TPTP Problem File: COM188^1.p
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%------------------------------------------------------------------------------
% File : COM188^1 : TPTP v8.2.0. Released v7.0.0.
% Domain : Computing Theory
% Problem : Grammars and languages 619
% 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__619.p [Bla16]
% Status : Theorem
% Rating : 0.67 v8.1.0, 0.75 v7.5.0, 1.00 v7.1.0
% Syntax : Number of formulae : 319 ( 122 unt; 54 typ; 0 def)
% Number of atoms : 682 ( 320 equ; 0 cnn)
% Maximal formula atoms : 7 ( 2 avg)
% Number of connectives : 4052 ( 105 ~; 15 |; 75 &;3563 @)
% ( 0 <=>; 294 =>; 0 <=; 0 <~>)
% Maximal formula depth : 21 ( 8 avg)
% Number of types : 4 ( 3 usr)
% Number of type conns : 260 ( 260 >; 0 *; 0 +; 0 <<)
% Number of symbols : 54 ( 51 usr; 12 con; 0-6 aty)
% Number of variables : 1096 ( 75 ^; 936 !; 48 ?;1096 :)
% ( 35 !>; 0 ?*; 0 @-; 2 @+)
% SPC : TH1_THM_EQU_NAR
% Comments : This file was generated by Isabelle (most likely Sledgehammer)
% 2016-07-13 14:43:44.227
%------------------------------------------------------------------------------
%----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 (48)
thf(sy_cl_Finite__Set_Ofinite,type,
finite_finite:
!>[A: $tType] : ( ( itself @ A ) > $o ) ).
thf(sy_c_BNF__Def_Ocollect,type,
bNF_collect:
!>[B: $tType,A: $tType] : ( ( set @ ( B > ( set @ A ) ) ) > B > ( set @ A ) ) ).
thf(sy_c_BNF__Greatest__Fixpoint_Ouniv,type,
bNF_Greatest_univ:
!>[B: $tType,A: $tType] : ( ( B > A ) > ( set @ B ) > A ) ).
thf(sy_c_Basic__BNFs_Osetl,type,
basic_setl:
!>[A: $tType,B: $tType] : ( ( sum_sum @ A @ B ) > ( set @ 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_DTree_Ounfold,type,
unfold:
!>[A: $tType] : ( ( A > n ) > ( A > ( set @ ( sum_sum @ t @ A ) ) ) > A > dtree ) ).
thf(sy_c_Finite__Set_Ofinite,type,
finite_finite2:
!>[A: $tType] : ( ( set @ A ) > $o ) ).
thf(sy_c_Fun_Ocomp,type,
comp:
!>[B: $tType,C: $tType,A: $tType] : ( ( B > C ) > ( A > B ) > A > C ) ).
thf(sy_c_Fun_Oinj__on,type,
inj_on:
!>[A: $tType,B: $tType] : ( ( A > B ) > ( set @ A ) > $o ) ).
thf(sy_c_Gram__Lang__Mirabelle__ojxrtuoybn_OFrr,type,
gram_L1556062726le_Frr: ( set @ n ) > dtree > ( set @ t ) ).
thf(sy_c_Gram__Lang__Mirabelle__ojxrtuoybn_Odeftr,type,
gram_L1231612515_deftr: n > dtree ).
thf(sy_c_Gram__Lang__Mirabelle__ojxrtuoybn_Ohsubst,type,
gram_L1004374585hsubst: dtree > dtree > dtree ).
thf(sy_c_Gram__Lang__Mirabelle__ojxrtuoybn_Ohsubst__c,type,
gram_L1905609002ubst_c: dtree > dtree > ( set @ ( sum_sum @ t @ dtree ) ) ).
thf(sy_c_Gram__Lang__Mirabelle__ojxrtuoybn_Ohsubst__r,type,
gram_L1905609017ubst_r: dtree > 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_OinFrr,type,
gram_L805317505_inFrr: ( 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_Osubtr2,type,
gram_L1283001940subtr2: ( set @ n ) > dtree > dtree > $o ).
thf(sy_c_Gram__Lang__Mirabelle__ojxrtuoybn_OsubtrOf,type,
gram_L1614515765ubtrOf: dtree > n > dtree ).
thf(sy_c_Gram__Lang__Mirabelle__ojxrtuoybn_Owf,type,
gram_L864798063lle_wf: dtree > $o ).
thf(sy_c_Groups_Ominus__class_Ominus,type,
minus_minus:
!>[A: $tType] : ( A > A > A ) ).
thf(sy_c_If,type,
if:
!>[A: $tType] : ( $o > A > A > A ) ).
thf(sy_c_Orderings_Obot__class_Obot,type,
bot_bot:
!>[A: $tType] : A ).
thf(sy_c_Pure_Otype,type,
type:
!>[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__singleton,type,
is_singleton:
!>[A: $tType] : ( ( set @ A ) > $o ) ).
thf(sy_c_Set_Oremove,type,
remove:
!>[A: $tType] : ( A > ( set @ A ) > ( set @ A ) ) ).
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_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_member,type,
member:
!>[A: $tType] : ( A > ( set @ A ) > $o ) ).
thf(sy_v_ns,type,
ns: set @ n ).
thf(sy_v_nsa____,type,
nsa: set @ n ).
thf(sy_v_t,type,
t2: t ).
thf(sy_v_ta____,type,
ta: t ).
thf(sy_v_thesis____,type,
thesis: $o ).
thf(sy_v_tr,type,
tr: dtree ).
thf(sy_v_tr0,type,
tr0: dtree ).
thf(sy_v_tr1_H____,type,
tr1: dtree ).
thf(sy_v_tr1a____,type,
tr1a: dtree ).
thf(sy_v_tra____,type,
tra: dtree ).
%----Relevant facts (256)
thf(fact_0_tr1_H__tr1,axiom,
member @ ( sum_sum @ t @ dtree ) @ ( sum_Inr @ dtree @ t @ tr1 ) @ ( cont @ tr1a ) ).
% tr1'_tr1
thf(fact_1_tr1,axiom,
( tr1a
= ( gram_L1004374585hsubst @ tr0 @ tra ) ) ).
% tr1
thf(fact_2_wf__hsubst,axiom,
! [Tr: dtree] :
( ( gram_L864798063lle_wf @ tr0 )
=> ( ( gram_L864798063lle_wf @ Tr )
=> ( gram_L864798063lle_wf @ ( gram_L1004374585hsubst @ tr0 @ Tr ) ) ) ) ).
% wf_hsubst
thf(fact_3_assms,axiom,
gram_L1333338417e_inFr @ ns @ ( gram_L1004374585hsubst @ tr0 @ tr ) @ t2 ).
% assms
thf(fact_4_sum_Oinject_I2_J,axiom,
! [A: $tType,B: $tType,X2: B,Y2: B] :
( ( ( sum_Inr @ B @ A @ X2 )
= ( sum_Inr @ B @ A @ Y2 ) )
= ( X2 = Y2 ) ) ).
% sum.inject(2)
thf(fact_5_old_Osum_Oinject_I2_J,axiom,
! [A: $tType,B: $tType,B2: B,B3: B] :
( ( ( sum_Inr @ B @ A @ B2 )
= ( sum_Inr @ B @ A @ B3 ) )
= ( B2 = B3 ) ) ).
% old.sum.inject(2)
thf(fact_6_hsubst__eq,axiom,
! [Tr: dtree] :
( ( ( root @ Tr )
= ( root @ tr0 ) )
=> ( ( gram_L1004374585hsubst @ tr0 @ Tr )
= ( gram_L1004374585hsubst @ tr0 @ tr0 ) ) ) ).
% hsubst_eq
thf(fact_7_hsubst__def,axiom,
( ( gram_L1004374585hsubst @ tr0 )
= ( unfold @ dtree @ gram_L1905609017ubst_r @ ( gram_L1905609002ubst_c @ tr0 ) ) ) ).
% hsubst_def
thf(fact_8_t__tr1_H,axiom,
gram_L1333338417e_inFr @ nsa @ tr1 @ ta ).
% t_tr1'
thf(fact_9_True,axiom,
( ( root @ tr1a )
= ( root @ tr0 ) ) ).
% True
thf(fact_10_Inr__inject,axiom,
! [A: $tType,B: $tType,X: B,Y: B] :
( ( ( sum_Inr @ B @ A @ X )
= ( sum_Inr @ B @ A @ Y ) )
=> ( X = Y ) ) ).
% Inr_inject
thf(fact_11_not__arg__cong__Inr,axiom,
! [B: $tType,A: $tType,X: A,Y: A] :
( ( X != Y )
=> ( ( sum_Inr @ A @ B @ X )
!= ( sum_Inr @ A @ B @ Y ) ) ) ).
% not_arg_cong_Inr
thf(fact_12_root__hsubst,axiom,
! [Tr: dtree] :
( ( root @ ( gram_L1004374585hsubst @ tr0 @ Tr ) )
= ( root @ Tr ) ) ).
% root_hsubst
thf(fact_13_hsubst__c__def,axiom,
! [Tr: dtree] :
( ( ( ( root @ Tr )
= ( root @ tr0 ) )
=> ( ( gram_L1905609002ubst_c @ tr0 @ Tr )
= ( cont @ tr0 ) ) )
& ( ( ( root @ Tr )
!= ( root @ tr0 ) )
=> ( ( gram_L1905609002ubst_c @ tr0 @ Tr )
= ( cont @ Tr ) ) ) ) ).
% hsubst_c_def
thf(fact_14_Ind_Ohyps_I1_J,axiom,
member @ n @ ( root @ tr1a ) @ nsa ).
% Ind.hyps(1)
thf(fact_15_rtr1,axiom,
( ( root @ tr1a )
= ( root @ tra ) ) ).
% rtr1
thf(fact_16_wf__inj,axiom,
! [Tr: dtree,Tr1: dtree,Tr2: dtree] :
( ( gram_L864798063lle_wf @ Tr )
=> ( ( member @ ( sum_sum @ t @ dtree ) @ ( sum_Inr @ dtree @ t @ Tr1 ) @ ( cont @ Tr ) )
=> ( ( member @ ( sum_sum @ t @ dtree ) @ ( sum_Inr @ dtree @ t @ Tr2 ) @ ( cont @ Tr ) )
=> ( ( ( root @ Tr1 )
= ( root @ Tr2 ) )
= ( Tr1 = Tr2 ) ) ) ) ) ).
% wf_inj
thf(fact_17_hsubst__r__def,axiom,
gram_L1905609017ubst_r = root ).
% hsubst_r_def
thf(fact_18_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_19_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_20_Gram__Lang__Mirabelle__ojxrtuoybn_Ohsubst__c__def,axiom,
( gram_L1905609002ubst_c
= ( ^ [Tr0: dtree,Tr3: dtree] :
( if @ ( set @ ( sum_sum @ t @ dtree ) )
@ ( ( root @ Tr3 )
= ( root @ Tr0 ) )
@ ( cont @ Tr0 )
@ ( cont @ Tr3 ) ) ) ) ).
% Gram_Lang_Mirabelle_ojxrtuoybn.hsubst_c_def
thf(fact_21_Gram__Lang__Mirabelle__ojxrtuoybn_Ohsubst__def,axiom,
( gram_L1004374585hsubst
= ( ^ [Tr0: dtree] : ( unfold @ dtree @ gram_L1905609017ubst_r @ ( gram_L1905609002ubst_c @ Tr0 ) ) ) ) ).
% Gram_Lang_Mirabelle_ojxrtuoybn.hsubst_def
thf(fact_22_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_23_Gram__Lang__Mirabelle__ojxrtuoybn_Oroot__hsubst,axiom,
! [Tr02: dtree,Tr: dtree] :
( ( root @ ( gram_L1004374585hsubst @ Tr02 @ Tr ) )
= ( root @ Tr ) ) ).
% Gram_Lang_Mirabelle_ojxrtuoybn.root_hsubst
thf(fact_24_Gram__Lang__Mirabelle__ojxrtuoybn_Ohsubst__eq,axiom,
! [Tr: dtree,Tr02: dtree] :
( ( ( root @ Tr )
= ( root @ Tr02 ) )
=> ( ( gram_L1004374585hsubst @ Tr02 @ Tr )
= ( gram_L1004374585hsubst @ Tr02 @ Tr02 ) ) ) ).
% Gram_Lang_Mirabelle_ojxrtuoybn.hsubst_eq
thf(fact_25_Gram__Lang__Mirabelle__ojxrtuoybn_Owf__hsubst,axiom,
! [Tr02: dtree,Tr: dtree] :
( ( gram_L864798063lle_wf @ Tr02 )
=> ( ( gram_L864798063lle_wf @ Tr )
=> ( gram_L864798063lle_wf @ ( gram_L1004374585hsubst @ Tr02 @ Tr ) ) ) ) ).
% Gram_Lang_Mirabelle_ojxrtuoybn.wf_hsubst
thf(fact_26_wf__cont,axiom,
! [Tr: dtree,Tr4: dtree] :
( ( gram_L864798063lle_wf @ Tr )
=> ( ( member @ ( sum_sum @ t @ dtree ) @ ( sum_Inr @ dtree @ t @ Tr4 ) @ ( cont @ Tr ) )
=> ( gram_L864798063lle_wf @ Tr4 ) ) ) ).
% wf_cont
thf(fact_27_subtrOf__root,axiom,
! [Tr: dtree,Tr4: dtree] :
( ( gram_L864798063lle_wf @ Tr )
=> ( ( member @ ( sum_sum @ t @ dtree ) @ ( sum_Inr @ dtree @ t @ Tr4 ) @ ( cont @ Tr ) )
=> ( ( gram_L1614515765ubtrOf @ Tr @ ( root @ Tr4 ) )
= Tr4 ) ) ) ).
% subtrOf_root
thf(fact_28_inFrr__def,axiom,
( gram_L805317505_inFrr
= ( ^ [Ns2: set @ n,Tr3: dtree,T3: t] :
? [Tr5: dtree] :
( ( member @ ( sum_sum @ t @ dtree ) @ ( sum_Inr @ dtree @ t @ Tr5 ) @ ( cont @ Tr3 ) )
& ( gram_L1333338417e_inFr @ Ns2 @ Tr5 @ T3 ) ) ) ) ).
