TPTP Problem File: COM185^1.p
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%------------------------------------------------------------------------------
% File : COM185^1 : TPTP v8.2.0. Released v7.0.0.
% Domain : Computing Theory
% Problem : Grammars and languages 307
% 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 : gram_lang__307.p [Bla16]
% Status : Theorem
% Rating : 1.00 v8.1.0, 0.75 v7.5.0, 0.33 v7.2.0, 0.50 v7.1.0
% Syntax : Number of formulae : 322 ( 143 unt; 47 typ; 0 def)
% Number of atoms : 731 ( 375 equ; 0 cnn)
% Maximal formula atoms : 7 ( 2 avg)
% Number of connectives : 4154 ( 82 ~; 10 |; 96 &;3652 @)
% ( 0 <=>; 314 =>; 0 <=; 0 <~>)
% Maximal formula depth : 21 ( 7 avg)
% Number of types : 4 ( 3 usr)
% Number of type conns : 293 ( 293 >; 0 *; 0 +; 0 <<)
% Number of symbols : 47 ( 44 usr; 4 con; 0-9 aty)
% Number of variables : 1282 ( 131 ^;1033 !; 71 ?;1282 :)
% ( 47 !>; 0 ?*; 0 @-; 0 @+)
% SPC : TH1_THM_EQU_NAR
% Comments : This file was generated by Isabelle (most likely Sledgehammer)
% 2016-07-13 14:42:04.312
%------------------------------------------------------------------------------
%----Could-be-implicit typings (6)
thf(ty_t_Sum__Type_Osum,type,
sum_sum: $tType > $tType > $tType ).
thf(ty_t_DTree_Odtree,type,
dtree: $tType ).
thf(ty_t_Set_Oset,type,
set: $tType > $tType ).
thf(ty_t_DTree_OT,type,
t: $tType ).
thf(ty_t_DTree_ON,type,
n: $tType ).
thf(ty_t_itself,type,
itself: $tType > $tType ).
%----Explicit typings (41)
thf(sy_cl_HOL_Otype,type,
type:
!>[A: $tType] : ( ( itself @ A ) > $o ) ).
thf(sy_cl_Orderings_Obot,type,
bot:
!>[A: $tType] : ( ( itself @ A ) > $o ) ).
thf(sy_cl_Complete__Lattices_OSup,type,
complete_Sup:
!>[A: $tType] : ( ( itself @ A ) > $o ) ).
thf(sy_cl_Complete__Partial__Order_Occpo,type,
comple1141879883l_ccpo:
!>[A: $tType] : ( ( itself @ A ) > $o ) ).
thf(sy_cl_Complete__Lattices_Ocomplete__lattice,type,
comple187826305attice:
!>[A: $tType] : ( ( itself @ A ) > $o ) ).
thf(sy_cl_Conditionally__Complete__Lattices_Oconditionally__complete__lattice,type,
condit378418413attice:
!>[A: $tType] : ( ( itself @ A ) > $o ) ).
thf(sy_c_BNF__Greatest__Fixpoint_Oimage2p,type,
bNF_Greatest_image2p:
!>[C: $tType,A: $tType,D: $tType,B: $tType] : ( ( C > A ) > ( D > B ) > ( C > D > $o ) > A > B > $o ) ).
thf(sy_c_Basic__BNFs_Osetl,type,
basic_setl:
!>[A: $tType,B: $tType] : ( ( sum_sum @ A @ B ) > ( set @ A ) ) ).
thf(sy_c_Basic__BNFs_Osetr,type,
basic_setr:
!>[A: $tType,B: $tType] : ( ( sum_sum @ A @ B ) > ( set @ B ) ) ).
thf(sy_c_Complete__Lattices_OSup__class_OSup,type,
complete_Sup_Sup:
!>[A: $tType] : ( ( set @ A ) > A ) ).
thf(sy_c_DTree_ONode,type,
node: n > ( set @ ( sum_sum @ t @ dtree ) ) > dtree ).
thf(sy_c_DTree_Ocont,type,
cont: dtree > ( set @ ( sum_sum @ t @ dtree ) ) ).
thf(sy_c_DTree_Odtree_Oroot,type,
root: dtree > n ).
thf(sy_c_Gram__Lang__Mirabelle__ojxrtuoybn_OFr,type,
gram_L861583724lle_Fr: ( set @ n ) > dtree > ( set @ t ) ).
thf(sy_c_Gram__Lang__Mirabelle__ojxrtuoybn_OItr,type,
gram_L1580978439le_Itr: ( set @ n ) > dtree > ( set @ n ) ).
thf(sy_c_Gram__Lang__Mirabelle__ojxrtuoybn_OinFr,type,
gram_L1333338417e_inFr: ( set @ n ) > dtree > t > $o ).
thf(sy_c_Gram__Lang__Mirabelle__ojxrtuoybn_OinFr2,type,
gram_L805317441_inFr2: ( set @ n ) > dtree > t > $o ).
thf(sy_c_Gram__Lang__Mirabelle__ojxrtuoybn_OinItr,type,
gram_L830233218_inItr: ( set @ n ) > dtree > n > $o ).
thf(sy_c_Gram__Lang__Mirabelle__ojxrtuoybn_Osubtr,type,
gram_L716654942_subtr: ( set @ n ) > dtree > dtree > $o ).
thf(sy_c_Gram__Lang__Mirabelle__ojxrtuoybn_Osubtr2,type,
gram_L1283001940subtr2: ( set @ n ) > dtree > dtree > $o ).
thf(sy_c_HOL_OThe,type,
the:
!>[A: $tType] : ( ( A > $o ) > A ) ).
thf(sy_c_If,type,
if:
!>[A: $tType] : ( $o > A > A > A ) ).
thf(sy_c_Lifting_Orel__pred__comp,type,
rel_pred_comp:
!>[A: $tType,B: $tType] : ( ( A > B > $o ) > ( B > $o ) > A > $o ) ).
thf(sy_c_Orderings_Obot__class_Obot,type,
bot_bot:
!>[A: $tType] : A ).
thf(sy_c_Pure_Otype,type,
type2:
!>[A: $tType] : ( itself @ A ) ).
thf(sy_c_Set_OCollect,type,
collect:
!>[A: $tType] : ( ( A > $o ) > ( set @ A ) ) ).
thf(sy_c_Set_Oimage,type,
image:
!>[A: $tType,B: $tType] : ( ( A > B ) > ( set @ A ) > ( set @ B ) ) ).
thf(sy_c_Set_Oinsert,type,
insert:
!>[A: $tType] : ( A > ( set @ A ) > ( set @ A ) ) ).
thf(sy_c_Set_Ois__empty,type,
is_empty:
!>[A: $tType] : ( ( set @ A ) > $o ) ).
thf(sy_c_Set_Ois__singleton,type,
is_singleton:
!>[A: $tType] : ( ( set @ A ) > $o ) ).
thf(sy_c_Set_Othe__elem,type,
the_elem:
!>[A: $tType] : ( ( set @ A ) > A ) ).
thf(sy_c_Set_Ovimage,type,
vimage:
!>[A: $tType,B: $tType] : ( ( A > B ) > ( set @ B ) > ( set @ A ) ) ).
thf(sy_c_Sum__Type_OInl,type,
sum_Inl:
!>[A: $tType,B: $tType] : ( A > ( sum_sum @ A @ B ) ) ).
thf(sy_c_Sum__Type_OInr,type,
sum_Inr:
!>[B: $tType,A: $tType] : ( B > ( sum_sum @ A @ B ) ) ).
thf(sy_c_Sum__Type_OSuml,type,
sum_Suml:
!>[A: $tType,C: $tType,B: $tType] : ( ( A > C ) > ( sum_sum @ A @ B ) > C ) ).
thf(sy_c_Sum__Type_Oold_Osum_Orec__set__sum,type,
sum_rec_set_sum:
!>[A: $tType,T: $tType,B: $tType] : ( ( A > T ) > ( B > T ) > ( sum_sum @ A @ B ) > T > $o ) ).
thf(sy_c_Sum__Type_Oold_Osum_Orec__sum,type,
sum_rec_sum:
!>[A: $tType,T: $tType,B: $tType] : ( ( A > T ) > ( B > T ) > ( sum_sum @ A @ B ) > T ) ).
thf(sy_c_Sum__Type_Osum_Ocase__sum,type,
sum_case_sum:
!>[A: $tType,C: $tType,B: $tType] : ( ( A > C ) > ( B > C ) > ( sum_sum @ A @ B ) > C ) ).
thf(sy_c_member,type,
member:
!>[A: $tType] : ( A > ( set @ A ) > $o ) ).
thf(sy_v_ns,type,
ns: set @ n ).
thf(sy_v_tr,type,
tr: dtree ).
%----Relevant facts (256)
thf(fact_0_inItr_OBase,axiom,
! [Tr: dtree,Ns: set @ n] :
( ( member @ n @ ( root @ Tr ) @ Ns )
=> ( gram_L830233218_inItr @ Ns @ Tr @ ( root @ Tr ) ) ) ).
% inItr.Base
thf(fact_1_subtr_ORefl,axiom,
! [Tr: dtree,Ns: set @ n] :
( ( member @ n @ ( root @ Tr ) @ Ns )
=> ( gram_L716654942_subtr @ Ns @ Tr @ Tr ) ) ).
% subtr.Refl
thf(fact_2_inItr__subtr,axiom,
! [Ns: set @ n,Tr: dtree,N: n] :
( ( gram_L830233218_inItr @ Ns @ Tr @ N )
=> ? [Tr2: dtree] :
( ( gram_L716654942_subtr @ Ns @ Tr2 @ Tr )
& ( ( root @ Tr2 )
= N ) ) ) ).
% inItr_subtr
thf(fact_3_subtr__inItr,axiom,
! [Ns: set @ n,Tr: dtree,N: n,Tr1: dtree] :
( ( gram_L830233218_inItr @ Ns @ Tr @ N )
=> ( ( gram_L716654942_subtr @ Ns @ Tr @ Tr1 )
=> ( gram_L830233218_inItr @ Ns @ Tr1 @ N ) ) ) ).
% subtr_inItr
thf(fact_4_subtr__trans,axiom,
! [Ns: set @ n,Tr1: dtree,Tr22: dtree,Tr3: dtree] :
( ( gram_L716654942_subtr @ Ns @ Tr1 @ Tr22 )
=> ( ( gram_L716654942_subtr @ Ns @ Tr22 @ Tr3 )
=> ( gram_L716654942_subtr @ Ns @ Tr1 @ Tr3 ) ) ) ).
% subtr_trans
thf(fact_5_inItr__root__in,axiom,
! [Ns: set @ n,Tr: dtree,N: n] :
( ( gram_L830233218_inItr @ Ns @ Tr @ N )
=> ( member @ n @ ( root @ Tr ) @ Ns ) ) ).
% inItr_root_in
thf(fact_6_subtr__rootL__in,axiom,
! [Ns: set @ n,Tr1: dtree,Tr22: dtree] :
( ( gram_L716654942_subtr @ Ns @ Tr1 @ Tr22 )
=> ( member @ n @ ( root @ Tr1 ) @ Ns ) ) ).
% subtr_rootL_in
thf(fact_7_subtr__rootR__in,axiom,
! [Ns: set @ n,Tr1: dtree,Tr22: dtree] :
( ( gram_L716654942_subtr @ Ns @ Tr1 @ Tr22 )
=> ( member @ n @ ( root @ Tr22 ) @ Ns ) ) ).
% subtr_rootR_in
thf(fact_8_Itr__def,axiom,
( gram_L1580978439le_Itr
= ( ^ [Ns2: set @ n,Tr4: dtree] : ( collect @ n @ ( gram_L830233218_inItr @ Ns2 @ Tr4 ) ) ) ) ).
% Itr_def
thf(fact_9_rel__pred__comp__def,axiom,
! [B: $tType,A: $tType] :
( ( rel_pred_comp @ A @ B )
= ( ^ [R: A > B > $o,P: B > $o,X: A] :
? [Y: B] :
( ( R @ X @ Y )
& ( P @ Y ) ) ) ) ).
% rel_pred_comp_def
thf(fact_10_Itr__subtr,axiom,
( gram_L1580978439le_Itr
= ( ^ [Ns2: set @ n,Tr4: dtree] :
( complete_Sup_Sup @ ( set @ n )
@ ( collect @ ( set @ n )
@ ^ [Uu: set @ n] :
? [Tr5: dtree] :
( ( Uu
= ( gram_L1580978439le_Itr @ Ns2 @ Tr5 ) )
& ( gram_L716654942_subtr @ Ns2 @ Tr5 @ Tr4 ) ) ) ) ) ) ).
% Itr_subtr
thf(fact_11_inFr2__root__in,axiom,
! [Ns: set @ n,Tr: dtree,T2: t] :
( ( gram_L805317441_inFr2 @ Ns @ Tr @ T2 )
=> ( member @ n @ ( root @ Tr ) @ Ns ) ) ).
% inFr2_root_in
thf(fact_12_Fr__subtr,axiom,
( gram_L861583724lle_Fr
= ( ^ [Ns2: set @ n,Tr4: dtree] :
( complete_Sup_Sup @ ( set @ t )
@ ( collect @ ( set @ t )
@ ^ [Uu: set @ t] :
? [Tr5: dtree] :
( ( Uu
= ( gram_L861583724lle_Fr @ Ns2 @ Tr5 ) )
& ( gram_L716654942_subtr @ Ns2 @ Tr5 @ Tr4 ) ) ) ) ) ) ).
% Fr_subtr
thf(fact_13_subtr__inFr,axiom,
! [Ns: set @ n,Tr: dtree,T2: t,Tr1: dtree] :
( ( gram_L1333338417e_inFr @ Ns @ Tr @ T2 )
=> ( ( gram_L716654942_subtr @ Ns @ Tr @ Tr1 )
=> ( gram_L1333338417e_inFr @ Ns @ Tr1 @ T2 ) ) ) ).
% subtr_inFr
thf(fact_14_subtr__subtr2,axiom,
gram_L716654942_subtr = gram_L1283001940subtr2 ).
% subtr_subtr2
thf(fact_15_inFr__root__in,axiom,
! [Ns: set @ n,Tr: dtree,T2: t] :
( ( gram_L1333338417e_inFr @ Ns @ Tr @ T2 )
=> ( member @ n @ ( root @ Tr ) @ Ns ) ) ).
% inFr_root_in
thf(fact_16_not__root__inFr,axiom,
! [Tr: dtree,Ns: set @ n,T2: t] :
( ~ ( member @ n @ ( root @ Tr ) @ Ns )
=> ~ ( gram_L1333338417e_inFr @ Ns @ Tr @ T2 ) ) ).
% not_root_inFr
thf(fact_17_subtr2_ORefl,axiom,
! [Tr: dtree,Ns: set @ n] :
( ( member @ n @ ( root @ Tr ) @ Ns )
=> ( gram_L1283001940subtr2 @ Ns @ Tr @ Tr ) ) ).
% subtr2.Refl
thf(fact_18_subtr2__rootL__in,axiom,
! [Ns: set @ n,Tr1: dtree,Tr22: dtree] :
( ( gram_L1283001940subtr2 @ Ns @ Tr1 @ Tr22 )
=> ( member @ n @ ( root @ Tr1 ) @ Ns ) ) ).
% subtr2_rootL_in
thf(fact_19_subtr2__rootR__in,axiom,
! [Ns: set @ n,Tr1: dtree,Tr22: dtree] :
( ( gram_L1283001940subtr2 @ Ns @ Tr1 @ Tr22 )
=> ( member @ n @ ( root @ Tr22 ) @ Ns ) ) ).
% subtr2_rootR_in
thf(fact_20_inFr__inFr2,axiom,
gram_L1333338417e_inFr = gram_L805317441_inFr2 ).
% inFr_inFr2
thf(fact_21_subtr2__trans,axiom,
! [Ns: set @ n,Tr1: dtree,Tr22: dtree,Tr3: dtree] :
( ( gram_L1283001940subtr2 @ Ns @ Tr1 @ Tr22 )
=> ( ( gram_L1283001940subtr2 @ Ns @ Tr22 @ Tr3 )
=> ( gram_L1283001940subtr2 @ Ns @ Tr1 @ Tr3 ) ) ) ).
