ [A4: int,B4: int] : ( ord_less_eq_int @ ( minus_minus_int @ A4 @ B4 ) @ zero_zero_int ) ) ) ).
% le_iff_diff_le_0
thf(fact_981_not__one__le__zero,axiom,
~ ( ord_less_eq_nat @ one_one_nat @ zero_zero_nat ) ).
% not_one_le_zero
thf(fact_982_not__one__le__zero,axiom,
~ ( ord_less_eq_int @ one_one_int @ zero_zero_int ) ).
% not_one_le_zero
thf(fact_983_linordered__nonzero__semiring__class_Ozero__le__one,axiom,
ord_less_eq_nat @ zero_zero_nat @ one_one_nat ).
% linordered_nonzero_semiring_class.zero_le_one
thf(fact_984_linordered__nonzero__semiring__class_Ozero__le__one,axiom,
ord_less_eq_int @ zero_zero_int @ one_one_int ).
% linordered_nonzero_semiring_class.zero_le_one
thf(fact_985_zero__less__one__class_Ozero__le__one,axiom,
ord_less_eq_nat @ zero_zero_nat @ one_one_nat ).
% zero_less_one_class.zero_le_one
thf(fact_986_zero__less__one__class_Ozero__le__one,axiom,
ord_less_eq_int @ zero_zero_int @ one_one_int ).
% zero_less_one_class.zero_le_one
thf(fact_987_not__numeral__less__neg__numeral,axiom,
! [M: num,N: num] :
~ ( ord_less_int @ ( numeral_numeral_int @ M ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ N ) ) ) ).
% not_numeral_less_neg_numeral
thf(fact_988_neg__numeral__less__numeral,axiom,
! [M: num,N: num] : ( ord_less_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ M ) ) @ ( numeral_numeral_int @ N ) ) ).
% neg_numeral_less_numeral
thf(fact_989_zero__less__power,axiom,
! [A: nat,N: nat] :
( ( ord_less_nat @ zero_zero_nat @ A )
=> ( ord_less_nat @ zero_zero_nat @ ( power_power_nat @ A @ N ) ) ) ).
% zero_less_power
thf(fact_990_zero__less__power,axiom,
! [A: int,N: nat] :
( ( ord_less_int @ zero_zero_int @ A )
=> ( ord_less_int @ zero_zero_int @ ( power_power_int @ A @ N ) ) ) ).
% zero_less_power
thf(fact_991_less__minus__one__simps_I4_J,axiom,
~ ( ord_less_int @ one_one_int @ ( uminus_uminus_int @ one_one_int ) ) ).
% less_minus_one_simps(4)
thf(fact_992_less__minus__one__simps_I2_J,axiom,
ord_less_int @ ( uminus_uminus_int @ one_one_int ) @ one_one_int ).
% less_minus_one_simps(2)
thf(fact_993_diff__le__eq,axiom,
! [A: int,B: int,C2: int] :
( ( ord_less_eq_int @ ( minus_minus_int @ A @ B ) @ C2 )
= ( ord_less_eq_int @ A @ ( plus_plus_int @ C2 @ B ) ) ) ).
% diff_le_eq
thf(fact_994_le__diff__eq,axiom,
! [A: int,C2: int,B: int] :
( ( ord_less_eq_int @ A @ ( minus_minus_int @ C2 @ B ) )
= ( ord_less_eq_int @ ( plus_plus_int @ A @ B ) @ C2 ) ) ).
% le_diff_eq
thf(fact_995_add__le__imp__le__diff,axiom,
! [I: nat,K: nat,N: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ I @ K ) @ N )
=> ( ord_less_eq_nat @ I @ ( minus_minus_nat @ N @ K ) ) ) ).
% add_le_imp_le_diff
thf(fact_996_add__le__imp__le__diff,axiom,
! [I: int,K: int,N: int] :
( ( ord_less_eq_int @ ( plus_plus_int @ I @ K ) @ N )
=> ( ord_less_eq_int @ I @ ( minus_minus_int @ N @ K ) ) ) ).
% add_le_imp_le_diff
thf(fact_997_ordered__cancel__comm__monoid__diff__class_Odiff__add,axiom,
! [A: nat,B: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( plus_plus_nat @ ( minus_minus_nat @ B @ A ) @ A )
= B ) ) ).
% ordered_cancel_comm_monoid_diff_class.diff_add
thf(fact_998_add__le__add__imp__diff__le,axiom,
! [I: nat,K: nat,N: nat,J: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ I @ K ) @ N )
=> ( ( ord_less_eq_nat @ N @ ( plus_plus_nat @ J @ K ) )
=> ( ( ord_less_eq_nat @ ( plus_plus_nat @ I @ K ) @ N )
=> ( ( ord_less_eq_nat @ N @ ( plus_plus_nat @ J @ K ) )
=> ( ord_less_eq_nat @ ( minus_minus_nat @ N @ K ) @ J ) ) ) ) ) ).
% add_le_add_imp_diff_le
thf(fact_999_add__le__add__imp__diff__le,axiom,
! [I: int,K: int,N: int,J: int] :
( ( ord_less_eq_int @ ( plus_plus_int @ I @ K ) @ N )
=> ( ( ord_less_eq_int @ N @ ( plus_plus_int @ J @ K ) )
=> ( ( ord_less_eq_int @ ( plus_plus_int @ I @ K ) @ N )
=> ( ( ord_less_eq_int @ N @ ( plus_plus_int @ J @ K ) )
=> ( ord_less_eq_int @ ( minus_minus_int @ N @ K ) @ J ) ) ) ) ) ).
% add_le_add_imp_diff_le
thf(fact_1000_le__add__diff,axiom,
! [A: nat,B: nat,C2: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ord_less_eq_nat @ C2 @ ( minus_minus_nat @ ( plus_plus_nat @ B @ C2 ) @ A ) ) ) ).
% le_add_diff
thf(fact_1001_ordered__cancel__comm__monoid__diff__class_Ole__diff__conv2,axiom,
! [A: nat,B: nat,C2: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ord_less_eq_nat @ C2 @ ( minus_minus_nat @ B @ A ) )
= ( ord_less_eq_nat @ ( plus_plus_nat @ C2 @ A ) @ B ) ) ) ).
% ordered_cancel_comm_monoid_diff_class.le_diff_conv2
thf(fact_1002_ordered__cancel__comm__monoid__diff__class_Oadd__diff__assoc,axiom,
! [A: nat,B: nat,C2: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( plus_plus_nat @ C2 @ ( minus_minus_nat @ B @ A ) )
= ( minus_minus_nat @ ( plus_plus_nat @ C2 @ B ) @ A ) ) ) ).
% ordered_cancel_comm_monoid_diff_class.add_diff_assoc
thf(fact_1003_ordered__cancel__comm__monoid__diff__class_Odiff__add__assoc,axiom,
! [A: nat,B: nat,C2: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( minus_minus_nat @ ( plus_plus_nat @ C2 @ B ) @ A )
= ( plus_plus_nat @ C2 @ ( minus_minus_nat @ B @ A ) ) ) ) ).
% ordered_cancel_comm_monoid_diff_class.diff_add_assoc
thf(fact_1004_ordered__cancel__comm__monoid__diff__class_Oadd__diff__assoc2,axiom,
! [A: nat,B: nat,C2: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( plus_plus_nat @ ( minus_minus_nat @ B @ A ) @ C2 )
= ( minus_minus_nat @ ( plus_plus_nat @ B @ C2 ) @ A ) ) ) ).
% ordered_cancel_comm_monoid_diff_class.add_diff_assoc2
thf(fact_1005_ordered__cancel__comm__monoid__diff__class_Odiff__add__assoc2,axiom,
! [A: nat,B: nat,C2: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( minus_minus_nat @ ( plus_plus_nat @ B @ C2 ) @ A )
= ( plus_plus_nat @ ( minus_minus_nat @ B @ A ) @ C2 ) ) ) ).
% ordered_cancel_comm_monoid_diff_class.diff_add_assoc2
thf(fact_1006_ordered__cancel__comm__monoid__diff__class_Odiff__diff__right,axiom,
! [A: nat,B: nat,C2: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( minus_minus_nat @ C2 @ ( minus_minus_nat @ B @ A ) )
= ( minus_minus_nat @ ( plus_plus_nat @ C2 @ A ) @ B ) ) ) ).
% ordered_cancel_comm_monoid_diff_class.diff_diff_right
thf(fact_1007_ordered__cancel__comm__monoid__diff__class_Oadd__diff__inverse,axiom,
! [A: nat,B: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( plus_plus_nat @ A @ ( minus_minus_nat @ B @ A ) )
= B ) ) ).
% ordered_cancel_comm_monoid_diff_class.add_diff_inverse
thf(fact_1008_ordered__cancel__comm__monoid__diff__class_Ole__imp__diff__is__add,axiom,
! [A: nat,B: nat,C2: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ord_less_eq_nat @ A @ B )
=> ( ( ( minus_minus_nat @ B @ A )
= C2 )
= ( B
= ( plus_plus_nat @ C2 @ A ) ) ) ) ) ).
% ordered_cancel_comm_monoid_diff_class.le_imp_diff_is_add
thf(fact_1009_one__le__numeral,axiom,
! [N: num] : ( ord_less_eq_nat @ one_one_nat @ ( numeral_numeral_nat @ N ) ) ).
% one_le_numeral
thf(fact_1010_one__le__numeral,axiom,
! [N: num] : ( ord_less_eq_int @ one_one_int @ ( numeral_numeral_int @ N ) ) ).
