TPTP Problem File: SLH0314^1.p
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- Solve Problem
%------------------------------------------------------------------------------
% File : SLH0000^1 : TPTP v8.2.0. Released v8.2.0.
% Domain : Archive of Formal Proofs
% Problem :
% Version : Especial.
% English :
% Refs : [Des23] Desharnais (2023), Email to Geoff Sutcliffe
% Source : [Des23]
% Names : Eval_FO/0005_Ailamazyan/prob_01589_059795__15713692_1 [Des23]
% Status : Theorem
% Rating : ? v8.2.0
% Syntax : Number of formulae : 1496 ( 539 unt; 224 typ; 0 def)
% Number of atoms : 4684 (1715 equ; 0 cnn)
% Maximal formula atoms : 44 ( 3 avg)
% Number of connectives : 13789 ( 646 ~; 144 |; 139 &;10361 @)
% ( 0 <=>;2499 =>; 0 <=; 0 <~>)
% Maximal formula depth : 26 ( 7 avg)
% Number of types : 53 ( 52 usr)
% Number of type conns : 661 ( 661 >; 0 *; 0 +; 0 <<)
% Number of symbols : 175 ( 172 usr; 28 con; 0-4 aty)
% Number of variables : 3884 ( 161 ^;3550 !; 173 ?;3884 :)
% SPC : TH0_THM_EQU_NAR
% Comments : This file was generated by Isabelle (most likely Sledgehammer)
% 2023-01-19 13:06:32.675
%------------------------------------------------------------------------------
% Could-be-implicit typings (52)
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thf(ty_n_t__Option__Ooption_It__Option__Ooption_It__Nat__Onat_J_J,type,
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thf(ty_n_t__Option__Ooption_It__Option__Ooption_It__Int__Oint_J_J,type,
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thf(ty_n_t__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J,type,
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thf(ty_n_t__Sum____Type__Osum_It__Nat__Onat_Mt__Nat__Onat_J,type,
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sum_sum_a_nat: $tType ).
thf(ty_n_t__Option__Ooption_It__Nat__Onat_J,type,
option_nat: $tType ).
thf(ty_n_t__Option__Ooption_It__Int__Oint_J,type,
option_int: $tType ).
thf(ty_n_t__Set__Oset_Itf__a_J,type,
set_a: $tType ).
thf(ty_n_t__Real__Oreal,type,
real: $tType ).
thf(ty_n_t__Nat__Onat,type,
nat: $tType ).
thf(ty_n_t__Int__Oint,type,
int: $tType ).
thf(ty_n_tf__a,type,
a: $tType ).
% Explicit typings (172)
thf(sy_c_Ailamazyan_Ofo__nmlz__rec_001tf__a,type,
fo_nmlz_rec_a: nat > ( sum_sum_a_nat > option_nat ) > set_a > list_Sum_sum_a_nat > list_Sum_sum_a_nat ).
thf(sy_c_Ailamazyan_Oid__map_001tf__a,type,
id_map_a: nat > sum_sum_a_nat > option_nat ).
thf(sy_c_Ailamazyan_Onall__tuples__rec_001tf__a,type,
nall_tuples_rec_a: set_a > nat > nat > set_li6526943997496501093_a_nat ).
thf(sy_c_Archimedean__Field_Oceiling_001t__Real__Oreal,type,
archim7802044766580827645g_real: real > int ).
thf(sy_c_BNF__Cardinal__Order__Relation_OrelChain_001t__Option__Ooption_It__Int__Oint_J_001t__Int__Oint,type,
bNF_Ca6594861251087899470nt_int: set_Pr5725982650624401465on_int > ( option_int > int ) > $o ).
thf(sy_c_BNF__Cardinal__Order__Relation_OrelChain_001t__Option__Ooption_It__Int__Oint_J_001t__Nat__Onat,type,
bNF_Ca6597351721596949746nt_nat: set_Pr5725982650624401465on_int > ( option_int > nat ) > $o ).
thf(sy_c_BNF__Cardinal__Order__Relation_OrelChain_001t__Option__Ooption_It__Int__Oint_J_001t__Real__Oreal,type,
bNF_Ca6297945793459226318t_real: set_Pr5725982650624401465on_int > ( option_int > real ) > $o ).
thf(sy_c_BNF__Cardinal__Order__Relation_OrelChain_001t__Option__Ooption_It__Nat__Onat_J_001t__Int__Oint,type,
bNF_Ca4407939676394877682at_int: set_Pr6588086440996610945on_nat > ( option_nat > int ) > $o ).
thf(sy_c_BNF__Cardinal__Order__Relation_OrelChain_001t__Option__Ooption_It__Nat__Onat_J_001t__Nat__Onat,type,
bNF_Ca4410430146903927958at_nat: set_Pr6588086440996610945on_nat > ( option_nat > nat ) > $o ).
thf(sy_c_BNF__Cardinal__Order__Relation_OrelChain_001t__Option__Ooption_It__Nat__Onat_J_001t__Real__Oreal,type,
bNF_Ca3079242199294856306t_real: set_Pr6588086440996610945on_nat > ( option_nat > real ) > $o ).
thf(sy_c_Collection__Order_Ocproper__interval__class_Ocproper__interval_001t__Int__Oint,type,
collec4138408100298950737al_int: option_int > option_int > $o ).
thf(sy_c_Collection__Order_Ocproper__interval__class_Ocproper__interval_001t__Nat__Onat,type,
collec4140898570808001013al_nat: option_nat > option_nat > $o ).
thf(sy_c_Collection__Order_Ocproper__interval__class_Ocproper__interval_001t__Option__Ooption_It__Int__Oint_J,type,
collec3052392823440695329on_int: option_option_int > option_option_int > $o ).
thf(sy_c_Collection__Order_Ocproper__interval__class_Ocproper__interval_001t__Option__Ooption_It__Nat__Onat_J,type,
collec7230243842949892037on_nat: option_option_nat > option_option_nat > $o ).
thf(sy_c_Collection__Order_Ocproper__interval__class_Ocproper__interval_001t__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J,type,
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thf(sy_c_Collection__Order_Ocproper__interval__class_Ocproper__interval_001t__Product____Type__Oprod_It__Int__Oint_Mt__Nat__Onat_J,type,
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thf(sy_c_Collection__Order_Ocproper__interval__class_Ocproper__interval_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Int__Oint_J,type,
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thf(sy_c_Collection__Order_Ocproper__interval__class_Ocproper__interval_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J,type,
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thf(sy_c_Collection__Order_Ocproper__interval__class_Ocproper__interval_001t__Sum____Type__Osum_It__Int__Oint_Mt__Int__Oint_J,type,
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thf(sy_c_Collection__Order_Ocproper__interval__class_Ocproper__interval_001t__Sum____Type__Osum_It__Int__Oint_Mt__Nat__Onat_J,type,
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thf(sy_c_Collection__Order_Ocproper__interval__class_Ocproper__interval_001t__Sum____Type__Osum_It__Nat__Onat_Mt__Nat__Onat_J,type,
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thf(sy_c_Collection__Order_Ocproper__interval__option__rel_001t__Int__Oint,type,
collec5418738772392851069el_int: produc420109317877091577on_int > produc420109317877091577on_int > $o ).
thf(sy_c_Collection__Order_Ocproper__interval__option__rel_001t__Nat__Onat,type,
collec5421229242901901345el_nat: produc6094937800015497793on_nat > produc6094937800015497793on_nat > $o ).
thf(sy_c_Collection__Order_Ocproper__interval__sum__rel_001t__Int__Oint_001t__Int__Oint,type,
collec5720877037025338812nt_int: produc6152922620594178031nt_int > produc6152922620594178031nt_int > $o ).
thf(sy_c_Collection__Order_Ocproper__interval__sum__rel_001t__Int__Oint_001t__Nat__Onat,type,
collec5723367507534389088nt_nat: produc334412093845648695nt_nat > produc334412093845648695nt_nat > $o ).
thf(sy_c_Collection__Order_Ocproper__interval__sum__rel_001t__Nat__Onat_001t__Int__Oint,type,
collec4721523325124284256at_int: produc6362011879373453623at_int > produc6362011879373453623at_int > $o ).
thf(sy_c_Collection__Order_Ocproper__interval__sum__rel_001t__Nat__Onat_001t__Nat__Onat,type,
collec4724013795633334532at_nat: produc543501352624924287at_nat > produc543501352624924287at_nat > $o ).
thf(sy_c_Conditionally__Complete__Lattices_Opreordering__bdd_001t__Int__Oint,type,
condit7933062003635074389dd_int: ( int > int > $o ) > ( int > int > $o ) > $o ).
thf(sy_c_Conditionally__Complete__Lattices_Opreordering__bdd_001t__Nat__Onat,type,
condit7935552474144124665dd_nat: ( nat > nat > $o ) > ( nat > nat > $o ) > $o ).
thf(sy_c_Conditionally__Complete__Lattices_Opreordering__bdd_001t__Real__Oreal,type,
condit1497324847667023189d_real: ( real > real > $o ) > ( real > real > $o ) > $o ).
thf(sy_c_Factorial_Osemiring__char__0__class_Ofact_001t__Nat__Onat,type,
semiri1408675320244567234ct_nat: nat > nat ).
thf(sy_c_Groups_Oabs__class_Oabs_001t__Int__Oint,type,
abs_abs_int: int > int ).
thf(sy_c_Groups_Oabs__class_Oabs_001t__Real__Oreal,type,
abs_abs_real: real > real ).
thf(sy_c_Groups_Ominus__class_Ominus_001t__Int__Oint,type,
minus_minus_int: int > int > int ).
thf(sy_c_Groups_Ominus__class_Ominus_001t__Nat__Onat,type,
minus_minus_nat: nat > nat > nat ).
thf(sy_c_Groups_Ominus__class_Ominus_001t__Real__Oreal,type,
minus_minus_real: real > real > real ).
thf(sy_c_Groups_Oone__class_Oone_001t__Int__Oint,type,
one_one_int: int ).
thf(sy_c_Groups_Oone__class_Oone_001t__Nat__Onat,type,
one_one_nat: nat ).
thf(sy_c_Groups_Oone__class_Oone_001t__Real__Oreal,type,
one_one_real: real ).
thf(sy_c_Groups_Oplus__class_Oplus_001t__Int__Oint,type,
plus_plus_int: int > int > int ).
thf(sy_c_Groups_Oplus__class_Oplus_001t__Nat__Onat,type,
plus_plus_nat: nat > nat > nat ).
thf(sy_c_Groups_Oplus__class_Oplus_001t__Real__Oreal,type,
plus_plus_real: real > real > real ).
thf(sy_c_Groups_Otimes__class_Otimes_001t__Int__Oint,type,
times_times_int: int > int > int ).
thf(sy_c_Groups_Otimes__class_Otimes_001t__Nat__Onat,type,
times_times_nat: nat > nat > nat ).
thf(sy_c_Groups_Otimes__class_Otimes_001t__Real__Oreal,type,
times_times_real: real > real > real ).
thf(sy_c_Groups_Ouminus__class_Ouminus_001t__Int__Oint,type,
uminus_uminus_int: int > int ).
thf(sy_c_Groups_Ouminus__class_Ouminus_001t__Real__Oreal,type,
uminus_uminus_real: real > real ).
thf(sy_c_Groups_Ozero__class_Ozero_001t__Int__Oint,type,
zero_zero_int: int ).
thf(sy_c_Groups_Ozero__class_Ozero_001t__Nat__Onat,type,
zero_zero_nat: nat ).
thf(sy_c_Groups_Ozero__class_Ozero_001t__Real__Oreal,type,
zero_zero_real: real ).
thf(sy_c_If_001t__Nat__Onat,type,
if_nat: $o > nat > nat > nat ).
thf(sy_c_Nat_OSuc,type,
suc: nat > nat ).
thf(sy_c_Nat_Osemiring__1__class_Oof__nat_001t__Int__Oint,type,
semiri1314217659103216013at_int: nat > int ).
thf(sy_c_Nat_Osemiring__1__class_Oof__nat_001t__Nat__Onat,type,
semiri1316708129612266289at_nat: nat > nat ).
thf(sy_c_Nat_Osemiring__1__class_Oof__nat_001t__Real__Oreal,type,
semiri5074537144036343181t_real: nat > real ).
thf(sy_c_NthRoot_Oroot,type,
root: nat > real > real ).
thf(sy_c_Option_Ooption_ONone_001t__Int__Oint,type,
none_int: option_int ).
thf(sy_c_Option_Ooption_ONone_001t__Nat__Onat,type,
none_nat: option_nat ).
thf(sy_c_Option_Ooption_ONone_001t__Option__Ooption_It__Int__Oint_J,type,
none_option_int: option_option_int ).
thf(sy_c_Option_Ooption_ONone_001t__Option__Ooption_It__Nat__Onat_J,type,
none_option_nat: option_option_nat ).
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thf(sy_c_Option_Ooption_ONone_001t__Sum____Type__Osum_It__Int__Oint_Mt__Int__Oint_J,type,
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none_Sum_sum_nat_nat: option3242699076814927183at_nat ).
thf(sy_c_Option_Ooption_OSome_001t__Int__Oint,type,
some_int: int > option_int ).
thf(sy_c_Option_Ooption_OSome_001t__Nat__Onat,type,
some_nat: nat > option_nat ).
thf(sy_c_Option_Ooption_OSome_001t__Option__Ooption_It__Int__Oint_J,type,
some_option_int: option_int > option_option_int ).
thf(sy_c_Option_Ooption_OSome_001t__Option__Ooption_It__Nat__Onat_J,type,
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thf(sy_c_Option_Ooption_OSome_001t__Sum____Type__Osum_It__Int__Oint_Mt__Nat__Onat_J,type,
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thf(sy_c_Option_Ooption_OSome_001t__Sum____Type__Osum_It__Nat__Onat_Mt__Nat__Onat_J,type,
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thf(sy_c_Orderings_Oord__class_Oless_001t__Int__Oint,type,
ord_less_int: int > int > $o ).
thf(sy_c_Orderings_Oord__class_Oless_001t__Nat__Onat,type,
ord_less_nat: nat > nat > $o ).
thf(sy_c_Orderings_Oord__class_Oless_001t__Real__Oreal,type,
ord_less_real: real > real > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001_062_I_Eo_Mt__Int__Oint_J,type,
ord_less_eq_o_int: ( $o > int ) > ( $o > int ) > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001_062_I_Eo_Mt__Nat__Onat_J,type,
ord_less_eq_o_nat: ( $o > nat ) > ( $o > nat ) > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001_062_I_Eo_Mt__Real__Oreal_J,type,
ord_less_eq_o_real: ( $o > real ) > ( $o > real ) > $o ).
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thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Int__Oint,type,
ord_less_eq_int: int > int > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Nat__Onat,type,
ord_less_eq_nat: nat > nat > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Real__Oreal,type,
ord_less_eq_real: real > real > $o ).
thf(sy_c_Orderings_Oorder__class_OGreatest_001t__Int__Oint,type,
order_Greatest_int: ( int > $o ) > int ).
thf(sy_c_Orderings_Oorder__class_OGreatest_001t__Nat__Onat,type,
order_Greatest_nat: ( nat > $o ) > nat ).
thf(sy_c_Orderings_Oorder__class_OGreatest_001t__Real__Oreal,type,
order_Greatest_real: ( real > $o ) > real ).
thf(sy_c_Power_Opower__class_Opower_001t__Int__Oint,type,
power_power_int: int > nat > int ).
thf(sy_c_Power_Opower__class_Opower_001t__Nat__Onat,type,
power_power_nat: nat > nat > nat ).
thf(sy_c_Power_Opower__class_Opower_001t__Real__Oreal,type,
power_power_real: real > nat > real ).
thf(sy_c_Product__Type_OPair_001t__Int__Oint_001t__Int__Oint,type,
product_Pair_int_int: int > int > product_prod_int_int ).
thf(sy_c_Product__Type_OPair_001t__Int__Oint_001t__Nat__Onat,type,
product_Pair_int_nat: int > nat > product_prod_int_nat ).
thf(sy_c_Product__Type_OPair_001t__Nat__Onat_001t__Int__Oint,type,
product_Pair_nat_int: nat > int > product_prod_nat_int ).
thf(sy_c_Product__Type_OPair_001t__Nat__Onat_001t__Nat__Onat,type,
product_Pair_nat_nat: nat > nat > product_prod_nat_nat ).
thf(sy_c_Product__Type_OPair_001t__Option__Ooption_It__Int__Oint_J_001t__Option__Ooption_It__Int__Oint_J,type,
produc6331568796615304721on_int: option_int > option_int > produc7874843650251301337on_int ).
thf(sy_c_Product__Type_OPair_001t__Option__Ooption_It__Int__Oint_J_001t__Option__Ooption_It__Nat__Onat_J,type,
produc1286047779269725621on_nat: option_int > option_nat > produc1141554758328252285on_nat ).
thf(sy_c_Product__Type_OPair_001t__Option__Ooption_It__Nat__Onat_J_001t__Option__Ooption_It__Int__Oint_J,type,
produc920486614911842229on_int: option_nat > option_int > produc2463761468547838845on_int ).
thf(sy_c_Product__Type_OPair_001t__Option__Ooption_It__Nat__Onat_J_001t__Option__Ooption_It__Nat__Onat_J,type,
produc5098337634421038937on_nat: option_nat > option_nat > produc4953844613479565601on_nat ).
thf(sy_c_Product__Type_OPair_001t__Option__Ooption_It__Option__Ooption_It__Int__Oint_J_J_001t__Option__Ooption_It__Option__Ooption_It__Int__Oint_J_J,type,
produc3564219861986917937on_int: option_option_int > option_option_int > produc420109317877091577on_int ).
thf(sy_c_Product__Type_OPair_001t__Option__Ooption_It__Option__Ooption_It__Nat__Onat_J_J_001t__Option__Ooption_It__Option__Ooption_It__Nat__Onat_J_J,type,
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thf(sy_c_Set__Linorder_Oproper__interval__int__rel,type,
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thf(sy_c_Sum__Type_OInl_001t__Int__Oint_001t__Int__Oint,type,
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thf(sy_c_Sum__Type_OInl_001t__Int__Oint_001t__Nat__Onat,type,
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thf(sy_c_Sum__Type_OInl_001t__Nat__Onat_001t__Int__Oint,type,
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thf(sy_c_Sum__Type_OInl_001t__Nat__Onat_001t__Nat__Onat,type,
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thf(sy_c_Sum__Type_OInl_001tf__a_001t__Nat__Onat,type,
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thf(sy_c_Sum__Type_OInr_001t__Int__Oint_001t__Int__Oint,type,
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thf(sy_c_Sum__Type_OInr_001t__Int__Oint_001t__Nat__Onat,type,
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thf(sy_c_Sum__Type_OInr_001t__Nat__Onat_001t__Int__Oint,type,
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thf(sy_c_Sum__Type_OInr_001t__Nat__Onat_001t__Nat__Onat,type,
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thf(sy_c_Sum__Type_OInr_001t__Nat__Onat_001tf__a,type,
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thf(sy_c_Wellfounded_Oaccp_001t__Product____Type__Oprod_It__Option__Ooption_It__Sum____Type__Osum_It__Int__Oint_Mt__Nat__Onat_J_J_Mt__Option__Ooption_It__Sum____Type__Osum_It__Int__Oint_Mt__Nat__Onat_J_J_J,type,
accp_P3675550703157275328nt_nat: ( produc334412093845648695nt_nat > produc334412093845648695nt_nat > $o ) > produc334412093845648695nt_nat > $o ).
thf(sy_c_Wellfounded_Oaccp_001t__Product____Type__Oprod_It__Option__Ooption_It__Sum____Type__Osum_It__Nat__Onat_Mt__Int__Oint_J_J_Mt__Option__Ooption_It__Sum____Type__Osum_It__Nat__Onat_Mt__Int__Oint_J_J_J,type,
accp_P479778451830304448at_int: ( produc6362011879373453623at_int > produc6362011879373453623at_int > $o ) > produc6362011879373453623at_int > $o ).
thf(sy_c_Wellfounded_Oaccp_001t__Product____Type__Oprod_It__Option__Ooption_It__Sum____Type__Osum_It__Nat__Onat_Mt__Nat__Onat_J_J_Mt__Option__Ooption_It__Sum____Type__Osum_It__Nat__Onat_Mt__Nat__Onat_J_J_J,type,
accp_P3884639961936550920at_nat: ( produc543501352624924287at_nat > produc543501352624924287at_nat > $o ) > produc543501352624924287at_nat > $o ).
thf(sy_c_Wellfounded_Olex__prod_001t__Option__Ooption_It__Int__Oint_J_001t__Option__Ooption_It__Int__Oint_J,type,
lex_pr4366770192587316777on_int: set_Pr5725982650624401465on_int > set_Pr5725982650624401465on_int > set_Pr4536470520447424327on_int ).
thf(sy_c_Wellfounded_Olex__prod_001t__Option__Ooption_It__Int__Oint_J_001t__Option__Ooption_It__Nat__Onat_J,type,
lex_pr8544621212096513485on_nat: set_Pr5725982650624401465on_int > set_Pr6588086440996610945on_nat > set_Pr8423666735169913927on_nat ).
thf(sy_c_Wellfounded_Olex__prod_001t__Option__Ooption_It__Nat__Onat_J_001t__Option__Ooption_It__Int__Oint_J,type,
lex_pr8179060047738630093on_int: set_Pr6588086440996610945on_nat > set_Pr5725982650624401465on_int > set_Pr7992093122888647751on_int ).
thf(sy_c_Wellfounded_Olex__prod_001t__Option__Ooption_It__Nat__Onat_J_001t__Option__Ooption_It__Nat__Onat_J,type,
lex_pr3133539030393050993on_nat: set_Pr6588086440996610945on_nat > set_Pr6588086440996610945on_nat > set_Pr2655917300756361543on_nat ).
thf(sy_c_Wellfounded_Omeasure_001t__Option__Ooption_It__Int__Oint_J,type,
measure_option_int: ( option_int > nat ) > set_Pr5725982650624401465on_int ).
thf(sy_c_Wellfounded_Omeasure_001t__Option__Ooption_It__Nat__Onat_J,type,
measure_option_nat: ( option_nat > nat ) > set_Pr6588086440996610945on_nat ).
thf(sy_c_Wellfounded_Omlex__prod_001t__Option__Ooption_It__Int__Oint_J,type,
mlex_prod_option_int: ( option_int > nat ) > set_Pr5725982650624401465on_int > set_Pr5725982650624401465on_int ).
thf(sy_c_Wellfounded_Omlex__prod_001t__Option__Ooption_It__Nat__Onat_J,type,
mlex_prod_option_nat: ( option_nat > nat ) > set_Pr6588086440996610945on_nat > set_Pr6588086440996610945on_nat ).
thf(sy_c_Wfrec_Osame__fst_001t__Option__Ooption_It__Int__Oint_J_001t__Option__Ooption_It__Int__Oint_J,type,
same_f901569133820327014on_int: ( option_int > $o ) > ( option_int > set_Pr5725982650624401465on_int ) > set_Pr4536470520447424327on_int ).
thf(sy_c_Wfrec_Osame__fst_001t__Option__Ooption_It__Nat__Onat_J_001t__Option__Ooption_It__Nat__Onat_J,type,
same_f8891710008480837038on_nat: ( option_nat > $o ) > ( option_nat > set_Pr6588086440996610945on_nat ) > set_Pr2655917300756361543on_nat ).
thf(sy_c_member_001t__List__Olist_It__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J_J,type,
member408289922725080238_a_nat: list_Sum_sum_a_nat > set_li6526943997496501093_a_nat > $o ).
thf(sy_c_member_001t__Product____Type__Oprod_It__Option__Ooption_It__Int__Oint_J_Mt__Option__Ooption_It__Int__Oint_J_J,type,
member7038936195297346946on_int: produc7874843650251301337on_int > set_Pr5725982650624401465on_int > $o ).
thf(sy_c_member_001t__Product____Type__Oprod_It__Option__Ooption_It__Nat__Onat_J_Mt__Option__Ooption_It__Nat__Onat_J_J,type,
member4117937158525611210on_nat: produc4953844613479565601on_nat > set_Pr6588086440996610945on_nat > $o ).
thf(sy_c_member_001t__Product____Type__Oprod_It__Product____Type__Oprod_It__Option__Ooption_It__Int__Oint_J_Mt__Option__Ooption_It__Int__Oint_J_J_Mt__Product____Type__Oprod_It__Option__Ooption_It__Int__Oint_J_Mt__Option__Ooption_It__Int__Oint_J_J_J,type,
member5505722672893847952on_int: produc9050054836243589735on_int > set_Pr4536470520447424327on_int > $o ).
thf(sy_c_member_001t__Product____Type__Oprod_It__Product____Type__Oprod_It__Option__Ooption_It__Int__Oint_J_Mt__Option__Ooption_It__Nat__Onat_J_J_Mt__Product____Type__Oprod_It__Option__Ooption_It__Int__Oint_J_Mt__Option__Ooption_It__Nat__Onat_J_J_J,type,
member846496771861665424on_nat: produc4390828935211407207on_nat > set_Pr8423666735169913927on_nat > $o ).
thf(sy_c_member_001t__Product____Type__Oprod_It__Product____Type__Oprod_It__Option__Ooption_It__Nat__Onat_J_Mt__Option__Ooption_It__Int__Oint_J_J_Mt__Product____Type__Oprod_It__Option__Ooption_It__Nat__Onat_J_Mt__Option__Ooption_It__Int__Oint_J_J_J,type,
member7430460011603515024on_int: produc1751420138098480999on_int > set_Pr7992093122888647751on_int > $o ).
thf(sy_c_member_001t__Product____Type__Oprod_It__Product____Type__Oprod_It__Option__Ooption_It__Nat__Onat_J_Mt__Option__Ooption_It__Nat__Onat_J_J_Mt__Product____Type__Oprod_It__Option__Ooption_It__Nat__Onat_J_Mt__Option__Ooption_It__Nat__Onat_J_J_J,type,
member2771234110571332496on_nat: produc6315566273921074279on_nat > set_Pr2655917300756361543on_nat > $o ).
thf(sy_v_AD,type,
ad: set_a ).
thf(sy_v_ia,type,
ia: nat ).
thf(sy_v_ja,type,
ja: nat ).
thf(sy_v_na,type,
na: nat ).
thf(sy_v_x,type,
x: list_Sum_sum_a_nat ).
thf(sy_v_xb,type,
xb: nat ).
% Relevant facts (1264)
thf(fact_0_id__mapD_I2_J,axiom,
! [J: nat,I: nat,X: nat] :
( ( ( id_map_a @ J @ ( sum_Inr_nat_a @ I ) )
= ( some_nat @ X ) )
=> ( ( ord_less_nat @ I @ J )
& ( I = X ) ) ) ).
% id_mapD(2)
thf(fact_1_sum_Oinject_I2_J,axiom,
! [X2: nat,Y2: nat] :
( ( ( sum_Inr_nat_a @ X2 )
= ( sum_Inr_nat_a @ Y2 ) )
= ( X2 = Y2 ) ) ).
% sum.inject(2)
thf(fact_2_old_Osum_Oinject_I2_J,axiom,
! [B: nat,B2: nat] :
( ( ( sum_Inr_nat_a @ B )
= ( sum_Inr_nat_a @ B2 ) )
= ( B = B2 ) ) ).
% old.sum.inject(2)
thf(fact_3_option_Oinject,axiom,
! [X2: nat,Y2: nat] :
( ( ( some_nat @ X2 )
= ( some_nat @ Y2 ) )
= ( X2 = Y2 ) ) ).
% option.inject
thf(fact_4_option_Oinject,axiom,
! [X2: int,Y2: int] :
( ( ( some_int @ X2 )
= ( some_int @ Y2 ) )
= ( X2 = Y2 ) ) ).
% option.inject
thf(fact_5_order__refl,axiom,
! [X: nat] : ( ord_less_eq_nat @ X @ X ) ).
% order_refl
thf(fact_6_order__refl,axiom,
! [X: int] : ( ord_less_eq_int @ X @ X ) ).
% order_refl
thf(fact_7_order__refl,axiom,
! [X: real] : ( ord_less_eq_real @ X @ X ) ).
% order_refl
thf(fact_8_dual__order_Orefl,axiom,
! [A: nat] : ( ord_less_eq_nat @ A @ A ) ).
% dual_order.refl
thf(fact_9_dual__order_Orefl,axiom,
! [A: int] : ( ord_less_eq_int @ A @ A ) ).
% dual_order.refl
thf(fact_10_dual__order_Orefl,axiom,
! [A: real] : ( ord_less_eq_real @ A @ A ) ).
% dual_order.refl
thf(fact_11_nat__less__le,axiom,
( ord_less_nat
= ( ^ [M: nat,N: nat] :
( ( ord_less_eq_nat @ M @ N )
& ( M != N ) ) ) ) ).
% nat_less_le
thf(fact_12_less__imp__le__nat,axiom,
! [M2: nat,N2: nat] :
( ( ord_less_nat @ M2 @ N2 )
=> ( ord_less_eq_nat @ M2 @ N2 ) ) ).
% less_imp_le_nat
thf(fact_13_le__eq__less__or__eq,axiom,
( ord_less_eq_nat
= ( ^ [M: nat,N: nat] :
( ( ord_less_nat @ M @ N )
| ( M = N ) ) ) ) ).
% le_eq_less_or_eq
thf(fact_14_less__or__eq__imp__le,axiom,
! [M2: nat,N2: nat] :
( ( ( ord_less_nat @ M2 @ N2 )
| ( M2 = N2 ) )
=> ( ord_less_eq_nat @ M2 @ N2 ) ) ).
% less_or_eq_imp_le
thf(fact_15_le__neq__implies__less,axiom,
! [M2: nat,N2: nat] :
( ( ord_less_eq_nat @ M2 @ N2 )
=> ( ( M2 != N2 )
=> ( ord_less_nat @ M2 @ N2 ) ) ) ).
% le_neq_implies_less
thf(fact_16_less__mono__imp__le__mono,axiom,
! [F: nat > nat,I: nat,J: nat] :
( ! [I2: nat,J2: nat] :
( ( ord_less_nat @ I2 @ J2 )
=> ( ord_less_nat @ ( F @ I2 ) @ ( F @ J2 ) ) )
=> ( ( ord_less_eq_nat @ I @ J )
=> ( ord_less_eq_nat @ ( F @ I ) @ ( F @ J ) ) ) ) ).
% less_mono_imp_le_mono
thf(fact_17_nat__descend__induct,axiom,
! [N2: nat,P: nat > $o,M2: nat] :
( ! [K: nat] :
( ( ord_less_nat @ N2 @ K )
=> ( P @ K ) )
=> ( ! [K: nat] :
( ( ord_less_eq_nat @ K @ N2 )
=> ( ! [I3: nat] :
( ( ord_less_nat @ K @ I3 )
=> ( P @ I3 ) )
=> ( P @ K ) ) )
=> ( P @ M2 ) ) ) ).
% nat_descend_induct
thf(fact_18_order__antisym__conv,axiom,
! [Y: nat,X: nat] :
( ( ord_less_eq_nat @ Y @ X )
=> ( ( ord_less_eq_nat @ X @ Y )
= ( X = Y ) ) ) ).
% order_antisym_conv
thf(fact_19_order__antisym__conv,axiom,
! [Y: int,X: int] :
( ( ord_less_eq_int @ Y @ X )
=> ( ( ord_less_eq_int @ X @ Y )
= ( X = Y ) ) ) ).
% order_antisym_conv
thf(fact_20_order__antisym__conv,axiom,
! [Y: real,X: real] :
( ( ord_less_eq_real @ Y @ X )
=> ( ( ord_less_eq_real @ X @ Y )
= ( X = Y ) ) ) ).
% order_antisym_conv
thf(fact_21_linorder__le__cases,axiom,
! [X: nat,Y: nat] :
( ~ ( ord_less_eq_nat @ X @ Y )
=> ( ord_less_eq_nat @ Y @ X ) ) ).
% linorder_le_cases
thf(fact_22_linorder__le__cases,axiom,
! [X: int,Y: int] :
( ~ ( ord_less_eq_int @ X @ Y )
=> ( ord_less_eq_int @ Y @ X ) ) ).
% linorder_le_cases
thf(fact_23_linorder__le__cases,axiom,
! [X: real,Y: real] :
( ~ ( ord_less_eq_real @ X @ Y )
=> ( ord_less_eq_real @ Y @ X ) ) ).
% linorder_le_cases
thf(fact_24_ord__le__eq__subst,axiom,
! [A: nat,B: nat,F: nat > nat,C: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X3: nat,Y3: nat] :
( ( ord_less_eq_nat @ X3 @ Y3 )
=> ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_nat @ ( F @ A ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_25_ord__le__eq__subst,axiom,
! [A: nat,B: nat,F: nat > int,C: int] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X3: nat,Y3: nat] :
( ( ord_less_eq_nat @ X3 @ Y3 )
=> ( ord_less_eq_int @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_int @ ( F @ A ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_26_ord__le__eq__subst,axiom,
! [A: nat,B: nat,F: nat > real,C: real] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X3: nat,Y3: nat] :
( ( ord_less_eq_nat @ X3 @ Y3 )
=> ( ord_less_eq_real @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_real @ ( F @ A ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_27_ord__le__eq__subst,axiom,
! [A: int,B: int,F: int > nat,C: nat] :
( ( ord_less_eq_int @ A @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X3: int,Y3: int] :
( ( ord_less_eq_int @ X3 @ Y3 )
=> ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_nat @ ( F @ A ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_28_ord__le__eq__subst,axiom,
! [A: int,B: int,F: int > int,C: int] :
( ( ord_less_eq_int @ A @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X3: int,Y3: int] :
( ( ord_less_eq_int @ X3 @ Y3 )
=> ( ord_less_eq_int @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_int @ ( F @ A ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_29_ord__le__eq__subst,axiom,
! [A: int,B: int,F: int > real,C: real] :
( ( ord_less_eq_int @ A @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X3: int,Y3: int] :
( ( ord_less_eq_int @ X3 @ Y3 )
=> ( ord_less_eq_real @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_real @ ( F @ A ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_30_ord__le__eq__subst,axiom,
! [A: real,B: real,F: real > nat,C: nat] :
( ( ord_less_eq_real @ A @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X3: real,Y3: real] :
( ( ord_less_eq_real @ X3 @ Y3 )
=> ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_nat @ ( F @ A ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_31_ord__le__eq__subst,axiom,
! [A: real,B: real,F: real > int,C: int] :
( ( ord_less_eq_real @ A @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X3: real,Y3: real] :
( ( ord_less_eq_real @ X3 @ Y3 )
=> ( ord_less_eq_int @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_int @ ( F @ A ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_32_ord__le__eq__subst,axiom,
! [A: real,B: real,F: real > real,C: real] :
( ( ord_less_eq_real @ A @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X3: real,Y3: real] :
( ( ord_less_eq_real @ X3 @ Y3 )
=> ( ord_less_eq_real @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_real @ ( F @ A ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_33_ord__eq__le__subst,axiom,
! [A: nat,F: nat > nat,B: nat,C: nat] :
( ( A
= ( F @ B ) )
=> ( ( ord_less_eq_nat @ B @ C )
=> ( ! [X3: nat,Y3: nat] :
( ( ord_less_eq_nat @ X3 @ Y3 )
=> ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_nat @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_34_ord__eq__le__subst,axiom,
! [A: int,F: nat > int,B: nat,C: nat] :
( ( A
= ( F @ B ) )
=> ( ( ord_less_eq_nat @ B @ C )
=> ( ! [X3: nat,Y3: nat] :
( ( ord_less_eq_nat @ X3 @ Y3 )
=> ( ord_less_eq_int @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_int @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_35_ord__eq__le__subst,axiom,
! [A: real,F: nat > real,B: nat,C: nat] :
( ( A
= ( F @ B ) )
=> ( ( ord_less_eq_nat @ B @ C )
=> ( ! [X3: nat,Y3: nat] :
( ( ord_less_eq_nat @ X3 @ Y3 )
=> ( ord_less_eq_real @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_real @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_36_ord__eq__le__subst,axiom,
! [A: nat,F: int > nat,B: int,C: int] :
( ( A
= ( F @ B ) )
=> ( ( ord_less_eq_int @ B @ C )
=> ( ! [X3: int,Y3: int] :
( ( ord_less_eq_int @ X3 @ Y3 )
=> ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_nat @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_37_ord__eq__le__subst,axiom,
! [A: int,F: int > int,B: int,C: int] :
( ( A
= ( F @ B ) )
=> ( ( ord_less_eq_int @ B @ C )
=> ( ! [X3: int,Y3: int] :
( ( ord_less_eq_int @ X3 @ Y3 )
=> ( ord_less_eq_int @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_int @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_38_ord__eq__le__subst,axiom,
! [A: real,F: int > real,B: int,C: int] :
( ( A
= ( F @ B ) )
=> ( ( ord_less_eq_int @ B @ C )
=> ( ! [X3: int,Y3: int] :
( ( ord_less_eq_int @ X3 @ Y3 )
=> ( ord_less_eq_real @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_real @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_39_ord__eq__le__subst,axiom,
! [A: nat,F: real > nat,B: real,C: real] :
( ( A
= ( F @ B ) )
=> ( ( ord_less_eq_real @ B @ C )
=> ( ! [X3: real,Y3: real] :
( ( ord_less_eq_real @ X3 @ Y3 )
=> ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_nat @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_40_ord__eq__le__subst,axiom,
! [A: int,F: real > int,B: real,C: real] :
( ( A
= ( F @ B ) )
=> ( ( ord_less_eq_real @ B @ C )
=> ( ! [X3: real,Y3: real] :
( ( ord_less_eq_real @ X3 @ Y3 )
=> ( ord_less_eq_int @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_int @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_41_ord__eq__le__subst,axiom,
! [A: real,F: real > real,B: real,C: real] :
( ( A
= ( F @ B ) )
=> ( ( ord_less_eq_real @ B @ C )
=> ( ! [X3: real,Y3: real] :
( ( ord_less_eq_real @ X3 @ Y3 )
=> ( ord_less_eq_real @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_real @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_42_linorder__linear,axiom,
! [X: nat,Y: nat] :
( ( ord_less_eq_nat @ X @ Y )
| ( ord_less_eq_nat @ Y @ X ) ) ).
% linorder_linear
thf(fact_43_linorder__linear,axiom,
! [X: int,Y: int] :
( ( ord_less_eq_int @ X @ Y )
| ( ord_less_eq_int @ Y @ X ) ) ).
% linorder_linear
thf(fact_44_linorder__linear,axiom,
! [X: real,Y: real] :
( ( ord_less_eq_real @ X @ Y )
| ( ord_less_eq_real @ Y @ X ) ) ).
% linorder_linear
thf(fact_45_order__eq__refl,axiom,
! [X: nat,Y: nat] :
( ( X = Y )
=> ( ord_less_eq_nat @ X @ Y ) ) ).
% order_eq_refl
thf(fact_46_order__eq__refl,axiom,
! [X: int,Y: int] :
( ( X = Y )
=> ( ord_less_eq_int @ X @ Y ) ) ).
% order_eq_refl
thf(fact_47_order__eq__refl,axiom,
! [X: real,Y: real] :
( ( X = Y )
=> ( ord_less_eq_real @ X @ Y ) ) ).
% order_eq_refl
thf(fact_48_order__subst2,axiom,
! [A: nat,B: nat,F: nat > nat,C: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ord_less_eq_nat @ ( F @ B ) @ C )
=> ( ! [X3: nat,Y3: nat] :
( ( ord_less_eq_nat @ X3 @ Y3 )
=> ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_nat @ ( F @ A ) @ C ) ) ) ) ).
% order_subst2
thf(fact_49_order__subst2,axiom,
! [A: nat,B: nat,F: nat > int,C: int] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ord_less_eq_int @ ( F @ B ) @ C )
=> ( ! [X3: nat,Y3: nat] :
( ( ord_less_eq_nat @ X3 @ Y3 )
=> ( ord_less_eq_int @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_int @ ( F @ A ) @ C ) ) ) ) ).
% order_subst2
thf(fact_50_order__subst2,axiom,
! [A: nat,B: nat,F: nat > real,C: real] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ord_less_eq_real @ ( F @ B ) @ C )
=> ( ! [X3: nat,Y3: nat] :
( ( ord_less_eq_nat @ X3 @ Y3 )
=> ( ord_less_eq_real @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_real @ ( F @ A ) @ C ) ) ) ) ).
% order_subst2
thf(fact_51_order__subst2,axiom,
! [A: int,B: int,F: int > nat,C: nat] :
( ( ord_less_eq_int @ A @ B )
=> ( ( ord_less_eq_nat @ ( F @ B ) @ C )
=> ( ! [X3: int,Y3: int] :
( ( ord_less_eq_int @ X3 @ Y3 )
=> ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_nat @ ( F @ A ) @ C ) ) ) ) ).
% order_subst2
thf(fact_52_order__subst2,axiom,
! [A: int,B: int,F: int > int,C: int] :
( ( ord_less_eq_int @ A @ B )
=> ( ( ord_less_eq_int @ ( F @ B ) @ C )
=> ( ! [X3: int,Y3: int] :
( ( ord_less_eq_int @ X3 @ Y3 )
=> ( ord_less_eq_int @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_int @ ( F @ A ) @ C ) ) ) ) ).
% order_subst2
thf(fact_53_order__subst2,axiom,
! [A: int,B: int,F: int > real,C: real] :
( ( ord_less_eq_int @ A @ B )
=> ( ( ord_less_eq_real @ ( F @ B ) @ C )
=> ( ! [X3: int,Y3: int] :
( ( ord_less_eq_int @ X3 @ Y3 )
=> ( ord_less_eq_real @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_real @ ( F @ A ) @ C ) ) ) ) ).
% order_subst2
thf(fact_54_order__subst2,axiom,
! [A: real,B: real,F: real > nat,C: nat] :
( ( ord_less_eq_real @ A @ B )
=> ( ( ord_less_eq_nat @ ( F @ B ) @ C )
=> ( ! [X3: real,Y3: real] :
( ( ord_less_eq_real @ X3 @ Y3 )
=> ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_nat @ ( F @ A ) @ C ) ) ) ) ).
% order_subst2
thf(fact_55_order__subst2,axiom,
! [A: real,B: real,F: real > int,C: int] :
( ( ord_less_eq_real @ A @ B )
=> ( ( ord_less_eq_int @ ( F @ B ) @ C )
=> ( ! [X3: real,Y3: real] :
( ( ord_less_eq_real @ X3 @ Y3 )
=> ( ord_less_eq_int @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_int @ ( F @ A ) @ C ) ) ) ) ).
% order_subst2
thf(fact_56_order__subst2,axiom,
! [A: real,B: real,F: real > real,C: real] :
( ( ord_less_eq_real @ A @ B )
=> ( ( ord_less_eq_real @ ( F @ B ) @ C )
=> ( ! [X3: real,Y3: real] :
( ( ord_less_eq_real @ X3 @ Y3 )
=> ( ord_less_eq_real @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_real @ ( F @ A ) @ C ) ) ) ) ).
% order_subst2
thf(fact_57_order__subst1,axiom,
! [A: nat,F: nat > nat,B: nat,C: nat] :
( ( ord_less_eq_nat @ A @ ( F @ B ) )
=> ( ( ord_less_eq_nat @ B @ C )
=> ( ! [X3: nat,Y3: nat] :
( ( ord_less_eq_nat @ X3 @ Y3 )
=> ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_nat @ A @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_58_order__subst1,axiom,
! [A: nat,F: int > nat,B: int,C: int] :
( ( ord_less_eq_nat @ A @ ( F @ B ) )
=> ( ( ord_less_eq_int @ B @ C )
=> ( ! [X3: int,Y3: int] :
( ( ord_less_eq_int @ X3 @ Y3 )
=> ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_nat @ A @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_59_order__subst1,axiom,
! [A: nat,F: real > nat,B: real,C: real] :
( ( ord_less_eq_nat @ A @ ( F @ B ) )
=> ( ( ord_less_eq_real @ B @ C )
=> ( ! [X3: real,Y3: real] :
( ( ord_less_eq_real @ X3 @ Y3 )
=> ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_nat @ A @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_60_order__subst1,axiom,
! [A: int,F: nat > int,B: nat,C: nat] :
( ( ord_less_eq_int @ A @ ( F @ B ) )
=> ( ( ord_less_eq_nat @ B @ C )
=> ( ! [X3: nat,Y3: nat] :
( ( ord_less_eq_nat @ X3 @ Y3 )
=> ( ord_less_eq_int @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_int @ A @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_61_order__subst1,axiom,
! [A: int,F: int > int,B: int,C: int] :
( ( ord_less_eq_int @ A @ ( F @ B ) )
=> ( ( ord_less_eq_int @ B @ C )
=> ( ! [X3: int,Y3: int] :
( ( ord_less_eq_int @ X3 @ Y3 )
=> ( ord_less_eq_int @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_int @ A @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_62_order__subst1,axiom,
! [A: int,F: real > int,B: real,C: real] :
( ( ord_less_eq_int @ A @ ( F @ B ) )
=> ( ( ord_less_eq_real @ B @ C )
=> ( ! [X3: real,Y3: real] :
( ( ord_less_eq_real @ X3 @ Y3 )
=> ( ord_less_eq_int @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_int @ A @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_63_order__subst1,axiom,
! [A: real,F: nat > real,B: nat,C: nat] :
( ( ord_less_eq_real @ A @ ( F @ B ) )
=> ( ( ord_less_eq_nat @ B @ C )
=> ( ! [X3: nat,Y3: nat] :
( ( ord_less_eq_nat @ X3 @ Y3 )
=> ( ord_less_eq_real @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_real @ A @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_64_order__subst1,axiom,
! [A: real,F: int > real,B: int,C: int] :
( ( ord_less_eq_real @ A @ ( F @ B ) )
=> ( ( ord_less_eq_int @ B @ C )
=> ( ! [X3: int,Y3: int] :
( ( ord_less_eq_int @ X3 @ Y3 )
=> ( ord_less_eq_real @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_real @ A @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_65_order__subst1,axiom,
! [A: real,F: real > real,B: real,C: real] :
( ( ord_less_eq_real @ A @ ( F @ B ) )
=> ( ( ord_less_eq_real @ B @ C )
=> ( ! [X3: real,Y3: real] :
( ( ord_less_eq_real @ X3 @ Y3 )
=> ( ord_less_eq_real @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_real @ A @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_66_Orderings_Oorder__eq__iff,axiom,
( ( ^ [Y4: nat,Z: nat] : ( Y4 = Z ) )
= ( ^ [A2: nat,B3: nat] :
( ( ord_less_eq_nat @ A2 @ B3 )
& ( ord_less_eq_nat @ B3 @ A2 ) ) ) ) ).
% Orderings.order_eq_iff
thf(fact_67_Orderings_Oorder__eq__iff,axiom,
( ( ^ [Y4: int,Z: int] : ( Y4 = Z ) )
= ( ^ [A2: int,B3: int] :
( ( ord_less_eq_int @ A2 @ B3 )
& ( ord_less_eq_int @ B3 @ A2 ) ) ) ) ).
% Orderings.order_eq_iff
thf(fact_68_Orderings_Oorder__eq__iff,axiom,
( ( ^ [Y4: real,Z: real] : ( Y4 = Z ) )
= ( ^ [A2: real,B3: real] :
( ( ord_less_eq_real @ A2 @ B3 )
& ( ord_less_eq_real @ B3 @ A2 ) ) ) ) ).
% Orderings.order_eq_iff
thf(fact_69_antisym,axiom,
! [A: nat,B: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ord_less_eq_nat @ B @ A )
=> ( A = B ) ) ) ).
% antisym
thf(fact_70_antisym,axiom,
! [A: int,B: int] :
( ( ord_less_eq_int @ A @ B )
=> ( ( ord_less_eq_int @ B @ A )
=> ( A = B ) ) ) ).
% antisym
thf(fact_71_antisym,axiom,
! [A: real,B: real] :
( ( ord_less_eq_real @ A @ B )
=> ( ( ord_less_eq_real @ B @ A )
=> ( A = B ) ) ) ).
% antisym
thf(fact_72_dual__order_Otrans,axiom,
! [B: nat,A: nat,C: nat] :
( ( ord_less_eq_nat @ B @ A )
=> ( ( ord_less_eq_nat @ C @ B )
=> ( ord_less_eq_nat @ C @ A ) ) ) ).
% dual_order.trans
thf(fact_73_dual__order_Otrans,axiom,
! [B: int,A: int,C: int] :
( ( ord_less_eq_int @ B @ A )
=> ( ( ord_less_eq_int @ C @ B )
=> ( ord_less_eq_int @ C @ A ) ) ) ).
% dual_order.trans
thf(fact_74_dual__order_Otrans,axiom,
! [B: real,A: real,C: real] :
( ( ord_less_eq_real @ B @ A )
=> ( ( ord_less_eq_real @ C @ B )
=> ( ord_less_eq_real @ C @ A ) ) ) ).
% dual_order.trans
thf(fact_75_dual__order_Oantisym,axiom,
! [B: nat,A: nat] :
( ( ord_less_eq_nat @ B @ A )
=> ( ( ord_less_eq_nat @ A @ B )
=> ( A = B ) ) ) ).
% dual_order.antisym
thf(fact_76_dual__order_Oantisym,axiom,
! [B: int,A: int] :
( ( ord_less_eq_int @ B @ A )
=> ( ( ord_less_eq_int @ A @ B )
=> ( A = B ) ) ) ).
% dual_order.antisym
thf(fact_77_dual__order_Oantisym,axiom,
! [B: real,A: real] :
( ( ord_less_eq_real @ B @ A )
=> ( ( ord_less_eq_real @ A @ B )
=> ( A = B ) ) ) ).
% dual_order.antisym
thf(fact_78_dual__order_Oeq__iff,axiom,
( ( ^ [Y4: nat,Z: nat] : ( Y4 = Z ) )
= ( ^ [A2: nat,B3: nat] :
( ( ord_less_eq_nat @ B3 @ A2 )
& ( ord_less_eq_nat @ A2 @ B3 ) ) ) ) ).
% dual_order.eq_iff
thf(fact_79_dual__order_Oeq__iff,axiom,
( ( ^ [Y4: int,Z: int] : ( Y4 = Z ) )
= ( ^ [A2: int,B3: int] :
( ( ord_less_eq_int @ B3 @ A2 )
& ( ord_less_eq_int @ A2 @ B3 ) ) ) ) ).
% dual_order.eq_iff
thf(fact_80_dual__order_Oeq__iff,axiom,
( ( ^ [Y4: real,Z: real] : ( Y4 = Z ) )
= ( ^ [A2: real,B3: real] :
( ( ord_less_eq_real @ B3 @ A2 )
& ( ord_less_eq_real @ A2 @ B3 ) ) ) ) ).
% dual_order.eq_iff
thf(fact_81_linorder__wlog,axiom,
! [P: nat > nat > $o,A: nat,B: nat] :
( ! [A3: nat,B4: nat] :
( ( ord_less_eq_nat @ A3 @ B4 )
=> ( P @ A3 @ B4 ) )
=> ( ! [A3: nat,B4: nat] :
( ( P @ B4 @ A3 )
=> ( P @ A3 @ B4 ) )
=> ( P @ A @ B ) ) ) ).
% linorder_wlog
thf(fact_82_linorder__wlog,axiom,
! [P: int > int > $o,A: int,B: int] :
( ! [A3: int,B4: int] :
( ( ord_less_eq_int @ A3 @ B4 )
=> ( P @ A3 @ B4 ) )
=> ( ! [A3: int,B4: int] :
( ( P @ B4 @ A3 )
=> ( P @ A3 @ B4 ) )
=> ( P @ A @ B ) ) ) ).
% linorder_wlog
thf(fact_83_linorder__wlog,axiom,
! [P: real > real > $o,A: real,B: real] :
( ! [A3: real,B4: real] :
( ( ord_less_eq_real @ A3 @ B4 )
=> ( P @ A3 @ B4 ) )
=> ( ! [A3: real,B4: real] :
( ( P @ B4 @ A3 )
=> ( P @ A3 @ B4 ) )
=> ( P @ A @ B ) ) ) ).
% linorder_wlog
thf(fact_84_order__trans,axiom,
! [X: nat,Y: nat,Z2: nat] :
( ( ord_less_eq_nat @ X @ Y )
=> ( ( ord_less_eq_nat @ Y @ Z2 )
=> ( ord_less_eq_nat @ X @ Z2 ) ) ) ).
% order_trans
thf(fact_85_order__trans,axiom,
! [X: int,Y: int,Z2: int] :
( ( ord_less_eq_int @ X @ Y )
=> ( ( ord_less_eq_int @ Y @ Z2 )
=> ( ord_less_eq_int @ X @ Z2 ) ) ) ).
% order_trans
thf(fact_86_order__trans,axiom,
! [X: real,Y: real,Z2: real] :
( ( ord_less_eq_real @ X @ Y )
=> ( ( ord_less_eq_real @ Y @ Z2 )
=> ( ord_less_eq_real @ X @ Z2 ) ) ) ).
% order_trans
thf(fact_87_order_Otrans,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ord_less_eq_nat @ B @ C )
=> ( ord_less_eq_nat @ A @ C ) ) ) ).
% order.trans
thf(fact_88_order_Otrans,axiom,
! [A: int,B: int,C: int] :
( ( ord_less_eq_int @ A @ B )
=> ( ( ord_less_eq_int @ B @ C )
=> ( ord_less_eq_int @ A @ C ) ) ) ).
% order.trans
thf(fact_89_order_Otrans,axiom,
! [A: real,B: real,C: real] :
( ( ord_less_eq_real @ A @ B )
=> ( ( ord_less_eq_real @ B @ C )
=> ( ord_less_eq_real @ A @ C ) ) ) ).
% order.trans
thf(fact_90_order__antisym,axiom,
! [X: nat,Y: nat] :
( ( ord_less_eq_nat @ X @ Y )
=> ( ( ord_less_eq_nat @ Y @ X )
=> ( X = Y ) ) ) ).
% order_antisym
thf(fact_91_order__antisym,axiom,
! [X: int,Y: int] :
( ( ord_less_eq_int @ X @ Y )
=> ( ( ord_less_eq_int @ Y @ X )
=> ( X = Y ) ) ) ).
% order_antisym
thf(fact_92_order__antisym,axiom,
! [X: real,Y: real] :
( ( ord_less_eq_real @ X @ Y )
=> ( ( ord_less_eq_real @ Y @ X )
=> ( X = Y ) ) ) ).
% order_antisym
thf(fact_93_ord__le__eq__trans,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( B = C )
=> ( ord_less_eq_nat @ A @ C ) ) ) ).
% ord_le_eq_trans
thf(fact_94_ord__le__eq__trans,axiom,
! [A: int,B: int,C: int] :
( ( ord_less_eq_int @ A @ B )
=> ( ( B = C )
=> ( ord_less_eq_int @ A @ C ) ) ) ).
% ord_le_eq_trans
thf(fact_95_ord__le__eq__trans,axiom,
! [A: real,B: real,C: real] :
( ( ord_less_eq_real @ A @ B )
=> ( ( B = C )
=> ( ord_less_eq_real @ A @ C ) ) ) ).
% ord_le_eq_trans
thf(fact_96_ord__eq__le__trans,axiom,
! [A: nat,B: nat,C: nat] :
( ( A = B )
=> ( ( ord_less_eq_nat @ B @ C )
=> ( ord_less_eq_nat @ A @ C ) ) ) ).
% ord_eq_le_trans
thf(fact_97_ord__eq__le__trans,axiom,
! [A: int,B: int,C: int] :
( ( A = B )
=> ( ( ord_less_eq_int @ B @ C )
=> ( ord_less_eq_int @ A @ C ) ) ) ).
% ord_eq_le_trans
thf(fact_98_ord__eq__le__trans,axiom,
! [A: real,B: real,C: real] :
( ( A = B )
=> ( ( ord_less_eq_real @ B @ C )
=> ( ord_less_eq_real @ A @ C ) ) ) ).
% ord_eq_le_trans
thf(fact_99_order__class_Oorder__eq__iff,axiom,
( ( ^ [Y4: nat,Z: nat] : ( Y4 = Z ) )
= ( ^ [X4: nat,Y5: nat] :
( ( ord_less_eq_nat @ X4 @ Y5 )
& ( ord_less_eq_nat @ Y5 @ X4 ) ) ) ) ).
% order_class.order_eq_iff
thf(fact_100_order__class_Oorder__eq__iff,axiom,
( ( ^ [Y4: int,Z: int] : ( Y4 = Z ) )
= ( ^ [X4: int,Y5: int] :
( ( ord_less_eq_int @ X4 @ Y5 )
& ( ord_less_eq_int @ Y5 @ X4 ) ) ) ) ).
% order_class.order_eq_iff
thf(fact_101_order__class_Oorder__eq__iff,axiom,
( ( ^ [Y4: real,Z: real] : ( Y4 = Z ) )
= ( ^ [X4: real,Y5: real] :
( ( ord_less_eq_real @ X4 @ Y5 )
& ( ord_less_eq_real @ Y5 @ X4 ) ) ) ) ).
% order_class.order_eq_iff
thf(fact_102_le__cases3,axiom,
! [X: nat,Y: nat,Z2: nat] :
( ( ( ord_less_eq_nat @ X @ Y )
=> ~ ( ord_less_eq_nat @ Y @ Z2 ) )
=> ( ( ( ord_less_eq_nat @ Y @ X )
=> ~ ( ord_less_eq_nat @ X @ Z2 ) )
=> ( ( ( ord_less_eq_nat @ X @ Z2 )
=> ~ ( ord_less_eq_nat @ Z2 @ Y ) )
=> ( ( ( ord_less_eq_nat @ Z2 @ Y )
=> ~ ( ord_less_eq_nat @ Y @ X ) )
=> ( ( ( ord_less_eq_nat @ Y @ Z2 )
=> ~ ( ord_less_eq_nat @ Z2 @ X ) )
=> ~ ( ( ord_less_eq_nat @ Z2 @ X )
=> ~ ( ord_less_eq_nat @ X @ Y ) ) ) ) ) ) ) ).
% le_cases3
thf(fact_103_le__cases3,axiom,
! [X: int,Y: int,Z2: int] :
( ( ( ord_less_eq_int @ X @ Y )
=> ~ ( ord_less_eq_int @ Y @ Z2 ) )
=> ( ( ( ord_less_eq_int @ Y @ X )
=> ~ ( ord_less_eq_int @ X @ Z2 ) )
=> ( ( ( ord_less_eq_int @ X @ Z2 )
=> ~ ( ord_less_eq_int @ Z2 @ Y ) )
=> ( ( ( ord_less_eq_int @ Z2 @ Y )
=> ~ ( ord_less_eq_int @ Y @ X ) )
=> ( ( ( ord_less_eq_int @ Y @ Z2 )
=> ~ ( ord_less_eq_int @ Z2 @ X ) )
=> ~ ( ( ord_less_eq_int @ Z2 @ X )
=> ~ ( ord_less_eq_int @ X @ Y ) ) ) ) ) ) ) ).
% le_cases3
thf(fact_104_le__cases3,axiom,
! [X: real,Y: real,Z2: real] :
( ( ( ord_less_eq_real @ X @ Y )
=> ~ ( ord_less_eq_real @ Y @ Z2 ) )
=> ( ( ( ord_less_eq_real @ Y @ X )
=> ~ ( ord_less_eq_real @ X @ Z2 ) )
=> ( ( ( ord_less_eq_real @ X @ Z2 )
=> ~ ( ord_less_eq_real @ Z2 @ Y ) )
=> ( ( ( ord_less_eq_real @ Z2 @ Y )
=> ~ ( ord_less_eq_real @ Y @ X ) )
=> ( ( ( ord_less_eq_real @ Y @ Z2 )
=> ~ ( ord_less_eq_real @ Z2 @ X ) )
=> ~ ( ( ord_less_eq_real @ Z2 @ X )
=> ~ ( ord_less_eq_real @ X @ Y ) ) ) ) ) ) ) ).
% le_cases3
thf(fact_105_nle__le,axiom,
! [A: nat,B: nat] :
( ( ~ ( ord_less_eq_nat @ A @ B ) )
= ( ( ord_less_eq_nat @ B @ A )
& ( B != A ) ) ) ).
% nle_le
thf(fact_106_nle__le,axiom,
! [A: int,B: int] :
( ( ~ ( ord_less_eq_int @ A @ B ) )
= ( ( ord_less_eq_int @ B @ A )
& ( B != A ) ) ) ).
% nle_le
thf(fact_107_nle__le,axiom,
! [A: real,B: real] :
( ( ~ ( ord_less_eq_real @ A @ B ) )
= ( ( ord_less_eq_real @ B @ A )
& ( B != A ) ) ) ).
% nle_le
thf(fact_108_order__less__imp__not__less,axiom,
! [X: nat,Y: nat] :
( ( ord_less_nat @ X @ Y )
=> ~ ( ord_less_nat @ Y @ X ) ) ).
% order_less_imp_not_less
thf(fact_109_order__less__imp__not__less,axiom,
! [X: int,Y: int] :
( ( ord_less_int @ X @ Y )
=> ~ ( ord_less_int @ Y @ X ) ) ).
% order_less_imp_not_less
thf(fact_110_order__less__imp__not__less,axiom,
! [X: real,Y: real] :
( ( ord_less_real @ X @ Y )
=> ~ ( ord_less_real @ Y @ X ) ) ).
% order_less_imp_not_less
thf(fact_111_order__less__imp__not__eq2,axiom,
! [X: nat,Y: nat] :
( ( ord_less_nat @ X @ Y )
=> ( Y != X ) ) ).
% order_less_imp_not_eq2
thf(fact_112_order__less__imp__not__eq2,axiom,
! [X: int,Y: int] :
( ( ord_less_int @ X @ Y )
=> ( Y != X ) ) ).
% order_less_imp_not_eq2
thf(fact_113_order__less__imp__not__eq2,axiom,
! [X: real,Y: real] :
( ( ord_less_real @ X @ Y )
=> ( Y != X ) ) ).
% order_less_imp_not_eq2
thf(fact_114_order__less__imp__not__eq,axiom,
! [X: nat,Y: nat] :
( ( ord_less_nat @ X @ Y )
=> ( X != Y ) ) ).
% order_less_imp_not_eq
thf(fact_115_order__less__imp__not__eq,axiom,
! [X: int,Y: int] :
( ( ord_less_int @ X @ Y )
=> ( X != Y ) ) ).
% order_less_imp_not_eq
thf(fact_116_order__less__imp__not__eq,axiom,
! [X: real,Y: real] :
( ( ord_less_real @ X @ Y )
=> ( X != Y ) ) ).
% order_less_imp_not_eq
thf(fact_117_linorder__less__linear,axiom,
! [X: nat,Y: nat] :
( ( ord_less_nat @ X @ Y )
| ( X = Y )
| ( ord_less_nat @ Y @ X ) ) ).
% linorder_less_linear
thf(fact_118_linorder__less__linear,axiom,
! [X: int,Y: int] :
( ( ord_less_int @ X @ Y )
| ( X = Y )
| ( ord_less_int @ Y @ X ) ) ).
% linorder_less_linear
thf(fact_119_linorder__less__linear,axiom,
! [X: real,Y: real] :
( ( ord_less_real @ X @ Y )
| ( X = Y )
| ( ord_less_real @ Y @ X ) ) ).
% linorder_less_linear
thf(fact_120_order__less__imp__triv,axiom,
! [X: nat,Y: nat,P: $o] :
( ( ord_less_nat @ X @ Y )
=> ( ( ord_less_nat @ Y @ X )
=> P ) ) ).
% order_less_imp_triv
thf(fact_121_order__less__imp__triv,axiom,
! [X: int,Y: int,P: $o] :
( ( ord_less_int @ X @ Y )
=> ( ( ord_less_int @ Y @ X )
=> P ) ) ).
% order_less_imp_triv
thf(fact_122_order__less__imp__triv,axiom,
! [X: real,Y: real,P: $o] :
( ( ord_less_real @ X @ Y )
=> ( ( ord_less_real @ Y @ X )
=> P ) ) ).
% order_less_imp_triv
thf(fact_123_order__less__not__sym,axiom,
! [X: nat,Y: nat] :
( ( ord_less_nat @ X @ Y )
=> ~ ( ord_less_nat @ Y @ X ) ) ).
% order_less_not_sym
thf(fact_124_order__less__not__sym,axiom,
! [X: int,Y: int] :
( ( ord_less_int @ X @ Y )
=> ~ ( ord_less_int @ Y @ X ) ) ).
% order_less_not_sym
thf(fact_125_order__less__not__sym,axiom,
! [X: real,Y: real] :
( ( ord_less_real @ X @ Y )
=> ~ ( ord_less_real @ Y @ X ) ) ).
% order_less_not_sym
thf(fact_126_mem__Collect__eq,axiom,
! [A: list_Sum_sum_a_nat,P: list_Sum_sum_a_nat > $o] :
( ( member408289922725080238_a_nat @ A @ ( collec7555443234367654128_a_nat @ P ) )
= ( P @ A ) ) ).
% mem_Collect_eq
thf(fact_127_Collect__mem__eq,axiom,
! [A4: set_li6526943997496501093_a_nat] :
( ( collec7555443234367654128_a_nat
@ ^ [X4: list_Sum_sum_a_nat] : ( member408289922725080238_a_nat @ X4 @ A4 ) )
= A4 ) ).
% Collect_mem_eq
thf(fact_128_order__less__subst2,axiom,
! [A: nat,B: nat,F: nat > nat,C: nat] :
( ( ord_less_nat @ A @ B )
=> ( ( ord_less_nat @ ( F @ B ) @ C )
=> ( ! [X3: nat,Y3: nat] :
( ( ord_less_nat @ X3 @ Y3 )
=> ( ord_less_nat @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_nat @ ( F @ A ) @ C ) ) ) ) ).
% order_less_subst2
thf(fact_129_order__less__subst2,axiom,
! [A: nat,B: nat,F: nat > int,C: int] :
( ( ord_less_nat @ A @ B )
=> ( ( ord_less_int @ ( F @ B ) @ C )
=> ( ! [X3: nat,Y3: nat] :
( ( ord_less_nat @ X3 @ Y3 )
=> ( ord_less_int @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_int @ ( F @ A ) @ C ) ) ) ) ).
% order_less_subst2
thf(fact_130_order__less__subst2,axiom,
! [A: nat,B: nat,F: nat > real,C: real] :
( ( ord_less_nat @ A @ B )
=> ( ( ord_less_real @ ( F @ B ) @ C )
=> ( ! [X3: nat,Y3: nat] :
( ( ord_less_nat @ X3 @ Y3 )
=> ( ord_less_real @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_real @ ( F @ A ) @ C ) ) ) ) ).
% order_less_subst2
thf(fact_131_order__less__subst2,axiom,
! [A: int,B: int,F: int > nat,C: nat] :
( ( ord_less_int @ A @ B )
=> ( ( ord_less_nat @ ( F @ B ) @ C )
=> ( ! [X3: int,Y3: int] :
( ( ord_less_int @ X3 @ Y3 )
=> ( ord_less_nat @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_nat @ ( F @ A ) @ C ) ) ) ) ).
% order_less_subst2
thf(fact_132_order__less__subst2,axiom,
! [A: int,B: int,F: int > int,C: int] :
( ( ord_less_int @ A @ B )
=> ( ( ord_less_int @ ( F @ B ) @ C )
=> ( ! [X3: int,Y3: int] :
( ( ord_less_int @ X3 @ Y3 )
=> ( ord_less_int @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_int @ ( F @ A ) @ C ) ) ) ) ).
% order_less_subst2
thf(fact_133_order__less__subst2,axiom,
! [A: int,B: int,F: int > real,C: real] :
( ( ord_less_int @ A @ B )
=> ( ( ord_less_real @ ( F @ B ) @ C )
=> ( ! [X3: int,Y3: int] :
( ( ord_less_int @ X3 @ Y3 )
=> ( ord_less_real @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_real @ ( F @ A ) @ C ) ) ) ) ).
% order_less_subst2
thf(fact_134_order__less__subst2,axiom,
! [A: real,B: real,F: real > nat,C: nat] :
( ( ord_less_real @ A @ B )
=> ( ( ord_less_nat @ ( F @ B ) @ C )
=> ( ! [X3: real,Y3: real] :
( ( ord_less_real @ X3 @ Y3 )
=> ( ord_less_nat @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_nat @ ( F @ A ) @ C ) ) ) ) ).
% order_less_subst2
thf(fact_135_order__less__subst2,axiom,
! [A: real,B: real,F: real > int,C: int] :
( ( ord_less_real @ A @ B )
=> ( ( ord_less_int @ ( F @ B ) @ C )
=> ( ! [X3: real,Y3: real] :
( ( ord_less_real @ X3 @ Y3 )
=> ( ord_less_int @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_int @ ( F @ A ) @ C ) ) ) ) ).
