TPTP Problem File: COM184^1.p
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%------------------------------------------------------------------------------
% File : COM184^1 : TPTP v8.2.0. Released v7.0.0.
% Domain : Computing Theory
% Problem : Grammars and languages 228
% 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__228.p [Bla16]
% Status : Theorem
% Rating : 0.33 v8.1.0, 0.50 v7.5.0, 0.33 v7.3.0, 0.00 v7.2.0, 0.25 v7.1.0
% Syntax : Number of formulae : 322 ( 109 unt; 51 typ; 0 def)
% Number of atoms : 749 ( 298 equ; 0 cnn)
% Maximal formula atoms : 7 ( 2 avg)
% Number of connectives : 4268 ( 106 ~; 13 |; 70 &;3713 @)
% ( 0 <=>; 366 =>; 0 <=; 0 <~>)
% Maximal formula depth : 21 ( 8 avg)
% Number of types : 4 ( 3 usr)
% Number of type conns : 243 ( 243 >; 0 *; 0 +; 0 <<)
% Number of symbols : 51 ( 48 usr; 6 con; 0-6 aty)
% Number of variables : 1189 ( 47 ^;1048 !; 47 ?;1189 :)
% ( 47 !>; 0 ?*; 0 @-; 0 @+)
% SPC : TH1_THM_EQU_NAR
% Comments : This file was generated by Isabelle (most likely Sledgehammer)
% 2016-07-13 14:41:22.472
%------------------------------------------------------------------------------
%----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 (45)
thf(sy_cl_HOL_Otype,type,
type:
!>[A: $tType] : ( ( itself @ A ) > $o ) ).
thf(sy_cl_Groups_Ominus,type,
minus:
!>[A: $tType] : ( ( itself @ A ) > $o ) ).
thf(sy_cl_Groups_Ogroup__add,type,
group_add:
!>[A: $tType] : ( ( itself @ A ) > $o ) ).
thf(sy_cl_Finite__Set_Ofinite,type,
finite_finite:
!>[A: $tType] : ( ( itself @ A ) > $o ) ).
thf(sy_cl_Groups_Oordered__ab__group__add,type,
ordered_ab_group_add:
!>[A: $tType] : ( ( itself @ A ) > $o ) ).
thf(sy_cl_Groups_Ocancel__ab__semigroup__add,type,
cancel146912293up_add:
!>[A: $tType] : ( ( itself @ A ) > $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_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_Ocorec,type,
corec:
!>[A: $tType] : ( ( A > n ) > ( A > ( set @ ( sum_sum @ t @ ( sum_sum @ dtree @ A ) ) ) ) > A > 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_Gram__Lang__Mirabelle__ojxrtuoybn_OFr,type,
gram_L861583724lle_Fr: ( set @ n ) > dtree > ( set @ t ) ).
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_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_Orderings_Oord__class_Oless,type,
ord_less:
!>[A: $tType] : ( A > A > $o ) ).
thf(sy_c_Orderings_Oord__class_Oless__eq,type,
ord_less_eq:
!>[A: $tType] : ( A > A > $o ) ).
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_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_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_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_OPlus,type,
sum_Plus:
!>[A: $tType,B: $tType] : ( ( set @ A ) > ( set @ B ) > ( set @ ( 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_OSumr,type,
sum_Sumr:
!>[B: $tType,C: $tType,A: $tType] : ( ( B > C ) > ( sum_sum @ A @ B ) > C ) ).
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_tr1,type,
tr1: dtree ).
thf(sy_v_tr2,type,
tr2: dtree ).
thf(sy_v_tr3,type,
tr3: dtree ).
%----Relevant facts (256)
thf(fact_0_subtr_ORefl,axiom,
! [Tr: dtree,Ns: set @ n] :
( ( member @ n @ ( root @ Tr ) @ Ns )
=> ( gram_L716654942_subtr @ Ns @ Tr @ Tr ) ) ).
% subtr.Refl
thf(fact_1_subtr_OStep,axiom,
! [Tr3: dtree,Ns: set @ n,Tr1: dtree,Tr2: dtree] :
( ( member @ n @ ( root @ Tr3 ) @ Ns )
=> ( ( gram_L716654942_subtr @ Ns @ Tr1 @ Tr2 )
=> ( ( member @ ( sum_sum @ t @ dtree ) @ ( sum_Inr @ dtree @ t @ Tr2 ) @ ( cont @ Tr3 ) )
=> ( gram_L716654942_subtr @ Ns @ Tr1 @ Tr3 ) ) ) ) ).
% subtr.Step
thf(fact_2_subtr_Ocases,axiom,
! [A1: set @ n,A2: dtree,A3: dtree] :
( ( gram_L716654942_subtr @ A1 @ A2 @ A3 )
=> ( ( ( A3 = A2 )
=> ~ ( member @ n @ ( root @ A2 ) @ A1 ) )
=> ~ ( ( member @ n @ ( root @ A3 ) @ A1 )
=> ! [Tr22: dtree] :
( ( gram_L716654942_subtr @ A1 @ A2 @ Tr22 )
=> ~ ( member @ ( sum_sum @ t @ dtree ) @ ( sum_Inr @ dtree @ t @ Tr22 ) @ ( cont @ A3 ) ) ) ) ) ) ).
% subtr.cases
thf(fact_3_subtr_Osimps,axiom,
( gram_L716654942_subtr
= ( ^ [A12: set @ n,A22: dtree,A32: dtree] :
( ? [Tr4: dtree,Ns2: set @ n] :
( ( A12 = Ns2 )
& ( A22 = Tr4 )
& ( A32 = Tr4 )
& ( member @ n @ ( root @ Tr4 ) @ Ns2 ) )
| ? [Tr32: dtree,Ns2: set @ n,Tr12: dtree,Tr23: dtree] :
( ( A12 = Ns2 )
& ( A22 = Tr12 )
& ( A32 = Tr32 )
& ( member @ n @ ( root @ Tr32 ) @ Ns2 )
& ( gram_L716654942_subtr @ Ns2 @ Tr12 @ Tr23 )
& ( member @ ( sum_sum @ t @ dtree ) @ ( sum_Inr @ dtree @ t @ Tr23 ) @ ( cont @ Tr32 ) ) ) ) ) ) ).
% subtr.simps
thf(fact_4_subtr2_ORefl,axiom,
! [Tr: dtree,Ns: set @ n] :
( ( member @ n @ ( root @ Tr ) @ Ns )
=> ( gram_L1283001940subtr2 @ Ns @ Tr @ Tr ) ) ).
% subtr2.Refl
thf(fact_5_subtr2_OStep,axiom,
! [Tr1: dtree,Ns: set @ n,Tr2: dtree,Tr3: dtree] :
( ( member @ n @ ( root @ Tr1 ) @ Ns )
=> ( ( member @ ( sum_sum @ t @ dtree ) @ ( sum_Inr @ dtree @ t @ Tr1 ) @ ( cont @ Tr2 ) )
=> ( ( gram_L1283001940subtr2 @ Ns @ Tr2 @ Tr3 )
=> ( gram_L1283001940subtr2 @ Ns @ Tr1 @ Tr3 ) ) ) ) ).
% subtr2.Step
thf(fact_6_subtr__StepL,axiom,
! [Tr1: dtree,Ns: set @ n,Tr2: dtree,Tr3: dtree] :
( ( member @ n @ ( root @ Tr1 ) @ Ns )
=> ( ( member @ ( sum_sum @ t @ dtree ) @ ( sum_Inr @ dtree @ t @ Tr1 ) @ ( cont @ Tr2 ) )
=> ( ( gram_L716654942_subtr @ Ns @ Tr2 @ Tr3 )
=> ( gram_L716654942_subtr @ Ns @ Tr1 @ Tr3 ) ) ) ) ).
% subtr_StepL
thf(fact_7_subtr__trans,axiom,
! [Ns: set @ n,Tr1: dtree,Tr2: dtree,Tr3: dtree] :
( ( gram_L716654942_subtr @ Ns @ Tr1 @ Tr2 )
=> ( ( gram_L716654942_subtr @ Ns @ Tr2 @ Tr3 )
=> ( gram_L716654942_subtr @ Ns @ Tr1 @ Tr3 ) ) ) ).
% subtr_trans
thf(fact_8_subtr2_Ocases,axiom,
! [A1: set @ n,A2: dtree,A3: dtree] :
( ( gram_L1283001940subtr2 @ A1 @ A2 @ A3 )
=> ( ( ( A3 = A2 )
=> ~ ( member @ n @ ( root @ A2 ) @ A1 ) )
=> ~ ( ( member @ n @ ( root @ A2 ) @ A1 )
=> ! [Tr22: dtree] :
( ( member @ ( sum_sum @ t @ dtree ) @ ( sum_Inr @ dtree @ t @ A2 ) @ ( cont @ Tr22 ) )
=> ~ ( gram_L1283001940subtr2 @ A1 @ Tr22 @ A3 ) ) ) ) ) ).
% subtr2.cases
thf(fact_9_subtr2_Osimps,axiom,
( gram_L1283001940subtr2
= ( ^ [A12: set @ n,A22: dtree,A32: dtree] :
( ? [Tr4: dtree,Ns2: set @ n] :
( ( A12 = Ns2 )
& ( A22 = Tr4 )
& ( A32 = Tr4 )
& ( member @ n @ ( root @ Tr4 ) @ Ns2 ) )
| ? [Tr12: dtree,Ns2: set @ n,Tr23: dtree,Tr32: dtree] :
( ( A12 = Ns2 )
& ( A22 = Tr12 )
& ( A32 = Tr32 )
& ( member @ n @ ( root @ Tr12 ) @ Ns2 )
& ( member @ ( sum_sum @ t @ dtree ) @ ( sum_Inr @ dtree @ t @ Tr12 ) @ ( cont @ Tr23 ) )
& ( gram_L1283001940subtr2 @ Ns2 @ Tr23 @ Tr32 ) ) ) ) ) ).
% subtr2.simps
thf(fact_10_subtr2__StepR,axiom,
! [Tr3: dtree,Ns: set @ n,Tr2: dtree,Tr1: dtree] :
( ( member @ n @ ( root @ Tr3 ) @ Ns )
=> ( ( member @ ( sum_sum @ t @ dtree ) @ ( sum_Inr @ dtree @ t @ Tr2 ) @ ( cont @ Tr3 ) )
=> ( ( gram_L1283001940subtr2 @ Ns @ Tr1 @ Tr2 )
=> ( gram_L1283001940subtr2 @ Ns @ Tr1 @ Tr3 ) ) ) ) ).
% subtr2_StepR
thf(fact_11_subtr2__trans,axiom,
! [Ns: set @ n,Tr1: dtree,Tr2: dtree,Tr3: dtree] :
( ( gram_L1283001940subtr2 @ Ns @ Tr1 @ Tr2 )
=> ( ( gram_L1283001940subtr2 @ Ns @ Tr2 @ Tr3 )
=> ( gram_L1283001940subtr2 @ Ns @ Tr1 @ Tr3 ) ) ) ).
% subtr2_trans
thf(fact_12_subtr_Oinducts,axiom,
! [X1: set @ n,X2: dtree,X3: dtree,P: ( set @ n ) > dtree > dtree > $o] :
( ( gram_L716654942_subtr @ X1 @ X2 @ X3 )
=> ( ! [Tr5: dtree,Ns3: set @ n] :
( ( member @ n @ ( root @ Tr5 ) @ Ns3 )
=> ( P @ Ns3 @ Tr5 @ Tr5 ) )
=> ( ! [Tr33: dtree,Ns3: set @ n,Tr13: dtree,Tr22: dtree] :
( ( member @ n @ ( root @ Tr33 ) @ Ns3 )
=> ( ( gram_L716654942_subtr @ Ns3 @ Tr13 @ Tr22 )
=> ( ( P @ Ns3 @ Tr13 @ Tr22 )
=> ( ( member @ ( sum_sum @ t @ dtree ) @ ( sum_Inr @ dtree @ t @ Tr22 ) @ ( cont @ Tr33 ) )
=> ( P @ Ns3 @ Tr13 @ Tr33 ) ) ) ) )
=> ( P @ X1 @ X2 @ X3 ) ) ) ) ).
