TPTP Problem File: COM189^1.p
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%------------------------------------------------------------------------------
% File : COM189^1 : TPTP v8.2.0. Released v7.0.0.
% Domain : Computing Theory
% Problem : Grammars and languages 706
% 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__706.p [Bla16]
% Status : Theorem
% Rating : 0.33 v8.1.0, 0.25 v7.5.0, 0.33 v7.2.0, 0.50 v7.1.0
% Syntax : Number of formulae : 331 ( 116 unt; 52 typ; 0 def)
% Number of atoms : 755 ( 265 equ; 0 cnn)
% Maximal formula atoms : 13 ( 2 avg)
% Number of connectives : 3930 ( 73 ~; 11 |; 65 &;3420 @)
% ( 0 <=>; 361 =>; 0 <=; 0 <~>)
% Maximal formula depth : 24 ( 8 avg)
% Number of types : 4 ( 3 usr)
% Number of type conns : 292 ( 292 >; 0 *; 0 +; 0 <<)
% Number of symbols : 52 ( 49 usr; 3 con; 0-6 aty)
% Number of variables : 1143 ( 62 ^;1001 !; 46 ?;1143 :)
% ( 34 !>; 0 ?*; 0 @-; 0 @+)
% SPC : TH1_THM_EQU_NAR
% Comments : This file was generated by Isabelle (most likely Sledgehammer)
% 2016-07-13 14:44:19.466
%------------------------------------------------------------------------------
%----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 (46)
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__top,type,
order_top:
!>[A: $tType] : ( ( itself @ A ) > $o ) ).
thf(sy_c_DTree_ONode,type,
node: n > ( set @ ( sum_sum @ t @ dtree ) ) > dtree ).
thf(sy_c_DTree_Ocont,type,
cont: dtree > ( set @ ( sum_sum @ t @ dtree ) ) ).
thf(sy_c_DTree_Ocorec,type,
corec:
!>[A: $tType] : ( ( A > n ) > ( A > ( set @ ( sum_sum @ t @ ( sum_sum @ dtree @ A ) ) ) ) > A > dtree ) ).
thf(sy_c_DTree_Odtree_Oroot,type,
root: dtree > n ).
thf(sy_c_DTree_Ounfold,type,
unfold:
!>[A: $tType] : ( ( A > n ) > ( A > ( set @ ( sum_sum @ t @ A ) ) ) > A > dtree ) ).
thf(sy_c_Fun_Ocomp,type,
comp:
!>[B: $tType,C: $tType,A: $tType] : ( ( B > C ) > ( A > B ) > A > C ) ).
thf(sy_c_Fun_Oid,type,
id:
!>[A: $tType] : ( A > A ) ).
thf(sy_c_Gram__Lang__Mirabelle__ojxrtuoybn_OS,type,
gram_L1451583635elle_S: n > ( set @ ( sum_sum @ t @ n ) ) ).
thf(sy_c_Gram__Lang__Mirabelle__ojxrtuoybn_Odeftr,type,
gram_L1231612515_deftr: n > dtree ).
thf(sy_c_Gram__Lang__Mirabelle__ojxrtuoybn_Ohsubst,type,
gram_L1004374585hsubst: dtree > dtree > dtree ).
thf(sy_c_Gram__Lang__Mirabelle__ojxrtuoybn_Ohsubst__c,type,
gram_L1905609002ubst_c: dtree > dtree > ( set @ ( sum_sum @ t @ dtree ) ) ).
thf(sy_c_Gram__Lang__Mirabelle__ojxrtuoybn_Ohsubst__r,type,
gram_L1905609017ubst_r: dtree > n ).
thf(sy_c_Gram__Lang__Mirabelle__ojxrtuoybn_OinFr,type,
gram_L1333338417e_inFr: ( set @ n ) > dtree > t > $o ).
thf(sy_c_Gram__Lang__Mirabelle__ojxrtuoybn_OinFr2,type,
gram_L805317441_inFr2: ( set @ n ) > dtree > t > $o ).
thf(sy_c_Gram__Lang__Mirabelle__ojxrtuoybn_OinFrr,type,
gram_L805317505_inFrr: ( set @ n ) > dtree > t > $o ).
thf(sy_c_Gram__Lang__Mirabelle__ojxrtuoybn_OinItr,type,
gram_L830233218_inItr: ( set @ n ) > dtree > n > $o ).
thf(sy_c_Gram__Lang__Mirabelle__ojxrtuoybn_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_Groups_Ominus__class_Ominus,type,
minus_minus:
!>[A: $tType] : ( A > A > A ) ).
thf(sy_c_If,type,
if:
!>[A: $tType] : ( $o > A > A > A ) ).
thf(sy_c_Orderings_Obot__class_Obot,type,
bot_bot:
!>[A: $tType] : A ).
thf(sy_c_Orderings_Oord__class_Oless__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_Set_OCollect,type,
collect:
!>[A: $tType] : ( ( A > $o ) > ( set @ A ) ) ).
thf(sy_c_Set_Oinsert,type,
insert:
!>[A: $tType] : ( A > ( set @ A ) > ( set @ A ) ) ).
thf(sy_c_Set_Ovimage,type,
vimage:
!>[A: $tType,B: $tType] : ( ( A > B ) > ( set @ B ) > ( set @ A ) ) ).
thf(sy_c_Sum__Type_OInl,type,
sum_Inl:
!>[A: $tType,B: $tType] : ( A > ( sum_sum @ A @ B ) ) ).
thf(sy_c_Sum__Type_OInr,type,
sum_Inr:
!>[B: $tType,A: $tType] : ( B > ( sum_sum @ A @ B ) ) ).
thf(sy_c_Sum__Type_Oold_Osum_Orec__sum,type,
sum_rec_sum:
!>[A: $tType,T: $tType,B: $tType] : ( ( A > T ) > ( B > T ) > ( sum_sum @ A @ B ) > T ) ).
thf(sy_c_Typedef_Otype__definition,type,
type_definition:
!>[B: $tType,A: $tType] : ( ( B > A ) > ( A > B ) > ( set @ A ) > $o ) ).
thf(sy_c_member,type,
member:
!>[A: $tType] : ( A > ( set @ A ) > $o ) ).
thf(sy_v_f,type,
f: n > dtree ).
thf(sy_v_tr,type,
tr: dtree ).
%----Relevant facts (256)
thf(fact_0_assms,axiom,
gram_L1918716148le_reg @ f @ tr ).
% assms
thf(fact_1_hsubst__r__def,axiom,
gram_L1905609017ubst_r = root ).
% hsubst_r_def
thf(fact_2_root__deftr,axiom,
! [N: n] :
( ( root @ ( gram_L1231612515_deftr @ N ) )
= N ) ).
% root_deftr
thf(fact_3_root__hsubst,axiom,
! [Tr0: dtree,Tr: dtree] :
( ( root @ ( gram_L1004374585hsubst @ Tr0 @ Tr ) )
= ( root @ Tr ) ) ).
% root_hsubst
thf(fact_4_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_5_inItr_OBase,axiom,
! [Tr: dtree,Ns: set @ n] :
( ( member @ n @ ( root @ Tr ) @ Ns )
=> ( gram_L830233218_inItr @ Ns @ Tr @ ( root @ Tr ) ) ) ).
% inItr.Base
thf(fact_6_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_7_regular__def2,axiom,
( gram_L646766332egular
= ( ^ [Tr2: dtree] :
? [F: n > dtree] :
( ( gram_L1918716148le_reg @ F @ Tr2 )
& ! [N2: n] :
( ( root @ ( F @ N2 ) )
= N2 ) ) ) ) ).
% regular_def2
thf(fact_8_hsubst__eq,axiom,
! [Tr: dtree,Tr0: dtree] :
( ( ( root @ Tr )
= ( root @ Tr0 ) )
=> ( ( gram_L1004374585hsubst @ Tr0 @ Tr )
= ( gram_L1004374585hsubst @ Tr0 @ Tr0 ) ) ) ).
% hsubst_eq
thf(fact_9_subtr2_ORefl,axiom,
! [Tr: dtree,Ns: set @ n] :
( ( member @ n @ ( root @ Tr ) @ Ns )
=> ( gram_L1283001940subtr2 @ Ns @ Tr @ Tr ) ) ).
% subtr2.Refl
thf(fact_10_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_11_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_12_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_13_inFr__inFr2,axiom,
gram_L1333338417e_inFr = gram_L805317441_inFr2 ).
% inFr_inFr2
thf(fact_14_regular__def,axiom,
( gram_L646766332egular
= ( ^ [Tr2: dtree] :
? [F: n > dtree] : ( gram_L1918716148le_reg @ F @ Tr2 ) ) ) ).
% regular_def
thf(fact_15_subtr2__trans,axiom,
! [Ns: set @ n,Tr1: dtree,Tr22: dtree,Tr3: dtree] :
( ( gram_L1283001940subtr2 @ Ns @ Tr1 @ Tr22 )
=> ( ( gram_L1283001940subtr2 @ Ns @ Tr22 @ Tr3 )
=> ( gram_L1283001940subtr2 @ Ns @ Tr1 @ Tr3 ) ) ) ).
% subtr2_trans
thf(fact_16_inFr__hsubst__notin,axiom,
! [Ns: set @ n,Tr: dtree,T2: t,Tr0: dtree] :
( ( gram_L1333338417e_inFr @ Ns @ Tr @ T2 )
=> ( ~ ( member @ n @ ( root @ Tr0 ) @ Ns )
=> ( gram_L1333338417e_inFr @ Ns @ ( gram_L1004374585hsubst @ Tr0 @ Tr ) @ T2 ) ) ) ).
% inFr_hsubst_notin
thf(fact_17_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_18_hsubst__def,axiom,
( gram_L1004374585hsubst
= ( ^ [Tr02: dtree] : ( unfold @ dtree @ gram_L1905609017ubst_r @ ( gram_L1905609002ubst_c @ Tr02 ) ) ) ) ).
% hsubst_def
thf(fact_19_reg__def2,axiom,
( gram_L1918716148le_reg
= ( ^ [F: n > dtree,Tr2: dtree] :
! [Ns2: set @ n,Tr4: dtree] :
( ( gram_L716654942_subtr @ Ns2 @ Tr4 @ Tr2 )
=> ( Tr4
= ( F @ ( root @ Tr4 ) ) ) ) ) ) ).
% reg_def2
thf(fact_20_inItr__subtr,axiom,
! [Ns: set @ n,Tr: dtree,N: n] :
( ( gram_L830233218_inItr @ Ns @ Tr @ N )
=> ? [Tr5: dtree] :
( ( gram_L716654942_subtr @ Ns @ Tr5 @ Tr )
& ( ( root @ Tr5 )
= N ) ) ) ).
% inItr_subtr
thf(fact_21_root__o__subst,axiom,
! [Tr0: dtree] :
( ( comp @ dtree @ n @ dtree @ root @ ( gram_L1004374585hsubst @ Tr0 ) )
= root ) ).
% root_o_subst
thf(fact_22_wf__deftr,axiom,
! [N: n] : ( gram_L864798063lle_wf @ ( gram_L1231612515_deftr @ N ) ) ).
% wf_deftr
thf(fact_23_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_24_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_25_subtr__subtr2,axiom,
gram_L716654942_subtr = gram_L1283001940subtr2 ).
% subtr_subtr2
thf(fact_26_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_27_root__Node,axiom,
! [N: n,As: set @ ( sum_sum @ t @ dtree )] :
( ( root @ ( node @ N @ As ) )
= N ) ).
% root_Node
thf(fact_28_inFr2__mono,axiom,
! [Ns: set @ n,Tr: dtree,T2: t,Ns3: set @ n] :
( ( gram_L805317441_inFr2 @ Ns @ Tr @ T2 )
=> ( ( ord_less_eq @ ( set @ n ) @ Ns @ Ns3 )
=> ( gram_L805317441_inFr2 @ Ns3 @ Tr @ T2 ) ) ) ).
