TPTP Problem File: COM193^1.p
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%------------------------------------------------------------------------------
% File : COM193^1 : TPTP v8.2.0. Released v7.0.0.
% Domain : Computing Theory
% Problem : Grammars and languages 1154
% 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__1154.p [Bla16]
% Status : Theorem
% Rating : 0.00 v7.2.0, 0.25 v7.1.0
% Syntax : Number of formulae : 340 ( 127 unt; 57 typ; 0 def)
% Number of atoms : 712 ( 215 equ; 0 cnn)
% Maximal formula atoms : 13 ( 2 avg)
% Number of connectives : 3215 ( 70 ~; 6 |; 38 &;2817 @)
% ( 0 <=>; 284 =>; 0 <=; 0 <~>)
% Maximal formula depth : 17 ( 7 avg)
% Number of types : 4 ( 3 usr)
% Number of type conns : 172 ( 172 >; 0 *; 0 +; 0 <<)
% Number of symbols : 56 ( 54 usr; 3 con; 0-6 aty)
% Number of variables : 847 ( 46 ^; 759 !; 11 ?; 847 :)
% ( 31 !>; 0 ?*; 0 @-; 0 @+)
% SPC : TH1_THM_EQU_NAR
% Comments : This file was generated by Isabelle (most likely Sledgehammer)
% 2016-07-13 14:47:18.227
%------------------------------------------------------------------------------
%----Could-be-implicit typings (6)
thf(ty_t_Sum__Type_Osum,type,
sum_sum: $tType > $tType > $tType ).
thf(ty_t_DTree_Odtree,type,
dtree: $tType ).
thf(ty_t_Set_Oset,type,
set: $tType > $tType ).
thf(ty_t_DTree_OT,type,
t: $tType ).
thf(ty_t_DTree_ON,type,
n: $tType ).
thf(ty_t_itself,type,
itself: $tType > $tType ).
%----Explicit typings (51)
thf(sy_cl_HOL_Otype,type,
type:
!>[A: $tType] : ( ( itself @ A ) > $o ) ).
thf(sy_cl_Orderings_Obot,type,
bot:
!>[A: $tType] : ( ( itself @ A ) > $o ) ).
thf(sy_cl_Orderings_Oord,type,
ord:
!>[A: $tType] : ( ( itself @ A ) > $o ) ).
thf(sy_cl_Orderings_Otop,type,
top:
!>[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_cl_Orderings_Oorder__bot,type,
order_bot:
!>[A: $tType] : ( ( itself @ A ) > $o ) ).
thf(sy_cl_Orderings_Oorder__top,type,
order_top:
!>[A: $tType] : ( ( itself @ A ) > $o ) ).
thf(sy_cl_Lattices_Oboolean__algebra,type,
boolean_algebra:
!>[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_c_DTree_ONode,type,
node: n > ( set @ ( sum_sum @ t @ dtree ) ) > dtree ).
thf(sy_c_DTree_Ocorec,type,
corec:
!>[A: $tType] : ( ( A > n ) > ( A > ( set @ ( sum_sum @ t @ ( sum_sum @ dtree @ A ) ) ) ) > A > dtree ) ).
thf(sy_c_DTree_Odtree_Oroot,type,
root: dtree > n ).
thf(sy_c_DTree_Ounfold,type,
unfold:
!>[A: $tType] : ( ( A > n ) > ( A > ( set @ ( sum_sum @ t @ A ) ) ) > A > dtree ) ).
thf(sy_c_Gram__Lang__Mirabelle__ojxrtuoybn_OFr,type,
gram_L861583724lle_Fr: ( set @ n ) > dtree > ( set @ t ) ).
thf(sy_c_Gram__Lang__Mirabelle__ojxrtuoybn_OH,type,
gram_L1451583624elle_H: dtree > n > dtree ).
thf(sy_c_Gram__Lang__Mirabelle__ojxrtuoybn_OH__c,type,
gram_L1221482011le_H_c: dtree > n > ( set @ ( sum_sum @ t @ n ) ) ).
thf(sy_c_Gram__Lang__Mirabelle__ojxrtuoybn_OH__r,type,
gram_L1221482026le_H_r: dtree > n > n ).
thf(sy_c_Gram__Lang__Mirabelle__ojxrtuoybn_OL,type,
gram_L1451583628elle_L: ( set @ n ) > n > ( set @ ( set @ t ) ) ).
thf(sy_c_Gram__Lang__Mirabelle__ojxrtuoybn_Odeftr,type,
gram_L1231612515_deftr: n > dtree ).
thf(sy_c_Gram__Lang__Mirabelle__ojxrtuoybn_Ohsubst,type,
gram_L1004374585hsubst: dtree > dtree > dtree ).
thf(sy_c_Gram__Lang__Mirabelle__ojxrtuoybn_Ohsubst__c,type,
gram_L1905609002ubst_c: dtree > dtree > ( set @ ( sum_sum @ t @ dtree ) ) ).
thf(sy_c_Gram__Lang__Mirabelle__ojxrtuoybn_Ohsubst__r,type,
gram_L1905609017ubst_r: dtree > n ).
thf(sy_c_Gram__Lang__Mirabelle__ojxrtuoybn_OinFr,type,
gram_L1333338417e_inFr: ( set @ n ) > dtree > t > $o ).
thf(sy_c_Gram__Lang__Mirabelle__ojxrtuoybn_OinItr,type,
gram_L830233218_inItr: ( set @ n ) > dtree > n > $o ).
thf(sy_c_Gram__Lang__Mirabelle__ojxrtuoybn_Opick,type,
gram_L315592705e_pick: dtree > n > dtree ).
thf(sy_c_Gram__Lang__Mirabelle__ojxrtuoybn_Orcut,type,
gram_L1828378864e_rcut: dtree > dtree ).
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_Owf,type,
gram_L864798063lle_wf: dtree > $o ).
thf(sy_c_Groups_Ominus__class_Ominus,type,
minus_minus:
!>[A: $tType] : ( A > A > A ) ).
thf(sy_c_Groups_Ouminus__class_Ouminus,type,
uminus_uminus:
!>[A: $tType] : ( 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_Orderings_Otop__class_Otop,type,
top_top:
!>[A: $tType] : A ).
thf(sy_c_Pure_Otype,type,
type2:
!>[A: $tType] : ( itself @ A ) ).
thf(sy_c_Relation_Oinv__imagep,type,
inv_imagep:
!>[B: $tType,A: $tType] : ( ( B > B > $o ) > ( A > B ) > A > A > $o ) ).
thf(sy_c_Set_OCollect,type,
collect:
!>[A: $tType] : ( ( A > $o ) > ( set @ A ) ) ).
thf(sy_c_Set_OPow,type,
pow:
!>[A: $tType] : ( ( set @ A ) > ( set @ ( 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_Opairwise,type,
pairwise:
!>[A: $tType] : ( ( A > A > $o ) > ( 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_member,type,
member:
!>[A: $tType] : ( A > ( set @ A ) > $o ) ).
thf(sy_v_n,type,
n2: n ).
thf(sy_v_ns,type,
ns: set @ n ).
%----Relevant facts (256)
thf(fact_0_assms,axiom,
~ ( member @ n @ n2 @ ns ) ).
% assms
thf(fact_1_not__root__Fr,axiom,
! [Tr: dtree,Ns: set @ n] :
( ~ ( member @ n @ ( root @ Tr ) @ Ns )
=> ( ( gram_L861583724lle_Fr @ Ns @ Tr )
= ( bot_bot @ ( set @ t ) ) ) ) ).
% not_root_Fr
thf(fact_2_empty__iff,axiom,
! [A: $tType,C: A] :
~ ( member @ A @ C @ ( bot_bot @ ( set @ A ) ) ) ).
% empty_iff
thf(fact_3_all__not__in__conv,axiom,
! [A: $tType,A2: set @ A] :
( ( ! [X: A] :
~ ( member @ A @ X @ A2 ) )
= ( A2
= ( bot_bot @ ( set @ A ) ) ) ) ).
% all_not_in_conv
thf(fact_4_Collect__empty__eq,axiom,
! [A: $tType,P: A > $o] :
( ( ( collect @ A @ P )
= ( bot_bot @ ( set @ A ) ) )
= ( ! [X: A] :
~ ( P @ X ) ) ) ).
% Collect_empty_eq
thf(fact_5_empty__Collect__eq,axiom,
! [A: $tType,P: A > $o] :
( ( ( bot_bot @ ( set @ A ) )
= ( collect @ A @ P ) )
= ( ! [X: A] :
~ ( P @ X ) ) ) ).
% empty_Collect_eq
thf(fact_6_bot__apply,axiom,
! [C2: $tType,D: $tType] :
( ( bot @ C2 @ ( type2 @ C2 ) )
=> ( ( bot_bot @ ( D > C2 ) )
= ( ^ [X: D] : ( bot_bot @ C2 ) ) ) ) ).
% bot_apply
thf(fact_7_hsubst__r__def,axiom,
gram_L1905609017ubst_r = root ).
% hsubst_r_def
thf(fact_8_root__rcut,axiom,
! [Tr0: dtree] :
( ( root @ ( gram_L1828378864e_rcut @ Tr0 ) )
= ( root @ Tr0 ) ) ).
% root_rcut
thf(fact_9_regular__def2,axiom,
( gram_L646766332egular
= ( ^ [Tr2: dtree] :
? [F: n > dtree] :
( ( gram_L1918716148le_reg @ F @ Tr2 )
& ! [N: n] :
( ( root @ ( F @ N ) )
= N ) ) ) ) ).
% regular_def2
thf(fact_10_emptyE,axiom,
! [A: $tType,A3: A] :
~ ( member @ A @ A3 @ ( bot_bot @ ( set @ A ) ) ) ).
% emptyE
thf(fact_11_equals0D,axiom,
! [A: $tType,A2: set @ A,A3: A] :
( ( A2
= ( bot_bot @ ( set @ A ) ) )
=> ~ ( member @ A @ A3 @ A2 ) ) ).
% equals0D
thf(fact_12_equals0I,axiom,
! [A: $tType,A2: set @ A] :
( ! [Y: A] :
~ ( member @ A @ Y @ A2 )
=> ( A2
= ( bot_bot @ ( set @ A ) ) ) ) ).
% equals0I
thf(fact_13_ex__in__conv,axiom,
! [A: $tType,A2: set @ A] :
( ( ? [X: A] : ( member @ A @ X @ A2 ) )
= ( A2
!= ( bot_bot @ ( set @ A ) ) ) ) ).
% ex_in_conv
thf(fact_14_bot__set__def,axiom,
! [A: $tType] :
( ( bot_bot @ ( set @ A ) )
= ( collect @ A @ ( bot_bot @ ( A > $o ) ) ) ) ).
% bot_set_def
thf(fact_15_reg__root,axiom,
! [F2: n > dtree,Tr: dtree] :
( ( gram_L1918716148le_reg @ F2 @ Tr )
=> ( ( F2 @ ( root @ Tr ) )
= Tr ) ) ).
% reg_root
thf(fact_16_regular__def,axiom,
( gram_L646766332egular
= ( ^ [Tr2: dtree] :
? [F: n > dtree] : ( gram_L1918716148le_reg @ F @ Tr2 ) ) ) ).
% regular_def
thf(fact_17_wf__rcut,axiom,
! [Tr0: dtree] :
( ( gram_L864798063lle_wf @ Tr0 )
=> ( gram_L864798063lle_wf @ ( gram_L1828378864e_rcut @ Tr0 ) ) ) ).
% wf_rcut
thf(fact_18_regular__rcut,axiom,
! [Tr0: dtree] : ( gram_L646766332egular @ ( gram_L1828378864e_rcut @ Tr0 ) ) ).
% regular_rcut
thf(fact_19_bot__fun__def,axiom,
! [B: $tType,A: $tType] :
( ( bot @ B @ ( type2 @ B ) )
=> ( ( bot_bot @ ( A > B ) )
= ( ^ [X: A] : ( bot_bot @ B ) ) ) ) ).
% bot_fun_def
thf(fact_20_Set_Ois__empty__def,axiom,
! [A: $tType] :
( ( is_empty @ A )
= ( ^ [A4: set @ A] :
( A4
= ( bot_bot @ ( set @ A ) ) ) ) ) ).
% Set.is_empty_def
thf(fact_21_rcut__eq,axiom,
! [Tr0: dtree] :
( ( ( gram_L1828378864e_rcut @ Tr0 )
= Tr0 )
= ( gram_L1918716148le_reg @ ( gram_L1451583624elle_H @ Tr0 ) @ Tr0 ) ) ).
