TPTP Problem File: DAT258^1.p
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%------------------------------------------------------------------------------
% File : DAT258^1 : TPTP v8.2.0. Released v7.0.0.
% Domain : Data Structures
% Problem : Data structure for translators from streams to streams 166
% 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 : stream_processor__166.p [Bla16]
% Status : Theorem
% Rating : 1.00 v7.1.0
% Syntax : Number of formulae : 315 ( 155 unt; 55 typ; 0 def)
% Number of atoms : 477 ( 337 equ; 0 cnn)
% Maximal formula atoms : 7 ( 1 avg)
% Number of connectives : 4864 ( 32 ~; 1 |; 11 &;4641 @)
% ( 0 <=>; 179 =>; 0 <=; 0 <~>)
% Maximal formula depth : 28 ( 9 avg)
% Number of types : 4 ( 3 usr)
% Number of type conns : 695 ( 695 >; 0 *; 0 +; 0 <<)
% Number of symbols : 55 ( 52 usr; 4 con; 0-10 aty)
% Number of variables : 1640 ( 80 ^;1447 !; 11 ?;1640 :)
% ( 102 !>; 0 ?*; 0 @-; 0 @+)
% SPC : TH1_THM_EQU_NAR
% Comments : This file was generated by Isabelle (most likely Sledgehammer)
% 2016-07-13 14:39:39.033
%------------------------------------------------------------------------------
%----Could-be-implicit typings (10)
thf(ty_t_Stream__Processor__Mirabelle__rrumbueyrq_Osp_092_060_094sub_062_092_060nu_062,type,
stream1273403375_sp_nu: $tType > $tType > $tType ).
thf(ty_t_Stream__Processor__Mirabelle__rrumbueyrq_Osp_092_060_094sub_062_092_060mu_062,type,
stream901396144_sp_mu: $tType > $tType > $tType > $tType ).
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_Stream_Ostream,type,
stream: $tType > $tType ).
thf(ty_t_Set_Oset,type,
set: $tType > $tType ).
thf(ty_t_itself,type,
itself: $tType > $tType ).
thf(ty_tf_c,type,
c: $tType ).
thf(ty_tf_b,type,
b: $tType ).
thf(ty_tf_a,type,
a: $tType ).
%----Explicit typings (45)
thf(sy_c_BNF__Composition_Oid__bnf,type,
bNF_id_bnf:
!>[A: $tType] : ( A > A ) ).
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__Def_Ovimage2p,type,
bNF_vimage2p:
!>[A: $tType,D: $tType,B: $tType,E: $tType,C: $tType] : ( ( A > D ) > ( B > E ) > ( D > E > C ) > A > B > C ) ).
thf(sy_c_Fun_Ocomp,type,
comp:
!>[B: $tType,C: $tType,A: $tType] : ( ( B > C ) > ( A > B ) > A > C ) ).
thf(sy_c_Fun_Ofcomp,type,
fcomp:
!>[A: $tType,B: $tType,C: $tType] : ( ( A > B ) > ( B > C ) > A > C ) ).
thf(sy_c_Fun_Ofun__upd,type,
fun_upd:
!>[A: $tType,B: $tType] : ( ( A > B ) > A > B > A > B ) ).
thf(sy_c_Fun_Oid,type,
id:
!>[A: $tType] : ( A > A ) ).
thf(sy_c_Fun_Omap__fun,type,
map_fun:
!>[C: $tType,A: $tType,B: $tType,D: $tType] : ( ( C > A ) > ( B > D ) > ( A > B ) > C > D ) ).
thf(sy_c_Fun_Oswap,type,
swap:
!>[A: $tType,B: $tType] : ( A > A > ( A > B ) > A > B ) ).
thf(sy_c_If,type,
if:
!>[A: $tType] : ( $o > A > A > A ) ).
thf(sy_c_Orderings_Oord__class_Oless__eq,type,
ord_less_eq:
!>[A: $tType] : ( A > A > $o ) ).
thf(sy_c_Orderings_Otop__class_Otop,type,
top_top:
!>[A: $tType] : A ).
thf(sy_c_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_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_Pure_Otype,type,
type:
!>[A: $tType] : ( itself @ A ) ).
thf(sy_c_Quotient_OQuotient3,type,
quotient3:
!>[A: $tType,B: $tType] : ( ( A > A > $o ) > ( A > B ) > ( B > A ) > $o ) ).
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_Obind,type,
bind:
!>[A: $tType,B: $tType] : ( ( set @ A ) > ( A > ( set @ B ) ) > ( set @ B ) ) ).
thf(sy_c_Set_Oimage,type,
image:
!>[A: $tType,B: $tType] : ( ( A > B ) > ( set @ A ) > ( set @ B ) ) ).
thf(sy_c_Stream__Processor__Mirabelle__rrumbueyrq_Ocopy,type,
stream2017582925e_copy:
!>[A: $tType] : ( stream1273403375_sp_nu @ A @ A ) ).
thf(sy_c_Stream__Processor__Mirabelle__rrumbueyrq_Orun_092_060_094sub_062_092_060nu_062,type,
stream377071682run_nu:
!>[A: $tType,B: $tType] : ( ( stream1273403375_sp_nu @ A @ B ) > ( stream @ A ) > ( stream @ B ) ) ).
thf(sy_c_Stream__Processor__Mirabelle__rrumbueyrq_Osp_092_060_094sub_062_092_060nu_062_Ocorec__sp_092_060_094sub_062_092_060nu_062,type,
stream937840132_sp_nu:
!>[E: $tType,A: $tType,B: $tType] : ( ( E > ( stream901396144_sp_mu @ A @ B @ ( sum_sum @ ( stream1273403375_sp_nu @ A @ B ) @ E ) ) ) > E > ( stream1273403375_sp_nu @ A @ B ) ) ).
thf(sy_c_Stream__Processor__Mirabelle__rrumbueyrq_Osp_092_060_094sub_062_092_060nu_062_Omap__sp_092_060_094sub_062_092_060nu_062,type,
stream151454380_sp_nu:
!>[B: $tType,Ba: $tType,A: $tType] : ( ( B > Ba ) > ( stream1273403375_sp_nu @ A @ B ) > ( stream1273403375_sp_nu @ A @ Ba ) ) ).
thf(sy_c_Stream__Processor__Mirabelle__rrumbueyrq_Osp_092_060_094sub_062_092_060nu_062_Opred__sp_092_060_094sub_062_092_060nu_062,type,
stream1465372679_sp_nu:
!>[A: $tType,C: $tType] : ( ( A > $o ) > ( stream1273403375_sp_nu @ C @ A ) > $o ) ).
thf(sy_c_Stream__Processor__Mirabelle__rrumbueyrq_Osp_092_060_094sub_062_092_060nu_062_Oset__sp_092_060_094sub_062_092_060nu_062,type,
stream1493736486_sp_nu:
!>[A: $tType,B: $tType] : ( ( stream1273403375_sp_nu @ A @ B ) > ( set @ B ) ) ).
thf(sy_c_Stream__Processor__Mirabelle__rrumbueyrq_Osp_092_060_094sub_062_092_060nu_062__comp,type,
stream1967106959u_comp:
!>[A: $tType,B: $tType,C: $tType] : ( ( stream1273403375_sp_nu @ A @ B ) > ( stream1273403375_sp_nu @ C @ A ) > ( stream1273403375_sp_nu @ C @ B ) ) ).
thf(sy_c_Stream__Processor__Mirabelle__rrumbueyrq_Osp_092_060_094sub_062_092_060nu_062__comp2,type,
stream104478819_comp2:
!>[A: $tType,B: $tType,C: $tType] : ( ( stream1273403375_sp_nu @ A @ B ) > ( stream1273403375_sp_nu @ C @ A ) > ( stream1273403375_sp_nu @ C @ B ) ) ).
thf(sy_c_Typedef_Otype__definition,type,
type_definition:
!>[B: $tType,A: $tType] : ( ( B > A ) > ( A > B ) > ( set @ A ) > $o ) ).
thf(sy_c_member,type,
member:
!>[A: $tType] : ( A > ( set @ A ) > $o ) ).
thf(sy_v_sp,type,
sp: stream1273403375_sp_nu @ c @ b ).
thf(sy_v_sp_H,type,
sp2: stream1273403375_sp_nu @ a @ c ).
%----Relevant facts (256)
thf(fact_0_run_092_060_094sub_062_092_060nu_062__sp_092_060_094sub_062_092_060nu_062__comp,axiom,
! [B: $tType,C: $tType,A: $tType,Sp: stream1273403375_sp_nu @ C @ B,Sp2: stream1273403375_sp_nu @ A @ C] :
( ( stream377071682run_nu @ A @ B @ ( stream1967106959u_comp @ C @ B @ A @ Sp @ Sp2 ) )
= ( comp @ ( stream @ C ) @ ( stream @ B ) @ ( stream @ A ) @ ( stream377071682run_nu @ C @ B @ Sp ) @ ( stream377071682run_nu @ A @ C @ Sp2 ) ) ) ).
% run\<^sub>\<nu>_sp\<^sub>\<nu>_comp
thf(fact_1_run_092_060_094sub_062_092_060nu_062__sp_092_060_094sub_062_092_060nu_062__comp2,axiom,
! [B: $tType,C: $tType,A: $tType,Sp: stream1273403375_sp_nu @ C @ B,Sp2: stream1273403375_sp_nu @ A @ C] :
( ( stream377071682run_nu @ A @ B @ ( stream104478819_comp2 @ C @ B @ A @ Sp @ Sp2 ) )
= ( comp @ ( stream @ C ) @ ( stream @ B ) @ ( stream @ A ) @ ( stream377071682run_nu @ C @ B @ Sp ) @ ( stream377071682run_nu @ A @ C @ Sp2 ) ) ) ).
% run\<^sub>\<nu>_sp\<^sub>\<nu>_comp2
thf(fact_2_sp_092_060_094sub_062_092_060nu_062_Ocorec__disc,axiom,
! [B: $tType,A: $tType,E: $tType] :
( ( stream937840132_sp_nu @ E @ A @ B )
= ( stream937840132_sp_nu @ E @ A @ B ) ) ).
% sp\<^sub>\<nu>.corec_disc
thf(fact_3_comp__apply,axiom,
! [C: $tType,A: $tType,B: $tType] :
( ( comp @ B @ A @ C )
= ( ^ [F: B > A,G: C > B,X: C] : ( F @ ( G @ X ) ) ) ) ).
% comp_apply
thf(fact_4_run_092_060_094sub_062_092_060nu_062__copy,axiom,
! [A: $tType,S: stream @ A] :
( ( stream377071682run_nu @ A @ A @ ( stream2017582925e_copy @ A ) @ S )
= S ) ).
% run\<^sub>\<nu>_copy
thf(fact_5_comp__def,axiom,
! [A: $tType,C: $tType,B: $tType] :
( ( comp @ B @ C @ A )
= ( ^ [F: B > C,G: A > B,X: A] : ( F @ ( G @ X ) ) ) ) ).
% comp_def
thf(fact_6_comp__assoc,axiom,
! [B: $tType,D: $tType,C: $tType,A: $tType,F2: D > B,G2: C > D,H: A > C] :
( ( comp @ C @ B @ A @ ( comp @ D @ B @ C @ F2 @ G2 ) @ H )
= ( comp @ D @ B @ A @ F2 @ ( comp @ C @ D @ A @ G2 @ H ) ) ) ).
% comp_assoc
thf(fact_7_comp__eq__dest,axiom,
! [C: $tType,B: $tType,D: $tType,A: $tType,A2: C > B,B2: A > C,C2: D > B,D2: A > D,V: A] :
( ( ( comp @ C @ B @ A @ A2 @ B2 )
= ( comp @ D @ B @ A @ C2 @ D2 ) )
=> ( ( A2 @ ( B2 @ V ) )
= ( C2 @ ( D2 @ V ) ) ) ) ).
% comp_eq_dest
thf(fact_8_comp__eq__elim,axiom,
! [C: $tType,B: $tType,D: $tType,A: $tType,A2: C > B,B2: A > C,C2: D > B,D2: A > D] :
( ( ( comp @ C @ B @ A @ A2 @ B2 )
= ( comp @ D @ B @ A @ C2 @ D2 ) )
=> ! [V2: A] :
( ( A2 @ ( B2 @ V2 ) )
= ( C2 @ ( D2 @ V2 ) ) ) ) ).
% comp_eq_elim
thf(fact_9_comp__cong,axiom,
! [C: $tType,B: $tType,D: $tType,A: $tType,E: $tType,F2: B > A,G2: C > B,X2: C,F3: D > A,G3: E > D,X3: E] :
( ( ( F2 @ ( G2 @ X2 ) )
= ( F3 @ ( G3 @ X3 ) ) )
=> ( ( comp @ B @ A @ C @ F2 @ G2 @ X2 )
= ( comp @ D @ A @ E @ F3 @ G3 @ X3 ) ) ) ).
% comp_cong
thf(fact_10_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_11_comp__apply__eq,axiom,
! [B: $tType,D: $tType,A: $tType,C: $tType,F2: B > A,G2: C > B,X2: C,H: D > A,K: C > D] :
( ( ( F2 @ ( G2 @ X2 ) )
= ( H @ ( K @ X2 ) ) )
=> ( ( comp @ B @ A @ C @ F2 @ G2 @ X2 )
= ( comp @ D @ A @ C @ H @ K @ X2 ) ) ) ).
% comp_apply_eq
thf(fact_12_fun_Omap__comp,axiom,
! [B: $tType,C: $tType,A: $tType,D: $tType,G2: B > C,F2: A > B,V: D > A] :
( ( comp @ B @ C @ D @ G2 @ ( comp @ A @ B @ D @ F2 @ V ) )
= ( comp @ A @ C @ D @ ( comp @ B @ C @ A @ G2 @ F2 ) @ V ) ) ).
% fun.map_comp
thf(fact_13_type__copy__map__cong0,axiom,
! [B: $tType,D: $tType,E: $tType,A: $tType,C: $tType,M: B > A,G2: C > B,X2: C,N: D > A,H: C > D,F2: A > E] :
( ( ( M @ ( G2 @ X2 ) )
= ( N @ ( H @ X2 ) ) )
=> ( ( comp @ B @ E @ C @ ( comp @ A @ E @ B @ F2 @ M ) @ G2 @ X2 )
= ( comp @ D @ E @ C @ ( comp @ A @ E @ D @ F2 @ N ) @ H @ X2 ) ) ) ).
% type_copy_map_cong0
thf(fact_14_rewriteL__comp__comp,axiom,
! [C: $tType,B: $tType,A: $tType,D: $tType,F2: C > B,G2: A > C,L: A > B,H: D > A] :
( ( ( comp @ C @ B @ A @ F2 @ G2 )
= L )
=> ( ( comp @ C @ B @ D @ F2 @ ( comp @ A @ C @ D @ G2 @ H ) )
= ( comp @ A @ B @ D @ L @ H ) ) ) ).
