TPTP Problem File: COM204^1.p
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%------------------------------------------------------------------------------
% File : COM204^1 : TPTP v8.2.0. Released v7.0.0.
% Domain : Computing Theory
% Problem : Parallel extension to grammars and languages 65
% 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__65.p [Bla16]
% Status : Theorem
% Rating : 0.33 v8.1.0, 0.25 v7.5.0, 1.00 v7.1.0
% Syntax : Number of formulae : 327 ( 153 unt; 55 typ; 0 def)
% Number of atoms : 495 ( 281 equ; 0 cnn)
% Maximal formula atoms : 7 ( 1 avg)
% Number of connectives : 4633 ( 69 ~; 9 |; 29 &;4304 @)
% ( 0 <=>; 222 =>; 0 <=; 0 <~>)
% Maximal formula depth : 28 ( 8 avg)
% Number of types : 4 ( 3 usr)
% Number of type conns : 403 ( 403 >; 0 *; 0 +; 0 <<)
% Number of symbols : 55 ( 52 usr; 5 con; 0-7 aty)
% Number of variables : 1454 ( 177 ^;1185 !; 16 ?;1454 :)
% ( 76 !>; 0 ?*; 0 @-; 0 @+)
% SPC : TH1_THM_EQU_NAR
% Comments : This file was generated by Isabelle (most likely Sledgehammer)
% 2016-07-13 14:38:53.195
%------------------------------------------------------------------------------
%----Could-be-implicit typings (7)
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_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 (48)
thf(sy_cl_Lattices_Obounded__lattice,type,
bounded_lattice:
!>[A: $tType] : ( ( itself @ A ) > $o ) ).
thf(sy_cl_HOL_Otype,type,
type:
!>[A: $tType] : ( ( itself @ A ) > $o ) ).
thf(sy_cl_Lattices_Olattice,type,
lattice:
!>[A: $tType] : ( ( itself @ A ) > $o ) ).
thf(sy_cl_Lattices_Osemilattice__sup,type,
semilattice_sup:
!>[A: $tType] : ( ( itself @ A ) > $o ) ).
thf(sy_cl_Lattices_Obounded__lattice__top,type,
bounded_lattice_top:
!>[A: $tType] : ( ( itself @ A ) > $o ) ).
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_Ocont,type,
cont: dtree > ( set @ ( sum_sum @ t @ dtree ) ) ).
thf(sy_c_DTree_Ounfold,type,
unfold:
!>[A: $tType] : ( ( A > n ) > ( A > ( set @ ( sum_sum @ t @ A ) ) ) > A > dtree ) ).
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_Lattices_Osup__class_Osup,type,
sup_sup:
!>[A: $tType] : ( A > A > A ) ).
thf(sy_c_Orderings_Otop__class_Otop,type,
top_top:
!>[A: $tType] : A ).
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__c__rel,type,
parall1418634510_c_rel: ( product_prod @ dtree @ dtree ) > ( product_prod @ dtree @ dtree ) > $o ).
thf(sy_c_Parallel__Mirabelle__hykpkoupgu_Opar__r,type,
parall1914194362_par_r: ( product_prod @ dtree @ dtree ) > n ).
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_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_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_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_Pure_Otype,type,
type2:
!>[A: $tType] : ( itself @ A ) ).
thf(sy_c_Relation_OId,type,
id:
!>[A: $tType] : ( set @ ( product_prod @ A @ A ) ) ).
thf(sy_c_Relation_Oantisym,type,
antisym:
!>[A: $tType] : ( ( set @ ( product_prod @ A @ A ) ) > $o ) ).
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_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_Sum__Type_OInl,type,
sum_Inl:
!>[A: $tType,B: $tType] : ( A > ( sum_sum @ A @ B ) ) ).
thf(sy_c_Sum__Type_OInr,type,
sum_Inr:
!>[B: $tType,A: $tType] : ( B > ( sum_sum @ A @ B ) ) ).
thf(sy_c_Sum__Type_OPlus,type,
sum_Plus:
!>[A: $tType,B: $tType] : ( ( set @ A ) > ( set @ B ) > ( set @ ( sum_sum @ A @ B ) ) ) ).
thf(sy_c_Sum__Type_OSuml,type,
sum_Suml:
!>[A: $tType,C: $tType,B: $tType] : ( ( A > C ) > ( sum_sum @ A @ B ) > C ) ).
thf(sy_c_Sum__Type_OSumr,type,
sum_Sumr:
!>[B: $tType,C: $tType,A: $tType] : ( ( B > C ) > ( sum_sum @ A @ B ) > C ) ).
thf(sy_c_Sum__Type_Oold_Osum_Orec__set__sum,type,
sum_rec_set_sum:
!>[A: $tType,T: $tType,B: $tType] : ( ( A > T ) > ( B > T ) > ( sum_sum @ A @ B ) > T > $o ) ).
thf(sy_c_Sum__Type_Oold_Osum_Orec__sum,type,
sum_rec_sum:
!>[A: $tType,T: $tType,B: $tType] : ( ( A > T ) > ( B > T ) > ( sum_sum @ A @ B ) > T ) ).
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_t,type,
t2: dtree ).
thf(sy_v_tr1,type,
tr1: dtree ).
thf(sy_v_tr2,type,
tr2: dtree ).
%----Relevant facts (256)
thf(fact_0_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_1_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_2__092_060open_062Inr_A_N_096_Acont_A_Itr1_A_092_060parallel_062_Atr2_J_A_061_Apar_A_096_A_IInr_A_N_096_Acont_Atr1_A_092_060times_062_AInr_A_N_096_Acont_Atr2_J_092_060close_062,axiom,
( ( vimage @ dtree @ ( sum_sum @ t @ dtree ) @ ( sum_Inr @ dtree @ t ) @ ( cont @ ( parall1899940088le_par @ ( product_Pair @ dtree @ dtree @ tr1 @ tr2 ) ) ) )
= ( image @ ( product_prod @ dtree @ dtree ) @ dtree @ parall1899940088le_par
@ ( product_Sigma @ dtree @ dtree @ ( vimage @ dtree @ ( sum_sum @ t @ dtree ) @ ( sum_Inr @ dtree @ t ) @ ( cont @ tr1 ) )
@ ^ [Uu: dtree] : ( vimage @ dtree @ ( sum_sum @ t @ dtree ) @ ( sum_Inr @ dtree @ t ) @ ( cont @ tr2 ) ) ) ) ) ).
% \<open>Inr -` cont (tr1 \<parallel> tr2) = par ` (Inr -` cont tr1 \<times> Inr -` cont tr2)\<close>
thf(fact_3_SigmaI,axiom,
! [B: $tType,A: $tType,A2: A,A3: set @ A,B2: B,B3: A > ( set @ B )] :
( ( member @ A @ A2 @ A3 )
=> ( ( member @ B @ B2 @ ( B3 @ A2 ) )
=> ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ A2 @ B2 ) @ ( product_Sigma @ A @ B @ A3 @ B3 ) ) ) ) ).
% SigmaI
thf(fact_4_mem__Sigma__iff,axiom,
! [B: $tType,A: $tType,A2: A,B2: B,A3: set @ A,B3: A > ( set @ B )] :
( ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ A2 @ B2 ) @ ( product_Sigma @ A @ B @ A3 @ B3 ) )
= ( ( member @ A @ A2 @ A3 )
& ( member @ B @ B2 @ ( B3 @ A2 ) ) ) ) ).
% mem_Sigma_iff
thf(fact_5_image__ident,axiom,
! [A: $tType,Y: set @ A] :
( ( image @ A @ A
@ ^ [X2: A] : X2
@ Y )
= Y ) ).
% image_ident
thf(fact_6_Inr__cont__par,axiom,
! [Tr12: dtree,Tr22: dtree] :
( ( vimage @ dtree @ ( sum_sum @ t @ dtree ) @ ( sum_Inr @ dtree @ t ) @ ( cont @ ( parall1899940088le_par @ ( product_Pair @ dtree @ dtree @ Tr12 @ Tr22 ) ) ) )
= ( image @ ( product_prod @ dtree @ dtree ) @ dtree @ parall1899940088le_par
@ ( product_Sigma @ dtree @ dtree @ ( vimage @ dtree @ ( sum_sum @ t @ dtree ) @ ( sum_Inr @ dtree @ t ) @ ( cont @ Tr12 ) )
@ ^ [Uu: dtree] : ( vimage @ dtree @ ( sum_sum @ t @ dtree ) @ ( sum_Inr @ dtree @ t ) @ ( cont @ Tr22 ) ) ) ) ) ).
% Inr_cont_par
thf(fact_7_sum_Oinject_I2_J,axiom,
! [A: $tType,B: $tType,X22: B,Y2: B] :
( ( ( sum_Inr @ B @ A @ X22 )
= ( sum_Inr @ B @ A @ Y2 ) )
= ( X22 = Y2 ) ) ).
% sum.inject(2)
thf(fact_8_old_Osum_Oinject_I2_J,axiom,
! [A: $tType,B: $tType,B2: B,B4: B] :
( ( ( sum_Inr @ B @ A @ B2 )
= ( sum_Inr @ B @ A @ B4 ) )
= ( B2 = B4 ) ) ).
% old.sum.inject(2)
thf(fact_9_image__eqI,axiom,
! [A: $tType,B: $tType,B2: A,F: B > A,X: B,A3: set @ B] :
( ( B2
= ( F @ X ) )
=> ( ( member @ B @ X @ A3 )
=> ( member @ A @ B2 @ ( image @ B @ A @ F @ A3 ) ) ) ) ).
% image_eqI
thf(fact_10_prod_Oinject,axiom,
! [A: $tType,B: $tType,X1: A,X22: B,Y1: A,Y2: B] :
( ( ( product_Pair @ A @ B @ X1 @ X22 )
= ( product_Pair @ A @ B @ Y1 @ Y2 ) )
= ( ( X1 = Y1 )
& ( X22 = Y2 ) ) ) ).
% prod.inject
thf(fact_11_old_Oprod_Oinject,axiom,
! [A: $tType,B: $tType,A2: A,B2: B,A4: A,B4: B] :
( ( ( product_Pair @ A @ B @ A2 @ B2 )
= ( product_Pair @ A @ B @ A4 @ B4 ) )
= ( ( A2 = A4 )
& ( B2 = B4 ) ) ) ).
% old.prod.inject
thf(fact_12_SigmaE,axiom,
! [A: $tType,B: $tType,C2: product_prod @ A @ B,A3: set @ A,B3: A > ( set @ B )] :
( ( member @ ( product_prod @ A @ B ) @ C2 @ ( product_Sigma @ A @ B @ A3 @ B3 ) )
=> ~ ! [X3: A] :
( ( member @ A @ X3 @ A3 )
=> ! [Y3: B] :
( ( member @ B @ Y3 @ ( B3 @ X3 ) )
=> ( C2
!= ( product_Pair @ A @ B @ X3 @ Y3 ) ) ) ) ) ).
% SigmaE
thf(fact_13_SigmaD1,axiom,
! [B: $tType,A: $tType,A2: A,B2: B,A3: set @ A,B3: A > ( set @ B )] :
( ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ A2 @ B2 ) @ ( product_Sigma @ A @ B @ A3 @ B3 ) )
=> ( member @ A @ A2 @ A3 ) ) ).
% SigmaD1
thf(fact_14_vimageI,axiom,
! [B: $tType,A: $tType,F: B > A,A2: B,B2: A,B3: set @ A] :
( ( ( F @ A2 )
= B2 )
=> ( ( member @ A @ B2 @ B3 )
=> ( member @ B @ A2 @ ( vimage @ B @ A @ F @ B3 ) ) ) ) ).
% vimageI
thf(fact_15_vimage__eq,axiom,
! [A: $tType,B: $tType,A2: A,F: A > B,B3: set @ B] :
( ( member @ A @ A2 @ ( vimage @ A @ B @ F @ B3 ) )
= ( member @ B @ ( F @ A2 ) @ B3 ) ) ).
% vimage_eq
thf(fact_16_vimage__ident,axiom,
! [A: $tType,Y: set @ A] :
( ( vimage @ A @ A
@ ^ [X2: A] : X2
@ Y )
= Y ) ).
% vimage_ident
thf(fact_17_vimage__Collect__eq,axiom,
! [B: $tType,A: $tType,F: A > B,P: B > $o] :
( ( vimage @ A @ B @ F @ ( collect @ B @ P ) )
= ( collect @ A
@ ^ [Y4: A] : ( P @ ( F @ Y4 ) ) ) ) ).
% vimage_Collect_eq
thf(fact_18_vimageD,axiom,
! [A: $tType,B: $tType,A2: A,F: A > B,A3: set @ B] :
( ( member @ A @ A2 @ ( vimage @ A @ B @ F @ A3 ) )
=> ( member @ B @ ( F @ A2 ) @ A3 ) ) ).
% vimageD
thf(fact_19_vimageE,axiom,
! [A: $tType,B: $tType,A2: A,F: A > B,B3: set @ B] :
( ( member @ A @ A2 @ ( vimage @ A @ B @ F @ B3 ) )
=> ( member @ B @ ( F @ A2 ) @ B3 ) ) ).
% vimageE
thf(fact_20_vimageI2,axiom,
! [B: $tType,A: $tType,F: B > A,A2: B,A3: set @ A] :
( ( member @ A @ ( F @ A2 ) @ A3 )
=> ( member @ B @ A2 @ ( vimage @ B @ A @ F @ A3 ) ) ) ).
% vimageI2
thf(fact_21_vimage__def,axiom,
! [B: $tType,A: $tType] :
( ( vimage @ A @ B )
= ( ^ [F2: A > B,B5: set @ B] :
( collect @ A
@ ^ [X2: A] : ( member @ B @ ( F2 @ X2 ) @ B5 ) ) ) ) ).