% inFrr_def
thf(fact_29_finite__hsubst__c,axiom,
! [N: dtree] : ( finite_finite2 @ ( sum_sum @ t @ dtree ) @ ( gram_L1905609002ubst_c @ tr0 @ N ) ) ).
% finite_hsubst_c
thf(fact_30_unfold_I1_J,axiom,
! [A: $tType,Rt: A > n,Ct: A > ( set @ ( sum_sum @ t @ A ) ),B2: A] :
( ( root @ ( unfold @ A @ Rt @ Ct @ B2 ) )
= ( Rt @ B2 ) ) ).
% unfold(1)
thf(fact_31_old_Osum_Osimps_I8_J,axiom,
! [A: $tType,T: $tType,B: $tType,F1: A > T,F2: B > T,B2: B] :
( ( sum_rec_sum @ A @ T @ B @ F1 @ F2 @ ( sum_Inr @ B @ A @ B2 ) )
= ( F2 @ B2 ) ) ).
% old.sum.simps(8)
thf(fact_32_dtree__cong,axiom,
! [Tr: dtree,Tr4: dtree] :
( ( ( root @ Tr )
= ( root @ Tr4 ) )
=> ( ( ( cont @ Tr )
= ( cont @ Tr4 ) )
=> ( Tr = Tr4 ) ) ) ).
% dtree_cong
thf(fact_33_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_34_inItr_Oinducts,axiom,
! [X1: set @ n,X2: dtree,X3: n,P: ( set @ n ) > dtree > n > $o] :
( ( gram_L830233218_inItr @ X1 @ X2 @ X3 )
=> ( ! [Tr6: dtree,Ns3: set @ n] :
( ( member @ n @ ( root @ Tr6 ) @ Ns3 )
=> ( P @ Ns3 @ Tr6 @ ( root @ Tr6 ) ) )
=> ( ! [Tr6: dtree,Ns3: set @ n,Tr12: dtree,N2: n] :
( ( member @ n @ ( root @ Tr6 ) @ Ns3 )
=> ( ( member @ ( sum_sum @ t @ dtree ) @ ( sum_Inr @ dtree @ t @ Tr12 ) @ ( cont @ Tr6 ) )
=> ( ( gram_L830233218_inItr @ Ns3 @ Tr12 @ N2 )
=> ( ( P @ Ns3 @ Tr12 @ N2 )
=> ( P @ Ns3 @ Tr6 @ N2 ) ) ) ) )
=> ( P @ X1 @ X2 @ X3 ) ) ) ) ).
% inItr.inducts
thf(fact_35_inItr_Osimps,axiom,
( gram_L830233218_inItr
= ( ^ [A1: set @ n,A2: dtree,A3: n] :
( ? [Tr3: dtree,Ns2: set @ n] :
( ( A1 = Ns2 )
& ( A2 = Tr3 )
& ( A3
= ( root @ Tr3 ) )
& ( member @ n @ ( root @ Tr3 ) @ Ns2 ) )
| ? [Tr3: dtree,Ns2: set @ n,Tr13: dtree,N3: n] :
( ( A1 = Ns2 )
& ( A2 = Tr3 )
& ( A3 = N3 )
& ( member @ n @ ( root @ Tr3 ) @ Ns2 )
& ( member @ ( sum_sum @ t @ dtree ) @ ( sum_Inr @ dtree @ t @ Tr13 ) @ ( cont @ Tr3 ) )
& ( gram_L830233218_inItr @ Ns2 @ Tr13 @ N3 ) ) ) ) ) ).
% inItr.simps
thf(fact_36_inItr_Ocases,axiom,
! [A12: set @ n,A22: dtree,A32: n] :
( ( gram_L830233218_inItr @ A12 @ A22 @ A32 )
=> ( ( ( A32
= ( root @ A22 ) )
=> ~ ( member @ n @ ( root @ A22 ) @ A12 ) )
=> ~ ( ( member @ n @ ( root @ A22 ) @ A12 )
=> ! [Tr12: dtree] :
( ( member @ ( sum_sum @ t @ dtree ) @ ( sum_Inr @ dtree @ t @ Tr12 ) @ ( cont @ A22 ) )
=> ~ ( gram_L830233218_inItr @ A12 @ Tr12 @ A32 ) ) ) ) ) ).
% inItr.cases
thf(fact_37_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_38_finite__cont,axiom,
! [Tr: dtree] : ( finite_finite2 @ ( sum_sum @ t @ dtree ) @ ( cont @ Tr ) ) ).
% finite_cont
thf(fact_39_Gram__Lang__Mirabelle__ojxrtuoybn_Ofinite__hsubst__c,axiom,
! [Tr02: dtree,N: dtree] : ( finite_finite2 @ ( sum_sum @ t @ dtree ) @ ( gram_L1905609002ubst_c @ Tr02 @ N ) ) ).
% Gram_Lang_Mirabelle_ojxrtuoybn.finite_hsubst_c
thf(fact_40_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_41_inItr_OBase,axiom,
! [Tr: dtree,Ns: set @ n] :
( ( member @ n @ ( root @ Tr ) @ Ns )
=> ( gram_L830233218_inItr @ Ns @ Tr @ ( root @ Tr ) ) ) ).
% inItr.Base
thf(fact_42_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_43_inFr__inFr2,axiom,
gram_L1333338417e_inFr = gram_L805317441_inFr2 ).
% inFr_inFr2
thf(fact_44_finite__code,axiom,
! [A: $tType] :
( ( finite_finite @ A @ ( type @ A ) )
=> ( ( finite_finite2 @ A )
= ( ^ [A4: set @ A] : $true ) ) ) ).
% finite_code
thf(fact_45_mem__Collect__eq,axiom,
! [A: $tType,A5: A,P: A > $o] :
( ( member @ A @ A5 @ ( collect @ A @ P ) )
= ( P @ A5 ) ) ).
% mem_Collect_eq
thf(fact_46_Collect__mem__eq,axiom,
! [A: $tType,A6: set @ A] :
( ( collect @ A
@ ^ [X4: A] : ( member @ A @ X4 @ A6 ) )
= A6 ) ).
% Collect_mem_eq
thf(fact_47_Collect__cong,axiom,
! [A: $tType,P: A > $o,Q: A > $o] :
( ! [X5: A] :
( ( P @ X5 )
= ( Q @ X5 ) )
=> ( ( collect @ A @ P )
= ( collect @ A @ Q ) ) ) ).
% Collect_cong
thf(fact_48_ext,axiom,
! [B: $tType,A: $tType,F: A > B,G: A > B] :
( ! [X5: A] :
( ( F @ X5 )
= ( G @ X5 ) )
=> ( F = G ) ) ).
% ext
thf(fact_49_cont__Node,axiom,
! [As: set @ ( sum_sum @ t @ dtree ),N: n] :
( ( finite_finite2 @ ( sum_sum @ t @ dtree ) @ As )
=> ( ( cont @ ( node @ N @ As ) )
= As ) ) ).
% cont_Node
thf(fact_50_Node__root__cont,axiom,
! [Tr: dtree] :
( ( node @ ( root @ Tr ) @ ( cont @ Tr ) )
= Tr ) ).
% Node_root_cont
thf(fact_51_dtree__simps,axiom,
! [As: set @ ( sum_sum @ t @ dtree ),As2: set @ ( sum_sum @ t @ dtree ),N: n,N4: n] :
( ( finite_finite2 @ ( sum_sum @ t @ dtree ) @ As )
=> ( ( finite_finite2 @ ( sum_sum @ t @ dtree ) @ As2 )
=> ( ( ( node @ N @ As )
= ( node @ N4 @ As2 ) )
= ( ( N = N4 )
& ( As = As2 ) ) ) ) ) ).
% dtree_simps
thf(fact_52_inFr2_OInd,axiom,
! [Tr1: dtree,Tr: dtree,Ns1: set @ n,T2: t] :
( ( member @ ( sum_sum @ t @ dtree ) @ ( sum_Inr @ dtree @ t @ Tr1 ) @ ( cont @ Tr ) )
=> ( ( gram_L805317441_inFr2 @ Ns1 @ Tr1 @ T2 )
=> ( gram_L805317441_inFr2 @ ( insert @ n @ ( root @ Tr ) @ Ns1 ) @ Tr @ T2 ) ) ) ).
% inFr2.Ind
thf(fact_53_subtr2_OStep,axiom,
! [Tr1: dtree,Ns: set @ n,Tr2: dtree,Tr32: dtree] :
( ( member @ n @ ( root @ Tr1 ) @ Ns )
=> ( ( member @ ( sum_sum @ t @ dtree ) @ ( sum_Inr @ dtree @ t @ Tr1 ) @ ( cont @ Tr2 ) )
=> ( ( gram_L1283001940subtr2 @ Ns @ Tr2 @ Tr32 )
=> ( gram_L1283001940subtr2 @ Ns @ Tr1 @ Tr32 ) ) ) ) ).
% subtr2.Step
thf(fact_54_subtr2_Ocases,axiom,
! [A12: set @ n,A22: dtree,A32: dtree] :
( ( gram_L1283001940subtr2 @ A12 @ A22 @ A32 )
=> ( ( ( A32 = A22 )
=> ~ ( member @ n @ ( root @ A22 ) @ A12 ) )
=> ~ ( ( member @ n @ ( root @ A22 ) @ A12 )
=> ! [Tr22: dtree] :
( ( member @ ( sum_sum @ t @ dtree ) @ ( sum_Inr @ dtree @ t @ A22 ) @ ( cont @ Tr22 ) )
=> ~ ( gram_L1283001940subtr2 @ A12 @ Tr22 @ A32 ) ) ) ) ) ).
% subtr2.cases
thf(fact_55_subtr2_Osimps,axiom,
( gram_L1283001940subtr2
= ( ^ [A1: set @ n,A2: dtree,A3: dtree] :
( ? [Tr3: dtree,Ns2: set @ n] :
( ( A1 = Ns2 )
& ( A2 = Tr3 )
& ( A3 = Tr3 )
& ( member @ n @ ( root @ Tr3 ) @ Ns2 ) )
| ? [Tr13: dtree,Ns2: set @ n,Tr23: dtree,Tr33: dtree] :
( ( A1 = Ns2 )
& ( A2 = Tr13 )
& ( A3 = Tr33 )
& ( member @ n @ ( root @ Tr13 ) @ Ns2 )
& ( member @ ( sum_sum @ t @ dtree ) @ ( sum_Inr @ dtree @ t @ Tr13 ) @ ( cont @ Tr23 ) )
& ( gram_L1283001940subtr2 @ Ns2 @ Tr23 @ Tr33 ) ) ) ) ) ).
% subtr2.simps
thf(fact_56_subtr2__StepR,axiom,
! [Tr32: dtree,Ns: set @ n,Tr2: dtree,Tr1: dtree] :
( ( member @ n @ ( root @ Tr32 ) @ Ns )
=> ( ( member @ ( sum_sum @ t @ dtree ) @ ( sum_Inr @ dtree @ t @ Tr2 ) @ ( cont @ Tr32 ) )
=> ( ( gram_L1283001940subtr2 @ Ns @ Tr1 @ Tr2 )
=> ( gram_L1283001940subtr2 @ Ns @ Tr1 @ Tr32 ) ) ) ) ).
% subtr2_StepR
thf(fact_57_finite__insert,axiom,
! [A: $tType,A5: A,A6: set @ A] :
( ( finite_finite2 @ A @ ( insert @ A @ A5 @ A6 ) )
= ( finite_finite2 @ A @ A6 ) ) ).
% finite_insert
thf(fact_58_root__Node,axiom,
! [N: n,As: set @ ( sum_sum @ t @ dtree )] :
( ( root @ ( node @ N @ As ) )
= N ) ).
% root_Node
thf(fact_59_finite_OinsertI,axiom,
! [A: $tType,A6: set @ A,A5: A] :
( ( finite_finite2 @ A @ A6 )
=> ( finite_finite2 @ A @ ( insert @ A @ A5 @ A6 ) ) ) ).
% finite.insertI
thf(fact_60_subtr2__trans,axiom,
! [Ns: set @ n,Tr1: dtree,Tr2: dtree,Tr32: dtree] :
( ( gram_L1283001940subtr2 @ Ns @ Tr1 @ Tr2 )
=> ( ( gram_L1283001940subtr2 @ Ns @ Tr2 @ Tr32 )
=> ( gram_L1283001940subtr2 @ Ns @ Tr1 @ Tr32 ) ) ) ).
% subtr2_trans
thf(fact_61_subtr2_ORefl,axiom,
! [Tr: dtree,Ns: set @ n] :
( ( member @ n @ ( root @ Tr ) @ Ns )
=> ( gram_L1283001940subtr2 @ Ns @ Tr @ Tr ) ) ).
% subtr2.Refl
thf(fact_62_subtr2__rootL__in,axiom,
! [Ns: set @ n,Tr1: dtree,Tr2: dtree] :
( ( gram_L1283001940subtr2 @ Ns @ Tr1 @ Tr2 )
=> ( member @ n @ ( root @ Tr1 ) @ Ns ) ) ).
% subtr2_rootL_in
thf(fact_63_subtr2__rootR__in,axiom,
! [Ns: set @ n,Tr1: dtree,Tr2: dtree] :
( ( gram_L1283001940subtr2 @ Ns @ Tr1 @ Tr2 )
=> ( member @ n @ ( root @ Tr2 ) @ Ns ) ) ).
% subtr2_rootR_in
thf(fact_64_finite__set__choice,axiom,
! [B: $tType,A: $tType,A6: set @ A,P: A > B > $o] :
( ( finite_finite2 @ A @ A6 )
=> ( ! [X5: A] :
( ( member @ A @ X5 @ A6 )
=> ? [X12: B] : ( P @ X5 @ X12 ) )
=> ? [F3: A > B] :
! [X6: A] :
( ( member @ A @ X6 @ A6 )
=> ( P @ X6 @ ( F3 @ X6 ) ) ) ) ) ).
