TPTP Problem File: COM203^1.p
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%------------------------------------------------------------------------------
% File : COM203^1 : TPTP v8.2.0. Released v7.0.0.
% Domain : Computing Theory
% Problem : Parallel extension to grammars and languages 44
% Version : [Bla16] axioms : Especial.
% English :
% Refs : [BH+14] Blanchette et al. (2014), Truly Modular (Co)datatypes
% : [RB15] Reynolds & Blanchette (2015), A Decision Procedure for
% : [Bla16] Blanchette (2016), Email to Geoff Sutcliffe
% Source : [Bla16]
% Names : parallel__44.p [Bla16]
% Status : Theorem
% Rating : 0.00 v7.2.0, 0.25 v7.1.0
% Syntax : Number of formulae : 329 ( 123 unt; 57 typ; 0 def)
% Number of atoms : 592 ( 250 equ; 0 cnn)
% Maximal formula atoms : 7 ( 2 avg)
% Number of connectives : 4381 ( 46 ~; 2 |; 27 &;4031 @)
% ( 0 <=>; 275 =>; 0 <=; 0 <~>)
% Maximal formula depth : 28 ( 9 avg)
% Number of types : 4 ( 3 usr)
% Number of type conns : 363 ( 363 >; 0 *; 0 +; 0 <<)
% Number of symbols : 56 ( 54 usr; 3 con; 0-7 aty)
% Number of variables : 1272 ( 44 ^;1137 !; 18 ?;1272 :)
% ( 73 !>; 0 ?*; 0 @-; 0 @+)
% SPC : TH1_THM_EQU_NAR
% Comments : This file was generated by Isabelle (most likely Sledgehammer)
% 2016-07-13 14:38:49.144
%------------------------------------------------------------------------------
%----Could-be-implicit typings (8)
thf(ty_t_Product__Type_Oprod,type,
product_prod: $tType > $tType > $tType ).
thf(ty_t_Sum__Type_Osum,type,
sum_sum: $tType > $tType > $tType ).
thf(ty_t_Option_Ooption,type,
option: $tType > $tType ).
thf(ty_t_DTree_Odtree,type,
dtree: $tType ).
thf(ty_t_Set_Oset,type,
set: $tType > $tType ).
thf(ty_t_DTree_OT,type,
t: $tType ).
thf(ty_t_DTree_ON,type,
n: $tType ).
thf(ty_t_itself,type,
itself: $tType > $tType ).
%----Explicit typings (49)
thf(sy_cl_HOL_Otype,type,
type:
!>[A: $tType] : ( ( itself @ A ) > $o ) ).
thf(sy_cl_Orderings_Oord,type,
ord:
!>[A: $tType] : ( ( itself @ A ) > $o ) ).
thf(sy_cl_Orderings_Otop,type,
top:
!>[A: $tType] : ( ( itself @ A ) > $o ) ).
thf(sy_cl_Finite__Set_Ofinite,type,
finite_finite:
!>[A: $tType] : ( ( itself @ A ) > $o ) ).
thf(sy_cl_Nat_Osemiring__char__0,type,
semiring_char_0:
!>[A: $tType] : ( ( itself @ A ) > $o ) ).
thf(sy_cl_Orderings_Oorder__top,type,
order_top:
!>[A: $tType] : ( ( itself @ A ) > $o ) ).
thf(sy_c_BNF__Cardinal__Order__Relation_OrelChain,type,
bNF_Ca1785829860lChain:
!>[A: $tType,B: $tType] : ( ( set @ ( product_prod @ A @ A ) ) > ( A > B ) > $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_DTree_ONode,type,
node: n > ( set @ ( sum_sum @ t @ dtree ) ) > dtree ).
thf(sy_c_DTree_Ocont,type,
cont: dtree > ( set @ ( sum_sum @ t @ dtree ) ) ).
thf(sy_c_DTree_Ocorec,type,
corec:
!>[A: $tType] : ( ( A > n ) > ( A > ( set @ ( sum_sum @ t @ ( sum_sum @ dtree @ A ) ) ) ) > A > dtree ) ).
thf(sy_c_DTree_Odtree_Oroot,type,
root: dtree > n ).
thf(sy_c_DTree_Ounfold,type,
unfold:
!>[A: $tType] : ( ( A > n ) > ( A > ( set @ ( sum_sum @ t @ A ) ) ) > A > dtree ) ).
thf(sy_c_Finite__Set_Ofinite,type,
finite_finite2:
!>[A: $tType] : ( ( set @ A ) > $o ) ).
thf(sy_c_Fun_Oinj__on,type,
inj_on:
!>[A: $tType,B: $tType] : ( ( A > B ) > ( set @ A ) > $o ) ).
thf(sy_c_Fun_Oswap,type,
swap:
!>[A: $tType,B: $tType] : ( A > A > ( A > B ) > A > B ) ).
thf(sy_c_Lattices_Oinf__class_Oinf,type,
inf_inf:
!>[A: $tType] : ( A > A > 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_Parallel__Mirabelle__hykpkoupgu_ONplus,type,
parall1518086719_Nplus: n > n > n ).
thf(sy_c_Parallel__Mirabelle__hykpkoupgu_Opar,type,
parall1899940088le_par: ( product_prod @ dtree @ dtree ) > dtree ).
thf(sy_c_Parallel__Mirabelle__hykpkoupgu_Opar__c,type,
parall1914194347_par_c: ( product_prod @ dtree @ dtree ) > ( set @ ( sum_sum @ t @ ( product_prod @ dtree @ dtree ) ) ) ).
thf(sy_c_Parallel__Mirabelle__hykpkoupgu_Opar__r,type,
parall1914194362_par_r: ( product_prod @ dtree @ dtree ) > n ).
thf(sy_c_Parallel__Mirabelle__hykpkoupgu_Opar__r__rel,type,
parall556292031_r_rel: ( product_prod @ dtree @ dtree ) > ( product_prod @ dtree @ dtree ) > $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_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_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_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_Pure_Otype,type,
type2:
!>[A: $tType] : ( itself @ 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_Set_OCollect,type,
collect:
!>[A: $tType] : ( ( A > $o ) > ( set @ A ) ) ).
thf(sy_c_Set_OPow,type,
pow:
!>[A: $tType] : ( ( set @ A ) > ( set @ ( set @ A ) ) ) ).
thf(sy_c_Set_Oimage,type,
image:
!>[A: $tType,B: $tType] : ( ( A > B ) > ( set @ A ) > ( set @ B ) ) ).
thf(sy_c_Set_Ovimage,type,
vimage:
!>[A: $tType,B: $tType] : ( ( A > B ) > ( set @ B ) > ( set @ A ) ) ).
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_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_tr1,type,
tr1: dtree ).
thf(sy_v_tr2,type,
tr2: dtree ).
%----Relevant facts (256)
thf(fact_0__092_060open_062root_A_Iunfold_Apar__r_Apar__c_A_Itr1_M_Atr2_J_J_A_061_Apar__r_A_Itr1_M_Atr2_J_092_060close_062,axiom,
( ( root @ ( unfold @ ( product_prod @ dtree @ dtree ) @ parall1914194362_par_r @ parall1914194347_par_c @ ( product_Pair @ dtree @ dtree @ tr1 @ tr2 ) ) )
= ( parall1914194362_par_r @ ( product_Pair @ dtree @ dtree @ tr1 @ tr2 ) ) ) ).
% \<open>root (unfold par_r par_c (tr1, tr2)) = par_r (tr1, tr2)\<close>
thf(fact_1_Nplus__comm,axiom,
( parall1518086719_Nplus
= ( ^ [A2: n,B2: n] : ( parall1518086719_Nplus @ B2 @ A2 ) ) ) ).
% Nplus_comm
thf(fact_2_Nplus__assoc,axiom,
! [A3: n,B3: n,C2: n] :
( ( parall1518086719_Nplus @ ( parall1518086719_Nplus @ A3 @ B3 ) @ C2 )
= ( parall1518086719_Nplus @ A3 @ ( parall1518086719_Nplus @ B3 @ C2 ) ) ) ).
% Nplus_assoc
thf(fact_3_par__r_Ocases,axiom,
! [X: product_prod @ dtree @ dtree] :
~ ! [Tr1: dtree,Tr2: dtree] :
( X
!= ( product_Pair @ dtree @ dtree @ Tr1 @ Tr2 ) ) ).
% par_r.cases
thf(fact_4_par__r_Oelims,axiom,
! [X: product_prod @ dtree @ dtree,Y: n] :
( ( ( parall1914194362_par_r @ X )
= Y )
=> ~ ! [Tr1: dtree,Tr2: dtree] :
( ( X
= ( product_Pair @ dtree @ dtree @ Tr1 @ Tr2 ) )
=> ( Y
!= ( parall1518086719_Nplus @ ( root @ Tr1 ) @ ( root @ Tr2 ) ) ) ) ) ).
% par_r.elims
thf(fact_5_par__r_Osimps,axiom,
! [Tr12: dtree,Tr22: dtree] :
( ( parall1914194362_par_r @ ( product_Pair @ dtree @ dtree @ Tr12 @ Tr22 ) )
= ( parall1518086719_Nplus @ ( root @ Tr12 ) @ ( root @ Tr22 ) ) ) ).
% par_r.simps
thf(fact_6_par__r_Oinduct,axiom,
! [P: ( product_prod @ dtree @ dtree ) > $o,A0: product_prod @ dtree @ dtree] :
( ! [Tr1: dtree,Tr2: dtree] : ( P @ ( product_Pair @ dtree @ dtree @ Tr1 @ Tr2 ) )
=> ( P @ A0 ) ) ).
% par_r.induct
thf(fact_7_par__def,axiom,
( parall1899940088le_par
= ( unfold @ ( product_prod @ dtree @ dtree ) @ parall1914194362_par_r @ parall1914194347_par_c ) ) ).
% par_def
thf(fact_8_unfold_I1_J,axiom,
! [A: $tType,Rt: A > n,Ct: A > ( set @ ( sum_sum @ t @ A ) ),B3: A] :
( ( root @ ( unfold @ A @ Rt @ Ct @ B3 ) )
= ( Rt @ B3 ) ) ).
% unfold(1)
thf(fact_9_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_10_old_Oprod_Oinject,axiom,
! [A: $tType,B: $tType,A3: A,B3: B,A4: A,B4: B] :
( ( ( product_Pair @ A @ B @ A3 @ B3 )
= ( product_Pair @ A @ B @ A4 @ B4 ) )
= ( ( A3 = A4 )
& ( B3 = B4 ) ) ) ).
% old.prod.inject
thf(fact_11_par__r_Opelims,axiom,
! [X: product_prod @ dtree @ dtree,Y: n] :
( ( ( parall1914194362_par_r @ X )
= Y )
=> ( ( accp @ ( product_prod @ dtree @ dtree ) @ parall556292031_r_rel @ X )
=> ~ ! [Tr1: dtree,Tr2: dtree] :
( ( X
= ( product_Pair @ dtree @ dtree @ Tr1 @ Tr2 ) )
=> ( ( Y
= ( parall1518086719_Nplus @ ( root @ Tr1 ) @ ( root @ Tr2 ) ) )
=> ~ ( accp @ ( product_prod @ dtree @ dtree ) @ parall556292031_r_rel @ ( product_Pair @ dtree @ dtree @ Tr1 @ Tr2 ) ) ) ) ) ) ).
% par_r.pelims
thf(fact_12_surj__pair,axiom,
! [A: $tType,B: $tType,P2: product_prod @ A @ B] :
? [X3: A,Y3: B] :
( P2
= ( product_Pair @ A @ B @ X3 @ Y3 ) ) ).
% surj_pair
thf(fact_13_prod__cases,axiom,
! [B: $tType,A: $tType,P: ( product_prod @ A @ B ) > $o,P2: product_prod @ A @ B] :
( ! [A5: A,B5: B] : ( P @ ( product_Pair @ A @ B @ A5 @ B5 ) )
=> ( P @ P2 ) ) ).
% prod_cases
thf(fact_14_Pair__inject,axiom,
! [A: $tType,B: $tType,A3: A,B3: B,A4: A,B4: B] :
( ( ( product_Pair @ A @ B @ A3 @ B3 )
= ( product_Pair @ A @ B @ A4 @ B4 ) )
=> ~ ( ( A3 = A4 )
=> ( B3 != B4 ) ) ) ).
% Pair_inject
thf(fact_15_prod__cases3,axiom,
! [A: $tType,B: $tType,C: $tType,Y: product_prod @ A @ ( product_prod @ B @ C )] :
~ ! [A5: A,B5: B,C3: C] :
( Y
!= ( product_Pair @ A @ ( product_prod @ B @ C ) @ A5 @ ( product_Pair @ B @ C @ B5 @ C3 ) ) ) ).
% prod_cases3
thf(fact_16_prod__cases4,axiom,
! [A: $tType,B: $tType,C: $tType,D: $tType,Y: product_prod @ A @ ( product_prod @ B @ ( product_prod @ C @ D ) )] :
~ ! [A5: A,B5: B,C3: C,D2: D] :
( Y
!= ( product_Pair @ A @ ( product_prod @ B @ ( product_prod @ C @ D ) ) @ A5 @ ( product_Pair @ B @ ( product_prod @ C @ D ) @ B5 @ ( product_Pair @ C @ D @ C3 @ D2 ) ) ) ) ).
% prod_cases4
thf(fact_17_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,B5: B,C3: 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 ) ) @ B5 @ ( product_Pair @ C @ ( product_prod @ D @ E ) @ C3 @ ( product_Pair @ D @ E @ D2 @ E2 ) ) ) ) ) ).
