TPTP Problem File: COM202^1.p
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%------------------------------------------------------------------------------
% File : COM202^1 : TPTP v8.2.0. Released v7.0.0.
% Domain : Computing Theory
% Problem : Parallel extension to grammars and languages 38
% 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__38.p [Bla16]
% Status : Theorem
% Rating : 1.00 v7.5.0, 0.33 v7.2.0, 0.50 v7.1.0
% Syntax : Number of formulae : 323 ( 139 unt; 46 typ; 0 def)
% Number of atoms : 637 ( 339 equ; 0 cnn)
% Maximal formula atoms : 7 ( 2 avg)
% Number of connectives : 4406 ( 106 ~; 22 |; 57 &;3981 @)
% ( 0 <=>; 240 =>; 0 <=; 0 <~>)
% Maximal formula depth : 28 ( 8 avg)
% Number of types : 4 ( 3 usr)
% Number of type conns : 261 ( 261 >; 0 *; 0 +; 0 <<)
% Number of symbols : 46 ( 43 usr; 4 con; 0-6 aty)
% Number of variables : 1268 ( 120 ^;1073 !; 24 ?;1268 :)
% ( 51 !>; 0 ?*; 0 @-; 0 @+)
% SPC : TH1_THM_EQU_NAR
% Comments : This file was generated by Isabelle (most likely Sledgehammer)
% 2016-07-13 14:38:45.901
%------------------------------------------------------------------------------
%----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 (39)
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_Finite__Set_Ofinite,type,
finite_finite:
!>[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__bot,type,
bounded_lattice_bot:
!>[A: $tType] : ( ( itself @ A ) > $o ) ).
thf(sy_cl_Lattices_Obounded__semilattice__sup__bot,type,
bounde1808546759up_bot:
!>[A: $tType] : ( ( itself @ A ) > $o ) ).
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_Odtree_Oroot,type,
root: dtree > n ).
thf(sy_c_Finite__Set_Ofinite,type,
finite_finite2:
!>[A: $tType] : ( ( set @ A ) > $o ) ).
thf(sy_c_Lattices_Osup__class_Osup,type,
sup_sup:
!>[A: $tType] : ( A > A > A ) ).
thf(sy_c_Orderings_Obot__class_Obot,type,
bot_bot:
!>[A: $tType] : A ).
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_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_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_Oold_Oprod_Orec__prod,type,
product_rec_prod:
!>[A: $tType,B: $tType,T: $tType] : ( ( A > B > T ) > ( product_prod @ A @ B ) > T ) ).
thf(sy_c_Product__Type_Oprod_Ofst,type,
product_fst:
!>[A: $tType,B: $tType] : ( ( product_prod @ A @ B ) > A ) ).
thf(sy_c_Product__Type_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_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_Oinsert,type,
insert:
!>[A: $tType] : ( A > ( set @ A ) > ( set @ A ) ) ).
thf(sy_c_Set_Ois__empty,type,
is_empty:
!>[A: $tType] : ( ( set @ A ) > $o ) ).
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__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_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_finite__SigmaI,axiom,
! [B: $tType,A: $tType,A2: set @ A,B2: A > ( set @ B )] :
( ( finite_finite2 @ A @ A2 )
=> ( ! [A3: A] :
( ( member @ A @ A3 @ A2 )
=> ( finite_finite2 @ B @ ( B2 @ A3 ) ) )
=> ( finite_finite2 @ ( product_prod @ A @ B ) @ ( product_Sigma @ A @ B @ A2 @ B2 ) ) ) ) ).
% finite_SigmaI
thf(fact_1_Inl__Inr__False,axiom,
! [A: $tType,B: $tType,X: A,Y: B] :
( ( sum_Inl @ A @ B @ X )
!= ( sum_Inr @ B @ A @ Y ) ) ).
% Inl_Inr_False
thf(fact_2_Inr__Inl__False,axiom,
! [B: $tType,A: $tType,X: B,Y: A] :
( ( sum_Inr @ B @ A @ X )
!= ( sum_Inl @ A @ B @ Y ) ) ).
% Inr_Inl_False
thf(fact_3_vimage__Un,axiom,
! [A: $tType,B: $tType,F: A > B,A2: set @ B,B2: set @ B] :
( ( vimage @ A @ B @ F @ ( sup_sup @ ( set @ B ) @ A2 @ B2 ) )
= ( sup_sup @ ( set @ A ) @ ( vimage @ A @ B @ F @ A2 ) @ ( vimage @ A @ B @ F @ B2 ) ) ) ).
% vimage_Un
thf(fact_4_finite__Un,axiom,
! [A: $tType,F2: set @ A,G: set @ A] :
( ( finite_finite2 @ A @ ( sup_sup @ ( set @ A ) @ F2 @ G ) )
= ( ( finite_finite2 @ A @ F2 )
& ( finite_finite2 @ A @ G ) ) ) ).
% finite_Un
thf(fact_5_finite__imageI,axiom,
! [B: $tType,A: $tType,F2: set @ A,H: A > B] :
( ( finite_finite2 @ A @ F2 )
=> ( finite_finite2 @ B @ ( image @ A @ B @ H @ F2 ) ) ) ).
% finite_imageI
thf(fact_6_par__c_Oelims,axiom,
! [X: product_prod @ dtree @ dtree,Y: set @ ( sum_sum @ t @ ( product_prod @ dtree @ dtree ) )] :
( ( ( parall1914194347_par_c @ X )
= Y )
=> ~ ! [Tr1: dtree,Tr2: dtree] :
( ( X
= ( product_Pair @ dtree @ dtree @ Tr1 @ Tr2 ) )
=> ( Y
!= ( 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_7_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_8_vimage__ident,axiom,
! [A: $tType,Y2: set @ A] :
( ( vimage @ A @ A
@ ^ [X2: A] : X2
@ Y2 )
= Y2 ) ).
% vimage_ident
thf(fact_9_vimage__Collect__eq,axiom,
! [B: $tType,A: $tType,F: A > B,P: B > $o] :
( ( vimage @ A @ B @ F @ ( collect @ B @ P ) )
= ( collect @ A
@ ^ [Y3: A] : ( P @ ( F @ Y3 ) ) ) ) ).
% vimage_Collect_eq
thf(fact_10_Inl__Inr__image__cong,axiom,
! [B: $tType,A: $tType,A2: set @ ( sum_sum @ A @ B ),B2: set @ ( sum_sum @ A @ B )] :
( ( ( vimage @ A @ ( sum_sum @ A @ B ) @ ( sum_Inl @ A @ B ) @ A2 )
= ( vimage @ A @ ( sum_sum @ A @ B ) @ ( sum_Inl @ A @ B ) @ B2 ) )
=> ( ( ( vimage @ B @ ( sum_sum @ A @ B ) @ ( sum_Inr @ B @ A ) @ A2 )
= ( vimage @ B @ ( sum_sum @ A @ B ) @ ( sum_Inr @ B @ A ) @ B2 ) )
=> ( A2 = B2 ) ) ) ).
% Inl_Inr_image_cong
thf(fact_11_finite__Collect__conjI,axiom,
! [A: $tType,P: A > $o,Q: A > $o] :
( ( ( finite_finite2 @ A @ ( collect @ A @ P ) )
| ( finite_finite2 @ A @ ( collect @ A @ Q ) ) )
=> ( finite_finite2 @ A
@ ( collect @ A
@ ^ [X2: A] :
( ( P @ X2 )
& ( Q @ X2 ) ) ) ) ) ).
% finite_Collect_conjI
thf(fact_12_image__eqI,axiom,
! [A: $tType,B: $tType,B3: A,F: B > A,X: B,A2: set @ B] :
( ( B3
= ( F @ X ) )
=> ( ( member @ B @ X @ A2 )
=> ( member @ A @ B3 @ ( image @ B @ A @ F @ A2 ) ) ) ) ).
% image_eqI
thf(fact_13_finite__code,axiom,
! [A: $tType] :
( ( finite_finite @ A @ ( type2 @ A ) )
=> ( ( finite_finite2 @ A )
= ( ^ [A4: set @ A] : $true ) ) ) ).
% finite_code
thf(fact_14_Un__iff,axiom,
! [A: $tType,C2: A,A2: set @ A,B2: set @ A] :
( ( member @ A @ C2 @ ( sup_sup @ ( set @ A ) @ A2 @ B2 ) )
= ( ( member @ A @ C2 @ A2 )
| ( member @ A @ C2 @ B2 ) ) ) ).
% Un_iff
thf(fact_15_UnCI,axiom,
! [A: $tType,C2: A,B2: set @ A,A2: set @ A] :
( ( ~ ( member @ A @ C2 @ B2 )
=> ( member @ A @ C2 @ A2 ) )
=> ( member @ A @ C2 @ ( sup_sup @ ( set @ A ) @ A2 @ B2 ) ) ) ).
% UnCI
thf(fact_16_vimage__eq,axiom,
! [A: $tType,B: $tType,A5: A,F: A > B,B2: set @ B] :
( ( member @ A @ A5 @ ( vimage @ A @ B @ F @ B2 ) )
= ( member @ B @ ( F @ A5 ) @ B2 ) ) ).
% vimage_eq
thf(fact_17_vimageI,axiom,
! [B: $tType,A: $tType,F: B > A,A5: B,B3: A,B2: set @ A] :
( ( ( F @ A5 )
= B3 )
=> ( ( member @ A @ B3 @ B2 )
=> ( member @ B @ A5 @ ( vimage @ B @ A @ F @ B2 ) ) ) ) ).
% vimageI
thf(fact_18_image__ident,axiom,
! [A: $tType,Y2: set @ A] :
( ( image @ A @ A
@ ^ [X2: A] : X2
@ Y2 )
= Y2 ) ).
% image_ident
thf(fact_19_finite__Collect__disjI,axiom,
! [A: $tType,P: A > $o,Q: A > $o] :
( ( finite_finite2 @ A
@ ( collect @ A
@ ^ [X2: A] :
( ( P @ X2 )
| ( Q @ X2 ) ) ) )
= ( ( finite_finite2 @ A @ ( collect @ A @ P ) )
& ( finite_finite2 @ A @ ( collect @ A @ Q ) ) ) ) ).
% finite_Collect_disjI
thf(fact_20_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_21_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_22_rev__image__eqI,axiom,
! [B: $tType,A: $tType,X: A,A2: set @ A,B3: B,F: A > B] :
( ( member @ A @ X @ A2 )
=> ( ( B3
= ( F @ X ) )
=> ( member @ B @ B3 @ ( image @ A @ B @ F @ A2 ) ) ) ) ).
% rev_image_eqI
thf(fact_23_ball__imageD,axiom,
! [A: $tType,B: $tType,F: B > A,A2: set @ B,P: A > $o] :
( ! [X3: A] :
( ( member @ A @ X3 @ ( image @ B @ A @ F @ A2 ) )
=> ( P @ X3 ) )
=> ! [X4: B] :
( ( member @ B @ X4 @ A2 )
=> ( P @ ( F @ X4 ) ) ) ) ).
% ball_imageD
thf(fact_24_image__cong,axiom,
! [B: $tType,A: $tType,M: set @ A,N: set @ A,F: A > B,G2: A > B] :
( ( M = N )
=> ( ! [X3: A] :
( ( member @ A @ X3 @ N )
=> ( ( F @ X3 )
= ( G2 @ X3 ) ) )
=> ( ( image @ A @ B @ F @ M )
= ( image @ A @ B @ G2 @ N ) ) ) ) ).
% image_cong
thf(fact_25_bex__imageD,axiom,
! [A: $tType,B: $tType,F: B > A,A2: set @ B,P: A > $o] :
( ? [X4: A] :
( ( member @ A @ X4 @ ( image @ B @ A @ F @ A2 ) )
& ( P @ X4 ) )
=> ? [X3: B] :
( ( member @ B @ X3 @ A2 )
& ( P @ ( F @ X3 ) ) ) ) ).
% bex_imageD
thf(fact_26_image__iff,axiom,
! [A: $tType,B: $tType,Z: A,F: B > A,A2: set @ B] :
( ( member @ A @ Z @ ( image @ B @ A @ F @ A2 ) )
= ( ? [X2: B] :
( ( member @ B @ X2 @ A2 )
& ( Z
= ( F @ X2 ) ) ) ) ) ).
% image_iff
thf(fact_27_imageI,axiom,
! [B: $tType,A: $tType,X: A,A2: set @ A,F: A > B] :
( ( member @ A @ X @ A2 )
=> ( member @ B @ ( F @ X ) @ ( image @ A @ B @ F @ A2 ) ) ) ).
% imageI
thf(fact_28_finite,axiom,
! [A: $tType] :
( ( finite_finite @ A @ ( type2 @ A ) )
=> ! [A2: set @ A] : ( finite_finite2 @ A @ A2 ) ) ).
% finite
thf(fact_29_finite__set__choice,axiom,
! [B: $tType,A: $tType,A2: set @ A,P: A > B > $o] :
( ( finite_finite2 @ A @ A2 )
=> ( ! [X3: A] :
( ( member @ A @ X3 @ A2 )
=> ? [X1: B] : ( P @ X3 @ X1 ) )
=> ? [F3: A > B] :
! [X4: A] :
( ( member @ A @ X4 @ A2 )
=> ( P @ X4 @ ( F3 @ X4 ) ) ) ) ) ).