% subtr.inducts
thf(fact_13_subtr2_Oinducts,axiom,
! [X1: set @ n,X2: dtree,X3: dtree,P: ( set @ n ) > dtree > dtree > $o] :
( ( gram_L1283001940subtr2 @ X1 @ X2 @ X3 )
=> ( ! [Tr5: dtree,Ns3: set @ n] :
( ( member @ n @ ( root @ Tr5 ) @ Ns3 )
=> ( P @ Ns3 @ Tr5 @ Tr5 ) )
=> ( ! [Tr13: dtree,Ns3: set @ n,Tr22: dtree,Tr33: dtree] :
( ( member @ n @ ( root @ Tr13 ) @ Ns3 )
=> ( ( member @ ( sum_sum @ t @ dtree ) @ ( sum_Inr @ dtree @ t @ Tr13 ) @ ( cont @ Tr22 ) )
=> ( ( gram_L1283001940subtr2 @ Ns3 @ Tr22 @ Tr33 )
=> ( ( P @ Ns3 @ Tr22 @ Tr33 )
=> ( P @ Ns3 @ Tr13 @ Tr33 ) ) ) ) )
=> ( P @ X1 @ X2 @ X3 ) ) ) ) ).
% subtr2.inducts
thf(fact_14_subtr__rootL__in,axiom,
! [Ns: set @ n,Tr1: dtree,Tr2: dtree] :
( ( gram_L716654942_subtr @ Ns @ Tr1 @ Tr2 )
=> ( member @ n @ ( root @ Tr1 ) @ Ns ) ) ).
% subtr_rootL_in
thf(fact_15_subtr__rootR__in,axiom,
! [Ns: set @ n,Tr1: dtree,Tr2: dtree] :
( ( gram_L716654942_subtr @ Ns @ Tr1 @ Tr2 )
=> ( member @ n @ ( root @ Tr2 ) @ Ns ) ) ).
% subtr_rootR_in
thf(fact_16_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_17_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_18_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_19_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_20_dtree__cong,axiom,
! [Tr: dtree,Tr6: dtree] :
( ( ( root @ Tr )
= ( root @ Tr6 ) )
=> ( ( ( cont @ Tr )
= ( cont @ Tr6 ) )
=> ( Tr = Tr6 ) ) ) ).
% dtree_cong
thf(fact_21_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_22_inItr_Ocases,axiom,
! [A1: set @ n,A2: dtree,A3: n] :
( ( gram_L830233218_inItr @ A1 @ A2 @ A3 )
=> ( ( ( A3
= ( root @ A2 ) )
=> ~ ( member @ n @ ( root @ A2 ) @ A1 ) )
=> ~ ( ( member @ n @ ( root @ A2 ) @ A1 )
=> ! [Tr13: dtree] :
( ( member @ ( sum_sum @ t @ dtree ) @ ( sum_Inr @ dtree @ t @ Tr13 ) @ ( cont @ A2 ) )
=> ~ ( gram_L830233218_inItr @ A1 @ Tr13 @ A3 ) ) ) ) ) ).
% inItr.cases
thf(fact_23_inItr_Osimps,axiom,
( gram_L830233218_inItr
= ( ^ [A12: set @ n,A22: dtree,A32: n] :
( ? [Tr4: dtree,Ns2: set @ n] :
( ( A12 = Ns2 )
& ( A22 = Tr4 )
& ( A32
= ( root @ Tr4 ) )
& ( member @ n @ ( root @ Tr4 ) @ Ns2 ) )
| ? [Tr4: dtree,Ns2: set @ n,Tr12: dtree,N2: n] :
( ( A12 = Ns2 )
& ( A22 = Tr4 )
& ( A32 = N2 )
& ( member @ n @ ( root @ Tr4 ) @ Ns2 )
& ( member @ ( sum_sum @ t @ dtree ) @ ( sum_Inr @ dtree @ t @ Tr12 ) @ ( cont @ Tr4 ) )
& ( gram_L830233218_inItr @ Ns2 @ Tr12 @ N2 ) ) ) ) ) ).
% inItr.simps
thf(fact_24_inItr_Oinducts,axiom,
! [X1: set @ n,X2: dtree,X3: n,P: ( set @ n ) > dtree > n > $o] :
( ( gram_L830233218_inItr @ X1 @ X2 @ X3 )
=> ( ! [Tr5: dtree,Ns3: set @ n] :
( ( member @ n @ ( root @ Tr5 ) @ Ns3 )
=> ( P @ Ns3 @ Tr5 @ ( root @ Tr5 ) ) )
=> ( ! [Tr5: dtree,Ns3: set @ n,Tr13: dtree,N3: n] :
( ( member @ n @ ( root @ Tr5 ) @ Ns3 )
=> ( ( member @ ( sum_sum @ t @ dtree ) @ ( sum_Inr @ dtree @ t @ Tr13 ) @ ( cont @ Tr5 ) )
=> ( ( gram_L830233218_inItr @ Ns3 @ Tr13 @ N3 )
=> ( ( P @ Ns3 @ Tr13 @ N3 )
=> ( P @ Ns3 @ Tr5 @ N3 ) ) ) ) )
=> ( P @ X1 @ X2 @ X3 ) ) ) ) ).
% inItr.inducts
thf(fact_25_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_26_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_27_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_28_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_29_Node__root__cont,axiom,
! [Tr: dtree] :
( ( node @ ( root @ Tr ) @ ( cont @ Tr ) )
= Tr ) ).
% Node_root_cont
thf(fact_30_root__Node,axiom,
! [N: n,As: set @ ( sum_sum @ t @ dtree )] :
( ( root @ ( node @ N @ As ) )
= N ) ).
% root_Node
thf(fact_31_inFr__inFr2,axiom,
gram_L1333338417e_inFr = gram_L805317441_inFr2 ).
% inFr_inFr2
thf(fact_32_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_33_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_34_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_35_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_36_inItr_OBase,axiom,
! [Tr: dtree,Ns: set @ n] :
( ( member @ n @ ( root @ Tr ) @ Ns )
=> ( gram_L830233218_inItr @ Ns @ Tr @ ( root @ Tr ) ) ) ).
% inItr.Base
thf(fact_37_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_38_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_39_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_40_inFr_Oinducts,axiom,
! [X1: set @ n,X2: dtree,X3: t,P: ( set @ n ) > dtree > t > $o] :
( ( gram_L1333338417e_inFr @ X1 @ X2 @ X3 )
=> ( ! [Tr5: dtree,Ns3: set @ n,T3: t] :
( ( member @ n @ ( root @ Tr5 ) @ Ns3 )
=> ( ( member @ ( sum_sum @ t @ dtree ) @ ( sum_Inl @ t @ dtree @ T3 ) @ ( cont @ Tr5 ) )
=> ( P @ Ns3 @ Tr5 @ T3 ) ) )
=> ( ! [Tr5: dtree,Ns3: set @ n,Tr13: dtree,T3: t] :
( ( member @ n @ ( root @ Tr5 ) @ Ns3 )
=> ( ( member @ ( sum_sum @ t @ dtree ) @ ( sum_Inr @ dtree @ t @ Tr13 ) @ ( cont @ Tr5 ) )
=> ( ( gram_L1333338417e_inFr @ Ns3 @ Tr13 @ T3 )
=> ( ( P @ Ns3 @ Tr13 @ T3 )
=> ( P @ Ns3 @ Tr5 @ T3 ) ) ) ) )
=> ( P @ X1 @ X2 @ X3 ) ) ) ) ).
% inFr.inducts
thf(fact_41_inFr_Osimps,axiom,
( gram_L1333338417e_inFr
= ( ^ [A12: set @ n,A22: dtree,A32: t] :
( ? [Tr4: dtree,Ns2: set @ n,T4: t] :
( ( A12 = Ns2 )
& ( A22 = Tr4 )
& ( A32 = T4 )
& ( member @ n @ ( root @ Tr4 ) @ Ns2 )
& ( member @ ( sum_sum @ t @ dtree ) @ ( sum_Inl @ t @ dtree @ T4 ) @ ( cont @ Tr4 ) ) )
| ? [Tr4: dtree,Ns2: set @ n,Tr12: dtree,T4: t] :
( ( A12 = Ns2 )
& ( A22 = Tr4 )
& ( A32 = T4 )
& ( member @ n @ ( root @ Tr4 ) @ Ns2 )
& ( member @ ( sum_sum @ t @ dtree ) @ ( sum_Inr @ dtree @ t @ Tr12 ) @ ( cont @ Tr4 ) )
& ( gram_L1333338417e_inFr @ Ns2 @ Tr12 @ T4 ) ) ) ) ) ).
% inFr.simps
thf(fact_42_inFr_Ocases,axiom,
! [A1: set @ n,A2: dtree,A3: t] :
( ( gram_L1333338417e_inFr @ A1 @ A2 @ A3 )
=> ( ( ( member @ n @ ( root @ A2 ) @ A1 )
=> ~ ( member @ ( sum_sum @ t @ dtree ) @ ( sum_Inl @ t @ dtree @ A3 ) @ ( cont @ A2 ) ) )
=> ~ ( ( member @ n @ ( root @ A2 ) @ A1 )
=> ! [Tr13: dtree] :
( ( member @ ( sum_sum @ t @ dtree ) @ ( sum_Inr @ dtree @ t @ Tr13 ) @ ( cont @ A2 ) )
=> ~ ( gram_L1333338417e_inFr @ A1 @ Tr13 @ A3 ) ) ) ) ) ).
% inFr.cases
thf(fact_43_Sumr_Osimps,axiom,
! [A: $tType,C: $tType,B: $tType,F: B > C,X: B] :
( ( sum_Sumr @ B @ C @ A @ F @ ( sum_Inr @ B @ A @ X ) )
= ( F @ X ) ) ).
% Sumr.simps
thf(fact_44_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_45_mem__Collect__eq,axiom,
! [A: $tType,A4: A,P: A > $o] :
( ( member @ A @ A4 @ ( collect @ A @ P ) )
= ( P @ A4 ) ) ).
% mem_Collect_eq
thf(fact_46_Collect__mem__eq,axiom,
! [A: $tType,A5: set @ A] :
( ( collect @ A
@ ^ [X4: A] : ( member @ A @ X4 @ A5 ) )
= A5 ) ).
% 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_corec_I1_J,axiom,
! [A: $tType,Rt: A > n,Ct: A > ( set @ ( sum_sum @ t @ ( sum_sum @ dtree @ A ) ) ),B2: A] :
( ( root @ ( corec @ A @ Rt @ Ct @ B2 ) )
= ( Rt @ B2 ) ) ).
% corec(1)
thf(fact_50_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_51_old_Osum_Oinject_I1_J,axiom,
! [B: $tType,A: $tType,A4: A,A6: A] :
( ( ( sum_Inl @ A @ B @ A4 )
= ( sum_Inl @ A @ B @ A6 ) )
= ( A4 = A6 ) ) ).
% old.sum.inject(1)
thf(fact_52_old_Osum_Osimps_I7_J,axiom,
! [B: $tType,T: $tType,A: $tType,F1: A > T,F2: B > T,A4: A] :
( ( sum_rec_sum @ A @ T @ B @ F1 @ F2 @ ( sum_Inl @ A @ B @ A4 ) )
= ( F1 @ A4 ) ) ).
% old.sum.simps(7)
thf(fact_53_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_54_Sumr__inject,axiom,
! [A: $tType,C: $tType,B: $tType,F: B > C,G: B > C] :
( ( ( sum_Sumr @ B @ C @ A @ F )
= ( sum_Sumr @ B @ C @ A @ G ) )
=> ( F = G ) ) ).
% Sumr_inject
thf(fact_55_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_56_old_Osum_Odistinct_I2_J,axiom,
! [B4: $tType,A7: $tType,B5: B4,A8: A7] :
( ( sum_Inr @ B4 @ A7 @ B5 )
!= ( sum_Inl @ A7 @ B4 @ A8 ) ) ).
% old.sum.distinct(2)
thf(fact_57_old_Osum_Odistinct_I1_J,axiom,
! [A: $tType,B: $tType,A4: A,B3: B] :
( ( sum_Inl @ A @ B @ A4 )
!= ( sum_Inr @ B @ A @ B3 ) ) ).
