TPTP Problem File: ITP137^2.p
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- Solve Problem
%------------------------------------------------------------------------------
% File : ITP137^2 : TPTP v8.2.0. Released v7.5.0.
% Domain : Interactive Theorem Proving
% Problem : Sledgehammer Paraconsistency problem prob_407__3265932_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_407__3265932_1 [Des21]
% Status : Theorem
% Rating : 0.33 v8.1.0, 0.25 v7.5.0
% Syntax : Number of formulae : 316 ( 180 unt; 56 typ; 0 def)
% Number of atoms : 531 ( 386 equ; 0 cnn)
% Maximal formula atoms : 16 ( 2 avg)
% Number of connectives : 4410 ( 99 ~; 8 |; 53 &;4048 @)
% ( 0 <=>; 202 =>; 0 <=; 0 <~>)
% Maximal formula depth : 28 ( 8 avg)
% Number of types : 6 ( 5 usr)
% Number of type conns : 471 ( 471 >; 0 *; 0 +; 0 <<)
% Number of symbols : 54 ( 51 usr; 4 con; 0-8 aty)
% Number of variables : 1493 ( 197 ^;1218 !; 13 ?;1493 :)
% ( 65 !>; 0 ?*; 0 @-; 0 @+)
% SPC : TH1_THM_EQU_NAR
% Comments : This file was generated by Sledgehammer 2021-02-23 16:21:07.434
%------------------------------------------------------------------------------
% Could-be-implicit typings (8)
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_Ounit,type,
product_unit: $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 (48)
thf(sy_c_Fun__Def_Oreduction__pair,type,
fun_reduction_pair:
!>[A: $tType] : ( ( product_prod @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ A @ A ) ) ) > $o ) ).
thf(sy_c_Fun__Def_Orp__inv__image,type,
fun_rp_inv_image:
!>[A: $tType,B: $tType] : ( ( product_prod @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ A @ A ) ) ) > ( B > A ) > ( product_prod @ ( set @ ( product_prod @ B @ B ) ) @ ( set @ ( product_prod @ B @ B ) ) ) ) ).
thf(sy_c_HOL_OThe,type,
the:
!>[A: $tType] : ( ( A > $o ) > A ) ).
thf(sy_c_If,type,
if:
!>[A: $tType] : ( $o > A > A > A ) ).
thf(sy_c_Order__Relation_Oabove,type,
order_above:
!>[A: $tType] : ( ( set @ ( product_prod @ A @ A ) ) > A > ( set @ A ) ) ).
thf(sy_c_Order__Relation_OaboveS,type,
order_aboveS:
!>[A: $tType] : ( ( set @ ( product_prod @ A @ A ) ) > A > ( set @ A ) ) ).
thf(sy_c_Order__Relation_Orelation__of,type,
order_relation_of:
!>[A: $tType] : ( ( A > A > $o ) > ( set @ A ) > ( set @ ( product_prod @ A @ A ) ) ) ).
thf(sy_c_Orderings_Obot__class_Obot,type,
bot_bot:
!>[A: $tType] : A ).
thf(sy_c_Orderings_Oord__class_Oless__eq,type,
ord_less_eq:
!>[A: $tType] : ( A > A > $o ) ).
thf(sy_c_Orderings_Otop__class_Otop,type,
top_top:
!>[A: $tType] : A ).
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_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_Product__Type_OPair,type,
product_Pair:
!>[A: $tType,B: $tType] : ( A > B > ( product_prod @ A @ B ) ) ).
thf(sy_c_Product__Type_OSigma,type,
product_Sigma:
!>[A: $tType,B: $tType] : ( ( set @ A ) > ( A > ( set @ B ) ) > ( set @ ( product_prod @ A @ B ) ) ) ).
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_Oold_Oprod_Orec__set__prod,type,
product_rec_set_prod:
!>[A: $tType,B: $tType,T: $tType] : ( ( A > B > T ) > ( product_prod @ A @ B ) > T > $o ) ).
thf(sy_c_Product__Type_Oold_Ounit_Orec__set__unit,type,
product_rec_set_unit:
!>[T: $tType] : ( T > product_unit > T > $o ) ).
thf(sy_c_Product__Type_Oold_Ounit_Orec__unit,type,
product_rec_unit:
!>[T: $tType] : ( T > product_unit > T ) ).
thf(sy_c_Product__Type_Oprod_Ocase__prod,type,
product_case_prod:
!>[A: $tType,B: $tType,C: $tType] : ( ( A > B > C ) > ( product_prod @ A @ B ) > C ) ).
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_Product__Type_Oproduct,type,
product_product:
!>[A: $tType,B: $tType] : ( ( set @ A ) > ( set @ B ) > ( set @ ( product_prod @ A @ B ) ) ) ).
thf(sy_c_Product__Type_Oscomp,type,
product_scomp:
!>[A: $tType,B: $tType,C: $tType,D: $tType] : ( ( A > ( product_prod @ B @ C ) ) > ( B > C > D ) > A > D ) ).
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_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_Set_Othe__elem,type,
the_elem:
!>[A: $tType] : ( ( set @ A ) > A ) ).
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_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_p,type,
p: paraco414474393lle_fm ).
% Relevant facts (256)
thf(fact_0_tv_Oinject_I1_J,axiom,
! [X1: $o,Y1: $o] :
( ( ( paraco2040174112le_Det @ X1 )
= ( paraco2040174112le_Det @ Y1 ) )
= ( X1 = Y1 ) ) ).
% tv.inject(1)
thf(fact_1_assms,axiom,
paraco769098683_valid @ ( paraco329115265le_Neg @ p ) ).
% assms
thf(fact_2_valid__def,axiom,
( paraco769098683_valid
= ( ^ [P: paraco414474393lle_fm] :
! [I: ( list @ char ) > paraco415392788lle_tv] :
( ( paraco876059933e_eval @ I @ P )
= ( paraco2040174112le_Det @ $true ) ) ) ) ).
% valid_def
thf(fact_3_eval_Osimps_I2_J,axiom,
! [I2: ( list @ char ) > paraco415392788lle_tv] :
( ( paraco876059933e_eval @ I2 @ paraco251304083_Truth )
= ( paraco2040174112le_Det @ $true ) ) ).
% eval.simps(2)
thf(fact_4_eval_Osimps_I4_J,axiom,
! [I2: ( list @ char ) > paraco415392788lle_tv,P2: paraco414474393lle_fm,Q: paraco414474393lle_fm] :
( ( ( ( paraco876059933e_eval @ I2 @ P2 )
= ( paraco876059933e_eval @ I2 @ Q ) )
=> ( ( paraco876059933e_eval @ I2 @ ( paraco2100061555le_Con @ P2 @ Q ) )
= ( paraco876059933e_eval @ I2 @ P2 ) ) )
& ( ( ( paraco876059933e_eval @ I2 @ P2 )
!= ( paraco876059933e_eval @ I2 @ Q ) )
=> ( ( ( ( paraco876059933e_eval @ I2 @ P2 )
= ( paraco2040174112le_Det @ $true ) )
=> ( ( paraco876059933e_eval @ I2 @ ( paraco2100061555le_Con @ P2 @ Q ) )
= ( paraco876059933e_eval @ I2 @ Q ) ) )
& ( ( ( paraco876059933e_eval @ I2 @ P2 )
!= ( paraco2040174112le_Det @ $true ) )
=> ( ( ( ( paraco876059933e_eval @ I2 @ Q )
= ( paraco2040174112le_Det @ $true ) )
=> ( ( paraco876059933e_eval @ I2 @ ( paraco2100061555le_Con @ P2 @ Q ) )
= ( paraco876059933e_eval @ I2 @ P2 ) ) )
& ( ( ( paraco876059933e_eval @ I2 @ Q )
!= ( paraco2040174112le_Det @ $true ) )
=> ( ( paraco876059933e_eval @ I2 @ ( paraco2100061555le_Con @ P2 @ Q ) )
= ( paraco2040174112le_Det @ $false ) ) ) ) ) ) ) ) ).
% eval.simps(4)
thf(fact_5_eval_Osimps_I5_J,axiom,
! [I2: ( list @ char ) > paraco415392788lle_tv,P2: paraco414474393lle_fm,Q: paraco414474393lle_fm] :
( ( ( ( paraco876059933e_eval @ I2 @ P2 )
= ( paraco876059933e_eval @ I2 @ Q ) )
=> ( ( paraco876059933e_eval @ I2 @ ( paraco2084319816le_Eql @ P2 @ Q ) )
= ( paraco2040174112le_Det @ $true ) ) )
& ( ( ( paraco876059933e_eval @ I2 @ P2 )
!= ( paraco876059933e_eval @ I2 @ Q ) )
=> ( ( paraco876059933e_eval @ I2 @ ( paraco2084319816le_Eql @ P2 @ Q ) )
= ( paraco2040174112le_Det @ $false ) ) ) ) ).
% eval.simps(5)
thf(fact_6_eval__equality__simplify,axiom,
! [I2: ( list @ char ) > paraco415392788lle_tv,P2: paraco414474393lle_fm,Q: paraco414474393lle_fm] :
( ( paraco876059933e_eval @ I2 @ ( paraco2084319816le_Eql @ P2 @ Q ) )
= ( paraco2040174112le_Det
@ ( ( paraco876059933e_eval @ I2 @ P2 )
= ( paraco876059933e_eval @ I2 @ Q ) ) ) ) ).
% eval_equality_simplify
thf(fact_7_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_8_eval__negation,axiom,
! [I2: ( list @ char ) > paraco415392788lle_tv,P2: paraco414474393lle_fm] :
( ( ( ( paraco876059933e_eval @ I2 @ P2 )
= ( paraco2040174112le_Det @ $false ) )
=> ( ( paraco876059933e_eval @ I2 @ ( paraco329115265le_Neg @ P2 ) )
= ( paraco2040174112le_Det @ $true ) ) )
& ( ( ( paraco876059933e_eval @ I2 @ P2 )
!= ( paraco2040174112le_Det @ $false ) )
=> ( ( ( ( paraco876059933e_eval @ I2 @ P2 )
= ( paraco2040174112le_Det @ $true ) )
=> ( ( paraco876059933e_eval @ I2 @ ( paraco329115265le_Neg @ P2 ) )
= ( paraco2040174112le_Det @ $false ) ) )
& ( ( ( paraco876059933e_eval @ I2 @ P2 )
!= ( paraco2040174112le_Det @ $true ) )
=> ( ( paraco876059933e_eval @ I2 @ ( paraco329115265le_Neg @ P2 ) )
= ( paraco876059933e_eval @ I2 @ P2 ) ) ) ) ) ) ).
% eval_negation
thf(fact_9_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_10_eval_Osimps_I1_J,axiom,
! [I2: ( list @ char ) > paraco415392788lle_tv,S: list @ char] :
( ( paraco876059933e_eval @ I2 @ ( paraco27778325le_Pro @ S ) )
= ( I2 @ S ) ) ).
% eval.simps(1)
thf(fact_11_tv_Odistinct_I1_J,axiom,
! [X1: $o,X2: nat] :
( ( paraco2040174112le_Det @ X1 )
!= ( paraco676387099_Indet @ X2 ) ) ).
% tv.distinct(1)
thf(fact_12_tv_Oinduct,axiom,
! [P3: paraco415392788lle_tv > $o,Tv: paraco415392788lle_tv] :
( ! [X: $o] : ( P3 @ ( paraco2040174112le_Det @ X ) )
=> ( ! [X: nat] : ( P3 @ ( paraco676387099_Indet @ X ) )
=> ( P3 @ Tv ) ) ) ).
% tv.induct
thf(fact_13_tv_Oexhaust,axiom,
! [Y: paraco415392788lle_tv] :
( ! [X12: $o] :
( Y
!= ( paraco2040174112le_Det @ X12 ) )
=> ~ ! [X22: nat] :
( Y
!= ( paraco676387099_Indet @ X22 ) ) ) ).
% tv.exhaust
thf(fact_14_fm_Oinject_I2_J,axiom,
! [X3: paraco414474393lle_fm,Y3: paraco414474393lle_fm] :
( ( ( paraco329115265le_Neg @ X3 )
= ( paraco329115265le_Neg @ Y3 ) )
= ( X3 = Y3 ) ) ).
% fm.inject(2)
thf(fact_15_tv_Oinject_I2_J,axiom,
! [X2: nat,Y2: nat] :
( ( ( paraco676387099_Indet @ X2 )
= ( paraco676387099_Indet @ Y2 ) )
= ( X2 = Y2 ) ) ).
% tv.inject(2)
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_Oinject_I1_J,axiom,
! [X1: list @ char,Y1: list @ char] :
( ( ( paraco27778325le_Pro @ X1 )
= ( paraco27778325le_Pro @ Y1 ) )
= ( X1 = Y1 ) ) ).
% fm.inject(1)
thf(fact_18_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_19_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_20_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_21_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_22_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_23_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_24_fm_Odistinct_I15_J,axiom,
! [X51: paraco414474393lle_fm,X52: paraco414474393lle_fm] :
( paraco251304083_Truth
!= ( paraco2084319816le_Eql @ X51 @ X52 ) ) ).
% fm.distinct(15)
thf(fact_25_fm_Odistinct_I13_J,axiom,
! [X41: paraco414474393lle_fm,X42: paraco414474393lle_fm] :
( paraco251304083_Truth
!= ( paraco2100061555le_Con @ X41 @ X42 ) ) ).
% fm.distinct(13)
thf(fact_26_fm_Odistinct_I11_J,axiom,
! [X3: paraco414474393lle_fm] :
( paraco251304083_Truth
!= ( paraco329115265le_Neg @ X3 ) ) ).
