TPTP Problem File: SLH0937^1.p
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%------------------------------------------------------------------------------
% File : SLH0000^1 : TPTP v8.2.0. Released v8.2.0.
% Domain : Archive of Formal Proofs
% Problem :
% Version : Especial.
% English :
% Refs : [Des23] Desharnais (2023), Email to Geoff Sutcliffe
% Source : [Des23]
% Names : Quasi_Borel_Spaces/0006_CoProduct_QuasiBorel/prob_00456_016631__15287364_1 [Des23]
% Status : Theorem
% Rating : ? v8.2.0
% Syntax : Number of formulae : 1667 ( 954 unt; 389 typ; 0 def)
% Number of atoms : 2780 (1987 equ; 0 cnn)
% Maximal formula atoms : 14 ( 2 avg)
% Number of connectives : 9323 ( 497 ~; 48 |; 137 &;7957 @)
% ( 0 <=>; 684 =>; 0 <=; 0 <~>)
% Maximal formula depth : 21 ( 4 avg)
% Number of types : 78 ( 77 usr)
% Number of type conns : 846 ( 846 >; 0 *; 0 +; 0 <<)
% Number of symbols : 315 ( 312 usr; 39 con; 0-4 aty)
% Number of variables : 2734 ( 113 ^;2581 !; 40 ?;2734 :)
% SPC : TH0_THM_EQU_NAR
% Comments : This file was generated by Isabelle (most likely Sledgehammer)
% 2023-01-19 14:09:44.012
%------------------------------------------------------------------------------
% Could-be-implicit typings (77)
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% Explicit typings (312)
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thf(sy_c_Product__Type_OPair_001t__Real__Oreal_001t__List__Olist_It__Product____Type__Oprod_It__Real__Oreal_Mt__Nat__Onat_J_J,type,
produc2239748315323221726al_nat: real > list_P6834414599653733731al_nat > produc626917917429841708al_nat ).
thf(sy_c_Product__Type_OPair_001t__Real__Oreal_001t__List__Olist_It__Product____Type__Oprod_It__Real__Oreal_Mt__Real__Oreal_J_J,type,
produc5056970835876393402l_real: real > list_P8689742595348180415l_real > produc2814942353377581704l_real ).
thf(sy_c_Product__Type_OPair_001t__Real__Oreal_001t__Nat__Onat,type,
produc3181502643871035669al_nat: real > nat > produc3741383161447143261al_nat ).
thf(sy_c_Product__Type_OPair_001t__Real__Oreal_001t__Num__Onum,type,
produc8962206466881590111al_num: real > num > produc4449627098395980967al_num ).
thf(sy_c_Product__Type_OPair_001t__Real__Oreal_001t__Product____Type__Oprod_I_062_It__Nat__Onat_Mt__Nat__Onat_J_Mt__List__Olist_It__Product____Type__Oprod_It__Real__Oreal_Mt__Nat__Onat_J_J_J,type,
produc8229845255737718302al_nat: real > produc7527230689940420825al_nat > produc3836125934878705638al_nat ).
thf(sy_c_Product__Type_OPair_001t__Real__Oreal_001t__Product____Type__Oprod_I_062_It__Real__Oreal_Mt__Real__Oreal_J_Mt__List__Olist_It__Product____Type__Oprod_It__Real__Oreal_Mt__Real__Oreal_J_J_J,type,
produc9156241399310741058l_real: real > produc5863196298257148029l_real > produc4144304610149905034l_real ).
thf(sy_c_Product__Type_OPair_001t__Real__Oreal_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Product____Type__Oprod_I_062_It__Nat__Onat_Mt__Nat__Onat_J_Mt__List__Olist_It__Product____Type__Oprod_It__Real__Oreal_Mt__Nat__Onat_J_J_J_J,type,
produc1739509229069810181al_nat: real > produc8914008544917701002al_nat > produc401355577171983827al_nat ).
thf(sy_c_Product__Type_OPair_001t__Real__Oreal_001t__Product____Type__Oprod_It__Real__Oreal_Mt__Product____Type__Oprod_I_062_It__Real__Oreal_Mt__Real__Oreal_J_Mt__List__Olist_It__Product____Type__Oprod_It__Real__Oreal_Mt__Real__Oreal_J_J_J_J,type,
produc6975477229195654405l_real: real > produc4144304610149905034l_real > produc5714321930622857939l_real ).
thf(sy_c_Product__Type_OPair_001t__Real__Oreal_001t__Real__Oreal,type,
produc4511245868158468465l_real: real > real > produc2422161461964618553l_real ).
thf(sy_c_Product__Type_Oprod_Ofst_001t__Extended____Nonnegative____Real__Oennreal_001t__Extended____Nonnegative____Real__Oennreal,type,
produc1455421958045324533nnreal: produc7414223468410354641nnreal > extend8495563244428889912nnreal ).
thf(sy_c_Product__Type_Oprod_Ofst_001t__Extended____Nonnegative____Real__Oennreal_001t__Int__Oint,type,
produc8046150099153143785al_int: produc8392006394566016197al_int > extend8495563244428889912nnreal ).
thf(sy_c_Product__Type_Oprod_Ofst_001t__Extended____Nonnegative____Real__Oennreal_001t__Nat__Onat,type,
produc8048640569662194061al_nat: produc3346485377220437097al_nat > extend8495563244428889912nnreal ).
thf(sy_c_Product__Type_Oprod_Ofst_001t__Extended____Nonnegative____Real__Oennreal_001t__Real__Oreal,type,
produc8390762859368617449l_real: produc1520197602750038597l_real > extend8495563244428889912nnreal ).
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__Int__Oint_001t__Real__Oreal,type,
product_fst_int_real: produc679980390762269497t_real > int ).
thf(sy_c_Product__Type_Oprod_Ofst_001t__Nat__Onat_001_062_It__Nat__Onat_Mt__Int__Oint_J,type,
produc522749003355801712at_int: produc6819000835714826316at_int > nat ).
thf(sy_c_Product__Type_Oprod_Ofst_001t__Nat__Onat_001_062_It__Nat__Onat_Mtf__a_J,type,
produc3194919578927588176_nat_a: produc7123000486447228170_nat_a > nat ).
thf(sy_c_Product__Type_Oprod_Ofst_001t__Nat__Onat_001t__Extended____Nonnegative____Real__Oennreal,type,
produc3274940128995334541nnreal: produc5192943231052834921nnreal > nat ).
thf(sy_c_Product__Type_Oprod_Ofst_001t__Nat__Onat_001t__Int__Oint,type,
product_fst_nat_int: product_prod_nat_int > nat ).
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__Nat__Onat_001t__Real__Oreal,type,
product_fst_nat_real: produc7716430852924023517t_real > 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_Product__Type_Oprod_Ofst_001t__Real__Oreal_001t__Extended____Nonnegative____Real__Oennreal,type,
produc1135137091964385897nnreal: produc1565745902094061253nnreal > real ).
thf(sy_c_Product__Type_Oprod_Ofst_001t__Real__Oreal_001t__Int__Oint,type,
product_fst_real_int: produc8786904178792722361al_int > real ).
thf(sy_c_Product__Type_Oprod_Ofst_001t__Real__Oreal_001t__Nat__Onat,type,
product_fst_real_nat: produc3741383161447143261al_nat > real ).
thf(sy_c_Product__Type_Oprod_Ofst_001t__Real__Oreal_001t__Real__Oreal,type,
produc5828954698716094813l_real: produc2422161461964618553l_real > real ).
thf(sy_c_Product__Type_Oprod_Osnd_001t__Extended____Nonnegative____Real__Oennreal_001t__Extended____Nonnegative____Real__Oennreal,type,
produc4758864974362570039nnreal: produc7414223468410354641nnreal > extend8495563244428889912nnreal ).
thf(sy_c_Product__Type_Oprod_Osnd_001t__Extended____Nonnegative____Real__Oennreal_001t__Int__Oint,type,
produc5711287753968775211al_int: produc8392006394566016197al_int > int ).
thf(sy_c_Product__Type_Oprod_Osnd_001t__Extended____Nonnegative____Real__Oennreal_001t__Nat__Onat,type,
produc5713778224477825487al_nat: produc3346485377220437097al_nat > nat ).
thf(sy_c_Product__Type_Oprod_Osnd_001t__Extended____Nonnegative____Real__Oennreal_001t__Real__Oreal,type,
produc2623728747760237867l_real: produc1520197602750038597l_real > real ).
thf(sy_c_Product__Type_Oprod_Osnd_001t__Int__Oint_001t__Int__Oint,type,
product_snd_int_int: product_prod_int_int > int ).
thf(sy_c_Product__Type_Oprod_Osnd_001t__Int__Oint_001t__Real__Oreal,type,
product_snd_int_real: produc679980390762269497t_real > real ).
thf(sy_c_Product__Type_Oprod_Osnd_001t__Nat__Onat_001_062_It__Nat__Onat_Mt__Int__Oint_J,type,
produc5442246252769801650at_int: produc6819000835714826316at_int > nat > int ).
thf(sy_c_Product__Type_Oprod_Osnd_001t__Nat__Onat_001_062_It__Nat__Onat_Mtf__a_J,type,
produc4809910040060592782_nat_a: produc7123000486447228170_nat_a > nat > a ).
thf(sy_c_Product__Type_Oprod_Osnd_001t__Nat__Onat_001t__Extended____Nonnegative____Real__Oennreal,type,
produc940077783810965967nnreal: produc5192943231052834921nnreal > extend8495563244428889912nnreal ).
thf(sy_c_Product__Type_Oprod_Osnd_001t__Nat__Onat_001t__Int__Oint,type,
product_snd_nat_int: product_prod_nat_int > int ).
thf(sy_c_Product__Type_Oprod_Osnd_001t__Nat__Onat_001t__Nat__Onat,type,
product_snd_nat_nat: product_prod_nat_nat > nat ).
thf(sy_c_Product__Type_Oprod_Osnd_001t__Nat__Onat_001t__Real__Oreal,type,
product_snd_nat_real: produc7716430852924023517t_real > real ).
thf(sy_c_Product__Type_Oprod_Osnd_001t__Num__Onum_001t__Num__Onum,type,
product_snd_num_num: product_prod_num_num > num ).
thf(sy_c_Product__Type_Oprod_Osnd_001t__Real__Oreal_001t__Extended____Nonnegative____Real__Oennreal,type,
produc4591475017210782123nnreal: produc1565745902094061253nnreal > extend8495563244428889912nnreal ).
thf(sy_c_Product__Type_Oprod_Osnd_001t__Real__Oreal_001t__Int__Oint,type,
product_snd_real_int: produc8786904178792722361al_int > int ).
thf(sy_c_Product__Type_Oprod_Osnd_001t__Real__Oreal_001t__Nat__Onat,type,
product_snd_real_nat: produc3741383161447143261al_nat > nat ).
thf(sy_c_Product__Type_Oprod_Osnd_001t__Real__Oreal_001t__Real__Oreal,type,
produc3484788084999411615l_real: produc2422161461964618553l_real > real ).
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_Odivide__class_Odivide_001t__Real__Oreal,type,
divide_divide_real: real > real > real ).
thf(sy_c_Set_OCollect_001t__Nat__Onat,type,
collect_nat: ( nat > $o ) > set_nat ).
thf(sy_c_StandardBorel_Or01__to__r01__r01,type,
r01_to_r01_r01: real > produc2422161461964618553l_real ).
thf(sy_c_Transcendental_Oarcosh_001t__Real__Oreal,type,
arcosh_real: real > real ).
thf(sy_c_Transcendental_Oarsinh_001t__Real__Oreal,type,
arsinh_real: real > real ).
thf(sy_c_Transcendental_Oartanh_001t__Real__Oreal,type,
artanh_real: real > real ).
thf(sy_c_Transcendental_Oln__class_Oln_001t__Real__Oreal,type,
ln_ln_real: real > real ).
thf(sy_c_Wellfounded_Oaccp_001t__List__Olist_It__Int__Oint_J,type,
accp_list_int: ( list_int > list_int > $o ) > list_int > $o ).
thf(sy_c_Wellfounded_Oaccp_001t__List__Olist_Itf__a_J,type,
accp_list_a: ( list_a > list_a > $o ) > list_a > $o ).
thf(sy_c_Wellfounded_Oaccp_001t__Product____Type__Oprod_It__Nat__Onat_M_062_It__Nat__Onat_Mt__Int__Oint_J_J,type,
accp_P5031299702865183317at_int: ( produc6819000835714826316at_int > produc6819000835714826316at_int > $o ) > produc6819000835714826316at_int > $o ).
thf(sy_c_Wellfounded_Oaccp_001t__Product____Type__Oprod_It__Nat__Onat_M_062_It__Nat__Onat_Mtf__a_J_J,type,
accp_P4997817491579065153_nat_a: ( produc7123000486447228170_nat_a > produc7123000486447228170_nat_a > $o ) > produc7123000486447228170_nat_a > $o ).
thf(sy_c_Wellfounded_Oaccp_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Product____Type__Oprod_I_062_I_062_It__Nat__Onat_Mtf__a_J_M_062_It__Nat__Onat_Mtf__a_J_J_Mt__List__Olist_It__Product____Type__Oprod_It__Nat__Onat_M_062_It__Nat__Onat_Mtf__a_J_J_J_J_J,type,
accp_P3452593570100577660_nat_a: ( produc4108149672839534661_nat_a > produc4108149672839534661_nat_a > $o ) > produc4108149672839534661_nat_a > $o ).
thf(sy_c_Wellfounded_Oaccp_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Product____Type__Oprod_I_062_It__Nat__Onat_Mtf__a_J_Mt__Product____Type__Oprod_I_062_I_062_It__Nat__Onat_Mtf__a_J_M_062_It__Nat__Onat_Mtf__a_J_J_Mt__List__Olist_It__Product____Type__Oprod_It__Nat__Onat_M_062_It__Nat__Onat_Mtf__a_J_J_J_J_J_J,type,
accp_P5156064751882252798_nat_a: ( produc4648422496819114567_nat_a > produc4648422496819114567_nat_a > $o ) > produc4648422496819114567_nat_a > $o ).
thf(sy_c_Wellfounded_Oaccp_001t__Product____Type__Oprod_It__Num__Onum_Mt__Product____Type__Oprod_I_062_It__Num__Onum_Mt__Num__Onum_J_Mt__List__Olist_It__Product____Type__Oprod_It__Num__Onum_Mt__Num__Onum_J_J_J_J,type,
accp_P3686700994160295809um_num: ( produc754433770195134328um_num > produc754433770195134328um_num > $o ) > produc754433770195134328um_num > $o ).
thf(sy_c_Wellfounded_Oaccp_001t__Product____Type__Oprod_It__Num__Onum_Mt__Product____Type__Oprod_It__Num__Onum_Mt__Product____Type__Oprod_I_062_It__Num__Onum_Mt__Num__Onum_J_Mt__List__Olist_It__Product____Type__Oprod_It__Num__Onum_Mt__Num__Onum_J_J_J_J_J,type,
accp_P4146607925373659146um_num: ( produc1289371440409268051um_num > produc1289371440409268051um_num > $o ) > produc1289371440409268051um_num > $o ).
thf(sy_c_Wellfounded_Oaccp_001t__Product____Type__Oprod_It__Real__Oreal_Mt__Product____Type__Oprod_I_062_It__Nat__Onat_Mt__Nat__Onat_J_Mt__List__Olist_It__Product____Type__Oprod_It__Real__Oreal_Mt__Nat__Onat_J_J_J_J,type,
accp_P5383833502838552303al_nat: ( produc3836125934878705638al_nat > produc3836125934878705638al_nat > $o ) > produc3836125934878705638al_nat > $o ).
thf(sy_c_Wellfounded_Oaccp_001t__Product____Type__Oprod_It__Real__Oreal_Mt__Product____Type__Oprod_I_062_It__Real__Oreal_Mt__Real__Oreal_J_Mt__List__Olist_It__Product____Type__Oprod_It__Real__Oreal_Mt__Real__Oreal_J_J_J_J,type,
accp_P2253066923555800083l_real: ( produc4144304610149905034l_real > produc4144304610149905034l_real > $o ) > produc4144304610149905034l_real > $o ).
thf(sy_c_Wellfounded_Oaccp_001t__Product____Type__Oprod_It__Real__Oreal_Mt__Product____Type__Oprod_It__Nat__Onat_Mt__Product____Type__Oprod_I_062_It__Nat__Onat_Mt__Nat__Onat_J_Mt__List__Olist_It__Product____Type__Oprod_It__Real__Oreal_Mt__Nat__Onat_J_J_J_J_J,type,
accp_P4161832113633214346al_nat: ( produc401355577171983827al_nat > produc401355577171983827al_nat > $o ) > produc401355577171983827al_nat > $o ).
thf(sy_c_Wellfounded_Oaccp_001t__Product____Type__Oprod_It__Real__Oreal_Mt__Product____Type__Oprod_It__Real__Oreal_Mt__Product____Type__Oprod_I_062_It__Real__Oreal_Mt__Real__Oreal_J_Mt__List__Olist_It__Product____Type__Oprod_It__Real__Oreal_Mt__Real__Oreal_J_J_J_J_J,type,
accp_P5976341164542609034l_real: ( produc5714321930622857939l_real > produc5714321930622857939l_real > $o ) > produc5714321930622857939l_real > $o ).
thf(sy_c_member_001t__Nat__Onat,type,
member_nat: nat > set_nat > $o ).
thf(sy_c_member_001t__Product____Type__Oprod_It__Num__Onum_Mt__Num__Onum_J,type,
member7279096912039735102um_num: product_prod_num_num > set_Pr8218934625190621173um_num > $o ).
thf(sy_c_member_001t__Product____Type__Oprod_It__Real__Oreal_Mt__Real__Oreal_J,type,
member7849222048561428706l_real: produc2422161461964618553l_real > set_Pr6218003697084177305l_real > $o ).
thf(sy_v_a,type,
a2: a ).
thf(sy_v_l,type,
l: produc7123000486447228170_nat_a ).
% Relevant facts (1264)
thf(fact_0_list__simp2_I2_J,axiom,
! [Al: produc7123000486447228170_nat_a,A: a,Bl: produc7123000486447228170_nat_a,B: a] :
( ( ( produc5292568359338195516_nat_a @ ( suc @ ( produc3194919578927588176_nat_a @ Al ) )
@ ^ [N: nat] : ( if_a @ ( N = zero_zero_nat ) @ A @ ( produc4809910040060592782_nat_a @ Al @ ( minus_minus_nat @ N @ one_one_nat ) ) ) )
= ( produc5292568359338195516_nat_a @ ( suc @ ( produc3194919578927588176_nat_a @ Bl ) )
@ ^ [N: nat] : ( if_a @ ( N = zero_zero_nat ) @ B @ ( produc4809910040060592782_nat_a @ Bl @ ( minus_minus_nat @ N @ one_one_nat ) ) ) ) )
=> ( Al = Bl ) ) ).
% list_simp2(2)
thf(fact_1_list__simp2_I1_J,axiom,
! [Al: produc7123000486447228170_nat_a,A: a,Bl: produc7123000486447228170_nat_a,B: a] :
( ( ( produc5292568359338195516_nat_a @ ( suc @ ( produc3194919578927588176_nat_a @ Al ) )
@ ^ [N: nat] : ( if_a @ ( N = zero_zero_nat ) @ A @ ( produc4809910040060592782_nat_a @ Al @ ( minus_minus_nat @ N @ one_one_nat ) ) ) )
= ( produc5292568359338195516_nat_a @ ( suc @ ( produc3194919578927588176_nat_a @ Bl ) )
@ ^ [N: nat] : ( if_a @ ( N = zero_zero_nat ) @ B @ ( produc4809910040060592782_nat_a @ Bl @ ( minus_minus_nat @ N @ one_one_nat ) ) ) ) )
=> ( A = B ) ) ).
% list_simp2(1)
thf(fact_2_to__list_H_Ocases,axiom,
! [X: produc7123000486447228170_nat_a] :
( ! [Uu: nat > a] :
( X
!= ( produc5292568359338195516_nat_a @ zero_zero_nat @ Uu ) )
=> ~ ! [N2: nat,F: nat > a] :
( X
!= ( produc5292568359338195516_nat_a @ ( suc @ N2 ) @ F ) ) ) ).
% to_list'.cases
thf(fact_3_diff__Suc__1,axiom,
! [N3: nat] :
( ( minus_minus_nat @ ( suc @ N3 ) @ one_one_nat )
= N3 ) ).
% diff_Suc_1
thf(fact_4_prod_Ocollapse,axiom,
! [Prod: produc3741383161447143261al_nat] :
( ( produc3181502643871035669al_nat @ ( product_fst_real_nat @ Prod ) @ ( product_snd_real_nat @ Prod ) )
= Prod ) ).
% prod.collapse
thf(fact_5_prod_Ocollapse,axiom,
! [Prod: product_prod_num_num] :
( ( product_Pair_num_num @ ( product_fst_num_num @ Prod ) @ ( product_snd_num_num @ Prod ) )
= Prod ) ).
% prod.collapse
thf(fact_6_prod_Ocollapse,axiom,
! [Prod: produc2422161461964618553l_real] :
( ( produc4511245868158468465l_real @ ( produc5828954698716094813l_real @ Prod ) @ ( produc3484788084999411615l_real @ Prod ) )
= Prod ) ).
% prod.collapse
thf(fact_7_prod_Ocollapse,axiom,
! [Prod: produc7123000486447228170_nat_a] :
( ( produc5292568359338195516_nat_a @ ( produc3194919578927588176_nat_a @ Prod ) @ ( produc4809910040060592782_nat_a @ Prod ) )
= Prod ) ).
% prod.collapse
thf(fact_8_diff__numeral__special_I9_J,axiom,
( ( minus_minus_real @ one_one_real @ one_one_real )
= zero_zero_real ) ).
% diff_numeral_special(9)
thf(fact_9_diff__numeral__special_I9_J,axiom,
( ( minus_minus_int @ one_one_int @ one_one_int )
= zero_zero_int ) ).
% diff_numeral_special(9)
thf(fact_10_list__simp3,axiom,
! [L: produc7123000486447228170_nat_a,A: a] :
( ( coProd6334972811210254351head_a
@ ( produc5292568359338195516_nat_a @ ( suc @ ( produc3194919578927588176_nat_a @ L ) )
@ ^ [N: nat] : ( if_a @ ( N = zero_zero_nat ) @ A @ ( produc4809910040060592782_nat_a @ L @ ( minus_minus_nat @ N @ one_one_nat ) ) ) ) )
= A ) ).
% list_simp3
thf(fact_11_list__tail__def,axiom,
( coProd8554194001436178207tail_a
= ( ^ [L2: produc7123000486447228170_nat_a] :
( produc5292568359338195516_nat_a @ ( minus_minus_nat @ ( produc3194919578927588176_nat_a @ L2 ) @ one_one_nat )
@ ^ [M: nat] : ( produc4809910040060592782_nat_a @ L2 @ ( suc @ M ) ) ) ) ) ).
% list_tail_def
thf(fact_12_diff__Suc__Suc,axiom,
! [M2: nat,N3: nat] :
( ( minus_minus_nat @ ( suc @ M2 ) @ ( suc @ N3 ) )
= ( minus_minus_nat @ M2 @ N3 ) ) ).
% diff_Suc_Suc
thf(fact_13_Suc__diff__diff,axiom,
! [M2: nat,N3: nat,K: nat] :
( ( minus_minus_nat @ ( minus_minus_nat @ ( suc @ M2 ) @ N3 ) @ ( suc @ K ) )
= ( minus_minus_nat @ ( minus_minus_nat @ M2 @ N3 ) @ K ) ) ).
% Suc_diff_diff
thf(fact_14_diff__0__eq__0,axiom,
! [N3: nat] :
( ( minus_minus_nat @ zero_zero_nat @ N3 )
= zero_zero_nat ) ).
% diff_0_eq_0
thf(fact_15_diff__self__eq__0,axiom,
! [M2: nat] :
( ( minus_minus_nat @ M2 @ M2 )
= zero_zero_nat ) ).
% diff_self_eq_0
thf(fact_16_snd__diff,axiom,
! [X: produc3741383161447143261al_nat,Y: produc3741383161447143261al_nat] :
( ( product_snd_real_nat @ ( minus_1582581163013509572al_nat @ X @ Y ) )
= ( minus_minus_nat @ ( product_snd_real_nat @ X ) @ ( product_snd_real_nat @ Y ) ) ) ).
% snd_diff
thf(fact_17_snd__diff,axiom,
! [X: produc2422161461964618553l_real,Y: produc2422161461964618553l_real] :
( ( produc3484788084999411615l_real @ ( minus_885040589139849760l_real @ X @ Y ) )
= ( minus_minus_real @ ( produc3484788084999411615l_real @ X ) @ ( produc3484788084999411615l_real @ Y ) ) ) ).
% snd_diff
thf(fact_18_fst__diff,axiom,
! [X: produc2422161461964618553l_real,Y: produc2422161461964618553l_real] :
( ( produc5828954698716094813l_real @ ( minus_885040589139849760l_real @ X @ Y ) )
= ( minus_minus_real @ ( produc5828954698716094813l_real @ X ) @ ( produc5828954698716094813l_real @ Y ) ) ) ).
% fst_diff
thf(fact_19_fst__diff,axiom,
! [X: produc3741383161447143261al_nat,Y: produc3741383161447143261al_nat] :
( ( product_fst_real_nat @ ( minus_1582581163013509572al_nat @ X @ Y ) )
= ( minus_minus_real @ ( product_fst_real_nat @ X ) @ ( product_fst_real_nat @ Y ) ) ) ).
% fst_diff
thf(fact_20_snd__zero,axiom,
( ( product_snd_real_nat @ zero_z5987101913011988884al_nat )
= zero_zero_nat ) ).
% snd_zero
thf(fact_21_snd__zero,axiom,
( ( produc3484788084999411615l_real @ zero_z1365759597461889520l_real )
= zero_zero_real ) ).
% snd_zero
thf(fact_22_old_Oprod_Oinject,axiom,
! [A: nat,B: nat > a,A2: nat,B2: nat > a] :
( ( ( produc5292568359338195516_nat_a @ A @ B )
= ( produc5292568359338195516_nat_a @ A2 @ B2 ) )
= ( ( A = A2 )
& ( B = B2 ) ) ) ).
% old.prod.inject
thf(fact_23_old_Oprod_Oinject,axiom,
! [A: real,B: nat,A2: real,B2: nat] :
( ( ( produc3181502643871035669al_nat @ A @ B )
= ( produc3181502643871035669al_nat @ A2 @ B2 ) )
= ( ( A = A2 )
& ( B = B2 ) ) ) ).
% old.prod.inject
thf(fact_24_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_25_old_Oprod_Oinject,axiom,
! [A: real,B: real,A2: real,B2: real] :
( ( ( produc4511245868158468465l_real @ A @ B )
= ( produc4511245868158468465l_real @ A2 @ B2 ) )
= ( ( A = A2 )
& ( B = B2 ) ) ) ).
% old.prod.inject
thf(fact_26_prod_Oinject,axiom,
! [X1: nat,X2: nat > a,Y1: nat,Y2: nat > a] :
( ( ( produc5292568359338195516_nat_a @ X1 @ X2 )
= ( produc5292568359338195516_nat_a @ Y1 @ Y2 ) )
= ( ( X1 = Y1 )
& ( X2 = Y2 ) ) ) ).
% prod.inject
thf(fact_27_prod_Oinject,axiom,
! [X1: real,X2: nat,Y1: real,Y2: nat] :
( ( ( produc3181502643871035669al_nat @ X1 @ X2 )
= ( produc3181502643871035669al_nat @ Y1 @ Y2 ) )
= ( ( X1 = Y1 )
& ( X2 = Y2 ) ) ) ).
% prod.inject
thf(fact_28_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_29_prod_Oinject,axiom,
! [X1: real,X2: real,Y1: real,Y2: real] :
( ( ( produc4511245868158468465l_real @ X1 @ X2 )
= ( produc4511245868158468465l_real @ Y1 @ Y2 ) )
= ( ( X1 = Y1 )
& ( X2 = Y2 ) ) ) ).
% prod.inject
thf(fact_30_old_Onat_Oinject,axiom,
! [Nat: nat,Nat2: nat] :
( ( ( suc @ Nat )
= ( suc @ Nat2 ) )
= ( Nat = Nat2 ) ) ).
% old.nat.inject
thf(fact_31_nat_Oinject,axiom,
! [X2: nat,Y2: nat] :
( ( ( suc @ X2 )
= ( suc @ Y2 ) )
= ( X2 = Y2 ) ) ).
% nat.inject
thf(fact_32_diff__Pair,axiom,
! [A: nat,B: nat,C: nat,D: nat] :
( ( minus_4365393887724441320at_nat @ ( product_Pair_nat_nat @ A @ B ) @ ( product_Pair_nat_nat @ C @ D ) )
= ( product_Pair_nat_nat @ ( minus_minus_nat @ A @ C ) @ ( minus_minus_nat @ B @ D ) ) ) ).
% diff_Pair
thf(fact_33_diff__Pair,axiom,
! [A: nat,B: extend8495563244428889912nnreal,C: nat,D: extend8495563244428889912nnreal] :
( ( minus_1545026942176751184nnreal @ ( produc5075389201112886689nnreal @ A @ B ) @ ( produc5075389201112886689nnreal @ C @ D ) )
= ( produc5075389201112886689nnreal @ ( minus_minus_nat @ A @ C ) @ ( minus_8429688780609304081nnreal @ B @ D ) ) ) ).
% diff_Pair
thf(fact_34_diff__Pair,axiom,
! [A: nat,B: real,C: nat,D: real] :
( ( minus_5557628854490389828t_real @ ( produc7837566107596912789t_real @ A @ B ) @ ( produc7837566107596912789t_real @ C @ D ) )
= ( produc7837566107596912789t_real @ ( minus_minus_nat @ A @ C ) @ ( minus_minus_real @ B @ D ) ) ) ).
% diff_Pair
thf(fact_35_diff__Pair,axiom,
! [A: nat,B: int,C: nat,D: int] :
( ( minus_187542868215244612at_int @ ( product_Pair_nat_int @ A @ B ) @ ( product_Pair_nat_int @ C @ D ) )
= ( product_Pair_nat_int @ ( minus_minus_nat @ A @ C ) @ ( minus_minus_int @ B @ D ) ) ) ).
% diff_Pair
thf(fact_36_diff__Pair,axiom,
! [A: extend8495563244428889912nnreal,B: nat,C: extend8495563244428889912nnreal,D: nat] :
( ( minus_8921941125199129168al_nat @ ( produc625717604924970401al_nat @ A @ B ) @ ( produc625717604924970401al_nat @ C @ D ) )
= ( produc625717604924970401al_nat @ ( minus_8429688780609304081nnreal @ A @ C ) @ ( minus_minus_nat @ B @ D ) ) ) ).
% diff_Pair
thf(fact_37_diff__Pair,axiom,
! [A: extend8495563244428889912nnreal,B: extend8495563244428889912nnreal,C: extend8495563244428889912nnreal,D: extend8495563244428889912nnreal] :
( ( minus_2620848512045058488nnreal @ ( produc344325839068023049nnreal @ A @ B ) @ ( produc344325839068023049nnreal @ C @ D ) )
= ( produc344325839068023049nnreal @ ( minus_8429688780609304081nnreal @ A @ C ) @ ( minus_8429688780609304081nnreal @ B @ D ) ) ) ).
% diff_Pair
thf(fact_38_diff__Pair,axiom,
! [A: extend8495563244428889912nnreal,B: real,C: extend8495563244428889912nnreal,D: real] :
( ( minus_7344577033118975148l_real @ ( produc2810268924804063229l_real @ A @ B ) @ ( produc2810268924804063229l_real @ C @ D ) )
= ( produc2810268924804063229l_real @ ( minus_8429688780609304081nnreal @ A @ C ) @ ( minus_minus_real @ B @ D ) ) ) ).
% diff_Pair
thf(fact_39_diff__Pair,axiom,
! [A: extend8495563244428889912nnreal,B: int,C: extend8495563244428889912nnreal,D: int] :
( ( minus_4744090105689932460al_int @ ( produc623227134415920125al_int @ A @ B ) @ ( produc623227134415920125al_int @ C @ D ) )
= ( produc623227134415920125al_int @ ( minus_8429688780609304081nnreal @ A @ C ) @ ( minus_minus_int @ B @ D ) ) ) ).
% diff_Pair
thf(fact_40_diff__Pair,axiom,
! [A: real,B: nat,C: real,D: nat] :
( ( minus_1582581163013509572al_nat @ ( produc3181502643871035669al_nat @ A @ B ) @ ( produc3181502643871035669al_nat @ C @ D ) )
= ( produc3181502643871035669al_nat @ ( minus_minus_real @ A @ C ) @ ( minus_minus_nat @ B @ D ) ) ) ).
% diff_Pair
thf(fact_41_diff__Pair,axiom,
! [A: real,B: extend8495563244428889912nnreal,C: real,D: extend8495563244428889912nnreal] :
( ( minus_7390125332462997804nnreal @ ( produc4778015194254607485nnreal @ A @ B ) @ ( produc4778015194254607485nnreal @ C @ D ) )
= ( produc4778015194254607485nnreal @ ( minus_minus_real @ A @ C ) @ ( minus_8429688780609304081nnreal @ B @ D ) ) ) ).
% diff_Pair
thf(fact_42_fst__zero,axiom,
( ( produc5828954698716094813l_real @ zero_z1365759597461889520l_real )
= zero_zero_real ) ).
% fst_zero
thf(fact_43_fst__zero,axiom,
( ( product_fst_real_nat @ zero_z5987101913011988884al_nat )
= zero_zero_real ) ).
% fst_zero
thf(fact_44_Pair__inject,axiom,
! [A: nat,B: nat > a,A2: nat,B2: nat > a] :
( ( ( produc5292568359338195516_nat_a @ A @ B )
= ( produc5292568359338195516_nat_a @ A2 @ B2 ) )
=> ~ ( ( A = A2 )
=> ( B != B2 ) ) ) ).
% Pair_inject
thf(fact_45_Pair__inject,axiom,
! [A: real,B: nat,A2: real,B2: nat] :
( ( ( produc3181502643871035669al_nat @ A @ B )
= ( produc3181502643871035669al_nat @ 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: real,B: real,A2: real,B2: real] :
( ( ( produc4511245868158468465l_real @ A @ B )
= ( produc4511245868158468465l_real @ A2 @ B2 ) )
=> ~ ( ( A = A2 )
=> ( B != B2 ) ) ) ).
% Pair_inject
thf(fact_48_prod__cases,axiom,
! [P: produc7123000486447228170_nat_a > $o,P2: produc7123000486447228170_nat_a] :
( ! [A3: nat,B3: nat > a] : ( P @ ( produc5292568359338195516_nat_a @ A3 @ B3 ) )
=> ( P @ P2 ) ) ).
% prod_cases
thf(fact_49_prod__cases,axiom,
! [P: produc3741383161447143261al_nat > $o,P2: produc3741383161447143261al_nat] :
( ! [A3: real,B3: nat] : ( P @ ( produc3181502643871035669al_nat @ A3 @ B3 ) )
=> ( P @ P2 ) ) ).
% prod_cases
thf(fact_50_prod__cases,axiom,
! [P: product_prod_num_num > $o,P2: product_prod_num_num] :
( ! [A3: num,B3: num] : ( P @ ( product_Pair_num_num @ A3 @ B3 ) )
=> ( P @ P2 ) ) ).
% prod_cases
thf(fact_51_prod__cases,axiom,
! [P: produc2422161461964618553l_real > $o,P2: produc2422161461964618553l_real] :
( ! [A3: real,B3: real] : ( P @ ( produc4511245868158468465l_real @ A3 @ B3 ) )
=> ( P @ P2 ) ) ).
% prod_cases
thf(fact_52_surj__pair,axiom,
! [P2: produc7123000486447228170_nat_a] :
? [X3: nat,Y3: nat > a] :
( P2
= ( produc5292568359338195516_nat_a @ X3 @ Y3 ) ) ).
% surj_pair
thf(fact_53_surj__pair,axiom,
! [P2: produc3741383161447143261al_nat] :
? [X3: real,Y3: nat] :
( P2
= ( produc3181502643871035669al_nat @ X3 @ Y3 ) ) ).
% surj_pair
thf(fact_54_surj__pair,axiom,
! [P2: product_prod_num_num] :
? [X3: num,Y3: num] :
( P2
= ( product_Pair_num_num @ X3 @ Y3 ) ) ).
% surj_pair
thf(fact_55_surj__pair,axiom,
! [P2: produc2422161461964618553l_real] :
? [X3: real,Y3: real] :
( P2
= ( produc4511245868158468465l_real @ X3 @ Y3 ) ) ).
% surj_pair
thf(fact_56_old_Oprod_Oexhaust,axiom,
! [Y: produc7123000486447228170_nat_a] :
~ ! [A3: nat,B3: nat > a] :
( Y
!= ( produc5292568359338195516_nat_a @ A3 @ B3 ) ) ).
% old.prod.exhaust
thf(fact_57_old_Oprod_Oexhaust,axiom,
! [Y: produc3741383161447143261al_nat] :
~ ! [A3: real,B3: nat] :
( Y
!= ( produc3181502643871035669al_nat @ A3 @ B3 ) ) ).
% old.prod.exhaust
thf(fact_58_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_59_old_Oprod_Oexhaust,axiom,
! [Y: produc2422161461964618553l_real] :
~ ! [A3: real,B3: real] :
( Y
!= ( produc4511245868158468465l_real @ A3 @ B3 ) ) ).
% old.prod.exhaust
thf(fact_60_n__not__Suc__n,axiom,
! [N3: nat] :
( N3
!= ( suc @ N3 ) ) ).
% n_not_Suc_n
thf(fact_61_Suc__inject,axiom,
! [X: nat,Y: nat] :
( ( ( suc @ X )
= ( suc @ Y ) )
=> ( X = Y ) ) ).
% Suc_inject
thf(fact_62_diff__commute,axiom,
! [I: nat,J: nat,K: nat] :
( ( minus_minus_nat @ ( minus_minus_nat @ I @ J ) @ K )
= ( minus_minus_nat @ ( minus_minus_nat @ I @ K ) @ J ) ) ).
% diff_commute
thf(fact_63_zero__prod__def,axiom,
( zero_z3979849011205770936at_nat
= ( product_Pair_nat_nat @ zero_zero_nat @ zero_zero_nat ) ) ).
% zero_prod_def
thf(fact_64_zero__prod__def,axiom,
( zero_z4425213877625096212d_enat
= ( produc3145321926189070141d_enat @ zero_zero_nat @ zero_z5237406670263579293d_enat ) ) ).
% zero_prod_def
thf(fact_65_zero__prod__def,axiom,
( zero_z6342917800447569440nnreal
= ( produc5075389201112886689nnreal @ zero_zero_nat @ zero_z7100319975126383169nnreal ) ) ).
% zero_prod_def
thf(fact_66_zero__prod__def,axiom,
( zero_z738777567634093332t_real
= ( produc7837566107596912789t_real @ zero_zero_nat @ zero_zero_real ) ) ).
% zero_prod_def
thf(fact_67_zero__prod__def,axiom,
( zero_z9025370028551350036at_int
= ( product_Pair_nat_int @ zero_zero_nat @ zero_zero_int ) ) ).
% zero_prod_def
thf(fact_68_zero__prod__def,axiom,
( zero_z3287715462348899822at_nat
= ( produc7287838018917372127at_nat @ zero_z5237406670263579293d_enat @ zero_zero_nat ) ) ).
% zero_prod_def
thf(fact_69_zero__prod__def,axiom,
( zero_z1477744776741765214d_enat
= ( produc5368287554538945559d_enat @ zero_z5237406670263579293d_enat @ zero_z5237406670263579293d_enat ) ) ).
% zero_prod_def
thf(fact_70_zero__prod__def,axiom,
( zero_z4477978591438550358nnreal
= ( produc5407049607088739399nnreal @ zero_z5237406670263579293d_enat @ zero_z7100319975126383169nnreal ) ) ).
% zero_prod_def
thf(fact_71_zero__prod__def,axiom,
( zero_z8351172300768850250t_real
= ( produc1106501630054063163t_real @ zero_z5237406670263579293d_enat @ zero_zero_real ) ) ).
% zero_prod_def
thf(fact_72_zero__prod__def,axiom,
( zero_z8333236479694478922at_int
= ( produc7285347548408321851at_int @ zero_z5237406670263579293d_enat @ zero_zero_int ) ) ).
% zero_prod_def
thf(fact_73_not0__implies__Suc,axiom,
! [N3: nat] :
( ( N3 != zero_zero_nat )
=> ? [M3: nat] :
( N3
= ( suc @ M3 ) ) ) ).
% not0_implies_Suc
thf(fact_74_Zero__not__Suc,axiom,
! [M2: nat] :
( zero_zero_nat
!= ( suc @ M2 ) ) ).
% Zero_not_Suc
thf(fact_75_mem__Collect__eq,axiom,
! [A: nat,P: nat > $o] :
( ( member_nat @ A @ ( collect_nat @ P ) )
= ( P @ A ) ) ).
% mem_Collect_eq
thf(fact_76_Collect__mem__eq,axiom,
! [A4: set_nat] :
( ( collect_nat
@ ^ [X4: nat] : ( member_nat @ X4 @ A4 ) )
= A4 ) ).
% Collect_mem_eq
thf(fact_77_Zero__neq__Suc,axiom,
! [M2: nat] :
( zero_zero_nat
!= ( suc @ M2 ) ) ).
% Zero_neq_Suc
thf(fact_78_Suc__neq__Zero,axiom,
! [M2: nat] :
( ( suc @ M2 )
!= zero_zero_nat ) ).
% Suc_neq_Zero
thf(fact_79_zero__induct,axiom,
! [P: nat > $o,K: nat] :
( ( P @ K )
=> ( ! [N2: nat] :
( ( P @ ( suc @ N2 ) )
=> ( P @ N2 ) )
=> ( P @ zero_zero_nat ) ) ) ).
% zero_induct
thf(fact_80_diff__induct,axiom,
! [P: nat > nat > $o,M2: nat,N3: nat] :
( ! [X3: nat] : ( P @ X3 @ zero_zero_nat )
=> ( ! [Y3: nat] : ( P @ zero_zero_nat @ ( suc @ Y3 ) )
=> ( ! [X3: nat,Y3: nat] :
( ( P @ X3 @ Y3 )
=> ( P @ ( suc @ X3 ) @ ( suc @ Y3 ) ) )
=> ( P @ M2 @ N3 ) ) ) ) ).
% diff_induct
thf(fact_81_nat__induct,axiom,
! [P: nat > $o,N3: nat] :
( ( P @ zero_zero_nat )
=> ( ! [N2: nat] :
( ( P @ N2 )
=> ( P @ ( suc @ N2 ) ) )
=> ( P @ N3 ) ) ) ).
% nat_induct
thf(fact_82_old_Onat_Oexhaust,axiom,
! [Y: nat] :
( ( Y != zero_zero_nat )
=> ~ ! [Nat3: nat] :
( Y
!= ( suc @ Nat3 ) ) ) ).
% old.nat.exhaust
thf(fact_83_nat_OdiscI,axiom,
! [Nat: nat,X2: nat] :
( ( Nat
= ( suc @ X2 ) )
=> ( Nat != zero_zero_nat ) ) ).
% nat.discI
thf(fact_84_old_Onat_Odistinct_I1_J,axiom,
! [Nat2: nat] :
( zero_zero_nat
!= ( suc @ Nat2 ) ) ).
% old.nat.distinct(1)
thf(fact_85_old_Onat_Odistinct_I2_J,axiom,
! [Nat2: nat] :
( ( suc @ Nat2 )
!= zero_zero_nat ) ).
% old.nat.distinct(2)
thf(fact_86_nat_Odistinct_I1_J,axiom,
! [X2: nat] :
( zero_zero_nat
!= ( suc @ X2 ) ) ).
% nat.distinct(1)
thf(fact_87_fst__conv,axiom,
! [X1: nat,X2: nat > a] :
( ( produc3194919578927588176_nat_a @ ( produc5292568359338195516_nat_a @ X1 @ X2 ) )
= X1 ) ).
% fst_conv
thf(fact_88_fst__conv,axiom,
! [X1: real,X2: real] :
( ( produc5828954698716094813l_real @ ( produc4511245868158468465l_real @ X1 @ X2 ) )
= X1 ) ).
% fst_conv
thf(fact_89_fst__conv,axiom,
! [X1: num,X2: num] :
( ( product_fst_num_num @ ( product_Pair_num_num @ X1 @ X2 ) )
= X1 ) ).
% fst_conv
thf(fact_90_fst__conv,axiom,
! [X1: real,X2: nat] :
( ( product_fst_real_nat @ ( produc3181502643871035669al_nat @ X1 @ X2 ) )
= X1 ) ).
% fst_conv
thf(fact_91_fst__eqD,axiom,
! [X: nat,Y: nat > a,A: nat] :
( ( ( produc3194919578927588176_nat_a @ ( produc5292568359338195516_nat_a @ X @ Y ) )
= A )
=> ( X = A ) ) ).
% fst_eqD
thf(fact_92_fst__eqD,axiom,
! [X: real,Y: real,A: real] :
( ( ( produc5828954698716094813l_real @ ( produc4511245868158468465l_real @ X @ Y ) )
= A )
=> ( X = A ) ) ).
% fst_eqD
thf(fact_93_fst__eqD,axiom,
! [X: num,Y: num,A: num] :
( ( ( product_fst_num_num @ ( product_Pair_num_num @ X @ Y ) )
= A )
=> ( X = A ) ) ).
% fst_eqD
thf(fact_94_fst__eqD,axiom,
! [X: real,Y: nat,A: real] :
( ( ( product_fst_real_nat @ ( produc3181502643871035669al_nat @ X @ Y ) )
= A )
=> ( X = A ) ) ).
% fst_eqD
thf(fact_95_snd__conv,axiom,
! [X1: nat,X2: nat > a] :
( ( produc4809910040060592782_nat_a @ ( produc5292568359338195516_nat_a @ X1 @ X2 ) )
= X2 ) ).
% snd_conv
thf(fact_96_snd__conv,axiom,
! [X1: real,X2: real] :
( ( produc3484788084999411615l_real @ ( produc4511245868158468465l_real @ X1 @ X2 ) )
= X2 ) ).
% snd_conv
thf(fact_97_snd__conv,axiom,
! [X1: num,X2: num] :
( ( product_snd_num_num @ ( product_Pair_num_num @ X1 @ X2 ) )
= X2 ) ).
% snd_conv
thf(fact_98_snd__conv,axiom,
! [X1: real,X2: nat] :
( ( product_snd_real_nat @ ( produc3181502643871035669al_nat @ X1 @ X2 ) )
= X2 ) ).
% snd_conv
thf(fact_99_snd__eqD,axiom,
! [X: nat,Y: nat > a,A: nat > a] :
( ( ( produc4809910040060592782_nat_a @ ( produc5292568359338195516_nat_a @ X @ Y ) )
= A )
=> ( Y = A ) ) ).
% snd_eqD
thf(fact_100_snd__eqD,axiom,
! [X: real,Y: real,A: real] :
( ( ( produc3484788084999411615l_real @ ( produc4511245868158468465l_real @ X @ Y ) )
= A )
=> ( Y = A ) ) ).
% snd_eqD
thf(fact_101_snd__eqD,axiom,
! [X: num,Y: num,A: num] :
( ( ( product_snd_num_num @ ( product_Pair_num_num @ X @ Y ) )
= A )
=> ( Y = A ) ) ).
% snd_eqD
thf(fact_102_snd__eqD,axiom,
! [X: real,Y: nat,A: nat] :
( ( ( product_snd_real_nat @ ( produc3181502643871035669al_nat @ X @ Y ) )
= A )
=> ( Y = A ) ) ).
% snd_eqD
thf(fact_103_diffs0__imp__equal,axiom,
! [M2: nat,N3: nat] :
( ( ( minus_minus_nat @ M2 @ N3 )
= zero_zero_nat )
=> ( ( ( minus_minus_nat @ N3 @ M2 )
= zero_zero_nat )
=> ( M2 = N3 ) ) ) ).
% diffs0_imp_equal
thf(fact_104_minus__nat_Odiff__0,axiom,
! [M2: nat] :
( ( minus_minus_nat @ M2 @ zero_zero_nat )
= M2 ) ).
% minus_nat.diff_0
thf(fact_105_zero__induct__lemma,axiom,
! [P: nat > $o,K: nat,I: nat] :
( ( P @ K )
=> ( ! [N2: nat] :
( ( P @ ( suc @ N2 ) )
=> ( P @ N2 ) )
=> ( P @ ( minus_minus_nat @ K @ I ) ) ) ) ).
% zero_induct_lemma
thf(fact_106_prod__eq__iff,axiom,
( ( ^ [Y4: produc7123000486447228170_nat_a,Z: produc7123000486447228170_nat_a] : ( Y4 = Z ) )
= ( ^ [S: produc7123000486447228170_nat_a,T: produc7123000486447228170_nat_a] :
( ( ( produc3194919578927588176_nat_a @ S )
= ( produc3194919578927588176_nat_a @ T ) )
& ( ( produc4809910040060592782_nat_a @ S )
= ( produc4809910040060592782_nat_a @ T ) ) ) ) ) ).
% prod_eq_iff
thf(fact_107_prod__eq__iff,axiom,
( ( ^ [Y4: produc2422161461964618553l_real,Z: produc2422161461964618553l_real] : ( Y4 = Z ) )
= ( ^ [S: produc2422161461964618553l_real,T: produc2422161461964618553l_real] :
( ( ( produc5828954698716094813l_real @ S )
= ( produc5828954698716094813l_real @ T ) )
& ( ( produc3484788084999411615l_real @ S )
= ( produc3484788084999411615l_real @ T ) ) ) ) ) ).
% prod_eq_iff
thf(fact_108_prod__eq__iff,axiom,
( ( ^ [Y4: product_prod_num_num,Z: product_prod_num_num] : ( Y4 = Z ) )
= ( ^ [S: product_prod_num_num,T: product_prod_num_num] :
( ( ( product_fst_num_num @ S )
= ( product_fst_num_num @ T ) )
& ( ( product_snd_num_num @ S )
= ( product_snd_num_num @ T ) ) ) ) ) ).
% prod_eq_iff
thf(fact_109_prod__eq__iff,axiom,
( ( ^ [Y4: produc3741383161447143261al_nat,Z: produc3741383161447143261al_nat] : ( Y4 = Z ) )
= ( ^ [S: produc3741383161447143261al_nat,T: produc3741383161447143261al_nat] :
( ( ( product_fst_real_nat @ S )
= ( product_fst_real_nat @ T ) )
& ( ( product_snd_real_nat @ S )
= ( product_snd_real_nat @ T ) ) ) ) ) ).
% prod_eq_iff
thf(fact_110_prod__eqI,axiom,
! [P2: produc7123000486447228170_nat_a,Q: produc7123000486447228170_nat_a] :
( ( ( produc3194919578927588176_nat_a @ P2 )
= ( produc3194919578927588176_nat_a @ Q ) )
=> ( ( ( produc4809910040060592782_nat_a @ P2 )
= ( produc4809910040060592782_nat_a @ Q ) )
=> ( P2 = Q ) ) ) ).
% prod_eqI
thf(fact_111_prod__eqI,axiom,
! [P2: produc2422161461964618553l_real,Q: produc2422161461964618553l_real] :
( ( ( produc5828954698716094813l_real @ P2 )
= ( produc5828954698716094813l_real @ Q ) )
=> ( ( ( produc3484788084999411615l_real @ P2 )
= ( produc3484788084999411615l_real @ Q ) )
=> ( P2 = Q ) ) ) ).
% prod_eqI
thf(fact_112_prod__eqI,axiom,
! [P2: product_prod_num_num,Q: product_prod_num_num] :
( ( ( product_fst_num_num @ P2 )
= ( product_fst_num_num @ Q ) )
=> ( ( ( product_snd_num_num @ P2 )
= ( product_snd_num_num @ Q ) )
=> ( P2 = Q ) ) ) ).
% prod_eqI
thf(fact_113_prod__eqI,axiom,
! [P2: produc3741383161447143261al_nat,Q: produc3741383161447143261al_nat] :
( ( ( product_fst_real_nat @ P2 )
= ( product_fst_real_nat @ Q ) )
=> ( ( ( product_snd_real_nat @ P2 )
= ( product_snd_real_nat @ Q ) )
=> ( P2 = Q ) ) ) ).
% prod_eqI
thf(fact_114_prod_Oexpand,axiom,
! [Prod: produc7123000486447228170_nat_a,Prod2: produc7123000486447228170_nat_a] :
( ( ( ( produc3194919578927588176_nat_a @ Prod )
= ( produc3194919578927588176_nat_a @ Prod2 ) )
& ( ( produc4809910040060592782_nat_a @ Prod )
= ( produc4809910040060592782_nat_a @ Prod2 ) ) )
=> ( Prod = Prod2 ) ) ).
% prod.expand
thf(fact_115_prod_Oexpand,axiom,
! [Prod: produc2422161461964618553l_real,Prod2: produc2422161461964618553l_real] :
( ( ( ( produc5828954698716094813l_real @ Prod )
= ( produc5828954698716094813l_real @ Prod2 ) )
& ( ( produc3484788084999411615l_real @ Prod )
= ( produc3484788084999411615l_real @ Prod2 ) ) )
=> ( Prod = Prod2 ) ) ).
% prod.expand
thf(fact_116_prod_Oexpand,axiom,
! [Prod: product_prod_num_num,Prod2: product_prod_num_num] :
( ( ( ( product_fst_num_num @ Prod )
= ( product_fst_num_num @ Prod2 ) )
& ( ( product_snd_num_num @ Prod )
= ( product_snd_num_num @ Prod2 ) ) )
=> ( Prod = Prod2 ) ) ).
% prod.expand
thf(fact_117_prod_Oexpand,axiom,
! [Prod: produc3741383161447143261al_nat,Prod2: produc3741383161447143261al_nat] :
( ( ( ( product_fst_real_nat @ Prod )
= ( product_fst_real_nat @ Prod2 ) )
& ( ( product_snd_real_nat @ Prod )
= ( product_snd_real_nat @ Prod2 ) ) )
=> ( Prod = Prod2 ) ) ).
% prod.expand
thf(fact_118_list__head__def,axiom,
( coProd6334972811210254351head_a
= ( ^ [L2: produc7123000486447228170_nat_a] : ( produc4809910040060592782_nat_a @ L2 @ zero_zero_nat ) ) ) ).