% order_less_subst2
thf(fact_136_order__less__subst2,axiom,
! [A: real,B: real,F: real > real,C: real] :
( ( ord_less_real @ A @ B )
=> ( ( ord_less_real @ ( F @ B ) @ C )
=> ( ! [X3: real,Y3: real] :
( ( ord_less_real @ X3 @ Y3 )
=> ( ord_less_real @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_real @ ( F @ A ) @ C ) ) ) ) ).
% order_less_subst2
thf(fact_137_order__less__subst1,axiom,
! [A: nat,F: nat > nat,B: nat,C: nat] :
( ( ord_less_nat @ A @ ( F @ B ) )
=> ( ( ord_less_nat @ B @ C )
=> ( ! [X3: nat,Y3: nat] :
( ( ord_less_nat @ X3 @ Y3 )
=> ( ord_less_nat @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_nat @ A @ ( F @ C ) ) ) ) ) ).
% order_less_subst1
thf(fact_138_order__less__subst1,axiom,
! [A: nat,F: int > nat,B: int,C: int] :
( ( ord_less_nat @ A @ ( F @ B ) )
=> ( ( ord_less_int @ B @ C )
=> ( ! [X3: int,Y3: int] :
( ( ord_less_int @ X3 @ Y3 )
=> ( ord_less_nat @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_nat @ A @ ( F @ C ) ) ) ) ) ).
% order_less_subst1
thf(fact_139_order__less__subst1,axiom,
! [A: nat,F: real > nat,B: real,C: real] :
( ( ord_less_nat @ A @ ( F @ B ) )
=> ( ( ord_less_real @ B @ C )
=> ( ! [X3: real,Y3: real] :
( ( ord_less_real @ X3 @ Y3 )
=> ( ord_less_nat @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_nat @ A @ ( F @ C ) ) ) ) ) ).
% order_less_subst1
thf(fact_140_order__less__subst1,axiom,
! [A: int,F: nat > int,B: nat,C: nat] :
( ( ord_less_int @ A @ ( F @ B ) )
=> ( ( ord_less_nat @ B @ C )
=> ( ! [X3: nat,Y3: nat] :
( ( ord_less_nat @ X3 @ Y3 )
=> ( ord_less_int @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_int @ A @ ( F @ C ) ) ) ) ) ).
% order_less_subst1
thf(fact_141_order__less__subst1,axiom,
! [A: int,F: int > int,B: int,C: int] :
( ( ord_less_int @ A @ ( F @ B ) )
=> ( ( ord_less_int @ B @ C )
=> ( ! [X3: int,Y3: int] :
( ( ord_less_int @ X3 @ Y3 )
=> ( ord_less_int @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_int @ A @ ( F @ C ) ) ) ) ) ).
% order_less_subst1
thf(fact_142_order__less__subst1,axiom,
! [A: int,F: real > int,B: real,C: real] :
( ( ord_less_int @ A @ ( F @ B ) )
=> ( ( ord_less_real @ B @ C )
=> ( ! [X3: real,Y3: real] :
( ( ord_less_real @ X3 @ Y3 )
=> ( ord_less_int @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_int @ A @ ( F @ C ) ) ) ) ) ).
% order_less_subst1
thf(fact_143_order__less__subst1,axiom,
! [A: real,F: nat > real,B: nat,C: nat] :
( ( ord_less_real @ A @ ( F @ B ) )
=> ( ( ord_less_nat @ B @ C )
=> ( ! [X3: nat,Y3: nat] :
( ( ord_less_nat @ X3 @ Y3 )
=> ( ord_less_real @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_real @ A @ ( F @ C ) ) ) ) ) ).
% order_less_subst1
thf(fact_144_order__less__subst1,axiom,
! [A: real,F: int > real,B: int,C: int] :
( ( ord_less_real @ A @ ( F @ B ) )
=> ( ( ord_less_int @ B @ C )
=> ( ! [X3: int,Y3: int] :
( ( ord_less_int @ X3 @ Y3 )
=> ( ord_less_real @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_real @ A @ ( F @ C ) ) ) ) ) ).
% order_less_subst1
thf(fact_145_order__less__subst1,axiom,
! [A: real,F: real > real,B: real,C: real] :
( ( ord_less_real @ A @ ( F @ B ) )
=> ( ( ord_less_real @ B @ C )
=> ( ! [X3: real,Y3: real] :
( ( ord_less_real @ X3 @ Y3 )
=> ( ord_less_real @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_real @ A @ ( F @ C ) ) ) ) ) ).
% order_less_subst1
thf(fact_146_order__less__irrefl,axiom,
! [X: nat] :
~ ( ord_less_nat @ X @ X ) ).
% order_less_irrefl
thf(fact_147_order__less__irrefl,axiom,
! [X: int] :
~ ( ord_less_int @ X @ X ) ).
% order_less_irrefl
thf(fact_148_order__less__irrefl,axiom,
! [X: real] :
~ ( ord_less_real @ X @ X ) ).
% order_less_irrefl
thf(fact_149_ord__less__eq__subst,axiom,
! [A: nat,B: nat,F: nat > nat,C: nat] :
( ( ord_less_nat @ A @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X3: nat,Y3: nat] :
( ( ord_less_nat @ X3 @ Y3 )
=> ( ord_less_nat @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_nat @ ( F @ A ) @ C ) ) ) ) ).
% ord_less_eq_subst
thf(fact_150_ord__less__eq__subst,axiom,
! [A: nat,B: nat,F: nat > int,C: int] :
( ( ord_less_nat @ A @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X3: nat,Y3: nat] :
( ( ord_less_nat @ X3 @ Y3 )
=> ( ord_less_int @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_int @ ( F @ A ) @ C ) ) ) ) ).
% ord_less_eq_subst
thf(fact_151_ord__less__eq__subst,axiom,
! [A: nat,B: nat,F: nat > real,C: real] :
( ( ord_less_nat @ A @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X3: nat,Y3: nat] :
( ( ord_less_nat @ X3 @ Y3 )
=> ( ord_less_real @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_real @ ( F @ A ) @ C ) ) ) ) ).
% ord_less_eq_subst
thf(fact_152_ord__less__eq__subst,axiom,
! [A: int,B: int,F: int > nat,C: nat] :
( ( ord_less_int @ A @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X3: int,Y3: int] :
( ( ord_less_int @ X3 @ Y3 )
=> ( ord_less_nat @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_nat @ ( F @ A ) @ C ) ) ) ) ).
% ord_less_eq_subst
thf(fact_153_ord__less__eq__subst,axiom,
! [A: int,B: int,F: int > int,C: int] :
( ( ord_less_int @ A @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X3: int,Y3: int] :
( ( ord_less_int @ X3 @ Y3 )
=> ( ord_less_int @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_int @ ( F @ A ) @ C ) ) ) ) ).
% ord_less_eq_subst
thf(fact_154_ord__less__eq__subst,axiom,
! [A: int,B: int,F: int > real,C: real] :
( ( ord_less_int @ A @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X3: int,Y3: int] :
( ( ord_less_int @ X3 @ Y3 )
=> ( ord_less_real @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_real @ ( F @ A ) @ C ) ) ) ) ).
% ord_less_eq_subst
thf(fact_155_ord__less__eq__subst,axiom,
! [A: real,B: real,F: real > nat,C: nat] :
( ( ord_less_real @ A @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X3: real,Y3: real] :
( ( ord_less_real @ X3 @ Y3 )
=> ( ord_less_nat @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_nat @ ( F @ A ) @ C ) ) ) ) ).
% ord_less_eq_subst
thf(fact_156_ord__less__eq__subst,axiom,
! [A: real,B: real,F: real > int,C: int] :
( ( ord_less_real @ A @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X3: real,Y3: real] :
( ( ord_less_real @ X3 @ Y3 )
=> ( ord_less_int @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_int @ ( F @ A ) @ C ) ) ) ) ).
% ord_less_eq_subst
thf(fact_157_ord__less__eq__subst,axiom,
! [A: real,B: real,F: real > real,C: real] :
( ( ord_less_real @ A @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X3: real,Y3: real] :
( ( ord_less_real @ X3 @ Y3 )
=> ( ord_less_real @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_real @ ( F @ A ) @ C ) ) ) ) ).
% ord_less_eq_subst
thf(fact_158_ord__eq__less__subst,axiom,
! [A: nat,F: nat > nat,B: nat,C: nat] :
( ( A
= ( F @ B ) )
=> ( ( ord_less_nat @ B @ C )
=> ( ! [X3: nat,Y3: nat] :
( ( ord_less_nat @ X3 @ Y3 )
=> ( ord_less_nat @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_nat @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_less_subst
thf(fact_159_ord__eq__less__subst,axiom,
! [A: int,F: nat > int,B: nat,C: nat] :
( ( A
= ( F @ B ) )
=> ( ( ord_less_nat @ B @ C )
=> ( ! [X3: nat,Y3: nat] :
( ( ord_less_nat @ X3 @ Y3 )
=> ( ord_less_int @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_int @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_less_subst
thf(fact_160_ord__eq__less__subst,axiom,
! [A: real,F: nat > real,B: nat,C: nat] :
( ( A
= ( F @ B ) )
=> ( ( ord_less_nat @ B @ C )
=> ( ! [X3: nat,Y3: nat] :
( ( ord_less_nat @ X3 @ Y3 )
=> ( ord_less_real @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_real @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_less_subst
thf(fact_161_ord__eq__less__subst,axiom,
! [A: nat,F: int > nat,B: int,C: int] :
( ( A
= ( F @ B ) )
=> ( ( ord_less_int @ B @ C )
=> ( ! [X3: int,Y3: int] :
( ( ord_less_int @ X3 @ Y3 )
=> ( ord_less_nat @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_nat @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_less_subst
thf(fact_162_ord__eq__less__subst,axiom,
! [A: int,F: int > int,B: int,C: int] :
( ( A
= ( F @ B ) )
=> ( ( ord_less_int @ B @ C )
=> ( ! [X3: int,Y3: int] :
( ( ord_less_int @ X3 @ Y3 )
=> ( ord_less_int @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_int @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_less_subst
thf(fact_163_ord__eq__less__subst,axiom,
! [A: real,F: int > real,B: int,C: int] :
( ( A
= ( F @ B ) )
=> ( ( ord_less_int @ B @ C )
=> ( ! [X3: int,Y3: int] :
( ( ord_less_int @ X3 @ Y3 )
=> ( ord_less_real @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_real @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_less_subst
thf(fact_164_ord__eq__less__subst,axiom,
! [A: nat,F: real > nat,B: real,C: real] :
( ( A
= ( F @ B ) )
=> ( ( ord_less_real @ B @ C )
=> ( ! [X3: real,Y3: real] :
( ( ord_less_real @ X3 @ Y3 )
=> ( ord_less_nat @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_nat @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_less_subst
thf(fact_165_ord__eq__less__subst,axiom,
! [A: int,F: real > int,B: real,C: real] :
( ( A
= ( F @ B ) )
=> ( ( ord_less_real @ B @ C )
=> ( ! [X3: real,Y3: real] :
( ( ord_less_real @ X3 @ Y3 )
=> ( ord_less_int @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_int @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_less_subst
thf(fact_166_ord__eq__less__subst,axiom,
! [A: real,F: real > real,B: real,C: real] :
( ( A
= ( F @ B ) )
=> ( ( ord_less_real @ B @ C )
=> ( ! [X3: real,Y3: real] :
( ( ord_less_real @ X3 @ Y3 )
=> ( ord_less_real @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_real @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_less_subst
thf(fact_167_order__less__trans,axiom,
! [X: nat,Y: nat,Z2: nat] :
( ( ord_less_nat @ X @ Y )
=> ( ( ord_less_nat @ Y @ Z2 )
=> ( ord_less_nat @ X @ Z2 ) ) ) ).
% order_less_trans
thf(fact_168_order__less__trans,axiom,
! [X: int,Y: int,Z2: int] :
( ( ord_less_int @ X @ Y )
=> ( ( ord_less_int @ Y @ Z2 )
=> ( ord_less_int @ X @ Z2 ) ) ) ).
% order_less_trans
thf(fact_169_order__less__trans,axiom,
! [X: real,Y: real,Z2: real] :
( ( ord_less_real @ X @ Y )
=> ( ( ord_less_real @ Y @ Z2 )
=> ( ord_less_real @ X @ Z2 ) ) ) ).
% order_less_trans
thf(fact_170_order__less__asym_H,axiom,
! [A: nat,B: nat] :
( ( ord_less_nat @ A @ B )
=> ~ ( ord_less_nat @ B @ A ) ) ).
% order_less_asym'
thf(fact_171_order__less__asym_H,axiom,
! [A: int,B: int] :
( ( ord_less_int @ A @ B )
=> ~ ( ord_less_int @ B @ A ) ) ).
% order_less_asym'
thf(fact_172_order__less__asym_H,axiom,
! [A: real,B: real] :
( ( ord_less_real @ A @ B )
=> ~ ( ord_less_real @ B @ A ) ) ).
% order_less_asym'
thf(fact_173_linorder__neq__iff,axiom,
! [X: nat,Y: nat] :
( ( X != Y )
= ( ( ord_less_nat @ X @ Y )
| ( ord_less_nat @ Y @ X ) ) ) ).
% linorder_neq_iff
thf(fact_174_linorder__neq__iff,axiom,
! [X: int,Y: int] :
( ( X != Y )
= ( ( ord_less_int @ X @ Y )
| ( ord_less_int @ Y @ X ) ) ) ).
% linorder_neq_iff
thf(fact_175_linorder__neq__iff,axiom,
! [X: real,Y: real] :
( ( X != Y )
= ( ( ord_less_real @ X @ Y )
| ( ord_less_real @ Y @ X ) ) ) ).
% linorder_neq_iff
thf(fact_176_order__less__asym,axiom,
! [X: nat,Y: nat] :
( ( ord_less_nat @ X @ Y )
=> ~ ( ord_less_nat @ Y @ X ) ) ).
% order_less_asym
thf(fact_177_order__less__asym,axiom,
! [X: int,Y: int] :
( ( ord_less_int @ X @ Y )
=> ~ ( ord_less_int @ Y @ X ) ) ).
% order_less_asym
thf(fact_178_order__less__asym,axiom,
! [X: real,Y: real] :
( ( ord_less_real @ X @ Y )
=> ~ ( ord_less_real @ Y @ X ) ) ).
% order_less_asym
thf(fact_179_linorder__neqE,axiom,
! [X: nat,Y: nat] :
( ( X != Y )
=> ( ~ ( ord_less_nat @ X @ Y )
=> ( ord_less_nat @ Y @ X ) ) ) ).
% linorder_neqE
thf(fact_180_linorder__neqE,axiom,
! [X: int,Y: int] :
( ( X != Y )
=> ( ~ ( ord_less_int @ X @ Y )
=> ( ord_less_int @ Y @ X ) ) ) ).
% linorder_neqE
thf(fact_181_linorder__neqE,axiom,
! [X: real,Y: real] :
( ( X != Y )
=> ( ~ ( ord_less_real @ X @ Y )
=> ( ord_less_real @ Y @ X ) ) ) ).
% linorder_neqE
thf(fact_182_dual__order_Ostrict__implies__not__eq,axiom,
! [B: nat,A: nat] :
( ( ord_less_nat @ B @ A )
=> ( A != B ) ) ).
% dual_order.strict_implies_not_eq
thf(fact_183_dual__order_Ostrict__implies__not__eq,axiom,
! [B: int,A: int] :
( ( ord_less_int @ B @ A )
=> ( A != B ) ) ).
% dual_order.strict_implies_not_eq
thf(fact_184_dual__order_Ostrict__implies__not__eq,axiom,
! [B: real,A: real] :
( ( ord_less_real @ B @ A )
=> ( A != B ) ) ).
% dual_order.strict_implies_not_eq
thf(fact_185_order_Ostrict__implies__not__eq,axiom,
! [A: nat,B: nat] :
( ( ord_less_nat @ A @ B )
=> ( A != B ) ) ).
% order.strict_implies_not_eq
thf(fact_186_order_Ostrict__implies__not__eq,axiom,
! [A: int,B: int] :
( ( ord_less_int @ A @ B )
=> ( A != B ) ) ).
% order.strict_implies_not_eq
thf(fact_187_order_Ostrict__implies__not__eq,axiom,
! [A: real,B: real] :
( ( ord_less_real @ A @ B )
=> ( A != B ) ) ).
% order.strict_implies_not_eq
thf(fact_188_dual__order_Ostrict__trans,axiom,
! [B: nat,A: nat,C: nat] :
( ( ord_less_nat @ B @ A )
=> ( ( ord_less_nat @ C @ B )
=> ( ord_less_nat @ C @ A ) ) ) ).
% dual_order.strict_trans
thf(fact_189_dual__order_Ostrict__trans,axiom,
! [B: int,A: int,C: int] :
( ( ord_less_int @ B @ A )
=> ( ( ord_less_int @ C @ B )
=> ( ord_less_int @ C @ A ) ) ) ).
% dual_order.strict_trans
thf(fact_190_dual__order_Ostrict__trans,axiom,
! [B: real,A: real,C: real] :
( ( ord_less_real @ B @ A )
=> ( ( ord_less_real @ C @ B )
=> ( ord_less_real @ C @ A ) ) ) ).
% dual_order.strict_trans
thf(fact_191_not__less__iff__gr__or__eq,axiom,
! [X: nat,Y: nat] :
( ( ~ ( ord_less_nat @ X @ Y ) )
= ( ( ord_less_nat @ Y @ X )
| ( X = Y ) ) ) ).
% not_less_iff_gr_or_eq
thf(fact_192_not__less__iff__gr__or__eq,axiom,
! [X: int,Y: int] :
( ( ~ ( ord_less_int @ X @ Y ) )
= ( ( ord_less_int @ Y @ X )
| ( X = Y ) ) ) ).
% not_less_iff_gr_or_eq
thf(fact_193_not__less__iff__gr__or__eq,axiom,
! [X: real,Y: real] :
( ( ~ ( ord_less_real @ X @ Y ) )
= ( ( ord_less_real @ Y @ X )
| ( X = Y ) ) ) ).
% not_less_iff_gr_or_eq
thf(fact_194_order_Ostrict__trans,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_nat @ A @ B )
=> ( ( ord_less_nat @ B @ C )
=> ( ord_less_nat @ A @ C ) ) ) ).
% order.strict_trans
thf(fact_195_order_Ostrict__trans,axiom,
! [A: int,B: int,C: int] :
( ( ord_less_int @ A @ B )
=> ( ( ord_less_int @ B @ C )
=> ( ord_less_int @ A @ C ) ) ) ).
% order.strict_trans
thf(fact_196_order_Ostrict__trans,axiom,
! [A: real,B: real,C: real] :
( ( ord_less_real @ A @ B )
=> ( ( ord_less_real @ B @ C )
=> ( ord_less_real @ A @ C ) ) ) ).
% order.strict_trans
thf(fact_197_linorder__less__wlog,axiom,
! [P: nat > nat > $o,A: nat,B: nat] :
( ! [A3: nat,B4: nat] :
( ( ord_less_nat @ A3 @ B4 )
=> ( P @ A3 @ B4 ) )
=> ( ! [A3: nat] : ( P @ A3 @ A3 )
=> ( ! [A3: nat,B4: nat] :
( ( P @ B4 @ A3 )
=> ( P @ A3 @ B4 ) )
=> ( P @ A @ B ) ) ) ) ).
% linorder_less_wlog
thf(fact_198_linorder__less__wlog,axiom,
! [P: int > int > $o,A: int,B: int] :
( ! [A3: int,B4: int] :
( ( ord_less_int @ A3 @ B4 )
=> ( P @ A3 @ B4 ) )
=> ( ! [A3: int] : ( P @ A3 @ A3 )
=> ( ! [A3: int,B4: int] :
( ( P @ B4 @ A3 )
=> ( P @ A3 @ B4 ) )
=> ( P @ A @ B ) ) ) ) ).
% linorder_less_wlog
thf(fact_199_linorder__less__wlog,axiom,
! [P: real > real > $o,A: real,B: real] :
( ! [A3: real,B4: real] :
( ( ord_less_real @ A3 @ B4 )
=> ( P @ A3 @ B4 ) )
=> ( ! [A3: real] : ( P @ A3 @ A3 )
=> ( ! [A3: real,B4: real] :
( ( P @ B4 @ A3 )
=> ( P @ A3 @ B4 ) )
=> ( P @ A @ B ) ) ) ) ).
% linorder_less_wlog
thf(fact_200_exists__least__iff,axiom,
( ( ^ [P2: nat > $o] :
? [X5: nat] : ( P2 @ X5 ) )
= ( ^ [P3: nat > $o] :
? [N: nat] :
( ( P3 @ N )
& ! [M: nat] :
( ( ord_less_nat @ M @ N )
=> ~ ( P3 @ M ) ) ) ) ) ).
% exists_least_iff
thf(fact_201_dual__order_Oirrefl,axiom,
! [A: nat] :
~ ( ord_less_nat @ A @ A ) ).
% dual_order.irrefl
thf(fact_202_dual__order_Oirrefl,axiom,
! [A: int] :
~ ( ord_less_int @ A @ A ) ).
% dual_order.irrefl
thf(fact_203_dual__order_Oirrefl,axiom,
! [A: real] :
~ ( ord_less_real @ A @ A ) ).
% dual_order.irrefl
thf(fact_204_dual__order_Oasym,axiom,
! [B: nat,A: nat] :
( ( ord_less_nat @ B @ A )
=> ~ ( ord_less_nat @ A @ B ) ) ).
% dual_order.asym
thf(fact_205_dual__order_Oasym,axiom,
! [B: int,A: int] :
( ( ord_less_int @ B @ A )
=> ~ ( ord_less_int @ A @ B ) ) ).
% dual_order.asym
thf(fact_206_dual__order_Oasym,axiom,
! [B: real,A: real] :
( ( ord_less_real @ B @ A )
=> ~ ( ord_less_real @ A @ B ) ) ).
% dual_order.asym
thf(fact_207_linorder__cases,axiom,
! [X: nat,Y: nat] :
( ~ ( ord_less_nat @ X @ Y )
=> ( ( X != Y )
=> ( ord_less_nat @ Y @ X ) ) ) ).
% linorder_cases
thf(fact_208_linorder__cases,axiom,
! [X: int,Y: int] :
( ~ ( ord_less_int @ X @ Y )
=> ( ( X != Y )
=> ( ord_less_int @ Y @ X ) ) ) ).
% linorder_cases
thf(fact_209_linorder__cases,axiom,
! [X: real,Y: real] :
( ~ ( ord_less_real @ X @ Y )
=> ( ( X != Y )
=> ( ord_less_real @ Y @ X ) ) ) ).
% linorder_cases
thf(fact_210_antisym__conv3,axiom,
! [Y: nat,X: nat] :
( ~ ( ord_less_nat @ Y @ X )
=> ( ( ~ ( ord_less_nat @ X @ Y ) )
= ( X = Y ) ) ) ).
% antisym_conv3
thf(fact_211_antisym__conv3,axiom,
! [Y: int,X: int] :
( ~ ( ord_less_int @ Y @ X )
=> ( ( ~ ( ord_less_int @ X @ Y ) )
= ( X = Y ) ) ) ).
% antisym_conv3
thf(fact_212_antisym__conv3,axiom,
! [Y: real,X: real] :
( ~ ( ord_less_real @ Y @ X )
=> ( ( ~ ( ord_less_real @ X @ Y ) )
= ( X = Y ) ) ) ).
% antisym_conv3
thf(fact_213_less__induct,axiom,
! [P: nat > $o,A: nat] :
( ! [X3: nat] :
( ! [Y6: nat] :
( ( ord_less_nat @ Y6 @ X3 )
=> ( P @ Y6 ) )
=> ( P @ X3 ) )
=> ( P @ A ) ) ).
% less_induct
thf(fact_214_ord__less__eq__trans,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_nat @ A @ B )
=> ( ( B = C )
=> ( ord_less_nat @ A @ C ) ) ) ).
% ord_less_eq_trans
thf(fact_215_ord__less__eq__trans,axiom,
! [A: int,B: int,C: int] :
( ( ord_less_int @ A @ B )
=> ( ( B = C )
=> ( ord_less_int @ A @ C ) ) ) ).
% ord_less_eq_trans
thf(fact_216_ord__less__eq__trans,axiom,
! [A: real,B: real,C: real] :
( ( ord_less_real @ A @ B )
=> ( ( B = C )
=> ( ord_less_real @ A @ C ) ) ) ).
% ord_less_eq_trans
thf(fact_217_ord__eq__less__trans,axiom,
! [A: nat,B: nat,C: nat] :
( ( A = B )
=> ( ( ord_less_nat @ B @ C )
=> ( ord_less_nat @ A @ C ) ) ) ).
% ord_eq_less_trans
thf(fact_218_ord__eq__less__trans,axiom,
! [A: int,B: int,C: int] :
( ( A = B )
=> ( ( ord_less_int @ B @ C )
=> ( ord_less_int @ A @ C ) ) ) ).
% ord_eq_less_trans
thf(fact_219_ord__eq__less__trans,axiom,
! [A: real,B: real,C: real] :
( ( A = B )
=> ( ( ord_less_real @ B @ C )
=> ( ord_less_real @ A @ C ) ) ) ).
% ord_eq_less_trans
thf(fact_220_order_Oasym,axiom,
! [A: nat,B: nat] :
( ( ord_less_nat @ A @ B )
=> ~ ( ord_less_nat @ B @ A ) ) ).
% order.asym
thf(fact_221_order_Oasym,axiom,
! [A: int,B: int] :
( ( ord_less_int @ A @ B )
=> ~ ( ord_less_int @ B @ A ) ) ).
% order.asym
thf(fact_222_order_Oasym,axiom,
! [A: real,B: real] :
( ( ord_less_real @ A @ B )
=> ~ ( ord_less_real @ B @ A ) ) ).
% order.asym
thf(fact_223_less__imp__neq,axiom,
! [X: nat,Y: nat] :
( ( ord_less_nat @ X @ Y )
=> ( X != Y ) ) ).
% less_imp_neq
thf(fact_224_less__imp__neq,axiom,
! [X: int,Y: int] :
( ( ord_less_int @ X @ Y )
=> ( X != Y ) ) ).
% less_imp_neq
thf(fact_225_less__imp__neq,axiom,
! [X: real,Y: real] :
( ( ord_less_real @ X @ Y )
=> ( X != Y ) ) ).
% less_imp_neq
thf(fact_226_dense,axiom,
! [X: real,Y: real] :
( ( ord_less_real @ X @ Y )
=> ? [Z3: real] :
( ( ord_less_real @ X @ Z3 )
& ( ord_less_real @ Z3 @ Y ) ) ) ).
% dense
thf(fact_227_gt__ex,axiom,
! [X: nat] :
? [X_1: nat] : ( ord_less_nat @ X @ X_1 ) ).
% gt_ex
thf(fact_228_gt__ex,axiom,
! [X: int] :
? [X_1: int] : ( ord_less_int @ X @ X_1 ) ).
% gt_ex
thf(fact_229_gt__ex,axiom,
! [X: real] :
? [X_1: real] : ( ord_less_real @ X @ X_1 ) ).
% gt_ex
thf(fact_230_lt__ex,axiom,
! [X: int] :
? [Y3: int] : ( ord_less_int @ Y3 @ X ) ).
% lt_ex
thf(fact_231_lt__ex,axiom,
! [X: real] :
? [Y3: real] : ( ord_less_real @ Y3 @ X ) ).
% lt_ex
thf(fact_232_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_233_infinite__descent,axiom,
! [P: nat > $o,N2: nat] :
( ! [N3: nat] :
( ~ ( P @ N3 )
=> ? [M3: nat] :
( ( ord_less_nat @ M3 @ N3 )
& ~ ( P @ M3 ) ) )
=> ( P @ N2 ) ) ).
% infinite_descent
thf(fact_234_nat__less__induct,axiom,
! [P: nat > $o,N2: nat] :
( ! [N3: nat] :
( ! [M3: nat] :
( ( ord_less_nat @ M3 @ N3 )
=> ( P @ M3 ) )
=> ( P @ N3 ) )
=> ( P @ N2 ) ) ).
% nat_less_induct
thf(fact_235_less__irrefl__nat,axiom,
! [N2: nat] :
~ ( ord_less_nat @ N2 @ N2 ) ).
% less_irrefl_nat
thf(fact_236_less__not__refl3,axiom,
! [S: nat,T: nat] :
( ( ord_less_nat @ S @ T )
=> ( S != T ) ) ).
% less_not_refl3
thf(fact_237_less__not__refl2,axiom,
! [N2: nat,M2: nat] :
( ( ord_less_nat @ N2 @ M2 )
=> ( M2 != N2 ) ) ).
% less_not_refl2
thf(fact_238_less__not__refl,axiom,
! [N2: nat] :
~ ( ord_less_nat @ N2 @ N2 ) ).
% less_not_refl
thf(fact_239_nat__neq__iff,axiom,
! [M2: nat,N2: nat] :
( ( M2 != N2 )
= ( ( ord_less_nat @ M2 @ N2 )
| ( ord_less_nat @ N2 @ M2 ) ) ) ).
% nat_neq_iff
thf(fact_240_Nat_Oex__has__greatest__nat,axiom,
! [P: nat > $o,K2: nat,B: nat] :
( ( P @ K2 )
=> ( ! [Y3: nat] :
( ( P @ Y3 )
=> ( ord_less_eq_nat @ Y3 @ B ) )
=> ? [X3: nat] :
( ( P @ X3 )
& ! [Y6: nat] :
( ( P @ Y6 )
=> ( ord_less_eq_nat @ Y6 @ X3 ) ) ) ) ) ).
% Nat.ex_has_greatest_nat
thf(fact_241_nat__le__linear,axiom,
! [M2: nat,N2: nat] :
( ( ord_less_eq_nat @ M2 @ N2 )
| ( ord_less_eq_nat @ N2 @ M2 ) ) ).
% nat_le_linear
thf(fact_242_le__antisym,axiom,
! [M2: nat,N2: nat] :
( ( ord_less_eq_nat @ M2 @ N2 )
=> ( ( ord_less_eq_nat @ N2 @ M2 )
=> ( M2 = N2 ) ) ) ).
% le_antisym
thf(fact_243_eq__imp__le,axiom,
! [M2: nat,N2: nat] :
( ( M2 = N2 )
=> ( ord_less_eq_nat @ M2 @ N2 ) ) ).
% eq_imp_le
thf(fact_244_le__trans,axiom,
! [I: nat,J: nat,K2: nat] :
( ( ord_less_eq_nat @ I @ J )
=> ( ( ord_less_eq_nat @ J @ K2 )
=> ( ord_less_eq_nat @ I @ K2 ) ) ) ).
% le_trans
thf(fact_245_le__refl,axiom,
! [N2: nat] : ( ord_less_eq_nat @ N2 @ N2 ) ).
% le_refl
thf(fact_246_Inr__inject,axiom,
! [X: nat,Y: nat] :
( ( ( sum_Inr_nat_a @ X )
= ( sum_Inr_nat_a @ Y ) )
=> ( X = Y ) ) ).
% Inr_inject
thf(fact_247_order__le__imp__less__or__eq,axiom,
! [X: nat,Y: nat] :
( ( ord_less_eq_nat @ X @ Y )
=> ( ( ord_less_nat @ X @ Y )
| ( X = Y ) ) ) ).
% order_le_imp_less_or_eq
thf(fact_248_order__le__imp__less__or__eq,axiom,
! [X: int,Y: int] :
( ( ord_less_eq_int @ X @ Y )
=> ( ( ord_less_int @ X @ Y )
| ( X = Y ) ) ) ).
% order_le_imp_less_or_eq
thf(fact_249_order__le__imp__less__or__eq,axiom,
! [X: real,Y: real] :
( ( ord_less_eq_real @ X @ Y )
=> ( ( ord_less_real @ X @ Y )
| ( X = Y ) ) ) ).
% order_le_imp_less_or_eq
thf(fact_250_linorder__le__less__linear,axiom,
! [X: nat,Y: nat] :
( ( ord_less_eq_nat @ X @ Y )
| ( ord_less_nat @ Y @ X ) ) ).
% linorder_le_less_linear
thf(fact_251_linorder__le__less__linear,axiom,
! [X: int,Y: int] :
( ( ord_less_eq_int @ X @ Y )
| ( ord_less_int @ Y @ X ) ) ).
% linorder_le_less_linear
thf(fact_252_linorder__le__less__linear,axiom,
! [X: real,Y: real] :
( ( ord_less_eq_real @ X @ Y )
| ( ord_less_real @ Y @ X ) ) ).
% linorder_le_less_linear
thf(fact_253_order__less__le__subst2,axiom,
! [A: nat,B: nat,F: nat > nat,C: nat] :
( ( ord_less_nat @ A @ B )
=> ( ( ord_less_eq_nat @ ( F @ B ) @ C )
=> ( ! [X3: nat,Y3: nat] :
( ( ord_less_nat @ X3 @ Y3 )
=> ( ord_less_nat @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_nat @ ( F @ A ) @ C ) ) ) ) ).
% order_less_le_subst2
thf(fact_254_order__less__le__subst2,axiom,
! [A: int,B: int,F: int > nat,C: nat] :
( ( ord_less_int @ A @ B )
=> ( ( ord_less_eq_nat @ ( F @ B ) @ C )
=> ( ! [X3: int,Y3: int] :
( ( ord_less_int @ X3 @ Y3 )
=> ( ord_less_nat @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_nat @ ( F @ A ) @ C ) ) ) ) ).
% order_less_le_subst2
thf(fact_255_order__less__le__subst2,axiom,
! [A: real,B: real,F: real > nat,C: nat] :
( ( ord_less_real @ A @ B )
=> ( ( ord_less_eq_nat @ ( F @ B ) @ C )
=> ( ! [X3: real,Y3: real] :
( ( ord_less_real @ X3 @ Y3 )
=> ( ord_less_nat @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_nat @ ( F @ A ) @ C ) ) ) ) ).
% order_less_le_subst2
thf(fact_256_order__less__le__subst2,axiom,
! [A: nat,B: nat,F: nat > int,C: int] :
( ( ord_less_nat @ A @ B )
=> ( ( ord_less_eq_int @ ( F @ B ) @ C )
=> ( ! [X3: nat,Y3: nat] :
( ( ord_less_nat @ X3 @ Y3 )
=> ( ord_less_int @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_int @ ( F @ A ) @ C ) ) ) ) ).
% order_less_le_subst2
thf(fact_257_order__less__le__subst2,axiom,
! [A: int,B: int,F: int > int,C: int] :
( ( ord_less_int @ A @ B )
=> ( ( ord_less_eq_int @ ( F @ B ) @ C )
=> ( ! [X3: int,Y3: int] :
( ( ord_less_int @ X3 @ Y3 )
=> ( ord_less_int @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_int @ ( F @ A ) @ C ) ) ) ) ).
% order_less_le_subst2
thf(fact_258_order__less__le__subst2,axiom,
! [A: real,B: real,F: real > int,C: int] :
( ( ord_less_real @ A @ B )
=> ( ( ord_less_eq_int @ ( F @ B ) @ C )
=> ( ! [X3: real,Y3: real] :
( ( ord_less_real @ X3 @ Y3 )
=> ( ord_less_int @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_int @ ( F @ A ) @ C ) ) ) ) ).
% order_less_le_subst2
thf(fact_259_order__less__le__subst2,axiom,
! [A: nat,B: nat,F: nat > real,C: real] :
( ( ord_less_nat @ A @ B )
=> ( ( ord_less_eq_real @ ( F @ B ) @ C )
=> ( ! [X3: nat,Y3: nat] :
( ( ord_less_nat @ X3 @ Y3 )
=> ( ord_less_real @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_real @ ( F @ A ) @ C ) ) ) ) ).
% order_less_le_subst2
thf(fact_260_order__less__le__subst2,axiom,
! [A: int,B: int,F: int > real,C: real] :
( ( ord_less_int @ A @ B )
=> ( ( ord_less_eq_real @ ( F @ B ) @ C )
=> ( ! [X3: int,Y3: int] :
( ( ord_less_int @ X3 @ Y3 )
=> ( ord_less_real @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_real @ ( F @ A ) @ C ) ) ) ) ).
% order_less_le_subst2
thf(fact_261_order__less__le__subst2,axiom,
! [A: real,B: real,F: real > real,C: real] :
( ( ord_less_real @ A @ B )
=> ( ( ord_less_eq_real @ ( F @ B ) @ C )
=> ( ! [X3: real,Y3: real] :
( ( ord_less_real @ X3 @ Y3 )
=> ( ord_less_real @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_real @ ( F @ A ) @ C ) ) ) ) ).
% order_less_le_subst2
thf(fact_262_order__less__le__subst1,axiom,
! [A: nat,F: nat > nat,B: nat,C: nat] :
( ( ord_less_nat @ A @ ( F @ B ) )
=> ( ( ord_less_eq_nat @ B @ C )
=> ( ! [X3: nat,Y3: nat] :
( ( ord_less_eq_nat @ X3 @ Y3 )
=> ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_nat @ A @ ( F @ C ) ) ) ) ) ).
% order_less_le_subst1
thf(fact_263_order__less__le__subst1,axiom,
! [A: int,F: nat > int,B: nat,C: nat] :
( ( ord_less_int @ A @ ( F @ B ) )
=> ( ( ord_less_eq_nat @ B @ C )
=> ( ! [X3: nat,Y3: nat] :
( ( ord_less_eq_nat @ X3 @ Y3 )
=> ( ord_less_eq_int @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_int @ A @ ( F @ C ) ) ) ) ) ).
% order_less_le_subst1
thf(fact_264_order__less__le__subst1,axiom,
! [A: real,F: nat > real,B: nat,C: nat] :
( ( ord_less_real @ A @ ( F @ B ) )
=> ( ( ord_less_eq_nat @ B @ C )
=> ( ! [X3: nat,Y3: nat] :
( ( ord_less_eq_nat @ X3 @ Y3 )
=> ( ord_less_eq_real @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_real @ A @ ( F @ C ) ) ) ) ) ).
% order_less_le_subst1
thf(fact_265_order__less__le__subst1,axiom,
! [A: nat,F: int > nat,B: int,C: int] :
( ( ord_less_nat @ A @ ( F @ B ) )
=> ( ( ord_less_eq_int @ B @ C )
=> ( ! [X3: int,Y3: int] :
( ( ord_less_eq_int @ X3 @ Y3 )
=> ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_nat @ A @ ( F @ C ) ) ) ) ) ).
% order_less_le_subst1
thf(fact_266_order__less__le__subst1,axiom,
! [A: int,F: int > int,B: int,C: int] :
( ( ord_less_int @ A @ ( F @ B ) )
=> ( ( ord_less_eq_int @ B @ C )
=> ( ! [X3: int,Y3: int] :
( ( ord_less_eq_int @ X3 @ Y3 )
=> ( ord_less_eq_int @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_int @ A @ ( F @ C ) ) ) ) ) ).
% order_less_le_subst1
thf(fact_267_order__less__le__subst1,axiom,
! [A: real,F: int > real,B: int,C: int] :
( ( ord_less_real @ A @ ( F @ B ) )
=> ( ( ord_less_eq_int @ B @ C )
=> ( ! [X3: int,Y3: int] :
( ( ord_less_eq_int @ X3 @ Y3 )
=> ( ord_less_eq_real @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_real @ A @ ( F @ C ) ) ) ) ) ).
% order_less_le_subst1
thf(fact_268_order__less__le__subst1,axiom,
! [A: nat,F: real > nat,B: real,C: real] :
( ( ord_less_nat @ A @ ( F @ B ) )
=> ( ( ord_less_eq_real @ B @ C )
=> ( ! [X3: real,Y3: real] :
( ( ord_less_eq_real @ X3 @ Y3 )
=> ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_nat @ A @ ( F @ C ) ) ) ) ) ).
% order_less_le_subst1
thf(fact_269_order__less__le__subst1,axiom,
! [A: int,F: real > int,B: real,C: real] :
( ( ord_less_int @ A @ ( F @ B ) )
=> ( ( ord_less_eq_real @ B @ C )
=> ( ! [X3: real,Y3: real] :
( ( ord_less_eq_real @ X3 @ Y3 )
=> ( ord_less_eq_int @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_int @ A @ ( F @ C ) ) ) ) ) ).
% order_less_le_subst1
thf(fact_270_order__less__le__subst1,axiom,
! [A: real,F: real > real,B: real,C: real] :
( ( ord_less_real @ A @ ( F @ B ) )
=> ( ( ord_less_eq_real @ B @ C )
=> ( ! [X3: real,Y3: real] :
( ( ord_less_eq_real @ X3 @ Y3 )
=> ( ord_less_eq_real @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_real @ A @ ( F @ C ) ) ) ) ) ).
% order_less_le_subst1
thf(fact_271_order__le__less__subst2,axiom,
! [A: nat,B: nat,F: nat > nat,C: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ord_less_nat @ ( F @ B ) @ C )
=> ( ! [X3: nat,Y3: nat] :
( ( ord_less_eq_nat @ X3 @ Y3 )
=> ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_nat @ ( F @ A ) @ C ) ) ) ) ).
% order_le_less_subst2
thf(fact_272_order__le__less__subst2,axiom,
! [A: nat,B: nat,F: nat > int,C: int] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ord_less_int @ ( F @ B ) @ C )
=> ( ! [X3: nat,Y3: nat] :
( ( ord_less_eq_nat @ X3 @ Y3 )
=> ( ord_less_eq_int @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_int @ ( F @ A ) @ C ) ) ) ) ).
% order_le_less_subst2
thf(fact_273_order__le__less__subst2,axiom,
! [A: nat,B: nat,F: nat > real,C: real] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ord_less_real @ ( F @ B ) @ C )
=> ( ! [X3: nat,Y3: nat] :
( ( ord_less_eq_nat @ X3 @ Y3 )
=> ( ord_less_eq_real @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_real @ ( F @ A ) @ C ) ) ) ) ).
% order_le_less_subst2
thf(fact_274_order__le__less__subst2,axiom,
! [A: int,B: int,F: int > nat,C: nat] :
( ( ord_less_eq_int @ A @ B )
=> ( ( ord_less_nat @ ( F @ B ) @ C )
=> ( ! [X3: int,Y3: int] :
( ( ord_less_eq_int @ X3 @ Y3 )
=> ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_nat @ ( F @ A ) @ C ) ) ) ) ).
% order_le_less_subst2
thf(fact_275_order__le__less__subst2,axiom,
! [A: int,B: int,F: int > int,C: int] :
( ( ord_less_eq_int @ A @ B )
=> ( ( ord_less_int @ ( F @ B ) @ C )
=> ( ! [X3: int,Y3: int] :
( ( ord_less_eq_int @ X3 @ Y3 )
=> ( ord_less_eq_int @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_int @ ( F @ A ) @ C ) ) ) ) ).
% order_le_less_subst2
thf(fact_276_order__le__less__subst2,axiom,
! [A: int,B: int,F: int > real,C: real] :
( ( ord_less_eq_int @ A @ B )
=> ( ( ord_less_real @ ( F @ B ) @ C )
=> ( ! [X3: int,Y3: int] :
( ( ord_less_eq_int @ X3 @ Y3 )
=> ( ord_less_eq_real @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_real @ ( F @ A ) @ C ) ) ) ) ).
% order_le_less_subst2
thf(fact_277_order__le__less__subst2,axiom,
! [A: real,B: real,F: real > nat,C: nat] :
( ( ord_less_eq_real @ A @ B )
=> ( ( ord_less_nat @ ( F @ B ) @ C )
=> ( ! [X3: real,Y3: real] :
( ( ord_less_eq_real @ X3 @ Y3 )
=> ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_nat @ ( F @ A ) @ C ) ) ) ) ).
% order_le_less_subst2
thf(fact_278_order__le__less__subst2,axiom,
! [A: real,B: real,F: real > int,C: int] :
( ( ord_less_eq_real @ A @ B )
=> ( ( ord_less_int @ ( F @ B ) @ C )
=> ( ! [X3: real,Y3: real] :
( ( ord_less_eq_real @ X3 @ Y3 )
=> ( ord_less_eq_int @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_int @ ( F @ A ) @ C ) ) ) ) ).
% order_le_less_subst2
thf(fact_279_order__le__less__subst2,axiom,
! [A: real,B: real,F: real > real,C: real] :
( ( ord_less_eq_real @ A @ B )
=> ( ( ord_less_real @ ( F @ B ) @ C )
=> ( ! [X3: real,Y3: real] :
( ( ord_less_eq_real @ X3 @ Y3 )
=> ( ord_less_eq_real @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_real @ ( F @ A ) @ C ) ) ) ) ).
% order_le_less_subst2
thf(fact_280_order__le__less__subst1,axiom,
! [A: nat,F: nat > nat,B: nat,C: nat] :
( ( ord_less_eq_nat @ A @ ( F @ B ) )
=> ( ( ord_less_nat @ B @ C )
=> ( ! [X3: nat,Y3: nat] :
( ( ord_less_nat @ X3 @ Y3 )
=> ( ord_less_nat @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_nat @ A @ ( F @ C ) ) ) ) ) ).
% order_le_less_subst1
thf(fact_281_order__le__less__subst1,axiom,
! [A: nat,F: int > nat,B: int,C: int] :
( ( ord_less_eq_nat @ A @ ( F @ B ) )
=> ( ( ord_less_int @ B @ C )
=> ( ! [X3: int,Y3: int] :
( ( ord_less_int @ X3 @ Y3 )
=> ( ord_less_nat @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_nat @ A @ ( F @ C ) ) ) ) ) ).
% order_le_less_subst1
thf(fact_282_order__le__less__subst1,axiom,
! [A: nat,F: real > nat,B: real,C: real] :
( ( ord_less_eq_nat @ A @ ( F @ B ) )
=> ( ( ord_less_real @ B @ C )
=> ( ! [X3: real,Y3: real] :
( ( ord_less_real @ X3 @ Y3 )
=> ( ord_less_nat @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_nat @ A @ ( F @ C ) ) ) ) ) ).
% order_le_less_subst1
thf(fact_283_order__le__less__subst1,axiom,
! [A: int,F: nat > int,B: nat,C: nat] :
( ( ord_less_eq_int @ A @ ( F @ B ) )
=> ( ( ord_less_nat @ B @ C )
=> ( ! [X3: nat,Y3: nat] :
( ( ord_less_nat @ X3 @ Y3 )
=> ( ord_less_int @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_int @ A @ ( F @ C ) ) ) ) ) ).
% order_le_less_subst1
thf(fact_284_order__le__less__subst1,axiom,
! [A: int,F: int > int,B: int,C: int] :
( ( ord_less_eq_int @ A @ ( F @ B ) )
=> ( ( ord_less_int @ B @ C )
=> ( ! [X3: int,Y3: int] :
( ( ord_less_int @ X3 @ Y3 )
=> ( ord_less_int @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_int @ A @ ( F @ C ) ) ) ) ) ).
% order_le_less_subst1
thf(fact_285_order__le__less__subst1,axiom,
! [A: int,F: real > int,B: real,C: real] :
( ( ord_less_eq_int @ A @ ( F @ B ) )
=> ( ( ord_less_real @ B @ C )
=> ( ! [X3: real,Y3: real] :
( ( ord_less_real @ X3 @ Y3 )
=> ( ord_less_int @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_int @ A @ ( F @ C ) ) ) ) ) ).
% order_le_less_subst1
thf(fact_286_order__le__less__subst1,axiom,
! [A: real,F: nat > real,B: nat,C: nat] :
( ( ord_less_eq_real @ A @ ( F @ B ) )
=> ( ( ord_less_nat @ B @ C )
=> ( ! [X3: nat,Y3: nat] :
( ( ord_less_nat @ X3 @ Y3 )
=> ( ord_less_real @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_real @ A @ ( F @ C ) ) ) ) ) ).
% order_le_less_subst1
thf(fact_287_order__le__less__subst1,axiom,
! [A: real,F: int > real,B: int,C: int] :
( ( ord_less_eq_real @ A @ ( F @ B ) )
=> ( ( ord_less_int @ B @ C )
=> ( ! [X3: int,Y3: int] :
( ( ord_less_int @ X3 @ Y3 )
=> ( ord_less_real @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_real @ A @ ( F @ C ) ) ) ) ) ).
% order_le_less_subst1
thf(fact_288_order__le__less__subst1,axiom,
! [A: real,F: real > real,B: real,C: real] :
( ( ord_less_eq_real @ A @ ( F @ B ) )
=> ( ( ord_less_real @ B @ C )
=> ( ! [X3: real,Y3: real] :
( ( ord_less_real @ X3 @ Y3 )
=> ( ord_less_real @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_real @ A @ ( F @ C ) ) ) ) ) ).
% order_le_less_subst1
thf(fact_289_order__less__le__trans,axiom,
! [X: nat,Y: nat,Z2: nat] :
( ( ord_less_nat @ X @ Y )
=> ( ( ord_less_eq_nat @ Y @ Z2 )
=> ( ord_less_nat @ X @ Z2 ) ) ) ).
% order_less_le_trans
thf(fact_290_order__less__le__trans,axiom,
! [X: int,Y: int,Z2: int] :
( ( ord_less_int @ X @ Y )
=> ( ( ord_less_eq_int @ Y @ Z2 )
=> ( ord_less_int @ X @ Z2 ) ) ) ).
% order_less_le_trans
thf(fact_291_order__less__le__trans,axiom,
! [X: real,Y: real,Z2: real] :
( ( ord_less_real @ X @ Y )
=> ( ( ord_less_eq_real @ Y @ Z2 )
=> ( ord_less_real @ X @ Z2 ) ) ) ).
% order_less_le_trans
thf(fact_292_order__le__less__trans,axiom,
! [X: nat,Y: nat,Z2: nat] :
( ( ord_less_eq_nat @ X @ Y )
=> ( ( ord_less_nat @ Y @ Z2 )
=> ( ord_less_nat @ X @ Z2 ) ) ) ).
% order_le_less_trans
thf(fact_293_order__le__less__trans,axiom,
! [X: int,Y: int,Z2: int] :
( ( ord_less_eq_int @ X @ Y )
=> ( ( ord_less_int @ Y @ Z2 )
=> ( ord_less_int @ X @ Z2 ) ) ) ).
% order_le_less_trans
thf(fact_294_order__le__less__trans,axiom,
! [X: real,Y: real,Z2: real] :
( ( ord_less_eq_real @ X @ Y )
=> ( ( ord_less_real @ Y @ Z2 )
=> ( ord_less_real @ X @ Z2 ) ) ) ).
% order_le_less_trans
thf(fact_295_order__neq__le__trans,axiom,
! [A: nat,B: nat] :
( ( A != B )
=> ( ( ord_less_eq_nat @ A @ B )
=> ( ord_less_nat @ A @ B ) ) ) ).
% order_neq_le_trans
thf(fact_296_order__neq__le__trans,axiom,
! [A: int,B: int] :
( ( A != B )
=> ( ( ord_less_eq_int @ A @ B )
=> ( ord_less_int @ A @ B ) ) ) ).
% order_neq_le_trans
thf(fact_297_order__neq__le__trans,axiom,
! [A: real,B: real] :
( ( A != B )
=> ( ( ord_less_eq_real @ A @ B )
=> ( ord_less_real @ A @ B ) ) ) ).
% order_neq_le_trans
thf(fact_298_order__le__neq__trans,axiom,
! [A: nat,B: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( A != B )
=> ( ord_less_nat @ A @ B ) ) ) ).
% order_le_neq_trans
thf(fact_299_order__le__neq__trans,axiom,
! [A: int,B: int] :
( ( ord_less_eq_int @ A @ B )
=> ( ( A != B )
=> ( ord_less_int @ A @ B ) ) ) ).
% order_le_neq_trans
thf(fact_300_order__le__neq__trans,axiom,
! [A: real,B: real] :
( ( ord_less_eq_real @ A @ B )
=> ( ( A != B )
=> ( ord_less_real @ A @ B ) ) ) ).
% order_le_neq_trans
thf(fact_301_order__less__imp__le,axiom,
! [X: nat,Y: nat] :
( ( ord_less_nat @ X @ Y )
=> ( ord_less_eq_nat @ X @ Y ) ) ).
% order_less_imp_le
thf(fact_302_order__less__imp__le,axiom,
! [X: int,Y: int] :
( ( ord_less_int @ X @ Y )
=> ( ord_less_eq_int @ X @ Y ) ) ).
% order_less_imp_le
thf(fact_303_order__less__imp__le,axiom,
! [X: real,Y: real] :
( ( ord_less_real @ X @ Y )
=> ( ord_less_eq_real @ X @ Y ) ) ).
% order_less_imp_le
thf(fact_304_linorder__not__less,axiom,
! [X: nat,Y: nat] :
( ( ~ ( ord_less_nat @ X @ Y ) )
= ( ord_less_eq_nat @ Y @ X ) ) ).
% linorder_not_less
thf(fact_305_linorder__not__less,axiom,
! [X: int,Y: int] :
( ( ~ ( ord_less_int @ X @ Y ) )
= ( ord_less_eq_int @ Y @ X ) ) ).
% linorder_not_less
thf(fact_306_linorder__not__less,axiom,
! [X: real,Y: real] :
( ( ~ ( ord_less_real @ X @ Y ) )
= ( ord_less_eq_real @ Y @ X ) ) ).
% linorder_not_less
thf(fact_307_linorder__not__le,axiom,
! [X: nat,Y: nat] :
( ( ~ ( ord_less_eq_nat @ X @ Y ) )
= ( ord_less_nat @ Y @ X ) ) ).
% linorder_not_le
thf(fact_308_linorder__not__le,axiom,
! [X: int,Y: int] :
( ( ~ ( ord_less_eq_int @ X @ Y ) )
= ( ord_less_int @ Y @ X ) ) ).
% linorder_not_le
thf(fact_309_linorder__not__le,axiom,
! [X: real,Y: real] :
( ( ~ ( ord_less_eq_real @ X @ Y ) )
= ( ord_less_real @ Y @ X ) ) ).
% linorder_not_le
thf(fact_310_order__less__le,axiom,
( ord_less_nat
= ( ^ [X4: nat,Y5: nat] :
( ( ord_less_eq_nat @ X4 @ Y5 )
& ( X4 != Y5 ) ) ) ) ).
% order_less_le
thf(fact_311_order__less__le,axiom,
( ord_less_int
= ( ^ [X4: int,Y5: int] :
( ( ord_less_eq_int @ X4 @ Y5 )
& ( X4 != Y5 ) ) ) ) ).
% order_less_le
thf(fact_312_order__less__le,axiom,
( ord_less_real
= ( ^ [X4: real,Y5: real] :
( ( ord_less_eq_real @ X4 @ Y5 )
& ( X4 != Y5 ) ) ) ) ).
% order_less_le
thf(fact_313_order__le__less,axiom,
( ord_less_eq_nat
= ( ^ [X4: nat,Y5: nat] :
( ( ord_less_nat @ X4 @ Y5 )
| ( X4 = Y5 ) ) ) ) ).
% order_le_less
thf(fact_314_order__le__less,axiom,
( ord_less_eq_int
= ( ^ [X4: int,Y5: int] :
( ( ord_less_int @ X4 @ Y5 )
| ( X4 = Y5 ) ) ) ) ).
% order_le_less
thf(fact_315_order__le__less,axiom,
( ord_less_eq_real
= ( ^ [X4: real,Y5: real] :
( ( ord_less_real @ X4 @ Y5 )
| ( X4 = Y5 ) ) ) ) ).
% order_le_less
thf(fact_316_dual__order_Ostrict__implies__order,axiom,
! [B: nat,A: nat] :
( ( ord_less_nat @ B @ A )
=> ( ord_less_eq_nat @ B @ A ) ) ).
% dual_order.strict_implies_order
thf(fact_317_dual__order_Ostrict__implies__order,axiom,
! [B: int,A: int] :
( ( ord_less_int @ B @ A )
=> ( ord_less_eq_int @ B @ A ) ) ).
% dual_order.strict_implies_order
thf(fact_318_dual__order_Ostrict__implies__order,axiom,
! [B: real,A: real] :
( ( ord_less_real @ B @ A )
=> ( ord_less_eq_real @ B @ A ) ) ).
% dual_order.strict_implies_order
thf(fact_319_order_Ostrict__implies__order,axiom,
! [A: nat,B: nat] :
( ( ord_less_nat @ A @ B )
=> ( ord_less_eq_nat @ A @ B ) ) ).
% order.strict_implies_order
thf(fact_320_order_Ostrict__implies__order,axiom,
! [A: int,B: int] :
( ( ord_less_int @ A @ B )
=> ( ord_less_eq_int @ A @ B ) ) ).
% order.strict_implies_order
thf(fact_321_order_Ostrict__implies__order,axiom,
! [A: real,B: real] :
( ( ord_less_real @ A @ B )
=> ( ord_less_eq_real @ A @ B ) ) ).
% order.strict_implies_order
thf(fact_322_dual__order_Ostrict__iff__not,axiom,
( ord_less_nat
= ( ^ [B3: nat,A2: nat] :
( ( ord_less_eq_nat @ B3 @ A2 )
& ~ ( ord_less_eq_nat @ A2 @ B3 ) ) ) ) ).
% dual_order.strict_iff_not
thf(fact_323_dual__order_Ostrict__iff__not,axiom,
( ord_less_int
= ( ^ [B3: int,A2: int] :
( ( ord_less_eq_int @ B3 @ A2 )
& ~ ( ord_less_eq_int @ A2 @ B3 ) ) ) ) ).
% dual_order.strict_iff_not
thf(fact_324_dual__order_Ostrict__iff__not,axiom,
( ord_less_real
= ( ^ [B3: real,A2: real] :
( ( ord_less_eq_real @ B3 @ A2 )
& ~ ( ord_less_eq_real @ A2 @ B3 ) ) ) ) ).
% dual_order.strict_iff_not
thf(fact_325_dual__order_Ostrict__trans2,axiom,
! [B: nat,A: nat,C: nat] :
( ( ord_less_nat @ B @ A )
=> ( ( ord_less_eq_nat @ C @ B )
=> ( ord_less_nat @ C @ A ) ) ) ).
% dual_order.strict_trans2
thf(fact_326_dual__order_Ostrict__trans2,axiom,
! [B: int,A: int,C: int] :
( ( ord_less_int @ B @ A )
=> ( ( ord_less_eq_int @ C @ B )
=> ( ord_less_int @ C @ A ) ) ) ).
% dual_order.strict_trans2
thf(fact_327_dual__order_Ostrict__trans2,axiom,
! [B: real,A: real,C: real] :
( ( ord_less_real @ B @ A )
=> ( ( ord_less_eq_real @ C @ B )
=> ( ord_less_real @ C @ A ) ) ) ).
% dual_order.strict_trans2
thf(fact_328_dual__order_Ostrict__trans1,axiom,
! [B: nat,A: nat,C: nat] :
( ( ord_less_eq_nat @ B @ A )
=> ( ( ord_less_nat @ C @ B )
=> ( ord_less_nat @ C @ A ) ) ) ).
% dual_order.strict_trans1
thf(fact_329_dual__order_Ostrict__trans1,axiom,
! [B: int,A: int,C: int] :
( ( ord_less_eq_int @ B @ A )
=> ( ( ord_less_int @ C @ B )
=> ( ord_less_int @ C @ A ) ) ) ).
% dual_order.strict_trans1
thf(fact_330_dual__order_Ostrict__trans1,axiom,
! [B: real,A: real,C: real] :
( ( ord_less_eq_real @ B @ A )
=> ( ( ord_less_real @ C @ B )
=> ( ord_less_real @ C @ A ) ) ) ).
% dual_order.strict_trans1
thf(fact_331_dual__order_Ostrict__iff__order,axiom,
( ord_less_nat
= ( ^ [B3: nat,A2: nat] :
( ( ord_less_eq_nat @ B3 @ A2 )
& ( A2 != B3 ) ) ) ) ).
% dual_order.strict_iff_order
thf(fact_332_dual__order_Ostrict__iff__order,axiom,
( ord_less_int
= ( ^ [B3: int,A2: int] :
( ( ord_less_eq_int @ B3 @ A2 )
& ( A2 != B3 ) ) ) ) ).
% dual_order.strict_iff_order
thf(fact_333_dual__order_Ostrict__iff__order,axiom,
( ord_less_real
= ( ^ [B3: real,A2: real] :
( ( ord_less_eq_real @ B3 @ A2 )
& ( A2 != B3 ) ) ) ) ).
% dual_order.strict_iff_order
thf(fact_334_dual__order_Oorder__iff__strict,axiom,
( ord_less_eq_nat
= ( ^ [B3: nat,A2: nat] :
( ( ord_less_nat @ B3 @ A2 )
| ( A2 = B3 ) ) ) ) ).
% dual_order.order_iff_strict
thf(fact_335_dual__order_Oorder__iff__strict,axiom,
( ord_less_eq_int
= ( ^ [B3: int,A2: int] :
( ( ord_less_int @ B3 @ A2 )
| ( A2 = B3 ) ) ) ) ).
% dual_order.order_iff_strict
thf(fact_336_dual__order_Oorder__iff__strict,axiom,
( ord_less_eq_real
= ( ^ [B3: real,A2: real] :
( ( ord_less_real @ B3 @ A2 )
| ( A2 = B3 ) ) ) ) ).
% dual_order.order_iff_strict
thf(fact_337_dense__le__bounded,axiom,
! [X: real,Y: real,Z2: real] :
( ( ord_less_real @ X @ Y )
=> ( ! [W: real] :
( ( ord_less_real @ X @ W )
=> ( ( ord_less_real @ W @ Y )
=> ( ord_less_eq_real @ W @ Z2 ) ) )
=> ( ord_less_eq_real @ Y @ Z2 ) ) ) ).
% dense_le_bounded
thf(fact_338_dense__ge__bounded,axiom,
! [Z2: real,X: real,Y: real] :
( ( ord_less_real @ Z2 @ X )
=> ( ! [W: real] :
( ( ord_less_real @ Z2 @ W )
=> ( ( ord_less_real @ W @ X )
=> ( ord_less_eq_real @ Y @ W ) ) )
=> ( ord_less_eq_real @ Y @ Z2 ) ) ) ).
% dense_ge_bounded
thf(fact_339_order_Ostrict__iff__not,axiom,
( ord_less_nat
= ( ^ [A2: nat,B3: nat] :
( ( ord_less_eq_nat @ A2 @ B3 )
& ~ ( ord_less_eq_nat @ B3 @ A2 ) ) ) ) ).
% order.strict_iff_not
thf(fact_340_order_Ostrict__iff__not,axiom,
( ord_less_int
= ( ^ [A2: int,B3: int] :
( ( ord_less_eq_int @ A2 @ B3 )
& ~ ( ord_less_eq_int @ B3 @ A2 ) ) ) ) ).
% order.strict_iff_not
thf(fact_341_order_Ostrict__iff__not,axiom,
( ord_less_real
= ( ^ [A2: real,B3: real] :
( ( ord_less_eq_real @ A2 @ B3 )
& ~ ( ord_less_eq_real @ B3 @ A2 ) ) ) ) ).
% order.strict_iff_not
thf(fact_342_order_Ostrict__trans2,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_nat @ A @ B )
=> ( ( ord_less_eq_nat @ B @ C )
=> ( ord_less_nat @ A @ C ) ) ) ).
% order.strict_trans2
thf(fact_343_order_Ostrict__trans2,axiom,
! [A: int,B: int,C: int] :
( ( ord_less_int @ A @ B )
=> ( ( ord_less_eq_int @ B @ C )
=> ( ord_less_int @ A @ C ) ) ) ).
% order.strict_trans2
thf(fact_344_order_Ostrict__trans2,axiom,
! [A: real,B: real,C: real] :
( ( ord_less_real @ A @ B )
=> ( ( ord_less_eq_real @ B @ C )
=> ( ord_less_real @ A @ C ) ) ) ).
% order.strict_trans2
thf(fact_345_order_Ostrict__trans1,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ord_less_nat @ B @ C )
=> ( ord_less_nat @ A @ C ) ) ) ).
% order.strict_trans1
thf(fact_346_order_Ostrict__trans1,axiom,
! [A: int,B: int,C: int] :
( ( ord_less_eq_int @ A @ B )
=> ( ( ord_less_int @ B @ C )
=> ( ord_less_int @ A @ C ) ) ) ).
% order.strict_trans1
thf(fact_347_order_Ostrict__trans1,axiom,
! [A: real,B: real,C: real] :
( ( ord_less_eq_real @ A @ B )
=> ( ( ord_less_real @ B @ C )
=> ( ord_less_real @ A @ C ) ) ) ).
% order.strict_trans1
thf(fact_348_order_Ostrict__iff__order,axiom,
( ord_less_nat
= ( ^ [A2: nat,B3: nat] :
( ( ord_less_eq_nat @ A2 @ B3 )
& ( A2 != B3 ) ) ) ) ).
% order.strict_iff_order
thf(fact_349_order_Ostrict__iff__order,axiom,
( ord_less_int
= ( ^ [A2: int,B3: int] :
( ( ord_less_eq_int @ A2 @ B3 )
& ( A2 != B3 ) ) ) ) ).
% order.strict_iff_order
thf(fact_350_order_Ostrict__iff__order,axiom,
( ord_less_real
= ( ^ [A2: real,B3: real] :
( ( ord_less_eq_real @ A2 @ B3 )
& ( A2 != B3 ) ) ) ) ).
% order.strict_iff_order
thf(fact_351_order_Oorder__iff__strict,axiom,
( ord_less_eq_nat
= ( ^ [A2: nat,B3: nat] :
( ( ord_less_nat @ A2 @ B3 )
| ( A2 = B3 ) ) ) ) ).
% order.order_iff_strict
thf(fact_352_order_Oorder__iff__strict,axiom,
( ord_less_eq_int
= ( ^ [A2: int,B3: int] :
( ( ord_less_int @ A2 @ B3 )
| ( A2 = B3 ) ) ) ) ).
% order.order_iff_strict
thf(fact_353_order_Oorder__iff__strict,axiom,
( ord_less_eq_real
= ( ^ [A2: real,B3: real] :
( ( ord_less_real @ A2 @ B3 )
| ( A2 = B3 ) ) ) ) ).
% order.order_iff_strict
thf(fact_354_not__le__imp__less,axiom,
! [Y: nat,X: nat] :
( ~ ( ord_less_eq_nat @ Y @ X )
=> ( ord_less_nat @ X @ Y ) ) ).
% not_le_imp_less
thf(fact_355_not__le__imp__less,axiom,
! [Y: int,X: int] :
( ~ ( ord_less_eq_int @ Y @ X )
=> ( ord_less_int @ X @ Y ) ) ).
% not_le_imp_less
thf(fact_356_not__le__imp__less,axiom,
! [Y: real,X: real] :
( ~ ( ord_less_eq_real @ Y @ X )
=> ( ord_less_real @ X @ Y ) ) ).
% not_le_imp_less
thf(fact_357_less__le__not__le,axiom,
( ord_less_nat
= ( ^ [X4: nat,Y5: nat] :
( ( ord_less_eq_nat @ X4 @ Y5 )
& ~ ( ord_less_eq_nat @ Y5 @ X4 ) ) ) ) ).
% less_le_not_le
thf(fact_358_less__le__not__le,axiom,
( ord_less_int
= ( ^ [X4: int,Y5: int] :
( ( ord_less_eq_int @ X4 @ Y5 )
& ~ ( ord_less_eq_int @ Y5 @ X4 ) ) ) ) ).
% less_le_not_le
thf(fact_359_less__le__not__le,axiom,
( ord_less_real
= ( ^ [X4: real,Y5: real] :
( ( ord_less_eq_real @ X4 @ Y5 )
& ~ ( ord_less_eq_real @ Y5 @ X4 ) ) ) ) ).
% less_le_not_le
thf(fact_360_dense__le,axiom,
! [Y: real,Z2: real] :
( ! [X3: real] :
( ( ord_less_real @ X3 @ Y )
=> ( ord_less_eq_real @ X3 @ Z2 ) )
=> ( ord_less_eq_real @ Y @ Z2 ) ) ).
% dense_le
thf(fact_361_dense__ge,axiom,
! [Z2: real,Y: real] :
( ! [X3: real] :
( ( ord_less_real @ Z2 @ X3 )
=> ( ord_less_eq_real @ Y @ X3 ) )
=> ( ord_less_eq_real @ Y @ Z2 ) ) ).
% dense_ge
thf(fact_362_antisym__conv2,axiom,
! [X: nat,Y: nat] :
( ( ord_less_eq_nat @ X @ Y )
=> ( ( ~ ( ord_less_nat @ X @ Y ) )
= ( X = Y ) ) ) ).
% antisym_conv2
thf(fact_363_antisym__conv2,axiom,
! [X: int,Y: int] :
( ( ord_less_eq_int @ X @ Y )
=> ( ( ~ ( ord_less_int @ X @ Y ) )
= ( X = Y ) ) ) ).
