TPTP Problem File: COM176^1.p
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%------------------------------------------------------------------------------
% File : COM176^1 : TPTP v8.2.0. Released v7.0.0.
% Domain : Computing Theory
% Problem : Koenig's lemma 131
% 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__131.p [Bla16]
% Status : Theorem
% Rating : 0.67 v8.1.0, 0.50 v7.5.0, 0.33 v7.2.0, 0.50 v7.1.0
% Syntax : Number of formulae : 327 ( 98 unt; 45 typ; 0 def)
% Number of atoms : 833 ( 217 equ; 0 cnn)
% Maximal formula atoms : 13 ( 2 avg)
% Number of connectives : 3825 ( 93 ~; 11 |; 43 &;3299 @)
% ( 0 <=>; 379 =>; 0 <=; 0 <~>)
% Maximal formula depth : 18 ( 8 avg)
% Number of types : 2 ( 1 usr)
% Number of type conns : 140 ( 140 >; 0 *; 0 +; 0 <<)
% Number of symbols : 45 ( 44 usr; 7 con; 0-4 aty)
% Number of variables : 947 ( 51 ^; 855 !; 8 ?; 947 :)
% ( 33 !>; 0 ?*; 0 @-; 0 @+)
% SPC : TH1_THM_EQU_NAR
% Comments : This file was generated by Isabelle (most likely Sledgehammer)
% 2016-07-13 14:46:41.177
%------------------------------------------------------------------------------
%----Could-be-implicit typings (4)
thf(ty_t_Coinductive__List_Ollist,type,
coinductive_llist: $tType > $tType ).
thf(ty_t_Set_Oset,type,
set: $tType > $tType ).
thf(ty_t_itself,type,
itself: $tType > $tType ).
thf(ty_tf_a,type,
a: $tType ).
%----Explicit typings (41)
thf(sy_cl_HOL_Otype,type,
type:
!>[A: $tType] : ( ( itself @ A ) > $o ) ).
thf(sy_cl_Groups_Ominus,type,
minus:
!>[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_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_Orderings_Olinorder,type,
linorder:
!>[A: $tType] : ( ( itself @ A ) > $o ) ).
thf(sy_cl_Orderings_Opreorder,type,
preorder:
!>[A: $tType] : ( ( itself @ A ) > $o ) ).
thf(sy_cl_Orderings_Oorder__bot,type,
order_bot:
!>[A: $tType] : ( ( itself @ A ) > $o ) ).
thf(sy_cl_Lattices_Oboolean__algebra,type,
boolean_algebra:
!>[A: $tType] : ( ( itself @ A ) > $o ) ).
thf(sy_cl_Orderings_Odense__linorder,type,
dense_linorder:
!>[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_cl_Groups_Ocancel__ab__semigroup__add,type,
cancel146912293up_add:
!>[A: $tType] : ( ( itself @ A ) > $o ) ).
thf(sy_c_Coinductive__List_Ogen__lset,type,
coinductive_gen_lset:
!>[A: $tType] : ( ( set @ A ) > ( coinductive_llist @ A ) > ( set @ A ) ) ).
thf(sy_c_Coinductive__List_Ollist_OLCons,type,
coinductive_LCons:
!>[A: $tType] : ( A > ( coinductive_llist @ A ) > ( coinductive_llist @ A ) ) ).
thf(sy_c_Coinductive__List_Ollist_Olset,type,
coinductive_lset:
!>[A: $tType] : ( ( coinductive_llist @ A ) > ( set @ A ) ) ).
thf(sy_c_Groups_Ominus__class_Ominus,type,
minus_minus:
!>[A: $tType] : ( A > A > A ) ).
thf(sy_c_Groups_Ouminus__class_Ouminus,type,
uminus_uminus:
!>[A: $tType] : ( A > A ) ).
thf(sy_c_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,type,
ord_less:
!>[A: $tType] : ( A > A > $o ) ).
thf(sy_c_Orderings_Oord__class_Oless__eq,type,
ord_less_eq:
!>[A: $tType] : ( A > A > $o ) ).
thf(sy_c_Pure_Otype,type,
type2:
!>[A: $tType] : ( itself @ A ) ).
thf(sy_c_Set_OCollect,type,
collect:
!>[A: $tType] : ( ( A > $o ) > ( set @ A ) ) ).
thf(sy_c_Set_Oinsert,type,
insert:
!>[A: $tType] : ( A > ( set @ A ) > ( set @ A ) ) ).
thf(sy_c_Set_Ois__empty,type,
is_empty:
!>[A: $tType] : ( ( set @ A ) > $o ) ).
thf(sy_c_Set_Ois__singleton,type,
is_singleton:
!>[A: $tType] : ( ( set @ A ) > $o ) ).
thf(sy_c_Set_Opairwise,type,
pairwise:
!>[A: $tType] : ( ( A > A > $o ) > ( set @ A ) > $o ) ).
thf(sy_c_Set_Oremove,type,
remove:
!>[A: $tType] : ( A > ( set @ A ) > ( set @ A ) ) ).
thf(sy_c_Set_Othe__elem,type,
the_elem:
!>[A: $tType] : ( ( set @ A ) > A ) ).
thf(sy_c_member,type,
member:
!>[A: $tType] : ( A > ( set @ A ) > $o ) ).
thf(sy_v_graph,type,
graph: a > a > $o ).
thf(sy_v_n,type,
n: a ).
thf(sy_v_n_H____,type,
n2: a ).
thf(sy_v_ns,type,
ns: set @ a ).
thf(sy_v_x____,type,
x: a ).
thf(sy_v_xs_H_H____,type,
xs: coinductive_llist @ a ).
thf(sy_v_xs_H____,type,
xs2: coinductive_llist @ a ).
thf(sy_v_xs____,type,
xs3: coinductive_llist @ a ).