% prod_cases5
thf(fact_18_prod__cases6,axiom,
! [A: $tType,B: $tType,C: $tType,D: $tType,E: $tType,F: $tType,Y: product_prod @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D @ ( product_prod @ E @ F ) ) ) )] :
~ ! [A5: A,B5: B,C3: C,D2: D,E2: E,F2: F] :
( Y
!= ( product_Pair @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D @ ( product_prod @ E @ F ) ) ) ) @ A5 @ ( product_Pair @ B @ ( product_prod @ C @ ( product_prod @ D @ ( product_prod @ E @ F ) ) ) @ B5 @ ( product_Pair @ C @ ( product_prod @ D @ ( product_prod @ E @ F ) ) @ C3 @ ( product_Pair @ D @ ( product_prod @ E @ F ) @ D2 @ ( product_Pair @ E @ F @ E2 @ F2 ) ) ) ) ) ) ).
% prod_cases6
thf(fact_19_old_Oprod_Oinducts,axiom,
! [B: $tType,A: $tType,P: ( product_prod @ A @ B ) > $o,Prod: product_prod @ A @ B] :
( ! [A5: A,B5: B] : ( P @ ( product_Pair @ A @ B @ A5 @ B5 ) )
=> ( P @ Prod ) ) ).
% old.prod.inducts
thf(fact_20_old_Oprod_Oexhaust,axiom,
! [A: $tType,B: $tType,Y: product_prod @ A @ B] :
~ ! [A5: A,B5: B] :
( Y
!= ( product_Pair @ A @ B @ A5 @ B5 ) ) ).
% old.prod.exhaust
thf(fact_21_prod__induct7,axiom,
! [G: $tType,F: $tType,E: $tType,D: $tType,C: $tType,B: $tType,A: $tType,P: ( product_prod @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D @ ( product_prod @ E @ ( product_prod @ F @ G ) ) ) ) ) ) > $o,X: product_prod @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D @ ( product_prod @ E @ ( product_prod @ F @ G ) ) ) ) )] :
( ! [A5: A,B5: B,C3: C,D2: D,E2: E,F2: F,G2: G] : ( P @ ( product_Pair @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D @ ( product_prod @ E @ ( product_prod @ F @ G ) ) ) ) ) @ A5 @ ( product_Pair @ B @ ( product_prod @ C @ ( product_prod @ D @ ( product_prod @ E @ ( product_prod @ F @ G ) ) ) ) @ B5 @ ( product_Pair @ C @ ( product_prod @ D @ ( product_prod @ E @ ( product_prod @ F @ G ) ) ) @ C3 @ ( product_Pair @ D @ ( product_prod @ E @ ( product_prod @ F @ G ) ) @ D2 @ ( product_Pair @ E @ ( product_prod @ F @ G ) @ E2 @ ( product_Pair @ F @ G @ F2 @ G2 ) ) ) ) ) ) )
=> ( P @ X ) ) ).
% prod_induct7
thf(fact_22_prod__induct6,axiom,
! [F: $tType,E: $tType,D: $tType,C: $tType,B: $tType,A: $tType,P: ( product_prod @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D @ ( product_prod @ E @ F ) ) ) ) ) > $o,X: product_prod @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D @ ( product_prod @ E @ F ) ) ) )] :
( ! [A5: A,B5: B,C3: C,D2: D,E2: E,F2: F] : ( P @ ( product_Pair @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D @ ( product_prod @ E @ F ) ) ) ) @ A5 @ ( product_Pair @ B @ ( product_prod @ C @ ( product_prod @ D @ ( product_prod @ E @ F ) ) ) @ B5 @ ( product_Pair @ C @ ( product_prod @ D @ ( product_prod @ E @ F ) ) @ C3 @ ( product_Pair @ D @ ( product_prod @ E @ F ) @ D2 @ ( product_Pair @ E @ F @ E2 @ F2 ) ) ) ) ) )
=> ( P @ X ) ) ).
% prod_induct6
thf(fact_23_prod__induct5,axiom,
! [E: $tType,D: $tType,C: $tType,B: $tType,A: $tType,P: ( 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,B5: B,C3: C,D2: D,E2: E] : ( P @ ( product_Pair @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D @ E ) ) ) @ A5 @ ( product_Pair @ B @ ( product_prod @ C @ ( product_prod @ D @ E ) ) @ B5 @ ( product_Pair @ C @ ( product_prod @ D @ E ) @ C3 @ ( product_Pair @ D @ E @ D2 @ E2 ) ) ) ) )
=> ( P @ X ) ) ).
% prod_induct5
thf(fact_24_prod__induct4,axiom,
! [D: $tType,C: $tType,B: $tType,A: $tType,P: ( product_prod @ A @ ( product_prod @ B @ ( product_prod @ C @ D ) ) ) > $o,X: product_prod @ A @ ( product_prod @ B @ ( product_prod @ C @ D ) )] :
( ! [A5: A,B5: B,C3: C,D2: D] : ( P @ ( product_Pair @ A @ ( product_prod @ B @ ( product_prod @ C @ D ) ) @ A5 @ ( product_Pair @ B @ ( product_prod @ C @ D ) @ B5 @ ( product_Pair @ C @ D @ C3 @ D2 ) ) ) )
=> ( P @ X ) ) ).
% prod_induct4
thf(fact_25_prod__induct3,axiom,
! [C: $tType,B: $tType,A: $tType,P: ( product_prod @ A @ ( product_prod @ B @ C ) ) > $o,X: product_prod @ A @ ( product_prod @ B @ C )] :
( ! [A5: A,B5: B,C3: C] : ( P @ ( product_Pair @ A @ ( product_prod @ B @ C ) @ A5 @ ( product_Pair @ B @ C @ B5 @ C3 ) ) )
=> ( P @ X ) ) ).
% prod_induct3
thf(fact_26_prod__cases7,axiom,
! [A: $tType,B: $tType,C: $tType,D: $tType,E: $tType,F: $tType,G: $tType,Y: product_prod @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D @ ( product_prod @ E @ ( product_prod @ F @ G ) ) ) ) )] :
~ ! [A5: A,B5: B,C3: C,D2: D,E2: E,F2: F,G2: G] :
( Y
!= ( product_Pair @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D @ ( product_prod @ E @ ( product_prod @ F @ G ) ) ) ) ) @ A5 @ ( product_Pair @ B @ ( product_prod @ C @ ( product_prod @ D @ ( product_prod @ E @ ( product_prod @ F @ G ) ) ) ) @ B5 @ ( product_Pair @ C @ ( product_prod @ D @ ( product_prod @ E @ ( product_prod @ F @ G ) ) ) @ C3 @ ( product_Pair @ D @ ( product_prod @ E @ ( product_prod @ F @ G ) ) @ D2 @ ( product_Pair @ E @ ( product_prod @ F @ G ) @ E2 @ ( product_Pair @ F @ G @ F2 @ G2 ) ) ) ) ) ) ) ).
% prod_cases7
thf(fact_27_old_Oprod_Orec,axiom,
! [A: $tType,T: $tType,B: $tType,F1: A > B > T,A3: A,B3: B] :
( ( product_rec_prod @ A @ B @ T @ F1 @ ( product_Pair @ A @ B @ A3 @ B3 ) )
= ( F1 @ A3 @ B3 ) ) ).
% old.prod.rec
thf(fact_28_internal__case__prod__conv,axiom,
! [B: $tType,A: $tType,C: $tType,C2: B > C > A,A3: B,B3: C] :
( ( produc2004651681e_prod @ B @ C @ A @ C2 @ ( product_Pair @ B @ C @ A3 @ B3 ) )
= ( C2 @ A3 @ B3 ) ) ).
% internal_case_prod_conv
thf(fact_29_root__Node,axiom,
! [N: n,As: set @ ( sum_sum @ t @ dtree )] :
( ( root @ ( node @ N @ As ) )
= N ) ).
% root_Node
thf(fact_30_accp_Ocases,axiom,
! [A: $tType,R: A > A > $o,A3: A] :
( ( accp @ A @ R @ A3 )
=> ! [Y4: A] :
( ( R @ Y4 @ A3 )
=> ( accp @ A @ R @ Y4 ) ) ) ).
% accp.cases
thf(fact_31_accp_Osimps,axiom,
! [A: $tType] :
( ( accp @ A )
= ( ^ [R2: A > A > $o,A2: A] :
? [X4: A] :
( ( A2 = X4 )
& ! [Y5: A] :
( ( R2 @ Y5 @ X4 )
=> ( accp @ A @ R2 @ Y5 ) ) ) ) ) ).
% accp.simps
thf(fact_32_accp_Ointros,axiom,
! [A: $tType,R: A > A > $o,X: A] :
( ! [Y3: A] :
( ( R @ Y3 @ X )
=> ( accp @ A @ R @ Y3 ) )
=> ( accp @ A @ R @ X ) ) ).
% accp.intros
thf(fact_33_accp__induct,axiom,
! [A: $tType,R: A > A > $o,A3: A,P: A > $o] :
( ( accp @ A @ R @ A3 )
=> ( ! [X3: A] :
( ( accp @ A @ R @ X3 )
=> ( ! [Y4: A] :
( ( R @ Y4 @ X3 )
=> ( P @ Y4 ) )
=> ( P @ X3 ) ) )
=> ( P @ A3 ) ) ) ).
% accp_induct
thf(fact_34_accp_Oinducts,axiom,
! [A: $tType,R: A > A > $o,X: A,P: A > $o] :
( ( accp @ A @ R @ X )
=> ( ! [X3: A] :
( ! [Y4: A] :
( ( R @ Y4 @ X3 )
=> ( accp @ A @ R @ Y4 ) )
=> ( ! [Y4: A] :
( ( R @ Y4 @ X3 )
=> ( P @ Y4 ) )
=> ( P @ X3 ) ) )
=> ( P @ X ) ) ) ).
% accp.inducts
thf(fact_35_accp__downward,axiom,
! [A: $tType,R: A > A > $o,B3: A,A3: A] :
( ( accp @ A @ R @ B3 )
=> ( ( R @ A3 @ B3 )
=> ( accp @ A @ R @ A3 ) ) ) ).
% accp_downward
thf(fact_36_not__accp__down,axiom,
! [A: $tType,R3: A > A > $o,X: A] :
( ~ ( accp @ A @ R3 @ X )
=> ~ ! [Z: A] :
( ( R3 @ Z @ X )
=> ( accp @ A @ R3 @ Z ) ) ) ).
% not_accp_down
thf(fact_37_accp__induct__rule,axiom,
! [A: $tType,R: A > A > $o,A3: A,P: A > $o] :
( ( accp @ A @ R @ A3 )
=> ( ! [X3: A] :
( ( accp @ A @ R @ X3 )
=> ( ! [Y4: A] :
( ( R @ Y4 @ X3 )
=> ( P @ Y4 ) )
=> ( P @ X3 ) ) )
=> ( P @ A3 ) ) ) ).
% accp_induct_rule
thf(fact_38_in__lex__prod,axiom,
! [A: $tType,B: $tType,A3: A,B3: B,A4: A,B4: 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 @ A3 @ B3 ) @ ( product_Pair @ A @ B @ A4 @ B4 ) ) @ ( lex_prod @ A @ B @ R @ S ) )
= ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A3 @ A4 ) @ R )
| ( ( A3 = A4 )
& ( member @ ( product_prod @ B @ B ) @ ( product_Pair @ B @ B @ B3 @ B4 ) @ S ) ) ) ) ).
% in_lex_prod
thf(fact_39_Node__root__cont,axiom,
! [Tr: dtree] :
( ( node @ ( root @ Tr ) @ ( cont @ Tr ) )
= Tr ) ).
% Node_root_cont
thf(fact_40_corec_I1_J,axiom,
! [A: $tType,Rt: A > n,Ct: A > ( set @ ( sum_sum @ t @ ( sum_sum @ dtree @ A ) ) ),B3: A] :
( ( root @ ( corec @ A @ Rt @ Ct @ B3 ) )
= ( Rt @ B3 ) ) ).
% corec(1)
thf(fact_41_ssubst__Pair__rhs,axiom,
! [B: $tType,A: $tType,R: A,S: B,R3: set @ ( product_prod @ A @ B ),S2: B] :
( ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ R @ S ) @ R3 )
=> ( ( S2 = S )
=> ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ R @ S2 ) @ R3 ) ) ) ).
% ssubst_Pair_rhs
thf(fact_42_curry__conv,axiom,
! [A: $tType,C: $tType,B: $tType] :
( ( product_curry @ B @ C @ A )
= ( ^ [F3: ( product_prod @ B @ C ) > A,A2: B,B2: C] : ( F3 @ ( product_Pair @ B @ C @ A2 @ B2 ) ) ) ) ).
% curry_conv
thf(fact_43_finite__par__c,axiom,
! [Tr12: dtree,Tr22: dtree] : ( finite_finite2 @ ( sum_sum @ t @ ( product_prod @ dtree @ dtree ) ) @ ( parall1914194347_par_c @ ( product_Pair @ dtree @ dtree @ Tr12 @ Tr22 ) ) ) ).
% finite_par_c
thf(fact_44_curryI,axiom,
! [A: $tType,B: $tType,F4: ( product_prod @ A @ B ) > $o,A3: A,B3: B] :
( ( F4 @ ( product_Pair @ A @ B @ A3 @ B3 ) )
=> ( product_curry @ A @ B @ $o @ F4 @ A3 @ B3 ) ) ).
% curryI
thf(fact_45_mem__Collect__eq,axiom,
! [A: $tType,A3: A,P: A > $o] :
( ( member @ A @ A3 @ ( collect @ A @ P ) )
= ( P @ A3 ) ) ).
% mem_Collect_eq
thf(fact_46_Collect__mem__eq,axiom,
! [A: $tType,A6: set @ A] :
( ( collect @ A
@ ^ [X4: A] : ( member @ A @ X4 @ A6 ) )
= A6 ) ).
