TPTP Problem File: DAT254^1.p
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%------------------------------------------------------------------------------
% File : DAT254^1 : TPTP v8.2.0. Released v7.0.0.
% Domain : Data Structures
% Problem : Data structure for translators from streams to streams 38
% Version : [Bla16] axioms : Especial.
% English :
% Refs : [BH+14] Blanchette et al. (2014), Truly Modular (Co)datatypes
% : [RB15] Reynolds & Blanchette (2015), A Decision Procedure for
% : [Bla16] Blanchette (2016), Email to Geoff Sutcliffe
% Source : [Bla16]
% Names : stream_processor__38.p [Bla16]
% Status : Theorem
% Rating : 1.00 v7.1.0
% Syntax : Number of formulae : 330 ( 140 unt; 57 typ; 0 def)
% Number of atoms : 798 ( 339 equ; 0 cnn)
% Maximal formula atoms : 15 ( 2 avg)
% Number of connectives : 6768 ( 41 ~; 3 |; 24 &;6387 @)
% ( 0 <=>; 313 =>; 0 <=; 0 <~>)
% Maximal formula depth : 28 ( 11 avg)
% Number of types : 3 ( 2 usr)
% Number of type conns : 1051 (1051 >; 0 *; 0 +; 0 <<)
% Number of symbols : 57 ( 55 usr; 2 con; 0-10 aty)
% Number of variables : 1874 ( 152 ^;1593 !; 22 ?;1874 :)
% ( 107 !>; 0 ?*; 0 @-; 0 @+)
% SPC : TH1_THM_EQU_NAR
% Comments : This file was generated by Isabelle (most likely Sledgehammer)
% 2016-07-13 14:39:09.330
%------------------------------------------------------------------------------
%----Could-be-implicit typings (9)
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_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_Lattices_Obounded__lattice,type,
bounded_lattice:
!>[A: $tType] : ( ( itself @ A ) > $o ) ).
thf(sy_cl_HOL_Otype,type,
type:
!>[A: $tType] : ( ( itself @ A ) > $o ) ).
thf(sy_cl_Orderings_Otop,type,
top:
!>[A: $tType] : ( ( itself @ A ) > $o ) ).
thf(sy_cl_Complete__Lattices_OSup,type,
complete_Sup:
!>[A: $tType] : ( ( itself @ A ) > $o ) ).
thf(sy_cl_Lattices_Obounded__lattice__top,type,
bounded_lattice_top:
!>[A: $tType] : ( ( itself @ A ) > $o ) ).
thf(sy_cl_Complete__Lattices_Ocomplete__lattice,type,
comple187826305attice:
!>[A: $tType] : ( ( itself @ A ) > $o ) ).
thf(sy_c_BNF__Composition_Oid__bnf,type,
bNF_id_bnf:
!>[A: $tType] : ( A > A ) ).
thf(sy_c_BNF__Def_Orel__fun,type,
bNF_rel_fun:
!>[A: $tType,C: $tType,B: $tType,D: $tType] : ( ( A > C > $o ) > ( B > D > $o ) > ( A > B ) > ( C > D ) > $o ) ).
thf(sy_c_BNF__Def_Orel__sum,type,
bNF_rel_sum:
!>[A: $tType,C: $tType,B: $tType,D: $tType] : ( ( A > C > $o ) > ( B > D > $o ) > ( sum_sum @ A @ B ) > ( sum_sum @ C @ D ) > $o ) ).
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_Basic__BNFs_Opred__prod,type,
basic_pred_prod:
!>[A: $tType,B: $tType] : ( ( A > $o ) > ( B > $o ) > ( product_prod @ A @ B ) > $o ) ).
thf(sy_c_Basic__BNFs_Opred__sum,type,
basic_pred_sum:
!>[A: $tType,B: $tType] : ( ( A > $o ) > ( B > $o ) > ( sum_sum @ A @ B ) > $o ) ).
thf(sy_c_Basic__BNFs_Orel__prod,type,
basic_rel_prod:
!>[A: $tType,B: $tType,C: $tType,D: $tType] : ( ( A > B > $o ) > ( C > D > $o ) > ( product_prod @ A @ C ) > ( product_prod @ B @ D ) > $o ) ).
thf(sy_c_Complete__Lattices_OSup__class_OSup,type,
complete_Sup_Sup:
!>[A: $tType] : ( ( set @ A ) > A ) ).
thf(sy_c_Fun_Ocomp,type,
comp:
!>[B: $tType,C: $tType,A: $tType] : ( ( B > C ) > ( A > B ) > A > C ) ).
thf(sy_c_Fun_Oswap,type,
swap:
!>[A: $tType,B: $tType] : ( A > A > ( A > B ) > A > B ) ).
thf(sy_c_Lattices_Osup__class_Osup,type,
sup_sup:
!>[A: $tType] : ( A > A > A ) ).
thf(sy_c_Orderings_Otop__class_Otop,type,
top_top:
!>[A: $tType] : A ).
thf(sy_c_Pure_Otype,type,
type2:
!>[A: $tType] : ( itself @ A ) ).
thf(sy_c_Set_OBall,type,
ball:
!>[A: $tType] : ( ( set @ A ) > ( A > $o ) > $o ) ).
thf(sy_c_Set_OCollect,type,
collect:
!>[A: $tType] : ( ( A > $o ) > ( set @ A ) ) ).
thf(sy_c_Set_OPow,type,
pow:
!>[A: $tType] : ( ( set @ A ) > ( set @ ( 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_Set_Oinsert,type,
insert:
!>[A: $tType] : ( A > ( set @ A ) > ( set @ A ) ) ).
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_060mu_062_OGet,type,
stream1294929701mu_Get:
!>[A: $tType,B: $tType,C: $tType] : ( ( A > ( stream901396144_sp_mu @ A @ B @ C ) ) > ( stream901396144_sp_mu @ A @ B @ C ) ) ).
thf(sy_c_Stream__Processor__Mirabelle__rrumbueyrq_Osp_092_060_094sub_062_092_060mu_062_OPut,type,
stream1370332830mu_Put:
!>[B: $tType,C: $tType,A: $tType] : ( B > C > ( stream901396144_sp_mu @ A @ B @ C ) ) ).
thf(sy_c_Stream__Processor__Mirabelle__rrumbueyrq_Osp_092_060_094sub_062_092_060mu_062_Ocase__sp_092_060_094sub_062_092_060mu_062,type,
stream160000856_sp_mu:
!>[A: $tType,B: $tType,C: $tType,D: $tType] : ( ( ( A > ( stream901396144_sp_mu @ A @ B @ C ) ) > D ) > ( B > C > D ) > ( stream901396144_sp_mu @ A @ B @ C ) > D ) ).
thf(sy_c_Stream__Processor__Mirabelle__rrumbueyrq_Osp_092_060_094sub_062_092_060mu_062_Omap__sp_092_060_094sub_062_092_060mu_062,type,
stream1058219438_sp_mu:
!>[B: $tType,Ba: $tType,C: $tType,Ca: $tType,A: $tType] : ( ( B > Ba ) > ( C > Ca ) > ( stream901396144_sp_mu @ A @ B @ C ) > ( stream901396144_sp_mu @ A @ Ba @ Ca ) ) ).
thf(sy_c_Stream__Processor__Mirabelle__rrumbueyrq_Osp_092_060_094sub_062_092_060mu_062_Opred__sp_092_060_094sub_062_092_060mu_062,type,
stream123009735_sp_mu:
!>[A: $tType,B: $tType,E: $tType] : ( ( A > $o ) > ( B > $o ) > ( stream901396144_sp_mu @ E @ A @ B ) > $o ) ).
thf(sy_c_Stream__Processor__Mirabelle__rrumbueyrq_Osp_092_060_094sub_062_092_060mu_062_Orec__sp_092_060_094sub_062_092_060mu_062,type,
stream674629690_sp_mu:
!>[A: $tType,B: $tType,C: $tType,G: $tType] : ( ( ( A > ( product_prod @ ( stream901396144_sp_mu @ A @ B @ C ) @ G ) ) > G ) > ( B > C > G ) > ( stream901396144_sp_mu @ A @ B @ C ) > G ) ).
thf(sy_c_Stream__Processor__Mirabelle__rrumbueyrq_Osp_092_060_094sub_062_092_060mu_062_Orel__sp_092_060_094sub_062_092_060mu_062,type,
stream1924447089_sp_mu:
!>[A: $tType,C: $tType,B: $tType,D: $tType,E: $tType] : ( ( A > C > $o ) > ( B > D > $o ) > ( stream901396144_sp_mu @ E @ A @ B ) > ( stream901396144_sp_mu @ E @ C @ D ) > $o ) ).
thf(sy_c_Stream__Processor__Mirabelle__rrumbueyrq_Osp_092_060_094sub_062_092_060mu_062_Oset1__sp_092_060_094sub_062_092_060mu_062,type,
stream2074884121_sp_mu:
!>[A: $tType,B: $tType,C: $tType] : ( ( stream901396144_sp_mu @ A @ B @ C ) > ( set @ B ) ) ).
thf(sy_c_Stream__Processor__Mirabelle__rrumbueyrq_Osp_092_060_094sub_062_092_060mu_062_Oset2__sp_092_060_094sub_062_092_060mu_062,type,
stream1259315544_sp_mu:
!>[A: $tType,B: $tType,C: $tType] : ( ( stream901396144_sp_mu @ A @ B @ C ) > ( set @ C ) ) ).
thf(sy_c_Stream__Processor__Mirabelle__rrumbueyrq_Osp_092_060_094sub_062_092_060mu_062_Osize__sp_092_060_094sub_062_092_060mu_062,type,
stream1362355943_sp_mu:
!>[A: $tType,B: $tType,C: $tType] : ( ( A > nat ) > ( B > nat ) > ( C > nat ) > ( stream901396144_sp_mu @ A @ B @ C ) > nat ) ).
thf(sy_c_Stream__Processor__Mirabelle__rrumbueyrq_Osp_092_060_094sub_062_092_060nu_062_OIn,type,
stream1686501333_nu_In:
!>[A: $tType,B: $tType] : ( ( stream901396144_sp_mu @ A @ B @ ( stream1273403375_sp_nu @ A @ B ) ) > ( stream1273403375_sp_nu @ A @ B ) ) ).
thf(sy_c_Stream__Processor__Mirabelle__rrumbueyrq_Osp_092_060_094sub_062_092_060nu_062_Ocase__sp_092_060_094sub_062_092_060nu_062,type,
stream1502363800_sp_nu:
!>[A: $tType,B: $tType,C: $tType] : ( ( ( stream901396144_sp_mu @ A @ B @ ( stream1273403375_sp_nu @ A @ B ) ) > C ) > ( stream1273403375_sp_nu @ A @ B ) > C ) ).
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_Oout,type,
stream1200245950nu_out:
!>[A: $tType,B: $tType] : ( ( stream1273403375_sp_nu @ A @ B ) > ( stream901396144_sp_mu @ A @ B @ ( stream1273403375_sp_nu @ A @ B ) ) ) ).
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_Orel__sp_092_060_094sub_062_092_060nu_062,type,
stream1017682031_sp_nu:
!>[A: $tType,B: $tType,C: $tType] : ( ( A > B > $o ) > ( stream1273403375_sp_nu @ C @ A ) > ( stream1273403375_sp_nu @ C @ B ) > $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_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_s,type,
s: stream @ a ).
%----Relevant facts (256)
thf(fact_0_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_1_sp_092_060_094sub_062_092_060nu_062_Oinject,axiom,
! [B: $tType,A: $tType,X: stream901396144_sp_mu @ A @ B @ ( stream1273403375_sp_nu @ A @ B ),Ya: stream901396144_sp_mu @ A @ B @ ( stream1273403375_sp_nu @ A @ B )] :
( ( ( stream1686501333_nu_In @ A @ B @ X )
= ( stream1686501333_nu_In @ A @ B @ Ya ) )
= ( X = Ya ) ) ).
% sp\<^sub>\<nu>.inject
thf(fact_2_sp_092_060_094sub_062_092_060mu_062_Oinject_I2_J,axiom,
! [A: $tType,B: $tType,C: $tType,X21: B,X22: C,Y21: B,Y22: C] :
( ( ( stream1370332830mu_Put @ B @ C @ A @ X21 @ X22 )
= ( stream1370332830mu_Put @ B @ C @ A @ Y21 @ Y22 ) )
= ( ( X21 = Y21 )
& ( X22 = Y22 ) ) ) ).
% sp\<^sub>\<mu>.inject(2)
thf(fact_3_sp_092_060_094sub_062_092_060nu_062_Oexhaust,axiom,
! [B: $tType,A: $tType,Y: stream1273403375_sp_nu @ A @ B] :
~ ! [X2: stream901396144_sp_mu @ A @ B @ ( stream1273403375_sp_nu @ A @ B )] :
( Y
!= ( stream1686501333_nu_In @ A @ B @ X2 ) ) ).
% sp\<^sub>\<nu>.exhaust
thf(fact_4_sp_092_060_094sub_062_092_060mu_062_Osimps_I6_J,axiom,
! [A: $tType,B: $tType,D: $tType,C: $tType,F1: ( A > ( stream901396144_sp_mu @ A @ B @ C ) ) > D,F2: B > C > D,X21: B,X22: C] :
( ( stream160000856_sp_mu @ A @ B @ C @ D @ F1 @ F2 @ ( stream1370332830mu_Put @ B @ C @ A @ X21 @ X22 ) )
= ( F2 @ X21 @ X22 ) ) ).
% sp\<^sub>\<mu>.simps(6)
thf(fact_5_sp_092_060_094sub_062_092_060nu_062_Ocase,axiom,
! [C: $tType,B: $tType,A: $tType,F: ( stream901396144_sp_mu @ A @ B @ ( stream1273403375_sp_nu @ A @ B ) ) > C,X: stream901396144_sp_mu @ A @ B @ ( stream1273403375_sp_nu @ A @ B )] :
( ( stream1502363800_sp_nu @ A @ B @ C @ F @ ( stream1686501333_nu_In @ A @ B @ X ) )
= ( F @ X ) ) ).
% sp\<^sub>\<nu>.case
thf(fact_6_sp_092_060_094sub_062_092_060mu_062_Osimps_I8_J,axiom,
! [A: $tType,B: $tType,G: $tType,C: $tType,F1: ( A > ( product_prod @ ( stream901396144_sp_mu @ A @ B @ C ) @ G ) ) > G,F2: B > C > G,X21: B,X22: C] :
( ( stream674629690_sp_mu @ A @ B @ C @ G @ F1 @ F2 @ ( stream1370332830mu_Put @ B @ C @ A @ X21 @ X22 ) )
= ( F2 @ X21 @ X22 ) ) ).
% sp\<^sub>\<mu>.simps(8)
thf(fact_7_sp_092_060_094sub_062_092_060mu_062_Opred__inject_I2_J,axiom,
! [A: $tType,B: $tType,C: $tType,P1: B > $o,P2: C > $o,A2: B,Aa: C] :
( ( stream123009735_sp_mu @ B @ C @ A @ P1 @ P2 @ ( stream1370332830mu_Put @ B @ C @ A @ A2 @ Aa ) )
= ( ( P1 @ A2 )
& ( P2 @ Aa ) ) ) ).
% sp\<^sub>\<mu>.pred_inject(2)
thf(fact_8_sp_092_060_094sub_062_092_060nu_062_Ocollapse,axiom,
! [B: $tType,A: $tType,Sp_nu: stream1273403375_sp_nu @ A @ B] :
( ( stream1686501333_nu_In @ A @ B @ ( stream1200245950nu_out @ A @ B @ Sp_nu ) )
= Sp_nu ) ).
% sp\<^sub>\<nu>.collapse
thf(fact_9_sp_092_060_094sub_062_092_060mu_062_Oset__intros_I4_J,axiom,
! [B: $tType,A: $tType,C: $tType,A2a: C,A1a: B] : ( member @ C @ A2a @ ( stream1259315544_sp_mu @ A @ B @ C @ ( stream1370332830mu_Put @ B @ C @ A @ A1a @ A2a ) ) ) ).
% sp\<^sub>\<mu>.set_intros(4)
thf(fact_10_sp_092_060_094sub_062_092_060mu_062_Oset__intros_I2_J,axiom,
! [B: $tType,A: $tType,C: $tType,A1: B,A22: C] : ( member @ B @ A1 @ ( stream2074884121_sp_mu @ A @ B @ C @ ( stream1370332830mu_Put @ B @ C @ A @ A1 @ A22 ) ) ) ).
% sp\<^sub>\<mu>.set_intros(2)
thf(fact_11_sp_092_060_094sub_062_092_060nu_062_Osel,axiom,
! [B: $tType,A: $tType,X: stream901396144_sp_mu @ A @ B @ ( stream1273403375_sp_nu @ A @ B )] :
( ( stream1200245950nu_out @ A @ B @ ( stream1686501333_nu_In @ A @ B @ X ) )
= X ) ).
% sp\<^sub>\<nu>.sel
thf(fact_12_sp_092_060_094sub_062_092_060nu_062_Oexhaust__sel,axiom,
! [B: $tType,A: $tType,Sp_nu: stream1273403375_sp_nu @ A @ B] :
( Sp_nu
= ( stream1686501333_nu_In @ A @ B @ ( stream1200245950nu_out @ A @ B @ Sp_nu ) ) ) ).
% sp\<^sub>\<nu>.exhaust_sel
thf(fact_13_sp_092_060_094sub_062_092_060mu_062_Osimps_I10_J,axiom,
! [B: $tType,E: $tType,A: $tType,F3: $tType,C: $tType,F1: B > E,F2: C > F3,X21: B,X22: C] :
( ( stream1058219438_sp_mu @ B @ E @ C @ F3 @ A @ F1 @ F2 @ ( stream1370332830mu_Put @ B @ C @ A @ X21 @ X22 ) )
= ( stream1370332830mu_Put @ E @ F3 @ A @ ( F1 @ X21 ) @ ( F2 @ X22 ) ) ) ).
% sp\<^sub>\<mu>.simps(10)
thf(fact_14_sp_092_060_094sub_062_092_060mu_062_Odistinct_I1_J,axiom,
! [B: $tType,A: $tType,C: $tType,X1: A > ( stream901396144_sp_mu @ A @ B @ C ),X21: B,X22: C] :
( ( stream1294929701mu_Get @ A @ B @ C @ X1 )
!= ( stream1370332830mu_Put @ B @ C @ A @ X21 @ X22 ) ) ).
% sp\<^sub>\<mu>.distinct(1)
thf(fact_15_sp_092_060_094sub_062_092_060mu_062_Oexhaust,axiom,
! [B: $tType,A: $tType,C: $tType,Y: stream901396144_sp_mu @ A @ B @ C] :
( ! [X12: A > ( stream901396144_sp_mu @ A @ B @ C )] :
( Y
!= ( stream1294929701mu_Get @ A @ B @ C @ X12 ) )
=> ~ ! [X212: B,X222: C] :
( Y
!= ( stream1370332830mu_Put @ B @ C @ A @ X212 @ X222 ) ) ) ).
% sp\<^sub>\<mu>.exhaust
thf(fact_16_sp_092_060_094sub_062_092_060mu_062_Oinject_I1_J,axiom,
! [C: $tType,B: $tType,A: $tType,X1: A > ( stream901396144_sp_mu @ A @ B @ C ),Y1: A > ( stream901396144_sp_mu @ A @ B @ C )] :
( ( ( stream1294929701mu_Get @ A @ B @ C @ X1 )
= ( stream1294929701mu_Get @ A @ B @ C @ Y1 ) )
= ( X1 = Y1 ) ) ).
% sp\<^sub>\<mu>.inject(1)
thf(fact_17_sp_092_060_094sub_062_092_060mu_062_Opred__mono__strong,axiom,
! [B: $tType,A: $tType,G: $tType,P1: A > $o,P2: B > $o,X: stream901396144_sp_mu @ G @ A @ B,P1a: A > $o,P2a: B > $o] :
( ( stream123009735_sp_mu @ A @ B @ G @ P1 @ P2 @ X )
=> ( ! [Z1: A] :
( ( member @ A @ Z1 @ ( stream2074884121_sp_mu @ G @ A @ B @ X ) )
=> ( ( P1 @ Z1 )
=> ( P1a @ Z1 ) ) )
=> ( ! [Z2: B] :
( ( member @ B @ Z2 @ ( stream1259315544_sp_mu @ G @ A @ B @ X ) )
=> ( ( P2 @ Z2 )
=> ( P2a @ Z2 ) ) )
=> ( stream123009735_sp_mu @ A @ B @ G @ P1a @ P2a @ X ) ) ) ) ).
% sp\<^sub>\<mu>.pred_mono_strong
thf(fact_18_sp_092_060_094sub_062_092_060mu_062_Oinj__map__strong,axiom,
! [C: $tType,D: $tType,B: $tType,A: $tType,G: $tType,X: stream901396144_sp_mu @ G @ A @ B,Xa: stream901396144_sp_mu @ G @ A @ B,F1: A > C,F1a: A > C,F2: B > D,F2a: B > D] :
( ! [Z1: A,Z1a: A] :
( ( member @ A @ Z1 @ ( stream2074884121_sp_mu @ G @ A @ B @ X ) )
=> ( ( member @ A @ Z1a @ ( stream2074884121_sp_mu @ G @ A @ B @ Xa ) )
=> ( ( ( F1 @ Z1 )
= ( F1a @ Z1a ) )
=> ( Z1 = Z1a ) ) ) )
=> ( ! [Z2: B,Z2a: B] :
( ( member @ B @ Z2 @ ( stream1259315544_sp_mu @ G @ A @ B @ X ) )
=> ( ( member @ B @ Z2a @ ( stream1259315544_sp_mu @ G @ A @ B @ Xa ) )
=> ( ( ( F2 @ Z2 )
= ( F2a @ Z2a ) )
=> ( Z2 = Z2a ) ) ) )
=> ( ( ( stream1058219438_sp_mu @ A @ C @ B @ D @ G @ F1 @ F2 @ X )
= ( stream1058219438_sp_mu @ A @ C @ B @ D @ G @ F1a @ F2a @ Xa ) )
=> ( X = Xa ) ) ) ) ).
% sp\<^sub>\<mu>.inj_map_strong
thf(fact_19_sp_092_060_094sub_062_092_060mu_062_Opred__cong,axiom,
! [B: $tType,A: $tType,G: $tType,X: stream901396144_sp_mu @ G @ A @ B,Ya: stream901396144_sp_mu @ G @ A @ B,P1: A > $o,P1a: A > $o,P2: B > $o,P2a: B > $o] :
( ( X = Ya )
=> ( ! [Z1: A] :
( ( member @ A @ Z1 @ ( stream2074884121_sp_mu @ G @ A @ B @ Ya ) )
=> ( ( P1 @ Z1 )
= ( P1a @ Z1 ) ) )
=> ( ! [Z2: B] :
( ( member @ B @ Z2 @ ( stream1259315544_sp_mu @ G @ A @ B @ Ya ) )
=> ( ( P2 @ Z2 )
= ( P2a @ Z2 ) ) )
=> ( ( stream123009735_sp_mu @ A @ B @ G @ P1 @ P2 @ X )
= ( stream123009735_sp_mu @ A @ B @ G @ P1a @ P2a @ Ya ) ) ) ) ) ).
