TPTP Problem File: DAT141^1.p
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%------------------------------------------------------------------------------
% File : DAT141^1 : TPTP v8.2.0. Released v7.0.0.
% Domain : Data Structures
% Problem : Coinductive list prefix 70
% Version : [Bla16] axioms : Especial.
% English :
% Refs : [Loc10] Lochbihler (2010), Coinductive
% : [RB15] Reynolds & Blanchette (2015), A Decision Procedure for
% : [Bla16] Blanchette (2016), Email to Geoff Sutcliffe
% Source : [Bla16]
% Names : coinductive_list_prefix__70.p [Bla16]
% Status : Theorem
% Rating : 0.67 v8.1.0, 0.75 v7.5.0, 0.33 v7.2.0, 0.50 v7.1.0
% Syntax : Number of formulae : 325 ( 181 unt; 55 typ; 0 def)
% Number of atoms : 445 ( 329 equ; 0 cnn)
% Maximal formula atoms : 6 ( 1 avg)
% Number of connectives : 4668 ( 43 ~; 1 |; 23 &;4439 @)
% ( 0 <=>; 162 =>; 0 <=; 0 <~>)
% Maximal formula depth : 28 ( 8 avg)
% Number of types : 4 ( 3 usr)
% Number of type conns : 629 ( 629 >; 0 *; 0 +; 0 <<)
% Number of symbols : 55 ( 52 usr; 3 con; 0-9 aty)
% Number of variables : 1597 ( 181 ^;1314 !; 7 ?;1597 :)
% ( 95 !>; 0 ?*; 0 @-; 0 @+)
% SPC : TH1_THM_EQU_NAR
% Comments : This file was generated by Isabelle (most likely Sledgehammer)
% 2016-07-13 14:40:51.031
%------------------------------------------------------------------------------
%----Could-be-implicit typings (7)
thf(ty_t_Coinductive__List_Ollist,type,
coinductive_llist: $tType > $tType ).
thf(ty_t_Product__Type_Ounit,type,
product_unit: $tType ).
thf(ty_t_Product__Type_Oprod,type,
product_prod: $tType > $tType > $tType ).
thf(ty_t_Set_Oset,type,
set: $tType > $tType ).
thf(ty_t_Nat_Onat,type,
nat: $tType ).
thf(ty_t_itself,type,
itself: $tType > $tType ).
thf(ty_tf_a,type,
a: $tType ).
%----Explicit typings (48)
thf(sy_cl_HOL_Otype,type,
type:
!>[A: $tType] : ( ( itself @ A ) > $o ) ).
thf(sy_cl_Lattices_Olattice,type,
lattice:
!>[A: $tType] : ( ( itself @ A ) > $o ) ).
thf(sy_cl_Lattices_Osemilattice__inf,type,
semilattice_inf:
!>[A: $tType] : ( ( itself @ A ) > $o ) ).
thf(sy_c_BNF__Def_OGrp,type,
bNF_Grp:
!>[A: $tType,B: $tType] : ( ( set @ A ) > ( A > B ) > A > B > $o ) ).
thf(sy_c_BNF__Def_Oconvol,type,
bNF_convol:
!>[A: $tType,B: $tType,C: $tType] : ( ( A > B ) > ( A > C ) > A > ( product_prod @ B @ C ) ) ).
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_Basic__BNF__LFPs_Oprod_Osize__prod,type,
basic_BNF_size_prod:
!>[A: $tType,B: $tType] : ( ( A > nat ) > ( B > nat ) > ( product_prod @ A @ B ) > nat ) ).
thf(sy_c_Basic__BNFs_Opred__fun,type,
basic_pred_fun:
!>[A: $tType,B: $tType] : ( ( A > $o ) > ( B > $o ) > ( A > B ) > $o ) ).
thf(sy_c_Coinductive__List_Ollist_OLNil,type,
coinductive_LNil:
!>[A: $tType] : ( coinductive_llist @ A ) ).
thf(sy_c_Coinductive__List_Ollist_Olhd,type,
coinductive_lhd:
!>[A: $tType] : ( ( coinductive_llist @ A ) > A ) ).
thf(sy_c_Coinductive__List_Ollist_Oltl,type,
coinductive_ltl:
!>[A: $tType] : ( ( coinductive_llist @ A ) > ( coinductive_llist @ A ) ) ).
thf(sy_c_Coinductive__List_Ounfold__llist,type,
coindu1441602521_llist:
!>[A: $tType,B: $tType] : ( ( A > $o ) > ( A > B ) > ( A > A ) > A > ( coinductive_llist @ B ) ) ).
thf(sy_c_Fun_Ocomp,type,
comp:
!>[B: $tType,C: $tType,A: $tType] : ( ( B > C ) > ( A > B ) > A > C ) ).
thf(sy_c_Fun_Oid,type,
id:
!>[A: $tType] : ( A > A ) ).
thf(sy_c_Fun__Def_Oreduction__pair,type,
fun_reduction_pair:
!>[A: $tType] : ( ( product_prod @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ A @ A ) ) ) > $o ) ).
thf(sy_c_Fun__Def_Orp__inv__image,type,
fun_rp_inv_image:
!>[A: $tType,B: $tType] : ( ( product_prod @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ A @ A ) ) ) > ( B > A ) > ( product_prod @ ( set @ ( product_prod @ B @ B ) ) @ ( set @ ( product_prod @ B @ B ) ) ) ) ).
thf(sy_c_HOL_OThe,type,
the:
!>[A: $tType] : ( ( A > $o ) > A ) ).
thf(sy_c_If,type,
if:
!>[A: $tType] : ( $o > A > A > A ) ).
thf(sy_c_Lattices_Oinf__class_Oinf,type,
inf_inf:
!>[A: $tType] : ( A > A > A ) ).
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_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_Obool_Orec__set__bool,type,
product_rec_set_bool:
!>[T: $tType] : ( T > T > $o > T > $o ) ).
thf(sy_c_Product__Type_Oold_Oprod_Orec__prod,type,
product_rec_prod:
!>[A: $tType,B: $tType,T: $tType] : ( ( A > B > T ) > ( product_prod @ A @ B ) > T ) ).
thf(sy_c_Product__Type_Oold_Oprod_Orec__set__prod,type,
product_rec_set_prod:
!>[A: $tType,B: $tType,T: $tType] : ( ( A > B > T ) > ( product_prod @ A @ B ) > T > $o ) ).
thf(sy_c_Product__Type_Oold_Ounit_Orec__set__unit,type,
product_rec_set_unit:
!>[T: $tType] : ( T > product_unit > T > $o ) ).
thf(sy_c_Product__Type_Oold_Ounit_Orec__unit,type,
product_rec_unit:
!>[T: $tType] : ( T > product_unit > T ) ).
thf(sy_c_Product__Type_Oprod_Ocase__prod,type,
product_case_prod:
!>[A: $tType,B: $tType,C: $tType] : ( ( A > B > C ) > ( product_prod @ A @ B ) > C ) ).
thf(sy_c_Product__Type_Oprod_Ofst,type,
product_fst:
!>[A: $tType,B: $tType] : ( ( product_prod @ A @ B ) > A ) ).
thf(sy_c_Product__Type_Oprod_Osnd,type,
product_snd:
!>[A: $tType,B: $tType] : ( ( product_prod @ A @ B ) > B ) ).
thf(sy_c_Product__Type_Oprod_Oswap,type,
product_swap:
!>[A: $tType,B: $tType] : ( ( product_prod @ A @ B ) > ( product_prod @ B @ A ) ) ).
thf(sy_c_Product__Type_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_Record_Oiso__tuple__update__accessor__cong__assist,type,
iso_tu2017585022assist:
!>[B: $tType,A: $tType] : ( ( ( B > B ) > A > A ) > ( A > B ) > $o ) ).
thf(sy_c_Record_Oiso__tuple__update__accessor__eq__assist,type,
iso_tu2011167877assist:
!>[B: $tType,A: $tType] : ( ( ( B > B ) > A > A ) > ( A > B ) > A > ( B > B ) > A > B > $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_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_xs,type,
xs: coinductive_llist @ a ).
%----Relevant facts (256)
thf(fact_0_local_Oinf__llist__def,axiom,
( ( inf_inf @ ( coinductive_llist @ a ) )
= ( ^ [Xs: coinductive_llist @ a,Ys: coinductive_llist @ a] :
( coindu1441602521_llist @ ( product_prod @ ( coinductive_llist @ a ) @ ( coinductive_llist @ a ) ) @ a
@ ( product_case_prod @ ( coinductive_llist @ a ) @ ( coinductive_llist @ a ) @ $o
@ ^ [Xt: coinductive_llist @ a,Yt: coinductive_llist @ a] :
( ( Xt
!= ( coinductive_LNil @ a ) )
=> ( ( Yt
!= ( coinductive_LNil @ a ) )
=> ( ( coinductive_lhd @ a @ Xt )
!= ( coinductive_lhd @ a @ Yt ) ) ) ) )
@ ( comp @ ( coinductive_llist @ a ) @ a @ ( product_prod @ ( coinductive_llist @ a ) @ ( coinductive_llist @ a ) ) @ ( coinductive_lhd @ a ) @ ( product_snd @ ( coinductive_llist @ a ) @ ( coinductive_llist @ a ) ) )
@ ( product_map_prod @ ( coinductive_llist @ a ) @ ( coinductive_llist @ a ) @ ( coinductive_llist @ a ) @ ( coinductive_llist @ a ) @ ( coinductive_ltl @ a ) @ ( coinductive_ltl @ a ) )
@ ( product_Pair @ ( coinductive_llist @ a ) @ ( coinductive_llist @ a ) @ Xs @ Ys ) ) ) ) ).
% local.inf_llist_def
thf(fact_1_snd__comp__map__prod,axiom,
! [D: $tType,C: $tType,B: $tType,A: $tType,F: A > D,G: B > C] :
( ( comp @ ( product_prod @ D @ C ) @ C @ ( product_prod @ A @ B ) @ ( product_snd @ D @ C ) @ ( product_map_prod @ A @ D @ B @ C @ F @ G ) )
= ( comp @ B @ C @ ( product_prod @ A @ B ) @ G @ ( product_snd @ A @ B ) ) ) ).
% snd_comp_map_prod
thf(fact_2_case__prodI,axiom,
! [A: $tType,B: $tType,F: A > B > $o,A2: A,B2: B] :
( ( F @ A2 @ B2 )
=> ( product_case_prod @ A @ B @ $o @ F @ ( product_Pair @ A @ B @ A2 @ B2 ) ) ) ).
% case_prodI
thf(fact_3_case__prodI2,axiom,
! [B: $tType,A: $tType,P: product_prod @ A @ B,C2: A > B > $o] :
( ! [A3: A,B3: B] :
( ( P
= ( product_Pair @ A @ B @ A3 @ B3 ) )
=> ( C2 @ A3 @ B3 ) )
=> ( product_case_prod @ A @ B @ $o @ C2 @ P ) ) ).
% case_prodI2
thf(fact_4_case__prod__conv,axiom,
! [B: $tType,A: $tType,C: $tType,F: B > C > A,A2: B,B2: C] :
( ( product_case_prod @ B @ C @ A @ F @ ( product_Pair @ B @ C @ A2 @ B2 ) )
= ( F @ A2 @ B2 ) ) ).
% case_prod_conv
thf(fact_5_snd__map__prod,axiom,
! [B: $tType,A: $tType,D: $tType,C: $tType,F: C > B,G: D > A,X: product_prod @ C @ D] :
( ( product_snd @ B @ A @ ( product_map_prod @ C @ B @ D @ A @ F @ G @ X ) )
= ( G @ ( product_snd @ C @ D @ X ) ) ) ).
% snd_map_prod
thf(fact_6_map__prod__simp,axiom,
! [C: $tType,A: $tType,B: $tType,D: $tType,F: C > A,G: D > B,A2: C,B2: D] :
( ( product_map_prod @ C @ A @ D @ B @ F @ G @ ( product_Pair @ C @ D @ A2 @ B2 ) )
= ( product_Pair @ A @ B @ ( F @ A2 ) @ ( G @ B2 ) ) ) ).
% map_prod_simp
thf(fact_7_Coinductive__List__Prefix__Mirabelle__yfrixeyiok_Oinf__llist__def,axiom,
! [A: $tType] :
( ( inf_inf @ ( coinductive_llist @ A ) )
= ( ^ [Xs: coinductive_llist @ A,Ys: coinductive_llist @ A] :
( coindu1441602521_llist @ ( product_prod @ ( coinductive_llist @ A ) @ ( coinductive_llist @ A ) ) @ A
@ ( product_case_prod @ ( coinductive_llist @ A ) @ ( coinductive_llist @ A ) @ $o
@ ^ [Xt: coinductive_llist @ A,Yt: coinductive_llist @ A] :
( ( Xt
!= ( coinductive_LNil @ A ) )
=> ( ( Yt
!= ( coinductive_LNil @ A ) )
=> ( ( coinductive_lhd @ A @ Xt )
!= ( coinductive_lhd @ A @ Yt ) ) ) ) )
@ ( comp @ ( coinductive_llist @ A ) @ A @ ( product_prod @ ( coinductive_llist @ A ) @ ( coinductive_llist @ A ) ) @ ( coinductive_lhd @ A ) @ ( product_snd @ ( coinductive_llist @ A ) @ ( coinductive_llist @ A ) ) )
@ ( product_map_prod @ ( coinductive_llist @ A ) @ ( coinductive_llist @ A ) @ ( coinductive_llist @ A ) @ ( coinductive_llist @ A ) @ ( coinductive_ltl @ A ) @ ( coinductive_ltl @ A ) )
@ ( product_Pair @ ( coinductive_llist @ A ) @ ( coinductive_llist @ A ) @ Xs @ Ys ) ) ) ) ).
% Coinductive_List_Prefix_Mirabelle_yfrixeyiok.inf_llist_def
thf(fact_8_map__prod__ident,axiom,
! [B: $tType,A: $tType] :
( ( product_map_prod @ A @ A @ B @ B
@ ^ [X2: A] : X2
@ ^ [Y: B] : Y )
= ( ^ [Z: product_prod @ A @ B] : Z ) ) ).
% map_prod_ident
thf(fact_9_split__part,axiom,
! [B: $tType,A: $tType,P2: $o,Q: A > B > $o] :
( ( product_case_prod @ A @ B @ $o
@ ^ [A4: A,B4: B] :
( P2
& ( Q @ A4 @ B4 ) ) )
= ( ^ [Ab: product_prod @ A @ B] :
( P2
& ( product_case_prod @ A @ B @ $o @ Q @ Ab ) ) ) ) ).
% split_part
thf(fact_10_ltl__unfold__llist,axiom,
! [A: $tType,B: $tType,IS_LNIL: B > $o,A2: B,LHD: B > A,LTL: B > B] :
( ( ( IS_LNIL @ A2 )
=> ( ( coinductive_ltl @ A @ ( coindu1441602521_llist @ B @ A @ IS_LNIL @ LHD @ LTL @ A2 ) )
= ( coinductive_LNil @ A ) ) )
& ( ~ ( IS_LNIL @ A2 )
=> ( ( coinductive_ltl @ A @ ( coindu1441602521_llist @ B @ A @ IS_LNIL @ LHD @ LTL @ A2 ) )
= ( coindu1441602521_llist @ B @ A @ IS_LNIL @ LHD @ LTL @ ( LTL @ A2 ) ) ) ) ) ).