% rcut_eq
thf(fact_22_rcut__reg,axiom,
! [Tr0: dtree] :
( ( gram_L1918716148le_reg @ ( gram_L1451583624elle_H @ Tr0 ) @ Tr0 )
=> ( ( gram_L1828378864e_rcut @ Tr0 )
= Tr0 ) ) ).
% rcut_reg
thf(fact_23_reg__rcut,axiom,
! [Tr0: dtree] : ( gram_L1918716148le_reg @ ( gram_L1451583624elle_H @ Tr0 ) @ ( gram_L1828378864e_rcut @ Tr0 ) ) ).
% reg_rcut
thf(fact_24_rcut__def,axiom,
( gram_L1828378864e_rcut
= ( ^ [Tr02: dtree] : ( gram_L1451583624elle_H @ Tr02 @ ( root @ Tr02 ) ) ) ) ).
% rcut_def
thf(fact_25_Collect__empty__eq__bot,axiom,
! [A: $tType,P: A > $o] :
( ( ( collect @ A @ P )
= ( bot_bot @ ( set @ A ) ) )
= ( P
= ( bot_bot @ ( A > $o ) ) ) ) ).
% Collect_empty_eq_bot
thf(fact_26_bot__empty__eq,axiom,
! [A: $tType] :
( ( bot_bot @ ( A > $o ) )
= ( ^ [X: A] : ( member @ A @ X @ ( bot_bot @ ( set @ A ) ) ) ) ) ).
% bot_empty_eq
thf(fact_27_root__deftr,axiom,
! [N2: n] :
( ( root @ ( gram_L1231612515_deftr @ N2 ) )
= N2 ) ).
% root_deftr
thf(fact_28_is__singletonI_H,axiom,
! [A: $tType,A2: set @ A] :
( ( A2
!= ( bot_bot @ ( set @ A ) ) )
=> ( ! [X2: A,Y: A] :
( ( member @ A @ X2 @ A2 )
=> ( ( member @ A @ Y @ A2 )
=> ( X2 = Y ) ) )
=> ( is_singleton @ A @ A2 ) ) ) ).
% is_singletonI'
thf(fact_29_reg__deftr,axiom,
! [N2: n] : ( gram_L1918716148le_reg @ gram_L1231612515_deftr @ ( gram_L1231612515_deftr @ N2 ) ) ).
% reg_deftr
thf(fact_30_reg__def2,axiom,
( gram_L1918716148le_reg
= ( ^ [F: n > dtree,Tr2: dtree] :
! [Ns2: set @ n,Tr3: dtree] :
( ( gram_L716654942_subtr @ Ns2 @ Tr3 @ Tr2 )
=> ( Tr3
= ( F @ ( root @ Tr3 ) ) ) ) ) ) ).
% reg_def2
thf(fact_31_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_32_subtr__deftr,axiom,
! [Ns: set @ n,Tr4: dtree,N2: n] :
( ( gram_L716654942_subtr @ Ns @ Tr4 @ ( gram_L1231612515_deftr @ N2 ) )
=> ( Tr4
= ( gram_L1231612515_deftr @ ( root @ Tr4 ) ) ) ) ).
% subtr_deftr
thf(fact_33_subtr_ORefl,axiom,
! [Tr: dtree,Ns: set @ n] :
( ( member @ n @ ( root @ Tr ) @ Ns )
=> ( gram_L716654942_subtr @ Ns @ Tr @ Tr ) ) ).
% subtr.Refl
thf(fact_34_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_35_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_36_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_37_reg__subtr,axiom,
! [F2: n > dtree,Tr: dtree,Ns: set @ n,Tr4: dtree] :
( ( gram_L1918716148le_reg @ F2 @ Tr )
=> ( ( gram_L716654942_subtr @ Ns @ Tr4 @ Tr )
=> ( gram_L1918716148le_reg @ F2 @ Tr4 ) ) ) ).
% reg_subtr
thf(fact_38_regular__subtr,axiom,
! [Tr: dtree,Ns: set @ n,Tr4: dtree] :
( ( gram_L646766332egular @ Tr )
=> ( ( gram_L716654942_subtr @ Ns @ Tr4 @ Tr )
=> ( gram_L646766332egular @ Tr4 ) ) ) ).
% regular_subtr
thf(fact_39_wf__deftr,axiom,
! [N2: n] : ( gram_L864798063lle_wf @ ( gram_L1231612515_deftr @ N2 ) ) ).
% wf_deftr
thf(fact_40_root__H__pick,axiom,
! [Tr0: dtree,N2: n] :
( ( root @ ( gram_L1451583624elle_H @ Tr0 @ N2 ) )
= ( root @ ( gram_L315592705e_pick @ Tr0 @ N2 ) ) ) ).
% root_H_pick
thf(fact_41_reg__def,axiom,
( gram_L1918716148le_reg
= ( ^ [F: n > dtree,Tr2: dtree] :
! [Tr3: dtree] :
( ( gram_L716654942_subtr @ ( top_top @ ( set @ n ) ) @ Tr3 @ Tr2 )
=> ( Tr3
= ( F @ ( root @ Tr3 ) ) ) ) ) ) ).
% reg_def
thf(fact_42_in__inv__imagep,axiom,
! [B: $tType,A: $tType] :
( ( inv_imagep @ A @ B )
= ( ^ [R: A > A > $o,F: B > A,X: B,Y2: B] : ( R @ ( F @ X ) @ ( F @ Y2 ) ) ) ) ).
% in_inv_imagep
thf(fact_43_is__singletonI,axiom,
! [A: $tType,X3: A] : ( is_singleton @ A @ ( insert @ A @ X3 @ ( bot_bot @ ( set @ A ) ) ) ) ).
% is_singletonI
thf(fact_44_inItr__subtr,axiom,
! [Ns: set @ n,Tr: dtree,N2: n] :
( ( gram_L830233218_inItr @ Ns @ Tr @ N2 )
=> ? [Tr5: dtree] :
( ( gram_L716654942_subtr @ Ns @ Tr5 @ Tr )
& ( ( root @ Tr5 )
= N2 ) ) ) ).
% inItr_subtr
thf(fact_45_mem__Collect__eq,axiom,
! [A: $tType,A3: A,P: A > $o] :
( ( member @ A @ A3 @ ( collect @ A @ P ) )
= ( P @ A3 ) ) ).
% mem_Collect_eq
thf(fact_46_Collect__mem__eq,axiom,
! [A: $tType,A2: set @ A] :
( ( collect @ A
@ ^ [X: A] : ( member @ A @ X @ A2 ) )
= A2 ) ).
% Collect_mem_eq
thf(fact_47_Collect__cong,axiom,
! [A: $tType,P: A > $o,Q: A > $o] :
( ! [X2: A] :
( ( P @ X2 )
= ( Q @ X2 ) )
=> ( ( collect @ A @ P )
= ( collect @ A @ Q ) ) ) ).
% Collect_cong
thf(fact_48_ext,axiom,
! [B: $tType,A: $tType,F2: A > B,G: A > B] :
( ! [X2: A] :
( ( F2 @ X2 )
= ( G @ X2 ) )
=> ( F2 = G ) ) ).
% ext
thf(fact_49_is__singleton__def,axiom,
! [A: $tType] :
( ( is_singleton @ A )
= ( ^ [A4: set @ A] :
? [X: A] :
( A4
= ( insert @ A @ X @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ).
% is_singleton_def
thf(fact_50_is__singletonE,axiom,
! [A: $tType,A2: set @ A] :
( ( is_singleton @ A @ A2 )
=> ~ ! [X2: A] :
( A2
!= ( insert @ A @ X2 @ ( bot_bot @ ( set @ A ) ) ) ) ) ).
% is_singletonE
thf(fact_51_root__hsubst,axiom,
! [Tr0: dtree,Tr: dtree] :
( ( root @ ( gram_L1004374585hsubst @ Tr0 @ Tr ) )
= ( root @ Tr ) ) ).
% root_hsubst
thf(fact_52_pairwise__empty,axiom,
! [A: $tType,P: A > A > $o] : ( pairwise @ A @ P @ ( bot_bot @ ( set @ A ) ) ) ).
% pairwise_empty
thf(fact_53_top__apply,axiom,
! [C2: $tType,D: $tType] :
( ( top @ C2 @ ( type2 @ C2 ) )
=> ( ( top_top @ ( D > C2 ) )
= ( ^ [X: D] : ( top_top @ C2 ) ) ) ) ).
% top_apply
thf(fact_54_UNIV__I,axiom,
! [A: $tType,X3: A] : ( member @ A @ X3 @ ( top_top @ ( set @ A ) ) ) ).
% UNIV_I
thf(fact_55_insert__absorb2,axiom,
! [A: $tType,X3: A,A2: set @ A] :
( ( insert @ A @ X3 @ ( insert @ A @ X3 @ A2 ) )
= ( insert @ A @ X3 @ A2 ) ) ).
% insert_absorb2
thf(fact_56_insert__iff,axiom,
! [A: $tType,A3: A,B2: A,A2: set @ A] :
( ( member @ A @ A3 @ ( insert @ A @ B2 @ A2 ) )
= ( ( A3 = B2 )
| ( member @ A @ A3 @ A2 ) ) ) ).
% insert_iff
thf(fact_57_insertCI,axiom,
! [A: $tType,A3: A,B3: set @ A,B2: A] :
( ( ~ ( member @ A @ A3 @ B3 )
=> ( A3 = B2 ) )
=> ( member @ A @ A3 @ ( insert @ A @ B2 @ B3 ) ) ) ).
% insertCI
thf(fact_58_singletonI,axiom,
! [A: $tType,A3: A] : ( member @ A @ A3 @ ( insert @ A @ A3 @ ( bot_bot @ ( set @ A ) ) ) ) ).
% singletonI
thf(fact_59_root__H,axiom,
! [Tr0: dtree,N2: n] :
( ( gram_L830233218_inItr @ ( top_top @ ( set @ n ) ) @ Tr0 @ N2 )
=> ( ( root @ ( gram_L1451583624elle_H @ Tr0 @ N2 ) )
= N2 ) ) ).
% root_H
thf(fact_60_subtr__pick,axiom,
! [Tr0: dtree,N2: n] :
( ( gram_L830233218_inItr @ ( top_top @ ( set @ n ) ) @ Tr0 @ N2 )
=> ( gram_L716654942_subtr @ ( top_top @ ( set @ n ) ) @ ( gram_L315592705e_pick @ Tr0 @ N2 ) @ Tr0 ) ) ).
% subtr_pick
thf(fact_61_pairwise__def,axiom,
! [A: $tType] :
( ( pairwise @ A )
= ( ^ [R2: A > A > $o,S: set @ A] :
! [X: A] :
( ( member @ A @ X @ S )
=> ! [Y2: A] :
( ( member @ A @ Y2 @ S )
=> ( ( X != Y2 )
=> ( R2 @ X @ Y2 ) ) ) ) ) ) ).
% pairwise_def
thf(fact_62_wf__pick,axiom,
! [Tr0: dtree,N2: n] :
( ( gram_L864798063lle_wf @ Tr0 )
=> ( ( gram_L830233218_inItr @ ( top_top @ ( set @ n ) ) @ Tr0 @ N2 )
=> ( gram_L864798063lle_wf @ ( gram_L315592705e_pick @ Tr0 @ N2 ) ) ) ) ).
% wf_pick
thf(fact_63_root__pick,axiom,
! [Tr0: dtree,N2: n] :
( ( gram_L830233218_inItr @ ( top_top @ ( set @ n ) ) @ Tr0 @ N2 )
=> ( ( root @ ( gram_L315592705e_pick @ Tr0 @ N2 ) )
= N2 ) ) ).
% root_pick
thf(fact_64_mk__disjoint__insert,axiom,
! [A: $tType,A3: A,A2: set @ A] :
( ( member @ A @ A3 @ A2 )
=> ? [B4: set @ A] :
( ( A2
= ( insert @ A @ A3 @ B4 ) )
& ~ ( member @ A @ A3 @ B4 ) ) ) ).
% mk_disjoint_insert
thf(fact_65_pairwise__insert,axiom,
! [A: $tType,R3: A > A > $o,X3: A,S2: set @ A] :
( ( pairwise @ A @ R3 @ ( insert @ A @ X3 @ S2 ) )
= ( ! [Y2: A] :
( ( ( member @ A @ Y2 @ S2 )
& ( Y2 != X3 ) )
=> ( ( R3 @ X3 @ Y2 )
& ( R3 @ Y2 @ X3 ) ) )
& ( pairwise @ A @ R3 @ S2 ) ) ) ).