% finite_set_choice
thf(fact_65_finite,axiom,
! [A: $tType] :
( ( finite_finite @ A @ ( type @ A ) )
=> ! [A6: set @ A] : ( finite_finite2 @ A @ A6 ) ) ).
% finite
thf(fact_66_dtree__cases,axiom,
! [Tr: dtree] :
~ ! [N2: n,As3: set @ ( sum_sum @ t @ dtree )] :
( ( finite_finite2 @ ( sum_sum @ t @ dtree ) @ As3 )
=> ( Tr
!= ( node @ N2 @ As3 ) ) ) ).
% dtree_cases
thf(fact_67_inFr__Ind__minus,axiom,
! [Ns1: set @ n,Tr1: dtree,T2: t,Tr: dtree] :
( ( gram_L1333338417e_inFr @ Ns1 @ Tr1 @ T2 )
=> ( ( member @ ( sum_sum @ t @ dtree ) @ ( sum_Inr @ dtree @ t @ Tr1 ) @ ( cont @ Tr ) )
=> ( gram_L1333338417e_inFr @ ( insert @ n @ ( root @ Tr ) @ Ns1 ) @ Tr @ T2 ) ) ) ).
% inFr_Ind_minus
thf(fact_68_subtr2_Oinducts,axiom,
! [X1: set @ n,X2: dtree,X3: dtree,P: ( set @ n ) > dtree > dtree > $o] :
( ( gram_L1283001940subtr2 @ X1 @ X2 @ X3 )
=> ( ! [Tr6: dtree,Ns3: set @ n] :
( ( member @ n @ ( root @ Tr6 ) @ Ns3 )
=> ( P @ Ns3 @ Tr6 @ Tr6 ) )
=> ( ! [Tr12: dtree,Ns3: set @ n,Tr22: dtree,Tr34: dtree] :
( ( member @ n @ ( root @ Tr12 ) @ Ns3 )
=> ( ( member @ ( sum_sum @ t @ dtree ) @ ( sum_Inr @ dtree @ t @ Tr12 ) @ ( cont @ Tr22 ) )
=> ( ( gram_L1283001940subtr2 @ Ns3 @ Tr22 @ Tr34 )
=> ( ( P @ Ns3 @ Tr22 @ Tr34 )
=> ( P @ Ns3 @ Tr12 @ Tr34 ) ) ) ) )
=> ( P @ X1 @ X2 @ X3 ) ) ) ) ).
% subtr2.inducts
thf(fact_69_insertCI,axiom,
! [A: $tType,A5: A,B4: set @ A,B2: A] :
( ( ~ ( member @ A @ A5 @ B4 )
=> ( A5 = B2 ) )
=> ( member @ A @ A5 @ ( insert @ A @ B2 @ B4 ) ) ) ).
% insertCI
thf(fact_70_insert__iff,axiom,
! [A: $tType,A5: A,B2: A,A6: set @ A] :
( ( member @ A @ A5 @ ( insert @ A @ B2 @ A6 ) )
= ( ( A5 = B2 )
| ( member @ A @ A5 @ A6 ) ) ) ).
% insert_iff
thf(fact_71_insert__absorb2,axiom,
! [A: $tType,X: A,A6: set @ A] :
( ( insert @ A @ X @ ( insert @ A @ X @ A6 ) )
= ( insert @ A @ X @ A6 ) ) ).
% insert_absorb2
thf(fact_72_inFr2_Oinducts,axiom,
! [X1: set @ n,X2: dtree,X3: t,P: ( set @ n ) > dtree > t > $o] :
( ( gram_L805317441_inFr2 @ X1 @ X2 @ X3 )
=> ( ! [Tr6: dtree,Ns3: set @ n,T4: t] :
( ( member @ n @ ( root @ Tr6 ) @ Ns3 )
=> ( ( member @ ( sum_sum @ t @ dtree ) @ ( sum_Inl @ t @ dtree @ T4 ) @ ( cont @ Tr6 ) )
=> ( P @ Ns3 @ Tr6 @ T4 ) ) )
=> ( ! [Tr12: dtree,Tr6: dtree,Ns12: set @ n,T4: t] :
( ( member @ ( sum_sum @ t @ dtree ) @ ( sum_Inr @ dtree @ t @ Tr12 ) @ ( cont @ Tr6 ) )
=> ( ( gram_L805317441_inFr2 @ Ns12 @ Tr12 @ T4 )
=> ( ( P @ Ns12 @ Tr12 @ T4 )
=> ( P @ ( insert @ n @ ( root @ Tr6 ) @ Ns12 ) @ Tr6 @ T4 ) ) ) )
=> ( P @ X1 @ X2 @ X3 ) ) ) ) ).
% inFr2.inducts
thf(fact_73_inFr2_Osimps,axiom,
( gram_L805317441_inFr2
= ( ^ [A1: set @ n,A2: dtree,A3: t] :
( ? [Tr3: dtree,Ns2: set @ n,T3: t] :
( ( A1 = Ns2 )
& ( A2 = Tr3 )
& ( A3 = T3 )
& ( member @ n @ ( root @ Tr3 ) @ Ns2 )
& ( member @ ( sum_sum @ t @ dtree ) @ ( sum_Inl @ t @ dtree @ T3 ) @ ( cont @ Tr3 ) ) )
| ? [Tr13: dtree,Tr3: dtree,Ns13: set @ n,T3: t] :
( ( A1
= ( insert @ n @ ( root @ Tr3 ) @ Ns13 ) )
& ( A2 = Tr3 )
& ( A3 = T3 )
& ( member @ ( sum_sum @ t @ dtree ) @ ( sum_Inr @ dtree @ t @ Tr13 ) @ ( cont @ Tr3 ) )
& ( gram_L805317441_inFr2 @ Ns13 @ Tr13 @ T3 ) ) ) ) ) ).
% inFr2.simps
thf(fact_74_inFr2_Ocases,axiom,
! [A12: set @ n,A22: dtree,A32: t] :
( ( gram_L805317441_inFr2 @ A12 @ A22 @ A32 )
=> ( ( ( member @ n @ ( root @ A22 ) @ A12 )
=> ~ ( member @ ( sum_sum @ t @ dtree ) @ ( sum_Inl @ t @ dtree @ A32 ) @ ( cont @ A22 ) ) )
=> ~ ! [Tr12: dtree,Tr6: dtree,Ns12: set @ n] :
( ( A12
= ( insert @ n @ ( root @ Tr6 ) @ Ns12 ) )
=> ( ( A22 = Tr6 )
=> ( ( member @ ( sum_sum @ t @ dtree ) @ ( sum_Inr @ dtree @ t @ Tr12 ) @ ( cont @ Tr6 ) )
=> ~ ( gram_L805317441_inFr2 @ Ns12 @ Tr12 @ A32 ) ) ) ) ) ) ).
% inFr2.cases
thf(fact_75_root__deftr,axiom,
! [N: n] :
( ( root @ ( gram_L1231612515_deftr @ N ) )
= N ) ).
% root_deftr
thf(fact_76_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_77_old_Osum_Oinject_I1_J,axiom,
! [B: $tType,A: $tType,A5: A,A7: A] :
( ( ( sum_Inl @ A @ B @ A5 )
= ( sum_Inl @ A @ B @ A7 ) )
= ( A5 = A7 ) ) ).
% old.sum.inject(1)
thf(fact_78_old_Osum_Osimps_I7_J,axiom,
! [B: $tType,T: $tType,A: $tType,F1: A > T,F2: B > T,A5: A] :
( ( sum_rec_sum @ A @ T @ B @ F1 @ F2 @ ( sum_Inl @ A @ B @ A5 ) )
= ( F1 @ A5 ) ) ).
% old.sum.simps(7)
thf(fact_79_Inl__inject,axiom,
! [B: $tType,A: $tType,X: A,Y: A] :
( ( ( sum_Inl @ A @ B @ X )
= ( sum_Inl @ A @ B @ Y ) )
=> ( X = Y ) ) ).
% Inl_inject
thf(fact_80_sum_Odistinct_I1_J,axiom,
! [A: $tType,B: $tType,X1: A,X2: B] :
( ( sum_Inl @ A @ B @ X1 )
!= ( sum_Inr @ B @ A @ X2 ) ) ).
% sum.distinct(1)
thf(fact_81_old_Osum_Odistinct_I2_J,axiom,
! [B5: $tType,A8: $tType,B6: B5,A9: A8] :
( ( sum_Inr @ B5 @ A8 @ B6 )
!= ( sum_Inl @ A8 @ B5 @ A9 ) ) ).
% old.sum.distinct(2)
thf(fact_82_old_Osum_Odistinct_I1_J,axiom,
! [A: $tType,B: $tType,A5: A,B3: B] :
( ( sum_Inl @ A @ B @ A5 )
!= ( sum_Inr @ B @ A @ B3 ) ) ).
% old.sum.distinct(1)
thf(fact_83_sumE,axiom,
! [A: $tType,B: $tType,S: sum_sum @ A @ B] :
( ! [X5: A] :
( S
!= ( sum_Inl @ A @ B @ X5 ) )
=> ~ ! [Y3: B] :
( S
!= ( sum_Inr @ B @ A @ Y3 ) ) ) ).
% sumE
thf(fact_84_Inr__not__Inl,axiom,
! [B: $tType,A: $tType,B2: B,A5: A] :
( ( sum_Inr @ B @ A @ B2 )
!= ( sum_Inl @ A @ B @ A5 ) ) ).
% Inr_not_Inl
thf(fact_85_split__sum__ex,axiom,
! [B: $tType,A: $tType] :
( ( ^ [P2: ( sum_sum @ A @ B ) > $o] :
? [X7: sum_sum @ A @ B] : ( P2 @ X7 ) )
= ( ^ [P3: ( sum_sum @ A @ B ) > $o] :
( ? [X4: A] : ( P3 @ ( sum_Inl @ A @ B @ X4 ) )
| ? [X4: B] : ( P3 @ ( sum_Inr @ B @ A @ X4 ) ) ) ) ) ).
% split_sum_ex
thf(fact_86_split__sum__all,axiom,
! [B: $tType,A: $tType] :
( ( ^ [P2: ( sum_sum @ A @ B ) > $o] :
! [X7: sum_sum @ A @ B] : ( P2 @ X7 ) )
= ( ^ [P3: ( sum_sum @ A @ B ) > $o] :
( ! [X4: A] : ( P3 @ ( sum_Inl @ A @ B @ X4 ) )
& ! [X4: B] : ( P3 @ ( sum_Inr @ B @ A @ X4 ) ) ) ) ) ).
% split_sum_all
thf(fact_87_old_Osum_Oexhaust,axiom,
! [A: $tType,B: $tType,Y: sum_sum @ A @ B] :
( ! [A10: A] :
( Y
!= ( sum_Inl @ A @ B @ A10 ) )
=> ~ ! [B7: B] :
( Y
!= ( sum_Inr @ B @ A @ B7 ) ) ) ).
% old.sum.exhaust
thf(fact_88_old_Osum_Oinducts,axiom,
! [B: $tType,A: $tType,P: ( sum_sum @ A @ B ) > $o,Sum: sum_sum @ A @ B] :
( ! [A10: A] : ( P @ ( sum_Inl @ A @ B @ A10 ) )
=> ( ! [B7: B] : ( P @ ( sum_Inr @ B @ A @ B7 ) )
=> ( P @ Sum ) ) ) ).
% old.sum.inducts
thf(fact_89_obj__sumE,axiom,
! [A: $tType,B: $tType,S: sum_sum @ A @ B] :
( ! [X5: A] :
( S
!= ( sum_Inl @ A @ B @ X5 ) )
=> ~ ! [X5: B] :
( S
!= ( sum_Inr @ B @ A @ X5 ) ) ) ).
% obj_sumE
thf(fact_90_wf__deftr,axiom,
! [N: n] : ( gram_L864798063lle_wf @ ( gram_L1231612515_deftr @ N ) ) ).
% wf_deftr
thf(fact_91_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_92_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_93_mk__disjoint__insert,axiom,
! [A: $tType,A5: A,A6: set @ A] :
( ( member @ A @ A5 @ A6 )
=> ? [B8: set @ A] :
( ( A6
= ( insert @ A @ A5 @ B8 ) )
& ~ ( member @ A @ A5 @ B8 ) ) ) ).
% mk_disjoint_insert
thf(fact_94_insert__commute,axiom,
! [A: $tType,X: A,Y: A,A6: set @ A] :
( ( insert @ A @ X @ ( insert @ A @ Y @ A6 ) )
= ( insert @ A @ Y @ ( insert @ A @ X @ A6 ) ) ) ).
% insert_commute
thf(fact_95_insert__eq__iff,axiom,
! [A: $tType,A5: A,A6: set @ A,B2: A,B4: set @ A] :
( ~ ( member @ A @ A5 @ A6 )
=> ( ~ ( member @ A @ B2 @ B4 )
=> ( ( ( insert @ A @ A5 @ A6 )
= ( insert @ A @ B2 @ B4 ) )
= ( ( ( A5 = B2 )
=> ( A6 = B4 ) )
& ( ( A5 != B2 )
=> ? [C2: set @ A] :
( ( A6
= ( insert @ A @ B2 @ C2 ) )
& ~ ( member @ A @ B2 @ C2 )
& ( B4
= ( insert @ A @ A5 @ C2 ) )
& ~ ( member @ A @ A5 @ C2 ) ) ) ) ) ) ) ).
% insert_eq_iff
thf(fact_96_insert__absorb,axiom,
! [A: $tType,A5: A,A6: set @ A] :
( ( member @ A @ A5 @ A6 )
=> ( ( insert @ A @ A5 @ A6 )
= A6 ) ) ).
% insert_absorb
thf(fact_97_insert__ident,axiom,
! [A: $tType,X: A,A6: set @ A,B4: set @ A] :
( ~ ( member @ A @ X @ A6 )
=> ( ~ ( member @ A @ X @ B4 )
=> ( ( ( insert @ A @ X @ A6 )
= ( insert @ A @ X @ B4 ) )
= ( A6 = B4 ) ) ) ) ).