% finite_set_choice
thf(fact_30_Un__left__commute,axiom,
! [A: $tType,A2: set @ A,B2: set @ A,C3: set @ A] :
( ( sup_sup @ ( set @ A ) @ A2 @ ( sup_sup @ ( set @ A ) @ B2 @ C3 ) )
= ( sup_sup @ ( set @ A ) @ B2 @ ( sup_sup @ ( set @ A ) @ A2 @ C3 ) ) ) ).
% Un_left_commute
thf(fact_31_Un__left__absorb,axiom,
! [A: $tType,A2: set @ A,B2: set @ A] :
( ( sup_sup @ ( set @ A ) @ A2 @ ( sup_sup @ ( set @ A ) @ A2 @ B2 ) )
= ( sup_sup @ ( set @ A ) @ A2 @ B2 ) ) ).
% Un_left_absorb
thf(fact_32_Un__commute,axiom,
! [A: $tType] :
( ( sup_sup @ ( set @ A ) )
= ( ^ [A4: set @ A,B4: set @ A] : ( sup_sup @ ( set @ A ) @ B4 @ A4 ) ) ) ).
% Un_commute
thf(fact_33_Un__absorb,axiom,
! [A: $tType,A2: set @ A] :
( ( sup_sup @ ( set @ A ) @ A2 @ A2 )
= A2 ) ).
% Un_absorb
thf(fact_34_Un__assoc,axiom,
! [A: $tType,A2: set @ A,B2: set @ A,C3: set @ A] :
( ( sup_sup @ ( set @ A ) @ ( sup_sup @ ( set @ A ) @ A2 @ B2 ) @ C3 )
= ( sup_sup @ ( set @ A ) @ A2 @ ( sup_sup @ ( set @ A ) @ B2 @ C3 ) ) ) ).
% Un_assoc
thf(fact_35_ball__Un,axiom,
! [A: $tType,A2: set @ A,B2: set @ A,P: A > $o] :
( ( ! [X2: A] :
( ( member @ A @ X2 @ ( sup_sup @ ( set @ A ) @ A2 @ B2 ) )
=> ( P @ X2 ) ) )
= ( ! [X2: A] :
( ( member @ A @ X2 @ A2 )
=> ( P @ X2 ) )
& ! [X2: A] :
( ( member @ A @ X2 @ B2 )
=> ( P @ X2 ) ) ) ) ).
% ball_Un
thf(fact_36_bex__Un,axiom,
! [A: $tType,A2: set @ A,B2: set @ A,P: A > $o] :
( ( ? [X2: A] :
( ( member @ A @ X2 @ ( sup_sup @ ( set @ A ) @ A2 @ B2 ) )
& ( P @ X2 ) ) )
= ( ? [X2: A] :
( ( member @ A @ X2 @ A2 )
& ( P @ X2 ) )
| ? [X2: A] :
( ( member @ A @ X2 @ B2 )
& ( P @ X2 ) ) ) ) ).
% bex_Un
thf(fact_37_UnI2,axiom,
! [A: $tType,C2: A,B2: set @ A,A2: set @ A] :
( ( member @ A @ C2 @ B2 )
=> ( member @ A @ C2 @ ( sup_sup @ ( set @ A ) @ A2 @ B2 ) ) ) ).
% UnI2
thf(fact_38_UnI1,axiom,
! [A: $tType,C2: A,A2: set @ A,B2: set @ A] :
( ( member @ A @ C2 @ A2 )
=> ( member @ A @ C2 @ ( sup_sup @ ( set @ A ) @ A2 @ B2 ) ) ) ).
% UnI1
thf(fact_39_UnE,axiom,
! [A: $tType,C2: A,A2: set @ A,B2: set @ A] :
( ( member @ A @ C2 @ ( sup_sup @ ( set @ A ) @ A2 @ B2 ) )
=> ( ~ ( member @ A @ C2 @ A2 )
=> ( member @ A @ C2 @ B2 ) ) ) ).
% UnE
thf(fact_40_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_41_vimageI2,axiom,
! [B: $tType,A: $tType,F: B > A,A5: B,A2: set @ A] :
( ( member @ A @ ( F @ A5 ) @ A2 )
=> ( member @ B @ A5 @ ( vimage @ B @ A @ F @ A2 ) ) ) ).
% vimageI2
thf(fact_42_vimageE,axiom,
! [A: $tType,B: $tType,A5: A,F: A > B,B2: set @ B] :
( ( member @ A @ A5 @ ( vimage @ A @ B @ F @ B2 ) )
=> ( member @ B @ ( F @ A5 ) @ B2 ) ) ).
% vimageE
thf(fact_43_vimageD,axiom,
! [A: $tType,B: $tType,A5: A,F: A > B,A2: set @ B] :
( ( member @ A @ A5 @ ( vimage @ A @ B @ F @ A2 ) )
=> ( member @ B @ ( F @ A5 ) @ A2 ) ) ).
% vimageD
thf(fact_44_Compr__image__eq,axiom,
! [A: $tType,B: $tType,F: B > A,A2: set @ B,P: A > $o] :
( ( collect @ A
@ ^ [X2: A] :
( ( member @ A @ X2 @ ( image @ B @ A @ F @ A2 ) )
& ( P @ X2 ) ) )
= ( image @ B @ A @ F
@ ( collect @ B
@ ^ [X2: B] :
( ( member @ B @ X2 @ A2 )
& ( P @ ( F @ X2 ) ) ) ) ) ) ).
% Compr_image_eq
thf(fact_45_mem__Collect__eq,axiom,
! [A: $tType,A5: A,P: A > $o] :
( ( member @ A @ A5 @ ( collect @ A @ P ) )
= ( P @ A5 ) ) ).
% mem_Collect_eq
thf(fact_46_Collect__mem__eq,axiom,
! [A: $tType,A2: set @ A] :
( ( collect @ A
@ ^ [X2: A] : ( member @ A @ X2 @ A2 ) )
= A2 ) ).
% 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,G2: A > B] :
( ! [X3: A] :
( ( F @ X3 )
= ( G2 @ X3 ) )
=> ( F = G2 ) ) ).
% ext
thf(fact_49_image__image,axiom,
! [A: $tType,B: $tType,C: $tType,F: B > A,G2: C > B,A2: set @ C] :
( ( image @ B @ A @ F @ ( image @ C @ B @ G2 @ A2 ) )
= ( image @ C @ A
@ ^ [X2: C] : ( F @ ( G2 @ X2 ) )
@ A2 ) ) ).
% image_image
thf(fact_50_imageE,axiom,
! [A: $tType,B: $tType,B3: A,F: B > A,A2: set @ B] :
( ( member @ A @ B3 @ ( image @ B @ A @ F @ A2 ) )
=> ~ ! [X3: B] :
( ( B3
= ( F @ X3 ) )
=> ~ ( member @ B @ X3 @ A2 ) ) ) ).
% imageE
thf(fact_51_pigeonhole__infinite__rel,axiom,
! [B: $tType,A: $tType,A2: set @ A,B2: set @ B,R: A > B > $o] :
( ~ ( finite_finite2 @ A @ A2 )
=> ( ( finite_finite2 @ B @ B2 )
=> ( ! [X3: A] :
( ( member @ A @ X3 @ A2 )
=> ? [Xa: B] :
( ( member @ B @ Xa @ B2 )
& ( R @ X3 @ Xa ) ) )
=> ? [X3: B] :
( ( member @ B @ X3 @ B2 )
& ~ ( finite_finite2 @ A
@ ( collect @ A
@ ^ [A6: A] :
( ( member @ A @ A6 @ A2 )
& ( R @ A6 @ X3 ) ) ) ) ) ) ) ) ).
% pigeonhole_infinite_rel
thf(fact_52_not__finite__existsD,axiom,
! [A: $tType,P: A > $o] :
( ~ ( finite_finite2 @ A @ ( collect @ A @ P ) )
=> ? [X12: A] : ( P @ X12 ) ) ).
% not_finite_existsD
thf(fact_53_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_54_Un__def,axiom,
! [A: $tType] :
( ( sup_sup @ ( set @ A ) )
= ( ^ [A4: set @ A,B4: set @ A] :
( collect @ A
@ ^ [X2: A] :
( ( member @ A @ X2 @ A4 )
| ( member @ A @ X2 @ B4 ) ) ) ) ) ).
% Un_def
thf(fact_55_vimage__def,axiom,
! [B: $tType,A: $tType] :
( ( vimage @ A @ B )
= ( ^ [F4: A > B,B4: set @ B] :
( collect @ A
@ ^ [X2: A] : ( member @ B @ ( F4 @ X2 ) @ B4 ) ) ) ) ).
% vimage_def
thf(fact_56_image__Un,axiom,
! [A: $tType,B: $tType,F: B > A,A2: set @ B,B2: set @ B] :
( ( image @ B @ A @ F @ ( sup_sup @ ( set @ B ) @ A2 @ B2 ) )
= ( sup_sup @ ( set @ A ) @ ( image @ B @ A @ F @ A2 ) @ ( image @ B @ A @ F @ B2 ) ) ) ).
% image_Un
thf(fact_57_infinite__Un,axiom,
! [A: $tType,S: set @ A,T2: set @ A] :
( ( ~ ( finite_finite2 @ A @ ( sup_sup @ ( set @ A ) @ S @ T2 ) ) )
= ( ~ ( finite_finite2 @ A @ S )
| ~ ( finite_finite2 @ A @ T2 ) ) ) ).
% infinite_Un
thf(fact_58_Un__infinite,axiom,
! [A: $tType,S: set @ A,T2: set @ A] :
( ~ ( finite_finite2 @ A @ S )
=> ~ ( finite_finite2 @ A @ ( sup_sup @ ( set @ A ) @ S @ T2 ) ) ) ).
% Un_infinite
thf(fact_59_finite__UnI,axiom,
! [A: $tType,F2: set @ A,G: set @ A] :
( ( finite_finite2 @ A @ F2 )
=> ( ( finite_finite2 @ A @ G )
=> ( finite_finite2 @ A @ ( sup_sup @ ( set @ A ) @ F2 @ G ) ) ) ) ).
% finite_UnI
thf(fact_60_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_61_pigeonhole__infinite,axiom,
! [B: $tType,A: $tType,A2: set @ A,F: A > B] :
( ~ ( finite_finite2 @ A @ A2 )
=> ( ( finite_finite2 @ B @ ( image @ A @ B @ F @ A2 ) )
=> ? [X3: A] :
( ( member @ A @ X3 @ A2 )
& ~ ( finite_finite2 @ A
@ ( collect @ A
@ ^ [A6: A] :
( ( member @ A @ A6 @ A2 )
& ( ( F @ A6 )
= ( F @ X3 ) ) ) ) ) ) ) ) ).
% pigeonhole_infinite
thf(fact_62_finite__cartesian__product,axiom,
! [B: $tType,A: $tType,A2: set @ A,B2: set @ B] :
( ( finite_finite2 @ A @ A2 )
=> ( ( finite_finite2 @ B @ B2 )
=> ( finite_finite2 @ ( product_prod @ A @ B )
@ ( product_Sigma @ A @ B @ A2
@ ^ [Uu: A] : B2 ) ) ) ) ).
% finite_cartesian_product
thf(fact_63_par__c_Opelims,axiom,
! [X: product_prod @ dtree @ dtree,Y: set @ ( sum_sum @ t @ ( product_prod @ dtree @ dtree ) )] :
( ( ( parall1914194347_par_c @ X )
= Y )
=> ( ( accp @ ( product_prod @ dtree @ dtree ) @ parall1418634510_c_rel @ X )
=> ~ ! [Tr1: dtree,Tr2: dtree] :
( ( X
= ( product_Pair @ dtree @ dtree @ Tr1 @ Tr2 ) )
=> ( ( Y
= ( 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_64_mem__Sigma__iff,axiom,
! [B: $tType,A: $tType,A5: A,B3: B,A2: set @ A,B2: A > ( set @ B )] :
( ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ A5 @ B3 ) @ ( product_Sigma @ A @ B @ A2 @ B2 ) )
= ( ( member @ A @ A5 @ A2 )
& ( member @ B @ B3 @ ( B2 @ A5 ) ) ) ) ).
% mem_Sigma_iff
thf(fact_65_SigmaI,axiom,
! [B: $tType,A: $tType,A5: A,A2: set @ A,B3: B,B2: A > ( set @ B )] :
( ( member @ A @ A5 @ A2 )
=> ( ( member @ B @ B3 @ ( B2 @ A5 ) )
=> ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ A5 @ B3 ) @ ( product_Sigma @ A @ B @ A2 @ B2 ) ) ) ) ).
% SigmaI
thf(fact_66_infinite__cartesian__product,axiom,
! [B: $tType,A: $tType,A2: set @ A,B2: set @ B] :
( ~ ( finite_finite2 @ A @ A2 )
=> ( ~ ( finite_finite2 @ B @ B2 )
=> ~ ( finite_finite2 @ ( product_prod @ A @ B )
@ ( product_Sigma @ A @ B @ A2
@ ^ [Uu: A] : B2 ) ) ) ) ).
