TPTP Problem File: COM191^1.p
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%------------------------------------------------------------------------------
% File : COM191^1 : TPTP v8.2.0. Released v7.0.0.
% Domain : Computing Theory
% Problem : Grammars and languages 882
% 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__882.p [Bla16]
% Status : Theorem
% Rating : 0.67 v8.1.0, 0.75 v7.5.0, 0.67 v7.2.0, 0.75 v7.1.0
% Syntax : Number of formulae : 333 ( 103 unt; 64 typ; 0 def)
% Number of atoms : 774 ( 247 equ; 0 cnn)
% Maximal formula atoms : 13 ( 2 avg)
% Number of connectives : 3932 ( 96 ~; 13 |; 86 &;3309 @)
% ( 0 <=>; 428 =>; 0 <=; 0 <~>)
% Maximal formula depth : 21 ( 8 avg)
% Number of types : 4 ( 3 usr)
% Number of type conns : 240 ( 240 >; 0 *; 0 +; 0 <<)
% Number of symbols : 64 ( 61 usr; 10 con; 0-6 aty)
% Number of variables : 1101 ( 82 ^; 908 !; 58 ?;1101 :)
% ( 40 !>; 0 ?*; 0 @-; 13 @+)
% SPC : TH1_THM_EQU_NAR
% Comments : This file was generated by Isabelle (most likely Sledgehammer)
% 2016-07-13 14:45:37.740
%------------------------------------------------------------------------------
%----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_List_Olist,type,
list: $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 (57)
thf(sy_cl_HOL_Otype,type,
type:
!>[A: $tType] : ( ( itself @ A ) > $o ) ).
thf(sy_cl_Orderings_Oord,type,
ord:
!>[A: $tType] : ( ( itself @ A ) > $o ) ).
thf(sy_cl_Orderings_Oorder,type,
order:
!>[A: $tType] : ( ( itself @ A ) > $o ) ).
thf(sy_cl_Orderings_Olinorder,type,
linorder:
!>[A: $tType] : ( ( itself @ A ) > $o ) ).
thf(sy_cl_Orderings_Opreorder,type,
preorder:
!>[A: $tType] : ( ( itself @ A ) > $o ) ).
thf(sy_c_BNF__Greatest__Fixpoint_Oshift,type,
bNF_Greatest_shift:
!>[A: $tType,B: $tType] : ( ( ( list @ A ) > B ) > A > ( list @ A ) > B ) ).
thf(sy_c_BNF__Greatest__Fixpoint_Ouniv,type,
bNF_Greatest_univ:
!>[B: $tType,A: $tType] : ( ( B > A ) > ( set @ B ) > A ) ).
thf(sy_c_DTree_ONode,type,
node: n > ( set @ ( sum_sum @ t @ dtree ) ) > dtree ).
thf(sy_c_DTree_Ocont,type,
cont: dtree > ( set @ ( sum_sum @ t @ dtree ) ) ).
thf(sy_c_DTree_Odtree_Oroot,type,
root: dtree > n ).
thf(sy_c_Gram__Lang__Mirabelle__ojxrtuoybn_OItr,type,
gram_L1580978439le_Itr: ( set @ n ) > dtree > ( set @ n ) ).
thf(sy_c_Gram__Lang__Mirabelle__ojxrtuoybn_Ohsubst__c,type,
gram_L1905609002ubst_c: dtree > dtree > ( set @ ( sum_sum @ t @ dtree ) ) ).
thf(sy_c_Gram__Lang__Mirabelle__ojxrtuoybn_Ohsubst__r,type,
gram_L1905609017ubst_r: dtree > n ).
thf(sy_c_Gram__Lang__Mirabelle__ojxrtuoybn_OinFr,type,
gram_L1333338417e_inFr: ( set @ n ) > dtree > t > $o ).
thf(sy_c_Gram__Lang__Mirabelle__ojxrtuoybn_OinFr2,type,
gram_L805317441_inFr2: ( set @ n ) > dtree > t > $o ).
thf(sy_c_Gram__Lang__Mirabelle__ojxrtuoybn_OinItr,type,
gram_L830233218_inItr: ( set @ n ) > dtree > n > $o ).
thf(sy_c_Gram__Lang__Mirabelle__ojxrtuoybn_Opath,type,
gram_L250615845e_path: ( n > dtree ) > ( list @ n ) > $o ).
thf(sy_c_Gram__Lang__Mirabelle__ojxrtuoybn_Oreg,type,
gram_L1918716148le_reg: ( n > dtree ) > dtree > $o ).
thf(sy_c_Gram__Lang__Mirabelle__ojxrtuoybn_Oregular,type,
gram_L646766332egular: dtree > $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_Gram__Lang__Mirabelle__ojxrtuoybn_OsubtrOf,type,
gram_L1614515765ubtrOf: dtree > n > dtree ).
thf(sy_c_Gram__Lang__Mirabelle__ojxrtuoybn_Owf,type,
gram_L864798063lle_wf: dtree > $o ).
thf(sy_c_Hilbert__Choice_OGreatest,type,
hilbert_Greatest:
!>[A: $tType] : ( ( A > $o ) > A ) ).
thf(sy_c_Hilbert__Choice_OGreatestM,type,
hilbert_GreatestM:
!>[A: $tType,B: $tType] : ( ( A > B ) > ( A > $o ) > A ) ).
thf(sy_c_Hilbert__Choice_OLeastM,type,
hilbert_LeastM:
!>[A: $tType,B: $tType] : ( ( A > B ) > ( A > $o ) > A ) ).
thf(sy_c_If,type,
if:
!>[A: $tType] : ( $o > A > A > A ) ).
thf(sy_c_List_Ocoset,type,
coset:
!>[A: $tType] : ( ( list @ A ) > ( set @ A ) ) ).
thf(sy_c_List_Odistinct,type,
distinct:
!>[A: $tType] : ( ( list @ A ) > $o ) ).
thf(sy_c_List_Oinsert,type,
insert:
!>[A: $tType] : ( A > ( list @ A ) > ( list @ A ) ) ).
thf(sy_c_List_Olast,type,
last:
!>[A: $tType] : ( ( list @ A ) > A ) ).
thf(sy_c_List_Olist_OCons,type,
cons:
!>[A: $tType] : ( A > ( list @ A ) > ( list @ A ) ) ).
thf(sy_c_List_Olist_ONil,type,
nil:
!>[A: $tType] : ( list @ A ) ).
thf(sy_c_List_Olist_Ohd,type,
hd:
!>[A: $tType] : ( ( list @ A ) > A ) ).
thf(sy_c_List_Olist_Oset,type,
set2:
!>[A: $tType] : ( ( list @ A ) > ( set @ A ) ) ).
thf(sy_c_List_Oproduct__lists,type,
product_lists:
!>[A: $tType] : ( ( list @ ( list @ A ) ) > ( list @ ( list @ A ) ) ) ).
thf(sy_c_List_Osublists,type,
sublists:
!>[A: $tType] : ( ( list @ A ) > ( list @ ( list @ A ) ) ) ).
thf(sy_c_List_Ounion,type,
union:
!>[A: $tType] : ( ( list @ A ) > ( list @ A ) > ( list @ A ) ) ).
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,
insert2:
!>[A: $tType] : ( A > ( set @ A ) > ( set @ A ) ) ).
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_Oold_Osum_Orec__sum,type,
sum_rec_sum:
!>[A: $tType,T: $tType,B: $tType] : ( ( A > T ) > ( B > T ) > ( sum_sum @ A @ B ) > T ) ).
thf(sy_c_member,type,
member:
!>[A: $tType] : ( A > ( set @ A ) > $o ) ).
thf(sy_v_f,type,
f: n > dtree ).
thf(sy_v_nl____,type,
nl: list @ n ).
thf(sy_v_ns,type,
ns: set @ n ).
thf(sy_v_nsa____,type,
nsa: set @ n ).
thf(sy_v_tr,type,
tr: dtree ).
thf(sy_v_tr1,type,
tr1: dtree ).
thf(sy_v_tr1a____,type,
tr1a: dtree ).
thf(sy_v_tr2____,type,
tr2: dtree ).
thf(sy_v_tra____,type,
tra: dtree ).
%----Relevant facts (255)
thf(fact_0_reg__root,axiom,
! [F: n > dtree,Tr: dtree] :
( ( gram_L1918716148le_reg @ F @ Tr )
=> ( ( F @ ( root @ Tr ) )
= Tr ) ) ).
% reg_root
thf(fact_1_tr,axiom,
gram_L1918716148le_reg @ f @ tra ).
% tr
thf(fact_2_tr1__tr,axiom,
member @ ( sum_sum @ t @ dtree ) @ ( sum_Inr @ dtree @ t @ tr1a ) @ ( cont @ tra ) ).
% tr1_tr
thf(fact_3_nl,axiom,
gram_L250615845e_path @ f @ nl ).
% nl
thf(fact_4_f,axiom,
gram_L1918716148le_reg @ f @ tr ).
% f
thf(fact_5_f__nl,axiom,
( ( f @ ( hd @ n @ nl ) )
= tr1a ) ).
% f_nl
thf(fact_6_rtr,axiom,
member @ n @ ( root @ tra ) @ nsa ).
% rtr
thf(fact_7_tr1,axiom,
gram_L1918716148le_reg @ f @ tr1a ).
% tr1
thf(fact_8_path__NE,axiom,
! [F: n > dtree,Nl: list @ n] :
( ( gram_L250615845e_path @ F @ Nl )
=> ( Nl
!= ( nil @ n ) ) ) ).
% path_NE
thf(fact_9_path_OInd,axiom,
! [F: n > dtree,N1: n,Nl: list @ n,N: n] :
( ( gram_L250615845e_path @ F @ ( cons @ n @ N1 @ Nl ) )
=> ( ( member @ ( sum_sum @ t @ dtree ) @ ( sum_Inr @ dtree @ t @ ( F @ N1 ) ) @ ( cont @ ( F @ N ) ) )
=> ( gram_L250615845e_path @ F @ ( cons @ n @ N @ ( cons @ n @ N1 @ Nl ) ) ) ) ) ).
% path.Ind
thf(fact_10_path_OBase,axiom,
! [F: n > dtree,N: n] : ( gram_L250615845e_path @ F @ ( cons @ n @ N @ ( nil @ n ) ) ) ).
% path.Base
thf(fact_11_path__post,axiom,
! [F: n > dtree,N: n,Nl: list @ n] :
( ( gram_L250615845e_path @ F @ ( cons @ n @ N @ Nl ) )
=> ( ( Nl
!= ( nil @ n ) )
=> ( gram_L250615845e_path @ F @ Nl ) ) ) ).
% path_post
thf(fact_12_path_Ocases,axiom,
! [F: n > dtree,A2: list @ n] :
( ( gram_L250615845e_path @ F @ A2 )
=> ( ! [N2: n] :
( A2
!= ( cons @ n @ N2 @ ( nil @ n ) ) )
=> ~ ! [N12: n,Nl2: list @ n,N2: n] :
( ( A2
= ( cons @ n @ N2 @ ( cons @ n @ N12 @ Nl2 ) ) )
=> ( ( gram_L250615845e_path @ F @ ( cons @ n @ N12 @ Nl2 ) )
=> ~ ( member @ ( sum_sum @ t @ dtree ) @ ( sum_Inr @ dtree @ t @ ( F @ N12 ) ) @ ( cont @ ( F @ N2 ) ) ) ) ) ) ) ).
% path.cases
thf(fact_13_path_Osimps,axiom,
( gram_L250615845e_path
= ( ^ [F2: n > dtree,A3: list @ n] :
( ? [N3: n] :
( A3
= ( cons @ n @ N3 @ ( nil @ n ) ) )
| ? [N13: n,Nl3: list @ n,N3: n] :
( ( A3
= ( cons @ n @ N3 @ ( cons @ n @ N13 @ Nl3 ) ) )
& ( gram_L250615845e_path @ F2 @ ( cons @ n @ N13 @ Nl3 ) )
& ( member @ ( sum_sum @ t @ dtree ) @ ( sum_Inr @ dtree @ t @ ( F2 @ N13 ) ) @ ( cont @ ( F2 @ N3 ) ) ) ) ) ) ) ).
% path.simps
thf(fact_14_path_Oinducts,axiom,
! [F: n > dtree,X: list @ n,P: ( list @ n ) > $o] :
( ( gram_L250615845e_path @ F @ X )
=> ( ! [N2: n] : ( P @ ( cons @ n @ N2 @ ( nil @ n ) ) )
=> ( ! [N12: n,Nl2: list @ n,N2: n] :
( ( gram_L250615845e_path @ F @ ( cons @ n @ N12 @ Nl2 ) )
=> ( ( P @ ( cons @ n @ N12 @ Nl2 ) )
=> ( ( member @ ( sum_sum @ t @ dtree ) @ ( sum_Inr @ dtree @ t @ ( F @ N12 ) ) @ ( cont @ ( F @ N2 ) ) )
=> ( P @ ( cons @ n @ N2 @ ( cons @ n @ N12 @ Nl2 ) ) ) ) ) )
=> ( P @ X ) ) ) ) ).
% path.inducts
thf(fact_15_last__nl,axiom,
( ( f @ ( last @ n @ nl ) )
= tr2 ) ).
% last_nl
thf(fact_16_reg__Inr__cont,axiom,
! [F: n > dtree,Tr: dtree,Tr2: dtree] :
( ( gram_L1918716148le_reg @ F @ Tr )
=> ( ( member @ ( sum_sum @ t @ dtree ) @ ( sum_Inr @ dtree @ t @ Tr2 ) @ ( cont @ Tr ) )
=> ( gram_L1918716148le_reg @ F @ Tr2 ) ) ) ).
% reg_Inr_cont
thf(fact_17_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_18_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_19_list_Oinject,axiom,
! [A: $tType,X21: A,X22: list @ A,Y21: A,Y22: list @ A] :
( ( ( cons @ A @ X21 @ X22 )
= ( cons @ A @ Y21 @ Y22 ) )
= ( ( X21 = Y21 )
& ( X22 = Y22 ) ) ) ).