% rewriteL_comp_comp
thf(fact_15_rewriteR__comp__comp2,axiom,
! [C: $tType,B: $tType,E: $tType,D: $tType,A: $tType,G2: C > B,H: A > C,R1: D > B,R2: A > D,F2: B > E,L: D > E] :
( ( ( comp @ C @ B @ A @ G2 @ H )
= ( comp @ D @ B @ A @ R1 @ R2 ) )
=> ( ( ( comp @ B @ E @ D @ F2 @ R1 )
= L )
=> ( ( comp @ C @ E @ A @ ( comp @ B @ E @ C @ F2 @ G2 ) @ H )
= ( comp @ D @ E @ A @ L @ R2 ) ) ) ) ).
% rewriteR_comp_comp2
thf(fact_16_rewriteL__comp__comp2,axiom,
! [A: $tType,C: $tType,B: $tType,D: $tType,E: $tType,F2: C > B,G2: A > C,L1: D > B,L2: A > D,H: E > A,R: E > D] :
( ( ( comp @ C @ B @ A @ F2 @ G2 )
= ( comp @ D @ B @ A @ L1 @ L2 ) )
=> ( ( ( comp @ A @ D @ E @ L2 @ H )
= R )
=> ( ( comp @ C @ B @ E @ F2 @ ( comp @ A @ C @ E @ G2 @ H ) )
= ( comp @ D @ B @ E @ L1 @ R ) ) ) ) ).
% rewriteL_comp_comp2
thf(fact_17_rewriteR__comp__comp,axiom,
! [C: $tType,D: $tType,B: $tType,A: $tType,G2: C > B,H: A > C,R: A > B,F2: B > D] :
( ( ( comp @ C @ B @ A @ G2 @ H )
= R )
=> ( ( comp @ C @ D @ A @ ( comp @ B @ D @ C @ F2 @ G2 ) @ H )
= ( comp @ B @ D @ A @ F2 @ R ) ) ) ).
% rewriteR_comp_comp
thf(fact_18_vimage2p__comp,axiom,
! [E: $tType,D: $tType,F4: $tType,A: $tType,C: $tType,B: $tType,G4: $tType,F1: F4 > A,F22: D > F4,G1: G4 > B,G22: E > G4] :
( ( bNF_vimage2p @ D @ A @ E @ B @ C @ ( comp @ F4 @ A @ D @ F1 @ F22 ) @ ( comp @ G4 @ B @ E @ G1 @ G22 ) )
= ( comp @ ( F4 > G4 > C ) @ ( D > E > C ) @ ( A > B > C ) @ ( bNF_vimage2p @ D @ F4 @ E @ G4 @ C @ F22 @ G22 ) @ ( bNF_vimage2p @ F4 @ A @ G4 @ B @ C @ F1 @ G1 ) ) ) ).
% vimage2p_comp
thf(fact_19_csquare__def,axiom,
! [B: $tType,C: $tType,D: $tType,A: $tType] :
( ( bNF_csquare @ A @ B @ C @ D )
= ( ^ [A3: set @ A,F12: B > C,F23: D > C,P1: A > B,P2: A > D] :
! [X: A] :
( ( member @ A @ X @ A3 )
=> ( ( F12 @ ( P1 @ X ) )
= ( F23 @ ( P2 @ X ) ) ) ) ) ) ).
% csquare_def
thf(fact_20_fcomp__comp,axiom,
! [B: $tType,C: $tType,A: $tType] :
( ( fcomp @ A @ C @ B )
= ( ^ [F: A > C,G: C > B] : ( comp @ C @ B @ A @ G @ F ) ) ) ).
% fcomp_comp
thf(fact_21_convol__o,axiom,
! [B: $tType,C: $tType,D: $tType,A: $tType,F2: D > B,G2: D > C,H: A > D] :
( ( comp @ D @ ( product_prod @ B @ C ) @ A @ ( bNF_convol @ D @ B @ C @ F2 @ G2 ) @ H )
= ( bNF_convol @ A @ B @ C @ ( comp @ D @ B @ A @ F2 @ H ) @ ( comp @ D @ C @ A @ G2 @ H ) ) ) ).
% convol_o
thf(fact_22_comp__swap,axiom,
! [B: $tType,C: $tType,A: $tType,F2: C > B,A2: A,B2: A,G2: A > C] :
( ( comp @ C @ B @ A @ F2 @ ( swap @ A @ C @ A2 @ B2 @ G2 ) )
= ( swap @ A @ B @ A2 @ B2 @ ( comp @ C @ B @ A @ F2 @ G2 ) ) ) ).
% comp_swap
thf(fact_23_fun__upd__comp,axiom,
! [A: $tType,B: $tType,C: $tType,F2: C > B,G2: A > C,X2: A,Y: C] :
( ( comp @ C @ B @ A @ F2 @ ( fun_upd @ A @ C @ G2 @ X2 @ Y ) )
= ( fun_upd @ A @ B @ ( comp @ C @ B @ A @ F2 @ G2 ) @ X2 @ ( F2 @ Y ) ) ) ).
% fun_upd_comp
thf(fact_24_sp_092_060_094sub_062_092_060nu_062_Omap__comp,axiom,
! [B: $tType,C: $tType,A: $tType,D: $tType,G2: B > C,F2: A > B,V: stream1273403375_sp_nu @ D @ A] :
( ( stream151454380_sp_nu @ B @ C @ D @ G2 @ ( stream151454380_sp_nu @ A @ B @ D @ F2 @ V ) )
= ( stream151454380_sp_nu @ A @ C @ D @ ( comp @ B @ C @ A @ G2 @ F2 ) @ V ) ) ).
% sp\<^sub>\<nu>.map_comp
thf(fact_25_map__fun_Ocompositionality,axiom,
! [D: $tType,F4: $tType,C: $tType,E: $tType,B: $tType,A: $tType,F2: E > C,G2: D > F4,H: C > A,I: B > D,Fun: A > B] :
( ( map_fun @ E @ C @ D @ F4 @ F2 @ G2 @ ( map_fun @ C @ A @ B @ D @ H @ I @ Fun ) )
= ( map_fun @ E @ A @ B @ F4 @ ( comp @ C @ A @ E @ H @ F2 ) @ ( comp @ D @ F4 @ B @ G2 @ I ) @ Fun ) ) ).
% map_fun.compositionality
thf(fact_26_map__fun_Ocomp,axiom,
! [E: $tType,C: $tType,A: $tType,F4: $tType,D: $tType,B: $tType,F2: E > C,G2: D > F4,H: C > A,I: B > D] :
( ( comp @ ( C > D ) @ ( E > F4 ) @ ( A > B ) @ ( map_fun @ E @ C @ D @ F4 @ F2 @ G2 ) @ ( map_fun @ C @ A @ B @ D @ H @ I ) )
= ( map_fun @ E @ A @ B @ F4 @ ( comp @ C @ A @ E @ H @ F2 ) @ ( comp @ D @ F4 @ B @ G2 @ I ) ) ) ).
% map_fun.comp
thf(fact_27_map__fun__def,axiom,
! [B: $tType,D: $tType,A: $tType,C: $tType] :
( ( map_fun @ C @ A @ B @ D )
= ( ^ [F: C > A,G: B > D,H2: A > B] : ( comp @ A @ D @ C @ ( comp @ B @ D @ A @ G @ H2 ) @ F ) ) ) ).
% map_fun_def
thf(fact_28_map__fun__apply,axiom,
! [D: $tType,A: $tType,C: $tType,B: $tType] :
( ( map_fun @ B @ C @ D @ A )
= ( ^ [F: B > C,G: D > A,H2: C > D,X: B] : ( G @ ( H2 @ ( F @ X ) ) ) ) ) ).
% map_fun_apply
thf(fact_29_fun__upd__apply,axiom,
! [A: $tType,B: $tType] :
( ( fun_upd @ B @ A )
= ( ^ [F: B > A,X: B,Y2: A,Z: B] : ( if @ A @ ( Z = X ) @ Y2 @ ( F @ Z ) ) ) ) ).
% fun_upd_apply
thf(fact_30_fun__upd__triv,axiom,
! [B: $tType,A: $tType,F2: A > B,X2: A] :
( ( fun_upd @ A @ B @ F2 @ X2 @ ( F2 @ X2 ) )
= F2 ) ).
% fun_upd_triv
thf(fact_31_fun__upd__upd,axiom,
! [A: $tType,B: $tType,F2: A > B,X2: A,Y: B,Z2: B] :
( ( fun_upd @ A @ B @ ( fun_upd @ A @ B @ F2 @ X2 @ Y ) @ X2 @ Z2 )
= ( fun_upd @ A @ B @ F2 @ X2 @ Z2 ) ) ).
% fun_upd_upd
thf(fact_32_swap__nilpotent,axiom,
! [B: $tType,A: $tType,A2: A,B2: A,F2: A > B] :
( ( swap @ A @ B @ A2 @ B2 @ ( swap @ A @ B @ A2 @ B2 @ F2 ) )
= F2 ) ).
% swap_nilpotent
thf(fact_33_swap__self,axiom,
! [B: $tType,A: $tType,A2: A,F2: A > B] :
( ( swap @ A @ B @ A2 @ A2 @ F2 )
= F2 ) ).
% swap_self
thf(fact_34_swap__apply_I1_J,axiom,
! [A: $tType,B: $tType,A2: B,B2: B,F2: B > A] :
( ( swap @ B @ A @ A2 @ B2 @ F2 @ A2 )
= ( F2 @ B2 ) ) ).
% swap_apply(1)
thf(fact_35_swap__apply_I2_J,axiom,
! [A: $tType,B: $tType,A2: B,B2: B,F2: B > A] :
( ( swap @ B @ A @ A2 @ B2 @ F2 @ B2 )
= ( F2 @ A2 ) ) ).
% swap_apply(2)
thf(fact_36_swap__apply_I3_J,axiom,
! [A: $tType,B: $tType,C2: B,A2: B,B2: B,F2: B > A] :
( ( C2 != A2 )
=> ( ( C2 != B2 )
=> ( ( swap @ B @ A @ A2 @ B2 @ F2 @ C2 )
= ( F2 @ C2 ) ) ) ) ).
% swap_apply(3)
thf(fact_37_fcomp__apply,axiom,
! [A: $tType,C: $tType,B: $tType] :
( ( fcomp @ B @ C @ A )
= ( ^ [F: B > C,G: C > A,X: B] : ( G @ ( F @ X ) ) ) ) ).
% fcomp_apply
thf(fact_38_fun__upd__idem__iff,axiom,
! [A: $tType,B: $tType,F2: A > B,X2: A,Y: B] :
( ( ( fun_upd @ A @ B @ F2 @ X2 @ Y )
= F2 )
= ( ( F2 @ X2 )
= Y ) ) ).
% fun_upd_idem_iff
thf(fact_39_fun__upd__twist,axiom,
! [A: $tType,B: $tType,A2: A,C2: A,M2: A > B,B2: B,D2: B] :
( ( A2 != C2 )
=> ( ( fun_upd @ A @ B @ ( fun_upd @ A @ B @ M2 @ A2 @ B2 ) @ C2 @ D2 )
= ( fun_upd @ A @ B @ ( fun_upd @ A @ B @ M2 @ C2 @ D2 ) @ A2 @ B2 ) ) ) ).
% fun_upd_twist
thf(fact_40_fun__upd__other,axiom,
! [B: $tType,A: $tType,Z2: A,X2: A,F2: A > B,Y: B] :
( ( Z2 != X2 )
=> ( ( fun_upd @ A @ B @ F2 @ X2 @ Y @ Z2 )
= ( F2 @ Z2 ) ) ) ).
% fun_upd_other
thf(fact_41_swap__commute,axiom,
! [B: $tType,A: $tType] :
( ( swap @ A @ B )
= ( ^ [A4: A,B3: A] : ( swap @ A @ B @ B3 @ A4 ) ) ) ).
% swap_commute
thf(fact_42_fun__upd__same,axiom,
! [B: $tType,A: $tType,F2: B > A,X2: B,Y: A] :
( ( fun_upd @ B @ A @ F2 @ X2 @ Y @ X2 )
= Y ) ).
% fun_upd_same
thf(fact_43_fun__upd__idem,axiom,
! [A: $tType,B: $tType,F2: B > A,X2: B,Y: A] :
( ( ( F2 @ X2 )
= Y )
=> ( ( fun_upd @ B @ A @ F2 @ X2 @ Y )
= F2 ) ) ).
% fun_upd_idem
thf(fact_44_swap__triple,axiom,
! [B: $tType,A: $tType,A2: A,C2: A,B2: A,F2: A > B] :
( ( A2 != C2 )
=> ( ( B2 != C2 )
=> ( ( swap @ A @ B @ A2 @ B2 @ ( swap @ A @ B @ B2 @ C2 @ ( swap @ A @ B @ A2 @ B2 @ F2 ) ) )
= ( swap @ A @ B @ A2 @ C2 @ F2 ) ) ) ) ).
% swap_triple
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,A5: set @ A] :
( ( collect @ A
@ ^ [X: A] : ( member @ A @ X @ A5 ) )
= A5 ) ).
% Collect_mem_eq
thf(fact_47_Collect__cong,axiom,
! [A: $tType,P: A > $o,Q: A > $o] :
( ! [X4: A] :
( ( P @ X4 )
= ( Q @ X4 ) )
=> ( ( collect @ A @ P )
= ( collect @ A @ Q ) ) ) ).
% Collect_cong
thf(fact_48_ext,axiom,
! [B: $tType,A: $tType,F2: A > B,G2: A > B] :
( ! [X4: A] :
( ( F2 @ X4 )
= ( G2 @ X4 ) )
=> ( F2 = G2 ) ) ).
% ext
thf(fact_49_fun__upd__eqD,axiom,
! [A: $tType,B: $tType,F2: A > B,X2: A,Y: B,G2: A > B,Z2: B] :
( ( ( fun_upd @ A @ B @ F2 @ X2 @ Y )
= ( fun_upd @ A @ B @ G2 @ X2 @ Z2 ) )
=> ( Y = Z2 ) ) ).
% fun_upd_eqD
thf(fact_50_fun__upd__def,axiom,
! [B: $tType,A: $tType] :
( ( fun_upd @ A @ B )
= ( ^ [F: A > B,A4: A,B3: B,X: A] : ( if @ B @ ( X = A4 ) @ B3 @ ( F @ X ) ) ) ) ).
% fun_upd_def
thf(fact_51_fcomp__assoc,axiom,
! [A: $tType,D: $tType,B: $tType,C: $tType,F2: A > D,G2: D > C,H: C > B] :
( ( fcomp @ A @ C @ B @ ( fcomp @ A @ D @ C @ F2 @ G2 ) @ H )
= ( fcomp @ A @ D @ B @ F2 @ ( fcomp @ D @ C @ B @ G2 @ H ) ) ) ).
% fcomp_assoc
thf(fact_52_fcomp__def,axiom,
! [C: $tType,B: $tType,A: $tType] :
( ( fcomp @ A @ B @ C )
= ( ^ [F: A > B,G: B > C,X: A] : ( G @ ( F @ X ) ) ) ) ).