% antisym_conv2
thf(fact_364_antisym__conv2,axiom,
! [X: real,Y: real] :
( ( ord_less_eq_real @ X @ Y )
=> ( ( ~ ( ord_less_real @ X @ Y ) )
= ( X = Y ) ) ) ).
% antisym_conv2
thf(fact_365_antisym__conv1,axiom,
! [X: nat,Y: nat] :
( ~ ( ord_less_nat @ X @ Y )
=> ( ( ord_less_eq_nat @ X @ Y )
= ( X = Y ) ) ) ).
% antisym_conv1
thf(fact_366_antisym__conv1,axiom,
! [X: int,Y: int] :
( ~ ( ord_less_int @ X @ Y )
=> ( ( ord_less_eq_int @ X @ Y )
= ( X = Y ) ) ) ).
% antisym_conv1
thf(fact_367_antisym__conv1,axiom,
! [X: real,Y: real] :
( ~ ( ord_less_real @ X @ Y )
=> ( ( ord_less_eq_real @ X @ Y )
= ( X = Y ) ) ) ).
% antisym_conv1
thf(fact_368_nless__le,axiom,
! [A: nat,B: nat] :
( ( ~ ( ord_less_nat @ A @ B ) )
= ( ~ ( ord_less_eq_nat @ A @ B )
| ( A = B ) ) ) ).
% nless_le
thf(fact_369_nless__le,axiom,
! [A: int,B: int] :
( ( ~ ( ord_less_int @ A @ B ) )
= ( ~ ( ord_less_eq_int @ A @ B )
| ( A = B ) ) ) ).
% nless_le
thf(fact_370_nless__le,axiom,
! [A: real,B: real] :
( ( ~ ( ord_less_real @ A @ B ) )
= ( ~ ( ord_less_eq_real @ A @ B )
| ( A = B ) ) ) ).
% nless_le
thf(fact_371_leI,axiom,
! [X: nat,Y: nat] :
( ~ ( ord_less_nat @ X @ Y )
=> ( ord_less_eq_nat @ Y @ X ) ) ).
% leI
thf(fact_372_leI,axiom,
! [X: int,Y: int] :
( ~ ( ord_less_int @ X @ Y )
=> ( ord_less_eq_int @ Y @ X ) ) ).
% leI
thf(fact_373_leI,axiom,
! [X: real,Y: real] :
( ~ ( ord_less_real @ X @ Y )
=> ( ord_less_eq_real @ Y @ X ) ) ).
% leI
thf(fact_374_leD,axiom,
! [Y: nat,X: nat] :
( ( ord_less_eq_nat @ Y @ X )
=> ~ ( ord_less_nat @ X @ Y ) ) ).
% leD
thf(fact_375_leD,axiom,
! [Y: int,X: int] :
( ( ord_less_eq_int @ Y @ X )
=> ~ ( ord_less_int @ X @ Y ) ) ).
% leD
thf(fact_376_leD,axiom,
! [Y: real,X: real] :
( ( ord_less_eq_real @ Y @ X )
=> ~ ( ord_less_real @ X @ Y ) ) ).
% leD
thf(fact_377_complete__interval,axiom,
! [A: nat,B: nat,P: nat > $o] :
( ( ord_less_nat @ A @ B )
=> ( ( P @ A )
=> ( ~ ( P @ B )
=> ? [C2: nat] :
( ( ord_less_eq_nat @ A @ C2 )
& ( ord_less_eq_nat @ C2 @ B )
& ! [X6: nat] :
( ( ( ord_less_eq_nat @ A @ X6 )
& ( ord_less_nat @ X6 @ C2 ) )
=> ( P @ X6 ) )
& ! [D: nat] :
( ! [X3: nat] :
( ( ( ord_less_eq_nat @ A @ X3 )
& ( ord_less_nat @ X3 @ D ) )
=> ( P @ X3 ) )
=> ( ord_less_eq_nat @ D @ C2 ) ) ) ) ) ) ).
% complete_interval
thf(fact_378_complete__interval,axiom,
! [A: int,B: int,P: int > $o] :
( ( ord_less_int @ A @ B )
=> ( ( P @ A )
=> ( ~ ( P @ B )
=> ? [C2: int] :
( ( ord_less_eq_int @ A @ C2 )
& ( ord_less_eq_int @ C2 @ B )
& ! [X6: int] :
( ( ( ord_less_eq_int @ A @ X6 )
& ( ord_less_int @ X6 @ C2 ) )
=> ( P @ X6 ) )
& ! [D: int] :
( ! [X3: int] :
( ( ( ord_less_eq_int @ A @ X3 )
& ( ord_less_int @ X3 @ D ) )
=> ( P @ X3 ) )
=> ( ord_less_eq_int @ D @ C2 ) ) ) ) ) ) ).
% complete_interval
thf(fact_379_complete__interval,axiom,
! [A: real,B: real,P: real > $o] :
( ( ord_less_real @ A @ B )
=> ( ( P @ A )
=> ( ~ ( P @ B )
=> ? [C2: real] :
( ( ord_less_eq_real @ A @ C2 )
& ( ord_less_eq_real @ C2 @ B )
& ! [X6: real] :
( ( ( ord_less_eq_real @ A @ X6 )
& ( ord_less_real @ X6 @ C2 ) )
=> ( P @ X6 ) )
& ! [D: real] :
( ! [X3: real] :
( ( ( ord_less_eq_real @ A @ X3 )
& ( ord_less_real @ X3 @ D ) )
=> ( P @ X3 ) )
=> ( ord_less_eq_real @ D @ C2 ) ) ) ) ) ) ).
% complete_interval
thf(fact_380_verit__comp__simplify1_I3_J,axiom,
! [B2: nat,A5: nat] :
( ( ~ ( ord_less_eq_nat @ B2 @ A5 ) )
= ( ord_less_nat @ A5 @ B2 ) ) ).
% verit_comp_simplify1(3)
thf(fact_381_verit__comp__simplify1_I3_J,axiom,
! [B2: int,A5: int] :
( ( ~ ( ord_less_eq_int @ B2 @ A5 ) )
= ( ord_less_int @ A5 @ B2 ) ) ).
% verit_comp_simplify1(3)
thf(fact_382_verit__comp__simplify1_I3_J,axiom,
! [B2: real,A5: real] :
( ( ~ ( ord_less_eq_real @ B2 @ A5 ) )
= ( ord_less_real @ A5 @ B2 ) ) ).
% verit_comp_simplify1(3)
thf(fact_383_pinf_I6_J,axiom,
! [T: nat] :
? [Z3: nat] :
! [X6: nat] :
( ( ord_less_nat @ Z3 @ X6 )
=> ~ ( ord_less_eq_nat @ X6 @ T ) ) ).
% pinf(6)
thf(fact_384_pinf_I6_J,axiom,
! [T: int] :
? [Z3: int] :
! [X6: int] :
( ( ord_less_int @ Z3 @ X6 )
=> ~ ( ord_less_eq_int @ X6 @ T ) ) ).
% pinf(6)
thf(fact_385_pinf_I6_J,axiom,
! [T: real] :
? [Z3: real] :
! [X6: real] :
( ( ord_less_real @ Z3 @ X6 )
=> ~ ( ord_less_eq_real @ X6 @ T ) ) ).
% pinf(6)
thf(fact_386_pinf_I8_J,axiom,
! [T: nat] :
? [Z3: nat] :
! [X6: nat] :
( ( ord_less_nat @ Z3 @ X6 )
=> ( ord_less_eq_nat @ T @ X6 ) ) ).
% pinf(8)
thf(fact_387_pinf_I8_J,axiom,
! [T: int] :
? [Z3: int] :
! [X6: int] :
( ( ord_less_int @ Z3 @ X6 )
=> ( ord_less_eq_int @ T @ X6 ) ) ).
% pinf(8)
thf(fact_388_pinf_I8_J,axiom,
! [T: real] :
? [Z3: real] :
! [X6: real] :
( ( ord_less_real @ Z3 @ X6 )
=> ( ord_less_eq_real @ T @ X6 ) ) ).
% pinf(8)
thf(fact_389_minf_I6_J,axiom,
! [T: nat] :
? [Z3: nat] :
! [X6: nat] :
( ( ord_less_nat @ X6 @ Z3 )
=> ( ord_less_eq_nat @ X6 @ T ) ) ).
% minf(6)
thf(fact_390_minf_I6_J,axiom,
! [T: int] :
? [Z3: int] :
! [X6: int] :
( ( ord_less_int @ X6 @ Z3 )
=> ( ord_less_eq_int @ X6 @ T ) ) ).
% minf(6)
thf(fact_391_minf_I6_J,axiom,
! [T: real] :
? [Z3: real] :
! [X6: real] :
( ( ord_less_real @ X6 @ Z3 )
=> ( ord_less_eq_real @ X6 @ T ) ) ).
% minf(6)
thf(fact_392_minf_I8_J,axiom,
! [T: nat] :
? [Z3: nat] :
! [X6: nat] :
( ( ord_less_nat @ X6 @ Z3 )
=> ~ ( ord_less_eq_nat @ T @ X6 ) ) ).
% minf(8)
thf(fact_393_minf_I8_J,axiom,
! [T: int] :
? [Z3: int] :
! [X6: int] :
( ( ord_less_int @ X6 @ Z3 )
=> ~ ( ord_less_eq_int @ T @ X6 ) ) ).
% minf(8)
thf(fact_394_minf_I8_J,axiom,
! [T: real] :
? [Z3: real] :
! [X6: real] :
( ( ord_less_real @ X6 @ Z3 )
=> ~ ( ord_less_eq_real @ T @ X6 ) ) ).
% minf(8)
thf(fact_395_id__mapD_I1_J,axiom,
! [J: nat,I: nat] :
( ( ( id_map_a @ J @ ( sum_Inr_nat_a @ I ) )
= none_nat )
=> ( ord_less_eq_nat @ J @ I ) ) ).
% id_mapD(1)
thf(fact_396_Greatest__equality,axiom,
! [P: int > $o,X: int] :
( ( P @ X )
=> ( ! [Y3: int] :
( ( P @ Y3 )
=> ( ord_less_eq_int @ Y3 @ X ) )
=> ( ( order_Greatest_int @ P )
= X ) ) ) ).
% Greatest_equality
thf(fact_397_Greatest__equality,axiom,
! [P: real > $o,X: real] :
( ( P @ X )
=> ( ! [Y3: real] :
( ( P @ Y3 )
=> ( ord_less_eq_real @ Y3 @ X ) )
=> ( ( order_Greatest_real @ P )
= X ) ) ) ).
% Greatest_equality
thf(fact_398_Greatest__equality,axiom,
! [P: nat > $o,X: nat] :
( ( P @ X )
=> ( ! [Y3: nat] :
( ( P @ Y3 )
=> ( ord_less_eq_nat @ Y3 @ X ) )
=> ( ( order_Greatest_nat @ P )
= X ) ) ) ).
% Greatest_equality
thf(fact_399_GreatestI2__order,axiom,
! [P: int > $o,X: int,Q: int > $o] :
( ( P @ X )
=> ( ! [Y3: int] :
( ( P @ Y3 )
=> ( ord_less_eq_int @ Y3 @ X ) )
=> ( ! [X3: int] :
( ( P @ X3 )
=> ( ! [Y6: int] :
( ( P @ Y6 )
=> ( ord_less_eq_int @ Y6 @ X3 ) )
=> ( Q @ X3 ) ) )
=> ( Q @ ( order_Greatest_int @ P ) ) ) ) ) ).
% GreatestI2_order
thf(fact_400_GreatestI2__order,axiom,
! [P: real > $o,X: real,Q: real > $o] :
( ( P @ X )
=> ( ! [Y3: real] :
( ( P @ Y3 )
=> ( ord_less_eq_real @ Y3 @ X ) )
=> ( ! [X3: real] :
( ( P @ X3 )
=> ( ! [Y6: real] :
( ( P @ Y6 )
=> ( ord_less_eq_real @ Y6 @ X3 ) )
=> ( Q @ X3 ) ) )
=> ( Q @ ( order_Greatest_real @ P ) ) ) ) ) ).
% GreatestI2_order
thf(fact_401_GreatestI2__order,axiom,
! [P: nat > $o,X: nat,Q: nat > $o] :
( ( P @ X )
=> ( ! [Y3: nat] :
( ( P @ Y3 )
=> ( ord_less_eq_nat @ Y3 @ X ) )
=> ( ! [X3: nat] :
( ( P @ X3 )
=> ( ! [Y6: nat] :
( ( P @ Y6 )
=> ( ord_less_eq_nat @ Y6 @ X3 ) )
=> ( Q @ X3 ) ) )
=> ( Q @ ( order_Greatest_nat @ P ) ) ) ) ) ).
% GreatestI2_order
thf(fact_402_not__None__eq,axiom,
! [X: option_nat] :
( ( X != none_nat )
= ( ? [Y5: nat] :
( X
= ( some_nat @ Y5 ) ) ) ) ).
% not_None_eq
thf(fact_403_not__None__eq,axiom,
! [X: option_int] :
( ( X != none_int )
= ( ? [Y5: int] :
( X
= ( some_int @ Y5 ) ) ) ) ).
% not_None_eq
thf(fact_404_not__Some__eq,axiom,
! [X: option_nat] :
( ( ! [Y5: nat] :
( X
!= ( some_nat @ Y5 ) ) )
= ( X = none_nat ) ) ).
% not_Some_eq
thf(fact_405_not__Some__eq,axiom,
! [X: option_int] :
( ( ! [Y5: int] :
( X
!= ( some_int @ Y5 ) ) )
= ( X = none_int ) ) ).
% not_Some_eq
thf(fact_406_option_Odistinct_I1_J,axiom,
! [X2: nat] :
( none_nat
!= ( some_nat @ X2 ) ) ).
% option.distinct(1)
thf(fact_407_option_Odistinct_I1_J,axiom,
! [X2: int] :
( none_int
!= ( some_int @ X2 ) ) ).
% option.distinct(1)
thf(fact_408_option_OdiscI,axiom,
! [Option: option_nat,X2: nat] :
( ( Option
= ( some_nat @ X2 ) )
=> ( Option != none_nat ) ) ).
% option.discI
thf(fact_409_option_OdiscI,axiom,
! [Option: option_int,X2: int] :
( ( Option
= ( some_int @ X2 ) )
=> ( Option != none_int ) ) ).
% option.discI
thf(fact_410_option_Oexhaust,axiom,
! [Y: option_nat] :
( ( Y != none_nat )
=> ~ ! [X22: nat] :
( Y
!= ( some_nat @ X22 ) ) ) ).
% option.exhaust
thf(fact_411_option_Oexhaust,axiom,
! [Y: option_int] :
( ( Y != none_int )
=> ~ ! [X22: int] :
( Y
!= ( some_int @ X22 ) ) ) ).
% option.exhaust
thf(fact_412_split__option__ex,axiom,
( ( ^ [P2: option_nat > $o] :
? [X5: option_nat] : ( P2 @ X5 ) )
= ( ^ [P3: option_nat > $o] :
( ( P3 @ none_nat )
| ? [X4: nat] : ( P3 @ ( some_nat @ X4 ) ) ) ) ) ).
% split_option_ex
thf(fact_413_split__option__ex,axiom,
( ( ^ [P2: option_int > $o] :
? [X5: option_int] : ( P2 @ X5 ) )
= ( ^ [P3: option_int > $o] :
( ( P3 @ none_int )
| ? [X4: int] : ( P3 @ ( some_int @ X4 ) ) ) ) ) ).
% split_option_ex
thf(fact_414_split__option__all,axiom,
( ( ^ [P2: option_nat > $o] :
! [X5: option_nat] : ( P2 @ X5 ) )
= ( ^ [P3: option_nat > $o] :
( ( P3 @ none_nat )
& ! [X4: nat] : ( P3 @ ( some_nat @ X4 ) ) ) ) ) ).
% split_option_all
thf(fact_415_split__option__all,axiom,
( ( ^ [P2: option_int > $o] :
! [X5: option_int] : ( P2 @ X5 ) )
= ( ^ [P3: option_int > $o] :
( ( P3 @ none_int )
& ! [X4: int] : ( P3 @ ( some_int @ X4 ) ) ) ) ) ).
% split_option_all
thf(fact_416_combine__options__cases,axiom,
! [X: option_nat,P: option_nat > option_nat > $o,Y: option_nat] :
( ( ( X = none_nat )
=> ( P @ X @ Y ) )
=> ( ( ( Y = none_nat )
=> ( P @ X @ Y ) )
=> ( ! [A3: nat,B4: nat] :
( ( X
= ( some_nat @ A3 ) )
=> ( ( Y
= ( some_nat @ B4 ) )
=> ( P @ X @ Y ) ) )
=> ( P @ X @ Y ) ) ) ) ).
% combine_options_cases
thf(fact_417_combine__options__cases,axiom,
! [X: option_nat,P: option_nat > option_int > $o,Y: option_int] :
( ( ( X = none_nat )
=> ( P @ X @ Y ) )
=> ( ( ( Y = none_int )
=> ( P @ X @ Y ) )
=> ( ! [A3: nat,B4: int] :
( ( X
= ( some_nat @ A3 ) )
=> ( ( Y
= ( some_int @ B4 ) )
=> ( P @ X @ Y ) ) )
=> ( P @ X @ Y ) ) ) ) ).
% combine_options_cases
thf(fact_418_combine__options__cases,axiom,
! [X: option_int,P: option_int > option_nat > $o,Y: option_nat] :
( ( ( X = none_int )
=> ( P @ X @ Y ) )
=> ( ( ( Y = none_nat )
=> ( P @ X @ Y ) )
=> ( ! [A3: int,B4: nat] :
( ( X
= ( some_int @ A3 ) )
=> ( ( Y
= ( some_nat @ B4 ) )
=> ( P @ X @ Y ) ) )
=> ( P @ X @ Y ) ) ) ) ).
% combine_options_cases
thf(fact_419_combine__options__cases,axiom,
! [X: option_int,P: option_int > option_int > $o,Y: option_int] :
( ( ( X = none_int )
=> ( P @ X @ Y ) )
=> ( ( ( Y = none_int )
=> ( P @ X @ Y ) )
=> ( ! [A3: int,B4: int] :
( ( X
= ( some_int @ A3 ) )
=> ( ( Y
= ( some_int @ B4 ) )
=> ( P @ X @ Y ) ) )
=> ( P @ X @ Y ) ) ) ) ).
% combine_options_cases
thf(fact_420_GreatestI__nat,axiom,
! [P: nat > $o,K2: nat,B: nat] :
( ( P @ K2 )
=> ( ! [Y3: nat] :
( ( P @ Y3 )
=> ( ord_less_eq_nat @ Y3 @ B ) )
=> ( P @ ( order_Greatest_nat @ P ) ) ) ) ).
% GreatestI_nat
thf(fact_421_Greatest__le__nat,axiom,
! [P: nat > $o,K2: nat,B: nat] :
( ( P @ K2 )
=> ( ! [Y3: nat] :
( ( P @ Y3 )
=> ( ord_less_eq_nat @ Y3 @ B ) )
=> ( ord_less_eq_nat @ K2 @ ( order_Greatest_nat @ P ) ) ) ) ).
% Greatest_le_nat
thf(fact_422_GreatestI__ex__nat,axiom,
! [P: nat > $o,B: nat] :
( ? [X_12: nat] : ( P @ X_12 )
=> ( ! [Y3: nat] :
( ( P @ Y3 )
=> ( ord_less_eq_nat @ Y3 @ B ) )
=> ( P @ ( order_Greatest_nat @ P ) ) ) ) ).
% GreatestI_ex_nat
thf(fact_423_verit__comp__simplify1_I2_J,axiom,
! [A: nat] : ( ord_less_eq_nat @ A @ A ) ).
% verit_comp_simplify1(2)
thf(fact_424_verit__comp__simplify1_I2_J,axiom,
! [A: int] : ( ord_less_eq_int @ A @ A ) ).
% verit_comp_simplify1(2)
thf(fact_425_verit__comp__simplify1_I2_J,axiom,
! [A: real] : ( ord_less_eq_real @ A @ A ) ).
% verit_comp_simplify1(2)
thf(fact_426_verit__la__disequality,axiom,
! [A: nat,B: nat] :
( ( A = B )
| ~ ( ord_less_eq_nat @ A @ B )
| ~ ( ord_less_eq_nat @ B @ A ) ) ).
% verit_la_disequality
thf(fact_427_verit__la__disequality,axiom,
! [A: int,B: int] :
( ( A = B )
| ~ ( ord_less_eq_int @ A @ B )
| ~ ( ord_less_eq_int @ B @ A ) ) ).
% verit_la_disequality
thf(fact_428_verit__la__disequality,axiom,
! [A: real,B: real] :
( ( A = B )
| ~ ( ord_less_eq_real @ A @ B )
| ~ ( ord_less_eq_real @ B @ A ) ) ).
% verit_la_disequality
thf(fact_429_minf_I7_J,axiom,
! [T: nat] :
? [Z3: nat] :
! [X6: nat] :
( ( ord_less_nat @ X6 @ Z3 )
=> ~ ( ord_less_nat @ T @ X6 ) ) ).
% minf(7)
thf(fact_430_minf_I7_J,axiom,
! [T: int] :
? [Z3: int] :
! [X6: int] :
( ( ord_less_int @ X6 @ Z3 )
=> ~ ( ord_less_int @ T @ X6 ) ) ).
% minf(7)
thf(fact_431_minf_I7_J,axiom,
! [T: real] :
? [Z3: real] :
! [X6: real] :
( ( ord_less_real @ X6 @ Z3 )
=> ~ ( ord_less_real @ T @ X6 ) ) ).
% minf(7)
thf(fact_432_minf_I5_J,axiom,
! [T: nat] :
? [Z3: nat] :
! [X6: nat] :
( ( ord_less_nat @ X6 @ Z3 )
=> ( ord_less_nat @ X6 @ T ) ) ).
% minf(5)
thf(fact_433_minf_I5_J,axiom,
! [T: int] :
? [Z3: int] :
! [X6: int] :
( ( ord_less_int @ X6 @ Z3 )
=> ( ord_less_int @ X6 @ T ) ) ).
% minf(5)
thf(fact_434_minf_I5_J,axiom,
! [T: real] :
? [Z3: real] :
! [X6: real] :
( ( ord_less_real @ X6 @ Z3 )
=> ( ord_less_real @ X6 @ T ) ) ).
% minf(5)
thf(fact_435_minf_I4_J,axiom,
! [T: nat] :
? [Z3: nat] :
! [X6: nat] :
( ( ord_less_nat @ X6 @ Z3 )
=> ( X6 != T ) ) ).
% minf(4)
thf(fact_436_minf_I4_J,axiom,
! [T: int] :
? [Z3: int] :
! [X6: int] :
( ( ord_less_int @ X6 @ Z3 )
=> ( X6 != T ) ) ).
% minf(4)
thf(fact_437_minf_I4_J,axiom,
! [T: real] :
? [Z3: real] :
! [X6: real] :
( ( ord_less_real @ X6 @ Z3 )
=> ( X6 != T ) ) ).
% minf(4)
thf(fact_438_minf_I3_J,axiom,
! [T: nat] :
? [Z3: nat] :
! [X6: nat] :
( ( ord_less_nat @ X6 @ Z3 )
=> ( X6 != T ) ) ).
% minf(3)
thf(fact_439_minf_I3_J,axiom,
! [T: int] :
? [Z3: int] :
! [X6: int] :
( ( ord_less_int @ X6 @ Z3 )
=> ( X6 != T ) ) ).
% minf(3)
thf(fact_440_minf_I3_J,axiom,
! [T: real] :
? [Z3: real] :
! [X6: real] :
( ( ord_less_real @ X6 @ Z3 )
=> ( X6 != T ) ) ).
% minf(3)
thf(fact_441_minf_I2_J,axiom,
! [P: nat > $o,P4: nat > $o,Q: nat > $o,Q2: nat > $o] :
( ? [Z4: nat] :
! [X3: nat] :
( ( ord_less_nat @ X3 @ Z4 )
=> ( ( P @ X3 )
= ( P4 @ X3 ) ) )
=> ( ? [Z4: nat] :
! [X3: nat] :
( ( ord_less_nat @ X3 @ Z4 )
=> ( ( Q @ X3 )
= ( Q2 @ X3 ) ) )
=> ? [Z3: nat] :
! [X6: nat] :
( ( ord_less_nat @ X6 @ Z3 )
=> ( ( ( P @ X6 )
| ( Q @ X6 ) )
= ( ( P4 @ X6 )
| ( Q2 @ X6 ) ) ) ) ) ) ).
% minf(2)
thf(fact_442_minf_I2_J,axiom,
! [P: int > $o,P4: int > $o,Q: int > $o,Q2: int > $o] :
( ? [Z4: int] :
! [X3: int] :
( ( ord_less_int @ X3 @ Z4 )
=> ( ( P @ X3 )
= ( P4 @ X3 ) ) )
=> ( ? [Z4: int] :
! [X3: int] :
( ( ord_less_int @ X3 @ Z4 )
=> ( ( Q @ X3 )
= ( Q2 @ X3 ) ) )
=> ? [Z3: int] :
! [X6: int] :
( ( ord_less_int @ X6 @ Z3 )
=> ( ( ( P @ X6 )
| ( Q @ X6 ) )
= ( ( P4 @ X6 )
| ( Q2 @ X6 ) ) ) ) ) ) ).
% minf(2)
thf(fact_443_minf_I2_J,axiom,
! [P: real > $o,P4: real > $o,Q: real > $o,Q2: real > $o] :
( ? [Z4: real] :
! [X3: real] :
( ( ord_less_real @ X3 @ Z4 )
=> ( ( P @ X3 )
= ( P4 @ X3 ) ) )
=> ( ? [Z4: real] :
! [X3: real] :
( ( ord_less_real @ X3 @ Z4 )
=> ( ( Q @ X3 )
= ( Q2 @ X3 ) ) )
=> ? [Z3: real] :
! [X6: real] :
( ( ord_less_real @ X6 @ Z3 )
=> ( ( ( P @ X6 )
| ( Q @ X6 ) )
= ( ( P4 @ X6 )
| ( Q2 @ X6 ) ) ) ) ) ) ).
% minf(2)
thf(fact_444_minf_I1_J,axiom,
! [P: nat > $o,P4: nat > $o,Q: nat > $o,Q2: nat > $o] :
( ? [Z4: nat] :
! [X3: nat] :
( ( ord_less_nat @ X3 @ Z4 )
=> ( ( P @ X3 )
= ( P4 @ X3 ) ) )
=> ( ? [Z4: nat] :
! [X3: nat] :
( ( ord_less_nat @ X3 @ Z4 )
=> ( ( Q @ X3 )
= ( Q2 @ X3 ) ) )
=> ? [Z3: nat] :
! [X6: nat] :
( ( ord_less_nat @ X6 @ Z3 )
=> ( ( ( P @ X6 )
& ( Q @ X6 ) )
= ( ( P4 @ X6 )
& ( Q2 @ X6 ) ) ) ) ) ) ).
% minf(1)
thf(fact_445_minf_I1_J,axiom,
! [P: int > $o,P4: int > $o,Q: int > $o,Q2: int > $o] :
( ? [Z4: int] :
! [X3: int] :
( ( ord_less_int @ X3 @ Z4 )
=> ( ( P @ X3 )
= ( P4 @ X3 ) ) )
=> ( ? [Z4: int] :
! [X3: int] :
( ( ord_less_int @ X3 @ Z4 )
=> ( ( Q @ X3 )
= ( Q2 @ X3 ) ) )
=> ? [Z3: int] :
! [X6: int] :
( ( ord_less_int @ X6 @ Z3 )
=> ( ( ( P @ X6 )
& ( Q @ X6 ) )
= ( ( P4 @ X6 )
& ( Q2 @ X6 ) ) ) ) ) ) ).
% minf(1)
thf(fact_446_minf_I1_J,axiom,
! [P: real > $o,P4: real > $o,Q: real > $o,Q2: real > $o] :
( ? [Z4: real] :
! [X3: real] :
( ( ord_less_real @ X3 @ Z4 )
=> ( ( P @ X3 )
= ( P4 @ X3 ) ) )
=> ( ? [Z4: real] :
! [X3: real] :
( ( ord_less_real @ X3 @ Z4 )
=> ( ( Q @ X3 )
= ( Q2 @ X3 ) ) )
=> ? [Z3: real] :
! [X6: real] :
( ( ord_less_real @ X6 @ Z3 )
=> ( ( ( P @ X6 )
& ( Q @ X6 ) )
= ( ( P4 @ X6 )
& ( Q2 @ X6 ) ) ) ) ) ) ).
% minf(1)
thf(fact_447_pinf_I7_J,axiom,
! [T: nat] :
? [Z3: nat] :
! [X6: nat] :
( ( ord_less_nat @ Z3 @ X6 )
=> ( ord_less_nat @ T @ X6 ) ) ).
% pinf(7)
thf(fact_448_pinf_I7_J,axiom,
! [T: int] :
? [Z3: int] :
! [X6: int] :
( ( ord_less_int @ Z3 @ X6 )
=> ( ord_less_int @ T @ X6 ) ) ).
% pinf(7)
thf(fact_449_pinf_I7_J,axiom,
! [T: real] :
? [Z3: real] :
! [X6: real] :
( ( ord_less_real @ Z3 @ X6 )
=> ( ord_less_real @ T @ X6 ) ) ).
% pinf(7)
thf(fact_450_pinf_I5_J,axiom,
! [T: nat] :
? [Z3: nat] :
! [X6: nat] :
( ( ord_less_nat @ Z3 @ X6 )
=> ~ ( ord_less_nat @ X6 @ T ) ) ).
% pinf(5)
thf(fact_451_pinf_I5_J,axiom,
! [T: int] :
? [Z3: int] :
! [X6: int] :
( ( ord_less_int @ Z3 @ X6 )
=> ~ ( ord_less_int @ X6 @ T ) ) ).
% pinf(5)
thf(fact_452_pinf_I5_J,axiom,
! [T: real] :
? [Z3: real] :
! [X6: real] :
( ( ord_less_real @ Z3 @ X6 )
=> ~ ( ord_less_real @ X6 @ T ) ) ).
% pinf(5)
thf(fact_453_pinf_I4_J,axiom,
! [T: nat] :
? [Z3: nat] :
! [X6: nat] :
( ( ord_less_nat @ Z3 @ X6 )
=> ( X6 != T ) ) ).
% pinf(4)
thf(fact_454_pinf_I4_J,axiom,
! [T: int] :
? [Z3: int] :
! [X6: int] :
( ( ord_less_int @ Z3 @ X6 )
=> ( X6 != T ) ) ).
% pinf(4)
thf(fact_455_pinf_I4_J,axiom,
! [T: real] :
? [Z3: real] :
! [X6: real] :
( ( ord_less_real @ Z3 @ X6 )
=> ( X6 != T ) ) ).
% pinf(4)
thf(fact_456_pinf_I3_J,axiom,
! [T: nat] :
? [Z3: nat] :
! [X6: nat] :
( ( ord_less_nat @ Z3 @ X6 )
=> ( X6 != T ) ) ).
% pinf(3)
thf(fact_457_pinf_I3_J,axiom,
! [T: int] :
? [Z3: int] :
! [X6: int] :
( ( ord_less_int @ Z3 @ X6 )
=> ( X6 != T ) ) ).
% pinf(3)
thf(fact_458_pinf_I3_J,axiom,
! [T: real] :
? [Z3: real] :
! [X6: real] :
( ( ord_less_real @ Z3 @ X6 )
=> ( X6 != T ) ) ).
% pinf(3)
thf(fact_459_pinf_I2_J,axiom,
! [P: nat > $o,P4: nat > $o,Q: nat > $o,Q2: nat > $o] :
( ? [Z4: nat] :
! [X3: nat] :
( ( ord_less_nat @ Z4 @ X3 )
=> ( ( P @ X3 )
= ( P4 @ X3 ) ) )
=> ( ? [Z4: nat] :
! [X3: nat] :
( ( ord_less_nat @ Z4 @ X3 )
=> ( ( Q @ X3 )
= ( Q2 @ X3 ) ) )
=> ? [Z3: nat] :
! [X6: nat] :
( ( ord_less_nat @ Z3 @ X6 )
=> ( ( ( P @ X6 )
| ( Q @ X6 ) )
= ( ( P4 @ X6 )
| ( Q2 @ X6 ) ) ) ) ) ) ).
% pinf(2)
thf(fact_460_pinf_I2_J,axiom,
! [P: int > $o,P4: int > $o,Q: int > $o,Q2: int > $o] :
( ? [Z4: int] :
! [X3: int] :
( ( ord_less_int @ Z4 @ X3 )
=> ( ( P @ X3 )
= ( P4 @ X3 ) ) )
=> ( ? [Z4: int] :
! [X3: int] :
( ( ord_less_int @ Z4 @ X3 )
=> ( ( Q @ X3 )
= ( Q2 @ X3 ) ) )
=> ? [Z3: int] :
! [X6: int] :
( ( ord_less_int @ Z3 @ X6 )
=> ( ( ( P @ X6 )
| ( Q @ X6 ) )
= ( ( P4 @ X6 )
| ( Q2 @ X6 ) ) ) ) ) ) ).
% pinf(2)
thf(fact_461_pinf_I2_J,axiom,
! [P: real > $o,P4: real > $o,Q: real > $o,Q2: real > $o] :
( ? [Z4: real] :
! [X3: real] :
( ( ord_less_real @ Z4 @ X3 )
=> ( ( P @ X3 )
= ( P4 @ X3 ) ) )
=> ( ? [Z4: real] :
! [X3: real] :
( ( ord_less_real @ Z4 @ X3 )
=> ( ( Q @ X3 )
= ( Q2 @ X3 ) ) )
=> ? [Z3: real] :
! [X6: real] :
( ( ord_less_real @ Z3 @ X6 )
=> ( ( ( P @ X6 )
| ( Q @ X6 ) )
= ( ( P4 @ X6 )
| ( Q2 @ X6 ) ) ) ) ) ) ).
% pinf(2)
thf(fact_462_pinf_I1_J,axiom,
! [P: nat > $o,P4: nat > $o,Q: nat > $o,Q2: nat > $o] :
( ? [Z4: nat] :
! [X3: nat] :
( ( ord_less_nat @ Z4 @ X3 )
=> ( ( P @ X3 )
= ( P4 @ X3 ) ) )
=> ( ? [Z4: nat] :
! [X3: nat] :
( ( ord_less_nat @ Z4 @ X3 )
=> ( ( Q @ X3 )
= ( Q2 @ X3 ) ) )
=> ? [Z3: nat] :
! [X6: nat] :
( ( ord_less_nat @ Z3 @ X6 )
=> ( ( ( P @ X6 )
& ( Q @ X6 ) )
= ( ( P4 @ X6 )
& ( Q2 @ X6 ) ) ) ) ) ) ).
% pinf(1)
thf(fact_463_pinf_I1_J,axiom,
! [P: int > $o,P4: int > $o,Q: int > $o,Q2: int > $o] :
( ? [Z4: int] :
! [X3: int] :
( ( ord_less_int @ Z4 @ X3 )
=> ( ( P @ X3 )
= ( P4 @ X3 ) ) )
=> ( ? [Z4: int] :
! [X3: int] :
( ( ord_less_int @ Z4 @ X3 )
=> ( ( Q @ X3 )
= ( Q2 @ X3 ) ) )
=> ? [Z3: int] :
! [X6: int] :
( ( ord_less_int @ Z3 @ X6 )
=> ( ( ( P @ X6 )
& ( Q @ X6 ) )
= ( ( P4 @ X6 )
& ( Q2 @ X6 ) ) ) ) ) ) ).
% pinf(1)
thf(fact_464_pinf_I1_J,axiom,
! [P: real > $o,P4: real > $o,Q: real > $o,Q2: real > $o] :
( ? [Z4: real] :
! [X3: real] :
( ( ord_less_real @ Z4 @ X3 )
=> ( ( P @ X3 )
= ( P4 @ X3 ) ) )
=> ( ? [Z4: real] :
! [X3: real] :
( ( ord_less_real @ Z4 @ X3 )
=> ( ( Q @ X3 )
= ( Q2 @ X3 ) ) )
=> ? [Z3: real] :
! [X6: real] :
( ( ord_less_real @ Z3 @ X6 )
=> ( ( ( P @ X6 )
& ( Q @ X6 ) )
= ( ( P4 @ X6 )
& ( Q2 @ X6 ) ) ) ) ) ) ).
% pinf(1)
thf(fact_465_verit__comp__simplify1_I1_J,axiom,
! [A: nat] :
~ ( ord_less_nat @ A @ A ) ).
% verit_comp_simplify1(1)
thf(fact_466_verit__comp__simplify1_I1_J,axiom,
! [A: int] :
~ ( ord_less_int @ A @ A ) ).
% verit_comp_simplify1(1)
thf(fact_467_verit__comp__simplify1_I1_J,axiom,
! [A: real] :
~ ( ord_less_real @ A @ A ) ).
% verit_comp_simplify1(1)
thf(fact_468_ex__gt__or__lt,axiom,
! [A: real] :
? [B4: real] :
( ( ord_less_real @ A @ B4 )
| ( ord_less_real @ B4 @ A ) ) ).
% ex_gt_or_lt
thf(fact_469_not__arg__cong__Inr,axiom,
! [X: nat,Y: nat] :
( ( X != Y )
=> ( ( sum_Inr_nat_a @ X )
!= ( sum_Inr_nat_a @ Y ) ) ) ).
% not_arg_cong_Inr
thf(fact_470_bounded__Max__nat,axiom,
! [P: nat > $o,X: nat,M4: nat] :
( ( P @ X )
=> ( ! [X3: nat] :
( ( P @ X3 )
=> ( ord_less_eq_nat @ X3 @ M4 ) )
=> ~ ! [M5: nat] :
( ( P @ M5 )
=> ~ ! [X6: nat] :
( ( P @ X6 )
=> ( ord_less_eq_nat @ X6 @ M5 ) ) ) ) ) ).
% bounded_Max_nat
thf(fact_471_wlog__le,axiom,
! [P: nat > nat > $o,B: nat,A: nat] :
( ! [A3: nat,B4: nat] :
( ( P @ A3 @ B4 )
=> ( P @ B4 @ A3 ) )
=> ( ! [A3: nat,B4: nat] :
( ( ord_less_eq_nat @ A3 @ B4 )
=> ( P @ A3 @ B4 ) )
=> ( P @ B @ A ) ) ) ).
% wlog_le
thf(fact_472_wlog__le,axiom,
! [P: int > int > $o,B: int,A: int] :
( ! [A3: int,B4: int] :
( ( P @ A3 @ B4 )
=> ( P @ B4 @ A3 ) )
=> ( ! [A3: int,B4: int] :
( ( ord_less_eq_int @ A3 @ B4 )
=> ( P @ A3 @ B4 ) )
=> ( P @ B @ A ) ) ) ).
% wlog_le
thf(fact_473_wlog__le,axiom,
! [P: real > real > $o,B: real,A: real] :
( ! [A3: real,B4: real] :
( ( P @ A3 @ B4 )
=> ( P @ B4 @ A3 ) )
=> ( ! [A3: real,B4: real] :
( ( ord_less_eq_real @ A3 @ B4 )
=> ( P @ A3 @ B4 ) )
=> ( P @ B @ A ) ) ) ).
% wlog_le
thf(fact_474_cproper__interval__option_Ocases,axiom,
! [X: produc6094937800015497793on_nat] :
( ( X
!= ( produc6915583070269132153on_nat @ none_option_nat @ none_option_nat ) )
=> ( ! [X3: option_nat] :
( X
!= ( produc6915583070269132153on_nat @ none_option_nat @ ( some_option_nat @ X3 ) ) )
=> ( ! [X3: option_nat] :
( X
!= ( produc6915583070269132153on_nat @ ( some_option_nat @ X3 ) @ none_option_nat ) )
=> ( ! [X3: option_nat] :
( X
!= ( produc6915583070269132153on_nat @ ( some_option_nat @ X3 ) @ ( some_option_nat @ none_nat ) ) )
=> ~ ! [X3: option_nat,Y3: nat] :
( X
!= ( produc6915583070269132153on_nat @ ( some_option_nat @ X3 ) @ ( some_option_nat @ ( some_nat @ Y3 ) ) ) ) ) ) ) ) ).
% cproper_interval_option.cases
thf(fact_475_cproper__interval__option_Ocases,axiom,
! [X: produc420109317877091577on_int] :
( ( X
!= ( produc3564219861986917937on_int @ none_option_int @ none_option_int ) )
=> ( ! [X3: option_int] :
( X
!= ( produc3564219861986917937on_int @ none_option_int @ ( some_option_int @ X3 ) ) )
=> ( ! [X3: option_int] :
( X
!= ( produc3564219861986917937on_int @ ( some_option_int @ X3 ) @ none_option_int ) )
=> ( ! [X3: option_int] :
( X
!= ( produc3564219861986917937on_int @ ( some_option_int @ X3 ) @ ( some_option_int @ none_int ) ) )
=> ~ ! [X3: option_int,Y3: int] :
( X
!= ( produc3564219861986917937on_int @ ( some_option_int @ X3 ) @ ( some_option_int @ ( some_int @ Y3 ) ) ) ) ) ) ) ) ).
% cproper_interval_option.cases
thf(fact_476_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_477_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_478_linordered__field__no__ub,axiom,
! [X6: real] :
? [X_1: real] : ( ord_less_real @ X6 @ X_1 ) ).
% linordered_field_no_ub
thf(fact_479_linordered__field__no__lb,axiom,
! [X6: real] :
? [Y3: real] : ( ord_less_real @ Y3 @ X6 ) ).
% linordered_field_no_lb
thf(fact_480_bdd__above_Opreordering__bdd__axioms,axiom,
condit7935552474144124665dd_nat @ ord_less_eq_nat @ ord_less_nat ).
% bdd_above.preordering_bdd_axioms
thf(fact_481_bdd__above_Opreordering__bdd__axioms,axiom,
condit7933062003635074389dd_int @ ord_less_eq_int @ ord_less_int ).
% bdd_above.preordering_bdd_axioms
thf(fact_482_bdd__above_Opreordering__bdd__axioms,axiom,
condit1497324847667023189d_real @ ord_less_eq_real @ ord_less_real ).
% bdd_above.preordering_bdd_axioms
thf(fact_483_le__rel__bool__arg__iff,axiom,
( ord_less_eq_o_nat
= ( ^ [X7: $o > nat,Y7: $o > nat] :
( ( ord_less_eq_nat @ ( X7 @ $false ) @ ( Y7 @ $false ) )
& ( ord_less_eq_nat @ ( X7 @ $true ) @ ( Y7 @ $true ) ) ) ) ) ).
% le_rel_bool_arg_iff
thf(fact_484_le__rel__bool__arg__iff,axiom,
( ord_less_eq_o_int
= ( ^ [X7: $o > int,Y7: $o > int] :
( ( ord_less_eq_int @ ( X7 @ $false ) @ ( Y7 @ $false ) )
& ( ord_less_eq_int @ ( X7 @ $true ) @ ( Y7 @ $true ) ) ) ) ) ).
% le_rel_bool_arg_iff
thf(fact_485_le__rel__bool__arg__iff,axiom,
( ord_less_eq_o_real
= ( ^ [X7: $o > real,Y7: $o > real] :
( ( ord_less_eq_real @ ( X7 @ $false ) @ ( Y7 @ $false ) )
& ( ord_less_eq_real @ ( X7 @ $true ) @ ( Y7 @ $true ) ) ) ) ) ).
% le_rel_bool_arg_iff
thf(fact_486_cproper__interval__option_Osimps_I2_J,axiom,
! [X: option_nat] :
( ( collec7230243842949892037on_nat @ none_option_nat @ ( some_option_nat @ X ) )
= ( X != none_nat ) ) ).
% cproper_interval_option.simps(2)
thf(fact_487_cproper__interval__option_Osimps_I2_J,axiom,
! [X: option_int] :
( ( collec3052392823440695329on_int @ none_option_int @ ( some_option_int @ X ) )
= ( X != none_int ) ) ).
% cproper_interval_option.simps(2)
thf(fact_488_proper__interval__simps_I2_J,axiom,
! [Y: nat] :
( ( set_pr3848668467136637714al_nat @ none_nat @ ( some_nat @ Y ) )
= ( ? [Z5: nat] : ( ord_less_nat @ Z5 @ Y ) ) ) ).
% proper_interval_simps(2)
thf(fact_489_proper__interval__simps_I2_J,axiom,
! [Y: int] :
( ( set_pr3846177996627587438al_int @ none_int @ ( some_int @ Y ) )
= ( ? [Z5: int] : ( ord_less_int @ Z5 @ Y ) ) ) ).
% proper_interval_simps(2)
thf(fact_490_cproper__interval__prod_Osimps_I3_J,axiom,
! [X1: option_nat,X2: option_nat] :
( ( collec4307506151574712252on_nat @ ( some_P1340420066877415558on_nat @ ( produc5098337634421038937on_nat @ X1 @ X2 ) ) @ none_P2954882734657813250on_nat )
= ( ( collec7230243842949892037on_nat @ ( some_option_nat @ X1 ) @ none_option_nat )
| ( collec7230243842949892037on_nat @ ( some_option_nat @ X2 ) @ none_option_nat ) ) ) ).
% cproper_interval_prod.simps(3)
thf(fact_491_cproper__interval__prod_Osimps_I3_J,axiom,
! [X1: option_int,X2: option_int] :
( ( collec7228505188346447988on_int @ ( some_P4261419103649151294on_int @ ( produc6331568796615304721on_int @ X1 @ X2 ) ) @ none_P5875881771429548986on_int )
= ( ( collec3052392823440695329on_int @ ( some_option_int @ X1 ) @ none_option_int )
| ( collec3052392823440695329on_int @ ( some_option_int @ X2 ) @ none_option_int ) ) ) ).
% cproper_interval_prod.simps(3)
thf(fact_492_cproper__interval__prod_Osimps_I3_J,axiom,
! [X1: nat,X2: nat] :
( ( collec8994931431046529180at_nat @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ X1 @ X2 ) ) @ none_P5556105721700978146at_nat )
= ( ( collec4140898570808001013al_nat @ ( some_nat @ X1 ) @ none_nat )
| ( collec4140898570808001013al_nat @ ( some_nat @ X2 ) @ none_nat ) ) ) ).
% cproper_interval_prod.simps(3)
thf(fact_493_cproper__interval__prod_Osimps_I3_J,axiom,
! [X1: nat,X2: int] :
( ( collec4817080411537332472at_int @ ( some_P3185539396519409602at_int @ ( product_Pair_nat_int @ X1 @ X2 ) ) @ none_P1378254702191781438at_int )
= ( ( collec4140898570808001013al_nat @ ( some_nat @ X1 ) @ none_nat )
| ( collec4138408100298950737al_int @ ( some_int @ X2 ) @ none_int ) ) ) ).
% cproper_interval_prod.simps(3)
thf(fact_494_cproper__interval__prod_Osimps_I3_J,axiom,
! [X1: int,X2: nat] :
( ( collec770913106092807928nt_nat @ ( some_P8362744127929660866nt_nat @ ( product_Pair_int_nat @ X1 @ X2 ) ) @ none_P6555459433602032702nt_nat )
= ( ( collec4138408100298950737al_int @ ( some_int @ X1 ) @ none_int )
| ( collec4140898570808001013al_nat @ ( some_nat @ X2 ) @ none_nat ) ) ) ).
% cproper_interval_prod.simps(3)
thf(fact_495_cproper__interval__prod_Osimps_I3_J,axiom,
! [X1: int,X2: int] :
( ( collec5816434123438387028nt_int @ ( some_P4184893108420464158nt_int @ ( product_Pair_int_int @ X1 @ X2 ) ) @ none_P2377608414092835994nt_int )
= ( ( collec4138408100298950737al_int @ ( some_int @ X1 ) @ none_int )
| ( collec4138408100298950737al_int @ ( some_int @ X2 ) @ none_int ) ) ) ).
% cproper_interval_prod.simps(3)
thf(fact_496_cproper__interval__prod_Osimps_I2_J,axiom,
! [Y1: option_nat,Y2: option_nat] :
( ( collec4307506151574712252on_nat @ none_P2954882734657813250on_nat @ ( some_P1340420066877415558on_nat @ ( produc5098337634421038937on_nat @ Y1 @ Y2 ) ) )
= ( ( collec7230243842949892037on_nat @ none_option_nat @ ( some_option_nat @ Y1 ) )
| ( collec7230243842949892037on_nat @ none_option_nat @ ( some_option_nat @ Y2 ) ) ) ) ).
% cproper_interval_prod.simps(2)
thf(fact_497_cproper__interval__prod_Osimps_I2_J,axiom,
! [Y1: option_int,Y2: option_int] :
( ( collec7228505188346447988on_int @ none_P5875881771429548986on_int @ ( some_P4261419103649151294on_int @ ( produc6331568796615304721on_int @ Y1 @ Y2 ) ) )
= ( ( collec3052392823440695329on_int @ none_option_int @ ( some_option_int @ Y1 ) )
| ( collec3052392823440695329on_int @ none_option_int @ ( some_option_int @ Y2 ) ) ) ) ).
% cproper_interval_prod.simps(2)
thf(fact_498_cproper__interval__prod_Osimps_I2_J,axiom,
! [Y1: nat,Y2: nat] :
( ( collec8994931431046529180at_nat @ none_P5556105721700978146at_nat @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Y1 @ Y2 ) ) )
= ( ( collec4140898570808001013al_nat @ none_nat @ ( some_nat @ Y1 ) )
| ( collec4140898570808001013al_nat @ none_nat @ ( some_nat @ Y2 ) ) ) ) ).
% cproper_interval_prod.simps(2)
thf(fact_499_cproper__interval__prod_Osimps_I2_J,axiom,
! [Y1: nat,Y2: int] :
( ( collec4817080411537332472at_int @ none_P1378254702191781438at_int @ ( some_P3185539396519409602at_int @ ( product_Pair_nat_int @ Y1 @ Y2 ) ) )
= ( ( collec4140898570808001013al_nat @ none_nat @ ( some_nat @ Y1 ) )
| ( collec4138408100298950737al_int @ none_int @ ( some_int @ Y2 ) ) ) ) ).
% cproper_interval_prod.simps(2)
thf(fact_500_cproper__interval__prod_Osimps_I2_J,axiom,
! [Y1: int,Y2: nat] :
( ( collec770913106092807928nt_nat @ none_P6555459433602032702nt_nat @ ( some_P8362744127929660866nt_nat @ ( product_Pair_int_nat @ Y1 @ Y2 ) ) )
= ( ( collec4138408100298950737al_int @ none_int @ ( some_int @ Y1 ) )
| ( collec4140898570808001013al_nat @ none_nat @ ( some_nat @ Y2 ) ) ) ) ).
% cproper_interval_prod.simps(2)
thf(fact_501_cproper__interval__prod_Osimps_I2_J,axiom,
! [Y1: int,Y2: int] :
( ( collec5816434123438387028nt_int @ none_P2377608414092835994nt_int @ ( some_P4184893108420464158nt_int @ ( product_Pair_int_int @ Y1 @ Y2 ) ) )
= ( ( collec4138408100298950737al_int @ none_int @ ( some_int @ Y1 ) )
| ( collec4138408100298950737al_int @ none_int @ ( some_int @ Y2 ) ) ) ) ).
% cproper_interval_prod.simps(2)
thf(fact_502_cproper__interval__prod_Ocases,axiom,
! [X: produc5426583320905391847on_nat] :
( ( X
!= ( produc1960425215797453655on_nat @ none_P2954882734657813250on_nat @ none_P2954882734657813250on_nat ) )
=> ( ! [Y12: option_nat,Y22: option_nat] :
( X
!= ( produc1960425215797453655on_nat @ none_P2954882734657813250on_nat @ ( some_P1340420066877415558on_nat @ ( produc5098337634421038937on_nat @ Y12 @ Y22 ) ) ) )
=> ( ! [X12: option_nat,X22: option_nat] :
( X
!= ( produc1960425215797453655on_nat @ ( some_P1340420066877415558on_nat @ ( produc5098337634421038937on_nat @ X12 @ X22 ) ) @ none_P2954882734657813250on_nat ) )
=> ~ ! [X12: option_nat,X22: option_nat,Y12: option_nat,Y22: option_nat] :
( X
!= ( produc1960425215797453655on_nat @ ( some_P1340420066877415558on_nat @ ( produc5098337634421038937on_nat @ X12 @ X22 ) ) @ ( some_P1340420066877415558on_nat @ ( produc5098337634421038937on_nat @ Y12 @ Y22 ) ) ) ) ) ) ) ).
% cproper_interval_prod.cases
thf(fact_503_cproper__interval__prod_Ocases,axiom,
! [X: produc5472250993411995879on_int] :
( ( X
!= ( produc4796250914959645015on_int @ none_P5875881771429548986on_int @ none_P5875881771429548986on_int ) )
=> ( ! [Y12: option_int,Y22: option_int] :
( X
!= ( produc4796250914959645015on_int @ none_P5875881771429548986on_int @ ( some_P4261419103649151294on_int @ ( produc6331568796615304721on_int @ Y12 @ Y22 ) ) ) )
=> ( ! [X12: option_int,X22: option_int] :
( X
!= ( produc4796250914959645015on_int @ ( some_P4261419103649151294on_int @ ( produc6331568796615304721on_int @ X12 @ X22 ) ) @ none_P5875881771429548986on_int ) )
=> ~ ! [X12: option_int,X22: option_int,Y12: option_int,Y22: option_int] :
( X
!= ( produc4796250914959645015on_int @ ( some_P4261419103649151294on_int @ ( produc6331568796615304721on_int @ X12 @ X22 ) ) @ ( some_P4261419103649151294on_int @ ( produc6331568796615304721on_int @ Y12 @ Y22 ) ) ) ) ) ) ) ).
% cproper_interval_prod.cases
thf(fact_504_sp__equiv__pair_Ocases,axiom,
! [X: produc6315566273921074279on_nat] :
~ ! [A3: option_nat,B4: option_nat,A6: option_nat,B5: option_nat] :
( X
!= ( produc2417174425200997591on_nat @ ( produc5098337634421038937on_nat @ A3 @ B4 ) @ ( produc5098337634421038937on_nat @ A6 @ B5 ) ) ) ).
% sp_equiv_pair.cases
thf(fact_505_sp__equiv__pair_Ocases,axiom,
! [X: produc9050054836243589735on_int] :
~ ! [A3: option_int,B4: option_int,A6: option_int,B5: option_int] :
( X
!= ( produc8934765814667458263on_int @ ( produc6331568796615304721on_int @ A3 @ B4 ) @ ( produc6331568796615304721on_int @ A6 @ B5 ) ) ) ).
% sp_equiv_pair.cases
thf(fact_506_cproper__interval__option_Osimps_I5_J,axiom,
! [X: option_nat,Y: nat] :
( ( collec7230243842949892037on_nat @ ( some_option_nat @ X ) @ ( some_option_nat @ ( some_nat @ Y ) ) )
= ( collec4140898570808001013al_nat @ X @ ( some_nat @ Y ) ) ) ).
% cproper_interval_option.simps(5)
thf(fact_507_cproper__interval__option_Osimps_I5_J,axiom,
! [X: option_int,Y: int] :
( ( collec3052392823440695329on_int @ ( some_option_int @ X ) @ ( some_option_int @ ( some_int @ Y ) ) )
= ( collec4138408100298950737al_int @ X @ ( some_int @ Y ) ) ) ).
% cproper_interval_option.simps(5)
thf(fact_508_proper__interval__simps_I1_J,axiom,
set_pr3848668467136637714al_nat @ none_nat @ none_nat ).
% proper_interval_simps(1)
thf(fact_509_proper__interval__simps_I1_J,axiom,
set_pr3846177996627587438al_int @ none_int @ none_int ).
% proper_interval_simps(1)
thf(fact_510_proper__interval__nat_Osimps_I1_J,axiom,
! [No: option_nat] : ( set_pr3848668467136637714al_nat @ No @ none_nat ) ).
% proper_interval_nat.simps(1)
thf(fact_511_cproper__interval__option_Osimps_I3_J,axiom,
! [X: option_nat] :
( ( collec7230243842949892037on_nat @ ( some_option_nat @ X ) @ none_option_nat )
= ( collec4140898570808001013al_nat @ X @ none_nat ) ) ).
% cproper_interval_option.simps(3)
thf(fact_512_cproper__interval__option_Osimps_I3_J,axiom,
! [X: option_int] :
( ( collec3052392823440695329on_int @ ( some_option_int @ X ) @ none_option_int )
= ( collec4138408100298950737al_int @ X @ none_int ) ) ).
% cproper_interval_option.simps(3)
thf(fact_513_cproper__interval__option_Oelims_I1_J,axiom,
! [X: option_option_nat,Xa: option_option_nat,Y: $o] :
( ( ( collec7230243842949892037on_nat @ X @ Xa )
= Y )
=> ( ( ( X = none_option_nat )
=> ( ( Xa = none_option_nat )
=> ~ Y ) )
=> ( ( ( X = none_option_nat )
=> ! [X3: option_nat] :
( ( Xa
= ( some_option_nat @ X3 ) )
=> ( Y
= ( X3 = none_nat ) ) ) )
=> ( ! [X3: option_nat] :
( ( X
= ( some_option_nat @ X3 ) )
=> ( ( Xa = none_option_nat )
=> ( Y
= ( ~ ( collec4140898570808001013al_nat @ X3 @ none_nat ) ) ) ) )
=> ( ( ? [X3: option_nat] :
( X
= ( some_option_nat @ X3 ) )
=> ( ( Xa
= ( some_option_nat @ none_nat ) )
=> Y ) )
=> ~ ! [X3: option_nat] :
( ( X
= ( some_option_nat @ X3 ) )
=> ! [Y3: nat] :
( ( Xa
= ( some_option_nat @ ( some_nat @ Y3 ) ) )
=> ( Y
= ( ~ ( collec4140898570808001013al_nat @ X3 @ ( some_nat @ Y3 ) ) ) ) ) ) ) ) ) ) ) ).
% cproper_interval_option.elims(1)
thf(fact_514_cproper__interval__option_Oelims_I1_J,axiom,
! [X: option_option_int,Xa: option_option_int,Y: $o] :
( ( ( collec3052392823440695329on_int @ X @ Xa )
= Y )
=> ( ( ( X = none_option_int )
=> ( ( Xa = none_option_int )
=> ~ Y ) )
=> ( ( ( X = none_option_int )
=> ! [X3: option_int] :
( ( Xa
= ( some_option_int @ X3 ) )
=> ( Y
= ( X3 = none_int ) ) ) )
=> ( ! [X3: option_int] :
( ( X
= ( some_option_int @ X3 ) )
=> ( ( Xa = none_option_int )
=> ( Y
= ( ~ ( collec4138408100298950737al_int @ X3 @ none_int ) ) ) ) )
=> ( ( ? [X3: option_int] :
( X
= ( some_option_int @ X3 ) )
=> ( ( Xa
= ( some_option_int @ none_int ) )
=> Y ) )
=> ~ ! [X3: option_int] :
( ( X
= ( some_option_int @ X3 ) )
=> ! [Y3: int] :
( ( Xa
= ( some_option_int @ ( some_int @ Y3 ) ) )
=> ( Y
= ( ~ ( collec4138408100298950737al_int @ X3 @ ( some_int @ Y3 ) ) ) ) ) ) ) ) ) ) ) ).
% cproper_interval_option.elims(1)
thf(fact_515_cproper__interval__option_Oelims_I2_J,axiom,
! [X: option_option_nat,Xa: option_option_nat] :
( ( collec7230243842949892037on_nat @ X @ Xa )
=> ( ( ( X = none_option_nat )
=> ( Xa != none_option_nat ) )
=> ( ( ( X = none_option_nat )
=> ! [X3: option_nat] :
( ( Xa
= ( some_option_nat @ X3 ) )
=> ( X3 = none_nat ) ) )
=> ( ! [X3: option_nat] :
( ( X
= ( some_option_nat @ X3 ) )
=> ( ( Xa = none_option_nat )
=> ~ ( collec4140898570808001013al_nat @ X3 @ none_nat ) ) )
=> ~ ! [X3: option_nat] :
( ( X
= ( some_option_nat @ X3 ) )
=> ! [Y3: nat] :
( ( Xa
= ( some_option_nat @ ( some_nat @ Y3 ) ) )
=> ~ ( collec4140898570808001013al_nat @ X3 @ ( some_nat @ Y3 ) ) ) ) ) ) ) ) ).
% cproper_interval_option.elims(2)
thf(fact_516_cproper__interval__option_Oelims_I2_J,axiom,
! [X: option_option_int,Xa: option_option_int] :
( ( collec3052392823440695329on_int @ X @ Xa )
=> ( ( ( X = none_option_int )
=> ( Xa != none_option_int ) )
=> ( ( ( X = none_option_int )
=> ! [X3: option_int] :
( ( Xa
= ( some_option_int @ X3 ) )
=> ( X3 = none_int ) ) )
=> ( ! [X3: option_int] :
( ( X
= ( some_option_int @ X3 ) )
=> ( ( Xa = none_option_int )
=> ~ ( collec4138408100298950737al_int @ X3 @ none_int ) ) )
=> ~ ! [X3: option_int] :
( ( X
= ( some_option_int @ X3 ) )
=> ! [Y3: int] :
( ( Xa
= ( some_option_int @ ( some_int @ Y3 ) ) )
=> ~ ( collec4138408100298950737al_int @ X3 @ ( some_int @ Y3 ) ) ) ) ) ) ) ) ).
% cproper_interval_option.elims(2)
thf(fact_517_cproper__interval__option_Oelims_I3_J,axiom,
! [X: option_option_nat,Xa: option_option_nat] :
( ~ ( collec7230243842949892037on_nat @ X @ Xa )
=> ( ( ( X = none_option_nat )
=> ! [X3: option_nat] :
( ( Xa
= ( some_option_nat @ X3 ) )
=> ( X3 != none_nat ) ) )
=> ( ! [X3: option_nat] :
( ( X
= ( some_option_nat @ X3 ) )
=> ( ( Xa = none_option_nat )
=> ( collec4140898570808001013al_nat @ X3 @ none_nat ) ) )
=> ( ( ? [X3: option_nat] :
( X
= ( some_option_nat @ X3 ) )
=> ( Xa
!= ( some_option_nat @ none_nat ) ) )
=> ~ ! [X3: option_nat] :
( ( X
= ( some_option_nat @ X3 ) )
=> ! [Y3: nat] :
( ( Xa
= ( some_option_nat @ ( some_nat @ Y3 ) ) )
=> ( collec4140898570808001013al_nat @ X3 @ ( some_nat @ Y3 ) ) ) ) ) ) ) ) ).
% cproper_interval_option.elims(3)
thf(fact_518_cproper__interval__option_Oelims_I3_J,axiom,
! [X: option_option_int,Xa: option_option_int] :
( ~ ( collec3052392823440695329on_int @ X @ Xa )
=> ( ( ( X = none_option_int )
=> ! [X3: option_int] :
( ( Xa
= ( some_option_int @ X3 ) )
=> ( X3 != none_int ) ) )
=> ( ! [X3: option_int] :
( ( X
= ( some_option_int @ X3 ) )
=> ( ( Xa = none_option_int )
=> ( collec4138408100298950737al_int @ X3 @ none_int ) ) )
=> ( ( ? [X3: option_int] :
( X
= ( some_option_int @ X3 ) )
=> ( Xa
!= ( some_option_int @ none_int ) ) )
=> ~ ! [X3: option_int] :
( ( X
= ( some_option_int @ X3 ) )
=> ! [Y3: int] :
( ( Xa
= ( some_option_int @ ( some_int @ Y3 ) ) )
=> ( collec4138408100298950737al_int @ X3 @ ( some_int @ Y3 ) ) ) ) ) ) ) ) ).
% cproper_interval_option.elims(3)
thf(fact_519_proper__interval__simps_I4_J,axiom,
! [X: nat,Y: nat] :
( ( set_pr3848668467136637714al_nat @ ( some_nat @ X ) @ ( some_nat @ Y ) )
= ( ? [Z5: nat] :
( ( ord_less_nat @ X @ Z5 )
& ( ord_less_nat @ Z5 @ Y ) ) ) ) ).
% proper_interval_simps(4)
thf(fact_520_proper__interval__simps_I4_J,axiom,
! [X: int,Y: int] :
( ( set_pr3846177996627587438al_int @ ( some_int @ X ) @ ( some_int @ Y ) )
= ( ? [Z5: int] :
( ( ord_less_int @ X @ Z5 )
& ( ord_less_int @ Z5 @ Y ) ) ) ) ).
% proper_interval_simps(4)
thf(fact_521_cproper__interval__option_Osimps_I4_J,axiom,
! [X: option_nat] :
~ ( collec7230243842949892037on_nat @ ( some_option_nat @ X ) @ ( some_option_nat @ none_nat ) ) ).
% cproper_interval_option.simps(4)
thf(fact_522_cproper__interval__option_Osimps_I4_J,axiom,
! [X: option_int] :
~ ( collec3052392823440695329on_int @ ( some_option_int @ X ) @ ( some_option_int @ none_int ) ) ).
% cproper_interval_option.simps(4)
thf(fact_523_proper__interval__simps_I3_J,axiom,
! [X: nat] :
( ( set_pr3848668467136637714al_nat @ ( some_nat @ X ) @ none_nat )
= ( ? [X7: nat] : ( ord_less_nat @ X @ X7 ) ) ) ).
% proper_interval_simps(3)
thf(fact_524_proper__interval__simps_I3_J,axiom,
! [X: int] :
( ( set_pr3846177996627587438al_int @ ( some_int @ X ) @ none_int )
= ( ? [X7: int] : ( ord_less_int @ X @ X7 ) ) ) ).
% proper_interval_simps(3)
thf(fact_525_old_Oprod_Oinject,axiom,
! [A: option_nat,B: option_nat,A5: option_nat,B2: option_nat] :
( ( ( produc5098337634421038937on_nat @ A @ B )
= ( produc5098337634421038937on_nat @ A5 @ B2 ) )
= ( ( A = A5 )
& ( B = B2 ) ) ) ).
% old.prod.inject
thf(fact_526_old_Oprod_Oinject,axiom,
! [A: option_int,B: option_int,A5: option_int,B2: option_int] :
( ( ( produc6331568796615304721on_int @ A @ B )
= ( produc6331568796615304721on_int @ A5 @ B2 ) )
= ( ( A = A5 )
& ( B = B2 ) ) ) ).
% old.prod.inject
thf(fact_527_prod_Oinject,axiom,
! [X1: option_nat,X2: option_nat,Y1: option_nat,Y2: option_nat] :
( ( ( produc5098337634421038937on_nat @ X1 @ X2 )
= ( produc5098337634421038937on_nat @ Y1 @ Y2 ) )
= ( ( X1 = Y1 )
& ( X2 = Y2 ) ) ) ).
% prod.inject
thf(fact_528_prod_Oinject,axiom,
! [X1: option_int,X2: option_int,Y1: option_int,Y2: option_int] :
( ( ( produc6331568796615304721on_int @ X1 @ X2 )
= ( produc6331568796615304721on_int @ Y1 @ Y2 ) )
= ( ( X1 = Y1 )
& ( X2 = Y2 ) ) ) ).
% prod.inject
thf(fact_529_cproper__interval__option_Opelims_I1_J,axiom,
! [X: option_option_nat,Xa: option_option_nat,Y: $o] :
( ( ( collec7230243842949892037on_nat @ X @ Xa )
= Y )
=> ( ( accp_P4406482645238171594on_nat @ collec5421229242901901345el_nat @ ( produc6915583070269132153on_nat @ X @ Xa ) )
=> ( ( ( X = none_option_nat )
=> ( ( Xa = none_option_nat )
=> ( Y
=> ~ ( accp_P4406482645238171594on_nat @ collec5421229242901901345el_nat @ ( produc6915583070269132153on_nat @ none_option_nat @ none_option_nat ) ) ) ) )
=> ( ( ( X = none_option_nat )
=> ! [X3: option_nat] :
( ( Xa
= ( some_option_nat @ X3 ) )
=> ( ( Y
= ( X3 != none_nat ) )
=> ~ ( accp_P4406482645238171594on_nat @ collec5421229242901901345el_nat @ ( produc6915583070269132153on_nat @ none_option_nat @ ( some_option_nat @ X3 ) ) ) ) ) )
=> ( ! [X3: option_nat] :
( ( X
= ( some_option_nat @ X3 ) )
=> ( ( Xa = none_option_nat )
=> ( ( Y
= ( collec4140898570808001013al_nat @ X3 @ none_nat ) )
=> ~ ( accp_P4406482645238171594on_nat @ collec5421229242901901345el_nat @ ( produc6915583070269132153on_nat @ ( some_option_nat @ X3 ) @ none_option_nat ) ) ) ) )
=> ( ! [X3: option_nat] :
( ( X
= ( some_option_nat @ X3 ) )
=> ( ( Xa
= ( some_option_nat @ none_nat ) )
=> ( ~ Y
=> ~ ( accp_P4406482645238171594on_nat @ collec5421229242901901345el_nat @ ( produc6915583070269132153on_nat @ ( some_option_nat @ X3 ) @ ( some_option_nat @ none_nat ) ) ) ) ) )
=> ~ ! [X3: option_nat] :
( ( X
= ( some_option_nat @ X3 ) )
=> ! [Y3: nat] :
( ( Xa
= ( some_option_nat @ ( some_nat @ Y3 ) ) )
=> ( ( Y
= ( collec4140898570808001013al_nat @ X3 @ ( some_nat @ Y3 ) ) )
=> ~ ( accp_P4406482645238171594on_nat @ collec5421229242901901345el_nat @ ( produc6915583070269132153on_nat @ ( some_option_nat @ X3 ) @ ( some_option_nat @ ( some_nat @ Y3 ) ) ) ) ) ) ) ) ) ) ) ) ) ).
