TPTP Problem File: COM211^1.p
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%------------------------------------------------------------------------------
% File : COM211^1 : TPTP v8.2.0. Released v7.0.0.
% Domain : Computing Theory
% Problem : Parallel extension to grammars and languages 144
% 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__144.p [Bla16]
% Status : Theorem
% Rating : 0.33 v8.1.0, 0.25 v7.5.0, 0.00 v7.2.0, 0.25 v7.1.0
% Syntax : Number of formulae : 339 ( 124 unt; 66 typ; 0 def)
% Number of atoms : 641 ( 246 equ; 0 cnn)
% Maximal formula atoms : 13 ( 2 avg)
% Number of connectives : 4584 ( 51 ~; 3 |; 23 &;4181 @)
% ( 0 <=>; 326 =>; 0 <=; 0 <~>)
% Maximal formula depth : 28 ( 9 avg)
% Number of types : 4 ( 3 usr)
% Number of type conns : 490 ( 490 >; 0 *; 0 +; 0 <<)
% Number of symbols : 66 ( 63 usr; 5 con; 0-9 aty)
% Number of variables : 1387 ( 59 ^;1218 !; 17 ?;1387 :)
% ( 93 !>; 0 ?*; 0 @-; 0 @+)
% SPC : TH1_THM_EQU_NAR
% Comments : This file was generated by Isabelle (most likely Sledgehammer)
% 2016-07-13 14:39:25.594
%------------------------------------------------------------------------------
%----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 (59)
thf(sy_cl_HOL_Otype,type,
type:
!>[A: $tType] : ( ( itself @ A ) > $o ) ).
thf(sy_cl_Orderings_Oord,type,
ord:
!>[A: $tType] : ( ( itself @ A ) > $o ) ).
thf(sy_cl_Orderings_Oorder,type,
order:
!>[A: $tType] : ( ( itself @ A ) > $o ) ).
thf(sy_cl_Orderings_Olinorder,type,
linorder:
!>[A: $tType] : ( ( itself @ A ) > $o ) ).
thf(sy_cl_Orderings_Opreorder,type,
preorder:
!>[A: $tType] : ( ( itself @ A ) > $o ) ).
thf(sy_cl_Orderings_Oorder__top,type,
order_top:
!>[A: $tType] : ( ( itself @ A ) > $o ) ).
thf(sy_cl_Lattices_Oboolean__algebra,type,
boolean_algebra:
!>[A: $tType] : ( ( itself @ A ) > $o ) ).
thf(sy_cl_Groups_Oordered__ab__group__add,type,
ordered_ab_group_add:
!>[A: $tType] : ( ( itself @ A ) > $o ) ).
thf(sy_c_BNF__Cardinal__Order__Relation_OrelChain,type,
bNF_Ca1785829860lChain:
!>[A: $tType,B: $tType] : ( ( set @ ( product_prod @ A @ A ) ) > ( A > B ) > $o ) ).
thf(sy_c_BNF__Def_Ocsquare,type,
bNF_csquare:
!>[A: $tType,B: $tType,C: $tType,D: $tType] : ( ( set @ A ) > ( B > C ) > ( D > C ) > ( A > B ) > ( A > D ) > $o ) ).
thf(sy_c_BNF__Def_OfstOp,type,
bNF_fstOp:
!>[A: $tType,B: $tType,C: $tType] : ( ( A > B > $o ) > ( B > C > $o ) > ( product_prod @ A @ C ) > ( product_prod @ A @ B ) ) ).
thf(sy_c_BNF__Def_Opick__middlep,type,
bNF_pick_middlep:
!>[B: $tType,A: $tType,C: $tType] : ( ( B > A > $o ) > ( A > C > $o ) > B > C > A ) ).
thf(sy_c_BNF__Def_OsndOp,type,
bNF_sndOp:
!>[C: $tType,A: $tType,B: $tType] : ( ( C > A > $o ) > ( A > B > $o ) > ( product_prod @ C @ B ) > ( product_prod @ A @ B ) ) ).
thf(sy_c_BNF__Greatest__Fixpoint_Oimage2,type,
bNF_Greatest_image2:
!>[C: $tType,A: $tType,B: $tType] : ( ( set @ C ) > ( C > A ) > ( C > B ) > ( set @ ( product_prod @ A @ B ) ) ) ).
thf(sy_c_DTree_ONode,type,
node: n > ( set @ ( sum_sum @ t @ dtree ) ) > dtree ).
thf(sy_c_DTree_Ocont,type,
cont: dtree > ( set @ ( sum_sum @ t @ dtree ) ) ).
thf(sy_c_DTree_Ocorec,type,
corec:
!>[A: $tType] : ( ( A > n ) > ( A > ( set @ ( sum_sum @ t @ ( sum_sum @ dtree @ A ) ) ) ) > A > dtree ) ).
thf(sy_c_DTree_Odtree_Oroot,type,
root: dtree > n ).
thf(sy_c_DTree_Ounfold,type,
unfold:
!>[A: $tType] : ( ( A > n ) > ( A > ( set @ ( sum_sum @ t @ A ) ) ) > A > dtree ) ).
thf(sy_c_Fun_Oinj__on,type,
inj_on:
!>[A: $tType,B: $tType] : ( ( A > B ) > ( set @ A ) > $o ) ).
thf(sy_c_Fun_Oswap,type,
swap:
!>[A: $tType,B: $tType] : ( A > A > ( A > B ) > A > B ) ).
thf(sy_c_Fun_Othe__inv__into,type,
the_inv_into:
!>[A: $tType,B: $tType] : ( ( set @ A ) > ( A > B ) > B > A ) ).
thf(sy_c_Fun__Def_Oin__rel,type,
fun_in_rel:
!>[A: $tType,B: $tType] : ( ( set @ ( product_prod @ A @ B ) ) > A > B > $o ) ).
thf(sy_c_Groups_Ouminus__class_Ouminus,type,
uminus_uminus:
!>[A: $tType] : ( A > A ) ).
thf(sy_c_Orderings_Oord__class_Oless__eq,type,
ord_less_eq:
!>[A: $tType] : ( A > A > $o ) ).
thf(sy_c_Orderings_Otop__class_Otop,type,
top_top:
!>[A: $tType] : A ).
thf(sy_c_Parallel__Mirabelle__hykpkoupgu_ONplus,type,
parall1518086719_Nplus: n > n > n ).
thf(sy_c_Parallel__Mirabelle__hykpkoupgu_Opar,type,
parall1899940088le_par: ( product_prod @ dtree @ dtree ) > dtree ).
thf(sy_c_Parallel__Mirabelle__hykpkoupgu_Opar__c,type,
parall1914194347_par_c: ( product_prod @ dtree @ dtree ) > ( set @ ( sum_sum @ t @ ( product_prod @ dtree @ dtree ) ) ) ).
thf(sy_c_Parallel__Mirabelle__hykpkoupgu_Opar__r,type,
parall1914194362_par_r: ( product_prod @ dtree @ dtree ) > n ).
thf(sy_c_Parallel__Mirabelle__hykpkoupgu_Opar__r__rel,type,
parall556292031_r_rel: ( product_prod @ dtree @ dtree ) > ( product_prod @ dtree @ dtree ) > $o ).
thf(sy_c_Product__Type_OPair,type,
product_Pair:
!>[A: $tType,B: $tType] : ( A > B > ( product_prod @ A @ B ) ) ).
thf(sy_c_Product__Type_Oapfst,type,
product_apfst:
!>[A: $tType,C: $tType,B: $tType] : ( ( A > C ) > ( product_prod @ A @ B ) > ( product_prod @ C @ B ) ) ).
thf(sy_c_Product__Type_Oapsnd,type,
product_apsnd:
!>[B: $tType,C: $tType,A: $tType] : ( ( B > C ) > ( product_prod @ A @ B ) > ( product_prod @ A @ C ) ) ).
thf(sy_c_Product__Type_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_Obool_Orec__bool,type,
product_rec_bool:
!>[T: $tType] : ( T > T > $o > T ) ).
thf(sy_c_Product__Type_Oold_Oprod_Orec__prod,type,
product_rec_prod:
!>[A: $tType,B: $tType,T: $tType] : ( ( A > B > T ) > ( product_prod @ A @ B ) > T ) ).
thf(sy_c_Product__Type_Oprod_Ocase__prod,type,
product_case_prod:
!>[A: $tType,B: $tType,C: $tType] : ( ( A > B > C ) > ( product_prod @ A @ B ) > C ) ).
thf(sy_c_Product__Type_Oprod_Ofst,type,
product_fst:
!>[A: $tType,B: $tType] : ( ( product_prod @ A @ B ) > A ) ).
thf(sy_c_Product__Type_Oprod_Osnd,type,
product_snd:
!>[A: $tType,B: $tType] : ( ( product_prod @ A @ B ) > B ) ).
thf(sy_c_Product__Type_Oprod_Oswap,type,
product_swap:
!>[A: $tType,B: $tType] : ( ( product_prod @ A @ B ) > ( product_prod @ B @ A ) ) ).
thf(sy_c_Product__Type_Oscomp,type,
product_scomp:
!>[A: $tType,B: $tType,C: $tType,D: $tType] : ( ( A > ( product_prod @ B @ C ) ) > ( B > C > D ) > A > D ) ).
thf(sy_c_Pure_Otype,type,
type2:
!>[A: $tType] : ( itself @ A ) ).
thf(sy_c_Relation_OPowp,type,
powp:
!>[A: $tType] : ( ( A > $o ) > ( set @ 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_Relation_Orelcompp,type,
relcompp:
!>[A: $tType,B: $tType,C: $tType] : ( ( A > B > $o ) > ( B > C > $o ) > A > C > $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_Sum__Type_OInl,type,
sum_Inl:
!>[A: $tType,B: $tType] : ( A > ( sum_sum @ A @ B ) ) ).
thf(sy_c_Wellfounded_Oaccp,type,
accp:
!>[A: $tType] : ( ( A > A > $o ) > A > $o ) ).
thf(sy_c_Wellfounded_Olex__prod,type,
lex_prod:
!>[A: $tType,B: $tType] : ( ( set @ ( product_prod @ A @ A ) ) > ( set @ ( product_prod @ B @ B ) ) > ( set @ ( product_prod @ ( product_prod @ A @ B ) @ ( product_prod @ A @ B ) ) ) ) ).
thf(sy_c_Wellfounded_Owf,type,
wf:
!>[A: $tType] : ( ( set @ ( product_prod @ A @ A ) ) > $o ) ).
thf(sy_c_Wfrec_Osame__fst,type,
same_fst:
!>[A: $tType,B: $tType] : ( ( A > $o ) > ( A > ( set @ ( product_prod @ B @ B ) ) ) > ( set @ ( product_prod @ ( product_prod @ A @ B ) @ ( product_prod @ A @ B ) ) ) ) ).
thf(sy_c_member,type,
member:
!>[A: $tType] : ( A > ( set @ A ) > $o ) ).
thf(sy_v_tr1,type,
tr1: dtree ).
thf(sy_v_tr2,type,
tr2: dtree ).
thf(sy_v_tr3,type,
tr3: dtree ).
%----Relevant facts (256)
thf(fact_0__092_060open_062_092_060And_062trB_AtrA_O_A_092_060exists_062tr1_Atr2_Atr3_O_AtrA_A_061_A_Itr1_A_092_060parallel_062_Atr2_J_A_092_060parallel_062_Atr3_A_092_060and_062_AtrB_A_061_Atr1_A_092_060parallel_062_Atr2_A_092_060parallel_062_Atr3_A_092_060Longrightarrow_062_AtrA_A_061_AtrB_092_060close_062,axiom,
! [TrA: dtree,TrB: dtree] :
( ? [Tr1: dtree,Tr2: dtree,Tr3: dtree] :
( ( TrA
= ( parall1899940088le_par @ ( product_Pair @ dtree @ dtree @ ( parall1899940088le_par @ ( product_Pair @ dtree @ dtree @ Tr1 @ Tr2 ) ) @ Tr3 ) ) )
& ( TrB
= ( parall1899940088le_par @ ( product_Pair @ dtree @ dtree @ Tr1 @ ( parall1899940088le_par @ ( product_Pair @ dtree @ dtree @ Tr2 @ Tr3 ) ) ) ) ) )
=> ( TrA = TrB ) ) ).
% \<open>\<And>trB trA. \<exists>tr1 tr2 tr3. trA = (tr1 \<parallel> tr2) \<parallel> tr3 \<and> trB = tr1 \<parallel> tr2 \<parallel> tr3 \<Longrightarrow> trA = trB\<close>
thf(fact_1_par__com,axiom,
! [Tr12: dtree,Tr22: dtree] :
( ( parall1899940088le_par @ ( product_Pair @ dtree @ dtree @ Tr12 @ Tr22 ) )
= ( parall1899940088le_par @ ( product_Pair @ dtree @ dtree @ Tr22 @ Tr12 ) ) ) ).
% par_com
thf(fact_2_par__r_Ocases,axiom,
! [X: product_prod @ dtree @ dtree] :
~ ! [Tr13: dtree,Tr23: dtree] :
( X
!= ( product_Pair @ dtree @ dtree @ Tr13 @ Tr23 ) ) ).
% par_r.cases
thf(fact_3_par__r_Oinduct,axiom,
! [P: ( product_prod @ dtree @ dtree ) > $o,A0: product_prod @ dtree @ dtree] :
( ! [Tr13: dtree,Tr23: dtree] : ( P @ ( product_Pair @ dtree @ dtree @ Tr13 @ Tr23 ) )
=> ( P @ A0 ) ) ).
% par_r.induct
thf(fact_4_prod_Oinject,axiom,
! [A: $tType,B: $tType,X1: A,X2: B,Y1: A,Y2: B] :
( ( ( product_Pair @ A @ B @ X1 @ X2 )
= ( product_Pair @ A @ B @ Y1 @ Y2 ) )
= ( ( X1 = Y1 )
& ( X2 = Y2 ) ) ) ).
% prod.inject
thf(fact_5_old_Oprod_Oinject,axiom,
! [A: $tType,B: $tType,A2: A,B2: B,A3: A,B3: B] :
( ( ( product_Pair @ A @ B @ A2 @ B2 )
= ( product_Pair @ A @ B @ A3 @ B3 ) )
= ( ( A2 = A3 )
& ( B2 = B3 ) ) ) ).
% old.prod.inject
thf(fact_6_surj__pair,axiom,
! [A: $tType,B: $tType,P2: product_prod @ A @ B] :
? [X3: A,Y: B] :
( P2
= ( product_Pair @ A @ B @ X3 @ Y ) ) ).
% surj_pair
thf(fact_7_prod__cases,axiom,
! [B: $tType,A: $tType,P: ( product_prod @ A @ B ) > $o,P2: product_prod @ A @ B] :
( ! [A4: A,B4: B] : ( P @ ( product_Pair @ A @ B @ A4 @ B4 ) )
=> ( P @ P2 ) ) ).
% prod_cases
thf(fact_8_Pair__inject,axiom,
! [A: $tType,B: $tType,A2: A,B2: B,A3: A,B3: B] :
( ( ( product_Pair @ A @ B @ A2 @ B2 )
= ( product_Pair @ A @ B @ A3 @ B3 ) )
=> ~ ( ( A2 = A3 )
=> ( B2 != B3 ) ) ) ).
% Pair_inject
thf(fact_9_prod__cases3,axiom,
! [A: $tType,B: $tType,C: $tType,Y3: product_prod @ A @ ( product_prod @ B @ C )] :
~ ! [A4: A,B4: B,C2: C] :
( Y3
!= ( product_Pair @ A @ ( product_prod @ B @ C ) @ A4 @ ( product_Pair @ B @ C @ B4 @ C2 ) ) ) ).
