TPTP Problem File: COM182^1.p
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%------------------------------------------------------------------------------
% File : COM182^1 : TPTP v9.0.0. Released v7.0.0.
% Domain : Computing Theory
% Problem : Grammars and languages 60
% 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__60.p [Bla16]
% Status : Theorem
% Rating : 0.00 v7.2.0, 0.25 v7.1.0
% Syntax : Number of formulae : 329 ( 122 unt; 53 typ; 0 def)
% Number of atoms : 715 ( 300 equ; 0 cnn)
% Maximal formula atoms : 7 ( 2 avg)
% Number of connectives : 4149 ( 106 ~; 11 |; 53 &;3632 @)
% ( 0 <=>; 347 =>; 0 <=; 0 <~>)
% Maximal formula depth : 21 ( 8 avg)
% Number of types : 4 ( 3 usr)
% Number of type conns : 250 ( 250 >; 0 *; 0 +; 0 <<)
% Number of symbols : 53 ( 50 usr; 5 con; 0-6 aty)
% Number of variables : 1170 ( 46 ^;1041 !; 32 ?;1170 :)
% ( 51 !>; 0 ?*; 0 @-; 0 @+)
% SPC : TH1_THM_EQU_NAR
% Comments : This file was generated by Isabelle (most likely Sledgehammer)
% 2016-07-13 14:40:08.012
%------------------------------------------------------------------------------
%----Could-be-implicit typings (7)
thf(ty_t_Sum__Type_Osum,type,
sum_sum: $tType > $tType > $tType ).
thf(ty_t_DTree_Odtree,type,
dtree: $tType ).
thf(ty_t_FSet_Ofset,type,
fset: $tType > $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 (46)
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_Orderings_Olinorder,type,
linorder:
!>[A: $tType] : ( ( itself @ A ) > $o ) ).
thf(sy_cl_Lattices_Osemilattice__sup,type,
semilattice_sup:
!>[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_ONNode,type,
nNode: n > ( fset @ ( sum_sum @ t @ dtree ) ) > dtree ).
thf(sy_c_DTree_Odtree_Ocase__dtree,type,
case_dtree:
!>[A: $tType] : ( ( n > ( fset @ ( sum_sum @ t @ dtree ) ) > A ) > dtree > A ) ).
thf(sy_c_DTree_Odtree_Occont,type,
ccont: dtree > ( fset @ ( 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_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_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_Lattices__Big_Osemilattice__sup__class_OSup__fin,type,
lattic477160up_fin:
!>[A: $tType] : ( ( set @ 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_nsa,type,
nsa: set @ n ).
thf(sy_v_ta,type,
ta: t ).
thf(sy_v_tra,type,
tra: dtree ).
%----Relevant facts (256)
thf(fact_0_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_1_old_Osum_Oinject_I1_J,axiom,
! [B: $tType,A: $tType,A2: A,A3: A] :
( ( ( sum_Inl @ A @ B @ A2 )
= ( sum_Inl @ A @ B @ A3 ) )
= ( A2 = A3 ) ) ).
% old.sum.inject(1)
thf(fact_2_dtree__cong,axiom,
! [Tr: dtree,Tr2: dtree] :
( ( ( root @ Tr )
= ( root @ Tr2 ) )
=> ( ( ( cont @ Tr )
= ( cont @ Tr2 ) )
=> ( Tr = Tr2 ) ) ) ).
% dtree_cong
thf(fact_3_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_4_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_5_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_6_Node__root__cont,axiom,
! [Tr: dtree] :
( ( node @ ( root @ Tr ) @ ( cont @ Tr ) )
= Tr ) ).
% Node_root_cont
thf(fact_7_inFr_Ocases,axiom,
! [A1: set @ n,A22: dtree,A32: t] :
( ( gram_L1333338417e_inFr @ A1 @ A22 @ A32 )
=> ( ( ( member @ n @ ( root @ A22 ) @ A1 )
=> ~ ( member @ ( sum_sum @ t @ dtree ) @ ( sum_Inl @ t @ dtree @ A32 ) @ ( cont @ A22 ) ) )
=> ~ ( ( member @ n @ ( root @ A22 ) @ A1 )
=> ! [Tr1: dtree] :
( ( member @ ( sum_sum @ t @ dtree ) @ ( sum_Inr @ dtree @ t @ Tr1 ) @ ( cont @ A22 ) )
=> ~ ( gram_L1333338417e_inFr @ A1 @ Tr1 @ A32 ) ) ) ) ) ).
% inFr.cases
thf(fact_8_inFr_Osimps,axiom,
( gram_L1333338417e_inFr
= ( ^ [A12: set @ n,A23: dtree,A33: t] :
( ? [Tr3: dtree,Ns2: set @ n,T3: t] :
( ( A12 = Ns2 )
& ( A23 = Tr3 )
& ( A33 = T3 )
& ( member @ n @ ( root @ Tr3 ) @ Ns2 )
& ( member @ ( sum_sum @ t @ dtree ) @ ( sum_Inl @ t @ dtree @ T3 ) @ ( cont @ Tr3 ) ) )
| ? [Tr3: dtree,Ns2: set @ n,Tr12: dtree,T3: t] :
( ( A12 = Ns2 )
& ( A23 = Tr3 )
& ( A33 = T3 )
& ( member @ n @ ( root @ Tr3 ) @ Ns2 )
& ( member @ ( sum_sum @ t @ dtree ) @ ( sum_Inr @ dtree @ t @ Tr12 ) @ ( cont @ Tr3 ) )
& ( gram_L1333338417e_inFr @ Ns2 @ Tr12 @ T3 ) ) ) ) ) ).
% inFr.simps
thf(fact_9_inFr_Oinducts,axiom,
! [X1: set @ n,X2: dtree,X3: t,P: ( set @ n ) > dtree > t > $o] :
( ( gram_L1333338417e_inFr @ X1 @ X2 @ X3 )
=> ( ! [Tr4: dtree,Ns3: set @ n,T4: t] :
( ( member @ n @ ( root @ Tr4 ) @ Ns3 )
=> ( ( member @ ( sum_sum @ t @ dtree ) @ ( sum_Inl @ t @ dtree @ T4 ) @ ( cont @ Tr4 ) )
=> ( P @ Ns3 @ Tr4 @ T4 ) ) )
=> ( ! [Tr4: dtree,Ns3: set @ n,Tr1: dtree,T4: t] :
( ( member @ n @ ( root @ Tr4 ) @ Ns3 )
=> ( ( member @ ( sum_sum @ t @ dtree ) @ ( sum_Inr @ dtree @ t @ Tr1 ) @ ( cont @ Tr4 ) )
=> ( ( gram_L1333338417e_inFr @ Ns3 @ Tr1 @ T4 )
=> ( ( P @ Ns3 @ Tr1 @ T4 )
=> ( P @ Ns3 @ Tr4 @ T4 ) ) ) ) )
=> ( P @ X1 @ X2 @ X3 ) ) ) ) ).
% inFr.inducts
thf(fact_10_old_Osum_Osimps_I7_J,axiom,
! [B: $tType,T: $tType,A: $tType,F1: A > T,F2: B > T,A2: A] :
( ( sum_rec_sum @ A @ T @ B @ F1 @ F2 @ ( sum_Inl @ A @ B @ A2 ) )
= ( F1 @ A2 ) ) ).
% old.sum.simps(7)
thf(fact_11_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_12_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_13_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_14_root__Node,axiom,
! [N: n,As: set @ ( sum_sum @ t @ dtree )] :
( ( root @ ( node @ N @ As ) )
= N ) ).
% root_Node
thf(fact_15_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_16_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_17_old_Osum_Oinducts,axiom,
! [B: $tType,A: $tType,P: ( sum_sum @ A @ B ) > $o,Sum: sum_sum @ A @ B] :
( ! [A4: A] : ( P @ ( sum_Inl @ A @ B @ A4 ) )
=> ( ! [B4: B] : ( P @ ( sum_Inr @ B @ A @ B4 ) )
=> ( P @ Sum ) ) ) ).
% old.sum.inducts
thf(fact_18_old_Osum_Oexhaust,axiom,
! [A: $tType,B: $tType,Y: sum_sum @ A @ B] :
( ! [A4: A] :
( Y
!= ( sum_Inl @ A @ B @ A4 ) )
=> ~ ! [B4: B] :
( Y
!= ( sum_Inr @ B @ A @ B4 ) ) ) ).
% old.sum.exhaust
thf(fact_19_split__sum__all,axiom,
! [B: $tType,A: $tType] :
( ( ^ [P2: ( sum_sum @ A @ B ) > $o] :
! [X4: sum_sum @ A @ B] : ( P2 @ X4 ) )
= ( ^ [P3: ( sum_sum @ A @ B ) > $o] :
( ! [X5: A] : ( P3 @ ( sum_Inl @ A @ B @ X5 ) )
& ! [X5: B] : ( P3 @ ( sum_Inr @ B @ A @ X5 ) ) ) ) ) ).
% split_sum_all
thf(fact_20_split__sum__ex,axiom,
! [B: $tType,A: $tType] :
( ( ^ [P2: ( sum_sum @ A @ B ) > $o] :
? [X4: sum_sum @ A @ B] : ( P2 @ X4 ) )
= ( ^ [P3: ( sum_sum @ A @ B ) > $o] :
( ? [X5: A] : ( P3 @ ( sum_Inl @ A @ B @ X5 ) )
| ? [X5: B] : ( P3 @ ( sum_Inr @ B @ A @ X5 ) ) ) ) ) ).
% split_sum_ex
thf(fact_21_Inr__not__Inl,axiom,
! [B: $tType,A: $tType,B2: B,A2: A] :
( ( sum_Inr @ B @ A @ B2 )
!= ( sum_Inl @ A @ B @ A2 ) ) ).
% Inr_not_Inl
thf(fact_22_sumE,axiom,
! [A: $tType,B: $tType,S: sum_sum @ A @ B] :
( ! [X6: A] :
( S
!= ( sum_Inl @ A @ B @ X6 ) )
=> ~ ! [Y3: B] :
( S
!= ( sum_Inr @ B @ A @ Y3 ) ) ) ).
% sumE
thf(fact_23_old_Osum_Odistinct_I1_J,axiom,
! [A: $tType,B: $tType,A2: A,B3: B] :
( ( sum_Inl @ A @ B @ A2 )
!= ( sum_Inr @ B @ A @ B3 ) ) ).
% old.sum.distinct(1)
thf(fact_24_old_Osum_Odistinct_I2_J,axiom,
! [B5: $tType,A5: $tType,B6: B5,A6: A5] :
( ( sum_Inr @ B5 @ A5 @ B6 )
!= ( sum_Inl @ A5 @ B5 @ A6 ) ) ).
% old.sum.distinct(2)
thf(fact_25_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_26_inFr_OInd,axiom,
! [Tr: dtree,Ns: set @ n,Tr13: dtree,T2: t] :
( ( member @ n @ ( root @ Tr ) @ Ns )
=> ( ( member @ ( sum_sum @ t @ dtree ) @ ( sum_Inr @ dtree @ t @ Tr13 ) @ ( cont @ Tr ) )
=> ( ( gram_L1333338417e_inFr @ Ns @ Tr13 @ T2 )
=> ( gram_L1333338417e_inFr @ Ns @ Tr @ T2 ) ) ) ) ).
% inFr.Ind
thf(fact_27_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_28_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_29_obj__sumE,axiom,
! [A: $tType,B: $tType,S: sum_sum @ A @ B] :
( ! [X6: A] :
( S
!= ( sum_Inl @ A @ B @ X6 ) )
=> ~ ! [X6: B] :
( S
!= ( sum_Inr @ B @ A @ X6 ) ) ) ).