% list_head_def
thf(fact_119_One__nat__def,axiom,
( one_one_nat
= ( suc @ zero_zero_nat ) ) ).
% One_nat_def
thf(fact_120_surjective__pairing,axiom,
! [T2: produc7123000486447228170_nat_a] :
( T2
= ( produc5292568359338195516_nat_a @ ( produc3194919578927588176_nat_a @ T2 ) @ ( produc4809910040060592782_nat_a @ T2 ) ) ) ).
% surjective_pairing
thf(fact_121_surjective__pairing,axiom,
! [T2: produc2422161461964618553l_real] :
( T2
= ( produc4511245868158468465l_real @ ( produc5828954698716094813l_real @ T2 ) @ ( produc3484788084999411615l_real @ T2 ) ) ) ).
% surjective_pairing
thf(fact_122_surjective__pairing,axiom,
! [T2: product_prod_num_num] :
( T2
= ( product_Pair_num_num @ ( product_fst_num_num @ T2 ) @ ( product_snd_num_num @ T2 ) ) ) ).
% surjective_pairing
thf(fact_123_surjective__pairing,axiom,
! [T2: produc3741383161447143261al_nat] :
( T2
= ( produc3181502643871035669al_nat @ ( product_fst_real_nat @ T2 ) @ ( product_snd_real_nat @ T2 ) ) ) ).
% surjective_pairing
thf(fact_124_prod_Oexhaust__sel,axiom,
! [Prod: produc7123000486447228170_nat_a] :
( Prod
= ( produc5292568359338195516_nat_a @ ( produc3194919578927588176_nat_a @ Prod ) @ ( produc4809910040060592782_nat_a @ Prod ) ) ) ).
% prod.exhaust_sel
thf(fact_125_prod_Oexhaust__sel,axiom,
! [Prod: produc2422161461964618553l_real] :
( Prod
= ( produc4511245868158468465l_real @ ( produc5828954698716094813l_real @ Prod ) @ ( produc3484788084999411615l_real @ Prod ) ) ) ).
% prod.exhaust_sel
thf(fact_126_prod_Oexhaust__sel,axiom,
! [Prod: product_prod_num_num] :
( Prod
= ( product_Pair_num_num @ ( product_fst_num_num @ Prod ) @ ( product_snd_num_num @ Prod ) ) ) ).
% prod.exhaust_sel
thf(fact_127_prod_Oexhaust__sel,axiom,
! [Prod: produc3741383161447143261al_nat] :
( Prod
= ( produc3181502643871035669al_nat @ ( product_fst_real_nat @ Prod ) @ ( product_snd_real_nat @ Prod ) ) ) ).
% prod.exhaust_sel
thf(fact_128_diff__Suc__eq__diff__pred,axiom,
! [M2: nat,N3: nat] :
( ( minus_minus_nat @ M2 @ ( suc @ N3 ) )
= ( minus_minus_nat @ ( minus_minus_nat @ M2 @ one_one_nat ) @ N3 ) ) ).
% diff_Suc_eq_diff_pred
thf(fact_129_minus__prod__def,axiom,
( minus_4365393887724441320at_nat
= ( ^ [X4: product_prod_nat_nat,Y5: product_prod_nat_nat] : ( product_Pair_nat_nat @ ( minus_minus_nat @ ( product_fst_nat_nat @ X4 ) @ ( product_fst_nat_nat @ Y5 ) ) @ ( minus_minus_nat @ ( product_snd_nat_nat @ X4 ) @ ( product_snd_nat_nat @ Y5 ) ) ) ) ) ).
% minus_prod_def
thf(fact_130_minus__prod__def,axiom,
( minus_1545026942176751184nnreal
= ( ^ [X4: produc5192943231052834921nnreal,Y5: produc5192943231052834921nnreal] : ( produc5075389201112886689nnreal @ ( minus_minus_nat @ ( produc3274940128995334541nnreal @ X4 ) @ ( produc3274940128995334541nnreal @ Y5 ) ) @ ( minus_8429688780609304081nnreal @ ( produc940077783810965967nnreal @ X4 ) @ ( produc940077783810965967nnreal @ Y5 ) ) ) ) ) ).
% minus_prod_def
thf(fact_131_minus__prod__def,axiom,
( minus_5557628854490389828t_real
= ( ^ [X4: produc7716430852924023517t_real,Y5: produc7716430852924023517t_real] : ( produc7837566107596912789t_real @ ( minus_minus_nat @ ( product_fst_nat_real @ X4 ) @ ( product_fst_nat_real @ Y5 ) ) @ ( minus_minus_real @ ( product_snd_nat_real @ X4 ) @ ( product_snd_nat_real @ Y5 ) ) ) ) ) ).
% minus_prod_def
thf(fact_132_minus__prod__def,axiom,
( minus_187542868215244612at_int
= ( ^ [X4: product_prod_nat_int,Y5: product_prod_nat_int] : ( product_Pair_nat_int @ ( minus_minus_nat @ ( product_fst_nat_int @ X4 ) @ ( product_fst_nat_int @ Y5 ) ) @ ( minus_minus_int @ ( product_snd_nat_int @ X4 ) @ ( product_snd_nat_int @ Y5 ) ) ) ) ) ).
% minus_prod_def
thf(fact_133_minus__prod__def,axiom,
( minus_8921941125199129168al_nat
= ( ^ [X4: produc3346485377220437097al_nat,Y5: produc3346485377220437097al_nat] : ( produc625717604924970401al_nat @ ( minus_8429688780609304081nnreal @ ( produc8048640569662194061al_nat @ X4 ) @ ( produc8048640569662194061al_nat @ Y5 ) ) @ ( minus_minus_nat @ ( produc5713778224477825487al_nat @ X4 ) @ ( produc5713778224477825487al_nat @ Y5 ) ) ) ) ) ).
% minus_prod_def
thf(fact_134_minus__prod__def,axiom,
( minus_2620848512045058488nnreal
= ( ^ [X4: produc7414223468410354641nnreal,Y5: produc7414223468410354641nnreal] : ( produc344325839068023049nnreal @ ( minus_8429688780609304081nnreal @ ( produc1455421958045324533nnreal @ X4 ) @ ( produc1455421958045324533nnreal @ Y5 ) ) @ ( minus_8429688780609304081nnreal @ ( produc4758864974362570039nnreal @ X4 ) @ ( produc4758864974362570039nnreal @ Y5 ) ) ) ) ) ).
% minus_prod_def
thf(fact_135_minus__prod__def,axiom,
( minus_7344577033118975148l_real
= ( ^ [X4: produc1520197602750038597l_real,Y5: produc1520197602750038597l_real] : ( produc2810268924804063229l_real @ ( minus_8429688780609304081nnreal @ ( produc8390762859368617449l_real @ X4 ) @ ( produc8390762859368617449l_real @ Y5 ) ) @ ( minus_minus_real @ ( produc2623728747760237867l_real @ X4 ) @ ( produc2623728747760237867l_real @ Y5 ) ) ) ) ) ).
% minus_prod_def
thf(fact_136_minus__prod__def,axiom,
( minus_4744090105689932460al_int
= ( ^ [X4: produc8392006394566016197al_int,Y5: produc8392006394566016197al_int] : ( produc623227134415920125al_int @ ( minus_8429688780609304081nnreal @ ( produc8046150099153143785al_int @ X4 ) @ ( produc8046150099153143785al_int @ Y5 ) ) @ ( minus_minus_int @ ( produc5711287753968775211al_int @ X4 ) @ ( produc5711287753968775211al_int @ Y5 ) ) ) ) ) ).
% minus_prod_def
thf(fact_137_minus__prod__def,axiom,
( minus_1582581163013509572al_nat
= ( ^ [X4: produc3741383161447143261al_nat,Y5: produc3741383161447143261al_nat] : ( produc3181502643871035669al_nat @ ( minus_minus_real @ ( product_fst_real_nat @ X4 ) @ ( product_fst_real_nat @ Y5 ) ) @ ( minus_minus_nat @ ( product_snd_real_nat @ X4 ) @ ( product_snd_real_nat @ Y5 ) ) ) ) ) ).
% minus_prod_def
thf(fact_138_minus__prod__def,axiom,
( minus_7390125332462997804nnreal
= ( ^ [X4: produc1565745902094061253nnreal,Y5: produc1565745902094061253nnreal] : ( produc4778015194254607485nnreal @ ( minus_minus_real @ ( produc1135137091964385897nnreal @ X4 ) @ ( produc1135137091964385897nnreal @ Y5 ) ) @ ( minus_8429688780609304081nnreal @ ( produc4591475017210782123nnreal @ X4 ) @ ( produc4591475017210782123nnreal @ Y5 ) ) ) ) ) ).
% minus_prod_def
thf(fact_139_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_140_cancel__comm__monoid__add__class_Odiff__cancel,axiom,
! [A: real] :
( ( minus_minus_real @ A @ A )
= zero_zero_real ) ).
% cancel_comm_monoid_add_class.diff_cancel
thf(fact_141_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_142_diff__zero,axiom,
! [A: nat] :
( ( minus_minus_nat @ A @ zero_zero_nat )
= A ) ).
% diff_zero
thf(fact_143_diff__zero,axiom,
! [A: real] :
( ( minus_minus_real @ A @ zero_zero_real )
= A ) ).
% diff_zero
thf(fact_144_diff__zero,axiom,
! [A: int] :
( ( minus_minus_int @ A @ zero_zero_int )
= A ) ).
% diff_zero
thf(fact_145_zero__diff,axiom,
! [A: nat] :
( ( minus_minus_nat @ zero_zero_nat @ A )
= zero_zero_nat ) ).
% zero_diff
thf(fact_146_diff__0__right,axiom,
! [A: real] :
( ( minus_minus_real @ A @ zero_zero_real )
= A ) ).
% diff_0_right
thf(fact_147_diff__0__right,axiom,
! [A: int] :
( ( minus_minus_int @ A @ zero_zero_int )
= A ) ).
% diff_0_right
thf(fact_148_diff__self,axiom,
! [A: real] :
( ( minus_minus_real @ A @ A )
= zero_zero_real ) ).
% diff_self
thf(fact_149_diff__self,axiom,
! [A: int] :
( ( minus_minus_int @ A @ A )
= zero_zero_int ) ).
% diff_self
thf(fact_150_from__list_Osimps_I2_J,axiom,
! [A: int,L: list_int] :
( ( coProd6425397259703555173st_int @ ( cons_int @ A @ L ) )
= ( produc3661665842610097796at_int @ ( suc @ ( produc522749003355801712at_int @ ( coProd6425397259703555173st_int @ L ) ) )
@ ^ [N: nat] : ( if_int @ ( N = zero_zero_nat ) @ A @ ( produc5442246252769801650at_int @ ( coProd6425397259703555173st_int @ L ) @ ( minus_minus_nat @ N @ one_one_nat ) ) ) ) ) ).
% from_list.simps(2)
thf(fact_151_from__list_Osimps_I2_J,axiom,
! [A: a,L: list_a] :
( ( coProd4444644356563764485list_a @ ( cons_a @ A @ L ) )
= ( produc5292568359338195516_nat_a @ ( suc @ ( produc3194919578927588176_nat_a @ ( coProd4444644356563764485list_a @ L ) ) )
@ ^ [N: nat] : ( if_a @ ( N = zero_zero_nat ) @ A @ ( produc4809910040060592782_nat_a @ ( coProd4444644356563764485list_a @ L ) @ ( minus_minus_nat @ N @ one_one_nat ) ) ) ) ) ).
% from_list.simps(2)
thf(fact_152_BNF__Greatest__Fixpoint_Osubst__Pair,axiom,
! [P: nat > ( nat > a ) > $o,X: nat,Y: nat > a,A: produc7123000486447228170_nat_a] :
( ( P @ X @ Y )
=> ( ( A
= ( produc5292568359338195516_nat_a @ X @ Y ) )
=> ( P @ ( produc3194919578927588176_nat_a @ A ) @ ( produc4809910040060592782_nat_a @ A ) ) ) ) ).
% BNF_Greatest_Fixpoint.subst_Pair
thf(fact_153_BNF__Greatest__Fixpoint_Osubst__Pair,axiom,
! [P: real > real > $o,X: real,Y: real,A: produc2422161461964618553l_real] :
( ( P @ X @ Y )
=> ( ( A
= ( produc4511245868158468465l_real @ X @ Y ) )
=> ( P @ ( produc5828954698716094813l_real @ A ) @ ( produc3484788084999411615l_real @ A ) ) ) ) ).
% BNF_Greatest_Fixpoint.subst_Pair
thf(fact_154_BNF__Greatest__Fixpoint_Osubst__Pair,axiom,
! [P: num > num > $o,X: num,Y: num,A: product_prod_num_num] :
( ( P @ X @ Y )
=> ( ( A
= ( product_Pair_num_num @ X @ Y ) )
=> ( P @ ( product_fst_num_num @ A ) @ ( product_snd_num_num @ A ) ) ) ) ).
% BNF_Greatest_Fixpoint.subst_Pair
thf(fact_155_BNF__Greatest__Fixpoint_Osubst__Pair,axiom,
! [P: real > nat > $o,X: real,Y: nat,A: produc3741383161447143261al_nat] :
( ( P @ X @ Y )
=> ( ( A
= ( produc3181502643871035669al_nat @ X @ Y ) )
=> ( P @ ( product_fst_real_nat @ A ) @ ( product_snd_real_nat @ A ) ) ) ) ).
% BNF_Greatest_Fixpoint.subst_Pair
thf(fact_156_conjI__realizer,axiom,
! [P: nat > $o,P2: nat,Q2: ( nat > a ) > $o,Q: nat > a] :
( ( P @ P2 )
=> ( ( Q2 @ Q )
=> ( ( P @ ( produc3194919578927588176_nat_a @ ( produc5292568359338195516_nat_a @ P2 @ Q ) ) )
& ( Q2 @ ( produc4809910040060592782_nat_a @ ( produc5292568359338195516_nat_a @ P2 @ Q ) ) ) ) ) ) ).
% conjI_realizer
thf(fact_157_conjI__realizer,axiom,
! [P: real > $o,P2: real,Q2: real > $o,Q: real] :
( ( P @ P2 )
=> ( ( Q2 @ Q )
=> ( ( P @ ( produc5828954698716094813l_real @ ( produc4511245868158468465l_real @ P2 @ Q ) ) )
& ( Q2 @ ( produc3484788084999411615l_real @ ( produc4511245868158468465l_real @ P2 @ Q ) ) ) ) ) ) ).
% conjI_realizer
thf(fact_158_conjI__realizer,axiom,
! [P: num > $o,P2: num,Q2: num > $o,Q: num] :
( ( P @ P2 )
=> ( ( Q2 @ Q )
=> ( ( P @ ( product_fst_num_num @ ( product_Pair_num_num @ P2 @ Q ) ) )
& ( Q2 @ ( product_snd_num_num @ ( product_Pair_num_num @ P2 @ Q ) ) ) ) ) ) ).
% conjI_realizer
thf(fact_159_conjI__realizer,axiom,
! [P: real > $o,P2: real,Q2: nat > $o,Q: nat] :
( ( P @ P2 )
=> ( ( Q2 @ Q )
=> ( ( P @ ( product_fst_real_nat @ ( produc3181502643871035669al_nat @ P2 @ Q ) ) )
& ( Q2 @ ( product_snd_real_nat @ ( produc3181502643871035669al_nat @ P2 @ Q ) ) ) ) ) ) ).
% conjI_realizer
thf(fact_160_split__pairs,axiom,
! [A4: nat,B4: nat > a,X5: produc7123000486447228170_nat_a] :
( ( ( produc5292568359338195516_nat_a @ A4 @ B4 )
= X5 )
= ( ( ( produc3194919578927588176_nat_a @ X5 )
= A4 )
& ( ( produc4809910040060592782_nat_a @ X5 )
= B4 ) ) ) ).
% split_pairs
thf(fact_161_split__pairs,axiom,
! [A4: real,B4: real,X5: produc2422161461964618553l_real] :
( ( ( produc4511245868158468465l_real @ A4 @ B4 )
= X5 )
= ( ( ( produc5828954698716094813l_real @ X5 )
= A4 )
& ( ( produc3484788084999411615l_real @ X5 )
= B4 ) ) ) ).
% split_pairs
thf(fact_162_split__pairs,axiom,
! [A4: num,B4: num,X5: product_prod_num_num] :
( ( ( product_Pair_num_num @ A4 @ B4 )
= X5 )
= ( ( ( product_fst_num_num @ X5 )
= A4 )
& ( ( product_snd_num_num @ X5 )
= B4 ) ) ) ).
% split_pairs
thf(fact_163_split__pairs,axiom,
! [A4: real,B4: nat,X5: produc3741383161447143261al_nat] :
( ( ( produc3181502643871035669al_nat @ A4 @ B4 )
= X5 )
= ( ( ( product_fst_real_nat @ X5 )
= A4 )
& ( ( product_snd_real_nat @ X5 )
= B4 ) ) ) ).
% split_pairs
thf(fact_164_exI__realizer,axiom,
! [P: ( nat > a ) > nat > $o,Y: nat > a,X: nat] :
( ( P @ Y @ X )
=> ( P @ ( produc4809910040060592782_nat_a @ ( produc5292568359338195516_nat_a @ X @ Y ) ) @ ( produc3194919578927588176_nat_a @ ( produc5292568359338195516_nat_a @ X @ Y ) ) ) ) ).
% exI_realizer
thf(fact_165_exI__realizer,axiom,
! [P: real > real > $o,Y: real,X: real] :
( ( P @ Y @ X )
=> ( P @ ( produc3484788084999411615l_real @ ( produc4511245868158468465l_real @ X @ Y ) ) @ ( produc5828954698716094813l_real @ ( produc4511245868158468465l_real @ X @ Y ) ) ) ) ).
% exI_realizer
thf(fact_166_exI__realizer,axiom,
! [P: num > num > $o,Y: num,X: num] :
( ( P @ Y @ X )
=> ( P @ ( product_snd_num_num @ ( product_Pair_num_num @ X @ Y ) ) @ ( product_fst_num_num @ ( product_Pair_num_num @ X @ Y ) ) ) ) ).
% exI_realizer
thf(fact_167_exI__realizer,axiom,
! [P: nat > real > $o,Y: nat,X: real] :
( ( P @ Y @ X )
=> ( P @ ( product_snd_real_nat @ ( produc3181502643871035669al_nat @ X @ Y ) ) @ ( product_fst_real_nat @ ( produc3181502643871035669al_nat @ X @ Y ) ) ) ) ).
% exI_realizer
thf(fact_168_list_Oinject,axiom,
! [X21: a,X22: list_a,Y21: a,Y22: list_a] :
( ( ( cons_a @ X21 @ X22 )
= ( cons_a @ Y21 @ Y22 ) )
= ( ( X21 = Y21 )
& ( X22 = Y22 ) ) ) ).
% list.inject
thf(fact_169_list_Oinject,axiom,
! [X21: int,X22: list_int,Y21: int,Y22: list_int] :
( ( ( cons_int @ X21 @ X22 )
= ( cons_int @ Y21 @ Y22 ) )
= ( ( X21 = Y21 )
& ( X22 = Y22 ) ) ) ).
% list.inject
thf(fact_170_zero__reorient,axiom,
! [X: nat] :
( ( zero_zero_nat = X )
= ( X = zero_zero_nat ) ) ).
% zero_reorient
thf(fact_171_zero__reorient,axiom,
! [X: extended_enat] :
( ( zero_z5237406670263579293d_enat = X )
= ( X = zero_z5237406670263579293d_enat ) ) ).
% zero_reorient
thf(fact_172_zero__reorient,axiom,
! [X: extend8495563244428889912nnreal] :
( ( zero_z7100319975126383169nnreal = X )
= ( X = zero_z7100319975126383169nnreal ) ) ).
% zero_reorient
thf(fact_173_zero__reorient,axiom,
! [X: real] :
( ( zero_zero_real = X )
= ( X = zero_zero_real ) ) ).
% zero_reorient
thf(fact_174_zero__reorient,axiom,
! [X: int] :
( ( zero_zero_int = X )
= ( X = zero_zero_int ) ) ).
% zero_reorient
thf(fact_175_one__reorient,axiom,
! [X: nat] :
( ( one_one_nat = X )
= ( X = one_one_nat ) ) ).
% one_reorient
thf(fact_176_one__reorient,axiom,
! [X: extend8495563244428889912nnreal] :
( ( one_on2969667320475766781nnreal = X )
= ( X = one_on2969667320475766781nnreal ) ) ).
% one_reorient
thf(fact_177_one__reorient,axiom,
! [X: real] :
( ( one_one_real = X )
= ( X = one_one_real ) ) ).
% one_reorient
thf(fact_178_one__reorient,axiom,
! [X: int] :
( ( one_one_int = X )
= ( X = one_one_int ) ) ).
% one_reorient
thf(fact_179_diff__eq__diff__eq,axiom,
! [A: real,B: real,C: real,D: real] :
( ( ( minus_minus_real @ A @ B )
= ( minus_minus_real @ C @ D ) )
=> ( ( A = B )
= ( C = D ) ) ) ).
% diff_eq_diff_eq
thf(fact_180_diff__eq__diff__eq,axiom,
! [A: int,B: int,C: int,D: int] :
( ( ( minus_minus_int @ A @ B )
= ( minus_minus_int @ C @ D ) )
=> ( ( A = B )
= ( C = D ) ) ) ).
% diff_eq_diff_eq
thf(fact_181_cancel__ab__semigroup__add__class_Odiff__right__commute,axiom,
! [A: nat,C: nat,B: nat] :
( ( minus_minus_nat @ ( minus_minus_nat @ A @ C ) @ B )
= ( minus_minus_nat @ ( minus_minus_nat @ A @ B ) @ C ) ) ).
% cancel_ab_semigroup_add_class.diff_right_commute
thf(fact_182_cancel__ab__semigroup__add__class_Odiff__right__commute,axiom,
! [A: real,C: real,B: real] :
( ( minus_minus_real @ ( minus_minus_real @ A @ C ) @ B )
= ( minus_minus_real @ ( minus_minus_real @ A @ B ) @ C ) ) ).
% cancel_ab_semigroup_add_class.diff_right_commute
thf(fact_183_cancel__ab__semigroup__add__class_Odiff__right__commute,axiom,
! [A: int,C: int,B: int] :
( ( minus_minus_int @ ( minus_minus_int @ A @ C ) @ B )
= ( minus_minus_int @ ( minus_minus_int @ A @ B ) @ C ) ) ).
% cancel_ab_semigroup_add_class.diff_right_commute
thf(fact_184_not__Cons__self2,axiom,
! [X: a,Xs: list_a] :
( ( cons_a @ X @ Xs )
!= Xs ) ).
% not_Cons_self2
thf(fact_185_not__Cons__self2,axiom,
! [X: int,Xs: list_int] :
( ( cons_int @ X @ Xs )
!= Xs ) ).
% not_Cons_self2
thf(fact_186_eq__iff__diff__eq__0,axiom,
( ( ^ [Y4: real,Z: real] : ( Y4 = Z ) )
= ( ^ [A5: real,B5: real] :
( ( minus_minus_real @ A5 @ B5 )
= zero_zero_real ) ) ) ).
% eq_iff_diff_eq_0
thf(fact_187_eq__iff__diff__eq__0,axiom,
( ( ^ [Y4: int,Z: int] : ( Y4 = Z ) )
= ( ^ [A5: int,B5: int] :
( ( minus_minus_int @ A5 @ B5 )
= zero_zero_int ) ) ) ).
% eq_iff_diff_eq_0
thf(fact_188_exE__realizer_H,axiom,
! [P: ( nat > a ) > nat > $o,P2: produc7123000486447228170_nat_a] :
( ( P @ ( produc4809910040060592782_nat_a @ P2 ) @ ( produc3194919578927588176_nat_a @ P2 ) )
=> ~ ! [X3: nat,Y3: nat > a] :
~ ( P @ Y3 @ X3 ) ) ).
% exE_realizer'
thf(fact_189_exE__realizer_H,axiom,
! [P: real > real > $o,P2: produc2422161461964618553l_real] :
( ( P @ ( produc3484788084999411615l_real @ P2 ) @ ( produc5828954698716094813l_real @ P2 ) )
=> ~ ! [X3: real,Y3: real] :
~ ( P @ Y3 @ X3 ) ) ).
% exE_realizer'
thf(fact_190_exE__realizer_H,axiom,
! [P: num > num > $o,P2: product_prod_num_num] :
( ( P @ ( product_snd_num_num @ P2 ) @ ( product_fst_num_num @ P2 ) )
=> ~ ! [X3: num,Y3: num] :
~ ( P @ Y3 @ X3 ) ) ).
% exE_realizer'
thf(fact_191_exE__realizer_H,axiom,
! [P: nat > real > $o,P2: produc3741383161447143261al_nat] :
( ( P @ ( product_snd_real_nat @ P2 ) @ ( product_fst_real_nat @ P2 ) )
=> ~ ! [X3: real,Y3: nat] :
~ ( P @ Y3 @ X3 ) ) ).
% exE_realizer'
thf(fact_192_list__simp1,axiom,
! [L: produc7123000486447228170_nat_a,X: a] :
( ( produc5292568359338195516_nat_a @ zero_zero_nat
@ ^ [N: nat] : undefined_a )
!= ( produc5292568359338195516_nat_a @ ( suc @ ( produc3194919578927588176_nat_a @ L ) )
@ ^ [N: nat] : ( if_a @ ( N = zero_zero_nat ) @ X @ ( produc4809910040060592782_nat_a @ L @ ( minus_minus_nat @ N @ one_one_nat ) ) ) ) ) ).
% list_simp1
thf(fact_193_sndI,axiom,
! [X: produc7123000486447228170_nat_a,Y: nat,Z2: nat > a] :
( ( X
= ( produc5292568359338195516_nat_a @ Y @ Z2 ) )
=> ( ( produc4809910040060592782_nat_a @ X )
= Z2 ) ) ).
% sndI
thf(fact_194_sndI,axiom,
! [X: produc2422161461964618553l_real,Y: real,Z2: real] :
( ( X
= ( produc4511245868158468465l_real @ Y @ Z2 ) )
=> ( ( produc3484788084999411615l_real @ X )
= Z2 ) ) ).
% sndI
thf(fact_195_sndI,axiom,
! [X: product_prod_num_num,Y: num,Z2: num] :
( ( X
= ( product_Pair_num_num @ Y @ Z2 ) )
=> ( ( product_snd_num_num @ X )
= Z2 ) ) ).
% sndI
thf(fact_196_sndI,axiom,
! [X: produc3741383161447143261al_nat,Y: real,Z2: nat] :
( ( X
= ( produc3181502643871035669al_nat @ Y @ Z2 ) )
=> ( ( product_snd_real_nat @ X )
= Z2 ) ) ).
% sndI
thf(fact_197_eq__snd__iff,axiom,
! [B: nat > a,P2: produc7123000486447228170_nat_a] :
( ( B
= ( produc4809910040060592782_nat_a @ P2 ) )
= ( ? [A5: nat] :
( P2
= ( produc5292568359338195516_nat_a @ A5 @ B ) ) ) ) ).
% eq_snd_iff
thf(fact_198_eq__snd__iff,axiom,
! [B: real,P2: produc2422161461964618553l_real] :
( ( B
= ( produc3484788084999411615l_real @ P2 ) )
= ( ? [A5: real] :
( P2
= ( produc4511245868158468465l_real @ A5 @ B ) ) ) ) ).
% eq_snd_iff
thf(fact_199_eq__snd__iff,axiom,
! [B: num,P2: product_prod_num_num] :
( ( B
= ( product_snd_num_num @ P2 ) )
= ( ? [A5: num] :
( P2
= ( product_Pair_num_num @ A5 @ B ) ) ) ) ).
% eq_snd_iff
thf(fact_200_eq__snd__iff,axiom,
! [B: nat,P2: produc3741383161447143261al_nat] :
( ( B
= ( product_snd_real_nat @ P2 ) )
= ( ? [A5: real] :
( P2
= ( produc3181502643871035669al_nat @ A5 @ B ) ) ) ) ).
% eq_snd_iff
thf(fact_201_fstI,axiom,
! [X: produc7123000486447228170_nat_a,Y: nat,Z2: nat > a] :
( ( X
= ( produc5292568359338195516_nat_a @ Y @ Z2 ) )
=> ( ( produc3194919578927588176_nat_a @ X )
= Y ) ) ).
% fstI
thf(fact_202_fstI,axiom,
! [X: produc2422161461964618553l_real,Y: real,Z2: real] :
( ( X
= ( produc4511245868158468465l_real @ Y @ Z2 ) )
=> ( ( produc5828954698716094813l_real @ X )
= Y ) ) ).
% fstI
thf(fact_203_fstI,axiom,
! [X: product_prod_num_num,Y: num,Z2: num] :
( ( X
= ( product_Pair_num_num @ Y @ Z2 ) )
=> ( ( product_fst_num_num @ X )
= Y ) ) ).
% fstI
thf(fact_204_fstI,axiom,
! [X: produc3741383161447143261al_nat,Y: real,Z2: nat] :
( ( X
= ( produc3181502643871035669al_nat @ Y @ Z2 ) )
=> ( ( product_fst_real_nat @ X )
= Y ) ) ).
% fstI
thf(fact_205_eq__fst__iff,axiom,
! [A: nat,P2: produc7123000486447228170_nat_a] :
( ( A
= ( produc3194919578927588176_nat_a @ P2 ) )
= ( ? [B5: nat > a] :
( P2
= ( produc5292568359338195516_nat_a @ A @ B5 ) ) ) ) ).
% eq_fst_iff
thf(fact_206_eq__fst__iff,axiom,
! [A: real,P2: produc2422161461964618553l_real] :
( ( A
= ( produc5828954698716094813l_real @ P2 ) )
= ( ? [B5: real] :
( P2
= ( produc4511245868158468465l_real @ A @ B5 ) ) ) ) ).
% eq_fst_iff
thf(fact_207_eq__fst__iff,axiom,
! [A: num,P2: product_prod_num_num] :
( ( A
= ( product_fst_num_num @ P2 ) )
= ( ? [B5: num] :
( P2
= ( product_Pair_num_num @ A @ B5 ) ) ) ) ).
% eq_fst_iff
thf(fact_208_eq__fst__iff,axiom,
! [A: real,P2: produc3741383161447143261al_nat] :
( ( A
= ( product_fst_real_nat @ P2 ) )
= ( ? [B5: nat] :
( P2
= ( produc3181502643871035669al_nat @ A @ B5 ) ) ) ) ).
% eq_fst_iff
thf(fact_209_list__decode_Ocases,axiom,
! [X: nat] :
( ( X != zero_zero_nat )
=> ~ ! [N2: nat] :
( X
!= ( suc @ N2 ) ) ) ).
% list_decode.cases
thf(fact_210_r01__binary__expansion_H_H_Ocases,axiom,
! [X: produc3741383161447143261al_nat] :
( ! [R: real] :
( X
!= ( produc3181502643871035669al_nat @ R @ zero_zero_nat ) )
=> ~ ! [R: real,N2: nat] :
( X
!= ( produc3181502643871035669al_nat @ R @ ( suc @ N2 ) ) ) ) ).
% r01_binary_expansion''.cases
thf(fact_211_from__list_Oelims,axiom,
! [X: list_int,Y: produc6819000835714826316at_int] :
( ( ( coProd6425397259703555173st_int @ X )
= Y )
=> ( ( ( X = nil_int )
=> ( Y
!= ( produc3661665842610097796at_int @ zero_zero_nat
@ ^ [N: nat] : undefined_int ) ) )
=> ~ ! [A3: int,L3: list_int] :
( ( X
= ( cons_int @ A3 @ L3 ) )
=> ( Y
!= ( produc3661665842610097796at_int @ ( suc @ ( produc522749003355801712at_int @ ( coProd6425397259703555173st_int @ L3 ) ) )
@ ^ [N: nat] : ( if_int @ ( N = zero_zero_nat ) @ A3 @ ( produc5442246252769801650at_int @ ( coProd6425397259703555173st_int @ L3 ) @ ( minus_minus_nat @ N @ one_one_nat ) ) ) ) ) ) ) ) ).
% from_list.elims
thf(fact_212_from__list_Oelims,axiom,
! [X: list_a,Y: produc7123000486447228170_nat_a] :
( ( ( coProd4444644356563764485list_a @ X )
= Y )
=> ( ( ( X = nil_a )
=> ( Y
!= ( produc5292568359338195516_nat_a @ zero_zero_nat
@ ^ [N: nat] : undefined_a ) ) )
=> ~ ! [A3: a,L3: list_a] :
( ( X
= ( cons_a @ A3 @ L3 ) )
=> ( Y
!= ( produc5292568359338195516_nat_a @ ( suc @ ( produc3194919578927588176_nat_a @ ( coProd4444644356563764485list_a @ L3 ) ) )
@ ^ [N: nat] : ( if_a @ ( N = zero_zero_nat ) @ A3 @ ( produc4809910040060592782_nat_a @ ( coProd4444644356563764485list_a @ L3 ) @ ( minus_minus_nat @ N @ one_one_nat ) ) ) ) ) ) ) ) ).
% from_list.elims
thf(fact_213_fps__right__inverse__constructor_Ocases,axiom,
! [X: produc2300234653916283166al_nat] :
( ! [F: formal3361831859752904756s_real,A3: real] :
( X
!= ( produc6513356265823010512al_nat @ F @ ( produc3181502643871035669al_nat @ A3 @ zero_zero_nat ) ) )
=> ~ ! [F: formal3361831859752904756s_real,A3: real,V: nat] :
( X
!= ( produc6513356265823010512al_nat @ F @ ( produc3181502643871035669al_nat @ A3 @ ( suc @ V ) ) ) ) ) ).
% fps_right_inverse_constructor.cases
thf(fact_214_exists__least__lemma,axiom,
! [P: nat > $o] :
( ~ ( P @ zero_zero_nat )
=> ( ? [X_1: nat] : ( P @ X_1 )
=> ? [N2: nat] :
( ~ ( P @ N2 )
& ( P @ ( suc @ N2 ) ) ) ) ) ).
% exists_least_lemma
thf(fact_215_zero__neq__one,axiom,
zero_zero_nat != one_one_nat ).
% zero_neq_one
thf(fact_216_zero__neq__one,axiom,
zero_z5237406670263579293d_enat != one_on7984719198319812577d_enat ).
% zero_neq_one
thf(fact_217_zero__neq__one,axiom,
zero_z7100319975126383169nnreal != one_on2969667320475766781nnreal ).
% zero_neq_one
thf(fact_218_zero__neq__one,axiom,
zero_zero_real != one_one_real ).
% zero_neq_one
thf(fact_219_zero__neq__one,axiom,
zero_zero_int != one_one_int ).
% zero_neq_one
thf(fact_220_to__list_H_Osimps_I2_J,axiom,
! [N3: nat,F2: nat > a] :
( ( coProduct_to_list_a2 @ ( suc @ N3 ) @ F2 )
= ( cons_a @ ( F2 @ zero_zero_nat )
@ ( coProduct_to_list_a2 @ N3
@ ^ [N: nat] : ( F2 @ ( suc @ N ) ) ) ) ) ).
% to_list'.simps(2)
thf(fact_221_to__list_H_Osimps_I2_J,axiom,
! [N3: nat,F2: nat > int] :
( ( coProd6641981980276419519st_int @ ( suc @ N3 ) @ F2 )
= ( cons_int @ ( F2 @ zero_zero_nat )
@ ( coProd6641981980276419519st_int @ N3
@ ^ [N: nat] : ( F2 @ ( suc @ N ) ) ) ) ) ).
% to_list'.simps(2)
thf(fact_222_transpose_Ocases,axiom,
! [X: list_list_a] :
( ( X != nil_list_a )
=> ( ! [Xss: list_list_a] :
( X
!= ( cons_list_a @ nil_a @ Xss ) )
=> ~ ! [X3: a,Xs2: list_a,Xss: list_list_a] :
( X
!= ( cons_list_a @ ( cons_a @ X3 @ Xs2 ) @ Xss ) ) ) ) ).
% transpose.cases
thf(fact_223_transpose_Ocases,axiom,
! [X: list_list_int] :
( ( X != nil_list_int )
=> ( ! [Xss: list_list_int] :
( X
!= ( cons_list_int @ nil_int @ Xss ) )
=> ~ ! [X3: int,Xs2: list_int,Xss: list_list_int] :
( X
!= ( cons_list_int @ ( cons_int @ X3 @ Xs2 ) @ Xss ) ) ) ) ).
% transpose.cases
thf(fact_224_list_Odistinct_I1_J,axiom,
! [X21: a,X22: list_a] :
( nil_a
!= ( cons_a @ X21 @ X22 ) ) ).
% list.distinct(1)
thf(fact_225_list_Odistinct_I1_J,axiom,
! [X21: int,X22: list_int] :
( nil_int
!= ( cons_int @ X21 @ X22 ) ) ).
% list.distinct(1)
thf(fact_226_list_OdiscI,axiom,
! [List: list_a,X21: a,X22: list_a] :
( ( List
= ( cons_a @ X21 @ X22 ) )
=> ( List != nil_a ) ) ).
% list.discI
thf(fact_227_list_OdiscI,axiom,
! [List: list_int,X21: int,X22: list_int] :
( ( List
= ( cons_int @ X21 @ X22 ) )
=> ( List != nil_int ) ) ).
% list.discI
thf(fact_228_list_Oexhaust,axiom,
! [Y: list_a] :
( ( Y != nil_a )
=> ~ ! [X212: a,X222: list_a] :
( Y
!= ( cons_a @ X212 @ X222 ) ) ) ).
% list.exhaust
thf(fact_229_list_Oexhaust,axiom,
! [Y: list_int] :
( ( Y != nil_int )
=> ~ ! [X212: int,X222: list_int] :
( Y
!= ( cons_int @ X212 @ X222 ) ) ) ).
% list.exhaust
thf(fact_230_min__list_Ocases,axiom,
! [X: list_int] :
( ! [X3: int,Xs2: list_int] :
( X
!= ( cons_int @ X3 @ Xs2 ) )
=> ( X = nil_int ) ) ).
% min_list.cases
thf(fact_231_remdups__adj_Ocases,axiom,
! [X: list_a] :
( ( X != nil_a )
=> ( ! [X3: a] :
( X
!= ( cons_a @ X3 @ nil_a ) )
=> ~ ! [X3: a,Y3: a,Xs2: list_a] :
( X
!= ( cons_a @ X3 @ ( cons_a @ Y3 @ Xs2 ) ) ) ) ) ).
% remdups_adj.cases
thf(fact_232_remdups__adj_Ocases,axiom,
! [X: list_int] :
( ( X != nil_int )
=> ( ! [X3: int] :
( X
!= ( cons_int @ X3 @ nil_int ) )
=> ~ ! [X3: int,Y3: int,Xs2: list_int] :
( X
!= ( cons_int @ X3 @ ( cons_int @ Y3 @ Xs2 ) ) ) ) ) ).
% remdups_adj.cases
thf(fact_233_neq__Nil__conv,axiom,
! [Xs: list_a] :
( ( Xs != nil_a )
= ( ? [Y5: a,Ys: list_a] :
( Xs
= ( cons_a @ Y5 @ Ys ) ) ) ) ).
% neq_Nil_conv
thf(fact_234_neq__Nil__conv,axiom,
! [Xs: list_int] :
( ( Xs != nil_int )
= ( ? [Y5: int,Ys: list_int] :
( Xs
= ( cons_int @ Y5 @ Ys ) ) ) ) ).
% neq_Nil_conv
thf(fact_235_list__induct2_H,axiom,
! [P: list_a > list_a > $o,Xs: list_a,Ys2: list_a] :
( ( P @ nil_a @ nil_a )
=> ( ! [X3: a,Xs2: list_a] : ( P @ ( cons_a @ X3 @ Xs2 ) @ nil_a )
=> ( ! [Y3: a,Ys3: list_a] : ( P @ nil_a @ ( cons_a @ Y3 @ Ys3 ) )
=> ( ! [X3: a,Xs2: list_a,Y3: a,Ys3: list_a] :
( ( P @ Xs2 @ Ys3 )
=> ( P @ ( cons_a @ X3 @ Xs2 ) @ ( cons_a @ Y3 @ Ys3 ) ) )
=> ( P @ Xs @ Ys2 ) ) ) ) ) ).
% list_induct2'
thf(fact_236_list__induct2_H,axiom,
! [P: list_a > list_int > $o,Xs: list_a,Ys2: list_int] :
( ( P @ nil_a @ nil_int )
=> ( ! [X3: a,Xs2: list_a] : ( P @ ( cons_a @ X3 @ Xs2 ) @ nil_int )
=> ( ! [Y3: int,Ys3: list_int] : ( P @ nil_a @ ( cons_int @ Y3 @ Ys3 ) )
=> ( ! [X3: a,Xs2: list_a,Y3: int,Ys3: list_int] :
( ( P @ Xs2 @ Ys3 )
=> ( P @ ( cons_a @ X3 @ Xs2 ) @ ( cons_int @ Y3 @ Ys3 ) ) )
=> ( P @ Xs @ Ys2 ) ) ) ) ) ).
% list_induct2'
thf(fact_237_list__induct2_H,axiom,
! [P: list_int > list_a > $o,Xs: list_int,Ys2: list_a] :
( ( P @ nil_int @ nil_a )
=> ( ! [X3: int,Xs2: list_int] : ( P @ ( cons_int @ X3 @ Xs2 ) @ nil_a )
=> ( ! [Y3: a,Ys3: list_a] : ( P @ nil_int @ ( cons_a @ Y3 @ Ys3 ) )
=> ( ! [X3: int,Xs2: list_int,Y3: a,Ys3: list_a] :
( ( P @ Xs2 @ Ys3 )
=> ( P @ ( cons_int @ X3 @ Xs2 ) @ ( cons_a @ Y3 @ Ys3 ) ) )
=> ( P @ Xs @ Ys2 ) ) ) ) ) ).
% list_induct2'
thf(fact_238_list__induct2_H,axiom,
! [P: list_int > list_int > $o,Xs: list_int,Ys2: list_int] :
( ( P @ nil_int @ nil_int )
=> ( ! [X3: int,Xs2: list_int] : ( P @ ( cons_int @ X3 @ Xs2 ) @ nil_int )
=> ( ! [Y3: int,Ys3: list_int] : ( P @ nil_int @ ( cons_int @ Y3 @ Ys3 ) )
=> ( ! [X3: int,Xs2: list_int,Y3: int,Ys3: list_int] :
( ( P @ Xs2 @ Ys3 )
=> ( P @ ( cons_int @ X3 @ Xs2 ) @ ( cons_int @ Y3 @ Ys3 ) ) )
=> ( P @ Xs @ Ys2 ) ) ) ) ) ).
% list_induct2'
thf(fact_239_list__nonempty__induct,axiom,
! [Xs: list_a,P: list_a > $o] :
( ( Xs != nil_a )
=> ( ! [X3: a] : ( P @ ( cons_a @ X3 @ nil_a ) )
=> ( ! [X3: a,Xs2: list_a] :
( ( Xs2 != nil_a )
=> ( ( P @ Xs2 )
=> ( P @ ( cons_a @ X3 @ Xs2 ) ) ) )
=> ( P @ Xs ) ) ) ) ).
% list_nonempty_induct
thf(fact_240_list__nonempty__induct,axiom,
! [Xs: list_int,P: list_int > $o] :
( ( Xs != nil_int )
=> ( ! [X3: int] : ( P @ ( cons_int @ X3 @ nil_int ) )
=> ( ! [X3: int,Xs2: list_int] :
( ( Xs2 != nil_int )
=> ( ( P @ Xs2 )
=> ( P @ ( cons_int @ X3 @ Xs2 ) ) ) )
=> ( P @ Xs ) ) ) ) ).
% list_nonempty_induct
thf(fact_241_successively_Ocases,axiom,
! [X: produc5032551385658279741list_a] :
( ! [P3: a > a > $o] :
( X
!= ( produc8111569692950616493list_a @ P3 @ nil_a ) )
=> ( ! [P3: a > a > $o,X3: a] :
( X
!= ( produc8111569692950616493list_a @ P3 @ ( cons_a @ X3 @ nil_a ) ) )
=> ~ ! [P3: a > a > $o,X3: a,Y3: a,Xs2: list_a] :
( X
!= ( produc8111569692950616493list_a @ P3 @ ( cons_a @ X3 @ ( cons_a @ Y3 @ Xs2 ) ) ) ) ) ) ).
% successively.cases
thf(fact_242_successively_Ocases,axiom,
! [X: produc5834231552977413017st_int] :
( ! [P3: int > int > $o] :
( X
!= ( produc8618682346314911123st_int @ P3 @ nil_int ) )
=> ( ! [P3: int > int > $o,X3: int] :
( X
!= ( produc8618682346314911123st_int @ P3 @ ( cons_int @ X3 @ nil_int ) ) )
=> ~ ! [P3: int > int > $o,X3: int,Y3: int,Xs2: list_int] :
( X
!= ( produc8618682346314911123st_int @ P3 @ ( cons_int @ X3 @ ( cons_int @ Y3 @ Xs2 ) ) ) ) ) ) ).
% successively.cases
thf(fact_243_sorted__wrt_Ocases,axiom,
! [X: produc5032551385658279741list_a] :
( ! [P3: a > a > $o] :
( X
!= ( produc8111569692950616493list_a @ P3 @ nil_a ) )
=> ~ ! [P3: a > a > $o,X3: a,Ys3: list_a] :
( X
!= ( produc8111569692950616493list_a @ P3 @ ( cons_a @ X3 @ Ys3 ) ) ) ) ).
% sorted_wrt.cases
thf(fact_244_sorted__wrt_Ocases,axiom,
! [X: produc5834231552977413017st_int] :
( ! [P3: int > int > $o] :
( X
!= ( produc8618682346314911123st_int @ P3 @ nil_int ) )
=> ~ ! [P3: int > int > $o,X3: int,Ys3: list_int] :
( X
!= ( produc8618682346314911123st_int @ P3 @ ( cons_int @ X3 @ Ys3 ) ) ) ) ).
% sorted_wrt.cases
thf(fact_245_shuffles_Ocases,axiom,
! [X: produc9164743771328383783list_a] :
( ! [Ys3: list_a] :
( X
!= ( produc6837034575241423639list_a @ nil_a @ Ys3 ) )
=> ( ! [Xs2: list_a] :
( X
!= ( produc6837034575241423639list_a @ Xs2 @ nil_a ) )
=> ~ ! [X3: a,Xs2: list_a,Y3: a,Ys3: list_a] :
( X
!= ( produc6837034575241423639list_a @ ( cons_a @ X3 @ Xs2 ) @ ( cons_a @ Y3 @ Ys3 ) ) ) ) ) ).
% shuffles.cases
thf(fact_246_shuffles_Ocases,axiom,
! [X: produc1186641810826059865st_int] :
( ! [Ys3: list_int] :
( X
!= ( produc364263696895485585st_int @ nil_int @ Ys3 ) )
=> ( ! [Xs2: list_int] :
( X
!= ( produc364263696895485585st_int @ Xs2 @ nil_int ) )
=> ~ ! [X3: int,Xs2: list_int,Y3: int,Ys3: list_int] :
( X
!= ( produc364263696895485585st_int @ ( cons_int @ X3 @ Xs2 ) @ ( cons_int @ Y3 @ Ys3 ) ) ) ) ) ).
% shuffles.cases
thf(fact_247_from__list_Ocases,axiom,
! [X: list_a] :
( ( X != nil_a )
=> ~ ! [A3: a,L3: list_a] :
( X
!= ( cons_a @ A3 @ L3 ) ) ) ).
% from_list.cases
thf(fact_248_from__list_Ocases,axiom,
! [X: list_int] :
( ( X != nil_int )
=> ~ ! [A3: int,L3: list_int] :
( X
!= ( cons_int @ A3 @ L3 ) ) ) ).
% from_list.cases
thf(fact_249_to__list_H_Oelims,axiom,
! [X: nat,Xa: nat > a,Y: list_a] :
( ( ( coProduct_to_list_a2 @ X @ Xa )
= Y )
=> ( ( ( X = zero_zero_nat )
=> ( Y != nil_a ) )
=> ~ ! [N2: nat] :
( ( X
= ( suc @ N2 ) )
=> ( Y
!= ( cons_a @ ( Xa @ zero_zero_nat )
@ ( coProduct_to_list_a2 @ N2
@ ^ [O: nat] : ( Xa @ ( suc @ O ) ) ) ) ) ) ) ) ).
% to_list'.elims
thf(fact_250_to__list_H_Oelims,axiom,
! [X: nat,Xa: nat > int,Y: list_int] :
( ( ( coProd6641981980276419519st_int @ X @ Xa )
= Y )
=> ( ( ( X = zero_zero_nat )
=> ( Y != nil_int ) )
=> ~ ! [N2: nat] :
( ( X
= ( suc @ N2 ) )
=> ( Y
!= ( cons_int @ ( Xa @ zero_zero_nat )
@ ( coProd6641981980276419519st_int @ N2
@ ^ [O: nat] : ( Xa @ ( suc @ O ) ) ) ) ) ) ) ) ).
% to_list'.elims
thf(fact_251_from__list_Osimps_I1_J,axiom,
( ( coProd4444644356563764485list_a @ nil_a )
= ( produc5292568359338195516_nat_a @ zero_zero_nat
@ ^ [N: nat] : undefined_a ) ) ).
% from_list.simps(1)
thf(fact_252_to__list__simp1,axiom,
( ( coProduct_to_list_a
@ ( produc5292568359338195516_nat_a @ zero_zero_nat
@ ^ [N: nat] : undefined_a ) )
= nil_a ) ).
% to_list_simp1
thf(fact_253_from__list_Opelims,axiom,
! [X: list_int,Y: produc6819000835714826316at_int] :
( ( ( coProd6425397259703555173st_int @ X )
= Y )
=> ( ( accp_list_int @ coProd5303891215289060398el_int @ X )
=> ( ( ( X = nil_int )
=> ( ( Y
= ( produc3661665842610097796at_int @ zero_zero_nat
@ ^ [N: nat] : undefined_int ) )
=> ~ ( accp_list_int @ coProd5303891215289060398el_int @ nil_int ) ) )
=> ~ ! [A3: int,L3: list_int] :
( ( X
= ( cons_int @ A3 @ L3 ) )
=> ( ( Y
= ( produc3661665842610097796at_int @ ( suc @ ( produc522749003355801712at_int @ ( coProd6425397259703555173st_int @ L3 ) ) )
@ ^ [N: nat] : ( if_int @ ( N = zero_zero_nat ) @ A3 @ ( produc5442246252769801650at_int @ ( coProd6425397259703555173st_int @ L3 ) @ ( minus_minus_nat @ N @ one_one_nat ) ) ) ) )
=> ~ ( accp_list_int @ coProd5303891215289060398el_int @ ( cons_int @ A3 @ L3 ) ) ) ) ) ) ) ).
% from_list.pelims
thf(fact_254_from__list_Opelims,axiom,
! [X: list_a,Y: produc7123000486447228170_nat_a] :
( ( ( coProd4444644356563764485list_a @ X )
= Y )
=> ( ( accp_list_a @ coProd170618752486124924_rel_a @ X )
=> ( ( ( X = nil_a )
=> ( ( Y
= ( produc5292568359338195516_nat_a @ zero_zero_nat
@ ^ [N: nat] : undefined_a ) )
=> ~ ( accp_list_a @ coProd170618752486124924_rel_a @ nil_a ) ) )
=> ~ ! [A3: a,L3: list_a] :
( ( X
= ( cons_a @ A3 @ L3 ) )
=> ( ( Y
= ( produc5292568359338195516_nat_a @ ( suc @ ( produc3194919578927588176_nat_a @ ( coProd4444644356563764485list_a @ L3 ) ) )
@ ^ [N: nat] : ( if_a @ ( N = zero_zero_nat ) @ A3 @ ( produc4809910040060592782_nat_a @ ( coProd4444644356563764485list_a @ L3 ) @ ( minus_minus_nat @ N @ one_one_nat ) ) ) ) )
=> ~ ( accp_list_a @ coProd170618752486124924_rel_a @ ( cons_a @ A3 @ L3 ) ) ) ) ) ) ) ).
% from_list.pelims
thf(fact_255_longest__common__prefix_Ocases,axiom,
! [X: produc9164743771328383783list_a] :
( ! [X3: a,Xs2: list_a,Y3: a,Ys3: list_a] :
( X
!= ( produc6837034575241423639list_a @ ( cons_a @ X3 @ Xs2 ) @ ( cons_a @ Y3 @ Ys3 ) ) )
=> ( ! [Uv: list_a] :
( X
!= ( produc6837034575241423639list_a @ nil_a @ Uv ) )
=> ~ ! [Uu: list_a] :
( X
!= ( produc6837034575241423639list_a @ Uu @ nil_a ) ) ) ) ).
% longest_common_prefix.cases
thf(fact_256_longest__common__prefix_Ocases,axiom,
! [X: produc1186641810826059865st_int] :
( ! [X3: int,Xs2: list_int,Y3: int,Ys3: list_int] :
( X
!= ( produc364263696895485585st_int @ ( cons_int @ X3 @ Xs2 ) @ ( cons_int @ Y3 @ Ys3 ) ) )
=> ( ! [Uv: list_int] :
( X
!= ( produc364263696895485585st_int @ nil_int @ Uv ) )
=> ~ ! [Uu: list_int] :
( X
!= ( produc364263696895485585st_int @ Uu @ nil_int ) ) ) ) ).
% longest_common_prefix.cases
thf(fact_257_subset__eq__mset__impl_Ocases,axiom,
! [X: produc9164743771328383783list_a] :
( ! [Ys3: list_a] :
( X
!= ( produc6837034575241423639list_a @ nil_a @ Ys3 ) )
=> ~ ! [X3: a,Xs2: list_a,Ys3: list_a] :
( X
!= ( produc6837034575241423639list_a @ ( cons_a @ X3 @ Xs2 ) @ Ys3 ) ) ) ).
% subset_eq_mset_impl.cases
thf(fact_258_subset__eq__mset__impl_Ocases,axiom,
! [X: produc1186641810826059865st_int] :
( ! [Ys3: list_int] :
( X
!= ( produc364263696895485585st_int @ nil_int @ Ys3 ) )
=> ~ ! [X3: int,Xs2: list_int,Ys3: list_int] :
( X
!= ( produc364263696895485585st_int @ ( cons_int @ X3 @ Xs2 ) @ Ys3 ) ) ) ).