%----Relevant facts (256)
thf(fact_0__092_060open_062n_H_A_092_060notin_062_Alset_Axs_H_H_092_060close_062,axiom,
~ ( member @ a @ n2 @ ( coinductive_lset @ a @ xs ) ) ).
% \<open>n' \<notin> lset xs''\<close>
thf(fact_1__092_060open_062_092_060And_062thesis_O_A_I_092_060And_062n_H_Axs_H_O_Axs_A_061_ALCons_An_H_Axs_H_A_092_060Longrightarrow_062_Athesis_J_A_092_060Longrightarrow_062_Athesis_092_060close_062,axiom,
~ ! [N: a,Xs: coinductive_llist @ a] :
( xs3
!= ( coinductive_LCons @ a @ N @ Xs ) ) ).
% \<open>\<And>thesis. (\<And>n' xs'. xs = LCons n' xs' \<Longrightarrow> thesis) \<Longrightarrow> thesis\<close>
thf(fact_2__092_060open_062x_A_092_060in_062_Alset_Axs_H_H_092_060close_062,axiom,
member @ a @ x @ ( coinductive_lset @ a @ xs ) ).
% \<open>x \<in> lset xs''\<close>
thf(fact_3_xs,axiom,
( xs3
= ( coinductive_LCons @ a @ n2 @ xs2 ) ) ).
% xs
thf(fact_4__092_060open_062lset_Axs_A_092_060subseteq_062_Ans_092_060close_062,axiom,
ord_less_eq @ ( set @ a ) @ ( coinductive_lset @ a @ xs3 ) @ ns ).
% \<open>lset xs \<subseteq> ns\<close>
thf(fact_5_False,axiom,
x != n2 ).
% False
thf(fact_6__092_060open_062x_A_092_060in_062_Alset_Axs_092_060close_062,axiom,
member @ a @ x @ ( coinductive_lset @ a @ xs3 ) ).
% \<open>x \<in> lset xs\<close>
thf(fact_7_path_H_H,axiom,
member @ ( coinductive_llist @ a ) @ ( coinductive_LCons @ a @ n2 @ xs ) @ ( koenig916195507_paths @ a @ graph ) ).
% path''
thf(fact_8__092_060open_062n_H_A_092_060in_062_Ans_092_060close_062,axiom,
member @ a @ n2 @ ns ).
% \<open>n' \<in> ns\<close>
thf(fact_9__092_060open_062x_A_092_060in_062_Alset_Axs_H_092_060close_062,axiom,
member @ a @ x @ ( coinductive_lset @ a @ xs2 ) ).
% \<open>x \<in> lset xs'\<close>
thf(fact_10__092_060open_062x_A_092_060in_062_Areachable__via_Agraph_Ans_An_092_060close_062,axiom,
member @ a @ x @ ( koenig317145564le_via @ a @ graph @ ns @ n ) ).
% \<open>x \<in> reachable_via graph ns n\<close>
thf(fact_11__092_060open_062graph_An_An_H_092_060close_062,axiom,
graph @ n @ n2 ).
% \<open>graph n n'\<close>
thf(fact_12_path_H,axiom,
member @ ( coinductive_llist @ a ) @ ( coinductive_LCons @ a @ n2 @ xs2 ) @ ( koenig916195507_paths @ a @ graph ) ).
% path'
thf(fact_13__092_060open_062lset_Axs_H_A_092_060subseteq_062_Ans_092_060close_062,axiom,
ord_less_eq @ ( set @ a ) @ ( coinductive_lset @ a @ xs2 ) @ ns ).
% \<open>lset xs' \<subseteq> ns\<close>
thf(fact_14_path,axiom,
member @ ( coinductive_llist @ a ) @ ( coinductive_LCons @ a @ n @ xs3 ) @ ( koenig916195507_paths @ a @ graph ) ).
% path
thf(fact_15_insert__Diff__single,axiom,
! [A: $tType,A2: A,A3: set @ A] :
( ( insert @ A @ A2 @ ( minus_minus @ ( set @ A ) @ A3 @ ( insert @ A @ A2 @ ( bot_bot @ ( set @ A ) ) ) ) )
= ( insert @ A @ A2 @ A3 ) ) ).
% insert_Diff_single
thf(fact_16_Diff__insert0,axiom,
! [A: $tType,X: A,A3: set @ A,B: set @ A] :
( ~ ( member @ A @ X @ A3 )
=> ( ( minus_minus @ ( set @ A ) @ A3 @ ( insert @ A @ X @ B ) )
= ( minus_minus @ ( set @ A ) @ A3 @ B ) ) ) ).
% Diff_insert0
thf(fact_17_insert__Diff1,axiom,
! [A: $tType,X: A,B: set @ A,A3: set @ A] :
( ( member @ A @ X @ B )
=> ( ( minus_minus @ ( set @ A ) @ ( insert @ A @ X @ A3 ) @ B )
= ( minus_minus @ ( set @ A ) @ A3 @ B ) ) ) ).
% insert_Diff1
thf(fact_18_Diff__empty,axiom,
! [A: $tType,A3: set @ A] :
( ( minus_minus @ ( set @ A ) @ A3 @ ( bot_bot @ ( set @ A ) ) )
= A3 ) ).
% Diff_empty
thf(fact_19_empty__Diff,axiom,
! [A: $tType,A3: set @ A] :
( ( minus_minus @ ( set @ A ) @ ( bot_bot @ ( set @ A ) ) @ A3 )
= ( bot_bot @ ( set @ A ) ) ) ).
% empty_Diff
thf(fact_20_Diff__cancel,axiom,
! [A: $tType,A3: set @ A] :
( ( minus_minus @ ( set @ A ) @ A3 @ A3 )
= ( bot_bot @ ( set @ A ) ) ) ).
% Diff_cancel
thf(fact_21_singletonI,axiom,
! [A: $tType,A2: A] : ( member @ A @ A2 @ ( insert @ A @ A2 @ ( bot_bot @ ( set @ A ) ) ) ) ).