% fcomp_def
thf(fact_53_swap__def,axiom,
! [B: $tType,A: $tType] :
( ( swap @ A @ B )
= ( ^ [A4: A,B3: A,F: A > B] : ( fun_upd @ A @ B @ ( fun_upd @ A @ B @ F @ A4 @ ( F @ B3 ) ) @ B3 @ ( F @ A4 ) ) ) ) ).
% swap_def
thf(fact_54_vimage2p__cong,axiom,
! [E: $tType,D: $tType,C: $tType,B: $tType,A: $tType,R3: A > B > C,S2: A > B > C,F2: D > A,G2: E > B] :
( ( R3 = S2 )
=> ( ( bNF_vimage2p @ D @ A @ E @ B @ C @ F2 @ G2 @ R3 )
= ( bNF_vimage2p @ D @ A @ E @ B @ C @ F2 @ G2 @ S2 ) ) ) ).
% vimage2p_cong
thf(fact_55_swap__comp__involutory,axiom,
! [B: $tType,A: $tType,A2: A,B2: A] :
( ( comp @ ( A > B ) @ ( A > B ) @ ( A > B ) @ ( swap @ A @ B @ A2 @ B2 ) @ ( swap @ A @ B @ A2 @ B2 ) )
= ( id @ ( A > B ) ) ) ).
% swap_comp_involutory
thf(fact_56_map__prod__o__convol,axiom,
! [D: $tType,B: $tType,C: $tType,E: $tType,A: $tType,H1: D > B,H22: E > C,F2: A > D,G2: A > E] :
( ( comp @ ( product_prod @ D @ E ) @ ( product_prod @ B @ C ) @ A @ ( product_map_prod @ D @ B @ E @ C @ H1 @ H22 ) @ ( bNF_convol @ A @ D @ E @ F2 @ G2 ) )
= ( bNF_convol @ A @ B @ C @ ( comp @ D @ B @ A @ H1 @ F2 ) @ ( comp @ E @ C @ A @ H22 @ G2 ) ) ) ).
% map_prod_o_convol
thf(fact_57_sp_092_060_094sub_062_092_060nu_062_Opred__map,axiom,
! [B: $tType,A: $tType,D: $tType,Q: B > $o,F2: A > B,X2: stream1273403375_sp_nu @ D @ A] :
( ( stream1465372679_sp_nu @ B @ D @ Q @ ( stream151454380_sp_nu @ A @ B @ D @ F2 @ X2 ) )
= ( stream1465372679_sp_nu @ A @ D @ ( comp @ B @ $o @ A @ Q @ F2 ) @ X2 ) ) ).
% sp\<^sub>\<nu>.pred_map
thf(fact_58_sp_092_060_094sub_062_092_060nu_062_Omap__cong,axiom,
! [B: $tType,A: $tType,D: $tType,X2: stream1273403375_sp_nu @ D @ A,Ya: stream1273403375_sp_nu @ D @ A,F2: A > B,G2: A > B] :
( ( X2 = Ya )
=> ( ! [Z3: A] :
( ( member @ A @ Z3 @ ( stream1493736486_sp_nu @ D @ A @ Ya ) )
=> ( ( F2 @ Z3 )
= ( G2 @ Z3 ) ) )
=> ( ( stream151454380_sp_nu @ A @ B @ D @ F2 @ X2 )
= ( stream151454380_sp_nu @ A @ B @ D @ G2 @ Ya ) ) ) ) ).
% sp\<^sub>\<nu>.map_cong
thf(fact_59_sp_092_060_094sub_062_092_060nu_062_Omap__cong0,axiom,
! [B: $tType,A: $tType,D: $tType,X2: stream1273403375_sp_nu @ D @ A,F2: A > B,G2: A > B] :
( ! [Z3: A] :
( ( member @ A @ Z3 @ ( stream1493736486_sp_nu @ D @ A @ X2 ) )
=> ( ( F2 @ Z3 )
= ( G2 @ Z3 ) ) )
=> ( ( stream151454380_sp_nu @ A @ B @ D @ F2 @ X2 )
= ( stream151454380_sp_nu @ A @ B @ D @ G2 @ X2 ) ) ) ).
% sp\<^sub>\<nu>.map_cong0
thf(fact_60_sp_092_060_094sub_062_092_060nu_062_Oinj__map__strong,axiom,
! [B: $tType,A: $tType,D: $tType,X2: stream1273403375_sp_nu @ D @ A,Xa: stream1273403375_sp_nu @ D @ A,F2: A > B,Fa: A > B] :
( ! [Z3: A,Za: A] :
( ( member @ A @ Z3 @ ( stream1493736486_sp_nu @ D @ A @ X2 ) )
=> ( ( member @ A @ Za @ ( stream1493736486_sp_nu @ D @ A @ Xa ) )
=> ( ( ( F2 @ Z3 )
= ( Fa @ Za ) )
=> ( Z3 = Za ) ) ) )
=> ( ( ( stream151454380_sp_nu @ A @ B @ D @ F2 @ X2 )
= ( stream151454380_sp_nu @ A @ B @ D @ Fa @ Xa ) )
=> ( X2 = Xa ) ) ) ).
% sp\<^sub>\<nu>.inj_map_strong
thf(fact_61_id__fcomp,axiom,
! [B: $tType,A: $tType,G2: A > B] :
( ( fcomp @ A @ A @ B @ ( id @ A ) @ G2 )
= G2 ) ).
% id_fcomp
thf(fact_62_fcomp__id,axiom,
! [B: $tType,A: $tType,F2: A > B] :
( ( fcomp @ A @ B @ B @ F2 @ ( id @ B ) )
= F2 ) ).
% fcomp_id
thf(fact_63_swap__image__eq,axiom,
! [B: $tType,A: $tType,A2: A,A5: set @ A,B2: A,F2: A > B] :
( ( member @ A @ A2 @ A5 )
=> ( ( member @ A @ B2 @ A5 )
=> ( ( image @ A @ B @ ( swap @ A @ B @ A2 @ B2 @ F2 ) @ A5 )
= ( image @ A @ B @ F2 @ A5 ) ) ) ) ).
% swap_image_eq
thf(fact_64_id__apply,axiom,
! [A: $tType] :
( ( id @ A )
= ( ^ [X: A] : X ) ) ).
% id_apply
thf(fact_65_image__id,axiom,
! [A: $tType] :
( ( image @ A @ A @ ( id @ A ) )
= ( id @ ( set @ A ) ) ) ).
% image_id
thf(fact_66_fun_Omap__id,axiom,
! [A: $tType,D: $tType,T2: D > A] :
( ( comp @ A @ A @ D @ ( id @ A ) @ T2 )
= T2 ) ).
% fun.map_id
thf(fact_67_id__comp,axiom,
! [B: $tType,A: $tType,G2: A > B] :
( ( comp @ B @ B @ A @ ( id @ B ) @ G2 )
= G2 ) ).
% id_comp
thf(fact_68_comp__id,axiom,
! [B: $tType,A: $tType,F2: A > B] :
( ( comp @ A @ B @ A @ F2 @ ( id @ A ) )
= F2 ) ).
% comp_id
thf(fact_69_fun_Omap__id0,axiom,
! [A: $tType,D: $tType] :
( ( comp @ A @ A @ D @ ( id @ A ) )
= ( id @ ( D > A ) ) ) ).
% fun.map_id0
thf(fact_70_sp_092_060_094sub_062_092_060nu_062_Omap__id0,axiom,
! [A: $tType,D: $tType] :
( ( stream151454380_sp_nu @ A @ A @ D @ ( id @ A ) )
= ( id @ ( stream1273403375_sp_nu @ D @ A ) ) ) ).
% sp\<^sub>\<nu>.map_id0
thf(fact_71_sp_092_060_094sub_062_092_060nu_062_Opred__mono__strong,axiom,
! [A: $tType,D: $tType,P: A > $o,X2: stream1273403375_sp_nu @ D @ A,Pa: A > $o] :
( ( stream1465372679_sp_nu @ A @ D @ P @ X2 )
=> ( ! [Z3: A] :
( ( member @ A @ Z3 @ ( stream1493736486_sp_nu @ D @ A @ X2 ) )
=> ( ( P @ Z3 )
=> ( Pa @ Z3 ) ) )
=> ( stream1465372679_sp_nu @ A @ D @ Pa @ X2 ) ) ) ).
% sp\<^sub>\<nu>.pred_mono_strong
thf(fact_72_sp_092_060_094sub_062_092_060nu_062_Opred__cong,axiom,
! [A: $tType,D: $tType,X2: stream1273403375_sp_nu @ D @ A,Ya: stream1273403375_sp_nu @ D @ A,P: A > $o,Pa: A > $o] :
( ( X2 = Ya )
=> ( ! [Z3: A] :
( ( member @ A @ Z3 @ ( stream1493736486_sp_nu @ D @ A @ Ya ) )
=> ( ( P @ Z3 )
= ( Pa @ Z3 ) ) )
=> ( ( stream1465372679_sp_nu @ A @ D @ P @ X2 )
= ( stream1465372679_sp_nu @ A @ D @ Pa @ Ya ) ) ) ) ).
% sp\<^sub>\<nu>.pred_cong
thf(fact_73_id__def,axiom,
! [A: $tType] :
( ( id @ A )
= ( ^ [X: A] : X ) ) ).
% id_def
thf(fact_74_eq__id__iff,axiom,
! [A: $tType,F2: A > A] :
( ( ! [X: A] :
( ( F2 @ X )
= X ) )
= ( F2
= ( id @ A ) ) ) ).
% eq_id_iff
thf(fact_75_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_76_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_77_sp_092_060_094sub_062_092_060nu_062_Oset__map,axiom,
! [B: $tType,A: $tType,D: $tType,F2: A > B,V: stream1273403375_sp_nu @ D @ A] :
( ( stream1493736486_sp_nu @ D @ B @ ( stream151454380_sp_nu @ A @ B @ D @ F2 @ V ) )
= ( image @ A @ B @ F2 @ ( stream1493736486_sp_nu @ D @ A @ V ) ) ) ).
% sp\<^sub>\<nu>.set_map
thf(fact_78_map__prod__o__convol__id,axiom,
! [B: $tType,A: $tType,C: $tType,F2: C > A,G2: C > B,X2: C] :
( ( comp @ ( product_prod @ C @ B ) @ ( product_prod @ A @ B ) @ C @ ( product_map_prod @ C @ A @ B @ B @ F2 @ ( id @ B ) ) @ ( bNF_convol @ C @ C @ B @ ( id @ C ) @ G2 ) @ X2 )
= ( bNF_convol @ C @ A @ B @ ( comp @ A @ A @ C @ ( id @ A ) @ F2 ) @ G2 @ X2 ) ) ).
% map_prod_o_convol_id
thf(fact_79_pointfree__idE,axiom,
! [B: $tType,A: $tType,F2: B > A,G2: A > B,X2: A] :
( ( ( comp @ B @ A @ A @ F2 @ G2 )
= ( id @ A ) )
=> ( ( F2 @ ( G2 @ X2 ) )
= X2 ) ) ).
% pointfree_idE
thf(fact_80_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_81_image__eq__imp__comp,axiom,
! [B: $tType,A: $tType,D: $tType,C: $tType,F2: B > A,A5: set @ B,G2: C > A,B4: set @ C,H: A > D] :
( ( ( image @ B @ A @ F2 @ A5 )
= ( image @ C @ A @ G2 @ B4 ) )
=> ( ( image @ B @ D @ ( comp @ A @ D @ B @ H @ F2 ) @ A5 )
= ( image @ C @ D @ ( comp @ A @ D @ C @ H @ G2 ) @ B4 ) ) ) ).
% image_eq_imp_comp
thf(fact_82_image__comp,axiom,
! [B: $tType,A: $tType,C: $tType,F2: B > A,G2: C > B,R: set @ C] :
( ( image @ B @ A @ F2 @ ( image @ C @ B @ G2 @ R ) )
= ( image @ C @ A @ ( comp @ B @ A @ C @ F2 @ G2 ) @ R ) ) ).
% image_comp
thf(fact_83_prod_Omap__comp,axiom,
! [D: $tType,F4: $tType,E: $tType,C: $tType,B: $tType,A: $tType,G1: C > E,G22: D > F4,F1: A > C,F22: B > D,V: product_prod @ A @ B] :
( ( product_map_prod @ C @ E @ D @ F4 @ G1 @ G22 @ ( product_map_prod @ A @ C @ B @ D @ F1 @ F22 @ V ) )
= ( product_map_prod @ A @ E @ B @ F4 @ ( comp @ C @ E @ A @ G1 @ F1 ) @ ( comp @ D @ F4 @ B @ G22 @ F22 ) @ V ) ) ).
% prod.map_comp
thf(fact_84_sp_092_060_094sub_062_092_060nu_062_Omap__id,axiom,
! [A: $tType,D: $tType,T2: stream1273403375_sp_nu @ D @ A] :
( ( stream151454380_sp_nu @ A @ A @ D @ ( id @ A ) @ T2 )
= T2 ) ).
% sp\<^sub>\<nu>.map_id
thf(fact_85_vimage2p__id,axiom,
! [C: $tType,B: $tType,A: $tType,R3: A > B > C] :
( ( bNF_vimage2p @ A @ A @ B @ B @ C @ ( id @ A ) @ ( id @ B ) @ R3 )
= R3 ) ).
% vimage2p_id
thf(fact_86_map__fun__id,axiom,
! [B: $tType,A: $tType] :
( ( map_fun @ A @ A @ B @ B @ ( id @ A ) @ ( id @ B ) )
= ( id @ ( A > B ) ) ) ).
% map_fun_id
thf(fact_87_image__eqI,axiom,
! [A: $tType,B: $tType,B2: A,F2: B > A,X2: B,A5: set @ B] :
( ( B2
= ( F2 @ X2 ) )
=> ( ( member @ B @ X2 @ A5 )
=> ( member @ A @ B2 @ ( image @ B @ A @ F2 @ A5 ) ) ) ) ).
% image_eqI
thf(fact_88_map__prod_Ocompositionality,axiom,
! [D: $tType,F4: $tType,E: $tType,C: $tType,B: $tType,A: $tType,F2: C > E,G2: D > F4,H: A > C,I: B > D,Prod: product_prod @ A @ B] :
( ( product_map_prod @ C @ E @ D @ F4 @ F2 @ G2 @ ( product_map_prod @ A @ C @ B @ D @ H @ I @ Prod ) )
= ( product_map_prod @ A @ E @ B @ F4 @ ( comp @ C @ E @ A @ F2 @ H ) @ ( comp @ D @ F4 @ B @ G2 @ I ) @ Prod ) ) ).
% map_prod.compositionality
thf(fact_89_map__prod__compose,axiom,
! [D: $tType,C: $tType,A: $tType,E: $tType,F4: $tType,B: $tType,F1: E > C,F22: A > E,G1: F4 > D,G22: B > F4] :
( ( product_map_prod @ A @ C @ B @ D @ ( comp @ E @ C @ A @ F1 @ F22 ) @ ( comp @ F4 @ D @ B @ G1 @ G22 ) )
= ( comp @ ( product_prod @ E @ F4 ) @ ( product_prod @ C @ D ) @ ( product_prod @ A @ B ) @ ( product_map_prod @ E @ C @ F4 @ D @ F1 @ G1 ) @ ( product_map_prod @ A @ E @ B @ F4 @ F22 @ G22 ) ) ) ).