% vimage_def
thf(fact_22_vimage__Collect,axiom,
! [B: $tType,A: $tType,P: B > $o,F: A > B,Q: A > $o] :
( ! [X3: A] :
( ( P @ ( F @ X3 ) )
= ( Q @ X3 ) )
=> ( ( vimage @ A @ B @ F @ ( collect @ B @ P ) )
= ( collect @ A @ Q ) ) ) ).
% vimage_Collect
thf(fact_23_old_Oprod_Oinducts,axiom,
! [B: $tType,A: $tType,P: ( product_prod @ A @ B ) > $o,Prod: product_prod @ A @ B] :
( ! [A5: A,B6: B] : ( P @ ( product_Pair @ A @ B @ A5 @ B6 ) )
=> ( P @ Prod ) ) ).
% old.prod.inducts
thf(fact_24_old_Oprod_Oexhaust,axiom,
! [A: $tType,B: $tType,Y5: product_prod @ A @ B] :
~ ! [A5: A,B6: B] :
( Y5
!= ( product_Pair @ A @ B @ A5 @ B6 ) ) ).
% old.prod.exhaust
thf(fact_25_prod__induct7,axiom,
! [G: $tType,F3: $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 @ F3 @ G ) ) ) ) ) ) > $o,X: product_prod @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D @ ( product_prod @ E @ ( product_prod @ F3 @ G ) ) ) ) )] :
( ! [A5: A,B6: B,C3: C,D2: D,E2: E,F4: F3,G2: G] : ( P @ ( product_Pair @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D @ ( product_prod @ E @ ( product_prod @ F3 @ G ) ) ) ) ) @ A5 @ ( product_Pair @ B @ ( product_prod @ C @ ( product_prod @ D @ ( product_prod @ E @ ( product_prod @ F3 @ G ) ) ) ) @ B6 @ ( product_Pair @ C @ ( product_prod @ D @ ( product_prod @ E @ ( product_prod @ F3 @ G ) ) ) @ C3 @ ( product_Pair @ D @ ( product_prod @ E @ ( product_prod @ F3 @ G ) ) @ D2 @ ( product_Pair @ E @ ( product_prod @ F3 @ G ) @ E2 @ ( product_Pair @ F3 @ G @ F4 @ G2 ) ) ) ) ) ) )
=> ( P @ X ) ) ).
% prod_induct7
thf(fact_26_prod__induct6,axiom,
! [F3: $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 @ F3 ) ) ) ) ) > $o,X: product_prod @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D @ ( product_prod @ E @ F3 ) ) ) )] :
( ! [A5: A,B6: B,C3: C,D2: D,E2: E,F4: F3] : ( P @ ( product_Pair @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D @ ( product_prod @ E @ F3 ) ) ) ) @ A5 @ ( product_Pair @ B @ ( product_prod @ C @ ( product_prod @ D @ ( product_prod @ E @ F3 ) ) ) @ B6 @ ( product_Pair @ C @ ( product_prod @ D @ ( product_prod @ E @ F3 ) ) @ C3 @ ( product_Pair @ D @ ( product_prod @ E @ F3 ) @ D2 @ ( product_Pair @ E @ F3 @ E2 @ F4 ) ) ) ) ) )
=> ( P @ X ) ) ).
% prod_induct6
thf(fact_27_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,B6: 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 ) ) @ B6 @ ( product_Pair @ C @ ( product_prod @ D @ E ) @ C3 @ ( product_Pair @ D @ E @ D2 @ E2 ) ) ) ) )
=> ( P @ X ) ) ).
% prod_induct5
thf(fact_28_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,B6: B,C3: C,D2: D] : ( P @ ( product_Pair @ A @ ( product_prod @ B @ ( product_prod @ C @ D ) ) @ A5 @ ( product_Pair @ B @ ( product_prod @ C @ D ) @ B6 @ ( product_Pair @ C @ D @ C3 @ D2 ) ) ) )
=> ( P @ X ) ) ).
% prod_induct4
thf(fact_29_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,B6: B,C3: C] : ( P @ ( product_Pair @ A @ ( product_prod @ B @ C ) @ A5 @ ( product_Pair @ B @ C @ B6 @ C3 ) ) )
=> ( P @ X ) ) ).
% prod_induct3
thf(fact_30_prod__cases7,axiom,
! [A: $tType,B: $tType,C: $tType,D: $tType,E: $tType,F3: $tType,G: $tType,Y5: product_prod @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D @ ( product_prod @ E @ ( product_prod @ F3 @ G ) ) ) ) )] :
~ ! [A5: A,B6: B,C3: C,D2: D,E2: E,F4: F3,G2: G] :
( Y5
!= ( product_Pair @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D @ ( product_prod @ E @ ( product_prod @ F3 @ G ) ) ) ) ) @ A5 @ ( product_Pair @ B @ ( product_prod @ C @ ( product_prod @ D @ ( product_prod @ E @ ( product_prod @ F3 @ G ) ) ) ) @ B6 @ ( product_Pair @ C @ ( product_prod @ D @ ( product_prod @ E @ ( product_prod @ F3 @ G ) ) ) @ C3 @ ( product_Pair @ D @ ( product_prod @ E @ ( product_prod @ F3 @ G ) ) @ D2 @ ( product_Pair @ E @ ( product_prod @ F3 @ G ) @ E2 @ ( product_Pair @ F3 @ G @ F4 @ G2 ) ) ) ) ) ) ) ).
% prod_cases7
thf(fact_31_prod__cases6,axiom,
! [A: $tType,B: $tType,C: $tType,D: $tType,E: $tType,F3: $tType,Y5: product_prod @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D @ ( product_prod @ E @ F3 ) ) ) )] :
~ ! [A5: A,B6: B,C3: C,D2: D,E2: E,F4: F3] :
( Y5
!= ( product_Pair @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D @ ( product_prod @ E @ F3 ) ) ) ) @ A5 @ ( product_Pair @ B @ ( product_prod @ C @ ( product_prod @ D @ ( product_prod @ E @ F3 ) ) ) @ B6 @ ( product_Pair @ C @ ( product_prod @ D @ ( product_prod @ E @ F3 ) ) @ C3 @ ( product_Pair @ D @ ( product_prod @ E @ F3 ) @ D2 @ ( product_Pair @ E @ F3 @ E2 @ F4 ) ) ) ) ) ) ).
% prod_cases6
thf(fact_32_prod__cases5,axiom,
! [A: $tType,B: $tType,C: $tType,D: $tType,E: $tType,Y5: product_prod @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D @ E ) ) )] :
~ ! [A5: A,B6: B,C3: C,D2: D,E2: E] :
( Y5
!= ( product_Pair @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D @ E ) ) ) @ A5 @ ( product_Pair @ B @ ( product_prod @ C @ ( product_prod @ D @ E ) ) @ B6 @ ( product_Pair @ C @ ( product_prod @ D @ E ) @ C3 @ ( product_Pair @ D @ E @ D2 @ E2 ) ) ) ) ) ).
% prod_cases5
thf(fact_33_prod__cases4,axiom,
! [A: $tType,B: $tType,C: $tType,D: $tType,Y5: product_prod @ A @ ( product_prod @ B @ ( product_prod @ C @ D ) )] :
~ ! [A5: A,B6: B,C3: C,D2: D] :
( Y5
!= ( product_Pair @ A @ ( product_prod @ B @ ( product_prod @ C @ D ) ) @ A5 @ ( product_Pair @ B @ ( product_prod @ C @ D ) @ B6 @ ( product_Pair @ C @ D @ C3 @ D2 ) ) ) ) ).
% prod_cases4
thf(fact_34_prod__cases3,axiom,
! [A: $tType,B: $tType,C: $tType,Y5: product_prod @ A @ ( product_prod @ B @ C )] :
~ ! [A5: A,B6: B,C3: C] :
( Y5
!= ( product_Pair @ A @ ( product_prod @ B @ C ) @ A5 @ ( product_Pair @ B @ C @ B6 @ C3 ) ) ) ).
% prod_cases3
thf(fact_35_Pair__inject,axiom,
! [A: $tType,B: $tType,A2: A,B2: B,A4: A,B4: B] :
( ( ( product_Pair @ A @ B @ A2 @ B2 )
= ( product_Pair @ A @ B @ A4 @ B4 ) )
=> ~ ( ( A2 = A4 )
=> ( B2 != B4 ) ) ) ).
% Pair_inject
thf(fact_36_prod__cases,axiom,
! [B: $tType,A: $tType,P: ( product_prod @ A @ B ) > $o,P2: product_prod @ A @ B] :
( ! [A5: A,B6: B] : ( P @ ( product_Pair @ A @ B @ A5 @ B6 ) )
=> ( P @ P2 ) ) ).
% prod_cases
thf(fact_37_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_38_rev__image__eqI,axiom,
! [B: $tType,A: $tType,X: A,A3: set @ A,B2: B,F: A > B] :
( ( member @ A @ X @ A3 )
=> ( ( B2
= ( F @ X ) )
=> ( member @ B @ B2 @ ( image @ A @ B @ F @ A3 ) ) ) ) ).
% rev_image_eqI
thf(fact_39_ball__imageD,axiom,
! [A: $tType,B: $tType,F: B > A,A3: set @ B,P: A > $o] :
( ! [X3: A] :
( ( member @ A @ X3 @ ( image @ B @ A @ F @ A3 ) )
=> ( P @ X3 ) )
=> ! [X4: B] :
( ( member @ B @ X4 @ A3 )
=> ( P @ ( F @ X4 ) ) ) ) ).
% ball_imageD
thf(fact_40_image__cong,axiom,
! [B: $tType,A: $tType,M: set @ A,N: set @ A,F: A > B,G3: A > B] :
( ( M = N )
=> ( ! [X3: A] :
( ( member @ A @ X3 @ N )
=> ( ( F @ X3 )
= ( G3 @ X3 ) ) )
=> ( ( image @ A @ B @ F @ M )
= ( image @ A @ B @ G3 @ N ) ) ) ) ).
% image_cong
thf(fact_41_bex__imageD,axiom,
! [A: $tType,B: $tType,F: B > A,A3: set @ B,P: A > $o] :
( ? [X4: A] :
( ( member @ A @ X4 @ ( image @ B @ A @ F @ A3 ) )
& ( P @ X4 ) )
=> ? [X3: B] :
( ( member @ B @ X3 @ A3 )
& ( P @ ( F @ X3 ) ) ) ) ).
% bex_imageD
thf(fact_42_image__iff,axiom,
! [A: $tType,B: $tType,Z: A,F: B > A,A3: set @ B] :
( ( member @ A @ Z @ ( image @ B @ A @ F @ A3 ) )
= ( ? [X2: B] :
( ( member @ B @ X2 @ A3 )
& ( Z
= ( F @ X2 ) ) ) ) ) ).
% image_iff
thf(fact_43_imageI,axiom,
! [B: $tType,A: $tType,X: A,A3: set @ A,F: A > B] :
( ( member @ A @ X @ A3 )
=> ( member @ B @ ( F @ X ) @ ( image @ A @ B @ F @ A3 ) ) ) ).
% imageI
thf(fact_44_Inr__inject,axiom,
! [A: $tType,B: $tType,X: B,Y5: B] :
( ( ( sum_Inr @ B @ A @ X )
= ( sum_Inr @ B @ A @ Y5 ) )
=> ( X = Y5 ) ) ).
% Inr_inject
thf(fact_45_mem__Collect__eq,axiom,
! [A: $tType,A2: A,P: A > $o] :
( ( member @ A @ A2 @ ( collect @ A @ P ) )
= ( P @ A2 ) ) ).
% mem_Collect_eq
thf(fact_46_Collect__mem__eq,axiom,
! [A: $tType,A3: set @ A] :
( ( collect @ A
@ ^ [X2: A] : ( member @ A @ X2 @ A3 ) )
= A3 ) ).
% 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,F: A > B,G3: A > B] :
( ! [X3: A] :
( ( F @ X3 )
= ( G3 @ X3 ) )
=> ( F = G3 ) ) ).
% ext
thf(fact_49_Compr__image__eq,axiom,
! [A: $tType,B: $tType,F: B > A,A3: set @ B,P: A > $o] :
( ( collect @ A
@ ^ [X2: A] :
( ( member @ A @ X2 @ ( image @ B @ A @ F @ A3 ) )
& ( P @ X2 ) ) )
= ( image @ B @ A @ F
@ ( collect @ B
@ ^ [X2: B] :
( ( member @ B @ X2 @ A3 )
& ( P @ ( F @ X2 ) ) ) ) ) ) ).
% Compr_image_eq
thf(fact_50_image__image,axiom,
! [A: $tType,B: $tType,C: $tType,F: B > A,G3: C > B,A3: set @ C] :
( ( image @ B @ A @ F @ ( image @ C @ B @ G3 @ A3 ) )
= ( image @ C @ A
@ ^ [X2: C] : ( F @ ( G3 @ X2 ) )
@ A3 ) ) ).
% image_image
thf(fact_51_imageE,axiom,
! [A: $tType,B: $tType,B2: A,F: B > A,A3: set @ B] :
( ( member @ A @ B2 @ ( image @ B @ A @ F @ A3 ) )
=> ~ ! [X3: B] :
( ( B2
= ( F @ X3 ) )
=> ~ ( member @ B @ X3 @ A3 ) ) ) ).
% imageE
thf(fact_52_Times__eq__cancel2,axiom,
! [A: $tType,B: $tType,X: A,C4: set @ A,A3: set @ B,B3: set @ B] :
( ( member @ A @ X @ C4 )
=> ( ( ( product_Sigma @ B @ A @ A3
@ ^ [Uu: B] : C4 )
= ( product_Sigma @ B @ A @ B3
@ ^ [Uu: B] : C4 ) )
= ( A3 = B3 ) ) ) ).
% Times_eq_cancel2
thf(fact_53_Sigma__cong,axiom,
! [B: $tType,A: $tType,A3: set @ A,B3: set @ A,C4: A > ( set @ B ),D3: A > ( set @ B )] :
( ( A3 = B3 )
=> ( ! [X3: A] :
( ( member @ A @ X3 @ B3 )
=> ( ( C4 @ X3 )
= ( D3 @ X3 ) ) )
=> ( ( product_Sigma @ A @ B @ A3 @ C4 )
= ( product_Sigma @ A @ B @ B3 @ D3 ) ) ) ) ).
