TPTP Problem File: COM181^1.p
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%------------------------------------------------------------------------------
% File : COM181^1 : TPTP v8.2.0. Released v7.0.0.
% Domain : Computing Theory
% Problem : Koenig's lemma 269
% Version : [Bla16] axioms : Especial.
% English :
% Refs : [Loc10] Lochbihler (2010), Coinductive
% : [RB15] Reynolds & Blanchette (2015), A Decision Procedure for
% : [Bla16] Blanchette (2016), Email to Geoff Sutcliffe
% Source : [Bla16]
% Names : koenigslemma__269.p [Bla16]
% Status : Theorem
% Rating : 0.67 v8.1.0, 0.75 v7.5.0, 0.00 v7.2.0, 0.25 v7.1.0
% Syntax : Number of formulae : 335 ( 127 unt; 47 typ; 0 def)
% Number of atoms : 691 ( 226 equ; 0 cnn)
% Maximal formula atoms : 13 ( 2 avg)
% Number of connectives : 3903 ( 114 ~; 7 |; 61 &;3419 @)
% ( 0 <=>; 302 =>; 0 <=; 0 <~>)
% Maximal formula depth : 28 ( 8 avg)
% Number of types : 2 ( 1 usr)
% Number of type conns : 243 ( 243 >; 0 *; 0 +; 0 <<)
% Number of symbols : 49 ( 46 usr; 6 con; 0-5 aty)
% Number of variables : 1100 ( 116 ^; 904 !; 24 ?;1100 :)
% ( 38 !>; 0 ?*; 0 @-; 18 @+)
% SPC : TH1_THM_EQU_NAR
% Comments : This file was generated by Isabelle (most likely Sledgehammer)
% 2016-07-13 14:48:04.439
%------------------------------------------------------------------------------
%----Could-be-implicit typings (7)
thf(ty_t_Coinductive__List_Ollist,type,
coinductive_llist: $tType > $tType ).
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_Option_Ooption,type,
option: $tType > $tType ).
thf(ty_t_Set_Oset,type,
set: $tType > $tType ).
thf(ty_t_itself,type,
itself: $tType > $tType ).
thf(ty_tf_node,type,
node: $tType ).
%----Explicit typings (40)
thf(sy_cl_HOL_Otype,type,
type:
!>[A: $tType] : ( ( itself @ A ) > $o ) ).
thf(sy_cl_Groups_Ouminus,type,
uminus:
!>[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_Groups_Ogroup__add,type,
group_add:
!>[A: $tType] : ( ( itself @ A ) > $o ) ).
thf(sy_cl_Finite__Set_Ofinite,type,
finite_finite:
!>[A: $tType] : ( ( itself @ A ) > $o ) ).
thf(sy_cl_Orderings_Olinorder,type,
linorder:
!>[A: $tType] : ( ( itself @ A ) > $o ) ).
thf(sy_cl_Orderings_Opreorder,type,
preorder:
!>[A: $tType] : ( ( itself @ A ) > $o ) ).
thf(sy_cl_Nat_Osemiring__char__0,type,
semiring_char_0:
!>[A: $tType] : ( ( itself @ A ) > $o ) ).
thf(sy_cl_Orderings_Oorder__bot,type,
order_bot:
!>[A: $tType] : ( ( itself @ A ) > $o ) ).
thf(sy_cl_Lattices_Oboolean__algebra,type,
boolean_algebra:
!>[A: $tType] : ( ( itself @ A ) > $o ) ).
thf(sy_cl_Groups_Oordered__ab__group__add,type,
ordered_ab_group_add:
!>[A: $tType] : ( ( itself @ A ) > $o ) ).
thf(sy_c_Coinductive__List_Olfinite,type,
coinductive_lfinite:
!>[A: $tType] : ( ( coinductive_llist @ A ) > $o ) ).
thf(sy_c_Coinductive__List_Ollist_Olnull,type,
coinductive_lnull:
!>[A: $tType] : ( ( coinductive_llist @ A ) > $o ) ).
thf(sy_c_Coinductive__List_Ollist_Olset,type,
coinductive_lset:
!>[A: $tType] : ( ( coinductive_llist @ A ) > ( set @ A ) ) ).
thf(sy_c_Coinductive__List_Ollist_Oltl,type,
coinductive_ltl:
!>[A: $tType] : ( ( coinductive_llist @ A ) > ( coinductive_llist @ A ) ) ).
thf(sy_c_Finite__Set_Ofinite,type,
finite_finite2:
!>[A: $tType] : ( ( set @ A ) > $o ) ).
thf(sy_c_Groups_Ouminus__class_Ouminus,type,
uminus_uminus:
!>[A: $tType] : ( A > A ) ).
thf(sy_c_Koenigslemma__Mirabelle__aepjeeakgn_Oconnected,type,
koenig793108494nected:
!>[Node: $tType] : ( ( Node > Node > $o ) > $o ) ).
thf(sy_c_Koenigslemma__Mirabelle__aepjeeakgn_Opaths,type,
koenig916195507_paths:
!>[Node: $tType] : ( ( Node > Node > $o ) > ( set @ ( coinductive_llist @ Node ) ) ) ).
thf(sy_c_Koenigslemma__Mirabelle__aepjeeakgn_Oreachable__via,type,
koenig317145564le_via:
!>[Node: $tType] : ( ( Node > Node > $o ) > ( set @ Node ) > Node > ( set @ Node ) ) ).
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_Product__Type_OPair,type,
product_Pair:
!>[A: $tType,B: $tType] : ( A > B > ( product_prod @ A @ B ) ) ).
thf(sy_c_Product__Type_Oold_Oprod_Orec__prod,type,
product_rec_prod:
!>[A: $tType,B: $tType,T: $tType] : ( ( A > B > T ) > ( product_prod @ A @ B ) > T ) ).
thf(sy_c_Product__Type_Oprod_Ocase__prod,type,
product_case_prod:
!>[A: $tType,B: $tType,C: $tType] : ( ( A > B > C ) > ( product_prod @ A @ B ) > C ) ).
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_member,type,
member:
!>[A: $tType] : ( A > ( set @ A ) > $o ) ).
thf(sy_v_LTL____,type,
ltl: ( product_prod @ node @ ( set @ node ) ) > ( product_prod @ node @ ( set @ node ) ) ).
thf(sy_v_f____,type,
f: ( product_prod @ node @ ( set @ node ) ) > ( coinductive_llist @ node ) ).
thf(sy_v_graph,type,
graph: node > node > $o ).
thf(sy_v_n,type,
n: node ).
thf(sy_v_na____,type,
na: node ).
thf(sy_v_ns____,type,
ns: set @ node ).
thf(sy_v_nsa____,type,
nsa: set @ node ).
%----Relevant facts (256)
thf(fact_0_connected,axiom,
koenig793108494nected @ node @ graph ).
% connected
thf(fact_1__092_060open_062finite_Ans_092_060close_062,axiom,
finite_finite2 @ node @ nsa ).
% \<open>finite ns\<close>
thf(fact_2_finite__branching,axiom,
! [N: node] : ( finite_finite2 @ node @ ( collect @ node @ ( graph @ N ) ) ) ).
% finite_branching
thf(fact_3_infinite,axiom,
~ ( finite_finite2 @ node @ ( top_top @ ( set @ node ) ) ) ).
% infinite
thf(fact_4__092_060open_062_092_060exists_062n_H_O_A_Icase_A_In_M_Ans_J_Aof_A_In_M_Ans_J_A_092_060Rightarrow_062_A_092_060lambda_062n_H_O_Agraph_An_An_H_A_092_060and_062_Ainfinite_A_Ireachable__via_Agraph_A_I_N_Ainsert_An_A_Iinsert_An_H_Ans_J_J_An_H_J_A_092_060and_062_An_H_A_092_060notin_062_Ainsert_An_Ans_J_An_H_092_060close_062,axiom,
? [X1: node] :
( product_case_prod @ node @ ( set @ node ) @ ( node > $o )
@ ^ [N2: node,Ns: set @ node,N3: node] :
( ( graph @ N2 @ N3 )
& ~ ( finite_finite2 @ node @ ( koenig317145564le_via @ node @ graph @ ( uminus_uminus @ ( set @ node ) @ ( insert @ node @ N2 @ ( insert @ node @ N3 @ Ns ) ) ) @ N3 ) )
& ~ ( member @ node @ N3 @ ( insert @ node @ N2 @ Ns ) ) )
@ ( product_Pair @ node @ ( set @ node ) @ na @ nsa )
@ X1 ) ).
% \<open>\<exists>n'. (case (n, ns) of (n, ns) \<Rightarrow> \<lambda>n'. graph n n' \<and> infinite (reachable_via graph (- insert n (insert n' ns)) n') \<and> n' \<notin> insert n ns) n'\<close>
thf(fact_5_ldistinct_I1_J,axiom,
( ~ ( finite_finite2 @ node @ ( koenig317145564le_via @ node @ graph @ ( uminus_uminus @ ( set @ node ) @ ( insert @ node @ na @ nsa ) ) @ na ) )
& ( finite_finite2 @ node @ nsa ) ) ).
% ldistinct(1)
thf(fact_6__092_060open_062infinite_A_Ireachable__via_Agraph_A_I_N_Ainsert_An_Ans_J_An_J_092_060close_062,axiom,
~ ( finite_finite2 @ node @ ( koenig317145564le_via @ node @ graph @ ( uminus_uminus @ ( set @ node ) @ ( insert @ node @ na @ nsa ) ) @ na ) ) ).
% \<open>infinite (reachable_via graph (- insert n ns) n)\<close>
thf(fact_7_ex__P__I,axiom,
! [Na: node,Nsa: set @ node] :
( ~ ( finite_finite2 @ node @ ( koenig317145564le_via @ node @ graph @ ( uminus_uminus @ ( set @ node ) @ ( insert @ node @ Na @ Nsa ) ) @ Na ) )
=> ? [X1: node] :
( product_case_prod @ node @ ( set @ node ) @ ( node > $o )
@ ^ [N2: node,Ns: set @ node,N3: node] :
( ( graph @ N2 @ N3 )
& ~ ( finite_finite2 @ node @ ( koenig317145564le_via @ node @ graph @ ( uminus_uminus @ ( set @ node ) @ ( insert @ node @ N2 @ ( insert @ node @ N3 @ Ns ) ) ) @ N3 ) )
& ~ ( member @ node @ N3 @ ( insert @ node @ N2 @ Ns ) ) )
@ ( product_Pair @ node @ ( set @ node ) @ Na @ Nsa )
@ X1 ) ) ).
% ex_P_I
thf(fact_8__092_060open_062_092_060And_062thesis_O_A_I_092_060lbrakk_062finite_Ans_059_Ainfinite_A_Ireachable__via_Agraph_A_I_N_Ainsert_An_Ans_J_An_J_092_060rbrakk_062_A_092_060Longrightarrow_062_Athesis_J_A_092_060Longrightarrow_062_Athesis_092_060close_062,axiom,
~ ( ( finite_finite2 @ node @ nsa )
=> ( finite_finite2 @ node @ ( koenig317145564le_via @ node @ graph @ ( uminus_uminus @ ( set @ node ) @ ( insert @ node @ na @ nsa ) ) @ na ) ) ) ).
% \<open>\<And>thesis. (\<lbrakk>finite ns; infinite (reachable_via graph (- insert n ns) n)\<rbrakk> \<Longrightarrow> thesis) \<Longrightarrow> thesis\<close>
thf(fact_9_LTL__def,axiom,
( ltl
= ( product_case_prod @ node @ ( set @ node ) @ ( product_prod @ node @ ( set @ node ) )
@ ^ [N2: node,Ns: set @ node] :
( product_Pair @ node @ ( set @ node )
@ ( ^ [P: node > $o] :
@+[X: node] : ( P @ X )
@ ( product_case_prod @ node @ ( set @ node ) @ ( node > $o )
@ ^ [O: node,Nt: set @ node,N3: node] :
( ( graph @ O @ N3 )
& ~ ( finite_finite2 @ node @ ( koenig317145564le_via @ node @ graph @ ( uminus_uminus @ ( set @ node ) @ ( insert @ node @ O @ ( insert @ node @ N3 @ Nt ) ) ) @ N3 ) )
& ~ ( member @ node @ N3 @ ( insert @ node @ O @ Nt ) ) )
@ ( product_Pair @ node @ ( set @ node ) @ N2 @ Ns ) ) )
@ ( insert @ node @ N2 @ Ns ) ) ) ) ).
% LTL_def
thf(fact_10_case__prodI2_H,axiom,
! [A: $tType,B: $tType,C: $tType,P2: product_prod @ A @ B,C2: A > B > C > $o,X2: C] :
( ! [A2: A,B2: B] :
( ( ( product_Pair @ A @ B @ A2 @ B2 )
= P2 )
=> ( C2 @ A2 @ B2 @ X2 ) )
=> ( product_case_prod @ A @ B @ ( C > $o ) @ C2 @ P2 @ X2 ) ) ).
% case_prodI2'
thf(fact_11_case__prod__conv,axiom,
! [B: $tType,A: $tType,C: $tType,F: B > C > A,A3: B,B3: C] :
( ( product_case_prod @ B @ C @ A @ F @ ( product_Pair @ B @ C @ A3 @ B3 ) )
= ( F @ A3 @ B3 ) ) ).
% case_prod_conv
thf(fact_12_finite__insert,axiom,
! [A: $tType,A3: A,A4: set @ A] :
( ( finite_finite2 @ A @ ( insert @ A @ A3 @ A4 ) )
= ( finite_finite2 @ A @ A4 ) ) ).
% finite_insert
thf(fact_13_some__equality,axiom,
! [A: $tType,P3: A > $o,A3: A] :
( ( P3 @ A3 )
=> ( ! [X3: A] :
( ( P3 @ X3 )
=> ( X3 = A3 ) )
=> ( ( ^ [P: A > $o] :
@+[X: A] : ( P @ X )
@ P3 )
= A3 ) ) ) ).
% some_equality
thf(fact_14_some__eq__trivial,axiom,
! [A: $tType,X2: A] :
( ( @+[Y: A] : Y = X2 )
= X2 ) ).
% some_eq_trivial
thf(fact_15_some__sym__eq__trivial,axiom,
! [A: $tType,X2: A] :
( ( ^ [P: A > $o] :
@+[X: A] : ( P @ X )
@ ( ^ [Y2: A,Z: A] : Y2 = Z
@ X2 ) )
= X2 ) ).
% some_sym_eq_trivial
thf(fact_16_finite__Collect__conjI,axiom,
! [A: $tType,P3: A > $o,Q: A > $o] :
( ( ( finite_finite2 @ A @ ( collect @ A @ P3 ) )
| ( finite_finite2 @ A @ ( collect @ A @ Q ) ) )
=> ( finite_finite2 @ A
@ ( collect @ A
@ ^ [X4: A] :
( ( P3 @ X4 )
& ( Q @ X4 ) ) ) ) ) ).