% one_le_numeral
thf(fact_1011_not__numeral__le__neg__numeral,axiom,
! [M: num,N: num] :
~ ( ord_less_eq_int @ ( numeral_numeral_int @ M ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ N ) ) ) ).
% not_numeral_le_neg_numeral
thf(fact_1012_neg__numeral__le__numeral,axiom,
! [M: num,N: num] : ( ord_less_eq_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ M ) ) @ ( numeral_numeral_int @ N ) ) ).
% neg_numeral_le_numeral
thf(fact_1013_power__mono,axiom,
! [A: nat,B: nat,N: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ A )
=> ( ord_less_eq_nat @ ( power_power_nat @ A @ N ) @ ( power_power_nat @ B @ N ) ) ) ) ).
% power_mono
thf(fact_1014_power__mono,axiom,
! [A: int,B: int,N: nat] :
( ( ord_less_eq_int @ A @ B )
=> ( ( ord_less_eq_int @ zero_zero_int @ A )
=> ( ord_less_eq_int @ ( power_power_int @ A @ N ) @ ( power_power_int @ B @ N ) ) ) ) ).
% power_mono
thf(fact_1015_zero__le__power,axiom,
! [A: nat,N: nat] :
( ( ord_less_eq_nat @ zero_zero_nat @ A )
=> ( ord_less_eq_nat @ zero_zero_nat @ ( power_power_nat @ A @ N ) ) ) ).
% zero_le_power
thf(fact_1016_zero__le__power,axiom,
! [A: int,N: nat] :
( ( ord_less_eq_int @ zero_zero_int @ A )
=> ( ord_less_eq_int @ zero_zero_int @ ( power_power_int @ A @ N ) ) ) ).
% zero_le_power
thf(fact_1017_le__minus__one__simps_I4_J,axiom,
~ ( ord_less_eq_int @ one_one_int @ ( uminus_uminus_int @ one_one_int ) ) ).
% le_minus_one_simps(4)
thf(fact_1018_le__minus__one__simps_I2_J,axiom,
ord_less_eq_int @ ( uminus_uminus_int @ one_one_int ) @ one_one_int ).
% le_minus_one_simps(2)
thf(fact_1019_one__le__power,axiom,
! [A: nat,N: nat] :
( ( ord_less_eq_nat @ one_one_nat @ A )
=> ( ord_less_eq_nat @ one_one_nat @ ( power_power_nat @ A @ N ) ) ) ).
% one_le_power
thf(fact_1020_one__le__power,axiom,
! [A: int,N: nat] :
( ( ord_less_eq_int @ one_one_int @ A )
=> ( ord_less_eq_int @ one_one_int @ ( power_power_int @ A @ N ) ) ) ).
% one_le_power
thf(fact_1021_less__imp__add__positive,axiom,
! [I: nat,J: nat] :
( ( ord_less_nat @ I @ J )
=> ? [K3: nat] :
( ( ord_less_nat @ zero_zero_nat @ K3 )
& ( ( plus_plus_nat @ I @ K3 )
= J ) ) ) ).
% less_imp_add_positive
thf(fact_1022_diff__less,axiom,
! [N: nat,M: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( ord_less_nat @ zero_zero_nat @ M )
=> ( ord_less_nat @ ( minus_minus_nat @ M @ N ) @ M ) ) ) ).
% diff_less
thf(fact_1023_zero__less__eq__of__bool,axiom,
! [P2: $o] : ( ord_less_eq_nat @ zero_zero_nat @ ( zero_n2687167440665602831ol_nat @ P2 ) ) ).
% zero_less_eq_of_bool
thf(fact_1024_zero__less__eq__of__bool,axiom,
! [P2: $o] : ( ord_less_eq_int @ zero_zero_int @ ( zero_n2684676970156552555ol_int @ P2 ) ) ).
% zero_less_eq_of_bool
thf(fact_1025_fact__gt__zero,axiom,
! [N: nat] : ( ord_less_nat @ zero_zero_nat @ ( semiri1408675320244567234ct_nat @ N ) ) ).
% fact_gt_zero
thf(fact_1026_fact__gt__zero,axiom,
! [N: nat] : ( ord_less_int @ zero_zero_int @ ( semiri1406184849735516958ct_int @ N ) ) ).
% fact_gt_zero
thf(fact_1027_fact__not__neg,axiom,
! [N: nat] :
~ ( ord_less_nat @ ( semiri1408675320244567234ct_nat @ N ) @ zero_zero_nat ) ).
% fact_not_neg
thf(fact_1028_fact__not__neg,axiom,
! [N: nat] :
~ ( ord_less_int @ ( semiri1406184849735516958ct_int @ N ) @ zero_zero_int ) ).
% fact_not_neg
thf(fact_1029_of__bool__less__eq__one,axiom,
! [P2: $o] : ( ord_less_eq_nat @ ( zero_n2687167440665602831ol_nat @ P2 ) @ one_one_nat ) ).
% of_bool_less_eq_one
thf(fact_1030_of__bool__less__eq__one,axiom,
! [P2: $o] : ( ord_less_eq_int @ ( zero_n2684676970156552555ol_int @ P2 ) @ one_one_int ) ).
% of_bool_less_eq_one
thf(fact_1031_fact__ge__zero,axiom,
! [N: nat] : ( ord_less_eq_nat @ zero_zero_nat @ ( semiri1408675320244567234ct_nat @ N ) ) ).
% fact_ge_zero
thf(fact_1032_fact__ge__zero,axiom,
! [N: nat] : ( ord_less_eq_int @ zero_zero_int @ ( semiri1406184849735516958ct_int @ N ) ) ).
% fact_ge_zero
thf(fact_1033_Euclidean__Division_Odiv__eq__0__iff,axiom,
! [M: nat,N: nat] :
( ( ( divide_divide_nat @ M @ N )
= zero_zero_nat )
= ( ( ord_less_nat @ M @ N )
| ( N = zero_zero_nat ) ) ) ).
% Euclidean_Division.div_eq_0_iff
thf(fact_1034_fact__ge__1,axiom,
! [N: nat] : ( ord_less_eq_nat @ one_one_nat @ ( semiri1408675320244567234ct_nat @ N ) ) ).
% fact_ge_1
thf(fact_1035_fact__ge__1,axiom,
! [N: nat] : ( ord_less_eq_int @ one_one_int @ ( semiri1406184849735516958ct_int @ N ) ) ).
% fact_ge_1
thf(fact_1036_fact__dvd,axiom,
! [N: nat,M: nat] :
( ( ord_less_eq_nat @ N @ M )
=> ( dvd_dvd_nat @ ( semiri1408675320244567234ct_nat @ N ) @ ( semiri1408675320244567234ct_nat @ M ) ) ) ).
% fact_dvd
thf(fact_1037_fact__dvd,axiom,
! [N: nat,M: nat] :
( ( ord_less_eq_nat @ N @ M )
=> ( dvd_dvd_int @ ( semiri1406184849735516958ct_int @ N ) @ ( semiri1406184849735516958ct_int @ M ) ) ) ).
% fact_dvd
thf(fact_1038_fact__div__fact__le__pow,axiom,
! [R: nat,N: nat] :
( ( ord_less_eq_nat @ R @ N )
=> ( ord_less_eq_nat @ ( divide_divide_nat @ ( semiri1408675320244567234ct_nat @ N ) @ ( semiri1408675320244567234ct_nat @ ( minus_minus_nat @ N @ R ) ) ) @ ( power_power_nat @ N @ R ) ) ) ).
% fact_div_fact_le_pow
thf(fact_1039_power2__less__imp__less,axiom,
! [X4: nat,Y: nat] :
( ( ord_less_nat @ ( power_power_nat @ X4 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_nat @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ Y )
=> ( ord_less_nat @ X4 @ Y ) ) ) ).
% power2_less_imp_less
thf(fact_1040_power2__less__imp__less,axiom,
! [X4: int,Y: int] :
( ( ord_less_int @ ( power_power_int @ X4 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_int @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
=> ( ( ord_less_eq_int @ zero_zero_int @ Y )
=> ( ord_less_int @ X4 @ Y ) ) ) ).
% power2_less_imp_less
thf(fact_1041_ex__power__ivl2,axiom,
! [B: nat,K: nat] :
( ( ord_less_eq_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ B )
=> ( ( ord_less_eq_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ K )
=> ? [N2: nat] :
( ( ord_less_nat @ ( power_power_nat @ B @ N2 ) @ K )
& ( ord_less_eq_nat @ K @ ( power_power_nat @ B @ ( plus_plus_nat @ N2 @ one_one_nat ) ) ) ) ) ) ).
% ex_power_ivl2
thf(fact_1042_ex__power__ivl1,axiom,
! [B: nat,K: nat] :
( ( ord_less_eq_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ B )
=> ( ( ord_less_eq_nat @ one_one_nat @ K )
=> ? [N2: nat] :
( ( ord_less_eq_nat @ ( power_power_nat @ B @ N2 ) @ K )
& ( ord_less_nat @ K @ ( power_power_nat @ B @ ( plus_plus_nat @ N2 @ one_one_nat ) ) ) ) ) ) ).
% ex_power_ivl1
thf(fact_1043_power__le__zero__eq,axiom,
! [A: int,N: nat] :
( ( ord_less_eq_int @ ( power_power_int @ A @ N ) @ zero_zero_int )
= ( ( ord_less_nat @ zero_zero_nat @ N )
& ( ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
& ( ord_less_eq_int @ A @ zero_zero_int ) )
| ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
& ( A = zero_zero_int ) ) ) ) ) ).
% power_le_zero_eq
thf(fact_1044_zero__less__two,axiom,
ord_less_nat @ zero_zero_nat @ ( plus_plus_nat @ one_one_nat @ one_one_nat ) ).