% cproper_interval_option.pelims(1)
thf(fact_530_cproper__interval__option_Opelims_I1_J,axiom,
! [X: option_option_int,Xa: option_option_int,Y: $o] :
( ( ( collec3052392823440695329on_int @ X @ Xa )
= Y )
=> ( ( accp_P7955026199954541186on_int @ collec5418738772392851069el_int @ ( produc3564219861986917937on_int @ X @ Xa ) )
=> ( ( ( X = none_option_int )
=> ( ( Xa = none_option_int )
=> ( Y
=> ~ ( accp_P7955026199954541186on_int @ collec5418738772392851069el_int @ ( produc3564219861986917937on_int @ none_option_int @ none_option_int ) ) ) ) )
=> ( ( ( X = none_option_int )
=> ! [X3: option_int] :
( ( Xa
= ( some_option_int @ X3 ) )
=> ( ( Y
= ( X3 != none_int ) )
=> ~ ( accp_P7955026199954541186on_int @ collec5418738772392851069el_int @ ( produc3564219861986917937on_int @ none_option_int @ ( some_option_int @ X3 ) ) ) ) ) )
=> ( ! [X3: option_int] :
( ( X
= ( some_option_int @ X3 ) )
=> ( ( Xa = none_option_int )
=> ( ( Y
= ( collec4138408100298950737al_int @ X3 @ none_int ) )
=> ~ ( accp_P7955026199954541186on_int @ collec5418738772392851069el_int @ ( produc3564219861986917937on_int @ ( some_option_int @ X3 ) @ none_option_int ) ) ) ) )
=> ( ! [X3: option_int] :
( ( X
= ( some_option_int @ X3 ) )
=> ( ( Xa
= ( some_option_int @ none_int ) )
=> ( ~ Y
=> ~ ( accp_P7955026199954541186on_int @ collec5418738772392851069el_int @ ( produc3564219861986917937on_int @ ( some_option_int @ X3 ) @ ( some_option_int @ none_int ) ) ) ) ) )
=> ~ ! [X3: option_int] :
( ( X
= ( some_option_int @ X3 ) )
=> ! [Y3: int] :
( ( Xa
= ( some_option_int @ ( some_int @ Y3 ) ) )
=> ( ( Y
= ( collec4138408100298950737al_int @ X3 @ ( some_int @ Y3 ) ) )
=> ~ ( accp_P7955026199954541186on_int @ collec5418738772392851069el_int @ ( produc3564219861986917937on_int @ ( some_option_int @ X3 ) @ ( some_option_int @ ( some_int @ Y3 ) ) ) ) ) ) ) ) ) ) ) ) ) ).
% cproper_interval_option.pelims(1)
thf(fact_531_cproper__interval__option_Opelims_I2_J,axiom,
! [X: option_option_nat,Xa: option_option_nat] :
( ( collec7230243842949892037on_nat @ X @ Xa )
=> ( ( accp_P4406482645238171594on_nat @ collec5421229242901901345el_nat @ ( produc6915583070269132153on_nat @ X @ Xa ) )
=> ( ( ( X = none_option_nat )
=> ( ( Xa = none_option_nat )
=> ~ ( accp_P4406482645238171594on_nat @ collec5421229242901901345el_nat @ ( produc6915583070269132153on_nat @ none_option_nat @ none_option_nat ) ) ) )
=> ( ( ( X = none_option_nat )
=> ! [X3: option_nat] :
( ( Xa
= ( some_option_nat @ X3 ) )
=> ( ( accp_P4406482645238171594on_nat @ collec5421229242901901345el_nat @ ( produc6915583070269132153on_nat @ none_option_nat @ ( some_option_nat @ X3 ) ) )
=> ( X3 = none_nat ) ) ) )
=> ( ! [X3: option_nat] :
( ( X
= ( some_option_nat @ X3 ) )
=> ( ( Xa = none_option_nat )
=> ( ( accp_P4406482645238171594on_nat @ collec5421229242901901345el_nat @ ( produc6915583070269132153on_nat @ ( some_option_nat @ X3 ) @ none_option_nat ) )
=> ~ ( collec4140898570808001013al_nat @ X3 @ none_nat ) ) ) )
=> ~ ! [X3: option_nat] :
( ( X
= ( some_option_nat @ X3 ) )
=> ! [Y3: nat] :
( ( Xa
= ( some_option_nat @ ( some_nat @ Y3 ) ) )
=> ( ( accp_P4406482645238171594on_nat @ collec5421229242901901345el_nat @ ( produc6915583070269132153on_nat @ ( some_option_nat @ X3 ) @ ( some_option_nat @ ( some_nat @ Y3 ) ) ) )
=> ~ ( collec4140898570808001013al_nat @ X3 @ ( some_nat @ Y3 ) ) ) ) ) ) ) ) ) ) ).
% cproper_interval_option.pelims(2)
thf(fact_532_cproper__interval__option_Opelims_I2_J,axiom,
! [X: option_option_int,Xa: option_option_int] :
( ( collec3052392823440695329on_int @ X @ Xa )
=> ( ( accp_P7955026199954541186on_int @ collec5418738772392851069el_int @ ( produc3564219861986917937on_int @ X @ Xa ) )
=> ( ( ( X = none_option_int )
=> ( ( Xa = none_option_int )
=> ~ ( accp_P7955026199954541186on_int @ collec5418738772392851069el_int @ ( produc3564219861986917937on_int @ none_option_int @ none_option_int ) ) ) )
=> ( ( ( X = none_option_int )
=> ! [X3: option_int] :
( ( Xa
= ( some_option_int @ X3 ) )
=> ( ( accp_P7955026199954541186on_int @ collec5418738772392851069el_int @ ( produc3564219861986917937on_int @ none_option_int @ ( some_option_int @ X3 ) ) )
=> ( X3 = none_int ) ) ) )
=> ( ! [X3: option_int] :
( ( X
= ( some_option_int @ X3 ) )
=> ( ( Xa = none_option_int )
=> ( ( accp_P7955026199954541186on_int @ collec5418738772392851069el_int @ ( produc3564219861986917937on_int @ ( some_option_int @ X3 ) @ none_option_int ) )
=> ~ ( collec4138408100298950737al_int @ X3 @ none_int ) ) ) )
=> ~ ! [X3: option_int] :
( ( X
= ( some_option_int @ X3 ) )
=> ! [Y3: int] :
( ( Xa
= ( some_option_int @ ( some_int @ Y3 ) ) )
=> ( ( accp_P7955026199954541186on_int @ collec5418738772392851069el_int @ ( produc3564219861986917937on_int @ ( some_option_int @ X3 ) @ ( some_option_int @ ( some_int @ Y3 ) ) ) )
=> ~ ( collec4138408100298950737al_int @ X3 @ ( some_int @ Y3 ) ) ) ) ) ) ) ) ) ) ).
% cproper_interval_option.pelims(2)
thf(fact_533_cproper__interval__option_Opelims_I3_J,axiom,
! [X: option_option_nat,Xa: option_option_nat] :
( ~ ( collec7230243842949892037on_nat @ X @ Xa )
=> ( ( accp_P4406482645238171594on_nat @ collec5421229242901901345el_nat @ ( produc6915583070269132153on_nat @ X @ Xa ) )
=> ( ( ( X = none_option_nat )
=> ! [X3: option_nat] :
( ( Xa
= ( some_option_nat @ X3 ) )
=> ( ( accp_P4406482645238171594on_nat @ collec5421229242901901345el_nat @ ( produc6915583070269132153on_nat @ none_option_nat @ ( some_option_nat @ X3 ) ) )
=> ( X3 != none_nat ) ) ) )
=> ( ! [X3: option_nat] :
( ( X
= ( some_option_nat @ X3 ) )
=> ( ( Xa = none_option_nat )
=> ( ( accp_P4406482645238171594on_nat @ collec5421229242901901345el_nat @ ( produc6915583070269132153on_nat @ ( some_option_nat @ X3 ) @ none_option_nat ) )
=> ( collec4140898570808001013al_nat @ X3 @ none_nat ) ) ) )
=> ( ! [X3: option_nat] :
( ( X
= ( some_option_nat @ X3 ) )
=> ( ( Xa
= ( some_option_nat @ none_nat ) )
=> ~ ( accp_P4406482645238171594on_nat @ collec5421229242901901345el_nat @ ( produc6915583070269132153on_nat @ ( some_option_nat @ X3 ) @ ( some_option_nat @ none_nat ) ) ) ) )
=> ~ ! [X3: option_nat] :
( ( X
= ( some_option_nat @ X3 ) )
=> ! [Y3: nat] :
( ( Xa
= ( some_option_nat @ ( some_nat @ Y3 ) ) )
=> ( ( accp_P4406482645238171594on_nat @ collec5421229242901901345el_nat @ ( produc6915583070269132153on_nat @ ( some_option_nat @ X3 ) @ ( some_option_nat @ ( some_nat @ Y3 ) ) ) )
=> ( collec4140898570808001013al_nat @ X3 @ ( some_nat @ Y3 ) ) ) ) ) ) ) ) ) ) ).
% cproper_interval_option.pelims(3)
thf(fact_534_cproper__interval__option_Opelims_I3_J,axiom,
! [X: option_option_int,Xa: option_option_int] :
( ~ ( collec3052392823440695329on_int @ X @ Xa )
=> ( ( accp_P7955026199954541186on_int @ collec5418738772392851069el_int @ ( produc3564219861986917937on_int @ X @ Xa ) )
=> ( ( ( X = none_option_int )
=> ! [X3: option_int] :
( ( Xa
= ( some_option_int @ X3 ) )
=> ( ( accp_P7955026199954541186on_int @ collec5418738772392851069el_int @ ( produc3564219861986917937on_int @ none_option_int @ ( some_option_int @ X3 ) ) )
=> ( X3 != none_int ) ) ) )
=> ( ! [X3: option_int] :
( ( X
= ( some_option_int @ X3 ) )
=> ( ( Xa = none_option_int )
=> ( ( accp_P7955026199954541186on_int @ collec5418738772392851069el_int @ ( produc3564219861986917937on_int @ ( some_option_int @ X3 ) @ none_option_int ) )
=> ( collec4138408100298950737al_int @ X3 @ none_int ) ) ) )
=> ( ! [X3: option_int] :
( ( X
= ( some_option_int @ X3 ) )
=> ( ( Xa
= ( some_option_int @ none_int ) )
=> ~ ( accp_P7955026199954541186on_int @ collec5418738772392851069el_int @ ( produc3564219861986917937on_int @ ( some_option_int @ X3 ) @ ( some_option_int @ none_int ) ) ) ) )
=> ~ ! [X3: option_int] :
( ( X
= ( some_option_int @ X3 ) )
=> ! [Y3: int] :
( ( Xa
= ( some_option_int @ ( some_int @ Y3 ) ) )
=> ( ( accp_P7955026199954541186on_int @ collec5418738772392851069el_int @ ( produc3564219861986917937on_int @ ( some_option_int @ X3 ) @ ( some_option_int @ ( some_int @ Y3 ) ) ) )
=> ( collec4138408100298950737al_int @ X3 @ ( some_int @ Y3 ) ) ) ) ) ) ) ) ) ) ).
% cproper_interval_option.pelims(3)
thf(fact_535_in__measure,axiom,
! [X: option_nat,Y: option_nat,F: option_nat > nat] :
( ( member4117937158525611210on_nat @ ( produc5098337634421038937on_nat @ X @ Y ) @ ( measure_option_nat @ F ) )
= ( ord_less_nat @ ( F @ X ) @ ( F @ Y ) ) ) ).
% in_measure
thf(fact_536_in__measure,axiom,
! [X: option_int,Y: option_int,F: option_int > nat] :
( ( member7038936195297346946on_int @ ( produc6331568796615304721on_int @ X @ Y ) @ ( measure_option_int @ F ) )
= ( ord_less_nat @ ( F @ X ) @ ( F @ Y ) ) ) ).
% in_measure
thf(fact_537_relChain__def,axiom,
( bNF_Ca4410430146903927958at_nat
= ( ^ [R: set_Pr6588086440996610945on_nat,As: option_nat > nat] :
! [I4: option_nat,J3: option_nat] :
( ( member4117937158525611210on_nat @ ( produc5098337634421038937on_nat @ I4 @ J3 ) @ R )
=> ( ord_less_eq_nat @ ( As @ I4 ) @ ( As @ J3 ) ) ) ) ) ).
% relChain_def
thf(fact_538_relChain__def,axiom,
( bNF_Ca6597351721596949746nt_nat
= ( ^ [R: set_Pr5725982650624401465on_int,As: option_int > nat] :
! [I4: option_int,J3: option_int] :
( ( member7038936195297346946on_int @ ( produc6331568796615304721on_int @ I4 @ J3 ) @ R )
=> ( ord_less_eq_nat @ ( As @ I4 ) @ ( As @ J3 ) ) ) ) ) ).
% relChain_def
thf(fact_539_relChain__def,axiom,
( bNF_Ca4407939676394877682at_int
= ( ^ [R: set_Pr6588086440996610945on_nat,As: option_nat > int] :
! [I4: option_nat,J3: option_nat] :
( ( member4117937158525611210on_nat @ ( produc5098337634421038937on_nat @ I4 @ J3 ) @ R )
=> ( ord_less_eq_int @ ( As @ I4 ) @ ( As @ J3 ) ) ) ) ) ).
% relChain_def
thf(fact_540_relChain__def,axiom,
( bNF_Ca6594861251087899470nt_int
= ( ^ [R: set_Pr5725982650624401465on_int,As: option_int > int] :
! [I4: option_int,J3: option_int] :
( ( member7038936195297346946on_int @ ( produc6331568796615304721on_int @ I4 @ J3 ) @ R )
=> ( ord_less_eq_int @ ( As @ I4 ) @ ( As @ J3 ) ) ) ) ) ).
% relChain_def
thf(fact_541_relChain__def,axiom,
( bNF_Ca3079242199294856306t_real
= ( ^ [R: set_Pr6588086440996610945on_nat,As: option_nat > real] :
! [I4: option_nat,J3: option_nat] :
( ( member4117937158525611210on_nat @ ( produc5098337634421038937on_nat @ I4 @ J3 ) @ R )
=> ( ord_less_eq_real @ ( As @ I4 ) @ ( As @ J3 ) ) ) ) ) ).
% relChain_def
thf(fact_542_relChain__def,axiom,
( bNF_Ca6297945793459226318t_real
= ( ^ [R: set_Pr5725982650624401465on_int,As: option_int > real] :
! [I4: option_int,J3: option_int] :
( ( member7038936195297346946on_int @ ( produc6331568796615304721on_int @ I4 @ J3 ) @ R )
=> ( ord_less_eq_real @ ( As @ I4 ) @ ( As @ J3 ) ) ) ) ) ).
% relChain_def
thf(fact_543_cproper__interval__sum_Oelims_I3_J,axiom,
! [X: option3242699076814927183at_nat,Xa: option3242699076814927183at_nat] :
( ~ ( collec7691303616722397604at_nat @ X @ Xa )
=> ( ( ( X = none_Sum_sum_nat_nat )
=> ! [X3: nat] :
( ( Xa
= ( some_Sum_sum_nat_nat @ ( sum_Inl_nat_nat @ X3 ) ) )
=> ( collec4140898570808001013al_nat @ none_nat @ ( some_nat @ X3 ) ) ) )
=> ( ! [X3: nat] :
( ( X
= ( some_Sum_sum_nat_nat @ ( sum_Inl_nat_nat @ X3 ) ) )
=> ! [Y3: nat] :
( ( Xa
= ( some_Sum_sum_nat_nat @ ( sum_Inl_nat_nat @ Y3 ) ) )
=> ( collec4140898570808001013al_nat @ ( some_nat @ X3 ) @ ( some_nat @ Y3 ) ) ) )
=> ( ! [X3: nat] :
( ( X
= ( some_Sum_sum_nat_nat @ ( sum_Inl_nat_nat @ X3 ) ) )
=> ! [Y3: nat] :
( ( Xa
= ( some_Sum_sum_nat_nat @ ( sum_Inr_nat_nat @ Y3 ) ) )
=> ( ( collec4140898570808001013al_nat @ ( some_nat @ X3 ) @ none_nat )
| ( collec4140898570808001013al_nat @ none_nat @ ( some_nat @ Y3 ) ) ) ) )
=> ( ! [Y3: nat] :
( ( X
= ( some_Sum_sum_nat_nat @ ( sum_Inr_nat_nat @ Y3 ) ) )
=> ( ( Xa = none_Sum_sum_nat_nat )
=> ( collec4140898570808001013al_nat @ ( some_nat @ Y3 ) @ none_nat ) ) )
=> ( ( ? [Y3: nat] :
( X
= ( some_Sum_sum_nat_nat @ ( sum_Inr_nat_nat @ Y3 ) ) )
=> ! [X3: nat] :
( Xa
!= ( some_Sum_sum_nat_nat @ ( sum_Inl_nat_nat @ X3 ) ) ) )
=> ~ ! [X3: nat] :
( ( X
= ( some_Sum_sum_nat_nat @ ( sum_Inr_nat_nat @ X3 ) ) )
=> ! [Y3: nat] :
( ( Xa
= ( some_Sum_sum_nat_nat @ ( sum_Inr_nat_nat @ Y3 ) ) )
=> ( collec4140898570808001013al_nat @ ( some_nat @ X3 ) @ ( some_nat @ Y3 ) ) ) ) ) ) ) ) ) ) ).
% cproper_interval_sum.elims(3)
thf(fact_544_cproper__interval__sum_Oelims_I3_J,axiom,
! [X: option752615931883200427at_int,Xa: option752615931883200427at_int] :
( ~ ( collec3513452597213200896at_int @ X @ Xa )
=> ( ( ( X = none_Sum_sum_nat_int )
=> ! [X3: nat] :
( ( Xa
= ( some_Sum_sum_nat_int @ ( sum_Inl_nat_int @ X3 ) ) )
=> ( collec4140898570808001013al_nat @ none_nat @ ( some_nat @ X3 ) ) ) )
=> ( ! [X3: nat] :
( ( X
= ( some_Sum_sum_nat_int @ ( sum_Inl_nat_int @ X3 ) ) )
=> ! [Y3: nat] :
( ( Xa
= ( some_Sum_sum_nat_int @ ( sum_Inl_nat_int @ Y3 ) ) )
=> ( collec4140898570808001013al_nat @ ( some_nat @ X3 ) @ ( some_nat @ Y3 ) ) ) )
=> ( ! [X3: nat] :
( ( X
= ( some_Sum_sum_nat_int @ ( sum_Inl_nat_int @ X3 ) ) )
=> ! [Y3: int] :
( ( Xa
= ( some_Sum_sum_nat_int @ ( sum_Inr_int_nat @ Y3 ) ) )
=> ( ( collec4140898570808001013al_nat @ ( some_nat @ X3 ) @ none_nat )
| ( collec4138408100298950737al_int @ none_int @ ( some_int @ Y3 ) ) ) ) )
=> ( ! [Y3: int] :
( ( X
= ( some_Sum_sum_nat_int @ ( sum_Inr_int_nat @ Y3 ) ) )
=> ( ( Xa = none_Sum_sum_nat_int )
=> ( collec4138408100298950737al_int @ ( some_int @ Y3 ) @ none_int ) ) )
=> ( ( ? [Y3: int] :
( X
= ( some_Sum_sum_nat_int @ ( sum_Inr_int_nat @ Y3 ) ) )
=> ! [X3: nat] :
( Xa
!= ( some_Sum_sum_nat_int @ ( sum_Inl_nat_int @ X3 ) ) ) )
=> ~ ! [X3: int] :
( ( X
= ( some_Sum_sum_nat_int @ ( sum_Inr_int_nat @ X3 ) ) )
=> ! [Y3: int] :
( ( Xa
= ( some_Sum_sum_nat_int @ ( sum_Inr_int_nat @ Y3 ) ) )
=> ( collec4138408100298950737al_int @ ( some_int @ X3 ) @ ( some_int @ Y3 ) ) ) ) ) ) ) ) ) ) ).
% cproper_interval_sum.elims(3)
thf(fact_545_cproper__interval__sum_Oelims_I3_J,axiom,
! [X: option5429620651507948971nt_nat,Xa: option5429620651507948971nt_nat] :
( ~ ( collec8690657328623452160nt_nat @ X @ Xa )
=> ( ( ( X = none_Sum_sum_int_nat )
=> ! [X3: int] :
( ( Xa
= ( some_Sum_sum_int_nat @ ( sum_Inl_int_nat @ X3 ) ) )
=> ( collec4138408100298950737al_int @ none_int @ ( some_int @ X3 ) ) ) )
=> ( ! [X3: int] :
( ( X
= ( some_Sum_sum_int_nat @ ( sum_Inl_int_nat @ X3 ) ) )
=> ! [Y3: int] :
( ( Xa
= ( some_Sum_sum_int_nat @ ( sum_Inl_int_nat @ Y3 ) ) )
=> ( collec4138408100298950737al_int @ ( some_int @ X3 ) @ ( some_int @ Y3 ) ) ) )
=> ( ! [X3: int] :
( ( X
= ( some_Sum_sum_int_nat @ ( sum_Inl_int_nat @ X3 ) ) )
=> ! [Y3: nat] :
( ( Xa
= ( some_Sum_sum_int_nat @ ( sum_Inr_nat_int @ Y3 ) ) )
=> ( ( collec4138408100298950737al_int @ ( some_int @ X3 ) @ none_int )
| ( collec4140898570808001013al_nat @ none_nat @ ( some_nat @ Y3 ) ) ) ) )
=> ( ! [Y3: nat] :
( ( X
= ( some_Sum_sum_int_nat @ ( sum_Inr_nat_int @ Y3 ) ) )
=> ( ( Xa = none_Sum_sum_int_nat )
=> ( collec4140898570808001013al_nat @ ( some_nat @ Y3 ) @ none_nat ) ) )
=> ( ( ? [Y3: nat] :
( X
= ( some_Sum_sum_int_nat @ ( sum_Inr_nat_int @ Y3 ) ) )
=> ! [X3: int] :
( Xa
!= ( some_Sum_sum_int_nat @ ( sum_Inl_int_nat @ X3 ) ) ) )
=> ~ ! [X3: nat] :
( ( X
= ( some_Sum_sum_int_nat @ ( sum_Inr_nat_int @ X3 ) ) )
=> ! [Y3: nat] :
( ( Xa
= ( some_Sum_sum_int_nat @ ( sum_Inr_nat_int @ Y3 ) ) )
=> ( collec4140898570808001013al_nat @ ( some_nat @ X3 ) @ ( some_nat @ Y3 ) ) ) ) ) ) ) ) ) ) ).
% cproper_interval_sum.elims(3)
thf(fact_546_cproper__interval__sum_Oelims_I3_J,axiom,
! [X: option2939537506576222215nt_int,Xa: option2939537506576222215nt_int] :
( ~ ( collec4512806309114255452nt_int @ X @ Xa )
=> ( ( ( X = none_Sum_sum_int_int )
=> ! [X3: int] :
( ( Xa
= ( some_Sum_sum_int_int @ ( sum_Inl_int_int @ X3 ) ) )
=> ( collec4138408100298950737al_int @ none_int @ ( some_int @ X3 ) ) ) )
=> ( ! [X3: int] :
( ( X
= ( some_Sum_sum_int_int @ ( sum_Inl_int_int @ X3 ) ) )
=> ! [Y3: int] :
( ( Xa
= ( some_Sum_sum_int_int @ ( sum_Inl_int_int @ Y3 ) ) )
=> ( collec4138408100298950737al_int @ ( some_int @ X3 ) @ ( some_int @ Y3 ) ) ) )
=> ( ! [X3: int] :
( ( X
= ( some_Sum_sum_int_int @ ( sum_Inl_int_int @ X3 ) ) )
=> ! [Y3: int] :
( ( Xa
= ( some_Sum_sum_int_int @ ( sum_Inr_int_int @ Y3 ) ) )
=> ( ( collec4138408100298950737al_int @ ( some_int @ X3 ) @ none_int )
| ( collec4138408100298950737al_int @ none_int @ ( some_int @ Y3 ) ) ) ) )
=> ( ! [Y3: int] :
( ( X
= ( some_Sum_sum_int_int @ ( sum_Inr_int_int @ Y3 ) ) )
=> ( ( Xa = none_Sum_sum_int_int )
=> ( collec4138408100298950737al_int @ ( some_int @ Y3 ) @ none_int ) ) )
=> ( ( ? [Y3: int] :
( X
= ( some_Sum_sum_int_int @ ( sum_Inr_int_int @ Y3 ) ) )
=> ! [X3: int] :
( Xa
!= ( some_Sum_sum_int_int @ ( sum_Inl_int_int @ X3 ) ) ) )
=> ~ ! [X3: int] :
( ( X
= ( some_Sum_sum_int_int @ ( sum_Inr_int_int @ X3 ) ) )
=> ! [Y3: int] :
( ( Xa
= ( some_Sum_sum_int_int @ ( sum_Inr_int_int @ Y3 ) ) )
=> ( collec4138408100298950737al_int @ ( some_int @ X3 ) @ ( some_int @ Y3 ) ) ) ) ) ) ) ) ) ) ).
% cproper_interval_sum.elims(3)
thf(fact_547_cproper__interval__nat__def,axiom,
collec4140898570808001013al_nat = set_pr3848668467136637714al_nat ).
% cproper_interval_nat_def
thf(fact_548_accp__subset__induct,axiom,
! [D2: produc4953844613479565601on_nat > $o,R2: produc4953844613479565601on_nat > produc4953844613479565601on_nat > $o,X: produc4953844613479565601on_nat,P: produc4953844613479565601on_nat > $o] :
( ( ord_le8126618931240741628_nat_o @ D2 @ ( accp_P8646395344606611882on_nat @ R2 ) )
=> ( ! [X3: produc4953844613479565601on_nat,Z3: produc4953844613479565601on_nat] :
( ( D2 @ X3 )
=> ( ( R2 @ Z3 @ X3 )
=> ( D2 @ Z3 ) ) )
=> ( ( D2 @ X )
=> ( ! [X3: produc4953844613479565601on_nat] :
( ( D2 @ X3 )
=> ( ! [Z4: produc4953844613479565601on_nat] :
( ( R2 @ Z4 @ X3 )
=> ( P @ Z4 ) )
=> ( P @ X3 ) ) )
=> ( P @ X ) ) ) ) ) ).
% accp_subset_induct
thf(fact_549_accp__subset__induct,axiom,
! [D2: produc7874843650251301337on_int > $o,R2: produc7874843650251301337on_int > produc7874843650251301337on_int > $o,X: produc7874843650251301337on_int,P: produc7874843650251301337on_int > $o] :
( ( ord_le6771420595097293892_int_o @ D2 @ ( accp_P2344022344523571810on_int @ R2 ) )
=> ( ! [X3: produc7874843650251301337on_int,Z3: produc7874843650251301337on_int] :
( ( D2 @ X3 )
=> ( ( R2 @ Z3 @ X3 )
=> ( D2 @ Z3 ) ) )
=> ( ( D2 @ X )
=> ( ! [X3: produc7874843650251301337on_int] :
( ( D2 @ X3 )
=> ( ! [Z4: produc7874843650251301337on_int] :
( ( R2 @ Z4 @ X3 )
=> ( P @ Z4 ) )
=> ( P @ X3 ) ) )
=> ( P @ X ) ) ) ) ) ).
% accp_subset_induct
thf(fact_550_accp__subset,axiom,
! [R1: produc4953844613479565601on_nat > produc4953844613479565601on_nat > $o,R22: produc4953844613479565601on_nat > produc4953844613479565601on_nat > $o] :
( ( ord_le7862513914298786254_nat_o @ R1 @ R22 )
=> ( ord_le8126618931240741628_nat_o @ ( accp_P8646395344606611882on_nat @ R22 ) @ ( accp_P8646395344606611882on_nat @ R1 ) ) ) ).
% accp_subset
thf(fact_551_accp__subset,axiom,
! [R1: produc7874843650251301337on_int > produc7874843650251301337on_int > $o,R22: produc7874843650251301337on_int > produc7874843650251301337on_int > $o] :
( ( ord_le3805017572999309262_int_o @ R1 @ R22 )
=> ( ord_le6771420595097293892_int_o @ ( accp_P2344022344523571810on_int @ R22 ) @ ( accp_P2344022344523571810on_int @ R1 ) ) ) ).
% accp_subset
thf(fact_552_accp_Ocases,axiom,
! [R3: produc4953844613479565601on_nat > produc4953844613479565601on_nat > $o,A: produc4953844613479565601on_nat] :
( ( accp_P8646395344606611882on_nat @ R3 @ A )
=> ! [Y6: produc4953844613479565601on_nat] :
( ( R3 @ Y6 @ A )
=> ( accp_P8646395344606611882on_nat @ R3 @ Y6 ) ) ) ).
% accp.cases
thf(fact_553_accp_Ocases,axiom,
! [R3: produc7874843650251301337on_int > produc7874843650251301337on_int > $o,A: produc7874843650251301337on_int] :
( ( accp_P2344022344523571810on_int @ R3 @ A )
=> ! [Y6: produc7874843650251301337on_int] :
( ( R3 @ Y6 @ A )
=> ( accp_P2344022344523571810on_int @ R3 @ Y6 ) ) ) ).
% accp.cases
thf(fact_554_accp_Osimps,axiom,
( accp_P8646395344606611882on_nat
= ( ^ [R: produc4953844613479565601on_nat > produc4953844613479565601on_nat > $o,A2: produc4953844613479565601on_nat] :
? [X4: produc4953844613479565601on_nat] :
( ( A2 = X4 )
& ! [Y5: produc4953844613479565601on_nat] :
( ( R @ Y5 @ X4 )
=> ( accp_P8646395344606611882on_nat @ R @ Y5 ) ) ) ) ) ).
% accp.simps
thf(fact_555_accp_Osimps,axiom,
( accp_P2344022344523571810on_int
= ( ^ [R: produc7874843650251301337on_int > produc7874843650251301337on_int > $o,A2: produc7874843650251301337on_int] :
? [X4: produc7874843650251301337on_int] :
( ( A2 = X4 )
& ! [Y5: produc7874843650251301337on_int] :
( ( R @ Y5 @ X4 )
=> ( accp_P2344022344523571810on_int @ R @ Y5 ) ) ) ) ) ).
% accp.simps
thf(fact_556_accpI,axiom,
! [R3: produc4953844613479565601on_nat > produc4953844613479565601on_nat > $o,X: produc4953844613479565601on_nat] :
( ! [Y3: produc4953844613479565601on_nat] :
( ( R3 @ Y3 @ X )
=> ( accp_P8646395344606611882on_nat @ R3 @ Y3 ) )
=> ( accp_P8646395344606611882on_nat @ R3 @ X ) ) ).
% accpI
thf(fact_557_accpI,axiom,
! [R3: produc7874843650251301337on_int > produc7874843650251301337on_int > $o,X: produc7874843650251301337on_int] :
( ! [Y3: produc7874843650251301337on_int] :
( ( R3 @ Y3 @ X )
=> ( accp_P2344022344523571810on_int @ R3 @ Y3 ) )
=> ( accp_P2344022344523571810on_int @ R3 @ X ) ) ).
% accpI
thf(fact_558_accp__induct,axiom,
! [R3: produc4953844613479565601on_nat > produc4953844613479565601on_nat > $o,A: produc4953844613479565601on_nat,P: produc4953844613479565601on_nat > $o] :
( ( accp_P8646395344606611882on_nat @ R3 @ A )
=> ( ! [X3: produc4953844613479565601on_nat] :
( ( accp_P8646395344606611882on_nat @ R3 @ X3 )
=> ( ! [Y6: produc4953844613479565601on_nat] :
( ( R3 @ Y6 @ X3 )
=> ( P @ Y6 ) )
=> ( P @ X3 ) ) )
=> ( P @ A ) ) ) ).
% accp_induct
thf(fact_559_accp__induct,axiom,
! [R3: produc7874843650251301337on_int > produc7874843650251301337on_int > $o,A: produc7874843650251301337on_int,P: produc7874843650251301337on_int > $o] :
( ( accp_P2344022344523571810on_int @ R3 @ A )
=> ( ! [X3: produc7874843650251301337on_int] :
( ( accp_P2344022344523571810on_int @ R3 @ X3 )
=> ( ! [Y6: produc7874843650251301337on_int] :
( ( R3 @ Y6 @ X3 )
=> ( P @ Y6 ) )
=> ( P @ X3 ) ) )
=> ( P @ A ) ) ) ).
% accp_induct
thf(fact_560_accp__downward,axiom,
! [R3: produc4953844613479565601on_nat > produc4953844613479565601on_nat > $o,B: produc4953844613479565601on_nat,A: produc4953844613479565601on_nat] :
( ( accp_P8646395344606611882on_nat @ R3 @ B )
=> ( ( R3 @ A @ B )
=> ( accp_P8646395344606611882on_nat @ R3 @ A ) ) ) ).
% accp_downward
thf(fact_561_accp__downward,axiom,
! [R3: produc7874843650251301337on_int > produc7874843650251301337on_int > $o,B: produc7874843650251301337on_int,A: produc7874843650251301337on_int] :
( ( accp_P2344022344523571810on_int @ R3 @ B )
=> ( ( R3 @ A @ B )
=> ( accp_P2344022344523571810on_int @ R3 @ A ) ) ) ).
% accp_downward
thf(fact_562_not__accp__down,axiom,
! [R2: produc4953844613479565601on_nat > produc4953844613479565601on_nat > $o,X: produc4953844613479565601on_nat] :
( ~ ( accp_P8646395344606611882on_nat @ R2 @ X )
=> ~ ! [Z3: produc4953844613479565601on_nat] :
( ( R2 @ Z3 @ X )
=> ( accp_P8646395344606611882on_nat @ R2 @ Z3 ) ) ) ).
% not_accp_down
thf(fact_563_not__accp__down,axiom,
! [R2: produc7874843650251301337on_int > produc7874843650251301337on_int > $o,X: produc7874843650251301337on_int] :
( ~ ( accp_P2344022344523571810on_int @ R2 @ X )
=> ~ ! [Z3: produc7874843650251301337on_int] :
( ( R2 @ Z3 @ X )
=> ( accp_P2344022344523571810on_int @ R2 @ Z3 ) ) ) ).
% not_accp_down
thf(fact_564_accp__induct__rule,axiom,
! [R3: produc4953844613479565601on_nat > produc4953844613479565601on_nat > $o,A: produc4953844613479565601on_nat,P: produc4953844613479565601on_nat > $o] :
( ( accp_P8646395344606611882on_nat @ R3 @ A )
=> ( ! [X3: produc4953844613479565601on_nat] :
( ( accp_P8646395344606611882on_nat @ R3 @ X3 )
=> ( ! [Y6: produc4953844613479565601on_nat] :
( ( R3 @ Y6 @ X3 )
=> ( P @ Y6 ) )
=> ( P @ X3 ) ) )
=> ( P @ A ) ) ) ).
% accp_induct_rule
thf(fact_565_accp__induct__rule,axiom,
! [R3: produc7874843650251301337on_int > produc7874843650251301337on_int > $o,A: produc7874843650251301337on_int,P: produc7874843650251301337on_int > $o] :
( ( accp_P2344022344523571810on_int @ R3 @ A )
=> ( ! [X3: produc7874843650251301337on_int] :
( ( accp_P2344022344523571810on_int @ R3 @ X3 )
=> ( ! [Y6: produc7874843650251301337on_int] :
( ( R3 @ Y6 @ X3 )
=> ( P @ Y6 ) )
=> ( P @ X3 ) ) )
=> ( P @ A ) ) ) ).
% accp_induct_rule
thf(fact_566_obj__sumE,axiom,
! [S: sum_sum_a_nat] :
( ! [X3: a] :
( S
!= ( sum_Inl_a_nat @ X3 ) )
=> ~ ! [X3: nat] :
( S
!= ( sum_Inr_nat_a @ X3 ) ) ) ).
% obj_sumE
thf(fact_567_split__sum__all,axiom,
( ( ^ [P2: sum_sum_a_nat > $o] :
! [X5: sum_sum_a_nat] : ( P2 @ X5 ) )
= ( ^ [P3: sum_sum_a_nat > $o] :
( ! [X4: a] : ( P3 @ ( sum_Inl_a_nat @ X4 ) )
& ! [X4: nat] : ( P3 @ ( sum_Inr_nat_a @ X4 ) ) ) ) ) ).
% split_sum_all
thf(fact_568_split__sum__ex,axiom,
( ( ^ [P2: sum_sum_a_nat > $o] :
? [X5: sum_sum_a_nat] : ( P2 @ X5 ) )
= ( ^ [P3: sum_sum_a_nat > $o] :
( ? [X4: a] : ( P3 @ ( sum_Inl_a_nat @ X4 ) )
| ? [X4: nat] : ( P3 @ ( sum_Inr_nat_a @ X4 ) ) ) ) ) ).
% split_sum_ex
thf(fact_569_Inr__not__Inl,axiom,
! [B: nat,A: a] :
( ( sum_Inr_nat_a @ B )
!= ( sum_Inl_a_nat @ A ) ) ).
% Inr_not_Inl
thf(fact_570_sumE,axiom,
! [S: sum_sum_a_nat] :
( ! [X3: a] :
( S
!= ( sum_Inl_a_nat @ X3 ) )
=> ~ ! [Y3: nat] :
( S
!= ( sum_Inr_nat_a @ Y3 ) ) ) ).
% sumE
thf(fact_571_old_Osum_Oexhaust,axiom,
! [Y: sum_sum_a_nat] :
( ! [A3: a] :
( Y
!= ( sum_Inl_a_nat @ A3 ) )
=> ~ ! [B4: nat] :
( Y
!= ( sum_Inr_nat_a @ B4 ) ) ) ).
% old.sum.exhaust
thf(fact_572_old_Osum_Odistinct_I1_J,axiom,
! [A: a,B2: nat] :
( ( sum_Inl_a_nat @ A )
!= ( sum_Inr_nat_a @ B2 ) ) ).
% old.sum.distinct(1)
thf(fact_573_old_Osum_Odistinct_I2_J,axiom,
! [B2: nat,A: a] :
( ( sum_Inr_nat_a @ B2 )
!= ( sum_Inl_a_nat @ A ) ) ).
% old.sum.distinct(2)
thf(fact_574_sum_Odistinct_I1_J,axiom,
! [X1: a,X2: nat] :
( ( sum_Inl_a_nat @ X1 )
!= ( sum_Inr_nat_a @ X2 ) ) ).
% sum.distinct(1)
thf(fact_575_proper__interval__nat_Ocases,axiom,
! [X: produc4953844613479565601on_nat] :
( ! [No2: option_nat] :
( X
!= ( produc5098337634421038937on_nat @ No2 @ none_nat ) )
=> ( ! [X3: nat] :
( X
!= ( produc5098337634421038937on_nat @ none_nat @ ( some_nat @ X3 ) ) )
=> ~ ! [X3: nat,Y3: nat] :
( X
!= ( produc5098337634421038937on_nat @ ( some_nat @ X3 ) @ ( some_nat @ Y3 ) ) ) ) ) ).
% proper_interval_nat.cases
thf(fact_576_old_Oprod_Oexhaust,axiom,
! [Y: produc4953844613479565601on_nat] :
~ ! [A3: option_nat,B4: option_nat] :
( Y
!= ( produc5098337634421038937on_nat @ A3 @ B4 ) ) ).
% old.prod.exhaust
thf(fact_577_old_Oprod_Oexhaust,axiom,
! [Y: produc7874843650251301337on_int] :
~ ! [A3: option_int,B4: option_int] :
( Y
!= ( produc6331568796615304721on_int @ A3 @ B4 ) ) ).
% old.prod.exhaust
thf(fact_578_surj__pair,axiom,
! [P5: produc4953844613479565601on_nat] :
? [X3: option_nat,Y3: option_nat] :
( P5
= ( produc5098337634421038937on_nat @ X3 @ Y3 ) ) ).
% surj_pair
thf(fact_579_surj__pair,axiom,
! [P5: produc7874843650251301337on_int] :
? [X3: option_int,Y3: option_int] :
( P5
= ( produc6331568796615304721on_int @ X3 @ Y3 ) ) ).
% surj_pair
thf(fact_580_prod__cases,axiom,
! [P: produc4953844613479565601on_nat > $o,P5: produc4953844613479565601on_nat] :
( ! [A3: option_nat,B4: option_nat] : ( P @ ( produc5098337634421038937on_nat @ A3 @ B4 ) )
=> ( P @ P5 ) ) ).
% prod_cases
thf(fact_581_prod__cases,axiom,
! [P: produc7874843650251301337on_int > $o,P5: produc7874843650251301337on_int] :
( ! [A3: option_int,B4: option_int] : ( P @ ( produc6331568796615304721on_int @ A3 @ B4 ) )
=> ( P @ P5 ) ) ).
% prod_cases
thf(fact_582_Pair__inject,axiom,
! [A: option_nat,B: option_nat,A5: option_nat,B2: option_nat] :
( ( ( produc5098337634421038937on_nat @ A @ B )
= ( produc5098337634421038937on_nat @ A5 @ B2 ) )
=> ~ ( ( A = A5 )
=> ( B != B2 ) ) ) ).
% Pair_inject
thf(fact_583_Pair__inject,axiom,
! [A: option_int,B: option_int,A5: option_int,B2: option_int] :
( ( ( produc6331568796615304721on_int @ A @ B )
= ( produc6331568796615304721on_int @ A5 @ B2 ) )
=> ~ ( ( A = A5 )
=> ( B != B2 ) ) ) ).
% Pair_inject
thf(fact_584_cproper__interval__sum_Osimps_I6_J,axiom,
! [X: nat,Y: nat] :
( ( collec7691303616722397604at_nat @ ( some_Sum_sum_nat_nat @ ( sum_Inl_nat_nat @ X ) ) @ ( some_Sum_sum_nat_nat @ ( sum_Inr_nat_nat @ Y ) ) )
= ( ( collec4140898570808001013al_nat @ ( some_nat @ X ) @ none_nat )
| ( collec4140898570808001013al_nat @ none_nat @ ( some_nat @ Y ) ) ) ) ).
% cproper_interval_sum.simps(6)
thf(fact_585_cproper__interval__sum_Osimps_I6_J,axiom,
! [X: nat,Y: int] :
( ( collec3513452597213200896at_int @ ( some_Sum_sum_nat_int @ ( sum_Inl_nat_int @ X ) ) @ ( some_Sum_sum_nat_int @ ( sum_Inr_int_nat @ Y ) ) )
= ( ( collec4140898570808001013al_nat @ ( some_nat @ X ) @ none_nat )
| ( collec4138408100298950737al_int @ none_int @ ( some_int @ Y ) ) ) ) ).
% cproper_interval_sum.simps(6)
thf(fact_586_cproper__interval__sum_Osimps_I6_J,axiom,
! [X: int,Y: nat] :
( ( collec8690657328623452160nt_nat @ ( some_Sum_sum_int_nat @ ( sum_Inl_int_nat @ X ) ) @ ( some_Sum_sum_int_nat @ ( sum_Inr_nat_int @ Y ) ) )
= ( ( collec4138408100298950737al_int @ ( some_int @ X ) @ none_int )
| ( collec4140898570808001013al_nat @ none_nat @ ( some_nat @ Y ) ) ) ) ).
% cproper_interval_sum.simps(6)
thf(fact_587_cproper__interval__sum_Osimps_I6_J,axiom,
! [X: int,Y: int] :
( ( collec4512806309114255452nt_int @ ( some_Sum_sum_int_int @ ( sum_Inl_int_int @ X ) ) @ ( some_Sum_sum_int_int @ ( sum_Inr_int_int @ Y ) ) )
= ( ( collec4138408100298950737al_int @ ( some_int @ X ) @ none_int )
| ( collec4138408100298950737al_int @ none_int @ ( some_int @ Y ) ) ) ) ).
% cproper_interval_sum.simps(6)
thf(fact_588_cproper__interval__sum_Oelims_I1_J,axiom,
! [X: option3242699076814927183at_nat,Xa: option3242699076814927183at_nat,Y: $o] :
( ( ( collec7691303616722397604at_nat @ X @ Xa )
= Y )
=> ( ( ( X = none_Sum_sum_nat_nat )
=> ( ( Xa = none_Sum_sum_nat_nat )
=> ~ Y ) )
=> ( ( ( X = none_Sum_sum_nat_nat )
=> ! [X3: nat] :
( ( Xa
= ( some_Sum_sum_nat_nat @ ( sum_Inl_nat_nat @ X3 ) ) )
=> ( Y
= ( ~ ( collec4140898570808001013al_nat @ none_nat @ ( some_nat @ X3 ) ) ) ) ) )
=> ( ( ( X = none_Sum_sum_nat_nat )
=> ( ? [Y3: nat] :
( Xa
= ( some_Sum_sum_nat_nat @ ( sum_Inr_nat_nat @ Y3 ) ) )
=> ~ Y ) )
=> ( ( ? [X3: nat] :
( X
= ( some_Sum_sum_nat_nat @ ( sum_Inl_nat_nat @ X3 ) ) )
=> ( ( Xa = none_Sum_sum_nat_nat )
=> ~ Y ) )
=> ( ! [X3: nat] :
( ( X
= ( some_Sum_sum_nat_nat @ ( sum_Inl_nat_nat @ X3 ) ) )
=> ! [Y3: nat] :
( ( Xa
= ( some_Sum_sum_nat_nat @ ( sum_Inl_nat_nat @ Y3 ) ) )
=> ( Y
= ( ~ ( collec4140898570808001013al_nat @ ( some_nat @ X3 ) @ ( some_nat @ Y3 ) ) ) ) ) )
=> ( ! [X3: nat] :
( ( X
= ( some_Sum_sum_nat_nat @ ( sum_Inl_nat_nat @ X3 ) ) )
=> ! [Y3: nat] :
( ( Xa
= ( some_Sum_sum_nat_nat @ ( sum_Inr_nat_nat @ Y3 ) ) )
=> ( Y
= ( ~ ( ( collec4140898570808001013al_nat @ ( some_nat @ X3 ) @ none_nat )
| ( collec4140898570808001013al_nat @ none_nat @ ( some_nat @ Y3 ) ) ) ) ) ) )
=> ( ! [Y3: nat] :
( ( X
= ( some_Sum_sum_nat_nat @ ( sum_Inr_nat_nat @ Y3 ) ) )
=> ( ( Xa = none_Sum_sum_nat_nat )
=> ( Y
= ( ~ ( collec4140898570808001013al_nat @ ( some_nat @ Y3 ) @ none_nat ) ) ) ) )
=> ( ( ? [Y3: nat] :
( X
= ( some_Sum_sum_nat_nat @ ( sum_Inr_nat_nat @ Y3 ) ) )
=> ( ? [X3: nat] :
( Xa
= ( some_Sum_sum_nat_nat @ ( sum_Inl_nat_nat @ X3 ) ) )
=> Y ) )
=> ~ ! [X3: nat] :
( ( X
= ( some_Sum_sum_nat_nat @ ( sum_Inr_nat_nat @ X3 ) ) )
=> ! [Y3: nat] :
( ( Xa
= ( some_Sum_sum_nat_nat @ ( sum_Inr_nat_nat @ Y3 ) ) )
=> ( Y
= ( ~ ( collec4140898570808001013al_nat @ ( some_nat @ X3 ) @ ( some_nat @ Y3 ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ).
% cproper_interval_sum.elims(1)
thf(fact_589_cproper__interval__sum_Oelims_I1_J,axiom,
! [X: option752615931883200427at_int,Xa: option752615931883200427at_int,Y: $o] :
( ( ( collec3513452597213200896at_int @ X @ Xa )
= Y )
=> ( ( ( X = none_Sum_sum_nat_int )
=> ( ( Xa = none_Sum_sum_nat_int )
=> ~ Y ) )
=> ( ( ( X = none_Sum_sum_nat_int )
=> ! [X3: nat] :
( ( Xa
= ( some_Sum_sum_nat_int @ ( sum_Inl_nat_int @ X3 ) ) )
=> ( Y
= ( ~ ( collec4140898570808001013al_nat @ none_nat @ ( some_nat @ X3 ) ) ) ) ) )
=> ( ( ( X = none_Sum_sum_nat_int )
=> ( ? [Y3: int] :
( Xa
= ( some_Sum_sum_nat_int @ ( sum_Inr_int_nat @ Y3 ) ) )
=> ~ Y ) )
=> ( ( ? [X3: nat] :
( X
= ( some_Sum_sum_nat_int @ ( sum_Inl_nat_int @ X3 ) ) )
=> ( ( Xa = none_Sum_sum_nat_int )
=> ~ Y ) )
=> ( ! [X3: nat] :
( ( X
= ( some_Sum_sum_nat_int @ ( sum_Inl_nat_int @ X3 ) ) )
=> ! [Y3: nat] :
( ( Xa
= ( some_Sum_sum_nat_int @ ( sum_Inl_nat_int @ Y3 ) ) )
=> ( Y
= ( ~ ( collec4140898570808001013al_nat @ ( some_nat @ X3 ) @ ( some_nat @ Y3 ) ) ) ) ) )
=> ( ! [X3: nat] :
( ( X
= ( some_Sum_sum_nat_int @ ( sum_Inl_nat_int @ X3 ) ) )
=> ! [Y3: int] :
( ( Xa
= ( some_Sum_sum_nat_int @ ( sum_Inr_int_nat @ Y3 ) ) )
=> ( Y
= ( ~ ( ( collec4140898570808001013al_nat @ ( some_nat @ X3 ) @ none_nat )
| ( collec4138408100298950737al_int @ none_int @ ( some_int @ Y3 ) ) ) ) ) ) )
=> ( ! [Y3: int] :
( ( X
= ( some_Sum_sum_nat_int @ ( sum_Inr_int_nat @ Y3 ) ) )
=> ( ( Xa = none_Sum_sum_nat_int )
=> ( Y
= ( ~ ( collec4138408100298950737al_int @ ( some_int @ Y3 ) @ none_int ) ) ) ) )
=> ( ( ? [Y3: int] :
( X
= ( some_Sum_sum_nat_int @ ( sum_Inr_int_nat @ Y3 ) ) )
=> ( ? [X3: nat] :
( Xa
= ( some_Sum_sum_nat_int @ ( sum_Inl_nat_int @ X3 ) ) )
=> Y ) )
=> ~ ! [X3: int] :
( ( X
= ( some_Sum_sum_nat_int @ ( sum_Inr_int_nat @ X3 ) ) )
=> ! [Y3: int] :
( ( Xa
= ( some_Sum_sum_nat_int @ ( sum_Inr_int_nat @ Y3 ) ) )
=> ( Y
= ( ~ ( collec4138408100298950737al_int @ ( some_int @ X3 ) @ ( some_int @ Y3 ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ).
% cproper_interval_sum.elims(1)
thf(fact_590_cproper__interval__sum_Oelims_I1_J,axiom,
! [X: option5429620651507948971nt_nat,Xa: option5429620651507948971nt_nat,Y: $o] :
( ( ( collec8690657328623452160nt_nat @ X @ Xa )
= Y )
=> ( ( ( X = none_Sum_sum_int_nat )
=> ( ( Xa = none_Sum_sum_int_nat )
=> ~ Y ) )
=> ( ( ( X = none_Sum_sum_int_nat )
=> ! [X3: int] :
( ( Xa
= ( some_Sum_sum_int_nat @ ( sum_Inl_int_nat @ X3 ) ) )
=> ( Y
= ( ~ ( collec4138408100298950737al_int @ none_int @ ( some_int @ X3 ) ) ) ) ) )
=> ( ( ( X = none_Sum_sum_int_nat )
=> ( ? [Y3: nat] :
( Xa
= ( some_Sum_sum_int_nat @ ( sum_Inr_nat_int @ Y3 ) ) )
=> ~ Y ) )
=> ( ( ? [X3: int] :
( X
= ( some_Sum_sum_int_nat @ ( sum_Inl_int_nat @ X3 ) ) )
=> ( ( Xa = none_Sum_sum_int_nat )
=> ~ Y ) )
=> ( ! [X3: int] :
( ( X
= ( some_Sum_sum_int_nat @ ( sum_Inl_int_nat @ X3 ) ) )
=> ! [Y3: int] :
( ( Xa
= ( some_Sum_sum_int_nat @ ( sum_Inl_int_nat @ Y3 ) ) )
=> ( Y
= ( ~ ( collec4138408100298950737al_int @ ( some_int @ X3 ) @ ( some_int @ Y3 ) ) ) ) ) )
=> ( ! [X3: int] :
( ( X
= ( some_Sum_sum_int_nat @ ( sum_Inl_int_nat @ X3 ) ) )
=> ! [Y3: nat] :
( ( Xa
= ( some_Sum_sum_int_nat @ ( sum_Inr_nat_int @ Y3 ) ) )
=> ( Y
= ( ~ ( ( collec4138408100298950737al_int @ ( some_int @ X3 ) @ none_int )
| ( collec4140898570808001013al_nat @ none_nat @ ( some_nat @ Y3 ) ) ) ) ) ) )
=> ( ! [Y3: nat] :
( ( X
= ( some_Sum_sum_int_nat @ ( sum_Inr_nat_int @ Y3 ) ) )
=> ( ( Xa = none_Sum_sum_int_nat )
=> ( Y
= ( ~ ( collec4140898570808001013al_nat @ ( some_nat @ Y3 ) @ none_nat ) ) ) ) )
=> ( ( ? [Y3: nat] :
( X
= ( some_Sum_sum_int_nat @ ( sum_Inr_nat_int @ Y3 ) ) )
=> ( ? [X3: int] :
( Xa
= ( some_Sum_sum_int_nat @ ( sum_Inl_int_nat @ X3 ) ) )
=> Y ) )
=> ~ ! [X3: nat] :
( ( X
= ( some_Sum_sum_int_nat @ ( sum_Inr_nat_int @ X3 ) ) )
=> ! [Y3: nat] :
( ( Xa
= ( some_Sum_sum_int_nat @ ( sum_Inr_nat_int @ Y3 ) ) )
=> ( Y
= ( ~ ( collec4140898570808001013al_nat @ ( some_nat @ X3 ) @ ( some_nat @ Y3 ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ).
% cproper_interval_sum.elims(1)
thf(fact_591_cproper__interval__sum_Oelims_I1_J,axiom,
! [X: option2939537506576222215nt_int,Xa: option2939537506576222215nt_int,Y: $o] :
( ( ( collec4512806309114255452nt_int @ X @ Xa )
= Y )
=> ( ( ( X = none_Sum_sum_int_int )
=> ( ( Xa = none_Sum_sum_int_int )
=> ~ Y ) )
=> ( ( ( X = none_Sum_sum_int_int )
=> ! [X3: int] :
( ( Xa
= ( some_Sum_sum_int_int @ ( sum_Inl_int_int @ X3 ) ) )
=> ( Y
= ( ~ ( collec4138408100298950737al_int @ none_int @ ( some_int @ X3 ) ) ) ) ) )
=> ( ( ( X = none_Sum_sum_int_int )
=> ( ? [Y3: int] :
( Xa
= ( some_Sum_sum_int_int @ ( sum_Inr_int_int @ Y3 ) ) )
=> ~ Y ) )
=> ( ( ? [X3: int] :
( X
= ( some_Sum_sum_int_int @ ( sum_Inl_int_int @ X3 ) ) )
=> ( ( Xa = none_Sum_sum_int_int )
=> ~ Y ) )
=> ( ! [X3: int] :
( ( X
= ( some_Sum_sum_int_int @ ( sum_Inl_int_int @ X3 ) ) )
=> ! [Y3: int] :
( ( Xa
= ( some_Sum_sum_int_int @ ( sum_Inl_int_int @ Y3 ) ) )
=> ( Y
= ( ~ ( collec4138408100298950737al_int @ ( some_int @ X3 ) @ ( some_int @ Y3 ) ) ) ) ) )
=> ( ! [X3: int] :
( ( X
= ( some_Sum_sum_int_int @ ( sum_Inl_int_int @ X3 ) ) )
=> ! [Y3: int] :
( ( Xa
= ( some_Sum_sum_int_int @ ( sum_Inr_int_int @ Y3 ) ) )
=> ( Y
= ( ~ ( ( collec4138408100298950737al_int @ ( some_int @ X3 ) @ none_int )
| ( collec4138408100298950737al_int @ none_int @ ( some_int @ Y3 ) ) ) ) ) ) )
=> ( ! [Y3: int] :
( ( X
= ( some_Sum_sum_int_int @ ( sum_Inr_int_int @ Y3 ) ) )
=> ( ( Xa = none_Sum_sum_int_int )
=> ( Y
= ( ~ ( collec4138408100298950737al_int @ ( some_int @ Y3 ) @ none_int ) ) ) ) )
=> ( ( ? [Y3: int] :
( X
= ( some_Sum_sum_int_int @ ( sum_Inr_int_int @ Y3 ) ) )
=> ( ? [X3: int] :
( Xa
= ( some_Sum_sum_int_int @ ( sum_Inl_int_int @ X3 ) ) )
=> Y ) )
=> ~ ! [X3: int] :
( ( X
= ( some_Sum_sum_int_int @ ( sum_Inr_int_int @ X3 ) ) )
=> ! [Y3: int] :
( ( Xa
= ( some_Sum_sum_int_int @ ( sum_Inr_int_int @ Y3 ) ) )
=> ( Y
= ( ~ ( collec4138408100298950737al_int @ ( some_int @ X3 ) @ ( some_int @ Y3 ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ).
% cproper_interval_sum.elims(1)
thf(fact_592_cproper__interval__sum_Oelims_I2_J,axiom,
! [X: option3242699076814927183at_nat,Xa: option3242699076814927183at_nat] :
( ( collec7691303616722397604at_nat @ X @ Xa )
=> ( ( ( X = none_Sum_sum_nat_nat )
=> ( Xa != none_Sum_sum_nat_nat ) )
=> ( ( ( X = none_Sum_sum_nat_nat )
=> ! [X3: nat] :
( ( Xa
= ( some_Sum_sum_nat_nat @ ( sum_Inl_nat_nat @ X3 ) ) )
=> ~ ( collec4140898570808001013al_nat @ none_nat @ ( some_nat @ X3 ) ) ) )
=> ( ( ( X = none_Sum_sum_nat_nat )
=> ! [Y3: nat] :
( Xa
!= ( some_Sum_sum_nat_nat @ ( sum_Inr_nat_nat @ Y3 ) ) ) )
=> ( ( ? [X3: nat] :
( X
= ( some_Sum_sum_nat_nat @ ( sum_Inl_nat_nat @ X3 ) ) )
=> ( Xa != none_Sum_sum_nat_nat ) )
=> ( ! [X3: nat] :
( ( X
= ( some_Sum_sum_nat_nat @ ( sum_Inl_nat_nat @ X3 ) ) )
=> ! [Y3: nat] :
( ( Xa
= ( some_Sum_sum_nat_nat @ ( sum_Inl_nat_nat @ Y3 ) ) )
=> ~ ( collec4140898570808001013al_nat @ ( some_nat @ X3 ) @ ( some_nat @ Y3 ) ) ) )
=> ( ! [X3: nat] :
( ( X
= ( some_Sum_sum_nat_nat @ ( sum_Inl_nat_nat @ X3 ) ) )
=> ! [Y3: nat] :
( ( Xa
= ( some_Sum_sum_nat_nat @ ( sum_Inr_nat_nat @ Y3 ) ) )
=> ~ ( ( collec4140898570808001013al_nat @ ( some_nat @ X3 ) @ none_nat )
| ( collec4140898570808001013al_nat @ none_nat @ ( some_nat @ Y3 ) ) ) ) )
=> ( ! [Y3: nat] :
( ( X
= ( some_Sum_sum_nat_nat @ ( sum_Inr_nat_nat @ Y3 ) ) )
=> ( ( Xa = none_Sum_sum_nat_nat )
=> ~ ( collec4140898570808001013al_nat @ ( some_nat @ Y3 ) @ none_nat ) ) )
=> ~ ! [X3: nat] :
( ( X
= ( some_Sum_sum_nat_nat @ ( sum_Inr_nat_nat @ X3 ) ) )
=> ! [Y3: nat] :
( ( Xa
= ( some_Sum_sum_nat_nat @ ( sum_Inr_nat_nat @ Y3 ) ) )
=> ~ ( collec4140898570808001013al_nat @ ( some_nat @ X3 ) @ ( some_nat @ Y3 ) ) ) ) ) ) ) ) ) ) ) ) ).
% cproper_interval_sum.elims(2)
thf(fact_593_cproper__interval__sum_Oelims_I2_J,axiom,
! [X: option752615931883200427at_int,Xa: option752615931883200427at_int] :
( ( collec3513452597213200896at_int @ X @ Xa )
=> ( ( ( X = none_Sum_sum_nat_int )
=> ( Xa != none_Sum_sum_nat_int ) )
=> ( ( ( X = none_Sum_sum_nat_int )
=> ! [X3: nat] :
( ( Xa
= ( some_Sum_sum_nat_int @ ( sum_Inl_nat_int @ X3 ) ) )
=> ~ ( collec4140898570808001013al_nat @ none_nat @ ( some_nat @ X3 ) ) ) )
=> ( ( ( X = none_Sum_sum_nat_int )
=> ! [Y3: int] :
( Xa
!= ( some_Sum_sum_nat_int @ ( sum_Inr_int_nat @ Y3 ) ) ) )
=> ( ( ? [X3: nat] :
( X
= ( some_Sum_sum_nat_int @ ( sum_Inl_nat_int @ X3 ) ) )
=> ( Xa != none_Sum_sum_nat_int ) )
=> ( ! [X3: nat] :
( ( X
= ( some_Sum_sum_nat_int @ ( sum_Inl_nat_int @ X3 ) ) )
=> ! [Y3: nat] :
( ( Xa
= ( some_Sum_sum_nat_int @ ( sum_Inl_nat_int @ Y3 ) ) )
=> ~ ( collec4140898570808001013al_nat @ ( some_nat @ X3 ) @ ( some_nat @ Y3 ) ) ) )
=> ( ! [X3: nat] :
( ( X
= ( some_Sum_sum_nat_int @ ( sum_Inl_nat_int @ X3 ) ) )
=> ! [Y3: int] :
( ( Xa
= ( some_Sum_sum_nat_int @ ( sum_Inr_int_nat @ Y3 ) ) )
=> ~ ( ( collec4140898570808001013al_nat @ ( some_nat @ X3 ) @ none_nat )
| ( collec4138408100298950737al_int @ none_int @ ( some_int @ Y3 ) ) ) ) )
=> ( ! [Y3: int] :
( ( X
= ( some_Sum_sum_nat_int @ ( sum_Inr_int_nat @ Y3 ) ) )
=> ( ( Xa = none_Sum_sum_nat_int )
=> ~ ( collec4138408100298950737al_int @ ( some_int @ Y3 ) @ none_int ) ) )
=> ~ ! [X3: int] :
( ( X
= ( some_Sum_sum_nat_int @ ( sum_Inr_int_nat @ X3 ) ) )
=> ! [Y3: int] :
( ( Xa
= ( some_Sum_sum_nat_int @ ( sum_Inr_int_nat @ Y3 ) ) )
=> ~ ( collec4138408100298950737al_int @ ( some_int @ X3 ) @ ( some_int @ Y3 ) ) ) ) ) ) ) ) ) ) ) ) ).
% cproper_interval_sum.elims(2)
thf(fact_594_cproper__interval__sum_Oelims_I2_J,axiom,
! [X: option5429620651507948971nt_nat,Xa: option5429620651507948971nt_nat] :
( ( collec8690657328623452160nt_nat @ X @ Xa )
=> ( ( ( X = none_Sum_sum_int_nat )
=> ( Xa != none_Sum_sum_int_nat ) )
=> ( ( ( X = none_Sum_sum_int_nat )
=> ! [X3: int] :
( ( Xa
= ( some_Sum_sum_int_nat @ ( sum_Inl_int_nat @ X3 ) ) )
=> ~ ( collec4138408100298950737al_int @ none_int @ ( some_int @ X3 ) ) ) )
=> ( ( ( X = none_Sum_sum_int_nat )
=> ! [Y3: nat] :
( Xa
!= ( some_Sum_sum_int_nat @ ( sum_Inr_nat_int @ Y3 ) ) ) )
=> ( ( ? [X3: int] :
( X
= ( some_Sum_sum_int_nat @ ( sum_Inl_int_nat @ X3 ) ) )
=> ( Xa != none_Sum_sum_int_nat ) )
=> ( ! [X3: int] :
( ( X
= ( some_Sum_sum_int_nat @ ( sum_Inl_int_nat @ X3 ) ) )
=> ! [Y3: int] :
( ( Xa
= ( some_Sum_sum_int_nat @ ( sum_Inl_int_nat @ Y3 ) ) )
=> ~ ( collec4138408100298950737al_int @ ( some_int @ X3 ) @ ( some_int @ Y3 ) ) ) )
=> ( ! [X3: int] :
( ( X
= ( some_Sum_sum_int_nat @ ( sum_Inl_int_nat @ X3 ) ) )
=> ! [Y3: nat] :
( ( Xa
= ( some_Sum_sum_int_nat @ ( sum_Inr_nat_int @ Y3 ) ) )
=> ~ ( ( collec4138408100298950737al_int @ ( some_int @ X3 ) @ none_int )
| ( collec4140898570808001013al_nat @ none_nat @ ( some_nat @ Y3 ) ) ) ) )
=> ( ! [Y3: nat] :
( ( X
= ( some_Sum_sum_int_nat @ ( sum_Inr_nat_int @ Y3 ) ) )
=> ( ( Xa = none_Sum_sum_int_nat )
=> ~ ( collec4140898570808001013al_nat @ ( some_nat @ Y3 ) @ none_nat ) ) )
=> ~ ! [X3: nat] :
( ( X
= ( some_Sum_sum_int_nat @ ( sum_Inr_nat_int @ X3 ) ) )
=> ! [Y3: nat] :
( ( Xa
= ( some_Sum_sum_int_nat @ ( sum_Inr_nat_int @ Y3 ) ) )
=> ~ ( collec4140898570808001013al_nat @ ( some_nat @ X3 ) @ ( some_nat @ Y3 ) ) ) ) ) ) ) ) ) ) ) ) ).