% map_prod_compose
thf(fact_90_map__prod_Ocomp,axiom,
! [A: $tType,C: $tType,E: $tType,F4: $tType,D: $tType,B: $tType,F2: C > E,G2: D > F4,H: A > C,I: B > D] :
( ( comp @ ( product_prod @ C @ D ) @ ( product_prod @ E @ F4 ) @ ( product_prod @ A @ B ) @ ( product_map_prod @ C @ E @ D @ F4 @ F2 @ G2 ) @ ( product_map_prod @ A @ C @ B @ D @ H @ I ) )
= ( product_map_prod @ A @ E @ B @ F4 @ ( comp @ C @ E @ A @ F2 @ H ) @ ( comp @ D @ F4 @ B @ G2 @ I ) ) ) ).
% map_prod.comp
thf(fact_91_Sup_OSUP__id__eq,axiom,
! [A: $tType,Sup: ( set @ A ) > A,A5: set @ A] :
( ( Sup @ ( image @ A @ A @ ( id @ A ) @ A5 ) )
= ( Sup @ A5 ) ) ).
% Sup.SUP_id_eq
thf(fact_92_Inf_OINF__id__eq,axiom,
! [A: $tType,Inf: ( set @ A ) > A,A5: set @ A] :
( ( Inf @ ( image @ A @ A @ ( id @ A ) @ A5 ) )
= ( Inf @ A5 ) ) ).
% Inf.INF_id_eq
thf(fact_93_imageI,axiom,
! [B: $tType,A: $tType,X2: A,A5: set @ A,F2: A > B] :
( ( member @ A @ X2 @ A5 )
=> ( member @ B @ ( F2 @ X2 ) @ ( image @ A @ B @ F2 @ A5 ) ) ) ).
% imageI
thf(fact_94_image__iff,axiom,
! [A: $tType,B: $tType,Z2: A,F2: B > A,A5: set @ B] :
( ( member @ A @ Z2 @ ( image @ B @ A @ F2 @ A5 ) )
= ( ? [X: B] :
( ( member @ B @ X @ A5 )
& ( Z2
= ( F2 @ X ) ) ) ) ) ).
% image_iff
thf(fact_95_bex__imageD,axiom,
! [A: $tType,B: $tType,F2: B > A,A5: set @ B,P: A > $o] :
( ? [X5: A] :
( ( member @ A @ X5 @ ( image @ B @ A @ F2 @ A5 ) )
& ( P @ X5 ) )
=> ? [X4: B] :
( ( member @ B @ X4 @ A5 )
& ( P @ ( F2 @ X4 ) ) ) ) ).
% bex_imageD
thf(fact_96_image__cong,axiom,
! [B: $tType,A: $tType,M: set @ A,N: set @ A,F2: A > B,G2: A > B] :
( ( M = N )
=> ( ! [X4: A] :
( ( member @ A @ X4 @ N )
=> ( ( F2 @ X4 )
= ( G2 @ X4 ) ) )
=> ( ( image @ A @ B @ F2 @ M )
= ( image @ A @ B @ G2 @ N ) ) ) ) ).
% image_cong
thf(fact_97_ball__imageD,axiom,
! [A: $tType,B: $tType,F2: B > A,A5: set @ B,P: A > $o] :
( ! [X4: A] :
( ( member @ A @ X4 @ ( image @ B @ A @ F2 @ A5 ) )
=> ( P @ X4 ) )
=> ! [X5: B] :
( ( member @ B @ X5 @ A5 )
=> ( P @ ( F2 @ X5 ) ) ) ) ).
% ball_imageD
thf(fact_98_rev__image__eqI,axiom,
! [B: $tType,A: $tType,X2: A,A5: set @ A,B2: B,F2: A > B] :
( ( member @ A @ X2 @ A5 )
=> ( ( B2
= ( F2 @ X2 ) )
=> ( member @ B @ B2 @ ( image @ A @ B @ F2 @ A5 ) ) ) ) ).
% rev_image_eqI
thf(fact_99_Inf_OINF__cong,axiom,
! [A: $tType,B: $tType,A5: set @ B,B4: set @ B,C3: B > A,D3: B > A,Inf: ( set @ A ) > A] :
( ( A5 = B4 )
=> ( ! [X4: B] :
( ( member @ B @ X4 @ B4 )
=> ( ( C3 @ X4 )
= ( D3 @ X4 ) ) )
=> ( ( Inf @ ( image @ B @ A @ C3 @ A5 ) )
= ( Inf @ ( image @ B @ A @ D3 @ B4 ) ) ) ) ) ).
% Inf.INF_cong
thf(fact_100_Sup_OSUP__cong,axiom,
! [A: $tType,B: $tType,A5: set @ B,B4: set @ B,C3: B > A,D3: B > A,Sup: ( set @ A ) > A] :
( ( A5 = B4 )
=> ( ! [X4: B] :
( ( member @ B @ X4 @ B4 )
=> ( ( C3 @ X4 )
= ( D3 @ X4 ) ) )
=> ( ( Sup @ ( image @ B @ A @ C3 @ A5 ) )
= ( Sup @ ( image @ B @ A @ D3 @ B4 ) ) ) ) ) ).
% Sup.SUP_cong
thf(fact_101_Sup_OSUP__image,axiom,
! [B: $tType,A: $tType,C: $tType,Sup: ( set @ A ) > A,G2: B > A,F2: C > B,A5: set @ C] :
( ( Sup @ ( image @ B @ A @ G2 @ ( image @ C @ B @ F2 @ A5 ) ) )
= ( Sup @ ( image @ C @ A @ ( comp @ B @ A @ C @ G2 @ F2 ) @ A5 ) ) ) ).
% Sup.SUP_image
thf(fact_102_Inf_OINF__image,axiom,
! [B: $tType,A: $tType,C: $tType,Inf: ( set @ A ) > A,G2: B > A,F2: C > B,A5: set @ C] :
( ( Inf @ ( image @ B @ A @ G2 @ ( image @ C @ B @ F2 @ A5 ) ) )
= ( Inf @ ( image @ C @ A @ ( comp @ B @ A @ C @ G2 @ F2 ) @ A5 ) ) ) ).
% Inf.INF_image
thf(fact_103_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_104_apsnd__id,axiom,
! [B: $tType,A: $tType] :
( ( product_apsnd @ B @ B @ A @ ( id @ B ) )
= ( id @ ( product_prod @ A @ B ) ) ) ).
% apsnd_id
thf(fact_105_apfst__id,axiom,
! [B: $tType,A: $tType] :
( ( product_apfst @ A @ A @ B @ ( id @ A ) )
= ( id @ ( product_prod @ A @ B ) ) ) ).
% apfst_id
thf(fact_106_fst__comp__map__prod,axiom,
! [D: $tType,C: $tType,B: $tType,A: $tType,F2: A > C,G2: B > D] :
( ( comp @ ( product_prod @ C @ D ) @ C @ ( product_prod @ A @ B ) @ ( product_fst @ C @ D ) @ ( product_map_prod @ A @ C @ B @ D @ F2 @ G2 ) )
= ( comp @ A @ C @ ( product_prod @ A @ B ) @ F2 @ ( product_fst @ A @ B ) ) ) ).
% fst_comp_map_prod
thf(fact_107_snd__comp__map__prod,axiom,
! [D: $tType,C: $tType,B: $tType,A: $tType,F2: A > D,G2: B > C] :
( ( comp @ ( product_prod @ D @ C ) @ C @ ( product_prod @ A @ B ) @ ( product_snd @ D @ C ) @ ( product_map_prod @ A @ D @ B @ C @ F2 @ G2 ) )
= ( comp @ B @ C @ ( product_prod @ A @ B ) @ G2 @ ( product_snd @ A @ B ) ) ) ).
% snd_comp_map_prod
thf(fact_108_swap__swap,axiom,
! [B: $tType,A: $tType,P3: product_prod @ A @ B] :
( ( product_swap @ B @ A @ ( product_swap @ A @ B @ P3 ) )
= P3 ) ).
% swap_swap
thf(fact_109_fst__map__prod,axiom,
! [B: $tType,A: $tType,D: $tType,C: $tType,F2: C > A,G2: D > B,X2: product_prod @ C @ D] :
( ( product_fst @ A @ B @ ( product_map_prod @ C @ A @ D @ B @ F2 @ G2 @ X2 ) )
= ( F2 @ ( product_fst @ C @ D @ X2 ) ) ) ).
% fst_map_prod
thf(fact_110_snd__map__prod,axiom,
! [B: $tType,A: $tType,D: $tType,C: $tType,F2: C > B,G2: D > A,X2: product_prod @ C @ D] :
( ( product_snd @ B @ A @ ( product_map_prod @ C @ B @ D @ A @ F2 @ G2 @ X2 ) )
= ( G2 @ ( product_snd @ C @ D @ X2 ) ) ) ).
% snd_map_prod
thf(fact_111_fst__apfst,axiom,
! [A: $tType,B: $tType,C: $tType,F2: C > A,X2: product_prod @ C @ B] :
( ( product_fst @ A @ B @ ( product_apfst @ C @ A @ B @ F2 @ X2 ) )
= ( F2 @ ( product_fst @ C @ B @ X2 ) ) ) ).
% fst_apfst
thf(fact_112_apfst__eq__conv,axiom,
! [A: $tType,B: $tType,C: $tType,F2: C > A,X2: product_prod @ C @ B,G2: C > A] :
( ( ( product_apfst @ C @ A @ B @ F2 @ X2 )
= ( product_apfst @ C @ A @ B @ G2 @ X2 ) )
= ( ( F2 @ ( product_fst @ C @ B @ X2 ) )
= ( G2 @ ( product_fst @ C @ B @ X2 ) ) ) ) ).
% apfst_eq_conv
thf(fact_113_snd__apfst,axiom,
! [B: $tType,A: $tType,C: $tType,F2: C > B,X2: product_prod @ C @ A] :
( ( product_snd @ B @ A @ ( product_apfst @ C @ B @ A @ F2 @ X2 ) )
= ( product_snd @ C @ A @ X2 ) ) ).
% snd_apfst
thf(fact_114_fst__apsnd,axiom,
! [B: $tType,C: $tType,A: $tType,F2: C > B,X2: product_prod @ A @ C] :
( ( product_fst @ A @ B @ ( product_apsnd @ C @ B @ A @ F2 @ X2 ) )
= ( product_fst @ A @ C @ X2 ) ) ).
% fst_apsnd
thf(fact_115_snd__apsnd,axiom,
! [A: $tType,C: $tType,B: $tType,F2: C > A,X2: product_prod @ B @ C] :
( ( product_snd @ B @ A @ ( product_apsnd @ C @ A @ B @ F2 @ X2 ) )
= ( F2 @ ( product_snd @ B @ C @ X2 ) ) ) ).
% snd_apsnd
thf(fact_116_apsnd__eq__conv,axiom,
! [B: $tType,C: $tType,A: $tType,F2: C > B,X2: product_prod @ A @ C,G2: C > B] :
( ( ( product_apsnd @ C @ B @ A @ F2 @ X2 )
= ( product_apsnd @ C @ B @ A @ G2 @ X2 ) )
= ( ( F2 @ ( product_snd @ A @ C @ X2 ) )
= ( G2 @ ( product_snd @ A @ C @ X2 ) ) ) ) ).
% apsnd_eq_conv
thf(fact_117_fst__swap,axiom,
! [A: $tType,B: $tType,X2: product_prod @ B @ A] :
( ( product_fst @ A @ B @ ( product_swap @ B @ A @ X2 ) )
= ( product_snd @ B @ A @ X2 ) ) ).
% fst_swap
thf(fact_118_snd__swap,axiom,
! [B: $tType,A: $tType,X2: product_prod @ A @ B] :
( ( product_snd @ B @ A @ ( product_swap @ A @ B @ X2 ) )
= ( product_fst @ A @ B @ X2 ) ) ).
% snd_swap
thf(fact_119_snd__comp__apfst,axiom,
! [C: $tType,B: $tType,A: $tType,F2: A > C] :
( ( comp @ ( product_prod @ C @ B ) @ B @ ( product_prod @ A @ B ) @ ( product_snd @ C @ B ) @ ( product_apfst @ A @ C @ B @ F2 ) )
= ( product_snd @ A @ B ) ) ).
% snd_comp_apfst
thf(fact_120_fst__comp__apsnd,axiom,
! [C: $tType,B: $tType,A: $tType,F2: B > C] :
( ( comp @ ( product_prod @ A @ C ) @ A @ ( product_prod @ A @ B ) @ ( product_fst @ A @ C ) @ ( product_apsnd @ B @ C @ A @ F2 ) )
= ( product_fst @ A @ B ) ) ).
% fst_comp_apsnd
thf(fact_121_fst__comp__apfst,axiom,
! [C: $tType,B: $tType,A: $tType,F2: A > C] :
( ( comp @ ( product_prod @ C @ B ) @ C @ ( product_prod @ A @ B ) @ ( product_fst @ C @ B ) @ ( product_apfst @ A @ C @ B @ F2 ) )
= ( comp @ A @ C @ ( product_prod @ A @ B ) @ F2 @ ( product_fst @ A @ B ) ) ) ).
% fst_comp_apfst
thf(fact_122_snd__comp__apsnd,axiom,
! [C: $tType,B: $tType,A: $tType,F2: B > C] :
( ( comp @ ( product_prod @ A @ C ) @ C @ ( product_prod @ A @ B ) @ ( product_snd @ A @ C ) @ ( product_apsnd @ B @ C @ A @ F2 ) )
= ( comp @ B @ C @ ( product_prod @ A @ B ) @ F2 @ ( product_snd @ A @ B ) ) ) ).
% snd_comp_apsnd
thf(fact_123_prod__eqI,axiom,
! [B: $tType,A: $tType,P3: product_prod @ A @ B,Q2: product_prod @ A @ B] :
( ( ( product_fst @ A @ B @ P3 )
= ( product_fst @ A @ B @ Q2 ) )
=> ( ( ( product_snd @ A @ B @ P3 )
= ( product_snd @ A @ B @ Q2 ) )
=> ( P3 = Q2 ) ) ) ).
% prod_eqI
thf(fact_124_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_125_prod__eq__iff,axiom,
! [B: $tType,A: $tType] :
( ( ^ [Y3: product_prod @ A @ B,Z4: product_prod @ A @ B] : Y3 = Z4 )
= ( ^ [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_126_apsnd__apfst__commute,axiom,
! [A: $tType,B: $tType,C: $tType,D: $tType,F2: C > B,G2: D > A,P3: product_prod @ D @ C] :
( ( product_apsnd @ C @ B @ A @ F2 @ ( product_apfst @ D @ A @ C @ G2 @ P3 ) )
= ( product_apfst @ D @ A @ B @ G2 @ ( product_apsnd @ C @ B @ D @ F2 @ P3 ) ) ) ).
