TPTP Problem File: ITP140^2.p
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%------------------------------------------------------------------------------
% File : ITP140^2 : TPTP v8.2.0. Released v7.5.0.
% Domain : Interactive Theorem Proving
% Problem : Sledgehammer Paraconsistency problem prob_947__3275214_1
% Version : Especial.
% English :
% Refs : [BH+15] Blanchette et al. (2015), Mining the Archive of Formal
% : [Des21] Desharnais (2021), Email to Geoff Sutcliffe
% Source : [Des21]
% Names : Paraconsistency/prob_947__3275214_1 [Des21]
% Status : Theorem
% Rating : 0.00 v7.5.0
% Syntax : Number of formulae : 343 ( 149 unt; 66 typ; 0 def)
% Number of atoms : 685 ( 354 equ; 0 cnn)
% Maximal formula atoms : 16 ( 2 avg)
% Number of connectives : 4244 ( 114 ~; 6 |; 38 &;3759 @)
% ( 0 <=>; 327 =>; 0 <=; 0 <~>)
% Maximal formula depth : 28 ( 8 avg)
% Number of types : 5 ( 4 usr)
% Number of type conns : 533 ( 533 >; 0 *; 0 +; 0 <<)
% Number of symbols : 65 ( 62 usr; 5 con; 0-8 aty)
% Number of variables : 1340 ( 44 ^;1206 !; 16 ?;1340 :)
% ( 74 !>; 0 ?*; 0 @-; 0 @+)
% SPC : TH1_THM_EQU_NAR
% Comments : This file was generated by Sledgehammer 2021-02-23 16:24:15.916
%------------------------------------------------------------------------------
% Could-be-implicit typings (7)
thf(ty_t_Paraconsistency__Mirabelle__pjyhsvfwkt_Otv,type,
paraco415392788lle_tv: $tType ).
thf(ty_t_Paraconsistency__Mirabelle__pjyhsvfwkt_Ofm,type,
paraco414474393lle_fm: $tType ).
thf(ty_t_Product__Type_Oprod,type,
product_prod: $tType > $tType > $tType ).
thf(ty_t_String_Ochar,type,
char: $tType ).
thf(ty_t_List_Olist,type,
list: $tType > $tType ).
thf(ty_t_Set_Oset,type,
set: $tType > $tType ).
thf(ty_t_Nat_Onat,type,
nat: $tType ).
% Explicit typings (59)
thf(sy_cl_HOL_Otype,type,
type:
!>[A: $tType] : $o ).
thf(sy_cl_Orderings_Oord,type,
ord:
!>[A: $tType] : $o ).
thf(sy_cl_Orderings_Otop,type,
top:
!>[A: $tType] : $o ).
thf(sy_cl_Orderings_Oorder,type,
order:
!>[A: $tType] : $o ).
thf(sy_cl_Orderings_Olinorder,type,
linorder:
!>[A: $tType] : $o ).
thf(sy_cl_Orderings_Opreorder,type,
preorder:
!>[A: $tType] : $o ).
thf(sy_c_BNF__Def_OfstOp,type,
bNF_fstOp:
!>[A: $tType,B: $tType,C: $tType] : ( ( A > B > $o ) > ( B > C > $o ) > ( product_prod @ A @ C ) > ( product_prod @ A @ B ) ) ).
thf(sy_c_BNF__Def_Opick__middlep,type,
bNF_pick_middlep:
!>[B: $tType,A: $tType,C: $tType] : ( ( B > A > $o ) > ( A > C > $o ) > B > C > A ) ).
thf(sy_c_BNF__Def_OsndOp,type,
bNF_sndOp:
!>[C: $tType,A: $tType,B: $tType] : ( ( C > A > $o ) > ( A > B > $o ) > ( product_prod @ C @ B ) > ( product_prod @ A @ B ) ) ).
thf(sy_c_BNF__Greatest__Fixpoint_Oimage2,type,
bNF_Greatest_image2:
!>[C: $tType,A: $tType,B: $tType] : ( ( set @ C ) > ( C > A ) > ( C > B ) > ( set @ ( product_prod @ A @ B ) ) ) ).
thf(sy_c_Fun_Oinj__on,type,
inj_on:
!>[A: $tType,B: $tType] : ( ( A > B ) > ( set @ A ) > $o ) ).
thf(sy_c_Fun_Othe__inv__into,type,
the_inv_into:
!>[A: $tType,B: $tType] : ( ( set @ A ) > ( A > B ) > B > A ) ).
thf(sy_c_If,type,
if:
!>[A: $tType] : ( $o > A > A > A ) ).
thf(sy_c_Orderings_Oord__class_Oless__eq,type,
ord_less_eq:
!>[A: $tType] : ( A > A > $o ) ).
thf(sy_c_Orderings_Oorder__class_Ostrict__mono,type,
order_strict_mono:
!>[A: $tType,B: $tType] : ( ( A > B ) > $o ) ).
thf(sy_c_Orderings_Otop__class_Otop,type,
top_top:
!>[A: $tType] : A ).
thf(sy_c_Paraconsistency__Mirabelle__pjyhsvfwkt_Ochange__int,type,
paraco753463838ge_int: ( nat > nat ) > ( ( list @ char ) > paraco415392788lle_tv ) > ( list @ char ) > paraco415392788lle_tv ).
thf(sy_c_Paraconsistency__Mirabelle__pjyhsvfwkt_Ochange__tv,type,
paraco1920534163nge_tv: ( nat > nat ) > paraco415392788lle_tv > paraco415392788lle_tv ).
thf(sy_c_Paraconsistency__Mirabelle__pjyhsvfwkt_Ochange__tv__rel,type,
paraco2077297190tv_rel: ( product_prod @ ( nat > nat ) @ paraco415392788lle_tv ) > ( product_prod @ ( nat > nat ) @ paraco415392788lle_tv ) > $o ).
thf(sy_c_Paraconsistency__Mirabelle__pjyhsvfwkt_Oeval,type,
paraco876059933e_eval: ( ( list @ char ) > paraco415392788lle_tv ) > paraco414474393lle_fm > paraco415392788lle_tv ).
thf(sy_c_Paraconsistency__Mirabelle__pjyhsvfwkt_Ofm_OCon_H,type,
paraco2100061555le_Con: paraco414474393lle_fm > paraco414474393lle_fm > paraco414474393lle_fm ).
thf(sy_c_Paraconsistency__Mirabelle__pjyhsvfwkt_Ofm_OEql,type,
paraco2084319816le_Eql: paraco414474393lle_fm > paraco414474393lle_fm > paraco414474393lle_fm ).
thf(sy_c_Paraconsistency__Mirabelle__pjyhsvfwkt_Ofm_OEql_H,type,
paraco1628874225le_Eql: paraco414474393lle_fm > paraco414474393lle_fm > paraco414474393lle_fm ).
thf(sy_c_Paraconsistency__Mirabelle__pjyhsvfwkt_Ofm_ONeg_H,type,
paraco329115265le_Neg: paraco414474393lle_fm > paraco414474393lle_fm ).
thf(sy_c_Paraconsistency__Mirabelle__pjyhsvfwkt_Ofm_OPro,type,
paraco27778325le_Pro: ( list @ char ) > paraco414474393lle_fm ).
thf(sy_c_Paraconsistency__Mirabelle__pjyhsvfwkt_Ofm_OTruth,type,
paraco251304083_Truth: paraco414474393lle_fm ).
thf(sy_c_Paraconsistency__Mirabelle__pjyhsvfwkt_Ofm_Ocase__fm,type,
paraco1246693743ase_fm:
!>[A: $tType] : ( ( ( list @ char ) > A ) > A > ( paraco414474393lle_fm > A ) > ( paraco414474393lle_fm > paraco414474393lle_fm > A ) > ( paraco414474393lle_fm > paraco414474393lle_fm > A ) > ( paraco414474393lle_fm > paraco414474393lle_fm > A ) > paraco414474393lle_fm > A ) ).
thf(sy_c_Paraconsistency__Mirabelle__pjyhsvfwkt_Ofm_Orec__fm,type,
paraco815940159rec_fm:
!>[A: $tType] : ( ( ( list @ char ) > A ) > A > ( paraco414474393lle_fm > A > A ) > ( paraco414474393lle_fm > paraco414474393lle_fm > A > A > A ) > ( paraco414474393lle_fm > paraco414474393lle_fm > A > A > A ) > ( paraco414474393lle_fm > paraco414474393lle_fm > A > A > A ) > paraco414474393lle_fm > A ) ).
thf(sy_c_Paraconsistency__Mirabelle__pjyhsvfwkt_Otv_ODet,type,
paraco2040174112le_Det: $o > paraco415392788lle_tv ).
thf(sy_c_Paraconsistency__Mirabelle__pjyhsvfwkt_Otv_OIndet,type,
paraco676387099_Indet: nat > paraco415392788lle_tv ).
thf(sy_c_Paraconsistency__Mirabelle__pjyhsvfwkt_Otv_Ocase__tv,type,
paraco490622181ase_tv:
!>[A: $tType] : ( ( $o > A ) > ( nat > A ) > paraco415392788lle_tv > A ) ).
thf(sy_c_Paraconsistency__Mirabelle__pjyhsvfwkt_Otv_Orec__tv,type,
paraco152590079rec_tv:
!>[A: $tType] : ( ( $o > A ) > ( nat > A ) > paraco415392788lle_tv > A ) ).
thf(sy_c_Paraconsistency__Mirabelle__pjyhsvfwkt_Ovalid,type,
paraco769098683_valid: paraco414474393lle_fm > $o ).
thf(sy_c_Paraconsistency__Mirabelle__pjyhsvfwkt_Ovalid__in,type,
paraco2086025920lid_in: ( set @ nat ) > paraco414474393lle_fm > $o ).
thf(sy_c_Product__Type_OPair,type,
product_Pair:
!>[A: $tType,B: $tType] : ( A > B > ( product_prod @ A @ B ) ) ).
thf(sy_c_Product__Type_Oapfst,type,
product_apfst:
!>[A: $tType,C: $tType,B: $tType] : ( ( A > C ) > ( product_prod @ A @ B ) > ( product_prod @ C @ B ) ) ).
thf(sy_c_Product__Type_Oapsnd,type,
product_apsnd:
!>[B: $tType,C: $tType,A: $tType] : ( ( B > C ) > ( product_prod @ A @ B ) > ( product_prod @ A @ C ) ) ).
thf(sy_c_Product__Type_Obool_Ocase__bool,type,
product_case_bool:
!>[A: $tType] : ( A > A > $o > A ) ).
thf(sy_c_Product__Type_Ocurry,type,
product_curry:
!>[A: $tType,B: $tType,C: $tType] : ( ( ( product_prod @ A @ B ) > C ) > A > B > C ) ).
thf(sy_c_Product__Type_Ointernal__case__prod,type,
produc2004651681e_prod:
!>[A: $tType,B: $tType,C: $tType] : ( ( A > B > C ) > ( product_prod @ A @ B ) > C ) ).
thf(sy_c_Product__Type_Omap__prod,type,
product_map_prod:
!>[A: $tType,C: $tType,B: $tType,D: $tType] : ( ( A > C ) > ( B > D ) > ( product_prod @ A @ B ) > ( product_prod @ C @ D ) ) ).
thf(sy_c_Product__Type_Oold_Oprod_Orec__prod,type,
product_rec_prod:
!>[A: $tType,B: $tType,T: $tType] : ( ( A > B > T ) > ( product_prod @ A @ B ) > T ) ).
thf(sy_c_Product__Type_Oprod_Ofst,type,
product_fst:
!>[A: $tType,B: $tType] : ( ( product_prod @ A @ B ) > A ) ).
thf(sy_c_Product__Type_Oprod_Osnd,type,
product_snd:
!>[A: $tType,B: $tType] : ( ( product_prod @ A @ B ) > B ) ).
thf(sy_c_Product__Type_Oprod_Oswap,type,
product_swap:
!>[A: $tType,B: $tType] : ( ( product_prod @ A @ B ) > ( product_prod @ B @ A ) ) ).
thf(sy_c_Relation_Oinv__image,type,
inv_image:
!>[B: $tType,A: $tType] : ( ( set @ ( product_prod @ B @ B ) ) > ( A > B ) > ( set @ ( product_prod @ A @ A ) ) ) ).
thf(sy_c_Relation_Oinv__imagep,type,
inv_imagep:
!>[B: $tType,A: $tType] : ( ( B > B > $o ) > ( A > B ) > A > A > $o ) ).
thf(sy_c_Relation_Ototal__on,type,
total_on:
!>[A: $tType] : ( ( set @ A ) > ( set @ ( product_prod @ A @ A ) ) > $o ) ).
thf(sy_c_Set_OCollect,type,
collect:
!>[A: $tType] : ( ( A > $o ) > ( set @ A ) ) ).
thf(sy_c_Set_Oimage,type,
image:
!>[A: $tType,B: $tType] : ( ( A > B ) > ( set @ A ) > ( set @ B ) ) ).
thf(sy_c_Wellfounded_Oaccp,type,
accp:
!>[A: $tType] : ( ( A > A > $o ) > A > $o ) ).
thf(sy_c_Wellfounded_Olex__prod,type,
lex_prod:
!>[A: $tType,B: $tType] : ( ( set @ ( product_prod @ A @ A ) ) > ( set @ ( product_prod @ B @ B ) ) > ( set @ ( product_prod @ ( product_prod @ A @ B ) @ ( product_prod @ A @ B ) ) ) ) ).
thf(sy_c_Wellfounded_Owf,type,
wf:
!>[A: $tType] : ( ( set @ ( product_prod @ A @ A ) ) > $o ) ).
thf(sy_c_Wfrec_Osame__fst,type,
same_fst:
!>[A: $tType,B: $tType] : ( ( A > $o ) > ( A > ( set @ ( product_prod @ B @ B ) ) ) > ( set @ ( product_prod @ ( product_prod @ A @ B ) @ ( product_prod @ A @ B ) ) ) ) ).
thf(sy_c_member,type,
member:
!>[A: $tType] : ( A > ( set @ A ) > $o ) ).
thf(sy_v_f,type,
f: nat > nat ).
thf(sy_v_i,type,
i: ( list @ char ) > paraco415392788lle_tv ).
thf(sy_v_p1____,type,
p1: paraco414474393lle_fm ).
thf(sy_v_p2____,type,
p2: paraco414474393lle_fm ).
% Relevant facts (256)
thf(fact_0_a_H_H,axiom,
( ( paraco876059933e_eval @ i @ p1 )
!= ( paraco2040174112le_Det @ $true ) ) ).
% a''
thf(fact_1_a,axiom,
( ( paraco876059933e_eval @ i @ p2 )
= ( paraco2040174112le_Det @ $true ) ) ).
% a
thf(fact_2_a_H,axiom,
( ( paraco876059933e_eval @ i @ p1 )
!= ( paraco876059933e_eval @ i @ p2 ) ) ).
% a'
thf(fact_3_fm_Oinject_I3_J,axiom,
! [X41: paraco414474393lle_fm,X42: paraco414474393lle_fm,Y41: paraco414474393lle_fm,Y42: paraco414474393lle_fm] :
( ( ( paraco2100061555le_Con @ X41 @ X42 )
= ( paraco2100061555le_Con @ Y41 @ Y42 ) )
= ( ( X41 = Y41 )
& ( X42 = Y42 ) ) ) ).
% fm.inject(3)
thf(fact_4_b_H,axiom,
( ( paraco876059933e_eval @ ( paraco753463838ge_int @ f @ i ) @ p1 )
!= ( paraco876059933e_eval @ ( paraco753463838ge_int @ f @ i ) @ p2 ) ) ).
% b'
thf(fact_5_eval_Osimps_I4_J,axiom,
! [I: ( list @ char ) > paraco415392788lle_tv,P: paraco414474393lle_fm,Q: paraco414474393lle_fm] :
( ( ( ( paraco876059933e_eval @ I @ P )
= ( paraco876059933e_eval @ I @ Q ) )
=> ( ( paraco876059933e_eval @ I @ ( paraco2100061555le_Con @ P @ Q ) )
= ( paraco876059933e_eval @ I @ P ) ) )
& ( ( ( paraco876059933e_eval @ I @ P )
!= ( paraco876059933e_eval @ I @ Q ) )
=> ( ( ( ( paraco876059933e_eval @ I @ P )
= ( paraco2040174112le_Det @ $true ) )
=> ( ( paraco876059933e_eval @ I @ ( paraco2100061555le_Con @ P @ Q ) )
= ( paraco876059933e_eval @ I @ Q ) ) )
& ( ( ( paraco876059933e_eval @ I @ P )
!= ( paraco2040174112le_Det @ $true ) )
=> ( ( ( ( paraco876059933e_eval @ I @ Q )
= ( paraco2040174112le_Det @ $true ) )
=> ( ( paraco876059933e_eval @ I @ ( paraco2100061555le_Con @ P @ Q ) )
= ( paraco876059933e_eval @ I @ P ) ) )
& ( ( ( paraco876059933e_eval @ I @ Q )
!= ( paraco2040174112le_Det @ $true ) )
=> ( ( paraco876059933e_eval @ I @ ( paraco2100061555le_Con @ P @ Q ) )
= ( paraco2040174112le_Det @ $false ) ) ) ) ) ) ) ) ).