% Collect_mem_eq
thf(fact_47_Collect__cong,axiom,
! [A: $tType,P: A > $o,Q: A > $o] :
( ! [X3: A] :
( ( P @ X3 )
= ( Q @ X3 ) )
=> ( ( collect @ A @ P )
= ( collect @ A @ Q ) ) ) ).
% Collect_cong
thf(fact_48_ext,axiom,
! [B: $tType,A: $tType,F4: A > B,G3: A > B] :
( ! [X3: A] :
( ( F4 @ X3 )
= ( G3 @ X3 ) )
=> ( F4 = G3 ) ) ).
% ext
thf(fact_49_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_50_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_51_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_52_curry__case__prod,axiom,
! [C: $tType,B: $tType,A: $tType,F4: A > B > C] :
( ( product_curry @ A @ B @ C @ ( product_case_prod @ A @ B @ C @ F4 ) )
= F4 ) ).
% curry_case_prod
thf(fact_53_case__prod__curry,axiom,
! [C: $tType,B: $tType,A: $tType,F4: ( product_prod @ A @ B ) > C] :
( ( product_case_prod @ A @ B @ C @ ( product_curry @ A @ B @ C @ F4 ) )
= F4 ) ).
% case_prod_curry
thf(fact_54_mem__case__prodE,axiom,
! [B: $tType,A: $tType,C: $tType,Z2: A,C2: B > C > ( set @ A ),P2: product_prod @ B @ C] :
( ( member @ A @ Z2 @ ( product_case_prod @ B @ C @ ( set @ A ) @ C2 @ P2 ) )
=> ~ ! [X3: B,Y3: C] :
( ( P2
= ( product_Pair @ B @ C @ X3 @ Y3 ) )
=> ~ ( member @ A @ Z2 @ ( C2 @ X3 @ Y3 ) ) ) ) ).
% mem_case_prodE
thf(fact_55_old_Oprod_Ocase,axiom,
! [A: $tType,C: $tType,B: $tType,F4: A > B > C,X1: A,X2: B] :
( ( product_case_prod @ A @ B @ C @ F4 @ ( product_Pair @ A @ B @ X1 @ X2 ) )
= ( F4 @ X1 @ X2 ) ) ).
% old.prod.case
thf(fact_56_dtree__cong,axiom,
! [Tr: dtree,Tr3: dtree] :
( ( ( root @ Tr )
= ( root @ Tr3 ) )
=> ( ( ( cont @ Tr )
= ( cont @ Tr3 ) )
=> ( Tr = Tr3 ) ) ) ).
% dtree_cong
thf(fact_57_curryE,axiom,
! [A: $tType,B: $tType,F4: ( product_prod @ A @ B ) > $o,A3: A,B3: B] :
( ( product_curry @ A @ B @ $o @ F4 @ A3 @ B3 )
=> ( F4 @ ( product_Pair @ A @ B @ A3 @ B3 ) ) ) ).
% curryE
thf(fact_58_curryD,axiom,
! [A: $tType,B: $tType,F4: ( product_prod @ A @ B ) > $o,A3: A,B3: B] :
( ( product_curry @ A @ B @ $o @ F4 @ A3 @ B3 )
=> ( F4 @ ( product_Pair @ A @ B @ A3 @ B3 ) ) ) ).
% curryD
thf(fact_59_finite__code,axiom,
! [A: $tType] :
( ( finite_finite @ A @ ( type2 @ A ) )
=> ( ( finite_finite2 @ A )
= ( ^ [A7: set @ A] : $true ) ) ) ).
% finite_code
thf(fact_60_split__cong,axiom,
! [C: $tType,B: $tType,A: $tType,Q2: product_prod @ A @ B,F4: A > B > C,G3: A > B > C,P2: product_prod @ A @ B] :
( ! [X3: A,Y3: B] :
( ( ( product_Pair @ A @ B @ X3 @ Y3 )
= Q2 )
=> ( ( F4 @ X3 @ Y3 )
= ( G3 @ X3 @ Y3 ) ) )
=> ( ( P2 = Q2 )
=> ( ( product_case_prod @ A @ B @ C @ F4 @ P2 )
= ( product_case_prod @ A @ B @ C @ G3 @ Q2 ) ) ) ) ).
% split_cong
thf(fact_61_same__fstI,axiom,
! [B: $tType,A: $tType,P: A > $o,X: A,Y6: B,Y: B,R3: A > ( set @ ( product_prod @ B @ B ) )] :
( ( P @ X )
=> ( ( member @ ( product_prod @ B @ B ) @ ( product_Pair @ B @ B @ Y6 @ Y ) @ ( R3 @ 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 @ Y6 ) @ ( product_Pair @ A @ B @ X @ Y ) ) @ ( same_fst @ A @ B @ P @ R3 ) ) ) ) ).
% same_fstI
thf(fact_62_cont__Node,axiom,
! [As: set @ ( sum_sum @ t @ dtree ),N: n] :
( ( finite_finite2 @ ( sum_sum @ t @ dtree ) @ As )
=> ( ( cont @ ( node @ N @ As ) )
= As ) ) ).
% cont_Node
thf(fact_63_finite__set__choice,axiom,
! [B: $tType,A: $tType,A6: set @ A,P: A > B > $o] :
( ( finite_finite2 @ A @ A6 )
=> ( ! [X3: A] :
( ( member @ A @ X3 @ A6 )
=> ? [X12: B] : ( P @ X3 @ X12 ) )
=> ? [F2: A > B] :
! [X5: A] :
( ( member @ A @ X5 @ A6 )
=> ( P @ X5 @ ( F2 @ X5 ) ) ) ) ) ).
% finite_set_choice
thf(fact_64_finite,axiom,
! [A: $tType] :
( ( finite_finite @ A @ ( type2 @ A ) )
=> ! [A6: set @ A] : ( finite_finite2 @ A @ A6 ) ) ).
% finite
thf(fact_65_scomp__apply,axiom,
! [A: $tType,D: $tType,C: $tType,B: $tType] :
( ( product_scomp @ B @ C @ D @ A )
= ( ^ [F3: B > ( product_prod @ C @ D ),G4: C > D > A,X4: B] : ( product_case_prod @ C @ D @ A @ G4 @ ( F3 @ X4 ) ) ) ) ).
% scomp_apply
thf(fact_66_pair__in__swap__image,axiom,
! [A: $tType,B: $tType,Y: A,X: B,A6: 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 ) @ A6 ) )
= ( member @ ( product_prod @ B @ A ) @ ( product_Pair @ B @ A @ X @ Y ) @ A6 ) ) ).
% pair_in_swap_image
thf(fact_67_finite__imageI,axiom,
! [B: $tType,A: $tType,F5: set @ A,H: A > B] :
( ( finite_finite2 @ A @ F5 )
=> ( finite_finite2 @ B @ ( image @ A @ B @ H @ F5 ) ) ) ).
% finite_imageI
thf(fact_68_dtree__simps,axiom,
! [As: set @ ( sum_sum @ t @ dtree ),As2: set @ ( sum_sum @ t @ dtree ),N: n,N2: n] :
( ( finite_finite2 @ ( sum_sum @ t @ dtree ) @ As )
=> ( ( finite_finite2 @ ( sum_sum @ t @ dtree ) @ As2 )
=> ( ( ( node @ N @ As )
= ( node @ N2 @ As2 ) )
= ( ( N = N2 )
& ( As = As2 ) ) ) ) ) ).
% dtree_simps
thf(fact_69_finite__cont,axiom,
! [Tr: dtree] : ( finite_finite2 @ ( sum_sum @ t @ dtree ) @ ( cont @ Tr ) ) ).
% finite_cont
thf(fact_70_dtree__cases,axiom,
! [Tr: dtree] :
~ ! [N3: n,As3: set @ ( sum_sum @ t @ dtree )] :
( ( finite_finite2 @ ( sum_sum @ t @ dtree ) @ As3 )
=> ( Tr
!= ( node @ N3 @ As3 ) ) ) ).
% dtree_cases
thf(fact_71_scomp__Pair,axiom,
! [C: $tType,B: $tType,A: $tType,X: A > ( product_prod @ B @ C )] :
( ( product_scomp @ A @ B @ C @ ( product_prod @ B @ C ) @ X @ ( product_Pair @ B @ C ) )
= X ) ).
% scomp_Pair
thf(fact_72_Pair__scomp,axiom,
! [A: $tType,B: $tType,C: $tType,X: C,F4: C > A > B] :
( ( product_scomp @ A @ C @ A @ B @ ( product_Pair @ C @ A @ X ) @ F4 )
= ( F4 @ X ) ) ).
% Pair_scomp
thf(fact_73_scomp__def,axiom,
! [D: $tType,C: $tType,B: $tType,A: $tType] :
( ( product_scomp @ A @ B @ C @ D )
= ( ^ [F3: A > ( product_prod @ B @ C ),G4: B > C > D,X4: A] : ( product_case_prod @ B @ C @ D @ G4 @ ( F3 @ X4 ) ) ) ) ).
% scomp_def
thf(fact_74_image__eqI,axiom,
! [A: $tType,B: $tType,B3: A,F4: B > A,X: B,A6: set @ B] :
( ( B3
= ( F4 @ X ) )
=> ( ( member @ B @ X @ A6 )
=> ( member @ A @ B3 @ ( image @ B @ A @ F4 @ A6 ) ) ) ) ).
% image_eqI
thf(fact_75_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_76_Sup_OSUP__cong,axiom,
! [A: $tType,B: $tType,A6: set @ B,B6: set @ B,C4: B > A,D3: B > A,Sup: ( set @ A ) > A] :
( ( A6 = B6 )
=> ( ! [X3: B] :
( ( member @ B @ X3 @ B6 )
=> ( ( C4 @ X3 )
= ( D3 @ X3 ) ) )
=> ( ( Sup @ ( image @ B @ A @ C4 @ A6 ) )
= ( Sup @ ( image @ B @ A @ D3 @ B6 ) ) ) ) ) ).
% Sup.SUP_cong
thf(fact_77_Inf_OINF__cong,axiom,
! [A: $tType,B: $tType,A6: set @ B,B6: set @ B,C4: B > A,D3: B > A,Inf: ( set @ A ) > A] :
( ( A6 = B6 )
=> ( ! [X3: B] :
( ( member @ B @ X3 @ B6 )
=> ( ( C4 @ X3 )
= ( D3 @ X3 ) ) )
=> ( ( Inf @ ( image @ B @ A @ C4 @ A6 ) )
= ( Inf @ ( image @ B @ A @ D3 @ B6 ) ) ) ) ) ).
% Inf.INF_cong
thf(fact_78_rev__image__eqI,axiom,
! [B: $tType,A: $tType,X: A,A6: set @ A,B3: B,F4: A > B] :
( ( member @ A @ X @ A6 )
=> ( ( B3
= ( F4 @ X ) )
=> ( member @ B @ B3 @ ( image @ A @ B @ F4 @ A6 ) ) ) ) ).
% rev_image_eqI
thf(fact_79_ball__imageD,axiom,
! [A: $tType,B: $tType,F4: B > A,A6: set @ B,P: A > $o] :
( ! [X3: A] :
( ( member @ A @ X3 @ ( image @ B @ A @ F4 @ A6 ) )
=> ( P @ X3 ) )
=> ! [X5: B] :
( ( member @ B @ X5 @ A6 )
=> ( P @ ( F4 @ X5 ) ) ) ) ).
% ball_imageD
thf(fact_80_image__cong,axiom,
! [B: $tType,A: $tType,M: set @ A,N4: set @ A,F4: A > B,G3: A > B] :
( ( M = N4 )
=> ( ! [X3: A] :
( ( member @ A @ X3 @ N4 )
=> ( ( F4 @ X3 )
= ( G3 @ X3 ) ) )
=> ( ( image @ A @ B @ F4 @ M )
= ( image @ A @ B @ G3 @ N4 ) ) ) ) ).
% image_cong
thf(fact_81_UNIV__I,axiom,
! [A: $tType,X: A] : ( member @ A @ X @ ( top_top @ ( set @ A ) ) ) ).
% UNIV_I
thf(fact_82_finite__Plus__UNIV__iff,axiom,
! [A: $tType,B: $tType] :
( ( finite_finite2 @ ( sum_sum @ A @ B ) @ ( top_top @ ( set @ ( sum_sum @ A @ B ) ) ) )
= ( ( finite_finite2 @ A @ ( top_top @ ( set @ A ) ) )
& ( finite_finite2 @ B @ ( top_top @ ( set @ B ) ) ) ) ) ).
% finite_Plus_UNIV_iff
thf(fact_83_Finite__Set_Ofinite__set,axiom,
! [A: $tType] :
( ( finite_finite2 @ ( set @ A ) @ ( top_top @ ( set @ ( set @ A ) ) ) )
= ( finite_finite2 @ A @ ( top_top @ ( set @ A ) ) ) ) ).
% Finite_Set.finite_set
thf(fact_84_finite__prod,axiom,
! [A: $tType,B: $tType] :
( ( finite_finite2 @ ( product_prod @ A @ B ) @ ( top_top @ ( set @ ( product_prod @ A @ B ) ) ) )
= ( ( finite_finite2 @ A @ ( top_top @ ( set @ A ) ) )
& ( finite_finite2 @ B @ ( top_top @ ( set @ B ) ) ) ) ) ).
% finite_prod
thf(fact_85_finite__Prod__UNIV,axiom,
! [B: $tType,A: $tType] :
( ( finite_finite2 @ A @ ( top_top @ ( set @ A ) ) )
=> ( ( finite_finite2 @ B @ ( top_top @ ( set @ B ) ) )
=> ( finite_finite2 @ ( product_prod @ A @ B ) @ ( top_top @ ( set @ ( product_prod @ A @ B ) ) ) ) ) ) ).
% finite_Prod_UNIV
thf(fact_86_finite__fun__UNIVD2,axiom,
! [A: $tType,B: $tType] :
( ( finite_finite2 @ ( A > B ) @ ( top_top @ ( set @ ( A > B ) ) ) )
=> ( finite_finite2 @ B @ ( top_top @ ( set @ B ) ) ) ) ).