% obj_sumE
thf(fact_30_obj__sumE__f,axiom,
! [A: $tType,C: $tType,B: $tType,S: B,F: ( sum_sum @ A @ C ) > B,P: $o] :
( ! [X6: A] :
( ( S
= ( F @ ( sum_Inl @ A @ C @ X6 ) ) )
=> P )
=> ( ! [X6: C] :
( ( S
= ( F @ ( sum_Inr @ C @ A @ X6 ) ) )
=> P )
=> ! [X7: sum_sum @ A @ C] :
( ( S
= ( F @ X7 ) )
=> P ) ) ) ).
% obj_sumE_f
thf(fact_31_inFr2_Oinducts,axiom,
! [X1: set @ n,X2: dtree,X3: t,P: ( set @ n ) > dtree > t > $o] :
( ( gram_L805317441_inFr2 @ X1 @ X2 @ X3 )
=> ( ! [Tr4: dtree,Ns3: set @ n,T4: t] :
( ( member @ n @ ( root @ Tr4 ) @ Ns3 )
=> ( ( member @ ( sum_sum @ t @ dtree ) @ ( sum_Inl @ t @ dtree @ T4 ) @ ( cont @ Tr4 ) )
=> ( P @ Ns3 @ Tr4 @ T4 ) ) )
=> ( ! [Tr1: dtree,Tr4: dtree,Ns1: set @ n,T4: t] :
( ( member @ ( sum_sum @ t @ dtree ) @ ( sum_Inr @ dtree @ t @ Tr1 ) @ ( cont @ Tr4 ) )
=> ( ( gram_L805317441_inFr2 @ Ns1 @ Tr1 @ T4 )
=> ( ( P @ Ns1 @ Tr1 @ T4 )
=> ( P @ ( insert @ n @ ( root @ Tr4 ) @ Ns1 ) @ Tr4 @ T4 ) ) ) )
=> ( P @ X1 @ X2 @ X3 ) ) ) ) ).
% inFr2.inducts
thf(fact_32_inFr2_Osimps,axiom,
( gram_L805317441_inFr2
= ( ^ [A12: set @ n,A23: dtree,A33: t] :
( ? [Tr3: dtree,Ns2: set @ n,T3: t] :
( ( A12 = Ns2 )
& ( A23 = Tr3 )
& ( A33 = T3 )
& ( member @ n @ ( root @ Tr3 ) @ Ns2 )
& ( member @ ( sum_sum @ t @ dtree ) @ ( sum_Inl @ t @ dtree @ T3 ) @ ( cont @ Tr3 ) ) )
| ? [Tr12: dtree,Tr3: dtree,Ns12: set @ n,T3: t] :
( ( A12
= ( insert @ n @ ( root @ Tr3 ) @ Ns12 ) )
& ( A23 = Tr3 )
& ( A33 = T3 )
& ( member @ ( sum_sum @ t @ dtree ) @ ( sum_Inr @ dtree @ t @ Tr12 ) @ ( cont @ Tr3 ) )
& ( gram_L805317441_inFr2 @ Ns12 @ Tr12 @ T3 ) ) ) ) ) ).
% inFr2.simps
thf(fact_33_inFr2_Ocases,axiom,
! [A1: set @ n,A22: dtree,A32: t] :
( ( gram_L805317441_inFr2 @ A1 @ A22 @ A32 )
=> ( ( ( member @ n @ ( root @ A22 ) @ A1 )
=> ~ ( member @ ( sum_sum @ t @ dtree ) @ ( sum_Inl @ t @ dtree @ A32 ) @ ( cont @ A22 ) ) )
=> ~ ! [Tr1: dtree,Tr4: dtree,Ns1: set @ n] :
( ( A1
= ( insert @ n @ ( root @ Tr4 ) @ Ns1 ) )
=> ( ( A22 = Tr4 )
=> ( ( member @ ( sum_sum @ t @ dtree ) @ ( sum_Inr @ dtree @ t @ Tr1 ) @ ( cont @ Tr4 ) )
=> ~ ( gram_L805317441_inFr2 @ Ns1 @ Tr1 @ A32 ) ) ) ) ) ) ).
% inFr2.cases
thf(fact_34_inFr2_OInd,axiom,
! [Tr13: dtree,Tr: dtree,Ns13: set @ n,T2: t] :
( ( member @ ( sum_sum @ t @ dtree ) @ ( sum_Inr @ dtree @ t @ Tr13 ) @ ( cont @ Tr ) )
=> ( ( gram_L805317441_inFr2 @ Ns13 @ Tr13 @ T2 )
=> ( gram_L805317441_inFr2 @ ( insert @ n @ ( root @ Tr ) @ Ns13 ) @ Tr @ T2 ) ) ) ).
% inFr2.Ind
thf(fact_35_inFr__Ind__minus,axiom,
! [Ns13: set @ n,Tr13: dtree,T2: t,Tr: dtree] :
( ( gram_L1333338417e_inFr @ Ns13 @ Tr13 @ T2 )
=> ( ( member @ ( sum_sum @ t @ dtree ) @ ( sum_Inr @ dtree @ t @ Tr13 ) @ ( cont @ Tr ) )
=> ( gram_L1333338417e_inFr @ ( insert @ n @ ( root @ Tr ) @ Ns13 ) @ Tr @ T2 ) ) ) ).
% inFr_Ind_minus
thf(fact_36_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_37_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_38_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_39_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_40_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_41_insert__absorb2,axiom,
! [A: $tType,X: A,A7: set @ A] :
( ( insert @ A @ X @ ( insert @ A @ X @ A7 ) )
= ( insert @ A @ X @ A7 ) ) ).
% insert_absorb2
thf(fact_42_insert__iff,axiom,
! [A: $tType,A2: A,B2: A,A7: set @ A] :
( ( member @ A @ A2 @ ( insert @ A @ B2 @ A7 ) )
= ( ( A2 = B2 )
| ( member @ A @ A2 @ A7 ) ) ) ).
% insert_iff
thf(fact_43_insertCI,axiom,
! [A: $tType,A2: A,B7: set @ A,B2: A] :
( ( ~ ( member @ A @ A2 @ B7 )
=> ( A2 = B2 ) )
=> ( member @ A @ A2 @ ( insert @ A @ B2 @ B7 ) ) ) ).
% insertCI
thf(fact_44_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_45_mem__Collect__eq,axiom,
! [A: $tType,A2: A,P: A > $o] :
( ( member @ A @ A2 @ ( collect @ A @ P ) )
= ( P @ A2 ) ) ).
% mem_Collect_eq
thf(fact_46_Collect__mem__eq,axiom,
! [A: $tType,A7: set @ A] :
( ( collect @ A
@ ^ [X5: A] : ( member @ A @ X5 @ A7 ) )
= A7 ) ).
% Collect_mem_eq
thf(fact_47_Collect__cong,axiom,
! [A: $tType,P: A > $o,Q: A > $o] :
( ! [X6: A] :
( ( P @ X6 )
= ( Q @ X6 ) )
=> ( ( collect @ A @ P )
= ( collect @ A @ Q ) ) ) ).
% Collect_cong
thf(fact_48_ext,axiom,
! [B: $tType,A: $tType,F: A > B,G: A > B] :
( ! [X6: A] :
( ( F @ X6 )
= ( G @ X6 ) )
=> ( F = G ) ) ).
% ext
thf(fact_49_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_50_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_51_mk__disjoint__insert,axiom,
! [A: $tType,A2: A,A7: set @ A] :
( ( member @ A @ A2 @ A7 )
=> ? [B8: set @ A] :
( ( A7
= ( insert @ A @ A2 @ B8 ) )
& ~ ( member @ A @ A2 @ B8 ) ) ) ).
% mk_disjoint_insert
thf(fact_52_insert__commute,axiom,
! [A: $tType,X: A,Y: A,A7: set @ A] :
( ( insert @ A @ X @ ( insert @ A @ Y @ A7 ) )
= ( insert @ A @ Y @ ( insert @ A @ X @ A7 ) ) ) ).
% insert_commute
thf(fact_53_insert__eq__iff,axiom,
! [A: $tType,A2: A,A7: set @ A,B2: A,B7: set @ A] :
( ~ ( member @ A @ A2 @ A7 )
=> ( ~ ( member @ A @ B2 @ B7 )
=> ( ( ( insert @ A @ A2 @ A7 )
= ( insert @ A @ B2 @ B7 ) )
= ( ( ( A2 = B2 )
=> ( A7 = B7 ) )
& ( ( A2 != B2 )
=> ? [C2: set @ A] :
( ( A7
= ( insert @ A @ B2 @ C2 ) )
& ~ ( member @ A @ B2 @ C2 )
& ( B7
= ( insert @ A @ A2 @ C2 ) )
& ~ ( member @ A @ A2 @ C2 ) ) ) ) ) ) ) ).
% insert_eq_iff
thf(fact_54_dtree__simps,axiom,
! [As: set @ ( sum_sum @ t @ dtree ),As2: set @ ( sum_sum @ t @ dtree ),N: n,N2: n] :
( ( finite_finite2 @ ( sum_sum @ t @ dtree ) @ As )
=> ( ( finite_finite2 @ ( sum_sum @ t @ dtree ) @ As2 )
=> ( ( ( node @ N @ As )
= ( node @ N2 @ As2 ) )
= ( ( N = N2 )
& ( As = As2 ) ) ) ) ) ).
% dtree_simps
thf(fact_55_finite__cont,axiom,
! [Tr: dtree] : ( finite_finite2 @ ( sum_sum @ t @ dtree ) @ ( cont @ Tr ) ) ).
% finite_cont
thf(fact_56_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_57_insertE,axiom,
! [A: $tType,A2: A,B2: A,A7: set @ A] :
( ( member @ A @ A2 @ ( insert @ A @ B2 @ A7 ) )
=> ( ( A2 != B2 )
=> ( member @ A @ A2 @ A7 ) ) ) ).
% insertE
thf(fact_58_insertI1,axiom,
! [A: $tType,A2: A,B7: set @ A] : ( member @ A @ A2 @ ( insert @ A @ A2 @ B7 ) ) ).
% insertI1
thf(fact_59_insertI2,axiom,
! [A: $tType,A2: A,B7: set @ A,B2: A] :
( ( member @ A @ A2 @ B7 )
=> ( member @ A @ A2 @ ( insert @ A @ B2 @ B7 ) ) ) ).
% insertI2
thf(fact_60_Set_Oset__insert,axiom,
! [A: $tType,X: A,A7: set @ A] :
( ( member @ A @ X @ A7 )
=> ~ ! [B8: set @ A] :
( ( A7
= ( insert @ A @ X @ B8 ) )
=> ( member @ A @ X @ B8 ) ) ) ).
% Set.set_insert
thf(fact_61_insert__ident,axiom,
! [A: $tType,X: A,A7: set @ A,B7: set @ A] :
( ~ ( member @ A @ X @ A7 )
=> ( ~ ( member @ A @ X @ B7 )
=> ( ( ( insert @ A @ X @ A7 )
= ( insert @ A @ X @ B7 ) )
= ( A7 = B7 ) ) ) ) ).
% insert_ident
thf(fact_62_insert__absorb,axiom,
! [A: $tType,A2: A,A7: set @ A] :
( ( member @ A @ A2 @ A7 )
=> ( ( insert @ A @ A2 @ A7 )
= A7 ) ) ).
% insert_absorb
thf(fact_63_finite__insert,axiom,
! [A: $tType,A2: A,A7: set @ A] :
( ( finite_finite2 @ A @ ( insert @ A @ A2 @ A7 ) )
= ( finite_finite2 @ A @ A7 ) ) ).
% finite_insert
thf(fact_64_finite__code,axiom,
! [A: $tType] :
( ( finite_finite @ A @ ( type2 @ A ) )
=> ( ( finite_finite2 @ A )
= ( ^ [A8: set @ A] : $true ) ) ) ).