% ltl_unfold_llist
thf(fact_11_comp__apply,axiom,
! [C: $tType,A: $tType,B: $tType] :
( ( comp @ B @ A @ C )
= ( ^ [F2: B > A,G2: C > B,X2: C] : ( F2 @ ( G2 @ X2 ) ) ) ) ).
% comp_apply
thf(fact_12_old_Oprod_Oinject,axiom,
! [A: $tType,B: $tType,A2: A,B2: B,A5: A,B5: B] :
( ( ( product_Pair @ A @ B @ A2 @ B2 )
= ( product_Pair @ A @ B @ A5 @ B5 ) )
= ( ( A2 = A5 )
& ( B2 = B5 ) ) ) ).
% old.prod.inject
thf(fact_13_prod_Oinject,axiom,
! [A: $tType,B: $tType,X1: A,X22: B,Y1: A,Y2: B] :
( ( ( product_Pair @ A @ B @ X1 @ X22 )
= ( product_Pair @ A @ B @ Y1 @ Y2 ) )
= ( ( X1 = Y1 )
& ( X22 = Y2 ) ) ) ).
% prod.inject
thf(fact_14_mem__case__prodI2,axiom,
! [C: $tType,B: $tType,A: $tType,P: product_prod @ A @ B,Z2: C,C2: A > B > ( set @ C )] :
( ! [A3: A,B3: B] :
( ( P
= ( product_Pair @ A @ B @ A3 @ B3 ) )
=> ( member @ C @ Z2 @ ( C2 @ A3 @ B3 ) ) )
=> ( member @ C @ Z2 @ ( product_case_prod @ A @ B @ ( set @ C ) @ C2 @ P ) ) ) ).
% mem_case_prodI2
thf(fact_15_mem__case__prodI,axiom,
! [A: $tType,B: $tType,C: $tType,Z2: A,C2: B > C > ( set @ A ),A2: B,B2: C] :
( ( member @ A @ Z2 @ ( C2 @ A2 @ B2 ) )
=> ( member @ A @ Z2 @ ( product_case_prod @ B @ C @ ( set @ A ) @ C2 @ ( product_Pair @ B @ C @ A2 @ B2 ) ) ) ) ).
% mem_case_prodI
thf(fact_16_case__prodI2_H,axiom,
! [A: $tType,B: $tType,C: $tType,P: product_prod @ A @ B,C2: A > B > C > $o,X: C] :
( ! [A3: A,B3: B] :
( ( ( product_Pair @ A @ B @ A3 @ B3 )
= P )
=> ( C2 @ A3 @ B3 @ X ) )
=> ( product_case_prod @ A @ B @ ( C > $o ) @ C2 @ P @ X ) ) ).
% case_prodI2'
thf(fact_17_mem__case__prodE,axiom,
! [B: $tType,A: $tType,C: $tType,Z2: A,C2: B > C > ( set @ A ),P: product_prod @ B @ C] :
( ( member @ A @ Z2 @ ( product_case_prod @ B @ C @ ( set @ A ) @ C2 @ P ) )
=> ~ ! [X3: B,Y3: C] :
( ( P
= ( product_Pair @ B @ C @ X3 @ Y3 ) )
=> ~ ( member @ A @ Z2 @ ( C2 @ X3 @ Y3 ) ) ) ) ).
% mem_case_prodE
thf(fact_18_case__prodE_H,axiom,
! [B: $tType,A: $tType,C: $tType,C2: A > B > C > $o,P: product_prod @ A @ B,Z2: C] :
( ( product_case_prod @ A @ B @ ( C > $o ) @ C2 @ P @ Z2 )
=> ~ ! [X3: A,Y3: B] :
( ( P
= ( product_Pair @ A @ B @ X3 @ Y3 ) )
=> ~ ( C2 @ X3 @ Y3 @ Z2 ) ) ) ).
% case_prodE'
thf(fact_19_case__prodD_H,axiom,
! [B: $tType,A: $tType,C: $tType,R: A > B > C > $o,A2: A,B2: B,C2: C] :
( ( product_case_prod @ A @ B @ ( C > $o ) @ R @ ( product_Pair @ A @ B @ A2 @ B2 ) @ C2 )
=> ( R @ A2 @ B2 @ C2 ) ) ).
% case_prodD'
thf(fact_20_map__prod__def,axiom,
! [B: $tType,D: $tType,C: $tType,A: $tType] :
( ( product_map_prod @ A @ C @ B @ D )
= ( ^ [F2: A > C,G2: B > D] :
( product_case_prod @ A @ B @ ( product_prod @ C @ D )
@ ^ [X2: A,Y: B] : ( product_Pair @ C @ D @ ( F2 @ X2 ) @ ( G2 @ Y ) ) ) ) ) ).
% map_prod_def
thf(fact_21_old_Oprod_Oinducts,axiom,
! [B: $tType,A: $tType,P2: ( product_prod @ A @ B ) > $o,Prod: product_prod @ A @ B] :
( ! [A3: A,B3: B] : ( P2 @ ( product_Pair @ A @ B @ A3 @ B3 ) )
=> ( P2 @ Prod ) ) ).
% old.prod.inducts
thf(fact_22_old_Oprod_Oexhaust,axiom,
! [A: $tType,B: $tType,Y4: product_prod @ A @ B] :
~ ! [A3: A,B3: B] :
( Y4
!= ( product_Pair @ A @ B @ A3 @ B3 ) ) ).
% old.prod.exhaust
thf(fact_23_prod__induct7,axiom,
! [G3: $tType,F3: $tType,E: $tType,D: $tType,C: $tType,B: $tType,A: $tType,P2: ( product_prod @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D @ ( product_prod @ E @ ( product_prod @ F3 @ G3 ) ) ) ) ) ) > $o,X: product_prod @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D @ ( product_prod @ E @ ( product_prod @ F3 @ G3 ) ) ) ) )] :
( ! [A3: A,B3: B,C3: C,D2: D,E2: E,F4: F3,G4: G3] : ( P2 @ ( product_Pair @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D @ ( product_prod @ E @ ( product_prod @ F3 @ G3 ) ) ) ) ) @ A3 @ ( product_Pair @ B @ ( product_prod @ C @ ( product_prod @ D @ ( product_prod @ E @ ( product_prod @ F3 @ G3 ) ) ) ) @ B3 @ ( product_Pair @ C @ ( product_prod @ D @ ( product_prod @ E @ ( product_prod @ F3 @ G3 ) ) ) @ C3 @ ( product_Pair @ D @ ( product_prod @ E @ ( product_prod @ F3 @ G3 ) ) @ D2 @ ( product_Pair @ E @ ( product_prod @ F3 @ G3 ) @ E2 @ ( product_Pair @ F3 @ G3 @ F4 @ G4 ) ) ) ) ) ) )
=> ( P2 @ X ) ) ).
% prod_induct7
thf(fact_24_prod__induct6,axiom,
! [F3: $tType,E: $tType,D: $tType,C: $tType,B: $tType,A: $tType,P2: ( product_prod @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D @ ( product_prod @ E @ F3 ) ) ) ) ) > $o,X: product_prod @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D @ ( product_prod @ E @ F3 ) ) ) )] :
( ! [A3: A,B3: B,C3: C,D2: D,E2: E,F4: F3] : ( P2 @ ( product_Pair @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D @ ( product_prod @ E @ F3 ) ) ) ) @ A3 @ ( product_Pair @ B @ ( product_prod @ C @ ( product_prod @ D @ ( product_prod @ E @ F3 ) ) ) @ B3 @ ( product_Pair @ C @ ( product_prod @ D @ ( product_prod @ E @ F3 ) ) @ C3 @ ( product_Pair @ D @ ( product_prod @ E @ F3 ) @ D2 @ ( product_Pair @ E @ F3 @ E2 @ F4 ) ) ) ) ) )
=> ( P2 @ X ) ) ).
% prod_induct6
thf(fact_25_prod__induct5,axiom,
! [E: $tType,D: $tType,C: $tType,B: $tType,A: $tType,P2: ( 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,B3: B,C3: C,D2: D,E2: E] : ( P2 @ ( product_Pair @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D @ E ) ) ) @ A3 @ ( product_Pair @ B @ ( product_prod @ C @ ( product_prod @ D @ E ) ) @ B3 @ ( product_Pair @ C @ ( product_prod @ D @ E ) @ C3 @ ( product_Pair @ D @ E @ D2 @ E2 ) ) ) ) )
=> ( P2 @ X ) ) ).
% prod_induct5
thf(fact_26_prod__induct4,axiom,
! [D: $tType,C: $tType,B: $tType,A: $tType,P2: ( product_prod @ A @ ( product_prod @ B @ ( product_prod @ C @ D ) ) ) > $o,X: product_prod @ A @ ( product_prod @ B @ ( product_prod @ C @ D ) )] :
( ! [A3: A,B3: B,C3: C,D2: D] : ( P2 @ ( product_Pair @ A @ ( product_prod @ B @ ( product_prod @ C @ D ) ) @ A3 @ ( product_Pair @ B @ ( product_prod @ C @ D ) @ B3 @ ( product_Pair @ C @ D @ C3 @ D2 ) ) ) )
=> ( P2 @ X ) ) ).
% prod_induct4
thf(fact_27_prod__induct3,axiom,
! [C: $tType,B: $tType,A: $tType,P2: ( product_prod @ A @ ( product_prod @ B @ C ) ) > $o,X: product_prod @ A @ ( product_prod @ B @ C )] :
( ! [A3: A,B3: B,C3: C] : ( P2 @ ( product_Pair @ A @ ( product_prod @ B @ C ) @ A3 @ ( product_Pair @ B @ C @ B3 @ C3 ) ) )
=> ( P2 @ X ) ) ).
% prod_induct3
thf(fact_28_prod__cases7,axiom,
! [A: $tType,B: $tType,C: $tType,D: $tType,E: $tType,F3: $tType,G3: $tType,Y4: product_prod @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D @ ( product_prod @ E @ ( product_prod @ F3 @ G3 ) ) ) ) )] :
~ ! [A3: A,B3: B,C3: C,D2: D,E2: E,F4: F3,G4: G3] :
( Y4
!= ( product_Pair @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D @ ( product_prod @ E @ ( product_prod @ F3 @ G3 ) ) ) ) ) @ A3 @ ( product_Pair @ B @ ( product_prod @ C @ ( product_prod @ D @ ( product_prod @ E @ ( product_prod @ F3 @ G3 ) ) ) ) @ B3 @ ( product_Pair @ C @ ( product_prod @ D @ ( product_prod @ E @ ( product_prod @ F3 @ G3 ) ) ) @ C3 @ ( product_Pair @ D @ ( product_prod @ E @ ( product_prod @ F3 @ G3 ) ) @ D2 @ ( product_Pair @ E @ ( product_prod @ F3 @ G3 ) @ E2 @ ( product_Pair @ F3 @ G3 @ F4 @ G4 ) ) ) ) ) ) ) ).
% prod_cases7
thf(fact_29_prod__cases6,axiom,
! [A: $tType,B: $tType,C: $tType,D: $tType,E: $tType,F3: $tType,Y4: product_prod @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D @ ( product_prod @ E @ F3 ) ) ) )] :
~ ! [A3: A,B3: B,C3: C,D2: D,E2: E,F4: F3] :
( Y4
!= ( product_Pair @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D @ ( product_prod @ E @ F3 ) ) ) ) @ A3 @ ( product_Pair @ B @ ( product_prod @ C @ ( product_prod @ D @ ( product_prod @ E @ F3 ) ) ) @ B3 @ ( product_Pair @ C @ ( product_prod @ D @ ( product_prod @ E @ F3 ) ) @ C3 @ ( product_Pair @ D @ ( product_prod @ E @ F3 ) @ D2 @ ( product_Pair @ E @ F3 @ E2 @ F4 ) ) ) ) ) ) ).
% prod_cases6
thf(fact_30_prod__cases5,axiom,
! [A: $tType,B: $tType,C: $tType,D: $tType,E: $tType,Y4: product_prod @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D @ E ) ) )] :
~ ! [A3: A,B3: B,C3: C,D2: D,E2: E] :
( Y4
!= ( product_Pair @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D @ E ) ) ) @ A3 @ ( product_Pair @ B @ ( product_prod @ C @ ( product_prod @ D @ E ) ) @ B3 @ ( product_Pair @ C @ ( product_prod @ D @ E ) @ C3 @ ( product_Pair @ D @ E @ D2 @ E2 ) ) ) ) ) ).
% prod_cases5
thf(fact_31_prod__cases4,axiom,
! [A: $tType,B: $tType,C: $tType,D: $tType,Y4: product_prod @ A @ ( product_prod @ B @ ( product_prod @ C @ D ) )] :
~ ! [A3: A,B3: B,C3: C,D2: D] :
( Y4
!= ( product_Pair @ A @ ( product_prod @ B @ ( product_prod @ C @ D ) ) @ A3 @ ( product_Pair @ B @ ( product_prod @ C @ D ) @ B3 @ ( product_Pair @ C @ D @ C3 @ D2 ) ) ) ) ).
% prod_cases4
thf(fact_32_prod__cases3,axiom,
! [A: $tType,B: $tType,C: $tType,Y4: product_prod @ A @ ( product_prod @ B @ C )] :
~ ! [A3: A,B3: B,C3: C] :
( Y4
!= ( product_Pair @ A @ ( product_prod @ B @ C ) @ A3 @ ( product_Pair @ B @ C @ B3 @ C3 ) ) ) ).
% prod_cases3
thf(fact_33_Pair__inject,axiom,
! [A: $tType,B: $tType,A2: A,B2: B,A5: A,B5: B] :
( ( ( product_Pair @ A @ B @ A2 @ B2 )
= ( product_Pair @ A @ B @ A5 @ B5 ) )
=> ~ ( ( A2 = A5 )
=> ( B2 != B5 ) ) ) ).
% Pair_inject
thf(fact_34_prod__cases,axiom,
! [B: $tType,A: $tType,P2: ( product_prod @ A @ B ) > $o,P: product_prod @ A @ B] :
( ! [A3: A,B3: B] : ( P2 @ ( product_Pair @ A @ B @ A3 @ B3 ) )
=> ( P2 @ P ) ) ).
% prod_cases
thf(fact_35_surj__pair,axiom,
! [A: $tType,B: $tType,P: product_prod @ A @ B] :
? [X3: A,Y3: B] :
( P
= ( product_Pair @ A @ B @ X3 @ Y3 ) ) ).
% surj_pair
thf(fact_36_comp__eq__dest__lhs,axiom,
! [C: $tType,B: $tType,A: $tType,A2: C > B,B2: A > C,C2: A > B,V: A] :
( ( ( comp @ C @ B @ A @ A2 @ B2 )
= C2 )
=> ( ( A2 @ ( B2 @ V ) )
= ( C2 @ V ) ) ) ).
% comp_eq_dest_lhs
thf(fact_37_comp__eq__elim,axiom,
! [C: $tType,B: $tType,D: $tType,A: $tType,A2: C > B,B2: A > C,C2: D > B,D3: A > D] :
( ( ( comp @ C @ B @ A @ A2 @ B2 )
= ( comp @ D @ B @ A @ C2 @ D3 ) )
=> ! [V2: A] :
( ( A2 @ ( B2 @ V2 ) )
= ( C2 @ ( D3 @ V2 ) ) ) ) ).