% zero_less_two
thf(fact_1045_zero__less__two,axiom,
ord_less_int @ zero_zero_int @ ( plus_plus_int @ one_one_int @ one_one_int ) ).
% zero_less_two
thf(fact_1046_neg__numeral__less__zero,axiom,
! [N: num] : ( ord_less_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ N ) ) @ zero_zero_int ) ).
% neg_numeral_less_zero
thf(fact_1047_not__zero__less__neg__numeral,axiom,
! [N: num] :
~ ( ord_less_int @ zero_zero_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ N ) ) ) ).
% not_zero_less_neg_numeral
thf(fact_1048_less__minus__one__simps_I3_J,axiom,
~ ( ord_less_int @ zero_zero_int @ ( uminus_uminus_int @ one_one_int ) ) ).
% less_minus_one_simps(3)
thf(fact_1049_less__minus__one__simps_I1_J,axiom,
ord_less_int @ ( uminus_uminus_int @ one_one_int ) @ zero_zero_int ).
% less_minus_one_simps(1)
thf(fact_1050_not__neg__one__less__neg__numeral,axiom,
! [M: num] :
~ ( ord_less_int @ ( uminus_uminus_int @ one_one_int ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ M ) ) ) ).
% not_neg_one_less_neg_numeral
thf(fact_1051_not__one__less__neg__numeral,axiom,
! [M: num] :
~ ( ord_less_int @ one_one_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ M ) ) ) ).
% not_one_less_neg_numeral
thf(fact_1052_not__numeral__less__neg__one,axiom,
! [M: num] :
~ ( ord_less_int @ ( numeral_numeral_int @ M ) @ ( uminus_uminus_int @ one_one_int ) ) ).
% not_numeral_less_neg_one
thf(fact_1053_neg__one__less__numeral,axiom,
! [M: num] : ( ord_less_int @ ( uminus_uminus_int @ one_one_int ) @ ( numeral_numeral_int @ M ) ) ).
% neg_one_less_numeral
thf(fact_1054_neg__numeral__less__one,axiom,
! [M: num] : ( ord_less_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ M ) ) @ one_one_int ) ).
% neg_numeral_less_one
thf(fact_1055_neg__numeral__le__zero,axiom,
! [N: num] : ( ord_less_eq_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ N ) ) @ zero_zero_int ) ).
% neg_numeral_le_zero
thf(fact_1056_not__zero__le__neg__numeral,axiom,
! [N: num] :
~ ( ord_less_eq_int @ zero_zero_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ N ) ) ) ).
% not_zero_le_neg_numeral
thf(fact_1057_le__minus__one__simps_I3_J,axiom,
~ ( ord_less_eq_int @ zero_zero_int @ ( uminus_uminus_int @ one_one_int ) ) ).
% le_minus_one_simps(3)
thf(fact_1058_le__minus__one__simps_I1_J,axiom,
ord_less_eq_int @ ( uminus_uminus_int @ one_one_int ) @ zero_zero_int ).
% le_minus_one_simps(1)
thf(fact_1059_not__one__le__neg__numeral,axiom,
! [M: num] :
~ ( ord_less_eq_int @ one_one_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ M ) ) ) ).
% not_one_le_neg_numeral
thf(fact_1060_not__numeral__le__neg__one,axiom,
! [M: num] :
~ ( ord_less_eq_int @ ( numeral_numeral_int @ M ) @ ( uminus_uminus_int @ one_one_int ) ) ).
% not_numeral_le_neg_one
thf(fact_1061_neg__numeral__le__neg__one,axiom,
! [M: num] : ( ord_less_eq_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ M ) ) @ ( uminus_uminus_int @ one_one_int ) ) ).
% neg_numeral_le_neg_one
thf(fact_1062_neg__one__le__numeral,axiom,
! [M: num] : ( ord_less_eq_int @ ( uminus_uminus_int @ one_one_int ) @ ( numeral_numeral_int @ M ) ) ).
% neg_one_le_numeral
thf(fact_1063_neg__numeral__le__one,axiom,
! [M: num] : ( ord_less_eq_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ M ) ) @ one_one_int ) ).
% neg_numeral_le_one
thf(fact_1064_power__le__one,axiom,
! [A: nat,N: nat] :
( ( ord_less_eq_nat @ zero_zero_nat @ A )
=> ( ( ord_less_eq_nat @ A @ one_one_nat )
=> ( ord_less_eq_nat @ ( power_power_nat @ A @ N ) @ one_one_nat ) ) ) ).
% power_le_one
thf(fact_1065_power__le__one,axiom,
! [A: int,N: nat] :
( ( ord_less_eq_int @ zero_zero_int @ A )
=> ( ( ord_less_eq_int @ A @ one_one_int )
=> ( ord_less_eq_int @ ( power_power_int @ A @ N ) @ one_one_int ) ) ) ).
% power_le_one
thf(fact_1066_zero__power,axiom,
! [N: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( power_power_int @ zero_zero_int @ N )
= zero_zero_int ) ) ).
% zero_power
thf(fact_1067_zero__power,axiom,
! [N: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( power_power_nat @ zero_zero_nat @ N )
= zero_zero_nat ) ) ).
% zero_power
thf(fact_1068_less__exp,axiom,
! [N: nat] : ( ord_less_nat @ N @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ).
% less_exp
thf(fact_1069_power2__nat__le__imp__le,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq_nat @ ( power_power_nat @ M @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ N )
=> ( ord_less_eq_nat @ M @ N ) ) ).
% power2_nat_le_imp_le
thf(fact_1070_power2__nat__le__eq__le,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq_nat @ ( power_power_nat @ M @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_nat @ N @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
= ( ord_less_eq_nat @ M @ N ) ) ).
% power2_nat_le_eq_le
thf(fact_1071_self__le__ge2__pow,axiom,
! [K: nat,M: nat] :
( ( ord_less_eq_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ K )
=> ( ord_less_eq_nat @ M @ ( power_power_nat @ K @ M ) ) ) ).
% self_le_ge2_pow
thf(fact_1072_dvd__power__iff__le,axiom,
! [K: nat,M: nat,N: nat] :
( ( ord_less_eq_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ K )
=> ( ( dvd_dvd_nat @ ( power_power_nat @ K @ M ) @ ( power_power_nat @ K @ N ) )
= ( ord_less_eq_nat @ M @ N ) ) ) ).
% dvd_power_iff_le
thf(fact_1073_nat__diff__split,axiom,
! [P2: nat > $o,A: nat,B: nat] :
( ( P2 @ ( minus_minus_nat @ A @ B ) )
= ( ( ( ord_less_nat @ A @ B )
=> ( P2 @ zero_zero_nat ) )
& ! [D2: nat] :
( ( A
= ( plus_plus_nat @ B @ D2 ) )
=> ( P2 @ D2 ) ) ) ) ).
% nat_diff_split
thf(fact_1074_nat__diff__split__asm,axiom,
! [P2: nat > $o,A: nat,B: nat] :
( ( P2 @ ( minus_minus_nat @ A @ B ) )
= ( ~ ( ( ( ord_less_nat @ A @ B )
& ~ ( P2 @ zero_zero_nat ) )
| ? [D2: nat] :
( ( A
= ( plus_plus_nat @ B @ D2 ) )
& ~ ( P2 @ D2 ) ) ) ) ) ).
% nat_diff_split_asm
thf(fact_1075_div__less__dividend,axiom,
! [N: nat,M: nat] :
( ( ord_less_nat @ one_one_nat @ N )
=> ( ( ord_less_nat @ zero_zero_nat @ M )
=> ( ord_less_nat @ ( divide_divide_nat @ M @ N ) @ M ) ) ) ).
% div_less_dividend
thf(fact_1076_div__eq__dividend__iff,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ zero_zero_nat @ M )
=> ( ( ( divide_divide_nat @ M @ N )
= M )
= ( N = one_one_nat ) ) ) ).
% div_eq_dividend_iff
thf(fact_1077_dvd__power__iff,axiom,
! [X4: int,M: nat,N: nat] :
( ( X4 != zero_zero_int )
=> ( ( dvd_dvd_int @ ( power_power_int @ X4 @ M ) @ ( power_power_int @ X4 @ N ) )
= ( ( dvd_dvd_int @ X4 @ one_one_int )
| ( ord_less_eq_nat @ M @ N ) ) ) ) ).
% dvd_power_iff
thf(fact_1078_dvd__power__iff,axiom,
! [X4: nat,M: nat,N: nat] :
( ( X4 != zero_zero_nat )
=> ( ( dvd_dvd_nat @ ( power_power_nat @ X4 @ M ) @ ( power_power_nat @ X4 @ N ) )
= ( ( dvd_dvd_nat @ X4 @ one_one_nat )
| ( ord_less_eq_nat @ M @ N ) ) ) ) ).
% dvd_power_iff
thf(fact_1079_odd__pos,axiom,
! [N: nat] :
( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
=> ( ord_less_nat @ zero_zero_nat @ N ) ) ).
% odd_pos
thf(fact_1080_dvd__power,axiom,
! [N: nat,X4: int] :
( ( ( ord_less_nat @ zero_zero_nat @ N )
| ( X4 = one_one_int ) )
=> ( dvd_dvd_int @ X4 @ ( power_power_int @ X4 @ N ) ) ) ).
% dvd_power
thf(fact_1081_dvd__power,axiom,
! [N: nat,X4: nat] :
( ( ( ord_less_nat @ zero_zero_nat @ N )
| ( X4 = one_one_nat ) )
=> ( dvd_dvd_nat @ X4 @ ( power_power_nat @ X4 @ N ) ) ) ).