% old.sum.distinct(1)
thf(fact_58_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_59_Inr__not__Inl,axiom,
! [B: $tType,A: $tType,B2: B,A4: A] :
( ( sum_Inr @ B @ A @ B2 )
!= ( sum_Inl @ A @ B @ A4 ) ) ).
% Inr_not_Inl
thf(fact_60_split__sum__ex,axiom,
! [B: $tType,A: $tType] :
( ( ^ [P2: ( sum_sum @ A @ B ) > $o] :
? [X6: sum_sum @ A @ B] : ( P2 @ X6 ) )
= ( ^ [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_61_split__sum__all,axiom,
! [B: $tType,A: $tType] :
( ( ^ [P2: ( sum_sum @ A @ B ) > $o] :
! [X6: sum_sum @ A @ B] : ( P2 @ X6 ) )
= ( ^ [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_62_old_Osum_Oexhaust,axiom,
! [A: $tType,B: $tType,Y: sum_sum @ A @ B] :
( ! [A9: A] :
( Y
!= ( sum_Inl @ A @ B @ A9 ) )
=> ~ ! [B6: B] :
( Y
!= ( sum_Inr @ B @ A @ B6 ) ) ) ).
% old.sum.exhaust
thf(fact_63_old_Osum_Oinducts,axiom,
! [B: $tType,A: $tType,P: ( sum_sum @ A @ B ) > $o,Sum: sum_sum @ A @ B] :
( ! [A9: A] : ( P @ ( sum_Inl @ A @ B @ A9 ) )
=> ( ! [B6: B] : ( P @ ( sum_Inr @ B @ A @ B6 ) )
=> ( P @ Sum ) ) ) ).
% old.sum.inducts
thf(fact_64_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_65_inFr2_Ocases,axiom,
! [A1: set @ n,A2: dtree,A3: t] :
( ( gram_L805317441_inFr2 @ A1 @ A2 @ A3 )
=> ( ( ( member @ n @ ( root @ A2 ) @ A1 )
=> ~ ( member @ ( sum_sum @ t @ dtree ) @ ( sum_Inl @ t @ dtree @ A3 ) @ ( cont @ A2 ) ) )
=> ~ ! [Tr13: dtree,Tr5: dtree,Ns12: set @ n] :
( ( A1
= ( insert @ n @ ( root @ Tr5 ) @ Ns12 ) )
=> ( ( A2 = Tr5 )
=> ( ( member @ ( sum_sum @ t @ dtree ) @ ( sum_Inr @ dtree @ t @ Tr13 ) @ ( cont @ Tr5 ) )
=> ~ ( gram_L805317441_inFr2 @ Ns12 @ Tr13 @ A3 ) ) ) ) ) ) ).
% inFr2.cases
thf(fact_66_inFr2_Osimps,axiom,
( gram_L805317441_inFr2
= ( ^ [A12: set @ n,A22: dtree,A32: t] :
( ? [Tr4: dtree,Ns2: set @ n,T4: t] :
( ( A12 = Ns2 )
& ( A22 = Tr4 )
& ( A32 = T4 )
& ( member @ n @ ( root @ Tr4 ) @ Ns2 )
& ( member @ ( sum_sum @ t @ dtree ) @ ( sum_Inl @ t @ dtree @ T4 ) @ ( cont @ Tr4 ) ) )
| ? [Tr12: dtree,Tr4: dtree,Ns13: set @ n,T4: t] :
( ( A12
= ( insert @ n @ ( root @ Tr4 ) @ Ns13 ) )
& ( A22 = Tr4 )
& ( A32 = T4 )
& ( member @ ( sum_sum @ t @ dtree ) @ ( sum_Inr @ dtree @ t @ Tr12 ) @ ( cont @ Tr4 ) )
& ( gram_L805317441_inFr2 @ Ns13 @ Tr12 @ T4 ) ) ) ) ) ).
% inFr2.simps
thf(fact_67_inFr2_Oinducts,axiom,
! [X1: set @ n,X2: dtree,X3: t,P: ( set @ n ) > dtree > t > $o] :
( ( gram_L805317441_inFr2 @ X1 @ X2 @ X3 )
=> ( ! [Tr5: dtree,Ns3: set @ n,T3: t] :
( ( member @ n @ ( root @ Tr5 ) @ Ns3 )
=> ( ( member @ ( sum_sum @ t @ dtree ) @ ( sum_Inl @ t @ dtree @ T3 ) @ ( cont @ Tr5 ) )
=> ( P @ Ns3 @ Tr5 @ T3 ) ) )
=> ( ! [Tr13: dtree,Tr5: dtree,Ns12: set @ n,T3: t] :
( ( member @ ( sum_sum @ t @ dtree ) @ ( sum_Inr @ dtree @ t @ Tr13 ) @ ( cont @ Tr5 ) )
=> ( ( gram_L805317441_inFr2 @ Ns12 @ Tr13 @ T3 )
=> ( ( P @ Ns12 @ Tr13 @ T3 )
=> ( P @ ( insert @ n @ ( root @ Tr5 ) @ Ns12 ) @ Tr5 @ T3 ) ) ) )
=> ( P @ X1 @ X2 @ X3 ) ) ) ) ).
% inFr2.inducts
thf(fact_68_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_69_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_70_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_71_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_72_insert__absorb2,axiom,
! [A: $tType,X: A,A5: set @ A] :
( ( insert @ A @ X @ ( insert @ A @ X @ A5 ) )
= ( insert @ A @ X @ A5 ) ) ).
% insert_absorb2
thf(fact_73_insert__iff,axiom,
! [A: $tType,A4: A,B2: A,A5: set @ A] :
( ( member @ A @ A4 @ ( insert @ A @ B2 @ A5 ) )
= ( ( A4 = B2 )
| ( member @ A @ A4 @ A5 ) ) ) ).
% insert_iff
thf(fact_74_insertCI,axiom,
! [A: $tType,A4: A,B7: set @ A,B2: A] :
( ( ~ ( member @ A @ A4 @ B7 )
=> ( A4 = B2 ) )
=> ( member @ A @ A4 @ ( insert @ A @ B2 @ B7 ) ) ) ).
% insertCI
thf(fact_75_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 )
=> ! [X7: sum_sum @ A @ C] :
( ( S
= ( F @ X7 ) )
=> P ) ) ) ).
% obj_sumE_f
thf(fact_76_Suml_Osimps,axiom,
! [B: $tType,C: $tType,A: $tType,F: A > C,X: A] :
( ( sum_Suml @ A @ C @ B @ F @ ( sum_Inl @ A @ B @ X ) )
= ( F @ X ) ) ).
% Suml.simps
thf(fact_77_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_78_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_79_Suml__inject,axiom,
! [B: $tType,C: $tType,A: $tType,F: A > C,G: A > C] :
( ( ( sum_Suml @ A @ C @ B @ F )
= ( sum_Suml @ A @ C @ B @ G ) )
=> ( F = G ) ) ).
% Suml_inject
thf(fact_80_finite__cont,axiom,
! [Tr: dtree] : ( finite_finite2 @ ( sum_sum @ t @ dtree ) @ ( cont @ Tr ) ) ).
% finite_cont
thf(fact_81_dtree__cases,axiom,
! [Tr: dtree] :
~ ! [N3: n,As3: set @ ( sum_sum @ t @ dtree )] :
( ( finite_finite2 @ ( sum_sum @ t @ dtree ) @ As3 )
=> ( Tr
!= ( node @ N3 @ As3 ) ) ) ).
% dtree_cases
thf(fact_82_insertE,axiom,
! [A: $tType,A4: A,B2: A,A5: set @ A] :
( ( member @ A @ A4 @ ( insert @ A @ B2 @ A5 ) )
=> ( ( A4 != B2 )
=> ( member @ A @ A4 @ A5 ) ) ) ).
% insertE
thf(fact_83_insertI1,axiom,
! [A: $tType,A4: A,B7: set @ A] : ( member @ A @ A4 @ ( insert @ A @ A4 @ B7 ) ) ).
% insertI1
thf(fact_84_insertI2,axiom,
! [A: $tType,A4: A,B7: set @ A,B2: A] :
( ( member @ A @ A4 @ B7 )
=> ( member @ A @ A4 @ ( insert @ A @ B2 @ B7 ) ) ) ).
% insertI2
thf(fact_85_Set_Oset__insert,axiom,
! [A: $tType,X: A,A5: set @ A] :
( ( member @ A @ X @ A5 )
=> ~ ! [B8: set @ A] :
( ( A5
= ( insert @ A @ X @ B8 ) )
=> ( member @ A @ X @ B8 ) ) ) ).
% Set.set_insert
thf(fact_86_insert__ident,axiom,
! [A: $tType,X: A,A5: set @ A,B7: set @ A] :
( ~ ( member @ A @ X @ A5 )
=> ( ~ ( member @ A @ X @ B7 )
=> ( ( ( insert @ A @ X @ A5 )
= ( insert @ A @ X @ B7 ) )
= ( A5 = B7 ) ) ) ) ).
% insert_ident
thf(fact_87_insert__absorb,axiom,
! [A: $tType,A4: A,A5: set @ A] :
( ( member @ A @ A4 @ A5 )
=> ( ( insert @ A @ A4 @ A5 )
= A5 ) ) ).
% insert_absorb
thf(fact_88_insert__eq__iff,axiom,
! [A: $tType,A4: A,A5: set @ A,B2: A,B7: set @ A] :
( ~ ( member @ A @ A4 @ A5 )
=> ( ~ ( member @ A @ B2 @ B7 )
=> ( ( ( insert @ A @ A4 @ A5 )
= ( insert @ A @ B2 @ B7 ) )
= ( ( ( A4 = B2 )
=> ( A5 = B7 ) )
& ( ( A4 != B2 )
=> ? [C2: set @ A] :
( ( A5
= ( insert @ A @ B2 @ C2 ) )
& ~ ( member @ A @ B2 @ C2 )
& ( B7
= ( insert @ A @ A4 @ C2 ) )
& ~ ( member @ A @ A4 @ C2 ) ) ) ) ) ) ) ).
% insert_eq_iff
thf(fact_89_insert__commute,axiom,
! [A: $tType,X: A,Y: A,A5: set @ A] :
( ( insert @ A @ X @ ( insert @ A @ Y @ A5 ) )
= ( insert @ A @ Y @ ( insert @ A @ X @ A5 ) ) ) ).
% insert_commute
thf(fact_90_mk__disjoint__insert,axiom,
! [A: $tType,A4: A,A5: set @ A] :
( ( member @ A @ A4 @ A5 )
=> ? [B8: set @ A] :
( ( A5
= ( insert @ A @ A4 @ B8 ) )
& ~ ( member @ A @ A4 @ B8 ) ) ) ).
% mk_disjoint_insert
thf(fact_91_finite__insert,axiom,
! [A: $tType,A4: A,A5: set @ A] :
( ( finite_finite2 @ A @ ( insert @ A @ A4 @ A5 ) )
= ( finite_finite2 @ A @ A5 ) ) ).
% finite_insert
thf(fact_92_finite__code,axiom,
! [A: $tType] :
( ( finite_finite @ A @ ( type2 @ A ) )
=> ( ( finite_finite2 @ A )
= ( ^ [A10: set @ A] : $true ) ) ) ).
% finite_code
thf(fact_93_finite_OinsertI,axiom,
! [A: $tType,A5: set @ A,A4: A] :
( ( finite_finite2 @ A @ A5 )
=> ( finite_finite2 @ A @ ( insert @ A @ A4 @ A5 ) ) ) ).
% finite.insertI
thf(fact_94_finite__set__choice,axiom,
! [B: $tType,A: $tType,A5: set @ A,P: A > B > $o] :
( ( finite_finite2 @ A @ A5 )
=> ( ! [X5: A] :
( ( member @ A @ X5 @ A5 )
=> ? [X12: B] : ( P @ X5 @ X12 ) )
=> ? [F3: A > B] :
! [X7: A] :
( ( member @ A @ X7 @ A5 )
=> ( P @ X7 @ ( F3 @ X7 ) ) ) ) ) ).