% fm.distinct(11)
thf(fact_27_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_28_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_29_fm_Odistinct_I3_J,axiom,
! [X1: list @ char,X3: paraco414474393lle_fm] :
( ( paraco27778325le_Pro @ X1 )
!= ( paraco329115265le_Neg @ X3 ) ) ).
% fm.distinct(3)
thf(fact_30_fm_Odistinct_I1_J,axiom,
! [X1: list @ char] :
( ( paraco27778325le_Pro @ X1 )
!= paraco251304083_Truth ) ).
% fm.distinct(1)
thf(fact_31_double__negation,axiom,
( paraco876059933e_eval
= ( ^ [I: ( list @ char ) > paraco415392788lle_tv,P: paraco414474393lle_fm] : ( paraco876059933e_eval @ I @ ( paraco329115265le_Neg @ ( paraco329115265le_Neg @ P ) ) ) ) ) ).
% double_negation
thf(fact_32_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_33_string__tv_Ocases,axiom,
! [X4: paraco415392788lle_tv] :
( ( X4
!= ( paraco2040174112le_Det @ $true ) )
=> ( ( X4
!= ( paraco2040174112le_Det @ $false ) )
=> ~ ! [N: nat] :
( X4
!= ( paraco676387099_Indet @ N ) ) ) ) ).
% string_tv.cases
thf(fact_34_fm_Oinduct,axiom,
! [P3: paraco414474393lle_fm > $o,Fm: paraco414474393lle_fm] :
( ! [X: list @ char] : ( P3 @ ( paraco27778325le_Pro @ X ) )
=> ( ( P3 @ paraco251304083_Truth )
=> ( ! [X: paraco414474393lle_fm] :
( ( P3 @ X )
=> ( P3 @ ( paraco329115265le_Neg @ X ) ) )
=> ( ! [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_35_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 ) )
=> ~ ! [X61: paraco414474393lle_fm,X62: paraco414474393lle_fm] :
( Y
!= ( paraco1628874225le_Eql @ X61 @ X62 ) ) ) ) ) ) ) ).
% fm.exhaust
thf(fact_36_eval_Osimps_I3_J,axiom,
! [I2: ( list @ char ) > paraco415392788lle_tv,P2: paraco414474393lle_fm] :
( ( paraco876059933e_eval @ I2 @ ( paraco329115265le_Neg @ P2 ) )
= ( paraco490622181ase_tv @ paraco415392788lle_tv @ ( product_case_bool @ paraco415392788lle_tv @ ( paraco2040174112le_Det @ $false ) @ ( paraco2040174112le_Det @ $true ) ) @ paraco676387099_Indet @ ( paraco876059933e_eval @ I2 @ P2 ) ) ) ).
% eval.simps(3)
thf(fact_37_eval__equality,axiom,
! [I2: ( list @ char ) > paraco415392788lle_tv,P2: paraco414474393lle_fm,Q: paraco414474393lle_fm] :
( ( ( ( paraco876059933e_eval @ I2 @ P2 )
= ( paraco876059933e_eval @ I2 @ Q ) )
=> ( ( paraco876059933e_eval @ I2 @ ( paraco1628874225le_Eql @ P2 @ Q ) )
= ( paraco2040174112le_Det @ $true ) ) )
& ( ( ( paraco876059933e_eval @ I2 @ P2 )
!= ( paraco876059933e_eval @ I2 @ Q ) )
=> ( ( ( ( paraco876059933e_eval @ I2 @ P2 )
= ( paraco2040174112le_Det @ $true ) )
=> ( ( paraco876059933e_eval @ I2 @ ( paraco1628874225le_Eql @ P2 @ Q ) )
= ( paraco876059933e_eval @ I2 @ Q ) ) )
& ( ( ( paraco876059933e_eval @ I2 @ P2 )
!= ( paraco2040174112le_Det @ $true ) )
=> ( ( ( ( paraco876059933e_eval @ I2 @ Q )
= ( paraco2040174112le_Det @ $true ) )
=> ( ( paraco876059933e_eval @ I2 @ ( paraco1628874225le_Eql @ P2 @ Q ) )
= ( paraco876059933e_eval @ I2 @ P2 ) ) )
& ( ( ( paraco876059933e_eval @ I2 @ Q )
!= ( paraco2040174112le_Det @ $true ) )
=> ( ( ( ( paraco876059933e_eval @ I2 @ P2 )
= ( paraco2040174112le_Det @ $false ) )
=> ( ( paraco876059933e_eval @ I2 @ ( paraco1628874225le_Eql @ P2 @ Q ) )
= ( paraco876059933e_eval @ I2 @ ( paraco329115265le_Neg @ Q ) ) ) )
& ( ( ( paraco876059933e_eval @ I2 @ P2 )
!= ( paraco2040174112le_Det @ $false ) )
=> ( ( ( ( paraco876059933e_eval @ I2 @ Q )
= ( paraco2040174112le_Det @ $false ) )
=> ( ( paraco876059933e_eval @ I2 @ ( paraco1628874225le_Eql @ P2 @ Q ) )
= ( paraco876059933e_eval @ I2 @ ( paraco329115265le_Neg @ P2 ) ) ) )
& ( ( ( paraco876059933e_eval @ I2 @ Q )
!= ( paraco2040174112le_Det @ $false ) )
=> ( ( paraco876059933e_eval @ I2 @ ( paraco1628874225le_Eql @ P2 @ Q ) )
= ( paraco2040174112le_Det @ $false ) ) ) ) ) ) ) ) ) ) ) ) ).
% eval_equality
thf(fact_38_eval_Ocases,axiom,
! [X4: product_prod @ ( ( list @ char ) > paraco415392788lle_tv ) @ paraco414474393lle_fm] :
( ! [I3: ( list @ char ) > paraco415392788lle_tv,S2: list @ char] :
( X4
!= ( product_Pair @ ( ( list @ char ) > paraco415392788lle_tv ) @ paraco414474393lle_fm @ I3 @ ( paraco27778325le_Pro @ S2 ) ) )
=> ( ! [I3: ( list @ char ) > paraco415392788lle_tv] :
( X4
!= ( product_Pair @ ( ( list @ char ) > paraco415392788lle_tv ) @ paraco414474393lle_fm @ I3 @ paraco251304083_Truth ) )
=> ( ! [I3: ( list @ char ) > paraco415392788lle_tv,P4: paraco414474393lle_fm] :
( X4
!= ( product_Pair @ ( ( list @ char ) > paraco415392788lle_tv ) @ paraco414474393lle_fm @ I3 @ ( paraco329115265le_Neg @ P4 ) ) )
=> ( ! [I3: ( list @ char ) > paraco415392788lle_tv,P4: paraco414474393lle_fm,Q2: paraco414474393lle_fm] :
( X4
!= ( product_Pair @ ( ( list @ char ) > paraco415392788lle_tv ) @ paraco414474393lle_fm @ I3 @ ( paraco2100061555le_Con @ P4 @ Q2 ) ) )
=> ( ! [I3: ( list @ char ) > paraco415392788lle_tv,P4: paraco414474393lle_fm,Q2: paraco414474393lle_fm] :
( X4
!= ( product_Pair @ ( ( list @ char ) > paraco415392788lle_tv ) @ paraco414474393lle_fm @ I3 @ ( paraco2084319816le_Eql @ P4 @ Q2 ) ) )
=> ~ ! [I3: ( list @ char ) > paraco415392788lle_tv,P4: paraco414474393lle_fm,Q2: paraco414474393lle_fm] :
( X4
!= ( product_Pair @ ( ( list @ char ) > paraco415392788lle_tv ) @ paraco414474393lle_fm @ I3 @ ( paraco1628874225le_Eql @ P4 @ Q2 ) ) ) ) ) ) ) ) ).
% eval.cases
thf(fact_39_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_40_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_41_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_42_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_43_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_44_fm_Odistinct_I17_J,axiom,
! [X612: paraco414474393lle_fm,X622: paraco414474393lle_fm] :
( paraco251304083_Truth
!= ( paraco1628874225le_Eql @ X612 @ X622 ) ) ).
% fm.distinct(17)
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
@ ^ [X5: A] : ( member @ A @ X5 @ A3 ) )
= A3 ) ).
% Collect_mem_eq
thf(fact_47_Collect__cong,axiom,
! [A: $tType,P3: A > $o,Q3: A > $o] :
( ! [X: A] :
( ( P3 @ X )
= ( Q3 @ X ) )
=> ( ( collect @ A @ P3 )
= ( collect @ A @ Q3 ) ) ) ).
% Collect_cong
thf(fact_48_ext,axiom,
! [B: $tType,A: $tType,F: A > B,G: A > B] :
( ! [X: A] :
( ( F @ X )
= ( G @ X ) )
=> ( F = G ) ) ).
% ext
thf(fact_49_fm_Oinject_I5_J,axiom,
! [X612: paraco414474393lle_fm,X622: paraco414474393lle_fm,Y61: paraco414474393lle_fm,Y62: paraco414474393lle_fm] :
( ( ( paraco1628874225le_Eql @ X612 @ X622 )
= ( paraco1628874225le_Eql @ Y61 @ Y62 ) )
= ( ( X612 = Y61 )
& ( X622 = Y62 ) ) ) ).
% fm.inject(5)
thf(fact_50_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,X612: paraco414474393lle_fm,X622: paraco414474393lle_fm] :
( ( paraco1246693743ase_fm @ A @ F1 @ F2 @ F3 @ F4 @ F5 @ F6 @ ( paraco1628874225le_Eql @ X612 @ X622 ) )
= ( F6 @ X612 @ X622 ) ) ).
% fm.simps(41)
thf(fact_51_fm_Odistinct_I23_J,axiom,
! [X3: paraco414474393lle_fm,X612: paraco414474393lle_fm,X622: paraco414474393lle_fm] :
( ( paraco329115265le_Neg @ X3 )
!= ( paraco1628874225le_Eql @ X612 @ X622 ) ) ).
% fm.distinct(23)
thf(fact_52_fm_Odistinct_I29_J,axiom,
! [X51: paraco414474393lle_fm,X52: paraco414474393lle_fm,X612: paraco414474393lle_fm,X622: paraco414474393lle_fm] :
( ( paraco2084319816le_Eql @ X51 @ X52 )
!= ( paraco1628874225le_Eql @ X612 @ X622 ) ) ).
% fm.distinct(29)
thf(fact_53_fm_Odistinct_I9_J,axiom,
! [X1: list @ char,X612: paraco414474393lle_fm,X622: paraco414474393lle_fm] :
( ( paraco27778325le_Pro @ X1 )
!= ( paraco1628874225le_Eql @ X612 @ X622 ) ) ).
% fm.distinct(9)
thf(fact_54_fm_Odistinct_I27_J,axiom,
! [X41: paraco414474393lle_fm,X42: paraco414474393lle_fm,X612: paraco414474393lle_fm,X622: paraco414474393lle_fm] :
( ( paraco2100061555le_Con @ X41 @ X42 )
!= ( paraco1628874225le_Eql @ X612 @ X622 ) ) ).
% fm.distinct(27)
thf(fact_55_old_Oprod_Oinject,axiom,
! [A: $tType,B: $tType,A2: A,B2: B,A4: A,B3: B] :
( ( ( product_Pair @ A @ B @ A2 @ B2 )
= ( product_Pair @ A @ B @ A4 @ B3 ) )
= ( ( A2 = A4 )
& ( B2 = B3 ) ) ) ).
% old.prod.inject
thf(fact_56_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_57_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_58_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_59_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_60_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_61_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_62_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_63_surj__pair,axiom,
! [A: $tType,B: $tType,P2: product_prod @ A @ B] :
? [X: A,Y4: B] :
( P2
= ( product_Pair @ A @ B @ X @ Y4 ) ) ).
% surj_pair
thf(fact_64_prod__cases,axiom,
! [B: $tType,A: $tType,P3: ( product_prod @ A @ B ) > $o,P2: product_prod @ A @ B] :
( ! [A5: A,B4: B] : ( P3 @ ( product_Pair @ A @ B @ A5 @ B4 ) )
=> ( P3 @ P2 ) ) ).
% prod_cases
thf(fact_65_Pair__inject,axiom,
! [A: $tType,B: $tType,A2: A,B2: B,A4: A,B3: B] :
( ( ( product_Pair @ A @ B @ A2 @ B2 )
= ( product_Pair @ A @ B @ A4 @ B3 ) )
=> ~ ( ( A2 = A4 )
=> ( B2 != B3 ) ) ) ).
% Pair_inject
thf(fact_66_prod__cases3,axiom,
! [A: $tType,B: $tType,C: $tType,Y: product_prod @ A @ ( product_prod @ B @ C )] :
~ ! [A5: A,B4: B,C2: C] :
( Y
!= ( product_Pair @ A @ ( product_prod @ B @ C ) @ A5 @ ( product_Pair @ B @ C @ B4 @ C2 ) ) ) ).
% prod_cases3
thf(fact_67_prod__cases4,axiom,
! [A: $tType,B: $tType,C: $tType,D: $tType,Y: product_prod @ A @ ( product_prod @ B @ ( product_prod @ C @ D ) )] :
~ ! [A5: A,B4: B,C2: C,D2: D] :
( Y
!= ( product_Pair @ A @ ( product_prod @ B @ ( product_prod @ C @ D ) ) @ A5 @ ( product_Pair @ B @ ( product_prod @ C @ D ) @ B4 @ ( product_Pair @ C @ D @ C2 @ D2 ) ) ) ) ).
% prod_cases4
thf(fact_68_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,B4: 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 ) ) @ B4 @ ( product_Pair @ C @ ( product_prod @ D @ E ) @ C2 @ ( product_Pair @ D @ E @ D2 @ E2 ) ) ) ) ) ).