% inFr2_mono
thf(fact_29_subtr__trans,axiom,
! [Ns: set @ n,Tr1: dtree,Tr22: dtree,Tr3: dtree] :
( ( gram_L716654942_subtr @ Ns @ Tr1 @ Tr22 )
=> ( ( gram_L716654942_subtr @ Ns @ Tr22 @ Tr3 )
=> ( gram_L716654942_subtr @ Ns @ Tr1 @ Tr3 ) ) ) ).
% subtr_trans
thf(fact_30_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_31_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_32_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_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_inFr__mono,axiom,
! [Ns: set @ n,Tr: dtree,T2: t,Ns3: set @ n] :
( ( gram_L1333338417e_inFr @ Ns @ Tr @ T2 )
=> ( ( ord_less_eq @ ( set @ n ) @ Ns @ Ns3 )
=> ( gram_L1333338417e_inFr @ Ns3 @ Tr @ T2 ) ) ) ).
% inFr_mono
thf(fact_37_subtr2__mono,axiom,
! [Ns: set @ n,Tr1: dtree,Tr22: dtree,Ns3: set @ n] :
( ( gram_L1283001940subtr2 @ Ns @ Tr1 @ Tr22 )
=> ( ( ord_less_eq @ ( set @ n ) @ Ns @ Ns3 )
=> ( gram_L1283001940subtr2 @ Ns3 @ Tr1 @ Tr22 ) ) ) ).
% subtr2_mono
thf(fact_38_inItr__mono,axiom,
! [Ns: set @ n,Tr: dtree,N: n,Ns3: set @ n] :
( ( gram_L830233218_inItr @ Ns @ Tr @ N )
=> ( ( ord_less_eq @ ( set @ n ) @ Ns @ Ns3 )
=> ( gram_L830233218_inItr @ Ns3 @ Tr @ N ) ) ) ).
% inItr_mono
thf(fact_39_comp__apply,axiom,
! [C: $tType,A: $tType,B: $tType] :
( ( comp @ B @ A @ C )
= ( ^ [F: B > A,G: C > B,X: C] : ( F @ ( G @ X ) ) ) ) ).
% comp_apply
thf(fact_40_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_41_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_42_order__refl,axiom,
! [A: $tType] :
( ( preorder @ A @ ( type2 @ A ) )
=> ! [X3: A] : ( ord_less_eq @ A @ X3 @ X3 ) ) ).
% order_refl
thf(fact_43_reg__def,axiom,
( gram_L1918716148le_reg
= ( ^ [F: n > dtree,Tr2: dtree] :
! [Tr4: dtree] :
( ( gram_L716654942_subtr @ ( top_top @ ( set @ n ) ) @ Tr4 @ Tr2 )
=> ( Tr4
= ( F @ ( root @ Tr4 ) ) ) ) ) ) ).
% reg_def
thf(fact_44_rewriteR__comp__comp2,axiom,
! [C: $tType,B: $tType,E: $tType,D: $tType,A: $tType,G2: C > B,H: A > C,R1: D > B,R2: A > D,F2: B > E,L: D > E] :
( ( ( comp @ C @ B @ A @ G2 @ H )
= ( comp @ D @ B @ A @ R1 @ R2 ) )
=> ( ( ( comp @ B @ E @ D @ F2 @ R1 )
= L )
=> ( ( comp @ C @ E @ A @ ( comp @ B @ E @ C @ F2 @ G2 ) @ H )
= ( comp @ D @ E @ A @ L @ R2 ) ) ) ) ).
% rewriteR_comp_comp2
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,G2: A > B] :
( ! [X2: A] :
( ( F2 @ X2 )
= ( G2 @ X2 ) )
=> ( F2 = G2 ) ) ).
% ext
thf(fact_49_rewriteL__comp__comp2,axiom,
! [A: $tType,C: $tType,B: $tType,D: $tType,E: $tType,F2: C > B,G2: A > C,L1: D > B,L2: A > D,H: E > A,R: E > D] :
( ( ( comp @ C @ B @ A @ F2 @ G2 )
= ( comp @ D @ B @ A @ L1 @ L2 ) )
=> ( ( ( comp @ A @ D @ E @ L2 @ H )
= R )
=> ( ( comp @ C @ B @ E @ F2 @ ( comp @ A @ C @ E @ G2 @ H ) )
= ( comp @ D @ B @ E @ L1 @ R ) ) ) ) ).
% rewriteL_comp_comp2
thf(fact_50_rewriteR__comp__comp,axiom,
! [C: $tType,D: $tType,B: $tType,A: $tType,G2: C > B,H: A > C,R: A > B,F2: B > D] :
( ( ( comp @ C @ B @ A @ G2 @ H )
= R )
=> ( ( comp @ C @ D @ A @ ( comp @ B @ D @ C @ F2 @ G2 ) @ H )
= ( comp @ B @ D @ A @ F2 @ R ) ) ) ).
% rewriteR_comp_comp
thf(fact_51_rewriteL__comp__comp,axiom,
! [C: $tType,B: $tType,A: $tType,D: $tType,F2: C > B,G2: A > C,L: A > B,H: D > A] :
( ( ( comp @ C @ B @ A @ F2 @ G2 )
= L )
=> ( ( comp @ C @ B @ D @ F2 @ ( comp @ A @ C @ D @ G2 @ H ) )
= ( comp @ A @ B @ D @ L @ H ) ) ) ).
% rewriteL_comp_comp
thf(fact_52_top__apply,axiom,
! [C: $tType,D: $tType] :
( ( top @ C @ ( type2 @ C ) )
=> ( ( top_top @ ( D > C ) )
= ( ^ [X: D] : ( top_top @ C ) ) ) ) ).
% top_apply
thf(fact_53_UNIV__I,axiom,
! [A: $tType,X3: A] : ( member @ A @ X3 @ ( top_top @ ( set @ A ) ) ) ).
% UNIV_I
thf(fact_54_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_55_UNIV__witness,axiom,
! [A: $tType] :
? [X2: A] : ( member @ A @ X2 @ ( top_top @ ( set @ A ) ) ) ).
% UNIV_witness
thf(fact_56_top__greatest,axiom,
! [A: $tType] :
( ( order_top @ A @ ( type2 @ A ) )
=> ! [A3: A] : ( ord_less_eq @ A @ A3 @ ( top_top @ A ) ) ) ).
% top_greatest
thf(fact_57_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_58_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_59_subset__UNIV,axiom,
! [A: $tType,A2: set @ A] : ( ord_less_eq @ ( set @ A ) @ A2 @ ( top_top @ ( set @ A ) ) ) ).
% subset_UNIV
thf(fact_60_le__funD,axiom,
! [B: $tType,A: $tType] :
( ( ord @ B @ ( type2 @ B ) )
=> ! [F2: A > B,G2: A > B,X3: A] :
( ( ord_less_eq @ ( A > B ) @ F2 @ G2 )
=> ( ord_less_eq @ B @ ( F2 @ X3 ) @ ( G2 @ X3 ) ) ) ) ).
% le_funD
thf(fact_61_le__funE,axiom,
! [B: $tType,A: $tType] :
( ( ord @ B @ ( type2 @ B ) )
=> ! [F2: A > B,G2: A > B,X3: A] :
( ( ord_less_eq @ ( A > B ) @ F2 @ G2 )
=> ( ord_less_eq @ B @ ( F2 @ X3 ) @ ( G2 @ X3 ) ) ) ) ).
% le_funE
thf(fact_62_le__funI,axiom,
! [B: $tType,A: $tType] :
( ( ord @ B @ ( type2 @ B ) )
=> ! [F2: A > B,G2: A > B] :
( ! [X2: A] : ( ord_less_eq @ B @ ( F2 @ X2 ) @ ( G2 @ X2 ) )
=> ( ord_less_eq @ ( A > B ) @ F2 @ G2 ) ) ) ).
% le_funI
thf(fact_63_le__fun__def,axiom,
! [B: $tType,A: $tType] :
( ( ord @ B @ ( type2 @ B ) )
=> ( ( ord_less_eq @ ( A > B ) )
= ( ^ [F: A > B,G: A > B] :
! [X: A] : ( ord_less_eq @ B @ ( F @ X ) @ ( G @ X ) ) ) ) ) ).
% le_fun_def
thf(fact_64_order__subst1,axiom,
! [A: $tType,B: $tType] :
( ( ( order @ B @ ( type2 @ B ) )
& ( order @ A @ ( type2 @ A ) ) )
=> ! [A3: A,F2: B > A,B2: B,C2: B] :
( ( ord_less_eq @ A @ A3 @ ( F2 @ B2 ) )
=> ( ( ord_less_eq @ B @ B2 @ C2 )
=> ( ! [X2: B,Y: B] :
( ( ord_less_eq @ B @ X2 @ Y )
=> ( ord_less_eq @ A @ ( F2 @ X2 ) @ ( F2 @ Y ) ) )
=> ( ord_less_eq @ A @ A3 @ ( F2 @ C2 ) ) ) ) ) ) ).
% order_subst1
thf(fact_65_order__subst2,axiom,
! [A: $tType,C: $tType] :
( ( ( order @ C @ ( type2 @ C ) )
& ( order @ A @ ( type2 @ A ) ) )
=> ! [A3: A,B2: A,F2: A > C,C2: C] :
( ( ord_less_eq @ A @ A3 @ B2 )
=> ( ( ord_less_eq @ C @ ( F2 @ B2 ) @ C2 )
=> ( ! [X2: A,Y: A] :
( ( ord_less_eq @ A @ X2 @ Y )
=> ( ord_less_eq @ C @ ( F2 @ X2 ) @ ( F2 @ Y ) ) )
=> ( ord_less_eq @ C @ ( F2 @ A3 ) @ C2 ) ) ) ) ) ).
% order_subst2
thf(fact_66_ord__eq__le__subst,axiom,
! [A: $tType,B: $tType] :
( ( ( ord @ B @ ( type2 @ B ) )
& ( ord @ A @ ( type2 @ A ) ) )
=> ! [A3: A,F2: B > A,B2: B,C2: B] :
( ( A3
= ( F2 @ B2 ) )
=> ( ( ord_less_eq @ B @ B2 @ C2 )
=> ( ! [X2: B,Y: B] :
( ( ord_less_eq @ B @ X2 @ Y )
=> ( ord_less_eq @ A @ ( F2 @ X2 ) @ ( F2 @ Y ) ) )
=> ( ord_less_eq @ A @ A3 @ ( F2 @ C2 ) ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_67_ord__le__eq__subst,axiom,
! [A: $tType,B: $tType] :
( ( ( ord @ B @ ( type2 @ B ) )
& ( ord @ A @ ( type2 @ A ) ) )
=> ! [A3: A,B2: A,F2: A > B,C2: B] :
( ( ord_less_eq @ A @ A3 @ B2 )
=> ( ( ( F2 @ B2 )
= C2 )
=> ( ! [X2: A,Y: A] :
( ( ord_less_eq @ A @ X2 @ Y )
=> ( ord_less_eq @ B @ ( F2 @ X2 ) @ ( F2 @ Y ) ) )
=> ( ord_less_eq @ B @ ( F2 @ A3 ) @ C2 ) ) ) ) ) ).
% ord_le_eq_subst
thf(fact_68_eq__iff,axiom,
! [A: $tType] :
( ( order @ A @ ( type2 @ A ) )
=> ( ( ^ [Y2: A,Z: A] : Y2 = Z )
= ( ^ [X: A,Y3: A] :
( ( ord_less_eq @ A @ X @ Y3 )
& ( ord_less_eq @ A @ Y3 @ X ) ) ) ) ) ).