% apsnd_apfst_commute
thf(fact_127_convol__expand__snd_H,axiom,
! [C: $tType,B: $tType,A: $tType,F2: A > ( product_prod @ B @ C ),G2: A > B,H: A > C] :
( ( ( comp @ ( product_prod @ B @ C ) @ B @ A @ ( product_fst @ B @ C ) @ F2 )
= G2 )
=> ( ( H
= ( comp @ ( product_prod @ B @ C ) @ C @ A @ ( product_snd @ B @ C ) @ F2 ) )
= ( ( bNF_convol @ A @ B @ C @ G2 @ H )
= F2 ) ) ) ).
% convol_expand_snd'
thf(fact_128_convol__expand__snd,axiom,
! [C: $tType,B: $tType,A: $tType,F2: A > ( product_prod @ B @ C ),G2: A > B] :
( ( ( comp @ ( product_prod @ B @ C ) @ B @ A @ ( product_fst @ B @ C ) @ F2 )
= G2 )
=> ( ( bNF_convol @ A @ B @ C @ G2 @ ( comp @ ( product_prod @ B @ C ) @ C @ A @ ( product_snd @ B @ C ) @ F2 ) )
= F2 ) ) ).
% convol_expand_snd
thf(fact_129_snd__convol_H,axiom,
! [B: $tType,A: $tType,C: $tType,F2: C > B,G2: C > A,X2: C] :
( ( product_snd @ B @ A @ ( bNF_convol @ C @ B @ A @ F2 @ G2 @ X2 ) )
= ( G2 @ X2 ) ) ).
% snd_convol'
thf(fact_130_fst__convol_H,axiom,
! [B: $tType,A: $tType,C: $tType,F2: C > A,G2: C > B,X2: C] :
( ( product_fst @ A @ B @ ( bNF_convol @ C @ A @ B @ F2 @ G2 @ X2 ) )
= ( F2 @ X2 ) ) ).
% fst_convol'
thf(fact_131_apfst__compose,axiom,
! [C: $tType,A: $tType,B: $tType,D: $tType,F2: C > A,G2: D > C,X2: product_prod @ D @ B] :
( ( product_apfst @ C @ A @ B @ F2 @ ( product_apfst @ D @ C @ B @ G2 @ X2 ) )
= ( product_apfst @ D @ A @ B @ ( comp @ C @ A @ D @ F2 @ G2 ) @ X2 ) ) ).
% apfst_compose
thf(fact_132_apsnd__compose,axiom,
! [C: $tType,B: $tType,D: $tType,A: $tType,F2: C > B,G2: D > C,X2: product_prod @ A @ D] :
( ( product_apsnd @ C @ B @ A @ F2 @ ( product_apsnd @ D @ C @ A @ G2 @ X2 ) )
= ( product_apsnd @ D @ B @ A @ ( comp @ C @ B @ D @ F2 @ G2 ) @ X2 ) ) ).
% apsnd_compose
thf(fact_133_snd__convol,axiom,
! [C: $tType,B: $tType,A: $tType,F2: A > C,G2: A > B] :
( ( comp @ ( product_prod @ C @ B ) @ B @ A @ ( product_snd @ C @ B ) @ ( bNF_convol @ A @ C @ B @ F2 @ G2 ) )
= G2 ) ).
% snd_convol
thf(fact_134_fst__convol,axiom,
! [C: $tType,B: $tType,A: $tType,F2: A > B,G2: A > C] :
( ( comp @ ( product_prod @ B @ C ) @ B @ A @ ( product_fst @ B @ C ) @ ( bNF_convol @ A @ B @ C @ F2 @ G2 ) )
= F2 ) ).
% fst_convol
thf(fact_135_apfst__def,axiom,
! [B: $tType,C: $tType,A: $tType] :
( ( product_apfst @ A @ C @ B )
= ( ^ [F: A > C] : ( product_map_prod @ A @ C @ B @ B @ F @ ( id @ B ) ) ) ) ).
% apfst_def
thf(fact_136_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_137_snd__sndOp,axiom,
! [B: $tType,A: $tType,C: $tType,P: 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 @ P @ Q ) ) ) ).
% snd_sndOp
thf(fact_138_fst__fstOp,axiom,
! [A: $tType,B: $tType,C: $tType,P: 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 @ P @ Q ) ) ) ).
% fst_fstOp
thf(fact_139_apfst__apsnd,axiom,
! [A: $tType,B: $tType,D: $tType,C: $tType,F2: C > A,G2: D > B,X2: product_prod @ C @ D] :
( ( product_apfst @ C @ A @ B @ F2 @ ( product_apsnd @ D @ B @ C @ G2 @ X2 ) )
= ( product_Pair @ A @ B @ ( F2 @ ( product_fst @ C @ D @ X2 ) ) @ ( G2 @ ( product_snd @ C @ D @ X2 ) ) ) ) ).
% apfst_apsnd
thf(fact_140_apsnd__apfst,axiom,
! [A: $tType,B: $tType,C: $tType,D: $tType,F2: C > B,G2: D > A,X2: product_prod @ D @ C] :
( ( product_apsnd @ C @ B @ A @ F2 @ ( product_apfst @ D @ A @ C @ G2 @ X2 ) )
= ( product_Pair @ A @ B @ ( G2 @ ( product_fst @ D @ C @ X2 ) ) @ ( F2 @ ( product_snd @ D @ C @ X2 ) ) ) ) ).
% apsnd_apfst
thf(fact_141_prod_Oswap__def,axiom,
! [B: $tType,A: $tType] :
( ( product_swap @ A @ B )
= ( ^ [P4: product_prod @ A @ B] : ( product_Pair @ B @ A @ ( product_snd @ A @ B @ P4 ) @ ( product_fst @ A @ B @ P4 ) ) ) ) ).
% prod.swap_def
thf(fact_142_old_Oprod_Oinject,axiom,
! [A: $tType,B: $tType,A2: A,B2: B,A6: A,B5: B] :
( ( ( product_Pair @ A @ B @ A2 @ B2 )
= ( product_Pair @ A @ B @ A6 @ B5 ) )
= ( ( A2 = A6 )
& ( B2 = B5 ) ) ) ).
% old.prod.inject
thf(fact_143_prod_Oinject,axiom,
! [A: $tType,B: $tType,X1: A,X22: B,Y1: A,Y22: B] :
( ( ( product_Pair @ A @ B @ X1 @ X22 )
= ( product_Pair @ A @ B @ Y1 @ Y22 ) )
= ( ( X1 = Y1 )
& ( X22 = Y22 ) ) ) ).
% prod.inject
thf(fact_144_map__prod__simp,axiom,
! [C: $tType,A: $tType,B: $tType,D: $tType,F2: C > A,G2: D > B,A2: C,B2: D] :
( ( product_map_prod @ C @ A @ D @ B @ F2 @ G2 @ ( product_Pair @ C @ D @ A2 @ B2 ) )
= ( product_Pair @ A @ B @ ( F2 @ A2 ) @ ( G2 @ B2 ) ) ) ).
% map_prod_simp
thf(fact_145_map__prod__imageI,axiom,
! [D: $tType,C: $tType,B: $tType,A: $tType,A2: A,B2: B,R3: set @ ( product_prod @ A @ B ),F2: A > C,G2: B > D] :
( ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ A2 @ B2 ) @ R3 )
=> ( member @ ( product_prod @ C @ D ) @ ( product_Pair @ C @ D @ ( F2 @ A2 ) @ ( G2 @ B2 ) ) @ ( image @ ( product_prod @ A @ B ) @ ( product_prod @ C @ D ) @ ( product_map_prod @ A @ C @ B @ D @ F2 @ G2 ) @ R3 ) ) ) ).
% map_prod_imageI
thf(fact_146_apfst__conv,axiom,
! [C: $tType,A: $tType,B: $tType,F2: C > A,X2: C,Y: B] :
( ( product_apfst @ C @ A @ B @ F2 @ ( product_Pair @ C @ B @ X2 @ Y ) )
= ( product_Pair @ A @ B @ ( F2 @ X2 ) @ Y ) ) ).
% apfst_conv
thf(fact_147_apsnd__conv,axiom,
! [A: $tType,B: $tType,C: $tType,F2: C > B,X2: A,Y: C] :
( ( product_apsnd @ C @ B @ A @ F2 @ ( product_Pair @ A @ C @ X2 @ Y ) )
= ( product_Pair @ A @ B @ X2 @ ( F2 @ Y ) ) ) ).
% apsnd_conv
thf(fact_148_swap__simp,axiom,
! [A: $tType,B: $tType,X2: B,Y: A] :
( ( product_swap @ B @ A @ ( product_Pair @ B @ A @ X2 @ Y ) )
= ( product_Pair @ A @ B @ Y @ X2 ) ) ).
% swap_simp
thf(fact_149_pair__in__swap__image,axiom,
! [A: $tType,B: $tType,Y: A,X2: B,A5: set @ ( product_prod @ B @ A )] :
( ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ Y @ X2 ) @ ( image @ ( product_prod @ B @ A ) @ ( product_prod @ A @ B ) @ ( product_swap @ B @ A ) @ A5 ) )
= ( member @ ( product_prod @ B @ A ) @ ( product_Pair @ B @ A @ X2 @ Y ) @ A5 ) ) ).
% pair_in_swap_image
thf(fact_150_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_151_old_Oprod_Oinducts,axiom,
! [B: $tType,A: $tType,P: ( product_prod @ A @ B ) > $o,Prod: product_prod @ A @ B] :
( ! [A7: A,B6: B] : ( P @ ( product_Pair @ A @ B @ A7 @ B6 ) )
=> ( P @ Prod ) ) ).
% old.prod.inducts
thf(fact_152_old_Oprod_Oexhaust,axiom,
! [A: $tType,B: $tType,Y: product_prod @ A @ B] :
~ ! [A7: A,B6: B] :
( Y
!= ( product_Pair @ A @ B @ A7 @ B6 ) ) ).
% old.prod.exhaust
thf(fact_153_prod__induct7,axiom,
! [G4: $tType,F4: $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 @ F4 @ G4 ) ) ) ) ) ) > $o,X2: product_prod @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D @ ( product_prod @ E @ ( product_prod @ F4 @ G4 ) ) ) ) )] :
( ! [A7: A,B6: B,C4: C,D4: D,E2: E,F5: F4,G5: G4] : ( P @ ( product_Pair @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D @ ( product_prod @ E @ ( product_prod @ F4 @ G4 ) ) ) ) ) @ A7 @ ( product_Pair @ B @ ( product_prod @ C @ ( product_prod @ D @ ( product_prod @ E @ ( product_prod @ F4 @ G4 ) ) ) ) @ B6 @ ( product_Pair @ C @ ( product_prod @ D @ ( product_prod @ E @ ( product_prod @ F4 @ G4 ) ) ) @ C4 @ ( product_Pair @ D @ ( product_prod @ E @ ( product_prod @ F4 @ G4 ) ) @ D4 @ ( product_Pair @ E @ ( product_prod @ F4 @ G4 ) @ E2 @ ( product_Pair @ F4 @ G4 @ F5 @ G5 ) ) ) ) ) ) )
=> ( P @ X2 ) ) ).
% prod_induct7
thf(fact_154_prod__induct6,axiom,
! [F4: $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 @ F4 ) ) ) ) ) > $o,X2: product_prod @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D @ ( product_prod @ E @ F4 ) ) ) )] :
( ! [A7: A,B6: B,C4: C,D4: D,E2: E,F5: F4] : ( P @ ( product_Pair @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D @ ( product_prod @ E @ F4 ) ) ) ) @ A7 @ ( product_Pair @ B @ ( product_prod @ C @ ( product_prod @ D @ ( product_prod @ E @ F4 ) ) ) @ B6 @ ( product_Pair @ C @ ( product_prod @ D @ ( product_prod @ E @ F4 ) ) @ C4 @ ( product_Pair @ D @ ( product_prod @ E @ F4 ) @ D4 @ ( product_Pair @ E @ F4 @ E2 @ F5 ) ) ) ) ) )
=> ( P @ X2 ) ) ).
% prod_induct6
thf(fact_155_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,X2: product_prod @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D @ E ) ) )] :
( ! [A7: A,B6: B,C4: C,D4: D,E2: E] : ( P @ ( product_Pair @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D @ E ) ) ) @ A7 @ ( product_Pair @ B @ ( product_prod @ C @ ( product_prod @ D @ E ) ) @ B6 @ ( product_Pair @ C @ ( product_prod @ D @ E ) @ C4 @ ( product_Pair @ D @ E @ D4 @ E2 ) ) ) ) )
=> ( P @ X2 ) ) ).
% prod_induct5
thf(fact_156_prod__induct4,axiom,
! [D: $tType,C: $tType,B: $tType,A: $tType,P: ( product_prod @ A @ ( product_prod @ B @ ( product_prod @ C @ D ) ) ) > $o,X2: product_prod @ A @ ( product_prod @ B @ ( product_prod @ C @ D ) )] :
( ! [A7: A,B6: B,C4: C,D4: D] : ( P @ ( product_Pair @ A @ ( product_prod @ B @ ( product_prod @ C @ D ) ) @ A7 @ ( product_Pair @ B @ ( product_prod @ C @ D ) @ B6 @ ( product_Pair @ C @ D @ C4 @ D4 ) ) ) )
=> ( P @ X2 ) ) ).
% prod_induct4
thf(fact_157_prod__induct3,axiom,
! [C: $tType,B: $tType,A: $tType,P: ( product_prod @ A @ ( product_prod @ B @ C ) ) > $o,X2: product_prod @ A @ ( product_prod @ B @ C )] :
( ! [A7: A,B6: B,C4: C] : ( P @ ( product_Pair @ A @ ( product_prod @ B @ C ) @ A7 @ ( product_Pair @ B @ C @ B6 @ C4 ) ) )
=> ( P @ X2 ) ) ).
% prod_induct3
thf(fact_158_prod__cases7,axiom,
! [A: $tType,B: $tType,C: $tType,D: $tType,E: $tType,F4: $tType,G4: $tType,Y: product_prod @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D @ ( product_prod @ E @ ( product_prod @ F4 @ G4 ) ) ) ) )] :
~ ! [A7: A,B6: B,C4: C,D4: D,E2: E,F5: F4,G5: G4] :
( Y
!= ( product_Pair @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D @ ( product_prod @ E @ ( product_prod @ F4 @ G4 ) ) ) ) ) @ A7 @ ( product_Pair @ B @ ( product_prod @ C @ ( product_prod @ D @ ( product_prod @ E @ ( product_prod @ F4 @ G4 ) ) ) ) @ B6 @ ( product_Pair @ C @ ( product_prod @ D @ ( product_prod @ E @ ( product_prod @ F4 @ G4 ) ) ) @ C4 @ ( product_Pair @ D @ ( product_prod @ E @ ( product_prod @ F4 @ G4 ) ) @ D4 @ ( product_Pair @ E @ ( product_prod @ F4 @ G4 ) @ E2 @ ( product_Pair @ F4 @ G4 @ F5 @ G5 ) ) ) ) ) ) ) ).