% finite_set_choice
thf(fact_95_finite,axiom,
! [A: $tType] :
( ( finite_finite @ A @ ( type2 @ A ) )
=> ! [A5: set @ A] : ( finite_finite2 @ A @ A5 ) ) ).
% finite
thf(fact_96_case__sum__if,axiom,
! [B: $tType,A: $tType,C: $tType,P4: $o,F: B > A,G: C > A,X: B,Y: C] :
( ( P4
=> ( ( sum_case_sum @ B @ A @ C @ F @ G @ ( if @ ( sum_sum @ B @ C ) @ P4 @ ( sum_Inl @ B @ C @ X ) @ ( sum_Inr @ C @ B @ Y ) ) )
= ( F @ X ) ) )
& ( ~ P4
=> ( ( sum_case_sum @ B @ A @ C @ F @ G @ ( if @ ( sum_sum @ B @ C ) @ P4 @ ( sum_Inl @ B @ C @ X ) @ ( sum_Inr @ C @ B @ Y ) ) )
= ( G @ Y ) ) ) ) ).
% case_sum_if
thf(fact_97_case__sum__step_I1_J,axiom,
! [D: $tType,A: $tType,C: $tType,B: $tType,F4: B > A,G2: C > A,G: D > A,P4: sum_sum @ B @ C] :
( ( sum_case_sum @ ( sum_sum @ B @ C ) @ A @ D @ ( sum_case_sum @ B @ A @ C @ F4 @ G2 ) @ G @ ( sum_Inl @ ( sum_sum @ B @ C ) @ D @ P4 ) )
= ( sum_case_sum @ B @ A @ C @ F4 @ G2 @ P4 ) ) ).
% case_sum_step(1)
thf(fact_98_case__sum__step_I2_J,axiom,
! [E: $tType,A: $tType,C: $tType,B: $tType,F: E > A,F4: B > A,G2: C > A,P4: sum_sum @ B @ C] :
( ( sum_case_sum @ E @ A @ ( sum_sum @ B @ C ) @ F @ ( sum_case_sum @ B @ A @ C @ F4 @ G2 ) @ ( sum_Inr @ ( sum_sum @ B @ C ) @ E @ P4 ) )
= ( sum_case_sum @ B @ A @ C @ F4 @ G2 @ P4 ) ) ).
% case_sum_step(2)
thf(fact_99_case__sum__inject,axiom,
! [A: $tType,C: $tType,B: $tType,F1: A > C,F2: B > C,G1: A > C,G22: B > C] :
( ( ( sum_case_sum @ A @ C @ B @ F1 @ F2 )
= ( sum_case_sum @ A @ C @ B @ G1 @ G22 ) )
=> ~ ( ( F1 = G1 )
=> ( F2 != G22 ) ) ) ).
% case_sum_inject
thf(fact_100_old_Osum_Osimps_I6_J,axiom,
! [A: $tType,C: $tType,B: $tType,F1: A > C,F2: B > C,X2: B] :
( ( sum_case_sum @ A @ C @ B @ F1 @ F2 @ ( sum_Inr @ B @ A @ X2 ) )
= ( F2 @ X2 ) ) ).
% old.sum.simps(6)
thf(fact_101_old_Osum_Osimps_I5_J,axiom,
! [B: $tType,C: $tType,A: $tType,F1: A > C,F2: B > C,X1: A] :
( ( sum_case_sum @ A @ C @ B @ F1 @ F2 @ ( sum_Inl @ A @ B @ X1 ) )
= ( F1 @ X1 ) ) ).
% old.sum.simps(5)
thf(fact_102_case__sum__expand__Inr,axiom,
! [B: $tType,C: $tType,A: $tType,F: ( sum_sum @ A @ C ) > B,G: A > B,X: sum_sum @ A @ C] :
( ( ( comp @ ( sum_sum @ A @ C ) @ B @ A @ F @ ( sum_Inl @ A @ C ) )
= G )
=> ( ( F @ X )
= ( sum_case_sum @ A @ B @ C @ G @ ( comp @ ( sum_sum @ A @ C ) @ B @ C @ F @ ( sum_Inr @ C @ A ) ) @ X ) ) ) ).
% case_sum_expand_Inr
thf(fact_103_case__sum__expand__Inr_H,axiom,
! [B: $tType,C: $tType,A: $tType,F: ( sum_sum @ A @ C ) > B,G: A > B,H: C > B] :
( ( ( comp @ ( sum_sum @ A @ C ) @ B @ A @ F @ ( sum_Inl @ A @ C ) )
= G )
=> ( ( H
= ( comp @ ( sum_sum @ A @ C ) @ B @ C @ F @ ( sum_Inr @ C @ A ) ) )
= ( ( sum_case_sum @ A @ B @ C @ G @ H )
= F ) ) ) ).
% case_sum_expand_Inr'
thf(fact_104_case__sum__expand__Inr__pointfree,axiom,
! [B: $tType,C: $tType,A: $tType,F: ( sum_sum @ A @ C ) > B,G: A > B] :
( ( ( comp @ ( sum_sum @ A @ C ) @ B @ A @ F @ ( sum_Inl @ A @ C ) )
= G )
=> ( ( sum_case_sum @ A @ B @ C @ G @ ( comp @ ( sum_sum @ A @ C ) @ B @ C @ F @ ( sum_Inr @ C @ A ) ) )
= F ) ) ).
% case_sum_expand_Inr_pointfree
thf(fact_105_finite__Plus__iff,axiom,
! [A: $tType,B: $tType,A5: set @ A,B7: set @ B] :
( ( finite_finite2 @ ( sum_sum @ A @ B ) @ ( sum_Plus @ A @ B @ A5 @ B7 ) )
= ( ( finite_finite2 @ A @ A5 )
& ( finite_finite2 @ B @ B7 ) ) ) ).
% finite_Plus_iff
thf(fact_106_finite__Diff__insert,axiom,
! [A: $tType,A5: set @ A,A4: A,B7: set @ A] :
( ( finite_finite2 @ A @ ( minus_minus @ ( set @ A ) @ A5 @ ( insert @ A @ A4 @ B7 ) ) )
= ( finite_finite2 @ A @ ( minus_minus @ ( set @ A ) @ A5 @ B7 ) ) ) ).
% finite_Diff_insert
thf(fact_107_Diff__idemp,axiom,
! [A: $tType,A5: set @ A,B7: set @ A] :
( ( minus_minus @ ( set @ A ) @ ( minus_minus @ ( set @ A ) @ A5 @ B7 ) @ B7 )
= ( minus_minus @ ( set @ A ) @ A5 @ B7 ) ) ).
% Diff_idemp
thf(fact_108_Diff__iff,axiom,
! [A: $tType,C3: A,A5: set @ A,B7: set @ A] :
( ( member @ A @ C3 @ ( minus_minus @ ( set @ A ) @ A5 @ B7 ) )
= ( ( member @ A @ C3 @ A5 )
& ~ ( member @ A @ C3 @ B7 ) ) ) ).
% Diff_iff
thf(fact_109_DiffI,axiom,
! [A: $tType,C3: A,A5: set @ A,B7: set @ A] :
( ( member @ A @ C3 @ A5 )
=> ( ~ ( member @ A @ C3 @ B7 )
=> ( member @ A @ C3 @ ( minus_minus @ ( set @ A ) @ A5 @ B7 ) ) ) ) ).
% DiffI
thf(fact_110_finite__Diff2,axiom,
! [A: $tType,B7: set @ A,A5: set @ A] :
( ( finite_finite2 @ A @ B7 )
=> ( ( finite_finite2 @ A @ ( minus_minus @ ( set @ A ) @ A5 @ B7 ) )
= ( finite_finite2 @ A @ A5 ) ) ) ).
% finite_Diff2
thf(fact_111_finite__Diff,axiom,
! [A: $tType,A5: set @ A,B7: set @ A] :
( ( finite_finite2 @ A @ A5 )
=> ( finite_finite2 @ A @ ( minus_minus @ ( set @ A ) @ A5 @ B7 ) ) ) ).
% finite_Diff
thf(fact_112_insert__Diff1,axiom,
! [A: $tType,X: A,B7: set @ A,A5: set @ A] :
( ( member @ A @ X @ B7 )
=> ( ( minus_minus @ ( set @ A ) @ ( insert @ A @ X @ A5 ) @ B7 )
= ( minus_minus @ ( set @ A ) @ A5 @ B7 ) ) ) ).
% insert_Diff1
thf(fact_113_Diff__insert0,axiom,
! [A: $tType,X: A,A5: set @ A,B7: set @ A] :
( ~ ( member @ A @ X @ A5 )
=> ( ( minus_minus @ ( set @ A ) @ A5 @ ( insert @ A @ X @ B7 ) )
= ( minus_minus @ ( set @ A ) @ A5 @ B7 ) ) ) ).
% Diff_insert0
thf(fact_114_InrI,axiom,
! [B: $tType,A: $tType,B2: A,B7: set @ A,A5: set @ B] :
( ( member @ A @ B2 @ B7 )
=> ( member @ ( sum_sum @ B @ A ) @ ( sum_Inr @ A @ B @ B2 ) @ ( sum_Plus @ B @ A @ A5 @ B7 ) ) ) ).
% InrI
thf(fact_115_InlI,axiom,
! [A: $tType,B: $tType,A4: A,A5: set @ A,B7: set @ B] :
( ( member @ A @ A4 @ A5 )
=> ( member @ ( sum_sum @ A @ B ) @ ( sum_Inl @ A @ B @ A4 ) @ ( sum_Plus @ A @ B @ A5 @ B7 ) ) ) ).
% InlI
thf(fact_116_o__case__sum,axiom,
! [A: $tType,C: $tType,D: $tType,B: $tType,H: D > C,F: A > D,G: B > D] :
( ( comp @ D @ C @ ( sum_sum @ A @ B ) @ H @ ( sum_case_sum @ A @ D @ B @ F @ G ) )
= ( sum_case_sum @ A @ C @ B @ ( comp @ D @ C @ A @ H @ F ) @ ( comp @ D @ C @ B @ H @ G ) ) ) ).
% o_case_sum
thf(fact_117_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_118_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_119_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_120_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_121_DiffD2,axiom,
! [A: $tType,C3: A,A5: set @ A,B7: set @ A] :
( ( member @ A @ C3 @ ( minus_minus @ ( set @ A ) @ A5 @ B7 ) )
=> ~ ( member @ A @ C3 @ B7 ) ) ).
% DiffD2
thf(fact_122_DiffD1,axiom,
! [A: $tType,C3: A,A5: set @ A,B7: set @ A] :
( ( member @ A @ C3 @ ( minus_minus @ ( set @ A ) @ A5 @ B7 ) )
=> ( member @ A @ C3 @ A5 ) ) ).
% DiffD1
thf(fact_123_DiffE,axiom,
! [A: $tType,C3: A,A5: set @ A,B7: set @ A] :
( ( member @ A @ C3 @ ( minus_minus @ ( set @ A ) @ A5 @ B7 ) )
=> ~ ( ( member @ A @ C3 @ A5 )
=> ( member @ A @ C3 @ B7 ) ) ) ).
% DiffE
thf(fact_124_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_125_insert__Diff__if,axiom,
! [A: $tType,X: A,B7: set @ A,A5: set @ A] :
( ( ( member @ A @ X @ B7 )
=> ( ( minus_minus @ ( set @ A ) @ ( insert @ A @ X @ A5 ) @ B7 )
= ( minus_minus @ ( set @ A ) @ A5 @ B7 ) ) )
& ( ~ ( member @ A @ X @ B7 )
=> ( ( minus_minus @ ( set @ A ) @ ( insert @ A @ X @ A5 ) @ B7 )
= ( insert @ A @ X @ ( minus_minus @ ( set @ A ) @ A5 @ B7 ) ) ) ) ) ).