% eq_iff
thf(fact_69_antisym,axiom,
! [A: $tType] :
( ( order @ A @ ( type2 @ A ) )
=> ! [X3: A,Y4: A] :
( ( ord_less_eq @ A @ X3 @ Y4 )
=> ( ( ord_less_eq @ A @ Y4 @ X3 )
=> ( X3 = Y4 ) ) ) ) ).
% antisym
thf(fact_70_linear,axiom,
! [A: $tType] :
( ( linorder @ A @ ( type2 @ A ) )
=> ! [X3: A,Y4: A] :
( ( ord_less_eq @ A @ X3 @ Y4 )
| ( ord_less_eq @ A @ Y4 @ X3 ) ) ) ).
% linear
thf(fact_71_eq__refl,axiom,
! [A: $tType] :
( ( preorder @ A @ ( type2 @ A ) )
=> ! [X3: A,Y4: A] :
( ( X3 = Y4 )
=> ( ord_less_eq @ A @ X3 @ Y4 ) ) ) ).
% eq_refl
thf(fact_72_le__cases,axiom,
! [A: $tType] :
( ( linorder @ A @ ( type2 @ A ) )
=> ! [X3: A,Y4: A] :
( ~ ( ord_less_eq @ A @ X3 @ Y4 )
=> ( ord_less_eq @ A @ Y4 @ X3 ) ) ) ).
% le_cases
thf(fact_73_order_Otrans,axiom,
! [A: $tType] :
( ( order @ A @ ( type2 @ A ) )
=> ! [A3: A,B2: A,C2: A] :
( ( ord_less_eq @ A @ A3 @ B2 )
=> ( ( ord_less_eq @ A @ B2 @ C2 )
=> ( ord_less_eq @ A @ A3 @ C2 ) ) ) ) ).
% order.trans
thf(fact_74_le__cases3,axiom,
! [A: $tType] :
( ( linorder @ A @ ( type2 @ A ) )
=> ! [X3: A,Y4: A,Z2: A] :
( ( ( ord_less_eq @ A @ X3 @ Y4 )
=> ~ ( ord_less_eq @ A @ Y4 @ Z2 ) )
=> ( ( ( ord_less_eq @ A @ Y4 @ X3 )
=> ~ ( ord_less_eq @ A @ X3 @ Z2 ) )
=> ( ( ( ord_less_eq @ A @ X3 @ Z2 )
=> ~ ( ord_less_eq @ A @ Z2 @ Y4 ) )
=> ( ( ( ord_less_eq @ A @ Z2 @ Y4 )
=> ~ ( ord_less_eq @ A @ Y4 @ X3 ) )
=> ( ( ( ord_less_eq @ A @ Y4 @ Z2 )
=> ~ ( ord_less_eq @ A @ Z2 @ X3 ) )
=> ~ ( ( ord_less_eq @ A @ Z2 @ X3 )
=> ~ ( ord_less_eq @ A @ X3 @ Y4 ) ) ) ) ) ) ) ) ).
% le_cases3
thf(fact_75_antisym__conv,axiom,
! [A: $tType] :
( ( order @ A @ ( type2 @ A ) )
=> ! [Y4: A,X3: A] :
( ( ord_less_eq @ A @ Y4 @ X3 )
=> ( ( ord_less_eq @ A @ X3 @ Y4 )
= ( X3 = Y4 ) ) ) ) ).
% antisym_conv
thf(fact_76_ord__eq__le__trans,axiom,
! [A: $tType] :
( ( ord @ A @ ( type2 @ A ) )
=> ! [A3: A,B2: A,C2: A] :
( ( A3 = B2 )
=> ( ( ord_less_eq @ A @ B2 @ C2 )
=> ( ord_less_eq @ A @ A3 @ C2 ) ) ) ) ).
% ord_eq_le_trans
thf(fact_77_ord__le__eq__trans,axiom,
! [A: $tType] :
( ( ord @ A @ ( type2 @ A ) )
=> ! [A3: A,B2: A,C2: A] :
( ( ord_less_eq @ A @ A3 @ B2 )
=> ( ( B2 = C2 )
=> ( ord_less_eq @ A @ A3 @ C2 ) ) ) ) ).
% ord_le_eq_trans
thf(fact_78_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_79_order__trans,axiom,
! [A: $tType] :
( ( preorder @ A @ ( type2 @ A ) )
=> ! [X3: A,Y4: A,Z2: A] :
( ( ord_less_eq @ A @ X3 @ Y4 )
=> ( ( ord_less_eq @ A @ Y4 @ Z2 )
=> ( ord_less_eq @ A @ X3 @ Z2 ) ) ) ) ).
% order_trans
thf(fact_80_dual__order_Orefl,axiom,
! [A: $tType] :
( ( order @ A @ ( type2 @ A ) )
=> ! [A3: A] : ( ord_less_eq @ A @ A3 @ A3 ) ) ).
% dual_order.refl
thf(fact_81_linorder__wlog,axiom,
! [A: $tType] :
( ( linorder @ A @ ( type2 @ A ) )
=> ! [P: A > A > $o,A3: A,B2: A] :
( ! [A4: A,B4: A] :
( ( ord_less_eq @ A @ A4 @ B4 )
=> ( P @ A4 @ B4 ) )
=> ( ! [A4: A,B4: A] :
( ( P @ B4 @ A4 )
=> ( P @ A4 @ B4 ) )
=> ( P @ A3 @ B2 ) ) ) ) ).
% linorder_wlog
thf(fact_82_dual__order_Otrans,axiom,
! [A: $tType] :
( ( order @ A @ ( type2 @ A ) )
=> ! [B2: A,A3: A,C2: A] :
( ( ord_less_eq @ A @ B2 @ A3 )
=> ( ( ord_less_eq @ A @ C2 @ B2 )
=> ( ord_less_eq @ A @ C2 @ A3 ) ) ) ) ).
% dual_order.trans
thf(fact_83_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_84_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_85_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_86_subsetD,axiom,
! [A: $tType,A2: set @ A,B3: set @ A,C2: A] :
( ( ord_less_eq @ ( set @ A ) @ A2 @ B3 )
=> ( ( member @ A @ C2 @ A2 )
=> ( member @ A @ C2 @ B3 ) ) ) ).
% subsetD
thf(fact_87_subsetCE,axiom,
! [A: $tType,A2: set @ A,B3: set @ A,C2: A] :
( ( ord_less_eq @ ( set @ A ) @ A2 @ B3 )
=> ( ( member @ A @ C2 @ A2 )
=> ( member @ A @ C2 @ B3 ) ) ) ).
% subsetCE
thf(fact_88_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_89_subset__eq,axiom,
! [A: $tType] :
( ( ord_less_eq @ ( set @ A ) )
= ( ^ [A5: set @ A,B5: set @ A] :
! [X: A] :
( ( member @ A @ X @ A5 )
=> ( member @ A @ X @ B5 ) ) ) ) ).
% subset_eq
thf(fact_90_equalityD1,axiom,
! [A: $tType,A2: set @ A,B3: set @ A] :
( ( A2 = B3 )
=> ( ord_less_eq @ ( set @ A ) @ A2 @ B3 ) ) ).
% equalityD1
thf(fact_91_equalityD2,axiom,
! [A: $tType,A2: set @ A,B3: set @ A] :
( ( A2 = B3 )
=> ( ord_less_eq @ ( set @ A ) @ B3 @ A2 ) ) ).
% equalityD2
thf(fact_92_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_93_subset__iff,axiom,
! [A: $tType] :
( ( ord_less_eq @ ( set @ A ) )
= ( ^ [A5: set @ A,B5: set @ A] :
! [T3: A] :
( ( member @ A @ T3 @ A5 )
=> ( member @ A @ T3 @ B5 ) ) ) ) ).
% subset_iff
thf(fact_94_rev__subsetD,axiom,
! [A: $tType,C2: A,A2: set @ A,B3: set @ A] :
( ( member @ A @ C2 @ A2 )
=> ( ( ord_less_eq @ ( set @ A ) @ A2 @ B3 )
=> ( member @ A @ C2 @ B3 ) ) ) ).
% rev_subsetD
thf(fact_95_subset__refl,axiom,
! [A: $tType,A2: set @ A] : ( ord_less_eq @ ( set @ A ) @ A2 @ A2 ) ).
% subset_refl
thf(fact_96_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_97_subset__trans,axiom,
! [A: $tType,A2: set @ A,B3: set @ A,C3: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ A2 @ B3 )
=> ( ( ord_less_eq @ ( set @ A ) @ B3 @ C3 )
=> ( ord_less_eq @ ( set @ A ) @ A2 @ C3 ) ) ) ).
% subset_trans
thf(fact_98_set__eq__subset,axiom,
! [A: $tType] :
( ( ^ [Y2: set @ A,Z: set @ A] : Y2 = Z )
= ( ^ [A5: set @ A,B5: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ A5 @ B5 )
& ( ord_less_eq @ ( set @ A ) @ B5 @ A5 ) ) ) ) ).
% set_eq_subset
thf(fact_99_contra__subsetD,axiom,
! [A: $tType,A2: set @ A,B3: set @ A,C2: A] :
( ( ord_less_eq @ ( set @ A ) @ A2 @ B3 )
=> ( ~ ( member @ A @ C2 @ B3 )
=> ~ ( member @ A @ C2 @ A2 ) ) ) ).
% contra_subsetD
thf(fact_100_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_101_comp__def,axiom,
! [A: $tType,C: $tType,B: $tType] :
( ( comp @ B @ C @ A )
= ( ^ [F: B > C,G: A > B,X: A] : ( F @ ( G @ X ) ) ) ) ).
% comp_def
thf(fact_102_comp__assoc,axiom,
! [B: $tType,D: $tType,C: $tType,A: $tType,F2: D > B,G2: C > D,H: A > C] :
( ( comp @ C @ B @ A @ ( comp @ D @ B @ C @ F2 @ G2 ) @ H )
= ( comp @ D @ B @ A @ F2 @ ( comp @ C @ D @ A @ G2 @ H ) ) ) ).
% comp_assoc
thf(fact_103_comp__eq__dest,axiom,
! [C: $tType,B: $tType,D: $tType,A: $tType,A3: C > B,B2: A > C,C2: D > B,D2: A > D,V: A] :
( ( ( comp @ C @ B @ A @ A3 @ B2 )
= ( comp @ D @ B @ A @ C2 @ D2 ) )
=> ( ( A3 @ ( B2 @ V ) )
= ( C2 @ ( D2 @ V ) ) ) ) ).
% comp_eq_dest
thf(fact_104_comp__eq__elim,axiom,
! [C: $tType,B: $tType,D: $tType,A: $tType,A3: C > B,B2: A > C,C2: D > B,D2: A > D] :
( ( ( comp @ C @ B @ A @ A3 @ B2 )
= ( comp @ D @ B @ A @ C2 @ D2 ) )
=> ! [V2: A] :
( ( A3 @ ( B2 @ V2 ) )
= ( C2 @ ( D2 @ V2 ) ) ) ) ).
% comp_eq_elim
thf(fact_105_comp__eq__dest__lhs,axiom,
! [C: $tType,B: $tType,A: $tType,A3: C > B,B2: A > C,C2: A > B,V: A] :
( ( ( comp @ C @ B @ A @ A3 @ B2 )
= C2 )
=> ( ( A3 @ ( B2 @ V ) )
= ( C2 @ V ) ) ) ).
% comp_eq_dest_lhs
thf(fact_106_iso__tuple__UNIV__I,axiom,
! [A: $tType,X3: A] : ( member @ A @ X3 @ ( top_top @ ( set @ A ) ) ) ).
% iso_tuple_UNIV_I
thf(fact_107_comp__cong,axiom,
! [C: $tType,B: $tType,D: $tType,A: $tType,E: $tType,F2: B > A,G2: C > B,X3: C,F3: D > A,G3: E > D,X4: E] :
( ( ( F2 @ ( G2 @ X3 ) )
= ( F3 @ ( G3 @ X4 ) ) )
=> ( ( comp @ B @ A @ C @ F2 @ G2 @ X3 )
= ( comp @ D @ A @ E @ F3 @ G3 @ X4 ) ) ) ).