% Sigma_cong
thf(fact_54_SigmaE2,axiom,
! [B: $tType,A: $tType,A2: A,B2: B,A3: set @ A,B3: A > ( set @ B )] :
( ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ A2 @ B2 ) @ ( product_Sigma @ A @ B @ A3 @ B3 ) )
=> ~ ( ( member @ A @ A2 @ A3 )
=> ~ ( member @ B @ B2 @ ( B3 @ A2 ) ) ) ) ).
% SigmaE2
thf(fact_55_SigmaD2,axiom,
! [B: $tType,A: $tType,A2: A,B2: B,A3: set @ A,B3: A > ( set @ B )] :
( ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ A2 @ B2 ) @ ( product_Sigma @ A @ B @ A3 @ B3 ) )
=> ( member @ B @ B2 @ ( B3 @ A2 ) ) ) ).
% SigmaD2
thf(fact_56_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_57_old_Osum_Osimps_I8_J,axiom,
! [A: $tType,T: $tType,B: $tType,F1: A > T,F22: B > T,B2: B] :
( ( sum_rec_sum @ A @ T @ B @ F1 @ F22 @ ( sum_Inr @ B @ A @ B2 ) )
= ( F22 @ B2 ) ) ).
% old.sum.simps(8)
thf(fact_58_member__product,axiom,
! [B: $tType,A: $tType,X: product_prod @ A @ B,A3: set @ A,B3: set @ B] :
( ( member @ ( product_prod @ A @ B ) @ X @ ( product_product @ A @ B @ A3 @ B3 ) )
= ( member @ ( product_prod @ A @ B ) @ X
@ ( product_Sigma @ A @ B @ A3
@ ^ [Uu: A] : B3 ) ) ) ).
% member_product
thf(fact_59_Product__Type_Oproduct__def,axiom,
! [B: $tType,A: $tType] :
( ( product_product @ A @ B )
= ( ^ [A6: set @ A,B5: set @ B] :
( product_Sigma @ A @ B @ A6
@ ^ [Uu: A] : B5 ) ) ) ).
% Product_Type.product_def
thf(fact_60_Sup_OSUP__identity__eq,axiom,
! [A: $tType,Sup: ( set @ A ) > A,A3: set @ A] :
( ( Sup
@ ( image @ A @ A
@ ^ [X2: A] : X2
@ A3 ) )
= ( Sup @ A3 ) ) ).
% Sup.SUP_identity_eq
thf(fact_61_Inf_OINF__identity__eq,axiom,
! [A: $tType,Inf: ( set @ A ) > A,A3: set @ A] :
( ( Inf
@ ( image @ A @ A
@ ^ [X2: A] : X2
@ A3 ) )
= ( Inf @ A3 ) ) ).
% Inf.INF_identity_eq
thf(fact_62_pred__equals__eq2,axiom,
! [B: $tType,A: $tType,R: set @ ( product_prod @ A @ B ),S: set @ ( product_prod @ A @ B )] :
( ( ( ^ [X2: A,Y4: B] : ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X2 @ Y4 ) @ R ) )
= ( ^ [X2: A,Y4: B] : ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X2 @ Y4 ) @ S ) ) )
= ( R = S ) ) ).
% pred_equals_eq2
thf(fact_63_internal__case__prod__conv,axiom,
! [B: $tType,A: $tType,C: $tType,C2: B > C > A,A2: B,B2: C] :
( ( produc2004651681e_prod @ B @ C @ A @ C2 @ ( product_Pair @ B @ C @ A2 @ B2 ) )
= ( C2 @ A2 @ B2 ) ) ).
% internal_case_prod_conv
thf(fact_64_Sumr_Osimps,axiom,
! [A: $tType,C: $tType,B: $tType,F: B > C,X: B] :
( ( sum_Sumr @ B @ C @ A @ F @ ( sum_Inr @ B @ A @ X ) )
= ( F @ X ) ) ).
% Sumr.simps
thf(fact_65_not__arg__cong__Inr,axiom,
! [B: $tType,A: $tType,X: A,Y5: A] :
( ( X != Y5 )
=> ( ( sum_Inr @ A @ B @ X )
!= ( sum_Inr @ A @ B @ Y5 ) ) ) ).
% not_arg_cong_Inr
thf(fact_66_Inl__in__cont__par,axiom,
! [T2: t,Tr12: dtree,Tr22: dtree] :
( ( member @ ( sum_sum @ t @ dtree ) @ ( sum_Inl @ t @ dtree @ T2 ) @ ( cont @ ( parall1899940088le_par @ ( product_Pair @ dtree @ dtree @ Tr12 @ Tr22 ) ) ) )
= ( ( member @ ( sum_sum @ t @ dtree ) @ ( sum_Inl @ t @ dtree @ T2 ) @ ( cont @ Tr12 ) )
| ( member @ ( sum_sum @ t @ dtree ) @ ( sum_Inl @ t @ dtree @ T2 ) @ ( cont @ Tr22 ) ) ) ) ).
% Inl_in_cont_par
thf(fact_67_sum_Oinject_I1_J,axiom,
! [B: $tType,A: $tType,X1: A,Y1: A] :
( ( ( sum_Inl @ A @ B @ X1 )
= ( sum_Inl @ A @ B @ Y1 ) )
= ( X1 = Y1 ) ) ).
% sum.inject(1)
thf(fact_68_old_Osum_Oinject_I1_J,axiom,
! [B: $tType,A: $tType,A2: A,A4: A] :
( ( ( sum_Inl @ A @ B @ A2 )
= ( sum_Inl @ A @ B @ A4 ) )
= ( A2 = A4 ) ) ).
% old.sum.inject(1)
thf(fact_69_old_Osum_Osimps_I7_J,axiom,
! [B: $tType,T: $tType,A: $tType,F1: A > T,F22: B > T,A2: A] :
( ( sum_rec_sum @ A @ T @ B @ F1 @ F22 @ ( sum_Inl @ A @ B @ A2 ) )
= ( F1 @ A2 ) ) ).
% old.sum.simps(7)
thf(fact_70_Inl__inject,axiom,
! [B: $tType,A: $tType,X: A,Y5: A] :
( ( ( sum_Inl @ A @ B @ X )
= ( sum_Inl @ A @ B @ Y5 ) )
=> ( X = Y5 ) ) ).
% Inl_inject
thf(fact_71_obj__sumE,axiom,
! [A: $tType,B: $tType,S2: sum_sum @ A @ B] :
( ! [X3: A] :
( S2
!= ( sum_Inl @ A @ B @ X3 ) )
=> ~ ! [X3: B] :
( S2
!= ( sum_Inr @ B @ A @ X3 ) ) ) ).
% obj_sumE
thf(fact_72_Sumr__inject,axiom,
! [A: $tType,C: $tType,B: $tType,F: B > C,G3: B > C] :
( ( ( sum_Sumr @ B @ C @ A @ F )
= ( sum_Sumr @ B @ C @ A @ G3 ) )
=> ( F = G3 ) ) ).
% Sumr_inject
thf(fact_73_sum_Odistinct_I1_J,axiom,
! [A: $tType,B: $tType,X1: A,X22: B] :
( ( sum_Inl @ A @ B @ X1 )
!= ( sum_Inr @ B @ A @ X22 ) ) ).
% sum.distinct(1)
thf(fact_74_old_Osum_Odistinct_I2_J,axiom,
! [B7: $tType,A7: $tType,B8: B7,A8: A7] :
( ( sum_Inr @ B7 @ A7 @ B8 )
!= ( sum_Inl @ A7 @ B7 @ A8 ) ) ).
% old.sum.distinct(2)
thf(fact_75_old_Osum_Odistinct_I1_J,axiom,
! [A: $tType,B: $tType,A2: A,B4: B] :
( ( sum_Inl @ A @ B @ A2 )
!= ( sum_Inr @ B @ A @ B4 ) ) ).
% old.sum.distinct(1)
thf(fact_76_sumE,axiom,
! [A: $tType,B: $tType,S2: sum_sum @ A @ B] :
( ! [X3: A] :
( S2
!= ( sum_Inl @ A @ B @ X3 ) )
=> ~ ! [Y3: B] :
( S2
!= ( sum_Inr @ B @ A @ Y3 ) ) ) ).
% sumE
thf(fact_77_Inr__not__Inl,axiom,
! [B: $tType,A: $tType,B2: B,A2: A] :
( ( sum_Inr @ B @ A @ B2 )
!= ( sum_Inl @ A @ B @ A2 ) ) ).
% Inr_not_Inl
thf(fact_78_split__sum__ex,axiom,
! [B: $tType,A: $tType] :
( ( ^ [P3: ( sum_sum @ A @ B ) > $o] :
? [X5: sum_sum @ A @ B] : ( P3 @ X5 ) )
= ( ^ [P4: ( sum_sum @ A @ B ) > $o] :
( ? [X2: A] : ( P4 @ ( sum_Inl @ A @ B @ X2 ) )
| ? [X2: B] : ( P4 @ ( sum_Inr @ B @ A @ X2 ) ) ) ) ) ).
% split_sum_ex
thf(fact_79_split__sum__all,axiom,
! [B: $tType,A: $tType] :
( ( ^ [P3: ( sum_sum @ A @ B ) > $o] :
! [X5: sum_sum @ A @ B] : ( P3 @ X5 ) )
= ( ^ [P4: ( sum_sum @ A @ B ) > $o] :
( ! [X2: A] : ( P4 @ ( sum_Inl @ A @ B @ X2 ) )
& ! [X2: B] : ( P4 @ ( sum_Inr @ B @ A @ X2 ) ) ) ) ) ).
% split_sum_all
thf(fact_80_old_Osum_Oexhaust,axiom,
! [A: $tType,B: $tType,Y5: sum_sum @ A @ B] :
( ! [A5: A] :
( Y5
!= ( sum_Inl @ A @ B @ A5 ) )
=> ~ ! [B6: B] :
( Y5
!= ( sum_Inr @ B @ A @ B6 ) ) ) ).
% old.sum.exhaust
thf(fact_81_old_Osum_Oinducts,axiom,
! [B: $tType,A: $tType,P: ( sum_sum @ A @ B ) > $o,Sum: sum_sum @ A @ B] :
( ! [A5: A] : ( P @ ( sum_Inl @ A @ B @ A5 ) )
=> ( ! [B6: B] : ( P @ ( sum_Inr @ B @ A @ B6 ) )
=> ( P @ Sum ) ) ) ).
% old.sum.inducts
thf(fact_82_Inf_OINF__cong,axiom,
! [A: $tType,B: $tType,A3: set @ B,B3: set @ B,C4: B > A,D3: B > A,Inf: ( set @ A ) > A] :
( ( A3 = B3 )
=> ( ! [X3: B] :
( ( member @ B @ X3 @ B3 )
=> ( ( C4 @ X3 )
= ( D3 @ X3 ) ) )
=> ( ( Inf @ ( image @ B @ A @ C4 @ A3 ) )
= ( Inf @ ( image @ B @ A @ D3 @ B3 ) ) ) ) ) ).
% Inf.INF_cong
thf(fact_83_Sup_OSUP__cong,axiom,
! [A: $tType,B: $tType,A3: set @ B,B3: set @ B,C4: B > A,D3: B > A,Sup: ( set @ A ) > A] :
( ( A3 = B3 )
=> ( ! [X3: B] :
( ( member @ B @ X3 @ B3 )
=> ( ( C4 @ X3 )
= ( D3 @ X3 ) ) )
=> ( ( Sup @ ( image @ B @ A @ C4 @ A3 ) )
= ( Sup @ ( image @ B @ A @ D3 @ B3 ) ) ) ) ) ).
% Sup.SUP_cong
thf(fact_84_Inl__Inr__False,axiom,
! [A: $tType,B: $tType,X: A,Y5: B] :
( ( sum_Inl @ A @ B @ X )
!= ( sum_Inr @ B @ A @ Y5 ) ) ).
% Inl_Inr_False
thf(fact_85_Inr__Inl__False,axiom,
! [B: $tType,A: $tType,X: B,Y5: A] :
( ( sum_Inr @ B @ A @ X )
!= ( sum_Inl @ A @ B @ Y5 ) ) ).
% Inr_Inl_False
thf(fact_86_Inl__Inr__image__cong,axiom,
! [B: $tType,A: $tType,A3: set @ ( sum_sum @ A @ B ),B3: set @ ( sum_sum @ A @ B )] :
( ( ( vimage @ A @ ( sum_sum @ A @ B ) @ ( sum_Inl @ A @ B ) @ A3 )
= ( vimage @ A @ ( sum_sum @ A @ B ) @ ( sum_Inl @ A @ B ) @ B3 ) )
=> ( ( ( vimage @ B @ ( sum_sum @ A @ B ) @ ( sum_Inr @ B @ A ) @ A3 )
= ( vimage @ B @ ( sum_sum @ A @ B ) @ ( sum_Inr @ B @ A ) @ B3 ) )
=> ( A3 = B3 ) ) ) ).
% Inl_Inr_image_cong
thf(fact_87_obj__sumE__f,axiom,
! [A: $tType,C: $tType,B: $tType,S2: B,F: ( sum_sum @ A @ C ) > B,P: $o] :
( ! [X3: A] :
( ( S2
= ( F @ ( sum_Inl @ A @ C @ X3 ) ) )
=> P )
=> ( ! [X3: C] :
( ( S2
= ( F @ ( sum_Inr @ C @ A @ X3 ) ) )
=> P )
=> ! [X4: sum_sum @ A @ C] :
( ( S2
= ( F @ X4 ) )
=> P ) ) ) ).