% prod_cases3
thf(fact_10_prod__cases4,axiom,
! [A: $tType,B: $tType,C: $tType,D: $tType,Y3: product_prod @ A @ ( product_prod @ B @ ( product_prod @ C @ D ) )] :
~ ! [A4: A,B4: B,C2: C,D2: D] :
( Y3
!= ( product_Pair @ A @ ( product_prod @ B @ ( product_prod @ C @ D ) ) @ A4 @ ( product_Pair @ B @ ( product_prod @ C @ D ) @ B4 @ ( product_Pair @ C @ D @ C2 @ D2 ) ) ) ) ).
% prod_cases4
thf(fact_11_prod__cases5,axiom,
! [A: $tType,B: $tType,C: $tType,D: $tType,E: $tType,Y3: product_prod @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D @ E ) ) )] :
~ ! [A4: A,B4: B,C2: C,D2: D,E2: E] :
( Y3
!= ( product_Pair @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D @ E ) ) ) @ A4 @ ( product_Pair @ B @ ( product_prod @ C @ ( product_prod @ D @ E ) ) @ B4 @ ( product_Pair @ C @ ( product_prod @ D @ E ) @ C2 @ ( product_Pair @ D @ E @ D2 @ E2 ) ) ) ) ) ).
% prod_cases5
thf(fact_12_prod__cases6,axiom,
! [A: $tType,B: $tType,C: $tType,D: $tType,E: $tType,F: $tType,Y3: product_prod @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D @ ( product_prod @ E @ F ) ) ) )] :
~ ! [A4: A,B4: B,C2: C,D2: D,E2: E,F2: F] :
( Y3
!= ( product_Pair @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D @ ( product_prod @ E @ F ) ) ) ) @ A4 @ ( product_Pair @ B @ ( product_prod @ C @ ( product_prod @ D @ ( product_prod @ E @ F ) ) ) @ B4 @ ( product_Pair @ C @ ( product_prod @ D @ ( product_prod @ E @ F ) ) @ C2 @ ( product_Pair @ D @ ( product_prod @ E @ F ) @ D2 @ ( product_Pair @ E @ F @ E2 @ F2 ) ) ) ) ) ) ).
% prod_cases6
thf(fact_13_prod__cases7,axiom,
! [A: $tType,B: $tType,C: $tType,D: $tType,E: $tType,F: $tType,G: $tType,Y3: product_prod @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D @ ( product_prod @ E @ ( product_prod @ F @ G ) ) ) ) )] :
~ ! [A4: A,B4: B,C2: C,D2: D,E2: E,F2: F,G2: G] :
( Y3
!= ( product_Pair @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D @ ( product_prod @ E @ ( product_prod @ F @ G ) ) ) ) ) @ A4 @ ( product_Pair @ B @ ( product_prod @ C @ ( product_prod @ D @ ( product_prod @ E @ ( product_prod @ F @ G ) ) ) ) @ B4 @ ( product_Pair @ C @ ( product_prod @ D @ ( product_prod @ E @ ( product_prod @ F @ G ) ) ) @ C2 @ ( product_Pair @ D @ ( product_prod @ E @ ( product_prod @ F @ G ) ) @ D2 @ ( product_Pair @ E @ ( product_prod @ F @ G ) @ E2 @ ( product_Pair @ F @ G @ F2 @ G2 ) ) ) ) ) ) ) ).
% prod_cases7
thf(fact_14_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 )] :
( ! [A4: A,B4: B,C2: C] : ( P @ ( product_Pair @ A @ ( product_prod @ B @ C ) @ A4 @ ( product_Pair @ B @ C @ B4 @ C2 ) ) )
=> ( P @ X ) ) ).
% prod_induct3
thf(fact_15_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 ) )] :
( ! [A4: A,B4: B,C2: C,D2: D] : ( P @ ( product_Pair @ A @ ( product_prod @ B @ ( product_prod @ C @ D ) ) @ A4 @ ( product_Pair @ B @ ( product_prod @ C @ D ) @ B4 @ ( product_Pair @ C @ D @ C2 @ D2 ) ) ) )
=> ( P @ X ) ) ).
% prod_induct4
thf(fact_16_old_Oprod_Oinducts,axiom,
! [B: $tType,A: $tType,P: ( product_prod @ A @ B ) > $o,Prod: product_prod @ A @ B] :
( ! [A4: A,B4: B] : ( P @ ( product_Pair @ A @ B @ A4 @ B4 ) )
=> ( P @ Prod ) ) ).
% old.prod.inducts
thf(fact_17_old_Oprod_Oexhaust,axiom,
! [A: $tType,B: $tType,Y3: product_prod @ A @ B] :
~ ! [A4: A,B4: B] :
( Y3
!= ( product_Pair @ A @ B @ A4 @ B4 ) ) ).
% old.prod.exhaust
thf(fact_18_prod__induct7,axiom,
! [G: $tType,F: $tType,E: $tType,D: $tType,C: $tType,B: $tType,A: $tType,P: ( product_prod @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D @ ( product_prod @ E @ ( product_prod @ F @ G ) ) ) ) ) ) > $o,X: product_prod @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D @ ( product_prod @ E @ ( product_prod @ F @ G ) ) ) ) )] :
( ! [A4: A,B4: B,C2: C,D2: D,E2: E,F2: F,G2: G] : ( P @ ( product_Pair @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D @ ( product_prod @ E @ ( product_prod @ F @ G ) ) ) ) ) @ A4 @ ( product_Pair @ B @ ( product_prod @ C @ ( product_prod @ D @ ( product_prod @ E @ ( product_prod @ F @ G ) ) ) ) @ B4 @ ( product_Pair @ C @ ( product_prod @ D @ ( product_prod @ E @ ( product_prod @ F @ G ) ) ) @ C2 @ ( product_Pair @ D @ ( product_prod @ E @ ( product_prod @ F @ G ) ) @ D2 @ ( product_Pair @ E @ ( product_prod @ F @ G ) @ E2 @ ( product_Pair @ F @ G @ F2 @ G2 ) ) ) ) ) ) )
=> ( P @ X ) ) ).
% prod_induct7
thf(fact_19_prod__induct6,axiom,
! [F: $tType,E: $tType,D: $tType,C: $tType,B: $tType,A: $tType,P: ( product_prod @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D @ ( product_prod @ E @ F ) ) ) ) ) > $o,X: product_prod @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D @ ( product_prod @ E @ F ) ) ) )] :
( ! [A4: A,B4: B,C2: C,D2: D,E2: E,F2: F] : ( P @ ( product_Pair @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D @ ( product_prod @ E @ F ) ) ) ) @ A4 @ ( product_Pair @ B @ ( product_prod @ C @ ( product_prod @ D @ ( product_prod @ E @ F ) ) ) @ B4 @ ( product_Pair @ C @ ( product_prod @ D @ ( product_prod @ E @ F ) ) @ C2 @ ( product_Pair @ D @ ( product_prod @ E @ F ) @ D2 @ ( product_Pair @ E @ F @ E2 @ F2 ) ) ) ) ) )
=> ( P @ X ) ) ).
% prod_induct6
thf(fact_20_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 ) ) )] :
( ! [A4: A,B4: B,C2: C,D2: D,E2: E] : ( P @ ( product_Pair @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D @ E ) ) ) @ A4 @ ( product_Pair @ B @ ( product_prod @ C @ ( product_prod @ D @ E ) ) @ B4 @ ( product_Pair @ C @ ( product_prod @ D @ E ) @ C2 @ ( product_Pair @ D @ E @ D2 @ E2 ) ) ) ) )
=> ( P @ X ) ) ).
% prod_induct5
thf(fact_21_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_22_internal__case__prod__conv,axiom,
! [B: $tType,A: $tType,C: $tType,C3: B > C > A,A2: B,B2: C] :
( ( produc2004651681e_prod @ B @ C @ A @ C3 @ ( product_Pair @ B @ C @ A2 @ B2 ) )
= ( C3 @ A2 @ B2 ) ) ).
% internal_case_prod_conv
thf(fact_23_ssubst__Pair__rhs,axiom,
! [B: $tType,A: $tType,R: A,S: B,R2: set @ ( product_prod @ A @ B ),S2: B] :
( ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ R @ S ) @ R2 )
=> ( ( S2 = S )
=> ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ R @ S2 ) @ R2 ) ) ) ).
% ssubst_Pair_rhs
thf(fact_24_curry__conv,axiom,
! [A: $tType,C: $tType,B: $tType] :
( ( product_curry @ B @ C @ A )
= ( ^ [F3: ( product_prod @ B @ C ) > A,A5: B,B5: C] : ( F3 @ ( product_Pair @ B @ C @ A5 @ B5 ) ) ) ) ).
% curry_conv
thf(fact_25_curryI,axiom,
! [A: $tType,B: $tType,F4: ( product_prod @ A @ B ) > $o,A2: A,B2: B] :
( ( F4 @ ( product_Pair @ A @ B @ A2 @ B2 ) )
=> ( product_curry @ A @ B @ $o @ F4 @ A2 @ B2 ) ) ).
% curryI
thf(fact_26_swap__simp,axiom,
! [A: $tType,B: $tType,X: B,Y3: A] :
( ( product_swap @ B @ A @ ( product_Pair @ B @ A @ X @ Y3 ) )
= ( product_Pair @ A @ B @ Y3 @ X ) ) ).
% swap_simp
thf(fact_27_apsnd__conv,axiom,
! [A: $tType,B: $tType,C: $tType,F4: C > B,X: A,Y3: C] :
( ( product_apsnd @ C @ B @ A @ F4 @ ( product_Pair @ A @ C @ X @ Y3 ) )
= ( product_Pair @ A @ B @ X @ ( F4 @ Y3 ) ) ) ).
% apsnd_conv
thf(fact_28_old_Obool_Osimps_I6_J,axiom,
! [T: $tType,F1: T,F22: T] :
( ( product_rec_bool @ T @ F1 @ F22 @ $false )
= F22 ) ).
% old.bool.simps(6)
thf(fact_29_old_Obool_Osimps_I5_J,axiom,
! [T: $tType,F1: T,F22: T] :
( ( product_rec_bool @ T @ F1 @ F22 @ $true )
= F1 ) ).
% old.bool.simps(5)
thf(fact_30_apfst__conv,axiom,
! [C: $tType,A: $tType,B: $tType,F4: C > A,X: C,Y3: B] :
( ( product_apfst @ C @ A @ B @ F4 @ ( product_Pair @ C @ B @ X @ Y3 ) )
= ( product_Pair @ A @ B @ ( F4 @ X ) @ Y3 ) ) ).
% apfst_conv
thf(fact_31_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_32_apsnd__apfst__commute,axiom,
! [A: $tType,B: $tType,C: $tType,D: $tType,F4: C > B,G3: D > A,P2: product_prod @ D @ C] :
( ( product_apsnd @ C @ B @ A @ F4 @ ( product_apfst @ D @ A @ C @ G3 @ P2 ) )
= ( product_apfst @ D @ A @ B @ G3 @ ( product_apsnd @ C @ B @ D @ F4 @ P2 ) ) ) ).
% apsnd_apfst_commute
thf(fact_33_curryD,axiom,
! [A: $tType,B: $tType,F4: ( product_prod @ A @ B ) > $o,A2: A,B2: B] :
( ( product_curry @ A @ B @ $o @ F4 @ A2 @ B2 )
=> ( F4 @ ( product_Pair @ A @ B @ A2 @ B2 ) ) ) ).
% curryD
thf(fact_34_curryE,axiom,
! [A: $tType,B: $tType,F4: ( product_prod @ A @ B ) > $o,A2: A,B2: B] :
( ( product_curry @ A @ B @ $o @ F4 @ A2 @ B2 )
=> ( F4 @ ( product_Pair @ A @ B @ A2 @ B2 ) ) ) ).
% curryE
thf(fact_35_pair__in__swap__image,axiom,
! [A: $tType,B: $tType,Y3: A,X: B,A6: set @ ( product_prod @ B @ A )] :
( ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ Y3 @ X ) @ ( image @ ( product_prod @ B @ A ) @ ( product_prod @ A @ B ) @ ( product_swap @ B @ A ) @ A6 ) )
= ( member @ ( product_prod @ B @ A ) @ ( product_Pair @ B @ A @ X @ Y3 ) @ A6 ) ) ).
% pair_in_swap_image
thf(fact_36_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_37_case__prod__curry,axiom,
! [C: $tType,B: $tType,A: $tType,F4: ( product_prod @ A @ B ) > C] :
( ( product_case_prod @ A @ B @ C @ ( product_curry @ A @ B @ C @ F4 ) )
= F4 ) ).
% case_prod_curry
thf(fact_38_curry__case__prod,axiom,
! [C: $tType,B: $tType,A: $tType,F4: A > B > C] :
( ( product_curry @ A @ B @ C @ ( product_case_prod @ A @ B @ C @ F4 ) )
= F4 ) ).
% curry_case_prod
thf(fact_39_Pair__scomp,axiom,
! [A: $tType,B: $tType,C: $tType,X: C,F4: C > A > B] :
( ( product_scomp @ A @ C @ A @ B @ ( product_Pair @ C @ A @ X ) @ F4 )
= ( F4 @ X ) ) ).
% Pair_scomp
thf(fact_40_same__fstI,axiom,
! [B: $tType,A: $tType,P: A > $o,X: A,Y4: B,Y3: B,R2: A > ( set @ ( product_prod @ B @ B ) )] :
( ( P @ X )
=> ( ( member @ ( product_prod @ B @ B ) @ ( product_Pair @ B @ B @ Y4 @ Y3 ) @ ( R2 @ X ) )
=> ( member @ ( product_prod @ ( product_prod @ A @ B ) @ ( product_prod @ A @ B ) ) @ ( product_Pair @ ( product_prod @ A @ B ) @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X @ Y4 ) @ ( product_Pair @ A @ B @ X @ Y3 ) ) @ ( same_fst @ A @ B @ P @ R2 ) ) ) ) ).
% same_fstI
thf(fact_41_root__par,axiom,
! [Tr12: dtree,Tr22: dtree] :
( ( root @ ( parall1899940088le_par @ ( product_Pair @ dtree @ dtree @ Tr12 @ Tr22 ) ) )
= ( parall1518086719_Nplus @ ( root @ Tr12 ) @ ( root @ Tr22 ) ) ) ).
% root_par
thf(fact_42_in__lex__prod,axiom,
! [A: $tType,B: $tType,A2: A,B2: B,A3: A,B3: B,R: set @ ( product_prod @ A @ A ),S: set @ ( product_prod @ B @ B )] :
( ( member @ ( product_prod @ ( product_prod @ A @ B ) @ ( product_prod @ A @ B ) ) @ ( product_Pair @ ( product_prod @ A @ B ) @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ A2 @ B2 ) @ ( product_Pair @ A @ B @ A3 @ B3 ) ) @ ( lex_prod @ A @ B @ R @ S ) )
= ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A2 @ A3 ) @ R )
| ( ( A2 = A3 )
& ( member @ ( product_prod @ B @ B ) @ ( product_Pair @ B @ B @ B2 @ B3 ) @ S ) ) ) ) ).
% in_lex_prod
thf(fact_43_apfst__apsnd,axiom,
! [A: $tType,B: $tType,D: $tType,C: $tType,F4: C > A,G3: D > B,X: product_prod @ C @ D] :
( ( product_apfst @ C @ A @ B @ F4 @ ( product_apsnd @ D @ B @ C @ G3 @ X ) )
= ( product_Pair @ A @ B @ ( F4 @ ( product_fst @ C @ D @ X ) ) @ ( G3 @ ( product_snd @ C @ D @ X ) ) ) ) ).
