TPTP Problem File: COM187^1.p
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%------------------------------------------------------------------------------
% File : COM187^1 : TPTP v8.2.0. Released v7.0.0.
% Domain : Computing Theory
% Problem : Grammars and languages 549
% 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__549.p [Bla16]
% Status : Theorem
% Rating : 0.00 v8.1.0, 0.25 v7.5.0, 0.00 v7.2.0, 0.25 v7.1.0
% Syntax : Number of formulae : 310 ( 109 unt; 45 typ; 0 def)
% Number of atoms : 723 ( 325 equ; 0 cnn)
% Maximal formula atoms : 7 ( 2 avg)
% Number of connectives : 4872 ( 79 ~; 12 |; 82 &;4348 @)
% ( 0 <=>; 351 =>; 0 <=; 0 <~>)
% Maximal formula depth : 21 ( 8 avg)
% Number of types : 4 ( 3 usr)
% Number of type conns : 286 ( 286 >; 0 *; 0 +; 0 <<)
% Number of symbols : 45 ( 42 usr; 5 con; 0-7 aty)
% Number of variables : 1157 ( 36 ^;1032 !; 57 ?;1157 :)
% ( 32 !>; 0 ?*; 0 @-; 0 @+)
% SPC : TH1_THM_EQU_NAR
% Comments : This file was generated by Isabelle (most likely Sledgehammer)
% 2016-07-13 14:43:13.600
%------------------------------------------------------------------------------
%----Could-be-implicit typings (7)
thf(ty_t_Product__Type_Oprod,type,
product_prod: $tType > $tType > $tType ).
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 (38)
thf(sy_cl_Finite__Set_Ofinite,type,
finite_finite:
!>[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_Odtree_Oroot,type,
root: dtree > n ).
thf(sy_c_DTree_Ounfold,type,
unfold:
!>[A: $tType] : ( ( A > n ) > ( A > ( set @ ( sum_sum @ t @ A ) ) ) > A > dtree ) ).
thf(sy_c_Finite__Set_Ofinite,type,
finite_finite2:
!>[A: $tType] : ( ( set @ A ) > $o ) ).
thf(sy_c_Fun_Ocomp,type,
comp:
!>[B: $tType,C: $tType,A: $tType] : ( ( B > C ) > ( A > B ) > A > C ) ).
thf(sy_c_Fun_Oid,type,
id:
!>[A: $tType] : ( A > A ) ).
thf(sy_c_Fun_Oinj__on,type,
inj_on:
!>[A: $tType,B: $tType] : ( ( A > B ) > ( set @ A ) > $o ) ).
thf(sy_c_Gram__Lang__Mirabelle__ojxrtuoybn_OP,type,
gram_L1451583632elle_P: set @ ( product_prod @ n @ ( set @ ( sum_sum @ t @ n ) ) ) ).
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_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_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_Product__Type_OPair,type,
product_Pair:
!>[A: $tType,B: $tType] : ( A > B > ( product_prod @ A @ B ) ) ).
thf(sy_c_Pure_Otype,type,
type:
!>[A: $tType] : ( itself @ A ) ).
thf(sy_c_Set_OCollect,type,
collect:
!>[A: $tType] : ( ( A > $o ) > ( set @ A ) ) ).
thf(sy_c_Set_Oimage,type,
image:
!>[A: $tType,B: $tType] : ( ( A > B ) > ( set @ A ) > ( set @ B ) ) ).
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_Omap__sum,type,
sum_map_sum:
!>[A: $tType,C: $tType,B: $tType,D: $tType] : ( ( A > C ) > ( B > D ) > ( sum_sum @ A @ B ) > ( sum_sum @ C @ D ) ) ).
thf(sy_c_Sum__Type_Oold_Osum_Orec__sum,type,
sum_rec_sum:
!>[A: $tType,T: $tType,B: $tType] : ( ( A > T ) > ( B > T ) > ( sum_sum @ A @ B ) > T ) ).
thf(sy_c_member,type,
member:
!>[A: $tType] : ( A > ( set @ A ) > $o ) ).
thf(sy_v_tr,type,
tr: dtree ).
thf(sy_v_tr0,type,
tr0: dtree ).
%----Relevant facts (256)
thf(fact_0__092_060open_062cont_A_Ilocal_Ohsubst_Atr_J_A_061_A_Iid_A_092_060oplus_062_Alocal_Ohsubst_J_A_096_Acont_Atr_092_060close_062,axiom,
( ( cont @ ( gram_L1004374585hsubst @ tr0 @ tr ) )
= ( image @ ( sum_sum @ t @ dtree ) @ ( sum_sum @ t @ dtree ) @ ( sum_map_sum @ t @ t @ dtree @ dtree @ ( id @ t ) @ ( gram_L1004374585hsubst @ tr0 ) ) @ ( cont @ tr ) ) ) ).
% \<open>cont (local.hsubst tr) = (id \<oplus> local.hsubst) ` cont tr\<close>
thf(fact_1_assms,axiom,
( ( root @ tr )
!= ( root @ tr0 ) ) ).
% assms
thf(fact_2_Inl__m__oplus,axiom,
! [B: $tType,C: $tType,A: $tType,F: C > B,Tns: set @ ( sum_sum @ A @ C )] :
( ( vimage @ A @ ( sum_sum @ A @ B ) @ ( sum_Inl @ A @ B ) @ ( image @ ( sum_sum @ A @ C ) @ ( sum_sum @ A @ B ) @ ( sum_map_sum @ A @ A @ C @ B @ ( id @ A ) @ F ) @ Tns ) )
= ( vimage @ A @ ( sum_sum @ A @ C ) @ ( sum_Inl @ A @ C ) @ Tns ) ) ).
% Inl_m_oplus
thf(fact_3_Inl__oplus__iff,axiom,
! [B: $tType,C: $tType,A: $tType,Tr: A,F: C > B,Tns: set @ ( sum_sum @ A @ C )] :
( ( member @ ( sum_sum @ A @ B ) @ ( sum_Inl @ A @ B @ Tr ) @ ( image @ ( sum_sum @ A @ C ) @ ( sum_sum @ A @ B ) @ ( sum_map_sum @ A @ A @ C @ B @ ( id @ A ) @ F ) @ Tns ) )
= ( member @ ( sum_sum @ A @ C ) @ ( sum_Inl @ A @ C @ Tr ) @ Tns ) ) ).
% Inl_oplus_iff
thf(fact_4_vimage__id,axiom,
! [A: $tType] :
( ( vimage @ A @ A @ ( id @ A ) )
= ( id @ ( set @ A ) ) ) ).
% vimage_id
thf(fact_5_image__id,axiom,
! [A: $tType] :
( ( image @ A @ A @ ( id @ A ) )
= ( id @ ( set @ A ) ) ) ).
% image_id
thf(fact_6_Inl__oplus__elim,axiom,
! [B: $tType,C: $tType,A: $tType,Tr: A,F: C > B,Tns: set @ ( sum_sum @ A @ C )] :
( ( member @ ( sum_sum @ A @ B ) @ ( sum_Inl @ A @ B @ Tr ) @ ( image @ ( sum_sum @ A @ C ) @ ( sum_sum @ A @ B ) @ ( sum_map_sum @ A @ A @ C @ B @ ( id @ A ) @ F ) @ Tns ) )
=> ( member @ ( sum_sum @ A @ C ) @ ( sum_Inl @ A @ C @ Tr ) @ Tns ) ) ).
% Inl_oplus_elim
thf(fact_7_cont__hsubst__eq,axiom,
! [Tr: dtree] :
( ( ( root @ Tr )
= ( root @ tr0 ) )
=> ( ( cont @ ( gram_L1004374585hsubst @ tr0 @ Tr ) )
= ( image @ ( sum_sum @ t @ dtree ) @ ( sum_sum @ t @ dtree ) @ ( sum_map_sum @ t @ t @ dtree @ dtree @ ( id @ t ) @ ( gram_L1004374585hsubst @ tr0 ) ) @ ( cont @ tr0 ) ) ) ) ).
% cont_hsubst_eq
thf(fact_8_cont__hsubst__neq,axiom,
! [Tr: dtree] :
( ( ( root @ Tr )
!= ( root @ tr0 ) )
=> ( ( cont @ ( gram_L1004374585hsubst @ tr0 @ Tr ) )
= ( image @ ( sum_sum @ t @ dtree ) @ ( sum_sum @ t @ dtree ) @ ( sum_map_sum @ t @ t @ dtree @ dtree @ ( id @ t ) @ ( gram_L1004374585hsubst @ tr0 ) ) @ ( cont @ Tr ) ) ) ) ).
% cont_hsubst_neq
thf(fact_9_sum_Oinject_I1_J,axiom,
! [B: $tType,A: $tType,X1: A,Y1: A] :
( ( ( sum_Inl @ A @ B @ X1 )
= ( sum_Inl @ A @ B @ Y1 ) )
= ( X1 = Y1 ) ) ).
% sum.inject(1)
thf(fact_10_old_Osum_Oinject_I1_J,axiom,
! [B: $tType,A: $tType,A2: A,A3: A] :
( ( ( sum_Inl @ A @ B @ A2 )
= ( sum_Inl @ A @ B @ A3 ) )
= ( A2 = A3 ) ) ).
% old.sum.inject(1)
thf(fact_11_vimageI,axiom,
! [B: $tType,A: $tType,F: B > A,A2: B,B2: A,B3: set @ A] :
( ( ( F @ A2 )
= B2 )
=> ( ( member @ A @ B2 @ B3 )
=> ( member @ B @ A2 @ ( vimage @ B @ A @ F @ B3 ) ) ) ) ).
% vimageI
thf(fact_12_vimage__eq,axiom,
! [A: $tType,B: $tType,A2: A,F: A > B,B3: set @ B] :
( ( member @ A @ A2 @ ( vimage @ A @ B @ F @ B3 ) )
= ( member @ B @ ( F @ A2 ) @ B3 ) ) ).
% vimage_eq
thf(fact_13_id__apply,axiom,
! [A: $tType] :
( ( id @ A )
= ( ^ [X: A] : X ) ) ).
% id_apply
thf(fact_14_hsubst__eq,axiom,
! [Tr: dtree] :
( ( ( root @ Tr )
= ( root @ tr0 ) )
=> ( ( gram_L1004374585hsubst @ tr0 @ Tr )
= ( gram_L1004374585hsubst @ tr0 @ tr0 ) ) ) ).
% hsubst_eq
thf(fact_15_image__eqI,axiom,
! [A: $tType,B: $tType,B2: A,F: B > A,X2: B,A4: set @ B] :
( ( B2
= ( F @ X2 ) )
=> ( ( member @ B @ X2 @ A4 )
=> ( member @ A @ B2 @ ( image @ B @ A @ F @ A4 ) ) ) ) ).
% image_eqI
thf(fact_16_root__hsubst,axiom,
! [Tr: dtree] :
( ( root @ ( gram_L1004374585hsubst @ tr0 @ Tr ) )
= ( root @ Tr ) ) ).
% root_hsubst
thf(fact_17_hsubst__c__def,axiom,
! [Tr: dtree] :
( ( ( ( root @ Tr )
= ( root @ tr0 ) )
=> ( ( gram_L1905609002ubst_c @ tr0 @ Tr )
= ( cont @ tr0 ) ) )
& ( ( ( root @ Tr )
!= ( root @ tr0 ) )
=> ( ( gram_L1905609002ubst_c @ tr0 @ Tr )
= ( cont @ Tr ) ) ) ) ).
% hsubst_c_def
thf(fact_18_Inl__cont__hsubst__eq,axiom,
! [Tr: 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_19_Inl__prodOf,axiom,
! [Tr: dtree] :
( ( vimage @ t @ ( sum_sum @ t @ n ) @ ( sum_Inl @ t @ n ) @ ( image @ ( sum_sum @ t @ dtree ) @ ( sum_sum @ t @ n ) @ ( sum_map_sum @ t @ t @ dtree @ n @ ( id @ t ) @ root ) @ ( cont @ Tr ) ) )
= ( vimage @ t @ ( sum_sum @ t @ dtree ) @ ( sum_Inl @ t @ dtree ) @ ( cont @ Tr ) ) ) ).
% Inl_prodOf
thf(fact_20_Gram__Lang__Mirabelle__ojxrtuoybn_Oroot__hsubst,axiom,
! [Tr0: dtree,Tr: dtree] :
( ( root @ ( gram_L1004374585hsubst @ Tr0 @ Tr ) )
= ( root @ Tr ) ) ).
% Gram_Lang_Mirabelle_ojxrtuoybn.root_hsubst
thf(fact_21_Gram__Lang__Mirabelle__ojxrtuoybn_Ohsubst__eq,axiom,
! [Tr: dtree,Tr0: dtree] :
( ( ( root @ Tr )
= ( root @ Tr0 ) )
=> ( ( gram_L1004374585hsubst @ Tr0 @ Tr )
= ( gram_L1004374585hsubst @ Tr0 @ Tr0 ) ) ) ).
% Gram_Lang_Mirabelle_ojxrtuoybn.hsubst_eq
thf(fact_22_rev__image__eqI,axiom,
! [B: $tType,A: $tType,X2: A,A4: set @ A,B2: B,F: A > B] :
( ( member @ A @ X2 @ A4 )
=> ( ( B2
= ( F @ X2 ) )
=> ( member @ B @ B2 @ ( image @ A @ B @ F @ A4 ) ) ) ) ).
% rev_image_eqI
thf(fact_23_ball__imageD,axiom,
! [A: $tType,B: $tType,F: B > A,A4: set @ B,P: A > $o] :
( ! [X3: A] :
( ( member @ A @ X3 @ ( image @ B @ A @ F @ A4 ) )
=> ( P @ X3 ) )
=> ! [X4: B] :
( ( member @ B @ X4 @ A4 )
=> ( P @ ( F @ X4 ) ) ) ) ).
% ball_imageD
thf(fact_24_image__cong,axiom,
! [B: $tType,A: $tType,M: set @ A,N: set @ A,F: A > B,G: A > B] :
( ( M = N )
=> ( ! [X3: A] :
( ( member @ A @ X3 @ N )
=> ( ( F @ X3 )
= ( G @ X3 ) ) )
=> ( ( image @ A @ B @ F @ M )
= ( image @ A @ B @ G @ N ) ) ) ) ).
% image_cong
thf(fact_25_bex__imageD,axiom,
! [A: $tType,B: $tType,F: B > A,A4: set @ B,P: A > $o] :
( ? [X4: A] :
( ( member @ A @ X4 @ ( image @ B @ A @ F @ A4 ) )
& ( P @ X4 ) )
=> ? [X3: B] :
( ( member @ B @ X3 @ A4 )
& ( P @ ( F @ X3 ) ) ) ) ).