% sp\<^sub>\<mu>.pred_cong
thf(fact_20_sp_092_060_094sub_062_092_060mu_062_Omap__cong0,axiom,
! [D: $tType,C: $tType,B: $tType,A: $tType,G: $tType,X: stream901396144_sp_mu @ G @ A @ B,F1: A > C,G1: A > C,F2: B > D,G2: B > D] :
( ! [Z1: A] :
( ( member @ A @ Z1 @ ( stream2074884121_sp_mu @ G @ A @ B @ X ) )
=> ( ( F1 @ Z1 )
= ( G1 @ Z1 ) ) )
=> ( ! [Z2: B] :
( ( member @ B @ Z2 @ ( stream1259315544_sp_mu @ G @ A @ B @ X ) )
=> ( ( F2 @ Z2 )
= ( G2 @ Z2 ) ) )
=> ( ( stream1058219438_sp_mu @ A @ C @ B @ D @ G @ F1 @ F2 @ X )
= ( stream1058219438_sp_mu @ A @ C @ B @ D @ G @ G1 @ G2 @ X ) ) ) ) ).
% sp\<^sub>\<mu>.map_cong0
thf(fact_21_sp_092_060_094sub_062_092_060mu_062_Omap__cong,axiom,
! [D: $tType,C: $tType,B: $tType,A: $tType,G: $tType,X: stream901396144_sp_mu @ G @ A @ B,Ya: stream901396144_sp_mu @ G @ A @ B,F1: A > C,G1: A > C,F2: B > D,G2: B > D] :
( ( X = Ya )
=> ( ! [Z1: A] :
( ( member @ A @ Z1 @ ( stream2074884121_sp_mu @ G @ A @ B @ Ya ) )
=> ( ( F1 @ Z1 )
= ( G1 @ Z1 ) ) )
=> ( ! [Z2: B] :
( ( member @ B @ Z2 @ ( stream1259315544_sp_mu @ G @ A @ B @ Ya ) )
=> ( ( F2 @ Z2 )
= ( G2 @ Z2 ) ) )
=> ( ( stream1058219438_sp_mu @ A @ C @ B @ D @ G @ F1 @ F2 @ X )
= ( stream1058219438_sp_mu @ A @ C @ B @ D @ G @ G1 @ G2 @ Ya ) ) ) ) ) ).
% sp\<^sub>\<mu>.map_cong
thf(fact_22_sp_092_060_094sub_062_092_060nu_062_Ocase__eq__if,axiom,
! [C: $tType,B: $tType,A: $tType] :
( ( stream1502363800_sp_nu @ A @ B @ C )
= ( ^ [F4: ( stream901396144_sp_mu @ A @ B @ ( stream1273403375_sp_nu @ A @ B ) ) > C,Sp_nu2: stream1273403375_sp_nu @ A @ B] : ( F4 @ ( stream1200245950nu_out @ A @ B @ Sp_nu2 ) ) ) ) ).
% sp\<^sub>\<nu>.case_eq_if
thf(fact_23_sp_092_060_094sub_062_092_060nu_062_Oexpand,axiom,
! [B: $tType,A: $tType,Sp_nu: stream1273403375_sp_nu @ A @ B,Sp_nu3: stream1273403375_sp_nu @ A @ B] :
( ( ( stream1200245950nu_out @ A @ B @ Sp_nu )
= ( stream1200245950nu_out @ A @ B @ Sp_nu3 ) )
=> ( Sp_nu = Sp_nu3 ) ) ).
% sp\<^sub>\<nu>.expand
thf(fact_24_sp_092_060_094sub_062_092_060mu_062_Osimps_I5_J,axiom,
! [D: $tType,C: $tType,B: $tType,A: $tType,F1: ( A > ( stream901396144_sp_mu @ A @ B @ C ) ) > D,F2: B > C > D,X1: A > ( stream901396144_sp_mu @ A @ B @ C )] :
( ( stream160000856_sp_mu @ A @ B @ C @ D @ F1 @ F2 @ ( stream1294929701mu_Get @ A @ B @ C @ X1 ) )
= ( F1 @ X1 ) ) ).
% sp\<^sub>\<mu>.simps(5)
thf(fact_25_sp_092_060_094sub_062_092_060nu_062_Osplit__sel__asm,axiom,
! [C: $tType,B: $tType,A: $tType,P: C > $o,F: ( stream901396144_sp_mu @ A @ B @ ( stream1273403375_sp_nu @ A @ B ) ) > C,Sp_nu: stream1273403375_sp_nu @ A @ B] :
( ( P @ ( stream1502363800_sp_nu @ A @ B @ C @ F @ Sp_nu ) )
= ( ~ ( ( Sp_nu
= ( stream1686501333_nu_In @ A @ B @ ( stream1200245950nu_out @ A @ B @ Sp_nu ) ) )
& ~ ( P @ ( F @ ( stream1200245950nu_out @ A @ B @ Sp_nu ) ) ) ) ) ) ).
% sp\<^sub>\<nu>.split_sel_asm
thf(fact_26_sp_092_060_094sub_062_092_060nu_062_Osplit__sel,axiom,
! [C: $tType,B: $tType,A: $tType,P: C > $o,F: ( stream901396144_sp_mu @ A @ B @ ( stream1273403375_sp_nu @ A @ B ) ) > C,Sp_nu: stream1273403375_sp_nu @ A @ B] :
( ( P @ ( stream1502363800_sp_nu @ A @ B @ C @ F @ Sp_nu ) )
= ( ( Sp_nu
= ( stream1686501333_nu_In @ A @ B @ ( stream1200245950nu_out @ A @ B @ Sp_nu ) ) )
=> ( P @ ( F @ ( stream1200245950nu_out @ A @ B @ Sp_nu ) ) ) ) ) ).
% sp\<^sub>\<nu>.split_sel
thf(fact_27_sp_092_060_094sub_062_092_060mu_062_Opred__set,axiom,
! [B: $tType,G: $tType,A: $tType] :
( ( stream123009735_sp_mu @ A @ B @ G )
= ( ^ [P12: A > $o,P22: B > $o,X3: stream901396144_sp_mu @ G @ A @ B] :
( ! [Y2: A] :
( ( member @ A @ Y2 @ ( stream2074884121_sp_mu @ G @ A @ B @ X3 ) )
=> ( P12 @ Y2 ) )
& ! [Y2: B] :
( ( member @ B @ Y2 @ ( stream1259315544_sp_mu @ G @ A @ B @ X3 ) )
=> ( P22 @ Y2 ) ) ) ) ) ).
% sp\<^sub>\<mu>.pred_set
thf(fact_28_sp_092_060_094sub_062_092_060nu_062_Opred__inject,axiom,
! [B: $tType,A: $tType,P: B > $o,A2: stream901396144_sp_mu @ A @ B @ ( stream1273403375_sp_nu @ A @ B )] :
( ( stream1465372679_sp_nu @ B @ A @ P @ ( stream1686501333_nu_In @ A @ B @ A2 ) )
= ( stream123009735_sp_mu @ B @ ( stream1273403375_sp_nu @ A @ B ) @ A @ P @ ( stream1465372679_sp_nu @ B @ A @ P ) @ A2 ) ) ).
% sp\<^sub>\<nu>.pred_inject
thf(fact_29_sp_092_060_094sub_062_092_060mu_062_Orel__refl__strong,axiom,
! [C: $tType,A: $tType,B: $tType,X: stream901396144_sp_mu @ B @ A @ C,R1a: A > A > $o,R2a: C > C > $o] :
( ! [Z1: A] :
( ( member @ A @ Z1 @ ( stream2074884121_sp_mu @ B @ A @ C @ X ) )
=> ( R1a @ Z1 @ Z1 ) )
=> ( ! [Z2: C] :
( ( member @ C @ Z2 @ ( stream1259315544_sp_mu @ B @ A @ C @ X ) )
=> ( R2a @ Z2 @ Z2 ) )
=> ( stream1924447089_sp_mu @ A @ A @ C @ C @ B @ R1a @ R2a @ X @ X ) ) ) ).
% sp\<^sub>\<mu>.rel_refl_strong
thf(fact_30_sp_092_060_094sub_062_092_060mu_062_Orel__mono__strong,axiom,
! [A: $tType,B: $tType,D: $tType,C: $tType,G: $tType,R1: A > C > $o,R2: B > D > $o,X: stream901396144_sp_mu @ G @ A @ B,Y: stream901396144_sp_mu @ G @ C @ D,R1a: A > C > $o,R2a: B > D > $o] :
( ( stream1924447089_sp_mu @ A @ C @ B @ D @ G @ R1 @ R2 @ X @ Y )
=> ( ! [Z1: A,Y12: C] :
( ( member @ A @ Z1 @ ( stream2074884121_sp_mu @ G @ A @ B @ X ) )
=> ( ( member @ C @ Y12 @ ( stream2074884121_sp_mu @ G @ C @ D @ Y ) )
=> ( ( R1 @ Z1 @ Y12 )
=> ( R1a @ Z1 @ Y12 ) ) ) )
=> ( ! [Z2: B,Y23: D] :
( ( member @ B @ Z2 @ ( stream1259315544_sp_mu @ G @ A @ B @ X ) )
=> ( ( member @ D @ Y23 @ ( stream1259315544_sp_mu @ G @ C @ D @ Y ) )
=> ( ( R2 @ Z2 @ Y23 )
=> ( R2a @ Z2 @ Y23 ) ) ) )
=> ( stream1924447089_sp_mu @ A @ C @ B @ D @ G @ R1a @ R2a @ X @ Y ) ) ) ) ).
% sp\<^sub>\<mu>.rel_mono_strong
thf(fact_31_sp_092_060_094sub_062_092_060mu_062_Orel__cong,axiom,
! [A: $tType,B: $tType,D: $tType,C: $tType,G: $tType,X: stream901396144_sp_mu @ G @ A @ B,Ya: stream901396144_sp_mu @ G @ A @ B,Y: stream901396144_sp_mu @ G @ C @ D,Xa: stream901396144_sp_mu @ G @ C @ D,R1: A > C > $o,R1a: A > C > $o,R2: B > D > $o,R2a: B > D > $o] :
( ( X = Ya )
=> ( ( Y = Xa )
=> ( ! [Z1: A,Y12: C] :
( ( member @ A @ Z1 @ ( stream2074884121_sp_mu @ G @ A @ B @ Ya ) )
=> ( ( member @ C @ Y12 @ ( stream2074884121_sp_mu @ G @ C @ D @ Xa ) )
=> ( ( R1 @ Z1 @ Y12 )
= ( R1a @ Z1 @ Y12 ) ) ) )
=> ( ! [Z2: B,Y23: D] :
( ( member @ B @ Z2 @ ( stream1259315544_sp_mu @ G @ A @ B @ Ya ) )
=> ( ( member @ D @ Y23 @ ( stream1259315544_sp_mu @ G @ C @ D @ Xa ) )
=> ( ( R2 @ Z2 @ Y23 )
= ( R2a @ Z2 @ Y23 ) ) ) )
=> ( ( stream1924447089_sp_mu @ A @ C @ B @ D @ G @ R1 @ R2 @ X @ Y )
= ( stream1924447089_sp_mu @ A @ C @ B @ D @ G @ R1a @ R2a @ Ya @ Xa ) ) ) ) ) ) ).
% sp\<^sub>\<mu>.rel_cong
thf(fact_32_sp_092_060_094sub_062_092_060mu_062_Orel__distinct_I1_J,axiom,
! [B: $tType,C: $tType,E: $tType,A: $tType,F3: $tType,R1: B > E > $o,R2: C > F3 > $o,X1: A > ( stream901396144_sp_mu @ A @ B @ C ),Y21: E,Y22: F3] :
~ ( stream1924447089_sp_mu @ B @ E @ C @ F3 @ A @ R1 @ R2 @ ( stream1294929701mu_Get @ A @ B @ C @ X1 ) @ ( stream1370332830mu_Put @ E @ F3 @ A @ Y21 @ Y22 ) ) ).
% sp\<^sub>\<mu>.rel_distinct(1)
thf(fact_33_sp_092_060_094sub_062_092_060mu_062_Orel__distinct_I2_J,axiom,
! [B: $tType,A: $tType,D: $tType,C: $tType,G: $tType,R1: A > C > $o,R2: B > D > $o,Y21: A,Y22: B,X1: G > ( stream901396144_sp_mu @ G @ C @ D )] :
~ ( stream1924447089_sp_mu @ A @ C @ B @ D @ G @ R1 @ R2 @ ( stream1370332830mu_Put @ A @ B @ G @ Y21 @ Y22 ) @ ( stream1294929701mu_Get @ G @ C @ D @ X1 ) ) ).
% sp\<^sub>\<mu>.rel_distinct(2)
thf(fact_34_sp_092_060_094sub_062_092_060mu_062_Opred__map,axiom,
! [D: $tType,C: $tType,B: $tType,A: $tType,G: $tType,Q1: C > $o,Q2: D > $o,F1: A > C,F2: B > D,X: stream901396144_sp_mu @ G @ A @ B] :
( ( stream123009735_sp_mu @ C @ D @ G @ Q1 @ Q2 @ ( stream1058219438_sp_mu @ A @ C @ B @ D @ G @ F1 @ F2 @ X ) )
= ( stream123009735_sp_mu @ A @ B @ G @ ( comp @ C @ $o @ A @ Q1 @ F1 ) @ ( comp @ D @ $o @ B @ Q2 @ F2 ) @ X ) ) ).
% sp\<^sub>\<mu>.pred_map
thf(fact_35_sp_092_060_094sub_062_092_060mu_062_Oset__map_I1_J,axiom,
! [D: $tType,C: $tType,B: $tType,A: $tType,G: $tType,F1: A > C,F2: B > D,V: stream901396144_sp_mu @ G @ A @ B] :
( ( stream2074884121_sp_mu @ G @ C @ D @ ( stream1058219438_sp_mu @ A @ C @ B @ D @ G @ F1 @ F2 @ V ) )
= ( image @ A @ C @ F1 @ ( stream2074884121_sp_mu @ G @ A @ B @ V ) ) ) ).
% sp\<^sub>\<mu>.set_map(1)
thf(fact_36_sp_092_060_094sub_062_092_060mu_062_Oset__map_I2_J,axiom,
! [C: $tType,D: $tType,B: $tType,A: $tType,G: $tType,F1: A > C,F2: B > D,V: stream901396144_sp_mu @ G @ A @ B] :
( ( stream1259315544_sp_mu @ G @ C @ D @ ( stream1058219438_sp_mu @ A @ C @ B @ D @ G @ F1 @ F2 @ V ) )
= ( image @ B @ D @ F2 @ ( stream1259315544_sp_mu @ G @ A @ B @ V ) ) ) ).
% sp\<^sub>\<mu>.set_map(2)
thf(fact_37_sp_092_060_094sub_062_092_060mu_062_Orel__eq,axiom,
! [B: $tType,A: $tType,G: $tType] :
( ( stream1924447089_sp_mu @ A @ A @ B @ B @ G
@ ^ [Y3: A,Z: A] : Y3 = Z
@ ^ [Y3: B,Z: B] : Y3 = Z )
= ( ^ [Y3: stream901396144_sp_mu @ G @ A @ B,Z: stream901396144_sp_mu @ G @ A @ B] : Y3 = Z ) ) ).
% sp\<^sub>\<mu>.rel_eq
thf(fact_38_sp_092_060_094sub_062_092_060mu_062_Orel__refl,axiom,
! [D: $tType,C: $tType,G: $tType,R1a: C > C > $o,R2a: D > D > $o,X: stream901396144_sp_mu @ G @ C @ D] :
( ! [X2: C] : ( R1a @ X2 @ X2 )
=> ( ! [X2: D] : ( R2a @ X2 @ X2 )
=> ( stream1924447089_sp_mu @ C @ C @ D @ D @ G @ R1a @ R2a @ X @ X ) ) ) ).
% sp\<^sub>\<mu>.rel_refl
thf(fact_39_sp_092_060_094sub_062_092_060mu_062_Omap__comp,axiom,
! [D: $tType,F3: $tType,E: $tType,C: $tType,B: $tType,A: $tType,G: $tType,G1: C > E,G2: D > F3,F1: A > C,F2: B > D,V: stream901396144_sp_mu @ G @ A @ B] :
( ( stream1058219438_sp_mu @ C @ E @ D @ F3 @ G @ G1 @ G2 @ ( stream1058219438_sp_mu @ A @ C @ B @ D @ G @ F1 @ F2 @ V ) )
= ( stream1058219438_sp_mu @ A @ E @ B @ F3 @ G @ ( comp @ C @ E @ A @ G1 @ F1 ) @ ( comp @ D @ F3 @ B @ G2 @ F2 ) @ V ) ) ).
% sp\<^sub>\<mu>.map_comp
thf(fact_40_sp_092_060_094sub_062_092_060nu_062_Ocoinduct,axiom,
! [B: $tType,A: $tType,R: ( stream1273403375_sp_nu @ A @ B ) > ( stream1273403375_sp_nu @ A @ B ) > $o,Sp_nu: stream1273403375_sp_nu @ A @ B,Sp_nu3: stream1273403375_sp_nu @ A @ B] :
( ( R @ Sp_nu @ Sp_nu3 )
=> ( ! [Sp_nu4: stream1273403375_sp_nu @ A @ B,Sp_nu5: stream1273403375_sp_nu @ A @ B] :
( ( R @ Sp_nu4 @ Sp_nu5 )
=> ( stream1924447089_sp_mu @ B @ B @ ( stream1273403375_sp_nu @ A @ B ) @ ( stream1273403375_sp_nu @ A @ B ) @ A
@ ^ [Y3: B,Z: B] : Y3 = Z
@ R
@ ( stream1200245950nu_out @ A @ B @ Sp_nu4 )
@ ( stream1200245950nu_out @ A @ B @ Sp_nu5 ) ) )
=> ( Sp_nu = Sp_nu3 ) ) ) ).
% sp\<^sub>\<nu>.coinduct
thf(fact_41_sp_092_060_094sub_062_092_060mu_062_Orel__inject_I2_J,axiom,
! [A: $tType,E: $tType,B: $tType,C: $tType,F3: $tType,R1: B > E > $o,R2: C > F3 > $o,X21: B,X22: C,Y21: E,Y22: F3] :
( ( stream1924447089_sp_mu @ B @ E @ C @ F3 @ A @ R1 @ R2 @ ( stream1370332830mu_Put @ B @ C @ A @ X21 @ X22 ) @ ( stream1370332830mu_Put @ E @ F3 @ A @ Y21 @ Y22 ) )
= ( ( R1 @ X21 @ Y21 )
& ( R2 @ X22 @ Y22 ) ) ) ).
% sp\<^sub>\<mu>.rel_inject(2)
thf(fact_42_sp_092_060_094sub_062_092_060mu_062_Orel__intros_I2_J,axiom,
! [C: $tType,B: $tType,E: $tType,A: $tType,F3: $tType,R1: B > E > $o,X21: B,Y21: E,R2: C > F3 > $o,X22: C,Y22: F3] :
( ( R1 @ X21 @ Y21 )
=> ( ( R2 @ X22 @ Y22 )
=> ( stream1924447089_sp_mu @ B @ E @ C @ F3 @ A @ R1 @ R2 @ ( stream1370332830mu_Put @ B @ C @ A @ X21 @ X22 ) @ ( stream1370332830mu_Put @ E @ F3 @ A @ Y21 @ Y22 ) ) ) ) ).
% sp\<^sub>\<mu>.rel_intros(2)
thf(fact_43_Ball__image__comp,axiom,
! [A: $tType,B: $tType,F: B > A,A3: set @ B,G3: A > $o] :
( ( ! [X3: A] :
( ( member @ A @ X3 @ ( image @ B @ A @ F @ A3 ) )
=> ( G3 @ X3 ) ) )
= ( ! [X3: B] :
( ( member @ B @ X3 @ A3 )
=> ( comp @ A @ $o @ B @ G3 @ F @ X3 ) ) ) ) ).
% Ball_image_comp
thf(fact_44_not__False__in__image__Ball,axiom,
! [A: $tType,P: A > $o,A3: set @ A] :
( ( ~ ( member @ $o @ $false @ ( image @ A @ $o @ P @ A3 ) ) )
= ( ! [X3: A] :
( ( member @ A @ X3 @ A3 )
=> ( P @ X3 ) ) ) ) ).
% not_False_in_image_Ball
thf(fact_45_mem__Collect__eq,axiom,
! [A: $tType,A2: A,P: A > $o] :
( ( member @ A @ A2 @ ( collect @ A @ P ) )
= ( P @ A2 ) ) ).
% mem_Collect_eq
thf(fact_46_Collect__mem__eq,axiom,
! [A: $tType,A3: set @ A] :
( ( collect @ A
@ ^ [X3: A] : ( member @ A @ X3 @ A3 ) )
= A3 ) ).
% Collect_mem_eq
thf(fact_47_Collect__cong,axiom,
! [A: $tType,P: A > $o,Q: A > $o] :
( ! [X2: A] :
( ( P @ X2 )
= ( Q @ X2 ) )
=> ( ( collect @ A @ P )
= ( collect @ A @ Q ) ) ) ).
% Collect_cong
thf(fact_48_ext,axiom,
! [B: $tType,A: $tType,F: A > B,G3: A > B] :
( ! [X2: A] :
( ( F @ X2 )
= ( G3 @ X2 ) )
=> ( F = G3 ) ) ).
% ext
thf(fact_49_comp__apply,axiom,
! [C: $tType,A: $tType,B: $tType] :
( ( comp @ B @ A @ C )
= ( ^ [F4: B > A,G4: C > B,X3: C] : ( F4 @ ( G4 @ X3 ) ) ) ) ).
% comp_apply
thf(fact_50_image__eqI,axiom,
! [A: $tType,B: $tType,B2: A,F: B > A,X: B,A3: set @ B] :
( ( B2
= ( F @ X ) )
=> ( ( member @ B @ X @ A3 )
=> ( member @ A @ B2 @ ( image @ B @ A @ F @ A3 ) ) ) ) ).
% image_eqI
thf(fact_51_image__comp,axiom,
! [B: $tType,A: $tType,C: $tType,F: B > A,G3: C > B,R3: set @ C] :
( ( image @ B @ A @ F @ ( image @ C @ B @ G3 @ R3 ) )
= ( image @ C @ A @ ( comp @ B @ A @ C @ F @ G3 ) @ R3 ) ) ).
% image_comp
thf(fact_52_image__eq__imp__comp,axiom,
! [B: $tType,A: $tType,D: $tType,C: $tType,F: B > A,A3: set @ B,G3: C > A,B3: set @ C,H: A > D] :
( ( ( image @ B @ A @ F @ A3 )
= ( image @ C @ A @ G3 @ B3 ) )
=> ( ( image @ B @ D @ ( comp @ A @ D @ B @ H @ F ) @ A3 )
= ( image @ C @ D @ ( comp @ A @ D @ C @ H @ G3 ) @ B3 ) ) ) ).
% image_eq_imp_comp
thf(fact_53_Inf_OINF__image,axiom,
! [B: $tType,A: $tType,C: $tType,Inf: ( set @ A ) > A,G3: B > A,F: C > B,A3: set @ C] :
( ( Inf @ ( image @ B @ A @ G3 @ ( image @ C @ B @ F @ A3 ) ) )
= ( Inf @ ( image @ C @ A @ ( comp @ B @ A @ C @ G3 @ F ) @ A3 ) ) ) ).