% apfst_apsnd
thf(fact_44_scomp__apply,axiom,
! [A: $tType,D: $tType,C: $tType,B: $tType] :
( ( product_scomp @ B @ C @ D @ A )
= ( ^ [F3: B > ( product_prod @ C @ D ),G4: C > D > A,X4: B] : ( product_case_prod @ C @ D @ A @ G4 @ ( F3 @ X4 ) ) ) ) ).
% scomp_apply
thf(fact_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,A6: set @ A] :
( ( collect @ A
@ ^ [X4: A] : ( member @ A @ X4 @ A6 ) )
= A6 ) ).
% Collect_mem_eq
thf(fact_47_Collect__cong,axiom,
! [A: $tType,P: A > $o,Q: A > $o] :
( ! [X3: A] :
( ( P @ X3 )
= ( Q @ X3 ) )
=> ( ( collect @ A @ P )
= ( collect @ A @ Q ) ) ) ).
% Collect_cong
thf(fact_48_ext,axiom,
! [B: $tType,A: $tType,F4: A > B,G3: A > B] :
( ! [X3: A] :
( ( F4 @ X3 )
= ( G3 @ X3 ) )
=> ( F4 = G3 ) ) ).
% ext
thf(fact_49_apfst__eq__conv,axiom,
! [A: $tType,B: $tType,C: $tType,F4: C > A,X: product_prod @ C @ B,G3: C > A] :
( ( ( product_apfst @ C @ A @ B @ F4 @ X )
= ( product_apfst @ C @ A @ B @ G3 @ X ) )
= ( ( F4 @ ( product_fst @ C @ B @ X ) )
= ( G3 @ ( product_fst @ C @ B @ X ) ) ) ) ).
% apfst_eq_conv
thf(fact_50_fst__apfst,axiom,
! [A: $tType,B: $tType,C: $tType,F4: C > A,X: product_prod @ C @ B] :
( ( product_fst @ A @ B @ ( product_apfst @ C @ A @ B @ F4 @ X ) )
= ( F4 @ ( product_fst @ C @ B @ X ) ) ) ).
% fst_apfst
thf(fact_51_snd__apfst,axiom,
! [B: $tType,A: $tType,C: $tType,F4: C > B,X: product_prod @ C @ A] :
( ( product_snd @ B @ A @ ( product_apfst @ C @ B @ A @ F4 @ X ) )
= ( product_snd @ C @ A @ X ) ) ).
% snd_apfst
thf(fact_52_fst__apsnd,axiom,
! [B: $tType,C: $tType,A: $tType,F4: C > B,X: product_prod @ A @ C] :
( ( product_fst @ A @ B @ ( product_apsnd @ C @ B @ A @ F4 @ X ) )
= ( product_fst @ A @ C @ X ) ) ).
% fst_apsnd
thf(fact_53_apsnd__eq__conv,axiom,
! [B: $tType,C: $tType,A: $tType,F4: C > B,X: product_prod @ A @ C,G3: C > B] :
( ( ( product_apsnd @ C @ B @ A @ F4 @ X )
= ( product_apsnd @ C @ B @ A @ G3 @ X ) )
= ( ( F4 @ ( product_snd @ A @ C @ X ) )
= ( G3 @ ( product_snd @ A @ C @ X ) ) ) ) ).
% apsnd_eq_conv
thf(fact_54_snd__apsnd,axiom,
! [A: $tType,C: $tType,B: $tType,F4: C > A,X: product_prod @ B @ C] :
( ( product_snd @ B @ A @ ( product_apsnd @ C @ A @ B @ F4 @ X ) )
= ( F4 @ ( product_snd @ B @ C @ X ) ) ) ).
% snd_apsnd
thf(fact_55_prod_Ocollapse,axiom,
! [B: $tType,A: $tType,Prod: product_prod @ A @ B] :
( ( product_Pair @ A @ B @ ( product_fst @ A @ B @ Prod ) @ ( product_snd @ A @ B @ Prod ) )
= Prod ) ).
% prod.collapse
thf(fact_56_snd__swap,axiom,
! [B: $tType,A: $tType,X: product_prod @ A @ B] :
( ( product_snd @ B @ A @ ( product_swap @ A @ B @ X ) )
= ( product_fst @ A @ B @ X ) ) ).
% snd_swap
thf(fact_57_fst__swap,axiom,
! [A: $tType,B: $tType,X: product_prod @ B @ A] :
( ( product_fst @ A @ B @ ( product_swap @ B @ A @ X ) )
= ( product_snd @ B @ A @ X ) ) ).
% fst_swap
thf(fact_58_apsnd__apfst,axiom,
! [A: $tType,B: $tType,C: $tType,D: $tType,F4: C > B,G3: D > A,X: product_prod @ D @ C] :
( ( product_apsnd @ C @ B @ A @ F4 @ ( product_apfst @ D @ A @ C @ G3 @ X ) )
= ( product_Pair @ A @ B @ ( G3 @ ( product_fst @ D @ C @ X ) ) @ ( F4 @ ( product_snd @ D @ C @ X ) ) ) ) ).
% apsnd_apfst
thf(fact_59_Nplus__comm,axiom,
( parall1518086719_Nplus
= ( ^ [A5: n,B5: n] : ( parall1518086719_Nplus @ B5 @ A5 ) ) ) ).
% Nplus_comm
thf(fact_60_Nplus__assoc,axiom,
! [A2: n,B2: n,C3: n] :
( ( parall1518086719_Nplus @ ( parall1518086719_Nplus @ A2 @ B2 ) @ C3 )
= ( parall1518086719_Nplus @ A2 @ ( parall1518086719_Nplus @ B2 @ C3 ) ) ) ).
% Nplus_assoc
thf(fact_61_surjective__pairing,axiom,
! [B: $tType,A: $tType,T2: product_prod @ A @ B] :
( T2
= ( product_Pair @ A @ B @ ( product_fst @ A @ B @ T2 ) @ ( product_snd @ A @ B @ T2 ) ) ) ).
% surjective_pairing
thf(fact_62_prod_Osplit__sel__asm,axiom,
! [C: $tType,B: $tType,A: $tType,P: C > $o,F4: A > B > C,Prod: product_prod @ A @ B] :
( ( P @ ( product_case_prod @ A @ B @ C @ F4 @ Prod ) )
= ( ~ ( ( Prod
= ( product_Pair @ A @ B @ ( product_fst @ A @ B @ Prod ) @ ( product_snd @ A @ B @ Prod ) ) )
& ~ ( P @ ( F4 @ ( product_fst @ A @ B @ Prod ) @ ( product_snd @ A @ B @ Prod ) ) ) ) ) ) ).
% prod.split_sel_asm
thf(fact_63_Product__Type_OCollect__case__prodD,axiom,
! [B: $tType,A: $tType,X: product_prod @ A @ B,A6: A > B > $o] :
( ( member @ ( product_prod @ A @ B ) @ X @ ( collect @ ( product_prod @ A @ B ) @ ( product_case_prod @ A @ B @ $o @ A6 ) ) )
=> ( A6 @ ( product_fst @ A @ B @ X ) @ ( product_snd @ A @ B @ X ) ) ) ).
% Product_Type.Collect_case_prodD
thf(fact_64_prod_Oexhaust__sel,axiom,
! [B: $tType,A: $tType,Prod: product_prod @ A @ B] :
( Prod
= ( product_Pair @ A @ B @ ( product_fst @ A @ B @ Prod ) @ ( product_snd @ A @ B @ Prod ) ) ) ).
% prod.exhaust_sel
thf(fact_65_prod_Ocase__eq__if,axiom,
! [C: $tType,B: $tType,A: $tType] :
( ( product_case_prod @ A @ B @ C )
= ( ^ [F3: A > B > C,Prod2: product_prod @ A @ B] : ( F3 @ ( product_fst @ A @ B @ Prod2 ) @ ( product_snd @ A @ B @ Prod2 ) ) ) ) ).
% prod.case_eq_if
thf(fact_66_prod_Osplit__sel,axiom,
! [C: $tType,B: $tType,A: $tType,P: C > $o,F4: A > B > C,Prod: product_prod @ A @ B] :
( ( P @ ( product_case_prod @ A @ B @ C @ F4 @ Prod ) )
= ( ( Prod
= ( product_Pair @ A @ B @ ( product_fst @ A @ B @ Prod ) @ ( product_snd @ A @ B @ Prod ) ) )
=> ( P @ ( F4 @ ( product_fst @ A @ B @ Prod ) @ ( product_snd @ A @ B @ Prod ) ) ) ) ) ).
% prod.split_sel
thf(fact_67_case__prod__beta,axiom,
! [A: $tType,C: $tType,B: $tType] :
( ( product_case_prod @ B @ C @ A )
= ( ^ [F3: B > C > A,P3: product_prod @ B @ C] : ( F3 @ ( product_fst @ B @ C @ P3 ) @ ( product_snd @ B @ C @ P3 ) ) ) ) ).
% case_prod_beta
thf(fact_68_prod__eq__iff,axiom,
! [B: $tType,A: $tType] :
( ( ^ [Y5: product_prod @ A @ B,Z: product_prod @ A @ B] : Y5 = Z )
= ( ^ [S3: product_prod @ A @ B,T3: product_prod @ A @ B] :
( ( ( product_fst @ A @ B @ S3 )
= ( product_fst @ A @ B @ T3 ) )
& ( ( product_snd @ A @ B @ S3 )
= ( product_snd @ A @ B @ T3 ) ) ) ) ) ).
% prod_eq_iff
thf(fact_69_prod_Oexpand,axiom,
! [B: $tType,A: $tType,Prod: product_prod @ A @ B,Prod3: product_prod @ A @ B] :
( ( ( ( product_fst @ A @ B @ Prod )
= ( product_fst @ A @ B @ Prod3 ) )
& ( ( product_snd @ A @ B @ Prod )
= ( product_snd @ A @ B @ Prod3 ) ) )
=> ( Prod = Prod3 ) ) ).
% prod.expand
thf(fact_70_scomp__def,axiom,
! [D: $tType,C: $tType,B: $tType,A: $tType] :
( ( product_scomp @ A @ B @ C @ D )
= ( ^ [F3: A > ( product_prod @ B @ C ),G4: B > C > D,X4: A] : ( product_case_prod @ B @ C @ D @ G4 @ ( F3 @ X4 ) ) ) ) ).
% scomp_def
thf(fact_71_prod__eqI,axiom,
! [B: $tType,A: $tType,P2: product_prod @ A @ B,Q2: product_prod @ A @ B] :
( ( ( product_fst @ A @ B @ P2 )
= ( product_fst @ A @ B @ Q2 ) )
=> ( ( ( product_snd @ A @ B @ P2 )
= ( product_snd @ A @ B @ Q2 ) )
=> ( P2 = Q2 ) ) ) ).
% prod_eqI
thf(fact_72_prod_Oswap__def,axiom,
! [B: $tType,A: $tType] :
( ( product_swap @ A @ B )
= ( ^ [P3: product_prod @ A @ B] : ( product_Pair @ B @ A @ ( product_snd @ A @ B @ P3 ) @ ( product_fst @ A @ B @ P3 ) ) ) ) ).
% prod.swap_def
thf(fact_73_snd__conv,axiom,
! [Aa: $tType,A: $tType,X1: Aa,X2: A] :
( ( product_snd @ Aa @ A @ ( product_Pair @ Aa @ A @ X1 @ X2 ) )
= X2 ) ).
% snd_conv
thf(fact_74_snd__eqD,axiom,
! [B: $tType,A: $tType,X: B,Y3: A,A2: A] :
( ( ( product_snd @ B @ A @ ( product_Pair @ B @ A @ X @ Y3 ) )
= A2 )
=> ( Y3 = A2 ) ) ).
% snd_eqD
thf(fact_75_fst__conv,axiom,
! [B: $tType,A: $tType,X1: A,X2: B] :
( ( product_fst @ A @ B @ ( product_Pair @ A @ B @ X1 @ X2 ) )
= X1 ) ).
% fst_conv
thf(fact_76_fst__eqD,axiom,
! [B: $tType,A: $tType,X: A,Y3: B,A2: A] :
( ( ( product_fst @ A @ B @ ( product_Pair @ A @ B @ X @ Y3 ) )
= A2 )
=> ( X = A2 ) ) ).
% fst_eqD
thf(fact_77_mem__case__prodE,axiom,
! [B: $tType,A: $tType,C: $tType,Z2: A,C3: B > C > ( set @ A ),P2: product_prod @ B @ C] :
( ( member @ A @ Z2 @ ( product_case_prod @ B @ C @ ( set @ A ) @ C3 @ P2 ) )
=> ~ ! [X3: B,Y: C] :
( ( P2
= ( product_Pair @ B @ C @ X3 @ Y ) )
=> ~ ( member @ A @ Z2 @ ( C3 @ X3 @ Y ) ) ) ) ).
% mem_case_prodE
thf(fact_78_old_Oprod_Ocase,axiom,
! [A: $tType,C: $tType,B: $tType,F4: A > B > C,X1: A,X2: B] :
( ( product_case_prod @ A @ B @ C @ F4 @ ( product_Pair @ A @ B @ X1 @ X2 ) )
= ( F4 @ X1 @ X2 ) ) ).
% old.prod.case
thf(fact_79_scomp__Pair,axiom,
! [C: $tType,B: $tType,A: $tType,X: A > ( product_prod @ B @ C )] :
( ( product_scomp @ A @ B @ C @ ( product_prod @ B @ C ) @ X @ ( product_Pair @ B @ C ) )
= X ) ).
% scomp_Pair
thf(fact_80_BNF__Greatest__Fixpoint_Osubst__Pair,axiom,
! [B: $tType,A: $tType,P: A > B > $o,X: A,Y3: B,A2: product_prod @ A @ B] :
( ( P @ X @ Y3 )
=> ( ( A2
= ( product_Pair @ A @ B @ X @ Y3 ) )
=> ( P @ ( product_fst @ A @ B @ A2 ) @ ( product_snd @ A @ B @ A2 ) ) ) ) ).
% BNF_Greatest_Fixpoint.subst_Pair
thf(fact_81_conjI__realizer,axiom,
! [A: $tType,B: $tType,P: A > $o,P2: A,Q: B > $o,Q2: B] :
( ( P @ P2 )
=> ( ( Q @ Q2 )
=> ( ( P @ ( product_fst @ A @ B @ ( product_Pair @ A @ B @ P2 @ Q2 ) ) )
& ( Q @ ( product_snd @ A @ B @ ( product_Pair @ A @ B @ P2 @ Q2 ) ) ) ) ) ) ).
% conjI_realizer
thf(fact_82_exI__realizer,axiom,
! [B: $tType,A: $tType,P: A > B > $o,Y3: A,X: B] :
( ( P @ Y3 @ X )
=> ( P @ ( product_snd @ B @ A @ ( product_Pair @ B @ A @ X @ Y3 ) ) @ ( product_fst @ B @ A @ ( product_Pair @ B @ A @ X @ Y3 ) ) ) ) ).
% exI_realizer
thf(fact_83_par__r_Osimps,axiom,
! [Tr12: dtree,Tr22: dtree] :
( ( parall1914194362_par_r @ ( product_Pair @ dtree @ dtree @ Tr12 @ Tr22 ) )
= ( parall1518086719_Nplus @ ( root @ Tr12 ) @ ( root @ Tr22 ) ) ) ).