% subtr2_trans
thf(fact_22_Fr__def,axiom,
( gram_L861583724lle_Fr
= ( ^ [Ns2: set @ n,Tr4: dtree] : ( collect @ t @ ( gram_L1333338417e_inFr @ Ns2 @ Tr4 ) ) ) ) ).
% Fr_def
thf(fact_23_UN__ball__bex__simps_I3_J,axiom,
! [D: $tType,A2: set @ ( set @ D ),P2: D > $o] :
( ( ? [X: D] :
( ( member @ D @ X @ ( complete_Sup_Sup @ ( set @ D ) @ A2 ) )
& ( P2 @ X ) ) )
= ( ? [X: set @ D] :
( ( member @ ( set @ D ) @ X @ A2 )
& ? [Y: D] :
( ( member @ D @ Y @ X )
& ( P2 @ Y ) ) ) ) ) ).
% UN_ball_bex_simps(3)
thf(fact_24_UN__ball__bex__simps_I1_J,axiom,
! [A: $tType,A2: set @ ( set @ A ),P2: A > $o] :
( ( ! [X: A] :
( ( member @ A @ X @ ( complete_Sup_Sup @ ( set @ A ) @ A2 ) )
=> ( P2 @ X ) ) )
= ( ! [X: set @ A] :
( ( member @ ( set @ A ) @ X @ A2 )
=> ! [Y: A] :
( ( member @ A @ Y @ X )
=> ( P2 @ Y ) ) ) ) ) ).
% UN_ball_bex_simps(1)
thf(fact_25_UnionI,axiom,
! [A: $tType,X2: set @ A,C2: set @ ( set @ A ),A2: A] :
( ( member @ ( set @ A ) @ X2 @ C2 )
=> ( ( member @ A @ A2 @ X2 )
=> ( member @ A @ A2 @ ( complete_Sup_Sup @ ( set @ A ) @ C2 ) ) ) ) ).
% UnionI
thf(fact_26_Union__iff,axiom,
! [A: $tType,A2: A,C2: set @ ( set @ A )] :
( ( member @ A @ A2 @ ( complete_Sup_Sup @ ( set @ A ) @ C2 ) )
= ( ? [X: set @ A] :
( ( member @ ( set @ A ) @ X @ C2 )
& ( member @ A @ A2 @ X ) ) ) ) ).
% Union_iff
thf(fact_27_Union__SetCompr__eq,axiom,
! [B: $tType,A: $tType,F: B > ( set @ A ),P2: B > $o] :
( ( complete_Sup_Sup @ ( set @ A )
@ ( collect @ ( set @ A )
@ ^ [Uu: set @ A] :
? [X: B] :
( ( Uu
= ( F @ X ) )
& ( P2 @ X ) ) ) )
= ( collect @ A
@ ^ [A3: A] :
? [X: B] :
( ( P2 @ X )
& ( member @ A @ A3 @ ( F @ X ) ) ) ) ) ).
% Union_SetCompr_eq
thf(fact_28_UnionE,axiom,
! [A: $tType,A2: A,C2: set @ ( set @ A )] :
( ( member @ A @ A2 @ ( complete_Sup_Sup @ ( set @ A ) @ C2 ) )
=> ~ ! [X3: set @ A] :
( ( member @ A @ A2 @ X3 )
=> ~ ( member @ ( set @ A ) @ X3 @ C2 ) ) ) ).
% UnionE
thf(fact_29_not__root__Fr,axiom,
! [Tr: dtree,Ns: set @ n] :
( ~ ( member @ n @ ( root @ Tr ) @ Ns )
=> ( ( gram_L861583724lle_Fr @ Ns @ Tr )
= ( bot_bot @ ( set @ t ) ) ) ) ).
% not_root_Fr
thf(fact_30_Fr__subtr__cont,axiom,
( gram_L861583724lle_Fr
= ( ^ [Ns2: set @ n,Tr4: dtree] :
( complete_Sup_Sup @ ( set @ t )
@ ( collect @ ( set @ t )
@ ^ [Uu: set @ t] :
? [Tr5: dtree] :
( ( Uu
= ( vimage @ t @ ( sum_sum @ t @ dtree ) @ ( sum_Inl @ t @ dtree ) @ ( cont @ Tr5 ) ) )
& ( gram_L716654942_subtr @ Ns2 @ Tr5 @ Tr4 ) ) ) ) ) ) ).
% Fr_subtr_cont
thf(fact_31_image2p__def,axiom,
! [D: $tType,B: $tType,A: $tType,C: $tType] :
( ( bNF_Greatest_image2p @ C @ A @ D @ B )
= ( ^ [F2: C > A,G: D > B,R: C > D > $o,X: A,Y: B] :
? [X4: C,Y2: D] :
( ( R @ X4 @ Y2 )
& ( ( F2 @ X4 )
= X )
& ( ( G @ Y2 )
= Y ) ) ) ) ).
% image2p_def
thf(fact_32_root__Node,axiom,
! [N: n,As: set @ ( sum_sum @ t @ dtree )] :
( ( root @ ( node @ N @ As ) )
= N ) ).
% root_Node
thf(fact_33_Sup__bot__conv_I1_J,axiom,
! [A: $tType] :
( ( comple187826305attice @ A @ ( type2 @ A ) )
=> ! [A2: set @ A] :
( ( ( complete_Sup_Sup @ A @ A2 )
= ( bot_bot @ A ) )
= ( ! [X: A] :
( ( member @ A @ X @ A2 )
=> ( X
= ( bot_bot @ A ) ) ) ) ) ) ).
% Sup_bot_conv(1)
thf(fact_34_Sup__bot__conv_I2_J,axiom,
! [A: $tType] :
( ( comple187826305attice @ A @ ( type2 @ A ) )
=> ! [A2: set @ A] :
( ( ( bot_bot @ A )
= ( complete_Sup_Sup @ A @ A2 ) )
= ( ! [X: A] :
( ( member @ A @ X @ A2 )
=> ( X
= ( bot_bot @ A ) ) ) ) ) ) ).
% Sup_bot_conv(2)
thf(fact_35_Sup__empty,axiom,
! [A: $tType] :
( ( comple187826305attice @ A @ ( type2 @ A ) )
=> ( ( complete_Sup_Sup @ A @ ( bot_bot @ ( set @ A ) ) )
= ( bot_bot @ A ) ) ) ).
% Sup_empty
thf(fact_36_Node__root__cont,axiom,
! [Tr: dtree] :
( ( node @ ( root @ Tr ) @ ( cont @ Tr ) )
= Tr ) ).
% Node_root_cont
thf(fact_37_Union__empty,axiom,
! [A: $tType] :
( ( complete_Sup_Sup @ ( set @ A ) @ ( bot_bot @ ( set @ ( set @ A ) ) ) )
= ( bot_bot @ ( set @ A ) ) ) ).
% Union_empty
thf(fact_38_image2pI,axiom,
! [A: $tType,C: $tType,D: $tType,B: $tType,R2: A > B > $o,X5: A,Y3: B,F: A > C,G2: B > D] :
( ( R2 @ X5 @ Y3 )
=> ( bNF_Greatest_image2p @ A @ C @ B @ D @ F @ G2 @ R2 @ ( F @ X5 ) @ ( G2 @ Y3 ) ) ) ).
% image2pI
thf(fact_39_image2pE,axiom,
! [D: $tType,B: $tType,A: $tType,C: $tType,F: A > B,G2: C > D,R2: A > C > $o,Fx: B,Gy: D] :
( ( bNF_Greatest_image2p @ A @ B @ C @ D @ F @ G2 @ R2 @ Fx @ Gy )
=> ~ ! [X6: A] :
( ( Fx
= ( F @ X6 ) )
=> ! [Y4: C] :
( ( Gy
= ( G2 @ Y4 ) )
=> ~ ( R2 @ X6 @ Y4 ) ) ) ) ).
% image2pE
thf(fact_40_empty__Union__conv,axiom,
! [A: $tType,A2: set @ ( set @ A )] :
( ( ( bot_bot @ ( set @ A ) )
= ( complete_Sup_Sup @ ( set @ A ) @ A2 ) )
= ( ! [X: set @ A] :
( ( member @ ( set @ A ) @ X @ A2 )
=> ( X
= ( bot_bot @ ( set @ A ) ) ) ) ) ) ).
% empty_Union_conv
thf(fact_41_Union__empty__conv,axiom,
! [A: $tType,A2: set @ ( set @ A )] :
( ( ( complete_Sup_Sup @ ( set @ A ) @ A2 )
= ( bot_bot @ ( set @ A ) ) )
= ( ! [X: set @ A] :
( ( member @ ( set @ A ) @ X @ A2 )
=> ( X
= ( bot_bot @ ( set @ A ) ) ) ) ) ) ).
% Union_empty_conv
thf(fact_42_dtree__cong,axiom,
! [Tr: dtree,Tr6: dtree] :
( ( ( root @ Tr )
= ( root @ Tr6 ) )
=> ( ( ( cont @ Tr )
= ( cont @ Tr6 ) )
=> ( Tr = Tr6 ) ) ) ).
% dtree_cong
thf(fact_43_inFr_OBase,axiom,
! [Tr: dtree,Ns: set @ n,T2: t] :
( ( member @ n @ ( root @ Tr ) @ Ns )
=> ( ( member @ ( sum_sum @ t @ dtree ) @ ( sum_Inl @ t @ dtree @ T2 ) @ ( cont @ Tr ) )
=> ( gram_L1333338417e_inFr @ Ns @ Tr @ T2 ) ) ) ).
% inFr.Base
thf(fact_44_inFr2_OBase,axiom,
! [Tr: dtree,Ns: set @ n,T2: t] :
( ( member @ n @ ( root @ Tr ) @ Ns )
=> ( ( member @ ( sum_sum @ t @ dtree ) @ ( sum_Inl @ t @ dtree @ T2 ) @ ( cont @ Tr ) )
=> ( gram_L805317441_inFr2 @ Ns @ Tr @ T2 ) ) ) ).
% inFr2.Base
thf(fact_45_mem__Collect__eq,axiom,
! [A: $tType,A4: A,P2: A > $o] :
( ( member @ A @ A4 @ ( collect @ A @ P2 ) )
= ( P2 @ A4 ) ) ).
% mem_Collect_eq
thf(fact_46_Collect__mem__eq,axiom,
! [A: $tType,A2: set @ A] :
( ( collect @ A
@ ^ [X: A] : ( member @ A @ X @ A2 ) )
= A2 ) ).
% Collect_mem_eq
thf(fact_47_Collect__cong,axiom,
! [A: $tType,P2: A > $o,Q: A > $o] :
( ! [X6: A] :
( ( P2 @ X6 )
= ( Q @ X6 ) )
=> ( ( collect @ A @ P2 )
= ( collect @ A @ Q ) ) ) ).
% Collect_cong
thf(fact_48_ext,axiom,
! [B: $tType,A: $tType,F: A > B,G2: A > B] :
( ! [X6: A] :
( ( F @ X6 )
= ( G2 @ X6 ) )
=> ( F = G2 ) ) ).
% ext
thf(fact_49_inFr__subtr,axiom,
! [Ns: set @ n,Tr: dtree,T2: t] :
( ( gram_L1333338417e_inFr @ Ns @ Tr @ T2 )
=> ? [Tr2: dtree] :
( ( gram_L716654942_subtr @ Ns @ Tr2 @ Tr )
& ( member @ ( sum_sum @ t @ dtree ) @ ( sum_Inl @ t @ dtree @ T2 ) @ ( cont @ Tr2 ) ) ) ) ).
% inFr_subtr
thf(fact_50_vimage__empty,axiom,
! [B: $tType,A: $tType,F: A > B] :
( ( vimage @ A @ B @ F @ ( bot_bot @ ( set @ B ) ) )
= ( bot_bot @ ( set @ A ) ) ) ).
% vimage_empty
thf(fact_51_vimage__Collect__eq,axiom,
! [B: $tType,A: $tType,F: A > B,P2: B > $o] :
( ( vimage @ A @ B @ F @ ( collect @ B @ P2 ) )
= ( collect @ A
@ ^ [Y: A] : ( P2 @ ( F @ Y ) ) ) ) ).
% vimage_Collect_eq
thf(fact_52_vimage__ident,axiom,
! [A: $tType,Y5: set @ A] :
( ( vimage @ A @ A
@ ^ [X: A] : X
@ Y5 )
= Y5 ) ).
% vimage_ident
thf(fact_53_vimage__eq,axiom,
! [A: $tType,B: $tType,A4: A,F: A > B,B2: set @ B] :
( ( member @ A @ A4 @ ( vimage @ A @ B @ F @ B2 ) )
= ( member @ B @ ( F @ A4 ) @ B2 ) ) ).
% vimage_eq
thf(fact_54_vimageI,axiom,
! [B: $tType,A: $tType,F: B > A,A4: B,B3: A,B2: set @ A] :
( ( ( F @ A4 )
= B3 )
=> ( ( member @ A @ B3 @ B2 )
=> ( member @ B @ A4 @ ( vimage @ B @ A @ F @ B2 ) ) ) ) ).
% vimageI
thf(fact_55_old_Osum_Oinject_I1_J,axiom,
! [B: $tType,A: $tType,A4: A,A5: A] :
( ( ( sum_Inl @ A @ B @ A4 )
= ( sum_Inl @ A @ B @ A5 ) )
= ( A4 = A5 ) ) ).
% old.sum.inject(1)
thf(fact_56_sum_Oinject_I1_J,axiom,
! [B: $tType,A: $tType,X1: A,Y1: A] :
( ( ( sum_Inl @ A @ B @ X1 )
= ( sum_Inl @ A @ B @ Y1 ) )
= ( X1 = Y1 ) ) ).
% sum.inject(1)
thf(fact_57_empty__Collect__eq,axiom,
! [A: $tType,P2: A > $o] :
( ( ( bot_bot @ ( set @ A ) )
= ( collect @ A @ P2 ) )
= ( ! [X: A] :
~ ( P2 @ X ) ) ) ).
% empty_Collect_eq
thf(fact_58_Collect__empty__eq,axiom,
! [A: $tType,P2: A > $o] :
( ( ( collect @ A @ P2 )
= ( bot_bot @ ( set @ A ) ) )
= ( ! [X: A] :
~ ( P2 @ X ) ) ) ).
% Collect_empty_eq
thf(fact_59_empty__iff,axiom,
! [A: $tType,C3: A] :
~ ( member @ A @ C3 @ ( bot_bot @ ( set @ A ) ) ) ).
% empty_iff
thf(fact_60_all__not__in__conv,axiom,
! [A: $tType,A2: set @ A] :
( ( ! [X: A] :
~ ( member @ A @ X @ A2 ) )
= ( A2
= ( bot_bot @ ( set @ A ) ) ) ) ).
% all_not_in_conv
thf(fact_61_bot__set__def,axiom,
! [A: $tType] :
( ( bot_bot @ ( set @ A ) )
= ( collect @ A @ ( bot_bot @ ( A > $o ) ) ) ) ).
% bot_set_def
thf(fact_62_emptyE,axiom,
! [A: $tType,A4: A] :
~ ( member @ A @ A4 @ ( bot_bot @ ( set @ A ) ) ) ).
% emptyE
thf(fact_63_equals0D,axiom,
! [A: $tType,A2: set @ A,A4: A] :
( ( A2
= ( bot_bot @ ( set @ A ) ) )
=> ~ ( member @ A @ A4 @ A2 ) ) ).
% equals0D
thf(fact_64_equals0I,axiom,
! [A: $tType,A2: set @ A] :
( ! [Y4: A] :
~ ( member @ A @ Y4 @ A2 )
=> ( A2
= ( bot_bot @ ( set @ A ) ) ) ) ).
% equals0I
thf(fact_65_ex__in__conv,axiom,
! [A: $tType,A2: set @ A] :
( ( ? [X: A] : ( member @ A @ X @ A2 ) )
= ( A2
!= ( bot_bot @ ( set @ A ) ) ) ) ).
% ex_in_conv
thf(fact_66_Inl__inject,axiom,
! [B: $tType,A: $tType,X5: A,Y3: A] :
( ( ( sum_Inl @ A @ B @ X5 )
= ( sum_Inl @ A @ B @ Y3 ) )
=> ( X5 = Y3 ) ) ).