% Inf.INF_image
thf(fact_54_Sup_OSUP__image,axiom,
! [B: $tType,A: $tType,C: $tType,Sup: ( set @ A ) > A,G3: B > A,F: C > B,A3: set @ C] :
( ( Sup @ ( image @ B @ A @ G3 @ ( image @ C @ B @ F @ A3 ) ) )
= ( Sup @ ( image @ C @ A @ ( comp @ B @ A @ C @ G3 @ F ) @ A3 ) ) ) ).
% Sup.SUP_image
thf(fact_55_sp_092_060_094sub_062_092_060mu_062_Oset__cases_I1_J,axiom,
! [B: $tType,A: $tType,C: $tType,E2: B,A2: stream901396144_sp_mu @ A @ B @ C] :
( ( member @ B @ E2 @ ( stream2074884121_sp_mu @ A @ B @ C @ A2 ) )
=> ( ! [Z3: A > ( stream901396144_sp_mu @ A @ B @ C )] :
( ( A2
= ( stream1294929701mu_Get @ A @ B @ C @ Z3 ) )
=> ! [X2: stream901396144_sp_mu @ A @ B @ C] :
( ( member @ ( stream901396144_sp_mu @ A @ B @ C ) @ X2 @ ( image @ A @ ( stream901396144_sp_mu @ A @ B @ C ) @ Z3 @ ( top_top @ ( set @ A ) ) ) )
=> ~ ( member @ B @ E2 @ ( stream2074884121_sp_mu @ A @ B @ C @ X2 ) ) ) )
=> ~ ! [Z2: C] :
( A2
!= ( stream1370332830mu_Put @ B @ C @ A @ E2 @ Z2 ) ) ) ) ).
% sp\<^sub>\<mu>.set_cases(1)
thf(fact_56_UNIV__I,axiom,
! [A: $tType,X: A] : ( member @ A @ X @ ( top_top @ ( set @ A ) ) ) ).
% UNIV_I
thf(fact_57_surjD,axiom,
! [A: $tType,B: $tType,F: B > A,Y: A] :
( ( ( image @ B @ A @ F @ ( top_top @ ( set @ B ) ) )
= ( top_top @ ( set @ A ) ) )
=> ? [X2: B] :
( Y
= ( F @ X2 ) ) ) ).
% surjD
thf(fact_58_surjE,axiom,
! [A: $tType,B: $tType,F: B > A,Y: A] :
( ( ( image @ B @ A @ F @ ( top_top @ ( set @ B ) ) )
= ( top_top @ ( set @ A ) ) )
=> ~ ! [X2: B] :
( Y
!= ( F @ X2 ) ) ) ).
% surjE
thf(fact_59_surjI,axiom,
! [B: $tType,A: $tType,G3: B > A,F: A > B] :
( ! [X2: A] :
( ( G3 @ ( F @ X2 ) )
= X2 )
=> ( ( image @ B @ A @ G3 @ ( top_top @ ( set @ B ) ) )
= ( top_top @ ( set @ A ) ) ) ) ).
% surjI
thf(fact_60_rangeI,axiom,
! [A: $tType,B: $tType,F: B > A,X: B] : ( member @ A @ ( F @ X ) @ ( image @ B @ A @ F @ ( top_top @ ( set @ B ) ) ) ) ).
% rangeI
thf(fact_61_surj__def,axiom,
! [B: $tType,A: $tType,F: B > A] :
( ( ( image @ B @ A @ F @ ( top_top @ ( set @ B ) ) )
= ( top_top @ ( set @ A ) ) )
= ( ! [Y2: A] :
? [X3: B] :
( Y2
= ( F @ X3 ) ) ) ) ).
% surj_def
thf(fact_62_range__eqI,axiom,
! [A: $tType,B: $tType,B2: A,F: B > A,X: B] :
( ( B2
= ( F @ X ) )
=> ( member @ A @ B2 @ ( image @ B @ A @ F @ ( top_top @ ( set @ B ) ) ) ) ) ).
% range_eqI
thf(fact_63_UNIV__witness,axiom,
! [A: $tType] :
? [X2: A] : ( member @ A @ X2 @ ( top_top @ ( set @ A ) ) ) ).
% UNIV_witness
thf(fact_64_UNIV__eq__I,axiom,
! [A: $tType,A3: set @ A] :
( ! [X2: A] : ( member @ A @ X2 @ A3 )
=> ( ( top_top @ ( set @ A ) )
= A3 ) ) ).
% UNIV_eq_I
thf(fact_65_comp__surj,axiom,
! [B: $tType,A: $tType,C: $tType,F: B > A,G3: A > C] :
( ( ( image @ B @ A @ F @ ( top_top @ ( set @ B ) ) )
= ( top_top @ ( set @ A ) ) )
=> ( ( ( image @ A @ C @ G3 @ ( top_top @ ( set @ A ) ) )
= ( top_top @ ( set @ C ) ) )
=> ( ( image @ B @ C @ ( comp @ A @ C @ B @ G3 @ F ) @ ( top_top @ ( set @ B ) ) )
= ( top_top @ ( set @ C ) ) ) ) ) ).
% comp_surj
thf(fact_66_sp_092_060_094sub_062_092_060mu_062_Osimps_I9_J,axiom,
! [F3: $tType,E: $tType,C: $tType,B: $tType,A: $tType,F1: B > E,F2: C > F3,X1: A > ( stream901396144_sp_mu @ A @ B @ C )] :
( ( stream1058219438_sp_mu @ B @ E @ C @ F3 @ A @ F1 @ F2 @ ( stream1294929701mu_Get @ A @ B @ C @ X1 ) )
= ( stream1294929701mu_Get @ A @ E @ F3 @ ( comp @ ( stream901396144_sp_mu @ A @ B @ C ) @ ( stream901396144_sp_mu @ A @ E @ F3 ) @ A @ ( stream1058219438_sp_mu @ B @ E @ C @ F3 @ A @ F1 @ F2 ) @ X1 ) ) ) ).
% sp\<^sub>\<mu>.simps(9)
thf(fact_67_sp_092_060_094sub_062_092_060mu_062_Oinduct,axiom,
! [C: $tType,B: $tType,A: $tType,P: ( stream901396144_sp_mu @ A @ B @ C ) > $o,Sp_mu: stream901396144_sp_mu @ A @ B @ C] :
( ! [X2: A > ( stream901396144_sp_mu @ A @ B @ C )] :
( ! [Xa2: stream901396144_sp_mu @ A @ B @ C] :
( ( member @ ( stream901396144_sp_mu @ A @ B @ C ) @ Xa2 @ ( image @ A @ ( stream901396144_sp_mu @ A @ B @ C ) @ X2 @ ( top_top @ ( set @ A ) ) ) )
=> ( P @ Xa2 ) )
=> ( P @ ( stream1294929701mu_Get @ A @ B @ C @ X2 ) ) )
=> ( ! [X1a: B,X23: C] : ( P @ ( stream1370332830mu_Put @ B @ C @ A @ X1a @ X23 ) )
=> ( P @ Sp_mu ) ) ) ).
% sp\<^sub>\<mu>.induct
thf(fact_68_sp_092_060_094sub_062_092_060mu_062_Oset__intros_I1_J,axiom,
! [C: $tType,B: $tType,A: $tType,X: stream901396144_sp_mu @ A @ B @ C,A2: A > ( stream901396144_sp_mu @ A @ B @ C ),Xa: B] :
( ( member @ ( stream901396144_sp_mu @ A @ B @ C ) @ X @ ( image @ A @ ( stream901396144_sp_mu @ A @ B @ C ) @ A2 @ ( top_top @ ( set @ A ) ) ) )
=> ( ( member @ B @ Xa @ ( stream2074884121_sp_mu @ A @ B @ C @ X ) )
=> ( member @ B @ Xa @ ( stream2074884121_sp_mu @ A @ B @ C @ ( stream1294929701mu_Get @ A @ B @ C @ A2 ) ) ) ) ) ).
% sp\<^sub>\<mu>.set_intros(1)
thf(fact_69_sp_092_060_094sub_062_092_060mu_062_Oset__intros_I3_J,axiom,
! [C: $tType,B: $tType,A: $tType,Xc: stream901396144_sp_mu @ A @ B @ C,Aa: A > ( stream901396144_sp_mu @ A @ B @ C ),Xe: C] :
( ( member @ ( stream901396144_sp_mu @ A @ B @ C ) @ Xc @ ( image @ A @ ( stream901396144_sp_mu @ A @ B @ C ) @ Aa @ ( top_top @ ( set @ A ) ) ) )
=> ( ( member @ C @ Xe @ ( stream1259315544_sp_mu @ A @ B @ C @ Xc ) )
=> ( member @ C @ Xe @ ( stream1259315544_sp_mu @ A @ B @ C @ ( stream1294929701mu_Get @ A @ B @ C @ Aa ) ) ) ) ) ).
% sp\<^sub>\<mu>.set_intros(3)
thf(fact_70_Sup_OSUP__cong,axiom,
! [A: $tType,B: $tType,A3: set @ B,B3: set @ B,C2: B > A,D2: B > A,Sup: ( set @ A ) > A] :
( ( A3 = B3 )
=> ( ! [X2: B] :
( ( member @ B @ X2 @ B3 )
=> ( ( C2 @ X2 )
= ( D2 @ X2 ) ) )
=> ( ( Sup @ ( image @ B @ A @ C2 @ A3 ) )
= ( Sup @ ( image @ B @ A @ D2 @ B3 ) ) ) ) ) ).
% Sup.SUP_cong
thf(fact_71_Inf_OINF__cong,axiom,
! [A: $tType,B: $tType,A3: set @ B,B3: set @ B,C2: B > A,D2: B > A,Inf: ( set @ A ) > A] :
( ( A3 = B3 )
=> ( ! [X2: B] :
( ( member @ B @ X2 @ B3 )
=> ( ( C2 @ X2 )
= ( D2 @ X2 ) ) )
=> ( ( Inf @ ( image @ B @ A @ C2 @ A3 ) )
= ( Inf @ ( image @ B @ A @ D2 @ B3 ) ) ) ) ) ).
% Inf.INF_cong
thf(fact_72_rev__image__eqI,axiom,
! [B: $tType,A: $tType,X: A,A3: set @ A,B2: B,F: A > B] :
( ( member @ A @ X @ A3 )
=> ( ( B2
= ( F @ X ) )
=> ( member @ B @ B2 @ ( image @ A @ B @ F @ A3 ) ) ) ) ).
% rev_image_eqI
thf(fact_73_ball__imageD,axiom,
! [A: $tType,B: $tType,F: B > A,A3: set @ B,P: A > $o] :
( ! [X2: A] :
( ( member @ A @ X2 @ ( image @ B @ A @ F @ A3 ) )
=> ( P @ X2 ) )
=> ! [X4: B] :
( ( member @ B @ X4 @ A3 )
=> ( P @ ( F @ X4 ) ) ) ) ).
% ball_imageD
thf(fact_74_image__cong,axiom,
! [B: $tType,A: $tType,M: set @ A,N: set @ A,F: A > B,G3: A > B] :
( ( M = N )
=> ( ! [X2: A] :
( ( member @ A @ X2 @ N )
=> ( ( F @ X2 )
= ( G3 @ X2 ) ) )
=> ( ( image @ A @ B @ F @ M )
= ( image @ A @ B @ G3 @ N ) ) ) ) ).
% image_cong
thf(fact_75_bex__imageD,axiom,
! [A: $tType,B: $tType,F: B > A,A3: set @ B,P: A > $o] :
( ? [X4: A] :
( ( member @ A @ X4 @ ( image @ B @ A @ F @ A3 ) )
& ( P @ X4 ) )
=> ? [X2: B] :
( ( member @ B @ X2 @ A3 )
& ( P @ ( F @ X2 ) ) ) ) ).
% bex_imageD
thf(fact_76_image__iff,axiom,
! [A: $tType,B: $tType,Z4: A,F: B > A,A3: set @ B] :
( ( member @ A @ Z4 @ ( image @ B @ A @ F @ A3 ) )
= ( ? [X3: B] :
( ( member @ B @ X3 @ A3 )
& ( Z4
= ( F @ X3 ) ) ) ) ) ).
% image_iff
thf(fact_77_imageI,axiom,
! [B: $tType,A: $tType,X: A,A3: set @ A,F: A > B] :
( ( member @ A @ X @ A3 )
=> ( member @ B @ ( F @ X ) @ ( image @ A @ B @ F @ A3 ) ) ) ).
% imageI
thf(fact_78_comp__apply__eq,axiom,
! [B: $tType,D: $tType,A: $tType,C: $tType,F: B > A,G3: C > B,X: C,H: D > A,K: C > D] :
( ( ( F @ ( G3 @ X ) )
= ( H @ ( K @ X ) ) )
=> ( ( comp @ B @ A @ C @ F @ G3 @ X )
= ( comp @ D @ A @ C @ H @ K @ X ) ) ) ).
% comp_apply_eq
thf(fact_79_comp__eq__dest__lhs,axiom,
! [C: $tType,B: $tType,A: $tType,A2: C > B,B2: A > C,C3: A > B,V: A] :
( ( ( comp @ C @ B @ A @ A2 @ B2 )
= C3 )
=> ( ( A2 @ ( B2 @ V ) )
= ( C3 @ V ) ) ) ).
% comp_eq_dest_lhs
thf(fact_80_comp__eq__elim,axiom,
! [C: $tType,B: $tType,D: $tType,A: $tType,A2: C > B,B2: A > C,C3: D > B,D3: A > D] :
( ( ( comp @ C @ B @ A @ A2 @ B2 )
= ( comp @ D @ B @ A @ C3 @ D3 ) )
=> ! [V2: A] :
( ( A2 @ ( B2 @ V2 ) )
= ( C3 @ ( D3 @ V2 ) ) ) ) ).
% comp_eq_elim
thf(fact_81_comp__eq__dest,axiom,
! [C: $tType,B: $tType,D: $tType,A: $tType,A2: C > B,B2: A > C,C3: D > B,D3: A > D,V: A] :
( ( ( comp @ C @ B @ A @ A2 @ B2 )
= ( comp @ D @ B @ A @ C3 @ D3 ) )
=> ( ( A2 @ ( B2 @ V ) )
= ( C3 @ ( D3 @ V ) ) ) ) ).
% comp_eq_dest
thf(fact_82_comp__assoc,axiom,
! [B: $tType,D: $tType,C: $tType,A: $tType,F: D > B,G3: C > D,H: A > C] :
( ( comp @ C @ B @ A @ ( comp @ D @ B @ C @ F @ G3 ) @ H )
= ( comp @ D @ B @ A @ F @ ( comp @ C @ D @ A @ G3 @ H ) ) ) ).
% comp_assoc
thf(fact_83_comp__def,axiom,
! [A: $tType,C: $tType,B: $tType] :
( ( comp @ B @ C @ A )
= ( ^ [F4: B > C,G4: A > B,X3: A] : ( F4 @ ( G4 @ X3 ) ) ) ) ).
% comp_def
thf(fact_84_Ball__def,axiom,
! [A: $tType] :
( ( ball @ A )
= ( ^ [A4: set @ A,P3: A > $o] :
! [X3: A] :
( ( member @ A @ X3 @ A4 )
=> ( P3 @ X3 ) ) ) ) ).
% Ball_def
thf(fact_85_sp_092_060_094sub_062_092_060mu_062_Oset__cases_I2_J,axiom,
! [B: $tType,A: $tType,C: $tType,E2: C,A2: stream901396144_sp_mu @ A @ B @ C] :
( ( member @ C @ E2 @ ( stream1259315544_sp_mu @ A @ B @ C @ A2 ) )
=> ( ! [Z3: A > ( stream901396144_sp_mu @ A @ B @ C )] :
( ( A2
= ( stream1294929701mu_Get @ A @ B @ C @ Z3 ) )
=> ! [X2: stream901396144_sp_mu @ A @ B @ C] :
( ( member @ ( stream901396144_sp_mu @ A @ B @ C ) @ X2 @ ( image @ A @ ( stream901396144_sp_mu @ A @ B @ C ) @ Z3 @ ( top_top @ ( set @ A ) ) ) )
=> ~ ( member @ C @ E2 @ ( stream1259315544_sp_mu @ A @ B @ C @ X2 ) ) ) )
=> ~ ! [Z1: B] :
( A2
!= ( stream1370332830mu_Put @ B @ C @ A @ Z1 @ E2 ) ) ) ) ).
% sp\<^sub>\<mu>.set_cases(2)
thf(fact_86_surj__fun__eq,axiom,
! [B: $tType,C: $tType,A: $tType,F: B > A,X5: set @ B,G1: A > C,G2: A > C] :
( ( ( image @ B @ A @ F @ X5 )
= ( top_top @ ( set @ A ) ) )
=> ( ! [X2: B] :
( ( member @ B @ X2 @ X5 )
=> ( ( comp @ A @ C @ B @ G1 @ F @ X2 )
= ( comp @ A @ C @ B @ G2 @ F @ X2 ) ) )
=> ( G1 = G2 ) ) ) ).
% surj_fun_eq
thf(fact_87_fun_Oinj__map__strong,axiom,
! [B: $tType,A: $tType,D: $tType,X: D > A,Xa: D > A,F: A > B,Fa: A > B] :
( ! [Z3: A,Za: A] :
( ( member @ A @ Z3 @ ( image @ D @ A @ X @ ( top_top @ ( set @ D ) ) ) )
=> ( ( member @ A @ Za @ ( image @ D @ A @ Xa @ ( top_top @ ( set @ D ) ) ) )
=> ( ( ( F @ Z3 )
= ( Fa @ Za ) )
=> ( Z3 = Za ) ) ) )
=> ( ( ( comp @ A @ B @ D @ F @ X )
= ( comp @ A @ B @ D @ Fa @ Xa ) )
=> ( X = Xa ) ) ) ).
% fun.inj_map_strong
thf(fact_88_fun_Omap__cong0,axiom,
! [B: $tType,A: $tType,D: $tType,X: D > A,F: A > B,G3: A > B] :
( ! [Z3: A] :
( ( member @ A @ Z3 @ ( image @ D @ A @ X @ ( top_top @ ( set @ D ) ) ) )
=> ( ( F @ Z3 )
= ( G3 @ Z3 ) ) )
=> ( ( comp @ A @ B @ D @ F @ X )
= ( comp @ A @ B @ D @ G3 @ X ) ) ) ).
% fun.map_cong0
thf(fact_89_fun_Omap__cong,axiom,
! [B: $tType,A: $tType,D: $tType,X: D > A,Ya: D > A,F: A > B,G3: A > B] :
( ( X = Ya )
=> ( ! [Z3: A] :
( ( member @ A @ Z3 @ ( image @ D @ A @ Ya @ ( top_top @ ( set @ D ) ) ) )
=> ( ( F @ Z3 )
= ( G3 @ Z3 ) ) )
=> ( ( comp @ A @ B @ D @ F @ X )
= ( comp @ A @ B @ D @ G3 @ Ya ) ) ) ) ).
% fun.map_cong
thf(fact_90_fun_Oset__map,axiom,
! [B: $tType,A: $tType,D: $tType,F: A > B,V: D > A] :
( ( image @ D @ B @ ( comp @ A @ B @ D @ F @ V ) @ ( top_top @ ( set @ D ) ) )
= ( image @ A @ B @ F @ ( image @ D @ A @ V @ ( top_top @ ( set @ D ) ) ) ) ) ).
% fun.set_map
thf(fact_91_iso__tuple__UNIV__I,axiom,
! [A: $tType,X: A] : ( member @ A @ X @ ( top_top @ ( set @ A ) ) ) ).
% iso_tuple_UNIV_I
thf(fact_92_top__apply,axiom,
! [C: $tType,D: $tType] :
( ( top @ C @ ( type2 @ C ) )
=> ( ( top_top @ ( D > C ) )
= ( ^ [X3: D] : ( top_top @ C ) ) ) ) ).
% top_apply
thf(fact_93_sp_092_060_094sub_062_092_060mu_062_Osimps_I15_J,axiom,
! [B: $tType,C: $tType,A: $tType,X1: A > ( stream901396144_sp_mu @ A @ B @ C )] :
( ( stream2074884121_sp_mu @ A @ B @ C @ ( stream1294929701mu_Get @ A @ B @ C @ X1 ) )
= ( complete_Sup_Sup @ ( set @ B ) @ ( image @ ( stream901396144_sp_mu @ A @ B @ C ) @ ( set @ B ) @ ( stream2074884121_sp_mu @ A @ B @ C ) @ ( image @ A @ ( stream901396144_sp_mu @ A @ B @ C ) @ X1 @ ( top_top @ ( set @ A ) ) ) ) ) ) ).
% sp\<^sub>\<mu>.simps(15)
thf(fact_94_Union__iff,axiom,
! [A: $tType,A3: A,C2: set @ ( set @ A )] :
( ( member @ A @ A3 @ ( complete_Sup_Sup @ ( set @ A ) @ C2 ) )
= ( ? [X3: set @ A] :
( ( member @ ( set @ A ) @ X3 @ C2 )
& ( member @ A @ A3 @ X3 ) ) ) ) ).
% Union_iff
thf(fact_95_UnionI,axiom,
! [A: $tType,X5: set @ A,C2: set @ ( set @ A ),A3: A] :
( ( member @ ( set @ A ) @ X5 @ C2 )
=> ( ( member @ A @ A3 @ X5 )
=> ( member @ A @ A3 @ ( complete_Sup_Sup @ ( set @ A ) @ C2 ) ) ) ) ).
% UnionI
thf(fact_96_UN__ball__bex__simps_I1_J,axiom,
! [A: $tType,A3: set @ ( set @ A ),P: A > $o] :
( ( ! [X3: A] :
( ( member @ A @ X3 @ ( complete_Sup_Sup @ ( set @ A ) @ A3 ) )
=> ( P @ X3 ) ) )
= ( ! [X3: set @ A] :
( ( member @ ( set @ A ) @ X3 @ A3 )
=> ! [Y2: A] :
( ( member @ A @ Y2 @ X3 )
=> ( P @ Y2 ) ) ) ) ) ).
% UN_ball_bex_simps(1)
thf(fact_97_UN__ball__bex__simps_I3_J,axiom,
! [D: $tType,A3: set @ ( set @ D ),P: D > $o] :
( ( ? [X3: D] :
( ( member @ D @ X3 @ ( complete_Sup_Sup @ ( set @ D ) @ A3 ) )
& ( P @ X3 ) ) )
= ( ? [X3: set @ D] :
( ( member @ ( set @ D ) @ X3 @ A3 )
& ? [Y2: D] :
( ( member @ D @ Y2 @ X3 )
& ( P @ Y2 ) ) ) ) ) ).
% UN_ball_bex_simps(3)
thf(fact_98_UN__ball__bex__simps_I4_J,axiom,
! [F3: $tType,E: $tType,B3: E > ( set @ F3 ),A3: set @ E,P: F3 > $o] :
( ( ? [X3: F3] :
( ( member @ F3 @ X3 @ ( complete_Sup_Sup @ ( set @ F3 ) @ ( image @ E @ ( set @ F3 ) @ B3 @ A3 ) ) )
& ( P @ X3 ) ) )
= ( ? [X3: E] :
( ( member @ E @ X3 @ A3 )
& ? [Y2: F3] :
( ( member @ F3 @ Y2 @ ( B3 @ X3 ) )
& ( P @ Y2 ) ) ) ) ) ).