% finite_fun_UNIVD2
thf(fact_87_range__eqI,axiom,
! [A: $tType,B: $tType,B3: A,F4: B > A,X: B] :
( ( B3
= ( F4 @ X ) )
=> ( member @ A @ B3 @ ( image @ B @ A @ F4 @ ( top_top @ ( set @ B ) ) ) ) ) ).
% range_eqI
thf(fact_88_rangeI,axiom,
! [A: $tType,B: $tType,F4: B > A,X: B] : ( member @ A @ ( F4 @ X ) @ ( image @ B @ A @ F4 @ ( top_top @ ( set @ B ) ) ) ) ).
% rangeI
thf(fact_89_UNIV__eq__I,axiom,
! [A: $tType,A6: set @ A] :
( ! [X3: A] : ( member @ A @ X3 @ A6 )
=> ( ( top_top @ ( set @ A ) )
= A6 ) ) ).
% UNIV_eq_I
thf(fact_90_UNIV__witness,axiom,
! [A: $tType] :
? [X3: A] : ( member @ A @ X3 @ ( top_top @ ( set @ A ) ) ) ).
% UNIV_witness
thf(fact_91_finite__UNIV,axiom,
! [A: $tType] :
( ( finite_finite @ A @ ( type2 @ A ) )
=> ( finite_finite2 @ A @ ( top_top @ ( set @ A ) ) ) ) ).
% finite_UNIV
thf(fact_92_infinite__UNIV__char__0,axiom,
! [A: $tType] :
( ( semiring_char_0 @ A @ ( type2 @ A ) )
=> ~ ( finite_finite2 @ A @ ( top_top @ ( set @ A ) ) ) ) ).
% infinite_UNIV_char_0
thf(fact_93_ex__new__if__finite,axiom,
! [A: $tType,A6: set @ A] :
( ~ ( finite_finite2 @ A @ ( top_top @ ( set @ A ) ) )
=> ( ( finite_finite2 @ A @ A6 )
=> ? [A5: A] :
~ ( member @ A @ A5 @ A6 ) ) ) ).
% ex_new_if_finite
thf(fact_94_imageI,axiom,
! [B: $tType,A: $tType,X: A,A6: set @ A,F4: A > B] :
( ( member @ A @ X @ A6 )
=> ( member @ B @ ( F4 @ X ) @ ( image @ A @ B @ F4 @ A6 ) ) ) ).
% imageI
thf(fact_95_image__iff,axiom,
! [A: $tType,B: $tType,Z2: A,F4: B > A,A6: set @ B] :
( ( member @ A @ Z2 @ ( image @ B @ A @ F4 @ A6 ) )
= ( ? [X4: B] :
( ( member @ B @ X4 @ A6 )
& ( Z2
= ( F4 @ X4 ) ) ) ) ) ).
% image_iff
thf(fact_96_bex__imageD,axiom,
! [A: $tType,B: $tType,F4: B > A,A6: set @ B,P: A > $o] :
( ? [X5: A] :
( ( member @ A @ X5 @ ( image @ B @ A @ F4 @ A6 ) )
& ( P @ X5 ) )
=> ? [X3: B] :
( ( member @ B @ X3 @ A6 )
& ( P @ ( F4 @ X3 ) ) ) ) ).
% bex_imageD
thf(fact_97_finite__option__UNIV,axiom,
! [A: $tType] :
( ( finite_finite2 @ ( option @ A ) @ ( top_top @ ( set @ ( option @ A ) ) ) )
= ( finite_finite2 @ A @ ( top_top @ ( set @ A ) ) ) ) ).
% finite_option_UNIV
thf(fact_98_iso__tuple__UNIV__I,axiom,
! [A: $tType,X: A] : ( member @ A @ X @ ( top_top @ ( set @ A ) ) ) ).
% iso_tuple_UNIV_I
thf(fact_99_top__apply,axiom,
! [C: $tType,D: $tType] :
( ( top @ C @ ( type2 @ C ) )
=> ( ( top_top @ ( D > C ) )
= ( ^ [X4: D] : ( top_top @ C ) ) ) ) ).
% top_apply
thf(fact_100_surj__def,axiom,
! [B: $tType,A: $tType,F4: B > A] :
( ( ( image @ B @ A @ F4 @ ( top_top @ ( set @ B ) ) )
= ( top_top @ ( set @ A ) ) )
= ( ! [Y5: A] :
? [X4: B] :
( Y5
= ( F4 @ X4 ) ) ) ) ).
% surj_def
thf(fact_101_surjI,axiom,
! [B: $tType,A: $tType,G3: B > A,F4: A > B] :
( ! [X3: A] :
( ( G3 @ ( F4 @ X3 ) )
= X3 )
=> ( ( image @ B @ A @ G3 @ ( top_top @ ( set @ B ) ) )
= ( top_top @ ( set @ A ) ) ) ) ).
% surjI
thf(fact_102_surjE,axiom,
! [A: $tType,B: $tType,F4: B > A,Y: A] :
( ( ( image @ B @ A @ F4 @ ( top_top @ ( set @ B ) ) )
= ( top_top @ ( set @ A ) ) )
=> ~ ! [X3: B] :
( Y
!= ( F4 @ X3 ) ) ) ).
% surjE
thf(fact_103_top__set__def,axiom,
! [A: $tType] :
( ( top_top @ ( set @ A ) )
= ( collect @ A @ ( top_top @ ( A > $o ) ) ) ) ).
% top_set_def
thf(fact_104_surjD,axiom,
! [A: $tType,B: $tType,F4: B > A,Y: A] :
( ( ( image @ B @ A @ F4 @ ( top_top @ ( set @ B ) ) )
= ( top_top @ ( set @ A ) ) )
=> ? [X3: B] :
( Y
= ( F4 @ X3 ) ) ) ).
% surjD
thf(fact_105_map__prod__surj,axiom,
! [A: $tType,C: $tType,D: $tType,B: $tType,F4: A > B,G3: C > D] :
( ( ( image @ A @ B @ F4 @ ( top_top @ ( set @ A ) ) )
= ( top_top @ ( set @ B ) ) )
=> ( ( ( image @ C @ D @ G3 @ ( top_top @ ( set @ C ) ) )
= ( top_top @ ( set @ D ) ) )
=> ( ( image @ ( product_prod @ A @ C ) @ ( product_prod @ B @ D ) @ ( product_map_prod @ A @ B @ C @ D @ F4 @ G3 ) @ ( top_top @ ( set @ ( product_prod @ A @ C ) ) ) )
= ( top_top @ ( set @ ( product_prod @ B @ D ) ) ) ) ) ) ).
% map_prod_surj
thf(fact_106_Pow__UNIV,axiom,
! [A: $tType] :
( ( pow @ A @ ( top_top @ ( set @ A ) ) )
= ( top_top @ ( set @ ( set @ A ) ) ) ) ).
% Pow_UNIV
thf(fact_107_surj__swap__iff,axiom,
! [B: $tType,A: $tType,A3: B,B3: B,F4: B > A] :
( ( ( image @ B @ A @ ( swap @ B @ A @ A3 @ B3 @ F4 ) @ ( top_top @ ( set @ B ) ) )
= ( top_top @ ( set @ A ) ) )
= ( ( image @ B @ A @ F4 @ ( top_top @ ( set @ B ) ) )
= ( top_top @ ( set @ A ) ) ) ) ).
% surj_swap_iff
thf(fact_108_top__empty__eq,axiom,
! [A: $tType] :
( ( top_top @ ( A > $o ) )
= ( ^ [X4: A] : ( member @ A @ X4 @ ( top_top @ ( set @ A ) ) ) ) ) ).
% top_empty_eq
thf(fact_109_finite__Pow__iff,axiom,
! [A: $tType,A6: set @ A] :
( ( finite_finite2 @ ( set @ A ) @ ( pow @ A @ A6 ) )
= ( finite_finite2 @ A @ A6 ) ) ).
% finite_Pow_iff
thf(fact_110_top1I,axiom,
! [A: $tType,X: A] : ( top_top @ ( A > $o ) @ X ) ).
% top1I
thf(fact_111_map__prod__simp,axiom,
! [C: $tType,A: $tType,B: $tType,D: $tType,F4: C > A,G3: D > B,A3: C,B3: D] :
( ( product_map_prod @ C @ A @ D @ B @ F4 @ G3 @ ( product_Pair @ C @ D @ A3 @ B3 ) )
= ( product_Pair @ A @ B @ ( F4 @ A3 ) @ ( G3 @ B3 ) ) ) ).
% map_prod_simp
thf(fact_112_swap__image__eq,axiom,
! [B: $tType,A: $tType,A3: A,A6: set @ A,B3: A,F4: A > B] :
( ( member @ A @ A3 @ A6 )
=> ( ( member @ A @ B3 @ A6 )
=> ( ( image @ A @ B @ ( swap @ A @ B @ A3 @ B3 @ F4 ) @ A6 )
= ( image @ A @ B @ F4 @ A6 ) ) ) ) ).
% swap_image_eq
thf(fact_113_map__prod__imageI,axiom,
! [D: $tType,C: $tType,B: $tType,A: $tType,A3: A,B3: B,R3: set @ ( product_prod @ A @ B ),F4: A > C,G3: B > D] :
( ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ A3 @ B3 ) @ R3 )
=> ( member @ ( product_prod @ C @ D ) @ ( product_Pair @ C @ D @ ( F4 @ A3 ) @ ( G3 @ B3 ) ) @ ( image @ ( product_prod @ A @ B ) @ ( product_prod @ C @ D ) @ ( product_map_prod @ A @ C @ B @ D @ F4 @ G3 ) @ R3 ) ) ) ).
% map_prod_imageI
thf(fact_114_Cantors__paradox,axiom,
! [A: $tType,A6: set @ A] :
~ ? [X5: A > ( set @ A )] :
( ( image @ A @ ( set @ A ) @ X5 @ A6 )
= ( pow @ A @ A6 ) ) ).
% Cantors_paradox
thf(fact_115_Pow__top,axiom,
! [A: $tType,A6: set @ A] : ( member @ ( set @ A ) @ A6 @ ( pow @ A @ A6 ) ) ).
% Pow_top
thf(fact_116_image__Pow__surj,axiom,
! [B: $tType,A: $tType,F4: B > A,A6: set @ B,B6: set @ A] :
( ( ( image @ B @ A @ F4 @ A6 )
= B6 )
=> ( ( image @ ( set @ B ) @ ( set @ A ) @ ( image @ B @ A @ F4 ) @ ( pow @ B @ A6 ) )
= ( pow @ A @ B6 ) ) ) ).
% image_Pow_surj
thf(fact_117_prod__fun__imageE,axiom,
! [B: $tType,A: $tType,D: $tType,C: $tType,C2: product_prod @ A @ B,F4: C > A,G3: D > B,R3: set @ ( product_prod @ C @ D )] :
( ( member @ ( product_prod @ A @ B ) @ C2 @ ( image @ ( product_prod @ C @ D ) @ ( product_prod @ A @ B ) @ ( product_map_prod @ C @ A @ D @ B @ F4 @ G3 ) @ R3 ) )
=> ~ ! [X3: C,Y3: D] :
( ( C2
= ( product_Pair @ A @ B @ ( F4 @ X3 ) @ ( G3 @ Y3 ) ) )
=> ~ ( member @ ( product_prod @ C @ D ) @ ( product_Pair @ C @ D @ X3 @ Y3 ) @ R3 ) ) ) ).
% prod_fun_imageE
thf(fact_118_surj__imp__surj__swap,axiom,
! [B: $tType,A: $tType,F4: B > A,A3: B,B3: B] :
( ( ( image @ B @ A @ F4 @ ( top_top @ ( set @ B ) ) )
= ( top_top @ ( set @ A ) ) )
=> ( ( image @ B @ A @ ( swap @ B @ A @ A3 @ B3 @ F4 ) @ ( top_top @ ( set @ B ) ) )
= ( top_top @ ( set @ A ) ) ) ) ).
% surj_imp_surj_swap
thf(fact_119_top__conj_I2_J,axiom,
! [A: $tType,P: $o,X: A] :
( ( P
& ( top_top @ ( A > $o ) @ X ) )
= P ) ).
% top_conj(2)
thf(fact_120_top__conj_I1_J,axiom,
! [A: $tType,X: A,P: $o] :
( ( ( top_top @ ( A > $o ) @ X )
& P )
= P ) ).
% top_conj(1)
thf(fact_121_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_122_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_123_in__inv__image,axiom,
! [A: $tType,B: $tType,X: A,Y: A,R: set @ ( product_prod @ B @ B ),F4: A > B] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X @ Y ) @ ( inv_image @ B @ A @ R @ F4 ) )
= ( member @ ( product_prod @ B @ B ) @ ( product_Pair @ B @ B @ ( F4 @ X ) @ ( F4 @ Y ) ) @ R ) ) ).
% in_inv_image
thf(fact_124_fst__map__prod,axiom,
! [B: $tType,A: $tType,D: $tType,C: $tType,F4: C > A,G3: D > B,X: product_prod @ C @ D] :
( ( product_fst @ A @ B @ ( product_map_prod @ C @ A @ D @ B @ F4 @ G3 @ X ) )
= ( F4 @ ( product_fst @ C @ D @ X ) ) ) ).
% fst_map_prod
thf(fact_125_snd__map__prod,axiom,
! [B: $tType,A: $tType,D: $tType,C: $tType,F4: C > B,G3: D > A,X: product_prod @ C @ D] :
( ( product_snd @ B @ A @ ( product_map_prod @ C @ B @ D @ A @ F4 @ G3 @ X ) )
= ( G3 @ ( product_snd @ C @ D @ X ) ) ) ).