% eval.simps(4)
thf(fact_6_ih2,axiom,
( ( paraco876059933e_eval @ ( paraco753463838ge_int @ f @ i ) @ p2 )
= ( paraco1920534163nge_tv @ f @ ( paraco876059933e_eval @ i @ p2 ) ) ) ).
% ih2
thf(fact_7_ih1,axiom,
( ( paraco876059933e_eval @ ( paraco753463838ge_int @ f @ i ) @ p1 )
= ( paraco1920534163nge_tv @ f @ ( paraco876059933e_eval @ i @ p1 ) ) ) ).
% ih1
thf(fact_8_b_H_H,axiom,
( ( paraco876059933e_eval @ ( paraco753463838ge_int @ f @ i ) @ p1 )
!= ( paraco2040174112le_Det @ $true ) ) ).
% b''
thf(fact_9_tv_Oinject_I1_J,axiom,
! [X1: $o,Y1: $o] :
( ( ( paraco2040174112le_Det @ X1 )
= ( paraco2040174112le_Det @ Y1 ) )
= ( X1 = Y1 ) ) ).
% tv.inject(1)
thf(fact_10_conjunction,axiom,
! [P: paraco414474393lle_fm,Q: paraco414474393lle_fm] :
( ( paraco769098683_valid @ ( paraco2100061555le_Con @ P @ Q ) )
= ( ( paraco769098683_valid @ P )
& ( paraco769098683_valid @ Q ) ) ) ).
% conjunction
thf(fact_11_conjunction__in,axiom,
! [U: set @ nat,P: paraco414474393lle_fm,Q: paraco414474393lle_fm] :
( ( paraco2086025920lid_in @ U @ ( paraco2100061555le_Con @ P @ Q ) )
= ( ( paraco2086025920lid_in @ U @ P )
& ( paraco2086025920lid_in @ U @ Q ) ) ) ).
% conjunction_in
thf(fact_12_fm_Osimps_I39_J,axiom,
! [A: $tType,F1: ( list @ char ) > A,F2: A,F3: paraco414474393lle_fm > A,F4: paraco414474393lle_fm > paraco414474393lle_fm > A,F5: paraco414474393lle_fm > paraco414474393lle_fm > A,F6: paraco414474393lle_fm > paraco414474393lle_fm > A,X41: paraco414474393lle_fm,X42: paraco414474393lle_fm] :
( ( paraco1246693743ase_fm @ A @ F1 @ F2 @ F3 @ F4 @ F5 @ F6 @ ( paraco2100061555le_Con @ X41 @ X42 ) )
= ( F4 @ X41 @ X42 ) ) ).
% fm.simps(39)
thf(fact_13_fm_Odistinct_I13_J,axiom,
! [X41: paraco414474393lle_fm,X42: paraco414474393lle_fm] :
( paraco251304083_Truth
!= ( paraco2100061555le_Con @ X41 @ X42 ) ) ).
% fm.distinct(13)
thf(fact_14_fm_Osimps_I45_J,axiom,
! [A: $tType,F1: ( list @ char ) > A,F2: A,F3: paraco414474393lle_fm > A > A,F4: paraco414474393lle_fm > paraco414474393lle_fm > A > A > A,F5: paraco414474393lle_fm > paraco414474393lle_fm > A > A > A,F6: paraco414474393lle_fm > paraco414474393lle_fm > A > A > A,X41: paraco414474393lle_fm,X42: paraco414474393lle_fm] :
( ( paraco815940159rec_fm @ A @ F1 @ F2 @ F3 @ F4 @ F5 @ F6 @ ( paraco2100061555le_Con @ X41 @ X42 ) )
= ( F4 @ X41 @ X42 @ ( paraco815940159rec_fm @ A @ F1 @ F2 @ F3 @ F4 @ F5 @ F6 @ X41 ) @ ( paraco815940159rec_fm @ A @ F1 @ F2 @ F3 @ F4 @ F5 @ F6 @ X42 ) ) ) ).
% fm.simps(45)
thf(fact_15_fm_Odistinct_I25_J,axiom,
! [X41: paraco414474393lle_fm,X42: paraco414474393lle_fm,X51: paraco414474393lle_fm,X52: paraco414474393lle_fm] :
( ( paraco2100061555le_Con @ X41 @ X42 )
!= ( paraco2084319816le_Eql @ X51 @ X52 ) ) ).
% fm.distinct(25)
thf(fact_16_fm_Oinject_I4_J,axiom,
! [X51: paraco414474393lle_fm,X52: paraco414474393lle_fm,Y51: paraco414474393lle_fm,Y52: paraco414474393lle_fm] :
( ( ( paraco2084319816le_Eql @ X51 @ X52 )
= ( paraco2084319816le_Eql @ Y51 @ Y52 ) )
= ( ( X51 = Y51 )
& ( X52 = Y52 ) ) ) ).
% fm.inject(4)
thf(fact_17_fm_Osimps_I46_J,axiom,
! [A: $tType,F1: ( list @ char ) > A,F2: A,F3: paraco414474393lle_fm > A > A,F4: paraco414474393lle_fm > paraco414474393lle_fm > A > A > A,F5: paraco414474393lle_fm > paraco414474393lle_fm > A > A > A,F6: paraco414474393lle_fm > paraco414474393lle_fm > A > A > A,X51: paraco414474393lle_fm,X52: paraco414474393lle_fm] :
( ( paraco815940159rec_fm @ A @ F1 @ F2 @ F3 @ F4 @ F5 @ F6 @ ( paraco2084319816le_Eql @ X51 @ X52 ) )
= ( F5 @ X51 @ X52 @ ( paraco815940159rec_fm @ A @ F1 @ F2 @ F3 @ F4 @ F5 @ F6 @ X51 ) @ ( paraco815940159rec_fm @ A @ F1 @ F2 @ F3 @ F4 @ F5 @ F6 @ X52 ) ) ) ).
% fm.simps(46)
thf(fact_18_fm_Osimps_I43_J,axiom,
! [A: $tType,F1: ( list @ char ) > A,F2: A,F3: paraco414474393lle_fm > A > A,F4: paraco414474393lle_fm > paraco414474393lle_fm > A > A > A,F5: paraco414474393lle_fm > paraco414474393lle_fm > A > A > A,F6: paraco414474393lle_fm > paraco414474393lle_fm > A > A > A] :
( ( paraco815940159rec_fm @ A @ F1 @ F2 @ F3 @ F4 @ F5 @ F6 @ paraco251304083_Truth )
= F2 ) ).
% fm.simps(43)
thf(fact_19_fm_Osimps_I40_J,axiom,
! [A: $tType,F1: ( list @ char ) > A,F2: A,F3: paraco414474393lle_fm > A,F4: paraco414474393lle_fm > paraco414474393lle_fm > A,F5: paraco414474393lle_fm > paraco414474393lle_fm > A,F6: paraco414474393lle_fm > paraco414474393lle_fm > A,X51: paraco414474393lle_fm,X52: paraco414474393lle_fm] :
( ( paraco1246693743ase_fm @ A @ F1 @ F2 @ F3 @ F4 @ F5 @ F6 @ ( paraco2084319816le_Eql @ X51 @ X52 ) )
= ( F5 @ X51 @ X52 ) ) ).
% fm.simps(40)
thf(fact_20_fm_Osimps_I37_J,axiom,
! [A: $tType,F1: ( list @ char ) > A,F2: A,F3: paraco414474393lle_fm > A,F4: paraco414474393lle_fm > paraco414474393lle_fm > A,F5: paraco414474393lle_fm > paraco414474393lle_fm > A,F6: paraco414474393lle_fm > paraco414474393lle_fm > A] :
( ( paraco1246693743ase_fm @ A @ F1 @ F2 @ F3 @ F4 @ F5 @ F6 @ paraco251304083_Truth )
= F2 ) ).
% fm.simps(37)
thf(fact_21_fm_Odistinct_I15_J,axiom,
! [X51: paraco414474393lle_fm,X52: paraco414474393lle_fm] :
( paraco251304083_Truth
!= ( paraco2084319816le_Eql @ X51 @ X52 ) ) ).
% fm.distinct(15)
thf(fact_22_change__tv_Osimps_I1_J,axiom,
! [F: nat > nat,B2: $o] :
( ( paraco1920534163nge_tv @ F @ ( paraco2040174112le_Det @ B2 ) )
= ( paraco2040174112le_Det @ B2 ) ) ).
% change_tv.simps(1)
thf(fact_23_transfer,axiom,
! [U: set @ nat,P: paraco414474393lle_fm] :
( ~ ( paraco2086025920lid_in @ U @ P )
=> ~ ( paraco769098683_valid @ P ) ) ).
% transfer
thf(fact_24_change__int__def,axiom,
( paraco753463838ge_int
= ( ^ [F7: nat > nat,I2: ( list @ char ) > paraco415392788lle_tv,S: list @ char] : ( paraco1920534163nge_tv @ F7 @ ( I2 @ S ) ) ) ) ).
% change_int_def
thf(fact_25_valid__valid__in,axiom,
! [P: paraco414474393lle_fm,U: set @ nat] :
( ( paraco769098683_valid @ P )
=> ( paraco2086025920lid_in @ U @ P ) ) ).
% valid_valid_in
thf(fact_26_eval__equality__simplify,axiom,
! [I: ( list @ char ) > paraco415392788lle_tv,P: paraco414474393lle_fm,Q: paraco414474393lle_fm] :
( ( paraco876059933e_eval @ I @ ( paraco2084319816le_Eql @ P @ Q ) )
= ( paraco2040174112le_Det
@ ( ( paraco876059933e_eval @ I @ P )
= ( paraco876059933e_eval @ I @ Q ) ) ) ) ).
% eval_equality_simplify
thf(fact_27_eval_Osimps_I5_J,axiom,
! [I: ( list @ char ) > paraco415392788lle_tv,P: paraco414474393lle_fm,Q: paraco414474393lle_fm] :
( ( ( ( paraco876059933e_eval @ I @ P )
= ( paraco876059933e_eval @ I @ Q ) )
=> ( ( paraco876059933e_eval @ I @ ( paraco2084319816le_Eql @ P @ Q ) )
= ( paraco2040174112le_Det @ $true ) ) )
& ( ( ( paraco876059933e_eval @ I @ P )
!= ( paraco876059933e_eval @ I @ Q ) )
=> ( ( paraco876059933e_eval @ I @ ( paraco2084319816le_Eql @ P @ Q ) )
= ( paraco2040174112le_Det @ $false ) ) ) ) ).
% eval.simps(5)
thf(fact_28_eval_Osimps_I2_J,axiom,
! [I: ( list @ char ) > paraco415392788lle_tv] :
( ( paraco876059933e_eval @ I @ paraco251304083_Truth )
= ( paraco2040174112le_Det @ $true ) ) ).
% eval.simps(2)
thf(fact_29_valid__def,axiom,
( paraco769098683_valid
= ( ^ [P2: paraco414474393lle_fm] :
! [I2: ( list @ char ) > paraco415392788lle_tv] :
( ( paraco876059933e_eval @ I2 @ P2 )
= ( paraco2040174112le_Det @ $true ) ) ) ) ).
% valid_def
thf(fact_30_tv_Osimps_I7_J,axiom,
! [A: $tType,F1: $o > A,F2: nat > A,X1: $o] :
( ( paraco152590079rec_tv @ A @ F1 @ F2 @ ( paraco2040174112le_Det @ X1 ) )
= ( F1 @ X1 ) ) ).
% tv.simps(7)
thf(fact_31_change__tv_Oelims,axiom,
! [X: nat > nat,Xa: paraco415392788lle_tv,Y: paraco415392788lle_tv] :
( ( ( paraco1920534163nge_tv @ X @ Xa )
= Y )
=> ( ! [B3: $o] :
( ( Xa
= ( paraco2040174112le_Det @ B3 ) )
=> ( Y
!= ( paraco2040174112le_Det @ B3 ) ) )
=> ~ ! [N: nat] :
( ( Xa
= ( paraco676387099_Indet @ N ) )
=> ( Y
!= ( paraco676387099_Indet @ ( X @ N ) ) ) ) ) ) ).
% change_tv.elims
thf(fact_32_tv_Osimps_I5_J,axiom,
! [A: $tType,F1: $o > A,F2: nat > A,X1: $o] :
( ( paraco490622181ase_tv @ A @ F1 @ F2 @ ( paraco2040174112le_Det @ X1 ) )
= ( F1 @ X1 ) ) ).
% tv.simps(5)
thf(fact_33_eval__negation,axiom,
! [I: ( list @ char ) > paraco415392788lle_tv,P: paraco414474393lle_fm] :
( ( ( ( paraco876059933e_eval @ I @ P )
= ( paraco2040174112le_Det @ $false ) )
=> ( ( paraco876059933e_eval @ I @ ( paraco329115265le_Neg @ P ) )
= ( paraco2040174112le_Det @ $true ) ) )
& ( ( ( paraco876059933e_eval @ I @ P )
!= ( paraco2040174112le_Det @ $false ) )
=> ( ( ( ( paraco876059933e_eval @ I @ P )
= ( paraco2040174112le_Det @ $true ) )
=> ( ( paraco876059933e_eval @ I @ ( paraco329115265le_Neg @ P ) )
= ( paraco2040174112le_Det @ $false ) ) )
& ( ( ( paraco876059933e_eval @ I @ P )
!= ( paraco2040174112le_Det @ $true ) )
=> ( ( paraco876059933e_eval @ I @ ( paraco329115265le_Neg @ P ) )
= ( paraco876059933e_eval @ I @ P ) ) ) ) ) ) ).
% eval_negation
thf(fact_34_assms,axiom,
inj_on @ nat @ nat @ f @ ( top_top @ ( set @ nat ) ) ).
% assms
thf(fact_35_change__tv_Osimps_I2_J,axiom,
! [F: nat > nat,N2: nat] :
( ( paraco1920534163nge_tv @ F @ ( paraco676387099_Indet @ N2 ) )
= ( paraco676387099_Indet @ ( F @ N2 ) ) ) ).
% change_tv.simps(2)
thf(fact_36_fm_Osimps_I41_J,axiom,
! [A: $tType,F1: ( list @ char ) > A,F2: A,F3: paraco414474393lle_fm > A,F4: paraco414474393lle_fm > paraco414474393lle_fm > A,F5: paraco414474393lle_fm > paraco414474393lle_fm > A,F6: paraco414474393lle_fm > paraco414474393lle_fm > A,X61: paraco414474393lle_fm,X62: paraco414474393lle_fm] :
( ( paraco1246693743ase_fm @ A @ F1 @ F2 @ F3 @ F4 @ F5 @ F6 @ ( paraco1628874225le_Eql @ X61 @ X62 ) )
= ( F6 @ X61 @ X62 ) ) ).
% fm.simps(41)
thf(fact_37_fm_Osimps_I47_J,axiom,
! [A: $tType,F1: ( list @ char ) > A,F2: A,F3: paraco414474393lle_fm > A > A,F4: paraco414474393lle_fm > paraco414474393lle_fm > A > A > A,F5: paraco414474393lle_fm > paraco414474393lle_fm > A > A > A,F6: paraco414474393lle_fm > paraco414474393lle_fm > A > A > A,X61: paraco414474393lle_fm,X62: paraco414474393lle_fm] :
( ( paraco815940159rec_fm @ A @ F1 @ F2 @ F3 @ F4 @ F5 @ F6 @ ( paraco1628874225le_Eql @ X61 @ X62 ) )
= ( F6 @ X61 @ X62 @ ( paraco815940159rec_fm @ A @ F1 @ F2 @ F3 @ F4 @ F5 @ F6 @ X61 ) @ ( paraco815940159rec_fm @ A @ F1 @ F2 @ F3 @ F4 @ F5 @ F6 @ X62 ) ) ) ).
% fm.simps(47)
thf(fact_38_fm_Osimps_I38_J,axiom,
! [A: $tType,F1: ( list @ char ) > A,F2: A,F3: paraco414474393lle_fm > A,F4: paraco414474393lle_fm > paraco414474393lle_fm > A,F5: paraco414474393lle_fm > paraco414474393lle_fm > A,F6: paraco414474393lle_fm > paraco414474393lle_fm > A,X3: paraco414474393lle_fm] :
( ( paraco1246693743ase_fm @ A @ F1 @ F2 @ F3 @ F4 @ F5 @ F6 @ ( paraco329115265le_Neg @ X3 ) )
= ( F3 @ X3 ) ) ).
% fm.simps(38)
thf(fact_39_fm_Odistinct_I17_J,axiom,
! [X61: paraco414474393lle_fm,X62: paraco414474393lle_fm] :
( paraco251304083_Truth
!= ( paraco1628874225le_Eql @ X61 @ X62 ) ) ).