% insert_ident
thf(fact_98_Set_Oset__insert,axiom,
! [A: $tType,X: A,A6: set @ A] :
( ( member @ A @ X @ A6 )
=> ~ ! [B8: set @ A] :
( ( A6
= ( insert @ A @ X @ B8 ) )
=> ( member @ A @ X @ B8 ) ) ) ).
% Set.set_insert
thf(fact_99_insertI2,axiom,
! [A: $tType,A5: A,B4: set @ A,B2: A] :
( ( member @ A @ A5 @ B4 )
=> ( member @ A @ A5 @ ( insert @ A @ B2 @ B4 ) ) ) ).
% insertI2
thf(fact_100_insertI1,axiom,
! [A: $tType,A5: A,B4: set @ A] : ( member @ A @ A5 @ ( insert @ A @ A5 @ B4 ) ) ).
% insertI1
thf(fact_101_insertE,axiom,
! [A: $tType,A5: A,B2: A,A6: set @ A] :
( ( member @ A @ A5 @ ( insert @ A @ B2 @ A6 ) )
=> ( ( A5 != B2 )
=> ( member @ A @ A5 @ A6 ) ) ) ).
% insertE
thf(fact_102_inFr_Ocases,axiom,
! [A12: set @ n,A22: dtree,A32: t] :
( ( gram_L1333338417e_inFr @ A12 @ A22 @ A32 )
=> ( ( ( member @ n @ ( root @ A22 ) @ A12 )
=> ~ ( member @ ( sum_sum @ t @ dtree ) @ ( sum_Inl @ t @ dtree @ A32 ) @ ( cont @ A22 ) ) )
=> ~ ( ( member @ n @ ( root @ A22 ) @ A12 )
=> ! [Tr12: dtree] :
( ( member @ ( sum_sum @ t @ dtree ) @ ( sum_Inr @ dtree @ t @ Tr12 ) @ ( cont @ A22 ) )
=> ~ ( gram_L1333338417e_inFr @ A12 @ Tr12 @ A32 ) ) ) ) ) ).
% inFr.cases
thf(fact_103_inFr_Osimps,axiom,
( gram_L1333338417e_inFr
= ( ^ [A1: set @ n,A2: dtree,A3: t] :
( ? [Tr3: dtree,Ns2: set @ n,T3: t] :
( ( A1 = Ns2 )
& ( A2 = Tr3 )
& ( A3 = T3 )
& ( member @ n @ ( root @ Tr3 ) @ Ns2 )
& ( member @ ( sum_sum @ t @ dtree ) @ ( sum_Inl @ t @ dtree @ T3 ) @ ( cont @ Tr3 ) ) )
| ? [Tr3: dtree,Ns2: set @ n,Tr13: dtree,T3: t] :
( ( A1 = Ns2 )
& ( A2 = Tr3 )
& ( A3 = T3 )
& ( member @ n @ ( root @ Tr3 ) @ Ns2 )
& ( member @ ( sum_sum @ t @ dtree ) @ ( sum_Inr @ dtree @ t @ Tr13 ) @ ( cont @ Tr3 ) )
& ( gram_L1333338417e_inFr @ Ns2 @ Tr13 @ T3 ) ) ) ) ) ).
% inFr.simps
thf(fact_104_inFr_Oinducts,axiom,
! [X1: set @ n,X2: dtree,X3: t,P: ( set @ n ) > dtree > t > $o] :
( ( gram_L1333338417e_inFr @ X1 @ X2 @ X3 )
=> ( ! [Tr6: dtree,Ns3: set @ n,T4: t] :
( ( member @ n @ ( root @ Tr6 ) @ Ns3 )
=> ( ( member @ ( sum_sum @ t @ dtree ) @ ( sum_Inl @ t @ dtree @ T4 ) @ ( cont @ Tr6 ) )
=> ( P @ Ns3 @ Tr6 @ T4 ) ) )
=> ( ! [Tr6: dtree,Ns3: set @ n,Tr12: dtree,T4: t] :
( ( member @ n @ ( root @ Tr6 ) @ Ns3 )
=> ( ( member @ ( sum_sum @ t @ dtree ) @ ( sum_Inr @ dtree @ t @ Tr12 ) @ ( cont @ Tr6 ) )
=> ( ( gram_L1333338417e_inFr @ Ns3 @ Tr12 @ T4 )
=> ( ( P @ Ns3 @ Tr12 @ T4 )
=> ( P @ Ns3 @ Tr6 @ T4 ) ) ) ) )
=> ( P @ X1 @ X2 @ X3 ) ) ) ) ).
% inFr.inducts
thf(fact_105_Inr__Inl__False,axiom,
! [B: $tType,A: $tType,X: B,Y: A] :
( ( sum_Inr @ B @ A @ X )
!= ( sum_Inl @ A @ B @ Y ) ) ).
% Inr_Inl_False
thf(fact_106_Inl__Inr__False,axiom,
! [A: $tType,B: $tType,X: A,Y: B] :
( ( sum_Inl @ A @ B @ X )
!= ( sum_Inr @ B @ A @ Y ) ) ).
% Inl_Inr_False
thf(fact_107_obj__sumE__f,axiom,
! [A: $tType,C: $tType,B: $tType,S: B,F: ( sum_sum @ A @ C ) > B,P: $o] :
( ! [X5: A] :
( ( S
= ( F @ ( sum_Inl @ A @ C @ X5 ) ) )
=> P )
=> ( ! [X5: C] :
( ( S
= ( F @ ( sum_Inr @ C @ A @ X5 ) ) )
=> P )
=> ! [X6: sum_sum @ A @ C] :
( ( S
= ( F @ X6 ) )
=> P ) ) ) ).
% obj_sumE_f
thf(fact_108_Inl__cont__hsubst__eq,axiom,
! [Tr: dtree] :
( ( ( root @ Tr )
= ( root @ tr0 ) )
=> ( ( vimage @ t @ ( sum_sum @ t @ dtree ) @ ( sum_Inl @ t @ dtree ) @ ( cont @ ( gram_L1004374585hsubst @ tr0 @ Tr ) ) )
= ( vimage @ t @ ( sum_sum @ t @ dtree ) @ ( sum_Inl @ t @ dtree ) @ ( cont @ tr0 ) ) ) ) ).
% Inl_cont_hsubst_eq
thf(fact_109_Inl__cont__hsubst__neq,axiom,
! [Tr: dtree] :
( ( ( root @ Tr )
!= ( root @ tr0 ) )
=> ( ( vimage @ t @ ( sum_sum @ t @ dtree ) @ ( sum_Inl @ t @ dtree ) @ ( cont @ ( gram_L1004374585hsubst @ tr0 @ Tr ) ) )
= ( vimage @ t @ ( sum_sum @ t @ dtree ) @ ( sum_Inl @ t @ dtree ) @ ( cont @ Tr ) ) ) ) ).
% Inl_cont_hsubst_neq
thf(fact_110_root__o__subst,axiom,
( ( comp @ dtree @ n @ dtree @ root @ ( gram_L1004374585hsubst @ tr0 ) )
= root ) ).
% root_o_subst
thf(fact_111_vimage__eq,axiom,
! [A: $tType,B: $tType,A5: A,F: A > B,B4: set @ B] :
( ( member @ A @ A5 @ ( vimage @ A @ B @ F @ B4 ) )
= ( member @ B @ ( F @ A5 ) @ B4 ) ) ).
% vimage_eq
thf(fact_112_vimageI,axiom,
! [B: $tType,A: $tType,F: B > A,A5: B,B2: A,B4: set @ A] :
( ( ( F @ A5 )
= B2 )
=> ( ( member @ A @ B2 @ B4 )
=> ( member @ B @ A5 @ ( vimage @ B @ A @ F @ B4 ) ) ) ) ).
% vimageI
thf(fact_113_vimage__Collect,axiom,
! [B: $tType,A: $tType,P: B > $o,F: A > B,Q: A > $o] :
( ! [X5: A] :
( ( P @ ( F @ X5 ) )
= ( Q @ X5 ) )
=> ( ( vimage @ A @ B @ F @ ( collect @ B @ P ) )
= ( collect @ A @ Q ) ) ) ).
% vimage_Collect
thf(fact_114_vimageI2,axiom,
! [B: $tType,A: $tType,F: B > A,A5: B,A6: set @ A] :
( ( member @ A @ ( F @ A5 ) @ A6 )
=> ( member @ B @ A5 @ ( vimage @ B @ A @ F @ A6 ) ) ) ).
% vimageI2
thf(fact_115_vimageE,axiom,
! [A: $tType,B: $tType,A5: A,F: A > B,B4: set @ B] :
( ( member @ A @ A5 @ ( vimage @ A @ B @ F @ B4 ) )
=> ( member @ B @ ( F @ A5 ) @ B4 ) ) ).
% vimageE
thf(fact_116_vimageD,axiom,
! [A: $tType,B: $tType,A5: A,F: A > B,A6: set @ B] :
( ( member @ A @ A5 @ ( vimage @ A @ B @ F @ A6 ) )
=> ( member @ B @ ( F @ A5 ) @ A6 ) ) ).
% vimageD
thf(fact_117_rewriteL__comp__comp,axiom,
! [C: $tType,B: $tType,A: $tType,D: $tType,F: C > B,G: A > C,L: A > B,H: D > A] :
( ( ( comp @ C @ B @ A @ F @ G )
= L )
=> ( ( comp @ C @ B @ D @ F @ ( comp @ A @ C @ D @ G @ H ) )
= ( comp @ A @ B @ D @ L @ H ) ) ) ).
% rewriteL_comp_comp
thf(fact_118_rewriteR__comp__comp,axiom,
! [C: $tType,D: $tType,B: $tType,A: $tType,G: C > B,H: A > C,R: A > B,F: B > D] :
( ( ( comp @ C @ B @ A @ G @ H )
= R )
=> ( ( comp @ C @ D @ A @ ( comp @ B @ D @ C @ F @ G ) @ H )
= ( comp @ B @ D @ A @ F @ R ) ) ) ).
% rewriteR_comp_comp
thf(fact_119_rewriteL__comp__comp2,axiom,
! [A: $tType,C: $tType,B: $tType,D: $tType,E: $tType,F: C > B,G: A > C,L1: D > B,L2: A > D,H: E > A,R: E > D] :
( ( ( comp @ C @ B @ A @ F @ G )
= ( comp @ D @ B @ A @ L1 @ L2 ) )
=> ( ( ( comp @ A @ D @ E @ L2 @ H )
= R )
=> ( ( comp @ C @ B @ E @ F @ ( comp @ A @ C @ E @ G @ H ) )
= ( comp @ D @ B @ E @ L1 @ R ) ) ) ) ).
% rewriteL_comp_comp2
thf(fact_120_rewriteR__comp__comp2,axiom,
! [C: $tType,B: $tType,E: $tType,D: $tType,A: $tType,G: C > B,H: A > C,R1: D > B,R2: A > D,F: B > E,L: D > E] :
( ( ( comp @ C @ B @ A @ G @ H )
= ( comp @ D @ B @ A @ R1 @ R2 ) )
=> ( ( ( comp @ B @ E @ D @ F @ R1 )
= L )
=> ( ( comp @ C @ E @ A @ ( comp @ B @ E @ C @ F @ G ) @ H )
= ( comp @ D @ E @ A @ L @ R2 ) ) ) ) ).
% rewriteR_comp_comp2
thf(fact_121_Gram__Lang__Mirabelle__ojxrtuoybn_Oroot__o__subst,axiom,
! [Tr02: dtree] :
( ( comp @ dtree @ n @ dtree @ root @ ( gram_L1004374585hsubst @ Tr02 ) )
= root ) ).
% Gram_Lang_Mirabelle_ojxrtuoybn.root_o_subst
thf(fact_122_Gram__Lang__Mirabelle__ojxrtuoybn_OInl__cont__hsubst__neq,axiom,
! [Tr: dtree,Tr02: dtree] :
( ( ( root @ Tr )
!= ( root @ Tr02 ) )
=> ( ( vimage @ t @ ( sum_sum @ t @ dtree ) @ ( sum_Inl @ t @ dtree ) @ ( cont @ ( gram_L1004374585hsubst @ Tr02 @ Tr ) ) )
= ( vimage @ t @ ( sum_sum @ t @ dtree ) @ ( sum_Inl @ t @ dtree ) @ ( cont @ Tr ) ) ) ) ).
% Gram_Lang_Mirabelle_ojxrtuoybn.Inl_cont_hsubst_neq
thf(fact_123_Gram__Lang__Mirabelle__ojxrtuoybn_OInl__cont__hsubst__eq,axiom,
! [Tr: dtree,Tr02: dtree] :
( ( ( root @ Tr )
= ( root @ Tr02 ) )
=> ( ( vimage @ t @ ( sum_sum @ t @ dtree ) @ ( sum_Inl @ t @ dtree ) @ ( cont @ ( gram_L1004374585hsubst @ Tr02 @ Tr ) ) )
= ( vimage @ t @ ( sum_sum @ t @ dtree ) @ ( sum_Inl @ t @ dtree ) @ ( cont @ Tr02 ) ) ) ) ).
% Gram_Lang_Mirabelle_ojxrtuoybn.Inl_cont_hsubst_eq
thf(fact_124_Inl__Inr__image__cong,axiom,
! [B: $tType,A: $tType,A6: set @ ( sum_sum @ A @ B ),B4: set @ ( sum_sum @ A @ B )] :
( ( ( vimage @ A @ ( sum_sum @ A @ B ) @ ( sum_Inl @ A @ B ) @ A6 )
= ( vimage @ A @ ( sum_sum @ A @ B ) @ ( sum_Inl @ A @ B ) @ B4 ) )
=> ( ( ( vimage @ B @ ( sum_sum @ A @ B ) @ ( sum_Inr @ B @ A ) @ A6 )
= ( vimage @ B @ ( sum_sum @ A @ B ) @ ( sum_Inr @ B @ A ) @ B4 ) )
=> ( A6 = B4 ) ) ) ).