% dvd_power
thf(fact_1082_power__mono__odd,axiom,
! [N: nat,A: int,B: int] :
( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
=> ( ( ord_less_eq_int @ A @ B )
=> ( ord_less_eq_int @ ( power_power_int @ A @ N ) @ ( power_power_int @ B @ N ) ) ) ) ).
% power_mono_odd
thf(fact_1083_power__diff,axiom,
! [A: int,N: nat,M: nat] :
( ( A != zero_zero_int )
=> ( ( ord_less_eq_nat @ N @ M )
=> ( ( power_power_int @ A @ ( minus_minus_nat @ M @ N ) )
= ( divide_divide_int @ ( power_power_int @ A @ M ) @ ( power_power_int @ A @ N ) ) ) ) ) ).
% power_diff
thf(fact_1084_power__diff,axiom,
! [A: nat,N: nat,M: nat] :
( ( A != zero_zero_nat )
=> ( ( ord_less_eq_nat @ N @ M )
=> ( ( power_power_nat @ A @ ( minus_minus_nat @ M @ N ) )
= ( divide_divide_nat @ ( power_power_nat @ A @ M ) @ ( power_power_nat @ A @ N ) ) ) ) ) ).
% power_diff
thf(fact_1085_diff__le__diff__pow,axiom,
! [K: nat,M: nat,N: nat] :
( ( ord_less_eq_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ K )
=> ( ord_less_eq_nat @ ( minus_minus_nat @ M @ N ) @ ( minus_minus_nat @ ( power_power_nat @ K @ M ) @ ( power_power_nat @ K @ N ) ) ) ) ).
% diff_le_diff_pow
thf(fact_1086_power2__less__0,axiom,
! [A: int] :
~ ( ord_less_int @ ( power_power_int @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ zero_zero_int ) ).
% power2_less_0
thf(fact_1087_power2__le__imp__le,axiom,
! [X4: nat,Y: nat] :
( ( ord_less_eq_nat @ ( power_power_nat @ X4 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_nat @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ Y )
=> ( ord_less_eq_nat @ X4 @ Y ) ) ) ).
% power2_le_imp_le
thf(fact_1088_power2__le__imp__le,axiom,
! [X4: int,Y: int] :
( ( ord_less_eq_int @ ( power_power_int @ X4 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_int @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
=> ( ( ord_less_eq_int @ zero_zero_int @ Y )
=> ( ord_less_eq_int @ X4 @ Y ) ) ) ).
% power2_le_imp_le
thf(fact_1089_power2__eq__imp__eq,axiom,
! [X4: nat,Y: nat] :
( ( ( power_power_nat @ X4 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
= ( power_power_nat @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ X4 )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ Y )
=> ( X4 = Y ) ) ) ) ).
% power2_eq_imp_eq
thf(fact_1090_power2__eq__imp__eq,axiom,
! [X4: int,Y: int] :
( ( ( power_power_int @ X4 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
= ( power_power_int @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
=> ( ( ord_less_eq_int @ zero_zero_int @ X4 )
=> ( ( ord_less_eq_int @ zero_zero_int @ Y )
=> ( X4 = Y ) ) ) ) ).
% power2_eq_imp_eq
thf(fact_1091_zero__le__power2,axiom,
! [A: int] : ( ord_less_eq_int @ zero_zero_int @ ( power_power_int @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ).
% zero_le_power2
thf(fact_1092_zero__le__even__power,axiom,
! [N: nat,A: int] :
( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
=> ( ord_less_eq_int @ zero_zero_int @ ( power_power_int @ A @ N ) ) ) ).
% zero_le_even_power
thf(fact_1093_zero__le__odd__power,axiom,
! [N: nat,A: int] :
( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
=> ( ( ord_less_eq_int @ zero_zero_int @ ( power_power_int @ A @ N ) )
= ( ord_less_eq_int @ zero_zero_int @ A ) ) ) ).
% zero_le_odd_power
thf(fact_1094_zero__le__power__eq,axiom,
! [A: int,N: nat] :
( ( ord_less_eq_int @ zero_zero_int @ ( power_power_int @ A @ N ) )
= ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
| ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
& ( ord_less_eq_int @ zero_zero_int @ A ) ) ) ) ).
% zero_le_power_eq
thf(fact_1095_power__diff__power__eq,axiom,
! [A: int,N: nat,M: nat] :
( ( A != zero_zero_int )
=> ( ( ( ord_less_eq_nat @ N @ M )
=> ( ( divide_divide_int @ ( power_power_int @ A @ M ) @ ( power_power_int @ A @ N ) )
= ( power_power_int @ A @ ( minus_minus_nat @ M @ N ) ) ) )
& ( ~ ( ord_less_eq_nat @ N @ M )
=> ( ( divide_divide_int @ ( power_power_int @ A @ M ) @ ( power_power_int @ A @ N ) )
= ( divide_divide_int @ one_one_int @ ( power_power_int @ A @ ( minus_minus_nat @ N @ M ) ) ) ) ) ) ) ).
% power_diff_power_eq
thf(fact_1096_power__diff__power__eq,axiom,
! [A: nat,N: nat,M: nat] :
( ( A != zero_zero_nat )
=> ( ( ( ord_less_eq_nat @ N @ M )
=> ( ( divide_divide_nat @ ( power_power_nat @ A @ M ) @ ( power_power_nat @ A @ N ) )
= ( power_power_nat @ A @ ( minus_minus_nat @ M @ N ) ) ) )
& ( ~ ( ord_less_eq_nat @ N @ M )
=> ( ( divide_divide_nat @ ( power_power_nat @ A @ M ) @ ( power_power_nat @ A @ N ) )
= ( divide_divide_nat @ one_one_nat @ ( power_power_nat @ A @ ( minus_minus_nat @ N @ M ) ) ) ) ) ) ) ).
% power_diff_power_eq
thf(fact_1097_neg__one__power__add__eq__neg__one__power__diff,axiom,
! [K: nat,N: nat] :
( ( ord_less_eq_nat @ K @ N )
=> ( ( power_power_int @ ( uminus_uminus_int @ one_one_int ) @ ( plus_plus_nat @ N @ K ) )
= ( power_power_int @ ( uminus_uminus_int @ one_one_int ) @ ( minus_minus_nat @ N @ K ) ) ) ) ).
% neg_one_power_add_eq_neg_one_power_diff
thf(fact_1098_sum__power2__gt__zero__iff,axiom,
! [X4: int,Y: int] :
( ( ord_less_int @ zero_zero_int @ ( plus_plus_int @ ( power_power_int @ X4 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_int @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) )
= ( ( X4 != zero_zero_int )
| ( Y != zero_zero_int ) ) ) ).
% sum_power2_gt_zero_iff
thf(fact_1099_not__sum__power2__lt__zero,axiom,
! [X4: int,Y: int] :
~ ( ord_less_int @ ( plus_plus_int @ ( power_power_int @ X4 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_int @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ zero_zero_int ) ).
% not_sum_power2_lt_zero
thf(fact_1100_zero__less__power__eq,axiom,
! [A: int,N: nat] :
( ( ord_less_int @ zero_zero_int @ ( power_power_int @ A @ N ) )
= ( ( N = zero_zero_nat )
| ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
& ( A != zero_zero_int ) )
| ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
& ( ord_less_int @ zero_zero_int @ A ) ) ) ) ).
% zero_less_power_eq
thf(fact_1101_sum__power2__le__zero__iff,axiom,
! [X4: int,Y: int] :
( ( ord_less_eq_int @ ( plus_plus_int @ ( power_power_int @ X4 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_int @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ zero_zero_int )
= ( ( X4 = zero_zero_int )
& ( Y = zero_zero_int ) ) ) ).
% sum_power2_le_zero_iff
thf(fact_1102_sum__power2__ge__zero,axiom,
! [X4: int,Y: int] : ( ord_less_eq_int @ zero_zero_int @ ( plus_plus_int @ ( power_power_int @ X4 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_int @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ).
% sum_power2_ge_zero
thf(fact_1103_square__le__1,axiom,
! [X4: int] :
( ( ord_less_eq_int @ ( uminus_uminus_int @ one_one_int ) @ X4 )
=> ( ( ord_less_eq_int @ X4 @ one_one_int )
=> ( ord_less_eq_int @ ( power_power_int @ X4 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ one_one_int ) ) ) ).
% square_le_1
thf(fact_1104_even__mask__div__iff_H,axiom,
! [M: nat,N: nat] :
( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( divide_divide_int @ ( minus_minus_int @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ M ) @ one_one_int ) @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) ) )
= ( ord_less_eq_nat @ M @ N ) ) ).
% even_mask_div_iff'
thf(fact_1105_even__mask__div__iff_H,axiom,
! [M: nat,N: nat] :
( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( minus_minus_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M ) @ one_one_nat ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) )
= ( ord_less_eq_nat @ M @ N ) ) ).
% even_mask_div_iff'
thf(fact_1106_pow__divides__pow__iff,axiom,
! [N: nat,A: int,B: int] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( dvd_dvd_int @ ( power_power_int @ A @ N ) @ ( power_power_int @ B @ N ) )
= ( dvd_dvd_int @ A @ B ) ) ) ).
% pow_divides_pow_iff
thf(fact_1107_pow__divides__pow__iff,axiom,
! [N: nat,A: nat,B: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( dvd_dvd_nat @ ( power_power_nat @ A @ N ) @ ( power_power_nat @ B @ N ) )
= ( dvd_dvd_nat @ A @ B ) ) ) ).