% fm.distinct(17)
thf(fact_40_fm_Oinject_I2_J,axiom,
! [X3: paraco414474393lle_fm,Y3: paraco414474393lle_fm] :
( ( ( paraco329115265le_Neg @ X3 )
= ( paraco329115265le_Neg @ Y3 ) )
= ( X3 = Y3 ) ) ).
% fm.inject(2)
thf(fact_41_fm_Oinject_I5_J,axiom,
! [X61: paraco414474393lle_fm,X62: paraco414474393lle_fm,Y61: paraco414474393lle_fm,Y62: paraco414474393lle_fm] :
( ( ( paraco1628874225le_Eql @ X61 @ X62 )
= ( paraco1628874225le_Eql @ Y61 @ Y62 ) )
= ( ( X61 = Y61 )
& ( X62 = Y62 ) ) ) ).
% fm.inject(5)
thf(fact_42_tv_Oinject_I2_J,axiom,
! [X2: nat,Y2: nat] :
( ( ( paraco676387099_Indet @ X2 )
= ( paraco676387099_Indet @ Y2 ) )
= ( X2 = Y2 ) ) ).
% tv.inject(2)
thf(fact_43_fm_Odistinct_I23_J,axiom,
! [X3: paraco414474393lle_fm,X61: paraco414474393lle_fm,X62: paraco414474393lle_fm] :
( ( paraco329115265le_Neg @ X3 )
!= ( paraco1628874225le_Eql @ X61 @ X62 ) ) ).
% fm.distinct(23)
thf(fact_44_tv_Osimps_I6_J,axiom,
! [A: $tType,F1: $o > A,F2: nat > A,X2: nat] :
( ( paraco490622181ase_tv @ A @ F1 @ F2 @ ( paraco676387099_Indet @ X2 ) )
= ( F2 @ X2 ) ) ).
% tv.simps(6)
thf(fact_45_mem__Collect__eq,axiom,
! [A: $tType,A2: A,P3: A > $o] :
( ( member @ A @ A2 @ ( collect @ A @ P3 ) )
= ( P3 @ A2 ) ) ).
% mem_Collect_eq
thf(fact_46_Collect__mem__eq,axiom,
! [A: $tType,A3: set @ A] :
( ( collect @ A
@ ^ [X4: A] : ( member @ A @ X4 @ A3 ) )
= A3 ) ).
% Collect_mem_eq
thf(fact_47_Collect__cong,axiom,
! [A: $tType,P3: A > $o,Q2: A > $o] :
( ! [X5: A] :
( ( P3 @ X5 )
= ( Q2 @ X5 ) )
=> ( ( collect @ A @ P3 )
= ( collect @ A @ Q2 ) ) ) ).
% Collect_cong
thf(fact_48_ext,axiom,
! [B: $tType,A: $tType,F: A > B,G: A > B] :
( ! [X5: A] :
( ( F @ X5 )
= ( G @ X5 ) )
=> ( F = G ) ) ).
% ext
thf(fact_49_tv_Osimps_I8_J,axiom,
! [A: $tType,F1: $o > A,F2: nat > A,X2: nat] :
( ( paraco152590079rec_tv @ A @ F1 @ F2 @ ( paraco676387099_Indet @ X2 ) )
= ( F2 @ X2 ) ) ).
% tv.simps(8)
thf(fact_50_eval__equality,axiom,
! [I: ( list @ char ) > paraco415392788lle_tv,P: paraco414474393lle_fm,Q: paraco414474393lle_fm] :
( ( ( ( paraco876059933e_eval @ I @ P )
= ( paraco876059933e_eval @ I @ Q ) )
=> ( ( paraco876059933e_eval @ I @ ( paraco1628874225le_Eql @ P @ Q ) )
= ( paraco2040174112le_Det @ $true ) ) )
& ( ( ( paraco876059933e_eval @ I @ P )
!= ( paraco876059933e_eval @ I @ Q ) )
=> ( ( ( ( paraco876059933e_eval @ I @ P )
= ( paraco2040174112le_Det @ $true ) )
=> ( ( paraco876059933e_eval @ I @ ( paraco1628874225le_Eql @ P @ Q ) )
= ( paraco876059933e_eval @ I @ Q ) ) )
& ( ( ( paraco876059933e_eval @ I @ P )
!= ( paraco2040174112le_Det @ $true ) )
=> ( ( ( ( paraco876059933e_eval @ I @ Q )
= ( paraco2040174112le_Det @ $true ) )
=> ( ( paraco876059933e_eval @ I @ ( paraco1628874225le_Eql @ P @ Q ) )
= ( paraco876059933e_eval @ I @ P ) ) )
& ( ( ( paraco876059933e_eval @ I @ Q )
!= ( paraco2040174112le_Det @ $true ) )
=> ( ( ( ( paraco876059933e_eval @ I @ P )
= ( paraco2040174112le_Det @ $false ) )
=> ( ( paraco876059933e_eval @ I @ ( paraco1628874225le_Eql @ P @ Q ) )
= ( paraco876059933e_eval @ I @ ( paraco329115265le_Neg @ Q ) ) ) )
& ( ( ( paraco876059933e_eval @ I @ P )
!= ( paraco2040174112le_Det @ $false ) )
=> ( ( ( ( paraco876059933e_eval @ I @ Q )
= ( paraco2040174112le_Det @ $false ) )
=> ( ( paraco876059933e_eval @ I @ ( paraco1628874225le_Eql @ P @ Q ) )
= ( paraco876059933e_eval @ I @ ( paraco329115265le_Neg @ P ) ) ) )
& ( ( ( paraco876059933e_eval @ I @ Q )
!= ( paraco2040174112le_Det @ $false ) )
=> ( ( paraco876059933e_eval @ I @ ( paraco1628874225le_Eql @ P @ Q ) )
= ( paraco2040174112le_Det @ $false ) ) ) ) ) ) ) ) ) ) ) ) ).
% eval_equality
thf(fact_51_double__negation,axiom,
( paraco876059933e_eval
= ( ^ [I2: ( list @ char ) > paraco415392788lle_tv,P2: paraco414474393lle_fm] : ( paraco876059933e_eval @ I2 @ ( paraco329115265le_Neg @ ( paraco329115265le_Neg @ P2 ) ) ) ) ) ).
% double_negation
thf(fact_52_fm_Odistinct_I19_J,axiom,
! [X3: paraco414474393lle_fm,X41: paraco414474393lle_fm,X42: paraco414474393lle_fm] :
( ( paraco329115265le_Neg @ X3 )
!= ( paraco2100061555le_Con @ X41 @ X42 ) ) ).
% fm.distinct(19)
thf(fact_53_string__tv_Oinduct,axiom,
! [P3: paraco415392788lle_tv > $o,A0: paraco415392788lle_tv] :
( ( P3 @ ( paraco2040174112le_Det @ $true ) )
=> ( ( P3 @ ( paraco2040174112le_Det @ $false ) )
=> ( ! [N: nat] : ( P3 @ ( paraco676387099_Indet @ N ) )
=> ( P3 @ A0 ) ) ) ) ).
% string_tv.induct
thf(fact_54_change__tv_Oinduct,axiom,
! [P3: ( nat > nat ) > paraco415392788lle_tv > $o,A0: nat > nat,A1: paraco415392788lle_tv] :
( ! [F8: nat > nat,B3: $o] : ( P3 @ F8 @ ( paraco2040174112le_Det @ B3 ) )
=> ( ! [F8: nat > nat,N: nat] : ( P3 @ F8 @ ( paraco676387099_Indet @ N ) )
=> ( P3 @ A0 @ A1 ) ) ) ).
% change_tv.induct
thf(fact_55_string__tv_Ocases,axiom,
! [X: paraco415392788lle_tv] :
( ( X
!= ( paraco2040174112le_Det @ $true ) )
=> ( ( X
!= ( paraco2040174112le_Det @ $false ) )
=> ~ ! [N: nat] :
( X
!= ( paraco676387099_Indet @ N ) ) ) ) ).
% string_tv.cases
thf(fact_56_tv_Oexhaust,axiom,
! [Y: paraco415392788lle_tv] :
( ! [X12: $o] :
( Y
!= ( paraco2040174112le_Det @ X12 ) )
=> ~ ! [X22: nat] :
( Y
!= ( paraco676387099_Indet @ X22 ) ) ) ).
% tv.exhaust
thf(fact_57_tv_Oinduct,axiom,
! [P3: paraco415392788lle_tv > $o,Tv: paraco415392788lle_tv] :
( ! [X5: $o] : ( P3 @ ( paraco2040174112le_Det @ X5 ) )
=> ( ! [X5: nat] : ( P3 @ ( paraco676387099_Indet @ X5 ) )
=> ( P3 @ Tv ) ) ) ).
% tv.induct
thf(fact_58_tv_Odistinct_I1_J,axiom,
! [X1: $o,X2: nat] :
( ( paraco2040174112le_Det @ X1 )
!= ( paraco676387099_Indet @ X2 ) ) ).
% tv.distinct(1)
thf(fact_59_fm_Odistinct_I21_J,axiom,
! [X3: paraco414474393lle_fm,X51: paraco414474393lle_fm,X52: paraco414474393lle_fm] :
( ( paraco329115265le_Neg @ X3 )
!= ( paraco2084319816le_Eql @ X51 @ X52 ) ) ).
% fm.distinct(21)
thf(fact_60_fm_Odistinct_I11_J,axiom,
! [X3: paraco414474393lle_fm] :
( paraco251304083_Truth
!= ( paraco329115265le_Neg @ X3 ) ) ).
% fm.distinct(11)
thf(fact_61_fm_Odistinct_I27_J,axiom,
! [X41: paraco414474393lle_fm,X42: paraco414474393lle_fm,X61: paraco414474393lle_fm,X62: paraco414474393lle_fm] :
( ( paraco2100061555le_Con @ X41 @ X42 )
!= ( paraco1628874225le_Eql @ X61 @ X62 ) ) ).
% fm.distinct(27)
thf(fact_62_fm_Osimps_I44_J,axiom,
! [A: $tType,F1: ( list @ char ) > A,F2: A,F3: paraco414474393lle_fm > A > A,F4: paraco414474393lle_fm > paraco414474393lle_fm > A > A > A,F5: paraco414474393lle_fm > paraco414474393lle_fm > A > A > A,F6: paraco414474393lle_fm > paraco414474393lle_fm > A > A > A,X3: paraco414474393lle_fm] :
( ( paraco815940159rec_fm @ A @ F1 @ F2 @ F3 @ F4 @ F5 @ F6 @ ( paraco329115265le_Neg @ X3 ) )
= ( F3 @ X3 @ ( paraco815940159rec_fm @ A @ F1 @ F2 @ F3 @ F4 @ F5 @ F6 @ X3 ) ) ) ).
% fm.simps(44)
thf(fact_63_fm_Odistinct_I29_J,axiom,
! [X51: paraco414474393lle_fm,X52: paraco414474393lle_fm,X61: paraco414474393lle_fm,X62: paraco414474393lle_fm] :
( ( paraco2084319816le_Eql @ X51 @ X52 )
!= ( paraco1628874225le_Eql @ X61 @ X62 ) ) ).
% fm.distinct(29)
thf(fact_64_iso__tuple__UNIV__I,axiom,
! [A: $tType,X: A] : ( member @ A @ X @ ( top_top @ ( set @ A ) ) ) ).
% iso_tuple_UNIV_I
thf(fact_65_UNIV__I,axiom,
! [A: $tType,X: A] : ( member @ A @ X @ ( top_top @ ( set @ A ) ) ) ).
% UNIV_I
thf(fact_66_top__apply,axiom,
! [C: $tType,D: $tType] :
( ( top @ C )
=> ( ( top_top @ ( D > C ) )
= ( ^ [X4: D] : ( top_top @ C ) ) ) ) ).
% top_apply
thf(fact_67_eval_Osimps_I3_J,axiom,
! [I: ( list @ char ) > paraco415392788lle_tv,P: paraco414474393lle_fm] :
( ( paraco876059933e_eval @ I @ ( paraco329115265le_Neg @ P ) )
= ( paraco490622181ase_tv @ paraco415392788lle_tv @ ( product_case_bool @ paraco415392788lle_tv @ ( paraco2040174112le_Det @ $false ) @ ( paraco2040174112le_Det @ $true ) ) @ paraco676387099_Indet @ ( paraco876059933e_eval @ I @ P ) ) ) ).
% eval.simps(3)
thf(fact_68_inj__def,axiom,
! [B: $tType,A: $tType,F: A > B] :
( ( inj_on @ A @ B @ F @ ( top_top @ ( set @ A ) ) )
= ( ! [X4: A,Y4: A] :
( ( ( F @ X4 )
= ( F @ Y4 ) )
=> ( X4 = Y4 ) ) ) ) ).
% inj_def
thf(fact_69_inj__eq,axiom,
! [B: $tType,A: $tType,F: A > B,X: A,Y: A] :
( ( inj_on @ A @ B @ F @ ( top_top @ ( set @ A ) ) )
=> ( ( ( F @ X )
= ( F @ Y ) )
= ( X = Y ) ) ) ).
% inj_eq
thf(fact_70_injI,axiom,
! [B: $tType,A: $tType,F: A > B] :
( ! [X5: A,Y5: A] :
( ( ( F @ X5 )
= ( F @ Y5 ) )
=> ( X5 = Y5 ) )
=> ( inj_on @ A @ B @ F @ ( top_top @ ( set @ A ) ) ) ) ).
% injI
thf(fact_71_injD,axiom,
! [B: $tType,A: $tType,F: A > B,X: A,Y: A] :
( ( inj_on @ A @ B @ F @ ( top_top @ ( set @ A ) ) )
=> ( ( ( F @ X )
= ( F @ Y ) )
=> ( X = Y ) ) ) ).
% injD
thf(fact_72_top__set__def,axiom,
! [A: $tType] :
( ( top_top @ ( set @ A ) )
= ( collect @ A @ ( top_top @ ( A > $o ) ) ) ) ).
% top_set_def
thf(fact_73_change__tv__injection,axiom,
! [F: nat > nat] :
( ( inj_on @ nat @ nat @ F @ ( top_top @ ( set @ nat ) ) )
=> ( inj_on @ paraco415392788lle_tv @ paraco415392788lle_tv @ ( paraco1920534163nge_tv @ F ) @ ( top_top @ ( set @ paraco415392788lle_tv ) ) ) ) ).
% change_tv_injection
thf(fact_74_UNIV__eq__I,axiom,
! [A: $tType,A3: set @ A] :
( ! [X5: A] : ( member @ A @ X5 @ A3 )
=> ( ( top_top @ ( set @ A ) )
= A3 ) ) ).
% UNIV_eq_I
thf(fact_75_UNIV__witness,axiom,
! [A: $tType] :
? [X5: A] : ( member @ A @ X5 @ ( top_top @ ( set @ A ) ) ) ).
% UNIV_witness
thf(fact_76_inj__onD,axiom,
! [B: $tType,A: $tType,F: A > B,A3: set @ A,X: A,Y: A] :
( ( inj_on @ A @ B @ F @ A3 )
=> ( ( ( F @ X )
= ( F @ Y ) )
=> ( ( member @ A @ X @ A3 )
=> ( ( member @ A @ Y @ A3 )
=> ( X = Y ) ) ) ) ) ).
% inj_onD
thf(fact_77_inj__onI,axiom,
! [B: $tType,A: $tType,A3: set @ A,F: A > B] :
( ! [X5: A,Y5: A] :
( ( member @ A @ X5 @ A3 )
=> ( ( member @ A @ Y5 @ A3 )
=> ( ( ( F @ X5 )
= ( F @ Y5 ) )
=> ( X5 = Y5 ) ) ) )
=> ( inj_on @ A @ B @ F @ A3 ) ) ).
% inj_onI
thf(fact_78_inj__on__def,axiom,
! [B: $tType,A: $tType] :
( ( inj_on @ A @ B )
= ( ^ [F7: A > B,A4: set @ A] :
! [X4: A] :
( ( member @ A @ X4 @ A4 )
=> ! [Y4: A] :
( ( member @ A @ Y4 @ A4 )
=> ( ( ( F7 @ X4 )
= ( F7 @ Y4 ) )
=> ( X4 = Y4 ) ) ) ) ) ) ).
% inj_on_def
thf(fact_79_inj__on__cong,axiom,
! [B: $tType,A: $tType,A3: set @ A,F: A > B,G: A > B] :
( ! [A5: A] :
( ( member @ A @ A5 @ A3 )
=> ( ( F @ A5 )
= ( G @ A5 ) ) )
=> ( ( inj_on @ A @ B @ F @ A3 )
= ( inj_on @ A @ B @ G @ A3 ) ) ) ).