% infinite_cartesian_product
thf(fact_67_old_Osum_Oinject_I1_J,axiom,
! [B: $tType,A: $tType,A5: A,A7: A] :
( ( ( sum_Inl @ A @ B @ A5 )
= ( sum_Inl @ A @ B @ A7 ) )
= ( A5 = A7 ) ) ).
% old.sum.inject(1)
thf(fact_68_sum_Oinject_I1_J,axiom,
! [B: $tType,A: $tType,X13: A,Y1: A] :
( ( ( sum_Inl @ A @ B @ X13 )
= ( sum_Inl @ A @ B @ Y1 ) )
= ( X13 = Y1 ) ) ).
% sum.inject(1)
thf(fact_69_old_Osum_Oinject_I2_J,axiom,
! [A: $tType,B: $tType,B3: B,B5: B] :
( ( ( sum_Inr @ B @ A @ B3 )
= ( sum_Inr @ B @ A @ B5 ) )
= ( B3 = B5 ) ) ).
% old.sum.inject(2)
thf(fact_70_sum_Oinject_I2_J,axiom,
! [A: $tType,B: $tType,X22: B,Y22: B] :
( ( ( sum_Inr @ B @ A @ X22 )
= ( sum_Inr @ B @ A @ Y22 ) )
= ( X22 = Y22 ) ) ).
% sum.inject(2)
thf(fact_71_sup_Oright__idem,axiom,
! [A: $tType] :
( ( semilattice_sup @ A @ ( type2 @ A ) )
=> ! [A5: A,B3: A] :
( ( sup_sup @ A @ ( sup_sup @ A @ A5 @ B3 ) @ B3 )
= ( sup_sup @ A @ A5 @ B3 ) ) ) ).
% sup.right_idem
thf(fact_72_sup__left__idem,axiom,
! [A: $tType] :
( ( semilattice_sup @ A @ ( type2 @ A ) )
=> ! [X: A,Y: A] :
( ( sup_sup @ A @ X @ ( sup_sup @ A @ X @ Y ) )
= ( sup_sup @ A @ X @ Y ) ) ) ).
% sup_left_idem
thf(fact_73_sup_Oleft__idem,axiom,
! [A: $tType] :
( ( semilattice_sup @ A @ ( type2 @ A ) )
=> ! [A5: A,B3: A] :
( ( sup_sup @ A @ A5 @ ( sup_sup @ A @ A5 @ B3 ) )
= ( sup_sup @ A @ A5 @ B3 ) ) ) ).
% sup.left_idem
thf(fact_74_old_Oprod_Oinject,axiom,
! [A: $tType,B: $tType,A5: A,B3: B,A7: A,B5: B] :
( ( ( product_Pair @ A @ B @ A5 @ B3 )
= ( product_Pair @ A @ B @ A7 @ B5 ) )
= ( ( A5 = A7 )
& ( B3 = B5 ) ) ) ).
% old.prod.inject
thf(fact_75_prod_Oinject,axiom,
! [A: $tType,B: $tType,X13: A,X22: B,Y1: A,Y22: B] :
( ( ( product_Pair @ A @ B @ X13 @ X22 )
= ( product_Pair @ A @ B @ Y1 @ Y22 ) )
= ( ( X13 = Y1 )
& ( X22 = Y22 ) ) ) ).
% prod.inject
thf(fact_76_sup__apply,axiom,
! [B: $tType,A: $tType] :
( ( semilattice_sup @ B @ ( type2 @ B ) )
=> ( ( sup_sup @ ( A > B ) )
= ( ^ [F4: A > B,G3: A > B,X2: A] : ( sup_sup @ B @ ( F4 @ X2 ) @ ( G3 @ X2 ) ) ) ) ) ).
% sup_apply
thf(fact_77_sup_Oidem,axiom,
! [A: $tType] :
( ( semilattice_sup @ A @ ( type2 @ A ) )
=> ! [A5: A] :
( ( sup_sup @ A @ A5 @ A5 )
= A5 ) ) ).
% sup.idem
thf(fact_78_sup__idem,axiom,
! [A: $tType] :
( ( semilattice_sup @ A @ ( type2 @ A ) )
=> ! [X: A] :
( ( sup_sup @ A @ X @ X )
= X ) ) ).
% sup_idem
thf(fact_79_sup__set__def,axiom,
! [A: $tType] :
( ( sup_sup @ ( set @ A ) )
= ( ^ [A4: set @ A,B4: set @ A] :
( collect @ A
@ ( sup_sup @ ( A > $o )
@ ^ [X2: A] : ( member @ A @ X2 @ A4 )
@ ^ [X2: A] : ( member @ A @ X2 @ B4 ) ) ) ) ) ).
% sup_set_def
thf(fact_80_old_Oprod_Oinducts,axiom,
! [B: $tType,A: $tType,P: ( product_prod @ A @ B ) > $o,Prod: product_prod @ A @ B] :
( ! [A3: A,B6: B] : ( P @ ( product_Pair @ A @ B @ A3 @ B6 ) )
=> ( P @ Prod ) ) ).
% old.prod.inducts
thf(fact_81_old_Oprod_Oexhaust,axiom,
! [A: $tType,B: $tType,Y: product_prod @ A @ B] :
~ ! [A3: A,B6: B] :
( Y
!= ( product_Pair @ A @ B @ A3 @ B6 ) ) ).
% old.prod.exhaust
thf(fact_82_prod__induct7,axiom,
! [G4: $tType,F5: $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 @ F5 @ G4 ) ) ) ) ) ) > $o,X: product_prod @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D @ ( product_prod @ E @ ( product_prod @ F5 @ G4 ) ) ) ) )] :
( ! [A3: A,B6: B,C4: C,D2: D,E2: E,F3: F5,G5: G4] : ( P @ ( product_Pair @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D @ ( product_prod @ E @ ( product_prod @ F5 @ G4 ) ) ) ) ) @ A3 @ ( product_Pair @ B @ ( product_prod @ C @ ( product_prod @ D @ ( product_prod @ E @ ( product_prod @ F5 @ G4 ) ) ) ) @ B6 @ ( product_Pair @ C @ ( product_prod @ D @ ( product_prod @ E @ ( product_prod @ F5 @ G4 ) ) ) @ C4 @ ( product_Pair @ D @ ( product_prod @ E @ ( product_prod @ F5 @ G4 ) ) @ D2 @ ( product_Pair @ E @ ( product_prod @ F5 @ G4 ) @ E2 @ ( product_Pair @ F5 @ G4 @ F3 @ G5 ) ) ) ) ) ) )
=> ( P @ X ) ) ).
% prod_induct7
thf(fact_83_prod__induct6,axiom,
! [F5: $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 @ F5 ) ) ) ) ) > $o,X: product_prod @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D @ ( product_prod @ E @ F5 ) ) ) )] :
( ! [A3: A,B6: B,C4: C,D2: D,E2: E,F3: F5] : ( P @ ( product_Pair @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D @ ( product_prod @ E @ F5 ) ) ) ) @ A3 @ ( product_Pair @ B @ ( product_prod @ C @ ( product_prod @ D @ ( product_prod @ E @ F5 ) ) ) @ B6 @ ( product_Pair @ C @ ( product_prod @ D @ ( product_prod @ E @ F5 ) ) @ C4 @ ( product_Pair @ D @ ( product_prod @ E @ F5 ) @ D2 @ ( product_Pair @ E @ F5 @ E2 @ F3 ) ) ) ) ) )
=> ( P @ X ) ) ).
% prod_induct6
thf(fact_84_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 ) ) )] :
( ! [A3: A,B6: B,C4: C,D2: D,E2: E] : ( P @ ( product_Pair @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D @ E ) ) ) @ A3 @ ( product_Pair @ B @ ( product_prod @ C @ ( product_prod @ D @ E ) ) @ B6 @ ( product_Pair @ C @ ( product_prod @ D @ E ) @ C4 @ ( product_Pair @ D @ E @ D2 @ E2 ) ) ) ) )
=> ( P @ X ) ) ).
% prod_induct5
thf(fact_85_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 ) )] :
( ! [A3: A,B6: B,C4: C,D2: D] : ( P @ ( product_Pair @ A @ ( product_prod @ B @ ( product_prod @ C @ D ) ) @ A3 @ ( product_Pair @ B @ ( product_prod @ C @ D ) @ B6 @ ( product_Pair @ C @ D @ C4 @ D2 ) ) ) )
=> ( P @ X ) ) ).
% prod_induct4
thf(fact_86_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 )] :
( ! [A3: A,B6: B,C4: C] : ( P @ ( product_Pair @ A @ ( product_prod @ B @ C ) @ A3 @ ( product_Pair @ B @ C @ B6 @ C4 ) ) )
=> ( P @ X ) ) ).
% prod_induct3
thf(fact_87_prod__cases7,axiom,
! [A: $tType,B: $tType,C: $tType,D: $tType,E: $tType,F5: $tType,G4: $tType,Y: product_prod @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D @ ( product_prod @ E @ ( product_prod @ F5 @ G4 ) ) ) ) )] :
~ ! [A3: A,B6: B,C4: C,D2: D,E2: E,F3: F5,G5: G4] :
( Y
!= ( product_Pair @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D @ ( product_prod @ E @ ( product_prod @ F5 @ G4 ) ) ) ) ) @ A3 @ ( product_Pair @ B @ ( product_prod @ C @ ( product_prod @ D @ ( product_prod @ E @ ( product_prod @ F5 @ G4 ) ) ) ) @ B6 @ ( product_Pair @ C @ ( product_prod @ D @ ( product_prod @ E @ ( product_prod @ F5 @ G4 ) ) ) @ C4 @ ( product_Pair @ D @ ( product_prod @ E @ ( product_prod @ F5 @ G4 ) ) @ D2 @ ( product_Pair @ E @ ( product_prod @ F5 @ G4 ) @ E2 @ ( product_Pair @ F5 @ G4 @ F3 @ G5 ) ) ) ) ) ) ) ).
% prod_cases7
thf(fact_88_prod__cases6,axiom,
! [A: $tType,B: $tType,C: $tType,D: $tType,E: $tType,F5: $tType,Y: product_prod @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D @ ( product_prod @ E @ F5 ) ) ) )] :
~ ! [A3: A,B6: B,C4: C,D2: D,E2: E,F3: F5] :
( Y
!= ( product_Pair @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D @ ( product_prod @ E @ F5 ) ) ) ) @ A3 @ ( product_Pair @ B @ ( product_prod @ C @ ( product_prod @ D @ ( product_prod @ E @ F5 ) ) ) @ B6 @ ( product_Pair @ C @ ( product_prod @ D @ ( product_prod @ E @ F5 ) ) @ C4 @ ( product_Pair @ D @ ( product_prod @ E @ F5 ) @ D2 @ ( product_Pair @ E @ F5 @ E2 @ F3 ) ) ) ) ) ) ).
% prod_cases6
thf(fact_89_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 ) ) )] :
~ ! [A3: A,B6: B,C4: C,D2: D,E2: E] :
( Y
!= ( product_Pair @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D @ E ) ) ) @ A3 @ ( product_Pair @ B @ ( product_prod @ C @ ( product_prod @ D @ E ) ) @ B6 @ ( product_Pair @ C @ ( product_prod @ D @ E ) @ C4 @ ( product_Pair @ D @ E @ D2 @ E2 ) ) ) ) ) ).
% prod_cases5
thf(fact_90_prod__cases4,axiom,
! [A: $tType,B: $tType,C: $tType,D: $tType,Y: product_prod @ A @ ( product_prod @ B @ ( product_prod @ C @ D ) )] :
~ ! [A3: A,B6: B,C4: C,D2: D] :
( Y
!= ( product_Pair @ A @ ( product_prod @ B @ ( product_prod @ C @ D ) ) @ A3 @ ( product_Pair @ B @ ( product_prod @ C @ D ) @ B6 @ ( product_Pair @ C @ D @ C4 @ D2 ) ) ) ) ).
% prod_cases4
thf(fact_91_prod__cases3,axiom,
! [A: $tType,B: $tType,C: $tType,Y: product_prod @ A @ ( product_prod @ B @ C )] :
~ ! [A3: A,B6: B,C4: C] :
( Y
!= ( product_Pair @ A @ ( product_prod @ B @ C ) @ A3 @ ( product_Pair @ B @ C @ B6 @ C4 ) ) ) ).
% prod_cases3
thf(fact_92_Pair__inject,axiom,
! [A: $tType,B: $tType,A5: A,B3: B,A7: A,B5: B] :
( ( ( product_Pair @ A @ B @ A5 @ B3 )
= ( product_Pair @ A @ B @ A7 @ B5 ) )
=> ~ ( ( A5 = A7 )
=> ( B3 != B5 ) ) ) ).
% Pair_inject
thf(fact_93_prod__cases,axiom,
! [B: $tType,A: $tType,P: ( product_prod @ A @ B ) > $o,P2: product_prod @ A @ B] :
( ! [A3: A,B6: B] : ( P @ ( product_Pair @ A @ B @ A3 @ B6 ) )
=> ( P @ P2 ) ) ).
% prod_cases
thf(fact_94_surj__pair,axiom,
! [A: $tType,B: $tType,P2: product_prod @ A @ B] :
? [X3: A,Y4: B] :
( P2
= ( product_Pair @ A @ B @ X3 @ Y4 ) ) ).