% finite_code
thf(fact_65_finite_OinsertI,axiom,
! [A: $tType,A7: set @ A,A2: A] :
( ( finite_finite2 @ A @ A7 )
=> ( finite_finite2 @ A @ ( insert @ A @ A2 @ A7 ) ) ) ).
% finite.insertI
thf(fact_66_finite__set__choice,axiom,
! [B: $tType,A: $tType,A7: set @ A,P: A > B > $o] :
( ( finite_finite2 @ A @ A7 )
=> ( ! [X6: A] :
( ( member @ A @ X6 @ A7 )
=> ? [X12: B] : ( P @ X6 @ X12 ) )
=> ? [F3: A > B] :
! [X7: A] :
( ( member @ A @ X7 @ A7 )
=> ( P @ X7 @ ( F3 @ X7 ) ) ) ) ) ).
% finite_set_choice
thf(fact_67_finite,axiom,
! [A: $tType] :
( ( finite_finite @ A @ ( type2 @ A ) )
=> ! [A7: set @ A] : ( finite_finite2 @ A @ A7 ) ) ).
% finite
thf(fact_68_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_69_PlusE,axiom,
! [A: $tType,B: $tType,U: sum_sum @ A @ B,A7: set @ A,B7: set @ B] :
( ( member @ ( sum_sum @ A @ B ) @ U @ ( sum_Plus @ A @ B @ A7 @ B7 ) )
=> ( ! [X6: A] :
( ( member @ A @ X6 @ A7 )
=> ( U
!= ( sum_Inl @ A @ B @ X6 ) ) )
=> ~ ! [Y3: B] :
( ( member @ B @ Y3 @ B7 )
=> ( U
!= ( sum_Inr @ B @ A @ Y3 ) ) ) ) ) ).
% PlusE
thf(fact_70_dtree_Osel_I1_J,axiom,
! [X1: n,X2: fset @ ( sum_sum @ t @ dtree )] :
( ( root @ ( nNode @ X1 @ X2 ) )
= X1 ) ).
% dtree.sel(1)
thf(fact_71_dtree_Oinject,axiom,
! [X1: n,X2: fset @ ( sum_sum @ t @ dtree ),Y1: n,Y2: fset @ ( sum_sum @ t @ dtree )] :
( ( ( nNode @ X1 @ X2 )
= ( nNode @ Y1 @ Y2 ) )
= ( ( X1 = Y1 )
& ( X2 = Y2 ) ) ) ).
% dtree.inject
thf(fact_72_InrI,axiom,
! [B: $tType,A: $tType,B2: A,B7: set @ A,A7: set @ B] :
( ( member @ A @ B2 @ B7 )
=> ( member @ ( sum_sum @ B @ A ) @ ( sum_Inr @ A @ B @ B2 ) @ ( sum_Plus @ B @ A @ A7 @ B7 ) ) ) ).
% InrI
thf(fact_73_InlI,axiom,
! [A: $tType,B: $tType,A2: A,A7: set @ A,B7: set @ B] :
( ( member @ A @ A2 @ A7 )
=> ( member @ ( sum_sum @ A @ B ) @ ( sum_Inl @ A @ B @ A2 ) @ ( sum_Plus @ A @ B @ A7 @ B7 ) ) ) ).
% InlI
thf(fact_74_finite__Plus__iff,axiom,
! [A: $tType,B: $tType,A7: set @ A,B7: set @ B] :
( ( finite_finite2 @ ( sum_sum @ A @ B ) @ ( sum_Plus @ A @ B @ A7 @ B7 ) )
= ( ( finite_finite2 @ A @ A7 )
& ( finite_finite2 @ B @ B7 ) ) ) ).
% finite_Plus_iff
thf(fact_75_dtree_Oexhaust,axiom,
! [Y: dtree] :
~ ! [X13: n,X22: fset @ ( sum_sum @ t @ dtree )] :
( Y
!= ( nNode @ X13 @ X22 ) ) ).
% dtree.exhaust
thf(fact_76_case__sum__inject,axiom,
! [A: $tType,C: $tType,B: $tType,F1: A > C,F2: B > C,G1: A > C,G2: B > C] :
( ( ( sum_case_sum @ A @ C @ B @ F1 @ F2 )
= ( sum_case_sum @ A @ C @ B @ G1 @ G2 ) )
=> ~ ( ( F1 = G1 )
=> ( F2 != G2 ) ) ) ).
% case_sum_inject
thf(fact_77_case__sum__step_I1_J,axiom,
! [D: $tType,A: $tType,C: $tType,B: $tType,F4: B > A,G3: 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 @ G3 ) @ G @ ( sum_Inl @ ( sum_sum @ B @ C ) @ D @ P4 ) )
= ( sum_case_sum @ B @ A @ C @ F4 @ G3 @ P4 ) ) ).
% case_sum_step(1)
thf(fact_78_case__sum__step_I2_J,axiom,
! [E: $tType,A: $tType,C: $tType,B: $tType,F: E > A,F4: B > A,G3: C > A,P4: sum_sum @ B @ C] :
( ( sum_case_sum @ E @ A @ ( sum_sum @ B @ C ) @ F @ ( sum_case_sum @ B @ A @ C @ F4 @ G3 ) @ ( sum_Inr @ ( sum_sum @ B @ C ) @ E @ P4 ) )
= ( sum_case_sum @ B @ A @ C @ F4 @ G3 @ P4 ) ) ).
% case_sum_step(2)
thf(fact_79_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_80_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_81_finite__Plus,axiom,
! [A: $tType,B: $tType,A7: set @ A,B7: set @ B] :
( ( finite_finite2 @ A @ A7 )
=> ( ( finite_finite2 @ B @ B7 )
=> ( finite_finite2 @ ( sum_sum @ A @ B ) @ ( sum_Plus @ A @ B @ A7 @ B7 ) ) ) ) ).
% finite_Plus
thf(fact_82_finite__PlusD_I1_J,axiom,
! [B: $tType,A: $tType,A7: set @ A,B7: set @ B] :
( ( finite_finite2 @ ( sum_sum @ A @ B ) @ ( sum_Plus @ A @ B @ A7 @ B7 ) )
=> ( finite_finite2 @ A @ A7 ) ) ).
% finite_PlusD(1)
thf(fact_83_finite__PlusD_I2_J,axiom,
! [A: $tType,B: $tType,A7: set @ A,B7: set @ B] :
( ( finite_finite2 @ ( sum_sum @ A @ B ) @ ( sum_Plus @ A @ B @ A7 @ B7 ) )
=> ( finite_finite2 @ B @ B7 ) ) ).
% finite_PlusD(2)
thf(fact_84_dtree_Ocollapse,axiom,
! [Dtree: dtree] :
( ( nNode @ ( root @ Dtree ) @ ( ccont @ Dtree ) )
= Dtree ) ).
% dtree.collapse
thf(fact_85_dtree_Oexhaust__sel,axiom,
! [Dtree: dtree] :
( Dtree
= ( nNode @ ( root @ Dtree ) @ ( ccont @ Dtree ) ) ) ).
% dtree.exhaust_sel
thf(fact_86_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_87_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_88_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_89_dtree_Ocase,axiom,
! [A: $tType,F: n > ( fset @ ( sum_sum @ t @ dtree ) ) > A,X1: n,X2: fset @ ( sum_sum @ t @ dtree )] :
( ( case_dtree @ A @ F @ ( nNode @ X1 @ X2 ) )
= ( F @ X1 @ X2 ) ) ).
% dtree.case
thf(fact_90_finite__Diff__insert,axiom,
! [A: $tType,A7: set @ A,A2: A,B7: set @ A] :
( ( finite_finite2 @ A @ ( minus_minus @ ( set @ A ) @ A7 @ ( insert @ A @ A2 @ B7 ) ) )
= ( finite_finite2 @ A @ ( minus_minus @ ( set @ A ) @ A7 @ B7 ) ) ) ).
% finite_Diff_insert
thf(fact_91_Diff__idemp,axiom,
! [A: $tType,A7: set @ A,B7: set @ A] :
( ( minus_minus @ ( set @ A ) @ ( minus_minus @ ( set @ A ) @ A7 @ B7 ) @ B7 )
= ( minus_minus @ ( set @ A ) @ A7 @ B7 ) ) ).
% Diff_idemp
thf(fact_92_Diff__iff,axiom,
! [A: $tType,C3: A,A7: set @ A,B7: set @ A] :
( ( member @ A @ C3 @ ( minus_minus @ ( set @ A ) @ A7 @ B7 ) )
= ( ( member @ A @ C3 @ A7 )
& ~ ( member @ A @ C3 @ B7 ) ) ) ).
% Diff_iff
thf(fact_93_DiffI,axiom,
! [A: $tType,C3: A,A7: set @ A,B7: set @ A] :
( ( member @ A @ C3 @ A7 )
=> ( ~ ( member @ A @ C3 @ B7 )
=> ( member @ A @ C3 @ ( minus_minus @ ( set @ A ) @ A7 @ B7 ) ) ) ) ).
% DiffI
thf(fact_94_finite__Diff2,axiom,
! [A: $tType,B7: set @ A,A7: set @ A] :
( ( finite_finite2 @ A @ B7 )
=> ( ( finite_finite2 @ A @ ( minus_minus @ ( set @ A ) @ A7 @ B7 ) )
= ( finite_finite2 @ A @ A7 ) ) ) ).
% finite_Diff2
thf(fact_95_finite__Diff,axiom,
! [A: $tType,A7: set @ A,B7: set @ A] :
( ( finite_finite2 @ A @ A7 )
=> ( finite_finite2 @ A @ ( minus_minus @ ( set @ A ) @ A7 @ B7 ) ) ) ).
% finite_Diff
thf(fact_96_Diff__insert0,axiom,
! [A: $tType,X: A,A7: set @ A,B7: set @ A] :
( ~ ( member @ A @ X @ A7 )
=> ( ( minus_minus @ ( set @ A ) @ A7 @ ( insert @ A @ X @ B7 ) )
= ( minus_minus @ ( set @ A ) @ A7 @ B7 ) ) ) ).
% Diff_insert0
thf(fact_97_insert__Diff1,axiom,
! [A: $tType,X: A,B7: set @ A,A7: set @ A] :
( ( member @ A @ X @ B7 )
=> ( ( minus_minus @ ( set @ A ) @ ( insert @ A @ X @ A7 ) @ B7 )
= ( minus_minus @ ( set @ A ) @ A7 @ B7 ) ) ) ).
% insert_Diff1
thf(fact_98_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_99_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_100_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_101_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_102_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_103_DiffD2,axiom,
! [A: $tType,C3: A,A7: set @ A,B7: set @ A] :
( ( member @ A @ C3 @ ( minus_minus @ ( set @ A ) @ A7 @ B7 ) )
=> ~ ( member @ A @ C3 @ B7 ) ) ).
% DiffD2
thf(fact_104_DiffD1,axiom,
! [A: $tType,C3: A,A7: set @ A,B7: set @ A] :
( ( member @ A @ C3 @ ( minus_minus @ ( set @ A ) @ A7 @ B7 ) )
=> ( member @ A @ C3 @ A7 ) ) ).
% DiffD1
thf(fact_105_DiffE,axiom,
! [A: $tType,C3: A,A7: set @ A,B7: set @ A] :
( ( member @ A @ C3 @ ( minus_minus @ ( set @ A ) @ A7 @ B7 ) )
=> ~ ( ( member @ A @ C3 @ A7 )
=> ( member @ A @ C3 @ B7 ) ) ) ).
% DiffE
thf(fact_106_dtree_Ocase__eq__if,axiom,
! [A: $tType] :
( ( case_dtree @ A )
= ( ^ [F5: n > ( fset @ ( sum_sum @ t @ dtree ) ) > A,Dtree2: dtree] : ( F5 @ ( root @ Dtree2 ) @ ( ccont @ Dtree2 ) ) ) ) ).