% subset_eq_mset_impl.cases
thf(fact_259_enumerate__simps_I2_J,axiom,
! [N3: nat,X: a,Xs: list_a] :
( ( enumerate_a @ N3 @ ( cons_a @ X @ Xs ) )
= ( cons_P8443330267410185325_nat_a @ ( product_Pair_nat_a @ N3 @ X ) @ ( enumerate_a @ ( suc @ N3 ) @ Xs ) ) ) ).
% enumerate_simps(2)
thf(fact_260_enumerate__simps_I2_J,axiom,
! [N3: nat,X: int,Xs: list_int] :
( ( enumerate_int @ N3 @ ( cons_int @ X @ Xs ) )
= ( cons_P2335045147070616083at_int @ ( product_Pair_nat_int @ N3 @ X ) @ ( enumerate_int @ ( suc @ N3 ) @ Xs ) ) ) ).
% enumerate_simps(2)
thf(fact_261_enumerate__simps_I2_J,axiom,
! [N3: nat,X: nat > a,Xs: list_nat_a] :
( ( enumerate_nat_a @ N3 @ ( cons_nat_a @ X @ Xs ) )
= ( cons_P6755270464639534804_nat_a @ ( produc5292568359338195516_nat_a @ N3 @ X ) @ ( enumerate_nat_a @ ( suc @ N3 ) @ Xs ) ) ) ).
% enumerate_simps(2)
thf(fact_262_dbl__inc__simps_I2_J,axiom,
( ( neg_nu8295874005876285629c_real @ zero_zero_real )
= one_one_real ) ).
% dbl_inc_simps(2)
thf(fact_263_dbl__inc__simps_I2_J,axiom,
( ( neg_nu5851722552734809277nc_int @ zero_zero_int )
= one_one_int ) ).
% dbl_inc_simps(2)
thf(fact_264_to__list_H_Opelims,axiom,
! [X: nat,Xa: nat > int,Y: list_int] :
( ( ( coProd6641981980276419519st_int @ X @ Xa )
= Y )
=> ( ( accp_P5031299702865183317at_int @ coProd2616336096964879700el_int @ ( produc3661665842610097796at_int @ X @ Xa ) )
=> ( ( ( X = zero_zero_nat )
=> ( ( Y = nil_int )
=> ~ ( accp_P5031299702865183317at_int @ coProd2616336096964879700el_int @ ( produc3661665842610097796at_int @ zero_zero_nat @ Xa ) ) ) )
=> ~ ! [N2: nat] :
( ( X
= ( suc @ N2 ) )
=> ( ( Y
= ( cons_int @ ( Xa @ zero_zero_nat )
@ ( coProd6641981980276419519st_int @ N2
@ ^ [O: nat] : ( Xa @ ( suc @ O ) ) ) ) )
=> ~ ( accp_P5031299702865183317at_int @ coProd2616336096964879700el_int @ ( produc3661665842610097796at_int @ ( suc @ N2 ) @ Xa ) ) ) ) ) ) ) ).
% to_list'.pelims
thf(fact_265_to__list_H_Opelims,axiom,
! [X: nat,Xa: nat > a,Y: list_a] :
( ( ( coProduct_to_list_a2 @ X @ Xa )
= Y )
=> ( ( accp_P4997817491579065153_nat_a @ coProd105341971985011158_rel_a @ ( produc5292568359338195516_nat_a @ X @ Xa ) )
=> ( ( ( X = zero_zero_nat )
=> ( ( Y = nil_a )
=> ~ ( accp_P4997817491579065153_nat_a @ coProd105341971985011158_rel_a @ ( produc5292568359338195516_nat_a @ zero_zero_nat @ Xa ) ) ) )
=> ~ ! [N2: nat] :
( ( X
= ( suc @ N2 ) )
=> ( ( Y
= ( cons_a @ ( Xa @ zero_zero_nat )
@ ( coProduct_to_list_a2 @ N2
@ ^ [O: nat] : ( Xa @ ( suc @ O ) ) ) ) )
=> ~ ( accp_P4997817491579065153_nat_a @ coProd105341971985011158_rel_a @ ( produc5292568359338195516_nat_a @ ( suc @ N2 ) @ Xa ) ) ) ) ) ) ) ).
% to_list'.pelims
thf(fact_266_Suc__diff__1,axiom,
! [N3: nat] :
( ( ord_less_nat @ zero_zero_nat @ N3 )
=> ( ( suc @ ( minus_minus_nat @ N3 @ one_one_nat ) )
= N3 ) ) ).
% Suc_diff_1
thf(fact_267_not__gr__zero,axiom,
! [N3: extend8495563244428889912nnreal] :
( ( ~ ( ord_le7381754540660121996nnreal @ zero_z7100319975126383169nnreal @ N3 ) )
= ( N3 = zero_z7100319975126383169nnreal ) ) ).
% not_gr_zero
thf(fact_268_not__gr__zero,axiom,
! [N3: extended_enat] :
( ( ~ ( ord_le72135733267957522d_enat @ zero_z5237406670263579293d_enat @ N3 ) )
= ( N3 = zero_z5237406670263579293d_enat ) ) ).
% not_gr_zero
thf(fact_269_not__gr__zero,axiom,
! [N3: nat] :
( ( ~ ( ord_less_nat @ zero_zero_nat @ N3 ) )
= ( N3 = zero_zero_nat ) ) ).
% not_gr_zero
thf(fact_270_less__nat__zero__code,axiom,
! [N3: nat] :
~ ( ord_less_nat @ N3 @ zero_zero_nat ) ).
% less_nat_zero_code
thf(fact_271_neq0__conv,axiom,
! [N3: nat] :
( ( N3 != zero_zero_nat )
= ( ord_less_nat @ zero_zero_nat @ N3 ) ) ).
% neq0_conv
thf(fact_272_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_273_Suc__less__eq,axiom,
! [M2: nat,N3: nat] :
( ( ord_less_nat @ ( suc @ M2 ) @ ( suc @ N3 ) )
= ( ord_less_nat @ M2 @ N3 ) ) ).
% Suc_less_eq
thf(fact_274_Suc__mono,axiom,
! [M2: nat,N3: nat] :
( ( ord_less_nat @ M2 @ N3 )
=> ( ord_less_nat @ ( suc @ M2 ) @ ( suc @ N3 ) ) ) ).
% Suc_mono
thf(fact_275_lessI,axiom,
! [N3: nat] : ( ord_less_nat @ N3 @ ( suc @ N3 ) ) ).
% lessI
thf(fact_276_diff__gt__0__iff__gt,axiom,
! [A: real,B: real] :
( ( ord_less_real @ zero_zero_real @ ( minus_minus_real @ A @ B ) )
= ( ord_less_real @ B @ A ) ) ).
% diff_gt_0_iff_gt
thf(fact_277_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_278_zero__less__Suc,axiom,
! [N3: nat] : ( ord_less_nat @ zero_zero_nat @ ( suc @ N3 ) ) ).
% zero_less_Suc
thf(fact_279_less__Suc0,axiom,
! [N3: nat] :
( ( ord_less_nat @ N3 @ ( suc @ zero_zero_nat ) )
= ( N3 = zero_zero_nat ) ) ).
% less_Suc0
thf(fact_280_less__one,axiom,
! [N3: nat] :
( ( ord_less_nat @ N3 @ one_one_nat )
= ( N3 = zero_zero_nat ) ) ).
% less_one
thf(fact_281_zero__less__diff,axiom,
! [N3: nat,M2: nat] :
( ( ord_less_nat @ zero_zero_nat @ ( minus_minus_nat @ N3 @ M2 ) )
= ( ord_less_nat @ M2 @ N3 ) ) ).
% zero_less_diff
thf(fact_282_Suc__pred,axiom,
! [N3: nat] :
( ( ord_less_nat @ zero_zero_nat @ N3 )
=> ( ( suc @ ( minus_minus_nat @ N3 @ ( suc @ zero_zero_nat ) ) )
= N3 ) ) ).
% Suc_pred
thf(fact_283_linorder__neqE__linordered__idom,axiom,
! [X: real,Y: real] :
( ( X != Y )
=> ( ~ ( ord_less_real @ X @ Y )
=> ( ord_less_real @ Y @ X ) ) ) ).
% linorder_neqE_linordered_idom
thf(fact_284_linorder__neqE__linordered__idom,axiom,
! [X: int,Y: int] :
( ( X != Y )
=> ( ~ ( ord_less_int @ X @ Y )
=> ( ord_less_int @ Y @ X ) ) ) ).
% linorder_neqE_linordered_idom
thf(fact_285_lift__Suc__mono__less__iff,axiom,
! [F2: nat > extended_enat,N3: nat,M2: nat] :
( ! [N2: nat] : ( ord_le72135733267957522d_enat @ ( F2 @ N2 ) @ ( F2 @ ( suc @ N2 ) ) )
=> ( ( ord_le72135733267957522d_enat @ ( F2 @ N3 ) @ ( F2 @ M2 ) )
= ( ord_less_nat @ N3 @ M2 ) ) ) ).
% lift_Suc_mono_less_iff
thf(fact_286_lift__Suc__mono__less__iff,axiom,
! [F2: nat > real,N3: nat,M2: nat] :
( ! [N2: nat] : ( ord_less_real @ ( F2 @ N2 ) @ ( F2 @ ( suc @ N2 ) ) )
=> ( ( ord_less_real @ ( F2 @ N3 ) @ ( F2 @ M2 ) )
= ( ord_less_nat @ N3 @ M2 ) ) ) ).
% lift_Suc_mono_less_iff
thf(fact_287_lift__Suc__mono__less__iff,axiom,
! [F2: nat > num,N3: nat,M2: nat] :
( ! [N2: nat] : ( ord_less_num @ ( F2 @ N2 ) @ ( F2 @ ( suc @ N2 ) ) )
=> ( ( ord_less_num @ ( F2 @ N3 ) @ ( F2 @ M2 ) )
= ( ord_less_nat @ N3 @ M2 ) ) ) ).
% lift_Suc_mono_less_iff
thf(fact_288_lift__Suc__mono__less__iff,axiom,
! [F2: nat > nat,N3: nat,M2: nat] :
( ! [N2: nat] : ( ord_less_nat @ ( F2 @ N2 ) @ ( F2 @ ( suc @ N2 ) ) )
=> ( ( ord_less_nat @ ( F2 @ N3 ) @ ( F2 @ M2 ) )
= ( ord_less_nat @ N3 @ M2 ) ) ) ).
% lift_Suc_mono_less_iff
thf(fact_289_lift__Suc__mono__less__iff,axiom,
! [F2: nat > int,N3: nat,M2: nat] :
( ! [N2: nat] : ( ord_less_int @ ( F2 @ N2 ) @ ( F2 @ ( suc @ N2 ) ) )
=> ( ( ord_less_int @ ( F2 @ N3 ) @ ( F2 @ M2 ) )
= ( ord_less_nat @ N3 @ M2 ) ) ) ).
% lift_Suc_mono_less_iff
thf(fact_290_lift__Suc__mono__less,axiom,
! [F2: nat > extended_enat,N3: nat,N4: nat] :
( ! [N2: nat] : ( ord_le72135733267957522d_enat @ ( F2 @ N2 ) @ ( F2 @ ( suc @ N2 ) ) )
=> ( ( ord_less_nat @ N3 @ N4 )
=> ( ord_le72135733267957522d_enat @ ( F2 @ N3 ) @ ( F2 @ N4 ) ) ) ) ).
% lift_Suc_mono_less
thf(fact_291_lift__Suc__mono__less,axiom,
! [F2: nat > real,N3: nat,N4: nat] :
( ! [N2: nat] : ( ord_less_real @ ( F2 @ N2 ) @ ( F2 @ ( suc @ N2 ) ) )
=> ( ( ord_less_nat @ N3 @ N4 )
=> ( ord_less_real @ ( F2 @ N3 ) @ ( F2 @ N4 ) ) ) ) ).
% lift_Suc_mono_less
thf(fact_292_lift__Suc__mono__less,axiom,
! [F2: nat > num,N3: nat,N4: nat] :
( ! [N2: nat] : ( ord_less_num @ ( F2 @ N2 ) @ ( F2 @ ( suc @ N2 ) ) )
=> ( ( ord_less_nat @ N3 @ N4 )
=> ( ord_less_num @ ( F2 @ N3 ) @ ( F2 @ N4 ) ) ) ) ).
% lift_Suc_mono_less
thf(fact_293_lift__Suc__mono__less,axiom,
! [F2: nat > nat,N3: nat,N4: nat] :
( ! [N2: nat] : ( ord_less_nat @ ( F2 @ N2 ) @ ( F2 @ ( suc @ N2 ) ) )
=> ( ( ord_less_nat @ N3 @ N4 )
=> ( ord_less_nat @ ( F2 @ N3 ) @ ( F2 @ N4 ) ) ) ) ).
% lift_Suc_mono_less
thf(fact_294_lift__Suc__mono__less,axiom,
! [F2: nat > int,N3: nat,N4: nat] :
( ! [N2: nat] : ( ord_less_int @ ( F2 @ N2 ) @ ( F2 @ ( suc @ N2 ) ) )
=> ( ( ord_less_nat @ N3 @ N4 )
=> ( ord_less_int @ ( F2 @ N3 ) @ ( F2 @ N4 ) ) ) ) ).
% lift_Suc_mono_less
thf(fact_295_linorder__neqE__nat,axiom,
! [X: nat,Y: nat] :
( ( X != Y )
=> ( ~ ( ord_less_nat @ X @ Y )
=> ( ord_less_nat @ Y @ X ) ) ) ).
% linorder_neqE_nat
thf(fact_296_infinite__descent,axiom,
! [P: nat > $o,N3: nat] :
( ! [N2: nat] :
( ~ ( P @ N2 )
=> ? [M4: nat] :
( ( ord_less_nat @ M4 @ N2 )
& ~ ( P @ M4 ) ) )
=> ( P @ N3 ) ) ).
% infinite_descent
thf(fact_297_nat__less__induct,axiom,
! [P: nat > $o,N3: nat] :
( ! [N2: nat] :
( ! [M4: nat] :
( ( ord_less_nat @ M4 @ N2 )
=> ( P @ M4 ) )
=> ( P @ N2 ) )
=> ( P @ N3 ) ) ).
% nat_less_induct
thf(fact_298_less__irrefl__nat,axiom,
! [N3: nat] :
~ ( ord_less_nat @ N3 @ N3 ) ).
% less_irrefl_nat
thf(fact_299_less__not__refl3,axiom,
! [S2: nat,T2: nat] :
( ( ord_less_nat @ S2 @ T2 )
=> ( S2 != T2 ) ) ).
% less_not_refl3
thf(fact_300_less__not__refl2,axiom,
! [N3: nat,M2: nat] :
( ( ord_less_nat @ N3 @ M2 )
=> ( M2 != N3 ) ) ).
% less_not_refl2
thf(fact_301_less__not__refl,axiom,
! [N3: nat] :
~ ( ord_less_nat @ N3 @ N3 ) ).
% less_not_refl
thf(fact_302_nat__neq__iff,axiom,
! [M2: nat,N3: nat] :
( ( M2 != N3 )
= ( ( ord_less_nat @ M2 @ N3 )
| ( ord_less_nat @ N3 @ M2 ) ) ) ).
% nat_neq_iff
thf(fact_303_less__numeral__extra_I3_J,axiom,
~ ( ord_le7381754540660121996nnreal @ zero_z7100319975126383169nnreal @ zero_z7100319975126383169nnreal ) ).
% less_numeral_extra(3)
thf(fact_304_less__numeral__extra_I3_J,axiom,
~ ( ord_le72135733267957522d_enat @ zero_z5237406670263579293d_enat @ zero_z5237406670263579293d_enat ) ).
% less_numeral_extra(3)
thf(fact_305_less__numeral__extra_I3_J,axiom,
~ ( ord_less_real @ zero_zero_real @ zero_zero_real ) ).
% less_numeral_extra(3)
thf(fact_306_less__numeral__extra_I3_J,axiom,
~ ( ord_less_nat @ zero_zero_nat @ zero_zero_nat ) ).
% less_numeral_extra(3)
thf(fact_307_less__numeral__extra_I3_J,axiom,
~ ( ord_less_int @ zero_zero_int @ zero_zero_int ) ).
% less_numeral_extra(3)
thf(fact_308_zero__less__iff__neq__zero,axiom,
! [N3: extend8495563244428889912nnreal] :
( ( ord_le7381754540660121996nnreal @ zero_z7100319975126383169nnreal @ N3 )
= ( N3 != zero_z7100319975126383169nnreal ) ) ).
% zero_less_iff_neq_zero
thf(fact_309_zero__less__iff__neq__zero,axiom,
! [N3: extended_enat] :
( ( ord_le72135733267957522d_enat @ zero_z5237406670263579293d_enat @ N3 )
= ( N3 != zero_z5237406670263579293d_enat ) ) ).
% zero_less_iff_neq_zero
thf(fact_310_zero__less__iff__neq__zero,axiom,
! [N3: nat] :
( ( ord_less_nat @ zero_zero_nat @ N3 )
= ( N3 != zero_zero_nat ) ) ).
% zero_less_iff_neq_zero
thf(fact_311_gr__implies__not__zero,axiom,
! [M2: extend8495563244428889912nnreal,N3: extend8495563244428889912nnreal] :
( ( ord_le7381754540660121996nnreal @ M2 @ N3 )
=> ( N3 != zero_z7100319975126383169nnreal ) ) ).
% gr_implies_not_zero
thf(fact_312_gr__implies__not__zero,axiom,
! [M2: extended_enat,N3: extended_enat] :
( ( ord_le72135733267957522d_enat @ M2 @ N3 )
=> ( N3 != zero_z5237406670263579293d_enat ) ) ).
% gr_implies_not_zero
thf(fact_313_gr__implies__not__zero,axiom,
! [M2: nat,N3: nat] :
( ( ord_less_nat @ M2 @ N3 )
=> ( N3 != zero_zero_nat ) ) ).
% gr_implies_not_zero
thf(fact_314_not__less__zero,axiom,
! [N3: extend8495563244428889912nnreal] :
~ ( ord_le7381754540660121996nnreal @ N3 @ zero_z7100319975126383169nnreal ) ).
% not_less_zero
thf(fact_315_not__less__zero,axiom,
! [N3: extended_enat] :
~ ( ord_le72135733267957522d_enat @ N3 @ zero_z5237406670263579293d_enat ) ).
% not_less_zero
thf(fact_316_not__less__zero,axiom,
! [N3: nat] :
~ ( ord_less_nat @ N3 @ zero_zero_nat ) ).
% not_less_zero
thf(fact_317_gr__zeroI,axiom,
! [N3: extend8495563244428889912nnreal] :
( ( N3 != zero_z7100319975126383169nnreal )
=> ( ord_le7381754540660121996nnreal @ zero_z7100319975126383169nnreal @ N3 ) ) ).
% gr_zeroI
thf(fact_318_gr__zeroI,axiom,
! [N3: extended_enat] :
( ( N3 != zero_z5237406670263579293d_enat )
=> ( ord_le72135733267957522d_enat @ zero_z5237406670263579293d_enat @ N3 ) ) ).
% gr_zeroI
thf(fact_319_gr__zeroI,axiom,
! [N3: nat] :
( ( N3 != zero_zero_nat )
=> ( ord_less_nat @ zero_zero_nat @ N3 ) ) ).
% gr_zeroI
thf(fact_320_less__numeral__extra_I4_J,axiom,
~ ( ord_le7381754540660121996nnreal @ one_on2969667320475766781nnreal @ one_on2969667320475766781nnreal ) ).
% less_numeral_extra(4)
thf(fact_321_less__numeral__extra_I4_J,axiom,
~ ( ord_le72135733267957522d_enat @ one_on7984719198319812577d_enat @ one_on7984719198319812577d_enat ) ).
% less_numeral_extra(4)
thf(fact_322_less__numeral__extra_I4_J,axiom,
~ ( ord_less_real @ one_one_real @ one_one_real ) ).
% less_numeral_extra(4)
thf(fact_323_less__numeral__extra_I4_J,axiom,
~ ( ord_less_nat @ one_one_nat @ one_one_nat ) ).
% less_numeral_extra(4)
thf(fact_324_less__numeral__extra_I4_J,axiom,
~ ( ord_less_int @ one_one_int @ one_one_int ) ).
% less_numeral_extra(4)
thf(fact_325_diff__strict__right__mono,axiom,
! [A: real,B: real,C: real] :
( ( ord_less_real @ A @ B )
=> ( ord_less_real @ ( minus_minus_real @ A @ C ) @ ( minus_minus_real @ B @ C ) ) ) ).
% diff_strict_right_mono
thf(fact_326_diff__strict__right__mono,axiom,
! [A: int,B: int,C: int] :
( ( ord_less_int @ A @ B )
=> ( ord_less_int @ ( minus_minus_int @ A @ C ) @ ( minus_minus_int @ B @ C ) ) ) ).
% diff_strict_right_mono
thf(fact_327_diff__strict__left__mono,axiom,
! [B: real,A: real,C: real] :
( ( ord_less_real @ B @ A )
=> ( ord_less_real @ ( minus_minus_real @ C @ A ) @ ( minus_minus_real @ C @ B ) ) ) ).
% diff_strict_left_mono
thf(fact_328_diff__strict__left__mono,axiom,
! [B: int,A: int,C: int] :
( ( ord_less_int @ B @ A )
=> ( ord_less_int @ ( minus_minus_int @ C @ A ) @ ( minus_minus_int @ C @ B ) ) ) ).
% diff_strict_left_mono
thf(fact_329_diff__eq__diff__less,axiom,
! [A: real,B: real,C: real,D: real] :
( ( ( minus_minus_real @ A @ B )
= ( minus_minus_real @ C @ D ) )
=> ( ( ord_less_real @ A @ B )
= ( ord_less_real @ C @ D ) ) ) ).
% diff_eq_diff_less
thf(fact_330_diff__eq__diff__less,axiom,
! [A: int,B: int,C: int,D: int] :
( ( ( minus_minus_int @ A @ B )
= ( minus_minus_int @ C @ D ) )
=> ( ( ord_less_int @ A @ B )
= ( ord_less_int @ C @ D ) ) ) ).
% diff_eq_diff_less
thf(fact_331_diff__strict__mono,axiom,
! [A: real,B: real,D: real,C: real] :
( ( ord_less_real @ A @ B )
=> ( ( ord_less_real @ D @ C )
=> ( ord_less_real @ ( minus_minus_real @ A @ C ) @ ( minus_minus_real @ B @ D ) ) ) ) ).
% diff_strict_mono
thf(fact_332_diff__strict__mono,axiom,
! [A: int,B: int,D: int,C: int] :
( ( ord_less_int @ A @ B )
=> ( ( ord_less_int @ D @ C )
=> ( ord_less_int @ ( minus_minus_int @ A @ C ) @ ( minus_minus_int @ B @ D ) ) ) ) ).
% diff_strict_mono
thf(fact_333_infinite__descent0,axiom,
! [P: nat > $o,N3: nat] :
( ( P @ zero_zero_nat )
=> ( ! [N2: nat] :
( ( ord_less_nat @ zero_zero_nat @ N2 )
=> ( ~ ( P @ N2 )
=> ? [M4: nat] :
( ( ord_less_nat @ M4 @ N2 )
& ~ ( P @ M4 ) ) ) )
=> ( P @ N3 ) ) ) ).
% infinite_descent0
thf(fact_334_gr__implies__not0,axiom,
! [M2: nat,N3: nat] :
( ( ord_less_nat @ M2 @ N3 )
=> ( N3 != zero_zero_nat ) ) ).
% gr_implies_not0
thf(fact_335_less__zeroE,axiom,
! [N3: nat] :
~ ( ord_less_nat @ N3 @ zero_zero_nat ) ).
% less_zeroE
thf(fact_336_not__less0,axiom,
! [N3: nat] :
~ ( ord_less_nat @ N3 @ zero_zero_nat ) ).
% not_less0
thf(fact_337_not__gr0,axiom,
! [N3: nat] :
( ( ~ ( ord_less_nat @ zero_zero_nat @ N3 ) )
= ( N3 = zero_zero_nat ) ) ).
% not_gr0
thf(fact_338_gr0I,axiom,
! [N3: nat] :
( ( N3 != zero_zero_nat )
=> ( ord_less_nat @ zero_zero_nat @ N3 ) ) ).
% gr0I
thf(fact_339_bot__nat__0_Oextremum__strict,axiom,
! [A: nat] :
~ ( ord_less_nat @ A @ zero_zero_nat ) ).
% bot_nat_0.extremum_strict
thf(fact_340_not__less__less__Suc__eq,axiom,
! [N3: nat,M2: nat] :
( ~ ( ord_less_nat @ N3 @ M2 )
=> ( ( ord_less_nat @ N3 @ ( suc @ M2 ) )
= ( N3 = M2 ) ) ) ).
% not_less_less_Suc_eq
thf(fact_341_strict__inc__induct,axiom,
! [I: nat,J: nat,P: nat > $o] :
( ( ord_less_nat @ I @ J )
=> ( ! [I2: nat] :
( ( J
= ( suc @ I2 ) )
=> ( P @ I2 ) )
=> ( ! [I2: nat] :
( ( ord_less_nat @ I2 @ J )
=> ( ( P @ ( suc @ I2 ) )
=> ( P @ I2 ) ) )
=> ( P @ I ) ) ) ) ).
% strict_inc_induct
thf(fact_342_less__Suc__induct,axiom,
! [I: nat,J: nat,P: nat > nat > $o] :
( ( ord_less_nat @ I @ J )
=> ( ! [I2: nat] : ( P @ I2 @ ( suc @ I2 ) )
=> ( ! [I2: nat,J2: nat,K2: nat] :
( ( ord_less_nat @ I2 @ J2 )
=> ( ( ord_less_nat @ J2 @ K2 )
=> ( ( P @ I2 @ J2 )
=> ( ( P @ J2 @ K2 )
=> ( P @ I2 @ K2 ) ) ) ) )
=> ( P @ I @ J ) ) ) ) ).
% less_Suc_induct
thf(fact_343_less__trans__Suc,axiom,
! [I: nat,J: nat,K: nat] :
( ( ord_less_nat @ I @ J )
=> ( ( ord_less_nat @ J @ K )
=> ( ord_less_nat @ ( suc @ I ) @ K ) ) ) ).
% less_trans_Suc
thf(fact_344_Suc__less__SucD,axiom,
! [M2: nat,N3: nat] :
( ( ord_less_nat @ ( suc @ M2 ) @ ( suc @ N3 ) )
=> ( ord_less_nat @ M2 @ N3 ) ) ).
% Suc_less_SucD
thf(fact_345_less__antisym,axiom,
! [N3: nat,M2: nat] :
( ~ ( ord_less_nat @ N3 @ M2 )
=> ( ( ord_less_nat @ N3 @ ( suc @ M2 ) )
=> ( M2 = N3 ) ) ) ).
% less_antisym
thf(fact_346_Suc__less__eq2,axiom,
! [N3: nat,M2: nat] :
( ( ord_less_nat @ ( suc @ N3 ) @ M2 )
= ( ? [M5: nat] :
( ( M2
= ( suc @ M5 ) )
& ( ord_less_nat @ N3 @ M5 ) ) ) ) ).
% Suc_less_eq2
thf(fact_347_All__less__Suc,axiom,
! [N3: nat,P: nat > $o] :
( ( ! [I3: nat] :
( ( ord_less_nat @ I3 @ ( suc @ N3 ) )
=> ( P @ I3 ) ) )
= ( ( P @ N3 )
& ! [I3: nat] :
( ( ord_less_nat @ I3 @ N3 )
=> ( P @ I3 ) ) ) ) ).
% All_less_Suc
thf(fact_348_not__less__eq,axiom,
! [M2: nat,N3: nat] :
( ( ~ ( ord_less_nat @ M2 @ N3 ) )
= ( ord_less_nat @ N3 @ ( suc @ M2 ) ) ) ).
% not_less_eq
thf(fact_349_less__Suc__eq,axiom,
! [M2: nat,N3: nat] :
( ( ord_less_nat @ M2 @ ( suc @ N3 ) )
= ( ( ord_less_nat @ M2 @ N3 )
| ( M2 = N3 ) ) ) ).
% less_Suc_eq
thf(fact_350_Ex__less__Suc,axiom,
! [N3: nat,P: nat > $o] :
( ( ? [I3: nat] :
( ( ord_less_nat @ I3 @ ( suc @ N3 ) )
& ( P @ I3 ) ) )
= ( ( P @ N3 )
| ? [I3: nat] :
( ( ord_less_nat @ I3 @ N3 )
& ( P @ I3 ) ) ) ) ).
% Ex_less_Suc
thf(fact_351_less__SucI,axiom,
! [M2: nat,N3: nat] :
( ( ord_less_nat @ M2 @ N3 )
=> ( ord_less_nat @ M2 @ ( suc @ N3 ) ) ) ).
% less_SucI
thf(fact_352_less__SucE,axiom,
! [M2: nat,N3: nat] :
( ( ord_less_nat @ M2 @ ( suc @ N3 ) )
=> ( ~ ( ord_less_nat @ M2 @ N3 )
=> ( M2 = N3 ) ) ) ).
% less_SucE
thf(fact_353_Suc__lessI,axiom,
! [M2: nat,N3: nat] :
( ( ord_less_nat @ M2 @ N3 )
=> ( ( ( suc @ M2 )
!= N3 )
=> ( ord_less_nat @ ( suc @ M2 ) @ N3 ) ) ) ).
% Suc_lessI
thf(fact_354_Suc__lessE,axiom,
! [I: nat,K: nat] :
( ( ord_less_nat @ ( suc @ I ) @ K )
=> ~ ! [J2: nat] :
( ( ord_less_nat @ I @ J2 )
=> ( K
!= ( suc @ J2 ) ) ) ) ).
% Suc_lessE
thf(fact_355_Suc__lessD,axiom,
! [M2: nat,N3: nat] :
( ( ord_less_nat @ ( suc @ M2 ) @ N3 )
=> ( ord_less_nat @ M2 @ N3 ) ) ).
% Suc_lessD
thf(fact_356_Nat_OlessE,axiom,
! [I: nat,K: nat] :
( ( ord_less_nat @ I @ K )
=> ( ( K
!= ( suc @ I ) )
=> ~ ! [J2: nat] :
( ( ord_less_nat @ I @ J2 )
=> ( K
!= ( suc @ J2 ) ) ) ) ) ).
% Nat.lessE
thf(fact_357_less__imp__diff__less,axiom,
! [J: nat,K: nat,N3: nat] :
( ( ord_less_nat @ J @ K )
=> ( ord_less_nat @ ( minus_minus_nat @ J @ N3 ) @ K ) ) ).
% less_imp_diff_less
thf(fact_358_diff__less__mono2,axiom,
! [M2: nat,N3: nat,L: nat] :
( ( ord_less_nat @ M2 @ N3 )
=> ( ( ord_less_nat @ M2 @ L )
=> ( ord_less_nat @ ( minus_minus_nat @ L @ N3 ) @ ( minus_minus_nat @ L @ M2 ) ) ) ) ).
% diff_less_mono2
thf(fact_359_less__numeral__extra_I1_J,axiom,
ord_le7381754540660121996nnreal @ zero_z7100319975126383169nnreal @ one_on2969667320475766781nnreal ).
% less_numeral_extra(1)
thf(fact_360_less__numeral__extra_I1_J,axiom,
ord_le72135733267957522d_enat @ zero_z5237406670263579293d_enat @ one_on7984719198319812577d_enat ).
% less_numeral_extra(1)
thf(fact_361_less__numeral__extra_I1_J,axiom,
ord_less_real @ zero_zero_real @ one_one_real ).
% less_numeral_extra(1)
thf(fact_362_less__numeral__extra_I1_J,axiom,
ord_less_nat @ zero_zero_nat @ one_one_nat ).
% less_numeral_extra(1)
thf(fact_363_less__numeral__extra_I1_J,axiom,
ord_less_int @ zero_zero_int @ one_one_int ).
% less_numeral_extra(1)
thf(fact_364_not__one__less__zero,axiom,
~ ( ord_le7381754540660121996nnreal @ one_on2969667320475766781nnreal @ zero_z7100319975126383169nnreal ) ).
% not_one_less_zero
thf(fact_365_not__one__less__zero,axiom,
~ ( ord_le72135733267957522d_enat @ one_on7984719198319812577d_enat @ zero_z5237406670263579293d_enat ) ).
% not_one_less_zero
thf(fact_366_not__one__less__zero,axiom,
~ ( ord_less_real @ one_one_real @ zero_zero_real ) ).
% not_one_less_zero
thf(fact_367_not__one__less__zero,axiom,
~ ( ord_less_nat @ one_one_nat @ zero_zero_nat ) ).
% not_one_less_zero
thf(fact_368_not__one__less__zero,axiom,
~ ( ord_less_int @ one_one_int @ zero_zero_int ) ).
% not_one_less_zero
thf(fact_369_zero__less__one,axiom,
ord_le7381754540660121996nnreal @ zero_z7100319975126383169nnreal @ one_on2969667320475766781nnreal ).
% zero_less_one
thf(fact_370_zero__less__one,axiom,
ord_le72135733267957522d_enat @ zero_z5237406670263579293d_enat @ one_on7984719198319812577d_enat ).
% zero_less_one
thf(fact_371_zero__less__one,axiom,
ord_less_real @ zero_zero_real @ one_one_real ).
% zero_less_one
thf(fact_372_zero__less__one,axiom,
ord_less_nat @ zero_zero_nat @ one_one_nat ).
% zero_less_one
thf(fact_373_zero__less__one,axiom,
ord_less_int @ zero_zero_int @ one_one_int ).
% zero_less_one
thf(fact_374_less__iff__diff__less__0,axiom,
( ord_less_real
= ( ^ [A5: real,B5: real] : ( ord_less_real @ ( minus_minus_real @ A5 @ B5 ) @ zero_zero_real ) ) ) ).
% less_iff_diff_less_0
thf(fact_375_less__iff__diff__less__0,axiom,
( ord_less_int
= ( ^ [A5: int,B5: int] : ( ord_less_int @ ( minus_minus_int @ A5 @ B5 ) @ zero_zero_int ) ) ) ).
% less_iff_diff_less_0
thf(fact_376_less__Suc__eq__0__disj,axiom,
! [M2: nat,N3: nat] :
( ( ord_less_nat @ M2 @ ( suc @ N3 ) )
= ( ( M2 = zero_zero_nat )
| ? [J3: nat] :
( ( M2
= ( suc @ J3 ) )
& ( ord_less_nat @ J3 @ N3 ) ) ) ) ).
% less_Suc_eq_0_disj
thf(fact_377_gr0__implies__Suc,axiom,
! [N3: nat] :
( ( ord_less_nat @ zero_zero_nat @ N3 )
=> ? [M3: nat] :
( N3
= ( suc @ M3 ) ) ) ).
% gr0_implies_Suc
thf(fact_378_All__less__Suc2,axiom,
! [N3: nat,P: nat > $o] :
( ( ! [I3: nat] :
( ( ord_less_nat @ I3 @ ( suc @ N3 ) )
=> ( P @ I3 ) ) )
= ( ( P @ zero_zero_nat )
& ! [I3: nat] :
( ( ord_less_nat @ I3 @ N3 )
=> ( P @ ( suc @ I3 ) ) ) ) ) ).
% All_less_Suc2
thf(fact_379_gr0__conv__Suc,axiom,
! [N3: nat] :
( ( ord_less_nat @ zero_zero_nat @ N3 )
= ( ? [M: nat] :
( N3
= ( suc @ M ) ) ) ) ).
% gr0_conv_Suc
thf(fact_380_Ex__less__Suc2,axiom,
! [N3: nat,P: nat > $o] :
( ( ? [I3: nat] :
( ( ord_less_nat @ I3 @ ( suc @ N3 ) )
& ( P @ I3 ) ) )
= ( ( P @ zero_zero_nat )
| ? [I3: nat] :
( ( ord_less_nat @ I3 @ N3 )
& ( P @ ( suc @ I3 ) ) ) ) ) ).
% Ex_less_Suc2
thf(fact_381_diff__less,axiom,
! [N3: nat,M2: nat] :
( ( ord_less_nat @ zero_zero_nat @ N3 )
=> ( ( ord_less_nat @ zero_zero_nat @ M2 )
=> ( ord_less_nat @ ( minus_minus_nat @ M2 @ N3 ) @ M2 ) ) ) ).
% diff_less
thf(fact_382_diff__less__Suc,axiom,
! [M2: nat,N3: nat] : ( ord_less_nat @ ( minus_minus_nat @ M2 @ N3 ) @ ( suc @ M2 ) ) ).
% diff_less_Suc
thf(fact_383_Suc__diff__Suc,axiom,
! [N3: nat,M2: nat] :
( ( ord_less_nat @ N3 @ M2 )
=> ( ( suc @ ( minus_minus_nat @ M2 @ ( suc @ N3 ) ) )
= ( minus_minus_nat @ M2 @ N3 ) ) ) ).
% Suc_diff_Suc
thf(fact_384_nat__induct__non__zero,axiom,
! [N3: nat,P: nat > $o] :
( ( ord_less_nat @ zero_zero_nat @ N3 )
=> ( ( P @ one_one_nat )
=> ( ! [N2: nat] :
( ( ord_less_nat @ zero_zero_nat @ N2 )
=> ( ( P @ N2 )
=> ( P @ ( suc @ N2 ) ) ) )
=> ( P @ N3 ) ) ) ) ).
% nat_induct_non_zero
thf(fact_385_diff__Suc__less,axiom,
! [N3: nat,I: nat] :
( ( ord_less_nat @ zero_zero_nat @ N3 )
=> ( ord_less_nat @ ( minus_minus_nat @ N3 @ ( suc @ I ) ) @ N3 ) ) ).
% diff_Suc_less
thf(fact_386_Suc__diff__eq__diff__pred,axiom,
! [N3: nat,M2: nat] :
( ( ord_less_nat @ zero_zero_nat @ N3 )
=> ( ( minus_minus_nat @ ( suc @ M2 ) @ N3 )
= ( minus_minus_nat @ M2 @ ( minus_minus_nat @ N3 @ one_one_nat ) ) ) ) ).
% Suc_diff_eq_diff_pred
thf(fact_387_Suc__pred_H,axiom,
! [N3: nat] :
( ( ord_less_nat @ zero_zero_nat @ N3 )
=> ( N3
= ( suc @ ( minus_minus_nat @ N3 @ one_one_nat ) ) ) ) ).
% Suc_pred'
thf(fact_388_inverse__permutation__of__list_Ocases,axiom,
! [X: produc934318026945906148um_num] :
( ! [X3: num] :
( X
!= ( produc1199376803376955102um_num @ nil_Pr4317560964547046811um_num @ X3 ) )
=> ~ ! [Y3: num,X6: num,Xs2: list_P3744719386663036955um_num,X3: num] :
( X
!= ( produc1199376803376955102um_num @ ( cons_P5351470506836119883um_num @ ( product_Pair_num_num @ Y3 @ X6 ) @ Xs2 ) @ X3 ) ) ) ).
% inverse_permutation_of_list.cases
thf(fact_389_inverse__permutation__of__list_Ocases,axiom,
! [X: produc3257586164597176986l_real] :
( ! [X3: real] :
( X
!= ( produc6063341862636304660l_real @ nil_Pr366592848263477823l_real @ X3 ) )
=> ~ ! [Y3: real,X6: real,Xs2: list_P8689742595348180415l_real,X3: real] :
( X
!= ( produc6063341862636304660l_real @ ( cons_P1861573166434266607l_real @ ( produc4511245868158468465l_real @ Y3 @ X6 ) @ Xs2 ) @ X3 ) ) ) ).
% inverse_permutation_of_list.cases
thf(fact_390_delete__aux_Ocases,axiom,
! [X: produc2908375388873924811_nat_a] :
( ! [K2: nat] :
( X
!= ( produc8034358855101601283_nat_a @ K2 @ nil_Pr2013076336424982148_nat_a ) )
=> ~ ! [K2: nat,K3: nat,V: nat > a,Xs2: list_P8194537880601713306_nat_a] :
( X
!= ( produc8034358855101601283_nat_a @ K2 @ ( cons_P6755270464639534804_nat_a @ ( produc5292568359338195516_nat_a @ K3 @ V ) @ Xs2 ) ) ) ) ).
% delete_aux.cases
thf(fact_391_delete__aux_Ocases,axiom,
! [X: produc626917917429841708al_nat] :
( ! [K2: real] :
( X
!= ( produc2239748315323221726al_nat @ K2 @ nil_Pr1917482104270272867al_nat ) )
=> ~ ! [K2: real,K3: real,V: nat,Xs2: list_P6834414599653733731al_nat] :
( X
!= ( produc2239748315323221726al_nat @ K2 @ ( cons_P500833500243608851al_nat @ ( produc3181502643871035669al_nat @ K3 @ V ) @ Xs2 ) ) ) ) ).
% delete_aux.cases
thf(fact_392_delete__aux_Ocases,axiom,
! [X: produc3308868277474639478um_num] :
( ! [K2: num] :
( X
!= ( produc3259339371196666152um_num @ K2 @ nil_Pr4317560964547046811um_num ) )
=> ~ ! [K2: num,K3: num,V: num,Xs2: list_P3744719386663036955um_num] :
( X
!= ( produc3259339371196666152um_num @ K2 @ ( cons_P5351470506836119883um_num @ ( product_Pair_num_num @ K3 @ V ) @ Xs2 ) ) ) ) ).
% delete_aux.cases
thf(fact_393_delete__aux_Ocases,axiom,
! [X: produc2814942353377581704l_real] :
( ! [K2: real] :
( X
!= ( produc5056970835876393402l_real @ K2 @ nil_Pr366592848263477823l_real ) )
=> ~ ! [K2: real,K3: real,V: real,Xs2: list_P8689742595348180415l_real] :
( X
!= ( produc5056970835876393402l_real @ K2 @ ( cons_P1861573166434266607l_real @ ( produc4511245868158468465l_real @ K3 @ V ) @ Xs2 ) ) ) ) ).
% delete_aux.cases
thf(fact_394_field__lbound__gt__zero,axiom,
! [D1: real,D2: real] :
( ( ord_less_real @ zero_zero_real @ D1 )
=> ( ( ord_less_real @ zero_zero_real @ D2 )
=> ? [E: real] :
( ( ord_less_real @ zero_zero_real @ E )
& ( ord_less_real @ E @ D1 )
& ( ord_less_real @ E @ D2 ) ) ) ) ).
% field_lbound_gt_zero
thf(fact_395_map__entry_Oelims,axiom,
! [X: nat,Xa: ( nat > a ) > nat > a,Xb: list_P8194537880601713306_nat_a,Y: list_P8194537880601713306_nat_a] :
( ( ( map_entry_nat_nat_a @ X @ Xa @ Xb )
= Y )
=> ( ( ( Xb = nil_Pr2013076336424982148_nat_a )
=> ( Y != nil_Pr2013076336424982148_nat_a ) )
=> ~ ! [P4: produc7123000486447228170_nat_a,Ps: list_P8194537880601713306_nat_a] :
( ( Xb
= ( cons_P6755270464639534804_nat_a @ P4 @ Ps ) )
=> ~ ( ( ( ( produc3194919578927588176_nat_a @ P4 )
= X )
=> ( Y
= ( cons_P6755270464639534804_nat_a @ ( produc5292568359338195516_nat_a @ X @ ( Xa @ ( produc4809910040060592782_nat_a @ P4 ) ) ) @ Ps ) ) )
& ( ( ( produc3194919578927588176_nat_a @ P4 )
!= X )
=> ( Y
= ( cons_P6755270464639534804_nat_a @ P4 @ ( map_entry_nat_nat_a @ X @ Xa @ Ps ) ) ) ) ) ) ) ) ).
% map_entry.elims
thf(fact_396_map__entry_Oelims,axiom,
! [X: real,Xa: real > real,Xb: list_P8689742595348180415l_real,Y: list_P8689742595348180415l_real] :
( ( ( map_entry_real_real @ X @ Xa @ Xb )
= Y )
=> ( ( ( Xb = nil_Pr366592848263477823l_real )
=> ( Y != nil_Pr366592848263477823l_real ) )
=> ~ ! [P4: produc2422161461964618553l_real,Ps: list_P8689742595348180415l_real] :
( ( Xb
= ( cons_P1861573166434266607l_real @ P4 @ Ps ) )
=> ~ ( ( ( ( produc5828954698716094813l_real @ P4 )
= X )
=> ( Y
= ( cons_P1861573166434266607l_real @ ( produc4511245868158468465l_real @ X @ ( Xa @ ( produc3484788084999411615l_real @ P4 ) ) ) @ Ps ) ) )
& ( ( ( produc5828954698716094813l_real @ P4 )
!= X )
=> ( Y
= ( cons_P1861573166434266607l_real @ P4 @ ( map_entry_real_real @ X @ Xa @ Ps ) ) ) ) ) ) ) ) ).
% map_entry.elims
thf(fact_397_map__entry_Oelims,axiom,
! [X: num,Xa: num > num,Xb: list_P3744719386663036955um_num,Y: list_P3744719386663036955um_num] :
( ( ( map_entry_num_num @ X @ Xa @ Xb )
= Y )
=> ( ( ( Xb = nil_Pr4317560964547046811um_num )
=> ( Y != nil_Pr4317560964547046811um_num ) )
=> ~ ! [P4: product_prod_num_num,Ps: list_P3744719386663036955um_num] :
( ( Xb
= ( cons_P5351470506836119883um_num @ P4 @ Ps ) )
=> ~ ( ( ( ( product_fst_num_num @ P4 )
= X )
=> ( Y
= ( cons_P5351470506836119883um_num @ ( product_Pair_num_num @ X @ ( Xa @ ( product_snd_num_num @ P4 ) ) ) @ Ps ) ) )
& ( ( ( product_fst_num_num @ P4 )
!= X )
=> ( Y
= ( cons_P5351470506836119883um_num @ P4 @ ( map_entry_num_num @ X @ Xa @ Ps ) ) ) ) ) ) ) ) ).
% map_entry.elims
thf(fact_398_map__entry_Oelims,axiom,
! [X: real,Xa: nat > nat,Xb: list_P6834414599653733731al_nat,Y: list_P6834414599653733731al_nat] :
( ( ( map_entry_real_nat @ X @ Xa @ Xb )
= Y )
=> ( ( ( Xb = nil_Pr1917482104270272867al_nat )
=> ( Y != nil_Pr1917482104270272867al_nat ) )
=> ~ ! [P4: produc3741383161447143261al_nat,Ps: list_P6834414599653733731al_nat] :
( ( Xb
= ( cons_P500833500243608851al_nat @ P4 @ Ps ) )
=> ~ ( ( ( ( product_fst_real_nat @ P4 )
= X )
=> ( Y
= ( cons_P500833500243608851al_nat @ ( produc3181502643871035669al_nat @ X @ ( Xa @ ( product_snd_real_nat @ P4 ) ) ) @ Ps ) ) )
& ( ( ( product_fst_real_nat @ P4 )
!= X )
=> ( Y
= ( cons_P500833500243608851al_nat @ P4 @ ( map_entry_real_nat @ X @ Xa @ Ps ) ) ) ) ) ) ) ) ).
% map_entry.elims
thf(fact_399_map__default_Oelims,axiom,
! [X: nat,Xa: nat > a,Xb: ( nat > a ) > nat > a,Xc: list_P8194537880601713306_nat_a,Y: list_P8194537880601713306_nat_a] :
( ( ( map_de630094633906692667_nat_a @ X @ Xa @ Xb @ Xc )
= Y )
=> ( ( ( Xc = nil_Pr2013076336424982148_nat_a )
=> ( Y
!= ( cons_P6755270464639534804_nat_a @ ( produc5292568359338195516_nat_a @ X @ Xa ) @ nil_Pr2013076336424982148_nat_a ) ) )
=> ~ ! [P4: produc7123000486447228170_nat_a,Ps: list_P8194537880601713306_nat_a] :
( ( Xc
= ( cons_P6755270464639534804_nat_a @ P4 @ Ps ) )
=> ~ ( ( ( ( produc3194919578927588176_nat_a @ P4 )
= X )
=> ( Y
= ( cons_P6755270464639534804_nat_a @ ( produc5292568359338195516_nat_a @ X @ ( Xb @ ( produc4809910040060592782_nat_a @ P4 ) ) ) @ Ps ) ) )
& ( ( ( produc3194919578927588176_nat_a @ P4 )
!= X )
=> ( Y
= ( cons_P6755270464639534804_nat_a @ P4 @ ( map_de630094633906692667_nat_a @ X @ Xa @ Xb @ Ps ) ) ) ) ) ) ) ) ).
% map_default.elims
thf(fact_400_map__default_Oelims,axiom,
! [X: real,Xa: real,Xb: real > real,Xc: list_P8689742595348180415l_real,Y: list_P8689742595348180415l_real] :
( ( ( map_de3913085918471371826l_real @ X @ Xa @ Xb @ Xc )
= Y )
=> ( ( ( Xc = nil_Pr366592848263477823l_real )
=> ( Y
!= ( cons_P1861573166434266607l_real @ ( produc4511245868158468465l_real @ X @ Xa ) @ nil_Pr366592848263477823l_real ) ) )
=> ~ ! [P4: produc2422161461964618553l_real,Ps: list_P8689742595348180415l_real] :
( ( Xc
= ( cons_P1861573166434266607l_real @ P4 @ Ps ) )
=> ~ ( ( ( ( produc5828954698716094813l_real @ P4 )
= X )
=> ( Y
= ( cons_P1861573166434266607l_real @ ( produc4511245868158468465l_real @ X @ ( Xb @ ( produc3484788084999411615l_real @ P4 ) ) ) @ Ps ) ) )
& ( ( ( produc5828954698716094813l_real @ P4 )
!= X )
=> ( Y
= ( cons_P1861573166434266607l_real @ P4 @ ( map_de3913085918471371826l_real @ X @ Xa @ Xb @ Ps ) ) ) ) ) ) ) ) ).
% map_default.elims
thf(fact_401_map__default_Oelims,axiom,
! [X: num,Xa: num,Xb: num > num,Xc: list_P3744719386663036955um_num,Y: list_P3744719386663036955um_num] :
( ( ( map_default_num_num @ X @ Xa @ Xb @ Xc )
= Y )
=> ( ( ( Xc = nil_Pr4317560964547046811um_num )
=> ( Y
!= ( cons_P5351470506836119883um_num @ ( product_Pair_num_num @ X @ Xa ) @ nil_Pr4317560964547046811um_num ) ) )
=> ~ ! [P4: product_prod_num_num,Ps: list_P3744719386663036955um_num] :
( ( Xc
= ( cons_P5351470506836119883um_num @ P4 @ Ps ) )
=> ~ ( ( ( ( product_fst_num_num @ P4 )
= X )
=> ( Y
= ( cons_P5351470506836119883um_num @ ( product_Pair_num_num @ X @ ( Xb @ ( product_snd_num_num @ P4 ) ) ) @ Ps ) ) )
& ( ( ( product_fst_num_num @ P4 )
!= X )
=> ( Y
= ( cons_P5351470506836119883um_num @ P4 @ ( map_default_num_num @ X @ Xa @ Xb @ Ps ) ) ) ) ) ) ) ) ).
% map_default.elims
thf(fact_402_map__default_Oelims,axiom,
! [X: real,Xa: nat,Xb: nat > nat,Xc: list_P6834414599653733731al_nat,Y: list_P6834414599653733731al_nat] :
( ( ( map_default_real_nat @ X @ Xa @ Xb @ Xc )
= Y )
=> ( ( ( Xc = nil_Pr1917482104270272867al_nat )
=> ( Y
!= ( cons_P500833500243608851al_nat @ ( produc3181502643871035669al_nat @ X @ Xa ) @ nil_Pr1917482104270272867al_nat ) ) )
=> ~ ! [P4: produc3741383161447143261al_nat,Ps: list_P6834414599653733731al_nat] :
( ( Xc
= ( cons_P500833500243608851al_nat @ P4 @ Ps ) )
=> ~ ( ( ( ( product_fst_real_nat @ P4 )
= X )
=> ( Y
= ( cons_P500833500243608851al_nat @ ( produc3181502643871035669al_nat @ X @ ( Xb @ ( product_snd_real_nat @ P4 ) ) ) @ Ps ) ) )
& ( ( ( product_fst_real_nat @ P4 )
!= X )
=> ( Y
= ( cons_P500833500243608851al_nat @ P4 @ ( map_default_real_nat @ X @ Xa @ Xb @ Ps ) ) ) ) ) ) ) ) ).
% map_default.elims
thf(fact_403_update__with__aux_Osimps_I2_J,axiom,
! [P2: produc7123000486447228170_nat_a,K: nat,V2: nat > a,F2: ( nat > a ) > nat > a,Ps2: list_P8194537880601713306_nat_a] :
( ( ( ( produc3194919578927588176_nat_a @ P2 )
= K )
=> ( ( update7324014601402209203_a_nat @ V2 @ K @ F2 @ ( cons_P6755270464639534804_nat_a @ P2 @ Ps2 ) )
= ( cons_P6755270464639534804_nat_a @ ( produc5292568359338195516_nat_a @ K @ ( F2 @ ( produc4809910040060592782_nat_a @ P2 ) ) ) @ Ps2 ) ) )
& ( ( ( produc3194919578927588176_nat_a @ P2 )
!= K )
=> ( ( update7324014601402209203_a_nat @ V2 @ K @ F2 @ ( cons_P6755270464639534804_nat_a @ P2 @ Ps2 ) )
= ( cons_P6755270464639534804_nat_a @ P2 @ ( update7324014601402209203_a_nat @ V2 @ K @ F2 @ Ps2 ) ) ) ) ) ).
% update_with_aux.simps(2)
thf(fact_404_update__with__aux_Osimps_I2_J,axiom,
! [P2: produc2422161461964618553l_real,K: real,V2: real,F2: real > real,Ps2: list_P8689742595348180415l_real] :
( ( ( ( produc5828954698716094813l_real @ P2 )
= K )
=> ( ( update4217048494389064540l_real @ V2 @ K @ F2 @ ( cons_P1861573166434266607l_real @ P2 @ Ps2 ) )
= ( cons_P1861573166434266607l_real @ ( produc4511245868158468465l_real @ K @ ( F2 @ ( produc3484788084999411615l_real @ P2 ) ) ) @ Ps2 ) ) )
& ( ( ( produc5828954698716094813l_real @ P2 )
!= K )
=> ( ( update4217048494389064540l_real @ V2 @ K @ F2 @ ( cons_P1861573166434266607l_real @ P2 @ Ps2 ) )
= ( cons_P1861573166434266607l_real @ P2 @ ( update4217048494389064540l_real @ V2 @ K @ F2 @ Ps2 ) ) ) ) ) ).