% Inl_Inr_image_cong
thf(fact_125_comp__apply,axiom,
! [C: $tType,A: $tType,B: $tType] :
( ( comp @ B @ A @ C )
= ( ^ [F4: B > A,G2: C > B,X4: C] : ( F4 @ ( G2 @ X4 ) ) ) ) ).
% comp_apply
thf(fact_126_set_Ocomp,axiom,
! [A: $tType,B: $tType,C: $tType,F: C > B,G: B > A] :
( ( comp @ ( set @ B ) @ ( set @ C ) @ ( set @ A ) @ ( vimage @ C @ B @ F ) @ ( vimage @ B @ A @ G ) )
= ( vimage @ C @ A @ ( comp @ B @ A @ C @ G @ F ) ) ) ).
% set.comp
thf(fact_127_vimage__comp,axiom,
! [A: $tType,B: $tType,C: $tType,F: A > B,G: B > C,X: set @ C] :
( ( vimage @ A @ B @ F @ ( vimage @ B @ C @ G @ X ) )
= ( vimage @ A @ C @ ( comp @ B @ C @ A @ G @ F ) @ X ) ) ).
% vimage_comp
thf(fact_128_set_Ocompositionality,axiom,
! [C: $tType,B: $tType,A: $tType,F: C > B,G: B > A,Set: set @ A] :
( ( vimage @ C @ B @ F @ ( vimage @ B @ A @ G @ Set ) )
= ( vimage @ C @ A @ ( comp @ B @ A @ C @ G @ F ) @ Set ) ) ).
% set.compositionality
thf(fact_129_comp__eq__dest__lhs,axiom,
! [C: $tType,B: $tType,A: $tType,A5: C > B,B2: A > C,C3: A > B,V: A] :
( ( ( comp @ C @ B @ A @ A5 @ B2 )
= C3 )
=> ( ( A5 @ ( B2 @ V ) )
= ( C3 @ V ) ) ) ).
% comp_eq_dest_lhs
thf(fact_130_comp__eq__elim,axiom,
! [C: $tType,B: $tType,D: $tType,A: $tType,A5: C > B,B2: A > C,C3: D > B,D2: A > D] :
( ( ( comp @ C @ B @ A @ A5 @ B2 )
= ( comp @ D @ B @ A @ C3 @ D2 ) )
=> ! [V2: A] :
( ( A5 @ ( B2 @ V2 ) )
= ( C3 @ ( D2 @ V2 ) ) ) ) ).
% comp_eq_elim
thf(fact_131_comp__eq__dest,axiom,
! [C: $tType,B: $tType,D: $tType,A: $tType,A5: C > B,B2: A > C,C3: D > B,D2: A > D,V: A] :
( ( ( comp @ C @ B @ A @ A5 @ B2 )
= ( comp @ D @ B @ A @ C3 @ D2 ) )
=> ( ( A5 @ ( B2 @ V ) )
= ( C3 @ ( D2 @ V ) ) ) ) ).
% comp_eq_dest
thf(fact_132_comp__assoc,axiom,
! [B: $tType,D: $tType,C: $tType,A: $tType,F: D > B,G: C > D,H: A > C] :
( ( comp @ C @ B @ A @ ( comp @ D @ B @ C @ F @ G ) @ H )
= ( comp @ D @ B @ A @ F @ ( comp @ C @ D @ A @ G @ H ) ) ) ).
% comp_assoc
thf(fact_133_comp__def,axiom,
! [A: $tType,C: $tType,B: $tType] :
( ( comp @ B @ C @ A )
= ( ^ [F4: B > C,G2: A > B,X4: A] : ( F4 @ ( G2 @ X4 ) ) ) ) ).
% comp_def
thf(fact_134_Inr__cont__hsubst__eq,axiom,
! [Tr: dtree] :
( ( ( root @ Tr )
= ( root @ tr0 ) )
=> ( ( vimage @ dtree @ ( sum_sum @ t @ dtree ) @ ( sum_Inr @ dtree @ t ) @ ( cont @ ( gram_L1004374585hsubst @ tr0 @ Tr ) ) )
= ( image @ dtree @ dtree @ ( gram_L1004374585hsubst @ tr0 ) @ ( vimage @ dtree @ ( sum_sum @ t @ dtree ) @ ( sum_Inr @ dtree @ t ) @ ( cont @ tr0 ) ) ) ) ) ).
% Inr_cont_hsubst_eq
thf(fact_135_Inr__cont__hsubst__neq,axiom,
! [Tr: dtree] :
( ( ( root @ Tr )
!= ( root @ tr0 ) )
=> ( ( vimage @ dtree @ ( sum_sum @ t @ dtree ) @ ( sum_Inr @ dtree @ t ) @ ( cont @ ( gram_L1004374585hsubst @ tr0 @ Tr ) ) )
= ( image @ dtree @ dtree @ ( gram_L1004374585hsubst @ tr0 ) @ ( vimage @ dtree @ ( sum_sum @ t @ dtree ) @ ( sum_Inr @ dtree @ t ) @ ( cont @ Tr ) ) ) ) ) ).
% Inr_cont_hsubst_neq
thf(fact_136_IH,axiom,
! [Tr: dtree] :
( ( tr1
= ( gram_L1004374585hsubst @ tr0 @ Tr ) )
=> ( ( member @ t @ ta @ ( vimage @ t @ ( sum_sum @ t @ dtree ) @ ( sum_Inl @ t @ dtree ) @ ( cont @ tr0 ) ) )
| ( gram_L805317505_inFrr @ ( minus_minus @ ( set @ n ) @ nsa @ ( insert @ n @ ( root @ tr0 ) @ ( bot_bot @ ( set @ n ) ) ) ) @ tr0 @ ta )
| ( gram_L1333338417e_inFr @ ( minus_minus @ ( set @ n ) @ nsa @ ( insert @ n @ ( root @ tr0 ) @ ( bot_bot @ ( set @ n ) ) ) ) @ Tr @ ta ) ) ) ).
% IH
thf(fact_137_Gram__Lang__Mirabelle__ojxrtuoybn_OInr__cont__hsubst__eq,axiom,
! [Tr: dtree,Tr02: dtree] :
( ( ( root @ Tr )
= ( root @ Tr02 ) )
=> ( ( vimage @ dtree @ ( sum_sum @ t @ dtree ) @ ( sum_Inr @ dtree @ t ) @ ( cont @ ( gram_L1004374585hsubst @ Tr02 @ Tr ) ) )
= ( image @ dtree @ dtree @ ( gram_L1004374585hsubst @ Tr02 ) @ ( vimage @ dtree @ ( sum_sum @ t @ dtree ) @ ( sum_Inr @ dtree @ t ) @ ( cont @ Tr02 ) ) ) ) ) ).
% Gram_Lang_Mirabelle_ojxrtuoybn.Inr_cont_hsubst_eq
thf(fact_138_image__eqI,axiom,
! [A: $tType,B: $tType,B2: A,F: B > A,X: B,A6: set @ B] :
( ( B2
= ( F @ X ) )
=> ( ( member @ B @ X @ A6 )
=> ( member @ A @ B2 @ ( image @ B @ A @ F @ A6 ) ) ) ) ).
% image_eqI
thf(fact_139_empty__Collect__eq,axiom,
! [A: $tType,P: A > $o] :
( ( ( bot_bot @ ( set @ A ) )
= ( collect @ A @ P ) )
= ( ! [X4: A] :
~ ( P @ X4 ) ) ) ).
% empty_Collect_eq
thf(fact_140_Collect__empty__eq,axiom,
! [A: $tType,P: A > $o] :
( ( ( collect @ A @ P )
= ( bot_bot @ ( set @ A ) ) )
= ( ! [X4: A] :
~ ( P @ X4 ) ) ) ).
% Collect_empty_eq
thf(fact_141_all__not__in__conv,axiom,
! [A: $tType,A6: set @ A] :
( ( ! [X4: A] :
~ ( member @ A @ X4 @ A6 ) )
= ( A6
= ( bot_bot @ ( set @ A ) ) ) ) ).
% all_not_in_conv
thf(fact_142_empty__iff,axiom,
! [A: $tType,C3: A] :
~ ( member @ A @ C3 @ ( bot_bot @ ( set @ A ) ) ) ).
% empty_iff
thf(fact_143_Diff__idemp,axiom,
! [A: $tType,A6: set @ A,B4: set @ A] :
( ( minus_minus @ ( set @ A ) @ ( minus_minus @ ( set @ A ) @ A6 @ B4 ) @ B4 )
= ( minus_minus @ ( set @ A ) @ A6 @ B4 ) ) ).
% Diff_idemp
thf(fact_144_Diff__iff,axiom,
! [A: $tType,C3: A,A6: set @ A,B4: set @ A] :
( ( member @ A @ C3 @ ( minus_minus @ ( set @ A ) @ A6 @ B4 ) )
= ( ( member @ A @ C3 @ A6 )
& ~ ( member @ A @ C3 @ B4 ) ) ) ).
% Diff_iff
thf(fact_145_DiffI,axiom,
! [A: $tType,C3: A,A6: set @ A,B4: set @ A] :
( ( member @ A @ C3 @ A6 )
=> ( ~ ( member @ A @ C3 @ B4 )
=> ( member @ A @ C3 @ ( minus_minus @ ( set @ A ) @ A6 @ B4 ) ) ) ) ).
% DiffI
thf(fact_146_image__ident,axiom,
! [A: $tType,Y4: set @ A] :
( ( image @ A @ A
@ ^ [X4: A] : X4
@ Y4 )
= Y4 ) ).
% image_ident
thf(fact_147_finite__Collect__conjI,axiom,
! [A: $tType,P: A > $o,Q: A > $o] :
( ( ( finite_finite2 @ A @ ( collect @ A @ P ) )
| ( finite_finite2 @ A @ ( collect @ A @ Q ) ) )
=> ( finite_finite2 @ A
@ ( collect @ A
@ ^ [X4: A] :
( ( P @ X4 )
& ( Q @ X4 ) ) ) ) ) ).
% finite_Collect_conjI
thf(fact_148_finite__Collect__disjI,axiom,
! [A: $tType,P: A > $o,Q: A > $o] :
( ( finite_finite2 @ A
@ ( collect @ A
@ ^ [X4: A] :
( ( P @ X4 )
| ( Q @ X4 ) ) ) )
= ( ( finite_finite2 @ A @ ( collect @ A @ P ) )
& ( finite_finite2 @ A @ ( collect @ A @ Q ) ) ) ) ).
% finite_Collect_disjI
thf(fact_149_vimage__Collect__eq,axiom,
! [B: $tType,A: $tType,F: A > B,P: B > $o] :
( ( vimage @ A @ B @ F @ ( collect @ B @ P ) )
= ( collect @ A
@ ^ [Y5: A] : ( P @ ( F @ Y5 ) ) ) ) ).
% vimage_Collect_eq
thf(fact_150_vimage__ident,axiom,
! [A: $tType,Y4: set @ A] :
( ( vimage @ A @ A
@ ^ [X4: A] : X4
@ Y4 )
= Y4 ) ).
% vimage_ident
thf(fact_151_image__is__empty,axiom,
! [A: $tType,B: $tType,F: B > A,A6: set @ B] :
( ( ( image @ B @ A @ F @ A6 )
= ( bot_bot @ ( set @ A ) ) )
= ( A6
= ( bot_bot @ ( set @ B ) ) ) ) ).
% image_is_empty
thf(fact_152_empty__is__image,axiom,
! [A: $tType,B: $tType,F: B > A,A6: set @ B] :
( ( ( bot_bot @ ( set @ A ) )
= ( image @ B @ A @ F @ A6 ) )
= ( A6
= ( bot_bot @ ( set @ B ) ) ) ) ).
% empty_is_image
thf(fact_153_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_154_finite__imageI,axiom,
! [B: $tType,A: $tType,F5: set @ A,H: A > B] :
( ( finite_finite2 @ A @ F5 )
=> ( finite_finite2 @ B @ ( image @ A @ B @ H @ F5 ) ) ) ).
% finite_imageI
thf(fact_155_image__insert,axiom,
! [A: $tType,B: $tType,F: B > A,A5: B,B4: set @ B] :
( ( image @ B @ A @ F @ ( insert @ B @ A5 @ B4 ) )
= ( insert @ A @ ( F @ A5 ) @ ( image @ B @ A @ F @ B4 ) ) ) ).
% image_insert
thf(fact_156_insert__image,axiom,
! [B: $tType,A: $tType,X: A,A6: set @ A,F: A > B] :
( ( member @ A @ X @ A6 )
=> ( ( insert @ B @ ( F @ X ) @ ( image @ A @ B @ F @ A6 ) )
= ( image @ A @ B @ F @ A6 ) ) ) ).
% insert_image
thf(fact_157_singletonI,axiom,
! [A: $tType,A5: A] : ( member @ A @ A5 @ ( insert @ A @ A5 @ ( bot_bot @ ( set @ A ) ) ) ) ).
% singletonI
thf(fact_158_Diff__cancel,axiom,
! [A: $tType,A6: set @ A] :
( ( minus_minus @ ( set @ A ) @ A6 @ A6 )
= ( bot_bot @ ( set @ A ) ) ) ).
% Diff_cancel
thf(fact_159_empty__Diff,axiom,
! [A: $tType,A6: set @ A] :
( ( minus_minus @ ( set @ A ) @ ( bot_bot @ ( set @ A ) ) @ A6 )
= ( bot_bot @ ( set @ A ) ) ) ).
% empty_Diff
thf(fact_160_Diff__empty,axiom,
! [A: $tType,A6: set @ A] :
( ( minus_minus @ ( set @ A ) @ A6 @ ( bot_bot @ ( set @ A ) ) )
= A6 ) ).
% Diff_empty
thf(fact_161_finite__Diff2,axiom,
! [A: $tType,B4: set @ A,A6: set @ A] :
( ( finite_finite2 @ A @ B4 )
=> ( ( finite_finite2 @ A @ ( minus_minus @ ( set @ A ) @ A6 @ B4 ) )
= ( finite_finite2 @ A @ A6 ) ) ) ).