% comp_cong
thf(fact_108_comp__apply__eq,axiom,
! [B: $tType,D: $tType,A: $tType,C: $tType,F2: B > A,G2: C > B,X3: C,H: D > A,K: C > D] :
( ( ( F2 @ ( G2 @ X3 ) )
= ( H @ ( K @ X3 ) ) )
=> ( ( comp @ B @ A @ C @ F2 @ G2 @ X3 )
= ( comp @ D @ A @ C @ H @ K @ X3 ) ) ) ).
% comp_apply_eq
thf(fact_109_fun_Omap__comp,axiom,
! [B: $tType,C: $tType,A: $tType,D: $tType,G2: B > C,F2: A > B,V: D > A] :
( ( comp @ B @ C @ D @ G2 @ ( comp @ A @ B @ D @ F2 @ V ) )
= ( comp @ A @ C @ D @ ( comp @ B @ C @ A @ G2 @ F2 ) @ V ) ) ).
% fun.map_comp
thf(fact_110_type__copy__map__cong0,axiom,
! [B: $tType,D: $tType,E: $tType,A: $tType,C: $tType,M: B > A,G2: C > B,X3: C,N3: D > A,H: C > D,F2: A > E] :
( ( ( M @ ( G2 @ X3 ) )
= ( N3 @ ( H @ X3 ) ) )
=> ( ( comp @ B @ E @ C @ ( comp @ A @ E @ B @ F2 @ M ) @ G2 @ X3 )
= ( comp @ D @ E @ C @ ( comp @ A @ E @ D @ F2 @ N3 ) @ H @ X3 ) ) ) ).
% type_copy_map_cong0
thf(fact_111_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_112_root__o__deftr,axiom,
( ( comp @ dtree @ n @ n @ root @ gram_L1231612515_deftr )
= ( id @ n ) ) ).
% root_o_deftr
thf(fact_113_Node__root__cont,axiom,
! [Tr: dtree] :
( ( node @ ( root @ Tr ) @ ( cont @ Tr ) )
= Tr ) ).
% Node_root_cont
thf(fact_114_id__apply,axiom,
! [A: $tType] :
( ( id @ A )
= ( ^ [X: A] : X ) ) ).
% id_apply
thf(fact_115_fun_Omap__id,axiom,
! [A: $tType,D: $tType,T2: D > A] :
( ( comp @ A @ A @ D @ ( id @ A ) @ T2 )
= T2 ) ).
% fun.map_id
thf(fact_116_id__comp,axiom,
! [B: $tType,A: $tType,G2: A > B] :
( ( comp @ B @ B @ A @ ( id @ B ) @ G2 )
= G2 ) ).
% id_comp
thf(fact_117_comp__id,axiom,
! [B: $tType,A: $tType,F2: A > B] :
( ( comp @ A @ B @ A @ F2 @ ( id @ A ) )
= F2 ) ).
% comp_id
thf(fact_118_fun_Omap__id0,axiom,
! [A: $tType,D: $tType] :
( ( comp @ A @ A @ D @ ( id @ A ) )
= ( id @ ( D > A ) ) ) ).
% fun.map_id0
thf(fact_119_id__def,axiom,
! [A: $tType] :
( ( id @ A )
= ( ^ [X: A] : X ) ) ).
% id_def
thf(fact_120_eq__id__iff,axiom,
! [A: $tType,F2: A > A] :
( ( ! [X: A] :
( ( F2 @ X )
= X ) )
= ( F2
= ( id @ A ) ) ) ).
% eq_id_iff
thf(fact_121_DEADID_Oin__rel,axiom,
! [B: $tType] :
( ( ^ [Y2: B,Z: B] : Y2 = Z )
= ( ^ [A6: B,B6: B] :
? [Z3: B] :
( ( member @ B @ Z3 @ ( top_top @ ( set @ B ) ) )
& ( ( id @ B @ Z3 )
= A6 )
& ( ( id @ B @ Z3 )
= B6 ) ) ) ) ).
% DEADID.in_rel
thf(fact_122_comp__eq__id__dest,axiom,
! [C: $tType,B: $tType,A: $tType,A3: C > B,B2: A > C,C2: A > B,V: A] :
( ( ( comp @ C @ B @ A @ A3 @ B2 )
= ( comp @ B @ B @ A @ ( id @ B ) @ C2 ) )
=> ( ( A3 @ ( B2 @ V ) )
= ( C2 @ V ) ) ) ).
% comp_eq_id_dest
thf(fact_123_pointfree__idE,axiom,
! [B: $tType,A: $tType,F2: B > A,G2: A > B,X3: A] :
( ( ( comp @ B @ A @ A @ F2 @ G2 )
= ( id @ A ) )
=> ( ( F2 @ ( G2 @ X3 ) )
= X3 ) ) ).
% pointfree_idE
thf(fact_124_dtree__cong,axiom,
! [Tr: dtree,Tr6: dtree] :
( ( ( root @ Tr )
= ( root @ Tr6 ) )
=> ( ( ( cont @ Tr )
= ( cont @ Tr6 ) )
=> ( Tr = Tr6 ) ) ) ).
% dtree_cong
thf(fact_125_top__set__def,axiom,
! [A: $tType] :
( ( top_top @ ( set @ A ) )
= ( collect @ A @ ( top_top @ ( A > $o ) ) ) ) ).
% top_set_def
thf(fact_126_hsubst__c__def,axiom,
( gram_L1905609002ubst_c
= ( ^ [Tr02: dtree,Tr2: dtree] :
( if @ ( set @ ( sum_sum @ t @ dtree ) )
@ ( ( root @ Tr2 )
= ( root @ Tr02 ) )
@ ( cont @ Tr02 )
@ ( cont @ Tr2 ) ) ) ) ).
% hsubst_c_def
thf(fact_127_deftr__def,axiom,
( gram_L1231612515_deftr
= ( unfold @ n @ ( id @ n ) @ gram_L1451583635elle_S ) ) ).
% deftr_def
thf(fact_128_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_129_subtr__UNIV__inductL,axiom,
! [Tr1: dtree,Tr22: dtree,Phi: dtree > dtree > $o] :
( ( gram_L716654942_subtr @ ( top_top @ ( set @ n ) ) @ Tr1 @ Tr22 )
=> ( ! [Tr7: dtree] : ( Phi @ Tr7 @ Tr7 )
=> ( ! [Tr12: dtree,Tr23: dtree,Tr32: dtree] :
( ( member @ ( sum_sum @ t @ dtree ) @ ( sum_Inr @ dtree @ t @ Tr12 ) @ ( cont @ Tr23 ) )
=> ( ( gram_L716654942_subtr @ ( top_top @ ( set @ n ) ) @ Tr23 @ Tr32 )
=> ( ( Phi @ Tr23 @ Tr32 )
=> ( Phi @ Tr12 @ Tr32 ) ) ) )
=> ( Phi @ Tr1 @ Tr22 ) ) ) ) ).
% subtr_UNIV_inductL
thf(fact_130_type__copy__map__id0,axiom,
! [B: $tType,A: $tType,Rep: A > B,Abs: B > A,M: B > B] :
( ( type_definition @ A @ B @ Rep @ Abs @ ( top_top @ ( set @ B ) ) )
=> ( ( M
= ( id @ B ) )
=> ( ( comp @ B @ A @ A @ ( comp @ B @ A @ B @ Abs @ M ) @ Rep )
= ( id @ A ) ) ) ) ).
% type_copy_map_id0
thf(fact_131_type__copy__Abs__o__Rep,axiom,
! [B: $tType,A: $tType,Rep: A > B,Abs: B > A] :
( ( type_definition @ A @ B @ Rep @ Abs @ ( top_top @ ( set @ B ) ) )
=> ( ( comp @ B @ A @ A @ Abs @ Rep )
= ( id @ A ) ) ) ).
% type_copy_Abs_o_Rep
thf(fact_132_top1I,axiom,
! [A: $tType,X3: A] : ( top_top @ ( A > $o ) @ X3 ) ).
% top1I
thf(fact_133_type__copy__wit,axiom,
! [A: $tType,C: $tType,B: $tType,Rep: A > B,Abs: B > A,X3: C,S: B > ( set @ C ),Y4: B] :
( ( type_definition @ A @ B @ Rep @ Abs @ ( top_top @ ( set @ B ) ) )
=> ( ( member @ C @ X3 @ ( comp @ B @ ( set @ C ) @ A @ S @ Rep @ ( Abs @ Y4 ) ) )
=> ( member @ C @ X3 @ ( S @ Y4 ) ) ) ) ).
% type_copy_wit
thf(fact_134_top__conj_I2_J,axiom,
! [A: $tType,P: $o,X3: A] :
( ( P
& ( top_top @ ( A > $o ) @ X3 ) )
= P ) ).
% top_conj(2)
thf(fact_135_top__conj_I1_J,axiom,
! [A: $tType,X3: A,P: $o] :
( ( ( top_top @ ( A > $o ) @ X3 )
& P )
= P ) ).
% top_conj(1)
thf(fact_136_type__copy__ex__RepI,axiom,
! [B: $tType,A: $tType,Rep: A > B,Abs: B > A,F4: B > $o] :
( ( type_definition @ A @ B @ Rep @ Abs @ ( top_top @ ( set @ B ) ) )
=> ( ( ^ [P2: B > $o] :
? [X5: B] : ( P2 @ X5 )
@ F4 )
= ( ? [B6: A] : ( F4 @ ( Rep @ B6 ) ) ) ) ) ).
% type_copy_ex_RepI
thf(fact_137_type__copy__obj__one__point__absE,axiom,
! [A: $tType,B: $tType,Rep: A > B,Abs: B > A,S2: A] :
( ( type_definition @ A @ B @ Rep @ Abs @ ( top_top @ ( set @ B ) ) )
=> ~ ! [X2: B] :
( S2
!= ( Abs @ X2 ) ) ) ).
% type_copy_obj_one_point_absE
thf(fact_138_wf__cont,axiom,
! [Tr: dtree,Tr6: dtree] :
( ( gram_L864798063lle_wf @ Tr )
=> ( ( member @ ( sum_sum @ t @ dtree ) @ ( sum_Inr @ dtree @ t @ Tr6 ) @ ( cont @ Tr ) )
=> ( gram_L864798063lle_wf @ Tr6 ) ) ) ).
% wf_cont
thf(fact_139_type__copy__map__comp0__undo,axiom,
! [E: $tType,A: $tType,C: $tType,B: $tType,D: $tType,F5: $tType,Rep: A > B,Abs: B > A,Rep2: C > D,Abs2: D > C,Rep3: E > F5,Abs3: F5 > E,M: F5 > D,M1: B > D,M2: F5 > B] :
( ( type_definition @ A @ B @ Rep @ Abs @ ( top_top @ ( set @ B ) ) )
=> ( ( type_definition @ C @ D @ Rep2 @ Abs2 @ ( top_top @ ( set @ D ) ) )
=> ( ( type_definition @ E @ F5 @ Rep3 @ Abs3 @ ( top_top @ ( set @ F5 ) ) )
=> ( ( ( comp @ F5 @ C @ E @ ( comp @ D @ C @ F5 @ Abs2 @ M ) @ Rep3 )
= ( comp @ A @ C @ E @ ( comp @ B @ C @ A @ ( comp @ D @ C @ B @ Abs2 @ M1 ) @ Rep ) @ ( comp @ F5 @ A @ E @ ( comp @ B @ A @ F5 @ Abs @ M2 ) @ Rep3 ) ) )
=> ( ( comp @ B @ D @ F5 @ M1 @ M2 )
= M ) ) ) ) ) ).