% par_r.simps
thf(fact_84_par__r_Oelims,axiom,
! [X: product_prod @ dtree @ dtree,Y3: n] :
( ( ( parall1914194362_par_r @ X )
= Y3 )
=> ~ ! [Tr13: dtree,Tr23: dtree] :
( ( X
= ( product_Pair @ dtree @ dtree @ Tr13 @ Tr23 ) )
=> ( Y3
!= ( parall1518086719_Nplus @ ( root @ Tr13 ) @ ( root @ Tr23 ) ) ) ) ) ).
% par_r.elims
thf(fact_85_image__eqI,axiom,
! [A: $tType,B: $tType,B2: A,F4: B > A,X: B,A6: set @ B] :
( ( B2
= ( F4 @ X ) )
=> ( ( member @ B @ X @ A6 )
=> ( member @ A @ B2 @ ( image @ B @ A @ F4 @ A6 ) ) ) ) ).
% image_eqI
thf(fact_86_exE__realizer_H,axiom,
! [A: $tType,B: $tType,P: A > B > $o,P2: product_prod @ B @ A] :
( ( P @ ( product_snd @ B @ A @ P2 ) @ ( product_fst @ B @ A @ P2 ) )
=> ~ ! [X3: B,Y: A] :
~ ( P @ Y @ X3 ) ) ).
% exE_realizer'
thf(fact_87_sndI,axiom,
! [A: $tType,B: $tType,X: product_prod @ A @ B,Y3: A,Z2: B] :
( ( X
= ( product_Pair @ A @ B @ Y3 @ Z2 ) )
=> ( ( product_snd @ A @ B @ X )
= Z2 ) ) ).
% sndI
thf(fact_88_rev__image__eqI,axiom,
! [B: $tType,A: $tType,X: A,A6: set @ A,B2: B,F4: A > B] :
( ( member @ A @ X @ A6 )
=> ( ( B2
= ( F4 @ X ) )
=> ( member @ B @ B2 @ ( image @ A @ B @ F4 @ A6 ) ) ) ) ).
% rev_image_eqI
thf(fact_89_ball__imageD,axiom,
! [A: $tType,B: $tType,F4: B > A,A6: set @ B,P: A > $o] :
( ! [X3: A] :
( ( member @ A @ X3 @ ( image @ B @ A @ F4 @ A6 ) )
=> ( P @ X3 ) )
=> ! [X5: B] :
( ( member @ B @ X5 @ A6 )
=> ( P @ ( F4 @ X5 ) ) ) ) ).
% ball_imageD
thf(fact_90_image__cong,axiom,
! [B: $tType,A: $tType,M: set @ A,N: set @ A,F4: A > B,G3: A > B] :
( ( M = N )
=> ( ! [X3: A] :
( ( member @ A @ X3 @ N )
=> ( ( F4 @ X3 )
= ( G3 @ X3 ) ) )
=> ( ( image @ A @ B @ F4 @ M )
= ( image @ A @ B @ G3 @ N ) ) ) ) ).
% image_cong
thf(fact_91_bex__imageD,axiom,
! [A: $tType,B: $tType,F4: B > A,A6: set @ B,P: A > $o] :
( ? [X5: A] :
( ( member @ A @ X5 @ ( image @ B @ A @ F4 @ A6 ) )
& ( P @ X5 ) )
=> ? [X3: B] :
( ( member @ B @ X3 @ A6 )
& ( P @ ( F4 @ X3 ) ) ) ) ).
% bex_imageD
thf(fact_92_image__iff,axiom,
! [A: $tType,B: $tType,Z2: A,F4: B > A,A6: set @ B] :
( ( member @ A @ Z2 @ ( image @ B @ A @ F4 @ A6 ) )
= ( ? [X4: B] :
( ( member @ B @ X4 @ A6 )
& ( Z2
= ( F4 @ X4 ) ) ) ) ) ).
% image_iff
thf(fact_93_imageI,axiom,
! [B: $tType,A: $tType,X: A,A6: set @ A,F4: A > B] :
( ( member @ A @ X @ A6 )
=> ( member @ B @ ( F4 @ X ) @ ( image @ A @ B @ F4 @ A6 ) ) ) ).
% imageI
thf(fact_94_fstI,axiom,
! [B: $tType,A: $tType,X: product_prod @ A @ B,Y3: A,Z2: B] :
( ( X
= ( product_Pair @ A @ B @ Y3 @ Z2 ) )
=> ( ( product_fst @ A @ B @ X )
= Y3 ) ) ).
% fstI
thf(fact_95_eq__snd__iff,axiom,
! [A: $tType,B: $tType,B2: A,P2: product_prod @ B @ A] :
( ( B2
= ( product_snd @ B @ A @ P2 ) )
= ( ? [A5: B] :
( P2
= ( product_Pair @ B @ A @ A5 @ B2 ) ) ) ) ).
% eq_snd_iff
thf(fact_96_eq__fst__iff,axiom,
! [A: $tType,B: $tType,A2: A,P2: product_prod @ A @ B] :
( ( A2
= ( product_fst @ A @ B @ P2 ) )
= ( ? [B5: B] :
( P2
= ( product_Pair @ A @ B @ A2 @ B5 ) ) ) ) ).
% eq_fst_iff
thf(fact_97_split__cong,axiom,
! [C: $tType,B: $tType,A: $tType,Q2: product_prod @ A @ B,F4: A > B > C,G3: A > B > C,P2: product_prod @ A @ B] :
( ! [X3: A,Y: B] :
( ( ( product_Pair @ A @ B @ X3 @ Y )
= Q2 )
=> ( ( F4 @ X3 @ Y )
= ( G3 @ X3 @ Y ) ) )
=> ( ( P2 = Q2 )
=> ( ( product_case_prod @ A @ B @ C @ F4 @ P2 )
= ( product_case_prod @ A @ B @ C @ G3 @ Q2 ) ) ) ) ).
% split_cong
thf(fact_98_par__r_Opelims,axiom,
! [X: product_prod @ dtree @ dtree,Y3: n] :
( ( ( parall1914194362_par_r @ X )
= Y3 )
=> ( ( accp @ ( product_prod @ dtree @ dtree ) @ parall556292031_r_rel @ X )
=> ~ ! [Tr13: dtree,Tr23: dtree] :
( ( X
= ( product_Pair @ dtree @ dtree @ Tr13 @ Tr23 ) )
=> ( ( Y3
= ( parall1518086719_Nplus @ ( root @ Tr13 ) @ ( root @ Tr23 ) ) )
=> ~ ( accp @ ( product_prod @ dtree @ dtree ) @ parall556292031_r_rel @ ( product_Pair @ dtree @ dtree @ Tr13 @ Tr23 ) ) ) ) ) ) ).
% par_r.pelims
thf(fact_99_Collect__split__mono__strong,axiom,
! [B: $tType,A: $tType,X6: set @ A,A6: set @ ( product_prod @ A @ B ),Y6: set @ B,P: A > B > $o,Q: A > B > $o] :
( ( X6
= ( image @ ( product_prod @ A @ B ) @ A @ ( product_fst @ A @ B ) @ A6 ) )
=> ( ( Y6
= ( image @ ( product_prod @ A @ B ) @ B @ ( product_snd @ A @ B ) @ A6 ) )
=> ( ! [X3: A] :
( ( member @ A @ X3 @ X6 )
=> ! [Xa: B] :
( ( member @ B @ Xa @ Y6 )
=> ( ( P @ X3 @ Xa )
=> ( Q @ X3 @ Xa ) ) ) )
=> ( ( ord_less_eq @ ( set @ ( product_prod @ A @ B ) ) @ A6 @ ( collect @ ( product_prod @ A @ B ) @ ( product_case_prod @ A @ B @ $o @ P ) ) )
=> ( ord_less_eq @ ( set @ ( product_prod @ A @ B ) ) @ A6 @ ( collect @ ( product_prod @ A @ B ) @ ( product_case_prod @ A @ B @ $o @ Q ) ) ) ) ) ) ) ).
% Collect_split_mono_strong
thf(fact_100_fstOp__def,axiom,
! [C: $tType,B: $tType,A: $tType] :
( ( bNF_fstOp @ A @ B @ C )
= ( ^ [P4: A > B > $o,Q3: B > C > $o,Ac: product_prod @ A @ C] : ( product_Pair @ A @ B @ ( product_fst @ A @ C @ Ac ) @ ( bNF_pick_middlep @ A @ B @ C @ P4 @ Q3 @ ( product_fst @ A @ C @ Ac ) @ ( product_snd @ A @ C @ Ac ) ) ) ) ) ).
% fstOp_def
thf(fact_101_subset__antisym,axiom,
! [A: $tType,A6: set @ A,B6: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ A6 @ B6 )
=> ( ( ord_less_eq @ ( set @ A ) @ B6 @ A6 )
=> ( A6 = B6 ) ) ) ).
% subset_antisym
thf(fact_102_subsetI,axiom,
! [A: $tType,A6: set @ A,B6: set @ A] :
( ! [X3: A] :
( ( member @ A @ X3 @ A6 )
=> ( member @ A @ X3 @ B6 ) )
=> ( ord_less_eq @ ( set @ A ) @ A6 @ B6 ) ) ).
% subsetI
thf(fact_103_Collect__mono__iff,axiom,
! [A: $tType,P: A > $o,Q: A > $o] :
( ( ord_less_eq @ ( set @ A ) @ ( collect @ A @ P ) @ ( collect @ A @ Q ) )
= ( ! [X4: A] :
( ( P @ X4 )
=> ( Q @ X4 ) ) ) ) ).
% Collect_mono_iff
thf(fact_104_contra__subsetD,axiom,
! [A: $tType,A6: set @ A,B6: set @ A,C3: A] :
( ( ord_less_eq @ ( set @ A ) @ A6 @ B6 )
=> ( ~ ( member @ A @ C3 @ B6 )
=> ~ ( member @ A @ C3 @ A6 ) ) ) ).
% contra_subsetD
thf(fact_105_set__eq__subset,axiom,
! [A: $tType] :
( ( ^ [Y5: set @ A,Z: set @ A] : Y5 = Z )
= ( ^ [A7: set @ A,B7: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ A7 @ B7 )
& ( ord_less_eq @ ( set @ A ) @ B7 @ A7 ) ) ) ) ).
% set_eq_subset
thf(fact_106_subset__trans,axiom,
! [A: $tType,A6: set @ A,B6: set @ A,C4: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ A6 @ B6 )
=> ( ( ord_less_eq @ ( set @ A ) @ B6 @ C4 )
=> ( ord_less_eq @ ( set @ A ) @ A6 @ C4 ) ) ) ).
% subset_trans
thf(fact_107_Collect__mono,axiom,
! [A: $tType,P: A > $o,Q: A > $o] :
( ! [X3: A] :
( ( P @ X3 )
=> ( Q @ X3 ) )
=> ( ord_less_eq @ ( set @ A ) @ ( collect @ A @ P ) @ ( collect @ A @ Q ) ) ) ).
% Collect_mono
thf(fact_108_subset__refl,axiom,
! [A: $tType,A6: set @ A] : ( ord_less_eq @ ( set @ A ) @ A6 @ A6 ) ).
% subset_refl
thf(fact_109_rev__subsetD,axiom,
! [A: $tType,C3: A,A6: set @ A,B6: set @ A] :
( ( member @ A @ C3 @ A6 )
=> ( ( ord_less_eq @ ( set @ A ) @ A6 @ B6 )
=> ( member @ A @ C3 @ B6 ) ) ) ).
% rev_subsetD
thf(fact_110_subset__iff,axiom,
! [A: $tType] :
( ( ord_less_eq @ ( set @ A ) )
= ( ^ [A7: set @ A,B7: set @ A] :
! [T3: A] :
( ( member @ A @ T3 @ A7 )
=> ( member @ A @ T3 @ B7 ) ) ) ) ).
% subset_iff
thf(fact_111_set__rev__mp,axiom,
! [A: $tType,X: A,A6: set @ A,B6: set @ A] :
( ( member @ A @ X @ A6 )
=> ( ( ord_less_eq @ ( set @ A ) @ A6 @ B6 )
=> ( member @ A @ X @ B6 ) ) ) ).
% set_rev_mp
thf(fact_112_equalityD2,axiom,
! [A: $tType,A6: set @ A,B6: set @ A] :
( ( A6 = B6 )
=> ( ord_less_eq @ ( set @ A ) @ B6 @ A6 ) ) ).
% equalityD2
thf(fact_113_equalityD1,axiom,
! [A: $tType,A6: set @ A,B6: set @ A] :
( ( A6 = B6 )
=> ( ord_less_eq @ ( set @ A ) @ A6 @ B6 ) ) ).
% equalityD1
thf(fact_114_subset__eq,axiom,
! [A: $tType] :
( ( ord_less_eq @ ( set @ A ) )
= ( ^ [A7: set @ A,B7: set @ A] :
! [X4: A] :
( ( member @ A @ X4 @ A7 )
=> ( member @ A @ X4 @ B7 ) ) ) ) ).
% subset_eq
thf(fact_115_equalityE,axiom,
! [A: $tType,A6: set @ A,B6: set @ A] :
( ( A6 = B6 )
=> ~ ( ( ord_less_eq @ ( set @ A ) @ A6 @ B6 )
=> ~ ( ord_less_eq @ ( set @ A ) @ B6 @ A6 ) ) ) ).
% equalityE
thf(fact_116_subsetCE,axiom,
! [A: $tType,A6: set @ A,B6: set @ A,C3: A] :
( ( ord_less_eq @ ( set @ A ) @ A6 @ B6 )
=> ( ( member @ A @ C3 @ A6 )
=> ( member @ A @ C3 @ B6 ) ) ) ).
% subsetCE
thf(fact_117_subsetD,axiom,
! [A: $tType,A6: set @ A,B6: set @ A,C3: A] :
( ( ord_less_eq @ ( set @ A ) @ A6 @ B6 )
=> ( ( member @ A @ C3 @ A6 )
=> ( member @ A @ C3 @ B6 ) ) ) ).
% subsetD
thf(fact_118_in__mono,axiom,
! [A: $tType,A6: set @ A,B6: set @ A,X: A] :
( ( ord_less_eq @ ( set @ A ) @ A6 @ B6 )
=> ( ( member @ A @ X @ A6 )
=> ( member @ A @ X @ B6 ) ) ) ).
% in_mono
thf(fact_119_set__mp,axiom,
! [A: $tType,A6: set @ A,B6: set @ A,X: A] :
( ( ord_less_eq @ ( set @ A ) @ A6 @ B6 )
=> ( ( member @ A @ X @ A6 )
=> ( member @ A @ X @ B6 ) ) ) ).
% set_mp
thf(fact_120_accp__induct__rule,axiom,
! [A: $tType,R: A > A > $o,A2: A,P: A > $o] :
( ( accp @ A @ R @ A2 )
=> ( ! [X3: A] :
( ( accp @ A @ R @ X3 )
=> ( ! [Y7: A] :
( ( R @ Y7 @ X3 )
=> ( P @ Y7 ) )
=> ( P @ X3 ) ) )
=> ( P @ A2 ) ) ) ).
% accp_induct_rule
thf(fact_121_not__accp__down,axiom,
! [A: $tType,R2: A > A > $o,X: A] :
( ~ ( accp @ A @ R2 @ X )
=> ~ ! [Z3: A] :
( ( R2 @ Z3 @ X )
=> ( accp @ A @ R2 @ Z3 ) ) ) ).
% not_accp_down
thf(fact_122_accp__downward,axiom,
! [A: $tType,R: A > A > $o,B2: A,A2: A] :
( ( accp @ A @ R @ B2 )
=> ( ( R @ A2 @ B2 )
=> ( accp @ A @ R @ A2 ) ) ) ).