% singletonI
thf(fact_22__092_060open_062n_H_A_092_060noteq_062_Ax_A_092_060Longrightarrow_062_A_092_060exists_062xs_Ha_O_ALCons_An_H_Axs_Ha_A_092_060in_062_Apaths_Agraph_A_092_060and_062_Alset_Axs_Ha_A_092_060subseteq_062_Alset_Axs_H_A_092_060and_062_Ax_A_092_060in_062_Alset_Axs_Ha_A_092_060and_062_An_H_A_092_060notin_062_Alset_Axs_Ha_092_060close_062,axiom,
( ( n2 != x )
=> ? [Xs: coinductive_llist @ a] :
( ( member @ ( coinductive_llist @ a ) @ ( coinductive_LCons @ a @ n2 @ Xs ) @ ( koenig916195507_paths @ a @ graph ) )
& ( ord_less_eq @ ( set @ a ) @ ( coinductive_lset @ a @ Xs ) @ ( coinductive_lset @ a @ xs2 ) )
& ( member @ a @ x @ ( coinductive_lset @ a @ Xs ) )
& ~ ( member @ a @ n2 @ ( coinductive_lset @ a @ Xs ) ) ) ) ).
% \<open>n' \<noteq> x \<Longrightarrow> \<exists>xs'a. LCons n' xs'a \<in> paths graph \<and> lset xs'a \<subseteq> lset xs' \<and> x \<in> lset xs'a \<and> n' \<notin> lset xs'a\<close>
thf(fact_23__092_060open_062lset_Axs_H_H_A_092_060subseteq_062_Alset_Axs_H_092_060close_062,axiom,
ord_less_eq @ ( set @ a ) @ ( coinductive_lset @ a @ xs ) @ ( coinductive_lset @ a @ xs2 ) ).
% \<open>lset xs'' \<subseteq> lset xs'\<close>
thf(fact_24_empty__Collect__eq,axiom,
! [A: $tType,P: A > $o] :
( ( ( bot_bot @ ( set @ A ) )
= ( collect @ A @ P ) )
= ( ! [X2: A] :
~ ( P @ X2 ) ) ) ).
% empty_Collect_eq
thf(fact_25_Collect__empty__eq,axiom,
! [A: $tType,P: A > $o] :
( ( ( collect @ A @ P )
= ( bot_bot @ ( set @ A ) ) )
= ( ! [X2: A] :
~ ( P @ X2 ) ) ) ).
% Collect_empty_eq
thf(fact_26_all__not__in__conv,axiom,
! [A: $tType,A3: set @ A] :
( ( ! [X2: A] :
~ ( member @ A @ X2 @ A3 ) )
= ( A3
= ( bot_bot @ ( set @ A ) ) ) ) ).
% all_not_in_conv
thf(fact_27_empty__iff,axiom,
! [A: $tType,C: A] :
~ ( member @ A @ C @ ( bot_bot @ ( set @ A ) ) ) ).
% empty_iff
thf(fact_28_subsetI,axiom,
! [A: $tType,A3: set @ A,B: set @ A] :
( ! [X3: A] :
( ( member @ A @ X3 @ A3 )
=> ( member @ A @ X3 @ B ) )
=> ( ord_less_eq @ ( set @ A ) @ A3 @ B ) ) ).
% subsetI
thf(fact_29_subset__antisym,axiom,
! [A: $tType,A3: set @ A,B: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ A3 @ B )
=> ( ( ord_less_eq @ ( set @ A ) @ B @ A3 )
=> ( A3 = B ) ) ) ).
% subset_antisym
thf(fact_30_insert__absorb2,axiom,
! [A: $tType,X: A,A3: set @ A] :
( ( insert @ A @ X @ ( insert @ A @ X @ A3 ) )
= ( insert @ A @ X @ A3 ) ) ).
% insert_absorb2
thf(fact_31_insert__iff,axiom,
! [A: $tType,A2: A,B2: A,A3: set @ A] :
( ( member @ A @ A2 @ ( insert @ A @ B2 @ A3 ) )
= ( ( A2 = B2 )
| ( member @ A @ A2 @ A3 ) ) ) ).
% insert_iff
thf(fact_32_insertCI,axiom,
! [A: $tType,A2: A,B: set @ A,B2: A] :
( ( ~ ( member @ A @ A2 @ B )
=> ( A2 = B2 ) )
=> ( member @ A @ A2 @ ( insert @ A @ B2 @ B ) ) ) ).
% insertCI
thf(fact_33_Diff__idemp,axiom,
! [A: $tType,A3: set @ A,B: set @ A] :
( ( minus_minus @ ( set @ A ) @ ( minus_minus @ ( set @ A ) @ A3 @ B ) @ B )
= ( minus_minus @ ( set @ A ) @ A3 @ B ) ) ).
% Diff_idemp
thf(fact_34_Diff__iff,axiom,
! [A: $tType,C: A,A3: set @ A,B: set @ A] :
( ( member @ A @ C @ ( minus_minus @ ( set @ A ) @ A3 @ B ) )
= ( ( member @ A @ C @ A3 )
& ~ ( member @ A @ C @ B ) ) ) ).
% Diff_iff
thf(fact_35_DiffI,axiom,
! [A: $tType,C: A,A3: set @ A,B: set @ A] :
( ( member @ A @ C @ A3 )
=> ( ~ ( member @ A @ C @ B )
=> ( member @ A @ C @ ( minus_minus @ ( set @ A ) @ A3 @ B ) ) ) ) ).
% DiffI
thf(fact_36_empty__subsetI,axiom,
! [A: $tType,A3: set @ A] : ( ord_less_eq @ ( set @ A ) @ ( bot_bot @ ( set @ A ) ) @ A3 ) ).
% empty_subsetI
thf(fact_37_subset__empty,axiom,
! [A: $tType,A3: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ A3 @ ( bot_bot @ ( set @ A ) ) )
= ( A3
= ( bot_bot @ ( set @ A ) ) ) ) ).