% pow_divides_pow_iff
thf(fact_1108_flip__bit__0,axiom,
! [A: nat] :
( ( bit_se2161824704523386999it_nat @ zero_zero_nat @ A )
= ( plus_plus_nat @ ( zero_n2687167440665602831ol_nat @ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A ) ) @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ).
% flip_bit_0
thf(fact_1109_flip__bit__0,axiom,
! [A: int] :
( ( bit_se2159334234014336723it_int @ zero_zero_nat @ A )
= ( plus_plus_int @ ( zero_n2684676970156552555ol_int @ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A ) ) @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( divide_divide_int @ A @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ) ) ).
% flip_bit_0
thf(fact_1110_dvd__pos__nat,axiom,
! [N: nat,M: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( dvd_dvd_nat @ M @ N )
=> ( ord_less_nat @ zero_zero_nat @ M ) ) ) ).
% dvd_pos_nat
thf(fact_1111_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_1112_semiring__norm_I12_J,axiom,
! [N: num] :
( ( times_times_num @ one @ N )
= N ) ).
% semiring_norm(12)
thf(fact_1113_semiring__norm_I11_J,axiom,
! [M: num] :
( ( times_times_num @ M @ one )
= M ) ).
% semiring_norm(11)
thf(fact_1114_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_1115_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_1116_mult__0__right,axiom,
! [M: nat] :
( ( times_times_nat @ M @ zero_zero_nat )
= zero_zero_nat ) ).
% mult_0_right
thf(fact_1117_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_1118_unset__bit__negative__int__iff,axiom,
! [N: nat,K: int] :
( ( ord_less_int @ ( bit_se4203085406695923979it_int @ N @ K ) @ zero_zero_int )
= ( ord_less_int @ K @ zero_zero_int ) ) ).
% unset_bit_negative_int_iff
thf(fact_1119_flip__bit__negative__int__iff,axiom,
! [N: nat,K: int] :
( ( ord_less_int @ ( bit_se2159334234014336723it_int @ N @ K ) @ zero_zero_int )
= ( ord_less_int @ K @ zero_zero_int ) ) ).
% flip_bit_negative_int_iff
thf(fact_1120_set__bit__negative__int__iff,axiom,
! [N: nat,K: int] :
( ( ord_less_int @ ( bit_se7879613467334960850it_int @ N @ K ) @ zero_zero_int )
= ( ord_less_int @ K @ zero_zero_int ) ) ).
% set_bit_negative_int_iff
thf(fact_1121_unset__bit__nonnegative__int__iff,axiom,
! [N: nat,K: int] :
( ( ord_less_eq_int @ zero_zero_int @ ( bit_se4203085406695923979it_int @ N @ K ) )
= ( ord_less_eq_int @ zero_zero_int @ K ) ) ).
% unset_bit_nonnegative_int_iff
thf(fact_1122_flip__bit__nonnegative__int__iff,axiom,
! [N: nat,K: int] :
( ( ord_less_eq_int @ zero_zero_int @ ( bit_se2159334234014336723it_int @ N @ K ) )
= ( ord_less_eq_int @ zero_zero_int @ K ) ) ).
% flip_bit_nonnegative_int_iff
thf(fact_1123_set__bit__nonnegative__int__iff,axiom,
! [N: nat,K: int] :
( ( ord_less_eq_int @ zero_zero_int @ ( bit_se7879613467334960850it_int @ N @ K ) )
= ( ord_less_eq_int @ zero_zero_int @ K ) ) ).
% set_bit_nonnegative_int_iff
thf(fact_1124_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_1125_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_1126_mult__zero__left,axiom,
! [A: nat] :
( ( times_times_nat @ zero_zero_nat @ A )
= zero_zero_nat ) ).
% mult_zero_left
thf(fact_1127_mult__zero__left,axiom,
! [A: int] :
( ( times_times_int @ zero_zero_int @ A )
= zero_zero_int ) ).
% mult_zero_left
thf(fact_1128_mult__zero__right,axiom,
! [A: nat] :
( ( times_times_nat @ A @ zero_zero_nat )
= zero_zero_nat ) ).
% mult_zero_right
thf(fact_1129_mult__zero__right,axiom,
! [A: int] :
( ( times_times_int @ A @ zero_zero_int )
= zero_zero_int ) ).
% mult_zero_right
thf(fact_1130_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_1131_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_1132_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_1133_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_1134_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_1135_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_1136_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_1137_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_1138_mult__numeral__left__semiring__numeral,axiom,
! [V: num,W: num,Z2: nat] :
( ( times_times_nat @ ( numeral_numeral_nat @ V ) @ ( times_times_nat @ ( numeral_numeral_nat @ W ) @ Z2 ) )
= ( times_times_nat @ ( numeral_numeral_nat @ ( times_times_num @ V @ W ) ) @ Z2 ) ) ).
% mult_numeral_left_semiring_numeral
thf(fact_1139_mult__numeral__left__semiring__numeral,axiom,
! [V: num,W: num,Z2: int] :
( ( times_times_int @ ( numeral_numeral_int @ V ) @ ( times_times_int @ ( numeral_numeral_int @ W ) @ Z2 ) )
= ( times_times_int @ ( numeral_numeral_int @ ( times_times_num @ V @ W ) ) @ Z2 ) ) ).
% mult_numeral_left_semiring_numeral
thf(fact_1140_mult__1,axiom,
! [A: nat] :
( ( times_times_nat @ one_one_nat @ A )
= A ) ).
% mult_1
thf(fact_1141_mult__1,axiom,
! [A: int] :
( ( times_times_int @ one_one_int @ A )
= A ) ).
% mult_1
thf(fact_1142_mult_Oright__neutral,axiom,
! [A: nat] :
( ( times_times_nat @ A @ one_one_nat )
= A ) ).
% mult.right_neutral
thf(fact_1143_mult_Oright__neutral,axiom,
! [A: int] :
( ( times_times_int @ A @ one_one_int )
= A ) ).
% mult.right_neutral
thf(fact_1144_num__double,axiom,
! [N: num] :
( ( times_times_num @ ( bit0 @ one ) @ N )
= ( bit0 @ N ) ) ).
% num_double
thf(fact_1145_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_1146_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_1147_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_1148_mult__less__cancel2,axiom,
! [M: nat,K: nat,N: nat] :
( ( ord_less_nat @ ( times_times_nat @ M @ K ) @ ( times_times_nat @ N @ K ) )
= ( ( ord_less_nat @ zero_zero_nat @ K )
& ( ord_less_nat @ M @ N ) ) ) ).
% mult_less_cancel2
thf(fact_1149_nat__0__less__mult__iff,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ zero_zero_nat @ ( times_times_nat @ M @ N ) )
= ( ( ord_less_nat @ zero_zero_nat @ M )
& ( ord_less_nat @ zero_zero_nat @ N ) ) ) ).
% nat_0_less_mult_iff
thf(fact_1150_nat__mult__less__cancel__disj,axiom,
! [K: nat,M: nat,N: nat] :
( ( ord_less_nat @ ( times_times_nat @ K @ M ) @ ( times_times_nat @ K @ N ) )
= ( ( ord_less_nat @ zero_zero_nat @ K )
& ( ord_less_nat @ M @ N ) ) ) ).
% nat_mult_less_cancel_disj
thf(fact_1151_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_1152_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_1153_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_1154_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_1155_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_1156_div__neg__neg__trivial,axiom,
! [K: int,L: int] :
( ( ord_less_eq_int @ K @ zero_zero_int )
=> ( ( ord_less_int @ L @ K )
=> ( ( divide_divide_int @ K @ L )
= zero_zero_int ) ) ) ).
% div_neg_neg_trivial
thf(fact_1157_div__pos__pos__trivial,axiom,
! [K: int,L: int] :
( ( ord_less_eq_int @ zero_zero_int @ K )
=> ( ( ord_less_int @ K @ L )
=> ( ( divide_divide_int @ K @ L )
= zero_zero_int ) ) ) ).
% div_pos_pos_trivial
thf(fact_1158_semiring__norm_I78_J,axiom,
! [M: num,N: num] :
( ( ord_less_num @ ( bit0 @ M ) @ ( bit0 @ N ) )
= ( ord_less_num @ M @ N ) ) ).
% semiring_norm(78)
thf(fact_1159_semiring__norm_I71_J,axiom,
! [M: num,N: num] :
( ( ord_less_eq_num @ ( bit0 @ M ) @ ( bit0 @ N ) )
= ( ord_less_eq_num @ M @ N ) ) ).
% semiring_norm(71)
thf(fact_1160_semiring__norm_I75_J,axiom,
! [M: num] :
~ ( ord_less_num @ M @ one ) ).
% semiring_norm(75)
thf(fact_1161_semiring__norm_I68_J,axiom,
! [N: num] : ( ord_less_eq_num @ one @ N ) ).
% semiring_norm(68)
thf(fact_1162_semiring__norm_I80_J,axiom,
! [M: num,N: num] :
( ( ord_less_num @ ( bit1 @ M ) @ ( bit1 @ N ) )
= ( ord_less_num @ M @ N ) ) ).
% semiring_norm(80)
thf(fact_1163_semiring__norm_I73_J,axiom,
! [M: num,N: num] :
( ( ord_less_eq_num @ ( bit1 @ M ) @ ( bit1 @ N ) )
= ( ord_less_eq_num @ M @ N ) ) ).
% semiring_norm(73)
thf(fact_1164_push__bit__negative__int__iff,axiom,
! [N: nat,K: int] :
( ( ord_less_int @ ( bit_se545348938243370406it_int @ N @ K ) @ zero_zero_int )
= ( ord_less_int @ K @ zero_zero_int ) ) ).