% bex_imageD
thf(fact_26_image__iff,axiom,
! [A: $tType,B: $tType,Z: A,F: B > A,A4: set @ B] :
( ( member @ A @ Z @ ( image @ B @ A @ F @ A4 ) )
= ( ? [X: B] :
( ( member @ B @ X @ A4 )
& ( Z
= ( F @ X ) ) ) ) ) ).
% image_iff
thf(fact_27_imageI,axiom,
! [B: $tType,A: $tType,X2: A,A4: set @ A,F: A > B] :
( ( member @ A @ X2 @ A4 )
=> ( member @ B @ ( F @ X2 ) @ ( image @ A @ B @ F @ A4 ) ) ) ).
% imageI
thf(fact_28_eq__id__iff,axiom,
! [A: $tType,F: A > A] :
( ( ! [X: A] :
( ( F @ X )
= X ) )
= ( F
= ( id @ A ) ) ) ).
% eq_id_iff
thf(fact_29_id__def,axiom,
! [A: $tType] :
( ( id @ A )
= ( ^ [X: A] : X ) ) ).
% id_def
thf(fact_30_vimage__Collect,axiom,
! [B: $tType,A: $tType,P: B > $o,F: A > B,Q: A > $o] :
( ! [X3: A] :
( ( P @ ( F @ X3 ) )
= ( Q @ X3 ) )
=> ( ( vimage @ A @ B @ F @ ( collect @ B @ P ) )
= ( collect @ A @ Q ) ) ) ).
% vimage_Collect
thf(fact_31_vimageI2,axiom,
! [B: $tType,A: $tType,F: B > A,A2: B,A4: set @ A] :
( ( member @ A @ ( F @ A2 ) @ A4 )
=> ( member @ B @ A2 @ ( vimage @ B @ A @ F @ A4 ) ) ) ).
% vimageI2
thf(fact_32_vimageE,axiom,
! [A: $tType,B: $tType,A2: A,F: A > B,B3: set @ B] :
( ( member @ A @ A2 @ ( vimage @ A @ B @ F @ B3 ) )
=> ( member @ B @ ( F @ A2 ) @ B3 ) ) ).
% vimageE
thf(fact_33_vimageD,axiom,
! [A: $tType,B: $tType,A2: A,F: A > B,A4: set @ B] :
( ( member @ A @ A2 @ ( vimage @ A @ B @ F @ A4 ) )
=> ( member @ B @ ( F @ A2 ) @ A4 ) ) ).
% vimageD
thf(fact_34_Inl__inject,axiom,
! [B: $tType,A: $tType,X2: A,Y: A] :
( ( ( sum_Inl @ A @ B @ X2 )
= ( sum_Inl @ A @ B @ Y ) )
=> ( X2 = Y ) ) ).
% Inl_inject
thf(fact_35_Gram__Lang__Mirabelle__ojxrtuoybn_OInl__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 ) ) ) ) ).
% Gram_Lang_Mirabelle_ojxrtuoybn.Inl_cont_hsubst_eq
thf(fact_36_Gram__Lang__Mirabelle__ojxrtuoybn_Ocont__hsubst__neq,axiom,
! [Tr: dtree,Tr0: dtree] :
( ( ( root @ Tr )
!= ( root @ Tr0 ) )
=> ( ( cont @ ( gram_L1004374585hsubst @ Tr0 @ Tr ) )
= ( image @ ( sum_sum @ t @ dtree ) @ ( sum_sum @ t @ dtree ) @ ( sum_map_sum @ t @ t @ dtree @ dtree @ ( id @ t ) @ ( gram_L1004374585hsubst @ Tr0 ) ) @ ( cont @ Tr ) ) ) ) ).
% Gram_Lang_Mirabelle_ojxrtuoybn.cont_hsubst_neq
thf(fact_37_Gram__Lang__Mirabelle__ojxrtuoybn_Ocont__hsubst__eq,axiom,
! [Tr: dtree,Tr0: dtree] :
( ( ( root @ Tr )
= ( root @ Tr0 ) )
=> ( ( cont @ ( gram_L1004374585hsubst @ Tr0 @ Tr ) )
= ( image @ ( sum_sum @ t @ dtree ) @ ( sum_sum @ t @ dtree ) @ ( sum_map_sum @ t @ t @ dtree @ dtree @ ( id @ t ) @ ( gram_L1004374585hsubst @ Tr0 ) ) @ ( cont @ Tr0 ) ) ) ) ).
% Gram_Lang_Mirabelle_ojxrtuoybn.cont_hsubst_eq
thf(fact_38_map__sum__InlD,axiom,
! [D: $tType,B: $tType,C: $tType,A: $tType,F: C > A,G: D > B,Z: sum_sum @ C @ D,X2: A] :
( ( ( sum_map_sum @ C @ A @ D @ B @ F @ G @ Z )
= ( sum_Inl @ A @ B @ X2 ) )
=> ? [Y2: C] :
( ( Z
= ( sum_Inl @ C @ D @ Y2 ) )
& ( ( F @ Y2 )
= X2 ) ) ) ).
% map_sum_InlD
thf(fact_39_map__sum_Osimps_I1_J,axiom,
! [B: $tType,D: $tType,C: $tType,A: $tType,F1: A > C,F2: B > D,A2: A] :
( ( sum_map_sum @ A @ C @ B @ D @ F1 @ F2 @ ( sum_Inl @ A @ B @ A2 ) )
= ( sum_Inl @ C @ D @ ( F1 @ A2 ) ) ) ).
% map_sum.simps(1)
thf(fact_40_old_Osum_Osimps_I7_J,axiom,
! [B: $tType,T: $tType,A: $tType,F1: A > T,F2: B > T,A2: A] :
( ( sum_rec_sum @ A @ T @ B @ F1 @ F2 @ ( sum_Inl @ A @ B @ A2 ) )
= ( F1 @ A2 ) ) ).
% old.sum.simps(7)
thf(fact_41_map__sum__if__distrib__then_I1_J,axiom,
! [D: $tType,B: $tType,C: $tType,A: $tType,E: $o,F: A > B,G: C > D,X2: A,Y: sum_sum @ A @ C] :
( ( E
=> ( ( sum_map_sum @ A @ B @ C @ D @ F @ G @ ( if @ ( sum_sum @ A @ C ) @ E @ ( sum_Inl @ A @ C @ X2 ) @ Y ) )
= ( sum_Inl @ B @ D @ ( F @ X2 ) ) ) )
& ( ~ E
=> ( ( sum_map_sum @ A @ B @ C @ D @ F @ G @ ( if @ ( sum_sum @ A @ C ) @ E @ ( sum_Inl @ A @ C @ X2 ) @ Y ) )
= ( sum_map_sum @ A @ B @ C @ D @ F @ G @ Y ) ) ) ) ).
% map_sum_if_distrib_then(1)
thf(fact_42_map__sum__if__distrib__else_I1_J,axiom,
! [C: $tType,D: $tType,B: $tType,A: $tType,E: $o,F: A > B,G: C > D,X2: sum_sum @ A @ C,Y: A] :
( ( E
=> ( ( sum_map_sum @ A @ B @ C @ D @ F @ G @ ( if @ ( sum_sum @ A @ C ) @ E @ X2 @ ( sum_Inl @ A @ C @ Y ) ) )
= ( sum_map_sum @ A @ B @ C @ D @ F @ G @ X2 ) ) )
& ( ~ E
=> ( ( sum_map_sum @ A @ B @ C @ D @ F @ G @ ( if @ ( sum_sum @ A @ C ) @ E @ X2 @ ( sum_Inl @ A @ C @ Y ) ) )
= ( sum_Inl @ B @ D @ ( F @ Y ) ) ) ) ) ).
% map_sum_if_distrib_else(1)
thf(fact_43_dtree__cong,axiom,
! [Tr: dtree,Tr2: dtree] :
( ( ( root @ Tr )
= ( root @ Tr2 ) )
=> ( ( ( cont @ Tr )
= ( cont @ Tr2 ) )
=> ( Tr = Tr2 ) ) ) ).
% dtree_cong
thf(fact_44_sum_Omap__id0,axiom,
! [B: $tType,A: $tType] :
( ( sum_map_sum @ A @ A @ B @ B @ ( id @ A ) @ ( id @ B ) )
= ( id @ ( sum_sum @ A @ B ) ) ) ).
% sum.map_id0
thf(fact_45_mem__Collect__eq,axiom,
! [A: $tType,A2: A,P: A > $o] :
( ( member @ A @ A2 @ ( collect @ A @ P ) )
= ( P @ A2 ) ) ).
% mem_Collect_eq
thf(fact_46_Collect__mem__eq,axiom,
! [A: $tType,A4: set @ A] :
( ( collect @ A
@ ^ [X: A] : ( member @ A @ X @ A4 ) )
= A4 ) ).
% Collect_mem_eq
thf(fact_47_Collect__cong,axiom,
! [A: $tType,P: A > $o,Q: A > $o] :
( ! [X3: A] :
( ( P @ X3 )
= ( Q @ X3 ) )
=> ( ( collect @ A @ P )
= ( collect @ A @ Q ) ) ) ).
% Collect_cong
thf(fact_48_ext,axiom,
! [B: $tType,A: $tType,F: A > B,G: A > B] :
( ! [X3: A] :
( ( F @ X3 )
= ( G @ X3 ) )
=> ( F = G ) ) ).
% ext
thf(fact_49_sum_Omap__id,axiom,
! [B: $tType,A: $tType,T2: sum_sum @ A @ B] :
( ( sum_map_sum @ A @ A @ B @ B @ ( id @ A ) @ ( id @ B ) @ T2 )
= T2 ) ).
% sum.map_id
thf(fact_50_Sup_OSUP__id__eq,axiom,
! [A: $tType,Sup: ( set @ A ) > A,A4: set @ A] :
( ( Sup @ ( image @ A @ A @ ( id @ A ) @ A4 ) )
= ( Sup @ A4 ) ) ).
% Sup.SUP_id_eq
thf(fact_51_Inf_OINF__id__eq,axiom,
! [A: $tType,Inf: ( set @ A ) > A,A4: set @ A] :
( ( Inf @ ( image @ A @ A @ ( id @ A ) @ A4 ) )
= ( Inf @ A4 ) ) ).
% Inf.INF_id_eq
thf(fact_52_Inr__m__oplus,axiom,
! [A: $tType,C: $tType,B: $tType,F: C > A,Tns: set @ ( sum_sum @ B @ C )] :
( ( vimage @ A @ ( sum_sum @ B @ A ) @ ( sum_Inr @ A @ B ) @ ( image @ ( sum_sum @ B @ C ) @ ( sum_sum @ B @ A ) @ ( sum_map_sum @ B @ B @ C @ A @ ( id @ B ) @ F ) @ Tns ) )
= ( image @ C @ A @ F @ ( vimage @ C @ ( sum_sum @ B @ C ) @ ( sum_Inr @ C @ B ) @ Tns ) ) ) ).
% Inr_m_oplus
thf(fact_53_Inr__cont__hsubst__eq,axiom,
! [Tr: dtree] :
( ( ( root @ Tr )
= ( root @ tr0 ) )
=> ( ( vimage @ dtree @ ( sum_sum @ t @ dtree ) @ ( sum_Inr @ dtree @ t ) @ ( cont @ ( gram_L1004374585hsubst @ tr0 @ Tr ) ) )
= ( image @ dtree @ dtree @ ( gram_L1004374585hsubst @ tr0 ) @ ( vimage @ dtree @ ( sum_sum @ t @ dtree ) @ ( sum_Inr @ dtree @ t ) @ ( cont @ tr0 ) ) ) ) ) ).
% Inr_cont_hsubst_eq
thf(fact_54_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_55_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_56_old_Osum_Oinject_I2_J,axiom,
! [A: $tType,B: $tType,B2: B,B4: B] :
( ( ( sum_Inr @ B @ A @ B2 )
= ( sum_Inr @ B @ A @ B4 ) )
= ( B2 = B4 ) ) ).
% old.sum.inject(2)
thf(fact_57_Inl__Inr__False,axiom,
! [A: $tType,B: $tType,X2: A,Y: B] :
( ( sum_Inl @ A @ B @ X2 )
!= ( sum_Inr @ B @ A @ Y ) ) ).
% Inl_Inr_False
thf(fact_58_Inr__Inl__False,axiom,
! [B: $tType,A: $tType,X2: B,Y: A] :
( ( sum_Inr @ B @ A @ X2 )
!= ( sum_Inl @ A @ B @ Y ) ) ).
% Inr_Inl_False
thf(fact_59_old_Osum_Osimps_I8_J,axiom,
! [A: $tType,T: $tType,B: $tType,F1: A > T,F2: B > T,B2: B] :
( ( sum_rec_sum @ A @ T @ B @ F1 @ F2 @ ( sum_Inr @ B @ A @ B2 ) )
= ( F2 @ B2 ) ) ).
% old.sum.simps(8)
thf(fact_60_Inr__oplus__iff,axiom,
! [B: $tType,A: $tType,C: $tType,Tr: B,F: C > B,Tns: set @ ( sum_sum @ A @ C )] :
( ( member @ ( sum_sum @ A @ B ) @ ( sum_Inr @ B @ A @ Tr ) @ ( image @ ( sum_sum @ A @ C ) @ ( sum_sum @ A @ B ) @ ( sum_map_sum @ A @ A @ C @ B @ ( id @ A ) @ F ) @ Tns ) )
= ( ? [N2: C] :
( ( member @ ( sum_sum @ A @ C ) @ ( sum_Inr @ C @ A @ N2 ) @ Tns )
& ( ( F @ N2 )
= Tr ) ) ) ) ).
% Inr_oplus_iff
thf(fact_61_Inr__inject,axiom,
! [A: $tType,B: $tType,X2: B,Y: B] :
( ( ( sum_Inr @ B @ A @ X2 )
= ( sum_Inr @ B @ A @ Y ) )
=> ( X2 = Y ) ) ).
% Inr_inject
thf(fact_62_obj__sumE__f,axiom,
! [A: $tType,C: $tType,B: $tType,S: B,F: ( sum_sum @ A @ C ) > B,P: $o] :
( ! [X3: A] :
( ( S
= ( F @ ( sum_Inl @ A @ C @ X3 ) ) )
=> P )
=> ( ! [X3: C] :
( ( S
= ( F @ ( sum_Inr @ C @ A @ X3 ) ) )
=> P )
=> ! [X4: sum_sum @ A @ C] :
( ( S
= ( F @ X4 ) )
=> P ) ) ) ).