% update_with_aux.simps(2)
thf(fact_405_update__with__aux_Osimps_I2_J,axiom,
! [P2: product_prod_num_num,K: num,V2: num,F2: num > num,Ps2: list_P3744719386663036955um_num] :
( ( ( ( product_fst_num_num @ P2 )
= K )
=> ( ( update4439271885653463992um_num @ V2 @ K @ F2 @ ( cons_P5351470506836119883um_num @ P2 @ Ps2 ) )
= ( cons_P5351470506836119883um_num @ ( product_Pair_num_num @ K @ ( F2 @ ( product_snd_num_num @ P2 ) ) ) @ Ps2 ) ) )
& ( ( ( product_fst_num_num @ P2 )
!= K )
=> ( ( update4439271885653463992um_num @ V2 @ K @ F2 @ ( cons_P5351470506836119883um_num @ P2 @ Ps2 ) )
= ( cons_P5351470506836119883um_num @ P2 @ ( update4439271885653463992um_num @ V2 @ K @ F2 @ Ps2 ) ) ) ) ) ).
% update_with_aux.simps(2)
thf(fact_406_update__with__aux_Osimps_I2_J,axiom,
! [P2: produc3741383161447143261al_nat,K: real,V2: nat,F2: nat > nat,Ps2: list_P6834414599653733731al_nat] :
( ( ( ( product_fst_real_nat @ P2 )
= K )
=> ( ( update1720414882749132800t_real @ V2 @ K @ F2 @ ( cons_P500833500243608851al_nat @ P2 @ Ps2 ) )
= ( cons_P500833500243608851al_nat @ ( produc3181502643871035669al_nat @ K @ ( F2 @ ( product_snd_real_nat @ P2 ) ) ) @ Ps2 ) ) )
& ( ( ( product_fst_real_nat @ P2 )
!= K )
=> ( ( update1720414882749132800t_real @ V2 @ K @ F2 @ ( cons_P500833500243608851al_nat @ P2 @ Ps2 ) )
= ( cons_P500833500243608851al_nat @ P2 @ ( update1720414882749132800t_real @ V2 @ K @ F2 @ Ps2 ) ) ) ) ) ).
% update_with_aux.simps(2)
thf(fact_407_map__entry_Osimps_I2_J,axiom,
! [P2: produc7123000486447228170_nat_a,K: nat,F2: ( nat > a ) > nat > a,Ps2: list_P8194537880601713306_nat_a] :
( ( ( ( produc3194919578927588176_nat_a @ P2 )
= K )
=> ( ( map_entry_nat_nat_a @ K @ F2 @ ( cons_P6755270464639534804_nat_a @ P2 @ Ps2 ) )
= ( cons_P6755270464639534804_nat_a @ ( produc5292568359338195516_nat_a @ K @ ( F2 @ ( produc4809910040060592782_nat_a @ P2 ) ) ) @ Ps2 ) ) )
& ( ( ( produc3194919578927588176_nat_a @ P2 )
!= K )
=> ( ( map_entry_nat_nat_a @ K @ F2 @ ( cons_P6755270464639534804_nat_a @ P2 @ Ps2 ) )
= ( cons_P6755270464639534804_nat_a @ P2 @ ( map_entry_nat_nat_a @ K @ F2 @ Ps2 ) ) ) ) ) ).
% map_entry.simps(2)
thf(fact_408_map__entry_Osimps_I2_J,axiom,
! [P2: produc2422161461964618553l_real,K: real,F2: real > real,Ps2: list_P8689742595348180415l_real] :
( ( ( ( produc5828954698716094813l_real @ P2 )
= K )
=> ( ( map_entry_real_real @ K @ F2 @ ( cons_P1861573166434266607l_real @ P2 @ Ps2 ) )
= ( cons_P1861573166434266607l_real @ ( produc4511245868158468465l_real @ K @ ( F2 @ ( produc3484788084999411615l_real @ P2 ) ) ) @ Ps2 ) ) )
& ( ( ( produc5828954698716094813l_real @ P2 )
!= K )
=> ( ( map_entry_real_real @ K @ F2 @ ( cons_P1861573166434266607l_real @ P2 @ Ps2 ) )
= ( cons_P1861573166434266607l_real @ P2 @ ( map_entry_real_real @ K @ F2 @ Ps2 ) ) ) ) ) ).
% map_entry.simps(2)
thf(fact_409_map__entry_Osimps_I2_J,axiom,
! [P2: product_prod_num_num,K: num,F2: num > num,Ps2: list_P3744719386663036955um_num] :
( ( ( ( product_fst_num_num @ P2 )
= K )
=> ( ( map_entry_num_num @ K @ F2 @ ( cons_P5351470506836119883um_num @ P2 @ Ps2 ) )
= ( cons_P5351470506836119883um_num @ ( product_Pair_num_num @ K @ ( F2 @ ( product_snd_num_num @ P2 ) ) ) @ Ps2 ) ) )
& ( ( ( product_fst_num_num @ P2 )
!= K )
=> ( ( map_entry_num_num @ K @ F2 @ ( cons_P5351470506836119883um_num @ P2 @ Ps2 ) )
= ( cons_P5351470506836119883um_num @ P2 @ ( map_entry_num_num @ K @ F2 @ Ps2 ) ) ) ) ) ).
% map_entry.simps(2)
thf(fact_410_map__entry_Osimps_I2_J,axiom,
! [P2: produc3741383161447143261al_nat,K: real,F2: nat > nat,Ps2: list_P6834414599653733731al_nat] :
( ( ( ( product_fst_real_nat @ P2 )
= K )
=> ( ( map_entry_real_nat @ K @ F2 @ ( cons_P500833500243608851al_nat @ P2 @ Ps2 ) )
= ( cons_P500833500243608851al_nat @ ( produc3181502643871035669al_nat @ K @ ( F2 @ ( product_snd_real_nat @ P2 ) ) ) @ Ps2 ) ) )
& ( ( ( product_fst_real_nat @ P2 )
!= K )
=> ( ( map_entry_real_nat @ K @ F2 @ ( cons_P500833500243608851al_nat @ P2 @ Ps2 ) )
= ( cons_P500833500243608851al_nat @ P2 @ ( map_entry_real_nat @ K @ F2 @ Ps2 ) ) ) ) ) ).
% map_entry.simps(2)
thf(fact_411_map__default_Osimps_I2_J,axiom,
! [P2: produc7123000486447228170_nat_a,K: nat,V2: nat > a,F2: ( nat > a ) > nat > a,Ps2: list_P8194537880601713306_nat_a] :
( ( ( ( produc3194919578927588176_nat_a @ P2 )
= K )
=> ( ( map_de630094633906692667_nat_a @ K @ V2 @ F2 @ ( cons_P6755270464639534804_nat_a @ P2 @ Ps2 ) )
= ( cons_P6755270464639534804_nat_a @ ( produc5292568359338195516_nat_a @ K @ ( F2 @ ( produc4809910040060592782_nat_a @ P2 ) ) ) @ Ps2 ) ) )
& ( ( ( produc3194919578927588176_nat_a @ P2 )
!= K )
=> ( ( map_de630094633906692667_nat_a @ K @ V2 @ F2 @ ( cons_P6755270464639534804_nat_a @ P2 @ Ps2 ) )
= ( cons_P6755270464639534804_nat_a @ P2 @ ( map_de630094633906692667_nat_a @ K @ V2 @ F2 @ Ps2 ) ) ) ) ) ).
% map_default.simps(2)
thf(fact_412_map__default_Osimps_I2_J,axiom,
! [P2: produc2422161461964618553l_real,K: real,V2: real,F2: real > real,Ps2: list_P8689742595348180415l_real] :
( ( ( ( produc5828954698716094813l_real @ P2 )
= K )
=> ( ( map_de3913085918471371826l_real @ K @ V2 @ F2 @ ( cons_P1861573166434266607l_real @ P2 @ Ps2 ) )
= ( cons_P1861573166434266607l_real @ ( produc4511245868158468465l_real @ K @ ( F2 @ ( produc3484788084999411615l_real @ P2 ) ) ) @ Ps2 ) ) )
& ( ( ( produc5828954698716094813l_real @ P2 )
!= K )
=> ( ( map_de3913085918471371826l_real @ K @ V2 @ F2 @ ( cons_P1861573166434266607l_real @ P2 @ Ps2 ) )
= ( cons_P1861573166434266607l_real @ P2 @ ( map_de3913085918471371826l_real @ K @ V2 @ F2 @ Ps2 ) ) ) ) ) ).
% map_default.simps(2)
thf(fact_413_map__default_Osimps_I2_J,axiom,
! [P2: product_prod_num_num,K: num,V2: num,F2: num > num,Ps2: list_P3744719386663036955um_num] :
( ( ( ( product_fst_num_num @ P2 )
= K )
=> ( ( map_default_num_num @ K @ V2 @ F2 @ ( cons_P5351470506836119883um_num @ P2 @ Ps2 ) )
= ( cons_P5351470506836119883um_num @ ( product_Pair_num_num @ K @ ( F2 @ ( product_snd_num_num @ P2 ) ) ) @ Ps2 ) ) )
& ( ( ( product_fst_num_num @ P2 )
!= K )
=> ( ( map_default_num_num @ K @ V2 @ F2 @ ( cons_P5351470506836119883um_num @ P2 @ Ps2 ) )
= ( cons_P5351470506836119883um_num @ P2 @ ( map_default_num_num @ K @ V2 @ F2 @ Ps2 ) ) ) ) ) ).
% map_default.simps(2)
thf(fact_414_map__default_Osimps_I2_J,axiom,
! [P2: produc3741383161447143261al_nat,K: real,V2: nat,F2: nat > nat,Ps2: list_P6834414599653733731al_nat] :
( ( ( ( product_fst_real_nat @ P2 )
= K )
=> ( ( map_default_real_nat @ K @ V2 @ F2 @ ( cons_P500833500243608851al_nat @ P2 @ Ps2 ) )
= ( cons_P500833500243608851al_nat @ ( produc3181502643871035669al_nat @ K @ ( F2 @ ( product_snd_real_nat @ P2 ) ) ) @ Ps2 ) ) )
& ( ( ( product_fst_real_nat @ P2 )
!= K )
=> ( ( map_default_real_nat @ K @ V2 @ F2 @ ( cons_P500833500243608851al_nat @ P2 @ Ps2 ) )
= ( cons_P500833500243608851al_nat @ P2 @ ( map_default_real_nat @ K @ V2 @ F2 @ Ps2 ) ) ) ) ) ).
% map_default.simps(2)
thf(fact_415_update__with__aux_Osimps_I1_J,axiom,
! [V2: nat > a,K: nat,F2: ( nat > a ) > nat > a] :
( ( update7324014601402209203_a_nat @ V2 @ K @ F2 @ nil_Pr2013076336424982148_nat_a )
= ( cons_P6755270464639534804_nat_a @ ( produc5292568359338195516_nat_a @ K @ ( F2 @ V2 ) ) @ nil_Pr2013076336424982148_nat_a ) ) ).
% update_with_aux.simps(1)
thf(fact_416_update__with__aux_Osimps_I1_J,axiom,
! [V2: nat,K: real,F2: nat > nat] :
( ( update1720414882749132800t_real @ V2 @ K @ F2 @ nil_Pr1917482104270272867al_nat )
= ( cons_P500833500243608851al_nat @ ( produc3181502643871035669al_nat @ K @ ( F2 @ V2 ) ) @ nil_Pr1917482104270272867al_nat ) ) ).
% update_with_aux.simps(1)
thf(fact_417_update__with__aux_Osimps_I1_J,axiom,
! [V2: num,K: num,F2: num > num] :
( ( update4439271885653463992um_num @ V2 @ K @ F2 @ nil_Pr4317560964547046811um_num )
= ( cons_P5351470506836119883um_num @ ( product_Pair_num_num @ K @ ( F2 @ V2 ) ) @ nil_Pr4317560964547046811um_num ) ) ).
% update_with_aux.simps(1)
thf(fact_418_update__with__aux_Osimps_I1_J,axiom,
! [V2: real,K: real,F2: real > real] :
( ( update4217048494389064540l_real @ V2 @ K @ F2 @ nil_Pr366592848263477823l_real )
= ( cons_P1861573166434266607l_real @ ( produc4511245868158468465l_real @ K @ ( F2 @ V2 ) ) @ nil_Pr366592848263477823l_real ) ) ).
% update_with_aux.simps(1)
thf(fact_419_map__ran__Cons__sel,axiom,
! [F2: nat > ( nat > a ) > nat > a,P2: produc7123000486447228170_nat_a,Ps2: list_P8194537880601713306_nat_a] :
( ( map_ra5122625054061645511_nat_a @ F2 @ ( cons_P6755270464639534804_nat_a @ P2 @ Ps2 ) )
= ( cons_P6755270464639534804_nat_a @ ( produc5292568359338195516_nat_a @ ( produc3194919578927588176_nat_a @ P2 ) @ ( F2 @ ( produc3194919578927588176_nat_a @ P2 ) @ ( produc4809910040060592782_nat_a @ P2 ) ) ) @ ( map_ra5122625054061645511_nat_a @ F2 @ Ps2 ) ) ) ).
% map_ran_Cons_sel
thf(fact_420_map__ran__Cons__sel,axiom,
! [F2: real > real > nat,P2: produc2422161461964618553l_real,Ps2: list_P8689742595348180415l_real] :
( ( map_ra134723660208699745al_nat @ F2 @ ( cons_P1861573166434266607l_real @ P2 @ Ps2 ) )
= ( cons_P500833500243608851al_nat @ ( produc3181502643871035669al_nat @ ( produc5828954698716094813l_real @ P2 ) @ ( F2 @ ( produc5828954698716094813l_real @ P2 ) @ ( produc3484788084999411615l_real @ P2 ) ) ) @ ( map_ra134723660208699745al_nat @ F2 @ Ps2 ) ) ) ).
% map_ran_Cons_sel
thf(fact_421_map__ran__Cons__sel,axiom,
! [F2: real > real > real,P2: produc2422161461964618553l_real,Ps2: list_P8689742595348180415l_real] :
( ( map_ra6600816322303468477l_real @ F2 @ ( cons_P1861573166434266607l_real @ P2 @ Ps2 ) )
= ( cons_P1861573166434266607l_real @ ( produc4511245868158468465l_real @ ( produc5828954698716094813l_real @ P2 ) @ ( F2 @ ( produc5828954698716094813l_real @ P2 ) @ ( produc3484788084999411615l_real @ P2 ) ) ) @ ( map_ra6600816322303468477l_real @ F2 @ Ps2 ) ) ) ).
% map_ran_Cons_sel
thf(fact_422_map__ran__Cons__sel,axiom,
! [F2: num > num > num,P2: product_prod_num_num,Ps2: list_P3744719386663036955um_num] :
( ( map_ran_num_num_num @ F2 @ ( cons_P5351470506836119883um_num @ P2 @ Ps2 ) )
= ( cons_P5351470506836119883um_num @ ( product_Pair_num_num @ ( product_fst_num_num @ P2 ) @ ( F2 @ ( product_fst_num_num @ P2 ) @ ( product_snd_num_num @ P2 ) ) ) @ ( map_ran_num_num_num @ F2 @ Ps2 ) ) ) ).
% map_ran_Cons_sel
thf(fact_423_map__ran__Cons__sel,axiom,
! [F2: real > nat > nat,P2: produc3741383161447143261al_nat,Ps2: list_P6834414599653733731al_nat] :
( ( map_ran_real_nat_nat @ F2 @ ( cons_P500833500243608851al_nat @ P2 @ Ps2 ) )
= ( cons_P500833500243608851al_nat @ ( produc3181502643871035669al_nat @ ( product_fst_real_nat @ P2 ) @ ( F2 @ ( product_fst_real_nat @ P2 ) @ ( product_snd_real_nat @ P2 ) ) ) @ ( map_ran_real_nat_nat @ F2 @ Ps2 ) ) ) ).
% map_ran_Cons_sel
thf(fact_424_map__ran__Cons__sel,axiom,
! [F2: real > nat > real,P2: produc3741383161447143261al_nat,Ps2: list_P6834414599653733731al_nat] :
( ( map_ra4790787123934576865t_real @ F2 @ ( cons_P500833500243608851al_nat @ P2 @ Ps2 ) )
= ( cons_P1861573166434266607l_real @ ( produc4511245868158468465l_real @ ( product_fst_real_nat @ P2 ) @ ( F2 @ ( product_fst_real_nat @ P2 ) @ ( product_snd_real_nat @ P2 ) ) ) @ ( map_ra4790787123934576865t_real @ F2 @ Ps2 ) ) ) ).
% map_ran_Cons_sel
thf(fact_425_map__ran__simps_I2_J,axiom,
! [F2: nat > ( nat > a ) > nat > a,K: nat,V2: nat > a,Ps2: list_P8194537880601713306_nat_a] :
( ( map_ra5122625054061645511_nat_a @ F2 @ ( cons_P6755270464639534804_nat_a @ ( produc5292568359338195516_nat_a @ K @ V2 ) @ Ps2 ) )
= ( cons_P6755270464639534804_nat_a @ ( produc5292568359338195516_nat_a @ K @ ( F2 @ K @ V2 ) ) @ ( map_ra5122625054061645511_nat_a @ F2 @ Ps2 ) ) ) ).
% map_ran_simps(2)
thf(fact_426_map__ran__simps_I2_J,axiom,
! [F2: real > nat > nat,K: real,V2: nat,Ps2: list_P6834414599653733731al_nat] :
( ( map_ran_real_nat_nat @ F2 @ ( cons_P500833500243608851al_nat @ ( produc3181502643871035669al_nat @ K @ V2 ) @ Ps2 ) )
= ( cons_P500833500243608851al_nat @ ( produc3181502643871035669al_nat @ K @ ( F2 @ K @ V2 ) ) @ ( map_ran_real_nat_nat @ F2 @ Ps2 ) ) ) ).
% map_ran_simps(2)
thf(fact_427_map__ran__simps_I2_J,axiom,
! [F2: real > nat > real,K: real,V2: nat,Ps2: list_P6834414599653733731al_nat] :
( ( map_ra4790787123934576865t_real @ F2 @ ( cons_P500833500243608851al_nat @ ( produc3181502643871035669al_nat @ K @ V2 ) @ Ps2 ) )
= ( cons_P1861573166434266607l_real @ ( produc4511245868158468465l_real @ K @ ( F2 @ K @ V2 ) ) @ ( map_ra4790787123934576865t_real @ F2 @ Ps2 ) ) ) ).
% map_ran_simps(2)
thf(fact_428_map__ran__simps_I2_J,axiom,
! [F2: num > num > num,K: num,V2: num,Ps2: list_P3744719386663036955um_num] :
( ( map_ran_num_num_num @ F2 @ ( cons_P5351470506836119883um_num @ ( product_Pair_num_num @ K @ V2 ) @ Ps2 ) )
= ( cons_P5351470506836119883um_num @ ( product_Pair_num_num @ K @ ( F2 @ K @ V2 ) ) @ ( map_ran_num_num_num @ F2 @ Ps2 ) ) ) ).
% map_ran_simps(2)
thf(fact_429_map__ran__simps_I2_J,axiom,
! [F2: real > real > nat,K: real,V2: real,Ps2: list_P8689742595348180415l_real] :
( ( map_ra134723660208699745al_nat @ F2 @ ( cons_P1861573166434266607l_real @ ( produc4511245868158468465l_real @ K @ V2 ) @ Ps2 ) )
= ( cons_P500833500243608851al_nat @ ( produc3181502643871035669al_nat @ K @ ( F2 @ K @ V2 ) ) @ ( map_ra134723660208699745al_nat @ F2 @ Ps2 ) ) ) ).
% map_ran_simps(2)
thf(fact_430_map__ran__simps_I2_J,axiom,
! [F2: real > real > real,K: real,V2: real,Ps2: list_P8689742595348180415l_real] :
( ( map_ra6600816322303468477l_real @ F2 @ ( cons_P1861573166434266607l_real @ ( produc4511245868158468465l_real @ K @ V2 ) @ Ps2 ) )
= ( cons_P1861573166434266607l_real @ ( produc4511245868158468465l_real @ K @ ( F2 @ K @ V2 ) ) @ ( map_ra6600816322303468477l_real @ F2 @ Ps2 ) ) ) ).
% map_ran_simps(2)
thf(fact_431_map__default_Osimps_I1_J,axiom,
! [K: nat,V2: nat > a,F2: ( nat > a ) > nat > a] :
( ( map_de630094633906692667_nat_a @ K @ V2 @ F2 @ nil_Pr2013076336424982148_nat_a )
= ( cons_P6755270464639534804_nat_a @ ( produc5292568359338195516_nat_a @ K @ V2 ) @ nil_Pr2013076336424982148_nat_a ) ) ).
% map_default.simps(1)
thf(fact_432_map__default_Osimps_I1_J,axiom,
! [K: real,V2: nat,F2: nat > nat] :
( ( map_default_real_nat @ K @ V2 @ F2 @ nil_Pr1917482104270272867al_nat )
= ( cons_P500833500243608851al_nat @ ( produc3181502643871035669al_nat @ K @ V2 ) @ nil_Pr1917482104270272867al_nat ) ) ).
% map_default.simps(1)
thf(fact_433_map__default_Osimps_I1_J,axiom,
! [K: num,V2: num,F2: num > num] :
( ( map_default_num_num @ K @ V2 @ F2 @ nil_Pr4317560964547046811um_num )
= ( cons_P5351470506836119883um_num @ ( product_Pair_num_num @ K @ V2 ) @ nil_Pr4317560964547046811um_num ) ) ).
% map_default.simps(1)
thf(fact_434_map__default_Osimps_I1_J,axiom,
! [K: real,V2: real,F2: real > real] :
( ( map_de3913085918471371826l_real @ K @ V2 @ F2 @ nil_Pr366592848263477823l_real )
= ( cons_P1861573166434266607l_real @ ( produc4511245868158468465l_real @ K @ V2 ) @ nil_Pr366592848263477823l_real ) ) ).
% map_default.simps(1)
thf(fact_435_map__default_Opelims,axiom,
! [X: nat,Xa: nat > a,Xb: ( nat > a ) > nat > a,Xc: list_P8194537880601713306_nat_a,Y: list_P8194537880601713306_nat_a] :
( ( ( map_de630094633906692667_nat_a @ X @ Xa @ Xb @ Xc )
= Y )
=> ( ( accp_P5156064751882252798_nat_a @ map_de1280766450222638898_nat_a @ ( produc7306357460058920569_nat_a @ X @ ( produc39007591599667346_nat_a @ Xa @ ( produc6488654407618133976_nat_a @ Xb @ Xc ) ) ) )
=> ( ( ( Xc = nil_Pr2013076336424982148_nat_a )
=> ( ( Y
= ( cons_P6755270464639534804_nat_a @ ( produc5292568359338195516_nat_a @ X @ Xa ) @ nil_Pr2013076336424982148_nat_a ) )
=> ~ ( accp_P5156064751882252798_nat_a @ map_de1280766450222638898_nat_a @ ( produc7306357460058920569_nat_a @ X @ ( produc39007591599667346_nat_a @ Xa @ ( produc6488654407618133976_nat_a @ Xb @ nil_Pr2013076336424982148_nat_a ) ) ) ) ) )
=> ~ ! [P4: produc7123000486447228170_nat_a,Ps: list_P8194537880601713306_nat_a] :
( ( Xc
= ( cons_P6755270464639534804_nat_a @ P4 @ Ps ) )
=> ( ( ( ( ( produc3194919578927588176_nat_a @ P4 )
= X )
=> ( Y
= ( cons_P6755270464639534804_nat_a @ ( produc5292568359338195516_nat_a @ X @ ( Xb @ ( produc4809910040060592782_nat_a @ P4 ) ) ) @ Ps ) ) )
& ( ( ( produc3194919578927588176_nat_a @ P4 )
!= X )
=> ( Y
= ( cons_P6755270464639534804_nat_a @ P4 @ ( map_de630094633906692667_nat_a @ X @ Xa @ Xb @ Ps ) ) ) ) )
=> ~ ( accp_P5156064751882252798_nat_a @ map_de1280766450222638898_nat_a @ ( produc7306357460058920569_nat_a @ X @ ( produc39007591599667346_nat_a @ Xa @ ( produc6488654407618133976_nat_a @ Xb @ ( cons_P6755270464639534804_nat_a @ P4 @ Ps ) ) ) ) ) ) ) ) ) ) ).
% map_default.pelims
thf(fact_436_map__default_Opelims,axiom,
! [X: real,Xa: real,Xb: real > real,Xc: list_P8689742595348180415l_real,Y: list_P8689742595348180415l_real] :
( ( ( map_de3913085918471371826l_real @ X @ Xa @ Xb @ Xc )
= Y )
=> ( ( accp_P5976341164542609034l_real @ map_de6606940589474017275l_real @ ( produc6975477229195654405l_real @ X @ ( produc9156241399310741058l_real @ Xa @ ( produc4799400148392655279l_real @ Xb @ Xc ) ) ) )
=> ( ( ( Xc = nil_Pr366592848263477823l_real )
=> ( ( Y
= ( cons_P1861573166434266607l_real @ ( produc4511245868158468465l_real @ X @ Xa ) @ nil_Pr366592848263477823l_real ) )
=> ~ ( accp_P5976341164542609034l_real @ map_de6606940589474017275l_real @ ( produc6975477229195654405l_real @ X @ ( produc9156241399310741058l_real @ Xa @ ( produc4799400148392655279l_real @ Xb @ nil_Pr366592848263477823l_real ) ) ) ) ) )
=> ~ ! [P4: produc2422161461964618553l_real,Ps: list_P8689742595348180415l_real] :
( ( Xc
= ( cons_P1861573166434266607l_real @ P4 @ Ps ) )
=> ( ( ( ( ( produc5828954698716094813l_real @ P4 )
= X )
=> ( Y
= ( cons_P1861573166434266607l_real @ ( produc4511245868158468465l_real @ X @ ( Xb @ ( produc3484788084999411615l_real @ P4 ) ) ) @ Ps ) ) )
& ( ( ( produc5828954698716094813l_real @ P4 )
!= X )
=> ( Y
= ( cons_P1861573166434266607l_real @ P4 @ ( map_de3913085918471371826l_real @ X @ Xa @ Xb @ Ps ) ) ) ) )
=> ~ ( accp_P5976341164542609034l_real @ map_de6606940589474017275l_real @ ( produc6975477229195654405l_real @ X @ ( produc9156241399310741058l_real @ Xa @ ( produc4799400148392655279l_real @ Xb @ ( cons_P1861573166434266607l_real @ P4 @ Ps ) ) ) ) ) ) ) ) ) ) ).
% map_default.pelims
thf(fact_437_map__default_Opelims,axiom,
! [X: num,Xa: num,Xb: num > num,Xc: list_P3744719386663036955um_num,Y: list_P3744719386663036955um_num] :
( ( ( map_default_num_num @ X @ Xa @ Xb @ Xc )
= Y )
=> ( ( accp_P4146607925373659146um_num @ map_de5457363097827823447um_num @ ( produc2135470696632816005um_num @ X @ ( produc7684608680066660528um_num @ Xa @ ( produc2892947491196644271um_num @ Xb @ Xc ) ) ) )
=> ( ( ( Xc = nil_Pr4317560964547046811um_num )
=> ( ( Y
= ( cons_P5351470506836119883um_num @ ( product_Pair_num_num @ X @ Xa ) @ nil_Pr4317560964547046811um_num ) )
=> ~ ( accp_P4146607925373659146um_num @ map_de5457363097827823447um_num @ ( produc2135470696632816005um_num @ X @ ( produc7684608680066660528um_num @ Xa @ ( produc2892947491196644271um_num @ Xb @ nil_Pr4317560964547046811um_num ) ) ) ) ) )
=> ~ ! [P4: product_prod_num_num,Ps: list_P3744719386663036955um_num] :
( ( Xc
= ( cons_P5351470506836119883um_num @ P4 @ Ps ) )
=> ( ( ( ( ( product_fst_num_num @ P4 )
= X )
=> ( Y
= ( cons_P5351470506836119883um_num @ ( product_Pair_num_num @ X @ ( Xb @ ( product_snd_num_num @ P4 ) ) ) @ Ps ) ) )
& ( ( ( product_fst_num_num @ P4 )
!= X )
=> ( Y
= ( cons_P5351470506836119883um_num @ P4 @ ( map_default_num_num @ X @ Xa @ Xb @ Ps ) ) ) ) )
=> ~ ( accp_P4146607925373659146um_num @ map_de5457363097827823447um_num @ ( produc2135470696632816005um_num @ X @ ( produc7684608680066660528um_num @ Xa @ ( produc2892947491196644271um_num @ Xb @ ( cons_P5351470506836119883um_num @ P4 @ Ps ) ) ) ) ) ) ) ) ) ) ).
% map_default.pelims
thf(fact_438_map__default_Opelims,axiom,
! [X: real,Xa: nat,Xb: nat > nat,Xc: list_P6834414599653733731al_nat,Y: list_P6834414599653733731al_nat] :
( ( ( map_default_real_nat @ X @ Xa @ Xb @ Xc )
= Y )
=> ( ( accp_P4161832113633214346al_nat @ map_de7945623465553822879al_nat @ ( produc1739509229069810181al_nat @ X @ ( produc1552154576430284610al_nat @ Xa @ ( produc7644893288092425099al_nat @ Xb @ Xc ) ) ) )
=> ( ( ( Xc = nil_Pr1917482104270272867al_nat )
=> ( ( Y
= ( cons_P500833500243608851al_nat @ ( produc3181502643871035669al_nat @ X @ Xa ) @ nil_Pr1917482104270272867al_nat ) )
=> ~ ( accp_P4161832113633214346al_nat @ map_de7945623465553822879al_nat @ ( produc1739509229069810181al_nat @ X @ ( produc1552154576430284610al_nat @ Xa @ ( produc7644893288092425099al_nat @ Xb @ nil_Pr1917482104270272867al_nat ) ) ) ) ) )
=> ~ ! [P4: produc3741383161447143261al_nat,Ps: list_P6834414599653733731al_nat] :
( ( Xc
= ( cons_P500833500243608851al_nat @ P4 @ Ps ) )
=> ( ( ( ( ( product_fst_real_nat @ P4 )
= X )
=> ( Y
= ( cons_P500833500243608851al_nat @ ( produc3181502643871035669al_nat @ X @ ( Xb @ ( product_snd_real_nat @ P4 ) ) ) @ Ps ) ) )
& ( ( ( product_fst_real_nat @ P4 )
!= X )
=> ( Y
= ( cons_P500833500243608851al_nat @ P4 @ ( map_default_real_nat @ X @ Xa @ Xb @ Ps ) ) ) ) )
=> ~ ( accp_P4161832113633214346al_nat @ map_de7945623465553822879al_nat @ ( produc1739509229069810181al_nat @ X @ ( produc1552154576430284610al_nat @ Xa @ ( produc7644893288092425099al_nat @ Xb @ ( cons_P500833500243608851al_nat @ P4 @ Ps ) ) ) ) ) ) ) ) ) ) ).
% map_default.pelims
thf(fact_439_map__entry_Opelims,axiom,
! [X: nat,Xa: ( nat > a ) > nat > a,Xb: list_P8194537880601713306_nat_a,Y: list_P8194537880601713306_nat_a] :
( ( ( map_entry_nat_nat_a @ X @ Xa @ Xb )
= Y )
=> ( ( accp_P3452593570100577660_nat_a @ map_en1310760600757422627_nat_a @ ( produc2720549996417486839_nat_a @ X @ ( produc6488654407618133976_nat_a @ Xa @ Xb ) ) )
=> ( ( ( Xb = nil_Pr2013076336424982148_nat_a )
=> ( ( Y = nil_Pr2013076336424982148_nat_a )
=> ~ ( accp_P3452593570100577660_nat_a @ map_en1310760600757422627_nat_a @ ( produc2720549996417486839_nat_a @ X @ ( produc6488654407618133976_nat_a @ Xa @ nil_Pr2013076336424982148_nat_a ) ) ) ) )
=> ~ ! [P4: produc7123000486447228170_nat_a,Ps: list_P8194537880601713306_nat_a] :
( ( Xb
= ( cons_P6755270464639534804_nat_a @ P4 @ Ps ) )
=> ( ( ( ( ( produc3194919578927588176_nat_a @ P4 )
= X )
=> ( Y
= ( cons_P6755270464639534804_nat_a @ ( produc5292568359338195516_nat_a @ X @ ( Xa @ ( produc4809910040060592782_nat_a @ P4 ) ) ) @ Ps ) ) )
& ( ( ( produc3194919578927588176_nat_a @ P4 )
!= X )
=> ( Y
= ( cons_P6755270464639534804_nat_a @ P4 @ ( map_entry_nat_nat_a @ X @ Xa @ Ps ) ) ) ) )
=> ~ ( accp_P3452593570100577660_nat_a @ map_en1310760600757422627_nat_a @ ( produc2720549996417486839_nat_a @ X @ ( produc6488654407618133976_nat_a @ Xa @ ( cons_P6755270464639534804_nat_a @ P4 @ Ps ) ) ) ) ) ) ) ) ) ).
% map_entry.pelims
thf(fact_440_map__entry_Opelims,axiom,
! [X: real,Xa: real > real,Xb: list_P8689742595348180415l_real,Y: list_P8689742595348180415l_real] :
( ( ( map_entry_real_real @ X @ Xa @ Xb )
= Y )
=> ( ( accp_P2253066923555800083l_real @ map_en2765764844231136842l_real @ ( produc9156241399310741058l_real @ X @ ( produc4799400148392655279l_real @ Xa @ Xb ) ) )
=> ( ( ( Xb = nil_Pr366592848263477823l_real )
=> ( ( Y = nil_Pr366592848263477823l_real )
=> ~ ( accp_P2253066923555800083l_real @ map_en2765764844231136842l_real @ ( produc9156241399310741058l_real @ X @ ( produc4799400148392655279l_real @ Xa @ nil_Pr366592848263477823l_real ) ) ) ) )
=> ~ ! [P4: produc2422161461964618553l_real,Ps: list_P8689742595348180415l_real] :
( ( Xb
= ( cons_P1861573166434266607l_real @ P4 @ Ps ) )
=> ( ( ( ( ( produc5828954698716094813l_real @ P4 )
= X )
=> ( Y
= ( cons_P1861573166434266607l_real @ ( produc4511245868158468465l_real @ X @ ( Xa @ ( produc3484788084999411615l_real @ P4 ) ) ) @ Ps ) ) )
& ( ( ( produc5828954698716094813l_real @ P4 )
!= X )
=> ( Y
= ( cons_P1861573166434266607l_real @ P4 @ ( map_entry_real_real @ X @ Xa @ Ps ) ) ) ) )
=> ~ ( accp_P2253066923555800083l_real @ map_en2765764844231136842l_real @ ( produc9156241399310741058l_real @ X @ ( produc4799400148392655279l_real @ Xa @ ( cons_P1861573166434266607l_real @ P4 @ Ps ) ) ) ) ) ) ) ) ) ).
% map_entry.pelims
thf(fact_441_map__entry_Opelims,axiom,
! [X: num,Xa: num > num,Xb: list_P3744719386663036955um_num,Y: list_P3744719386663036955um_num] :
( ( ( map_entry_num_num @ X @ Xa @ Xb )
= Y )
=> ( ( accp_P3686700994160295809um_num @ map_en979851596782478502um_num @ ( produc7684608680066660528um_num @ X @ ( produc2892947491196644271um_num @ Xa @ Xb ) ) )
=> ( ( ( Xb = nil_Pr4317560964547046811um_num )
=> ( ( Y = nil_Pr4317560964547046811um_num )
=> ~ ( accp_P3686700994160295809um_num @ map_en979851596782478502um_num @ ( produc7684608680066660528um_num @ X @ ( produc2892947491196644271um_num @ Xa @ nil_Pr4317560964547046811um_num ) ) ) ) )
=> ~ ! [P4: product_prod_num_num,Ps: list_P3744719386663036955um_num] :
( ( Xb
= ( cons_P5351470506836119883um_num @ P4 @ Ps ) )
=> ( ( ( ( ( product_fst_num_num @ P4 )
= X )
=> ( Y
= ( cons_P5351470506836119883um_num @ ( product_Pair_num_num @ X @ ( Xa @ ( product_snd_num_num @ P4 ) ) ) @ Ps ) ) )
& ( ( ( product_fst_num_num @ P4 )
!= X )
=> ( Y
= ( cons_P5351470506836119883um_num @ P4 @ ( map_entry_num_num @ X @ Xa @ Ps ) ) ) ) )
=> ~ ( accp_P3686700994160295809um_num @ map_en979851596782478502um_num @ ( produc7684608680066660528um_num @ X @ ( produc2892947491196644271um_num @ Xa @ ( cons_P5351470506836119883um_num @ P4 @ Ps ) ) ) ) ) ) ) ) ) ).
% map_entry.pelims
thf(fact_442_map__entry_Opelims,axiom,
! [X: real,Xa: nat > nat,Xb: list_P6834414599653733731al_nat,Y: list_P6834414599653733731al_nat] :
( ( ( map_entry_real_nat @ X @ Xa @ Xb )
= Y )
=> ( ( accp_P5383833502838552303al_nat @ map_en7983188651647148654al_nat @ ( produc8229845255737718302al_nat @ X @ ( produc7644893288092425099al_nat @ Xa @ Xb ) ) )
=> ( ( ( Xb = nil_Pr1917482104270272867al_nat )
=> ( ( Y = nil_Pr1917482104270272867al_nat )
=> ~ ( accp_P5383833502838552303al_nat @ map_en7983188651647148654al_nat @ ( produc8229845255737718302al_nat @ X @ ( produc7644893288092425099al_nat @ Xa @ nil_Pr1917482104270272867al_nat ) ) ) ) )
=> ~ ! [P4: produc3741383161447143261al_nat,Ps: list_P6834414599653733731al_nat] :
( ( Xb
= ( cons_P500833500243608851al_nat @ P4 @ Ps ) )
=> ( ( ( ( ( product_fst_real_nat @ P4 )
= X )
=> ( Y
= ( cons_P500833500243608851al_nat @ ( produc3181502643871035669al_nat @ X @ ( Xa @ ( product_snd_real_nat @ P4 ) ) ) @ Ps ) ) )
& ( ( ( product_fst_real_nat @ P4 )
!= X )
=> ( Y
= ( cons_P500833500243608851al_nat @ P4 @ ( map_entry_real_nat @ X @ Xa @ Ps ) ) ) ) )
=> ~ ( accp_P5383833502838552303al_nat @ map_en7983188651647148654al_nat @ ( produc8229845255737718302al_nat @ X @ ( produc7644893288092425099al_nat @ Xa @ ( cons_P500833500243608851al_nat @ P4 @ Ps ) ) ) ) ) ) ) ) ) ).
% map_entry.pelims
thf(fact_443_in__measures_I2_J,axiom,
! [X: num,Y: num,F2: num > nat,Fs: list_num_nat] :
( ( member7279096912039735102um_num @ ( product_Pair_num_num @ X @ Y ) @ ( measures_num @ ( cons_num_nat @ F2 @ Fs ) ) )
= ( ( ord_less_nat @ ( F2 @ X ) @ ( F2 @ Y ) )
| ( ( ( F2 @ X )
= ( F2 @ Y ) )
& ( member7279096912039735102um_num @ ( product_Pair_num_num @ X @ Y ) @ ( measures_num @ Fs ) ) ) ) ) ).
% in_measures(2)
thf(fact_444_in__measures_I2_J,axiom,
! [X: real,Y: real,F2: real > nat,Fs: list_real_nat] :
( ( member7849222048561428706l_real @ ( produc4511245868158468465l_real @ X @ Y ) @ ( measures_real @ ( cons_real_nat @ F2 @ Fs ) ) )
= ( ( ord_less_nat @ ( F2 @ X ) @ ( F2 @ Y ) )
| ( ( ( F2 @ X )
= ( F2 @ Y ) )
& ( member7849222048561428706l_real @ ( produc4511245868158468465l_real @ X @ Y ) @ ( measures_real @ Fs ) ) ) ) ) ).
% in_measures(2)
thf(fact_445_nth__Cons__pos,axiom,
! [N3: nat,X: a,Xs: list_a] :
( ( ord_less_nat @ zero_zero_nat @ N3 )
=> ( ( nth_a @ ( cons_a @ X @ Xs ) @ N3 )
= ( nth_a @ Xs @ ( minus_minus_nat @ N3 @ one_one_nat ) ) ) ) ).
% nth_Cons_pos
thf(fact_446_nth__Cons__pos,axiom,
! [N3: nat,X: int,Xs: list_int] :
( ( ord_less_nat @ zero_zero_nat @ N3 )
=> ( ( nth_int @ ( cons_int @ X @ Xs ) @ N3 )
= ( nth_int @ Xs @ ( minus_minus_nat @ N3 @ one_one_nat ) ) ) ) ).
% nth_Cons_pos
thf(fact_447_delete__aux_Oelims,axiom,
! [X: nat,Xa: list_P8194537880601713306_nat_a,Y: list_P8194537880601713306_nat_a] :
( ( ( delete_aux_nat_nat_a @ X @ Xa )
= Y )
=> ( ( ( Xa = nil_Pr2013076336424982148_nat_a )
=> ( Y != nil_Pr2013076336424982148_nat_a ) )
=> ~ ! [K3: nat,V: nat > a,Xs2: list_P8194537880601713306_nat_a] :
( ( Xa
= ( cons_P6755270464639534804_nat_a @ ( produc5292568359338195516_nat_a @ K3 @ V ) @ Xs2 ) )
=> ~ ( ( ( X = K3 )
=> ( Y = Xs2 ) )
& ( ( X != K3 )
=> ( Y
= ( cons_P6755270464639534804_nat_a @ ( produc5292568359338195516_nat_a @ K3 @ V ) @ ( delete_aux_nat_nat_a @ X @ Xs2 ) ) ) ) ) ) ) ) ).
% delete_aux.elims
thf(fact_448_delete__aux_Oelims,axiom,
! [X: real,Xa: list_P6834414599653733731al_nat,Y: list_P6834414599653733731al_nat] :
( ( ( delete_aux_real_nat @ X @ Xa )
= Y )
=> ( ( ( Xa = nil_Pr1917482104270272867al_nat )
=> ( Y != nil_Pr1917482104270272867al_nat ) )
=> ~ ! [K3: real,V: nat,Xs2: list_P6834414599653733731al_nat] :
( ( Xa
= ( cons_P500833500243608851al_nat @ ( produc3181502643871035669al_nat @ K3 @ V ) @ Xs2 ) )
=> ~ ( ( ( X = K3 )
=> ( Y = Xs2 ) )
& ( ( X != K3 )
=> ( Y
= ( cons_P500833500243608851al_nat @ ( produc3181502643871035669al_nat @ K3 @ V ) @ ( delete_aux_real_nat @ X @ Xs2 ) ) ) ) ) ) ) ) ).
% delete_aux.elims
thf(fact_449_delete__aux_Oelims,axiom,
! [X: num,Xa: list_P3744719386663036955um_num,Y: list_P3744719386663036955um_num] :
( ( ( delete_aux_num_num @ X @ Xa )
= Y )
=> ( ( ( Xa = nil_Pr4317560964547046811um_num )
=> ( Y != nil_Pr4317560964547046811um_num ) )
=> ~ ! [K3: num,V: num,Xs2: list_P3744719386663036955um_num] :
( ( Xa
= ( cons_P5351470506836119883um_num @ ( product_Pair_num_num @ K3 @ V ) @ Xs2 ) )
=> ~ ( ( ( X = K3 )
=> ( Y = Xs2 ) )
& ( ( X != K3 )
=> ( Y
= ( cons_P5351470506836119883um_num @ ( product_Pair_num_num @ K3 @ V ) @ ( delete_aux_num_num @ X @ Xs2 ) ) ) ) ) ) ) ) ).
% delete_aux.elims
thf(fact_450_delete__aux_Oelims,axiom,
! [X: real,Xa: list_P8689742595348180415l_real,Y: list_P8689742595348180415l_real] :
( ( ( delete_aux_real_real @ X @ Xa )
= Y )
=> ( ( ( Xa = nil_Pr366592848263477823l_real )
=> ( Y != nil_Pr366592848263477823l_real ) )
=> ~ ! [K3: real,V: real,Xs2: list_P8689742595348180415l_real] :
( ( Xa
= ( cons_P1861573166434266607l_real @ ( produc4511245868158468465l_real @ K3 @ V ) @ Xs2 ) )
=> ~ ( ( ( X = K3 )
=> ( Y = Xs2 ) )
& ( ( X != K3 )
=> ( Y
= ( cons_P1861573166434266607l_real @ ( produc4511245868158468465l_real @ K3 @ V ) @ ( delete_aux_real_real @ X @ Xs2 ) ) ) ) ) ) ) ) ).
% delete_aux.elims
thf(fact_451_delete__aux__eq__Nil__conv,axiom,
! [K: nat,Ts: list_P8194537880601713306_nat_a] :
( ( ( delete_aux_nat_nat_a @ K @ Ts )
= nil_Pr2013076336424982148_nat_a )
= ( ( Ts = nil_Pr2013076336424982148_nat_a )
| ? [V3: nat > a] :
( Ts
= ( cons_P6755270464639534804_nat_a @ ( produc5292568359338195516_nat_a @ K @ V3 ) @ nil_Pr2013076336424982148_nat_a ) ) ) ) ).
% delete_aux_eq_Nil_conv
thf(fact_452_delete__aux__eq__Nil__conv,axiom,
! [K: real,Ts: list_P6834414599653733731al_nat] :
( ( ( delete_aux_real_nat @ K @ Ts )
= nil_Pr1917482104270272867al_nat )
= ( ( Ts = nil_Pr1917482104270272867al_nat )
| ? [V3: nat] :
( Ts
= ( cons_P500833500243608851al_nat @ ( produc3181502643871035669al_nat @ K @ V3 ) @ nil_Pr1917482104270272867al_nat ) ) ) ) ).
% delete_aux_eq_Nil_conv
thf(fact_453_delete__aux__eq__Nil__conv,axiom,
! [K: num,Ts: list_P3744719386663036955um_num] :
( ( ( delete_aux_num_num @ K @ Ts )
= nil_Pr4317560964547046811um_num )
= ( ( Ts = nil_Pr4317560964547046811um_num )
| ? [V3: num] :
( Ts
= ( cons_P5351470506836119883um_num @ ( product_Pair_num_num @ K @ V3 ) @ nil_Pr4317560964547046811um_num ) ) ) ) ).
% delete_aux_eq_Nil_conv
thf(fact_454_delete__aux__eq__Nil__conv,axiom,
! [K: real,Ts: list_P8689742595348180415l_real] :
( ( ( delete_aux_real_real @ K @ Ts )
= nil_Pr366592848263477823l_real )
= ( ( Ts = nil_Pr366592848263477823l_real )
| ? [V3: real] :
( Ts
= ( cons_P1861573166434266607l_real @ ( produc4511245868158468465l_real @ K @ V3 ) @ nil_Pr366592848263477823l_real ) ) ) ) ).
% delete_aux_eq_Nil_conv
thf(fact_455_in__measures_I1_J,axiom,
! [X: num,Y: num] :
~ ( member7279096912039735102um_num @ ( product_Pair_num_num @ X @ Y ) @ ( measures_num @ nil_num_nat ) ) ).
% in_measures(1)
thf(fact_456_in__measures_I1_J,axiom,
! [X: real,Y: real] :
~ ( member7849222048561428706l_real @ ( produc4511245868158468465l_real @ X @ Y ) @ ( measures_real @ nil_real_nat ) ) ).
% in_measures(1)
thf(fact_457_nth__Cons__0,axiom,
! [X: a,Xs: list_a] :
( ( nth_a @ ( cons_a @ X @ Xs ) @ zero_zero_nat )
= X ) ).
% nth_Cons_0
thf(fact_458_nth__Cons__0,axiom,
! [X: int,Xs: list_int] :
( ( nth_int @ ( cons_int @ X @ Xs ) @ zero_zero_nat )
= X ) ).
% nth_Cons_0
thf(fact_459_nth__Cons__Suc,axiom,
! [X: a,Xs: list_a,N3: nat] :
( ( nth_a @ ( cons_a @ X @ Xs ) @ ( suc @ N3 ) )
= ( nth_a @ Xs @ N3 ) ) ).
% nth_Cons_Suc
thf(fact_460_nth__Cons__Suc,axiom,
! [X: int,Xs: list_int,N3: nat] :
( ( nth_int @ ( cons_int @ X @ Xs ) @ ( suc @ N3 ) )
= ( nth_int @ Xs @ N3 ) ) ).
% nth_Cons_Suc
thf(fact_461_delete__aux_Osimps_I2_J,axiom,
! [K: nat,K4: nat,V2: nat > a,Xs: list_P8194537880601713306_nat_a] :
( ( ( K = K4 )
=> ( ( delete_aux_nat_nat_a @ K @ ( cons_P6755270464639534804_nat_a @ ( produc5292568359338195516_nat_a @ K4 @ V2 ) @ Xs ) )
= Xs ) )
& ( ( K != K4 )
=> ( ( delete_aux_nat_nat_a @ K @ ( cons_P6755270464639534804_nat_a @ ( produc5292568359338195516_nat_a @ K4 @ V2 ) @ Xs ) )
= ( cons_P6755270464639534804_nat_a @ ( produc5292568359338195516_nat_a @ K4 @ V2 ) @ ( delete_aux_nat_nat_a @ K @ Xs ) ) ) ) ) ).
% delete_aux.simps(2)
thf(fact_462_delete__aux_Osimps_I2_J,axiom,
! [K: real,K4: real,V2: nat,Xs: list_P6834414599653733731al_nat] :
( ( ( K = K4 )
=> ( ( delete_aux_real_nat @ K @ ( cons_P500833500243608851al_nat @ ( produc3181502643871035669al_nat @ K4 @ V2 ) @ Xs ) )
= Xs ) )
& ( ( K != K4 )
=> ( ( delete_aux_real_nat @ K @ ( cons_P500833500243608851al_nat @ ( produc3181502643871035669al_nat @ K4 @ V2 ) @ Xs ) )
= ( cons_P500833500243608851al_nat @ ( produc3181502643871035669al_nat @ K4 @ V2 ) @ ( delete_aux_real_nat @ K @ Xs ) ) ) ) ) ).
% delete_aux.simps(2)
thf(fact_463_delete__aux_Osimps_I2_J,axiom,
! [K: num,K4: num,V2: num,Xs: list_P3744719386663036955um_num] :
( ( ( K = K4 )
=> ( ( delete_aux_num_num @ K @ ( cons_P5351470506836119883um_num @ ( product_Pair_num_num @ K4 @ V2 ) @ Xs ) )
= Xs ) )
& ( ( K != K4 )
=> ( ( delete_aux_num_num @ K @ ( cons_P5351470506836119883um_num @ ( product_Pair_num_num @ K4 @ V2 ) @ Xs ) )
= ( cons_P5351470506836119883um_num @ ( product_Pair_num_num @ K4 @ V2 ) @ ( delete_aux_num_num @ K @ Xs ) ) ) ) ) ).
% delete_aux.simps(2)
thf(fact_464_delete__aux_Osimps_I2_J,axiom,
! [K: real,K4: real,V2: real,Xs: list_P8689742595348180415l_real] :
( ( ( K = K4 )
=> ( ( delete_aux_real_real @ K @ ( cons_P1861573166434266607l_real @ ( produc4511245868158468465l_real @ K4 @ V2 ) @ Xs ) )
= Xs ) )
& ( ( K != K4 )
=> ( ( delete_aux_real_real @ K @ ( cons_P1861573166434266607l_real @ ( produc4511245868158468465l_real @ K4 @ V2 ) @ Xs ) )
= ( cons_P1861573166434266607l_real @ ( produc4511245868158468465l_real @ K4 @ V2 ) @ ( delete_aux_real_real @ K @ Xs ) ) ) ) ) ).
% delete_aux.simps(2)
thf(fact_465_nth__Cons_H,axiom,
! [N3: nat,X: a,Xs: list_a] :
( ( ( N3 = zero_zero_nat )
=> ( ( nth_a @ ( cons_a @ X @ Xs ) @ N3 )
= X ) )
& ( ( N3 != zero_zero_nat )
=> ( ( nth_a @ ( cons_a @ X @ Xs ) @ N3 )
= ( nth_a @ Xs @ ( minus_minus_nat @ N3 @ one_one_nat ) ) ) ) ) ).
% nth_Cons'
thf(fact_466_nth__Cons_H,axiom,
! [N3: nat,X: int,Xs: list_int] :
( ( ( N3 = zero_zero_nat )
=> ( ( nth_int @ ( cons_int @ X @ Xs ) @ N3 )
= X ) )
& ( ( N3 != zero_zero_nat )
=> ( ( nth_int @ ( cons_int @ X @ Xs ) @ N3 )
= ( nth_int @ Xs @ ( minus_minus_nat @ N3 @ one_one_nat ) ) ) ) ) ).
% nth_Cons'
thf(fact_467_measures__less,axiom,
! [F2: num > nat,X: num,Y: num,Fs: list_num_nat] :
( ( ord_less_nat @ ( F2 @ X ) @ ( F2 @ Y ) )
=> ( member7279096912039735102um_num @ ( product_Pair_num_num @ X @ Y ) @ ( measures_num @ ( cons_num_nat @ F2 @ Fs ) ) ) ) ).
% measures_less
thf(fact_468_measures__less,axiom,
! [F2: real > nat,X: real,Y: real,Fs: list_real_nat] :
( ( ord_less_nat @ ( F2 @ X ) @ ( F2 @ Y ) )
=> ( member7849222048561428706l_real @ ( produc4511245868158468465l_real @ X @ Y ) @ ( measures_real @ ( cons_real_nat @ F2 @ Fs ) ) ) ) ).
% measures_less
thf(fact_469_nth__non__equal__first__eq,axiom,
! [X: a,Y: a,Xs: list_a,N3: nat] :
( ( X != Y )
=> ( ( ( nth_a @ ( cons_a @ X @ Xs ) @ N3 )
= Y )
= ( ( ( nth_a @ Xs @ ( minus_minus_nat @ N3 @ one_one_nat ) )
= Y )
& ( ord_less_nat @ zero_zero_nat @ N3 ) ) ) ) ).