% accp_downward
thf(fact_123_accp_Oinducts,axiom,
! [A: $tType,R: A > A > $o,X: A,P: A > $o] :
( ( accp @ A @ R @ X )
=> ( ! [X3: A] :
( ! [Y7: A] :
( ( R @ Y7 @ X3 )
=> ( accp @ A @ R @ Y7 ) )
=> ( ! [Y7: A] :
( ( R @ Y7 @ X3 )
=> ( P @ Y7 ) )
=> ( P @ X3 ) ) )
=> ( P @ X ) ) ) ).
% accp.inducts
thf(fact_124_accp__induct,axiom,
! [A: $tType,R: A > A > $o,A2: A,P: A > $o] :
( ( accp @ A @ R @ A2 )
=> ( ! [X3: A] :
( ( accp @ A @ R @ X3 )
=> ( ! [Y7: A] :
( ( R @ Y7 @ X3 )
=> ( P @ Y7 ) )
=> ( P @ X3 ) ) )
=> ( P @ A2 ) ) ) ).
% accp_induct
thf(fact_125_accp_Ointros,axiom,
! [A: $tType,R: A > A > $o,X: A] :
( ! [Y: A] :
( ( R @ Y @ X )
=> ( accp @ A @ R @ Y ) )
=> ( accp @ A @ R @ X ) ) ).
% accp.intros
thf(fact_126_accp_Osimps,axiom,
! [A: $tType] :
( ( accp @ A )
= ( ^ [R3: A > A > $o,A5: A] :
? [X4: A] :
( ( A5 = X4 )
& ! [Y8: A] :
( ( R3 @ Y8 @ X4 )
=> ( accp @ A @ R3 @ Y8 ) ) ) ) ) ).
% accp.simps
thf(fact_127_accp_Ocases,axiom,
! [A: $tType,R: A > A > $o,A2: A] :
( ( accp @ A @ R @ A2 )
=> ! [Y7: A] :
( ( R @ Y7 @ A2 )
=> ( accp @ A @ R @ Y7 ) ) ) ).
% accp.cases
thf(fact_128_subset__image__iff,axiom,
! [A: $tType,B: $tType,B6: set @ A,F4: B > A,A6: set @ B] :
( ( ord_less_eq @ ( set @ A ) @ B6 @ ( image @ B @ A @ F4 @ A6 ) )
= ( ? [AA: set @ B] :
( ( ord_less_eq @ ( set @ B ) @ AA @ A6 )
& ( B6
= ( image @ B @ A @ F4 @ AA ) ) ) ) ) ).
% subset_image_iff
thf(fact_129_image__subset__iff,axiom,
! [A: $tType,B: $tType,F4: B > A,A6: set @ B,B6: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ ( image @ B @ A @ F4 @ A6 ) @ B6 )
= ( ! [X4: B] :
( ( member @ B @ X4 @ A6 )
=> ( member @ A @ ( F4 @ X4 ) @ B6 ) ) ) ) ).
% image_subset_iff
thf(fact_130_subset__imageE,axiom,
! [A: $tType,B: $tType,B6: set @ A,F4: B > A,A6: set @ B] :
( ( ord_less_eq @ ( set @ A ) @ B6 @ ( image @ B @ A @ F4 @ A6 ) )
=> ~ ! [C5: set @ B] :
( ( ord_less_eq @ ( set @ B ) @ C5 @ A6 )
=> ( B6
!= ( image @ B @ A @ F4 @ C5 ) ) ) ) ).
% subset_imageE
thf(fact_131_image__subsetI,axiom,
! [A: $tType,B: $tType,A6: set @ A,F4: A > B,B6: set @ B] :
( ! [X3: A] :
( ( member @ A @ X3 @ A6 )
=> ( member @ B @ ( F4 @ X3 ) @ B6 ) )
=> ( ord_less_eq @ ( set @ B ) @ ( image @ A @ B @ F4 @ A6 ) @ B6 ) ) ).
% image_subsetI
thf(fact_132_image__mono,axiom,
! [B: $tType,A: $tType,A6: set @ A,B6: set @ A,F4: A > B] :
( ( ord_less_eq @ ( set @ A ) @ A6 @ B6 )
=> ( ord_less_eq @ ( set @ B ) @ ( image @ A @ B @ F4 @ A6 ) @ ( image @ A @ B @ F4 @ B6 ) ) ) ).
% image_mono
thf(fact_133_sndOp__def,axiom,
! [B: $tType,A: $tType,C: $tType] :
( ( bNF_sndOp @ C @ A @ B )
= ( ^ [P4: C > A > $o,Q3: A > B > $o,Ac: product_prod @ C @ B] : ( product_Pair @ A @ B @ ( bNF_pick_middlep @ C @ A @ B @ P4 @ Q3 @ ( product_fst @ C @ B @ Ac ) @ ( product_snd @ C @ B @ Ac ) ) @ ( product_snd @ C @ B @ Ac ) ) ) ) ).
% sndOp_def
thf(fact_134_order__refl,axiom,
! [A: $tType] :
( ( preorder @ A @ ( type2 @ A ) )
=> ! [X: A] : ( ord_less_eq @ A @ X @ X ) ) ).
% order_refl
thf(fact_135_subrelI,axiom,
! [B: $tType,A: $tType,R: set @ ( product_prod @ A @ B ),S: set @ ( product_prod @ A @ B )] :
( ! [X3: A,Y: B] :
( ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X3 @ Y ) @ R )
=> ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X3 @ Y ) @ S ) )
=> ( ord_less_eq @ ( set @ ( product_prod @ A @ B ) ) @ R @ S ) ) ).
% subrelI
thf(fact_136_relChain__def,axiom,
! [B: $tType,A: $tType] :
( ( ord @ B @ ( type2 @ B ) )
=> ( ( bNF_Ca1785829860lChain @ A @ B )
= ( ^ [R3: set @ ( product_prod @ A @ A ),As: A > B] :
! [I: A,J: A] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ I @ J ) @ R3 )
=> ( ord_less_eq @ B @ ( As @ I ) @ ( As @ J ) ) ) ) ) ) ).
% relChain_def
thf(fact_137_Collect__case__prod__in__rel__leE,axiom,
! [B: $tType,A: $tType,X6: set @ ( product_prod @ A @ B ),Y6: set @ ( product_prod @ A @ B )] :
( ( ord_less_eq @ ( set @ ( product_prod @ A @ B ) ) @ X6 @ ( collect @ ( product_prod @ A @ B ) @ ( product_case_prod @ A @ B @ $o @ ( fun_in_rel @ A @ B @ Y6 ) ) ) )
=> ( ord_less_eq @ ( set @ ( product_prod @ A @ B ) ) @ X6 @ Y6 ) ) ).
% Collect_case_prod_in_rel_leE
thf(fact_138_accp__subset,axiom,
! [A: $tType,R1: A > A > $o,R22: A > A > $o] :
( ( ord_less_eq @ ( A > A > $o ) @ R1 @ R22 )
=> ( ord_less_eq @ ( A > $o ) @ ( accp @ A @ R22 ) @ ( accp @ A @ R1 ) ) ) ).
% accp_subset
thf(fact_139_accp__subset__induct,axiom,
! [A: $tType,D3: A > $o,R2: A > A > $o,X: A,P: A > $o] :
( ( ord_less_eq @ ( A > $o ) @ D3 @ ( accp @ A @ R2 ) )
=> ( ! [X3: A,Z3: A] :
( ( D3 @ X3 )
=> ( ( R2 @ Z3 @ X3 )
=> ( D3 @ Z3 ) ) )
=> ( ( D3 @ X )
=> ( ! [X3: A] :
( ( D3 @ X3 )
=> ( ! [Z4: A] :
( ( R2 @ Z4 @ X3 )
=> ( P @ Z4 ) )
=> ( P @ X3 ) ) )
=> ( P @ X ) ) ) ) ) ).
% accp_subset_induct
thf(fact_140_in__rel__def,axiom,
! [B: $tType,A: $tType] :
( ( fun_in_rel @ A @ B )
= ( ^ [R4: set @ ( product_prod @ A @ B ),X4: A,Y8: B] : ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X4 @ Y8 ) @ R4 ) ) ) ).
% in_rel_def
thf(fact_141_in__rel__Collect__case__prod__eq,axiom,
! [B: $tType,A: $tType,X6: A > B > $o] :
( ( fun_in_rel @ A @ B @ ( collect @ ( product_prod @ A @ B ) @ ( product_case_prod @ A @ B @ $o @ X6 ) ) )
= X6 ) ).
% in_rel_Collect_case_prod_eq
thf(fact_142_Collect__case__prod__mono,axiom,
! [B: $tType,A: $tType,A6: A > B > $o,B6: A > B > $o] :
( ( ord_less_eq @ ( A > B > $o ) @ A6 @ B6 )
=> ( ord_less_eq @ ( set @ ( product_prod @ A @ B ) ) @ ( collect @ ( product_prod @ A @ B ) @ ( product_case_prod @ A @ B @ $o @ A6 ) ) @ ( collect @ ( product_prod @ A @ B ) @ ( product_case_prod @ A @ B @ $o @ B6 ) ) ) ) ).
% Collect_case_prod_mono
thf(fact_143_dual__order_Oantisym,axiom,
! [A: $tType] :
( ( order @ A @ ( type2 @ A ) )
=> ! [B2: A,A2: A] :
( ( ord_less_eq @ A @ B2 @ A2 )
=> ( ( ord_less_eq @ A @ A2 @ B2 )
=> ( A2 = B2 ) ) ) ) ).
% dual_order.antisym
thf(fact_144_dual__order_Otrans,axiom,
! [A: $tType] :
( ( order @ A @ ( type2 @ A ) )
=> ! [B2: A,A2: A,C3: A] :
( ( ord_less_eq @ A @ B2 @ A2 )
=> ( ( ord_less_eq @ A @ C3 @ B2 )
=> ( ord_less_eq @ A @ C3 @ A2 ) ) ) ) ).
% dual_order.trans
thf(fact_145_linorder__wlog,axiom,
! [A: $tType] :
( ( linorder @ A @ ( type2 @ A ) )
=> ! [P: A > A > $o,A2: A,B2: A] :
( ! [A4: A,B4: A] :
( ( ord_less_eq @ A @ A4 @ B4 )
=> ( P @ A4 @ B4 ) )
=> ( ! [A4: A,B4: A] :
( ( P @ B4 @ A4 )
=> ( P @ A4 @ B4 ) )
=> ( P @ A2 @ B2 ) ) ) ) ).
% linorder_wlog
thf(fact_146_dual__order_Orefl,axiom,
! [A: $tType] :
( ( order @ A @ ( type2 @ A ) )
=> ! [A2: A] : ( ord_less_eq @ A @ A2 @ A2 ) ) ).
% dual_order.refl
thf(fact_147_order__trans,axiom,
! [A: $tType] :
( ( preorder @ A @ ( type2 @ A ) )
=> ! [X: A,Y3: A,Z2: A] :
( ( ord_less_eq @ A @ X @ Y3 )
=> ( ( ord_less_eq @ A @ Y3 @ Z2 )
=> ( ord_less_eq @ A @ X @ Z2 ) ) ) ) ).
% order_trans
thf(fact_148_order__class_Oorder_Oantisym,axiom,
! [A: $tType] :
( ( order @ A @ ( type2 @ A ) )
=> ! [A2: A,B2: A] :
( ( ord_less_eq @ A @ A2 @ B2 )
=> ( ( ord_less_eq @ A @ B2 @ A2 )
=> ( A2 = B2 ) ) ) ) ).
% order_class.order.antisym
thf(fact_149_ord__le__eq__trans,axiom,
! [A: $tType] :
( ( ord @ A @ ( type2 @ A ) )
=> ! [A2: A,B2: A,C3: A] :
( ( ord_less_eq @ A @ A2 @ B2 )
=> ( ( B2 = C3 )
=> ( ord_less_eq @ A @ A2 @ C3 ) ) ) ) ).
% ord_le_eq_trans
thf(fact_150_ord__eq__le__trans,axiom,
! [A: $tType] :
( ( ord @ A @ ( type2 @ A ) )
=> ! [A2: A,B2: A,C3: A] :
( ( A2 = B2 )
=> ( ( ord_less_eq @ A @ B2 @ C3 )
=> ( ord_less_eq @ A @ A2 @ C3 ) ) ) ) ).
% ord_eq_le_trans
thf(fact_151_antisym__conv,axiom,
! [A: $tType] :
( ( order @ A @ ( type2 @ A ) )
=> ! [Y3: A,X: A] :
( ( ord_less_eq @ A @ Y3 @ X )
=> ( ( ord_less_eq @ A @ X @ Y3 )
= ( X = Y3 ) ) ) ) ).
% antisym_conv
thf(fact_152_le__cases3,axiom,
! [A: $tType] :
( ( linorder @ A @ ( type2 @ A ) )
=> ! [X: A,Y3: A,Z2: A] :
( ( ( ord_less_eq @ A @ X @ Y3 )
=> ~ ( ord_less_eq @ A @ Y3 @ Z2 ) )
=> ( ( ( ord_less_eq @ A @ Y3 @ X )
=> ~ ( ord_less_eq @ A @ X @ Z2 ) )
=> ( ( ( ord_less_eq @ A @ X @ Z2 )
=> ~ ( ord_less_eq @ A @ Z2 @ Y3 ) )
=> ( ( ( ord_less_eq @ A @ Z2 @ Y3 )
=> ~ ( ord_less_eq @ A @ Y3 @ X ) )
=> ( ( ( ord_less_eq @ A @ Y3 @ Z2 )
=> ~ ( ord_less_eq @ A @ Z2 @ X ) )
=> ~ ( ( ord_less_eq @ A @ Z2 @ X )
=> ~ ( ord_less_eq @ A @ X @ Y3 ) ) ) ) ) ) ) ) ).
% le_cases3
thf(fact_153_order_Otrans,axiom,
! [A: $tType] :
( ( order @ A @ ( type2 @ A ) )
=> ! [A2: A,B2: A,C3: A] :
( ( ord_less_eq @ A @ A2 @ B2 )
=> ( ( ord_less_eq @ A @ B2 @ C3 )
=> ( ord_less_eq @ A @ A2 @ C3 ) ) ) ) ).
% order.trans
thf(fact_154_le__cases,axiom,
! [A: $tType] :
( ( linorder @ A @ ( type2 @ A ) )
=> ! [X: A,Y3: A] :
( ~ ( ord_less_eq @ A @ X @ Y3 )
=> ( ord_less_eq @ A @ Y3 @ X ) ) ) ).
% le_cases
thf(fact_155_eq__refl,axiom,
! [A: $tType] :
( ( preorder @ A @ ( type2 @ A ) )
=> ! [X: A,Y3: A] :
( ( X = Y3 )
=> ( ord_less_eq @ A @ X @ Y3 ) ) ) ).
% eq_refl
thf(fact_156_linear,axiom,
! [A: $tType] :
( ( linorder @ A @ ( type2 @ A ) )
=> ! [X: A,Y3: A] :
( ( ord_less_eq @ A @ X @ Y3 )
| ( ord_less_eq @ A @ Y3 @ X ) ) ) ).
% linear
thf(fact_157_antisym,axiom,
! [A: $tType] :
( ( order @ A @ ( type2 @ A ) )
=> ! [X: A,Y3: A] :
( ( ord_less_eq @ A @ X @ Y3 )
=> ( ( ord_less_eq @ A @ Y3 @ X )
=> ( X = Y3 ) ) ) ) ).