% type_copy_map_comp0_undo
thf(fact_140_type__copy__map__comp0,axiom,
! [F5: $tType,D: $tType,B: $tType,A: $tType,C: $tType,E: $tType,Rep: A > B,Abs: B > A,M: C > D,M1: B > D,M2: C > B,F2: D > F5,G2: E > C] :
( ( type_definition @ A @ B @ Rep @ Abs @ ( top_top @ ( set @ B ) ) )
=> ( ( M
= ( comp @ B @ D @ C @ M1 @ M2 ) )
=> ( ( comp @ C @ F5 @ E @ ( comp @ D @ F5 @ C @ F2 @ M ) @ G2 )
= ( comp @ A @ F5 @ E @ ( comp @ B @ F5 @ A @ ( comp @ D @ F5 @ B @ F2 @ M1 ) @ Rep ) @ ( comp @ C @ A @ E @ ( comp @ B @ A @ C @ Abs @ M2 ) @ G2 ) ) ) ) ) ).
% type_copy_map_comp0
thf(fact_141_subtr_OStep,axiom,
! [Tr3: dtree,Ns: set @ n,Tr1: dtree,Tr22: dtree] :
( ( member @ n @ ( root @ Tr3 ) @ Ns )
=> ( ( gram_L716654942_subtr @ Ns @ Tr1 @ Tr22 )
=> ( ( member @ ( sum_sum @ t @ dtree ) @ ( sum_Inr @ dtree @ t @ Tr22 ) @ ( cont @ Tr3 ) )
=> ( gram_L716654942_subtr @ Ns @ Tr1 @ Tr3 ) ) ) ) ).
% subtr.Step
thf(fact_142_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_143_subtr_Osimps,axiom,
( gram_L716654942_subtr
= ( ^ [A12: set @ n,A23: dtree,A33: dtree] :
( ? [Tr2: dtree,Ns2: set @ n] :
( ( A12 = Ns2 )
& ( A23 = Tr2 )
& ( A33 = Tr2 )
& ( member @ n @ ( root @ Tr2 ) @ Ns2 ) )
| ? [Tr33: dtree,Ns2: set @ n,Tr13: dtree,Tr24: dtree] :
( ( A12 = Ns2 )
& ( A23 = Tr13 )
& ( A33 = Tr33 )
& ( member @ n @ ( root @ Tr33 ) @ Ns2 )
& ( gram_L716654942_subtr @ Ns2 @ Tr13 @ Tr24 )
& ( member @ ( sum_sum @ t @ dtree ) @ ( sum_Inr @ dtree @ t @ Tr24 ) @ ( cont @ Tr33 ) ) ) ) ) ) ).
% subtr.simps
thf(fact_144_subtr__StepL,axiom,
! [Tr1: dtree,Ns: set @ n,Tr22: dtree,Tr3: dtree] :
( ( member @ n @ ( root @ Tr1 ) @ Ns )
=> ( ( member @ ( sum_sum @ t @ dtree ) @ ( sum_Inr @ dtree @ t @ Tr1 ) @ ( cont @ Tr22 ) )
=> ( ( gram_L716654942_subtr @ Ns @ Tr22 @ Tr3 )
=> ( gram_L716654942_subtr @ Ns @ Tr1 @ Tr3 ) ) ) ) ).
% subtr_StepL
thf(fact_145_subtr_Oinducts,axiom,
! [X1: set @ n,X22: dtree,X32: dtree,P: ( set @ n ) > dtree > dtree > $o] :
( ( gram_L716654942_subtr @ X1 @ X22 @ X32 )
=> ( ! [Tr7: dtree,Ns4: set @ n] :
( ( member @ n @ ( root @ Tr7 ) @ Ns4 )
=> ( P @ Ns4 @ Tr7 @ Tr7 ) )
=> ( ! [Tr32: dtree,Ns4: set @ n,Tr12: dtree,Tr23: dtree] :
( ( member @ n @ ( root @ Tr32 ) @ Ns4 )
=> ( ( gram_L716654942_subtr @ Ns4 @ Tr12 @ Tr23 )
=> ( ( P @ Ns4 @ Tr12 @ Tr23 )
=> ( ( member @ ( sum_sum @ t @ dtree ) @ ( sum_Inr @ dtree @ t @ Tr23 ) @ ( cont @ Tr32 ) )
=> ( P @ Ns4 @ Tr12 @ Tr32 ) ) ) ) )
=> ( P @ X1 @ X22 @ X32 ) ) ) ) ).
% subtr.inducts
thf(fact_146_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,Tr7: dtree] : ( Phi @ Ns4 @ Tr7 @ Tr7 )
=> ( ! [Ns4: set @ n,Tr12: dtree,Tr23: dtree,Tr32: dtree] :
( ( member @ n @ ( root @ Tr12 ) @ Ns4 )
=> ( ( member @ ( sum_sum @ t @ dtree ) @ ( sum_Inr @ dtree @ t @ Tr12 ) @ ( cont @ Tr23 ) )
=> ( ( gram_L716654942_subtr @ Ns4 @ Tr23 @ Tr32 )
=> ( ( Phi @ Ns4 @ Tr23 @ Tr32 )
=> ( Phi @ Ns4 @ Tr12 @ Tr32 ) ) ) ) )
=> ( Phi @ Ns @ Tr1 @ Tr22 ) ) ) ) ).
% subtr_inductL
thf(fact_147_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_148_subtr2_OStep,axiom,
! [Tr1: dtree,Ns: set @ n,Tr22: dtree,Tr3: dtree] :
( ( member @ n @ ( root @ Tr1 ) @ Ns )
=> ( ( member @ ( sum_sum @ t @ dtree ) @ ( sum_Inr @ dtree @ t @ Tr1 ) @ ( cont @ Tr22 ) )
=> ( ( gram_L1283001940subtr2 @ Ns @ Tr22 @ Tr3 )
=> ( gram_L1283001940subtr2 @ Ns @ Tr1 @ Tr3 ) ) ) ) ).
% subtr2.Step
thf(fact_149_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_150_subtr2_Osimps,axiom,
( gram_L1283001940subtr2
= ( ^ [A12: set @ n,A23: dtree,A33: dtree] :
( ? [Tr2: dtree,Ns2: set @ n] :
( ( A12 = Ns2 )
& ( A23 = Tr2 )
& ( A33 = Tr2 )
& ( member @ n @ ( root @ Tr2 ) @ Ns2 ) )
| ? [Tr13: dtree,Ns2: set @ n,Tr24: dtree,Tr33: dtree] :
( ( A12 = Ns2 )
& ( A23 = Tr13 )
& ( A33 = Tr33 )
& ( member @ n @ ( root @ Tr13 ) @ Ns2 )
& ( member @ ( sum_sum @ t @ dtree ) @ ( sum_Inr @ dtree @ t @ Tr13 ) @ ( cont @ Tr24 ) )
& ( gram_L1283001940subtr2 @ Ns2 @ Tr24 @ Tr33 ) ) ) ) ) ).
% subtr2.simps
thf(fact_151_subtr2__StepR,axiom,
! [Tr3: dtree,Ns: set @ n,Tr22: dtree,Tr1: dtree] :
( ( member @ n @ ( root @ Tr3 ) @ Ns )
=> ( ( member @ ( sum_sum @ t @ dtree ) @ ( sum_Inr @ dtree @ t @ Tr22 ) @ ( cont @ Tr3 ) )
=> ( ( gram_L1283001940subtr2 @ Ns @ Tr1 @ Tr22 )
=> ( gram_L1283001940subtr2 @ Ns @ Tr1 @ Tr3 ) ) ) ) ).
% subtr2_StepR
thf(fact_152_subtr2_Oinducts,axiom,
! [X1: set @ n,X22: dtree,X32: dtree,P: ( set @ n ) > dtree > dtree > $o] :
( ( gram_L1283001940subtr2 @ X1 @ X22 @ X32 )
=> ( ! [Tr7: dtree,Ns4: set @ n] :
( ( member @ n @ ( root @ Tr7 ) @ Ns4 )
=> ( P @ Ns4 @ Tr7 @ Tr7 ) )
=> ( ! [Tr12: dtree,Ns4: set @ n,Tr23: dtree,Tr32: dtree] :
( ( member @ n @ ( root @ Tr12 ) @ Ns4 )
=> ( ( member @ ( sum_sum @ t @ dtree ) @ ( sum_Inr @ dtree @ t @ Tr12 ) @ ( cont @ Tr23 ) )
=> ( ( gram_L1283001940subtr2 @ Ns4 @ Tr23 @ Tr32 )
=> ( ( P @ Ns4 @ Tr23 @ Tr32 )
=> ( P @ Ns4 @ Tr12 @ Tr32 ) ) ) ) )
=> ( P @ X1 @ X22 @ X32 ) ) ) ) ).
% subtr2.inducts
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 )
=> ! [Tr12: dtree] :
( ( member @ ( sum_sum @ t @ dtree ) @ ( sum_Inr @ dtree @ t @ Tr12 ) @ ( cont @ A22 ) )
=> ~ ( gram_L830233218_inItr @ A1 @ Tr12 @ A32 ) ) ) ) ) ).
% inItr.cases
thf(fact_155_inItr_Osimps,axiom,
( gram_L830233218_inItr
= ( ^ [A12: set @ n,A23: dtree,A33: n] :
( ? [Tr2: dtree,Ns2: set @ n] :
( ( A12 = Ns2 )
& ( A23 = Tr2 )
& ( A33
= ( root @ Tr2 ) )
& ( member @ n @ ( root @ Tr2 ) @ Ns2 ) )
| ? [Tr2: dtree,Ns2: set @ n,Tr13: dtree,N2: n] :
( ( A12 = Ns2 )
& ( A23 = Tr2 )
& ( A33 = N2 )
& ( member @ n @ ( root @ Tr2 ) @ Ns2 )
& ( member @ ( sum_sum @ t @ dtree ) @ ( sum_Inr @ dtree @ t @ Tr13 ) @ ( cont @ Tr2 ) )
& ( gram_L830233218_inItr @ Ns2 @ Tr13 @ N2 ) ) ) ) ) ).
% inItr.simps
thf(fact_156_inItr_Oinducts,axiom,
! [X1: set @ n,X22: dtree,X32: n,P: ( set @ n ) > dtree > n > $o] :
( ( gram_L830233218_inItr @ X1 @ X22 @ X32 )
=> ( ! [Tr7: dtree,Ns4: set @ n] :
( ( member @ n @ ( root @ Tr7 ) @ Ns4 )
=> ( P @ Ns4 @ Tr7 @ ( root @ Tr7 ) ) )
=> ( ! [Tr7: dtree,Ns4: set @ n,Tr12: dtree,N4: n] :
( ( member @ n @ ( root @ Tr7 ) @ Ns4 )
=> ( ( member @ ( sum_sum @ t @ dtree ) @ ( sum_Inr @ dtree @ t @ Tr12 ) @ ( cont @ Tr7 ) )
=> ( ( gram_L830233218_inItr @ Ns4 @ Tr12 @ N4 )
=> ( ( P @ Ns4 @ Tr12 @ N4 )
=> ( P @ Ns4 @ Tr7 @ N4 ) ) ) ) )
=> ( P @ X1 @ X22 @ X32 ) ) ) ) ).
% inItr.inducts
thf(fact_157_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_158_type__copy__Rep__o__Abs,axiom,
! [A: $tType,B: $tType,Rep: A > B,Abs: B > A] :
( ( type_definition @ A @ B @ Rep @ Abs @ ( top_top @ ( set @ B ) ) )
=> ( ( comp @ A @ B @ B @ Rep @ Abs )
= ( id @ B ) ) ) ).
% type_copy_Rep_o_Abs
thf(fact_159_subtrOf__root,axiom,
! [Tr: dtree,Tr6: dtree] :
( ( gram_L864798063lle_wf @ Tr )
=> ( ( member @ ( sum_sum @ t @ dtree ) @ ( sum_Inr @ dtree @ t @ Tr6 ) @ ( cont @ Tr ) )
=> ( ( gram_L1614515765ubtrOf @ Tr @ ( root @ Tr6 ) )
= Tr6 ) ) ) ).