% pairwise_insert
thf(fact_66_insert__commute,axiom,
! [A: $tType,X3: A,Y3: A,A2: set @ A] :
( ( insert @ A @ X3 @ ( insert @ A @ Y3 @ A2 ) )
= ( insert @ A @ Y3 @ ( insert @ A @ X3 @ A2 ) ) ) ).
% insert_commute
thf(fact_67_insert__eq__iff,axiom,
! [A: $tType,A3: A,A2: set @ A,B2: A,B3: set @ A] :
( ~ ( member @ A @ A3 @ A2 )
=> ( ~ ( member @ A @ B2 @ B3 )
=> ( ( ( insert @ A @ A3 @ A2 )
= ( insert @ A @ B2 @ B3 ) )
= ( ( ( A3 = B2 )
=> ( A2 = B3 ) )
& ( ( A3 != B2 )
=> ? [C3: set @ A] :
( ( A2
= ( insert @ A @ B2 @ C3 ) )
& ~ ( member @ A @ B2 @ C3 )
& ( B3
= ( insert @ A @ A3 @ C3 ) )
& ~ ( member @ A @ A3 @ C3 ) ) ) ) ) ) ) ).
% insert_eq_iff
thf(fact_68_insert__absorb,axiom,
! [A: $tType,A3: A,A2: set @ A] :
( ( member @ A @ A3 @ A2 )
=> ( ( insert @ A @ A3 @ A2 )
= A2 ) ) ).
% insert_absorb
thf(fact_69_insert__ident,axiom,
! [A: $tType,X3: A,A2: set @ A,B3: set @ A] :
( ~ ( member @ A @ X3 @ A2 )
=> ( ~ ( member @ A @ X3 @ B3 )
=> ( ( ( insert @ A @ X3 @ A2 )
= ( insert @ A @ X3 @ B3 ) )
= ( A2 = B3 ) ) ) ) ).
% insert_ident
thf(fact_70_UNIV__witness,axiom,
! [A: $tType] :
? [X2: A] : ( member @ A @ X2 @ ( top_top @ ( set @ A ) ) ) ).
% UNIV_witness
thf(fact_71_insert__UNIV,axiom,
! [A: $tType,X3: A] :
( ( insert @ A @ X3 @ ( top_top @ ( set @ A ) ) )
= ( top_top @ ( set @ A ) ) ) ).
% insert_UNIV
thf(fact_72_Set_Oset__insert,axiom,
! [A: $tType,X3: A,A2: set @ A] :
( ( member @ A @ X3 @ A2 )
=> ~ ! [B4: set @ A] :
( ( A2
= ( insert @ A @ X3 @ B4 ) )
=> ( member @ A @ X3 @ B4 ) ) ) ).
% Set.set_insert
thf(fact_73_UNIV__eq__I,axiom,
! [A: $tType,A2: set @ A] :
( ! [X2: A] : ( member @ A @ X2 @ A2 )
=> ( ( top_top @ ( set @ A ) )
= A2 ) ) ).
% UNIV_eq_I
thf(fact_74_insertI2,axiom,
! [A: $tType,A3: A,B3: set @ A,B2: A] :
( ( member @ A @ A3 @ B3 )
=> ( member @ A @ A3 @ ( insert @ A @ B2 @ B3 ) ) ) ).
% insertI2
thf(fact_75_insertI1,axiom,
! [A: $tType,A3: A,B3: set @ A] : ( member @ A @ A3 @ ( insert @ A @ A3 @ B3 ) ) ).
% insertI1
thf(fact_76_insertE,axiom,
! [A: $tType,A3: A,B2: A,A2: set @ A] :
( ( member @ A @ A3 @ ( insert @ A @ B2 @ A2 ) )
=> ( ( A3 != B2 )
=> ( member @ A @ A3 @ A2 ) ) ) ).
% insertE
thf(fact_77_pairwise__singleton,axiom,
! [A: $tType,P: A > A > $o,A2: A] : ( pairwise @ A @ P @ ( insert @ A @ A2 @ ( bot_bot @ ( set @ A ) ) ) ) ).
% pairwise_singleton
thf(fact_78_pick,axiom,
! [Tr0: dtree,N2: n] :
( ( gram_L830233218_inItr @ ( top_top @ ( set @ n ) ) @ Tr0 @ N2 )
=> ( ( gram_L716654942_subtr @ ( top_top @ ( set @ n ) ) @ ( gram_L315592705e_pick @ Tr0 @ N2 ) @ Tr0 )
& ( ( root @ ( gram_L315592705e_pick @ Tr0 @ N2 ) )
= N2 ) ) ) ).
% pick
thf(fact_79_singleton__inject,axiom,
! [A: $tType,A3: A,B2: A] :
( ( ( insert @ A @ A3 @ ( bot_bot @ ( set @ A ) ) )
= ( insert @ A @ B2 @ ( bot_bot @ ( set @ A ) ) ) )
=> ( A3 = B2 ) ) ).
% singleton_inject
thf(fact_80_insert__not__empty,axiom,
! [A: $tType,A3: A,A2: set @ A] :
( ( insert @ A @ A3 @ A2 )
!= ( bot_bot @ ( set @ A ) ) ) ).
% insert_not_empty
thf(fact_81_doubleton__eq__iff,axiom,
! [A: $tType,A3: A,B2: A,C: A,D2: A] :
( ( ( insert @ A @ A3 @ ( insert @ A @ B2 @ ( bot_bot @ ( set @ A ) ) ) )
= ( insert @ A @ C @ ( insert @ A @ D2 @ ( bot_bot @ ( set @ A ) ) ) ) )
= ( ( ( A3 = C )
& ( B2 = D2 ) )
| ( ( A3 = D2 )
& ( B2 = C ) ) ) ) ).
% doubleton_eq_iff
thf(fact_82_singleton__iff,axiom,
! [A: $tType,B2: A,A3: A] :
( ( member @ A @ B2 @ ( insert @ A @ A3 @ ( bot_bot @ ( set @ A ) ) ) )
= ( B2 = A3 ) ) ).
% singleton_iff
thf(fact_83_singletonD,axiom,
! [A: $tType,B2: A,A3: A] :
( ( member @ A @ B2 @ ( insert @ A @ A3 @ ( bot_bot @ ( set @ A ) ) ) )
=> ( B2 = A3 ) ) ).
% singletonD
thf(fact_84_empty__not__UNIV,axiom,
! [A: $tType] :
( ( bot_bot @ ( set @ A ) )
!= ( top_top @ ( set @ A ) ) ) ).
% empty_not_UNIV
thf(fact_85_subtr__H,axiom,
! [Tr0: dtree,N2: n,Tr1: dtree] :
( ( gram_L830233218_inItr @ ( top_top @ ( set @ n ) ) @ Tr0 @ N2 )
=> ( ( gram_L716654942_subtr @ ( top_top @ ( set @ n ) ) @ Tr1 @ ( gram_L1451583624elle_H @ Tr0 @ N2 ) )
=> ? [N1: n] :
( ( gram_L830233218_inItr @ ( top_top @ ( set @ n ) ) @ Tr0 @ N1 )
& ( Tr1
= ( gram_L1451583624elle_H @ Tr0 @ N1 ) ) ) ) ) ).
% subtr_H
thf(fact_86_wf__H,axiom,
! [Tr0: dtree,N2: n] :
( ( gram_L864798063lle_wf @ Tr0 )
=> ( ( gram_L830233218_inItr @ ( top_top @ ( set @ n ) ) @ Tr0 @ N2 )
=> ( gram_L864798063lle_wf @ ( gram_L1451583624elle_H @ Tr0 @ N2 ) ) ) ) ).
% wf_H
thf(fact_87_inItr__root__in,axiom,
! [Ns: set @ n,Tr: dtree,N2: n] :
( ( gram_L830233218_inItr @ Ns @ Tr @ N2 )
=> ( member @ n @ ( root @ Tr ) @ Ns ) ) ).
% inItr_root_in
thf(fact_88_inItr_OBase,axiom,
! [Tr: dtree,Ns: set @ n] :
( ( member @ n @ ( root @ Tr ) @ Ns )
=> ( gram_L830233218_inItr @ Ns @ Tr @ ( root @ Tr ) ) ) ).
% inItr.Base
thf(fact_89_hsubst__eq,axiom,
! [Tr: dtree,Tr0: dtree] :
( ( ( root @ Tr )
= ( root @ Tr0 ) )
=> ( ( gram_L1004374585hsubst @ Tr0 @ Tr )
= ( gram_L1004374585hsubst @ Tr0 @ Tr0 ) ) ) ).
% hsubst_eq
thf(fact_90_subtr__inItr,axiom,
! [Ns: set @ n,Tr: dtree,N2: n,Tr1: dtree] :
( ( gram_L830233218_inItr @ Ns @ Tr @ N2 )
=> ( ( gram_L716654942_subtr @ Ns @ Tr @ Tr1 )
=> ( gram_L830233218_inItr @ Ns @ Tr1 @ N2 ) ) ) ).
% subtr_inItr
thf(fact_91_wf__hsubst,axiom,
! [Tr0: dtree,Tr: dtree] :
( ( gram_L864798063lle_wf @ Tr0 )
=> ( ( gram_L864798063lle_wf @ Tr )
=> ( gram_L864798063lle_wf @ ( gram_L1004374585hsubst @ Tr0 @ Tr ) ) ) ) ).
% wf_hsubst
thf(fact_92_H__r__def,axiom,
( gram_L1221482026le_H_r
= ( ^ [Tr02: dtree,N: n] : ( root @ ( gram_L315592705e_pick @ Tr02 @ N ) ) ) ) ).
% H_r_def
thf(fact_93_L__rec__notin,axiom,
! [N2: n,Ns: set @ n] :
( ~ ( member @ n @ N2 @ Ns )
=> ( ( gram_L1451583628elle_L @ Ns @ N2 )
= ( insert @ ( set @ t ) @ ( bot_bot @ ( set @ t ) ) @ ( bot_bot @ ( set @ ( set @ t ) ) ) ) ) ) ).
% L_rec_notin
thf(fact_94_iso__tuple__UNIV__I,axiom,
! [A: $tType,X3: A] : ( member @ A @ X3 @ ( top_top @ ( set @ A ) ) ) ).
% iso_tuple_UNIV_I
thf(fact_95_is__singleton__the__elem,axiom,
! [A: $tType] :
( ( is_singleton @ A )
= ( ^ [A4: set @ A] :
( A4
= ( insert @ A @ ( the_elem @ A @ A4 ) @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ).
% is_singleton_the_elem
thf(fact_96_the__elem__eq,axiom,
! [A: $tType,X3: A] :
( ( the_elem @ A @ ( insert @ A @ X3 @ ( bot_bot @ ( set @ A ) ) ) )
= X3 ) ).
% the_elem_eq
thf(fact_97_hsubst__def,axiom,
( gram_L1004374585hsubst
= ( ^ [Tr02: dtree] : ( unfold @ dtree @ gram_L1905609017ubst_r @ ( gram_L1905609002ubst_c @ Tr02 ) ) ) ) ).
% hsubst_def
thf(fact_98_Fr__rcut,axiom,
! [Tr0: dtree] : ( ord_less_eq @ ( set @ t ) @ ( gram_L861583724lle_Fr @ ( top_top @ ( set @ n ) ) @ ( gram_L1828378864e_rcut @ Tr0 ) ) @ ( gram_L861583724lle_Fr @ ( top_top @ ( set @ n ) ) @ Tr0 ) ) ).
% Fr_rcut
thf(fact_99_inFr__hsubst__notin,axiom,
! [Ns: set @ n,Tr: dtree,T: t,Tr0: dtree] :
( ( gram_L1333338417e_inFr @ Ns @ Tr @ T )
=> ( ~ ( member @ n @ ( root @ Tr0 ) @ Ns )
=> ( gram_L1333338417e_inFr @ Ns @ ( gram_L1004374585hsubst @ Tr0 @ Tr ) @ T ) ) ) ).
% inFr_hsubst_notin
thf(fact_100_order__refl,axiom,
! [A: $tType] :
( ( preorder @ A @ ( type2 @ A ) )
=> ! [X3: A] : ( ord_less_eq @ A @ X3 @ X3 ) ) ).
% order_refl
thf(fact_101_subsetI,axiom,
! [A: $tType,A2: set @ A,B3: set @ A] :
( ! [X2: A] :
( ( member @ A @ X2 @ A2 )
=> ( member @ A @ X2 @ B3 ) )
=> ( ord_less_eq @ ( set @ A ) @ A2 @ B3 ) ) ).
% subsetI
thf(fact_102_subset__antisym,axiom,
! [A: $tType,A2: set @ A,B3: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ A2 @ B3 )
=> ( ( ord_less_eq @ ( set @ A ) @ B3 @ A2 )
=> ( A2 = B3 ) ) ) ).