% Inl_inject
thf(fact_67_vimageD,axiom,
! [A: $tType,B: $tType,A4: A,F: A > B,A2: set @ B] :
( ( member @ A @ A4 @ ( vimage @ A @ B @ F @ A2 ) )
=> ( member @ B @ ( F @ A4 ) @ A2 ) ) ).
% vimageD
thf(fact_68_vimageE,axiom,
! [A: $tType,B: $tType,A4: A,F: A > B,B2: set @ B] :
( ( member @ A @ A4 @ ( vimage @ A @ B @ F @ B2 ) )
=> ( member @ B @ ( F @ A4 ) @ B2 ) ) ).
% vimageE
thf(fact_69_vimageI2,axiom,
! [B: $tType,A: $tType,F: B > A,A4: B,A2: set @ A] :
( ( member @ A @ ( F @ A4 ) @ A2 )
=> ( member @ B @ A4 @ ( vimage @ B @ A @ F @ A2 ) ) ) ).
% vimageI2
thf(fact_70_vimage__Collect,axiom,
! [B: $tType,A: $tType,P2: B > $o,F: A > B,Q: A > $o] :
( ! [X6: A] :
( ( P2 @ ( F @ X6 ) )
= ( Q @ X6 ) )
=> ( ( vimage @ A @ B @ F @ ( collect @ B @ P2 ) )
= ( collect @ A @ Q ) ) ) ).
% vimage_Collect
thf(fact_71_empty__def,axiom,
! [A: $tType] :
( ( bot_bot @ ( set @ A ) )
= ( collect @ A
@ ^ [X: A] : $false ) ) ).
% empty_def
thf(fact_72_vimage__def,axiom,
! [B: $tType,A: $tType] :
( ( vimage @ A @ B )
= ( ^ [F2: A > B,B4: set @ B] :
( collect @ A
@ ^ [X: A] : ( member @ B @ ( F2 @ X ) @ B4 ) ) ) ) ).
% vimage_def
thf(fact_73_bot__apply,axiom,
! [C: $tType,D: $tType] :
( ( bot @ C @ ( type2 @ C ) )
=> ( ( bot_bot @ ( D > C ) )
= ( ^ [X: D] : ( bot_bot @ C ) ) ) ) ).
% bot_apply
thf(fact_74_old_Osum_Osimps_I7_J,axiom,
! [B: $tType,T: $tType,A: $tType,F1: A > T,F22: B > T,A4: A] :
( ( sum_rec_sum @ A @ T @ B @ F1 @ F22 @ ( sum_Inl @ A @ B @ A4 ) )
= ( F1 @ A4 ) ) ).
% old.sum.simps(7)
thf(fact_75_Set_Ois__empty__def,axiom,
! [A: $tType] :
( ( is_empty @ A )
= ( ^ [A6: set @ A] :
( A6
= ( bot_bot @ ( set @ A ) ) ) ) ) ).
% Set.is_empty_def
thf(fact_76_inFr_Ocases,axiom,
! [A1: set @ n,A22: dtree,A32: t] :
( ( gram_L1333338417e_inFr @ A1 @ A22 @ A32 )
=> ( ( ( member @ n @ ( root @ A22 ) @ A1 )
=> ~ ( member @ ( sum_sum @ t @ dtree ) @ ( sum_Inl @ t @ dtree @ A32 ) @ ( cont @ A22 ) ) )
=> ~ ( ( member @ n @ ( root @ A22 ) @ A1 )
=> ! [Tr12: dtree] :
( ( member @ ( sum_sum @ t @ dtree ) @ ( sum_Inr @ dtree @ t @ Tr12 ) @ ( cont @ A22 ) )
=> ~ ( gram_L1333338417e_inFr @ A1 @ Tr12 @ A32 ) ) ) ) ) ).
% inFr.cases
thf(fact_77_inFr_Osimps,axiom,
( gram_L1333338417e_inFr
= ( ^ [A12: set @ n,A23: dtree,A33: t] :
( ? [Tr4: dtree,Ns2: set @ n,T3: t] :
( ( A12 = Ns2 )
& ( A23 = Tr4 )
& ( A33 = T3 )
& ( member @ n @ ( root @ Tr4 ) @ Ns2 )
& ( member @ ( sum_sum @ t @ dtree ) @ ( sum_Inl @ t @ dtree @ T3 ) @ ( cont @ Tr4 ) ) )
| ? [Tr4: dtree,Ns2: set @ n,Tr13: dtree,T3: t] :
( ( A12 = Ns2 )
& ( A23 = Tr4 )
& ( A33 = T3 )
& ( member @ n @ ( root @ Tr4 ) @ Ns2 )
& ( member @ ( sum_sum @ t @ dtree ) @ ( sum_Inr @ dtree @ t @ Tr13 ) @ ( cont @ Tr4 ) )
& ( gram_L1333338417e_inFr @ Ns2 @ Tr13 @ T3 ) ) ) ) ) ).
% inFr.simps
thf(fact_78_inFr_Oinducts,axiom,
! [X1: set @ n,X22: dtree,X32: t,P2: ( set @ n ) > dtree > t > $o] :
( ( gram_L1333338417e_inFr @ X1 @ X22 @ X32 )
=> ( ! [Tr7: dtree,Ns3: set @ n,T4: t] :
( ( member @ n @ ( root @ Tr7 ) @ Ns3 )
=> ( ( member @ ( sum_sum @ t @ dtree ) @ ( sum_Inl @ t @ dtree @ T4 ) @ ( cont @ Tr7 ) )
=> ( P2 @ Ns3 @ Tr7 @ T4 ) ) )
=> ( ! [Tr7: dtree,Ns3: set @ n,Tr12: dtree,T4: t] :
( ( member @ n @ ( root @ Tr7 ) @ Ns3 )
=> ( ( member @ ( sum_sum @ t @ dtree ) @ ( sum_Inr @ dtree @ t @ Tr12 ) @ ( cont @ Tr7 ) )
=> ( ( gram_L1333338417e_inFr @ Ns3 @ Tr12 @ T4 )
=> ( ( P2 @ Ns3 @ Tr12 @ T4 )
=> ( P2 @ Ns3 @ Tr7 @ T4 ) ) ) ) )
=> ( P2 @ X1 @ X22 @ X32 ) ) ) ) ).
% inFr.inducts
thf(fact_79_Suml_Osimps,axiom,
! [B: $tType,C: $tType,A: $tType,F: A > C,X5: A] :
( ( sum_Suml @ A @ C @ B @ F @ ( sum_Inl @ A @ B @ X5 ) )
= ( F @ X5 ) ) ).
% Suml.simps
thf(fact_80_sum_Oinject_I2_J,axiom,
! [A: $tType,B: $tType,X22: B,Y22: B] :
( ( ( sum_Inr @ B @ A @ X22 )
= ( sum_Inr @ B @ A @ Y22 ) )
= ( X22 = Y22 ) ) ).
% sum.inject(2)
thf(fact_81_old_Osum_Oinject_I2_J,axiom,
! [A: $tType,B: $tType,B3: B,B5: B] :
( ( ( sum_Inr @ B @ A @ B3 )
= ( sum_Inr @ B @ A @ B5 ) )
= ( B3 = B5 ) ) ).
% old.sum.inject(2)
thf(fact_82_old_Osum_Osimps_I8_J,axiom,
! [A: $tType,T: $tType,B: $tType,F1: A > T,F22: B > T,B3: B] :
( ( sum_rec_sum @ A @ T @ B @ F1 @ F22 @ ( sum_Inr @ B @ A @ B3 ) )
= ( F22 @ B3 ) ) ).
% old.sum.simps(8)
thf(fact_83_Inr__inject,axiom,
! [A: $tType,B: $tType,X5: B,Y3: B] :
( ( ( sum_Inr @ B @ A @ X5 )
= ( sum_Inr @ B @ A @ Y3 ) )
=> ( X5 = Y3 ) ) ).
% Inr_inject
thf(fact_84_not__arg__cong__Inr,axiom,
! [B: $tType,A: $tType,X5: A,Y3: A] :
( ( X5 != Y3 )
=> ( ( sum_Inr @ A @ B @ X5 )
!= ( sum_Inr @ A @ B @ Y3 ) ) ) ).
% not_arg_cong_Inr
thf(fact_85_Suml__inject,axiom,
! [B: $tType,C: $tType,A: $tType,F: A > C,G2: A > C] :
( ( ( sum_Suml @ A @ C @ B @ F )
= ( sum_Suml @ A @ C @ B @ G2 ) )
=> ( F = G2 ) ) ).
% Suml_inject
thf(fact_86_obj__sumE,axiom,
! [A: $tType,B: $tType,S: sum_sum @ A @ B] :
( ! [X6: A] :
( S
!= ( sum_Inl @ A @ B @ X6 ) )
=> ~ ! [X6: B] :
( S
!= ( sum_Inr @ B @ A @ X6 ) ) ) ).
% obj_sumE
thf(fact_87_sum_Odistinct_I1_J,axiom,
! [A: $tType,B: $tType,X1: A,X22: B] :
( ( sum_Inl @ A @ B @ X1 )
!= ( sum_Inr @ B @ A @ X22 ) ) ).
% sum.distinct(1)
thf(fact_88_old_Osum_Odistinct_I2_J,axiom,
! [B6: $tType,A7: $tType,B7: B6,A8: A7] :
( ( sum_Inr @ B6 @ A7 @ B7 )
!= ( sum_Inl @ A7 @ B6 @ A8 ) ) ).
% old.sum.distinct(2)
thf(fact_89_old_Osum_Odistinct_I1_J,axiom,
! [A: $tType,B: $tType,A4: A,B5: B] :
( ( sum_Inl @ A @ B @ A4 )
!= ( sum_Inr @ B @ A @ B5 ) ) ).
% old.sum.distinct(1)
thf(fact_90_sumE,axiom,
! [A: $tType,B: $tType,S: sum_sum @ A @ B] :
( ! [X6: A] :
( S
!= ( sum_Inl @ A @ B @ X6 ) )
=> ~ ! [Y4: B] :
( S
!= ( sum_Inr @ B @ A @ Y4 ) ) ) ).
% sumE
thf(fact_91_Inr__not__Inl,axiom,
! [B: $tType,A: $tType,B3: B,A4: A] :
( ( sum_Inr @ B @ A @ B3 )
!= ( sum_Inl @ A @ B @ A4 ) ) ).
% Inr_not_Inl
thf(fact_92_split__sum__ex,axiom,
! [B: $tType,A: $tType] :
( ( ^ [P3: ( sum_sum @ A @ B ) > $o] :
? [X7: sum_sum @ A @ B] : ( P3 @ X7 ) )
= ( ^ [P: ( sum_sum @ A @ B ) > $o] :
( ? [X: A] : ( P @ ( sum_Inl @ A @ B @ X ) )
| ? [X: B] : ( P @ ( sum_Inr @ B @ A @ X ) ) ) ) ) ).
% split_sum_ex
thf(fact_93_split__sum__all,axiom,
! [B: $tType,A: $tType] :
( ( ^ [P3: ( sum_sum @ A @ B ) > $o] :
! [X7: sum_sum @ A @ B] : ( P3 @ X7 ) )
= ( ^ [P: ( sum_sum @ A @ B ) > $o] :
( ! [X: A] : ( P @ ( sum_Inl @ A @ B @ X ) )
& ! [X: B] : ( P @ ( sum_Inr @ B @ A @ X ) ) ) ) ) ).
% split_sum_all
thf(fact_94_old_Osum_Oexhaust,axiom,
! [A: $tType,B: $tType,Y3: sum_sum @ A @ B] :
( ! [A9: A] :
( Y3
!= ( sum_Inl @ A @ B @ A9 ) )
=> ~ ! [B8: B] :
( Y3
!= ( sum_Inr @ B @ A @ B8 ) ) ) ).
% old.sum.exhaust
thf(fact_95_old_Osum_Oinducts,axiom,
! [B: $tType,A: $tType,P2: ( sum_sum @ A @ B ) > $o,Sum: sum_sum @ A @ B] :
( ! [A9: A] : ( P2 @ ( sum_Inl @ A @ B @ A9 ) )
=> ( ! [B8: B] : ( P2 @ ( sum_Inr @ B @ A @ B8 ) )
=> ( P2 @ Sum ) ) ) ).
% old.sum.inducts
thf(fact_96_subtr_OStep,axiom,
! [Tr3: dtree,Ns: set @ n,Tr1: dtree,Tr22: dtree] :
( ( member @ n @ ( root @ Tr3 ) @ Ns )
=> ( ( gram_L716654942_subtr @ Ns @ Tr1 @ Tr22 )
=> ( ( member @ ( sum_sum @ t @ dtree ) @ ( sum_Inr @ dtree @ t @ Tr22 ) @ ( cont @ Tr3 ) )
=> ( gram_L716654942_subtr @ Ns @ Tr1 @ Tr3 ) ) ) ) ).
% subtr.Step
thf(fact_97_subtr_Ocases,axiom,
! [A1: set @ n,A22: dtree,A32: dtree] :
( ( gram_L716654942_subtr @ A1 @ A22 @ A32 )
=> ( ( ( A32 = A22 )
=> ~ ( member @ n @ ( root @ A22 ) @ A1 ) )
=> ~ ( ( member @ n @ ( root @ A32 ) @ A1 )
=> ! [Tr23: dtree] :
( ( gram_L716654942_subtr @ A1 @ A22 @ Tr23 )
=> ~ ( member @ ( sum_sum @ t @ dtree ) @ ( sum_Inr @ dtree @ t @ Tr23 ) @ ( cont @ A32 ) ) ) ) ) ) ).
% subtr.cases
thf(fact_98_subtr_Osimps,axiom,
( gram_L716654942_subtr
= ( ^ [A12: set @ n,A23: dtree,A33: dtree] :
( ? [Tr4: dtree,Ns2: set @ n] :
( ( A12 = Ns2 )
& ( A23 = Tr4 )
& ( A33 = Tr4 )
& ( member @ n @ ( root @ Tr4 ) @ Ns2 ) )
| ? [Tr32: dtree,Ns2: set @ n,Tr13: dtree,Tr24: dtree] :
( ( A12 = Ns2 )
& ( A23 = Tr13 )
& ( A33 = Tr32 )
& ( member @ n @ ( root @ Tr32 ) @ Ns2 )
& ( gram_L716654942_subtr @ Ns2 @ Tr13 @ Tr24 )
& ( member @ ( sum_sum @ t @ dtree ) @ ( sum_Inr @ dtree @ t @ Tr24 ) @ ( cont @ Tr32 ) ) ) ) ) ) ).
% subtr.simps
thf(fact_99_subtr__StepL,axiom,
! [Tr1: dtree,Ns: set @ n,Tr22: dtree,Tr3: dtree] :
( ( member @ n @ ( root @ Tr1 ) @ Ns )
=> ( ( member @ ( sum_sum @ t @ dtree ) @ ( sum_Inr @ dtree @ t @ Tr1 ) @ ( cont @ Tr22 ) )
=> ( ( gram_L716654942_subtr @ Ns @ Tr22 @ Tr3 )
=> ( gram_L716654942_subtr @ Ns @ Tr1 @ Tr3 ) ) ) ) ).
% subtr_StepL
thf(fact_100_subtr_Oinducts,axiom,
! [X1: set @ n,X22: dtree,X32: dtree,P2: ( set @ n ) > dtree > dtree > $o] :
( ( gram_L716654942_subtr @ X1 @ X22 @ X32 )
=> ( ! [Tr7: dtree,Ns3: set @ n] :
( ( member @ n @ ( root @ Tr7 ) @ Ns3 )
=> ( P2 @ Ns3 @ Tr7 @ Tr7 ) )
=> ( ! [Tr33: dtree,Ns3: set @ n,Tr12: dtree,Tr23: dtree] :
( ( member @ n @ ( root @ Tr33 ) @ Ns3 )
=> ( ( gram_L716654942_subtr @ Ns3 @ Tr12 @ Tr23 )
=> ( ( P2 @ Ns3 @ Tr12 @ Tr23 )
=> ( ( member @ ( sum_sum @ t @ dtree ) @ ( sum_Inr @ dtree @ t @ Tr23 ) @ ( cont @ Tr33 ) )
=> ( P2 @ Ns3 @ Tr12 @ Tr33 ) ) ) ) )
=> ( P2 @ X1 @ X22 @ X32 ) ) ) ) ).