% prod_cases7
thf(fact_159_prod__cases6,axiom,
! [A: $tType,B: $tType,C: $tType,D: $tType,E: $tType,F4: $tType,Y: product_prod @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D @ ( product_prod @ E @ F4 ) ) ) )] :
~ ! [A7: A,B6: B,C4: C,D4: D,E2: E,F5: F4] :
( Y
!= ( product_Pair @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D @ ( product_prod @ E @ F4 ) ) ) ) @ A7 @ ( product_Pair @ B @ ( product_prod @ C @ ( product_prod @ D @ ( product_prod @ E @ F4 ) ) ) @ B6 @ ( product_Pair @ C @ ( product_prod @ D @ ( product_prod @ E @ F4 ) ) @ C4 @ ( product_Pair @ D @ ( product_prod @ E @ F4 ) @ D4 @ ( product_Pair @ E @ F4 @ E2 @ F5 ) ) ) ) ) ) ).
% prod_cases6
thf(fact_160_prod__cases5,axiom,
! [A: $tType,B: $tType,C: $tType,D: $tType,E: $tType,Y: product_prod @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D @ E ) ) )] :
~ ! [A7: A,B6: B,C4: C,D4: D,E2: E] :
( Y
!= ( product_Pair @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D @ E ) ) ) @ A7 @ ( product_Pair @ B @ ( product_prod @ C @ ( product_prod @ D @ E ) ) @ B6 @ ( product_Pair @ C @ ( product_prod @ D @ E ) @ C4 @ ( product_Pair @ D @ E @ D4 @ E2 ) ) ) ) ) ).
% prod_cases5
thf(fact_161_prod__cases4,axiom,
! [A: $tType,B: $tType,C: $tType,D: $tType,Y: product_prod @ A @ ( product_prod @ B @ ( product_prod @ C @ D ) )] :
~ ! [A7: A,B6: B,C4: C,D4: D] :
( Y
!= ( product_Pair @ A @ ( product_prod @ B @ ( product_prod @ C @ D ) ) @ A7 @ ( product_Pair @ B @ ( product_prod @ C @ D ) @ B6 @ ( product_Pair @ C @ D @ C4 @ D4 ) ) ) ) ).
% prod_cases4
thf(fact_162_prod__cases3,axiom,
! [A: $tType,B: $tType,C: $tType,Y: product_prod @ A @ ( product_prod @ B @ C )] :
~ ! [A7: A,B6: B,C4: C] :
( Y
!= ( product_Pair @ A @ ( product_prod @ B @ C ) @ A7 @ ( product_Pair @ B @ C @ B6 @ C4 ) ) ) ).
% prod_cases3
thf(fact_163_Pair__inject,axiom,
! [A: $tType,B: $tType,A2: A,B2: B,A6: A,B5: B] :
( ( ( product_Pair @ A @ B @ A2 @ B2 )
= ( product_Pair @ A @ B @ A6 @ B5 ) )
=> ~ ( ( A2 = A6 )
=> ( B2 != B5 ) ) ) ).
% Pair_inject
thf(fact_164_prod__cases,axiom,
! [B: $tType,A: $tType,P: ( product_prod @ A @ B ) > $o,P3: product_prod @ A @ B] :
( ! [A7: A,B6: B] : ( P @ ( product_Pair @ A @ B @ A7 @ B6 ) )
=> ( P @ P3 ) ) ).
% prod_cases
thf(fact_165_surj__pair,axiom,
! [A: $tType,B: $tType,P3: product_prod @ A @ B] :
? [X4: A,Y4: B] :
( P3
= ( product_Pair @ A @ B @ X4 @ Y4 ) ) ).
% surj_pair
thf(fact_166_fstI,axiom,
! [B: $tType,A: $tType,X2: product_prod @ A @ B,Y: A,Z2: B] :
( ( X2
= ( product_Pair @ A @ B @ Y @ Z2 ) )
=> ( ( product_fst @ A @ B @ X2 )
= Y ) ) ).
% fstI
thf(fact_167_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_168_fst__eqD,axiom,
! [B: $tType,A: $tType,X2: A,Y: B,A2: A] :
( ( ( product_fst @ A @ B @ ( product_Pair @ A @ B @ X2 @ Y ) )
= A2 )
=> ( X2 = A2 ) ) ).
% fst_eqD
thf(fact_169_sndI,axiom,
! [A: $tType,B: $tType,X2: product_prod @ A @ B,Y: A,Z2: B] :
( ( X2
= ( product_Pair @ A @ B @ Y @ Z2 ) )
=> ( ( product_snd @ A @ B @ X2 )
= Z2 ) ) ).
% sndI
thf(fact_170_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_171_snd__eqD,axiom,
! [B: $tType,A: $tType,X2: B,Y: A,A2: A] :
( ( ( product_snd @ B @ A @ ( product_Pair @ B @ A @ X2 @ Y ) )
= A2 )
=> ( Y = A2 ) ) ).
% snd_eqD
thf(fact_172_prod__fun__imageE,axiom,
! [B: $tType,A: $tType,D: $tType,C: $tType,C2: product_prod @ A @ B,F2: C > A,G2: D > B,R3: set @ ( product_prod @ C @ D )] :
( ( member @ ( product_prod @ A @ B ) @ C2 @ ( image @ ( product_prod @ C @ D ) @ ( product_prod @ A @ B ) @ ( product_map_prod @ C @ A @ D @ B @ F2 @ G2 ) @ R3 ) )
=> ~ ! [X4: C,Y4: D] :
( ( C2
= ( product_Pair @ A @ B @ ( F2 @ X4 ) @ ( G2 @ Y4 ) ) )
=> ~ ( member @ ( product_prod @ C @ D ) @ ( product_Pair @ C @ D @ X4 @ Y4 ) @ R3 ) ) ) ).
% prod_fun_imageE
thf(fact_173_convol__def,axiom,
! [C: $tType,B: $tType,A: $tType] :
( ( bNF_convol @ A @ B @ C )
= ( ^ [F: A > B,G: A > C,A4: A] : ( product_Pair @ B @ C @ ( F @ A4 ) @ ( G @ A4 ) ) ) ) ).
% convol_def
thf(fact_174_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_175_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_176_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_177_fstOp__def,axiom,
! [C: $tType,B: $tType,A: $tType] :
( ( bNF_fstOp @ A @ B @ C )
= ( ^ [P5: 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 @ P5 @ Q3 @ ( product_fst @ A @ C @ Ac ) @ ( product_snd @ A @ C @ Ac ) ) ) ) ) ).
% fstOp_def
thf(fact_178_sndOp__def,axiom,
! [B: $tType,A: $tType,C: $tType] :
( ( bNF_sndOp @ C @ A @ B )
= ( ^ [P5: C > A > $o,Q3: A > B > $o,Ac: product_prod @ C @ B] : ( product_Pair @ A @ B @ ( bNF_pick_middlep @ C @ A @ B @ P5 @ Q3 @ ( product_fst @ C @ B @ Ac ) @ ( product_snd @ C @ B @ Ac ) ) @ ( product_snd @ C @ B @ Ac ) ) ) ) ).
% sndOp_def
thf(fact_179_csquare__fstOp__sndOp,axiom,
! [A: $tType,B: $tType,C: $tType,F2: ( 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 ) @ ( F2 @ ( 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_180_pick__middlep,axiom,
! [B: $tType,A: $tType,C: $tType,P: A > B > $o,Q: B > C > $o,A2: A,C2: C] :
( ( relcompp @ A @ B @ C @ P @ Q @ A2 @ C2 )
=> ( ( P @ A2 @ ( bNF_pick_middlep @ A @ B @ C @ P @ Q @ A2 @ C2 ) )
& ( Q @ ( bNF_pick_middlep @ A @ B @ C @ P @ Q @ A2 @ C2 ) @ C2 ) ) ) ).
% pick_middlep
thf(fact_181_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_182_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_183_o__prs_I2_J,axiom,
! [F4: $tType,E: $tType,C: $tType,D: $tType,A: $tType,B: $tType,H3: $tType,G4: $tType,R12: A > A > $o,Abs1: A > B,Rep1: B > A,R22: C > C > $o,Abs2: C > D,Rep2: D > C,R32: E > E > $o,Abs3: E > F4,Rep3: F4 > E] :
( ( quotient3 @ A @ B @ R12 @ Abs1 @ Rep1 )
=> ( ( quotient3 @ C @ D @ R22 @ Abs2 @ Rep2 )
=> ( ( quotient3 @ E @ F4 @ R32 @ Abs3 @ Rep3 )
=> ( ( map_fun @ ( G4 > H3 ) @ ( G4 > H3 ) @ ( ( A > G4 ) > A > H3 ) @ ( ( B > G4 ) > B > H3 ) @ ( id @ ( G4 > H3 ) ) @ ( map_fun @ ( B > G4 ) @ ( A > G4 ) @ ( A > H3 ) @ ( B > H3 ) @ ( map_fun @ A @ B @ G4 @ G4 @ Abs1 @ ( id @ G4 ) ) @ ( map_fun @ B @ A @ H3 @ H3 @ Rep1 @ ( id @ H3 ) ) ) @ ( comp @ G4 @ H3 @ A ) )
= ( comp @ G4 @ H3 @ B ) ) ) ) ) ).
% o_prs(2)
thf(fact_184_UNIV__I,axiom,
! [A: $tType,X2: A] : ( member @ A @ X2 @ ( top_top @ ( set @ A ) ) ) ).
% UNIV_I
thf(fact_185_surj__swap__iff,axiom,
! [B: $tType,A: $tType,A2: B,B2: B,F2: B > A] :
( ( ( image @ B @ A @ ( swap @ B @ A @ A2 @ B2 @ F2 ) @ ( top_top @ ( set @ B ) ) )
= ( top_top @ ( set @ A ) ) )
= ( ( image @ B @ A @ F2 @ ( top_top @ ( set @ B ) ) )
= ( top_top @ ( set @ A ) ) ) ) ).
% surj_swap_iff
thf(fact_186_OOO__quotient3,axiom,
! [A: $tType,B: $tType,C: $tType,R12: A > A > $o,Abs1: A > B,Rep1: B > A,R22: B > B > $o,Abs2: B > C,Rep2: C > B,R23: A > A > $o] :
( ( quotient3 @ A @ B @ R12 @ Abs1 @ Rep1 )
=> ( ( quotient3 @ B @ C @ R22 @ Abs2 @ Rep2 )
=> ( ! [X4: A,Y4: A] :
( ( R23 @ X4 @ Y4 )
=> ( ( R12 @ X4 @ X4 )
=> ( ( R12 @ Y4 @ Y4 )
=> ( R22 @ ( Abs1 @ X4 ) @ ( Abs1 @ Y4 ) ) ) ) )
=> ( ! [X4: B,Y4: B] :
( ( R22 @ X4 @ Y4 )
=> ( R23 @ ( Rep1 @ X4 ) @ ( Rep1 @ Y4 ) ) )
=> ( quotient3 @ A @ C @ ( relcompp @ A @ A @ A @ R12 @ ( relcompp @ A @ A @ A @ R23 @ R12 ) ) @ ( comp @ B @ C @ A @ Abs2 @ Abs1 ) @ ( comp @ B @ A @ C @ Rep1 @ Rep2 ) ) ) ) ) ) ).
% OOO_quotient3
thf(fact_187_OOO__eq__quotient3,axiom,
! [A: $tType,B: $tType,C: $tType,R12: A > A > $o,Abs1: A > B,Rep1: B > A,Abs2: B > C,Rep2: C > B] :
( ( quotient3 @ A @ B @ R12 @ Abs1 @ Rep1 )
=> ( ( quotient3 @ B @ C
@ ^ [Y3: B,Z4: B] : Y3 = Z4
@ Abs2
@ Rep2 )
=> ( quotient3 @ A @ C
@ ( relcompp @ A @ A @ A @ R12
@ ( relcompp @ A @ A @ A
@ ^ [Y3: A,Z4: A] : Y3 = Z4
@ R12 ) )
@ ( comp @ B @ C @ A @ Abs2 @ Abs1 )
@ ( comp @ B @ A @ C @ Rep1 @ Rep2 ) ) ) ) ).
% OOO_eq_quotient3
thf(fact_188_range__eqI,axiom,
! [A: $tType,B: $tType,B2: A,F2: B > A,X2: B] :
( ( B2
= ( F2 @ X2 ) )
=> ( member @ A @ B2 @ ( image @ B @ A @ F2 @ ( top_top @ ( set @ B ) ) ) ) ) ).
% range_eqI
thf(fact_189_rangeI,axiom,
! [A: $tType,B: $tType,F2: B > A,X2: B] : ( member @ A @ ( F2 @ X2 ) @ ( image @ B @ A @ F2 @ ( top_top @ ( set @ B ) ) ) ) ).
% rangeI
thf(fact_190_surj__def,axiom,
! [B: $tType,A: $tType,F2: B > A] :
( ( ( image @ B @ A @ F2 @ ( top_top @ ( set @ B ) ) )
= ( top_top @ ( set @ A ) ) )
= ( ! [Y2: A] :
? [X: B] :
( Y2
= ( F2 @ X ) ) ) ) ).
% surj_def
thf(fact_191_surjI,axiom,
! [B: $tType,A: $tType,G2: B > A,F2: A > B] :
( ! [X4: A] :
( ( G2 @ ( F2 @ X4 ) )
= X4 )
=> ( ( image @ B @ A @ G2 @ ( top_top @ ( set @ B ) ) )
= ( top_top @ ( set @ A ) ) ) ) ).
% surjI
thf(fact_192_surjE,axiom,
! [A: $tType,B: $tType,F2: B > A,Y: A] :
( ( ( image @ B @ A @ F2 @ ( top_top @ ( set @ B ) ) )
= ( top_top @ ( set @ A ) ) )
=> ~ ! [X4: B] :
( Y
!= ( F2 @ X4 ) ) ) ).
% surjE
thf(fact_193_surjD,axiom,
! [A: $tType,B: $tType,F2: B > A,Y: A] :
( ( ( image @ B @ A @ F2 @ ( top_top @ ( set @ B ) ) )
= ( top_top @ ( set @ A ) ) )
=> ? [X4: B] :
( Y
= ( F2 @ X4 ) ) ) ).
% surjD
thf(fact_194_map__prod__surj,axiom,
! [A: $tType,C: $tType,D: $tType,B: $tType,F2: A > B,G2: C > D] :
( ( ( image @ A @ B @ F2 @ ( top_top @ ( set @ A ) ) )
= ( top_top @ ( set @ B ) ) )
=> ( ( ( image @ C @ D @ G2 @ ( top_top @ ( set @ C ) ) )
= ( top_top @ ( set @ D ) ) )
=> ( ( image @ ( product_prod @ A @ C ) @ ( product_prod @ B @ D ) @ ( product_map_prod @ A @ B @ C @ D @ F2 @ G2 ) @ ( top_top @ ( set @ ( product_prod @ A @ C ) ) ) )
= ( top_top @ ( set @ ( product_prod @ B @ D ) ) ) ) ) ) ).