% antisym
thf(fact_158_eq__iff,axiom,
! [A: $tType] :
( ( order @ A @ ( type2 @ A ) )
=> ( ( ^ [Y5: A,Z: A] : Y5 = Z )
= ( ^ [X4: A,Y8: A] :
( ( ord_less_eq @ A @ X4 @ Y8 )
& ( ord_less_eq @ A @ Y8 @ X4 ) ) ) ) ) ).
% eq_iff
thf(fact_159_ord__le__eq__subst,axiom,
! [A: $tType,B: $tType] :
( ( ( ord @ B @ ( type2 @ B ) )
& ( ord @ A @ ( type2 @ A ) ) )
=> ! [A2: A,B2: A,F4: A > B,C3: B] :
( ( ord_less_eq @ A @ A2 @ B2 )
=> ( ( ( F4 @ B2 )
= C3 )
=> ( ! [X3: A,Y: A] :
( ( ord_less_eq @ A @ X3 @ Y )
=> ( ord_less_eq @ B @ ( F4 @ X3 ) @ ( F4 @ Y ) ) )
=> ( ord_less_eq @ B @ ( F4 @ A2 ) @ C3 ) ) ) ) ) ).
% ord_le_eq_subst
thf(fact_160_ord__eq__le__subst,axiom,
! [A: $tType,B: $tType] :
( ( ( ord @ B @ ( type2 @ B ) )
& ( ord @ A @ ( type2 @ A ) ) )
=> ! [A2: A,F4: B > A,B2: B,C3: B] :
( ( A2
= ( F4 @ B2 ) )
=> ( ( ord_less_eq @ B @ B2 @ C3 )
=> ( ! [X3: B,Y: B] :
( ( ord_less_eq @ B @ X3 @ Y )
=> ( ord_less_eq @ A @ ( F4 @ X3 ) @ ( F4 @ Y ) ) )
=> ( ord_less_eq @ A @ A2 @ ( F4 @ C3 ) ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_161_order__subst2,axiom,
! [A: $tType,C: $tType] :
( ( ( order @ C @ ( type2 @ C ) )
& ( order @ A @ ( type2 @ A ) ) )
=> ! [A2: A,B2: A,F4: A > C,C3: C] :
( ( ord_less_eq @ A @ A2 @ B2 )
=> ( ( ord_less_eq @ C @ ( F4 @ B2 ) @ C3 )
=> ( ! [X3: A,Y: A] :
( ( ord_less_eq @ A @ X3 @ Y )
=> ( ord_less_eq @ C @ ( F4 @ X3 ) @ ( F4 @ Y ) ) )
=> ( ord_less_eq @ C @ ( F4 @ A2 ) @ C3 ) ) ) ) ) ).
% order_subst2
thf(fact_162_order__subst1,axiom,
! [A: $tType,B: $tType] :
( ( ( order @ B @ ( type2 @ B ) )
& ( order @ A @ ( type2 @ A ) ) )
=> ! [A2: A,F4: B > A,B2: B,C3: B] :
( ( ord_less_eq @ A @ A2 @ ( F4 @ B2 ) )
=> ( ( ord_less_eq @ B @ B2 @ C3 )
=> ( ! [X3: B,Y: B] :
( ( ord_less_eq @ B @ X3 @ Y )
=> ( ord_less_eq @ A @ ( F4 @ X3 ) @ ( F4 @ Y ) ) )
=> ( ord_less_eq @ A @ A2 @ ( F4 @ C3 ) ) ) ) ) ) ).
% order_subst1
thf(fact_163_le__fun__def,axiom,
! [B: $tType,A: $tType] :
( ( ord @ B @ ( type2 @ B ) )
=> ( ( ord_less_eq @ ( A > B ) )
= ( ^ [F3: A > B,G4: A > B] :
! [X4: A] : ( ord_less_eq @ B @ ( F3 @ X4 ) @ ( G4 @ X4 ) ) ) ) ) ).
% le_fun_def
thf(fact_164_le__funI,axiom,
! [B: $tType,A: $tType] :
( ( ord @ B @ ( type2 @ B ) )
=> ! [F4: A > B,G3: A > B] :
( ! [X3: A] : ( ord_less_eq @ B @ ( F4 @ X3 ) @ ( G3 @ X3 ) )
=> ( ord_less_eq @ ( A > B ) @ F4 @ G3 ) ) ) ).
% le_funI
thf(fact_165_le__funE,axiom,
! [B: $tType,A: $tType] :
( ( ord @ B @ ( type2 @ B ) )
=> ! [F4: A > B,G3: A > B,X: A] :
( ( ord_less_eq @ ( A > B ) @ F4 @ G3 )
=> ( ord_less_eq @ B @ ( F4 @ X ) @ ( G3 @ X ) ) ) ) ).
% le_funE
thf(fact_166_le__funD,axiom,
! [B: $tType,A: $tType] :
( ( ord @ B @ ( type2 @ B ) )
=> ! [F4: A > B,G3: A > B,X: A] :
( ( ord_less_eq @ ( A > B ) @ F4 @ G3 )
=> ( ord_less_eq @ B @ ( F4 @ X ) @ ( G3 @ X ) ) ) ) ).
% le_funD
thf(fact_167_Collect__case__prod__in__rel__leI,axiom,
! [B: $tType,A: $tType,X6: set @ ( product_prod @ A @ B ),Y6: set @ ( product_prod @ A @ B )] :
( ( ord_less_eq @ ( set @ ( product_prod @ A @ B ) ) @ X6 @ Y6 )
=> ( ord_less_eq @ ( set @ ( product_prod @ A @ B ) ) @ X6 @ ( collect @ ( product_prod @ A @ B ) @ ( product_case_prod @ A @ B @ $o @ ( fun_in_rel @ A @ B @ Y6 ) ) ) ) ) ).
% Collect_case_prod_in_rel_leI
thf(fact_168_csquare__fstOp__sndOp,axiom,
! [A: $tType,B: $tType,C: $tType,F4: ( A > B > $o ) > ( product_prod @ A @ B ) > $o,P: A > C > $o,Q: C > B > $o] : ( bNF_csquare @ ( product_prod @ A @ B ) @ ( product_prod @ A @ C ) @ C @ ( product_prod @ C @ B ) @ ( collect @ ( product_prod @ A @ B ) @ ( F4 @ ( relcompp @ A @ C @ B @ P @ Q ) ) ) @ ( product_snd @ A @ C ) @ ( product_fst @ C @ B ) @ ( bNF_fstOp @ A @ C @ B @ P @ Q ) @ ( bNF_sndOp @ A @ C @ B @ P @ Q ) ) ).
% csquare_fstOp_sndOp
thf(fact_169_image2__eqI,axiom,
! [A: $tType,C: $tType,B: $tType,B2: A,F4: B > A,X: B,C3: C,G3: B > C,A6: set @ B] :
( ( B2
= ( F4 @ X ) )
=> ( ( C3
= ( G3 @ X ) )
=> ( ( member @ B @ X @ A6 )
=> ( member @ ( product_prod @ A @ C ) @ ( product_Pair @ A @ C @ B2 @ C3 ) @ ( bNF_Greatest_image2 @ B @ A @ C @ A6 @ F4 @ G3 ) ) ) ) ) ).
% image2_eqI
thf(fact_170_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_171_par__def,axiom,
( parall1899940088le_par
= ( unfold @ ( product_prod @ dtree @ dtree ) @ parall1914194362_par_r @ parall1914194347_par_c ) ) ).
% par_def
thf(fact_172_UNIV__I,axiom,
! [A: $tType,X: A] : ( member @ A @ X @ ( top_top @ ( set @ A ) ) ) ).
% UNIV_I
thf(fact_173_predicate2I,axiom,
! [B: $tType,A: $tType,P: A > B > $o,Q: A > B > $o] :
( ! [X3: A,Y: B] :
( ( P @ X3 @ Y )
=> ( Q @ X3 @ Y ) )
=> ( ord_less_eq @ ( A > B > $o ) @ P @ Q ) ) ).
% predicate2I
thf(fact_174_predicate1I,axiom,
! [A: $tType,P: A > $o,Q: A > $o] :
( ! [X3: A] :
( ( P @ X3 )
=> ( Q @ X3 ) )
=> ( ord_less_eq @ ( A > $o ) @ P @ Q ) ) ).
% predicate1I
thf(fact_175_range__fst,axiom,
! [B: $tType,A: $tType] :
( ( image @ ( product_prod @ A @ B ) @ A @ ( product_fst @ A @ B ) @ ( top_top @ ( set @ ( product_prod @ A @ B ) ) ) )
= ( top_top @ ( set @ A ) ) ) ).
% range_fst
thf(fact_176_range__snd,axiom,
! [B: $tType,A: $tType] :
( ( image @ ( product_prod @ B @ A ) @ A @ ( product_snd @ B @ A ) @ ( top_top @ ( set @ ( product_prod @ B @ A ) ) ) )
= ( top_top @ ( set @ A ) ) ) ).
% range_snd
thf(fact_177_refl__ge__eq,axiom,
! [A: $tType,R2: A > A > $o] :
( ! [X3: A] : ( R2 @ X3 @ X3 )
=> ( ord_less_eq @ ( A > A > $o )
@ ^ [Y5: A,Z: A] : Y5 = Z
@ R2 ) ) ).
% refl_ge_eq
thf(fact_178_ge__eq__refl,axiom,
! [A: $tType,R2: A > A > $o,X: A] :
( ( ord_less_eq @ ( A > A > $o )
@ ^ [Y5: A,Z: A] : Y5 = Z
@ R2 )
=> ( R2 @ X @ X ) ) ).
% ge_eq_refl
thf(fact_179_leq__OOI,axiom,
! [A: $tType,R2: A > A > $o] :
( ( R2
= ( ^ [Y5: A,Z: A] : Y5 = Z ) )
=> ( ord_less_eq @ ( A > A > $o ) @ R2 @ ( relcompp @ A @ A @ A @ R2 @ R2 ) ) ) ).
% leq_OOI
thf(fact_180_predicate1D,axiom,
! [A: $tType,P: A > $o,Q: A > $o,X: A] :
( ( ord_less_eq @ ( A > $o ) @ P @ Q )
=> ( ( P @ X )
=> ( Q @ X ) ) ) ).
% predicate1D
thf(fact_181_predicate2D,axiom,
! [A: $tType,B: $tType,P: A > B > $o,Q: A > B > $o,X: A,Y3: B] :
( ( ord_less_eq @ ( A > B > $o ) @ P @ Q )
=> ( ( P @ X @ Y3 )
=> ( Q @ X @ Y3 ) ) ) ).
% predicate2D
thf(fact_182_relcompp__mono,axiom,
! [A: $tType,C: $tType,B: $tType,R5: A > B > $o,R: A > B > $o,S2: B > C > $o,S: B > C > $o] :
( ( ord_less_eq @ ( A > B > $o ) @ R5 @ R )
=> ( ( ord_less_eq @ ( B > C > $o ) @ S2 @ S )
=> ( ord_less_eq @ ( A > C > $o ) @ ( relcompp @ A @ B @ C @ R5 @ S2 ) @ ( relcompp @ A @ B @ C @ R @ S ) ) ) ) ).
% relcompp_mono
thf(fact_183_rev__predicate1D,axiom,
! [A: $tType,P: A > $o,X: A,Q: A > $o] :
( ( P @ X )
=> ( ( ord_less_eq @ ( A > $o ) @ P @ Q )
=> ( Q @ X ) ) ) ).
% rev_predicate1D
thf(fact_184_rev__predicate2D,axiom,
! [A: $tType,B: $tType,P: A > B > $o,X: A,Y3: B,Q: A > B > $o] :
( ( P @ X @ Y3 )
=> ( ( ord_less_eq @ ( A > B > $o ) @ P @ Q )
=> ( Q @ X @ Y3 ) ) ) ).
% rev_predicate2D
thf(fact_185_top__greatest,axiom,
! [A: $tType] :
( ( order_top @ A @ ( type2 @ A ) )
=> ! [A2: A] : ( ord_less_eq @ A @ A2 @ ( top_top @ A ) ) ) ).
% top_greatest
thf(fact_186_top_Oextremum__unique,axiom,
! [A: $tType] :
( ( order_top @ A @ ( type2 @ A ) )
=> ! [A2: A] :
( ( ord_less_eq @ A @ ( top_top @ A ) @ A2 )
= ( A2
= ( top_top @ A ) ) ) ) ).
% top.extremum_unique
thf(fact_187_top_Oextremum__uniqueI,axiom,
! [A: $tType] :
( ( order_top @ A @ ( type2 @ A ) )
=> ! [A2: A] :
( ( ord_less_eq @ A @ ( top_top @ A ) @ A2 )
=> ( A2
= ( top_top @ A ) ) ) ) ).
% top.extremum_uniqueI
thf(fact_188_rangeI,axiom,
! [A: $tType,B: $tType,F4: B > A,X: B] : ( member @ A @ ( F4 @ X ) @ ( image @ B @ A @ F4 @ ( top_top @ ( set @ B ) ) ) ) ).
% rangeI
thf(fact_189_range__eqI,axiom,
! [A: $tType,B: $tType,B2: A,F4: B > A,X: B] :
( ( B2
= ( F4 @ X ) )
=> ( member @ A @ B2 @ ( image @ B @ A @ F4 @ ( top_top @ ( set @ B ) ) ) ) ) ).
% range_eqI
thf(fact_190_subset__UNIV,axiom,
! [A: $tType,A6: set @ A] : ( ord_less_eq @ ( set @ A ) @ A6 @ ( top_top @ ( set @ A ) ) ) ).
% subset_UNIV
thf(fact_191_pick__middlep,axiom,
! [B: $tType,A: $tType,C: $tType,P: A > B > $o,Q: B > C > $o,A2: A,C3: C] :
( ( relcompp @ A @ B @ C @ P @ Q @ A2 @ C3 )
=> ( ( P @ A2 @ ( bNF_pick_middlep @ A @ B @ C @ P @ Q @ A2 @ C3 ) )
& ( Q @ ( bNF_pick_middlep @ A @ B @ C @ P @ Q @ A2 @ C3 ) @ C3 ) ) ) ).
% pick_middlep
thf(fact_192_UNIV__witness,axiom,
! [A: $tType] :
? [X3: A] : ( member @ A @ X3 @ ( top_top @ ( set @ A ) ) ) ).
% UNIV_witness
thf(fact_193_UNIV__eq__I,axiom,
! [A: $tType,A6: set @ A] :
( ! [X3: A] : ( member @ A @ X3 @ A6 )
=> ( ( top_top @ ( set @ A ) )
= A6 ) ) ).
% UNIV_eq_I
thf(fact_194_nchotomy__relcomppE,axiom,
! [C: $tType,B: $tType,A: $tType,D: $tType,F4: B > A,R: C > A > $o,S: A > D > $o,A2: C,C3: D] :
( ! [Y: A] :
? [X5: B] :
( Y
= ( F4 @ X5 ) )
=> ( ( relcompp @ C @ A @ D @ R @ S @ A2 @ C3 )
=> ~ ! [B4: B] :
( ( R @ A2 @ ( F4 @ B4 ) )
=> ~ ( S @ ( F4 @ B4 ) @ C3 ) ) ) ) ).
% nchotomy_relcomppE
thf(fact_195_csquare__def,axiom,
! [B: $tType,C: $tType,D: $tType,A: $tType] :
( ( bNF_csquare @ A @ B @ C @ D )
= ( ^ [A7: set @ A,F12: B > C,F23: D > C,P1: A > B,P22: A > D] :
! [X4: A] :
( ( member @ A @ X4 @ A7 )
=> ( ( F12 @ ( P1 @ X4 ) )
= ( F23 @ ( P22 @ X4 ) ) ) ) ) ) ).