% obj_sumE_f
thf(fact_63_map__sum__if__distrib__else_I2_J,axiom,
! [E2: $tType,F3: $tType,H: $tType,G2: $tType,E: $o,F: E2 > F3,G: G2 > H,X2: sum_sum @ E2 @ G2,Y: G2] :
( ( E
=> ( ( sum_map_sum @ E2 @ F3 @ G2 @ H @ F @ G @ ( if @ ( sum_sum @ E2 @ G2 ) @ E @ X2 @ ( sum_Inr @ G2 @ E2 @ Y ) ) )
= ( sum_map_sum @ E2 @ F3 @ G2 @ H @ F @ G @ X2 ) ) )
& ( ~ E
=> ( ( sum_map_sum @ E2 @ F3 @ G2 @ H @ F @ G @ ( if @ ( sum_sum @ E2 @ G2 ) @ E @ X2 @ ( sum_Inr @ G2 @ E2 @ Y ) ) )
= ( sum_Inr @ H @ F3 @ ( G @ Y ) ) ) ) ) ).
% map_sum_if_distrib_else(2)
thf(fact_64_map__sum__if__distrib__then_I2_J,axiom,
! [H: $tType,F3: $tType,G2: $tType,E2: $tType,E: $o,F: E2 > F3,G: G2 > H,X2: G2,Y: sum_sum @ E2 @ G2] :
( ( E
=> ( ( sum_map_sum @ E2 @ F3 @ G2 @ H @ F @ G @ ( if @ ( sum_sum @ E2 @ G2 ) @ E @ ( sum_Inr @ G2 @ E2 @ X2 ) @ Y ) )
= ( sum_Inr @ H @ F3 @ ( G @ X2 ) ) ) )
& ( ~ E
=> ( ( sum_map_sum @ E2 @ F3 @ G2 @ H @ F @ G @ ( if @ ( sum_sum @ E2 @ G2 ) @ E @ ( sum_Inr @ G2 @ E2 @ X2 ) @ Y ) )
= ( sum_map_sum @ E2 @ F3 @ G2 @ H @ F @ G @ Y ) ) ) ) ).
% map_sum_if_distrib_then(2)
thf(fact_65_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_66_old_Osum_Odistinct_I2_J,axiom,
! [B5: $tType,A5: $tType,B6: B5,A6: A5] :
( ( sum_Inr @ B5 @ A5 @ B6 )
!= ( sum_Inl @ A5 @ B5 @ A6 ) ) ).
% old.sum.distinct(2)
thf(fact_67_old_Osum_Odistinct_I1_J,axiom,
! [A: $tType,B: $tType,A2: A,B4: B] :
( ( sum_Inl @ A @ B @ A2 )
!= ( sum_Inr @ B @ A @ B4 ) ) ).
% old.sum.distinct(1)
thf(fact_68_sumE,axiom,
! [A: $tType,B: $tType,S: sum_sum @ A @ B] :
( ! [X3: A] :
( S
!= ( sum_Inl @ A @ B @ X3 ) )
=> ~ ! [Y2: B] :
( S
!= ( sum_Inr @ B @ A @ Y2 ) ) ) ).
% sumE
thf(fact_69_Inr__not__Inl,axiom,
! [B: $tType,A: $tType,B2: B,A2: A] :
( ( sum_Inr @ B @ A @ B2 )
!= ( sum_Inl @ A @ B @ A2 ) ) ).
% Inr_not_Inl
thf(fact_70_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_71_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_72_old_Osum_Oexhaust,axiom,
! [A: $tType,B: $tType,Y: sum_sum @ A @ B] :
( ! [A7: A] :
( Y
!= ( sum_Inl @ A @ B @ A7 ) )
=> ~ ! [B7: B] :
( Y
!= ( sum_Inr @ B @ A @ B7 ) ) ) ).
% old.sum.exhaust
thf(fact_73_old_Osum_Oinducts,axiom,
! [B: $tType,A: $tType,P: ( sum_sum @ A @ B ) > $o,Sum: sum_sum @ A @ B] :
( ! [A7: A] : ( P @ ( sum_Inl @ A @ B @ A7 ) )
=> ( ! [B7: B] : ( P @ ( sum_Inr @ B @ A @ B7 ) )
=> ( P @ Sum ) ) ) ).
% old.sum.inducts
thf(fact_74_map__sum_Osimps_I2_J,axiom,
! [A: $tType,C: $tType,D: $tType,B: $tType,F1: A > C,F2: B > D,A2: B] :
( ( sum_map_sum @ A @ C @ B @ D @ F1 @ F2 @ ( sum_Inr @ B @ A @ A2 ) )
= ( sum_Inr @ D @ C @ ( F2 @ A2 ) ) ) ).
% map_sum.simps(2)
thf(fact_75_map__sum__InrD,axiom,
! [C: $tType,A: $tType,D: $tType,B: $tType,F: C > A,G: D > B,Z: sum_sum @ C @ D,X2: B] :
( ( ( sum_map_sum @ C @ A @ D @ B @ F @ G @ Z )
= ( sum_Inr @ B @ A @ X2 ) )
=> ? [Y2: D] :
( ( Z
= ( sum_Inr @ D @ C @ Y2 ) )
& ( ( G @ Y2 )
= X2 ) ) ) ).
% map_sum_InrD
thf(fact_76_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_77_root__prodOf,axiom,
! [Tr2: dtree,Tr: dtree] :
( ( member @ ( sum_sum @ t @ dtree ) @ ( sum_Inr @ dtree @ t @ Tr2 ) @ ( cont @ Tr ) )
=> ( member @ ( sum_sum @ t @ n ) @ ( sum_Inr @ n @ t @ ( root @ Tr2 ) ) @ ( image @ ( sum_sum @ t @ dtree ) @ ( sum_sum @ t @ n ) @ ( sum_map_sum @ t @ t @ dtree @ n @ ( id @ t ) @ root ) @ ( cont @ Tr ) ) ) ) ).
% root_prodOf
thf(fact_78_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_79_Inl__Inr__image__cong,axiom,
! [B: $tType,A: $tType,A4: set @ ( sum_sum @ A @ B ),B3: set @ ( sum_sum @ A @ B )] :
( ( ( vimage @ A @ ( sum_sum @ A @ B ) @ ( sum_Inl @ A @ B ) @ A4 )
= ( vimage @ A @ ( sum_sum @ A @ B ) @ ( sum_Inl @ A @ B ) @ B3 ) )
=> ( ( ( vimage @ B @ ( sum_sum @ A @ B ) @ ( sum_Inr @ B @ A ) @ A4 )
= ( vimage @ B @ ( sum_sum @ A @ B ) @ ( sum_Inr @ B @ A ) @ B3 ) )
=> ( A4 = B3 ) ) ) ).
% Inl_Inr_image_cong
thf(fact_80_Inr__oplus__elim,axiom,
! [A: $tType,C: $tType,B: $tType,Tr: B,F: C > B,Tns: set @ ( sum_sum @ A @ C )] :
( ( member @ ( sum_sum @ A @ B ) @ ( sum_Inr @ B @ A @ Tr ) @ ( image @ ( sum_sum @ A @ C ) @ ( sum_sum @ A @ B ) @ ( sum_map_sum @ A @ A @ C @ B @ ( id @ A ) @ F ) @ Tns ) )
=> ? [N3: C] :
( ( member @ ( sum_sum @ A @ C ) @ ( sum_Inr @ C @ A @ N3 ) @ Tns )
& ( ( F @ N3 )
= Tr ) ) ) ).
% Inr_oplus_elim
thf(fact_81_Gram__Lang__Mirabelle__ojxrtuoybn_OInr__cont__hsubst__eq,axiom,
! [Tr: dtree,Tr0: dtree] :
( ( ( root @ Tr )
= ( root @ Tr0 ) )
=> ( ( vimage @ dtree @ ( sum_sum @ t @ dtree ) @ ( sum_Inr @ dtree @ t ) @ ( cont @ ( gram_L1004374585hsubst @ Tr0 @ Tr ) ) )
= ( image @ dtree @ dtree @ ( gram_L1004374585hsubst @ Tr0 ) @ ( vimage @ dtree @ ( sum_sum @ t @ dtree ) @ ( sum_Inr @ dtree @ t ) @ ( cont @ Tr0 ) ) ) ) ) ).
% Gram_Lang_Mirabelle_ojxrtuoybn.Inr_cont_hsubst_eq
thf(fact_82_Inf_OINF__cong,axiom,
! [A: $tType,B: $tType,A4: set @ B,B3: set @ B,C2: B > A,D2: B > A,Inf: ( set @ A ) > A] :
( ( A4 = B3 )
=> ( ! [X3: B] :
( ( member @ B @ X3 @ B3 )
=> ( ( C2 @ X3 )
= ( D2 @ X3 ) ) )
=> ( ( Inf @ ( image @ B @ A @ C2 @ A4 ) )
= ( Inf @ ( image @ B @ A @ D2 @ B3 ) ) ) ) ) ).
% Inf.INF_cong
thf(fact_83_Sup_OSUP__cong,axiom,
! [A: $tType,B: $tType,A4: set @ B,B3: set @ B,C2: B > A,D2: B > A,Sup: ( set @ A ) > A] :
( ( A4 = B3 )
=> ( ! [X3: B] :
( ( member @ B @ X3 @ B3 )
=> ( ( C2 @ X3 )
= ( D2 @ X3 ) ) )
=> ( ( Sup @ ( image @ B @ A @ C2 @ A4 ) )
= ( Sup @ ( image @ B @ A @ D2 @ B3 ) ) ) ) ) ).
% Sup.SUP_cong
thf(fact_84_Gram__Lang__Mirabelle__ojxrtuoybn_Ohsubst__c__def,axiom,
( gram_L1905609002ubst_c
= ( ^ [Tr02: dtree,Tr3: dtree] :
( if @ ( set @ ( sum_sum @ t @ dtree ) )
@ ( ( root @ Tr3 )
= ( root @ Tr02 ) )
@ ( cont @ Tr02 )
@ ( cont @ Tr3 ) ) ) ) ).
% Gram_Lang_Mirabelle_ojxrtuoybn.hsubst_c_def
thf(fact_85_hsubst__def,axiom,
( ( gram_L1004374585hsubst @ tr0 )
= ( unfold @ dtree @ gram_L1905609017ubst_r @ ( gram_L1905609002ubst_c @ tr0 ) ) ) ).
% hsubst_def
thf(fact_86_root__subtrOf,axiom,
! [N4: n,Tr: dtree] :
( ( member @ ( sum_sum @ t @ n ) @ ( sum_Inr @ n @ t @ N4 ) @ ( image @ ( sum_sum @ t @ dtree ) @ ( sum_sum @ t @ n ) @ ( sum_map_sum @ t @ t @ dtree @ n @ ( id @ t ) @ root ) @ ( cont @ Tr ) ) )
=> ( ( root @ ( gram_L1614515765ubtrOf @ Tr @ N4 ) )
= N4 ) ) ).
% root_subtrOf
thf(fact_87_Inr__subtrOf,axiom,
! [N4: n,Tr: dtree] :
( ( member @ ( sum_sum @ t @ n ) @ ( sum_Inr @ n @ t @ N4 ) @ ( image @ ( sum_sum @ t @ dtree ) @ ( sum_sum @ t @ n ) @ ( sum_map_sum @ t @ t @ dtree @ n @ ( id @ t ) @ root ) @ ( cont @ Tr ) ) )
=> ( member @ ( sum_sum @ t @ dtree ) @ ( sum_Inr @ dtree @ t @ ( gram_L1614515765ubtrOf @ Tr @ N4 ) ) @ ( cont @ Tr ) ) ) ).
% Inr_subtrOf
thf(fact_88_subtrOf,axiom,
! [N4: n,Tr: dtree] :
( ( member @ ( sum_sum @ t @ n ) @ ( sum_Inr @ n @ t @ N4 ) @ ( image @ ( sum_sum @ t @ dtree ) @ ( sum_sum @ t @ n ) @ ( sum_map_sum @ t @ t @ dtree @ n @ ( id @ t ) @ root ) @ ( cont @ Tr ) ) )
=> ( ( member @ ( sum_sum @ t @ dtree ) @ ( sum_Inr @ dtree @ t @ ( gram_L1614515765ubtrOf @ Tr @ N4 ) ) @ ( cont @ Tr ) )
& ( ( root @ ( gram_L1614515765ubtrOf @ Tr @ N4 ) )
= N4 ) ) ) ).
% subtrOf
thf(fact_89_finite__hsubst__c,axiom,
! [N4: dtree] : ( finite_finite2 @ ( sum_sum @ t @ dtree ) @ ( gram_L1905609002ubst_c @ tr0 @ N4 ) ) ).
% finite_hsubst_c
thf(fact_90_obj__sumE,axiom,
! [A: $tType,B: $tType,S: sum_sum @ A @ B] :
( ! [X3: A] :
( S
!= ( sum_Inl @ A @ B @ X3 ) )
=> ~ ! [X3: B] :
( S
!= ( sum_Inr @ B @ A @ X3 ) ) ) ).
% obj_sumE
thf(fact_91_Node__root__cont,axiom,
! [Tr: dtree] :
( ( node @ ( root @ Tr ) @ ( cont @ Tr ) )
= Tr ) ).
% Node_root_cont
thf(fact_92_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_93_root__Node,axiom,
! [N4: n,As: set @ ( sum_sum @ t @ dtree )] :
( ( root @ ( node @ N4 @ As ) )
= N4 ) ).
% root_Node
thf(fact_94_dtree__simps,axiom,
! [As: set @ ( sum_sum @ t @ dtree ),As2: set @ ( sum_sum @ t @ dtree ),N4: n,N5: n] :
( ( finite_finite2 @ ( sum_sum @ t @ dtree ) @ As )
=> ( ( finite_finite2 @ ( sum_sum @ t @ dtree ) @ As2 )
=> ( ( ( node @ N4 @ As )
= ( node @ N5 @ As2 ) )
= ( ( N4 = N5 )
& ( As = As2 ) ) ) ) ) ).
% dtree_simps
thf(fact_95_cont__Node,axiom,
! [As: set @ ( sum_sum @ t @ dtree ),N4: n] :
( ( finite_finite2 @ ( sum_sum @ t @ dtree ) @ As )
=> ( ( cont @ ( node @ N4 @ As ) )
= As ) ) ).
% cont_Node
thf(fact_96_dtree__cases,axiom,
! [Tr: dtree] :
~ ! [N3: n,As3: set @ ( sum_sum @ t @ dtree )] :
( ( finite_finite2 @ ( sum_sum @ t @ dtree ) @ As3 )
=> ( Tr
!= ( node @ N3 @ As3 ) ) ) ).
% dtree_cases
thf(fact_97_finite__cont,axiom,
! [Tr: dtree] : ( finite_finite2 @ ( sum_sum @ t @ dtree ) @ ( cont @ Tr ) ) ).
% finite_cont
thf(fact_98_Gram__Lang__Mirabelle__ojxrtuoybn_Ofinite__hsubst__c,axiom,
! [Tr0: dtree,N4: dtree] : ( finite_finite2 @ ( sum_sum @ t @ dtree ) @ ( gram_L1905609002ubst_c @ Tr0 @ N4 ) ) ).