% map_prod_surj
thf(fact_195_let__prs,axiom,
! [C: $tType,A: $tType,B: $tType,D: $tType,R12: A > A > $o,Abs1: A > B,Rep1: B > A,R22: C > C > $o,Abs2: C > D,Rep2: D > C] :
( ( quotient3 @ A @ B @ R12 @ Abs1 @ Rep1 )
=> ( ( quotient3 @ C @ D @ R22 @ Abs2 @ Rep2 )
=> ( ( map_fun @ D @ C @ ( ( C > A ) > A ) @ ( ( D > B ) > B ) @ Rep2 @ ( map_fun @ ( D > B ) @ ( C > A ) @ A @ B @ ( map_fun @ C @ D @ B @ A @ Abs2 @ Rep1 ) @ Abs1 )
@ ^ [S3: C,F: C > A] : ( F @ S3 ) )
= ( ^ [S3: D,F: D > B] : ( F @ S3 ) ) ) ) ) ).
% let_prs
thf(fact_196_UNIV__witness,axiom,
! [A: $tType] :
? [X4: A] : ( member @ A @ X4 @ ( top_top @ ( set @ A ) ) ) ).
% UNIV_witness
thf(fact_197_UNIV__eq__I,axiom,
! [A: $tType,A5: set @ A] :
( ! [X4: A] : ( member @ A @ X4 @ A5 )
=> ( ( top_top @ ( set @ A ) )
= A5 ) ) ).
% UNIV_eq_I
thf(fact_198_identity__quotient3,axiom,
! [A: $tType] :
( quotient3 @ A @ A
@ ^ [Y3: A,Z4: A] : Y3 = Z4
@ ( id @ A )
@ ( id @ A ) ) ).
% identity_quotient3
thf(fact_199_if__prs,axiom,
! [A: $tType,B: $tType,R3: A > A > $o,Abs: A > B,Rep: B > A] :
( ( quotient3 @ A @ B @ R3 @ Abs @ Rep )
=> ( ( map_fun @ $o @ $o @ ( A > A > A ) @ ( B > B > B ) @ ( id @ $o ) @ ( map_fun @ B @ A @ ( A > A ) @ ( B > B ) @ Rep @ ( map_fun @ B @ A @ A @ B @ Rep @ Abs ) ) @ ( if @ A ) )
= ( if @ B ) ) ) ).
% if_prs
thf(fact_200_id__prs,axiom,
! [A: $tType,B: $tType,R3: A > A > $o,Abs: A > B,Rep: B > A] :
( ( quotient3 @ A @ B @ R3 @ Abs @ Rep )
=> ( ( map_fun @ B @ A @ A @ B @ Rep @ Abs @ ( id @ A ) )
= ( id @ B ) ) ) ).
% id_prs
thf(fact_201_o__prs_I1_J,axiom,
! [C: $tType,E: $tType,A: $tType,B: $tType,F4: $tType,D: $tType,R12: A > A > $o,Abs1: A > B,Rep1: B > A,R22: C > C > $o,Abs2: C > D,Rep2: D > C,R32: E > E > $o,Abs3: E > F4,Rep3: F4 > E] :
( ( quotient3 @ A @ B @ R12 @ Abs1 @ Rep1 )
=> ( ( quotient3 @ C @ D @ R22 @ Abs2 @ Rep2 )
=> ( ( quotient3 @ E @ F4 @ R32 @ Abs3 @ Rep3 )
=> ( ( map_fun @ ( D > F4 ) @ ( C > E ) @ ( ( A > C ) > A > E ) @ ( ( B > D ) > B > F4 ) @ ( map_fun @ C @ D @ F4 @ E @ Abs2 @ Rep3 ) @ ( map_fun @ ( B > D ) @ ( A > C ) @ ( A > E ) @ ( B > F4 ) @ ( map_fun @ A @ B @ D @ C @ Abs1 @ Rep2 ) @ ( map_fun @ B @ A @ E @ F4 @ Rep1 @ Abs3 ) ) @ ( comp @ C @ E @ A ) )
= ( comp @ D @ F4 @ B ) ) ) ) ) ).
% o_prs(1)
thf(fact_202_abs__o__rep,axiom,
! [A: $tType,B: $tType,R3: A > A > $o,Abs: A > B,Rep: B > A] :
( ( quotient3 @ A @ B @ R3 @ Abs @ Rep )
=> ( ( comp @ A @ B @ B @ Abs @ Rep )
= ( id @ B ) ) ) ).
% abs_o_rep
thf(fact_203_DEADID_Oin__rel,axiom,
! [B: $tType] :
( ( ^ [Y3: B,Z4: B] : Y3 = Z4 )
= ( ^ [A4: B,B3: B] :
? [Z: B] :
( ( member @ B @ Z @ ( top_top @ ( set @ B ) ) )
& ( ( id @ B @ Z )
= A4 )
& ( ( id @ B @ Z )
= B3 ) ) ) ) ).
% DEADID.in_rel
thf(fact_204_comp__surj,axiom,
! [B: $tType,A: $tType,C: $tType,F2: B > A,G2: A > C] :
( ( ( image @ B @ A @ F2 @ ( top_top @ ( set @ B ) ) )
= ( top_top @ ( set @ A ) ) )
=> ( ( ( image @ A @ C @ G2 @ ( top_top @ ( set @ A ) ) )
= ( top_top @ ( set @ C ) ) )
=> ( ( image @ B @ C @ ( comp @ A @ C @ B @ G2 @ F2 ) @ ( top_top @ ( set @ B ) ) )
= ( top_top @ ( set @ C ) ) ) ) ) ).
% comp_surj
thf(fact_205_fun_Oset__map,axiom,
! [B: $tType,A: $tType,D: $tType,F2: A > B,V: D > A] :
( ( image @ D @ B @ ( comp @ A @ B @ D @ F2 @ V ) @ ( top_top @ ( set @ D ) ) )
= ( image @ A @ B @ F2 @ ( image @ D @ A @ V @ ( top_top @ ( set @ D ) ) ) ) ) ).
% fun.set_map
thf(fact_206_fun_Omap__cong,axiom,
! [B: $tType,A: $tType,D: $tType,X2: D > A,Ya: D > A,F2: A > B,G2: A > B] :
( ( X2 = Ya )
=> ( ! [Z3: A] :
( ( member @ A @ Z3 @ ( image @ D @ A @ Ya @ ( top_top @ ( set @ D ) ) ) )
=> ( ( F2 @ Z3 )
= ( G2 @ Z3 ) ) )
=> ( ( comp @ A @ B @ D @ F2 @ X2 )
= ( comp @ A @ B @ D @ G2 @ Ya ) ) ) ) ).
% fun.map_cong
thf(fact_207_fun_Omap__cong0,axiom,
! [B: $tType,A: $tType,D: $tType,X2: D > A,F2: A > B,G2: A > B] :
( ! [Z3: A] :
( ( member @ A @ Z3 @ ( image @ D @ A @ X2 @ ( top_top @ ( set @ D ) ) ) )
=> ( ( F2 @ Z3 )
= ( G2 @ Z3 ) ) )
=> ( ( comp @ A @ B @ D @ F2 @ X2 )
= ( comp @ A @ B @ D @ G2 @ X2 ) ) ) ).
% fun.map_cong0
thf(fact_208_fun_Oinj__map__strong,axiom,
! [B: $tType,A: $tType,D: $tType,X2: D > A,Xa: D > A,F2: A > B,Fa: A > B] :
( ! [Z3: A,Za: A] :
( ( member @ A @ Z3 @ ( image @ D @ A @ X2 @ ( top_top @ ( set @ D ) ) ) )
=> ( ( member @ A @ Za @ ( image @ D @ A @ Xa @ ( top_top @ ( set @ D ) ) ) )
=> ( ( ( F2 @ Z3 )
= ( Fa @ Za ) )
=> ( Z3 = Za ) ) ) )
=> ( ( ( comp @ A @ B @ D @ F2 @ X2 )
= ( comp @ A @ B @ D @ Fa @ Xa ) )
=> ( X2 = Xa ) ) ) ).
% fun.inj_map_strong
thf(fact_209_surj__id,axiom,
! [A: $tType] :
( ( image @ A @ A @ ( id @ A ) @ ( top_top @ ( set @ A ) ) )
= ( top_top @ ( set @ A ) ) ) ).
% surj_id
thf(fact_210_surj__imp__surj__swap,axiom,
! [B: $tType,A: $tType,F2: B > A,A2: B,B2: B] :
( ( ( image @ B @ A @ F2 @ ( top_top @ ( set @ B ) ) )
= ( top_top @ ( set @ A ) ) )
=> ( ( image @ B @ A @ ( swap @ B @ A @ A2 @ B2 @ F2 ) @ ( top_top @ ( set @ B ) ) )
= ( top_top @ ( set @ A ) ) ) ) ).
% surj_imp_surj_swap
thf(fact_211_surj__fun__eq,axiom,
! [B: $tType,C: $tType,A: $tType,F2: B > A,X6: set @ B,G1: A > C,G22: A > C] :
( ( ( image @ B @ A @ F2 @ X6 )
= ( top_top @ ( set @ A ) ) )
=> ( ! [X4: B] :
( ( member @ B @ X4 @ X6 )
=> ( ( comp @ A @ C @ B @ G1 @ F2 @ X4 )
= ( comp @ A @ C @ B @ G22 @ F2 @ X4 ) ) )
=> ( G1 = G22 ) ) ) ).
% surj_fun_eq
thf(fact_212_type__copy__map__id0,axiom,
! [B: $tType,A: $tType,Rep: A > B,Abs: B > A,M: B > B] :
( ( type_definition @ A @ B @ Rep @ Abs @ ( top_top @ ( set @ B ) ) )
=> ( ( M
= ( id @ B ) )
=> ( ( comp @ B @ A @ A @ ( comp @ B @ A @ B @ Abs @ M ) @ Rep )
= ( id @ A ) ) ) ) ).
% type_copy_map_id0
thf(fact_213_type__copy__Abs__o__Rep,axiom,
! [B: $tType,A: $tType,Rep: A > B,Abs: B > A] :
( ( type_definition @ A @ B @ Rep @ Abs @ ( top_top @ ( set @ B ) ) )
=> ( ( comp @ B @ A @ A @ Abs @ Rep )
= ( id @ A ) ) ) ).
% type_copy_Abs_o_Rep
thf(fact_214_type__copy__obj__one__point__absE,axiom,
! [A: $tType,B: $tType,Rep: A > B,Abs: B > A,S: A] :
( ( type_definition @ A @ B @ Rep @ Abs @ ( top_top @ ( set @ B ) ) )
=> ~ ! [X4: B] :
( S
!= ( Abs @ X4 ) ) ) ).
% type_copy_obj_one_point_absE
thf(fact_215_type__copy__ex__RepI,axiom,
! [B: $tType,A: $tType,Rep: A > B,Abs: B > A,F6: B > $o] :
( ( type_definition @ A @ B @ Rep @ Abs @ ( top_top @ ( set @ B ) ) )
=> ( ( ^ [P6: B > $o] :
? [X7: B] : ( P6 @ X7 )
@ F6 )
= ( ? [B3: A] : ( F6 @ ( Rep @ B3 ) ) ) ) ) ).
% type_copy_ex_RepI
thf(fact_216_type__copy__wit,axiom,
! [A: $tType,C: $tType,B: $tType,Rep: A > B,Abs: B > A,X2: C,S2: B > ( set @ C ),Y: B] :
( ( type_definition @ A @ B @ Rep @ Abs @ ( top_top @ ( set @ B ) ) )
=> ( ( member @ C @ X2 @ ( comp @ B @ ( set @ C ) @ A @ S2 @ Rep @ ( Abs @ Y ) ) )
=> ( member @ C @ X2 @ ( S2 @ Y ) ) ) ) ).
% type_copy_wit
thf(fact_217_type__copy__map__comp0__undo,axiom,
! [E: $tType,A: $tType,C: $tType,B: $tType,D: $tType,F4: $tType,Rep: A > B,Abs: B > A,Rep4: C > D,Abs4: D > C,Rep5: E > F4,Abs5: F4 > E,M: F4 > D,M1: B > D,M22: F4 > B] :
( ( type_definition @ A @ B @ Rep @ Abs @ ( top_top @ ( set @ B ) ) )
=> ( ( type_definition @ C @ D @ Rep4 @ Abs4 @ ( top_top @ ( set @ D ) ) )
=> ( ( type_definition @ E @ F4 @ Rep5 @ Abs5 @ ( top_top @ ( set @ F4 ) ) )
=> ( ( ( comp @ F4 @ C @ E @ ( comp @ D @ C @ F4 @ Abs4 @ M ) @ Rep5 )
= ( comp @ A @ C @ E @ ( comp @ B @ C @ A @ ( comp @ D @ C @ B @ Abs4 @ M1 ) @ Rep ) @ ( comp @ F4 @ A @ E @ ( comp @ B @ A @ F4 @ Abs @ M22 ) @ Rep5 ) ) )
=> ( ( comp @ B @ D @ F4 @ M1 @ M22 )
= M ) ) ) ) ) ).
% type_copy_map_comp0_undo
thf(fact_218_type__copy__map__comp0,axiom,
! [F4: $tType,D: $tType,B: $tType,A: $tType,C: $tType,E: $tType,Rep: A > B,Abs: B > A,M: C > D,M1: B > D,M22: C > B,F2: D > F4,G2: E > C] :
( ( type_definition @ A @ B @ Rep @ Abs @ ( top_top @ ( set @ B ) ) )
=> ( ( M
= ( comp @ B @ D @ C @ M1 @ M22 ) )
=> ( ( comp @ C @ F4 @ E @ ( comp @ D @ F4 @ C @ F2 @ M ) @ G2 )
= ( comp @ A @ F4 @ E @ ( comp @ B @ F4 @ A @ ( comp @ D @ F4 @ B @ F2 @ M1 ) @ Rep ) @ ( comp @ C @ A @ E @ ( comp @ B @ A @ C @ Abs @ M22 ) @ G2 ) ) ) ) ) ).
% type_copy_map_comp0
thf(fact_219_type__copy__set__map0,axiom,
! [A: $tType,B: $tType,D: $tType,E: $tType,C: $tType,F4: $tType,Rep: A > B,Abs: B > A,S2: B > ( set @ D ),M: C > B,F2: E > D,S4: C > ( set @ E ),G2: F4 > C] :
( ( type_definition @ A @ B @ Rep @ Abs @ ( top_top @ ( set @ B ) ) )
=> ( ( ( comp @ B @ ( set @ D ) @ C @ S2 @ M )
= ( comp @ ( set @ E ) @ ( set @ D ) @ C @ ( image @ E @ D @ F2 ) @ S4 ) )
=> ( ( comp @ A @ ( set @ D ) @ F4 @ ( comp @ B @ ( set @ D ) @ A @ S2 @ Rep ) @ ( comp @ C @ A @ F4 @ ( comp @ B @ A @ C @ Abs @ M ) @ G2 ) )
= ( comp @ ( set @ E ) @ ( set @ D ) @ F4 @ ( image @ E @ D @ F2 ) @ ( comp @ C @ ( set @ E ) @ F4 @ S4 @ G2 ) ) ) ) ) ).