% nth_non_equal_first_eq
thf(fact_470_nth__non__equal__first__eq,axiom,
! [X: int,Y: int,Xs: list_int,N3: nat] :
( ( X != Y )
=> ( ( ( nth_int @ ( cons_int @ X @ Xs ) @ N3 )
= Y )
= ( ( ( nth_int @ Xs @ ( minus_minus_nat @ N3 @ one_one_nat ) )
= Y )
& ( ord_less_nat @ zero_zero_nat @ N3 ) ) ) ) ).
% nth_non_equal_first_eq
thf(fact_471_nth__Cons__numeral,axiom,
! [X: a,Xs: list_a,V2: num] :
( ( nth_a @ ( cons_a @ X @ Xs ) @ ( numeral_numeral_nat @ V2 ) )
= ( nth_a @ Xs @ ( minus_minus_nat @ ( numeral_numeral_nat @ V2 ) @ one_one_nat ) ) ) ).
% nth_Cons_numeral
thf(fact_472_nth__Cons__numeral,axiom,
! [X: int,Xs: list_int,V2: num] :
( ( nth_int @ ( cons_int @ X @ Xs ) @ ( numeral_numeral_nat @ V2 ) )
= ( nth_int @ Xs @ ( minus_minus_nat @ ( numeral_numeral_nat @ V2 ) @ one_one_nat ) ) ) ).
% nth_Cons_numeral
thf(fact_473_inverse__permutation__of__list_Oelims,axiom,
! [X: list_P3744719386663036955um_num,Xa: num,Y: num] :
( ( ( invers4287837041963334271st_num @ X @ Xa )
= Y )
=> ( ( ( X = nil_Pr4317560964547046811um_num )
=> ( Y != Xa ) )
=> ~ ! [Y3: num,X6: num,Xs2: list_P3744719386663036955um_num] :
( ( X
= ( cons_P5351470506836119883um_num @ ( product_Pair_num_num @ Y3 @ X6 ) @ Xs2 ) )
=> ~ ( ( ( Xa = X6 )
=> ( Y = Y3 ) )
& ( ( Xa != X6 )
=> ( Y
= ( invers4287837041963334271st_num @ Xs2 @ Xa ) ) ) ) ) ) ) ).
% inverse_permutation_of_list.elims
thf(fact_474_inverse__permutation__of__list_Oelims,axiom,
! [X: list_P8689742595348180415l_real,Xa: real,Y: real] :
( ( ( invers7512574637932550289t_real @ X @ Xa )
= Y )
=> ( ( ( X = nil_Pr366592848263477823l_real )
=> ( Y != Xa ) )
=> ~ ! [Y3: real,X6: real,Xs2: list_P8689742595348180415l_real] :
( ( X
= ( cons_P1861573166434266607l_real @ ( produc4511245868158468465l_real @ Y3 @ X6 ) @ Xs2 ) )
=> ~ ( ( ( Xa = X6 )
=> ( Y = Y3 ) )
& ( ( Xa != X6 )
=> ( Y
= ( invers7512574637932550289t_real @ Xs2 @ Xa ) ) ) ) ) ) ) ).
% inverse_permutation_of_list.elims
thf(fact_475_gen__length__code_I2_J,axiom,
! [N3: nat,X: a,Xs: list_a] :
( ( gen_length_a @ N3 @ ( cons_a @ X @ Xs ) )
= ( gen_length_a @ ( suc @ N3 ) @ Xs ) ) ).
% gen_length_code(2)
thf(fact_476_gen__length__code_I2_J,axiom,
! [N3: nat,X: int,Xs: list_int] :
( ( gen_length_int @ N3 @ ( cons_int @ X @ Xs ) )
= ( gen_length_int @ ( suc @ N3 ) @ Xs ) ) ).
% gen_length_code(2)
thf(fact_477_radical__0,axiom,
! [N3: nat,R2: nat > real > real,A: formal3361831859752904756s_real] :
( ( ord_less_nat @ zero_zero_nat @ N3 )
=> ( ( formal8005797870169972230l_real @ R2 @ zero_zero_nat @ A @ N3 )
= zero_zero_real ) ) ).
% radical_0
thf(fact_478_eval__permutation__of__list_I3_J,axiom,
! [X: num,X7: num,Y6: num,Xs: list_P3744719386663036955um_num] :
( ( X != X7 )
=> ( ( permut9062699409770291517st_num @ ( cons_P5351470506836119883um_num @ ( product_Pair_num_num @ X7 @ Y6 ) @ Xs ) @ X )
= ( permut9062699409770291517st_num @ Xs @ X ) ) ) ).
% eval_permutation_of_list(3)
thf(fact_479_eval__permutation__of__list_I3_J,axiom,
! [X: real,X7: real,Y6: real,Xs: list_P8689742595348180415l_real] :
( ( X != X7 )
=> ( ( permut5466000449812578383t_real @ ( cons_P1861573166434266607l_real @ ( produc4511245868158468465l_real @ X7 @ Y6 ) @ Xs ) @ X )
= ( permut5466000449812578383t_real @ Xs @ X ) ) ) ).
% eval_permutation_of_list(3)
thf(fact_480_eval__permutation__of__list_I2_J,axiom,
! [X: num,X7: num,Y: num,Xs: list_P3744719386663036955um_num] :
( ( X = X7 )
=> ( ( permut9062699409770291517st_num @ ( cons_P5351470506836119883um_num @ ( product_Pair_num_num @ X7 @ Y ) @ Xs ) @ X )
= Y ) ) ).
% eval_permutation_of_list(2)
thf(fact_481_eval__permutation__of__list_I2_J,axiom,
! [X: real,X7: real,Y: real,Xs: list_P8689742595348180415l_real] :
( ( X = X7 )
=> ( ( permut5466000449812578383t_real @ ( cons_P1861573166434266607l_real @ ( produc4511245868158468465l_real @ X7 @ Y ) @ Xs ) @ X )
= Y ) ) ).
% eval_permutation_of_list(2)
thf(fact_482_numeral__eq__iff,axiom,
! [M2: num,N3: num] :
( ( ( numera1916890842035813515d_enat @ M2 )
= ( numera1916890842035813515d_enat @ N3 ) )
= ( M2 = N3 ) ) ).
% numeral_eq_iff
thf(fact_483_numeral__eq__iff,axiom,
! [M2: num,N3: num] :
( ( ( numeral_numeral_nat @ M2 )
= ( numeral_numeral_nat @ N3 ) )
= ( M2 = N3 ) ) ).
% numeral_eq_iff
thf(fact_484_numeral__eq__iff,axiom,
! [M2: num,N3: num] :
( ( ( numera4658534427948366547nnreal @ M2 )
= ( numera4658534427948366547nnreal @ N3 ) )
= ( M2 = N3 ) ) ).
% numeral_eq_iff
thf(fact_485_numeral__eq__iff,axiom,
! [M2: num,N3: num] :
( ( ( numeral_numeral_real @ M2 )
= ( numeral_numeral_real @ N3 ) )
= ( M2 = N3 ) ) ).
% numeral_eq_iff
thf(fact_486_numeral__eq__iff,axiom,
! [M2: num,N3: num] :
( ( ( numeral_numeral_int @ M2 )
= ( numeral_numeral_int @ N3 ) )
= ( M2 = N3 ) ) ).
% numeral_eq_iff
thf(fact_487_numeral__less__iff,axiom,
! [M2: num,N3: num] :
( ( ord_le72135733267957522d_enat @ ( numera1916890842035813515d_enat @ M2 ) @ ( numera1916890842035813515d_enat @ N3 ) )
= ( ord_less_num @ M2 @ N3 ) ) ).
% numeral_less_iff
thf(fact_488_numeral__less__iff,axiom,
! [M2: num,N3: num] :
( ( ord_less_nat @ ( numeral_numeral_nat @ M2 ) @ ( numeral_numeral_nat @ N3 ) )
= ( ord_less_num @ M2 @ N3 ) ) ).
% numeral_less_iff
thf(fact_489_numeral__less__iff,axiom,
! [M2: num,N3: num] :
( ( ord_le7381754540660121996nnreal @ ( numera4658534427948366547nnreal @ M2 ) @ ( numera4658534427948366547nnreal @ N3 ) )
= ( ord_less_num @ M2 @ N3 ) ) ).
% numeral_less_iff
thf(fact_490_numeral__less__iff,axiom,
! [M2: num,N3: num] :
( ( ord_less_real @ ( numeral_numeral_real @ M2 ) @ ( numeral_numeral_real @ N3 ) )
= ( ord_less_num @ M2 @ N3 ) ) ).
% numeral_less_iff
thf(fact_491_numeral__less__iff,axiom,
! [M2: num,N3: num] :
( ( ord_less_int @ ( numeral_numeral_int @ M2 ) @ ( numeral_numeral_int @ N3 ) )
= ( ord_less_num @ M2 @ N3 ) ) ).
% numeral_less_iff
thf(fact_492_eval__inverse__permutation__of__list_I2_J,axiom,
! [X: num,X7: num,Y: num,Xs: list_P3744719386663036955um_num] :
( ( X = X7 )
=> ( ( invers4287837041963334271st_num @ ( cons_P5351470506836119883um_num @ ( product_Pair_num_num @ Y @ X7 ) @ Xs ) @ X )
= Y ) ) ).
% eval_inverse_permutation_of_list(2)
thf(fact_493_eval__inverse__permutation__of__list_I2_J,axiom,
! [X: real,X7: real,Y: real,Xs: list_P8689742595348180415l_real] :
( ( X = X7 )
=> ( ( invers7512574637932550289t_real @ ( cons_P1861573166434266607l_real @ ( produc4511245868158468465l_real @ Y @ X7 ) @ Xs ) @ X )
= Y ) ) ).
% eval_inverse_permutation_of_list(2)
thf(fact_494_eval__inverse__permutation__of__list_I3_J,axiom,
! [X: num,X7: num,Y6: num,Xs: list_P3744719386663036955um_num] :
( ( X != X7 )
=> ( ( invers4287837041963334271st_num @ ( cons_P5351470506836119883um_num @ ( product_Pair_num_num @ Y6 @ X7 ) @ Xs ) @ X )
= ( invers4287837041963334271st_num @ Xs @ X ) ) ) ).
% eval_inverse_permutation_of_list(3)
thf(fact_495_eval__inverse__permutation__of__list_I3_J,axiom,
! [X: real,X7: real,Y6: real,Xs: list_P8689742595348180415l_real] :
( ( X != X7 )
=> ( ( invers7512574637932550289t_real @ ( cons_P1861573166434266607l_real @ ( produc4511245868158468465l_real @ Y6 @ X7 ) @ Xs ) @ X )
= ( invers7512574637932550289t_real @ Xs @ X ) ) ) ).
% eval_inverse_permutation_of_list(3)
thf(fact_496_zero__neq__numeral,axiom,
! [N3: num] :
( zero_z5237406670263579293d_enat
!= ( numera1916890842035813515d_enat @ N3 ) ) ).
% zero_neq_numeral
thf(fact_497_zero__neq__numeral,axiom,
! [N3: num] :
( zero_zero_nat
!= ( numeral_numeral_nat @ N3 ) ) ).
% zero_neq_numeral
thf(fact_498_zero__neq__numeral,axiom,
! [N3: num] :
( zero_z7100319975126383169nnreal
!= ( numera4658534427948366547nnreal @ N3 ) ) ).
% zero_neq_numeral
thf(fact_499_zero__neq__numeral,axiom,
! [N3: num] :
( zero_zero_real
!= ( numeral_numeral_real @ N3 ) ) ).
% zero_neq_numeral
thf(fact_500_zero__neq__numeral,axiom,
! [N3: num] :
( zero_zero_int
!= ( numeral_numeral_int @ N3 ) ) ).
% zero_neq_numeral
thf(fact_501_not__numeral__less__zero,axiom,
! [N3: num] :
~ ( ord_le72135733267957522d_enat @ ( numera1916890842035813515d_enat @ N3 ) @ zero_z5237406670263579293d_enat ) ).
% not_numeral_less_zero
thf(fact_502_not__numeral__less__zero,axiom,
! [N3: num] :
~ ( ord_less_nat @ ( numeral_numeral_nat @ N3 ) @ zero_zero_nat ) ).
% not_numeral_less_zero
thf(fact_503_not__numeral__less__zero,axiom,
! [N3: num] :
~ ( ord_le7381754540660121996nnreal @ ( numera4658534427948366547nnreal @ N3 ) @ zero_z7100319975126383169nnreal ) ).
% not_numeral_less_zero
thf(fact_504_not__numeral__less__zero,axiom,
! [N3: num] :
~ ( ord_less_real @ ( numeral_numeral_real @ N3 ) @ zero_zero_real ) ).
% not_numeral_less_zero
thf(fact_505_not__numeral__less__zero,axiom,
! [N3: num] :
~ ( ord_less_int @ ( numeral_numeral_int @ N3 ) @ zero_zero_int ) ).
% not_numeral_less_zero
thf(fact_506_zero__less__numeral,axiom,
! [N3: num] : ( ord_le72135733267957522d_enat @ zero_z5237406670263579293d_enat @ ( numera1916890842035813515d_enat @ N3 ) ) ).
% zero_less_numeral
thf(fact_507_zero__less__numeral,axiom,
! [N3: num] : ( ord_less_nat @ zero_zero_nat @ ( numeral_numeral_nat @ N3 ) ) ).
% zero_less_numeral
thf(fact_508_zero__less__numeral,axiom,
! [N3: num] : ( ord_le7381754540660121996nnreal @ zero_z7100319975126383169nnreal @ ( numera4658534427948366547nnreal @ N3 ) ) ).
% zero_less_numeral
thf(fact_509_zero__less__numeral,axiom,
! [N3: num] : ( ord_less_real @ zero_zero_real @ ( numeral_numeral_real @ N3 ) ) ).
% zero_less_numeral
thf(fact_510_zero__less__numeral,axiom,
! [N3: num] : ( ord_less_int @ zero_zero_int @ ( numeral_numeral_int @ N3 ) ) ).
% zero_less_numeral
thf(fact_511_not__numeral__less__one,axiom,
! [N3: num] :
~ ( ord_le72135733267957522d_enat @ ( numera1916890842035813515d_enat @ N3 ) @ one_on7984719198319812577d_enat ) ).
% not_numeral_less_one
thf(fact_512_not__numeral__less__one,axiom,
! [N3: num] :
~ ( ord_less_nat @ ( numeral_numeral_nat @ N3 ) @ one_one_nat ) ).
% not_numeral_less_one
thf(fact_513_not__numeral__less__one,axiom,
! [N3: num] :
~ ( ord_le7381754540660121996nnreal @ ( numera4658534427948366547nnreal @ N3 ) @ one_on2969667320475766781nnreal ) ).
% not_numeral_less_one
thf(fact_514_not__numeral__less__one,axiom,
! [N3: num] :
~ ( ord_less_real @ ( numeral_numeral_real @ N3 ) @ one_one_real ) ).
% not_numeral_less_one
thf(fact_515_not__numeral__less__one,axiom,
! [N3: num] :
~ ( ord_less_int @ ( numeral_numeral_int @ N3 ) @ one_one_int ) ).
% not_numeral_less_one
thf(fact_516_radical_Osimps_I1_J,axiom,
! [R2: nat > real > real,A: formal3361831859752904756s_real] :
( ( formal8005797870169972230l_real @ R2 @ zero_zero_nat @ A @ zero_zero_nat )
= one_one_real ) ).
% radical.simps(1)
thf(fact_517_inverse__permutation__of__list_Osimps_I2_J,axiom,
! [X: num,X7: num,Y: num,Xs: list_P3744719386663036955um_num] :
( ( ( X = X7 )
=> ( ( invers4287837041963334271st_num @ ( cons_P5351470506836119883um_num @ ( product_Pair_num_num @ Y @ X7 ) @ Xs ) @ X )
= Y ) )
& ( ( X != X7 )
=> ( ( invers4287837041963334271st_num @ ( cons_P5351470506836119883um_num @ ( product_Pair_num_num @ Y @ X7 ) @ Xs ) @ X )
= ( invers4287837041963334271st_num @ Xs @ X ) ) ) ) ).
% inverse_permutation_of_list.simps(2)
thf(fact_518_inverse__permutation__of__list_Osimps_I2_J,axiom,
! [X: real,X7: real,Y: real,Xs: list_P8689742595348180415l_real] :
( ( ( X = X7 )
=> ( ( invers7512574637932550289t_real @ ( cons_P1861573166434266607l_real @ ( produc4511245868158468465l_real @ Y @ X7 ) @ Xs ) @ X )
= Y ) )
& ( ( X != X7 )
=> ( ( invers7512574637932550289t_real @ ( cons_P1861573166434266607l_real @ ( produc4511245868158468465l_real @ Y @ X7 ) @ Xs ) @ X )
= ( invers7512574637932550289t_real @ Xs @ X ) ) ) ) ).
% inverse_permutation_of_list.simps(2)
thf(fact_519_permutation__of__list__Cons,axiom,
! [X: num,X7: num,Y: num,Xs: list_P3744719386663036955um_num] :
( ( ( X = X7 )
=> ( ( permut9062699409770291517st_num @ ( cons_P5351470506836119883um_num @ ( product_Pair_num_num @ X @ Y ) @ Xs ) @ X7 )
= Y ) )
& ( ( X != X7 )
=> ( ( permut9062699409770291517st_num @ ( cons_P5351470506836119883um_num @ ( product_Pair_num_num @ X @ Y ) @ Xs ) @ X7 )
= ( permut9062699409770291517st_num @ Xs @ X7 ) ) ) ) ).
% permutation_of_list_Cons
thf(fact_520_permutation__of__list__Cons,axiom,
! [X: real,X7: real,Y: real,Xs: list_P8689742595348180415l_real] :
( ( ( X = X7 )
=> ( ( permut5466000449812578383t_real @ ( cons_P1861573166434266607l_real @ ( produc4511245868158468465l_real @ X @ Y ) @ Xs ) @ X7 )
= Y ) )
& ( ( X != X7 )
=> ( ( permut5466000449812578383t_real @ ( cons_P1861573166434266607l_real @ ( produc4511245868158468465l_real @ X @ Y ) @ Xs ) @ X7 )
= ( permut5466000449812578383t_real @ Xs @ X7 ) ) ) ) ).
% permutation_of_list_Cons
thf(fact_521_radical_Osimps_I2_J,axiom,
! [R2: nat > real > real,A: formal3361831859752904756s_real,N3: nat] :
( ( formal8005797870169972230l_real @ R2 @ zero_zero_nat @ A @ ( suc @ N3 ) )
= zero_zero_real ) ).
% radical.simps(2)
thf(fact_522_insert__Nil,axiom,
! [X: a] :
( ( insert_a @ X @ nil_a )
= ( cons_a @ X @ nil_a ) ) ).
% insert_Nil
thf(fact_523_insert__Nil,axiom,
! [X: int] :
( ( insert_int @ X @ nil_int )
= ( cons_int @ X @ nil_int ) ) ).
% insert_Nil
thf(fact_524_Cons__replicate__eq,axiom,
! [X: a,Xs: list_a,N3: nat,Y: a] :
( ( ( cons_a @ X @ Xs )
= ( replicate_a @ N3 @ Y ) )
= ( ( X = Y )
& ( ord_less_nat @ zero_zero_nat @ N3 )
& ( Xs
= ( replicate_a @ ( minus_minus_nat @ N3 @ one_one_nat ) @ X ) ) ) ) ).
% Cons_replicate_eq
thf(fact_525_Cons__replicate__eq,axiom,
! [X: int,Xs: list_int,N3: nat,Y: int] :
( ( ( cons_int @ X @ Xs )
= ( replicate_int @ N3 @ Y ) )
= ( ( X = Y )
& ( ord_less_nat @ zero_zero_nat @ N3 )
& ( Xs
= ( replicate_int @ ( minus_minus_nat @ N3 @ one_one_nat ) @ X ) ) ) ) ).
% Cons_replicate_eq
thf(fact_526_arcosh__1,axiom,
( ( arcosh_real @ one_one_real )
= zero_zero_real ) ).
% arcosh_1
thf(fact_527_filter_Osimps_I2_J,axiom,
! [P: a > $o,X: a,Xs: list_a] :
( ( ( P @ X )
=> ( ( filter_a @ P @ ( cons_a @ X @ Xs ) )
= ( cons_a @ X @ ( filter_a @ P @ Xs ) ) ) )
& ( ~ ( P @ X )
=> ( ( filter_a @ P @ ( cons_a @ X @ Xs ) )
= ( filter_a @ P @ Xs ) ) ) ) ).
% filter.simps(2)
thf(fact_528_filter_Osimps_I2_J,axiom,
! [P: int > $o,X: int,Xs: list_int] :
( ( ( P @ X )
=> ( ( filter_int @ P @ ( cons_int @ X @ Xs ) )
= ( cons_int @ X @ ( filter_int @ P @ Xs ) ) ) )
& ( ~ ( P @ X )
=> ( ( filter_int @ P @ ( cons_int @ X @ Xs ) )
= ( filter_int @ P @ Xs ) ) ) ) ).
% filter.simps(2)
thf(fact_529_replicate__Suc,axiom,
! [N3: nat,X: a] :
( ( replicate_a @ ( suc @ N3 ) @ X )
= ( cons_a @ X @ ( replicate_a @ N3 @ X ) ) ) ).
% replicate_Suc
thf(fact_530_replicate__Suc,axiom,
! [N3: nat,X: int] :
( ( replicate_int @ ( suc @ N3 ) @ X )
= ( cons_int @ X @ ( replicate_int @ N3 @ X ) ) ) ).
% replicate_Suc
thf(fact_531_map__ran__filter,axiom,
! [F2: nat > ( nat > a ) > nat > a,A: nat,Ps2: list_P8194537880601713306_nat_a] :
( ( map_ra5122625054061645511_nat_a @ F2
@ ( filter5971802444331671245_nat_a
@ ^ [P5: produc7123000486447228170_nat_a] :
( ( produc3194919578927588176_nat_a @ P5 )
!= A )
@ Ps2 ) )
= ( filter5971802444331671245_nat_a
@ ^ [P5: produc7123000486447228170_nat_a] :
( ( produc3194919578927588176_nat_a @ P5 )
!= A )
@ ( map_ra5122625054061645511_nat_a @ F2 @ Ps2 ) ) ) ).
% map_ran_filter
thf(fact_532_map__ran__filter,axiom,
! [F2: real > real > real,A: real,Ps2: list_P8689742595348180415l_real] :
( ( map_ra6600816322303468477l_real @ F2
@ ( filter4198271037049850166l_real
@ ^ [P5: produc2422161461964618553l_real] :
( ( produc5828954698716094813l_real @ P5 )
!= A )
@ Ps2 ) )
= ( filter4198271037049850166l_real
@ ^ [P5: produc2422161461964618553l_real] :
( ( produc5828954698716094813l_real @ P5 )
!= A )
@ ( map_ra6600816322303468477l_real @ F2 @ Ps2 ) ) ) ).
% map_ran_filter
thf(fact_533_map__ran__filter,axiom,
! [F2: real > real > nat,A: real,Ps2: list_P8689742595348180415l_real] :
( ( map_ra134723660208699745al_nat @ F2
@ ( filter4198271037049850166l_real
@ ^ [P5: produc2422161461964618553l_real] :
( ( produc5828954698716094813l_real @ P5 )
!= A )
@ Ps2 ) )
= ( filter3407683625666204634al_nat
@ ^ [P5: produc3741383161447143261al_nat] :
( ( product_fst_real_nat @ P5 )
!= A )
@ ( map_ra134723660208699745al_nat @ F2 @ Ps2 ) ) ) ).
% map_ran_filter
thf(fact_534_map__ran__filter,axiom,
! [F2: num > num > num,A: num,Ps2: list_P3744719386663036955um_num] :
( ( map_ran_num_num_num @ F2
@ ( filter5211065455625245586um_num
@ ^ [P5: product_prod_num_num] :
( ( product_fst_num_num @ P5 )
!= A )
@ Ps2 ) )
= ( filter5211065455625245586um_num
@ ^ [P5: product_prod_num_num] :
( ( product_fst_num_num @ P5 )
!= A )
@ ( map_ran_num_num_num @ F2 @ Ps2 ) ) ) ).
% map_ran_filter
thf(fact_535_map__ran__filter,axiom,
! [F2: real > nat > real,A: real,Ps2: list_P6834414599653733731al_nat] :
( ( map_ra4790787123934576865t_real @ F2
@ ( filter3407683625666204634al_nat
@ ^ [P5: produc3741383161447143261al_nat] :
( ( product_fst_real_nat @ P5 )
!= A )
@ Ps2 ) )
= ( filter4198271037049850166l_real
@ ^ [P5: produc2422161461964618553l_real] :
( ( produc5828954698716094813l_real @ P5 )
!= A )
@ ( map_ra4790787123934576865t_real @ F2 @ Ps2 ) ) ) ).
% map_ran_filter
thf(fact_536_map__ran__filter,axiom,
! [F2: real > nat > nat,A: real,Ps2: list_P6834414599653733731al_nat] :
( ( map_ran_real_nat_nat @ F2
@ ( filter3407683625666204634al_nat
@ ^ [P5: produc3741383161447143261al_nat] :
( ( product_fst_real_nat @ P5 )
!= A )
@ Ps2 ) )
= ( filter3407683625666204634al_nat
@ ^ [P5: produc3741383161447143261al_nat] :
( ( product_fst_real_nat @ P5 )
!= A )
@ ( map_ran_real_nat_nat @ F2 @ Ps2 ) ) ) ).
% map_ran_filter
thf(fact_537_enat__ord__number_I2_J,axiom,
! [M2: num,N3: num] :
( ( ord_le72135733267957522d_enat @ ( numera1916890842035813515d_enat @ M2 ) @ ( numera1916890842035813515d_enat @ N3 ) )
= ( ord_less_nat @ ( numeral_numeral_nat @ M2 ) @ ( numeral_numeral_nat @ N3 ) ) ) ).
% enat_ord_number(2)
thf(fact_538_arsinh__0,axiom,
( ( arsinh_real @ zero_zero_real )
= zero_zero_real ) ).
% arsinh_0
thf(fact_539_artanh__0,axiom,
( ( artanh_real @ zero_zero_real )
= zero_zero_real ) ).
% artanh_0
thf(fact_540_one__less__numeral__iff,axiom,
! [N3: num] :
( ( ord_le72135733267957522d_enat @ one_on7984719198319812577d_enat @ ( numera1916890842035813515d_enat @ N3 ) )
= ( ord_less_num @ one @ N3 ) ) ).
% one_less_numeral_iff
thf(fact_541_one__less__numeral__iff,axiom,
! [N3: num] :
( ( ord_less_nat @ one_one_nat @ ( numeral_numeral_nat @ N3 ) )
= ( ord_less_num @ one @ N3 ) ) ).
% one_less_numeral_iff
thf(fact_542_one__less__numeral__iff,axiom,
! [N3: num] :
( ( ord_le7381754540660121996nnreal @ one_on2969667320475766781nnreal @ ( numera4658534427948366547nnreal @ N3 ) )
= ( ord_less_num @ one @ N3 ) ) ).
% one_less_numeral_iff
thf(fact_543_one__less__numeral__iff,axiom,
! [N3: num] :
( ( ord_less_real @ one_one_real @ ( numeral_numeral_real @ N3 ) )
= ( ord_less_num @ one @ N3 ) ) ).
% one_less_numeral_iff
thf(fact_544_one__less__numeral__iff,axiom,
! [N3: num] :
( ( ord_less_int @ one_one_int @ ( numeral_numeral_int @ N3 ) )
= ( ord_less_num @ one @ N3 ) ) ).
% one_less_numeral_iff
thf(fact_545_ln__one,axiom,
( ( ln_ln_real @ one_one_real )
= zero_zero_real ) ).
% ln_one
thf(fact_546_i0__less,axiom,
! [N3: extended_enat] :
( ( ord_le72135733267957522d_enat @ zero_z5237406670263579293d_enat @ N3 )
= ( N3 != zero_z5237406670263579293d_enat ) ) ).
% i0_less
thf(fact_547_one__eq__numeral__iff,axiom,
! [N3: num] :
( ( one_on7984719198319812577d_enat
= ( numera1916890842035813515d_enat @ N3 ) )
= ( one = N3 ) ) ).
% one_eq_numeral_iff
thf(fact_548_one__eq__numeral__iff,axiom,
! [N3: num] :
( ( one_one_nat
= ( numeral_numeral_nat @ N3 ) )
= ( one = N3 ) ) ).
% one_eq_numeral_iff
thf(fact_549_one__eq__numeral__iff,axiom,
! [N3: num] :
( ( one_on2969667320475766781nnreal
= ( numera4658534427948366547nnreal @ N3 ) )
= ( one = N3 ) ) ).
% one_eq_numeral_iff
thf(fact_550_one__eq__numeral__iff,axiom,
! [N3: num] :
( ( one_one_real
= ( numeral_numeral_real @ N3 ) )
= ( one = N3 ) ) ).
% one_eq_numeral_iff
thf(fact_551_one__eq__numeral__iff,axiom,
! [N3: num] :
( ( one_one_int
= ( numeral_numeral_int @ N3 ) )
= ( one = N3 ) ) ).
% one_eq_numeral_iff
thf(fact_552_numeral__eq__one__iff,axiom,
! [N3: num] :
( ( ( numera1916890842035813515d_enat @ N3 )
= one_on7984719198319812577d_enat )
= ( N3 = one ) ) ).
% numeral_eq_one_iff
thf(fact_553_numeral__eq__one__iff,axiom,
! [N3: num] :
( ( ( numeral_numeral_nat @ N3 )
= one_one_nat )
= ( N3 = one ) ) ).
% numeral_eq_one_iff
thf(fact_554_numeral__eq__one__iff,axiom,
! [N3: num] :
( ( ( numera4658534427948366547nnreal @ N3 )
= one_on2969667320475766781nnreal )
= ( N3 = one ) ) ).
% numeral_eq_one_iff
thf(fact_555_numeral__eq__one__iff,axiom,
! [N3: num] :
( ( ( numeral_numeral_real @ N3 )
= one_one_real )
= ( N3 = one ) ) ).
% numeral_eq_one_iff
thf(fact_556_numeral__eq__one__iff,axiom,
! [N3: num] :
( ( ( numeral_numeral_int @ N3 )
= one_one_int )
= ( N3 = one ) ) ).
% numeral_eq_one_iff
thf(fact_557_not__iless0,axiom,
! [N3: extended_enat] :
~ ( ord_le72135733267957522d_enat @ N3 @ zero_z5237406670263579293d_enat ) ).
% not_iless0
thf(fact_558_enat__less__induct,axiom,
! [P: extended_enat > $o,N3: extended_enat] :
( ! [N2: extended_enat] :
( ! [M4: extended_enat] :
( ( ord_le72135733267957522d_enat @ M4 @ N2 )
=> ( P @ M4 ) )
=> ( P @ N2 ) )
=> ( P @ N3 ) ) ).
% enat_less_induct
thf(fact_559_numeral__One,axiom,
( ( numera1916890842035813515d_enat @ one )
= one_on7984719198319812577d_enat ) ).
% numeral_One
thf(fact_560_numeral__One,axiom,
( ( numeral_numeral_nat @ one )
= one_one_nat ) ).
% numeral_One
thf(fact_561_numeral__One,axiom,
( ( numera4658534427948366547nnreal @ one )
= one_on2969667320475766781nnreal ) ).
% numeral_One
thf(fact_562_numeral__One,axiom,
( ( numeral_numeral_real @ one )
= one_one_real ) ).
% numeral_One
thf(fact_563_numeral__One,axiom,
( ( numeral_numeral_int @ one )
= one_one_int ) ).
% numeral_One
thf(fact_564_numerals_I1_J,axiom,
( ( numeral_numeral_nat @ one )
= one_one_nat ) ).
% numerals(1)
thf(fact_565_numeral__1__eq__Suc__0,axiom,
( ( numeral_numeral_nat @ one )
= ( suc @ zero_zero_nat ) ) ).
% numeral_1_eq_Suc_0
thf(fact_566_semiring__norm_I75_J,axiom,
! [M2: num] :
~ ( ord_less_num @ M2 @ one ) ).
% semiring_norm(75)
thf(fact_567_one__less__numeral,axiom,
! [N3: num] :
( ( ord_le7381754540660121996nnreal @ one_on2969667320475766781nnreal @ ( numera4658534427948366547nnreal @ N3 ) )
= ( ord_less_num @ one @ N3 ) ) ).
% one_less_numeral
thf(fact_568_diff__numeral__special_I2_J,axiom,
! [M2: num] :
( ( minus_minus_real @ ( numeral_numeral_real @ M2 ) @ one_one_real )
= ( neg_numeral_sub_real @ M2 @ one ) ) ).
% diff_numeral_special(2)
thf(fact_569_diff__numeral__special_I2_J,axiom,
! [M2: num] :
( ( minus_minus_int @ ( numeral_numeral_int @ M2 ) @ one_one_int )
= ( neg_numeral_sub_int @ M2 @ one ) ) ).
% diff_numeral_special(2)
thf(fact_570_diff__numeral__special_I1_J,axiom,
! [N3: num] :
( ( minus_minus_real @ one_one_real @ ( numeral_numeral_real @ N3 ) )
= ( neg_numeral_sub_real @ one @ N3 ) ) ).
% diff_numeral_special(1)
thf(fact_571_diff__numeral__special_I1_J,axiom,
! [N3: num] :
( ( minus_minus_int @ one_one_int @ ( numeral_numeral_int @ N3 ) )
= ( neg_numeral_sub_int @ one @ N3 ) ) ).
% diff_numeral_special(1)
thf(fact_572_dbl__inc__simps_I3_J,axiom,
( ( neg_nu8295874005876285629c_real @ one_one_real )
= ( numeral_numeral_real @ ( bit1 @ one ) ) ) ).
% dbl_inc_simps(3)
thf(fact_573_dbl__inc__simps_I3_J,axiom,
( ( neg_nu5851722552734809277nc_int @ one_one_int )
= ( numeral_numeral_int @ ( bit1 @ one ) ) ) ).
% dbl_inc_simps(3)
thf(fact_574_less__2__cases__iff,axiom,
! [N3: nat] :
( ( ord_less_nat @ N3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
= ( ( N3 = zero_zero_nat )
| ( N3
= ( suc @ zero_zero_nat ) ) ) ) ).
% less_2_cases_iff
thf(fact_575_semiring__norm_I78_J,axiom,
! [M2: num,N3: num] :
( ( ord_less_num @ ( bit0 @ M2 ) @ ( bit0 @ N3 ) )
= ( ord_less_num @ M2 @ N3 ) ) ).
% semiring_norm(78)
thf(fact_576_semiring__norm_I80_J,axiom,
! [M2: num,N3: num] :
( ( ord_less_num @ ( bit1 @ M2 ) @ ( bit1 @ N3 ) )
= ( ord_less_num @ M2 @ N3 ) ) ).
% semiring_norm(80)
thf(fact_577_sub__num__simps_I1_J,axiom,
( ( neg_numeral_sub_real @ one @ one )
= zero_zero_real ) ).
% sub_num_simps(1)
thf(fact_578_sub__num__simps_I1_J,axiom,
( ( neg_numeral_sub_int @ one @ one )
= zero_zero_int ) ).
% sub_num_simps(1)
thf(fact_579_diff__numeral__simps_I1_J,axiom,
! [M2: num,N3: num] :
( ( minus_minus_real @ ( numeral_numeral_real @ M2 ) @ ( numeral_numeral_real @ N3 ) )
= ( neg_numeral_sub_real @ M2 @ N3 ) ) ).
% diff_numeral_simps(1)
thf(fact_580_diff__numeral__simps_I1_J,axiom,
! [M2: num,N3: num] :
( ( minus_minus_int @ ( numeral_numeral_int @ M2 ) @ ( numeral_numeral_int @ N3 ) )
= ( neg_numeral_sub_int @ M2 @ N3 ) ) ).
% diff_numeral_simps(1)
thf(fact_581_semiring__norm_I76_J,axiom,
! [N3: num] : ( ord_less_num @ one @ ( bit0 @ N3 ) ) ).
% semiring_norm(76)
thf(fact_582_semiring__norm_I81_J,axiom,
! [M2: num,N3: num] :
( ( ord_less_num @ ( bit1 @ M2 ) @ ( bit0 @ N3 ) )
= ( ord_less_num @ M2 @ N3 ) ) ).
% semiring_norm(81)
thf(fact_583_semiring__norm_I77_J,axiom,
! [N3: num] : ( ord_less_num @ one @ ( bit1 @ N3 ) ) ).
% semiring_norm(77)
thf(fact_584_dbl__inc__simps_I5_J,axiom,
! [K: num] :
( ( neg_nu8295874005876285629c_real @ ( numeral_numeral_real @ K ) )
= ( numeral_numeral_real @ ( bit1 @ K ) ) ) ).
% dbl_inc_simps(5)
thf(fact_585_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_586_Suc__1,axiom,
( ( suc @ one_one_nat )
= ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ).
% Suc_1
thf(fact_587_sub__num__simps_I5_J,axiom,
! [K: num] :
( ( neg_numeral_sub_real @ ( bit1 @ K ) @ one )
= ( numeral_numeral_real @ ( bit0 @ K ) ) ) ).
% sub_num_simps(5)
thf(fact_588_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_589_diff__gr0__ennreal,axiom,
! [B: extend8495563244428889912nnreal,A: extend8495563244428889912nnreal] :
( ( ord_le7381754540660121996nnreal @ B @ A )
=> ( ord_le7381754540660121996nnreal @ zero_z7100319975126383169nnreal @ ( minus_8429688780609304081nnreal @ A @ B ) ) ) ).
% diff_gr0_ennreal
thf(fact_590_ennreal__zero__less__one,axiom,
ord_le7381754540660121996nnreal @ zero_z7100319975126383169nnreal @ one_on2969667320475766781nnreal ).
% ennreal_zero_less_one
thf(fact_591_num_Oexhaust,axiom,
! [Y: num] :
( ( Y != one )
=> ( ! [X23: num] :
( Y
!= ( bit0 @ X23 ) )
=> ~ ! [X32: num] :
( Y
!= ( bit1 @ X32 ) ) ) ) ).
% num.exhaust
thf(fact_592_eval__nat__numeral_I3_J,axiom,
! [N3: num] :
( ( numeral_numeral_nat @ ( bit1 @ N3 ) )
= ( suc @ ( numeral_numeral_nat @ ( bit0 @ N3 ) ) ) ) ).
% eval_nat_numeral(3)
thf(fact_593_neg__numeral__class_Osub__def,axiom,
( neg_numeral_sub_real
= ( ^ [K5: num,L2: num] : ( minus_minus_real @ ( numeral_numeral_real @ K5 ) @ ( numeral_numeral_real @ L2 ) ) ) ) ).
% neg_numeral_class.sub_def
thf(fact_594_neg__numeral__class_Osub__def,axiom,
( neg_numeral_sub_int
= ( ^ [K5: num,L2: num] : ( minus_minus_int @ ( numeral_numeral_int @ K5 ) @ ( numeral_numeral_int @ L2 ) ) ) ) ).
% neg_numeral_class.sub_def
thf(fact_595_sub__negative,axiom,
! [N3: num,M2: num] :
( ( ord_less_real @ ( neg_numeral_sub_real @ N3 @ M2 ) @ zero_zero_real )
= ( ord_less_num @ N3 @ M2 ) ) ).
% sub_negative
thf(fact_596_sub__negative,axiom,
! [N3: num,M2: num] :
( ( ord_less_int @ ( neg_numeral_sub_int @ N3 @ M2 ) @ zero_zero_int )
= ( ord_less_num @ N3 @ M2 ) ) ).
% sub_negative
thf(fact_597_sub__positive,axiom,
! [N3: num,M2: num] :
( ( ord_less_real @ zero_zero_real @ ( neg_numeral_sub_real @ N3 @ M2 ) )
= ( ord_less_num @ M2 @ N3 ) ) ).
% sub_positive
thf(fact_598_sub__positive,axiom,
! [N3: num,M2: num] :
( ( ord_less_int @ zero_zero_int @ ( neg_numeral_sub_int @ N3 @ M2 ) )
= ( ord_less_num @ M2 @ N3 ) ) ).
% sub_positive
thf(fact_599_numeral__3__eq__3,axiom,
( ( numeral_numeral_nat @ ( bit1 @ one ) )
= ( suc @ ( suc @ ( suc @ zero_zero_nat ) ) ) ) ).
% numeral_3_eq_3
thf(fact_600_numeral__2__eq__2,axiom,
( ( numeral_numeral_nat @ ( bit0 @ one ) )
= ( suc @ ( suc @ zero_zero_nat ) ) ) ).
% numeral_2_eq_2
thf(fact_601_less__2__cases,axiom,
! [N3: nat] :
( ( ord_less_nat @ N3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
=> ( ( N3 = zero_zero_nat )
| ( N3
= ( suc @ zero_zero_nat ) ) ) ) ).
% less_2_cases
thf(fact_602_pos2,axiom,
ord_less_nat @ zero_zero_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ).
% pos2
thf(fact_603_dbl__simps_I3_J,axiom,
( ( neg_numeral_dbl_real @ one_one_real )
= ( numeral_numeral_real @ ( bit0 @ one ) ) ) ).
% dbl_simps(3)
thf(fact_604_dbl__simps_I3_J,axiom,
( ( neg_numeral_dbl_int @ one_one_int )
= ( numeral_numeral_int @ ( bit0 @ one ) ) ) ).
% dbl_simps(3)
thf(fact_605_xor__num_Ocases,axiom,
! [X: product_prod_num_num] :
( ( X
!= ( product_Pair_num_num @ one @ one ) )
=> ( ! [N2: num] :
( X
!= ( product_Pair_num_num @ one @ ( bit0 @ N2 ) ) )
=> ( ! [N2: num] :
( X
!= ( product_Pair_num_num @ one @ ( bit1 @ N2 ) ) )
=> ( ! [M3: num] :
( X
!= ( product_Pair_num_num @ ( bit0 @ M3 ) @ one ) )
=> ( ! [M3: num,N2: num] :
( X
!= ( product_Pair_num_num @ ( bit0 @ M3 ) @ ( bit0 @ N2 ) ) )
=> ( ! [M3: num,N2: num] :
( X
!= ( product_Pair_num_num @ ( bit0 @ M3 ) @ ( bit1 @ N2 ) ) )
=> ( ! [M3: num] :
( X
!= ( product_Pair_num_num @ ( bit1 @ M3 ) @ one ) )
=> ( ! [M3: num,N2: num] :
( X
!= ( product_Pair_num_num @ ( bit1 @ M3 ) @ ( bit0 @ N2 ) ) )
=> ~ ! [M3: num,N2: num] :
( X
!= ( product_Pair_num_num @ ( bit1 @ M3 ) @ ( bit1 @ N2 ) ) ) ) ) ) ) ) ) ) ) ).
% xor_num.cases
thf(fact_606_ln3__gt__1,axiom,
ord_less_real @ one_one_real @ ( ln_ln_real @ ( numeral_numeral_real @ ( bit1 @ one ) ) ) ).
% ln3_gt_1
thf(fact_607_ln__2__less__1,axiom,
ord_less_real @ ( ln_ln_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) @ one_one_real ).
% ln_2_less_1
thf(fact_608_ln__inj__iff,axiom,
! [X: real,Y: real] :
( ( ord_less_real @ zero_zero_real @ X )
=> ( ( ord_less_real @ zero_zero_real @ Y )
=> ( ( ( ln_ln_real @ X )
= ( ln_ln_real @ Y ) )
= ( X = Y ) ) ) ) ).
% ln_inj_iff
thf(fact_609_ln__less__cancel__iff,axiom,
! [X: real,Y: real] :
( ( ord_less_real @ zero_zero_real @ X )
=> ( ( ord_less_real @ zero_zero_real @ Y )
=> ( ( ord_less_real @ ( ln_ln_real @ X ) @ ( ln_ln_real @ Y ) )
= ( ord_less_real @ X @ Y ) ) ) ) ).
% ln_less_cancel_iff
thf(fact_610_dbl__simps_I2_J,axiom,
( ( neg_numeral_dbl_real @ zero_zero_real )
= zero_zero_real ) ).
% dbl_simps(2)
thf(fact_611_dbl__simps_I2_J,axiom,
( ( neg_numeral_dbl_int @ zero_zero_int )
= zero_zero_int ) ).
% dbl_simps(2)
thf(fact_612_dbl__dec__simps_I3_J,axiom,
( ( neg_nu6075765906172075777c_real @ one_one_real )
= one_one_real ) ).
% dbl_dec_simps(3)
thf(fact_613_dbl__dec__simps_I3_J,axiom,
( ( neg_nu3811975205180677377ec_int @ one_one_int )
= one_one_int ) ).
% dbl_dec_simps(3)
thf(fact_614_ln__eq__zero__iff,axiom,
! [X: real] :
( ( ord_less_real @ zero_zero_real @ X )
=> ( ( ( ln_ln_real @ X )
= zero_zero_real )
= ( X = one_one_real ) ) ) ).
% ln_eq_zero_iff
thf(fact_615_ln__gt__zero__iff,axiom,
! [X: real] :
( ( ord_less_real @ zero_zero_real @ X )
=> ( ( ord_less_real @ zero_zero_real @ ( ln_ln_real @ X ) )
= ( ord_less_real @ one_one_real @ X ) ) ) ).
% ln_gt_zero_iff
thf(fact_616_ln__less__zero__iff,axiom,
! [X: real] :
( ( ord_less_real @ zero_zero_real @ X )
=> ( ( ord_less_real @ ( ln_ln_real @ X ) @ zero_zero_real )
= ( ord_less_real @ X @ one_one_real ) ) ) ).
% ln_less_zero_iff
thf(fact_617_dbl__simps_I5_J,axiom,
! [K: num] :
( ( neg_numeral_dbl_real @ ( numeral_numeral_real @ K ) )
= ( numeral_numeral_real @ ( bit0 @ K ) ) ) ).
% dbl_simps(5)
thf(fact_618_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_619_ln__eq__minus__one,axiom,
! [X: real] :
( ( ord_less_real @ zero_zero_real @ X )
=> ( ( ( ln_ln_real @ X )
= ( minus_minus_real @ X @ one_one_real ) )
=> ( X = one_one_real ) ) ) ).
% ln_eq_minus_one
thf(fact_620_ln__gt__zero,axiom,
! [X: real] :
( ( ord_less_real @ one_one_real @ X )
=> ( ord_less_real @ zero_zero_real @ ( ln_ln_real @ X ) ) ) ).
% ln_gt_zero
thf(fact_621_ln__less__self,axiom,
! [X: real] :
( ( ord_less_real @ zero_zero_real @ X )
=> ( ord_less_real @ ( ln_ln_real @ X ) @ X ) ) ).
% ln_less_self
thf(fact_622_ln__less__zero,axiom,
! [X: real] :
( ( ord_less_real @ zero_zero_real @ X )
=> ( ( ord_less_real @ X @ one_one_real )
=> ( ord_less_real @ ( ln_ln_real @ X ) @ zero_zero_real ) ) ) ).
% ln_less_zero
thf(fact_623_ln__gt__zero__imp__gt__one,axiom,
! [X: real] :
( ( ord_less_real @ zero_zero_real @ ( ln_ln_real @ X ) )
=> ( ( ord_less_real @ zero_zero_real @ X )
=> ( ord_less_real @ one_one_real @ X ) ) ) ).
% ln_gt_zero_imp_gt_one
thf(fact_624_ln__272__gt__1,axiom,
ord_less_real @ one_one_real @ ( ln_ln_real @ ( divide_divide_real @ ( numeral_numeral_real @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit1 @ ( bit0 @ ( bit0 @ ( bit0 @ one ) ) ) ) ) ) ) ) ) @ ( numeral_numeral_real @ ( bit0 @ ( bit0 @ ( bit1 @ ( bit0 @ ( bit0 @ ( bit1 @ one ) ) ) ) ) ) ) ) ) ).
% ln_272_gt_1
thf(fact_625_minus__one__less,axiom,
! [X: real] : ( ord_less_real @ ( minus_minus_real @ X @ one_one_real ) @ X ) ).
% minus_one_less
thf(fact_626_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_627_dbl__dec__simps_I4_J,axiom,
( ( neg_nu6075765906172075777c_real @ ( uminus_uminus_real @ one_one_real ) )
= ( uminus_uminus_real @ ( numeral_numeral_real @ ( bit1 @ one ) ) ) ) ).
% dbl_dec_simps(4)
thf(fact_628_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_629_dbl__simps_I4_J,axiom,
( ( neg_numeral_dbl_real @ ( uminus_uminus_real @ one_one_real ) )
= ( uminus_uminus_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ).
% dbl_simps(4)
thf(fact_630_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_631_sub__num__simps_I3_J,axiom,
! [L: num] :
( ( neg_numeral_sub_real @ one @ ( bit1 @ L ) )
= ( uminus_uminus_real @ ( numeral_numeral_real @ ( bit0 @ L ) ) ) ) ).
% sub_num_simps(3)
thf(fact_632_add_Oinverse__inverse,axiom,
! [A: int] :
( ( uminus_uminus_int @ ( uminus_uminus_int @ A ) )
= A ) ).
% add.inverse_inverse
thf(fact_633_add_Oinverse__inverse,axiom,
! [A: real] :
( ( uminus_uminus_real @ ( uminus_uminus_real @ A ) )
= A ) ).
% add.inverse_inverse
thf(fact_634_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_635_neg__equal__iff__equal,axiom,
! [A: real,B: real] :
( ( ( uminus_uminus_real @ A )
= ( uminus_uminus_real @ B ) )
= ( A = B ) ) ).
% neg_equal_iff_equal
thf(fact_636_add_Oinverse__neutral,axiom,
( ( uminus_uminus_int @ zero_zero_int )
= zero_zero_int ) ).
% add.inverse_neutral
thf(fact_637_add_Oinverse__neutral,axiom,
( ( uminus_uminus_real @ zero_zero_real )
= zero_zero_real ) ).
% add.inverse_neutral
thf(fact_638_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_639_neg__0__equal__iff__equal,axiom,
! [A: real] :
( ( zero_zero_real
= ( uminus_uminus_real @ A ) )
= ( zero_zero_real = A ) ) ).
% neg_0_equal_iff_equal
thf(fact_640_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_641_neg__equal__0__iff__equal,axiom,
! [A: real] :
( ( ( uminus_uminus_real @ A )
= zero_zero_real )
= ( A = zero_zero_real ) ) ).
% neg_equal_0_iff_equal
thf(fact_642_equal__neg__zero,axiom,
! [A: int] :
( ( A
= ( uminus_uminus_int @ A ) )
= ( A = zero_zero_int ) ) ).
% equal_neg_zero
thf(fact_643_equal__neg__zero,axiom,
! [A: real] :
( ( A
= ( uminus_uminus_real @ A ) )
= ( A = zero_zero_real ) ) ).
% equal_neg_zero
thf(fact_644_neg__equal__zero,axiom,
! [A: int] :
( ( ( uminus_uminus_int @ A )
= A )
= ( A = zero_zero_int ) ) ).
% neg_equal_zero
thf(fact_645_neg__equal__zero,axiom,
! [A: real] :
( ( ( uminus_uminus_real @ A )
= A )
= ( A = zero_zero_real ) ) ).
% neg_equal_zero
thf(fact_646_div__0,axiom,
! [A: real] :
( ( divide_divide_real @ zero_zero_real @ A )
= zero_zero_real ) ).
% div_0
thf(fact_647_div__0,axiom,
! [A: nat] :
( ( divide_divide_nat @ zero_zero_nat @ A )
= zero_zero_nat ) ).
% div_0
thf(fact_648_div__0,axiom,
! [A: int] :
( ( divide_divide_int @ zero_zero_int @ A )
= zero_zero_int ) ).
% div_0
thf(fact_649_div__by__0,axiom,
! [A: real] :
( ( divide_divide_real @ A @ zero_zero_real )
= zero_zero_real ) ).
% div_by_0
thf(fact_650_div__by__0,axiom,
! [A: nat] :
( ( divide_divide_nat @ A @ zero_zero_nat )
= zero_zero_nat ) ).
% div_by_0
thf(fact_651_div__by__0,axiom,
! [A: int] :
( ( divide_divide_int @ A @ zero_zero_int )
= zero_zero_int ) ).
% div_by_0
thf(fact_652_bits__div__0,axiom,
! [A: nat] :
( ( divide_divide_nat @ zero_zero_nat @ A )
= zero_zero_nat ) ).
% bits_div_0
thf(fact_653_bits__div__0,axiom,
! [A: int] :
( ( divide_divide_int @ zero_zero_int @ A )
= zero_zero_int ) ).
% bits_div_0
thf(fact_654_bits__div__by__0,axiom,
! [A: nat] :
( ( divide_divide_nat @ A @ zero_zero_nat )
= zero_zero_nat ) ).
% bits_div_by_0
thf(fact_655_bits__div__by__0,axiom,
! [A: int] :
( ( divide_divide_int @ A @ zero_zero_int )
= zero_zero_int ) ).
% bits_div_by_0
thf(fact_656_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_657_neg__less__iff__less,axiom,
! [B: real,A: real] :
( ( ord_less_real @ ( uminus_uminus_real @ B ) @ ( uminus_uminus_real @ A ) )
= ( ord_less_real @ A @ B ) ) ).
% neg_less_iff_less
thf(fact_658_neg__numeral__eq__iff,axiom,
! [M2: num,N3: num] :
( ( ( uminus_uminus_int @ ( numeral_numeral_int @ M2 ) )
= ( uminus_uminus_int @ ( numeral_numeral_int @ N3 ) ) )
= ( M2 = N3 ) ) ).
% neg_numeral_eq_iff
thf(fact_659_neg__numeral__eq__iff,axiom,
! [M2: num,N3: num] :
( ( ( uminus_uminus_real @ ( numeral_numeral_real @ M2 ) )
= ( uminus_uminus_real @ ( numeral_numeral_real @ N3 ) ) )
= ( M2 = N3 ) ) ).
% neg_numeral_eq_iff
thf(fact_660_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_661_minus__diff__eq,axiom,
! [A: real,B: real] :
( ( uminus_uminus_real @ ( minus_minus_real @ A @ B ) )
= ( minus_minus_real @ B @ A ) ) ).
% minus_diff_eq
thf(fact_662_div__by__1,axiom,
! [A: real] :
( ( divide_divide_real @ A @ one_one_real )
= A ) ).
% div_by_1
thf(fact_663_div__by__1,axiom,
! [A: nat] :
( ( divide_divide_nat @ A @ one_one_nat )
= A ) ).
% div_by_1
thf(fact_664_div__by__1,axiom,
! [A: int] :
( ( divide_divide_int @ A @ one_one_int )
= A ) ).