% finite_Collect_conjI
thf(fact_17_finite__Collect__disjI,axiom,
! [A: $tType,P3: A > $o,Q: A > $o] :
( ( finite_finite2 @ A
@ ( collect @ A
@ ^ [X4: A] :
( ( P3 @ X4 )
| ( Q @ X4 ) ) ) )
= ( ( finite_finite2 @ A @ ( collect @ A @ P3 ) )
& ( finite_finite2 @ A @ ( collect @ A @ Q ) ) ) ) ).
% finite_Collect_disjI
thf(fact_18_ComplI,axiom,
! [A: $tType,C2: A,A4: set @ A] :
( ~ ( member @ A @ C2 @ A4 )
=> ( member @ A @ C2 @ ( uminus_uminus @ ( set @ A ) @ A4 ) ) ) ).
% ComplI
thf(fact_19_Compl__iff,axiom,
! [A: $tType,C2: A,A4: set @ A] :
( ( member @ A @ C2 @ ( uminus_uminus @ ( set @ A ) @ A4 ) )
= ( ~ ( member @ A @ C2 @ A4 ) ) ) ).
% Compl_iff
thf(fact_20_Compl__eq__Compl__iff,axiom,
! [A: $tType,A4: set @ A,B4: set @ A] :
( ( ( uminus_uminus @ ( set @ A ) @ A4 )
= ( uminus_uminus @ ( set @ A ) @ B4 ) )
= ( A4 = B4 ) ) ).
% Compl_eq_Compl_iff
thf(fact_21_old_Oprod_Oinject,axiom,
! [A: $tType,B: $tType,A3: A,B3: B,A5: A,B5: B] :
( ( ( product_Pair @ A @ B @ A3 @ B3 )
= ( product_Pair @ A @ B @ A5 @ B5 ) )
= ( ( A3 = A5 )
& ( B3 = B5 ) ) ) ).
% old.prod.inject
thf(fact_22_prod_Oinject,axiom,
! [A: $tType,B: $tType,X12: A,X22: B,Y1: A,Y22: B] :
( ( ( product_Pair @ A @ B @ X12 @ X22 )
= ( product_Pair @ A @ B @ Y1 @ Y22 ) )
= ( ( X12 = Y1 )
& ( X22 = Y22 ) ) ) ).
% prod.inject
thf(fact_23_UNIV__I,axiom,
! [A: $tType,X2: A] : ( member @ A @ X2 @ ( top_top @ ( set @ A ) ) ) ).
% UNIV_I
thf(fact_24_finite__code,axiom,
! [A: $tType] :
( ( finite_finite @ A @ ( type2 @ A ) )
=> ( ( finite_finite2 @ A )
= ( ^ [A6: set @ A] : $true ) ) ) ).
% finite_code
thf(fact_25_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_26_insert__iff,axiom,
! [A: $tType,A3: A,B3: A,A4: set @ A] :
( ( member @ A @ A3 @ ( insert @ A @ B3 @ A4 ) )
= ( ( A3 = B3 )
| ( member @ A @ A3 @ A4 ) ) ) ).
% insert_iff
thf(fact_27_insertCI,axiom,
! [A: $tType,A3: A,B4: set @ A,B3: A] :
( ( ~ ( member @ A @ A3 @ B4 )
=> ( A3 = B3 ) )
=> ( member @ A @ A3 @ ( insert @ A @ B3 @ B4 ) ) ) ).
% insertCI
thf(fact_28_Eps__case__prod__eq,axiom,
! [A: $tType,B: $tType,X2: A,Y3: B] :
( ( ^ [P: ( product_prod @ A @ B ) > $o] :
@+[X: product_prod @ A @ B] : ( P @ X )
@ ( product_case_prod @ A @ B @ $o
@ ^ [X5: A,Y4: B] :
( ( X2 = X5 )
& ( Y3 = Y4 ) ) ) )
= ( product_Pair @ A @ B @ X2 @ Y3 ) ) ).
% Eps_case_prod_eq
thf(fact_29_mem__case__prodI2,axiom,
! [C: $tType,B: $tType,A: $tType,P2: product_prod @ A @ B,Z2: C,C2: A > B > ( set @ C )] :
( ! [A2: A,B2: B] :
( ( P2
= ( product_Pair @ A @ B @ A2 @ B2 ) )
=> ( member @ C @ Z2 @ ( C2 @ A2 @ B2 ) ) )
=> ( member @ C @ Z2 @ ( product_case_prod @ A @ B @ ( set @ C ) @ C2 @ P2 ) ) ) ).
% mem_case_prodI2
thf(fact_30_mem__case__prodI,axiom,
! [A: $tType,B: $tType,C: $tType,Z2: A,C2: B > C > ( set @ A ),A3: B,B3: C] :
( ( member @ A @ Z2 @ ( C2 @ A3 @ B3 ) )
=> ( member @ A @ Z2 @ ( product_case_prod @ B @ C @ ( set @ A ) @ C2 @ ( product_Pair @ B @ C @ A3 @ B3 ) ) ) ) ).
% mem_case_prodI
thf(fact_31_case__prodI2,axiom,
! [B: $tType,A: $tType,P2: product_prod @ A @ B,C2: A > B > $o] :
( ! [A2: A,B2: B] :
( ( P2
= ( product_Pair @ A @ B @ A2 @ B2 ) )
=> ( C2 @ A2 @ B2 ) )
=> ( product_case_prod @ A @ B @ $o @ C2 @ P2 ) ) ).
% case_prodI2
thf(fact_32_case__prodI,axiom,
! [A: $tType,B: $tType,F: A > B > $o,A3: A,B3: B] :
( ( F @ A3 @ B3 )
=> ( product_case_prod @ A @ B @ $o @ F @ ( product_Pair @ A @ B @ A3 @ B3 ) ) ) ).
% case_prodI
thf(fact_33_finite__Plus__UNIV__iff,axiom,
! [A: $tType,B: $tType] :
( ( finite_finite2 @ ( sum_sum @ A @ B ) @ ( top_top @ ( set @ ( sum_sum @ A @ B ) ) ) )
= ( ( finite_finite2 @ A @ ( top_top @ ( set @ A ) ) )
& ( finite_finite2 @ B @ ( top_top @ ( set @ B ) ) ) ) ) ).
% finite_Plus_UNIV_iff
thf(fact_34_finite__Collect__not,axiom,
! [A: $tType,P3: A > $o] :
( ( finite_finite2 @ A @ ( collect @ A @ P3 ) )
=> ( ( finite_finite2 @ A
@ ( collect @ A
@ ^ [X4: A] :
~ ( P3 @ X4 ) ) )
= ( finite_finite2 @ A @ ( top_top @ ( set @ A ) ) ) ) ) ).
% finite_Collect_not
thf(fact_35_finite__compl,axiom,
! [A: $tType,A4: set @ A] :
( ( finite_finite2 @ A @ A4 )
=> ( ( finite_finite2 @ A @ ( uminus_uminus @ ( set @ A ) @ A4 ) )
= ( finite_finite2 @ A @ ( top_top @ ( set @ A ) ) ) ) ) ).
% finite_compl
thf(fact_36_UNIV__def,axiom,
! [A: $tType] :
( ( top_top @ ( set @ A ) )
= ( collect @ A
@ ^ [X4: A] : $true ) ) ).
% UNIV_def
thf(fact_37_UNIV__eq__I,axiom,
! [A: $tType,A4: set @ A] :
( ! [X3: A] : ( member @ A @ X3 @ A4 )
=> ( ( top_top @ ( set @ A ) )
= A4 ) ) ).
% UNIV_eq_I
thf(fact_38_UNIV__witness,axiom,
! [A: $tType] :
? [X3: A] : ( member @ A @ X3 @ ( top_top @ ( set @ A ) ) ) ).
% UNIV_witness
thf(fact_39_Finite__Set_Ofinite__set,axiom,
! [A: $tType] :
( ( finite_finite2 @ ( set @ A ) @ ( top_top @ ( set @ ( set @ A ) ) ) )
= ( finite_finite2 @ A @ ( top_top @ ( set @ A ) ) ) ) ).
% Finite_Set.finite_set
thf(fact_40_finite__prod,axiom,
! [A: $tType,B: $tType] :
( ( finite_finite2 @ ( product_prod @ A @ B ) @ ( top_top @ ( set @ ( product_prod @ A @ B ) ) ) )
= ( ( finite_finite2 @ A @ ( top_top @ ( set @ A ) ) )
& ( finite_finite2 @ B @ ( top_top @ ( set @ B ) ) ) ) ) ).
% finite_prod
thf(fact_41_finite__Prod__UNIV,axiom,
! [B: $tType,A: $tType] :
( ( finite_finite2 @ A @ ( top_top @ ( set @ A ) ) )
=> ( ( finite_finite2 @ B @ ( top_top @ ( set @ B ) ) )
=> ( finite_finite2 @ ( product_prod @ A @ B ) @ ( top_top @ ( set @ ( product_prod @ A @ B ) ) ) ) ) ) ).
% finite_Prod_UNIV
thf(fact_42_finite__fun__UNIVD2,axiom,
! [A: $tType,B: $tType] :
( ( finite_finite2 @ ( A > B ) @ ( top_top @ ( set @ ( A > B ) ) ) )
=> ( finite_finite2 @ B @ ( top_top @ ( set @ B ) ) ) ) ).
% finite_fun_UNIVD2
thf(fact_43_split__paired__Eps,axiom,
! [B: $tType,A: $tType] :
( ( ^ [P: ( product_prod @ A @ B ) > $o] :
@+[X: product_prod @ A @ B] : ( P @ X ) )
= ( ^ [P4: ( product_prod @ A @ B ) > $o] :
( ^ [P: ( product_prod @ A @ B ) > $o] :
@+[X: product_prod @ A @ B] : ( P @ X )
@ ( product_case_prod @ A @ B @ $o
@ ^ [A7: A,B6: B] : ( P4 @ ( product_Pair @ A @ B @ A7 @ B6 ) ) ) ) ) ) ).
% split_paired_Eps
thf(fact_44_mem__case__prodE,axiom,
! [B: $tType,A: $tType,C: $tType,Z2: A,C2: B > C > ( set @ A ),P2: product_prod @ B @ C] :
( ( member @ A @ Z2 @ ( product_case_prod @ B @ C @ ( set @ A ) @ C2 @ P2 ) )
=> ~ ! [X3: B,Y5: C] :
( ( P2
= ( product_Pair @ B @ C @ X3 @ Y5 ) )
=> ~ ( member @ A @ Z2 @ ( C2 @ X3 @ Y5 ) ) ) ) ).
% mem_case_prodE
thf(fact_45_mem__Collect__eq,axiom,
! [A: $tType,A3: A,P3: A > $o] :
( ( member @ A @ A3 @ ( collect @ A @ P3 ) )
= ( P3 @ A3 ) ) ).
% mem_Collect_eq
thf(fact_46_Collect__mem__eq,axiom,
! [A: $tType,A4: set @ A] :
( ( collect @ A
@ ^ [X4: A] : ( member @ A @ X4 @ A4 ) )
= A4 ) ).
% Collect_mem_eq
thf(fact_47_Collect__cong,axiom,
! [A: $tType,P3: A > $o,Q: A > $o] :
( ! [X3: A] :
( ( P3 @ X3 )
= ( Q @ X3 ) )
=> ( ( collect @ A @ P3 )
= ( 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_finite__UNIV,axiom,
! [A: $tType] :
( ( finite_finite @ A @ ( type2 @ A ) )
=> ( finite_finite2 @ A @ ( top_top @ ( set @ A ) ) ) ) ).
% finite_UNIV
thf(fact_50_infinite__UNIV__char__0,axiom,
! [A: $tType] :
( ( semiring_char_0 @ A @ ( type2 @ A ) )
=> ~ ( finite_finite2 @ A @ ( top_top @ ( set @ A ) ) ) ) ).
% infinite_UNIV_char_0
thf(fact_51_ex__new__if__finite,axiom,
! [A: $tType,A4: set @ A] :
( ~ ( finite_finite2 @ A @ ( top_top @ ( set @ A ) ) )
=> ( ( finite_finite2 @ A @ A4 )
=> ? [A2: A] :
~ ( member @ A @ A2 @ A4 ) ) ) ).
% ex_new_if_finite
thf(fact_52_insert__UNIV,axiom,
! [A: $tType,X2: A] :
( ( insert @ A @ X2 @ ( top_top @ ( set @ A ) ) )
= ( top_top @ ( set @ A ) ) ) ).
% insert_UNIV
thf(fact_53_case__prodE,axiom,
! [A: $tType,B: $tType,C2: A > B > $o,P2: product_prod @ A @ B] :
( ( product_case_prod @ A @ B @ $o @ C2 @ P2 )
=> ~ ! [X3: A,Y5: B] :
( ( P2
= ( product_Pair @ A @ B @ X3 @ Y5 ) )
=> ~ ( C2 @ X3 @ Y5 ) ) ) ).
% case_prodE
thf(fact_54_case__prodD,axiom,
! [A: $tType,B: $tType,F: A > B > $o,A3: A,B3: B] :
( ( product_case_prod @ A @ B @ $o @ F @ ( product_Pair @ A @ B @ A3 @ B3 ) )
=> ( F @ A3 @ B3 ) ) ).
% case_prodD
thf(fact_55_old_Oprod_Oinducts,axiom,
! [B: $tType,A: $tType,P3: ( product_prod @ A @ B ) > $o,Prod: product_prod @ A @ B] :
( ! [A2: A,B2: B] : ( P3 @ ( product_Pair @ A @ B @ A2 @ B2 ) )
=> ( P3 @ Prod ) ) ).
% old.prod.inducts
thf(fact_56_old_Oprod_Oexhaust,axiom,
! [A: $tType,B: $tType,Y3: product_prod @ A @ B] :
~ ! [A2: A,B2: B] :
( Y3
!= ( product_Pair @ A @ B @ A2 @ B2 ) ) ).