% obj_sumE_f
thf(fact_88_Inl__cont__par,axiom,
! [Tr12: dtree,Tr22: dtree] :
( ( vimage @ t @ ( sum_sum @ t @ dtree ) @ ( sum_Inl @ t @ dtree ) @ ( cont @ ( parall1899940088le_par @ ( product_Pair @ dtree @ dtree @ Tr12 @ Tr22 ) ) ) )
= ( vimage @ t @ ( sum_sum @ t @ dtree ) @ ( sum_Inl @ t @ dtree ) @ ( sup_sup @ ( set @ ( sum_sum @ t @ dtree ) ) @ ( cont @ Tr12 ) @ ( cont @ Tr22 ) ) ) ) ).
% Inl_cont_par
thf(fact_89_Suml_Osimps,axiom,
! [B: $tType,C: $tType,A: $tType,F: A > C,X: A] :
( ( sum_Suml @ A @ C @ B @ F @ ( sum_Inl @ A @ B @ X ) )
= ( F @ X ) ) ).
% Suml.simps
thf(fact_90_image2__eqI,axiom,
! [A: $tType,C: $tType,B: $tType,B2: A,F: B > A,X: B,C2: C,G3: B > C,A3: set @ B] :
( ( B2
= ( F @ X ) )
=> ( ( C2
= ( G3 @ X ) )
=> ( ( member @ B @ X @ A3 )
=> ( member @ ( product_prod @ A @ C ) @ ( product_Pair @ A @ C @ B2 @ C2 ) @ ( bNF_Greatest_image2 @ B @ A @ C @ A3 @ F @ G3 ) ) ) ) ) ).
% image2_eqI
thf(fact_91_Un__iff,axiom,
! [A: $tType,C2: A,A3: set @ A,B3: set @ A] :
( ( member @ A @ C2 @ ( sup_sup @ ( set @ A ) @ A3 @ B3 ) )
= ( ( member @ A @ C2 @ A3 )
| ( member @ A @ C2 @ B3 ) ) ) ).
% Un_iff
thf(fact_92_UnCI,axiom,
! [A: $tType,C2: A,B3: set @ A,A3: set @ A] :
( ( ~ ( member @ A @ C2 @ B3 )
=> ( member @ A @ C2 @ A3 ) )
=> ( member @ A @ C2 @ ( sup_sup @ ( set @ A ) @ A3 @ B3 ) ) ) ).
% UnCI
thf(fact_93_vimage__Un,axiom,
! [A: $tType,B: $tType,F: A > B,A3: set @ B,B3: set @ B] :
( ( vimage @ A @ B @ F @ ( sup_sup @ ( set @ B ) @ A3 @ B3 ) )
= ( sup_sup @ ( set @ A ) @ ( vimage @ A @ B @ F @ A3 ) @ ( vimage @ A @ B @ F @ B3 ) ) ) ).
% vimage_Un
thf(fact_94_Sigma__Un__distrib1,axiom,
! [B: $tType,A: $tType,I: set @ A,J: set @ A,C4: A > ( set @ B )] :
( ( product_Sigma @ A @ B @ ( sup_sup @ ( set @ A ) @ I @ J ) @ C4 )
= ( sup_sup @ ( set @ ( product_prod @ A @ B ) ) @ ( product_Sigma @ A @ B @ I @ C4 ) @ ( product_Sigma @ A @ B @ J @ C4 ) ) ) ).
% Sigma_Un_distrib1
thf(fact_95_Collect__disj__eq,axiom,
! [A: $tType,P: A > $o,Q: A > $o] :
( ( collect @ A
@ ^ [X2: A] :
( ( P @ X2 )
| ( Q @ X2 ) ) )
= ( sup_sup @ ( set @ A ) @ ( collect @ A @ P ) @ ( collect @ A @ Q ) ) ) ).
% Collect_disj_eq
thf(fact_96_Un__def,axiom,
! [A: $tType] :
( ( sup_sup @ ( set @ A ) )
= ( ^ [A6: set @ A,B5: set @ A] :
( collect @ A
@ ^ [X2: A] :
( ( member @ A @ X2 @ A6 )
| ( member @ A @ X2 @ B5 ) ) ) ) ) ).
% Un_def
thf(fact_97_Sigma__Un__distrib2,axiom,
! [B: $tType,A: $tType,I: set @ A,A3: A > ( set @ B ),B3: A > ( set @ B )] :
( ( product_Sigma @ A @ B @ I
@ ^ [I2: A] : ( sup_sup @ ( set @ B ) @ ( A3 @ I2 ) @ ( B3 @ I2 ) ) )
= ( sup_sup @ ( set @ ( product_prod @ A @ B ) ) @ ( product_Sigma @ A @ B @ I @ A3 ) @ ( product_Sigma @ A @ B @ I @ B3 ) ) ) ).
% Sigma_Un_distrib2
thf(fact_98_Times__Un__distrib1,axiom,
! [B: $tType,A: $tType,A3: set @ A,B3: set @ A,C4: set @ B] :
( ( product_Sigma @ A @ B @ ( sup_sup @ ( set @ A ) @ A3 @ B3 )
@ ^ [Uu: A] : C4 )
= ( sup_sup @ ( set @ ( product_prod @ A @ B ) )
@ ( product_Sigma @ A @ B @ A3
@ ^ [Uu: A] : C4 )
@ ( product_Sigma @ A @ B @ B3
@ ^ [Uu: A] : C4 ) ) ) ).
% Times_Un_distrib1
thf(fact_99_Un__left__commute,axiom,
! [A: $tType,A3: set @ A,B3: set @ A,C4: set @ A] :
( ( sup_sup @ ( set @ A ) @ A3 @ ( sup_sup @ ( set @ A ) @ B3 @ C4 ) )
= ( sup_sup @ ( set @ A ) @ B3 @ ( sup_sup @ ( set @ A ) @ A3 @ C4 ) ) ) ).
% Un_left_commute
thf(fact_100_Un__left__absorb,axiom,
! [A: $tType,A3: set @ A,B3: set @ A] :
( ( sup_sup @ ( set @ A ) @ A3 @ ( sup_sup @ ( set @ A ) @ A3 @ B3 ) )
= ( sup_sup @ ( set @ A ) @ A3 @ B3 ) ) ).
% Un_left_absorb
thf(fact_101_Un__commute,axiom,
! [A: $tType] :
( ( sup_sup @ ( set @ A ) )
= ( ^ [A6: set @ A,B5: set @ A] : ( sup_sup @ ( set @ A ) @ B5 @ A6 ) ) ) ).
% Un_commute
thf(fact_102_Un__absorb,axiom,
! [A: $tType,A3: set @ A] :
( ( sup_sup @ ( set @ A ) @ A3 @ A3 )
= A3 ) ).
% Un_absorb
thf(fact_103_Un__assoc,axiom,
! [A: $tType,A3: set @ A,B3: set @ A,C4: set @ A] :
( ( sup_sup @ ( set @ A ) @ ( sup_sup @ ( set @ A ) @ A3 @ B3 ) @ C4 )
= ( sup_sup @ ( set @ A ) @ A3 @ ( sup_sup @ ( set @ A ) @ B3 @ C4 ) ) ) ).
% Un_assoc
thf(fact_104_ball__Un,axiom,
! [A: $tType,A3: set @ A,B3: set @ A,P: A > $o] :
( ( ! [X2: A] :
( ( member @ A @ X2 @ ( sup_sup @ ( set @ A ) @ A3 @ B3 ) )
=> ( P @ X2 ) ) )
= ( ! [X2: A] :
( ( member @ A @ X2 @ A3 )
=> ( P @ X2 ) )
& ! [X2: A] :
( ( member @ A @ X2 @ B3 )
=> ( P @ X2 ) ) ) ) ).
% ball_Un
thf(fact_105_bex__Un,axiom,
! [A: $tType,A3: set @ A,B3: set @ A,P: A > $o] :
( ( ? [X2: A] :
( ( member @ A @ X2 @ ( sup_sup @ ( set @ A ) @ A3 @ B3 ) )
& ( P @ X2 ) ) )
= ( ? [X2: A] :
( ( member @ A @ X2 @ A3 )
& ( P @ X2 ) )
| ? [X2: A] :
( ( member @ A @ X2 @ B3 )
& ( P @ X2 ) ) ) ) ).
% bex_Un
thf(fact_106_UnI2,axiom,
! [A: $tType,C2: A,B3: set @ A,A3: set @ A] :
( ( member @ A @ C2 @ B3 )
=> ( member @ A @ C2 @ ( sup_sup @ ( set @ A ) @ A3 @ B3 ) ) ) ).
% UnI2
thf(fact_107_UnI1,axiom,
! [A: $tType,C2: A,A3: set @ A,B3: set @ A] :
( ( member @ A @ C2 @ A3 )
=> ( member @ A @ C2 @ ( sup_sup @ ( set @ A ) @ A3 @ B3 ) ) ) ).
% UnI1
thf(fact_108_UnE,axiom,
! [A: $tType,C2: A,A3: set @ A,B3: set @ A] :
( ( member @ A @ C2 @ ( sup_sup @ ( set @ A ) @ A3 @ B3 ) )
=> ( ~ ( member @ A @ C2 @ A3 )
=> ( member @ A @ C2 @ B3 ) ) ) ).
% UnE
thf(fact_109_Suml__inject,axiom,
! [B: $tType,C: $tType,A: $tType,F: A > C,G3: A > C] :
( ( ( sum_Suml @ A @ C @ B @ F )
= ( sum_Suml @ A @ C @ B @ G3 ) )
=> ( F = G3 ) ) ).
% Suml_inject
thf(fact_110_image__Un,axiom,
! [A: $tType,B: $tType,F: B > A,A3: set @ B,B3: set @ B] :
( ( image @ B @ A @ F @ ( sup_sup @ ( set @ B ) @ A3 @ B3 ) )
= ( sup_sup @ ( set @ A ) @ ( image @ B @ A @ F @ A3 ) @ ( image @ B @ A @ F @ B3 ) ) ) ).
% image_Un
thf(fact_111_sup__apply,axiom,
! [B: $tType,A: $tType] :
( ( semilattice_sup @ B @ ( type2 @ B ) )
=> ( ( sup_sup @ ( A > B ) )
= ( ^ [F2: A > B,G4: A > B,X2: A] : ( sup_sup @ B @ ( F2 @ X2 ) @ ( G4 @ X2 ) ) ) ) ) ).
% sup_apply
thf(fact_112_sup_Oidem,axiom,
! [A: $tType] :
( ( semilattice_sup @ A @ ( type2 @ A ) )
=> ! [A2: A] :
( ( sup_sup @ A @ A2 @ A2 )
= A2 ) ) ).
% sup.idem
thf(fact_113_sup__idem,axiom,
! [A: $tType] :
( ( semilattice_sup @ A @ ( type2 @ A ) )
=> ! [X: A] :
( ( sup_sup @ A @ X @ X )
= X ) ) ).
% sup_idem
thf(fact_114_sup_Oleft__idem,axiom,
! [A: $tType] :
( ( semilattice_sup @ A @ ( type2 @ A ) )
=> ! [A2: A,B2: A] :
( ( sup_sup @ A @ A2 @ ( sup_sup @ A @ A2 @ B2 ) )
= ( sup_sup @ A @ A2 @ B2 ) ) ) ).
% sup.left_idem
thf(fact_115_sup__left__idem,axiom,
! [A: $tType] :
( ( semilattice_sup @ A @ ( type2 @ A ) )
=> ! [X: A,Y5: A] :
( ( sup_sup @ A @ X @ ( sup_sup @ A @ X @ Y5 ) )
= ( sup_sup @ A @ X @ Y5 ) ) ) ).
% sup_left_idem
thf(fact_116_sup_Oright__idem,axiom,
! [A: $tType] :
( ( semilattice_sup @ A @ ( type2 @ A ) )
=> ! [A2: A,B2: A] :
( ( sup_sup @ A @ ( sup_sup @ A @ A2 @ B2 ) @ B2 )
= ( sup_sup @ A @ A2 @ B2 ) ) ) ).
% sup.right_idem
thf(fact_117_sup__Un__eq2,axiom,
! [B: $tType,A: $tType,R: set @ ( product_prod @ A @ B ),S: set @ ( product_prod @ A @ B )] :
( ( sup_sup @ ( A > B > $o )
@ ^ [X2: A,Y4: B] : ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X2 @ Y4 ) @ R )
@ ^ [X2: A,Y4: B] : ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X2 @ Y4 ) @ S ) )
= ( ^ [X2: A,Y4: B] : ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X2 @ Y4 ) @ ( sup_sup @ ( set @ ( product_prod @ A @ B ) ) @ R @ S ) ) ) ) ).
% sup_Un_eq2
thf(fact_118_sup__set__def,axiom,
! [A: $tType] :
( ( sup_sup @ ( set @ A ) )
= ( ^ [A6: set @ A,B5: set @ A] :
( collect @ A
@ ( sup_sup @ ( A > $o )
@ ^ [X2: A] : ( member @ A @ X2 @ A6 )
@ ^ [X2: A] : ( member @ A @ X2 @ B5 ) ) ) ) ) ).
% sup_set_def
thf(fact_119_sup__Un__eq,axiom,
! [A: $tType,R: set @ A,S: set @ A] :
( ( sup_sup @ ( A > $o )
@ ^ [X2: A] : ( member @ A @ X2 @ R )
@ ^ [X2: A] : ( member @ A @ X2 @ S ) )
= ( ^ [X2: A] : ( member @ A @ X2 @ ( sup_sup @ ( set @ A ) @ R @ S ) ) ) ) ).
% sup_Un_eq
thf(fact_120_sup__left__commute,axiom,
! [A: $tType] :
( ( semilattice_sup @ A @ ( type2 @ A ) )
=> ! [X: A,Y5: A,Z: A] :
( ( sup_sup @ A @ X @ ( sup_sup @ A @ Y5 @ Z ) )
= ( sup_sup @ A @ Y5 @ ( sup_sup @ A @ X @ Z ) ) ) ) ).