% surj_pair
thf(fact_95_inf__sup__aci_I8_J,axiom,
! [A: $tType] :
( ( lattice @ A @ ( type2 @ A ) )
=> ! [X: A,Y: A] :
( ( sup_sup @ A @ X @ ( sup_sup @ A @ X @ Y ) )
= ( sup_sup @ A @ X @ Y ) ) ) ).
% inf_sup_aci(8)
thf(fact_96_inf__sup__aci_I7_J,axiom,
! [A: $tType] :
( ( lattice @ A @ ( type2 @ A ) )
=> ! [X: A,Y: A,Z: A] :
( ( sup_sup @ A @ X @ ( sup_sup @ A @ Y @ Z ) )
= ( sup_sup @ A @ Y @ ( sup_sup @ A @ X @ Z ) ) ) ) ).
% inf_sup_aci(7)
thf(fact_97_inf__sup__aci_I6_J,axiom,
! [A: $tType] :
( ( lattice @ A @ ( type2 @ A ) )
=> ! [X: A,Y: A,Z: A] :
( ( sup_sup @ A @ ( sup_sup @ A @ X @ Y ) @ Z )
= ( sup_sup @ A @ X @ ( sup_sup @ A @ Y @ Z ) ) ) ) ).
% inf_sup_aci(6)
thf(fact_98_inf__sup__aci_I5_J,axiom,
! [A: $tType] :
( ( lattice @ A @ ( type2 @ A ) )
=> ( ( sup_sup @ A )
= ( ^ [X2: A,Y3: A] : ( sup_sup @ A @ Y3 @ X2 ) ) ) ) ).
% inf_sup_aci(5)
thf(fact_99_sup__fun__def,axiom,
! [B: $tType,A: $tType] :
( ( semilattice_sup @ B @ ( type2 @ B ) )
=> ( ( sup_sup @ ( A > B ) )
= ( ^ [F4: A > B,G3: A > B,X2: A] : ( sup_sup @ B @ ( F4 @ X2 ) @ ( G3 @ X2 ) ) ) ) ) ).
% sup_fun_def
thf(fact_100_sup_Oassoc,axiom,
! [A: $tType] :
( ( semilattice_sup @ A @ ( type2 @ A ) )
=> ! [A5: A,B3: A,C2: A] :
( ( sup_sup @ A @ ( sup_sup @ A @ A5 @ B3 ) @ C2 )
= ( sup_sup @ A @ A5 @ ( sup_sup @ A @ B3 @ C2 ) ) ) ) ).
% sup.assoc
thf(fact_101_sup__assoc,axiom,
! [A: $tType] :
( ( semilattice_sup @ A @ ( type2 @ A ) )
=> ! [X: A,Y: A,Z: A] :
( ( sup_sup @ A @ ( sup_sup @ A @ X @ Y ) @ Z )
= ( sup_sup @ A @ X @ ( sup_sup @ A @ Y @ Z ) ) ) ) ).
% sup_assoc
thf(fact_102_sup_Ocommute,axiom,
! [A: $tType] :
( ( semilattice_sup @ A @ ( type2 @ A ) )
=> ( ( sup_sup @ A )
= ( ^ [A6: A,B7: A] : ( sup_sup @ A @ B7 @ A6 ) ) ) ) ).
% sup.commute
thf(fact_103_sup__commute,axiom,
! [A: $tType] :
( ( semilattice_sup @ A @ ( type2 @ A ) )
=> ( ( sup_sup @ A )
= ( ^ [X2: A,Y3: A] : ( sup_sup @ A @ Y3 @ X2 ) ) ) ) ).
% sup_commute
thf(fact_104_sup_Oleft__commute,axiom,
! [A: $tType] :
( ( semilattice_sup @ A @ ( type2 @ A ) )
=> ! [B3: A,A5: A,C2: A] :
( ( sup_sup @ A @ B3 @ ( sup_sup @ A @ A5 @ C2 ) )
= ( sup_sup @ A @ A5 @ ( sup_sup @ A @ B3 @ C2 ) ) ) ) ).
% sup.left_commute
thf(fact_105_sup__left__commute,axiom,
! [A: $tType] :
( ( semilattice_sup @ A @ ( type2 @ A ) )
=> ! [X: A,Y: A,Z: A] :
( ( sup_sup @ A @ X @ ( sup_sup @ A @ Y @ Z ) )
= ( sup_sup @ A @ Y @ ( sup_sup @ A @ X @ Z ) ) ) ) ).
% sup_left_commute
thf(fact_106_Inr__inject,axiom,
! [A: $tType,B: $tType,X: B,Y: B] :
( ( ( sum_Inr @ B @ A @ X )
= ( sum_Inr @ B @ A @ Y ) )
=> ( X = Y ) ) ).
% Inr_inject
thf(fact_107_Inl__inject,axiom,
! [B: $tType,A: $tType,X: A,Y: A] :
( ( ( sum_Inl @ A @ B @ X )
= ( sum_Inl @ A @ B @ Y ) )
=> ( X = Y ) ) ).
% Inl_inject
thf(fact_108_Sigma__cong,axiom,
! [B: $tType,A: $tType,A2: set @ A,B2: set @ A,C3: A > ( set @ B ),D3: A > ( set @ B )] :
( ( A2 = B2 )
=> ( ! [X3: A] :
( ( member @ A @ X3 @ B2 )
=> ( ( C3 @ X3 )
= ( D3 @ X3 ) ) )
=> ( ( product_Sigma @ A @ B @ A2 @ C3 )
= ( product_Sigma @ A @ B @ B2 @ D3 ) ) ) ) ).
% Sigma_cong
thf(fact_109_Times__eq__cancel2,axiom,
! [A: $tType,B: $tType,X: A,C3: set @ A,A2: set @ B,B2: set @ B] :
( ( member @ A @ X @ C3 )
=> ( ( ( product_Sigma @ B @ A @ A2
@ ^ [Uu: B] : C3 )
= ( product_Sigma @ B @ A @ B2
@ ^ [Uu: B] : C3 ) )
= ( A2 = B2 ) ) ) ).
% Times_eq_cancel2
thf(fact_110_SigmaE,axiom,
! [A: $tType,B: $tType,C2: product_prod @ A @ B,A2: set @ A,B2: A > ( set @ B )] :
( ( member @ ( product_prod @ A @ B ) @ C2 @ ( product_Sigma @ A @ B @ A2 @ B2 ) )
=> ~ ! [X3: A] :
( ( member @ A @ X3 @ A2 )
=> ! [Y4: B] :
( ( member @ B @ Y4 @ ( B2 @ X3 ) )
=> ( C2
!= ( product_Pair @ A @ B @ X3 @ Y4 ) ) ) ) ) ).
% SigmaE
thf(fact_111_SigmaD1,axiom,
! [B: $tType,A: $tType,A5: A,B3: B,A2: set @ A,B2: A > ( set @ B )] :
( ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ A5 @ B3 ) @ ( product_Sigma @ A @ B @ A2 @ B2 ) )
=> ( member @ A @ A5 @ A2 ) ) ).
% SigmaD1
thf(fact_112_SigmaD2,axiom,
! [B: $tType,A: $tType,A5: A,B3: B,A2: set @ A,B2: A > ( set @ B )] :
( ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ A5 @ B3 ) @ ( product_Sigma @ A @ B @ A2 @ B2 ) )
=> ( member @ B @ B3 @ ( B2 @ A5 ) ) ) ).
% SigmaD2
thf(fact_113_SigmaE2,axiom,
! [B: $tType,A: $tType,A5: A,B3: B,A2: set @ A,B2: A > ( set @ B )] :
( ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ A5 @ B3 ) @ ( product_Sigma @ A @ B @ A2 @ B2 ) )
=> ~ ( ( member @ A @ A5 @ A2 )
=> ~ ( member @ B @ B3 @ ( B2 @ A5 ) ) ) ) ).
% SigmaE2
thf(fact_114_Sigma__Un__distrib1,axiom,
! [B: $tType,A: $tType,I: set @ A,J: set @ A,C3: A > ( set @ B )] :
( ( product_Sigma @ A @ B @ ( sup_sup @ ( set @ A ) @ I @ J ) @ C3 )
= ( sup_sup @ ( set @ ( product_prod @ A @ B ) ) @ ( product_Sigma @ A @ B @ I @ C3 ) @ ( product_Sigma @ A @ B @ J @ C3 ) ) ) ).
% Sigma_Un_distrib1
thf(fact_115_sum_Odistinct_I1_J,axiom,
! [A: $tType,B: $tType,X13: A,X22: B] :
( ( sum_Inl @ A @ B @ X13 )
!= ( sum_Inr @ B @ A @ X22 ) ) ).
% sum.distinct(1)
thf(fact_116_old_Osum_Odistinct_I2_J,axiom,
! [B8: $tType,A8: $tType,B9: B8,A9: A8] :
( ( sum_Inr @ B8 @ A8 @ B9 )
!= ( sum_Inl @ A8 @ B8 @ A9 ) ) ).
% old.sum.distinct(2)
thf(fact_117_old_Osum_Odistinct_I1_J,axiom,
! [A: $tType,B: $tType,A5: A,B5: B] :
( ( sum_Inl @ A @ B @ A5 )
!= ( sum_Inr @ B @ A @ B5 ) ) ).
% old.sum.distinct(1)
thf(fact_118_sumE,axiom,
! [A: $tType,B: $tType,S2: sum_sum @ A @ B] :
( ! [X3: A] :
( S2
!= ( sum_Inl @ A @ B @ X3 ) )
=> ~ ! [Y4: B] :
( S2
!= ( sum_Inr @ B @ A @ Y4 ) ) ) ).
% sumE
thf(fact_119_Inr__not__Inl,axiom,
! [B: $tType,A: $tType,B3: B,A5: A] :
( ( sum_Inr @ B @ A @ B3 )
!= ( sum_Inl @ A @ B @ A5 ) ) ).
% Inr_not_Inl
thf(fact_120_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_121_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_122_old_Osum_Oexhaust,axiom,
! [A: $tType,B: $tType,Y: sum_sum @ A @ B] :
( ! [A3: A] :
( Y
!= ( sum_Inl @ A @ B @ A3 ) )
=> ~ ! [B6: B] :
( Y
!= ( sum_Inr @ B @ A @ B6 ) ) ) ).
% old.sum.exhaust
thf(fact_123_old_Osum_Oinducts,axiom,
! [B: $tType,A: $tType,P: ( sum_sum @ A @ B ) > $o,Sum: sum_sum @ A @ B] :
( ! [A3: A] : ( P @ ( sum_Inl @ A @ B @ A3 ) )
=> ( ! [B6: B] : ( P @ ( sum_Inr @ B @ A @ B6 ) )
=> ( P @ Sum ) ) ) ).
% old.sum.inducts
thf(fact_124_Sigma__Un__distrib2,axiom,
! [B: $tType,A: $tType,I: set @ A,A2: A > ( set @ B ),B2: A > ( set @ B )] :
( ( product_Sigma @ A @ B @ I
@ ^ [I2: A] : ( sup_sup @ ( set @ B ) @ ( A2 @ I2 ) @ ( B2 @ I2 ) ) )
= ( sup_sup @ ( set @ ( product_prod @ A @ B ) ) @ ( product_Sigma @ A @ B @ I @ A2 ) @ ( product_Sigma @ A @ B @ I @ B2 ) ) ) ).
% Sigma_Un_distrib2
thf(fact_125_Times__Un__distrib1,axiom,
! [B: $tType,A: $tType,A2: set @ A,B2: set @ A,C3: set @ B] :
( ( product_Sigma @ A @ B @ ( sup_sup @ ( set @ A ) @ A2 @ B2 )
@ ^ [Uu: A] : C3 )
= ( sup_sup @ ( set @ ( product_prod @ A @ B ) )
@ ( product_Sigma @ A @ B @ A2
@ ^ [Uu: A] : C3 )
@ ( product_Sigma @ A @ B @ B2
@ ^ [Uu: A] : C3 ) ) ) ).
% Times_Un_distrib1
thf(fact_126_old_Oprod_Orec,axiom,
! [A: $tType,T: $tType,B: $tType,F1: A > B > T,A5: A,B3: B] :
( ( product_rec_prod @ A @ B @ T @ F1 @ ( product_Pair @ A @ B @ A5 @ B3 ) )
= ( F1 @ A5 @ B3 ) ) ).
% old.prod.rec
thf(fact_127_old_Osum_Osimps_I7_J,axiom,
! [B: $tType,T: $tType,A: $tType,F1: A > T,F22: B > T,A5: A] :
( ( sum_rec_sum @ A @ T @ B @ F1 @ F22 @ ( sum_Inl @ A @ B @ A5 ) )
= ( F1 @ A5 ) ) ).
% old.sum.simps(7)
thf(fact_128_old_Osum_Osimps_I8_J,axiom,
! [A: $tType,T: $tType,B: $tType,F1: A > T,F22: B > T,B3: B] :
( ( sum_rec_sum @ A @ T @ B @ F1 @ F22 @ ( sum_Inr @ B @ A @ B3 ) )
= ( F22 @ B3 ) ) ).