% finite_Diff2
thf(fact_162_finite__Diff,axiom,
! [A: $tType,A6: set @ A,B4: set @ A] :
( ( finite_finite2 @ A @ A6 )
=> ( finite_finite2 @ A @ ( minus_minus @ ( set @ A ) @ A6 @ B4 ) ) ) ).
% finite_Diff
thf(fact_163_Diff__insert0,axiom,
! [A: $tType,X: A,A6: set @ A,B4: set @ A] :
( ~ ( member @ A @ X @ A6 )
=> ( ( minus_minus @ ( set @ A ) @ A6 @ ( insert @ A @ X @ B4 ) )
= ( minus_minus @ ( set @ A ) @ A6 @ B4 ) ) ) ).
% Diff_insert0
thf(fact_164_insert__Diff1,axiom,
! [A: $tType,X: A,B4: set @ A,A6: set @ A] :
( ( member @ A @ X @ B4 )
=> ( ( minus_minus @ ( set @ A ) @ ( insert @ A @ X @ A6 ) @ B4 )
= ( minus_minus @ ( set @ A ) @ A6 @ B4 ) ) ) ).
% insert_Diff1
thf(fact_165_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_166_singleton__conv,axiom,
! [A: $tType,A5: A] :
( ( collect @ A
@ ^ [X4: A] : X4 = A5 )
= ( insert @ A @ A5 @ ( bot_bot @ ( set @ A ) ) ) ) ).
% singleton_conv
thf(fact_167_singleton__conv2,axiom,
! [A: $tType,A5: A] :
( ( collect @ A
@ ( ^ [Y6: A,Z: A] : Y6 = Z
@ A5 ) )
= ( insert @ A @ A5 @ ( bot_bot @ ( set @ A ) ) ) ) ).
% singleton_conv2
thf(fact_168_insert__Diff__single,axiom,
! [A: $tType,A5: A,A6: set @ A] :
( ( insert @ A @ A5 @ ( minus_minus @ ( set @ A ) @ A6 @ ( insert @ A @ A5 @ ( bot_bot @ ( set @ A ) ) ) ) )
= ( insert @ A @ A5 @ A6 ) ) ).
% insert_Diff_single
thf(fact_169_finite__Diff__insert,axiom,
! [A: $tType,A6: set @ A,A5: A,B4: set @ A] :
( ( finite_finite2 @ A @ ( minus_minus @ ( set @ A ) @ A6 @ ( insert @ A @ A5 @ B4 ) ) )
= ( finite_finite2 @ A @ ( minus_minus @ ( set @ A ) @ A6 @ B4 ) ) ) ).
% finite_Diff_insert
thf(fact_170_vimage__Diff,axiom,
! [A: $tType,B: $tType,F: A > B,A6: set @ B,B4: set @ B] :
( ( vimage @ A @ B @ F @ ( minus_minus @ ( set @ B ) @ A6 @ B4 ) )
= ( minus_minus @ ( set @ A ) @ ( vimage @ A @ B @ F @ A6 ) @ ( vimage @ A @ B @ F @ B4 ) ) ) ).
% vimage_Diff
thf(fact_171_vimage__def,axiom,
! [B: $tType,A: $tType] :
( ( vimage @ A @ B )
= ( ^ [F4: A > B,B9: set @ B] :
( collect @ A
@ ^ [X4: A] : ( member @ B @ ( F4 @ X4 ) @ B9 ) ) ) ) ).
% vimage_def
thf(fact_172_image__comp,axiom,
! [B: $tType,A: $tType,C: $tType,F: B > A,G: C > B,R: set @ C] :
( ( image @ B @ A @ F @ ( image @ C @ B @ G @ R ) )
= ( image @ C @ A @ ( comp @ B @ A @ C @ F @ G ) @ R ) ) ).
% image_comp
thf(fact_173_image__eq__imp__comp,axiom,
! [B: $tType,A: $tType,D: $tType,C: $tType,F: B > A,A6: set @ B,G: C > A,B4: set @ C,H: A > D] :
( ( ( image @ B @ A @ F @ A6 )
= ( image @ C @ A @ G @ B4 ) )
=> ( ( image @ B @ D @ ( comp @ A @ D @ B @ H @ F ) @ A6 )
= ( image @ C @ D @ ( comp @ A @ D @ C @ H @ G ) @ B4 ) ) ) ).
% image_eq_imp_comp
thf(fact_174_Diff__insert,axiom,
! [A: $tType,A6: set @ A,A5: A,B4: set @ A] :
( ( minus_minus @ ( set @ A ) @ A6 @ ( insert @ A @ A5 @ B4 ) )
= ( minus_minus @ ( set @ A ) @ ( minus_minus @ ( set @ A ) @ A6 @ B4 ) @ ( insert @ A @ A5 @ ( bot_bot @ ( set @ A ) ) ) ) ) ).
% Diff_insert
thf(fact_175_insert__Diff,axiom,
! [A: $tType,A5: A,A6: set @ A] :
( ( member @ A @ A5 @ A6 )
=> ( ( insert @ A @ A5 @ ( minus_minus @ ( set @ A ) @ A6 @ ( insert @ A @ A5 @ ( bot_bot @ ( set @ A ) ) ) ) )
= A6 ) ) ).
% insert_Diff
thf(fact_176_Diff__insert2,axiom,
! [A: $tType,A6: set @ A,A5: A,B4: set @ A] :
( ( minus_minus @ ( set @ A ) @ A6 @ ( insert @ A @ A5 @ B4 ) )
= ( minus_minus @ ( set @ A ) @ ( minus_minus @ ( set @ A ) @ A6 @ ( insert @ A @ A5 @ ( bot_bot @ ( set @ A ) ) ) ) @ B4 ) ) ).
% Diff_insert2
thf(fact_177_insert__compr,axiom,
! [A: $tType] :
( ( insert @ A )
= ( ^ [A11: A,B9: set @ A] :
( collect @ A
@ ^ [X4: A] :
( ( X4 = A11 )
| ( member @ A @ X4 @ B9 ) ) ) ) ) ).
% insert_compr
thf(fact_178_image__constant,axiom,
! [A: $tType,B: $tType,X: A,A6: set @ A,C3: B] :
( ( member @ A @ X @ A6 )
=> ( ( image @ A @ B
@ ^ [X4: A] : C3
@ A6 )
= ( insert @ B @ C3 @ ( bot_bot @ ( set @ B ) ) ) ) ) ).
% image_constant
thf(fact_179_insert__Collect,axiom,
! [A: $tType,A5: A,P: A > $o] :
( ( insert @ A @ A5 @ ( collect @ A @ P ) )
= ( collect @ A
@ ^ [U: A] :
( ( U != A5 )
=> ( P @ U ) ) ) ) ).
% insert_Collect
thf(fact_180_Collect__conv__if,axiom,
! [A: $tType,P: A > $o,A5: A] :
( ( ( P @ A5 )
=> ( ( collect @ A
@ ^ [X4: A] :
( ( X4 = A5 )
& ( P @ X4 ) ) )
= ( insert @ A @ A5 @ ( bot_bot @ ( set @ A ) ) ) ) )
& ( ~ ( P @ A5 )
=> ( ( collect @ A
@ ^ [X4: A] :
( ( X4 = A5 )
& ( P @ X4 ) ) )
= ( bot_bot @ ( set @ A ) ) ) ) ) ).
% Collect_conv_if
thf(fact_181_Collect__conv__if2,axiom,
! [A: $tType,P: A > $o,A5: A] :
( ( ( P @ A5 )
=> ( ( collect @ A
@ ^ [X4: A] :
( ( A5 = X4 )
& ( P @ X4 ) ) )
= ( insert @ A @ A5 @ ( bot_bot @ ( set @ A ) ) ) ) )
& ( ~ ( P @ A5 )
=> ( ( collect @ A
@ ^ [X4: A] :
( ( A5 = X4 )
& ( P @ X4 ) ) )
= ( bot_bot @ ( set @ A ) ) ) ) ) ).
% Collect_conv_if2
thf(fact_182_Diff__insert__absorb,axiom,
! [A: $tType,X: A,A6: set @ A] :
( ~ ( member @ A @ X @ A6 )
=> ( ( minus_minus @ ( set @ A ) @ ( insert @ A @ X @ A6 ) @ ( insert @ A @ X @ ( bot_bot @ ( set @ A ) ) ) )
= A6 ) ) ).
% Diff_insert_absorb
thf(fact_183_image__constant__conv,axiom,
! [B: $tType,A: $tType,A6: set @ B,C3: A] :
( ( ( A6
= ( bot_bot @ ( set @ B ) ) )
=> ( ( image @ B @ A
@ ^ [X4: B] : C3
@ A6 )
= ( bot_bot @ ( set @ A ) ) ) )
& ( ( A6
!= ( bot_bot @ ( set @ B ) ) )
=> ( ( image @ B @ A
@ ^ [X4: B] : C3
@ A6 )
= ( insert @ A @ C3 @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ).
% image_constant_conv
thf(fact_184_Compr__image__eq,axiom,
! [A: $tType,B: $tType,F: B > A,A6: set @ B,P: A > $o] :
( ( collect @ A
@ ^ [X4: A] :
( ( member @ A @ X4 @ ( image @ B @ A @ F @ A6 ) )
& ( P @ X4 ) ) )
= ( image @ B @ A @ F
@ ( collect @ B
@ ^ [X4: B] :
( ( member @ B @ X4 @ A6 )
& ( P @ ( F @ X4 ) ) ) ) ) ) ).
% Compr_image_eq
thf(fact_185_rev__image__eqI,axiom,
! [B: $tType,A: $tType,X: A,A6: set @ A,B2: B,F: A > B] :
( ( member @ A @ X @ A6 )
=> ( ( B2
= ( F @ X ) )
=> ( member @ B @ B2 @ ( image @ A @ B @ F @ A6 ) ) ) ) ).
% rev_image_eqI
thf(fact_186_set__diff__eq,axiom,
! [A: $tType] :
( ( minus_minus @ ( set @ A ) )
= ( ^ [A4: set @ A,B9: set @ A] :
( collect @ A
@ ^ [X4: A] :
( ( member @ A @ X4 @ A4 )
& ~ ( member @ A @ X4 @ B9 ) ) ) ) ) ).
% set_diff_eq
thf(fact_187_image__image,axiom,
! [A: $tType,B: $tType,C: $tType,F: B > A,G: C > B,A6: set @ C] :
( ( image @ B @ A @ F @ ( image @ C @ B @ G @ A6 ) )
= ( image @ C @ A
@ ^ [X4: C] : ( F @ ( G @ X4 ) )
@ A6 ) ) ).
% image_image
thf(fact_188_ball__imageD,axiom,
! [A: $tType,B: $tType,F: B > A,A6: set @ B,P: A > $o] :
( ! [X5: A] :
( ( member @ A @ X5 @ ( image @ B @ A @ F @ A6 ) )
=> ( P @ X5 ) )
=> ! [X6: B] :
( ( member @ B @ X6 @ A6 )
=> ( P @ ( F @ X6 ) ) ) ) ).
% ball_imageD
thf(fact_189_image__cong,axiom,
! [B: $tType,A: $tType,M: set @ A,N5: set @ A,F: A > B,G: A > B] :
( ( M = N5 )
=> ( ! [X5: A] :
( ( member @ A @ X5 @ N5 )
=> ( ( F @ X5 )
= ( G @ X5 ) ) )
=> ( ( image @ A @ B @ F @ M )
= ( image @ A @ B @ G @ N5 ) ) ) ) ).
% image_cong
thf(fact_190_ex__in__conv,axiom,
! [A: $tType,A6: set @ A] :
( ( ? [X4: A] : ( member @ A @ X4 @ A6 ) )
= ( A6
!= ( bot_bot @ ( set @ A ) ) ) ) ).
% ex_in_conv
thf(fact_191_bex__imageD,axiom,
! [A: $tType,B: $tType,F: B > A,A6: set @ B,P: A > $o] :
( ? [X6: A] :
( ( member @ A @ X6 @ ( image @ B @ A @ F @ A6 ) )
& ( P @ X6 ) )
=> ? [X5: B] :
( ( member @ B @ X5 @ A6 )
& ( P @ ( F @ X5 ) ) ) ) ).
% bex_imageD
thf(fact_192_image__iff,axiom,
! [A: $tType,B: $tType,Z2: A,F: B > A,A6: set @ B] :
( ( member @ A @ Z2 @ ( image @ B @ A @ F @ A6 ) )
= ( ? [X4: B] :
( ( member @ B @ X4 @ A6 )
& ( Z2
= ( F @ X4 ) ) ) ) ) ).
% image_iff
thf(fact_193_empty__def,axiom,
! [A: $tType] :
( ( bot_bot @ ( set @ A ) )
= ( collect @ A
@ ^ [X4: A] : $false ) ) ).
% empty_def
thf(fact_194_equals0I,axiom,
! [A: $tType,A6: set @ A] :
( ! [Y3: A] :
~ ( member @ A @ Y3 @ A6 )
=> ( A6
= ( bot_bot @ ( set @ A ) ) ) ) ).
% equals0I
thf(fact_195_equals0D,axiom,
! [A: $tType,A6: set @ A,A5: A] :
( ( A6
= ( bot_bot @ ( set @ A ) ) )
=> ~ ( member @ A @ A5 @ A6 ) ) ).
% equals0D
thf(fact_196_imageI,axiom,
! [B: $tType,A: $tType,X: A,A6: set @ A,F: A > B] :
( ( member @ A @ X @ A6 )
=> ( member @ B @ ( F @ X ) @ ( image @ A @ B @ F @ A6 ) ) ) ).
% imageI
thf(fact_197_imageE,axiom,
! [A: $tType,B: $tType,B2: A,F: B > A,A6: set @ B] :
( ( member @ A @ B2 @ ( image @ B @ A @ F @ A6 ) )
=> ~ ! [X5: B] :
( ( B2
= ( F @ X5 ) )
=> ~ ( member @ B @ X5 @ A6 ) ) ) ).