% sup_left_commute
thf(fact_121_sup_Oleft__commute,axiom,
! [A: $tType] :
( ( semilattice_sup @ A @ ( type2 @ A ) )
=> ! [B2: A,A2: A,C2: A] :
( ( sup_sup @ A @ B2 @ ( sup_sup @ A @ A2 @ C2 ) )
= ( sup_sup @ A @ A2 @ ( sup_sup @ A @ B2 @ C2 ) ) ) ) ).
% sup.left_commute
thf(fact_122_sup__commute,axiom,
! [A: $tType] :
( ( semilattice_sup @ A @ ( type2 @ A ) )
=> ( ( sup_sup @ A )
= ( ^ [X2: A,Y4: A] : ( sup_sup @ A @ Y4 @ X2 ) ) ) ) ).
% sup_commute
thf(fact_123_sup_Ocommute,axiom,
! [A: $tType] :
( ( semilattice_sup @ A @ ( type2 @ A ) )
=> ( ( sup_sup @ A )
= ( ^ [A9: A,B9: A] : ( sup_sup @ A @ B9 @ A9 ) ) ) ) ).
% sup.commute
thf(fact_124_sup__assoc,axiom,
! [A: $tType] :
( ( semilattice_sup @ A @ ( type2 @ A ) )
=> ! [X: A,Y5: A,Z: A] :
( ( sup_sup @ A @ ( sup_sup @ A @ X @ Y5 ) @ Z )
= ( sup_sup @ A @ X @ ( sup_sup @ A @ Y5 @ Z ) ) ) ) ).
% sup_assoc
thf(fact_125_sup_Oassoc,axiom,
! [A: $tType] :
( ( semilattice_sup @ A @ ( type2 @ A ) )
=> ! [A2: A,B2: A,C2: A] :
( ( sup_sup @ A @ ( sup_sup @ A @ A2 @ B2 ) @ C2 )
= ( sup_sup @ A @ A2 @ ( sup_sup @ A @ B2 @ C2 ) ) ) ) ).
% sup.assoc
thf(fact_126_sup__fun__def,axiom,
! [B: $tType,A: $tType] :
( ( semilattice_sup @ B @ ( type2 @ B ) )
=> ( ( sup_sup @ ( A > B ) )
= ( ^ [F2: A > B,G4: A > B,X2: A] : ( sup_sup @ B @ ( F2 @ X2 ) @ ( G4 @ X2 ) ) ) ) ) ).
% sup_fun_def
thf(fact_127_inf__sup__aci_I5_J,axiom,
! [A: $tType] :
( ( lattice @ A @ ( type2 @ A ) )
=> ( ( sup_sup @ A )
= ( ^ [X2: A,Y4: A] : ( sup_sup @ A @ Y4 @ X2 ) ) ) ) ).
% inf_sup_aci(5)
thf(fact_128_inf__sup__aci_I6_J,axiom,
! [A: $tType] :
( ( lattice @ A @ ( type2 @ A ) )
=> ! [X: A,Y5: A,Z: A] :
( ( sup_sup @ A @ ( sup_sup @ A @ X @ Y5 ) @ Z )
= ( sup_sup @ A @ X @ ( sup_sup @ A @ Y5 @ Z ) ) ) ) ).
% inf_sup_aci(6)
thf(fact_129_inf__sup__aci_I7_J,axiom,
! [A: $tType] :
( ( lattice @ A @ ( type2 @ A ) )
=> ! [X: A,Y5: A,Z: A] :
( ( sup_sup @ A @ X @ ( sup_sup @ A @ Y5 @ Z ) )
= ( sup_sup @ A @ Y5 @ ( sup_sup @ A @ X @ Z ) ) ) ) ).
% inf_sup_aci(7)
thf(fact_130_inf__sup__aci_I8_J,axiom,
! [A: $tType] :
( ( lattice @ A @ ( type2 @ A ) )
=> ! [X: A,Y5: A] :
( ( sup_sup @ A @ X @ ( sup_sup @ A @ X @ Y5 ) )
= ( sup_sup @ A @ X @ Y5 ) ) ) ).
% inf_sup_aci(8)
thf(fact_131_par__c_Oelims,axiom,
! [X: product_prod @ dtree @ dtree,Y5: set @ ( sum_sum @ t @ ( product_prod @ dtree @ dtree ) )] :
( ( ( parall1914194347_par_c @ X )
= Y5 )
=> ~ ! [Tr1: dtree,Tr2: dtree] :
( ( X
= ( product_Pair @ dtree @ dtree @ Tr1 @ Tr2 ) )
=> ( Y5
!= ( sup_sup @ ( set @ ( sum_sum @ t @ ( product_prod @ dtree @ dtree ) ) ) @ ( image @ t @ ( sum_sum @ t @ ( product_prod @ dtree @ dtree ) ) @ ( sum_Inl @ t @ ( product_prod @ dtree @ dtree ) ) @ ( vimage @ t @ ( sum_sum @ t @ dtree ) @ ( sum_Inl @ t @ dtree ) @ ( sup_sup @ ( set @ ( sum_sum @ t @ dtree ) ) @ ( cont @ Tr1 ) @ ( cont @ Tr2 ) ) ) )
@ ( image @ ( product_prod @ dtree @ dtree ) @ ( sum_sum @ t @ ( product_prod @ dtree @ dtree ) ) @ ( sum_Inr @ ( product_prod @ dtree @ dtree ) @ t )
@ ( product_Sigma @ dtree @ dtree @ ( vimage @ dtree @ ( sum_sum @ t @ dtree ) @ ( sum_Inr @ dtree @ t ) @ ( cont @ Tr1 ) )
@ ^ [Uu: dtree] : ( vimage @ dtree @ ( sum_sum @ t @ dtree ) @ ( sum_Inr @ dtree @ t ) @ ( cont @ Tr2 ) ) ) ) ) ) ) ) ).
% par_c.elims
thf(fact_132_par__c_Osimps,axiom,
! [Tr12: dtree,Tr22: dtree] :
( ( parall1914194347_par_c @ ( product_Pair @ dtree @ dtree @ Tr12 @ Tr22 ) )
= ( sup_sup @ ( set @ ( sum_sum @ t @ ( product_prod @ dtree @ dtree ) ) ) @ ( image @ t @ ( sum_sum @ t @ ( product_prod @ dtree @ dtree ) ) @ ( sum_Inl @ t @ ( product_prod @ dtree @ dtree ) ) @ ( vimage @ t @ ( sum_sum @ t @ dtree ) @ ( sum_Inl @ t @ dtree ) @ ( sup_sup @ ( set @ ( sum_sum @ t @ dtree ) ) @ ( cont @ Tr12 ) @ ( cont @ Tr22 ) ) ) )
@ ( image @ ( product_prod @ dtree @ dtree ) @ ( sum_sum @ t @ ( product_prod @ dtree @ dtree ) ) @ ( sum_Inr @ ( product_prod @ dtree @ dtree ) @ t )
@ ( product_Sigma @ dtree @ dtree @ ( vimage @ dtree @ ( sum_sum @ t @ dtree ) @ ( sum_Inr @ dtree @ t ) @ ( cont @ Tr12 ) )
@ ^ [Uu: dtree] : ( vimage @ dtree @ ( sum_sum @ t @ dtree ) @ ( sum_Inr @ dtree @ t ) @ ( cont @ Tr22 ) ) ) ) ) ) ).
% par_c.simps
thf(fact_133_par__c_Opelims,axiom,
! [X: product_prod @ dtree @ dtree,Y5: set @ ( sum_sum @ t @ ( product_prod @ dtree @ dtree ) )] :
( ( ( parall1914194347_par_c @ X )
= Y5 )
=> ( ( accp @ ( product_prod @ dtree @ dtree ) @ parall1418634510_c_rel @ X )
=> ~ ! [Tr1: dtree,Tr2: dtree] :
( ( X
= ( product_Pair @ dtree @ dtree @ Tr1 @ Tr2 ) )
=> ( ( Y5
= ( sup_sup @ ( set @ ( sum_sum @ t @ ( product_prod @ dtree @ dtree ) ) ) @ ( image @ t @ ( sum_sum @ t @ ( product_prod @ dtree @ dtree ) ) @ ( sum_Inl @ t @ ( product_prod @ dtree @ dtree ) ) @ ( vimage @ t @ ( sum_sum @ t @ dtree ) @ ( sum_Inl @ t @ dtree ) @ ( sup_sup @ ( set @ ( sum_sum @ t @ dtree ) ) @ ( cont @ Tr1 ) @ ( cont @ Tr2 ) ) ) )
@ ( image @ ( product_prod @ dtree @ dtree ) @ ( sum_sum @ t @ ( product_prod @ dtree @ dtree ) ) @ ( sum_Inr @ ( product_prod @ dtree @ dtree ) @ t )
@ ( product_Sigma @ dtree @ dtree @ ( vimage @ dtree @ ( sum_sum @ t @ dtree ) @ ( sum_Inr @ dtree @ t ) @ ( cont @ Tr1 ) )
@ ^ [Uu: dtree] : ( vimage @ dtree @ ( sum_sum @ t @ dtree ) @ ( sum_Inr @ dtree @ t ) @ ( cont @ Tr2 ) ) ) ) ) )
=> ~ ( accp @ ( product_prod @ dtree @ dtree ) @ parall1418634510_c_rel @ ( product_Pair @ dtree @ dtree @ Tr1 @ Tr2 ) ) ) ) ) ) ).
% par_c.pelims
thf(fact_134_Plus__def,axiom,
! [B: $tType,A: $tType] :
( ( sum_Plus @ A @ B )
= ( ^ [A6: set @ A,B5: set @ B] : ( sup_sup @ ( set @ ( sum_sum @ A @ B ) ) @ ( image @ A @ ( sum_sum @ A @ B ) @ ( sum_Inl @ A @ B ) @ A6 ) @ ( image @ B @ ( sum_sum @ A @ B ) @ ( sum_Inr @ B @ A ) @ B5 ) ) ) ) ).
% Plus_def
thf(fact_135_sup2CI,axiom,
! [A: $tType,B: $tType,B3: A > B > $o,X: A,Y5: B,A3: A > B > $o] :
( ( ~ ( B3 @ X @ Y5 )
=> ( A3 @ X @ Y5 ) )
=> ( sup_sup @ ( A > B > $o ) @ A3 @ B3 @ X @ Y5 ) ) ).
% sup2CI
thf(fact_136_sup1CI,axiom,
! [A: $tType,B3: A > $o,X: A,A3: A > $o] :
( ( ~ ( B3 @ X )
=> ( A3 @ X ) )
=> ( sup_sup @ ( A > $o ) @ A3 @ B3 @ X ) ) ).
% sup1CI
thf(fact_137_InrI,axiom,
! [B: $tType,A: $tType,B2: A,B3: set @ A,A3: set @ B] :
( ( member @ A @ B2 @ B3 )
=> ( member @ ( sum_sum @ B @ A ) @ ( sum_Inr @ A @ B @ B2 ) @ ( sum_Plus @ B @ A @ A3 @ B3 ) ) ) ).
% InrI
thf(fact_138_InlI,axiom,
! [A: $tType,B: $tType,A2: A,A3: set @ A,B3: set @ B] :
( ( member @ A @ A2 @ A3 )
=> ( member @ ( sum_sum @ A @ B ) @ ( sum_Inl @ A @ B @ A2 ) @ ( sum_Plus @ A @ B @ A3 @ B3 ) ) ) ).
% InlI
thf(fact_139_sup1E,axiom,
! [A: $tType,A3: A > $o,B3: A > $o,X: A] :
( ( sup_sup @ ( A > $o ) @ A3 @ B3 @ X )
=> ( ~ ( A3 @ X )
=> ( B3 @ X ) ) ) ).
% sup1E
thf(fact_140_sup2E,axiom,
! [A: $tType,B: $tType,A3: A > B > $o,B3: A > B > $o,X: A,Y5: B] :
( ( sup_sup @ ( A > B > $o ) @ A3 @ B3 @ X @ Y5 )
=> ( ~ ( A3 @ X @ Y5 )
=> ( B3 @ X @ Y5 ) ) ) ).
% sup2E
thf(fact_141_sup1I1,axiom,
! [A: $tType,A3: A > $o,X: A,B3: A > $o] :
( ( A3 @ X )
=> ( sup_sup @ ( A > $o ) @ A3 @ B3 @ X ) ) ).
% sup1I1
thf(fact_142_sup1I2,axiom,
! [A: $tType,B3: A > $o,X: A,A3: A > $o] :
( ( B3 @ X )
=> ( sup_sup @ ( A > $o ) @ A3 @ B3 @ X ) ) ).
% sup1I2
thf(fact_143_sup2I1,axiom,
! [A: $tType,B: $tType,A3: A > B > $o,X: A,Y5: B,B3: A > B > $o] :
( ( A3 @ X @ Y5 )
=> ( sup_sup @ ( A > B > $o ) @ A3 @ B3 @ X @ Y5 ) ) ).
% sup2I1
thf(fact_144_sup2I2,axiom,
! [A: $tType,B: $tType,B3: A > B > $o,X: A,Y5: B,A3: A > B > $o] :
( ( B3 @ X @ Y5 )
=> ( sup_sup @ ( A > B > $o ) @ A3 @ B3 @ X @ Y5 ) ) ).
% sup2I2
thf(fact_145_PlusE,axiom,
! [A: $tType,B: $tType,U: sum_sum @ A @ B,A3: set @ A,B3: set @ B] :
( ( member @ ( sum_sum @ A @ B ) @ U @ ( sum_Plus @ A @ B @ A3 @ B3 ) )
=> ( ! [X3: A] :
( ( member @ A @ X3 @ A3 )
=> ( U
!= ( sum_Inl @ A @ B @ X3 ) ) )
=> ~ ! [Y3: B] :
( ( member @ B @ Y3 @ B3 )
=> ( U
!= ( sum_Inr @ B @ A @ Y3 ) ) ) ) ) ).