% csquare_def
thf(fact_196_fstOp__in,axiom,
! [B: $tType,C: $tType,A: $tType,Ac2: product_prod @ A @ B,P: A > C > $o,Q: C > B > $o] :
( ( member @ ( product_prod @ A @ B ) @ Ac2 @ ( collect @ ( product_prod @ A @ B ) @ ( product_case_prod @ A @ B @ $o @ ( relcompp @ A @ C @ B @ P @ Q ) ) ) )
=> ( member @ ( product_prod @ A @ C ) @ ( bNF_fstOp @ A @ C @ B @ P @ Q @ Ac2 ) @ ( collect @ ( product_prod @ A @ C ) @ ( product_case_prod @ A @ C @ $o @ P ) ) ) ) ).
% fstOp_in
thf(fact_197_sndOp__in,axiom,
! [A: $tType,B: $tType,C: $tType,Ac2: product_prod @ A @ B,P: A > C > $o,Q: C > B > $o] :
( ( member @ ( product_prod @ A @ B ) @ Ac2 @ ( collect @ ( product_prod @ A @ B ) @ ( product_case_prod @ A @ B @ $o @ ( relcompp @ A @ C @ B @ P @ Q ) ) ) )
=> ( member @ ( product_prod @ C @ B ) @ ( bNF_sndOp @ A @ C @ B @ P @ Q @ Ac2 ) @ ( collect @ ( product_prod @ C @ B ) @ ( product_case_prod @ C @ B @ $o @ Q ) ) ) ) ).
% sndOp_in
thf(fact_198_unfold_I1_J,axiom,
! [A: $tType,Rt: A > n,Ct: A > ( set @ ( sum_sum @ t @ A ) ),B2: A] :
( ( root @ ( unfold @ A @ Rt @ Ct @ B2 ) )
= ( Rt @ B2 ) ) ).
% unfold(1)
thf(fact_199_surj__def,axiom,
! [B: $tType,A: $tType,F4: B > A] :
( ( ( image @ B @ A @ F4 @ ( top_top @ ( set @ B ) ) )
= ( top_top @ ( set @ A ) ) )
= ( ! [Y8: A] :
? [X4: B] :
( Y8
= ( F4 @ X4 ) ) ) ) ).
% surj_def
thf(fact_200_surjI,axiom,
! [B: $tType,A: $tType,G3: B > A,F4: A > B] :
( ! [X3: A] :
( ( G3 @ ( F4 @ X3 ) )
= X3 )
=> ( ( image @ B @ A @ G3 @ ( top_top @ ( set @ B ) ) )
= ( top_top @ ( set @ A ) ) ) ) ).
% surjI
thf(fact_201_top__set__def,axiom,
! [A: $tType] :
( ( top_top @ ( set @ A ) )
= ( collect @ A @ ( top_top @ ( A > $o ) ) ) ) ).
% top_set_def
thf(fact_202_surjD,axiom,
! [A: $tType,B: $tType,F4: B > A,Y3: A] :
( ( ( image @ B @ A @ F4 @ ( top_top @ ( set @ B ) ) )
= ( top_top @ ( set @ A ) ) )
=> ? [X3: B] :
( Y3
= ( F4 @ X3 ) ) ) ).
% surjD
thf(fact_203_surjE,axiom,
! [A: $tType,B: $tType,F4: B > A,Y3: A] :
( ( ( image @ B @ A @ F4 @ ( top_top @ ( set @ B ) ) )
= ( top_top @ ( set @ A ) ) )
=> ~ ! [X3: B] :
( Y3
!= ( F4 @ X3 ) ) ) ).
% surjE
thf(fact_204_map__prod__surj,axiom,
! [A: $tType,C: $tType,D: $tType,B: $tType,F4: A > B,G3: C > D] :
( ( ( image @ A @ B @ F4 @ ( top_top @ ( set @ A ) ) )
= ( top_top @ ( set @ B ) ) )
=> ( ( ( image @ C @ D @ G3 @ ( top_top @ ( set @ C ) ) )
= ( top_top @ ( set @ D ) ) )
=> ( ( image @ ( product_prod @ A @ C ) @ ( product_prod @ B @ D ) @ ( product_map_prod @ A @ B @ C @ D @ F4 @ G3 ) @ ( top_top @ ( set @ ( product_prod @ A @ C ) ) ) )
= ( top_top @ ( set @ ( product_prod @ B @ D ) ) ) ) ) ) ).
% map_prod_surj
thf(fact_205_root__Node,axiom,
! [N2: n,As2: set @ ( sum_sum @ t @ dtree )] :
( ( root @ ( node @ N2 @ As2 ) )
= N2 ) ).
% root_Node
thf(fact_206_map__prod__simp,axiom,
! [C: $tType,A: $tType,B: $tType,D: $tType,F4: C > A,G3: D > B,A2: C,B2: D] :
( ( product_map_prod @ C @ A @ D @ B @ F4 @ G3 @ ( product_Pair @ C @ D @ A2 @ B2 ) )
= ( product_Pair @ A @ B @ ( F4 @ A2 ) @ ( G3 @ B2 ) ) ) ).
% map_prod_simp
thf(fact_207_fst__map__prod,axiom,
! [B: $tType,A: $tType,D: $tType,C: $tType,F4: C > A,G3: D > B,X: product_prod @ C @ D] :
( ( product_fst @ A @ B @ ( product_map_prod @ C @ A @ D @ B @ F4 @ G3 @ X ) )
= ( F4 @ ( product_fst @ C @ D @ X ) ) ) ).
% fst_map_prod
thf(fact_208_snd__map__prod,axiom,
! [B: $tType,A: $tType,D: $tType,C: $tType,F4: C > B,G3: D > A,X: product_prod @ C @ D] :
( ( product_snd @ B @ A @ ( product_map_prod @ C @ B @ D @ A @ F4 @ G3 @ X ) )
= ( G3 @ ( product_snd @ C @ D @ X ) ) ) ).
% snd_map_prod
thf(fact_209_map__prod__imageI,axiom,
! [D: $tType,C: $tType,B: $tType,A: $tType,A2: A,B2: B,R2: set @ ( product_prod @ A @ B ),F4: A > C,G3: B > D] :
( ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ A2 @ B2 ) @ R2 )
=> ( member @ ( product_prod @ C @ D ) @ ( product_Pair @ C @ D @ ( F4 @ A2 ) @ ( G3 @ B2 ) ) @ ( image @ ( product_prod @ A @ B ) @ ( product_prod @ C @ D ) @ ( product_map_prod @ A @ C @ B @ D @ F4 @ G3 ) @ R2 ) ) ) ).
% map_prod_imageI
thf(fact_210_prod__fun__imageE,axiom,
! [B: $tType,A: $tType,D: $tType,C: $tType,C3: product_prod @ A @ B,F4: C > A,G3: D > B,R2: set @ ( product_prod @ C @ D )] :
( ( member @ ( product_prod @ A @ B ) @ C3 @ ( image @ ( product_prod @ C @ D ) @ ( product_prod @ A @ B ) @ ( product_map_prod @ C @ A @ D @ B @ F4 @ G3 ) @ R2 ) )
=> ~ ! [X3: C,Y: D] :
( ( C3
= ( product_Pair @ A @ B @ ( F4 @ X3 ) @ ( G3 @ Y ) ) )
=> ~ ( member @ ( product_prod @ C @ D ) @ ( product_Pair @ C @ D @ X3 @ Y ) @ R2 ) ) ) ).
% prod_fun_imageE
thf(fact_211_Node__root__cont,axiom,
! [Tr: dtree] :
( ( node @ ( root @ Tr ) @ ( cont @ Tr ) )
= Tr ) ).
% Node_root_cont
thf(fact_212_Powp__mono,axiom,
! [A: $tType,A6: A > $o,B6: A > $o] :
( ( ord_less_eq @ ( A > $o ) @ A6 @ B6 )
=> ( ord_less_eq @ ( ( set @ A ) > $o ) @ ( powp @ A @ A6 ) @ ( powp @ A @ B6 ) ) ) ).
% Powp_mono
thf(fact_213_dtree__cong,axiom,
! [Tr: dtree,Tr4: dtree] :
( ( ( root @ Tr )
= ( root @ Tr4 ) )
=> ( ( ( cont @ Tr )
= ( cont @ Tr4 ) )
=> ( Tr = Tr4 ) ) ) ).
% dtree_cong
thf(fact_214_predicate2D__conj,axiom,
! [A: $tType,B: $tType,P: A > B > $o,Q: A > B > $o,R2: $o,X: A,Y3: B] :
( ( ( ord_less_eq @ ( A > B > $o ) @ P @ Q )
& R2 )
=> ( R2
& ( ( P @ X @ Y3 )
=> ( Q @ X @ Y3 ) ) ) ) ).
% predicate2D_conj
thf(fact_215_corec_I1_J,axiom,
! [A: $tType,Rt: A > n,Ct: A > ( set @ ( sum_sum @ t @ ( sum_sum @ dtree @ A ) ) ),B2: A] :
( ( root @ ( corec @ A @ Rt @ Ct @ B2 ) )
= ( Rt @ B2 ) ) ).
% corec(1)
thf(fact_216_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_217_surj__swap__iff,axiom,
! [B: $tType,A: $tType,A2: B,B2: B,F4: B > A] :
( ( ( image @ B @ A @ ( swap @ B @ A @ A2 @ B2 @ F4 ) @ ( top_top @ ( set @ B ) ) )
= ( top_top @ ( set @ A ) ) )
= ( ( image @ B @ A @ F4 @ ( top_top @ ( set @ B ) ) )
= ( top_top @ ( set @ A ) ) ) ) ).
% surj_swap_iff
thf(fact_218_swap__image__eq,axiom,
! [B: $tType,A: $tType,A2: A,A6: set @ A,B2: A,F4: A > B] :
( ( member @ A @ A2 @ A6 )
=> ( ( member @ A @ B2 @ A6 )
=> ( ( image @ A @ B @ ( swap @ A @ B @ A2 @ B2 @ F4 ) @ A6 )
= ( image @ A @ B @ F4 @ A6 ) ) ) ) ).
% swap_image_eq
thf(fact_219_surj__imp__surj__swap,axiom,
! [B: $tType,A: $tType,F4: B > A,A2: B,B2: B] :
( ( ( image @ B @ A @ F4 @ ( top_top @ ( set @ B ) ) )
= ( top_top @ ( set @ A ) ) )
=> ( ( image @ B @ A @ ( swap @ B @ A @ A2 @ B2 @ F4 ) @ ( top_top @ ( set @ B ) ) )
= ( top_top @ ( set @ A ) ) ) ) ).
% surj_imp_surj_swap
thf(fact_220_Sup_OSUP__cong,axiom,
! [A: $tType,B: $tType,A6: set @ B,B6: set @ B,C4: B > A,D3: B > A,Sup: ( set @ A ) > A] :
( ( A6 = B6 )
=> ( ! [X3: B] :
( ( member @ B @ X3 @ B6 )
=> ( ( C4 @ X3 )
= ( D3 @ X3 ) ) )
=> ( ( Sup @ ( image @ B @ A @ C4 @ A6 ) )
= ( Sup @ ( image @ B @ A @ D3 @ B6 ) ) ) ) ) ).
% Sup.SUP_cong
thf(fact_221_Inf_OINF__cong,axiom,
! [A: $tType,B: $tType,A6: set @ B,B6: set @ B,C4: B > A,D3: B > A,Inf: ( set @ A ) > A] :
( ( A6 = B6 )
=> ( ! [X3: B] :
( ( member @ B @ X3 @ B6 )
=> ( ( C4 @ X3 )
= ( D3 @ X3 ) ) )
=> ( ( Inf @ ( image @ B @ A @ C4 @ A6 ) )
= ( Inf @ ( image @ B @ A @ D3 @ B6 ) ) ) ) ) ).
% Inf.INF_cong
thf(fact_222_surj__Compl__image__subset,axiom,
! [A: $tType,B: $tType,F4: B > A,A6: set @ B] :
( ( ( image @ B @ A @ F4 @ ( top_top @ ( set @ B ) ) )
= ( top_top @ ( set @ A ) ) )
=> ( ord_less_eq @ ( set @ A ) @ ( uminus_uminus @ ( set @ A ) @ ( image @ B @ A @ F4 @ A6 ) ) @ ( image @ B @ A @ F4 @ ( uminus_uminus @ ( set @ B ) @ A6 ) ) ) ) ).
% surj_Compl_image_subset
thf(fact_223_in__inv__image,axiom,
! [A: $tType,B: $tType,X: A,Y3: A,R: set @ ( product_prod @ B @ B ),F4: A > B] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X @ Y3 ) @ ( inv_image @ B @ A @ R @ F4 ) )
= ( member @ ( product_prod @ B @ B ) @ ( product_Pair @ B @ B @ ( F4 @ X ) @ ( F4 @ Y3 ) ) @ R ) ) ).
% in_inv_image
thf(fact_224_ComplI,axiom,
! [A: $tType,C3: A,A6: set @ A] :
( ~ ( member @ A @ C3 @ A6 )
=> ( member @ A @ C3 @ ( uminus_uminus @ ( set @ A ) @ A6 ) ) ) ).
% ComplI
thf(fact_225_Compl__iff,axiom,
! [A: $tType,C3: A,A6: set @ A] :
( ( member @ A @ C3 @ ( uminus_uminus @ ( set @ A ) @ A6 ) )
= ( ~ ( member @ A @ C3 @ A6 ) ) ) ).
% Compl_iff
thf(fact_226_Compl__eq__Compl__iff,axiom,
! [A: $tType,A6: set @ A,B6: set @ A] :
( ( ( uminus_uminus @ ( set @ A ) @ A6 )
= ( uminus_uminus @ ( set @ A ) @ B6 ) )
= ( A6 = B6 ) ) ).
% Compl_eq_Compl_iff
thf(fact_227_Compl__subset__Compl__iff,axiom,
! [A: $tType,A6: set @ A,B6: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ ( uminus_uminus @ ( set @ A ) @ A6 ) @ ( uminus_uminus @ ( set @ A ) @ B6 ) )
= ( ord_less_eq @ ( set @ A ) @ B6 @ A6 ) ) ).
% Compl_subset_Compl_iff
thf(fact_228_Compl__anti__mono,axiom,
! [A: $tType,A6: set @ A,B6: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ A6 @ B6 )
=> ( ord_less_eq @ ( set @ A ) @ ( uminus_uminus @ ( set @ A ) @ B6 ) @ ( uminus_uminus @ ( set @ A ) @ A6 ) ) ) ).
% Compl_anti_mono
thf(fact_229_ComplD,axiom,
! [A: $tType,C3: A,A6: set @ A] :
( ( member @ A @ C3 @ ( uminus_uminus @ ( set @ A ) @ A6 ) )
=> ~ ( member @ A @ C3 @ A6 ) ) ).
% ComplD
thf(fact_230_double__complement,axiom,
! [A: $tType,A6: set @ A] :
( ( uminus_uminus @ ( set @ A ) @ ( uminus_uminus @ ( set @ A ) @ A6 ) )
= A6 ) ).
% double_complement
thf(fact_231_compl__le__compl__iff,axiom,
! [A: $tType] :
( ( boolean_algebra @ A @ ( type2 @ A ) )
=> ! [X: A,Y3: A] :
( ( ord_less_eq @ A @ ( uminus_uminus @ A @ X ) @ ( uminus_uminus @ A @ Y3 ) )
= ( ord_less_eq @ A @ Y3 @ X ) ) ) ).