% subset_antisym
thf(fact_103_subset__empty,axiom,
! [A: $tType,A2: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ A2 @ ( bot_bot @ ( set @ A ) ) )
= ( A2
= ( bot_bot @ ( set @ A ) ) ) ) ).
% subset_empty
thf(fact_104_empty__subsetI,axiom,
! [A: $tType,A2: set @ A] : ( ord_less_eq @ ( set @ A ) @ ( bot_bot @ ( set @ A ) ) @ A2 ) ).
% empty_subsetI
thf(fact_105_insert__subset,axiom,
! [A: $tType,X3: A,A2: set @ A,B3: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ ( insert @ A @ X3 @ A2 ) @ B3 )
= ( ( member @ A @ X3 @ B3 )
& ( ord_less_eq @ ( set @ A ) @ A2 @ B3 ) ) ) ).
% insert_subset
thf(fact_106_singleton__insert__inj__eq,axiom,
! [A: $tType,B2: A,A3: A,A2: set @ A] :
( ( ( insert @ A @ B2 @ ( bot_bot @ ( set @ A ) ) )
= ( insert @ A @ A3 @ A2 ) )
= ( ( A3 = B2 )
& ( ord_less_eq @ ( set @ A ) @ A2 @ ( insert @ A @ B2 @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ).
% singleton_insert_inj_eq
thf(fact_107_singleton__insert__inj__eq_H,axiom,
! [A: $tType,A3: A,A2: set @ A,B2: A] :
( ( ( insert @ A @ A3 @ A2 )
= ( insert @ A @ B2 @ ( bot_bot @ ( set @ A ) ) ) )
= ( ( A3 = B2 )
& ( ord_less_eq @ ( set @ A ) @ A2 @ ( insert @ A @ B2 @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ).
% singleton_insert_inj_eq'
thf(fact_108_set__mp,axiom,
! [A: $tType,A2: set @ A,B3: set @ A,X3: A] :
( ( ord_less_eq @ ( set @ A ) @ A2 @ B3 )
=> ( ( member @ A @ X3 @ A2 )
=> ( member @ A @ X3 @ B3 ) ) ) ).
% set_mp
thf(fact_109_in__mono,axiom,
! [A: $tType,A2: set @ A,B3: set @ A,X3: A] :
( ( ord_less_eq @ ( set @ A ) @ A2 @ B3 )
=> ( ( member @ A @ X3 @ A2 )
=> ( member @ A @ X3 @ B3 ) ) ) ).
% in_mono
thf(fact_110_subsetD,axiom,
! [A: $tType,A2: set @ A,B3: set @ A,C: A] :
( ( ord_less_eq @ ( set @ A ) @ A2 @ B3 )
=> ( ( member @ A @ C @ A2 )
=> ( member @ A @ C @ B3 ) ) ) ).
% subsetD
thf(fact_111_subsetCE,axiom,
! [A: $tType,A2: set @ A,B3: set @ A,C: A] :
( ( ord_less_eq @ ( set @ A ) @ A2 @ B3 )
=> ( ( member @ A @ C @ A2 )
=> ( member @ A @ C @ B3 ) ) ) ).
% subsetCE
thf(fact_112_equalityE,axiom,
! [A: $tType,A2: set @ A,B3: set @ A] :
( ( A2 = B3 )
=> ~ ( ( ord_less_eq @ ( set @ A ) @ A2 @ B3 )
=> ~ ( ord_less_eq @ ( set @ A ) @ B3 @ A2 ) ) ) ).
% equalityE
thf(fact_113_subset__eq,axiom,
! [A: $tType] :
( ( ord_less_eq @ ( set @ A ) )
= ( ^ [A4: set @ A,B5: set @ A] :
! [X: A] :
( ( member @ A @ X @ A4 )
=> ( member @ A @ X @ B5 ) ) ) ) ).
% subset_eq
thf(fact_114_equalityD1,axiom,
! [A: $tType,A2: set @ A,B3: set @ A] :
( ( A2 = B3 )
=> ( ord_less_eq @ ( set @ A ) @ A2 @ B3 ) ) ).
% equalityD1
thf(fact_115_equalityD2,axiom,
! [A: $tType,A2: set @ A,B3: set @ A] :
( ( A2 = B3 )
=> ( ord_less_eq @ ( set @ A ) @ B3 @ A2 ) ) ).
% equalityD2
thf(fact_116_set__rev__mp,axiom,
! [A: $tType,X3: A,A2: set @ A,B3: set @ A] :
( ( member @ A @ X3 @ A2 )
=> ( ( ord_less_eq @ ( set @ A ) @ A2 @ B3 )
=> ( member @ A @ X3 @ B3 ) ) ) ).
% set_rev_mp
thf(fact_117_subset__iff,axiom,
! [A: $tType] :
( ( ord_less_eq @ ( set @ A ) )
= ( ^ [A4: set @ A,B5: set @ A] :
! [T2: A] :
( ( member @ A @ T2 @ A4 )
=> ( member @ A @ T2 @ B5 ) ) ) ) ).
% subset_iff
thf(fact_118_rev__subsetD,axiom,
! [A: $tType,C: A,A2: set @ A,B3: set @ A] :
( ( member @ A @ C @ A2 )
=> ( ( ord_less_eq @ ( set @ A ) @ A2 @ B3 )
=> ( member @ A @ C @ B3 ) ) ) ).
% rev_subsetD
thf(fact_119_subset__refl,axiom,
! [A: $tType,A2: set @ A] : ( ord_less_eq @ ( set @ A ) @ A2 @ A2 ) ).
% subset_refl
thf(fact_120_Collect__mono,axiom,
! [A: $tType,P: A > $o,Q: A > $o] :
( ! [X2: A] :
( ( P @ X2 )
=> ( Q @ X2 ) )
=> ( ord_less_eq @ ( set @ A ) @ ( collect @ A @ P ) @ ( collect @ A @ Q ) ) ) ).
% Collect_mono
thf(fact_121_subset__trans,axiom,
! [A: $tType,A2: set @ A,B3: set @ A,C4: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ A2 @ B3 )
=> ( ( ord_less_eq @ ( set @ A ) @ B3 @ C4 )
=> ( ord_less_eq @ ( set @ A ) @ A2 @ C4 ) ) ) ).
% subset_trans
thf(fact_122_le__funD,axiom,
! [B: $tType,A: $tType] :
( ( ord @ B @ ( type2 @ B ) )
=> ! [F2: A > B,G: A > B,X3: A] :
( ( ord_less_eq @ ( A > B ) @ F2 @ G )
=> ( ord_less_eq @ B @ ( F2 @ X3 ) @ ( G @ X3 ) ) ) ) ).
% le_funD
thf(fact_123_le__funE,axiom,
! [B: $tType,A: $tType] :
( ( ord @ B @ ( type2 @ B ) )
=> ! [F2: A > B,G: A > B,X3: A] :
( ( ord_less_eq @ ( A > B ) @ F2 @ G )
=> ( ord_less_eq @ B @ ( F2 @ X3 ) @ ( G @ X3 ) ) ) ) ).
% le_funE
thf(fact_124_le__funI,axiom,
! [B: $tType,A: $tType] :
( ( ord @ B @ ( type2 @ B ) )
=> ! [F2: A > B,G: A > B] :
( ! [X2: A] : ( ord_less_eq @ B @ ( F2 @ X2 ) @ ( G @ X2 ) )
=> ( ord_less_eq @ ( A > B ) @ F2 @ G ) ) ) ).
% le_funI
thf(fact_125_set__eq__subset,axiom,
! [A: $tType] :
( ( ^ [Y4: set @ A,Z: set @ A] : Y4 = Z )
= ( ^ [A4: set @ A,B5: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ A4 @ B5 )
& ( ord_less_eq @ ( set @ A ) @ B5 @ A4 ) ) ) ) ).
% set_eq_subset
thf(fact_126_contra__subsetD,axiom,
! [A: $tType,A2: set @ A,B3: set @ A,C: A] :
( ( ord_less_eq @ ( set @ A ) @ A2 @ B3 )
=> ( ~ ( member @ A @ C @ B3 )
=> ~ ( member @ A @ C @ A2 ) ) ) ).
% contra_subsetD
thf(fact_127_le__fun__def,axiom,
! [B: $tType,A: $tType] :
( ( ord @ B @ ( type2 @ B ) )
=> ( ( ord_less_eq @ ( A > B ) )
= ( ^ [F: A > B,G2: A > B] :
! [X: A] : ( ord_less_eq @ B @ ( F @ X ) @ ( G2 @ X ) ) ) ) ) ).
% le_fun_def
thf(fact_128_Collect__mono__iff,axiom,
! [A: $tType,P: A > $o,Q: A > $o] :
( ( ord_less_eq @ ( set @ A ) @ ( collect @ A @ P ) @ ( collect @ A @ Q ) )
= ( ! [X: A] :
( ( P @ X )
=> ( Q @ X ) ) ) ) ).
% Collect_mono_iff
thf(fact_129_order__subst1,axiom,
! [A: $tType,B: $tType] :
( ( ( order @ B @ ( type2 @ B ) )
& ( order @ A @ ( type2 @ A ) ) )
=> ! [A3: A,F2: B > A,B2: B,C: B] :
( ( ord_less_eq @ A @ A3 @ ( F2 @ B2 ) )
=> ( ( ord_less_eq @ B @ B2 @ C )
=> ( ! [X2: B,Y: B] :
( ( ord_less_eq @ B @ X2 @ Y )
=> ( ord_less_eq @ A @ ( F2 @ X2 ) @ ( F2 @ Y ) ) )
=> ( ord_less_eq @ A @ A3 @ ( F2 @ C ) ) ) ) ) ) ).
% order_subst1
thf(fact_130_order__subst2,axiom,
! [A: $tType,C2: $tType] :
( ( ( order @ C2 @ ( type2 @ C2 ) )
& ( order @ A @ ( type2 @ A ) ) )
=> ! [A3: A,B2: A,F2: A > C2,C: C2] :
( ( ord_less_eq @ A @ A3 @ B2 )
=> ( ( ord_less_eq @ C2 @ ( F2 @ B2 ) @ C )
=> ( ! [X2: A,Y: A] :
( ( ord_less_eq @ A @ X2 @ Y )
=> ( ord_less_eq @ C2 @ ( F2 @ X2 ) @ ( F2 @ Y ) ) )
=> ( ord_less_eq @ C2 @ ( F2 @ A3 ) @ C ) ) ) ) ) ).
% order_subst2
thf(fact_131_ord__eq__le__subst,axiom,
! [A: $tType,B: $tType] :
( ( ( ord @ B @ ( type2 @ B ) )
& ( ord @ A @ ( type2 @ A ) ) )
=> ! [A3: A,F2: B > A,B2: B,C: B] :
( ( A3
= ( F2 @ B2 ) )
=> ( ( ord_less_eq @ B @ B2 @ C )
=> ( ! [X2: B,Y: B] :
( ( ord_less_eq @ B @ X2 @ Y )
=> ( ord_less_eq @ A @ ( F2 @ X2 ) @ ( F2 @ Y ) ) )
=> ( ord_less_eq @ A @ A3 @ ( F2 @ C ) ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_132_ord__le__eq__subst,axiom,
! [A: $tType,B: $tType] :
( ( ( ord @ B @ ( type2 @ B ) )
& ( ord @ A @ ( type2 @ A ) ) )
=> ! [A3: A,B2: A,F2: A > B,C: B] :
( ( ord_less_eq @ A @ A3 @ B2 )
=> ( ( ( F2 @ B2 )
= C )
=> ( ! [X2: A,Y: A] :
( ( ord_less_eq @ A @ X2 @ Y )
=> ( ord_less_eq @ B @ ( F2 @ X2 ) @ ( F2 @ Y ) ) )
=> ( ord_less_eq @ B @ ( F2 @ A3 ) @ C ) ) ) ) ) ).
% ord_le_eq_subst
thf(fact_133_eq__iff,axiom,
! [A: $tType] :
( ( order @ A @ ( type2 @ A ) )
=> ( ( ^ [Y4: A,Z: A] : Y4 = Z )
= ( ^ [X: A,Y2: A] :
( ( ord_less_eq @ A @ X @ Y2 )
& ( ord_less_eq @ A @ Y2 @ X ) ) ) ) ) ).
% eq_iff
thf(fact_134_antisym,axiom,
! [A: $tType] :
( ( order @ A @ ( type2 @ A ) )
=> ! [X3: A,Y3: A] :
( ( ord_less_eq @ A @ X3 @ Y3 )
=> ( ( ord_less_eq @ A @ Y3 @ X3 )
=> ( X3 = Y3 ) ) ) ) ).