% Gram_Lang_Mirabelle_ojxrtuoybn.finite_hsubst_c
thf(fact_99_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_100_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_101_inFr__inFr2,axiom,
gram_L1333338417e_inFr = gram_L805317441_inFr2 ).
% inFr_inFr2
thf(fact_102_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_103_Gram__Lang__Mirabelle__ojxrtuoybn_Ohsubst__def,axiom,
( gram_L1004374585hsubst
= ( ^ [Tr02: dtree] : ( unfold @ dtree @ gram_L1905609017ubst_r @ ( gram_L1905609002ubst_c @ Tr02 ) ) ) ) ).
% Gram_Lang_Mirabelle_ojxrtuoybn.hsubst_def
thf(fact_104_not__arg__cong__Inr,axiom,
! [B: $tType,A: $tType,X2: A,Y: A] :
( ( X2 != Y )
=> ( ( sum_Inr @ A @ B @ X2 )
!= ( sum_Inr @ A @ B @ Y ) ) ) ).
% not_arg_cong_Inr
thf(fact_105_unfold_I2_J,axiom,
! [A: $tType,Ct: A > ( set @ ( sum_sum @ t @ A ) ),B2: A,Rt: A > n] :
( ( finite_finite2 @ ( sum_sum @ t @ A ) @ ( Ct @ B2 ) )
=> ( ( cont @ ( unfold @ A @ Rt @ Ct @ B2 ) )
= ( image @ ( sum_sum @ t @ A ) @ ( sum_sum @ t @ dtree ) @ ( sum_map_sum @ t @ t @ A @ dtree @ ( id @ t ) @ ( unfold @ A @ Rt @ Ct ) ) @ ( Ct @ B2 ) ) ) ) ).
% unfold(2)
thf(fact_106_hsubst__r__def,axiom,
gram_L1905609017ubst_r = root ).
% hsubst_r_def
thf(fact_107_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_108_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_109_inFr_Oinducts,axiom,
! [X1: set @ n,X22: dtree,X32: t,P: ( set @ n ) > dtree > t > $o] :
( ( gram_L1333338417e_inFr @ X1 @ X22 @ X32 )
=> ( ! [Tr4: dtree,Ns2: set @ n,T3: t] :
( ( member @ n @ ( root @ Tr4 ) @ Ns2 )
=> ( ( member @ ( sum_sum @ t @ dtree ) @ ( sum_Inl @ t @ dtree @ T3 ) @ ( cont @ Tr4 ) )
=> ( P @ Ns2 @ Tr4 @ T3 ) ) )
=> ( ! [Tr4: dtree,Ns2: set @ n,Tr12: dtree,T3: t] :
( ( member @ n @ ( root @ Tr4 ) @ Ns2 )
=> ( ( member @ ( sum_sum @ t @ dtree ) @ ( sum_Inr @ dtree @ t @ Tr12 ) @ ( cont @ Tr4 ) )
=> ( ( gram_L1333338417e_inFr @ Ns2 @ Tr12 @ T3 )
=> ( ( P @ Ns2 @ Tr12 @ T3 )
=> ( P @ Ns2 @ Tr4 @ T3 ) ) ) ) )
=> ( P @ X1 @ X22 @ X32 ) ) ) ) ).
% inFr.inducts
thf(fact_110_inFr_Osimps,axiom,
( gram_L1333338417e_inFr
= ( ^ [A12: set @ n,A23: dtree,A33: t] :
( ? [Tr3: dtree,Ns3: set @ n,T4: t] :
( ( A12 = Ns3 )
& ( A23 = Tr3 )
& ( A33 = T4 )
& ( member @ n @ ( root @ Tr3 ) @ Ns3 )
& ( member @ ( sum_sum @ t @ dtree ) @ ( sum_Inl @ t @ dtree @ T4 ) @ ( cont @ Tr3 ) ) )
| ? [Tr3: dtree,Ns3: set @ n,Tr13: dtree,T4: t] :
( ( A12 = Ns3 )
& ( A23 = Tr3 )
& ( A33 = T4 )
& ( member @ n @ ( root @ Tr3 ) @ Ns3 )
& ( member @ ( sum_sum @ t @ dtree ) @ ( sum_Inr @ dtree @ t @ Tr13 ) @ ( cont @ Tr3 ) )
& ( gram_L1333338417e_inFr @ Ns3 @ Tr13 @ T4 ) ) ) ) ) ).
% inFr.simps
thf(fact_111_inFrr__def,axiom,
( gram_L805317505_inFrr
= ( ^ [Ns3: set @ n,Tr3: dtree,T4: t] :
? [Tr5: dtree] :
( ( member @ ( sum_sum @ t @ dtree ) @ ( sum_Inr @ dtree @ t @ Tr5 ) @ ( cont @ Tr3 ) )
& ( gram_L1333338417e_inFr @ Ns3 @ Tr5 @ T4 ) ) ) ) ).
% inFrr_def
thf(fact_112_finite__imageI,axiom,
! [B: $tType,A: $tType,F4: set @ A,H2: A > B] :
( ( finite_finite2 @ A @ F4 )
=> ( finite_finite2 @ B @ ( image @ A @ B @ H2 @ F4 ) ) ) ).
% finite_imageI
thf(fact_113_surj__subtrOf,axiom,
! [Tr: dtree,Tr2: dtree] :
( ( gram_L864798063lle_wf @ Tr )
=> ( ( member @ ( sum_sum @ t @ dtree ) @ ( sum_Inr @ dtree @ t @ Tr2 ) @ ( cont @ Tr ) )
=> ? [N3: n] :
( ( member @ ( sum_sum @ t @ n ) @ ( sum_Inr @ n @ t @ N3 ) @ ( image @ ( sum_sum @ t @ dtree ) @ ( sum_sum @ t @ n ) @ ( sum_map_sum @ t @ t @ dtree @ n @ ( id @ t ) @ root ) @ ( cont @ Tr ) ) )
& ( ( gram_L1614515765ubtrOf @ Tr @ N3 )
= Tr2 ) ) ) ) ).
% surj_subtrOf
thf(fact_114_finite__code,axiom,
! [A: $tType] :
( ( finite_finite @ A @ ( type @ A ) )
=> ( ( finite_finite2 @ A )
= ( ^ [A8: set @ A] : $true ) ) ) ).
% finite_code
thf(fact_115_subtrOf__root,axiom,
! [Tr: dtree,Tr2: dtree] :
( ( gram_L864798063lle_wf @ Tr )
=> ( ( member @ ( sum_sum @ t @ dtree ) @ ( sum_Inr @ dtree @ t @ Tr2 ) @ ( cont @ Tr ) )
=> ( ( gram_L1614515765ubtrOf @ Tr @ ( root @ Tr2 ) )
= Tr2 ) ) ) ).
% subtrOf_root
thf(fact_116_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_117_wf__cont,axiom,
! [Tr: dtree,Tr2: dtree] :
( ( gram_L864798063lle_wf @ Tr )
=> ( ( member @ ( sum_sum @ t @ dtree ) @ ( sum_Inr @ dtree @ t @ Tr2 ) @ ( cont @ Tr ) )
=> ( gram_L864798063lle_wf @ Tr2 ) ) ) ).
% wf_cont
thf(fact_118_finite,axiom,
! [A: $tType] :
( ( finite_finite @ A @ ( type @ A ) )
=> ! [A4: set @ A] : ( finite_finite2 @ A @ A4 ) ) ).
% finite
thf(fact_119_finite__set__choice,axiom,
! [B: $tType,A: $tType,A4: set @ A,P: A > B > $o] :
( ( finite_finite2 @ A @ A4 )
=> ( ! [X3: A] :
( ( member @ A @ X3 @ A4 )
=> ? [X12: B] : ( P @ X3 @ X12 ) )
=> ? [F5: A > B] :
! [X4: A] :
( ( member @ A @ X4 @ A4 )
=> ( P @ X4 @ ( F5 @ X4 ) ) ) ) ) ).
% finite_set_choice
thf(fact_120_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,Tr4: dtree,Ns1: set @ n] :
( ( A1
= ( insert @ n @ ( root @ Tr4 ) @ Ns1 ) )
=> ( ( A22 = Tr4 )
=> ( ( member @ ( sum_sum @ t @ dtree ) @ ( sum_Inr @ dtree @ t @ Tr12 ) @ ( cont @ Tr4 ) )
=> ~ ( gram_L805317441_inFr2 @ Ns1 @ Tr12 @ A32 ) ) ) ) ) ) ).
% inFr2.cases
thf(fact_121_inFr2_Osimps,axiom,
( gram_L805317441_inFr2
= ( ^ [A12: set @ n,A23: dtree,A33: t] :
( ? [Tr3: dtree,Ns3: set @ n,T4: t] :
( ( A12 = Ns3 )
& ( A23 = Tr3 )
& ( A33 = T4 )
& ( member @ n @ ( root @ Tr3 ) @ Ns3 )
& ( member @ ( sum_sum @ t @ dtree ) @ ( sum_Inl @ t @ dtree @ T4 ) @ ( cont @ Tr3 ) ) )
| ? [Tr13: dtree,Tr3: dtree,Ns12: set @ n,T4: t] :
( ( A12
= ( insert @ n @ ( root @ Tr3 ) @ Ns12 ) )
& ( A23 = Tr3 )
& ( A33 = T4 )
& ( member @ ( sum_sum @ t @ dtree ) @ ( sum_Inr @ dtree @ t @ Tr13 ) @ ( cont @ Tr3 ) )
& ( gram_L805317441_inFr2 @ Ns12 @ Tr13 @ T4 ) ) ) ) ) ).
% inFr2.simps
thf(fact_122_inFr2_Oinducts,axiom,
! [X1: set @ n,X22: dtree,X32: t,P: ( set @ n ) > dtree > t > $o] :
( ( gram_L805317441_inFr2 @ X1 @ X22 @ X32 )
=> ( ! [Tr4: dtree,Ns2: set @ n,T3: t] :
( ( member @ n @ ( root @ Tr4 ) @ Ns2 )
=> ( ( member @ ( sum_sum @ t @ dtree ) @ ( sum_Inl @ t @ dtree @ T3 ) @ ( cont @ Tr4 ) )
=> ( P @ Ns2 @ Tr4 @ T3 ) ) )
=> ( ! [Tr12: dtree,Tr4: dtree,Ns1: set @ n,T3: t] :
( ( member @ ( sum_sum @ t @ dtree ) @ ( sum_Inr @ dtree @ t @ Tr12 ) @ ( cont @ Tr4 ) )
=> ( ( gram_L805317441_inFr2 @ Ns1 @ Tr12 @ T3 )
=> ( ( P @ Ns1 @ Tr12 @ T3 )
=> ( P @ ( insert @ n @ ( root @ Tr4 ) @ Ns1 ) @ Tr4 @ T3 ) ) ) )
=> ( P @ X1 @ X22 @ X32 ) ) ) ) ).
% inFr2.inducts
thf(fact_123_wf__inj__on,axiom,
! [Tr: dtree] :
( ( gram_L864798063lle_wf @ Tr )
=> ( inj_on @ dtree @ n @ root @ ( vimage @ dtree @ ( sum_sum @ t @ dtree ) @ ( sum_Inr @ dtree @ t ) @ ( cont @ Tr ) ) ) ) ).
% wf_inj_on
thf(fact_124_insert__absorb2,axiom,
! [A: $tType,X2: A,A4: set @ A] :
( ( insert @ A @ X2 @ ( insert @ A @ X2 @ A4 ) )
= ( insert @ A @ X2 @ A4 ) ) ).
% insert_absorb2
thf(fact_125_insert__iff,axiom,
! [A: $tType,A2: A,B2: A,A4: set @ A] :
( ( member @ A @ A2 @ ( insert @ A @ B2 @ A4 ) )
= ( ( A2 = B2 )
| ( member @ A @ A2 @ A4 ) ) ) ).
% insert_iff
thf(fact_126_insertCI,axiom,
! [A: $tType,A2: A,B3: set @ A,B2: A] :
( ( ~ ( member @ A @ A2 @ B3 )
=> ( A2 = B2 ) )
=> ( member @ A @ A2 @ ( insert @ A @ B2 @ B3 ) ) ) ).
% insertCI
thf(fact_127_insert__image,axiom,
! [B: $tType,A: $tType,X2: A,A4: set @ A,F: A > B] :
( ( member @ A @ X2 @ A4 )
=> ( ( insert @ B @ ( F @ X2 ) @ ( image @ A @ B @ F @ A4 ) )
= ( image @ A @ B @ F @ A4 ) ) ) ).
% insert_image
thf(fact_128_image__insert,axiom,
! [A: $tType,B: $tType,F: B > A,A2: B,B3: set @ B] :
( ( image @ B @ A @ F @ ( insert @ B @ A2 @ B3 ) )
= ( insert @ A @ ( F @ A2 ) @ ( image @ B @ A @ F @ B3 ) ) ) ).
% image_insert
thf(fact_129_finite__insert,axiom,
! [A: $tType,A2: A,A4: set @ A] :
( ( finite_finite2 @ A @ ( insert @ A @ A2 @ A4 ) )
= ( finite_finite2 @ A @ A4 ) ) ).
% finite_insert
thf(fact_130_finite_OinsertI,axiom,
! [A: $tType,A4: set @ A,A2: A] :
( ( finite_finite2 @ A @ A4 )
=> ( finite_finite2 @ A @ ( insert @ A @ A2 @ A4 ) ) ) ).
% finite.insertI
thf(fact_131_inj__on__image__iff,axiom,
! [B: $tType,A: $tType,A4: set @ A,G: A > B,F: A > A] :
( ! [X3: A] :
( ( member @ A @ X3 @ A4 )
=> ! [Xa: A] :
( ( member @ A @ Xa @ A4 )
=> ( ( ( G @ ( F @ X3 ) )
= ( G @ ( F @ Xa ) ) )
= ( ( G @ X3 )
= ( G @ Xa ) ) ) ) )
=> ( ( inj_on @ A @ A @ F @ A4 )
=> ( ( inj_on @ A @ B @ G @ ( image @ A @ A @ F @ A4 ) )
= ( inj_on @ A @ B @ G @ A4 ) ) ) ) ).
% inj_on_image_iff
thf(fact_132_inj__img__insertE,axiom,
! [B: $tType,A: $tType,F: A > B,A4: set @ A,X2: B,B3: set @ B] :
( ( inj_on @ A @ B @ F @ A4 )
=> ( ~ ( member @ B @ X2 @ B3 )
=> ( ( ( insert @ B @ X2 @ B3 )
= ( image @ A @ B @ F @ A4 ) )
=> ~ ! [X6: A,A9: set @ A] :
( ~ ( member @ A @ X6 @ A9 )
=> ( ( A4
= ( insert @ A @ X6 @ A9 ) )
=> ( ( X2
= ( F @ X6 ) )
=> ( B3
!= ( image @ A @ B @ F @ A9 ) ) ) ) ) ) ) ) ).