% snd_map_prod
thf(fact_126_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_127_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_128_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_129_prod_Osplit__sel,axiom,
! [C: $tType,B: $tType,A: $tType,P: C > $o,F4: A > B > C,Prod: product_prod @ A @ B] :
( ( P @ ( product_case_prod @ A @ B @ C @ F4 @ Prod ) )
= ( ( Prod
= ( product_Pair @ A @ B @ ( product_fst @ A @ B @ Prod ) @ ( product_snd @ A @ B @ Prod ) ) )
=> ( P @ ( F4 @ ( product_fst @ A @ B @ Prod ) @ ( product_snd @ A @ B @ Prod ) ) ) ) ) ).
% prod.split_sel
thf(fact_130_prod_Osplit__sel__asm,axiom,
! [C: $tType,B: $tType,A: $tType,P: C > $o,F4: A > B > C,Prod: product_prod @ A @ B] :
( ( P @ ( product_case_prod @ A @ B @ C @ F4 @ Prod ) )
= ( ~ ( ( Prod
= ( product_Pair @ A @ B @ ( product_fst @ A @ B @ Prod ) @ ( product_snd @ A @ B @ Prod ) ) )
& ~ ( P @ ( F4 @ ( product_fst @ A @ B @ Prod ) @ ( product_snd @ A @ B @ Prod ) ) ) ) ) ) ).
% prod.split_sel_asm
thf(fact_131_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_132_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_133_prod__eqI,axiom,
! [B: $tType,A: $tType,P2: product_prod @ A @ B,Q2: product_prod @ A @ B] :
( ( ( product_fst @ A @ B @ P2 )
= ( product_fst @ A @ B @ Q2 ) )
=> ( ( ( product_snd @ A @ B @ P2 )
= ( product_snd @ A @ B @ Q2 ) )
=> ( P2 = Q2 ) ) ) ).
% prod_eqI
thf(fact_134_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_135_prod__eq__iff,axiom,
! [B: $tType,A: $tType] :
( ( ^ [Y7: product_prod @ A @ B,Z3: product_prod @ A @ B] : Y7 = Z3 )
= ( ^ [S3: product_prod @ A @ B,T3: product_prod @ A @ B] :
( ( ( product_fst @ A @ B @ S3 )
= ( product_fst @ A @ B @ T3 ) )
& ( ( product_snd @ A @ B @ S3 )
= ( product_snd @ A @ B @ T3 ) ) ) ) ) ).
% prod_eq_iff
thf(fact_136_case__prod__beta,axiom,
! [A: $tType,C: $tType,B: $tType] :
( ( product_case_prod @ B @ C @ A )
= ( ^ [F3: B > C > A,P3: product_prod @ B @ C] : ( F3 @ ( product_fst @ B @ C @ P3 ) @ ( product_snd @ B @ C @ P3 ) ) ) ) ).
% case_prod_beta
thf(fact_137_prod_Ocase__eq__if,axiom,
! [C: $tType,B: $tType,A: $tType] :
( ( product_case_prod @ A @ B @ C )
= ( ^ [F3: A > B > C,Prod3: product_prod @ A @ B] : ( F3 @ ( product_fst @ A @ B @ Prod3 ) @ ( product_snd @ A @ B @ Prod3 ) ) ) ) ).
% prod.case_eq_if
thf(fact_138_Product__Type_OCollect__case__prodD,axiom,
! [B: $tType,A: $tType,X: product_prod @ A @ B,A6: A > B > $o] :
( ( member @ ( product_prod @ A @ B ) @ X @ ( collect @ ( product_prod @ A @ B ) @ ( product_case_prod @ A @ B @ $o @ A6 ) ) )
=> ( A6 @ ( product_fst @ A @ B @ X ) @ ( product_snd @ A @ B @ X ) ) ) ).
% Product_Type.Collect_case_prodD
thf(fact_139_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_140_fst__eqD,axiom,
! [B: $tType,A: $tType,X: A,Y: B,A3: A] :
( ( ( product_fst @ A @ B @ ( product_Pair @ A @ B @ X @ Y ) )
= A3 )
=> ( X = A3 ) ) ).
% fst_eqD
thf(fact_141_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_142_snd__eqD,axiom,
! [B: $tType,A: $tType,X: B,Y: A,A3: A] :
( ( ( product_snd @ B @ A @ ( product_Pair @ B @ A @ X @ Y ) )
= A3 )
=> ( Y = A3 ) ) ).
% snd_eqD
thf(fact_143_prod_Oswap__def,axiom,
! [B: $tType,A: $tType] :
( ( product_swap @ A @ B )
= ( ^ [P3: product_prod @ A @ B] : ( product_Pair @ B @ A @ ( product_snd @ A @ B @ P3 ) @ ( product_fst @ A @ B @ P3 ) ) ) ) ).
% prod.swap_def
thf(fact_144_exI__realizer,axiom,
! [B: $tType,A: $tType,P: A > B > $o,Y: A,X: B] :
( ( P @ Y @ X )
=> ( P @ ( product_snd @ B @ A @ ( product_Pair @ B @ A @ X @ Y ) ) @ ( product_fst @ B @ A @ ( product_Pair @ B @ A @ X @ Y ) ) ) ) ).
% exI_realizer
thf(fact_145_conjI__realizer,axiom,
! [A: $tType,B: $tType,P: A > $o,P2: A,Q: B > $o,Q2: B] :
( ( P @ P2 )
=> ( ( Q @ Q2 )
=> ( ( P @ ( product_fst @ A @ B @ ( product_Pair @ A @ B @ P2 @ Q2 ) ) )
& ( Q @ ( product_snd @ A @ B @ ( product_Pair @ A @ B @ P2 @ Q2 ) ) ) ) ) ) ).
% conjI_realizer
thf(fact_146_BNF__Greatest__Fixpoint_Osubst__Pair,axiom,
! [B: $tType,A: $tType,P: A > B > $o,X: A,Y: B,A3: product_prod @ A @ B] :
( ( P @ X @ Y )
=> ( ( A3
= ( product_Pair @ A @ B @ X @ Y ) )
=> ( P @ ( product_fst @ A @ B @ A3 ) @ ( product_snd @ A @ B @ A3 ) ) ) ) ).
% BNF_Greatest_Fixpoint.subst_Pair
thf(fact_147_sndI,axiom,
! [A: $tType,B: $tType,X: product_prod @ A @ B,Y: A,Z2: B] :
( ( X
= ( product_Pair @ A @ B @ Y @ Z2 ) )
=> ( ( product_snd @ A @ B @ X )
= Z2 ) ) ).
% sndI
thf(fact_148_fstI,axiom,
! [B: $tType,A: $tType,X: product_prod @ A @ B,Y: A,Z2: B] :
( ( X
= ( product_Pair @ A @ B @ Y @ Z2 ) )
=> ( ( product_fst @ A @ B @ X )
= Y ) ) ).
% fstI
thf(fact_149_eq__snd__iff,axiom,
! [A: $tType,B: $tType,B3: A,P2: product_prod @ B @ A] :
( ( B3
= ( product_snd @ B @ A @ P2 ) )
= ( ? [A2: B] :
( P2
= ( product_Pair @ B @ A @ A2 @ B3 ) ) ) ) ).
% eq_snd_iff
thf(fact_150_eq__fst__iff,axiom,
! [A: $tType,B: $tType,A3: A,P2: product_prod @ A @ B] :
( ( A3
= ( product_fst @ A @ B @ P2 ) )
= ( ? [B2: B] :
( P2
= ( product_Pair @ A @ B @ A3 @ B2 ) ) ) ) ).
% eq_fst_iff
thf(fact_151_Collect__split__mono__strong,axiom,
! [B: $tType,A: $tType,X6: set @ A,A6: set @ ( product_prod @ A @ B ),Y8: set @ B,P: A > B > $o,Q: A > B > $o] :
( ( X6
= ( image @ ( product_prod @ A @ B ) @ A @ ( product_fst @ A @ B ) @ A6 ) )
=> ( ( Y8
= ( image @ ( product_prod @ A @ B ) @ B @ ( product_snd @ A @ B ) @ A6 ) )
=> ( ! [X3: A] :
( ( member @ A @ X3 @ X6 )
=> ! [Xa: B] :
( ( member @ B @ Xa @ Y8 )
=> ( ( P @ X3 @ Xa )
=> ( Q @ X3 @ Xa ) ) ) )
=> ( ( ord_less_eq @ ( set @ ( product_prod @ A @ B ) ) @ A6 @ ( collect @ ( product_prod @ A @ B ) @ ( product_case_prod @ A @ B @ $o @ P ) ) )
=> ( ord_less_eq @ ( set @ ( product_prod @ A @ B ) ) @ A6 @ ( collect @ ( product_prod @ A @ B ) @ ( product_case_prod @ A @ B @ $o @ Q ) ) ) ) ) ) ) ).
% Collect_split_mono_strong
thf(fact_152_subsetI,axiom,
! [A: $tType,A6: set @ A,B6: set @ A] :
( ! [X3: A] :
( ( member @ A @ X3 @ A6 )
=> ( member @ A @ X3 @ B6 ) )
=> ( ord_less_eq @ ( set @ A ) @ A6 @ B6 ) ) ).
% subsetI
thf(fact_153_subset__antisym,axiom,
! [A: $tType,A6: set @ A,B6: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ A6 @ B6 )
=> ( ( ord_less_eq @ ( set @ A ) @ B6 @ A6 )
=> ( A6 = B6 ) ) ) ).
% subset_antisym
thf(fact_154_Pow__iff,axiom,
! [A: $tType,A6: set @ A,B6: set @ A] :
( ( member @ ( set @ A ) @ A6 @ ( pow @ A @ B6 ) )
= ( ord_less_eq @ ( set @ A ) @ A6 @ B6 ) ) ).
% Pow_iff
thf(fact_155_PowI,axiom,
! [A: $tType,A6: set @ A,B6: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ A6 @ B6 )
=> ( member @ ( set @ A ) @ A6 @ ( pow @ A @ B6 ) ) ) ).
% PowI
thf(fact_156_subrelI,axiom,
! [B: $tType,A: $tType,R: set @ ( product_prod @ A @ B ),S: set @ ( product_prod @ A @ B )] :
( ! [X3: A,Y3: B] :
( ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X3 @ Y3 ) @ R )
=> ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X3 @ Y3 ) @ S ) )
=> ( ord_less_eq @ ( set @ ( product_prod @ A @ B ) ) @ R @ S ) ) ).
% subrelI
thf(fact_157_Pow__mono,axiom,
! [A: $tType,A6: set @ A,B6: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ A6 @ B6 )
=> ( ord_less_eq @ ( set @ ( set @ A ) ) @ ( pow @ A @ A6 ) @ ( pow @ A @ B6 ) ) ) ).
% Pow_mono
thf(fact_158_top_Oextremum__uniqueI,axiom,
! [A: $tType] :
( ( order_top @ A @ ( type2 @ A ) )
=> ! [A3: A] :
( ( ord_less_eq @ A @ ( top_top @ A ) @ A3 )
=> ( A3
= ( top_top @ A ) ) ) ) ).
% top.extremum_uniqueI
thf(fact_159_top_Oextremum__unique,axiom,
! [A: $tType] :
( ( order_top @ A @ ( type2 @ A ) )
=> ! [A3: A] :
( ( ord_less_eq @ A @ ( top_top @ A ) @ A3 )
= ( A3
= ( top_top @ A ) ) ) ) ).
% top.extremum_unique
thf(fact_160_top__greatest,axiom,
! [A: $tType] :
( ( order_top @ A @ ( type2 @ A ) )
=> ! [A3: A] : ( ord_less_eq @ A @ A3 @ ( top_top @ A ) ) ) ).
% top_greatest
thf(fact_161_rev__finite__subset,axiom,
! [A: $tType,B6: set @ A,A6: set @ A] :
( ( finite_finite2 @ A @ B6 )
=> ( ( ord_less_eq @ ( set @ A ) @ A6 @ B6 )
=> ( finite_finite2 @ A @ A6 ) ) ) ).
% rev_finite_subset
thf(fact_162_infinite__super,axiom,
! [A: $tType,S4: set @ A,T4: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ S4 @ T4 )
=> ( ~ ( finite_finite2 @ A @ S4 )
=> ~ ( finite_finite2 @ A @ T4 ) ) ) ).
% infinite_super
thf(fact_163_finite__subset,axiom,
! [A: $tType,A6: set @ A,B6: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ A6 @ B6 )
=> ( ( finite_finite2 @ A @ B6 )
=> ( finite_finite2 @ A @ A6 ) ) ) ).
% finite_subset
thf(fact_164_subset__UNIV,axiom,
! [A: $tType,A6: set @ A] : ( ord_less_eq @ ( set @ A ) @ A6 @ ( top_top @ ( set @ A ) ) ) ).
% subset_UNIV
thf(fact_165_set__mp,axiom,
! [A: $tType,A6: set @ A,B6: set @ A,X: A] :
( ( ord_less_eq @ ( set @ A ) @ A6 @ B6 )
=> ( ( member @ A @ X @ A6 )
=> ( member @ A @ X @ B6 ) ) ) ).
% set_mp
thf(fact_166_in__mono,axiom,
! [A: $tType,A6: set @ A,B6: set @ A,X: A] :
( ( ord_less_eq @ ( set @ A ) @ A6 @ B6 )
=> ( ( member @ A @ X @ A6 )
=> ( member @ A @ X @ B6 ) ) ) ).
% in_mono
thf(fact_167_subsetD,axiom,
! [A: $tType,A6: set @ A,B6: set @ A,C2: A] :
( ( ord_less_eq @ ( set @ A ) @ A6 @ B6 )
=> ( ( member @ A @ C2 @ A6 )
=> ( member @ A @ C2 @ B6 ) ) ) ).