% inj_on_cong
thf(fact_80_inj__on__eq__iff,axiom,
! [B: $tType,A: $tType,F: A > B,A3: set @ A,X: A,Y: A] :
( ( inj_on @ A @ B @ F @ A3 )
=> ( ( member @ A @ X @ A3 )
=> ( ( member @ A @ Y @ A3 )
=> ( ( ( F @ X )
= ( F @ Y ) )
= ( X = Y ) ) ) ) ) ).
% inj_on_eq_iff
thf(fact_81_inj__on__contraD,axiom,
! [B: $tType,A: $tType,F: A > B,A3: set @ A,X: A,Y: A] :
( ( inj_on @ A @ B @ F @ A3 )
=> ( ( X != Y )
=> ( ( member @ A @ X @ A3 )
=> ( ( member @ A @ Y @ A3 )
=> ( ( F @ X )
!= ( F @ Y ) ) ) ) ) ) ).
% inj_on_contraD
thf(fact_82_inj__on__inverseI,axiom,
! [B: $tType,A: $tType,A3: set @ A,G: B > A,F: A > B] :
( ! [X5: A] :
( ( member @ A @ X5 @ A3 )
=> ( ( G @ ( F @ X5 ) )
= X5 ) )
=> ( inj_on @ A @ B @ F @ A3 ) ) ).
% inj_on_inverseI
thf(fact_83_fm_Oinduct,axiom,
! [P3: paraco414474393lle_fm > $o,Fm: paraco414474393lle_fm] :
( ! [X5: list @ char] : ( P3 @ ( paraco27778325le_Pro @ X5 ) )
=> ( ( P3 @ paraco251304083_Truth )
=> ( ! [X5: paraco414474393lle_fm] :
( ( P3 @ X5 )
=> ( P3 @ ( paraco329115265le_Neg @ X5 ) ) )
=> ( ! [X1a: paraco414474393lle_fm,X22: paraco414474393lle_fm] :
( ( P3 @ X1a )
=> ( ( P3 @ X22 )
=> ( P3 @ ( paraco2100061555le_Con @ X1a @ X22 ) ) ) )
=> ( ! [X1a: paraco414474393lle_fm,X22: paraco414474393lle_fm] :
( ( P3 @ X1a )
=> ( ( P3 @ X22 )
=> ( P3 @ ( paraco2084319816le_Eql @ X1a @ X22 ) ) ) )
=> ( ! [X1a: paraco414474393lle_fm,X22: paraco414474393lle_fm] :
( ( P3 @ X1a )
=> ( ( P3 @ X22 )
=> ( P3 @ ( paraco1628874225le_Eql @ X1a @ X22 ) ) ) )
=> ( P3 @ Fm ) ) ) ) ) ) ) ).
% fm.induct
thf(fact_84_fm_Oexhaust,axiom,
! [Y: paraco414474393lle_fm] :
( ! [X12: list @ char] :
( Y
!= ( paraco27778325le_Pro @ X12 ) )
=> ( ( Y != paraco251304083_Truth )
=> ( ! [X32: paraco414474393lle_fm] :
( Y
!= ( paraco329115265le_Neg @ X32 ) )
=> ( ! [X412: paraco414474393lle_fm,X422: paraco414474393lle_fm] :
( Y
!= ( paraco2100061555le_Con @ X412 @ X422 ) )
=> ( ! [X512: paraco414474393lle_fm,X522: paraco414474393lle_fm] :
( Y
!= ( paraco2084319816le_Eql @ X512 @ X522 ) )
=> ~ ! [X612: paraco414474393lle_fm,X622: paraco414474393lle_fm] :
( Y
!= ( paraco1628874225le_Eql @ X612 @ X622 ) ) ) ) ) ) ) ).
% fm.exhaust
thf(fact_85_props_Ocases,axiom,
! [X: paraco414474393lle_fm] :
( ( X != paraco251304083_Truth )
=> ( ! [S2: list @ char] :
( X
!= ( paraco27778325le_Pro @ S2 ) )
=> ( ! [P4: paraco414474393lle_fm] :
( X
!= ( paraco329115265le_Neg @ P4 ) )
=> ( ! [P4: paraco414474393lle_fm,Q3: paraco414474393lle_fm] :
( X
!= ( paraco2100061555le_Con @ P4 @ Q3 ) )
=> ( ! [P4: paraco414474393lle_fm,Q3: paraco414474393lle_fm] :
( X
!= ( paraco2084319816le_Eql @ P4 @ Q3 ) )
=> ~ ! [P4: paraco414474393lle_fm,Q3: paraco414474393lle_fm] :
( X
!= ( paraco1628874225le_Eql @ P4 @ Q3 ) ) ) ) ) ) ) ).
% props.cases
thf(fact_86_props_Oinduct,axiom,
! [P3: paraco414474393lle_fm > $o,A0: paraco414474393lle_fm] :
( ( P3 @ paraco251304083_Truth )
=> ( ! [S2: list @ char] : ( P3 @ ( paraco27778325le_Pro @ S2 ) )
=> ( ! [P4: paraco414474393lle_fm] :
( ( P3 @ P4 )
=> ( P3 @ ( paraco329115265le_Neg @ P4 ) ) )
=> ( ! [P4: paraco414474393lle_fm,Q3: paraco414474393lle_fm] :
( ( P3 @ P4 )
=> ( ( P3 @ Q3 )
=> ( P3 @ ( paraco2100061555le_Con @ P4 @ Q3 ) ) ) )
=> ( ! [P4: paraco414474393lle_fm,Q3: paraco414474393lle_fm] :
( ( P3 @ P4 )
=> ( ( P3 @ Q3 )
=> ( P3 @ ( paraco2084319816le_Eql @ P4 @ Q3 ) ) ) )
=> ( ! [P4: paraco414474393lle_fm,Q3: paraco414474393lle_fm] :
( ( P3 @ P4 )
=> ( ( P3 @ Q3 )
=> ( P3 @ ( paraco1628874225le_Eql @ P4 @ Q3 ) ) ) )
=> ( P3 @ A0 ) ) ) ) ) ) ) ).
% props.induct
thf(fact_87_old_Obool_Osimps_I4_J,axiom,
! [A: $tType,F1: A,F2: A] :
( ( product_case_bool @ A @ F1 @ F2 @ $false )
= F2 ) ).
% old.bool.simps(4)
thf(fact_88_old_Obool_Osimps_I3_J,axiom,
! [A: $tType,F1: A,F2: A] :
( ( product_case_bool @ A @ F1 @ F2 @ $true )
= F1 ) ).
% old.bool.simps(3)
thf(fact_89_bool_Osplit__sel,axiom,
! [A: $tType,P3: A > $o,F1: A,F2: A,Bool: $o] :
( ( P3 @ ( product_case_bool @ A @ F1 @ F2 @ Bool ) )
= ( ( Bool
=> ( P3 @ F1 ) )
& ( ~ Bool
=> ( P3 @ F2 ) ) ) ) ).
% bool.split_sel
thf(fact_90_fm_Oinject_I1_J,axiom,
! [X1: list @ char,Y1: list @ char] :
( ( ( paraco27778325le_Pro @ X1 )
= ( paraco27778325le_Pro @ Y1 ) )
= ( X1 = Y1 ) ) ).
% fm.inject(1)
thf(fact_91_top1I,axiom,
! [A: $tType,X: A] : ( top_top @ ( A > $o ) @ X ) ).
% top1I
thf(fact_92_eval_Osimps_I1_J,axiom,
! [I: ( list @ char ) > paraco415392788lle_tv,S3: list @ char] :
( ( paraco876059933e_eval @ I @ ( paraco27778325le_Pro @ S3 ) )
= ( I @ S3 ) ) ).
% eval.simps(1)
thf(fact_93_fm_Odistinct_I3_J,axiom,
! [X1: list @ char,X3: paraco414474393lle_fm] :
( ( paraco27778325le_Pro @ X1 )
!= ( paraco329115265le_Neg @ X3 ) ) ).
% fm.distinct(3)
thf(fact_94_fm_Odistinct_I5_J,axiom,
! [X1: list @ char,X41: paraco414474393lle_fm,X42: paraco414474393lle_fm] :
( ( paraco27778325le_Pro @ X1 )
!= ( paraco2100061555le_Con @ X41 @ X42 ) ) ).
% fm.distinct(5)
thf(fact_95_fm_Odistinct_I9_J,axiom,
! [X1: list @ char,X61: paraco414474393lle_fm,X62: paraco414474393lle_fm] :
( ( paraco27778325le_Pro @ X1 )
!= ( paraco1628874225le_Eql @ X61 @ X62 ) ) ).
% fm.distinct(9)
thf(fact_96_fm_Odistinct_I7_J,axiom,
! [X1: list @ char,X51: paraco414474393lle_fm,X52: paraco414474393lle_fm] :
( ( paraco27778325le_Pro @ X1 )
!= ( paraco2084319816le_Eql @ X51 @ X52 ) ) ).
% fm.distinct(7)
thf(fact_97_fm_Odistinct_I1_J,axiom,
! [X1: list @ char] :
( ( paraco27778325le_Pro @ X1 )
!= paraco251304083_Truth ) ).
% fm.distinct(1)
thf(fact_98_fm_Osimps_I42_J,axiom,
! [A: $tType,F1: ( list @ char ) > A,F2: A,F3: paraco414474393lle_fm > A > A,F4: paraco414474393lle_fm > paraco414474393lle_fm > A > A > A,F5: paraco414474393lle_fm > paraco414474393lle_fm > A > A > A,F6: paraco414474393lle_fm > paraco414474393lle_fm > A > A > A,X1: list @ char] :
( ( paraco815940159rec_fm @ A @ F1 @ F2 @ F3 @ F4 @ F5 @ F6 @ ( paraco27778325le_Pro @ X1 ) )
= ( F1 @ X1 ) ) ).
% fm.simps(42)
thf(fact_99_fm_Osimps_I36_J,axiom,
! [A: $tType,F1: ( list @ char ) > A,F2: A,F3: paraco414474393lle_fm > A,F4: paraco414474393lle_fm > paraco414474393lle_fm > A,F5: paraco414474393lle_fm > paraco414474393lle_fm > A,F6: paraco414474393lle_fm > paraco414474393lle_fm > A,X1: list @ char] :
( ( paraco1246693743ase_fm @ A @ F1 @ F2 @ F3 @ F4 @ F5 @ F6 @ ( paraco27778325le_Pro @ X1 ) )
= ( F1 @ X1 ) ) ).
% fm.simps(36)
thf(fact_100_bool_Osplit__sel__asm,axiom,
! [A: $tType,P3: A > $o,F1: A,F2: A,Bool: $o] :
( ( P3 @ ( product_case_bool @ A @ F1 @ F2 @ Bool ) )
= ( ~ ( ( Bool
& ~ ( P3 @ F1 ) )
| ( ~ Bool
& ~ ( P3 @ F2 ) ) ) ) ) ).
% bool.split_sel_asm
thf(fact_101_bool_Ocase__distrib,axiom,
! [A: $tType,B: $tType,H: A > B,F1: A,F2: A,Bool: $o] :
( ( H @ ( product_case_bool @ A @ F1 @ F2 @ Bool ) )
= ( product_case_bool @ B @ ( H @ F1 ) @ ( H @ F2 ) @ Bool ) ) ).
% bool.case_distrib
thf(fact_102_bool_Ocase__eq__if,axiom,
! [A: $tType] :
( ( product_case_bool @ A )
= ( ^ [F12: A,F22: A,Bool2: $o] : ( if @ A @ Bool2 @ F12 @ F22 ) ) ) ).
% bool.case_eq_if
thf(fact_103_top__empty__eq,axiom,
! [A: $tType] :
( ( top_top @ ( A > $o ) )
= ( ^ [X4: A] : ( member @ A @ X4 @ ( top_top @ ( set @ A ) ) ) ) ) ).
% top_empty_eq
thf(fact_104_eval_Ocases,axiom,
! [X: product_prod @ ( ( list @ char ) > paraco415392788lle_tv ) @ paraco414474393lle_fm] :
( ! [I3: ( list @ char ) > paraco415392788lle_tv,S2: list @ char] :
( X
!= ( product_Pair @ ( ( list @ char ) > paraco415392788lle_tv ) @ paraco414474393lle_fm @ I3 @ ( paraco27778325le_Pro @ S2 ) ) )
=> ( ! [I3: ( list @ char ) > paraco415392788lle_tv] :
( X
!= ( product_Pair @ ( ( list @ char ) > paraco415392788lle_tv ) @ paraco414474393lle_fm @ I3 @ paraco251304083_Truth ) )
=> ( ! [I3: ( list @ char ) > paraco415392788lle_tv,P4: paraco414474393lle_fm] :
( X
!= ( product_Pair @ ( ( list @ char ) > paraco415392788lle_tv ) @ paraco414474393lle_fm @ I3 @ ( paraco329115265le_Neg @ P4 ) ) )
=> ( ! [I3: ( list @ char ) > paraco415392788lle_tv,P4: paraco414474393lle_fm,Q3: paraco414474393lle_fm] :
( X
!= ( product_Pair @ ( ( list @ char ) > paraco415392788lle_tv ) @ paraco414474393lle_fm @ I3 @ ( paraco2100061555le_Con @ P4 @ Q3 ) ) )
=> ( ! [I3: ( list @ char ) > paraco415392788lle_tv,P4: paraco414474393lle_fm,Q3: paraco414474393lle_fm] :
( X
!= ( product_Pair @ ( ( list @ char ) > paraco415392788lle_tv ) @ paraco414474393lle_fm @ I3 @ ( paraco2084319816le_Eql @ P4 @ Q3 ) ) )
=> ~ ! [I3: ( list @ char ) > paraco415392788lle_tv,P4: paraco414474393lle_fm,Q3: paraco414474393lle_fm] :
( X
!= ( product_Pair @ ( ( list @ char ) > paraco415392788lle_tv ) @ paraco414474393lle_fm @ I3 @ ( paraco1628874225le_Eql @ P4 @ Q3 ) ) ) ) ) ) ) ) ).
% eval.cases
thf(fact_105_top__conj_I1_J,axiom,
! [A: $tType,X: A,P3: $o] :
( ( ( top_top @ ( A > $o ) @ X )
& P3 )
= P3 ) ).
% top_conj(1)
thf(fact_106_top__conj_I2_J,axiom,
! [A: $tType,P3: $o,X: A] :
( ( P3
& ( top_top @ ( A > $o ) @ X ) )
= P3 ) ).
% top_conj(2)
thf(fact_107_inj__apsnd,axiom,
! [A: $tType,C: $tType,B: $tType,F: B > C] :
( ( inj_on @ ( product_prod @ A @ B ) @ ( product_prod @ A @ C ) @ ( product_apsnd @ B @ C @ A @ F ) @ ( top_top @ ( set @ ( product_prod @ A @ B ) ) ) )
= ( inj_on @ B @ C @ F @ ( top_top @ ( set @ B ) ) ) ) ).
% inj_apsnd
thf(fact_108_inj__apfst,axiom,
! [B: $tType,C: $tType,A: $tType,F: A > C] :
( ( inj_on @ ( product_prod @ A @ B ) @ ( product_prod @ C @ B ) @ ( product_apfst @ A @ C @ B @ F ) @ ( top_top @ ( set @ ( product_prod @ A @ B ) ) ) )
= ( inj_on @ A @ C @ F @ ( top_top @ ( set @ A ) ) ) ) ).
% inj_apfst
thf(fact_109_old_Oprod_Oinject,axiom,
! [A: $tType,B: $tType,A2: A,B2: B,A6: A,B4: B] :
( ( ( product_Pair @ A @ B @ A2 @ B2 )
= ( product_Pair @ A @ B @ A6 @ B4 ) )
= ( ( A2 = A6 )
& ( B2 = B4 ) ) ) ).
% old.prod.inject
thf(fact_110_prod_Oinject,axiom,
! [A: $tType,B: $tType,X1: A,X2: B,Y1: A,Y2: B] :
( ( ( product_Pair @ A @ B @ X1 @ X2 )
= ( product_Pair @ A @ B @ Y1 @ Y2 ) )
= ( ( X1 = Y1 )
& ( X2 = Y2 ) ) ) ).
% prod.inject
thf(fact_111_apfst__conv,axiom,
! [C: $tType,A: $tType,B: $tType,F: C > A,X: C,Y: B] :
( ( product_apfst @ C @ A @ B @ F @ ( product_Pair @ C @ B @ X @ Y ) )
= ( product_Pair @ A @ B @ ( F @ X ) @ Y ) ) ).
% apfst_conv
thf(fact_112_apsnd__conv,axiom,
! [A: $tType,B: $tType,C: $tType,F: C > B,X: A,Y: C] :
( ( product_apsnd @ C @ B @ A @ F @ ( product_Pair @ A @ C @ X @ Y ) )
= ( product_Pair @ A @ B @ X @ ( F @ Y ) ) ) ).
% apsnd_conv
thf(fact_113_apsnd__apfst__commute,axiom,
! [A: $tType,B: $tType,C: $tType,D: $tType,F: C > B,G: D > A,P: product_prod @ D @ C] :
( ( product_apsnd @ C @ B @ A @ F @ ( product_apfst @ D @ A @ C @ G @ P ) )
= ( product_apfst @ D @ A @ B @ G @ ( product_apsnd @ C @ B @ D @ F @ P ) ) ) ).