% prod_cases5
thf(fact_69_prod__cases6,axiom,
! [A: $tType,B: $tType,C: $tType,D: $tType,E: $tType,F7: $tType,Y: product_prod @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D @ ( product_prod @ E @ F7 ) ) ) )] :
~ ! [A5: A,B4: B,C2: C,D2: D,E2: E,F8: F7] :
( Y
!= ( product_Pair @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D @ ( product_prod @ E @ F7 ) ) ) ) @ A5 @ ( product_Pair @ B @ ( product_prod @ C @ ( product_prod @ D @ ( product_prod @ E @ F7 ) ) ) @ B4 @ ( product_Pair @ C @ ( product_prod @ D @ ( product_prod @ E @ F7 ) ) @ C2 @ ( product_Pair @ D @ ( product_prod @ E @ F7 ) @ D2 @ ( product_Pair @ E @ F7 @ E2 @ F8 ) ) ) ) ) ) ).
% prod_cases6
thf(fact_70_prod__cases7,axiom,
! [A: $tType,B: $tType,C: $tType,D: $tType,E: $tType,F7: $tType,G2: $tType,Y: product_prod @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D @ ( product_prod @ E @ ( product_prod @ F7 @ G2 ) ) ) ) )] :
~ ! [A5: A,B4: B,C2: C,D2: D,E2: E,F8: F7,G3: G2] :
( Y
!= ( product_Pair @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D @ ( product_prod @ E @ ( product_prod @ F7 @ G2 ) ) ) ) ) @ A5 @ ( product_Pair @ B @ ( product_prod @ C @ ( product_prod @ D @ ( product_prod @ E @ ( product_prod @ F7 @ G2 ) ) ) ) @ B4 @ ( product_Pair @ C @ ( product_prod @ D @ ( product_prod @ E @ ( product_prod @ F7 @ G2 ) ) ) @ C2 @ ( product_Pair @ D @ ( product_prod @ E @ ( product_prod @ F7 @ G2 ) ) @ D2 @ ( product_Pair @ E @ ( product_prod @ F7 @ G2 ) @ E2 @ ( product_Pair @ F7 @ G2 @ F8 @ G3 ) ) ) ) ) ) ) ).
% prod_cases7
thf(fact_71_prod__induct3,axiom,
! [C: $tType,B: $tType,A: $tType,P3: ( product_prod @ A @ ( product_prod @ B @ C ) ) > $o,X4: product_prod @ A @ ( product_prod @ B @ C )] :
( ! [A5: A,B4: B,C2: C] : ( P3 @ ( product_Pair @ A @ ( product_prod @ B @ C ) @ A5 @ ( product_Pair @ B @ C @ B4 @ C2 ) ) )
=> ( P3 @ X4 ) ) ).
% prod_induct3
thf(fact_72_prod__induct4,axiom,
! [D: $tType,C: $tType,B: $tType,A: $tType,P3: ( product_prod @ A @ ( product_prod @ B @ ( product_prod @ C @ D ) ) ) > $o,X4: product_prod @ A @ ( product_prod @ B @ ( product_prod @ C @ D ) )] :
( ! [A5: A,B4: B,C2: C,D2: D] : ( P3 @ ( product_Pair @ A @ ( product_prod @ B @ ( product_prod @ C @ D ) ) @ A5 @ ( product_Pair @ B @ ( product_prod @ C @ D ) @ B4 @ ( product_Pair @ C @ D @ C2 @ D2 ) ) ) )
=> ( P3 @ X4 ) ) ).
% prod_induct4
thf(fact_73_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,X4: product_prod @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D @ E ) ) )] :
( ! [A5: A,B4: 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 ) ) @ B4 @ ( product_Pair @ C @ ( product_prod @ D @ E ) @ C2 @ ( product_Pair @ D @ E @ D2 @ E2 ) ) ) ) )
=> ( P3 @ X4 ) ) ).
% prod_induct5
thf(fact_74_prod__induct6,axiom,
! [F7: $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 @ F7 ) ) ) ) ) > $o,X4: product_prod @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D @ ( product_prod @ E @ F7 ) ) ) )] :
( ! [A5: A,B4: B,C2: C,D2: D,E2: E,F8: F7] : ( P3 @ ( product_Pair @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D @ ( product_prod @ E @ F7 ) ) ) ) @ A5 @ ( product_Pair @ B @ ( product_prod @ C @ ( product_prod @ D @ ( product_prod @ E @ F7 ) ) ) @ B4 @ ( product_Pair @ C @ ( product_prod @ D @ ( product_prod @ E @ F7 ) ) @ C2 @ ( product_Pair @ D @ ( product_prod @ E @ F7 ) @ D2 @ ( product_Pair @ E @ F7 @ E2 @ F8 ) ) ) ) ) )
=> ( P3 @ X4 ) ) ).
% prod_induct6
thf(fact_75_prod__induct7,axiom,
! [G2: $tType,F7: $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 @ F7 @ G2 ) ) ) ) ) ) > $o,X4: product_prod @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D @ ( product_prod @ E @ ( product_prod @ F7 @ G2 ) ) ) ) )] :
( ! [A5: A,B4: B,C2: C,D2: D,E2: E,F8: F7,G3: G2] : ( P3 @ ( product_Pair @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D @ ( product_prod @ E @ ( product_prod @ F7 @ G2 ) ) ) ) ) @ A5 @ ( product_Pair @ B @ ( product_prod @ C @ ( product_prod @ D @ ( product_prod @ E @ ( product_prod @ F7 @ G2 ) ) ) ) @ B4 @ ( product_Pair @ C @ ( product_prod @ D @ ( product_prod @ E @ ( product_prod @ F7 @ G2 ) ) ) @ C2 @ ( product_Pair @ D @ ( product_prod @ E @ ( product_prod @ F7 @ G2 ) ) @ D2 @ ( product_Pair @ E @ ( product_prod @ F7 @ G2 ) @ E2 @ ( product_Pair @ F7 @ G2 @ F8 @ G3 ) ) ) ) ) ) )
=> ( P3 @ X4 ) ) ).
% prod_induct7
thf(fact_76_old_Oprod_Oexhaust,axiom,
! [A: $tType,B: $tType,Y: product_prod @ A @ B] :
~ ! [A5: A,B4: B] :
( Y
!= ( product_Pair @ A @ B @ A5 @ B4 ) ) ).
% old.prod.exhaust
thf(fact_77_old_Oprod_Oinducts,axiom,
! [B: $tType,A: $tType,P3: ( product_prod @ A @ B ) > $o,Prod: product_prod @ A @ B] :
( ! [A5: A,B4: B] : ( P3 @ ( product_Pair @ A @ B @ A5 @ B4 ) )
=> ( P3 @ Prod ) ) ).
% old.prod.inducts
thf(fact_78_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_79_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_80_ssubst__Pair__rhs,axiom,
! [B: $tType,A: $tType,R: A,S: B,R2: set @ ( product_prod @ A @ B ),S3: B] :
( ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ R @ S ) @ R2 )
=> ( ( S3 = S )
=> ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ R @ S3 ) @ R2 ) ) ) ).
% ssubst_Pair_rhs
thf(fact_81_curry__conv,axiom,
! [A: $tType,C: $tType,B: $tType] :
( ( product_curry @ B @ C @ A )
= ( ^ [F9: ( product_prod @ B @ C ) > A,A6: B,B5: C] : ( F9 @ ( product_Pair @ B @ C @ A6 @ B5 ) ) ) ) ).
% curry_conv
thf(fact_82_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_83_swap__simp,axiom,
! [A: $tType,B: $tType,X4: B,Y: A] :
( ( product_swap @ B @ A @ ( product_Pair @ B @ A @ X4 @ Y ) )
= ( product_Pair @ A @ B @ Y @ X4 ) ) ).
% swap_simp
thf(fact_84_eval_Osimps_I6_J,axiom,
! [I2: ( list @ char ) > paraco415392788lle_tv,P2: paraco414474393lle_fm,Q: paraco414474393lle_fm] :
( ( ( ( paraco876059933e_eval @ I2 @ P2 )
= ( paraco876059933e_eval @ I2 @ Q ) )
=> ( ( paraco876059933e_eval @ I2 @ ( paraco1628874225le_Eql @ P2 @ Q ) )
= ( paraco2040174112le_Det @ $true ) ) )
& ( ( ( paraco876059933e_eval @ I2 @ P2 )
!= ( paraco876059933e_eval @ I2 @ Q ) )
=> ( ( paraco876059933e_eval @ I2 @ ( paraco1628874225le_Eql @ P2 @ Q ) )
= ( product_case_prod @ paraco415392788lle_tv @ paraco415392788lle_tv @ paraco415392788lle_tv
@ ^ [X5: paraco415392788lle_tv,Y5: paraco415392788lle_tv] :
( paraco490622181ase_tv @ paraco415392788lle_tv
@ ( product_case_bool @ paraco415392788lle_tv @ ( paraco876059933e_eval @ I2 @ Q )
@ ( paraco490622181ase_tv @ paraco415392788lle_tv @ ( product_case_bool @ paraco415392788lle_tv @ ( paraco876059933e_eval @ I2 @ P2 ) @ ( paraco490622181ase_tv @ paraco415392788lle_tv @ ( product_case_bool @ paraco415392788lle_tv @ ( paraco2040174112le_Det @ $false ) @ ( paraco2040174112le_Det @ $true ) ) @ paraco676387099_Indet @ ( paraco876059933e_eval @ I2 @ Q ) ) )
@ ^ [Nat: nat] : ( paraco490622181ase_tv @ paraco415392788lle_tv @ ( product_case_bool @ paraco415392788lle_tv @ ( paraco2040174112le_Det @ $false ) @ ( paraco2040174112le_Det @ $true ) ) @ paraco676387099_Indet @ ( paraco876059933e_eval @ I2 @ Q ) )
@ Y5 ) )
@ ^ [Nat: nat] :
( paraco490622181ase_tv @ paraco415392788lle_tv @ ( product_case_bool @ paraco415392788lle_tv @ ( paraco876059933e_eval @ I2 @ P2 ) @ ( paraco490622181ase_tv @ paraco415392788lle_tv @ ( product_case_bool @ paraco415392788lle_tv @ ( paraco2040174112le_Det @ $false ) @ ( paraco2040174112le_Det @ $true ) ) @ paraco676387099_Indet @ ( paraco876059933e_eval @ I2 @ P2 ) ) )
@ ^ [Nata: nat] : ( paraco2040174112le_Det @ $false )
@ Y5 )
@ X5 )
@ ( product_Pair @ paraco415392788lle_tv @ paraco415392788lle_tv @ ( paraco876059933e_eval @ I2 @ P2 ) @ ( paraco876059933e_eval @ I2 @ Q ) ) ) ) ) ) ).
% eval.simps(6)
thf(fact_85_swap__swap,axiom,
! [B: $tType,A: $tType,P2: product_prod @ A @ B] :
( ( product_swap @ B @ A @ ( product_swap @ A @ B @ P2 ) )
= P2 ) ).
% swap_swap
thf(fact_86_case__prod__conv,axiom,
! [B: $tType,A: $tType,C: $tType,F: B > C > A,A2: B,B2: C] :
( ( product_case_prod @ B @ C @ A @ F @ ( product_Pair @ B @ C @ A2 @ B2 ) )
= ( F @ A2 @ B2 ) ) ).
% case_prod_conv
thf(fact_87_case__prod__curry,axiom,
! [C: $tType,B: $tType,A: $tType,F: ( product_prod @ A @ B ) > C] :
( ( product_case_prod @ A @ B @ C @ ( product_curry @ A @ B @ C @ F ) )
= F ) ).
% case_prod_curry
thf(fact_88_curry__case__prod,axiom,
! [C: $tType,B: $tType,A: $tType,F: A > B > C] :
( ( product_curry @ A @ B @ C @ ( product_case_prod @ A @ B @ C @ F ) )
= F ) ).
% curry_case_prod
thf(fact_89_case__swap,axiom,
! [A: $tType,B: $tType,C: $tType,F: C > B > A,P2: product_prod @ C @ B] :
( ( product_case_prod @ B @ C @ A
@ ^ [Y5: B,X5: C] : ( F @ X5 @ Y5 )
@ ( product_swap @ C @ B @ P2 ) )
= ( product_case_prod @ C @ B @ A @ F @ P2 ) ) ).
% case_swap
thf(fact_90_cond__case__prod__eta,axiom,
! [C: $tType,B: $tType,A: $tType,F: A > B > C,G: ( product_prod @ A @ B ) > C] :
( ! [X: A,Y4: B] :
( ( F @ X @ Y4 )
= ( G @ ( product_Pair @ A @ B @ X @ Y4 ) ) )
=> ( ( product_case_prod @ A @ B @ C @ F )
= G ) ) ).
% cond_case_prod_eta
thf(fact_91_case__prod__eta,axiom,
! [C: $tType,B: $tType,A: $tType,F: ( product_prod @ A @ B ) > C] :
( ( product_case_prod @ A @ B @ C
@ ^ [X5: A,Y5: B] : ( F @ ( product_Pair @ A @ B @ X5 @ Y5 ) ) )
= F ) ).
% case_prod_eta
thf(fact_92_case__prodE2,axiom,
! [B: $tType,A: $tType,C: $tType,Q3: A > $o,P3: B > C > A,Z: product_prod @ B @ C] :
( ( Q3 @ ( product_case_prod @ B @ C @ A @ P3 @ Z ) )
=> ~ ! [X: B,Y4: C] :
( ( Z
= ( product_Pair @ B @ C @ X @ Y4 ) )
=> ~ ( Q3 @ ( P3 @ X @ Y4 ) ) ) ) ).