% subsetD
thf(fact_168_subsetCE,axiom,
! [A: $tType,A6: set @ A,B6: set @ A,C2: A] :
( ( ord_less_eq @ ( set @ A ) @ A6 @ B6 )
=> ( ( member @ A @ C2 @ A6 )
=> ( member @ A @ C2 @ B6 ) ) ) ).
% subsetCE
thf(fact_169_equalityE,axiom,
! [A: $tType,A6: set @ A,B6: set @ A] :
( ( A6 = B6 )
=> ~ ( ( ord_less_eq @ ( set @ A ) @ A6 @ B6 )
=> ~ ( ord_less_eq @ ( set @ A ) @ B6 @ A6 ) ) ) ).
% equalityE
thf(fact_170_subset__eq,axiom,
! [A: $tType] :
( ( ord_less_eq @ ( set @ A ) )
= ( ^ [A7: set @ A,B7: set @ A] :
! [X4: A] :
( ( member @ A @ X4 @ A7 )
=> ( member @ A @ X4 @ B7 ) ) ) ) ).
% subset_eq
thf(fact_171_equalityD1,axiom,
! [A: $tType,A6: set @ A,B6: set @ A] :
( ( A6 = B6 )
=> ( ord_less_eq @ ( set @ A ) @ A6 @ B6 ) ) ).
% equalityD1
thf(fact_172_equalityD2,axiom,
! [A: $tType,A6: set @ A,B6: set @ A] :
( ( A6 = B6 )
=> ( ord_less_eq @ ( set @ A ) @ B6 @ A6 ) ) ).
% equalityD2
thf(fact_173_set__rev__mp,axiom,
! [A: $tType,X: A,A6: set @ A,B6: set @ A] :
( ( member @ A @ X @ A6 )
=> ( ( ord_less_eq @ ( set @ A ) @ A6 @ B6 )
=> ( member @ A @ X @ B6 ) ) ) ).
% set_rev_mp
thf(fact_174_subset__iff,axiom,
! [A: $tType] :
( ( ord_less_eq @ ( set @ A ) )
= ( ^ [A7: set @ A,B7: set @ A] :
! [T3: A] :
( ( member @ A @ T3 @ A7 )
=> ( member @ A @ T3 @ B7 ) ) ) ) ).
% subset_iff
thf(fact_175_rev__subsetD,axiom,
! [A: $tType,C2: A,A6: set @ A,B6: set @ A] :
( ( member @ A @ C2 @ A6 )
=> ( ( ord_less_eq @ ( set @ A ) @ A6 @ B6 )
=> ( member @ A @ C2 @ B6 ) ) ) ).
% rev_subsetD
thf(fact_176_subset__refl,axiom,
! [A: $tType,A6: set @ A] : ( ord_less_eq @ ( set @ A ) @ A6 @ A6 ) ).
% subset_refl
thf(fact_177_Collect__mono,axiom,
! [A: $tType,P: A > $o,Q: A > $o] :
( ! [X3: A] :
( ( P @ X3 )
=> ( Q @ X3 ) )
=> ( ord_less_eq @ ( set @ A ) @ ( collect @ A @ P ) @ ( collect @ A @ Q ) ) ) ).
% Collect_mono
thf(fact_178_subset__trans,axiom,
! [A: $tType,A6: set @ A,B6: set @ A,C4: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ A6 @ B6 )
=> ( ( ord_less_eq @ ( set @ A ) @ B6 @ C4 )
=> ( ord_less_eq @ ( set @ A ) @ A6 @ C4 ) ) ) ).
% subset_trans
thf(fact_179_set__eq__subset,axiom,
! [A: $tType] :
( ( ^ [Y7: set @ A,Z3: set @ A] : Y7 = Z3 )
= ( ^ [A7: set @ A,B7: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ A7 @ B7 )
& ( ord_less_eq @ ( set @ A ) @ B7 @ A7 ) ) ) ) ).
% set_eq_subset
thf(fact_180_contra__subsetD,axiom,
! [A: $tType,A6: set @ A,B6: set @ A,C2: A] :
( ( ord_less_eq @ ( set @ A ) @ A6 @ B6 )
=> ( ~ ( member @ A @ C2 @ B6 )
=> ~ ( member @ A @ C2 @ A6 ) ) ) ).
% contra_subsetD
thf(fact_181_Collect__mono__iff,axiom,
! [A: $tType,P: A > $o,Q: A > $o] :
( ( ord_less_eq @ ( set @ A ) @ ( collect @ A @ P ) @ ( collect @ A @ Q ) )
= ( ! [X4: A] :
( ( P @ X4 )
=> ( Q @ X4 ) ) ) ) ).
% Collect_mono_iff
thf(fact_182_PowD,axiom,
! [A: $tType,A6: set @ A,B6: set @ A] :
( ( member @ ( set @ A ) @ A6 @ ( pow @ A @ B6 ) )
=> ( ord_less_eq @ ( set @ A ) @ A6 @ B6 ) ) ).
% PowD
thf(fact_183_image__mono,axiom,
! [B: $tType,A: $tType,A6: set @ A,B6: set @ A,F4: A > B] :
( ( ord_less_eq @ ( set @ A ) @ A6 @ B6 )
=> ( ord_less_eq @ ( set @ B ) @ ( image @ A @ B @ F4 @ A6 ) @ ( image @ A @ B @ F4 @ B6 ) ) ) ).
% image_mono
thf(fact_184_image__subsetI,axiom,
! [A: $tType,B: $tType,A6: set @ A,F4: A > B,B6: set @ B] :
( ! [X3: A] :
( ( member @ A @ X3 @ A6 )
=> ( member @ B @ ( F4 @ X3 ) @ B6 ) )
=> ( ord_less_eq @ ( set @ B ) @ ( image @ A @ B @ F4 @ A6 ) @ B6 ) ) ).
% image_subsetI
thf(fact_185_subset__imageE,axiom,
! [A: $tType,B: $tType,B6: set @ A,F4: B > A,A6: set @ B] :
( ( ord_less_eq @ ( set @ A ) @ B6 @ ( image @ B @ A @ F4 @ A6 ) )
=> ~ ! [C5: set @ B] :
( ( ord_less_eq @ ( set @ B ) @ C5 @ A6 )
=> ( B6
!= ( image @ B @ A @ F4 @ C5 ) ) ) ) ).
% subset_imageE
thf(fact_186_image__subset__iff,axiom,
! [A: $tType,B: $tType,F4: B > A,A6: set @ B,B6: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ ( image @ B @ A @ F4 @ A6 ) @ B6 )
= ( ! [X4: B] :
( ( member @ B @ X4 @ A6 )
=> ( member @ A @ ( F4 @ X4 ) @ B6 ) ) ) ) ).
% image_subset_iff
thf(fact_187_subset__image__iff,axiom,
! [A: $tType,B: $tType,B6: set @ A,F4: B > A,A6: set @ B] :
( ( ord_less_eq @ ( set @ A ) @ B6 @ ( image @ B @ A @ F4 @ A6 ) )
= ( ? [AA: set @ B] :
( ( ord_less_eq @ ( set @ B ) @ AA @ A6 )
& ( B6
= ( image @ B @ A @ F4 @ AA ) ) ) ) ) ).
% subset_image_iff
thf(fact_188_image__Pow__mono,axiom,
! [B: $tType,A: $tType,F4: B > A,A6: set @ B,B6: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ ( image @ B @ A @ F4 @ A6 ) @ B6 )
=> ( ord_less_eq @ ( set @ ( set @ A ) ) @ ( image @ ( set @ B ) @ ( set @ A ) @ ( image @ B @ A @ F4 ) @ ( pow @ B @ A6 ) ) @ ( pow @ A @ B6 ) ) ) ).
% image_Pow_mono
thf(fact_189_finite__surj,axiom,
! [A: $tType,B: $tType,A6: set @ A,B6: set @ B,F4: A > B] :
( ( finite_finite2 @ A @ A6 )
=> ( ( ord_less_eq @ ( set @ B ) @ B6 @ ( image @ A @ B @ F4 @ A6 ) )
=> ( finite_finite2 @ B @ B6 ) ) ) ).
% finite_surj
thf(fact_190_finite__subset__image,axiom,
! [A: $tType,B: $tType,B6: set @ A,F4: B > A,A6: set @ B] :
( ( finite_finite2 @ A @ B6 )
=> ( ( ord_less_eq @ ( set @ A ) @ B6 @ ( image @ B @ A @ F4 @ A6 ) )
=> ? [C5: set @ B] :
( ( ord_less_eq @ ( set @ B ) @ C5 @ A6 )
& ( finite_finite2 @ B @ C5 )
& ( B6
= ( image @ B @ A @ F4 @ C5 ) ) ) ) ) ).
% finite_subset_image
thf(fact_191_relChain__def,axiom,
! [B: $tType,A: $tType] :
( ( ord @ B @ ( type2 @ B ) )
=> ( ( bNF_Ca1785829860lChain @ A @ B )
= ( ^ [R2: set @ ( product_prod @ A @ A ),As4: A > B] :
! [I: A,J: A] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ I @ J ) @ R2 )
=> ( ord_less_eq @ B @ ( As4 @ I ) @ ( As4 @ J ) ) ) ) ) ) ).
% relChain_def
thf(fact_192_fstOp__def,axiom,
! [C: $tType,B: $tType,A: $tType] :
( ( bNF_fstOp @ A @ B @ C )
= ( ^ [P4: A > B > $o,Q3: B > C > $o,Ac: product_prod @ A @ C] : ( product_Pair @ A @ B @ ( product_fst @ A @ C @ Ac ) @ ( bNF_pick_middlep @ A @ B @ C @ P4 @ Q3 @ ( product_fst @ A @ C @ Ac ) @ ( product_snd @ A @ C @ Ac ) ) ) ) ) ).
% fstOp_def
thf(fact_193_sndOp__def,axiom,
! [B: $tType,A: $tType,C: $tType] :
( ( bNF_sndOp @ C @ A @ B )
= ( ^ [P4: C > A > $o,Q3: A > B > $o,Ac: product_prod @ C @ B] : ( product_Pair @ A @ B @ ( bNF_pick_middlep @ C @ A @ B @ P4 @ Q3 @ ( product_fst @ C @ B @ Ac ) @ ( product_snd @ C @ B @ Ac ) ) @ ( product_snd @ C @ B @ Ac ) ) ) ) ).
% sndOp_def
thf(fact_194_accp__subset,axiom,
! [A: $tType,R1: A > A > $o,R22: A > A > $o] :
( ( ord_less_eq @ ( A > A > $o ) @ R1 @ R22 )
=> ( ord_less_eq @ ( A > $o ) @ ( accp @ A @ R22 ) @ ( accp @ A @ R1 ) ) ) ).
% accp_subset
thf(fact_195_accp__subset__induct,axiom,
! [A: $tType,D3: A > $o,R3: A > A > $o,X: A,P: A > $o] :
( ( ord_less_eq @ ( A > $o ) @ D3 @ ( accp @ A @ R3 ) )
=> ( ! [X3: A,Z: A] :
( ( D3 @ X3 )
=> ( ( R3 @ Z @ X3 )
=> ( D3 @ Z ) ) )
=> ( ( D3 @ X )
=> ( ! [X3: A] :
( ( D3 @ X3 )
=> ( ! [Z4: A] :
( ( R3 @ Z4 @ X3 )
=> ( P @ Z4 ) )
=> ( P @ X3 ) ) )
=> ( P @ X ) ) ) ) ) ).
% accp_subset_induct
thf(fact_196_Collect__case__prod__mono,axiom,
! [B: $tType,A: $tType,A6: A > B > $o,B6: A > B > $o] :
( ( ord_less_eq @ ( A > B > $o ) @ A6 @ B6 )
=> ( ord_less_eq @ ( set @ ( product_prod @ A @ B ) ) @ ( collect @ ( product_prod @ A @ B ) @ ( product_case_prod @ A @ B @ $o @ A6 ) ) @ ( collect @ ( product_prod @ A @ B ) @ ( product_case_prod @ A @ B @ $o @ B6 ) ) ) ) ).
% Collect_case_prod_mono
thf(fact_197_finite__vimageD_H,axiom,
! [A: $tType,B: $tType,F4: A > B,A6: set @ B] :
( ( finite_finite2 @ A @ ( vimage @ A @ B @ F4 @ A6 ) )
=> ( ( ord_less_eq @ ( set @ B ) @ A6 @ ( image @ A @ B @ F4 @ ( top_top @ ( set @ A ) ) ) )
=> ( finite_finite2 @ B @ A6 ) ) ) ).
% finite_vimageD'
thf(fact_198_image2__eqI,axiom,
! [A: $tType,C: $tType,B: $tType,B3: A,F4: B > A,X: B,C2: C,G3: B > C,A6: set @ B] :
( ( B3
= ( F4 @ X ) )
=> ( ( C2
= ( G3 @ X ) )
=> ( ( member @ B @ X @ A6 )
=> ( member @ ( product_prod @ A @ C ) @ ( product_Pair @ A @ C @ B3 @ C2 ) @ ( bNF_Greatest_image2 @ B @ A @ C @ A6 @ F4 @ G3 ) ) ) ) ) ).
% image2_eqI
thf(fact_199_vimageI,axiom,
! [B: $tType,A: $tType,F4: B > A,A3: B,B3: A,B6: set @ A] :
( ( ( F4 @ A3 )
= B3 )
=> ( ( member @ A @ B3 @ B6 )
=> ( member @ B @ A3 @ ( vimage @ B @ A @ F4 @ B6 ) ) ) ) ).
% vimageI
thf(fact_200_vimage__eq,axiom,
! [A: $tType,B: $tType,A3: A,F4: A > B,B6: set @ B] :
( ( member @ A @ A3 @ ( vimage @ A @ B @ F4 @ B6 ) )
= ( member @ B @ ( F4 @ A3 ) @ B6 ) ) ).