% comp_eq_elim
thf(fact_38_comp__eq__dest,axiom,
! [C: $tType,B: $tType,D: $tType,A: $tType,A2: C > B,B2: A > C,C2: D > B,D3: A > D,V: A] :
( ( ( comp @ C @ B @ A @ A2 @ B2 )
= ( comp @ D @ B @ A @ C2 @ D3 ) )
=> ( ( A2 @ ( B2 @ V ) )
= ( C2 @ ( D3 @ V ) ) ) ) ).
% comp_eq_dest
thf(fact_39_comp__assoc,axiom,
! [B: $tType,D: $tType,C: $tType,A: $tType,F: D > B,G: C > D,H: A > C] :
( ( comp @ C @ B @ A @ ( comp @ D @ B @ C @ F @ G ) @ H )
= ( comp @ D @ B @ A @ F @ ( comp @ C @ D @ A @ G @ H ) ) ) ).
% comp_assoc
thf(fact_40_comp__def,axiom,
! [A: $tType,C: $tType,B: $tType] :
( ( comp @ B @ C @ A )
= ( ^ [F2: B > C,G2: A > B,X2: A] : ( F2 @ ( G2 @ X2 ) ) ) ) ).
% comp_def
thf(fact_41_prod_Ocase__distrib,axiom,
! [C: $tType,D: $tType,B: $tType,A: $tType,H: C > D,F: A > B > C,Prod: product_prod @ A @ B] :
( ( H @ ( product_case_prod @ A @ B @ C @ F @ Prod ) )
= ( product_case_prod @ A @ B @ D
@ ^ [X12: A,X23: B] : ( H @ ( F @ X12 @ X23 ) )
@ Prod ) ) ).
% prod.case_distrib
thf(fact_42_prod_Odisc__eq__case,axiom,
! [B: $tType,A: $tType,Prod: product_prod @ A @ B] :
( product_case_prod @ A @ B @ $o
@ ^ [Uu: A,Uv: B] : $true
@ Prod ) ).
% prod.disc_eq_case
thf(fact_43_old_Oprod_Ocase,axiom,
! [A: $tType,C: $tType,B: $tType,F: A > B > C,X1: A,X22: B] :
( ( product_case_prod @ A @ B @ C @ F @ ( product_Pair @ A @ B @ X1 @ X22 ) )
= ( F @ X1 @ X22 ) ) ).
% old.prod.case
thf(fact_44_snd__conv,axiom,
! [Aa: $tType,A: $tType,X1: Aa,X22: A] :
( ( product_snd @ Aa @ A @ ( product_Pair @ Aa @ A @ X1 @ X22 ) )
= X22 ) ).
% snd_conv
thf(fact_45_mem__Collect__eq,axiom,
! [A: $tType,A2: A,P2: A > $o] :
( ( member @ A @ A2 @ ( collect @ A @ P2 ) )
= ( P2 @ A2 ) ) ).
% mem_Collect_eq
thf(fact_46_Collect__mem__eq,axiom,
! [A: $tType,A6: set @ A] :
( ( collect @ A
@ ^ [X2: A] : ( member @ A @ X2 @ A6 ) )
= A6 ) ).
% Collect_mem_eq
thf(fact_47_Collect__cong,axiom,
! [A: $tType,P2: A > $o,Q: A > $o] :
( ! [X3: A] :
( ( P2 @ X3 )
= ( Q @ X3 ) )
=> ( ( collect @ A @ P2 )
= ( collect @ A @ Q ) ) ) ).
% Collect_cong
thf(fact_48_ext,axiom,
! [B: $tType,A: $tType,F: A > B,G: A > B] :
( ! [X3: A] :
( ( F @ X3 )
= ( G @ X3 ) )
=> ( F = G ) ) ).
% ext
thf(fact_49_snd__eqD,axiom,
! [B: $tType,A: $tType,X: B,Y4: A,A2: A] :
( ( ( product_snd @ B @ A @ ( product_Pair @ B @ A @ X @ Y4 ) )
= A2 )
=> ( Y4 = A2 ) ) ).
% snd_eqD
thf(fact_50_ltl__simps_I1_J,axiom,
! [A: $tType] :
( ( coinductive_ltl @ A @ ( coinductive_LNil @ A ) )
= ( coinductive_LNil @ A ) ) ).
% ltl_simps(1)
thf(fact_51_map__prod_Ocompositionality,axiom,
! [D: $tType,F3: $tType,E: $tType,C: $tType,B: $tType,A: $tType,F: C > E,G: D > F3,H: A > C,I: B > D,Prod: product_prod @ A @ B] :
( ( product_map_prod @ C @ E @ D @ F3 @ F @ G @ ( product_map_prod @ A @ C @ B @ D @ H @ I @ Prod ) )
= ( product_map_prod @ A @ E @ B @ F3 @ ( comp @ C @ E @ A @ F @ H ) @ ( comp @ D @ F3 @ B @ G @ I ) @ Prod ) ) ).
% map_prod.compositionality
thf(fact_52_map__prod__compose,axiom,
! [D: $tType,C: $tType,A: $tType,E: $tType,F3: $tType,B: $tType,F1: E > C,F22: A > E,G1: F3 > D,G22: B > F3] :
( ( product_map_prod @ A @ C @ B @ D @ ( comp @ E @ C @ A @ F1 @ F22 ) @ ( comp @ F3 @ D @ B @ G1 @ G22 ) )
= ( comp @ ( product_prod @ E @ F3 ) @ ( product_prod @ C @ D ) @ ( product_prod @ A @ B ) @ ( product_map_prod @ E @ C @ F3 @ D @ F1 @ G1 ) @ ( product_map_prod @ A @ E @ B @ F3 @ F22 @ G22 ) ) ) ).
% map_prod_compose
thf(fact_53_map__prod_Ocomp,axiom,
! [A: $tType,C: $tType,E: $tType,F3: $tType,D: $tType,B: $tType,F: C > E,G: D > F3,H: A > C,I: B > D] :
( ( comp @ ( product_prod @ C @ D ) @ ( product_prod @ E @ F3 ) @ ( product_prod @ A @ B ) @ ( product_map_prod @ C @ E @ D @ F3 @ F @ G ) @ ( product_map_prod @ A @ C @ B @ D @ H @ I ) )
= ( product_map_prod @ A @ E @ B @ F3 @ ( comp @ C @ E @ A @ F @ H ) @ ( comp @ D @ F3 @ B @ G @ I ) ) ) ).
% map_prod.comp
thf(fact_54_unfold__llist__ltl__unroll,axiom,
! [A: $tType,B: $tType,IS_LNIL: B > $o,LHD: B > A,LTL: B > B,B2: B] :
( ( coindu1441602521_llist @ B @ A @ IS_LNIL @ LHD @ LTL @ ( LTL @ B2 ) )
= ( coindu1441602521_llist @ B @ A @ ( comp @ B @ $o @ B @ IS_LNIL @ LTL ) @ ( comp @ B @ A @ B @ LHD @ LTL ) @ LTL @ B2 ) ) ).
% unfold_llist_ltl_unroll
thf(fact_55_unfold__llist_Octr_I1_J,axiom,
! [A: $tType,B: $tType,P: A > $o,A2: A,G21: A > B,G222: A > A] :
( ( P @ A2 )
=> ( ( coindu1441602521_llist @ A @ B @ P @ G21 @ G222 @ A2 )
= ( coinductive_LNil @ B ) ) ) ).
% unfold_llist.ctr(1)
thf(fact_56_unfold__llist_Osimps_I4_J,axiom,
! [B: $tType,A: $tType,P: A > $o,A2: A,G21: A > B,G222: A > A] :
( ~ ( P @ A2 )
=> ( ( coinductive_ltl @ B @ ( coindu1441602521_llist @ A @ B @ P @ G21 @ G222 @ A2 ) )
= ( coindu1441602521_llist @ A @ B @ P @ G21 @ G222 @ ( G222 @ A2 ) ) ) ) ).
% unfold_llist.simps(4)
thf(fact_57_unfold__llist_Osimps_I3_J,axiom,
! [B: $tType,A: $tType,P: A > $o,A2: A,G21: A > B,G222: A > A] :
( ~ ( P @ A2 )
=> ( ( coinductive_lhd @ B @ ( coindu1441602521_llist @ A @ B @ P @ G21 @ G222 @ A2 ) )
= ( G21 @ A2 ) ) ) ).
% unfold_llist.simps(3)
thf(fact_58_cond__case__prod__eta,axiom,
! [C: $tType,B: $tType,A: $tType,F: A > B > C,G: ( product_prod @ A @ B ) > C] :
( ! [X3: A,Y3: B] :
( ( F @ X3 @ Y3 )
= ( G @ ( product_Pair @ A @ B @ X3 @ Y3 ) ) )
=> ( ( product_case_prod @ A @ B @ C @ F )
= G ) ) ).
% cond_case_prod_eta
thf(fact_59_case__prod__eta,axiom,
! [C: $tType,B: $tType,A: $tType,F: ( product_prod @ A @ B ) > C] :
( ( product_case_prod @ A @ B @ C
@ ^ [X2: A,Y: B] : ( F @ ( product_Pair @ A @ B @ X2 @ Y ) ) )
= F ) ).
% case_prod_eta
thf(fact_60_case__prodE2,axiom,
! [B: $tType,A: $tType,C: $tType,Q: A > $o,P2: B > C > A,Z2: product_prod @ B @ C] :
( ( Q @ ( product_case_prod @ B @ C @ A @ P2 @ Z2 ) )
=> ~ ! [X3: B,Y3: C] :
( ( Z2
= ( product_Pair @ B @ C @ X3 @ Y3 ) )
=> ~ ( Q @ ( P2 @ X3 @ Y3 ) ) ) ) ).
% case_prodE2
thf(fact_61_case__prodE,axiom,
! [A: $tType,B: $tType,C2: A > B > $o,P: product_prod @ A @ B] :
( ( product_case_prod @ A @ B @ $o @ C2 @ P )
=> ~ ! [X3: A,Y3: B] :
( ( P
= ( product_Pair @ A @ B @ X3 @ Y3 ) )
=> ~ ( C2 @ X3 @ Y3 ) ) ) ).
% case_prodE
thf(fact_62_case__prodD,axiom,
! [A: $tType,B: $tType,F: A > B > $o,A2: A,B2: B] :
( ( product_case_prod @ A @ B @ $o @ F @ ( product_Pair @ A @ B @ A2 @ B2 ) )
=> ( F @ A2 @ B2 ) ) ).
% case_prodD
thf(fact_63_snd__def,axiom,
! [B: $tType,A: $tType] :
( ( product_snd @ A @ B )
= ( product_case_prod @ A @ B @ B
@ ^ [X12: A,X23: B] : X23 ) ) ).
% snd_def
thf(fact_64_case__prod__o__map__prod,axiom,
! [B: $tType,D: $tType,C: $tType,E: $tType,A: $tType,F: D > E > C,G1: A > D,G22: B > E] :
( ( comp @ ( product_prod @ D @ E ) @ C @ ( product_prod @ A @ B ) @ ( product_case_prod @ D @ E @ C @ F ) @ ( product_map_prod @ A @ D @ B @ E @ G1 @ G22 ) )
= ( product_case_prod @ A @ B @ C
@ ^ [L: A,R2: B] : ( F @ ( G1 @ L ) @ ( G22 @ R2 ) ) ) ) ).
% case_prod_o_map_prod
thf(fact_65_snd__diag__snd,axiom,
! [B: $tType,A: $tType] :
( ( comp @ ( product_prod @ B @ B ) @ B @ ( product_prod @ A @ B ) @ ( product_snd @ B @ B )
@ ( comp @ B @ ( product_prod @ B @ B ) @ ( product_prod @ A @ B )
@ ^ [X2: B] : ( product_Pair @ B @ B @ X2 @ X2 )
@ ( product_snd @ A @ B ) ) )
= ( product_snd @ A @ B ) ) ).
% snd_diag_snd
thf(fact_66_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_67_case__prod__map__prod,axiom,
! [C: $tType,A: $tType,B: $tType,E: $tType,D: $tType,H: B > C > A,F: D > B,G: E > C,X: product_prod @ D @ E] :
( ( product_case_prod @ B @ C @ A @ H @ ( product_map_prod @ D @ B @ E @ C @ F @ G @ X ) )
= ( product_case_prod @ D @ E @ A
@ ^ [L: D,R2: E] : ( H @ ( F @ L ) @ ( G @ R2 ) )
@ X ) ) ).
% case_prod_map_prod
thf(fact_68_inf__apply,axiom,
! [B: $tType,A: $tType] :
( ( semilattice_inf @ B @ ( type2 @ B ) )
=> ( ( inf_inf @ ( A > B ) )
= ( ^ [F2: A > B,G2: A > B,X2: A] : ( inf_inf @ B @ ( F2 @ X2 ) @ ( G2 @ X2 ) ) ) ) ) ).
% inf_apply
thf(fact_69_inf_Oidem,axiom,
! [A: $tType] :
( ( semilattice_inf @ A @ ( type2 @ A ) )
=> ! [A2: A] :
( ( inf_inf @ A @ A2 @ A2 )
= A2 ) ) ).
% inf.idem
thf(fact_70_inf__idem,axiom,
! [A: $tType] :
( ( semilattice_inf @ A @ ( type2 @ A ) )
=> ! [X: A] :
( ( inf_inf @ A @ X @ X )
= X ) ) ).
% inf_idem
thf(fact_71_inf_Oleft__idem,axiom,
! [A: $tType] :
( ( semilattice_inf @ A @ ( type2 @ A ) )
=> ! [A2: A,B2: A] :
( ( inf_inf @ A @ A2 @ ( inf_inf @ A @ A2 @ B2 ) )
= ( inf_inf @ A @ A2 @ B2 ) ) ) ).
% inf.left_idem
thf(fact_72_inf__left__idem,axiom,
! [A: $tType] :
( ( semilattice_inf @ A @ ( type2 @ A ) )
=> ! [X: A,Y4: A] :
( ( inf_inf @ A @ X @ ( inf_inf @ A @ X @ Y4 ) )
= ( inf_inf @ A @ X @ Y4 ) ) ) ).
% inf_left_idem
thf(fact_73_inf_Oright__idem,axiom,
! [A: $tType] :
( ( semilattice_inf @ A @ ( type2 @ A ) )
=> ! [A2: A,B2: A] :
( ( inf_inf @ A @ ( inf_inf @ A @ A2 @ B2 ) @ B2 )
= ( inf_inf @ A @ A2 @ B2 ) ) ) ).
% inf.right_idem
thf(fact_74_inf__right__idem,axiom,
! [A: $tType] :
( ( semilattice_inf @ A @ ( type2 @ A ) )
=> ! [X: A,Y4: A] :
( ( inf_inf @ A @ ( inf_inf @ A @ X @ Y4 ) @ Y4 )
= ( inf_inf @ A @ X @ Y4 ) ) ) ).
% inf_right_idem
thf(fact_75_inf__left__commute,axiom,
! [A: $tType] :
( ( semilattice_inf @ A @ ( type2 @ A ) )
=> ! [X: A,Y4: A,Z2: A] :
( ( inf_inf @ A @ X @ ( inf_inf @ A @ Y4 @ Z2 ) )
= ( inf_inf @ A @ Y4 @ ( inf_inf @ A @ X @ Z2 ) ) ) ) ).