% apsnd_apfst_commute
thf(fact_114_old_Oprod_Oinducts,axiom,
! [B: $tType,A: $tType,P3: ( product_prod @ A @ B ) > $o,Prod: product_prod @ A @ B] :
( ! [A5: A,B3: B] : ( P3 @ ( product_Pair @ A @ B @ A5 @ B3 ) )
=> ( P3 @ Prod ) ) ).
% old.prod.inducts
thf(fact_115_old_Oprod_Oexhaust,axiom,
! [A: $tType,B: $tType,Y: product_prod @ A @ B] :
~ ! [A5: A,B3: B] :
( Y
!= ( product_Pair @ A @ B @ A5 @ B3 ) ) ).
% old.prod.exhaust
thf(fact_116_prod__induct7,axiom,
! [G2: $tType,F9: $tType,E: $tType,D: $tType,C: $tType,B: $tType,A: $tType,P3: ( product_prod @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D @ ( product_prod @ E @ ( product_prod @ F9 @ G2 ) ) ) ) ) ) > $o,X: product_prod @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D @ ( product_prod @ E @ ( product_prod @ F9 @ G2 ) ) ) ) )] :
( ! [A5: A,B3: B,C2: C,D2: D,E2: E,F8: F9,G3: G2] : ( P3 @ ( product_Pair @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D @ ( product_prod @ E @ ( product_prod @ F9 @ G2 ) ) ) ) ) @ A5 @ ( product_Pair @ B @ ( product_prod @ C @ ( product_prod @ D @ ( product_prod @ E @ ( product_prod @ F9 @ G2 ) ) ) ) @ B3 @ ( product_Pair @ C @ ( product_prod @ D @ ( product_prod @ E @ ( product_prod @ F9 @ G2 ) ) ) @ C2 @ ( product_Pair @ D @ ( product_prod @ E @ ( product_prod @ F9 @ G2 ) ) @ D2 @ ( product_Pair @ E @ ( product_prod @ F9 @ G2 ) @ E2 @ ( product_Pair @ F9 @ G2 @ F8 @ G3 ) ) ) ) ) ) )
=> ( P3 @ X ) ) ).
% prod_induct7
thf(fact_117_prod__induct6,axiom,
! [F9: $tType,E: $tType,D: $tType,C: $tType,B: $tType,A: $tType,P3: ( product_prod @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D @ ( product_prod @ E @ F9 ) ) ) ) ) > $o,X: product_prod @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D @ ( product_prod @ E @ F9 ) ) ) )] :
( ! [A5: A,B3: B,C2: C,D2: D,E2: E,F8: F9] : ( P3 @ ( product_Pair @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D @ ( product_prod @ E @ F9 ) ) ) ) @ A5 @ ( product_Pair @ B @ ( product_prod @ C @ ( product_prod @ D @ ( product_prod @ E @ F9 ) ) ) @ B3 @ ( product_Pair @ C @ ( product_prod @ D @ ( product_prod @ E @ F9 ) ) @ C2 @ ( product_Pair @ D @ ( product_prod @ E @ F9 ) @ D2 @ ( product_Pair @ E @ F9 @ E2 @ F8 ) ) ) ) ) )
=> ( P3 @ X ) ) ).
% prod_induct6
thf(fact_118_prod__induct5,axiom,
! [E: $tType,D: $tType,C: $tType,B: $tType,A: $tType,P3: ( product_prod @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D @ E ) ) ) ) > $o,X: product_prod @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D @ E ) ) )] :
( ! [A5: A,B3: B,C2: C,D2: D,E2: E] : ( P3 @ ( product_Pair @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D @ E ) ) ) @ A5 @ ( product_Pair @ B @ ( product_prod @ C @ ( product_prod @ D @ E ) ) @ B3 @ ( product_Pair @ C @ ( product_prod @ D @ E ) @ C2 @ ( product_Pair @ D @ E @ D2 @ E2 ) ) ) ) )
=> ( P3 @ X ) ) ).
% prod_induct5
thf(fact_119_prod__induct4,axiom,
! [D: $tType,C: $tType,B: $tType,A: $tType,P3: ( product_prod @ A @ ( product_prod @ B @ ( product_prod @ C @ D ) ) ) > $o,X: product_prod @ A @ ( product_prod @ B @ ( product_prod @ C @ D ) )] :
( ! [A5: A,B3: B,C2: C,D2: D] : ( P3 @ ( product_Pair @ A @ ( product_prod @ B @ ( product_prod @ C @ D ) ) @ A5 @ ( product_Pair @ B @ ( product_prod @ C @ D ) @ B3 @ ( product_Pair @ C @ D @ C2 @ D2 ) ) ) )
=> ( P3 @ X ) ) ).
% prod_induct4
thf(fact_120_prod__induct3,axiom,
! [C: $tType,B: $tType,A: $tType,P3: ( product_prod @ A @ ( product_prod @ B @ C ) ) > $o,X: product_prod @ A @ ( product_prod @ B @ C )] :
( ! [A5: A,B3: B,C2: C] : ( P3 @ ( product_Pair @ A @ ( product_prod @ B @ C ) @ A5 @ ( product_Pair @ B @ C @ B3 @ C2 ) ) )
=> ( P3 @ X ) ) ).
% prod_induct3
thf(fact_121_prod__cases7,axiom,
! [A: $tType,B: $tType,C: $tType,D: $tType,E: $tType,F9: $tType,G2: $tType,Y: product_prod @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D @ ( product_prod @ E @ ( product_prod @ F9 @ G2 ) ) ) ) )] :
~ ! [A5: A,B3: B,C2: C,D2: D,E2: E,F8: F9,G3: G2] :
( Y
!= ( product_Pair @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D @ ( product_prod @ E @ ( product_prod @ F9 @ G2 ) ) ) ) ) @ A5 @ ( product_Pair @ B @ ( product_prod @ C @ ( product_prod @ D @ ( product_prod @ E @ ( product_prod @ F9 @ G2 ) ) ) ) @ B3 @ ( product_Pair @ C @ ( product_prod @ D @ ( product_prod @ E @ ( product_prod @ F9 @ G2 ) ) ) @ C2 @ ( product_Pair @ D @ ( product_prod @ E @ ( product_prod @ F9 @ G2 ) ) @ D2 @ ( product_Pair @ E @ ( product_prod @ F9 @ G2 ) @ E2 @ ( product_Pair @ F9 @ G2 @ F8 @ G3 ) ) ) ) ) ) ) ).
% prod_cases7
thf(fact_122_prod__cases6,axiom,
! [A: $tType,B: $tType,C: $tType,D: $tType,E: $tType,F9: $tType,Y: product_prod @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D @ ( product_prod @ E @ F9 ) ) ) )] :
~ ! [A5: A,B3: B,C2: C,D2: D,E2: E,F8: F9] :
( Y
!= ( product_Pair @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D @ ( product_prod @ E @ F9 ) ) ) ) @ A5 @ ( product_Pair @ B @ ( product_prod @ C @ ( product_prod @ D @ ( product_prod @ E @ F9 ) ) ) @ B3 @ ( product_Pair @ C @ ( product_prod @ D @ ( product_prod @ E @ F9 ) ) @ C2 @ ( product_Pair @ D @ ( product_prod @ E @ F9 ) @ D2 @ ( product_Pair @ E @ F9 @ E2 @ F8 ) ) ) ) ) ) ).
% prod_cases6
thf(fact_123_prod__cases5,axiom,
! [A: $tType,B: $tType,C: $tType,D: $tType,E: $tType,Y: product_prod @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D @ E ) ) )] :
~ ! [A5: A,B3: B,C2: C,D2: D,E2: E] :
( Y
!= ( product_Pair @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D @ E ) ) ) @ A5 @ ( product_Pair @ B @ ( product_prod @ C @ ( product_prod @ D @ E ) ) @ B3 @ ( product_Pair @ C @ ( product_prod @ D @ E ) @ C2 @ ( product_Pair @ D @ E @ D2 @ E2 ) ) ) ) ) ).
% prod_cases5
thf(fact_124_prod__cases4,axiom,
! [A: $tType,B: $tType,C: $tType,D: $tType,Y: product_prod @ A @ ( product_prod @ B @ ( product_prod @ C @ D ) )] :
~ ! [A5: A,B3: B,C2: C,D2: D] :
( Y
!= ( product_Pair @ A @ ( product_prod @ B @ ( product_prod @ C @ D ) ) @ A5 @ ( product_Pair @ B @ ( product_prod @ C @ D ) @ B3 @ ( product_Pair @ C @ D @ C2 @ D2 ) ) ) ) ).
% prod_cases4
thf(fact_125_prod__cases3,axiom,
! [A: $tType,B: $tType,C: $tType,Y: product_prod @ A @ ( product_prod @ B @ C )] :
~ ! [A5: A,B3: B,C2: C] :
( Y
!= ( product_Pair @ A @ ( product_prod @ B @ C ) @ A5 @ ( product_Pair @ B @ C @ B3 @ C2 ) ) ) ).
% prod_cases3
thf(fact_126_Pair__inject,axiom,
! [A: $tType,B: $tType,A2: A,B2: B,A6: A,B4: B] :
( ( ( product_Pair @ A @ B @ A2 @ B2 )
= ( product_Pair @ A @ B @ A6 @ B4 ) )
=> ~ ( ( A2 = A6 )
=> ( B2 != B4 ) ) ) ).
% Pair_inject
thf(fact_127_prod__cases,axiom,
! [B: $tType,A: $tType,P3: ( product_prod @ A @ B ) > $o,P: product_prod @ A @ B] :
( ! [A5: A,B3: B] : ( P3 @ ( product_Pair @ A @ B @ A5 @ B3 ) )
=> ( P3 @ P ) ) ).
% prod_cases
thf(fact_128_surj__pair,axiom,
! [A: $tType,B: $tType,P: product_prod @ A @ B] :
? [X5: A,Y5: B] :
( P
= ( product_Pair @ A @ B @ X5 @ Y5 ) ) ).
% surj_pair
thf(fact_129_old_Oprod_Orec,axiom,
! [A: $tType,T: $tType,B: $tType,F1: A > B > T,A2: A,B2: B] :
( ( product_rec_prod @ A @ B @ T @ F1 @ ( product_Pair @ A @ B @ A2 @ B2 ) )
= ( F1 @ A2 @ B2 ) ) ).
% old.prod.rec
thf(fact_130_internal__case__prod__conv,axiom,
! [B: $tType,A: $tType,C: $tType,C3: B > C > A,A2: B,B2: C] :
( ( produc2004651681e_prod @ B @ C @ A @ C3 @ ( product_Pair @ B @ C @ A2 @ B2 ) )
= ( C3 @ A2 @ B2 ) ) ).
% internal_case_prod_conv
thf(fact_131_prod_Oinj__map,axiom,
! [D: $tType,C: $tType,B: $tType,A: $tType,F1: A > C,F2: B > D] :
( ( inj_on @ A @ C @ F1 @ ( top_top @ ( set @ A ) ) )
=> ( ( inj_on @ B @ D @ F2 @ ( top_top @ ( set @ B ) ) )
=> ( inj_on @ ( product_prod @ A @ B ) @ ( product_prod @ C @ D ) @ ( product_map_prod @ A @ C @ B @ D @ F1 @ F2 ) @ ( top_top @ ( set @ ( product_prod @ A @ B ) ) ) ) ) ) ).
% prod.inj_map
thf(fact_132_ssubst__Pair__rhs,axiom,
! [B: $tType,A: $tType,R: A,S3: B,R2: set @ ( product_prod @ A @ B ),S4: B] :
( ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ R @ S3 ) @ R2 )
=> ( ( S4 = S3 )
=> ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ R @ S4 ) @ R2 ) ) ) ).
% ssubst_Pair_rhs
thf(fact_133_curry__conv,axiom,
! [A: $tType,C: $tType,B: $tType] :
( ( product_curry @ B @ C @ A )
= ( ^ [F7: ( product_prod @ B @ C ) > A,A7: B,B5: C] : ( F7 @ ( product_Pair @ B @ C @ A7 @ B5 ) ) ) ) ).
% curry_conv
thf(fact_134_curryI,axiom,
! [A: $tType,B: $tType,F: ( product_prod @ A @ B ) > $o,A2: A,B2: B] :
( ( F @ ( product_Pair @ A @ B @ A2 @ B2 ) )
=> ( product_curry @ A @ B @ $o @ F @ A2 @ B2 ) ) ).
% curryI
thf(fact_135_map__prod__simp,axiom,
! [C: $tType,A: $tType,B: $tType,D: $tType,F: C > A,G: D > B,A2: C,B2: D] :
( ( product_map_prod @ C @ A @ D @ B @ F @ G @ ( product_Pair @ C @ D @ A2 @ B2 ) )
= ( product_Pair @ A @ B @ ( F @ A2 ) @ ( G @ B2 ) ) ) ).
% map_prod_simp
thf(fact_136_curryD,axiom,
! [A: $tType,B: $tType,F: ( product_prod @ A @ B ) > $o,A2: A,B2: B] :
( ( product_curry @ A @ B @ $o @ F @ A2 @ B2 )
=> ( F @ ( product_Pair @ A @ B @ A2 @ B2 ) ) ) ).
% curryD
thf(fact_137_curryE,axiom,
! [A: $tType,B: $tType,F: ( product_prod @ A @ B ) > $o,A2: A,B2: B] :
( ( product_curry @ A @ B @ $o @ F @ A2 @ B2 )
=> ( F @ ( product_Pair @ A @ B @ A2 @ B2 ) ) ) ).
% curryE
thf(fact_138_change__tv_Ocases,axiom,
! [X: product_prod @ ( nat > nat ) @ paraco415392788lle_tv] :
( ! [F8: nat > nat,B3: $o] :
( X
!= ( product_Pair @ ( nat > nat ) @ paraco415392788lle_tv @ F8 @ ( paraco2040174112le_Det @ B3 ) ) )
=> ~ ! [F8: nat > nat,N: nat] :
( X
!= ( product_Pair @ ( nat > nat ) @ paraco415392788lle_tv @ F8 @ ( paraco676387099_Indet @ N ) ) ) ) ).
% change_tv.cases
thf(fact_139_in__inv__imagep,axiom,
! [B: $tType,A: $tType] :
( ( inv_imagep @ A @ B )
= ( ^ [R3: A > A > $o,F7: B > A,X4: B,Y4: B] : ( R3 @ ( F7 @ X4 ) @ ( F7 @ Y4 ) ) ) ) ).
% in_inv_imagep
thf(fact_140_the__inv__f__f,axiom,
! [B: $tType,A: $tType,F: A > B,X: A] :
( ( inj_on @ A @ B @ F @ ( top_top @ ( set @ A ) ) )
=> ( ( the_inv_into @ A @ B @ ( top_top @ ( set @ A ) ) @ F @ ( F @ X ) )
= X ) ) ).
% the_inv_f_f
thf(fact_141_change__tv_Opelims,axiom,
! [X: nat > nat,Xa: paraco415392788lle_tv,Y: paraco415392788lle_tv] :
( ( ( paraco1920534163nge_tv @ X @ Xa )
= Y )
=> ( ( accp @ ( product_prod @ ( nat > nat ) @ paraco415392788lle_tv ) @ paraco2077297190tv_rel @ ( product_Pair @ ( nat > nat ) @ paraco415392788lle_tv @ X @ Xa ) )
=> ( ! [B3: $o] :
( ( Xa
= ( paraco2040174112le_Det @ B3 ) )
=> ( ( Y
= ( paraco2040174112le_Det @ B3 ) )
=> ~ ( accp @ ( product_prod @ ( nat > nat ) @ paraco415392788lle_tv ) @ paraco2077297190tv_rel @ ( product_Pair @ ( nat > nat ) @ paraco415392788lle_tv @ X @ ( paraco2040174112le_Det @ B3 ) ) ) ) )
=> ~ ! [N: nat] :
( ( Xa
= ( paraco676387099_Indet @ N ) )
=> ( ( Y
= ( paraco676387099_Indet @ ( X @ N ) ) )
=> ~ ( accp @ ( product_prod @ ( nat > nat ) @ paraco415392788lle_tv ) @ paraco2077297190tv_rel @ ( product_Pair @ ( nat > nat ) @ paraco415392788lle_tv @ X @ ( paraco676387099_Indet @ N ) ) ) ) ) ) ) ) ).
% change_tv.pelims
thf(fact_142_in__inv__image,axiom,
! [A: $tType,B: $tType,X: A,Y: A,R: set @ ( product_prod @ B @ B ),F: A > B] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X @ Y ) @ ( inv_image @ B @ A @ R @ F ) )
= ( member @ ( product_prod @ B @ B ) @ ( product_Pair @ B @ B @ ( F @ X ) @ ( F @ Y ) ) @ R ) ) ).