% dtree.case_eq_if
thf(fact_107_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_108_insert__Diff__if,axiom,
! [A: $tType,X: A,B7: set @ A,A7: set @ A] :
( ( ( member @ A @ X @ B7 )
=> ( ( minus_minus @ ( set @ A ) @ ( insert @ A @ X @ A7 ) @ B7 )
= ( minus_minus @ ( set @ A ) @ A7 @ B7 ) ) )
& ( ~ ( member @ A @ X @ B7 )
=> ( ( minus_minus @ ( set @ A ) @ ( insert @ A @ X @ A7 ) @ B7 )
= ( insert @ A @ X @ ( minus_minus @ ( set @ A ) @ A7 @ B7 ) ) ) ) ) ).
% insert_Diff_if
thf(fact_109_dtree_Oexpand,axiom,
! [Dtree: dtree,Dtree3: dtree] :
( ( ( ( root @ Dtree )
= ( root @ Dtree3 ) )
& ( ( ccont @ Dtree )
= ( ccont @ Dtree3 ) ) )
=> ( Dtree = Dtree3 ) ) ).
% dtree.expand
thf(fact_110_dtree_Osplit__sel__asm,axiom,
! [A: $tType,P: A > $o,F: n > ( fset @ ( sum_sum @ t @ dtree ) ) > A,Dtree: dtree] :
( ( P @ ( case_dtree @ A @ F @ Dtree ) )
= ( ~ ( ( Dtree
= ( nNode @ ( root @ Dtree ) @ ( ccont @ Dtree ) ) )
& ~ ( P @ ( F @ ( root @ Dtree ) @ ( ccont @ Dtree ) ) ) ) ) ) ).
% dtree.split_sel_asm
thf(fact_111_dtree_Osplit__sel,axiom,
! [A: $tType,P: A > $o,F: n > ( fset @ ( sum_sum @ t @ dtree ) ) > A,Dtree: dtree] :
( ( P @ ( case_dtree @ A @ F @ Dtree ) )
= ( ( Dtree
= ( nNode @ ( root @ Dtree ) @ ( ccont @ Dtree ) ) )
=> ( P @ ( F @ ( root @ Dtree ) @ ( ccont @ Dtree ) ) ) ) ) ).
% dtree.split_sel
thf(fact_112_dtree_Osel_I2_J,axiom,
! [X1: n,X2: fset @ ( sum_sum @ t @ dtree )] :
( ( ccont @ ( nNode @ X1 @ X2 ) )
= X2 ) ).
% dtree.sel(2)
thf(fact_113_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_114_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_115_comp__apply,axiom,
! [C: $tType,A: $tType,B: $tType] :
( ( comp @ B @ A @ C )
= ( ^ [F5: B > A,G4: C > B,X5: C] : ( F5 @ ( G4 @ X5 ) ) ) ) ).
% comp_apply
thf(fact_116_minus__apply,axiom,
! [B: $tType,A: $tType] :
( ( minus @ B @ ( type2 @ B ) )
=> ( ( minus_minus @ ( A > B ) )
= ( ^ [A8: A > B,B9: A > B,X5: A] : ( minus_minus @ B @ ( A8 @ X5 ) @ ( B9 @ X5 ) ) ) ) ) ).
% minus_apply
thf(fact_117_comp__def,axiom,
! [A: $tType,C: $tType,B: $tType] :
( ( comp @ B @ C @ A )
= ( ^ [F5: B > C,G4: A > B,X5: A] : ( F5 @ ( G4 @ X5 ) ) ) ) ).
% comp_def
thf(fact_118_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_119_fun__diff__def,axiom,
! [B: $tType,A: $tType] :
( ( minus @ B @ ( type2 @ B ) )
=> ( ( minus_minus @ ( A > B ) )
= ( ^ [A8: A > B,B9: A > B,X5: A] : ( minus_minus @ B @ ( A8 @ X5 ) @ ( B9 @ X5 ) ) ) ) ) ).
% fun_diff_def
thf(fact_120_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_121_comp__eq__dest__lhs,axiom,
! [C: $tType,B: $tType,A: $tType,A2: C > B,B2: A > C,C3: A > B,V: A] :
( ( ( comp @ C @ B @ A @ A2 @ B2 )
= C3 )
=> ( ( A2 @ ( B2 @ V ) )
= ( C3 @ V ) ) ) ).
% comp_eq_dest_lhs
thf(fact_122_comp__eq__elim,axiom,
! [C: $tType,B: $tType,D: $tType,A: $tType,A2: C > B,B2: A > C,C3: D > B,D2: A > D] :
( ( ( comp @ C @ B @ A @ A2 @ B2 )
= ( comp @ D @ B @ A @ C3 @ D2 ) )
=> ! [V2: A] :
( ( A2 @ ( B2 @ V2 ) )
= ( C3 @ ( D2 @ V2 ) ) ) ) ).
% comp_eq_elim
thf(fact_123_comp__eq__dest,axiom,
! [C: $tType,B: $tType,D: $tType,A: $tType,A2: C > B,B2: A > C,C3: D > B,D2: A > D,V: A] :
( ( ( comp @ C @ B @ A @ A2 @ B2 )
= ( comp @ D @ B @ A @ C3 @ D2 ) )
=> ( ( A2 @ ( B2 @ V ) )
= ( C3 @ ( D2 @ V ) ) ) ) ).
% comp_eq_dest
thf(fact_124_type__copy__map__cong0,axiom,
! [B: $tType,D: $tType,E: $tType,A: $tType,C: $tType,M: B > A,G: C > B,X: C,N4: D > A,H: C > D,F: A > E] :
( ( ( M @ ( G @ X ) )
= ( N4 @ ( H @ X ) ) )
=> ( ( comp @ B @ E @ C @ ( comp @ A @ E @ B @ F @ M ) @ G @ X )
= ( comp @ D @ E @ C @ ( comp @ A @ E @ D @ F @ N4 ) @ H @ X ) ) ) ).
% type_copy_map_cong0
thf(fact_125_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_126_comp__cong,axiom,
! [C: $tType,B: $tType,D: $tType,A: $tType,E: $tType,F: B > A,G: C > B,X: C,F4: D > A,G3: E > D,X8: E] :
( ( ( F @ ( G @ X ) )
= ( F4 @ ( G3 @ X8 ) ) )
=> ( ( comp @ B @ A @ C @ F @ G @ X )
= ( comp @ D @ A @ E @ F4 @ G3 @ X8 ) ) ) ).
% comp_cong
thf(fact_127_finite__empty__induct,axiom,
! [A: $tType,A7: set @ A,P: ( set @ A ) > $o] :
( ( finite_finite2 @ A @ A7 )
=> ( ( P @ A7 )
=> ( ! [A4: A,A9: set @ A] :
( ( finite_finite2 @ A @ A9 )
=> ( ( member @ A @ A4 @ A9 )
=> ( ( P @ A9 )
=> ( P @ ( minus_minus @ ( set @ A ) @ A9 @ ( insert @ A @ A4 @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ) )
=> ( P @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ).
% finite_empty_induct
thf(fact_128_empty__Collect__eq,axiom,
! [A: $tType,P: A > $o] :
( ( ( bot_bot @ ( set @ A ) )
= ( collect @ A @ P ) )
= ( ! [X5: A] :
~ ( P @ X5 ) ) ) ).
% empty_Collect_eq
thf(fact_129_Collect__empty__eq,axiom,
! [A: $tType,P: A > $o] :
( ( ( collect @ A @ P )
= ( bot_bot @ ( set @ A ) ) )
= ( ! [X5: A] :
~ ( P @ X5 ) ) ) ).
% Collect_empty_eq
thf(fact_130_all__not__in__conv,axiom,
! [A: $tType,A7: set @ A] :
( ( ! [X5: A] :
~ ( member @ A @ X5 @ A7 ) )
= ( A7
= ( bot_bot @ ( set @ A ) ) ) ) ).
% all_not_in_conv
thf(fact_131_empty__iff,axiom,
! [A: $tType,C3: A] :
~ ( member @ A @ C3 @ ( bot_bot @ ( set @ A ) ) ) ).
% empty_iff
thf(fact_132_singletonI,axiom,
! [A: $tType,A2: A] : ( member @ A @ A2 @ ( insert @ A @ A2 @ ( bot_bot @ ( set @ A ) ) ) ) ).
% singletonI
thf(fact_133_Diff__cancel,axiom,
! [A: $tType,A7: set @ A] :
( ( minus_minus @ ( set @ A ) @ A7 @ A7 )
= ( bot_bot @ ( set @ A ) ) ) ).
% Diff_cancel
thf(fact_134_empty__Diff,axiom,
! [A: $tType,A7: set @ A] :
( ( minus_minus @ ( set @ A ) @ ( bot_bot @ ( set @ A ) ) @ A7 )
= ( bot_bot @ ( set @ A ) ) ) ).
% empty_Diff
thf(fact_135_Diff__empty,axiom,
! [A: $tType,A7: set @ A] :
( ( minus_minus @ ( set @ A ) @ A7 @ ( bot_bot @ ( set @ A ) ) )
= A7 ) ).
% Diff_empty
thf(fact_136_Plus__eq__empty__conv,axiom,
! [A: $tType,B: $tType,A7: set @ A,B7: set @ B] :
( ( ( sum_Plus @ A @ B @ A7 @ B7 )
= ( bot_bot @ ( set @ ( sum_sum @ A @ B ) ) ) )
= ( ( A7
= ( bot_bot @ ( set @ A ) ) )
& ( B7
= ( bot_bot @ ( set @ B ) ) ) ) ) ).
% Plus_eq_empty_conv
thf(fact_137_insert__Diff__single,axiom,
! [A: $tType,A2: A,A7: set @ A] :
( ( insert @ A @ A2 @ ( minus_minus @ ( set @ A ) @ A7 @ ( insert @ A @ A2 @ ( bot_bot @ ( set @ A ) ) ) ) )
= ( insert @ A @ A2 @ A7 ) ) ).
% insert_Diff_single
thf(fact_138_finite_OemptyI,axiom,
! [A: $tType] : ( finite_finite2 @ A @ ( bot_bot @ ( set @ A ) ) ) ).
% finite.emptyI
thf(fact_139_infinite__imp__nonempty,axiom,
! [A: $tType,S2: set @ A] :
( ~ ( finite_finite2 @ A @ S2 )
=> ( S2
!= ( bot_bot @ ( set @ A ) ) ) ) ).
% infinite_imp_nonempty
thf(fact_140_singletonD,axiom,
! [A: $tType,B2: A,A2: A] :
( ( member @ A @ B2 @ ( insert @ A @ A2 @ ( bot_bot @ ( set @ A ) ) ) )
=> ( B2 = A2 ) ) ).
% singletonD
thf(fact_141_singleton__iff,axiom,
! [A: $tType,B2: A,A2: A] :
( ( member @ A @ B2 @ ( insert @ A @ A2 @ ( bot_bot @ ( set @ A ) ) ) )
= ( B2 = A2 ) ) ).
% singleton_iff
thf(fact_142_doubleton__eq__iff,axiom,
! [A: $tType,A2: A,B2: A,C3: A,D2: A] :
( ( ( insert @ A @ A2 @ ( insert @ A @ B2 @ ( bot_bot @ ( set @ A ) ) ) )
= ( insert @ A @ C3 @ ( insert @ A @ D2 @ ( bot_bot @ ( set @ A ) ) ) ) )
= ( ( ( A2 = C3 )
& ( B2 = D2 ) )
| ( ( A2 = D2 )
& ( B2 = C3 ) ) ) ) ).