% UN_ball_bex_simps(4)
thf(fact_99_UN__ball__bex__simps_I2_J,axiom,
! [C: $tType,B: $tType,B3: B > ( set @ C ),A3: set @ B,P: C > $o] :
( ( ! [X3: C] :
( ( member @ C @ X3 @ ( complete_Sup_Sup @ ( set @ C ) @ ( image @ B @ ( set @ C ) @ B3 @ A3 ) ) )
=> ( P @ X3 ) ) )
= ( ! [X3: B] :
( ( member @ B @ X3 @ A3 )
=> ! [Y2: C] :
( ( member @ C @ Y2 @ ( B3 @ X3 ) )
=> ( P @ Y2 ) ) ) ) ) ).
% UN_ball_bex_simps(2)
thf(fact_100_bex__UN,axiom,
! [A: $tType,B: $tType,B3: B > ( set @ A ),A3: set @ B,P: A > $o] :
( ( ? [X3: A] :
( ( member @ A @ X3 @ ( complete_Sup_Sup @ ( set @ A ) @ ( image @ B @ ( set @ A ) @ B3 @ A3 ) ) )
& ( P @ X3 ) ) )
= ( ? [X3: B] :
( ( member @ B @ X3 @ A3 )
& ? [Y2: A] :
( ( member @ A @ Y2 @ ( B3 @ X3 ) )
& ( P @ Y2 ) ) ) ) ) ).
% bex_UN
thf(fact_101_ball__UN,axiom,
! [A: $tType,B: $tType,B3: B > ( set @ A ),A3: set @ B,P: A > $o] :
( ( ! [X3: A] :
( ( member @ A @ X3 @ ( complete_Sup_Sup @ ( set @ A ) @ ( image @ B @ ( set @ A ) @ B3 @ A3 ) ) )
=> ( P @ X3 ) ) )
= ( ! [X3: B] :
( ( member @ B @ X3 @ A3 )
=> ! [Y2: A] :
( ( member @ A @ Y2 @ ( B3 @ X3 ) )
=> ( P @ Y2 ) ) ) ) ) ).
% ball_UN
thf(fact_102_SUP__image,axiom,
! [B: $tType,A: $tType,C: $tType] :
( ( complete_Sup @ A @ ( type2 @ A ) )
=> ! [G3: B > A,F: C > B,A3: set @ C] :
( ( complete_Sup_Sup @ A @ ( image @ B @ A @ G3 @ ( image @ C @ B @ F @ A3 ) ) )
= ( complete_Sup_Sup @ A @ ( image @ C @ A @ ( comp @ B @ A @ C @ G3 @ F ) @ A3 ) ) ) ) ).
% SUP_image
thf(fact_103_Sup__UNIV,axiom,
! [A: $tType] :
( ( comple187826305attice @ A @ ( type2 @ A ) )
=> ( ( complete_Sup_Sup @ A @ ( top_top @ ( set @ A ) ) )
= ( top_top @ A ) ) ) ).
% Sup_UNIV
thf(fact_104_UnionE,axiom,
! [A: $tType,A3: A,C2: set @ ( set @ A )] :
( ( member @ A @ A3 @ ( complete_Sup_Sup @ ( set @ A ) @ C2 ) )
=> ~ ! [X6: set @ A] :
( ( member @ A @ A3 @ X6 )
=> ~ ( member @ ( set @ A ) @ X6 @ C2 ) ) ) ).
% UnionE
thf(fact_105_SUP__cong,axiom,
! [A: $tType,B: $tType] :
( ( complete_Sup @ A @ ( type2 @ A ) )
=> ! [A3: set @ B,B3: set @ B,C2: B > A,D2: B > A] :
( ( A3 = B3 )
=> ( ! [X2: B] :
( ( member @ B @ X2 @ B3 )
=> ( ( C2 @ X2 )
= ( D2 @ X2 ) ) )
=> ( ( complete_Sup_Sup @ A @ ( image @ B @ A @ C2 @ A3 ) )
= ( complete_Sup_Sup @ A @ ( image @ B @ A @ D2 @ B3 ) ) ) ) ) ) ).
% SUP_cong
thf(fact_106_top__set__def,axiom,
! [A: $tType] :
( ( top_top @ ( set @ A ) )
= ( collect @ A @ ( top_top @ ( A > $o ) ) ) ) ).
% top_set_def
thf(fact_107_Union__UNIV,axiom,
! [A: $tType] :
( ( complete_Sup_Sup @ ( set @ A ) @ ( top_top @ ( set @ ( set @ A ) ) ) )
= ( top_top @ ( set @ A ) ) ) ).
% Union_UNIV
thf(fact_108_fun_Omap__comp,axiom,
! [B: $tType,C: $tType,A: $tType,D: $tType,G3: B > C,F: A > B,V: D > A] :
( ( comp @ B @ C @ D @ G3 @ ( comp @ A @ B @ D @ F @ V ) )
= ( comp @ A @ C @ D @ ( comp @ B @ C @ A @ G3 @ F ) @ V ) ) ).
% fun.map_comp
thf(fact_109_sp_092_060_094sub_062_092_060mu_062_Osimps_I17_J,axiom,
! [B: $tType,C: $tType,A: $tType,X1: A > ( stream901396144_sp_mu @ A @ B @ C )] :
( ( stream1259315544_sp_mu @ A @ B @ C @ ( stream1294929701mu_Get @ A @ B @ C @ X1 ) )
= ( complete_Sup_Sup @ ( set @ C ) @ ( image @ ( stream901396144_sp_mu @ A @ B @ C ) @ ( set @ C ) @ ( stream1259315544_sp_mu @ A @ B @ C ) @ ( image @ A @ ( stream901396144_sp_mu @ A @ B @ C ) @ X1 @ ( top_top @ ( set @ A ) ) ) ) ) ) ).
% sp\<^sub>\<mu>.simps(17)
thf(fact_110_member__bind,axiom,
! [A: $tType,B: $tType,X: A,P: set @ B,F: B > ( set @ A )] :
( ( member @ A @ X @ ( bind @ B @ A @ P @ F ) )
= ( member @ A @ X @ ( complete_Sup_Sup @ ( set @ A ) @ ( image @ B @ ( set @ A ) @ F @ P ) ) ) ) ).
% member_bind
thf(fact_111_sp_092_060_094sub_062_092_060mu_062_Osize__gen__o__map,axiom,
! [B: $tType,E: $tType,A: $tType,F3: $tType,C: $tType,F: A > nat,Fa: E > nat,Fb: F3 > nat,Ga: B > E,Gb: C > F3] :
( ( comp @ ( stream901396144_sp_mu @ A @ E @ F3 ) @ nat @ ( stream901396144_sp_mu @ A @ B @ C ) @ ( stream1362355943_sp_mu @ A @ E @ F3 @ F @ Fa @ Fb ) @ ( stream1058219438_sp_mu @ B @ E @ C @ F3 @ A @ Ga @ Gb ) )
= ( stream1362355943_sp_mu @ A @ B @ C @ F @ ( comp @ E @ nat @ B @ Fa @ Ga ) @ ( comp @ F3 @ nat @ C @ Fb @ Gb ) ) ) ).
% sp\<^sub>\<mu>.size_gen_o_map
thf(fact_112_surj__swap__iff,axiom,
! [B: $tType,A: $tType,A2: B,B2: B,F: B > A] :
( ( ( image @ B @ A @ ( swap @ B @ A @ A2 @ B2 @ F ) @ ( top_top @ ( set @ B ) ) )
= ( top_top @ ( set @ A ) ) )
= ( ( image @ B @ A @ F @ ( top_top @ ( set @ B ) ) )
= ( top_top @ ( set @ A ) ) ) ) ).
% surj_swap_iff
thf(fact_113_ball__reg,axiom,
! [A: $tType,R: set @ A,P: A > $o,Q: A > $o] :
( ! [X2: A] :
( ( member @ A @ X2 @ R )
=> ( ( P @ X2 )
=> ( Q @ X2 ) ) )
=> ( ! [X2: A] :
( ( member @ A @ X2 @ R )
=> ( P @ X2 ) )
=> ! [X4: A] :
( ( member @ A @ X4 @ R )
=> ( Q @ X4 ) ) ) ) ).
% ball_reg
thf(fact_114_comp__cong,axiom,
! [C: $tType,B: $tType,D: $tType,A: $tType,E: $tType,F: B > A,G3: C > B,X: C,F5: D > A,G5: E > D,X7: E] :
( ( ( F @ ( G3 @ X ) )
= ( F5 @ ( G5 @ X7 ) ) )
=> ( ( comp @ B @ A @ C @ F @ G3 @ X )
= ( comp @ D @ A @ E @ F5 @ G5 @ X7 ) ) ) ).
% comp_cong
thf(fact_115_swap__apply_I3_J,axiom,
! [A: $tType,B: $tType,C3: B,A2: B,B2: B,F: B > A] :
( ( C3 != A2 )
=> ( ( C3 != B2 )
=> ( ( swap @ B @ A @ A2 @ B2 @ F @ C3 )
= ( F @ C3 ) ) ) ) ).
% swap_apply(3)
thf(fact_116_swap__apply_I2_J,axiom,
! [A: $tType,B: $tType,A2: B,B2: B,F: B > A] :
( ( swap @ B @ A @ A2 @ B2 @ F @ B2 )
= ( F @ A2 ) ) ).
% swap_apply(2)
thf(fact_117_swap__apply_I1_J,axiom,
! [A: $tType,B: $tType,A2: B,B2: B,F: B > A] :
( ( swap @ B @ A @ A2 @ B2 @ F @ A2 )
= ( F @ B2 ) ) ).
% swap_apply(1)
thf(fact_118_swap__self,axiom,
! [B: $tType,A: $tType,A2: A,F: A > B] :
( ( swap @ A @ B @ A2 @ A2 @ F )
= F ) ).
% swap_self
thf(fact_119_swap__nilpotent,axiom,
! [B: $tType,A: $tType,A2: A,B2: A,F: A > B] :
( ( swap @ A @ B @ A2 @ B2 @ ( swap @ A @ B @ A2 @ B2 @ F ) )
= F ) ).
% swap_nilpotent
thf(fact_120_top1I,axiom,
! [A: $tType,X: A] : ( top_top @ ( A > $o ) @ X ) ).
% top1I
thf(fact_121_swap__image__eq,axiom,
! [B: $tType,A: $tType,A2: A,A3: set @ A,B2: A,F: A > B] :
( ( member @ A @ A2 @ A3 )
=> ( ( member @ A @ B2 @ A3 )
=> ( ( image @ A @ B @ ( swap @ A @ B @ A2 @ B2 @ F ) @ A3 )
= ( image @ A @ B @ F @ A3 ) ) ) ) ).
% swap_image_eq
thf(fact_122_swap__triple,axiom,
! [B: $tType,A: $tType,A2: A,C3: A,B2: A,F: A > B] :
( ( A2 != C3 )
=> ( ( B2 != C3 )
=> ( ( swap @ A @ B @ A2 @ B2 @ ( swap @ A @ B @ B2 @ C3 @ ( swap @ A @ B @ A2 @ B2 @ F ) ) )
= ( swap @ A @ B @ A2 @ C3 @ F ) ) ) ) ).
% swap_triple
thf(fact_123_swap__commute,axiom,
! [B: $tType,A: $tType] :
( ( swap @ A @ B )
= ( ^ [A5: A,B4: A] : ( swap @ A @ B @ B4 @ A5 ) ) ) ).
% swap_commute
thf(fact_124_comp__swap,axiom,
! [B: $tType,C: $tType,A: $tType,F: C > B,A2: A,B2: A,G3: A > C] :
( ( comp @ C @ B @ A @ F @ ( swap @ A @ C @ A2 @ B2 @ G3 ) )
= ( swap @ A @ B @ A2 @ B2 @ ( comp @ C @ B @ A @ F @ G3 ) ) ) ).
% comp_swap
thf(fact_125_bind__image,axiom,
! [A: $tType,B: $tType,C: $tType,F: C > B,A3: set @ C,G3: B > ( set @ A )] :
( ( bind @ B @ A @ ( image @ C @ B @ F @ A3 ) @ G3 )
= ( bind @ C @ A @ A3 @ ( comp @ B @ ( set @ A ) @ C @ G3 @ F ) ) ) ).
% bind_image
thf(fact_126_image__bind,axiom,
! [A: $tType,B: $tType,C: $tType,F: B > A,A3: set @ C,G3: C > ( set @ B )] :
( ( image @ B @ A @ F @ ( bind @ C @ B @ A3 @ G3 ) )
= ( bind @ C @ A @ A3 @ ( comp @ ( set @ B ) @ ( set @ A ) @ C @ ( image @ B @ A @ F ) @ G3 ) ) ) ).
% image_bind
thf(fact_127_surj__imp__surj__swap,axiom,
! [B: $tType,A: $tType,F: B > A,A2: B,B2: B] :
( ( ( image @ B @ A @ F @ ( top_top @ ( set @ B ) ) )
= ( top_top @ ( set @ A ) ) )
=> ( ( image @ B @ A @ ( swap @ B @ A @ A2 @ B2 @ F ) @ ( top_top @ ( set @ B ) ) )
= ( top_top @ ( set @ A ) ) ) ) ).
% surj_imp_surj_swap
thf(fact_128_bind__UNION,axiom,
! [A: $tType,B: $tType] :
( ( bind @ B @ A )
= ( ^ [A4: set @ B,F4: B > ( set @ A )] : ( complete_Sup_Sup @ ( set @ A ) @ ( image @ B @ ( set @ A ) @ F4 @ A4 ) ) ) ) ).
% bind_UNION
thf(fact_129_top__empty__eq,axiom,
! [A: $tType] :
( ( top_top @ ( A > $o ) )
= ( ^ [X3: A] : ( member @ A @ X3 @ ( top_top @ ( set @ A ) ) ) ) ) ).
% top_empty_eq
thf(fact_130_top__conj_I1_J,axiom,
! [A: $tType,X: A,P: $o] :
( ( ( top_top @ ( A > $o ) @ X )
& P )
= P ) ).
% top_conj(1)
thf(fact_131_top__conj_I2_J,axiom,
! [A: $tType,P: $o,X: A] :
( ( P
& ( top_top @ ( A > $o ) @ X ) )
= P ) ).
% top_conj(2)
thf(fact_132_Pow__UNIV,axiom,
! [A: $tType] :
( ( pow @ A @ ( top_top @ ( set @ A ) ) )
= ( top_top @ ( set @ ( set @ A ) ) ) ) ).
% Pow_UNIV
thf(fact_133_Union__Pow__eq,axiom,
! [A: $tType,A3: set @ A] :
( ( complete_Sup_Sup @ ( set @ A ) @ ( pow @ A @ A3 ) )
= A3 ) ).
% Union_Pow_eq
thf(fact_134_image__Pow__surj,axiom,
! [B: $tType,A: $tType,F: B > A,A3: set @ B,B3: set @ A] :
( ( ( image @ B @ A @ F @ A3 )
= B3 )
=> ( ( image @ ( set @ B ) @ ( set @ A ) @ ( image @ B @ A @ F ) @ ( pow @ B @ A3 ) )
= ( pow @ A @ B3 ) ) ) ).
% image_Pow_surj
thf(fact_135_Cantors__paradox,axiom,
! [A: $tType,A3: set @ A] :
~ ? [X4: A > ( set @ A )] :
( ( image @ A @ ( set @ A ) @ X4 @ A3 )
= ( pow @ A @ A3 ) ) ).
% Cantors_paradox
thf(fact_136_Pow__top,axiom,
! [A: $tType,A3: set @ A] : ( member @ ( set @ A ) @ A3 @ ( pow @ A @ A3 ) ) ).
% Pow_top
thf(fact_137_rewriteL__comp__comp,axiom,
! [C: $tType,B: $tType,A: $tType,D: $tType,F: C > B,G3: A > C,L: A > B,H: D > A] :
( ( ( comp @ C @ B @ A @ F @ G3 )
= L )
=> ( ( comp @ C @ B @ D @ F @ ( comp @ A @ C @ D @ G3 @ H ) )
= ( comp @ A @ B @ D @ L @ H ) ) ) ).
% rewriteL_comp_comp
thf(fact_138_rewriteR__comp__comp,axiom,
! [C: $tType,D: $tType,B: $tType,A: $tType,G3: C > B,H: A > C,R3: A > B,F: B > D] :
( ( ( comp @ C @ B @ A @ G3 @ H )
= R3 )
=> ( ( comp @ C @ D @ A @ ( comp @ B @ D @ C @ F @ G3 ) @ H )
= ( comp @ B @ D @ A @ F @ R3 ) ) ) ).
% rewriteR_comp_comp
thf(fact_139_rewriteL__comp__comp2,axiom,
! [A: $tType,C: $tType,B: $tType,D: $tType,E: $tType,F: C > B,G3: A > C,L1: D > B,L2: A > D,H: E > A,R3: E > D] :
( ( ( comp @ C @ B @ A @ F @ G3 )
= ( comp @ D @ B @ A @ L1 @ L2 ) )
=> ( ( ( comp @ A @ D @ E @ L2 @ H )
= R3 )
=> ( ( comp @ C @ B @ E @ F @ ( comp @ A @ C @ E @ G3 @ H ) )
= ( comp @ D @ B @ E @ L1 @ R3 ) ) ) ) ).
% rewriteL_comp_comp2
thf(fact_140_rewriteR__comp__comp2,axiom,
! [C: $tType,B: $tType,E: $tType,D: $tType,A: $tType,G3: C > B,H: A > C,R12: D > B,R22: A > D,F: B > E,L: D > E] :
( ( ( comp @ C @ B @ A @ G3 @ H )
= ( comp @ D @ B @ A @ R12 @ R22 ) )
=> ( ( ( comp @ B @ E @ D @ F @ R12 )
= L )
=> ( ( comp @ C @ E @ A @ ( comp @ B @ E @ C @ F @ G3 ) @ H )
= ( comp @ D @ E @ A @ L @ R22 ) ) ) ) ).
% rewriteR_comp_comp2
thf(fact_141_Sup__SUP__eq,axiom,
! [A: $tType] :
( ( complete_Sup_Sup @ ( A > $o ) )
= ( ^ [S: set @ ( A > $o ),X3: A] : ( member @ A @ X3 @ ( complete_Sup_Sup @ ( set @ A ) @ ( image @ ( A > $o ) @ ( set @ A ) @ ( collect @ A ) @ S ) ) ) ) ) ).
% Sup_SUP_eq
thf(fact_142_type__copy__set__map0,axiom,
! [A: $tType,B: $tType,D: $tType,E: $tType,C: $tType,F3: $tType,Rep: A > B,Abs: B > A,S2: B > ( set @ D ),M: C > B,F: E > D,S3: C > ( set @ E ),G3: F3 > 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 @ F ) @ S3 ) )
=> ( ( comp @ A @ ( set @ D ) @ F3 @ ( comp @ B @ ( set @ D ) @ A @ S2 @ Rep ) @ ( comp @ C @ A @ F3 @ ( comp @ B @ A @ C @ Abs @ M ) @ G3 ) )
= ( comp @ ( set @ E ) @ ( set @ D ) @ F3 @ ( image @ E @ D @ F ) @ ( comp @ C @ ( set @ E ) @ F3 @ S3 @ G3 ) ) ) ) ) ).
% type_copy_set_map0
thf(fact_143_type__copy__map__cong0,axiom,
! [B: $tType,D: $tType,E: $tType,A: $tType,C: $tType,M: B > A,G3: C > B,X: C,N: D > A,H: C > D,F: A > E] :
( ( ( M @ ( G3 @ X ) )
= ( N @ ( H @ X ) ) )
=> ( ( comp @ B @ E @ C @ ( comp @ A @ E @ B @ F @ M ) @ G3 @ X )
= ( comp @ D @ E @ C @ ( comp @ A @ E @ D @ F @ N ) @ H @ X ) ) ) ).
% type_copy_map_cong0
thf(fact_144_sp_092_060_094sub_062_092_060nu_062_Orel__inject,axiom,
! [B: $tType,D: $tType,A: $tType,R: B > D > $o,X: stream901396144_sp_mu @ A @ B @ ( stream1273403375_sp_nu @ A @ B ),Y: stream901396144_sp_mu @ A @ D @ ( stream1273403375_sp_nu @ A @ D )] :
( ( stream1017682031_sp_nu @ B @ D @ A @ R @ ( stream1686501333_nu_In @ A @ B @ X ) @ ( stream1686501333_nu_In @ A @ D @ Y ) )
= ( stream1924447089_sp_mu @ B @ D @ ( stream1273403375_sp_nu @ A @ B ) @ ( stream1273403375_sp_nu @ A @ D ) @ A @ R @ ( stream1017682031_sp_nu @ B @ D @ A @ R ) @ X @ Y ) ) ).
% sp\<^sub>\<nu>.rel_inject
thf(fact_145_sp_092_060_094sub_062_092_060mu_062_Orel__induct,axiom,
! [B: $tType,C: $tType,F3: $tType,E: $tType,A: $tType,R2: B > E > $o,R32: C > F3 > $o,X: stream901396144_sp_mu @ A @ B @ C,Y: stream901396144_sp_mu @ A @ E @ F3,Q: ( stream901396144_sp_mu @ A @ B @ C ) > ( stream901396144_sp_mu @ A @ E @ F3 ) > $o] :
( ( stream1924447089_sp_mu @ B @ E @ C @ F3 @ A @ R2 @ R32 @ X @ Y )
=> ( ! [A12: A > ( stream901396144_sp_mu @ A @ B @ C ),B1: A > ( stream901396144_sp_mu @ A @ E @ F3 )] :
( ( bNF_rel_fun @ A @ A @ ( stream901396144_sp_mu @ A @ B @ C ) @ ( stream901396144_sp_mu @ A @ E @ F3 )
@ ^ [Y3: A,Z: A] : Y3 = Z
@ Q
@ A12
@ B1 )
=> ( Q @ ( stream1294929701mu_Get @ A @ B @ C @ A12 ) @ ( stream1294929701mu_Get @ A @ E @ F3 @ B1 ) ) )
=> ( ! [A21: B,A222: C,B21: E,B22: F3] :
( ( R2 @ A21 @ B21 )
=> ( ( R32 @ A222 @ B22 )
=> ( Q @ ( stream1370332830mu_Put @ B @ C @ A @ A21 @ A222 ) @ ( stream1370332830mu_Put @ E @ F3 @ A @ B21 @ B22 ) ) ) )
=> ( Q @ X @ Y ) ) ) ) ).
% sp\<^sub>\<mu>.rel_induct
thf(fact_146_rel__funI,axiom,
! [A: $tType,C: $tType,D: $tType,B: $tType,A3: A > B > $o,B3: C > D > $o,F: A > C,G3: B > D] :
( ! [X2: A,Y4: B] :
( ( A3 @ X2 @ Y4 )
=> ( B3 @ ( F @ X2 ) @ ( G3 @ Y4 ) ) )
=> ( bNF_rel_fun @ A @ B @ C @ D @ A3 @ B3 @ F @ G3 ) ) ).