% in_inv_image
thf(fact_143_the__inv__into__f__f,axiom,
! [B: $tType,A: $tType,F: A > B,A3: set @ A,X: A] :
( ( inj_on @ A @ B @ F @ A3 )
=> ( ( member @ A @ X @ A3 )
=> ( ( the_inv_into @ A @ B @ A3 @ F @ ( F @ X ) )
= X ) ) ) ).
% the_inv_into_f_f
thf(fact_144_the__inv__into__f__eq,axiom,
! [B: $tType,A: $tType,F: A > B,A3: set @ A,X: A,Y: B] :
( ( inj_on @ A @ B @ F @ A3 )
=> ( ( ( F @ X )
= Y )
=> ( ( member @ A @ X @ A3 )
=> ( ( the_inv_into @ A @ B @ A3 @ F @ Y )
= X ) ) ) ) ).
% the_inv_into_f_eq
thf(fact_145_total__inv__image,axiom,
! [B: $tType,A: $tType,F: A > B,R: set @ ( product_prod @ B @ B )] :
( ( inj_on @ A @ B @ F @ ( top_top @ ( set @ A ) ) )
=> ( ( total_on @ B @ ( top_top @ ( set @ B ) ) @ R )
=> ( total_on @ A @ ( top_top @ ( set @ A ) ) @ ( inv_image @ B @ A @ R @ F ) ) ) ) ).
% total_inv_image
thf(fact_146_the__inv__into__onto,axiom,
! [B: $tType,A: $tType,F: A > B,A3: set @ A] :
( ( inj_on @ A @ B @ F @ A3 )
=> ( ( image @ B @ A @ ( the_inv_into @ A @ B @ A3 @ F ) @ ( image @ A @ B @ F @ A3 ) )
= A3 ) ) ).
% the_inv_into_onto
thf(fact_147_map__prod__imageI,axiom,
! [D: $tType,C: $tType,B: $tType,A: $tType,A2: A,B2: B,R2: set @ ( product_prod @ A @ B ),F: A > C,G: B > D] :
( ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ A2 @ B2 ) @ R2 )
=> ( member @ ( product_prod @ C @ D ) @ ( product_Pair @ C @ D @ ( F @ A2 ) @ ( G @ B2 ) ) @ ( image @ ( product_prod @ A @ B ) @ ( product_prod @ C @ D ) @ ( product_map_prod @ A @ C @ B @ D @ F @ G ) @ R2 ) ) ) ).
% map_prod_imageI
thf(fact_148_inj__swap,axiom,
! [B: $tType,A: $tType,A3: set @ ( product_prod @ A @ B )] : ( inj_on @ ( product_prod @ A @ B ) @ ( product_prod @ B @ A ) @ ( product_swap @ A @ B ) @ A3 ) ).
% inj_swap
thf(fact_149_image__eqI,axiom,
! [A: $tType,B: $tType,B2: A,F: B > A,X: B,A3: set @ B] :
( ( B2
= ( F @ X ) )
=> ( ( member @ B @ X @ A3 )
=> ( member @ A @ B2 @ ( image @ B @ A @ F @ A3 ) ) ) ) ).
% image_eqI
thf(fact_150_swap__swap,axiom,
! [B: $tType,A: $tType,P: product_prod @ A @ B] :
( ( product_swap @ B @ A @ ( product_swap @ A @ B @ P ) )
= P ) ).
% swap_swap
thf(fact_151_swap__simp,axiom,
! [A: $tType,B: $tType,X: B,Y: A] :
( ( product_swap @ B @ A @ ( product_Pair @ B @ A @ X @ Y ) )
= ( product_Pair @ A @ B @ Y @ X ) ) ).
% swap_simp
thf(fact_152_pair__in__swap__image,axiom,
! [A: $tType,B: $tType,Y: A,X: B,A3: set @ ( product_prod @ B @ A )] :
( ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ Y @ X ) @ ( image @ ( product_prod @ B @ A ) @ ( product_prod @ A @ B ) @ ( product_swap @ B @ A ) @ A3 ) )
= ( member @ ( product_prod @ B @ A ) @ ( product_Pair @ B @ A @ X @ Y ) @ A3 ) ) ).
% pair_in_swap_image
thf(fact_153_surj__swap,axiom,
! [B: $tType,A: $tType] :
( ( image @ ( product_prod @ B @ A ) @ ( product_prod @ A @ B ) @ ( product_swap @ B @ A ) @ ( top_top @ ( set @ ( product_prod @ B @ A ) ) ) )
= ( top_top @ ( set @ ( product_prod @ A @ B ) ) ) ) ).
% surj_swap
thf(fact_154_total__on__def,axiom,
! [A: $tType] :
( ( total_on @ A )
= ( ^ [A4: set @ A,R3: set @ ( product_prod @ A @ A )] :
! [X4: A] :
( ( member @ A @ X4 @ A4 )
=> ! [Y4: A] :
( ( member @ A @ Y4 @ A4 )
=> ( ( X4 != Y4 )
=> ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X4 @ Y4 ) @ R3 )
| ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Y4 @ X4 ) @ R3 ) ) ) ) ) ) ) ).
% total_on_def
thf(fact_155_total__onI,axiom,
! [A: $tType,A3: set @ A,R: set @ ( product_prod @ A @ A )] :
( ! [X5: A,Y5: A] :
( ( member @ A @ X5 @ A3 )
=> ( ( member @ A @ Y5 @ A3 )
=> ( ( X5 != Y5 )
=> ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X5 @ Y5 ) @ R )
| ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Y5 @ X5 ) @ R ) ) ) ) )
=> ( total_on @ A @ A3 @ R ) ) ).
% total_onI
thf(fact_156_map__prod__surj,axiom,
! [A: $tType,C: $tType,D: $tType,B: $tType,F: A > B,G: C > D] :
( ( ( image @ A @ B @ F @ ( top_top @ ( set @ A ) ) )
= ( top_top @ ( set @ B ) ) )
=> ( ( ( image @ C @ D @ G @ ( top_top @ ( set @ C ) ) )
= ( top_top @ ( set @ D ) ) )
=> ( ( image @ ( product_prod @ A @ C ) @ ( product_prod @ B @ D ) @ ( product_map_prod @ A @ B @ C @ D @ F @ G ) @ ( top_top @ ( set @ ( product_prod @ A @ C ) ) ) )
= ( top_top @ ( set @ ( product_prod @ B @ D ) ) ) ) ) ) ).
% map_prod_surj
thf(fact_157_range__eqI,axiom,
! [A: $tType,B: $tType,B2: A,F: B > A,X: B] :
( ( B2
= ( F @ X ) )
=> ( member @ A @ B2 @ ( image @ B @ A @ F @ ( top_top @ ( set @ B ) ) ) ) ) ).
% range_eqI
thf(fact_158_surj__def,axiom,
! [B: $tType,A: $tType,F: B > A] :
( ( ( image @ B @ A @ F @ ( top_top @ ( set @ B ) ) )
= ( top_top @ ( set @ A ) ) )
= ( ! [Y4: A] :
? [X4: B] :
( Y4
= ( F @ X4 ) ) ) ) ).
% surj_def
thf(fact_159_rangeI,axiom,
! [A: $tType,B: $tType,F: B > A,X: B] : ( member @ A @ ( F @ X ) @ ( image @ B @ A @ F @ ( top_top @ ( set @ B ) ) ) ) ).
% rangeI
thf(fact_160_surjI,axiom,
! [B: $tType,A: $tType,G: B > A,F: A > B] :
( ! [X5: A] :
( ( G @ ( F @ X5 ) )
= X5 )
=> ( ( image @ B @ A @ G @ ( top_top @ ( set @ B ) ) )
= ( top_top @ ( set @ A ) ) ) ) ).
% surjI
thf(fact_161_surjE,axiom,
! [A: $tType,B: $tType,F: B > A,Y: A] :
( ( ( image @ B @ A @ F @ ( top_top @ ( set @ B ) ) )
= ( top_top @ ( set @ A ) ) )
=> ~ ! [X5: B] :
( Y
!= ( F @ X5 ) ) ) ).
% surjE
thf(fact_162_surjD,axiom,
! [A: $tType,B: $tType,F: B > A,Y: A] :
( ( ( image @ B @ A @ F @ ( top_top @ ( set @ B ) ) )
= ( top_top @ ( set @ A ) ) )
=> ? [X5: B] :
( Y
= ( F @ X5 ) ) ) ).
% surjD
thf(fact_163_inj__on__image__iff,axiom,
! [B: $tType,A: $tType,A3: set @ A,G: A > B,F: A > A] :
( ! [X5: A] :
( ( member @ A @ X5 @ A3 )
=> ! [Xa2: A] :
( ( member @ A @ Xa2 @ A3 )
=> ( ( ( G @ ( F @ X5 ) )
= ( G @ ( F @ Xa2 ) ) )
= ( ( G @ X5 )
= ( G @ Xa2 ) ) ) ) )
=> ( ( inj_on @ A @ A @ F @ A3 )
=> ( ( inj_on @ A @ B @ G @ ( image @ A @ A @ F @ A3 ) )
= ( inj_on @ A @ B @ G @ A3 ) ) ) ) ).
% inj_on_image_iff
thf(fact_164_imageI,axiom,
! [B: $tType,A: $tType,X: A,A3: set @ A,F: A > B] :
( ( member @ A @ X @ A3 )
=> ( member @ B @ ( F @ X ) @ ( image @ A @ B @ F @ A3 ) ) ) ).
% imageI
thf(fact_165_image__iff,axiom,
! [A: $tType,B: $tType,Z: A,F: B > A,A3: set @ B] :
( ( member @ A @ Z @ ( image @ B @ A @ F @ A3 ) )
= ( ? [X4: B] :
( ( member @ B @ X4 @ A3 )
& ( Z
= ( F @ X4 ) ) ) ) ) ).
% image_iff
thf(fact_166_bex__imageD,axiom,
! [A: $tType,B: $tType,F: B > A,A3: set @ B,P3: A > $o] :
( ? [X6: A] :
( ( member @ A @ X6 @ ( image @ B @ A @ F @ A3 ) )
& ( P3 @ X6 ) )
=> ? [X5: B] :
( ( member @ B @ X5 @ A3 )
& ( P3 @ ( F @ X5 ) ) ) ) ).
% bex_imageD
thf(fact_167_image__cong,axiom,
! [B: $tType,A: $tType,M: set @ A,N3: set @ A,F: A > B,G: A > B] :
( ( M = N3 )
=> ( ! [X5: A] :
( ( member @ A @ X5 @ N3 )
=> ( ( F @ X5 )
= ( G @ X5 ) ) )
=> ( ( image @ A @ B @ F @ M )
= ( image @ A @ B @ G @ N3 ) ) ) ) ).
% image_cong
thf(fact_168_ball__imageD,axiom,
! [A: $tType,B: $tType,F: B > A,A3: set @ B,P3: A > $o] :
( ! [X5: A] :
( ( member @ A @ X5 @ ( image @ B @ A @ F @ A3 ) )
=> ( P3 @ X5 ) )
=> ! [X6: B] :
( ( member @ B @ X6 @ A3 )
=> ( P3 @ ( F @ X6 ) ) ) ) ).
% ball_imageD
thf(fact_169_rev__image__eqI,axiom,
! [B: $tType,A: $tType,X: A,A3: set @ A,B2: B,F: A > B] :
( ( member @ A @ X @ A3 )
=> ( ( B2
= ( F @ X ) )
=> ( member @ B @ B2 @ ( image @ A @ B @ F @ A3 ) ) ) ) ).
% rev_image_eqI
thf(fact_170_inj__image__mem__iff,axiom,
! [B: $tType,A: $tType,F: A > B,A2: A,A3: set @ A] :
( ( inj_on @ A @ B @ F @ ( top_top @ ( set @ A ) ) )
=> ( ( member @ B @ ( F @ A2 ) @ ( image @ A @ B @ F @ A3 ) )
= ( member @ A @ A2 @ A3 ) ) ) ).
% inj_image_mem_iff
thf(fact_171_inj__image__eq__iff,axiom,
! [B: $tType,A: $tType,F: A > B,A3: set @ A,B6: set @ A] :
( ( inj_on @ A @ B @ F @ ( top_top @ ( set @ A ) ) )
=> ( ( ( image @ A @ B @ F @ A3 )
= ( image @ A @ B @ F @ B6 ) )
= ( A3 = B6 ) ) ) ).
% inj_image_eq_iff
thf(fact_172_range__ex1__eq,axiom,
! [B: $tType,A: $tType,F: A > B,B2: B] :
( ( inj_on @ A @ B @ F @ ( top_top @ ( set @ A ) ) )
=> ( ( member @ B @ B2 @ ( image @ A @ B @ F @ ( top_top @ ( set @ A ) ) ) )
= ( ? [X4: A] :
( ( B2
= ( F @ X4 ) )
& ! [Y4: A] :
( ( B2
= ( F @ Y4 ) )
=> ( Y4 = X4 ) ) ) ) ) ) ).
% range_ex1_eq
thf(fact_173_prod__fun__imageE,axiom,
! [B: $tType,A: $tType,D: $tType,C: $tType,C3: product_prod @ A @ B,F: C > A,G: D > B,R2: set @ ( product_prod @ C @ D )] :
( ( member @ ( product_prod @ A @ B ) @ C3 @ ( image @ ( product_prod @ C @ D ) @ ( product_prod @ A @ B ) @ ( product_map_prod @ C @ A @ D @ B @ F @ G ) @ R2 ) )
=> ~ ! [X5: C,Y5: D] :
( ( C3
= ( product_Pair @ A @ B @ ( F @ X5 ) @ ( G @ Y5 ) ) )
=> ~ ( member @ ( product_prod @ C @ D ) @ ( product_Pair @ C @ D @ X5 @ Y5 ) @ R2 ) ) ) ).
% prod_fun_imageE
thf(fact_174_inj__on__the__inv__into,axiom,
! [B: $tType,A: $tType,F: A > B,A3: set @ A] :
( ( inj_on @ A @ B @ F @ A3 )
=> ( inj_on @ B @ A @ ( the_inv_into @ A @ B @ A3 @ F ) @ ( image @ A @ B @ F @ A3 ) ) ) ).
% inj_on_the_inv_into
thf(fact_175_f__the__inv__into__f,axiom,
! [A: $tType,B: $tType,F: A > B,A3: set @ A,Y: B] :
( ( inj_on @ A @ B @ F @ A3 )
=> ( ( member @ B @ Y @ ( image @ A @ B @ F @ A3 ) )
=> ( ( F @ ( the_inv_into @ A @ B @ A3 @ F @ Y ) )
= Y ) ) ) ).
% f_the_inv_into_f
thf(fact_176_total__lex__prod,axiom,
! [A: $tType,B: $tType,R: set @ ( product_prod @ A @ A ),S3: set @ ( product_prod @ B @ B )] :
( ( total_on @ A @ ( top_top @ ( set @ A ) ) @ R )
=> ( ( total_on @ B @ ( top_top @ ( set @ B ) ) @ S3 )
=> ( total_on @ ( product_prod @ A @ B ) @ ( top_top @ ( set @ ( product_prod @ A @ B ) ) ) @ ( lex_prod @ A @ B @ R @ S3 ) ) ) ) ).
% total_lex_prod
thf(fact_177_range__fst,axiom,
! [B: $tType,A: $tType] :
( ( image @ ( product_prod @ A @ B ) @ A @ ( product_fst @ A @ B ) @ ( top_top @ ( set @ ( product_prod @ A @ B ) ) ) )
= ( top_top @ ( set @ A ) ) ) ).
% range_fst
thf(fact_178_range__snd,axiom,
! [B: $tType,A: $tType] :
( ( image @ ( product_prod @ B @ A ) @ A @ ( product_snd @ B @ A ) @ ( top_top @ ( set @ ( product_prod @ B @ A ) ) ) )
= ( top_top @ ( set @ A ) ) ) ).
% range_snd
thf(fact_179_fst__map__prod,axiom,
! [B: $tType,A: $tType,D: $tType,C: $tType,F: C > A,G: D > B,X: product_prod @ C @ D] :
( ( product_fst @ A @ B @ ( product_map_prod @ C @ A @ D @ B @ F @ G @ X ) )
= ( F @ ( product_fst @ C @ D @ X ) ) ) ).
% fst_map_prod
thf(fact_180_snd__map__prod,axiom,
! [B: $tType,A: $tType,D: $tType,C: $tType,F: C > B,G: D > A,X: product_prod @ C @ D] :
( ( product_snd @ B @ A @ ( product_map_prod @ C @ B @ D @ A @ F @ G @ X ) )
= ( G @ ( product_snd @ C @ D @ X ) ) ) ).
% snd_map_prod
thf(fact_181_fst__apfst,axiom,
! [A: $tType,B: $tType,C: $tType,F: C > A,X: product_prod @ C @ B] :
( ( product_fst @ A @ B @ ( product_apfst @ C @ A @ B @ F @ X ) )
= ( F @ ( product_fst @ C @ B @ X ) ) ) ).