% cproper_interval_sum.elims(2)
thf(fact_595_cproper__interval__sum_Oelims_I2_J,axiom,
! [X: option2939537506576222215nt_int,Xa: option2939537506576222215nt_int] :
( ( collec4512806309114255452nt_int @ X @ Xa )
=> ( ( ( X = none_Sum_sum_int_int )
=> ( Xa != none_Sum_sum_int_int ) )
=> ( ( ( X = none_Sum_sum_int_int )
=> ! [X3: int] :
( ( Xa
= ( some_Sum_sum_int_int @ ( sum_Inl_int_int @ X3 ) ) )
=> ~ ( collec4138408100298950737al_int @ none_int @ ( some_int @ X3 ) ) ) )
=> ( ( ( X = none_Sum_sum_int_int )
=> ! [Y3: int] :
( Xa
!= ( some_Sum_sum_int_int @ ( sum_Inr_int_int @ Y3 ) ) ) )
=> ( ( ? [X3: int] :
( X
= ( some_Sum_sum_int_int @ ( sum_Inl_int_int @ X3 ) ) )
=> ( Xa != none_Sum_sum_int_int ) )
=> ( ! [X3: int] :
( ( X
= ( some_Sum_sum_int_int @ ( sum_Inl_int_int @ X3 ) ) )
=> ! [Y3: int] :
( ( Xa
= ( some_Sum_sum_int_int @ ( sum_Inl_int_int @ Y3 ) ) )
=> ~ ( collec4138408100298950737al_int @ ( some_int @ X3 ) @ ( some_int @ Y3 ) ) ) )
=> ( ! [X3: int] :
( ( X
= ( some_Sum_sum_int_int @ ( sum_Inl_int_int @ X3 ) ) )
=> ! [Y3: int] :
( ( Xa
= ( some_Sum_sum_int_int @ ( sum_Inr_int_int @ Y3 ) ) )
=> ~ ( ( collec4138408100298950737al_int @ ( some_int @ X3 ) @ none_int )
| ( collec4138408100298950737al_int @ none_int @ ( some_int @ Y3 ) ) ) ) )
=> ( ! [Y3: int] :
( ( X
= ( some_Sum_sum_int_int @ ( sum_Inr_int_int @ Y3 ) ) )
=> ( ( Xa = none_Sum_sum_int_int )
=> ~ ( collec4138408100298950737al_int @ ( some_int @ Y3 ) @ none_int ) ) )
=> ~ ! [X3: int] :
( ( X
= ( some_Sum_sum_int_int @ ( sum_Inr_int_int @ X3 ) ) )
=> ! [Y3: int] :
( ( Xa
= ( some_Sum_sum_int_int @ ( sum_Inr_int_int @ Y3 ) ) )
=> ~ ( collec4138408100298950737al_int @ ( some_int @ X3 ) @ ( some_int @ Y3 ) ) ) ) ) ) ) ) ) ) ) ) ).
% cproper_interval_sum.elims(2)
thf(fact_596_Inl__Inr__False,axiom,
! [X: a,Y: nat] :
( ( sum_Inl_a_nat @ X )
!= ( sum_Inr_nat_a @ Y ) ) ).
% Inl_Inr_False
thf(fact_597_Inr__Inl__False,axiom,
! [X: nat,Y: a] :
( ( sum_Inr_nat_a @ X )
!= ( sum_Inl_a_nat @ Y ) ) ).
% Inr_Inl_False
thf(fact_598_proper__interval__nat_Osimps_I2_J,axiom,
! [X: nat] :
( ( set_pr3848668467136637714al_nat @ none_nat @ ( some_nat @ X ) )
= ( ord_less_nat @ zero_zero_nat @ X ) ) ).
% proper_interval_nat.simps(2)
thf(fact_599_in__lex__prod,axiom,
! [A: option_nat,B: option_int,A5: option_nat,B2: option_int,R3: set_Pr6588086440996610945on_nat,S: set_Pr5725982650624401465on_int] :
( ( member7430460011603515024on_int @ ( produc176493563234596311on_int @ ( produc920486614911842229on_int @ A @ B ) @ ( produc920486614911842229on_int @ A5 @ B2 ) ) @ ( lex_pr8179060047738630093on_int @ R3 @ S ) )
= ( ( member4117937158525611210on_nat @ ( produc5098337634421038937on_nat @ A @ A5 ) @ R3 )
| ( ( A = A5 )
& ( member7038936195297346946on_int @ ( produc6331568796615304721on_int @ B @ B2 ) @ S ) ) ) ) ).
% in_lex_prod
thf(fact_600_in__lex__prod,axiom,
! [A: option_int,B: option_nat,A5: option_int,B2: option_nat,R3: set_Pr5725982650624401465on_int,S: set_Pr6588086440996610945on_nat] :
( ( member846496771861665424on_nat @ ( produc1952074639779083735on_nat @ ( produc1286047779269725621on_nat @ A @ B ) @ ( produc1286047779269725621on_nat @ A5 @ B2 ) ) @ ( lex_pr8544621212096513485on_nat @ R3 @ S ) )
= ( ( member7038936195297346946on_int @ ( produc6331568796615304721on_int @ A @ A5 ) @ R3 )
| ( ( A = A5 )
& ( member4117937158525611210on_nat @ ( produc5098337634421038937on_nat @ B @ B2 ) @ S ) ) ) ) ).
% in_lex_prod
thf(fact_601_in__lex__prod,axiom,
! [A: option_nat,B: option_nat,A5: option_nat,B2: option_nat,R3: set_Pr6588086440996610945on_nat,S: set_Pr6588086440996610945on_nat] :
( ( member2771234110571332496on_nat @ ( produc2417174425200997591on_nat @ ( produc5098337634421038937on_nat @ A @ B ) @ ( produc5098337634421038937on_nat @ A5 @ B2 ) ) @ ( lex_pr3133539030393050993on_nat @ R3 @ S ) )
= ( ( member4117937158525611210on_nat @ ( produc5098337634421038937on_nat @ A @ A5 ) @ R3 )
| ( ( A = A5 )
& ( member4117937158525611210on_nat @ ( produc5098337634421038937on_nat @ B @ B2 ) @ S ) ) ) ) ).
% in_lex_prod
thf(fact_602_in__lex__prod,axiom,
! [A: option_int,B: option_int,A5: option_int,B2: option_int,R3: set_Pr5725982650624401465on_int,S: set_Pr5725982650624401465on_int] :
( ( member5505722672893847952on_int @ ( produc8934765814667458263on_int @ ( produc6331568796615304721on_int @ A @ B ) @ ( produc6331568796615304721on_int @ A5 @ B2 ) ) @ ( lex_pr4366770192587316777on_int @ R3 @ S ) )
= ( ( member7038936195297346946on_int @ ( produc6331568796615304721on_int @ A @ A5 ) @ R3 )
| ( ( A = A5 )
& ( member7038936195297346946on_int @ ( produc6331568796615304721on_int @ B @ B2 ) @ S ) ) ) ) ).
% in_lex_prod
thf(fact_603_cproper__interval__sum_Opelims_I1_J,axiom,
! [X: option3242699076814927183at_nat,Xa: option3242699076814927183at_nat,Y: $o] :
( ( ( collec7691303616722397604at_nat @ X @ Xa )
= Y )
=> ( ( accp_P3884639961936550920at_nat @ collec4724013795633334532at_nat @ ( produc7249471109196833079at_nat @ X @ Xa ) )
=> ( ( ( X = none_Sum_sum_nat_nat )
=> ( ( Xa = none_Sum_sum_nat_nat )
=> ( Y
=> ~ ( accp_P3884639961936550920at_nat @ collec4724013795633334532at_nat @ ( produc7249471109196833079at_nat @ none_Sum_sum_nat_nat @ none_Sum_sum_nat_nat ) ) ) ) )
=> ( ( ( X = none_Sum_sum_nat_nat )
=> ! [X3: nat] :
( ( Xa
= ( some_Sum_sum_nat_nat @ ( sum_Inl_nat_nat @ X3 ) ) )
=> ( ( Y
= ( collec4140898570808001013al_nat @ none_nat @ ( some_nat @ X3 ) ) )
=> ~ ( accp_P3884639961936550920at_nat @ collec4724013795633334532at_nat @ ( produc7249471109196833079at_nat @ none_Sum_sum_nat_nat @ ( some_Sum_sum_nat_nat @ ( sum_Inl_nat_nat @ X3 ) ) ) ) ) ) )
=> ( ( ( X = none_Sum_sum_nat_nat )
=> ! [Y3: nat] :
( ( Xa
= ( some_Sum_sum_nat_nat @ ( sum_Inr_nat_nat @ Y3 ) ) )
=> ( Y
=> ~ ( accp_P3884639961936550920at_nat @ collec4724013795633334532at_nat @ ( produc7249471109196833079at_nat @ none_Sum_sum_nat_nat @ ( some_Sum_sum_nat_nat @ ( sum_Inr_nat_nat @ Y3 ) ) ) ) ) ) )
=> ( ! [X3: nat] :
( ( X
= ( some_Sum_sum_nat_nat @ ( sum_Inl_nat_nat @ X3 ) ) )
=> ( ( Xa = none_Sum_sum_nat_nat )
=> ( Y
=> ~ ( accp_P3884639961936550920at_nat @ collec4724013795633334532at_nat @ ( produc7249471109196833079at_nat @ ( some_Sum_sum_nat_nat @ ( sum_Inl_nat_nat @ X3 ) ) @ none_Sum_sum_nat_nat ) ) ) ) )
=> ( ! [X3: nat] :
( ( X
= ( some_Sum_sum_nat_nat @ ( sum_Inl_nat_nat @ X3 ) ) )
=> ! [Y3: nat] :
( ( Xa
= ( some_Sum_sum_nat_nat @ ( sum_Inl_nat_nat @ Y3 ) ) )
=> ( ( Y
= ( collec4140898570808001013al_nat @ ( some_nat @ X3 ) @ ( some_nat @ Y3 ) ) )
=> ~ ( accp_P3884639961936550920at_nat @ collec4724013795633334532at_nat @ ( produc7249471109196833079at_nat @ ( some_Sum_sum_nat_nat @ ( sum_Inl_nat_nat @ X3 ) ) @ ( some_Sum_sum_nat_nat @ ( sum_Inl_nat_nat @ Y3 ) ) ) ) ) ) )
=> ( ! [X3: nat] :
( ( X
= ( some_Sum_sum_nat_nat @ ( sum_Inl_nat_nat @ X3 ) ) )
=> ! [Y3: nat] :
( ( Xa
= ( some_Sum_sum_nat_nat @ ( sum_Inr_nat_nat @ Y3 ) ) )
=> ( ( Y
= ( ( collec4140898570808001013al_nat @ ( some_nat @ X3 ) @ none_nat )
| ( collec4140898570808001013al_nat @ none_nat @ ( some_nat @ Y3 ) ) ) )
=> ~ ( accp_P3884639961936550920at_nat @ collec4724013795633334532at_nat @ ( produc7249471109196833079at_nat @ ( some_Sum_sum_nat_nat @ ( sum_Inl_nat_nat @ X3 ) ) @ ( some_Sum_sum_nat_nat @ ( sum_Inr_nat_nat @ Y3 ) ) ) ) ) ) )
=> ( ! [Y3: nat] :
( ( X
= ( some_Sum_sum_nat_nat @ ( sum_Inr_nat_nat @ Y3 ) ) )
=> ( ( Xa = none_Sum_sum_nat_nat )
=> ( ( Y
= ( collec4140898570808001013al_nat @ ( some_nat @ Y3 ) @ none_nat ) )
=> ~ ( accp_P3884639961936550920at_nat @ collec4724013795633334532at_nat @ ( produc7249471109196833079at_nat @ ( some_Sum_sum_nat_nat @ ( sum_Inr_nat_nat @ Y3 ) ) @ none_Sum_sum_nat_nat ) ) ) ) )
=> ( ! [Y3: nat] :
( ( X
= ( some_Sum_sum_nat_nat @ ( sum_Inr_nat_nat @ Y3 ) ) )
=> ! [X3: nat] :
( ( Xa
= ( some_Sum_sum_nat_nat @ ( sum_Inl_nat_nat @ X3 ) ) )
=> ( ~ Y
=> ~ ( accp_P3884639961936550920at_nat @ collec4724013795633334532at_nat @ ( produc7249471109196833079at_nat @ ( some_Sum_sum_nat_nat @ ( sum_Inr_nat_nat @ Y3 ) ) @ ( some_Sum_sum_nat_nat @ ( sum_Inl_nat_nat @ X3 ) ) ) ) ) ) )
=> ~ ! [X3: nat] :
( ( X
= ( some_Sum_sum_nat_nat @ ( sum_Inr_nat_nat @ X3 ) ) )
=> ! [Y3: nat] :
( ( Xa
= ( some_Sum_sum_nat_nat @ ( sum_Inr_nat_nat @ Y3 ) ) )
=> ( ( Y
= ( collec4140898570808001013al_nat @ ( some_nat @ X3 ) @ ( some_nat @ Y3 ) ) )
=> ~ ( accp_P3884639961936550920at_nat @ collec4724013795633334532at_nat @ ( produc7249471109196833079at_nat @ ( some_Sum_sum_nat_nat @ ( sum_Inr_nat_nat @ X3 ) ) @ ( some_Sum_sum_nat_nat @ ( sum_Inr_nat_nat @ Y3 ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ).
% cproper_interval_sum.pelims(1)
thf(fact_604_cproper__interval__sum_Opelims_I1_J,axiom,
! [X: option752615931883200427at_int,Xa: option752615931883200427at_int,Y: $o] :
( ( ( collec3513452597213200896at_int @ X @ Xa )
= Y )
=> ( ( accp_P479778451830304448at_int @ collec4721523325124284256at_int @ ( produc6168074872946778607at_int @ X @ Xa ) )
=> ( ( ( X = none_Sum_sum_nat_int )
=> ( ( Xa = none_Sum_sum_nat_int )
=> ( Y
=> ~ ( accp_P479778451830304448at_int @ collec4721523325124284256at_int @ ( produc6168074872946778607at_int @ none_Sum_sum_nat_int @ none_Sum_sum_nat_int ) ) ) ) )
=> ( ( ( X = none_Sum_sum_nat_int )
=> ! [X3: nat] :
( ( Xa
= ( some_Sum_sum_nat_int @ ( sum_Inl_nat_int @ X3 ) ) )
=> ( ( Y
= ( collec4140898570808001013al_nat @ none_nat @ ( some_nat @ X3 ) ) )
=> ~ ( accp_P479778451830304448at_int @ collec4721523325124284256at_int @ ( produc6168074872946778607at_int @ none_Sum_sum_nat_int @ ( some_Sum_sum_nat_int @ ( sum_Inl_nat_int @ X3 ) ) ) ) ) ) )
=> ( ( ( X = none_Sum_sum_nat_int )
=> ! [Y3: int] :
( ( Xa
= ( some_Sum_sum_nat_int @ ( sum_Inr_int_nat @ Y3 ) ) )
=> ( Y
=> ~ ( accp_P479778451830304448at_int @ collec4721523325124284256at_int @ ( produc6168074872946778607at_int @ none_Sum_sum_nat_int @ ( some_Sum_sum_nat_int @ ( sum_Inr_int_nat @ Y3 ) ) ) ) ) ) )
=> ( ! [X3: nat] :
( ( X
= ( some_Sum_sum_nat_int @ ( sum_Inl_nat_int @ X3 ) ) )
=> ( ( Xa = none_Sum_sum_nat_int )
=> ( Y
=> ~ ( accp_P479778451830304448at_int @ collec4721523325124284256at_int @ ( produc6168074872946778607at_int @ ( some_Sum_sum_nat_int @ ( sum_Inl_nat_int @ X3 ) ) @ none_Sum_sum_nat_int ) ) ) ) )
=> ( ! [X3: nat] :
( ( X
= ( some_Sum_sum_nat_int @ ( sum_Inl_nat_int @ X3 ) ) )
=> ! [Y3: nat] :
( ( Xa
= ( some_Sum_sum_nat_int @ ( sum_Inl_nat_int @ Y3 ) ) )
=> ( ( Y
= ( collec4140898570808001013al_nat @ ( some_nat @ X3 ) @ ( some_nat @ Y3 ) ) )
=> ~ ( accp_P479778451830304448at_int @ collec4721523325124284256at_int @ ( produc6168074872946778607at_int @ ( some_Sum_sum_nat_int @ ( sum_Inl_nat_int @ X3 ) ) @ ( some_Sum_sum_nat_int @ ( sum_Inl_nat_int @ Y3 ) ) ) ) ) ) )
=> ( ! [X3: nat] :
( ( X
= ( some_Sum_sum_nat_int @ ( sum_Inl_nat_int @ X3 ) ) )
=> ! [Y3: int] :
( ( Xa
= ( some_Sum_sum_nat_int @ ( sum_Inr_int_nat @ Y3 ) ) )
=> ( ( Y
= ( ( collec4140898570808001013al_nat @ ( some_nat @ X3 ) @ none_nat )
| ( collec4138408100298950737al_int @ none_int @ ( some_int @ Y3 ) ) ) )
=> ~ ( accp_P479778451830304448at_int @ collec4721523325124284256at_int @ ( produc6168074872946778607at_int @ ( some_Sum_sum_nat_int @ ( sum_Inl_nat_int @ X3 ) ) @ ( some_Sum_sum_nat_int @ ( sum_Inr_int_nat @ Y3 ) ) ) ) ) ) )
=> ( ! [Y3: int] :
( ( X
= ( some_Sum_sum_nat_int @ ( sum_Inr_int_nat @ Y3 ) ) )
=> ( ( Xa = none_Sum_sum_nat_int )
=> ( ( Y
= ( collec4138408100298950737al_int @ ( some_int @ Y3 ) @ none_int ) )
=> ~ ( accp_P479778451830304448at_int @ collec4721523325124284256at_int @ ( produc6168074872946778607at_int @ ( some_Sum_sum_nat_int @ ( sum_Inr_int_nat @ Y3 ) ) @ none_Sum_sum_nat_int ) ) ) ) )
=> ( ! [Y3: int] :
( ( X
= ( some_Sum_sum_nat_int @ ( sum_Inr_int_nat @ Y3 ) ) )
=> ! [X3: nat] :
( ( Xa
= ( some_Sum_sum_nat_int @ ( sum_Inl_nat_int @ X3 ) ) )
=> ( ~ Y
=> ~ ( accp_P479778451830304448at_int @ collec4721523325124284256at_int @ ( produc6168074872946778607at_int @ ( some_Sum_sum_nat_int @ ( sum_Inr_int_nat @ Y3 ) ) @ ( some_Sum_sum_nat_int @ ( sum_Inl_nat_int @ X3 ) ) ) ) ) ) )
=> ~ ! [X3: int] :
( ( X
= ( some_Sum_sum_nat_int @ ( sum_Inr_int_nat @ X3 ) ) )
=> ! [Y3: int] :
( ( Xa
= ( some_Sum_sum_nat_int @ ( sum_Inr_int_nat @ Y3 ) ) )
=> ( ( Y
= ( collec4138408100298950737al_int @ ( some_int @ X3 ) @ ( some_int @ Y3 ) ) )
=> ~ ( accp_P479778451830304448at_int @ collec4721523325124284256at_int @ ( produc6168074872946778607at_int @ ( some_Sum_sum_nat_int @ ( sum_Inr_int_nat @ X3 ) ) @ ( some_Sum_sum_nat_int @ ( sum_Inr_int_nat @ Y3 ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ).
% cproper_interval_sum.pelims(1)
thf(fact_605_cproper__interval__sum_Opelims_I1_J,axiom,
! [X: option5429620651507948971nt_nat,Xa: option5429620651507948971nt_nat,Y: $o] :
( ( ( collec8690657328623452160nt_nat @ X @ Xa )
= Y )
=> ( ( accp_P3675550703157275328nt_nat @ collec5723367507534389088nt_nat @ ( produc6008599830946209263nt_nat @ X @ Xa ) )
=> ( ( ( X = none_Sum_sum_int_nat )
=> ( ( Xa = none_Sum_sum_int_nat )
=> ( Y
=> ~ ( accp_P3675550703157275328nt_nat @ collec5723367507534389088nt_nat @ ( produc6008599830946209263nt_nat @ none_Sum_sum_int_nat @ none_Sum_sum_int_nat ) ) ) ) )
=> ( ( ( X = none_Sum_sum_int_nat )
=> ! [X3: int] :
( ( Xa
= ( some_Sum_sum_int_nat @ ( sum_Inl_int_nat @ X3 ) ) )
=> ( ( Y
= ( collec4138408100298950737al_int @ none_int @ ( some_int @ X3 ) ) )
=> ~ ( accp_P3675550703157275328nt_nat @ collec5723367507534389088nt_nat @ ( produc6008599830946209263nt_nat @ none_Sum_sum_int_nat @ ( some_Sum_sum_int_nat @ ( sum_Inl_int_nat @ X3 ) ) ) ) ) ) )
=> ( ( ( X = none_Sum_sum_int_nat )
=> ! [Y3: nat] :
( ( Xa
= ( some_Sum_sum_int_nat @ ( sum_Inr_nat_int @ Y3 ) ) )
=> ( Y
=> ~ ( accp_P3675550703157275328nt_nat @ collec5723367507534389088nt_nat @ ( produc6008599830946209263nt_nat @ none_Sum_sum_int_nat @ ( some_Sum_sum_int_nat @ ( sum_Inr_nat_int @ Y3 ) ) ) ) ) ) )
=> ( ! [X3: int] :
( ( X
= ( some_Sum_sum_int_nat @ ( sum_Inl_int_nat @ X3 ) ) )
=> ( ( Xa = none_Sum_sum_int_nat )
=> ( Y
=> ~ ( accp_P3675550703157275328nt_nat @ collec5723367507534389088nt_nat @ ( produc6008599830946209263nt_nat @ ( some_Sum_sum_int_nat @ ( sum_Inl_int_nat @ X3 ) ) @ none_Sum_sum_int_nat ) ) ) ) )
=> ( ! [X3: int] :
( ( X
= ( some_Sum_sum_int_nat @ ( sum_Inl_int_nat @ X3 ) ) )
=> ! [Y3: int] :
( ( Xa
= ( some_Sum_sum_int_nat @ ( sum_Inl_int_nat @ Y3 ) ) )
=> ( ( Y
= ( collec4138408100298950737al_int @ ( some_int @ X3 ) @ ( some_int @ Y3 ) ) )
=> ~ ( accp_P3675550703157275328nt_nat @ collec5723367507534389088nt_nat @ ( produc6008599830946209263nt_nat @ ( some_Sum_sum_int_nat @ ( sum_Inl_int_nat @ X3 ) ) @ ( some_Sum_sum_int_nat @ ( sum_Inl_int_nat @ Y3 ) ) ) ) ) ) )
=> ( ! [X3: int] :
( ( X
= ( some_Sum_sum_int_nat @ ( sum_Inl_int_nat @ X3 ) ) )
=> ! [Y3: nat] :
( ( Xa
= ( some_Sum_sum_int_nat @ ( sum_Inr_nat_int @ Y3 ) ) )
=> ( ( Y
= ( ( collec4138408100298950737al_int @ ( some_int @ X3 ) @ none_int )
| ( collec4140898570808001013al_nat @ none_nat @ ( some_nat @ Y3 ) ) ) )
=> ~ ( accp_P3675550703157275328nt_nat @ collec5723367507534389088nt_nat @ ( produc6008599830946209263nt_nat @ ( some_Sum_sum_int_nat @ ( sum_Inl_int_nat @ X3 ) ) @ ( some_Sum_sum_int_nat @ ( sum_Inr_nat_int @ Y3 ) ) ) ) ) ) )
=> ( ! [Y3: nat] :
( ( X
= ( some_Sum_sum_int_nat @ ( sum_Inr_nat_int @ Y3 ) ) )
=> ( ( Xa = none_Sum_sum_int_nat )
=> ( ( Y
= ( collec4140898570808001013al_nat @ ( some_nat @ Y3 ) @ none_nat ) )
=> ~ ( accp_P3675550703157275328nt_nat @ collec5723367507534389088nt_nat @ ( produc6008599830946209263nt_nat @ ( some_Sum_sum_int_nat @ ( sum_Inr_nat_int @ Y3 ) ) @ none_Sum_sum_int_nat ) ) ) ) )
=> ( ! [Y3: nat] :
( ( X
= ( some_Sum_sum_int_nat @ ( sum_Inr_nat_int @ Y3 ) ) )
=> ! [X3: int] :
( ( Xa
= ( some_Sum_sum_int_nat @ ( sum_Inl_int_nat @ X3 ) ) )
=> ( ~ Y
=> ~ ( accp_P3675550703157275328nt_nat @ collec5723367507534389088nt_nat @ ( produc6008599830946209263nt_nat @ ( some_Sum_sum_int_nat @ ( sum_Inr_nat_int @ Y3 ) ) @ ( some_Sum_sum_int_nat @ ( sum_Inl_int_nat @ X3 ) ) ) ) ) ) )
=> ~ ! [X3: nat] :
( ( X
= ( some_Sum_sum_int_nat @ ( sum_Inr_nat_int @ X3 ) ) )
=> ! [Y3: nat] :
( ( Xa
= ( some_Sum_sum_int_nat @ ( sum_Inr_nat_int @ Y3 ) ) )
=> ( ( Y
= ( collec4140898570808001013al_nat @ ( some_nat @ X3 ) @ ( some_nat @ Y3 ) ) )
=> ~ ( accp_P3675550703157275328nt_nat @ collec5723367507534389088nt_nat @ ( produc6008599830946209263nt_nat @ ( some_Sum_sum_int_nat @ ( sum_Inr_nat_int @ X3 ) ) @ ( some_Sum_sum_int_nat @ ( sum_Inr_nat_int @ Y3 ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ).
% cproper_interval_sum.pelims(1)
thf(fact_606_cproper__interval__sum_Opelims_I1_J,axiom,
! [X: option2939537506576222215nt_int,Xa: option2939537506576222215nt_int,Y: $o] :
( ( ( collec4512806309114255452nt_int @ X @ Xa )
= Y )
=> ( ( accp_P270689193051028856nt_int @ collec5720877037025338812nt_int @ ( produc4927203594696154791nt_int @ X @ Xa ) )
=> ( ( ( X = none_Sum_sum_int_int )
=> ( ( Xa = none_Sum_sum_int_int )
=> ( Y
=> ~ ( accp_P270689193051028856nt_int @ collec5720877037025338812nt_int @ ( produc4927203594696154791nt_int @ none_Sum_sum_int_int @ none_Sum_sum_int_int ) ) ) ) )
=> ( ( ( X = none_Sum_sum_int_int )
=> ! [X3: int] :
( ( Xa
= ( some_Sum_sum_int_int @ ( sum_Inl_int_int @ X3 ) ) )
=> ( ( Y
= ( collec4138408100298950737al_int @ none_int @ ( some_int @ X3 ) ) )
=> ~ ( accp_P270689193051028856nt_int @ collec5720877037025338812nt_int @ ( produc4927203594696154791nt_int @ none_Sum_sum_int_int @ ( some_Sum_sum_int_int @ ( sum_Inl_int_int @ X3 ) ) ) ) ) ) )
=> ( ( ( X = none_Sum_sum_int_int )
=> ! [Y3: int] :
( ( Xa
= ( some_Sum_sum_int_int @ ( sum_Inr_int_int @ Y3 ) ) )
=> ( Y
=> ~ ( accp_P270689193051028856nt_int @ collec5720877037025338812nt_int @ ( produc4927203594696154791nt_int @ none_Sum_sum_int_int @ ( some_Sum_sum_int_int @ ( sum_Inr_int_int @ Y3 ) ) ) ) ) ) )
=> ( ! [X3: int] :
( ( X
= ( some_Sum_sum_int_int @ ( sum_Inl_int_int @ X3 ) ) )
=> ( ( Xa = none_Sum_sum_int_int )
=> ( Y
=> ~ ( accp_P270689193051028856nt_int @ collec5720877037025338812nt_int @ ( produc4927203594696154791nt_int @ ( some_Sum_sum_int_int @ ( sum_Inl_int_int @ X3 ) ) @ none_Sum_sum_int_int ) ) ) ) )
=> ( ! [X3: int] :
( ( X
= ( some_Sum_sum_int_int @ ( sum_Inl_int_int @ X3 ) ) )
=> ! [Y3: int] :
( ( Xa
= ( some_Sum_sum_int_int @ ( sum_Inl_int_int @ Y3 ) ) )
=> ( ( Y
= ( collec4138408100298950737al_int @ ( some_int @ X3 ) @ ( some_int @ Y3 ) ) )
=> ~ ( accp_P270689193051028856nt_int @ collec5720877037025338812nt_int @ ( produc4927203594696154791nt_int @ ( some_Sum_sum_int_int @ ( sum_Inl_int_int @ X3 ) ) @ ( some_Sum_sum_int_int @ ( sum_Inl_int_int @ Y3 ) ) ) ) ) ) )
=> ( ! [X3: int] :
( ( X
= ( some_Sum_sum_int_int @ ( sum_Inl_int_int @ X3 ) ) )
=> ! [Y3: int] :
( ( Xa
= ( some_Sum_sum_int_int @ ( sum_Inr_int_int @ Y3 ) ) )
=> ( ( Y
= ( ( collec4138408100298950737al_int @ ( some_int @ X3 ) @ none_int )
| ( collec4138408100298950737al_int @ none_int @ ( some_int @ Y3 ) ) ) )
=> ~ ( accp_P270689193051028856nt_int @ collec5720877037025338812nt_int @ ( produc4927203594696154791nt_int @ ( some_Sum_sum_int_int @ ( sum_Inl_int_int @ X3 ) ) @ ( some_Sum_sum_int_int @ ( sum_Inr_int_int @ Y3 ) ) ) ) ) ) )
=> ( ! [Y3: int] :
( ( X
= ( some_Sum_sum_int_int @ ( sum_Inr_int_int @ Y3 ) ) )
=> ( ( Xa = none_Sum_sum_int_int )
=> ( ( Y
= ( collec4138408100298950737al_int @ ( some_int @ Y3 ) @ none_int ) )
=> ~ ( accp_P270689193051028856nt_int @ collec5720877037025338812nt_int @ ( produc4927203594696154791nt_int @ ( some_Sum_sum_int_int @ ( sum_Inr_int_int @ Y3 ) ) @ none_Sum_sum_int_int ) ) ) ) )
=> ( ! [Y3: int] :
( ( X
= ( some_Sum_sum_int_int @ ( sum_Inr_int_int @ Y3 ) ) )
=> ! [X3: int] :
( ( Xa
= ( some_Sum_sum_int_int @ ( sum_Inl_int_int @ X3 ) ) )
=> ( ~ Y
=> ~ ( accp_P270689193051028856nt_int @ collec5720877037025338812nt_int @ ( produc4927203594696154791nt_int @ ( some_Sum_sum_int_int @ ( sum_Inr_int_int @ Y3 ) ) @ ( some_Sum_sum_int_int @ ( sum_Inl_int_int @ X3 ) ) ) ) ) ) )
=> ~ ! [X3: int] :
( ( X
= ( some_Sum_sum_int_int @ ( sum_Inr_int_int @ X3 ) ) )
=> ! [Y3: int] :
( ( Xa
= ( some_Sum_sum_int_int @ ( sum_Inr_int_int @ Y3 ) ) )
=> ( ( Y
= ( collec4138408100298950737al_int @ ( some_int @ X3 ) @ ( some_int @ Y3 ) ) )
=> ~ ( accp_P270689193051028856nt_int @ collec5720877037025338812nt_int @ ( produc4927203594696154791nt_int @ ( some_Sum_sum_int_int @ ( sum_Inr_int_int @ X3 ) ) @ ( some_Sum_sum_int_int @ ( sum_Inr_int_int @ Y3 ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ).
% cproper_interval_sum.pelims(1)
thf(fact_607_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_608_neq0__conv,axiom,
! [N2: nat] :
( ( N2 != zero_zero_nat )
= ( ord_less_nat @ zero_zero_nat @ N2 ) ) ).
% neq0_conv
thf(fact_609_less__nat__zero__code,axiom,
! [N2: nat] :
~ ( ord_less_nat @ N2 @ zero_zero_nat ) ).
% less_nat_zero_code
thf(fact_610_bot__nat__0_Oextremum,axiom,
! [A: nat] : ( ord_less_eq_nat @ zero_zero_nat @ A ) ).
% bot_nat_0.extremum
thf(fact_611_le0,axiom,
! [N2: nat] : ( ord_less_eq_nat @ zero_zero_nat @ N2 ) ).
% le0
thf(fact_612_bot__nat__0_Oextremum__strict,axiom,
! [A: nat] :
~ ( ord_less_nat @ A @ zero_zero_nat ) ).
% bot_nat_0.extremum_strict
thf(fact_613_gr0I,axiom,
! [N2: nat] :
( ( N2 != zero_zero_nat )
=> ( ord_less_nat @ zero_zero_nat @ N2 ) ) ).
% gr0I
thf(fact_614_not__gr0,axiom,
! [N2: nat] :
( ( ~ ( ord_less_nat @ zero_zero_nat @ N2 ) )
= ( N2 = zero_zero_nat ) ) ).
% not_gr0
thf(fact_615_not__less0,axiom,
! [N2: nat] :
~ ( ord_less_nat @ N2 @ zero_zero_nat ) ).
% not_less0
thf(fact_616_less__zeroE,axiom,
! [N2: nat] :
~ ( ord_less_nat @ N2 @ zero_zero_nat ) ).
% less_zeroE
thf(fact_617_gr__implies__not0,axiom,
! [M2: nat,N2: nat] :
( ( ord_less_nat @ M2 @ N2 )
=> ( N2 != zero_zero_nat ) ) ).
% gr_implies_not0
thf(fact_618_infinite__descent0,axiom,
! [P: nat > $o,N2: nat] :
( ( P @ zero_zero_nat )
=> ( ! [N3: nat] :
( ( ord_less_nat @ zero_zero_nat @ N3 )
=> ( ~ ( P @ N3 )
=> ? [M3: nat] :
( ( ord_less_nat @ M3 @ N3 )
& ~ ( P @ M3 ) ) ) )
=> ( P @ N2 ) ) ) ).
% infinite_descent0
thf(fact_619_less__eq__nat_Osimps_I1_J,axiom,
! [N2: nat] : ( ord_less_eq_nat @ zero_zero_nat @ N2 ) ).
% less_eq_nat.simps(1)
thf(fact_620_bot__nat__0_Oextremum__unique,axiom,
! [A: nat] :
( ( ord_less_eq_nat @ A @ zero_zero_nat )
= ( A = zero_zero_nat ) ) ).
% bot_nat_0.extremum_unique
thf(fact_621_bot__nat__0_Oextremum__uniqueI,axiom,
! [A: nat] :
( ( ord_less_eq_nat @ A @ zero_zero_nat )
=> ( A = zero_zero_nat ) ) ).
% bot_nat_0.extremum_uniqueI
thf(fact_622_le__0__eq,axiom,
! [N2: nat] :
( ( ord_less_eq_nat @ N2 @ zero_zero_nat )
= ( N2 = zero_zero_nat ) ) ).
% le_0_eq
thf(fact_623_ex__least__nat__le,axiom,
! [P: nat > $o,N2: nat] :
( ( P @ N2 )
=> ( ~ ( P @ zero_zero_nat )
=> ? [K: nat] :
( ( ord_less_eq_nat @ K @ N2 )
& ! [I3: nat] :
( ( ord_less_nat @ I3 @ K )
=> ~ ( P @ I3 ) )
& ( P @ K ) ) ) ) ).
% ex_least_nat_le
thf(fact_624_cproper__interval__sum_Opelims_I3_J,axiom,
! [X: option3242699076814927183at_nat,Xa: option3242699076814927183at_nat] :
( ~ ( collec7691303616722397604at_nat @ X @ Xa )
=> ( ( accp_P3884639961936550920at_nat @ collec4724013795633334532at_nat @ ( produc7249471109196833079at_nat @ X @ Xa ) )
=> ( ( ( X = none_Sum_sum_nat_nat )
=> ! [X3: nat] :
( ( Xa
= ( some_Sum_sum_nat_nat @ ( sum_Inl_nat_nat @ X3 ) ) )
=> ( ( accp_P3884639961936550920at_nat @ collec4724013795633334532at_nat @ ( produc7249471109196833079at_nat @ none_Sum_sum_nat_nat @ ( some_Sum_sum_nat_nat @ ( sum_Inl_nat_nat @ X3 ) ) ) )
=> ( collec4140898570808001013al_nat @ none_nat @ ( some_nat @ X3 ) ) ) ) )
=> ( ! [X3: nat] :
( ( X
= ( some_Sum_sum_nat_nat @ ( sum_Inl_nat_nat @ X3 ) ) )
=> ! [Y3: nat] :
( ( Xa
= ( some_Sum_sum_nat_nat @ ( sum_Inl_nat_nat @ Y3 ) ) )
=> ( ( accp_P3884639961936550920at_nat @ collec4724013795633334532at_nat @ ( produc7249471109196833079at_nat @ ( some_Sum_sum_nat_nat @ ( sum_Inl_nat_nat @ X3 ) ) @ ( some_Sum_sum_nat_nat @ ( sum_Inl_nat_nat @ Y3 ) ) ) )
=> ( collec4140898570808001013al_nat @ ( some_nat @ X3 ) @ ( some_nat @ Y3 ) ) ) ) )
=> ( ! [X3: nat] :
( ( X
= ( some_Sum_sum_nat_nat @ ( sum_Inl_nat_nat @ X3 ) ) )
=> ! [Y3: nat] :
( ( Xa
= ( some_Sum_sum_nat_nat @ ( sum_Inr_nat_nat @ Y3 ) ) )
=> ( ( accp_P3884639961936550920at_nat @ collec4724013795633334532at_nat @ ( produc7249471109196833079at_nat @ ( some_Sum_sum_nat_nat @ ( sum_Inl_nat_nat @ X3 ) ) @ ( some_Sum_sum_nat_nat @ ( sum_Inr_nat_nat @ Y3 ) ) ) )
=> ( ( collec4140898570808001013al_nat @ ( some_nat @ X3 ) @ none_nat )
| ( collec4140898570808001013al_nat @ none_nat @ ( some_nat @ Y3 ) ) ) ) ) )
=> ( ! [Y3: nat] :
( ( X
= ( some_Sum_sum_nat_nat @ ( sum_Inr_nat_nat @ Y3 ) ) )
=> ( ( Xa = none_Sum_sum_nat_nat )
=> ( ( accp_P3884639961936550920at_nat @ collec4724013795633334532at_nat @ ( produc7249471109196833079at_nat @ ( some_Sum_sum_nat_nat @ ( sum_Inr_nat_nat @ Y3 ) ) @ none_Sum_sum_nat_nat ) )
=> ( collec4140898570808001013al_nat @ ( some_nat @ Y3 ) @ none_nat ) ) ) )
=> ( ! [Y3: nat] :
( ( X
= ( some_Sum_sum_nat_nat @ ( sum_Inr_nat_nat @ Y3 ) ) )
=> ! [X3: nat] :
( ( Xa
= ( some_Sum_sum_nat_nat @ ( sum_Inl_nat_nat @ X3 ) ) )
=> ~ ( accp_P3884639961936550920at_nat @ collec4724013795633334532at_nat @ ( produc7249471109196833079at_nat @ ( some_Sum_sum_nat_nat @ ( sum_Inr_nat_nat @ Y3 ) ) @ ( some_Sum_sum_nat_nat @ ( sum_Inl_nat_nat @ X3 ) ) ) ) ) )
=> ~ ! [X3: nat] :
( ( X
= ( some_Sum_sum_nat_nat @ ( sum_Inr_nat_nat @ X3 ) ) )
=> ! [Y3: nat] :
( ( Xa
= ( some_Sum_sum_nat_nat @ ( sum_Inr_nat_nat @ Y3 ) ) )
=> ( ( accp_P3884639961936550920at_nat @ collec4724013795633334532at_nat @ ( produc7249471109196833079at_nat @ ( some_Sum_sum_nat_nat @ ( sum_Inr_nat_nat @ X3 ) ) @ ( some_Sum_sum_nat_nat @ ( sum_Inr_nat_nat @ Y3 ) ) ) )
=> ( collec4140898570808001013al_nat @ ( some_nat @ X3 ) @ ( some_nat @ Y3 ) ) ) ) ) ) ) ) ) ) ) ) ).
% cproper_interval_sum.pelims(3)
thf(fact_625_cproper__interval__sum_Opelims_I3_J,axiom,
! [X: option752615931883200427at_int,Xa: option752615931883200427at_int] :
( ~ ( collec3513452597213200896at_int @ X @ Xa )
=> ( ( accp_P479778451830304448at_int @ collec4721523325124284256at_int @ ( produc6168074872946778607at_int @ X @ Xa ) )
=> ( ( ( X = none_Sum_sum_nat_int )
=> ! [X3: nat] :
( ( Xa
= ( some_Sum_sum_nat_int @ ( sum_Inl_nat_int @ X3 ) ) )
=> ( ( accp_P479778451830304448at_int @ collec4721523325124284256at_int @ ( produc6168074872946778607at_int @ none_Sum_sum_nat_int @ ( some_Sum_sum_nat_int @ ( sum_Inl_nat_int @ X3 ) ) ) )
=> ( collec4140898570808001013al_nat @ none_nat @ ( some_nat @ X3 ) ) ) ) )
=> ( ! [X3: nat] :
( ( X
= ( some_Sum_sum_nat_int @ ( sum_Inl_nat_int @ X3 ) ) )
=> ! [Y3: nat] :
( ( Xa
= ( some_Sum_sum_nat_int @ ( sum_Inl_nat_int @ Y3 ) ) )
=> ( ( accp_P479778451830304448at_int @ collec4721523325124284256at_int @ ( produc6168074872946778607at_int @ ( some_Sum_sum_nat_int @ ( sum_Inl_nat_int @ X3 ) ) @ ( some_Sum_sum_nat_int @ ( sum_Inl_nat_int @ Y3 ) ) ) )
=> ( collec4140898570808001013al_nat @ ( some_nat @ X3 ) @ ( some_nat @ Y3 ) ) ) ) )
=> ( ! [X3: nat] :
( ( X
= ( some_Sum_sum_nat_int @ ( sum_Inl_nat_int @ X3 ) ) )
=> ! [Y3: int] :
( ( Xa
= ( some_Sum_sum_nat_int @ ( sum_Inr_int_nat @ Y3 ) ) )
=> ( ( accp_P479778451830304448at_int @ collec4721523325124284256at_int @ ( produc6168074872946778607at_int @ ( some_Sum_sum_nat_int @ ( sum_Inl_nat_int @ X3 ) ) @ ( some_Sum_sum_nat_int @ ( sum_Inr_int_nat @ Y3 ) ) ) )
=> ( ( collec4140898570808001013al_nat @ ( some_nat @ X3 ) @ none_nat )
| ( collec4138408100298950737al_int @ none_int @ ( some_int @ Y3 ) ) ) ) ) )
=> ( ! [Y3: int] :
( ( X
= ( some_Sum_sum_nat_int @ ( sum_Inr_int_nat @ Y3 ) ) )
=> ( ( Xa = none_Sum_sum_nat_int )
=> ( ( accp_P479778451830304448at_int @ collec4721523325124284256at_int @ ( produc6168074872946778607at_int @ ( some_Sum_sum_nat_int @ ( sum_Inr_int_nat @ Y3 ) ) @ none_Sum_sum_nat_int ) )
=> ( collec4138408100298950737al_int @ ( some_int @ Y3 ) @ none_int ) ) ) )
=> ( ! [Y3: int] :
( ( X
= ( some_Sum_sum_nat_int @ ( sum_Inr_int_nat @ Y3 ) ) )
=> ! [X3: nat] :
( ( Xa
= ( some_Sum_sum_nat_int @ ( sum_Inl_nat_int @ X3 ) ) )
=> ~ ( accp_P479778451830304448at_int @ collec4721523325124284256at_int @ ( produc6168074872946778607at_int @ ( some_Sum_sum_nat_int @ ( sum_Inr_int_nat @ Y3 ) ) @ ( some_Sum_sum_nat_int @ ( sum_Inl_nat_int @ X3 ) ) ) ) ) )
=> ~ ! [X3: int] :
( ( X
= ( some_Sum_sum_nat_int @ ( sum_Inr_int_nat @ X3 ) ) )
=> ! [Y3: int] :
( ( Xa
= ( some_Sum_sum_nat_int @ ( sum_Inr_int_nat @ Y3 ) ) )
=> ( ( accp_P479778451830304448at_int @ collec4721523325124284256at_int @ ( produc6168074872946778607at_int @ ( some_Sum_sum_nat_int @ ( sum_Inr_int_nat @ X3 ) ) @ ( some_Sum_sum_nat_int @ ( sum_Inr_int_nat @ Y3 ) ) ) )
=> ( collec4138408100298950737al_int @ ( some_int @ X3 ) @ ( some_int @ Y3 ) ) ) ) ) ) ) ) ) ) ) ) ).
% cproper_interval_sum.pelims(3)
thf(fact_626_cproper__interval__sum_Opelims_I3_J,axiom,
! [X: option5429620651507948971nt_nat,Xa: option5429620651507948971nt_nat] :
( ~ ( collec8690657328623452160nt_nat @ X @ Xa )
=> ( ( accp_P3675550703157275328nt_nat @ collec5723367507534389088nt_nat @ ( produc6008599830946209263nt_nat @ X @ Xa ) )
=> ( ( ( X = none_Sum_sum_int_nat )
=> ! [X3: int] :
( ( Xa
= ( some_Sum_sum_int_nat @ ( sum_Inl_int_nat @ X3 ) ) )
=> ( ( accp_P3675550703157275328nt_nat @ collec5723367507534389088nt_nat @ ( produc6008599830946209263nt_nat @ none_Sum_sum_int_nat @ ( some_Sum_sum_int_nat @ ( sum_Inl_int_nat @ X3 ) ) ) )
=> ( collec4138408100298950737al_int @ none_int @ ( some_int @ X3 ) ) ) ) )
=> ( ! [X3: int] :
( ( X
= ( some_Sum_sum_int_nat @ ( sum_Inl_int_nat @ X3 ) ) )
=> ! [Y3: int] :
( ( Xa
= ( some_Sum_sum_int_nat @ ( sum_Inl_int_nat @ Y3 ) ) )
=> ( ( accp_P3675550703157275328nt_nat @ collec5723367507534389088nt_nat @ ( produc6008599830946209263nt_nat @ ( some_Sum_sum_int_nat @ ( sum_Inl_int_nat @ X3 ) ) @ ( some_Sum_sum_int_nat @ ( sum_Inl_int_nat @ Y3 ) ) ) )
=> ( collec4138408100298950737al_int @ ( some_int @ X3 ) @ ( some_int @ Y3 ) ) ) ) )
=> ( ! [X3: int] :
( ( X
= ( some_Sum_sum_int_nat @ ( sum_Inl_int_nat @ X3 ) ) )
=> ! [Y3: nat] :
( ( Xa
= ( some_Sum_sum_int_nat @ ( sum_Inr_nat_int @ Y3 ) ) )
=> ( ( accp_P3675550703157275328nt_nat @ collec5723367507534389088nt_nat @ ( produc6008599830946209263nt_nat @ ( some_Sum_sum_int_nat @ ( sum_Inl_int_nat @ X3 ) ) @ ( some_Sum_sum_int_nat @ ( sum_Inr_nat_int @ Y3 ) ) ) )
=> ( ( collec4138408100298950737al_int @ ( some_int @ X3 ) @ none_int )
| ( collec4140898570808001013al_nat @ none_nat @ ( some_nat @ Y3 ) ) ) ) ) )
=> ( ! [Y3: nat] :
( ( X
= ( some_Sum_sum_int_nat @ ( sum_Inr_nat_int @ Y3 ) ) )
=> ( ( Xa = none_Sum_sum_int_nat )
=> ( ( accp_P3675550703157275328nt_nat @ collec5723367507534389088nt_nat @ ( produc6008599830946209263nt_nat @ ( some_Sum_sum_int_nat @ ( sum_Inr_nat_int @ Y3 ) ) @ none_Sum_sum_int_nat ) )
=> ( collec4140898570808001013al_nat @ ( some_nat @ Y3 ) @ none_nat ) ) ) )
=> ( ! [Y3: nat] :
( ( X
= ( some_Sum_sum_int_nat @ ( sum_Inr_nat_int @ Y3 ) ) )
=> ! [X3: int] :
( ( Xa
= ( some_Sum_sum_int_nat @ ( sum_Inl_int_nat @ X3 ) ) )
=> ~ ( accp_P3675550703157275328nt_nat @ collec5723367507534389088nt_nat @ ( produc6008599830946209263nt_nat @ ( some_Sum_sum_int_nat @ ( sum_Inr_nat_int @ Y3 ) ) @ ( some_Sum_sum_int_nat @ ( sum_Inl_int_nat @ X3 ) ) ) ) ) )
=> ~ ! [X3: nat] :
( ( X
= ( some_Sum_sum_int_nat @ ( sum_Inr_nat_int @ X3 ) ) )
=> ! [Y3: nat] :
( ( Xa
= ( some_Sum_sum_int_nat @ ( sum_Inr_nat_int @ Y3 ) ) )
=> ( ( accp_P3675550703157275328nt_nat @ collec5723367507534389088nt_nat @ ( produc6008599830946209263nt_nat @ ( some_Sum_sum_int_nat @ ( sum_Inr_nat_int @ X3 ) ) @ ( some_Sum_sum_int_nat @ ( sum_Inr_nat_int @ Y3 ) ) ) )
=> ( collec4140898570808001013al_nat @ ( some_nat @ X3 ) @ ( some_nat @ Y3 ) ) ) ) ) ) ) ) ) ) ) ) ).
% cproper_interval_sum.pelims(3)
thf(fact_627_cproper__interval__sum_Opelims_I3_J,axiom,
! [X: option2939537506576222215nt_int,Xa: option2939537506576222215nt_int] :
( ~ ( collec4512806309114255452nt_int @ X @ Xa )
=> ( ( accp_P270689193051028856nt_int @ collec5720877037025338812nt_int @ ( produc4927203594696154791nt_int @ X @ Xa ) )
=> ( ( ( X = none_Sum_sum_int_int )
=> ! [X3: int] :
( ( Xa
= ( some_Sum_sum_int_int @ ( sum_Inl_int_int @ X3 ) ) )
=> ( ( accp_P270689193051028856nt_int @ collec5720877037025338812nt_int @ ( produc4927203594696154791nt_int @ none_Sum_sum_int_int @ ( some_Sum_sum_int_int @ ( sum_Inl_int_int @ X3 ) ) ) )
=> ( collec4138408100298950737al_int @ none_int @ ( some_int @ X3 ) ) ) ) )
=> ( ! [X3: int] :
( ( X
= ( some_Sum_sum_int_int @ ( sum_Inl_int_int @ X3 ) ) )
=> ! [Y3: int] :
( ( Xa
= ( some_Sum_sum_int_int @ ( sum_Inl_int_int @ Y3 ) ) )
=> ( ( accp_P270689193051028856nt_int @ collec5720877037025338812nt_int @ ( produc4927203594696154791nt_int @ ( some_Sum_sum_int_int @ ( sum_Inl_int_int @ X3 ) ) @ ( some_Sum_sum_int_int @ ( sum_Inl_int_int @ Y3 ) ) ) )
=> ( collec4138408100298950737al_int @ ( some_int @ X3 ) @ ( some_int @ Y3 ) ) ) ) )
=> ( ! [X3: int] :
( ( X
= ( some_Sum_sum_int_int @ ( sum_Inl_int_int @ X3 ) ) )
=> ! [Y3: int] :
( ( Xa
= ( some_Sum_sum_int_int @ ( sum_Inr_int_int @ Y3 ) ) )
=> ( ( accp_P270689193051028856nt_int @ collec5720877037025338812nt_int @ ( produc4927203594696154791nt_int @ ( some_Sum_sum_int_int @ ( sum_Inl_int_int @ X3 ) ) @ ( some_Sum_sum_int_int @ ( sum_Inr_int_int @ Y3 ) ) ) )
=> ( ( collec4138408100298950737al_int @ ( some_int @ X3 ) @ none_int )
| ( collec4138408100298950737al_int @ none_int @ ( some_int @ Y3 ) ) ) ) ) )
=> ( ! [Y3: int] :
( ( X
= ( some_Sum_sum_int_int @ ( sum_Inr_int_int @ Y3 ) ) )
=> ( ( Xa = none_Sum_sum_int_int )
=> ( ( accp_P270689193051028856nt_int @ collec5720877037025338812nt_int @ ( produc4927203594696154791nt_int @ ( some_Sum_sum_int_int @ ( sum_Inr_int_int @ Y3 ) ) @ none_Sum_sum_int_int ) )
=> ( collec4138408100298950737al_int @ ( some_int @ Y3 ) @ none_int ) ) ) )
=> ( ! [Y3: int] :
( ( X
= ( some_Sum_sum_int_int @ ( sum_Inr_int_int @ Y3 ) ) )
=> ! [X3: int] :
( ( Xa
= ( some_Sum_sum_int_int @ ( sum_Inl_int_int @ X3 ) ) )
=> ~ ( accp_P270689193051028856nt_int @ collec5720877037025338812nt_int @ ( produc4927203594696154791nt_int @ ( some_Sum_sum_int_int @ ( sum_Inr_int_int @ Y3 ) ) @ ( some_Sum_sum_int_int @ ( sum_Inl_int_int @ X3 ) ) ) ) ) )
=> ~ ! [X3: int] :
( ( X
= ( some_Sum_sum_int_int @ ( sum_Inr_int_int @ X3 ) ) )
=> ! [Y3: int] :
( ( Xa
= ( some_Sum_sum_int_int @ ( sum_Inr_int_int @ Y3 ) ) )
=> ( ( accp_P270689193051028856nt_int @ collec5720877037025338812nt_int @ ( produc4927203594696154791nt_int @ ( some_Sum_sum_int_int @ ( sum_Inr_int_int @ X3 ) ) @ ( some_Sum_sum_int_int @ ( sum_Inr_int_int @ Y3 ) ) ) )
=> ( collec4138408100298950737al_int @ ( some_int @ X3 ) @ ( some_int @ Y3 ) ) ) ) ) ) ) ) ) ) ) ) ).
% cproper_interval_sum.pelims(3)
thf(fact_628_cproper__interval__sum_Opelims_I2_J,axiom,
! [X: option3242699076814927183at_nat,Xa: option3242699076814927183at_nat] :
( ( collec7691303616722397604at_nat @ X @ Xa )
=> ( ( accp_P3884639961936550920at_nat @ collec4724013795633334532at_nat @ ( produc7249471109196833079at_nat @ X @ Xa ) )
=> ( ( ( X = none_Sum_sum_nat_nat )
=> ( ( Xa = none_Sum_sum_nat_nat )
=> ~ ( accp_P3884639961936550920at_nat @ collec4724013795633334532at_nat @ ( produc7249471109196833079at_nat @ none_Sum_sum_nat_nat @ none_Sum_sum_nat_nat ) ) ) )
=> ( ( ( X = none_Sum_sum_nat_nat )
=> ! [X3: nat] :
( ( Xa
= ( some_Sum_sum_nat_nat @ ( sum_Inl_nat_nat @ X3 ) ) )
=> ( ( accp_P3884639961936550920at_nat @ collec4724013795633334532at_nat @ ( produc7249471109196833079at_nat @ none_Sum_sum_nat_nat @ ( some_Sum_sum_nat_nat @ ( sum_Inl_nat_nat @ X3 ) ) ) )
=> ~ ( collec4140898570808001013al_nat @ none_nat @ ( some_nat @ X3 ) ) ) ) )
=> ( ( ( X = none_Sum_sum_nat_nat )
=> ! [Y3: nat] :
( ( Xa
= ( some_Sum_sum_nat_nat @ ( sum_Inr_nat_nat @ Y3 ) ) )
=> ~ ( accp_P3884639961936550920at_nat @ collec4724013795633334532at_nat @ ( produc7249471109196833079at_nat @ none_Sum_sum_nat_nat @ ( some_Sum_sum_nat_nat @ ( sum_Inr_nat_nat @ Y3 ) ) ) ) ) )
=> ( ! [X3: nat] :
( ( X
= ( some_Sum_sum_nat_nat @ ( sum_Inl_nat_nat @ X3 ) ) )
=> ( ( Xa = none_Sum_sum_nat_nat )
=> ~ ( accp_P3884639961936550920at_nat @ collec4724013795633334532at_nat @ ( produc7249471109196833079at_nat @ ( some_Sum_sum_nat_nat @ ( sum_Inl_nat_nat @ X3 ) ) @ none_Sum_sum_nat_nat ) ) ) )
=> ( ! [X3: nat] :
( ( X
= ( some_Sum_sum_nat_nat @ ( sum_Inl_nat_nat @ X3 ) ) )
=> ! [Y3: nat] :
( ( Xa
= ( some_Sum_sum_nat_nat @ ( sum_Inl_nat_nat @ Y3 ) ) )
=> ( ( accp_P3884639961936550920at_nat @ collec4724013795633334532at_nat @ ( produc7249471109196833079at_nat @ ( some_Sum_sum_nat_nat @ ( sum_Inl_nat_nat @ X3 ) ) @ ( some_Sum_sum_nat_nat @ ( sum_Inl_nat_nat @ Y3 ) ) ) )
=> ~ ( collec4140898570808001013al_nat @ ( some_nat @ X3 ) @ ( some_nat @ Y3 ) ) ) ) )
=> ( ! [X3: nat] :
( ( X
= ( some_Sum_sum_nat_nat @ ( sum_Inl_nat_nat @ X3 ) ) )
=> ! [Y3: nat] :
( ( Xa
= ( some_Sum_sum_nat_nat @ ( sum_Inr_nat_nat @ Y3 ) ) )
=> ( ( accp_P3884639961936550920at_nat @ collec4724013795633334532at_nat @ ( produc7249471109196833079at_nat @ ( some_Sum_sum_nat_nat @ ( sum_Inl_nat_nat @ X3 ) ) @ ( some_Sum_sum_nat_nat @ ( sum_Inr_nat_nat @ Y3 ) ) ) )
=> ~ ( ( collec4140898570808001013al_nat @ ( some_nat @ X3 ) @ none_nat )
| ( collec4140898570808001013al_nat @ none_nat @ ( some_nat @ Y3 ) ) ) ) ) )
=> ( ! [Y3: nat] :
( ( X
= ( some_Sum_sum_nat_nat @ ( sum_Inr_nat_nat @ Y3 ) ) )
=> ( ( Xa = none_Sum_sum_nat_nat )
=> ( ( accp_P3884639961936550920at_nat @ collec4724013795633334532at_nat @ ( produc7249471109196833079at_nat @ ( some_Sum_sum_nat_nat @ ( sum_Inr_nat_nat @ Y3 ) ) @ none_Sum_sum_nat_nat ) )
=> ~ ( collec4140898570808001013al_nat @ ( some_nat @ Y3 ) @ none_nat ) ) ) )
=> ~ ! [X3: nat] :
( ( X
= ( some_Sum_sum_nat_nat @ ( sum_Inr_nat_nat @ X3 ) ) )
=> ! [Y3: nat] :
( ( Xa
= ( some_Sum_sum_nat_nat @ ( sum_Inr_nat_nat @ Y3 ) ) )
=> ( ( accp_P3884639961936550920at_nat @ collec4724013795633334532at_nat @ ( produc7249471109196833079at_nat @ ( some_Sum_sum_nat_nat @ ( sum_Inr_nat_nat @ X3 ) ) @ ( some_Sum_sum_nat_nat @ ( sum_Inr_nat_nat @ Y3 ) ) ) )
=> ~ ( collec4140898570808001013al_nat @ ( some_nat @ X3 ) @ ( some_nat @ Y3 ) ) ) ) ) ) ) ) ) ) ) ) ) ) ).
% cproper_interval_sum.pelims(2)
thf(fact_629_cproper__interval__sum_Opelims_I2_J,axiom,
! [X: option752615931883200427at_int,Xa: option752615931883200427at_int] :
( ( collec3513452597213200896at_int @ X @ Xa )
=> ( ( accp_P479778451830304448at_int @ collec4721523325124284256at_int @ ( produc6168074872946778607at_int @ X @ Xa ) )
=> ( ( ( X = none_Sum_sum_nat_int )
=> ( ( Xa = none_Sum_sum_nat_int )
=> ~ ( accp_P479778451830304448at_int @ collec4721523325124284256at_int @ ( produc6168074872946778607at_int @ none_Sum_sum_nat_int @ none_Sum_sum_nat_int ) ) ) )
=> ( ( ( X = none_Sum_sum_nat_int )
=> ! [X3: nat] :
( ( Xa
= ( some_Sum_sum_nat_int @ ( sum_Inl_nat_int @ X3 ) ) )
=> ( ( accp_P479778451830304448at_int @ collec4721523325124284256at_int @ ( produc6168074872946778607at_int @ none_Sum_sum_nat_int @ ( some_Sum_sum_nat_int @ ( sum_Inl_nat_int @ X3 ) ) ) )
=> ~ ( collec4140898570808001013al_nat @ none_nat @ ( some_nat @ X3 ) ) ) ) )
=> ( ( ( X = none_Sum_sum_nat_int )
=> ! [Y3: int] :
( ( Xa
= ( some_Sum_sum_nat_int @ ( sum_Inr_int_nat @ Y3 ) ) )
=> ~ ( accp_P479778451830304448at_int @ collec4721523325124284256at_int @ ( produc6168074872946778607at_int @ none_Sum_sum_nat_int @ ( some_Sum_sum_nat_int @ ( sum_Inr_int_nat @ Y3 ) ) ) ) ) )
=> ( ! [X3: nat] :
( ( X
= ( some_Sum_sum_nat_int @ ( sum_Inl_nat_int @ X3 ) ) )
=> ( ( Xa = none_Sum_sum_nat_int )
=> ~ ( accp_P479778451830304448at_int @ collec4721523325124284256at_int @ ( produc6168074872946778607at_int @ ( some_Sum_sum_nat_int @ ( sum_Inl_nat_int @ X3 ) ) @ none_Sum_sum_nat_int ) ) ) )
=> ( ! [X3: nat] :
( ( X
= ( some_Sum_sum_nat_int @ ( sum_Inl_nat_int @ X3 ) ) )
=> ! [Y3: nat] :
( ( Xa
= ( some_Sum_sum_nat_int @ ( sum_Inl_nat_int @ Y3 ) ) )
=> ( ( accp_P479778451830304448at_int @ collec4721523325124284256at_int @ ( produc6168074872946778607at_int @ ( some_Sum_sum_nat_int @ ( sum_Inl_nat_int @ X3 ) ) @ ( some_Sum_sum_nat_int @ ( sum_Inl_nat_int @ Y3 ) ) ) )
=> ~ ( collec4140898570808001013al_nat @ ( some_nat @ X3 ) @ ( some_nat @ Y3 ) ) ) ) )
=> ( ! [X3: nat] :
( ( X
= ( some_Sum_sum_nat_int @ ( sum_Inl_nat_int @ X3 ) ) )
=> ! [Y3: int] :
( ( Xa
= ( some_Sum_sum_nat_int @ ( sum_Inr_int_nat @ Y3 ) ) )
=> ( ( accp_P479778451830304448at_int @ collec4721523325124284256at_int @ ( produc6168074872946778607at_int @ ( some_Sum_sum_nat_int @ ( sum_Inl_nat_int @ X3 ) ) @ ( some_Sum_sum_nat_int @ ( sum_Inr_int_nat @ Y3 ) ) ) )
=> ~ ( ( collec4140898570808001013al_nat @ ( some_nat @ X3 ) @ none_nat )
| ( collec4138408100298950737al_int @ none_int @ ( some_int @ Y3 ) ) ) ) ) )
=> ( ! [Y3: int] :
( ( X
= ( some_Sum_sum_nat_int @ ( sum_Inr_int_nat @ Y3 ) ) )
=> ( ( Xa = none_Sum_sum_nat_int )
=> ( ( accp_P479778451830304448at_int @ collec4721523325124284256at_int @ ( produc6168074872946778607at_int @ ( some_Sum_sum_nat_int @ ( sum_Inr_int_nat @ Y3 ) ) @ none_Sum_sum_nat_int ) )
=> ~ ( collec4138408100298950737al_int @ ( some_int @ Y3 ) @ none_int ) ) ) )
=> ~ ! [X3: int] :
( ( X
= ( some_Sum_sum_nat_int @ ( sum_Inr_int_nat @ X3 ) ) )
=> ! [Y3: int] :
( ( Xa
= ( some_Sum_sum_nat_int @ ( sum_Inr_int_nat @ Y3 ) ) )
=> ( ( accp_P479778451830304448at_int @ collec4721523325124284256at_int @ ( produc6168074872946778607at_int @ ( some_Sum_sum_nat_int @ ( sum_Inr_int_nat @ X3 ) ) @ ( some_Sum_sum_nat_int @ ( sum_Inr_int_nat @ Y3 ) ) ) )
=> ~ ( collec4138408100298950737al_int @ ( some_int @ X3 ) @ ( some_int @ Y3 ) ) ) ) ) ) ) ) ) ) ) ) ) ) ).
% cproper_interval_sum.pelims(2)
thf(fact_630_cproper__interval__sum_Opelims_I2_J,axiom,
! [X: option5429620651507948971nt_nat,Xa: option5429620651507948971nt_nat] :
( ( collec8690657328623452160nt_nat @ X @ Xa )
=> ( ( accp_P3675550703157275328nt_nat @ collec5723367507534389088nt_nat @ ( produc6008599830946209263nt_nat @ X @ Xa ) )
=> ( ( ( X = none_Sum_sum_int_nat )
=> ( ( Xa = none_Sum_sum_int_nat )
=> ~ ( accp_P3675550703157275328nt_nat @ collec5723367507534389088nt_nat @ ( produc6008599830946209263nt_nat @ none_Sum_sum_int_nat @ none_Sum_sum_int_nat ) ) ) )
=> ( ( ( X = none_Sum_sum_int_nat )
=> ! [X3: int] :
( ( Xa
= ( some_Sum_sum_int_nat @ ( sum_Inl_int_nat @ X3 ) ) )
=> ( ( accp_P3675550703157275328nt_nat @ collec5723367507534389088nt_nat @ ( produc6008599830946209263nt_nat @ none_Sum_sum_int_nat @ ( some_Sum_sum_int_nat @ ( sum_Inl_int_nat @ X3 ) ) ) )
=> ~ ( collec4138408100298950737al_int @ none_int @ ( some_int @ X3 ) ) ) ) )
=> ( ( ( X = none_Sum_sum_int_nat )
=> ! [Y3: nat] :
( ( Xa
= ( some_Sum_sum_int_nat @ ( sum_Inr_nat_int @ Y3 ) ) )
=> ~ ( accp_P3675550703157275328nt_nat @ collec5723367507534389088nt_nat @ ( produc6008599830946209263nt_nat @ none_Sum_sum_int_nat @ ( some_Sum_sum_int_nat @ ( sum_Inr_nat_int @ Y3 ) ) ) ) ) )
=> ( ! [X3: int] :
( ( X
= ( some_Sum_sum_int_nat @ ( sum_Inl_int_nat @ X3 ) ) )
=> ( ( Xa = none_Sum_sum_int_nat )
=> ~ ( accp_P3675550703157275328nt_nat @ collec5723367507534389088nt_nat @ ( produc6008599830946209263nt_nat @ ( some_Sum_sum_int_nat @ ( sum_Inl_int_nat @ X3 ) ) @ none_Sum_sum_int_nat ) ) ) )
=> ( ! [X3: int] :
( ( X
= ( some_Sum_sum_int_nat @ ( sum_Inl_int_nat @ X3 ) ) )
=> ! [Y3: int] :
( ( Xa
= ( some_Sum_sum_int_nat @ ( sum_Inl_int_nat @ Y3 ) ) )
=> ( ( accp_P3675550703157275328nt_nat @ collec5723367507534389088nt_nat @ ( produc6008599830946209263nt_nat @ ( some_Sum_sum_int_nat @ ( sum_Inl_int_nat @ X3 ) ) @ ( some_Sum_sum_int_nat @ ( sum_Inl_int_nat @ Y3 ) ) ) )
=> ~ ( collec4138408100298950737al_int @ ( some_int @ X3 ) @ ( some_int @ Y3 ) ) ) ) )
=> ( ! [X3: int] :
( ( X
= ( some_Sum_sum_int_nat @ ( sum_Inl_int_nat @ X3 ) ) )
=> ! [Y3: nat] :
( ( Xa
= ( some_Sum_sum_int_nat @ ( sum_Inr_nat_int @ Y3 ) ) )
=> ( ( accp_P3675550703157275328nt_nat @ collec5723367507534389088nt_nat @ ( produc6008599830946209263nt_nat @ ( some_Sum_sum_int_nat @ ( sum_Inl_int_nat @ X3 ) ) @ ( some_Sum_sum_int_nat @ ( sum_Inr_nat_int @ Y3 ) ) ) )
=> ~ ( ( collec4138408100298950737al_int @ ( some_int @ X3 ) @ none_int )
| ( collec4140898570808001013al_nat @ none_nat @ ( some_nat @ Y3 ) ) ) ) ) )
=> ( ! [Y3: nat] :
( ( X
= ( some_Sum_sum_int_nat @ ( sum_Inr_nat_int @ Y3 ) ) )
=> ( ( Xa = none_Sum_sum_int_nat )
=> ( ( accp_P3675550703157275328nt_nat @ collec5723367507534389088nt_nat @ ( produc6008599830946209263nt_nat @ ( some_Sum_sum_int_nat @ ( sum_Inr_nat_int @ Y3 ) ) @ none_Sum_sum_int_nat ) )
=> ~ ( collec4140898570808001013al_nat @ ( some_nat @ Y3 ) @ none_nat ) ) ) )
=> ~ ! [X3: nat] :
( ( X
= ( some_Sum_sum_int_nat @ ( sum_Inr_nat_int @ X3 ) ) )
=> ! [Y3: nat] :
( ( Xa
= ( some_Sum_sum_int_nat @ ( sum_Inr_nat_int @ Y3 ) ) )
=> ( ( accp_P3675550703157275328nt_nat @ collec5723367507534389088nt_nat @ ( produc6008599830946209263nt_nat @ ( some_Sum_sum_int_nat @ ( sum_Inr_nat_int @ X3 ) ) @ ( some_Sum_sum_int_nat @ ( sum_Inr_nat_int @ Y3 ) ) ) )
=> ~ ( collec4140898570808001013al_nat @ ( some_nat @ X3 ) @ ( some_nat @ Y3 ) ) ) ) ) ) ) ) ) ) ) ) ) ) ).