% insert_Diff_if
thf(fact_126_PlusE,axiom,
! [A: $tType,B: $tType,U: sum_sum @ A @ B,A5: set @ A,B7: set @ B] :
( ( member @ ( sum_sum @ A @ B ) @ U @ ( sum_Plus @ A @ B @ A5 @ B7 ) )
=> ( ! [X5: A] :
( ( member @ A @ X5 @ A5 )
=> ( U
!= ( sum_Inl @ A @ B @ X5 ) ) )
=> ~ ! [Y3: B] :
( ( member @ B @ Y3 @ B7 )
=> ( U
!= ( sum_Inr @ B @ A @ Y3 ) ) ) ) ) ).
% PlusE
thf(fact_127_finite__PlusD_I2_J,axiom,
! [A: $tType,B: $tType,A5: set @ A,B7: set @ B] :
( ( finite_finite2 @ ( sum_sum @ A @ B ) @ ( sum_Plus @ A @ B @ A5 @ B7 ) )
=> ( finite_finite2 @ B @ B7 ) ) ).
% finite_PlusD(2)
thf(fact_128_finite__PlusD_I1_J,axiom,
! [B: $tType,A: $tType,A5: set @ A,B7: set @ B] :
( ( finite_finite2 @ ( sum_sum @ A @ B ) @ ( sum_Plus @ A @ B @ A5 @ B7 ) )
=> ( finite_finite2 @ A @ A5 ) ) ).
% finite_PlusD(1)
thf(fact_129_finite__Plus,axiom,
! [A: $tType,B: $tType,A5: set @ A,B7: set @ B] :
( ( finite_finite2 @ A @ A5 )
=> ( ( finite_finite2 @ B @ B7 )
=> ( finite_finite2 @ ( sum_sum @ A @ B ) @ ( sum_Plus @ A @ B @ A5 @ B7 ) ) ) ) ).
% finite_Plus
thf(fact_130_case__sum__o__inj_I1_J,axiom,
! [C: $tType,B: $tType,A: $tType,F: A > B,G: C > B] :
( ( comp @ ( sum_sum @ A @ C ) @ B @ A @ ( sum_case_sum @ A @ B @ C @ F @ G ) @ ( sum_Inl @ A @ C ) )
= F ) ).
% case_sum_o_inj(1)
thf(fact_131_case__sum__o__inj_I2_J,axiom,
! [A: $tType,B: $tType,C: $tType,F: A > B,G: C > B] :
( ( comp @ ( sum_sum @ A @ C ) @ B @ C @ ( sum_case_sum @ A @ B @ C @ F @ G ) @ ( sum_Inr @ C @ A ) )
= G ) ).
% case_sum_o_inj(2)
thf(fact_132_comp__apply,axiom,
! [C: $tType,A: $tType,B: $tType] :
( ( comp @ B @ A @ C )
= ( ^ [F5: B > A,G3: C > B,X4: C] : ( F5 @ ( G3 @ X4 ) ) ) ) ).
% comp_apply
thf(fact_133_minus__apply,axiom,
! [B: $tType,A: $tType] :
( ( minus @ B @ ( type2 @ B ) )
=> ( ( minus_minus @ ( A > B ) )
= ( ^ [A10: A > B,B9: A > B,X4: A] : ( minus_minus @ B @ ( A10 @ X4 ) @ ( B9 @ X4 ) ) ) ) ) ).
% minus_apply
thf(fact_134_comp__def,axiom,
! [A: $tType,C: $tType,B: $tType] :
( ( comp @ B @ C @ A )
= ( ^ [F5: B > C,G3: A > B,X4: A] : ( F5 @ ( G3 @ X4 ) ) ) ) ).
% comp_def
thf(fact_135_fun__diff__def,axiom,
! [B: $tType,A: $tType] :
( ( minus @ B @ ( type2 @ B ) )
=> ( ( minus_minus @ ( A > B ) )
= ( ^ [A10: A > B,B9: A > B,X4: A] : ( minus_minus @ B @ ( A10 @ X4 ) @ ( B9 @ X4 ) ) ) ) ) ).
% fun_diff_def
thf(fact_136_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_137_comp__eq__dest__lhs,axiom,
! [C: $tType,B: $tType,A: $tType,A4: C > B,B2: A > C,C3: A > B,V: A] :
( ( ( comp @ C @ B @ A @ A4 @ B2 )
= C3 )
=> ( ( A4 @ ( B2 @ V ) )
= ( C3 @ V ) ) ) ).
% comp_eq_dest_lhs
thf(fact_138_comp__eq__elim,axiom,
! [C: $tType,B: $tType,D: $tType,A: $tType,A4: C > B,B2: A > C,C3: D > B,D2: A > D] :
( ( ( comp @ C @ B @ A @ A4 @ B2 )
= ( comp @ D @ B @ A @ C3 @ D2 ) )
=> ! [V2: A] :
( ( A4 @ ( B2 @ V2 ) )
= ( C3 @ ( D2 @ V2 ) ) ) ) ).
% comp_eq_elim
thf(fact_139_comp__eq__dest,axiom,
! [C: $tType,B: $tType,D: $tType,A: $tType,A4: C > B,B2: A > C,C3: D > B,D2: A > D,V: A] :
( ( ( comp @ C @ B @ A @ A4 @ B2 )
= ( comp @ D @ B @ A @ C3 @ D2 ) )
=> ( ( A4 @ ( B2 @ V ) )
= ( C3 @ ( D2 @ V ) ) ) ) ).
% comp_eq_dest
thf(fact_140_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_141_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_142_fun_Omap__comp,axiom,
! [B: $tType,C: $tType,A: $tType,D: $tType,G: B > C,F: A > B,V: D > A] :
( ( comp @ B @ C @ D @ G @ ( comp @ A @ B @ D @ F @ V ) )
= ( comp @ A @ C @ D @ ( comp @ B @ C @ A @ G @ F ) @ V ) ) ).
% fun.map_comp
thf(fact_143_comp__cong,axiom,
! [C: $tType,B: $tType,D: $tType,A: $tType,E: $tType,F: B > A,G: C > B,X: C,F4: D > A,G2: E > D,X8: E] :
( ( ( F @ ( G @ X ) )
= ( F4 @ ( G2 @ X8 ) ) )
=> ( ( comp @ B @ A @ C @ F @ G @ X )
= ( comp @ D @ A @ E @ F4 @ G2 @ X8 ) ) ) ).
% comp_cong
thf(fact_144_infinite__remove,axiom,
! [A: $tType,S2: set @ A,A4: A] :
( ~ ( finite_finite2 @ A @ S2 )
=> ~ ( finite_finite2 @ A @ ( minus_minus @ ( set @ A ) @ S2 @ ( insert @ A @ A4 @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ).
% infinite_remove
thf(fact_145_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_146_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_147_all__not__in__conv,axiom,
! [A: $tType,A5: set @ A] :
( ( ! [X4: A] :
~ ( member @ A @ X4 @ A5 ) )
= ( A5
= ( bot_bot @ ( set @ A ) ) ) ) ).
% all_not_in_conv
thf(fact_148_empty__iff,axiom,
! [A: $tType,C3: A] :
~ ( member @ A @ C3 @ ( bot_bot @ ( set @ A ) ) ) ).
% empty_iff
thf(fact_149_singletonI,axiom,
! [A: $tType,A4: A] : ( member @ A @ A4 @ ( insert @ A @ A4 @ ( bot_bot @ ( set @ A ) ) ) ) ).
% singletonI
thf(fact_150_Diff__cancel,axiom,
! [A: $tType,A5: set @ A] :
( ( minus_minus @ ( set @ A ) @ A5 @ A5 )
= ( bot_bot @ ( set @ A ) ) ) ).
% Diff_cancel
thf(fact_151_empty__Diff,axiom,
! [A: $tType,A5: set @ A] :
( ( minus_minus @ ( set @ A ) @ ( bot_bot @ ( set @ A ) ) @ A5 )
= ( bot_bot @ ( set @ A ) ) ) ).
% empty_Diff
thf(fact_152_Diff__empty,axiom,
! [A: $tType,A5: set @ A] :
( ( minus_minus @ ( set @ A ) @ A5 @ ( bot_bot @ ( set @ A ) ) )
= A5 ) ).
% Diff_empty
thf(fact_153_Plus__eq__empty__conv,axiom,
! [A: $tType,B: $tType,A5: set @ A,B7: set @ B] :
( ( ( sum_Plus @ A @ B @ A5 @ B7 )
= ( bot_bot @ ( set @ ( sum_sum @ A @ B ) ) ) )
= ( ( A5
= ( bot_bot @ ( set @ A ) ) )
& ( B7
= ( bot_bot @ ( set @ B ) ) ) ) ) ).
% Plus_eq_empty_conv
thf(fact_154_insert__Diff__single,axiom,
! [A: $tType,A4: A,A5: set @ A] :
( ( insert @ A @ A4 @ ( minus_minus @ ( set @ A ) @ A5 @ ( insert @ A @ A4 @ ( bot_bot @ ( set @ A ) ) ) ) )
= ( insert @ A @ A4 @ A5 ) ) ).
% insert_Diff_single
thf(fact_155_finite_OemptyI,axiom,
! [A: $tType] : ( finite_finite2 @ A @ ( bot_bot @ ( set @ A ) ) ) ).
% finite.emptyI
thf(fact_156_infinite__imp__nonempty,axiom,
! [A: $tType,S2: set @ A] :
( ~ ( finite_finite2 @ A @ S2 )
=> ( S2
!= ( bot_bot @ ( set @ A ) ) ) ) ).
% infinite_imp_nonempty
thf(fact_157_singletonD,axiom,
! [A: $tType,B2: A,A4: A] :
( ( member @ A @ B2 @ ( insert @ A @ A4 @ ( bot_bot @ ( set @ A ) ) ) )
=> ( B2 = A4 ) ) ).
% singletonD
thf(fact_158_singleton__iff,axiom,
! [A: $tType,B2: A,A4: A] :
( ( member @ A @ B2 @ ( insert @ A @ A4 @ ( bot_bot @ ( set @ A ) ) ) )
= ( B2 = A4 ) ) ).
% singleton_iff
thf(fact_159_doubleton__eq__iff,axiom,
! [A: $tType,A4: A,B2: A,C3: A,D2: A] :
( ( ( insert @ A @ A4 @ ( insert @ A @ B2 @ ( bot_bot @ ( set @ A ) ) ) )
= ( insert @ A @ C3 @ ( insert @ A @ D2 @ ( bot_bot @ ( set @ A ) ) ) ) )
= ( ( ( A4 = C3 )
& ( B2 = D2 ) )
| ( ( A4 = D2 )
& ( B2 = C3 ) ) ) ) ).
% doubleton_eq_iff
thf(fact_160_insert__not__empty,axiom,
! [A: $tType,A4: A,A5: set @ A] :
( ( insert @ A @ A4 @ A5 )
!= ( bot_bot @ ( set @ A ) ) ) ).
% insert_not_empty
thf(fact_161_singleton__inject,axiom,
! [A: $tType,A4: A,B2: A] :
( ( ( insert @ A @ A4 @ ( bot_bot @ ( set @ A ) ) )
= ( insert @ A @ B2 @ ( bot_bot @ ( set @ A ) ) ) )
=> ( A4 = B2 ) ) ).
% singleton_inject
thf(fact_162_ex__in__conv,axiom,
! [A: $tType,A5: set @ A] :
( ( ? [X4: A] : ( member @ A @ X4 @ A5 ) )
= ( A5
!= ( bot_bot @ ( set @ A ) ) ) ) ).
% ex_in_conv
thf(fact_163_equals0I,axiom,
! [A: $tType,A5: set @ A] :
( ! [Y3: A] :
~ ( member @ A @ Y3 @ A5 )
=> ( A5
= ( bot_bot @ ( set @ A ) ) ) ) ).
% equals0I
thf(fact_164_equals0D,axiom,
! [A: $tType,A5: set @ A,A4: A] :
( ( A5
= ( bot_bot @ ( set @ A ) ) )
=> ~ ( member @ A @ A4 @ A5 ) ) ).
% equals0D
thf(fact_165_emptyE,axiom,
! [A: $tType,A4: A] :
~ ( member @ A @ A4 @ ( bot_bot @ ( set @ A ) ) ) ).
% emptyE
thf(fact_166_infinite__finite__induct,axiom,
! [A: $tType,P: ( set @ A ) > $o,A5: set @ A] :
( ! [A11: set @ A] :
( ~ ( finite_finite2 @ A @ A11 )
=> ( P @ A11 ) )
=> ( ( 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 @ A5 ) ) ) ) ).