% inf_left_commute
thf(fact_76_inf_Oleft__commute,axiom,
! [A: $tType] :
( ( semilattice_inf @ A @ ( type2 @ A ) )
=> ! [B2: A,A2: A,C2: A] :
( ( inf_inf @ A @ B2 @ ( inf_inf @ A @ A2 @ C2 ) )
= ( inf_inf @ A @ A2 @ ( inf_inf @ A @ B2 @ C2 ) ) ) ) ).
% inf.left_commute
thf(fact_77_inf__commute,axiom,
! [A: $tType] :
( ( semilattice_inf @ A @ ( type2 @ A ) )
=> ( ( inf_inf @ A )
= ( ^ [X2: A,Y: A] : ( inf_inf @ A @ Y @ X2 ) ) ) ) ).
% inf_commute
thf(fact_78_inf_Ocommute,axiom,
! [A: $tType] :
( ( semilattice_inf @ A @ ( type2 @ A ) )
=> ( ( inf_inf @ A )
= ( ^ [A4: A,B4: A] : ( inf_inf @ A @ B4 @ A4 ) ) ) ) ).
% inf.commute
thf(fact_79_inf__assoc,axiom,
! [A: $tType] :
( ( semilattice_inf @ A @ ( type2 @ A ) )
=> ! [X: A,Y4: A,Z2: A] :
( ( inf_inf @ A @ ( inf_inf @ A @ X @ Y4 ) @ Z2 )
= ( inf_inf @ A @ X @ ( inf_inf @ A @ Y4 @ Z2 ) ) ) ) ).
% inf_assoc
thf(fact_80_inf_Oassoc,axiom,
! [A: $tType] :
( ( semilattice_inf @ A @ ( type2 @ A ) )
=> ! [A2: A,B2: A,C2: A] :
( ( inf_inf @ A @ ( inf_inf @ A @ A2 @ B2 ) @ C2 )
= ( inf_inf @ A @ A2 @ ( inf_inf @ A @ B2 @ C2 ) ) ) ) ).
% inf.assoc
thf(fact_81_inf__fun__def,axiom,
! [B: $tType,A: $tType] :
( ( semilattice_inf @ B @ ( type2 @ B ) )
=> ( ( inf_inf @ ( A > B ) )
= ( ^ [F2: A > B,G2: A > B,X2: A] : ( inf_inf @ B @ ( F2 @ X2 ) @ ( G2 @ X2 ) ) ) ) ) ).
% inf_fun_def
thf(fact_82_inf__sup__aci_I1_J,axiom,
! [A: $tType] :
( ( lattice @ A @ ( type2 @ A ) )
=> ( ( inf_inf @ A )
= ( ^ [X2: A,Y: A] : ( inf_inf @ A @ Y @ X2 ) ) ) ) ).
% inf_sup_aci(1)
thf(fact_83_inf__sup__aci_I2_J,axiom,
! [A: $tType] :
( ( lattice @ A @ ( type2 @ A ) )
=> ! [X: A,Y4: A,Z2: A] :
( ( inf_inf @ A @ ( inf_inf @ A @ X @ Y4 ) @ Z2 )
= ( inf_inf @ A @ X @ ( inf_inf @ A @ Y4 @ Z2 ) ) ) ) ).
% inf_sup_aci(2)
thf(fact_84_inf__sup__aci_I3_J,axiom,
! [A: $tType] :
( ( lattice @ A @ ( type2 @ A ) )
=> ! [X: A,Y4: A,Z2: A] :
( ( inf_inf @ A @ X @ ( inf_inf @ A @ Y4 @ Z2 ) )
= ( inf_inf @ A @ Y4 @ ( inf_inf @ A @ X @ Z2 ) ) ) ) ).
% inf_sup_aci(3)
thf(fact_85_inf__sup__aci_I4_J,axiom,
! [A: $tType] :
( ( lattice @ A @ ( type2 @ A ) )
=> ! [X: A,Y4: A] :
( ( inf_inf @ A @ X @ ( inf_inf @ A @ X @ Y4 ) )
= ( inf_inf @ A @ X @ Y4 ) ) ) ).
% inf_sup_aci(4)
thf(fact_86_rewriteL__comp__comp,axiom,
! [C: $tType,B: $tType,A: $tType,D: $tType,F: C > B,G: A > C,L2: A > B,H: D > A] :
( ( ( comp @ C @ B @ A @ F @ G )
= L2 )
=> ( ( comp @ C @ B @ D @ F @ ( comp @ A @ C @ D @ G @ H ) )
= ( comp @ A @ B @ D @ L2 @ H ) ) ) ).
% rewriteL_comp_comp
thf(fact_87_rewriteR__comp__comp,axiom,
! [C: $tType,D: $tType,B: $tType,A: $tType,G: C > B,H: A > C,R3: A > B,F: B > D] :
( ( ( comp @ C @ B @ A @ G @ H )
= R3 )
=> ( ( comp @ C @ D @ A @ ( comp @ B @ D @ C @ F @ G ) @ H )
= ( comp @ B @ D @ A @ F @ R3 ) ) ) ).
% rewriteR_comp_comp
thf(fact_88_rewriteL__comp__comp2,axiom,
! [A: $tType,C: $tType,B: $tType,D: $tType,E: $tType,F: C > B,G: A > C,L1: D > B,L22: A > D,H: E > A,R3: E > D] :
( ( ( comp @ C @ B @ A @ F @ G )
= ( comp @ D @ B @ A @ L1 @ L22 ) )
=> ( ( ( comp @ A @ D @ E @ L22 @ H )
= R3 )
=> ( ( comp @ C @ B @ E @ F @ ( comp @ A @ C @ E @ G @ H ) )
= ( comp @ D @ B @ E @ L1 @ R3 ) ) ) ) ).
% rewriteL_comp_comp2
thf(fact_89_rewriteR__comp__comp2,axiom,
! [C: $tType,B: $tType,E: $tType,D: $tType,A: $tType,G: C > B,H: A > C,R1: D > B,R22: A > D,F: B > E,L2: D > E] :
( ( ( comp @ C @ B @ A @ G @ H )
= ( comp @ D @ B @ A @ R1 @ R22 ) )
=> ( ( ( comp @ B @ E @ D @ F @ R1 )
= L2 )
=> ( ( comp @ C @ E @ A @ ( comp @ B @ E @ C @ F @ G ) @ H )
= ( comp @ D @ E @ A @ L2 @ R22 ) ) ) ) ).
% rewriteR_comp_comp2
thf(fact_90_case__prod__Pair__iden,axiom,
! [B: $tType,A: $tType,P: product_prod @ A @ B] :
( ( product_case_prod @ A @ B @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B ) @ P )
= P ) ).
% case_prod_Pair_iden
thf(fact_91_case__prod__app,axiom,
! [A: $tType,D: $tType,C: $tType,B: $tType] :
( ( product_case_prod @ B @ C @ ( D > A ) )
= ( ^ [F2: B > C > D > A,X2: product_prod @ B @ C,Y: D] :
( product_case_prod @ B @ C @ A
@ ^ [L: B,R2: C] : ( F2 @ L @ R2 @ Y )
@ X2 ) ) ) ).
% case_prod_app
thf(fact_92_prod_Omap__comp,axiom,
! [D: $tType,F3: $tType,E: $tType,C: $tType,B: $tType,A: $tType,G1: C > E,G22: D > F3,F1: A > C,F22: B > D,V: product_prod @ A @ B] :
( ( product_map_prod @ C @ E @ D @ F3 @ G1 @ G22 @ ( product_map_prod @ A @ C @ B @ D @ F1 @ F22 @ V ) )
= ( product_map_prod @ A @ E @ B @ F3 @ ( comp @ C @ E @ A @ G1 @ F1 ) @ ( comp @ D @ F3 @ B @ G22 @ F22 ) @ V ) ) ).
% prod.map_comp
thf(fact_93_sndI,axiom,
! [A: $tType,B: $tType,X: product_prod @ A @ B,Y4: A,Z2: B] :
( ( X
= ( product_Pair @ A @ B @ Y4 @ Z2 ) )
=> ( ( product_snd @ A @ B @ X )
= Z2 ) ) ).
% sndI
thf(fact_94_eq__snd__iff,axiom,
! [A: $tType,B: $tType,B2: A,P: product_prod @ B @ A] :
( ( B2
= ( product_snd @ B @ A @ P ) )
= ( ? [A4: B] :
( P
= ( product_Pair @ B @ A @ A4 @ B2 ) ) ) ) ).
% eq_snd_iff
thf(fact_95_split__cong,axiom,
! [C: $tType,B: $tType,A: $tType,Q2: product_prod @ A @ B,F: A > B > C,G: A > B > C,P: product_prod @ A @ B] :
( ! [X3: A,Y3: B] :
( ( ( product_Pair @ A @ B @ X3 @ Y3 )
= Q2 )
=> ( ( F @ X3 @ Y3 )
= ( G @ X3 @ Y3 ) ) )
=> ( ( P = Q2 )
=> ( ( product_case_prod @ A @ B @ C @ F @ P )
= ( product_case_prod @ A @ B @ C @ G @ Q2 ) ) ) ) ).
% split_cong
thf(fact_96_snd__comp__apsnd,axiom,
! [C: $tType,B: $tType,A: $tType,F: B > C] :
( ( comp @ ( product_prod @ A @ C ) @ C @ ( product_prod @ A @ B ) @ ( product_snd @ A @ C ) @ ( product_apsnd @ B @ C @ A @ F ) )
= ( comp @ B @ C @ ( product_prod @ A @ B ) @ F @ ( product_snd @ A @ B ) ) ) ).
% snd_comp_apsnd
thf(fact_97_map__prod__o__convol,axiom,
! [D: $tType,B: $tType,C: $tType,E: $tType,A: $tType,H1: D > B,H2: E > C,F: A > D,G: A > E] :
( ( comp @ ( product_prod @ D @ E ) @ ( product_prod @ B @ C ) @ A @ ( product_map_prod @ D @ B @ E @ C @ H1 @ H2 ) @ ( bNF_convol @ A @ D @ E @ F @ G ) )
= ( bNF_convol @ A @ B @ C @ ( comp @ D @ B @ A @ H1 @ F ) @ ( comp @ E @ C @ A @ H2 @ G ) ) ) ).
% map_prod_o_convol
thf(fact_98_apsnd__conv,axiom,
! [A: $tType,B: $tType,C: $tType,F: C > B,X: A,Y4: C] :
( ( product_apsnd @ C @ B @ A @ F @ ( product_Pair @ A @ C @ X @ Y4 ) )
= ( product_Pair @ A @ B @ X @ ( F @ Y4 ) ) ) ).
% apsnd_conv
thf(fact_99_apsnd__eq__conv,axiom,
! [B: $tType,C: $tType,A: $tType,F: C > B,X: product_prod @ A @ C,G: C > B] :
( ( ( product_apsnd @ C @ B @ A @ F @ X )
= ( product_apsnd @ C @ B @ A @ G @ X ) )
= ( ( F @ ( product_snd @ A @ C @ X ) )
= ( G @ ( product_snd @ A @ C @ X ) ) ) ) ).
% apsnd_eq_conv
thf(fact_100_snd__apsnd,axiom,
! [A: $tType,C: $tType,B: $tType,F: C > A,X: product_prod @ B @ C] :
( ( product_snd @ B @ A @ ( product_apsnd @ C @ A @ B @ F @ X ) )
= ( F @ ( product_snd @ B @ C @ X ) ) ) ).
% snd_apsnd
thf(fact_101_convol__def,axiom,
! [C: $tType,B: $tType,A: $tType] :
( ( bNF_convol @ A @ B @ C )
= ( ^ [F2: A > B,G2: A > C,A4: A] : ( product_Pair @ B @ C @ ( F2 @ A4 ) @ ( G2 @ A4 ) ) ) ) ).
% convol_def
thf(fact_102_apsnd__compose,axiom,
! [C: $tType,B: $tType,D: $tType,A: $tType,F: C > B,G: D > C,X: product_prod @ A @ D] :
( ( product_apsnd @ C @ B @ A @ F @ ( product_apsnd @ D @ C @ A @ G @ X ) )
= ( product_apsnd @ D @ B @ A @ ( comp @ C @ B @ D @ F @ G ) @ X ) ) ).
% apsnd_compose
thf(fact_103_snd__convol_H,axiom,
! [B: $tType,A: $tType,C: $tType,F: C > B,G: C > A,X: C] :
( ( product_snd @ B @ A @ ( bNF_convol @ C @ B @ A @ F @ G @ X ) )
= ( G @ X ) ) ).
% snd_convol'
thf(fact_104_snd__convol,axiom,
! [C: $tType,B: $tType,A: $tType,F: A > C,G: A > B] :
( ( comp @ ( product_prod @ C @ B ) @ B @ A @ ( product_snd @ C @ B ) @ ( bNF_convol @ A @ C @ B @ F @ G ) )
= G ) ).
% snd_convol
thf(fact_105_fun_Omap__comp,axiom,
! [B: $tType,C: $tType,A: $tType,D: $tType,G: B > C,F: A > B,V: D > A] :
( ( comp @ B @ C @ D @ G @ ( comp @ A @ B @ D @ F @ V ) )
= ( comp @ A @ C @ D @ ( comp @ B @ C @ A @ G @ F ) @ V ) ) ).
% fun.map_comp
thf(fact_106_comp__apply__eq,axiom,
! [B: $tType,D: $tType,A: $tType,C: $tType,F: B > A,G: C > B,X: C,H: D > A,K: C > D] :
( ( ( F @ ( G @ X ) )
= ( H @ ( K @ X ) ) )
=> ( ( comp @ B @ A @ C @ F @ G @ X )
= ( comp @ D @ A @ C @ H @ K @ X ) ) ) ).
% comp_apply_eq
thf(fact_107_comp__cong,axiom,
! [C: $tType,B: $tType,D: $tType,A: $tType,E: $tType,F: B > A,G: C > B,X: C,F5: D > A,G5: E > D,X4: E] :
( ( ( F @ ( G @ X ) )
= ( F5 @ ( G5 @ X4 ) ) )
=> ( ( comp @ B @ A @ C @ F @ G @ X )
= ( comp @ D @ A @ E @ F5 @ G5 @ X4 ) ) ) ).
% comp_cong
thf(fact_108_convol__o,axiom,
! [B: $tType,C: $tType,D: $tType,A: $tType,F: D > B,G: D > C,H: A > D] :
( ( comp @ D @ ( product_prod @ B @ C ) @ A @ ( bNF_convol @ D @ B @ C @ F @ G ) @ H )
= ( bNF_convol @ A @ B @ C @ ( comp @ D @ B @ A @ F @ H ) @ ( comp @ D @ C @ A @ G @ H ) ) ) ).
% convol_o
thf(fact_109_fun_Omap__ident,axiom,
! [A: $tType,D: $tType,T2: D > A] :
( ( comp @ A @ A @ D
@ ^ [X2: A] : X2
@ T2 )
= T2 ) ).
% fun.map_ident
thf(fact_110_prod_Omap__ident,axiom,
! [B: $tType,A: $tType,T2: product_prod @ A @ B] :
( ( product_map_prod @ A @ A @ B @ B
@ ^ [X2: A] : X2
@ ^ [X2: B] : X2
@ T2 )
= T2 ) ).
% prod.map_ident
thf(fact_111_snd__sndOp,axiom,
! [B: $tType,A: $tType,C: $tType,P2: B > C > $o,Q: C > A > $o] :
( ( product_snd @ B @ A )
= ( comp @ ( product_prod @ C @ A ) @ A @ ( product_prod @ B @ A ) @ ( product_snd @ C @ A ) @ ( bNF_sndOp @ B @ C @ A @ P2 @ Q ) ) ) ).