% fst_apfst
thf(fact_182_apfst__eq__conv,axiom,
! [A: $tType,B: $tType,C: $tType,F: C > A,X: product_prod @ C @ B,G: C > A] :
( ( ( product_apfst @ C @ A @ B @ F @ X )
= ( product_apfst @ C @ A @ B @ G @ X ) )
= ( ( F @ ( product_fst @ C @ B @ X ) )
= ( G @ ( product_fst @ C @ B @ X ) ) ) ) ).
% apfst_eq_conv
thf(fact_183_snd__apfst,axiom,
! [B: $tType,A: $tType,C: $tType,F: C > B,X: product_prod @ C @ A] :
( ( product_snd @ B @ A @ ( product_apfst @ C @ B @ A @ F @ X ) )
= ( product_snd @ C @ A @ X ) ) ).
% snd_apfst
thf(fact_184_fst__apsnd,axiom,
! [B: $tType,C: $tType,A: $tType,F: C > B,X: product_prod @ A @ C] :
( ( product_fst @ A @ B @ ( product_apsnd @ C @ B @ A @ F @ X ) )
= ( product_fst @ A @ C @ X ) ) ).
% fst_apsnd
thf(fact_185_snd__apsnd,axiom,
! [A: $tType,C: $tType,B: $tType,F: C > A,X: product_prod @ B @ C] :
( ( product_snd @ B @ A @ ( product_apsnd @ C @ A @ B @ F @ X ) )
= ( F @ ( product_snd @ B @ C @ X ) ) ) ).
% snd_apsnd
thf(fact_186_apsnd__eq__conv,axiom,
! [B: $tType,C: $tType,A: $tType,F: C > B,X: product_prod @ A @ C,G: C > B] :
( ( ( product_apsnd @ C @ B @ A @ F @ X )
= ( product_apsnd @ C @ B @ A @ G @ X ) )
= ( ( F @ ( product_snd @ A @ C @ X ) )
= ( G @ ( product_snd @ A @ C @ X ) ) ) ) ).
% apsnd_eq_conv
thf(fact_187_in__lex__prod,axiom,
! [A: $tType,B: $tType,A2: A,B2: B,A6: A,B4: B,R: set @ ( product_prod @ A @ A ),S3: set @ ( product_prod @ B @ B )] :
( ( member @ ( product_prod @ ( product_prod @ A @ B ) @ ( product_prod @ A @ B ) ) @ ( product_Pair @ ( product_prod @ A @ B ) @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ A2 @ B2 ) @ ( product_Pair @ A @ B @ A6 @ B4 ) ) @ ( lex_prod @ A @ B @ R @ S3 ) )
= ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A2 @ A6 ) @ R )
| ( ( A2 = A6 )
& ( member @ ( product_prod @ B @ B ) @ ( product_Pair @ B @ B @ B2 @ B4 ) @ S3 ) ) ) ) ).
% in_lex_prod
thf(fact_188_prod_Ocollapse,axiom,
! [B: $tType,A: $tType,Prod: product_prod @ A @ B] :
( ( product_Pair @ A @ B @ ( product_fst @ A @ B @ Prod ) @ ( product_snd @ A @ B @ Prod ) )
= Prod ) ).
% prod.collapse
thf(fact_189_fst__swap,axiom,
! [A: $tType,B: $tType,X: product_prod @ B @ A] :
( ( product_fst @ A @ B @ ( product_swap @ B @ A @ X ) )
= ( product_snd @ B @ A @ X ) ) ).
% fst_swap
thf(fact_190_snd__swap,axiom,
! [B: $tType,A: $tType,X: product_prod @ A @ B] :
( ( product_snd @ B @ A @ ( product_swap @ A @ B @ X ) )
= ( product_fst @ A @ B @ X ) ) ).
% snd_swap
thf(fact_191_apsnd__apfst,axiom,
! [A: $tType,B: $tType,C: $tType,D: $tType,F: C > B,G: D > A,X: product_prod @ D @ C] :
( ( product_apsnd @ C @ B @ A @ F @ ( product_apfst @ D @ A @ C @ G @ X ) )
= ( product_Pair @ A @ B @ ( G @ ( product_fst @ D @ C @ X ) ) @ ( F @ ( product_snd @ D @ C @ X ) ) ) ) ).
% apsnd_apfst
thf(fact_192_apfst__apsnd,axiom,
! [A: $tType,B: $tType,D: $tType,C: $tType,F: C > A,G: D > B,X: product_prod @ C @ D] :
( ( product_apfst @ C @ A @ B @ F @ ( product_apsnd @ D @ B @ C @ G @ X ) )
= ( product_Pair @ A @ B @ ( F @ ( product_fst @ C @ D @ X ) ) @ ( G @ ( product_snd @ C @ D @ X ) ) ) ) ).
% apfst_apsnd
thf(fact_193_prod_Oswap__def,axiom,
! [B: $tType,A: $tType] :
( ( product_swap @ A @ B )
= ( ^ [P2: product_prod @ A @ B] : ( product_Pair @ B @ A @ ( product_snd @ A @ B @ P2 ) @ ( product_fst @ A @ B @ P2 ) ) ) ) ).
% prod.swap_def
thf(fact_194_prod__eq__iff,axiom,
! [B: $tType,A: $tType] :
( ( ^ [Y6: product_prod @ A @ B,Z2: product_prod @ A @ B] : Y6 = Z2 )
= ( ^ [S: product_prod @ A @ B,T2: product_prod @ A @ B] :
( ( ( product_fst @ A @ B @ S )
= ( product_fst @ A @ B @ T2 ) )
& ( ( product_snd @ A @ B @ S )
= ( product_snd @ A @ B @ T2 ) ) ) ) ) ).
% prod_eq_iff
thf(fact_195_prod_Oexpand,axiom,
! [B: $tType,A: $tType,Prod: product_prod @ A @ B,Prod2: product_prod @ A @ B] :
( ( ( ( product_fst @ A @ B @ Prod )
= ( product_fst @ A @ B @ Prod2 ) )
& ( ( product_snd @ A @ B @ Prod )
= ( product_snd @ A @ B @ Prod2 ) ) )
=> ( Prod = Prod2 ) ) ).
% prod.expand
thf(fact_196_prod__eqI,axiom,
! [B: $tType,A: $tType,P: product_prod @ A @ B,Q: product_prod @ A @ B] :
( ( ( product_fst @ A @ B @ P )
= ( product_fst @ A @ B @ Q ) )
=> ( ( ( product_snd @ A @ B @ P )
= ( product_snd @ A @ B @ Q ) )
=> ( P = Q ) ) ) ).
% prod_eqI
thf(fact_197_fst__eqD,axiom,
! [B: $tType,A: $tType,X: A,Y: B,A2: A] :
( ( ( product_fst @ A @ B @ ( product_Pair @ A @ B @ X @ Y ) )
= A2 )
=> ( X = A2 ) ) ).
% fst_eqD
thf(fact_198_snd__eqD,axiom,
! [B: $tType,A: $tType,X: B,Y: A,A2: A] :
( ( ( product_snd @ B @ A @ ( product_Pair @ B @ A @ X @ Y ) )
= A2 )
=> ( Y = A2 ) ) ).
% snd_eqD
thf(fact_199_fst__conv,axiom,
! [B: $tType,A: $tType,X1: A,X2: B] :
( ( product_fst @ A @ B @ ( product_Pair @ A @ B @ X1 @ X2 ) )
= X1 ) ).
% fst_conv
thf(fact_200_snd__conv,axiom,
! [Aa: $tType,A: $tType,X1: Aa,X2: A] :
( ( product_snd @ Aa @ A @ ( product_Pair @ Aa @ A @ X1 @ X2 ) )
= X2 ) ).
% snd_conv
thf(fact_201_prod_Oexhaust__sel,axiom,
! [B: $tType,A: $tType,Prod: product_prod @ A @ B] :
( Prod
= ( product_Pair @ A @ B @ ( product_fst @ A @ B @ Prod ) @ ( product_snd @ A @ B @ Prod ) ) ) ).
% prod.exhaust_sel
thf(fact_202_surjective__pairing,axiom,
! [B: $tType,A: $tType,T3: product_prod @ A @ B] :
( T3
= ( product_Pair @ A @ B @ ( product_fst @ A @ B @ T3 ) @ ( product_snd @ A @ B @ T3 ) ) ) ).
% surjective_pairing
thf(fact_203_BNF__Greatest__Fixpoint_Osubst__Pair,axiom,
! [B: $tType,A: $tType,P3: A > B > $o,X: A,Y: B,A2: product_prod @ A @ B] :
( ( P3 @ X @ Y )
=> ( ( A2
= ( product_Pair @ A @ B @ X @ Y ) )
=> ( P3 @ ( product_fst @ A @ B @ A2 ) @ ( product_snd @ A @ B @ A2 ) ) ) ) ).
% BNF_Greatest_Fixpoint.subst_Pair
thf(fact_204_conjI__realizer,axiom,
! [A: $tType,B: $tType,P3: A > $o,P: A,Q2: B > $o,Q: B] :
( ( P3 @ P )
=> ( ( Q2 @ Q )
=> ( ( P3 @ ( product_fst @ A @ B @ ( product_Pair @ A @ B @ P @ Q ) ) )
& ( Q2 @ ( product_snd @ A @ B @ ( product_Pair @ A @ B @ P @ Q ) ) ) ) ) ) ).
% conjI_realizer
thf(fact_205_exI__realizer,axiom,
! [B: $tType,A: $tType,P3: A > B > $o,Y: A,X: B] :
( ( P3 @ Y @ X )
=> ( P3 @ ( product_snd @ B @ A @ ( product_Pair @ B @ A @ X @ Y ) ) @ ( product_fst @ B @ A @ ( product_Pair @ B @ A @ X @ Y ) ) ) ) ).
% exI_realizer
thf(fact_206_sndI,axiom,
! [A: $tType,B: $tType,X: product_prod @ A @ B,Y: A,Z: B] :
( ( X
= ( product_Pair @ A @ B @ Y @ Z ) )
=> ( ( product_snd @ A @ B @ X )
= Z ) ) ).
% sndI
thf(fact_207_eq__snd__iff,axiom,
! [A: $tType,B: $tType,B2: A,P: product_prod @ B @ A] :
( ( B2
= ( product_snd @ B @ A @ P ) )
= ( ? [A7: B] :
( P
= ( product_Pair @ B @ A @ A7 @ B2 ) ) ) ) ).
% eq_snd_iff
thf(fact_208_eq__fst__iff,axiom,
! [A: $tType,B: $tType,A2: A,P: product_prod @ A @ B] :
( ( A2
= ( product_fst @ A @ B @ P ) )
= ( ? [B5: B] :
( P
= ( product_Pair @ A @ B @ A2 @ B5 ) ) ) ) ).
% eq_fst_iff
thf(fact_209_fstI,axiom,
! [B: $tType,A: $tType,X: product_prod @ A @ B,Y: A,Z: B] :
( ( X
= ( product_Pair @ A @ B @ Y @ Z ) )
=> ( ( product_fst @ A @ B @ X )
= Y ) ) ).
% fstI
thf(fact_210_same__fstI,axiom,
! [B: $tType,A: $tType,P3: A > $o,X: A,Y7: B,Y: B,R2: A > ( set @ ( product_prod @ B @ B ) )] :
( ( P3 @ X )
=> ( ( member @ ( product_prod @ B @ B ) @ ( product_Pair @ B @ B @ Y7 @ Y ) @ ( R2 @ X ) )
=> ( member @ ( product_prod @ ( product_prod @ A @ B ) @ ( product_prod @ A @ B ) ) @ ( product_Pair @ ( product_prod @ A @ B ) @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X @ Y7 ) @ ( product_Pair @ A @ B @ X @ Y ) ) @ ( same_fst @ A @ B @ P3 @ R2 ) ) ) ) ).
% same_fstI
thf(fact_211_wf__map__prod__image,axiom,
! [B: $tType,A: $tType,R: set @ ( product_prod @ A @ A ),F: A > B] :
( ( wf @ A @ R )
=> ( ( inj_on @ A @ B @ F @ ( top_top @ ( set @ A ) ) )
=> ( wf @ B @ ( image @ ( product_prod @ A @ A ) @ ( product_prod @ B @ B ) @ ( product_map_prod @ A @ B @ A @ B @ F @ F ) @ R ) ) ) ) ).
% wf_map_prod_image
thf(fact_212_wf__def,axiom,
! [A: $tType] :
( ( wf @ A )
= ( ^ [R3: set @ ( product_prod @ A @ A )] :
! [P5: A > $o] :
( ! [X4: A] :
( ! [Y4: A] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Y4 @ X4 ) @ R3 )
=> ( P5 @ Y4 ) )
=> ( P5 @ X4 ) )
=> ! [X7: A] : ( P5 @ X7 ) ) ) ) ).
% wf_def
thf(fact_213_wfE__min,axiom,
! [A: $tType,R2: set @ ( product_prod @ A @ A ),X: A,Q2: set @ A] :
( ( wf @ A @ R2 )
=> ( ( member @ A @ X @ Q2 )
=> ~ ! [Z3: A] :
( ( member @ A @ Z3 @ Q2 )
=> ~ ! [Y8: A] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Y8 @ Z3 ) @ R2 )
=> ~ ( member @ A @ Y8 @ Q2 ) ) ) ) ) ).
% wfE_min
thf(fact_214_wfI__min,axiom,
! [A: $tType,R2: set @ ( product_prod @ A @ A )] :
( ! [X5: A,Q4: set @ A] :
( ( member @ A @ X5 @ Q4 )
=> ? [Xa3: A] :
( ( member @ A @ Xa3 @ Q4 )
& ! [Y5: A] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Y5 @ Xa3 ) @ R2 )
=> ~ ( member @ A @ Y5 @ Q4 ) ) ) )
=> ( wf @ A @ R2 ) ) ).
% wfI_min
thf(fact_215_wfUNIVI,axiom,
! [A: $tType,R: set @ ( product_prod @ A @ A )] :
( ! [P6: A > $o,X5: A] :
( ! [Xa3: A] :
( ! [Y5: A] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Y5 @ Xa3 ) @ R )
=> ( P6 @ Y5 ) )
=> ( P6 @ Xa3 ) )
=> ( P6 @ X5 ) )
=> ( wf @ A @ R ) ) ).
% wfUNIVI
thf(fact_216_wf__asym,axiom,
! [A: $tType,R: set @ ( product_prod @ A @ A ),A2: A,X: A] :
( ( wf @ A @ R )
=> ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A2 @ X ) @ R )
=> ~ ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X @ A2 ) @ R ) ) ) ).
% wf_asym
thf(fact_217_wf__induct,axiom,
! [A: $tType,R: set @ ( product_prod @ A @ A ),P3: A > $o,A2: A] :
( ( wf @ A @ R )
=> ( ! [X5: A] :
( ! [Y8: A] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Y8 @ X5 ) @ R )
=> ( P3 @ Y8 ) )
=> ( P3 @ X5 ) )
=> ( P3 @ A2 ) ) ) ).
% wf_induct
thf(fact_218_wf__irrefl,axiom,
! [A: $tType,R: set @ ( product_prod @ A @ A ),A2: A] :
( ( wf @ A @ R )
=> ~ ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A2 @ A2 ) @ R ) ) ).
% wf_irrefl
thf(fact_219_wf__not__sym,axiom,
! [A: $tType,R: set @ ( product_prod @ A @ A ),A2: A,X: A] :
( ( wf @ A @ R )
=> ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A2 @ X ) @ R )
=> ~ ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X @ A2 ) @ R ) ) ) ).
% wf_not_sym
thf(fact_220_wf__not__refl,axiom,
! [A: $tType,R: set @ ( product_prod @ A @ A ),A2: A] :
( ( wf @ A @ R )
=> ~ ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A2 @ A2 ) @ R ) ) ).
% wf_not_refl
thf(fact_221_wf__eq__minimal,axiom,
! [A: $tType] :
( ( wf @ A )
= ( ^ [R3: set @ ( product_prod @ A @ A )] :
! [Q5: set @ A] :
( ? [X4: A] : ( member @ A @ X4 @ Q5 )
=> ? [X4: A] :
( ( member @ A @ X4 @ Q5 )
& ! [Y4: A] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Y4 @ X4 ) @ R3 )
=> ~ ( member @ A @ Y4 @ Q5 ) ) ) ) ) ) ).
% wf_eq_minimal
thf(fact_222_wf__induct__rule,axiom,
! [A: $tType,R: set @ ( product_prod @ A @ A ),P3: A > $o,A2: A] :
( ( wf @ A @ R )
=> ( ! [X5: A] :
( ! [Y8: A] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Y8 @ X5 ) @ R )
=> ( P3 @ Y8 ) )
=> ( P3 @ X5 ) )
=> ( P3 @ A2 ) ) ) ).