% old.prod.exhaust
thf(fact_57_prod__induct7,axiom,
! [G2: $tType,F2: $tType,E: $tType,D: $tType,C: $tType,B: $tType,A: $tType,P3: ( product_prod @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D @ ( product_prod @ E @ ( product_prod @ F2 @ G2 ) ) ) ) ) ) > $o,X2: product_prod @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D @ ( product_prod @ E @ ( product_prod @ F2 @ G2 ) ) ) ) )] :
( ! [A2: A,B2: B,C3: C,D2: D,E2: E,F3: F2,G3: G2] : ( P3 @ ( product_Pair @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D @ ( product_prod @ E @ ( product_prod @ F2 @ G2 ) ) ) ) ) @ A2 @ ( product_Pair @ B @ ( product_prod @ C @ ( product_prod @ D @ ( product_prod @ E @ ( product_prod @ F2 @ G2 ) ) ) ) @ B2 @ ( product_Pair @ C @ ( product_prod @ D @ ( product_prod @ E @ ( product_prod @ F2 @ G2 ) ) ) @ C3 @ ( product_Pair @ D @ ( product_prod @ E @ ( product_prod @ F2 @ G2 ) ) @ D2 @ ( product_Pair @ E @ ( product_prod @ F2 @ G2 ) @ E2 @ ( product_Pair @ F2 @ G2 @ F3 @ G3 ) ) ) ) ) ) )
=> ( P3 @ X2 ) ) ).
% prod_induct7
thf(fact_58_prod__induct6,axiom,
! [F2: $tType,E: $tType,D: $tType,C: $tType,B: $tType,A: $tType,P3: ( product_prod @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D @ ( product_prod @ E @ F2 ) ) ) ) ) > $o,X2: product_prod @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D @ ( product_prod @ E @ F2 ) ) ) )] :
( ! [A2: A,B2: B,C3: C,D2: D,E2: E,F3: F2] : ( P3 @ ( product_Pair @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D @ ( product_prod @ E @ F2 ) ) ) ) @ A2 @ ( product_Pair @ B @ ( product_prod @ C @ ( product_prod @ D @ ( product_prod @ E @ F2 ) ) ) @ B2 @ ( product_Pair @ C @ ( product_prod @ D @ ( product_prod @ E @ F2 ) ) @ C3 @ ( product_Pair @ D @ ( product_prod @ E @ F2 ) @ D2 @ ( product_Pair @ E @ F2 @ E2 @ F3 ) ) ) ) ) )
=> ( P3 @ X2 ) ) ).
% prod_induct6
thf(fact_59_prod__induct5,axiom,
! [E: $tType,D: $tType,C: $tType,B: $tType,A: $tType,P3: ( product_prod @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D @ E ) ) ) ) > $o,X2: product_prod @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D @ E ) ) )] :
( ! [A2: A,B2: B,C3: C,D2: D,E2: E] : ( P3 @ ( product_Pair @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D @ E ) ) ) @ A2 @ ( product_Pair @ B @ ( product_prod @ C @ ( product_prod @ D @ E ) ) @ B2 @ ( product_Pair @ C @ ( product_prod @ D @ E ) @ C3 @ ( product_Pair @ D @ E @ D2 @ E2 ) ) ) ) )
=> ( P3 @ X2 ) ) ).
% prod_induct5
thf(fact_60_prod__induct4,axiom,
! [D: $tType,C: $tType,B: $tType,A: $tType,P3: ( product_prod @ A @ ( product_prod @ B @ ( product_prod @ C @ D ) ) ) > $o,X2: product_prod @ A @ ( product_prod @ B @ ( product_prod @ C @ D ) )] :
( ! [A2: A,B2: B,C3: C,D2: D] : ( P3 @ ( product_Pair @ A @ ( product_prod @ B @ ( product_prod @ C @ D ) ) @ A2 @ ( product_Pair @ B @ ( product_prod @ C @ D ) @ B2 @ ( product_Pair @ C @ D @ C3 @ D2 ) ) ) )
=> ( P3 @ X2 ) ) ).
% prod_induct4
thf(fact_61_prod__induct3,axiom,
! [C: $tType,B: $tType,A: $tType,P3: ( product_prod @ A @ ( product_prod @ B @ C ) ) > $o,X2: product_prod @ A @ ( product_prod @ B @ C )] :
( ! [A2: A,B2: B,C3: C] : ( P3 @ ( product_Pair @ A @ ( product_prod @ B @ C ) @ A2 @ ( product_Pair @ B @ C @ B2 @ C3 ) ) )
=> ( P3 @ X2 ) ) ).
% prod_induct3
thf(fact_62_prod__cases7,axiom,
! [A: $tType,B: $tType,C: $tType,D: $tType,E: $tType,F2: $tType,G2: $tType,Y3: product_prod @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D @ ( product_prod @ E @ ( product_prod @ F2 @ G2 ) ) ) ) )] :
~ ! [A2: A,B2: B,C3: C,D2: D,E2: E,F3: F2,G3: G2] :
( Y3
!= ( product_Pair @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D @ ( product_prod @ E @ ( product_prod @ F2 @ G2 ) ) ) ) ) @ A2 @ ( product_Pair @ B @ ( product_prod @ C @ ( product_prod @ D @ ( product_prod @ E @ ( product_prod @ F2 @ G2 ) ) ) ) @ B2 @ ( product_Pair @ C @ ( product_prod @ D @ ( product_prod @ E @ ( product_prod @ F2 @ G2 ) ) ) @ C3 @ ( product_Pair @ D @ ( product_prod @ E @ ( product_prod @ F2 @ G2 ) ) @ D2 @ ( product_Pair @ E @ ( product_prod @ F2 @ G2 ) @ E2 @ ( product_Pair @ F2 @ G2 @ F3 @ G3 ) ) ) ) ) ) ) ).
% prod_cases7
thf(fact_63_prod__cases6,axiom,
! [A: $tType,B: $tType,C: $tType,D: $tType,E: $tType,F2: $tType,Y3: product_prod @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D @ ( product_prod @ E @ F2 ) ) ) )] :
~ ! [A2: A,B2: B,C3: C,D2: D,E2: E,F3: F2] :
( Y3
!= ( product_Pair @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D @ ( product_prod @ E @ F2 ) ) ) ) @ A2 @ ( product_Pair @ B @ ( product_prod @ C @ ( product_prod @ D @ ( product_prod @ E @ F2 ) ) ) @ B2 @ ( product_Pair @ C @ ( product_prod @ D @ ( product_prod @ E @ F2 ) ) @ C3 @ ( product_Pair @ D @ ( product_prod @ E @ F2 ) @ D2 @ ( product_Pair @ E @ F2 @ E2 @ F3 ) ) ) ) ) ) ).
% prod_cases6
thf(fact_64_prod__cases5,axiom,
! [A: $tType,B: $tType,C: $tType,D: $tType,E: $tType,Y3: product_prod @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D @ E ) ) )] :
~ ! [A2: A,B2: B,C3: C,D2: D,E2: E] :
( Y3
!= ( product_Pair @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D @ E ) ) ) @ A2 @ ( product_Pair @ B @ ( product_prod @ C @ ( product_prod @ D @ E ) ) @ B2 @ ( product_Pair @ C @ ( product_prod @ D @ E ) @ C3 @ ( product_Pair @ D @ E @ D2 @ E2 ) ) ) ) ) ).
% prod_cases5
thf(fact_65_prod__cases4,axiom,
! [A: $tType,B: $tType,C: $tType,D: $tType,Y3: product_prod @ A @ ( product_prod @ B @ ( product_prod @ C @ D ) )] :
~ ! [A2: A,B2: B,C3: C,D2: D] :
( Y3
!= ( product_Pair @ A @ ( product_prod @ B @ ( product_prod @ C @ D ) ) @ A2 @ ( product_Pair @ B @ ( product_prod @ C @ D ) @ B2 @ ( product_Pair @ C @ D @ C3 @ D2 ) ) ) ) ).
% prod_cases4
thf(fact_66_prod__cases3,axiom,
! [A: $tType,B: $tType,C: $tType,Y3: product_prod @ A @ ( product_prod @ B @ C )] :
~ ! [A2: A,B2: B,C3: C] :
( Y3
!= ( product_Pair @ A @ ( product_prod @ B @ C ) @ A2 @ ( product_Pair @ B @ C @ B2 @ C3 ) ) ) ).
% prod_cases3
thf(fact_67_Pair__inject,axiom,
! [A: $tType,B: $tType,A3: A,B3: B,A5: A,B5: B] :
( ( ( product_Pair @ A @ B @ A3 @ B3 )
= ( product_Pair @ A @ B @ A5 @ B5 ) )
=> ~ ( ( A3 = A5 )
=> ( B3 != B5 ) ) ) ).
% Pair_inject
thf(fact_68_prod__cases,axiom,
! [B: $tType,A: $tType,P3: ( product_prod @ A @ B ) > $o,P2: product_prod @ A @ B] :
( ! [A2: A,B2: B] : ( P3 @ ( product_Pair @ A @ B @ A2 @ B2 ) )
=> ( P3 @ P2 ) ) ).
% prod_cases
thf(fact_69_surj__pair,axiom,
! [A: $tType,B: $tType,P2: product_prod @ A @ B] :
? [X3: A,Y5: B] :
( P2
= ( product_Pair @ A @ B @ X3 @ Y5 ) ) ).
% surj_pair
thf(fact_70_finite,axiom,
! [A: $tType] :
( ( finite_finite @ A @ ( type2 @ A ) )
=> ! [A4: set @ A] : ( finite_finite2 @ A @ A4 ) ) ).
% finite
thf(fact_71_finite__set__choice,axiom,
! [B: $tType,A: $tType,A4: set @ A,P3: A > B > $o] :
( ( finite_finite2 @ A @ A4 )
=> ( ! [X3: A] :
( ( member @ A @ X3 @ A4 )
=> ? [X13: B] : ( P3 @ X3 @ X13 ) )
=> ? [F3: A > B] :
! [X6: A] :
( ( member @ A @ X6 @ A4 )
=> ( P3 @ X6 @ ( F3 @ X6 ) ) ) ) ) ).
% finite_set_choice
thf(fact_72_mk__disjoint__insert,axiom,
! [A: $tType,A3: A,A4: set @ A] :
( ( member @ A @ A3 @ A4 )
=> ? [B7: set @ A] :
( ( A4
= ( insert @ A @ A3 @ B7 ) )
& ~ ( member @ A @ A3 @ B7 ) ) ) ).
% mk_disjoint_insert
thf(fact_73_insert__commute,axiom,
! [A: $tType,X2: A,Y3: A,A4: set @ A] :
( ( insert @ A @ X2 @ ( insert @ A @ Y3 @ A4 ) )
= ( insert @ A @ Y3 @ ( insert @ A @ X2 @ A4 ) ) ) ).
% insert_commute
thf(fact_74_insert__eq__iff,axiom,
! [A: $tType,A3: A,A4: set @ A,B3: A,B4: set @ A] :
( ~ ( member @ A @ A3 @ A4 )
=> ( ~ ( member @ A @ B3 @ B4 )
=> ( ( ( insert @ A @ A3 @ A4 )
= ( insert @ A @ B3 @ B4 ) )
= ( ( ( A3 = B3 )
=> ( A4 = B4 ) )
& ( ( A3 != B3 )
=> ? [C4: set @ A] :
( ( A4
= ( insert @ A @ B3 @ C4 ) )
& ~ ( member @ A @ B3 @ C4 )
& ( B4
= ( insert @ A @ A3 @ C4 ) )
& ~ ( member @ A @ A3 @ C4 ) ) ) ) ) ) ) ).
% insert_eq_iff
thf(fact_75_insert__absorb,axiom,
! [A: $tType,A3: A,A4: set @ A] :
( ( member @ A @ A3 @ A4 )
=> ( ( insert @ A @ A3 @ A4 )
= A4 ) ) ).
% insert_absorb
thf(fact_76_insert__ident,axiom,
! [A: $tType,X2: A,A4: set @ A,B4: set @ A] :
( ~ ( member @ A @ X2 @ A4 )
=> ( ~ ( member @ A @ X2 @ B4 )
=> ( ( ( insert @ A @ X2 @ A4 )
= ( insert @ A @ X2 @ B4 ) )
= ( A4 = B4 ) ) ) ) ).
% insert_ident
thf(fact_77_Set_Oset__insert,axiom,
! [A: $tType,X2: A,A4: set @ A] :
( ( member @ A @ X2 @ A4 )
=> ~ ! [B7: set @ A] :
( ( A4
= ( insert @ A @ X2 @ B7 ) )
=> ( member @ A @ X2 @ B7 ) ) ) ).
% Set.set_insert
thf(fact_78_insertI2,axiom,
! [A: $tType,A3: A,B4: set @ A,B3: A] :
( ( member @ A @ A3 @ B4 )
=> ( member @ A @ A3 @ ( insert @ A @ B3 @ B4 ) ) ) ).
% insertI2
thf(fact_79_insertI1,axiom,
! [A: $tType,A3: A,B4: set @ A] : ( member @ A @ A3 @ ( insert @ A @ A3 @ B4 ) ) ).
% insertI1
thf(fact_80_insertE,axiom,
! [A: $tType,A3: A,B3: A,A4: set @ A] :
( ( member @ A @ A3 @ ( insert @ A @ B3 @ A4 ) )
=> ( ( A3 != B3 )
=> ( member @ A @ A3 @ A4 ) ) ) ).
% insertE
thf(fact_81_double__complement,axiom,
! [A: $tType,A4: set @ A] :
( ( uminus_uminus @ ( set @ A ) @ ( uminus_uminus @ ( set @ A ) @ A4 ) )
= A4 ) ).
% double_complement
thf(fact_82_ComplD,axiom,
! [A: $tType,C2: A,A4: set @ A] :
( ( member @ A @ C2 @ ( uminus_uminus @ ( set @ A ) @ A4 ) )
=> ~ ( member @ A @ C2 @ A4 ) ) ).
% ComplD
thf(fact_83_tfl__some,axiom,
! [A: $tType,P5: A > $o,X6: A] :
( ( P5 @ X6 )
=> ( P5
@ ( ^ [P: A > $o] :
@+[X: A] : ( P @ X )
@ P5 ) ) ) ).
% tfl_some
thf(fact_84_someI,axiom,
! [A: $tType,P3: A > $o,X2: A] :
( ( P3 @ X2 )
=> ( P3
@ ( ^ [P: A > $o] :
@+[X: A] : ( P @ X )
@ P3 ) ) ) ).
% someI
thf(fact_85_pigeonhole__infinite__rel,axiom,
! [B: $tType,A: $tType,A4: set @ A,B4: set @ B,R: A > B > $o] :
( ~ ( finite_finite2 @ A @ A4 )
=> ( ( finite_finite2 @ B @ B4 )
=> ( ! [X3: A] :
( ( member @ A @ X3 @ A4 )
=> ? [Xa: B] :
( ( member @ B @ Xa @ B4 )
& ( R @ X3 @ Xa ) ) )
=> ? [X3: B] :
( ( member @ B @ X3 @ B4 )
& ~ ( finite_finite2 @ A
@ ( collect @ A
@ ^ [A7: A] :
( ( member @ A @ A7 @ A4 )
& ( R @ A7 @ X3 ) ) ) ) ) ) ) ) ).