% imageE
thf(fact_198_emptyE,axiom,
! [A: $tType,A5: A] :
~ ( member @ A @ A5 @ ( bot_bot @ ( set @ A ) ) ) ).
% emptyE
thf(fact_199_DiffD2,axiom,
! [A: $tType,C3: A,A6: set @ A,B4: set @ A] :
( ( member @ A @ C3 @ ( minus_minus @ ( set @ A ) @ A6 @ B4 ) )
=> ~ ( member @ A @ C3 @ B4 ) ) ).
% DiffD2
thf(fact_200_DiffD1,axiom,
! [A: $tType,C3: A,A6: set @ A,B4: set @ A] :
( ( member @ A @ C3 @ ( minus_minus @ ( set @ A ) @ A6 @ B4 ) )
=> ( member @ A @ C3 @ A6 ) ) ).
% DiffD1
thf(fact_201_DiffE,axiom,
! [A: $tType,C3: A,A6: set @ A,B4: set @ A] :
( ( member @ A @ C3 @ ( minus_minus @ ( set @ A ) @ A6 @ B4 ) )
=> ~ ( ( member @ A @ C3 @ A6 )
=> ( member @ A @ C3 @ B4 ) ) ) ).
% DiffE
thf(fact_202_not__finite__existsD,axiom,
! [A: $tType,P: A > $o] :
( ~ ( finite_finite2 @ A @ ( collect @ A @ P ) )
=> ? [X13: A] : ( P @ X13 ) ) ).
% not_finite_existsD
thf(fact_203_pigeonhole__infinite,axiom,
! [B: $tType,A: $tType,A6: set @ A,F: A > B] :
( ~ ( finite_finite2 @ A @ A6 )
=> ( ( finite_finite2 @ B @ ( image @ A @ B @ F @ A6 ) )
=> ? [X5: A] :
( ( member @ A @ X5 @ A6 )
& ~ ( finite_finite2 @ A
@ ( collect @ A
@ ^ [A11: A] :
( ( member @ A @ A11 @ A6 )
& ( ( F @ A11 )
= ( F @ X5 ) ) ) ) ) ) ) ) ).
% pigeonhole_infinite
thf(fact_204_pigeonhole__infinite__rel,axiom,
! [B: $tType,A: $tType,A6: set @ A,B4: set @ B,R3: A > B > $o] :
( ~ ( finite_finite2 @ A @ A6 )
=> ( ( finite_finite2 @ B @ B4 )
=> ( ! [X5: A] :
( ( member @ A @ X5 @ A6 )
=> ? [Xa: B] :
( ( member @ B @ Xa @ B4 )
& ( R3 @ X5 @ Xa ) ) )
=> ? [X5: B] :
( ( member @ B @ X5 @ B4 )
& ~ ( finite_finite2 @ A
@ ( collect @ A
@ ^ [A11: A] :
( ( member @ A @ A11 @ A6 )
& ( R3 @ A11 @ X5 ) ) ) ) ) ) ) ) ).
% pigeonhole_infinite_rel
thf(fact_205_singletonD,axiom,
! [A: $tType,B2: A,A5: A] :
( ( member @ A @ B2 @ ( insert @ A @ A5 @ ( bot_bot @ ( set @ A ) ) ) )
=> ( B2 = A5 ) ) ).
% singletonD
thf(fact_206_singleton__iff,axiom,
! [A: $tType,B2: A,A5: A] :
( ( member @ A @ B2 @ ( insert @ A @ A5 @ ( bot_bot @ ( set @ A ) ) ) )
= ( B2 = A5 ) ) ).
% singleton_iff
thf(fact_207_doubleton__eq__iff,axiom,
! [A: $tType,A5: A,B2: A,C3: A,D2: A] :
( ( ( insert @ A @ A5 @ ( insert @ A @ B2 @ ( bot_bot @ ( set @ A ) ) ) )
= ( insert @ A @ C3 @ ( insert @ A @ D2 @ ( bot_bot @ ( set @ A ) ) ) ) )
= ( ( ( A5 = C3 )
& ( B2 = D2 ) )
| ( ( A5 = D2 )
& ( B2 = C3 ) ) ) ) ).
% doubleton_eq_iff
thf(fact_208_insert__not__empty,axiom,
! [A: $tType,A5: A,A6: set @ A] :
( ( insert @ A @ A5 @ A6 )
!= ( bot_bot @ ( set @ A ) ) ) ).
% insert_not_empty
thf(fact_209_singleton__inject,axiom,
! [A: $tType,A5: A,B2: A] :
( ( ( insert @ A @ A5 @ ( bot_bot @ ( set @ A ) ) )
= ( insert @ A @ B2 @ ( bot_bot @ ( set @ A ) ) ) )
=> ( A5 = B2 ) ) ).
% singleton_inject
thf(fact_210_finite_OemptyI,axiom,
! [A: $tType] : ( finite_finite2 @ A @ ( bot_bot @ ( set @ A ) ) ) ).
% finite.emptyI
thf(fact_211_infinite__imp__nonempty,axiom,
! [A: $tType,S2: set @ A] :
( ~ ( finite_finite2 @ A @ S2 )
=> ( S2
!= ( bot_bot @ ( set @ A ) ) ) ) ).
% infinite_imp_nonempty
thf(fact_212_insert__Diff__if,axiom,
! [A: $tType,X: A,B4: set @ A,A6: set @ A] :
( ( ( member @ A @ X @ B4 )
=> ( ( minus_minus @ ( set @ A ) @ ( insert @ A @ X @ A6 ) @ B4 )
= ( minus_minus @ ( set @ A ) @ A6 @ B4 ) ) )
& ( ~ ( member @ A @ X @ B4 )
=> ( ( minus_minus @ ( set @ A ) @ ( insert @ A @ X @ A6 ) @ B4 )
= ( insert @ A @ X @ ( minus_minus @ ( set @ A ) @ A6 @ B4 ) ) ) ) ) ).
% insert_Diff_if
thf(fact_213_Diff__infinite__finite,axiom,
! [A: $tType,T5: set @ A,S2: set @ A] :
( ( finite_finite2 @ A @ T5 )
=> ( ~ ( finite_finite2 @ A @ S2 )
=> ~ ( finite_finite2 @ A @ ( minus_minus @ ( set @ A ) @ S2 @ T5 ) ) ) ) ).
% Diff_infinite_finite
thf(fact_214_infinite__remove,axiom,
! [A: $tType,S2: set @ A,A5: A] :
( ~ ( finite_finite2 @ A @ S2 )
=> ~ ( finite_finite2 @ A @ ( minus_minus @ ( set @ A ) @ S2 @ ( insert @ A @ A5 @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ).
% infinite_remove
thf(fact_215_infinite__coinduct,axiom,
! [A: $tType,X8: ( set @ A ) > $o,A6: set @ A] :
( ( X8 @ A6 )
=> ( ! [A13: set @ A] :
( ( X8 @ A13 )
=> ? [X6: A] :
( ( member @ A @ X6 @ A13 )
& ( ( X8 @ ( minus_minus @ ( set @ A ) @ A13 @ ( insert @ A @ X6 @ ( bot_bot @ ( set @ A ) ) ) ) )
| ~ ( finite_finite2 @ A @ ( minus_minus @ ( set @ A ) @ A13 @ ( insert @ A @ X6 @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ) )
=> ~ ( finite_finite2 @ A @ A6 ) ) ) ).
% infinite_coinduct
thf(fact_216_finite__empty__induct,axiom,
! [A: $tType,A6: set @ A,P: ( set @ A ) > $o] :
( ( finite_finite2 @ A @ A6 )
=> ( ( P @ A6 )
=> ( ! [A10: A,A13: set @ A] :
( ( finite_finite2 @ A @ A13 )
=> ( ( member @ A @ A10 @ A13 )
=> ( ( P @ A13 )
=> ( P @ ( minus_minus @ ( set @ A ) @ A13 @ ( insert @ A @ A10 @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ) )
=> ( P @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ).
% finite_empty_induct
thf(fact_217_inf__img__fin__domE,axiom,
! [B: $tType,A: $tType,F: B > A,A6: set @ B] :
( ( finite_finite2 @ A @ ( image @ B @ A @ F @ A6 ) )
=> ( ~ ( finite_finite2 @ B @ A6 )
=> ~ ! [Y3: A] :
( ( member @ A @ Y3 @ ( image @ B @ A @ F @ A6 ) )
=> ( finite_finite2 @ B @ ( vimage @ B @ A @ F @ ( insert @ A @ Y3 @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ) ) ).
% inf_img_fin_domE
thf(fact_218_inf__img__fin__dom,axiom,
! [B: $tType,A: $tType,F: B > A,A6: set @ B] :
( ( finite_finite2 @ A @ ( image @ B @ A @ F @ A6 ) )
=> ( ~ ( finite_finite2 @ B @ A6 )
=> ? [X5: A] :
( ( member @ A @ X5 @ ( image @ B @ A @ F @ A6 ) )
& ~ ( finite_finite2 @ B @ ( vimage @ B @ A @ F @ ( insert @ A @ X5 @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ) ) ).
% inf_img_fin_dom
thf(fact_219_infinite__finite__induct,axiom,
! [A: $tType,P: ( set @ A ) > $o,A6: set @ A] :
( ! [A13: set @ A] :
( ~ ( finite_finite2 @ A @ A13 )
=> ( P @ A13 ) )
=> ( ( P @ ( bot_bot @ ( set @ A ) ) )
=> ( ! [X5: A,F6: set @ A] :
( ( finite_finite2 @ A @ F6 )
=> ( ~ ( member @ A @ X5 @ F6 )
=> ( ( P @ F6 )
=> ( P @ ( insert @ A @ X5 @ F6 ) ) ) ) )
=> ( P @ A6 ) ) ) ) ).
% infinite_finite_induct
thf(fact_220_finite__ne__induct,axiom,
! [A: $tType,F5: set @ A,P: ( set @ A ) > $o] :
( ( finite_finite2 @ A @ F5 )
=> ( ( F5
!= ( bot_bot @ ( set @ A ) ) )
=> ( ! [X5: A] : ( P @ ( insert @ A @ X5 @ ( bot_bot @ ( set @ A ) ) ) )
=> ( ! [X5: A,F6: set @ A] :
( ( finite_finite2 @ A @ F6 )
=> ( ( F6
!= ( bot_bot @ ( set @ A ) ) )
=> ( ~ ( member @ A @ X5 @ F6 )
=> ( ( P @ F6 )
=> ( P @ ( insert @ A @ X5 @ F6 ) ) ) ) ) )
=> ( P @ F5 ) ) ) ) ) ).
% finite_ne_induct
thf(fact_221_finite_Oinducts,axiom,
! [A: $tType,X: set @ A,P: ( set @ A ) > $o] :
( ( finite_finite2 @ A @ X )
=> ( ( P @ ( bot_bot @ ( set @ A ) ) )
=> ( ! [A13: set @ A,A10: A] :
( ( finite_finite2 @ A @ A13 )
=> ( ( P @ A13 )
=> ( P @ ( insert @ A @ A10 @ A13 ) ) ) )
=> ( P @ X ) ) ) ) ).
% finite.inducts
thf(fact_222_finite__induct,axiom,
! [A: $tType,F5: set @ A,P: ( set @ A ) > $o] :
( ( finite_finite2 @ A @ F5 )
=> ( ( P @ ( bot_bot @ ( set @ A ) ) )
=> ( ! [X5: A,F6: set @ A] :
( ( finite_finite2 @ A @ F6 )
=> ( ~ ( member @ A @ X5 @ F6 )
=> ( ( P @ F6 )
=> ( P @ ( insert @ A @ X5 @ F6 ) ) ) ) )
=> ( P @ F5 ) ) ) ) ).
% finite_induct
thf(fact_223_finite_Osimps,axiom,
! [A: $tType] :
( ( finite_finite2 @ A )
= ( ^ [A11: set @ A] :
( ( A11
= ( bot_bot @ ( set @ A ) ) )
| ? [A4: set @ A,B10: A] :
( ( A11
= ( insert @ A @ B10 @ A4 ) )
& ( finite_finite2 @ A @ A4 ) ) ) ) ) ).
% finite.simps
thf(fact_224_finite_Ocases,axiom,
! [A: $tType,A5: set @ A] :
( ( finite_finite2 @ A @ A5 )
=> ( ( A5
!= ( bot_bot @ ( set @ A ) ) )
=> ~ ! [A13: set @ A] :
( ? [A10: A] :
( A5
= ( insert @ A @ A10 @ A13 ) )
=> ~ ( finite_finite2 @ A @ A13 ) ) ) ) ).
% finite.cases
thf(fact_225_vimage__singleton__eq,axiom,
! [A: $tType,B: $tType,A5: A,F: A > B,B2: B] :
( ( member @ A @ A5 @ ( vimage @ A @ B @ F @ ( insert @ B @ B2 @ ( bot_bot @ ( set @ B ) ) ) ) )
= ( ( F @ A5 )
= B2 ) ) ).
% vimage_singleton_eq
thf(fact_226_Gram__Lang__Mirabelle__ojxrtuoybn_OInr__cont__hsubst__neq,axiom,
! [Tr: dtree,Tr02: dtree] :
( ( ( root @ Tr )
!= ( root @ Tr02 ) )
=> ( ( vimage @ dtree @ ( sum_sum @ t @ dtree ) @ ( sum_Inr @ dtree @ t ) @ ( cont @ ( gram_L1004374585hsubst @ Tr02 @ Tr ) ) )
= ( image @ dtree @ dtree @ ( gram_L1004374585hsubst @ Tr02 ) @ ( vimage @ dtree @ ( sum_sum @ t @ dtree ) @ ( sum_Inr @ dtree @ t ) @ ( cont @ Tr ) ) ) ) ) ).