% antisym
thf(fact_135_linear,axiom,
! [A: $tType] :
( ( linorder @ A @ ( type2 @ A ) )
=> ! [X3: A,Y3: A] :
( ( ord_less_eq @ A @ X3 @ Y3 )
| ( ord_less_eq @ A @ Y3 @ X3 ) ) ) ).
% linear
thf(fact_136_eq__refl,axiom,
! [A: $tType] :
( ( preorder @ A @ ( type2 @ A ) )
=> ! [X3: A,Y3: A] :
( ( X3 = Y3 )
=> ( ord_less_eq @ A @ X3 @ Y3 ) ) ) ).
% eq_refl
thf(fact_137_le__cases,axiom,
! [A: $tType] :
( ( linorder @ A @ ( type2 @ A ) )
=> ! [X3: A,Y3: A] :
( ~ ( ord_less_eq @ A @ X3 @ Y3 )
=> ( ord_less_eq @ A @ Y3 @ X3 ) ) ) ).
% le_cases
thf(fact_138_order_Otrans,axiom,
! [A: $tType] :
( ( order @ A @ ( type2 @ A ) )
=> ! [A3: A,B2: A,C: A] :
( ( ord_less_eq @ A @ A3 @ B2 )
=> ( ( ord_less_eq @ A @ B2 @ C )
=> ( ord_less_eq @ A @ A3 @ C ) ) ) ) ).
% order.trans
thf(fact_139_le__cases3,axiom,
! [A: $tType] :
( ( linorder @ A @ ( type2 @ A ) )
=> ! [X3: A,Y3: A,Z2: A] :
( ( ( ord_less_eq @ A @ X3 @ Y3 )
=> ~ ( ord_less_eq @ A @ Y3 @ Z2 ) )
=> ( ( ( ord_less_eq @ A @ Y3 @ X3 )
=> ~ ( ord_less_eq @ A @ X3 @ Z2 ) )
=> ( ( ( ord_less_eq @ A @ X3 @ Z2 )
=> ~ ( ord_less_eq @ A @ Z2 @ Y3 ) )
=> ( ( ( ord_less_eq @ A @ Z2 @ Y3 )
=> ~ ( ord_less_eq @ A @ Y3 @ X3 ) )
=> ( ( ( ord_less_eq @ A @ Y3 @ Z2 )
=> ~ ( ord_less_eq @ A @ Z2 @ X3 ) )
=> ~ ( ( ord_less_eq @ A @ Z2 @ X3 )
=> ~ ( ord_less_eq @ A @ X3 @ Y3 ) ) ) ) ) ) ) ) ).
% le_cases3
thf(fact_140_antisym__conv,axiom,
! [A: $tType] :
( ( order @ A @ ( type2 @ A ) )
=> ! [Y3: A,X3: A] :
( ( ord_less_eq @ A @ Y3 @ X3 )
=> ( ( ord_less_eq @ A @ X3 @ Y3 )
= ( X3 = Y3 ) ) ) ) ).
% antisym_conv
thf(fact_141_ord__eq__le__trans,axiom,
! [A: $tType] :
( ( ord @ A @ ( type2 @ A ) )
=> ! [A3: A,B2: A,C: A] :
( ( A3 = B2 )
=> ( ( ord_less_eq @ A @ B2 @ C )
=> ( ord_less_eq @ A @ A3 @ C ) ) ) ) ).
% ord_eq_le_trans
thf(fact_142_ord__le__eq__trans,axiom,
! [A: $tType] :
( ( ord @ A @ ( type2 @ A ) )
=> ! [A3: A,B2: A,C: A] :
( ( ord_less_eq @ A @ A3 @ B2 )
=> ( ( B2 = C )
=> ( ord_less_eq @ A @ A3 @ C ) ) ) ) ).
% ord_le_eq_trans
thf(fact_143_order__class_Oorder_Oantisym,axiom,
! [A: $tType] :
( ( order @ A @ ( type2 @ A ) )
=> ! [A3: A,B2: A] :
( ( ord_less_eq @ A @ A3 @ B2 )
=> ( ( ord_less_eq @ A @ B2 @ A3 )
=> ( A3 = B2 ) ) ) ) ).
% order_class.order.antisym
thf(fact_144_order__trans,axiom,
! [A: $tType] :
( ( preorder @ A @ ( type2 @ A ) )
=> ! [X3: A,Y3: A,Z2: A] :
( ( ord_less_eq @ A @ X3 @ Y3 )
=> ( ( ord_less_eq @ A @ Y3 @ Z2 )
=> ( ord_less_eq @ A @ X3 @ Z2 ) ) ) ) ).
% order_trans
thf(fact_145_dual__order_Orefl,axiom,
! [A: $tType] :
( ( order @ A @ ( type2 @ A ) )
=> ! [A3: A] : ( ord_less_eq @ A @ A3 @ A3 ) ) ).
% dual_order.refl
thf(fact_146_linorder__wlog,axiom,
! [A: $tType] :
( ( linorder @ A @ ( type2 @ A ) )
=> ! [P: A > A > $o,A3: A,B2: A] :
( ! [A5: A,B6: A] :
( ( ord_less_eq @ A @ A5 @ B6 )
=> ( P @ A5 @ B6 ) )
=> ( ! [A5: A,B6: A] :
( ( P @ B6 @ A5 )
=> ( P @ A5 @ B6 ) )
=> ( P @ A3 @ B2 ) ) ) ) ).
% linorder_wlog
thf(fact_147_dual__order_Otrans,axiom,
! [A: $tType] :
( ( order @ A @ ( type2 @ A ) )
=> ! [B2: A,A3: A,C: A] :
( ( ord_less_eq @ A @ B2 @ A3 )
=> ( ( ord_less_eq @ A @ C @ B2 )
=> ( ord_less_eq @ A @ C @ A3 ) ) ) ) ).
% dual_order.trans
thf(fact_148_dual__order_Oantisym,axiom,
! [A: $tType] :
( ( order @ A @ ( type2 @ A ) )
=> ! [B2: A,A3: A] :
( ( ord_less_eq @ A @ B2 @ A3 )
=> ( ( ord_less_eq @ A @ A3 @ B2 )
=> ( A3 = B2 ) ) ) ) ).
% dual_order.antisym
thf(fact_149_top__set__def,axiom,
! [A: $tType] :
( ( top_top @ ( set @ A ) )
= ( collect @ A @ ( top_top @ ( A > $o ) ) ) ) ).
% top_set_def
thf(fact_150_top__empty__eq,axiom,
! [A: $tType] :
( ( top_top @ ( A > $o ) )
= ( ^ [X: A] : ( member @ A @ X @ ( top_top @ ( set @ A ) ) ) ) ) ).
% top_empty_eq
thf(fact_151_bot_Oextremum,axiom,
! [A: $tType] :
( ( order_bot @ A @ ( type2 @ A ) )
=> ! [A3: A] : ( ord_less_eq @ A @ ( bot_bot @ A ) @ A3 ) ) ).
% bot.extremum
thf(fact_152_bot_Oextremum__unique,axiom,
! [A: $tType] :
( ( order_bot @ A @ ( type2 @ A ) )
=> ! [A3: A] :
( ( ord_less_eq @ A @ A3 @ ( bot_bot @ A ) )
= ( A3
= ( bot_bot @ A ) ) ) ) ).
% bot.extremum_unique
thf(fact_153_bot_Oextremum__uniqueI,axiom,
! [A: $tType] :
( ( order_bot @ A @ ( type2 @ A ) )
=> ! [A3: A] :
( ( ord_less_eq @ A @ A3 @ ( bot_bot @ A ) )
=> ( A3
= ( bot_bot @ A ) ) ) ) ).
% bot.extremum_uniqueI
thf(fact_154_top__greatest,axiom,
! [A: $tType] :
( ( order_top @ A @ ( type2 @ A ) )
=> ! [A3: A] : ( ord_less_eq @ A @ A3 @ ( top_top @ A ) ) ) ).
% top_greatest
thf(fact_155_top_Oextremum__unique,axiom,
! [A: $tType] :
( ( order_top @ A @ ( type2 @ A ) )
=> ! [A3: A] :
( ( ord_less_eq @ A @ ( top_top @ A ) @ A3 )
= ( A3
= ( top_top @ A ) ) ) ) ).
% top.extremum_unique
thf(fact_156_top_Oextremum__uniqueI,axiom,
! [A: $tType] :
( ( order_top @ A @ ( type2 @ A ) )
=> ! [A3: A] :
( ( ord_less_eq @ A @ ( top_top @ A ) @ A3 )
=> ( A3
= ( top_top @ A ) ) ) ) ).
% top.extremum_uniqueI
thf(fact_157_subset__UNIV,axiom,
! [A: $tType,A2: set @ A] : ( ord_less_eq @ ( set @ A ) @ A2 @ ( top_top @ ( set @ A ) ) ) ).
% subset_UNIV
thf(fact_158_insert__mono,axiom,
! [A: $tType,C4: set @ A,D3: set @ A,A3: A] :
( ( ord_less_eq @ ( set @ A ) @ C4 @ D3 )
=> ( ord_less_eq @ ( set @ A ) @ ( insert @ A @ A3 @ C4 ) @ ( insert @ A @ A3 @ D3 ) ) ) ).
% insert_mono
thf(fact_159_subset__insert,axiom,
! [A: $tType,X3: A,A2: set @ A,B3: set @ A] :
( ~ ( member @ A @ X3 @ A2 )
=> ( ( ord_less_eq @ ( set @ A ) @ A2 @ ( insert @ A @ X3 @ B3 ) )
= ( ord_less_eq @ ( set @ A ) @ A2 @ B3 ) ) ) ).
% subset_insert
thf(fact_160_subset__insertI,axiom,
! [A: $tType,B3: set @ A,A3: A] : ( ord_less_eq @ ( set @ A ) @ B3 @ ( insert @ A @ A3 @ B3 ) ) ).
% subset_insertI
thf(fact_161_subset__insertI2,axiom,
! [A: $tType,A2: set @ A,B3: set @ A,B2: A] :
( ( ord_less_eq @ ( set @ A ) @ A2 @ B3 )
=> ( ord_less_eq @ ( set @ A ) @ A2 @ ( insert @ A @ B2 @ B3 ) ) ) ).
% subset_insertI2
thf(fact_162_inFr__root__in,axiom,
! [Ns: set @ n,Tr: dtree,T: t] :
( ( gram_L1333338417e_inFr @ Ns @ Tr @ T )
=> ( member @ n @ ( root @ Tr ) @ Ns ) ) ).
% inFr_root_in
thf(fact_163_not__root__inFr,axiom,
! [Tr: dtree,Ns: set @ n,T: t] :
( ~ ( member @ n @ ( root @ Tr ) @ Ns )
=> ~ ( gram_L1333338417e_inFr @ Ns @ Tr @ T ) ) ).
% not_root_inFr
thf(fact_164_subtr__inFr,axiom,
! [Ns: set @ n,Tr: dtree,T: t,Tr1: dtree] :
( ( gram_L1333338417e_inFr @ Ns @ Tr @ T )
=> ( ( gram_L716654942_subtr @ Ns @ Tr @ Tr1 )
=> ( gram_L1333338417e_inFr @ Ns @ Tr1 @ T ) ) ) ).
% subtr_inFr
thf(fact_165_pairwise__subset,axiom,
! [A: $tType,P: A > A > $o,S3: set @ A,T3: set @ A] :
( ( pairwise @ A @ P @ S3 )
=> ( ( ord_less_eq @ ( set @ A ) @ T3 @ S3 )
=> ( pairwise @ A @ P @ T3 ) ) ) ).
% pairwise_subset
thf(fact_166_subset__singletonD,axiom,
! [A: $tType,A2: set @ A,X3: A] :
( ( ord_less_eq @ ( set @ A ) @ A2 @ ( insert @ A @ X3 @ ( bot_bot @ ( set @ A ) ) ) )
=> ( ( A2
= ( bot_bot @ ( set @ A ) ) )
| ( A2
= ( insert @ A @ X3 @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ).
% subset_singletonD
thf(fact_167_subset__singleton__iff,axiom,
! [A: $tType,X4: set @ A,A3: A] :
( ( ord_less_eq @ ( set @ A ) @ X4 @ ( insert @ A @ A3 @ ( bot_bot @ ( set @ A ) ) ) )
= ( ( X4
= ( bot_bot @ ( set @ A ) ) )
| ( X4
= ( insert @ A @ A3 @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ).