% compl_le_compl_iff
thf(fact_232_neg__le__iff__le,axiom,
! [A: $tType] :
( ( ordered_ab_group_add @ A @ ( type2 @ A ) )
=> ! [B2: A,A2: A] :
( ( ord_less_eq @ A @ ( uminus_uminus @ A @ B2 ) @ ( uminus_uminus @ A @ A2 ) )
= ( ord_less_eq @ A @ A2 @ B2 ) ) ) ).
% neg_le_iff_le
thf(fact_233_compl__mono,axiom,
! [A: $tType] :
( ( boolean_algebra @ A @ ( type2 @ A ) )
=> ! [X: A,Y3: A] :
( ( ord_less_eq @ A @ X @ Y3 )
=> ( ord_less_eq @ A @ ( uminus_uminus @ A @ Y3 ) @ ( uminus_uminus @ A @ X ) ) ) ) ).
% compl_mono
thf(fact_234_compl__le__swap1,axiom,
! [A: $tType] :
( ( boolean_algebra @ A @ ( type2 @ A ) )
=> ! [Y3: A,X: A] :
( ( ord_less_eq @ A @ Y3 @ ( uminus_uminus @ A @ X ) )
=> ( ord_less_eq @ A @ X @ ( uminus_uminus @ A @ Y3 ) ) ) ) ).
% compl_le_swap1
thf(fact_235_compl__le__swap2,axiom,
! [A: $tType] :
( ( boolean_algebra @ A @ ( type2 @ A ) )
=> ! [Y3: A,X: A] :
( ( ord_less_eq @ A @ ( uminus_uminus @ A @ Y3 ) @ X )
=> ( ord_less_eq @ A @ ( uminus_uminus @ A @ X ) @ Y3 ) ) ) ).
% compl_le_swap2
thf(fact_236_le__minus__iff,axiom,
! [A: $tType] :
( ( ordered_ab_group_add @ A @ ( type2 @ A ) )
=> ! [A2: A,B2: A] :
( ( ord_less_eq @ A @ A2 @ ( uminus_uminus @ A @ B2 ) )
= ( ord_less_eq @ A @ B2 @ ( uminus_uminus @ A @ A2 ) ) ) ) ).
% le_minus_iff
thf(fact_237_minus__le__iff,axiom,
! [A: $tType] :
( ( ordered_ab_group_add @ A @ ( type2 @ A ) )
=> ! [A2: A,B2: A] :
( ( ord_less_eq @ A @ ( uminus_uminus @ A @ A2 ) @ B2 )
= ( ord_less_eq @ A @ ( uminus_uminus @ A @ B2 ) @ A2 ) ) ) ).
% minus_le_iff
thf(fact_238_le__imp__neg__le,axiom,
! [A: $tType] :
( ( ordered_ab_group_add @ A @ ( type2 @ A ) )
=> ! [A2: A,B2: A] :
( ( ord_less_eq @ A @ A2 @ B2 )
=> ( ord_less_eq @ A @ ( uminus_uminus @ A @ B2 ) @ ( uminus_uminus @ A @ A2 ) ) ) ) ).
% le_imp_neg_le
thf(fact_239_inj__image__Compl__subset,axiom,
! [B: $tType,A: $tType,F4: A > B,A6: set @ A] :
( ( inj_on @ A @ B @ F4 @ ( top_top @ ( set @ A ) ) )
=> ( ord_less_eq @ ( set @ B ) @ ( image @ A @ B @ F4 @ ( uminus_uminus @ ( set @ A ) @ A6 ) ) @ ( uminus_uminus @ ( set @ B ) @ ( image @ A @ B @ F4 @ A6 ) ) ) ) ).
% inj_image_Compl_subset
thf(fact_240_inj__apfst,axiom,
! [B: $tType,C: $tType,A: $tType,F4: A > C] :
( ( inj_on @ ( product_prod @ A @ B ) @ ( product_prod @ C @ B ) @ ( product_apfst @ A @ C @ B @ F4 ) @ ( top_top @ ( set @ ( product_prod @ A @ B ) ) ) )
= ( inj_on @ A @ C @ F4 @ ( top_top @ ( set @ A ) ) ) ) ).
% inj_apfst
thf(fact_241_inj__apsnd,axiom,
! [A: $tType,C: $tType,B: $tType,F4: B > C] :
( ( inj_on @ ( product_prod @ A @ B ) @ ( product_prod @ A @ C ) @ ( product_apsnd @ B @ C @ A @ F4 ) @ ( top_top @ ( set @ ( product_prod @ A @ B ) ) ) )
= ( inj_on @ B @ C @ F4 @ ( top_top @ ( set @ B ) ) ) ) ).
% inj_apsnd
thf(fact_242_inj__image__subset__iff,axiom,
! [B: $tType,A: $tType,F4: A > B,A6: set @ A,B6: set @ A] :
( ( inj_on @ A @ B @ F4 @ ( top_top @ ( set @ A ) ) )
=> ( ( ord_less_eq @ ( set @ B ) @ ( image @ A @ B @ F4 @ A6 ) @ ( image @ A @ B @ F4 @ B6 ) )
= ( ord_less_eq @ ( set @ A ) @ A6 @ B6 ) ) ) ).
% inj_image_subset_iff
thf(fact_243_inj__image__mem__iff,axiom,
! [B: $tType,A: $tType,F4: A > B,A2: A,A6: set @ A] :
( ( inj_on @ A @ B @ F4 @ ( top_top @ ( set @ A ) ) )
=> ( ( member @ B @ ( F4 @ A2 ) @ ( image @ A @ B @ F4 @ A6 ) )
= ( member @ A @ A2 @ A6 ) ) ) ).
% inj_image_mem_iff
thf(fact_244_inj__image__eq__iff,axiom,
! [B: $tType,A: $tType,F4: A > B,A6: set @ A,B6: set @ A] :
( ( inj_on @ A @ B @ F4 @ ( top_top @ ( set @ A ) ) )
=> ( ( ( image @ A @ B @ F4 @ A6 )
= ( image @ A @ B @ F4 @ B6 ) )
= ( A6 = B6 ) ) ) ).
% inj_image_eq_iff
thf(fact_245_range__ex1__eq,axiom,
! [B: $tType,A: $tType,F4: A > B,B2: B] :
( ( inj_on @ A @ B @ F4 @ ( top_top @ ( set @ A ) ) )
=> ( ( member @ B @ B2 @ ( image @ A @ B @ F4 @ ( top_top @ ( set @ A ) ) ) )
= ( ? [X4: A] :
( ( B2
= ( F4 @ X4 ) )
& ! [Y8: A] :
( ( B2
= ( F4 @ Y8 ) )
=> ( Y8 = X4 ) ) ) ) ) ) ).
% range_ex1_eq
thf(fact_246_inj__swap,axiom,
! [B: $tType,A: $tType,A6: set @ ( product_prod @ A @ B )] : ( inj_on @ ( product_prod @ A @ B ) @ ( product_prod @ B @ A ) @ ( product_swap @ A @ B ) @ A6 ) ).
% inj_swap
thf(fact_247_inj__on__image__iff,axiom,
! [B: $tType,A: $tType,A6: set @ A,G3: A > B,F4: A > A] :
( ! [X3: A] :
( ( member @ A @ X3 @ A6 )
=> ! [Xa: A] :
( ( member @ A @ Xa @ A6 )
=> ( ( ( G3 @ ( F4 @ X3 ) )
= ( G3 @ ( F4 @ Xa ) ) )
= ( ( G3 @ X3 )
= ( G3 @ Xa ) ) ) ) )
=> ( ( inj_on @ A @ A @ F4 @ A6 )
=> ( ( inj_on @ A @ B @ G3 @ ( image @ A @ A @ F4 @ A6 ) )
= ( inj_on @ A @ B @ G3 @ A6 ) ) ) ) ).
% inj_on_image_iff
thf(fact_248_inj__on__subset,axiom,
! [B: $tType,A: $tType,F4: A > B,A6: set @ A,B6: set @ A] :
( ( inj_on @ A @ B @ F4 @ A6 )
=> ( ( ord_less_eq @ ( set @ A ) @ B6 @ A6 )
=> ( inj_on @ A @ B @ F4 @ B6 ) ) ) ).
% inj_on_subset
thf(fact_249_subset__inj__on,axiom,
! [B: $tType,A: $tType,F4: A > B,B6: set @ A,A6: set @ A] :
( ( inj_on @ A @ B @ F4 @ B6 )
=> ( ( ord_less_eq @ ( set @ A ) @ A6 @ B6 )
=> ( inj_on @ A @ B @ F4 @ A6 ) ) ) ).
% subset_inj_on
thf(fact_250_inj__on__image__mem__iff__alt,axiom,
! [B: $tType,A: $tType,F4: A > B,B6: set @ A,A6: set @ A,A2: A] :
( ( inj_on @ A @ B @ F4 @ B6 )
=> ( ( ord_less_eq @ ( set @ A ) @ A6 @ B6 )
=> ( ( member @ B @ ( F4 @ A2 ) @ ( image @ A @ B @ F4 @ A6 ) )
=> ( ( member @ A @ A2 @ B6 )
=> ( member @ A @ A2 @ A6 ) ) ) ) ) ).
% inj_on_image_mem_iff_alt
thf(fact_251_inj__on__image__mem__iff,axiom,
! [B: $tType,A: $tType,F4: A > B,B6: set @ A,A2: A,A6: set @ A] :
( ( inj_on @ A @ B @ F4 @ B6 )
=> ( ( member @ A @ A2 @ B6 )
=> ( ( ord_less_eq @ ( set @ A ) @ A6 @ B6 )
=> ( ( member @ B @ ( F4 @ A2 ) @ ( image @ A @ B @ F4 @ A6 ) )
= ( member @ A @ A2 @ A6 ) ) ) ) ) ).
% inj_on_image_mem_iff
thf(fact_252_inj__on__image__eq__iff,axiom,
! [B: $tType,A: $tType,F4: A > B,C4: set @ A,A6: set @ A,B6: set @ A] :
( ( inj_on @ A @ B @ F4 @ C4 )
=> ( ( ord_less_eq @ ( set @ A ) @ A6 @ C4 )
=> ( ( ord_less_eq @ ( set @ A ) @ B6 @ C4 )
=> ( ( ( image @ A @ B @ F4 @ A6 )
= ( image @ A @ B @ F4 @ B6 ) )
= ( A6 = B6 ) ) ) ) ) ).
% inj_on_image_eq_iff
thf(fact_253_the__inv__into__into,axiom,
! [B: $tType,A: $tType,F4: A > B,A6: set @ A,X: B,B6: set @ A] :
( ( inj_on @ A @ B @ F4 @ A6 )
=> ( ( member @ B @ X @ ( image @ A @ B @ F4 @ A6 ) )
=> ( ( ord_less_eq @ ( set @ A ) @ A6 @ B6 )
=> ( member @ A @ ( the_inv_into @ A @ B @ A6 @ F4 @ X ) @ B6 ) ) ) ) ).
% the_inv_into_into
thf(fact_254_wf__map__prod__image,axiom,
! [B: $tType,A: $tType,R: set @ ( product_prod @ A @ A ),F4: A > B] :
( ( wf @ A @ R )
=> ( ( inj_on @ A @ B @ F4 @ ( top_top @ ( set @ A ) ) )
=> ( wf @ B @ ( image @ ( product_prod @ A @ A ) @ ( product_prod @ B @ B ) @ ( product_map_prod @ A @ B @ A @ B @ F4 @ F4 ) @ R ) ) ) ) ).
% wf_map_prod_image
thf(fact_255_wf__inv__image,axiom,
! [B: $tType,A: $tType,R: set @ ( product_prod @ B @ B ),F4: A > B] :
( ( wf @ B @ R )
=> ( wf @ A @ ( inv_image @ B @ A @ R @ F4 ) ) ) ).
% wf_inv_image
%----Type constructors (16)
thf(tcon_fun___Lattices_Oboolean__algebra,axiom,
! [A8: $tType,A9: $tType] :
( ( boolean_algebra @ A9 @ ( type2 @ A9 ) )
=> ( boolean_algebra @ ( A8 > A9 ) @ ( type2 @ ( A8 > A9 ) ) ) ) ).
thf(tcon_fun___Orderings_Oorder__top,axiom,
! [A8: $tType,A9: $tType] :
( ( order_top @ A9 @ ( type2 @ A9 ) )
=> ( order_top @ ( A8 > A9 ) @ ( type2 @ ( A8 > A9 ) ) ) ) ).
thf(tcon_fun___Orderings_Opreorder,axiom,
! [A8: $tType,A9: $tType] :
( ( preorder @ A9 @ ( type2 @ A9 ) )
=> ( preorder @ ( A8 > A9 ) @ ( type2 @ ( A8 > A9 ) ) ) ) ).
thf(tcon_fun___Orderings_Oorder,axiom,
! [A8: $tType,A9: $tType] :
( ( order @ A9 @ ( type2 @ A9 ) )
=> ( order @ ( A8 > A9 ) @ ( type2 @ ( A8 > A9 ) ) ) ) ).
thf(tcon_fun___Orderings_Oord,axiom,
! [A8: $tType,A9: $tType] :
( ( ord @ A9 @ ( type2 @ A9 ) )
=> ( ord @ ( A8 > A9 ) @ ( type2 @ ( A8 > A9 ) ) ) ) ).
thf(tcon_Set_Oset___Lattices_Oboolean__algebra_1,axiom,
! [A8: $tType] : ( boolean_algebra @ ( set @ A8 ) @ ( type2 @ ( set @ A8 ) ) ) ).
thf(tcon_Set_Oset___Orderings_Oorder__top_2,axiom,
! [A8: $tType] : ( order_top @ ( set @ A8 ) @ ( type2 @ ( set @ A8 ) ) ) ).
thf(tcon_Set_Oset___Orderings_Opreorder_3,axiom,
! [A8: $tType] : ( preorder @ ( set @ A8 ) @ ( type2 @ ( set @ A8 ) ) ) ).
thf(tcon_Set_Oset___Orderings_Oorder_4,axiom,
! [A8: $tType] : ( order @ ( set @ A8 ) @ ( type2 @ ( set @ A8 ) ) ) ).
thf(tcon_Set_Oset___Orderings_Oord_5,axiom,
! [A8: $tType] : ( ord @ ( set @ A8 ) @ ( type2 @ ( set @ A8 ) ) ) ).
thf(tcon_HOL_Obool___Lattices_Oboolean__algebra_6,axiom,
boolean_algebra @ $o @ ( type2 @ $o ) ).
thf(tcon_HOL_Obool___Orderings_Oorder__top_7,axiom,
order_top @ $o @ ( type2 @ $o ) ).
thf(tcon_HOL_Obool___Orderings_Opreorder_8,axiom,
preorder @ $o @ ( type2 @ $o ) ).
thf(tcon_HOL_Obool___Orderings_Olinorder,axiom,
linorder @ $o @ ( type2 @ $o ) ).
thf(tcon_HOL_Obool___Orderings_Oorder_9,axiom,
order @ $o @ ( type2 @ $o ) ).
thf(tcon_HOL_Obool___Orderings_Oord_10,axiom,
ord @ $o @ ( type2 @ $o ) ).
%----Conjectures (1)
thf(conj_0,conjecture,
( ( parall1899940088le_par @ ( product_Pair @ dtree @ dtree @ ( parall1899940088le_par @ ( product_Pair @ dtree @ dtree @ tr1 @ tr2 ) ) @ tr3 ) )
= ( parall1899940088le_par @ ( product_Pair @ dtree @ dtree @ tr1 @ ( parall1899940088le_par @ ( product_Pair @ dtree @ dtree @ tr2 @ tr3 ) ) ) ) ) ).
%------------------------------------------------------------------------------