% pigeonhole_infinite_rel
thf(fact_86_not__finite__existsD,axiom,
! [A: $tType,P3: A > $o] :
( ~ ( finite_finite2 @ A @ ( collect @ A @ P3 ) )
=> ? [X1: A] : ( P3 @ X1 ) ) ).
% not_finite_existsD
thf(fact_87_prod_Ocase__distrib,axiom,
! [C: $tType,D: $tType,B: $tType,A: $tType,H: C > D,F: A > B > C,Prod: product_prod @ A @ B] :
( ( H @ ( product_case_prod @ A @ B @ C @ F @ Prod ) )
= ( product_case_prod @ A @ B @ D
@ ^ [X14: A,X23: B] : ( H @ ( F @ X14 @ X23 ) )
@ Prod ) ) ).
% prod.case_distrib
thf(fact_88_insert__Collect,axiom,
! [A: $tType,A3: A,P3: A > $o] :
( ( insert @ A @ A3 @ ( collect @ A @ P3 ) )
= ( collect @ A
@ ^ [U: A] :
( ( U != A3 )
=> ( P3 @ U ) ) ) ) ).
% insert_Collect
thf(fact_89_insert__compr,axiom,
! [A: $tType] :
( ( insert @ A )
= ( ^ [A7: A,B8: set @ A] :
( collect @ A
@ ^ [X4: A] :
( ( X4 = A7 )
| ( member @ A @ X4 @ B8 ) ) ) ) ) ).
% insert_compr
thf(fact_90_Collect__neg__eq,axiom,
! [A: $tType,P3: A > $o] :
( ( collect @ A
@ ^ [X4: A] :
~ ( P3 @ X4 ) )
= ( uminus_uminus @ ( set @ A ) @ ( collect @ A @ P3 ) ) ) ).
% Collect_neg_eq
thf(fact_91_Compl__eq,axiom,
! [A: $tType] :
( ( uminus_uminus @ ( set @ A ) )
= ( ^ [A6: set @ A] :
( collect @ A
@ ^ [X4: A] :
~ ( member @ A @ X4 @ A6 ) ) ) ) ).
% Compl_eq
thf(fact_92_some1__equality,axiom,
! [A: $tType,P3: A > $o,A3: A] :
( ? [X6: A] :
( ( P3 @ X6 )
& ! [Y5: A] :
( ( P3 @ Y5 )
=> ( Y5 = X6 ) ) )
=> ( ( P3 @ A3 )
=> ( ( ^ [P: A > $o] :
@+[X: A] : ( P @ X )
@ P3 )
= A3 ) ) ) ).
% some1_equality
thf(fact_93_some__eq__ex,axiom,
! [A: $tType,P3: A > $o] :
( ( P3
@ ( ^ [P: A > $o] :
@+[X: A] : ( P @ X )
@ P3 ) )
= ( ^ [P: A > $o] :
? [X: A] : ( P @ X )
@ P3 ) ) ).
% some_eq_ex
thf(fact_94_someI2__bex,axiom,
! [A: $tType,A4: set @ A,P3: A > $o,Q: A > $o] :
( ? [X6: A] :
( ( member @ A @ X6 @ A4 )
& ( P3 @ X6 ) )
=> ( ! [X3: A] :
( ( ( member @ A @ X3 @ A4 )
& ( P3 @ X3 ) )
=> ( Q @ X3 ) )
=> ( Q
@ @+[X4: A] :
( ( member @ A @ X4 @ A4 )
& ( P3 @ X4 ) ) ) ) ) ).
% someI2_bex
thf(fact_95_someI2__ex,axiom,
! [A: $tType,P3: A > $o,Q: A > $o] :
( ? [X13: A] : ( P3 @ X13 )
=> ( ! [X3: A] :
( ( P3 @ X3 )
=> ( Q @ X3 ) )
=> ( Q
@ ( ^ [P: A > $o] :
@+[X: A] : ( P @ X )
@ P3 ) ) ) ) ).
% someI2_ex
thf(fact_96_someI__ex,axiom,
! [A: $tType,P3: A > $o] :
( ? [X13: A] : ( P3 @ X13 )
=> ( P3
@ ( ^ [P: A > $o] :
@+[X: A] : ( P @ X )
@ P3 ) ) ) ).
% someI_ex
thf(fact_97_someI2,axiom,
! [A: $tType,P3: A > $o,A3: A,Q: A > $o] :
( ( P3 @ A3 )
=> ( ! [X3: A] :
( ( P3 @ X3 )
=> ( Q @ X3 ) )
=> ( Q
@ ( ^ [P: A > $o] :
@+[X: A] : ( P @ X )
@ P3 ) ) ) ) ).
% someI2
thf(fact_98_old_Oprod_Ocase,axiom,
! [A: $tType,C: $tType,B: $tType,F: A > B > C,X12: A,X22: B] :
( ( product_case_prod @ A @ B @ C @ F @ ( product_Pair @ A @ B @ X12 @ X22 ) )
= ( F @ X12 @ X22 ) ) ).
% old.prod.case
thf(fact_99_finite_OinsertI,axiom,
! [A: $tType,A4: set @ A,A3: A] :
( ( finite_finite2 @ A @ A4 )
=> ( finite_finite2 @ A @ ( insert @ A @ A3 @ A4 ) ) ) ).
% finite.insertI
thf(fact_100_cond__case__prod__eta,axiom,
! [C: $tType,B: $tType,A: $tType,F: A > B > C,G: ( product_prod @ A @ B ) > C] :
( ! [X3: A,Y5: B] :
( ( F @ X3 @ Y5 )
= ( G @ ( product_Pair @ A @ B @ X3 @ Y5 ) ) )
=> ( ( product_case_prod @ A @ B @ C @ F )
= G ) ) ).
% cond_case_prod_eta
thf(fact_101_case__prod__eta,axiom,
! [C: $tType,B: $tType,A: $tType,F: ( product_prod @ A @ B ) > C] :
( ( product_case_prod @ A @ B @ C
@ ^ [X4: A,Y: B] : ( F @ ( product_Pair @ A @ B @ X4 @ Y ) ) )
= F ) ).
% case_prod_eta
thf(fact_102_case__prodE2,axiom,
! [B: $tType,A: $tType,C: $tType,Q: A > $o,P3: B > C > A,Z2: product_prod @ B @ C] :
( ( Q @ ( product_case_prod @ B @ C @ A @ P3 @ Z2 ) )
=> ~ ! [X3: B,Y5: C] :
( ( Z2
= ( product_Pair @ B @ C @ X3 @ Y5 ) )
=> ~ ( Q @ ( P3 @ X3 @ Y5 ) ) ) ) ).
% case_prodE2
thf(fact_103_case__prodE_H,axiom,
! [B: $tType,A: $tType,C: $tType,C2: A > B > C > $o,P2: product_prod @ A @ B,Z2: C] :
( ( product_case_prod @ A @ B @ ( C > $o ) @ C2 @ P2 @ Z2 )
=> ~ ! [X3: A,Y5: B] :
( ( P2
= ( product_Pair @ A @ B @ X3 @ Y5 ) )
=> ~ ( C2 @ X3 @ Y5 @ Z2 ) ) ) ).
% case_prodE'
thf(fact_104_case__prodD_H,axiom,
! [B: $tType,A: $tType,C: $tType,R: A > B > C > $o,A3: A,B3: B,C2: C] :
( ( product_case_prod @ A @ B @ ( C > $o ) @ R @ ( product_Pair @ A @ B @ A3 @ B3 ) @ C2 )
=> ( R @ A3 @ B3 @ C2 ) ) ).
% case_prodD'
thf(fact_105_finite__option__UNIV,axiom,
! [A: $tType] :
( ( finite_finite2 @ ( option @ A ) @ ( top_top @ ( set @ ( option @ A ) ) ) )
= ( finite_finite2 @ A @ ( top_top @ ( set @ A ) ) ) ) ).
% finite_option_UNIV
thf(fact_106_old_Oprod_Orec,axiom,
! [A: $tType,T: $tType,B: $tType,F1: A > B > T,A3: A,B3: B] :
( ( product_rec_prod @ A @ B @ T @ F1 @ ( product_Pair @ A @ B @ A3 @ B3 ) )
= ( F1 @ A3 @ B3 ) ) ).
% old.prod.rec
thf(fact_107_case__prod__Pair__iden,axiom,
! [B: $tType,A: $tType,P2: product_prod @ A @ B] :
( ( product_case_prod @ A @ B @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B ) @ P2 )
= P2 ) ).
% case_prod_Pair_iden
thf(fact_108_iso__tuple__UNIV__I,axiom,
! [A: $tType,X2: A] : ( member @ A @ X2 @ ( top_top @ ( set @ A ) ) ) ).
% iso_tuple_UNIV_I
thf(fact_109_compl__eq__compl__iff,axiom,
! [A: $tType] :
( ( boolean_algebra @ A @ ( type2 @ A ) )
=> ! [X2: A,Y3: A] :
( ( ( uminus_uminus @ A @ X2 )
= ( uminus_uminus @ A @ Y3 ) )
= ( X2 = Y3 ) ) ) ).
% compl_eq_compl_iff
thf(fact_110_double__compl,axiom,
! [A: $tType] :
( ( boolean_algebra @ A @ ( type2 @ A ) )
=> ! [X2: A] :
( ( uminus_uminus @ A @ ( uminus_uminus @ A @ X2 ) )
= X2 ) ) ).
% double_compl
thf(fact_111_neg__equal__iff__equal,axiom,
! [A: $tType] :
( ( group_add @ A @ ( type2 @ A ) )
=> ! [A3: A,B3: A] :
( ( ( uminus_uminus @ A @ A3 )
= ( uminus_uminus @ A @ B3 ) )
= ( A3 = B3 ) ) ) ).
% neg_equal_iff_equal
thf(fact_112_add_Oinverse__inverse,axiom,
! [A: $tType] :
( ( group_add @ A @ ( type2 @ A ) )
=> ! [A3: A] :
( ( uminus_uminus @ A @ ( uminus_uminus @ A @ A3 ) )
= A3 ) ) ).
% add.inverse_inverse
thf(fact_113_uminus__apply,axiom,
! [B: $tType,A: $tType] :
( ( uminus @ B @ ( type2 @ B ) )
=> ( ( uminus_uminus @ ( A > B ) )
= ( ^ [A6: A > B,X4: A] : ( uminus_uminus @ B @ ( A6 @ X4 ) ) ) ) ) ).
% uminus_apply
thf(fact_114_top__apply,axiom,
! [C: $tType,D: $tType] :
( ( top @ C @ ( type2 @ C ) )
=> ( ( top_top @ ( D > C ) )
= ( ^ [X4: D] : ( top_top @ C ) ) ) ) ).
% top_apply
thf(fact_115_split__part,axiom,
! [B: $tType,A: $tType,P3: $o,Q: A > B > $o] :
( ( product_case_prod @ A @ B @ $o
@ ^ [A7: A,B6: B] :
( P3
& ( Q @ A7 @ B6 ) ) )
= ( ^ [Ab: product_prod @ A @ B] :
( P3
& ( product_case_prod @ A @ B @ $o @ Q @ Ab ) ) ) ) ).
% split_part
thf(fact_116_prod_Odisc__eq__case,axiom,
! [B: $tType,A: $tType,Prod: product_prod @ A @ B] :
( product_case_prod @ A @ B @ $o
@ ^ [Uu: A,Uv: B] : $true
@ Prod ) ).
% prod.disc_eq_case
thf(fact_117_top__set__def,axiom,
! [A: $tType] :
( ( top_top @ ( set @ A ) )
= ( collect @ A @ ( top_top @ ( A > $o ) ) ) ) ).
% top_set_def
thf(fact_118_uminus__set__def,axiom,
! [A: $tType] :
( ( uminus_uminus @ ( set @ A ) )
= ( ^ [A6: set @ A] :
( collect @ A
@ ( uminus_uminus @ ( A > $o )
@ ^ [X4: A] : ( member @ A @ X4 @ A6 ) ) ) ) ) ).
% uminus_set_def
thf(fact_119_fun__Compl__def,axiom,
! [B: $tType,A: $tType] :
( ( uminus @ B @ ( type2 @ B ) )
=> ( ( uminus_uminus @ ( A > B ) )
= ( ^ [A6: A > B,X4: A] : ( uminus_uminus @ B @ ( A6 @ X4 ) ) ) ) ) ).
% fun_Compl_def
thf(fact_120_equation__minus__iff,axiom,
! [A: $tType] :
( ( group_add @ A @ ( type2 @ A ) )
=> ! [A3: A,B3: A] :
( ( A3
= ( uminus_uminus @ A @ B3 ) )
= ( B3
= ( uminus_uminus @ A @ A3 ) ) ) ) ).
% equation_minus_iff
thf(fact_121_minus__equation__iff,axiom,
! [A: $tType] :
( ( group_add @ A @ ( type2 @ A ) )
=> ! [A3: A,B3: A] :
( ( ( uminus_uminus @ A @ A3 )
= B3 )
= ( ( uminus_uminus @ A @ B3 )
= A3 ) ) ) ).
% minus_equation_iff
thf(fact_122_case__prod__app,axiom,
! [A: $tType,D: $tType,C: $tType,B: $tType] :
( ( product_case_prod @ B @ C @ ( D > A ) )
= ( ^ [F4: B > C > D > A,X4: product_prod @ B @ C,Y: D] :
( product_case_prod @ B @ C @ A
@ ^ [L: B,R2: C] : ( F4 @ L @ R2 @ Y )
@ X4 ) ) ) ).
% case_prod_app
thf(fact_123_top__empty__eq2,axiom,
! [B: $tType,A: $tType] :
( ( top_top @ ( A > B > $o ) )
= ( ^ [X4: A,Y: B] : ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X4 @ Y ) @ ( top_top @ ( set @ ( product_prod @ A @ B ) ) ) ) ) ) ).
% top_empty_eq2
thf(fact_124__092_060open_062infinite_A_Ireachable__via_Agraph_A_I_N_Ainsert_An_Ans_____J_An_J_A_092_060and_062_Afinite_Ans_____092_060close_062,axiom,
( ~ ( finite_finite2 @ node @ ( koenig317145564le_via @ node @ graph @ ( uminus_uminus @ ( set @ node ) @ ( insert @ node @ n @ ns ) ) @ n ) )
& ( finite_finite2 @ node @ ns ) ) ).
% \<open>infinite (reachable_via graph (- insert n ns__) n) \<and> finite ns__\<close>
thf(fact_125_ldistinct_I2_J,axiom,
~ ( coinductive_lnull @ node @ ( f @ ( product_Pair @ node @ ( set @ node ) @ na @ nsa ) ) ) ).