% infinite_finite_induct
thf(fact_167_finite__ne__induct,axiom,
! [A: $tType,F7: set @ A,P: ( set @ A ) > $o] :
( ( finite_finite2 @ A @ F7 )
=> ( ( F7
!= ( 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 @ F7 ) ) ) ) ) ).
% finite_ne_induct
thf(fact_168_finite_Oinducts,axiom,
! [A: $tType,X: set @ A,P: ( set @ A ) > $o] :
( ( finite_finite2 @ A @ X )
=> ( ( P @ ( bot_bot @ ( set @ A ) ) )
=> ( ! [A11: set @ A,A9: A] :
( ( finite_finite2 @ A @ A11 )
=> ( ( P @ A11 )
=> ( P @ ( insert @ A @ A9 @ A11 ) ) ) )
=> ( P @ X ) ) ) ) ).
% finite.inducts
thf(fact_169_finite__induct,axiom,
! [A: $tType,F7: set @ A,P: ( set @ A ) > $o] :
( ( finite_finite2 @ A @ F7 )
=> ( ( 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 @ F7 ) ) ) ) ).
% finite_induct
thf(fact_170_finite_Osimps,axiom,
! [A: $tType] :
( ( finite_finite2 @ A )
= ( ^ [A13: set @ A] :
( ( A13
= ( bot_bot @ ( set @ A ) ) )
| ? [A10: set @ A,B10: A] :
( ( A13
= ( insert @ A @ B10 @ A10 ) )
& ( finite_finite2 @ A @ A10 ) ) ) ) ) ).
% finite.simps
thf(fact_171_finite_Ocases,axiom,
! [A: $tType,A4: set @ A] :
( ( finite_finite2 @ A @ A4 )
=> ( ( A4
!= ( bot_bot @ ( set @ A ) ) )
=> ~ ! [A11: set @ A] :
( ? [A9: A] :
( A4
= ( insert @ A @ A9 @ A11 ) )
=> ~ ( finite_finite2 @ A @ A11 ) ) ) ) ).
% finite.cases
thf(fact_172_Diff__insert,axiom,
! [A: $tType,A5: set @ A,A4: A,B7: set @ A] :
( ( minus_minus @ ( set @ A ) @ A5 @ ( insert @ A @ A4 @ B7 ) )
= ( minus_minus @ ( set @ A ) @ ( minus_minus @ ( set @ A ) @ A5 @ B7 ) @ ( insert @ A @ A4 @ ( bot_bot @ ( set @ A ) ) ) ) ) ).
% Diff_insert
thf(fact_173_insert__Diff,axiom,
! [A: $tType,A4: A,A5: set @ A] :
( ( member @ A @ A4 @ A5 )
=> ( ( insert @ A @ A4 @ ( minus_minus @ ( set @ A ) @ A5 @ ( insert @ A @ A4 @ ( bot_bot @ ( set @ A ) ) ) ) )
= A5 ) ) ).
% insert_Diff
thf(fact_174_Diff__insert2,axiom,
! [A: $tType,A5: set @ A,A4: A,B7: set @ A] :
( ( minus_minus @ ( set @ A ) @ A5 @ ( insert @ A @ A4 @ B7 ) )
= ( minus_minus @ ( set @ A ) @ ( minus_minus @ ( set @ A ) @ A5 @ ( insert @ A @ A4 @ ( bot_bot @ ( set @ A ) ) ) ) @ B7 ) ) ).
% Diff_insert2
thf(fact_175_Diff__insert__absorb,axiom,
! [A: $tType,X: A,A5: set @ A] :
( ~ ( member @ A @ X @ A5 )
=> ( ( minus_minus @ ( set @ A ) @ ( insert @ A @ X @ A5 ) @ ( insert @ A @ X @ ( bot_bot @ ( set @ A ) ) ) )
= A5 ) ) ).
% Diff_insert_absorb
thf(fact_176_finite__empty__induct,axiom,
! [A: $tType,A5: set @ A,P: ( set @ A ) > $o] :
( ( finite_finite2 @ A @ A5 )
=> ( ( P @ A5 )
=> ( ! [A9: A,A11: set @ A] :
( ( finite_finite2 @ A @ A11 )
=> ( ( member @ A @ A9 @ A11 )
=> ( ( P @ A11 )
=> ( P @ ( minus_minus @ ( set @ A ) @ A11 @ ( insert @ A @ A9 @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ) )
=> ( P @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ).
% finite_empty_induct
thf(fact_177_infinite__coinduct,axiom,
! [A: $tType,X9: ( set @ A ) > $o,A5: set @ A] :
( ( X9 @ A5 )
=> ( ! [A11: set @ A] :
( ( X9 @ A11 )
=> ? [X7: A] :
( ( member @ A @ X7 @ A11 )
& ( ( X9 @ ( minus_minus @ ( set @ A ) @ A11 @ ( insert @ A @ X7 @ ( bot_bot @ ( set @ A ) ) ) ) )
| ~ ( finite_finite2 @ A @ ( minus_minus @ ( set @ A ) @ A11 @ ( insert @ A @ X7 @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ) )
=> ~ ( finite_finite2 @ A @ A5 ) ) ) ).
% infinite_coinduct
thf(fact_178_is__singletonI,axiom,
! [A: $tType,X: A] : ( is_singleton @ A @ ( insert @ A @ X @ ( bot_bot @ ( set @ A ) ) ) ) ).
% is_singletonI
thf(fact_179_remove__def,axiom,
! [A: $tType] :
( ( remove @ A )
= ( ^ [X4: A,A10: set @ A] : ( minus_minus @ ( set @ A ) @ A10 @ ( insert @ A @ X4 @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ).
% remove_def
thf(fact_180_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_181_member__remove,axiom,
! [A: $tType,X: A,Y: A,A5: set @ A] :
( ( member @ A @ X @ ( remove @ A @ Y @ A5 ) )
= ( ( member @ A @ X @ A5 )
& ( X != Y ) ) ) ).
% member_remove
thf(fact_182_is__singletonI_H,axiom,
! [A: $tType,A5: set @ A] :
( ( A5
!= ( bot_bot @ ( set @ A ) ) )
=> ( ! [X5: A,Y3: A] :
( ( member @ A @ X5 @ A5 )
=> ( ( member @ A @ Y3 @ A5 )
=> ( X5 = Y3 ) ) )
=> ( is_singleton @ A @ A5 ) ) ) ).
% is_singletonI'
thf(fact_183_is__singleton__def,axiom,
! [A: $tType] :
( ( is_singleton @ A )
= ( ^ [A10: set @ A] :
? [X4: A] :
( A10
= ( insert @ A @ X4 @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ).
% is_singleton_def
thf(fact_184_is__singletonE,axiom,
! [A: $tType,A5: set @ A] :
( ( is_singleton @ A @ A5 )
=> ~ ! [X5: A] :
( A5
!= ( insert @ A @ X5 @ ( bot_bot @ ( set @ A ) ) ) ) ) ).
% is_singletonE
thf(fact_185_is__singleton__the__elem,axiom,
! [A: $tType] :
( ( is_singleton @ A )
= ( ^ [A10: set @ A] :
( A10
= ( insert @ A @ ( the_elem @ A @ A10 ) @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ).
% is_singleton_the_elem
thf(fact_186_the__elem__eq,axiom,
! [A: $tType,X: A] :
( ( the_elem @ A @ ( insert @ A @ X @ ( bot_bot @ ( set @ A ) ) ) )
= X ) ).
% the_elem_eq
thf(fact_187_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_188_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_189_setl_Oinducts,axiom,
! [B: $tType,A: $tType,X: A,S: sum_sum @ A @ B,P: A > $o] :
( ( member @ A @ X @ ( basic_setl @ A @ B @ S ) )
=> ( ! [X5: A] :
( ( S
= ( sum_Inl @ A @ B @ X5 ) )
=> ( P @ X5 ) )
=> ( P @ X ) ) ) ).
% setl.inducts
thf(fact_190_setl_Ointros,axiom,
! [B: $tType,A: $tType,S: sum_sum @ A @ B,X: A] :
( ( S
= ( sum_Inl @ A @ B @ X ) )
=> ( member @ A @ X @ ( basic_setl @ A @ B @ S ) ) ) ).
% setl.intros
thf(fact_191_setl_Osimps,axiom,
! [B: $tType,A: $tType,A4: A,S: sum_sum @ A @ B] :
( ( member @ A @ A4 @ ( basic_setl @ A @ B @ S ) )
= ( ? [X4: A] :
( ( A4 = X4 )
& ( S
= ( sum_Inl @ A @ B @ X4 ) ) ) ) ) ).
% setl.simps
thf(fact_192_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_193_sum__set__simps_I4_J,axiom,
! [E: $tType,A: $tType,X: A] :
( ( basic_setr @ E @ A @ ( sum_Inr @ A @ E @ X ) )
= ( insert @ A @ X @ ( bot_bot @ ( set @ A ) ) ) ) ).
% sum_set_simps(4)
thf(fact_194_Set_Ois__empty__def,axiom,
! [A: $tType] :
( ( is_empty @ A )
= ( ^ [A10: set @ A] :
( A10
= ( bot_bot @ ( set @ A ) ) ) ) ) ).
% Set.is_empty_def
thf(fact_195_sum__set__simps_I3_J,axiom,
! [A: $tType,D: $tType,X: A] :
( ( basic_setr @ A @ D @ ( sum_Inl @ A @ D @ X ) )
= ( bot_bot @ ( set @ D ) ) ) ).
% sum_set_simps(3)
thf(fact_196_setr_Oinducts,axiom,
! [A: $tType,B: $tType,X: B,S: sum_sum @ A @ B,P: B > $o] :
( ( member @ B @ X @ ( basic_setr @ A @ B @ S ) )
=> ( ! [X5: B] :
( ( S
= ( sum_Inr @ B @ A @ X5 ) )
=> ( P @ X5 ) )
=> ( P @ X ) ) ) ).
% setr.inducts
thf(fact_197_setr_Ointros,axiom,
! [B: $tType,A: $tType,S: sum_sum @ A @ B,X: B] :
( ( S
= ( sum_Inr @ B @ A @ X ) )
=> ( member @ B @ X @ ( basic_setr @ A @ B @ S ) ) ) ).
% setr.intros
thf(fact_198_setr_Osimps,axiom,
! [A: $tType,B: $tType,A4: B,S: sum_sum @ A @ B] :
( ( member @ B @ A4 @ ( basic_setr @ A @ B @ S ) )
= ( ? [X4: B] :
( ( A4 = X4 )
& ( S
= ( sum_Inr @ B @ A @ X4 ) ) ) ) ) ).
% setr.simps
thf(fact_199_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_200_remove__induct,axiom,
! [A: $tType,P: ( set @ A ) > $o,B7: set @ A] :
( ( P @ ( bot_bot @ ( set @ A ) ) )
=> ( ( ~ ( finite_finite2 @ A @ B7 )
=> ( P @ B7 ) )
=> ( ! [A11: set @ A] :
( ( finite_finite2 @ A @ A11 )
=> ( ( A11
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( ord_less_eq @ ( set @ A ) @ A11 @ B7 )
=> ( ! [X7: A] :
( ( member @ A @ X7 @ A11 )
=> ( P @ ( minus_minus @ ( set @ A ) @ A11 @ ( insert @ A @ X7 @ ( bot_bot @ ( set @ A ) ) ) ) ) )
=> ( P @ A11 ) ) ) ) )
=> ( P @ B7 ) ) ) ) ).
% remove_induct
thf(fact_201_finite__remove__induct,axiom,
! [A: $tType,B7: set @ A,P: ( set @ A ) > $o] :
( ( finite_finite2 @ A @ B7 )
=> ( ( P @ ( bot_bot @ ( set @ A ) ) )
=> ( ! [A11: set @ A] :
( ( finite_finite2 @ A @ A11 )
=> ( ( A11
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( ord_less_eq @ ( set @ A ) @ A11 @ B7 )
=> ( ! [X7: A] :
( ( member @ A @ X7 @ A11 )
=> ( P @ ( minus_minus @ ( set @ A ) @ A11 @ ( insert @ A @ X7 @ ( bot_bot @ ( set @ A ) ) ) ) ) )
=> ( P @ A11 ) ) ) ) )
=> ( P @ B7 ) ) ) ) ).