% list.inject
thf(fact_20_dtree__cong,axiom,
! [Tr: dtree,Tr2: dtree] :
( ( ( root @ Tr )
= ( root @ Tr2 ) )
=> ( ( ( cont @ Tr )
= ( cont @ Tr2 ) )
=> ( Tr = Tr2 ) ) ) ).
% dtree_cong
thf(fact_21_hsubst__c__def,axiom,
( gram_L1905609002ubst_c
= ( ^ [Tr0: dtree,Tr3: dtree] :
( if @ ( set @ ( sum_sum @ t @ dtree ) )
@ ( ( root @ Tr3 )
= ( root @ Tr0 ) )
@ ( cont @ Tr0 )
@ ( cont @ Tr3 ) ) ) ) ).
% hsubst_c_def
thf(fact_22_list_Odistinct_I1_J,axiom,
! [A: $tType,X21: A,X22: list @ A] :
( ( nil @ A )
!= ( cons @ A @ X21 @ X22 ) ) ).
% list.distinct(1)
thf(fact_23_list_OdiscI,axiom,
! [A: $tType,List: list @ A,X21: A,X22: list @ A] :
( ( List
= ( cons @ A @ X21 @ X22 ) )
=> ( List
!= ( nil @ A ) ) ) ).
% list.discI
thf(fact_24_list_Oexhaust,axiom,
! [A: $tType,Y: list @ A] :
( ( Y
!= ( nil @ A ) )
=> ~ ! [X212: A,X222: list @ A] :
( Y
!= ( cons @ A @ X212 @ X222 ) ) ) ).
% list.exhaust
thf(fact_25_list_Oinducts,axiom,
! [A: $tType,P: ( list @ A ) > $o,List: list @ A] :
( ( P @ ( nil @ A ) )
=> ( ! [X1: A,X23: list @ A] :
( ( P @ X23 )
=> ( P @ ( cons @ A @ X1 @ X23 ) ) )
=> ( P @ List ) ) ) ).
% list.inducts
thf(fact_26_neq__Nil__conv,axiom,
! [A: $tType,Xs: list @ A] :
( ( Xs
!= ( nil @ A ) )
= ( ? [Y3: A,Ys: list @ A] :
( Xs
= ( cons @ A @ Y3 @ Ys ) ) ) ) ).
% neq_Nil_conv
thf(fact_27_tr2__tr1,axiom,
gram_L716654942_subtr @ nsa @ tr2 @ tr1a ).
% tr2_tr1
thf(fact_28_last__ConsR,axiom,
! [A: $tType,Xs: list @ A,X: A] :
( ( Xs
!= ( nil @ A ) )
=> ( ( last @ A @ ( cons @ A @ X @ Xs ) )
= ( last @ A @ Xs ) ) ) ).
% last_ConsR
thf(fact_29_last__ConsL,axiom,
! [A: $tType,Xs: list @ A,X: A] :
( ( Xs
= ( nil @ A ) )
=> ( ( last @ A @ ( cons @ A @ X @ Xs ) )
= X ) ) ).
% last_ConsL
thf(fact_30_last_Osimps,axiom,
! [A: $tType,Xs: list @ A,X: A] :
( ( ( Xs
= ( nil @ A ) )
=> ( ( last @ A @ ( cons @ A @ X @ Xs ) )
= X ) )
& ( ( Xs
!= ( nil @ A ) )
=> ( ( last @ A @ ( cons @ A @ X @ Xs ) )
= ( last @ A @ Xs ) ) ) ) ).
% last.simps
thf(fact_31_list_Osel_I1_J,axiom,
! [A: $tType,X21: A,X22: list @ A] :
( ( hd @ A @ ( cons @ A @ X21 @ X22 ) )
= X21 ) ).
% list.sel(1)
thf(fact_32_not__Cons__self2,axiom,
! [A: $tType,X: A,Xs: list @ A] :
( ( cons @ A @ X @ Xs )
!= Xs ) ).
% not_Cons_self2
thf(fact_33_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_34_map__tailrec__rev_Oinduct,axiom,
! [A: $tType,B: $tType,P: ( A > B ) > ( list @ A ) > ( list @ B ) > $o,A0: A > B,A1: list @ A,A22: list @ B] :
( ! [F3: A > B,X1: list @ B] : ( P @ F3 @ ( nil @ A ) @ X1 )
=> ( ! [F3: A > B,A4: A,As: list @ A,Bs: list @ B] :
( ( P @ F3 @ As @ ( cons @ B @ ( F3 @ A4 ) @ Bs ) )
=> ( P @ F3 @ ( cons @ A @ A4 @ As ) @ Bs ) )
=> ( P @ A0 @ A1 @ A22 ) ) ) ).
% map_tailrec_rev.induct
thf(fact_35_list__nonempty__induct,axiom,
! [A: $tType,Xs: list @ A,P: ( list @ A ) > $o] :
( ( Xs
!= ( nil @ A ) )
=> ( ! [X3: A] : ( P @ ( cons @ A @ X3 @ ( nil @ A ) ) )
=> ( ! [X3: A,Xs2: list @ A] :
( ( Xs2
!= ( nil @ A ) )
=> ( ( P @ Xs2 )
=> ( P @ ( cons @ A @ X3 @ Xs2 ) ) ) )
=> ( P @ Xs ) ) ) ) ).
% list_nonempty_induct
thf(fact_36_remdups__adj_Oinduct,axiom,
! [A: $tType,P: ( list @ A ) > $o,A0: list @ A] :
( ( P @ ( nil @ A ) )
=> ( ! [X3: A] : ( P @ ( cons @ A @ X3 @ ( nil @ A ) ) )
=> ( ! [X3: A,Y4: A,Xs2: list @ A] :
( ( ( X3 = Y4 )
=> ( P @ ( cons @ A @ X3 @ Xs2 ) ) )
=> ( ( ( X3 != Y4 )
=> ( P @ ( cons @ A @ Y4 @ Xs2 ) ) )
=> ( P @ ( cons @ A @ X3 @ ( cons @ A @ Y4 @ Xs2 ) ) ) ) )
=> ( P @ A0 ) ) ) ) ).
% remdups_adj.induct
thf(fact_37_remdups__adj_Ocases,axiom,
! [A: $tType,X: list @ A] :
( ( X
!= ( nil @ A ) )
=> ( ! [X3: A] :
( X
!= ( cons @ A @ X3 @ ( nil @ A ) ) )
=> ~ ! [X3: A,Y4: A,Xs2: list @ A] :
( X
!= ( cons @ A @ X3 @ ( cons @ A @ Y4 @ Xs2 ) ) ) ) ) ).
% remdups_adj.cases
thf(fact_38_transpose_Ocases,axiom,
! [A: $tType,X: list @ ( list @ A )] :
( ( X
!= ( nil @ ( list @ A ) ) )
=> ( ! [Xss: list @ ( list @ A )] :
( X
!= ( cons @ ( list @ A ) @ ( nil @ A ) @ Xss ) )
=> ~ ! [X3: A,Xs2: list @ A,Xss: list @ ( list @ A )] :
( X
!= ( cons @ ( list @ A ) @ ( cons @ A @ X3 @ Xs2 ) @ Xss ) ) ) ) ).
% transpose.cases
thf(fact_39_splice_Oinduct,axiom,
! [A: $tType,P: ( list @ A ) > ( list @ A ) > $o,A0: list @ A,A1: list @ A] :
( ! [X1: list @ A] : ( P @ ( nil @ A ) @ X1 )
=> ( ! [V: A,Va: list @ A] : ( P @ ( cons @ A @ V @ Va ) @ ( nil @ A ) )
=> ( ! [X3: A,Xs2: list @ A,Y4: A,Ys2: list @ A] :
( ( P @ Xs2 @ Ys2 )
=> ( P @ ( cons @ A @ X3 @ Xs2 ) @ ( cons @ A @ Y4 @ Ys2 ) ) )
=> ( P @ A0 @ A1 ) ) ) ) ).
% splice.induct
thf(fact_40_list__induct2_H,axiom,
! [A: $tType,B: $tType,P: ( list @ A ) > ( list @ B ) > $o,Xs: list @ A,Ys3: list @ B] :
( ( P @ ( nil @ A ) @ ( nil @ B ) )
=> ( ! [X3: A,Xs2: list @ A] : ( P @ ( cons @ A @ X3 @ Xs2 ) @ ( nil @ B ) )
=> ( ! [Y4: B,Ys2: list @ B] : ( P @ ( nil @ A ) @ ( cons @ B @ Y4 @ Ys2 ) )
=> ( ! [X3: A,Xs2: list @ A,Y4: B,Ys2: list @ B] :
( ( P @ Xs2 @ Ys2 )
=> ( P @ ( cons @ A @ X3 @ Xs2 ) @ ( cons @ B @ Y4 @ Ys2 ) ) )
=> ( P @ Xs @ Ys3 ) ) ) ) ) ).
% list_induct2'
thf(fact_41_n,axiom,
gram_L716654942_subtr @ ns @ tr1 @ tr ).
% n
thf(fact_42__092_060open_062_092_060And_062thesis_O_A_I_092_060And_062nl_O_A_092_060lbrakk_062path_Af_Anl_059_Af_A_Ihd_Anl_J_A_061_Atr1_059_Af_A_Ilast_Anl_J_A_061_Atr2_059_Aset_Anl_A_092_060subseteq_062_Ans_092_060rbrakk_062_A_092_060Longrightarrow_062_Athesis_J_A_092_060Longrightarrow_062_Athesis_092_060close_062,axiom,
~ ! [Nl2: list @ n] :
( ( gram_L250615845e_path @ f @ Nl2 )
=> ( ( ( f @ ( hd @ n @ Nl2 ) )
= tr1a )
=> ( ( ( f @ ( last @ n @ Nl2 ) )
= tr2 )
=> ~ ( ord_less_eq @ ( set @ n ) @ ( set2 @ n @ Nl2 ) @ nsa ) ) ) ) ).
% \<open>\<And>thesis. (\<And>nl. \<lbrakk>path f nl; f (hd nl) = tr1; f (last nl) = tr2; set nl \<subseteq> ns\<rbrakk> \<Longrightarrow> thesis) \<Longrightarrow> thesis\<close>
thf(fact_43_old_Osum_Osimps_I8_J,axiom,
! [A: $tType,T: $tType,B: $tType,F1: A > T,F22: B > T,B2: B] :
( ( sum_rec_sum @ A @ T @ B @ F1 @ F22 @ ( sum_Inr @ B @ A @ B2 ) )
= ( F22 @ B2 ) ) ).
% old.sum.simps(8)
thf(fact_44_mem__Collect__eq,axiom,
! [A: $tType,A2: A,P: A > $o] :
( ( member @ A @ A2 @ ( collect @ A @ P ) )
= ( P @ A2 ) ) ).
% mem_Collect_eq
thf(fact_45_Collect__mem__eq,axiom,
! [A: $tType,A5: set @ A] :
( ( collect @ A
@ ^ [X4: A] : ( member @ A @ X4 @ A5 ) )
= A5 ) ).
% Collect_mem_eq
thf(fact_46_Collect__cong,axiom,
! [A: $tType,P: A > $o,Q: A > $o] :
( ! [X3: A] :
( ( P @ X3 )
= ( Q @ X3 ) )
=> ( ( collect @ A @ P )
= ( collect @ A @ Q ) ) ) ).
% Collect_cong
thf(fact_47_ext,axiom,
! [B: $tType,A: $tType,F: A > B,G: A > B] :
( ! [X3: A] :
( ( F @ X3 )
= ( G @ X3 ) )
=> ( F = G ) ) ).
% ext
thf(fact_48_Node__root__cont,axiom,
! [Tr: dtree] :
( ( node @ ( root @ Tr ) @ ( cont @ Tr ) )
= Tr ) ).
% Node_root_cont
thf(fact_49_regular__def2,axiom,
( gram_L646766332egular
= ( ^ [Tr3: dtree] :
? [F2: n > dtree] :
( ( gram_L1918716148le_reg @ F2 @ Tr3 )
& ! [N3: n] :
( ( root @ ( F2 @ N3 ) )
= N3 ) ) ) ) ).
% regular_def2
thf(fact_50_set,axiom,
ord_less_eq @ ( set @ n ) @ ( set2 @ n @ nl ) @ nsa ).
% set
thf(fact_51_Step_Oprems,axiom,
gram_L1918716148le_reg @ f @ tra ).
% Step.prems
thf(fact_52_wf__inj,axiom,
! [Tr: dtree,Tr1: dtree,Tr22: dtree] :
( ( gram_L864798063lle_wf @ Tr )
=> ( ( member @ ( sum_sum @ t @ dtree ) @ ( sum_Inr @ dtree @ t @ Tr1 ) @ ( cont @ Tr ) )
=> ( ( member @ ( sum_sum @ t @ dtree ) @ ( sum_Inr @ dtree @ t @ Tr22 ) @ ( cont @ Tr ) )
=> ( ( ( root @ Tr1 )
= ( root @ Tr22 ) )
= ( Tr1 = Tr22 ) ) ) ) ) ).
% wf_inj
thf(fact_53_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_54_Step_Ohyps_I3_J,axiom,
( ( gram_L1918716148le_reg @ f @ tr1a )
=> ? [Nl2: list @ n] :
( ( gram_L250615845e_path @ f @ Nl2 )
& ( ( f @ ( hd @ n @ Nl2 ) )
= tr1a )
& ( ( f @ ( last @ n @ Nl2 ) )
= tr2 )
& ( ord_less_eq @ ( set @ n ) @ ( set2 @ n @ Nl2 ) @ nsa ) ) ) ).
% Step.hyps(3)
thf(fact_55_root__Node,axiom,
! [N: n,As2: set @ ( sum_sum @ t @ dtree )] :
( ( root @ ( node @ N @ As2 ) )
= N ) ).