% inj_img_insertE
thf(fact_133_inj__on__id,axiom,
! [A: $tType,A4: set @ A] : ( inj_on @ A @ A @ ( id @ A ) @ A4 ) ).
% inj_on_id
thf(fact_134_mk__disjoint__insert,axiom,
! [A: $tType,A2: A,A4: set @ A] :
( ( member @ A @ A2 @ A4 )
=> ? [B8: set @ A] :
( ( A4
= ( insert @ A @ A2 @ B8 ) )
& ~ ( member @ A @ A2 @ B8 ) ) ) ).
% mk_disjoint_insert
thf(fact_135_inj__on__inverseI,axiom,
! [B: $tType,A: $tType,A4: set @ A,G: B > A,F: A > B] :
( ! [X3: A] :
( ( member @ A @ X3 @ A4 )
=> ( ( G @ ( F @ X3 ) )
= X3 ) )
=> ( inj_on @ A @ B @ F @ A4 ) ) ).
% inj_on_inverseI
thf(fact_136_insert__commute,axiom,
! [A: $tType,X2: A,Y: A,A4: set @ A] :
( ( insert @ A @ X2 @ ( insert @ A @ Y @ A4 ) )
= ( insert @ A @ Y @ ( insert @ A @ X2 @ A4 ) ) ) ).
% insert_commute
thf(fact_137_inj__on__contraD,axiom,
! [B: $tType,A: $tType,F: A > B,A4: set @ A,X2: A,Y: A] :
( ( inj_on @ A @ B @ F @ A4 )
=> ( ( X2 != Y )
=> ( ( member @ A @ X2 @ A4 )
=> ( ( member @ A @ Y @ A4 )
=> ( ( F @ X2 )
!= ( F @ Y ) ) ) ) ) ) ).
% inj_on_contraD
thf(fact_138_insert__eq__iff,axiom,
! [A: $tType,A2: A,A4: set @ A,B2: A,B3: set @ A] :
( ~ ( member @ A @ A2 @ A4 )
=> ( ~ ( member @ A @ B2 @ B3 )
=> ( ( ( insert @ A @ A2 @ A4 )
= ( insert @ A @ B2 @ B3 ) )
= ( ( ( A2 = B2 )
=> ( A4 = B3 ) )
& ( ( A2 != B2 )
=> ? [C3: set @ A] :
( ( A4
= ( insert @ A @ B2 @ C3 ) )
& ~ ( member @ A @ B2 @ C3 )
& ( B3
= ( insert @ A @ A2 @ C3 ) )
& ~ ( member @ A @ A2 @ C3 ) ) ) ) ) ) ) ).
% insert_eq_iff
thf(fact_139_insert__absorb,axiom,
! [A: $tType,A2: A,A4: set @ A] :
( ( member @ A @ A2 @ A4 )
=> ( ( insert @ A @ A2 @ A4 )
= A4 ) ) ).
% insert_absorb
thf(fact_140_inj__on__eq__iff,axiom,
! [B: $tType,A: $tType,F: A > B,A4: set @ A,X2: A,Y: A] :
( ( inj_on @ A @ B @ F @ A4 )
=> ( ( member @ A @ X2 @ A4 )
=> ( ( member @ A @ Y @ A4 )
=> ( ( ( F @ X2 )
= ( F @ Y ) )
= ( X2 = Y ) ) ) ) ) ).
% inj_on_eq_iff
thf(fact_141_insert__ident,axiom,
! [A: $tType,X2: A,A4: set @ A,B3: set @ A] :
( ~ ( member @ A @ X2 @ A4 )
=> ( ~ ( member @ A @ X2 @ B3 )
=> ( ( ( insert @ A @ X2 @ A4 )
= ( insert @ A @ X2 @ B3 ) )
= ( A4 = B3 ) ) ) ) ).
% insert_ident
thf(fact_142_inj__on__cong,axiom,
! [B: $tType,A: $tType,A4: set @ A,F: A > B,G: A > B] :
( ! [A7: A] :
( ( member @ A @ A7 @ A4 )
=> ( ( F @ A7 )
= ( G @ A7 ) ) )
=> ( ( inj_on @ A @ B @ F @ A4 )
= ( inj_on @ A @ B @ G @ A4 ) ) ) ).
% inj_on_cong
thf(fact_143_Set_Oset__insert,axiom,
! [A: $tType,X2: A,A4: set @ A] :
( ( member @ A @ X2 @ A4 )
=> ~ ! [B8: set @ A] :
( ( A4
= ( insert @ A @ X2 @ B8 ) )
=> ( member @ A @ X2 @ B8 ) ) ) ).
% Set.set_insert
thf(fact_144_inj__on__def,axiom,
! [B: $tType,A: $tType] :
( ( inj_on @ A @ B )
= ( ^ [F6: A > B,A8: set @ A] :
! [X: A] :
( ( member @ A @ X @ A8 )
=> ! [Y3: A] :
( ( member @ A @ Y3 @ A8 )
=> ( ( ( F6 @ X )
= ( F6 @ Y3 ) )
=> ( X = Y3 ) ) ) ) ) ) ).
% inj_on_def
thf(fact_145_insertI2,axiom,
! [A: $tType,A2: A,B3: set @ A,B2: A] :
( ( member @ A @ A2 @ B3 )
=> ( member @ A @ A2 @ ( insert @ A @ B2 @ B3 ) ) ) ).
% insertI2
thf(fact_146_insertI1,axiom,
! [A: $tType,A2: A,B3: set @ A] : ( member @ A @ A2 @ ( insert @ A @ A2 @ B3 ) ) ).
% insertI1
thf(fact_147_insertE,axiom,
! [A: $tType,A2: A,B2: A,A4: set @ A] :
( ( member @ A @ A2 @ ( insert @ A @ B2 @ A4 ) )
=> ( ( A2 != B2 )
=> ( member @ A @ A2 @ A4 ) ) ) ).
% insertE
thf(fact_148_inj__onI,axiom,
! [B: $tType,A: $tType,A4: set @ A,F: A > B] :
( ! [X3: A,Y2: A] :
( ( member @ A @ X3 @ A4 )
=> ( ( member @ A @ Y2 @ A4 )
=> ( ( ( F @ X3 )
= ( F @ Y2 ) )
=> ( X3 = Y2 ) ) ) )
=> ( inj_on @ A @ B @ F @ A4 ) ) ).
% inj_onI
thf(fact_149_inj__onD,axiom,
! [B: $tType,A: $tType,F: A > B,A4: set @ A,X2: A,Y: A] :
( ( inj_on @ A @ B @ F @ A4 )
=> ( ( ( F @ X2 )
= ( F @ Y ) )
=> ( ( member @ A @ X2 @ A4 )
=> ( ( member @ A @ Y @ A4 )
=> ( X2 = Y ) ) ) ) ) ).
% inj_onD
thf(fact_150_finite__imageD,axiom,
! [A: $tType,B: $tType,F: B > A,A4: set @ B] :
( ( finite_finite2 @ A @ ( image @ B @ A @ F @ A4 ) )
=> ( ( inj_on @ B @ A @ F @ A4 )
=> ( finite_finite2 @ B @ A4 ) ) ) ).
% finite_imageD
thf(fact_151_finite__image__iff,axiom,
! [B: $tType,A: $tType,F: A > B,A4: set @ A] :
( ( inj_on @ A @ B @ F @ A4 )
=> ( ( finite_finite2 @ B @ ( image @ A @ B @ F @ A4 ) )
= ( finite_finite2 @ A @ A4 ) ) ) ).
% finite_image_iff
thf(fact_152_inFr__Ind__minus,axiom,
! [Ns13: set @ n,Tr1: dtree,T2: t,Tr: dtree] :
( ( gram_L1333338417e_inFr @ Ns13 @ Tr1 @ T2 )
=> ( ( member @ ( sum_sum @ t @ dtree ) @ ( sum_Inr @ dtree @ t @ Tr1 ) @ ( cont @ Tr ) )
=> ( gram_L1333338417e_inFr @ ( insert @ n @ ( root @ Tr ) @ Ns13 ) @ Tr @ T2 ) ) ) ).
% inFr_Ind_minus
thf(fact_153_inFr2_OInd,axiom,
! [Tr1: dtree,Tr: dtree,Ns13: set @ n,T2: t] :
( ( member @ ( sum_sum @ t @ dtree ) @ ( sum_Inr @ dtree @ t @ Tr1 ) @ ( cont @ Tr ) )
=> ( ( gram_L805317441_inFr2 @ Ns13 @ Tr1 @ T2 )
=> ( gram_L805317441_inFr2 @ ( insert @ n @ ( root @ Tr ) @ Ns13 ) @ Tr @ T2 ) ) ) ).
% inFr2.Ind
thf(fact_154_wf__subtr__inj__on,axiom,
! [Tr1: dtree,Ns: set @ n,Tr: dtree] :
( ( gram_L864798063lle_wf @ Tr1 )
=> ( ( gram_L716654942_subtr @ Ns @ Tr @ Tr1 )
=> ( inj_on @ dtree @ n @ root @ ( vimage @ dtree @ ( sum_sum @ t @ dtree ) @ ( sum_Inr @ dtree @ t ) @ ( cont @ Tr ) ) ) ) ) ).
% wf_subtr_inj_on
thf(fact_155_wf_Ocases,axiom,
! [A2: dtree] :
( ( gram_L864798063lle_wf @ A2 )
=> ~ ( ( member @ ( product_prod @ n @ ( set @ ( sum_sum @ t @ n ) ) ) @ ( product_Pair @ n @ ( set @ ( sum_sum @ t @ n ) ) @ ( root @ A2 ) @ ( image @ ( sum_sum @ t @ dtree ) @ ( sum_sum @ t @ n ) @ ( sum_map_sum @ t @ t @ dtree @ n @ ( id @ t ) @ root ) @ ( cont @ A2 ) ) ) @ gram_L1451583632elle_P )
=> ( ( inj_on @ dtree @ n @ root @ ( vimage @ dtree @ ( sum_sum @ t @ dtree ) @ ( sum_Inr @ dtree @ t ) @ ( cont @ A2 ) ) )
=> ~ ! [Tr6: dtree] :
( ( member @ dtree @ Tr6 @ ( vimage @ dtree @ ( sum_sum @ t @ dtree ) @ ( sum_Inr @ dtree @ t ) @ ( cont @ A2 ) ) )
=> ( gram_L864798063lle_wf @ Tr6 ) ) ) ) ) ).
% wf.cases
thf(fact_156_wf_Osimps,axiom,
( gram_L864798063lle_wf
= ( ^ [A10: dtree] :
? [Tr3: dtree] :
( ( A10 = Tr3 )
& ( member @ ( product_prod @ n @ ( set @ ( sum_sum @ t @ n ) ) ) @ ( product_Pair @ n @ ( set @ ( sum_sum @ t @ n ) ) @ ( root @ Tr3 ) @ ( image @ ( sum_sum @ t @ dtree ) @ ( sum_sum @ t @ n ) @ ( sum_map_sum @ t @ t @ dtree @ n @ ( id @ t ) @ root ) @ ( cont @ Tr3 ) ) ) @ gram_L1451583632elle_P )
& ( inj_on @ dtree @ n @ root @ ( vimage @ dtree @ ( sum_sum @ t @ dtree ) @ ( sum_Inr @ dtree @ t ) @ ( cont @ Tr3 ) ) )
& ! [X: dtree] :
( ( member @ dtree @ X @ ( vimage @ dtree @ ( sum_sum @ t @ dtree ) @ ( sum_Inr @ dtree @ t ) @ ( cont @ Tr3 ) ) )
=> ( gram_L864798063lle_wf @ X ) ) ) ) ) ).
% wf.simps
thf(fact_157_wf__coind,axiom,
! [Phi: dtree > $o,Tr: dtree] :
( ( Phi @ Tr )
=> ( ! [Tr4: dtree] :
( ( Phi @ Tr4 )
=> ( ( member @ ( product_prod @ n @ ( set @ ( sum_sum @ t @ n ) ) ) @ ( product_Pair @ n @ ( set @ ( sum_sum @ t @ n ) ) @ ( root @ Tr4 ) @ ( image @ ( sum_sum @ t @ dtree ) @ ( sum_sum @ t @ n ) @ ( sum_map_sum @ t @ t @ dtree @ n @ ( id @ t ) @ root ) @ ( cont @ Tr4 ) ) ) @ gram_L1451583632elle_P )
& ( inj_on @ dtree @ n @ root @ ( vimage @ dtree @ ( sum_sum @ t @ dtree ) @ ( sum_Inr @ dtree @ t ) @ ( cont @ Tr4 ) ) )
& ! [X3: dtree] :
( ( member @ dtree @ X3 @ ( vimage @ dtree @ ( sum_sum @ t @ dtree ) @ ( sum_Inr @ dtree @ t ) @ ( cont @ Tr4 ) ) )
=> ( ( Phi @ X3 )
| ( gram_L864798063lle_wf @ X3 ) ) ) ) )
=> ( gram_L864798063lle_wf @ Tr ) ) ) ).
% wf_coind
thf(fact_158_inj__Inr,axiom,
! [B: $tType,A: $tType,A4: set @ A] : ( inj_on @ A @ ( sum_sum @ B @ A ) @ ( sum_Inr @ A @ B ) @ A4 ) ).
% inj_Inr
thf(fact_159_inj__Inl,axiom,
! [B: $tType,A: $tType,A4: set @ A] : ( inj_on @ A @ ( sum_sum @ A @ B ) @ ( sum_Inl @ A @ B ) @ A4 ) ).
% inj_Inl
thf(fact_160_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_161_used,axiom,
! [N4: n] :
? [Tns2: set @ ( sum_sum @ t @ n )] : ( member @ ( product_prod @ n @ ( set @ ( sum_sum @ t @ n ) ) ) @ ( product_Pair @ n @ ( set @ ( sum_sum @ t @ n ) ) @ N4 @ Tns2 ) @ gram_L1451583632elle_P ) ).
% used
thf(fact_162_finite__in__P,axiom,
! [N4: n,Tns: set @ ( sum_sum @ t @ n )] :
( ( member @ ( product_prod @ n @ ( set @ ( sum_sum @ t @ n ) ) ) @ ( product_Pair @ n @ ( set @ ( sum_sum @ t @ n ) ) @ N4 @ Tns ) @ gram_L1451583632elle_P )
=> ( finite_finite2 @ ( sum_sum @ t @ n ) @ Tns ) ) ).
% finite_in_P
thf(fact_163_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_164_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_165_subtr_ORefl,axiom,
! [Tr: dtree,Ns: set @ n] :
( ( member @ n @ ( root @ Tr ) @ Ns )
=> ( gram_L716654942_subtr @ Ns @ Tr @ Tr ) ) ).