% subtrOf_root
thf(fact_160_inFrr__def,axiom,
( gram_L805317505_inFrr
= ( ^ [Ns2: set @ n,Tr2: dtree,T3: t] :
? [Tr4: dtree] :
( ( member @ ( sum_sum @ t @ dtree ) @ ( sum_Inr @ dtree @ t @ Tr4 ) @ ( cont @ Tr2 ) )
& ( gram_L1333338417e_inFr @ Ns2 @ Tr4 @ T3 ) ) ) ) ).
% inFrr_def
thf(fact_161_sum_Oinject_I2_J,axiom,
! [A: $tType,B: $tType,X22: B,Y22: B] :
( ( ( sum_Inr @ B @ A @ X22 )
= ( sum_Inr @ B @ A @ Y22 ) )
= ( X22 = Y22 ) ) ).
% sum.inject(2)
thf(fact_162_old_Osum_Oinject_I2_J,axiom,
! [A: $tType,B: $tType,B2: B,B7: B] :
( ( ( sum_Inr @ B @ A @ B2 )
= ( sum_Inr @ B @ A @ B7 ) )
= ( B2 = B7 ) ) ).
% old.sum.inject(2)
thf(fact_163_Inr__inject,axiom,
! [A: $tType,B: $tType,X3: B,Y4: B] :
( ( ( sum_Inr @ B @ A @ X3 )
= ( sum_Inr @ B @ A @ Y4 ) )
=> ( X3 = Y4 ) ) ).
% Inr_inject
thf(fact_164_top__empty__eq,axiom,
! [A: $tType] :
( ( top_top @ ( A > $o ) )
= ( ^ [X: A] : ( member @ A @ X @ ( top_top @ ( set @ A ) ) ) ) ) ).
% top_empty_eq
thf(fact_165_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_166_inFr2_OInd,axiom,
! [Tr1: dtree,Tr: dtree,Ns1: set @ n,T2: t] :
( ( member @ ( sum_sum @ t @ dtree ) @ ( sum_Inr @ dtree @ t @ Tr1 ) @ ( cont @ Tr ) )
=> ( ( gram_L805317441_inFr2 @ Ns1 @ Tr1 @ T2 )
=> ( gram_L805317441_inFr2 @ ( insert @ n @ ( root @ Tr ) @ Ns1 ) @ Tr @ T2 ) ) ) ).
% inFr2.Ind
thf(fact_167_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_168_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_169_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_170_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_171_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_172_subset__insertI,axiom,
! [A: $tType,B3: set @ A,A3: A] : ( ord_less_eq @ ( set @ A ) @ B3 @ ( insert @ A @ A3 @ B3 ) ) ).
% subset_insertI
thf(fact_173_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_174_insert__mono,axiom,
! [A: $tType,C3: set @ A,D3: set @ A,A3: A] :
( ( ord_less_eq @ ( set @ A ) @ C3 @ D3 )
=> ( ord_less_eq @ ( set @ A ) @ ( insert @ A @ A3 @ C3 ) @ ( insert @ A @ A3 @ D3 ) ) ) ).
% insert_mono
thf(fact_175_mk__disjoint__insert,axiom,
! [A: $tType,A3: A,A2: set @ A] :
( ( member @ A @ A3 @ A2 )
=> ? [B8: set @ A] :
( ( A2
= ( insert @ A @ A3 @ B8 ) )
& ~ ( member @ A @ A3 @ B8 ) ) ) ).
% mk_disjoint_insert
thf(fact_176_insert__commute,axiom,
! [A: $tType,X3: A,Y4: A,A2: set @ A] :
( ( insert @ A @ X3 @ ( insert @ A @ Y4 @ A2 ) )
= ( insert @ A @ Y4 @ ( insert @ A @ X3 @ A2 ) ) ) ).
% insert_commute
thf(fact_177_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 )
=> ? [C4: set @ A] :
( ( A2
= ( insert @ A @ B2 @ C4 ) )
& ~ ( member @ A @ B2 @ C4 )
& ( B3
= ( insert @ A @ A3 @ C4 ) )
& ~ ( member @ A @ A3 @ C4 ) ) ) ) ) ) ) ).
% insert_eq_iff
thf(fact_178_insert__absorb,axiom,
! [A: $tType,A3: A,A2: set @ A] :
( ( member @ A @ A3 @ A2 )
=> ( ( insert @ A @ A3 @ A2 )
= A2 ) ) ).
% insert_absorb
thf(fact_179_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_180_Set_Oset__insert,axiom,
! [A: $tType,X3: A,A2: set @ A] :
( ( member @ A @ X3 @ A2 )
=> ~ ! [B8: set @ A] :
( ( A2
= ( insert @ A @ X3 @ B8 ) )
=> ( member @ A @ X3 @ B8 ) ) ) ).
% Set.set_insert
thf(fact_181_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_182_insertI1,axiom,
! [A: $tType,A3: A,B3: set @ A] : ( member @ A @ A3 @ ( insert @ A @ A3 @ B3 ) ) ).
% insertI1
thf(fact_183_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_184_insert__UNIV,axiom,
! [A: $tType,X3: A] :
( ( insert @ A @ X3 @ ( top_top @ ( set @ A ) ) )
= ( top_top @ ( set @ A ) ) ) ).
% insert_UNIV
thf(fact_185_inFr__Ind__minus,axiom,
! [Ns1: set @ n,Tr1: dtree,T2: t,Tr: dtree] :
( ( gram_L1333338417e_inFr @ Ns1 @ Tr1 @ T2 )
=> ( ( member @ ( sum_sum @ t @ dtree ) @ ( sum_Inr @ dtree @ t @ Tr1 ) @ ( cont @ Tr ) )
=> ( gram_L1333338417e_inFr @ ( insert @ n @ ( root @ Tr ) @ Ns1 ) @ Tr @ T2 ) ) ) ).
% inFr_Ind_minus
thf(fact_186_insert__subsetI,axiom,
! [A: $tType,X3: A,A2: set @ A,X6: set @ A] :
( ( member @ A @ X3 @ A2 )
=> ( ( ord_less_eq @ ( set @ A ) @ X6 @ A2 )
=> ( ord_less_eq @ ( set @ A ) @ ( insert @ A @ X3 @ X6 ) @ A2 ) ) ) ).
% insert_subsetI
thf(fact_187_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 ) ) )
=> ~ ! [Tr12: dtree,Tr7: dtree,Ns12: set @ n] :
( ( A1
= ( insert @ n @ ( root @ Tr7 ) @ Ns12 ) )
=> ( ( A22 = Tr7 )
=> ( ( member @ ( sum_sum @ t @ dtree ) @ ( sum_Inr @ dtree @ t @ Tr12 ) @ ( cont @ Tr7 ) )
=> ~ ( gram_L805317441_inFr2 @ Ns12 @ Tr12 @ A32 ) ) ) ) ) ) ).
% inFr2.cases
thf(fact_188_inFr2_Osimps,axiom,
( gram_L805317441_inFr2
= ( ^ [A12: set @ n,A23: dtree,A33: t] :
( ? [Tr2: dtree,Ns2: set @ n,T3: t] :
( ( A12 = Ns2 )
& ( A23 = Tr2 )
& ( A33 = T3 )
& ( member @ n @ ( root @ Tr2 ) @ Ns2 )
& ( member @ ( sum_sum @ t @ dtree ) @ ( sum_Inl @ t @ dtree @ T3 ) @ ( cont @ Tr2 ) ) )
| ? [Tr13: dtree,Tr2: dtree,Ns13: set @ n,T3: t] :
( ( A12
= ( insert @ n @ ( root @ Tr2 ) @ Ns13 ) )
& ( A23 = Tr2 )
& ( A33 = T3 )
& ( member @ ( sum_sum @ t @ dtree ) @ ( sum_Inr @ dtree @ t @ Tr13 ) @ ( cont @ Tr2 ) )
& ( gram_L805317441_inFr2 @ Ns13 @ Tr13 @ T3 ) ) ) ) ) ).
% inFr2.simps
thf(fact_189_Inl__Inr__False,axiom,
! [A: $tType,B: $tType,X3: A,Y4: B] :
( ( sum_Inl @ A @ B @ X3 )
!= ( sum_Inr @ B @ A @ Y4 ) ) ).
% Inl_Inr_False
thf(fact_190_Inr__Inl__False,axiom,
! [B: $tType,A: $tType,X3: B,Y4: A] :
( ( sum_Inr @ B @ A @ X3 )
!= ( sum_Inl @ A @ B @ Y4 ) ) ).
% Inr_Inl_False
thf(fact_191_obj__sumE__f,axiom,
! [A: $tType,C: $tType,B: $tType,S2: B,F2: ( sum_sum @ A @ C ) > B,P: $o] :
( ! [X2: A] :
( ( S2
= ( F2 @ ( sum_Inl @ A @ C @ X2 ) ) )
=> P )
=> ( ! [X2: C] :
( ( S2
= ( F2 @ ( sum_Inr @ C @ A @ X2 ) ) )
=> P )
=> ! [X7: sum_sum @ A @ C] :
( ( S2
= ( F2 @ X7 ) )
=> P ) ) ) ).
% obj_sumE_f
thf(fact_192_old_Osum_Oinducts,axiom,
! [B: $tType,A: $tType,P: ( sum_sum @ A @ B ) > $o,Sum: sum_sum @ A @ B] :
( ! [A4: A] : ( P @ ( sum_Inl @ A @ B @ A4 ) )
=> ( ! [B4: B] : ( P @ ( sum_Inr @ B @ A @ B4 ) )
=> ( P @ Sum ) ) ) ).
% old.sum.inducts
thf(fact_193_old_Osum_Oexhaust,axiom,
! [A: $tType,B: $tType,Y4: sum_sum @ A @ B] :
( ! [A4: A] :
( Y4
!= ( sum_Inl @ A @ B @ A4 ) )
=> ~ ! [B4: B] :
( Y4
!= ( sum_Inr @ B @ A @ B4 ) ) ) ).
% old.sum.exhaust
thf(fact_194_split__sum__all,axiom,
! [B: $tType,A: $tType] :
( ( ^ [P2: ( sum_sum @ A @ B ) > $o] :
! [X5: sum_sum @ A @ B] : ( P2 @ X5 ) )
= ( ^ [P3: ( sum_sum @ A @ B ) > $o] :
( ! [X: A] : ( P3 @ ( sum_Inl @ A @ B @ X ) )
& ! [X: B] : ( P3 @ ( sum_Inr @ B @ A @ X ) ) ) ) ) ).
% split_sum_all
thf(fact_195_split__sum__ex,axiom,
! [B: $tType,A: $tType] :
( ( ^ [P2: ( sum_sum @ A @ B ) > $o] :
? [X5: sum_sum @ A @ B] : ( P2 @ X5 ) )
= ( ^ [P3: ( sum_sum @ A @ B ) > $o] :
( ? [X: A] : ( P3 @ ( sum_Inl @ A @ B @ X ) )
| ? [X: B] : ( P3 @ ( sum_Inr @ B @ A @ X ) ) ) ) ) ).
% split_sum_ex
thf(fact_196_Inr__not__Inl,axiom,
! [B: $tType,A: $tType,B2: B,A3: A] :
( ( sum_Inr @ B @ A @ B2 )
!= ( sum_Inl @ A @ B @ A3 ) ) ).
% Inr_not_Inl
thf(fact_197_sumE,axiom,
! [A: $tType,B: $tType,S2: sum_sum @ A @ B] :
( ! [X2: A] :
( S2
!= ( sum_Inl @ A @ B @ X2 ) )
=> ~ ! [Y: B] :
( S2
!= ( sum_Inr @ B @ A @ Y ) ) ) ).