% root_Node
thf(fact_56_subset__code_I1_J,axiom,
! [A: $tType,Xs: list @ A,B4: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ ( set2 @ A @ Xs ) @ B4 )
= ( ! [X4: A] :
( ( member @ A @ X4 @ ( set2 @ A @ Xs ) )
=> ( member @ A @ X4 @ B4 ) ) ) ) ).
% subset_code(1)
thf(fact_57_set__subset__Cons,axiom,
! [A: $tType,Xs: list @ A,X: A] : ( ord_less_eq @ ( set @ A ) @ ( set2 @ A @ Xs ) @ ( set2 @ A @ ( cons @ A @ X @ Xs ) ) ) ).
% set_subset_Cons
thf(fact_58_regular__subtr,axiom,
! [Tr: dtree,Ns: set @ n,Tr2: dtree] :
( ( gram_L646766332egular @ Tr )
=> ( ( gram_L716654942_subtr @ Ns @ Tr2 @ Tr )
=> ( gram_L646766332egular @ Tr2 ) ) ) ).
% regular_subtr
thf(fact_59_subtr__trans,axiom,
! [Ns: set @ n,Tr1: dtree,Tr22: dtree,Tr32: dtree] :
( ( gram_L716654942_subtr @ Ns @ Tr1 @ Tr22 )
=> ( ( gram_L716654942_subtr @ Ns @ Tr22 @ Tr32 )
=> ( gram_L716654942_subtr @ Ns @ Tr1 @ Tr32 ) ) ) ).
% subtr_trans
thf(fact_60_subtr__mono,axiom,
! [Ns: set @ n,Tr1: dtree,Tr22: dtree,Ns2: set @ n] :
( ( gram_L716654942_subtr @ Ns @ Tr1 @ Tr22 )
=> ( ( ord_less_eq @ ( set @ n ) @ Ns @ Ns2 )
=> ( gram_L716654942_subtr @ Ns2 @ Tr1 @ Tr22 ) ) ) ).
% subtr_mono
thf(fact_61_inFr2__mono,axiom,
! [Ns: set @ n,Tr: dtree,T2: t,Ns2: set @ n] :
( ( gram_L805317441_inFr2 @ Ns @ Tr @ T2 )
=> ( ( ord_less_eq @ ( set @ n ) @ Ns @ Ns2 )
=> ( gram_L805317441_inFr2 @ Ns2 @ Tr @ T2 ) ) ) ).
% inFr2_mono
thf(fact_62_wf__subtr,axiom,
! [Tr1: dtree,Ns: set @ n,Tr: dtree] :
( ( gram_L864798063lle_wf @ Tr1 )
=> ( ( gram_L716654942_subtr @ Ns @ Tr @ Tr1 )
=> ( gram_L864798063lle_wf @ Tr ) ) ) ).
% wf_subtr
thf(fact_63_list_Oset__intros_I2_J,axiom,
! [A: $tType,X: A,A22: list @ A,A1: A] :
( ( member @ A @ X @ ( set2 @ A @ A22 ) )
=> ( member @ A @ X @ ( set2 @ A @ ( cons @ A @ A1 @ A22 ) ) ) ) ).
% list.set_intros(2)
thf(fact_64_list_Oset__intros_I1_J,axiom,
! [A: $tType,A1: A,A22: list @ A] : ( member @ A @ A1 @ ( set2 @ A @ ( cons @ A @ A1 @ A22 ) ) ) ).
% list.set_intros(1)
thf(fact_65_set__ConsD,axiom,
! [A: $tType,Y: A,X: A,Xs: list @ A] :
( ( member @ A @ Y @ ( set2 @ A @ ( cons @ A @ X @ Xs ) ) )
=> ( ( Y = X )
| ( member @ A @ Y @ ( set2 @ A @ Xs ) ) ) ) ).
% set_ConsD
thf(fact_66_list_Oset__cases,axiom,
! [A: $tType,E: A,A2: list @ A] :
( ( member @ A @ E @ ( set2 @ A @ A2 ) )
=> ( ! [Z2: list @ A] :
( A2
!= ( cons @ A @ E @ Z2 ) )
=> ~ ! [Z1: A,Z2: list @ A] :
( ( A2
= ( cons @ A @ Z1 @ Z2 ) )
=> ~ ( member @ A @ E @ ( set2 @ A @ Z2 ) ) ) ) ) ).
% list.set_cases
thf(fact_67_subtr_ORefl,axiom,
! [Tr: dtree,Ns: set @ n] :
( ( member @ n @ ( root @ Tr ) @ Ns )
=> ( gram_L716654942_subtr @ Ns @ Tr @ Tr ) ) ).
% subtr.Refl
thf(fact_68_subtr__rootL__in,axiom,
! [Ns: set @ n,Tr1: dtree,Tr22: dtree] :
( ( gram_L716654942_subtr @ Ns @ Tr1 @ Tr22 )
=> ( member @ n @ ( root @ Tr1 ) @ Ns ) ) ).
% subtr_rootL_in
thf(fact_69_subtr__rootR__in,axiom,
! [Ns: set @ n,Tr1: dtree,Tr22: dtree] :
( ( gram_L716654942_subtr @ Ns @ Tr1 @ Tr22 )
=> ( member @ n @ ( root @ Tr22 ) @ Ns ) ) ).
% subtr_rootR_in
thf(fact_70_reg__subtr,axiom,
! [F: n > dtree,Tr: dtree,Ns: set @ n,Tr2: dtree] :
( ( gram_L1918716148le_reg @ F @ Tr )
=> ( ( gram_L716654942_subtr @ Ns @ Tr2 @ Tr )
=> ( gram_L1918716148le_reg @ F @ Tr2 ) ) ) ).
% reg_subtr
thf(fact_71_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_72_list_Oset__sel_I1_J,axiom,
! [A: $tType,A2: list @ A] :
( ( A2
!= ( nil @ A ) )
=> ( member @ A @ ( hd @ A @ A2 ) @ ( set2 @ A @ A2 ) ) ) ).
% list.set_sel(1)
thf(fact_73_hd__in__set,axiom,
! [A: $tType,Xs: list @ A] :
( ( Xs
!= ( nil @ A ) )
=> ( member @ A @ ( hd @ A @ Xs ) @ ( set2 @ A @ Xs ) ) ) ).
% hd_in_set
thf(fact_74_last__in__set,axiom,
! [A: $tType,As2: list @ A] :
( ( As2
!= ( nil @ A ) )
=> ( member @ A @ ( last @ A @ As2 ) @ ( set2 @ A @ As2 ) ) ) ).
% last_in_set
thf(fact_75_path__subtr,axiom,
! [F: n > dtree,Nl: list @ n] :
( ! [N2: n] :
( ( root @ ( F @ N2 ) )
= N2 )
=> ( ( gram_L250615845e_path @ F @ Nl )
=> ( gram_L716654942_subtr @ ( set2 @ n @ Nl ) @ ( F @ ( last @ n @ Nl ) ) @ ( F @ ( hd @ n @ Nl ) ) ) ) ) ).
% path_subtr
thf(fact_76_reg__def2,axiom,
( gram_L1918716148le_reg
= ( ^ [F2: n > dtree,Tr3: dtree] :
! [Ns3: set @ n,Tr4: dtree] :
( ( gram_L716654942_subtr @ Ns3 @ Tr4 @ Tr3 )
=> ( Tr4
= ( F2 @ ( root @ Tr4 ) ) ) ) ) ) ).
% reg_def2
thf(fact_77_wf__cont,axiom,
! [Tr: dtree,Tr2: dtree] :
( ( gram_L864798063lle_wf @ Tr )
=> ( ( member @ ( sum_sum @ t @ dtree ) @ ( sum_Inr @ dtree @ t @ Tr2 ) @ ( cont @ Tr ) )
=> ( gram_L864798063lle_wf @ Tr2 ) ) ) ).
% wf_cont
thf(fact_78_subtr_OStep,axiom,
! [Tr32: dtree,Ns: set @ n,Tr1: dtree,Tr22: dtree] :
( ( member @ n @ ( root @ Tr32 ) @ Ns )
=> ( ( gram_L716654942_subtr @ Ns @ Tr1 @ Tr22 )
=> ( ( member @ ( sum_sum @ t @ dtree ) @ ( sum_Inr @ dtree @ t @ Tr22 ) @ ( cont @ Tr32 ) )
=> ( gram_L716654942_subtr @ Ns @ Tr1 @ Tr32 ) ) ) ) ).
% subtr.Step
thf(fact_79_subtr_Ocases,axiom,
! [A1: set @ n,A22: dtree,A32: dtree] :
( ( gram_L716654942_subtr @ A1 @ A22 @ A32 )
=> ( ( ( A32 = A22 )
=> ~ ( member @ n @ ( root @ A22 ) @ A1 ) )
=> ~ ( ( member @ n @ ( root @ A32 ) @ A1 )
=> ! [Tr23: dtree] :
( ( gram_L716654942_subtr @ A1 @ A22 @ Tr23 )
=> ~ ( member @ ( sum_sum @ t @ dtree ) @ ( sum_Inr @ dtree @ t @ Tr23 ) @ ( cont @ A32 ) ) ) ) ) ) ).
% subtr.cases
thf(fact_80_subtr_Osimps,axiom,
( gram_L716654942_subtr
= ( ^ [A12: set @ n,A23: dtree,A33: dtree] :
( ? [Tr3: dtree,Ns3: set @ n] :
( ( A12 = Ns3 )
& ( A23 = Tr3 )
& ( A33 = Tr3 )
& ( member @ n @ ( root @ Tr3 ) @ Ns3 ) )
| ? [Tr33: dtree,Ns3: set @ n,Tr12: dtree,Tr24: dtree] :
( ( A12 = Ns3 )
& ( A23 = Tr12 )
& ( A33 = Tr33 )
& ( member @ n @ ( root @ Tr33 ) @ Ns3 )
& ( gram_L716654942_subtr @ Ns3 @ Tr12 @ Tr24 )
& ( member @ ( sum_sum @ t @ dtree ) @ ( sum_Inr @ dtree @ t @ Tr24 ) @ ( cont @ Tr33 ) ) ) ) ) ) ).
% subtr.simps
thf(fact_81_subtr__StepL,axiom,
! [Tr1: dtree,Ns: set @ n,Tr22: dtree,Tr32: dtree] :
( ( member @ n @ ( root @ Tr1 ) @ Ns )
=> ( ( member @ ( sum_sum @ t @ dtree ) @ ( sum_Inr @ dtree @ t @ Tr1 ) @ ( cont @ Tr22 ) )
=> ( ( gram_L716654942_subtr @ Ns @ Tr22 @ Tr32 )
=> ( gram_L716654942_subtr @ Ns @ Tr1 @ Tr32 ) ) ) ) ).
% subtr_StepL
thf(fact_82_subtr_Oinducts,axiom,
! [X12: set @ n,X2: dtree,X32: dtree,P: ( set @ n ) > dtree > dtree > $o] :
( ( gram_L716654942_subtr @ X12 @ X2 @ X32 )
=> ( ! [Tr5: dtree,Ns4: set @ n] :
( ( member @ n @ ( root @ Tr5 ) @ Ns4 )
=> ( P @ Ns4 @ Tr5 @ Tr5 ) )
=> ( ! [Tr34: dtree,Ns4: set @ n,Tr13: dtree,Tr23: dtree] :
( ( member @ n @ ( root @ Tr34 ) @ Ns4 )
=> ( ( gram_L716654942_subtr @ Ns4 @ Tr13 @ Tr23 )
=> ( ( P @ Ns4 @ Tr13 @ Tr23 )
=> ( ( member @ ( sum_sum @ t @ dtree ) @ ( sum_Inr @ dtree @ t @ Tr23 ) @ ( cont @ Tr34 ) )
=> ( P @ Ns4 @ Tr13 @ Tr34 ) ) ) ) )
=> ( P @ X12 @ X2 @ X32 ) ) ) ) ).
% subtr.inducts
thf(fact_83_subtr__inductL,axiom,
! [Ns: set @ n,Tr1: dtree,Tr22: dtree,Phi: ( set @ n ) > dtree > dtree > $o] :
( ( gram_L716654942_subtr @ Ns @ Tr1 @ Tr22 )
=> ( ! [Ns4: set @ n,Tr5: dtree] : ( Phi @ Ns4 @ Tr5 @ Tr5 )
=> ( ! [Ns4: set @ n,Tr13: dtree,Tr23: dtree,Tr34: dtree] :
( ( member @ n @ ( root @ Tr13 ) @ Ns4 )
=> ( ( member @ ( sum_sum @ t @ dtree ) @ ( sum_Inr @ dtree @ t @ Tr13 ) @ ( cont @ Tr23 ) )
=> ( ( gram_L716654942_subtr @ Ns4 @ Tr23 @ Tr34 )
=> ( ( Phi @ Ns4 @ Tr23 @ Tr34 )
=> ( Phi @ Ns4 @ Tr13 @ Tr34 ) ) ) ) )
=> ( Phi @ Ns @ Tr1 @ Tr22 ) ) ) ) ).
% subtr_inductL
thf(fact_84_regular__def,axiom,
( gram_L646766332egular
= ( ^ [Tr3: dtree] :
? [F2: n > dtree] : ( gram_L1918716148le_reg @ F2 @ Tr3 ) ) ) ).
% regular_def
thf(fact_85_subtrOf__root,axiom,
! [Tr: dtree,Tr2: dtree] :
( ( gram_L864798063lle_wf @ Tr )
=> ( ( member @ ( sum_sum @ t @ dtree ) @ ( sum_Inr @ dtree @ t @ Tr2 ) @ ( cont @ Tr ) )
=> ( ( gram_L1614515765ubtrOf @ Tr @ ( root @ Tr2 ) )
= Tr2 ) ) ) ).