% snd_sndOp
thf(fact_112_conj__comp__iff,axiom,
! [B: $tType,A: $tType,P2: B > $o,Q: B > $o,G: A > B] :
( ( comp @ B @ $o @ A
@ ^ [X2: B] :
( ( P2 @ X2 )
& ( Q @ X2 ) )
@ G )
= ( ^ [X2: A] :
( ( comp @ B @ $o @ A @ P2 @ G @ X2 )
& ( comp @ B @ $o @ A @ Q @ G @ X2 ) ) ) ) ).
% conj_comp_iff
thf(fact_113_snd__diag__fst,axiom,
! [B: $tType,A: $tType] :
( ( comp @ ( product_prod @ A @ A ) @ A @ ( product_prod @ A @ B ) @ ( product_snd @ A @ A )
@ ( comp @ A @ ( product_prod @ A @ A ) @ ( product_prod @ A @ B )
@ ^ [X2: A] : ( product_Pair @ A @ A @ X2 @ X2 )
@ ( product_fst @ A @ B ) ) )
= ( product_fst @ A @ B ) ) ).
% snd_diag_fst
thf(fact_114_fst__diag__snd,axiom,
! [B: $tType,A: $tType] :
( ( comp @ ( product_prod @ B @ B ) @ B @ ( product_prod @ A @ B ) @ ( product_fst @ B @ B )
@ ( comp @ B @ ( product_prod @ B @ B ) @ ( product_prod @ A @ B )
@ ^ [X2: B] : ( product_Pair @ B @ B @ X2 @ X2 )
@ ( product_snd @ A @ B ) ) )
= ( product_snd @ A @ B ) ) ).
% fst_diag_snd
thf(fact_115_K__record__comp,axiom,
! [C: $tType,B: $tType,A: $tType,C2: B,F: A > C] :
( ( comp @ C @ B @ A
@ ^ [X2: C] : C2
@ F )
= ( ^ [X2: A] : C2 ) ) ).
% K_record_comp
thf(fact_116_fst__map__prod,axiom,
! [B: $tType,A: $tType,D: $tType,C: $tType,F: C > A,G: D > B,X: product_prod @ C @ D] :
( ( product_fst @ A @ B @ ( product_map_prod @ C @ A @ D @ B @ F @ G @ X ) )
= ( F @ ( product_fst @ C @ D @ X ) ) ) ).
% fst_map_prod
thf(fact_117_fst__apsnd,axiom,
! [B: $tType,C: $tType,A: $tType,F: C > B,X: product_prod @ A @ C] :
( ( product_fst @ A @ B @ ( product_apsnd @ C @ B @ A @ F @ X ) )
= ( product_fst @ A @ C @ X ) ) ).
% fst_apsnd
thf(fact_118_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_119_fst__comp__apsnd,axiom,
! [C: $tType,B: $tType,A: $tType,F: B > C] :
( ( comp @ ( product_prod @ A @ C ) @ A @ ( product_prod @ A @ B ) @ ( product_fst @ A @ C ) @ ( product_apsnd @ B @ C @ A @ F ) )
= ( product_fst @ A @ B ) ) ).
% fst_comp_apsnd
thf(fact_120_fst__comp__map__prod,axiom,
! [D: $tType,C: $tType,B: $tType,A: $tType,F: A > C,G: B > D] :
( ( comp @ ( product_prod @ C @ D ) @ C @ ( product_prod @ A @ B ) @ ( product_fst @ C @ D ) @ ( product_map_prod @ A @ C @ B @ D @ F @ G ) )
= ( comp @ A @ C @ ( product_prod @ A @ B ) @ F @ ( product_fst @ A @ B ) ) ) ).
% fst_comp_map_prod
thf(fact_121_fstI,axiom,
! [B: $tType,A: $tType,X: product_prod @ A @ B,Y4: A,Z2: B] :
( ( X
= ( product_Pair @ A @ B @ Y4 @ Z2 ) )
=> ( ( product_fst @ A @ B @ X )
= Y4 ) ) ).
% fstI
thf(fact_122_eq__fst__iff,axiom,
! [A: $tType,B: $tType,A2: A,P: product_prod @ A @ B] :
( ( A2
= ( product_fst @ A @ B @ P ) )
= ( ? [B4: B] :
( P
= ( product_Pair @ A @ B @ A2 @ B4 ) ) ) ) ).
% eq_fst_iff
thf(fact_123_fst__conv,axiom,
! [B: $tType,A: $tType,X1: A,X22: B] :
( ( product_fst @ A @ B @ ( product_Pair @ A @ B @ X1 @ X22 ) )
= X1 ) ).
% fst_conv
thf(fact_124_fst__eqD,axiom,
! [B: $tType,A: $tType,X: A,Y4: B,A2: A] :
( ( ( product_fst @ A @ B @ ( product_Pair @ A @ B @ X @ Y4 ) )
= A2 )
=> ( X = A2 ) ) ).
% fst_eqD
thf(fact_125_prod__eq__iff,axiom,
! [B: $tType,A: $tType] :
( ( ^ [Y5: product_prod @ A @ B,Z3: product_prod @ A @ B] : Y5 = Z3 )
= ( ^ [S: product_prod @ A @ B,T3: product_prod @ A @ B] :
( ( ( product_fst @ A @ B @ S )
= ( product_fst @ A @ B @ T3 ) )
& ( ( product_snd @ A @ B @ S )
= ( product_snd @ A @ B @ T3 ) ) ) ) ) ).
% prod_eq_iff
thf(fact_126_prod_Oexpand,axiom,
! [B: $tType,A: $tType,Prod: product_prod @ A @ B,Prod2: product_prod @ A @ B] :
( ( ( ( product_fst @ A @ B @ Prod )
= ( product_fst @ A @ B @ Prod2 ) )
& ( ( product_snd @ A @ B @ Prod )
= ( product_snd @ A @ B @ Prod2 ) ) )
=> ( Prod = Prod2 ) ) ).
% prod.expand
thf(fact_127_prod__eqI,axiom,
! [B: $tType,A: $tType,P: product_prod @ A @ B,Q2: product_prod @ A @ B] :
( ( ( product_fst @ A @ B @ P )
= ( product_fst @ A @ B @ Q2 ) )
=> ( ( ( product_snd @ A @ B @ P )
= ( product_snd @ A @ B @ Q2 ) )
=> ( P = Q2 ) ) ) ).
% prod_eqI
thf(fact_128_fst__convol_H,axiom,
! [B: $tType,A: $tType,C: $tType,F: C > A,G: C > B,X: C] :
( ( product_fst @ A @ B @ ( bNF_convol @ C @ A @ B @ F @ G @ X ) )
= ( F @ X ) ) ).
% fst_convol'
thf(fact_129_fst__def,axiom,
! [B: $tType,A: $tType] :
( ( product_fst @ A @ B )
= ( product_case_prod @ A @ B @ A
@ ^ [X12: A,X23: B] : X12 ) ) ).
% fst_def
thf(fact_130_BNF__Greatest__Fixpoint_Osubst__Pair,axiom,
! [B: $tType,A: $tType,P2: A > B > $o,X: A,Y4: B,A2: product_prod @ A @ B] :
( ( P2 @ X @ Y4 )
=> ( ( A2
= ( product_Pair @ A @ B @ X @ Y4 ) )
=> ( P2 @ ( product_fst @ A @ B @ A2 ) @ ( product_snd @ A @ B @ A2 ) ) ) ) ).
% BNF_Greatest_Fixpoint.subst_Pair
thf(fact_131_prod_Oexhaust__sel,axiom,
! [B: $tType,A: $tType,Prod: product_prod @ A @ B] :
( Prod
= ( product_Pair @ A @ B @ ( product_fst @ A @ B @ Prod ) @ ( product_snd @ A @ B @ Prod ) ) ) ).
% prod.exhaust_sel
thf(fact_132_surjective__pairing,axiom,
! [B: $tType,A: $tType,T2: product_prod @ A @ B] :
( T2
= ( product_Pair @ A @ B @ ( product_fst @ A @ B @ T2 ) @ ( product_snd @ A @ B @ T2 ) ) ) ).
% surjective_pairing
thf(fact_133_case__prod__beta,axiom,
! [A: $tType,C: $tType,B: $tType] :
( ( product_case_prod @ B @ C @ A )
= ( ^ [F2: B > C > A,P3: product_prod @ B @ C] : ( F2 @ ( product_fst @ B @ C @ P3 ) @ ( product_snd @ B @ C @ P3 ) ) ) ) ).
% case_prod_beta
thf(fact_134_prod_Ocase__eq__if,axiom,
! [C: $tType,B: $tType,A: $tType] :
( ( product_case_prod @ A @ B @ C )
= ( ^ [F2: A > B > C,Prod3: product_prod @ A @ B] : ( F2 @ ( product_fst @ A @ B @ Prod3 ) @ ( product_snd @ A @ B @ Prod3 ) ) ) ) ).
% prod.case_eq_if
thf(fact_135_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_136_split__comp__eq,axiom,
! [A: $tType,C: $tType,B: $tType,D: $tType,F: A > B > C,G: D > A] :
( ( ^ [U: product_prod @ D @ B] : ( F @ ( G @ ( product_fst @ D @ B @ U ) ) @ ( product_snd @ D @ B @ U ) ) )
= ( product_case_prod @ D @ B @ C
@ ^ [X2: D] : ( F @ ( G @ X2 ) ) ) ) ).
% split_comp_eq
thf(fact_137_case__prod__beta_H,axiom,
! [C: $tType,B: $tType,A: $tType] :
( ( product_case_prod @ A @ B @ C )
= ( ^ [F2: A > B > C,X2: product_prod @ A @ B] : ( F2 @ ( product_fst @ A @ B @ X2 ) @ ( product_snd @ A @ B @ X2 ) ) ) ) ).
% case_prod_beta'
thf(fact_138_case__prod__unfold,axiom,
! [C: $tType,B: $tType,A: $tType] :
( ( product_case_prod @ A @ B @ C )
= ( ^ [C4: A > B > C,P3: product_prod @ A @ B] : ( C4 @ ( product_fst @ A @ B @ P3 ) @ ( product_snd @ A @ B @ P3 ) ) ) ) ).
% case_prod_unfold
thf(fact_139_case__prod__comp,axiom,
! [D: $tType,A: $tType,C: $tType,B: $tType,F: D > C > A,G: B > D,X: product_prod @ B @ C] :
( ( product_case_prod @ B @ C @ A @ ( comp @ D @ ( C > A ) @ B @ F @ G ) @ X )
= ( F @ ( G @ ( product_fst @ B @ C @ X ) ) @ ( product_snd @ B @ C @ X ) ) ) ).
% case_prod_comp
thf(fact_140_fst__convol,axiom,
! [C: $tType,B: $tType,A: $tType,F: A > B,G: A > C] :
( ( comp @ ( product_prod @ B @ C ) @ B @ A @ ( product_fst @ B @ C ) @ ( bNF_convol @ A @ B @ C @ F @ G ) )
= F ) ).
% fst_convol
thf(fact_141_prod_Osplit__sel,axiom,
! [C: $tType,B: $tType,A: $tType,P2: C > $o,F: A > B > C,Prod: product_prod @ A @ B] :
( ( P2 @ ( product_case_prod @ A @ B @ C @ F @ Prod ) )
= ( ( Prod
= ( product_Pair @ A @ B @ ( product_fst @ A @ B @ Prod ) @ ( product_snd @ A @ B @ Prod ) ) )
=> ( P2 @ ( F @ ( product_fst @ A @ B @ Prod ) @ ( product_snd @ A @ B @ Prod ) ) ) ) ) ).
% prod.split_sel
thf(fact_142_prod_Osplit__sel__asm,axiom,
! [C: $tType,B: $tType,A: $tType,P2: C > $o,F: A > B > C,Prod: product_prod @ A @ B] :
( ( P2 @ ( product_case_prod @ A @ B @ C @ F @ Prod ) )
= ( ~ ( ( Prod
= ( product_Pair @ A @ B @ ( product_fst @ A @ B @ Prod ) @ ( product_snd @ A @ B @ Prod ) ) )
& ~ ( P2 @ ( F @ ( product_fst @ A @ B @ Prod ) @ ( product_snd @ A @ B @ Prod ) ) ) ) ) ) ).
% prod.split_sel_asm
thf(fact_143_type__copy__map__cong0,axiom,
! [B: $tType,D: $tType,E: $tType,A: $tType,C: $tType,M: B > A,G: C > B,X: C,N: D > A,H: C > D,F: A > E] :
( ( ( M @ ( G @ X ) )
= ( N @ ( H @ X ) ) )
=> ( ( comp @ B @ E @ C @ ( comp @ A @ E @ B @ F @ M ) @ G @ X )
= ( comp @ D @ E @ C @ ( comp @ A @ E @ D @ F @ N ) @ H @ X ) ) ) ).
% type_copy_map_cong0
thf(fact_144_convol__expand__snd_H,axiom,
! [C: $tType,B: $tType,A: $tType,F: A > ( product_prod @ B @ C ),G: A > B,H: A > C] :
( ( ( comp @ ( product_prod @ B @ C ) @ B @ A @ ( product_fst @ B @ C ) @ F )
= G )
=> ( ( H
= ( comp @ ( product_prod @ B @ C ) @ C @ A @ ( product_snd @ B @ C ) @ F ) )
= ( ( bNF_convol @ A @ B @ C @ G @ H )
= F ) ) ) ).
% convol_expand_snd'
thf(fact_145_convol__expand__snd,axiom,
! [C: $tType,B: $tType,A: $tType,F: A > ( product_prod @ B @ C ),G: A > B] :
( ( ( comp @ ( product_prod @ B @ C ) @ B @ A @ ( product_fst @ B @ C ) @ F )
= G )
=> ( ( bNF_convol @ A @ B @ C @ G @ ( comp @ ( product_prod @ B @ C ) @ C @ A @ ( product_snd @ B @ C ) @ F ) )
= F ) ) ).
% convol_expand_snd
thf(fact_146_fst__diag__fst,axiom,
! [B: $tType,A: $tType] :
( ( comp @ ( product_prod @ A @ A ) @ A @ ( product_prod @ A @ B ) @ ( product_fst @ A @ A )
@ ( comp @ A @ ( product_prod @ A @ A ) @ ( product_prod @ A @ B )
@ ^ [X2: A] : ( product_Pair @ A @ A @ X2 @ X2 )
@ ( product_fst @ A @ B ) ) )
= ( product_fst @ A @ B ) ) ).
% fst_diag_fst
thf(fact_147_fst__snd__flip,axiom,
! [B: $tType,A: $tType] :
( ( product_fst @ A @ B )
= ( comp @ ( product_prod @ B @ A ) @ A @ ( product_prod @ A @ B ) @ ( product_snd @ B @ A )
@ ( product_case_prod @ A @ B @ ( product_prod @ B @ A )
@ ^ [X2: A,Y: B] : ( product_Pair @ B @ A @ Y @ X2 ) ) ) ) ).
% fst_snd_flip
thf(fact_148_snd__fst__flip,axiom,
! [A: $tType,B: $tType] :
( ( product_snd @ B @ A )
= ( comp @ ( product_prod @ A @ B ) @ A @ ( product_prod @ B @ A ) @ ( product_fst @ A @ B )
@ ( product_case_prod @ B @ A @ ( product_prod @ A @ B )
@ ^ [X2: B,Y: A] : ( product_Pair @ A @ B @ Y @ X2 ) ) ) ) ).