% subtr.Refl
thf(fact_166_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_167_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_168_wf__subtr__P,axiom,
! [Tr1: dtree,Ns: set @ n,Tr: dtree] :
( ( gram_L864798063lle_wf @ Tr1 )
=> ( ( gram_L716654942_subtr @ Ns @ Tr @ Tr1 )
=> ( member @ ( product_prod @ n @ ( set @ ( sum_sum @ t @ n ) ) ) @ ( product_Pair @ n @ ( set @ ( sum_sum @ t @ n ) ) @ ( root @ Tr ) @ ( image @ ( sum_sum @ t @ dtree ) @ ( sum_sum @ t @ n ) @ ( sum_map_sum @ t @ t @ dtree @ n @ ( id @ t ) @ root ) @ ( cont @ Tr ) ) ) @ gram_L1451583632elle_P ) ) ) ).
% wf_subtr_P
thf(fact_169_subtr__inductL,axiom,
! [Ns: set @ n,Tr1: dtree,Tr22: dtree,Phi: ( set @ n ) > dtree > dtree > $o] :
( ( gram_L716654942_subtr @ Ns @ Tr1 @ Tr22 )
=> ( ! [Ns2: set @ n,Tr4: dtree] : ( Phi @ Ns2 @ Tr4 @ Tr4 )
=> ( ! [Ns2: set @ n,Tr12: dtree,Tr23: dtree,Tr33: dtree] :
( ( member @ n @ ( root @ Tr12 ) @ Ns2 )
=> ( ( member @ ( sum_sum @ t @ dtree ) @ ( sum_Inr @ dtree @ t @ Tr12 ) @ ( cont @ Tr23 ) )
=> ( ( gram_L716654942_subtr @ Ns2 @ Tr23 @ Tr33 )
=> ( ( Phi @ Ns2 @ Tr23 @ Tr33 )
=> ( Phi @ Ns2 @ Tr12 @ Tr33 ) ) ) ) )
=> ( Phi @ Ns @ Tr1 @ Tr22 ) ) ) ) ).
% subtr_inductL
thf(fact_170_subtr_Oinducts,axiom,
! [X1: set @ n,X22: dtree,X32: dtree,P: ( set @ n ) > dtree > dtree > $o] :
( ( gram_L716654942_subtr @ X1 @ X22 @ X32 )
=> ( ! [Tr4: dtree,Ns2: set @ n] :
( ( member @ n @ ( root @ Tr4 ) @ Ns2 )
=> ( P @ Ns2 @ Tr4 @ Tr4 ) )
=> ( ! [Tr33: dtree,Ns2: set @ n,Tr12: dtree,Tr23: dtree] :
( ( member @ n @ ( root @ Tr33 ) @ Ns2 )
=> ( ( gram_L716654942_subtr @ Ns2 @ Tr12 @ Tr23 )
=> ( ( P @ Ns2 @ Tr12 @ Tr23 )
=> ( ( member @ ( sum_sum @ t @ dtree ) @ ( sum_Inr @ dtree @ t @ Tr23 ) @ ( cont @ Tr33 ) )
=> ( P @ Ns2 @ Tr12 @ Tr33 ) ) ) ) )
=> ( P @ X1 @ X22 @ X32 ) ) ) ) ).
% subtr.inducts
thf(fact_171_subtr__StepL,axiom,
! [Tr1: dtree,Ns: set @ n,Tr22: dtree,Tr32: dtree] :
( ( member @ n @ ( root @ Tr1 ) @ Ns )
=> ( ( member @ ( sum_sum @ t @ dtree ) @ ( sum_Inr @ dtree @ t @ Tr1 ) @ ( cont @ Tr22 ) )
=> ( ( gram_L716654942_subtr @ Ns @ Tr22 @ Tr32 )
=> ( gram_L716654942_subtr @ Ns @ Tr1 @ Tr32 ) ) ) ) ).
% subtr_StepL
thf(fact_172_subtr_Osimps,axiom,
( gram_L716654942_subtr
= ( ^ [A12: set @ n,A23: dtree,A33: dtree] :
( ? [Tr3: dtree,Ns3: set @ n] :
( ( A12 = Ns3 )
& ( A23 = Tr3 )
& ( A33 = Tr3 )
& ( member @ n @ ( root @ Tr3 ) @ Ns3 ) )
| ? [Tr34: dtree,Ns3: set @ n,Tr13: dtree,Tr24: dtree] :
( ( A12 = Ns3 )
& ( A23 = Tr13 )
& ( A33 = Tr34 )
& ( member @ n @ ( root @ Tr34 ) @ Ns3 )
& ( gram_L716654942_subtr @ Ns3 @ Tr13 @ Tr24 )
& ( member @ ( sum_sum @ t @ dtree ) @ ( sum_Inr @ dtree @ t @ Tr24 ) @ ( cont @ Tr34 ) ) ) ) ) ) ).
% subtr.simps
thf(fact_173_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_174_subtr_OStep,axiom,
! [Tr32: dtree,Ns: set @ n,Tr1: dtree,Tr22: dtree] :
( ( member @ n @ ( root @ Tr32 ) @ Ns )
=> ( ( gram_L716654942_subtr @ Ns @ Tr1 @ Tr22 )
=> ( ( member @ ( sum_sum @ t @ dtree ) @ ( sum_Inr @ dtree @ t @ Tr22 ) @ ( cont @ Tr32 ) )
=> ( gram_L716654942_subtr @ Ns @ Tr1 @ Tr32 ) ) ) ) ).
% subtr.Step
thf(fact_175_inFr__subtr,axiom,
! [Ns: set @ n,Tr: dtree,T2: t] :
( ( gram_L1333338417e_inFr @ Ns @ Tr @ T2 )
=> ? [Tr7: dtree] :
( ( gram_L716654942_subtr @ Ns @ Tr7 @ Tr )
& ( member @ ( sum_sum @ t @ dtree ) @ ( sum_Inl @ t @ dtree @ T2 ) @ ( cont @ Tr7 ) ) ) ) ).
% inFr_subtr
thf(fact_176_wf__P,axiom,
! [Tr: dtree] :
( ( gram_L864798063lle_wf @ Tr )
=> ( member @ ( product_prod @ n @ ( set @ ( sum_sum @ t @ n ) ) ) @ ( product_Pair @ n @ ( set @ ( sum_sum @ t @ n ) ) @ ( root @ Tr ) @ ( image @ ( sum_sum @ t @ dtree ) @ ( sum_sum @ t @ n ) @ ( sum_map_sum @ t @ t @ dtree @ n @ ( id @ t ) @ root ) @ ( cont @ Tr ) ) ) @ gram_L1451583632elle_P ) ) ).
% wf_P
thf(fact_177_wf__raw__coind,axiom,
! [Phi: dtree > $o,Tr: dtree] :
( ( Phi @ Tr )
=> ( ! [Tr4: dtree] :
( ( Phi @ Tr4 )
=> ( ( member @ ( product_prod @ n @ ( set @ ( sum_sum @ t @ n ) ) ) @ ( product_Pair @ n @ ( set @ ( sum_sum @ t @ n ) ) @ ( root @ Tr4 ) @ ( image @ ( sum_sum @ t @ dtree ) @ ( sum_sum @ t @ n ) @ ( sum_map_sum @ t @ t @ dtree @ n @ ( id @ t ) @ root ) @ ( cont @ Tr4 ) ) ) @ gram_L1451583632elle_P )
& ( inj_on @ dtree @ n @ root @ ( vimage @ dtree @ ( sum_sum @ t @ dtree ) @ ( sum_Inr @ dtree @ t ) @ ( cont @ Tr4 ) ) )
& ! [X3: dtree] :
( ( member @ dtree @ X3 @ ( vimage @ dtree @ ( sum_sum @ t @ dtree ) @ ( sum_Inr @ dtree @ t ) @ ( cont @ Tr4 ) ) )
=> ( Phi @ X3 ) ) ) )
=> ( gram_L864798063lle_wf @ Tr ) ) ) ).
% wf_raw_coind
thf(fact_178_wf_Ocoinduct,axiom,
! [X7: dtree > $o,X2: dtree] :
( ( X7 @ X2 )
=> ( ! [X3: dtree] :
( ( X7 @ X3 )
=> ? [Tr8: dtree] :
( ( X3 = Tr8 )
& ( member @ ( product_prod @ n @ ( set @ ( sum_sum @ t @ n ) ) ) @ ( product_Pair @ n @ ( set @ ( sum_sum @ t @ n ) ) @ ( root @ Tr8 ) @ ( image @ ( sum_sum @ t @ dtree ) @ ( sum_sum @ t @ n ) @ ( sum_map_sum @ t @ t @ dtree @ n @ ( id @ t ) @ root ) @ ( cont @ Tr8 ) ) ) @ gram_L1451583632elle_P )
& ( inj_on @ dtree @ n @ root @ ( vimage @ dtree @ ( sum_sum @ t @ dtree ) @ ( sum_Inr @ dtree @ t ) @ ( cont @ Tr8 ) ) )
& ! [Xa: dtree] :
( ( member @ dtree @ Xa @ ( vimage @ dtree @ ( sum_sum @ t @ dtree ) @ ( sum_Inr @ dtree @ t ) @ ( cont @ Tr8 ) ) )
=> ( ( X7 @ Xa )
| ( gram_L864798063lle_wf @ Xa ) ) ) ) )
=> ( gram_L864798063lle_wf @ X2 ) ) ) ).
% wf.coinduct
thf(fact_179_wf_Ointros,axiom,
! [Tr: dtree] :
( ( member @ ( product_prod @ n @ ( set @ ( sum_sum @ t @ n ) ) ) @ ( product_Pair @ n @ ( set @ ( sum_sum @ t @ n ) ) @ ( root @ Tr ) @ ( image @ ( sum_sum @ t @ dtree ) @ ( sum_sum @ t @ n ) @ ( sum_map_sum @ t @ t @ dtree @ n @ ( id @ t ) @ root ) @ ( cont @ Tr ) ) ) @ gram_L1451583632elle_P )
=> ( ( inj_on @ dtree @ n @ root @ ( vimage @ dtree @ ( sum_sum @ t @ dtree ) @ ( sum_Inr @ dtree @ t ) @ ( cont @ Tr ) ) )
=> ( ! [Tr7: dtree] :
( ( member @ dtree @ Tr7 @ ( vimage @ dtree @ ( sum_sum @ t @ dtree ) @ ( sum_Inr @ dtree @ t ) @ ( cont @ Tr ) ) )
=> ( gram_L864798063lle_wf @ Tr7 ) )
=> ( gram_L864798063lle_wf @ Tr ) ) ) ) ).
% wf.intros
thf(fact_180_S__P,axiom,
! [N4: n] : ( member @ ( product_prod @ n @ ( set @ ( sum_sum @ t @ n ) ) ) @ ( product_Pair @ n @ ( set @ ( sum_sum @ t @ n ) ) @ N4 @ ( gram_L1451583635elle_S @ N4 ) ) @ gram_L1451583632elle_P ) ).
% S_P
thf(fact_181_inItr_Oinducts,axiom,
! [X1: set @ n,X22: dtree,X32: n,P: ( set @ n ) > dtree > n > $o] :
( ( gram_L830233218_inItr @ X1 @ X22 @ X32 )
=> ( ! [Tr4: dtree,Ns2: set @ n] :
( ( member @ n @ ( root @ Tr4 ) @ Ns2 )
=> ( P @ Ns2 @ Tr4 @ ( root @ Tr4 ) ) )
=> ( ! [Tr4: dtree,Ns2: set @ n,Tr12: dtree,N3: n] :
( ( member @ n @ ( root @ Tr4 ) @ Ns2 )
=> ( ( member @ ( sum_sum @ t @ dtree ) @ ( sum_Inr @ dtree @ t @ Tr12 ) @ ( cont @ Tr4 ) )
=> ( ( gram_L830233218_inItr @ Ns2 @ Tr12 @ N3 )
=> ( ( P @ Ns2 @ Tr12 @ N3 )
=> ( P @ Ns2 @ Tr4 @ N3 ) ) ) ) )
=> ( P @ X1 @ X22 @ X32 ) ) ) ) ).
% inItr.inducts
thf(fact_182_inItr_Osimps,axiom,
( gram_L830233218_inItr
= ( ^ [A12: set @ n,A23: dtree,A33: n] :
( ? [Tr3: dtree,Ns3: set @ n] :
( ( A12 = Ns3 )
& ( A23 = Tr3 )
& ( A33
= ( root @ Tr3 ) )
& ( member @ n @ ( root @ Tr3 ) @ Ns3 ) )
| ? [Tr3: dtree,Ns3: set @ n,Tr13: dtree,N2: n] :
( ( A12 = Ns3 )
& ( A23 = Tr3 )
& ( A33 = N2 )
& ( member @ n @ ( root @ Tr3 ) @ Ns3 )
& ( member @ ( sum_sum @ t @ dtree ) @ ( sum_Inr @ dtree @ t @ Tr13 ) @ ( cont @ Tr3 ) )
& ( gram_L830233218_inItr @ Ns3 @ Tr13 @ N2 ) ) ) ) ) ).
% inItr.simps
thf(fact_183_finite__S,axiom,
! [N4: n] : ( finite_finite2 @ ( sum_sum @ t @ n ) @ ( gram_L1451583635elle_S @ N4 ) ) ).
% finite_S
thf(fact_184_inItr_OBase,axiom,
! [Tr: dtree,Ns: set @ n] :
( ( member @ n @ ( root @ Tr ) @ Ns )
=> ( gram_L830233218_inItr @ Ns @ Tr @ ( root @ Tr ) ) ) ).
% inItr.Base
thf(fact_185_inItr__root__in,axiom,
! [Ns: set @ n,Tr: dtree,N4: n] :
( ( gram_L830233218_inItr @ Ns @ Tr @ N4 )
=> ( member @ n @ ( root @ Tr ) @ Ns ) ) ).
% inItr_root_in
thf(fact_186_subtr__inItr,axiom,
! [Ns: set @ n,Tr: dtree,N4: n,Tr1: dtree] :
( ( gram_L830233218_inItr @ Ns @ Tr @ N4 )
=> ( ( gram_L716654942_subtr @ Ns @ Tr @ Tr1 )
=> ( gram_L830233218_inItr @ Ns @ Tr1 @ N4 ) ) ) ).
% subtr_inItr
thf(fact_187_inItr__subtr,axiom,
! [Ns: set @ n,Tr: dtree,N4: n] :
( ( gram_L830233218_inItr @ Ns @ Tr @ N4 )
=> ? [Tr7: dtree] :
( ( gram_L716654942_subtr @ Ns @ Tr7 @ Tr )
& ( ( root @ Tr7 )
= N4 ) ) ) ).