% case_prodE2
thf(fact_93_curry__def,axiom,
! [C: $tType,B: $tType,A: $tType] :
( ( product_curry @ A @ B @ C )
= ( ^ [C4: ( product_prod @ A @ B ) > C,X5: A,Y5: B] : ( C4 @ ( product_Pair @ A @ B @ X5 @ Y5 ) ) ) ) ).
% curry_def
thf(fact_94_curry__K,axiom,
! [B: $tType,C: $tType,A: $tType,C3: C] :
( ( product_curry @ A @ B @ C
@ ^ [X5: product_prod @ A @ B] : C3 )
= ( ^ [X5: A,Y5: B] : C3 ) ) ).
% curry_K
thf(fact_95_prod_Ocase__distrib,axiom,
! [C: $tType,D: $tType,B: $tType,A: $tType,H: C > D,F: A > B > C,Prod: product_prod @ A @ B] :
( ( H @ ( product_case_prod @ A @ B @ C @ F @ Prod ) )
= ( product_case_prod @ A @ B @ D
@ ^ [X13: A,X23: B] : ( H @ ( F @ X13 @ X23 ) )
@ Prod ) ) ).
% prod.case_distrib
thf(fact_96_internal__case__prod__def,axiom,
! [C: $tType,B: $tType,A: $tType] :
( ( produc2004651681e_prod @ A @ B @ C )
= ( product_case_prod @ A @ B @ C ) ) ).
% internal_case_prod_def
thf(fact_97_tv_Ocase__distrib,axiom,
! [B: $tType,A: $tType,H: A > B,F1: $o > A,F2: nat > A,Tv: paraco415392788lle_tv] :
( ( H @ ( paraco490622181ase_tv @ A @ F1 @ F2 @ Tv ) )
= ( paraco490622181ase_tv @ B
@ ^ [X5: $o] : ( H @ ( F1 @ X5 ) )
@ ^ [X5: nat] : ( H @ ( F2 @ X5 ) )
@ Tv ) ) ).
% tv.case_distrib
thf(fact_98_fm_Ocase__distrib,axiom,
! [A: $tType,B: $tType,H: A > B,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,Fm: paraco414474393lle_fm] :
( ( H @ ( paraco1246693743ase_fm @ A @ F1 @ F2 @ F3 @ F4 @ F5 @ F6 @ Fm ) )
= ( paraco1246693743ase_fm @ B
@ ^ [X5: list @ char] : ( H @ ( F1 @ X5 ) )
@ ( H @ F2 )
@ ^ [X5: paraco414474393lle_fm] : ( H @ ( F3 @ X5 ) )
@ ^ [X13: paraco414474393lle_fm,X23: paraco414474393lle_fm] : ( H @ ( F4 @ X13 @ X23 ) )
@ ^ [X13: paraco414474393lle_fm,X23: paraco414474393lle_fm] : ( H @ ( F5 @ X13 @ X23 ) )
@ ^ [X13: paraco414474393lle_fm,X23: paraco414474393lle_fm] : ( H @ ( F6 @ X13 @ X23 ) )
@ Fm ) ) ).
% fm.case_distrib
thf(fact_99_old_Oprod_Ocase,axiom,
! [A: $tType,C: $tType,B: $tType,F: A > B > C,X1: A,X2: B] :
( ( product_case_prod @ A @ B @ C @ F @ ( product_Pair @ A @ B @ X1 @ X2 ) )
= ( F @ X1 @ X2 ) ) ).
% old.prod.case
thf(fact_100_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_101_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_102_split__cong,axiom,
! [C: $tType,B: $tType,A: $tType,Q: product_prod @ A @ B,F: A > B > C,G: A > B > C,P2: product_prod @ A @ B] :
( ! [X: A,Y4: B] :
( ( ( product_Pair @ A @ B @ X @ Y4 )
= Q )
=> ( ( F @ X @ Y4 )
= ( G @ X @ Y4 ) ) )
=> ( ( P2 = Q )
=> ( ( product_case_prod @ A @ B @ C @ F @ P2 )
= ( product_case_prod @ A @ B @ C @ G @ Q ) ) ) ) ).
% split_cong
thf(fact_103_case__bool__if,axiom,
! [A: $tType] :
( ( product_case_bool @ A )
= ( ^ [X5: A,Y5: A,P5: $o] : ( if @ A @ P5 @ X5 @ Y5 ) ) ) ).
% case_bool_if
thf(fact_104_case__prod__app,axiom,
! [A: $tType,D: $tType,C: $tType,B: $tType] :
( ( product_case_prod @ B @ C @ ( D > A ) )
= ( ^ [F9: B > C > D > A,X5: product_prod @ B @ C,Y5: D] :
( product_case_prod @ B @ C @ A
@ ^ [L: B,R3: C] : ( F9 @ L @ R3 @ Y5 )
@ X5 ) ) ) ).
% case_prod_app
thf(fact_105_pred__equals__eq2,axiom,
! [B: $tType,A: $tType,R2: set @ ( product_prod @ A @ B ),S4: set @ ( product_prod @ A @ B )] :
( ( ( ^ [X5: A,Y5: B] : ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X5 @ Y5 ) @ R2 ) )
= ( ^ [X5: A,Y5: B] : ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X5 @ Y5 ) @ S4 ) ) )
= ( R2 = S4 ) ) ).
% pred_equals_eq2
thf(fact_106_pair__imageI,axiom,
! [C: $tType,B: $tType,A: $tType,A2: A,B2: B,A3: set @ ( product_prod @ A @ B ),F: A > B > C] :
( ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ A2 @ B2 ) @ A3 )
=> ( member @ C @ ( F @ A2 @ B2 ) @ ( image @ ( product_prod @ A @ B ) @ C @ ( product_case_prod @ A @ B @ C @ F ) @ A3 ) ) ) ).
% pair_imageI
thf(fact_107_scomp__apply,axiom,
! [A: $tType,D: $tType,C: $tType,B: $tType] :
( ( product_scomp @ B @ C @ D @ A )
= ( ^ [F9: B > ( product_prod @ C @ D ),G4: C > D > A,X5: B] : ( product_case_prod @ C @ D @ A @ G4 @ ( F9 @ X5 ) ) ) ) ).
% scomp_apply
thf(fact_108_case__prodI,axiom,
! [A: $tType,B: $tType,F: A > B > $o,A2: A,B2: B] :
( ( F @ A2 @ B2 )
=> ( product_case_prod @ A @ B @ $o @ F @ ( product_Pair @ A @ B @ A2 @ B2 ) ) ) ).
% case_prodI
thf(fact_109_case__prodI2,axiom,
! [B: $tType,A: $tType,P2: product_prod @ A @ B,C3: A > B > $o] :
( ! [A5: A,B4: B] :
( ( P2
= ( product_Pair @ A @ B @ A5 @ B4 ) )
=> ( C3 @ A5 @ B4 ) )
=> ( product_case_prod @ A @ B @ $o @ C3 @ P2 ) ) ).
% case_prodI2
thf(fact_110_mem__case__prodI,axiom,
! [A: $tType,B: $tType,C: $tType,Z: A,C3: B > C > ( set @ A ),A2: B,B2: C] :
( ( member @ A @ Z @ ( C3 @ A2 @ B2 ) )
=> ( member @ A @ Z @ ( product_case_prod @ B @ C @ ( set @ A ) @ C3 @ ( product_Pair @ B @ C @ A2 @ B2 ) ) ) ) ).
% mem_case_prodI
thf(fact_111_mem__case__prodI2,axiom,
! [C: $tType,B: $tType,A: $tType,P2: product_prod @ A @ B,Z: C,C3: A > B > ( set @ C )] :
( ! [A5: A,B4: B] :
( ( P2
= ( product_Pair @ A @ B @ A5 @ B4 ) )
=> ( member @ C @ Z @ ( C3 @ A5 @ B4 ) ) )
=> ( member @ C @ Z @ ( product_case_prod @ A @ B @ ( set @ C ) @ C3 @ P2 ) ) ) ).
% mem_case_prodI2
thf(fact_112_case__prodI2_H,axiom,
! [A: $tType,B: $tType,C: $tType,P2: product_prod @ A @ B,C3: A > B > C > $o,X4: C] :
( ! [A5: A,B4: B] :
( ( ( product_Pair @ A @ B @ A5 @ B4 )
= P2 )
=> ( C3 @ A5 @ B4 @ X4 ) )
=> ( product_case_prod @ A @ B @ ( C > $o ) @ C3 @ P2 @ X4 ) ) ).
% case_prodI2'
thf(fact_113_pair__in__swap__image,axiom,
! [A: $tType,B: $tType,Y: A,X4: B,A3: set @ ( product_prod @ B @ A )] :
( ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ Y @ X4 ) @ ( image @ ( product_prod @ B @ A ) @ ( product_prod @ A @ B ) @ ( product_swap @ B @ A ) @ A3 ) )
= ( member @ ( product_prod @ B @ A ) @ ( product_Pair @ B @ A @ X4 @ Y ) @ A3 ) ) ).
% pair_in_swap_image
thf(fact_114_case__prod__Pair__iden,axiom,
! [B: $tType,A: $tType,P2: product_prod @ A @ B] :
( ( product_case_prod @ A @ B @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B ) @ P2 )
= P2 ) ).
% case_prod_Pair_iden
thf(fact_115_scomp__scomp,axiom,
! [A: $tType,C: $tType,D: $tType,B: $tType,F7: $tType,E: $tType,F: A > ( product_prod @ E @ F7 ),G: E > F7 > ( product_prod @ C @ D ),H: C > D > B] :
( ( product_scomp @ A @ C @ D @ B @ ( product_scomp @ A @ E @ F7 @ ( product_prod @ C @ D ) @ F @ G ) @ H )
= ( product_scomp @ A @ E @ F7 @ B @ F
@ ^ [X5: E] : ( product_scomp @ F7 @ C @ D @ B @ ( G @ X5 ) @ H ) ) ) ).
% scomp_scomp
thf(fact_116_mem__case__prodE,axiom,
! [B: $tType,A: $tType,C: $tType,Z: A,C3: B > C > ( set @ A ),P2: product_prod @ B @ C] :
( ( member @ A @ Z @ ( product_case_prod @ B @ C @ ( set @ A ) @ C3 @ P2 ) )
=> ~ ! [X: B,Y4: C] :
( ( P2
= ( product_Pair @ B @ C @ X @ Y4 ) )
=> ~ ( member @ A @ Z @ ( C3 @ X @ Y4 ) ) ) ) ).
% mem_case_prodE
thf(fact_117_case__prodD,axiom,
! [A: $tType,B: $tType,F: A > B > $o,A2: A,B2: B] :
( ( product_case_prod @ A @ B @ $o @ F @ ( product_Pair @ A @ B @ A2 @ B2 ) )
=> ( F @ A2 @ B2 ) ) ).
% case_prodD
thf(fact_118_case__prodE,axiom,
! [A: $tType,B: $tType,C3: A > B > $o,P2: product_prod @ A @ B] :
( ( product_case_prod @ A @ B @ $o @ C3 @ P2 )
=> ~ ! [X: A,Y4: B] :
( ( P2
= ( product_Pair @ A @ B @ X @ Y4 ) )
=> ~ ( C3 @ X @ Y4 ) ) ) ).
% case_prodE
thf(fact_119_case__prodD_H,axiom,
! [B: $tType,A: $tType,C: $tType,R2: A > B > C > $o,A2: A,B2: B,C3: C] :
( ( product_case_prod @ A @ B @ ( C > $o ) @ R2 @ ( product_Pair @ A @ B @ A2 @ B2 ) @ C3 )
=> ( R2 @ A2 @ B2 @ C3 ) ) ).
% case_prodD'
thf(fact_120_case__prodE_H,axiom,
! [B: $tType,A: $tType,C: $tType,C3: A > B > C > $o,P2: product_prod @ A @ B,Z: C] :
( ( product_case_prod @ A @ B @ ( C > $o ) @ C3 @ P2 @ Z )
=> ~ ! [X: A,Y4: B] :
( ( P2
= ( product_Pair @ A @ B @ X @ Y4 ) )
=> ~ ( C3 @ X @ Y4 @ Z ) ) ) ).
% case_prodE'
thf(fact_121_scomp__Pair,axiom,
! [C: $tType,B: $tType,A: $tType,X4: A > ( product_prod @ B @ C )] :
( ( product_scomp @ A @ B @ C @ ( product_prod @ B @ C ) @ X4 @ ( product_Pair @ B @ C ) )
= X4 ) ).
% scomp_Pair
thf(fact_122_Pair__scomp,axiom,
! [A: $tType,B: $tType,C: $tType,X4: C,F: C > A > B] :
( ( product_scomp @ A @ C @ A @ B @ ( product_Pair @ C @ A @ X4 ) @ F )
= ( F @ X4 ) ) ).
% Pair_scomp
thf(fact_123_scomp__def,axiom,
! [D: $tType,C: $tType,B: $tType,A: $tType] :
( ( product_scomp @ A @ B @ C @ D )
= ( ^ [F9: A > ( product_prod @ B @ C ),G4: B > C > D,X5: A] : ( product_case_prod @ B @ C @ D @ G4 @ ( F9 @ X5 ) ) ) ) ).
% scomp_def
thf(fact_124_image__ident,axiom,
! [A: $tType,Y6: set @ A] :
( ( image @ A @ A
@ ^ [X5: A] : X5
@ Y6 )
= Y6 ) ).
% image_ident
thf(fact_125_image__eqI,axiom,
! [A: $tType,B: $tType,B2: A,F: B > A,X4: B,A3: set @ B] :
( ( B2
= ( F @ X4 ) )
=> ( ( member @ B @ X4 @ A3 )
=> ( member @ A @ B2 @ ( image @ B @ A @ F @ A3 ) ) ) ) ).