% Gram_Lang_Mirabelle_ojxrtuoybn.Inr_cont_hsubst_neq
thf(fact_227_empty__natural,axiom,
! [C: $tType,B: $tType,D: $tType,A: $tType,F: A > C,G: D > B] :
( ( comp @ C @ ( set @ B ) @ A
@ ^ [Uu: C] : ( bot_bot @ ( set @ B ) )
@ F )
= ( comp @ ( set @ D ) @ ( set @ B ) @ A @ ( image @ D @ B @ G )
@ ^ [Uu: A] : ( bot_bot @ ( set @ D ) ) ) ) ).
% empty_natural
thf(fact_228_Frr,axiom,
( gram_L1556062726le_Frr
= ( ^ [Ns2: set @ n,Tr3: dtree] : ( collect @ t @ ( gram_L805317505_inFrr @ Ns2 @ Tr3 ) ) ) ) ).
% Frr
thf(fact_229_is__singletonI,axiom,
! [A: $tType,X: A] : ( is_singleton @ A @ ( insert @ A @ X @ ( bot_bot @ ( set @ A ) ) ) ) ).
% is_singletonI
thf(fact_230_Inf_OINF__image,axiom,
! [B: $tType,A: $tType,C: $tType,Inf: ( set @ A ) > A,G: B > A,F: C > B,A6: set @ C] :
( ( Inf @ ( image @ B @ A @ G @ ( image @ C @ B @ F @ A6 ) ) )
= ( Inf @ ( image @ C @ A @ ( comp @ B @ A @ C @ G @ F ) @ A6 ) ) ) ).
% Inf.INF_image
thf(fact_231_is__singletonI_H,axiom,
! [A: $tType,A6: set @ A] :
( ( A6
!= ( bot_bot @ ( set @ A ) ) )
=> ( ! [X5: A,Y3: A] :
( ( member @ A @ X5 @ A6 )
=> ( ( member @ A @ Y3 @ A6 )
=> ( X5 = Y3 ) ) )
=> ( is_singleton @ A @ A6 ) ) ) ).
% is_singletonI'
thf(fact_232_in__image__insert__iff,axiom,
! [A: $tType,B4: set @ ( set @ A ),X: A,A6: set @ A] :
( ! [C4: set @ A] :
( ( member @ ( set @ A ) @ C4 @ B4 )
=> ~ ( member @ A @ X @ C4 ) )
=> ( ( member @ ( set @ A ) @ A6 @ ( image @ ( set @ A ) @ ( set @ A ) @ ( insert @ A @ X ) @ B4 ) )
= ( ( member @ A @ X @ A6 )
& ( member @ ( set @ A ) @ ( minus_minus @ ( set @ A ) @ A6 @ ( insert @ A @ X @ ( bot_bot @ ( set @ A ) ) ) ) @ B4 ) ) ) ) ).
% in_image_insert_iff
thf(fact_233_type__copy__map__cong0,axiom,
! [B: $tType,D: $tType,E: $tType,A: $tType,C: $tType,M: B > A,G: C > B,X: C,N5: D > A,H: C > D,F: A > E] :
( ( ( M @ ( G @ X ) )
= ( N5 @ ( H @ X ) ) )
=> ( ( comp @ B @ E @ C @ ( comp @ A @ E @ B @ F @ M ) @ G @ X )
= ( comp @ D @ E @ C @ ( comp @ A @ E @ D @ F @ N5 ) @ H @ X ) ) ) ).
% type_copy_map_cong0
thf(fact_234_is__singleton__def,axiom,
! [A: $tType] :
( ( is_singleton @ A )
= ( ^ [A4: set @ A] :
? [X4: A] :
( A4
= ( insert @ A @ X4 @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ).
% is_singleton_def
thf(fact_235_is__singletonE,axiom,
! [A: $tType,A6: set @ A] :
( ( is_singleton @ A @ A6 )
=> ~ ! [X5: A] :
( A6
!= ( insert @ A @ X5 @ ( bot_bot @ ( set @ A ) ) ) ) ) ).
% is_singletonE
thf(fact_236_Sup_OSUP__image,axiom,
! [B: $tType,A: $tType,C: $tType,Sup: ( set @ A ) > A,G: B > A,F: C > B,A6: set @ C] :
( ( Sup @ ( image @ B @ A @ G @ ( image @ C @ B @ F @ A6 ) ) )
= ( Sup @ ( image @ C @ A @ ( comp @ B @ A @ C @ G @ F ) @ A6 ) ) ) ).
% Sup.SUP_image
thf(fact_237_image__o__collect,axiom,
! [B: $tType,C: $tType,A: $tType,G: C > B,F5: set @ ( A > ( set @ C ) )] :
( ( bNF_collect @ A @ B @ ( image @ ( A > ( set @ C ) ) @ ( A > ( set @ B ) ) @ ( comp @ ( set @ C ) @ ( set @ B ) @ A @ ( image @ C @ B @ G ) ) @ F5 ) )
= ( comp @ ( set @ C ) @ ( set @ B ) @ A @ ( image @ C @ B @ G ) @ ( bNF_collect @ A @ C @ F5 ) ) ) ).
% image_o_collect
thf(fact_238_is__singleton__the__elem,axiom,
! [A: $tType] :
( ( is_singleton @ A )
= ( ^ [A4: set @ A] :
( A4
= ( insert @ A @ ( the_elem @ A @ A4 ) @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ).
% is_singleton_the_elem
thf(fact_239_the__elem__eq,axiom,
! [A: $tType,X: A] :
( ( the_elem @ A @ ( insert @ A @ X @ ( bot_bot @ ( set @ A ) ) ) )
= X ) ).
% the_elem_eq
thf(fact_240_the__elem__image__unique,axiom,
! [B: $tType,A: $tType,A6: set @ A,F: A > B,X: A] :
( ( A6
!= ( bot_bot @ ( set @ A ) ) )
=> ( ! [Y3: A] :
( ( member @ A @ Y3 @ A6 )
=> ( ( F @ Y3 )
= ( F @ X ) ) )
=> ( ( the_elem @ B @ ( image @ A @ B @ F @ A6 ) )
= ( F @ X ) ) ) ) ).
% the_elem_image_unique
thf(fact_241_collect__comp,axiom,
! [A: $tType,B: $tType,C: $tType,F5: set @ ( C > ( set @ B ) ),G: A > C] :
( ( comp @ C @ ( set @ B ) @ A @ ( bNF_collect @ C @ B @ F5 ) @ G )
= ( bNF_collect @ A @ B
@ ( image @ ( C > ( set @ B ) ) @ ( A > ( set @ B ) )
@ ^ [F4: C > ( set @ B )] : ( comp @ C @ ( set @ B ) @ A @ F4 @ G )
@ F5 ) ) ) ).
% collect_comp
thf(fact_242_subtrOf__def,axiom,
( gram_L1614515765ubtrOf
= ( ^ [Tr3: dtree,N3: n] :
@+[Tr5: dtree] :
( ( member @ ( sum_sum @ t @ dtree ) @ ( sum_Inr @ dtree @ t @ Tr5 ) @ ( cont @ Tr3 ) )
& ( ( root @ Tr5 )
= N3 ) ) ) ) ).
% subtrOf_def
thf(fact_243_comp__apply__eq,axiom,
! [B: $tType,D: $tType,A: $tType,C: $tType,F: B > A,G: C > B,X: C,H: D > A,K: C > D] :
( ( ( F @ ( G @ X ) )
= ( H @ ( K @ X ) ) )
=> ( ( comp @ B @ A @ C @ F @ G @ X )
= ( comp @ D @ A @ C @ H @ K @ X ) ) ) ).
% comp_apply_eq
thf(fact_244_remove__def,axiom,
! [A: $tType] :
( ( remove @ A )
= ( ^ [X4: A,A4: set @ A] : ( minus_minus @ ( set @ A ) @ A4 @ ( insert @ A @ X4 @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ).
% remove_def
thf(fact_245_univ__def,axiom,
! [A: $tType,B: $tType] :
( ( bNF_Greatest_univ @ B @ A )
= ( ^ [F4: B > A,X9: set @ B] :
( F4
@ @+[X4: B] : ( member @ B @ X4 @ X9 ) ) ) ) ).
% univ_def
thf(fact_246_member__remove,axiom,
! [A: $tType,X: A,Y: A,A6: set @ A] :
( ( member @ A @ X @ ( remove @ A @ Y @ A6 ) )
= ( ( member @ A @ X @ A6 )
& ( X != Y ) ) ) ).
% member_remove
thf(fact_247_wf__inj__on,axiom,
! [Tr: dtree] :
( ( gram_L864798063lle_wf @ Tr )
=> ( inj_on @ dtree @ n @ root @ ( vimage @ dtree @ ( sum_sum @ t @ dtree ) @ ( sum_Inr @ dtree @ t ) @ ( cont @ Tr ) ) ) ) ).
% wf_inj_on
thf(fact_248_sum__set__simps_I1_J,axiom,
! [B: $tType,A: $tType,X: A] :
( ( basic_setl @ A @ B @ ( sum_Inl @ A @ B @ X ) )
= ( insert @ A @ X @ ( bot_bot @ ( set @ A ) ) ) ) ).
% sum_set_simps(1)
thf(fact_249_sum__set__simps_I2_J,axiom,
! [A: $tType,C: $tType,X: A] :
( ( basic_setl @ C @ A @ ( sum_Inr @ A @ C @ X ) )
= ( bot_bot @ ( set @ C ) ) ) ).
% sum_set_simps(2)
thf(fact_250_inj__on__insert,axiom,
! [B: $tType,A: $tType,F: A > B,A5: A,A6: set @ A] :
( ( inj_on @ A @ B @ F @ ( insert @ A @ A5 @ A6 ) )
= ( ( inj_on @ A @ B @ F @ A6 )
& ~ ( member @ B @ ( F @ A5 ) @ ( image @ A @ B @ F @ ( minus_minus @ ( set @ A ) @ A6 @ ( insert @ A @ A5 @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ) ) ).
% inj_on_insert
thf(fact_251_finite__image__iff,axiom,
! [B: $tType,A: $tType,F: A > B,A6: set @ A] :
( ( inj_on @ A @ B @ F @ A6 )
=> ( ( finite_finite2 @ B @ ( image @ A @ B @ F @ A6 ) )
= ( finite_finite2 @ A @ A6 ) ) ) ).
% finite_image_iff
thf(fact_252_finite__imageD,axiom,
! [A: $tType,B: $tType,F: B > A,A6: set @ B] :
( ( finite_finite2 @ A @ ( image @ B @ A @ F @ A6 ) )
=> ( ( inj_on @ B @ A @ F @ A6 )
=> ( finite_finite2 @ B @ A6 ) ) ) ).
% finite_imageD
thf(fact_253_comp__inj__on__iff,axiom,
! [B: $tType,C: $tType,A: $tType,F: A > B,A6: set @ A,F7: B > C] :
( ( inj_on @ A @ B @ F @ A6 )
=> ( ( inj_on @ B @ C @ F7 @ ( image @ A @ B @ F @ A6 ) )
= ( inj_on @ A @ C @ ( comp @ B @ C @ A @ F7 @ F ) @ A6 ) ) ) ).
% comp_inj_on_iff
thf(fact_254_inj__on__imageI,axiom,
! [B: $tType,C: $tType,A: $tType,G: C > B,F: A > C,A6: set @ A] :
( ( inj_on @ A @ B @ ( comp @ C @ B @ A @ G @ F ) @ A6 )
=> ( inj_on @ C @ B @ G @ ( image @ A @ C @ F @ A6 ) ) ) ).
% inj_on_imageI
thf(fact_255_comp__inj__on,axiom,
! [B: $tType,C: $tType,A: $tType,F: A > B,A6: set @ A,G: B > C] :
( ( inj_on @ A @ B @ F @ A6 )
=> ( ( inj_on @ B @ C @ G @ ( image @ A @ B @ F @ A6 ) )
=> ( inj_on @ A @ C @ ( comp @ B @ C @ A @ G @ F ) @ A6 ) ) ) ).
% comp_inj_on
%----Type constructors (4)
thf(tcon_fun___Finite__Set_Ofinite,axiom,
! [A8: $tType,A14: $tType] :
( ( ( finite_finite @ A8 @ ( type @ A8 ) )
& ( finite_finite @ A14 @ ( type @ A14 ) ) )
=> ( finite_finite @ ( A8 > A14 ) @ ( type @ ( A8 > A14 ) ) ) ) ).
thf(tcon_Set_Oset___Finite__Set_Ofinite_1,axiom,
! [A8: $tType] :
( ( finite_finite @ A8 @ ( type @ A8 ) )
=> ( finite_finite @ ( set @ A8 ) @ ( type @ ( set @ A8 ) ) ) ) ).
thf(tcon_HOL_Obool___Finite__Set_Ofinite_2,axiom,
finite_finite @ $o @ ( type @ $o ) ).
thf(tcon_Sum__Type_Osum___Finite__Set_Ofinite_3,axiom,
! [A8: $tType,A14: $tType] :
( ( ( finite_finite @ A8 @ ( type @ A8 ) )
& ( finite_finite @ A14 @ ( type @ A14 ) ) )
=> ( finite_finite @ ( sum_sum @ A8 @ A14 ) @ ( type @ ( sum_sum @ A8 @ A14 ) ) ) ) ).
%----Helper facts (3)
thf(help_If_3_1_T,axiom,
! [P: $o] :
( ( P = $true )
| ( P = $false ) ) ).
thf(help_If_2_1_T,axiom,
! [A: $tType,X: A,Y: A] :
( ( if @ A @ $false @ X @ Y )
= Y ) ).
thf(help_If_1_1_T,axiom,
! [A: $tType,X: A,Y: A] :
( ( if @ A @ $true @ X @ Y )
= X ) ).
%----Conjectures (2)
thf(conj_0,hypothesis,
! [Tr7: dtree] :
( ( member @ ( sum_sum @ t @ dtree ) @ ( sum_Inr @ dtree @ t @ Tr7 ) @ ( cont @ tr0 ) )
=> ( ( tr1
= ( gram_L1004374585hsubst @ tr0 @ Tr7 ) )
=> thesis ) ) ).
thf(conj_1,conjecture,
thesis ).
%------------------------------------------------------------------------------