% old.sum.simps(8)
thf(fact_129_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_130_Product__Type_Oproduct__def,axiom,
! [B: $tType,A: $tType] :
( ( product_product @ A @ B )
= ( ^ [A4: set @ A,B4: set @ B] :
( product_Sigma @ A @ B @ A4
@ ^ [Uu: A] : B4 ) ) ) ).
% Product_Type.product_def
thf(fact_131_member__product,axiom,
! [B: $tType,A: $tType,X: product_prod @ A @ B,A2: set @ A,B2: set @ B] :
( ( member @ ( product_prod @ A @ B ) @ X @ ( product_product @ A @ B @ A2 @ B2 ) )
= ( member @ ( product_prod @ A @ B ) @ X
@ ( product_Sigma @ A @ B @ A2
@ ^ [Uu: A] : B2 ) ) ) ).
% member_product
thf(fact_132_sup1CI,axiom,
! [A: $tType,B2: A > $o,X: A,A2: A > $o] :
( ( ~ ( B2 @ X )
=> ( A2 @ X ) )
=> ( sup_sup @ ( A > $o ) @ A2 @ B2 @ X ) ) ).
% sup1CI
thf(fact_133_sup1E,axiom,
! [A: $tType,A2: A > $o,B2: A > $o,X: A] :
( ( sup_sup @ ( A > $o ) @ A2 @ B2 @ X )
=> ( ~ ( A2 @ X )
=> ( B2 @ X ) ) ) ).
% sup1E
thf(fact_134_sup1I1,axiom,
! [A: $tType,A2: A > $o,X: A,B2: A > $o] :
( ( A2 @ X )
=> ( sup_sup @ ( A > $o ) @ A2 @ B2 @ X ) ) ).
% sup1I1
thf(fact_135_sup1I2,axiom,
! [A: $tType,B2: A > $o,X: A,A2: A > $o] :
( ( B2 @ X )
=> ( sup_sup @ ( A > $o ) @ A2 @ B2 @ X ) ) ).
% sup1I2
thf(fact_136_not__arg__cong__Inr,axiom,
! [B: $tType,A: $tType,X: A,Y: A] :
( ( X != Y )
=> ( ( sum_Inr @ A @ B @ X )
!= ( sum_Inr @ A @ B @ Y ) ) ) ).
% not_arg_cong_Inr
thf(fact_137_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_138_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,Y3: B] : ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X2 @ Y3 ) @ R )
@ ^ [X2: A,Y3: B] : ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X2 @ Y3 ) @ S ) )
= ( ^ [X2: A,Y3: B] : ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X2 @ Y3 ) @ ( sup_sup @ ( set @ ( product_prod @ A @ B ) ) @ R @ S ) ) ) ) ).
% sup_Un_eq2
thf(fact_139_Plus__def,axiom,
! [B: $tType,A: $tType] :
( ( sum_Plus @ A @ B )
= ( ^ [A4: set @ A,B4: set @ B] : ( sup_sup @ ( set @ ( sum_sum @ A @ B ) ) @ ( image @ A @ ( sum_sum @ A @ B ) @ ( sum_Inl @ A @ B ) @ A4 ) @ ( image @ B @ ( sum_sum @ A @ B ) @ ( sum_Inr @ B @ A ) @ B4 ) ) ) ) ).
% Plus_def
thf(fact_140_Sup_OSUP__identity__eq,axiom,
! [A: $tType,Sup: ( set @ A ) > A,A2: set @ A] :
( ( Sup
@ ( image @ A @ A
@ ^ [X2: A] : X2
@ A2 ) )
= ( Sup @ A2 ) ) ).
% Sup.SUP_identity_eq
thf(fact_141_sup2CI,axiom,
! [A: $tType,B: $tType,B2: A > B > $o,X: A,Y: B,A2: A > B > $o] :
( ( ~ ( B2 @ X @ Y )
=> ( A2 @ X @ Y ) )
=> ( sup_sup @ ( A > B > $o ) @ A2 @ B2 @ X @ Y ) ) ).
% sup2CI
thf(fact_142_InrI,axiom,
! [B: $tType,A: $tType,B3: A,B2: set @ A,A2: set @ B] :
( ( member @ A @ B3 @ B2 )
=> ( member @ ( sum_sum @ B @ A ) @ ( sum_Inr @ A @ B @ B3 ) @ ( sum_Plus @ B @ A @ A2 @ B2 ) ) ) ).
% InrI
thf(fact_143_InlI,axiom,
! [A: $tType,B: $tType,A5: A,A2: set @ A,B2: set @ B] :
( ( member @ A @ A5 @ A2 )
=> ( member @ ( sum_sum @ A @ B ) @ ( sum_Inl @ A @ B @ A5 ) @ ( sum_Plus @ A @ B @ A2 @ B2 ) ) ) ).
% InlI
thf(fact_144_finite__Plus__iff,axiom,
! [A: $tType,B: $tType,A2: set @ A,B2: set @ B] :
( ( finite_finite2 @ ( sum_sum @ A @ B ) @ ( sum_Plus @ A @ B @ A2 @ B2 ) )
= ( ( finite_finite2 @ A @ A2 )
& ( finite_finite2 @ B @ B2 ) ) ) ).
% finite_Plus_iff
thf(fact_145_sup2E,axiom,
! [A: $tType,B: $tType,A2: A > B > $o,B2: A > B > $o,X: A,Y: B] :
( ( sup_sup @ ( A > B > $o ) @ A2 @ B2 @ X @ Y )
=> ( ~ ( A2 @ X @ Y )
=> ( B2 @ X @ Y ) ) ) ).
% sup2E
thf(fact_146_sup2I1,axiom,
! [A: $tType,B: $tType,A2: A > B > $o,X: A,Y: B,B2: A > B > $o] :
( ( A2 @ X @ Y )
=> ( sup_sup @ ( A > B > $o ) @ A2 @ B2 @ X @ Y ) ) ).
% sup2I1
thf(fact_147_sup2I2,axiom,
! [A: $tType,B: $tType,B2: A > B > $o,X: A,Y: B,A2: A > B > $o] :
( ( B2 @ X @ Y )
=> ( sup_sup @ ( A > B > $o ) @ A2 @ B2 @ X @ Y ) ) ).
% sup2I2
thf(fact_148_PlusE,axiom,
! [A: $tType,B: $tType,U: sum_sum @ A @ B,A2: set @ A,B2: set @ B] :
( ( member @ ( sum_sum @ A @ B ) @ U @ ( sum_Plus @ A @ B @ A2 @ B2 ) )
=> ( ! [X3: A] :
( ( member @ A @ X3 @ A2 )
=> ( U
!= ( sum_Inl @ A @ B @ X3 ) ) )
=> ~ ! [Y4: B] :
( ( member @ B @ Y4 @ B2 )
=> ( U
!= ( sum_Inr @ B @ A @ Y4 ) ) ) ) ) ).
% PlusE
thf(fact_149_finite__PlusD_I2_J,axiom,
! [A: $tType,B: $tType,A2: set @ A,B2: set @ B] :
( ( finite_finite2 @ ( sum_sum @ A @ B ) @ ( sum_Plus @ A @ B @ A2 @ B2 ) )
=> ( finite_finite2 @ B @ B2 ) ) ).
% finite_PlusD(2)
thf(fact_150_finite__PlusD_I1_J,axiom,
! [B: $tType,A: $tType,A2: set @ A,B2: set @ B] :
( ( finite_finite2 @ ( sum_sum @ A @ B ) @ ( sum_Plus @ A @ B @ A2 @ B2 ) )
=> ( finite_finite2 @ A @ A2 ) ) ).
% finite_PlusD(1)
thf(fact_151_finite__Plus,axiom,
! [A: $tType,B: $tType,A2: set @ A,B2: set @ B] :
( ( finite_finite2 @ A @ A2 )
=> ( ( finite_finite2 @ B @ B2 )
=> ( finite_finite2 @ ( sum_sum @ A @ B ) @ ( sum_Plus @ A @ B @ A2 @ B2 ) ) ) ) ).
% finite_Plus
thf(fact_152_Inf_OINF__cong,axiom,
! [A: $tType,B: $tType,A2: set @ B,B2: set @ B,C3: B > A,D3: B > A,Inf: ( set @ A ) > A] :
( ( A2 = B2 )
=> ( ! [X3: B] :
( ( member @ B @ X3 @ B2 )
=> ( ( C3 @ X3 )
= ( D3 @ X3 ) ) )
=> ( ( Inf @ ( image @ B @ A @ C3 @ A2 ) )
= ( Inf @ ( image @ B @ A @ D3 @ B2 ) ) ) ) ) ).
% Inf.INF_cong
thf(fact_153_Sup_OSUP__cong,axiom,
! [A: $tType,B: $tType,A2: set @ B,B2: set @ B,C3: B > A,D3: B > A,Sup: ( set @ A ) > A] :
( ( A2 = B2 )
=> ( ! [X3: B] :
( ( member @ B @ X3 @ B2 )
=> ( ( C3 @ X3 )
= ( D3 @ X3 ) ) )
=> ( ( Sup @ ( image @ B @ A @ C3 @ A2 ) )
= ( Sup @ ( image @ B @ A @ D3 @ B2 ) ) ) ) ) ).
% Sup.SUP_cong
thf(fact_154_pred__equals__eq2,axiom,
! [B: $tType,A: $tType,R: set @ ( product_prod @ A @ B ),S: set @ ( product_prod @ A @ B )] :
( ( ( ^ [X2: A,Y3: B] : ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X2 @ Y3 ) @ R ) )
= ( ^ [X2: A,Y3: B] : ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X2 @ Y3 ) @ S ) ) )
= ( R = S ) ) ).
% pred_equals_eq2
thf(fact_155_Inf_OINF__identity__eq,axiom,
! [A: $tType,Inf: ( set @ A ) > A,A2: set @ A] :
( ( Inf
@ ( image @ A @ A
@ ^ [X2: A] : X2
@ A2 ) )
= ( Inf @ A2 ) ) ).
% Inf.INF_identity_eq
thf(fact_156_reflclp__idemp,axiom,
! [A: $tType,P: A > A > $o] :
( ( sup_sup @ ( A > A > $o )
@ ( sup_sup @ ( A > A > $o ) @ P
@ ^ [Y5: A,Z2: A] : Y5 = Z2 )
@ ^ [Y5: A,Z2: A] : Y5 = Z2 )
= ( sup_sup @ ( A > A > $o ) @ P
@ ^ [Y5: A,Z2: A] : Y5 = Z2 ) ) ).
% reflclp_idemp
thf(fact_157_internal__case__prod__conv,axiom,
! [B: $tType,A: $tType,C: $tType,C2: B > C > A,A5: B,B3: C] :
( ( produc2004651681e_prod @ B @ C @ A @ C2 @ ( product_Pair @ B @ C @ A5 @ B3 ) )
= ( C2 @ A5 @ B3 ) ) ).
% internal_case_prod_conv
thf(fact_158_finite__cont,axiom,
! [Tr: dtree] : ( finite_finite2 @ ( sum_sum @ t @ dtree ) @ ( cont @ Tr ) ) ).
% finite_cont
thf(fact_159_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_160_Suml__inject,axiom,
! [B: $tType,C: $tType,A: $tType,F: A > C,G2: A > C] :
( ( ( sum_Suml @ A @ C @ B @ F )
= ( sum_Suml @ A @ C @ B @ G2 ) )
=> ( F = G2 ) ) ).
% Suml_inject
thf(fact_161_cont__Node,axiom,
! [As: set @ ( sum_sum @ t @ dtree ),N2: n] :
( ( finite_finite2 @ ( sum_sum @ t @ dtree ) @ As )
=> ( ( cont @ ( node @ N2 @ As ) )
= As ) ) ).
% cont_Node
thf(fact_162_dtree__simps,axiom,
! [As: set @ ( sum_sum @ t @ dtree ),As2: set @ ( sum_sum @ t @ dtree ),N2: n,N3: n] :
( ( finite_finite2 @ ( sum_sum @ t @ dtree ) @ As )
=> ( ( finite_finite2 @ ( sum_sum @ t @ dtree ) @ As2 )
=> ( ( ( node @ N2 @ As )
= ( node @ N3 @ As2 ) )
= ( ( N2 = N3 )
& ( As = As2 ) ) ) ) ) ).
% dtree_simps
thf(fact_163_reflcl__set__eq,axiom,
! [A: $tType,R2: set @ ( product_prod @ A @ A )] :
( ( sup_sup @ ( A > A > $o )
@ ^ [X2: A,Y3: A] : ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X2 @ Y3 ) @ R2 )
@ ^ [Y5: A,Z2: A] : Y5 = Z2 )
= ( ^ [X2: A,Y3: A] : ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X2 @ Y3 ) @ ( sup_sup @ ( set @ ( product_prod @ A @ A ) ) @ R2 @ ( id @ A ) ) ) ) ) ).
% reflcl_set_eq
thf(fact_164_IdI,axiom,
! [A: $tType,A5: A] : ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A5 @ A5 ) @ ( id @ A ) ) ).
% IdI
thf(fact_165_pair__in__Id__conv,axiom,
! [A: $tType,A5: A,B3: A] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A5 @ B3 ) @ ( id @ A ) )
= ( A5 = B3 ) ) ).