% subset_singleton_iff
thf(fact_168_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_169_insert__subsetI,axiom,
! [A: $tType,X3: A,A2: set @ A,X4: set @ A] :
( ( member @ A @ X3 @ A2 )
=> ( ( ord_less_eq @ ( set @ A ) @ X4 @ A2 )
=> ( ord_less_eq @ ( set @ A ) @ ( insert @ A @ X3 @ X4 ) @ A2 ) ) ) ).
% insert_subsetI
thf(fact_170_subset__emptyI,axiom,
! [A: $tType,A2: set @ A] :
( ! [X2: A] :
~ ( member @ A @ X2 @ A2 )
=> ( ord_less_eq @ ( set @ A ) @ A2 @ ( bot_bot @ ( set @ A ) ) ) ) ).
% subset_emptyI
thf(fact_171_Pow__empty,axiom,
! [A: $tType] :
( ( pow @ A @ ( bot_bot @ ( set @ A ) ) )
= ( insert @ ( set @ A ) @ ( bot_bot @ ( set @ A ) ) @ ( bot_bot @ ( set @ ( set @ A ) ) ) ) ) ).
% Pow_empty
thf(fact_172_Pow__singleton__iff,axiom,
! [A: $tType,X4: set @ A,Y5: set @ A] :
( ( ( pow @ A @ X4 )
= ( insert @ ( set @ A ) @ Y5 @ ( bot_bot @ ( set @ ( set @ A ) ) ) ) )
= ( ( X4
= ( bot_bot @ ( set @ A ) ) )
& ( Y5
= ( bot_bot @ ( set @ A ) ) ) ) ) ).
% Pow_singleton_iff
thf(fact_173_top1I,axiom,
! [A: $tType,X3: A] : ( top_top @ ( A > $o ) @ X3 ) ).
% top1I
thf(fact_174_Pow__iff,axiom,
! [A: $tType,A2: set @ A,B3: set @ A] :
( ( member @ ( set @ A ) @ A2 @ ( pow @ A @ B3 ) )
= ( ord_less_eq @ ( set @ A ) @ A2 @ B3 ) ) ).
% Pow_iff
thf(fact_175_PowI,axiom,
! [A: $tType,A2: set @ A,B3: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ A2 @ B3 )
=> ( member @ ( set @ A ) @ A2 @ ( pow @ A @ B3 ) ) ) ).
% PowI
thf(fact_176_Pow__UNIV,axiom,
! [A: $tType] :
( ( pow @ A @ ( top_top @ ( set @ A ) ) )
= ( top_top @ ( set @ ( set @ A ) ) ) ) ).
% Pow_UNIV
thf(fact_177_Pow__top,axiom,
! [A: $tType,A2: set @ A] : ( member @ ( set @ A ) @ A2 @ ( pow @ A @ A2 ) ) ).
% Pow_top
thf(fact_178_Pow__bottom,axiom,
! [A: $tType,B3: set @ A] : ( member @ ( set @ A ) @ ( bot_bot @ ( set @ A ) ) @ ( pow @ A @ B3 ) ) ).
% Pow_bottom
thf(fact_179_Pow__mono,axiom,
! [A: $tType,A2: set @ A,B3: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ A2 @ B3 )
=> ( ord_less_eq @ ( set @ ( set @ A ) ) @ ( pow @ A @ A2 ) @ ( pow @ A @ B3 ) ) ) ).
% Pow_mono
thf(fact_180_PowD,axiom,
! [A: $tType,A2: set @ A,B3: set @ A] :
( ( member @ ( set @ A ) @ A2 @ ( pow @ A @ B3 ) )
=> ( ord_less_eq @ ( set @ A ) @ A2 @ B3 ) ) ).
% PowD
thf(fact_181_subtr__mono,axiom,
! [Ns: set @ n,Tr1: dtree,Tr22: dtree,Ns3: set @ n] :
( ( gram_L716654942_subtr @ Ns @ Tr1 @ Tr22 )
=> ( ( ord_less_eq @ ( set @ n ) @ Ns @ Ns3 )
=> ( gram_L716654942_subtr @ Ns3 @ Tr1 @ Tr22 ) ) ) ).
% subtr_mono
thf(fact_182_inFr__mono,axiom,
! [Ns: set @ n,Tr: dtree,T: t,Ns3: set @ n] :
( ( gram_L1333338417e_inFr @ Ns @ Tr @ T )
=> ( ( ord_less_eq @ ( set @ n ) @ Ns @ Ns3 )
=> ( gram_L1333338417e_inFr @ Ns3 @ Tr @ T ) ) ) ).
% inFr_mono
thf(fact_183_inItr__mono,axiom,
! [Ns: set @ n,Tr: dtree,N2: n,Ns3: set @ n] :
( ( gram_L830233218_inItr @ Ns @ Tr @ N2 )
=> ( ( ord_less_eq @ ( set @ n ) @ Ns @ Ns3 )
=> ( gram_L830233218_inItr @ Ns3 @ Tr @ N2 ) ) ) ).
% inItr_mono
thf(fact_184_Pow__not__empty,axiom,
! [A: $tType,A2: set @ A] :
( ( pow @ A @ A2 )
!= ( bot_bot @ ( set @ ( set @ A ) ) ) ) ).
% Pow_not_empty
thf(fact_185_H__def,axiom,
( gram_L1451583624elle_H
= ( ^ [Tr02: dtree] : ( unfold @ n @ ( gram_L1221482026le_H_r @ Tr02 ) @ ( gram_L1221482011le_H_c @ Tr02 ) ) ) ) ).
% H_def
thf(fact_186_top__conj_I2_J,axiom,
! [A: $tType,P: $o,X3: A] :
( ( P
& ( top_top @ ( A > $o ) @ X3 ) )
= P ) ).
% top_conj(2)
thf(fact_187_top__conj_I1_J,axiom,
! [A: $tType,X3: A,P: $o] :
( ( ( top_top @ ( A > $o ) @ X3 )
& P )
= P ) ).
% top_conj(1)
thf(fact_188_root__Node,axiom,
! [N2: n,As: set @ ( sum_sum @ t @ dtree )] :
( ( root @ ( node @ N2 @ As ) )
= N2 ) ).
% root_Node
thf(fact_189_subset__Compl__singleton,axiom,
! [A: $tType,A2: set @ A,B2: A] :
( ( ord_less_eq @ ( set @ A ) @ A2 @ ( uminus_uminus @ ( set @ A ) @ ( insert @ A @ B2 @ ( bot_bot @ ( set @ A ) ) ) ) )
= ( ~ ( member @ A @ B2 @ A2 ) ) ) ).
% subset_Compl_singleton
thf(fact_190_Compl__eq__Compl__iff,axiom,
! [A: $tType,A2: set @ A,B3: set @ A] :
( ( ( uminus_uminus @ ( set @ A ) @ A2 )
= ( uminus_uminus @ ( set @ A ) @ B3 ) )
= ( A2 = B3 ) ) ).
% Compl_eq_Compl_iff
thf(fact_191_Compl__iff,axiom,
! [A: $tType,C: A,A2: set @ A] :
( ( member @ A @ C @ ( uminus_uminus @ ( set @ A ) @ A2 ) )
= ( ~ ( member @ A @ C @ A2 ) ) ) ).
% Compl_iff
thf(fact_192_ComplI,axiom,
! [A: $tType,C: A,A2: set @ A] :
( ~ ( member @ A @ C @ A2 )
=> ( member @ A @ C @ ( uminus_uminus @ ( set @ A ) @ A2 ) ) ) ).
% ComplI
thf(fact_193_Compl__subset__Compl__iff,axiom,
! [A: $tType,A2: set @ A,B3: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ ( uminus_uminus @ ( set @ A ) @ A2 ) @ ( uminus_uminus @ ( set @ A ) @ B3 ) )
= ( ord_less_eq @ ( set @ A ) @ B3 @ A2 ) ) ).
% Compl_subset_Compl_iff
thf(fact_194_Compl__anti__mono,axiom,
! [A: $tType,A2: set @ A,B3: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ A2 @ B3 )
=> ( ord_less_eq @ ( set @ A ) @ ( uminus_uminus @ ( set @ A ) @ B3 ) @ ( uminus_uminus @ ( set @ A ) @ A2 ) ) ) ).
% Compl_anti_mono
thf(fact_195_subset__Compl__self__eq,axiom,
! [A: $tType,A2: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ A2 @ ( uminus_uminus @ ( set @ A ) @ A2 ) )
= ( A2
= ( bot_bot @ ( set @ A ) ) ) ) ).
% subset_Compl_self_eq
thf(fact_196_Compl__UNIV__eq,axiom,
! [A: $tType] :
( ( uminus_uminus @ ( set @ A ) @ ( top_top @ ( set @ A ) ) )
= ( bot_bot @ ( set @ A ) ) ) ).
% Compl_UNIV_eq
thf(fact_197_Compl__empty__eq,axiom,
! [A: $tType] :
( ( uminus_uminus @ ( set @ A ) @ ( bot_bot @ ( set @ A ) ) )
= ( top_top @ ( set @ A ) ) ) ).
% Compl_empty_eq
thf(fact_198_double__complement,axiom,
! [A: $tType,A2: set @ A] :
( ( uminus_uminus @ ( set @ A ) @ ( uminus_uminus @ ( set @ A ) @ A2 ) )
= A2 ) ).
% double_complement
thf(fact_199_ComplD,axiom,
! [A: $tType,C: A,A2: set @ A] :
( ( member @ A @ C @ ( uminus_uminus @ ( set @ A ) @ A2 ) )
=> ~ ( member @ A @ C @ A2 ) ) ).
% ComplD
thf(fact_200_compl__bot__eq,axiom,
! [A: $tType] :
( ( boolean_algebra @ A @ ( type2 @ A ) )
=> ( ( uminus_uminus @ A @ ( bot_bot @ A ) )
= ( top_top @ A ) ) ) ).
% compl_bot_eq
thf(fact_201_compl__top__eq,axiom,
! [A: $tType] :
( ( boolean_algebra @ A @ ( type2 @ A ) )
=> ( ( uminus_uminus @ A @ ( top_top @ A ) )
= ( bot_bot @ A ) ) ) ).
% compl_top_eq
thf(fact_202_compl__le__compl__iff,axiom,
! [A: $tType] :
( ( boolean_algebra @ A @ ( type2 @ A ) )
=> ! [X3: A,Y3: A] :
( ( ord_less_eq @ A @ ( uminus_uminus @ A @ X3 ) @ ( uminus_uminus @ A @ Y3 ) )
= ( ord_less_eq @ A @ Y3 @ X3 ) ) ) ).
% compl_le_compl_iff
thf(fact_203_compl__mono,axiom,
! [A: $tType] :
( ( boolean_algebra @ A @ ( type2 @ A ) )
=> ! [X3: A,Y3: A] :
( ( ord_less_eq @ A @ X3 @ Y3 )
=> ( ord_less_eq @ A @ ( uminus_uminus @ A @ Y3 ) @ ( uminus_uminus @ A @ X3 ) ) ) ) ).
% compl_mono
thf(fact_204_compl__le__swap1,axiom,
! [A: $tType] :
( ( boolean_algebra @ A @ ( type2 @ A ) )
=> ! [Y3: A,X3: A] :
( ( ord_less_eq @ A @ Y3 @ ( uminus_uminus @ A @ X3 ) )
=> ( ord_less_eq @ A @ X3 @ ( uminus_uminus @ A @ Y3 ) ) ) ) ).
% compl_le_swap1
thf(fact_205_compl__le__swap2,axiom,
! [A: $tType] :
( ( boolean_algebra @ A @ ( type2 @ A ) )
=> ! [Y3: A,X3: A] :
( ( ord_less_eq @ A @ ( uminus_uminus @ A @ Y3 ) @ X3 )
=> ( ord_less_eq @ A @ ( uminus_uminus @ A @ X3 ) @ Y3 ) ) ) ).
% compl_le_swap2
thf(fact_206_neg__le__iff__le,axiom,
! [A: $tType] :
( ( ordered_ab_group_add @ A @ ( type2 @ A ) )
=> ! [B2: A,A3: A] :
( ( ord_less_eq @ A @ ( uminus_uminus @ A @ B2 ) @ ( uminus_uminus @ A @ A3 ) )
= ( ord_less_eq @ A @ A3 @ B2 ) ) ) ).
% neg_le_iff_le
thf(fact_207_le__minus__iff,axiom,
! [A: $tType] :
( ( ordered_ab_group_add @ A @ ( type2 @ A ) )
=> ! [A3: A,B2: A] :
( ( ord_less_eq @ A @ A3 @ ( uminus_uminus @ A @ B2 ) )
= ( ord_less_eq @ A @ B2 @ ( uminus_uminus @ A @ A3 ) ) ) ) ).