% doubleton_eq_iff
thf(fact_143_insert__not__empty,axiom,
! [A: $tType,A2: A,A7: set @ A] :
( ( insert @ A @ A2 @ A7 )
!= ( bot_bot @ ( set @ A ) ) ) ).
% insert_not_empty
thf(fact_144_singleton__inject,axiom,
! [A: $tType,A2: A,B2: A] :
( ( ( insert @ A @ A2 @ ( bot_bot @ ( set @ A ) ) )
= ( insert @ A @ B2 @ ( bot_bot @ ( set @ A ) ) ) )
=> ( A2 = B2 ) ) ).
% singleton_inject
thf(fact_145_ex__in__conv,axiom,
! [A: $tType,A7: set @ A] :
( ( ? [X5: A] : ( member @ A @ X5 @ A7 ) )
= ( A7
!= ( bot_bot @ ( set @ A ) ) ) ) ).
% ex_in_conv
thf(fact_146_equals0I,axiom,
! [A: $tType,A7: set @ A] :
( ! [Y3: A] :
~ ( member @ A @ Y3 @ A7 )
=> ( A7
= ( bot_bot @ ( set @ A ) ) ) ) ).
% equals0I
thf(fact_147_equals0D,axiom,
! [A: $tType,A7: set @ A,A2: A] :
( ( A7
= ( bot_bot @ ( set @ A ) ) )
=> ~ ( member @ A @ A2 @ A7 ) ) ).
% equals0D
thf(fact_148_emptyE,axiom,
! [A: $tType,A2: A] :
~ ( member @ A @ A2 @ ( bot_bot @ ( set @ A ) ) ) ).
% emptyE
thf(fact_149_infinite__finite__induct,axiom,
! [A: $tType,P: ( set @ A ) > $o,A7: set @ A] :
( ! [A9: set @ A] :
( ~ ( finite_finite2 @ A @ A9 )
=> ( P @ A9 ) )
=> ( ( P @ ( bot_bot @ ( set @ A ) ) )
=> ( ! [X6: A,F6: set @ A] :
( ( finite_finite2 @ A @ F6 )
=> ( ~ ( member @ A @ X6 @ F6 )
=> ( ( P @ F6 )
=> ( P @ ( insert @ A @ X6 @ F6 ) ) ) ) )
=> ( P @ A7 ) ) ) ) ).
% infinite_finite_induct
thf(fact_150_finite__ne__induct,axiom,
! [A: $tType,F7: set @ A,P: ( set @ A ) > $o] :
( ( finite_finite2 @ A @ F7 )
=> ( ( F7
!= ( bot_bot @ ( set @ A ) ) )
=> ( ! [X6: A] : ( P @ ( insert @ A @ X6 @ ( bot_bot @ ( set @ A ) ) ) )
=> ( ! [X6: A,F6: set @ A] :
( ( finite_finite2 @ A @ F6 )
=> ( ( F6
!= ( bot_bot @ ( set @ A ) ) )
=> ( ~ ( member @ A @ X6 @ F6 )
=> ( ( P @ F6 )
=> ( P @ ( insert @ A @ X6 @ F6 ) ) ) ) ) )
=> ( P @ F7 ) ) ) ) ) ).
% finite_ne_induct
thf(fact_151_finite_Oinducts,axiom,
! [A: $tType,X: set @ A,P: ( set @ A ) > $o] :
( ( finite_finite2 @ A @ X )
=> ( ( P @ ( bot_bot @ ( set @ A ) ) )
=> ( ! [A9: set @ A,A4: A] :
( ( finite_finite2 @ A @ A9 )
=> ( ( P @ A9 )
=> ( P @ ( insert @ A @ A4 @ A9 ) ) ) )
=> ( P @ X ) ) ) ) ).
% finite.inducts
thf(fact_152_finite__induct,axiom,
! [A: $tType,F7: set @ A,P: ( set @ A ) > $o] :
( ( finite_finite2 @ A @ F7 )
=> ( ( P @ ( bot_bot @ ( set @ A ) ) )
=> ( ! [X6: A,F6: set @ A] :
( ( finite_finite2 @ A @ F6 )
=> ( ~ ( member @ A @ X6 @ F6 )
=> ( ( P @ F6 )
=> ( P @ ( insert @ A @ X6 @ F6 ) ) ) ) )
=> ( P @ F7 ) ) ) ) ).
% finite_induct
thf(fact_153_finite_Osimps,axiom,
! [A: $tType] :
( ( finite_finite2 @ A )
= ( ^ [A10: set @ A] :
( ( A10
= ( bot_bot @ ( set @ A ) ) )
| ? [A8: set @ A,B10: A] :
( ( A10
= ( insert @ A @ B10 @ A8 ) )
& ( finite_finite2 @ A @ A8 ) ) ) ) ) ).
% finite.simps
thf(fact_154_finite_Ocases,axiom,
! [A: $tType,A2: set @ A] :
( ( finite_finite2 @ A @ A2 )
=> ( ( A2
!= ( bot_bot @ ( set @ A ) ) )
=> ~ ! [A9: set @ A] :
( ? [A4: A] :
( A2
= ( insert @ A @ A4 @ A9 ) )
=> ~ ( finite_finite2 @ A @ A9 ) ) ) ) ).
% finite.cases
thf(fact_155_Diff__insert,axiom,
! [A: $tType,A7: set @ A,A2: A,B7: set @ A] :
( ( minus_minus @ ( set @ A ) @ A7 @ ( insert @ A @ A2 @ B7 ) )
= ( minus_minus @ ( set @ A ) @ ( minus_minus @ ( set @ A ) @ A7 @ B7 ) @ ( insert @ A @ A2 @ ( bot_bot @ ( set @ A ) ) ) ) ) ).
% Diff_insert
thf(fact_156_insert__Diff,axiom,
! [A: $tType,A2: A,A7: set @ A] :
( ( member @ A @ A2 @ A7 )
=> ( ( insert @ A @ A2 @ ( minus_minus @ ( set @ A ) @ A7 @ ( insert @ A @ A2 @ ( bot_bot @ ( set @ A ) ) ) ) )
= A7 ) ) ).
% insert_Diff
thf(fact_157_Diff__insert2,axiom,
! [A: $tType,A7: set @ A,A2: A,B7: set @ A] :
( ( minus_minus @ ( set @ A ) @ A7 @ ( insert @ A @ A2 @ B7 ) )
= ( minus_minus @ ( set @ A ) @ ( minus_minus @ ( set @ A ) @ A7 @ ( insert @ A @ A2 @ ( bot_bot @ ( set @ A ) ) ) ) @ B7 ) ) ).
% Diff_insert2
thf(fact_158_Diff__insert__absorb,axiom,
! [A: $tType,X: A,A7: set @ A] :
( ~ ( member @ A @ X @ A7 )
=> ( ( minus_minus @ ( set @ A ) @ ( insert @ A @ X @ A7 ) @ ( insert @ A @ X @ ( bot_bot @ ( set @ A ) ) ) )
= A7 ) ) ).
% Diff_insert_absorb
thf(fact_159_infinite__remove,axiom,
! [A: $tType,S2: set @ A,A2: A] :
( ~ ( finite_finite2 @ A @ S2 )
=> ~ ( finite_finite2 @ A @ ( minus_minus @ ( set @ A ) @ S2 @ ( insert @ A @ A2 @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ).
% infinite_remove
thf(fact_160_infinite__coinduct,axiom,
! [A: $tType,X9: ( set @ A ) > $o,A7: set @ A] :
( ( X9 @ A7 )
=> ( ! [A9: set @ A] :
( ( X9 @ A9 )
=> ? [X7: A] :
( ( member @ A @ X7 @ A9 )
& ( ( X9 @ ( minus_minus @ ( set @ A ) @ A9 @ ( insert @ A @ X7 @ ( bot_bot @ ( set @ A ) ) ) ) )
| ~ ( finite_finite2 @ A @ ( minus_minus @ ( set @ A ) @ A9 @ ( insert @ A @ X7 @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ) )
=> ~ ( finite_finite2 @ A @ A7 ) ) ) ).
% infinite_coinduct
thf(fact_161_is__singletonI,axiom,
! [A: $tType,X: A] : ( is_singleton @ A @ ( insert @ A @ X @ ( bot_bot @ ( set @ A ) ) ) ) ).
% is_singletonI
thf(fact_162_remove__def,axiom,
! [A: $tType] :
( ( remove @ A )
= ( ^ [X5: A,A8: set @ A] : ( minus_minus @ ( set @ A ) @ A8 @ ( insert @ A @ X5 @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ).
% remove_def
thf(fact_163_the__elem__eq,axiom,
! [A: $tType,X: A] :
( ( the_elem @ A @ ( insert @ A @ X @ ( bot_bot @ ( set @ A ) ) ) )
= X ) ).
% the_elem_eq
thf(fact_164_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_165_member__remove,axiom,
! [A: $tType,X: A,Y: A,A7: set @ A] :
( ( member @ A @ X @ ( remove @ A @ Y @ A7 ) )
= ( ( member @ A @ X @ A7 )
& ( X != Y ) ) ) ).
% member_remove
thf(fact_166_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_167_setl_Ocases,axiom,
! [B: $tType,A: $tType,A2: A,S: sum_sum @ A @ B] :
( ( member @ A @ A2 @ ( basic_setl @ A @ B @ S ) )
=> ( S
= ( sum_Inl @ A @ B @ A2 ) ) ) ).
% setl.cases
thf(fact_168_setl_Osimps,axiom,
! [B: $tType,A: $tType,A2: A,S: sum_sum @ A @ B] :
( ( member @ A @ A2 @ ( basic_setl @ A @ B @ S ) )
= ( ? [X5: A] :
( ( A2 = X5 )
& ( S
= ( sum_Inl @ A @ B @ X5 ) ) ) ) ) ).
% setl.simps
thf(fact_169_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_170_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 ) )
=> ( ! [X6: A] :
( ( S
= ( sum_Inl @ A @ B @ X6 ) )
=> ( P @ X6 ) )
=> ( P @ X ) ) ) ).
% setl.inducts
thf(fact_171_is__singleton__the__elem,axiom,
! [A: $tType] :
( ( is_singleton @ A )
= ( ^ [A8: set @ A] :
( A8
= ( insert @ A @ ( the_elem @ A @ A8 ) @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ).
% is_singleton_the_elem
thf(fact_172_is__singletonI_H,axiom,
! [A: $tType,A7: set @ A] :
( ( A7
!= ( bot_bot @ ( set @ A ) ) )
=> ( ! [X6: A,Y3: A] :
( ( member @ A @ X6 @ A7 )
=> ( ( member @ A @ Y3 @ A7 )
=> ( X6 = Y3 ) ) )
=> ( is_singleton @ A @ A7 ) ) ) ).
% is_singletonI'
thf(fact_173_is__singleton__def,axiom,
! [A: $tType] :
( ( is_singleton @ A )
= ( ^ [A8: set @ A] :
? [X5: A] :
( A8
= ( insert @ A @ X5 @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ).
% is_singleton_def
thf(fact_174_is__singletonE,axiom,
! [A: $tType,A7: set @ A] :
( ( is_singleton @ A @ A7 )
=> ~ ! [X6: A] :
( A7
!= ( insert @ A @ X6 @ ( bot_bot @ ( set @ A ) ) ) ) ) ).
% is_singletonE
thf(fact_175_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_176_Set_Ois__empty__def,axiom,
! [A: $tType] :
( ( is_empty @ A )
= ( ^ [A8: set @ A] :
( A8
= ( bot_bot @ ( set @ A ) ) ) ) ) ).