% subtrOf_root
thf(fact_86_subset__antisym,axiom,
! [A: $tType,A5: set @ A,B4: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ A5 @ B4 )
=> ( ( ord_less_eq @ ( set @ A ) @ B4 @ A5 )
=> ( A5 = B4 ) ) ) ).
% subset_antisym
thf(fact_87_subsetI,axiom,
! [A: $tType,A5: set @ A,B4: set @ A] :
( ! [X3: A] :
( ( member @ A @ X3 @ A5 )
=> ( member @ A @ X3 @ B4 ) )
=> ( ord_less_eq @ ( set @ A ) @ A5 @ B4 ) ) ).
% subsetI
thf(fact_88_order__refl,axiom,
! [A: $tType] :
( ( preorder @ A @ ( type2 @ A ) )
=> ! [X: A] : ( ord_less_eq @ A @ X @ X ) ) ).
% order_refl
thf(fact_89_sublists_Osimps_I1_J,axiom,
! [A: $tType] :
( ( sublists @ A @ ( nil @ A ) )
= ( cons @ ( list @ A ) @ ( nil @ A ) @ ( nil @ ( list @ A ) ) ) ) ).
% sublists.simps(1)
thf(fact_90_path__distinct,axiom,
! [F: n > dtree,Nl: list @ n] :
( ( gram_L250615845e_path @ F @ Nl )
=> ? [Nl4: list @ n] :
( ( gram_L250615845e_path @ F @ Nl4 )
& ( ( hd @ n @ Nl4 )
= ( hd @ n @ Nl ) )
& ( ( last @ n @ Nl4 )
= ( last @ n @ Nl ) )
& ( ord_less_eq @ ( set @ n ) @ ( set2 @ n @ Nl4 ) @ ( set2 @ n @ Nl ) )
& ( distinct @ n @ Nl4 ) ) ) ).
% path_distinct
thf(fact_91_product__lists_Osimps_I1_J,axiom,
! [A: $tType] :
( ( product_lists @ A @ ( nil @ ( list @ A ) ) )
= ( cons @ ( list @ A ) @ ( nil @ A ) @ ( nil @ ( list @ A ) ) ) ) ).
% product_lists.simps(1)
thf(fact_92_the__elem__set,axiom,
! [A: $tType,X: A] :
( ( the_elem @ A @ ( set2 @ A @ ( cons @ A @ X @ ( nil @ A ) ) ) )
= X ) ).
% the_elem_set
thf(fact_93_distinct__product__lists,axiom,
! [A: $tType,Xss2: list @ ( list @ A )] :
( ! [X3: list @ A] :
( ( member @ ( list @ A ) @ X3 @ ( set2 @ ( list @ A ) @ Xss2 ) )
=> ( distinct @ A @ X3 ) )
=> ( distinct @ ( list @ A ) @ ( product_lists @ A @ Xss2 ) ) ) ).
% distinct_product_lists
thf(fact_94_less__eq__set__def,axiom,
! [A: $tType] :
( ( ord_less_eq @ ( set @ A ) )
= ( ^ [A6: set @ A,B5: set @ A] :
( ord_less_eq @ ( A > $o )
@ ^ [X4: A] : ( member @ A @ X4 @ A6 )
@ ^ [X4: A] : ( member @ A @ X4 @ B5 ) ) ) ) ).
% less_eq_set_def
thf(fact_95_distinct__length__2__or__more,axiom,
! [A: $tType,A2: A,B2: A,Xs: list @ A] :
( ( distinct @ A @ ( cons @ A @ A2 @ ( cons @ A @ B2 @ Xs ) ) )
= ( ( A2 != B2 )
& ( distinct @ A @ ( cons @ A @ A2 @ Xs ) )
& ( distinct @ A @ ( cons @ A @ B2 @ Xs ) ) ) ) ).
% distinct_length_2_or_more
thf(fact_96_distinct_Osimps_I1_J,axiom,
! [A: $tType] : ( distinct @ A @ ( nil @ A ) ) ).
% distinct.simps(1)
thf(fact_97_distinct__singleton,axiom,
! [A: $tType,X: A] : ( distinct @ A @ ( cons @ A @ X @ ( nil @ A ) ) ) ).
% distinct_singleton
thf(fact_98_distinct_Osimps_I2_J,axiom,
! [A: $tType,X: A,Xs: list @ A] :
( ( distinct @ A @ ( cons @ A @ X @ Xs ) )
= ( ~ ( member @ A @ X @ ( set2 @ A @ Xs ) )
& ( distinct @ A @ Xs ) ) ) ).
% distinct.simps(2)
thf(fact_99_le__funD,axiom,
! [B: $tType,A: $tType] :
( ( ord @ B @ ( type2 @ B ) )
=> ! [F: A > B,G: A > B,X: A] :
( ( ord_less_eq @ ( A > B ) @ F @ G )
=> ( ord_less_eq @ B @ ( F @ X ) @ ( G @ X ) ) ) ) ).
% le_funD
thf(fact_100_le__funE,axiom,
! [B: $tType,A: $tType] :
( ( ord @ B @ ( type2 @ B ) )
=> ! [F: A > B,G: A > B,X: A] :
( ( ord_less_eq @ ( A > B ) @ F @ G )
=> ( ord_less_eq @ B @ ( F @ X ) @ ( G @ X ) ) ) ) ).
% le_funE
thf(fact_101_le__funI,axiom,
! [B: $tType,A: $tType] :
( ( ord @ B @ ( type2 @ B ) )
=> ! [F: A > B,G: A > B] :
( ! [X3: A] : ( ord_less_eq @ B @ ( F @ X3 ) @ ( G @ X3 ) )
=> ( ord_less_eq @ ( A > B ) @ F @ G ) ) ) ).
% le_funI
thf(fact_102_le__fun__def,axiom,
! [B: $tType,A: $tType] :
( ( ord @ B @ ( type2 @ B ) )
=> ( ( ord_less_eq @ ( A > B ) )
= ( ^ [F2: A > B,G2: A > B] :
! [X4: A] : ( ord_less_eq @ B @ ( F2 @ X4 ) @ ( G2 @ X4 ) ) ) ) ) ).
% le_fun_def
thf(fact_103_order__subst1,axiom,
! [A: $tType,B: $tType] :
( ( ( order @ B @ ( type2 @ B ) )
& ( order @ A @ ( type2 @ A ) ) )
=> ! [A2: A,F: B > A,B2: B,C: B] :
( ( ord_less_eq @ A @ A2 @ ( F @ B2 ) )
=> ( ( ord_less_eq @ B @ B2 @ C )
=> ( ! [X3: B,Y4: B] :
( ( ord_less_eq @ B @ X3 @ Y4 )
=> ( ord_less_eq @ A @ ( F @ X3 ) @ ( F @ Y4 ) ) )
=> ( ord_less_eq @ A @ A2 @ ( F @ C ) ) ) ) ) ) ).
% order_subst1
thf(fact_104_order__subst2,axiom,
! [A: $tType,C2: $tType] :
( ( ( order @ C2 @ ( type2 @ C2 ) )
& ( order @ A @ ( type2 @ A ) ) )
=> ! [A2: A,B2: A,F: A > C2,C: C2] :
( ( ord_less_eq @ A @ A2 @ B2 )
=> ( ( ord_less_eq @ C2 @ ( F @ B2 ) @ C )
=> ( ! [X3: A,Y4: A] :
( ( ord_less_eq @ A @ X3 @ Y4 )
=> ( ord_less_eq @ C2 @ ( F @ X3 ) @ ( F @ Y4 ) ) )
=> ( ord_less_eq @ C2 @ ( F @ A2 ) @ C ) ) ) ) ) ).
% order_subst2
thf(fact_105_ord__eq__le__subst,axiom,
! [A: $tType,B: $tType] :
( ( ( ord @ B @ ( type2 @ B ) )
& ( ord @ A @ ( type2 @ A ) ) )
=> ! [A2: A,F: B > A,B2: B,C: B] :
( ( A2
= ( F @ B2 ) )
=> ( ( ord_less_eq @ B @ B2 @ C )
=> ( ! [X3: B,Y4: B] :
( ( ord_less_eq @ B @ X3 @ Y4 )
=> ( ord_less_eq @ A @ ( F @ X3 ) @ ( F @ Y4 ) ) )
=> ( ord_less_eq @ A @ A2 @ ( F @ C ) ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_106_ord__le__eq__subst,axiom,
! [A: $tType,B: $tType] :
( ( ( ord @ B @ ( type2 @ B ) )
& ( ord @ A @ ( type2 @ A ) ) )
=> ! [A2: A,B2: A,F: A > B,C: B] :
( ( ord_less_eq @ A @ A2 @ B2 )
=> ( ( ( F @ B2 )
= C )
=> ( ! [X3: A,Y4: A] :
( ( ord_less_eq @ A @ X3 @ Y4 )
=> ( ord_less_eq @ B @ ( F @ X3 ) @ ( F @ Y4 ) ) )
=> ( ord_less_eq @ B @ ( F @ A2 ) @ C ) ) ) ) ) ).
% ord_le_eq_subst
thf(fact_107_eq__iff,axiom,
! [A: $tType] :
( ( order @ A @ ( type2 @ A ) )
=> ( ( ^ [Y5: A,Z: A] : Y5 = Z )
= ( ^ [X4: A,Y3: A] :
( ( ord_less_eq @ A @ X4 @ Y3 )
& ( ord_less_eq @ A @ Y3 @ X4 ) ) ) ) ) ).
% eq_iff
thf(fact_108_antisym,axiom,
! [A: $tType] :
( ( order @ A @ ( type2 @ A ) )
=> ! [X: A,Y: A] :
( ( ord_less_eq @ A @ X @ Y )
=> ( ( ord_less_eq @ A @ Y @ X )
=> ( X = Y ) ) ) ) ).
% antisym
thf(fact_109_linear,axiom,
! [A: $tType] :
( ( linorder @ A @ ( type2 @ A ) )
=> ! [X: A,Y: A] :
( ( ord_less_eq @ A @ X @ Y )
| ( ord_less_eq @ A @ Y @ X ) ) ) ).
% linear
thf(fact_110_eq__refl,axiom,
! [A: $tType] :
( ( preorder @ A @ ( type2 @ A ) )
=> ! [X: A,Y: A] :
( ( X = Y )
=> ( ord_less_eq @ A @ X @ Y ) ) ) ).
% eq_refl
thf(fact_111_le__cases,axiom,
! [A: $tType] :
( ( linorder @ A @ ( type2 @ A ) )
=> ! [X: A,Y: A] :
( ~ ( ord_less_eq @ A @ X @ Y )
=> ( ord_less_eq @ A @ Y @ X ) ) ) ).
% le_cases
thf(fact_112_order_Otrans,axiom,
! [A: $tType] :
( ( order @ A @ ( type2 @ A ) )
=> ! [A2: A,B2: A,C: A] :
( ( ord_less_eq @ A @ A2 @ B2 )
=> ( ( ord_less_eq @ A @ B2 @ C )
=> ( ord_less_eq @ A @ A2 @ C ) ) ) ) ).
% order.trans
thf(fact_113_le__cases3,axiom,
! [A: $tType] :
( ( linorder @ A @ ( type2 @ A ) )
=> ! [X: A,Y: A,Z3: A] :
( ( ( ord_less_eq @ A @ X @ Y )
=> ~ ( ord_less_eq @ A @ Y @ Z3 ) )
=> ( ( ( ord_less_eq @ A @ Y @ X )
=> ~ ( ord_less_eq @ A @ X @ Z3 ) )
=> ( ( ( ord_less_eq @ A @ X @ Z3 )
=> ~ ( ord_less_eq @ A @ Z3 @ Y ) )
=> ( ( ( ord_less_eq @ A @ Z3 @ Y )
=> ~ ( ord_less_eq @ A @ Y @ X ) )
=> ( ( ( ord_less_eq @ A @ Y @ Z3 )
=> ~ ( ord_less_eq @ A @ Z3 @ X ) )
=> ~ ( ( ord_less_eq @ A @ Z3 @ X )
=> ~ ( ord_less_eq @ A @ X @ Y ) ) ) ) ) ) ) ) ).
% le_cases3
thf(fact_114_antisym__conv,axiom,
! [A: $tType] :
( ( order @ A @ ( type2 @ A ) )
=> ! [Y: A,X: A] :
( ( ord_less_eq @ A @ Y @ X )
=> ( ( ord_less_eq @ A @ X @ Y )
= ( X = Y ) ) ) ) ).
% antisym_conv
thf(fact_115_ord__eq__le__trans,axiom,
! [A: $tType] :
( ( ord @ A @ ( type2 @ A ) )
=> ! [A2: A,B2: A,C: A] :
( ( A2 = B2 )
=> ( ( ord_less_eq @ A @ B2 @ C )
=> ( ord_less_eq @ A @ A2 @ C ) ) ) ) ).
% ord_eq_le_trans
thf(fact_116_ord__le__eq__trans,axiom,
! [A: $tType] :
( ( ord @ A @ ( type2 @ A ) )
=> ! [A2: A,B2: A,C: A] :
( ( ord_less_eq @ A @ A2 @ B2 )
=> ( ( B2 = C )
=> ( ord_less_eq @ A @ A2 @ C ) ) ) ) ).
% ord_le_eq_trans
thf(fact_117_order__class_Oorder_Oantisym,axiom,
! [A: $tType] :
( ( order @ A @ ( type2 @ A ) )
=> ! [A2: A,B2: A] :
( ( ord_less_eq @ A @ A2 @ B2 )
=> ( ( ord_less_eq @ A @ B2 @ A2 )
=> ( A2 = B2 ) ) ) ) ).
% order_class.order.antisym
thf(fact_118_order__trans,axiom,
! [A: $tType] :
( ( preorder @ A @ ( type2 @ A ) )
=> ! [X: A,Y: A,Z3: A] :
( ( ord_less_eq @ A @ X @ Y )
=> ( ( ord_less_eq @ A @ Y @ Z3 )
=> ( ord_less_eq @ A @ X @ Z3 ) ) ) ) ).