% vimage_eq
thf(fact_201_vimage__UNIV,axiom,
! [B: $tType,A: $tType,F4: A > B] :
( ( vimage @ A @ B @ F4 @ ( top_top @ ( set @ B ) ) )
= ( top_top @ ( set @ A ) ) ) ).
% vimage_UNIV
thf(fact_202_vimage__mono,axiom,
! [B: $tType,A: $tType,A6: set @ A,B6: set @ A,F4: B > A] :
( ( ord_less_eq @ ( set @ A ) @ A6 @ B6 )
=> ( ord_less_eq @ ( set @ B ) @ ( vimage @ B @ A @ F4 @ A6 ) @ ( vimage @ B @ A @ F4 @ B6 ) ) ) ).
% vimage_mono
thf(fact_203_image__subset__iff__subset__vimage,axiom,
! [B: $tType,A: $tType,F4: B > A,A6: set @ B,B6: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ ( image @ B @ A @ F4 @ A6 ) @ B6 )
= ( ord_less_eq @ ( set @ B ) @ A6 @ ( vimage @ B @ A @ F4 @ B6 ) ) ) ).
% image_subset_iff_subset_vimage
thf(fact_204_image__vimage__subset,axiom,
! [B: $tType,A: $tType,F4: B > A,A6: set @ A] : ( ord_less_eq @ ( set @ A ) @ ( image @ B @ A @ F4 @ ( vimage @ B @ A @ F4 @ A6 ) ) @ A6 ) ).
% image_vimage_subset
thf(fact_205_surj__image__vimage__eq,axiom,
! [B: $tType,A: $tType,F4: B > A,A6: set @ A] :
( ( ( image @ B @ A @ F4 @ ( top_top @ ( set @ B ) ) )
= ( top_top @ ( set @ A ) ) )
=> ( ( image @ B @ A @ F4 @ ( vimage @ B @ A @ F4 @ A6 ) )
= A6 ) ) ).
% surj_image_vimage_eq
thf(fact_206_vimageD,axiom,
! [A: $tType,B: $tType,A3: A,F4: A > B,A6: set @ B] :
( ( member @ A @ A3 @ ( vimage @ A @ B @ F4 @ A6 ) )
=> ( member @ B @ ( F4 @ A3 ) @ A6 ) ) ).
% vimageD
thf(fact_207_vimageE,axiom,
! [A: $tType,B: $tType,A3: A,F4: A > B,B6: set @ B] :
( ( member @ A @ A3 @ ( vimage @ A @ B @ F4 @ B6 ) )
=> ( member @ B @ ( F4 @ A3 ) @ B6 ) ) ).
% vimageE
thf(fact_208_vimageI2,axiom,
! [B: $tType,A: $tType,F4: B > A,A3: B,A6: set @ A] :
( ( member @ A @ ( F4 @ A3 ) @ A6 )
=> ( member @ B @ A3 @ ( vimage @ B @ A @ F4 @ A6 ) ) ) ).
% vimageI2
thf(fact_209_vimage__Collect,axiom,
! [B: $tType,A: $tType,P: B > $o,F4: A > B,Q: A > $o] :
( ! [X3: A] :
( ( P @ ( F4 @ X3 ) )
= ( Q @ X3 ) )
=> ( ( vimage @ A @ B @ F4 @ ( collect @ B @ P ) )
= ( collect @ A @ Q ) ) ) ).
% vimage_Collect
thf(fact_210_finite__vimageD,axiom,
! [A: $tType,B: $tType,H: A > B,F5: set @ B] :
( ( finite_finite2 @ A @ ( vimage @ A @ B @ H @ F5 ) )
=> ( ( ( image @ A @ B @ H @ ( top_top @ ( set @ A ) ) )
= ( top_top @ ( set @ B ) ) )
=> ( finite_finite2 @ B @ F5 ) ) ) ).
% finite_vimageD
thf(fact_211_vimage__subsetD,axiom,
! [A: $tType,B: $tType,F4: B > A,B6: set @ A,A6: set @ B] :
( ( ( image @ B @ A @ F4 @ ( top_top @ ( set @ B ) ) )
= ( top_top @ ( set @ A ) ) )
=> ( ( ord_less_eq @ ( set @ B ) @ ( vimage @ B @ A @ F4 @ B6 ) @ A6 )
=> ( ord_less_eq @ ( set @ A ) @ B6 @ ( image @ B @ A @ F4 @ A6 ) ) ) ) ).
% vimage_subsetD
thf(fact_212_image__vimage__eq,axiom,
! [A: $tType,B: $tType,F4: B > A,A6: set @ A] :
( ( image @ B @ A @ F4 @ ( vimage @ B @ A @ F4 @ A6 ) )
= ( inf_inf @ ( set @ A ) @ A6 @ ( image @ B @ A @ F4 @ ( top_top @ ( set @ B ) ) ) ) ) ).
% image_vimage_eq
thf(fact_213_vimage__subsetI,axiom,
! [B: $tType,A: $tType,F4: A > B,B6: set @ B,A6: set @ A] :
( ( inj_on @ A @ B @ F4 @ ( top_top @ ( set @ A ) ) )
=> ( ( ord_less_eq @ ( set @ B ) @ B6 @ ( image @ A @ B @ F4 @ A6 ) )
=> ( ord_less_eq @ ( set @ A ) @ ( vimage @ A @ B @ F4 @ B6 ) @ A6 ) ) ) ).
% vimage_subsetI
thf(fact_214_Int__iff,axiom,
! [A: $tType,C2: A,A6: set @ A,B6: set @ A] :
( ( member @ A @ C2 @ ( inf_inf @ ( set @ A ) @ A6 @ B6 ) )
= ( ( member @ A @ C2 @ A6 )
& ( member @ A @ C2 @ B6 ) ) ) ).
% Int_iff
thf(fact_215_IntI,axiom,
! [A: $tType,C2: A,A6: set @ A,B6: set @ A] :
( ( member @ A @ C2 @ A6 )
=> ( ( member @ A @ C2 @ B6 )
=> ( member @ A @ C2 @ ( inf_inf @ ( set @ A ) @ A6 @ B6 ) ) ) ) ).
% IntI
thf(fact_216_finite__Int,axiom,
! [A: $tType,F5: set @ A,G5: set @ A] :
( ( ( finite_finite2 @ A @ F5 )
| ( finite_finite2 @ A @ G5 ) )
=> ( finite_finite2 @ A @ ( inf_inf @ ( set @ A ) @ F5 @ G5 ) ) ) ).
% finite_Int
thf(fact_217_Int__subset__iff,axiom,
! [A: $tType,C4: set @ A,A6: set @ A,B6: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ C4 @ ( inf_inf @ ( set @ A ) @ A6 @ B6 ) )
= ( ( ord_less_eq @ ( set @ A ) @ C4 @ A6 )
& ( ord_less_eq @ ( set @ A ) @ C4 @ B6 ) ) ) ).
% Int_subset_iff
thf(fact_218_Int__UNIV,axiom,
! [A: $tType,A6: set @ A,B6: set @ A] :
( ( ( inf_inf @ ( set @ A ) @ A6 @ B6 )
= ( top_top @ ( set @ A ) ) )
= ( ( A6
= ( top_top @ ( set @ A ) ) )
& ( B6
= ( top_top @ ( set @ A ) ) ) ) ) ).
% Int_UNIV
thf(fact_219_vimage__Int,axiom,
! [A: $tType,B: $tType,F4: A > B,A6: set @ B,B6: set @ B] :
( ( vimage @ A @ B @ F4 @ ( inf_inf @ ( set @ B ) @ A6 @ B6 ) )
= ( inf_inf @ ( set @ A ) @ ( vimage @ A @ B @ F4 @ A6 ) @ ( vimage @ A @ B @ F4 @ B6 ) ) ) ).
% vimage_Int
thf(fact_220_Pow__Int__eq,axiom,
! [A: $tType,A6: set @ A,B6: set @ A] :
( ( pow @ A @ ( inf_inf @ ( set @ A ) @ A6 @ B6 ) )
= ( inf_inf @ ( set @ ( set @ A ) ) @ ( pow @ A @ A6 ) @ ( pow @ A @ B6 ) ) ) ).
% Pow_Int_eq
thf(fact_221_Int__mono,axiom,
! [A: $tType,A6: set @ A,C4: set @ A,B6: set @ A,D3: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ A6 @ C4 )
=> ( ( ord_less_eq @ ( set @ A ) @ B6 @ D3 )
=> ( ord_less_eq @ ( set @ A ) @ ( inf_inf @ ( set @ A ) @ A6 @ B6 ) @ ( inf_inf @ ( set @ A ) @ C4 @ D3 ) ) ) ) ).
% Int_mono
thf(fact_222_Int__lower1,axiom,
! [A: $tType,A6: set @ A,B6: set @ A] : ( ord_less_eq @ ( set @ A ) @ ( inf_inf @ ( set @ A ) @ A6 @ B6 ) @ A6 ) ).
% Int_lower1
thf(fact_223_Int__lower2,axiom,
! [A: $tType,A6: set @ A,B6: set @ A] : ( ord_less_eq @ ( set @ A ) @ ( inf_inf @ ( set @ A ) @ A6 @ B6 ) @ B6 ) ).
% Int_lower2
thf(fact_224_Int__absorb1,axiom,
! [A: $tType,B6: set @ A,A6: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ B6 @ A6 )
=> ( ( inf_inf @ ( set @ A ) @ A6 @ B6 )
= B6 ) ) ).
% Int_absorb1
thf(fact_225_Int__absorb2,axiom,
! [A: $tType,A6: set @ A,B6: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ A6 @ B6 )
=> ( ( inf_inf @ ( set @ A ) @ A6 @ B6 )
= A6 ) ) ).
% Int_absorb2
thf(fact_226_Int__greatest,axiom,
! [A: $tType,C4: set @ A,A6: set @ A,B6: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ C4 @ A6 )
=> ( ( ord_less_eq @ ( set @ A ) @ C4 @ B6 )
=> ( ord_less_eq @ ( set @ A ) @ C4 @ ( inf_inf @ ( set @ A ) @ A6 @ B6 ) ) ) ) ).
% Int_greatest
thf(fact_227_Int__Collect__mono,axiom,
! [A: $tType,A6: set @ A,B6: set @ A,P: A > $o,Q: A > $o] :
( ( ord_less_eq @ ( set @ A ) @ A6 @ B6 )
=> ( ! [X3: A] :
( ( member @ A @ X3 @ A6 )
=> ( ( P @ X3 )
=> ( Q @ X3 ) ) )
=> ( ord_less_eq @ ( set @ A ) @ ( inf_inf @ ( set @ A ) @ A6 @ ( collect @ A @ P ) ) @ ( inf_inf @ ( set @ A ) @ B6 @ ( collect @ A @ Q ) ) ) ) ) ).
% Int_Collect_mono
thf(fact_228_inj__on__image__Int,axiom,
! [B: $tType,A: $tType,F4: A > B,C4: set @ A,A6: set @ A,B6: set @ A] :
( ( inj_on @ A @ B @ F4 @ C4 )
=> ( ( ord_less_eq @ ( set @ A ) @ A6 @ C4 )
=> ( ( ord_less_eq @ ( set @ A ) @ B6 @ C4 )
=> ( ( image @ A @ B @ F4 @ ( inf_inf @ ( set @ A ) @ A6 @ B6 ) )
= ( inf_inf @ ( set @ B ) @ ( image @ A @ B @ F4 @ A6 ) @ ( image @ A @ B @ F4 @ B6 ) ) ) ) ) ) ).
% inj_on_image_Int
thf(fact_229_inj__on__image__eq__iff,axiom,
! [B: $tType,A: $tType,F4: A > B,C4: set @ A,A6: set @ A,B6: set @ A] :
( ( inj_on @ A @ B @ F4 @ C4 )
=> ( ( ord_less_eq @ ( set @ A ) @ A6 @ C4 )
=> ( ( ord_less_eq @ ( set @ A ) @ B6 @ C4 )
=> ( ( ( image @ A @ B @ F4 @ A6 )
= ( image @ A @ B @ F4 @ B6 ) )
= ( A6 = B6 ) ) ) ) ) ).
% inj_on_image_eq_iff
thf(fact_230_inj__on__image__mem__iff,axiom,
! [B: $tType,A: $tType,F4: A > B,B6: set @ A,A3: A,A6: set @ A] :
( ( inj_on @ A @ B @ F4 @ B6 )
=> ( ( member @ A @ A3 @ B6 )
=> ( ( ord_less_eq @ ( set @ A ) @ A6 @ B6 )
=> ( ( member @ B @ ( F4 @ A3 ) @ ( image @ A @ B @ F4 @ A6 ) )
= ( member @ A @ A3 @ A6 ) ) ) ) ) ).
% inj_on_image_mem_iff
thf(fact_231_inj__on__image__mem__iff__alt,axiom,
! [B: $tType,A: $tType,F4: A > B,B6: set @ A,A6: set @ A,A3: A] :
( ( inj_on @ A @ B @ F4 @ B6 )
=> ( ( ord_less_eq @ ( set @ A ) @ A6 @ B6 )
=> ( ( member @ B @ ( F4 @ A3 ) @ ( image @ A @ B @ F4 @ A6 ) )
=> ( ( member @ A @ A3 @ B6 )
=> ( member @ A @ A3 @ A6 ) ) ) ) ) ).
% inj_on_image_mem_iff_alt
thf(fact_232_image__Int__subset,axiom,
! [A: $tType,B: $tType,F4: B > A,A6: set @ B,B6: set @ B] : ( ord_less_eq @ ( set @ A ) @ ( image @ B @ A @ F4 @ ( inf_inf @ ( set @ B ) @ A6 @ B6 ) ) @ ( inf_inf @ ( set @ A ) @ ( image @ B @ A @ F4 @ A6 ) @ ( image @ B @ A @ F4 @ B6 ) ) ) ).