% inItr_subtr
thf(fact_188_inItr_OInd,axiom,
! [Tr: dtree,Ns: set @ n,Tr1: dtree,N4: n] :
( ( member @ n @ ( root @ Tr ) @ Ns )
=> ( ( member @ ( sum_sum @ t @ dtree ) @ ( sum_Inr @ dtree @ t @ Tr1 ) @ ( cont @ Tr ) )
=> ( ( gram_L830233218_inItr @ Ns @ Tr1 @ N4 )
=> ( gram_L830233218_inItr @ Ns @ Tr @ N4 ) ) ) ) ).
% inItr.Ind
thf(fact_189_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_190_cont__deftr,axiom,
! [N4: n] :
( ( cont @ ( gram_L1231612515_deftr @ N4 ) )
= ( image @ ( sum_sum @ t @ n ) @ ( sum_sum @ t @ dtree ) @ ( sum_map_sum @ t @ t @ n @ dtree @ ( id @ t ) @ gram_L1231612515_deftr ) @ ( gram_L1451583635elle_S @ N4 ) ) ) ).
% cont_deftr
thf(fact_191_subtr2_OStep,axiom,
! [Tr1: dtree,Ns: set @ n,Tr22: dtree,Tr32: dtree] :
( ( member @ n @ ( root @ Tr1 ) @ Ns )
=> ( ( member @ ( sum_sum @ t @ dtree ) @ ( sum_Inr @ dtree @ t @ Tr1 ) @ ( cont @ Tr22 ) )
=> ( ( gram_L1283001940subtr2 @ Ns @ Tr22 @ Tr32 )
=> ( gram_L1283001940subtr2 @ Ns @ Tr1 @ Tr32 ) ) ) ) ).
% subtr2.Step
thf(fact_192_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_193_root__deftr,axiom,
! [N4: n] :
( ( root @ ( gram_L1231612515_deftr @ N4 ) )
= N4 ) ).
% root_deftr
thf(fact_194_subtr__subtr2,axiom,
gram_L716654942_subtr = gram_L1283001940subtr2 ).
% subtr_subtr2
thf(fact_195_subtr2__trans,axiom,
! [Ns: set @ n,Tr1: dtree,Tr22: dtree,Tr32: dtree] :
( ( gram_L1283001940subtr2 @ Ns @ Tr1 @ Tr22 )
=> ( ( gram_L1283001940subtr2 @ Ns @ Tr22 @ Tr32 )
=> ( gram_L1283001940subtr2 @ Ns @ Tr1 @ Tr32 ) ) ) ).
% subtr2_trans
thf(fact_196_subtr2_ORefl,axiom,
! [Tr: dtree,Ns: set @ n] :
( ( member @ n @ ( root @ Tr ) @ Ns )
=> ( gram_L1283001940subtr2 @ Ns @ Tr @ Tr ) ) ).
% subtr2.Refl
thf(fact_197_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_198_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_199_wf__deftr,axiom,
! [N4: n] : ( gram_L864798063lle_wf @ ( gram_L1231612515_deftr @ N4 ) ) ).
% wf_deftr
thf(fact_200_deftr__def,axiom,
( gram_L1231612515_deftr
= ( unfold @ n @ ( id @ n ) @ gram_L1451583635elle_S ) ) ).
% deftr_def
thf(fact_201_subtr2_Oinducts,axiom,
! [X1: set @ n,X22: dtree,X32: dtree,P: ( set @ n ) > dtree > dtree > $o] :
( ( gram_L1283001940subtr2 @ X1 @ X22 @ X32 )
=> ( ! [Tr4: dtree,Ns2: set @ n] :
( ( member @ n @ ( root @ Tr4 ) @ Ns2 )
=> ( P @ Ns2 @ Tr4 @ Tr4 ) )
=> ( ! [Tr12: dtree,Ns2: set @ n,Tr23: dtree,Tr33: dtree] :
( ( member @ n @ ( root @ Tr12 ) @ Ns2 )
=> ( ( member @ ( sum_sum @ t @ dtree ) @ ( sum_Inr @ dtree @ t @ Tr12 ) @ ( cont @ Tr23 ) )
=> ( ( gram_L1283001940subtr2 @ Ns2 @ Tr23 @ Tr33 )
=> ( ( P @ Ns2 @ Tr23 @ Tr33 )
=> ( P @ Ns2 @ Tr12 @ Tr33 ) ) ) ) )
=> ( P @ X1 @ X22 @ X32 ) ) ) ) ).
% subtr2.inducts
thf(fact_202_subtr2__StepR,axiom,
! [Tr32: dtree,Ns: set @ n,Tr22: dtree,Tr1: dtree] :
( ( member @ n @ ( root @ Tr32 ) @ Ns )
=> ( ( member @ ( sum_sum @ t @ dtree ) @ ( sum_Inr @ dtree @ t @ Tr22 ) @ ( cont @ Tr32 ) )
=> ( ( gram_L1283001940subtr2 @ Ns @ Tr1 @ Tr22 )
=> ( gram_L1283001940subtr2 @ Ns @ Tr1 @ Tr32 ) ) ) ) ).
% subtr2_StepR
thf(fact_203_subtr2_Osimps,axiom,
( gram_L1283001940subtr2
= ( ^ [A12: set @ n,A23: dtree,A33: dtree] :
( ? [Tr3: dtree,Ns3: set @ n] :
( ( A12 = Ns3 )
& ( A23 = Tr3 )
& ( A33 = Tr3 )
& ( member @ n @ ( root @ Tr3 ) @ Ns3 ) )
| ? [Tr13: dtree,Ns3: set @ n,Tr24: dtree,Tr34: dtree] :
( ( A12 = Ns3 )
& ( A23 = Tr13 )
& ( A33 = Tr34 )
& ( member @ n @ ( root @ Tr13 ) @ Ns3 )
& ( member @ ( sum_sum @ t @ dtree ) @ ( sum_Inr @ dtree @ t @ Tr13 ) @ ( cont @ Tr24 ) )
& ( gram_L1283001940subtr2 @ Ns3 @ Tr24 @ Tr34 ) ) ) ) ) ).
% subtr2.simps
thf(fact_204_root__o__subst,axiom,
( ( comp @ dtree @ n @ dtree @ root @ ( gram_L1004374585hsubst @ tr0 ) )
= root ) ).
% root_o_subst
thf(fact_205_inf__img__fin__dom,axiom,
! [B: $tType,A: $tType,F: B > A,A4: set @ B] :
( ( finite_finite2 @ A @ ( image @ B @ A @ F @ A4 ) )
=> ( ~ ( finite_finite2 @ B @ A4 )
=> ? [X3: A] :
( ( member @ A @ X3 @ ( image @ B @ A @ F @ A4 ) )
& ~ ( finite_finite2 @ B @ ( vimage @ B @ A @ F @ ( insert @ A @ X3 @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ) ) ).
% inf_img_fin_dom
thf(fact_206_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_207_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_208_all__not__in__conv,axiom,
! [A: $tType,A4: set @ A] :
( ( ! [X: A] :
~ ( member @ A @ X @ A4 ) )
= ( A4
= ( bot_bot @ ( set @ A ) ) ) ) ).
% all_not_in_conv
thf(fact_209_empty__iff,axiom,
! [A: $tType,C4: A] :
~ ( member @ A @ C4 @ ( bot_bot @ ( set @ A ) ) ) ).
% empty_iff
thf(fact_210_comp__apply,axiom,
! [C: $tType,A: $tType,B: $tType] :
( ( comp @ B @ A @ C )
= ( ^ [F6: B > A,G3: C > B,X: C] : ( F6 @ ( G3 @ X ) ) ) ) ).
% comp_apply
thf(fact_211_image__is__empty,axiom,
! [A: $tType,B: $tType,F: B > A,A4: set @ B] :
( ( ( image @ B @ A @ F @ A4 )
= ( bot_bot @ ( set @ A ) ) )
= ( A4
= ( bot_bot @ ( set @ B ) ) ) ) ).
% image_is_empty
thf(fact_212_empty__is__image,axiom,
! [A: $tType,B: $tType,F: B > A,A4: set @ B] :
( ( ( bot_bot @ ( set @ A ) )
= ( image @ B @ A @ F @ A4 ) )
= ( A4
= ( bot_bot @ ( set @ B ) ) ) ) ).
% empty_is_image
thf(fact_213_image__empty,axiom,
! [B: $tType,A: $tType,F: B > A] :
( ( image @ B @ A @ F @ ( bot_bot @ ( set @ B ) ) )
= ( bot_bot @ ( set @ A ) ) ) ).
% image_empty
thf(fact_214_singletonI,axiom,
! [A: $tType,A2: A] : ( member @ A @ A2 @ ( insert @ A @ A2 @ ( bot_bot @ ( set @ A ) ) ) ) ).
% singletonI
thf(fact_215_inj__on__empty,axiom,
! [B: $tType,A: $tType,F: A > B] : ( inj_on @ A @ B @ F @ ( bot_bot @ ( set @ A ) ) ) ).
% inj_on_empty
thf(fact_216_fun_Omap__id,axiom,
! [A: $tType,D: $tType,T2: D > A] :
( ( comp @ A @ A @ D @ ( id @ A ) @ T2 )
= T2 ) ).
% fun.map_id
thf(fact_217_comp__id,axiom,
! [B: $tType,A: $tType,F: A > B] :
( ( comp @ A @ B @ A @ F @ ( id @ A ) )
= F ) ).
% comp_id
thf(fact_218_id__comp,axiom,
! [B: $tType,A: $tType,G: A > B] :
( ( comp @ B @ B @ A @ ( id @ B ) @ G )
= G ) ).
% id_comp
thf(fact_219_vimage__empty,axiom,
! [B: $tType,A: $tType,F: A > B] :
( ( vimage @ A @ B @ F @ ( bot_bot @ ( set @ B ) ) )
= ( bot_bot @ ( set @ A ) ) ) ).
% vimage_empty
thf(fact_220_comp__inj__on__iff,axiom,
! [B: $tType,C: $tType,A: $tType,F: A > B,A4: set @ A,F7: B > C] :
( ( inj_on @ A @ B @ F @ A4 )
=> ( ( inj_on @ B @ C @ F7 @ ( image @ A @ B @ F @ A4 ) )
= ( inj_on @ A @ C @ ( comp @ B @ C @ A @ F7 @ F ) @ A4 ) ) ) ).
% comp_inj_on_iff
thf(fact_221_inj__on__imageI,axiom,
! [B: $tType,C: $tType,A: $tType,G: C > B,F: A > C,A4: set @ A] :
( ( inj_on @ A @ B @ ( comp @ C @ B @ A @ G @ F ) @ A4 )
=> ( inj_on @ C @ B @ G @ ( image @ A @ C @ F @ A4 ) ) ) ).
% inj_on_imageI
thf(fact_222_comp__inj__on,axiom,
! [B: $tType,C: $tType,A: $tType,F: A > B,A4: set @ A,G: B > C] :
( ( inj_on @ A @ B @ F @ A4 )
=> ( ( inj_on @ B @ C @ G @ ( image @ A @ B @ F @ A4 ) )
=> ( inj_on @ A @ C @ ( comp @ B @ C @ A @ G @ F ) @ A4 ) ) ) ).
% comp_inj_on
thf(fact_223_map__sum_Ocompositionality,axiom,
! [D: $tType,F3: $tType,E2: $tType,C: $tType,B: $tType,A: $tType,F: C > E2,G: D > F3,H2: A > C,I: B > D,Sum: sum_sum @ A @ B] :
( ( sum_map_sum @ C @ E2 @ D @ F3 @ F @ G @ ( sum_map_sum @ A @ C @ B @ D @ H2 @ I @ Sum ) )
= ( sum_map_sum @ A @ E2 @ B @ F3 @ ( comp @ C @ E2 @ A @ F @ H2 ) @ ( comp @ D @ F3 @ B @ G @ I ) @ Sum ) ) ).
% map_sum.compositionality
thf(fact_224_sum_Omap__comp,axiom,
! [D: $tType,F3: $tType,E2: $tType,C: $tType,B: $tType,A: $tType,G1: C > E2,G22: D > F3,F1: A > C,F2: B > D,V: sum_sum @ A @ B] :
( ( sum_map_sum @ C @ E2 @ D @ F3 @ G1 @ G22 @ ( sum_map_sum @ A @ C @ B @ D @ F1 @ F2 @ V ) )
= ( sum_map_sum @ A @ E2 @ B @ F3 @ ( comp @ C @ E2 @ A @ G1 @ F1 ) @ ( comp @ D @ F3 @ B @ G22 @ F2 ) @ V ) ) ).
% sum.map_comp
thf(fact_225_map__sum_Ocomp,axiom,
! [A: $tType,C: $tType,E2: $tType,F3: $tType,D: $tType,B: $tType,F: C > E2,G: D > F3,H2: A > C,I: B > D] :
( ( comp @ ( sum_sum @ C @ D ) @ ( sum_sum @ E2 @ F3 ) @ ( sum_sum @ A @ B ) @ ( sum_map_sum @ C @ E2 @ D @ F3 @ F @ G ) @ ( sum_map_sum @ A @ C @ B @ D @ H2 @ I ) )
= ( sum_map_sum @ A @ E2 @ B @ F3 @ ( comp @ C @ E2 @ A @ F @ H2 ) @ ( comp @ D @ F3 @ B @ G @ I ) ) ) ).
% map_sum.comp
thf(fact_226_Inf_OINF__image,axiom,
! [B: $tType,A: $tType,C: $tType,Inf: ( set @ A ) > A,G: B > A,F: C > B,A4: set @ C] :
( ( Inf @ ( image @ B @ A @ G @ ( image @ C @ B @ F @ A4 ) ) )
= ( Inf @ ( image @ C @ A @ ( comp @ B @ A @ C @ G @ F ) @ A4 ) ) ) ).
% Inf.INF_image
thf(fact_227_Sup_OSUP__image,axiom,
! [B: $tType,A: $tType,C: $tType,Sup: ( set @ A ) > A,G: B > A,F: C > B,A4: set @ C] :
( ( Sup @ ( image @ B @ A @ G @ ( image @ C @ B @ F @ A4 ) ) )
= ( Sup @ ( image @ C @ A @ ( comp @ B @ A @ C @ G @ F ) @ A4 ) ) ) ).
% Sup.SUP_image
thf(fact_228_image__comp,axiom,
! [B: $tType,A: $tType,C: $tType,F: B > A,G: C > B,R: set @ C] :
( ( image @ B @ A @ F @ ( image @ C @ B @ G @ R ) )
= ( image @ C @ A @ ( comp @ B @ A @ C @ F @ G ) @ R ) ) ).
% image_comp
thf(fact_229_image__eq__imp__comp,axiom,
! [B: $tType,A: $tType,D: $tType,C: $tType,F: B > A,A4: set @ B,G: C > A,B3: set @ C,H2: A > D] :
( ( ( image @ B @ A @ F @ A4 )
= ( image @ C @ A @ G @ B3 ) )
=> ( ( image @ B @ D @ ( comp @ A @ D @ B @ H2 @ F ) @ A4 )
= ( image @ C @ D @ ( comp @ A @ D @ C @ H2 @ G ) @ B3 ) ) ) ).