% ldistinct(2)
thf(fact_126_f__simps_I3_J,axiom,
! [Na: node,Nsa: set @ node] :
( ( coinductive_ltl @ node @ ( f @ ( product_Pair @ node @ ( set @ node ) @ Na @ Nsa ) ) )
= ( f
@ ( product_Pair @ node @ ( set @ node )
@ ( ^ [P: node > $o] :
@+[X: node] : ( P @ X )
@ ( product_case_prod @ node @ ( set @ node ) @ ( node > $o )
@ ^ [N2: node,Ns: set @ node,N3: node] :
( ( graph @ N2 @ N3 )
& ~ ( finite_finite2 @ node @ ( koenig317145564le_via @ node @ graph @ ( uminus_uminus @ ( set @ node ) @ ( insert @ node @ N2 @ ( insert @ node @ N3 @ Ns ) ) ) @ N3 ) )
& ~ ( member @ node @ N3 @ ( insert @ node @ N2 @ Ns ) ) )
@ ( product_Pair @ node @ ( set @ node ) @ Na @ Nsa ) ) )
@ ( insert @ node @ Na @ Nsa ) ) ) ) ).
% f_simps(3)
thf(fact_127_f__simps_I1_J,axiom,
! [Na: node,Nsa: set @ node] :
~ ( coinductive_lnull @ node @ ( f @ ( product_Pair @ node @ ( set @ node ) @ Na @ Nsa ) ) ) ).
% f_simps(1)
thf(fact_128_top1I,axiom,
! [A: $tType,X2: A] : ( top_top @ ( A > $o ) @ X2 ) ).
% top1I
thf(fact_129_ns__def,axiom,
( ns
= ( bot_bot @ ( set @ node ) ) ) ).
% ns_def
thf(fact_130_top2I,axiom,
! [A: $tType,B: $tType,X2: A,Y3: B] : ( top_top @ ( A > B > $o ) @ X2 @ Y3 ) ).
% top2I
thf(fact_131_top__conj_I2_J,axiom,
! [A: $tType,P3: $o,X2: A] :
( ( P3
& ( top_top @ ( A > $o ) @ X2 ) )
= P3 ) ).
% top_conj(2)
thf(fact_132_top__conj_I1_J,axiom,
! [A: $tType,X2: A,P3: $o] :
( ( ( top_top @ ( A > $o ) @ X2 )
& P3 )
= P3 ) ).
% top_conj(1)
thf(fact_133_top__empty__eq,axiom,
! [A: $tType] :
( ( top_top @ ( A > $o ) )
= ( ^ [X4: A] : ( member @ A @ X4 @ ( top_top @ ( set @ A ) ) ) ) ) ).
% top_empty_eq
thf(fact_134_pred__equals__eq2,axiom,
! [B: $tType,A: $tType,R: set @ ( product_prod @ A @ B ),S: set @ ( product_prod @ A @ B )] :
( ( ( ^ [X4: A,Y: B] : ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X4 @ Y ) @ R ) )
= ( ^ [X4: A,Y: B] : ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X4 @ Y ) @ S ) ) )
= ( R = S ) ) ).
% pred_equals_eq2
thf(fact_135_X,axiom,
? [N4: node,Ns2: set @ node] :
( ( ( f @ ( product_Pair @ node @ ( set @ node ) @ n @ ( bot_bot @ ( set @ node ) ) ) )
= ( f @ ( product_Pair @ node @ ( set @ node ) @ N4 @ Ns2 ) ) )
& ( finite_finite2 @ node @ Ns2 )
& ~ ( finite_finite2 @ node @ ( koenig317145564le_via @ node @ graph @ ( uminus_uminus @ ( set @ node ) @ ( insert @ node @ N4 @ Ns2 ) ) @ N4 ) ) ) ).
% X
thf(fact_136__092_060open_062infinite_A_Ireachable__via_Agraph_A_I_N_A_123n_125_J_An_J_092_060close_062,axiom,
~ ( finite_finite2 @ node @ ( koenig317145564le_via @ node @ graph @ ( uminus_uminus @ ( set @ node ) @ ( insert @ node @ n @ ( bot_bot @ ( set @ node ) ) ) ) @ n ) ) ).
% \<open>infinite (reachable_via graph (- {n}) n)\<close>
thf(fact_137_lset,axiom,
! [X7: node,Na: node,Nsa: set @ node] :
( ( member @ node @ X7 @ ( coinductive_lset @ node @ ( f @ ( product_Pair @ node @ ( set @ node ) @ Na @ Nsa ) ) ) )
=> ( ~ ( member @ node @ Na @ Nsa )
=> ( ( finite_finite2 @ node @ Nsa )
=> ( ~ ( finite_finite2 @ node @ ( koenig317145564le_via @ node @ graph @ ( uminus_uminus @ ( set @ node ) @ ( insert @ node @ Na @ Nsa ) ) @ Na ) )
=> ~ ( member @ node @ X7 @ Nsa ) ) ) ) ) ).
% lset
thf(fact_138__092_060open_062n_A_092_060in_062_Alset_A_If_A_In_M_Ans_____J_J_092_060close_062,axiom,
member @ node @ n @ ( coinductive_lset @ node @ ( f @ ( product_Pair @ node @ ( set @ node ) @ n @ ns ) ) ) ).
% \<open>n \<in> lset (f (n, ns__))\<close>
thf(fact_139_bot__apply,axiom,
! [C: $tType,D: $tType] :
( ( bot @ C @ ( type2 @ C ) )
=> ( ( bot_bot @ ( D > C ) )
= ( ^ [X4: D] : ( bot_bot @ C ) ) ) ) ).
% bot_apply
thf(fact_140_empty__iff,axiom,
! [A: $tType,C2: A] :
~ ( member @ A @ C2 @ ( bot_bot @ ( set @ A ) ) ) ).
% empty_iff
thf(fact_141_all__not__in__conv,axiom,
! [A: $tType,A4: set @ A] :
( ( ! [X4: A] :
~ ( member @ A @ X4 @ A4 ) )
= ( A4
= ( bot_bot @ ( set @ A ) ) ) ) ).
% all_not_in_conv
thf(fact_142_Collect__empty__eq,axiom,
! [A: $tType,P3: A > $o] :
( ( ( collect @ A @ P3 )
= ( bot_bot @ ( set @ A ) ) )
= ( ! [X4: A] :
~ ( P3 @ X4 ) ) ) ).
% Collect_empty_eq
thf(fact_143_empty__Collect__eq,axiom,
! [A: $tType,P3: A > $o] :
( ( ( bot_bot @ ( set @ A ) )
= ( collect @ A @ P3 ) )
= ( ! [X4: A] :
~ ( P3 @ X4 ) ) ) ).
% empty_Collect_eq
thf(fact_144_singletonI,axiom,
! [A: $tType,A3: A] : ( member @ A @ A3 @ ( insert @ A @ A3 @ ( bot_bot @ ( set @ A ) ) ) ) ).
% singletonI
thf(fact_145_Collect__const,axiom,
! [A: $tType,P3: $o] :
( ( P3
=> ( ( collect @ A
@ ^ [S2: A] : P3 )
= ( top_top @ ( set @ A ) ) ) )
& ( ~ P3
=> ( ( collect @ A
@ ^ [S2: A] : P3 )
= ( bot_bot @ ( set @ A ) ) ) ) ) ).
% Collect_const
thf(fact_146_singleton__conv,axiom,
! [A: $tType,A3: A] :
( ( collect @ A
@ ^ [X4: A] : X4 = A3 )
= ( insert @ A @ A3 @ ( bot_bot @ ( set @ A ) ) ) ) ).
% singleton_conv
thf(fact_147_singleton__conv2,axiom,
! [A: $tType,A3: A] :
( ( collect @ A
@ ( ^ [Y2: A,Z: A] : Y2 = Z
@ A3 ) )
= ( insert @ A @ A3 @ ( bot_bot @ ( set @ A ) ) ) ) ).
% singleton_conv2
thf(fact_148_compl__top__eq,axiom,
! [A: $tType] :
( ( boolean_algebra @ A @ ( type2 @ A ) )
=> ( ( uminus_uminus @ A @ ( top_top @ A ) )
= ( bot_bot @ A ) ) ) ).
% compl_top_eq
thf(fact_149_compl__bot__eq,axiom,
! [A: $tType] :
( ( boolean_algebra @ A @ ( type2 @ A ) )
=> ( ( uminus_uminus @ A @ ( bot_bot @ A ) )
= ( top_top @ A ) ) ) ).
% compl_bot_eq
thf(fact_150_empty__not__UNIV,axiom,
! [A: $tType] :
( ( bot_bot @ ( set @ A ) )
!= ( top_top @ ( set @ A ) ) ) ).
% empty_not_UNIV
thf(fact_151_emptyE,axiom,
! [A: $tType,A3: A] :
~ ( member @ A @ A3 @ ( bot_bot @ ( set @ A ) ) ) ).
% emptyE
thf(fact_152_equals0D,axiom,
! [A: $tType,A4: set @ A,A3: A] :
( ( A4
= ( bot_bot @ ( set @ A ) ) )
=> ~ ( member @ A @ A3 @ A4 ) ) ).
% equals0D
thf(fact_153_equals0I,axiom,
! [A: $tType,A4: set @ A] :
( ! [Y5: A] :
~ ( member @ A @ Y5 @ A4 )
=> ( A4
= ( bot_bot @ ( set @ A ) ) ) ) ).
% equals0I
thf(fact_154_ex__in__conv,axiom,
! [A: $tType,A4: set @ A] :
( ( ? [X4: A] : ( member @ A @ X4 @ A4 ) )
= ( A4
!= ( bot_bot @ ( set @ A ) ) ) ) ).
% ex_in_conv
thf(fact_155_singleton__inject,axiom,
! [A: $tType,A3: A,B3: A] :
( ( ( insert @ A @ A3 @ ( bot_bot @ ( set @ A ) ) )
= ( insert @ A @ B3 @ ( bot_bot @ ( set @ A ) ) ) )
=> ( A3 = B3 ) ) ).
% singleton_inject
thf(fact_156_insert__not__empty,axiom,
! [A: $tType,A3: A,A4: set @ A] :
( ( insert @ A @ A3 @ A4 )
!= ( bot_bot @ ( set @ A ) ) ) ).
% insert_not_empty
thf(fact_157_doubleton__eq__iff,axiom,
! [A: $tType,A3: A,B3: A,C2: A,D3: A] :
( ( ( insert @ A @ A3 @ ( insert @ A @ B3 @ ( bot_bot @ ( set @ A ) ) ) )
= ( insert @ A @ C2 @ ( insert @ A @ D3 @ ( bot_bot @ ( set @ A ) ) ) ) )
= ( ( ( A3 = C2 )
& ( B3 = D3 ) )
| ( ( A3 = D3 )
& ( B3 = C2 ) ) ) ) ).
% doubleton_eq_iff
thf(fact_158_singleton__iff,axiom,
! [A: $tType,B3: A,A3: A] :
( ( member @ A @ B3 @ ( insert @ A @ A3 @ ( bot_bot @ ( set @ A ) ) ) )
= ( B3 = A3 ) ) ).
% singleton_iff
thf(fact_159_singletonD,axiom,
! [A: $tType,B3: A,A3: A] :
( ( member @ A @ B3 @ ( insert @ A @ A3 @ ( bot_bot @ ( set @ A ) ) ) )
=> ( B3 = A3 ) ) ).
% singletonD
thf(fact_160_infinite__imp__nonempty,axiom,
! [A: $tType,S: set @ A] :
( ~ ( finite_finite2 @ A @ S )
=> ( S
!= ( bot_bot @ ( set @ A ) ) ) ) ).
% infinite_imp_nonempty
thf(fact_161_finite_OemptyI,axiom,
! [A: $tType] : ( finite_finite2 @ A @ ( bot_bot @ ( set @ A ) ) ) ).
% finite.emptyI
thf(fact_162_empty__def,axiom,
! [A: $tType] :
( ( bot_bot @ ( set @ A ) )
= ( collect @ A
@ ^ [X4: A] : $false ) ) ).
% empty_def
thf(fact_163_bot__fun__def,axiom,
! [B: $tType,A: $tType] :
( ( bot @ B @ ( type2 @ B ) )
=> ( ( bot_bot @ ( A > B ) )
= ( ^ [X4: A] : ( bot_bot @ B ) ) ) ) ).
% bot_fun_def
thf(fact_164_Collect__conv__if,axiom,
! [A: $tType,P3: A > $o,A3: A] :
( ( ( P3 @ A3 )
=> ( ( collect @ A
@ ^ [X4: A] :
( ( X4 = A3 )
& ( P3 @ X4 ) ) )
= ( insert @ A @ A3 @ ( bot_bot @ ( set @ A ) ) ) ) )
& ( ~ ( P3 @ A3 )
=> ( ( collect @ A
@ ^ [X4: A] :
( ( X4 = A3 )
& ( P3 @ X4 ) ) )
= ( bot_bot @ ( set @ A ) ) ) ) ) ).
% Collect_conv_if
thf(fact_165_Collect__conv__if2,axiom,
! [A: $tType,P3: A > $o,A3: A] :
( ( ( P3 @ A3 )
=> ( ( collect @ A
@ ^ [X4: A] :
( ( A3 = X4 )
& ( P3 @ X4 ) ) )
= ( insert @ A @ A3 @ ( bot_bot @ ( set @ A ) ) ) ) )
& ( ~ ( P3 @ A3 )
=> ( ( collect @ A
@ ^ [X4: A] :
( ( A3 = X4 )
& ( P3 @ X4 ) ) )
= ( bot_bot @ ( set @ A ) ) ) ) ) ).
% Collect_conv_if2
thf(fact_166_some__in__eq,axiom,
! [A: $tType,A4: set @ A] :
( ( member @ A
@ @+[X4: A] : ( member @ A @ X4 @ A4 )
@ A4 )
= ( A4
!= ( bot_bot @ ( set @ A ) ) ) ) ).
% some_in_eq
thf(fact_167_finite_Ocases,axiom,
! [A: $tType,A3: set @ A] :
( ( finite_finite2 @ A @ A3 )
=> ( ( A3
!= ( bot_bot @ ( set @ A ) ) )
=> ~ ! [A8: set @ A] :
( ? [A2: A] :
( A3
= ( insert @ A @ A2 @ A8 ) )
=> ~ ( finite_finite2 @ A @ A8 ) ) ) ) ).
% finite.cases
thf(fact_168_finite_Osimps,axiom,
! [A: $tType] :
( ( finite_finite2 @ A )
= ( ^ [A7: set @ A] :
( ( A7
= ( bot_bot @ ( set @ A ) ) )
| ? [A6: set @ A,B6: A] :
( ( A7
= ( insert @ A @ B6 @ A6 ) )
& ( finite_finite2 @ A @ A6 ) ) ) ) ) ).