% image_eqI
thf(fact_126_Sup_OSUP__identity__eq,axiom,
! [A: $tType,Sup: ( set @ A ) > A,A3: set @ A] :
( ( Sup
@ ( image @ A @ A
@ ^ [X5: A] : X5
@ A3 ) )
= ( Sup @ A3 ) ) ).
% Sup.SUP_identity_eq
thf(fact_127_Inf_OINF__identity__eq,axiom,
! [A: $tType,Inf: ( set @ A ) > A,A3: set @ A] :
( ( Inf
@ ( image @ A @ A
@ ^ [X5: A] : X5
@ A3 ) )
= ( Inf @ A3 ) ) ).
% Inf.INF_identity_eq
thf(fact_128_Compr__image__eq,axiom,
! [A: $tType,B: $tType,F: B > A,A3: set @ B,P3: A > $o] :
( ( collect @ A
@ ^ [X5: A] :
( ( member @ A @ X5 @ ( image @ B @ A @ F @ A3 ) )
& ( P3 @ X5 ) ) )
= ( image @ B @ A @ F
@ ( collect @ B
@ ^ [X5: B] :
( ( member @ B @ X5 @ A3 )
& ( P3 @ ( F @ X5 ) ) ) ) ) ) ).
% Compr_image_eq
thf(fact_129_split__part,axiom,
! [B: $tType,A: $tType,P3: $o,Q3: A > B > $o] :
( ( product_case_prod @ A @ B @ $o
@ ^ [A6: A,B5: B] :
( P3
& ( Q3 @ A6 @ B5 ) ) )
= ( ^ [Ab: product_prod @ A @ B] :
( P3
& ( product_case_prod @ A @ B @ $o @ Q3 @ Ab ) ) ) ) ).
% split_part
thf(fact_130_prod_Odisc__eq__case,axiom,
! [B: $tType,A: $tType,Prod: product_prod @ A @ B] :
( product_case_prod @ A @ B @ $o
@ ^ [Uu: A,Uv: B] : $true
@ Prod ) ).
% prod.disc_eq_case
thf(fact_131_Sup_OSUP__cong,axiom,
! [A: $tType,B: $tType,A3: set @ B,B6: set @ B,C5: B > A,D3: B > A,Sup: ( set @ A ) > A] :
( ( A3 = B6 )
=> ( ! [X: B] :
( ( member @ B @ X @ B6 )
=> ( ( C5 @ X )
= ( D3 @ X ) ) )
=> ( ( Sup @ ( image @ B @ A @ C5 @ A3 ) )
= ( Sup @ ( image @ B @ A @ D3 @ B6 ) ) ) ) ) ).
% Sup.SUP_cong
thf(fact_132_Inf_OINF__cong,axiom,
! [A: $tType,B: $tType,A3: set @ B,B6: set @ B,C5: B > A,D3: B > A,Inf: ( set @ A ) > A] :
( ( A3 = B6 )
=> ( ! [X: B] :
( ( member @ B @ X @ B6 )
=> ( ( C5 @ X )
= ( D3 @ X ) ) )
=> ( ( Inf @ ( image @ B @ A @ C5 @ A3 ) )
= ( Inf @ ( image @ B @ A @ D3 @ B6 ) ) ) ) ) ).
% Inf.INF_cong
thf(fact_133_rev__image__eqI,axiom,
! [B: $tType,A: $tType,X4: A,A3: set @ A,B2: B,F: A > B] :
( ( member @ A @ X4 @ A3 )
=> ( ( B2
= ( F @ X4 ) )
=> ( member @ B @ B2 @ ( image @ A @ B @ F @ A3 ) ) ) ) ).
% rev_image_eqI
thf(fact_134_ball__imageD,axiom,
! [A: $tType,B: $tType,F: B > A,A3: set @ B,P3: A > $o] :
( ! [X: A] :
( ( member @ A @ X @ ( image @ B @ A @ F @ A3 ) )
=> ( P3 @ X ) )
=> ! [X6: B] :
( ( member @ B @ X6 @ A3 )
=> ( P3 @ ( F @ X6 ) ) ) ) ).
% ball_imageD
thf(fact_135_image__cong,axiom,
! [B: $tType,A: $tType,M: set @ A,N2: set @ A,F: A > B,G: A > B] :
( ( M = N2 )
=> ( ! [X: A] :
( ( member @ A @ X @ N2 )
=> ( ( F @ X )
= ( G @ X ) ) )
=> ( ( image @ A @ B @ F @ M )
= ( image @ A @ B @ G @ N2 ) ) ) ) ).
% image_cong
thf(fact_136_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 ) )
=> ? [X: B] :
( ( member @ B @ X @ A3 )
& ( P3 @ ( F @ X ) ) ) ) ).
% bex_imageD
thf(fact_137_image__iff,axiom,
! [A: $tType,B: $tType,Z: A,F: B > A,A3: set @ B] :
( ( member @ A @ Z @ ( image @ B @ A @ F @ A3 ) )
= ( ? [X5: B] :
( ( member @ B @ X5 @ A3 )
& ( Z
= ( F @ X5 ) ) ) ) ) ).
% image_iff
thf(fact_138_imageI,axiom,
! [B: $tType,A: $tType,X4: A,A3: set @ A,F: A > B] :
( ( member @ A @ X4 @ A3 )
=> ( member @ B @ ( F @ X4 ) @ ( image @ A @ B @ F @ A3 ) ) ) ).
% imageI
thf(fact_139_imageE,axiom,
! [A: $tType,B: $tType,B2: A,F: B > A,A3: set @ B] :
( ( member @ A @ B2 @ ( image @ B @ A @ F @ A3 ) )
=> ~ ! [X: B] :
( ( B2
= ( F @ X ) )
=> ~ ( member @ B @ X @ A3 ) ) ) ).
% imageE
thf(fact_140_image__image,axiom,
! [A: $tType,B: $tType,C: $tType,F: B > A,G: C > B,A3: set @ C] :
( ( image @ B @ A @ F @ ( image @ C @ B @ G @ A3 ) )
= ( image @ C @ A
@ ^ [X5: C] : ( F @ ( G @ X5 ) )
@ A3 ) ) ).
% image_image
thf(fact_141_image__paired__Times,axiom,
! [C: $tType,D: $tType,B: $tType,A: $tType,F: C > A,G: D > B,A3: set @ C,B6: set @ D] :
( ( image @ ( product_prod @ C @ D ) @ ( product_prod @ A @ B )
@ ( product_case_prod @ C @ D @ ( product_prod @ A @ B )
@ ^ [X5: C,Y5: D] : ( product_Pair @ A @ B @ ( F @ X5 ) @ ( G @ Y5 ) ) )
@ ( product_Sigma @ C @ D @ A3
@ ^ [Uu: C] : B6 ) )
= ( product_Sigma @ A @ B @ ( image @ C @ A @ F @ A3 )
@ ^ [Uu: A] : ( image @ D @ B @ G @ B6 ) ) ) ).
% image_paired_Times
thf(fact_142_inv__image__def,axiom,
! [A: $tType,B: $tType] :
( ( inv_image @ B @ A )
= ( ^ [R3: set @ ( product_prod @ B @ B ),F9: A > B] :
( collect @ ( product_prod @ A @ A )
@ ( product_case_prod @ A @ A @ $o
@ ^ [X5: A,Y5: A] : ( member @ ( product_prod @ B @ B ) @ ( product_Pair @ B @ B @ ( F9 @ X5 ) @ ( F9 @ Y5 ) ) @ R3 ) ) ) ) ) ).
% inv_image_def
thf(fact_143_same__fst__def,axiom,
! [B: $tType,A: $tType] :
( ( same_fst @ A @ B )
= ( ^ [P5: A > $o,R4: A > ( set @ ( product_prod @ B @ B ) )] :
( collect @ ( product_prod @ ( product_prod @ A @ B ) @ ( product_prod @ A @ B ) )
@ ( product_case_prod @ ( product_prod @ A @ B ) @ ( product_prod @ A @ B ) @ $o
@ ( product_case_prod @ A @ B @ ( ( product_prod @ A @ B ) > $o )
@ ^ [X7: A,Y7: B] :
( product_case_prod @ A @ B @ $o
@ ^ [X5: A,Y5: B] :
( ( X7 = X5 )
& ( P5 @ X5 )
& ( member @ ( product_prod @ B @ B ) @ ( product_Pair @ B @ B @ Y7 @ Y5 ) @ ( R4 @ X5 ) ) ) ) ) ) ) ) ) ).
% same_fst_def
thf(fact_144_swap__product,axiom,
! [B: $tType,A: $tType,A3: set @ B,B6: set @ A] :
( ( image @ ( product_prod @ B @ A ) @ ( product_prod @ A @ B )
@ ( product_case_prod @ B @ A @ ( product_prod @ A @ B )
@ ^ [I: B,J: A] : ( product_Pair @ A @ B @ J @ I ) )
@ ( product_Sigma @ B @ A @ A3
@ ^ [Uu: B] : B6 ) )
= ( product_Sigma @ A @ B @ B6
@ ^ [Uu: A] : A3 ) ) ).
% swap_product
thf(fact_145_SigmaI,axiom,
! [B: $tType,A: $tType,A2: A,A3: set @ A,B2: B,B6: A > ( set @ B )] :
( ( member @ A @ A2 @ A3 )
=> ( ( member @ B @ B2 @ ( B6 @ A2 ) )
=> ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ A2 @ B2 ) @ ( product_Sigma @ A @ B @ A3 @ B6 ) ) ) ) ).
% SigmaI
thf(fact_146_mem__Sigma__iff,axiom,
! [B: $tType,A: $tType,A2: A,B2: B,A3: set @ A,B6: A > ( set @ B )] :
( ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ A2 @ B2 ) @ ( product_Sigma @ A @ B @ A3 @ B6 ) )
= ( ( member @ A @ A2 @ A3 )
& ( member @ B @ B2 @ ( B6 @ A2 ) ) ) ) ).
% mem_Sigma_iff
thf(fact_147_in__inv__image,axiom,
! [A: $tType,B: $tType,X4: A,Y: A,R: set @ ( product_prod @ B @ B ),F: A > B] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X4 @ Y ) @ ( inv_image @ B @ A @ R @ F ) )
= ( member @ ( product_prod @ B @ B ) @ ( product_Pair @ B @ B @ ( F @ X4 ) @ ( F @ Y ) ) @ R ) ) ).
% in_inv_image
thf(fact_148_Collect__case__prod,axiom,
! [B: $tType,A: $tType,P3: A > $o,Q3: B > $o] :
( ( collect @ ( product_prod @ A @ B )
@ ( product_case_prod @ A @ B @ $o
@ ^ [A6: A,B5: B] :
( ( P3 @ A6 )
& ( Q3 @ B5 ) ) ) )
= ( product_Sigma @ A @ B @ ( collect @ A @ P3 )
@ ^ [Uu: A] : ( collect @ B @ Q3 ) ) ) ).
% Collect_case_prod
thf(fact_149_same__fstI,axiom,
! [B: $tType,A: $tType,P3: A > $o,X4: A,Y8: B,Y: B,R2: A > ( set @ ( product_prod @ B @ B ) )] :
( ( P3 @ X4 )
=> ( ( member @ ( product_prod @ B @ B ) @ ( product_Pair @ B @ B @ Y8 @ Y ) @ ( R2 @ X4 ) )
=> ( 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 @ X4 @ Y8 ) @ ( product_Pair @ A @ B @ X4 @ Y ) ) @ ( same_fst @ A @ B @ P3 @ R2 ) ) ) ) ).
% same_fstI
thf(fact_150_Collect__case__prod__Sigma,axiom,
! [B: $tType,A: $tType,P3: A > $o,Q3: A > B > $o] :
( ( collect @ ( product_prod @ A @ B )
@ ( product_case_prod @ A @ B @ $o
@ ^ [X5: A,Y5: B] :
( ( P3 @ X5 )
& ( Q3 @ X5 @ Y5 ) ) ) )
= ( product_Sigma @ A @ B @ ( collect @ A @ P3 )
@ ^ [X5: A] : ( collect @ B @ ( Q3 @ X5 ) ) ) ) ).
% Collect_case_prod_Sigma
thf(fact_151_Sigma__cong,axiom,
! [B: $tType,A: $tType,A3: set @ A,B6: set @ A,C5: A > ( set @ B ),D3: A > ( set @ B )] :
( ( A3 = B6 )
=> ( ! [X: A] :
( ( member @ A @ X @ B6 )
=> ( ( C5 @ X )
= ( D3 @ X ) ) )
=> ( ( product_Sigma @ A @ B @ A3 @ C5 )
= ( product_Sigma @ A @ B @ B6 @ D3 ) ) ) ) ).
% Sigma_cong
thf(fact_152_Times__eq__cancel2,axiom,
! [A: $tType,B: $tType,X4: A,C5: set @ A,A3: set @ B,B6: set @ B] :
( ( member @ A @ X4 @ C5 )
=> ( ( ( product_Sigma @ B @ A @ A3
@ ^ [Uu: B] : C5 )
= ( product_Sigma @ B @ A @ B6
@ ^ [Uu: B] : C5 ) )
= ( A3 = B6 ) ) ) ).
% Times_eq_cancel2
thf(fact_153_SigmaE,axiom,
! [A: $tType,B: $tType,C3: product_prod @ A @ B,A3: set @ A,B6: A > ( set @ B )] :
( ( member @ ( product_prod @ A @ B ) @ C3 @ ( product_Sigma @ A @ B @ A3 @ B6 ) )
=> ~ ! [X: A] :
( ( member @ A @ X @ A3 )
=> ! [Y4: B] :
( ( member @ B @ Y4 @ ( B6 @ X ) )
=> ( C3
!= ( product_Pair @ A @ B @ X @ Y4 ) ) ) ) ) ).