% image_eq_imp_comp
thf(fact_230_Gram__Lang__Mirabelle__ojxrtuoybn_Oroot__o__subst,axiom,
! [Tr0: dtree] :
( ( comp @ dtree @ n @ dtree @ root @ ( gram_L1004374585hsubst @ Tr0 ) )
= root ) ).
% Gram_Lang_Mirabelle_ojxrtuoybn.root_o_subst
thf(fact_231_infinite__finite__induct,axiom,
! [A: $tType,P: ( set @ A ) > $o,A4: set @ A] :
( ! [A11: set @ A] :
( ~ ( finite_finite2 @ A @ A11 )
=> ( P @ A11 ) )
=> ( ( P @ ( bot_bot @ ( set @ A ) ) )
=> ( ! [X3: A,F8: set @ A] :
( ( finite_finite2 @ A @ F8 )
=> ( ~ ( member @ A @ X3 @ F8 )
=> ( ( P @ F8 )
=> ( P @ ( insert @ A @ X3 @ F8 ) ) ) ) )
=> ( P @ A4 ) ) ) ) ).
% infinite_finite_induct
thf(fact_232_finite__ne__induct,axiom,
! [A: $tType,F4: set @ A,P: ( set @ A ) > $o] :
( ( finite_finite2 @ A @ F4 )
=> ( ( F4
!= ( bot_bot @ ( set @ A ) ) )
=> ( ! [X3: A] : ( P @ ( insert @ A @ X3 @ ( bot_bot @ ( set @ A ) ) ) )
=> ( ! [X3: A,F8: set @ A] :
( ( finite_finite2 @ A @ F8 )
=> ( ( F8
!= ( bot_bot @ ( set @ A ) ) )
=> ( ~ ( member @ A @ X3 @ F8 )
=> ( ( P @ F8 )
=> ( P @ ( insert @ A @ X3 @ F8 ) ) ) ) ) )
=> ( P @ F4 ) ) ) ) ) ).
% finite_ne_induct
thf(fact_233_finite_Oinducts,axiom,
! [A: $tType,X2: set @ A,P: ( set @ A ) > $o] :
( ( finite_finite2 @ A @ X2 )
=> ( ( P @ ( bot_bot @ ( set @ A ) ) )
=> ( ! [A11: set @ A,A7: A] :
( ( finite_finite2 @ A @ A11 )
=> ( ( P @ A11 )
=> ( P @ ( insert @ A @ A7 @ A11 ) ) ) )
=> ( P @ X2 ) ) ) ) ).
% finite.inducts
thf(fact_234_finite__induct,axiom,
! [A: $tType,F4: set @ A,P: ( set @ A ) > $o] :
( ( finite_finite2 @ A @ F4 )
=> ( ( P @ ( bot_bot @ ( set @ A ) ) )
=> ( ! [X3: A,F8: set @ A] :
( ( finite_finite2 @ A @ F8 )
=> ( ~ ( member @ A @ X3 @ F8 )
=> ( ( P @ F8 )
=> ( P @ ( insert @ A @ X3 @ F8 ) ) ) ) )
=> ( P @ F4 ) ) ) ) ).
% finite_induct
thf(fact_235_finite_Osimps,axiom,
! [A: $tType] :
( ( finite_finite2 @ A )
= ( ^ [A10: set @ A] :
( ( A10
= ( bot_bot @ ( set @ A ) ) )
| ? [A8: set @ A,B9: A] :
( ( A10
= ( insert @ A @ B9 @ A8 ) )
& ( finite_finite2 @ A @ A8 ) ) ) ) ) ).
% finite.simps
thf(fact_236_finite_Ocases,axiom,
! [A: $tType,A2: set @ A] :
( ( finite_finite2 @ A @ A2 )
=> ( ( A2
!= ( bot_bot @ ( set @ A ) ) )
=> ~ ! [A11: set @ A] :
( ? [A7: A] :
( A2
= ( insert @ A @ A7 @ A11 ) )
=> ~ ( finite_finite2 @ A @ A11 ) ) ) ) ).
% finite.cases
thf(fact_237_singleton__inject,axiom,
! [A: $tType,A2: A,B2: A] :
( ( ( insert @ A @ A2 @ ( bot_bot @ ( set @ A ) ) )
= ( insert @ A @ B2 @ ( bot_bot @ ( set @ A ) ) ) )
=> ( A2 = B2 ) ) ).
% singleton_inject
thf(fact_238_insert__not__empty,axiom,
! [A: $tType,A2: A,A4: set @ A] :
( ( insert @ A @ A2 @ A4 )
!= ( bot_bot @ ( set @ A ) ) ) ).
% insert_not_empty
thf(fact_239_doubleton__eq__iff,axiom,
! [A: $tType,A2: A,B2: A,C4: A,D3: A] :
( ( ( insert @ A @ A2 @ ( insert @ A @ B2 @ ( bot_bot @ ( set @ A ) ) ) )
= ( insert @ A @ C4 @ ( insert @ A @ D3 @ ( bot_bot @ ( set @ A ) ) ) ) )
= ( ( ( A2 = C4 )
& ( B2 = D3 ) )
| ( ( A2 = D3 )
& ( B2 = C4 ) ) ) ) ).
% doubleton_eq_iff
thf(fact_240_singleton__iff,axiom,
! [A: $tType,B2: A,A2: A] :
( ( member @ A @ B2 @ ( insert @ A @ A2 @ ( bot_bot @ ( set @ A ) ) ) )
= ( B2 = A2 ) ) ).
% singleton_iff
thf(fact_241_singletonD,axiom,
! [A: $tType,B2: A,A2: A] :
( ( member @ A @ B2 @ ( insert @ A @ A2 @ ( bot_bot @ ( set @ A ) ) ) )
=> ( B2 = A2 ) ) ).
% singletonD
thf(fact_242_fun_Omap__id0,axiom,
! [A: $tType,D: $tType] :
( ( comp @ A @ A @ D @ ( id @ A ) )
= ( id @ ( D > A ) ) ) ).
% fun.map_id0
thf(fact_243_pointfree__idE,axiom,
! [B: $tType,A: $tType,F: B > A,G: A > B,X2: A] :
( ( ( comp @ B @ A @ A @ F @ G )
= ( id @ A ) )
=> ( ( F @ ( G @ X2 ) )
= X2 ) ) ).
% pointfree_idE
thf(fact_244_comp__eq__id__dest,axiom,
! [C: $tType,B: $tType,A: $tType,A2: C > B,B2: A > C,C4: A > B,V: A] :
( ( ( comp @ C @ B @ A @ A2 @ B2 )
= ( comp @ B @ B @ A @ ( id @ B ) @ C4 ) )
=> ( ( A2 @ ( B2 @ V ) )
= ( C4 @ V ) ) ) ).
% comp_eq_id_dest
thf(fact_245_vimage__comp,axiom,
! [A: $tType,B: $tType,C: $tType,F: A > B,G: B > C,X2: set @ C] :
( ( vimage @ A @ B @ F @ ( vimage @ B @ C @ G @ X2 ) )
= ( vimage @ A @ C @ ( comp @ B @ C @ A @ G @ F ) @ X2 ) ) ).
% vimage_comp
thf(fact_246_set_Ocompositionality,axiom,
! [C: $tType,B: $tType,A: $tType,F: C > B,G: B > A,Set: set @ A] :
( ( vimage @ C @ B @ F @ ( vimage @ B @ A @ G @ Set ) )
= ( vimage @ C @ A @ ( comp @ B @ A @ C @ G @ F ) @ Set ) ) ).
% set.compositionality
thf(fact_247_rewriteR__comp__comp2,axiom,
! [C: $tType,B: $tType,E2: $tType,D: $tType,A: $tType,G: C > B,H2: A > C,R1: D > B,R2: A > D,F: B > E2,L: D > E2] :
( ( ( comp @ C @ B @ A @ G @ H2 )
= ( comp @ D @ B @ A @ R1 @ R2 ) )
=> ( ( ( comp @ B @ E2 @ D @ F @ R1 )
= L )
=> ( ( comp @ C @ E2 @ A @ ( comp @ B @ E2 @ C @ F @ G ) @ H2 )
= ( comp @ D @ E2 @ A @ L @ R2 ) ) ) ) ).
% rewriteR_comp_comp2
thf(fact_248_rewriteL__comp__comp2,axiom,
! [A: $tType,C: $tType,B: $tType,D: $tType,E2: $tType,F: C > B,G: A > C,L1: D > B,L2: A > D,H2: E2 > A,R: E2 > D] :
( ( ( comp @ C @ B @ A @ F @ G )
= ( comp @ D @ B @ A @ L1 @ L2 ) )
=> ( ( ( comp @ A @ D @ E2 @ L2 @ H2 )
= R )
=> ( ( comp @ C @ B @ E2 @ F @ ( comp @ A @ C @ E2 @ G @ H2 ) )
= ( comp @ D @ B @ E2 @ L1 @ R ) ) ) ) ).
% rewriteL_comp_comp2
thf(fact_249_rewriteR__comp__comp,axiom,
! [C: $tType,D: $tType,B: $tType,A: $tType,G: C > B,H2: A > C,R: A > B,F: B > D] :
( ( ( comp @ C @ B @ A @ G @ H2 )
= R )
=> ( ( comp @ C @ D @ A @ ( comp @ B @ D @ C @ F @ G ) @ H2 )
= ( comp @ B @ D @ A @ F @ R ) ) ) ).
% rewriteR_comp_comp
thf(fact_250_rewriteL__comp__comp,axiom,
! [C: $tType,B: $tType,A: $tType,D: $tType,F: C > B,G: A > C,L: A > B,H2: D > A] :
( ( ( comp @ C @ B @ A @ F @ G )
= L )
=> ( ( comp @ C @ B @ D @ F @ ( comp @ A @ C @ D @ G @ H2 ) )
= ( comp @ A @ B @ D @ L @ H2 ) ) ) ).
% rewriteL_comp_comp
thf(fact_251_fun_Omap__comp,axiom,
! [B: $tType,C: $tType,A: $tType,D: $tType,G: B > C,F: A > B,V: D > A] :
( ( comp @ B @ C @ D @ G @ ( comp @ A @ B @ D @ F @ V ) )
= ( comp @ A @ C @ D @ ( comp @ B @ C @ A @ G @ F ) @ V ) ) ).
% fun.map_comp
thf(fact_252_set_Ocomp,axiom,
! [A: $tType,B: $tType,C: $tType,F: C > B,G: B > A] :
( ( comp @ ( set @ B ) @ ( set @ C ) @ ( set @ A ) @ ( vimage @ C @ B @ F ) @ ( vimage @ B @ A @ G ) )
= ( vimage @ C @ A @ ( comp @ B @ A @ C @ G @ F ) ) ) ).
% set.comp
thf(fact_253_comp__eq__dest__lhs,axiom,
! [C: $tType,B: $tType,A: $tType,A2: C > B,B2: A > C,C4: A > B,V: A] :
( ( ( comp @ C @ B @ A @ A2 @ B2 )
= C4 )
=> ( ( A2 @ ( B2 @ V ) )
= ( C4 @ V ) ) ) ).
% comp_eq_dest_lhs
thf(fact_254_comp__eq__elim,axiom,
! [C: $tType,B: $tType,D: $tType,A: $tType,A2: C > B,B2: A > C,C4: D > B,D3: A > D] :
( ( ( comp @ C @ B @ A @ A2 @ B2 )
= ( comp @ D @ B @ A @ C4 @ D3 ) )
=> ! [V2: A] :
( ( A2 @ ( B2 @ V2 ) )
= ( C4 @ ( D3 @ V2 ) ) ) ) ).
% comp_eq_elim
thf(fact_255_comp__eq__dest,axiom,
! [C: $tType,B: $tType,D: $tType,A: $tType,A2: C > B,B2: A > C,C4: D > B,D3: A > D,V: A] :
( ( ( comp @ C @ B @ A @ A2 @ B2 )
= ( comp @ D @ B @ A @ C4 @ D3 ) )
=> ( ( A2 @ ( B2 @ V ) )
= ( C4 @ ( D3 @ V ) ) ) ) ).
% comp_eq_dest
%----Type constructors (5)
thf(tcon_fun___Finite__Set_Ofinite,axiom,
! [A5: $tType,A13: $tType] :
( ( ( finite_finite @ A5 @ ( type @ A5 ) )
& ( finite_finite @ A13 @ ( type @ A13 ) ) )
=> ( finite_finite @ ( A5 > A13 ) @ ( type @ ( A5 > A13 ) ) ) ) ).
thf(tcon_Set_Oset___Finite__Set_Ofinite_1,axiom,
! [A5: $tType] :
( ( finite_finite @ A5 @ ( type @ A5 ) )
=> ( finite_finite @ ( set @ A5 ) @ ( type @ ( set @ A5 ) ) ) ) ).
thf(tcon_HOL_Obool___Finite__Set_Ofinite_2,axiom,
finite_finite @ $o @ ( type @ $o ) ).
thf(tcon_Sum__Type_Osum___Finite__Set_Ofinite_3,axiom,
! [A5: $tType,A13: $tType] :
( ( ( finite_finite @ A5 @ ( type @ A5 ) )
& ( finite_finite @ A13 @ ( type @ A13 ) ) )
=> ( finite_finite @ ( sum_sum @ A5 @ A13 ) @ ( type @ ( sum_sum @ A5 @ A13 ) ) ) ) ).
thf(tcon_Product__Type_Oprod___Finite__Set_Ofinite_4,axiom,
! [A5: $tType,A13: $tType] :
( ( ( finite_finite @ A5 @ ( type @ A5 ) )
& ( finite_finite @ A13 @ ( type @ A13 ) ) )
=> ( finite_finite @ ( product_prod @ A5 @ A13 ) @ ( type @ ( product_prod @ A5 @ A13 ) ) ) ) ).
%----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,X2: A,Y: A] :
( ( if @ A @ $false @ X2 @ Y )
= Y ) ).
thf(help_If_1_1_T,axiom,
! [A: $tType,X2: A,Y: A] :
( ( if @ A @ $true @ X2 @ Y )
= X2 ) ).
%----Conjectures (1)
thf(conj_0,conjecture,
( ( vimage @ t @ ( sum_sum @ t @ dtree ) @ ( sum_Inl @ t @ dtree ) @ ( image @ ( sum_sum @ t @ dtree ) @ ( sum_sum @ t @ dtree ) @ ( sum_map_sum @ t @ t @ dtree @ dtree @ ( id @ t ) @ ( gram_L1004374585hsubst @ tr0 ) ) @ ( cont @ tr ) ) )
= ( vimage @ t @ ( sum_sum @ t @ dtree ) @ ( sum_Inl @ t @ dtree ) @ ( cont @ tr ) ) ) ).
%------------------------------------------------------------------------------