% Set.is_empty_def
thf(fact_177_remove__induct,axiom,
! [A: $tType,P: ( set @ A ) > $o,B7: set @ A] :
( ( P @ ( bot_bot @ ( set @ A ) ) )
=> ( ( ~ ( finite_finite2 @ A @ B7 )
=> ( P @ B7 ) )
=> ( ! [A9: set @ A] :
( ( finite_finite2 @ A @ A9 )
=> ( ( A9
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( ord_less_eq @ ( set @ A ) @ A9 @ B7 )
=> ( ! [X7: A] :
( ( member @ A @ X7 @ A9 )
=> ( P @ ( minus_minus @ ( set @ A ) @ A9 @ ( insert @ A @ X7 @ ( bot_bot @ ( set @ A ) ) ) ) ) )
=> ( P @ A9 ) ) ) ) )
=> ( P @ B7 ) ) ) ) ).
% remove_induct
thf(fact_178_subset__antisym,axiom,
! [A: $tType,A7: set @ A,B7: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ A7 @ B7 )
=> ( ( ord_less_eq @ ( set @ A ) @ B7 @ A7 )
=> ( A7 = B7 ) ) ) ).
% subset_antisym
thf(fact_179_subsetI,axiom,
! [A: $tType,A7: set @ A,B7: set @ A] :
( ! [X6: A] :
( ( member @ A @ X6 @ A7 )
=> ( member @ A @ X6 @ B7 ) )
=> ( ord_less_eq @ ( set @ A ) @ A7 @ B7 ) ) ).
% subsetI
thf(fact_180_subset__empty,axiom,
! [A: $tType,A7: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ A7 @ ( bot_bot @ ( set @ A ) ) )
= ( A7
= ( bot_bot @ ( set @ A ) ) ) ) ).
% subset_empty
thf(fact_181_empty__subsetI,axiom,
! [A: $tType,A7: set @ A] : ( ord_less_eq @ ( set @ A ) @ ( bot_bot @ ( set @ A ) ) @ A7 ) ).
% empty_subsetI
thf(fact_182_insert__subset,axiom,
! [A: $tType,X: A,A7: set @ A,B7: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ ( insert @ A @ X @ A7 ) @ B7 )
= ( ( member @ A @ X @ B7 )
& ( ord_less_eq @ ( set @ A ) @ A7 @ B7 ) ) ) ).
% insert_subset
thf(fact_183_singleton__insert__inj__eq,axiom,
! [A: $tType,B2: A,A2: A,A7: set @ A] :
( ( ( insert @ A @ B2 @ ( bot_bot @ ( set @ A ) ) )
= ( insert @ A @ A2 @ A7 ) )
= ( ( A2 = B2 )
& ( ord_less_eq @ ( set @ A ) @ A7 @ ( insert @ A @ B2 @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ).
% singleton_insert_inj_eq
thf(fact_184_singleton__insert__inj__eq_H,axiom,
! [A: $tType,A2: A,A7: set @ A,B2: A] :
( ( ( insert @ A @ A2 @ A7 )
= ( insert @ A @ B2 @ ( bot_bot @ ( set @ A ) ) ) )
= ( ( A2 = B2 )
& ( ord_less_eq @ ( set @ A ) @ A7 @ ( insert @ A @ B2 @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ).
% singleton_insert_inj_eq'
thf(fact_185_Diff__eq__empty__iff,axiom,
! [A: $tType,A7: set @ A,B7: set @ A] :
( ( ( minus_minus @ ( set @ A ) @ A7 @ B7 )
= ( bot_bot @ ( set @ A ) ) )
= ( ord_less_eq @ ( set @ A ) @ A7 @ B7 ) ) ).
% Diff_eq_empty_iff
thf(fact_186_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_187_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_188_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 ) )
=> ( ! [X6: B] :
( ( S
= ( sum_Inr @ B @ A @ X6 ) )
=> ( P @ X6 ) )
=> ( P @ X ) ) ) ).
% setr.inducts
thf(fact_189_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_190_setr_Osimps,axiom,
! [A: $tType,B: $tType,A2: B,S: sum_sum @ A @ B] :
( ( member @ B @ A2 @ ( basic_setr @ A @ B @ S ) )
= ( ? [X5: B] :
( ( A2 = X5 )
& ( S
= ( sum_Inr @ B @ A @ X5 ) ) ) ) ) ).
% setr.simps
thf(fact_191_setr_Ocases,axiom,
! [A: $tType,B: $tType,A2: B,S: sum_sum @ A @ B] :
( ( member @ B @ A2 @ ( basic_setr @ A @ B @ S ) )
=> ( S
= ( sum_Inr @ B @ A @ A2 ) ) ) ).
% setr.cases
thf(fact_192_finite__subset,axiom,
! [A: $tType,A7: set @ A,B7: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ A7 @ B7 )
=> ( ( finite_finite2 @ A @ B7 )
=> ( finite_finite2 @ A @ A7 ) ) ) ).
% finite_subset
thf(fact_193_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_194_rev__finite__subset,axiom,
! [A: $tType,B7: set @ A,A7: set @ A] :
( ( finite_finite2 @ A @ B7 )
=> ( ( ord_less_eq @ ( set @ A ) @ A7 @ B7 )
=> ( finite_finite2 @ A @ A7 ) ) ) ).
% rev_finite_subset
thf(fact_195_insert__mono,axiom,
! [A: $tType,C4: set @ A,D3: set @ A,A2: A] :
( ( ord_less_eq @ ( set @ A ) @ C4 @ D3 )
=> ( ord_less_eq @ ( set @ A ) @ ( insert @ A @ A2 @ C4 ) @ ( insert @ A @ A2 @ D3 ) ) ) ).
% insert_mono
thf(fact_196_subset__insert,axiom,
! [A: $tType,X: A,A7: set @ A,B7: set @ A] :
( ~ ( member @ A @ X @ A7 )
=> ( ( ord_less_eq @ ( set @ A ) @ A7 @ ( insert @ A @ X @ B7 ) )
= ( ord_less_eq @ ( set @ A ) @ A7 @ B7 ) ) ) ).
% subset_insert
thf(fact_197_subset__insertI,axiom,
! [A: $tType,B7: set @ A,A2: A] : ( ord_less_eq @ ( set @ A ) @ B7 @ ( insert @ A @ A2 @ B7 ) ) ).
% subset_insertI
thf(fact_198_subset__insertI2,axiom,
! [A: $tType,A7: set @ A,B7: set @ A,B2: A] :
( ( ord_less_eq @ ( set @ A ) @ A7 @ B7 )
=> ( ord_less_eq @ ( set @ A ) @ A7 @ ( insert @ A @ B2 @ B7 ) ) ) ).
% subset_insertI2
thf(fact_199_Diff__mono,axiom,
! [A: $tType,A7: set @ A,C4: set @ A,D3: set @ A,B7: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ A7 @ C4 )
=> ( ( ord_less_eq @ ( set @ A ) @ D3 @ B7 )
=> ( ord_less_eq @ ( set @ A ) @ ( minus_minus @ ( set @ A ) @ A7 @ B7 ) @ ( minus_minus @ ( set @ A ) @ C4 @ D3 ) ) ) ) ).
% Diff_mono
thf(fact_200_Diff__subset,axiom,
! [A: $tType,A7: set @ A,B7: set @ A] : ( ord_less_eq @ ( set @ A ) @ ( minus_minus @ ( set @ A ) @ A7 @ B7 ) @ A7 ) ).
% Diff_subset
thf(fact_201_double__diff,axiom,
! [A: $tType,A7: set @ A,B7: set @ A,C4: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ A7 @ B7 )
=> ( ( ord_less_eq @ ( set @ A ) @ B7 @ C4 )
=> ( ( minus_minus @ ( set @ A ) @ B7 @ ( minus_minus @ ( set @ A ) @ C4 @ A7 ) )
= A7 ) ) ) ).
% double_diff
thf(fact_202_Collect__mono__iff,axiom,
! [A: $tType,P: A > $o,Q: A > $o] :
( ( ord_less_eq @ ( set @ A ) @ ( collect @ A @ P ) @ ( collect @ A @ Q ) )
= ( ! [X5: A] :
( ( P @ X5 )
=> ( Q @ X5 ) ) ) ) ).
% Collect_mono_iff
thf(fact_203_contra__subsetD,axiom,
! [A: $tType,A7: set @ A,B7: set @ A,C3: A] :
( ( ord_less_eq @ ( set @ A ) @ A7 @ B7 )
=> ( ~ ( member @ A @ C3 @ B7 )
=> ~ ( member @ A @ C3 @ A7 ) ) ) ).
% contra_subsetD
thf(fact_204_set__eq__subset,axiom,
! [A: $tType] :
( ( ^ [Y4: set @ A,Z: set @ A] : ( Y4 = Z ) )
= ( ^ [A8: set @ A,B9: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ A8 @ B9 )
& ( ord_less_eq @ ( set @ A ) @ B9 @ A8 ) ) ) ) ).
% set_eq_subset
thf(fact_205_subset__trans,axiom,
! [A: $tType,A7: set @ A,B7: set @ A,C4: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ A7 @ B7 )
=> ( ( ord_less_eq @ ( set @ A ) @ B7 @ C4 )
=> ( ord_less_eq @ ( set @ A ) @ A7 @ C4 ) ) ) ).
% subset_trans
thf(fact_206_Collect__mono,axiom,
! [A: $tType,P: A > $o,Q: A > $o] :
( ! [X6: A] :
( ( P @ X6 )
=> ( Q @ X6 ) )
=> ( ord_less_eq @ ( set @ A ) @ ( collect @ A @ P ) @ ( collect @ A @ Q ) ) ) ).
% Collect_mono
thf(fact_207_subset__refl,axiom,
! [A: $tType,A7: set @ A] : ( ord_less_eq @ ( set @ A ) @ A7 @ A7 ) ).
% subset_refl
thf(fact_208_rev__subsetD,axiom,
! [A: $tType,C3: A,A7: set @ A,B7: set @ A] :
( ( member @ A @ C3 @ A7 )
=> ( ( ord_less_eq @ ( set @ A ) @ A7 @ B7 )
=> ( member @ A @ C3 @ B7 ) ) ) ).
% rev_subsetD
thf(fact_209_subset__iff,axiom,
! [A: $tType] :
( ( ord_less_eq @ ( set @ A ) )
= ( ^ [A8: set @ A,B9: set @ A] :
! [T3: A] :
( ( member @ A @ T3 @ A8 )
=> ( member @ A @ T3 @ B9 ) ) ) ) ).
% subset_iff
thf(fact_210_set__rev__mp,axiom,
! [A: $tType,X: A,A7: set @ A,B7: set @ A] :
( ( member @ A @ X @ A7 )
=> ( ( ord_less_eq @ ( set @ A ) @ A7 @ B7 )
=> ( member @ A @ X @ B7 ) ) ) ).
% set_rev_mp
thf(fact_211_equalityD2,axiom,
! [A: $tType,A7: set @ A,B7: set @ A] :
( ( A7 = B7 )
=> ( ord_less_eq @ ( set @ A ) @ B7 @ A7 ) ) ).
% equalityD2
thf(fact_212_equalityD1,axiom,
! [A: $tType,A7: set @ A,B7: set @ A] :
( ( A7 = B7 )
=> ( ord_less_eq @ ( set @ A ) @ A7 @ B7 ) ) ).
% equalityD1
thf(fact_213_subset__eq,axiom,
! [A: $tType] :
( ( ord_less_eq @ ( set @ A ) )
= ( ^ [A8: set @ A,B9: set @ A] :
! [X5: A] :
( ( member @ A @ X5 @ A8 )
=> ( member @ A @ X5 @ B9 ) ) ) ) ).
% subset_eq
thf(fact_214_equalityE,axiom,
! [A: $tType,A7: set @ A,B7: set @ A] :
( ( A7 = B7 )
=> ~ ( ( ord_less_eq @ ( set @ A ) @ A7 @ B7 )
=> ~ ( ord_less_eq @ ( set @ A ) @ B7 @ A7 ) ) ) ).