% order_trans
thf(fact_119_dual__order_Orefl,axiom,
! [A: $tType] :
( ( order @ A @ ( type2 @ A ) )
=> ! [A2: A] : ( ord_less_eq @ A @ A2 @ A2 ) ) ).
% dual_order.refl
thf(fact_120_linorder__wlog,axiom,
! [A: $tType] :
( ( linorder @ A @ ( type2 @ A ) )
=> ! [P: A > A > $o,A2: A,B2: A] :
( ! [A4: A,B6: A] :
( ( ord_less_eq @ A @ A4 @ B6 )
=> ( P @ A4 @ B6 ) )
=> ( ! [A4: A,B6: A] :
( ( P @ B6 @ A4 )
=> ( P @ A4 @ B6 ) )
=> ( P @ A2 @ B2 ) ) ) ) ).
% linorder_wlog
thf(fact_121_dual__order_Otrans,axiom,
! [A: $tType] :
( ( order @ A @ ( type2 @ A ) )
=> ! [B2: A,A2: A,C: A] :
( ( ord_less_eq @ A @ B2 @ A2 )
=> ( ( ord_less_eq @ A @ C @ B2 )
=> ( ord_less_eq @ A @ C @ A2 ) ) ) ) ).
% dual_order.trans
thf(fact_122_dual__order_Oantisym,axiom,
! [A: $tType] :
( ( order @ A @ ( type2 @ A ) )
=> ! [B2: A,A2: A] :
( ( ord_less_eq @ A @ B2 @ A2 )
=> ( ( ord_less_eq @ A @ A2 @ B2 )
=> ( A2 = B2 ) ) ) ) ).
% dual_order.antisym
thf(fact_123_set__mp,axiom,
! [A: $tType,A5: set @ A,B4: set @ A,X: A] :
( ( ord_less_eq @ ( set @ A ) @ A5 @ B4 )
=> ( ( member @ A @ X @ A5 )
=> ( member @ A @ X @ B4 ) ) ) ).
% set_mp
thf(fact_124_in__mono,axiom,
! [A: $tType,A5: set @ A,B4: set @ A,X: A] :
( ( ord_less_eq @ ( set @ A ) @ A5 @ B4 )
=> ( ( member @ A @ X @ A5 )
=> ( member @ A @ X @ B4 ) ) ) ).
% in_mono
thf(fact_125_subsetD,axiom,
! [A: $tType,A5: set @ A,B4: set @ A,C: A] :
( ( ord_less_eq @ ( set @ A ) @ A5 @ B4 )
=> ( ( member @ A @ C @ A5 )
=> ( member @ A @ C @ B4 ) ) ) ).
% subsetD
thf(fact_126_subsetCE,axiom,
! [A: $tType,A5: set @ A,B4: set @ A,C: A] :
( ( ord_less_eq @ ( set @ A ) @ A5 @ B4 )
=> ( ( member @ A @ C @ A5 )
=> ( member @ A @ C @ B4 ) ) ) ).
% subsetCE
thf(fact_127_equalityE,axiom,
! [A: $tType,A5: set @ A,B4: set @ A] :
( ( A5 = B4 )
=> ~ ( ( ord_less_eq @ ( set @ A ) @ A5 @ B4 )
=> ~ ( ord_less_eq @ ( set @ A ) @ B4 @ A5 ) ) ) ).
% equalityE
thf(fact_128_subset__eq,axiom,
! [A: $tType] :
( ( ord_less_eq @ ( set @ A ) )
= ( ^ [A6: set @ A,B5: set @ A] :
! [X4: A] :
( ( member @ A @ X4 @ A6 )
=> ( member @ A @ X4 @ B5 ) ) ) ) ).
% subset_eq
thf(fact_129_equalityD1,axiom,
! [A: $tType,A5: set @ A,B4: set @ A] :
( ( A5 = B4 )
=> ( ord_less_eq @ ( set @ A ) @ A5 @ B4 ) ) ).
% equalityD1
thf(fact_130_equalityD2,axiom,
! [A: $tType,A5: set @ A,B4: set @ A] :
( ( A5 = B4 )
=> ( ord_less_eq @ ( set @ A ) @ B4 @ A5 ) ) ).
% equalityD2
thf(fact_131_set__rev__mp,axiom,
! [A: $tType,X: A,A5: set @ A,B4: set @ A] :
( ( member @ A @ X @ A5 )
=> ( ( ord_less_eq @ ( set @ A ) @ A5 @ B4 )
=> ( member @ A @ X @ B4 ) ) ) ).
% set_rev_mp
thf(fact_132_subset__iff,axiom,
! [A: $tType] :
( ( ord_less_eq @ ( set @ A ) )
= ( ^ [A6: set @ A,B5: set @ A] :
! [T3: A] :
( ( member @ A @ T3 @ A6 )
=> ( member @ A @ T3 @ B5 ) ) ) ) ).
% subset_iff
thf(fact_133_rev__subsetD,axiom,
! [A: $tType,C: A,A5: set @ A,B4: set @ A] :
( ( member @ A @ C @ A5 )
=> ( ( ord_less_eq @ ( set @ A ) @ A5 @ B4 )
=> ( member @ A @ C @ B4 ) ) ) ).
% rev_subsetD
thf(fact_134_subset__refl,axiom,
! [A: $tType,A5: set @ A] : ( ord_less_eq @ ( set @ A ) @ A5 @ A5 ) ).
% subset_refl
thf(fact_135_Collect__mono,axiom,
! [A: $tType,P: A > $o,Q: A > $o] :
( ! [X3: A] :
( ( P @ X3 )
=> ( Q @ X3 ) )
=> ( ord_less_eq @ ( set @ A ) @ ( collect @ A @ P ) @ ( collect @ A @ Q ) ) ) ).
% Collect_mono
thf(fact_136_subset__trans,axiom,
! [A: $tType,A5: set @ A,B4: set @ A,C3: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ A5 @ B4 )
=> ( ( ord_less_eq @ ( set @ A ) @ B4 @ C3 )
=> ( ord_less_eq @ ( set @ A ) @ A5 @ C3 ) ) ) ).
% subset_trans
thf(fact_137_set__eq__subset,axiom,
! [A: $tType] :
( ( ^ [Y5: set @ A,Z: set @ A] : Y5 = Z )
= ( ^ [A6: set @ A,B5: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ A6 @ B5 )
& ( ord_less_eq @ ( set @ A ) @ B5 @ A6 ) ) ) ) ).
% set_eq_subset
thf(fact_138_contra__subsetD,axiom,
! [A: $tType,A5: set @ A,B4: set @ A,C: A] :
( ( ord_less_eq @ ( set @ A ) @ A5 @ B4 )
=> ( ~ ( member @ A @ C @ B4 )
=> ~ ( member @ A @ C @ A5 ) ) ) ).
% contra_subsetD
thf(fact_139_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_140_distinct__union,axiom,
! [A: $tType,Xs: list @ A,Ys3: list @ A] :
( ( distinct @ A @ ( union @ A @ Xs @ Ys3 ) )
= ( distinct @ A @ Ys3 ) ) ).
% distinct_union
thf(fact_141_subtrOf__def,axiom,
( gram_L1614515765ubtrOf
= ( ^ [Tr3: dtree,N3: n] :
@+[Tr4: dtree] :
( ( member @ ( sum_sum @ t @ dtree ) @ ( sum_Inr @ dtree @ t @ Tr4 ) @ ( cont @ Tr3 ) )
& ( ( root @ Tr4 )
= N3 ) ) ) ) ).
% subtrOf_def
thf(fact_142_hsubst__r__def,axiom,
gram_L1905609017ubst_r = root ).
% hsubst_r_def
thf(fact_143_inItr_Oinducts,axiom,
! [X12: set @ n,X2: dtree,X32: n,P: ( set @ n ) > dtree > n > $o] :
( ( gram_L830233218_inItr @ X12 @ X2 @ X32 )
=> ( ! [Tr5: dtree,Ns4: set @ n] :
( ( member @ n @ ( root @ Tr5 ) @ Ns4 )
=> ( P @ Ns4 @ Tr5 @ ( root @ Tr5 ) ) )
=> ( ! [Tr5: dtree,Ns4: set @ n,Tr13: dtree,N2: n] :
( ( member @ n @ ( root @ Tr5 ) @ Ns4 )
=> ( ( member @ ( sum_sum @ t @ dtree ) @ ( sum_Inr @ dtree @ t @ Tr13 ) @ ( cont @ Tr5 ) )
=> ( ( gram_L830233218_inItr @ Ns4 @ Tr13 @ N2 )
=> ( ( P @ Ns4 @ Tr13 @ N2 )
=> ( P @ Ns4 @ Tr5 @ N2 ) ) ) ) )
=> ( P @ X12 @ X2 @ X32 ) ) ) ) ).
% inItr.inducts
thf(fact_144_inItr_Osimps,axiom,
( gram_L830233218_inItr
= ( ^ [A12: set @ n,A23: dtree,A33: n] :
( ? [Tr3: dtree,Ns3: set @ n] :
( ( A12 = Ns3 )
& ( A23 = Tr3 )
& ( A33
= ( root @ Tr3 ) )
& ( member @ n @ ( root @ Tr3 ) @ Ns3 ) )
| ? [Tr3: dtree,Ns3: set @ n,Tr12: dtree,N3: n] :
( ( A12 = Ns3 )
& ( A23 = Tr3 )
& ( A33 = N3 )
& ( member @ n @ ( root @ Tr3 ) @ Ns3 )
& ( member @ ( sum_sum @ t @ dtree ) @ ( sum_Inr @ dtree @ t @ Tr12 ) @ ( cont @ Tr3 ) )
& ( gram_L830233218_inItr @ Ns3 @ Tr12 @ N3 ) ) ) ) ) ).
% inItr.simps
thf(fact_145_predicate1I,axiom,
! [A: $tType,P: A > $o,Q: A > $o] :
( ! [X3: A] :
( ( P @ X3 )
=> ( Q @ X3 ) )
=> ( ord_less_eq @ ( A > $o ) @ P @ Q ) ) ).
% predicate1I
thf(fact_146_rev__predicate1D,axiom,
! [A: $tType,P: A > $o,X: A,Q: A > $o] :
( ( P @ X )
=> ( ( ord_less_eq @ ( A > $o ) @ P @ Q )
=> ( Q @ X ) ) ) ).
% rev_predicate1D
thf(fact_147_predicate1D,axiom,
! [A: $tType,P: A > $o,Q: A > $o,X: A] :
( ( ord_less_eq @ ( A > $o ) @ P @ Q )
=> ( ( P @ X )
=> ( Q @ X ) ) ) ).
% predicate1D
thf(fact_148_inItr_OBase,axiom,
! [Tr: dtree,Ns: set @ n] :
( ( member @ n @ ( root @ Tr ) @ Ns )
=> ( gram_L830233218_inItr @ Ns @ Tr @ ( root @ Tr ) ) ) ).
% inItr.Base
thf(fact_149_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_150_subtr__inItr,axiom,
! [Ns: set @ n,Tr: dtree,N: n,Tr1: dtree] :
( ( gram_L830233218_inItr @ Ns @ Tr @ N )
=> ( ( gram_L716654942_subtr @ Ns @ Tr @ Tr1 )
=> ( gram_L830233218_inItr @ Ns @ Tr1 @ N ) ) ) ).
% subtr_inItr
thf(fact_151_inItr__mono,axiom,
! [Ns: set @ n,Tr: dtree,N: n,Ns2: set @ n] :
( ( gram_L830233218_inItr @ Ns @ Tr @ N )
=> ( ( ord_less_eq @ ( set @ n ) @ Ns @ Ns2 )
=> ( gram_L830233218_inItr @ Ns2 @ Tr @ N ) ) ) ).
% inItr_mono
thf(fact_152_inItr__subtr,axiom,
! [Ns: set @ n,Tr: dtree,N: n] :
( ( gram_L830233218_inItr @ Ns @ Tr @ N )
=> ? [Tr6: dtree] :
( ( gram_L716654942_subtr @ Ns @ Tr6 @ Tr )
& ( ( root @ Tr6 )
= N ) ) ) ).
% inItr_subtr
thf(fact_153_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_154_inItr_Ocases,axiom,
! [A1: set @ n,A22: dtree,A32: n] :
( ( gram_L830233218_inItr @ A1 @ A22 @ A32 )
=> ( ( ( A32
= ( root @ A22 ) )
=> ~ ( member @ n @ ( root @ A22 ) @ A1 ) )
=> ~ ( ( member @ n @ ( root @ A22 ) @ A1 )
=> ! [Tr13: dtree] :
( ( member @ ( sum_sum @ t @ dtree ) @ ( sum_Inr @ dtree @ t @ Tr13 ) @ ( cont @ A22 ) )
=> ~ ( gram_L830233218_inItr @ A1 @ Tr13 @ A32 ) ) ) ) ) ).
% inItr.cases
thf(fact_155_some__equality,axiom,
! [A: $tType,P: A > $o,A2: A] :
( ( P @ A2 )
=> ( ! [X3: A] :
( ( P @ X3 )
=> ( X3 = A2 ) )
=> ( ( ^ [P2: A > $o] :
@+[X5: A] : ( P2 @ X5 )
@ P )
= A2 ) ) ) ).
% some_equality
thf(fact_156_some__eq__trivial,axiom,
! [A: $tType,X: A] :
( ( @+[Y3: A] : Y3 = X )
= X ) ).
% some_eq_trivial
thf(fact_157_some__sym__eq__trivial,axiom,
! [A: $tType,X: A] :
( ( ^ [P2: A > $o] :
@+[X5: A] : ( P2 @ X5 )
@ ( ^ [Y5: A,Z: A] : Y5 = Z
@ X ) )
= X ) ).