% wf_induct_rule
thf(fact_223_dependent__wf__choice,axiom,
! [B: $tType,A: $tType,R2: set @ ( product_prod @ A @ A ),P3: ( A > B ) > A > B > $o] :
( ( wf @ A @ R2 )
=> ( ! [F8: A > B,G3: A > B,X5: A,R4: B] :
( ! [Z4: A] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Z4 @ X5 ) @ R2 )
=> ( ( F8 @ Z4 )
= ( G3 @ Z4 ) ) )
=> ( ( P3 @ F8 @ X5 @ R4 )
= ( P3 @ G3 @ X5 @ R4 ) ) )
=> ( ! [X5: A,F8: A > B] :
( ! [Y8: A] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Y8 @ X5 ) @ R2 )
=> ( P3 @ F8 @ Y8 @ ( F8 @ Y8 ) ) )
=> ? [X_1: B] : ( P3 @ F8 @ X5 @ X_1 ) )
=> ? [F8: A > B] :
! [X6: A] : ( P3 @ F8 @ X6 @ ( F8 @ X6 ) ) ) ) ) ).
% dependent_wf_choice
thf(fact_224_sndOp__def,axiom,
! [B: $tType,A: $tType,C: $tType] :
( ( bNF_sndOp @ C @ A @ B )
= ( ^ [P5: C > A > $o,Q5: A > B > $o,Ac: product_prod @ C @ B] : ( product_Pair @ A @ B @ ( bNF_pick_middlep @ C @ A @ B @ P5 @ Q5 @ ( product_fst @ C @ B @ Ac ) @ ( product_snd @ C @ B @ Ac ) ) @ ( product_snd @ C @ B @ Ac ) ) ) ) ).
% sndOp_def
thf(fact_225_fstOp__def,axiom,
! [C: $tType,B: $tType,A: $tType] :
( ( bNF_fstOp @ A @ B @ C )
= ( ^ [P5: A > B > $o,Q5: B > C > $o,Ac: product_prod @ A @ C] : ( product_Pair @ A @ B @ ( product_fst @ A @ C @ Ac ) @ ( bNF_pick_middlep @ A @ B @ C @ P5 @ Q5 @ ( product_fst @ A @ C @ Ac ) @ ( product_snd @ A @ C @ Ac ) ) ) ) ) ).
% fstOp_def
thf(fact_226_image2__eqI,axiom,
! [A: $tType,C: $tType,B: $tType,B2: A,F: B > A,X: B,C3: C,G: B > C,A3: set @ B] :
( ( B2
= ( F @ X ) )
=> ( ( C3
= ( G @ X ) )
=> ( ( member @ B @ X @ A3 )
=> ( member @ ( product_prod @ A @ C ) @ ( product_Pair @ A @ C @ B2 @ C3 ) @ ( bNF_Greatest_image2 @ B @ A @ C @ A3 @ F @ G ) ) ) ) ) ).
% image2_eqI
thf(fact_227_the__inv__into__into,axiom,
! [B: $tType,A: $tType,F: A > B,A3: set @ A,X: B,B6: set @ A] :
( ( inj_on @ A @ B @ F @ A3 )
=> ( ( member @ B @ X @ ( image @ A @ B @ F @ A3 ) )
=> ( ( ord_less_eq @ ( set @ A ) @ A3 @ B6 )
=> ( member @ A @ ( the_inv_into @ A @ B @ A3 @ F @ X ) @ B6 ) ) ) ) ).
% the_inv_into_into
thf(fact_228_strict__mono__inv,axiom,
! [A: $tType,B: $tType] :
( ( ( linorder @ B )
& ( linorder @ A ) )
=> ! [F: A > B,G: B > A] :
( ( order_strict_mono @ A @ B @ F )
=> ( ( ( image @ A @ B @ F @ ( top_top @ ( set @ A ) ) )
= ( top_top @ ( set @ B ) ) )
=> ( ! [X5: A] :
( ( G @ ( F @ X5 ) )
= X5 )
=> ( order_strict_mono @ B @ A @ G ) ) ) ) ) ).
% strict_mono_inv
thf(fact_229_order__refl,axiom,
! [A: $tType] :
( ( preorder @ A )
=> ! [X: A] : ( ord_less_eq @ A @ X @ X ) ) ).
% order_refl
thf(fact_230_subset__antisym,axiom,
! [A: $tType,A3: set @ A,B6: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ A3 @ B6 )
=> ( ( ord_less_eq @ ( set @ A ) @ B6 @ A3 )
=> ( A3 = B6 ) ) ) ).
% subset_antisym
thf(fact_231_subsetI,axiom,
! [A: $tType,A3: set @ A,B6: set @ A] :
( ! [X5: A] :
( ( member @ A @ X5 @ A3 )
=> ( member @ A @ X5 @ B6 ) )
=> ( ord_less_eq @ ( set @ A ) @ A3 @ B6 ) ) ).
% subsetI
thf(fact_232_subrelI,axiom,
! [B: $tType,A: $tType,R: set @ ( product_prod @ A @ B ),S3: set @ ( product_prod @ A @ B )] :
( ! [X5: A,Y5: B] :
( ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X5 @ Y5 ) @ R )
=> ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X5 @ Y5 ) @ S3 ) )
=> ( ord_less_eq @ ( set @ ( product_prod @ A @ B ) ) @ R @ S3 ) ) ).
% subrelI
thf(fact_233_inj__on__image__mem__iff,axiom,
! [B: $tType,A: $tType,F: A > B,B6: set @ A,A2: A,A3: set @ A] :
( ( inj_on @ A @ B @ F @ B6 )
=> ( ( member @ A @ A2 @ B6 )
=> ( ( ord_less_eq @ ( set @ A ) @ A3 @ B6 )
=> ( ( member @ B @ ( F @ A2 ) @ ( image @ A @ B @ F @ A3 ) )
= ( member @ A @ A2 @ A3 ) ) ) ) ) ).
% inj_on_image_mem_iff
thf(fact_234_inj__on__image__eq__iff,axiom,
! [B: $tType,A: $tType,F: A > B,C4: set @ A,A3: set @ A,B6: set @ A] :
( ( inj_on @ A @ B @ F @ C4 )
=> ( ( ord_less_eq @ ( set @ A ) @ A3 @ C4 )
=> ( ( ord_less_eq @ ( set @ A ) @ B6 @ C4 )
=> ( ( ( image @ A @ B @ F @ A3 )
= ( image @ A @ B @ F @ B6 ) )
= ( A3 = B6 ) ) ) ) ) ).
% inj_on_image_eq_iff
thf(fact_235_inj__on__subset,axiom,
! [B: $tType,A: $tType,F: A > B,A3: set @ A,B6: set @ A] :
( ( inj_on @ A @ B @ F @ A3 )
=> ( ( ord_less_eq @ ( set @ A ) @ B6 @ A3 )
=> ( inj_on @ A @ B @ F @ B6 ) ) ) ).
% inj_on_subset
thf(fact_236_subset__inj__on,axiom,
! [B: $tType,A: $tType,F: A > B,B6: set @ A,A3: set @ A] :
( ( inj_on @ A @ B @ F @ B6 )
=> ( ( ord_less_eq @ ( set @ A ) @ A3 @ B6 )
=> ( inj_on @ A @ B @ F @ A3 ) ) ) ).
% subset_inj_on
thf(fact_237_strict__mono__less__eq,axiom,
! [B: $tType,A: $tType] :
( ( ( linorder @ A )
& ( order @ B ) )
=> ! [F: A > B,X: A,Y: A] :
( ( order_strict_mono @ A @ B @ F )
=> ( ( ord_less_eq @ B @ ( F @ X ) @ ( F @ Y ) )
= ( ord_less_eq @ A @ X @ Y ) ) ) ) ).
% strict_mono_less_eq
thf(fact_238_dual__order_Oantisym,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [B2: A,A2: A] :
( ( ord_less_eq @ A @ B2 @ A2 )
=> ( ( ord_less_eq @ A @ A2 @ B2 )
=> ( A2 = B2 ) ) ) ) ).
% dual_order.antisym
thf(fact_239_dual__order_Oeq__iff,axiom,
! [A: $tType] :
( ( order @ A )
=> ( ( ^ [Y6: A,Z2: A] : Y6 = Z2 )
= ( ^ [A7: A,B5: A] :
( ( ord_less_eq @ A @ B5 @ A7 )
& ( ord_less_eq @ A @ A7 @ B5 ) ) ) ) ) ).
% dual_order.eq_iff
thf(fact_240_strict__mono__eq,axiom,
! [B: $tType,A: $tType] :
( ( ( linorder @ A )
& ( order @ B ) )
=> ! [F: A > B,X: A,Y: A] :
( ( order_strict_mono @ A @ B @ F )
=> ( ( ( F @ X )
= ( F @ Y ) )
= ( X = Y ) ) ) ) ).
% strict_mono_eq
thf(fact_241_dual__order_Otrans,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [B2: A,A2: A,C3: A] :
( ( ord_less_eq @ A @ B2 @ A2 )
=> ( ( ord_less_eq @ A @ C3 @ B2 )
=> ( ord_less_eq @ A @ C3 @ A2 ) ) ) ) ).
% dual_order.trans
thf(fact_242_linorder__wlog,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [P3: A > A > $o,A2: A,B2: A] :
( ! [A5: A,B3: A] :
( ( ord_less_eq @ A @ A5 @ B3 )
=> ( P3 @ A5 @ B3 ) )
=> ( ! [A5: A,B3: A] :
( ( P3 @ B3 @ A5 )
=> ( P3 @ A5 @ B3 ) )
=> ( P3 @ A2 @ B2 ) ) ) ) ).
% linorder_wlog
thf(fact_243_dual__order_Orefl,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [A2: A] : ( ord_less_eq @ A @ A2 @ A2 ) ) ).
% dual_order.refl
thf(fact_244_order__trans,axiom,
! [A: $tType] :
( ( preorder @ A )
=> ! [X: A,Y: A,Z: A] :
( ( ord_less_eq @ A @ X @ Y )
=> ( ( ord_less_eq @ A @ Y @ Z )
=> ( ord_less_eq @ A @ X @ Z ) ) ) ) ).
% order_trans
thf(fact_245_order__class_Oorder_Oantisym,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [A2: A,B2: A] :
( ( ord_less_eq @ A @ A2 @ B2 )
=> ( ( ord_less_eq @ A @ B2 @ A2 )
=> ( A2 = B2 ) ) ) ) ).
% order_class.order.antisym
thf(fact_246_ord__le__eq__trans,axiom,
! [A: $tType] :
( ( ord @ A )
=> ! [A2: A,B2: A,C3: A] :
( ( ord_less_eq @ A @ A2 @ B2 )
=> ( ( B2 = C3 )
=> ( ord_less_eq @ A @ A2 @ C3 ) ) ) ) ).
% ord_le_eq_trans
thf(fact_247_ord__eq__le__trans,axiom,
! [A: $tType] :
( ( ord @ A )
=> ! [A2: A,B2: A,C3: A] :
( ( A2 = B2 )
=> ( ( ord_less_eq @ A @ B2 @ C3 )
=> ( ord_less_eq @ A @ A2 @ C3 ) ) ) ) ).
% ord_eq_le_trans
thf(fact_248_order__class_Oorder_Oeq__iff,axiom,
! [A: $tType] :
( ( order @ A )
=> ( ( ^ [Y6: A,Z2: A] : Y6 = Z2 )
= ( ^ [A7: A,B5: A] :
( ( ord_less_eq @ A @ A7 @ B5 )
& ( ord_less_eq @ A @ B5 @ A7 ) ) ) ) ) ).
% order_class.order.eq_iff
thf(fact_249_antisym__conv,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [Y: A,X: A] :
( ( ord_less_eq @ A @ Y @ X )
=> ( ( ord_less_eq @ A @ X @ Y )
= ( X = Y ) ) ) ) ).
% antisym_conv
thf(fact_250_le__cases3,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [X: A,Y: A,Z: A] :
( ( ( ord_less_eq @ A @ X @ Y )
=> ~ ( ord_less_eq @ A @ Y @ Z ) )
=> ( ( ( ord_less_eq @ A @ Y @ X )
=> ~ ( ord_less_eq @ A @ X @ Z ) )
=> ( ( ( ord_less_eq @ A @ X @ Z )
=> ~ ( ord_less_eq @ A @ Z @ Y ) )
=> ( ( ( ord_less_eq @ A @ Z @ Y )
=> ~ ( ord_less_eq @ A @ Y @ X ) )
=> ( ( ( ord_less_eq @ A @ Y @ Z )
=> ~ ( ord_less_eq @ A @ Z @ X ) )
=> ~ ( ( ord_less_eq @ A @ Z @ X )
=> ~ ( ord_less_eq @ A @ X @ Y ) ) ) ) ) ) ) ) ).
% le_cases3
thf(fact_251_order_Otrans,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [A2: A,B2: A,C3: A] :
( ( ord_less_eq @ A @ A2 @ B2 )
=> ( ( ord_less_eq @ A @ B2 @ C3 )
=> ( ord_less_eq @ A @ A2 @ C3 ) ) ) ) ).
% order.trans
thf(fact_252_le__cases,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [X: A,Y: A] :
( ~ ( ord_less_eq @ A @ X @ Y )
=> ( ord_less_eq @ A @ Y @ X ) ) ) ).
% le_cases
thf(fact_253_eq__refl,axiom,
! [A: $tType] :
( ( preorder @ A )
=> ! [X: A,Y: A] :
( ( X = Y )
=> ( ord_less_eq @ A @ X @ Y ) ) ) ).
% eq_refl
thf(fact_254_linear,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [X: A,Y: A] :
( ( ord_less_eq @ A @ X @ Y )
| ( ord_less_eq @ A @ Y @ X ) ) ) ).
% linear
thf(fact_255_antisym,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [X: A,Y: A] :
( ( ord_less_eq @ A @ X @ Y )
=> ( ( ord_less_eq @ A @ Y @ X )
=> ( X = Y ) ) ) ) ).
% antisym
% Type constructors (17)
thf(tcon_fun___Orderings_Opreorder,axiom,
! [A8: $tType,A9: $tType] :
( ( preorder @ A9 )
=> ( preorder @ ( A8 > A9 ) ) ) ).
thf(tcon_fun___Orderings_Oorder,axiom,
! [A8: $tType,A9: $tType] :
( ( order @ A9 )
=> ( order @ ( A8 > A9 ) ) ) ).
thf(tcon_fun___Orderings_Otop,axiom,
! [A8: $tType,A9: $tType] :
( ( top @ A9 )
=> ( top @ ( A8 > A9 ) ) ) ).
thf(tcon_fun___Orderings_Oord,axiom,
! [A8: $tType,A9: $tType] :
( ( ord @ A9 )
=> ( ord @ ( A8 > A9 ) ) ) ).
thf(tcon_Nat_Onat___Orderings_Opreorder_1,axiom,
preorder @ nat ).
thf(tcon_Nat_Onat___Orderings_Olinorder,axiom,
linorder @ nat ).
thf(tcon_Nat_Onat___Orderings_Oorder_2,axiom,
order @ nat ).
thf(tcon_Nat_Onat___Orderings_Oord_3,axiom,
ord @ nat ).
thf(tcon_Set_Oset___Orderings_Opreorder_4,axiom,
! [A8: $tType] : ( preorder @ ( set @ A8 ) ) ).
thf(tcon_Set_Oset___Orderings_Oorder_5,axiom,
! [A8: $tType] : ( order @ ( set @ A8 ) ) ).
thf(tcon_Set_Oset___Orderings_Otop_6,axiom,
! [A8: $tType] : ( top @ ( set @ A8 ) ) ).
thf(tcon_Set_Oset___Orderings_Oord_7,axiom,
! [A8: $tType] : ( ord @ ( set @ A8 ) ) ).
thf(tcon_HOL_Obool___Orderings_Opreorder_8,axiom,
preorder @ $o ).
thf(tcon_HOL_Obool___Orderings_Olinorder_9,axiom,
linorder @ $o ).
thf(tcon_HOL_Obool___Orderings_Oorder_10,axiom,
order @ $o ).
thf(tcon_HOL_Obool___Orderings_Otop_11,axiom,
top @ $o ).
thf(tcon_HOL_Obool___Orderings_Oord_12,axiom,
ord @ $o ).
% Helper facts (3)
thf(help_If_3_1_T,axiom,
! [P3: $o] :
( ( P3 = $true )
| ( P3 = $false ) ) ).
thf(help_If_2_1_T,axiom,
! [A: $tType,X: A,Y: A] :
( ( if @ A @ $false @ X @ Y )
= Y ) ).
thf(help_If_1_1_T,axiom,
! [A: $tType,X: A,Y: A] :
( ( if @ A @ $true @ X @ Y )
= X ) ).
% Conjectures (1)
thf(conj_0,conjecture,
( ( paraco876059933e_eval @ i @ ( paraco2100061555le_Con @ p1 @ p2 ) )
= ( paraco876059933e_eval @ i @ p1 ) ) ).
%------------------------------------------------------------------------------