% equalityE
thf(fact_215_subsetCE,axiom,
! [A: $tType,A7: set @ A,B7: set @ A,C3: A] :
( ( ord_less_eq @ ( set @ A ) @ A7 @ B7 )
=> ( ( member @ A @ C3 @ A7 )
=> ( member @ A @ C3 @ B7 ) ) ) ).
% subsetCE
thf(fact_216_subsetD,axiom,
! [A: $tType,A7: set @ A,B7: set @ A,C3: A] :
( ( ord_less_eq @ ( set @ A ) @ A7 @ B7 )
=> ( ( member @ A @ C3 @ A7 )
=> ( member @ A @ C3 @ B7 ) ) ) ).
% subsetD
thf(fact_217_in__mono,axiom,
! [A: $tType,A7: set @ A,B7: set @ A,X: A] :
( ( ord_less_eq @ ( set @ A ) @ A7 @ B7 )
=> ( ( member @ A @ X @ A7 )
=> ( member @ A @ X @ B7 ) ) ) ).
% in_mono
thf(fact_218_set__mp,axiom,
! [A: $tType,A7: set @ A,B7: set @ A,X: A] :
( ( ord_less_eq @ ( set @ A ) @ A7 @ B7 )
=> ( ( member @ A @ X @ A7 )
=> ( member @ A @ X @ B7 ) ) ) ).
% set_mp
thf(fact_219_subset__singletonD,axiom,
! [A: $tType,A7: set @ A,X: A] :
( ( ord_less_eq @ ( set @ A ) @ A7 @ ( insert @ A @ X @ ( bot_bot @ ( set @ A ) ) ) )
=> ( ( A7
= ( bot_bot @ ( set @ A ) ) )
| ( A7
= ( insert @ A @ X @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ).
% subset_singletonD
thf(fact_220_subset__singleton__iff,axiom,
! [A: $tType,X9: set @ A,A2: A] :
( ( ord_less_eq @ ( set @ A ) @ X9 @ ( insert @ A @ A2 @ ( bot_bot @ ( set @ A ) ) ) )
= ( ( X9
= ( bot_bot @ ( set @ A ) ) )
| ( X9
= ( insert @ A @ A2 @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ).
% subset_singleton_iff
thf(fact_221_subset__Diff__insert,axiom,
! [A: $tType,A7: set @ A,B7: set @ A,X: A,C4: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ A7 @ ( minus_minus @ ( set @ A ) @ B7 @ ( insert @ A @ X @ C4 ) ) )
= ( ( ord_less_eq @ ( set @ A ) @ A7 @ ( minus_minus @ ( set @ A ) @ B7 @ C4 ) )
& ~ ( member @ A @ X @ A7 ) ) ) ).
% subset_Diff_insert
thf(fact_222_finite__subset__induct,axiom,
! [A: $tType,F7: set @ A,A7: set @ A,P: ( set @ A ) > $o] :
( ( finite_finite2 @ A @ F7 )
=> ( ( ord_less_eq @ ( set @ A ) @ F7 @ A7 )
=> ( ( P @ ( bot_bot @ ( set @ A ) ) )
=> ( ! [A4: A,F6: set @ A] :
( ( finite_finite2 @ A @ F6 )
=> ( ( member @ A @ A4 @ A7 )
=> ( ~ ( member @ A @ A4 @ F6 )
=> ( ( P @ F6 )
=> ( P @ ( insert @ A @ A4 @ F6 ) ) ) ) ) )
=> ( P @ F7 ) ) ) ) ) ).
% finite_subset_induct
thf(fact_223_subset__insert__iff,axiom,
! [A: $tType,A7: set @ A,X: A,B7: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ A7 @ ( insert @ A @ X @ B7 ) )
= ( ( ( member @ A @ X @ A7 )
=> ( ord_less_eq @ ( set @ A ) @ ( minus_minus @ ( set @ A ) @ A7 @ ( insert @ A @ X @ ( bot_bot @ ( set @ A ) ) ) ) @ B7 ) )
& ( ~ ( member @ A @ X @ A7 )
=> ( ord_less_eq @ ( set @ A ) @ A7 @ B7 ) ) ) ) ).
% subset_insert_iff
thf(fact_224_Diff__single__insert,axiom,
! [A: $tType,A7: set @ A,X: A,B7: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ ( minus_minus @ ( set @ A ) @ A7 @ ( insert @ A @ X @ ( bot_bot @ ( set @ A ) ) ) ) @ B7 )
=> ( ord_less_eq @ ( set @ A ) @ A7 @ ( insert @ A @ X @ B7 ) ) ) ).
% Diff_single_insert
thf(fact_225_finite__remove__induct,axiom,
! [A: $tType,B7: set @ A,P: ( set @ A ) > $o] :
( ( finite_finite2 @ A @ B7 )
=> ( ( P @ ( bot_bot @ ( set @ A ) ) )
=> ( ! [A9: set @ A] :
( ( finite_finite2 @ A @ A9 )
=> ( ( A9
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( ord_less_eq @ ( set @ A ) @ A9 @ B7 )
=> ( ! [X7: A] :
( ( member @ A @ X7 @ A9 )
=> ( P @ ( minus_minus @ ( set @ A ) @ A9 @ ( insert @ A @ X7 @ ( bot_bot @ ( set @ A ) ) ) ) ) )
=> ( P @ A9 ) ) ) ) )
=> ( P @ B7 ) ) ) ) ).
% finite_remove_induct
thf(fact_226_insert__subsetI,axiom,
! [A: $tType,X: A,A7: set @ A,X9: set @ A] :
( ( member @ A @ X @ A7 )
=> ( ( ord_less_eq @ ( set @ A ) @ X9 @ A7 )
=> ( ord_less_eq @ ( set @ A ) @ ( insert @ A @ X @ X9 ) @ A7 ) ) ) ).
% insert_subsetI
thf(fact_227_diff__mono,axiom,
! [A: $tType] :
( ( ordered_ab_group_add @ A @ ( type2 @ A ) )
=> ! [A2: A,B2: A,D2: A,C3: A] :
( ( ord_less_eq @ A @ A2 @ B2 )
=> ( ( ord_less_eq @ A @ D2 @ C3 )
=> ( ord_less_eq @ A @ ( minus_minus @ A @ A2 @ C3 ) @ ( minus_minus @ A @ B2 @ D2 ) ) ) ) ) ).
% diff_mono
thf(fact_228_diff__left__mono,axiom,
! [A: $tType] :
( ( ordered_ab_group_add @ A @ ( type2 @ A ) )
=> ! [B2: A,A2: A,C3: A] :
( ( ord_less_eq @ A @ B2 @ A2 )
=> ( ord_less_eq @ A @ ( minus_minus @ A @ C3 @ A2 ) @ ( minus_minus @ A @ C3 @ B2 ) ) ) ) ).
% diff_left_mono
thf(fact_229_diff__right__commute,axiom,
! [A: $tType] :
( ( cancel146912293up_add @ A @ ( type2 @ A ) )
=> ! [A2: A,C3: A,B2: A] :
( ( minus_minus @ A @ ( minus_minus @ A @ A2 @ C3 ) @ B2 )
= ( minus_minus @ A @ ( minus_minus @ A @ A2 @ B2 ) @ C3 ) ) ) ).
% diff_right_commute
thf(fact_230_diff__eq__diff__eq,axiom,
! [A: $tType] :
( ( group_add @ A @ ( type2 @ A ) )
=> ! [A2: A,B2: A,C3: A,D2: A] :
( ( ( minus_minus @ A @ A2 @ B2 )
= ( minus_minus @ A @ C3 @ D2 ) )
=> ( ( A2 = B2 )
= ( C3 = D2 ) ) ) ) ).
% diff_eq_diff_eq
thf(fact_231_diff__eq__diff__less__eq,axiom,
! [A: $tType] :
( ( ordered_ab_group_add @ A @ ( type2 @ A ) )
=> ! [A2: A,B2: A,C3: A,D2: A] :
( ( ( minus_minus @ A @ A2 @ B2 )
= ( minus_minus @ A @ C3 @ D2 ) )
=> ( ( ord_less_eq @ A @ A2 @ B2 )
= ( ord_less_eq @ A @ C3 @ D2 ) ) ) ) ).
% diff_eq_diff_less_eq
thf(fact_232_diff__right__mono,axiom,
! [A: $tType] :
( ( ordered_ab_group_add @ A @ ( type2 @ A ) )
=> ! [A2: A,B2: A,C3: A] :
( ( ord_less_eq @ A @ A2 @ B2 )
=> ( ord_less_eq @ A @ ( minus_minus @ A @ A2 @ C3 ) @ ( minus_minus @ A @ B2 @ C3 ) ) ) ) ).
% diff_right_mono
thf(fact_233_psubset__insert__iff,axiom,
! [A: $tType,A7: set @ A,X: A,B7: set @ A] :
( ( ord_less @ ( set @ A ) @ A7 @ ( insert @ A @ X @ B7 ) )
= ( ( ( member @ A @ X @ B7 )
=> ( ord_less @ ( set @ A ) @ A7 @ B7 ) )
& ( ~ ( member @ A @ X @ B7 )
=> ( ( ( member @ A @ X @ A7 )
=> ( ord_less @ ( set @ A ) @ ( minus_minus @ ( set @ A ) @ A7 @ ( insert @ A @ X @ ( bot_bot @ ( set @ A ) ) ) ) @ B7 ) )
& ( ~ ( member @ A @ X @ A7 )
=> ( ord_less_eq @ ( set @ A ) @ A7 @ B7 ) ) ) ) ) ) ).
% psubset_insert_iff
thf(fact_234_finite__induct__select,axiom,
! [A: $tType,S2: set @ A,P: ( set @ A ) > $o] :
( ( finite_finite2 @ A @ S2 )
=> ( ( P @ ( bot_bot @ ( set @ A ) ) )
=> ( ! [T6: set @ A] :
( ( ord_less @ ( set @ A ) @ T6 @ S2 )
=> ( ( P @ T6 )
=> ? [X7: A] :
( ( member @ A @ X7 @ ( minus_minus @ ( set @ A ) @ S2 @ T6 ) )
& ( P @ ( insert @ A @ X7 @ T6 ) ) ) ) )
=> ( P @ S2 ) ) ) ) ).
% finite_induct_select
thf(fact_235_psubsetI,axiom,
! [A: $tType,A7: set @ A,B7: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ A7 @ B7 )
=> ( ( A7 != B7 )
=> ( ord_less @ ( set @ A ) @ A7 @ B7 ) ) ) ).
% psubsetI
thf(fact_236_subset__iff__psubset__eq,axiom,
! [A: $tType] :
( ( ord_less_eq @ ( set @ A ) )
= ( ^ [A8: set @ A,B9: set @ A] :
( ( ord_less @ ( set @ A ) @ A8 @ B9 )
| ( A8 = B9 ) ) ) ) ).
% subset_iff_psubset_eq
thf(fact_237_subset__psubset__trans,axiom,
! [A: $tType,A7: set @ A,B7: set @ A,C4: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ A7 @ B7 )
=> ( ( ord_less @ ( set @ A ) @ B7 @ C4 )
=> ( ord_less @ ( set @ A ) @ A7 @ C4 ) ) ) ).
% subset_psubset_trans
thf(fact_238_subset__not__subset__eq,axiom,
! [A: $tType] :
( ( ord_less @ ( set @ A ) )
= ( ^ [A8: set @ A,B9: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ A8 @ B9 )
& ~ ( ord_less_eq @ ( set @ A ) @ B9 @ A8 ) ) ) ) ).