% PlusE
thf(fact_146_reflclp__idemp,axiom,
! [A: $tType,P: A > A > $o] :
( ( sup_sup @ ( A > A > $o )
@ ( sup_sup @ ( A > A > $o ) @ P
@ ^ [Y6: A,Z2: A] : Y6 = Z2 )
@ ^ [Y6: A,Z2: A] : Y6 = Z2 )
= ( sup_sup @ ( A > A > $o ) @ P
@ ^ [Y6: A,Z2: A] : Y6 = Z2 ) ) ).
% reflclp_idemp
thf(fact_147_accp_Ocases,axiom,
! [A: $tType,R2: A > A > $o,A2: A] :
( ( accp @ A @ R2 @ A2 )
=> ! [Y7: A] :
( ( R2 @ Y7 @ A2 )
=> ( accp @ A @ R2 @ Y7 ) ) ) ).
% accp.cases
thf(fact_148_accp_Osimps,axiom,
! [A: $tType] :
( ( accp @ A )
= ( ^ [R3: A > A > $o,A9: A] :
? [X2: A] :
( ( A9 = X2 )
& ! [Y4: A] :
( ( R3 @ Y4 @ X2 )
=> ( accp @ A @ R3 @ Y4 ) ) ) ) ) ).
% accp.simps
thf(fact_149_accp_Ointros,axiom,
! [A: $tType,R2: A > A > $o,X: A] :
( ! [Y3: A] :
( ( R2 @ Y3 @ X )
=> ( accp @ A @ R2 @ Y3 ) )
=> ( accp @ A @ R2 @ X ) ) ).
% accp.intros
thf(fact_150_accp__induct__rule,axiom,
! [A: $tType,R2: A > A > $o,A2: A,P: A > $o] :
( ( accp @ A @ R2 @ A2 )
=> ( ! [X3: A] :
( ( accp @ A @ R2 @ X3 )
=> ( ! [Y7: A] :
( ( R2 @ Y7 @ X3 )
=> ( P @ Y7 ) )
=> ( P @ X3 ) ) )
=> ( P @ A2 ) ) ) ).
% accp_induct_rule
thf(fact_151_not__accp__down,axiom,
! [A: $tType,R: A > A > $o,X: A] :
( ~ ( accp @ A @ R @ X )
=> ~ ! [Z3: A] :
( ( R @ Z3 @ X )
=> ( accp @ A @ R @ Z3 ) ) ) ).
% not_accp_down
thf(fact_152_accp__downward,axiom,
! [A: $tType,R2: A > A > $o,B2: A,A2: A] :
( ( accp @ A @ R2 @ B2 )
=> ( ( R2 @ A2 @ B2 )
=> ( accp @ A @ R2 @ A2 ) ) ) ).
% accp_downward
thf(fact_153_accp_Oinducts,axiom,
! [A: $tType,R2: A > A > $o,X: A,P: A > $o] :
( ( accp @ A @ R2 @ X )
=> ( ! [X3: A] :
( ! [Y7: A] :
( ( R2 @ Y7 @ X3 )
=> ( accp @ A @ R2 @ Y7 ) )
=> ( ! [Y7: A] :
( ( R2 @ Y7 @ X3 )
=> ( P @ Y7 ) )
=> ( P @ X3 ) ) )
=> ( P @ X ) ) ) ).
% accp.inducts
thf(fact_154_accp__induct,axiom,
! [A: $tType,R2: A > A > $o,A2: A,P: A > $o] :
( ( accp @ A @ R2 @ A2 )
=> ( ! [X3: A] :
( ( accp @ A @ R2 @ X3 )
=> ( ! [Y7: A] :
( ( R2 @ Y7 @ X3 )
=> ( P @ Y7 ) )
=> ( P @ X3 ) ) )
=> ( P @ A2 ) ) ) ).
% accp_induct
thf(fact_155_in__lex__prod,axiom,
! [A: $tType,B: $tType,A2: A,B2: B,A4: A,B4: B,R2: set @ ( product_prod @ A @ A ),S2: 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 @ B4 ) ) @ ( lex_prod @ A @ B @ R2 @ S2 ) )
= ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A2 @ A4 ) @ R2 )
| ( ( A2 = A4 )
& ( member @ ( product_prod @ B @ B ) @ ( product_Pair @ B @ B @ B2 @ B4 ) @ S2 ) ) ) ) ).
% in_lex_prod
thf(fact_156_reflcl__set__eq,axiom,
! [A: $tType,R2: set @ ( product_prod @ A @ A )] :
( ( sup_sup @ ( A > A > $o )
@ ^ [X2: A,Y4: A] : ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X2 @ Y4 ) @ R2 )
@ ^ [Y6: A,Z2: A] : Y6 = Z2 )
= ( ^ [X2: A,Y4: A] : ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X2 @ Y4 ) @ ( sup_sup @ ( set @ ( product_prod @ A @ A ) ) @ R2 @ ( id @ A ) ) ) ) ) ).
% reflcl_set_eq
thf(fact_157_par__def,axiom,
( parall1899940088le_par
= ( unfold @ ( product_prod @ dtree @ dtree ) @ parall1914194362_par_r @ parall1914194347_par_c ) ) ).
% par_def
thf(fact_158_IdI,axiom,
! [A: $tType,A2: A] : ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A2 @ A2 ) @ ( id @ A ) ) ).
% IdI
thf(fact_159_pair__in__Id__conv,axiom,
! [A: $tType,A2: A,B2: A] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A2 @ B2 ) @ ( id @ A ) )
= ( A2 = B2 ) ) ).
% pair_in_Id_conv
thf(fact_160_IdE,axiom,
! [A: $tType,P2: product_prod @ A @ A] :
( ( member @ ( product_prod @ A @ A ) @ P2 @ ( id @ A ) )
=> ~ ! [X3: A] :
( P2
!= ( product_Pair @ A @ A @ X3 @ X3 ) ) ) ).
% IdE
thf(fact_161_IdD,axiom,
! [A: $tType,A2: A,B2: A] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A2 @ B2 ) @ ( id @ A ) )
=> ( A2 = B2 ) ) ).
% IdD
thf(fact_162_same__fstI,axiom,
! [B: $tType,A: $tType,P: A > $o,X: A,Y8: B,Y5: B,R: A > ( set @ ( product_prod @ B @ B ) )] :
( ( P @ X )
=> ( ( member @ ( product_prod @ B @ B ) @ ( product_Pair @ B @ B @ Y8 @ Y5 ) @ ( R @ 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 @ Y8 ) @ ( product_Pair @ A @ B @ X @ Y5 ) ) @ ( same_fst @ A @ B @ P @ R ) ) ) ) ).
% same_fstI
thf(fact_163_antisym__reflcl,axiom,
! [A: $tType,R2: set @ ( product_prod @ A @ A )] :
( ( antisym @ A @ ( sup_sup @ ( set @ ( product_prod @ A @ A ) ) @ R2 @ ( id @ A ) ) )
= ( antisym @ A @ R2 ) ) ).
% antisym_reflcl
thf(fact_164_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_165_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_166_case__prodI2,axiom,
! [B: $tType,A: $tType,P2: product_prod @ A @ B,C2: A > B > $o] :
( ! [A5: A,B6: B] :
( ( P2
= ( product_Pair @ A @ B @ A5 @ B6 ) )
=> ( C2 @ A5 @ B6 ) )
=> ( product_case_prod @ A @ B @ $o @ C2 @ P2 ) ) ).
% case_prodI2
thf(fact_167_mem__case__prodI,axiom,
! [A: $tType,B: $tType,C: $tType,Z: A,C2: B > C > ( set @ A ),A2: B,B2: C] :
( ( member @ A @ Z @ ( C2 @ A2 @ B2 ) )
=> ( member @ A @ Z @ ( product_case_prod @ B @ C @ ( set @ A ) @ C2 @ ( product_Pair @ B @ C @ A2 @ B2 ) ) ) ) ).
% mem_case_prodI
thf(fact_168_mem__case__prodI2,axiom,
! [C: $tType,B: $tType,A: $tType,P2: product_prod @ A @ B,Z: C,C2: A > B > ( set @ C )] :
( ! [A5: A,B6: B] :
( ( P2
= ( product_Pair @ A @ B @ A5 @ B6 ) )
=> ( member @ C @ Z @ ( C2 @ A5 @ B6 ) ) )
=> ( member @ C @ Z @ ( product_case_prod @ A @ B @ ( set @ C ) @ C2 @ P2 ) ) ) ).
% mem_case_prodI2
thf(fact_169_case__prodI2_H,axiom,
! [A: $tType,B: $tType,C: $tType,P2: product_prod @ A @ B,C2: A > B > C > $o,X: C] :
( ! [A5: A,B6: B] :
( ( ( product_Pair @ A @ B @ A5 @ B6 )
= P2 )
=> ( C2 @ A5 @ B6 @ X ) )
=> ( product_case_prod @ A @ B @ ( C > $o ) @ C2 @ P2 @ X ) ) ).
% case_prodI2'
thf(fact_170_Collect__case__prod,axiom,
! [B: $tType,A: $tType,P: A > $o,Q: B > $o] :
( ( collect @ ( product_prod @ A @ B )
@ ( product_case_prod @ A @ B @ $o
@ ^ [A9: A,B9: B] :
( ( P @ A9 )
& ( Q @ B9 ) ) ) )
= ( product_Sigma @ A @ B @ ( collect @ A @ P )
@ ^ [Uu: A] : ( collect @ B @ Q ) ) ) ).
% Collect_case_prod
thf(fact_171_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_172_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 )
@ ^ [A9: A,B9: B] :
( product_case_prod @ A @ B @ $o
@ ^ [A10: A,B10: B] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A9 @ A10 ) @ Ra )
| ( ( A9 = A10 )
& ( member @ ( product_prod @ B @ B ) @ ( product_Pair @ B @ B @ B9 @ B10 ) @ Rb ) ) ) ) ) ) ) ) ) ).
% lex_prod_def
thf(fact_173_case__prodD_H,axiom,
! [B: $tType,A: $tType,C: $tType,R: A > B > C > $o,A2: A,B2: B,C2: C] :
( ( product_case_prod @ A @ B @ ( C > $o ) @ R @ ( product_Pair @ A @ B @ A2 @ B2 ) @ C2 )
=> ( R @ A2 @ B2 @ C2 ) ) ).
% case_prodD'
thf(fact_174_case__prodE_H,axiom,
! [B: $tType,A: $tType,C: $tType,C2: A > B > C > $o,P2: product_prod @ A @ B,Z: C] :
( ( product_case_prod @ A @ B @ ( C > $o ) @ C2 @ P2 @ Z )
=> ~ ! [X3: A,Y3: B] :
( ( P2
= ( product_Pair @ A @ B @ X3 @ Y3 ) )
=> ~ ( C2 @ X3 @ Y3 @ Z ) ) ) ).
% case_prodE'
thf(fact_175_case__prodE2,axiom,
! [B: $tType,A: $tType,C: $tType,Q: A > $o,P: B > C > A,Z: product_prod @ B @ C] :
( ( Q @ ( product_case_prod @ B @ C @ A @ P @ Z ) )
=> ~ ! [X3: B,Y3: C] :
( ( Z
= ( product_Pair @ B @ C @ X3 @ Y3 ) )
=> ~ ( Q @ ( P @ X3 @ Y3 ) ) ) ) ).
% case_prodE2
thf(fact_176_case__prod__eta,axiom,
! [C: $tType,B: $tType,A: $tType,F: ( product_prod @ A @ B ) > C] :
( ( product_case_prod @ A @ B @ C
@ ^ [X2: A,Y4: B] : ( F @ ( product_Pair @ A @ B @ X2 @ Y4 ) ) )
= F ) ).
% case_prod_eta
thf(fact_177_cond__case__prod__eta,axiom,
! [C: $tType,B: $tType,A: $tType,F: A > B > C,G3: ( product_prod @ A @ B ) > C] :
( ! [X3: A,Y3: B] :
( ( F @ X3 @ Y3 )
= ( G3 @ ( product_Pair @ A @ B @ X3 @ Y3 ) ) )
=> ( ( product_case_prod @ A @ B @ C @ F )
= G3 ) ) ).
% cond_case_prod_eta
thf(fact_178_mem__case__prodE,axiom,
! [B: $tType,A: $tType,C: $tType,Z: A,C2: B > C > ( set @ A ),P2: product_prod @ B @ C] :
( ( member @ A @ Z @ ( product_case_prod @ B @ C @ ( set @ A ) @ C2 @ P2 ) )
=> ~ ! [X3: B,Y3: C] :
( ( P2
= ( product_Pair @ B @ C @ X3 @ Y3 ) )
=> ~ ( member @ A @ Z @ ( C2 @ X3 @ Y3 ) ) ) ) ).
% mem_case_prodE
thf(fact_179_old_Oprod_Ocase,axiom,
! [A: $tType,C: $tType,B: $tType,F: A > B > C,X1: A,X22: B] :
( ( product_case_prod @ A @ B @ C @ F @ ( product_Pair @ A @ B @ X1 @ X22 ) )
= ( F @ X1 @ X22 ) ) ).
% old.prod.case
thf(fact_180_same__fst__def,axiom,
! [B: $tType,A: $tType] :
( ( same_fst @ A @ B )
= ( ^ [P4: 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 )
@ ^ [X6: A,Y9: B] :
( product_case_prod @ A @ B @ $o
@ ^ [X2: A,Y4: B] :
( ( X6 = X2 )
& ( P4 @ X2 )
& ( member @ ( product_prod @ B @ B ) @ ( product_Pair @ B @ B @ Y9 @ Y4 ) @ ( R4 @ X2 ) ) ) ) ) ) ) ) ) ).
% same_fst_def
thf(fact_181_antisym__Id,axiom,
! [A: $tType] : ( antisym @ A @ ( id @ A ) ) ).
% antisym_Id
thf(fact_182_antisymD,axiom,
! [A: $tType,R2: set @ ( product_prod @ A @ A ),A2: A,B2: A] :
( ( antisym @ A @ R2 )
=> ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A2 @ B2 ) @ R2 )
=> ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ B2 @ A2 ) @ R2 )
=> ( A2 = B2 ) ) ) ) ).