% rel_funI
thf(fact_147_Sup1__I,axiom,
! [A: $tType,P: A > $o,A3: set @ ( A > $o ),A2: A] :
( ( member @ ( A > $o ) @ P @ A3 )
=> ( ( P @ A2 )
=> ( complete_Sup_Sup @ ( A > $o ) @ A3 @ A2 ) ) ) ).
% Sup1_I
thf(fact_148_o__rsp_I2_J,axiom,
! [E: $tType,F3: $tType,H2: $tType,G: $tType,R1: E > F3 > $o] :
( bNF_rel_fun @ ( G > H2 ) @ ( G > H2 ) @ ( ( E > G ) > E > H2 ) @ ( ( F3 > G ) > F3 > H2 )
@ ^ [Y3: G > H2,Z: G > H2] : Y3 = Z
@ ( bNF_rel_fun @ ( E > G ) @ ( F3 > G ) @ ( E > H2 ) @ ( F3 > H2 )
@ ( bNF_rel_fun @ E @ F3 @ G @ G @ R1
@ ^ [Y3: G,Z: G] : Y3 = Z )
@ ( bNF_rel_fun @ E @ F3 @ H2 @ H2 @ R1
@ ^ [Y3: H2,Z: H2] : Y3 = Z ) )
@ ( comp @ G @ H2 @ E )
@ ( comp @ G @ H2 @ F3 ) ) ).
% o_rsp(2)
thf(fact_149_o__rsp_I1_J,axiom,
! [A: $tType,B: $tType,E: $tType,F3: $tType,D: $tType,C: $tType,R2: A > C > $o,R32: B > D > $o,R1: E > F3 > $o] : ( bNF_rel_fun @ ( A > B ) @ ( C > D ) @ ( ( E > A ) > E > B ) @ ( ( F3 > C ) > F3 > D ) @ ( bNF_rel_fun @ A @ C @ B @ D @ R2 @ R32 ) @ ( bNF_rel_fun @ ( E > A ) @ ( F3 > C ) @ ( E > B ) @ ( F3 > D ) @ ( bNF_rel_fun @ E @ F3 @ A @ C @ R1 @ R2 ) @ ( bNF_rel_fun @ E @ F3 @ B @ D @ R1 @ R32 ) ) @ ( comp @ A @ B @ E ) @ ( comp @ C @ D @ F3 ) ) ).
% o_rsp(1)
thf(fact_150_fun_Omap__transfer,axiom,
! [A: $tType,B: $tType,D: $tType,G: $tType,F3: $tType,Rb: A > F3 > $o,Sd: B > G > $o] :
( bNF_rel_fun @ ( A > B ) @ ( F3 > G ) @ ( ( D > A ) > D > B ) @ ( ( D > F3 ) > D > G ) @ ( bNF_rel_fun @ A @ F3 @ B @ G @ Rb @ Sd )
@ ( bNF_rel_fun @ ( D > A ) @ ( D > F3 ) @ ( D > B ) @ ( D > G )
@ ( bNF_rel_fun @ D @ D @ A @ F3
@ ^ [Y3: D,Z: D] : Y3 = Z
@ Rb )
@ ( bNF_rel_fun @ D @ D @ B @ G
@ ^ [Y3: D,Z: D] : Y3 = Z
@ Sd ) )
@ ( comp @ A @ B @ D )
@ ( comp @ F3 @ G @ D ) ) ).
% fun.map_transfer
thf(fact_151_comp__transfer,axiom,
! [A: $tType,B: $tType,E: $tType,F3: $tType,D: $tType,C: $tType,B3: A > C > $o,C2: B > D > $o,A3: E > F3 > $o] : ( bNF_rel_fun @ ( A > B ) @ ( C > D ) @ ( ( E > A ) > E > B ) @ ( ( F3 > C ) > F3 > D ) @ ( bNF_rel_fun @ A @ C @ B @ D @ B3 @ C2 ) @ ( bNF_rel_fun @ ( E > A ) @ ( F3 > C ) @ ( E > B ) @ ( F3 > D ) @ ( bNF_rel_fun @ E @ F3 @ A @ C @ A3 @ B3 ) @ ( bNF_rel_fun @ E @ F3 @ B @ D @ A3 @ C2 ) ) @ ( comp @ A @ B @ E ) @ ( comp @ C @ D @ F3 ) ) ).
% comp_transfer
thf(fact_152_type__copy__obj__one__point__absE,axiom,
! [A: $tType,B: $tType,Rep: A > B,Abs: B > A,S4: A] :
( ( type_definition @ A @ B @ Rep @ Abs @ ( top_top @ ( set @ B ) ) )
=> ~ ! [X2: B] :
( S4
!= ( Abs @ X2 ) ) ) ).
% type_copy_obj_one_point_absE
thf(fact_153_type__copy__map__comp0,axiom,
! [F3: $tType,D: $tType,B: $tType,A: $tType,C: $tType,E: $tType,Rep: A > B,Abs: B > A,M: C > D,M1: B > D,M2: C > B,F: D > F3,G3: E > C] :
( ( type_definition @ A @ B @ Rep @ Abs @ ( top_top @ ( set @ B ) ) )
=> ( ( M
= ( comp @ B @ D @ C @ M1 @ M2 ) )
=> ( ( comp @ C @ F3 @ E @ ( comp @ D @ F3 @ C @ F @ M ) @ G3 )
= ( comp @ A @ F3 @ E @ ( comp @ B @ F3 @ A @ ( comp @ D @ F3 @ B @ F @ M1 ) @ Rep ) @ ( comp @ C @ A @ E @ ( comp @ B @ A @ C @ Abs @ M2 ) @ G3 ) ) ) ) ) ).
% type_copy_map_comp0
thf(fact_154_fun_Orel__eq,axiom,
! [A: $tType,D: $tType] :
( ( bNF_rel_fun @ D @ D @ A @ A
@ ^ [Y3: D,Z: D] : Y3 = Z
@ ^ [Y3: A,Z: A] : Y3 = Z )
= ( ^ [Y3: D > A,Z: D > A] : Y3 = Z ) ) ).
% fun.rel_eq
thf(fact_155_fun_Orel__refl,axiom,
! [B: $tType,D: $tType,Ra: B > B > $o,X: D > B] :
( ! [X2: B] : ( Ra @ X2 @ X2 )
=> ( bNF_rel_fun @ D @ D @ B @ B
@ ^ [Y3: D,Z: D] : Y3 = Z
@ Ra
@ X
@ X ) ) ).
% fun.rel_refl
thf(fact_156_fun_Orel__transfer,axiom,
! [B: $tType,A: $tType,C: $tType,E: $tType,D: $tType,Sa: A > C > $o,Sc: B > E > $o] :
( bNF_rel_fun @ ( A > B > $o ) @ ( C > E > $o ) @ ( ( D > A ) > ( D > B ) > $o ) @ ( ( D > C ) > ( D > E ) > $o )
@ ( bNF_rel_fun @ A @ C @ ( B > $o ) @ ( E > $o ) @ Sa
@ ( bNF_rel_fun @ B @ E @ $o @ $o @ Sc
@ ^ [Y3: $o,Z: $o] : Y3 = Z ) )
@ ( bNF_rel_fun @ ( D > A ) @ ( D > C ) @ ( ( D > B ) > $o ) @ ( ( D > E ) > $o )
@ ( bNF_rel_fun @ D @ D @ A @ C
@ ^ [Y3: D,Z: D] : Y3 = Z
@ Sa )
@ ( bNF_rel_fun @ ( D > B ) @ ( D > E ) @ $o @ $o
@ ( bNF_rel_fun @ D @ D @ B @ E
@ ^ [Y3: D,Z: D] : Y3 = Z
@ Sc )
@ ^ [Y3: $o,Z: $o] : Y3 = Z ) )
@ ( bNF_rel_fun @ D @ D @ A @ B
@ ^ [Y3: D,Z: D] : Y3 = Z )
@ ( bNF_rel_fun @ D @ D @ C @ E
@ ^ [Y3: D,Z: D] : Y3 = Z ) ) ).
% fun.rel_transfer
thf(fact_157_sp_092_060_094sub_062_092_060mu_062_Octr__transfer_I1_J,axiom,
! [B: $tType,C: $tType,F3: $tType,E: $tType,A: $tType,R1: B > E > $o,R2: C > F3 > $o] :
( bNF_rel_fun @ ( A > ( stream901396144_sp_mu @ A @ B @ C ) ) @ ( A > ( stream901396144_sp_mu @ A @ E @ F3 ) ) @ ( stream901396144_sp_mu @ A @ B @ C ) @ ( stream901396144_sp_mu @ A @ E @ F3 )
@ ( bNF_rel_fun @ A @ A @ ( stream901396144_sp_mu @ A @ B @ C ) @ ( stream901396144_sp_mu @ A @ E @ F3 )
@ ^ [Y3: A,Z: A] : Y3 = Z
@ ( stream1924447089_sp_mu @ B @ E @ C @ F3 @ A @ R1 @ R2 ) )
@ ( stream1924447089_sp_mu @ B @ E @ C @ F3 @ A @ R1 @ R2 )
@ ( stream1294929701mu_Get @ A @ B @ C )
@ ( stream1294929701mu_Get @ A @ E @ F3 ) ) ).
% sp\<^sub>\<mu>.ctr_transfer(1)
thf(fact_158_sp_092_060_094sub_062_092_060nu_062_Orel__refl,axiom,
! [B: $tType,D: $tType,Ra: B > B > $o,X: stream1273403375_sp_nu @ D @ B] :
( ! [X2: B] : ( Ra @ X2 @ X2 )
=> ( stream1017682031_sp_nu @ B @ B @ D @ Ra @ X @ X ) ) ).
% sp\<^sub>\<nu>.rel_refl
thf(fact_159_sp_092_060_094sub_062_092_060nu_062_Orel__eq,axiom,
! [A: $tType,D: $tType] :
( ( stream1017682031_sp_nu @ A @ A @ D
@ ^ [Y3: A,Z: A] : Y3 = Z )
= ( ^ [Y3: stream1273403375_sp_nu @ D @ A,Z: stream1273403375_sp_nu @ D @ A] : Y3 = Z ) ) ).
% sp\<^sub>\<nu>.rel_eq
thf(fact_160_rel__funD,axiom,
! [A: $tType,C: $tType,D: $tType,B: $tType,A3: A > B > $o,B3: C > D > $o,F: A > C,G3: B > D,X: A,Y: B] :
( ( bNF_rel_fun @ A @ B @ C @ D @ A3 @ B3 @ F @ G3 )
=> ( ( A3 @ X @ Y )
=> ( B3 @ ( F @ X ) @ ( G3 @ Y ) ) ) ) ).
% rel_funD
thf(fact_161_rel__fun__mono,axiom,
! [A: $tType,C: $tType,D: $tType,B: $tType,X5: A > B > $o,A3: C > D > $o,F: A > C,G3: B > D,Y5: A > B > $o,B3: C > D > $o] :
( ( bNF_rel_fun @ A @ B @ C @ D @ X5 @ A3 @ F @ G3 )
=> ( ! [X2: A,Y4: B] :
( ( Y5 @ X2 @ Y4 )
=> ( X5 @ X2 @ Y4 ) )
=> ( ! [X2: C,Y4: D] :
( ( A3 @ X2 @ Y4 )
=> ( B3 @ X2 @ Y4 ) )
=> ( bNF_rel_fun @ A @ B @ C @ D @ Y5 @ B3 @ F @ G3 ) ) ) ) ).
% rel_fun_mono
thf(fact_162_rel__fun__mono_H,axiom,
! [A: $tType,C: $tType,D: $tType,B: $tType,Y5: A > B > $o,X5: A > B > $o,A3: C > D > $o,B3: C > D > $o,F: A > C,G3: B > D] :
( ! [X2: A,Y4: B] :
( ( Y5 @ X2 @ Y4 )
=> ( X5 @ X2 @ Y4 ) )
=> ( ! [X2: C,Y4: D] :
( ( A3 @ X2 @ Y4 )
=> ( B3 @ X2 @ Y4 ) )
=> ( ( bNF_rel_fun @ A @ B @ C @ D @ X5 @ A3 @ F @ G3 )
=> ( bNF_rel_fun @ A @ B @ C @ D @ Y5 @ B3 @ F @ G3 ) ) ) ) ).
% rel_fun_mono'
thf(fact_163_sp_092_060_094sub_062_092_060mu_062_Ocase__transfer,axiom,
! [B: $tType,C: $tType,G: $tType,H2: $tType,F3: $tType,E: $tType,A: $tType,R1: B > E > $o,R2: C > F3 > $o,S2: G > H2 > $o] :
( bNF_rel_fun @ ( ( A > ( stream901396144_sp_mu @ A @ B @ C ) ) > G ) @ ( ( A > ( stream901396144_sp_mu @ A @ E @ F3 ) ) > H2 ) @ ( ( B > C > G ) > ( stream901396144_sp_mu @ A @ B @ C ) > G ) @ ( ( E > F3 > H2 ) > ( stream901396144_sp_mu @ A @ E @ F3 ) > H2 )
@ ( bNF_rel_fun @ ( A > ( stream901396144_sp_mu @ A @ B @ C ) ) @ ( A > ( stream901396144_sp_mu @ A @ E @ F3 ) ) @ G @ H2
@ ( bNF_rel_fun @ A @ A @ ( stream901396144_sp_mu @ A @ B @ C ) @ ( stream901396144_sp_mu @ A @ E @ F3 )
@ ^ [Y3: A,Z: A] : Y3 = Z
@ ( stream1924447089_sp_mu @ B @ E @ C @ F3 @ A @ R1 @ R2 ) )
@ S2 )
@ ( bNF_rel_fun @ ( B > C > G ) @ ( E > F3 > H2 ) @ ( ( stream901396144_sp_mu @ A @ B @ C ) > G ) @ ( ( stream901396144_sp_mu @ A @ E @ F3 ) > H2 ) @ ( bNF_rel_fun @ B @ E @ ( C > G ) @ ( F3 > H2 ) @ R1 @ ( bNF_rel_fun @ C @ F3 @ G @ H2 @ R2 @ S2 ) ) @ ( bNF_rel_fun @ ( stream901396144_sp_mu @ A @ B @ C ) @ ( stream901396144_sp_mu @ A @ E @ F3 ) @ G @ H2 @ ( stream1924447089_sp_mu @ B @ E @ C @ F3 @ A @ R1 @ R2 ) @ S2 ) )
@ ( stream160000856_sp_mu @ A @ B @ C @ G )
@ ( stream160000856_sp_mu @ A @ E @ F3 @ H2 ) ) ).
% sp\<^sub>\<mu>.case_transfer
thf(fact_164_sp_092_060_094sub_062_092_060mu_062_Omap__transfer,axiom,
! [A: $tType,C: $tType,B: $tType,D: $tType,G: $tType,M3: $tType,K2: $tType,L3: $tType,J: $tType,R1b: A > J > $o,S1d: C > L3 > $o,R2b: B > K2 > $o,S2d: D > M3 > $o] : ( bNF_rel_fun @ ( A > C ) @ ( J > L3 ) @ ( ( B > D ) > ( stream901396144_sp_mu @ G @ A @ B ) > ( stream901396144_sp_mu @ G @ C @ D ) ) @ ( ( K2 > M3 ) > ( stream901396144_sp_mu @ G @ J @ K2 ) > ( stream901396144_sp_mu @ G @ L3 @ M3 ) ) @ ( bNF_rel_fun @ A @ J @ C @ L3 @ R1b @ S1d ) @ ( bNF_rel_fun @ ( B > D ) @ ( K2 > M3 ) @ ( ( stream901396144_sp_mu @ G @ A @ B ) > ( stream901396144_sp_mu @ G @ C @ D ) ) @ ( ( stream901396144_sp_mu @ G @ J @ K2 ) > ( stream901396144_sp_mu @ G @ L3 @ M3 ) ) @ ( bNF_rel_fun @ B @ K2 @ D @ M3 @ R2b @ S2d ) @ ( bNF_rel_fun @ ( stream901396144_sp_mu @ G @ A @ B ) @ ( stream901396144_sp_mu @ G @ J @ K2 ) @ ( stream901396144_sp_mu @ G @ C @ D ) @ ( stream901396144_sp_mu @ G @ L3 @ M3 ) @ ( stream1924447089_sp_mu @ A @ J @ B @ K2 @ G @ R1b @ R2b ) @ ( stream1924447089_sp_mu @ C @ L3 @ D @ M3 @ G @ S1d @ S2d ) ) ) @ ( stream1058219438_sp_mu @ A @ C @ B @ D @ G ) @ ( stream1058219438_sp_mu @ J @ L3 @ K2 @ M3 @ G ) ) ).
% sp\<^sub>\<mu>.map_transfer
thf(fact_165_sp_092_060_094sub_062_092_060nu_062_Osel__transfer,axiom,
! [B: $tType,D: $tType,A: $tType,R: B > D > $o] : ( bNF_rel_fun @ ( stream1273403375_sp_nu @ A @ B ) @ ( stream1273403375_sp_nu @ A @ D ) @ ( stream901396144_sp_mu @ A @ B @ ( stream1273403375_sp_nu @ A @ B ) ) @ ( stream901396144_sp_mu @ A @ D @ ( stream1273403375_sp_nu @ A @ D ) ) @ ( stream1017682031_sp_nu @ B @ D @ A @ R ) @ ( stream1924447089_sp_mu @ B @ D @ ( stream1273403375_sp_nu @ A @ B ) @ ( stream1273403375_sp_nu @ A @ D ) @ A @ R @ ( stream1017682031_sp_nu @ B @ D @ A @ R ) ) @ ( stream1200245950nu_out @ A @ B ) @ ( stream1200245950nu_out @ A @ D ) ) ).
% sp\<^sub>\<nu>.sel_transfer
thf(fact_166_Sup1__E,axiom,
! [A: $tType,A3: set @ ( A > $o ),A2: A] :
( ( complete_Sup_Sup @ ( A > $o ) @ A3 @ A2 )
=> ~ ! [P4: A > $o] :
( ( member @ ( A > $o ) @ P4 @ A3 )
=> ~ ( P4 @ A2 ) ) ) ).
% Sup1_E
thf(fact_167_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 ) ) )
=> ( ( ^ [P5: B > $o] :
? [X8: B] : ( P5 @ X8 )
@ F6 )
= ( ? [B4: A] : ( F6 @ ( Rep @ B4 ) ) ) ) ) ).
% type_copy_ex_RepI
thf(fact_168_type__copy__wit,axiom,
! [A: $tType,C: $tType,B: $tType,Rep: A > B,Abs: B > A,X: C,S2: B > ( set @ C ),Y: B] :
( ( type_definition @ A @ B @ Rep @ Abs @ ( top_top @ ( set @ B ) ) )
=> ( ( member @ C @ X @ ( comp @ B @ ( set @ C ) @ A @ S2 @ Rep @ ( Abs @ Y ) ) )
=> ( member @ C @ X @ ( S2 @ Y ) ) ) ) ).
% type_copy_wit
thf(fact_169_fun_Orel__cong,axiom,
! [A: $tType,B: $tType,D: $tType,X: D > A,Ya: D > A,Y: D > B,Xa: D > B,R: A > B > $o,Ra: A > B > $o] :
( ( X = Ya )
=> ( ( Y = Xa )
=> ( ! [Z3: A,Yb: B] :
( ( member @ A @ Z3 @ ( image @ D @ A @ Ya @ ( top_top @ ( set @ D ) ) ) )
=> ( ( member @ B @ Yb @ ( image @ D @ B @ Xa @ ( top_top @ ( set @ D ) ) ) )
=> ( ( R @ Z3 @ Yb )
= ( Ra @ Z3 @ Yb ) ) ) )
=> ( ( bNF_rel_fun @ D @ D @ A @ B
@ ^ [Y3: D,Z: D] : Y3 = Z
@ R
@ X
@ Y )
= ( bNF_rel_fun @ D @ D @ A @ B
@ ^ [Y3: D,Z: D] : Y3 = Z
@ Ra
@ Ya
@ Xa ) ) ) ) ) ).
% fun.rel_cong
thf(fact_170_fun_Orel__mono__strong,axiom,
! [A: $tType,B: $tType,D: $tType,R: A > B > $o,X: D > A,Y: D > B,Ra: A > B > $o] :
( ( bNF_rel_fun @ D @ D @ A @ B
@ ^ [Y3: D,Z: D] : Y3 = Z
@ R
@ X
@ Y )
=> ( ! [Z3: A,Yb: B] :
( ( member @ A @ Z3 @ ( image @ D @ A @ X @ ( top_top @ ( set @ D ) ) ) )
=> ( ( member @ B @ Yb @ ( image @ D @ B @ Y @ ( top_top @ ( set @ D ) ) ) )
=> ( ( R @ Z3 @ Yb )
=> ( Ra @ Z3 @ Yb ) ) ) )
=> ( bNF_rel_fun @ D @ D @ A @ B
@ ^ [Y3: D,Z: D] : Y3 = Z
@ Ra
@ X
@ Y ) ) ) ).
% fun.rel_mono_strong
thf(fact_171_fun_Orel__refl__strong,axiom,
! [A: $tType,B: $tType,X: B > A,Ra: A > A > $o] :
( ! [Z3: A] :
( ( member @ A @ Z3 @ ( image @ B @ A @ X @ ( top_top @ ( set @ B ) ) ) )
=> ( Ra @ Z3 @ Z3 ) )
=> ( bNF_rel_fun @ B @ B @ A @ A
@ ^ [Y3: B,Z: B] : Y3 = Z
@ Ra
@ X
@ X ) ) ).
% fun.rel_refl_strong
thf(fact_172_sp_092_060_094sub_062_092_060mu_062_Orel__intros_I1_J,axiom,
! [B: $tType,C: $tType,F3: $tType,E: $tType,A: $tType,R1: B > E > $o,R2: C > F3 > $o,X1: A > ( stream901396144_sp_mu @ A @ B @ C ),Y1: A > ( stream901396144_sp_mu @ A @ E @ F3 )] :
( ( bNF_rel_fun @ A @ A @ ( stream901396144_sp_mu @ A @ B @ C ) @ ( stream901396144_sp_mu @ A @ E @ F3 )
@ ^ [Y3: A,Z: A] : Y3 = Z
@ ( stream1924447089_sp_mu @ B @ E @ C @ F3 @ A @ R1 @ R2 )
@ X1
@ Y1 )
=> ( stream1924447089_sp_mu @ B @ E @ C @ F3 @ A @ R1 @ R2 @ ( stream1294929701mu_Get @ A @ B @ C @ X1 ) @ ( stream1294929701mu_Get @ A @ E @ F3 @ Y1 ) ) ) ).
% sp\<^sub>\<mu>.rel_intros(1)
thf(fact_173_sp_092_060_094sub_062_092_060mu_062_Orel__inject_I1_J,axiom,
! [B: $tType,C: $tType,F3: $tType,E: $tType,A: $tType,R1: B > E > $o,R2: C > F3 > $o,X1: A > ( stream901396144_sp_mu @ A @ B @ C ),Y1: A > ( stream901396144_sp_mu @ A @ E @ F3 )] :
( ( stream1924447089_sp_mu @ B @ E @ C @ F3 @ A @ R1 @ R2 @ ( stream1294929701mu_Get @ A @ B @ C @ X1 ) @ ( stream1294929701mu_Get @ A @ E @ F3 @ Y1 ) )
= ( bNF_rel_fun @ A @ A @ ( stream901396144_sp_mu @ A @ B @ C ) @ ( stream901396144_sp_mu @ A @ E @ F3 )
@ ^ [Y3: A,Z: A] : Y3 = Z
@ ( stream1924447089_sp_mu @ B @ E @ C @ F3 @ A @ R1 @ R2 )
@ X1
@ Y1 ) ) ).