% finite.simps
thf(fact_169_finite__induct,axiom,
! [A: $tType,F5: set @ A,P3: ( set @ A ) > $o] :
( ( finite_finite2 @ A @ F5 )
=> ( ( P3 @ ( bot_bot @ ( set @ A ) ) )
=> ( ! [X3: A,F6: set @ A] :
( ( finite_finite2 @ A @ F6 )
=> ( ~ ( member @ A @ X3 @ F6 )
=> ( ( P3 @ F6 )
=> ( P3 @ ( insert @ A @ X3 @ F6 ) ) ) ) )
=> ( P3 @ F5 ) ) ) ) ).
% finite_induct
thf(fact_170_finite_Oinducts,axiom,
! [A: $tType,X2: set @ A,P3: ( set @ A ) > $o] :
( ( finite_finite2 @ A @ X2 )
=> ( ( P3 @ ( bot_bot @ ( set @ A ) ) )
=> ( ! [A8: set @ A,A2: A] :
( ( finite_finite2 @ A @ A8 )
=> ( ( P3 @ A8 )
=> ( P3 @ ( insert @ A @ A2 @ A8 ) ) ) )
=> ( P3 @ X2 ) ) ) ) ).
% finite.inducts
thf(fact_171_finite__ne__induct,axiom,
! [A: $tType,F5: set @ A,P3: ( set @ A ) > $o] :
( ( finite_finite2 @ A @ F5 )
=> ( ( F5
!= ( bot_bot @ ( set @ A ) ) )
=> ( ! [X3: A] : ( P3 @ ( insert @ A @ X3 @ ( bot_bot @ ( set @ A ) ) ) )
=> ( ! [X3: A,F6: set @ A] :
( ( finite_finite2 @ A @ F6 )
=> ( ( F6
!= ( bot_bot @ ( set @ A ) ) )
=> ( ~ ( member @ A @ X3 @ F6 )
=> ( ( P3 @ F6 )
=> ( P3 @ ( insert @ A @ X3 @ F6 ) ) ) ) ) )
=> ( P3 @ F5 ) ) ) ) ) ).
% finite_ne_induct
thf(fact_172_infinite__finite__induct,axiom,
! [A: $tType,P3: ( set @ A ) > $o,A4: set @ A] :
( ! [A8: set @ A] :
( ~ ( finite_finite2 @ A @ A8 )
=> ( P3 @ A8 ) )
=> ( ( P3 @ ( bot_bot @ ( set @ A ) ) )
=> ( ! [X3: A,F6: set @ A] :
( ( finite_finite2 @ A @ F6 )
=> ( ~ ( member @ A @ X3 @ F6 )
=> ( ( P3 @ F6 )
=> ( P3 @ ( insert @ A @ X3 @ F6 ) ) ) ) )
=> ( P3 @ A4 ) ) ) ) ).
% infinite_finite_induct
thf(fact_173_Compl__empty__eq,axiom,
! [A: $tType] :
( ( uminus_uminus @ ( set @ A ) @ ( bot_bot @ ( set @ A ) ) )
= ( top_top @ ( set @ A ) ) ) ).
% Compl_empty_eq
thf(fact_174_Compl__UNIV__eq,axiom,
! [A: $tType] :
( ( uminus_uminus @ ( set @ A ) @ ( top_top @ ( set @ A ) ) )
= ( bot_bot @ ( set @ A ) ) ) ).
% Compl_UNIV_eq
thf(fact_175__092_060open_062f_A_In_M_A_123_125_J_A_092_060in_062_Apaths_Agraph_092_060close_062,axiom,
member @ ( coinductive_llist @ node ) @ ( f @ ( product_Pair @ node @ ( set @ node ) @ n @ ( bot_bot @ ( set @ node ) ) ) ) @ ( koenig916195507_paths @ node @ graph ) ).
% \<open>f (n, {}) \<in> paths graph\<close>
thf(fact_176_lset__eq__empty,axiom,
! [A: $tType,Xs: coinductive_llist @ A] :
( ( ( coinductive_lset @ A @ Xs )
= ( bot_bot @ ( set @ A ) ) )
= ( coinductive_lnull @ A @ Xs ) ) ).
% lset_eq_empty
thf(fact_177__092_060open_062_N_A_123n_125_A_092_060subseteq_062_Areachable__via_Agraph_A_I_N_A_123n_125_J_An_092_060close_062,axiom,
ord_less_eq @ ( set @ node ) @ ( uminus_uminus @ ( set @ node ) @ ( insert @ node @ n @ ( bot_bot @ ( set @ node ) ) ) ) @ ( koenig317145564le_via @ node @ graph @ ( uminus_uminus @ ( set @ node ) @ ( insert @ node @ n @ ( bot_bot @ ( set @ node ) ) ) ) @ n ) ).
% \<open>- {n} \<subseteq> reachable_via graph (- {n}) n\<close>
thf(fact_178__092_060open_062_092_060not_062_Alfinite_A_If_A_In_M_Ans_____J_J_092_060close_062,axiom,
~ ( coinductive_lfinite @ node @ ( f @ ( product_Pair @ node @ ( set @ node ) @ n @ ns ) ) ) ).
% \<open>\<not> lfinite (f (n, ns__))\<close>
thf(fact_179_order__refl,axiom,
! [A: $tType] :
( ( preorder @ A @ ( type2 @ A ) )
=> ! [X2: A] : ( ord_less_eq @ A @ X2 @ X2 ) ) ).
% order_refl
thf(fact_180_subset__antisym,axiom,
! [A: $tType,A4: set @ A,B4: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ A4 @ B4 )
=> ( ( ord_less_eq @ ( set @ A ) @ B4 @ A4 )
=> ( A4 = B4 ) ) ) ).
% subset_antisym
thf(fact_181_subsetI,axiom,
! [A: $tType,A4: set @ A,B4: set @ A] :
( ! [X3: A] :
( ( member @ A @ X3 @ A4 )
=> ( member @ A @ X3 @ B4 ) )
=> ( ord_less_eq @ ( set @ A ) @ A4 @ B4 ) ) ).
% subsetI
thf(fact_182_neg__le__iff__le,axiom,
! [A: $tType] :
( ( ordered_ab_group_add @ A @ ( type2 @ A ) )
=> ! [B3: A,A3: A] :
( ( ord_less_eq @ A @ ( uminus_uminus @ A @ B3 ) @ ( uminus_uminus @ A @ A3 ) )
= ( ord_less_eq @ A @ A3 @ B3 ) ) ) ).
% neg_le_iff_le
thf(fact_183_compl__le__compl__iff,axiom,
! [A: $tType] :
( ( boolean_algebra @ A @ ( type2 @ A ) )
=> ! [X2: A,Y3: A] :
( ( ord_less_eq @ A @ ( uminus_uminus @ A @ X2 ) @ ( uminus_uminus @ A @ Y3 ) )
= ( ord_less_eq @ A @ Y3 @ X2 ) ) ) ).
% compl_le_compl_iff
thf(fact_184_subset__empty,axiom,
! [A: $tType,A4: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ A4 @ ( bot_bot @ ( set @ A ) ) )
= ( A4
= ( bot_bot @ ( set @ A ) ) ) ) ).
% subset_empty
thf(fact_185_empty__subsetI,axiom,
! [A: $tType,A4: set @ A] : ( ord_less_eq @ ( set @ A ) @ ( bot_bot @ ( set @ A ) ) @ A4 ) ).
% empty_subsetI
thf(fact_186_insert__subset,axiom,
! [A: $tType,X2: A,A4: set @ A,B4: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ ( insert @ A @ X2 @ A4 ) @ B4 )
= ( ( member @ A @ X2 @ B4 )
& ( ord_less_eq @ ( set @ A ) @ A4 @ B4 ) ) ) ).
% insert_subset
thf(fact_187_Compl__subset__Compl__iff,axiom,
! [A: $tType,A4: set @ A,B4: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ ( uminus_uminus @ ( set @ A ) @ A4 ) @ ( uminus_uminus @ ( set @ A ) @ B4 ) )
= ( ord_less_eq @ ( set @ A ) @ B4 @ A4 ) ) ).
% Compl_subset_Compl_iff
thf(fact_188_Compl__anti__mono,axiom,
! [A: $tType,A4: set @ A,B4: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ A4 @ B4 )
=> ( ord_less_eq @ ( set @ A ) @ ( uminus_uminus @ ( set @ A ) @ B4 ) @ ( uminus_uminus @ ( set @ A ) @ A4 ) ) ) ).
% Compl_anti_mono
thf(fact_189_lfinite__ltl,axiom,
! [A: $tType,Xs: coinductive_llist @ A] :
( ( coinductive_lfinite @ A @ ( coinductive_ltl @ A @ Xs ) )
= ( coinductive_lfinite @ A @ Xs ) ) ).
% lfinite_ltl
thf(fact_190_singleton__insert__inj__eq_H,axiom,
! [A: $tType,A3: A,A4: set @ A,B3: A] :
( ( ( insert @ A @ A3 @ A4 )
= ( insert @ A @ B3 @ ( bot_bot @ ( set @ A ) ) ) )
= ( ( A3 = B3 )
& ( ord_less_eq @ ( set @ A ) @ A4 @ ( insert @ A @ B3 @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ).
% singleton_insert_inj_eq'
thf(fact_191_singleton__insert__inj__eq,axiom,
! [A: $tType,B3: A,A3: A,A4: set @ A] :
( ( ( insert @ A @ B3 @ ( bot_bot @ ( set @ A ) ) )
= ( insert @ A @ A3 @ A4 ) )
= ( ( A3 = B3 )
& ( ord_less_eq @ ( set @ A ) @ A4 @ ( insert @ A @ B3 @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ).
% singleton_insert_inj_eq
thf(fact_192_finite__Collect__subsets,axiom,
! [A: $tType,A4: set @ A] :
( ( finite_finite2 @ A @ A4 )
=> ( finite_finite2 @ ( set @ A )
@ ( collect @ ( set @ A )
@ ^ [B8: set @ A] : ( ord_less_eq @ ( set @ A ) @ B8 @ A4 ) ) ) ) ).
% finite_Collect_subsets
thf(fact_193_subset__Compl__singleton,axiom,
! [A: $tType,A4: set @ A,B3: A] :
( ( ord_less_eq @ ( set @ A ) @ A4 @ ( uminus_uminus @ ( set @ A ) @ ( insert @ A @ B3 @ ( bot_bot @ ( set @ A ) ) ) ) )
= ( ~ ( member @ A @ B3 @ A4 ) ) ) ).
% subset_Compl_singleton
thf(fact_194_bot__empty__eq2,axiom,
! [B: $tType,A: $tType] :
( ( bot_bot @ ( A > B > $o ) )
= ( ^ [X4: A,Y: B] : ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X4 @ Y ) @ ( bot_bot @ ( set @ ( product_prod @ A @ B ) ) ) ) ) ) ).
% bot_empty_eq2
thf(fact_195_bot__empty__eq,axiom,
! [A: $tType] :
( ( bot_bot @ ( A > $o ) )
= ( ^ [X4: A] : ( member @ A @ X4 @ ( bot_bot @ ( set @ A ) ) ) ) ) ).
% bot_empty_eq
thf(fact_196_lset__ltl,axiom,
! [A: $tType,Xs: coinductive_llist @ A] : ( ord_less_eq @ ( set @ A ) @ ( coinductive_lset @ A @ ( coinductive_ltl @ A @ Xs ) ) @ ( coinductive_lset @ A @ Xs ) ) ).
% lset_ltl
thf(fact_197_bot__set__def,axiom,
! [A: $tType] :
( ( bot_bot @ ( set @ A ) )
= ( collect @ A @ ( bot_bot @ ( A > $o ) ) ) ) ).
% bot_set_def
thf(fact_198_bot_Oextremum,axiom,
! [A: $tType] :
( ( order_bot @ A @ ( type2 @ A ) )
=> ! [A3: A] : ( ord_less_eq @ A @ ( bot_bot @ A ) @ A3 ) ) ).
% bot.extremum
thf(fact_199_bot_Oextremum__unique,axiom,
! [A: $tType] :
( ( order_bot @ A @ ( type2 @ A ) )
=> ! [A3: A] :
( ( ord_less_eq @ A @ A3 @ ( bot_bot @ A ) )
= ( A3
= ( bot_bot @ A ) ) ) ) ).
% bot.extremum_unique
thf(fact_200_bot_Oextremum__uniqueI,axiom,
! [A: $tType] :
( ( order_bot @ A @ ( type2 @ A ) )
=> ! [A3: A] :
( ( ord_less_eq @ A @ A3 @ ( bot_bot @ A ) )
=> ( A3
= ( bot_bot @ A ) ) ) ) ).
% bot.extremum_uniqueI
thf(fact_201_lfinite__imp__finite__lset,axiom,
! [A: $tType,Xs: coinductive_llist @ A] :
( ( coinductive_lfinite @ A @ Xs )
=> ( finite_finite2 @ A @ ( coinductive_lset @ A @ Xs ) ) ) ).
% lfinite_imp_finite_lset
thf(fact_202_dual__order_Oantisym,axiom,
! [A: $tType] :
( ( order @ A @ ( type2 @ A ) )
=> ! [B3: A,A3: A] :
( ( ord_less_eq @ A @ B3 @ A3 )
=> ( ( ord_less_eq @ A @ A3 @ B3 )
=> ( A3 = B3 ) ) ) ) ).
% dual_order.antisym
thf(fact_203_dual__order_Otrans,axiom,
! [A: $tType] :
( ( order @ A @ ( type2 @ A ) )
=> ! [B3: A,A3: A,C2: A] :
( ( ord_less_eq @ A @ B3 @ A3 )
=> ( ( ord_less_eq @ A @ C2 @ B3 )
=> ( ord_less_eq @ A @ C2 @ A3 ) ) ) ) ).
% dual_order.trans
thf(fact_204_linorder__wlog,axiom,
! [A: $tType] :
( ( linorder @ A @ ( type2 @ A ) )
=> ! [P3: A > A > $o,A3: A,B3: A] :
( ! [A2: A,B2: A] :
( ( ord_less_eq @ A @ A2 @ B2 )
=> ( P3 @ A2 @ B2 ) )
=> ( ! [A2: A,B2: A] :
( ( P3 @ B2 @ A2 )
=> ( P3 @ A2 @ B2 ) )
=> ( P3 @ A3 @ B3 ) ) ) ) ).
% linorder_wlog
thf(fact_205_dual__order_Orefl,axiom,
! [A: $tType] :
( ( order @ A @ ( type2 @ A ) )
=> ! [A3: A] : ( ord_less_eq @ A @ A3 @ A3 ) ) ).