% le_minus_iff
thf(fact_208_le__imp__neg__le,axiom,
! [A: $tType] :
( ( ordered_ab_group_add @ A @ ( type2 @ A ) )
=> ! [A3: A,B2: A] :
( ( ord_less_eq @ A @ A3 @ B2 )
=> ( ord_less_eq @ A @ ( uminus_uminus @ A @ B2 ) @ ( uminus_uminus @ A @ A3 ) ) ) ) ).
% le_imp_neg_le
thf(fact_209_minus__le__iff,axiom,
! [A: $tType] :
( ( ordered_ab_group_add @ A @ ( type2 @ A ) )
=> ! [A3: A,B2: A] :
( ( ord_less_eq @ A @ ( uminus_uminus @ A @ A3 ) @ B2 )
= ( ord_less_eq @ A @ ( uminus_uminus @ A @ B2 ) @ A3 ) ) ) ).
% minus_le_iff
thf(fact_210_inFr__hsubst__minus,axiom,
! [Ns: set @ n,Tr0: dtree,Tr: dtree,T: t] :
( ( gram_L1333338417e_inFr @ ( minus_minus @ ( set @ n ) @ Ns @ ( insert @ n @ ( root @ Tr0 ) @ ( bot_bot @ ( set @ n ) ) ) ) @ Tr @ T )
=> ( gram_L1333338417e_inFr @ Ns @ ( gram_L1004374585hsubst @ Tr0 @ Tr ) @ T ) ) ).
% inFr_hsubst_minus
thf(fact_211_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_212_DiffI,axiom,
! [A: $tType,C: A,A2: set @ A,B3: set @ A] :
( ( member @ A @ C @ A2 )
=> ( ~ ( member @ A @ C @ B3 )
=> ( member @ A @ C @ ( minus_minus @ ( set @ A ) @ A2 @ B3 ) ) ) ) ).
% DiffI
thf(fact_213_Diff__iff,axiom,
! [A: $tType,C: A,A2: set @ A,B3: set @ A] :
( ( member @ A @ C @ ( minus_minus @ ( set @ A ) @ A2 @ B3 ) )
= ( ( member @ A @ C @ A2 )
& ~ ( member @ A @ C @ B3 ) ) ) ).
% Diff_iff
thf(fact_214_Diff__idemp,axiom,
! [A: $tType,A2: set @ A,B3: set @ A] :
( ( minus_minus @ ( set @ A ) @ ( minus_minus @ ( set @ A ) @ A2 @ B3 ) @ B3 )
= ( minus_minus @ ( set @ A ) @ A2 @ B3 ) ) ).
% Diff_idemp
thf(fact_215_Diff__cancel,axiom,
! [A: $tType,A2: set @ A] :
( ( minus_minus @ ( set @ A ) @ A2 @ A2 )
= ( bot_bot @ ( set @ A ) ) ) ).
% Diff_cancel
thf(fact_216_empty__Diff,axiom,
! [A: $tType,A2: set @ A] :
( ( minus_minus @ ( set @ A ) @ ( bot_bot @ ( set @ A ) ) @ A2 )
= ( bot_bot @ ( set @ A ) ) ) ).
% empty_Diff
thf(fact_217_Diff__empty,axiom,
! [A: $tType,A2: set @ A] :
( ( minus_minus @ ( set @ A ) @ A2 @ ( bot_bot @ ( set @ A ) ) )
= A2 ) ).
% Diff_empty
thf(fact_218_insert__Diff1,axiom,
! [A: $tType,X3: A,B3: set @ A,A2: set @ A] :
( ( member @ A @ X3 @ B3 )
=> ( ( minus_minus @ ( set @ A ) @ ( insert @ A @ X3 @ A2 ) @ B3 )
= ( minus_minus @ ( set @ A ) @ A2 @ B3 ) ) ) ).
% insert_Diff1
thf(fact_219_Diff__insert0,axiom,
! [A: $tType,X3: A,A2: set @ A,B3: set @ A] :
( ~ ( member @ A @ X3 @ A2 )
=> ( ( minus_minus @ ( set @ A ) @ A2 @ ( insert @ A @ X3 @ B3 ) )
= ( minus_minus @ ( set @ A ) @ A2 @ B3 ) ) ) ).
% Diff_insert0
thf(fact_220_Diff__eq__empty__iff,axiom,
! [A: $tType,A2: set @ A,B3: set @ A] :
( ( ( minus_minus @ ( set @ A ) @ A2 @ B3 )
= ( bot_bot @ ( set @ A ) ) )
= ( ord_less_eq @ ( set @ A ) @ A2 @ B3 ) ) ).
% Diff_eq_empty_iff
thf(fact_221_Diff__UNIV,axiom,
! [A: $tType,A2: set @ A] :
( ( minus_minus @ ( set @ A ) @ A2 @ ( top_top @ ( set @ A ) ) )
= ( bot_bot @ ( set @ A ) ) ) ).
% Diff_UNIV
thf(fact_222_insert__Diff__single,axiom,
! [A: $tType,A3: A,A2: set @ A] :
( ( insert @ A @ A3 @ ( minus_minus @ ( set @ A ) @ A2 @ ( insert @ A @ A3 @ ( bot_bot @ ( set @ A ) ) ) ) )
= ( insert @ A @ A3 @ A2 ) ) ).
% insert_Diff_single
thf(fact_223_diff__eq__diff__less__eq,axiom,
! [A: $tType] :
( ( ordered_ab_group_add @ A @ ( type2 @ A ) )
=> ! [A3: A,B2: A,C: A,D2: A] :
( ( ( minus_minus @ A @ A3 @ B2 )
= ( minus_minus @ A @ C @ D2 ) )
=> ( ( ord_less_eq @ A @ A3 @ B2 )
= ( ord_less_eq @ A @ C @ D2 ) ) ) ) ).
% diff_eq_diff_less_eq
thf(fact_224_diff__right__mono,axiom,
! [A: $tType] :
( ( ordered_ab_group_add @ A @ ( type2 @ A ) )
=> ! [A3: A,B2: A,C: A] :
( ( ord_less_eq @ A @ A3 @ B2 )
=> ( ord_less_eq @ A @ ( minus_minus @ A @ A3 @ C ) @ ( minus_minus @ A @ B2 @ C ) ) ) ) ).
% diff_right_mono
thf(fact_225_diff__left__mono,axiom,
! [A: $tType] :
( ( ordered_ab_group_add @ A @ ( type2 @ A ) )
=> ! [B2: A,A3: A,C: A] :
( ( ord_less_eq @ A @ B2 @ A3 )
=> ( ord_less_eq @ A @ ( minus_minus @ A @ C @ A3 ) @ ( minus_minus @ A @ C @ B2 ) ) ) ) ).
% diff_left_mono
thf(fact_226_diff__mono,axiom,
! [A: $tType] :
( ( ordered_ab_group_add @ A @ ( type2 @ A ) )
=> ! [A3: A,B2: A,D2: A,C: A] :
( ( ord_less_eq @ A @ A3 @ B2 )
=> ( ( ord_less_eq @ A @ D2 @ C )
=> ( ord_less_eq @ A @ ( minus_minus @ A @ A3 @ C ) @ ( minus_minus @ A @ B2 @ D2 ) ) ) ) ) ).
% diff_mono
thf(fact_227_insert__Diff__if,axiom,
! [A: $tType,X3: A,B3: set @ A,A2: set @ A] :
( ( ( member @ A @ X3 @ B3 )
=> ( ( minus_minus @ ( set @ A ) @ ( insert @ A @ X3 @ A2 ) @ B3 )
= ( minus_minus @ ( set @ A ) @ A2 @ B3 ) ) )
& ( ~ ( member @ A @ X3 @ B3 )
=> ( ( minus_minus @ ( set @ A ) @ ( insert @ A @ X3 @ A2 ) @ B3 )
= ( insert @ A @ X3 @ ( minus_minus @ ( set @ A ) @ A2 @ B3 ) ) ) ) ) ).
% insert_Diff_if
thf(fact_228_DiffE,axiom,
! [A: $tType,C: A,A2: set @ A,B3: set @ A] :
( ( member @ A @ C @ ( minus_minus @ ( set @ A ) @ A2 @ B3 ) )
=> ~ ( ( member @ A @ C @ A2 )
=> ( member @ A @ C @ B3 ) ) ) ).
% DiffE
thf(fact_229_DiffD1,axiom,
! [A: $tType,C: A,A2: set @ A,B3: set @ A] :
( ( member @ A @ C @ ( minus_minus @ ( set @ A ) @ A2 @ B3 ) )
=> ( member @ A @ C @ A2 ) ) ).
% DiffD1
thf(fact_230_DiffD2,axiom,
! [A: $tType,C: A,A2: set @ A,B3: set @ A] :
( ( member @ A @ C @ ( minus_minus @ ( set @ A ) @ A2 @ B3 ) )
=> ~ ( member @ A @ C @ B3 ) ) ).
% DiffD2
thf(fact_231_subset__Diff__insert,axiom,
! [A: $tType,A2: set @ A,B3: set @ A,X3: A,C4: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ A2 @ ( minus_minus @ ( set @ A ) @ B3 @ ( insert @ A @ X3 @ C4 ) ) )
= ( ( ord_less_eq @ ( set @ A ) @ A2 @ ( minus_minus @ ( set @ A ) @ B3 @ C4 ) )
& ~ ( member @ A @ X3 @ A2 ) ) ) ).
% subset_Diff_insert
thf(fact_232_Diff__insert__absorb,axiom,
! [A: $tType,X3: A,A2: set @ A] :
( ~ ( member @ A @ X3 @ A2 )
=> ( ( minus_minus @ ( set @ A ) @ ( insert @ A @ X3 @ A2 ) @ ( insert @ A @ X3 @ ( bot_bot @ ( set @ A ) ) ) )
= A2 ) ) ).
% Diff_insert_absorb
thf(fact_233_Diff__insert2,axiom,
! [A: $tType,A2: set @ A,A3: A,B3: set @ A] :
( ( minus_minus @ ( set @ A ) @ A2 @ ( insert @ A @ A3 @ B3 ) )
= ( minus_minus @ ( set @ A ) @ ( minus_minus @ ( set @ A ) @ A2 @ ( insert @ A @ A3 @ ( bot_bot @ ( set @ A ) ) ) ) @ B3 ) ) ).
% Diff_insert2
thf(fact_234_insert__Diff,axiom,
! [A: $tType,A3: A,A2: set @ A] :
( ( member @ A @ A3 @ A2 )
=> ( ( insert @ A @ A3 @ ( minus_minus @ ( set @ A ) @ A2 @ ( insert @ A @ A3 @ ( bot_bot @ ( set @ A ) ) ) ) )
= A2 ) ) ).
% insert_Diff
thf(fact_235_Diff__insert,axiom,
! [A: $tType,A2: set @ A,A3: A,B3: set @ A] :
( ( minus_minus @ ( set @ A ) @ A2 @ ( insert @ A @ A3 @ B3 ) )
= ( minus_minus @ ( set @ A ) @ ( minus_minus @ ( set @ A ) @ A2 @ B3 ) @ ( insert @ A @ A3 @ ( bot_bot @ ( set @ A ) ) ) ) ) ).
% Diff_insert
thf(fact_236_Diff__mono,axiom,
! [A: $tType,A2: set @ A,C4: set @ A,D3: set @ A,B3: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ A2 @ C4 )
=> ( ( ord_less_eq @ ( set @ A ) @ D3 @ B3 )
=> ( ord_less_eq @ ( set @ A ) @ ( minus_minus @ ( set @ A ) @ A2 @ B3 ) @ ( minus_minus @ ( set @ A ) @ C4 @ D3 ) ) ) ) ).
% Diff_mono
thf(fact_237_Diff__subset,axiom,
! [A: $tType,A2: set @ A,B3: set @ A] : ( ord_less_eq @ ( set @ A ) @ ( minus_minus @ ( set @ A ) @ A2 @ B3 ) @ A2 ) ).
% Diff_subset
thf(fact_238_double__diff,axiom,
! [A: $tType,A2: set @ A,B3: set @ A,C4: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ A2 @ B3 )
=> ( ( ord_less_eq @ ( set @ A ) @ B3 @ C4 )
=> ( ( minus_minus @ ( set @ A ) @ B3 @ ( minus_minus @ ( set @ A ) @ C4 @ A2 ) )
= A2 ) ) ) ).