% snd_fst_flip
thf(fact_149_exE__realizer,axiom,
! [C: $tType,A: $tType,B: $tType,P2: A > B > $o,P: product_prod @ B @ A,Q: C > $o,F: B > A > C] :
( ( P2 @ ( product_snd @ B @ A @ P ) @ ( product_fst @ B @ A @ P ) )
=> ( ! [X3: B,Y3: A] :
( ( P2 @ Y3 @ X3 )
=> ( Q @ ( F @ X3 @ Y3 ) ) )
=> ( Q @ ( product_case_prod @ B @ A @ C @ F @ P ) ) ) ) ).
% exE_realizer
thf(fact_150_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_151_exI__realizer,axiom,
! [B: $tType,A: $tType,P2: A > B > $o,Y4: A,X: B] :
( ( P2 @ Y4 @ X )
=> ( P2 @ ( product_snd @ B @ A @ ( product_Pair @ B @ A @ X @ Y4 ) ) @ ( product_fst @ B @ A @ ( product_Pair @ B @ A @ X @ Y4 ) ) ) ) ).
% exI_realizer
thf(fact_152_conjI__realizer,axiom,
! [A: $tType,B: $tType,P2: A > $o,P: A,Q: B > $o,Q2: B] :
( ( P2 @ P )
=> ( ( Q @ Q2 )
=> ( ( P2 @ ( product_fst @ A @ B @ ( product_Pair @ A @ B @ P @ Q2 ) ) )
& ( Q @ ( product_snd @ A @ B @ ( product_Pair @ A @ B @ P @ Q2 ) ) ) ) ) ) ).
% conjI_realizer
thf(fact_153_exE__realizer_H,axiom,
! [A: $tType,B: $tType,P2: A > B > $o,P: product_prod @ B @ A] :
( ( P2 @ ( product_snd @ B @ A @ P ) @ ( product_fst @ B @ A @ P ) )
=> ~ ! [X3: B,Y3: A] :
~ ( P2 @ Y3 @ X3 ) ) ).
% exE_realizer'
thf(fact_154_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_155_apfst__apsnd,axiom,
! [A: $tType,B: $tType,D: $tType,C: $tType,F: C > A,G: D > B,X: product_prod @ C @ D] :
( ( product_apfst @ C @ A @ B @ F @ ( product_apsnd @ D @ B @ C @ G @ X ) )
= ( product_Pair @ A @ B @ ( F @ ( product_fst @ C @ D @ X ) ) @ ( G @ ( product_snd @ C @ D @ X ) ) ) ) ).
% apfst_apsnd
thf(fact_156_apsnd__apfst,axiom,
! [A: $tType,B: $tType,C: $tType,D: $tType,F: C > B,G: D > A,X: product_prod @ D @ C] :
( ( product_apsnd @ C @ B @ A @ F @ ( product_apfst @ D @ A @ C @ G @ X ) )
= ( product_Pair @ A @ B @ ( G @ ( product_fst @ D @ C @ X ) ) @ ( F @ ( product_snd @ D @ C @ X ) ) ) ) ).
% apsnd_apfst
thf(fact_157_apfst__conv,axiom,
! [C: $tType,A: $tType,B: $tType,F: C > A,X: C,Y4: B] :
( ( product_apfst @ C @ A @ B @ F @ ( product_Pair @ C @ B @ X @ Y4 ) )
= ( product_Pair @ A @ B @ ( F @ X ) @ Y4 ) ) ).
% apfst_conv
thf(fact_158_snd__apfst,axiom,
! [B: $tType,A: $tType,C: $tType,F: C > B,X: product_prod @ C @ A] :
( ( product_snd @ B @ A @ ( product_apfst @ C @ B @ A @ F @ X ) )
= ( product_snd @ C @ A @ X ) ) ).
% snd_apfst
thf(fact_159_fst__apfst,axiom,
! [A: $tType,B: $tType,C: $tType,F: C > A,X: product_prod @ C @ B] :
( ( product_fst @ A @ B @ ( product_apfst @ C @ A @ B @ F @ X ) )
= ( F @ ( product_fst @ C @ B @ X ) ) ) ).
% fst_apfst
thf(fact_160_apfst__eq__conv,axiom,
! [A: $tType,B: $tType,C: $tType,F: C > A,X: product_prod @ C @ B,G: C > A] :
( ( ( product_apfst @ C @ A @ B @ F @ X )
= ( product_apfst @ C @ A @ B @ G @ X ) )
= ( ( F @ ( product_fst @ C @ B @ X ) )
= ( G @ ( product_fst @ C @ B @ X ) ) ) ) ).
% apfst_eq_conv
thf(fact_161_snd__comp__apfst,axiom,
! [C: $tType,B: $tType,A: $tType,F: A > C] :
( ( comp @ ( product_prod @ C @ B ) @ B @ ( product_prod @ A @ B ) @ ( product_snd @ C @ B ) @ ( product_apfst @ A @ C @ B @ F ) )
= ( product_snd @ A @ B ) ) ).
% snd_comp_apfst
thf(fact_162_fst__comp__apfst,axiom,
! [C: $tType,B: $tType,A: $tType,F: A > C] :
( ( comp @ ( product_prod @ C @ B ) @ C @ ( product_prod @ A @ B ) @ ( product_fst @ C @ B ) @ ( product_apfst @ A @ C @ B @ F ) )
= ( comp @ A @ C @ ( product_prod @ A @ B ) @ F @ ( product_fst @ A @ B ) ) ) ).
% fst_comp_apfst
thf(fact_163_apfst__compose,axiom,
! [C: $tType,A: $tType,B: $tType,D: $tType,F: C > A,G: D > C,X: product_prod @ D @ B] :
( ( product_apfst @ C @ A @ B @ F @ ( product_apfst @ D @ C @ B @ G @ X ) )
= ( product_apfst @ D @ A @ B @ ( comp @ C @ A @ D @ F @ G ) @ X ) ) ).
% apfst_compose
thf(fact_164_apsnd__apfst__commute,axiom,
! [A: $tType,B: $tType,C: $tType,D: $tType,F: C > B,G: D > A,P: product_prod @ D @ C] :
( ( product_apsnd @ C @ B @ A @ F @ ( product_apfst @ D @ A @ C @ G @ P ) )
= ( product_apfst @ D @ A @ B @ G @ ( product_apsnd @ C @ B @ D @ F @ P ) ) ) ).
% apsnd_apfst_commute
thf(fact_165_fst__fstOp,axiom,
! [A: $tType,B: $tType,C: $tType,P2: A > C > $o,Q: C > B > $o] :
( ( product_fst @ A @ B )
= ( comp @ ( product_prod @ A @ C ) @ A @ ( product_prod @ A @ B ) @ ( product_fst @ A @ C ) @ ( bNF_fstOp @ A @ C @ B @ P2 @ Q ) ) ) ).
% fst_fstOp
thf(fact_166_apfst__convE,axiom,
! [C: $tType,A: $tType,B: $tType,Q2: product_prod @ A @ B,F: C > A,P: product_prod @ C @ B] :
( ( Q2
= ( product_apfst @ C @ A @ B @ F @ P ) )
=> ~ ! [X3: C,Y3: B] :
( ( P
= ( product_Pair @ C @ B @ X3 @ Y3 ) )
=> ( Q2
!= ( product_Pair @ A @ B @ ( F @ X3 ) @ Y3 ) ) ) ) ).
% apfst_convE
thf(fact_167_scomp__unfold,axiom,
! [D: $tType,C: $tType,B: $tType,A: $tType] :
( ( product_scomp @ A @ B @ C @ D )
= ( ^ [F2: A > ( product_prod @ B @ C ),G2: B > C > D,X2: A] : ( G2 @ ( product_fst @ B @ C @ ( F2 @ X2 ) ) @ ( product_snd @ B @ C @ ( F2 @ X2 ) ) ) ) ) ).
% scomp_unfold
thf(fact_168_Collect__case__prod__Grp__eqD,axiom,
! [B: $tType,A: $tType,Z2: product_prod @ A @ B,A6: set @ A,F: A > B] :
( ( member @ ( product_prod @ A @ B ) @ Z2 @ ( collect @ ( product_prod @ A @ B ) @ ( product_case_prod @ A @ B @ $o @ ( bNF_Grp @ A @ B @ A6 @ F ) ) ) )
=> ( ( comp @ A @ B @ ( product_prod @ A @ B ) @ F @ ( product_fst @ A @ B ) @ Z2 )
= ( product_snd @ A @ B @ Z2 ) ) ) ).
% Collect_case_prod_Grp_eqD
thf(fact_169_scomp__apply,axiom,
! [A: $tType,D: $tType,C: $tType,B: $tType] :
( ( product_scomp @ B @ C @ D @ A )
= ( ^ [F2: B > ( product_prod @ C @ D ),G2: C > D > A,X2: B] : ( product_case_prod @ C @ D @ A @ G2 @ ( F2 @ X2 ) ) ) ) ).
% scomp_apply
thf(fact_170_scomp__scomp,axiom,
! [A: $tType,C: $tType,D: $tType,B: $tType,F3: $tType,E: $tType,F: A > ( product_prod @ E @ F3 ),G: E > F3 > ( product_prod @ C @ D ),H: C > D > B] :
( ( product_scomp @ A @ C @ D @ B @ ( product_scomp @ A @ E @ F3 @ ( product_prod @ C @ D ) @ F @ G ) @ H )
= ( product_scomp @ A @ E @ F3 @ B @ F
@ ^ [X2: E] : ( product_scomp @ F3 @ C @ D @ B @ ( G @ X2 ) @ H ) ) ) ).
% scomp_scomp
thf(fact_171_Grp__def,axiom,
! [B: $tType,A: $tType] :
( ( bNF_Grp @ A @ B )
= ( ^ [A7: set @ A,F2: A > B,A4: A,B4: B] :
( ( B4
= ( F2 @ A4 ) )
& ( member @ A @ A4 @ A7 ) ) ) ) ).
% Grp_def
thf(fact_172_GrpI,axiom,
! [B: $tType,A: $tType,F: B > A,X: B,Y4: A,A6: set @ B] :
( ( ( F @ X )
= Y4 )
=> ( ( member @ B @ X @ A6 )
=> ( bNF_Grp @ B @ A @ A6 @ F @ X @ Y4 ) ) ) ).
% GrpI
thf(fact_173_GrpE,axiom,
! [B: $tType,A: $tType,A6: set @ A,F: A > B,X: A,Y4: B] :
( ( bNF_Grp @ A @ B @ A6 @ F @ X @ Y4 )
=> ~ ( ( ( F @ X )
= Y4 )
=> ~ ( member @ A @ X @ A6 ) ) ) ).
% GrpE
thf(fact_174_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_175_Pair__scomp,axiom,
! [A: $tType,B: $tType,C: $tType,X: C,F: C > A > B] :
( ( product_scomp @ A @ C @ A @ B @ ( product_Pair @ C @ A @ X ) @ F )
= ( F @ X ) ) ).
% Pair_scomp
thf(fact_176_scomp__def,axiom,
! [D: $tType,C: $tType,B: $tType,A: $tType] :
( ( product_scomp @ A @ B @ C @ D )
= ( ^ [F2: A > ( product_prod @ B @ C ),G2: B > C > D,X2: A] : ( product_case_prod @ B @ C @ D @ G2 @ ( F2 @ X2 ) ) ) ) ).
% scomp_def
thf(fact_177_Collect__case__prod__Grp__in,axiom,
! [B: $tType,A: $tType,Z2: product_prod @ A @ B,A6: set @ A,F: A > B] :
( ( member @ ( product_prod @ A @ B ) @ Z2 @ ( collect @ ( product_prod @ A @ B ) @ ( product_case_prod @ A @ B @ $o @ ( bNF_Grp @ A @ B @ A6 @ F ) ) ) )
=> ( member @ A @ ( product_fst @ A @ B @ Z2 ) @ A6 ) ) ).
% Collect_case_prod_Grp_in
thf(fact_178_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_179_map__prod__o__convol__id,axiom,
! [B: $tType,A: $tType,C: $tType,F: C > A,G: C > B,X: C] :
( ( comp @ ( product_prod @ C @ B ) @ ( product_prod @ A @ B ) @ C @ ( product_map_prod @ C @ A @ B @ B @ F @ ( id @ B ) ) @ ( bNF_convol @ C @ C @ B @ ( id @ C ) @ G ) @ X )
= ( bNF_convol @ C @ A @ B @ ( comp @ A @ A @ C @ ( id @ A ) @ F ) @ G @ X ) ) ).
% map_prod_o_convol_id
thf(fact_180_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_181_id__apply,axiom,
! [A: $tType] :
( ( id @ A )
= ( ^ [X2: A] : X2 ) ) ).
% id_apply
thf(fact_182_swap__swap,axiom,
! [B: $tType,A: $tType,P: product_prod @ A @ B] :
( ( product_swap @ B @ A @ ( product_swap @ A @ B @ P ) )
= P ) ).
% swap_swap
thf(fact_183_comp__id,axiom,
! [B: $tType,A: $tType,F: A > B] :
( ( comp @ A @ B @ A @ F @ ( id @ A ) )
= F ) ).
% comp_id
thf(fact_184_id__comp,axiom,
! [B: $tType,A: $tType,G: A > B] :
( ( comp @ B @ B @ A @ ( id @ B ) @ G )
= G ) ).
% id_comp
thf(fact_185_fun_Omap__id,axiom,
! [A: $tType,D: $tType,T2: D > A] :
( ( comp @ A @ A @ D @ ( id @ A ) @ T2 )
= T2 ) ).
% fun.map_id
thf(fact_186_case__prod__Pair,axiom,
! [B: $tType,A: $tType] :
( ( product_case_prod @ A @ B @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B ) )
= ( id @ ( product_prod @ A @ B ) ) ) ).
% case_prod_Pair
thf(fact_187_swap__simp,axiom,
! [A: $tType,B: $tType,X: B,Y4: A] :
( ( product_swap @ B @ A @ ( product_Pair @ B @ A @ X @ Y4 ) )
= ( product_Pair @ A @ B @ Y4 @ X ) ) ).
% swap_simp
thf(fact_188_apfst__id,axiom,
! [B: $tType,A: $tType] :
( ( product_apfst @ A @ A @ B @ ( id @ A ) )
= ( id @ ( product_prod @ A @ B ) ) ) ).
% apfst_id
thf(fact_189_apsnd__id,axiom,
! [B: $tType,A: $tType] :
( ( product_apsnd @ B @ B @ A @ ( id @ B ) )
= ( id @ ( product_prod @ A @ B ) ) ) ).
% apsnd_id
thf(fact_190_case__swap,axiom,
! [A: $tType,B: $tType,C: $tType,F: C > B > A,P: product_prod @ C @ B] :
( ( product_case_prod @ B @ C @ A
@ ^ [Y: B,X2: C] : ( F @ X2 @ Y )
@ ( product_swap @ C @ B @ P ) )
= ( product_case_prod @ C @ B @ A @ F @ P ) ) ).