% pair_in_Id_conv
thf(fact_166_IdD,axiom,
! [A: $tType,A5: A,B3: A] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A5 @ B3 ) @ ( id @ A ) )
=> ( A5 = B3 ) ) ).
% IdD
thf(fact_167_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_168_dtree__cases,axiom,
! [Tr: dtree] :
~ ! [N4: n,As3: set @ ( sum_sum @ t @ dtree )] :
( ( finite_finite2 @ ( sum_sum @ t @ dtree ) @ As3 )
=> ( Tr
!= ( node @ N4 @ As3 ) ) ) ).
% dtree_cases
thf(fact_169_Node__root__cont,axiom,
! [Tr: dtree] :
( ( node @ ( root @ Tr ) @ ( cont @ Tr ) )
= Tr ) ).
% Node_root_cont
thf(fact_170_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_171_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_172_root__Node,axiom,
! [N2: n,As: set @ ( sum_sum @ t @ dtree )] :
( ( root @ ( node @ N2 @ As ) )
= N2 ) ).
% root_Node
thf(fact_173_antisym__Id,axiom,
! [A: $tType] : ( antisym @ A @ ( id @ A ) ) ).
% antisym_Id
thf(fact_174_antisym__def,axiom,
! [A: $tType] :
( ( antisym @ A )
= ( ^ [R3: set @ ( product_prod @ A @ A )] :
! [X2: A,Y3: A] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X2 @ Y3 ) @ R3 )
=> ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Y3 @ X2 ) @ R3 )
=> ( X2 = Y3 ) ) ) ) ) ).
% antisym_def
thf(fact_175_antisymI,axiom,
! [A: $tType,R2: set @ ( product_prod @ A @ A )] :
( ! [X3: A,Y4: A] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X3 @ Y4 ) @ R2 )
=> ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Y4 @ X3 ) @ R2 )
=> ( X3 = Y4 ) ) )
=> ( antisym @ A @ R2 ) ) ).
% antisymI
thf(fact_176_antisymD,axiom,
! [A: $tType,R2: set @ ( product_prod @ A @ A ),A5: A,B3: A] :
( ( antisym @ A @ R2 )
=> ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A5 @ B3 ) @ R2 )
=> ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ B3 @ A5 ) @ R2 )
=> ( A5 = B3 ) ) ) ) ).
% antisymD
thf(fact_177_Sumr__inject,axiom,
! [A: $tType,C: $tType,B: $tType,F: B > C,G2: B > C] :
( ( ( sum_Sumr @ B @ C @ A @ F )
= ( sum_Sumr @ B @ C @ A @ G2 ) )
=> ( F = G2 ) ) ).
% Sumr_inject
thf(fact_178_dtree__cong,axiom,
! [Tr: dtree,Tr3: dtree] :
( ( ( root @ Tr )
= ( root @ Tr3 ) )
=> ( ( ( cont @ Tr )
= ( cont @ Tr3 ) )
=> ( Tr = Tr3 ) ) ) ).
% dtree_cong
thf(fact_179_finite__cartesian__product__iff,axiom,
! [A: $tType,B: $tType,A2: set @ A,B2: set @ B] :
( ( finite_finite2 @ ( product_prod @ A @ B )
@ ( product_Sigma @ A @ B @ A2
@ ^ [Uu: A] : B2 ) )
= ( ( A2
= ( bot_bot @ ( set @ A ) ) )
| ( B2
= ( bot_bot @ ( set @ B ) ) )
| ( ( finite_finite2 @ A @ A2 )
& ( finite_finite2 @ B @ B2 ) ) ) ) ).
% finite_cartesian_product_iff
thf(fact_180_finite__cartesian__productD2,axiom,
! [A: $tType,B: $tType,A2: set @ A,B2: set @ B] :
( ( finite_finite2 @ ( product_prod @ A @ B )
@ ( product_Sigma @ A @ B @ A2
@ ^ [Uu: A] : B2 ) )
=> ( ( A2
!= ( bot_bot @ ( set @ A ) ) )
=> ( finite_finite2 @ B @ B2 ) ) ) ).
% finite_cartesian_productD2
thf(fact_181_finite__cartesian__productD1,axiom,
! [B: $tType,A: $tType,A2: set @ A,B2: set @ B] :
( ( finite_finite2 @ ( product_prod @ A @ B )
@ ( product_Sigma @ A @ B @ A2
@ ^ [Uu: A] : B2 ) )
=> ( ( B2
!= ( bot_bot @ ( set @ B ) ) )
=> ( finite_finite2 @ A @ A2 ) ) ) ).
% finite_cartesian_productD1
thf(fact_182_empty__iff,axiom,
! [A: $tType,C2: A] :
~ ( member @ A @ C2 @ ( bot_bot @ ( set @ A ) ) ) ).
% empty_iff
thf(fact_183_all__not__in__conv,axiom,
! [A: $tType,A2: set @ A] :
( ( ! [X2: A] :
~ ( member @ A @ X2 @ A2 ) )
= ( A2
= ( bot_bot @ ( set @ A ) ) ) ) ).
% all_not_in_conv
thf(fact_184_Collect__empty__eq,axiom,
! [A: $tType,P: A > $o] :
( ( ( collect @ A @ P )
= ( bot_bot @ ( set @ A ) ) )
= ( ! [X2: A] :
~ ( P @ X2 ) ) ) ).
% Collect_empty_eq
thf(fact_185_empty__Collect__eq,axiom,
! [A: $tType,P: A > $o] :
( ( ( bot_bot @ ( set @ A ) )
= ( collect @ A @ P ) )
= ( ! [X2: A] :
~ ( P @ X2 ) ) ) ).
% empty_Collect_eq
thf(fact_186_image__is__empty,axiom,
! [A: $tType,B: $tType,F: B > A,A2: set @ B] :
( ( ( image @ B @ A @ F @ A2 )
= ( bot_bot @ ( set @ A ) ) )
= ( A2
= ( bot_bot @ ( set @ B ) ) ) ) ).
% image_is_empty
thf(fact_187_empty__is__image,axiom,
! [A: $tType,B: $tType,F: B > A,A2: set @ B] :
( ( ( bot_bot @ ( set @ A ) )
= ( image @ B @ A @ F @ A2 ) )
= ( A2
= ( bot_bot @ ( set @ B ) ) ) ) ).
% empty_is_image
thf(fact_188_image__empty,axiom,
! [B: $tType,A: $tType,F: B > A] :
( ( image @ B @ A @ F @ ( bot_bot @ ( set @ B ) ) )
= ( bot_bot @ ( set @ A ) ) ) ).
% image_empty
thf(fact_189_bot__eq__sup__iff,axiom,
! [A: $tType] :
( ( bounded_lattice_bot @ A @ ( type2 @ A ) )
=> ! [X: A,Y: A] :
( ( ( bot_bot @ A )
= ( sup_sup @ A @ X @ Y ) )
= ( ( X
= ( bot_bot @ A ) )
& ( Y
= ( bot_bot @ A ) ) ) ) ) ).
% bot_eq_sup_iff
thf(fact_190_sup__eq__bot__iff,axiom,
! [A: $tType] :
( ( bounded_lattice_bot @ A @ ( type2 @ A ) )
=> ! [X: A,Y: A] :
( ( ( sup_sup @ A @ X @ Y )
= ( bot_bot @ A ) )
= ( ( X
= ( bot_bot @ A ) )
& ( Y
= ( bot_bot @ A ) ) ) ) ) ).
% sup_eq_bot_iff
thf(fact_191_sup__bot_Oleft__neutral,axiom,
! [A: $tType] :
( ( bounde1808546759up_bot @ A @ ( type2 @ A ) )
=> ! [A5: A] :
( ( sup_sup @ A @ ( bot_bot @ A ) @ A5 )
= A5 ) ) ).
% sup_bot.left_neutral
thf(fact_192_sup__bot_Oright__neutral,axiom,
! [A: $tType] :
( ( bounde1808546759up_bot @ A @ ( type2 @ A ) )
=> ! [A5: A] :
( ( sup_sup @ A @ A5 @ ( bot_bot @ A ) )
= A5 ) ) ).
% sup_bot.right_neutral
thf(fact_193_Un__empty,axiom,
! [A: $tType,A2: set @ A,B2: set @ A] :
( ( ( sup_sup @ ( set @ A ) @ A2 @ B2 )
= ( bot_bot @ ( set @ A ) ) )
= ( ( A2
= ( bot_bot @ ( set @ A ) ) )
& ( B2
= ( bot_bot @ ( set @ A ) ) ) ) ) ).
% Un_empty
thf(fact_194_Sigma__empty1,axiom,
! [B: $tType,A: $tType,B2: A > ( set @ B )] :
( ( product_Sigma @ A @ B @ ( bot_bot @ ( set @ A ) ) @ B2 )
= ( bot_bot @ ( set @ ( product_prod @ A @ B ) ) ) ) ).
% Sigma_empty1
thf(fact_195_vimage__empty,axiom,
! [B: $tType,A: $tType,F: A > B] :
( ( vimage @ A @ B @ F @ ( bot_bot @ ( set @ B ) ) )
= ( bot_bot @ ( set @ A ) ) ) ).
% vimage_empty
thf(fact_196_Plus__eq__empty__conv,axiom,
! [A: $tType,B: $tType,A2: set @ A,B2: set @ B] :
( ( ( sum_Plus @ A @ B @ A2 @ B2 )
= ( bot_bot @ ( set @ ( sum_sum @ A @ B ) ) ) )
= ( ( A2
= ( bot_bot @ ( set @ A ) ) )
& ( B2
= ( bot_bot @ ( set @ B ) ) ) ) ) ).
% Plus_eq_empty_conv
thf(fact_197_Sigma__empty2,axiom,
! [B: $tType,A: $tType,A2: set @ A] :
( ( product_Sigma @ A @ B @ A2
@ ^ [Uu: A] : ( bot_bot @ ( set @ B ) ) )
= ( bot_bot @ ( set @ ( product_prod @ A @ B ) ) ) ) ).
% Sigma_empty2
thf(fact_198_Times__empty,axiom,
! [A: $tType,B: $tType,A2: set @ A,B2: set @ B] :
( ( ( product_Sigma @ A @ B @ A2
@ ^ [Uu: A] : B2 )
= ( bot_bot @ ( set @ ( product_prod @ A @ B ) ) ) )
= ( ( A2
= ( bot_bot @ ( set @ A ) ) )
| ( B2
= ( bot_bot @ ( set @ B ) ) ) ) ) ).
% Times_empty
thf(fact_199_antisym__empty,axiom,
! [A: $tType] : ( antisym @ A @ ( bot_bot @ ( set @ ( product_prod @ A @ A ) ) ) ) ).
% antisym_empty
thf(fact_200_finite_OemptyI,axiom,
! [A: $tType] : ( finite_finite2 @ A @ ( bot_bot @ ( set @ A ) ) ) ).
% finite.emptyI
thf(fact_201_infinite__imp__nonempty,axiom,
! [A: $tType,S: set @ A] :
( ~ ( finite_finite2 @ A @ S )
=> ( S
!= ( bot_bot @ ( set @ A ) ) ) ) ).
% infinite_imp_nonempty
thf(fact_202_times__eq__iff,axiom,
! [A: $tType,B: $tType,A2: set @ A,B2: set @ B,C3: set @ A,D3: set @ B] :
( ( ( product_Sigma @ A @ B @ A2
@ ^ [Uu: A] : B2 )
= ( product_Sigma @ A @ B @ C3
@ ^ [Uu: A] : D3 ) )
= ( ( ( A2 = C3 )
& ( B2 = D3 ) )
| ( ( ( A2
= ( bot_bot @ ( set @ A ) ) )
| ( B2
= ( bot_bot @ ( set @ B ) ) ) )
& ( ( C3
= ( bot_bot @ ( set @ A ) ) )
| ( D3
= ( bot_bot @ ( set @ B ) ) ) ) ) ) ) ).
% times_eq_iff
thf(fact_203_emptyE,axiom,
! [A: $tType,A5: A] :
~ ( member @ A @ A5 @ ( bot_bot @ ( set @ A ) ) ) ).
% emptyE
thf(fact_204_equals0D,axiom,
! [A: $tType,A2: set @ A,A5: A] :
( ( A2
= ( bot_bot @ ( set @ A ) ) )
=> ~ ( member @ A @ A5 @ A2 ) ) ).
% equals0D
thf(fact_205_equals0I,axiom,
! [A: $tType,A2: set @ A] :
( ! [Y4: A] :
~ ( member @ A @ Y4 @ A2 )
=> ( A2
= ( bot_bot @ ( set @ A ) ) ) ) ).
% equals0I
thf(fact_206_ex__in__conv,axiom,
! [A: $tType,A2: set @ A] :
( ( ? [X2: A] : ( member @ A @ X2 @ A2 ) )
= ( A2
!= ( bot_bot @ ( set @ A ) ) ) ) ).
% ex_in_conv
thf(fact_207_empty__def,axiom,
! [A: $tType] :
( ( bot_bot @ ( set @ A ) )
= ( collect @ A
@ ^ [X2: A] : $false ) ) ).