% sp\<^sub>\<mu>.rel_inject(1)
thf(fact_174_type__copy__map__comp0__undo,axiom,
! [E: $tType,A: $tType,C: $tType,B: $tType,D: $tType,F3: $tType,Rep: A > B,Abs: B > A,Rep2: C > D,Abs2: D > C,Rep3: E > F3,Abs3: F3 > E,M: F3 > D,M1: B > D,M2: F3 > B] :
( ( type_definition @ A @ B @ Rep @ Abs @ ( top_top @ ( set @ B ) ) )
=> ( ( type_definition @ C @ D @ Rep2 @ Abs2 @ ( top_top @ ( set @ D ) ) )
=> ( ( type_definition @ E @ F3 @ Rep3 @ Abs3 @ ( top_top @ ( set @ F3 ) ) )
=> ( ( ( comp @ F3 @ C @ E @ ( comp @ D @ C @ F3 @ Abs2 @ M ) @ Rep3 )
= ( comp @ A @ C @ E @ ( comp @ B @ C @ A @ ( comp @ D @ C @ B @ Abs2 @ M1 ) @ Rep ) @ ( comp @ F3 @ A @ E @ ( comp @ B @ A @ F3 @ Abs @ M2 ) @ Rep3 ) ) )
=> ( ( comp @ B @ D @ F3 @ M1 @ M2 )
= M ) ) ) ) ) ).
% type_copy_map_comp0_undo
thf(fact_175_sp_092_060_094sub_062_092_060nu_062_Orel__coinduct,axiom,
! [B: $tType,D: $tType,A: $tType,P: ( stream1273403375_sp_nu @ A @ B ) > ( stream1273403375_sp_nu @ A @ D ) > $o,X: stream1273403375_sp_nu @ A @ B,Y: stream1273403375_sp_nu @ A @ D,R2: B > D > $o] :
( ( P @ X @ Y )
=> ( ! [Sp_nu4: stream1273403375_sp_nu @ A @ B,Sp_nu5: stream1273403375_sp_nu @ A @ D] :
( ( P @ Sp_nu4 @ Sp_nu5 )
=> ( stream1924447089_sp_mu @ B @ D @ ( stream1273403375_sp_nu @ A @ B ) @ ( stream1273403375_sp_nu @ A @ D ) @ A @ R2 @ P @ ( stream1200245950nu_out @ A @ B @ Sp_nu4 ) @ ( stream1200245950nu_out @ A @ D @ Sp_nu5 ) ) )
=> ( stream1017682031_sp_nu @ B @ D @ A @ R2 @ X @ Y ) ) ) ).
% sp\<^sub>\<nu>.rel_coinduct
thf(fact_176_sp_092_060_094sub_062_092_060nu_062_Orel__sel,axiom,
! [A: $tType,D: $tType,B: $tType] :
( ( stream1017682031_sp_nu @ B @ D @ A )
= ( ^ [R4: B > D > $o,A5: stream1273403375_sp_nu @ A @ B,B4: stream1273403375_sp_nu @ A @ D] : ( stream1924447089_sp_mu @ B @ D @ ( stream1273403375_sp_nu @ A @ B ) @ ( stream1273403375_sp_nu @ A @ D ) @ A @ R4 @ ( stream1017682031_sp_nu @ B @ D @ A @ R4 ) @ ( stream1200245950nu_out @ A @ B @ A5 ) @ ( stream1200245950nu_out @ A @ D @ B4 ) ) ) ) ).
% sp\<^sub>\<nu>.rel_sel
thf(fact_177_sp_092_060_094sub_062_092_060nu_062_Orel__intros,axiom,
! [B: $tType,D: $tType,A: $tType,R: B > D > $o,X: stream901396144_sp_mu @ A @ B @ ( stream1273403375_sp_nu @ A @ B ),Y: stream901396144_sp_mu @ A @ D @ ( stream1273403375_sp_nu @ A @ D )] :
( ( stream1924447089_sp_mu @ B @ D @ ( stream1273403375_sp_nu @ A @ B ) @ ( stream1273403375_sp_nu @ A @ D ) @ A @ R @ ( stream1017682031_sp_nu @ B @ D @ A @ R ) @ X @ Y )
=> ( stream1017682031_sp_nu @ B @ D @ A @ R @ ( stream1686501333_nu_In @ A @ B @ X ) @ ( stream1686501333_nu_In @ A @ D @ Y ) ) ) ).
% sp\<^sub>\<nu>.rel_intros
thf(fact_178_sp_092_060_094sub_062_092_060nu_062_Orel__cases,axiom,
! [B: $tType,D: $tType,A: $tType,R: B > D > $o,A2: stream1273403375_sp_nu @ A @ B,B2: stream1273403375_sp_nu @ A @ D] :
( ( stream1017682031_sp_nu @ B @ D @ A @ R @ A2 @ B2 )
=> ~ ! [Xa3: stream901396144_sp_mu @ A @ B @ ( stream1273403375_sp_nu @ A @ B )] :
( ( A2
= ( stream1686501333_nu_In @ A @ B @ Xa3 ) )
=> ! [Ya2: stream901396144_sp_mu @ A @ D @ ( stream1273403375_sp_nu @ A @ D )] :
( ( B2
= ( stream1686501333_nu_In @ A @ D @ Ya2 ) )
=> ~ ( stream1924447089_sp_mu @ B @ D @ ( stream1273403375_sp_nu @ A @ B ) @ ( stream1273403375_sp_nu @ A @ D ) @ A @ R @ ( stream1017682031_sp_nu @ B @ D @ A @ R ) @ Xa3 @ Ya2 ) ) ) ) ).
% sp\<^sub>\<nu>.rel_cases
thf(fact_179_sp_092_060_094sub_062_092_060mu_062_Orel__cases,axiom,
! [E: $tType,B: $tType,A: $tType,C: $tType,F3: $tType,R1: B > E > $o,R2: C > F3 > $o,A2: stream901396144_sp_mu @ A @ B @ C,B2: stream901396144_sp_mu @ A @ E @ F3] :
( ( stream1924447089_sp_mu @ B @ E @ C @ F3 @ A @ R1 @ R2 @ A2 @ B2 )
=> ( ! [X2: A > ( stream901396144_sp_mu @ A @ B @ C )] :
( ( A2
= ( stream1294929701mu_Get @ A @ B @ C @ X2 ) )
=> ! [Y4: A > ( stream901396144_sp_mu @ A @ E @ F3 )] :
( ( B2
= ( stream1294929701mu_Get @ A @ E @ F3 @ Y4 ) )
=> ~ ( bNF_rel_fun @ A @ A @ ( stream901396144_sp_mu @ A @ B @ C ) @ ( stream901396144_sp_mu @ A @ E @ F3 )
@ ^ [Y3: A,Z: A] : Y3 = Z
@ ( stream1924447089_sp_mu @ B @ E @ C @ F3 @ A @ R1 @ R2 )
@ X2
@ Y4 ) ) )
=> ~ ! [X1a: B,X23: C] :
( ( A2
= ( stream1370332830mu_Put @ B @ C @ A @ X1a @ X23 ) )
=> ! [Y1a: E,Y23: F3] :
( ( B2
= ( stream1370332830mu_Put @ E @ F3 @ A @ Y1a @ Y23 ) )
=> ( ( R1 @ X1a @ Y1a )
=> ~ ( R2 @ X23 @ Y23 ) ) ) ) ) ) ).
% sp\<^sub>\<mu>.rel_cases
thf(fact_180_type__definition_ORep__range,axiom,
! [B: $tType,A: $tType,Rep: B > A,Abs: A > B,A3: set @ A] :
( ( type_definition @ B @ A @ Rep @ Abs @ A3 )
=> ( ( image @ B @ A @ Rep @ ( top_top @ ( set @ B ) ) )
= A3 ) ) ).
% type_definition.Rep_range
thf(fact_181_type__definition_OAbs__image,axiom,
! [A: $tType,B: $tType,Rep: B > A,Abs: A > B,A3: set @ A] :
( ( type_definition @ B @ A @ Rep @ Abs @ A3 )
=> ( ( image @ A @ B @ Abs @ A3 )
= ( top_top @ ( set @ B ) ) ) ) ).
% type_definition.Abs_image
thf(fact_182_Abs__transfer,axiom,
! [B: $tType,A: $tType,C: $tType,D: $tType,Rep1: A > B,Abs1: B > A,Rep22: C > D,Abs22: D > C,R: B > D > $o] :
( ( type_definition @ A @ B @ Rep1 @ Abs1 @ ( top_top @ ( set @ B ) ) )
=> ( ( type_definition @ C @ D @ Rep22 @ Abs22 @ ( top_top @ ( set @ D ) ) )
=> ( bNF_rel_fun @ B @ D @ A @ C @ R @ ( bNF_vimage2p @ A @ B @ C @ D @ $o @ Rep1 @ Rep22 @ R ) @ Abs1 @ Abs22 ) ) ) ).
% Abs_transfer
thf(fact_183_sp_092_060_094sub_062_092_060nu_062_Opred__transfer,axiom,
! [A: $tType,D: $tType,B: $tType,R: A > B > $o] :
( bNF_rel_fun @ ( A > $o ) @ ( B > $o ) @ ( ( stream1273403375_sp_nu @ D @ A ) > $o ) @ ( ( stream1273403375_sp_nu @ D @ B ) > $o )
@ ( bNF_rel_fun @ A @ B @ $o @ $o @ R
@ ^ [Y3: $o,Z: $o] : Y3 = Z )
@ ( bNF_rel_fun @ ( stream1273403375_sp_nu @ D @ A ) @ ( stream1273403375_sp_nu @ D @ B ) @ $o @ $o @ ( stream1017682031_sp_nu @ A @ B @ D @ R )
@ ^ [Y3: $o,Z: $o] : Y3 = Z )
@ ( stream1465372679_sp_nu @ A @ D )
@ ( stream1465372679_sp_nu @ B @ D ) ) ).
% sp\<^sub>\<nu>.pred_transfer
thf(fact_184_sp_092_060_094sub_062_092_060nu_062_Orel__transfer,axiom,
! [A: $tType,B: $tType,D: $tType,E: $tType,C: $tType,Sa: A > C > $o,Sc: B > E > $o] :
( bNF_rel_fun @ ( A > B > $o ) @ ( C > E > $o ) @ ( ( stream1273403375_sp_nu @ D @ A ) > ( stream1273403375_sp_nu @ D @ B ) > $o ) @ ( ( stream1273403375_sp_nu @ D @ C ) > ( stream1273403375_sp_nu @ D @ E ) > $o )
@ ( bNF_rel_fun @ A @ C @ ( B > $o ) @ ( E > $o ) @ Sa
@ ( bNF_rel_fun @ B @ E @ $o @ $o @ Sc
@ ^ [Y3: $o,Z: $o] : Y3 = Z ) )
@ ( bNF_rel_fun @ ( stream1273403375_sp_nu @ D @ A ) @ ( stream1273403375_sp_nu @ D @ C ) @ ( ( stream1273403375_sp_nu @ D @ B ) > $o ) @ ( ( stream1273403375_sp_nu @ D @ E ) > $o ) @ ( stream1017682031_sp_nu @ A @ C @ D @ Sa )
@ ( bNF_rel_fun @ ( stream1273403375_sp_nu @ D @ B ) @ ( stream1273403375_sp_nu @ D @ E ) @ $o @ $o @ ( stream1017682031_sp_nu @ B @ E @ D @ Sc )
@ ^ [Y3: $o,Z: $o] : Y3 = Z ) )
@ ( stream1017682031_sp_nu @ A @ B @ D )
@ ( stream1017682031_sp_nu @ C @ E @ D ) ) ).
% sp\<^sub>\<nu>.rel_transfer
thf(fact_185_sp_092_060_094sub_062_092_060mu_062_Orel__transfer,axiom,
! [A: $tType,C: $tType,B: $tType,D: $tType,G: $tType,I: $tType,F3: $tType,H2: $tType,E: $tType,S1a: A > E > $o,S1c: C > H2 > $o,S2a: B > F3 > $o,S2c: D > I > $o] :
( bNF_rel_fun @ ( A > C > $o ) @ ( E > H2 > $o ) @ ( ( B > D > $o ) > ( stream901396144_sp_mu @ G @ A @ B ) > ( stream901396144_sp_mu @ G @ C @ D ) > $o ) @ ( ( F3 > I > $o ) > ( stream901396144_sp_mu @ G @ E @ F3 ) > ( stream901396144_sp_mu @ G @ H2 @ I ) > $o )
@ ( bNF_rel_fun @ A @ E @ ( C > $o ) @ ( H2 > $o ) @ S1a
@ ( bNF_rel_fun @ C @ H2 @ $o @ $o @ S1c
@ ^ [Y3: $o,Z: $o] : Y3 = Z ) )
@ ( bNF_rel_fun @ ( B > D > $o ) @ ( F3 > I > $o ) @ ( ( stream901396144_sp_mu @ G @ A @ B ) > ( stream901396144_sp_mu @ G @ C @ D ) > $o ) @ ( ( stream901396144_sp_mu @ G @ E @ F3 ) > ( stream901396144_sp_mu @ G @ H2 @ I ) > $o )
@ ( bNF_rel_fun @ B @ F3 @ ( D > $o ) @ ( I > $o ) @ S2a
@ ( bNF_rel_fun @ D @ I @ $o @ $o @ S2c
@ ^ [Y3: $o,Z: $o] : Y3 = Z ) )
@ ( bNF_rel_fun @ ( stream901396144_sp_mu @ G @ A @ B ) @ ( stream901396144_sp_mu @ G @ E @ F3 ) @ ( ( stream901396144_sp_mu @ G @ C @ D ) > $o ) @ ( ( stream901396144_sp_mu @ G @ H2 @ I ) > $o ) @ ( stream1924447089_sp_mu @ A @ E @ B @ F3 @ G @ S1a @ S2a )
@ ( bNF_rel_fun @ ( stream901396144_sp_mu @ G @ C @ D ) @ ( stream901396144_sp_mu @ G @ H2 @ I ) @ $o @ $o @ ( stream1924447089_sp_mu @ C @ H2 @ D @ I @ G @ S1c @ S2c )
@ ^ [Y3: $o,Z: $o] : Y3 = Z ) ) )
@ ( stream1924447089_sp_mu @ A @ C @ B @ D @ G )
@ ( stream1924447089_sp_mu @ E @ H2 @ F3 @ I @ G ) ) ).
% sp\<^sub>\<mu>.rel_transfer
thf(fact_186_sp_092_060_094sub_062_092_060nu_062_Ocase__transfer,axiom,
! [B: $tType,E: $tType,F3: $tType,D: $tType,A: $tType,R: B > D > $o,S2: E > F3 > $o] : ( bNF_rel_fun @ ( ( stream901396144_sp_mu @ A @ B @ ( stream1273403375_sp_nu @ A @ B ) ) > E ) @ ( ( stream901396144_sp_mu @ A @ D @ ( stream1273403375_sp_nu @ A @ D ) ) > F3 ) @ ( ( stream1273403375_sp_nu @ A @ B ) > E ) @ ( ( stream1273403375_sp_nu @ A @ D ) > F3 ) @ ( bNF_rel_fun @ ( stream901396144_sp_mu @ A @ B @ ( stream1273403375_sp_nu @ A @ B ) ) @ ( stream901396144_sp_mu @ A @ D @ ( stream1273403375_sp_nu @ A @ D ) ) @ E @ F3 @ ( stream1924447089_sp_mu @ B @ D @ ( stream1273403375_sp_nu @ A @ B ) @ ( stream1273403375_sp_nu @ A @ D ) @ A @ R @ ( stream1017682031_sp_nu @ B @ D @ A @ R ) ) @ S2 ) @ ( bNF_rel_fun @ ( stream1273403375_sp_nu @ A @ B ) @ ( stream1273403375_sp_nu @ A @ D ) @ E @ F3 @ ( stream1017682031_sp_nu @ B @ D @ A @ R ) @ S2 ) @ ( stream1502363800_sp_nu @ A @ B @ E ) @ ( stream1502363800_sp_nu @ A @ D @ F3 ) ) ).
% sp\<^sub>\<nu>.case_transfer
thf(fact_187_sp_092_060_094sub_062_092_060mu_062_Opred__transfer,axiom,
! [A: $tType,B: $tType,G: $tType,D: $tType,C: $tType,R1: A > C > $o,R2: B > D > $o] :
( bNF_rel_fun @ ( A > $o ) @ ( C > $o ) @ ( ( B > $o ) > ( stream901396144_sp_mu @ G @ A @ B ) > $o ) @ ( ( D > $o ) > ( stream901396144_sp_mu @ G @ C @ D ) > $o )
@ ( bNF_rel_fun @ A @ C @ $o @ $o @ R1
@ ^ [Y3: $o,Z: $o] : Y3 = Z )
@ ( bNF_rel_fun @ ( B > $o ) @ ( D > $o ) @ ( ( stream901396144_sp_mu @ G @ A @ B ) > $o ) @ ( ( stream901396144_sp_mu @ G @ C @ D ) > $o )
@ ( bNF_rel_fun @ B @ D @ $o @ $o @ R2
@ ^ [Y3: $o,Z: $o] : Y3 = Z )
@ ( bNF_rel_fun @ ( stream901396144_sp_mu @ G @ A @ B ) @ ( stream901396144_sp_mu @ G @ C @ D ) @ $o @ $o @ ( stream1924447089_sp_mu @ A @ C @ B @ D @ G @ R1 @ R2 )
@ ^ [Y3: $o,Z: $o] : Y3 = Z ) )
@ ( stream123009735_sp_mu @ A @ B @ G )
@ ( stream123009735_sp_mu @ C @ D @ G ) ) ).
% sp\<^sub>\<mu>.pred_transfer
thf(fact_188_vimage2pI,axiom,
! [C: $tType,B: $tType,A: $tType,D: $tType,R: A > B > $o,F: C > A,X: C,G3: D > B,Y: D] :
( ( R @ ( F @ X ) @ ( G3 @ Y ) )
=> ( bNF_vimage2p @ C @ A @ D @ B @ $o @ F @ G3 @ R @ X @ Y ) ) ).
% vimage2pI
thf(fact_189_vimage2p__comp,axiom,
! [E: $tType,D: $tType,F3: $tType,A: $tType,C: $tType,B: $tType,G: $tType,F1: F3 > A,F2: D > F3,G1: G > B,G2: E > G] :
( ( bNF_vimage2p @ D @ A @ E @ B @ C @ ( comp @ F3 @ A @ D @ F1 @ F2 ) @ ( comp @ G @ B @ E @ G1 @ G2 ) )
= ( comp @ ( F3 > G > C ) @ ( D > E > C ) @ ( A > B > C ) @ ( bNF_vimage2p @ D @ F3 @ E @ G @ C @ F2 @ G2 ) @ ( bNF_vimage2p @ F3 @ A @ G @ B @ C @ F1 @ G1 ) ) ) ).
% vimage2p_comp
thf(fact_190_sp_092_060_094sub_062_092_060mu_062_Octr__transfer_I2_J,axiom,
! [B: $tType,C: $tType,A: $tType,F3: $tType,E: $tType,R1: B > E > $o,R2: C > F3 > $o] : ( bNF_rel_fun @ B @ E @ ( C > ( stream901396144_sp_mu @ A @ B @ C ) ) @ ( F3 > ( stream901396144_sp_mu @ A @ E @ F3 ) ) @ R1 @ ( bNF_rel_fun @ C @ F3 @ ( stream901396144_sp_mu @ A @ B @ C ) @ ( stream901396144_sp_mu @ A @ E @ F3 ) @ R2 @ ( stream1924447089_sp_mu @ B @ E @ C @ F3 @ A @ R1 @ R2 ) ) @ ( stream1370332830mu_Put @ B @ C @ A ) @ ( stream1370332830mu_Put @ E @ F3 @ A ) ) ).
% sp\<^sub>\<mu>.ctr_transfer(2)
thf(fact_191_sp_092_060_094sub_062_092_060nu_062_Octr__transfer,axiom,
! [B: $tType,D: $tType,A: $tType,R: B > D > $o] : ( bNF_rel_fun @ ( stream901396144_sp_mu @ A @ B @ ( stream1273403375_sp_nu @ A @ B ) ) @ ( stream901396144_sp_mu @ A @ D @ ( stream1273403375_sp_nu @ A @ D ) ) @ ( stream1273403375_sp_nu @ A @ B ) @ ( stream1273403375_sp_nu @ A @ D ) @ ( stream1924447089_sp_mu @ B @ D @ ( stream1273403375_sp_nu @ A @ B ) @ ( stream1273403375_sp_nu @ A @ D ) @ A @ R @ ( stream1017682031_sp_nu @ B @ D @ A @ R ) ) @ ( stream1017682031_sp_nu @ B @ D @ A @ R ) @ ( stream1686501333_nu_In @ A @ B ) @ ( stream1686501333_nu_In @ A @ D ) ) ).
% sp\<^sub>\<nu>.ctr_transfer
thf(fact_192_sp_092_060_094sub_062_092_060mu_062_Orec__transfer,axiom,
! [B: $tType,C: $tType,G: $tType,H2: $tType,F3: $tType,E: $tType,A: $tType,R1: B > E > $o,R2: C > F3 > $o,S2: G > H2 > $o] :
( bNF_rel_fun @ ( ( A > ( product_prod @ ( stream901396144_sp_mu @ A @ B @ C ) @ G ) ) > G ) @ ( ( A > ( product_prod @ ( stream901396144_sp_mu @ A @ E @ F3 ) @ H2 ) ) > H2 ) @ ( ( B > C > G ) > ( stream901396144_sp_mu @ A @ B @ C ) > G ) @ ( ( E > F3 > H2 ) > ( stream901396144_sp_mu @ A @ E @ F3 ) > H2 )
@ ( bNF_rel_fun @ ( A > ( product_prod @ ( stream901396144_sp_mu @ A @ B @ C ) @ G ) ) @ ( A > ( product_prod @ ( stream901396144_sp_mu @ A @ E @ F3 ) @ H2 ) ) @ G @ H2
@ ( bNF_rel_fun @ A @ A @ ( product_prod @ ( stream901396144_sp_mu @ A @ B @ C ) @ G ) @ ( product_prod @ ( stream901396144_sp_mu @ A @ E @ F3 ) @ H2 )
@ ^ [Y3: A,Z: A] : Y3 = Z
@ ( basic_rel_prod @ ( stream901396144_sp_mu @ A @ B @ C ) @ ( stream901396144_sp_mu @ A @ E @ F3 ) @ G @ H2 @ ( stream1924447089_sp_mu @ B @ E @ C @ F3 @ A @ R1 @ R2 ) @ S2 ) )
@ S2 )
@ ( bNF_rel_fun @ ( B > C > G ) @ ( E > F3 > H2 ) @ ( ( stream901396144_sp_mu @ A @ B @ C ) > G ) @ ( ( stream901396144_sp_mu @ A @ E @ F3 ) > H2 ) @ ( bNF_rel_fun @ B @ E @ ( C > G ) @ ( F3 > H2 ) @ R1 @ ( bNF_rel_fun @ C @ F3 @ G @ H2 @ R2 @ S2 ) ) @ ( bNF_rel_fun @ ( stream901396144_sp_mu @ A @ B @ C ) @ ( stream901396144_sp_mu @ A @ E @ F3 ) @ G @ H2 @ ( stream1924447089_sp_mu @ B @ E @ C @ F3 @ A @ R1 @ R2 ) @ S2 ) )
@ ( stream674629690_sp_mu @ A @ B @ C @ G )
@ ( stream674629690_sp_mu @ A @ E @ F3 @ H2 ) ) ).