% subset_empty
thf(fact_38_insert__subset,axiom,
! [A: $tType,X: A,A3: set @ A,B: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ ( insert @ A @ X @ A3 ) @ B )
= ( ( member @ A @ X @ B )
& ( ord_less_eq @ ( set @ A ) @ A3 @ B ) ) ) ).
% insert_subset
thf(fact_39_singleton__insert__inj__eq_H,axiom,
! [A: $tType,A2: A,A3: set @ A,B2: A] :
( ( ( insert @ A @ A2 @ A3 )
= ( insert @ A @ B2 @ ( bot_bot @ ( set @ A ) ) ) )
= ( ( A2 = B2 )
& ( ord_less_eq @ ( set @ A ) @ A3 @ ( insert @ A @ B2 @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ).
% singleton_insert_inj_eq'
thf(fact_40_singleton__insert__inj__eq,axiom,
! [A: $tType,B2: A,A2: A,A3: set @ A] :
( ( ( insert @ A @ B2 @ ( bot_bot @ ( set @ A ) ) )
= ( insert @ A @ A2 @ A3 ) )
= ( ( A2 = B2 )
& ( ord_less_eq @ ( set @ A ) @ A3 @ ( insert @ A @ B2 @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ).
% singleton_insert_inj_eq
thf(fact_41_Diff__eq__empty__iff,axiom,
! [A: $tType,A3: set @ A,B: set @ A] :
( ( ( minus_minus @ ( set @ A ) @ A3 @ B )
= ( bot_bot @ ( set @ A ) ) )
= ( ord_less_eq @ ( set @ A ) @ A3 @ B ) ) ).
% Diff_eq_empty_iff
thf(fact_42__092_060open_062_092_060And_062thesis_O_A_I_092_060And_062xs_H_H_O_A_092_060lbrakk_062LCons_An_H_Axs_H_H_A_092_060in_062_Apaths_Agraph_059_Alset_Axs_H_H_A_092_060subseteq_062_Alset_Axs_H_059_Ax_A_092_060in_062_Alset_Axs_H_H_059_An_H_A_092_060notin_062_Alset_Axs_H_H_092_060rbrakk_062_A_092_060Longrightarrow_062_Athesis_J_A_092_060Longrightarrow_062_Athesis_092_060close_062,axiom,
~ ! [Xs2: coinductive_llist @ a] :
( ( member @ ( coinductive_llist @ a ) @ ( coinductive_LCons @ a @ n2 @ Xs2 ) @ ( koenig916195507_paths @ a @ graph ) )
=> ( ( ord_less_eq @ ( set @ a ) @ ( coinductive_lset @ a @ Xs2 ) @ ( coinductive_lset @ a @ xs2 ) )
=> ( ( member @ a @ x @ ( coinductive_lset @ a @ Xs2 ) )
=> ( member @ a @ n2 @ ( coinductive_lset @ a @ Xs2 ) ) ) ) ) ).
% \<open>\<And>thesis. (\<And>xs''. \<lbrakk>LCons n' xs'' \<in> paths graph; lset xs'' \<subseteq> lset xs'; x \<in> lset xs''; n' \<notin> lset xs''\<rbrakk> \<Longrightarrow> thesis) \<Longrightarrow> thesis\<close>
thf(fact_43__092_060open_062_092_060And_062thesis_O_A_I_092_060And_062xs_O_A_092_060lbrakk_062LCons_An_Axs_A_092_060in_062_Apaths_Agraph_059_Ax_A_092_060in_062_Alset_Axs_059_Alset_Axs_A_092_060subseteq_062_Ans_092_060rbrakk_062_A_092_060Longrightarrow_062_Athesis_J_A_092_060Longrightarrow_062_Athesis_092_060close_062,axiom,
~ ! [Xs3: coinductive_llist @ a] :
( ( member @ ( coinductive_llist @ a ) @ ( coinductive_LCons @ a @ n @ Xs3 ) @ ( koenig916195507_paths @ a @ graph ) )
=> ( ( member @ a @ x @ ( coinductive_lset @ a @ Xs3 ) )
=> ~ ( ord_less_eq @ ( set @ a ) @ ( coinductive_lset @ a @ Xs3 ) @ ns ) ) ) ).
% \<open>\<And>thesis. (\<And>xs. \<lbrakk>LCons n xs \<in> paths graph; x \<in> lset xs; lset xs \<subseteq> ns\<rbrakk> \<Longrightarrow> thesis) \<Longrightarrow> thesis\<close>
thf(fact_44_set__mp,axiom,
! [A: $tType,A3: set @ A,B: set @ A,X: A] :
( ( ord_less_eq @ ( set @ A ) @ A3 @ B )
=> ( ( member @ A @ X @ A3 )
=> ( member @ A @ X @ B ) ) ) ).
% set_mp
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,A3: set @ A] :
( ( collect @ A
@ ^ [X2: A] : ( member @ A @ X2 @ A3 ) )
= A3 ) ).
% 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,
! [B3: $tType,A: $tType,F: A > B3,G: A > B3] :
( ! [X3: A] :
( ( F @ X3 )
= ( G @ X3 ) )
=> ( F = G ) ) ).
% ext
thf(fact_49_in__mono,axiom,
! [A: $tType,A3: set @ A,B: set @ A,X: A] :
( ( ord_less_eq @ ( set @ A ) @ A3 @ B )
=> ( ( member @ A @ X @ A3 )
=> ( member @ A @ X @ B ) ) ) ).
% in_mono
thf(fact_50_subsetD,axiom,
! [A: $tType,A3: set @ A,B: set @ A,C: A] :
( ( ord_less_eq @ ( set @ A ) @ A3 @ B )
=> ( ( member @ A @ C @ A3 )
=> ( member @ A @ C @ B ) ) ) ).