% dual_order.refl
thf(fact_206_order__trans,axiom,
! [A: $tType] :
( ( preorder @ A @ ( type2 @ A ) )
=> ! [X2: A,Y3: A,Z2: A] :
( ( ord_less_eq @ A @ X2 @ Y3 )
=> ( ( ord_less_eq @ A @ Y3 @ Z2 )
=> ( ord_less_eq @ A @ X2 @ Z2 ) ) ) ) ).
% order_trans
thf(fact_207_order__class_Oorder_Oantisym,axiom,
! [A: $tType] :
( ( order @ A @ ( type2 @ A ) )
=> ! [A3: A,B3: A] :
( ( ord_less_eq @ A @ A3 @ B3 )
=> ( ( ord_less_eq @ A @ B3 @ A3 )
=> ( A3 = B3 ) ) ) ) ).
% order_class.order.antisym
thf(fact_208_ord__le__eq__trans,axiom,
! [A: $tType] :
( ( ord @ A @ ( type2 @ A ) )
=> ! [A3: A,B3: A,C2: A] :
( ( ord_less_eq @ A @ A3 @ B3 )
=> ( ( B3 = C2 )
=> ( ord_less_eq @ A @ A3 @ C2 ) ) ) ) ).
% ord_le_eq_trans
thf(fact_209_ord__eq__le__trans,axiom,
! [A: $tType] :
( ( ord @ A @ ( type2 @ A ) )
=> ! [A3: A,B3: A,C2: A] :
( ( A3 = B3 )
=> ( ( ord_less_eq @ A @ B3 @ C2 )
=> ( ord_less_eq @ A @ A3 @ C2 ) ) ) ) ).
% ord_eq_le_trans
thf(fact_210_antisym__conv,axiom,
! [A: $tType] :
( ( order @ A @ ( type2 @ A ) )
=> ! [Y3: A,X2: A] :
( ( ord_less_eq @ A @ Y3 @ X2 )
=> ( ( ord_less_eq @ A @ X2 @ Y3 )
= ( X2 = Y3 ) ) ) ) ).
% antisym_conv
thf(fact_211_le__cases3,axiom,
! [A: $tType] :
( ( linorder @ A @ ( type2 @ A ) )
=> ! [X2: A,Y3: A,Z2: A] :
( ( ( ord_less_eq @ A @ X2 @ Y3 )
=> ~ ( ord_less_eq @ A @ Y3 @ Z2 ) )
=> ( ( ( ord_less_eq @ A @ Y3 @ X2 )
=> ~ ( ord_less_eq @ A @ X2 @ Z2 ) )
=> ( ( ( ord_less_eq @ A @ X2 @ Z2 )
=> ~ ( ord_less_eq @ A @ Z2 @ Y3 ) )
=> ( ( ( ord_less_eq @ A @ Z2 @ Y3 )
=> ~ ( ord_less_eq @ A @ Y3 @ X2 ) )
=> ( ( ( ord_less_eq @ A @ Y3 @ Z2 )
=> ~ ( ord_less_eq @ A @ Z2 @ X2 ) )
=> ~ ( ( ord_less_eq @ A @ Z2 @ X2 )
=> ~ ( ord_less_eq @ A @ X2 @ Y3 ) ) ) ) ) ) ) ) ).
% le_cases3
thf(fact_212_order_Otrans,axiom,
! [A: $tType] :
( ( order @ A @ ( type2 @ A ) )
=> ! [A3: A,B3: A,C2: A] :
( ( ord_less_eq @ A @ A3 @ B3 )
=> ( ( ord_less_eq @ A @ B3 @ C2 )
=> ( ord_less_eq @ A @ A3 @ C2 ) ) ) ) ).
% order.trans
thf(fact_213_le__cases,axiom,
! [A: $tType] :
( ( linorder @ A @ ( type2 @ A ) )
=> ! [X2: A,Y3: A] :
( ~ ( ord_less_eq @ A @ X2 @ Y3 )
=> ( ord_less_eq @ A @ Y3 @ X2 ) ) ) ).
% le_cases
thf(fact_214_eq__refl,axiom,
! [A: $tType] :
( ( preorder @ A @ ( type2 @ A ) )
=> ! [X2: A,Y3: A] :
( ( X2 = Y3 )
=> ( ord_less_eq @ A @ X2 @ Y3 ) ) ) ).
% eq_refl
thf(fact_215_linear,axiom,
! [A: $tType] :
( ( linorder @ A @ ( type2 @ A ) )
=> ! [X2: A,Y3: A] :
( ( ord_less_eq @ A @ X2 @ Y3 )
| ( ord_less_eq @ A @ Y3 @ X2 ) ) ) ).
% linear
thf(fact_216_antisym,axiom,
! [A: $tType] :
( ( order @ A @ ( type2 @ A ) )
=> ! [X2: A,Y3: A] :
( ( ord_less_eq @ A @ X2 @ Y3 )
=> ( ( ord_less_eq @ A @ Y3 @ X2 )
=> ( X2 = Y3 ) ) ) ) ).
% antisym
thf(fact_217_eq__iff,axiom,
! [A: $tType] :
( ( order @ A @ ( type2 @ A ) )
=> ( ( ^ [Y2: A,Z: A] : Y2 = Z )
= ( ^ [X4: A,Y: A] :
( ( ord_less_eq @ A @ X4 @ Y )
& ( ord_less_eq @ A @ Y @ X4 ) ) ) ) ) ).
% eq_iff
thf(fact_218_ord__le__eq__subst,axiom,
! [A: $tType,B: $tType] :
( ( ( ord @ B @ ( type2 @ B ) )
& ( ord @ A @ ( type2 @ A ) ) )
=> ! [A3: A,B3: A,F: A > B,C2: B] :
( ( ord_less_eq @ A @ A3 @ B3 )
=> ( ( ( F @ B3 )
= C2 )
=> ( ! [X3: A,Y5: A] :
( ( ord_less_eq @ A @ X3 @ Y5 )
=> ( ord_less_eq @ B @ ( F @ X3 ) @ ( F @ Y5 ) ) )
=> ( ord_less_eq @ B @ ( F @ A3 ) @ C2 ) ) ) ) ) ).
% ord_le_eq_subst
thf(fact_219_ord__eq__le__subst,axiom,
! [A: $tType,B: $tType] :
( ( ( ord @ B @ ( type2 @ B ) )
& ( ord @ A @ ( type2 @ A ) ) )
=> ! [A3: A,F: B > A,B3: B,C2: B] :
( ( A3
= ( F @ B3 ) )
=> ( ( ord_less_eq @ B @ B3 @ C2 )
=> ( ! [X3: B,Y5: B] :
( ( ord_less_eq @ B @ X3 @ Y5 )
=> ( ord_less_eq @ A @ ( F @ X3 ) @ ( F @ Y5 ) ) )
=> ( ord_less_eq @ A @ A3 @ ( F @ C2 ) ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_220_order__subst2,axiom,
! [A: $tType,C: $tType] :
( ( ( order @ C @ ( type2 @ C ) )
& ( order @ A @ ( type2 @ A ) ) )
=> ! [A3: A,B3: A,F: A > C,C2: C] :
( ( ord_less_eq @ A @ A3 @ B3 )
=> ( ( ord_less_eq @ C @ ( F @ B3 ) @ C2 )
=> ( ! [X3: A,Y5: A] :
( ( ord_less_eq @ A @ X3 @ Y5 )
=> ( ord_less_eq @ C @ ( F @ X3 ) @ ( F @ Y5 ) ) )
=> ( ord_less_eq @ C @ ( F @ A3 ) @ C2 ) ) ) ) ) ).
% order_subst2
thf(fact_221_order__subst1,axiom,
! [A: $tType,B: $tType] :
( ( ( order @ B @ ( type2 @ B ) )
& ( order @ A @ ( type2 @ A ) ) )
=> ! [A3: A,F: B > A,B3: B,C2: B] :
( ( ord_less_eq @ A @ A3 @ ( F @ B3 ) )
=> ( ( ord_less_eq @ B @ B3 @ C2 )
=> ( ! [X3: B,Y5: B] :
( ( ord_less_eq @ B @ X3 @ Y5 )
=> ( ord_less_eq @ A @ ( F @ X3 ) @ ( F @ Y5 ) ) )
=> ( ord_less_eq @ A @ A3 @ ( F @ C2 ) ) ) ) ) ) ).
% order_subst1
thf(fact_222_le__fun__def,axiom,
! [B: $tType,A: $tType] :
( ( ord @ B @ ( type2 @ B ) )
=> ( ( ord_less_eq @ ( A > B ) )
= ( ^ [F4: A > B,G4: A > B] :
! [X4: A] : ( ord_less_eq @ B @ ( F4 @ X4 ) @ ( G4 @ X4 ) ) ) ) ) ).
% le_fun_def
thf(fact_223_le__funI,axiom,
! [B: $tType,A: $tType] :
( ( ord @ B @ ( type2 @ B ) )
=> ! [F: A > B,G: A > B] :
( ! [X3: A] : ( ord_less_eq @ B @ ( F @ X3 ) @ ( G @ X3 ) )
=> ( ord_less_eq @ ( A > B ) @ F @ G ) ) ) ).
% le_funI
thf(fact_224_le__funE,axiom,
! [B: $tType,A: $tType] :
( ( ord @ B @ ( type2 @ B ) )
=> ! [F: A > B,G: A > B,X2: A] :
( ( ord_less_eq @ ( A > B ) @ F @ G )
=> ( ord_less_eq @ B @ ( F @ X2 ) @ ( G @ X2 ) ) ) ) ).
% le_funE
thf(fact_225_le__funD,axiom,
! [B: $tType,A: $tType] :
( ( ord @ B @ ( type2 @ B ) )
=> ! [F: A > B,G: A > B,X2: A] :
( ( ord_less_eq @ ( A > B ) @ F @ G )
=> ( ord_less_eq @ B @ ( F @ X2 ) @ ( G @ X2 ) ) ) ) ).
% le_funD
thf(fact_226_finite__subset,axiom,
! [A: $tType,A4: set @ A,B4: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ A4 @ B4 )
=> ( ( finite_finite2 @ A @ B4 )
=> ( finite_finite2 @ A @ A4 ) ) ) ).
% finite_subset
thf(fact_227_infinite__super,axiom,
! [A: $tType,S: set @ A,T2: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ S @ T2 )
=> ( ~ ( finite_finite2 @ A @ S )
=> ~ ( finite_finite2 @ A @ T2 ) ) ) ).
% infinite_super
thf(fact_228_rev__finite__subset,axiom,
! [A: $tType,B4: set @ A,A4: set @ A] :
( ( finite_finite2 @ A @ B4 )
=> ( ( ord_less_eq @ ( set @ A ) @ A4 @ B4 )
=> ( finite_finite2 @ A @ A4 ) ) ) ).
% rev_finite_subset
thf(fact_229_Set_Oinsert__mono,axiom,
! [A: $tType,C5: set @ A,D4: set @ A,A3: A] :
( ( ord_less_eq @ ( set @ A ) @ C5 @ D4 )
=> ( ord_less_eq @ ( set @ A ) @ ( insert @ A @ A3 @ C5 ) @ ( insert @ A @ A3 @ D4 ) ) ) ).
% Set.insert_mono
thf(fact_230_subset__insert,axiom,
! [A: $tType,X2: A,A4: set @ A,B4: set @ A] :
( ~ ( member @ A @ X2 @ A4 )
=> ( ( ord_less_eq @ ( set @ A ) @ A4 @ ( insert @ A @ X2 @ B4 ) )
= ( ord_less_eq @ ( set @ A ) @ A4 @ B4 ) ) ) ).
% subset_insert
thf(fact_231_subset__insertI,axiom,
! [A: $tType,B4: set @ A,A3: A] : ( ord_less_eq @ ( set @ A ) @ B4 @ ( insert @ A @ A3 @ B4 ) ) ).
% subset_insertI
thf(fact_232_subset__insertI2,axiom,
! [A: $tType,A4: set @ A,B4: set @ A,B3: A] :
( ( ord_less_eq @ ( set @ A ) @ A4 @ B4 )
=> ( ord_less_eq @ ( set @ A ) @ A4 @ ( insert @ A @ B3 @ B4 ) ) ) ).
% subset_insertI2
thf(fact_233_wlog__linorder__le,axiom,
! [A: $tType] :
( ( linorder @ A @ ( type2 @ A ) )
=> ! [P3: A > A > $o,B3: A,A3: A] :
( ! [A2: A,B2: A] :
( ( ord_less_eq @ A @ A2 @ B2 )
=> ( P3 @ A2 @ B2 ) )
=> ( ( ( P3 @ B3 @ A3 )
=> ( P3 @ A3 @ B3 ) )
=> ( P3 @ A3 @ B3 ) ) ) ) ).
% wlog_linorder_le
thf(fact_234_Collect__mono__iff,axiom,
! [A: $tType,P3: A > $o,Q: A > $o] :
( ( ord_less_eq @ ( set @ A ) @ ( collect @ A @ P3 ) @ ( collect @ A @ Q ) )
= ( ! [X4: A] :
( ( P3 @ X4 )
=> ( Q @ X4 ) ) ) ) ).
% Collect_mono_iff
thf(fact_235_contra__subsetD,axiom,
! [A: $tType,A4: set @ A,B4: set @ A,C2: A] :
( ( ord_less_eq @ ( set @ A ) @ A4 @ B4 )
=> ( ~ ( member @ A @ C2 @ B4 )
=> ~ ( member @ A @ C2 @ A4 ) ) ) ).
% contra_subsetD
thf(fact_236_set__eq__subset,axiom,
! [A: $tType] :
( ( ^ [Y2: set @ A,Z: set @ A] : Y2 = Z )
= ( ^ [A6: set @ A,B8: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ A6 @ B8 )
& ( ord_less_eq @ ( set @ A ) @ B8 @ A6 ) ) ) ) ).
% set_eq_subset
thf(fact_237_subset__trans,axiom,
! [A: $tType,A4: set @ A,B4: set @ A,C5: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ A4 @ B4 )
=> ( ( ord_less_eq @ ( set @ A ) @ B4 @ C5 )
=> ( ord_less_eq @ ( set @ A ) @ A4 @ C5 ) ) ) ).
% subset_trans
thf(fact_238_Collect__mono,axiom,
! [A: $tType,P3: A > $o,Q: A > $o] :
( ! [X3: A] :
( ( P3 @ X3 )
=> ( Q @ X3 ) )
=> ( ord_less_eq @ ( set @ A ) @ ( collect @ A @ P3 ) @ ( collect @ A @ Q ) ) ) ).