% some_sym_eq_trivial
thf(fact_158_someI,axiom,
! [A: $tType,P: A > $o,X: A] :
( ( P @ X )
=> ( P
@ ( ^ [P2: A > $o] :
@+[X5: A] : ( P2 @ X5 )
@ P ) ) ) ).
% someI
thf(fact_159_tfl__some,axiom,
! [A: $tType,P3: A > $o,X6: A] :
( ( P3 @ X6 )
=> ( P3
@ ( ^ [P2: A > $o] :
@+[X5: A] : ( P2 @ X5 )
@ P3 ) ) ) ).
% tfl_some
thf(fact_160_some1__equality,axiom,
! [A: $tType,P: A > $o,A2: A] :
( ? [X6: A] :
( ( P @ X6 )
& ! [Y4: A] :
( ( P @ Y4 )
=> ( Y4 = X6 ) ) )
=> ( ( P @ A2 )
=> ( ( ^ [P2: A > $o] :
@+[X5: A] : ( P2 @ X5 )
@ P )
= A2 ) ) ) ).
% some1_equality
thf(fact_161_some__eq__ex,axiom,
! [A: $tType,P: A > $o] :
( ( P
@ ( ^ [P2: A > $o] :
@+[X5: A] : ( P2 @ X5 )
@ P ) )
= ( ^ [P2: A > $o] :
? [X5: A] : ( P2 @ X5 )
@ P ) ) ).
% some_eq_ex
thf(fact_162_someI2__bex,axiom,
! [A: $tType,A5: set @ A,P: A > $o,Q: A > $o] :
( ? [X6: A] :
( ( member @ A @ X6 @ A5 )
& ( P @ X6 ) )
=> ( ! [X3: A] :
( ( ( member @ A @ X3 @ A5 )
& ( P @ X3 ) )
=> ( Q @ X3 ) )
=> ( Q
@ @+[X4: A] :
( ( member @ A @ X4 @ A5 )
& ( P @ X4 ) ) ) ) ) ).
% someI2_bex
thf(fact_163_someI2__ex,axiom,
! [A: $tType,P: A > $o,Q: A > $o] :
( ? [X13: A] : ( P @ X13 )
=> ( ! [X3: A] :
( ( P @ X3 )
=> ( Q @ X3 ) )
=> ( Q
@ ( ^ [P2: A > $o] :
@+[X5: A] : ( P2 @ X5 )
@ P ) ) ) ) ).
% someI2_ex
thf(fact_164_someI__ex,axiom,
! [A: $tType,P: A > $o] :
( ? [X13: A] : ( P @ X13 )
=> ( P
@ ( ^ [P2: A > $o] :
@+[X5: A] : ( P2 @ X5 )
@ P ) ) ) ).
% someI_ex
thf(fact_165_someI2,axiom,
! [A: $tType,P: A > $o,A2: A,Q: A > $o] :
( ( P @ A2 )
=> ( ! [X3: A] :
( ( P @ X3 )
=> ( Q @ X3 ) )
=> ( Q
@ ( ^ [P2: A > $o] :
@+[X5: A] : ( P2 @ X5 )
@ P ) ) ) ) ).
% someI2
thf(fact_166_pred__subset__eq,axiom,
! [A: $tType,R: set @ A,S: set @ A] :
( ( ord_less_eq @ ( A > $o )
@ ^ [X4: A] : ( member @ A @ X4 @ R )
@ ^ [X4: A] : ( member @ A @ X4 @ S ) )
= ( ord_less_eq @ ( set @ A ) @ R @ S ) ) ).
% pred_subset_eq
thf(fact_167_Greatest__equality,axiom,
! [A: $tType] :
( ( order @ A @ ( type2 @ A ) )
=> ! [P: A > $o,K: A] :
( ( P @ K )
=> ( ! [X3: A] :
( ( P @ X3 )
=> ( ord_less_eq @ A @ X3 @ K ) )
=> ( ( hilbert_Greatest @ A @ P )
= K ) ) ) ) ).
% Greatest_equality
thf(fact_168_subset__code_I3_J,axiom,
! [C2: $tType] :
~ ( ord_less_eq @ ( set @ C2 ) @ ( coset @ C2 @ ( nil @ C2 ) ) @ ( set2 @ C2 @ ( nil @ C2 ) ) ) ).
% subset_code(3)
thf(fact_169_subset__code_I2_J,axiom,
! [B: $tType,A5: set @ B,Ys3: list @ B] :
( ( ord_less_eq @ ( set @ B ) @ A5 @ ( coset @ B @ Ys3 ) )
= ( ! [X4: B] :
( ( member @ B @ X4 @ ( set2 @ B @ Ys3 ) )
=> ~ ( member @ B @ X4 @ A5 ) ) ) ) ).
% subset_code(2)
thf(fact_170_Greatest__def,axiom,
! [A: $tType] :
( ( ord @ A @ ( type2 @ A ) )
=> ( ( hilbert_Greatest @ A )
= ( hilbert_GreatestM @ A @ A
@ ^ [X4: A] : X4 ) ) ) ).
% Greatest_def
thf(fact_171_Itr__def,axiom,
( gram_L1580978439le_Itr
= ( ^ [Ns3: set @ n,Tr3: dtree] : ( collect @ n @ ( gram_L830233218_inItr @ Ns3 @ Tr3 ) ) ) ) ).
% Itr_def
thf(fact_172_not__in__set__insert,axiom,
! [A: $tType,X: A,Xs: list @ A] :
( ~ ( member @ A @ X @ ( set2 @ A @ Xs ) )
=> ( ( insert @ A @ X @ Xs )
= ( cons @ A @ X @ Xs ) ) ) ).
% not_in_set_insert
thf(fact_173_in__set__insert,axiom,
! [A: $tType,X: A,Xs: list @ A] :
( ( member @ A @ X @ ( set2 @ A @ Xs ) )
=> ( ( insert @ A @ X @ Xs )
= Xs ) ) ).
% in_set_insert
thf(fact_174_distinct__insert,axiom,
! [A: $tType,X: A,Xs: list @ A] :
( ( distinct @ A @ ( insert @ A @ X @ Xs ) )
= ( distinct @ A @ Xs ) ) ).
% distinct_insert
thf(fact_175_insert__Nil,axiom,
! [A: $tType,X: A] :
( ( insert @ A @ X @ ( nil @ A ) )
= ( cons @ A @ X @ ( nil @ A ) ) ) ).
% insert_Nil
thf(fact_176_GreatestMI2,axiom,
! [B: $tType,A: $tType] :
( ( ord @ B @ ( type2 @ B ) )
=> ! [P: A > $o,X: A,M: A > B,Q: A > $o] :
( ( P @ X )
=> ( ! [Y4: A] :
( ( P @ Y4 )
=> ( ord_less_eq @ B @ ( M @ Y4 ) @ ( M @ X ) ) )
=> ( ! [X3: A] :
( ( P @ X3 )
=> ( ! [Y6: A] :
( ( P @ Y6 )
=> ( ord_less_eq @ B @ ( M @ Y6 ) @ ( M @ X3 ) ) )
=> ( Q @ X3 ) ) )
=> ( Q @ ( hilbert_GreatestM @ A @ B @ M @ P ) ) ) ) ) ) ).
% GreatestMI2
thf(fact_177_List_Oinsert__def,axiom,
! [A: $tType] :
( ( insert @ A )
= ( ^ [X4: A,Xs3: list @ A] : ( if @ ( list @ A ) @ ( member @ A @ X4 @ ( set2 @ A @ Xs3 ) ) @ Xs3 @ ( cons @ A @ X4 @ Xs3 ) ) ) ) ).
% List.insert_def
thf(fact_178_GreatestM__equality,axiom,
! [A: $tType,B: $tType] :
( ( order @ A @ ( type2 @ A ) )
=> ! [P: B > $o,K: B,M: B > A] :
( ( P @ K )
=> ( ! [X3: B] :
( ( P @ X3 )
=> ( ord_less_eq @ A @ ( M @ X3 ) @ ( M @ K ) ) )
=> ( ( M @ ( hilbert_GreatestM @ B @ A @ M @ P ) )
= ( M @ K ) ) ) ) ) ).
% GreatestM_equality
thf(fact_179_remove__code_I2_J,axiom,
! [A: $tType,X: A,Xs: list @ A] :
( ( remove @ A @ X @ ( coset @ A @ Xs ) )
= ( coset @ A @ ( insert @ A @ X @ Xs ) ) ) ).
% remove_code(2)
thf(fact_180_conj__subset__def,axiom,
! [A: $tType,A5: set @ A,P: A > $o,Q: A > $o] :
( ( ord_less_eq @ ( set @ A ) @ A5
@ ( collect @ A
@ ^ [X4: A] :
( ( P @ X4 )
& ( Q @ X4 ) ) ) )
= ( ( ord_less_eq @ ( set @ A ) @ A5 @ ( collect @ A @ P ) )
& ( ord_less_eq @ ( set @ A ) @ A5 @ ( collect @ A @ Q ) ) ) ) ).
% conj_subset_def
thf(fact_181_prop__restrict,axiom,
! [A: $tType,X: A,Z4: set @ A,X7: set @ A,P: A > $o] :
( ( member @ A @ X @ Z4 )
=> ( ( ord_less_eq @ ( set @ A ) @ Z4
@ ( collect @ A
@ ^ [X4: A] :
( ( member @ A @ X4 @ X7 )
& ( P @ X4 ) ) ) )
=> ( P @ X ) ) ) ).
% prop_restrict
thf(fact_182_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_183_Collect__restrict,axiom,
! [A: $tType,X7: set @ A,P: A > $o] :
( ord_less_eq @ ( set @ A )
@ ( collect @ A
@ ^ [X4: A] :
( ( member @ A @ X4 @ X7 )
& ( P @ X4 ) ) )
@ X7 ) ).
% Collect_restrict
thf(fact_184_subset__Collect__iff,axiom,
! [A: $tType,B4: set @ A,A5: set @ A,P: A > $o] :
( ( ord_less_eq @ ( set @ A ) @ B4 @ A5 )
=> ( ( ord_less_eq @ ( set @ A ) @ B4
@ ( collect @ A
@ ^ [X4: A] :
( ( member @ A @ X4 @ A5 )
& ( P @ X4 ) ) ) )
= ( ! [X4: A] :
( ( member @ A @ X4 @ B4 )
=> ( P @ X4 ) ) ) ) ) ).
% subset_Collect_iff
thf(fact_185_subset__CollectI,axiom,
! [A: $tType,B4: set @ A,A5: set @ A,Q: A > $o,P: A > $o] :
( ( ord_less_eq @ ( set @ A ) @ B4 @ A5 )
=> ( ! [X3: A] :
( ( member @ A @ X3 @ B4 )
=> ( ( Q @ X3 )
=> ( P @ X3 ) ) )
=> ( ord_less_eq @ ( set @ A )
@ ( collect @ A
@ ^ [X4: A] :
( ( member @ A @ X4 @ B4 )
& ( Q @ X4 ) ) )
@ ( collect @ A
@ ^ [X4: A] :
( ( member @ A @ X4 @ A5 )
& ( P @ X4 ) ) ) ) ) ) ).
% subset_CollectI
thf(fact_186_LeastM__equality,axiom,
! [A: $tType,B: $tType] :
( ( order @ A @ ( type2 @ A ) )
=> ! [P: B > $o,K: B,M: B > A] :
( ( P @ K )
=> ( ! [X3: B] :
( ( P @ X3 )
=> ( ord_less_eq @ A @ ( M @ K ) @ ( M @ X3 ) ) )
=> ( ( M @ ( hilbert_LeastM @ B @ A @ M @ P ) )
= ( M @ K ) ) ) ) ) ).
% LeastM_equality
thf(fact_187_LeastMI2,axiom,
! [B: $tType,A: $tType] :
( ( ord @ B @ ( type2 @ B ) )
=> ! [P: A > $o,X: A,M: A > B,Q: A > $o] :
( ( P @ X )
=> ( ! [Y4: A] :
( ( P @ Y4 )
=> ( ord_less_eq @ B @ ( M @ X ) @ ( M @ Y4 ) ) )
=> ( ! [X3: A] :
( ( P @ X3 )
=> ( ! [Y6: A] :
( ( P @ Y6 )
=> ( ord_less_eq @ B @ ( M @ X3 ) @ ( M @ Y6 ) ) )
=> ( Q @ X3 ) ) )
=> ( Q @ ( hilbert_LeastM @ A @ B @ M @ P ) ) ) ) ) ) ).
% LeastMI2
thf(fact_188_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_189_shift__def,axiom,
! [B: $tType,A: $tType] :
( ( bNF_Greatest_shift @ A @ B )
= ( ^ [Lab: ( list @ A ) > B,K2: A,Kl: list @ A] : ( Lab @ ( cons @ A @ K2 @ Kl ) ) ) ) ).
% shift_def
thf(fact_190_univ__def,axiom,
! [A: $tType,B: $tType] :
( ( bNF_Greatest_univ @ B @ A )
= ( ^ [F2: B > A,X8: set @ B] :
( F2
@ @+[X4: B] : ( member @ B @ X4 @ X8 ) ) ) ) ).
% univ_def
thf(fact_191_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 @ ( insert2 @ n @ ( root @ Tr ) @ Ns1 ) @ Tr @ T2 ) ) ) ).
% inFr2.Ind
thf(fact_192_subtr2_OStep,axiom,
! [Tr1: dtree,Ns: set @ n,Tr22: dtree,Tr32: dtree] :
( ( member @ n @ ( root @ Tr1 ) @ Ns )
=> ( ( member @ ( sum_sum @ t @ dtree ) @ ( sum_Inr @ dtree @ t @ Tr1 ) @ ( cont @ Tr22 ) )
=> ( ( gram_L1283001940subtr2 @ Ns @ Tr22 @ Tr32 )
=> ( gram_L1283001940subtr2 @ Ns @ Tr1 @ Tr32 ) ) ) ) ).