% SigmaE
thf(fact_154_SigmaD1,axiom,
! [B: $tType,A: $tType,A2: A,B2: B,A3: set @ A,B6: A > ( set @ B )] :
( ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ A2 @ B2 ) @ ( product_Sigma @ A @ B @ A3 @ B6 ) )
=> ( member @ A @ A2 @ A3 ) ) ).
% SigmaD1
thf(fact_155_SigmaD2,axiom,
! [B: $tType,A: $tType,A2: A,B2: B,A3: set @ A,B6: A > ( set @ B )] :
( ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ A2 @ B2 ) @ ( product_Sigma @ A @ B @ A3 @ B6 ) )
=> ( member @ B @ B2 @ ( B6 @ A2 ) ) ) ).
% SigmaD2
thf(fact_156_SigmaE2,axiom,
! [B: $tType,A: $tType,A2: A,B2: B,A3: set @ A,B6: A > ( set @ B )] :
( ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ A2 @ B2 ) @ ( product_Sigma @ A @ B @ A3 @ B6 ) )
=> ~ ( ( member @ A @ A2 @ A3 )
=> ~ ( member @ B @ B2 @ ( B6 @ A2 ) ) ) ) ).
% SigmaE2
thf(fact_157_product__swap,axiom,
! [B: $tType,A: $tType,A3: set @ B,B6: set @ A] :
( ( image @ ( product_prod @ B @ A ) @ ( product_prod @ A @ B ) @ ( product_swap @ B @ A )
@ ( product_Sigma @ B @ A @ A3
@ ^ [Uu: B] : B6 ) )
= ( product_Sigma @ A @ B @ B6
@ ^ [Uu: A] : A3 ) ) ).
% product_swap
thf(fact_158_rp__inv__image__def,axiom,
! [B: $tType,A: $tType] :
( ( fun_rp_inv_image @ A @ B )
= ( product_case_prod @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ A @ A ) ) @ ( ( B > A ) > ( product_prod @ ( set @ ( product_prod @ B @ B ) ) @ ( set @ ( product_prod @ B @ B ) ) ) )
@ ^ [R4: set @ ( product_prod @ A @ A ),S5: set @ ( product_prod @ A @ A ),F9: B > A] : ( product_Pair @ ( set @ ( product_prod @ B @ B ) ) @ ( set @ ( product_prod @ B @ B ) ) @ ( inv_image @ A @ B @ R4 @ F9 ) @ ( inv_image @ A @ B @ S5 @ F9 ) ) ) ) ).
% rp_inv_image_def
thf(fact_159_lex__prod__def,axiom,
! [B: $tType,A: $tType] :
( ( lex_prod @ A @ B )
= ( ^ [Ra: set @ ( product_prod @ A @ A ),Rb: set @ ( product_prod @ B @ B )] :
( collect @ ( product_prod @ ( product_prod @ A @ B ) @ ( product_prod @ A @ B ) )
@ ( product_case_prod @ ( product_prod @ A @ B ) @ ( product_prod @ A @ B ) @ $o
@ ( product_case_prod @ A @ B @ ( ( product_prod @ A @ B ) > $o )
@ ^ [A6: A,B5: B] :
( product_case_prod @ A @ B @ $o
@ ^ [A7: A,B7: B] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A6 @ A7 ) @ Ra )
| ( ( A6 = A7 )
& ( member @ ( product_prod @ B @ B ) @ ( product_Pair @ B @ B @ B5 @ B7 ) @ Rb ) ) ) ) ) ) ) ) ) ).
% lex_prod_def
thf(fact_160_Product__Type_Oproduct__def,axiom,
! [B: $tType,A: $tType] :
( ( product_product @ A @ B )
= ( ^ [A8: set @ A,B8: set @ B] :
( product_Sigma @ A @ B @ A8
@ ^ [Uu: A] : B8 ) ) ) ).
% Product_Type.product_def
thf(fact_161_in__lex__prod,axiom,
! [A: $tType,B: $tType,A2: A,B2: B,A4: A,B3: B,R: set @ ( product_prod @ A @ A ),S: 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 @ A4 @ B3 ) ) @ ( lex_prod @ A @ B @ R @ S ) )
= ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A2 @ A4 ) @ R )
| ( ( A2 = A4 )
& ( member @ ( product_prod @ B @ B ) @ ( product_Pair @ B @ B @ B2 @ B3 ) @ S ) ) ) ) ).
% in_lex_prod
thf(fact_162_member__product,axiom,
! [B: $tType,A: $tType,X4: product_prod @ A @ B,A3: set @ A,B6: set @ B] :
( ( member @ ( product_prod @ A @ B ) @ X4 @ ( product_product @ A @ B @ A3 @ B6 ) )
= ( member @ ( product_prod @ A @ B ) @ X4
@ ( product_Sigma @ A @ B @ A3
@ ^ [Uu: A] : B6 ) ) ) ).
% member_product
thf(fact_163_rp__inv__image__rp,axiom,
! [A: $tType,B: $tType,P3: product_prod @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ A @ A ) ),F: B > A] :
( ( fun_reduction_pair @ A @ P3 )
=> ( fun_reduction_pair @ B @ ( fun_rp_inv_image @ A @ B @ P3 @ F ) ) ) ).
% rp_inv_image_rp
thf(fact_164_map__prod__surj__on,axiom,
! [B: $tType,D: $tType,C: $tType,A: $tType,F: B > A,A3: set @ B,A9: set @ A,G: D > C,B6: set @ D,B9: set @ C] :
( ( ( image @ B @ A @ F @ A3 )
= A9 )
=> ( ( ( image @ D @ C @ G @ B6 )
= B9 )
=> ( ( image @ ( product_prod @ B @ D ) @ ( product_prod @ A @ C ) @ ( product_map_prod @ B @ A @ D @ C @ F @ G )
@ ( product_Sigma @ B @ D @ A3
@ ^ [Uu: B] : B6 ) )
= ( product_Sigma @ A @ C @ A9
@ ^ [Uu: A] : B9 ) ) ) ) ).
% map_prod_surj_on
thf(fact_165_relation__of__def,axiom,
! [A: $tType] :
( ( order_relation_of @ A )
= ( ^ [P5: A > A > $o,A8: set @ A] :
( collect @ ( product_prod @ A @ A )
@ ( product_case_prod @ A @ A @ $o
@ ^ [A6: A,B5: A] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A6 @ B5 )
@ ( product_Sigma @ A @ A @ A8
@ ^ [Uu: A] : A8 ) )
& ( P5 @ A6 @ B5 ) ) ) ) ) ) ).
% relation_of_def
thf(fact_166_map__prod__ident,axiom,
! [B: $tType,A: $tType] :
( ( product_map_prod @ A @ A @ B @ B
@ ^ [X5: A] : X5
@ ^ [Y5: B] : Y5 )
= ( ^ [Z2: product_prod @ A @ B] : Z2 ) ) ).
% map_prod_ident
thf(fact_167_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_168_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_169_case__prod__map__prod,axiom,
! [C: $tType,A: $tType,B: $tType,E: $tType,D: $tType,H: B > C > A,F: D > B,G: E > C,X4: product_prod @ D @ E] :
( ( product_case_prod @ B @ C @ A @ H @ ( product_map_prod @ D @ B @ E @ C @ F @ G @ X4 ) )
= ( product_case_prod @ D @ E @ A
@ ^ [L: D,R3: E] : ( H @ ( F @ L ) @ ( G @ R3 ) )
@ X4 ) ) ).
% case_prod_map_prod
thf(fact_170_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 ) )
=> ~ ! [X: C,Y4: D] :
( ( C3
= ( product_Pair @ A @ B @ ( F @ X ) @ ( G @ Y4 ) ) )
=> ~ ( member @ ( product_prod @ C @ D ) @ ( product_Pair @ C @ D @ X @ Y4 ) @ R2 ) ) ) ).
% prod_fun_imageE
thf(fact_171_map__prod__def,axiom,
! [B: $tType,D: $tType,C: $tType,A: $tType] :
( ( product_map_prod @ A @ C @ B @ D )
= ( ^ [F9: A > C,G4: B > D] :
( product_case_prod @ A @ B @ ( product_prod @ C @ D )
@ ^ [X5: A,Y5: B] : ( product_Pair @ C @ D @ ( F9 @ X5 ) @ ( G4 @ Y5 ) ) ) ) ) ).
% map_prod_def
thf(fact_172_prod_Omap__ident,axiom,
! [B: $tType,A: $tType,T2: product_prod @ A @ B] :
( ( product_map_prod @ A @ A @ B @ B
@ ^ [X5: A] : X5
@ ^ [X5: B] : X5
@ T2 )
= T2 ) ).
% prod.map_ident
thf(fact_173_aboveS__def,axiom,
! [A: $tType] :
( ( order_aboveS @ A )
= ( ^ [R3: set @ ( product_prod @ A @ A ),A6: A] :
( collect @ A
@ ^ [B5: A] :
( ( B5 != A6 )
& ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A6 @ B5 ) @ R3 ) ) ) ) ) ).
% aboveS_def
thf(fact_174_above__def,axiom,
! [A: $tType] :
( ( order_above @ A )
= ( ^ [R3: set @ ( product_prod @ A @ A ),A6: A] :
( collect @ A
@ ^ [B5: A] : ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A6 @ B5 ) @ R3 ) ) ) ) ).
% above_def
thf(fact_175_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_176_The__split__eq,axiom,
! [A: $tType,B: $tType,X4: A,Y: B] :
( ( the @ ( product_prod @ A @ B )
@ ( product_case_prod @ A @ B @ $o
@ ^ [X7: A,Y7: B] :
( ( X4 = X7 )
& ( Y = Y7 ) ) ) )
= ( product_Pair @ A @ B @ X4 @ Y ) ) ).
% The_split_eq
thf(fact_177_in__inv__imagep,axiom,
! [B: $tType,A: $tType] :
( ( inv_imagep @ A @ B )
= ( ^ [R3: A > A > $o,F9: B > A,X5: B,Y5: B] : ( R3 @ ( F9 @ X5 ) @ ( F9 @ Y5 ) ) ) ) ).
% in_inv_imagep
thf(fact_178_UNIV__Times__UNIV,axiom,
! [B: $tType,A: $tType] :
( ( product_Sigma @ A @ B @ ( top_top @ ( set @ A ) )
@ ^ [Uu: A] : ( top_top @ ( set @ B ) ) )
= ( top_top @ ( set @ ( product_prod @ A @ B ) ) ) ) ).
% UNIV_Times_UNIV
thf(fact_179_rangeE,axiom,
! [A: $tType,B: $tType,B2: A,F: B > A] :
( ( member @ A @ B2 @ ( image @ B @ A @ F @ ( top_top @ ( set @ B ) ) ) )
=> ~ ! [X: B] :
( B2
!= ( F @ X ) ) ) ).
% rangeE
thf(fact_180_range__composition,axiom,
! [A: $tType,C: $tType,B: $tType,F: C > A,G: B > C] :
( ( image @ B @ A
@ ^ [X5: B] : ( F @ ( G @ X5 ) )
@ ( top_top @ ( set @ B ) ) )
= ( image @ C @ A @ F @ ( image @ B @ C @ G @ ( top_top @ ( set @ B ) ) ) ) ) ).
% range_composition
thf(fact_181_UNIV__def,axiom,
! [A: $tType] :
( ( top_top @ ( set @ A ) )
= ( collect @ A
@ ^ [X5: A] : $true ) ) ).
% UNIV_def
thf(fact_182_rangeI,axiom,
! [A: $tType,B: $tType,F: B > A,X4: B] : ( member @ A @ ( F @ X4 ) @ ( image @ B @ A @ F @ ( top_top @ ( set @ B ) ) ) ) ).
% rangeI
thf(fact_183_range__eqI,axiom,
! [A: $tType,B: $tType,B2: A,F: B > A,X4: B] :
( ( B2
= ( F @ X4 ) )
=> ( member @ A @ B2 @ ( image @ B @ A @ F @ ( top_top @ ( set @ B ) ) ) ) ) ).
% range_eqI
thf(fact_184_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_185_inv__imagep__def,axiom,
! [A: $tType,B: $tType] :
( ( inv_imagep @ B @ A )
= ( ^ [R3: B > B > $o,F9: A > B,X5: A,Y5: A] : ( R3 @ ( F9 @ X5 ) @ ( F9 @ Y5 ) ) ) ) ).
% inv_imagep_def
thf(fact_186_old_Orec__prod__def,axiom,
! [T: $tType,B: $tType,A: $tType] :
( ( product_rec_prod @ A @ B @ T )
= ( ^ [F12: A > B > T,X5: product_prod @ A @ B] : ( the @ T @ ( product_rec_set_prod @ A @ B @ T @ F12 @ X5 ) ) ) ) ).
% old.rec_prod_def
thf(fact_187_the__equality,axiom,
! [A: $tType,P3: A > $o,A2: A] :
( ( P3 @ A2 )
=> ( ! [X: A] :
( ( P3 @ X )
=> ( X = A2 ) )
=> ( ( the @ A @ P3 )
= A2 ) ) ) ).
% the_equality
thf(fact_188_the__eq__trivial,axiom,
! [A: $tType,A2: A] :
( ( the @ A
@ ^ [X5: A] : X5 = A2 )
= A2 ) ).
% the_eq_trivial
thf(fact_189_the__sym__eq__trivial,axiom,
! [A: $tType,X4: A] :
( ( the @ A
@ ( ^ [Y9: A,Z3: A] : Y9 = Z3
@ X4 ) )
= X4 ) ).