% subtr.inducts
thf(fact_101_subtr__inductL,axiom,
! [Ns: set @ n,Tr1: dtree,Tr22: dtree,Phi: ( set @ n ) > dtree > dtree > $o] :
( ( gram_L716654942_subtr @ Ns @ Tr1 @ Tr22 )
=> ( ! [Ns3: set @ n,Tr7: dtree] : ( Phi @ Ns3 @ Tr7 @ Tr7 )
=> ( ! [Ns3: set @ n,Tr12: dtree,Tr23: dtree,Tr33: dtree] :
( ( member @ n @ ( root @ Tr12 ) @ Ns3 )
=> ( ( member @ ( sum_sum @ t @ dtree ) @ ( sum_Inr @ dtree @ t @ Tr12 ) @ ( cont @ Tr23 ) )
=> ( ( gram_L716654942_subtr @ Ns3 @ Tr23 @ Tr33 )
=> ( ( Phi @ Ns3 @ Tr23 @ Tr33 )
=> ( Phi @ Ns3 @ Tr12 @ Tr33 ) ) ) ) )
=> ( Phi @ Ns @ Tr1 @ Tr22 ) ) ) ) ).
% subtr_inductL
thf(fact_102_bot__fun__def,axiom,
! [B: $tType,A: $tType] :
( ( bot @ B @ ( type2 @ B ) )
=> ( ( bot_bot @ ( A > B ) )
= ( ^ [X: A] : ( bot_bot @ B ) ) ) ) ).
% bot_fun_def
thf(fact_103_subtr2_Oinducts,axiom,
! [X1: set @ n,X22: dtree,X32: dtree,P2: ( set @ n ) > dtree > dtree > $o] :
( ( gram_L1283001940subtr2 @ X1 @ X22 @ X32 )
=> ( ! [Tr7: dtree,Ns3: set @ n] :
( ( member @ n @ ( root @ Tr7 ) @ Ns3 )
=> ( P2 @ Ns3 @ Tr7 @ Tr7 ) )
=> ( ! [Tr12: dtree,Ns3: set @ n,Tr23: dtree,Tr33: dtree] :
( ( member @ n @ ( root @ Tr12 ) @ Ns3 )
=> ( ( member @ ( sum_sum @ t @ dtree ) @ ( sum_Inr @ dtree @ t @ Tr12 ) @ ( cont @ Tr23 ) )
=> ( ( gram_L1283001940subtr2 @ Ns3 @ Tr23 @ Tr33 )
=> ( ( P2 @ Ns3 @ Tr23 @ Tr33 )
=> ( P2 @ Ns3 @ Tr12 @ Tr33 ) ) ) ) )
=> ( P2 @ X1 @ X22 @ X32 ) ) ) ) ).
% subtr2.inducts
thf(fact_104_subtr2__StepR,axiom,
! [Tr3: dtree,Ns: set @ n,Tr22: dtree,Tr1: dtree] :
( ( member @ n @ ( root @ Tr3 ) @ Ns )
=> ( ( member @ ( sum_sum @ t @ dtree ) @ ( sum_Inr @ dtree @ t @ Tr22 ) @ ( cont @ Tr3 ) )
=> ( ( gram_L1283001940subtr2 @ Ns @ Tr1 @ Tr22 )
=> ( gram_L1283001940subtr2 @ Ns @ Tr1 @ Tr3 ) ) ) ) ).
% subtr2_StepR
thf(fact_105_subtr2_Osimps,axiom,
( gram_L1283001940subtr2
= ( ^ [A12: set @ n,A23: dtree,A33: dtree] :
( ? [Tr4: dtree,Ns2: set @ n] :
( ( A12 = Ns2 )
& ( A23 = Tr4 )
& ( A33 = Tr4 )
& ( member @ n @ ( root @ Tr4 ) @ Ns2 ) )
| ? [Tr13: dtree,Ns2: set @ n,Tr24: dtree,Tr32: dtree] :
( ( A12 = Ns2 )
& ( A23 = Tr13 )
& ( A33 = Tr32 )
& ( member @ n @ ( root @ Tr13 ) @ Ns2 )
& ( member @ ( sum_sum @ t @ dtree ) @ ( sum_Inr @ dtree @ t @ Tr13 ) @ ( cont @ Tr24 ) )
& ( gram_L1283001940subtr2 @ Ns2 @ Tr24 @ Tr32 ) ) ) ) ) ).
% subtr2.simps
thf(fact_106_subtr2_Ocases,axiom,
! [A1: set @ n,A22: dtree,A32: dtree] :
( ( gram_L1283001940subtr2 @ A1 @ A22 @ A32 )
=> ( ( ( A32 = A22 )
=> ~ ( member @ n @ ( root @ A22 ) @ A1 ) )
=> ~ ( ( member @ n @ ( root @ A22 ) @ A1 )
=> ! [Tr23: dtree] :
( ( member @ ( sum_sum @ t @ dtree ) @ ( sum_Inr @ dtree @ t @ A22 ) @ ( cont @ Tr23 ) )
=> ~ ( gram_L1283001940subtr2 @ A1 @ Tr23 @ A32 ) ) ) ) ) ).
% subtr2.cases
thf(fact_107_subtr2_OStep,axiom,
! [Tr1: dtree,Ns: set @ n,Tr22: dtree,Tr3: dtree] :
( ( member @ n @ ( root @ Tr1 ) @ Ns )
=> ( ( member @ ( sum_sum @ t @ dtree ) @ ( sum_Inr @ dtree @ t @ Tr1 ) @ ( cont @ Tr22 ) )
=> ( ( gram_L1283001940subtr2 @ Ns @ Tr22 @ Tr3 )
=> ( gram_L1283001940subtr2 @ Ns @ Tr1 @ Tr3 ) ) ) ) ).
% subtr2.Step
thf(fact_108_inFr_OInd,axiom,
! [Tr: dtree,Ns: set @ n,Tr1: dtree,T2: t] :
( ( member @ n @ ( root @ Tr ) @ Ns )
=> ( ( member @ ( sum_sum @ t @ dtree ) @ ( sum_Inr @ dtree @ t @ Tr1 ) @ ( cont @ Tr ) )
=> ( ( gram_L1333338417e_inFr @ Ns @ Tr1 @ T2 )
=> ( gram_L1333338417e_inFr @ Ns @ Tr @ T2 ) ) ) ) ).
% inFr.Ind
thf(fact_109_inItr_OInd,axiom,
! [Tr: dtree,Ns: set @ n,Tr1: dtree,N: n] :
( ( member @ n @ ( root @ Tr ) @ Ns )
=> ( ( member @ ( sum_sum @ t @ dtree ) @ ( sum_Inr @ dtree @ t @ Tr1 ) @ ( cont @ Tr ) )
=> ( ( gram_L830233218_inItr @ Ns @ Tr1 @ N )
=> ( gram_L830233218_inItr @ Ns @ Tr @ N ) ) ) ) ).
% inItr.Ind
thf(fact_110_inItr_Ocases,axiom,
! [A1: set @ n,A22: dtree,A32: n] :
( ( gram_L830233218_inItr @ A1 @ A22 @ A32 )
=> ( ( ( A32
= ( root @ A22 ) )
=> ~ ( member @ n @ ( root @ A22 ) @ A1 ) )
=> ~ ( ( member @ n @ ( root @ A22 ) @ A1 )
=> ! [Tr12: dtree] :
( ( member @ ( sum_sum @ t @ dtree ) @ ( sum_Inr @ dtree @ t @ Tr12 ) @ ( cont @ A22 ) )
=> ~ ( gram_L830233218_inItr @ A1 @ Tr12 @ A32 ) ) ) ) ) ).
% inItr.cases
thf(fact_111_inItr_Osimps,axiom,
( gram_L830233218_inItr
= ( ^ [A12: set @ n,A23: dtree,A33: n] :
( ? [Tr4: dtree,Ns2: set @ n] :
( ( A12 = Ns2 )
& ( A23 = Tr4 )
& ( A33
= ( root @ Tr4 ) )
& ( member @ n @ ( root @ Tr4 ) @ Ns2 ) )
| ? [Tr4: dtree,Ns2: set @ n,Tr13: dtree,N2: n] :
( ( A12 = Ns2 )
& ( A23 = Tr4 )
& ( A33 = N2 )
& ( member @ n @ ( root @ Tr4 ) @ Ns2 )
& ( member @ ( sum_sum @ t @ dtree ) @ ( sum_Inr @ dtree @ t @ Tr13 ) @ ( cont @ Tr4 ) )
& ( gram_L830233218_inItr @ Ns2 @ Tr13 @ N2 ) ) ) ) ) ).
% inItr.simps
thf(fact_112_inItr_Oinducts,axiom,
! [X1: set @ n,X22: dtree,X32: n,P2: ( set @ n ) > dtree > n > $o] :
( ( gram_L830233218_inItr @ X1 @ X22 @ X32 )
=> ( ! [Tr7: dtree,Ns3: set @ n] :
( ( member @ n @ ( root @ Tr7 ) @ Ns3 )
=> ( P2 @ Ns3 @ Tr7 @ ( root @ Tr7 ) ) )
=> ( ! [Tr7: dtree,Ns3: set @ n,Tr12: dtree,N3: n] :
( ( member @ n @ ( root @ Tr7 ) @ Ns3 )
=> ( ( member @ ( sum_sum @ t @ dtree ) @ ( sum_Inr @ dtree @ t @ Tr12 ) @ ( cont @ Tr7 ) )
=> ( ( gram_L830233218_inItr @ Ns3 @ Tr12 @ N3 )
=> ( ( P2 @ Ns3 @ Tr12 @ N3 )
=> ( P2 @ Ns3 @ Tr7 @ N3 ) ) ) ) )
=> ( P2 @ X1 @ X22 @ X32 ) ) ) ) ).
% inItr.inducts
thf(fact_113_inFr2__Ind,axiom,
! [Ns: set @ n,Tr1: dtree,T2: t,Tr: dtree] :
( ( gram_L805317441_inFr2 @ Ns @ Tr1 @ T2 )
=> ( ( member @ n @ ( root @ Tr ) @ Ns )
=> ( ( member @ ( sum_sum @ t @ dtree ) @ ( sum_Inr @ dtree @ t @ Tr1 ) @ ( cont @ Tr ) )
=> ( gram_L805317441_inFr2 @ Ns @ Tr @ T2 ) ) ) ) ).
% inFr2_Ind
thf(fact_114_Inl__Inr__False,axiom,
! [A: $tType,B: $tType,X5: A,Y3: B] :
( ( sum_Inl @ A @ B @ X5 )
!= ( sum_Inr @ B @ A @ Y3 ) ) ).
% Inl_Inr_False
thf(fact_115_Inr__Inl__False,axiom,
! [B: $tType,A: $tType,X5: B,Y3: A] :
( ( sum_Inr @ B @ A @ X5 )
!= ( sum_Inl @ A @ B @ Y3 ) ) ).
% Inr_Inl_False
thf(fact_116_Inl__Inr__image__cong,axiom,
! [B: $tType,A: $tType,A2: set @ ( sum_sum @ A @ B ),B2: set @ ( sum_sum @ A @ B )] :
( ( ( vimage @ A @ ( sum_sum @ A @ B ) @ ( sum_Inl @ A @ B ) @ A2 )
= ( vimage @ A @ ( sum_sum @ A @ B ) @ ( sum_Inl @ A @ B ) @ B2 ) )
=> ( ( ( vimage @ B @ ( sum_sum @ A @ B ) @ ( sum_Inr @ B @ A ) @ A2 )
= ( vimage @ B @ ( sum_sum @ A @ B ) @ ( sum_Inr @ B @ A ) @ B2 ) )
=> ( A2 = B2 ) ) ) ).
% Inl_Inr_image_cong
thf(fact_117_bot__empty__eq,axiom,
! [A: $tType] :
( ( bot_bot @ ( A > $o ) )
= ( ^ [X: A] : ( member @ A @ X @ ( bot_bot @ ( set @ A ) ) ) ) ) ).
% bot_empty_eq
thf(fact_118_Collect__empty__eq__bot,axiom,
! [A: $tType,P2: A > $o] :
( ( ( collect @ A @ P2 )
= ( bot_bot @ ( set @ A ) ) )
= ( P2
= ( bot_bot @ ( A > $o ) ) ) ) ).
% Collect_empty_eq_bot
thf(fact_119_obj__sumE__f,axiom,
! [A: $tType,C: $tType,B: $tType,S: B,F: ( sum_sum @ A @ C ) > B,P2: $o] :
( ! [X6: A] :
( ( S
= ( F @ ( sum_Inl @ A @ C @ X6 ) ) )
=> P2 )
=> ( ! [X6: C] :
( ( S
= ( F @ ( sum_Inr @ C @ A @ X6 ) ) )
=> P2 )
=> ! [X8: sum_sum @ A @ C] :
( ( S
= ( F @ X8 ) )
=> P2 ) ) ) ).
% obj_sumE_f
thf(fact_120_inFr2_Oinducts,axiom,
! [X1: set @ n,X22: dtree,X32: t,P2: ( set @ n ) > dtree > t > $o] :
( ( gram_L805317441_inFr2 @ X1 @ X22 @ X32 )
=> ( ! [Tr7: dtree,Ns3: set @ n,T4: t] :
( ( member @ n @ ( root @ Tr7 ) @ Ns3 )
=> ( ( member @ ( sum_sum @ t @ dtree ) @ ( sum_Inl @ t @ dtree @ T4 ) @ ( cont @ Tr7 ) )
=> ( P2 @ Ns3 @ Tr7 @ T4 ) ) )
=> ( ! [Tr12: dtree,Tr7: dtree,Ns1: set @ n,T4: t] :
( ( member @ ( sum_sum @ t @ dtree ) @ ( sum_Inr @ dtree @ t @ Tr12 ) @ ( cont @ Tr7 ) )
=> ( ( gram_L805317441_inFr2 @ Ns1 @ Tr12 @ T4 )
=> ( ( P2 @ Ns1 @ Tr12 @ T4 )
=> ( P2 @ ( insert @ n @ ( root @ Tr7 ) @ Ns1 ) @ Tr7 @ T4 ) ) ) )
=> ( P2 @ X1 @ X22 @ X32 ) ) ) ) ).
% inFr2.inducts
thf(fact_121_inFr2_Osimps,axiom,
( gram_L805317441_inFr2
= ( ^ [A12: set @ n,A23: dtree,A33: t] :
( ? [Tr4: dtree,Ns2: set @ n,T3: t] :
( ( A12 = Ns2 )
& ( A23 = Tr4 )
& ( A33 = T3 )
& ( member @ n @ ( root @ Tr4 ) @ Ns2 )
& ( member @ ( sum_sum @ t @ dtree ) @ ( sum_Inl @ t @ dtree @ T3 ) @ ( cont @ Tr4 ) ) )
| ? [Tr13: dtree,Tr4: dtree,Ns12: set @ n,T3: t] :
( ( A12
= ( insert @ n @ ( root @ Tr4 ) @ Ns12 ) )
& ( A23 = Tr4 )
& ( A33 = T3 )
& ( member @ ( sum_sum @ t @ dtree ) @ ( sum_Inr @ dtree @ t @ Tr13 ) @ ( cont @ Tr4 ) )
& ( gram_L805317441_inFr2 @ Ns12 @ Tr13 @ T3 ) ) ) ) ) ).
% inFr2.simps
thf(fact_122_inFr2_Ocases,axiom,
! [A1: set @ n,A22: dtree,A32: t] :
( ( gram_L805317441_inFr2 @ A1 @ A22 @ A32 )
=> ( ( ( member @ n @ ( root @ A22 ) @ A1 )
=> ~ ( member @ ( sum_sum @ t @ dtree ) @ ( sum_Inl @ t @ dtree @ A32 ) @ ( cont @ A22 ) ) )
=> ~ ! [Tr12: dtree,Tr7: dtree,Ns1: set @ n] :
( ( A1
= ( insert @ n @ ( root @ Tr7 ) @ Ns1 ) )
=> ( ( A22 = Tr7 )
=> ( ( member @ ( sum_sum @ t @ dtree ) @ ( sum_Inr @ dtree @ t @ Tr12 ) @ ( cont @ Tr7 ) )
=> ~ ( gram_L805317441_inFr2 @ Ns1 @ Tr12 @ A32 ) ) ) ) ) ) ).