% push_bit_negative_int_iff
thf(fact_1165_push__bit__nonnegative__int__iff,axiom,
! [N: nat,K: int] :
( ( ord_less_eq_int @ zero_zero_int @ ( bit_se545348938243370406it_int @ N @ K ) )
= ( ord_less_eq_int @ zero_zero_int @ K ) ) ).
% push_bit_nonnegative_int_iff
thf(fact_1166_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_1167_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_1168_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_1169_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_1170_distrib__left__numeral,axiom,
! [V: num,B: nat,C2: nat] :
( ( times_times_nat @ ( numeral_numeral_nat @ V ) @ ( plus_plus_nat @ B @ C2 ) )
= ( plus_plus_nat @ ( times_times_nat @ ( numeral_numeral_nat @ V ) @ B ) @ ( times_times_nat @ ( numeral_numeral_nat @ V ) @ C2 ) ) ) ).
% distrib_left_numeral
thf(fact_1171_distrib__left__numeral,axiom,
! [V: num,B: int,C2: int] :
( ( times_times_int @ ( numeral_numeral_int @ V ) @ ( plus_plus_int @ B @ C2 ) )
= ( plus_plus_int @ ( times_times_int @ ( numeral_numeral_int @ V ) @ B ) @ ( times_times_int @ ( numeral_numeral_int @ V ) @ C2 ) ) ) ).
% distrib_left_numeral
thf(fact_1172_distrib__right__numeral,axiom,
! [A: nat,B: nat,V: num] :
( ( times_times_nat @ ( plus_plus_nat @ A @ B ) @ ( numeral_numeral_nat @ V ) )
= ( plus_plus_nat @ ( times_times_nat @ A @ ( numeral_numeral_nat @ V ) ) @ ( times_times_nat @ B @ ( numeral_numeral_nat @ V ) ) ) ) ).
% distrib_right_numeral
thf(fact_1173_distrib__right__numeral,axiom,
! [A: int,B: int,V: num] :
( ( times_times_int @ ( plus_plus_int @ A @ B ) @ ( numeral_numeral_int @ V ) )
= ( plus_plus_int @ ( times_times_int @ A @ ( numeral_numeral_int @ V ) ) @ ( times_times_int @ B @ ( numeral_numeral_int @ V ) ) ) ) ).
% distrib_right_numeral
thf(fact_1174_left__diff__distrib__numeral,axiom,
! [A: int,B: int,V: num] :
( ( times_times_int @ ( minus_minus_int @ A @ B ) @ ( numeral_numeral_int @ V ) )
= ( minus_minus_int @ ( times_times_int @ A @ ( numeral_numeral_int @ V ) ) @ ( times_times_int @ B @ ( numeral_numeral_int @ V ) ) ) ) ).
% left_diff_distrib_numeral
thf(fact_1175_right__diff__distrib__numeral,axiom,
! [V: num,B: int,C2: int] :
( ( times_times_int @ ( numeral_numeral_int @ V ) @ ( minus_minus_int @ B @ C2 ) )
= ( minus_minus_int @ ( times_times_int @ ( numeral_numeral_int @ V ) @ B ) @ ( times_times_int @ ( numeral_numeral_int @ V ) @ C2 ) ) ) ).
% right_diff_distrib_numeral
thf(fact_1176_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_1177_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_1178_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_1179_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_1180_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_1181_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_1182_div__mult__mult1,axiom,
! [C2: int,A: int,B: int] :
( ( C2 != zero_zero_int )
=> ( ( divide_divide_int @ ( times_times_int @ C2 @ A ) @ ( times_times_int @ C2 @ B ) )
= ( divide_divide_int @ A @ B ) ) ) ).
% div_mult_mult1
thf(fact_1183_div__mult__mult1,axiom,
! [C2: nat,A: nat,B: nat] :
( ( C2 != zero_zero_nat )
=> ( ( divide_divide_nat @ ( times_times_nat @ C2 @ A ) @ ( times_times_nat @ C2 @ B ) )
= ( divide_divide_nat @ A @ B ) ) ) ).
% div_mult_mult1
thf(fact_1184_div__mult__mult2,axiom,
! [C2: int,A: int,B: int] :
( ( C2 != zero_zero_int )
=> ( ( divide_divide_int @ ( times_times_int @ A @ C2 ) @ ( times_times_int @ B @ C2 ) )
= ( divide_divide_int @ A @ B ) ) ) ).
% div_mult_mult2
thf(fact_1185_div__mult__mult2,axiom,
! [C2: nat,A: nat,B: nat] :
( ( C2 != zero_zero_nat )
=> ( ( divide_divide_nat @ ( times_times_nat @ A @ C2 ) @ ( times_times_nat @ B @ C2 ) )
= ( divide_divide_nat @ A @ B ) ) ) ).
% div_mult_mult2
thf(fact_1186_div__mult__mult1__if,axiom,
! [C2: int,A: int,B: int] :
( ( ( C2 = zero_zero_int )
=> ( ( divide_divide_int @ ( times_times_int @ C2 @ A ) @ ( times_times_int @ C2 @ B ) )
= zero_zero_int ) )
& ( ( C2 != zero_zero_int )
=> ( ( divide_divide_int @ ( times_times_int @ C2 @ A ) @ ( times_times_int @ C2 @ B ) )
= ( divide_divide_int @ A @ B ) ) ) ) ).
% div_mult_mult1_if
thf(fact_1187_div__mult__mult1__if,axiom,
! [C2: nat,A: nat,B: nat] :
( ( ( C2 = zero_zero_nat )
=> ( ( divide_divide_nat @ ( times_times_nat @ C2 @ A ) @ ( times_times_nat @ C2 @ B ) )
= zero_zero_nat ) )
& ( ( C2 != zero_zero_nat )
=> ( ( divide_divide_nat @ ( times_times_nat @ C2 @ A ) @ ( times_times_nat @ C2 @ B ) )
= ( divide_divide_nat @ A @ B ) ) ) ) ).
% div_mult_mult1_if
thf(fact_1188_nonzero__mult__div__cancel__right,axiom,
! [B: int,A: int] :
( ( B != zero_zero_int )
=> ( ( divide_divide_int @ ( times_times_int @ A @ B ) @ B )
= A ) ) ).
% nonzero_mult_div_cancel_right
thf(fact_1189_nonzero__mult__div__cancel__right,axiom,
! [B: nat,A: nat] :
( ( B != zero_zero_nat )
=> ( ( divide_divide_nat @ ( times_times_nat @ A @ B ) @ B )
= A ) ) ).
% nonzero_mult_div_cancel_right
thf(fact_1190_nonzero__mult__div__cancel__left,axiom,
! [A: int,B: int] :
( ( A != zero_zero_int )
=> ( ( divide_divide_int @ ( times_times_int @ A @ B ) @ A )
= B ) ) ).
% nonzero_mult_div_cancel_left
thf(fact_1191_nonzero__mult__div__cancel__left,axiom,
! [A: nat,B: nat] :
( ( A != zero_zero_nat )
=> ( ( divide_divide_nat @ ( times_times_nat @ A @ B ) @ A )
= B ) ) ).
% nonzero_mult_div_cancel_left
thf(fact_1192_semiring__norm_I171_J,axiom,
! [V: num,W: num,Y: int] :
( ( times_times_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ V ) ) @ ( times_times_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ W ) ) @ Y ) )
= ( times_times_int @ ( numeral_numeral_int @ ( times_times_num @ V @ W ) ) @ Y ) ) ).
% semiring_norm(171)
thf(fact_1193_semiring__norm_I170_J,axiom,
! [V: num,W: num,Y: int] :
( ( times_times_int @ ( numeral_numeral_int @ V ) @ ( times_times_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ W ) ) @ Y ) )
= ( times_times_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ ( times_times_num @ V @ W ) ) ) @ Y ) ) ).
% semiring_norm(170)
thf(fact_1194_semiring__norm_I169_J,axiom,
! [V: num,W: num,Y: int] :
( ( times_times_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ V ) ) @ ( times_times_int @ ( numeral_numeral_int @ W ) @ Y ) )
= ( times_times_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ ( times_times_num @ V @ W ) ) ) @ Y ) ) ).
% semiring_norm(169)
thf(fact_1195_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_1196_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_1197_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_1198_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_1199_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_1200_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_1201_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_1202_mult__minus1__right,axiom,
! [Z2: int] :
( ( times_times_int @ Z2 @ ( uminus_uminus_int @ one_one_int ) )
= ( uminus_uminus_int @ Z2 ) ) ).
% mult_minus1_right
thf(fact_1203_mult__minus1,axiom,
! [Z2: int] :
( ( times_times_int @ ( uminus_uminus_int @ one_one_int ) @ Z2 )
= ( uminus_uminus_int @ Z2 ) ) ).
% mult_minus1
thf(fact_1204_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_1205_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_1206_dvd__mult__div__cancel,axiom,
! [A: int,B: int] :
( ( dvd_dvd_int @ A @ B )
=> ( ( times_times_int @ A @ ( divide_divide_int @ B @ A ) )
= B ) ) ).
% dvd_mult_div_cancel
thf(fact_1207_dvd__mult__div__cancel,axiom,
! [A: nat,B: nat] :
( ( dvd_dvd_nat @ A @ B )
=> ( ( times_times_nat @ A @ ( divide_divide_nat @ B @ A ) )
= B ) ) ).
% dvd_mult_div_cancel
thf(fact_1208_dvd__div__mult__self,axiom,
! [A: int,B: int] :
( ( dvd_dvd_int @ A @ B )
=> ( ( times_times_int @ ( divide_divide_int @ B @ A ) @ A )
= B ) ) ).