% image_Int_subset
thf(fact_233_vimage__inter__cong,axiom,
! [B: $tType,A: $tType,S4: set @ A,F4: A > B,G3: A > B,Y: set @ B] :
( ! [W: A] :
( ( member @ A @ W @ S4 )
=> ( ( F4 @ W )
= ( G3 @ W ) ) )
=> ( ( inf_inf @ ( set @ A ) @ ( vimage @ A @ B @ F4 @ Y ) @ S4 )
= ( inf_inf @ ( set @ A ) @ ( vimage @ A @ B @ G3 @ Y ) @ S4 ) ) ) ).
% vimage_inter_cong
thf(fact_234_finite__vimage__IntI,axiom,
! [A: $tType,B: $tType,F5: set @ A,H: B > A,A6: set @ B] :
( ( finite_finite2 @ A @ F5 )
=> ( ( inj_on @ B @ A @ H @ A6 )
=> ( finite_finite2 @ B @ ( inf_inf @ ( set @ B ) @ ( vimage @ B @ A @ H @ F5 ) @ A6 ) ) ) ) ).
% finite_vimage_IntI
thf(fact_235_image__Int,axiom,
! [B: $tType,A: $tType,F4: A > B,A6: set @ A,B6: set @ A] :
( ( inj_on @ A @ B @ F4 @ ( top_top @ ( set @ A ) ) )
=> ( ( image @ A @ B @ F4 @ ( inf_inf @ ( set @ A ) @ A6 @ B6 ) )
= ( inf_inf @ ( set @ B ) @ ( image @ A @ B @ F4 @ A6 ) @ ( image @ A @ B @ F4 @ B6 ) ) ) ) ).
% image_Int
thf(fact_236_inj__on__image__iff,axiom,
! [B: $tType,A: $tType,A6: set @ A,G3: A > B,F4: A > A] :
( ! [X3: A] :
( ( member @ A @ X3 @ A6 )
=> ! [Xa: A] :
( ( member @ A @ Xa @ A6 )
=> ( ( ( G3 @ ( F4 @ X3 ) )
= ( G3 @ ( F4 @ Xa ) ) )
= ( ( G3 @ X3 )
= ( G3 @ Xa ) ) ) ) )
=> ( ( inj_on @ A @ A @ F4 @ A6 )
=> ( ( inj_on @ A @ B @ G3 @ ( image @ A @ A @ F4 @ A6 ) )
= ( inj_on @ A @ B @ G3 @ A6 ) ) ) ) ).
% inj_on_image_iff
thf(fact_237_inj__swap,axiom,
! [B: $tType,A: $tType,A6: set @ ( product_prod @ A @ B )] : ( inj_on @ ( product_prod @ A @ B ) @ ( product_prod @ B @ A ) @ ( product_swap @ A @ B ) @ A6 ) ).
% inj_swap
thf(fact_238_Int__left__commute,axiom,
! [A: $tType,A6: set @ A,B6: set @ A,C4: set @ A] :
( ( inf_inf @ ( set @ A ) @ A6 @ ( inf_inf @ ( set @ A ) @ B6 @ C4 ) )
= ( inf_inf @ ( set @ A ) @ B6 @ ( inf_inf @ ( set @ A ) @ A6 @ C4 ) ) ) ).
% Int_left_commute
thf(fact_239_Int__left__absorb,axiom,
! [A: $tType,A6: set @ A,B6: set @ A] :
( ( inf_inf @ ( set @ A ) @ A6 @ ( inf_inf @ ( set @ A ) @ A6 @ B6 ) )
= ( inf_inf @ ( set @ A ) @ A6 @ B6 ) ) ).
% Int_left_absorb
thf(fact_240_Int__commute,axiom,
! [A: $tType] :
( ( inf_inf @ ( set @ A ) )
= ( ^ [A7: set @ A,B7: set @ A] : ( inf_inf @ ( set @ A ) @ B7 @ A7 ) ) ) ).
% Int_commute
thf(fact_241_Int__absorb,axiom,
! [A: $tType,A6: set @ A] :
( ( inf_inf @ ( set @ A ) @ A6 @ A6 )
= A6 ) ).
% Int_absorb
thf(fact_242_Int__assoc,axiom,
! [A: $tType,A6: set @ A,B6: set @ A,C4: set @ A] :
( ( inf_inf @ ( set @ A ) @ ( inf_inf @ ( set @ A ) @ A6 @ B6 ) @ C4 )
= ( inf_inf @ ( set @ A ) @ A6 @ ( inf_inf @ ( set @ A ) @ B6 @ C4 ) ) ) ).
% Int_assoc
thf(fact_243_IntD2,axiom,
! [A: $tType,C2: A,A6: set @ A,B6: set @ A] :
( ( member @ A @ C2 @ ( inf_inf @ ( set @ A ) @ A6 @ B6 ) )
=> ( member @ A @ C2 @ B6 ) ) ).
% IntD2
thf(fact_244_IntD1,axiom,
! [A: $tType,C2: A,A6: set @ A,B6: set @ A] :
( ( member @ A @ C2 @ ( inf_inf @ ( set @ A ) @ A6 @ B6 ) )
=> ( member @ A @ C2 @ A6 ) ) ).
% IntD1
thf(fact_245_IntE,axiom,
! [A: $tType,C2: A,A6: set @ A,B6: set @ A] :
( ( member @ A @ C2 @ ( inf_inf @ ( set @ A ) @ A6 @ B6 ) )
=> ~ ( ( member @ A @ C2 @ A6 )
=> ~ ( member @ A @ C2 @ B6 ) ) ) ).
% IntE
thf(fact_246_Int__UNIV__left,axiom,
! [A: $tType,B6: set @ A] :
( ( inf_inf @ ( set @ A ) @ ( top_top @ ( set @ A ) ) @ B6 )
= B6 ) ).
% Int_UNIV_left
thf(fact_247_Int__UNIV__right,axiom,
! [A: $tType,A6: set @ A] :
( ( inf_inf @ ( set @ A ) @ A6 @ ( top_top @ ( set @ A ) ) )
= A6 ) ).
% Int_UNIV_right
thf(fact_248_inj__eq,axiom,
! [B: $tType,A: $tType,F4: A > B,X: A,Y: A] :
( ( inj_on @ A @ B @ F4 @ ( top_top @ ( set @ A ) ) )
=> ( ( ( F4 @ X )
= ( F4 @ Y ) )
= ( X = Y ) ) ) ).
% inj_eq
thf(fact_249_injI,axiom,
! [B: $tType,A: $tType,F4: A > B] :
( ! [X3: A,Y3: A] :
( ( ( F4 @ X3 )
= ( F4 @ Y3 ) )
=> ( X3 = Y3 ) )
=> ( inj_on @ A @ B @ F4 @ ( top_top @ ( set @ A ) ) ) ) ).
% injI
thf(fact_250_injD,axiom,
! [B: $tType,A: $tType,F4: A > B,X: A,Y: A] :
( ( inj_on @ A @ B @ F4 @ ( top_top @ ( set @ A ) ) )
=> ( ( ( F4 @ X )
= ( F4 @ Y ) )
=> ( X = Y ) ) ) ).
% injD
thf(fact_251_range__ex1__eq,axiom,
! [B: $tType,A: $tType,F4: A > B,B3: B] :
( ( inj_on @ A @ B @ F4 @ ( top_top @ ( set @ A ) ) )
=> ( ( member @ B @ B3 @ ( image @ A @ B @ F4 @ ( top_top @ ( set @ A ) ) ) )
= ( ? [X4: A] :
( ( B3
= ( F4 @ X4 ) )
& ! [Y5: A] :
( ( B3
= ( F4 @ Y5 ) )
=> ( Y5 = X4 ) ) ) ) ) ) ).
% range_ex1_eq
thf(fact_252_inj__image__eq__iff,axiom,
! [B: $tType,A: $tType,F4: A > B,A6: set @ A,B6: set @ A] :
( ( inj_on @ A @ B @ F4 @ ( top_top @ ( set @ A ) ) )
=> ( ( ( image @ A @ B @ F4 @ A6 )
= ( image @ A @ B @ F4 @ B6 ) )
= ( A6 = B6 ) ) ) ).
% inj_image_eq_iff
thf(fact_253_inj__image__mem__iff,axiom,
! [B: $tType,A: $tType,F4: A > B,A3: A,A6: set @ A] :
( ( inj_on @ A @ B @ F4 @ ( top_top @ ( set @ A ) ) )
=> ( ( member @ B @ ( F4 @ A3 ) @ ( image @ A @ B @ F4 @ A6 ) )
= ( member @ A @ A3 @ A6 ) ) ) ).
% inj_image_mem_iff
thf(fact_254_finite__imageD,axiom,
! [A: $tType,B: $tType,F4: B > A,A6: set @ B] :
( ( finite_finite2 @ A @ ( image @ B @ A @ F4 @ A6 ) )
=> ( ( inj_on @ B @ A @ F4 @ A6 )
=> ( finite_finite2 @ B @ A6 ) ) ) ).
% finite_imageD
thf(fact_255_finite__image__iff,axiom,
! [B: $tType,A: $tType,F4: A > B,A6: set @ A] :
( ( inj_on @ A @ B @ F4 @ A6 )
=> ( ( finite_finite2 @ B @ ( image @ A @ B @ F4 @ A6 ) )
= ( finite_finite2 @ A @ A6 ) ) ) ).
% finite_image_iff
%----Type constructors (15)
thf(tcon_fun___Orderings_Oorder__top,axiom,
! [A8: $tType,A9: $tType] :
( ( order_top @ A9 @ ( type2 @ A9 ) )
=> ( order_top @ ( A8 > A9 ) @ ( type2 @ ( A8 > A9 ) ) ) ) ).
thf(tcon_fun___Finite__Set_Ofinite,axiom,
! [A8: $tType,A9: $tType] :
( ( ( finite_finite @ A8 @ ( type2 @ A8 ) )
& ( finite_finite @ A9 @ ( type2 @ A9 ) ) )
=> ( finite_finite @ ( A8 > A9 ) @ ( type2 @ ( A8 > A9 ) ) ) ) ).
thf(tcon_fun___Orderings_Otop,axiom,
! [A8: $tType,A9: $tType] :
( ( top @ A9 @ ( type2 @ A9 ) )
=> ( top @ ( A8 > A9 ) @ ( type2 @ ( A8 > A9 ) ) ) ) ).
thf(tcon_fun___Orderings_Oord,axiom,
! [A8: $tType,A9: $tType] :
( ( ord @ A9 @ ( type2 @ A9 ) )
=> ( ord @ ( A8 > A9 ) @ ( type2 @ ( A8 > A9 ) ) ) ) ).
thf(tcon_Set_Oset___Orderings_Oorder__top_1,axiom,
! [A8: $tType] : ( order_top @ ( set @ A8 ) @ ( type2 @ ( set @ A8 ) ) ) ).
thf(tcon_Set_Oset___Finite__Set_Ofinite_2,axiom,
! [A8: $tType] :
( ( finite_finite @ A8 @ ( type2 @ A8 ) )
=> ( finite_finite @ ( set @ A8 ) @ ( type2 @ ( set @ A8 ) ) ) ) ).
thf(tcon_Set_Oset___Orderings_Otop_3,axiom,
! [A8: $tType] : ( top @ ( set @ A8 ) @ ( type2 @ ( set @ A8 ) ) ) ).
thf(tcon_Set_Oset___Orderings_Oord_4,axiom,
! [A8: $tType] : ( ord @ ( set @ A8 ) @ ( type2 @ ( set @ A8 ) ) ) ).
thf(tcon_HOL_Obool___Orderings_Oorder__top_5,axiom,
order_top @ $o @ ( type2 @ $o ) ).
thf(tcon_HOL_Obool___Finite__Set_Ofinite_6,axiom,
finite_finite @ $o @ ( type2 @ $o ) ).
thf(tcon_HOL_Obool___Orderings_Otop_7,axiom,
top @ $o @ ( type2 @ $o ) ).
thf(tcon_HOL_Obool___Orderings_Oord_8,axiom,
ord @ $o @ ( type2 @ $o ) ).
thf(tcon_Sum__Type_Osum___Finite__Set_Ofinite_9,axiom,
! [A8: $tType,A9: $tType] :
( ( ( finite_finite @ A8 @ ( type2 @ A8 ) )
& ( finite_finite @ A9 @ ( type2 @ A9 ) ) )
=> ( finite_finite @ ( sum_sum @ A8 @ A9 ) @ ( type2 @ ( sum_sum @ A8 @ A9 ) ) ) ) ).
thf(tcon_Option_Ooption___Finite__Set_Ofinite_10,axiom,
! [A8: $tType] :
( ( finite_finite @ A8 @ ( type2 @ A8 ) )
=> ( finite_finite @ ( option @ A8 ) @ ( type2 @ ( option @ A8 ) ) ) ) ).
thf(tcon_Product__Type_Oprod___Finite__Set_Ofinite_11,axiom,
! [A8: $tType,A9: $tType] :
( ( ( finite_finite @ A8 @ ( type2 @ A8 ) )
& ( finite_finite @ A9 @ ( type2 @ A9 ) ) )
=> ( finite_finite @ ( product_prod @ A8 @ A9 ) @ ( type2 @ ( product_prod @ A8 @ A9 ) ) ) ) ).
%----Conjectures (1)
thf(conj_0,conjecture,
( ( root @ ( unfold @ ( product_prod @ dtree @ dtree ) @ parall1914194362_par_r @ parall1914194347_par_c @ ( product_Pair @ dtree @ dtree @ tr1 @ tr2 ) ) )
= ( parall1518086719_Nplus @ ( root @ tr1 ) @ ( root @ tr2 ) ) ) ).
%------------------------------------------------------------------------------