% empty_def
thf(fact_208_Sigma__empty__iff,axiom,
! [B: $tType,A: $tType,I: set @ A,X6: A > ( set @ B )] :
( ( ( product_Sigma @ A @ B @ I @ X6 )
= ( bot_bot @ ( set @ ( product_prod @ A @ B ) ) ) )
= ( ! [X2: A] :
( ( member @ A @ X2 @ I )
=> ( ( X6 @ X2 )
= ( bot_bot @ ( set @ B ) ) ) ) ) ) ).
% Sigma_empty_iff
thf(fact_209_Un__empty__left,axiom,
! [A: $tType,B2: set @ A] :
( ( sup_sup @ ( set @ A ) @ ( bot_bot @ ( set @ A ) ) @ B2 )
= B2 ) ).
% Un_empty_left
thf(fact_210_Un__empty__right,axiom,
! [A: $tType,A2: set @ A] :
( ( sup_sup @ ( set @ A ) @ A2 @ ( bot_bot @ ( set @ A ) ) )
= A2 ) ).
% Un_empty_right
thf(fact_211_sup__bot__left,axiom,
! [A: $tType] :
( ( bounded_lattice_bot @ A @ ( type2 @ A ) )
=> ! [X: A] :
( ( sup_sup @ A @ ( bot_bot @ A ) @ X )
= X ) ) ).
% sup_bot_left
thf(fact_212_sup__bot__right,axiom,
! [A: $tType] :
( ( bounded_lattice_bot @ A @ ( type2 @ A ) )
=> ! [X: A] :
( ( sup_sup @ A @ X @ ( bot_bot @ A ) )
= X ) ) ).
% sup_bot_right
thf(fact_213_Pair__vimage__Sigma,axiom,
! [B: $tType,A: $tType,X: B,A2: set @ B,F: B > ( set @ A )] :
( ( ( member @ B @ X @ A2 )
=> ( ( vimage @ A @ ( product_prod @ B @ A ) @ ( product_Pair @ B @ A @ X ) @ ( product_Sigma @ B @ A @ A2 @ F ) )
= ( F @ X ) ) )
& ( ~ ( member @ B @ X @ A2 )
=> ( ( vimage @ A @ ( product_prod @ B @ A ) @ ( product_Pair @ B @ A @ X ) @ ( product_Sigma @ B @ A @ A2 @ F ) )
= ( bot_bot @ ( set @ A ) ) ) ) ) ).
% Pair_vimage_Sigma
thf(fact_214_finite__SigmaI2,axiom,
! [B: $tType,A: $tType,A2: set @ A,B2: A > ( set @ B )] :
( ( finite_finite2 @ A
@ ( collect @ A
@ ^ [X2: A] :
( ( member @ A @ X2 @ A2 )
& ( ( B2 @ X2 )
!= ( bot_bot @ ( set @ B ) ) ) ) ) )
=> ( ! [A3: A] :
( ( member @ A @ A3 @ A2 )
=> ( finite_finite2 @ B @ ( B2 @ A3 ) ) )
=> ( finite_finite2 @ ( product_prod @ A @ B ) @ ( product_Sigma @ A @ B @ A2 @ B2 ) ) ) ) ).
% finite_SigmaI2
thf(fact_215_Set_Ois__empty__def,axiom,
! [A: $tType] :
( ( is_empty @ A )
= ( ^ [A4: set @ A] :
( A4
= ( bot_bot @ ( set @ A ) ) ) ) ) ).
% Set.is_empty_def
thf(fact_216_fst__image__times,axiom,
! [B: $tType,A: $tType,B2: set @ B,A2: set @ A] :
( ( ( B2
= ( bot_bot @ ( set @ B ) ) )
=> ( ( image @ ( product_prod @ A @ B ) @ A @ ( product_fst @ A @ B )
@ ( product_Sigma @ A @ B @ A2
@ ^ [Uu: A] : B2 ) )
= ( bot_bot @ ( set @ A ) ) ) )
& ( ( B2
!= ( bot_bot @ ( set @ B ) ) )
=> ( ( image @ ( product_prod @ A @ B ) @ A @ ( product_fst @ A @ B )
@ ( product_Sigma @ A @ B @ A2
@ ^ [Uu: A] : B2 ) )
= A2 ) ) ) ).
% fst_image_times
thf(fact_217_inf__img__fin__dom,axiom,
! [B: $tType,A: $tType,F: B > A,A2: set @ B] :
( ( finite_finite2 @ A @ ( image @ B @ A @ F @ A2 ) )
=> ( ~ ( finite_finite2 @ B @ A2 )
=> ? [X3: A] :
( ( member @ A @ X3 @ ( image @ B @ A @ F @ A2 ) )
& ~ ( finite_finite2 @ B @ ( vimage @ B @ A @ F @ ( insert @ A @ X3 @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ) ) ).
% inf_img_fin_dom
thf(fact_218_insert__absorb2,axiom,
! [A: $tType,X: A,A2: set @ A] :
( ( insert @ A @ X @ ( insert @ A @ X @ A2 ) )
= ( insert @ A @ X @ A2 ) ) ).
% insert_absorb2
thf(fact_219_insert__iff,axiom,
! [A: $tType,A5: A,B3: A,A2: set @ A] :
( ( member @ A @ A5 @ ( insert @ A @ B3 @ A2 ) )
= ( ( A5 = B3 )
| ( member @ A @ A5 @ A2 ) ) ) ).
% insert_iff
thf(fact_220_insertCI,axiom,
! [A: $tType,A5: A,B2: set @ A,B3: A] :
( ( ~ ( member @ A @ A5 @ B2 )
=> ( A5 = B3 ) )
=> ( member @ A @ A5 @ ( insert @ A @ B3 @ B2 ) ) ) ).
% insertCI
thf(fact_221_insert__image,axiom,
! [B: $tType,A: $tType,X: A,A2: set @ A,F: A > B] :
( ( member @ A @ X @ A2 )
=> ( ( insert @ B @ ( F @ X ) @ ( image @ A @ B @ F @ A2 ) )
= ( image @ A @ B @ F @ A2 ) ) ) ).
% insert_image
thf(fact_222_image__insert,axiom,
! [A: $tType,B: $tType,F: B > A,A5: B,B2: set @ B] :
( ( image @ B @ A @ F @ ( insert @ B @ A5 @ B2 ) )
= ( insert @ A @ ( F @ A5 ) @ ( image @ B @ A @ F @ B2 ) ) ) ).
% image_insert
thf(fact_223_singletonI,axiom,
! [A: $tType,A5: A] : ( member @ A @ A5 @ ( insert @ A @ A5 @ ( bot_bot @ ( set @ A ) ) ) ) ).
% singletonI
thf(fact_224_finite__insert,axiom,
! [A: $tType,A5: A,A2: set @ A] :
( ( finite_finite2 @ A @ ( insert @ A @ A5 @ A2 ) )
= ( finite_finite2 @ A @ A2 ) ) ).
% finite_insert
thf(fact_225_Un__insert__right,axiom,
! [A: $tType,A2: set @ A,A5: A,B2: set @ A] :
( ( sup_sup @ ( set @ A ) @ A2 @ ( insert @ A @ A5 @ B2 ) )
= ( insert @ A @ A5 @ ( sup_sup @ ( set @ A ) @ A2 @ B2 ) ) ) ).
% Un_insert_right
thf(fact_226_Un__insert__left,axiom,
! [A: $tType,A5: A,B2: set @ A,C3: set @ A] :
( ( sup_sup @ ( set @ A ) @ ( insert @ A @ A5 @ B2 ) @ C3 )
= ( insert @ A @ A5 @ ( sup_sup @ ( set @ A ) @ B2 @ C3 ) ) ) ).
% Un_insert_left
thf(fact_227_singleton__conv,axiom,
! [A: $tType,A5: A] :
( ( collect @ A
@ ^ [X2: A] : X2 = A5 )
= ( insert @ A @ A5 @ ( bot_bot @ ( set @ A ) ) ) ) ).
% singleton_conv
thf(fact_228_singleton__conv2,axiom,
! [A: $tType,A5: A] :
( ( collect @ A
@ ( ^ [Y5: A,Z2: A] : Y5 = Z2
@ A5 ) )
= ( insert @ A @ A5 @ ( bot_bot @ ( set @ A ) ) ) ) ).
% singleton_conv2
thf(fact_229_insert__times__insert,axiom,
! [B: $tType,A: $tType,A5: A,A2: set @ A,B3: B,B2: set @ B] :
( ( product_Sigma @ A @ B @ ( insert @ A @ A5 @ A2 )
@ ^ [Uu: A] : ( insert @ B @ B3 @ B2 ) )
= ( insert @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ A5 @ B3 )
@ ( sup_sup @ ( set @ ( product_prod @ A @ B ) )
@ ( product_Sigma @ A @ B @ A2
@ ^ [Uu: A] : ( insert @ B @ B3 @ B2 ) )
@ ( product_Sigma @ A @ B @ ( insert @ A @ A5 @ A2 )
@ ^ [Uu: A] : B2 ) ) ) ) ).
% insert_times_insert
thf(fact_230_infinite__finite__induct,axiom,
! [A: $tType,P: ( set @ A ) > $o,A2: set @ A] :
( ! [A10: set @ A] :
( ~ ( finite_finite2 @ A @ A10 )
=> ( P @ A10 ) )
=> ( ( P @ ( bot_bot @ ( set @ A ) ) )
=> ( ! [X3: A,F6: set @ A] :
( ( finite_finite2 @ A @ F6 )
=> ( ~ ( member @ A @ X3 @ F6 )
=> ( ( P @ F6 )
=> ( P @ ( insert @ A @ X3 @ F6 ) ) ) ) )
=> ( P @ A2 ) ) ) ) ).
% infinite_finite_induct
thf(fact_231_finite__ne__induct,axiom,
! [A: $tType,F2: set @ A,P: ( set @ A ) > $o] :
( ( finite_finite2 @ A @ F2 )
=> ( ( F2
!= ( bot_bot @ ( set @ A ) ) )
=> ( ! [X3: A] : ( P @ ( insert @ A @ X3 @ ( bot_bot @ ( set @ A ) ) ) )
=> ( ! [X3: A,F6: set @ A] :
( ( finite_finite2 @ A @ F6 )
=> ( ( F6
!= ( bot_bot @ ( set @ A ) ) )
=> ( ~ ( member @ A @ X3 @ F6 )
=> ( ( P @ F6 )
=> ( P @ ( insert @ A @ X3 @ F6 ) ) ) ) ) )
=> ( P @ F2 ) ) ) ) ) ).
% finite_ne_induct
thf(fact_232_finite_Oinducts,axiom,
! [A: $tType,X: set @ A,P: ( set @ A ) > $o] :
( ( finite_finite2 @ A @ X )
=> ( ( P @ ( bot_bot @ ( set @ A ) ) )
=> ( ! [A10: set @ A,A3: A] :
( ( finite_finite2 @ A @ A10 )
=> ( ( P @ A10 )
=> ( P @ ( insert @ A @ A3 @ A10 ) ) ) )
=> ( P @ X ) ) ) ) ).
% finite.inducts
thf(fact_233_finite__induct,axiom,
! [A: $tType,F2: set @ A,P: ( set @ A ) > $o] :
( ( finite_finite2 @ A @ F2 )
=> ( ( P @ ( bot_bot @ ( set @ A ) ) )
=> ( ! [X3: A,F6: set @ A] :
( ( finite_finite2 @ A @ F6 )
=> ( ~ ( member @ A @ X3 @ F6 )
=> ( ( P @ F6 )
=> ( P @ ( insert @ A @ X3 @ F6 ) ) ) ) )
=> ( P @ F2 ) ) ) ) ).
% finite_induct
thf(fact_234_finite_Osimps,axiom,
! [A: $tType] :
( ( finite_finite2 @ A )
= ( ^ [A6: set @ A] :
( ( A6
= ( bot_bot @ ( set @ A ) ) )
| ? [A4: set @ A,B7: A] :
( ( A6
= ( insert @ A @ B7 @ A4 ) )
& ( finite_finite2 @ A @ A4 ) ) ) ) ) ).
% finite.simps
thf(fact_235_finite_Ocases,axiom,
! [A: $tType,A5: set @ A] :
( ( finite_finite2 @ A @ A5 )
=> ( ( A5
!= ( bot_bot @ ( set @ A ) ) )
=> ~ ! [A10: set @ A] :
( ? [A3: A] :
( A5
= ( insert @ A @ A3 @ A10 ) )
=> ~ ( finite_finite2 @ A @ A10 ) ) ) ) ).
% finite.cases
thf(fact_236_singletonD,axiom,
! [A: $tType,B3: A,A5: A] :
( ( member @ A @ B3 @ ( insert @ A @ A5 @ ( bot_bot @ ( set @ A ) ) ) )
=> ( B3 = A5 ) ) ).
% singletonD
thf(fact_237_bot__set__def,axiom,
! [A: $tType] :
( ( bot_bot @ ( set @ A ) )
= ( collect @ A @ ( bot_bot @ ( A > $o ) ) ) ) ).
% bot_set_def
thf(fact_238_singleton__iff,axiom,
! [A: $tType,B3: A,A5: A] :
( ( member @ A @ B3 @ ( insert @ A @ A5 @ ( bot_bot @ ( set @ A ) ) ) )
= ( B3 = A5 ) ) ).