% dvd_div_mult_self
thf(fact_1209_dvd__div__mult__self,axiom,
! [A: nat,B: nat] :
( ( dvd_dvd_nat @ A @ B )
=> ( ( times_times_nat @ ( divide_divide_nat @ B @ A ) @ A )
= B ) ) ).
% dvd_div_mult_self
thf(fact_1210_mult__le__cancel2,axiom,
! [M: nat,K: nat,N: nat] :
( ( ord_less_eq_nat @ ( times_times_nat @ M @ K ) @ ( times_times_nat @ N @ K ) )
= ( ( ord_less_nat @ zero_zero_nat @ K )
=> ( ord_less_eq_nat @ M @ N ) ) ) ).
% mult_le_cancel2
thf(fact_1211_nat__mult__le__cancel__disj,axiom,
! [K: nat,M: nat,N: nat] :
( ( ord_less_eq_nat @ ( times_times_nat @ K @ M ) @ ( times_times_nat @ K @ N ) )
= ( ( ord_less_nat @ zero_zero_nat @ K )
=> ( ord_less_eq_nat @ M @ N ) ) ) ).
% nat_mult_le_cancel_disj
thf(fact_1212_div__mult__self__is__m,axiom,
! [N: nat,M: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( divide_divide_nat @ ( times_times_nat @ M @ N ) @ N )
= M ) ) ).
% div_mult_self_is_m
thf(fact_1213_div__mult__self1__is__m,axiom,
! [N: nat,M: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( divide_divide_nat @ ( times_times_nat @ N @ M ) @ N )
= M ) ) ).
% div_mult_self1_is_m
thf(fact_1214_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_1215_semiring__norm_I76_J,axiom,
! [N: num] : ( ord_less_num @ one @ ( bit0 @ N ) ) ).
% semiring_norm(76)
thf(fact_1216_semiring__norm_I69_J,axiom,
! [M: num] :
~ ( ord_less_eq_num @ ( bit0 @ M ) @ one ) ).
% semiring_norm(69)
thf(fact_1217_semiring__norm_I81_J,axiom,
! [M: num,N: num] :
( ( ord_less_num @ ( bit1 @ M ) @ ( bit0 @ N ) )
= ( ord_less_num @ M @ N ) ) ).
% semiring_norm(81)
thf(fact_1218_semiring__norm_I72_J,axiom,
! [M: num,N: num] :
( ( ord_less_eq_num @ ( bit0 @ M ) @ ( bit1 @ N ) )
= ( ord_less_eq_num @ M @ N ) ) ).
% semiring_norm(72)
thf(fact_1219_semiring__norm_I77_J,axiom,
! [N: num] : ( ord_less_num @ one @ ( bit1 @ N ) ) ).
% semiring_norm(77)
thf(fact_1220_semiring__norm_I70_J,axiom,
! [M: num] :
~ ( ord_less_eq_num @ ( bit1 @ M ) @ one ) ).
% semiring_norm(70)
thf(fact_1221_div__mult__self4,axiom,
! [B: int,C2: int,A: int] :
( ( B != zero_zero_int )
=> ( ( divide_divide_int @ ( plus_plus_int @ ( times_times_int @ B @ C2 ) @ A ) @ B )
= ( plus_plus_int @ C2 @ ( divide_divide_int @ A @ B ) ) ) ) ).
% div_mult_self4
thf(fact_1222_div__mult__self4,axiom,
! [B: nat,C2: nat,A: nat] :
( ( B != zero_zero_nat )
=> ( ( divide_divide_nat @ ( plus_plus_nat @ ( times_times_nat @ B @ C2 ) @ A ) @ B )
= ( plus_plus_nat @ C2 @ ( divide_divide_nat @ A @ B ) ) ) ) ).
% div_mult_self4
thf(fact_1223_div__mult__self3,axiom,
! [B: int,C2: int,A: int] :
( ( B != zero_zero_int )
=> ( ( divide_divide_int @ ( plus_plus_int @ ( times_times_int @ C2 @ B ) @ A ) @ B )
= ( plus_plus_int @ C2 @ ( divide_divide_int @ A @ B ) ) ) ) ).
% div_mult_self3
thf(fact_1224_div__mult__self3,axiom,
! [B: nat,C2: nat,A: nat] :
( ( B != zero_zero_nat )
=> ( ( divide_divide_nat @ ( plus_plus_nat @ ( times_times_nat @ C2 @ B ) @ A ) @ B )
= ( plus_plus_nat @ C2 @ ( divide_divide_nat @ A @ B ) ) ) ) ).
% div_mult_self3
thf(fact_1225_div__mult__self2,axiom,
! [B: int,A: int,C2: int] :
( ( B != zero_zero_int )
=> ( ( divide_divide_int @ ( plus_plus_int @ A @ ( times_times_int @ B @ C2 ) ) @ B )
= ( plus_plus_int @ C2 @ ( divide_divide_int @ A @ B ) ) ) ) ).
% div_mult_self2
thf(fact_1226_div__mult__self2,axiom,
! [B: nat,A: nat,C2: nat] :
( ( B != zero_zero_nat )
=> ( ( divide_divide_nat @ ( plus_plus_nat @ A @ ( times_times_nat @ B @ C2 ) ) @ B )
= ( plus_plus_nat @ C2 @ ( divide_divide_nat @ A @ B ) ) ) ) ).
% div_mult_self2
thf(fact_1227_div__mult__self1,axiom,
! [B: int,A: int,C2: int] :
( ( B != zero_zero_int )
=> ( ( divide_divide_int @ ( plus_plus_int @ A @ ( times_times_int @ C2 @ B ) ) @ B )
= ( plus_plus_int @ C2 @ ( divide_divide_int @ A @ B ) ) ) ) ).
% div_mult_self1
thf(fact_1228_div__mult__self1,axiom,
! [B: nat,A: nat,C2: nat] :
( ( B != zero_zero_nat )
=> ( ( divide_divide_nat @ ( plus_plus_nat @ A @ ( times_times_nat @ C2 @ B ) ) @ B )
= ( plus_plus_nat @ C2 @ ( divide_divide_nat @ A @ B ) ) ) ) ).
% div_mult_self1
thf(fact_1229_unit__div__mult__self,axiom,
! [A: int,B: int] :
( ( dvd_dvd_int @ A @ one_one_int )
=> ( ( times_times_int @ ( divide_divide_int @ B @ A ) @ A )
= B ) ) ).
% unit_div_mult_self
thf(fact_1230_unit__div__mult__self,axiom,
! [A: nat,B: nat] :
( ( dvd_dvd_nat @ A @ one_one_nat )
=> ( ( times_times_nat @ ( divide_divide_nat @ B @ A ) @ A )
= B ) ) ).
% unit_div_mult_self
thf(fact_1231_unit__mult__div__div,axiom,
! [A: int,B: int] :
( ( dvd_dvd_int @ A @ one_one_int )
=> ( ( times_times_int @ B @ ( divide_divide_int @ one_one_int @ A ) )
= ( divide_divide_int @ B @ A ) ) ) ).
% unit_mult_div_div
thf(fact_1232_unit__mult__div__div,axiom,
! [A: nat,B: nat] :
( ( dvd_dvd_nat @ A @ one_one_nat )
=> ( ( times_times_nat @ B @ ( divide_divide_nat @ one_one_nat @ A ) )
= ( divide_divide_nat @ B @ A ) ) ) ).
% unit_mult_div_div
thf(fact_1233_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_1234_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_1235_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_1236_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_1237_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_1238_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_1239_semiring__norm_I79_J,axiom,
! [M: num,N: num] :
( ( ord_less_num @ ( bit0 @ M ) @ ( bit1 @ N ) )
= ( ord_less_eq_num @ M @ N ) ) ).
% semiring_norm(79)
thf(fact_1240_semiring__norm_I74_J,axiom,
! [M: num,N: num] :
( ( ord_less_eq_num @ ( bit1 @ M ) @ ( bit0 @ N ) )
= ( ord_less_num @ M @ N ) ) ).
% semiring_norm(74)
thf(fact_1241_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_1242_half__negative__int__iff,axiom,
! [K: int] :
( ( ord_less_int @ ( divide_divide_int @ K @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) @ zero_zero_int )
= ( ord_less_int @ K @ zero_zero_int ) ) ).
% half_negative_int_iff
thf(fact_1243_half__nonnegative__int__iff,axiom,
! [K: int] :
( ( ord_less_eq_int @ zero_zero_int @ ( divide_divide_int @ K @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) )
= ( ord_less_eq_int @ zero_zero_int @ K ) ) ).
% half_nonnegative_int_iff
thf(fact_1244_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_1245_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_1246_mult__0,axiom,
! [N: nat] :
( ( times_times_nat @ zero_zero_nat @ N )
= zero_zero_nat ) ).
% mult_0
thf(fact_1247_split__zdiv,axiom,
! [P2: int > $o,N: int,K: int] :
( ( P2 @ ( divide_divide_int @ N @ K ) )
= ( ( ( K = zero_zero_int )
=> ( P2 @ zero_zero_int ) )
& ( ( ord_less_int @ zero_zero_int @ K )
=> ! [I3: int,J2: int] :
( ( ( ord_less_eq_int @ zero_zero_int @ J2 )
& ( ord_less_int @ J2 @ K )
& ( N
= ( plus_plus_int @ ( times_times_int @ K @ I3 ) @ J2 ) ) )
=> ( P2 @ I3 ) ) )
& ( ( ord_less_int @ K @ zero_zero_int )
=> ! [I3: int,J2: int] :
( ( ( ord_less_int @ K @ J2 )
& ( ord_less_eq_int @ J2 @ zero_zero_int )
& ( N
= ( plus_plus_int @ ( times_times_int @ K @ I3 ) @ J2 ) ) )
=> ( P2 @ I3 ) ) ) ) ) ).