% sp\<^sub>\<mu>.rec_transfer
thf(fact_193_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_194_sp_092_060_094sub_062_092_060nu_062_Opred__map,axiom,
! [B: $tType,A: $tType,D: $tType,Q: B > $o,F: A > B,X: stream1273403375_sp_nu @ D @ A] :
( ( stream1465372679_sp_nu @ B @ D @ Q @ ( stream151454380_sp_nu @ A @ B @ D @ F @ X ) )
= ( stream1465372679_sp_nu @ A @ D @ ( comp @ B @ $o @ A @ Q @ F ) @ X ) ) ).
% sp\<^sub>\<nu>.pred_map
thf(fact_195_prod_Orel__transfer,axiom,
! [A: $tType,C: $tType,B: $tType,D: $tType,H2: $tType,F3: $tType,G: $tType,E: $tType,S1a: A > E > $o,S1c: C > G > $o,S2a: B > F3 > $o,S2c: D > H2 > $o] :
( bNF_rel_fun @ ( A > C > $o ) @ ( E > G > $o ) @ ( ( B > D > $o ) > ( product_prod @ A @ B ) > ( product_prod @ C @ D ) > $o ) @ ( ( F3 > H2 > $o ) > ( product_prod @ E @ F3 ) > ( product_prod @ G @ H2 ) > $o )
@ ( bNF_rel_fun @ A @ E @ ( C > $o ) @ ( G > $o ) @ S1a
@ ( bNF_rel_fun @ C @ G @ $o @ $o @ S1c
@ ^ [Y3: $o,Z: $o] : Y3 = Z ) )
@ ( bNF_rel_fun @ ( B > D > $o ) @ ( F3 > H2 > $o ) @ ( ( product_prod @ A @ B ) > ( product_prod @ C @ D ) > $o ) @ ( ( product_prod @ E @ F3 ) > ( product_prod @ G @ H2 ) > $o )
@ ( bNF_rel_fun @ B @ F3 @ ( D > $o ) @ ( H2 > $o ) @ S2a
@ ( bNF_rel_fun @ D @ H2 @ $o @ $o @ S2c
@ ^ [Y3: $o,Z: $o] : Y3 = Z ) )
@ ( bNF_rel_fun @ ( product_prod @ A @ B ) @ ( product_prod @ E @ F3 ) @ ( ( product_prod @ C @ D ) > $o ) @ ( ( product_prod @ G @ H2 ) > $o ) @ ( basic_rel_prod @ A @ E @ B @ F3 @ S1a @ S2a )
@ ( bNF_rel_fun @ ( product_prod @ C @ D ) @ ( product_prod @ G @ H2 ) @ $o @ $o @ ( basic_rel_prod @ C @ G @ D @ H2 @ S1c @ S2c )
@ ^ [Y3: $o,Z: $o] : Y3 = Z ) ) )
@ ( basic_rel_prod @ A @ C @ B @ D )
@ ( basic_rel_prod @ E @ G @ F3 @ H2 ) ) ).
% prod.rel_transfer
thf(fact_196_sp_092_060_094sub_062_092_060nu_062_Omap__comp,axiom,
! [B: $tType,C: $tType,A: $tType,D: $tType,G3: B > C,F: A > B,V: stream1273403375_sp_nu @ D @ A] :
( ( stream151454380_sp_nu @ B @ C @ D @ G3 @ ( stream151454380_sp_nu @ A @ B @ D @ F @ V ) )
= ( stream151454380_sp_nu @ A @ C @ D @ ( comp @ B @ C @ A @ G3 @ F ) @ V ) ) ).
% sp\<^sub>\<nu>.map_comp
thf(fact_197_prod_Orel__eq,axiom,
! [B: $tType,A: $tType] :
( ( basic_rel_prod @ A @ A @ B @ B
@ ^ [Y3: A,Z: A] : Y3 = Z
@ ^ [Y3: B,Z: B] : Y3 = Z )
= ( ^ [Y3: product_prod @ A @ B,Z: product_prod @ A @ B] : Y3 = Z ) ) ).
% prod.rel_eq
thf(fact_198_prod_Orel__refl,axiom,
! [D: $tType,C: $tType,R1a: C > C > $o,R2a: D > D > $o,X: product_prod @ C @ D] :
( ! [X2: C] : ( R1a @ X2 @ X2 )
=> ( ! [X2: D] : ( R2a @ X2 @ X2 )
=> ( basic_rel_prod @ C @ C @ D @ D @ R1a @ R2a @ X @ X ) ) ) ).
% prod.rel_refl
thf(fact_199_sp_092_060_094sub_062_092_060nu_062_Omap,axiom,
! [D: $tType,B: $tType,A: $tType,F: B > D,X: stream901396144_sp_mu @ A @ B @ ( stream1273403375_sp_nu @ A @ B )] :
( ( stream151454380_sp_nu @ B @ D @ A @ F @ ( stream1686501333_nu_In @ A @ B @ X ) )
= ( stream1686501333_nu_In @ A @ D @ ( stream1058219438_sp_mu @ B @ D @ ( stream1273403375_sp_nu @ A @ B ) @ ( stream1273403375_sp_nu @ A @ D ) @ A @ F @ ( stream151454380_sp_nu @ B @ D @ A @ F ) @ X ) ) ) ).
% sp\<^sub>\<nu>.map
thf(fact_200_sp_092_060_094sub_062_092_060nu_062_Omap__sel,axiom,
! [D: $tType,B: $tType,A: $tType,F: B > D,A2: stream1273403375_sp_nu @ A @ B] :
( ( stream1200245950nu_out @ A @ D @ ( stream151454380_sp_nu @ B @ D @ A @ F @ A2 ) )
= ( stream1058219438_sp_mu @ B @ D @ ( stream1273403375_sp_nu @ A @ B ) @ ( stream1273403375_sp_nu @ A @ D ) @ A @ F @ ( stream151454380_sp_nu @ B @ D @ A @ F ) @ ( stream1200245950nu_out @ A @ B @ A2 ) ) ) ).
% sp\<^sub>\<nu>.map_sel
thf(fact_201_ID_Opred__map,axiom,
! [B: $tType,A: $tType,Q: B > $o,F: A > B,X: A] :
( ( bNF_id_bnf @ ( B > $o ) @ Q @ ( bNF_id_bnf @ ( A > B ) @ F @ X ) )
= ( bNF_id_bnf @ ( A > $o ) @ ( comp @ B @ $o @ A @ Q @ F ) @ X ) ) ).
% ID.pred_map
thf(fact_202_sp_092_060_094sub_062_092_060nu_062_Omap__transfer,axiom,
! [A: $tType,B: $tType,D: $tType,G: $tType,F3: $tType,Rb: A > F3 > $o,Sd: B > G > $o] : ( bNF_rel_fun @ ( A > B ) @ ( F3 > G ) @ ( ( stream1273403375_sp_nu @ D @ A ) > ( stream1273403375_sp_nu @ D @ B ) ) @ ( ( stream1273403375_sp_nu @ D @ F3 ) > ( stream1273403375_sp_nu @ D @ G ) ) @ ( bNF_rel_fun @ A @ F3 @ B @ G @ Rb @ Sd ) @ ( bNF_rel_fun @ ( stream1273403375_sp_nu @ D @ A ) @ ( stream1273403375_sp_nu @ D @ F3 ) @ ( stream1273403375_sp_nu @ D @ B ) @ ( stream1273403375_sp_nu @ D @ G ) @ ( stream1017682031_sp_nu @ A @ F3 @ D @ Rb ) @ ( stream1017682031_sp_nu @ B @ G @ D @ Sd ) ) @ ( stream151454380_sp_nu @ A @ B @ D ) @ ( stream151454380_sp_nu @ F3 @ G @ D ) ) ).
% sp\<^sub>\<nu>.map_transfer
thf(fact_203_sp_092_060_094sub_062_092_060nu_062_Ocorec__transfer,axiom,
! [E: $tType,B: $tType,D: $tType,A: $tType,F3: $tType,S2: E > F3 > $o,R: B > D > $o] : ( bNF_rel_fun @ ( E > ( stream901396144_sp_mu @ A @ B @ ( sum_sum @ ( stream1273403375_sp_nu @ A @ B ) @ E ) ) ) @ ( F3 > ( stream901396144_sp_mu @ A @ D @ ( sum_sum @ ( stream1273403375_sp_nu @ A @ D ) @ F3 ) ) ) @ ( E > ( stream1273403375_sp_nu @ A @ B ) ) @ ( F3 > ( stream1273403375_sp_nu @ A @ D ) ) @ ( bNF_rel_fun @ E @ F3 @ ( stream901396144_sp_mu @ A @ B @ ( sum_sum @ ( stream1273403375_sp_nu @ A @ B ) @ E ) ) @ ( stream901396144_sp_mu @ A @ D @ ( sum_sum @ ( stream1273403375_sp_nu @ A @ D ) @ F3 ) ) @ S2 @ ( stream1924447089_sp_mu @ B @ D @ ( sum_sum @ ( stream1273403375_sp_nu @ A @ B ) @ E ) @ ( sum_sum @ ( stream1273403375_sp_nu @ A @ D ) @ F3 ) @ A @ R @ ( bNF_rel_sum @ ( stream1273403375_sp_nu @ A @ B ) @ ( stream1273403375_sp_nu @ A @ D ) @ E @ F3 @ ( stream1017682031_sp_nu @ B @ D @ A @ R ) @ S2 ) ) ) @ ( bNF_rel_fun @ E @ F3 @ ( stream1273403375_sp_nu @ A @ B ) @ ( stream1273403375_sp_nu @ A @ D ) @ S2 @ ( stream1017682031_sp_nu @ B @ D @ A @ R ) ) @ ( stream937840132_sp_nu @ E @ A @ B ) @ ( stream937840132_sp_nu @ F3 @ A @ D ) ) ).
% sp\<^sub>\<nu>.corec_transfer
thf(fact_204_sp_092_060_094sub_062_092_060nu_062_Opred__set,axiom,
! [D: $tType,A: $tType] :
( ( stream1465372679_sp_nu @ A @ D )
= ( ^ [P3: A > $o,X3: stream1273403375_sp_nu @ D @ A] :
! [Y2: A] :
( ( member @ A @ Y2 @ ( stream1493736486_sp_nu @ D @ A @ X3 ) )
=> ( P3 @ Y2 ) ) ) ) ).
% sp\<^sub>\<nu>.pred_set
thf(fact_205_sp_092_060_094sub_062_092_060nu_062_Oinj__map__strong,axiom,
! [B: $tType,A: $tType,D: $tType,X: stream1273403375_sp_nu @ D @ A,Xa: stream1273403375_sp_nu @ D @ A,F: A > B,Fa: A > B] :
( ! [Z3: A,Za: A] :
( ( member @ A @ Z3 @ ( stream1493736486_sp_nu @ D @ A @ X ) )
=> ( ( member @ A @ Za @ ( stream1493736486_sp_nu @ D @ A @ Xa ) )
=> ( ( ( F @ Z3 )
= ( Fa @ Za ) )
=> ( Z3 = Za ) ) ) )
=> ( ( ( stream151454380_sp_nu @ A @ B @ D @ F @ X )
= ( stream151454380_sp_nu @ A @ B @ D @ Fa @ Xa ) )
=> ( X = Xa ) ) ) ).
% sp\<^sub>\<nu>.inj_map_strong
thf(fact_206_sp_092_060_094sub_062_092_060nu_062_Omap__cong0,axiom,
! [B: $tType,A: $tType,D: $tType,X: stream1273403375_sp_nu @ D @ A,F: A > B,G3: A > B] :
( ! [Z3: A] :
( ( member @ A @ Z3 @ ( stream1493736486_sp_nu @ D @ A @ X ) )
=> ( ( F @ Z3 )
= ( G3 @ Z3 ) ) )
=> ( ( stream151454380_sp_nu @ A @ B @ D @ F @ X )
= ( stream151454380_sp_nu @ A @ B @ D @ G3 @ X ) ) ) ).
% sp\<^sub>\<nu>.map_cong0
thf(fact_207_sp_092_060_094sub_062_092_060nu_062_Omap__cong,axiom,
! [B: $tType,A: $tType,D: $tType,X: stream1273403375_sp_nu @ D @ A,Ya: stream1273403375_sp_nu @ D @ A,F: A > B,G3: A > B] :
( ( X = Ya )
=> ( ! [Z3: A] :
( ( member @ A @ Z3 @ ( stream1493736486_sp_nu @ D @ A @ Ya ) )
=> ( ( F @ Z3 )
= ( G3 @ Z3 ) ) )
=> ( ( stream151454380_sp_nu @ A @ B @ D @ F @ X )
= ( stream151454380_sp_nu @ A @ B @ D @ G3 @ Ya ) ) ) ) ).
% sp\<^sub>\<nu>.map_cong
thf(fact_208_sum_Orel__transfer,axiom,
! [A: $tType,C: $tType,B: $tType,D: $tType,H2: $tType,F3: $tType,G: $tType,E: $tType,S1a: A > E > $o,S1c: C > G > $o,S2a: B > F3 > $o,S2c: D > H2 > $o] :
( bNF_rel_fun @ ( A > C > $o ) @ ( E > G > $o ) @ ( ( B > D > $o ) > ( sum_sum @ A @ B ) > ( sum_sum @ C @ D ) > $o ) @ ( ( F3 > H2 > $o ) > ( sum_sum @ E @ F3 ) > ( sum_sum @ G @ H2 ) > $o )
@ ( bNF_rel_fun @ A @ E @ ( C > $o ) @ ( G > $o ) @ S1a
@ ( bNF_rel_fun @ C @ G @ $o @ $o @ S1c
@ ^ [Y3: $o,Z: $o] : Y3 = Z ) )
@ ( bNF_rel_fun @ ( B > D > $o ) @ ( F3 > H2 > $o ) @ ( ( sum_sum @ A @ B ) > ( sum_sum @ C @ D ) > $o ) @ ( ( sum_sum @ E @ F3 ) > ( sum_sum @ G @ H2 ) > $o )
@ ( bNF_rel_fun @ B @ F3 @ ( D > $o ) @ ( H2 > $o ) @ S2a
@ ( bNF_rel_fun @ D @ H2 @ $o @ $o @ S2c
@ ^ [Y3: $o,Z: $o] : Y3 = Z ) )
@ ( bNF_rel_fun @ ( sum_sum @ A @ B ) @ ( sum_sum @ E @ F3 ) @ ( ( sum_sum @ C @ D ) > $o ) @ ( ( sum_sum @ G @ H2 ) > $o ) @ ( bNF_rel_sum @ A @ E @ B @ F3 @ S1a @ S2a )
@ ( bNF_rel_fun @ ( sum_sum @ C @ D ) @ ( sum_sum @ G @ H2 ) @ $o @ $o @ ( bNF_rel_sum @ C @ G @ D @ H2 @ S1c @ S2c )
@ ^ [Y3: $o,Z: $o] : Y3 = Z ) ) )
@ ( bNF_rel_sum @ A @ C @ B @ D )
@ ( bNF_rel_sum @ E @ G @ F3 @ H2 ) ) ).
% sum.rel_transfer
thf(fact_209_sp_092_060_094sub_062_092_060nu_062_Orel__refl__strong,axiom,
! [A: $tType,B: $tType,X: stream1273403375_sp_nu @ B @ A,Ra: A > A > $o] :
( ! [Z3: A] :
( ( member @ A @ Z3 @ ( stream1493736486_sp_nu @ B @ A @ X ) )
=> ( Ra @ Z3 @ Z3 ) )
=> ( stream1017682031_sp_nu @ A @ A @ B @ Ra @ X @ X ) ) ).
% sp\<^sub>\<nu>.rel_refl_strong
thf(fact_210_sp_092_060_094sub_062_092_060nu_062_Orel__mono__strong,axiom,
! [A: $tType,B: $tType,D: $tType,R: A > B > $o,X: stream1273403375_sp_nu @ D @ A,Y: stream1273403375_sp_nu @ D @ B,Ra: A > B > $o] :
( ( stream1017682031_sp_nu @ A @ B @ D @ R @ X @ Y )
=> ( ! [Z3: A,Yb: B] :
( ( member @ A @ Z3 @ ( stream1493736486_sp_nu @ D @ A @ X ) )
=> ( ( member @ B @ Yb @ ( stream1493736486_sp_nu @ D @ B @ Y ) )
=> ( ( R @ Z3 @ Yb )
=> ( Ra @ Z3 @ Yb ) ) ) )
=> ( stream1017682031_sp_nu @ A @ B @ D @ Ra @ X @ Y ) ) ) ).
% sp\<^sub>\<nu>.rel_mono_strong
thf(fact_211_sp_092_060_094sub_062_092_060nu_062_Orel__cong,axiom,
! [A: $tType,B: $tType,D: $tType,X: stream1273403375_sp_nu @ D @ A,Ya: stream1273403375_sp_nu @ D @ A,Y: stream1273403375_sp_nu @ D @ B,Xa: stream1273403375_sp_nu @ D @ B,R: A > B > $o,Ra: A > B > $o] :
( ( X = Ya )
=> ( ( Y = Xa )
=> ( ! [Z3: A,Yb: B] :
( ( member @ A @ Z3 @ ( stream1493736486_sp_nu @ D @ A @ Ya ) )
=> ( ( member @ B @ Yb @ ( stream1493736486_sp_nu @ D @ B @ Xa ) )
=> ( ( R @ Z3 @ Yb )
= ( Ra @ Z3 @ Yb ) ) ) )
=> ( ( stream1017682031_sp_nu @ A @ B @ D @ R @ X @ Y )
= ( stream1017682031_sp_nu @ A @ B @ D @ Ra @ Ya @ Xa ) ) ) ) ) ).
% sp\<^sub>\<nu>.rel_cong
thf(fact_212_sum_Orel__refl,axiom,
! [D: $tType,C: $tType,R1a: C > C > $o,R2a: D > D > $o,X: sum_sum @ C @ D] :
( ! [X2: C] : ( R1a @ X2 @ X2 )
=> ( ! [X2: D] : ( R2a @ X2 @ X2 )
=> ( bNF_rel_sum @ C @ C @ D @ D @ R1a @ R2a @ X @ X ) ) ) ).
% sum.rel_refl
thf(fact_213_sum_Orel__eq,axiom,
! [B: $tType,A: $tType] :
( ( bNF_rel_sum @ A @ A @ B @ B
@ ^ [Y3: A,Z: A] : Y3 = Z
@ ^ [Y3: B,Z: B] : Y3 = Z )
= ( ^ [Y3: sum_sum @ A @ B,Z: sum_sum @ A @ B] : Y3 = Z ) ) ).
% sum.rel_eq
thf(fact_214_sp_092_060_094sub_062_092_060nu_062_Opred__mono__strong,axiom,
! [A: $tType,D: $tType,P: A > $o,X: stream1273403375_sp_nu @ D @ A,Pa: A > $o] :
( ( stream1465372679_sp_nu @ A @ D @ P @ X )
=> ( ! [Z3: A] :
( ( member @ A @ Z3 @ ( stream1493736486_sp_nu @ D @ A @ X ) )
=> ( ( P @ Z3 )
=> ( Pa @ Z3 ) ) )
=> ( stream1465372679_sp_nu @ A @ D @ Pa @ X ) ) ) ).
% sp\<^sub>\<nu>.pred_mono_strong
thf(fact_215_sp_092_060_094sub_062_092_060nu_062_Opred__cong,axiom,
! [A: $tType,D: $tType,X: stream1273403375_sp_nu @ D @ A,Ya: stream1273403375_sp_nu @ D @ A,P: A > $o,Pa: A > $o] :
( ( X = Ya )
=> ( ! [Z3: A] :
( ( member @ A @ Z3 @ ( stream1493736486_sp_nu @ D @ A @ Ya ) )
=> ( ( P @ Z3 )
= ( Pa @ Z3 ) ) )
=> ( ( stream1465372679_sp_nu @ A @ D @ P @ X )
= ( stream1465372679_sp_nu @ A @ D @ Pa @ Ya ) ) ) ) ).
% sp\<^sub>\<nu>.pred_cong
thf(fact_216_sp_092_060_094sub_062_092_060nu_062_Oset__induct,axiom,
! [A: $tType,Aa2: $tType,X: A,A2: stream1273403375_sp_nu @ Aa2 @ A,P: A > ( stream1273403375_sp_nu @ Aa2 @ A ) > $o] :
( ( member @ A @ X @ ( stream1493736486_sp_nu @ Aa2 @ A @ A2 ) )
=> ( ! [Z3: stream901396144_sp_mu @ Aa2 @ A @ ( stream1273403375_sp_nu @ Aa2 @ A ),Xa3: A] :
( ( member @ A @ Xa3 @ ( stream2074884121_sp_mu @ Aa2 @ A @ ( stream1273403375_sp_nu @ Aa2 @ A ) @ Z3 ) )
=> ( P @ Xa3 @ ( stream1686501333_nu_In @ Aa2 @ A @ Z3 ) ) )
=> ( ! [Z3: stream901396144_sp_mu @ Aa2 @ A @ ( stream1273403375_sp_nu @ Aa2 @ A ),Xb: stream1273403375_sp_nu @ Aa2 @ A,Xc2: A] :
( ( member @ ( stream1273403375_sp_nu @ Aa2 @ A ) @ Xb @ ( stream1259315544_sp_mu @ Aa2 @ A @ ( stream1273403375_sp_nu @ Aa2 @ A ) @ Z3 ) )
=> ( ( member @ A @ Xc2 @ ( stream1493736486_sp_nu @ Aa2 @ A @ Xb ) )
=> ( ( P @ Xc2 @ Xb )
=> ( P @ Xc2 @ ( stream1686501333_nu_In @ Aa2 @ A @ Z3 ) ) ) ) )
=> ( P @ X @ A2 ) ) ) ) ).