% antisymD
thf(fact_183_antisymI,axiom,
! [A: $tType,R2: set @ ( product_prod @ A @ A )] :
( ! [X3: A,Y3: A] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X3 @ Y3 ) @ R2 )
=> ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Y3 @ X3 ) @ R2 )
=> ( X3 = Y3 ) ) )
=> ( antisym @ A @ R2 ) ) ).
% antisymI
thf(fact_184_antisym__def,axiom,
! [A: $tType] :
( ( antisym @ A )
= ( ^ [R3: set @ ( product_prod @ A @ A )] :
! [X2: A,Y4: A] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X2 @ Y4 ) @ R3 )
=> ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Y4 @ X2 ) @ R3 )
=> ( X2 = Y4 ) ) ) ) ) ).
% antisym_def
thf(fact_185_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_186_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_187_case__prodE,axiom,
! [A: $tType,B: $tType,C2: A > B > $o,P2: product_prod @ A @ B] :
( ( product_case_prod @ A @ B @ $o @ C2 @ P2 )
=> ~ ! [X3: A,Y3: B] :
( ( P2
= ( product_Pair @ A @ B @ X3 @ Y3 ) )
=> ~ ( C2 @ X3 @ Y3 ) ) ) ).
% case_prodE
thf(fact_188_Collect__case__prod__Sigma,axiom,
! [B: $tType,A: $tType,P: A > $o,Q: A > B > $o] :
( ( collect @ ( product_prod @ A @ B )
@ ( product_case_prod @ A @ B @ $o
@ ^ [X2: A,Y4: B] :
( ( P @ X2 )
& ( Q @ X2 @ Y4 ) ) ) )
= ( product_Sigma @ A @ B @ ( collect @ A @ P )
@ ^ [X2: A] : ( collect @ B @ ( Q @ X2 ) ) ) ) ).
% Collect_case_prod_Sigma
thf(fact_189_case__prod__app,axiom,
! [A: $tType,D: $tType,C: $tType,B: $tType] :
( ( product_case_prod @ B @ C @ ( D > A ) )
= ( ^ [F2: B > C > D > A,X2: product_prod @ B @ C,Y4: D] :
( product_case_prod @ B @ C @ A
@ ^ [L: B,R3: C] : ( F2 @ L @ R3 @ Y4 )
@ X2 ) ) ) ).
% case_prod_app
thf(fact_190_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
@ ^ [X12: A,X23: B] : ( H @ ( F @ X12 @ X23 ) )
@ Prod ) ) ).
% prod.case_distrib
thf(fact_191_antisymP__equality,axiom,
! [A: $tType] :
( antisym @ A
@ ( collect @ ( product_prod @ A @ A )
@ ( product_case_prod @ A @ A @ $o
@ ^ [Y6: A,Z2: A] : Y6 = Z2 ) ) ) ).
% antisymP_equality
thf(fact_192_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_193_swap__product,axiom,
! [B: $tType,A: $tType,A3: set @ B,B3: set @ A] :
( ( image @ ( product_prod @ B @ A ) @ ( product_prod @ A @ B )
@ ( product_case_prod @ B @ A @ ( product_prod @ A @ B )
@ ^ [I2: B,J2: A] : ( product_Pair @ A @ B @ J2 @ I2 ) )
@ ( product_Sigma @ B @ A @ A3
@ ^ [Uu: B] : B3 ) )
= ( product_Sigma @ A @ B @ B3
@ ^ [Uu: A] : A3 ) ) ).
% swap_product
thf(fact_194_split__cong,axiom,
! [C: $tType,B: $tType,A: $tType,Q2: product_prod @ A @ B,F: A > B > C,G3: A > B > C,P2: product_prod @ A @ B] :
( ! [X3: A,Y3: B] :
( ( ( product_Pair @ A @ B @ X3 @ Y3 )
= Q2 )
=> ( ( F @ X3 @ Y3 )
= ( G3 @ X3 @ Y3 ) ) )
=> ( ( P2 = Q2 )
=> ( ( product_case_prod @ A @ B @ C @ F @ P2 )
= ( product_case_prod @ A @ B @ C @ G3 @ Q2 ) ) ) ) ).
% split_cong
thf(fact_195_case__swap,axiom,
! [A: $tType,B: $tType,C: $tType,F: C > B > A,P2: product_prod @ C @ B] :
( ( product_case_prod @ B @ C @ A
@ ^ [Y4: B,X2: C] : ( F @ X2 @ Y4 )
@ ( product_swap @ C @ B @ P2 ) )
= ( product_case_prod @ C @ B @ A @ F @ P2 ) ) ).
% case_swap
thf(fact_196_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_197_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_198_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_199_split__part,axiom,
! [B: $tType,A: $tType,P: $o,Q: A > B > $o] :
( ( product_case_prod @ A @ B @ $o
@ ^ [A9: A,B9: B] :
( P
& ( Q @ A9 @ B9 ) ) )
= ( ^ [Ab: product_prod @ A @ B] :
( P
& ( product_case_prod @ A @ B @ $o @ Q @ Ab ) ) ) ) ).
% split_part
thf(fact_200_swap__simp,axiom,
! [A: $tType,B: $tType,X: B,Y5: A] :
( ( product_swap @ B @ A @ ( product_Pair @ B @ A @ X @ Y5 ) )
= ( product_Pair @ A @ B @ Y5 @ X ) ) ).
% swap_simp
thf(fact_201_curry__conv,axiom,
! [A: $tType,C: $tType,B: $tType] :
( ( product_curry @ B @ C @ A )
= ( ^ [F2: ( product_prod @ B @ C ) > A,A9: B,B9: C] : ( F2 @ ( product_Pair @ B @ C @ A9 @ B9 ) ) ) ) ).
% curry_conv
thf(fact_202_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_203_pair__in__swap__image,axiom,
! [A: $tType,B: $tType,Y5: A,X: B,A3: set @ ( product_prod @ B @ A )] :
( ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ Y5 @ X ) @ ( image @ ( product_prod @ B @ A ) @ ( product_prod @ A @ B ) @ ( product_swap @ B @ A ) @ A3 ) )
= ( member @ ( product_prod @ B @ A ) @ ( product_Pair @ B @ A @ X @ Y5 ) @ A3 ) ) ).
% pair_in_swap_image
thf(fact_204_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_205_curry__K,axiom,
! [B: $tType,C: $tType,A: $tType,C2: C] :
( ( product_curry @ A @ B @ C
@ ^ [X2: product_prod @ A @ B] : C2 )
= ( ^ [X2: A,Y4: B] : C2 ) ) ).
% curry_K
thf(fact_206_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_207_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_208_product__swap,axiom,
! [B: $tType,A: $tType,A3: set @ B,B3: set @ A] :
( ( image @ ( product_prod @ B @ A ) @ ( product_prod @ A @ B ) @ ( product_swap @ B @ A )
@ ( product_Sigma @ B @ A @ A3
@ ^ [Uu: B] : B3 ) )
= ( product_Sigma @ A @ B @ B3
@ ^ [Uu: A] : A3 ) ) ).
% product_swap
thf(fact_209_curry__def,axiom,
! [C: $tType,B: $tType,A: $tType] :
( ( product_curry @ A @ B @ C )
= ( ^ [C5: ( product_prod @ A @ B ) > C,X2: A,Y4: B] : ( C5 @ ( product_Pair @ A @ B @ X2 @ Y4 ) ) ) ) ).
% curry_def
thf(fact_210_inv__image__def,axiom,
! [A: $tType,B: $tType] :
( ( inv_image @ B @ A )
= ( ^ [R3: set @ ( product_prod @ B @ B ),F2: A > B] :
( collect @ ( product_prod @ A @ A )
@ ( product_case_prod @ A @ A @ $o
@ ^ [X2: A,Y4: A] : ( member @ ( product_prod @ B @ B ) @ ( product_Pair @ B @ B @ ( F2 @ X2 ) @ ( F2 @ Y4 ) ) @ R3 ) ) ) ) ) ).
% inv_image_def
thf(fact_211_map__prod__surj__on,axiom,
! [B: $tType,D: $tType,C: $tType,A: $tType,F: B > A,A3: set @ B,A11: set @ A,G3: D > C,B3: set @ D,B11: set @ C] :
( ( ( image @ B @ A @ F @ A3 )
= A11 )
=> ( ( ( image @ D @ C @ G3 @ B3 )
= B11 )
=> ( ( image @ ( product_prod @ B @ D ) @ ( product_prod @ A @ C ) @ ( product_map_prod @ B @ A @ D @ C @ F @ G3 )
@ ( product_Sigma @ B @ D @ A3
@ ^ [Uu: B] : B3 ) )
= ( product_Sigma @ A @ C @ A11
@ ^ [Uu: A] : B11 ) ) ) ) ).
% map_prod_surj_on
thf(fact_212_map__prod__ident,axiom,
! [B: $tType,A: $tType] :
( ( product_map_prod @ A @ A @ B @ B
@ ^ [X2: A] : X2
@ ^ [Y4: B] : Y4 )
= ( ^ [Z4: product_prod @ A @ B] : Z4 ) ) ).
% map_prod_ident
thf(fact_213_map__prod__simp,axiom,
! [C: $tType,A: $tType,B: $tType,D: $tType,F: C > A,G3: D > B,A2: C,B2: D] :
( ( product_map_prod @ C @ A @ D @ B @ F @ G3 @ ( product_Pair @ C @ D @ A2 @ B2 ) )
= ( product_Pair @ A @ B @ ( F @ A2 ) @ ( G3 @ B2 ) ) ) ).
% map_prod_simp
thf(fact_214_in__inv__image,axiom,
! [A: $tType,B: $tType,X: A,Y5: A,R2: set @ ( product_prod @ B @ B ),F: A > B] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X @ Y5 ) @ ( inv_image @ B @ A @ R2 @ F ) )
= ( member @ ( product_prod @ B @ B ) @ ( product_Pair @ B @ B @ ( F @ X ) @ ( F @ Y5 ) ) @ R2 ) ) ).
% in_inv_image
thf(fact_215_map__prod__imageI,axiom,
! [D: $tType,C: $tType,B: $tType,A: $tType,A2: A,B2: B,R: set @ ( product_prod @ A @ B ),F: A > C,G3: B > D] :
( ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ A2 @ B2 ) @ R )
=> ( member @ ( product_prod @ C @ D ) @ ( product_Pair @ C @ D @ ( F @ A2 ) @ ( G3 @ B2 ) ) @ ( image @ ( product_prod @ A @ B ) @ ( product_prod @ C @ D ) @ ( product_map_prod @ A @ C @ B @ D @ F @ G3 ) @ R ) ) ) ).
% map_prod_imageI
thf(fact_216_case__prod__map__prod,axiom,
! [C: $tType,A: $tType,B: $tType,E: $tType,D: $tType,H: B > C > A,F: D > B,G3: E > C,X: product_prod @ D @ E] :
( ( product_case_prod @ B @ C @ A @ H @ ( product_map_prod @ D @ B @ E @ C @ F @ G3 @ X ) )
= ( product_case_prod @ D @ E @ A
@ ^ [L: D,R3: E] : ( H @ ( F @ L ) @ ( G3 @ R3 ) )
@ X ) ) ).
% case_prod_map_prod
thf(fact_217_prod__fun__imageE,axiom,
! [B: $tType,A: $tType,D: $tType,C: $tType,C2: product_prod @ A @ B,F: C > A,G3: D > B,R: 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 @ F @ G3 ) @ R ) )
=> ~ ! [X3: C,Y3: D] :
( ( C2
= ( product_Pair @ A @ B @ ( F @ X3 ) @ ( G3 @ Y3 ) ) )
=> ~ ( member @ ( product_prod @ C @ D ) @ ( product_Pair @ C @ D @ X3 @ Y3 ) @ R ) ) ) ).
% prod_fun_imageE
thf(fact_218_map__prod__def,axiom,
! [B: $tType,D: $tType,C: $tType,A: $tType] :
( ( product_map_prod @ A @ C @ B @ D )
= ( ^ [F2: A > C,G4: B > D] :
( product_case_prod @ A @ B @ ( product_prod @ C @ D )
@ ^ [X2: A,Y4: B] : ( product_Pair @ C @ D @ ( F2 @ X2 ) @ ( G4 @ Y4 ) ) ) ) ) ).
% map_prod_def
thf(fact_219_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 ),S3: set @ ( product_prod @ A @ A ),F2: B > A] : ( product_Pair @ ( set @ ( product_prod @ B @ B ) ) @ ( set @ ( product_prod @ B @ B ) ) @ ( inv_image @ A @ B @ R4 @ F2 ) @ ( inv_image @ A @ B @ S3 @ F2 ) ) ) ) ).
% rp_inv_image_def
thf(fact_220_prod_Omap__ident,axiom,
! [B: $tType,A: $tType,T2: product_prod @ A @ B] :
( ( product_map_prod @ A @ A @ B @ B
@ ^ [X2: A] : X2
@ ^ [X2: B] : X2
@ T2 )
= T2 ) ).
% prod.map_ident
thf(fact_221_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_222_The__split__eq,axiom,
! [A: $tType,B: $tType,X: A,Y5: B] :
( ( the @ ( product_prod @ A @ B )
@ ( product_case_prod @ A @ B @ $o
@ ^ [X6: A,Y9: B] :
( ( X = X6 )
& ( Y5 = Y9 ) ) ) )
= ( product_Pair @ A @ B @ X @ Y5 ) ) ).
% The_split_eq
thf(fact_223_UNIV__I,axiom,
! [A: $tType,X: A] : ( member @ A @ X @ ( top_top @ ( set @ A ) ) ) ).
% UNIV_I
thf(fact_224_sup__top__right,axiom,
! [A: $tType] :
( ( bounded_lattice_top @ A @ ( type2 @ A ) )
=> ! [X: A] :
( ( sup_sup @ A @ X @ ( top_top @ A ) )
= ( top_top @ A ) ) ) ).