% split_zdiv
thf(fact_1248_int__div__neg__eq,axiom,
! [A: int,B: int,Q: int,R: int] :
( ( A
= ( plus_plus_int @ ( times_times_int @ B @ Q ) @ R ) )
=> ( ( ord_less_eq_int @ R @ zero_zero_int )
=> ( ( ord_less_int @ B @ R )
=> ( ( divide_divide_int @ A @ B )
= Q ) ) ) ) ).
% int_div_neg_eq
thf(fact_1249_int__div__pos__eq,axiom,
! [A: int,B: int,Q: int,R: int] :
( ( A
= ( plus_plus_int @ ( times_times_int @ B @ Q ) @ R ) )
=> ( ( ord_less_eq_int @ zero_zero_int @ R )
=> ( ( ord_less_int @ R @ B )
=> ( ( divide_divide_int @ A @ B )
= Q ) ) ) ) ).
% int_div_pos_eq
thf(fact_1250_zdiv__zmult2__eq,axiom,
! [C2: int,A: int,B: int] :
( ( ord_less_eq_int @ zero_zero_int @ C2 )
=> ( ( divide_divide_int @ A @ ( times_times_int @ B @ C2 ) )
= ( divide_divide_int @ ( divide_divide_int @ A @ B ) @ C2 ) ) ) ).
% zdiv_zmult2_eq
thf(fact_1251_zdiv__mono1,axiom,
! [A: int,A2: int,B: int] :
( ( ord_less_eq_int @ A @ A2 )
=> ( ( ord_less_int @ zero_zero_int @ B )
=> ( ord_less_eq_int @ ( divide_divide_int @ A @ B ) @ ( divide_divide_int @ A2 @ B ) ) ) ) ).
% zdiv_mono1
thf(fact_1252_zdiv__mono2,axiom,
! [A: int,B2: int,B: int] :
( ( ord_less_eq_int @ zero_zero_int @ A )
=> ( ( ord_less_int @ zero_zero_int @ B2 )
=> ( ( ord_less_eq_int @ B2 @ B )
=> ( ord_less_eq_int @ ( divide_divide_int @ A @ B ) @ ( divide_divide_int @ A @ B2 ) ) ) ) ) ).
% zdiv_mono2
thf(fact_1253_zdiv__eq__0__iff,axiom,
! [I: int,K: int] :
( ( ( divide_divide_int @ I @ K )
= zero_zero_int )
= ( ( K = zero_zero_int )
| ( ( ord_less_eq_int @ zero_zero_int @ I )
& ( ord_less_int @ I @ K ) )
| ( ( ord_less_eq_int @ I @ zero_zero_int )
& ( ord_less_int @ K @ I ) ) ) ) ).
% zdiv_eq_0_iff
thf(fact_1254_zdiv__mono1__neg,axiom,
! [A: int,A2: int,B: int] :
( ( ord_less_eq_int @ A @ A2 )
=> ( ( ord_less_int @ B @ zero_zero_int )
=> ( ord_less_eq_int @ ( divide_divide_int @ A2 @ B ) @ ( divide_divide_int @ A @ B ) ) ) ) ).
% zdiv_mono1_neg
thf(fact_1255_zdiv__mono2__neg,axiom,
! [A: int,B2: int,B: int] :
( ( ord_less_int @ A @ zero_zero_int )
=> ( ( ord_less_int @ zero_zero_int @ B2 )
=> ( ( ord_less_eq_int @ B2 @ B )
=> ( ord_less_eq_int @ ( divide_divide_int @ A @ B2 ) @ ( divide_divide_int @ A @ B ) ) ) ) ) ).
% zdiv_mono2_neg
thf(fact_1256_div__int__pos__iff,axiom,
! [K: int,L: int] :
( ( ord_less_eq_int @ zero_zero_int @ ( divide_divide_int @ K @ L ) )
= ( ( K = zero_zero_int )
| ( L = zero_zero_int )
| ( ( ord_less_eq_int @ zero_zero_int @ K )
& ( ord_less_eq_int @ zero_zero_int @ L ) )
| ( ( ord_less_int @ K @ zero_zero_int )
& ( ord_less_int @ L @ zero_zero_int ) ) ) ) ).
% div_int_pos_iff
thf(fact_1257_div__nonneg__neg__le0,axiom,
! [A: int,B: int] :
( ( ord_less_eq_int @ zero_zero_int @ A )
=> ( ( ord_less_int @ B @ zero_zero_int )
=> ( ord_less_eq_int @ ( divide_divide_int @ A @ B ) @ zero_zero_int ) ) ) ).
% div_nonneg_neg_le0
thf(fact_1258_div__nonpos__pos__le0,axiom,
! [A: int,B: int] :
( ( ord_less_eq_int @ A @ zero_zero_int )
=> ( ( ord_less_int @ zero_zero_int @ B )
=> ( ord_less_eq_int @ ( divide_divide_int @ A @ B ) @ zero_zero_int ) ) ) ).
% div_nonpos_pos_le0
thf(fact_1259_pos__imp__zdiv__pos__iff,axiom,
! [K: int,I: int] :
( ( ord_less_int @ zero_zero_int @ K )
=> ( ( ord_less_int @ zero_zero_int @ ( divide_divide_int @ I @ K ) )
= ( ord_less_eq_int @ K @ I ) ) ) ).
% pos_imp_zdiv_pos_iff
thf(fact_1260_neg__imp__zdiv__nonneg__iff,axiom,
! [B: int,A: int] :
( ( ord_less_int @ B @ zero_zero_int )
=> ( ( ord_less_eq_int @ zero_zero_int @ ( divide_divide_int @ A @ B ) )
= ( ord_less_eq_int @ A @ zero_zero_int ) ) ) ).
% neg_imp_zdiv_nonneg_iff
thf(fact_1261_pos__imp__zdiv__nonneg__iff,axiom,
! [B: int,A: int] :
( ( ord_less_int @ zero_zero_int @ B )
=> ( ( ord_less_eq_int @ zero_zero_int @ ( divide_divide_int @ A @ B ) )
= ( ord_less_eq_int @ zero_zero_int @ A ) ) ) ).
% pos_imp_zdiv_nonneg_iff
thf(fact_1262_nonneg1__imp__zdiv__pos__iff,axiom,
! [A: int,B: int] :
( ( ord_less_eq_int @ zero_zero_int @ A )
=> ( ( ord_less_int @ zero_zero_int @ ( divide_divide_int @ A @ B ) )
= ( ( ord_less_eq_int @ B @ A )
& ( ord_less_int @ zero_zero_int @ B ) ) ) ) ).
% nonneg1_imp_zdiv_pos_iff
thf(fact_1263_le__num__One__iff,axiom,
! [X4: num] :
( ( ord_less_eq_num @ X4 @ one )
= ( X4 = one ) ) ).
% le_num_One_iff
thf(fact_1264_left__add__mult__distrib,axiom,
! [I: nat,U: nat,J: nat,K: nat] :
( ( plus_plus_nat @ ( times_times_nat @ I @ U ) @ ( plus_plus_nat @ ( times_times_nat @ J @ U ) @ K ) )
= ( plus_plus_nat @ ( times_times_nat @ ( plus_plus_nat @ I @ J ) @ U ) @ K ) ) ).
% left_add_mult_distrib
thf(fact_1265_nat__mult__1,axiom,
! [N: nat] :
( ( times_times_nat @ one_one_nat @ N )
= N ) ).
% nat_mult_1
thf(fact_1266_nat__mult__1__right,axiom,
! [N: nat] :
( ( times_times_nat @ N @ one_one_nat )
= N ) ).
% nat_mult_1_right
thf(fact_1267_div__mult2__eq,axiom,
! [M: nat,N: nat,Q: nat] :
( ( divide_divide_nat @ M @ ( times_times_nat @ N @ Q ) )
= ( divide_divide_nat @ ( divide_divide_nat @ M @ N ) @ Q ) ) ).
% div_mult2_eq
thf(fact_1268_verit__less__mono__div__int2,axiom,
! [A5: int,B5: int,N: int] :
( ( ord_less_eq_int @ A5 @ B5 )
=> ( ( ord_less_int @ zero_zero_int @ ( uminus_uminus_int @ N ) )
=> ( ord_less_eq_int @ ( divide_divide_int @ B5 @ N ) @ ( divide_divide_int @ A5 @ N ) ) ) ) ).
% verit_less_mono_div_int2
% Helper facts (5)
thf(help_If_2_1_If_001t__Int__Oint_T,axiom,
! [X4: int,Y: int] :
( ( if_int @ $false @ X4 @ Y )
= Y ) ).
thf(help_If_1_1_If_001t__Int__Oint_T,axiom,
! [X4: int,Y: int] :
( ( if_int @ $true @ X4 @ Y )
= X4 ) ).
thf(help_If_3_1_If_001t__Nat__Onat_T,axiom,
! [P2: $o] :
( ( P2 = $true )
| ( P2 = $false ) ) ).
thf(help_If_2_1_If_001t__Nat__Onat_T,axiom,
! [X4: nat,Y: nat] :
( ( if_nat @ $false @ X4 @ Y )
= Y ) ).
thf(help_If_1_1_If_001t__Nat__Onat_T,axiom,
! [X4: nat,Y: nat] :
( ( if_nat @ $true @ X4 @ Y )
= X4 ) ).
% Conjectures (1)
thf(conj_0,conjecture,
( ( propos4736024705303644274ropVar @ one2 )
= ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ).
%------------------------------------------------------------------------------