% sp\<^sub>\<nu>.set_induct
thf(fact_217_sp_092_060_094sub_062_092_060nu_062_Oset__cases,axiom,
! [B: $tType,A: $tType,E2: B,A2: stream1273403375_sp_nu @ A @ B] :
( ( member @ B @ E2 @ ( stream1493736486_sp_nu @ A @ B @ A2 ) )
=> ( ! [Z3: stream901396144_sp_mu @ A @ B @ ( stream1273403375_sp_nu @ A @ B )] :
( ( A2
= ( stream1686501333_nu_In @ A @ B @ Z3 ) )
=> ~ ( member @ B @ E2 @ ( stream2074884121_sp_mu @ A @ B @ ( stream1273403375_sp_nu @ A @ B ) @ Z3 ) ) )
=> ~ ! [Z3: stream901396144_sp_mu @ A @ B @ ( stream1273403375_sp_nu @ A @ B )] :
( ( A2
= ( stream1686501333_nu_In @ A @ B @ Z3 ) )
=> ! [Xa3: stream1273403375_sp_nu @ A @ B] :
( ( member @ ( stream1273403375_sp_nu @ A @ B ) @ Xa3 @ ( stream1259315544_sp_mu @ A @ B @ ( stream1273403375_sp_nu @ A @ B ) @ Z3 ) )
=> ~ ( member @ B @ E2 @ ( stream1493736486_sp_nu @ A @ B @ Xa3 ) ) ) ) ) ) ).
% sp\<^sub>\<nu>.set_cases
thf(fact_218_sp_092_060_094sub_062_092_060nu_062_Oset__intros_I1_J,axiom,
! [B: $tType,A: $tType,X: B,A2: stream901396144_sp_mu @ A @ B @ ( stream1273403375_sp_nu @ A @ B )] :
( ( member @ B @ X @ ( stream2074884121_sp_mu @ A @ B @ ( stream1273403375_sp_nu @ A @ B ) @ A2 ) )
=> ( member @ B @ X @ ( stream1493736486_sp_nu @ A @ B @ ( stream1686501333_nu_In @ A @ B @ A2 ) ) ) ) ).
% sp\<^sub>\<nu>.set_intros(1)
thf(fact_219_sp_092_060_094sub_062_092_060nu_062_Oset__intros_I2_J,axiom,
! [B: $tType,A: $tType,Xa: stream1273403375_sp_nu @ A @ B,A2: stream901396144_sp_mu @ A @ B @ ( stream1273403375_sp_nu @ A @ B ),Xb2: B] :
( ( member @ ( stream1273403375_sp_nu @ A @ B ) @ Xa @ ( stream1259315544_sp_mu @ A @ B @ ( stream1273403375_sp_nu @ A @ B ) @ A2 ) )
=> ( ( member @ B @ Xb2 @ ( stream1493736486_sp_nu @ A @ B @ Xa ) )
=> ( member @ B @ Xb2 @ ( stream1493736486_sp_nu @ A @ B @ ( stream1686501333_nu_In @ A @ B @ A2 ) ) ) ) ) ).
% sp\<^sub>\<nu>.set_intros(2)
thf(fact_220_sp_092_060_094sub_062_092_060nu_062_Oset__sel_I1_J,axiom,
! [B: $tType,A: $tType,Xa: B,A2: stream1273403375_sp_nu @ A @ B] :
( ( member @ B @ Xa @ ( stream2074884121_sp_mu @ A @ B @ ( stream1273403375_sp_nu @ A @ B ) @ ( stream1200245950nu_out @ A @ B @ A2 ) ) )
=> ( member @ B @ Xa @ ( stream1493736486_sp_nu @ A @ B @ A2 ) ) ) ).
% sp\<^sub>\<nu>.set_sel(1)
thf(fact_221_sp_092_060_094sub_062_092_060nu_062_Oset__sel_I2_J,axiom,
! [B: $tType,A: $tType,Xb2: stream1273403375_sp_nu @ A @ B,A2: stream1273403375_sp_nu @ A @ B,Xc: B] :
( ( member @ ( stream1273403375_sp_nu @ A @ B ) @ Xb2 @ ( stream1259315544_sp_mu @ A @ B @ ( stream1273403375_sp_nu @ A @ B ) @ ( stream1200245950nu_out @ A @ B @ A2 ) ) )
=> ( ( member @ B @ Xc @ ( stream1493736486_sp_nu @ A @ B @ Xb2 ) )
=> ( member @ B @ Xc @ ( stream1493736486_sp_nu @ A @ B @ A2 ) ) ) ) ).
% sp\<^sub>\<nu>.set_sel(2)
thf(fact_222_sp_092_060_094sub_062_092_060nu_062_Oset__map,axiom,
! [B: $tType,A: $tType,D: $tType,F: A > B,V: stream1273403375_sp_nu @ D @ A] :
( ( stream1493736486_sp_nu @ D @ B @ ( stream151454380_sp_nu @ A @ B @ D @ F @ V ) )
= ( image @ A @ B @ F @ ( stream1493736486_sp_nu @ D @ A @ V ) ) ) ).
% sp\<^sub>\<nu>.set_map
thf(fact_223_prod_Opred__transfer,axiom,
! [A: $tType,B: $tType,D: $tType,C: $tType,R1: A > C > $o,R2: B > D > $o] :
( bNF_rel_fun @ ( A > $o ) @ ( C > $o ) @ ( ( B > $o ) > ( product_prod @ A @ B ) > $o ) @ ( ( D > $o ) > ( product_prod @ C @ D ) > $o )
@ ( bNF_rel_fun @ A @ C @ $o @ $o @ R1
@ ^ [Y3: $o,Z: $o] : Y3 = Z )
@ ( bNF_rel_fun @ ( B > $o ) @ ( D > $o ) @ ( ( product_prod @ A @ B ) > $o ) @ ( ( product_prod @ C @ D ) > $o )
@ ( bNF_rel_fun @ B @ D @ $o @ $o @ R2
@ ^ [Y3: $o,Z: $o] : Y3 = Z )
@ ( bNF_rel_fun @ ( product_prod @ A @ B ) @ ( product_prod @ C @ D ) @ $o @ $o @ ( basic_rel_prod @ A @ C @ B @ D @ R1 @ R2 )
@ ^ [Y3: $o,Z: $o] : Y3 = Z ) )
@ ( basic_pred_prod @ A @ B )
@ ( basic_pred_prod @ C @ D ) ) ).
% prod.pred_transfer
thf(fact_224_sp_092_060_094sub_062_092_060nu_062_Oset,axiom,
! [B: $tType,A: $tType,X: stream901396144_sp_mu @ A @ B @ ( stream1273403375_sp_nu @ A @ B )] :
( ( stream1493736486_sp_nu @ A @ B @ ( stream1686501333_nu_In @ A @ B @ X ) )
= ( sup_sup @ ( set @ B ) @ ( stream2074884121_sp_mu @ A @ B @ ( stream1273403375_sp_nu @ A @ B ) @ X ) @ ( complete_Sup_Sup @ ( set @ B ) @ ( image @ ( stream1273403375_sp_nu @ A @ B ) @ ( set @ B ) @ ( stream1493736486_sp_nu @ A @ B ) @ ( stream1259315544_sp_mu @ A @ B @ ( stream1273403375_sp_nu @ A @ B ) @ X ) ) ) ) ) ).
% sp\<^sub>\<nu>.set
thf(fact_225_UnCI,axiom,
! [A: $tType,C3: A,B3: set @ A,A3: set @ A] :
( ( ~ ( member @ A @ C3 @ B3 )
=> ( member @ A @ C3 @ A3 ) )
=> ( member @ A @ C3 @ ( sup_sup @ ( set @ A ) @ A3 @ B3 ) ) ) ).
% UnCI
thf(fact_226_Un__iff,axiom,
! [A: $tType,C3: A,A3: set @ A,B3: set @ A] :
( ( member @ A @ C3 @ ( sup_sup @ ( set @ A ) @ A3 @ B3 ) )
= ( ( member @ A @ C3 @ A3 )
| ( member @ A @ C3 @ B3 ) ) ) ).
% Un_iff
thf(fact_227_Union__Un__distrib,axiom,
! [A: $tType,A3: set @ ( set @ A ),B3: set @ ( set @ A )] :
( ( complete_Sup_Sup @ ( set @ A ) @ ( sup_sup @ ( set @ ( set @ A ) ) @ A3 @ B3 ) )
= ( sup_sup @ ( set @ A ) @ ( complete_Sup_Sup @ ( set @ A ) @ A3 ) @ ( complete_Sup_Sup @ ( set @ A ) @ B3 ) ) ) ).
% Union_Un_distrib
thf(fact_228_UnE,axiom,
! [A: $tType,C3: A,A3: set @ A,B3: set @ A] :
( ( member @ A @ C3 @ ( sup_sup @ ( set @ A ) @ A3 @ B3 ) )
=> ( ~ ( member @ A @ C3 @ A3 )
=> ( member @ A @ C3 @ B3 ) ) ) ).
% UnE
thf(fact_229_UnI1,axiom,
! [A: $tType,C3: A,A3: set @ A,B3: set @ A] :
( ( member @ A @ C3 @ A3 )
=> ( member @ A @ C3 @ ( sup_sup @ ( set @ A ) @ A3 @ B3 ) ) ) ).
% UnI1
thf(fact_230_UnI2,axiom,
! [A: $tType,C3: A,B3: set @ A,A3: set @ A] :
( ( member @ A @ C3 @ B3 )
=> ( member @ A @ C3 @ ( sup_sup @ ( set @ A ) @ A3 @ B3 ) ) ) ).
% UnI2
thf(fact_231_bex__Un,axiom,
! [A: $tType,A3: set @ A,B3: set @ A,P: A > $o] :
( ( ? [X3: A] :
( ( member @ A @ X3 @ ( sup_sup @ ( set @ A ) @ A3 @ B3 ) )
& ( P @ X3 ) ) )
= ( ? [X3: A] :
( ( member @ A @ X3 @ A3 )
& ( P @ X3 ) )
| ? [X3: A] :
( ( member @ A @ X3 @ B3 )
& ( P @ X3 ) ) ) ) ).
% bex_Un
thf(fact_232_ball__Un,axiom,
! [A: $tType,A3: set @ A,B3: set @ A,P: A > $o] :
( ( ! [X3: A] :
( ( member @ A @ X3 @ ( sup_sup @ ( set @ A ) @ A3 @ B3 ) )
=> ( P @ X3 ) ) )
= ( ! [X3: A] :
( ( member @ A @ X3 @ A3 )
=> ( P @ X3 ) )
& ! [X3: A] :
( ( member @ A @ X3 @ B3 )
=> ( P @ X3 ) ) ) ) ).
% ball_Un
thf(fact_233_Un__assoc,axiom,
! [A: $tType,A3: set @ A,B3: set @ A,C2: set @ A] :
( ( sup_sup @ ( set @ A ) @ ( sup_sup @ ( set @ A ) @ A3 @ B3 ) @ C2 )
= ( sup_sup @ ( set @ A ) @ A3 @ ( sup_sup @ ( set @ A ) @ B3 @ C2 ) ) ) ).
% Un_assoc
thf(fact_234_Un__absorb,axiom,
! [A: $tType,A3: set @ A] :
( ( sup_sup @ ( set @ A ) @ A3 @ A3 )
= A3 ) ).
% Un_absorb
thf(fact_235_Un__commute,axiom,
! [A: $tType] :
( ( sup_sup @ ( set @ A ) )
= ( ^ [A4: set @ A,B5: set @ A] : ( sup_sup @ ( set @ A ) @ B5 @ A4 ) ) ) ).
% Un_commute
thf(fact_236_Un__left__absorb,axiom,
! [A: $tType,A3: set @ A,B3: set @ A] :
( ( sup_sup @ ( set @ A ) @ A3 @ ( sup_sup @ ( set @ A ) @ A3 @ B3 ) )
= ( sup_sup @ ( set @ A ) @ A3 @ B3 ) ) ).
% Un_left_absorb
thf(fact_237_Un__left__commute,axiom,
! [A: $tType,A3: set @ A,B3: set @ A,C2: set @ A] :
( ( sup_sup @ ( set @ A ) @ A3 @ ( sup_sup @ ( set @ A ) @ B3 @ C2 ) )
= ( sup_sup @ ( set @ A ) @ B3 @ ( sup_sup @ ( set @ A ) @ A3 @ C2 ) ) ) ).
% Un_left_commute
thf(fact_238_Un__UNIV__left,axiom,
! [A: $tType,B3: set @ A] :
( ( sup_sup @ ( set @ A ) @ ( top_top @ ( set @ A ) ) @ B3 )
= ( top_top @ ( set @ A ) ) ) ).
% Un_UNIV_left
thf(fact_239_Un__UNIV__right,axiom,
! [A: $tType,A3: set @ A] :
( ( sup_sup @ ( set @ A ) @ A3 @ ( top_top @ ( set @ A ) ) )
= ( top_top @ ( set @ A ) ) ) ).
% Un_UNIV_right
thf(fact_240_Sup__union__distrib,axiom,
! [A: $tType] :
( ( comple187826305attice @ A @ ( type2 @ A ) )
=> ! [A3: set @ A,B3: set @ A] :
( ( complete_Sup_Sup @ A @ ( sup_sup @ ( set @ A ) @ A3 @ B3 ) )
= ( sup_sup @ A @ ( complete_Sup_Sup @ A @ A3 ) @ ( complete_Sup_Sup @ A @ B3 ) ) ) ) ).
% Sup_union_distrib
thf(fact_241_image__Un,axiom,
! [A: $tType,B: $tType,F: B > A,A3: set @ B,B3: set @ B] :
( ( image @ B @ A @ F @ ( sup_sup @ ( set @ B ) @ A3 @ B3 ) )
= ( sup_sup @ ( set @ A ) @ ( image @ B @ A @ F @ A3 ) @ ( image @ B @ A @ F @ B3 ) ) ) ).
% image_Un
thf(fact_242_sup__top__left,axiom,
! [A: $tType] :
( ( bounded_lattice_top @ A @ ( type2 @ A ) )
=> ! [X: A] :
( ( sup_sup @ A @ ( top_top @ A ) @ X )
= ( top_top @ A ) ) ) ).
% sup_top_left
thf(fact_243_sup__top__right,axiom,
! [A: $tType] :
( ( bounded_lattice_top @ A @ ( type2 @ A ) )
=> ! [X: A] :
( ( sup_sup @ A @ X @ ( top_top @ A ) )
= ( top_top @ A ) ) ) ).
% sup_top_right
thf(fact_244_sum_Opred__transfer,axiom,
! [A: $tType,B: $tType,D: $tType,C: $tType,R1: A > C > $o,R2: B > D > $o] :
( bNF_rel_fun @ ( A > $o ) @ ( C > $o ) @ ( ( B > $o ) > ( sum_sum @ A @ B ) > $o ) @ ( ( D > $o ) > ( sum_sum @ C @ D ) > $o )
@ ( bNF_rel_fun @ A @ C @ $o @ $o @ R1
@ ^ [Y3: $o,Z: $o] : Y3 = Z )
@ ( bNF_rel_fun @ ( B > $o ) @ ( D > $o ) @ ( ( sum_sum @ A @ B ) > $o ) @ ( ( sum_sum @ C @ D ) > $o )
@ ( bNF_rel_fun @ B @ D @ $o @ $o @ R2
@ ^ [Y3: $o,Z: $o] : Y3 = Z )
@ ( bNF_rel_fun @ ( sum_sum @ A @ B ) @ ( sum_sum @ C @ D ) @ $o @ $o @ ( bNF_rel_sum @ A @ C @ B @ D @ R1 @ R2 )
@ ^ [Y3: $o,Z: $o] : Y3 = Z ) )
@ ( basic_pred_sum @ A @ B )
@ ( basic_pred_sum @ C @ D ) ) ).
% sum.pred_transfer
thf(fact_245_Pow__insert,axiom,
! [A: $tType,A2: A,A3: set @ A] :
( ( pow @ A @ ( insert @ A @ A2 @ A3 ) )
= ( sup_sup @ ( set @ ( set @ A ) ) @ ( pow @ A @ A3 ) @ ( image @ ( set @ A ) @ ( set @ A ) @ ( insert @ A @ A2 ) @ ( pow @ A @ A3 ) ) ) ) ).
% Pow_insert
thf(fact_246_insertCI,axiom,
! [A: $tType,A2: A,B3: set @ A,B2: A] :
( ( ~ ( member @ A @ A2 @ B3 )
=> ( A2 = B2 ) )
=> ( member @ A @ A2 @ ( insert @ A @ B2 @ B3 ) ) ) ).
% insertCI
thf(fact_247_insert__iff,axiom,
! [A: $tType,A2: A,B2: A,A3: set @ A] :
( ( member @ A @ A2 @ ( insert @ A @ B2 @ A3 ) )
= ( ( A2 = B2 )
| ( member @ A @ A2 @ A3 ) ) ) ).
% insert_iff
thf(fact_248_insert__absorb2,axiom,
! [A: $tType,X: A,A3: set @ A] :
( ( insert @ A @ X @ ( insert @ A @ X @ A3 ) )
= ( insert @ A @ X @ A3 ) ) ).
% insert_absorb2
thf(fact_249_image__insert,axiom,
! [A: $tType,B: $tType,F: B > A,A2: B,B3: set @ B] :
( ( image @ B @ A @ F @ ( insert @ B @ A2 @ B3 ) )
= ( insert @ A @ ( F @ A2 ) @ ( image @ B @ A @ F @ B3 ) ) ) ).
% image_insert
thf(fact_250_insert__image,axiom,
! [B: $tType,A: $tType,X: A,A3: set @ A,F: A > B] :
( ( member @ A @ X @ A3 )
=> ( ( insert @ B @ ( F @ X ) @ ( image @ A @ B @ F @ A3 ) )
= ( image @ A @ B @ F @ A3 ) ) ) ).
% insert_image
thf(fact_251_Un__insert__left,axiom,
! [A: $tType,A2: A,B3: set @ A,C2: set @ A] :
( ( sup_sup @ ( set @ A ) @ ( insert @ A @ A2 @ B3 ) @ C2 )
= ( insert @ A @ A2 @ ( sup_sup @ ( set @ A ) @ B3 @ C2 ) ) ) ).
% Un_insert_left
thf(fact_252_Un__insert__right,axiom,
! [A: $tType,A3: set @ A,A2: A,B3: set @ A] :
( ( sup_sup @ ( set @ A ) @ A3 @ ( insert @ A @ A2 @ B3 ) )
= ( insert @ A @ A2 @ ( sup_sup @ ( set @ A ) @ A3 @ B3 ) ) ) ).
% Un_insert_right
thf(fact_253_Sup__insert,axiom,
! [A: $tType] :
( ( comple187826305attice @ A @ ( type2 @ A ) )
=> ! [A2: A,A3: set @ A] :
( ( complete_Sup_Sup @ A @ ( insert @ A @ A2 @ A3 ) )
= ( sup_sup @ A @ A2 @ ( complete_Sup_Sup @ A @ A3 ) ) ) ) ).
% Sup_insert
thf(fact_254_Union__insert,axiom,
! [A: $tType,A2: set @ A,B3: set @ ( set @ A )] :
( ( complete_Sup_Sup @ ( set @ A ) @ ( insert @ ( set @ A ) @ A2 @ B3 ) )
= ( sup_sup @ ( set @ A ) @ A2 @ ( complete_Sup_Sup @ ( set @ A ) @ B3 ) ) ) ).
% Union_insert
thf(fact_255_Union__image__insert,axiom,
! [A: $tType,B: $tType,F: B > ( set @ A ),A2: B,B3: set @ B] :
( ( complete_Sup_Sup @ ( set @ A ) @ ( image @ B @ ( set @ A ) @ F @ ( insert @ B @ A2 @ B3 ) ) )
= ( sup_sup @ ( set @ A ) @ ( F @ A2 ) @ ( complete_Sup_Sup @ ( set @ A ) @ ( image @ B @ ( set @ A ) @ F @ B3 ) ) ) ) ).
% Union_image_insert
%----Type constructors (16)
thf(tcon_HOL_Obool___Lattices_Obounded__lattice,axiom,
bounded_lattice @ $o @ ( type2 @ $o ) ).
thf(tcon_Set_Oset___Lattices_Obounded__lattice_1,axiom,
! [A6: $tType] : ( bounded_lattice @ ( set @ A6 ) @ ( type2 @ ( set @ A6 ) ) ) ).
thf(tcon_fun___Lattices_Obounded__lattice_2,axiom,
! [A6: $tType,A7: $tType] :
( ( bounded_lattice @ A7 @ ( type2 @ A7 ) )
=> ( bounded_lattice @ ( A6 > A7 ) @ ( type2 @ ( A6 > A7 ) ) ) ) ).
thf(tcon_fun___Complete__Lattices_Ocomplete__lattice,axiom,
! [A6: $tType,A7: $tType] :
( ( comple187826305attice @ A7 @ ( type2 @ A7 ) )
=> ( comple187826305attice @ ( A6 > A7 ) @ ( type2 @ ( A6 > A7 ) ) ) ) ).
thf(tcon_fun___Lattices_Obounded__lattice__top,axiom,
! [A6: $tType,A7: $tType] :
( ( bounded_lattice @ A7 @ ( type2 @ A7 ) )
=> ( bounded_lattice_top @ ( A6 > A7 ) @ ( type2 @ ( A6 > A7 ) ) ) ) ).
thf(tcon_fun___Complete__Lattices_OSup,axiom,
! [A6: $tType,A7: $tType] :
( ( complete_Sup @ A7 @ ( type2 @ A7 ) )
=> ( complete_Sup @ ( A6 > A7 ) @ ( type2 @ ( A6 > A7 ) ) ) ) ).
thf(tcon_fun___Orderings_Otop,axiom,
! [A6: $tType,A7: $tType] :
( ( top @ A7 @ ( type2 @ A7 ) )
=> ( top @ ( A6 > A7 ) @ ( type2 @ ( A6 > A7 ) ) ) ) ).
thf(tcon_Nat_Onat___Complete__Lattices_OSup_3,axiom,
complete_Sup @ nat @ ( type2 @ nat ) ).
thf(tcon_Set_Oset___Complete__Lattices_Ocomplete__lattice_4,axiom,
! [A6: $tType] : ( comple187826305attice @ ( set @ A6 ) @ ( type2 @ ( set @ A6 ) ) ) ).
thf(tcon_Set_Oset___Lattices_Obounded__lattice__top_5,axiom,
! [A6: $tType] : ( bounded_lattice_top @ ( set @ A6 ) @ ( type2 @ ( set @ A6 ) ) ) ).
thf(tcon_Set_Oset___Complete__Lattices_OSup_6,axiom,
! [A6: $tType] : ( complete_Sup @ ( set @ A6 ) @ ( type2 @ ( set @ A6 ) ) ) ).
thf(tcon_Set_Oset___Orderings_Otop_7,axiom,
! [A6: $tType] : ( top @ ( set @ A6 ) @ ( type2 @ ( set @ A6 ) ) ) ).
thf(tcon_HOL_Obool___Complete__Lattices_Ocomplete__lattice_8,axiom,
comple187826305attice @ $o @ ( type2 @ $o ) ).
thf(tcon_HOL_Obool___Lattices_Obounded__lattice__top_9,axiom,
bounded_lattice_top @ $o @ ( type2 @ $o ) ).
thf(tcon_HOL_Obool___Complete__Lattices_OSup_10,axiom,
complete_Sup @ $o @ ( type2 @ $o ) ).
thf(tcon_HOL_Obool___Orderings_Otop_11,axiom,
top @ $o @ ( type2 @ $o ) ).
%----Conjectures (1)
thf(conj_0,conjecture,
( ( stream377071682run_nu @ a @ a @ ( stream2017582925e_copy @ a ) @ s )
= s ) ).
%------------------------------------------------------------------------------