% subtr2.Step
thf(fact_193_insert__absorb2,axiom,
! [A: $tType,X: A,A5: set @ A] :
( ( insert2 @ A @ X @ ( insert2 @ A @ X @ A5 ) )
= ( insert2 @ A @ X @ A5 ) ) ).
% insert_absorb2
thf(fact_194_insert__iff,axiom,
! [A: $tType,A2: A,B2: A,A5: set @ A] :
( ( member @ A @ A2 @ ( insert2 @ A @ B2 @ A5 ) )
= ( ( A2 = B2 )
| ( member @ A @ A2 @ A5 ) ) ) ).
% insert_iff
thf(fact_195_insertCI,axiom,
! [A: $tType,A2: A,B4: set @ A,B2: A] :
( ( ~ ( member @ A @ A2 @ B4 )
=> ( A2 = B2 ) )
=> ( member @ A @ A2 @ ( insert2 @ A @ B2 @ B4 ) ) ) ).
% insertCI
thf(fact_196_insert__subset,axiom,
! [A: $tType,X: A,A5: set @ A,B4: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ ( insert2 @ A @ X @ A5 ) @ B4 )
= ( ( member @ A @ X @ B4 )
& ( ord_less_eq @ ( set @ A ) @ A5 @ B4 ) ) ) ).
% insert_subset
thf(fact_197_list_Osimps_I15_J,axiom,
! [A: $tType,X21: A,X22: list @ A] :
( ( set2 @ A @ ( cons @ A @ X21 @ X22 ) )
= ( insert2 @ A @ X21 @ ( set2 @ A @ X22 ) ) ) ).
% list.simps(15)
thf(fact_198_List_Oset__insert,axiom,
! [A: $tType,X: A,Xs: list @ A] :
( ( set2 @ A @ ( insert @ A @ X @ Xs ) )
= ( insert2 @ A @ X @ ( set2 @ A @ Xs ) ) ) ).
% List.set_insert
thf(fact_199_insert__subsetI,axiom,
! [A: $tType,X: A,A5: set @ A,X7: set @ A] :
( ( member @ A @ X @ A5 )
=> ( ( ord_less_eq @ ( set @ A ) @ X7 @ A5 )
=> ( ord_less_eq @ ( set @ A ) @ ( insert2 @ A @ X @ X7 ) @ A5 ) ) ) ).
% insert_subsetI
thf(fact_200_subtr2__mono,axiom,
! [Ns: set @ n,Tr1: dtree,Tr22: dtree,Ns2: set @ n] :
( ( gram_L1283001940subtr2 @ Ns @ Tr1 @ Tr22 )
=> ( ( ord_less_eq @ ( set @ n ) @ Ns @ Ns2 )
=> ( gram_L1283001940subtr2 @ Ns2 @ Tr1 @ Tr22 ) ) ) ).
% subtr2_mono
thf(fact_201_insert__mono,axiom,
! [A: $tType,C3: set @ A,D: set @ A,A2: A] :
( ( ord_less_eq @ ( set @ A ) @ C3 @ D )
=> ( ord_less_eq @ ( set @ A ) @ ( insert2 @ A @ A2 @ C3 ) @ ( insert2 @ A @ A2 @ D ) ) ) ).
% insert_mono
thf(fact_202_subset__insert,axiom,
! [A: $tType,X: A,A5: set @ A,B4: set @ A] :
( ~ ( member @ A @ X @ A5 )
=> ( ( ord_less_eq @ ( set @ A ) @ A5 @ ( insert2 @ A @ X @ B4 ) )
= ( ord_less_eq @ ( set @ A ) @ A5 @ B4 ) ) ) ).
% subset_insert
thf(fact_203_subset__insertI,axiom,
! [A: $tType,B4: set @ A,A2: A] : ( ord_less_eq @ ( set @ A ) @ B4 @ ( insert2 @ A @ A2 @ B4 ) ) ).
% subset_insertI
thf(fact_204_subset__insertI2,axiom,
! [A: $tType,A5: set @ A,B4: set @ A,B2: A] :
( ( ord_less_eq @ ( set @ A ) @ A5 @ B4 )
=> ( ord_less_eq @ ( set @ A ) @ A5 @ ( insert2 @ A @ B2 @ B4 ) ) ) ).
% subset_insertI2
thf(fact_205_subtr2__trans,axiom,
! [Ns: set @ n,Tr1: dtree,Tr22: dtree,Tr32: dtree] :
( ( gram_L1283001940subtr2 @ Ns @ Tr1 @ Tr22 )
=> ( ( gram_L1283001940subtr2 @ Ns @ Tr22 @ Tr32 )
=> ( gram_L1283001940subtr2 @ Ns @ Tr1 @ Tr32 ) ) ) ).
% subtr2_trans
thf(fact_206_mk__disjoint__insert,axiom,
! [A: $tType,A2: A,A5: set @ A] :
( ( member @ A @ A2 @ A5 )
=> ? [B7: set @ A] :
( ( A5
= ( insert2 @ A @ A2 @ B7 ) )
& ~ ( member @ A @ A2 @ B7 ) ) ) ).
% mk_disjoint_insert
thf(fact_207_insert__commute,axiom,
! [A: $tType,X: A,Y: A,A5: set @ A] :
( ( insert2 @ A @ X @ ( insert2 @ A @ Y @ A5 ) )
= ( insert2 @ A @ Y @ ( insert2 @ A @ X @ A5 ) ) ) ).
% insert_commute
thf(fact_208_insert__eq__iff,axiom,
! [A: $tType,A2: A,A5: set @ A,B2: A,B4: set @ A] :
( ~ ( member @ A @ A2 @ A5 )
=> ( ~ ( member @ A @ B2 @ B4 )
=> ( ( ( insert2 @ A @ A2 @ A5 )
= ( insert2 @ A @ B2 @ B4 ) )
= ( ( ( A2 = B2 )
=> ( A5 = B4 ) )
& ( ( A2 != B2 )
=> ? [C4: set @ A] :
( ( A5
= ( insert2 @ A @ B2 @ C4 ) )
& ~ ( member @ A @ B2 @ C4 )
& ( B4
= ( insert2 @ A @ A2 @ C4 ) )
& ~ ( member @ A @ A2 @ C4 ) ) ) ) ) ) ) ).
% insert_eq_iff
thf(fact_209_insert__absorb,axiom,
! [A: $tType,A2: A,A5: set @ A] :
( ( member @ A @ A2 @ A5 )
=> ( ( insert2 @ A @ A2 @ A5 )
= A5 ) ) ).
% insert_absorb
thf(fact_210_insert__ident,axiom,
! [A: $tType,X: A,A5: set @ A,B4: set @ A] :
( ~ ( member @ A @ X @ A5 )
=> ( ~ ( member @ A @ X @ B4 )
=> ( ( ( insert2 @ A @ X @ A5 )
= ( insert2 @ A @ X @ B4 ) )
= ( A5 = B4 ) ) ) ) ).
% insert_ident
thf(fact_211_Set_Oset__insert,axiom,
! [A: $tType,X: A,A5: set @ A] :
( ( member @ A @ X @ A5 )
=> ~ ! [B7: set @ A] :
( ( A5
= ( insert2 @ A @ X @ B7 ) )
=> ( member @ A @ X @ B7 ) ) ) ).
% Set.set_insert
thf(fact_212_insertI2,axiom,
! [A: $tType,A2: A,B4: set @ A,B2: A] :
( ( member @ A @ A2 @ B4 )
=> ( member @ A @ A2 @ ( insert2 @ A @ B2 @ B4 ) ) ) ).
% insertI2
thf(fact_213_insertI1,axiom,
! [A: $tType,A2: A,B4: set @ A] : ( member @ A @ A2 @ ( insert2 @ A @ A2 @ B4 ) ) ).
% insertI1
thf(fact_214_insertE,axiom,
! [A: $tType,A2: A,B2: A,A5: set @ A] :
( ( member @ A @ A2 @ ( insert2 @ A @ B2 @ A5 ) )
=> ( ( A2 != B2 )
=> ( member @ A @ A2 @ A5 ) ) ) ).
% insertE
thf(fact_215_insert__compr,axiom,
! [A: $tType] :
( ( insert2 @ A )
= ( ^ [A3: A,B5: set @ A] :
( collect @ A
@ ^ [X4: A] :
( ( X4 = A3 )
| ( member @ A @ X4 @ B5 ) ) ) ) ) ).
% insert_compr
thf(fact_216_insert__Collect,axiom,
! [A: $tType,A2: A,P: A > $o] :
( ( insert2 @ A @ A2 @ ( collect @ A @ P ) )
= ( collect @ A
@ ^ [U: A] :
( ( U != A2 )
=> ( P @ U ) ) ) ) ).
% insert_Collect
thf(fact_217_subtr__subtr2,axiom,
gram_L716654942_subtr = gram_L1283001940subtr2 ).
% subtr_subtr2
thf(fact_218_subtr2_ORefl,axiom,
! [Tr: dtree,Ns: set @ n] :
( ( member @ n @ ( root @ Tr ) @ Ns )
=> ( gram_L1283001940subtr2 @ Ns @ Tr @ Tr ) ) ).
% subtr2.Refl
thf(fact_219_subtr2__rootL__in,axiom,
! [Ns: set @ n,Tr1: dtree,Tr22: dtree] :
( ( gram_L1283001940subtr2 @ Ns @ Tr1 @ Tr22 )
=> ( member @ n @ ( root @ Tr1 ) @ Ns ) ) ).
% subtr2_rootL_in
thf(fact_220_subtr2__rootR__in,axiom,
! [Ns: set @ n,Tr1: dtree,Tr22: dtree] :
( ( gram_L1283001940subtr2 @ Ns @ Tr1 @ Tr22 )
=> ( member @ n @ ( root @ Tr22 ) @ Ns ) ) ).
% subtr2_rootR_in
thf(fact_221_subtr2_Oinducts,axiom,
! [X12: set @ n,X2: dtree,X32: dtree,P: ( set @ n ) > dtree > dtree > $o] :
( ( gram_L1283001940subtr2 @ X12 @ X2 @ X32 )
=> ( ! [Tr5: dtree,Ns4: set @ n] :
( ( member @ n @ ( root @ Tr5 ) @ Ns4 )
=> ( P @ Ns4 @ Tr5 @ Tr5 ) )
=> ( ! [Tr13: dtree,Ns4: set @ n,Tr23: dtree,Tr34: dtree] :
( ( member @ n @ ( root @ Tr13 ) @ Ns4 )
=> ( ( member @ ( sum_sum @ t @ dtree ) @ ( sum_Inr @ dtree @ t @ Tr13 ) @ ( cont @ Tr23 ) )
=> ( ( gram_L1283001940subtr2 @ Ns4 @ Tr23 @ Tr34 )
=> ( ( P @ Ns4 @ Tr23 @ Tr34 )
=> ( P @ Ns4 @ Tr13 @ Tr34 ) ) ) ) )
=> ( P @ X12 @ X2 @ X32 ) ) ) ) ).
% subtr2.inducts
thf(fact_222_subtr2__StepR,axiom,
! [Tr32: dtree,Ns: set @ n,Tr22: dtree,Tr1: dtree] :
( ( member @ n @ ( root @ Tr32 ) @ Ns )
=> ( ( member @ ( sum_sum @ t @ dtree ) @ ( sum_Inr @ dtree @ t @ Tr22 ) @ ( cont @ Tr32 ) )
=> ( ( gram_L1283001940subtr2 @ Ns @ Tr1 @ Tr22 )
=> ( gram_L1283001940subtr2 @ Ns @ Tr1 @ Tr32 ) ) ) ) ).
% subtr2_StepR
thf(fact_223_subtr2_Osimps,axiom,
( gram_L1283001940subtr2
= ( ^ [A12: set @ n,A23: dtree,A33: dtree] :
( ? [Tr3: dtree,Ns3: set @ n] :
( ( A12 = Ns3 )
& ( A23 = Tr3 )
& ( A33 = Tr3 )
& ( member @ n @ ( root @ Tr3 ) @ Ns3 ) )
| ? [Tr12: dtree,Ns3: set @ n,Tr24: dtree,Tr33: dtree] :
( ( A12 = Ns3 )
& ( A23 = Tr12 )
& ( A33 = Tr33 )
& ( member @ n @ ( root @ Tr12 ) @ Ns3 )
& ( member @ ( sum_sum @ t @ dtree ) @ ( sum_Inr @ dtree @ t @ Tr12 ) @ ( cont @ Tr24 ) )
& ( gram_L1283001940subtr2 @ Ns3 @ Tr24 @ Tr33 ) ) ) ) ) ).
% subtr2.simps
thf(fact_224_subtr2_Ocases,axiom,
! [A1: set @ n,A22: dtree,A32: dtree] :
( ( gram_L1283001940subtr2 @ A1 @ A22 @ A32 )
=> ( ( ( A32 = A22 )
=> ~ ( member @ n @ ( root @ A22 ) @ A1 ) )
=> ~ ( ( member @ n @ ( root @ A22 ) @ A1 )
=> ! [Tr23: dtree] :
( ( member @ ( sum_sum @ t @ dtree ) @ ( sum_Inr @ dtree @ t @ A22 ) @ ( cont @ Tr23 ) )
=> ~ ( gram_L1283001940subtr2 @ A1 @ Tr23 @ A32 ) ) ) ) ) ).
% subtr2.cases
thf(fact_225_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 @ ( insert2 @ n @ ( root @ Tr ) @ Ns1 ) @ Tr @ T2 ) ) ) ).
% inFr_Ind_minus
thf(fact_226_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 ) ) )
=> ~ ! [Tr13: dtree,Tr5: dtree,Ns12: set @ n] :
( ( A1
= ( insert2 @ n @ ( root @ Tr5 ) @ Ns12 ) )
=> ( ( A22 = Tr5 )
=> ( ( member @ ( sum_sum @ t @ dtree ) @ ( sum_Inr @ dtree @ t @ Tr13 ) @ ( cont @ Tr5 ) )
=> ~ ( gram_L805317441_inFr2 @ Ns12 @ Tr13 @ A32 ) ) ) ) ) ) ).