% sup_top_right
thf(fact_225_sup__top__left,axiom,
! [A: $tType] :
( ( bounded_lattice_top @ A @ ( type2 @ A ) )
=> ! [X: A] :
( ( sup_sup @ A @ ( top_top @ A ) @ X )
= ( top_top @ A ) ) ) ).
% sup_top_left
thf(fact_226_vimage__UNIV,axiom,
! [B: $tType,A: $tType,F: A > B] :
( ( vimage @ A @ B @ F @ ( top_top @ ( set @ B ) ) )
= ( top_top @ ( set @ A ) ) ) ).
% vimage_UNIV
thf(fact_227_UNIV__Plus__UNIV,axiom,
! [B: $tType,A: $tType] :
( ( sum_Plus @ A @ B @ ( top_top @ ( set @ A ) ) @ ( top_top @ ( set @ B ) ) )
= ( top_top @ ( set @ ( sum_sum @ A @ B ) ) ) ) ).
% UNIV_Plus_UNIV
thf(fact_228_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_229_UNIV__def,axiom,
! [A: $tType] :
( ( top_top @ ( set @ A ) )
= ( collect @ A
@ ^ [X2: A] : $true ) ) ).
% UNIV_def
thf(fact_230_rangeE,axiom,
! [A: $tType,B: $tType,B2: A,F: B > A] :
( ( member @ A @ B2 @ ( image @ B @ A @ F @ ( top_top @ ( set @ B ) ) ) )
=> ~ ! [X3: B] :
( B2
!= ( F @ X3 ) ) ) ).
% rangeE
thf(fact_231_range__composition,axiom,
! [A: $tType,C: $tType,B: $tType,F: C > A,G3: B > C] :
( ( image @ B @ A
@ ^ [X2: B] : ( F @ ( G3 @ X2 ) )
@ ( top_top @ ( set @ B ) ) )
= ( image @ C @ A @ F @ ( image @ B @ C @ G3 @ ( top_top @ ( set @ B ) ) ) ) ) ).
% range_composition
thf(fact_232_rangeI,axiom,
! [A: $tType,B: $tType,F: B > A,X: B] : ( member @ A @ ( F @ X ) @ ( image @ B @ A @ F @ ( top_top @ ( set @ B ) ) ) ) ).
% rangeI
thf(fact_233_range__eqI,axiom,
! [A: $tType,B: $tType,B2: A,F: B > A,X: B] :
( ( B2
= ( F @ X ) )
=> ( member @ A @ B2 @ ( image @ B @ A @ F @ ( top_top @ ( set @ B ) ) ) ) ) ).
% range_eqI
thf(fact_234_Un__UNIV__left,axiom,
! [A: $tType,B3: set @ A] :
( ( sup_sup @ ( set @ A ) @ ( top_top @ ( set @ A ) ) @ B3 )
= ( top_top @ ( set @ A ) ) ) ).
% Un_UNIV_left
thf(fact_235_Un__UNIV__right,axiom,
! [A: $tType,A3: set @ A] :
( ( sup_sup @ ( set @ A ) @ A3 @ ( top_top @ ( set @ A ) ) )
= ( top_top @ ( set @ A ) ) ) ).
% Un_UNIV_right
thf(fact_236_UNIV__eq__I,axiom,
! [A: $tType,A3: set @ A] :
( ! [X3: A] : ( member @ A @ X3 @ A3 )
=> ( ( top_top @ ( set @ A ) )
= A3 ) ) ).
% UNIV_eq_I
thf(fact_237_UNIV__witness,axiom,
! [A: $tType] :
? [X3: A] : ( member @ A @ X3 @ ( top_top @ ( set @ A ) ) ) ).
% UNIV_witness
thf(fact_238_UNIV__sum,axiom,
! [A: $tType,B: $tType] :
( ( top_top @ ( set @ ( sum_sum @ A @ B ) ) )
= ( sup_sup @ ( set @ ( sum_sum @ A @ B ) ) @ ( image @ A @ ( sum_sum @ A @ B ) @ ( sum_Inl @ A @ B ) @ ( top_top @ ( set @ A ) ) ) @ ( image @ B @ ( sum_sum @ A @ B ) @ ( sum_Inr @ B @ A ) @ ( top_top @ ( set @ B ) ) ) ) ) ).
% UNIV_sum
thf(fact_239_map__prod__surj,axiom,
! [A: $tType,C: $tType,D: $tType,B: $tType,F: A > B,G3: C > D] :
( ( ( image @ A @ B @ F @ ( 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 @ F @ G3 ) @ ( top_top @ ( set @ ( product_prod @ A @ C ) ) ) )
= ( top_top @ ( set @ ( product_prod @ B @ D ) ) ) ) ) ) ).
% map_prod_surj
thf(fact_240_old_Orec__sum__def,axiom,
! [B: $tType,T: $tType,A: $tType] :
( ( sum_rec_sum @ A @ T @ B )
= ( ^ [F12: A > T,F23: B > T,X2: sum_sum @ A @ B] : ( the @ T @ ( sum_rec_set_sum @ A @ T @ B @ F12 @ F23 @ X2 ) ) ) ) ).
% old.rec_sum_def
thf(fact_241_old_Orec__prod__def,axiom,
! [T: $tType,B: $tType,A: $tType] :
( ( product_rec_prod @ A @ B @ T )
= ( ^ [F12: A > B > T,X2: product_prod @ A @ B] : ( the @ T @ ( product_rec_set_prod @ A @ B @ T @ F12 @ X2 ) ) ) ) ).
% old.rec_prod_def
thf(fact_242_top__empty__eq2,axiom,
! [B: $tType,A: $tType] :
( ( top_top @ ( A > B > $o ) )
= ( ^ [X2: A,Y4: B] : ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X2 @ Y4 ) @ ( top_top @ ( set @ ( product_prod @ A @ B ) ) ) ) ) ) ).
% top_empty_eq2
thf(fact_243_top__set__def,axiom,
! [A: $tType] :
( ( top_top @ ( set @ A ) )
= ( collect @ A @ ( top_top @ ( A > $o ) ) ) ) ).
% top_set_def
thf(fact_244_top__empty__eq,axiom,
! [A: $tType] :
( ( top_top @ ( A > $o ) )
= ( ^ [X2: A] : ( member @ A @ X2 @ ( top_top @ ( set @ A ) ) ) ) ) ).
% top_empty_eq
thf(fact_245_the__sym__eq__trivial,axiom,
! [A: $tType,X: A] :
( ( the @ A
@ ( ^ [Y6: A,Z2: A] : Y6 = Z2
@ X ) )
= X ) ).
% the_sym_eq_trivial
thf(fact_246_the__eq__trivial,axiom,
! [A: $tType,A2: A] :
( ( the @ A
@ ^ [X2: A] : X2 = A2 )
= A2 ) ).
% the_eq_trivial
thf(fact_247_the__equality,axiom,
! [A: $tType,P: A > $o,A2: A] :
( ( P @ A2 )
=> ( ! [X3: A] :
( ( P @ X3 )
=> ( X3 = A2 ) )
=> ( ( the @ A @ P )
= A2 ) ) ) ).
% the_equality
thf(fact_248_top__conj_I2_J,axiom,
! [A: $tType,P: $o,X: A] :
( ( P
& ( top_top @ ( A > $o ) @ X ) )
= P ) ).
% top_conj(2)
thf(fact_249_top__conj_I1_J,axiom,
! [A: $tType,X: A,P: $o] :
( ( ( top_top @ ( A > $o ) @ X )
& P )
= P ) ).
% top_conj(1)
thf(fact_250_theI,axiom,
! [A: $tType,P: A > $o,A2: A] :
( ( P @ A2 )
=> ( ! [X3: A] :
( ( P @ X3 )
=> ( X3 = A2 ) )
=> ( P @ ( the @ A @ P ) ) ) ) ).
% theI
thf(fact_251_theI_H,axiom,
! [A: $tType,P: A > $o] :
( ? [X4: A] :
( ( P @ X4 )
& ! [Y3: A] :
( ( P @ Y3 )
=> ( Y3 = X4 ) ) )
=> ( P @ ( the @ A @ P ) ) ) ).
% theI'
thf(fact_252_theI2,axiom,
! [A: $tType,P: A > $o,A2: A,Q: A > $o] :
( ( P @ A2 )
=> ( ! [X3: A] :
( ( P @ X3 )
=> ( X3 = A2 ) )
=> ( ! [X3: A] :
( ( P @ X3 )
=> ( Q @ X3 ) )
=> ( Q @ ( the @ A @ P ) ) ) ) ) ).
% theI2
thf(fact_253_If__def,axiom,
! [A: $tType] :
( ( if @ A )
= ( ^ [P4: $o,X2: A,Y4: A] :
( the @ A
@ ^ [Z4: A] :
( ( P4
=> ( Z4 = X2 ) )
& ( ~ P4
=> ( Z4 = Y4 ) ) ) ) ) ) ).
% If_def
thf(fact_254_the1I2,axiom,
! [A: $tType,P: A > $o,Q: A > $o] :
( ? [X4: A] :
( ( P @ X4 )
& ! [Y3: A] :
( ( P @ Y3 )
=> ( Y3 = X4 ) ) )
=> ( ! [X3: A] :
( ( P @ X3 )
=> ( Q @ X3 ) )
=> ( Q @ ( the @ A @ P ) ) ) ) ).
% the1I2
thf(fact_255_the1__equality,axiom,
! [A: $tType,P: A > $o,A2: A] :
( ? [X4: A] :
( ( P @ X4 )
& ! [Y3: A] :
( ( P @ Y3 )
=> ( Y3 = X4 ) ) )
=> ( ( P @ A2 )
=> ( ( the @ A @ P )
= A2 ) ) ) ).
% the1_equality
%----Type constructors (12)
thf(tcon_HOL_Obool___Lattices_Obounded__lattice,axiom,
bounded_lattice @ $o @ ( type2 @ $o ) ).
thf(tcon_Set_Oset___Lattices_Obounded__lattice_1,axiom,
! [A7: $tType] : ( bounded_lattice @ ( set @ A7 ) @ ( type2 @ ( set @ A7 ) ) ) ).
thf(tcon_fun___Lattices_Obounded__lattice_2,axiom,
! [A7: $tType,A12: $tType] :
( ( bounded_lattice @ A12 @ ( type2 @ A12 ) )
=> ( bounded_lattice @ ( A7 > A12 ) @ ( type2 @ ( A7 > A12 ) ) ) ) ).
thf(tcon_fun___Lattices_Obounded__lattice__top,axiom,
! [A7: $tType,A12: $tType] :
( ( bounded_lattice @ A12 @ ( type2 @ A12 ) )
=> ( bounded_lattice_top @ ( A7 > A12 ) @ ( type2 @ ( A7 > A12 ) ) ) ) ).
thf(tcon_fun___Lattices_Osemilattice__sup,axiom,
! [A7: $tType,A12: $tType] :
( ( semilattice_sup @ A12 @ ( type2 @ A12 ) )
=> ( semilattice_sup @ ( A7 > A12 ) @ ( type2 @ ( A7 > A12 ) ) ) ) ).
thf(tcon_fun___Lattices_Olattice,axiom,
! [A7: $tType,A12: $tType] :
( ( lattice @ A12 @ ( type2 @ A12 ) )
=> ( lattice @ ( A7 > A12 ) @ ( type2 @ ( A7 > A12 ) ) ) ) ).
thf(tcon_Set_Oset___Lattices_Obounded__lattice__top_3,axiom,
! [A7: $tType] : ( bounded_lattice_top @ ( set @ A7 ) @ ( type2 @ ( set @ A7 ) ) ) ).
thf(tcon_Set_Oset___Lattices_Osemilattice__sup_4,axiom,
! [A7: $tType] : ( semilattice_sup @ ( set @ A7 ) @ ( type2 @ ( set @ A7 ) ) ) ).
thf(tcon_Set_Oset___Lattices_Olattice_5,axiom,
! [A7: $tType] : ( lattice @ ( set @ A7 ) @ ( type2 @ ( set @ A7 ) ) ) ).
thf(tcon_HOL_Obool___Lattices_Obounded__lattice__top_6,axiom,
bounded_lattice_top @ $o @ ( type2 @ $o ) ).
thf(tcon_HOL_Obool___Lattices_Osemilattice__sup_7,axiom,
semilattice_sup @ $o @ ( type2 @ $o ) ).
thf(tcon_HOL_Obool___Lattices_Olattice_8,axiom,
lattice @ $o @ ( type2 @ $o ) ).
%----Helper facts (3)
thf(help_If_3_1_T,axiom,
! [P: $o] :
( ( P = $true )
| ( P = $false ) ) ).
thf(help_If_2_1_T,axiom,
! [A: $tType,X: A,Y5: A] :
( ( if @ A @ $false @ X @ Y5 )
= Y5 ) ).
thf(help_If_1_1_T,axiom,
! [A: $tType,X: A,Y5: A] :
( ( if @ A @ $true @ X @ Y5 )
= X ) ).
%----Conjectures (1)
thf(conj_0,conjecture,
( ( member @ ( sum_sum @ t @ dtree ) @ ( sum_Inr @ dtree @ t @ t2 ) @ ( cont @ ( parall1899940088le_par @ ( product_Pair @ dtree @ dtree @ tr1 @ tr2 ) ) ) )
= ( member @ dtree @ t2
@ ( image @ ( product_prod @ dtree @ dtree ) @ dtree @ parall1899940088le_par
@ ( product_Sigma @ dtree @ dtree
@ ( collect @ dtree
@ ^ [X2: dtree] : ( member @ ( sum_sum @ t @ dtree ) @ ( sum_Inr @ dtree @ t @ X2 ) @ ( cont @ tr1 ) ) )
@ ^ [Uu: dtree] :
( collect @ dtree
@ ^ [X2: dtree] : ( member @ ( sum_sum @ t @ dtree ) @ ( sum_Inr @ dtree @ t @ X2 ) @ ( cont @ tr2 ) ) ) ) ) ) ) ).
%------------------------------------------------------------------------------