% div_by_1
thf(fact_665_bits__div__by__1,axiom,
! [A: nat] :
( ( divide_divide_nat @ A @ one_one_nat )
= A ) ).
% bits_div_by_1
thf(fact_666_bits__div__by__1,axiom,
! [A: int] :
( ( divide_divide_int @ A @ one_one_int )
= A ) ).
% bits_div_by_1
thf(fact_667_uminus__Pair,axiom,
! [A: int,B: int] :
( ( uminus7465043315890402864nt_int @ ( product_Pair_int_int @ A @ B ) )
= ( product_Pair_int_int @ ( uminus_uminus_int @ A ) @ ( uminus_uminus_int @ B ) ) ) ).
% uminus_Pair
thf(fact_668_uminus__Pair,axiom,
! [A: int,B: real] :
( ( uminus6458443906363700784t_real @ ( produc801115645435158769t_real @ A @ B ) )
= ( produc801115645435158769t_real @ ( uminus_uminus_int @ A ) @ ( uminus_uminus_real @ B ) ) ) ).
% uminus_Pair
thf(fact_669_uminus__Pair,axiom,
! [A: real,B: int] :
( ( uminus5341995657539377840al_int @ ( produc3179012173361985393al_int @ A @ B ) )
= ( produc3179012173361985393al_int @ ( uminus_uminus_real @ A ) @ ( uminus_uminus_int @ B ) ) ) ).
% uminus_Pair
thf(fact_670_uminus__Pair,axiom,
! [A: real,B: real] :
( ( uminus2141826702334040752l_real @ ( produc4511245868158468465l_real @ A @ B ) )
= ( produc4511245868158468465l_real @ ( uminus_uminus_real @ A ) @ ( uminus_uminus_real @ B ) ) ) ).
% uminus_Pair
thf(fact_671_fst__uminus,axiom,
! [X: produc2422161461964618553l_real] :
( ( produc5828954698716094813l_real @ ( uminus2141826702334040752l_real @ X ) )
= ( uminus_uminus_real @ ( produc5828954698716094813l_real @ X ) ) ) ).
% fst_uminus
thf(fact_672_snd__uminus,axiom,
! [X: produc2422161461964618553l_real] :
( ( produc3484788084999411615l_real @ ( uminus2141826702334040752l_real @ X ) )
= ( uminus_uminus_real @ ( produc3484788084999411615l_real @ X ) ) ) ).
% snd_uminus
thf(fact_673_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_674_neg__less__0__iff__less,axiom,
! [A: real] :
( ( ord_less_real @ ( uminus_uminus_real @ A ) @ zero_zero_real )
= ( ord_less_real @ zero_zero_real @ A ) ) ).
% neg_less_0_iff_less
thf(fact_675_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_676_neg__0__less__iff__less,axiom,
! [A: real] :
( ( ord_less_real @ zero_zero_real @ ( uminus_uminus_real @ A ) )
= ( ord_less_real @ A @ zero_zero_real ) ) ).
% neg_0_less_iff_less
thf(fact_677_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_678_neg__less__pos,axiom,
! [A: real] :
( ( ord_less_real @ ( uminus_uminus_real @ A ) @ A )
= ( ord_less_real @ zero_zero_real @ A ) ) ).
% neg_less_pos
thf(fact_679_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_680_less__neg__neg,axiom,
! [A: real] :
( ( ord_less_real @ A @ ( uminus_uminus_real @ A ) )
= ( ord_less_real @ A @ zero_zero_real ) ) ).
% less_neg_neg
thf(fact_681_diff__0,axiom,
! [A: int] :
( ( minus_minus_int @ zero_zero_int @ A )
= ( uminus_uminus_int @ A ) ) ).
% diff_0
thf(fact_682_diff__0,axiom,
! [A: real] :
( ( minus_minus_real @ zero_zero_real @ A )
= ( uminus_uminus_real @ A ) ) ).
% diff_0
thf(fact_683_div__self,axiom,
! [A: real] :
( ( A != zero_zero_real )
=> ( ( divide_divide_real @ A @ A )
= one_one_real ) ) ).
% div_self
thf(fact_684_div__self,axiom,
! [A: nat] :
( ( A != zero_zero_nat )
=> ( ( divide_divide_nat @ A @ A )
= one_one_nat ) ) ).
% div_self
thf(fact_685_div__self,axiom,
! [A: int] :
( ( A != zero_zero_int )
=> ( ( divide_divide_int @ A @ A )
= one_one_int ) ) ).
% div_self
thf(fact_686_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_687_dbl__simps_I1_J,axiom,
! [K: num] :
( ( neg_numeral_dbl_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ K ) ) )
= ( uminus_uminus_real @ ( neg_numeral_dbl_real @ ( numeral_numeral_real @ K ) ) ) ) ).
% dbl_simps(1)
thf(fact_688_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_689_dbl__inc__simps_I4_J,axiom,
( ( neg_nu8295874005876285629c_real @ ( uminus_uminus_real @ one_one_real ) )
= ( uminus_uminus_real @ one_one_real ) ) ).
% dbl_inc_simps(4)
thf(fact_690_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_691_diff__numeral__special_I12_J,axiom,
( ( minus_minus_real @ ( uminus_uminus_real @ one_one_real ) @ ( uminus_uminus_real @ one_one_real ) )
= zero_zero_real ) ).
% diff_numeral_special(12)
thf(fact_692_neg__one__eq__numeral__iff,axiom,
! [N3: num] :
( ( ( uminus_uminus_int @ one_one_int )
= ( uminus_uminus_int @ ( numeral_numeral_int @ N3 ) ) )
= ( N3 = one ) ) ).
% neg_one_eq_numeral_iff
thf(fact_693_neg__one__eq__numeral__iff,axiom,
! [N3: num] :
( ( ( uminus_uminus_real @ one_one_real )
= ( uminus_uminus_real @ ( numeral_numeral_real @ N3 ) ) )
= ( N3 = one ) ) ).
% neg_one_eq_numeral_iff
thf(fact_694_numeral__eq__neg__one__iff,axiom,
! [N3: num] :
( ( ( uminus_uminus_int @ ( numeral_numeral_int @ N3 ) )
= ( uminus_uminus_int @ one_one_int ) )
= ( N3 = one ) ) ).
% numeral_eq_neg_one_iff
thf(fact_695_numeral__eq__neg__one__iff,axiom,
! [N3: num] :
( ( ( uminus_uminus_real @ ( numeral_numeral_real @ N3 ) )
= ( uminus_uminus_real @ one_one_real ) )
= ( N3 = one ) ) ).
% numeral_eq_neg_one_iff
thf(fact_696_diff__numeral__simps_I4_J,axiom,
! [M2: num,N3: num] :
( ( minus_minus_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ M2 ) ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ N3 ) ) )
= ( neg_numeral_sub_int @ N3 @ M2 ) ) ).
% diff_numeral_simps(4)
thf(fact_697_diff__numeral__simps_I4_J,axiom,
! [M2: num,N3: num] :
( ( minus_minus_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ M2 ) ) @ ( uminus_uminus_real @ ( numeral_numeral_real @ N3 ) ) )
= ( neg_numeral_sub_real @ N3 @ M2 ) ) ).
% diff_numeral_simps(4)
thf(fact_698_neg__numeral__less__iff,axiom,
! [M2: num,N3: num] :
( ( ord_less_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ M2 ) ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ N3 ) ) )
= ( ord_less_num @ N3 @ M2 ) ) ).
% neg_numeral_less_iff
thf(fact_699_neg__numeral__less__iff,axiom,
! [M2: num,N3: num] :
( ( ord_less_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ M2 ) ) @ ( uminus_uminus_real @ ( numeral_numeral_real @ N3 ) ) )
= ( ord_less_num @ N3 @ M2 ) ) ).
% neg_numeral_less_iff
thf(fact_700_dbl__dec__simps_I2_J,axiom,
( ( neg_nu3811975205180677377ec_int @ zero_zero_int )
= ( uminus_uminus_int @ one_one_int ) ) ).
% dbl_dec_simps(2)
thf(fact_701_dbl__dec__simps_I2_J,axiom,
( ( neg_nu6075765906172075777c_real @ zero_zero_real )
= ( uminus_uminus_real @ one_one_real ) ) ).
% dbl_dec_simps(2)
thf(fact_702_neg__numeral__less__neg__one__iff,axiom,
! [M2: num] :
( ( ord_less_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ M2 ) ) @ ( uminus_uminus_int @ one_one_int ) )
= ( M2 != one ) ) ).
% neg_numeral_less_neg_one_iff
thf(fact_703_neg__numeral__less__neg__one__iff,axiom,
! [M2: num] :
( ( ord_less_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ M2 ) ) @ ( uminus_uminus_real @ one_one_real ) )
= ( M2 != one ) ) ).
% neg_numeral_less_neg_one_iff
thf(fact_704_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_705_dbl__inc__simps_I1_J,axiom,
! [K: num] :
( ( neg_nu8295874005876285629c_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ K ) ) )
= ( uminus_uminus_real @ ( neg_nu6075765906172075777c_real @ ( numeral_numeral_real @ K ) ) ) ) ).
% dbl_inc_simps(1)
thf(fact_706_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_707_dbl__dec__simps_I1_J,axiom,
! [K: num] :
( ( neg_nu6075765906172075777c_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ K ) ) )
= ( uminus_uminus_real @ ( neg_nu8295874005876285629c_real @ ( numeral_numeral_real @ K ) ) ) ) ).
% dbl_dec_simps(1)
thf(fact_708_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_709_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_710_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_711_diff__numeral__special_I10_J,axiom,
( ( minus_minus_real @ ( uminus_uminus_real @ one_one_real ) @ one_one_real )
= ( uminus_uminus_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ).
% diff_numeral_special(10)
thf(fact_712_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_713_diff__numeral__special_I11_J,axiom,
( ( minus_minus_real @ one_one_real @ ( uminus_uminus_real @ one_one_real ) )
= ( numeral_numeral_real @ ( bit0 @ one ) ) ) ).
% diff_numeral_special(11)
thf(fact_714_minus__sub__one__diff__one,axiom,
! [M2: num] :
( ( minus_minus_int @ ( uminus_uminus_int @ ( neg_numeral_sub_int @ M2 @ one ) ) @ one_one_int )
= ( uminus_uminus_int @ ( numeral_numeral_int @ M2 ) ) ) ).
% minus_sub_one_diff_one
thf(fact_715_minus__sub__one__diff__one,axiom,
! [M2: num] :
( ( minus_minus_real @ ( uminus_uminus_real @ ( neg_numeral_sub_real @ M2 @ one ) ) @ one_one_real )
= ( uminus_uminus_real @ ( numeral_numeral_real @ M2 ) ) ) ).
% minus_sub_one_diff_one
thf(fact_716_diff__numeral__special_I7_J,axiom,
! [N3: num] :
( ( minus_minus_int @ ( uminus_uminus_int @ one_one_int ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ N3 ) ) )
= ( neg_numeral_sub_int @ N3 @ one ) ) ).
% diff_numeral_special(7)
thf(fact_717_diff__numeral__special_I7_J,axiom,
! [N3: num] :
( ( minus_minus_real @ ( uminus_uminus_real @ one_one_real ) @ ( uminus_uminus_real @ ( numeral_numeral_real @ N3 ) ) )
= ( neg_numeral_sub_real @ N3 @ one ) ) ).
% diff_numeral_special(7)
thf(fact_718_diff__numeral__special_I8_J,axiom,
! [M2: num] :
( ( minus_minus_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ M2 ) ) @ ( uminus_uminus_int @ one_one_int ) )
= ( neg_numeral_sub_int @ one @ M2 ) ) ).
% diff_numeral_special(8)
thf(fact_719_diff__numeral__special_I8_J,axiom,
! [M2: num] :
( ( minus_minus_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ M2 ) ) @ ( uminus_uminus_real @ one_one_real ) )
= ( neg_numeral_sub_real @ one @ M2 ) ) ).
% diff_numeral_special(8)
thf(fact_720_equation__minus__iff,axiom,
! [A: int,B: int] :
( ( A
= ( uminus_uminus_int @ B ) )
= ( B
= ( uminus_uminus_int @ A ) ) ) ).
% equation_minus_iff
thf(fact_721_equation__minus__iff,axiom,
! [A: real,B: real] :
( ( A
= ( uminus_uminus_real @ B ) )
= ( B
= ( uminus_uminus_real @ A ) ) ) ).
% equation_minus_iff
thf(fact_722_minus__equation__iff,axiom,
! [A: int,B: int] :
( ( ( uminus_uminus_int @ A )
= B )
= ( ( uminus_uminus_int @ B )
= A ) ) ).
% minus_equation_iff
thf(fact_723_minus__equation__iff,axiom,
! [A: real,B: real] :
( ( ( uminus_uminus_real @ A )
= B )
= ( ( uminus_uminus_real @ B )
= A ) ) ).
% minus_equation_iff
thf(fact_724_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_725_less__minus__iff,axiom,
! [A: real,B: real] :
( ( ord_less_real @ A @ ( uminus_uminus_real @ B ) )
= ( ord_less_real @ B @ ( uminus_uminus_real @ A ) ) ) ).
% less_minus_iff
thf(fact_726_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_727_minus__less__iff,axiom,
! [A: real,B: real] :
( ( ord_less_real @ ( uminus_uminus_real @ A ) @ B )
= ( ord_less_real @ ( uminus_uminus_real @ B ) @ A ) ) ).
% minus_less_iff
thf(fact_728_one__neq__neg__one,axiom,
( one_one_int
!= ( uminus_uminus_int @ one_one_int ) ) ).
% one_neq_neg_one
thf(fact_729_one__neq__neg__one,axiom,
( one_one_real
!= ( uminus_uminus_real @ one_one_real ) ) ).
% one_neq_neg_one
thf(fact_730_neg__numeral__neq__numeral,axiom,
! [M2: num,N3: num] :
( ( uminus_uminus_int @ ( numeral_numeral_int @ M2 ) )
!= ( numeral_numeral_int @ N3 ) ) ).
% neg_numeral_neq_numeral
thf(fact_731_neg__numeral__neq__numeral,axiom,
! [M2: num,N3: num] :
( ( uminus_uminus_real @ ( numeral_numeral_real @ M2 ) )
!= ( numeral_numeral_real @ N3 ) ) ).
% neg_numeral_neq_numeral
thf(fact_732_numeral__neq__neg__numeral,axiom,
! [M2: num,N3: num] :
( ( numeral_numeral_int @ M2 )
!= ( uminus_uminus_int @ ( numeral_numeral_int @ N3 ) ) ) ).
% numeral_neq_neg_numeral
thf(fact_733_numeral__neq__neg__numeral,axiom,
! [M2: num,N3: num] :
( ( numeral_numeral_real @ M2 )
!= ( uminus_uminus_real @ ( numeral_numeral_real @ N3 ) ) ) ).
% numeral_neq_neg_numeral
thf(fact_734_minus__diff__minus,axiom,
! [A: int,B: int] :
( ( minus_minus_int @ ( uminus_uminus_int @ A ) @ ( uminus_uminus_int @ B ) )
= ( uminus_uminus_int @ ( minus_minus_int @ A @ B ) ) ) ).
% minus_diff_minus
thf(fact_735_minus__diff__minus,axiom,
! [A: real,B: real] :
( ( minus_minus_real @ ( uminus_uminus_real @ A ) @ ( uminus_uminus_real @ B ) )
= ( uminus_uminus_real @ ( minus_minus_real @ A @ B ) ) ) ).
% minus_diff_minus
thf(fact_736_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_737_minus__diff__commute,axiom,
! [B: real,A: real] :
( ( minus_minus_real @ ( uminus_uminus_real @ B ) @ A )
= ( minus_minus_real @ ( uminus_uminus_real @ A ) @ B ) ) ).
% minus_diff_commute
thf(fact_738_uminus__prod__def,axiom,
( uminus7465043315890402864nt_int
= ( ^ [X4: product_prod_int_int] : ( product_Pair_int_int @ ( uminus_uminus_int @ ( product_fst_int_int @ X4 ) ) @ ( uminus_uminus_int @ ( product_snd_int_int @ X4 ) ) ) ) ) ).
% uminus_prod_def
thf(fact_739_uminus__prod__def,axiom,
( uminus6458443906363700784t_real
= ( ^ [X4: produc679980390762269497t_real] : ( produc801115645435158769t_real @ ( uminus_uminus_int @ ( product_fst_int_real @ X4 ) ) @ ( uminus_uminus_real @ ( product_snd_int_real @ X4 ) ) ) ) ) ).
% uminus_prod_def
thf(fact_740_uminus__prod__def,axiom,
( uminus5341995657539377840al_int
= ( ^ [X4: produc8786904178792722361al_int] : ( produc3179012173361985393al_int @ ( uminus_uminus_real @ ( product_fst_real_int @ X4 ) ) @ ( uminus_uminus_int @ ( product_snd_real_int @ X4 ) ) ) ) ) ).
% uminus_prod_def
thf(fact_741_uminus__prod__def,axiom,
( uminus2141826702334040752l_real
= ( ^ [X4: produc2422161461964618553l_real] : ( produc4511245868158468465l_real @ ( uminus_uminus_real @ ( produc5828954698716094813l_real @ X4 ) ) @ ( uminus_uminus_real @ ( produc3484788084999411615l_real @ X4 ) ) ) ) ) ).
% uminus_prod_def
thf(fact_742_zero__neq__neg__one,axiom,
( zero_zero_int
!= ( uminus_uminus_int @ one_one_int ) ) ).
% zero_neq_neg_one
thf(fact_743_zero__neq__neg__one,axiom,
( zero_zero_real
!= ( uminus_uminus_real @ one_one_real ) ) ).
% zero_neq_neg_one
thf(fact_744_zero__neq__neg__numeral,axiom,
! [N3: num] :
( zero_zero_int
!= ( uminus_uminus_int @ ( numeral_numeral_int @ N3 ) ) ) ).
% zero_neq_neg_numeral
thf(fact_745_zero__neq__neg__numeral,axiom,
! [N3: num] :
( zero_zero_real
!= ( uminus_uminus_real @ ( numeral_numeral_real @ N3 ) ) ) ).
% zero_neq_neg_numeral
thf(fact_746_divide__numeral__1,axiom,
! [A: real] :
( ( divide_divide_real @ A @ ( numeral_numeral_real @ one ) )
= A ) ).
% divide_numeral_1
thf(fact_747_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_748_less__minus__one__simps_I4_J,axiom,
~ ( ord_less_real @ one_one_real @ ( uminus_uminus_real @ one_one_real ) ) ).
% less_minus_one_simps(4)
thf(fact_749_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_750_less__minus__one__simps_I2_J,axiom,
ord_less_real @ ( uminus_uminus_real @ one_one_real ) @ one_one_real ).
% less_minus_one_simps(2)
thf(fact_751_neg__numeral__less__numeral,axiom,
! [M2: num,N3: num] : ( ord_less_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ M2 ) ) @ ( numeral_numeral_int @ N3 ) ) ).
% neg_numeral_less_numeral
thf(fact_752_neg__numeral__less__numeral,axiom,
! [M2: num,N3: num] : ( ord_less_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ M2 ) ) @ ( numeral_numeral_real @ N3 ) ) ).
% neg_numeral_less_numeral
thf(fact_753_not__numeral__less__neg__numeral,axiom,
! [M2: num,N3: num] :
~ ( ord_less_int @ ( numeral_numeral_int @ M2 ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ N3 ) ) ) ).
% not_numeral_less_neg_numeral
thf(fact_754_not__numeral__less__neg__numeral,axiom,
! [M2: num,N3: num] :
~ ( ord_less_real @ ( numeral_numeral_real @ M2 ) @ ( uminus_uminus_real @ ( numeral_numeral_real @ N3 ) ) ) ).
% not_numeral_less_neg_numeral
thf(fact_755_numeral__neq__neg__one,axiom,
! [N3: num] :
( ( numeral_numeral_int @ N3 )
!= ( uminus_uminus_int @ one_one_int ) ) ).
% numeral_neq_neg_one
thf(fact_756_numeral__neq__neg__one,axiom,
! [N3: num] :
( ( numeral_numeral_real @ N3 )
!= ( uminus_uminus_real @ one_one_real ) ) ).
% numeral_neq_neg_one
thf(fact_757_one__neq__neg__numeral,axiom,
! [N3: num] :
( one_one_int
!= ( uminus_uminus_int @ ( numeral_numeral_int @ N3 ) ) ) ).
% one_neq_neg_numeral
thf(fact_758_one__neq__neg__numeral,axiom,
! [N3: num] :
( one_one_real
!= ( uminus_uminus_real @ ( numeral_numeral_real @ N3 ) ) ) ).
% one_neq_neg_numeral
thf(fact_759_ln__div,axiom,
! [X: real,Y: real] :
( ( ord_less_real @ zero_zero_real @ X )
=> ( ( ord_less_real @ zero_zero_real @ Y )
=> ( ( ln_ln_real @ ( divide_divide_real @ X @ Y ) )
= ( minus_minus_real @ ( ln_ln_real @ X ) @ ( ln_ln_real @ Y ) ) ) ) ) ).
% ln_div
thf(fact_760_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_761_less__minus__one__simps_I3_J,axiom,
~ ( ord_less_real @ zero_zero_real @ ( uminus_uminus_real @ one_one_real ) ) ).
% less_minus_one_simps(3)
thf(fact_762_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_763_less__minus__one__simps_I1_J,axiom,
ord_less_real @ ( uminus_uminus_real @ one_one_real ) @ zero_zero_real ).
% less_minus_one_simps(1)
thf(fact_764_neg__numeral__less__zero,axiom,
! [N3: num] : ( ord_less_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ N3 ) ) @ zero_zero_int ) ).
% neg_numeral_less_zero
thf(fact_765_neg__numeral__less__zero,axiom,
! [N3: num] : ( ord_less_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ N3 ) ) @ zero_zero_real ) ).
% neg_numeral_less_zero
thf(fact_766_not__zero__less__neg__numeral,axiom,
! [N3: num] :
~ ( ord_less_int @ zero_zero_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ N3 ) ) ) ).
% not_zero_less_neg_numeral
thf(fact_767_not__zero__less__neg__numeral,axiom,
! [N3: num] :
~ ( ord_less_real @ zero_zero_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ N3 ) ) ) ).
% not_zero_less_neg_numeral
thf(fact_768_neg__numeral__less__one,axiom,
! [M2: num] : ( ord_less_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ M2 ) ) @ one_one_int ) ).
% neg_numeral_less_one
thf(fact_769_neg__numeral__less__one,axiom,
! [M2: num] : ( ord_less_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ M2 ) ) @ one_one_real ) ).
% neg_numeral_less_one
thf(fact_770_neg__one__less__numeral,axiom,
! [M2: num] : ( ord_less_int @ ( uminus_uminus_int @ one_one_int ) @ ( numeral_numeral_int @ M2 ) ) ).
% neg_one_less_numeral
thf(fact_771_neg__one__less__numeral,axiom,
! [M2: num] : ( ord_less_real @ ( uminus_uminus_real @ one_one_real ) @ ( numeral_numeral_real @ M2 ) ) ).
% neg_one_less_numeral
thf(fact_772_not__numeral__less__neg__one,axiom,
! [M2: num] :
~ ( ord_less_int @ ( numeral_numeral_int @ M2 ) @ ( uminus_uminus_int @ one_one_int ) ) ).
% not_numeral_less_neg_one
thf(fact_773_not__numeral__less__neg__one,axiom,
! [M2: num] :
~ ( ord_less_real @ ( numeral_numeral_real @ M2 ) @ ( uminus_uminus_real @ one_one_real ) ) ).
% not_numeral_less_neg_one
thf(fact_774_not__one__less__neg__numeral,axiom,
! [M2: num] :
~ ( ord_less_int @ one_one_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ M2 ) ) ) ).
% not_one_less_neg_numeral
thf(fact_775_not__one__less__neg__numeral,axiom,
! [M2: num] :
~ ( ord_less_real @ one_one_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ M2 ) ) ) ).
% not_one_less_neg_numeral
thf(fact_776_not__neg__one__less__neg__numeral,axiom,
! [M2: num] :
~ ( ord_less_int @ ( uminus_uminus_int @ one_one_int ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ M2 ) ) ) ).
% not_neg_one_less_neg_numeral
thf(fact_777_not__neg__one__less__neg__numeral,axiom,
! [M2: num] :
~ ( ord_less_real @ ( uminus_uminus_real @ one_one_real ) @ ( uminus_uminus_real @ ( numeral_numeral_real @ M2 ) ) ) ).
% not_neg_one_less_neg_numeral
thf(fact_778_uminus__numeral__One,axiom,
( ( uminus_uminus_int @ ( numeral_numeral_int @ one ) )
= ( uminus_uminus_int @ one_one_int ) ) ).
% uminus_numeral_One
thf(fact_779_uminus__numeral__One,axiom,
( ( uminus_uminus_real @ ( numeral_numeral_real @ one ) )
= ( uminus_uminus_real @ one_one_real ) ) ).
% uminus_numeral_One
thf(fact_780_half__gt__zero__iff,axiom,
! [A: real] :
( ( ord_less_real @ zero_zero_real @ ( divide_divide_real @ A @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
= ( ord_less_real @ zero_zero_real @ A ) ) ).
% half_gt_zero_iff
thf(fact_781_half__gt__zero,axiom,
! [A: real] :
( ( ord_less_real @ zero_zero_real @ A )
=> ( ord_less_real @ zero_zero_real @ ( divide_divide_real @ A @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ) ).
% half_gt_zero
thf(fact_782_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_783_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_784_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_785_divide__less__0__1__iff,axiom,
! [A: real] :
( ( ord_less_real @ ( divide_divide_real @ one_one_real @ A ) @ zero_zero_real )
= ( ord_less_real @ A @ zero_zero_real ) ) ).
% divide_less_0_1_iff
thf(fact_786_divide__less__eq__1__neg,axiom,
! [A: real,B: real] :
( ( ord_less_real @ A @ zero_zero_real )
=> ( ( ord_less_real @ ( divide_divide_real @ B @ A ) @ one_one_real )
= ( ord_less_real @ A @ B ) ) ) ).
% divide_less_eq_1_neg
thf(fact_787_divide__less__eq__1__pos,axiom,
! [A: real,B: real] :
( ( ord_less_real @ zero_zero_real @ A )
=> ( ( ord_less_real @ ( divide_divide_real @ B @ A ) @ one_one_real )
= ( ord_less_real @ B @ A ) ) ) ).
% divide_less_eq_1_pos
thf(fact_788_divide__eq__0__iff,axiom,
! [A: real,B: real] :
( ( ( divide_divide_real @ A @ B )
= zero_zero_real )
= ( ( A = zero_zero_real )
| ( B = zero_zero_real ) ) ) ).
% divide_eq_0_iff
thf(fact_789_divide__cancel__left,axiom,
! [C: real,A: real,B: real] :
( ( ( divide_divide_real @ C @ A )
= ( divide_divide_real @ C @ B ) )
= ( ( C = zero_zero_real )
| ( A = B ) ) ) ).
% divide_cancel_left
thf(fact_790_divide__cancel__right,axiom,
! [A: real,C: real,B: real] :
( ( ( divide_divide_real @ A @ C )
= ( divide_divide_real @ B @ C ) )
= ( ( C = zero_zero_real )
| ( A = B ) ) ) ).
% divide_cancel_right
thf(fact_791_division__ring__divide__zero,axiom,
! [A: real] :
( ( divide_divide_real @ A @ zero_zero_real )
= zero_zero_real ) ).
% division_ring_divide_zero
thf(fact_792_zero__eq__1__divide__iff,axiom,
! [A: real] :
( ( zero_zero_real
= ( divide_divide_real @ one_one_real @ A ) )
= ( A = zero_zero_real ) ) ).
% zero_eq_1_divide_iff
thf(fact_793_one__divide__eq__0__iff,axiom,
! [A: real] :
( ( ( divide_divide_real @ one_one_real @ A )
= zero_zero_real )
= ( A = zero_zero_real ) ) ).
% one_divide_eq_0_iff
thf(fact_794_eq__divide__eq__1,axiom,
! [B: real,A: real] :
( ( one_one_real
= ( divide_divide_real @ B @ A ) )
= ( ( A != zero_zero_real )
& ( A = B ) ) ) ).
% eq_divide_eq_1
thf(fact_795_divide__eq__eq__1,axiom,
! [B: real,A: real] :
( ( ( divide_divide_real @ B @ A )
= one_one_real )
= ( ( A != zero_zero_real )
& ( A = B ) ) ) ).
% divide_eq_eq_1
thf(fact_796_divide__self__if,axiom,
! [A: real] :
( ( ( A = zero_zero_real )
=> ( ( divide_divide_real @ A @ A )
= zero_zero_real ) )
& ( ( A != zero_zero_real )
=> ( ( divide_divide_real @ A @ A )
= one_one_real ) ) ) ).
% divide_self_if
thf(fact_797_divide__self,axiom,
! [A: real] :
( ( A != zero_zero_real )
=> ( ( divide_divide_real @ A @ A )
= one_one_real ) ) ).
% divide_self
thf(fact_798_one__eq__divide__iff,axiom,
! [A: real,B: real] :
( ( one_one_real
= ( divide_divide_real @ A @ B ) )
= ( ( B != zero_zero_real )
& ( A = B ) ) ) ).
% one_eq_divide_iff
thf(fact_799_divide__eq__1__iff,axiom,
! [A: real,B: real] :
( ( ( divide_divide_real @ A @ B )
= one_one_real )
= ( ( B != zero_zero_real )
& ( A = B ) ) ) ).
% divide_eq_1_iff
thf(fact_800_divide__minus1,axiom,
! [X: real] :
( ( divide_divide_real @ X @ ( uminus_uminus_real @ one_one_real ) )
= ( uminus_uminus_real @ X ) ) ).
% divide_minus1
thf(fact_801_zero__less__divide__1__iff,axiom,
! [A: real] :
( ( ord_less_real @ zero_zero_real @ ( divide_divide_real @ one_one_real @ A ) )
= ( ord_less_real @ zero_zero_real @ A ) ) ).
% zero_less_divide_1_iff
thf(fact_802_less__divide__eq__1__pos,axiom,
! [A: real,B: real] :
( ( ord_less_real @ zero_zero_real @ A )
=> ( ( ord_less_real @ one_one_real @ ( divide_divide_real @ B @ A ) )
= ( ord_less_real @ A @ B ) ) ) ).
% less_divide_eq_1_pos
thf(fact_803_less__divide__eq__1__neg,axiom,
! [A: real,B: real] :
( ( ord_less_real @ A @ zero_zero_real )
=> ( ( ord_less_real @ one_one_real @ ( divide_divide_real @ B @ A ) )
= ( ord_less_real @ B @ A ) ) ) ).
% less_divide_eq_1_neg
thf(fact_804_Suc__0__div__numeral_I2_J,axiom,
! [N3: num] :
( ( divide_divide_nat @ ( suc @ zero_zero_nat ) @ ( numeral_numeral_nat @ ( bit0 @ N3 ) ) )
= zero_zero_nat ) ).
% Suc_0_div_numeral(2)
thf(fact_805_Suc__0__div__numeral_I3_J,axiom,
! [N3: num] :
( ( divide_divide_nat @ ( suc @ zero_zero_nat ) @ ( numeral_numeral_nat @ ( bit1 @ N3 ) ) )
= zero_zero_nat ) ).
% Suc_0_div_numeral(3)
thf(fact_806_Suc__0__div__numeral_I1_J,axiom,
( ( divide_divide_nat @ ( suc @ zero_zero_nat ) @ ( numeral_numeral_nat @ one ) )
= one_one_nat ) ).
% Suc_0_div_numeral(1)
thf(fact_807_linordered__field__no__lb,axiom,
! [X8: real] :
? [Y3: real] : ( ord_less_real @ Y3 @ X8 ) ).
% linordered_field_no_lb
thf(fact_808_linordered__field__no__ub,axiom,
! [X8: real] :
? [X_12: real] : ( ord_less_real @ X8 @ X_12 ) ).
% linordered_field_no_ub
thf(fact_809_diff__divide__distrib,axiom,
! [A: real,B: real,C: real] :
( ( divide_divide_real @ ( minus_minus_real @ A @ B ) @ C )
= ( minus_minus_real @ ( divide_divide_real @ A @ C ) @ ( divide_divide_real @ B @ C ) ) ) ).
% diff_divide_distrib
thf(fact_810_divide__strict__right__mono__neg,axiom,
! [B: real,A: real,C: real] :
( ( ord_less_real @ B @ A )
=> ( ( ord_less_real @ C @ zero_zero_real )
=> ( ord_less_real @ ( divide_divide_real @ A @ C ) @ ( divide_divide_real @ B @ C ) ) ) ) ).
% divide_strict_right_mono_neg
thf(fact_811_divide__strict__right__mono,axiom,
! [A: real,B: real,C: real] :
( ( ord_less_real @ A @ B )
=> ( ( ord_less_real @ zero_zero_real @ C )
=> ( ord_less_real @ ( divide_divide_real @ A @ C ) @ ( divide_divide_real @ B @ C ) ) ) ) ).
% divide_strict_right_mono
thf(fact_812_zero__less__divide__iff,axiom,
! [A: real,B: real] :
( ( ord_less_real @ zero_zero_real @ ( divide_divide_real @ A @ B ) )
= ( ( ( ord_less_real @ zero_zero_real @ A )
& ( ord_less_real @ zero_zero_real @ B ) )
| ( ( ord_less_real @ A @ zero_zero_real )
& ( ord_less_real @ B @ zero_zero_real ) ) ) ) ).
% zero_less_divide_iff
thf(fact_813_divide__less__cancel,axiom,
! [A: real,C: real,B: real] :
( ( ord_less_real @ ( divide_divide_real @ A @ C ) @ ( divide_divide_real @ B @ C ) )
= ( ( ( ord_less_real @ zero_zero_real @ C )
=> ( ord_less_real @ A @ B ) )
& ( ( ord_less_real @ C @ zero_zero_real )
=> ( ord_less_real @ B @ A ) )
& ( C != zero_zero_real ) ) ) ).
% divide_less_cancel
thf(fact_814_divide__less__0__iff,axiom,
! [A: real,B: real] :
( ( ord_less_real @ ( divide_divide_real @ A @ B ) @ zero_zero_real )
= ( ( ( ord_less_real @ zero_zero_real @ A )
& ( ord_less_real @ B @ zero_zero_real ) )
| ( ( ord_less_real @ A @ zero_zero_real )
& ( ord_less_real @ zero_zero_real @ B ) ) ) ) ).
% divide_less_0_iff
thf(fact_815_divide__pos__pos,axiom,
! [X: real,Y: real] :
( ( ord_less_real @ zero_zero_real @ X )
=> ( ( ord_less_real @ zero_zero_real @ Y )
=> ( ord_less_real @ zero_zero_real @ ( divide_divide_real @ X @ Y ) ) ) ) ).
% divide_pos_pos
thf(fact_816_divide__pos__neg,axiom,
! [X: real,Y: real] :
( ( ord_less_real @ zero_zero_real @ X )
=> ( ( ord_less_real @ Y @ zero_zero_real )
=> ( ord_less_real @ ( divide_divide_real @ X @ Y ) @ zero_zero_real ) ) ) ).
% divide_pos_neg
thf(fact_817_divide__neg__pos,axiom,
! [X: real,Y: real] :
( ( ord_less_real @ X @ zero_zero_real )
=> ( ( ord_less_real @ zero_zero_real @ Y )
=> ( ord_less_real @ ( divide_divide_real @ X @ Y ) @ zero_zero_real ) ) ) ).
% divide_neg_pos
thf(fact_818_divide__neg__neg,axiom,
! [X: real,Y: real] :
( ( ord_less_real @ X @ zero_zero_real )
=> ( ( ord_less_real @ Y @ zero_zero_real )
=> ( ord_less_real @ zero_zero_real @ ( divide_divide_real @ X @ Y ) ) ) ) ).
% divide_neg_neg
thf(fact_819_right__inverse__eq,axiom,
! [B: real,A: real] :
( ( B != zero_zero_real )
=> ( ( ( divide_divide_real @ A @ B )
= one_one_real )
= ( A = B ) ) ) ).
% right_inverse_eq
thf(fact_820_nonzero__minus__divide__divide,axiom,
! [B: real,A: real] :
( ( B != zero_zero_real )
=> ( ( divide_divide_real @ ( uminus_uminus_real @ A ) @ ( uminus_uminus_real @ B ) )
= ( divide_divide_real @ A @ B ) ) ) ).
% nonzero_minus_divide_divide
thf(fact_821_nonzero__minus__divide__right,axiom,
! [B: real,A: real] :
( ( B != zero_zero_real )
=> ( ( uminus_uminus_real @ ( divide_divide_real @ A @ B ) )
= ( divide_divide_real @ A @ ( uminus_uminus_real @ B ) ) ) ) ).
% nonzero_minus_divide_right
thf(fact_822_less__divide__eq__1,axiom,
! [B: real,A: real] :
( ( ord_less_real @ one_one_real @ ( divide_divide_real @ B @ A ) )
= ( ( ( ord_less_real @ zero_zero_real @ A )
& ( ord_less_real @ A @ B ) )
| ( ( ord_less_real @ A @ zero_zero_real )
& ( ord_less_real @ B @ A ) ) ) ) ).
% less_divide_eq_1
thf(fact_823_divide__less__eq__1,axiom,
! [B: real,A: real] :
( ( ord_less_real @ ( divide_divide_real @ B @ A ) @ one_one_real )
= ( ( ( ord_less_real @ zero_zero_real @ A )
& ( ord_less_real @ B @ A ) )
| ( ( ord_less_real @ A @ zero_zero_real )
& ( ord_less_real @ A @ B ) )
| ( A = zero_zero_real ) ) ) ).
% divide_less_eq_1
thf(fact_824_divide__eq__minus__1__iff,axiom,
! [A: real,B: real] :
( ( ( divide_divide_real @ A @ B )
= ( uminus_uminus_real @ one_one_real ) )
= ( ( B != zero_zero_real )
& ( A
= ( uminus_uminus_real @ B ) ) ) ) ).
% divide_eq_minus_1_iff
thf(fact_825_div2__Suc__Suc,axiom,
! [M2: nat] :
( ( divide_divide_nat @ ( suc @ ( suc @ M2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
= ( suc @ ( divide_divide_nat @ M2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ).
% div2_Suc_Suc
thf(fact_826_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_827_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_828_verit__minus__simplify_I3_J,axiom,
! [B: real] :
( ( minus_minus_real @ zero_zero_real @ B )
= ( uminus_uminus_real @ B ) ) ).
% verit_minus_simplify(3)
thf(fact_829_Suc__n__div__2__gt__zero,axiom,
! [N3: nat] :
( ( ord_less_nat @ zero_zero_nat @ N3 )
=> ( ord_less_nat @ zero_zero_nat @ ( divide_divide_nat @ ( suc @ N3 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ).
% Suc_n_div_2_gt_zero
thf(fact_830_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_831_div__by__Suc__0,axiom,
! [M2: nat] :
( ( divide_divide_nat @ M2 @ ( suc @ zero_zero_nat ) )
= M2 ) ).
% div_by_Suc_0
thf(fact_832_div__less,axiom,
! [M2: nat,N3: nat] :
( ( ord_less_nat @ M2 @ N3 )
=> ( ( divide_divide_nat @ M2 @ N3 )
= zero_zero_nat ) ) ).
% div_less
thf(fact_833_verit__comp__simplify1_I1_J,axiom,
! [A: extended_enat] :
~ ( ord_le72135733267957522d_enat @ A @ A ) ).
% verit_comp_simplify1(1)
thf(fact_834_verit__comp__simplify1_I1_J,axiom,
! [A: real] :
~ ( ord_less_real @ A @ A ) ).
% verit_comp_simplify1(1)
thf(fact_835_verit__comp__simplify1_I1_J,axiom,
! [A: num] :
~ ( ord_less_num @ A @ A ) ).
% verit_comp_simplify1(1)
thf(fact_836_verit__comp__simplify1_I1_J,axiom,
! [A: nat] :
~ ( ord_less_nat @ A @ A ) ).
% verit_comp_simplify1(1)
thf(fact_837_verit__comp__simplify1_I1_J,axiom,
! [A: int] :
~ ( ord_less_int @ A @ A ) ).
% verit_comp_simplify1(1)
thf(fact_838_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_839_verit__negate__coefficient_I2_J,axiom,
! [A: real,B: real] :
( ( ord_less_real @ A @ B )
=> ( ord_less_real @ ( uminus_uminus_real @ B ) @ ( uminus_uminus_real @ A ) ) ) ).
% verit_negate_coefficient(2)
thf(fact_840_Euclidean__Division_Odiv__eq__0__iff,axiom,
! [M2: nat,N3: nat] :
( ( ( divide_divide_nat @ M2 @ N3 )
= zero_zero_nat )
= ( ( ord_less_nat @ M2 @ N3 )
| ( N3 = zero_zero_nat ) ) ) ).
% Euclidean_Division.div_eq_0_iff
thf(fact_841_div__less__dividend,axiom,
! [N3: nat,M2: nat] :
( ( ord_less_nat @ one_one_nat @ N3 )
=> ( ( ord_less_nat @ zero_zero_nat @ M2 )
=> ( ord_less_nat @ ( divide_divide_nat @ M2 @ N3 ) @ M2 ) ) ) ).
% div_less_dividend
thf(fact_842_div__eq__dividend__iff,axiom,
! [M2: nat,N3: nat] :
( ( ord_less_nat @ zero_zero_nat @ M2 )
=> ( ( ( divide_divide_nat @ M2 @ N3 )
= M2 )
= ( N3 = one_one_nat ) ) ) ).
% div_eq_dividend_iff
thf(fact_843_div__if,axiom,
( divide_divide_nat
= ( ^ [M: nat,N: nat] :
( if_nat
@ ( ( ord_less_nat @ M @ N )
| ( N = zero_zero_nat ) )
@ zero_zero_nat
@ ( suc @ ( divide_divide_nat @ ( minus_minus_nat @ M @ N ) @ N ) ) ) ) ) ).
% div_if
thf(fact_844_div__2__gt__zero,axiom,
! [N3: nat] :
( ( ord_less_nat @ ( suc @ zero_zero_nat ) @ N3 )
=> ( ord_less_nat @ zero_zero_nat @ ( divide_divide_nat @ N3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ).
% div_2_gt_zero
thf(fact_845_r01__to__r01__r01__3over4,axiom,
( ( r01_to_r01_r01 @ ( divide_divide_real @ ( numeral_numeral_real @ ( bit1 @ one ) ) @ ( numeral_numeral_real @ ( bit0 @ ( bit0 @ one ) ) ) ) )
= ( produc4511245868158468465l_real @ ( divide_divide_real @ one_one_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) @ ( divide_divide_real @ one_one_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ) ).
% r01_to_r01_r01_3over4
thf(fact_846_divides__aux__eq,axiom,
! [Q: nat,R2: nat] :
( ( unique5332122412489317741ux_nat @ ( product_Pair_nat_nat @ Q @ R2 ) )
= ( R2 = zero_zero_nat ) ) ).
% divides_aux_eq
thf(fact_847_divides__aux__eq,axiom,
! [Q: int,R2: int] :
( ( unique5329631941980267465ux_int @ ( product_Pair_int_int @ Q @ R2 ) )
= ( R2 = zero_zero_int ) ) ).
% divides_aux_eq
thf(fact_848_div__eq__minus1,axiom,
! [B: int] :
( ( ord_less_int @ zero_zero_int @ B )
=> ( ( divide_divide_int @ ( uminus_uminus_int @ one_one_int ) @ B )
= ( uminus_uminus_int @ one_one_int ) ) ) ).
% div_eq_minus1
thf(fact_849_div__neg__pos__less0,axiom,
! [A: int,B: int] :
( ( ord_less_int @ A @ zero_zero_int )
=> ( ( ord_less_int @ zero_zero_int @ B )
=> ( ord_less_int @ ( divide_divide_int @ A @ B ) @ zero_zero_int ) ) ) ).
% div_neg_pos_less0
thf(fact_850_int__div__less__self,axiom,
! [X: int,K: int] :
( ( ord_less_int @ zero_zero_int @ X )
=> ( ( ord_less_int @ one_one_int @ K )
=> ( ord_less_int @ ( divide_divide_int @ X @ K ) @ X ) ) ) ).
% int_div_less_self
thf(fact_851_neg__imp__zdiv__neg__iff,axiom,
! [B: int,A: int] :
( ( ord_less_int @ B @ zero_zero_int )
=> ( ( ord_less_int @ ( divide_divide_int @ A @ B ) @ zero_zero_int )
= ( ord_less_int @ zero_zero_int @ A ) ) ) ).
% neg_imp_zdiv_neg_iff
thf(fact_852_pos__imp__zdiv__neg__iff,axiom,
! [B: int,A: int] :
( ( ord_less_int @ zero_zero_int @ B )
=> ( ( ord_less_int @ ( divide_divide_int @ A @ B ) @ zero_zero_int )
= ( ord_less_int @ A @ zero_zero_int ) ) ) ).
% pos_imp_zdiv_neg_iff
thf(fact_853_divides__aux__def,axiom,
( unique5332122412489317741ux_nat
= ( ^ [Qr: product_prod_nat_nat] :
( ( product_snd_nat_nat @ Qr )
= zero_zero_nat ) ) ) ).
% divides_aux_def
thf(fact_854_divides__aux__def,axiom,
( unique5329631941980267465ux_int
= ( ^ [Qr: product_prod_int_int] :
( ( product_snd_int_int @ Qr )
= zero_zero_int ) ) ) ).
% divides_aux_def
thf(fact_855_set__decode__Suc,axiom,
! [N3: nat,X: nat] :
( ( member_nat @ ( suc @ N3 ) @ ( nat_set_decode @ X ) )
= ( member_nat @ N3 @ ( nat_set_decode @ ( divide_divide_nat @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ).
% set_decode_Suc
thf(fact_856_log__half,axiom,
! [N3: nat] :
( ( log @ ( divide_divide_nat @ N3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
= ( minus_minus_nat @ ( log @ N3 ) @ one_one_nat ) ) ).
% log_half
thf(fact_857_Discrete_Olog_Osimps,axiom,
( log
= ( ^ [N: nat] : ( if_nat @ ( ord_less_nat @ N @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ zero_zero_nat @ ( suc @ ( log @ ( divide_divide_nat @ N @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ) ).
% Discrete.log.simps
thf(fact_858_Discrete_Olog_Oelims,axiom,
! [X: nat,Y: nat] :
( ( ( log @ X )
= Y )
=> ( ( ( ord_less_nat @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
=> ( Y = zero_zero_nat ) )
& ( ~ ( ord_less_nat @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
=> ( Y
= ( suc @ ( log @ ( divide_divide_nat @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ) ) ).
% Discrete.log.elims
thf(fact_859_log__zero,axiom,
( ( log @ zero_zero_nat )
= zero_zero_nat ) ).
% log_zero
thf(fact_860_log__Suc__zero,axiom,
( ( log @ ( suc @ zero_zero_nat ) )
= zero_zero_nat ) ).
% log_Suc_zero
thf(fact_861_Discrete_Olog__one,axiom,
( ( log @ one_one_nat )
= zero_zero_nat ) ).
% Discrete.log_one
thf(fact_862_one__div__numeral,axiom,
! [N3: num] :
( ( divide_divide_nat @ one_one_nat @ ( numeral_numeral_nat @ N3 ) )
= ( product_fst_nat_nat @ ( unique5405566460079783412od_nat @ one @ N3 ) ) ) ).
% one_div_numeral
thf(fact_863_one__div__numeral,axiom,
! [N3: num] :
( ( divide_divide_int @ one_one_int @ ( numeral_numeral_int @ N3 ) )
= ( product_fst_int_int @ ( unique5403075989570733136od_int @ one @ N3 ) ) ) ).
% one_div_numeral
thf(fact_864_divmod__algorithm__code_I3_J,axiom,
! [N3: num] :
( ( unique5405566460079783412od_nat @ one @ ( bit1 @ N3 ) )
= ( product_Pair_nat_nat @ zero_zero_nat @ ( numeral_numeral_nat @ one ) ) ) ).
% divmod_algorithm_code(3)
thf(fact_865_divmod__algorithm__code_I3_J,axiom,
! [N3: num] :
( ( unique5403075989570733136od_int @ one @ ( bit1 @ N3 ) )
= ( product_Pair_int_int @ zero_zero_int @ ( numeral_numeral_int @ one ) ) ) ).
% divmod_algorithm_code(3)
thf(fact_866_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_867_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_868_divmod__algorithm__code_I1_J,axiom,
! [M2: num] :
( ( unique5405566460079783412od_nat @ M2 @ one )
= ( product_Pair_nat_nat @ ( numeral_numeral_nat @ M2 ) @ zero_zero_nat ) ) ).
% divmod_algorithm_code(1)
thf(fact_869_divmod__algorithm__code_I1_J,axiom,
! [M2: num] :
( ( unique5403075989570733136od_int @ M2 @ one )
= ( product_Pair_int_int @ ( numeral_numeral_int @ M2 ) @ zero_zero_int ) ) ).
% divmod_algorithm_code(1)
thf(fact_870_divmod__algorithm__code_I2_J,axiom,
! [N3: num] :
( ( unique5405566460079783412od_nat @ one @ ( bit0 @ N3 ) )
= ( product_Pair_nat_nat @ zero_zero_nat @ ( numeral_numeral_nat @ one ) ) ) ).
% divmod_algorithm_code(2)
thf(fact_871_divmod__algorithm__code_I2_J,axiom,
! [N3: num] :
( ( unique5403075989570733136od_int @ one @ ( bit0 @ N3 ) )
= ( product_Pair_int_int @ zero_zero_int @ ( numeral_numeral_int @ one ) ) ) ).
% divmod_algorithm_code(2)
thf(fact_872_fst__divmod,axiom,
! [M2: num,N3: num] :
( ( product_fst_nat_nat @ ( unique5405566460079783412od_nat @ M2 @ N3 ) )
= ( divide_divide_nat @ ( numeral_numeral_nat @ M2 ) @ ( numeral_numeral_nat @ N3 ) ) ) ).
% fst_divmod
thf(fact_873_fst__divmod,axiom,
! [M2: num,N3: num] :
( ( product_fst_int_int @ ( unique5403075989570733136od_int @ M2 @ N3 ) )
= ( divide_divide_int @ ( numeral_numeral_int @ M2 ) @ ( numeral_numeral_int @ N3 ) ) ) ).
% fst_divmod
thf(fact_874_divmod__algorithm__code_I7_J,axiom,
! [M2: num,N3: num] :
( ( ( ord_less_num @ M2 @ N3 )
=> ( ( unique5405566460079783412od_nat @ ( bit1 @ M2 ) @ ( bit1 @ N3 ) )
= ( product_Pair_nat_nat @ zero_zero_nat @ ( numeral_numeral_nat @ ( bit1 @ M2 ) ) ) ) )
& ( ~ ( ord_less_num @ M2 @ N3 )
=> ( ( unique5405566460079783412od_nat @ ( bit1 @ M2 ) @ ( bit1 @ N3 ) )
= ( unique4036640087844771520ep_nat @ ( numeral_numeral_nat @ ( bit1 @ N3 ) ) @ ( unique5405566460079783412od_nat @ ( bit1 @ M2 ) @ ( bit0 @ ( bit1 @ N3 ) ) ) ) ) ) ) ).
% divmod_algorithm_code(7)
thf(fact_875_divmod__algorithm__code_I7_J,axiom,
! [M2: num,N3: num] :
( ( ( ord_less_num @ M2 @ N3 )
=> ( ( unique5403075989570733136od_int @ ( bit1 @ M2 ) @ ( bit1 @ N3 ) )
= ( product_Pair_int_int @ zero_zero_int @ ( numeral_numeral_int @ ( bit1 @ M2 ) ) ) ) )
& ( ~ ( ord_less_num @ M2 @ N3 )
=> ( ( unique5403075989570733136od_int @ ( bit1 @ M2 ) @ ( bit1 @ N3 ) )
= ( unique4034149617335721244ep_int @ ( numeral_numeral_int @ ( bit1 @ N3 ) ) @ ( unique5403075989570733136od_int @ ( bit1 @ M2 ) @ ( bit0 @ ( bit1 @ N3 ) ) ) ) ) ) ) ).
% divmod_algorithm_code(7)
thf(fact_876_divmod__divmod__step,axiom,
( unique5405566460079783412od_nat
= ( ^ [M: num,N: num] : ( if_Pro6206227464963214023at_nat @ ( ord_less_num @ M @ N ) @ ( product_Pair_nat_nat @ zero_zero_nat @ ( numeral_numeral_nat @ M ) ) @ ( unique4036640087844771520ep_nat @ ( numeral_numeral_nat @ N ) @ ( unique5405566460079783412od_nat @ M @ ( bit0 @ N ) ) ) ) ) ) ).
% divmod_divmod_step
thf(fact_877_divmod__divmod__step,axiom,
( unique5403075989570733136od_int
= ( ^ [M: num,N: num] : ( if_Pro3027730157355071871nt_int @ ( ord_less_num @ M @ N ) @ ( product_Pair_int_int @ zero_zero_int @ ( numeral_numeral_int @ M ) ) @ ( unique4034149617335721244ep_int @ ( numeral_numeral_int @ N ) @ ( unique5403075989570733136od_int @ M @ ( bit0 @ N ) ) ) ) ) ) ).
% divmod_divmod_step
thf(fact_878_int__less__induct,axiom,
! [I: int,K: int,P: int > $o] :
( ( ord_less_int @ I @ K )
=> ( ( P @ ( minus_minus_int @ K @ one_one_int ) )
=> ( ! [I2: int] :
( ( ord_less_int @ I2 @ K )
=> ( ( P @ I2 )
=> ( P @ ( minus_minus_int @ I2 @ one_one_int ) ) ) )
=> ( P @ I ) ) ) ) ).
% int_less_induct
thf(fact_879_upto__aux__rec,axiom,
( upto_aux
= ( ^ [I3: int,J3: int,Js: list_int] : ( if_list_int @ ( ord_less_int @ J3 @ I3 ) @ Js @ ( upto_aux @ I3 @ ( minus_minus_int @ J3 @ one_one_int ) @ ( cons_int @ J3 @ Js ) ) ) ) ) ).
% upto_aux_rec
thf(fact_880_less__int__code_I1_J,axiom,
~ ( ord_less_int @ zero_zero_int @ zero_zero_int ) ).