% cproper_interval_sum.pelims(2)
thf(fact_631_cproper__interval__sum_Opelims_I2_J,axiom,
! [X: option2939537506576222215nt_int,Xa: option2939537506576222215nt_int] :
( ( collec4512806309114255452nt_int @ X @ Xa )
=> ( ( accp_P270689193051028856nt_int @ collec5720877037025338812nt_int @ ( produc4927203594696154791nt_int @ X @ Xa ) )
=> ( ( ( X = none_Sum_sum_int_int )
=> ( ( Xa = none_Sum_sum_int_int )
=> ~ ( accp_P270689193051028856nt_int @ collec5720877037025338812nt_int @ ( produc4927203594696154791nt_int @ none_Sum_sum_int_int @ none_Sum_sum_int_int ) ) ) )
=> ( ( ( X = none_Sum_sum_int_int )
=> ! [X3: int] :
( ( Xa
= ( some_Sum_sum_int_int @ ( sum_Inl_int_int @ X3 ) ) )
=> ( ( accp_P270689193051028856nt_int @ collec5720877037025338812nt_int @ ( produc4927203594696154791nt_int @ none_Sum_sum_int_int @ ( some_Sum_sum_int_int @ ( sum_Inl_int_int @ X3 ) ) ) )
=> ~ ( collec4138408100298950737al_int @ none_int @ ( some_int @ X3 ) ) ) ) )
=> ( ( ( X = none_Sum_sum_int_int )
=> ! [Y3: int] :
( ( Xa
= ( some_Sum_sum_int_int @ ( sum_Inr_int_int @ Y3 ) ) )
=> ~ ( accp_P270689193051028856nt_int @ collec5720877037025338812nt_int @ ( produc4927203594696154791nt_int @ none_Sum_sum_int_int @ ( some_Sum_sum_int_int @ ( sum_Inr_int_int @ Y3 ) ) ) ) ) )
=> ( ! [X3: int] :
( ( X
= ( some_Sum_sum_int_int @ ( sum_Inl_int_int @ X3 ) ) )
=> ( ( Xa = none_Sum_sum_int_int )
=> ~ ( accp_P270689193051028856nt_int @ collec5720877037025338812nt_int @ ( produc4927203594696154791nt_int @ ( some_Sum_sum_int_int @ ( sum_Inl_int_int @ X3 ) ) @ none_Sum_sum_int_int ) ) ) )
=> ( ! [X3: int] :
( ( X
= ( some_Sum_sum_int_int @ ( sum_Inl_int_int @ X3 ) ) )
=> ! [Y3: int] :
( ( Xa
= ( some_Sum_sum_int_int @ ( sum_Inl_int_int @ Y3 ) ) )
=> ( ( accp_P270689193051028856nt_int @ collec5720877037025338812nt_int @ ( produc4927203594696154791nt_int @ ( some_Sum_sum_int_int @ ( sum_Inl_int_int @ X3 ) ) @ ( some_Sum_sum_int_int @ ( sum_Inl_int_int @ Y3 ) ) ) )
=> ~ ( collec4138408100298950737al_int @ ( some_int @ X3 ) @ ( some_int @ Y3 ) ) ) ) )
=> ( ! [X3: int] :
( ( X
= ( some_Sum_sum_int_int @ ( sum_Inl_int_int @ X3 ) ) )
=> ! [Y3: int] :
( ( Xa
= ( some_Sum_sum_int_int @ ( sum_Inr_int_int @ Y3 ) ) )
=> ( ( accp_P270689193051028856nt_int @ collec5720877037025338812nt_int @ ( produc4927203594696154791nt_int @ ( some_Sum_sum_int_int @ ( sum_Inl_int_int @ X3 ) ) @ ( some_Sum_sum_int_int @ ( sum_Inr_int_int @ Y3 ) ) ) )
=> ~ ( ( collec4138408100298950737al_int @ ( some_int @ X3 ) @ none_int )
| ( collec4138408100298950737al_int @ none_int @ ( some_int @ Y3 ) ) ) ) ) )
=> ( ! [Y3: int] :
( ( X
= ( some_Sum_sum_int_int @ ( sum_Inr_int_int @ Y3 ) ) )
=> ( ( Xa = none_Sum_sum_int_int )
=> ( ( accp_P270689193051028856nt_int @ collec5720877037025338812nt_int @ ( produc4927203594696154791nt_int @ ( some_Sum_sum_int_int @ ( sum_Inr_int_int @ Y3 ) ) @ none_Sum_sum_int_int ) )
=> ~ ( collec4138408100298950737al_int @ ( some_int @ Y3 ) @ none_int ) ) ) )
=> ~ ! [X3: int] :
( ( X
= ( some_Sum_sum_int_int @ ( sum_Inr_int_int @ X3 ) ) )
=> ! [Y3: int] :
( ( Xa
= ( some_Sum_sum_int_int @ ( sum_Inr_int_int @ Y3 ) ) )
=> ( ( accp_P270689193051028856nt_int @ collec5720877037025338812nt_int @ ( produc4927203594696154791nt_int @ ( some_Sum_sum_int_int @ ( sum_Inr_int_int @ X3 ) ) @ ( some_Sum_sum_int_int @ ( sum_Inr_int_int @ Y3 ) ) ) )
=> ~ ( collec4138408100298950737al_int @ ( some_int @ X3 ) @ ( some_int @ Y3 ) ) ) ) ) ) ) ) ) ) ) ) ) ) ).
% cproper_interval_sum.pelims(2)
thf(fact_632_not__gr__zero,axiom,
! [N2: nat] :
( ( ~ ( ord_less_nat @ zero_zero_nat @ N2 ) )
= ( N2 = zero_zero_nat ) ) ).
% not_gr_zero
thf(fact_633_le__zero__eq,axiom,
! [N2: nat] :
( ( ord_less_eq_nat @ N2 @ zero_zero_nat )
= ( N2 = zero_zero_nat ) ) ).
% le_zero_eq
thf(fact_634_forall__finite_I1_J,axiom,
! [P: nat > $o,I3: nat] :
( ( ord_less_nat @ I3 @ zero_zero_nat )
=> ( P @ I3 ) ) ).
% forall_finite(1)
thf(fact_635_less__numeral__extra_I3_J,axiom,
~ ( ord_less_nat @ zero_zero_nat @ zero_zero_nat ) ).
% less_numeral_extra(3)
thf(fact_636_less__numeral__extra_I3_J,axiom,
~ ( ord_less_int @ zero_zero_int @ zero_zero_int ) ).
% less_numeral_extra(3)
thf(fact_637_less__numeral__extra_I3_J,axiom,
~ ( ord_less_real @ zero_zero_real @ zero_zero_real ) ).
% less_numeral_extra(3)
thf(fact_638_gr__zeroI,axiom,
! [N2: nat] :
( ( N2 != zero_zero_nat )
=> ( ord_less_nat @ zero_zero_nat @ N2 ) ) ).
% gr_zeroI
thf(fact_639_not__less__zero,axiom,
! [N2: nat] :
~ ( ord_less_nat @ N2 @ zero_zero_nat ) ).
% not_less_zero
thf(fact_640_zero__le,axiom,
! [X: nat] : ( ord_less_eq_nat @ zero_zero_nat @ X ) ).
% zero_le
thf(fact_641_le__numeral__extra_I3_J,axiom,
ord_less_eq_nat @ zero_zero_nat @ zero_zero_nat ).
% le_numeral_extra(3)
thf(fact_642_le__numeral__extra_I3_J,axiom,
ord_less_eq_int @ zero_zero_int @ zero_zero_int ).
% le_numeral_extra(3)
thf(fact_643_le__numeral__extra_I3_J,axiom,
ord_less_eq_real @ zero_zero_real @ zero_zero_real ).
% le_numeral_extra(3)
thf(fact_644_zero__less__iff__neq__zero,axiom,
! [N2: nat] :
( ( ord_less_nat @ zero_zero_nat @ N2 )
= ( N2 != zero_zero_nat ) ) ).
% zero_less_iff_neq_zero
thf(fact_645_gr__implies__not__zero,axiom,
! [M2: nat,N2: nat] :
( ( ord_less_nat @ M2 @ N2 )
=> ( N2 != zero_zero_nat ) ) ).
% gr_implies_not_zero
thf(fact_646_field__lbound__gt__zero,axiom,
! [D1: real,D22: real] :
( ( ord_less_real @ zero_zero_real @ D1 )
=> ( ( ord_less_real @ zero_zero_real @ D22 )
=> ? [E: real] :
( ( ord_less_real @ zero_zero_real @ E )
& ( ord_less_real @ E @ D1 )
& ( ord_less_real @ E @ D22 ) ) ) ) ).
% field_lbound_gt_zero
thf(fact_647_of__nat__0__less__iff,axiom,
! [N2: nat] :
( ( ord_less_nat @ zero_zero_nat @ ( semiri1316708129612266289at_nat @ N2 ) )
= ( ord_less_nat @ zero_zero_nat @ N2 ) ) ).
% of_nat_0_less_iff
thf(fact_648_of__nat__0__less__iff,axiom,
! [N2: nat] :
( ( ord_less_int @ zero_zero_int @ ( semiri1314217659103216013at_int @ N2 ) )
= ( ord_less_nat @ zero_zero_nat @ N2 ) ) ).
% of_nat_0_less_iff
thf(fact_649_of__nat__0__less__iff,axiom,
! [N2: nat] :
( ( ord_less_real @ zero_zero_real @ ( semiri5074537144036343181t_real @ N2 ) )
= ( ord_less_nat @ zero_zero_nat @ N2 ) ) ).
% of_nat_0_less_iff
thf(fact_650_mlex__leq,axiom,
! [F: option_nat > nat,X: option_nat,Y: option_nat,R2: set_Pr6588086440996610945on_nat] :
( ( ord_less_eq_nat @ ( F @ X ) @ ( F @ Y ) )
=> ( ( member4117937158525611210on_nat @ ( produc5098337634421038937on_nat @ X @ Y ) @ R2 )
=> ( member4117937158525611210on_nat @ ( produc5098337634421038937on_nat @ X @ Y ) @ ( mlex_prod_option_nat @ F @ R2 ) ) ) ) ).
% mlex_leq
thf(fact_651_mlex__leq,axiom,
! [F: option_int > nat,X: option_int,Y: option_int,R2: set_Pr5725982650624401465on_int] :
( ( ord_less_eq_nat @ ( F @ X ) @ ( F @ Y ) )
=> ( ( member7038936195297346946on_int @ ( produc6331568796615304721on_int @ X @ Y ) @ R2 )
=> ( member7038936195297346946on_int @ ( produc6331568796615304721on_int @ X @ Y ) @ ( mlex_prod_option_int @ F @ R2 ) ) ) ) ).
% mlex_leq
thf(fact_652_mlex__iff,axiom,
! [X: option_nat,Y: option_nat,F: option_nat > nat,R2: set_Pr6588086440996610945on_nat] :
( ( member4117937158525611210on_nat @ ( produc5098337634421038937on_nat @ X @ Y ) @ ( mlex_prod_option_nat @ F @ R2 ) )
= ( ( ord_less_nat @ ( F @ X ) @ ( F @ Y ) )
| ( ( ( F @ X )
= ( F @ Y ) )
& ( member4117937158525611210on_nat @ ( produc5098337634421038937on_nat @ X @ Y ) @ R2 ) ) ) ) ).
% mlex_iff
thf(fact_653_mlex__iff,axiom,
! [X: option_int,Y: option_int,F: option_int > nat,R2: set_Pr5725982650624401465on_int] :
( ( member7038936195297346946on_int @ ( produc6331568796615304721on_int @ X @ Y ) @ ( mlex_prod_option_int @ F @ R2 ) )
= ( ( ord_less_nat @ ( F @ X ) @ ( F @ Y ) )
| ( ( ( F @ X )
= ( F @ Y ) )
& ( member7038936195297346946on_int @ ( produc6331568796615304721on_int @ X @ Y ) @ R2 ) ) ) ) ).
% mlex_iff
thf(fact_654_mlex__less,axiom,
! [F: option_nat > nat,X: option_nat,Y: option_nat,R2: set_Pr6588086440996610945on_nat] :
( ( ord_less_nat @ ( F @ X ) @ ( F @ Y ) )
=> ( member4117937158525611210on_nat @ ( produc5098337634421038937on_nat @ X @ Y ) @ ( mlex_prod_option_nat @ F @ R2 ) ) ) ).
% mlex_less
thf(fact_655_mlex__less,axiom,
! [F: option_int > nat,X: option_int,Y: option_int,R2: set_Pr5725982650624401465on_int] :
( ( ord_less_nat @ ( F @ X ) @ ( F @ Y ) )
=> ( member7038936195297346946on_int @ ( produc6331568796615304721on_int @ X @ Y ) @ ( mlex_prod_option_int @ F @ R2 ) ) ) ).
% mlex_less
thf(fact_656_of__nat__eq__iff,axiom,
! [M2: nat,N2: nat] :
( ( ( semiri1314217659103216013at_int @ M2 )
= ( semiri1314217659103216013at_int @ N2 ) )
= ( M2 = N2 ) ) ).
% of_nat_eq_iff
thf(fact_657_of__nat__eq__iff,axiom,
! [M2: nat,N2: nat] :
( ( ( semiri5074537144036343181t_real @ M2 )
= ( semiri5074537144036343181t_real @ N2 ) )
= ( M2 = N2 ) ) ).
% of_nat_eq_iff
thf(fact_658_of__nat__0,axiom,
( ( semiri1316708129612266289at_nat @ zero_zero_nat )
= zero_zero_nat ) ).
% of_nat_0
thf(fact_659_of__nat__0,axiom,
( ( semiri1314217659103216013at_int @ zero_zero_nat )
= zero_zero_int ) ).
% of_nat_0
thf(fact_660_of__nat__0,axiom,
( ( semiri5074537144036343181t_real @ zero_zero_nat )
= zero_zero_real ) ).
% of_nat_0
thf(fact_661_of__nat__0__eq__iff,axiom,
! [N2: nat] :
( ( zero_zero_nat
= ( semiri1316708129612266289at_nat @ N2 ) )
= ( zero_zero_nat = N2 ) ) ).
% of_nat_0_eq_iff
thf(fact_662_of__nat__0__eq__iff,axiom,
! [N2: nat] :
( ( zero_zero_int
= ( semiri1314217659103216013at_int @ N2 ) )
= ( zero_zero_nat = N2 ) ) ).
% of_nat_0_eq_iff
thf(fact_663_of__nat__0__eq__iff,axiom,
! [N2: nat] :
( ( zero_zero_real
= ( semiri5074537144036343181t_real @ N2 ) )
= ( zero_zero_nat = N2 ) ) ).
% of_nat_0_eq_iff
thf(fact_664_of__nat__eq__0__iff,axiom,
! [M2: nat] :
( ( ( semiri1316708129612266289at_nat @ M2 )
= zero_zero_nat )
= ( M2 = zero_zero_nat ) ) ).
% of_nat_eq_0_iff
thf(fact_665_of__nat__eq__0__iff,axiom,
! [M2: nat] :
( ( ( semiri1314217659103216013at_int @ M2 )
= zero_zero_int )
= ( M2 = zero_zero_nat ) ) ).
% of_nat_eq_0_iff
thf(fact_666_of__nat__eq__0__iff,axiom,
! [M2: nat] :
( ( ( semiri5074537144036343181t_real @ M2 )
= zero_zero_real )
= ( M2 = zero_zero_nat ) ) ).
% of_nat_eq_0_iff
thf(fact_667_of__nat__le__iff,axiom,
! [M2: nat,N2: nat] :
( ( ord_less_eq_nat @ ( semiri1316708129612266289at_nat @ M2 ) @ ( semiri1316708129612266289at_nat @ N2 ) )
= ( ord_less_eq_nat @ M2 @ N2 ) ) ).
% of_nat_le_iff
thf(fact_668_of__nat__le__iff,axiom,
! [M2: nat,N2: nat] :
( ( ord_less_eq_int @ ( semiri1314217659103216013at_int @ M2 ) @ ( semiri1314217659103216013at_int @ N2 ) )
= ( ord_less_eq_nat @ M2 @ N2 ) ) ).
% of_nat_le_iff
thf(fact_669_of__nat__le__iff,axiom,
! [M2: nat,N2: nat] :
( ( ord_less_eq_real @ ( semiri5074537144036343181t_real @ M2 ) @ ( semiri5074537144036343181t_real @ N2 ) )
= ( ord_less_eq_nat @ M2 @ N2 ) ) ).
% of_nat_le_iff
thf(fact_670_of__nat__less__iff,axiom,
! [M2: nat,N2: nat] :
( ( ord_less_nat @ ( semiri1316708129612266289at_nat @ M2 ) @ ( semiri1316708129612266289at_nat @ N2 ) )
= ( ord_less_nat @ M2 @ N2 ) ) ).
% of_nat_less_iff
thf(fact_671_of__nat__less__iff,axiom,
! [M2: nat,N2: nat] :
( ( ord_less_int @ ( semiri1314217659103216013at_int @ M2 ) @ ( semiri1314217659103216013at_int @ N2 ) )
= ( ord_less_nat @ M2 @ N2 ) ) ).
% of_nat_less_iff
thf(fact_672_of__nat__less__iff,axiom,
! [M2: nat,N2: nat] :
( ( ord_less_real @ ( semiri5074537144036343181t_real @ M2 ) @ ( semiri5074537144036343181t_real @ N2 ) )
= ( ord_less_nat @ M2 @ N2 ) ) ).
% of_nat_less_iff
thf(fact_673_of__nat__le__0__iff,axiom,
! [M2: nat] :
( ( ord_less_eq_nat @ ( semiri1316708129612266289at_nat @ M2 ) @ zero_zero_nat )
= ( M2 = zero_zero_nat ) ) ).
% of_nat_le_0_iff
thf(fact_674_of__nat__le__0__iff,axiom,
! [M2: nat] :
( ( ord_less_eq_int @ ( semiri1314217659103216013at_int @ M2 ) @ zero_zero_int )
= ( M2 = zero_zero_nat ) ) ).
% of_nat_le_0_iff
thf(fact_675_of__nat__le__0__iff,axiom,
! [M2: nat] :
( ( ord_less_eq_real @ ( semiri5074537144036343181t_real @ M2 ) @ zero_zero_real )
= ( M2 = zero_zero_nat ) ) ).
% of_nat_le_0_iff
thf(fact_676_int__ops_I1_J,axiom,
( ( semiri1314217659103216013at_int @ zero_zero_nat )
= zero_zero_int ) ).
% int_ops(1)
thf(fact_677_nat__int__comparison_I2_J,axiom,
( ord_less_nat
= ( ^ [A2: nat,B3: nat] : ( ord_less_int @ ( semiri1314217659103216013at_int @ A2 ) @ ( semiri1314217659103216013at_int @ B3 ) ) ) ) ).
% nat_int_comparison(2)
thf(fact_678_nat__int__comparison_I3_J,axiom,
( ord_less_eq_nat
= ( ^ [A2: nat,B3: nat] : ( ord_less_eq_int @ ( semiri1314217659103216013at_int @ A2 ) @ ( semiri1314217659103216013at_int @ B3 ) ) ) ) ).
% nat_int_comparison(3)
thf(fact_679_of__nat__0__le__iff,axiom,
! [N2: nat] : ( ord_less_eq_nat @ zero_zero_nat @ ( semiri1316708129612266289at_nat @ N2 ) ) ).
% of_nat_0_le_iff
thf(fact_680_of__nat__0__le__iff,axiom,
! [N2: nat] : ( ord_less_eq_int @ zero_zero_int @ ( semiri1314217659103216013at_int @ N2 ) ) ).
% of_nat_0_le_iff
thf(fact_681_of__nat__0__le__iff,axiom,
! [N2: nat] : ( ord_less_eq_real @ zero_zero_real @ ( semiri5074537144036343181t_real @ N2 ) ) ).
% of_nat_0_le_iff
thf(fact_682_of__nat__less__0__iff,axiom,
! [M2: nat] :
~ ( ord_less_nat @ ( semiri1316708129612266289at_nat @ M2 ) @ zero_zero_nat ) ).
% of_nat_less_0_iff
thf(fact_683_of__nat__less__0__iff,axiom,
! [M2: nat] :
~ ( ord_less_int @ ( semiri1314217659103216013at_int @ M2 ) @ zero_zero_int ) ).
% of_nat_less_0_iff
thf(fact_684_of__nat__less__0__iff,axiom,
! [M2: nat] :
~ ( ord_less_real @ ( semiri5074537144036343181t_real @ M2 ) @ zero_zero_real ) ).
% of_nat_less_0_iff
thf(fact_685_of__nat__mono,axiom,
! [I: nat,J: nat] :
( ( ord_less_eq_nat @ I @ J )
=> ( ord_less_eq_nat @ ( semiri1316708129612266289at_nat @ I ) @ ( semiri1316708129612266289at_nat @ J ) ) ) ).
% of_nat_mono
thf(fact_686_of__nat__mono,axiom,
! [I: nat,J: nat] :
( ( ord_less_eq_nat @ I @ J )
=> ( ord_less_eq_int @ ( semiri1314217659103216013at_int @ I ) @ ( semiri1314217659103216013at_int @ J ) ) ) ).
% of_nat_mono
thf(fact_687_of__nat__mono,axiom,
! [I: nat,J: nat] :
( ( ord_less_eq_nat @ I @ J )
=> ( ord_less_eq_real @ ( semiri5074537144036343181t_real @ I ) @ ( semiri5074537144036343181t_real @ J ) ) ) ).
% of_nat_mono
thf(fact_688_less__imp__of__nat__less,axiom,
! [M2: nat,N2: nat] :
( ( ord_less_nat @ M2 @ N2 )
=> ( ord_less_nat @ ( semiri1316708129612266289at_nat @ M2 ) @ ( semiri1316708129612266289at_nat @ N2 ) ) ) ).
% less_imp_of_nat_less
thf(fact_689_less__imp__of__nat__less,axiom,
! [M2: nat,N2: nat] :
( ( ord_less_nat @ M2 @ N2 )
=> ( ord_less_int @ ( semiri1314217659103216013at_int @ M2 ) @ ( semiri1314217659103216013at_int @ N2 ) ) ) ).
% less_imp_of_nat_less
thf(fact_690_less__imp__of__nat__less,axiom,
! [M2: nat,N2: nat] :
( ( ord_less_nat @ M2 @ N2 )
=> ( ord_less_real @ ( semiri5074537144036343181t_real @ M2 ) @ ( semiri5074537144036343181t_real @ N2 ) ) ) ).
% less_imp_of_nat_less
thf(fact_691_of__nat__less__imp__less,axiom,
! [M2: nat,N2: nat] :
( ( ord_less_nat @ ( semiri1316708129612266289at_nat @ M2 ) @ ( semiri1316708129612266289at_nat @ N2 ) )
=> ( ord_less_nat @ M2 @ N2 ) ) ).
% of_nat_less_imp_less
thf(fact_692_of__nat__less__imp__less,axiom,
! [M2: nat,N2: nat] :
( ( ord_less_int @ ( semiri1314217659103216013at_int @ M2 ) @ ( semiri1314217659103216013at_int @ N2 ) )
=> ( ord_less_nat @ M2 @ N2 ) ) ).
% of_nat_less_imp_less
thf(fact_693_of__nat__less__imp__less,axiom,
! [M2: nat,N2: nat] :
( ( ord_less_real @ ( semiri5074537144036343181t_real @ M2 ) @ ( semiri5074537144036343181t_real @ N2 ) )
=> ( ord_less_nat @ M2 @ N2 ) ) ).
% of_nat_less_imp_less
thf(fact_694_pos__int__cases,axiom,
! [K2: int] :
( ( ord_less_int @ zero_zero_int @ K2 )
=> ~ ! [N3: nat] :
( ( K2
= ( semiri1314217659103216013at_int @ N3 ) )
=> ~ ( ord_less_nat @ zero_zero_nat @ N3 ) ) ) ).
% pos_int_cases
thf(fact_695_zero__less__imp__eq__int,axiom,
! [K2: int] :
( ( ord_less_int @ zero_zero_int @ K2 )
=> ? [N3: nat] :
( ( ord_less_nat @ zero_zero_nat @ N3 )
& ( K2
= ( semiri1314217659103216013at_int @ N3 ) ) ) ) ).
% zero_less_imp_eq_int
thf(fact_696_reals__Archimedean2,axiom,
! [X: real] :
? [N3: nat] : ( ord_less_real @ X @ ( semiri5074537144036343181t_real @ N3 ) ) ).
% reals_Archimedean2
thf(fact_697_real__arch__simple,axiom,
! [X: real] :
? [N3: nat] : ( ord_less_eq_real @ X @ ( semiri5074537144036343181t_real @ N3 ) ) ).
% real_arch_simple
thf(fact_698_same__fstI,axiom,
! [P: option_nat > $o,X: option_nat,Y8: option_nat,Y: option_nat,R2: option_nat > set_Pr6588086440996610945on_nat] :
( ( P @ X )
=> ( ( member4117937158525611210on_nat @ ( produc5098337634421038937on_nat @ Y8 @ Y ) @ ( R2 @ X ) )
=> ( member2771234110571332496on_nat @ ( produc2417174425200997591on_nat @ ( produc5098337634421038937on_nat @ X @ Y8 ) @ ( produc5098337634421038937on_nat @ X @ Y ) ) @ ( same_f8891710008480837038on_nat @ P @ R2 ) ) ) ) ).
% same_fstI
thf(fact_699_same__fstI,axiom,
! [P: option_int > $o,X: option_int,Y8: option_int,Y: option_int,R2: option_int > set_Pr5725982650624401465on_int] :
( ( P @ X )
=> ( ( member7038936195297346946on_int @ ( produc6331568796615304721on_int @ Y8 @ Y ) @ ( R2 @ X ) )
=> ( member5505722672893847952on_int @ ( produc8934765814667458263on_int @ ( produc6331568796615304721on_int @ X @ Y8 ) @ ( produc6331568796615304721on_int @ X @ Y ) ) @ ( same_f901569133820327014on_int @ P @ R2 ) ) ) ) ).
% same_fstI
thf(fact_700_of__nat__zero__less__power__iff,axiom,
! [X: nat,N2: nat] :
( ( ord_less_nat @ zero_zero_nat @ ( power_power_nat @ ( semiri1316708129612266289at_nat @ X ) @ N2 ) )
= ( ( ord_less_nat @ zero_zero_nat @ X )
| ( N2 = zero_zero_nat ) ) ) ).
% of_nat_zero_less_power_iff
thf(fact_701_of__nat__zero__less__power__iff,axiom,
! [X: nat,N2: nat] :
( ( ord_less_int @ zero_zero_int @ ( power_power_int @ ( semiri1314217659103216013at_int @ X ) @ N2 ) )
= ( ( ord_less_nat @ zero_zero_nat @ X )
| ( N2 = zero_zero_nat ) ) ) ).
% of_nat_zero_less_power_iff
thf(fact_702_of__nat__zero__less__power__iff,axiom,
! [X: nat,N2: nat] :
( ( ord_less_real @ zero_zero_real @ ( power_power_real @ ( semiri5074537144036343181t_real @ X ) @ N2 ) )
= ( ( ord_less_nat @ zero_zero_nat @ X )
| ( N2 = zero_zero_nat ) ) ) ).
% of_nat_zero_less_power_iff
thf(fact_703_nat__zero__less__power__iff,axiom,
! [X: nat,N2: nat] :
( ( ord_less_nat @ zero_zero_nat @ ( power_power_nat @ X @ N2 ) )
= ( ( ord_less_nat @ zero_zero_nat @ X )
| ( N2 = zero_zero_nat ) ) ) ).
% nat_zero_less_power_iff
thf(fact_704_power__eq__0__iff,axiom,
! [A: int,N2: nat] :
( ( ( power_power_int @ A @ N2 )
= zero_zero_int )
= ( ( A = zero_zero_int )
& ( ord_less_nat @ zero_zero_nat @ N2 ) ) ) ).
% power_eq_0_iff
thf(fact_705_power__eq__0__iff,axiom,
! [A: nat,N2: nat] :
( ( ( power_power_nat @ A @ N2 )
= zero_zero_nat )
= ( ( A = zero_zero_nat )
& ( ord_less_nat @ zero_zero_nat @ N2 ) ) ) ).
% power_eq_0_iff
thf(fact_706_power__eq__0__iff,axiom,
! [A: real,N2: nat] :
( ( ( power_power_real @ A @ N2 )
= zero_zero_real )
= ( ( A = zero_zero_real )
& ( ord_less_nat @ zero_zero_nat @ N2 ) ) ) ).
% power_eq_0_iff
thf(fact_707_of__nat__le__of__nat__power__cancel__iff,axiom,
! [B: nat,W2: nat,X: nat] :
( ( ord_less_eq_nat @ ( power_power_nat @ ( semiri1316708129612266289at_nat @ B ) @ W2 ) @ ( semiri1316708129612266289at_nat @ X ) )
= ( ord_less_eq_nat @ ( power_power_nat @ B @ W2 ) @ X ) ) ).
% of_nat_le_of_nat_power_cancel_iff
thf(fact_708_of__nat__le__of__nat__power__cancel__iff,axiom,
! [B: nat,W2: nat,X: nat] :
( ( ord_less_eq_int @ ( power_power_int @ ( semiri1314217659103216013at_int @ B ) @ W2 ) @ ( semiri1314217659103216013at_int @ X ) )
= ( ord_less_eq_nat @ ( power_power_nat @ B @ W2 ) @ X ) ) ).
% of_nat_le_of_nat_power_cancel_iff
thf(fact_709_of__nat__le__of__nat__power__cancel__iff,axiom,
! [B: nat,W2: nat,X: nat] :
( ( ord_less_eq_real @ ( power_power_real @ ( semiri5074537144036343181t_real @ B ) @ W2 ) @ ( semiri5074537144036343181t_real @ X ) )
= ( ord_less_eq_nat @ ( power_power_nat @ B @ W2 ) @ X ) ) ).
% of_nat_le_of_nat_power_cancel_iff
thf(fact_710_of__nat__power__le__of__nat__cancel__iff,axiom,
! [X: nat,B: nat,W2: nat] :
( ( ord_less_eq_nat @ ( semiri1316708129612266289at_nat @ X ) @ ( power_power_nat @ ( semiri1316708129612266289at_nat @ B ) @ W2 ) )
= ( ord_less_eq_nat @ X @ ( power_power_nat @ B @ W2 ) ) ) ).
% of_nat_power_le_of_nat_cancel_iff
thf(fact_711_of__nat__power__le__of__nat__cancel__iff,axiom,
! [X: nat,B: nat,W2: nat] :
( ( ord_less_eq_int @ ( semiri1314217659103216013at_int @ X ) @ ( power_power_int @ ( semiri1314217659103216013at_int @ B ) @ W2 ) )
= ( ord_less_eq_nat @ X @ ( power_power_nat @ B @ W2 ) ) ) ).
% of_nat_power_le_of_nat_cancel_iff
thf(fact_712_of__nat__power__le__of__nat__cancel__iff,axiom,
! [X: nat,B: nat,W2: nat] :
( ( ord_less_eq_real @ ( semiri5074537144036343181t_real @ X ) @ ( power_power_real @ ( semiri5074537144036343181t_real @ B ) @ W2 ) )
= ( ord_less_eq_nat @ X @ ( power_power_nat @ B @ W2 ) ) ) ).
% of_nat_power_le_of_nat_cancel_iff
thf(fact_713_of__nat__power__less__of__nat__cancel__iff,axiom,
! [X: nat,B: nat,W2: nat] :
( ( ord_less_nat @ ( semiri1316708129612266289at_nat @ X ) @ ( power_power_nat @ ( semiri1316708129612266289at_nat @ B ) @ W2 ) )
= ( ord_less_nat @ X @ ( power_power_nat @ B @ W2 ) ) ) ).
% of_nat_power_less_of_nat_cancel_iff
thf(fact_714_of__nat__power__less__of__nat__cancel__iff,axiom,
! [X: nat,B: nat,W2: nat] :
( ( ord_less_int @ ( semiri1314217659103216013at_int @ X ) @ ( power_power_int @ ( semiri1314217659103216013at_int @ B ) @ W2 ) )
= ( ord_less_nat @ X @ ( power_power_nat @ B @ W2 ) ) ) ).
% of_nat_power_less_of_nat_cancel_iff
thf(fact_715_of__nat__power__less__of__nat__cancel__iff,axiom,
! [X: nat,B: nat,W2: nat] :
( ( ord_less_real @ ( semiri5074537144036343181t_real @ X ) @ ( power_power_real @ ( semiri5074537144036343181t_real @ B ) @ W2 ) )
= ( ord_less_nat @ X @ ( power_power_nat @ B @ W2 ) ) ) ).
% of_nat_power_less_of_nat_cancel_iff
thf(fact_716_of__nat__less__of__nat__power__cancel__iff,axiom,
! [B: nat,W2: nat,X: nat] :
( ( ord_less_nat @ ( power_power_nat @ ( semiri1316708129612266289at_nat @ B ) @ W2 ) @ ( semiri1316708129612266289at_nat @ X ) )
= ( ord_less_nat @ ( power_power_nat @ B @ W2 ) @ X ) ) ).
% of_nat_less_of_nat_power_cancel_iff
thf(fact_717_of__nat__less__of__nat__power__cancel__iff,axiom,
! [B: nat,W2: nat,X: nat] :
( ( ord_less_int @ ( power_power_int @ ( semiri1314217659103216013at_int @ B ) @ W2 ) @ ( semiri1314217659103216013at_int @ X ) )
= ( ord_less_nat @ ( power_power_nat @ B @ W2 ) @ X ) ) ).
% of_nat_less_of_nat_power_cancel_iff
thf(fact_718_of__nat__less__of__nat__power__cancel__iff,axiom,
! [B: nat,W2: nat,X: nat] :
( ( ord_less_real @ ( power_power_real @ ( semiri5074537144036343181t_real @ B ) @ W2 ) @ ( semiri5074537144036343181t_real @ X ) )
= ( ord_less_nat @ ( power_power_nat @ B @ W2 ) @ X ) ) ).
% of_nat_less_of_nat_power_cancel_iff
thf(fact_719_power__mono__iff,axiom,
! [A: nat,B: nat,N2: nat] :
( ( ord_less_eq_nat @ zero_zero_nat @ A )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ B )
=> ( ( ord_less_nat @ zero_zero_nat @ N2 )
=> ( ( ord_less_eq_nat @ ( power_power_nat @ A @ N2 ) @ ( power_power_nat @ B @ N2 ) )
= ( ord_less_eq_nat @ A @ B ) ) ) ) ) ).
% power_mono_iff
thf(fact_720_power__mono__iff,axiom,
! [A: int,B: int,N2: nat] :
( ( ord_less_eq_int @ zero_zero_int @ A )
=> ( ( ord_less_eq_int @ zero_zero_int @ B )
=> ( ( ord_less_nat @ zero_zero_nat @ N2 )
=> ( ( ord_less_eq_int @ ( power_power_int @ A @ N2 ) @ ( power_power_int @ B @ N2 ) )
= ( ord_less_eq_int @ A @ B ) ) ) ) ) ).
% power_mono_iff
thf(fact_721_power__mono__iff,axiom,
! [A: real,B: real,N2: nat] :
( ( ord_less_eq_real @ zero_zero_real @ A )
=> ( ( ord_less_eq_real @ zero_zero_real @ B )
=> ( ( ord_less_nat @ zero_zero_nat @ N2 )
=> ( ( ord_less_eq_real @ ( power_power_real @ A @ N2 ) @ ( power_power_real @ B @ N2 ) )
= ( ord_less_eq_real @ A @ B ) ) ) ) ) ).
% power_mono_iff
thf(fact_722_nat__int__comparison_I1_J,axiom,
( ( ^ [Y4: nat,Z: nat] : ( Y4 = Z ) )
= ( ^ [A2: nat,B3: nat] :
( ( semiri1314217659103216013at_int @ A2 )
= ( semiri1314217659103216013at_int @ B3 ) ) ) ) ).
% nat_int_comparison(1)
thf(fact_723_int__if,axiom,
! [P: $o,A: nat,B: nat] :
( ( P
=> ( ( semiri1314217659103216013at_int @ ( if_nat @ P @ A @ B ) )
= ( semiri1314217659103216013at_int @ A ) ) )
& ( ~ P
=> ( ( semiri1314217659103216013at_int @ ( if_nat @ P @ A @ B ) )
= ( semiri1314217659103216013at_int @ B ) ) ) ) ).
% int_if
thf(fact_724_conj__le__cong,axiom,
! [X: int,X8: int,P: $o,P4: $o] :
( ( X = X8 )
=> ( ( ( ord_less_eq_int @ zero_zero_int @ X8 )
=> ( P = P4 ) )
=> ( ( ( ord_less_eq_int @ zero_zero_int @ X )
& P )
= ( ( ord_less_eq_int @ zero_zero_int @ X8 )
& P4 ) ) ) ) ).
% conj_le_cong
thf(fact_725_imp__le__cong,axiom,
! [X: int,X8: int,P: $o,P4: $o] :
( ( X = X8 )
=> ( ( ( ord_less_eq_int @ zero_zero_int @ X8 )
=> ( P = P4 ) )
=> ( ( ( ord_less_eq_int @ zero_zero_int @ X )
=> P )
= ( ( ord_less_eq_int @ zero_zero_int @ X8 )
=> P4 ) ) ) ) ).
% imp_le_cong
thf(fact_726_zero__le__power,axiom,
! [A: nat,N2: nat] :
( ( ord_less_eq_nat @ zero_zero_nat @ A )
=> ( ord_less_eq_nat @ zero_zero_nat @ ( power_power_nat @ A @ N2 ) ) ) ).
% zero_le_power
thf(fact_727_zero__le__power,axiom,
! [A: int,N2: nat] :
( ( ord_less_eq_int @ zero_zero_int @ A )
=> ( ord_less_eq_int @ zero_zero_int @ ( power_power_int @ A @ N2 ) ) ) ).
% zero_le_power
thf(fact_728_zero__le__power,axiom,
! [A: real,N2: nat] :
( ( ord_less_eq_real @ zero_zero_real @ A )
=> ( ord_less_eq_real @ zero_zero_real @ ( power_power_real @ A @ N2 ) ) ) ).
% zero_le_power
thf(fact_729_power__mono,axiom,
! [A: nat,B: nat,N2: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ A )
=> ( ord_less_eq_nat @ ( power_power_nat @ A @ N2 ) @ ( power_power_nat @ B @ N2 ) ) ) ) ).
% power_mono
thf(fact_730_power__mono,axiom,
! [A: int,B: int,N2: nat] :
( ( ord_less_eq_int @ A @ B )
=> ( ( ord_less_eq_int @ zero_zero_int @ A )
=> ( ord_less_eq_int @ ( power_power_int @ A @ N2 ) @ ( power_power_int @ B @ N2 ) ) ) ) ).
% power_mono
thf(fact_731_power__mono,axiom,
! [A: real,B: real,N2: nat] :
( ( ord_less_eq_real @ A @ B )
=> ( ( ord_less_eq_real @ zero_zero_real @ A )
=> ( ord_less_eq_real @ ( power_power_real @ A @ N2 ) @ ( power_power_real @ B @ N2 ) ) ) ) ).
% power_mono
thf(fact_732_zero__less__power,axiom,
! [A: nat,N2: nat] :
( ( ord_less_nat @ zero_zero_nat @ A )
=> ( ord_less_nat @ zero_zero_nat @ ( power_power_nat @ A @ N2 ) ) ) ).
% zero_less_power
thf(fact_733_zero__less__power,axiom,
! [A: int,N2: nat] :
( ( ord_less_int @ zero_zero_int @ A )
=> ( ord_less_int @ zero_zero_int @ ( power_power_int @ A @ N2 ) ) ) ).
% zero_less_power
thf(fact_734_zero__less__power,axiom,
! [A: real,N2: nat] :
( ( ord_less_real @ zero_zero_real @ A )
=> ( ord_less_real @ zero_zero_real @ ( power_power_real @ A @ N2 ) ) ) ).
% zero_less_power
thf(fact_735_nat__power__less__imp__less,axiom,
! [I: nat,M2: nat,N2: nat] :
( ( ord_less_nat @ zero_zero_nat @ I )
=> ( ( ord_less_nat @ ( power_power_nat @ I @ M2 ) @ ( power_power_nat @ I @ N2 ) )
=> ( ord_less_nat @ M2 @ N2 ) ) ) ).
% nat_power_less_imp_less
thf(fact_736_power__less__imp__less__base,axiom,
! [A: nat,N2: nat,B: nat] :
( ( ord_less_nat @ ( power_power_nat @ A @ N2 ) @ ( power_power_nat @ B @ N2 ) )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ B )
=> ( ord_less_nat @ A @ B ) ) ) ).
% power_less_imp_less_base
thf(fact_737_power__less__imp__less__base,axiom,
! [A: int,N2: nat,B: int] :
( ( ord_less_int @ ( power_power_int @ A @ N2 ) @ ( power_power_int @ B @ N2 ) )
=> ( ( ord_less_eq_int @ zero_zero_int @ B )
=> ( ord_less_int @ A @ B ) ) ) ).
% power_less_imp_less_base
thf(fact_738_power__less__imp__less__base,axiom,
! [A: real,N2: nat,B: real] :
( ( ord_less_real @ ( power_power_real @ A @ N2 ) @ ( power_power_real @ B @ N2 ) )
=> ( ( ord_less_eq_real @ zero_zero_real @ B )
=> ( ord_less_real @ A @ B ) ) ) ).
% power_less_imp_less_base
thf(fact_739_zero__power,axiom,
! [N2: nat] :
( ( ord_less_nat @ zero_zero_nat @ N2 )
=> ( ( power_power_int @ zero_zero_int @ N2 )
= zero_zero_int ) ) ).
% zero_power
thf(fact_740_zero__power,axiom,
! [N2: nat] :
( ( ord_less_nat @ zero_zero_nat @ N2 )
=> ( ( power_power_nat @ zero_zero_nat @ N2 )
= zero_zero_nat ) ) ).
% zero_power
thf(fact_741_zero__power,axiom,
! [N2: nat] :
( ( ord_less_nat @ zero_zero_nat @ N2 )
=> ( ( power_power_real @ zero_zero_real @ N2 )
= zero_zero_real ) ) ).
% zero_power
thf(fact_742_power__eq__iff__eq__base,axiom,
! [N2: nat,A: nat,B: nat] :
( ( ord_less_nat @ zero_zero_nat @ N2 )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ A )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ B )
=> ( ( ( power_power_nat @ A @ N2 )
= ( power_power_nat @ B @ N2 ) )
= ( A = B ) ) ) ) ) ).
% power_eq_iff_eq_base
thf(fact_743_power__eq__iff__eq__base,axiom,
! [N2: nat,A: int,B: int] :
( ( ord_less_nat @ zero_zero_nat @ N2 )
=> ( ( ord_less_eq_int @ zero_zero_int @ A )
=> ( ( ord_less_eq_int @ zero_zero_int @ B )
=> ( ( ( power_power_int @ A @ N2 )
= ( power_power_int @ B @ N2 ) )
= ( A = B ) ) ) ) ) ).
% power_eq_iff_eq_base
thf(fact_744_power__eq__iff__eq__base,axiom,
! [N2: nat,A: real,B: real] :
( ( ord_less_nat @ zero_zero_nat @ N2 )
=> ( ( ord_less_eq_real @ zero_zero_real @ A )
=> ( ( ord_less_eq_real @ zero_zero_real @ B )
=> ( ( ( power_power_real @ A @ N2 )
= ( power_power_real @ B @ N2 ) )
= ( A = B ) ) ) ) ) ).
% power_eq_iff_eq_base
thf(fact_745_power__eq__imp__eq__base,axiom,
! [A: nat,N2: nat,B: nat] :
( ( ( power_power_nat @ A @ N2 )
= ( power_power_nat @ B @ N2 ) )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ A )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ B )
=> ( ( ord_less_nat @ zero_zero_nat @ N2 )
=> ( A = B ) ) ) ) ) ).
% power_eq_imp_eq_base
thf(fact_746_power__eq__imp__eq__base,axiom,
! [A: int,N2: nat,B: int] :
( ( ( power_power_int @ A @ N2 )
= ( power_power_int @ B @ N2 ) )
=> ( ( ord_less_eq_int @ zero_zero_int @ A )
=> ( ( ord_less_eq_int @ zero_zero_int @ B )
=> ( ( ord_less_nat @ zero_zero_nat @ N2 )
=> ( A = B ) ) ) ) ) ).
% power_eq_imp_eq_base
thf(fact_747_power__eq__imp__eq__base,axiom,
! [A: real,N2: nat,B: real] :
( ( ( power_power_real @ A @ N2 )
= ( power_power_real @ B @ N2 ) )
=> ( ( ord_less_eq_real @ zero_zero_real @ A )
=> ( ( ord_less_eq_real @ zero_zero_real @ B )
=> ( ( ord_less_nat @ zero_zero_nat @ N2 )
=> ( A = B ) ) ) ) ) ).
% power_eq_imp_eq_base
thf(fact_748_zle__int,axiom,
! [M2: nat,N2: nat] :
( ( ord_less_eq_int @ ( semiri1314217659103216013at_int @ M2 ) @ ( semiri1314217659103216013at_int @ N2 ) )
= ( ord_less_eq_nat @ M2 @ N2 ) ) ).
% zle_int
thf(fact_749_power__strict__mono,axiom,
! [A: nat,B: nat,N2: nat] :
( ( ord_less_nat @ A @ B )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ A )
=> ( ( ord_less_nat @ zero_zero_nat @ N2 )
=> ( ord_less_nat @ ( power_power_nat @ A @ N2 ) @ ( power_power_nat @ B @ N2 ) ) ) ) ) ).
% power_strict_mono
thf(fact_750_power__strict__mono,axiom,
! [A: int,B: int,N2: nat] :
( ( ord_less_int @ A @ B )
=> ( ( ord_less_eq_int @ zero_zero_int @ A )
=> ( ( ord_less_nat @ zero_zero_nat @ N2 )
=> ( ord_less_int @ ( power_power_int @ A @ N2 ) @ ( power_power_int @ B @ N2 ) ) ) ) ) ).
% power_strict_mono
thf(fact_751_power__strict__mono,axiom,
! [A: real,B: real,N2: nat] :
( ( ord_less_real @ A @ B )
=> ( ( ord_less_eq_real @ zero_zero_real @ A )
=> ( ( ord_less_nat @ zero_zero_nat @ N2 )
=> ( ord_less_real @ ( power_power_real @ A @ N2 ) @ ( power_power_real @ B @ N2 ) ) ) ) ) ).
% power_strict_mono
thf(fact_752_realpow__pos__nth__unique,axiom,
! [N2: nat,A: real] :
( ( ord_less_nat @ zero_zero_nat @ N2 )
=> ( ( ord_less_real @ zero_zero_real @ A )
=> ? [X3: real] :
( ( ord_less_real @ zero_zero_real @ X3 )
& ( ( power_power_real @ X3 @ N2 )
= A )
& ! [Y6: real] :
( ( ( ord_less_real @ zero_zero_real @ Y6 )
& ( ( power_power_real @ Y6 @ N2 )
= A ) )
=> ( Y6 = X3 ) ) ) ) ) ).
% realpow_pos_nth_unique
thf(fact_753_realpow__pos__nth,axiom,
! [N2: nat,A: real] :
( ( ord_less_nat @ zero_zero_nat @ N2 )
=> ( ( ord_less_real @ zero_zero_real @ A )
=> ? [R4: real] :
( ( ord_less_real @ zero_zero_real @ R4 )
& ( ( power_power_real @ R4 @ N2 )
= A ) ) ) ) ).
% realpow_pos_nth
thf(fact_754_power__decreasing__iff,axiom,
! [B: nat,M2: nat,N2: nat] :
( ( ord_less_nat @ zero_zero_nat @ B )
=> ( ( ord_less_nat @ B @ one_one_nat )
=> ( ( ord_less_eq_nat @ ( power_power_nat @ B @ M2 ) @ ( power_power_nat @ B @ N2 ) )
= ( ord_less_eq_nat @ N2 @ M2 ) ) ) ) ).
% power_decreasing_iff
thf(fact_755_power__decreasing__iff,axiom,
! [B: int,M2: nat,N2: nat] :
( ( ord_less_int @ zero_zero_int @ B )
=> ( ( ord_less_int @ B @ one_one_int )
=> ( ( ord_less_eq_int @ ( power_power_int @ B @ M2 ) @ ( power_power_int @ B @ N2 ) )
= ( ord_less_eq_nat @ N2 @ M2 ) ) ) ) ).
% power_decreasing_iff
thf(fact_756_power__decreasing__iff,axiom,
! [B: real,M2: nat,N2: nat] :
( ( ord_less_real @ zero_zero_real @ B )
=> ( ( ord_less_real @ B @ one_one_real )
=> ( ( ord_less_eq_real @ ( power_power_real @ B @ M2 ) @ ( power_power_real @ B @ N2 ) )
= ( ord_less_eq_nat @ N2 @ M2 ) ) ) ) ).
% power_decreasing_iff
thf(fact_757_neg__int__cases,axiom,
! [K2: int] :
( ( ord_less_int @ K2 @ zero_zero_int )
=> ~ ! [N3: nat] :
( ( K2
= ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ N3 ) ) )
=> ~ ( ord_less_nat @ zero_zero_nat @ N3 ) ) ) ).
% neg_int_cases
thf(fact_758_power__increasing__iff,axiom,
! [B: nat,X: nat,Y: nat] :
( ( ord_less_nat @ one_one_nat @ B )
=> ( ( ord_less_eq_nat @ ( power_power_nat @ B @ X ) @ ( power_power_nat @ B @ Y ) )
= ( ord_less_eq_nat @ X @ Y ) ) ) ).
% power_increasing_iff
thf(fact_759_power__increasing__iff,axiom,
! [B: int,X: nat,Y: nat] :
( ( ord_less_int @ one_one_int @ B )
=> ( ( ord_less_eq_int @ ( power_power_int @ B @ X ) @ ( power_power_int @ B @ Y ) )
= ( ord_less_eq_nat @ X @ Y ) ) ) ).
% power_increasing_iff
thf(fact_760_power__increasing__iff,axiom,
! [B: real,X: nat,Y: nat] :
( ( ord_less_real @ one_one_real @ B )
=> ( ( ord_less_eq_real @ ( power_power_real @ B @ X ) @ ( power_power_real @ B @ Y ) )
= ( ord_less_eq_nat @ X @ Y ) ) ) ).
% power_increasing_iff
thf(fact_761_power__strict__decreasing__iff,axiom,
! [B: nat,M2: nat,N2: nat] :
( ( ord_less_nat @ zero_zero_nat @ B )
=> ( ( ord_less_nat @ B @ one_one_nat )
=> ( ( ord_less_nat @ ( power_power_nat @ B @ M2 ) @ ( power_power_nat @ B @ N2 ) )
= ( ord_less_nat @ N2 @ M2 ) ) ) ) ).
% power_strict_decreasing_iff
thf(fact_762_power__strict__decreasing__iff,axiom,
! [B: int,M2: nat,N2: nat] :
( ( ord_less_int @ zero_zero_int @ B )
=> ( ( ord_less_int @ B @ one_one_int )
=> ( ( ord_less_int @ ( power_power_int @ B @ M2 ) @ ( power_power_int @ B @ N2 ) )
= ( ord_less_nat @ N2 @ M2 ) ) ) ) ).
% power_strict_decreasing_iff
thf(fact_763_power__strict__decreasing__iff,axiom,
! [B: real,M2: nat,N2: nat] :
( ( ord_less_real @ zero_zero_real @ B )
=> ( ( ord_less_real @ B @ one_one_real )
=> ( ( ord_less_real @ ( power_power_real @ B @ M2 ) @ ( power_power_real @ B @ N2 ) )
= ( ord_less_nat @ N2 @ M2 ) ) ) ) ).
% power_strict_decreasing_iff
thf(fact_764_verit__minus__simplify_I4_J,axiom,
! [B: int] :
( ( uminus_uminus_int @ ( uminus_uminus_int @ B ) )
= B ) ).
% verit_minus_simplify(4)
thf(fact_765_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_766_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_767_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_768_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_769_of__nat__1,axiom,
( ( semiri1316708129612266289at_nat @ one_one_nat )
= one_one_nat ) ).
% of_nat_1
thf(fact_770_of__nat__1,axiom,
( ( semiri1314217659103216013at_int @ one_one_nat )
= one_one_int ) ).
% of_nat_1
thf(fact_771_of__nat__1,axiom,
( ( semiri5074537144036343181t_real @ one_one_nat )
= one_one_real ) ).
% of_nat_1
thf(fact_772_of__nat__1__eq__iff,axiom,
! [N2: nat] :
( ( one_one_nat
= ( semiri1316708129612266289at_nat @ N2 ) )
= ( N2 = one_one_nat ) ) ).
% of_nat_1_eq_iff
thf(fact_773_of__nat__1__eq__iff,axiom,
! [N2: nat] :
( ( one_one_int
= ( semiri1314217659103216013at_int @ N2 ) )
= ( N2 = one_one_nat ) ) ).
% of_nat_1_eq_iff
thf(fact_774_of__nat__1__eq__iff,axiom,
! [N2: nat] :
( ( one_one_real
= ( semiri5074537144036343181t_real @ N2 ) )
= ( N2 = one_one_nat ) ) ).
% of_nat_1_eq_iff
thf(fact_775_of__nat__eq__1__iff,axiom,
! [N2: nat] :
( ( ( semiri1316708129612266289at_nat @ N2 )
= one_one_nat )
= ( N2 = one_one_nat ) ) ).
% of_nat_eq_1_iff
thf(fact_776_of__nat__eq__1__iff,axiom,
! [N2: nat] :
( ( ( semiri1314217659103216013at_int @ N2 )
= one_one_int )
= ( N2 = one_one_nat ) ) ).
% of_nat_eq_1_iff
thf(fact_777_of__nat__eq__1__iff,axiom,
! [N2: nat] :
( ( ( semiri5074537144036343181t_real @ N2 )
= one_one_real )
= ( N2 = one_one_nat ) ) ).
% of_nat_eq_1_iff
thf(fact_778_less__one,axiom,
! [N2: nat] :
( ( ord_less_nat @ N2 @ one_one_nat )
= ( N2 = zero_zero_nat ) ) ).
% less_one
thf(fact_779_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_780_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_781_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_782_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_783_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_784_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_785_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_786_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_787_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_788_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_789_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_790_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_791_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_792_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_793_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_794_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_795_power__inject__exp,axiom,
! [A: nat,M2: nat,N2: nat] :
( ( ord_less_nat @ one_one_nat @ A )
=> ( ( ( power_power_nat @ A @ M2 )
= ( power_power_nat @ A @ N2 ) )
= ( M2 = N2 ) ) ) ).
% power_inject_exp
thf(fact_796_power__inject__exp,axiom,
! [A: int,M2: nat,N2: nat] :
( ( ord_less_int @ one_one_int @ A )
=> ( ( ( power_power_int @ A @ M2 )
= ( power_power_int @ A @ N2 ) )
= ( M2 = N2 ) ) ) ).
% power_inject_exp
thf(fact_797_power__inject__exp,axiom,
! [A: real,M2: nat,N2: nat] :
( ( ord_less_real @ one_one_real @ A )
=> ( ( ( power_power_real @ A @ M2 )
= ( power_power_real @ A @ N2 ) )
= ( M2 = N2 ) ) ) ).
% power_inject_exp
thf(fact_798_power__strict__increasing__iff,axiom,
! [B: nat,X: nat,Y: nat] :
( ( ord_less_nat @ one_one_nat @ B )
=> ( ( ord_less_nat @ ( power_power_nat @ B @ X ) @ ( power_power_nat @ B @ Y ) )
= ( ord_less_nat @ X @ Y ) ) ) ).
% power_strict_increasing_iff
thf(fact_799_power__strict__increasing__iff,axiom,
! [B: int,X: nat,Y: nat] :
( ( ord_less_int @ one_one_int @ B )
=> ( ( ord_less_int @ ( power_power_int @ B @ X ) @ ( power_power_int @ B @ Y ) )
= ( ord_less_nat @ X @ Y ) ) ) ).
% power_strict_increasing_iff
thf(fact_800_power__strict__increasing__iff,axiom,
! [B: real,X: nat,Y: nat] :
( ( ord_less_real @ one_one_real @ B )
=> ( ( ord_less_real @ ( power_power_real @ B @ X ) @ ( power_power_real @ B @ Y ) )
= ( ord_less_nat @ X @ Y ) ) ) ).
% power_strict_increasing_iff
thf(fact_801_int__ops_I2_J,axiom,
( ( semiri1314217659103216013at_int @ one_one_nat )
= one_one_int ) ).
% int_ops(2)
thf(fact_802_le__minus__one__simps_I1_J,axiom,
ord_less_eq_int @ ( uminus_uminus_int @ one_one_int ) @ zero_zero_int ).
% le_minus_one_simps(1)
thf(fact_803_le__minus__one__simps_I1_J,axiom,
ord_less_eq_real @ ( uminus_uminus_real @ one_one_real ) @ zero_zero_real ).
% le_minus_one_simps(1)
thf(fact_804_le__minus__one__simps_I3_J,axiom,
~ ( ord_less_eq_int @ zero_zero_int @ ( uminus_uminus_int @ one_one_int ) ) ).
% le_minus_one_simps(3)
thf(fact_805_le__minus__one__simps_I3_J,axiom,
~ ( ord_less_eq_real @ zero_zero_real @ ( uminus_uminus_real @ one_one_real ) ) ).
% le_minus_one_simps(3)
thf(fact_806_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_807_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_808_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_809_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_810_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_811_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_812_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_813_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_814_le__minus__one__simps_I2_J,axiom,
ord_less_eq_int @ ( uminus_uminus_int @ one_one_int ) @ one_one_int ).
% le_minus_one_simps(2)
thf(fact_815_le__minus__one__simps_I2_J,axiom,
ord_less_eq_real @ ( uminus_uminus_real @ one_one_real ) @ one_one_real ).
% le_minus_one_simps(2)
thf(fact_816_le__minus__one__simps_I4_J,axiom,
~ ( ord_less_eq_int @ one_one_int @ ( uminus_uminus_int @ one_one_int ) ) ).
% le_minus_one_simps(4)
thf(fact_817_le__minus__one__simps_I4_J,axiom,
~ ( ord_less_eq_real @ one_one_real @ ( uminus_uminus_real @ one_one_real ) ) ).
% le_minus_one_simps(4)
thf(fact_818_verit__negate__coefficient_I3_J,axiom,
! [A: int,B: int] :
( ( A = B )
=> ( ( uminus_uminus_int @ A )
= ( uminus_uminus_int @ B ) ) ) ).
% verit_negate_coefficient(3)
thf(fact_819_le__imp__neg__le,axiom,
! [A: int,B: int] :
( ( ord_less_eq_int @ A @ B )
=> ( ord_less_eq_int @ ( uminus_uminus_int @ B ) @ ( uminus_uminus_int @ A ) ) ) ).
% le_imp_neg_le
thf(fact_820_le__imp__neg__le,axiom,
! [A: real,B: real] :
( ( ord_less_eq_real @ A @ B )
=> ( ord_less_eq_real @ ( uminus_uminus_real @ B ) @ ( uminus_uminus_real @ A ) ) ) ).
% le_imp_neg_le
thf(fact_821_minus__le__iff,axiom,
! [A: int,B: int] :
( ( ord_less_eq_int @ ( uminus_uminus_int @ A ) @ B )
= ( ord_less_eq_int @ ( uminus_uminus_int @ B ) @ A ) ) ).
% minus_le_iff
thf(fact_822_minus__le__iff,axiom,
! [A: real,B: real] :
( ( ord_less_eq_real @ ( uminus_uminus_real @ A ) @ B )
= ( ord_less_eq_real @ ( uminus_uminus_real @ B ) @ A ) ) ).
% minus_le_iff
thf(fact_823_le__minus__iff,axiom,
! [A: int,B: int] :
( ( ord_less_eq_int @ A @ ( uminus_uminus_int @ B ) )
= ( ord_less_eq_int @ B @ ( uminus_uminus_int @ A ) ) ) ).
% le_minus_iff
thf(fact_824_le__minus__iff,axiom,
! [A: real,B: real] :
( ( ord_less_eq_real @ A @ ( uminus_uminus_real @ B ) )
= ( ord_less_eq_real @ B @ ( uminus_uminus_real @ A ) ) ) ).
% le_minus_iff
thf(fact_825_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_826_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_827_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_828_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_829_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_830_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_831_le__numeral__extra_I4_J,axiom,
ord_less_eq_nat @ one_one_nat @ one_one_nat ).
% le_numeral_extra(4)
thf(fact_832_le__numeral__extra_I4_J,axiom,
ord_less_eq_int @ one_one_int @ one_one_int ).
% le_numeral_extra(4)
thf(fact_833_le__numeral__extra_I4_J,axiom,
ord_less_eq_real @ one_one_real @ one_one_real ).
% le_numeral_extra(4)
thf(fact_834_zero__neq__one,axiom,
zero_zero_nat != one_one_nat ).
% zero_neq_one
thf(fact_835_zero__neq__one,axiom,
zero_zero_int != one_one_int ).
% zero_neq_one
thf(fact_836_zero__neq__one,axiom,
zero_zero_real != one_one_real ).
% zero_neq_one
thf(fact_837_less__numeral__extra_I4_J,axiom,
~ ( ord_less_nat @ one_one_nat @ one_one_nat ) ).
% less_numeral_extra(4)
thf(fact_838_less__numeral__extra_I4_J,axiom,
~ ( ord_less_int @ one_one_int @ one_one_int ) ).
% less_numeral_extra(4)
thf(fact_839_less__numeral__extra_I4_J,axiom,
~ ( ord_less_real @ one_one_real @ one_one_real ) ).
% less_numeral_extra(4)
thf(fact_840_zero__less__one__class_Ozero__le__one,axiom,
ord_less_eq_nat @ zero_zero_nat @ one_one_nat ).
% zero_less_one_class.zero_le_one
thf(fact_841_zero__less__one__class_Ozero__le__one,axiom,
ord_less_eq_int @ zero_zero_int @ one_one_int ).
% zero_less_one_class.zero_le_one
thf(fact_842_zero__less__one__class_Ozero__le__one,axiom,
ord_less_eq_real @ zero_zero_real @ one_one_real ).
% zero_less_one_class.zero_le_one
thf(fact_843_linordered__nonzero__semiring__class_Ozero__le__one,axiom,
ord_less_eq_nat @ zero_zero_nat @ one_one_nat ).
% linordered_nonzero_semiring_class.zero_le_one
thf(fact_844_linordered__nonzero__semiring__class_Ozero__le__one,axiom,
ord_less_eq_int @ zero_zero_int @ one_one_int ).
% linordered_nonzero_semiring_class.zero_le_one
thf(fact_845_linordered__nonzero__semiring__class_Ozero__le__one,axiom,
ord_less_eq_real @ zero_zero_real @ one_one_real ).
% linordered_nonzero_semiring_class.zero_le_one
thf(fact_846_not__one__le__zero,axiom,
~ ( ord_less_eq_nat @ one_one_nat @ zero_zero_nat ) ).
% not_one_le_zero
thf(fact_847_not__one__le__zero,axiom,
~ ( ord_less_eq_int @ one_one_int @ zero_zero_int ) ).
% not_one_le_zero
thf(fact_848_not__one__le__zero,axiom,
~ ( ord_less_eq_real @ one_one_real @ zero_zero_real ) ).
% not_one_le_zero
thf(fact_849_zero__less__one,axiom,
ord_less_nat @ zero_zero_nat @ one_one_nat ).
% zero_less_one
thf(fact_850_zero__less__one,axiom,
ord_less_int @ zero_zero_int @ one_one_int ).
% zero_less_one
thf(fact_851_zero__less__one,axiom,
ord_less_real @ zero_zero_real @ one_one_real ).
% zero_less_one
thf(fact_852_not__one__less__zero,axiom,
~ ( ord_less_nat @ one_one_nat @ zero_zero_nat ) ).
% not_one_less_zero
thf(fact_853_not__one__less__zero,axiom,
~ ( ord_less_int @ one_one_int @ zero_zero_int ) ).
% not_one_less_zero
thf(fact_854_not__one__less__zero,axiom,
~ ( ord_less_real @ one_one_real @ zero_zero_real ) ).
% not_one_less_zero
thf(fact_855_less__numeral__extra_I1_J,axiom,
ord_less_nat @ zero_zero_nat @ one_one_nat ).
% less_numeral_extra(1)
thf(fact_856_less__numeral__extra_I1_J,axiom,
ord_less_int @ zero_zero_int @ one_one_int ).
% less_numeral_extra(1)
thf(fact_857_less__numeral__extra_I1_J,axiom,
ord_less_real @ zero_zero_real @ one_one_real ).
% less_numeral_extra(1)
thf(fact_858_one__le__power,axiom,
! [A: nat,N2: nat] :
( ( ord_less_eq_nat @ one_one_nat @ A )
=> ( ord_less_eq_nat @ one_one_nat @ ( power_power_nat @ A @ N2 ) ) ) ).
% one_le_power
thf(fact_859_one__le__power,axiom,
! [A: int,N2: nat] :
( ( ord_less_eq_int @ one_one_int @ A )
=> ( ord_less_eq_int @ one_one_int @ ( power_power_int @ A @ N2 ) ) ) ).
% one_le_power
thf(fact_860_one__le__power,axiom,
! [A: real,N2: nat] :
( ( ord_less_eq_real @ one_one_real @ A )
=> ( ord_less_eq_real @ one_one_real @ ( power_power_real @ A @ N2 ) ) ) ).
% one_le_power
thf(fact_861_int__cases4,axiom,
! [M2: int] :
( ! [N3: nat] :
( M2
!= ( semiri1314217659103216013at_int @ N3 ) )
=> ~ ! [N3: nat] :
( ( ord_less_nat @ zero_zero_nat @ N3 )
=> ( M2
!= ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ N3 ) ) ) ) ) ).
% int_cases4
thf(fact_862_power__le__one,axiom,
! [A: nat,N2: nat] :
( ( ord_less_eq_nat @ zero_zero_nat @ A )
=> ( ( ord_less_eq_nat @ A @ one_one_nat )
=> ( ord_less_eq_nat @ ( power_power_nat @ A @ N2 ) @ one_one_nat ) ) ) ).
% power_le_one
thf(fact_863_power__le__one,axiom,
! [A: int,N2: nat] :
( ( ord_less_eq_int @ zero_zero_int @ A )
=> ( ( ord_less_eq_int @ A @ one_one_int )
=> ( ord_less_eq_int @ ( power_power_int @ A @ N2 ) @ one_one_int ) ) ) ).
% power_le_one
thf(fact_864_power__le__one,axiom,
! [A: real,N2: nat] :
( ( ord_less_eq_real @ zero_zero_real @ A )
=> ( ( ord_less_eq_real @ A @ one_one_real )
=> ( ord_less_eq_real @ ( power_power_real @ A @ N2 ) @ one_one_real ) ) ) ).
% power_le_one
thf(fact_865_power__increasing,axiom,
! [N2: nat,N4: nat,A: nat] :
( ( ord_less_eq_nat @ N2 @ N4 )
=> ( ( ord_less_eq_nat @ one_one_nat @ A )
=> ( ord_less_eq_nat @ ( power_power_nat @ A @ N2 ) @ ( power_power_nat @ A @ N4 ) ) ) ) ).
% power_increasing
thf(fact_866_power__increasing,axiom,
! [N2: nat,N4: nat,A: int] :
( ( ord_less_eq_nat @ N2 @ N4 )
=> ( ( ord_less_eq_int @ one_one_int @ A )
=> ( ord_less_eq_int @ ( power_power_int @ A @ N2 ) @ ( power_power_int @ A @ N4 ) ) ) ) ).
% power_increasing
thf(fact_867_power__increasing,axiom,
! [N2: nat,N4: nat,A: real] :
( ( ord_less_eq_nat @ N2 @ N4 )
=> ( ( ord_less_eq_real @ one_one_real @ A )
=> ( ord_less_eq_real @ ( power_power_real @ A @ N2 ) @ ( power_power_real @ A @ N4 ) ) ) ) ).