% subset_not_subset_eq
thf(fact_239_psubset__subset__trans,axiom,
! [A: $tType,A7: set @ A,B7: set @ A,C4: set @ A] :
( ( ord_less @ ( set @ A ) @ A7 @ B7 )
=> ( ( ord_less_eq @ ( set @ A ) @ B7 @ C4 )
=> ( ord_less @ ( set @ A ) @ A7 @ C4 ) ) ) ).
% psubset_subset_trans
thf(fact_240_psubset__imp__subset,axiom,
! [A: $tType,A7: set @ A,B7: set @ A] :
( ( ord_less @ ( set @ A ) @ A7 @ B7 )
=> ( ord_less_eq @ ( set @ A ) @ A7 @ B7 ) ) ).
% psubset_imp_subset
thf(fact_241_psubset__eq,axiom,
! [A: $tType] :
( ( ord_less @ ( set @ A ) )
= ( ^ [A8: set @ A,B9: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ A8 @ B9 )
& ( A8 != B9 ) ) ) ) ).
% psubset_eq
thf(fact_242_psubsetE,axiom,
! [A: $tType,A7: set @ A,B7: set @ A] :
( ( ord_less @ ( set @ A ) @ A7 @ B7 )
=> ~ ( ( ord_less_eq @ ( set @ A ) @ A7 @ B7 )
=> ( ord_less_eq @ ( set @ A ) @ B7 @ A7 ) ) ) ).
% psubsetE
thf(fact_243_diff__strict__right__mono,axiom,
! [A: $tType] :
( ( ordered_ab_group_add @ A @ ( type2 @ A ) )
=> ! [A2: A,B2: A,C3: A] :
( ( ord_less @ A @ A2 @ B2 )
=> ( ord_less @ A @ ( minus_minus @ A @ A2 @ C3 ) @ ( minus_minus @ A @ B2 @ C3 ) ) ) ) ).
% diff_strict_right_mono
thf(fact_244_diff__strict__left__mono,axiom,
! [A: $tType] :
( ( ordered_ab_group_add @ A @ ( type2 @ A ) )
=> ! [B2: A,A2: A,C3: A] :
( ( ord_less @ A @ B2 @ A2 )
=> ( ord_less @ A @ ( minus_minus @ A @ C3 @ A2 ) @ ( minus_minus @ A @ C3 @ B2 ) ) ) ) ).
% diff_strict_left_mono
thf(fact_245_diff__eq__diff__less,axiom,
! [A: $tType] :
( ( ordered_ab_group_add @ A @ ( type2 @ A ) )
=> ! [A2: A,B2: A,C3: A,D2: A] :
( ( ( minus_minus @ A @ A2 @ B2 )
= ( minus_minus @ A @ C3 @ D2 ) )
=> ( ( ord_less @ A @ A2 @ B2 )
= ( ord_less @ A @ C3 @ D2 ) ) ) ) ).
% diff_eq_diff_less
thf(fact_246_diff__strict__mono,axiom,
! [A: $tType] :
( ( ordered_ab_group_add @ A @ ( type2 @ A ) )
=> ! [A2: A,B2: A,D2: A,C3: A] :
( ( ord_less @ A @ A2 @ B2 )
=> ( ( ord_less @ A @ D2 @ C3 )
=> ( ord_less @ A @ ( minus_minus @ A @ A2 @ C3 ) @ ( minus_minus @ A @ B2 @ D2 ) ) ) ) ) ).
% diff_strict_mono
thf(fact_247_not__psubset__empty,axiom,
! [A: $tType,A7: set @ A] :
~ ( ord_less @ ( set @ A ) @ A7 @ ( bot_bot @ ( set @ A ) ) ) ).
% not_psubset_empty
thf(fact_248_finite__psubset__induct,axiom,
! [A: $tType,A7: set @ A,P: ( set @ A ) > $o] :
( ( finite_finite2 @ A @ A7 )
=> ( ! [A9: set @ A] :
( ( finite_finite2 @ A @ A9 )
=> ( ! [B11: set @ A] :
( ( ord_less @ ( set @ A ) @ B11 @ A9 )
=> ( P @ B11 ) )
=> ( P @ A9 ) ) )
=> ( P @ A7 ) ) ) ).
% finite_psubset_induct
thf(fact_249_psubset__imp__ex__mem,axiom,
! [A: $tType,A7: set @ A,B7: set @ A] :
( ( ord_less @ ( set @ A ) @ A7 @ B7 )
=> ? [B4: A] : ( member @ A @ B4 @ ( minus_minus @ ( set @ A ) @ B7 @ A7 ) ) ) ).
% psubset_imp_ex_mem
thf(fact_250_psubset__trans,axiom,
! [A: $tType,A7: set @ A,B7: set @ A,C4: set @ A] :
( ( ord_less @ ( set @ A ) @ A7 @ B7 )
=> ( ( ord_less @ ( set @ A ) @ B7 @ C4 )
=> ( ord_less @ ( set @ A ) @ A7 @ C4 ) ) ) ).
% psubset_trans
thf(fact_251_psubsetD,axiom,
! [A: $tType,A7: set @ A,B7: set @ A,C3: A] :
( ( ord_less @ ( set @ A ) @ A7 @ B7 )
=> ( ( member @ A @ C3 @ A7 )
=> ( member @ A @ C3 @ B7 ) ) ) ).
% psubsetD
thf(fact_252_finite__linorder__min__induct,axiom,
! [A: $tType] :
( ( linorder @ A @ ( type2 @ A ) )
=> ! [A7: set @ A,P: ( set @ A ) > $o] :
( ( finite_finite2 @ A @ A7 )
=> ( ( P @ ( bot_bot @ ( set @ A ) ) )
=> ( ! [B4: A,A9: set @ A] :
( ( finite_finite2 @ A @ A9 )
=> ( ! [X7: A] :
( ( member @ A @ X7 @ A9 )
=> ( ord_less @ A @ B4 @ X7 ) )
=> ( ( P @ A9 )
=> ( P @ ( insert @ A @ B4 @ A9 ) ) ) ) )
=> ( P @ A7 ) ) ) ) ) ).
% finite_linorder_min_induct
thf(fact_253_finite__linorder__max__induct,axiom,
! [A: $tType] :
( ( linorder @ A @ ( type2 @ A ) )
=> ! [A7: set @ A,P: ( set @ A ) > $o] :
( ( finite_finite2 @ A @ A7 )
=> ( ( P @ ( bot_bot @ ( set @ A ) ) )
=> ( ! [B4: A,A9: set @ A] :
( ( finite_finite2 @ A @ A9 )
=> ( ! [X7: A] :
( ( member @ A @ X7 @ A9 )
=> ( ord_less @ A @ X7 @ B4 ) )
=> ( ( P @ A9 )
=> ( P @ ( insert @ A @ B4 @ A9 ) ) ) ) )
=> ( P @ A7 ) ) ) ) ) ).
% finite_linorder_max_induct
thf(fact_254_infinite__growing,axiom,
! [A: $tType] :
( ( linorder @ A @ ( type2 @ A ) )
=> ! [X9: set @ A] :
( ( X9
!= ( bot_bot @ ( set @ A ) ) )
=> ( ! [X6: A] :
( ( member @ A @ X6 @ X9 )
=> ? [Xa: A] :
( ( member @ A @ Xa @ X9 )
& ( ord_less @ A @ X6 @ Xa ) ) )
=> ~ ( finite_finite2 @ A @ X9 ) ) ) ) ).
% infinite_growing
thf(fact_255_Sup__fin_Oantimono,axiom,
! [A: $tType] :
( ( semilattice_sup @ A @ ( type2 @ A ) )
=> ! [A7: set @ A,B7: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ A7 @ B7 )
=> ( ( A7
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( finite_finite2 @ A @ B7 )
=> ( ord_less_eq @ A @ ( lattic477160up_fin @ A @ A7 ) @ ( lattic477160up_fin @ A @ B7 ) ) ) ) ) ) ).
% Sup_fin.antimono
%----Type constructors (14)
thf(tcon_fun___Lattices_Osemilattice__sup,axiom,
! [A5: $tType,A11: $tType] :
( ( semilattice_sup @ A11 @ ( type2 @ A11 ) )
=> ( semilattice_sup @ ( A5 > A11 ) @ ( type2 @ ( A5 > A11 ) ) ) ) ).
thf(tcon_fun___Finite__Set_Ofinite,axiom,
! [A5: $tType,A11: $tType] :
( ( ( finite_finite @ A5 @ ( type2 @ A5 ) )
& ( finite_finite @ A11 @ ( type2 @ A11 ) ) )
=> ( finite_finite @ ( A5 > A11 ) @ ( type2 @ ( A5 > A11 ) ) ) ) ).
thf(tcon_fun___Groups_Ominus,axiom,
! [A5: $tType,A11: $tType] :
( ( minus @ A11 @ ( type2 @ A11 ) )
=> ( minus @ ( A5 > A11 ) @ ( type2 @ ( A5 > A11 ) ) ) ) ).
thf(tcon_Set_Oset___Lattices_Osemilattice__sup_1,axiom,
! [A5: $tType] : ( semilattice_sup @ ( set @ A5 ) @ ( type2 @ ( set @ A5 ) ) ) ).
thf(tcon_Set_Oset___Finite__Set_Ofinite_2,axiom,
! [A5: $tType] :
( ( finite_finite @ A5 @ ( type2 @ A5 ) )
=> ( finite_finite @ ( set @ A5 ) @ ( type2 @ ( set @ A5 ) ) ) ) ).
thf(tcon_Set_Oset___Groups_Ominus_3,axiom,
! [A5: $tType] : ( minus @ ( set @ A5 ) @ ( type2 @ ( set @ A5 ) ) ) ).
thf(tcon_HOL_Obool___Lattices_Osemilattice__sup_4,axiom,
semilattice_sup @ $o @ ( type2 @ $o ) ).
thf(tcon_HOL_Obool___Orderings_Olinorder,axiom,
linorder @ $o @ ( type2 @ $o ) ).
thf(tcon_HOL_Obool___Finite__Set_Ofinite_5,axiom,
finite_finite @ $o @ ( type2 @ $o ) ).
thf(tcon_HOL_Obool___Groups_Ominus_6,axiom,
minus @ $o @ ( type2 @ $o ) ).
thf(tcon_FSet_Ofset___Lattices_Osemilattice__sup_7,axiom,
! [A5: $tType] : ( semilattice_sup @ ( fset @ A5 ) @ ( type2 @ ( fset @ A5 ) ) ) ).
thf(tcon_FSet_Ofset___Finite__Set_Ofinite_8,axiom,
! [A5: $tType] :
( ( finite_finite @ A5 @ ( type2 @ A5 ) )
=> ( finite_finite @ ( fset @ A5 ) @ ( type2 @ ( fset @ A5 ) ) ) ) ).
thf(tcon_FSet_Ofset___Groups_Ominus_9,axiom,
! [A5: $tType] : ( minus @ ( fset @ A5 ) @ ( type2 @ ( fset @ A5 ) ) ) ).
thf(tcon_Sum__Type_Osum___Finite__Set_Ofinite_10,axiom,
! [A5: $tType,A11: $tType] :
( ( ( finite_finite @ A5 @ ( type2 @ A5 ) )
& ( finite_finite @ A11 @ ( type2 @ A11 ) ) )
=> ( finite_finite @ ( sum_sum @ A5 @ A11 ) @ ( type2 @ ( sum_sum @ A5 @ A11 ) ) ) ) ).
%----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 (3)
thf(conj_0,hypothesis,
member @ n @ ( root @ tra ) @ nsa ).
thf(conj_1,hypothesis,
member @ ( sum_sum @ t @ dtree ) @ ( sum_Inl @ t @ dtree @ ta ) @ ( cont @ tra ) ).
thf(conj_2,conjecture,
member @ n @ ( root @ tra ) @ nsa ).
%------------------------------------------------------------------------------