% subsetD
thf(fact_51_subsetCE,axiom,
! [A: $tType,A3: set @ A,B: set @ A,C: A] :
( ( ord_less_eq @ ( set @ A ) @ A3 @ B )
=> ( ( member @ A @ C @ A3 )
=> ( member @ A @ C @ B ) ) ) ).
% subsetCE
thf(fact_52_equalityE,axiom,
! [A: $tType,A3: set @ A,B: set @ A] :
( ( A3 = B )
=> ~ ( ( ord_less_eq @ ( set @ A ) @ A3 @ B )
=> ~ ( ord_less_eq @ ( set @ A ) @ B @ A3 ) ) ) ).
% equalityE
thf(fact_53_subset__eq,axiom,
! [A: $tType] :
( ( ord_less_eq @ ( set @ A ) )
= ( ^ [A4: set @ A,B4: set @ A] :
! [X2: A] :
( ( member @ A @ X2 @ A4 )
=> ( member @ A @ X2 @ B4 ) ) ) ) ).
% subset_eq
thf(fact_54_equalityD1,axiom,
! [A: $tType,A3: set @ A,B: set @ A] :
( ( A3 = B )
=> ( ord_less_eq @ ( set @ A ) @ A3 @ B ) ) ).
% equalityD1
thf(fact_55_equalityD2,axiom,
! [A: $tType,A3: set @ A,B: set @ A] :
( ( A3 = B )
=> ( ord_less_eq @ ( set @ A ) @ B @ A3 ) ) ).
% equalityD2
thf(fact_56_set__rev__mp,axiom,
! [A: $tType,X: A,A3: set @ A,B: set @ A] :
( ( member @ A @ X @ A3 )
=> ( ( ord_less_eq @ ( set @ A ) @ A3 @ B )
=> ( member @ A @ X @ B ) ) ) ).
% set_rev_mp
thf(fact_57_subset__iff,axiom,
! [A: $tType] :
( ( ord_less_eq @ ( set @ A ) )
= ( ^ [A4: set @ A,B4: set @ A] :
! [T: A] :
( ( member @ A @ T @ A4 )
=> ( member @ A @ T @ B4 ) ) ) ) ).
% subset_iff
thf(fact_58_rev__subsetD,axiom,
! [A: $tType,C: A,A3: set @ A,B: set @ A] :
( ( member @ A @ C @ A3 )
=> ( ( ord_less_eq @ ( set @ A ) @ A3 @ B )
=> ( member @ A @ C @ B ) ) ) ).
% rev_subsetD
thf(fact_59_subset__refl,axiom,
! [A: $tType,A3: set @ A] : ( ord_less_eq @ ( set @ A ) @ A3 @ A3 ) ).
% subset_refl
thf(fact_60_Collect__mono,axiom,
! [A: $tType,P: A > $o,Q: A > $o] :
( ! [X3: A] :
( ( P @ X3 )
=> ( Q @ X3 ) )
=> ( ord_less_eq @ ( set @ A ) @ ( collect @ A @ P ) @ ( collect @ A @ Q ) ) ) ).
% Collect_mono
thf(fact_61_subset__trans,axiom,
! [A: $tType,A3: set @ A,B: set @ A,C2: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ A3 @ B )
=> ( ( ord_less_eq @ ( set @ A ) @ B @ C2 )
=> ( ord_less_eq @ ( set @ A ) @ A3 @ C2 ) ) ) ).
% subset_trans
thf(fact_62_set__eq__subset,axiom,
! [A: $tType] :
( ( ^ [Y: set @ A,Z: set @ A] : Y = Z )
= ( ^ [A4: set @ A,B4: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ A4 @ B4 )
& ( ord_less_eq @ ( set @ A ) @ B4 @ A4 ) ) ) ) ).
% set_eq_subset
thf(fact_63_contra__subsetD,axiom,
! [A: $tType,A3: set @ A,B: set @ A,C: A] :
( ( ord_less_eq @ ( set @ A ) @ A3 @ B )
=> ( ~ ( member @ A @ C @ B )
=> ~ ( member @ A @ C @ A3 ) ) ) ).
% contra_subsetD
thf(fact_64_Collect__mono__iff,axiom,
! [A: $tType,P: A > $o,Q: A > $o] :
( ( ord_less_eq @ ( set @ A ) @ ( collect @ A @ P ) @ ( collect @ A @ Q ) )
= ( ! [X2: A] :
( ( P @ X2 )
=> ( Q @ X2 ) ) ) ) ).
% Collect_mono_iff
thf(fact_65_path__avoid__node,axiom,
! [A: $tType,N2: A,Xs4: coinductive_llist @ A,Graph: A > A > $o,X: A] :
( ( member @ ( coinductive_llist @ A ) @ ( coinductive_LCons @ A @ N2 @ Xs4 ) @ ( koenig916195507_paths @ A @ Graph ) )
=> ( ( member @ A @ X @ ( coinductive_lset @ A @ Xs4 ) )
=> ( ( N2 != X )
=> ? [Xs: coinductive_llist @ A] :
( ( member @ ( coinductive_llist @ A ) @ ( coinductive_LCons @ A @ N2 @ Xs ) @ ( koenig916195507_paths @ A @ Graph ) )
& ( ord_less_eq @ ( set @ A ) @ ( coinductive_lset @ A @ Xs ) @ ( coinductive_lset @ A @ Xs4 ) )
& ( member @ A @ X @ ( coinductive_lset @ A @ Xs ) )
& ~ ( member @ A @ N2 @ ( coinductive_lset @ A @ Xs ) ) ) ) ) ) ).
% path_avoid_node
thf(fact_66_paths__LConsD,axiom,
! [A: $tType,X: A,Xs4: coinductive_llist @ A,Graph: A > A > $o] :
( ( member @ ( coinductive_llist @ A ) @ ( coinductive_LCons @ A @ X @ Xs4 ) @ ( koenig916195507_paths @ A @ Graph ) )
=> ( member @ ( coinductive_llist @ A ) @ Xs4 @ ( koenig916195507_paths @ A @ Graph ) ) ) ).