% inFr2.cases
thf(fact_123_inFr2_OInd,axiom,
! [Tr1: dtree,Tr: dtree,Ns13: set @ n,T2: t] :
( ( member @ ( sum_sum @ t @ dtree ) @ ( sum_Inr @ dtree @ t @ Tr1 ) @ ( cont @ Tr ) )
=> ( ( gram_L805317441_inFr2 @ Ns13 @ Tr1 @ T2 )
=> ( gram_L805317441_inFr2 @ ( insert @ n @ ( root @ Tr ) @ Ns13 ) @ Tr @ T2 ) ) ) ).
% inFr2.Ind
thf(fact_124_insertCI,axiom,
! [A: $tType,A4: A,B2: set @ A,B3: A] :
( ( ~ ( member @ A @ A4 @ B2 )
=> ( A4 = B3 ) )
=> ( member @ A @ A4 @ ( insert @ A @ B3 @ B2 ) ) ) ).
% insertCI
thf(fact_125_insert__iff,axiom,
! [A: $tType,A4: A,B3: A,A2: set @ A] :
( ( member @ A @ A4 @ ( insert @ A @ B3 @ A2 ) )
= ( ( A4 = B3 )
| ( member @ A @ A4 @ A2 ) ) ) ).
% insert_iff
thf(fact_126_insert__absorb2,axiom,
! [A: $tType,X5: A,A2: set @ A] :
( ( insert @ A @ X5 @ ( insert @ A @ X5 @ A2 ) )
= ( insert @ A @ X5 @ A2 ) ) ).
% insert_absorb2
thf(fact_127_singletonI,axiom,
! [A: $tType,A4: A] : ( member @ A @ A4 @ ( insert @ A @ A4 @ ( bot_bot @ ( set @ A ) ) ) ) ).
% singletonI
thf(fact_128_singleton__conv2,axiom,
! [A: $tType,A4: A] :
( ( collect @ A
@ ( ^ [Y6: A,Z: A] : Y6 = Z
@ A4 ) )
= ( insert @ A @ A4 @ ( bot_bot @ ( set @ A ) ) ) ) ).
% singleton_conv2
thf(fact_129_singleton__conv,axiom,
! [A: $tType,A4: A] :
( ( collect @ A
@ ^ [X: A] : X = A4 )
= ( insert @ A @ A4 @ ( bot_bot @ ( set @ A ) ) ) ) ).
% singleton_conv
thf(fact_130_singleton__inject,axiom,
! [A: $tType,A4: A,B3: A] :
( ( ( insert @ A @ A4 @ ( bot_bot @ ( set @ A ) ) )
= ( insert @ A @ B3 @ ( bot_bot @ ( set @ A ) ) ) )
=> ( A4 = B3 ) ) ).
% singleton_inject
thf(fact_131_insert__not__empty,axiom,
! [A: $tType,A4: A,A2: set @ A] :
( ( insert @ A @ A4 @ A2 )
!= ( bot_bot @ ( set @ A ) ) ) ).
% insert_not_empty
thf(fact_132_doubleton__eq__iff,axiom,
! [A: $tType,A4: A,B3: A,C3: A,D2: A] :
( ( ( insert @ A @ A4 @ ( insert @ A @ B3 @ ( bot_bot @ ( set @ A ) ) ) )
= ( insert @ A @ C3 @ ( insert @ A @ D2 @ ( bot_bot @ ( set @ A ) ) ) ) )
= ( ( ( A4 = C3 )
& ( B3 = D2 ) )
| ( ( A4 = D2 )
& ( B3 = C3 ) ) ) ) ).
% doubleton_eq_iff
thf(fact_133_singleton__iff,axiom,
! [A: $tType,B3: A,A4: A] :
( ( member @ A @ B3 @ ( insert @ A @ A4 @ ( bot_bot @ ( set @ A ) ) ) )
= ( B3 = A4 ) ) ).
% singleton_iff
thf(fact_134_singletonD,axiom,
! [A: $tType,B3: A,A4: A] :
( ( member @ A @ B3 @ ( insert @ A @ A4 @ ( bot_bot @ ( set @ A ) ) ) )
=> ( B3 = A4 ) ) ).
% singletonD
thf(fact_135_insertE,axiom,
! [A: $tType,A4: A,B3: A,A2: set @ A] :
( ( member @ A @ A4 @ ( insert @ A @ B3 @ A2 ) )
=> ( ( A4 != B3 )
=> ( member @ A @ A4 @ A2 ) ) ) ).
% insertE
thf(fact_136_insertI1,axiom,
! [A: $tType,A4: A,B2: set @ A] : ( member @ A @ A4 @ ( insert @ A @ A4 @ B2 ) ) ).
% insertI1
thf(fact_137_insertI2,axiom,
! [A: $tType,A4: A,B2: set @ A,B3: A] :
( ( member @ A @ A4 @ B2 )
=> ( member @ A @ A4 @ ( insert @ A @ B3 @ B2 ) ) ) ).
% insertI2
thf(fact_138_Set_Oset__insert,axiom,
! [A: $tType,X5: A,A2: set @ A] :
( ( member @ A @ X5 @ A2 )
=> ~ ! [B9: set @ A] :
( ( A2
= ( insert @ A @ X5 @ B9 ) )
=> ( member @ A @ X5 @ B9 ) ) ) ).
% Set.set_insert
thf(fact_139_insert__compr,axiom,
! [A: $tType] :
( ( insert @ A )
= ( ^ [A3: A,B4: set @ A] :
( collect @ A
@ ^ [X: A] :
( ( X = A3 )
| ( member @ A @ X @ B4 ) ) ) ) ) ).
% insert_compr
thf(fact_140_insert__ident,axiom,
! [A: $tType,X5: A,A2: set @ A,B2: set @ A] :
( ~ ( member @ A @ X5 @ A2 )
=> ( ~ ( member @ A @ X5 @ B2 )
=> ( ( ( insert @ A @ X5 @ A2 )
= ( insert @ A @ X5 @ B2 ) )
= ( A2 = B2 ) ) ) ) ).
% insert_ident
thf(fact_141_insert__absorb,axiom,
! [A: $tType,A4: A,A2: set @ A] :
( ( member @ A @ A4 @ A2 )
=> ( ( insert @ A @ A4 @ A2 )
= A2 ) ) ).
% insert_absorb
thf(fact_142_insert__eq__iff,axiom,
! [A: $tType,A4: A,A2: set @ A,B3: A,B2: set @ A] :
( ~ ( member @ A @ A4 @ A2 )
=> ( ~ ( member @ A @ B3 @ B2 )
=> ( ( ( insert @ A @ A4 @ A2 )
= ( insert @ A @ B3 @ B2 ) )
= ( ( ( A4 = B3 )
=> ( A2 = B2 ) )
& ( ( A4 != B3 )
=> ? [C4: set @ A] :
( ( A2
= ( insert @ A @ B3 @ C4 ) )
& ~ ( member @ A @ B3 @ C4 )
& ( B2
= ( insert @ A @ A4 @ C4 ) )
& ~ ( member @ A @ A4 @ C4 ) ) ) ) ) ) ) ).
% insert_eq_iff
thf(fact_143_insert__Collect,axiom,
! [A: $tType,A4: A,P2: A > $o] :
( ( insert @ A @ A4 @ ( collect @ A @ P2 ) )
= ( collect @ A
@ ^ [U: A] :
( ( U != A4 )
=> ( P2 @ U ) ) ) ) ).
% insert_Collect
thf(fact_144_insert__commute,axiom,
! [A: $tType,X5: A,Y3: A,A2: set @ A] :
( ( insert @ A @ X5 @ ( insert @ A @ Y3 @ A2 ) )
= ( insert @ A @ Y3 @ ( insert @ A @ X5 @ A2 ) ) ) ).
% insert_commute
thf(fact_145_mk__disjoint__insert,axiom,
! [A: $tType,A4: A,A2: set @ A] :
( ( member @ A @ A4 @ A2 )
=> ? [B9: set @ A] :
( ( A2
= ( insert @ A @ A4 @ B9 ) )
& ~ ( member @ A @ A4 @ B9 ) ) ) ).
% mk_disjoint_insert
thf(fact_146_Collect__conv__if2,axiom,
! [A: $tType,P2: A > $o,A4: A] :
( ( ( P2 @ A4 )
=> ( ( collect @ A
@ ^ [X: A] :
( ( A4 = X )
& ( P2 @ X ) ) )
= ( insert @ A @ A4 @ ( bot_bot @ ( set @ A ) ) ) ) )
& ( ~ ( P2 @ A4 )
=> ( ( collect @ A
@ ^ [X: A] :
( ( A4 = X )
& ( P2 @ X ) ) )
= ( bot_bot @ ( set @ A ) ) ) ) ) ).
% Collect_conv_if2
thf(fact_147_Collect__conv__if,axiom,
! [A: $tType,P2: A > $o,A4: A] :
( ( ( P2 @ A4 )
=> ( ( collect @ A
@ ^ [X: A] :
( ( X = A4 )
& ( P2 @ X ) ) )
= ( insert @ A @ A4 @ ( bot_bot @ ( set @ A ) ) ) ) )
& ( ~ ( P2 @ A4 )
=> ( ( collect @ A
@ ^ [X: A] :
( ( X = A4 )
& ( P2 @ X ) ) )
= ( bot_bot @ ( set @ A ) ) ) ) ) ).
% Collect_conv_if
thf(fact_148_vimage__singleton__eq,axiom,
! [A: $tType,B: $tType,A4: A,F: A > B,B3: B] :
( ( member @ A @ A4 @ ( vimage @ A @ B @ F @ ( insert @ B @ B3 @ ( bot_bot @ ( set @ B ) ) ) ) )
= ( ( F @ A4 )
= B3 ) ) ).
% vimage_singleton_eq
thf(fact_149_inFr__Ind__minus,axiom,
! [Ns13: set @ n,Tr1: dtree,T2: t,Tr: dtree] :
( ( gram_L1333338417e_inFr @ Ns13 @ Tr1 @ T2 )
=> ( ( member @ ( sum_sum @ t @ dtree ) @ ( sum_Inr @ dtree @ t @ Tr1 ) @ ( cont @ Tr ) )
=> ( gram_L1333338417e_inFr @ ( insert @ n @ ( root @ Tr ) @ Ns13 ) @ Tr @ T2 ) ) ) ).
% inFr_Ind_minus
thf(fact_150_ccpo__Sup__singleton,axiom,
! [A: $tType] :
( ( comple1141879883l_ccpo @ A @ ( type2 @ A ) )
=> ! [X5: A] :
( ( complete_Sup_Sup @ A @ ( insert @ A @ X5 @ ( bot_bot @ ( set @ A ) ) ) )
= X5 ) ) ).
% ccpo_Sup_singleton
thf(fact_151_cSup__singleton,axiom,
! [A: $tType] :
( ( condit378418413attice @ A @ ( type2 @ A ) )
=> ! [X5: A] :
( ( complete_Sup_Sup @ A @ ( insert @ A @ X5 @ ( bot_bot @ ( set @ A ) ) ) )
= X5 ) ) ).
% cSup_singleton
thf(fact_152_is__singletonI,axiom,
! [A: $tType,X5: A] : ( is_singleton @ A @ ( insert @ A @ X5 @ ( bot_bot @ ( set @ A ) ) ) ) ).
% is_singletonI
thf(fact_153_the__elem__eq,axiom,
! [A: $tType,X5: A] :
( ( the_elem @ A @ ( insert @ A @ X5 @ ( bot_bot @ ( set @ A ) ) ) )
= X5 ) ).
% the_elem_eq
thf(fact_154_is__singleton__the__elem,axiom,
! [A: $tType] :
( ( is_singleton @ A )
= ( ^ [A6: set @ A] :
( A6
= ( insert @ A @ ( the_elem @ A @ A6 ) @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ).
% is_singleton_the_elem
thf(fact_155_is__singletonI_H,axiom,
! [A: $tType,A2: set @ A] :
( ( A2
!= ( bot_bot @ ( set @ A ) ) )
=> ( ! [X6: A,Y4: A] :
( ( member @ A @ X6 @ A2 )
=> ( ( member @ A @ Y4 @ A2 )
=> ( X6 = Y4 ) ) )
=> ( is_singleton @ A @ A2 ) ) ) ).
% is_singletonI'
thf(fact_156_is__singleton__def,axiom,
! [A: $tType] :
( ( is_singleton @ A )
= ( ^ [A6: set @ A] :
? [X: A] :
( A6
= ( insert @ A @ X @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ).
% is_singleton_def
thf(fact_157_is__singletonE,axiom,
! [A: $tType,A2: set @ A] :
( ( is_singleton @ A @ A2 )
=> ~ ! [X6: A] :
( A2
!= ( insert @ A @ X6 @ ( bot_bot @ ( set @ A ) ) ) ) ) ).
% is_singletonE
thf(fact_158_sum__set__simps_I1_J,axiom,
! [B: $tType,A: $tType,X5: A] :
( ( basic_setl @ A @ B @ ( sum_Inl @ A @ B @ X5 ) )
= ( insert @ A @ X5 @ ( bot_bot @ ( set @ A ) ) ) ) ).
% sum_set_simps(1)
thf(fact_159_sum__set__simps_I4_J,axiom,
! [E: $tType,A: $tType,X5: A] :
( ( basic_setr @ E @ A @ ( sum_Inr @ A @ E @ X5 ) )
= ( insert @ A @ X5 @ ( bot_bot @ ( set @ A ) ) ) ) ).
% sum_set_simps(4)
thf(fact_160_the__elem__def,axiom,
! [A: $tType] :
( ( the_elem @ A )
= ( ^ [X9: set @ A] :
( the @ A
@ ^ [X: A] :
( X9
= ( insert @ A @ X @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ) ).
% the_elem_def
thf(fact_161_sum__set__simps_I2_J,axiom,
! [A: $tType,C: $tType,X5: A] :
( ( basic_setl @ C @ A @ ( sum_Inr @ A @ C @ X5 ) )
= ( bot_bot @ ( set @ C ) ) ) ).
% sum_set_simps(2)
thf(fact_162_sum__set__simps_I3_J,axiom,
! [A: $tType,D: $tType,X5: A] :
( ( basic_setr @ A @ D @ ( sum_Inl @ A @ D @ X5 ) )
= ( bot_bot @ ( set @ D ) ) ) ).
% sum_set_simps(3)
thf(fact_163_old_Orec__sum__def,axiom,
! [B: $tType,T: $tType,A: $tType] :
( ( sum_rec_sum @ A @ T @ B )
= ( ^ [F12: A > T,F23: B > T,X: sum_sum @ A @ B] : ( the @ T @ ( sum_rec_set_sum @ A @ T @ B @ F12 @ F23 @ X ) ) ) ) ).
% old.rec_sum_def
thf(fact_164_the__sym__eq__trivial,axiom,
! [A: $tType,X5: A] :
( ( the @ A
@ ( ^ [Y6: A,Z: A] : Y6 = Z
@ X5 ) )
= X5 ) ).
% the_sym_eq_trivial
thf(fact_165_the__eq__trivial,axiom,
! [A: $tType,A4: A] :
( ( the @ A
@ ^ [X: A] : X = A4 )
= A4 ) ).
% the_eq_trivial
thf(fact_166_the__equality,axiom,
! [A: $tType,P2: A > $o,A4: A] :
( ( P2 @ A4 )
=> ( ! [X6: A] :
( ( P2 @ X6 )
=> ( X6 = A4 ) )
=> ( ( the @ A @ P2 )
= A4 ) ) ) ).
% the_equality
thf(fact_167_theI,axiom,
! [A: $tType,P2: A > $o,A4: A] :
( ( P2 @ A4 )
=> ( ! [X6: A] :
( ( P2 @ X6 )
=> ( X6 = A4 ) )
=> ( P2 @ ( the @ A @ P2 ) ) ) ) ).