% case_swap
thf(fact_191_swap__comp__swap,axiom,
! [B: $tType,A: $tType] :
( ( comp @ ( product_prod @ B @ A ) @ ( product_prod @ A @ B ) @ ( product_prod @ A @ B ) @ ( product_swap @ B @ A ) @ ( product_swap @ A @ B ) )
= ( id @ ( product_prod @ A @ B ) ) ) ).
% swap_comp_swap
thf(fact_192_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_193_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_194_map__prod_Oidentity,axiom,
! [B: $tType,A: $tType] :
( ( product_map_prod @ A @ A @ B @ B
@ ^ [X2: A] : X2
@ ^ [X2: B] : X2 )
= ( id @ ( product_prod @ A @ B ) ) ) ).
% map_prod.identity
thf(fact_195_fun_Omap__id0,axiom,
! [A: $tType,D: $tType] :
( ( comp @ A @ A @ D @ ( id @ A ) )
= ( id @ ( D > A ) ) ) ).
% fun.map_id0
thf(fact_196_comp__eq__id__dest,axiom,
! [C: $tType,B: $tType,A: $tType,A2: C > B,B2: A > C,C2: A > B,V: A] :
( ( ( comp @ C @ B @ A @ A2 @ B2 )
= ( comp @ B @ B @ A @ ( id @ B ) @ C2 ) )
=> ( ( A2 @ ( B2 @ V ) )
= ( C2 @ V ) ) ) ).
% comp_eq_id_dest
thf(fact_197_pointfree__idE,axiom,
! [B: $tType,A: $tType,F: B > A,G: A > B,X: A] :
( ( ( comp @ B @ A @ A @ F @ G )
= ( id @ A ) )
=> ( ( F @ ( G @ X ) )
= X ) ) ).
% pointfree_idE
thf(fact_198_prod_Omap__id0,axiom,
! [B: $tType,A: $tType] :
( ( product_map_prod @ A @ A @ B @ B @ ( id @ A ) @ ( id @ B ) )
= ( id @ ( product_prod @ A @ B ) ) ) ).
% prod.map_id0
thf(fact_199_id__def,axiom,
! [A: $tType] :
( ( id @ A )
= ( ^ [X2: A] : X2 ) ) ).
% id_def
thf(fact_200_eq__id__iff,axiom,
! [A: $tType,F: A > A] :
( ( ! [X2: A] :
( ( F @ X2 )
= X2 ) )
= ( F
= ( id @ A ) ) ) ).
% eq_id_iff
thf(fact_201_prod_Omap__id,axiom,
! [B: $tType,A: $tType,T2: product_prod @ A @ B] :
( ( product_map_prod @ A @ A @ B @ B @ ( id @ A ) @ ( id @ B ) @ T2 )
= T2 ) ).
% prod.map_id
thf(fact_202_apfst__def,axiom,
! [B: $tType,C: $tType,A: $tType] :
( ( product_apfst @ A @ C @ B )
= ( ^ [F2: A > C] : ( product_map_prod @ A @ C @ B @ B @ F2 @ ( id @ B ) ) ) ) ).
% apfst_def
thf(fact_203_apsnd__def,axiom,
! [B: $tType,C: $tType,A: $tType] :
( ( product_apsnd @ B @ C @ A )
= ( product_map_prod @ A @ A @ B @ C @ ( id @ A ) ) ) ).
% apsnd_def
thf(fact_204_convol__mem__GrpI,axiom,
! [B: $tType,A: $tType,X: A,A6: set @ A,G: A > B] :
( ( member @ A @ X @ A6 )
=> ( member @ ( product_prod @ A @ B ) @ ( bNF_convol @ A @ A @ B @ ( id @ A ) @ G @ X ) @ ( collect @ ( product_prod @ A @ B ) @ ( product_case_prod @ A @ B @ $o @ ( bNF_Grp @ A @ B @ A6 @ G ) ) ) ) ) ).
% convol_mem_GrpI
thf(fact_205_snd__diag__id,axiom,
! [A: $tType,Z2: A] :
( ( comp @ ( product_prod @ A @ A ) @ A @ A @ ( product_snd @ A @ A )
@ ^ [X2: A] : ( product_Pair @ A @ A @ X2 @ X2 )
@ Z2 )
= ( id @ A @ Z2 ) ) ).
% snd_diag_id
thf(fact_206_fst__diag__id,axiom,
! [A: $tType,Z2: A] :
( ( comp @ ( product_prod @ A @ A ) @ A @ A @ ( product_fst @ A @ A )
@ ^ [X2: A] : ( product_Pair @ A @ A @ X2 @ X2 )
@ Z2 )
= ( id @ A @ Z2 ) ) ).
% fst_diag_id
thf(fact_207_internal__case__prod__conv,axiom,
! [B: $tType,A: $tType,C: $tType,C2: B > C > A,A2: B,B2: C] :
( ( produc2004651681e_prod @ B @ C @ A @ C2 @ ( product_Pair @ B @ C @ A2 @ B2 ) )
= ( C2 @ A2 @ B2 ) ) ).
% internal_case_prod_conv
thf(fact_208_inf__Int__eq2,axiom,
! [B: $tType,A: $tType,R: set @ ( product_prod @ A @ B ),S2: set @ ( product_prod @ A @ B )] :
( ( inf_inf @ ( A > B > $o )
@ ^ [X2: A,Y: B] : ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X2 @ Y ) @ R )
@ ^ [X2: A,Y: B] : ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X2 @ Y ) @ S2 ) )
= ( ^ [X2: A,Y: B] : ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X2 @ Y ) @ ( inf_inf @ ( set @ ( product_prod @ A @ B ) ) @ R @ S2 ) ) ) ) ).
% inf_Int_eq2
thf(fact_209_pred__equals__eq2,axiom,
! [B: $tType,A: $tType,R: set @ ( product_prod @ A @ B ),S2: set @ ( product_prod @ A @ B )] :
( ( ( ^ [X2: A,Y: B] : ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X2 @ Y ) @ R ) )
= ( ^ [X2: A,Y: B] : ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X2 @ Y ) @ S2 ) ) )
= ( R = S2 ) ) ).
% pred_equals_eq2
thf(fact_210_prod_Osize__gen__o__map,axiom,
! [A: $tType,C: $tType,D: $tType,B: $tType,F: C > nat,Fa: D > nat,G: A > C,Ga: B > D] :
( ( comp @ ( product_prod @ C @ D ) @ nat @ ( product_prod @ A @ B ) @ ( basic_BNF_size_prod @ C @ D @ F @ Fa ) @ ( product_map_prod @ A @ C @ B @ D @ G @ Ga ) )
= ( basic_BNF_size_prod @ A @ B @ ( comp @ C @ nat @ A @ F @ G ) @ ( comp @ D @ nat @ B @ Fa @ Ga ) ) ) ).
% prod.size_gen_o_map
thf(fact_211_inv__image__def,axiom,
! [A: $tType,B: $tType] :
( ( inv_image @ B @ A )
= ( ^ [R2: set @ ( product_prod @ B @ B ),F2: A > B] :
( collect @ ( product_prod @ A @ A )
@ ( product_case_prod @ A @ A @ $o
@ ^ [X2: A,Y: A] : ( member @ ( product_prod @ B @ B ) @ ( product_Pair @ B @ B @ ( F2 @ X2 ) @ ( F2 @ Y ) ) @ R2 ) ) ) ) ) ).
% inv_image_def
thf(fact_212_iso__tuple__update__accessor__eq__assist__idI,axiom,
! [A: $tType,V3: A,F: A > A,V: A] :
( ( V3
= ( F @ V ) )
=> ( iso_tu2011167877assist @ A @ A @ ( id @ ( A > A ) ) @ ( id @ A ) @ V @ F @ V3 @ V ) ) ).
% iso_tuple_update_accessor_eq_assist_idI
thf(fact_213_in__inv__image,axiom,
! [A: $tType,B: $tType,X: A,Y4: A,R3: set @ ( product_prod @ B @ B ),F: A > B] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X @ Y4 ) @ ( inv_image @ B @ A @ R3 @ F ) )
= ( member @ ( product_prod @ B @ B ) @ ( product_Pair @ B @ B @ ( F @ X ) @ ( F @ Y4 ) ) @ R3 ) ) ).
% in_inv_image
thf(fact_214_update__accessor__updator__eqE,axiom,
! [A: $tType,B: $tType,Upd: ( A > A ) > B > B,Ac2: B > A,V: B,F: A > A,V3: B,X: A] :
( ( iso_tu2011167877assist @ A @ B @ Upd @ Ac2 @ V @ F @ V3 @ X )
=> ( ( Upd @ F @ V )
= V3 ) ) ).
% update_accessor_updator_eqE
thf(fact_215_update__accessor__accessor__eqE,axiom,
! [B: $tType,A: $tType,Upd: ( A > A ) > B > B,Ac2: B > A,V: B,F: A > A,V3: B,X: A] :
( ( iso_tu2011167877assist @ A @ B @ Upd @ Ac2 @ V @ F @ V3 @ X )
=> ( ( Ac2 @ V )
= X ) ) ).
% update_accessor_accessor_eqE
thf(fact_216_iso__tuple__update__accessor__eq__assist__triv,axiom,
! [B: $tType,A: $tType,Upd: ( A > A ) > B > B,Ac2: B > A,V: B,F: A > A,V3: B,X: A] :
( ( iso_tu2011167877assist @ A @ B @ Upd @ Ac2 @ V @ F @ V3 @ X )
=> ( iso_tu2011167877assist @ A @ B @ Upd @ Ac2 @ V @ F @ V3 @ X ) ) ).
% iso_tuple_update_accessor_eq_assist_triv
thf(fact_217_rp__inv__image__def,axiom,
! [B: $tType,A: $tType] :
( ( fun_rp_inv_image @ A @ B )
= ( product_case_prod @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ A @ A ) ) @ ( ( B > A ) > ( product_prod @ ( set @ ( product_prod @ B @ B ) ) @ ( set @ ( product_prod @ B @ B ) ) ) )
@ ^ [R4: set @ ( product_prod @ A @ A ),S3: set @ ( product_prod @ A @ A ),F2: B > A] : ( product_Pair @ ( set @ ( product_prod @ B @ B ) ) @ ( set @ ( product_prod @ B @ B ) ) @ ( inv_image @ A @ B @ R4 @ F2 ) @ ( inv_image @ A @ B @ S3 @ F2 ) ) ) ) ).
% rp_inv_image_def
thf(fact_218_fun_Opred__map,axiom,
! [B: $tType,A: $tType,D: $tType,Q: B > $o,F: A > B,X: D > A] :
( ( basic_pred_fun @ D @ B
@ ^ [Uu: D] : $true
@ Q
@ ( comp @ A @ B @ D @ F @ X ) )
= ( basic_pred_fun @ D @ A
@ ^ [Uu: D] : $true
@ ( comp @ B @ $o @ A @ Q @ F )
@ X ) ) ).
% fun.pred_map
thf(fact_219_pred__funI,axiom,
! [B: $tType,A: $tType,A6: A > $o,B6: B > $o,F: A > B] :
( ! [X3: A] :
( ( A6 @ X3 )
=> ( B6 @ ( F @ X3 ) ) )
=> ( basic_pred_fun @ A @ B @ A6 @ B6 @ F ) ) ).
% pred_funI
thf(fact_220_fun_Opred__True,axiom,
! [A: $tType,D: $tType] :
( ( basic_pred_fun @ D @ A
@ ^ [Uu: D] : $true
@ ^ [Uu: A] : $true )
= ( ^ [Uu: D > A] : $true ) ) ).
% fun.pred_True
thf(fact_221_fun_Omap__cong__pred,axiom,
! [B: $tType,A: $tType,D: $tType,X: D > A,Ya: D > A,F: A > B,G: A > B] :
( ( X = Ya )
=> ( ( basic_pred_fun @ D @ A
@ ^ [Uu: D] : $true
@ ^ [Z: A] :
( ( F @ Z )
= ( G @ Z ) )
@ Ya )
=> ( ( comp @ A @ B @ D @ F @ X )
= ( comp @ A @ B @ D @ G @ Ya ) ) ) ) ).
% fun.map_cong_pred
thf(fact_222_rp__inv__image__rp,axiom,
! [A: $tType,B: $tType,P2: product_prod @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ A @ A ) ),F: B > A] :
( ( fun_reduction_pair @ A @ P2 )
=> ( fun_reduction_pair @ B @ ( fun_rp_inv_image @ A @ B @ P2 @ F ) ) ) ).
% rp_inv_image_rp
thf(fact_223_csquare__fstOp__sndOp,axiom,
! [A: $tType,B: $tType,C: $tType,F: ( A > B > $o ) > ( product_prod @ A @ B ) > $o,P2: 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 ) @ ( F @ ( relcompp @ A @ C @ B @ P2 @ Q ) ) ) @ ( product_snd @ A @ C ) @ ( product_fst @ C @ B ) @ ( bNF_fstOp @ A @ C @ B @ P2 @ Q ) @ ( bNF_sndOp @ A @ C @ B @ P2 @ Q ) ) ).
% csquare_fstOp_sndOp
thf(fact_224_fstOp__in,axiom,
! [B: $tType,C: $tType,A: $tType,Ac2: product_prod @ A @ B,P2: 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 @ P2 @ Q ) ) ) )
=> ( member @ ( product_prod @ A @ C ) @ ( bNF_fstOp @ A @ C @ B @ P2 @ Q @ Ac2 ) @ ( collect @ ( product_prod @ A @ C ) @ ( product_case_prod @ A @ C @ $o @ P2 ) ) ) ) ).
% fstOp_in
thf(fact_225_sndOp__in,axiom,
! [A: $tType,B: $tType,C: $tType,Ac2: product_prod @ A @ B,P2: 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 @ P2 @ Q ) ) ) )
=> ( member @ ( product_prod @ C @ B ) @ ( bNF_sndOp @ A @ C @ B @ P2 @ Q @ Ac2 ) @ ( collect @ ( product_prod @ C @ B ) @ ( product_case_prod @ C @ B @ $o @ Q ) ) ) ) ).
% sndOp_in
thf(fact_226_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] :
! [X2: A] :
( ( member @ A @ X2 @ A7 )
=> ( ( F12 @ ( P1 @ X2 ) )
= ( F23 @ ( P22 @ X2 ) ) ) ) ) ) ).
% csquare_def
thf(fact_227_pick__middlep,axiom,
! [B: $tType,A: $tType,C: $tType,P2: A > B > $o,Q: B > C > $o,A2: A,C2: C] :
( ( relcompp @ A @ B @ C @ P2 @ Q @ A2 @ C2 )
=> ( ( P2 @ A2 @ ( bNF_pick_middlep @ A @ B @ C @ P2 @ Q @ A2 @ C2 ) )
& ( Q @ ( bNF_pick_middlep @ A @ B @ C @ P2 @ Q @ A2 @ C2 ) @ C2 ) ) ) ).
% pick_middlep
thf(fact_228_update__accessor__cong__assist__idI,axiom,
! [A: $tType] : ( iso_tu2017585022assist @ A @ A @ ( id @ ( A > A ) ) @ ( id @ A ) ) ).
% update_accessor_cong_assist_idI
thf(fact_229_The__case__prod,axiom,
! [B: $tType,A: $tType,P2: A > B > $o] :
( ( the @ ( product_prod @ A @ B ) @ ( product_case_prod @ A @ B @ $o @ P2 ) )
= ( the @ ( product_prod @ A @ B )
@ ^ [Xy: product_prod @ A @ B] : ( P2 @ ( product_fst @ A @ B @ Xy ) @ ( product_snd @ A @ B @ Xy ) ) ) ) ).