% paths_LConsD
thf(fact_67_paths_OLCons,axiom,
! [Node: $tType,Graph: Node > Node > $o,X: Node,Y2: Node,Xs4: coinductive_llist @ Node] :
( ( Graph @ X @ Y2 )
=> ( ( member @ ( coinductive_llist @ Node ) @ ( coinductive_LCons @ Node @ Y2 @ Xs4 ) @ ( koenig916195507_paths @ Node @ Graph ) )
=> ( member @ ( coinductive_llist @ Node ) @ ( coinductive_LCons @ Node @ X @ ( coinductive_LCons @ Node @ Y2 @ Xs4 ) ) @ ( koenig916195507_paths @ Node @ Graph ) ) ) ) ).
% paths.LCons
thf(fact_68_subset__insertI2,axiom,
! [A: $tType,A3: set @ A,B: set @ A,B2: A] :
( ( ord_less_eq @ ( set @ A ) @ A3 @ B )
=> ( ord_less_eq @ ( set @ A ) @ A3 @ ( insert @ A @ B2 @ B ) ) ) ).
% subset_insertI2
thf(fact_69_subset__insertI,axiom,
! [A: $tType,B: set @ A,A2: A] : ( ord_less_eq @ ( set @ A ) @ B @ ( insert @ A @ A2 @ B ) ) ).
% subset_insertI
thf(fact_70_subset__insert,axiom,
! [A: $tType,X: A,A3: set @ A,B: set @ A] :
( ~ ( member @ A @ X @ A3 )
=> ( ( ord_less_eq @ ( set @ A ) @ A3 @ ( insert @ A @ X @ B ) )
= ( ord_less_eq @ ( set @ A ) @ A3 @ B ) ) ) ).
% subset_insert
thf(fact_71_Set_Oinsert__mono,axiom,
! [A: $tType,C2: set @ A,D: set @ A,A2: A] :
( ( ord_less_eq @ ( set @ A ) @ C2 @ D )
=> ( ord_less_eq @ ( set @ A ) @ ( insert @ A @ A2 @ C2 ) @ ( insert @ A @ A2 @ D ) ) ) ).
% Set.insert_mono
thf(fact_72_reachable__via_Oinducts,axiom,
! [Node: $tType,X: Node,Graph: Node > Node > $o,Ns: set @ Node,N2: Node,P: Node > $o] :
( ( member @ Node @ X @ ( koenig317145564le_via @ Node @ Graph @ Ns @ N2 ) )
=> ( ! [Xs3: coinductive_llist @ Node,N: Node] :
( ( member @ ( coinductive_llist @ Node ) @ ( coinductive_LCons @ Node @ N2 @ Xs3 ) @ ( koenig916195507_paths @ Node @ Graph ) )
=> ( ( member @ Node @ N @ ( coinductive_lset @ Node @ Xs3 ) )
=> ( ( ord_less_eq @ ( set @ Node ) @ ( coinductive_lset @ Node @ Xs3 ) @ Ns )
=> ( P @ N ) ) ) )
=> ( P @ X ) ) ) ).
% reachable_via.inducts
thf(fact_73_reachable__via_Ointros,axiom,
! [Node: $tType,N2: Node,Xs4: coinductive_llist @ Node,Graph: Node > Node > $o,N3: Node,Ns: set @ Node] :
( ( member @ ( coinductive_llist @ Node ) @ ( coinductive_LCons @ Node @ N2 @ Xs4 ) @ ( koenig916195507_paths @ Node @ Graph ) )
=> ( ( member @ Node @ N3 @ ( coinductive_lset @ Node @ Xs4 ) )
=> ( ( ord_less_eq @ ( set @ Node ) @ ( coinductive_lset @ Node @ Xs4 ) @ Ns )
=> ( member @ Node @ N3 @ ( koenig317145564le_via @ Node @ Graph @ Ns @ N2 ) ) ) ) ) ).
% reachable_via.intros
thf(fact_74_reachable__via_Osimps,axiom,
! [Node: $tType,A2: Node,Graph: Node > Node > $o,Ns: set @ Node,N2: Node] :
( ( member @ Node @ A2 @ ( koenig317145564le_via @ Node @ Graph @ Ns @ N2 ) )
= ( ? [Xs5: coinductive_llist @ Node,N4: Node] :
( ( A2 = N4 )
& ( member @ ( coinductive_llist @ Node ) @ ( coinductive_LCons @ Node @ N2 @ Xs5 ) @ ( koenig916195507_paths @ Node @ Graph ) )
& ( member @ Node @ N4 @ ( coinductive_lset @ Node @ Xs5 ) )
& ( ord_less_eq @ ( set @ Node ) @ ( coinductive_lset @ Node @ Xs5 ) @ Ns ) ) ) ) ).
% reachable_via.simps
thf(fact_75_reachable__via_Ocases,axiom,
! [Node: $tType,A2: Node,Graph: Node > Node > $o,Ns: set @ Node,N2: Node] :
( ( member @ Node @ A2 @ ( koenig317145564le_via @ Node @ Graph @ Ns @ N2 ) )
=> ~ ! [Xs3: coinductive_llist @ Node] :
( ( member @ ( coinductive_llist @ Node ) @ ( coinductive_LCons @ Node @ N2 @ Xs3 ) @ ( koenig916195507_paths @ Node @ Graph ) )
=> ( ( member @ Node @ A2 @ ( coinductive_lset @ Node @ Xs3 ) )
=> ~ ( ord_less_eq @ ( set @ Node ) @ ( coinductive_lset @ Node @ Xs3 ) @ Ns ) ) ) ) ).