% power_increasing
thf(fact_868_power__strict__increasing,axiom,
! [N2: nat,N4: nat,A: nat] :
( ( ord_less_nat @ N2 @ N4 )
=> ( ( ord_less_nat @ one_one_nat @ A )
=> ( ord_less_nat @ ( power_power_nat @ A @ N2 ) @ ( power_power_nat @ A @ N4 ) ) ) ) ).
% power_strict_increasing
thf(fact_869_power__strict__increasing,axiom,
! [N2: nat,N4: nat,A: int] :
( ( ord_less_nat @ N2 @ N4 )
=> ( ( ord_less_int @ one_one_int @ A )
=> ( ord_less_int @ ( power_power_int @ A @ N2 ) @ ( power_power_int @ A @ N4 ) ) ) ) ).
% power_strict_increasing
thf(fact_870_power__strict__increasing,axiom,
! [N2: nat,N4: nat,A: real] :
( ( ord_less_nat @ N2 @ N4 )
=> ( ( ord_less_real @ one_one_real @ A )
=> ( ord_less_real @ ( power_power_real @ A @ N2 ) @ ( power_power_real @ A @ N4 ) ) ) ) ).
% power_strict_increasing
thf(fact_871_power__less__imp__less__exp,axiom,
! [A: nat,M2: nat,N2: nat] :
( ( ord_less_nat @ one_one_nat @ A )
=> ( ( ord_less_nat @ ( power_power_nat @ A @ M2 ) @ ( power_power_nat @ A @ N2 ) )
=> ( ord_less_nat @ M2 @ N2 ) ) ) ).
% power_less_imp_less_exp
thf(fact_872_power__less__imp__less__exp,axiom,
! [A: int,M2: nat,N2: nat] :
( ( ord_less_int @ one_one_int @ A )
=> ( ( ord_less_int @ ( power_power_int @ A @ M2 ) @ ( power_power_int @ A @ N2 ) )
=> ( ord_less_nat @ M2 @ N2 ) ) ) ).
% power_less_imp_less_exp
thf(fact_873_power__less__imp__less__exp,axiom,
! [A: real,M2: nat,N2: nat] :
( ( ord_less_real @ one_one_real @ A )
=> ( ( ord_less_real @ ( power_power_real @ A @ M2 ) @ ( power_power_real @ A @ N2 ) )
=> ( ord_less_nat @ M2 @ N2 ) ) ) ).
% power_less_imp_less_exp
thf(fact_874_int__cases3,axiom,
! [K2: int] :
( ( K2 != zero_zero_int )
=> ( ! [N3: nat] :
( ( K2
= ( semiri1314217659103216013at_int @ N3 ) )
=> ~ ( ord_less_nat @ zero_zero_nat @ N3 ) )
=> ~ ! [N3: nat] :
( ( K2
= ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ N3 ) ) )
=> ~ ( ord_less_nat @ zero_zero_nat @ N3 ) ) ) ) ).
% int_cases3
thf(fact_875_power__decreasing,axiom,
! [N2: nat,N4: nat,A: nat] :
( ( ord_less_eq_nat @ N2 @ N4 )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ A )
=> ( ( ord_less_eq_nat @ A @ one_one_nat )
=> ( ord_less_eq_nat @ ( power_power_nat @ A @ N4 ) @ ( power_power_nat @ A @ N2 ) ) ) ) ) ).
% power_decreasing
thf(fact_876_power__decreasing,axiom,
! [N2: nat,N4: nat,A: int] :
( ( ord_less_eq_nat @ N2 @ N4 )
=> ( ( ord_less_eq_int @ zero_zero_int @ A )
=> ( ( ord_less_eq_int @ A @ one_one_int )
=> ( ord_less_eq_int @ ( power_power_int @ A @ N4 ) @ ( power_power_int @ A @ N2 ) ) ) ) ) ).
% power_decreasing
thf(fact_877_power__decreasing,axiom,
! [N2: nat,N4: nat,A: real] :
( ( ord_less_eq_nat @ N2 @ N4 )
=> ( ( ord_less_eq_real @ zero_zero_real @ A )
=> ( ( ord_less_eq_real @ A @ one_one_real )
=> ( ord_less_eq_real @ ( power_power_real @ A @ N4 ) @ ( power_power_real @ A @ N2 ) ) ) ) ) ).
% power_decreasing
thf(fact_878_power__strict__decreasing,axiom,
! [N2: nat,N4: nat,A: nat] :
( ( ord_less_nat @ N2 @ N4 )
=> ( ( ord_less_nat @ zero_zero_nat @ A )
=> ( ( ord_less_nat @ A @ one_one_nat )
=> ( ord_less_nat @ ( power_power_nat @ A @ N4 ) @ ( power_power_nat @ A @ N2 ) ) ) ) ) ).
% power_strict_decreasing
thf(fact_879_power__strict__decreasing,axiom,
! [N2: nat,N4: nat,A: int] :
( ( ord_less_nat @ N2 @ N4 )
=> ( ( ord_less_int @ zero_zero_int @ A )
=> ( ( ord_less_int @ A @ one_one_int )
=> ( ord_less_int @ ( power_power_int @ A @ N4 ) @ ( power_power_int @ A @ N2 ) ) ) ) ) ).
% power_strict_decreasing
thf(fact_880_power__strict__decreasing,axiom,
! [N2: nat,N4: nat,A: real] :
( ( ord_less_nat @ N2 @ N4 )
=> ( ( ord_less_real @ zero_zero_real @ A )
=> ( ( ord_less_real @ A @ one_one_real )
=> ( ord_less_real @ ( power_power_real @ A @ N4 ) @ ( power_power_real @ A @ N2 ) ) ) ) ) ).
% power_strict_decreasing
thf(fact_881_power__le__imp__le__exp,axiom,
! [A: nat,M2: nat,N2: nat] :
( ( ord_less_nat @ one_one_nat @ A )
=> ( ( ord_less_eq_nat @ ( power_power_nat @ A @ M2 ) @ ( power_power_nat @ A @ N2 ) )
=> ( ord_less_eq_nat @ M2 @ N2 ) ) ) ).
% power_le_imp_le_exp
thf(fact_882_power__le__imp__le__exp,axiom,
! [A: int,M2: nat,N2: nat] :
( ( ord_less_int @ one_one_int @ A )
=> ( ( ord_less_eq_int @ ( power_power_int @ A @ M2 ) @ ( power_power_int @ A @ N2 ) )
=> ( ord_less_eq_nat @ M2 @ N2 ) ) ) ).
% power_le_imp_le_exp
thf(fact_883_power__le__imp__le__exp,axiom,
! [A: real,M2: nat,N2: nat] :
( ( ord_less_real @ one_one_real @ A )
=> ( ( ord_less_eq_real @ ( power_power_real @ A @ M2 ) @ ( power_power_real @ A @ N2 ) )
=> ( ord_less_eq_nat @ M2 @ N2 ) ) ) ).
% power_le_imp_le_exp
thf(fact_884_self__le__power,axiom,
! [A: nat,N2: nat] :
( ( ord_less_eq_nat @ one_one_nat @ A )
=> ( ( ord_less_nat @ zero_zero_nat @ N2 )
=> ( ord_less_eq_nat @ A @ ( power_power_nat @ A @ N2 ) ) ) ) ).
% self_le_power
thf(fact_885_self__le__power,axiom,
! [A: int,N2: nat] :
( ( ord_less_eq_int @ one_one_int @ A )
=> ( ( ord_less_nat @ zero_zero_nat @ N2 )
=> ( ord_less_eq_int @ A @ ( power_power_int @ A @ N2 ) ) ) ) ).
% self_le_power
thf(fact_886_self__le__power,axiom,
! [A: real,N2: nat] :
( ( ord_less_eq_real @ one_one_real @ A )
=> ( ( ord_less_nat @ zero_zero_nat @ N2 )
=> ( ord_less_eq_real @ A @ ( power_power_real @ A @ N2 ) ) ) ) ).
% self_le_power
thf(fact_887_one__less__power,axiom,
! [A: nat,N2: nat] :
( ( ord_less_nat @ one_one_nat @ A )
=> ( ( ord_less_nat @ zero_zero_nat @ N2 )
=> ( ord_less_nat @ one_one_nat @ ( power_power_nat @ A @ N2 ) ) ) ) ).
% one_less_power
thf(fact_888_one__less__power,axiom,
! [A: int,N2: nat] :
( ( ord_less_int @ one_one_int @ A )
=> ( ( ord_less_nat @ zero_zero_nat @ N2 )
=> ( ord_less_int @ one_one_int @ ( power_power_int @ A @ N2 ) ) ) ) ).
% one_less_power
thf(fact_889_one__less__power,axiom,
! [A: real,N2: nat] :
( ( ord_less_real @ one_one_real @ A )
=> ( ( ord_less_nat @ zero_zero_nat @ N2 )
=> ( ord_less_real @ one_one_real @ ( power_power_real @ A @ N2 ) ) ) ) ).
% one_less_power
thf(fact_890_proper__interval__nat_Oelims_I3_J,axiom,
! [X: option_nat,Xa: option_nat] :
( ~ ( set_pr3848668467136637714al_nat @ X @ Xa )
=> ( ( ( X = none_nat )
=> ! [X3: nat] :
( ( Xa
= ( some_nat @ X3 ) )
=> ( ord_less_nat @ zero_zero_nat @ X3 ) ) )
=> ~ ! [X3: nat] :
( ( X
= ( some_nat @ X3 ) )
=> ! [Y3: nat] :
( ( Xa
= ( some_nat @ Y3 ) )
=> ( ord_less_nat @ one_one_nat @ ( minus_minus_nat @ Y3 @ X3 ) ) ) ) ) ) ).
% proper_interval_nat.elims(3)
thf(fact_891_proper__interval__nat_Oelims_I2_J,axiom,
! [X: option_nat,Xa: option_nat] :
( ( set_pr3848668467136637714al_nat @ X @ Xa )
=> ( ( Xa != none_nat )
=> ( ( ( X = none_nat )
=> ! [X3: nat] :
( ( Xa
= ( some_nat @ X3 ) )
=> ~ ( ord_less_nat @ zero_zero_nat @ X3 ) ) )
=> ~ ! [X3: nat] :
( ( X
= ( some_nat @ X3 ) )
=> ! [Y3: nat] :
( ( Xa
= ( some_nat @ Y3 ) )
=> ~ ( ord_less_nat @ one_one_nat @ ( minus_minus_nat @ Y3 @ X3 ) ) ) ) ) ) ) ).
% proper_interval_nat.elims(2)
thf(fact_892_proper__interval__nat_Oelims_I1_J,axiom,
! [X: option_nat,Xa: option_nat,Y: $o] :
( ( ( set_pr3848668467136637714al_nat @ X @ Xa )
= Y )
=> ( ( ( Xa = none_nat )
=> ~ Y )
=> ( ( ( X = none_nat )
=> ! [X3: nat] :
( ( Xa
= ( some_nat @ X3 ) )
=> ( Y
= ( ~ ( ord_less_nat @ zero_zero_nat @ X3 ) ) ) ) )
=> ~ ! [X3: nat] :
( ( X
= ( some_nat @ X3 ) )
=> ! [Y3: nat] :
( ( Xa
= ( some_nat @ Y3 ) )
=> ( Y
= ( ~ ( ord_less_nat @ one_one_nat @ ( minus_minus_nat @ Y3 @ X3 ) ) ) ) ) ) ) ) ) ).
% proper_interval_nat.elims(1)
thf(fact_893_zero__le__ceiling,axiom,
! [X: real] :
( ( ord_less_eq_int @ zero_zero_int @ ( archim7802044766580827645g_real @ X ) )
= ( ord_less_real @ ( uminus_uminus_real @ one_one_real ) @ X ) ) ).
% zero_le_ceiling
thf(fact_894_ceiling__less__zero,axiom,
! [X: real] :
( ( ord_less_int @ ( archim7802044766580827645g_real @ X ) @ zero_zero_int )
= ( ord_less_eq_real @ X @ ( uminus_uminus_real @ one_one_real ) ) ) ).
% ceiling_less_zero
thf(fact_895_diff__0__eq__0,axiom,
! [N2: nat] :
( ( minus_minus_nat @ zero_zero_nat @ N2 )
= zero_zero_nat ) ).
% diff_0_eq_0
thf(fact_896_diff__self__eq__0,axiom,
! [M2: nat] :
( ( minus_minus_nat @ M2 @ M2 )
= zero_zero_nat ) ).
% diff_self_eq_0
thf(fact_897_diff__diff__cancel,axiom,
! [I: nat,N2: nat] :
( ( ord_less_eq_nat @ I @ N2 )
=> ( ( minus_minus_nat @ N2 @ ( minus_minus_nat @ N2 @ I ) )
= I ) ) ).
% diff_diff_cancel
thf(fact_898_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_899_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_900_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_901_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_902_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_903_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_904_zero__less__diff,axiom,
! [N2: nat,M2: nat] :
( ( ord_less_nat @ zero_zero_nat @ ( minus_minus_nat @ N2 @ M2 ) )
= ( ord_less_nat @ M2 @ N2 ) ) ).
% zero_less_diff
thf(fact_905_diff__is__0__eq_H,axiom,
! [M2: nat,N2: nat] :
( ( ord_less_eq_nat @ M2 @ N2 )
=> ( ( minus_minus_nat @ M2 @ N2 )
= zero_zero_nat ) ) ).
% diff_is_0_eq'
thf(fact_906_diff__is__0__eq,axiom,
! [M2: nat,N2: nat] :
( ( ( minus_minus_nat @ M2 @ N2 )
= zero_zero_nat )
= ( ord_less_eq_nat @ M2 @ N2 ) ) ).
% diff_is_0_eq
thf(fact_907_ceiling__le__zero,axiom,
! [X: real] :
( ( ord_less_eq_int @ ( archim7802044766580827645g_real @ X ) @ zero_zero_int )
= ( ord_less_eq_real @ X @ zero_zero_real ) ) ).
% ceiling_le_zero
thf(fact_908_zero__less__ceiling,axiom,
! [X: real] :
( ( ord_less_int @ zero_zero_int @ ( archim7802044766580827645g_real @ X ) )
= ( ord_less_real @ zero_zero_real @ X ) ) ).
% zero_less_ceiling
thf(fact_909_ceiling__less__one,axiom,
! [X: real] :
( ( ord_less_int @ ( archim7802044766580827645g_real @ X ) @ one_one_int )
= ( ord_less_eq_real @ X @ zero_zero_real ) ) ).
% ceiling_less_one
thf(fact_910_one__le__ceiling,axiom,
! [X: real] :
( ( ord_less_eq_int @ one_one_int @ ( archim7802044766580827645g_real @ X ) )
= ( ord_less_real @ zero_zero_real @ X ) ) ).
% one_le_ceiling
thf(fact_911_ceiling__le__one,axiom,
! [X: real] :
( ( ord_less_eq_int @ ( archim7802044766580827645g_real @ X ) @ one_one_int )
= ( ord_less_eq_real @ X @ one_one_real ) ) ).
% ceiling_le_one
thf(fact_912_one__less__ceiling,axiom,
! [X: real] :
( ( ord_less_int @ one_one_int @ ( archim7802044766580827645g_real @ X ) )
= ( ord_less_real @ one_one_real @ X ) ) ).
% one_less_ceiling
thf(fact_913_diff__mono,axiom,
! [A: int,B: int,D3: int,C: int] :
( ( ord_less_eq_int @ A @ B )
=> ( ( ord_less_eq_int @ D3 @ C )
=> ( ord_less_eq_int @ ( minus_minus_int @ A @ C ) @ ( minus_minus_int @ B @ D3 ) ) ) ) ).
% diff_mono
thf(fact_914_diff__mono,axiom,
! [A: real,B: real,D3: real,C: real] :
( ( ord_less_eq_real @ A @ B )
=> ( ( ord_less_eq_real @ D3 @ C )
=> ( ord_less_eq_real @ ( minus_minus_real @ A @ C ) @ ( minus_minus_real @ B @ D3 ) ) ) ) ).
% diff_mono
thf(fact_915_diff__left__mono,axiom,
! [B: int,A: int,C: int] :
( ( ord_less_eq_int @ B @ A )
=> ( ord_less_eq_int @ ( minus_minus_int @ C @ A ) @ ( minus_minus_int @ C @ B ) ) ) ).
% diff_left_mono
thf(fact_916_diff__left__mono,axiom,
! [B: real,A: real,C: real] :
( ( ord_less_eq_real @ B @ A )
=> ( ord_less_eq_real @ ( minus_minus_real @ C @ A ) @ ( minus_minus_real @ C @ B ) ) ) ).
% diff_left_mono
thf(fact_917_diff__right__mono,axiom,
! [A: int,B: int,C: int] :
( ( ord_less_eq_int @ A @ B )
=> ( ord_less_eq_int @ ( minus_minus_int @ A @ C ) @ ( minus_minus_int @ B @ C ) ) ) ).
% diff_right_mono
thf(fact_918_diff__right__mono,axiom,
! [A: real,B: real,C: real] :
( ( ord_less_eq_real @ A @ B )
=> ( ord_less_eq_real @ ( minus_minus_real @ A @ C ) @ ( minus_minus_real @ B @ C ) ) ) ).
% diff_right_mono
thf(fact_919_diff__eq__diff__less__eq,axiom,
! [A: int,B: int,C: int,D3: int] :
( ( ( minus_minus_int @ A @ B )
= ( minus_minus_int @ C @ D3 ) )
=> ( ( ord_less_eq_int @ A @ B )
= ( ord_less_eq_int @ C @ D3 ) ) ) ).
% diff_eq_diff_less_eq
thf(fact_920_diff__eq__diff__less__eq,axiom,
! [A: real,B: real,C: real,D3: real] :
( ( ( minus_minus_real @ A @ B )
= ( minus_minus_real @ C @ D3 ) )
=> ( ( ord_less_eq_real @ A @ B )
= ( ord_less_eq_real @ C @ D3 ) ) ) ).
% diff_eq_diff_less_eq
thf(fact_921_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_922_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_923_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_924_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_925_diff__eq__diff__less,axiom,
! [A: int,B: int,C: int,D3: int] :
( ( ( minus_minus_int @ A @ B )
= ( minus_minus_int @ C @ D3 ) )
=> ( ( ord_less_int @ A @ B )
= ( ord_less_int @ C @ D3 ) ) ) ).
% diff_eq_diff_less
thf(fact_926_diff__eq__diff__less,axiom,
! [A: real,B: real,C: real,D3: real] :
( ( ( minus_minus_real @ A @ B )
= ( minus_minus_real @ C @ D3 ) )
=> ( ( ord_less_real @ A @ B )
= ( ord_less_real @ C @ D3 ) ) ) ).
% diff_eq_diff_less
thf(fact_927_diff__strict__mono,axiom,
! [A: int,B: int,D3: int,C: int] :
( ( ord_less_int @ A @ B )
=> ( ( ord_less_int @ D3 @ C )
=> ( ord_less_int @ ( minus_minus_int @ A @ C ) @ ( minus_minus_int @ B @ D3 ) ) ) ) ).
% diff_strict_mono
thf(fact_928_diff__strict__mono,axiom,
! [A: real,B: real,D3: real,C: real] :
( ( ord_less_real @ A @ B )
=> ( ( ord_less_real @ D3 @ C )
=> ( ord_less_real @ ( minus_minus_real @ A @ C ) @ ( minus_minus_real @ B @ D3 ) ) ) ) ).
% diff_strict_mono
thf(fact_929_diffs0__imp__equal,axiom,
! [M2: nat,N2: nat] :
( ( ( minus_minus_nat @ M2 @ N2 )
= zero_zero_nat )
=> ( ( ( minus_minus_nat @ N2 @ M2 )
= zero_zero_nat )
=> ( M2 = N2 ) ) ) ).
% diffs0_imp_equal
thf(fact_930_minus__nat_Odiff__0,axiom,
! [M2: nat] :
( ( minus_minus_nat @ M2 @ zero_zero_nat )
= M2 ) ).
% minus_nat.diff_0
thf(fact_931_less__imp__diff__less,axiom,
! [J: nat,K2: nat,N2: nat] :
( ( ord_less_nat @ J @ K2 )
=> ( ord_less_nat @ ( minus_minus_nat @ J @ N2 ) @ K2 ) ) ).
% less_imp_diff_less
thf(fact_932_diff__less__mono2,axiom,
! [M2: nat,N2: nat,L: nat] :
( ( ord_less_nat @ M2 @ N2 )
=> ( ( ord_less_nat @ M2 @ L )
=> ( ord_less_nat @ ( minus_minus_nat @ L @ N2 ) @ ( minus_minus_nat @ L @ M2 ) ) ) ) ).
% diff_less_mono2
thf(fact_933_diff__le__mono2,axiom,
! [M2: nat,N2: nat,L: nat] :
( ( ord_less_eq_nat @ M2 @ N2 )
=> ( ord_less_eq_nat @ ( minus_minus_nat @ L @ N2 ) @ ( minus_minus_nat @ L @ M2 ) ) ) ).
% diff_le_mono2
thf(fact_934_le__diff__iff_H,axiom,
! [A: nat,C: nat,B: nat] :
( ( ord_less_eq_nat @ A @ C )
=> ( ( ord_less_eq_nat @ B @ C )
=> ( ( ord_less_eq_nat @ ( minus_minus_nat @ C @ A ) @ ( minus_minus_nat @ C @ B ) )
= ( ord_less_eq_nat @ B @ A ) ) ) ) ).
% le_diff_iff'
thf(fact_935_diff__le__self,axiom,
! [M2: nat,N2: nat] : ( ord_less_eq_nat @ ( minus_minus_nat @ M2 @ N2 ) @ M2 ) ).
% diff_le_self
thf(fact_936_diff__le__mono,axiom,
! [M2: nat,N2: nat,L: nat] :
( ( ord_less_eq_nat @ M2 @ N2 )
=> ( ord_less_eq_nat @ ( minus_minus_nat @ M2 @ L ) @ ( minus_minus_nat @ N2 @ L ) ) ) ).
% diff_le_mono
thf(fact_937_Nat_Odiff__diff__eq,axiom,
! [K2: nat,M2: nat,N2: nat] :
( ( ord_less_eq_nat @ K2 @ M2 )
=> ( ( ord_less_eq_nat @ K2 @ N2 )
=> ( ( minus_minus_nat @ ( minus_minus_nat @ M2 @ K2 ) @ ( minus_minus_nat @ N2 @ K2 ) )
= ( minus_minus_nat @ M2 @ N2 ) ) ) ) ).
% Nat.diff_diff_eq
thf(fact_938_le__diff__iff,axiom,
! [K2: nat,M2: nat,N2: nat] :
( ( ord_less_eq_nat @ K2 @ M2 )
=> ( ( ord_less_eq_nat @ K2 @ N2 )
=> ( ( ord_less_eq_nat @ ( minus_minus_nat @ M2 @ K2 ) @ ( minus_minus_nat @ N2 @ K2 ) )
= ( ord_less_eq_nat @ M2 @ N2 ) ) ) ) ).
% le_diff_iff
thf(fact_939_eq__diff__iff,axiom,
! [K2: nat,M2: nat,N2: nat] :
( ( ord_less_eq_nat @ K2 @ M2 )
=> ( ( ord_less_eq_nat @ K2 @ N2 )
=> ( ( ( minus_minus_nat @ M2 @ K2 )
= ( minus_minus_nat @ N2 @ K2 ) )
= ( M2 = N2 ) ) ) ) ).
% eq_diff_iff
thf(fact_940_of__nat__diff,axiom,
! [N2: nat,M2: nat] :
( ( ord_less_eq_nat @ N2 @ M2 )
=> ( ( semiri1316708129612266289at_nat @ ( minus_minus_nat @ M2 @ N2 ) )
= ( minus_minus_nat @ ( semiri1316708129612266289at_nat @ M2 ) @ ( semiri1316708129612266289at_nat @ N2 ) ) ) ) ).
% of_nat_diff
thf(fact_941_of__nat__diff,axiom,
! [N2: nat,M2: nat] :
( ( ord_less_eq_nat @ N2 @ M2 )
=> ( ( semiri1314217659103216013at_int @ ( minus_minus_nat @ M2 @ N2 ) )
= ( minus_minus_int @ ( semiri1314217659103216013at_int @ M2 ) @ ( semiri1314217659103216013at_int @ N2 ) ) ) ) ).
% of_nat_diff
thf(fact_942_of__nat__diff,axiom,
! [N2: nat,M2: nat] :
( ( ord_less_eq_nat @ N2 @ M2 )
=> ( ( semiri5074537144036343181t_real @ ( minus_minus_nat @ M2 @ N2 ) )
= ( minus_minus_real @ ( semiri5074537144036343181t_real @ M2 ) @ ( semiri5074537144036343181t_real @ N2 ) ) ) ) ).
% of_nat_diff
thf(fact_943_diff__commute,axiom,
! [I: nat,J: nat,K2: nat] :
( ( minus_minus_nat @ ( minus_minus_nat @ I @ J ) @ K2 )
= ( minus_minus_nat @ ( minus_minus_nat @ I @ K2 ) @ J ) ) ).
% diff_commute
thf(fact_944_ceiling__mono,axiom,
! [Y: real,X: real] :
( ( ord_less_eq_real @ Y @ X )
=> ( ord_less_eq_int @ ( archim7802044766580827645g_real @ Y ) @ ( archim7802044766580827645g_real @ X ) ) ) ).
% ceiling_mono
thf(fact_945_ceiling__less__cancel,axiom,
! [X: real,Y: real] :
( ( ord_less_int @ ( archim7802044766580827645g_real @ X ) @ ( archim7802044766580827645g_real @ Y ) )
=> ( ord_less_real @ X @ Y ) ) ).
% ceiling_less_cancel
thf(fact_946_le__iff__diff__le__0,axiom,
( ord_less_eq_int
= ( ^ [A2: int,B3: int] : ( ord_less_eq_int @ ( minus_minus_int @ A2 @ B3 ) @ zero_zero_int ) ) ) ).
% le_iff_diff_le_0
thf(fact_947_le__iff__diff__le__0,axiom,
( ord_less_eq_real
= ( ^ [A2: real,B3: real] : ( ord_less_eq_real @ ( minus_minus_real @ A2 @ B3 ) @ zero_zero_real ) ) ) ).
% le_iff_diff_le_0
thf(fact_948_less__iff__diff__less__0,axiom,
( ord_less_int
= ( ^ [A2: int,B3: int] : ( ord_less_int @ ( minus_minus_int @ A2 @ B3 ) @ zero_zero_int ) ) ) ).
% less_iff_diff_less_0
thf(fact_949_less__iff__diff__less__0,axiom,
( ord_less_real
= ( ^ [A2: real,B3: real] : ( ord_less_real @ ( minus_minus_real @ A2 @ B3 ) @ zero_zero_real ) ) ) ).
% less_iff_diff_less_0
thf(fact_950_diff__less,axiom,
! [N2: nat,M2: nat] :
( ( ord_less_nat @ zero_zero_nat @ N2 )
=> ( ( ord_less_nat @ zero_zero_nat @ M2 )
=> ( ord_less_nat @ ( minus_minus_nat @ M2 @ N2 ) @ M2 ) ) ) ).
% diff_less
thf(fact_951_less__diff__iff,axiom,
! [K2: nat,M2: nat,N2: nat] :
( ( ord_less_eq_nat @ K2 @ M2 )
=> ( ( ord_less_eq_nat @ K2 @ N2 )
=> ( ( ord_less_nat @ ( minus_minus_nat @ M2 @ K2 ) @ ( minus_minus_nat @ N2 @ K2 ) )
= ( ord_less_nat @ M2 @ N2 ) ) ) ) ).
% less_diff_iff
thf(fact_952_diff__less__mono,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_nat @ A @ B )
=> ( ( ord_less_eq_nat @ C @ A )
=> ( ord_less_nat @ ( minus_minus_nat @ A @ C ) @ ( minus_minus_nat @ B @ C ) ) ) ) ).
% diff_less_mono
thf(fact_953_proper__interval__nat_Osimps_I3_J,axiom,
! [X: nat,Y: nat] :
( ( set_pr3848668467136637714al_nat @ ( some_nat @ X ) @ ( some_nat @ Y ) )
= ( ord_less_nat @ one_one_nat @ ( minus_minus_nat @ Y @ X ) ) ) ).
% proper_interval_nat.simps(3)
thf(fact_954_proper__interval__nat_Opelims_I1_J,axiom,
! [X: option_nat,Xa: option_nat,Y: $o] :
( ( ( set_pr3848668467136637714al_nat @ X @ Xa )
= Y )
=> ( ( accp_P8646395344606611882on_nat @ set_pr8346368467067310656at_rel @ ( produc5098337634421038937on_nat @ X @ Xa ) )
=> ( ( ( Xa = none_nat )
=> ( Y
=> ~ ( accp_P8646395344606611882on_nat @ set_pr8346368467067310656at_rel @ ( produc5098337634421038937on_nat @ X @ none_nat ) ) ) )
=> ( ( ( X = none_nat )
=> ! [X3: nat] :
( ( Xa
= ( some_nat @ X3 ) )
=> ( ( Y
= ( ord_less_nat @ zero_zero_nat @ X3 ) )
=> ~ ( accp_P8646395344606611882on_nat @ set_pr8346368467067310656at_rel @ ( produc5098337634421038937on_nat @ none_nat @ ( some_nat @ X3 ) ) ) ) ) )
=> ~ ! [X3: nat] :
( ( X
= ( some_nat @ X3 ) )
=> ! [Y3: nat] :
( ( Xa
= ( some_nat @ Y3 ) )
=> ( ( Y
= ( ord_less_nat @ one_one_nat @ ( minus_minus_nat @ Y3 @ X3 ) ) )
=> ~ ( accp_P8646395344606611882on_nat @ set_pr8346368467067310656at_rel @ ( produc5098337634421038937on_nat @ ( some_nat @ X3 ) @ ( some_nat @ Y3 ) ) ) ) ) ) ) ) ) ) ).
% proper_interval_nat.pelims(1)
thf(fact_955_proper__interval__nat_Opelims_I2_J,axiom,
! [X: option_nat,Xa: option_nat] :
( ( set_pr3848668467136637714al_nat @ X @ Xa )
=> ( ( accp_P8646395344606611882on_nat @ set_pr8346368467067310656at_rel @ ( produc5098337634421038937on_nat @ X @ Xa ) )
=> ( ( ( Xa = none_nat )
=> ~ ( accp_P8646395344606611882on_nat @ set_pr8346368467067310656at_rel @ ( produc5098337634421038937on_nat @ X @ none_nat ) ) )
=> ( ( ( X = none_nat )
=> ! [X3: nat] :
( ( Xa
= ( some_nat @ X3 ) )
=> ( ( accp_P8646395344606611882on_nat @ set_pr8346368467067310656at_rel @ ( produc5098337634421038937on_nat @ none_nat @ ( some_nat @ X3 ) ) )
=> ~ ( ord_less_nat @ zero_zero_nat @ X3 ) ) ) )
=> ~ ! [X3: nat] :
( ( X
= ( some_nat @ X3 ) )
=> ! [Y3: nat] :
( ( Xa
= ( some_nat @ Y3 ) )
=> ( ( accp_P8646395344606611882on_nat @ set_pr8346368467067310656at_rel @ ( produc5098337634421038937on_nat @ ( some_nat @ X3 ) @ ( some_nat @ Y3 ) ) )
=> ~ ( ord_less_nat @ one_one_nat @ ( minus_minus_nat @ Y3 @ X3 ) ) ) ) ) ) ) ) ) ).
% proper_interval_nat.pelims(2)
thf(fact_956_proper__interval__nat_Opelims_I3_J,axiom,
! [X: option_nat,Xa: option_nat] :
( ~ ( set_pr3848668467136637714al_nat @ X @ Xa )
=> ( ( accp_P8646395344606611882on_nat @ set_pr8346368467067310656at_rel @ ( produc5098337634421038937on_nat @ X @ Xa ) )
=> ( ( ( X = none_nat )
=> ! [X3: nat] :
( ( Xa
= ( some_nat @ X3 ) )
=> ( ( accp_P8646395344606611882on_nat @ set_pr8346368467067310656at_rel @ ( produc5098337634421038937on_nat @ none_nat @ ( some_nat @ X3 ) ) )
=> ( ord_less_nat @ zero_zero_nat @ X3 ) ) ) )
=> ~ ! [X3: nat] :
( ( X
= ( some_nat @ X3 ) )
=> ! [Y3: nat] :
( ( Xa
= ( some_nat @ Y3 ) )
=> ( ( accp_P8646395344606611882on_nat @ set_pr8346368467067310656at_rel @ ( produc5098337634421038937on_nat @ ( some_nat @ X3 ) @ ( some_nat @ Y3 ) ) )
=> ( ord_less_nat @ one_one_nat @ ( minus_minus_nat @ Y3 @ X3 ) ) ) ) ) ) ) ) ).
% proper_interval_nat.pelims(3)
thf(fact_957_power__minus__mult,axiom,
! [N2: nat,A: nat] :
( ( ord_less_nat @ zero_zero_nat @ N2 )
=> ( ( times_times_nat @ ( power_power_nat @ A @ ( minus_minus_nat @ N2 @ one_one_nat ) ) @ A )
= ( power_power_nat @ A @ N2 ) ) ) ).
% power_minus_mult
thf(fact_958_power__minus__mult,axiom,
! [N2: nat,A: int] :
( ( ord_less_nat @ zero_zero_nat @ N2 )
=> ( ( times_times_int @ ( power_power_int @ A @ ( minus_minus_nat @ N2 @ one_one_nat ) ) @ A )
= ( power_power_int @ A @ N2 ) ) ) ).
% power_minus_mult
thf(fact_959_power__minus__mult,axiom,
! [N2: nat,A: real] :
( ( ord_less_nat @ zero_zero_nat @ N2 )
=> ( ( times_times_real @ ( power_power_real @ A @ ( minus_minus_nat @ N2 @ one_one_nat ) ) @ A )
= ( power_power_real @ A @ N2 ) ) ) ).
% power_minus_mult
thf(fact_960_zdiff__int__split,axiom,
! [P: int > $o,X: nat,Y: nat] :
( ( P @ ( semiri1314217659103216013at_int @ ( minus_minus_nat @ X @ Y ) ) )
= ( ( ( ord_less_eq_nat @ Y @ X )
=> ( P @ ( minus_minus_int @ ( semiri1314217659103216013at_int @ X ) @ ( semiri1314217659103216013at_int @ Y ) ) ) )
& ( ( ord_less_nat @ X @ Y )
=> ( P @ zero_zero_int ) ) ) ) ).
% zdiff_int_split
thf(fact_961_mult__cancel2,axiom,
! [M2: nat,K2: nat,N2: nat] :
( ( ( times_times_nat @ M2 @ K2 )
= ( times_times_nat @ N2 @ K2 ) )
= ( ( M2 = N2 )
| ( K2 = zero_zero_nat ) ) ) ).
% mult_cancel2
thf(fact_962_mult__cancel1,axiom,
! [K2: nat,M2: nat,N2: nat] :
( ( ( times_times_nat @ K2 @ M2 )
= ( times_times_nat @ K2 @ N2 ) )
= ( ( M2 = N2 )
| ( K2 = zero_zero_nat ) ) ) ).
% mult_cancel1
thf(fact_963_mult__0__right,axiom,
! [M2: nat] :
( ( times_times_nat @ M2 @ zero_zero_nat )
= zero_zero_nat ) ).
% mult_0_right
thf(fact_964_mult__is__0,axiom,
! [M2: nat,N2: nat] :
( ( ( times_times_nat @ M2 @ N2 )
= zero_zero_nat )
= ( ( M2 = zero_zero_nat )
| ( N2 = zero_zero_nat ) ) ) ).
% mult_is_0
thf(fact_965_nat__1__eq__mult__iff,axiom,
! [M2: nat,N2: nat] :
( ( one_one_nat
= ( times_times_nat @ M2 @ N2 ) )
= ( ( M2 = one_one_nat )
& ( N2 = one_one_nat ) ) ) ).
% nat_1_eq_mult_iff
thf(fact_966_nat__mult__eq__1__iff,axiom,
! [M2: nat,N2: nat] :
( ( ( times_times_nat @ M2 @ N2 )
= one_one_nat )
= ( ( M2 = one_one_nat )
& ( N2 = one_one_nat ) ) ) ).
% nat_mult_eq_1_iff
thf(fact_967_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_968_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_969_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_970_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_971_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_972_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_973_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_974_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_975_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_976_mult__zero__right,axiom,
! [A: nat] :
( ( times_times_nat @ A @ zero_zero_nat )
= zero_zero_nat ) ).
% mult_zero_right
thf(fact_977_mult__zero__right,axiom,
! [A: int] :
( ( times_times_int @ A @ zero_zero_int )
= zero_zero_int ) ).
% mult_zero_right
thf(fact_978_mult__zero__right,axiom,
! [A: real] :
( ( times_times_real @ A @ zero_zero_real )
= zero_zero_real ) ).
% mult_zero_right
thf(fact_979_mult__zero__left,axiom,
! [A: nat] :
( ( times_times_nat @ zero_zero_nat @ A )
= zero_zero_nat ) ).
% mult_zero_left
thf(fact_980_mult__zero__left,axiom,
! [A: int] :
( ( times_times_int @ zero_zero_int @ A )
= zero_zero_int ) ).
% mult_zero_left
thf(fact_981_mult__zero__left,axiom,
! [A: real] :
( ( times_times_real @ zero_zero_real @ A )
= zero_zero_real ) ).
% mult_zero_left
thf(fact_982_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_983_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_984_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_985_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_986_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_987_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_988_of__nat__mult,axiom,
! [M2: nat,N2: nat] :
( ( semiri1316708129612266289at_nat @ ( times_times_nat @ M2 @ N2 ) )
= ( times_times_nat @ ( semiri1316708129612266289at_nat @ M2 ) @ ( semiri1316708129612266289at_nat @ N2 ) ) ) ).
% of_nat_mult
thf(fact_989_of__nat__mult,axiom,
! [M2: nat,N2: nat] :
( ( semiri1314217659103216013at_int @ ( times_times_nat @ M2 @ N2 ) )
= ( times_times_int @ ( semiri1314217659103216013at_int @ M2 ) @ ( semiri1314217659103216013at_int @ N2 ) ) ) ).
% of_nat_mult
thf(fact_990_of__nat__mult,axiom,
! [M2: nat,N2: nat] :
( ( semiri5074537144036343181t_real @ ( times_times_nat @ M2 @ N2 ) )
= ( times_times_real @ ( semiri5074537144036343181t_real @ M2 ) @ ( semiri5074537144036343181t_real @ N2 ) ) ) ).
% of_nat_mult
thf(fact_991_mult__less__cancel2,axiom,
! [M2: nat,K2: nat,N2: nat] :
( ( ord_less_nat @ ( times_times_nat @ M2 @ K2 ) @ ( times_times_nat @ N2 @ K2 ) )
= ( ( ord_less_nat @ zero_zero_nat @ K2 )
& ( ord_less_nat @ M2 @ N2 ) ) ) ).
% mult_less_cancel2
thf(fact_992_nat__0__less__mult__iff,axiom,
! [M2: nat,N2: nat] :
( ( ord_less_nat @ zero_zero_nat @ ( times_times_nat @ M2 @ N2 ) )
= ( ( ord_less_nat @ zero_zero_nat @ M2 )
& ( ord_less_nat @ zero_zero_nat @ N2 ) ) ) ).
% nat_0_less_mult_iff
thf(fact_993_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_994_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_995_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_996_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_997_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_998_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_999_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_1000_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_1001_mult__le__cancel2,axiom,
! [M2: nat,K2: nat,N2: nat] :
( ( ord_less_eq_nat @ ( times_times_nat @ M2 @ K2 ) @ ( times_times_nat @ N2 @ K2 ) )
= ( ( ord_less_nat @ zero_zero_nat @ K2 )
=> ( ord_less_eq_nat @ M2 @ N2 ) ) ) ).
% mult_le_cancel2
thf(fact_1002_diff__mult__distrib,axiom,
! [M2: nat,N2: nat,K2: nat] :
( ( times_times_nat @ ( minus_minus_nat @ M2 @ N2 ) @ K2 )
= ( minus_minus_nat @ ( times_times_nat @ M2 @ K2 ) @ ( times_times_nat @ N2 @ K2 ) ) ) ).
% diff_mult_distrib
thf(fact_1003_diff__mult__distrib2,axiom,
! [K2: nat,M2: nat,N2: nat] :
( ( times_times_nat @ K2 @ ( minus_minus_nat @ M2 @ N2 ) )
= ( minus_minus_nat @ ( times_times_nat @ K2 @ M2 ) @ ( times_times_nat @ K2 @ N2 ) ) ) ).
% diff_mult_distrib2
thf(fact_1004_inf__period_I2_J,axiom,
! [P: int > $o,D2: int,Q: int > $o] :
( ! [X3: int,K: int] :
( ( P @ X3 )
= ( P @ ( minus_minus_int @ X3 @ ( times_times_int @ K @ D2 ) ) ) )
=> ( ! [X3: int,K: int] :
( ( Q @ X3 )
= ( Q @ ( minus_minus_int @ X3 @ ( times_times_int @ K @ D2 ) ) ) )
=> ! [X6: int,K3: int] :
( ( ( P @ X6 )
| ( Q @ X6 ) )
= ( ( P @ ( minus_minus_int @ X6 @ ( times_times_int @ K3 @ D2 ) ) )
| ( Q @ ( minus_minus_int @ X6 @ ( times_times_int @ K3 @ D2 ) ) ) ) ) ) ) ).
% inf_period(2)
thf(fact_1005_inf__period_I2_J,axiom,
! [P: real > $o,D2: real,Q: real > $o] :
( ! [X3: real,K: real] :
( ( P @ X3 )
= ( P @ ( minus_minus_real @ X3 @ ( times_times_real @ K @ D2 ) ) ) )
=> ( ! [X3: real,K: real] :
( ( Q @ X3 )
= ( Q @ ( minus_minus_real @ X3 @ ( times_times_real @ K @ D2 ) ) ) )
=> ! [X6: real,K3: real] :
( ( ( P @ X6 )
| ( Q @ X6 ) )
= ( ( P @ ( minus_minus_real @ X6 @ ( times_times_real @ K3 @ D2 ) ) )
| ( Q @ ( minus_minus_real @ X6 @ ( times_times_real @ K3 @ D2 ) ) ) ) ) ) ) ).
% inf_period(2)
thf(fact_1006_inf__period_I1_J,axiom,
! [P: int > $o,D2: int,Q: int > $o] :
( ! [X3: int,K: int] :
( ( P @ X3 )
= ( P @ ( minus_minus_int @ X3 @ ( times_times_int @ K @ D2 ) ) ) )
=> ( ! [X3: int,K: int] :
( ( Q @ X3 )
= ( Q @ ( minus_minus_int @ X3 @ ( times_times_int @ K @ D2 ) ) ) )
=> ! [X6: int,K3: int] :
( ( ( P @ X6 )
& ( Q @ X6 ) )
= ( ( P @ ( minus_minus_int @ X6 @ ( times_times_int @ K3 @ D2 ) ) )
& ( Q @ ( minus_minus_int @ X6 @ ( times_times_int @ K3 @ D2 ) ) ) ) ) ) ) ).
% inf_period(1)
thf(fact_1007_inf__period_I1_J,axiom,
! [P: real > $o,D2: real,Q: real > $o] :
( ! [X3: real,K: real] :
( ( P @ X3 )
= ( P @ ( minus_minus_real @ X3 @ ( times_times_real @ K @ D2 ) ) ) )
=> ( ! [X3: real,K: real] :
( ( Q @ X3 )
= ( Q @ ( minus_minus_real @ X3 @ ( times_times_real @ K @ D2 ) ) ) )
=> ! [X6: real,K3: real] :
( ( ( P @ X6 )
& ( Q @ X6 ) )
= ( ( P @ ( minus_minus_real @ X6 @ ( times_times_real @ K3 @ D2 ) ) )
& ( Q @ ( minus_minus_real @ X6 @ ( times_times_real @ K3 @ D2 ) ) ) ) ) ) ) ).
% inf_period(1)
thf(fact_1008_left__diff__distrib,axiom,
! [A: int,B: int,C: int] :
( ( times_times_int @ ( minus_minus_int @ A @ B ) @ C )
= ( minus_minus_int @ ( times_times_int @ A @ C ) @ ( times_times_int @ B @ C ) ) ) ).
% left_diff_distrib
thf(fact_1009_left__diff__distrib,axiom,
! [A: real,B: real,C: real] :
( ( times_times_real @ ( minus_minus_real @ A @ B ) @ C )
= ( minus_minus_real @ ( times_times_real @ A @ C ) @ ( times_times_real @ B @ C ) ) ) ).
% left_diff_distrib
thf(fact_1010_right__diff__distrib,axiom,
! [A: int,B: int,C: int] :
( ( times_times_int @ A @ ( minus_minus_int @ B @ C ) )
= ( minus_minus_int @ ( times_times_int @ A @ B ) @ ( times_times_int @ A @ C ) ) ) ).
% right_diff_distrib
thf(fact_1011_right__diff__distrib,axiom,
! [A: real,B: real,C: real] :
( ( times_times_real @ A @ ( minus_minus_real @ B @ C ) )
= ( minus_minus_real @ ( times_times_real @ A @ B ) @ ( times_times_real @ A @ C ) ) ) ).
% right_diff_distrib
thf(fact_1012_left__diff__distrib_H,axiom,
! [B: int,C: int,A: int] :
( ( times_times_int @ ( minus_minus_int @ B @ C ) @ A )
= ( minus_minus_int @ ( times_times_int @ B @ A ) @ ( times_times_int @ C @ A ) ) ) ).
% left_diff_distrib'
thf(fact_1013_left__diff__distrib_H,axiom,
! [B: real,C: real,A: real] :
( ( times_times_real @ ( minus_minus_real @ B @ C ) @ A )
= ( minus_minus_real @ ( times_times_real @ B @ A ) @ ( times_times_real @ C @ A ) ) ) ).
% left_diff_distrib'
thf(fact_1014_minusinfinity,axiom,
! [D3: int,P1: int > $o,P: int > $o] :
( ( ord_less_int @ zero_zero_int @ D3 )
=> ( ! [X3: int,K: int] :
( ( P1 @ X3 )
= ( P1 @ ( minus_minus_int @ X3 @ ( times_times_int @ K @ D3 ) ) ) )
=> ( ? [Z4: int] :
! [X3: int] :
( ( ord_less_int @ X3 @ Z4 )
=> ( ( P @ X3 )
= ( P1 @ X3 ) ) )
=> ( ? [X_12: int] : ( P1 @ X_12 )
=> ? [X_1: int] : ( P @ X_1 ) ) ) ) ) ).
% minusinfinity
thf(fact_1015_plusinfinity,axiom,
! [D3: int,P4: int > $o,P: int > $o] :
( ( ord_less_int @ zero_zero_int @ D3 )
=> ( ! [X3: int,K: int] :
( ( P4 @ X3 )
= ( P4 @ ( minus_minus_int @ X3 @ ( times_times_int @ K @ D3 ) ) ) )
=> ( ? [Z4: int] :
! [X3: int] :
( ( ord_less_int @ Z4 @ X3 )
=> ( ( P @ X3 )
= ( P4 @ X3 ) ) )
=> ( ? [X_12: int] : ( P4 @ X_12 )
=> ? [X_1: int] : ( P @ X_1 ) ) ) ) ) ).
% plusinfinity
thf(fact_1016_int__ops_I7_J,axiom,
! [A: nat,B: nat] :
( ( semiri1314217659103216013at_int @ ( times_times_nat @ A @ B ) )
= ( times_times_int @ ( semiri1314217659103216013at_int @ A ) @ ( semiri1314217659103216013at_int @ B ) ) ) ).
% int_ops(7)
thf(fact_1017_le__cube,axiom,
! [M2: nat] : ( ord_less_eq_nat @ M2 @ ( times_times_nat @ M2 @ ( times_times_nat @ M2 @ M2 ) ) ) ).
% le_cube
thf(fact_1018_le__square,axiom,
! [M2: nat] : ( ord_less_eq_nat @ M2 @ ( times_times_nat @ M2 @ M2 ) ) ).
% le_square
thf(fact_1019_mult__le__mono,axiom,
! [I: nat,J: nat,K2: nat,L: nat] :
( ( ord_less_eq_nat @ I @ J )
=> ( ( ord_less_eq_nat @ K2 @ L )
=> ( ord_less_eq_nat @ ( times_times_nat @ I @ K2 ) @ ( times_times_nat @ J @ L ) ) ) ) ).
% mult_le_mono
thf(fact_1020_mult__le__mono1,axiom,
! [I: nat,J: nat,K2: nat] :
( ( ord_less_eq_nat @ I @ J )
=> ( ord_less_eq_nat @ ( times_times_nat @ I @ K2 ) @ ( times_times_nat @ J @ K2 ) ) ) ).
% mult_le_mono1
thf(fact_1021_mult__le__mono2,axiom,
! [I: nat,J: nat,K2: nat] :
( ( ord_less_eq_nat @ I @ J )
=> ( ord_less_eq_nat @ ( times_times_nat @ K2 @ I ) @ ( times_times_nat @ K2 @ J ) ) ) ).
% mult_le_mono2
thf(fact_1022_mult__0,axiom,
! [N2: nat] :
( ( times_times_nat @ zero_zero_nat @ N2 )
= zero_zero_nat ) ).
% mult_0
thf(fact_1023_nat__mult__1__right,axiom,
! [N2: nat] :
( ( times_times_nat @ N2 @ one_one_nat )
= N2 ) ).
% nat_mult_1_right
thf(fact_1024_nat__mult__1,axiom,
! [N2: nat] :
( ( times_times_nat @ one_one_nat @ N2 )
= N2 ) ).
% nat_mult_1
thf(fact_1025_decr__mult__lemma,axiom,
! [D3: int,P: int > $o,K2: int] :
( ( ord_less_int @ zero_zero_int @ D3 )
=> ( ! [X3: int] :
( ( P @ X3 )
=> ( P @ ( minus_minus_int @ X3 @ D3 ) ) )
=> ( ( ord_less_eq_int @ zero_zero_int @ K2 )
=> ! [X6: int] :
( ( P @ X6 )
=> ( P @ ( minus_minus_int @ X6 @ ( times_times_int @ K2 @ D3 ) ) ) ) ) ) ) ).
% decr_mult_lemma
thf(fact_1026_mult__less__mono1,axiom,
! [I: nat,J: nat,K2: nat] :
( ( ord_less_nat @ I @ J )
=> ( ( ord_less_nat @ zero_zero_nat @ K2 )
=> ( ord_less_nat @ ( times_times_nat @ I @ K2 ) @ ( times_times_nat @ J @ K2 ) ) ) ) ).
% mult_less_mono1
thf(fact_1027_mult__less__mono2,axiom,
! [I: nat,J: nat,K2: nat] :
( ( ord_less_nat @ I @ J )
=> ( ( ord_less_nat @ zero_zero_nat @ K2 )
=> ( ord_less_nat @ ( times_times_nat @ K2 @ I ) @ ( times_times_nat @ K2 @ J ) ) ) ) ).
% mult_less_mono2
thf(fact_1028_mult__eq__self__implies__10,axiom,
! [M2: nat,N2: nat] :
( ( M2
= ( times_times_nat @ M2 @ N2 ) )
=> ( ( N2 = one_one_nat )
| ( M2 = zero_zero_nat ) ) ) ).
% mult_eq_self_implies_10
thf(fact_1029_proper__interval__int_Oelims_I3_J,axiom,
! [X: option_int,Xa: option_int] :
( ~ ( set_pr3846177996627587438al_int @ X @ Xa )
=> ~ ! [X3: int] :
( ( X
= ( some_int @ X3 ) )
=> ! [Y3: int] :
( ( Xa
= ( some_int @ Y3 ) )
=> ( ord_less_int @ one_one_int @ ( minus_minus_int @ Y3 @ X3 ) ) ) ) ) ).
% proper_interval_int.elims(3)
thf(fact_1030_proper__interval__int_Oelims_I2_J,axiom,
! [X: option_int,Xa: option_int] :
( ( set_pr3846177996627587438al_int @ X @ Xa )
=> ( ! [X3: int] :
( ( X
= ( some_int @ X3 ) )
=> ! [Y3: int] :
( ( Xa
= ( some_int @ Y3 ) )
=> ~ ( ord_less_int @ one_one_int @ ( minus_minus_int @ Y3 @ X3 ) ) ) )
=> ( ( X != none_int )
=> ( Xa = none_int ) ) ) ) ).
% proper_interval_int.elims(2)
thf(fact_1031_proper__interval__int_Oelims_I1_J,axiom,
! [X: option_int,Xa: option_int,Y: $o] :
( ( ( set_pr3846177996627587438al_int @ X @ Xa )
= Y )
=> ( ! [X3: int] :
( ( X
= ( some_int @ X3 ) )
=> ! [Y3: int] :
( ( Xa
= ( some_int @ Y3 ) )
=> ( Y
= ( ~ ( ord_less_int @ one_one_int @ ( minus_minus_int @ Y3 @ X3 ) ) ) ) ) )
=> ( ( ( X = none_int )
=> ~ Y )
=> ~ ( ( Xa = none_int )
=> ~ Y ) ) ) ) ).
% proper_interval_int.elims(1)
thf(fact_1032_proper__interval__int_Osimps_I1_J,axiom,
! [X: int,Y: int] :
( ( set_pr3846177996627587438al_int @ ( some_int @ X ) @ ( some_int @ Y ) )
= ( ord_less_int @ one_one_int @ ( minus_minus_int @ Y @ X ) ) ) ).
% proper_interval_int.simps(1)
thf(fact_1033_real__archimedian__rdiv__eq__0,axiom,
! [X: real,C: real] :
( ( ord_less_eq_real @ zero_zero_real @ X )
=> ( ( ord_less_eq_real @ zero_zero_real @ C )
=> ( ! [M5: nat] :
( ( ord_less_nat @ zero_zero_nat @ M5 )
=> ( ord_less_eq_real @ ( times_times_real @ ( semiri5074537144036343181t_real @ M5 ) @ X ) @ C ) )
=> ( X = zero_zero_real ) ) ) ) ).
% real_archimedian_rdiv_eq_0
thf(fact_1034_int__ops_I6_J,axiom,
! [A: nat,B: nat] :
( ( ( ord_less_int @ ( semiri1314217659103216013at_int @ A ) @ ( semiri1314217659103216013at_int @ B ) )
=> ( ( semiri1314217659103216013at_int @ ( minus_minus_nat @ A @ B ) )
= zero_zero_int ) )
& ( ~ ( ord_less_int @ ( semiri1314217659103216013at_int @ A ) @ ( semiri1314217659103216013at_int @ B ) )
=> ( ( semiri1314217659103216013at_int @ ( minus_minus_nat @ A @ B ) )
= ( minus_minus_int @ ( semiri1314217659103216013at_int @ A ) @ ( semiri1314217659103216013at_int @ B ) ) ) ) ) ).
% int_ops(6)
thf(fact_1035_zmult__zless__mono2__lemma,axiom,
! [I: int,J: int,K2: nat] :
( ( ord_less_int @ I @ J )
=> ( ( ord_less_nat @ zero_zero_nat @ K2 )
=> ( ord_less_int @ ( times_times_int @ ( semiri1314217659103216013at_int @ K2 ) @ I ) @ ( times_times_int @ ( semiri1314217659103216013at_int @ K2 ) @ J ) ) ) ) ).
% zmult_zless_mono2_lemma
thf(fact_1036_nat__mult__le__cancel__disj,axiom,
! [K2: nat,M2: nat,N2: nat] :
( ( ord_less_eq_nat @ ( times_times_nat @ K2 @ M2 ) @ ( times_times_nat @ K2 @ N2 ) )
= ( ( ord_less_nat @ zero_zero_nat @ K2 )
=> ( ord_less_eq_nat @ M2 @ N2 ) ) ) ).
% nat_mult_le_cancel_disj
thf(fact_1037_nat__mult__less__cancel__disj,axiom,
! [K2: nat,M2: nat,N2: nat] :
( ( ord_less_nat @ ( times_times_nat @ K2 @ M2 ) @ ( times_times_nat @ K2 @ N2 ) )
= ( ( ord_less_nat @ zero_zero_nat @ K2 )
& ( ord_less_nat @ M2 @ N2 ) ) ) ).
% nat_mult_less_cancel_disj
thf(fact_1038_proper__interval__int_Osimps_I2_J,axiom,
! [Uv: option_int] : ( set_pr3846177996627587438al_int @ none_int @ Uv ) ).
% proper_interval_int.simps(2)
thf(fact_1039_proper__interval__int_Osimps_I3_J,axiom,
! [Uu: option_int] : ( set_pr3846177996627587438al_int @ Uu @ none_int ) ).
% proper_interval_int.simps(3)
thf(fact_1040_cproper__interval__int__def,axiom,
collec4138408100298950737al_int = set_pr3846177996627587438al_int ).
% cproper_interval_int_def
thf(fact_1041_proper__interval__int_Ocases,axiom,
! [X: produc7874843650251301337on_int] :
( ! [X3: int,Y3: int] :
( X
!= ( produc6331568796615304721on_int @ ( some_int @ X3 ) @ ( some_int @ Y3 ) ) )
=> ( ! [Uv2: option_int] :
( X
!= ( produc6331568796615304721on_int @ none_int @ Uv2 ) )
=> ~ ! [Uu2: option_int] :
( X
!= ( produc6331568796615304721on_int @ Uu2 @ none_int ) ) ) ) ).
% proper_interval_int.cases
thf(fact_1042_nat__mult__less__cancel1,axiom,
! [K2: nat,M2: nat,N2: nat] :
( ( ord_less_nat @ zero_zero_nat @ K2 )
=> ( ( ord_less_nat @ ( times_times_nat @ K2 @ M2 ) @ ( times_times_nat @ K2 @ N2 ) )
= ( ord_less_nat @ M2 @ N2 ) ) ) ).
% nat_mult_less_cancel1
thf(fact_1043_nat__mult__eq__cancel1,axiom,
! [K2: nat,M2: nat,N2: nat] :
( ( ord_less_nat @ zero_zero_nat @ K2 )
=> ( ( ( times_times_nat @ K2 @ M2 )
= ( times_times_nat @ K2 @ N2 ) )
= ( M2 = N2 ) ) ) ).
% nat_mult_eq_cancel1
thf(fact_1044_nat__mult__le__cancel1,axiom,
! [K2: nat,M2: nat,N2: nat] :
( ( ord_less_nat @ zero_zero_nat @ K2 )
=> ( ( ord_less_eq_nat @ ( times_times_nat @ K2 @ M2 ) @ ( times_times_nat @ K2 @ N2 ) )
= ( ord_less_eq_nat @ M2 @ N2 ) ) ) ).
% nat_mult_le_cancel1
thf(fact_1045_proper__interval__int_Opelims_I1_J,axiom,
! [X: option_int,Xa: option_int,Y: $o] :
( ( ( set_pr3846177996627587438al_int @ X @ Xa )
= Y )
=> ( ( accp_P2344022344523571810on_int @ set_pr8513825800341370386nt_rel @ ( produc6331568796615304721on_int @ X @ Xa ) )
=> ( ! [X3: int] :
( ( X
= ( some_int @ X3 ) )
=> ! [Y3: int] :
( ( Xa
= ( some_int @ Y3 ) )
=> ( ( Y
= ( ord_less_int @ one_one_int @ ( minus_minus_int @ Y3 @ X3 ) ) )
=> ~ ( accp_P2344022344523571810on_int @ set_pr8513825800341370386nt_rel @ ( produc6331568796615304721on_int @ ( some_int @ X3 ) @ ( some_int @ Y3 ) ) ) ) ) )
=> ( ( ( X = none_int )
=> ( Y
=> ~ ( accp_P2344022344523571810on_int @ set_pr8513825800341370386nt_rel @ ( produc6331568796615304721on_int @ none_int @ Xa ) ) ) )
=> ~ ( ( Xa = none_int )
=> ( Y
=> ~ ( accp_P2344022344523571810on_int @ set_pr8513825800341370386nt_rel @ ( produc6331568796615304721on_int @ X @ none_int ) ) ) ) ) ) ) ) ).
% proper_interval_int.pelims(1)
thf(fact_1046_proper__interval__int_Opelims_I2_J,axiom,
! [X: option_int,Xa: option_int] :
( ( set_pr3846177996627587438al_int @ X @ Xa )
=> ( ( accp_P2344022344523571810on_int @ set_pr8513825800341370386nt_rel @ ( produc6331568796615304721on_int @ X @ Xa ) )
=> ( ! [X3: int] :
( ( X
= ( some_int @ X3 ) )
=> ! [Y3: int] :
( ( Xa
= ( some_int @ Y3 ) )
=> ( ( accp_P2344022344523571810on_int @ set_pr8513825800341370386nt_rel @ ( produc6331568796615304721on_int @ ( some_int @ X3 ) @ ( some_int @ Y3 ) ) )
=> ~ ( ord_less_int @ one_one_int @ ( minus_minus_int @ Y3 @ X3 ) ) ) ) )
=> ( ( ( X = none_int )
=> ~ ( accp_P2344022344523571810on_int @ set_pr8513825800341370386nt_rel @ ( produc6331568796615304721on_int @ none_int @ Xa ) ) )
=> ~ ( ( Xa = none_int )
=> ~ ( accp_P2344022344523571810on_int @ set_pr8513825800341370386nt_rel @ ( produc6331568796615304721on_int @ X @ none_int ) ) ) ) ) ) ) ).
% proper_interval_int.pelims(2)
thf(fact_1047_proper__interval__int_Opelims_I3_J,axiom,
! [X: option_int,Xa: option_int] :
( ~ ( set_pr3846177996627587438al_int @ X @ Xa )
=> ( ( accp_P2344022344523571810on_int @ set_pr8513825800341370386nt_rel @ ( produc6331568796615304721on_int @ X @ Xa ) )
=> ~ ! [X3: int] :
( ( X
= ( some_int @ X3 ) )
=> ! [Y3: int] :
( ( Xa
= ( some_int @ Y3 ) )
=> ( ( accp_P2344022344523571810on_int @ set_pr8513825800341370386nt_rel @ ( produc6331568796615304721on_int @ ( some_int @ X3 ) @ ( some_int @ Y3 ) ) )
=> ( ord_less_int @ one_one_int @ ( minus_minus_int @ Y3 @ X3 ) ) ) ) ) ) ) ).
% proper_interval_int.pelims(3)
thf(fact_1048_real__root__pow__pos2,axiom,
! [N2: nat,X: real] :
( ( ord_less_nat @ zero_zero_nat @ N2 )
=> ( ( ord_less_eq_real @ zero_zero_real @ X )
=> ( ( power_power_real @ ( root @ N2 @ X ) @ N2 )
= X ) ) ) ).
% real_root_pow_pos2
thf(fact_1049_Suc__diff__1,axiom,
! [N2: nat] :
( ( ord_less_nat @ zero_zero_nat @ N2 )
=> ( ( suc @ ( minus_minus_nat @ N2 @ one_one_nat ) )
= N2 ) ) ).
% Suc_diff_1
thf(fact_1050_nat_Oinject,axiom,
! [X2: nat,Y2: nat] :
( ( ( suc @ X2 )
= ( suc @ Y2 ) )
= ( X2 = Y2 ) ) ).
% nat.inject
thf(fact_1051_old_Onat_Oinject,axiom,
! [Nat: nat,Nat2: nat] :
( ( ( suc @ Nat )
= ( suc @ Nat2 ) )
= ( Nat = Nat2 ) ) ).
% old.nat.inject
thf(fact_1052_Suc__less__eq,axiom,
! [M2: nat,N2: nat] :
( ( ord_less_nat @ ( suc @ M2 ) @ ( suc @ N2 ) )
= ( ord_less_nat @ M2 @ N2 ) ) ).
% Suc_less_eq
thf(fact_1053_Suc__mono,axiom,
! [M2: nat,N2: nat] :
( ( ord_less_nat @ M2 @ N2 )
=> ( ord_less_nat @ ( suc @ M2 ) @ ( suc @ N2 ) ) ) ).
% Suc_mono
thf(fact_1054_lessI,axiom,
! [N2: nat] : ( ord_less_nat @ N2 @ ( suc @ N2 ) ) ).
% lessI
thf(fact_1055_Suc__le__mono,axiom,
! [N2: nat,M2: nat] :
( ( ord_less_eq_nat @ ( suc @ N2 ) @ ( suc @ M2 ) )
= ( ord_less_eq_nat @ N2 @ M2 ) ) ).
% Suc_le_mono
thf(fact_1056_Suc__diff__diff,axiom,
! [M2: nat,N2: nat,K2: nat] :
( ( minus_minus_nat @ ( minus_minus_nat @ ( suc @ M2 ) @ N2 ) @ ( suc @ K2 ) )
= ( minus_minus_nat @ ( minus_minus_nat @ M2 @ N2 ) @ K2 ) ) ).
% Suc_diff_diff
thf(fact_1057_diff__Suc__Suc,axiom,
! [M2: nat,N2: nat] :
( ( minus_minus_nat @ ( suc @ M2 ) @ ( suc @ N2 ) )
= ( minus_minus_nat @ M2 @ N2 ) ) ).
% diff_Suc_Suc
thf(fact_1058_zero__less__Suc,axiom,
! [N2: nat] : ( ord_less_nat @ zero_zero_nat @ ( suc @ N2 ) ) ).
% zero_less_Suc
thf(fact_1059_less__Suc0,axiom,
! [N2: nat] :
( ( ord_less_nat @ N2 @ ( suc @ zero_zero_nat ) )
= ( N2 = zero_zero_nat ) ) ).
% less_Suc0
thf(fact_1060_mult__eq__1__iff,axiom,
! [M2: nat,N2: nat] :
( ( ( times_times_nat @ M2 @ N2 )
= ( suc @ zero_zero_nat ) )
= ( ( M2
= ( suc @ zero_zero_nat ) )
& ( N2
= ( suc @ zero_zero_nat ) ) ) ) ).
% mult_eq_1_iff
thf(fact_1061_one__eq__mult__iff,axiom,
! [M2: nat,N2: nat] :
( ( ( suc @ zero_zero_nat )
= ( times_times_nat @ M2 @ N2 ) )
= ( ( M2
= ( suc @ zero_zero_nat ) )
& ( N2
= ( suc @ zero_zero_nat ) ) ) ) ).
% one_eq_mult_iff
thf(fact_1062_diff__Suc__1,axiom,
! [N2: nat] :
( ( minus_minus_nat @ ( suc @ N2 ) @ one_one_nat )
= N2 ) ).
% diff_Suc_1
thf(fact_1063_real__root__one,axiom,
! [N2: nat] :
( ( ord_less_nat @ zero_zero_nat @ N2 )
=> ( ( root @ N2 @ one_one_real )
= one_one_real ) ) ).
% real_root_one
thf(fact_1064_real__root__eq__1__iff,axiom,
! [N2: nat,X: real] :
( ( ord_less_nat @ zero_zero_nat @ N2 )
=> ( ( ( root @ N2 @ X )
= one_one_real )
= ( X = one_one_real ) ) ) ).
% real_root_eq_1_iff
thf(fact_1065_real__root__eq__iff,axiom,
! [N2: nat,X: real,Y: real] :
( ( ord_less_nat @ zero_zero_nat @ N2 )
=> ( ( ( root @ N2 @ X )
= ( root @ N2 @ Y ) )
= ( X = Y ) ) ) ).
% real_root_eq_iff
thf(fact_1066_Suc__pred,axiom,
! [N2: nat] :
( ( ord_less_nat @ zero_zero_nat @ N2 )
=> ( ( suc @ ( minus_minus_nat @ N2 @ ( suc @ zero_zero_nat ) ) )
= N2 ) ) ).
% Suc_pred
thf(fact_1067_one__le__mult__iff,axiom,
! [M2: nat,N2: nat] :
( ( ord_less_eq_nat @ ( suc @ zero_zero_nat ) @ ( times_times_nat @ M2 @ N2 ) )
= ( ( ord_less_eq_nat @ ( suc @ zero_zero_nat ) @ M2 )
& ( ord_less_eq_nat @ ( suc @ zero_zero_nat ) @ N2 ) ) ) ).
% one_le_mult_iff
thf(fact_1068_real__root__eq__0__iff,axiom,
! [N2: nat,X: real] :
( ( ord_less_nat @ zero_zero_nat @ N2 )
=> ( ( ( root @ N2 @ X )
= zero_zero_real )
= ( X = zero_zero_real ) ) ) ).
% real_root_eq_0_iff
thf(fact_1069_real__root__gt__1__iff,axiom,
! [N2: nat,Y: real] :
( ( ord_less_nat @ zero_zero_nat @ N2 )
=> ( ( ord_less_real @ one_one_real @ ( root @ N2 @ Y ) )
= ( ord_less_real @ one_one_real @ Y ) ) ) ).