% Collect_mono
thf(fact_239_subset__refl,axiom,
! [A: $tType,A4: set @ A] : ( ord_less_eq @ ( set @ A ) @ A4 @ A4 ) ).
% subset_refl
thf(fact_240_rev__subsetD,axiom,
! [A: $tType,C2: A,A4: set @ A,B4: set @ A] :
( ( member @ A @ C2 @ A4 )
=> ( ( ord_less_eq @ ( set @ A ) @ A4 @ B4 )
=> ( member @ A @ C2 @ B4 ) ) ) ).
% rev_subsetD
thf(fact_241_subset__iff,axiom,
! [A: $tType] :
( ( ord_less_eq @ ( set @ A ) )
= ( ^ [A6: set @ A,B8: set @ A] :
! [T3: A] :
( ( member @ A @ T3 @ A6 )
=> ( member @ A @ T3 @ B8 ) ) ) ) ).
% subset_iff
thf(fact_242_set__rev__mp,axiom,
! [A: $tType,X2: A,A4: set @ A,B4: set @ A] :
( ( member @ A @ X2 @ A4 )
=> ( ( ord_less_eq @ ( set @ A ) @ A4 @ B4 )
=> ( member @ A @ X2 @ B4 ) ) ) ).
% set_rev_mp
thf(fact_243_equalityD2,axiom,
! [A: $tType,A4: set @ A,B4: set @ A] :
( ( A4 = B4 )
=> ( ord_less_eq @ ( set @ A ) @ B4 @ A4 ) ) ).
% equalityD2
thf(fact_244_equalityD1,axiom,
! [A: $tType,A4: set @ A,B4: set @ A] :
( ( A4 = B4 )
=> ( ord_less_eq @ ( set @ A ) @ A4 @ B4 ) ) ).
% equalityD1
thf(fact_245_subset__eq,axiom,
! [A: $tType] :
( ( ord_less_eq @ ( set @ A ) )
= ( ^ [A6: set @ A,B8: set @ A] :
! [X4: A] :
( ( member @ A @ X4 @ A6 )
=> ( member @ A @ X4 @ B8 ) ) ) ) ).
% subset_eq
thf(fact_246_equalityE,axiom,
! [A: $tType,A4: set @ A,B4: set @ A] :
( ( A4 = B4 )
=> ~ ( ( ord_less_eq @ ( set @ A ) @ A4 @ B4 )
=> ~ ( ord_less_eq @ ( set @ A ) @ B4 @ A4 ) ) ) ).
% equalityE
thf(fact_247_subsetCE,axiom,
! [A: $tType,A4: set @ A,B4: set @ A,C2: A] :
( ( ord_less_eq @ ( set @ A ) @ A4 @ B4 )
=> ( ( member @ A @ C2 @ A4 )
=> ( member @ A @ C2 @ B4 ) ) ) ).
% subsetCE
thf(fact_248_subsetD,axiom,
! [A: $tType,A4: set @ A,B4: set @ A,C2: A] :
( ( ord_less_eq @ ( set @ A ) @ A4 @ B4 )
=> ( ( member @ A @ C2 @ A4 )
=> ( member @ A @ C2 @ B4 ) ) ) ).
% subsetD
thf(fact_249_in__mono,axiom,
! [A: $tType,A4: set @ A,B4: set @ A,X2: A] :
( ( ord_less_eq @ ( set @ A ) @ A4 @ B4 )
=> ( ( member @ A @ X2 @ A4 )
=> ( member @ A @ X2 @ B4 ) ) ) ).
% in_mono
thf(fact_250_set__mp,axiom,
! [A: $tType,A4: set @ A,B4: set @ A,X2: A] :
( ( ord_less_eq @ ( set @ A ) @ A4 @ B4 )
=> ( ( member @ A @ X2 @ A4 )
=> ( member @ A @ X2 @ B4 ) ) ) ).
% set_mp
thf(fact_251_lnull__imp__lfinite,axiom,
! [A: $tType,Xs: coinductive_llist @ A] :
( ( coinductive_lnull @ A @ Xs )
=> ( coinductive_lfinite @ A @ Xs ) ) ).
% lnull_imp_lfinite
thf(fact_252_lfinite__induct,axiom,
! [A: $tType,Xs: coinductive_llist @ A,P3: ( coinductive_llist @ A ) > $o] :
( ( coinductive_lfinite @ A @ Xs )
=> ( ! [Xs2: coinductive_llist @ A] :
( ( coinductive_lnull @ A @ Xs2 )
=> ( P3 @ Xs2 ) )
=> ( ! [Xs2: coinductive_llist @ A] :
( ( coinductive_lfinite @ A @ Xs2 )
=> ( ~ ( coinductive_lnull @ A @ Xs2 )
=> ( ( P3 @ ( coinductive_ltl @ A @ Xs2 ) )
=> ( P3 @ Xs2 ) ) ) )
=> ( P3 @ Xs ) ) ) ) ).
% lfinite_induct
thf(fact_253_subset__UNIV,axiom,
! [A: $tType,A4: set @ A] : ( ord_less_eq @ ( set @ A ) @ A4 @ ( top_top @ ( set @ A ) ) ) ).
% subset_UNIV
thf(fact_254_le__imp__neg__le,axiom,
! [A: $tType] :
( ( ordered_ab_group_add @ A @ ( type2 @ A ) )
=> ! [A3: A,B3: A] :
( ( ord_less_eq @ A @ A3 @ B3 )
=> ( ord_less_eq @ A @ ( uminus_uminus @ A @ B3 ) @ ( uminus_uminus @ A @ A3 ) ) ) ) ).
% le_imp_neg_le
thf(fact_255_minus__le__iff,axiom,
! [A: $tType] :
( ( ordered_ab_group_add @ A @ ( type2 @ A ) )
=> ! [A3: A,B3: A] :
( ( ord_less_eq @ A @ ( uminus_uminus @ A @ A3 ) @ B3 )
= ( ord_less_eq @ A @ ( uminus_uminus @ A @ B3 ) @ A3 ) ) ) ).
% minus_le_iff
%----Type constructors (31)
thf(tcon_fun___Lattices_Oboolean__algebra,axiom,
! [A9: $tType,A10: $tType] :
( ( boolean_algebra @ A10 @ ( type2 @ A10 ) )
=> ( boolean_algebra @ ( A9 > A10 ) @ ( type2 @ ( A9 > A10 ) ) ) ) ).
thf(tcon_fun___Orderings_Oorder__bot,axiom,
! [A9: $tType,A10: $tType] :
( ( order_bot @ A10 @ ( type2 @ A10 ) )
=> ( order_bot @ ( A9 > A10 ) @ ( type2 @ ( A9 > A10 ) ) ) ) ).
thf(tcon_fun___Orderings_Opreorder,axiom,
! [A9: $tType,A10: $tType] :
( ( preorder @ A10 @ ( type2 @ A10 ) )
=> ( preorder @ ( A9 > A10 ) @ ( type2 @ ( A9 > A10 ) ) ) ) ).
thf(tcon_fun___Finite__Set_Ofinite,axiom,
! [A9: $tType,A10: $tType] :
( ( ( finite_finite @ A9 @ ( type2 @ A9 ) )
& ( finite_finite @ A10 @ ( type2 @ A10 ) ) )
=> ( finite_finite @ ( A9 > A10 ) @ ( type2 @ ( A9 > A10 ) ) ) ) ).
thf(tcon_fun___Orderings_Oorder,axiom,
! [A9: $tType,A10: $tType] :
( ( order @ A10 @ ( type2 @ A10 ) )
=> ( order @ ( A9 > A10 ) @ ( type2 @ ( A9 > A10 ) ) ) ) ).
thf(tcon_fun___Orderings_Otop,axiom,
! [A9: $tType,A10: $tType] :
( ( top @ A10 @ ( type2 @ A10 ) )
=> ( top @ ( A9 > A10 ) @ ( type2 @ ( A9 > A10 ) ) ) ) ).
thf(tcon_fun___Orderings_Oord,axiom,
! [A9: $tType,A10: $tType] :
( ( ord @ A10 @ ( type2 @ A10 ) )
=> ( ord @ ( A9 > A10 ) @ ( type2 @ ( A9 > A10 ) ) ) ) ).
thf(tcon_fun___Orderings_Obot,axiom,
! [A9: $tType,A10: $tType] :
( ( bot @ A10 @ ( type2 @ A10 ) )
=> ( bot @ ( A9 > A10 ) @ ( type2 @ ( A9 > A10 ) ) ) ) ).
thf(tcon_fun___Groups_Ouminus,axiom,
! [A9: $tType,A10: $tType] :
( ( uminus @ A10 @ ( type2 @ A10 ) )
=> ( uminus @ ( A9 > A10 ) @ ( type2 @ ( A9 > A10 ) ) ) ) ).
thf(tcon_Set_Oset___Lattices_Oboolean__algebra_1,axiom,
! [A9: $tType] : ( boolean_algebra @ ( set @ A9 ) @ ( type2 @ ( set @ A9 ) ) ) ).
thf(tcon_Set_Oset___Orderings_Oorder__bot_2,axiom,
! [A9: $tType] : ( order_bot @ ( set @ A9 ) @ ( type2 @ ( set @ A9 ) ) ) ).
thf(tcon_Set_Oset___Orderings_Opreorder_3,axiom,
! [A9: $tType] : ( preorder @ ( set @ A9 ) @ ( type2 @ ( set @ A9 ) ) ) ).
thf(tcon_Set_Oset___Finite__Set_Ofinite_4,axiom,
! [A9: $tType] :
( ( finite_finite @ A9 @ ( type2 @ A9 ) )
=> ( finite_finite @ ( set @ A9 ) @ ( type2 @ ( set @ A9 ) ) ) ) ).
thf(tcon_Set_Oset___Orderings_Oorder_5,axiom,
! [A9: $tType] : ( order @ ( set @ A9 ) @ ( type2 @ ( set @ A9 ) ) ) ).
thf(tcon_Set_Oset___Orderings_Otop_6,axiom,
! [A9: $tType] : ( top @ ( set @ A9 ) @ ( type2 @ ( set @ A9 ) ) ) ).
thf(tcon_Set_Oset___Orderings_Oord_7,axiom,
! [A9: $tType] : ( ord @ ( set @ A9 ) @ ( type2 @ ( set @ A9 ) ) ) ).
thf(tcon_Set_Oset___Orderings_Obot_8,axiom,
! [A9: $tType] : ( bot @ ( set @ A9 ) @ ( type2 @ ( set @ A9 ) ) ) ).
thf(tcon_Set_Oset___Groups_Ouminus_9,axiom,
! [A9: $tType] : ( uminus @ ( set @ A9 ) @ ( type2 @ ( set @ A9 ) ) ) ).
thf(tcon_HOL_Obool___Lattices_Oboolean__algebra_10,axiom,
boolean_algebra @ $o @ ( type2 @ $o ) ).
thf(tcon_HOL_Obool___Orderings_Oorder__bot_11,axiom,
order_bot @ $o @ ( type2 @ $o ) ).
thf(tcon_HOL_Obool___Orderings_Opreorder_12,axiom,
preorder @ $o @ ( type2 @ $o ) ).
thf(tcon_HOL_Obool___Orderings_Olinorder,axiom,
linorder @ $o @ ( type2 @ $o ) ).
thf(tcon_HOL_Obool___Finite__Set_Ofinite_13,axiom,
finite_finite @ $o @ ( type2 @ $o ) ).
thf(tcon_HOL_Obool___Orderings_Oorder_14,axiom,
order @ $o @ ( type2 @ $o ) ).
thf(tcon_HOL_Obool___Orderings_Otop_15,axiom,
top @ $o @ ( type2 @ $o ) ).
thf(tcon_HOL_Obool___Orderings_Oord_16,axiom,
ord @ $o @ ( type2 @ $o ) ).
thf(tcon_HOL_Obool___Orderings_Obot_17,axiom,
bot @ $o @ ( type2 @ $o ) ).
thf(tcon_HOL_Obool___Groups_Ouminus_18,axiom,
uminus @ $o @ ( type2 @ $o ) ).
thf(tcon_Sum__Type_Osum___Finite__Set_Ofinite_19,axiom,
! [A9: $tType,A10: $tType] :
( ( ( finite_finite @ A9 @ ( type2 @ A9 ) )
& ( finite_finite @ A10 @ ( type2 @ A10 ) ) )
=> ( finite_finite @ ( sum_sum @ A9 @ A10 ) @ ( type2 @ ( sum_sum @ A9 @ A10 ) ) ) ) ).
thf(tcon_Option_Ooption___Finite__Set_Ofinite_20,axiom,
! [A9: $tType] :
( ( finite_finite @ A9 @ ( type2 @ A9 ) )
=> ( finite_finite @ ( option @ A9 ) @ ( type2 @ ( option @ A9 ) ) ) ) ).
thf(tcon_Product__Type_Oprod___Finite__Set_Ofinite_21,axiom,
! [A9: $tType,A10: $tType] :
( ( ( finite_finite @ A9 @ ( type2 @ A9 ) )
& ( finite_finite @ A10 @ ( type2 @ A10 ) ) )
=> ( finite_finite @ ( product_prod @ A9 @ A10 ) @ ( type2 @ ( product_prod @ A9 @ A10 ) ) ) ) ).
%----Conjectures (1)
thf(conj_0,conjecture,
( product_case_prod @ node @ ( set @ node ) @ ( node > $o )
@ ^ [N2: node,Ns: set @ node,N3: node] :
( ( graph @ N2 @ N3 )
& ~ ( finite_finite2 @ node @ ( koenig317145564le_via @ node @ graph @ ( uminus_uminus @ ( set @ node ) @ ( insert @ node @ N2 @ ( insert @ node @ N3 @ Ns ) ) ) @ N3 ) )
& ~ ( member @ node @ N3 @ ( insert @ node @ N2 @ Ns ) ) )
@ ( product_Pair @ node @ ( set @ node ) @ na @ nsa )
@ ( ^ [P: node > $o] :
@+[X: node] : ( P @ X )
@ ( product_case_prod @ node @ ( set @ node ) @ ( node > $o )
@ ^ [N2: node,Ns: set @ node,N3: node] :
( ( graph @ N2 @ N3 )
& ~ ( finite_finite2 @ node @ ( koenig317145564le_via @ node @ graph @ ( uminus_uminus @ ( set @ node ) @ ( insert @ node @ N2 @ ( insert @ node @ N3 @ Ns ) ) ) @ N3 ) )
& ~ ( member @ node @ N3 @ ( insert @ node @ N2 @ Ns ) ) )
@ ( product_Pair @ node @ ( set @ node ) @ na @ nsa ) ) ) ) ).
%------------------------------------------------------------------------------