% reachable_via.cases
thf(fact_76_double__diff,axiom,
! [A: $tType,A3: set @ A,B: set @ A,C2: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ A3 @ B )
=> ( ( ord_less_eq @ ( set @ A ) @ B @ C2 )
=> ( ( minus_minus @ ( set @ A ) @ B @ ( minus_minus @ ( set @ A ) @ C2 @ A3 ) )
= A3 ) ) ) ).
% double_diff
thf(fact_77_Diff__subset,axiom,
! [A: $tType,A3: set @ A,B: set @ A] : ( ord_less_eq @ ( set @ A ) @ ( minus_minus @ ( set @ A ) @ A3 @ B ) @ A3 ) ).
% Diff_subset
thf(fact_78_Diff__mono,axiom,
! [A: $tType,A3: set @ A,C2: set @ A,D: set @ A,B: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ A3 @ C2 )
=> ( ( ord_less_eq @ ( set @ A ) @ D @ B )
=> ( ord_less_eq @ ( set @ A ) @ ( minus_minus @ ( set @ A ) @ A3 @ B ) @ ( minus_minus @ ( set @ A ) @ C2 @ D ) ) ) ) ).
% Diff_mono
thf(fact_79_ex__in__conv,axiom,
! [A: $tType,A3: set @ A] :
( ( ? [X2: A] : ( member @ A @ X2 @ A3 ) )
= ( A3
!= ( bot_bot @ ( set @ A ) ) ) ) ).
% ex_in_conv
thf(fact_80_equals0I,axiom,
! [A: $tType,A3: set @ A] :
( ! [Y3: A] :
~ ( member @ A @ Y3 @ A3 )
=> ( A3
= ( bot_bot @ ( set @ A ) ) ) ) ).
% equals0I
thf(fact_81_equals0D,axiom,
! [A: $tType,A3: set @ A,A2: A] :
( ( A3
= ( bot_bot @ ( set @ A ) ) )
=> ~ ( member @ A @ A2 @ A3 ) ) ).
% equals0D
thf(fact_82_emptyE,axiom,
! [A: $tType,A2: A] :
~ ( member @ A @ A2 @ ( bot_bot @ ( set @ A ) ) ) ).
% emptyE
thf(fact_83_mk__disjoint__insert,axiom,
! [A: $tType,A2: A,A3: set @ A] :
( ( member @ A @ A2 @ A3 )
=> ? [B5: set @ A] :
( ( A3
= ( insert @ A @ A2 @ B5 ) )
& ~ ( member @ A @ A2 @ B5 ) ) ) ).
% mk_disjoint_insert
thf(fact_84_insert__commute,axiom,
! [A: $tType,X: A,Y2: A,A3: set @ A] :
( ( insert @ A @ X @ ( insert @ A @ Y2 @ A3 ) )
= ( insert @ A @ Y2 @ ( insert @ A @ X @ A3 ) ) ) ).
% insert_commute
thf(fact_85_insert__eq__iff,axiom,
! [A: $tType,A2: A,A3: set @ A,B2: A,B: set @ A] :
( ~ ( member @ A @ A2 @ A3 )
=> ( ~ ( member @ A @ B2 @ B )
=> ( ( ( insert @ A @ A2 @ A3 )
= ( insert @ A @ B2 @ B ) )
= ( ( ( A2 = B2 )
=> ( A3 = B ) )
& ( ( A2 != B2 )
=> ? [C3: set @ A] :
( ( A3
= ( insert @ A @ B2 @ C3 ) )
& ~ ( member @ A @ B2 @ C3 )
& ( B
= ( insert @ A @ A2 @ C3 ) )
& ~ ( member @ A @ A2 @ C3 ) ) ) ) ) ) ) ).
% insert_eq_iff
thf(fact_86_insert__absorb,axiom,
! [A: $tType,A2: A,A3: set @ A] :
( ( member @ A @ A2 @ A3 )
=> ( ( insert @ A @ A2 @ A3 )
= A3 ) ) ).
% insert_absorb
thf(fact_87_insert__ident,axiom,
! [A: $tType,X: A,A3: set @ A,B: set @ A] :
( ~ ( member @ A @ X @ A3 )
=> ( ~ ( member @ A @ X @ B )
=> ( ( ( insert @ A @ X @ A3 )
= ( insert @ A @ X @ B ) )
= ( A3 = B ) ) ) ) ).
% insert_ident
thf(fact_88_Set_Oset__insert,axiom,
! [A: $tType,X: A,A3: set @ A] :
( ( member @ A @ X @ A3 )
=> ~ ! [B5: set @ A] :
( ( A3
= ( insert @ A @ X @ B5 ) )
=> ( member @ A @ X @ B5 ) ) ) ).
% Set.set_insert
thf(fact_89_insertI2,axiom,
! [A: $tType,A2: A,B: set @ A,B2: A] :
( ( member @ A @ A2 @ B )
=> ( member @ A @ A2 @ ( insert @ A @ B2 @ B ) ) ) ).
% insertI2
thf(fact_90_insertI1,axiom,
! [A: $tType,A2: A,B: set @ A] : ( member @ A @ A2 @ ( insert @ A @ A2 @ B ) ) ).
% insertI1
thf(fact_91_insertE,axiom,
! [A: $tType,A2: A,B2: A,A3: set @ A] :
( ( member @ A @ A2 @ ( insert @ A @ B2 @ A3 ) )
=> ( ( A2 != B2 )
=> ( member @ A @ A2 @ A3 ) ) ) ).
% insertE
thf(fact_92_DiffD2,axiom,
! [A: $tType,C: A,A3: set @ A,B: set @ A] :
( ( member @ A @ C @ ( minus_minus @ ( set @ A ) @ A3 @ B ) )
=> ~ ( member @ A @ C @ B ) ) ).
% DiffD2
thf(fact_93_DiffD1,axiom,
! [A: $tType,C: A,A3: set @ A,B: set @ A] :
( ( member @ A @ C @ ( minus_minus @ ( set @ A ) @ A3 @ B ) )
=> ( member @ A @ C @ A3 ) ) ).