% theI
thf(fact_168_theI_H,axiom,
! [A: $tType,P2: A > $o] :
( ? [X8: A] :
( ( P2 @ X8 )
& ! [Y4: A] :
( ( P2 @ Y4 )
=> ( Y4 = X8 ) ) )
=> ( P2 @ ( the @ A @ P2 ) ) ) ).
% theI'
thf(fact_169_theI2,axiom,
! [A: $tType,P2: A > $o,A4: A,Q: A > $o] :
( ( P2 @ A4 )
=> ( ! [X6: A] :
( ( P2 @ X6 )
=> ( X6 = A4 ) )
=> ( ! [X6: A] :
( ( P2 @ X6 )
=> ( Q @ X6 ) )
=> ( Q @ ( the @ A @ P2 ) ) ) ) ) ).
% theI2
thf(fact_170_If__def,axiom,
! [A: $tType] :
( ( if @ A )
= ( ^ [P: $o,X: A,Y: A] :
( the @ A
@ ^ [Z2: A] :
( ( P
=> ( Z2 = X ) )
& ( ~ P
=> ( Z2 = Y ) ) ) ) ) ) ).
% If_def
thf(fact_171_the1I2,axiom,
! [A: $tType,P2: A > $o,Q: A > $o] :
( ? [X8: A] :
( ( P2 @ X8 )
& ! [Y4: A] :
( ( P2 @ Y4 )
=> ( Y4 = X8 ) ) )
=> ( ! [X6: A] :
( ( P2 @ X6 )
=> ( Q @ X6 ) )
=> ( Q @ ( the @ A @ P2 ) ) ) ) ).
% the1I2
thf(fact_172_the1__equality,axiom,
! [A: $tType,P2: A > $o,A4: A] :
( ? [X8: A] :
( ( P2 @ X8 )
& ! [Y4: A] :
( ( P2 @ Y4 )
=> ( Y4 = X8 ) ) )
=> ( ( P2 @ A4 )
=> ( ( the @ A @ P2 )
= A4 ) ) ) ).
% the1_equality
thf(fact_173_setl_Oinducts,axiom,
! [B: $tType,A: $tType,X5: A,S: sum_sum @ A @ B,P2: A > $o] :
( ( member @ A @ X5 @ ( basic_setl @ A @ B @ S ) )
=> ( ! [X6: A] :
( ( S
= ( sum_Inl @ A @ B @ X6 ) )
=> ( P2 @ X6 ) )
=> ( P2 @ X5 ) ) ) ).
% setl.inducts
thf(fact_174_setl_Ointros,axiom,
! [B: $tType,A: $tType,S: sum_sum @ A @ B,X5: A] :
( ( S
= ( sum_Inl @ A @ B @ X5 ) )
=> ( member @ A @ X5 @ ( basic_setl @ A @ B @ S ) ) ) ).
% setl.intros
thf(fact_175_setl_Osimps,axiom,
! [B: $tType,A: $tType,A4: A,S: sum_sum @ A @ B] :
( ( member @ A @ A4 @ ( basic_setl @ A @ B @ S ) )
= ( ? [X: A] :
( ( A4 = X )
& ( S
= ( sum_Inl @ A @ B @ X ) ) ) ) ) ).
% setl.simps
thf(fact_176_setr_Oinducts,axiom,
! [A: $tType,B: $tType,X5: B,S: sum_sum @ A @ B,P2: B > $o] :
( ( member @ B @ X5 @ ( basic_setr @ A @ B @ S ) )
=> ( ! [X6: B] :
( ( S
= ( sum_Inr @ B @ A @ X6 ) )
=> ( P2 @ X6 ) )
=> ( P2 @ X5 ) ) ) ).
% setr.inducts
thf(fact_177_setr_Ointros,axiom,
! [B: $tType,A: $tType,S: sum_sum @ A @ B,X5: B] :
( ( S
= ( sum_Inr @ B @ A @ X5 ) )
=> ( member @ B @ X5 @ ( basic_setr @ A @ B @ S ) ) ) ).
% setr.intros
thf(fact_178_setr_Osimps,axiom,
! [A: $tType,B: $tType,A4: B,S: sum_sum @ A @ B] :
( ( member @ B @ A4 @ ( basic_setr @ A @ B @ S ) )
= ( ? [X: B] :
( ( A4 = X )
& ( S
= ( sum_Inr @ B @ A @ X ) ) ) ) ) ).
% setr.simps
thf(fact_179_setr_Ocases,axiom,
! [A: $tType,B: $tType,A4: B,S: sum_sum @ A @ B] :
( ( member @ B @ A4 @ ( basic_setr @ A @ B @ S ) )
=> ( S
= ( sum_Inr @ B @ A @ A4 ) ) ) ).
% setr.cases
thf(fact_180_setl_Ocases,axiom,
! [B: $tType,A: $tType,A4: A,S: sum_sum @ A @ B] :
( ( member @ A @ A4 @ ( basic_setl @ A @ B @ S ) )
=> ( S
= ( sum_Inl @ A @ B @ A4 ) ) ) ).
% setl.cases
thf(fact_181_vimage__eq__UN,axiom,
! [B: $tType,A: $tType] :
( ( vimage @ A @ B )
= ( ^ [F2: A > B,B4: set @ B] :
( complete_Sup_Sup @ ( set @ A )
@ ( image @ B @ ( set @ A )
@ ^ [Y: B] : ( vimage @ A @ B @ F2 @ ( insert @ B @ Y @ ( bot_bot @ ( set @ B ) ) ) )
@ B4 ) ) ) ) ).
% vimage_eq_UN
thf(fact_182_sum__set__defs_I2_J,axiom,
! [C: $tType,D: $tType] :
( ( basic_setr @ C @ D )
= ( sum_case_sum @ C @ ( set @ D ) @ D
@ ^ [A3: C] : ( bot_bot @ ( set @ D ) )
@ ^ [Z2: D] : ( insert @ D @ Z2 @ ( bot_bot @ ( set @ D ) ) ) ) ) ).
% sum_set_defs(2)
thf(fact_183_sum__set__defs_I1_J,axiom,
! [A: $tType,B: $tType] :
( ( basic_setl @ A @ B )
= ( sum_case_sum @ A @ ( set @ A ) @ B
@ ^ [Z2: A] : ( insert @ A @ Z2 @ ( bot_bot @ ( set @ A ) ) )
@ ^ [B10: B] : ( bot_bot @ ( set @ A ) ) ) ) ).
% sum_set_defs(1)
thf(fact_184_image__eqI,axiom,
! [A: $tType,B: $tType,B3: A,F: B > A,X5: B,A2: set @ B] :
( ( B3
= ( F @ X5 ) )
=> ( ( member @ B @ X5 @ A2 )
=> ( member @ A @ B3 @ ( image @ B @ A @ F @ A2 ) ) ) ) ).
% image_eqI
thf(fact_185_image__ident,axiom,
! [A: $tType,Y5: set @ A] :
( ( image @ A @ A
@ ^ [X: A] : X
@ Y5 )
= Y5 ) ).
% image_ident
thf(fact_186_case__sum__KK,axiom,
! [C: $tType,B: $tType,A: $tType,A4: C] :
( ( sum_case_sum @ A @ C @ B
@ ^ [X: A] : A4
@ ^ [X: B] : A4 )
= ( ^ [X: sum_sum @ A @ B] : A4 ) ) ).
% case_sum_KK
thf(fact_187_image__is__empty,axiom,
! [A: $tType,B: $tType,F: B > A,A2: set @ B] :
( ( ( image @ B @ A @ F @ A2 )
= ( bot_bot @ ( set @ A ) ) )
= ( A2
= ( bot_bot @ ( set @ B ) ) ) ) ).
% image_is_empty
thf(fact_188_empty__is__image,axiom,
! [A: $tType,B: $tType,F: B > A,A2: set @ B] :
( ( ( bot_bot @ ( set @ A ) )
= ( image @ B @ A @ F @ A2 ) )
= ( A2
= ( bot_bot @ ( set @ B ) ) ) ) ).
% empty_is_image
thf(fact_189_image__empty,axiom,
! [B: $tType,A: $tType,F: B > A] :
( ( image @ B @ A @ F @ ( bot_bot @ ( set @ B ) ) )
= ( bot_bot @ ( set @ A ) ) ) ).
% image_empty
thf(fact_190_insert__image,axiom,
! [B: $tType,A: $tType,X5: A,A2: set @ A,F: A > B] :
( ( member @ A @ X5 @ A2 )
=> ( ( insert @ B @ ( F @ X5 ) @ ( image @ A @ B @ F @ A2 ) )
= ( image @ A @ B @ F @ A2 ) ) ) ).
% insert_image
thf(fact_191_image__insert,axiom,
! [A: $tType,B: $tType,F: B > A,A4: B,B2: set @ B] :
( ( image @ B @ A @ F @ ( insert @ B @ A4 @ B2 ) )
= ( insert @ A @ ( F @ A4 ) @ ( image @ B @ A @ F @ B2 ) ) ) ).
% image_insert
thf(fact_192_ball__UN,axiom,
! [A: $tType,B: $tType,B2: B > ( set @ A ),A2: set @ B,P2: A > $o] :
( ( ! [X: A] :
( ( member @ A @ X @ ( complete_Sup_Sup @ ( set @ A ) @ ( image @ B @ ( set @ A ) @ B2 @ A2 ) ) )
=> ( P2 @ X ) ) )
= ( ! [X: B] :
( ( member @ B @ X @ A2 )
=> ! [Y: A] :
( ( member @ A @ Y @ ( B2 @ X ) )
=> ( P2 @ Y ) ) ) ) ) ).
% ball_UN
thf(fact_193_bex__UN,axiom,
! [A: $tType,B: $tType,B2: B > ( set @ A ),A2: set @ B,P2: A > $o] :
( ( ? [X: A] :
( ( member @ A @ X @ ( complete_Sup_Sup @ ( set @ A ) @ ( image @ B @ ( set @ A ) @ B2 @ A2 ) ) )
& ( P2 @ X ) ) )
= ( ? [X: B] :
( ( member @ B @ X @ A2 )
& ? [Y: A] :
( ( member @ A @ Y @ ( B2 @ X ) )
& ( P2 @ Y ) ) ) ) ) ).
% bex_UN
thf(fact_194_UN__ball__bex__simps_I2_J,axiom,
! [C: $tType,B: $tType,B2: B > ( set @ C ),A2: set @ B,P2: C > $o] :
( ( ! [X: C] :
( ( member @ C @ X @ ( complete_Sup_Sup @ ( set @ C ) @ ( image @ B @ ( set @ C ) @ B2 @ A2 ) ) )
=> ( P2 @ X ) ) )
= ( ! [X: B] :
( ( member @ B @ X @ A2 )
=> ! [Y: C] :
( ( member @ C @ Y @ ( B2 @ X ) )
=> ( P2 @ Y ) ) ) ) ) ).
% UN_ball_bex_simps(2)
thf(fact_195_UN__ball__bex__simps_I4_J,axiom,
! [F3: $tType,E: $tType,B2: E > ( set @ F3 ),A2: set @ E,P2: F3 > $o] :
( ( ? [X: F3] :
( ( member @ F3 @ X @ ( complete_Sup_Sup @ ( set @ F3 ) @ ( image @ E @ ( set @ F3 ) @ B2 @ A2 ) ) )
& ( P2 @ X ) ) )
= ( ? [X: E] :
( ( member @ E @ X @ A2 )
& ? [Y: F3] :
( ( member @ F3 @ Y @ ( B2 @ X ) )
& ( P2 @ Y ) ) ) ) ) ).
% UN_ball_bex_simps(4)
thf(fact_196_SUP__identity__eq,axiom,
! [A: $tType] :
( ( complete_Sup @ A @ ( type2 @ A ) )
=> ! [A2: set @ A] :
( ( complete_Sup_Sup @ A
@ ( image @ A @ A
@ ^ [X: A] : X
@ A2 ) )
= ( complete_Sup_Sup @ A @ A2 ) ) ) ).
% SUP_identity_eq
thf(fact_197_SUP__apply,axiom,
! [A: $tType,B: $tType,C: $tType] :
( ( complete_Sup @ A @ ( type2 @ A ) )
=> ! [F: C > B > A,A2: set @ C,X5: B] :
( ( complete_Sup_Sup @ ( B > A ) @ ( image @ C @ ( B > A ) @ F @ A2 ) @ X5 )
= ( complete_Sup_Sup @ A
@ ( image @ C @ A
@ ^ [Y: C] : ( F @ Y @ X5 )
@ A2 ) ) ) ) ).
% SUP_apply
thf(fact_198_UN__I,axiom,
! [B: $tType,A: $tType,A4: A,A2: set @ A,B3: B,B2: A > ( set @ B )] :
( ( member @ A @ A4 @ A2 )
=> ( ( member @ B @ B3 @ ( B2 @ A4 ) )
=> ( member @ B @ B3 @ ( complete_Sup_Sup @ ( set @ B ) @ ( image @ A @ ( set @ B ) @ B2 @ A2 ) ) ) ) ) ).
% UN_I
thf(fact_199_UN__iff,axiom,
! [A: $tType,B: $tType,B3: A,B2: B > ( set @ A ),A2: set @ B] :
( ( member @ A @ B3 @ ( complete_Sup_Sup @ ( set @ A ) @ ( image @ B @ ( set @ A ) @ B2 @ A2 ) ) )
= ( ? [X: B] :
( ( member @ B @ X @ A2 )
& ( member @ A @ B3 @ ( B2 @ X ) ) ) ) ) ).
% UN_iff
thf(fact_200_SUP__bot,axiom,
! [B: $tType,A: $tType] :
( ( comple187826305attice @ A @ ( type2 @ A ) )
=> ! [A2: set @ B] :
( ( complete_Sup_Sup @ A
@ ( image @ B @ A
@ ^ [X: B] : ( bot_bot @ A )
@ A2 ) )
= ( bot_bot @ A ) ) ) ).
% SUP_bot
thf(fact_201_SUP__bot__conv_I1_J,axiom,
! [A: $tType,B: $tType] :
( ( comple187826305attice @ A @ ( type2 @ A ) )
=> ! [B2: B > A,A2: set @ B] :
( ( ( complete_Sup_Sup @ A @ ( image @ B @ A @ B2 @ A2 ) )
= ( bot_bot @ A ) )
= ( ! [X: B] :
( ( member @ B @ X @ A2 )
=> ( ( B2 @ X )
= ( bot_bot @ A ) ) ) ) ) ) ).
% SUP_bot_conv(1)
thf(fact_202_SUP__bot__conv_I2_J,axiom,
! [A: $tType,B: $tType] :
( ( comple187826305attice @ A @ ( type2 @ A ) )
=> ! [B2: B > A,A2: set @ B] :
( ( ( bot_bot @ A )
= ( complete_Sup_Sup @ A @ ( image @ B @ A @ B2 @ A2 ) ) )
= ( ! [X: B] :
( ( member @ B @ X @ A2 )
=> ( ( B2 @ X )
= ( bot_bot @ A ) ) ) ) ) ) ).
% SUP_bot_conv(2)
thf(fact_203_cSUP__const,axiom,
! [B: $tType,A: $tType] :
( ( condit378418413attice @ A @ ( type2 @ A ) )
=> ! [A2: set @ B,C3: A] :
( ( A2
!= ( bot_bot @ ( set @ B ) ) )
=> ( ( complete_Sup_Sup @ A
@ ( image @ B @ A
@ ^ [X: B] : C3
@ A2 ) )
= C3 ) ) ) ).
% cSUP_const
thf(fact_204_SUP__const,axiom,
! [B: $tType,A: $tType] :
( ( comple187826305attice @ A @ ( type2 @ A ) )
=> ! [A2: set @ B,F: A] :
( ( A2
!= ( bot_bot @ ( set @ B ) ) )
=> ( ( complete_Sup_Sup @ A
@ ( image @ B @ A
@ ^ [I: B] : F
@ A2 ) )
= F ) ) ) ).