% sumE
thf(fact_198_old_Osum_Odistinct_I1_J,axiom,
! [A: $tType,B: $tType,A3: A,B7: B] :
( ( sum_Inl @ A @ B @ A3 )
!= ( sum_Inr @ B @ A @ B7 ) ) ).
% old.sum.distinct(1)
thf(fact_199_old_Osum_Odistinct_I2_J,axiom,
! [B9: $tType,A7: $tType,B10: B9,A8: A7] :
( ( sum_Inr @ B9 @ A7 @ B10 )
!= ( sum_Inl @ A7 @ B9 @ A8 ) ) ).
% old.sum.distinct(2)
thf(fact_200_sum_Odistinct_I1_J,axiom,
! [A: $tType,B: $tType,X1: A,X22: B] :
( ( sum_Inl @ A @ B @ X1 )
!= ( sum_Inr @ B @ A @ X22 ) ) ).
% sum.distinct(1)
thf(fact_201_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_202_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_203_inFr__subtr,axiom,
! [Ns: set @ n,Tr: dtree,T2: t] :
( ( gram_L1333338417e_inFr @ Ns @ Tr @ T2 )
=> ? [Tr5: dtree] :
( ( gram_L716654942_subtr @ Ns @ Tr5 @ Tr )
& ( member @ ( sum_sum @ t @ dtree ) @ ( sum_Inl @ t @ dtree @ T2 ) @ ( cont @ Tr5 ) ) ) ) ).
% inFr_subtr
thf(fact_204_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 )
=> ! [Tr12: dtree] :
( ( member @ ( sum_sum @ t @ dtree ) @ ( sum_Inr @ dtree @ t @ Tr12 ) @ ( cont @ A22 ) )
=> ~ ( gram_L1333338417e_inFr @ A1 @ Tr12 @ A32 ) ) ) ) ) ).
% inFr.cases
thf(fact_205_inFr_Osimps,axiom,
( gram_L1333338417e_inFr
= ( ^ [A12: set @ n,A23: dtree,A33: t] :
( ? [Tr2: dtree,Ns2: set @ n,T3: t] :
( ( A12 = Ns2 )
& ( A23 = Tr2 )
& ( A33 = T3 )
& ( member @ n @ ( root @ Tr2 ) @ Ns2 )
& ( member @ ( sum_sum @ t @ dtree ) @ ( sum_Inl @ t @ dtree @ T3 ) @ ( cont @ Tr2 ) ) )
| ? [Tr2: dtree,Ns2: set @ n,Tr13: dtree,T3: t] :
( ( A12 = Ns2 )
& ( A23 = Tr2 )
& ( A33 = T3 )
& ( member @ n @ ( root @ Tr2 ) @ Ns2 )
& ( member @ ( sum_sum @ t @ dtree ) @ ( sum_Inr @ dtree @ t @ Tr13 ) @ ( cont @ Tr2 ) )
& ( gram_L1333338417e_inFr @ Ns2 @ Tr13 @ T3 ) ) ) ) ) ).
% inFr.simps
thf(fact_206_inFr_Oinducts,axiom,
! [X1: set @ n,X22: dtree,X32: t,P: ( set @ n ) > dtree > t > $o] :
( ( gram_L1333338417e_inFr @ X1 @ X22 @ X32 )
=> ( ! [Tr7: dtree,Ns4: set @ n,T4: t] :
( ( member @ n @ ( root @ Tr7 ) @ Ns4 )
=> ( ( member @ ( sum_sum @ t @ dtree ) @ ( sum_Inl @ t @ dtree @ T4 ) @ ( cont @ Tr7 ) )
=> ( P @ Ns4 @ Tr7 @ T4 ) ) )
=> ( ! [Tr7: dtree,Ns4: set @ n,Tr12: dtree,T4: t] :
( ( member @ n @ ( root @ Tr7 ) @ Ns4 )
=> ( ( member @ ( sum_sum @ t @ dtree ) @ ( sum_Inr @ dtree @ t @ Tr12 ) @ ( cont @ Tr7 ) )
=> ( ( gram_L1333338417e_inFr @ Ns4 @ Tr12 @ T4 )
=> ( ( P @ Ns4 @ Tr12 @ T4 )
=> ( P @ Ns4 @ Tr7 @ T4 ) ) ) ) )
=> ( P @ X1 @ X22 @ X32 ) ) ) ) ).
% inFr.inducts
thf(fact_207_inFr2_Oinducts,axiom,
! [X1: set @ n,X22: dtree,X32: t,P: ( set @ n ) > dtree > t > $o] :
( ( gram_L805317441_inFr2 @ X1 @ X22 @ X32 )
=> ( ! [Tr7: dtree,Ns4: set @ n,T4: t] :
( ( member @ n @ ( root @ Tr7 ) @ Ns4 )
=> ( ( member @ ( sum_sum @ t @ dtree ) @ ( sum_Inl @ t @ dtree @ T4 ) @ ( cont @ Tr7 ) )
=> ( P @ Ns4 @ Tr7 @ T4 ) ) )
=> ( ! [Tr12: dtree,Tr7: dtree,Ns12: set @ n,T4: t] :
( ( member @ ( sum_sum @ t @ dtree ) @ ( sum_Inr @ dtree @ t @ Tr12 ) @ ( cont @ Tr7 ) )
=> ( ( gram_L805317441_inFr2 @ Ns12 @ Tr12 @ T4 )
=> ( ( P @ Ns12 @ Tr12 @ T4 )
=> ( P @ ( insert @ n @ ( root @ Tr7 ) @ Ns12 ) @ Tr7 @ T4 ) ) ) )
=> ( P @ X1 @ X22 @ X32 ) ) ) ) ).
% inFr2.inducts
thf(fact_208_obj__sumE,axiom,
! [A: $tType,B: $tType,S2: sum_sum @ A @ B] :
( ! [X2: A] :
( S2
!= ( sum_Inl @ A @ B @ X2 ) )
=> ~ ! [X2: B] :
( S2
!= ( sum_Inr @ B @ A @ X2 ) ) ) ).
% obj_sumE
thf(fact_209_Inl__cont__hsubst__neq,axiom,
! [Tr: dtree,Tr0: dtree] :
( ( ( root @ Tr )
!= ( root @ Tr0 ) )
=> ( ( vimage @ t @ ( sum_sum @ t @ dtree ) @ ( sum_Inl @ t @ dtree ) @ ( cont @ ( gram_L1004374585hsubst @ Tr0 @ Tr ) ) )
= ( vimage @ t @ ( sum_sum @ t @ dtree ) @ ( sum_Inl @ t @ dtree ) @ ( cont @ Tr ) ) ) ) ).
% Inl_cont_hsubst_neq
thf(fact_210_vimage__eq,axiom,
! [A: $tType,B: $tType,A3: A,F2: A > B,B3: set @ B] :
( ( member @ A @ A3 @ ( vimage @ A @ B @ F2 @ B3 ) )
= ( member @ B @ ( F2 @ A3 ) @ B3 ) ) ).
% vimage_eq
thf(fact_211_vimageI,axiom,
! [B: $tType,A: $tType,F2: B > A,A3: B,B2: A,B3: set @ A] :
( ( ( F2 @ A3 )
= B2 )
=> ( ( member @ A @ B2 @ B3 )
=> ( member @ B @ A3 @ ( vimage @ B @ A @ F2 @ B3 ) ) ) ) ).
% vimageI
thf(fact_212_vimage__UNIV,axiom,
! [B: $tType,A: $tType,F2: A > B] :
( ( vimage @ A @ B @ F2 @ ( top_top @ ( set @ B ) ) )
= ( top_top @ ( set @ A ) ) ) ).
% vimage_UNIV
thf(fact_213_vimage__id,axiom,
! [A: $tType] :
( ( vimage @ A @ A @ ( id @ A ) )
= ( id @ ( set @ A ) ) ) ).
% vimage_id
thf(fact_214_Inl__cont__hsubst__eq,axiom,
! [Tr: dtree,Tr0: dtree] :
( ( ( root @ Tr )
= ( root @ Tr0 ) )
=> ( ( vimage @ t @ ( sum_sum @ t @ dtree ) @ ( sum_Inl @ t @ dtree ) @ ( cont @ ( gram_L1004374585hsubst @ Tr0 @ Tr ) ) )
= ( vimage @ t @ ( sum_sum @ t @ dtree ) @ ( sum_Inl @ t @ dtree ) @ ( cont @ Tr0 ) ) ) ) ).
% Inl_cont_hsubst_eq
thf(fact_215_set_Ocompositionality,axiom,
! [C: $tType,B: $tType,A: $tType,F2: C > B,G2: B > A,Set: set @ A] :
( ( vimage @ C @ B @ F2 @ ( vimage @ B @ A @ G2 @ Set ) )
= ( vimage @ C @ A @ ( comp @ B @ A @ C @ G2 @ F2 ) @ Set ) ) ).
% set.compositionality
thf(fact_216_vimage__comp,axiom,
! [A: $tType,B: $tType,C: $tType,F2: A > B,G2: B > C,X3: set @ C] :
( ( vimage @ A @ B @ F2 @ ( vimage @ B @ C @ G2 @ X3 ) )
= ( vimage @ A @ C @ ( comp @ B @ C @ A @ G2 @ F2 ) @ X3 ) ) ).
% vimage_comp
thf(fact_217_set_Ocomp,axiom,
! [A: $tType,B: $tType,C: $tType,F2: C > B,G2: B > A] :
( ( comp @ ( set @ B ) @ ( set @ C ) @ ( set @ A ) @ ( vimage @ C @ B @ F2 ) @ ( vimage @ B @ A @ G2 ) )
= ( vimage @ C @ A @ ( comp @ B @ A @ C @ G2 @ F2 ) ) ) ).
% set.comp
thf(fact_218_vimage__mono,axiom,
! [B: $tType,A: $tType,A2: set @ A,B3: set @ A,F2: B > A] :
( ( ord_less_eq @ ( set @ A ) @ A2 @ B3 )
=> ( ord_less_eq @ ( set @ B ) @ ( vimage @ B @ A @ F2 @ A2 ) @ ( vimage @ B @ A @ F2 @ B3 ) ) ) ).
% vimage_mono
thf(fact_219_vimage__Collect,axiom,
! [B: $tType,A: $tType,P: B > $o,F2: A > B,Q: A > $o] :
( ! [X2: A] :
( ( P @ ( F2 @ X2 ) )
= ( Q @ X2 ) )
=> ( ( vimage @ A @ B @ F2 @ ( collect @ B @ P ) )
= ( collect @ A @ Q ) ) ) ).
% vimage_Collect
thf(fact_220_vimageI2,axiom,
! [B: $tType,A: $tType,F2: B > A,A3: B,A2: set @ A] :
( ( member @ A @ ( F2 @ A3 ) @ A2 )
=> ( member @ B @ A3 @ ( vimage @ B @ A @ F2 @ A2 ) ) ) ).
% vimageI2
thf(fact_221_vimageE,axiom,
! [A: $tType,B: $tType,A3: A,F2: A > B,B3: set @ B] :
( ( member @ A @ A3 @ ( vimage @ A @ B @ F2 @ B3 ) )
=> ( member @ B @ ( F2 @ A3 ) @ B3 ) ) ).
% vimageE
thf(fact_222_vimageD,axiom,
! [A: $tType,B: $tType,A3: A,F2: A > B,A2: set @ B] :
( ( member @ A @ A3 @ ( vimage @ A @ B @ F2 @ A2 ) )
=> ( member @ B @ ( F2 @ A3 ) @ A2 ) ) ).
% vimageD
thf(fact_223_not__arg__cong__Inr,axiom,
! [B: $tType,A: $tType,X3: A,Y4: A] :
( ( X3 != Y4 )
=> ( ( sum_Inr @ A @ B @ X3 )
!= ( sum_Inr @ A @ B @ Y4 ) ) ) ).