% less_int_code(1)
thf(fact_881_divmod__algorithm__code_I6_J,axiom,
! [M2: num,N3: num] :
( ( ( ord_less_eq_num @ M2 @ N3 )
=> ( ( unique5405566460079783412od_nat @ ( bit0 @ M2 ) @ ( bit1 @ N3 ) )
= ( product_Pair_nat_nat @ zero_zero_nat @ ( numeral_numeral_nat @ ( bit0 @ M2 ) ) ) ) )
& ( ~ ( ord_less_eq_num @ M2 @ N3 )
=> ( ( unique5405566460079783412od_nat @ ( bit0 @ M2 ) @ ( bit1 @ N3 ) )
= ( unique4036640087844771520ep_nat @ ( numeral_numeral_nat @ ( bit1 @ N3 ) ) @ ( unique5405566460079783412od_nat @ ( bit0 @ M2 ) @ ( bit0 @ ( bit1 @ N3 ) ) ) ) ) ) ) ).
% divmod_algorithm_code(6)
thf(fact_882_divmod__algorithm__code_I6_J,axiom,
! [M2: num,N3: num] :
( ( ( ord_less_eq_num @ M2 @ N3 )
=> ( ( unique5403075989570733136od_int @ ( bit0 @ M2 ) @ ( bit1 @ N3 ) )
= ( product_Pair_int_int @ zero_zero_int @ ( numeral_numeral_int @ ( bit0 @ M2 ) ) ) ) )
& ( ~ ( ord_less_eq_num @ M2 @ N3 )
=> ( ( unique5403075989570733136od_int @ ( bit0 @ M2 ) @ ( bit1 @ N3 ) )
= ( unique4034149617335721244ep_int @ ( numeral_numeral_int @ ( bit1 @ N3 ) ) @ ( unique5403075989570733136od_int @ ( bit0 @ M2 ) @ ( bit0 @ ( bit1 @ N3 ) ) ) ) ) ) ) ).
% divmod_algorithm_code(6)
thf(fact_883_artanh__def,axiom,
( artanh_real
= ( ^ [X4: real] : ( divide_divide_real @ ( ln_ln_real @ ( divide_divide_real @ ( plus_plus_real @ one_one_real @ X4 ) @ ( minus_minus_real @ one_one_real @ X4 ) ) ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ) ).
% artanh_def
thf(fact_884_log__twice,axiom,
! [N3: nat] :
( ( N3 != zero_zero_nat )
=> ( ( log @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N3 ) )
= ( suc @ ( log @ N3 ) ) ) ) ).
% log_twice
thf(fact_885_round__neg__numeral,axiom,
! [N3: num] :
( ( archim8280529875227126926d_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ N3 ) ) )
= ( uminus_uminus_int @ ( numeral_numeral_int @ N3 ) ) ) ).
% round_neg_numeral
thf(fact_886_add__right__cancel,axiom,
! [B: nat,A: nat,C: nat] :
( ( ( plus_plus_nat @ B @ A )
= ( plus_plus_nat @ C @ A ) )
= ( B = C ) ) ).
% add_right_cancel
thf(fact_887_add__right__cancel,axiom,
! [B: real,A: real,C: real] :
( ( ( plus_plus_real @ B @ A )
= ( plus_plus_real @ C @ A ) )
= ( B = C ) ) ).
% add_right_cancel
thf(fact_888_add__left__cancel,axiom,
! [A: nat,B: nat,C: nat] :
( ( ( plus_plus_nat @ A @ B )
= ( plus_plus_nat @ A @ C ) )
= ( B = C ) ) ).
% add_left_cancel
thf(fact_889_add__left__cancel,axiom,
! [A: real,B: real,C: real] :
( ( ( plus_plus_real @ A @ B )
= ( plus_plus_real @ A @ C ) )
= ( B = C ) ) ).
% add_left_cancel
thf(fact_890_add__is__0,axiom,
! [M2: nat,N3: nat] :
( ( ( plus_plus_nat @ M2 @ N3 )
= zero_zero_nat )
= ( ( M2 = zero_zero_nat )
& ( N3 = zero_zero_nat ) ) ) ).
% add_is_0
thf(fact_891_Nat_Oadd__0__right,axiom,
! [M2: nat] :
( ( plus_plus_nat @ M2 @ zero_zero_nat )
= M2 ) ).
% Nat.add_0_right
thf(fact_892_add__Suc__right,axiom,
! [M2: nat,N3: nat] :
( ( plus_plus_nat @ M2 @ ( suc @ N3 ) )
= ( suc @ ( plus_plus_nat @ M2 @ N3 ) ) ) ).
% add_Suc_right
thf(fact_893_nat__add__left__cancel__less,axiom,
! [K: nat,M2: nat,N3: nat] :
( ( ord_less_nat @ ( plus_plus_nat @ K @ M2 ) @ ( plus_plus_nat @ K @ N3 ) )
= ( ord_less_nat @ M2 @ N3 ) ) ).
% nat_add_left_cancel_less
thf(fact_894_diff__diff__left,axiom,
! [I: nat,J: nat,K: nat] :
( ( minus_minus_nat @ ( minus_minus_nat @ I @ J ) @ K )
= ( minus_minus_nat @ I @ ( plus_plus_nat @ J @ K ) ) ) ).
% diff_diff_left
thf(fact_895_le__zero__eq,axiom,
! [N3: nat] :
( ( ord_less_eq_nat @ N3 @ zero_zero_nat )
= ( N3 = zero_zero_nat ) ) ).
% le_zero_eq
thf(fact_896_le__zero__eq,axiom,
! [N3: extended_enat] :
( ( ord_le2932123472753598470d_enat @ N3 @ zero_z5237406670263579293d_enat )
= ( N3 = zero_z5237406670263579293d_enat ) ) ).
% le_zero_eq
thf(fact_897_le__zero__eq,axiom,
! [N3: extend8495563244428889912nnreal] :
( ( ord_le3935885782089961368nnreal @ N3 @ zero_z7100319975126383169nnreal )
= ( N3 = zero_z7100319975126383169nnreal ) ) ).
% le_zero_eq
thf(fact_898_mult__cancel__right,axiom,
! [A: real,C: real,B: real] :
( ( ( times_times_real @ A @ C )
= ( times_times_real @ B @ C ) )
= ( ( C = zero_zero_real )
| ( A = B ) ) ) ).
% mult_cancel_right
thf(fact_899_mult__cancel__right,axiom,
! [A: int,C: int,B: int] :
( ( ( times_times_int @ A @ C )
= ( times_times_int @ B @ C ) )
= ( ( C = zero_zero_int )
| ( A = B ) ) ) ).
% mult_cancel_right
thf(fact_900_mult__cancel__right,axiom,
! [A: nat,C: nat,B: nat] :
( ( ( times_times_nat @ A @ C )
= ( times_times_nat @ B @ C ) )
= ( ( C = zero_zero_nat )
| ( A = B ) ) ) ).
% mult_cancel_right
thf(fact_901_mult__cancel__left,axiom,
! [C: real,A: real,B: real] :
( ( ( times_times_real @ C @ A )
= ( times_times_real @ C @ B ) )
= ( ( C = zero_zero_real )
| ( A = B ) ) ) ).
% mult_cancel_left
thf(fact_902_mult__cancel__left,axiom,
! [C: int,A: int,B: int] :
( ( ( times_times_int @ C @ A )
= ( times_times_int @ C @ B ) )
= ( ( C = zero_zero_int )
| ( A = B ) ) ) ).
% mult_cancel_left
thf(fact_903_mult__cancel__left,axiom,
! [C: nat,A: nat,B: nat] :
( ( ( times_times_nat @ C @ A )
= ( times_times_nat @ C @ B ) )
= ( ( C = zero_zero_nat )
| ( A = B ) ) ) ).
% mult_cancel_left
thf(fact_904_mult__eq__0__iff,axiom,
! [A: extended_enat,B: extended_enat] :
( ( ( times_7803423173614009249d_enat @ A @ B )
= zero_z5237406670263579293d_enat )
= ( ( A = zero_z5237406670263579293d_enat )
| ( B = zero_z5237406670263579293d_enat ) ) ) ).
% mult_eq_0_iff
thf(fact_905_mult__eq__0__iff,axiom,
! [A: extend8495563244428889912nnreal,B: extend8495563244428889912nnreal] :
( ( ( times_1893300245718287421nnreal @ A @ B )
= zero_z7100319975126383169nnreal )
= ( ( A = zero_z7100319975126383169nnreal )
| ( B = zero_z7100319975126383169nnreal ) ) ) ).
% mult_eq_0_iff
thf(fact_906_mult__eq__0__iff,axiom,
! [A: real,B: real] :
( ( ( times_times_real @ A @ B )
= zero_zero_real )
= ( ( A = zero_zero_real )
| ( B = zero_zero_real ) ) ) ).
% mult_eq_0_iff
thf(fact_907_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_908_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_909_mult__zero__right,axiom,
! [A: extended_enat] :
( ( times_7803423173614009249d_enat @ A @ zero_z5237406670263579293d_enat )
= zero_z5237406670263579293d_enat ) ).
% mult_zero_right
thf(fact_910_mult__zero__right,axiom,
! [A: extend8495563244428889912nnreal] :
( ( times_1893300245718287421nnreal @ A @ zero_z7100319975126383169nnreal )
= zero_z7100319975126383169nnreal ) ).
% mult_zero_right
thf(fact_911_mult__zero__right,axiom,
! [A: real] :
( ( times_times_real @ A @ zero_zero_real )
= zero_zero_real ) ).
% mult_zero_right
thf(fact_912_mult__zero__right,axiom,
! [A: int] :
( ( times_times_int @ A @ zero_zero_int )
= zero_zero_int ) ).
% mult_zero_right
thf(fact_913_mult__zero__right,axiom,
! [A: nat] :
( ( times_times_nat @ A @ zero_zero_nat )
= zero_zero_nat ) ).
% mult_zero_right
thf(fact_914_mult__zero__left,axiom,
! [A: extended_enat] :
( ( times_7803423173614009249d_enat @ zero_z5237406670263579293d_enat @ A )
= zero_z5237406670263579293d_enat ) ).
% mult_zero_left
thf(fact_915_mult__zero__left,axiom,
! [A: extend8495563244428889912nnreal] :
( ( times_1893300245718287421nnreal @ zero_z7100319975126383169nnreal @ A )
= zero_z7100319975126383169nnreal ) ).
% mult_zero_left
thf(fact_916_mult__zero__left,axiom,
! [A: real] :
( ( times_times_real @ zero_zero_real @ A )
= zero_zero_real ) ).
% mult_zero_left
thf(fact_917_mult__zero__left,axiom,
! [A: int] :
( ( times_times_int @ zero_zero_int @ A )
= zero_zero_int ) ).
% mult_zero_left
thf(fact_918_mult__zero__left,axiom,
! [A: nat] :
( ( times_times_nat @ zero_zero_nat @ A )
= zero_zero_nat ) ).
% mult_zero_left
thf(fact_919_add__le__cancel__right,axiom,
! [A: nat,C: nat,B: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ A @ C ) @ ( plus_plus_nat @ B @ C ) )
= ( ord_less_eq_nat @ A @ B ) ) ).
% add_le_cancel_right
thf(fact_920_add__le__cancel__right,axiom,
! [A: real,C: real,B: real] :
( ( ord_less_eq_real @ ( plus_plus_real @ A @ C ) @ ( plus_plus_real @ B @ C ) )
= ( ord_less_eq_real @ A @ B ) ) ).
% add_le_cancel_right
thf(fact_921_add__le__cancel__left,axiom,
! [C: nat,A: nat,B: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ C @ A ) @ ( plus_plus_nat @ C @ B ) )
= ( ord_less_eq_nat @ A @ B ) ) ).
% add_le_cancel_left
thf(fact_922_add__le__cancel__left,axiom,
! [C: real,A: real,B: real] :
( ( ord_less_eq_real @ ( plus_plus_real @ C @ A ) @ ( plus_plus_real @ C @ B ) )
= ( ord_less_eq_real @ A @ B ) ) ).
% add_le_cancel_left
thf(fact_923_double__eq__0__iff,axiom,
! [A: real] :
( ( ( plus_plus_real @ A @ A )
= zero_zero_real )
= ( A = zero_zero_real ) ) ).
% double_eq_0_iff
thf(fact_924_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_925_add__0,axiom,
! [A: nat] :
( ( plus_plus_nat @ zero_zero_nat @ A )
= A ) ).
% add_0
thf(fact_926_add__0,axiom,
! [A: extended_enat] :
( ( plus_p3455044024723400733d_enat @ zero_z5237406670263579293d_enat @ A )
= A ) ).
% add_0
thf(fact_927_add__0,axiom,
! [A: extend8495563244428889912nnreal] :
( ( plus_p1859984266308609217nnreal @ zero_z7100319975126383169nnreal @ A )
= A ) ).
% add_0
thf(fact_928_add__0,axiom,
! [A: real] :
( ( plus_plus_real @ zero_zero_real @ A )
= A ) ).
% add_0
thf(fact_929_add__0,axiom,
! [A: int] :
( ( plus_plus_int @ zero_zero_int @ A )
= A ) ).
% add_0
thf(fact_930_zero__eq__add__iff__both__eq__0,axiom,
! [X: nat,Y: nat] :
( ( zero_zero_nat
= ( plus_plus_nat @ X @ Y ) )
= ( ( X = zero_zero_nat )
& ( Y = zero_zero_nat ) ) ) ).
% zero_eq_add_iff_both_eq_0
thf(fact_931_zero__eq__add__iff__both__eq__0,axiom,
! [X: extended_enat,Y: extended_enat] :
( ( zero_z5237406670263579293d_enat
= ( plus_p3455044024723400733d_enat @ X @ Y ) )
= ( ( X = zero_z5237406670263579293d_enat )
& ( Y = zero_z5237406670263579293d_enat ) ) ) ).
% zero_eq_add_iff_both_eq_0
thf(fact_932_zero__eq__add__iff__both__eq__0,axiom,
! [X: extend8495563244428889912nnreal,Y: extend8495563244428889912nnreal] :
( ( zero_z7100319975126383169nnreal
= ( plus_p1859984266308609217nnreal @ X @ Y ) )
= ( ( X = zero_z7100319975126383169nnreal )
& ( Y = zero_z7100319975126383169nnreal ) ) ) ).
% zero_eq_add_iff_both_eq_0
thf(fact_933_add__eq__0__iff__both__eq__0,axiom,
! [X: nat,Y: nat] :
( ( ( plus_plus_nat @ X @ Y )
= zero_zero_nat )
= ( ( X = zero_zero_nat )
& ( Y = zero_zero_nat ) ) ) ).
% add_eq_0_iff_both_eq_0
thf(fact_934_add__eq__0__iff__both__eq__0,axiom,
! [X: extended_enat,Y: extended_enat] :
( ( ( plus_p3455044024723400733d_enat @ X @ Y )
= zero_z5237406670263579293d_enat )
= ( ( X = zero_z5237406670263579293d_enat )
& ( Y = zero_z5237406670263579293d_enat ) ) ) ).
% add_eq_0_iff_both_eq_0
thf(fact_935_add__eq__0__iff__both__eq__0,axiom,
! [X: extend8495563244428889912nnreal,Y: extend8495563244428889912nnreal] :
( ( ( plus_p1859984266308609217nnreal @ X @ Y )
= zero_z7100319975126383169nnreal )
= ( ( X = zero_z7100319975126383169nnreal )
& ( Y = zero_z7100319975126383169nnreal ) ) ) ).
% add_eq_0_iff_both_eq_0
thf(fact_936_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_937_add__cancel__right__right,axiom,
! [A: real,B: real] :
( ( A
= ( plus_plus_real @ A @ B ) )
= ( B = zero_zero_real ) ) ).
% add_cancel_right_right
thf(fact_938_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_939_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_940_add__cancel__right__left,axiom,
! [A: real,B: real] :
( ( A
= ( plus_plus_real @ B @ A ) )
= ( B = zero_zero_real ) ) ).
% add_cancel_right_left
thf(fact_941_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_942_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_943_add__cancel__left__right,axiom,
! [A: real,B: real] :
( ( ( plus_plus_real @ A @ B )
= A )
= ( B = zero_zero_real ) ) ).
% add_cancel_left_right
thf(fact_944_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_945_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_946_add__cancel__left__left,axiom,
! [B: real,A: real] :
( ( ( plus_plus_real @ B @ A )
= A )
= ( B = zero_zero_real ) ) ).
% add_cancel_left_left
thf(fact_947_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_948_double__zero__sym,axiom,
! [A: real] :
( ( zero_zero_real
= ( plus_plus_real @ A @ A ) )
= ( A = zero_zero_real ) ) ).
% double_zero_sym
thf(fact_949_double__zero__sym,axiom,
! [A: int] :
( ( zero_zero_int
= ( plus_plus_int @ A @ A ) )
= ( A = zero_zero_int ) ) ).
% double_zero_sym
thf(fact_950_add_Oright__neutral,axiom,
! [A: nat] :
( ( plus_plus_nat @ A @ zero_zero_nat )
= A ) ).
% add.right_neutral
thf(fact_951_add_Oright__neutral,axiom,
! [A: extended_enat] :
( ( plus_p3455044024723400733d_enat @ A @ zero_z5237406670263579293d_enat )
= A ) ).
% add.right_neutral
thf(fact_952_add_Oright__neutral,axiom,
! [A: extend8495563244428889912nnreal] :
( ( plus_p1859984266308609217nnreal @ A @ zero_z7100319975126383169nnreal )
= A ) ).
% add.right_neutral
thf(fact_953_add_Oright__neutral,axiom,
! [A: real] :
( ( plus_plus_real @ A @ zero_zero_real )
= A ) ).
% add.right_neutral
thf(fact_954_add_Oright__neutral,axiom,
! [A: int] :
( ( plus_plus_int @ A @ zero_zero_int )
= A ) ).
% add.right_neutral
thf(fact_955_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_956_neg__le__iff__le,axiom,
! [B: real,A: real] :
( ( ord_less_eq_real @ ( uminus_uminus_real @ B ) @ ( uminus_uminus_real @ A ) )
= ( ord_less_eq_real @ A @ B ) ) ).
% neg_le_iff_le
thf(fact_957_add__less__cancel__right,axiom,
! [A: real,C: real,B: real] :
( ( ord_less_real @ ( plus_plus_real @ A @ C ) @ ( plus_plus_real @ B @ C ) )
= ( ord_less_real @ A @ B ) ) ).
% add_less_cancel_right
thf(fact_958_add__less__cancel__right,axiom,
! [A: nat,C: nat,B: nat] :
( ( ord_less_nat @ ( plus_plus_nat @ A @ C ) @ ( plus_plus_nat @ B @ C ) )
= ( ord_less_nat @ A @ B ) ) ).
% add_less_cancel_right
thf(fact_959_add__less__cancel__right,axiom,
! [A: int,C: int,B: int] :
( ( ord_less_int @ ( plus_plus_int @ A @ C ) @ ( plus_plus_int @ B @ C ) )
= ( ord_less_int @ A @ B ) ) ).
% add_less_cancel_right
thf(fact_960_add__less__cancel__left,axiom,
! [C: real,A: real,B: real] :
( ( ord_less_real @ ( plus_plus_real @ C @ A ) @ ( plus_plus_real @ C @ B ) )
= ( ord_less_real @ A @ B ) ) ).
% add_less_cancel_left
thf(fact_961_add__less__cancel__left,axiom,
! [C: nat,A: nat,B: nat] :
( ( ord_less_nat @ ( plus_plus_nat @ C @ A ) @ ( plus_plus_nat @ C @ B ) )
= ( ord_less_nat @ A @ B ) ) ).
% add_less_cancel_left
thf(fact_962_add__less__cancel__left,axiom,
! [C: int,A: int,B: int] :
( ( ord_less_int @ ( plus_plus_int @ C @ A ) @ ( plus_plus_int @ C @ B ) )
= ( ord_less_int @ A @ B ) ) ).
% add_less_cancel_left
thf(fact_963_mult__1,axiom,
! [A: extend8495563244428889912nnreal] :
( ( times_1893300245718287421nnreal @ one_on2969667320475766781nnreal @ A )
= A ) ).
% mult_1
thf(fact_964_mult__1,axiom,
! [A: real] :
( ( times_times_real @ one_one_real @ A )
= A ) ).
% mult_1
thf(fact_965_mult__1,axiom,
! [A: int] :
( ( times_times_int @ one_one_int @ A )
= A ) ).
% mult_1
thf(fact_966_mult__1,axiom,
! [A: nat] :
( ( times_times_nat @ one_one_nat @ A )
= A ) ).
% mult_1
thf(fact_967_mult_Oright__neutral,axiom,
! [A: extend8495563244428889912nnreal] :
( ( times_1893300245718287421nnreal @ A @ one_on2969667320475766781nnreal )
= A ) ).
% mult.right_neutral
thf(fact_968_mult_Oright__neutral,axiom,
! [A: real] :
( ( times_times_real @ A @ one_one_real )
= A ) ).
% mult.right_neutral
thf(fact_969_mult_Oright__neutral,axiom,
! [A: int] :
( ( times_times_int @ A @ one_one_int )
= A ) ).
% mult.right_neutral
thf(fact_970_mult_Oright__neutral,axiom,
! [A: nat] :
( ( times_times_nat @ A @ one_one_nat )
= A ) ).
% mult.right_neutral
thf(fact_971_mult__numeral__left__semiring__numeral,axiom,
! [V2: num,W: num,Z2: extended_enat] :
( ( times_7803423173614009249d_enat @ ( numera1916890842035813515d_enat @ V2 ) @ ( times_7803423173614009249d_enat @ ( numera1916890842035813515d_enat @ W ) @ Z2 ) )
= ( times_7803423173614009249d_enat @ ( numera1916890842035813515d_enat @ ( times_times_num @ V2 @ W ) ) @ Z2 ) ) ).
% mult_numeral_left_semiring_numeral
thf(fact_972_mult__numeral__left__semiring__numeral,axiom,
! [V2: num,W: num,Z2: nat] :
( ( times_times_nat @ ( numeral_numeral_nat @ V2 ) @ ( times_times_nat @ ( numeral_numeral_nat @ W ) @ Z2 ) )
= ( times_times_nat @ ( numeral_numeral_nat @ ( times_times_num @ V2 @ W ) ) @ Z2 ) ) ).
% mult_numeral_left_semiring_numeral
thf(fact_973_mult__numeral__left__semiring__numeral,axiom,
! [V2: num,W: num,Z2: extend8495563244428889912nnreal] :
( ( times_1893300245718287421nnreal @ ( numera4658534427948366547nnreal @ V2 ) @ ( times_1893300245718287421nnreal @ ( numera4658534427948366547nnreal @ W ) @ Z2 ) )
= ( times_1893300245718287421nnreal @ ( numera4658534427948366547nnreal @ ( times_times_num @ V2 @ W ) ) @ Z2 ) ) ).
% mult_numeral_left_semiring_numeral
thf(fact_974_mult__numeral__left__semiring__numeral,axiom,
! [V2: num,W: num,Z2: real] :
( ( times_times_real @ ( numeral_numeral_real @ V2 ) @ ( times_times_real @ ( numeral_numeral_real @ W ) @ Z2 ) )
= ( times_times_real @ ( numeral_numeral_real @ ( times_times_num @ V2 @ W ) ) @ Z2 ) ) ).
% mult_numeral_left_semiring_numeral
thf(fact_975_mult__numeral__left__semiring__numeral,axiom,
! [V2: num,W: num,Z2: int] :
( ( times_times_int @ ( numeral_numeral_int @ V2 ) @ ( times_times_int @ ( numeral_numeral_int @ W ) @ Z2 ) )
= ( times_times_int @ ( numeral_numeral_int @ ( times_times_num @ V2 @ W ) ) @ Z2 ) ) ).
% mult_numeral_left_semiring_numeral
thf(fact_976_numeral__times__numeral,axiom,
! [M2: num,N3: num] :
( ( times_7803423173614009249d_enat @ ( numera1916890842035813515d_enat @ M2 ) @ ( numera1916890842035813515d_enat @ N3 ) )
= ( numera1916890842035813515d_enat @ ( times_times_num @ M2 @ N3 ) ) ) ).
% numeral_times_numeral
thf(fact_977_numeral__times__numeral,axiom,
! [M2: num,N3: num] :
( ( times_times_nat @ ( numeral_numeral_nat @ M2 ) @ ( numeral_numeral_nat @ N3 ) )
= ( numeral_numeral_nat @ ( times_times_num @ M2 @ N3 ) ) ) ).
% numeral_times_numeral
thf(fact_978_numeral__times__numeral,axiom,
! [M2: num,N3: num] :
( ( times_1893300245718287421nnreal @ ( numera4658534427948366547nnreal @ M2 ) @ ( numera4658534427948366547nnreal @ N3 ) )
= ( numera4658534427948366547nnreal @ ( times_times_num @ M2 @ N3 ) ) ) ).
% numeral_times_numeral
thf(fact_979_numeral__times__numeral,axiom,
! [M2: num,N3: num] :
( ( times_times_real @ ( numeral_numeral_real @ M2 ) @ ( numeral_numeral_real @ N3 ) )
= ( numeral_numeral_real @ ( times_times_num @ M2 @ N3 ) ) ) ).
% numeral_times_numeral
thf(fact_980_numeral__times__numeral,axiom,
! [M2: num,N3: num] :
( ( times_times_int @ ( numeral_numeral_int @ M2 ) @ ( numeral_numeral_int @ N3 ) )
= ( numeral_numeral_int @ ( times_times_num @ M2 @ N3 ) ) ) ).
% numeral_times_numeral
thf(fact_981_add__numeral__left,axiom,
! [V2: num,W: num,Z2: extended_enat] :
( ( plus_p3455044024723400733d_enat @ ( numera1916890842035813515d_enat @ V2 ) @ ( plus_p3455044024723400733d_enat @ ( numera1916890842035813515d_enat @ W ) @ Z2 ) )
= ( plus_p3455044024723400733d_enat @ ( numera1916890842035813515d_enat @ ( plus_plus_num @ V2 @ W ) ) @ Z2 ) ) ).
% add_numeral_left
thf(fact_982_add__numeral__left,axiom,
! [V2: num,W: num,Z2: nat] :
( ( plus_plus_nat @ ( numeral_numeral_nat @ V2 ) @ ( plus_plus_nat @ ( numeral_numeral_nat @ W ) @ Z2 ) )
= ( plus_plus_nat @ ( numeral_numeral_nat @ ( plus_plus_num @ V2 @ W ) ) @ Z2 ) ) ).
% add_numeral_left
thf(fact_983_add__numeral__left,axiom,
! [V2: num,W: num,Z2: extend8495563244428889912nnreal] :
( ( plus_p1859984266308609217nnreal @ ( numera4658534427948366547nnreal @ V2 ) @ ( plus_p1859984266308609217nnreal @ ( numera4658534427948366547nnreal @ W ) @ Z2 ) )
= ( plus_p1859984266308609217nnreal @ ( numera4658534427948366547nnreal @ ( plus_plus_num @ V2 @ W ) ) @ Z2 ) ) ).
% add_numeral_left
thf(fact_984_add__numeral__left,axiom,
! [V2: num,W: num,Z2: real] :
( ( plus_plus_real @ ( numeral_numeral_real @ V2 ) @ ( plus_plus_real @ ( numeral_numeral_real @ W ) @ Z2 ) )
= ( plus_plus_real @ ( numeral_numeral_real @ ( plus_plus_num @ V2 @ W ) ) @ Z2 ) ) ).
% add_numeral_left
thf(fact_985_add__numeral__left,axiom,
! [V2: num,W: num,Z2: int] :
( ( plus_plus_int @ ( numeral_numeral_int @ V2 ) @ ( plus_plus_int @ ( numeral_numeral_int @ W ) @ Z2 ) )
= ( plus_plus_int @ ( numeral_numeral_int @ ( plus_plus_num @ V2 @ W ) ) @ Z2 ) ) ).
% add_numeral_left
thf(fact_986_numeral__plus__numeral,axiom,
! [M2: num,N3: num] :
( ( plus_p3455044024723400733d_enat @ ( numera1916890842035813515d_enat @ M2 ) @ ( numera1916890842035813515d_enat @ N3 ) )
= ( numera1916890842035813515d_enat @ ( plus_plus_num @ M2 @ N3 ) ) ) ).
% numeral_plus_numeral
thf(fact_987_numeral__plus__numeral,axiom,
! [M2: num,N3: num] :
( ( plus_plus_nat @ ( numeral_numeral_nat @ M2 ) @ ( numeral_numeral_nat @ N3 ) )
= ( numeral_numeral_nat @ ( plus_plus_num @ M2 @ N3 ) ) ) ).
% numeral_plus_numeral
thf(fact_988_numeral__plus__numeral,axiom,
! [M2: num,N3: num] :
( ( plus_p1859984266308609217nnreal @ ( numera4658534427948366547nnreal @ M2 ) @ ( numera4658534427948366547nnreal @ N3 ) )
= ( numera4658534427948366547nnreal @ ( plus_plus_num @ M2 @ N3 ) ) ) ).
% numeral_plus_numeral
thf(fact_989_numeral__plus__numeral,axiom,
! [M2: num,N3: num] :
( ( plus_plus_real @ ( numeral_numeral_real @ M2 ) @ ( numeral_numeral_real @ N3 ) )
= ( numeral_numeral_real @ ( plus_plus_num @ M2 @ N3 ) ) ) ).
% numeral_plus_numeral
thf(fact_990_numeral__plus__numeral,axiom,
! [M2: num,N3: num] :
( ( plus_plus_int @ ( numeral_numeral_int @ M2 ) @ ( numeral_numeral_int @ N3 ) )
= ( numeral_numeral_int @ ( plus_plus_num @ M2 @ N3 ) ) ) ).
% numeral_plus_numeral
thf(fact_991_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_992_add__diff__cancel__right_H,axiom,
! [A: real,B: real] :
( ( minus_minus_real @ ( plus_plus_real @ A @ B ) @ B )
= A ) ).
% add_diff_cancel_right'
thf(fact_993_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_994_add__diff__cancel__right,axiom,
! [A: nat,C: nat,B: nat] :
( ( minus_minus_nat @ ( plus_plus_nat @ A @ C ) @ ( plus_plus_nat @ B @ C ) )
= ( minus_minus_nat @ A @ B ) ) ).
% add_diff_cancel_right
thf(fact_995_add__diff__cancel__right,axiom,
! [A: real,C: real,B: real] :
( ( minus_minus_real @ ( plus_plus_real @ A @ C ) @ ( plus_plus_real @ B @ C ) )
= ( minus_minus_real @ A @ B ) ) ).
% add_diff_cancel_right
thf(fact_996_add__diff__cancel__right,axiom,
! [A: int,C: int,B: int] :
( ( minus_minus_int @ ( plus_plus_int @ A @ C ) @ ( plus_plus_int @ B @ C ) )
= ( minus_minus_int @ A @ B ) ) ).
% add_diff_cancel_right
thf(fact_997_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_998_add__diff__cancel__left_H,axiom,
! [A: real,B: real] :
( ( minus_minus_real @ ( plus_plus_real @ A @ B ) @ A )
= B ) ).
% add_diff_cancel_left'
thf(fact_999_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_1000_add__diff__cancel__left,axiom,
! [C: nat,A: nat,B: nat] :
( ( minus_minus_nat @ ( plus_plus_nat @ C @ A ) @ ( plus_plus_nat @ C @ B ) )
= ( minus_minus_nat @ A @ B ) ) ).
% add_diff_cancel_left
thf(fact_1001_add__diff__cancel__left,axiom,
! [C: real,A: real,B: real] :
( ( minus_minus_real @ ( plus_plus_real @ C @ A ) @ ( plus_plus_real @ C @ B ) )
= ( minus_minus_real @ A @ B ) ) ).
% add_diff_cancel_left
thf(fact_1002_add__diff__cancel__left,axiom,
! [C: int,A: int,B: int] :
( ( minus_minus_int @ ( plus_plus_int @ C @ A ) @ ( plus_plus_int @ C @ B ) )
= ( minus_minus_int @ A @ B ) ) ).
% add_diff_cancel_left
thf(fact_1003_diff__add__cancel,axiom,
! [A: real,B: real] :
( ( plus_plus_real @ ( minus_minus_real @ A @ B ) @ B )
= A ) ).
% diff_add_cancel
thf(fact_1004_diff__add__cancel,axiom,
! [A: int,B: int] :
( ( plus_plus_int @ ( minus_minus_int @ A @ B ) @ B )
= A ) ).
% diff_add_cancel
thf(fact_1005_add__diff__cancel,axiom,
! [A: real,B: real] :
( ( minus_minus_real @ ( plus_plus_real @ A @ B ) @ B )
= A ) ).
% add_diff_cancel
thf(fact_1006_add__diff__cancel,axiom,
! [A: int,B: int] :
( ( minus_minus_int @ ( plus_plus_int @ A @ B ) @ B )
= A ) ).
% add_diff_cancel
thf(fact_1007_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_1008_mult__minus__right,axiom,
! [A: real,B: real] :
( ( times_times_real @ A @ ( uminus_uminus_real @ B ) )
= ( uminus_uminus_real @ ( times_times_real @ A @ B ) ) ) ).
% mult_minus_right
thf(fact_1009_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_1010_minus__mult__minus,axiom,
! [A: real,B: real] :
( ( times_times_real @ ( uminus_uminus_real @ A ) @ ( uminus_uminus_real @ B ) )
= ( times_times_real @ A @ B ) ) ).
% minus_mult_minus
thf(fact_1011_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_1012_mult__minus__left,axiom,
! [A: real,B: real] :
( ( times_times_real @ ( uminus_uminus_real @ A ) @ B )
= ( uminus_uminus_real @ ( times_times_real @ A @ B ) ) ) ).
% mult_minus_left
thf(fact_1013_add__Pair,axiom,
! [A: nat,B: nat,C: nat,D: nat] :
( ( plus_p9057090461656269880at_nat @ ( product_Pair_nat_nat @ A @ B ) @ ( product_Pair_nat_nat @ C @ D ) )
= ( product_Pair_nat_nat @ ( plus_plus_nat @ A @ C ) @ ( plus_plus_nat @ B @ D ) ) ) ).
% add_Pair
thf(fact_1014_add__Pair,axiom,
! [A: nat,B: num,C: nat,D: num] :
( ( plus_p541962361750331778at_num @ ( product_Pair_nat_num @ A @ B ) @ ( product_Pair_nat_num @ C @ D ) )
= ( product_Pair_nat_num @ ( plus_plus_nat @ A @ C ) @ ( plus_plus_num @ B @ D ) ) ) ).
% add_Pair
thf(fact_1015_add__Pair,axiom,
! [A: nat,B: real,C: nat,D: real] :
( ( plus_p8900843186509212308t_real @ ( produc7837566107596912789t_real @ A @ B ) @ ( produc7837566107596912789t_real @ C @ D ) )
= ( produc7837566107596912789t_real @ ( plus_plus_nat @ A @ C ) @ ( plus_plus_real @ B @ D ) ) ) ).
% add_Pair
thf(fact_1016_add__Pair,axiom,
! [A: num,B: nat,C: num,D: nat] :
( ( plus_p7187420864963739266um_nat @ ( product_Pair_num_nat @ A @ B ) @ ( product_Pair_num_nat @ C @ D ) )
= ( product_Pair_num_nat @ ( plus_plus_num @ A @ C ) @ ( plus_plus_nat @ B @ D ) ) ) ).
% add_Pair
thf(fact_1017_add__Pair,axiom,
! [A: num,B: num,C: num,D: num] :
( ( plus_p7895664801912576972um_num @ ( product_Pair_num_num @ A @ B ) @ ( product_Pair_num_num @ C @ D ) )
= ( product_Pair_num_num @ ( plus_plus_num @ A @ C ) @ ( plus_plus_num @ B @ D ) ) ) ).
% add_Pair
thf(fact_1018_add__Pair,axiom,
! [A: num,B: real,C: num,D: real] :
( ( plus_p1695866243620610014m_real @ ( produc632589164708310495m_real @ A @ B ) @ ( produc632589164708310495m_real @ C @ D ) )
= ( produc632589164708310495m_real @ ( plus_plus_num @ A @ C ) @ ( plus_plus_real @ B @ D ) ) ) ).
% add_Pair
thf(fact_1019_add__Pair,axiom,
! [A: real,B: nat,C: real,D: nat] :
( ( plus_p4925795495032332052al_nat @ ( produc3181502643871035669al_nat @ A @ B ) @ ( produc3181502643871035669al_nat @ C @ D ) )
= ( produc3181502643871035669al_nat @ ( plus_plus_real @ A @ C ) @ ( plus_plus_nat @ B @ D ) ) ) ).
% add_Pair
thf(fact_1020_add__Pair,axiom,
! [A: real,B: num,C: real,D: num] :
( ( plus_p5634039431981169758al_num @ ( produc8962206466881590111al_num @ A @ B ) @ ( produc8962206466881590111al_num @ C @ D ) )
= ( produc8962206466881590111al_num @ ( plus_plus_real @ A @ C ) @ ( plus_plus_num @ B @ D ) ) ) ).
% add_Pair
thf(fact_1021_add__Pair,axiom,
! [A: real,B: real,C: real,D: real] :
( ( plus_p1196244663705802608l_real @ ( produc4511245868158468465l_real @ A @ B ) @ ( produc4511245868158468465l_real @ C @ D ) )
= ( produc4511245868158468465l_real @ ( plus_plus_real @ A @ C ) @ ( plus_plus_real @ B @ D ) ) ) ).
% add_Pair
thf(fact_1022_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_1023_minus__add__distrib,axiom,
! [A: real,B: real] :
( ( uminus_uminus_real @ ( plus_plus_real @ A @ B ) )
= ( plus_plus_real @ ( uminus_uminus_real @ A ) @ ( uminus_uminus_real @ B ) ) ) ).
% minus_add_distrib
thf(fact_1024_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_1025_minus__add__cancel,axiom,
! [A: real,B: real] :
( ( plus_plus_real @ ( uminus_uminus_real @ A ) @ ( plus_plus_real @ A @ B ) )
= B ) ).
% minus_add_cancel
thf(fact_1026_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_1027_add__minus__cancel,axiom,
! [A: real,B: real] :
( ( plus_plus_real @ A @ ( plus_plus_real @ ( uminus_uminus_real @ A ) @ B ) )
= B ) ).
% add_minus_cancel
thf(fact_1028_add__gr__0,axiom,
! [M2: nat,N3: nat] :
( ( ord_less_nat @ zero_zero_nat @ ( plus_plus_nat @ M2 @ N3 ) )
= ( ( ord_less_nat @ zero_zero_nat @ M2 )
| ( ord_less_nat @ zero_zero_nat @ N3 ) ) ) ).
% add_gr_0
thf(fact_1029_fst__add,axiom,
! [X: product_prod_num_num,Y: product_prod_num_num] :
( ( product_fst_num_num @ ( plus_p7895664801912576972um_num @ X @ Y ) )
= ( plus_plus_num @ ( product_fst_num_num @ X ) @ ( product_fst_num_num @ Y ) ) ) ).
% fst_add
thf(fact_1030_fst__add,axiom,
! [X: produc2422161461964618553l_real,Y: produc2422161461964618553l_real] :
( ( produc5828954698716094813l_real @ ( plus_p1196244663705802608l_real @ X @ Y ) )
= ( plus_plus_real @ ( produc5828954698716094813l_real @ X ) @ ( produc5828954698716094813l_real @ Y ) ) ) ).
% fst_add
thf(fact_1031_fst__add,axiom,
! [X: produc3741383161447143261al_nat,Y: produc3741383161447143261al_nat] :
( ( product_fst_real_nat @ ( plus_p4925795495032332052al_nat @ X @ Y ) )
= ( plus_plus_real @ ( product_fst_real_nat @ X ) @ ( product_fst_real_nat @ Y ) ) ) ).
% fst_add
thf(fact_1032_snd__add,axiom,
! [X: produc3741383161447143261al_nat,Y: produc3741383161447143261al_nat] :
( ( product_snd_real_nat @ ( plus_p4925795495032332052al_nat @ X @ Y ) )
= ( plus_plus_nat @ ( product_snd_real_nat @ X ) @ ( product_snd_real_nat @ Y ) ) ) ).
% snd_add
thf(fact_1033_snd__add,axiom,
! [X: product_prod_num_num,Y: product_prod_num_num] :
( ( product_snd_num_num @ ( plus_p7895664801912576972um_num @ X @ Y ) )
= ( plus_plus_num @ ( product_snd_num_num @ X ) @ ( product_snd_num_num @ Y ) ) ) ).
% snd_add
thf(fact_1034_snd__add,axiom,
! [X: produc2422161461964618553l_real,Y: produc2422161461964618553l_real] :
( ( produc3484788084999411615l_real @ ( plus_p1196244663705802608l_real @ X @ Y ) )
= ( plus_plus_real @ ( produc3484788084999411615l_real @ X ) @ ( produc3484788084999411615l_real @ Y ) ) ) ).
% snd_add
thf(fact_1035_mult__is__0,axiom,
! [M2: nat,N3: nat] :
( ( ( times_times_nat @ M2 @ N3 )
= zero_zero_nat )
= ( ( M2 = zero_zero_nat )
| ( N3 = zero_zero_nat ) ) ) ).
% mult_is_0
thf(fact_1036_mult__0__right,axiom,
! [M2: nat] :
( ( times_times_nat @ M2 @ zero_zero_nat )
= zero_zero_nat ) ).
% mult_0_right
thf(fact_1037_mult__cancel1,axiom,
! [K: nat,M2: nat,N3: nat] :
( ( ( times_times_nat @ K @ M2 )
= ( times_times_nat @ K @ N3 ) )
= ( ( M2 = N3 )
| ( K = zero_zero_nat ) ) ) ).
% mult_cancel1
thf(fact_1038_mult__cancel2,axiom,
! [M2: nat,K: nat,N3: nat] :
( ( ( times_times_nat @ M2 @ K )
= ( times_times_nat @ N3 @ K ) )
= ( ( M2 = N3 )
| ( K = zero_zero_nat ) ) ) ).
% mult_cancel2
thf(fact_1039_mult__Suc__right,axiom,
! [M2: nat,N3: nat] :
( ( times_times_nat @ M2 @ ( suc @ N3 ) )
= ( plus_plus_nat @ M2 @ ( times_times_nat @ M2 @ N3 ) ) ) ).
% mult_Suc_right
thf(fact_1040_nat__1__eq__mult__iff,axiom,
! [M2: nat,N3: nat] :
( ( one_one_nat
= ( times_times_nat @ M2 @ N3 ) )
= ( ( M2 = one_one_nat )
& ( N3 = one_one_nat ) ) ) ).
% nat_1_eq_mult_iff
thf(fact_1041_nat__mult__eq__1__iff,axiom,
! [M2: nat,N3: nat] :
( ( ( times_times_nat @ M2 @ N3 )
= one_one_nat )
= ( ( M2 = one_one_nat )
& ( N3 = one_one_nat ) ) ) ).
% nat_mult_eq_1_iff
thf(fact_1042_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_1043_add__le__same__cancel1,axiom,
! [B: real,A: real] :
( ( ord_less_eq_real @ ( plus_plus_real @ B @ A ) @ B )
= ( ord_less_eq_real @ A @ zero_zero_real ) ) ).
% add_le_same_cancel1
thf(fact_1044_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_1045_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_1046_add__le__same__cancel2,axiom,
! [A: real,B: real] :
( ( ord_less_eq_real @ ( plus_plus_real @ A @ B ) @ B )
= ( ord_less_eq_real @ A @ zero_zero_real ) ) ).
% add_le_same_cancel2
thf(fact_1047_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_1048_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_1049_le__add__same__cancel1,axiom,
! [A: real,B: real] :
( ( ord_less_eq_real @ A @ ( plus_plus_real @ A @ B ) )
= ( ord_less_eq_real @ zero_zero_real @ B ) ) ).
% le_add_same_cancel1
thf(fact_1050_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_1051_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_1052_le__add__same__cancel2,axiom,
! [A: real,B: real] :
( ( ord_less_eq_real @ A @ ( plus_plus_real @ B @ A ) )
= ( ord_less_eq_real @ zero_zero_real @ B ) ) ).
% le_add_same_cancel2
thf(fact_1053_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_1054_double__add__le__zero__iff__single__add__le__zero,axiom,
! [A: real] :
( ( ord_less_eq_real @ ( plus_plus_real @ A @ A ) @ zero_zero_real )
= ( ord_less_eq_real @ A @ zero_zero_real ) ) ).
% double_add_le_zero_iff_single_add_le_zero
thf(fact_1055_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_1056_zero__le__double__add__iff__zero__le__single__add,axiom,
! [A: real] :
( ( ord_less_eq_real @ zero_zero_real @ ( plus_plus_real @ A @ A ) )
= ( ord_less_eq_real @ zero_zero_real @ A ) ) ).
% zero_le_double_add_iff_zero_le_single_add
thf(fact_1057_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_1058_diff__ge__0__iff__ge,axiom,
! [A: real,B: real] :
( ( ord_less_eq_real @ zero_zero_real @ ( minus_minus_real @ A @ B ) )
= ( ord_less_eq_real @ B @ A ) ) ).
% diff_ge_0_iff_ge
thf(fact_1059_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_1060_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_1061_neg__0__le__iff__le,axiom,
! [A: real] :
( ( ord_less_eq_real @ zero_zero_real @ ( uminus_uminus_real @ A ) )
= ( ord_less_eq_real @ A @ zero_zero_real ) ) ).
% neg_0_le_iff_le
thf(fact_1062_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_1063_neg__le__0__iff__le,axiom,
! [A: real] :
( ( ord_less_eq_real @ ( uminus_uminus_real @ A ) @ zero_zero_real )
= ( ord_less_eq_real @ zero_zero_real @ A ) ) ).
% neg_le_0_iff_le
thf(fact_1064_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_1065_less__eq__neg__nonpos,axiom,
! [A: real] :
( ( ord_less_eq_real @ A @ ( uminus_uminus_real @ A ) )
= ( ord_less_eq_real @ A @ zero_zero_real ) ) ).
% less_eq_neg_nonpos
thf(fact_1066_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_1067_neg__less__eq__nonneg,axiom,
! [A: real] :
( ( ord_less_eq_real @ ( uminus_uminus_real @ A ) @ A )
= ( ord_less_eq_real @ zero_zero_real @ A ) ) ).
% neg_less_eq_nonneg
thf(fact_1068_add__less__same__cancel1,axiom,
! [B: real,A: real] :
( ( ord_less_real @ ( plus_plus_real @ B @ A ) @ B )
= ( ord_less_real @ A @ zero_zero_real ) ) ).
% add_less_same_cancel1
thf(fact_1069_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_1070_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_1071_add__less__same__cancel2,axiom,
! [A: real,B: real] :
( ( ord_less_real @ ( plus_plus_real @ A @ B ) @ B )
= ( ord_less_real @ A @ zero_zero_real ) ) ).
% add_less_same_cancel2
thf(fact_1072_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_1073_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_1074_less__add__same__cancel1,axiom,
! [A: real,B: real] :
( ( ord_less_real @ A @ ( plus_plus_real @ A @ B ) )
= ( ord_less_real @ zero_zero_real @ B ) ) ).
% less_add_same_cancel1
thf(fact_1075_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_1076_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_1077_less__add__same__cancel2,axiom,
! [A: real,B: real] :
( ( ord_less_real @ A @ ( plus_plus_real @ B @ A ) )
= ( ord_less_real @ zero_zero_real @ B ) ) ).
% less_add_same_cancel2
thf(fact_1078_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_1079_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_1080_double__add__less__zero__iff__single__add__less__zero,axiom,
! [A: real] :
( ( ord_less_real @ ( plus_plus_real @ A @ A ) @ zero_zero_real )
= ( ord_less_real @ A @ zero_zero_real ) ) ).
% double_add_less_zero_iff_single_add_less_zero
thf(fact_1081_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_1082_zero__less__double__add__iff__zero__less__single__add,axiom,
! [A: real] :
( ( ord_less_real @ zero_zero_real @ ( plus_plus_real @ A @ A ) )
= ( ord_less_real @ zero_zero_real @ A ) ) ).
% zero_less_double_add_iff_zero_less_single_add
thf(fact_1083_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_1084_mult__cancel__right2,axiom,
! [A: real,C: real] :
( ( ( times_times_real @ A @ C )
= C )
= ( ( C = zero_zero_real )
| ( A = one_one_real ) ) ) ).
% mult_cancel_right2
thf(fact_1085_mult__cancel__right2,axiom,
! [A: int,C: int] :
( ( ( times_times_int @ A @ C )
= C )
= ( ( C = zero_zero_int )
| ( A = one_one_int ) ) ) ).
% mult_cancel_right2
thf(fact_1086_mult__cancel__right1,axiom,
! [C: real,B: real] :
( ( C
= ( times_times_real @ B @ C ) )
= ( ( C = zero_zero_real )
| ( B = one_one_real ) ) ) ).
% mult_cancel_right1
thf(fact_1087_mult__cancel__right1,axiom,
! [C: int,B: int] :
( ( C
= ( times_times_int @ B @ C ) )
= ( ( C = zero_zero_int )
| ( B = one_one_int ) ) ) ).
% mult_cancel_right1
thf(fact_1088_mult__cancel__left2,axiom,
! [C: real,A: real] :
( ( ( times_times_real @ C @ A )
= C )
= ( ( C = zero_zero_real )
| ( A = one_one_real ) ) ) ).
% mult_cancel_left2
thf(fact_1089_mult__cancel__left2,axiom,
! [C: int,A: int] :
( ( ( times_times_int @ C @ A )
= C )
= ( ( C = zero_zero_int )
| ( A = one_one_int ) ) ) ).
% mult_cancel_left2
thf(fact_1090_mult__cancel__left1,axiom,
! [C: real,B: real] :
( ( C
= ( times_times_real @ C @ B ) )
= ( ( C = zero_zero_real )
| ( B = one_one_real ) ) ) ).
% mult_cancel_left1
thf(fact_1091_mult__cancel__left1,axiom,
! [C: int,B: int] :
( ( C
= ( times_times_int @ C @ B ) )
= ( ( C = zero_zero_int )
| ( B = one_one_int ) ) ) ).
% mult_cancel_left1
thf(fact_1092_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_1093_le__add__diff__inverse,axiom,
! [B: real,A: real] :
( ( ord_less_eq_real @ B @ A )
=> ( ( plus_plus_real @ B @ ( minus_minus_real @ A @ B ) )
= A ) ) ).
% le_add_diff_inverse
thf(fact_1094_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_1095_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_1096_le__add__diff__inverse2,axiom,
! [B: real,A: real] :
( ( ord_less_eq_real @ B @ A )
=> ( ( plus_plus_real @ ( minus_minus_real @ A @ B ) @ B )
= A ) ) ).
% le_add_diff_inverse2
thf(fact_1097_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_1098_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_1099_nonzero__mult__divide__mult__cancel__right2,axiom,
! [C: real,A: real,B: real] :
( ( C != zero_zero_real )
=> ( ( divide_divide_real @ ( times_times_real @ A @ C ) @ ( times_times_real @ C @ B ) )
= ( divide_divide_real @ A @ B ) ) ) ).
% nonzero_mult_divide_mult_cancel_right2
thf(fact_1100_nonzero__mult__divide__mult__cancel__right,axiom,
! [C: real,A: real,B: real] :
( ( C != zero_zero_real )
=> ( ( divide_divide_real @ ( times_times_real @ A @ C ) @ ( times_times_real @ B @ C ) )
= ( divide_divide_real @ A @ B ) ) ) ).
% nonzero_mult_divide_mult_cancel_right
thf(fact_1101_nonzero__mult__divide__mult__cancel__left2,axiom,
! [C: real,A: real,B: real] :
( ( C != zero_zero_real )
=> ( ( divide_divide_real @ ( times_times_real @ C @ A ) @ ( times_times_real @ B @ C ) )
= ( divide_divide_real @ A @ B ) ) ) ).
% nonzero_mult_divide_mult_cancel_left2
thf(fact_1102_nonzero__mult__divide__mult__cancel__left,axiom,
! [C: real,A: real,B: real] :
( ( C != zero_zero_real )
=> ( ( divide_divide_real @ ( times_times_real @ C @ A ) @ ( times_times_real @ C @ B ) )
= ( divide_divide_real @ A @ B ) ) ) ).
% nonzero_mult_divide_mult_cancel_left
thf(fact_1103_mult__divide__mult__cancel__left__if,axiom,
! [C: real,A: real,B: real] :
( ( ( C = zero_zero_real )
=> ( ( divide_divide_real @ ( times_times_real @ C @ A ) @ ( times_times_real @ C @ B ) )
= zero_zero_real ) )
& ( ( C != zero_zero_real )
=> ( ( divide_divide_real @ ( times_times_real @ C @ A ) @ ( times_times_real @ C @ B ) )
= ( divide_divide_real @ A @ B ) ) ) ) ).
% mult_divide_mult_cancel_left_if
thf(fact_1104_nonzero__mult__div__cancel__left,axiom,
! [A: real,B: real] :
( ( A != zero_zero_real )
=> ( ( divide_divide_real @ ( times_times_real @ A @ B ) @ A )
= B ) ) ).
% nonzero_mult_div_cancel_left
thf(fact_1105_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_1106_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_1107_nonzero__mult__div__cancel__right,axiom,
! [B: real,A: real] :
( ( B != zero_zero_real )
=> ( ( divide_divide_real @ ( times_times_real @ A @ B ) @ B )
= A ) ) ).
% nonzero_mult_div_cancel_right
thf(fact_1108_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_1109_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_1110_div__mult__mult1__if,axiom,
! [C: nat,A: nat,B: nat] :
( ( ( C = zero_zero_nat )
=> ( ( divide_divide_nat @ ( times_times_nat @ C @ A ) @ ( times_times_nat @ C @ B ) )
= zero_zero_nat ) )
& ( ( C != zero_zero_nat )
=> ( ( divide_divide_nat @ ( times_times_nat @ C @ A ) @ ( times_times_nat @ C @ B ) )
= ( divide_divide_nat @ A @ B ) ) ) ) ).
% div_mult_mult1_if
thf(fact_1111_div__mult__mult1__if,axiom,
! [C: int,A: int,B: int] :
( ( ( C = zero_zero_int )
=> ( ( divide_divide_int @ ( times_times_int @ C @ A ) @ ( times_times_int @ C @ B ) )
= zero_zero_int ) )
& ( ( C != zero_zero_int )
=> ( ( divide_divide_int @ ( times_times_int @ C @ A ) @ ( times_times_int @ C @ B ) )
= ( divide_divide_int @ A @ B ) ) ) ) ).