% the_sym_eq_trivial
thf(fact_190_the1__equality,axiom,
! [A: $tType,P3: A > $o,A2: A] :
( ? [X6: A] :
( ( P3 @ X6 )
& ! [Y4: A] :
( ( P3 @ Y4 )
=> ( Y4 = X6 ) ) )
=> ( ( P3 @ A2 )
=> ( ( the @ A @ P3 )
= A2 ) ) ) ).
% the1_equality
thf(fact_191_the1I2,axiom,
! [A: $tType,P3: A > $o,Q3: A > $o] :
( ? [X6: A] :
( ( P3 @ X6 )
& ! [Y4: A] :
( ( P3 @ Y4 )
=> ( Y4 = X6 ) ) )
=> ( ! [X: A] :
( ( P3 @ X )
=> ( Q3 @ X ) )
=> ( Q3 @ ( the @ A @ P3 ) ) ) ) ).
% the1I2
thf(fact_192_If__def,axiom,
! [A: $tType] :
( ( if @ A )
= ( ^ [P5: $o,X5: A,Y5: A] :
( the @ A
@ ^ [Z2: A] :
( ( P5
=> ( Z2 = X5 ) )
& ( ~ P5
=> ( Z2 = Y5 ) ) ) ) ) ) ).
% If_def
thf(fact_193_theI2,axiom,
! [A: $tType,P3: A > $o,A2: A,Q3: A > $o] :
( ( P3 @ A2 )
=> ( ! [X: A] :
( ( P3 @ X )
=> ( X = A2 ) )
=> ( ! [X: A] :
( ( P3 @ X )
=> ( Q3 @ X ) )
=> ( Q3 @ ( the @ A @ P3 ) ) ) ) ) ).
% theI2
thf(fact_194_theI_H,axiom,
! [A: $tType,P3: A > $o] :
( ? [X6: A] :
( ( P3 @ X6 )
& ! [Y4: A] :
( ( P3 @ Y4 )
=> ( Y4 = X6 ) ) )
=> ( P3 @ ( the @ A @ P3 ) ) ) ).
% theI'
thf(fact_195_theI,axiom,
! [A: $tType,P3: A > $o,A2: A] :
( ( P3 @ A2 )
=> ( ! [X: A] :
( ( P3 @ X )
=> ( X = A2 ) )
=> ( P3 @ ( the @ A @ P3 ) ) ) ) ).
% theI
thf(fact_196_surj__def,axiom,
! [B: $tType,A: $tType,F: B > A] :
( ( ( image @ B @ A @ F @ ( top_top @ ( set @ B ) ) )
= ( top_top @ ( set @ A ) ) )
= ( ! [Y5: A] :
? [X5: B] :
( Y5
= ( F @ X5 ) ) ) ) ).
% surj_def
thf(fact_197_surjI,axiom,
! [B: $tType,A: $tType,G: B > A,F: A > B] :
( ! [X: A] :
( ( G @ ( F @ X ) )
= X )
=> ( ( image @ B @ A @ G @ ( top_top @ ( set @ B ) ) )
= ( top_top @ ( set @ A ) ) ) ) ).
% surjI
thf(fact_198_surjD,axiom,
! [A: $tType,B: $tType,F: B > A,Y: A] :
( ( ( image @ B @ A @ F @ ( top_top @ ( set @ B ) ) )
= ( top_top @ ( set @ A ) ) )
=> ? [X: B] :
( Y
= ( F @ X ) ) ) ).
% surjD
thf(fact_199_surjE,axiom,
! [A: $tType,B: $tType,F: B > A,Y: A] :
( ( ( image @ B @ A @ F @ ( top_top @ ( set @ B ) ) )
= ( top_top @ ( set @ A ) ) )
=> ~ ! [X: B] :
( Y
!= ( F @ X ) ) ) ).
% surjE
thf(fact_200_old_Orec__unit__def,axiom,
! [T: $tType] :
( ( product_rec_unit @ T )
= ( ^ [F12: T,X5: product_unit] : ( the @ T @ ( product_rec_set_unit @ T @ F12 @ X5 ) ) ) ) ).
% old.rec_unit_def
thf(fact_201_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_202_fst__map__prod,axiom,
! [B: $tType,A: $tType,D: $tType,C: $tType,F: C > A,G: D > B,X4: product_prod @ C @ D] :
( ( product_fst @ A @ B @ ( product_map_prod @ C @ A @ D @ B @ F @ G @ X4 ) )
= ( F @ ( product_fst @ C @ D @ X4 ) ) ) ).
% fst_map_prod
thf(fact_203_fst__eqD,axiom,
! [B: $tType,A: $tType,X4: A,Y: B,A2: A] :
( ( ( product_fst @ A @ B @ ( product_Pair @ A @ B @ X4 @ Y ) )
= A2 )
=> ( X4 = A2 ) ) ).
% fst_eqD
thf(fact_204_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_205_fst__def,axiom,
! [B: $tType,A: $tType] :
( ( product_fst @ A @ B )
= ( product_case_prod @ A @ B @ A
@ ^ [X13: A,X23: B] : X13 ) ) ).
% fst_def
thf(fact_206_eq__fst__iff,axiom,
! [A: $tType,B: $tType,A2: A,P2: product_prod @ A @ B] :
( ( A2
= ( product_fst @ A @ B @ P2 ) )
= ( ? [B5: B] :
( P2
= ( product_Pair @ A @ B @ A2 @ B5 ) ) ) ) ).
% eq_fst_iff
thf(fact_207_fstI,axiom,
! [B: $tType,A: $tType,X4: product_prod @ A @ B,Y: A,Z: B] :
( ( X4
= ( product_Pair @ A @ B @ Y @ Z ) )
=> ( ( product_fst @ A @ B @ X4 )
= Y ) ) ).
% fstI
thf(fact_208_fst__image__times,axiom,
! [B: $tType,A: $tType,B6: set @ B,A3: set @ A] :
( ( ( B6
= ( bot_bot @ ( set @ B ) ) )
=> ( ( image @ ( product_prod @ A @ B ) @ A @ ( product_fst @ A @ B )
@ ( product_Sigma @ A @ B @ A3
@ ^ [Uu: A] : B6 ) )
= ( bot_bot @ ( set @ A ) ) ) )
& ( ( B6
!= ( bot_bot @ ( set @ B ) ) )
=> ( ( image @ ( product_prod @ A @ B ) @ A @ ( product_fst @ A @ B )
@ ( product_Sigma @ A @ B @ A3
@ ^ [Uu: A] : B6 ) )
= A3 ) ) ) ).
% fst_image_times
thf(fact_209_The__case__prod,axiom,
! [B: $tType,A: $tType,P3: A > B > $o] :
( ( the @ ( product_prod @ A @ B ) @ ( product_case_prod @ A @ B @ $o @ P3 ) )
= ( the @ ( product_prod @ A @ B )
@ ^ [Xy: product_prod @ A @ B] : ( P3 @ ( product_fst @ A @ B @ Xy ) @ ( product_snd @ A @ B @ Xy ) ) ) ) ).
% The_case_prod
thf(fact_210_image__is__empty,axiom,
! [A: $tType,B: $tType,F: B > A,A3: set @ B] :
( ( ( image @ B @ A @ F @ A3 )
= ( bot_bot @ ( set @ A ) ) )
= ( A3
= ( bot_bot @ ( set @ B ) ) ) ) ).
% image_is_empty
thf(fact_211_empty__is__image,axiom,
! [A: $tType,B: $tType,F: B > A,A3: set @ B] :
( ( ( bot_bot @ ( set @ A ) )
= ( image @ B @ A @ F @ A3 ) )
= ( A3
= ( bot_bot @ ( set @ B ) ) ) ) ).
% empty_is_image
thf(fact_212_image__empty,axiom,
! [B: $tType,A: $tType,F: B > A] :
( ( image @ B @ A @ F @ ( bot_bot @ ( set @ B ) ) )
= ( bot_bot @ ( set @ A ) ) ) ).
% image_empty
thf(fact_213_Sigma__empty1,axiom,
! [B: $tType,A: $tType,B6: A > ( set @ B )] :
( ( product_Sigma @ A @ B @ ( bot_bot @ ( set @ A ) ) @ B6 )
= ( bot_bot @ ( set @ ( product_prod @ A @ B ) ) ) ) ).
% Sigma_empty1
thf(fact_214_snd__map__prod,axiom,
! [B: $tType,A: $tType,D: $tType,C: $tType,F: C > B,G: D > A,X4: product_prod @ C @ D] :
( ( product_snd @ B @ A @ ( product_map_prod @ C @ B @ D @ A @ F @ G @ X4 ) )
= ( G @ ( product_snd @ C @ D @ X4 ) ) ) ).
% snd_map_prod
thf(fact_215_Collect__const,axiom,
! [A: $tType,P3: $o] :
( ( P3
=> ( ( collect @ A
@ ^ [S6: A] : P3 )
= ( top_top @ ( set @ A ) ) ) )
& ( ~ P3
=> ( ( collect @ A
@ ^ [S6: A] : P3 )
= ( bot_bot @ ( set @ A ) ) ) ) ) ).
% Collect_const
thf(fact_216_Sigma__empty2,axiom,
! [B: $tType,A: $tType,A3: set @ A] :
( ( product_Sigma @ A @ B @ A3
@ ^ [Uu: A] : ( bot_bot @ ( set @ B ) ) )
= ( bot_bot @ ( set @ ( product_prod @ A @ B ) ) ) ) ).
% Sigma_empty2
thf(fact_217_Times__empty,axiom,
! [A: $tType,B: $tType,A3: set @ A,B6: set @ B] :
( ( ( product_Sigma @ A @ B @ A3
@ ^ [Uu: A] : B6 )
= ( bot_bot @ ( set @ ( product_prod @ A @ B ) ) ) )
= ( ( A3
= ( bot_bot @ ( set @ A ) ) )
| ( B6
= ( bot_bot @ ( set @ B ) ) ) ) ) ).
% Times_empty
thf(fact_218_Collect__const__case__prod,axiom,
! [B: $tType,A: $tType,P3: $o] :
( ( P3
=> ( ( collect @ ( product_prod @ A @ B )
@ ( product_case_prod @ A @ B @ $o
@ ^ [A6: A,B5: B] : P3 ) )
= ( top_top @ ( set @ ( product_prod @ A @ B ) ) ) ) )
& ( ~ P3
=> ( ( collect @ ( product_prod @ A @ B )
@ ( product_case_prod @ A @ B @ $o
@ ^ [A6: A,B5: B] : P3 ) )
= ( bot_bot @ ( set @ ( product_prod @ A @ B ) ) ) ) ) ) ).
% Collect_const_case_prod
thf(fact_219_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_220_snd__swap,axiom,
! [B: $tType,A: $tType,X4: product_prod @ A @ B] :
( ( product_snd @ B @ A @ ( product_swap @ A @ B @ X4 ) )
= ( product_fst @ A @ B @ X4 ) ) ).
% snd_swap
thf(fact_221_fst__swap,axiom,
! [A: $tType,B: $tType,X4: product_prod @ B @ A] :
( ( product_fst @ A @ B @ ( product_swap @ B @ A @ X4 ) )
= ( product_snd @ B @ A @ X4 ) ) ).
% fst_swap
thf(fact_222_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_223_snd__image__times,axiom,
! [B: $tType,A: $tType,A3: set @ B,B6: set @ A] :
( ( ( A3
= ( bot_bot @ ( set @ B ) ) )
=> ( ( image @ ( product_prod @ B @ A ) @ A @ ( product_snd @ B @ A )
@ ( product_Sigma @ B @ A @ A3
@ ^ [Uu: B] : B6 ) )
= ( bot_bot @ ( set @ A ) ) ) )
& ( ( A3
!= ( bot_bot @ ( set @ B ) ) )
=> ( ( image @ ( product_prod @ B @ A ) @ A @ ( product_snd @ B @ A )
@ ( product_Sigma @ B @ A @ A3
@ ^ [Uu: B] : B6 ) )
= B6 ) ) ) ).
% snd_image_times
thf(fact_224_sndI,axiom,
! [A: $tType,B: $tType,X4: product_prod @ A @ B,Y: A,Z: B] :
( ( X4
= ( product_Pair @ A @ B @ Y @ Z ) )
=> ( ( product_snd @ A @ B @ X4 )
= Z ) ) ).
% sndI
thf(fact_225_eq__snd__iff,axiom,
! [A: $tType,B: $tType,B2: A,P2: product_prod @ B @ A] :
( ( B2
= ( product_snd @ B @ A @ P2 ) )
= ( ? [A6: B] :
( P2
= ( product_Pair @ B @ A @ A6 @ B2 ) ) ) ) ).
% eq_snd_iff
thf(fact_226_times__eq__iff,axiom,
! [A: $tType,B: $tType,A3: set @ A,B6: set @ B,C5: set @ A,D3: set @ B] :
( ( ( product_Sigma @ A @ B @ A3
@ ^ [Uu: A] : B6 )
= ( product_Sigma @ A @ B @ C5
@ ^ [Uu: A] : D3 ) )
= ( ( ( A3 = C5 )
& ( B6 = D3 ) )
| ( ( ( A3
= ( bot_bot @ ( set @ A ) ) )
| ( B6
= ( bot_bot @ ( set @ B ) ) ) )
& ( ( C5
= ( bot_bot @ ( set @ A ) ) )
| ( D3
= ( bot_bot @ ( set @ B ) ) ) ) ) ) ) ).
% times_eq_iff
thf(fact_227_empty__def,axiom,
! [A: $tType] :
( ( bot_bot @ ( set @ A ) )
= ( collect @ A
@ ^ [X5: A] : $false ) ) ).
% empty_def
thf(fact_228_snd__eqD,axiom,
! [B: $tType,A: $tType,X4: B,Y: A,A2: A] :
( ( ( product_snd @ B @ A @ ( product_Pair @ B @ A @ X4 @ Y ) )
= A2 )
=> ( Y = A2 ) ) ).