% SUP_const
thf(fact_205_UN__constant,axiom,
! [B: $tType,A: $tType,A2: set @ B,C3: set @ A] :
( ( ( A2
= ( bot_bot @ ( set @ B ) ) )
=> ( ( complete_Sup_Sup @ ( set @ A )
@ ( image @ B @ ( set @ A )
@ ^ [Y: B] : C3
@ A2 ) )
= ( bot_bot @ ( set @ A ) ) ) )
& ( ( A2
!= ( bot_bot @ ( set @ B ) ) )
=> ( ( complete_Sup_Sup @ ( set @ A )
@ ( image @ B @ ( set @ A )
@ ^ [Y: B] : C3
@ A2 ) )
= C3 ) ) ) ).
% UN_constant
thf(fact_206_SUP__empty,axiom,
! [B: $tType,A: $tType] :
( ( comple187826305attice @ A @ ( type2 @ A ) )
=> ! [F: B > A] :
( ( complete_Sup_Sup @ A @ ( image @ B @ A @ F @ ( bot_bot @ ( set @ B ) ) ) )
= ( bot_bot @ A ) ) ) ).
% SUP_empty
thf(fact_207_UN__simps_I1_J,axiom,
! [A: $tType,B: $tType,C2: set @ B,A4: A,B2: B > ( set @ A )] :
( ( ( C2
= ( bot_bot @ ( set @ B ) ) )
=> ( ( complete_Sup_Sup @ ( set @ A )
@ ( image @ B @ ( set @ A )
@ ^ [X: B] : ( insert @ A @ A4 @ ( B2 @ X ) )
@ C2 ) )
= ( bot_bot @ ( set @ A ) ) ) )
& ( ( C2
!= ( bot_bot @ ( set @ B ) ) )
=> ( ( complete_Sup_Sup @ ( set @ A )
@ ( image @ B @ ( set @ A )
@ ^ [X: B] : ( insert @ A @ A4 @ ( B2 @ X ) )
@ C2 ) )
= ( insert @ A @ A4 @ ( complete_Sup_Sup @ ( set @ A ) @ ( image @ B @ ( set @ A ) @ B2 @ C2 ) ) ) ) ) ) ).
% UN_simps(1)
thf(fact_208_UN__singleton,axiom,
! [A: $tType,A2: set @ A] :
( ( complete_Sup_Sup @ ( set @ A )
@ ( image @ A @ ( set @ A )
@ ^ [X: A] : ( insert @ A @ X @ ( bot_bot @ ( set @ A ) ) )
@ A2 ) )
= A2 ) ).
% UN_singleton
thf(fact_209_SUP__eq__const,axiom,
! [B: $tType,A: $tType] :
( ( comple187826305attice @ A @ ( type2 @ A ) )
=> ! [I2: set @ B,F: B > A,X5: A] :
( ( I2
!= ( bot_bot @ ( set @ B ) ) )
=> ( ! [I3: B] :
( ( member @ B @ I3 @ I2 )
=> ( ( F @ I3 )
= X5 ) )
=> ( ( complete_Sup_Sup @ A @ ( image @ B @ A @ F @ I2 ) )
= X5 ) ) ) ) ).
% SUP_eq_const
thf(fact_210_Sup_OSUP__cong,axiom,
! [A: $tType,B: $tType,A2: set @ B,B2: set @ B,C2: B > A,D3: B > A,Sup: ( set @ A ) > A] :
( ( A2 = B2 )
=> ( ! [X6: B] :
( ( member @ B @ X6 @ B2 )
=> ( ( C2 @ X6 )
= ( D3 @ X6 ) ) )
=> ( ( Sup @ ( image @ B @ A @ C2 @ A2 ) )
= ( Sup @ ( image @ B @ A @ D3 @ B2 ) ) ) ) ) ).
% Sup.SUP_cong
thf(fact_211_Inf_OINF__cong,axiom,
! [A: $tType,B: $tType,A2: set @ B,B2: set @ B,C2: B > A,D3: B > A,Inf: ( set @ A ) > A] :
( ( A2 = B2 )
=> ( ! [X6: B] :
( ( member @ B @ X6 @ B2 )
=> ( ( C2 @ X6 )
= ( D3 @ X6 ) ) )
=> ( ( Inf @ ( image @ B @ A @ C2 @ A2 ) )
= ( Inf @ ( image @ B @ A @ D3 @ B2 ) ) ) ) ) ).
% Inf.INF_cong
thf(fact_212_case__sum__inject,axiom,
! [A: $tType,C: $tType,B: $tType,F1: A > C,F22: B > C,G1: A > C,G22: B > C] :
( ( ( sum_case_sum @ A @ C @ B @ F1 @ F22 )
= ( sum_case_sum @ A @ C @ B @ G1 @ G22 ) )
=> ~ ( ( F1 = G1 )
=> ( F22 != G22 ) ) ) ).
% case_sum_inject
thf(fact_213_rev__image__eqI,axiom,
! [B: $tType,A: $tType,X5: A,A2: set @ A,B3: B,F: A > B] :
( ( member @ A @ X5 @ A2 )
=> ( ( B3
= ( F @ X5 ) )
=> ( member @ B @ B3 @ ( image @ A @ B @ F @ A2 ) ) ) ) ).
% rev_image_eqI
thf(fact_214_ball__imageD,axiom,
! [A: $tType,B: $tType,F: B > A,A2: set @ B,P2: A > $o] :
( ! [X6: A] :
( ( member @ A @ X6 @ ( image @ B @ A @ F @ A2 ) )
=> ( P2 @ X6 ) )
=> ! [X8: B] :
( ( member @ B @ X8 @ A2 )
=> ( P2 @ ( F @ X8 ) ) ) ) ).
% ball_imageD
thf(fact_215_image__cong,axiom,
! [B: $tType,A: $tType,M: set @ A,N4: set @ A,F: A > B,G2: A > B] :
( ( M = N4 )
=> ( ! [X6: A] :
( ( member @ A @ X6 @ N4 )
=> ( ( F @ X6 )
= ( G2 @ X6 ) ) )
=> ( ( image @ A @ B @ F @ M )
= ( image @ A @ B @ G2 @ N4 ) ) ) ) ).
% image_cong
thf(fact_216_bex__imageD,axiom,
! [A: $tType,B: $tType,F: B > A,A2: set @ B,P2: A > $o] :
( ? [X8: A] :
( ( member @ A @ X8 @ ( image @ B @ A @ F @ A2 ) )
& ( P2 @ X8 ) )
=> ? [X6: B] :
( ( member @ B @ X6 @ A2 )
& ( P2 @ ( F @ X6 ) ) ) ) ).
% bex_imageD
thf(fact_217_image__iff,axiom,
! [A: $tType,B: $tType,Z3: A,F: B > A,A2: set @ B] :
( ( member @ A @ Z3 @ ( image @ B @ A @ F @ A2 ) )
= ( ? [X: B] :
( ( member @ B @ X @ A2 )
& ( Z3
= ( F @ X ) ) ) ) ) ).
% image_iff
thf(fact_218_imageI,axiom,
! [B: $tType,A: $tType,X5: A,A2: set @ A,F: A > B] :
( ( member @ A @ X5 @ A2 )
=> ( member @ B @ ( F @ X5 ) @ ( image @ A @ B @ F @ A2 ) ) ) ).
% imageI
thf(fact_219_SUP__cong,axiom,
! [A: $tType,B: $tType] :
( ( complete_Sup @ A @ ( type2 @ A ) )
=> ! [A2: set @ B,B2: set @ B,C2: B > A,D3: B > A] :
( ( A2 = B2 )
=> ( ! [X6: B] :
( ( member @ B @ X6 @ B2 )
=> ( ( C2 @ X6 )
= ( D3 @ X6 ) ) )
=> ( ( complete_Sup_Sup @ A @ ( image @ B @ A @ C2 @ A2 ) )
= ( complete_Sup_Sup @ A @ ( image @ B @ A @ D3 @ B2 ) ) ) ) ) ) ).
% SUP_cong
thf(fact_220_Sup__SUP__eq,axiom,
! [A: $tType] :
( ( complete_Sup_Sup @ ( A > $o ) )
= ( ^ [S2: set @ ( A > $o ),X: A] : ( member @ A @ X @ ( complete_Sup_Sup @ ( set @ A ) @ ( image @ ( A > $o ) @ ( set @ A ) @ ( collect @ A ) @ S2 ) ) ) ) ) ).
% Sup_SUP_eq
thf(fact_221_case__sum__step_I1_J,axiom,
! [D: $tType,A: $tType,C: $tType,B: $tType,F4: B > A,G3: C > A,G2: D > A,P4: sum_sum @ B @ C] :
( ( sum_case_sum @ ( sum_sum @ B @ C ) @ A @ D @ ( sum_case_sum @ B @ A @ C @ F4 @ G3 ) @ G2 @ ( sum_Inl @ ( sum_sum @ B @ C ) @ D @ P4 ) )
= ( sum_case_sum @ B @ A @ C @ F4 @ G3 @ P4 ) ) ).
% case_sum_step(1)
thf(fact_222_case__sum__step_I2_J,axiom,
! [E: $tType,A: $tType,C: $tType,B: $tType,F: E > A,F4: B > A,G3: C > A,P4: sum_sum @ B @ C] :
( ( sum_case_sum @ E @ A @ ( sum_sum @ B @ C ) @ F @ ( sum_case_sum @ B @ A @ C @ F4 @ G3 ) @ ( sum_Inr @ ( sum_sum @ B @ C ) @ E @ P4 ) )
= ( sum_case_sum @ B @ A @ C @ F4 @ G3 @ P4 ) ) ).
% case_sum_step(2)
thf(fact_223_old_Osum_Osimps_I5_J,axiom,
! [B: $tType,C: $tType,A: $tType,F1: A > C,F22: B > C,X1: A] :
( ( sum_case_sum @ A @ C @ B @ F1 @ F22 @ ( sum_Inl @ A @ B @ X1 ) )
= ( F1 @ X1 ) ) ).
% old.sum.simps(5)
thf(fact_224_old_Osum_Osimps_I6_J,axiom,
! [A: $tType,C: $tType,B: $tType,F1: A > C,F22: B > C,X22: B] :
( ( sum_case_sum @ A @ C @ B @ F1 @ F22 @ ( sum_Inr @ B @ A @ X22 ) )
= ( F22 @ X22 ) ) ).
% old.sum.simps(6)
thf(fact_225_setcompr__eq__image,axiom,
! [A: $tType,B: $tType,F: B > A,P2: B > $o] :
( ( collect @ A
@ ^ [Uu: A] :
? [X: B] :
( ( Uu
= ( F @ X ) )
& ( P2 @ X ) ) )
= ( image @ B @ A @ F @ ( collect @ B @ P2 ) ) ) ).
% setcompr_eq_image
thf(fact_226_Setcompr__eq__image,axiom,
! [A: $tType,B: $tType,F: B > A,A2: set @ B] :
( ( collect @ A
@ ^ [Uu: A] :
? [X: B] :
( ( Uu
= ( F @ X ) )
& ( member @ B @ X @ A2 ) ) )
= ( image @ B @ A @ F @ A2 ) ) ).
% Setcompr_eq_image
thf(fact_227_SUP__commute,axiom,
! [A: $tType,B: $tType,C: $tType] :
( ( comple187826305attice @ A @ ( type2 @ A ) )
=> ! [F: B > C > A,B2: set @ C,A2: set @ B] :
( ( complete_Sup_Sup @ A
@ ( image @ B @ A
@ ^ [I: B] : ( complete_Sup_Sup @ A @ ( image @ C @ A @ ( F @ I ) @ B2 ) )
@ A2 ) )
= ( complete_Sup_Sup @ A
@ ( image @ C @ A
@ ^ [J: C] :
( complete_Sup_Sup @ A
@ ( image @ B @ A
@ ^ [I: B] : ( F @ I @ J )
@ A2 ) )
@ B2 ) ) ) ) ).
% SUP_commute
thf(fact_228_UN__extend__simps_I9_J,axiom,
! [S3: $tType,R3: $tType,Q2: $tType,C2: R3 > ( set @ S3 ),B2: Q2 > ( set @ R3 ),A2: set @ Q2] :
( ( complete_Sup_Sup @ ( set @ S3 )
@ ( image @ Q2 @ ( set @ S3 )
@ ^ [X: Q2] : ( complete_Sup_Sup @ ( set @ S3 ) @ ( image @ R3 @ ( set @ S3 ) @ C2 @ ( B2 @ X ) ) )
@ A2 ) )
= ( complete_Sup_Sup @ ( set @ S3 ) @ ( image @ R3 @ ( set @ S3 ) @ C2 @ ( complete_Sup_Sup @ ( set @ R3 ) @ ( image @ Q2 @ ( set @ R3 ) @ B2 @ A2 ) ) ) ) ) ).
% UN_extend_simps(9)
thf(fact_229_UN__E,axiom,
! [A: $tType,B: $tType,B3: A,B2: B > ( set @ A ),A2: set @ B] :
( ( member @ A @ B3 @ ( complete_Sup_Sup @ ( set @ A ) @ ( image @ B @ ( set @ A ) @ B2 @ A2 ) ) )
=> ~ ! [X6: B] :
( ( member @ B @ X6 @ A2 )
=> ~ ( member @ A @ B3 @ ( B2 @ X6 ) ) ) ) ).
% UN_E
thf(fact_230_UN__UN__flatten,axiom,
! [A: $tType,B: $tType,C: $tType,C2: B > ( set @ A ),B2: C > ( set @ B ),A2: set @ C] :
( ( complete_Sup_Sup @ ( set @ A ) @ ( image @ B @ ( set @ A ) @ C2 @ ( complete_Sup_Sup @ ( set @ B ) @ ( image @ C @ ( set @ B ) @ B2 @ A2 ) ) ) )
= ( complete_Sup_Sup @ ( set @ A )
@ ( image @ C @ ( set @ A )
@ ^ [Y: C] : ( complete_Sup_Sup @ ( set @ A ) @ ( image @ B @ ( set @ A ) @ C2 @ ( B2 @ Y ) ) )
@ A2 ) ) ) ).
% UN_UN_flatten
thf(fact_231_SUP__UNION,axiom,
! [A: $tType,B: $tType,C: $tType] :
( ( comple187826305attice @ A @ ( type2 @ A ) )
=> ! [F: B > A,G2: C > ( set @ B ),A2: set @ C] :
( ( complete_Sup_Sup @ A @ ( image @ B @ A @ F @ ( complete_Sup_Sup @ ( set @ B ) @ ( image @ C @ ( set @ B ) @ G2 @ A2 ) ) ) )
= ( complete_Sup_Sup @ A
@ ( image @ C @ A
@ ^ [Y: C] : ( complete_Sup_Sup @ A @ ( image @ B @ A @ F @ ( G2 @ Y ) ) )
@ A2 ) ) ) ) ).
% SUP_UNION
thf(fact_232_imageE,axiom,
! [A: $tType,B: $tType,B3: A,F: B > A,A2: set @ B] :
( ( member @ A @ B3 @ ( image @ B @ A @ F @ A2 ) )
=> ~ ! [X6: B] :
( ( B3
= ( F @ X6 ) )
=> ~ ( member @ B @ X6 @ A2 ) ) ) ).
% imageE
thf(fact_233_image__image,axiom,
! [A: $tType,B: $tType,C: $tType,F: B > A,G2: C > B,A2: set @ C] :
( ( image @ B @ A @ F @ ( image @ C @ B @ G2 @ A2 ) )
= ( image @ C @ A
@ ^ [X: C] : ( F @ ( G2 @ X ) )
@ A2 ) ) ).
% image_image
thf(fact_234_Compr__image__eq,axiom,
! [A: $tType,B: $tType,F: B > A,A2: set @ B,P2: A > $o] :
( ( collect @ A
@ ^ [X: A] :
( ( member @ A @ X @ ( image @ B @ A @ F @ A2 ) )
& ( P2 @ X ) ) )
= ( image @ B @ A @ F
@ ( collect @ B
@ ^ [X: B] :
( ( member @ B @ X @ A2 )
& ( P2 @ ( F @ X ) ) ) ) ) ) ).