% singleton_iff
thf(fact_239_Collect__conv__if,axiom,
! [A: $tType,P: A > $o,A5: A] :
( ( ( P @ A5 )
=> ( ( collect @ A
@ ^ [X2: A] :
( ( X2 = A5 )
& ( P @ X2 ) ) )
= ( insert @ A @ A5 @ ( bot_bot @ ( set @ A ) ) ) ) )
& ( ~ ( P @ A5 )
=> ( ( collect @ A
@ ^ [X2: A] :
( ( X2 = A5 )
& ( P @ X2 ) ) )
= ( bot_bot @ ( set @ A ) ) ) ) ) ).
% Collect_conv_if
thf(fact_240_Collect__conv__if2,axiom,
! [A: $tType,P: A > $o,A5: A] :
( ( ( P @ A5 )
=> ( ( collect @ A
@ ^ [X2: A] :
( ( A5 = X2 )
& ( P @ X2 ) ) )
= ( insert @ A @ A5 @ ( bot_bot @ ( set @ A ) ) ) ) )
& ( ~ ( P @ A5 )
=> ( ( collect @ A
@ ^ [X2: A] :
( ( A5 = X2 )
& ( P @ X2 ) ) )
= ( bot_bot @ ( set @ A ) ) ) ) ) ).
% Collect_conv_if2
thf(fact_241_doubleton__eq__iff,axiom,
! [A: $tType,A5: A,B3: A,C2: A,D4: A] :
( ( ( insert @ A @ A5 @ ( insert @ A @ B3 @ ( bot_bot @ ( set @ A ) ) ) )
= ( insert @ A @ C2 @ ( insert @ A @ D4 @ ( bot_bot @ ( set @ A ) ) ) ) )
= ( ( ( A5 = C2 )
& ( B3 = D4 ) )
| ( ( A5 = D4 )
& ( B3 = C2 ) ) ) ) ).
% doubleton_eq_iff
thf(fact_242_insert__not__empty,axiom,
! [A: $tType,A5: A,A2: set @ A] :
( ( insert @ A @ A5 @ A2 )
!= ( bot_bot @ ( set @ A ) ) ) ).
% insert_not_empty
thf(fact_243_singleton__inject,axiom,
! [A: $tType,A5: A,B3: A] :
( ( ( insert @ A @ A5 @ ( bot_bot @ ( set @ A ) ) )
= ( insert @ A @ B3 @ ( bot_bot @ ( set @ A ) ) ) )
=> ( A5 = B3 ) ) ).
% singleton_inject
thf(fact_244_bot__empty__eq,axiom,
! [A: $tType] :
( ( bot_bot @ ( A > $o ) )
= ( ^ [X2: A] : ( member @ A @ X2 @ ( bot_bot @ ( set @ A ) ) ) ) ) ).
% bot_empty_eq
thf(fact_245_insert__is__Un,axiom,
! [A: $tType] :
( ( insert @ A )
= ( ^ [A6: A] : ( sup_sup @ ( set @ A ) @ ( insert @ A @ A6 @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ).
% insert_is_Un
thf(fact_246_Un__singleton__iff,axiom,
! [A: $tType,A2: set @ A,B2: set @ A,X: A] :
( ( ( sup_sup @ ( set @ A ) @ A2 @ B2 )
= ( insert @ A @ X @ ( bot_bot @ ( set @ A ) ) ) )
= ( ( ( A2
= ( bot_bot @ ( set @ A ) ) )
& ( B2
= ( insert @ A @ X @ ( bot_bot @ ( set @ A ) ) ) ) )
| ( ( A2
= ( insert @ A @ X @ ( bot_bot @ ( set @ A ) ) ) )
& ( B2
= ( bot_bot @ ( set @ A ) ) ) )
| ( ( A2
= ( insert @ A @ X @ ( bot_bot @ ( set @ A ) ) ) )
& ( B2
= ( insert @ A @ X @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ) ).
% Un_singleton_iff
thf(fact_247_singleton__Un__iff,axiom,
! [A: $tType,X: A,A2: set @ A,B2: set @ A] :
( ( ( insert @ A @ X @ ( bot_bot @ ( set @ A ) ) )
= ( sup_sup @ ( set @ A ) @ A2 @ B2 ) )
= ( ( ( A2
= ( bot_bot @ ( set @ A ) ) )
& ( B2
= ( insert @ A @ X @ ( bot_bot @ ( set @ A ) ) ) ) )
| ( ( A2
= ( insert @ A @ X @ ( bot_bot @ ( set @ A ) ) ) )
& ( B2
= ( bot_bot @ ( set @ A ) ) ) )
| ( ( A2
= ( insert @ A @ X @ ( bot_bot @ ( set @ A ) ) ) )
& ( B2
= ( insert @ A @ X @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ) ).
% singleton_Un_iff
thf(fact_248_bot__empty__eq2,axiom,
! [B: $tType,A: $tType] :
( ( bot_bot @ ( A > B > $o ) )
= ( ^ [X2: A,Y3: B] : ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X2 @ Y3 ) @ ( bot_bot @ ( set @ ( product_prod @ A @ B ) ) ) ) ) ) ).
% bot_empty_eq2
thf(fact_249_insert__def,axiom,
! [A: $tType] :
( ( insert @ A )
= ( ^ [A6: A] :
( sup_sup @ ( set @ A )
@ ( collect @ A
@ ^ [X2: A] : X2 = A6 ) ) ) ) ).
% insert_def
thf(fact_250_insert__Collect,axiom,
! [A: $tType,A5: A,P: A > $o] :
( ( insert @ A @ A5 @ ( collect @ A @ P ) )
= ( collect @ A
@ ^ [U2: A] :
( ( U2 != A5 )
=> ( P @ U2 ) ) ) ) ).
% insert_Collect
thf(fact_251_insert__compr,axiom,
! [A: $tType] :
( ( insert @ A )
= ( ^ [A6: A,B4: set @ A] :
( collect @ A
@ ^ [X2: A] :
( ( X2 = A6 )
| ( member @ A @ X2 @ B4 ) ) ) ) ) ).
% insert_compr
thf(fact_252_mk__disjoint__insert,axiom,
! [A: $tType,A5: A,A2: set @ A] :
( ( member @ A @ A5 @ A2 )
=> ? [B10: set @ A] :
( ( A2
= ( insert @ A @ A5 @ B10 ) )
& ~ ( member @ A @ A5 @ B10 ) ) ) ).
% mk_disjoint_insert
thf(fact_253_insert__commute,axiom,
! [A: $tType,X: A,Y: A,A2: set @ A] :
( ( insert @ A @ X @ ( insert @ A @ Y @ A2 ) )
= ( insert @ A @ Y @ ( insert @ A @ X @ A2 ) ) ) ).
% insert_commute
thf(fact_254_insert__eq__iff,axiom,
! [A: $tType,A5: A,A2: set @ A,B3: A,B2: set @ A] :
( ~ ( member @ A @ A5 @ A2 )
=> ( ~ ( member @ A @ B3 @ B2 )
=> ( ( ( insert @ A @ A5 @ A2 )
= ( insert @ A @ B3 @ B2 ) )
= ( ( ( A5 = B3 )
=> ( A2 = B2 ) )
& ( ( A5 != B3 )
=> ? [C5: set @ A] :
( ( A2
= ( insert @ A @ B3 @ C5 ) )
& ~ ( member @ A @ B3 @ C5 )
& ( B2
= ( insert @ A @ A5 @ C5 ) )
& ~ ( member @ A @ A5 @ C5 ) ) ) ) ) ) ) ).
% insert_eq_iff
thf(fact_255_insert__absorb,axiom,
! [A: $tType,A5: A,A2: set @ A] :
( ( member @ A @ A5 @ A2 )
=> ( ( insert @ A @ A5 @ A2 )
= A2 ) ) ).
% insert_absorb
%----Type constructors (20)
thf(tcon_HOL_Obool___Lattices_Obounded__lattice,axiom,
bounded_lattice @ $o @ ( type2 @ $o ) ).
thf(tcon_Set_Oset___Lattices_Obounded__lattice_1,axiom,
! [A8: $tType] : ( bounded_lattice @ ( set @ A8 ) @ ( type2 @ ( set @ A8 ) ) ) ).
thf(tcon_fun___Lattices_Obounded__lattice_2,axiom,
! [A8: $tType,A11: $tType] :
( ( bounded_lattice @ A11 @ ( type2 @ A11 ) )
=> ( bounded_lattice @ ( A8 > A11 ) @ ( type2 @ ( A8 > A11 ) ) ) ) ).
thf(tcon_fun___Lattices_Obounded__semilattice__sup__bot,axiom,
! [A8: $tType,A11: $tType] :
( ( bounded_lattice @ A11 @ ( type2 @ A11 ) )
=> ( bounde1808546759up_bot @ ( A8 > A11 ) @ ( type2 @ ( A8 > A11 ) ) ) ) ).
thf(tcon_fun___Lattices_Obounded__lattice__bot,axiom,
! [A8: $tType,A11: $tType] :
( ( bounded_lattice @ A11 @ ( type2 @ A11 ) )
=> ( bounded_lattice_bot @ ( A8 > A11 ) @ ( type2 @ ( A8 > A11 ) ) ) ) ).
thf(tcon_fun___Lattices_Osemilattice__sup,axiom,
! [A8: $tType,A11: $tType] :
( ( semilattice_sup @ A11 @ ( type2 @ A11 ) )
=> ( semilattice_sup @ ( A8 > A11 ) @ ( type2 @ ( A8 > A11 ) ) ) ) ).
thf(tcon_fun___Finite__Set_Ofinite,axiom,
! [A8: $tType,A11: $tType] :
( ( ( finite_finite @ A8 @ ( type2 @ A8 ) )
& ( finite_finite @ A11 @ ( type2 @ A11 ) ) )
=> ( finite_finite @ ( A8 > A11 ) @ ( type2 @ ( A8 > A11 ) ) ) ) ).
thf(tcon_fun___Lattices_Olattice,axiom,
! [A8: $tType,A11: $tType] :
( ( lattice @ A11 @ ( type2 @ A11 ) )
=> ( lattice @ ( A8 > A11 ) @ ( type2 @ ( A8 > A11 ) ) ) ) ).
thf(tcon_Set_Oset___Lattices_Obounded__semilattice__sup__bot_3,axiom,
! [A8: $tType] : ( bounde1808546759up_bot @ ( set @ A8 ) @ ( type2 @ ( set @ A8 ) ) ) ).
thf(tcon_Set_Oset___Lattices_Obounded__lattice__bot_4,axiom,
! [A8: $tType] : ( bounded_lattice_bot @ ( set @ A8 ) @ ( type2 @ ( set @ A8 ) ) ) ).
thf(tcon_Set_Oset___Lattices_Osemilattice__sup_5,axiom,
! [A8: $tType] : ( semilattice_sup @ ( set @ A8 ) @ ( type2 @ ( set @ A8 ) ) ) ).
thf(tcon_Set_Oset___Finite__Set_Ofinite_6,axiom,
! [A8: $tType] :
( ( finite_finite @ A8 @ ( type2 @ A8 ) )
=> ( finite_finite @ ( set @ A8 ) @ ( type2 @ ( set @ A8 ) ) ) ) ).
thf(tcon_Set_Oset___Lattices_Olattice_7,axiom,
! [A8: $tType] : ( lattice @ ( set @ A8 ) @ ( type2 @ ( set @ A8 ) ) ) ).
thf(tcon_HOL_Obool___Lattices_Obounded__semilattice__sup__bot_8,axiom,
bounde1808546759up_bot @ $o @ ( type2 @ $o ) ).
thf(tcon_HOL_Obool___Lattices_Obounded__lattice__bot_9,axiom,
bounded_lattice_bot @ $o @ ( type2 @ $o ) ).
thf(tcon_HOL_Obool___Lattices_Osemilattice__sup_10,axiom,
semilattice_sup @ $o @ ( type2 @ $o ) ).
thf(tcon_HOL_Obool___Finite__Set_Ofinite_11,axiom,
finite_finite @ $o @ ( type2 @ $o ) ).
thf(tcon_HOL_Obool___Lattices_Olattice_12,axiom,
lattice @ $o @ ( type2 @ $o ) ).
thf(tcon_Sum__Type_Osum___Finite__Set_Ofinite_13,axiom,
! [A8: $tType,A11: $tType] :
( ( ( finite_finite @ A8 @ ( type2 @ A8 ) )
& ( finite_finite @ A11 @ ( type2 @ A11 ) ) )
=> ( finite_finite @ ( sum_sum @ A8 @ A11 ) @ ( type2 @ ( sum_sum @ A8 @ A11 ) ) ) ) ).
thf(tcon_Product__Type_Oprod___Finite__Set_Ofinite_14,axiom,
! [A8: $tType,A11: $tType] :
( ( ( finite_finite @ A8 @ ( type2 @ A8 ) )
& ( finite_finite @ A11 @ ( type2 @ A11 ) ) )
=> ( finite_finite @ ( product_prod @ A8 @ A11 ) @ ( type2 @ ( product_prod @ A8 @ A11 ) ) ) ) ).
%----Conjectures (1)
thf(conj_0,conjecture,
( finite_finite2 @ ( sum_sum @ t @ ( product_prod @ dtree @ dtree ) )
@ ( 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 ) ) ) ) ) ) ).
%------------------------------------------------------------------------------