% snd_eqD
thf(fact_229_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_230_Sigma__empty__iff,axiom,
! [B: $tType,A: $tType,I4: set @ A,X8: A > ( set @ B )] :
( ( ( product_Sigma @ A @ B @ I4 @ X8 )
= ( bot_bot @ ( set @ ( product_prod @ A @ B ) ) ) )
= ( ! [X5: A] :
( ( member @ A @ X5 @ I4 )
=> ( ( X8 @ X5 )
= ( bot_bot @ ( set @ B ) ) ) ) ) ) ).
% Sigma_empty_iff
thf(fact_231_snd__def,axiom,
! [B: $tType,A: $tType] :
( ( product_snd @ A @ B )
= ( product_case_prod @ A @ B @ B
@ ^ [X13: A,X23: B] : X23 ) ) ).
% snd_def
thf(fact_232_split__comp__eq,axiom,
! [A: $tType,C: $tType,B: $tType,D: $tType,F: A > B > C,G: D > A] :
( ( ^ [U: product_prod @ D @ B] : ( F @ ( G @ ( product_fst @ D @ B @ U ) ) @ ( product_snd @ D @ B @ U ) ) )
= ( product_case_prod @ D @ B @ C
@ ^ [X5: D] : ( F @ ( G @ X5 ) ) ) ) ).
% split_comp_eq
thf(fact_233_case__prod__beta,axiom,
! [A: $tType,C: $tType,B: $tType] :
( ( product_case_prod @ B @ C @ A )
= ( ^ [F9: B > C > A,P: product_prod @ B @ C] : ( F9 @ ( product_fst @ B @ C @ P ) @ ( product_snd @ B @ C @ P ) ) ) ) ).
% case_prod_beta
thf(fact_234_case__prod__beta_H,axiom,
! [C: $tType,B: $tType,A: $tType] :
( ( product_case_prod @ A @ B @ C )
= ( ^ [F9: A > B > C,X5: product_prod @ A @ B] : ( F9 @ ( product_fst @ A @ B @ X5 ) @ ( product_snd @ A @ B @ X5 ) ) ) ) ).
% case_prod_beta'
thf(fact_235_prod_Ocase__eq__if,axiom,
! [C: $tType,B: $tType,A: $tType] :
( ( product_case_prod @ A @ B @ C )
= ( ^ [F9: A > B > C,Prod2: product_prod @ A @ B] : ( F9 @ ( product_fst @ A @ B @ Prod2 ) @ ( product_snd @ A @ B @ Prod2 ) ) ) ) ).
% prod.case_eq_if
thf(fact_236_case__prod__unfold,axiom,
! [C: $tType,B: $tType,A: $tType] :
( ( product_case_prod @ A @ B @ C )
= ( ^ [C4: A > B > C,P: product_prod @ A @ B] : ( C4 @ ( product_fst @ A @ B @ P ) @ ( product_snd @ A @ B @ P ) ) ) ) ).
% case_prod_unfold
thf(fact_237_Product__Type_OCollect__case__prodD,axiom,
! [B: $tType,A: $tType,X4: product_prod @ A @ B,A3: A > B > $o] :
( ( member @ ( product_prod @ A @ B ) @ X4 @ ( collect @ ( product_prod @ A @ B ) @ ( product_case_prod @ A @ B @ $o @ A3 ) ) )
=> ( A3 @ ( product_fst @ A @ B @ X4 ) @ ( product_snd @ A @ B @ X4 ) ) ) ).
% Product_Type.Collect_case_prodD
thf(fact_238_mem__Times__iff,axiom,
! [A: $tType,B: $tType,X4: product_prod @ A @ B,A3: set @ A,B6: set @ B] :
( ( member @ ( product_prod @ A @ B ) @ X4
@ ( product_Sigma @ A @ B @ A3
@ ^ [Uu: A] : B6 ) )
= ( ( member @ A @ ( product_fst @ A @ B @ X4 ) @ A3 )
& ( member @ B @ ( product_snd @ A @ B @ X4 ) @ B6 ) ) ) ).
% mem_Times_iff
thf(fact_239_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_240_surjective__pairing,axiom,
! [B: $tType,A: $tType,T2: product_prod @ A @ B] :
( T2
= ( product_Pair @ A @ B @ ( product_fst @ A @ B @ T2 ) @ ( product_snd @ A @ B @ T2 ) ) ) ).
% surjective_pairing
thf(fact_241_prod__eq__iff,axiom,
! [B: $tType,A: $tType] :
( ( ^ [Y9: product_prod @ A @ B,Z3: product_prod @ A @ B] : Y9 = Z3 )
= ( ^ [S6: product_prod @ A @ B,T3: product_prod @ A @ B] :
( ( ( product_fst @ A @ B @ S6 )
= ( product_fst @ A @ B @ T3 ) )
& ( ( product_snd @ A @ B @ S6 )
= ( product_snd @ A @ B @ T3 ) ) ) ) ) ).
% prod_eq_iff
thf(fact_242_prod_Oexpand,axiom,
! [B: $tType,A: $tType,Prod: product_prod @ A @ B,Prod3: product_prod @ A @ B] :
( ( ( ( product_fst @ A @ B @ Prod )
= ( product_fst @ A @ B @ Prod3 ) )
& ( ( product_snd @ A @ B @ Prod )
= ( product_snd @ A @ B @ Prod3 ) ) )
=> ( Prod = Prod3 ) ) ).
% prod.expand
thf(fact_243_prod__eqI,axiom,
! [B: $tType,A: $tType,P2: product_prod @ A @ B,Q: product_prod @ A @ B] :
( ( ( product_fst @ A @ B @ P2 )
= ( product_fst @ A @ B @ Q ) )
=> ( ( ( product_snd @ A @ B @ P2 )
= ( product_snd @ A @ B @ Q ) )
=> ( P2 = Q ) ) ) ).
% prod_eqI
thf(fact_244_prod_Osplit__sel,axiom,
! [C: $tType,B: $tType,A: $tType,P3: C > $o,F: A > B > C,Prod: product_prod @ A @ B] :
( ( P3 @ ( product_case_prod @ A @ B @ C @ F @ Prod ) )
= ( ( Prod
= ( product_Pair @ A @ B @ ( product_fst @ A @ B @ Prod ) @ ( product_snd @ A @ B @ Prod ) ) )
=> ( P3 @ ( F @ ( product_fst @ A @ B @ Prod ) @ ( product_snd @ A @ B @ Prod ) ) ) ) ) ).
% prod.split_sel
thf(fact_245_prod_Osplit__sel__asm,axiom,
! [C: $tType,B: $tType,A: $tType,P3: C > $o,F: A > B > C,Prod: product_prod @ A @ B] :
( ( P3 @ ( product_case_prod @ A @ B @ C @ F @ Prod ) )
= ( ~ ( ( Prod
= ( product_Pair @ A @ B @ ( product_fst @ A @ B @ Prod ) @ ( product_snd @ A @ B @ Prod ) ) )
& ~ ( P3 @ ( F @ ( product_fst @ A @ B @ Prod ) @ ( product_snd @ A @ B @ Prod ) ) ) ) ) ) ).
% prod.split_sel_asm
thf(fact_246_scomp__unfold,axiom,
! [D: $tType,C: $tType,B: $tType,A: $tType] :
( ( product_scomp @ A @ B @ C @ D )
= ( ^ [F9: A > ( product_prod @ B @ C ),G4: B > C > D,X5: A] : ( G4 @ ( product_fst @ B @ C @ ( F9 @ X5 ) ) @ ( product_snd @ B @ C @ ( F9 @ X5 ) ) ) ) ) ).
% scomp_unfold
thf(fact_247_prod_Oswap__def,axiom,
! [B: $tType,A: $tType] :
( ( product_swap @ A @ B )
= ( ^ [P: product_prod @ A @ B] : ( product_Pair @ B @ A @ ( product_snd @ A @ B @ P ) @ ( product_fst @ A @ B @ P ) ) ) ) ).
% prod.swap_def
thf(fact_248_fst__image__Sigma,axiom,
! [B: $tType,A: $tType,A3: set @ A,B6: A > ( set @ B )] :
( ( image @ ( product_prod @ A @ B ) @ A @ ( product_fst @ A @ B ) @ ( product_Sigma @ A @ B @ A3 @ B6 ) )
= ( collect @ A
@ ^ [X5: A] :
( ( member @ A @ X5 @ A3 )
& ( ( B6 @ X5 )
!= ( bot_bot @ ( set @ B ) ) ) ) ) ) ).
% fst_image_Sigma
thf(fact_249_exE__realizer,axiom,
! [C: $tType,A: $tType,B: $tType,P3: A > B > $o,P2: product_prod @ B @ A,Q3: C > $o,F: B > A > C] :
( ( P3 @ ( product_snd @ B @ A @ P2 ) @ ( product_fst @ B @ A @ P2 ) )
=> ( ! [X: B,Y4: A] :
( ( P3 @ Y4 @ X )
=> ( Q3 @ ( F @ X @ Y4 ) ) )
=> ( Q3 @ ( product_case_prod @ B @ A @ C @ F @ P2 ) ) ) ) ).
% exE_realizer
thf(fact_250_exI__realizer,axiom,
! [B: $tType,A: $tType,P3: A > B > $o,Y: A,X4: B] :
( ( P3 @ Y @ X4 )
=> ( P3 @ ( product_snd @ B @ A @ ( product_Pair @ B @ A @ X4 @ Y ) ) @ ( product_fst @ B @ A @ ( product_Pair @ B @ A @ X4 @ Y ) ) ) ) ).
% exI_realizer
thf(fact_251_conjI__realizer,axiom,
! [A: $tType,B: $tType,P3: A > $o,P2: A,Q3: B > $o,Q: B] :
( ( P3 @ P2 )
=> ( ( Q3 @ Q )
=> ( ( P3 @ ( product_fst @ A @ B @ ( product_Pair @ A @ B @ P2 @ Q ) ) )
& ( Q3 @ ( product_snd @ A @ B @ ( product_Pair @ A @ B @ P2 @ Q ) ) ) ) ) ) ).
% conjI_realizer
thf(fact_252_BNF__Greatest__Fixpoint_Osubst__Pair,axiom,
! [B: $tType,A: $tType,P3: A > B > $o,X4: A,Y: B,A2: product_prod @ A @ B] :
( ( P3 @ X4 @ Y )
=> ( ( A2
= ( product_Pair @ A @ B @ X4 @ Y ) )
=> ( P3 @ ( product_fst @ A @ B @ A2 ) @ ( product_snd @ A @ B @ A2 ) ) ) ) ).
% BNF_Greatest_Fixpoint.subst_Pair
thf(fact_253_the__elem__image__unique,axiom,
! [B: $tType,A: $tType,A3: set @ A,F: A > B,X4: A] :
( ( A3
!= ( bot_bot @ ( set @ A ) ) )
=> ( ! [Y4: A] :
( ( member @ A @ Y4 @ A3 )
=> ( ( F @ Y4 )
= ( F @ X4 ) ) )
=> ( ( the_elem @ B @ ( image @ A @ B @ F @ A3 ) )
= ( F @ X4 ) ) ) ) ).
% the_elem_image_unique
thf(fact_254_subset__fst__snd,axiom,
! [B: $tType,A: $tType,A3: set @ ( product_prod @ A @ B )] :
( ord_less_eq @ ( set @ ( product_prod @ A @ B ) ) @ A3
@ ( product_Sigma @ A @ B @ ( image @ ( product_prod @ A @ B ) @ A @ ( product_fst @ A @ B ) @ A3 )
@ ^ [Uu: A] : ( image @ ( product_prod @ A @ B ) @ B @ ( product_snd @ A @ B ) @ A3 ) ) ) ).
% subset_fst_snd
thf(fact_255_Collect__split__mono__strong,axiom,
! [B: $tType,A: $tType,X8: set @ A,A3: set @ ( product_prod @ A @ B ),Y6: set @ B,P3: A > B > $o,Q3: A > B > $o] :
( ( X8
= ( image @ ( product_prod @ A @ B ) @ A @ ( product_fst @ A @ B ) @ A3 ) )
=> ( ( Y6
= ( image @ ( product_prod @ A @ B ) @ B @ ( product_snd @ A @ B ) @ A3 ) )
=> ( ! [X: A] :
( ( member @ A @ X @ X8 )
=> ! [Xa: B] :
( ( member @ B @ Xa @ Y6 )
=> ( ( P3 @ X @ Xa )
=> ( Q3 @ X @ Xa ) ) ) )
=> ( ( ord_less_eq @ ( set @ ( product_prod @ A @ B ) ) @ A3 @ ( collect @ ( product_prod @ A @ B ) @ ( product_case_prod @ A @ B @ $o @ P3 ) ) )
=> ( ord_less_eq @ ( set @ ( product_prod @ A @ B ) ) @ A3 @ ( collect @ ( product_prod @ A @ B ) @ ( product_case_prod @ A @ B @ $o @ Q3 ) ) ) ) ) ) ) ).
% Collect_split_mono_strong
% 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,X4: A,Y: A] :
( ( if @ A @ $false @ X4 @ Y )
= Y ) ).
thf(help_If_1_1_T,axiom,
! [A: $tType,X4: A,Y: A] :
( ( if @ A @ $true @ X4 @ Y )
= X4 ) ).
% Conjectures (1)
thf(conj_0,conjecture,
? [I5: ( list @ char ) > paraco415392788lle_tv] :
( ( paraco876059933e_eval @ I5 @ p )
!= ( paraco2040174112le_Det @ $true ) ) ).
%------------------------------------------------------------------------------