% finite_remove_induct
thf(fact_202_subset__antisym,axiom,
! [A: $tType,A5: set @ A,B7: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ A5 @ B7 )
=> ( ( ord_less_eq @ ( set @ A ) @ B7 @ A5 )
=> ( A5 = B7 ) ) ) ).
% subset_antisym
thf(fact_203_subsetI,axiom,
! [A: $tType,A5: set @ A,B7: set @ A] :
( ! [X5: A] :
( ( member @ A @ X5 @ A5 )
=> ( member @ A @ X5 @ B7 ) )
=> ( ord_less_eq @ ( set @ A ) @ A5 @ B7 ) ) ).
% subsetI
thf(fact_204_empty__subsetI,axiom,
! [A: $tType,A5: set @ A] : ( ord_less_eq @ ( set @ A ) @ ( bot_bot @ ( set @ A ) ) @ A5 ) ).
% empty_subsetI
thf(fact_205_subset__empty,axiom,
! [A: $tType,A5: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ A5 @ ( bot_bot @ ( set @ A ) ) )
= ( A5
= ( bot_bot @ ( set @ A ) ) ) ) ).
% subset_empty
thf(fact_206_insert__subset,axiom,
! [A: $tType,X: A,A5: set @ A,B7: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ ( insert @ A @ X @ A5 ) @ B7 )
= ( ( member @ A @ X @ B7 )
& ( ord_less_eq @ ( set @ A ) @ A5 @ B7 ) ) ) ).
% insert_subset
thf(fact_207_singleton__insert__inj__eq,axiom,
! [A: $tType,B2: A,A4: A,A5: set @ A] :
( ( ( insert @ A @ B2 @ ( bot_bot @ ( set @ A ) ) )
= ( insert @ A @ A4 @ A5 ) )
= ( ( A4 = B2 )
& ( ord_less_eq @ ( set @ A ) @ A5 @ ( insert @ A @ B2 @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ).
% singleton_insert_inj_eq
thf(fact_208_singleton__insert__inj__eq_H,axiom,
! [A: $tType,A4: A,A5: set @ A,B2: A] :
( ( ( insert @ A @ A4 @ A5 )
= ( insert @ A @ B2 @ ( bot_bot @ ( set @ A ) ) ) )
= ( ( A4 = B2 )
& ( ord_less_eq @ ( set @ A ) @ A5 @ ( insert @ A @ B2 @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ).
% singleton_insert_inj_eq'
thf(fact_209_Diff__eq__empty__iff,axiom,
! [A: $tType,A5: set @ A,B7: set @ A] :
( ( ( minus_minus @ ( set @ A ) @ A5 @ B7 )
= ( bot_bot @ ( set @ A ) ) )
= ( ord_less_eq @ ( set @ A ) @ A5 @ B7 ) ) ).
% Diff_eq_empty_iff
thf(fact_210_Diff__mono,axiom,
! [A: $tType,A5: set @ A,C4: set @ A,D3: set @ A,B7: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ A5 @ C4 )
=> ( ( ord_less_eq @ ( set @ A ) @ D3 @ B7 )
=> ( ord_less_eq @ ( set @ A ) @ ( minus_minus @ ( set @ A ) @ A5 @ B7 ) @ ( minus_minus @ ( set @ A ) @ C4 @ D3 ) ) ) ) ).
% Diff_mono
thf(fact_211_Diff__subset,axiom,
! [A: $tType,A5: set @ A,B7: set @ A] : ( ord_less_eq @ ( set @ A ) @ ( minus_minus @ ( set @ A ) @ A5 @ B7 ) @ A5 ) ).
% Diff_subset
thf(fact_212_double__diff,axiom,
! [A: $tType,A5: set @ A,B7: set @ A,C4: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ A5 @ B7 )
=> ( ( ord_less_eq @ ( set @ A ) @ B7 @ C4 )
=> ( ( minus_minus @ ( set @ A ) @ B7 @ ( minus_minus @ ( set @ A ) @ C4 @ A5 ) )
= A5 ) ) ) ).
% double_diff
thf(fact_213_subtr__mono,axiom,
! [Ns: set @ n,Tr1: dtree,Tr2: dtree,Ns4: set @ n] :
( ( gram_L716654942_subtr @ Ns @ Tr1 @ Tr2 )
=> ( ( ord_less_eq @ ( set @ n ) @ Ns @ Ns4 )
=> ( gram_L716654942_subtr @ Ns4 @ Tr1 @ Tr2 ) ) ) ).
% subtr_mono
thf(fact_214_subtr2__mono,axiom,
! [Ns: set @ n,Tr1: dtree,Tr2: dtree,Ns4: set @ n] :
( ( gram_L1283001940subtr2 @ Ns @ Tr1 @ Tr2 )
=> ( ( ord_less_eq @ ( set @ n ) @ Ns @ Ns4 )
=> ( gram_L1283001940subtr2 @ Ns4 @ Tr1 @ Tr2 ) ) ) ).
% subtr2_mono
thf(fact_215_inFr__mono,axiom,
! [Ns: set @ n,Tr: dtree,T2: t,Ns4: set @ n] :
( ( gram_L1333338417e_inFr @ Ns @ Tr @ T2 )
=> ( ( ord_less_eq @ ( set @ n ) @ Ns @ Ns4 )
=> ( gram_L1333338417e_inFr @ Ns4 @ Tr @ T2 ) ) ) ).
% inFr_mono
thf(fact_216_inFr2__mono,axiom,
! [Ns: set @ n,Tr: dtree,T2: t,Ns4: set @ n] :
( ( gram_L805317441_inFr2 @ Ns @ Tr @ T2 )
=> ( ( ord_less_eq @ ( set @ n ) @ Ns @ Ns4 )
=> ( gram_L805317441_inFr2 @ Ns4 @ Tr @ T2 ) ) ) ).
% inFr2_mono
thf(fact_217_Collect__mono__iff,axiom,
! [A: $tType,P: A > $o,Q: A > $o] :
( ( ord_less_eq @ ( set @ A ) @ ( collect @ A @ P ) @ ( collect @ A @ Q ) )
= ( ! [X4: A] :
( ( P @ X4 )
=> ( Q @ X4 ) ) ) ) ).
% Collect_mono_iff
thf(fact_218_contra__subsetD,axiom,
! [A: $tType,A5: set @ A,B7: set @ A,C3: A] :
( ( ord_less_eq @ ( set @ A ) @ A5 @ B7 )
=> ( ~ ( member @ A @ C3 @ B7 )
=> ~ ( member @ A @ C3 @ A5 ) ) ) ).
% contra_subsetD
thf(fact_219_set__eq__subset,axiom,
! [A: $tType] :
( ( ^ [Y4: set @ A,Z: set @ A] : Y4 = Z )
= ( ^ [A10: set @ A,B9: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ A10 @ B9 )
& ( ord_less_eq @ ( set @ A ) @ B9 @ A10 ) ) ) ) ).
% set_eq_subset
thf(fact_220_subset__trans,axiom,
! [A: $tType,A5: set @ A,B7: set @ A,C4: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ A5 @ B7 )
=> ( ( ord_less_eq @ ( set @ A ) @ B7 @ C4 )
=> ( ord_less_eq @ ( set @ A ) @ A5 @ C4 ) ) ) ).
% subset_trans
thf(fact_221_Collect__mono,axiom,
! [A: $tType,P: A > $o,Q: A > $o] :
( ! [X5: A] :
( ( P @ X5 )
=> ( Q @ X5 ) )
=> ( ord_less_eq @ ( set @ A ) @ ( collect @ A @ P ) @ ( collect @ A @ Q ) ) ) ).
% Collect_mono
thf(fact_222_subset__refl,axiom,
! [A: $tType,A5: set @ A] : ( ord_less_eq @ ( set @ A ) @ A5 @ A5 ) ).
% subset_refl
thf(fact_223_rev__subsetD,axiom,
! [A: $tType,C3: A,A5: set @ A,B7: set @ A] :
( ( member @ A @ C3 @ A5 )
=> ( ( ord_less_eq @ ( set @ A ) @ A5 @ B7 )
=> ( member @ A @ C3 @ B7 ) ) ) ).
% rev_subsetD
thf(fact_224_subset__iff,axiom,
! [A: $tType] :
( ( ord_less_eq @ ( set @ A ) )
= ( ^ [A10: set @ A,B9: set @ A] :
! [T4: A] :
( ( member @ A @ T4 @ A10 )
=> ( member @ A @ T4 @ B9 ) ) ) ) ).
% subset_iff
thf(fact_225_set__rev__mp,axiom,
! [A: $tType,X: A,A5: set @ A,B7: set @ A] :
( ( member @ A @ X @ A5 )
=> ( ( ord_less_eq @ ( set @ A ) @ A5 @ B7 )
=> ( member @ A @ X @ B7 ) ) ) ).
% set_rev_mp
thf(fact_226_equalityD2,axiom,
! [A: $tType,A5: set @ A,B7: set @ A] :
( ( A5 = B7 )
=> ( ord_less_eq @ ( set @ A ) @ B7 @ A5 ) ) ).
% equalityD2
thf(fact_227_equalityD1,axiom,
! [A: $tType,A5: set @ A,B7: set @ A] :
( ( A5 = B7 )
=> ( ord_less_eq @ ( set @ A ) @ A5 @ B7 ) ) ).
% equalityD1
thf(fact_228_subset__eq,axiom,
! [A: $tType] :
( ( ord_less_eq @ ( set @ A ) )
= ( ^ [A10: set @ A,B9: set @ A] :
! [X4: A] :
( ( member @ A @ X4 @ A10 )
=> ( member @ A @ X4 @ B9 ) ) ) ) ).
% subset_eq
thf(fact_229_equalityE,axiom,
! [A: $tType,A5: set @ A,B7: set @ A] :
( ( A5 = B7 )
=> ~ ( ( ord_less_eq @ ( set @ A ) @ A5 @ B7 )
=> ~ ( ord_less_eq @ ( set @ A ) @ B7 @ A5 ) ) ) ).
% equalityE
thf(fact_230_subsetCE,axiom,
! [A: $tType,A5: set @ A,B7: set @ A,C3: A] :
( ( ord_less_eq @ ( set @ A ) @ A5 @ B7 )
=> ( ( member @ A @ C3 @ A5 )
=> ( member @ A @ C3 @ B7 ) ) ) ).
% subsetCE
thf(fact_231_subsetD,axiom,
! [A: $tType,A5: set @ A,B7: set @ A,C3: A] :
( ( ord_less_eq @ ( set @ A ) @ A5 @ B7 )
=> ( ( member @ A @ C3 @ A5 )
=> ( member @ A @ C3 @ B7 ) ) ) ).
% subsetD
thf(fact_232_in__mono,axiom,
! [A: $tType,A5: set @ A,B7: set @ A,X: A] :
( ( ord_less_eq @ ( set @ A ) @ A5 @ B7 )
=> ( ( member @ A @ X @ A5 )
=> ( member @ A @ X @ B7 ) ) ) ).
% in_mono
thf(fact_233_set__mp,axiom,
! [A: $tType,A5: set @ A,B7: set @ A,X: A] :
( ( ord_less_eq @ ( set @ A ) @ A5 @ B7 )
=> ( ( member @ A @ X @ A5 )
=> ( member @ A @ X @ B7 ) ) ) ).
% set_mp
thf(fact_234_insert__mono,axiom,
! [A: $tType,C4: set @ A,D3: set @ A,A4: A] :
( ( ord_less_eq @ ( set @ A ) @ C4 @ D3 )
=> ( ord_less_eq @ ( set @ A ) @ ( insert @ A @ A4 @ C4 ) @ ( insert @ A @ A4 @ D3 ) ) ) ).