% real_root_gt_1_iff
thf(fact_1070_real__root__lt__1__iff,axiom,
! [N2: nat,X: real] :
( ( ord_less_nat @ zero_zero_nat @ N2 )
=> ( ( ord_less_real @ ( root @ N2 @ X ) @ one_one_real )
= ( ord_less_real @ X @ one_one_real ) ) ) ).
% real_root_lt_1_iff
thf(fact_1071_real__root__less__iff,axiom,
! [N2: nat,X: real,Y: real] :
( ( ord_less_nat @ zero_zero_nat @ N2 )
=> ( ( ord_less_real @ ( root @ N2 @ X ) @ ( root @ N2 @ Y ) )
= ( ord_less_real @ X @ Y ) ) ) ).
% real_root_less_iff
thf(fact_1072_real__root__ge__1__iff,axiom,
! [N2: nat,Y: real] :
( ( ord_less_nat @ zero_zero_nat @ N2 )
=> ( ( ord_less_eq_real @ one_one_real @ ( root @ N2 @ Y ) )
= ( ord_less_eq_real @ one_one_real @ Y ) ) ) ).
% real_root_ge_1_iff
thf(fact_1073_real__root__le__1__iff,axiom,
! [N2: nat,X: real] :
( ( ord_less_nat @ zero_zero_nat @ N2 )
=> ( ( ord_less_eq_real @ ( root @ N2 @ X ) @ one_one_real )
= ( ord_less_eq_real @ X @ one_one_real ) ) ) ).
% real_root_le_1_iff
thf(fact_1074_real__root__le__iff,axiom,
! [N2: nat,X: real,Y: real] :
( ( ord_less_nat @ zero_zero_nat @ N2 )
=> ( ( ord_less_eq_real @ ( root @ N2 @ X ) @ ( root @ N2 @ Y ) )
= ( ord_less_eq_real @ X @ Y ) ) ) ).
% real_root_le_iff
thf(fact_1075_real__root__gt__0__iff,axiom,
! [N2: nat,Y: real] :
( ( ord_less_nat @ zero_zero_nat @ N2 )
=> ( ( ord_less_real @ zero_zero_real @ ( root @ N2 @ Y ) )
= ( ord_less_real @ zero_zero_real @ Y ) ) ) ).
% real_root_gt_0_iff
thf(fact_1076_real__root__lt__0__iff,axiom,
! [N2: nat,X: real] :
( ( ord_less_nat @ zero_zero_nat @ N2 )
=> ( ( ord_less_real @ ( root @ N2 @ X ) @ zero_zero_real )
= ( ord_less_real @ X @ zero_zero_real ) ) ) ).
% real_root_lt_0_iff
thf(fact_1077_real__root__ge__0__iff,axiom,
! [N2: nat,Y: real] :
( ( ord_less_nat @ zero_zero_nat @ N2 )
=> ( ( ord_less_eq_real @ zero_zero_real @ ( root @ N2 @ Y ) )
= ( ord_less_eq_real @ zero_zero_real @ Y ) ) ) ).
% real_root_ge_0_iff
thf(fact_1078_real__root__le__0__iff,axiom,
! [N2: nat,X: real] :
( ( ord_less_nat @ zero_zero_nat @ N2 )
=> ( ( ord_less_eq_real @ ( root @ N2 @ X ) @ zero_zero_real )
= ( ord_less_eq_real @ X @ zero_zero_real ) ) ) ).
% real_root_le_0_iff
thf(fact_1079_zero__induct__lemma,axiom,
! [P: nat > $o,K2: nat,I: nat] :
( ( P @ K2 )
=> ( ! [N3: nat] :
( ( P @ ( suc @ N3 ) )
=> ( P @ N3 ) )
=> ( P @ ( minus_minus_nat @ K2 @ I ) ) ) ) ).
% zero_induct_lemma
thf(fact_1080_not0__implies__Suc,axiom,
! [N2: nat] :
( ( N2 != zero_zero_nat )
=> ? [M5: nat] :
( N2
= ( suc @ M5 ) ) ) ).
% not0_implies_Suc
thf(fact_1081_Zero__not__Suc,axiom,
! [M2: nat] :
( zero_zero_nat
!= ( suc @ M2 ) ) ).
% Zero_not_Suc
thf(fact_1082_Zero__neq__Suc,axiom,
! [M2: nat] :
( zero_zero_nat
!= ( suc @ M2 ) ) ).
% Zero_neq_Suc
thf(fact_1083_Suc__neq__Zero,axiom,
! [M2: nat] :
( ( suc @ M2 )
!= zero_zero_nat ) ).
% Suc_neq_Zero
thf(fact_1084_zero__induct,axiom,
! [P: nat > $o,K2: nat] :
( ( P @ K2 )
=> ( ! [N3: nat] :
( ( P @ ( suc @ N3 ) )
=> ( P @ N3 ) )
=> ( P @ zero_zero_nat ) ) ) ).
% zero_induct
thf(fact_1085_diff__induct,axiom,
! [P: nat > nat > $o,M2: nat,N2: 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 @ N2 ) ) ) ) ).
% diff_induct
thf(fact_1086_nat__induct,axiom,
! [P: nat > $o,N2: nat] :
( ( P @ zero_zero_nat )
=> ( ! [N3: nat] :
( ( P @ N3 )
=> ( P @ ( suc @ N3 ) ) )
=> ( P @ N2 ) ) ) ).
% nat_induct
thf(fact_1087_old_Onat_Oexhaust,axiom,
! [Y: nat] :
( ( Y != zero_zero_nat )
=> ~ ! [Nat3: nat] :
( Y
!= ( suc @ Nat3 ) ) ) ).
% old.nat.exhaust
thf(fact_1088_nat_OdiscI,axiom,
! [Nat: nat,X2: nat] :
( ( Nat
= ( suc @ X2 ) )
=> ( Nat != zero_zero_nat ) ) ).
% nat.discI
thf(fact_1089_old_Onat_Odistinct_I1_J,axiom,
! [Nat2: nat] :
( zero_zero_nat
!= ( suc @ Nat2 ) ) ).
% old.nat.distinct(1)
thf(fact_1090_old_Onat_Odistinct_I2_J,axiom,
! [Nat2: nat] :
( ( suc @ Nat2 )
!= zero_zero_nat ) ).
% old.nat.distinct(2)
thf(fact_1091_nat_Odistinct_I1_J,axiom,
! [X2: nat] :
( zero_zero_nat
!= ( suc @ X2 ) ) ).
% nat.distinct(1)
thf(fact_1092_not__less__less__Suc__eq,axiom,
! [N2: nat,M2: nat] :
( ~ ( ord_less_nat @ N2 @ M2 )
=> ( ( ord_less_nat @ N2 @ ( suc @ M2 ) )
= ( N2 = M2 ) ) ) ).
% not_less_less_Suc_eq
thf(fact_1093_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_1094_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,K: nat] :
( ( ord_less_nat @ I2 @ J2 )
=> ( ( ord_less_nat @ J2 @ K )
=> ( ( P @ I2 @ J2 )
=> ( ( P @ J2 @ K )
=> ( P @ I2 @ K ) ) ) ) )
=> ( P @ I @ J ) ) ) ) ).
% less_Suc_induct
thf(fact_1095_less__trans__Suc,axiom,
! [I: nat,J: nat,K2: nat] :
( ( ord_less_nat @ I @ J )
=> ( ( ord_less_nat @ J @ K2 )
=> ( ord_less_nat @ ( suc @ I ) @ K2 ) ) ) ).
% less_trans_Suc
thf(fact_1096_Suc__less__SucD,axiom,
! [M2: nat,N2: nat] :
( ( ord_less_nat @ ( suc @ M2 ) @ ( suc @ N2 ) )
=> ( ord_less_nat @ M2 @ N2 ) ) ).
% Suc_less_SucD
thf(fact_1097_less__antisym,axiom,
! [N2: nat,M2: nat] :
( ~ ( ord_less_nat @ N2 @ M2 )
=> ( ( ord_less_nat @ N2 @ ( suc @ M2 ) )
=> ( M2 = N2 ) ) ) ).
% less_antisym
thf(fact_1098_Suc__less__eq2,axiom,
! [N2: nat,M2: nat] :
( ( ord_less_nat @ ( suc @ N2 ) @ M2 )
= ( ? [M6: nat] :
( ( M2
= ( suc @ M6 ) )
& ( ord_less_nat @ N2 @ M6 ) ) ) ) ).
% Suc_less_eq2
thf(fact_1099_Nat_OAll__less__Suc,axiom,
! [N2: nat,P: nat > $o] :
( ( ! [I4: nat] :
( ( ord_less_nat @ I4 @ ( suc @ N2 ) )
=> ( P @ I4 ) ) )
= ( ( P @ N2 )
& ! [I4: nat] :
( ( ord_less_nat @ I4 @ N2 )
=> ( P @ I4 ) ) ) ) ).
% Nat.All_less_Suc
thf(fact_1100_not__less__eq,axiom,
! [M2: nat,N2: nat] :
( ( ~ ( ord_less_nat @ M2 @ N2 ) )
= ( ord_less_nat @ N2 @ ( suc @ M2 ) ) ) ).
% not_less_eq
thf(fact_1101_less__Suc__eq,axiom,
! [M2: nat,N2: nat] :
( ( ord_less_nat @ M2 @ ( suc @ N2 ) )
= ( ( ord_less_nat @ M2 @ N2 )
| ( M2 = N2 ) ) ) ).
% less_Suc_eq
thf(fact_1102_Ex__less__Suc,axiom,
! [N2: nat,P: nat > $o] :
( ( ? [I4: nat] :
( ( ord_less_nat @ I4 @ ( suc @ N2 ) )
& ( P @ I4 ) ) )
= ( ( P @ N2 )
| ? [I4: nat] :
( ( ord_less_nat @ I4 @ N2 )
& ( P @ I4 ) ) ) ) ).
% Ex_less_Suc
thf(fact_1103_less__SucI,axiom,
! [M2: nat,N2: nat] :
( ( ord_less_nat @ M2 @ N2 )
=> ( ord_less_nat @ M2 @ ( suc @ N2 ) ) ) ).
% less_SucI
thf(fact_1104_less__SucE,axiom,
! [M2: nat,N2: nat] :
( ( ord_less_nat @ M2 @ ( suc @ N2 ) )
=> ( ~ ( ord_less_nat @ M2 @ N2 )
=> ( M2 = N2 ) ) ) ).
% less_SucE
thf(fact_1105_Suc__lessI,axiom,
! [M2: nat,N2: nat] :
( ( ord_less_nat @ M2 @ N2 )
=> ( ( ( suc @ M2 )
!= N2 )
=> ( ord_less_nat @ ( suc @ M2 ) @ N2 ) ) ) ).
% Suc_lessI
thf(fact_1106_Suc__lessE,axiom,
! [I: nat,K2: nat] :
( ( ord_less_nat @ ( suc @ I ) @ K2 )
=> ~ ! [J2: nat] :
( ( ord_less_nat @ I @ J2 )
=> ( K2
!= ( suc @ J2 ) ) ) ) ).
% Suc_lessE
thf(fact_1107_Suc__lessD,axiom,
! [M2: nat,N2: nat] :
( ( ord_less_nat @ ( suc @ M2 ) @ N2 )
=> ( ord_less_nat @ M2 @ N2 ) ) ).
% Suc_lessD
thf(fact_1108_Nat_OlessE,axiom,
! [I: nat,K2: nat] :
( ( ord_less_nat @ I @ K2 )
=> ( ( K2
!= ( suc @ I ) )
=> ~ ! [J2: nat] :
( ( ord_less_nat @ I @ J2 )
=> ( K2
!= ( suc @ J2 ) ) ) ) ) ).
% Nat.lessE
thf(fact_1109_Suc__inject,axiom,
! [X: nat,Y: nat] :
( ( ( suc @ X )
= ( suc @ Y ) )
=> ( X = Y ) ) ).
% Suc_inject
thf(fact_1110_n__not__Suc__n,axiom,
! [N2: nat] :
( N2
!= ( suc @ N2 ) ) ).
% n_not_Suc_n
thf(fact_1111_fact__ge__Suc__0__nat,axiom,
! [N2: nat] : ( ord_less_eq_nat @ ( suc @ zero_zero_nat ) @ ( semiri1408675320244567234ct_nat @ N2 ) ) ).
% fact_ge_Suc_0_nat
thf(fact_1112_fact__mono__nat,axiom,
! [M2: nat,N2: nat] :
( ( ord_less_eq_nat @ M2 @ N2 )
=> ( ord_less_eq_nat @ ( semiri1408675320244567234ct_nat @ M2 ) @ ( semiri1408675320244567234ct_nat @ N2 ) ) ) ).
% fact_mono_nat
thf(fact_1113_fact__ge__self,axiom,
! [N2: nat] : ( ord_less_eq_nat @ N2 @ ( semiri1408675320244567234ct_nat @ N2 ) ) ).
% fact_ge_self
thf(fact_1114_Suc__leD,axiom,
! [M2: nat,N2: nat] :
( ( ord_less_eq_nat @ ( suc @ M2 ) @ N2 )
=> ( ord_less_eq_nat @ M2 @ N2 ) ) ).
% Suc_leD
thf(fact_1115_le__SucE,axiom,
! [M2: nat,N2: nat] :
( ( ord_less_eq_nat @ M2 @ ( suc @ N2 ) )
=> ( ~ ( ord_less_eq_nat @ M2 @ N2 )
=> ( M2
= ( suc @ N2 ) ) ) ) ).
% le_SucE
thf(fact_1116_le__SucI,axiom,
! [M2: nat,N2: nat] :
( ( ord_less_eq_nat @ M2 @ N2 )
=> ( ord_less_eq_nat @ M2 @ ( suc @ N2 ) ) ) ).
% le_SucI
thf(fact_1117_Suc__le__D,axiom,
! [N2: nat,M7: nat] :
( ( ord_less_eq_nat @ ( suc @ N2 ) @ M7 )
=> ? [M5: nat] :
( M7
= ( suc @ M5 ) ) ) ).
% Suc_le_D
thf(fact_1118_le__Suc__eq,axiom,
! [M2: nat,N2: nat] :
( ( ord_less_eq_nat @ M2 @ ( suc @ N2 ) )
= ( ( ord_less_eq_nat @ M2 @ N2 )
| ( M2
= ( suc @ N2 ) ) ) ) ).
% le_Suc_eq
thf(fact_1119_Suc__n__not__le__n,axiom,
! [N2: nat] :
~ ( ord_less_eq_nat @ ( suc @ N2 ) @ N2 ) ).
% Suc_n_not_le_n
thf(fact_1120_not__less__eq__eq,axiom,
! [M2: nat,N2: nat] :
( ( ~ ( ord_less_eq_nat @ M2 @ N2 ) )
= ( ord_less_eq_nat @ ( suc @ N2 ) @ M2 ) ) ).
% not_less_eq_eq
thf(fact_1121_full__nat__induct,axiom,
! [P: nat > $o,N2: nat] :
( ! [N3: nat] :
( ! [M3: nat] :
( ( ord_less_eq_nat @ ( suc @ M3 ) @ N3 )
=> ( P @ M3 ) )
=> ( P @ N3 ) )
=> ( P @ N2 ) ) ).
% full_nat_induct
thf(fact_1122_nat__induct__at__least,axiom,
! [M2: nat,N2: nat,P: nat > $o] :
( ( ord_less_eq_nat @ M2 @ N2 )
=> ( ( P @ M2 )
=> ( ! [N3: nat] :
( ( ord_less_eq_nat @ M2 @ N3 )
=> ( ( P @ N3 )
=> ( P @ ( suc @ N3 ) ) ) )
=> ( P @ N2 ) ) ) ) ).
% nat_induct_at_least
thf(fact_1123_transitive__stepwise__le,axiom,
! [M2: nat,N2: nat,R2: nat > nat > $o] :
( ( ord_less_eq_nat @ M2 @ N2 )
=> ( ! [X3: nat] : ( R2 @ X3 @ X3 )
=> ( ! [X3: nat,Y3: nat,Z3: nat] :
( ( R2 @ X3 @ Y3 )
=> ( ( R2 @ Y3 @ Z3 )
=> ( R2 @ X3 @ Z3 ) ) )
=> ( ! [N3: nat] : ( R2 @ N3 @ ( suc @ N3 ) )
=> ( R2 @ M2 @ N2 ) ) ) ) ) ).
% transitive_stepwise_le
thf(fact_1124_Suc__mult__cancel1,axiom,
! [K2: nat,M2: nat,N2: nat] :
( ( ( times_times_nat @ ( suc @ K2 ) @ M2 )
= ( times_times_nat @ ( suc @ K2 ) @ N2 ) )
= ( M2 = N2 ) ) ).
% Suc_mult_cancel1
thf(fact_1125_fact__less__mono__nat,axiom,
! [M2: nat,N2: nat] :
( ( ord_less_nat @ zero_zero_nat @ M2 )
=> ( ( ord_less_nat @ M2 @ N2 )
=> ( ord_less_nat @ ( semiri1408675320244567234ct_nat @ M2 ) @ ( semiri1408675320244567234ct_nat @ N2 ) ) ) ) ).
% fact_less_mono_nat
thf(fact_1126_less__Suc__eq__0__disj,axiom,
! [M2: nat,N2: nat] :
( ( ord_less_nat @ M2 @ ( suc @ N2 ) )
= ( ( M2 = zero_zero_nat )
| ? [J3: nat] :
( ( M2
= ( suc @ J3 ) )
& ( ord_less_nat @ J3 @ N2 ) ) ) ) ).
% less_Suc_eq_0_disj
thf(fact_1127_gr0__implies__Suc,axiom,
! [N2: nat] :
( ( ord_less_nat @ zero_zero_nat @ N2 )
=> ? [M5: nat] :
( N2
= ( suc @ M5 ) ) ) ).
% gr0_implies_Suc
thf(fact_1128_All__less__Suc2,axiom,
! [N2: nat,P: nat > $o] :
( ( ! [I4: nat] :
( ( ord_less_nat @ I4 @ ( suc @ N2 ) )
=> ( P @ I4 ) ) )
= ( ( P @ zero_zero_nat )
& ! [I4: nat] :
( ( ord_less_nat @ I4 @ N2 )
=> ( P @ ( suc @ I4 ) ) ) ) ) ).
% All_less_Suc2
thf(fact_1129_gr0__conv__Suc,axiom,
! [N2: nat] :
( ( ord_less_nat @ zero_zero_nat @ N2 )
= ( ? [M: nat] :
( N2
= ( suc @ M ) ) ) ) ).
% gr0_conv_Suc
thf(fact_1130_Ex__less__Suc2,axiom,
! [N2: nat,P: nat > $o] :
( ( ? [I4: nat] :
( ( ord_less_nat @ I4 @ ( suc @ N2 ) )
& ( P @ I4 ) ) )
= ( ( P @ zero_zero_nat )
| ? [I4: nat] :
( ( ord_less_nat @ I4 @ N2 )
& ( P @ ( suc @ I4 ) ) ) ) ) ).
% Ex_less_Suc2
thf(fact_1131_Comparator__Generator_OAll__less__Suc,axiom,
! [X: nat,P: nat > $o] :
( ( ! [I4: nat] :
( ( ord_less_nat @ I4 @ ( suc @ X ) )
=> ( P @ I4 ) ) )
= ( ( P @ zero_zero_nat )
& ! [I4: nat] :
( ( ord_less_nat @ I4 @ X )
=> ( P @ ( suc @ I4 ) ) ) ) ) ).
% Comparator_Generator.All_less_Suc
thf(fact_1132_forall__finite_I2_J,axiom,
! [P: nat > $o] :
( ( ! [I4: nat] :
( ( ord_less_nat @ I4 @ ( suc @ zero_zero_nat ) )
=> ( P @ I4 ) ) )
= ( P @ zero_zero_nat ) ) ).
% forall_finite(2)
thf(fact_1133_forall__finite_I3_J,axiom,
! [X: nat,P: nat > $o] :
( ( ! [I4: nat] :
( ( ord_less_nat @ I4 @ ( suc @ ( suc @ X ) ) )
=> ( P @ I4 ) ) )
= ( ( P @ zero_zero_nat )
& ! [I4: nat] :
( ( ord_less_nat @ I4 @ ( suc @ X ) )
=> ( P @ ( suc @ I4 ) ) ) ) ) ).
% forall_finite(3)
thf(fact_1134_Suc__leI,axiom,
! [M2: nat,N2: nat] :
( ( ord_less_nat @ M2 @ N2 )
=> ( ord_less_eq_nat @ ( suc @ M2 ) @ N2 ) ) ).
% Suc_leI
thf(fact_1135_Suc__le__eq,axiom,
! [M2: nat,N2: nat] :
( ( ord_less_eq_nat @ ( suc @ M2 ) @ N2 )
= ( ord_less_nat @ M2 @ N2 ) ) ).
% Suc_le_eq
thf(fact_1136_dec__induct,axiom,
! [I: nat,J: nat,P: nat > $o] :
( ( ord_less_eq_nat @ I @ J )
=> ( ( P @ I )
=> ( ! [N3: nat] :
( ( ord_less_eq_nat @ I @ N3 )
=> ( ( ord_less_nat @ N3 @ J )
=> ( ( P @ N3 )
=> ( P @ ( suc @ N3 ) ) ) ) )
=> ( P @ J ) ) ) ) ).
% dec_induct
thf(fact_1137_inc__induct,axiom,
! [I: nat,J: nat,P: nat > $o] :
( ( ord_less_eq_nat @ I @ J )
=> ( ( P @ J )
=> ( ! [N3: nat] :
( ( ord_less_eq_nat @ I @ N3 )
=> ( ( ord_less_nat @ N3 @ J )
=> ( ( P @ ( suc @ N3 ) )
=> ( P @ N3 ) ) ) )
=> ( P @ I ) ) ) ) ).
% inc_induct
thf(fact_1138_Suc__le__lessD,axiom,
! [M2: nat,N2: nat] :
( ( ord_less_eq_nat @ ( suc @ M2 ) @ N2 )
=> ( ord_less_nat @ M2 @ N2 ) ) ).
% Suc_le_lessD
thf(fact_1139_le__less__Suc__eq,axiom,
! [M2: nat,N2: nat] :
( ( ord_less_eq_nat @ M2 @ N2 )
=> ( ( ord_less_nat @ N2 @ ( suc @ M2 ) )
= ( N2 = M2 ) ) ) ).
% le_less_Suc_eq
thf(fact_1140_less__Suc__eq__le,axiom,
! [M2: nat,N2: nat] :
( ( ord_less_nat @ M2 @ ( suc @ N2 ) )
= ( ord_less_eq_nat @ M2 @ N2 ) ) ).
% less_Suc_eq_le
thf(fact_1141_less__eq__Suc__le,axiom,
( ord_less_nat
= ( ^ [N: nat] : ( ord_less_eq_nat @ ( suc @ N ) ) ) ) ).
% less_eq_Suc_le
thf(fact_1142_le__imp__less__Suc,axiom,
! [M2: nat,N2: nat] :
( ( ord_less_eq_nat @ M2 @ N2 )
=> ( ord_less_nat @ M2 @ ( suc @ N2 ) ) ) ).
% le_imp_less_Suc
thf(fact_1143_One__nat__def,axiom,
( one_one_nat
= ( suc @ zero_zero_nat ) ) ).
% One_nat_def
thf(fact_1144_diff__less__Suc,axiom,
! [M2: nat,N2: nat] : ( ord_less_nat @ ( minus_minus_nat @ M2 @ N2 ) @ ( suc @ M2 ) ) ).
% diff_less_Suc
thf(fact_1145_Suc__diff__Suc,axiom,
! [N2: nat,M2: nat] :
( ( ord_less_nat @ N2 @ M2 )
=> ( ( suc @ ( minus_minus_nat @ M2 @ ( suc @ N2 ) ) )
= ( minus_minus_nat @ M2 @ N2 ) ) ) ).
% Suc_diff_Suc
thf(fact_1146_Suc__diff__le,axiom,
! [N2: nat,M2: nat] :
( ( ord_less_eq_nat @ N2 @ M2 )
=> ( ( minus_minus_nat @ ( suc @ M2 ) @ N2 )
= ( suc @ ( minus_minus_nat @ M2 @ N2 ) ) ) ) ).
% Suc_diff_le
thf(fact_1147_Suc__mult__less__cancel1,axiom,
! [K2: nat,M2: nat,N2: nat] :
( ( ord_less_nat @ ( times_times_nat @ ( suc @ K2 ) @ M2 ) @ ( times_times_nat @ ( suc @ K2 ) @ N2 ) )
= ( ord_less_nat @ M2 @ N2 ) ) ).
% Suc_mult_less_cancel1
thf(fact_1148_Suc__mult__le__cancel1,axiom,
! [K2: nat,M2: nat,N2: nat] :
( ( ord_less_eq_nat @ ( times_times_nat @ ( suc @ K2 ) @ M2 ) @ ( times_times_nat @ ( suc @ K2 ) @ N2 ) )
= ( ord_less_eq_nat @ M2 @ N2 ) ) ).
% Suc_mult_le_cancel1
thf(fact_1149_diff__Suc__eq__diff__pred,axiom,
! [M2: nat,N2: nat] :
( ( minus_minus_nat @ M2 @ ( suc @ N2 ) )
= ( minus_minus_nat @ ( minus_minus_nat @ M2 @ one_one_nat ) @ N2 ) ) ).
% diff_Suc_eq_diff_pred
thf(fact_1150_real__root__strict__decreasing,axiom,
! [N2: nat,N4: nat,X: real] :
( ( ord_less_nat @ zero_zero_nat @ N2 )
=> ( ( ord_less_nat @ N2 @ N4 )
=> ( ( ord_less_real @ one_one_real @ X )
=> ( ord_less_real @ ( root @ N4 @ X ) @ ( root @ N2 @ X ) ) ) ) ) ).
% real_root_strict_decreasing
thf(fact_1151_real__root__less__mono,axiom,
! [N2: nat,X: real,Y: real] :
( ( ord_less_nat @ zero_zero_nat @ N2 )
=> ( ( ord_less_real @ X @ Y )
=> ( ord_less_real @ ( root @ N2 @ X ) @ ( root @ N2 @ Y ) ) ) ) ).
% real_root_less_mono
thf(fact_1152_real__root__le__mono,axiom,
! [N2: nat,X: real,Y: real] :
( ( ord_less_nat @ zero_zero_nat @ N2 )
=> ( ( ord_less_eq_real @ X @ Y )
=> ( ord_less_eq_real @ ( root @ N2 @ X ) @ ( root @ N2 @ Y ) ) ) ) ).
% real_root_le_mono
thf(fact_1153_real__root__power,axiom,
! [N2: nat,X: real,K2: nat] :
( ( ord_less_nat @ zero_zero_nat @ N2 )
=> ( ( root @ N2 @ ( power_power_real @ X @ K2 ) )
= ( power_power_real @ ( root @ N2 @ X ) @ K2 ) ) ) ).
% real_root_power
thf(fact_1154_ex__least__nat__less,axiom,
! [P: nat > $o,N2: nat] :
( ( P @ N2 )
=> ( ~ ( P @ zero_zero_nat )
=> ? [K: nat] :
( ( ord_less_nat @ K @ N2 )
& ! [I3: nat] :
( ( ord_less_eq_nat @ I3 @ K )
=> ~ ( P @ I3 ) )
& ( P @ ( suc @ K ) ) ) ) ) ).
% ex_least_nat_less
thf(fact_1155_diff__Suc__less,axiom,
! [N2: nat,I: nat] :
( ( ord_less_nat @ zero_zero_nat @ N2 )
=> ( ord_less_nat @ ( minus_minus_nat @ N2 @ ( suc @ I ) ) @ N2 ) ) ).
% diff_Suc_less
thf(fact_1156_nat__induct__non__zero,axiom,
! [N2: nat,P: nat > $o] :
( ( ord_less_nat @ zero_zero_nat @ N2 )
=> ( ( P @ one_one_nat )
=> ( ! [N3: nat] :
( ( ord_less_nat @ zero_zero_nat @ N3 )
=> ( ( P @ N3 )
=> ( P @ ( suc @ N3 ) ) ) )
=> ( P @ N2 ) ) ) ) ).
% nat_induct_non_zero
thf(fact_1157_one__less__mult,axiom,
! [N2: nat,M2: nat] :
( ( ord_less_nat @ ( suc @ zero_zero_nat ) @ N2 )
=> ( ( ord_less_nat @ ( suc @ zero_zero_nat ) @ M2 )
=> ( ord_less_nat @ ( suc @ zero_zero_nat ) @ ( times_times_nat @ M2 @ N2 ) ) ) ) ).
% one_less_mult
thf(fact_1158_n__less__m__mult__n,axiom,
! [N2: nat,M2: nat] :
( ( ord_less_nat @ zero_zero_nat @ N2 )
=> ( ( ord_less_nat @ ( suc @ zero_zero_nat ) @ M2 )
=> ( ord_less_nat @ N2 @ ( times_times_nat @ M2 @ N2 ) ) ) ) ).
% n_less_m_mult_n
thf(fact_1159_n__less__n__mult__m,axiom,
! [N2: nat,M2: nat] :
( ( ord_less_nat @ zero_zero_nat @ N2 )
=> ( ( ord_less_nat @ ( suc @ zero_zero_nat ) @ M2 )
=> ( ord_less_nat @ N2 @ ( times_times_nat @ N2 @ M2 ) ) ) ) ).
% n_less_n_mult_m
thf(fact_1160_power__gt__expt,axiom,
! [N2: nat,K2: nat] :
( ( ord_less_nat @ ( suc @ zero_zero_nat ) @ N2 )
=> ( ord_less_nat @ K2 @ ( power_power_nat @ N2 @ K2 ) ) ) ).
% power_gt_expt
thf(fact_1161_nat__one__le__power,axiom,
! [I: nat,N2: nat] :
( ( ord_less_eq_nat @ ( suc @ zero_zero_nat ) @ I )
=> ( ord_less_eq_nat @ ( suc @ zero_zero_nat ) @ ( power_power_nat @ I @ N2 ) ) ) ).
% nat_one_le_power
thf(fact_1162_real__root__strict__increasing,axiom,
! [N2: nat,N4: nat,X: real] :
( ( ord_less_nat @ zero_zero_nat @ N2 )
=> ( ( ord_less_nat @ N2 @ N4 )
=> ( ( ord_less_real @ zero_zero_real @ X )
=> ( ( ord_less_real @ X @ one_one_real )
=> ( ord_less_real @ ( root @ N2 @ X ) @ ( root @ N4 @ X ) ) ) ) ) ) ).
% real_root_strict_increasing
thf(fact_1163_real__root__gt__zero,axiom,
! [N2: nat,X: real] :
( ( ord_less_nat @ zero_zero_nat @ N2 )
=> ( ( ord_less_real @ zero_zero_real @ X )
=> ( ord_less_real @ zero_zero_real @ ( root @ N2 @ X ) ) ) ) ).
% real_root_gt_zero
thf(fact_1164_real__root__decreasing,axiom,
! [N2: nat,N4: nat,X: real] :
( ( ord_less_nat @ zero_zero_nat @ N2 )
=> ( ( ord_less_eq_nat @ N2 @ N4 )
=> ( ( ord_less_eq_real @ one_one_real @ X )
=> ( ord_less_eq_real @ ( root @ N4 @ X ) @ ( root @ N2 @ X ) ) ) ) ) ).
% real_root_decreasing
thf(fact_1165_Suc__pred_H,axiom,
! [N2: nat] :
( ( ord_less_nat @ zero_zero_nat @ N2 )
=> ( N2
= ( suc @ ( minus_minus_nat @ N2 @ one_one_nat ) ) ) ) ).
% Suc_pred'
thf(fact_1166_Suc__diff__eq__diff__pred,axiom,
! [N2: nat,M2: nat] :
( ( ord_less_nat @ zero_zero_nat @ N2 )
=> ( ( minus_minus_nat @ ( suc @ M2 ) @ N2 )
= ( minus_minus_nat @ M2 @ ( minus_minus_nat @ N2 @ one_one_nat ) ) ) ) ).
% Suc_diff_eq_diff_pred
thf(fact_1167_real__root__pos__pos,axiom,
! [N2: nat,X: real] :
( ( ord_less_nat @ zero_zero_nat @ N2 )
=> ( ( ord_less_real @ zero_zero_real @ X )
=> ( ord_less_eq_real @ zero_zero_real @ ( root @ N2 @ X ) ) ) ) ).
% real_root_pos_pos
thf(fact_1168_real__root__increasing,axiom,
! [N2: nat,N4: nat,X: real] :
( ( ord_less_nat @ zero_zero_nat @ N2 )
=> ( ( ord_less_eq_nat @ N2 @ N4 )
=> ( ( ord_less_eq_real @ zero_zero_real @ X )
=> ( ( ord_less_eq_real @ X @ one_one_real )
=> ( ord_less_eq_real @ ( root @ N2 @ X ) @ ( root @ N4 @ X ) ) ) ) ) ) ).
% real_root_increasing
thf(fact_1169_real__root__pow__pos,axiom,
! [N2: nat,X: real] :
( ( ord_less_nat @ zero_zero_nat @ N2 )
=> ( ( ord_less_real @ zero_zero_real @ X )
=> ( ( power_power_real @ ( root @ N2 @ X ) @ N2 )
= X ) ) ) ).
% real_root_pow_pos
thf(fact_1170_real__root__power__cancel,axiom,
! [N2: nat,X: real] :
( ( ord_less_nat @ zero_zero_nat @ N2 )
=> ( ( ord_less_eq_real @ zero_zero_real @ X )
=> ( ( root @ N2 @ ( power_power_real @ X @ N2 ) )
= X ) ) ) ).
% real_root_power_cancel
thf(fact_1171_real__root__pos__unique,axiom,
! [N2: nat,Y: real,X: real] :
( ( ord_less_nat @ zero_zero_nat @ N2 )
=> ( ( ord_less_eq_real @ zero_zero_real @ Y )
=> ( ( ( power_power_real @ Y @ N2 )
= X )
=> ( ( root @ N2 @ X )
= Y ) ) ) ) ).
% real_root_pos_unique
thf(fact_1172_fact__diff__Suc,axiom,
! [N2: nat,M2: nat] :
( ( ord_less_nat @ N2 @ ( suc @ M2 ) )
=> ( ( semiri1408675320244567234ct_nat @ ( minus_minus_nat @ ( suc @ M2 ) @ N2 ) )
= ( times_times_nat @ ( minus_minus_nat @ ( suc @ M2 ) @ N2 ) @ ( semiri1408675320244567234ct_nat @ ( minus_minus_nat @ M2 @ N2 ) ) ) ) ) ).
% fact_diff_Suc
thf(fact_1173_nat__ivt__aux,axiom,
! [N2: nat,F: nat > int,K2: int] :
( ! [I2: nat] :
( ( ord_less_nat @ I2 @ N2 )
=> ( ord_less_eq_int @ ( abs_abs_int @ ( minus_minus_int @ ( F @ ( suc @ I2 ) ) @ ( F @ I2 ) ) ) @ one_one_int ) )
=> ( ( ord_less_eq_int @ ( F @ zero_zero_nat ) @ K2 )
=> ( ( ord_less_eq_int @ K2 @ ( F @ N2 ) )
=> ? [I2: nat] :
( ( ord_less_eq_nat @ I2 @ N2 )
& ( ( F @ I2 )
= K2 ) ) ) ) ) ).
% nat_ivt_aux
thf(fact_1174_nat__intermed__int__val,axiom,
! [M2: nat,N2: nat,F: nat > int,K2: int] :
( ! [I2: nat] :
( ( ( ord_less_eq_nat @ M2 @ I2 )
& ( ord_less_nat @ I2 @ N2 ) )
=> ( ord_less_eq_int @ ( abs_abs_int @ ( minus_minus_int @ ( F @ ( suc @ I2 ) ) @ ( F @ I2 ) ) ) @ one_one_int ) )
=> ( ( ord_less_eq_nat @ M2 @ N2 )
=> ( ( ord_less_eq_int @ ( F @ M2 ) @ K2 )
=> ( ( ord_less_eq_int @ K2 @ ( F @ N2 ) )
=> ? [I2: nat] :
( ( ord_less_eq_nat @ M2 @ I2 )
& ( ord_less_eq_nat @ I2 @ N2 )
& ( ( F @ I2 )
= K2 ) ) ) ) ) ) ).
% nat_intermed_int_val
thf(fact_1175_nat0__intermed__int__val,axiom,
! [N2: nat,F: nat > int,K2: int] :
( ! [I2: nat] :
( ( ord_less_nat @ I2 @ N2 )
=> ( ord_less_eq_int @ ( abs_abs_int @ ( minus_minus_int @ ( F @ ( plus_plus_nat @ I2 @ one_one_nat ) ) @ ( F @ I2 ) ) ) @ one_one_int ) )
=> ( ( ord_less_eq_int @ ( F @ zero_zero_nat ) @ K2 )
=> ( ( ord_less_eq_int @ K2 @ ( F @ N2 ) )
=> ? [I2: nat] :
( ( ord_less_eq_nat @ I2 @ N2 )
& ( ( F @ I2 )
= K2 ) ) ) ) ) ).
% nat0_intermed_int_val
thf(fact_1176_Nat_Oadd__0__right,axiom,
! [M2: nat] :
( ( plus_plus_nat @ M2 @ zero_zero_nat )
= M2 ) ).
% Nat.add_0_right
thf(fact_1177_add__is__0,axiom,
! [M2: nat,N2: nat] :
( ( ( plus_plus_nat @ M2 @ N2 )
= zero_zero_nat )
= ( ( M2 = zero_zero_nat )
& ( N2 = zero_zero_nat ) ) ) ).
% add_is_0
thf(fact_1178_add__Suc__right,axiom,
! [M2: nat,N2: nat] :
( ( plus_plus_nat @ M2 @ ( suc @ N2 ) )
= ( suc @ ( plus_plus_nat @ M2 @ N2 ) ) ) ).
% add_Suc_right
thf(fact_1179_nat__add__left__cancel__less,axiom,
! [K2: nat,M2: nat,N2: nat] :
( ( ord_less_nat @ ( plus_plus_nat @ K2 @ M2 ) @ ( plus_plus_nat @ K2 @ N2 ) )
= ( ord_less_nat @ M2 @ N2 ) ) ).
% nat_add_left_cancel_less
thf(fact_1180_nat__add__left__cancel__le,axiom,
! [K2: nat,M2: nat,N2: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ K2 @ M2 ) @ ( plus_plus_nat @ K2 @ N2 ) )
= ( ord_less_eq_nat @ M2 @ N2 ) ) ).
% nat_add_left_cancel_le
thf(fact_1181_diff__diff__left,axiom,
! [I: nat,J: nat,K2: nat] :
( ( minus_minus_nat @ ( minus_minus_nat @ I @ J ) @ K2 )
= ( minus_minus_nat @ I @ ( plus_plus_nat @ J @ K2 ) ) ) ).
% diff_diff_left
thf(fact_1182_add__gr__0,axiom,
! [M2: nat,N2: nat] :
( ( ord_less_nat @ zero_zero_nat @ ( plus_plus_nat @ M2 @ N2 ) )
= ( ( ord_less_nat @ zero_zero_nat @ M2 )
| ( ord_less_nat @ zero_zero_nat @ N2 ) ) ) ).
% add_gr_0
thf(fact_1183_Nat_Odiff__diff__right,axiom,
! [K2: nat,J: nat,I: nat] :
( ( ord_less_eq_nat @ K2 @ J )
=> ( ( minus_minus_nat @ I @ ( minus_minus_nat @ J @ K2 ) )
= ( minus_minus_nat @ ( plus_plus_nat @ I @ K2 ) @ J ) ) ) ).
% Nat.diff_diff_right
thf(fact_1184_Nat_Oadd__diff__assoc2,axiom,
! [K2: nat,J: nat,I: nat] :
( ( ord_less_eq_nat @ K2 @ J )
=> ( ( plus_plus_nat @ ( minus_minus_nat @ J @ K2 ) @ I )
= ( minus_minus_nat @ ( plus_plus_nat @ J @ I ) @ K2 ) ) ) ).
% Nat.add_diff_assoc2
thf(fact_1185_Nat_Oadd__diff__assoc,axiom,
! [K2: nat,J: nat,I: nat] :
( ( ord_less_eq_nat @ K2 @ J )
=> ( ( plus_plus_nat @ I @ ( minus_minus_nat @ J @ K2 ) )
= ( minus_minus_nat @ ( plus_plus_nat @ I @ J ) @ K2 ) ) ) ).
% Nat.add_diff_assoc
thf(fact_1186_mult__Suc__right,axiom,
! [M2: nat,N2: nat] :
( ( times_times_nat @ M2 @ ( suc @ N2 ) )
= ( plus_plus_nat @ M2 @ ( times_times_nat @ M2 @ N2 ) ) ) ).
% mult_Suc_right
thf(fact_1187_diff__Suc__diff__eq2,axiom,
! [K2: nat,J: nat,I: nat] :
( ( ord_less_eq_nat @ K2 @ J )
=> ( ( minus_minus_nat @ ( suc @ ( minus_minus_nat @ J @ K2 ) ) @ I )
= ( minus_minus_nat @ ( suc @ J ) @ ( plus_plus_nat @ K2 @ I ) ) ) ) ).
% diff_Suc_diff_eq2
thf(fact_1188_diff__Suc__diff__eq1,axiom,
! [K2: nat,J: nat,I: nat] :
( ( ord_less_eq_nat @ K2 @ J )
=> ( ( minus_minus_nat @ I @ ( suc @ ( minus_minus_nat @ J @ K2 ) ) )
= ( minus_minus_nat @ ( plus_plus_nat @ I @ K2 ) @ ( suc @ J ) ) ) ) ).
% diff_Suc_diff_eq1
thf(fact_1189_one__is__add,axiom,
! [M2: nat,N2: nat] :
( ( ( suc @ zero_zero_nat )
= ( plus_plus_nat @ M2 @ N2 ) )
= ( ( ( M2
= ( suc @ zero_zero_nat ) )
& ( N2 = zero_zero_nat ) )
| ( ( M2 = zero_zero_nat )
& ( N2
= ( suc @ zero_zero_nat ) ) ) ) ) ).
% one_is_add
thf(fact_1190_add__is__1,axiom,
! [M2: nat,N2: nat] :
( ( ( plus_plus_nat @ M2 @ N2 )
= ( suc @ zero_zero_nat ) )
= ( ( ( M2
= ( suc @ zero_zero_nat ) )
& ( N2 = zero_zero_nat ) )
| ( ( M2 = zero_zero_nat )
& ( N2
= ( suc @ zero_zero_nat ) ) ) ) ) ).
% add_is_1
thf(fact_1191_less__imp__add__positive,axiom,
! [I: nat,J: nat] :
( ( ord_less_nat @ I @ J )
=> ? [K: nat] :
( ( ord_less_nat @ zero_zero_nat @ K )
& ( ( plus_plus_nat @ I @ K )
= J ) ) ) ).
% less_imp_add_positive
thf(fact_1192_less__natE,axiom,
! [M2: nat,N2: nat] :
( ( ord_less_nat @ M2 @ N2 )
=> ~ ! [Q3: nat] :
( N2
!= ( suc @ ( plus_plus_nat @ M2 @ Q3 ) ) ) ) ).
% less_natE
thf(fact_1193_less__add__Suc1,axiom,
! [I: nat,M2: nat] : ( ord_less_nat @ I @ ( suc @ ( plus_plus_nat @ I @ M2 ) ) ) ).
% less_add_Suc1
thf(fact_1194_less__add__Suc2,axiom,
! [I: nat,M2: nat] : ( ord_less_nat @ I @ ( suc @ ( plus_plus_nat @ M2 @ I ) ) ) ).
% less_add_Suc2
thf(fact_1195_less__iff__Suc__add,axiom,
( ord_less_nat
= ( ^ [M: nat,N: nat] :
? [K4: nat] :
( N
= ( suc @ ( plus_plus_nat @ M @ K4 ) ) ) ) ) ).
% less_iff_Suc_add
thf(fact_1196_less__imp__Suc__add,axiom,
! [M2: nat,N2: nat] :
( ( ord_less_nat @ M2 @ N2 )
=> ? [K: nat] :
( N2
= ( suc @ ( plus_plus_nat @ M2 @ K ) ) ) ) ).
% less_imp_Suc_add
thf(fact_1197_mono__nat__linear__lb,axiom,
! [F: nat > nat,M2: nat,K2: nat] :
( ! [M5: nat,N3: nat] :
( ( ord_less_nat @ M5 @ N3 )
=> ( ord_less_nat @ ( F @ M5 ) @ ( F @ N3 ) ) )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ ( F @ M2 ) @ K2 ) @ ( F @ ( plus_plus_nat @ M2 @ K2 ) ) ) ) ).
% mono_nat_linear_lb
thf(fact_1198_diff__add__0,axiom,
! [N2: nat,M2: nat] :
( ( minus_minus_nat @ N2 @ ( plus_plus_nat @ N2 @ M2 ) )
= zero_zero_nat ) ).
% diff_add_0
thf(fact_1199_Suc__eq__plus1__left,axiom,
( suc
= ( plus_plus_nat @ one_one_nat ) ) ).
% Suc_eq_plus1_left
thf(fact_1200_plus__1__eq__Suc,axiom,
( ( plus_plus_nat @ one_one_nat )
= suc ) ).
% plus_1_eq_Suc
thf(fact_1201_Suc__eq__plus1,axiom,
( suc
= ( ^ [N: nat] : ( plus_plus_nat @ N @ one_one_nat ) ) ) ).
% Suc_eq_plus1
thf(fact_1202_less__diff__conv,axiom,
! [I: nat,J: nat,K2: nat] :
( ( ord_less_nat @ I @ ( minus_minus_nat @ J @ K2 ) )
= ( ord_less_nat @ ( plus_plus_nat @ I @ K2 ) @ J ) ) ).
% less_diff_conv
thf(fact_1203_add__diff__inverse__nat,axiom,
! [M2: nat,N2: nat] :
( ~ ( ord_less_nat @ M2 @ N2 )
=> ( ( plus_plus_nat @ N2 @ ( minus_minus_nat @ M2 @ N2 ) )
= M2 ) ) ).
% add_diff_inverse_nat
thf(fact_1204_le__diff__conv,axiom,
! [J: nat,K2: nat,I: nat] :
( ( ord_less_eq_nat @ ( minus_minus_nat @ J @ K2 ) @ I )
= ( ord_less_eq_nat @ J @ ( plus_plus_nat @ I @ K2 ) ) ) ).
% le_diff_conv
thf(fact_1205_Nat_Ole__diff__conv2,axiom,
! [K2: nat,J: nat,I: nat] :
( ( ord_less_eq_nat @ K2 @ J )
=> ( ( ord_less_eq_nat @ I @ ( minus_minus_nat @ J @ K2 ) )
= ( ord_less_eq_nat @ ( plus_plus_nat @ I @ K2 ) @ J ) ) ) ).
% Nat.le_diff_conv2
thf(fact_1206_Nat_Odiff__add__assoc,axiom,
! [K2: nat,J: nat,I: nat] :
( ( ord_less_eq_nat @ K2 @ J )
=> ( ( minus_minus_nat @ ( plus_plus_nat @ I @ J ) @ K2 )
= ( plus_plus_nat @ I @ ( minus_minus_nat @ J @ K2 ) ) ) ) ).
% Nat.diff_add_assoc
thf(fact_1207_Nat_Odiff__add__assoc2,axiom,
! [K2: nat,J: nat,I: nat] :
( ( ord_less_eq_nat @ K2 @ J )
=> ( ( minus_minus_nat @ ( plus_plus_nat @ J @ I ) @ K2 )
= ( plus_plus_nat @ ( minus_minus_nat @ J @ K2 ) @ I ) ) ) ).
% Nat.diff_add_assoc2
thf(fact_1208_Nat_Ole__imp__diff__is__add,axiom,
! [I: nat,J: nat,K2: nat] :
( ( ord_less_eq_nat @ I @ J )
=> ( ( ( minus_minus_nat @ J @ I )
= K2 )
= ( J
= ( plus_plus_nat @ K2 @ I ) ) ) ) ).
% Nat.le_imp_diff_is_add
thf(fact_1209_mult__Suc,axiom,
! [M2: nat,N2: nat] :
( ( times_times_nat @ ( suc @ M2 ) @ N2 )
= ( plus_plus_nat @ N2 @ ( times_times_nat @ M2 @ N2 ) ) ) ).
% mult_Suc
thf(fact_1210_Nat_Odiff__cancel,axiom,
! [K2: nat,M2: nat,N2: nat] :
( ( minus_minus_nat @ ( plus_plus_nat @ K2 @ M2 ) @ ( plus_plus_nat @ K2 @ N2 ) )
= ( minus_minus_nat @ M2 @ N2 ) ) ).
% Nat.diff_cancel
thf(fact_1211_diff__cancel2,axiom,
! [M2: nat,K2: nat,N2: nat] :
( ( minus_minus_nat @ ( plus_plus_nat @ M2 @ K2 ) @ ( plus_plus_nat @ N2 @ K2 ) )
= ( minus_minus_nat @ M2 @ N2 ) ) ).
% diff_cancel2
thf(fact_1212_diff__add__inverse,axiom,
! [N2: nat,M2: nat] :
( ( minus_minus_nat @ ( plus_plus_nat @ N2 @ M2 ) @ N2 )
= M2 ) ).
% diff_add_inverse
thf(fact_1213_diff__add__inverse2,axiom,
! [M2: nat,N2: nat] :
( ( minus_minus_nat @ ( plus_plus_nat @ M2 @ N2 ) @ N2 )
= M2 ) ).
% diff_add_inverse2
thf(fact_1214_add__eq__self__zero,axiom,
! [M2: nat,N2: nat] :
( ( ( plus_plus_nat @ M2 @ N2 )
= M2 )
=> ( N2 = zero_zero_nat ) ) ).
% add_eq_self_zero
thf(fact_1215_plus__nat_Oadd__0,axiom,
! [N2: nat] :
( ( plus_plus_nat @ zero_zero_nat @ N2 )
= N2 ) ).
% plus_nat.add_0
thf(fact_1216_less__add__eq__less,axiom,
! [K2: nat,L: nat,M2: nat,N2: nat] :
( ( ord_less_nat @ K2 @ L )
=> ( ( ( plus_plus_nat @ M2 @ L )
= ( plus_plus_nat @ K2 @ N2 ) )
=> ( ord_less_nat @ M2 @ N2 ) ) ) ).
% less_add_eq_less
thf(fact_1217_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_1218_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_1219_add__less__mono1,axiom,
! [I: nat,J: nat,K2: nat] :
( ( ord_less_nat @ I @ J )
=> ( ord_less_nat @ ( plus_plus_nat @ I @ K2 ) @ ( plus_plus_nat @ J @ K2 ) ) ) ).
% add_less_mono1
thf(fact_1220_not__add__less2,axiom,
! [J: nat,I: nat] :
~ ( ord_less_nat @ ( plus_plus_nat @ J @ I ) @ I ) ).
% not_add_less2
thf(fact_1221_not__add__less1,axiom,
! [I: nat,J: nat] :
~ ( ord_less_nat @ ( plus_plus_nat @ I @ J ) @ I ) ).
% not_add_less1
thf(fact_1222_add__less__mono,axiom,
! [I: nat,J: nat,K2: nat,L: nat] :
( ( ord_less_nat @ I @ J )
=> ( ( ord_less_nat @ K2 @ L )
=> ( ord_less_nat @ ( plus_plus_nat @ I @ K2 ) @ ( plus_plus_nat @ J @ L ) ) ) ) ).
% add_less_mono
thf(fact_1223_add__lessD1,axiom,
! [I: nat,J: nat,K2: nat] :
( ( ord_less_nat @ ( plus_plus_nat @ I @ J ) @ K2 )
=> ( ord_less_nat @ I @ K2 ) ) ).
% add_lessD1
thf(fact_1224_add__leE,axiom,
! [M2: nat,K2: nat,N2: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ M2 @ K2 ) @ N2 )
=> ~ ( ( ord_less_eq_nat @ M2 @ N2 )
=> ~ ( ord_less_eq_nat @ K2 @ N2 ) ) ) ).
% add_leE
thf(fact_1225_le__add1,axiom,
! [N2: nat,M2: nat] : ( ord_less_eq_nat @ N2 @ ( plus_plus_nat @ N2 @ M2 ) ) ).
% le_add1
thf(fact_1226_le__add2,axiom,
! [N2: nat,M2: nat] : ( ord_less_eq_nat @ N2 @ ( plus_plus_nat @ M2 @ N2 ) ) ).
% le_add2
thf(fact_1227_add__leD1,axiom,
! [M2: nat,K2: nat,N2: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ M2 @ K2 ) @ N2 )
=> ( ord_less_eq_nat @ M2 @ N2 ) ) ).
% add_leD1
thf(fact_1228_add__leD2,axiom,
! [M2: nat,K2: nat,N2: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ M2 @ K2 ) @ N2 )
=> ( ord_less_eq_nat @ K2 @ N2 ) ) ).
% add_leD2
thf(fact_1229_le__Suc__ex,axiom,
! [K2: nat,L: nat] :
( ( ord_less_eq_nat @ K2 @ L )
=> ? [N3: nat] :
( L
= ( plus_plus_nat @ K2 @ N3 ) ) ) ).
% le_Suc_ex
thf(fact_1230_add__le__mono,axiom,
! [I: nat,J: nat,K2: nat,L: nat] :
( ( ord_less_eq_nat @ I @ J )
=> ( ( ord_less_eq_nat @ K2 @ L )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ I @ K2 ) @ ( plus_plus_nat @ J @ L ) ) ) ) ).
% add_le_mono
thf(fact_1231_add__le__mono1,axiom,
! [I: nat,J: nat,K2: nat] :
( ( ord_less_eq_nat @ I @ J )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ I @ K2 ) @ ( plus_plus_nat @ J @ K2 ) ) ) ).
% add_le_mono1
thf(fact_1232_trans__le__add1,axiom,
! [I: nat,J: nat,M2: nat] :
( ( ord_less_eq_nat @ I @ J )
=> ( ord_less_eq_nat @ I @ ( plus_plus_nat @ J @ M2 ) ) ) ).
% trans_le_add1
thf(fact_1233_trans__le__add2,axiom,
! [I: nat,J: nat,M2: nat] :
( ( ord_less_eq_nat @ I @ J )
=> ( ord_less_eq_nat @ I @ ( plus_plus_nat @ M2 @ J ) ) ) ).
% trans_le_add2
thf(fact_1234_nat__le__iff__add,axiom,
( ord_less_eq_nat
= ( ^ [M: nat,N: nat] :
? [K4: nat] :
( N
= ( plus_plus_nat @ M @ K4 ) ) ) ) ).
% nat_le_iff_add
thf(fact_1235_add__mult__distrib,axiom,
! [M2: nat,N2: nat,K2: nat] :
( ( times_times_nat @ ( plus_plus_nat @ M2 @ N2 ) @ K2 )
= ( plus_plus_nat @ ( times_times_nat @ M2 @ K2 ) @ ( times_times_nat @ N2 @ K2 ) ) ) ).
% add_mult_distrib
thf(fact_1236_add__mult__distrib2,axiom,
! [K2: nat,M2: nat,N2: nat] :
( ( times_times_nat @ K2 @ ( plus_plus_nat @ M2 @ N2 ) )
= ( plus_plus_nat @ ( times_times_nat @ K2 @ M2 ) @ ( times_times_nat @ K2 @ N2 ) ) ) ).
% add_mult_distrib2
thf(fact_1237_add__Suc__shift,axiom,
! [M2: nat,N2: nat] :
( ( plus_plus_nat @ ( suc @ M2 ) @ N2 )
= ( plus_plus_nat @ M2 @ ( suc @ N2 ) ) ) ).
% add_Suc_shift
thf(fact_1238_add__Suc,axiom,
! [M2: nat,N2: nat] :
( ( plus_plus_nat @ ( suc @ M2 ) @ N2 )
= ( suc @ ( plus_plus_nat @ M2 @ N2 ) ) ) ).
% add_Suc
thf(fact_1239_nat__arith_Osuc1,axiom,
! [A4: nat,K2: nat,A: nat] :
( ( A4
= ( plus_plus_nat @ K2 @ A ) )
=> ( ( suc @ A4 )
= ( plus_plus_nat @ K2 @ ( suc @ A ) ) ) ) ).
% nat_arith.suc1
thf(fact_1240_real__root__abs,axiom,
! [N2: nat,X: real] :
( ( ord_less_nat @ zero_zero_nat @ N2 )
=> ( ( root @ N2 @ ( abs_abs_real @ X ) )
= ( abs_abs_real @ ( root @ N2 @ X ) ) ) ) ).
% real_root_abs
thf(fact_1241_nat__diff__split,axiom,
! [P: nat > $o,A: nat,B: nat] :
( ( P @ ( minus_minus_nat @ A @ B ) )
= ( ( ( ord_less_nat @ A @ B )
=> ( P @ zero_zero_nat ) )
& ! [D4: nat] :
( ( A
= ( plus_plus_nat @ B @ D4 ) )
=> ( P @ D4 ) ) ) ) ).
% nat_diff_split
thf(fact_1242_nat__diff__split__asm,axiom,
! [P: nat > $o,A: nat,B: nat] :
( ( P @ ( minus_minus_nat @ A @ B ) )
= ( ~ ( ( ( ord_less_nat @ A @ B )
& ~ ( P @ zero_zero_nat ) )
| ? [D4: nat] :
( ( A
= ( plus_plus_nat @ B @ D4 ) )
& ~ ( P @ D4 ) ) ) ) ) ).
% nat_diff_split_asm
thf(fact_1243_less__diff__conv2,axiom,
! [K2: nat,J: nat,I: nat] :
( ( ord_less_eq_nat @ K2 @ J )
=> ( ( ord_less_nat @ ( minus_minus_nat @ J @ K2 ) @ I )
= ( ord_less_nat @ J @ ( plus_plus_nat @ I @ K2 ) ) ) ) ).
% less_diff_conv2
thf(fact_1244_nat__eq__add__iff1,axiom,
! [J: nat,I: nat,U: nat,M2: nat,N2: nat] :
( ( ord_less_eq_nat @ J @ I )
=> ( ( ( plus_plus_nat @ ( times_times_nat @ I @ U ) @ M2 )
= ( plus_plus_nat @ ( times_times_nat @ J @ U ) @ N2 ) )
= ( ( plus_plus_nat @ ( times_times_nat @ ( minus_minus_nat @ I @ J ) @ U ) @ M2 )
= N2 ) ) ) ).
% nat_eq_add_iff1
thf(fact_1245_nat__eq__add__iff2,axiom,
! [I: nat,J: nat,U: nat,M2: nat,N2: nat] :
( ( ord_less_eq_nat @ I @ J )
=> ( ( ( plus_plus_nat @ ( times_times_nat @ I @ U ) @ M2 )
= ( plus_plus_nat @ ( times_times_nat @ J @ U ) @ N2 ) )
= ( M2
= ( plus_plus_nat @ ( times_times_nat @ ( minus_minus_nat @ J @ I ) @ U ) @ N2 ) ) ) ) ).
% nat_eq_add_iff2
thf(fact_1246_nat__le__add__iff1,axiom,
! [J: nat,I: nat,U: nat,M2: nat,N2: nat] :
( ( ord_less_eq_nat @ J @ I )
=> ( ( ord_less_eq_nat @ ( plus_plus_nat @ ( times_times_nat @ I @ U ) @ M2 ) @ ( plus_plus_nat @ ( times_times_nat @ J @ U ) @ N2 ) )
= ( ord_less_eq_nat @ ( plus_plus_nat @ ( times_times_nat @ ( minus_minus_nat @ I @ J ) @ U ) @ M2 ) @ N2 ) ) ) ).
% nat_le_add_iff1
thf(fact_1247_nat__le__add__iff2,axiom,
! [I: nat,J: nat,U: nat,M2: nat,N2: nat] :
( ( ord_less_eq_nat @ I @ J )
=> ( ( ord_less_eq_nat @ ( plus_plus_nat @ ( times_times_nat @ I @ U ) @ M2 ) @ ( plus_plus_nat @ ( times_times_nat @ J @ U ) @ N2 ) )
= ( ord_less_eq_nat @ M2 @ ( plus_plus_nat @ ( times_times_nat @ ( minus_minus_nat @ J @ I ) @ U ) @ N2 ) ) ) ) ).
% nat_le_add_iff2
thf(fact_1248_nat__diff__add__eq1,axiom,
! [J: nat,I: nat,U: nat,M2: nat,N2: nat] :
( ( ord_less_eq_nat @ J @ I )
=> ( ( minus_minus_nat @ ( plus_plus_nat @ ( times_times_nat @ I @ U ) @ M2 ) @ ( plus_plus_nat @ ( times_times_nat @ J @ U ) @ N2 ) )
= ( minus_minus_nat @ ( plus_plus_nat @ ( times_times_nat @ ( minus_minus_nat @ I @ J ) @ U ) @ M2 ) @ N2 ) ) ) ).
% nat_diff_add_eq1
thf(fact_1249_nat__diff__add__eq2,axiom,
! [I: nat,J: nat,U: nat,M2: nat,N2: nat] :
( ( ord_less_eq_nat @ I @ J )
=> ( ( minus_minus_nat @ ( plus_plus_nat @ ( times_times_nat @ I @ U ) @ M2 ) @ ( plus_plus_nat @ ( times_times_nat @ J @ U ) @ N2 ) )
= ( minus_minus_nat @ M2 @ ( plus_plus_nat @ ( times_times_nat @ ( minus_minus_nat @ J @ I ) @ U ) @ N2 ) ) ) ) ).
% nat_diff_add_eq2
thf(fact_1250_root__abs__power,axiom,
! [N2: nat,Y: real] :
( ( ord_less_nat @ zero_zero_nat @ N2 )
=> ( ( abs_abs_real @ ( root @ N2 @ ( power_power_real @ Y @ N2 ) ) )
= ( abs_abs_real @ Y ) ) ) ).
% root_abs_power
thf(fact_1251_add__eq__if,axiom,
( plus_plus_nat
= ( ^ [M: nat,N: nat] : ( if_nat @ ( M = zero_zero_nat ) @ N @ ( suc @ ( plus_plus_nat @ ( minus_minus_nat @ M @ one_one_nat ) @ N ) ) ) ) ) ).
% add_eq_if
thf(fact_1252_nat__less__add__iff2,axiom,
! [I: nat,J: nat,U: nat,M2: nat,N2: nat] :
( ( ord_less_eq_nat @ I @ J )
=> ( ( ord_less_nat @ ( plus_plus_nat @ ( times_times_nat @ I @ U ) @ M2 ) @ ( plus_plus_nat @ ( times_times_nat @ J @ U ) @ N2 ) )
= ( ord_less_nat @ M2 @ ( plus_plus_nat @ ( times_times_nat @ ( minus_minus_nat @ J @ I ) @ U ) @ N2 ) ) ) ) ).
% nat_less_add_iff2
thf(fact_1253_nat__less__add__iff1,axiom,
! [J: nat,I: nat,U: nat,M2: nat,N2: nat] :
( ( ord_less_eq_nat @ J @ I )
=> ( ( ord_less_nat @ ( plus_plus_nat @ ( times_times_nat @ I @ U ) @ M2 ) @ ( plus_plus_nat @ ( times_times_nat @ J @ U ) @ N2 ) )
= ( ord_less_nat @ ( plus_plus_nat @ ( times_times_nat @ ( minus_minus_nat @ I @ J ) @ U ) @ M2 ) @ N2 ) ) ) ).
% nat_less_add_iff1
thf(fact_1254_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_1255_int__plus,axiom,
! [N2: nat,M2: nat] :
( ( semiri1314217659103216013at_int @ ( plus_plus_nat @ N2 @ M2 ) )
= ( plus_plus_int @ ( semiri1314217659103216013at_int @ N2 ) @ ( semiri1314217659103216013at_int @ M2 ) ) ) ).
% int_plus
thf(fact_1256_int__ops_I5_J,axiom,
! [A: nat,B: nat] :
( ( semiri1314217659103216013at_int @ ( plus_plus_nat @ A @ B ) )
= ( plus_plus_int @ ( semiri1314217659103216013at_int @ A ) @ ( semiri1314217659103216013at_int @ B ) ) ) ).
% int_ops(5)
thf(fact_1257_int__ops_I4_J,axiom,
! [A: nat] :
( ( semiri1314217659103216013at_int @ ( suc @ A ) )
= ( plus_plus_int @ ( semiri1314217659103216013at_int @ A ) @ one_one_int ) ) ).
% int_ops(4)
thf(fact_1258_int__Suc,axiom,
! [N2: nat] :
( ( semiri1314217659103216013at_int @ ( suc @ N2 ) )
= ( plus_plus_int @ ( semiri1314217659103216013at_int @ N2 ) @ one_one_int ) ) ).
% int_Suc
thf(fact_1259_nat__le__real__less,axiom,
( ord_less_eq_nat
= ( ^ [N: nat,M: nat] : ( ord_less_real @ ( semiri5074537144036343181t_real @ N ) @ ( plus_plus_real @ ( semiri5074537144036343181t_real @ M ) @ one_one_real ) ) ) ) ).
% nat_le_real_less
thf(fact_1260_nat__less__real__le,axiom,
( ord_less_nat
= ( ^ [N: nat,M: nat] : ( ord_less_eq_real @ ( plus_plus_real @ ( semiri5074537144036343181t_real @ N ) @ one_one_real ) @ ( semiri5074537144036343181t_real @ M ) ) ) ) ).
% nat_less_real_le
thf(fact_1261_incr__mult__lemma,axiom,
! [D3: int,P: int > $o,K2: int] :
( ( ord_less_int @ zero_zero_int @ D3 )
=> ( ! [X3: int] :
( ( P @ X3 )
=> ( P @ ( plus_plus_int @ X3 @ D3 ) ) )
=> ( ( ord_less_eq_int @ zero_zero_int @ K2 )
=> ! [X6: int] :
( ( P @ X6 )
=> ( P @ ( plus_plus_int @ X6 @ ( times_times_int @ K2 @ D3 ) ) ) ) ) ) ) ).
% incr_mult_lemma
thf(fact_1262_decr__lemma,axiom,
! [D3: int,X: int,Z2: int] :
( ( ord_less_int @ zero_zero_int @ D3 )
=> ( ord_less_int @ ( minus_minus_int @ X @ ( times_times_int @ ( plus_plus_int @ ( abs_abs_int @ ( minus_minus_int @ X @ Z2 ) ) @ one_one_int ) @ D3 ) ) @ Z2 ) ) ).
% decr_lemma
thf(fact_1263_incr__lemma,axiom,
! [D3: int,Z2: int,X: int] :
( ( ord_less_int @ zero_zero_int @ D3 )
=> ( ord_less_int @ Z2 @ ( plus_plus_int @ X @ ( times_times_int @ ( plus_plus_int @ ( abs_abs_int @ ( minus_minus_int @ X @ Z2 ) ) @ one_one_int ) @ D3 ) ) ) ) ).
% incr_lemma
% Helper facts (3)
thf(help_If_3_1_If_001t__Nat__Onat_T,axiom,
! [P: $o] :
( ( P = $true )
| ( P = $false ) ) ).
thf(help_If_2_1_If_001t__Nat__Onat_T,axiom,
! [X: nat,Y: nat] :
( ( if_nat @ $false @ X @ Y )
= Y ) ).
thf(help_If_1_1_If_001t__Nat__Onat_T,axiom,
! [X: nat,Y: nat] :
( ( if_nat @ $true @ X @ Y )
= X ) ).
% Conjectures (5)
thf(conj_0,hypothesis,
! [I3: nat,J4: nat,Xs: list_Sum_sum_a_nat] :
( ( ord_less_eq_nat @ I3 @ J4 )
=> ( ( member408289922725080238_a_nat @ Xs @ ( nall_tuples_rec_a @ ad @ I3 @ na ) )
=> ( ( fo_nmlz_rec_a @ J4 @ ( id_map_a @ J4 ) @ ad @ Xs )
= Xs ) ) ) ).
thf(conj_1,hypothesis,
ord_less_eq_nat @ ia @ ja ).
thf(conj_2,hypothesis,
member408289922725080238_a_nat @ x @ ( nall_tuples_rec_a @ ad @ ia @ na ) ).
thf(conj_3,hypothesis,
ord_less_nat @ xb @ ia ).
thf(conj_4,conjecture,
? [Y6: nat] :
( ( id_map_a @ ja @ ( sum_Inr_nat_a @ xb ) )
= ( some_nat @ Y6 ) ) ).
%------------------------------------------------------------------------------