% not_arg_cong_Inr
thf(fact_224_Inl__Inr__image__cong,axiom,
! [B: $tType,A: $tType,A2: set @ ( sum_sum @ A @ B ),B3: set @ ( sum_sum @ A @ B )] :
( ( ( vimage @ A @ ( sum_sum @ A @ B ) @ ( sum_Inl @ A @ B ) @ A2 )
= ( vimage @ A @ ( sum_sum @ A @ B ) @ ( sum_Inl @ A @ B ) @ B3 ) )
=> ( ( ( vimage @ B @ ( sum_sum @ A @ B ) @ ( sum_Inr @ B @ A ) @ A2 )
= ( vimage @ B @ ( sum_sum @ A @ B ) @ ( sum_Inr @ B @ A ) @ B3 ) )
=> ( A2 = B3 ) ) ) ).
% Inl_Inr_image_cong
thf(fact_225_inFr__hsubst__imp,axiom,
! [Ns: set @ n,Tr0: dtree,Tr: dtree,T2: t] :
( ( gram_L1333338417e_inFr @ Ns @ ( gram_L1004374585hsubst @ Tr0 @ Tr ) @ T2 )
=> ( ( member @ t @ T2 @ ( vimage @ t @ ( sum_sum @ t @ dtree ) @ ( sum_Inl @ t @ dtree ) @ ( cont @ Tr0 ) ) )
| ( gram_L805317505_inFrr @ ( minus_minus @ ( set @ n ) @ Ns @ ( insert @ n @ ( root @ Tr0 ) @ ( bot_bot @ ( set @ n ) ) ) ) @ Tr0 @ T2 )
| ( gram_L1333338417e_inFr @ ( minus_minus @ ( set @ n ) @ Ns @ ( insert @ n @ ( root @ Tr0 ) @ ( bot_bot @ ( set @ n ) ) ) ) @ Tr @ T2 ) ) ) ).
% inFr_hsubst_imp
thf(fact_226_bot__apply,axiom,
! [C: $tType,D: $tType] :
( ( bot @ C @ ( type2 @ C ) )
=> ( ( bot_bot @ ( D > C ) )
= ( ^ [X: D] : ( bot_bot @ C ) ) ) ) ).
% bot_apply
thf(fact_227_empty__iff,axiom,
! [A: $tType,C2: A] :
~ ( member @ A @ C2 @ ( bot_bot @ ( set @ A ) ) ) ).
% empty_iff
thf(fact_228_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_229_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_230_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_231_DiffI,axiom,
! [A: $tType,C2: A,A2: set @ A,B3: set @ A] :
( ( member @ A @ C2 @ A2 )
=> ( ~ ( member @ A @ C2 @ B3 )
=> ( member @ A @ C2 @ ( minus_minus @ ( set @ A ) @ A2 @ B3 ) ) ) ) ).
% DiffI
thf(fact_232_Diff__iff,axiom,
! [A: $tType,C2: A,A2: set @ A,B3: set @ A] :
( ( member @ A @ C2 @ ( minus_minus @ ( set @ A ) @ A2 @ B3 ) )
= ( ( member @ A @ C2 @ A2 )
& ~ ( member @ A @ C2 @ B3 ) ) ) ).
% Diff_iff
thf(fact_233_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_234_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_235_empty__subsetI,axiom,
! [A: $tType,A2: set @ A] : ( ord_less_eq @ ( set @ A ) @ ( bot_bot @ ( set @ A ) ) @ A2 ) ).
% empty_subsetI
thf(fact_236_singletonI,axiom,
! [A: $tType,A3: A] : ( member @ A @ A3 @ ( insert @ A @ A3 @ ( bot_bot @ ( set @ A ) ) ) ) ).
% singletonI
thf(fact_237_Diff__empty,axiom,
! [A: $tType,A2: set @ A] :
( ( minus_minus @ ( set @ A ) @ A2 @ ( bot_bot @ ( set @ A ) ) )
= A2 ) ).
% Diff_empty
thf(fact_238_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_239_Diff__cancel,axiom,
! [A: $tType,A2: set @ A] :
( ( minus_minus @ ( set @ A ) @ A2 @ A2 )
= ( bot_bot @ ( set @ A ) ) ) ).
% Diff_cancel
thf(fact_240_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_241_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_242_vimage__empty,axiom,
! [B: $tType,A: $tType,F2: A > B] :
( ( vimage @ A @ B @ F2 @ ( bot_bot @ ( set @ B ) ) )
= ( bot_bot @ ( set @ A ) ) ) ).
% vimage_empty
thf(fact_243_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_244_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_245_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_246_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_247_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_248_vimage__singleton__eq,axiom,
! [A: $tType,B: $tType,A3: A,F2: A > B,B2: B] :
( ( member @ A @ A3 @ ( vimage @ A @ B @ F2 @ ( insert @ B @ B2 @ ( bot_bot @ ( set @ B ) ) ) ) )
= ( ( F2 @ A3 )
= B2 ) ) ).
% vimage_singleton_eq
thf(fact_249_vimage__Diff,axiom,
! [A: $tType,B: $tType,F2: A > B,A2: set @ B,B3: set @ B] :
( ( vimage @ A @ B @ F2 @ ( minus_minus @ ( set @ B ) @ A2 @ B3 ) )
= ( minus_minus @ ( set @ A ) @ ( vimage @ A @ B @ F2 @ A2 ) @ ( vimage @ A @ B @ F2 @ B3 ) ) ) ).
% vimage_Diff
thf(fact_250_inFr__hsubst__minus,axiom,
! [Ns: set @ n,Tr0: dtree,Tr: dtree,T2: t] :
( ( gram_L1333338417e_inFr @ ( minus_minus @ ( set @ n ) @ Ns @ ( insert @ n @ ( root @ Tr0 ) @ ( bot_bot @ ( set @ n ) ) ) ) @ Tr @ T2 )
=> ( gram_L1333338417e_inFr @ Ns @ ( gram_L1004374585hsubst @ Tr0 @ Tr ) @ T2 ) ) ).
% inFr_hsubst_minus
thf(fact_251_empty__not__UNIV,axiom,
! [A: $tType] :
( ( bot_bot @ ( set @ A ) )
!= ( top_top @ ( set @ A ) ) ) ).
% empty_not_UNIV
thf(fact_252_DiffE,axiom,
! [A: $tType,C2: A,A2: set @ A,B3: set @ A] :
( ( member @ A @ C2 @ ( minus_minus @ ( set @ A ) @ A2 @ B3 ) )
=> ~ ( ( member @ A @ C2 @ A2 )
=> ( member @ A @ C2 @ B3 ) ) ) ).
% DiffE
thf(fact_253_DiffD1,axiom,
! [A: $tType,C2: A,A2: set @ A,B3: set @ A] :
( ( member @ A @ C2 @ ( minus_minus @ ( set @ A ) @ A2 @ B3 ) )
=> ( member @ A @ C2 @ A2 ) ) ).
% DiffD1
thf(fact_254_DiffD2,axiom,
! [A: $tType,C2: A,A2: set @ A,B3: set @ A] :
( ( member @ A @ C2 @ ( minus_minus @ ( set @ A ) @ A2 @ B3 ) )
=> ~ ( member @ A @ C2 @ B3 ) ) ).
% DiffD2
thf(fact_255_emptyE,axiom,
! [A: $tType,A3: A] :
~ ( member @ A @ A3 @ ( bot_bot @ ( set @ A ) ) ) ).
% emptyE
%----Type constructors (19)
thf(tcon_fun___Orderings_Oorder__top,axiom,
! [A7: $tType,A9: $tType] :
( ( order_top @ A9 @ ( type2 @ A9 ) )
=> ( order_top @ ( A7 > A9 ) @ ( type2 @ ( A7 > A9 ) ) ) ) ).
thf(tcon_fun___Orderings_Opreorder,axiom,
! [A7: $tType,A9: $tType] :
( ( preorder @ A9 @ ( type2 @ A9 ) )
=> ( preorder @ ( A7 > A9 ) @ ( type2 @ ( A7 > A9 ) ) ) ) ).
thf(tcon_fun___Orderings_Oorder,axiom,
! [A7: $tType,A9: $tType] :
( ( order @ A9 @ ( type2 @ A9 ) )
=> ( order @ ( A7 > A9 ) @ ( type2 @ ( A7 > A9 ) ) ) ) ).
thf(tcon_fun___Orderings_Otop,axiom,
! [A7: $tType,A9: $tType] :
( ( top @ A9 @ ( type2 @ A9 ) )
=> ( top @ ( A7 > A9 ) @ ( type2 @ ( A7 > A9 ) ) ) ) ).
thf(tcon_fun___Orderings_Oord,axiom,
! [A7: $tType,A9: $tType] :
( ( ord @ A9 @ ( type2 @ A9 ) )
=> ( ord @ ( A7 > A9 ) @ ( type2 @ ( A7 > A9 ) ) ) ) ).
thf(tcon_fun___Orderings_Obot,axiom,
! [A7: $tType,A9: $tType] :
( ( bot @ A9 @ ( type2 @ A9 ) )
=> ( bot @ ( A7 > A9 ) @ ( type2 @ ( A7 > A9 ) ) ) ) ).
thf(tcon_Set_Oset___Orderings_Oorder__top_1,axiom,
! [A7: $tType] : ( order_top @ ( set @ A7 ) @ ( type2 @ ( set @ A7 ) ) ) ).
thf(tcon_Set_Oset___Orderings_Opreorder_2,axiom,
! [A7: $tType] : ( preorder @ ( set @ A7 ) @ ( type2 @ ( set @ A7 ) ) ) ).
thf(tcon_Set_Oset___Orderings_Oorder_3,axiom,
! [A7: $tType] : ( order @ ( set @ A7 ) @ ( type2 @ ( set @ A7 ) ) ) ).
thf(tcon_Set_Oset___Orderings_Otop_4,axiom,
! [A7: $tType] : ( top @ ( set @ A7 ) @ ( type2 @ ( set @ A7 ) ) ) ).
thf(tcon_Set_Oset___Orderings_Oord_5,axiom,
! [A7: $tType] : ( ord @ ( set @ A7 ) @ ( type2 @ ( set @ A7 ) ) ) ).
thf(tcon_Set_Oset___Orderings_Obot_6,axiom,
! [A7: $tType] : ( bot @ ( set @ A7 ) @ ( type2 @ ( set @ A7 ) ) ) ).
thf(tcon_HOL_Obool___Orderings_Oorder__top_7,axiom,
order_top @ $o @ ( type2 @ $o ) ).
thf(tcon_HOL_Obool___Orderings_Opreorder_8,axiom,
preorder @ $o @ ( type2 @ $o ) ).
thf(tcon_HOL_Obool___Orderings_Olinorder,axiom,
linorder @ $o @ ( type2 @ $o ) ).
thf(tcon_HOL_Obool___Orderings_Oorder_9,axiom,
order @ $o @ ( type2 @ $o ) ).
thf(tcon_HOL_Obool___Orderings_Otop_10,axiom,
top @ $o @ ( type2 @ $o ) ).
thf(tcon_HOL_Obool___Orderings_Oord_11,axiom,
ord @ $o @ ( type2 @ $o ) ).
thf(tcon_HOL_Obool___Orderings_Obot_12,axiom,
bot @ $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,X3: A,Y4: A] :
( ( if @ A @ $false @ X3 @ Y4 )
= Y4 ) ).
thf(help_If_1_1_T,axiom,
! [A: $tType,X3: A,Y4: A] :
( ( if @ A @ $true @ X3 @ Y4 )
= X3 ) ).
%----Conjectures (1)
thf(conj_0,conjecture,
( ( f @ ( root @ tr ) )
= tr ) ).
%------------------------------------------------------------------------------