% double_diff
thf(fact_239_Compl__eq__Diff__UNIV,axiom,
! [A: $tType] :
( ( uminus_uminus @ ( set @ A ) )
= ( minus_minus @ ( set @ A ) @ ( top_top @ ( set @ A ) ) ) ) ).
% Compl_eq_Diff_UNIV
thf(fact_240_Diff__single__insert,axiom,
! [A: $tType,A2: set @ A,X3: A,B3: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ ( minus_minus @ ( set @ A ) @ A2 @ ( insert @ A @ X3 @ ( bot_bot @ ( set @ A ) ) ) ) @ B3 )
=> ( ord_less_eq @ ( set @ A ) @ A2 @ ( insert @ A @ X3 @ B3 ) ) ) ).
% Diff_single_insert
thf(fact_241_subset__insert__iff,axiom,
! [A: $tType,A2: set @ A,X3: A,B3: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ A2 @ ( insert @ A @ X3 @ B3 ) )
= ( ( ( member @ A @ X3 @ A2 )
=> ( ord_less_eq @ ( set @ A ) @ ( minus_minus @ ( set @ A ) @ A2 @ ( insert @ A @ X3 @ ( bot_bot @ ( set @ A ) ) ) ) @ B3 ) )
& ( ~ ( member @ A @ X3 @ A2 )
=> ( ord_less_eq @ ( set @ A ) @ A2 @ B3 ) ) ) ) ).
% subset_insert_iff
thf(fact_242_Compl__insert,axiom,
! [A: $tType,X3: A,A2: set @ A] :
( ( uminus_uminus @ ( set @ A ) @ ( insert @ A @ X3 @ A2 ) )
= ( minus_minus @ ( set @ A ) @ ( uminus_uminus @ ( set @ A ) @ A2 ) @ ( insert @ A @ X3 @ ( bot_bot @ ( set @ A ) ) ) ) ) ).
% Compl_insert
thf(fact_243_remove__def,axiom,
! [A: $tType] :
( ( remove @ A )
= ( ^ [X: A,A4: set @ A] : ( minus_minus @ ( set @ A ) @ A4 @ ( insert @ A @ X @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ).
% remove_def
thf(fact_244_psubset__insert__iff,axiom,
! [A: $tType,A2: set @ A,X3: A,B3: set @ A] :
( ( ord_less @ ( set @ A ) @ A2 @ ( insert @ A @ X3 @ B3 ) )
= ( ( ( member @ A @ X3 @ B3 )
=> ( ord_less @ ( set @ A ) @ A2 @ B3 ) )
& ( ~ ( member @ A @ X3 @ B3 )
=> ( ( ( member @ A @ X3 @ A2 )
=> ( ord_less @ ( set @ A ) @ ( minus_minus @ ( set @ A ) @ A2 @ ( insert @ A @ X3 @ ( bot_bot @ ( set @ A ) ) ) ) @ B3 ) )
& ( ~ ( member @ A @ X3 @ A2 )
=> ( ord_less_eq @ ( set @ A ) @ A2 @ B3 ) ) ) ) ) ) ).
% psubset_insert_iff
thf(fact_245_member__remove,axiom,
! [A: $tType,X3: A,Y3: A,A2: set @ A] :
( ( member @ A @ X3 @ ( remove @ A @ Y3 @ A2 ) )
= ( ( member @ A @ X3 @ A2 )
& ( X3 != Y3 ) ) ) ).
% member_remove
thf(fact_246_psubsetI,axiom,
! [A: $tType,A2: set @ A,B3: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ A2 @ B3 )
=> ( ( A2 != B3 )
=> ( ord_less @ ( set @ A ) @ A2 @ B3 ) ) ) ).
% psubsetI
thf(fact_247_bot_Onot__eq__extremum,axiom,
! [A: $tType] :
( ( order_bot @ A @ ( type2 @ A ) )
=> ! [A3: A] :
( ( A3
!= ( bot_bot @ A ) )
= ( ord_less @ A @ ( bot_bot @ A ) @ A3 ) ) ) ).
% bot.not_eq_extremum
thf(fact_248_bot_Oextremum__strict,axiom,
! [A: $tType] :
( ( order_bot @ A @ ( type2 @ A ) )
=> ! [A3: A] :
~ ( ord_less @ A @ A3 @ ( bot_bot @ A ) ) ) ).
% bot.extremum_strict
thf(fact_249_not__psubset__empty,axiom,
! [A: $tType,A2: set @ A] :
~ ( ord_less @ ( set @ A ) @ A2 @ ( bot_bot @ ( set @ A ) ) ) ).
% not_psubset_empty
thf(fact_250_psubset__imp__ex__mem,axiom,
! [A: $tType,A2: set @ A,B3: set @ A] :
( ( ord_less @ ( set @ A ) @ A2 @ B3 )
=> ? [B6: A] : ( member @ A @ B6 @ ( minus_minus @ ( set @ A ) @ B3 @ A2 ) ) ) ).
% psubset_imp_ex_mem
thf(fact_251_leD,axiom,
! [A: $tType] :
( ( linorder @ A @ ( type2 @ A ) )
=> ! [Y3: A,X3: A] :
( ( ord_less_eq @ A @ Y3 @ X3 )
=> ~ ( ord_less @ A @ X3 @ Y3 ) ) ) ).
% leD
thf(fact_252_leI,axiom,
! [A: $tType] :
( ( linorder @ A @ ( type2 @ A ) )
=> ! [X3: A,Y3: A] :
( ~ ( ord_less @ A @ X3 @ Y3 )
=> ( ord_less_eq @ A @ Y3 @ X3 ) ) ) ).
% leI
thf(fact_253_le__less,axiom,
! [A: $tType] :
( ( order @ A @ ( type2 @ A ) )
=> ( ( ord_less_eq @ A )
= ( ^ [X: A,Y2: A] :
( ( ord_less @ A @ X @ Y2 )
| ( X = Y2 ) ) ) ) ) ).
% le_less
thf(fact_254_less__le,axiom,
! [A: $tType] :
( ( order @ A @ ( type2 @ A ) )
=> ( ( ord_less @ A )
= ( ^ [X: A,Y2: A] :
( ( ord_less_eq @ A @ X @ Y2 )
& ( X != Y2 ) ) ) ) ) ).
% less_le
thf(fact_255_order__le__less__subst1,axiom,
! [A: $tType,B: $tType] :
( ( ( order @ B @ ( type2 @ B ) )
& ( order @ A @ ( type2 @ A ) ) )
=> ! [A3: A,F2: B > A,B2: B,C: B] :
( ( ord_less_eq @ A @ A3 @ ( F2 @ B2 ) )
=> ( ( ord_less @ B @ B2 @ C )
=> ( ! [X2: B,Y: B] :
( ( ord_less @ B @ X2 @ Y )
=> ( ord_less @ A @ ( F2 @ X2 ) @ ( F2 @ Y ) ) )
=> ( ord_less @ A @ A3 @ ( F2 @ C ) ) ) ) ) ) ).
% order_le_less_subst1
%----Type constructors (25)
thf(tcon_fun___Lattices_Oboolean__algebra,axiom,
! [A6: $tType,A7: $tType] :
( ( boolean_algebra @ A7 @ ( type2 @ A7 ) )
=> ( boolean_algebra @ ( A6 > A7 ) @ ( type2 @ ( A6 > A7 ) ) ) ) ).
thf(tcon_fun___Orderings_Oorder__top,axiom,
! [A6: $tType,A7: $tType] :
( ( order_top @ A7 @ ( type2 @ A7 ) )
=> ( order_top @ ( A6 > A7 ) @ ( type2 @ ( A6 > A7 ) ) ) ) ).
thf(tcon_fun___Orderings_Oorder__bot,axiom,
! [A6: $tType,A7: $tType] :
( ( order_bot @ A7 @ ( type2 @ A7 ) )
=> ( order_bot @ ( A6 > A7 ) @ ( type2 @ ( A6 > A7 ) ) ) ) ).
thf(tcon_fun___Orderings_Opreorder,axiom,
! [A6: $tType,A7: $tType] :
( ( preorder @ A7 @ ( type2 @ A7 ) )
=> ( preorder @ ( A6 > A7 ) @ ( type2 @ ( A6 > A7 ) ) ) ) ).
thf(tcon_fun___Orderings_Oorder,axiom,
! [A6: $tType,A7: $tType] :
( ( order @ A7 @ ( type2 @ A7 ) )
=> ( order @ ( A6 > A7 ) @ ( type2 @ ( A6 > A7 ) ) ) ) ).
thf(tcon_fun___Orderings_Otop,axiom,
! [A6: $tType,A7: $tType] :
( ( top @ A7 @ ( type2 @ A7 ) )
=> ( top @ ( A6 > A7 ) @ ( type2 @ ( A6 > A7 ) ) ) ) ).
thf(tcon_fun___Orderings_Oord,axiom,
! [A6: $tType,A7: $tType] :
( ( ord @ A7 @ ( type2 @ A7 ) )
=> ( ord @ ( A6 > A7 ) @ ( type2 @ ( A6 > A7 ) ) ) ) ).
thf(tcon_fun___Orderings_Obot,axiom,
! [A6: $tType,A7: $tType] :
( ( bot @ A7 @ ( type2 @ A7 ) )
=> ( bot @ ( A6 > A7 ) @ ( type2 @ ( A6 > A7 ) ) ) ) ).
thf(tcon_Set_Oset___Lattices_Oboolean__algebra_1,axiom,
! [A6: $tType] : ( boolean_algebra @ ( set @ A6 ) @ ( type2 @ ( set @ A6 ) ) ) ).
thf(tcon_Set_Oset___Orderings_Oorder__top_2,axiom,
! [A6: $tType] : ( order_top @ ( set @ A6 ) @ ( type2 @ ( set @ A6 ) ) ) ).
thf(tcon_Set_Oset___Orderings_Oorder__bot_3,axiom,
! [A6: $tType] : ( order_bot @ ( set @ A6 ) @ ( type2 @ ( set @ A6 ) ) ) ).
thf(tcon_Set_Oset___Orderings_Opreorder_4,axiom,
! [A6: $tType] : ( preorder @ ( set @ A6 ) @ ( type2 @ ( set @ A6 ) ) ) ).
thf(tcon_Set_Oset___Orderings_Oorder_5,axiom,
! [A6: $tType] : ( order @ ( set @ A6 ) @ ( type2 @ ( set @ A6 ) ) ) ).
thf(tcon_Set_Oset___Orderings_Otop_6,axiom,
! [A6: $tType] : ( top @ ( set @ A6 ) @ ( type2 @ ( set @ A6 ) ) ) ).
thf(tcon_Set_Oset___Orderings_Oord_7,axiom,
! [A6: $tType] : ( ord @ ( set @ A6 ) @ ( type2 @ ( set @ A6 ) ) ) ).
thf(tcon_Set_Oset___Orderings_Obot_8,axiom,
! [A6: $tType] : ( bot @ ( set @ A6 ) @ ( type2 @ ( set @ A6 ) ) ) ).
thf(tcon_HOL_Obool___Lattices_Oboolean__algebra_9,axiom,
boolean_algebra @ $o @ ( type2 @ $o ) ).
thf(tcon_HOL_Obool___Orderings_Oorder__top_10,axiom,
order_top @ $o @ ( type2 @ $o ) ).
thf(tcon_HOL_Obool___Orderings_Oorder__bot_11,axiom,
order_bot @ $o @ ( type2 @ $o ) ).
thf(tcon_HOL_Obool___Orderings_Opreorder_12,axiom,
preorder @ $o @ ( type2 @ $o ) ).
thf(tcon_HOL_Obool___Orderings_Olinorder,axiom,
linorder @ $o @ ( type2 @ $o ) ).
thf(tcon_HOL_Obool___Orderings_Oorder_13,axiom,
order @ $o @ ( type2 @ $o ) ).
thf(tcon_HOL_Obool___Orderings_Otop_14,axiom,
top @ $o @ ( type2 @ $o ) ).
thf(tcon_HOL_Obool___Orderings_Oord_15,axiom,
ord @ $o @ ( type2 @ $o ) ).
thf(tcon_HOL_Obool___Orderings_Obot_16,axiom,
bot @ $o @ ( type2 @ $o ) ).
%----Conjectures (2)
thf(conj_0,hypothesis,
$true ).
thf(conj_1,conjecture,
? [Tr6: dtree] :
( ( ( bot_bot @ ( set @ t ) )
= ( gram_L861583724lle_Fr @ ns @ Tr6 ) )
& ( gram_L864798063lle_wf @ Tr6 )
& ( ( root @ Tr6 )
= n2 )
& ( gram_L646766332egular @ Tr6 ) ) ).
%------------------------------------------------------------------------------