% inFr2.cases
thf(fact_227_sum_Oinject_I1_J,axiom,
! [B: $tType,A: $tType,X12: A,Y1: A] :
( ( ( sum_Inl @ A @ B @ X12 )
= ( sum_Inl @ A @ B @ Y1 ) )
= ( X12 = Y1 ) ) ).
% sum.inject(1)
thf(fact_228_old_Osum_Oinject_I1_J,axiom,
! [B: $tType,A: $tType,A2: A,A7: A] :
( ( ( sum_Inl @ A @ B @ A2 )
= ( sum_Inl @ A @ B @ A7 ) )
= ( A2 = A7 ) ) ).
% old.sum.inject(1)
thf(fact_229_old_Osum_Osimps_I7_J,axiom,
! [B: $tType,T: $tType,A: $tType,F1: A > T,F22: B > T,A2: A] :
( ( sum_rec_sum @ A @ T @ B @ F1 @ F22 @ ( sum_Inl @ A @ B @ A2 ) )
= ( F1 @ A2 ) ) ).
% old.sum.simps(7)
thf(fact_230_obj__sumE,axiom,
! [A: $tType,B: $tType,S2: sum_sum @ A @ B] :
( ! [X3: A] :
( S2
!= ( sum_Inl @ A @ B @ X3 ) )
=> ~ ! [X3: B] :
( S2
!= ( sum_Inr @ B @ A @ X3 ) ) ) ).
% obj_sumE
thf(fact_231_inFr__mono,axiom,
! [Ns: set @ n,Tr: dtree,T2: t,Ns2: set @ n] :
( ( gram_L1333338417e_inFr @ Ns @ Tr @ T2 )
=> ( ( ord_less_eq @ ( set @ n ) @ Ns @ Ns2 )
=> ( gram_L1333338417e_inFr @ Ns2 @ Tr @ T2 ) ) ) ).
% inFr_mono
thf(fact_232_subtr__inFr,axiom,
! [Ns: set @ n,Tr: dtree,T2: t,Tr1: dtree] :
( ( gram_L1333338417e_inFr @ Ns @ Tr @ T2 )
=> ( ( gram_L716654942_subtr @ Ns @ Tr @ Tr1 )
=> ( gram_L1333338417e_inFr @ Ns @ Tr1 @ T2 ) ) ) ).
% subtr_inFr
thf(fact_233_inFr_Oinducts,axiom,
! [X12: set @ n,X2: dtree,X32: t,P: ( set @ n ) > dtree > t > $o] :
( ( gram_L1333338417e_inFr @ X12 @ X2 @ X32 )
=> ( ! [Tr5: dtree,Ns4: set @ n,T4: t] :
( ( member @ n @ ( root @ Tr5 ) @ Ns4 )
=> ( ( member @ ( sum_sum @ t @ dtree ) @ ( sum_Inl @ t @ dtree @ T4 ) @ ( cont @ Tr5 ) )
=> ( P @ Ns4 @ Tr5 @ T4 ) ) )
=> ( ! [Tr5: dtree,Ns4: set @ n,Tr13: dtree,T4: t] :
( ( member @ n @ ( root @ Tr5 ) @ Ns4 )
=> ( ( member @ ( sum_sum @ t @ dtree ) @ ( sum_Inr @ dtree @ t @ Tr13 ) @ ( cont @ Tr5 ) )
=> ( ( gram_L1333338417e_inFr @ Ns4 @ Tr13 @ T4 )
=> ( ( P @ Ns4 @ Tr13 @ T4 )
=> ( P @ Ns4 @ Tr5 @ T4 ) ) ) ) )
=> ( P @ X12 @ X2 @ X32 ) ) ) ) ).
% inFr.inducts
thf(fact_234_inFr_Osimps,axiom,
( gram_L1333338417e_inFr
= ( ^ [A12: set @ n,A23: dtree,A33: t] :
( ? [Tr3: dtree,Ns3: set @ n,T3: t] :
( ( A12 = Ns3 )
& ( A23 = Tr3 )
& ( A33 = T3 )
& ( member @ n @ ( root @ Tr3 ) @ Ns3 )
& ( member @ ( sum_sum @ t @ dtree ) @ ( sum_Inl @ t @ dtree @ T3 ) @ ( cont @ Tr3 ) ) )
| ? [Tr3: dtree,Ns3: set @ n,Tr12: dtree,T3: t] :
( ( A12 = Ns3 )
& ( A23 = Tr3 )
& ( A33 = T3 )
& ( member @ n @ ( root @ Tr3 ) @ Ns3 )
& ( member @ ( sum_sum @ t @ dtree ) @ ( sum_Inr @ dtree @ t @ Tr12 ) @ ( cont @ Tr3 ) )
& ( gram_L1333338417e_inFr @ Ns3 @ Tr12 @ T3 ) ) ) ) ) ).
% inFr.simps
thf(fact_235_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 )
=> ! [Tr13: dtree] :
( ( member @ ( sum_sum @ t @ dtree ) @ ( sum_Inr @ dtree @ t @ Tr13 ) @ ( cont @ A22 ) )
=> ~ ( gram_L1333338417e_inFr @ A1 @ Tr13 @ A32 ) ) ) ) ) ).
% inFr.cases
thf(fact_236_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_237_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_238_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_239_inFr__subtr,axiom,
! [Ns: set @ n,Tr: dtree,T2: t] :
( ( gram_L1333338417e_inFr @ Ns @ Tr @ T2 )
=> ? [Tr6: dtree] :
( ( gram_L716654942_subtr @ Ns @ Tr6 @ Tr )
& ( member @ ( sum_sum @ t @ dtree ) @ ( sum_Inl @ t @ dtree @ T2 ) @ ( cont @ Tr6 ) ) ) ) ).
% inFr_subtr
thf(fact_240_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_241_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 ) )
=> ( ! [B6: B] : ( P @ ( sum_Inr @ B @ A @ B6 ) )
=> ( P @ Sum ) ) ) ).
% old.sum.inducts
thf(fact_242_old_Osum_Oexhaust,axiom,
! [A: $tType,B: $tType,Y: sum_sum @ A @ B] :
( ! [A4: A] :
( Y
!= ( sum_Inl @ A @ B @ A4 ) )
=> ~ ! [B6: B] :
( Y
!= ( sum_Inr @ B @ A @ B6 ) ) ) ).
% old.sum.exhaust
thf(fact_243_split__sum__all,axiom,
! [B: $tType,A: $tType] :
( ( ^ [P2: ( sum_sum @ A @ B ) > $o] :
! [X5: sum_sum @ A @ B] : ( P2 @ X5 ) )
= ( ^ [P4: ( sum_sum @ A @ B ) > $o] :
( ! [X4: A] : ( P4 @ ( sum_Inl @ A @ B @ X4 ) )
& ! [X4: B] : ( P4 @ ( sum_Inr @ B @ A @ X4 ) ) ) ) ) ).
% split_sum_all
thf(fact_244_split__sum__ex,axiom,
! [B: $tType,A: $tType] :
( ( ^ [P2: ( sum_sum @ A @ B ) > $o] :
? [X5: sum_sum @ A @ B] : ( P2 @ X5 ) )
= ( ^ [P4: ( sum_sum @ A @ B ) > $o] :
( ? [X4: A] : ( P4 @ ( sum_Inl @ A @ B @ X4 ) )
| ? [X4: B] : ( P4 @ ( sum_Inr @ B @ A @ X4 ) ) ) ) ) ).
% split_sum_ex
thf(fact_245_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_246_sumE,axiom,
! [A: $tType,B: $tType,S2: sum_sum @ A @ B] :
( ! [X3: A] :
( S2
!= ( sum_Inl @ A @ B @ X3 ) )
=> ~ ! [Y4: B] :
( S2
!= ( sum_Inr @ B @ A @ Y4 ) ) ) ).
% sumE
thf(fact_247_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_248_old_Osum_Odistinct_I2_J,axiom,
! [B8: $tType,A8: $tType,B9: B8,A9: A8] :
( ( sum_Inr @ B8 @ A8 @ B9 )
!= ( sum_Inl @ A8 @ B8 @ A9 ) ) ).
% old.sum.distinct(2)
thf(fact_249_sum_Odistinct_I1_J,axiom,
! [A: $tType,B: $tType,X12: A,X2: B] :
( ( sum_Inl @ A @ B @ X12 )
!= ( sum_Inr @ B @ A @ X2 ) ) ).
% sum.distinct(1)
thf(fact_250_inFr__inFr2,axiom,
gram_L1333338417e_inFr = gram_L805317441_inFr2 ).
% inFr_inFr2
thf(fact_251_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_252_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_253_inFr2_Oinducts,axiom,
! [X12: set @ n,X2: dtree,X32: t,P: ( set @ n ) > dtree > t > $o] :
( ( gram_L805317441_inFr2 @ X12 @ X2 @ X32 )
=> ( ! [Tr5: dtree,Ns4: set @ n,T4: t] :
( ( member @ n @ ( root @ Tr5 ) @ Ns4 )
=> ( ( member @ ( sum_sum @ t @ dtree ) @ ( sum_Inl @ t @ dtree @ T4 ) @ ( cont @ Tr5 ) )
=> ( P @ Ns4 @ Tr5 @ T4 ) ) )
=> ( ! [Tr13: dtree,Tr5: dtree,Ns12: set @ n,T4: t] :
( ( member @ ( sum_sum @ t @ dtree ) @ ( sum_Inr @ dtree @ t @ Tr13 ) @ ( cont @ Tr5 ) )
=> ( ( gram_L805317441_inFr2 @ Ns12 @ Tr13 @ T4 )
=> ( ( P @ Ns12 @ Tr13 @ T4 )
=> ( P @ ( insert2 @ n @ ( root @ Tr5 ) @ Ns12 ) @ Tr5 @ T4 ) ) ) )
=> ( P @ X12 @ X2 @ X32 ) ) ) ) ).
% inFr2.inducts
thf(fact_254_inFr2_Osimps,axiom,
( gram_L805317441_inFr2
= ( ^ [A12: set @ n,A23: dtree,A33: t] :
( ? [Tr3: dtree,Ns3: set @ n,T3: t] :
( ( A12 = Ns3 )
& ( A23 = Tr3 )
& ( A33 = T3 )
& ( member @ n @ ( root @ Tr3 ) @ Ns3 )
& ( member @ ( sum_sum @ t @ dtree ) @ ( sum_Inl @ t @ dtree @ T3 ) @ ( cont @ Tr3 ) ) )
| ? [Tr12: dtree,Tr3: dtree,Ns13: set @ n,T3: t] :
( ( A12
= ( insert2 @ n @ ( root @ Tr3 ) @ Ns13 ) )
& ( A23 = Tr3 )
& ( A33 = T3 )
& ( member @ ( sum_sum @ t @ dtree ) @ ( sum_Inr @ dtree @ t @ Tr12 ) @ ( cont @ Tr3 ) )
& ( gram_L805317441_inFr2 @ Ns13 @ Tr12 @ T3 ) ) ) ) ) ).
% inFr2.simps
%----Type constructors (10)
thf(tcon_fun___Orderings_Opreorder,axiom,
! [A8: $tType,A10: $tType] :
( ( preorder @ A10 @ ( type2 @ A10 ) )
=> ( preorder @ ( A8 > A10 ) @ ( type2 @ ( A8 > A10 ) ) ) ) ).
thf(tcon_fun___Orderings_Oorder,axiom,
! [A8: $tType,A10: $tType] :
( ( order @ A10 @ ( type2 @ A10 ) )
=> ( order @ ( A8 > A10 ) @ ( type2 @ ( A8 > A10 ) ) ) ) ).
thf(tcon_fun___Orderings_Oord,axiom,
! [A8: $tType,A10: $tType] :
( ( ord @ A10 @ ( type2 @ A10 ) )
=> ( ord @ ( A8 > A10 ) @ ( type2 @ ( A8 > A10 ) ) ) ) ).
thf(tcon_Set_Oset___Orderings_Opreorder_1,axiom,
! [A8: $tType] : ( preorder @ ( set @ A8 ) @ ( type2 @ ( set @ A8 ) ) ) ).
thf(tcon_Set_Oset___Orderings_Oorder_2,axiom,
! [A8: $tType] : ( order @ ( set @ A8 ) @ ( type2 @ ( set @ A8 ) ) ) ).
thf(tcon_Set_Oset___Orderings_Oord_3,axiom,
! [A8: $tType] : ( ord @ ( set @ A8 ) @ ( type2 @ ( set @ A8 ) ) ) ).
thf(tcon_HOL_Obool___Orderings_Opreorder_4,axiom,
preorder @ $o @ ( type2 @ $o ) ).
thf(tcon_HOL_Obool___Orderings_Olinorder,axiom,
linorder @ $o @ ( type2 @ $o ) ).
thf(tcon_HOL_Obool___Orderings_Oorder_5,axiom,
order @ $o @ ( type2 @ $o ) ).
thf(tcon_HOL_Obool___Orderings_Oord_6,axiom,
ord @ $o @ ( type2 @ $o ) ).
%----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 (1)
thf(conj_0,conjecture,
( ? [N4: n] :
( ( cons @ n @ ( root @ tra ) @ nl )
= ( cons @ n @ N4 @ ( nil @ n ) ) )
| ? [N14: n,Nl5: list @ n,N4: n] :
( ( ( cons @ n @ ( root @ tra ) @ nl )
= ( cons @ n @ N4 @ ( cons @ n @ N14 @ Nl5 ) ) )
& ( gram_L250615845e_path @ f @ ( cons @ n @ N14 @ Nl5 ) )
& ( member @ ( sum_sum @ t @ dtree ) @ ( sum_Inr @ dtree @ t @ ( f @ N14 ) ) @ ( cont @ ( f @ N4 ) ) ) ) ) ).
%------------------------------------------------------------------------------