% div_mult_mult1_if
thf(fact_1112_div__mult__mult2,axiom,
! [C: nat,A: nat,B: nat] :
( ( C != zero_zero_nat )
=> ( ( divide_divide_nat @ ( times_times_nat @ A @ C ) @ ( times_times_nat @ B @ C ) )
= ( divide_divide_nat @ A @ B ) ) ) ).
% div_mult_mult2
thf(fact_1113_div__mult__mult2,axiom,
! [C: int,A: int,B: int] :
( ( C != zero_zero_int )
=> ( ( divide_divide_int @ ( times_times_int @ A @ C ) @ ( times_times_int @ B @ C ) )
= ( divide_divide_int @ A @ B ) ) ) ).
% div_mult_mult2
thf(fact_1114_div__mult__mult1,axiom,
! [C: nat,A: nat,B: nat] :
( ( C != zero_zero_nat )
=> ( ( divide_divide_nat @ ( times_times_nat @ C @ A ) @ ( times_times_nat @ C @ B ) )
= ( divide_divide_nat @ A @ B ) ) ) ).
% div_mult_mult1
thf(fact_1115_div__mult__mult1,axiom,
! [C: int,A: int,B: int] :
( ( C != zero_zero_int )
=> ( ( divide_divide_int @ ( times_times_int @ C @ A ) @ ( times_times_int @ C @ B ) )
= ( divide_divide_int @ A @ B ) ) ) ).
% div_mult_mult1
thf(fact_1116_add_Oright__inverse,axiom,
! [A: int] :
( ( plus_plus_int @ A @ ( uminus_uminus_int @ A ) )
= zero_zero_int ) ).
% add.right_inverse
thf(fact_1117_add_Oright__inverse,axiom,
! [A: real] :
( ( plus_plus_real @ A @ ( uminus_uminus_real @ A ) )
= zero_zero_real ) ).
% add.right_inverse
thf(fact_1118_ab__left__minus,axiom,
! [A: int] :
( ( plus_plus_int @ ( uminus_uminus_int @ A ) @ A )
= zero_zero_int ) ).
% ab_left_minus
thf(fact_1119_ab__left__minus,axiom,
! [A: real] :
( ( plus_plus_real @ ( uminus_uminus_real @ A ) @ A )
= zero_zero_real ) ).
% ab_left_minus
thf(fact_1120_distrib__right__numeral,axiom,
! [A: extended_enat,B: extended_enat,V2: num] :
( ( times_7803423173614009249d_enat @ ( plus_p3455044024723400733d_enat @ A @ B ) @ ( numera1916890842035813515d_enat @ V2 ) )
= ( plus_p3455044024723400733d_enat @ ( times_7803423173614009249d_enat @ A @ ( numera1916890842035813515d_enat @ V2 ) ) @ ( times_7803423173614009249d_enat @ B @ ( numera1916890842035813515d_enat @ V2 ) ) ) ) ).
% distrib_right_numeral
thf(fact_1121_distrib__right__numeral,axiom,
! [A: nat,B: nat,V2: num] :
( ( times_times_nat @ ( plus_plus_nat @ A @ B ) @ ( numeral_numeral_nat @ V2 ) )
= ( plus_plus_nat @ ( times_times_nat @ A @ ( numeral_numeral_nat @ V2 ) ) @ ( times_times_nat @ B @ ( numeral_numeral_nat @ V2 ) ) ) ) ).
% distrib_right_numeral
thf(fact_1122_distrib__right__numeral,axiom,
! [A: extend8495563244428889912nnreal,B: extend8495563244428889912nnreal,V2: num] :
( ( times_1893300245718287421nnreal @ ( plus_p1859984266308609217nnreal @ A @ B ) @ ( numera4658534427948366547nnreal @ V2 ) )
= ( plus_p1859984266308609217nnreal @ ( times_1893300245718287421nnreal @ A @ ( numera4658534427948366547nnreal @ V2 ) ) @ ( times_1893300245718287421nnreal @ B @ ( numera4658534427948366547nnreal @ V2 ) ) ) ) ).
% distrib_right_numeral
thf(fact_1123_distrib__right__numeral,axiom,
! [A: real,B: real,V2: num] :
( ( times_times_real @ ( plus_plus_real @ A @ B ) @ ( numeral_numeral_real @ V2 ) )
= ( plus_plus_real @ ( times_times_real @ A @ ( numeral_numeral_real @ V2 ) ) @ ( times_times_real @ B @ ( numeral_numeral_real @ V2 ) ) ) ) ).
% distrib_right_numeral
thf(fact_1124_distrib__right__numeral,axiom,
! [A: int,B: int,V2: num] :
( ( times_times_int @ ( plus_plus_int @ A @ B ) @ ( numeral_numeral_int @ V2 ) )
= ( plus_plus_int @ ( times_times_int @ A @ ( numeral_numeral_int @ V2 ) ) @ ( times_times_int @ B @ ( numeral_numeral_int @ V2 ) ) ) ) ).
% distrib_right_numeral
thf(fact_1125_distrib__left__numeral,axiom,
! [V2: num,B: extended_enat,C: extended_enat] :
( ( times_7803423173614009249d_enat @ ( numera1916890842035813515d_enat @ V2 ) @ ( plus_p3455044024723400733d_enat @ B @ C ) )
= ( plus_p3455044024723400733d_enat @ ( times_7803423173614009249d_enat @ ( numera1916890842035813515d_enat @ V2 ) @ B ) @ ( times_7803423173614009249d_enat @ ( numera1916890842035813515d_enat @ V2 ) @ C ) ) ) ).
% distrib_left_numeral
thf(fact_1126_distrib__left__numeral,axiom,
! [V2: num,B: nat,C: nat] :
( ( times_times_nat @ ( numeral_numeral_nat @ V2 ) @ ( plus_plus_nat @ B @ C ) )
= ( plus_plus_nat @ ( times_times_nat @ ( numeral_numeral_nat @ V2 ) @ B ) @ ( times_times_nat @ ( numeral_numeral_nat @ V2 ) @ C ) ) ) ).
% distrib_left_numeral
thf(fact_1127_distrib__left__numeral,axiom,
! [V2: num,B: extend8495563244428889912nnreal,C: extend8495563244428889912nnreal] :
( ( times_1893300245718287421nnreal @ ( numera4658534427948366547nnreal @ V2 ) @ ( plus_p1859984266308609217nnreal @ B @ C ) )
= ( plus_p1859984266308609217nnreal @ ( times_1893300245718287421nnreal @ ( numera4658534427948366547nnreal @ V2 ) @ B ) @ ( times_1893300245718287421nnreal @ ( numera4658534427948366547nnreal @ V2 ) @ C ) ) ) ).
% distrib_left_numeral
thf(fact_1128_distrib__left__numeral,axiom,
! [V2: num,B: real,C: real] :
( ( times_times_real @ ( numeral_numeral_real @ V2 ) @ ( plus_plus_real @ B @ C ) )
= ( plus_plus_real @ ( times_times_real @ ( numeral_numeral_real @ V2 ) @ B ) @ ( times_times_real @ ( numeral_numeral_real @ V2 ) @ C ) ) ) ).
% distrib_left_numeral
thf(fact_1129_distrib__left__numeral,axiom,
! [V2: num,B: int,C: int] :
( ( times_times_int @ ( numeral_numeral_int @ V2 ) @ ( plus_plus_int @ B @ C ) )
= ( plus_plus_int @ ( times_times_int @ ( numeral_numeral_int @ V2 ) @ B ) @ ( times_times_int @ ( numeral_numeral_int @ V2 ) @ C ) ) ) ).
% distrib_left_numeral
thf(fact_1130_right__diff__distrib__numeral,axiom,
! [V2: num,B: real,C: real] :
( ( times_times_real @ ( numeral_numeral_real @ V2 ) @ ( minus_minus_real @ B @ C ) )
= ( minus_minus_real @ ( times_times_real @ ( numeral_numeral_real @ V2 ) @ B ) @ ( times_times_real @ ( numeral_numeral_real @ V2 ) @ C ) ) ) ).
% right_diff_distrib_numeral
thf(fact_1131_right__diff__distrib__numeral,axiom,
! [V2: num,B: int,C: int] :
( ( times_times_int @ ( numeral_numeral_int @ V2 ) @ ( minus_minus_int @ B @ C ) )
= ( minus_minus_int @ ( times_times_int @ ( numeral_numeral_int @ V2 ) @ B ) @ ( times_times_int @ ( numeral_numeral_int @ V2 ) @ C ) ) ) ).
% right_diff_distrib_numeral
thf(fact_1132_left__diff__distrib__numeral,axiom,
! [A: real,B: real,V2: num] :
( ( times_times_real @ ( minus_minus_real @ A @ B ) @ ( numeral_numeral_real @ V2 ) )
= ( minus_minus_real @ ( times_times_real @ A @ ( numeral_numeral_real @ V2 ) ) @ ( times_times_real @ B @ ( numeral_numeral_real @ V2 ) ) ) ) ).
% left_diff_distrib_numeral
thf(fact_1133_left__diff__distrib__numeral,axiom,
! [A: int,B: int,V2: num] :
( ( times_times_int @ ( minus_minus_int @ A @ B ) @ ( numeral_numeral_int @ V2 ) )
= ( minus_minus_int @ ( times_times_int @ A @ ( numeral_numeral_int @ V2 ) ) @ ( times_times_int @ B @ ( numeral_numeral_int @ V2 ) ) ) ) ).
% left_diff_distrib_numeral
thf(fact_1134_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_1135_mult__minus1__right,axiom,
! [Z2: real] :
( ( times_times_real @ Z2 @ ( uminus_uminus_real @ one_one_real ) )
= ( uminus_uminus_real @ Z2 ) ) ).
% mult_minus1_right
thf(fact_1136_mult__minus1,axiom,
! [Z2: int] :
( ( times_times_int @ ( uminus_uminus_int @ one_one_int ) @ Z2 )
= ( uminus_uminus_int @ Z2 ) ) ).
% mult_minus1
thf(fact_1137_mult__minus1,axiom,
! [Z2: real] :
( ( times_times_real @ ( uminus_uminus_real @ one_one_real ) @ Z2 )
= ( uminus_uminus_real @ Z2 ) ) ).
% mult_minus1
thf(fact_1138_mult__neg__numeral__simps_I3_J,axiom,
! [M2: num,N3: num] :
( ( times_times_int @ ( numeral_numeral_int @ M2 ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ N3 ) ) )
= ( uminus_uminus_int @ ( numeral_numeral_int @ ( times_times_num @ M2 @ N3 ) ) ) ) ).
% mult_neg_numeral_simps(3)
thf(fact_1139_mult__neg__numeral__simps_I3_J,axiom,
! [M2: num,N3: num] :
( ( times_times_real @ ( numeral_numeral_real @ M2 ) @ ( uminus_uminus_real @ ( numeral_numeral_real @ N3 ) ) )
= ( uminus_uminus_real @ ( numeral_numeral_real @ ( times_times_num @ M2 @ N3 ) ) ) ) ).
% mult_neg_numeral_simps(3)
thf(fact_1140_mult__neg__numeral__simps_I2_J,axiom,
! [M2: num,N3: num] :
( ( times_times_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ M2 ) ) @ ( numeral_numeral_int @ N3 ) )
= ( uminus_uminus_int @ ( numeral_numeral_int @ ( times_times_num @ M2 @ N3 ) ) ) ) ).
% mult_neg_numeral_simps(2)
thf(fact_1141_mult__neg__numeral__simps_I2_J,axiom,
! [M2: num,N3: num] :
( ( times_times_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ M2 ) ) @ ( numeral_numeral_real @ N3 ) )
= ( uminus_uminus_real @ ( numeral_numeral_real @ ( times_times_num @ M2 @ N3 ) ) ) ) ).
% mult_neg_numeral_simps(2)
thf(fact_1142_mult__neg__numeral__simps_I1_J,axiom,
! [M2: num,N3: num] :
( ( times_times_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ M2 ) ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ N3 ) ) )
= ( numeral_numeral_int @ ( times_times_num @ M2 @ N3 ) ) ) ).
% mult_neg_numeral_simps(1)
thf(fact_1143_mult__neg__numeral__simps_I1_J,axiom,
! [M2: num,N3: num] :
( ( times_times_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ M2 ) ) @ ( uminus_uminus_real @ ( numeral_numeral_real @ N3 ) ) )
= ( numeral_numeral_real @ ( times_times_num @ M2 @ N3 ) ) ) ).
% mult_neg_numeral_simps(1)
thf(fact_1144_add__neg__numeral__simps_I3_J,axiom,
! [M2: num,N3: num] :
( ( plus_plus_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ M2 ) ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ N3 ) ) )
= ( uminus_uminus_int @ ( plus_plus_int @ ( numeral_numeral_int @ M2 ) @ ( numeral_numeral_int @ N3 ) ) ) ) ).
% add_neg_numeral_simps(3)
thf(fact_1145_add__neg__numeral__simps_I3_J,axiom,
! [M2: num,N3: num] :
( ( plus_plus_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ M2 ) ) @ ( uminus_uminus_real @ ( numeral_numeral_real @ N3 ) ) )
= ( uminus_uminus_real @ ( plus_plus_real @ ( numeral_numeral_real @ M2 ) @ ( numeral_numeral_real @ N3 ) ) ) ) ).
% add_neg_numeral_simps(3)
thf(fact_1146_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_1147_uminus__add__conv__diff,axiom,
! [A: real,B: real] :
( ( plus_plus_real @ ( uminus_uminus_real @ A ) @ B )
= ( minus_minus_real @ B @ A ) ) ).
% uminus_add_conv_diff
thf(fact_1148_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_1149_diff__minus__eq__add,axiom,
! [A: real,B: real] :
( ( minus_minus_real @ A @ ( uminus_uminus_real @ B ) )
= ( plus_plus_real @ A @ B ) ) ).
% diff_minus_eq_add
thf(fact_1150_diff__numeral__simps_I2_J,axiom,
! [M2: num,N3: num] :
( ( minus_minus_int @ ( numeral_numeral_int @ M2 ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ N3 ) ) )
= ( numeral_numeral_int @ ( plus_plus_num @ M2 @ N3 ) ) ) ).
% diff_numeral_simps(2)
thf(fact_1151_diff__numeral__simps_I2_J,axiom,
! [M2: num,N3: num] :
( ( minus_minus_real @ ( numeral_numeral_real @ M2 ) @ ( uminus_uminus_real @ ( numeral_numeral_real @ N3 ) ) )
= ( numeral_numeral_real @ ( plus_plus_num @ M2 @ N3 ) ) ) ).
% diff_numeral_simps(2)
thf(fact_1152_diff__numeral__simps_I3_J,axiom,
! [M2: num,N3: num] :
( ( minus_minus_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ M2 ) ) @ ( numeral_numeral_int @ N3 ) )
= ( uminus_uminus_int @ ( numeral_numeral_int @ ( plus_plus_num @ M2 @ N3 ) ) ) ) ).
% diff_numeral_simps(3)
thf(fact_1153_diff__numeral__simps_I3_J,axiom,
! [M2: num,N3: num] :
( ( minus_minus_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ M2 ) ) @ ( numeral_numeral_real @ N3 ) )
= ( uminus_uminus_real @ ( numeral_numeral_real @ ( plus_plus_num @ M2 @ N3 ) ) ) ) ).
% diff_numeral_simps(3)
thf(fact_1154_Suc__numeral,axiom,
! [N3: num] :
( ( suc @ ( numeral_numeral_nat @ N3 ) )
= ( numeral_numeral_nat @ ( plus_plus_num @ N3 @ one ) ) ) ).
% Suc_numeral
thf(fact_1155_one__eq__mult__iff,axiom,
! [M2: nat,N3: nat] :
( ( ( suc @ zero_zero_nat )
= ( times_times_nat @ M2 @ N3 ) )
= ( ( M2
= ( suc @ zero_zero_nat ) )
& ( N3
= ( suc @ zero_zero_nat ) ) ) ) ).
% one_eq_mult_iff
thf(fact_1156_mult__eq__1__iff,axiom,
! [M2: nat,N3: nat] :
( ( ( times_times_nat @ M2 @ N3 )
= ( suc @ zero_zero_nat ) )
= ( ( M2
= ( suc @ zero_zero_nat ) )
& ( N3
= ( suc @ zero_zero_nat ) ) ) ) ).
% mult_eq_1_iff
thf(fact_1157_mult__less__cancel2,axiom,
! [M2: nat,K: nat,N3: nat] :
( ( ord_less_nat @ ( times_times_nat @ M2 @ K ) @ ( times_times_nat @ N3 @ K ) )
= ( ( ord_less_nat @ zero_zero_nat @ K )
& ( ord_less_nat @ M2 @ N3 ) ) ) ).
% mult_less_cancel2
thf(fact_1158_nat__0__less__mult__iff,axiom,
! [M2: nat,N3: nat] :
( ( ord_less_nat @ zero_zero_nat @ ( times_times_nat @ M2 @ N3 ) )
= ( ( ord_less_nat @ zero_zero_nat @ M2 )
& ( ord_less_nat @ zero_zero_nat @ N3 ) ) ) ).
% nat_0_less_mult_iff
thf(fact_1159_nat__mult__less__cancel__disj,axiom,
! [K: nat,M2: nat,N3: nat] :
( ( ord_less_nat @ ( times_times_nat @ K @ M2 ) @ ( times_times_nat @ K @ N3 ) )
= ( ( ord_less_nat @ zero_zero_nat @ K )
& ( ord_less_nat @ M2 @ N3 ) ) ) ).
% nat_mult_less_cancel_disj
thf(fact_1160_numeral__le__iff,axiom,
! [M2: num,N3: num] :
( ( ord_le2932123472753598470d_enat @ ( numera1916890842035813515d_enat @ M2 ) @ ( numera1916890842035813515d_enat @ N3 ) )
= ( ord_less_eq_num @ M2 @ N3 ) ) ).
% numeral_le_iff
thf(fact_1161_numeral__le__iff,axiom,
! [M2: num,N3: num] :
( ( ord_less_eq_nat @ ( numeral_numeral_nat @ M2 ) @ ( numeral_numeral_nat @ N3 ) )
= ( ord_less_eq_num @ M2 @ N3 ) ) ).
% numeral_le_iff
thf(fact_1162_numeral__le__iff,axiom,
! [M2: num,N3: num] :
( ( ord_le3935885782089961368nnreal @ ( numera4658534427948366547nnreal @ M2 ) @ ( numera4658534427948366547nnreal @ N3 ) )
= ( ord_less_eq_num @ M2 @ N3 ) ) ).
% numeral_le_iff
thf(fact_1163_numeral__le__iff,axiom,
! [M2: num,N3: num] :
( ( ord_less_eq_real @ ( numeral_numeral_real @ M2 ) @ ( numeral_numeral_real @ N3 ) )
= ( ord_less_eq_num @ M2 @ N3 ) ) ).
% numeral_le_iff
thf(fact_1164_numeral__le__iff,axiom,
! [M2: num,N3: num] :
( ( ord_less_eq_int @ ( numeral_numeral_int @ M2 ) @ ( numeral_numeral_int @ N3 ) )
= ( ord_less_eq_num @ M2 @ N3 ) ) ).
% numeral_le_iff
thf(fact_1165_round__0,axiom,
( ( archim8280529875227126926d_real @ zero_zero_real )
= zero_zero_int ) ).
% round_0
thf(fact_1166_round__numeral,axiom,
! [N3: num] :
( ( archim8280529875227126926d_real @ ( numeral_numeral_real @ N3 ) )
= ( numeral_numeral_int @ N3 ) ) ).
% round_numeral
thf(fact_1167_round__1,axiom,
( ( archim8280529875227126926d_real @ one_one_real )
= one_one_int ) ).
% round_1
thf(fact_1168_divide__le__0__1__iff,axiom,
! [A: real] :
( ( ord_less_eq_real @ ( divide_divide_real @ one_one_real @ A ) @ zero_zero_real )
= ( ord_less_eq_real @ A @ zero_zero_real ) ) ).
% divide_le_0_1_iff
thf(fact_1169_zero__le__divide__1__iff,axiom,
! [A: real] :
( ( ord_less_eq_real @ zero_zero_real @ ( divide_divide_real @ one_one_real @ A ) )
= ( ord_less_eq_real @ zero_zero_real @ A ) ) ).
% zero_le_divide_1_iff
thf(fact_1170_nonzero__divide__mult__cancel__left,axiom,
! [A: real,B: real] :
( ( A != zero_zero_real )
=> ( ( divide_divide_real @ A @ ( times_times_real @ A @ B ) )
= ( divide_divide_real @ one_one_real @ B ) ) ) ).
% nonzero_divide_mult_cancel_left
thf(fact_1171_nonzero__divide__mult__cancel__right,axiom,
! [B: real,A: real] :
( ( B != zero_zero_real )
=> ( ( divide_divide_real @ B @ ( times_times_real @ A @ B ) )
= ( divide_divide_real @ one_one_real @ A ) ) ) ).
% nonzero_divide_mult_cancel_right
thf(fact_1172_divide__le__eq__numeral1_I1_J,axiom,
! [B: real,W: num,A: real] :
( ( ord_less_eq_real @ ( divide_divide_real @ B @ ( numeral_numeral_real @ W ) ) @ A )
= ( ord_less_eq_real @ B @ ( times_times_real @ A @ ( numeral_numeral_real @ W ) ) ) ) ).
% divide_le_eq_numeral1(1)
thf(fact_1173_le__divide__eq__numeral1_I1_J,axiom,
! [A: real,B: real,W: num] :
( ( ord_less_eq_real @ A @ ( divide_divide_real @ B @ ( numeral_numeral_real @ W ) ) )
= ( ord_less_eq_real @ ( times_times_real @ A @ ( numeral_numeral_real @ W ) ) @ B ) ) ).
% le_divide_eq_numeral1(1)
thf(fact_1174_div__mult__self4,axiom,
! [B: nat,C: nat,A: nat] :
( ( B != zero_zero_nat )
=> ( ( divide_divide_nat @ ( plus_plus_nat @ ( times_times_nat @ B @ C ) @ A ) @ B )
= ( plus_plus_nat @ C @ ( divide_divide_nat @ A @ B ) ) ) ) ).
% div_mult_self4
thf(fact_1175_div__mult__self4,axiom,
! [B: int,C: int,A: int] :
( ( B != zero_zero_int )
=> ( ( divide_divide_int @ ( plus_plus_int @ ( times_times_int @ B @ C ) @ A ) @ B )
= ( plus_plus_int @ C @ ( divide_divide_int @ A @ B ) ) ) ) ).
% div_mult_self4
thf(fact_1176_div__mult__self3,axiom,
! [B: nat,C: nat,A: nat] :
( ( B != zero_zero_nat )
=> ( ( divide_divide_nat @ ( plus_plus_nat @ ( times_times_nat @ C @ B ) @ A ) @ B )
= ( plus_plus_nat @ C @ ( divide_divide_nat @ A @ B ) ) ) ) ).
% div_mult_self3
thf(fact_1177_div__mult__self3,axiom,
! [B: int,C: int,A: int] :
( ( B != zero_zero_int )
=> ( ( divide_divide_int @ ( plus_plus_int @ ( times_times_int @ C @ B ) @ A ) @ B )
= ( plus_plus_int @ C @ ( divide_divide_int @ A @ B ) ) ) ) ).
% div_mult_self3
thf(fact_1178_div__mult__self2,axiom,
! [B: nat,A: nat,C: nat] :
( ( B != zero_zero_nat )
=> ( ( divide_divide_nat @ ( plus_plus_nat @ A @ ( times_times_nat @ B @ C ) ) @ B )
= ( plus_plus_nat @ C @ ( divide_divide_nat @ A @ B ) ) ) ) ).
% div_mult_self2
thf(fact_1179_div__mult__self2,axiom,
! [B: int,A: int,C: int] :
( ( B != zero_zero_int )
=> ( ( divide_divide_int @ ( plus_plus_int @ A @ ( times_times_int @ B @ C ) ) @ B )
= ( plus_plus_int @ C @ ( divide_divide_int @ A @ B ) ) ) ) ).
% div_mult_self2
thf(fact_1180_div__mult__self1,axiom,
! [B: nat,A: nat,C: nat] :
( ( B != zero_zero_nat )
=> ( ( divide_divide_nat @ ( plus_plus_nat @ A @ ( times_times_nat @ C @ B ) ) @ B )
= ( plus_plus_nat @ C @ ( divide_divide_nat @ A @ B ) ) ) ) ).
% div_mult_self1
thf(fact_1181_div__mult__self1,axiom,
! [B: int,A: int,C: int] :
( ( B != zero_zero_int )
=> ( ( divide_divide_int @ ( plus_plus_int @ A @ ( times_times_int @ C @ B ) ) @ B )
= ( plus_plus_int @ C @ ( divide_divide_int @ A @ B ) ) ) ) ).
% div_mult_self1
thf(fact_1182_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_1183_add__neg__numeral__special_I8_J,axiom,
( ( plus_plus_real @ ( uminus_uminus_real @ one_one_real ) @ one_one_real )
= zero_zero_real ) ).
% add_neg_numeral_special(8)
thf(fact_1184_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_1185_add__neg__numeral__special_I7_J,axiom,
( ( plus_plus_real @ one_one_real @ ( uminus_uminus_real @ one_one_real ) )
= zero_zero_real ) ).
% add_neg_numeral_special(7)
thf(fact_1186_divide__eq__eq__numeral1_I1_J,axiom,
! [B: real,W: num,A: real] :
( ( ( divide_divide_real @ B @ ( numeral_numeral_real @ W ) )
= A )
= ( ( ( ( numeral_numeral_real @ W )
!= zero_zero_real )
=> ( B
= ( times_times_real @ A @ ( numeral_numeral_real @ W ) ) ) )
& ( ( ( numeral_numeral_real @ W )
= zero_zero_real )
=> ( A = zero_zero_real ) ) ) ) ).
% divide_eq_eq_numeral1(1)
thf(fact_1187_eq__divide__eq__numeral1_I1_J,axiom,
! [A: real,B: real,W: num] :
( ( A
= ( divide_divide_real @ B @ ( numeral_numeral_real @ W ) ) )
= ( ( ( ( numeral_numeral_real @ W )
!= zero_zero_real )
=> ( ( times_times_real @ A @ ( numeral_numeral_real @ W ) )
= B ) )
& ( ( ( numeral_numeral_real @ W )
= zero_zero_real )
=> ( A = zero_zero_real ) ) ) ) ).
% eq_divide_eq_numeral1(1)
thf(fact_1188_divide__less__eq__numeral1_I1_J,axiom,
! [B: real,W: num,A: real] :
( ( ord_less_real @ ( divide_divide_real @ B @ ( numeral_numeral_real @ W ) ) @ A )
= ( ord_less_real @ B @ ( times_times_real @ A @ ( numeral_numeral_real @ W ) ) ) ) ).
% divide_less_eq_numeral1(1)
thf(fact_1189_less__divide__eq__numeral1_I1_J,axiom,
! [A: real,B: real,W: num] :
( ( ord_less_real @ A @ ( divide_divide_real @ B @ ( numeral_numeral_real @ W ) ) )
= ( ord_less_real @ ( times_times_real @ A @ ( numeral_numeral_real @ W ) ) @ B ) ) ).
% less_divide_eq_numeral1(1)
thf(fact_1190_one__plus__numeral,axiom,
! [N3: num] :
( ( plus_p3455044024723400733d_enat @ one_on7984719198319812577d_enat @ ( numera1916890842035813515d_enat @ N3 ) )
= ( numera1916890842035813515d_enat @ ( plus_plus_num @ one @ N3 ) ) ) ).
% one_plus_numeral
thf(fact_1191_one__plus__numeral,axiom,
! [N3: num] :
( ( plus_plus_nat @ one_one_nat @ ( numeral_numeral_nat @ N3 ) )
= ( numeral_numeral_nat @ ( plus_plus_num @ one @ N3 ) ) ) ).
% one_plus_numeral
thf(fact_1192_one__plus__numeral,axiom,
! [N3: num] :
( ( plus_p1859984266308609217nnreal @ one_on2969667320475766781nnreal @ ( numera4658534427948366547nnreal @ N3 ) )
= ( numera4658534427948366547nnreal @ ( plus_plus_num @ one @ N3 ) ) ) ).
% one_plus_numeral
thf(fact_1193_one__plus__numeral,axiom,
! [N3: num] :
( ( plus_plus_real @ one_one_real @ ( numeral_numeral_real @ N3 ) )
= ( numeral_numeral_real @ ( plus_plus_num @ one @ N3 ) ) ) ).
% one_plus_numeral
thf(fact_1194_one__plus__numeral,axiom,
! [N3: num] :
( ( plus_plus_int @ one_one_int @ ( numeral_numeral_int @ N3 ) )
= ( numeral_numeral_int @ ( plus_plus_num @ one @ N3 ) ) ) ).
% one_plus_numeral
thf(fact_1195_numeral__plus__one,axiom,
! [N3: num] :
( ( plus_p3455044024723400733d_enat @ ( numera1916890842035813515d_enat @ N3 ) @ one_on7984719198319812577d_enat )
= ( numera1916890842035813515d_enat @ ( plus_plus_num @ N3 @ one ) ) ) ).
% numeral_plus_one
thf(fact_1196_numeral__plus__one,axiom,
! [N3: num] :
( ( plus_plus_nat @ ( numeral_numeral_nat @ N3 ) @ one_one_nat )
= ( numeral_numeral_nat @ ( plus_plus_num @ N3 @ one ) ) ) ).
% numeral_plus_one
thf(fact_1197_numeral__plus__one,axiom,
! [N3: num] :
( ( plus_p1859984266308609217nnreal @ ( numera4658534427948366547nnreal @ N3 ) @ one_on2969667320475766781nnreal )
= ( numera4658534427948366547nnreal @ ( plus_plus_num @ N3 @ one ) ) ) ).
% numeral_plus_one
thf(fact_1198_numeral__plus__one,axiom,
! [N3: num] :
( ( plus_plus_real @ ( numeral_numeral_real @ N3 ) @ one_one_real )
= ( numeral_numeral_real @ ( plus_plus_num @ N3 @ one ) ) ) ).
% numeral_plus_one
thf(fact_1199_numeral__plus__one,axiom,
! [N3: num] :
( ( plus_plus_int @ ( numeral_numeral_int @ N3 ) @ one_one_int )
= ( numeral_numeral_int @ ( plus_plus_num @ N3 @ one ) ) ) ).
% numeral_plus_one
thf(fact_1200_add__2__eq__Suc_H,axiom,
! [N3: nat] :
( ( plus_plus_nat @ N3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
= ( suc @ ( suc @ N3 ) ) ) ).
% add_2_eq_Suc'
thf(fact_1201_add__2__eq__Suc,axiom,
! [N3: nat] :
( ( plus_plus_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N3 )
= ( suc @ ( suc @ N3 ) ) ) ).
% add_2_eq_Suc
thf(fact_1202_add__neg__numeral__simps_I2_J,axiom,
! [M2: num,N3: num] :
( ( plus_plus_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ M2 ) ) @ ( numeral_numeral_int @ N3 ) )
= ( neg_numeral_sub_int @ N3 @ M2 ) ) ).
% add_neg_numeral_simps(2)
thf(fact_1203_add__neg__numeral__simps_I2_J,axiom,
! [M2: num,N3: num] :
( ( plus_plus_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ M2 ) ) @ ( numeral_numeral_real @ N3 ) )
= ( neg_numeral_sub_real @ N3 @ M2 ) ) ).
% add_neg_numeral_simps(2)
thf(fact_1204_add__neg__numeral__simps_I1_J,axiom,
! [M2: num,N3: num] :
( ( plus_plus_int @ ( numeral_numeral_int @ M2 ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ N3 ) ) )
= ( neg_numeral_sub_int @ M2 @ N3 ) ) ).
% add_neg_numeral_simps(1)
thf(fact_1205_add__neg__numeral__simps_I1_J,axiom,
! [M2: num,N3: num] :
( ( plus_plus_real @ ( numeral_numeral_real @ M2 ) @ ( uminus_uminus_real @ ( numeral_numeral_real @ N3 ) ) )
= ( neg_numeral_sub_real @ M2 @ N3 ) ) ).
% add_neg_numeral_simps(1)
thf(fact_1206_div__mult__self1__is__m,axiom,
! [N3: nat,M2: nat] :
( ( ord_less_nat @ zero_zero_nat @ N3 )
=> ( ( divide_divide_nat @ ( times_times_nat @ N3 @ M2 ) @ N3 )
= M2 ) ) ).
% div_mult_self1_is_m
thf(fact_1207_div__mult__self__is__m,axiom,
! [N3: nat,M2: nat] :
( ( ord_less_nat @ zero_zero_nat @ N3 )
=> ( ( divide_divide_nat @ ( times_times_nat @ M2 @ N3 ) @ N3 )
= M2 ) ) ).
% div_mult_self_is_m
thf(fact_1208_neg__numeral__le__iff,axiom,
! [M2: num,N3: num] :
( ( ord_less_eq_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ M2 ) ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ N3 ) ) )
= ( ord_less_eq_num @ N3 @ M2 ) ) ).
% neg_numeral_le_iff
thf(fact_1209_neg__numeral__le__iff,axiom,
! [M2: num,N3: num] :
( ( ord_less_eq_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ M2 ) ) @ ( uminus_uminus_real @ ( numeral_numeral_real @ N3 ) ) )
= ( ord_less_eq_num @ N3 @ M2 ) ) ).
% neg_numeral_le_iff
thf(fact_1210_semiring__norm_I79_J,axiom,
! [M2: num,N3: num] :
( ( ord_less_num @ ( bit0 @ M2 ) @ ( bit1 @ N3 ) )
= ( ord_less_eq_num @ M2 @ N3 ) ) ).
% semiring_norm(79)
thf(fact_1211_semiring__norm_I74_J,axiom,
! [M2: num,N3: num] :
( ( ord_less_eq_num @ ( bit1 @ M2 ) @ ( bit0 @ N3 ) )
= ( ord_less_num @ M2 @ N3 ) ) ).
% semiring_norm(74)
thf(fact_1212_divide__le__eq__1__neg,axiom,
! [A: real,B: real] :
( ( ord_less_real @ A @ zero_zero_real )
=> ( ( ord_less_eq_real @ ( divide_divide_real @ B @ A ) @ one_one_real )
= ( ord_less_eq_real @ A @ B ) ) ) ).
% divide_le_eq_1_neg
thf(fact_1213_divide__le__eq__1__pos,axiom,
! [A: real,B: real] :
( ( ord_less_real @ zero_zero_real @ A )
=> ( ( ord_less_eq_real @ ( divide_divide_real @ B @ A ) @ one_one_real )
= ( ord_less_eq_real @ B @ A ) ) ) ).
% divide_le_eq_1_pos
thf(fact_1214_le__divide__eq__1__neg,axiom,
! [A: real,B: real] :
( ( ord_less_real @ A @ zero_zero_real )
=> ( ( ord_less_eq_real @ one_one_real @ ( divide_divide_real @ B @ A ) )
= ( ord_less_eq_real @ B @ A ) ) ) ).
% le_divide_eq_1_neg
thf(fact_1215_le__divide__eq__1__pos,axiom,
! [A: real,B: real] :
( ( ord_less_real @ zero_zero_real @ A )
=> ( ( ord_less_eq_real @ one_one_real @ ( divide_divide_real @ B @ A ) )
= ( ord_less_eq_real @ A @ B ) ) ) ).
% le_divide_eq_1_pos
thf(fact_1216_not__neg__one__le__neg__numeral__iff,axiom,
! [M2: num] :
( ( ~ ( ord_less_eq_int @ ( uminus_uminus_int @ one_one_int ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ M2 ) ) ) )
= ( M2 != one ) ) ).
% not_neg_one_le_neg_numeral_iff
thf(fact_1217_not__neg__one__le__neg__numeral__iff,axiom,
! [M2: num] :
( ( ~ ( ord_less_eq_real @ ( uminus_uminus_real @ one_one_real ) @ ( uminus_uminus_real @ ( numeral_numeral_real @ M2 ) ) ) )
= ( M2 != one ) ) ).
% not_neg_one_le_neg_numeral_iff
thf(fact_1218_divide__le__eq__numeral1_I2_J,axiom,
! [B: real,W: num,A: real] :
( ( ord_less_eq_real @ ( divide_divide_real @ B @ ( uminus_uminus_real @ ( numeral_numeral_real @ W ) ) ) @ A )
= ( ord_less_eq_real @ ( times_times_real @ A @ ( uminus_uminus_real @ ( numeral_numeral_real @ W ) ) ) @ B ) ) ).
% divide_le_eq_numeral1(2)
thf(fact_1219_le__divide__eq__numeral1_I2_J,axiom,
! [A: real,B: real,W: num] :
( ( ord_less_eq_real @ A @ ( divide_divide_real @ B @ ( uminus_uminus_real @ ( numeral_numeral_real @ W ) ) ) )
= ( ord_less_eq_real @ B @ ( times_times_real @ A @ ( uminus_uminus_real @ ( numeral_numeral_real @ W ) ) ) ) ) ).
% le_divide_eq_numeral1(2)
thf(fact_1220_divide__eq__eq__numeral1_I2_J,axiom,
! [B: real,W: num,A: real] :
( ( ( divide_divide_real @ B @ ( uminus_uminus_real @ ( numeral_numeral_real @ W ) ) )
= A )
= ( ( ( ( uminus_uminus_real @ ( numeral_numeral_real @ W ) )
!= zero_zero_real )
=> ( B
= ( times_times_real @ A @ ( uminus_uminus_real @ ( numeral_numeral_real @ W ) ) ) ) )
& ( ( ( uminus_uminus_real @ ( numeral_numeral_real @ W ) )
= zero_zero_real )
=> ( A = zero_zero_real ) ) ) ) ).
% divide_eq_eq_numeral1(2)
thf(fact_1221_eq__divide__eq__numeral1_I2_J,axiom,
! [A: real,B: real,W: num] :
( ( A
= ( divide_divide_real @ B @ ( uminus_uminus_real @ ( numeral_numeral_real @ W ) ) ) )
= ( ( ( ( uminus_uminus_real @ ( numeral_numeral_real @ W ) )
!= zero_zero_real )
=> ( ( times_times_real @ A @ ( uminus_uminus_real @ ( numeral_numeral_real @ W ) ) )
= B ) )
& ( ( ( uminus_uminus_real @ ( numeral_numeral_real @ W ) )
= zero_zero_real )
=> ( A = zero_zero_real ) ) ) ) ).
% eq_divide_eq_numeral1(2)
thf(fact_1222_divide__less__eq__numeral1_I2_J,axiom,
! [B: real,W: num,A: real] :
( ( ord_less_real @ ( divide_divide_real @ B @ ( uminus_uminus_real @ ( numeral_numeral_real @ W ) ) ) @ A )
= ( ord_less_real @ ( times_times_real @ A @ ( uminus_uminus_real @ ( numeral_numeral_real @ W ) ) ) @ B ) ) ).
% divide_less_eq_numeral1(2)
thf(fact_1223_less__divide__eq__numeral1_I2_J,axiom,
! [A: real,B: real,W: num] :
( ( ord_less_real @ A @ ( divide_divide_real @ B @ ( uminus_uminus_real @ ( numeral_numeral_real @ W ) ) ) )
= ( ord_less_real @ B @ ( times_times_real @ A @ ( uminus_uminus_real @ ( numeral_numeral_real @ W ) ) ) ) ) ).
% less_divide_eq_numeral1(2)
thf(fact_1224_one__add__one,axiom,
( ( plus_plus_real @ one_one_real @ one_one_real )
= ( numeral_numeral_real @ ( bit0 @ one ) ) ) ).
% one_add_one
thf(fact_1225_one__add__one,axiom,
( ( plus_plus_int @ one_one_int @ one_one_int )
= ( numeral_numeral_int @ ( bit0 @ one ) ) ) ).
% one_add_one
thf(fact_1226_Suc__div__eq__add3__div__numeral,axiom,
! [M2: nat,V2: num] :
( ( divide_divide_nat @ ( suc @ ( suc @ ( suc @ M2 ) ) ) @ ( numeral_numeral_nat @ V2 ) )
= ( divide_divide_nat @ ( plus_plus_nat @ ( numeral_numeral_nat @ ( bit1 @ one ) ) @ M2 ) @ ( numeral_numeral_nat @ V2 ) ) ) ).
% Suc_div_eq_add3_div_numeral
thf(fact_1227_div__Suc__eq__div__add3,axiom,
! [M2: nat,N3: nat] :
( ( divide_divide_nat @ M2 @ ( suc @ ( suc @ ( suc @ N3 ) ) ) )
= ( divide_divide_nat @ M2 @ ( plus_plus_nat @ ( numeral_numeral_nat @ ( bit1 @ one ) ) @ N3 ) ) ) ).
% div_Suc_eq_div_add3
thf(fact_1228_nat__mult__1__right,axiom,
! [N3: nat] :
( ( times_times_nat @ N3 @ one_one_nat )
= N3 ) ).
% nat_mult_1_right
thf(fact_1229_nat__mult__1,axiom,
! [N3: nat] :
( ( times_times_nat @ one_one_nat @ N3 )
= N3 ) ).
% nat_mult_1
thf(fact_1230_split__div,axiom,
! [P: nat > $o,M2: nat,N3: nat] :
( ( P @ ( divide_divide_nat @ M2 @ N3 ) )
= ( ( ( N3 = zero_zero_nat )
=> ( P @ zero_zero_nat ) )
& ( ( N3 != zero_zero_nat )
=> ! [I3: nat,J3: nat] :
( ( ( ord_less_nat @ J3 @ N3 )
& ( M2
= ( plus_plus_nat @ ( times_times_nat @ N3 @ I3 ) @ J3 ) ) )
=> ( P @ I3 ) ) ) ) ) ).
% split_div
thf(fact_1231_dividend__less__div__times,axiom,
! [N3: nat,M2: nat] :
( ( ord_less_nat @ zero_zero_nat @ N3 )
=> ( ord_less_nat @ M2 @ ( plus_plus_nat @ N3 @ ( times_times_nat @ ( divide_divide_nat @ M2 @ N3 ) @ N3 ) ) ) ) ).
% dividend_less_div_times
thf(fact_1232_dividend__less__times__div,axiom,
! [N3: nat,M2: nat] :
( ( ord_less_nat @ zero_zero_nat @ N3 )
=> ( ord_less_nat @ M2 @ ( plus_plus_nat @ N3 @ ( times_times_nat @ N3 @ ( divide_divide_nat @ M2 @ N3 ) ) ) ) ) ).
% dividend_less_times_div
thf(fact_1233_mult__Suc,axiom,
! [M2: nat,N3: nat] :
( ( times_times_nat @ ( suc @ M2 ) @ N3 )
= ( plus_plus_nat @ N3 @ ( times_times_nat @ M2 @ N3 ) ) ) ).
% mult_Suc
thf(fact_1234_add__mult__distrib2,axiom,
! [K: nat,M2: nat,N3: nat] :
( ( times_times_nat @ K @ ( plus_plus_nat @ M2 @ N3 ) )
= ( plus_plus_nat @ ( times_times_nat @ K @ M2 ) @ ( times_times_nat @ K @ N3 ) ) ) ).
% add_mult_distrib2
thf(fact_1235_add__mult__distrib,axiom,
! [M2: nat,N3: nat,K: nat] :
( ( times_times_nat @ ( plus_plus_nat @ M2 @ N3 ) @ K )
= ( plus_plus_nat @ ( times_times_nat @ M2 @ K ) @ ( times_times_nat @ N3 @ K ) ) ) ).
% add_mult_distrib
thf(fact_1236_Suc__mult__cancel1,axiom,
! [K: nat,M2: nat,N3: nat] :
( ( ( times_times_nat @ ( suc @ K ) @ M2 )
= ( times_times_nat @ ( suc @ K ) @ N3 ) )
= ( M2 = N3 ) ) ).
% Suc_mult_cancel1
thf(fact_1237_nat__arith_Osuc1,axiom,
! [A4: nat,K: nat,A: nat] :
( ( A4
= ( plus_plus_nat @ K @ A ) )
=> ( ( suc @ A4 )
= ( plus_plus_nat @ K @ ( suc @ A ) ) ) ) ).
% nat_arith.suc1
thf(fact_1238_add__Suc,axiom,
! [M2: nat,N3: nat] :
( ( plus_plus_nat @ ( suc @ M2 ) @ N3 )
= ( suc @ ( plus_plus_nat @ M2 @ N3 ) ) ) ).
% add_Suc
thf(fact_1239_add__Suc__shift,axiom,
! [M2: nat,N3: nat] :
( ( plus_plus_nat @ ( suc @ M2 ) @ N3 )
= ( plus_plus_nat @ M2 @ ( suc @ N3 ) ) ) ).
% add_Suc_shift
thf(fact_1240_le__num__One__iff,axiom,
! [X: num] :
( ( ord_less_eq_num @ X @ one )
= ( X = one ) ) ).
% le_num_One_iff
thf(fact_1241_add__One__commute,axiom,
! [N3: num] :
( ( plus_plus_num @ one @ N3 )
= ( plus_plus_num @ N3 @ one ) ) ).
% add_One_commute
thf(fact_1242_Suc__nat__number__of__add,axiom,
! [V2: num,N3: nat] :
( ( suc @ ( plus_plus_nat @ ( numeral_numeral_nat @ V2 ) @ N3 ) )
= ( plus_plus_nat @ ( numeral_numeral_nat @ ( plus_plus_num @ V2 @ one ) ) @ N3 ) ) ).
% Suc_nat_number_of_add
thf(fact_1243_add__eq__self__zero,axiom,
! [M2: nat,N3: nat] :
( ( ( plus_plus_nat @ M2 @ N3 )
= M2 )
=> ( N3 = zero_zero_nat ) ) ).
% add_eq_self_zero
thf(fact_1244_plus__nat_Oadd__0,axiom,
! [N3: nat] :
( ( plus_plus_nat @ zero_zero_nat @ N3 )
= N3 ) ).
% plus_nat.add_0
thf(fact_1245_mult__0,axiom,
! [N3: nat] :
( ( times_times_nat @ zero_zero_nat @ N3 )
= zero_zero_nat ) ).
% mult_0
thf(fact_1246_nat__mult__eq__cancel__disj,axiom,
! [K: nat,M2: nat,N3: nat] :
( ( ( times_times_nat @ K @ M2 )
= ( times_times_nat @ K @ N3 ) )
= ( ( K = zero_zero_nat )
| ( M2 = N3 ) ) ) ).
% nat_mult_eq_cancel_disj
thf(fact_1247_less__add__eq__less,axiom,
! [K: nat,L: nat,M2: nat,N3: nat] :
( ( ord_less_nat @ K @ L )
=> ( ( ( plus_plus_nat @ M2 @ L )
= ( plus_plus_nat @ K @ N3 ) )
=> ( ord_less_nat @ M2 @ N3 ) ) ) ).
% less_add_eq_less
thf(fact_1248_trans__less__add2,axiom,
! [I: nat,J: nat,M2: nat] :
( ( ord_less_nat @ I @ J )
=> ( ord_less_nat @ I @ ( plus_plus_nat @ M2 @ J ) ) ) ).
% trans_less_add2
thf(fact_1249_trans__less__add1,axiom,
! [I: nat,J: nat,M2: nat] :
( ( ord_less_nat @ I @ J )
=> ( ord_less_nat @ I @ ( plus_plus_nat @ J @ M2 ) ) ) ).
% trans_less_add1
thf(fact_1250_add__less__mono1,axiom,
! [I: nat,J: nat,K: nat] :
( ( ord_less_nat @ I @ J )
=> ( ord_less_nat @ ( plus_plus_nat @ I @ K ) @ ( plus_plus_nat @ J @ K ) ) ) ).
% add_less_mono1
thf(fact_1251_not__add__less2,axiom,
! [J: nat,I: nat] :
~ ( ord_less_nat @ ( plus_plus_nat @ J @ I ) @ I ) ).
% not_add_less2
thf(fact_1252_not__add__less1,axiom,
! [I: nat,J: nat] :
~ ( ord_less_nat @ ( plus_plus_nat @ I @ J ) @ I ) ).
% not_add_less1
thf(fact_1253_add__less__mono,axiom,
! [I: nat,J: nat,K: nat,L: nat] :
( ( ord_less_nat @ I @ J )
=> ( ( ord_less_nat @ K @ L )
=> ( ord_less_nat @ ( plus_plus_nat @ I @ K ) @ ( plus_plus_nat @ J @ L ) ) ) ) ).
% add_less_mono
thf(fact_1254_add__lessD1,axiom,
! [I: nat,J: nat,K: nat] :
( ( ord_less_nat @ ( plus_plus_nat @ I @ J ) @ K )
=> ( ord_less_nat @ I @ K ) ) ).
% add_lessD1
thf(fact_1255_diff__add__inverse2,axiom,
! [M2: nat,N3: nat] :
( ( minus_minus_nat @ ( plus_plus_nat @ M2 @ N3 ) @ N3 )
= M2 ) ).
% diff_add_inverse2
thf(fact_1256_diff__add__inverse,axiom,
! [N3: nat,M2: nat] :
( ( minus_minus_nat @ ( plus_plus_nat @ N3 @ M2 ) @ N3 )
= M2 ) ).
% diff_add_inverse
thf(fact_1257_diff__cancel2,axiom,
! [M2: nat,K: nat,N3: nat] :
( ( minus_minus_nat @ ( plus_plus_nat @ M2 @ K ) @ ( plus_plus_nat @ N3 @ K ) )
= ( minus_minus_nat @ M2 @ N3 ) ) ).
% diff_cancel2
thf(fact_1258_Nat_Odiff__cancel,axiom,
! [K: nat,M2: nat,N3: nat] :
( ( minus_minus_nat @ ( plus_plus_nat @ K @ M2 ) @ ( plus_plus_nat @ K @ N3 ) )
= ( minus_minus_nat @ M2 @ N3 ) ) ).
% Nat.diff_cancel
thf(fact_1259_diff__mult__distrib2,axiom,
! [K: nat,M2: nat,N3: nat] :
( ( times_times_nat @ K @ ( minus_minus_nat @ M2 @ N3 ) )
= ( minus_minus_nat @ ( times_times_nat @ K @ M2 ) @ ( times_times_nat @ K @ N3 ) ) ) ).
% diff_mult_distrib2
thf(fact_1260_diff__mult__distrib,axiom,
! [M2: nat,N3: nat,K: nat] :
( ( times_times_nat @ ( minus_minus_nat @ M2 @ N3 ) @ K )
= ( minus_minus_nat @ ( times_times_nat @ M2 @ K ) @ ( times_times_nat @ N3 @ K ) ) ) ).
% diff_mult_distrib
thf(fact_1261_mult__eq__if,axiom,
( times_times_nat
= ( ^ [M: nat,N: nat] : ( if_nat @ ( M = zero_zero_nat ) @ zero_zero_nat @ ( plus_plus_nat @ N @ ( times_times_nat @ ( minus_minus_nat @ M @ one_one_nat ) @ N ) ) ) ) ) ).
% mult_eq_if
thf(fact_1262_real__add__less__0__iff,axiom,
! [X: real,Y: real] :
( ( ord_less_real @ ( plus_plus_real @ X @ Y ) @ zero_zero_real )
= ( ord_less_real @ Y @ ( uminus_uminus_real @ X ) ) ) ).
% real_add_less_0_iff
thf(fact_1263_real__0__less__add__iff,axiom,
! [X: real,Y: real] :
( ( ord_less_real @ zero_zero_real @ ( plus_plus_real @ X @ Y ) )
= ( ord_less_real @ ( uminus_uminus_real @ X ) @ Y ) ) ).
% real_0_less_add_iff
% Helper facts (13)
thf(help_If_2_1_If_001tf__a_T,axiom,
! [X: a,Y: a] :
( ( if_a @ $false @ X @ Y )
= Y ) ).
thf(help_If_1_1_If_001tf__a_T,axiom,
! [X: a,Y: a] :
( ( if_a @ $true @ X @ Y )
= X ) ).
thf(help_If_2_1_If_001t__Int__Oint_T,axiom,
! [X: int,Y: int] :
( ( if_int @ $false @ X @ Y )
= Y ) ).
thf(help_If_1_1_If_001t__Int__Oint_T,axiom,
! [X: int,Y: int] :
( ( if_int @ $true @ X @ Y )
= X ) ).
thf(help_If_2_1_If_001t__Nat__Onat_T,axiom,
! [X: nat,Y: nat] :
( ( if_nat @ $false @ X @ Y )
= Y ) ).
thf(help_If_1_1_If_001t__Nat__Onat_T,axiom,
! [X: nat,Y: nat] :
( ( if_nat @ $true @ X @ Y )
= X ) ).
thf(help_If_2_1_If_001t__List__Olist_It__Int__Oint_J_T,axiom,
! [X: list_int,Y: list_int] :
( ( if_list_int @ $false @ X @ Y )
= Y ) ).
thf(help_If_1_1_If_001t__List__Olist_It__Int__Oint_J_T,axiom,
! [X: list_int,Y: list_int] :
( ( if_list_int @ $true @ X @ Y )
= X ) ).
thf(help_If_2_1_If_001t__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J_T,axiom,
! [X: product_prod_int_int,Y: product_prod_int_int] :
( ( if_Pro3027730157355071871nt_int @ $false @ X @ Y )
= Y ) ).
thf(help_If_1_1_If_001t__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J_T,axiom,
! [X: product_prod_int_int,Y: product_prod_int_int] :
( ( if_Pro3027730157355071871nt_int @ $true @ X @ Y )
= X ) ).
thf(help_If_3_1_If_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_T,axiom,
! [P: $o] :
( ( P = $true )
| ( P = $false ) ) ).
thf(help_If_2_1_If_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_T,axiom,
! [X: product_prod_nat_nat,Y: product_prod_nat_nat] :
( ( if_Pro6206227464963214023at_nat @ $false @ X @ Y )
= Y ) ).
thf(help_If_1_1_If_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_T,axiom,
! [X: product_prod_nat_nat,Y: product_prod_nat_nat] :
( ( if_Pro6206227464963214023at_nat @ $true @ X @ Y )
= X ) ).
% Conjectures (1)
thf(conj_0,conjecture,
( ( coProduct_to_list_a
@ ( produc5292568359338195516_nat_a @ ( suc @ ( produc3194919578927588176_nat_a @ l ) )
@ ^ [N: nat] : ( if_a @ ( N = zero_zero_nat ) @ a2 @ ( produc4809910040060592782_nat_a @ l @ ( minus_minus_nat @ N @ one_one_nat ) ) ) ) )
= ( cons_a @ a2 @ ( coProduct_to_list_a @ l ) ) ) ).
%------------------------------------------------------------------------------