% insert_mono
thf(fact_235_subset__insert,axiom,
! [A: $tType,X: A,A5: set @ A,B7: set @ A] :
( ~ ( member @ A @ X @ A5 )
=> ( ( ord_less_eq @ ( set @ A ) @ A5 @ ( insert @ A @ X @ B7 ) )
= ( ord_less_eq @ ( set @ A ) @ A5 @ B7 ) ) ) ).
% subset_insert
thf(fact_236_subset__insertI,axiom,
! [A: $tType,B7: set @ A,A4: A] : ( ord_less_eq @ ( set @ A ) @ B7 @ ( insert @ A @ A4 @ B7 ) ) ).
% subset_insertI
thf(fact_237_subset__insertI2,axiom,
! [A: $tType,A5: set @ A,B7: set @ A,B2: A] :
( ( ord_less_eq @ ( set @ A ) @ A5 @ B7 )
=> ( ord_less_eq @ ( set @ A ) @ A5 @ ( insert @ A @ B2 @ B7 ) ) ) ).
% subset_insertI2
thf(fact_238_finite__subset,axiom,
! [A: $tType,A5: set @ A,B7: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ A5 @ B7 )
=> ( ( finite_finite2 @ A @ B7 )
=> ( finite_finite2 @ A @ A5 ) ) ) ).
% finite_subset
thf(fact_239_infinite__super,axiom,
! [A: $tType,S2: set @ A,T5: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ S2 @ T5 )
=> ( ~ ( finite_finite2 @ A @ S2 )
=> ~ ( finite_finite2 @ A @ T5 ) ) ) ).
% infinite_super
thf(fact_240_rev__finite__subset,axiom,
! [A: $tType,B7: set @ A,A5: set @ A] :
( ( finite_finite2 @ A @ B7 )
=> ( ( ord_less_eq @ ( set @ A ) @ A5 @ B7 )
=> ( finite_finite2 @ A @ A5 ) ) ) ).
% rev_finite_subset
thf(fact_241_inItr__mono,axiom,
! [Ns: set @ n,Tr: dtree,N: n,Ns4: set @ n] :
( ( gram_L830233218_inItr @ Ns @ Tr @ N )
=> ( ( ord_less_eq @ ( set @ n ) @ Ns @ Ns4 )
=> ( gram_L830233218_inItr @ Ns4 @ Tr @ N ) ) ) ).
% inItr_mono
thf(fact_242_subset__Diff__insert,axiom,
! [A: $tType,A5: set @ A,B7: set @ A,X: A,C4: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ A5 @ ( minus_minus @ ( set @ A ) @ B7 @ ( insert @ A @ X @ C4 ) ) )
= ( ( ord_less_eq @ ( set @ A ) @ A5 @ ( minus_minus @ ( set @ A ) @ B7 @ C4 ) )
& ~ ( member @ A @ X @ A5 ) ) ) ).
% subset_Diff_insert
thf(fact_243_subset__singleton__iff,axiom,
! [A: $tType,X9: set @ A,A4: A] :
( ( ord_less_eq @ ( set @ A ) @ X9 @ ( insert @ A @ A4 @ ( bot_bot @ ( set @ A ) ) ) )
= ( ( X9
= ( bot_bot @ ( set @ A ) ) )
| ( X9
= ( insert @ A @ A4 @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ).
% subset_singleton_iff
thf(fact_244_subset__singletonD,axiom,
! [A: $tType,A5: set @ A,X: A] :
( ( ord_less_eq @ ( set @ A ) @ A5 @ ( insert @ A @ X @ ( bot_bot @ ( set @ A ) ) ) )
=> ( ( A5
= ( bot_bot @ ( set @ A ) ) )
| ( A5
= ( insert @ A @ X @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ).
% subset_singletonD
thf(fact_245_finite__subset__induct,axiom,
! [A: $tType,F7: set @ A,A5: set @ A,P: ( set @ A ) > $o] :
( ( finite_finite2 @ A @ F7 )
=> ( ( ord_less_eq @ ( set @ A ) @ F7 @ A5 )
=> ( ( P @ ( bot_bot @ ( set @ A ) ) )
=> ( ! [A9: A,F6: set @ A] :
( ( finite_finite2 @ A @ F6 )
=> ( ( member @ A @ A9 @ A5 )
=> ( ~ ( member @ A @ A9 @ F6 )
=> ( ( P @ F6 )
=> ( P @ ( insert @ A @ A9 @ F6 ) ) ) ) ) )
=> ( P @ F7 ) ) ) ) ) ).
% finite_subset_induct
thf(fact_246_subset__insert__iff,axiom,
! [A: $tType,A5: set @ A,X: A,B7: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ A5 @ ( insert @ A @ X @ B7 ) )
= ( ( ( member @ A @ X @ A5 )
=> ( ord_less_eq @ ( set @ A ) @ ( minus_minus @ ( set @ A ) @ A5 @ ( insert @ A @ X @ ( bot_bot @ ( set @ A ) ) ) ) @ B7 ) )
& ( ~ ( member @ A @ X @ A5 )
=> ( ord_less_eq @ ( set @ A ) @ A5 @ B7 ) ) ) ) ).
% subset_insert_iff
thf(fact_247_Diff__single__insert,axiom,
! [A: $tType,A5: set @ A,X: A,B7: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ ( minus_minus @ ( set @ A ) @ A5 @ ( insert @ A @ X @ ( bot_bot @ ( set @ A ) ) ) ) @ B7 )
=> ( ord_less_eq @ ( set @ A ) @ A5 @ ( insert @ A @ X @ B7 ) ) ) ).
% Diff_single_insert
thf(fact_248_insert__subsetI,axiom,
! [A: $tType,X: A,A5: set @ A,X9: set @ A] :
( ( member @ A @ X @ A5 )
=> ( ( ord_less_eq @ ( set @ A ) @ X9 @ A5 )
=> ( ord_less_eq @ ( set @ A ) @ ( insert @ A @ X @ X9 ) @ A5 ) ) ) ).
% insert_subsetI
thf(fact_249_diff__mono,axiom,
! [A: $tType] :
( ( ordered_ab_group_add @ A @ ( type2 @ A ) )
=> ! [A4: A,B2: A,D2: A,C3: A] :
( ( ord_less_eq @ A @ A4 @ B2 )
=> ( ( ord_less_eq @ A @ D2 @ C3 )
=> ( ord_less_eq @ A @ ( minus_minus @ A @ A4 @ C3 ) @ ( minus_minus @ A @ B2 @ D2 ) ) ) ) ) ).
% diff_mono
thf(fact_250_diff__right__commute,axiom,
! [A: $tType] :
( ( cancel146912293up_add @ A @ ( type2 @ A ) )
=> ! [A4: A,C3: A,B2: A] :
( ( minus_minus @ A @ ( minus_minus @ A @ A4 @ C3 ) @ B2 )
= ( minus_minus @ A @ ( minus_minus @ A @ A4 @ B2 ) @ C3 ) ) ) ).
% diff_right_commute
thf(fact_251_diff__eq__diff__eq,axiom,
! [A: $tType] :
( ( group_add @ A @ ( type2 @ A ) )
=> ! [A4: A,B2: A,C3: A,D2: A] :
( ( ( minus_minus @ A @ A4 @ B2 )
= ( minus_minus @ A @ C3 @ D2 ) )
=> ( ( A4 = B2 )
= ( C3 = D2 ) ) ) ) ).
% diff_eq_diff_eq
thf(fact_252_diff__eq__diff__less__eq,axiom,
! [A: $tType] :
( ( ordered_ab_group_add @ A @ ( type2 @ A ) )
=> ! [A4: A,B2: A,C3: A,D2: A] :
( ( ( minus_minus @ A @ A4 @ B2 )
= ( minus_minus @ A @ C3 @ D2 ) )
=> ( ( ord_less_eq @ A @ A4 @ B2 )
= ( ord_less_eq @ A @ C3 @ D2 ) ) ) ) ).
% diff_eq_diff_less_eq
thf(fact_253_diff__right__mono,axiom,
! [A: $tType] :
( ( ordered_ab_group_add @ A @ ( type2 @ A ) )
=> ! [A4: A,B2: A,C3: A] :
( ( ord_less_eq @ A @ A4 @ B2 )
=> ( ord_less_eq @ A @ ( minus_minus @ A @ A4 @ C3 ) @ ( minus_minus @ A @ B2 @ C3 ) ) ) ) ).
% diff_right_mono
thf(fact_254_diff__left__mono,axiom,
! [A: $tType] :
( ( ordered_ab_group_add @ A @ ( type2 @ A ) )
=> ! [B2: A,A4: A,C3: A] :
( ( ord_less_eq @ A @ B2 @ A4 )
=> ( ord_less_eq @ A @ ( minus_minus @ A @ C3 @ A4 ) @ ( minus_minus @ A @ C3 @ B2 ) ) ) ) ).
% diff_left_mono
thf(fact_255_psubset__insert__iff,axiom,
! [A: $tType,A5: set @ A,X: A,B7: set @ A] :
( ( ord_less @ ( set @ A ) @ A5 @ ( insert @ A @ X @ B7 ) )
= ( ( ( member @ A @ X @ B7 )
=> ( ord_less @ ( set @ A ) @ A5 @ B7 ) )
& ( ~ ( member @ A @ X @ B7 )
=> ( ( ( member @ A @ X @ A5 )
=> ( ord_less @ ( set @ A ) @ ( minus_minus @ ( set @ A ) @ A5 @ ( insert @ A @ X @ ( bot_bot @ ( set @ A ) ) ) ) @ B7 ) )
& ( ~ ( member @ A @ X @ A5 )
=> ( ord_less_eq @ ( set @ A ) @ A5 @ B7 ) ) ) ) ) ) ).
% psubset_insert_iff
%----Type constructors (7)
thf(tcon_fun___Finite__Set_Ofinite,axiom,
! [A7: $tType,A14: $tType] :
( ( ( finite_finite @ A7 @ ( type2 @ A7 ) )
& ( finite_finite @ A14 @ ( type2 @ A14 ) ) )
=> ( finite_finite @ ( A7 > A14 ) @ ( type2 @ ( A7 > A14 ) ) ) ) ).
thf(tcon_fun___Groups_Ominus,axiom,
! [A7: $tType,A14: $tType] :
( ( minus @ A14 @ ( type2 @ A14 ) )
=> ( minus @ ( A7 > A14 ) @ ( type2 @ ( A7 > A14 ) ) ) ) ).
thf(tcon_Set_Oset___Finite__Set_Ofinite_1,axiom,
! [A7: $tType] :
( ( finite_finite @ A7 @ ( type2 @ A7 ) )
=> ( finite_finite @ ( set @ A7 ) @ ( type2 @ ( set @ A7 ) ) ) ) ).
thf(tcon_Set_Oset___Groups_Ominus_2,axiom,
! [A7: $tType] : ( minus @ ( set @ A7 ) @ ( type2 @ ( set @ A7 ) ) ) ).
thf(tcon_HOL_Obool___Finite__Set_Ofinite_3,axiom,
finite_finite @ $o @ ( type2 @ $o ) ).
thf(tcon_HOL_Obool___Groups_Ominus_4,axiom,
minus @ $o @ ( type2 @ $o ) ).
thf(tcon_Sum__Type_Osum___Finite__Set_Ofinite_5,axiom,
! [A7: $tType,A14: $tType] :
( ( ( finite_finite @ A7 @ ( type2 @ A7 ) )
& ( finite_finite @ A14 @ ( type2 @ A14 ) ) )
=> ( finite_finite @ ( sum_sum @ A7 @ A14 ) @ ( type2 @ ( sum_sum @ A7 @ 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 (5)
thf(conj_0,hypothesis,
member @ n @ ( root @ tr1 ) @ ns ).
thf(conj_1,hypothesis,
member @ ( sum_sum @ t @ dtree ) @ ( sum_Inr @ dtree @ t @ tr1 ) @ ( cont @ tr2 ) ).
thf(conj_2,hypothesis,
gram_L1283001940subtr2 @ ns @ tr2 @ tr3 ).
thf(conj_3,hypothesis,
gram_L716654942_subtr @ ns @ tr2 @ tr3 ).
thf(conj_4,conjecture,
gram_L716654942_subtr @ ns @ tr1 @ tr3 ).
%------------------------------------------------------------------------------