% type_copy_set_map0
thf(fact_220_type__copy__Rep__o__Abs,axiom,
! [A: $tType,B: $tType,Rep: A > B,Abs: B > A] :
( ( type_definition @ A @ B @ Rep @ Abs @ ( top_top @ ( set @ B ) ) )
=> ( ( comp @ A @ B @ B @ Rep @ Abs )
= ( id @ B ) ) ) ).
% type_copy_Rep_o_Abs
thf(fact_221_type__definition_OAbs__image,axiom,
! [A: $tType,B: $tType,Rep: B > A,Abs: A > B,A5: set @ A] :
( ( type_definition @ B @ A @ Rep @ Abs @ A5 )
=> ( ( image @ A @ B @ Abs @ A5 )
= ( top_top @ ( set @ B ) ) ) ) ).
% type_definition.Abs_image
thf(fact_222_type__definition_ORep__range,axiom,
! [B: $tType,A: $tType,Rep: B > A,Abs: A > B,A5: set @ A] :
( ( type_definition @ B @ A @ Rep @ Abs @ A5 )
=> ( ( image @ B @ A @ Rep @ ( top_top @ ( set @ B ) ) )
= A5 ) ) ).
% type_definition.Rep_range
thf(fact_223_BNF__Composition_Otype__definition__id__bnf__UNIV,axiom,
! [A: $tType] : ( type_definition @ A @ A @ ( bNF_id_bnf @ A ) @ ( bNF_id_bnf @ A ) @ ( top_top @ ( set @ A ) ) ) ).
% BNF_Composition.type_definition_id_bnf_UNIV
thf(fact_224_image__bind,axiom,
! [A: $tType,B: $tType,C: $tType,F2: B > A,A5: set @ C,G2: C > ( set @ B )] :
( ( image @ B @ A @ F2 @ ( bind @ C @ B @ A5 @ G2 ) )
= ( bind @ C @ A @ A5 @ ( comp @ ( set @ B ) @ ( set @ A ) @ C @ ( image @ B @ A @ F2 ) @ G2 ) ) ) ).
% image_bind
thf(fact_225_ID_Opred__map,axiom,
! [B: $tType,A: $tType,Q: B > $o,F2: A > B,X2: A] :
( ( bNF_id_bnf @ ( B > $o ) @ Q @ ( bNF_id_bnf @ ( A > B ) @ F2 @ X2 ) )
= ( bNF_id_bnf @ ( A > $o ) @ ( comp @ B @ $o @ A @ Q @ F2 ) @ X2 ) ) ).
% ID.pred_map
thf(fact_226_BNF__Composition_Oid__bnf__def,axiom,
! [A: $tType] :
( ( bNF_id_bnf @ A )
= ( ^ [X: A] : X ) ) ).
% BNF_Composition.id_bnf_def
thf(fact_227_id__bnf__apply,axiom,
! [A: $tType] :
( ( bNF_id_bnf @ A )
= ( ^ [X: A] : X ) ) ).
% id_bnf_apply
thf(fact_228_bind__image,axiom,
! [A: $tType,B: $tType,C: $tType,F2: C > B,A5: set @ C,G2: B > ( set @ A )] :
( ( bind @ B @ A @ ( image @ C @ B @ F2 @ A5 ) @ G2 )
= ( bind @ C @ A @ A5 @ ( comp @ B @ ( set @ A ) @ C @ G2 @ F2 ) ) ) ).
% bind_image
thf(fact_229_prod_Osplit__sel__asm,axiom,
! [C: $tType,B: $tType,A: $tType,P: C > $o,F2: A > B > C,Prod: product_prod @ A @ B] :
( ( P @ ( product_case_prod @ A @ B @ C @ F2 @ Prod ) )
= ( ~ ( ( Prod
= ( product_Pair @ A @ B @ ( product_fst @ A @ B @ Prod ) @ ( product_snd @ A @ B @ Prod ) ) )
& ~ ( P @ ( F2 @ ( product_fst @ A @ B @ Prod ) @ ( product_snd @ A @ B @ Prod ) ) ) ) ) ) ).
% prod.split_sel_asm
thf(fact_230_prod_Osplit__sel,axiom,
! [C: $tType,B: $tType,A: $tType,P: C > $o,F2: A > B > C,Prod: product_prod @ A @ B] :
( ( P @ ( product_case_prod @ A @ B @ C @ F2 @ Prod ) )
= ( ( Prod
= ( product_Pair @ A @ B @ ( product_fst @ A @ B @ Prod ) @ ( product_snd @ A @ B @ Prod ) ) )
=> ( P @ ( F2 @ ( product_fst @ A @ B @ Prod ) @ ( product_snd @ A @ B @ Prod ) ) ) ) ) ).
% prod.split_sel
thf(fact_231_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_232_case__prod__beta,axiom,
! [A: $tType,C: $tType,B: $tType] :
( ( product_case_prod @ B @ C @ A )
= ( ^ [F: B > C > A,P4: product_prod @ B @ C] : ( F @ ( product_fst @ B @ C @ P4 ) @ ( product_snd @ B @ C @ P4 ) ) ) ) ).
% case_prod_beta
thf(fact_233_prod_Ocase__eq__if,axiom,
! [C: $tType,B: $tType,A: $tType] :
( ( product_case_prod @ A @ B @ C )
= ( ^ [F: A > B > C,Prod3: product_prod @ A @ B] : ( F @ ( product_fst @ A @ B @ Prod3 ) @ ( product_snd @ A @ B @ Prod3 ) ) ) ) ).
% prod.case_eq_if
thf(fact_234_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_235_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_236_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_237_split__cong,axiom,
! [C: $tType,B: $tType,A: $tType,Q2: product_prod @ A @ B,F2: A > B > C,G2: A > B > C,P3: product_prod @ A @ B] :
( ! [X4: A,Y4: B] :
( ( ( product_Pair @ A @ B @ X4 @ Y4 )
= Q2 )
=> ( ( F2 @ X4 @ Y4 )
= ( G2 @ X4 @ Y4 ) ) )
=> ( ( P3 = Q2 )
=> ( ( product_case_prod @ A @ B @ C @ F2 @ P3 )
= ( product_case_prod @ A @ B @ C @ G2 @ Q2 ) ) ) ) ).
% split_cong
thf(fact_238_mem__case__prodE,axiom,
! [B: $tType,A: $tType,C: $tType,Z2: A,C2: B > C > ( set @ A ),P3: product_prod @ B @ C] :
( ( member @ A @ Z2 @ ( product_case_prod @ B @ C @ ( set @ A ) @ C2 @ P3 ) )
=> ~ ! [X4: B,Y4: C] :
( ( P3
= ( product_Pair @ B @ C @ X4 @ Y4 ) )
=> ~ ( member @ A @ Z2 @ ( C2 @ X4 @ Y4 ) ) ) ) ).
% mem_case_prodE
thf(fact_239_old_Oprod_Ocase,axiom,
! [A: $tType,C: $tType,B: $tType,F2: A > B > C,X1: A,X22: B] :
( ( product_case_prod @ A @ B @ C @ F2 @ ( product_Pair @ A @ B @ X1 @ X22 ) )
= ( F2 @ X1 @ X22 ) ) ).
% old.prod.case
thf(fact_240_Product__Type_OCollect__case__prodD,axiom,
! [B: $tType,A: $tType,X2: product_prod @ A @ B,A5: A > B > $o] :
( ( member @ ( product_prod @ A @ B ) @ X2 @ ( collect @ ( product_prod @ A @ B ) @ ( product_case_prod @ A @ B @ $o @ A5 ) ) )
=> ( A5 @ ( product_fst @ A @ B @ X2 ) @ ( product_snd @ A @ B @ X2 ) ) ) ).
% Product_Type.Collect_case_prodD
thf(fact_241_convol__image__vimage2p,axiom,
! [C: $tType,D: $tType,B: $tType,A: $tType,F2: C > A,G2: D > B,R3: A > B > $o] : ( ord_less_eq @ ( set @ ( product_prod @ A @ B ) ) @ ( image @ ( product_prod @ C @ D ) @ ( product_prod @ A @ B ) @ ( bNF_convol @ ( product_prod @ C @ D ) @ A @ B @ ( comp @ C @ A @ ( product_prod @ C @ D ) @ F2 @ ( product_fst @ C @ D ) ) @ ( comp @ D @ B @ ( product_prod @ C @ D ) @ G2 @ ( product_snd @ C @ D ) ) ) @ ( collect @ ( product_prod @ C @ D ) @ ( product_case_prod @ C @ D @ $o @ ( bNF_vimage2p @ C @ A @ D @ B @ $o @ F2 @ G2 @ R3 ) ) ) ) @ ( collect @ ( product_prod @ A @ B ) @ ( product_case_prod @ A @ B @ $o @ R3 ) ) ) ).
% convol_image_vimage2p
thf(fact_242_Collect__case__prod__Grp__eqD,axiom,
! [B: $tType,A: $tType,Z2: product_prod @ A @ B,A5: set @ A,F2: A > B] :
( ( member @ ( product_prod @ A @ B ) @ Z2 @ ( collect @ ( product_prod @ A @ B ) @ ( product_case_prod @ A @ B @ $o @ ( bNF_Grp @ A @ B @ A5 @ F2 ) ) ) )
=> ( ( comp @ A @ B @ ( product_prod @ A @ B ) @ F2 @ ( product_fst @ A @ B ) @ Z2 )
= ( product_snd @ A @ B @ Z2 ) ) ) ).
% Collect_case_prod_Grp_eqD
thf(fact_243_subsetI,axiom,
! [A: $tType,A5: set @ A,B4: set @ A] :
( ! [X4: A] :
( ( member @ A @ X4 @ A5 )
=> ( member @ A @ X4 @ B4 ) )
=> ( ord_less_eq @ ( set @ A ) @ A5 @ B4 ) ) ).
% subsetI
thf(fact_244_subset__antisym,axiom,
! [A: $tType,A5: set @ A,B4: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ A5 @ B4 )
=> ( ( ord_less_eq @ ( set @ A ) @ B4 @ A5 )
=> ( A5 = B4 ) ) ) ).
% subset_antisym
thf(fact_245_convol__mem__GrpI,axiom,
! [B: $tType,A: $tType,X2: A,A5: set @ A,G2: A > B] :
( ( member @ A @ X2 @ A5 )
=> ( member @ ( product_prod @ A @ B ) @ ( bNF_convol @ A @ A @ B @ ( id @ A ) @ G2 @ X2 ) @ ( collect @ ( product_prod @ A @ B ) @ ( product_case_prod @ A @ B @ $o @ ( bNF_Grp @ A @ B @ A5 @ G2 ) ) ) ) ) ).
% convol_mem_GrpI
thf(fact_246_Collect__case__prod__Grp__in,axiom,
! [B: $tType,A: $tType,Z2: product_prod @ A @ B,A5: set @ A,F2: A > B] :
( ( member @ ( product_prod @ A @ B ) @ Z2 @ ( collect @ ( product_prod @ A @ B ) @ ( product_case_prod @ A @ B @ $o @ ( bNF_Grp @ A @ B @ A5 @ F2 ) ) ) )
=> ( member @ A @ ( product_fst @ A @ B @ Z2 ) @ A5 ) ) ).
% Collect_case_prod_Grp_in
thf(fact_247_set__mp,axiom,
! [A: $tType,A5: set @ A,B4: set @ A,X2: A] :
( ( ord_less_eq @ ( set @ A ) @ A5 @ B4 )
=> ( ( member @ A @ X2 @ A5 )
=> ( member @ A @ X2 @ B4 ) ) ) ).
% set_mp
thf(fact_248_in__mono,axiom,
! [A: $tType,A5: set @ A,B4: set @ A,X2: A] :
( ( ord_less_eq @ ( set @ A ) @ A5 @ B4 )
=> ( ( member @ A @ X2 @ A5 )
=> ( member @ A @ X2 @ B4 ) ) ) ).
% in_mono
thf(fact_249_subsetD,axiom,
! [A: $tType,A5: set @ A,B4: set @ A,C2: A] :
( ( ord_less_eq @ ( set @ A ) @ A5 @ B4 )
=> ( ( member @ A @ C2 @ A5 )
=> ( member @ A @ C2 @ B4 ) ) ) ).
% subsetD
thf(fact_250_subsetCE,axiom,
! [A: $tType,A5: set @ A,B4: set @ A,C2: A] :
( ( ord_less_eq @ ( set @ A ) @ A5 @ B4 )
=> ( ( member @ A @ C2 @ A5 )
=> ( member @ A @ C2 @ B4 ) ) ) ).
% subsetCE
thf(fact_251_equalityE,axiom,
! [A: $tType,A5: set @ A,B4: set @ A] :
( ( A5 = B4 )
=> ~ ( ( ord_less_eq @ ( set @ A ) @ A5 @ B4 )
=> ~ ( ord_less_eq @ ( set @ A ) @ B4 @ A5 ) ) ) ).
% equalityE
thf(fact_252_subset__eq,axiom,
! [A: $tType] :
( ( ord_less_eq @ ( set @ A ) )
= ( ^ [A3: set @ A,B7: set @ A] :
! [X: A] :
( ( member @ A @ X @ A3 )
=> ( member @ A @ X @ B7 ) ) ) ) ).
% subset_eq
thf(fact_253_equalityD1,axiom,
! [A: $tType,A5: set @ A,B4: set @ A] :
( ( A5 = B4 )
=> ( ord_less_eq @ ( set @ A ) @ A5 @ B4 ) ) ).
% equalityD1
thf(fact_254_equalityD2,axiom,
! [A: $tType,A5: set @ A,B4: set @ A] :
( ( A5 = B4 )
=> ( ord_less_eq @ ( set @ A ) @ B4 @ A5 ) ) ).
% equalityD2
thf(fact_255_set__rev__mp,axiom,
! [A: $tType,X2: A,A5: set @ A,B4: set @ A] :
( ( member @ A @ X2 @ A5 )
=> ( ( ord_less_eq @ ( set @ A ) @ A5 @ B4 )
=> ( member @ A @ X2 @ B4 ) ) ) ).
% set_rev_mp
%----Helper facts (3)
thf(help_If_3_1_T,axiom,
! [P: $o] :
( ( P = $true )
| ( P = $false ) ) ).
thf(help_If_2_1_T,axiom,
! [A: $tType,X2: A,Y: A] :
( ( if @ A @ $false @ X2 @ Y )
= Y ) ).
thf(help_If_1_1_T,axiom,
! [A: $tType,X2: A,Y: A] :
( ( if @ A @ $true @ X2 @ Y )
= X2 ) ).
%----Conjectures (1)
thf(conj_0,conjecture,
( ( stream1967106959u_comp @ c @ b @ a @ sp @ sp2 )
= ( stream104478819_comp2 @ c @ b @ a @ sp @ sp2 ) ) ).
%------------------------------------------------------------------------------