% Compr_image_eq
thf(fact_235_sum_Ocase__distrib,axiom,
! [D: $tType,C: $tType,B: $tType,A: $tType,H: C > D,F1: A > C,F22: B > C,Sum: sum_sum @ A @ B] :
( ( H @ ( sum_case_sum @ A @ C @ B @ F1 @ F22 @ Sum ) )
= ( sum_case_sum @ A @ D @ B
@ ^ [X: A] : ( H @ ( F1 @ X ) )
@ ^ [X: B] : ( H @ ( F22 @ X ) )
@ Sum ) ) ).
% sum.case_distrib
thf(fact_236_image__UN,axiom,
! [A: $tType,B: $tType,C: $tType,F: B > A,B2: C > ( set @ B ),A2: set @ C] :
( ( image @ B @ A @ F @ ( complete_Sup_Sup @ ( set @ B ) @ ( image @ C @ ( set @ B ) @ B2 @ A2 ) ) )
= ( complete_Sup_Sup @ ( set @ A )
@ ( image @ C @ ( set @ A )
@ ^ [X: C] : ( image @ B @ A @ F @ ( B2 @ X ) )
@ A2 ) ) ) ).
% image_UN
thf(fact_237_UN__extend__simps_I10_J,axiom,
! [V: $tType,U2: $tType,T: $tType,B2: U2 > ( set @ V ),F: T > U2,A2: set @ T] :
( ( complete_Sup_Sup @ ( set @ V )
@ ( image @ T @ ( set @ V )
@ ^ [A3: T] : ( B2 @ ( F @ A3 ) )
@ A2 ) )
= ( complete_Sup_Sup @ ( set @ V ) @ ( image @ U2 @ ( set @ V ) @ B2 @ ( image @ T @ U2 @ F @ A2 ) ) ) ) ).
% UN_extend_simps(10)
thf(fact_238_image__Union,axiom,
! [A: $tType,B: $tType,F: B > A,S4: set @ ( set @ B )] :
( ( image @ B @ A @ F @ ( complete_Sup_Sup @ ( set @ B ) @ S4 ) )
= ( complete_Sup_Sup @ ( set @ A ) @ ( image @ ( set @ B ) @ ( set @ A ) @ ( image @ B @ A @ F ) @ S4 ) ) ) ).
% image_Union
thf(fact_239_UN__extend__simps_I8_J,axiom,
! [P5: $tType,O: $tType,B2: O > ( set @ P5 ),A2: set @ ( set @ O )] :
( ( complete_Sup_Sup @ ( set @ P5 )
@ ( image @ ( set @ O ) @ ( set @ P5 )
@ ^ [Y: set @ O] : ( complete_Sup_Sup @ ( set @ P5 ) @ ( image @ O @ ( set @ P5 ) @ B2 @ Y ) )
@ A2 ) )
= ( complete_Sup_Sup @ ( set @ P5 ) @ ( image @ O @ ( set @ P5 ) @ B2 @ ( complete_Sup_Sup @ ( set @ O ) @ A2 ) ) ) ) ).
% UN_extend_simps(8)
thf(fact_240_Inf_OINF__identity__eq,axiom,
! [A: $tType,Inf: ( set @ A ) > A,A2: set @ A] :
( ( Inf
@ ( image @ A @ A
@ ^ [X: A] : X
@ A2 ) )
= ( Inf @ A2 ) ) ).
% Inf.INF_identity_eq
thf(fact_241_Sup_OSUP__identity__eq,axiom,
! [A: $tType,Sup: ( set @ A ) > A,A2: set @ A] :
( ( Sup
@ ( image @ A @ A
@ ^ [X: A] : X
@ A2 ) )
= ( Sup @ A2 ) ) ).
% Sup.SUP_identity_eq
thf(fact_242_the__elem__image__unique,axiom,
! [B: $tType,A: $tType,A2: set @ A,F: A > B,X5: A] :
( ( A2
!= ( bot_bot @ ( set @ A ) ) )
=> ( ! [Y4: A] :
( ( member @ A @ Y4 @ A2 )
=> ( ( F @ Y4 )
= ( F @ X5 ) ) )
=> ( ( the_elem @ B @ ( image @ A @ B @ F @ A2 ) )
= ( F @ X5 ) ) ) ) ).
% the_elem_image_unique
thf(fact_243_image__constant__conv,axiom,
! [B: $tType,A: $tType,A2: set @ B,C3: A] :
( ( ( A2
= ( bot_bot @ ( set @ B ) ) )
=> ( ( image @ B @ A
@ ^ [X: B] : C3
@ A2 )
= ( bot_bot @ ( set @ A ) ) ) )
& ( ( A2
!= ( bot_bot @ ( set @ B ) ) )
=> ( ( image @ B @ A
@ ^ [X: B] : C3
@ A2 )
= ( insert @ A @ C3 @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ).
% image_constant_conv
thf(fact_244_image__constant,axiom,
! [A: $tType,B: $tType,X5: A,A2: set @ A,C3: B] :
( ( member @ A @ X5 @ A2 )
=> ( ( image @ A @ B
@ ^ [X: A] : C3
@ A2 )
= ( insert @ B @ C3 @ ( bot_bot @ ( set @ B ) ) ) ) ) ).
% image_constant
thf(fact_245_UN__empty2,axiom,
! [B: $tType,A: $tType,A2: set @ B] :
( ( complete_Sup_Sup @ ( set @ A )
@ ( image @ B @ ( set @ A )
@ ^ [X: B] : ( bot_bot @ ( set @ A ) )
@ A2 ) )
= ( bot_bot @ ( set @ A ) ) ) ).
% UN_empty2
thf(fact_246_UN__empty,axiom,
! [B: $tType,A: $tType,B2: B > ( set @ A )] :
( ( complete_Sup_Sup @ ( set @ A ) @ ( image @ B @ ( set @ A ) @ B2 @ ( bot_bot @ ( set @ B ) ) ) )
= ( bot_bot @ ( set @ A ) ) ) ).
% UN_empty
thf(fact_247_UNION__empty__conv_I1_J,axiom,
! [A: $tType,B: $tType,B2: B > ( set @ A ),A2: set @ B] :
( ( ( bot_bot @ ( set @ A ) )
= ( complete_Sup_Sup @ ( set @ A ) @ ( image @ B @ ( set @ A ) @ B2 @ A2 ) ) )
= ( ! [X: B] :
( ( member @ B @ X @ A2 )
=> ( ( B2 @ X )
= ( bot_bot @ ( set @ A ) ) ) ) ) ) ).
% UNION_empty_conv(1)
thf(fact_248_UNION__empty__conv_I2_J,axiom,
! [A: $tType,B: $tType,B2: B > ( set @ A ),A2: set @ B] :
( ( ( complete_Sup_Sup @ ( set @ A ) @ ( image @ B @ ( set @ A ) @ B2 @ A2 ) )
= ( bot_bot @ ( set @ A ) ) )
= ( ! [X: B] :
( ( member @ B @ X @ A2 )
=> ( ( B2 @ X )
= ( bot_bot @ ( set @ A ) ) ) ) ) ) ).
% UNION_empty_conv(2)
thf(fact_249_UN__insert__distrib,axiom,
! [B: $tType,A: $tType,U3: A,A2: set @ A,A4: B,B2: A > ( set @ B )] :
( ( member @ A @ U3 @ A2 )
=> ( ( complete_Sup_Sup @ ( set @ B )
@ ( image @ A @ ( set @ B )
@ ^ [X: A] : ( insert @ B @ A4 @ ( B2 @ X ) )
@ A2 ) )
= ( insert @ B @ A4 @ ( complete_Sup_Sup @ ( set @ B ) @ ( image @ A @ ( set @ B ) @ B2 @ A2 ) ) ) ) ) ).
% UN_insert_distrib
thf(fact_250_vimage__UN,axiom,
! [A: $tType,B: $tType,C: $tType,F: A > B,B2: C > ( set @ B ),A2: set @ C] :
( ( vimage @ A @ B @ F @ ( complete_Sup_Sup @ ( set @ B ) @ ( image @ C @ ( set @ B ) @ B2 @ A2 ) ) )
= ( complete_Sup_Sup @ ( set @ A )
@ ( image @ C @ ( set @ A )
@ ^ [X: C] : ( vimage @ A @ B @ F @ ( B2 @ X ) )
@ A2 ) ) ) ).
% vimage_UN
thf(fact_251_vimage__Union,axiom,
! [A: $tType,B: $tType,F: A > B,A2: set @ ( set @ B )] :
( ( vimage @ A @ B @ F @ ( complete_Sup_Sup @ ( set @ B ) @ A2 ) )
= ( complete_Sup_Sup @ ( set @ A ) @ ( image @ ( set @ B ) @ ( set @ A ) @ ( vimage @ A @ B @ F ) @ A2 ) ) ) ).
% vimage_Union
thf(fact_252_case__sum__if,axiom,
! [B: $tType,A: $tType,C: $tType,P4: $o,F: B > A,G2: C > A,X5: B,Y3: C] :
( ( P4
=> ( ( sum_case_sum @ B @ A @ C @ F @ G2 @ ( if @ ( sum_sum @ B @ C ) @ P4 @ ( sum_Inl @ B @ C @ X5 ) @ ( sum_Inr @ C @ B @ Y3 ) ) )
= ( F @ X5 ) ) )
& ( ~ P4
=> ( ( sum_case_sum @ B @ A @ C @ F @ G2 @ ( if @ ( sum_sum @ B @ C ) @ P4 @ ( sum_Inl @ B @ C @ X5 ) @ ( sum_Inr @ C @ B @ Y3 ) ) )
= ( G2 @ Y3 ) ) ) ) ).
% case_sum_if
thf(fact_253_surjective__sum,axiom,
! [C: $tType,B: $tType,A: $tType,F: ( sum_sum @ A @ B ) > C] :
( ( sum_case_sum @ A @ C @ B
@ ^ [X: A] : ( F @ ( sum_Inl @ A @ B @ X ) )
@ ^ [Y: B] : ( F @ ( sum_Inr @ B @ A @ Y ) ) )
= F ) ).
% surjective_sum
thf(fact_254_SUP__constant,axiom,
! [B: $tType,A: $tType] :
( ( comple187826305attice @ A @ ( type2 @ A ) )
=> ! [A2: set @ B,C3: A] :
( ( ( A2
= ( bot_bot @ ( set @ B ) ) )
=> ( ( complete_Sup_Sup @ A
@ ( image @ B @ A
@ ^ [Y: B] : C3
@ A2 ) )
= ( bot_bot @ A ) ) )
& ( ( A2
!= ( bot_bot @ ( set @ B ) ) )
=> ( ( complete_Sup_Sup @ A
@ ( image @ B @ A
@ ^ [Y: B] : C3
@ A2 ) )
= C3 ) ) ) ) ).
% SUP_constant
thf(fact_255_UN__extend__simps_I1_J,axiom,
! [A: $tType,B: $tType,C2: set @ B,A4: A,B2: B > ( set @ A )] :
( ( ( C2
= ( bot_bot @ ( set @ B ) ) )
=> ( ( insert @ A @ A4 @ ( complete_Sup_Sup @ ( set @ A ) @ ( image @ B @ ( set @ A ) @ B2 @ C2 ) ) )
= ( insert @ A @ A4 @ ( bot_bot @ ( set @ A ) ) ) ) )
& ( ( C2
!= ( bot_bot @ ( set @ B ) ) )
=> ( ( insert @ A @ A4 @ ( complete_Sup_Sup @ ( set @ A ) @ ( image @ B @ ( set @ A ) @ B2 @ C2 ) ) )
= ( complete_Sup_Sup @ ( set @ A )
@ ( image @ B @ ( set @ A )
@ ^ [X: B] : ( insert @ A @ A4 @ ( B2 @ X ) )
@ C2 ) ) ) ) ) ).
% UN_extend_simps(1)
%----Type constructors (15)
thf(tcon_fun___Conditionally__Complete__Lattices_Oconditionally__complete__lattice,axiom,
! [A7: $tType,A10: $tType] :
( ( comple187826305attice @ A10 @ ( type2 @ A10 ) )
=> ( condit378418413attice @ ( A7 > A10 ) @ ( type2 @ ( A7 > A10 ) ) ) ) ).
thf(tcon_fun___Complete__Lattices_Ocomplete__lattice,axiom,
! [A7: $tType,A10: $tType] :
( ( comple187826305attice @ A10 @ ( type2 @ A10 ) )
=> ( comple187826305attice @ ( A7 > A10 ) @ ( type2 @ ( A7 > A10 ) ) ) ) ).
thf(tcon_fun___Complete__Partial__Order_Occpo,axiom,
! [A7: $tType,A10: $tType] :
( ( comple187826305attice @ A10 @ ( type2 @ A10 ) )
=> ( comple1141879883l_ccpo @ ( A7 > A10 ) @ ( type2 @ ( A7 > A10 ) ) ) ) ).
thf(tcon_fun___Complete__Lattices_OSup,axiom,
! [A7: $tType,A10: $tType] :
( ( complete_Sup @ A10 @ ( type2 @ A10 ) )
=> ( complete_Sup @ ( A7 > A10 ) @ ( type2 @ ( A7 > A10 ) ) ) ) ).
thf(tcon_fun___Orderings_Obot,axiom,
! [A7: $tType,A10: $tType] :
( ( bot @ A10 @ ( type2 @ A10 ) )
=> ( bot @ ( A7 > A10 ) @ ( type2 @ ( A7 > A10 ) ) ) ) ).
thf(tcon_Set_Oset___Conditionally__Complete__Lattices_Oconditionally__complete__lattice_1,axiom,
! [A7: $tType] : ( condit378418413attice @ ( set @ A7 ) @ ( type2 @ ( set @ A7 ) ) ) ).
thf(tcon_Set_Oset___Complete__Lattices_Ocomplete__lattice_2,axiom,
! [A7: $tType] : ( comple187826305attice @ ( set @ A7 ) @ ( type2 @ ( set @ A7 ) ) ) ).
thf(tcon_Set_Oset___Complete__Partial__Order_Occpo_3,axiom,
! [A7: $tType] : ( comple1141879883l_ccpo @ ( set @ A7 ) @ ( type2 @ ( set @ A7 ) ) ) ).
thf(tcon_Set_Oset___Complete__Lattices_OSup_4,axiom,
! [A7: $tType] : ( complete_Sup @ ( set @ A7 ) @ ( type2 @ ( set @ A7 ) ) ) ).
thf(tcon_Set_Oset___Orderings_Obot_5,axiom,
! [A7: $tType] : ( bot @ ( set @ A7 ) @ ( type2 @ ( set @ A7 ) ) ) ).
thf(tcon_HOL_Obool___Conditionally__Complete__Lattices_Oconditionally__complete__lattice_6,axiom,
condit378418413attice @ $o @ ( type2 @ $o ) ).
thf(tcon_HOL_Obool___Complete__Lattices_Ocomplete__lattice_7,axiom,
comple187826305attice @ $o @ ( type2 @ $o ) ).
thf(tcon_HOL_Obool___Complete__Partial__Order_Occpo_8,axiom,
comple1141879883l_ccpo @ $o @ ( type2 @ $o ) ).
thf(tcon_HOL_Obool___Complete__Lattices_OSup_9,axiom,
complete_Sup @ $o @ ( type2 @ $o ) ).
thf(tcon_HOL_Obool___Orderings_Obot_10,axiom,
bot @ $o @ ( type2 @ $o ) ).
%----Helper facts (3)
thf(help_If_3_1_T,axiom,
! [P2: $o] :
( ( P2 = $true )
| ( P2 = $false ) ) ).
thf(help_If_2_1_T,axiom,
! [A: $tType,X5: A,Y3: A] :
( ( if @ A @ $false @ X5 @ Y3 )
= Y3 ) ).
thf(help_If_1_1_T,axiom,
! [A: $tType,X5: A,Y3: A] :
( ( if @ A @ $true @ X5 @ Y3 )
= X5 ) ).
%----Conjectures (1)
thf(conj_0,conjecture,
( ( collect @ n @ ( gram_L830233218_inItr @ ns @ tr ) )
= ( collect @ n
@ ^ [Uu: n] :
? [Tr5: dtree] :
( ( Uu
= ( root @ Tr5 ) )
& ( gram_L716654942_subtr @ ns @ Tr5 @ tr ) ) ) ) ).
%------------------------------------------------------------------------------