% The_case_prod
thf(fact_230_The__split__eq,axiom,
! [A: $tType,B: $tType,X: A,Y4: B] :
( ( the @ ( product_prod @ A @ B )
@ ( product_case_prod @ A @ B @ $o
@ ^ [X5: A,Y6: B] :
( ( X = X5 )
& ( Y4 = Y6 ) ) ) )
= ( product_Pair @ A @ B @ X @ Y4 ) ) ).
% The_split_eq
thf(fact_231_iso__tuple__update__accessor__cong__from__eq,axiom,
! [A: $tType,B: $tType,Upd: ( A > A ) > B > B,Ac2: B > A,V: B,F: A > A,V3: B,X: A] :
( ( iso_tu2011167877assist @ A @ B @ Upd @ Ac2 @ V @ F @ V3 @ X )
=> ( iso_tu2017585022assist @ A @ B @ Upd @ Ac2 ) ) ).
% iso_tuple_update_accessor_cong_from_eq
thf(fact_232_iso__tuple__update__accessor__eq__assist__def,axiom,
! [A: $tType,B: $tType] :
( ( iso_tu2011167877assist @ B @ A )
= ( ^ [Upd2: ( B > B ) > A > A,Ac: A > B,V4: A,F2: B > B,V5: A,X2: B] :
( ( ( Upd2 @ F2 @ V4 )
= V5 )
& ( ( Ac @ V4 )
= X2 )
& ( iso_tu2017585022assist @ B @ A @ Upd2 @ Ac ) ) ) ) ).
% iso_tuple_update_accessor_eq_assist_def
thf(fact_233_update__accessor__congruence__unfoldE,axiom,
! [A: $tType,B: $tType,Upd: ( A > A ) > B > B,Ac2: B > A,R3: B,R5: B,V3: A,F: A > A,F5: A > A] :
( ( iso_tu2017585022assist @ A @ B @ Upd @ Ac2 )
=> ( ( R3 = R5 )
=> ( ( ( Ac2 @ R5 )
= V3 )
=> ( ! [V6: A] :
( ( V6 = V3 )
=> ( ( F @ V6 )
= ( F5 @ V6 ) ) )
=> ( ( Upd @ F @ R3 )
= ( Upd @ F5 @ R5 ) ) ) ) ) ) ).
% update_accessor_congruence_unfoldE
thf(fact_234_update__accessor__congruence__foldE,axiom,
! [A: $tType,B: $tType,Upd: ( A > A ) > B > B,Ac2: B > A,R3: B,R5: B,V3: A,F: A > A,F5: A > A] :
( ( iso_tu2017585022assist @ A @ B @ Upd @ Ac2 )
=> ( ( R3 = R5 )
=> ( ( ( Ac2 @ R5 )
= V3 )
=> ( ! [V6: A] :
( ( V3 = V6 )
=> ( ( F @ V6 )
= ( F5 @ V6 ) ) )
=> ( ( Upd @ F @ R3 )
= ( Upd @ F5 @ R5 ) ) ) ) ) ) ).
% update_accessor_congruence_foldE
thf(fact_235_update__accessor__cong__assist__triv,axiom,
! [A: $tType,B: $tType,Upd: ( A > A ) > B > B,Ac2: B > A] :
( ( iso_tu2017585022assist @ A @ B @ Upd @ Ac2 )
=> ( iso_tu2017585022assist @ A @ B @ Upd @ Ac2 ) ) ).
% update_accessor_cong_assist_triv
thf(fact_236_update__accessor__noopE,axiom,
! [A: $tType,B: $tType,Upd: ( A > A ) > B > B,Ac2: B > A,F: A > A,X: B] :
( ( iso_tu2017585022assist @ A @ B @ Upd @ Ac2 )
=> ( ( ( F @ ( Ac2 @ X ) )
= ( Ac2 @ X ) )
=> ( ( Upd @ F @ X )
= X ) ) ) ).
% update_accessor_noopE
thf(fact_237_iso__tuple__update__accessor__cong__assist__def,axiom,
! [A: $tType,B: $tType] :
( ( iso_tu2017585022assist @ B @ A )
= ( ^ [Upd2: ( B > B ) > A > A,Ac: A > B] :
( ! [F2: B > B,V4: A] :
( ( Upd2
@ ^ [X2: B] : ( F2 @ ( Ac @ V4 ) )
@ V4 )
= ( Upd2 @ F2 @ V4 ) )
& ! [V4: A] :
( ( Upd2 @ ( id @ B ) @ V4 )
= V4 ) ) ) ) ).
% iso_tuple_update_accessor_cong_assist_def
thf(fact_238_update__accessor__noop__compE,axiom,
! [A: $tType,B: $tType,Upd: ( A > A ) > B > B,Ac2: B > A,F: A > A,X: B,G: A > A] :
( ( iso_tu2017585022assist @ A @ B @ Upd @ Ac2 )
=> ( ( ( F @ ( Ac2 @ X ) )
= ( Ac2 @ X ) )
=> ( ( Upd @ ( comp @ A @ A @ A @ G @ F ) @ X )
= ( Upd @ G @ X ) ) ) ) ).
% update_accessor_noop_compE
thf(fact_239_iso__tuple__update__accessor__cong__assist__id,axiom,
! [A: $tType,B: $tType,Upd: ( A > A ) > B > B,Ac2: B > A] :
( ( iso_tu2017585022assist @ A @ B @ Upd @ Ac2 )
=> ( ( Upd @ ( id @ A ) )
= ( id @ B ) ) ) ).
% iso_tuple_update_accessor_cong_assist_id
thf(fact_240_old_Orec__prod__def,axiom,
! [T: $tType,B: $tType,A: $tType] :
( ( product_rec_prod @ A @ B @ T )
= ( ^ [F12: A > B > T,X2: product_prod @ A @ B] : ( the @ T @ ( product_rec_set_prod @ A @ B @ T @ F12 @ X2 ) ) ) ) ).
% old.rec_prod_def
thf(fact_241_the__sym__eq__trivial,axiom,
! [A: $tType,X: A] :
( ( the @ A
@ ( ^ [Y5: A,Z3: A] : Y5 = Z3
@ X ) )
= X ) ).
% the_sym_eq_trivial
thf(fact_242_the__equality,axiom,
! [A: $tType,P2: A > $o,A2: A] :
( ( P2 @ A2 )
=> ( ! [X3: A] :
( ( P2 @ X3 )
=> ( X3 = A2 ) )
=> ( ( the @ A @ P2 )
= A2 ) ) ) ).
% the_equality
thf(fact_243_the__eq__trivial,axiom,
! [A: $tType,A2: A] :
( ( the @ A
@ ^ [X2: A] : X2 = A2 )
= A2 ) ).
% the_eq_trivial
thf(fact_244_theI,axiom,
! [A: $tType,P2: A > $o,A2: A] :
( ( P2 @ A2 )
=> ( ! [X3: A] :
( ( P2 @ X3 )
=> ( X3 = A2 ) )
=> ( P2 @ ( the @ A @ P2 ) ) ) ) ).
% theI
thf(fact_245_theI_H,axiom,
! [A: $tType,P2: A > $o] :
( ? [X6: A] :
( ( P2 @ X6 )
& ! [Y3: A] :
( ( P2 @ Y3 )
=> ( Y3 = X6 ) ) )
=> ( P2 @ ( the @ A @ P2 ) ) ) ).
% theI'
thf(fact_246_theI2,axiom,
! [A: $tType,P2: A > $o,A2: A,Q: A > $o] :
( ( P2 @ A2 )
=> ( ! [X3: A] :
( ( P2 @ X3 )
=> ( X3 = A2 ) )
=> ( ! [X3: A] :
( ( P2 @ X3 )
=> ( Q @ X3 ) )
=> ( Q @ ( the @ A @ P2 ) ) ) ) ) ).
% theI2
thf(fact_247_If__def,axiom,
! [A: $tType] :
( ( if @ A )
= ( ^ [P4: $o,X2: A,Y: A] :
( the @ A
@ ^ [Z: A] :
( ( P4
=> ( Z = X2 ) )
& ( ~ P4
=> ( Z = Y ) ) ) ) ) ) ).
% If_def
thf(fact_248_the1I2,axiom,
! [A: $tType,P2: A > $o,Q: A > $o] :
( ? [X6: A] :
( ( P2 @ X6 )
& ! [Y3: A] :
( ( P2 @ Y3 )
=> ( Y3 = X6 ) ) )
=> ( ! [X3: A] :
( ( P2 @ X3 )
=> ( Q @ X3 ) )
=> ( Q @ ( the @ A @ P2 ) ) ) ) ).
% the1I2
thf(fact_249_the1__equality,axiom,
! [A: $tType,P2: A > $o,A2: A] :
( ? [X6: A] :
( ( P2 @ X6 )
& ! [Y3: A] :
( ( P2 @ Y3 )
=> ( Y3 = X6 ) ) )
=> ( ( P2 @ A2 )
=> ( ( the @ A @ P2 )
= A2 ) ) ) ).
% the1_equality
thf(fact_250_old_Orec__unit__def,axiom,
! [T: $tType] :
( ( product_rec_unit @ T )
= ( ^ [F12: T,X2: product_unit] : ( the @ T @ ( product_rec_set_unit @ T @ F12 @ X2 ) ) ) ) ).
% old.rec_unit_def
thf(fact_251_image2__eqI,axiom,
! [A: $tType,C: $tType,B: $tType,B2: A,F: B > A,X: B,C2: C,G: B > C,A6: set @ B] :
( ( B2
= ( F @ X ) )
=> ( ( C2
= ( G @ X ) )
=> ( ( member @ B @ X @ A6 )
=> ( member @ ( product_prod @ A @ C ) @ ( product_Pair @ A @ C @ B2 @ C2 ) @ ( bNF_Greatest_image2 @ B @ A @ C @ A6 @ F @ G ) ) ) ) ) ).
% image2_eqI
thf(fact_252_same__fst__def,axiom,
! [B: $tType,A: $tType] :
( ( same_fst @ A @ B )
= ( ^ [P4: A > $o,R4: A > ( set @ ( product_prod @ B @ B ) )] :
( collect @ ( product_prod @ ( product_prod @ A @ B ) @ ( product_prod @ A @ B ) )
@ ( product_case_prod @ ( product_prod @ A @ B ) @ ( product_prod @ A @ B ) @ $o
@ ( product_case_prod @ A @ B @ ( ( product_prod @ A @ B ) > $o )
@ ^ [X5: A,Y6: B] :
( product_case_prod @ A @ B @ $o
@ ^ [X2: A,Y: B] :
( ( X5 = X2 )
& ( P4 @ X2 )
& ( member @ ( product_prod @ B @ B ) @ ( product_Pair @ B @ B @ Y6 @ Y ) @ ( R4 @ X2 ) ) ) ) ) ) ) ) ) ).
% same_fst_def
thf(fact_253_old_Orec__bool__def,axiom,
! [T: $tType] :
( ( product_rec_bool @ T )
= ( ^ [F12: T,F23: T,X2: $o] : ( the @ T @ ( product_rec_set_bool @ T @ F12 @ F23 @ X2 ) ) ) ) ).
% old.rec_bool_def
thf(fact_254_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_255_old_Obool_Osimps_I5_J,axiom,
! [T: $tType,F1: T,F22: T] :
( ( product_rec_bool @ T @ F1 @ F22 @ $true )
= F1 ) ).
% old.bool.simps(5)
%----Type constructors (10)
thf(tcon_fun___Lattices_Osemilattice__inf,axiom,
! [A8: $tType,A9: $tType] :
( ( semilattice_inf @ A9 @ ( type2 @ A9 ) )
=> ( semilattice_inf @ ( A8 > A9 ) @ ( type2 @ ( A8 > A9 ) ) ) ) ).
thf(tcon_fun___Lattices_Olattice,axiom,
! [A8: $tType,A9: $tType] :
( ( lattice @ A9 @ ( type2 @ A9 ) )
=> ( lattice @ ( A8 > A9 ) @ ( type2 @ ( A8 > A9 ) ) ) ) ).
thf(tcon_Nat_Onat___Lattices_Osemilattice__inf_1,axiom,
semilattice_inf @ nat @ ( type2 @ nat ) ).
thf(tcon_Nat_Onat___Lattices_Olattice_2,axiom,
lattice @ nat @ ( type2 @ nat ) ).
thf(tcon_Set_Oset___Lattices_Osemilattice__inf_3,axiom,
! [A8: $tType] : ( semilattice_inf @ ( set @ A8 ) @ ( type2 @ ( set @ A8 ) ) ) ).
thf(tcon_Set_Oset___Lattices_Olattice_4,axiom,
! [A8: $tType] : ( lattice @ ( set @ A8 ) @ ( type2 @ ( set @ A8 ) ) ) ).
thf(tcon_HOL_Obool___Lattices_Osemilattice__inf_5,axiom,
semilattice_inf @ $o @ ( type2 @ $o ) ).
thf(tcon_HOL_Obool___Lattices_Olattice_6,axiom,
lattice @ $o @ ( type2 @ $o ) ).
thf(tcon_Product__Type_Ounit___Lattices_Osemilattice__inf_7,axiom,
semilattice_inf @ product_unit @ ( type2 @ product_unit ) ).
thf(tcon_Product__Type_Ounit___Lattices_Olattice_8,axiom,
lattice @ product_unit @ ( type2 @ product_unit ) ).
%----Helper facts (3)
thf(help_If_3_1_T,axiom,
! [P2: $o] :
( ( P2 = $true )
| ( P2 = $false ) ) ).
thf(help_If_2_1_T,axiom,
! [A: $tType,X: A,Y4: A] :
( ( if @ A @ $false @ X @ Y4 )
= Y4 ) ).
thf(help_If_1_1_T,axiom,
! [A: $tType,X: A,Y4: A] :
( ( if @ A @ $true @ X @ Y4 )
= X ) ).
%----Conjectures (1)
thf(conj_0,conjecture,
( ( coindu1441602521_llist @ ( product_prod @ ( coinductive_llist @ a ) @ ( coinductive_llist @ a ) ) @ a
@ ( product_case_prod @ ( coinductive_llist @ a ) @ ( coinductive_llist @ a ) @ $o
@ ^ [Xs: coinductive_llist @ a,Ys: coinductive_llist @ a] :
( ( Xs
!= ( coinductive_LNil @ a ) )
=> ( ( Ys
!= ( coinductive_LNil @ a ) )
=> ( ( coinductive_lhd @ a @ Xs )
!= ( coinductive_lhd @ a @ Ys ) ) ) ) )
@ ( comp @ ( coinductive_llist @ a ) @ a @ ( product_prod @ ( coinductive_llist @ a ) @ ( coinductive_llist @ a ) ) @ ( coinductive_lhd @ a ) @ ( product_snd @ ( coinductive_llist @ a ) @ ( coinductive_llist @ a ) ) )
@ ( product_map_prod @ ( coinductive_llist @ a ) @ ( coinductive_llist @ a ) @ ( coinductive_llist @ a ) @ ( coinductive_llist @ a ) @ ( coinductive_ltl @ a ) @ ( coinductive_ltl @ a ) )
@ ( product_Pair @ ( coinductive_llist @ a ) @ ( coinductive_llist @ a ) @ ( coinductive_LNil @ a ) @ xs ) )
= ( coinductive_LNil @ a ) ) ).
%------------------------------------------------------------------------------