% DiffD1
thf(fact_94_DiffE,axiom,
! [A: $tType,C: A,A3: set @ A,B: set @ A] :
( ( member @ A @ C @ ( minus_minus @ ( set @ A ) @ A3 @ B ) )
=> ~ ( ( member @ A @ C @ A3 )
=> ( member @ A @ C @ B ) ) ) ).
% DiffE
thf(fact_95_subset__singleton__iff,axiom,
! [A: $tType,X4: set @ A,A2: A] :
( ( ord_less_eq @ ( set @ A ) @ X4 @ ( insert @ A @ A2 @ ( bot_bot @ ( set @ A ) ) ) )
= ( ( X4
= ( bot_bot @ ( set @ A ) ) )
| ( X4
= ( insert @ A @ A2 @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ).
% subset_singleton_iff
thf(fact_96_subset__singletonD,axiom,
! [A: $tType,A3: set @ A,X: A] :
( ( ord_less_eq @ ( set @ A ) @ A3 @ ( insert @ A @ X @ ( bot_bot @ ( set @ A ) ) ) )
=> ( ( A3
= ( bot_bot @ ( set @ A ) ) )
| ( A3
= ( insert @ A @ X @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ).
% subset_singletonD
thf(fact_97_subset__Diff__insert,axiom,
! [A: $tType,A3: set @ A,B: set @ A,X: A,C2: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ A3 @ ( minus_minus @ ( set @ A ) @ B @ ( insert @ A @ X @ C2 ) ) )
= ( ( ord_less_eq @ ( set @ A ) @ A3 @ ( minus_minus @ ( set @ A ) @ B @ C2 ) )
& ~ ( member @ A @ X @ A3 ) ) ) ).
% subset_Diff_insert
thf(fact_98_Diff__single__insert,axiom,
! [A: $tType,A3: set @ A,X: A,B: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ ( minus_minus @ ( set @ A ) @ A3 @ ( insert @ A @ X @ ( bot_bot @ ( set @ A ) ) ) ) @ B )
=> ( ord_less_eq @ ( set @ A ) @ A3 @ ( insert @ A @ X @ B ) ) ) ).
% Diff_single_insert
thf(fact_99_subset__insert__iff,axiom,
! [A: $tType,A3: set @ A,X: A,B: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ A3 @ ( insert @ A @ X @ B ) )
= ( ( ( member @ A @ X @ A3 )
=> ( ord_less_eq @ ( set @ A ) @ ( minus_minus @ ( set @ A ) @ A3 @ ( insert @ A @ X @ ( bot_bot @ ( set @ A ) ) ) ) @ B ) )
& ( ~ ( member @ A @ X @ A3 )
=> ( ord_less_eq @ ( set @ A ) @ A3 @ B ) ) ) ) ).
% subset_insert_iff
thf(fact_100_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_101_insert__not__empty,axiom,
! [A: $tType,A2: A,A3: set @ A] :
( ( insert @ A @ A2 @ A3 )
!= ( bot_bot @ ( set @ A ) ) ) ).
% insert_not_empty
thf(fact_102_doubleton__eq__iff,axiom,
! [A: $tType,A2: A,B2: A,C: A,D2: A] :
( ( ( insert @ A @ A2 @ ( insert @ A @ B2 @ ( bot_bot @ ( set @ A ) ) ) )
= ( insert @ A @ C @ ( insert @ A @ D2 @ ( bot_bot @ ( set @ A ) ) ) ) )
= ( ( ( A2 = C )
& ( B2 = D2 ) )
| ( ( A2 = D2 )
& ( B2 = C ) ) ) ) ).
% doubleton_eq_iff
thf(fact_103_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_104_singletonD,axiom,
! [A: $tType,B2: A,A2: A] :
( ( member @ A @ B2 @ ( insert @ A @ A2 @ ( bot_bot @ ( set @ A ) ) ) )
=> ( B2 = A2 ) ) ).
% singletonD
thf(fact_105_insert__Diff__if,axiom,
! [A: $tType,X: A,B: set @ A,A3: set @ A] :
( ( ( member @ A @ X @ B )
=> ( ( minus_minus @ ( set @ A ) @ ( insert @ A @ X @ A3 ) @ B )
= ( minus_minus @ ( set @ A ) @ A3 @ B ) ) )
& ( ~ ( member @ A @ X @ B )
=> ( ( minus_minus @ ( set @ A ) @ ( insert @ A @ X @ A3 ) @ B )
= ( insert @ A @ X @ ( minus_minus @ ( set @ A ) @ A3 @ B ) ) ) ) ) ).
% insert_Diff_if
thf(fact_106_Diff__insert__absorb,axiom,
! [A: $tType,X: A,A3: set @ A] :
( ~ ( member @ A @ X @ A3 )
=> ( ( minus_minus @ ( set @ A ) @ ( insert @ A @ X @ A3 ) @ ( insert @ A @ X @ ( bot_bot @ ( set @ A ) ) ) )
= A3 ) ) ).
% Diff_insert_absorb
thf(fact_107_Diff__insert2,axiom,
! [A: $tType,A3: set @ A,A2: A,B: set @ A] :
( ( minus_minus @ ( set @ A ) @ A3 @ ( insert @ A @ A2 @ B ) )
= ( minus_minus @ ( set @ A ) @ ( minus_minus @ ( set @ A ) @ A3 @ ( insert @ A @ A2 @ ( bot_bot @ ( set @ A ) ) ) ) @ B ) ) ).
% Diff_insert2
thf(fact_108_insert__Diff,axiom,
! [A: $tType,A2: A,A3: set @ A] :
( ( member @ A @ A2 @ A3 )
=> ( ( insert @ A @ A2 @ ( minus_minus @ ( set @ A ) @ A3 @ ( insert @ A @ A2 @ ( bot_bot @ ( set @ A ) ) ) ) )
= A3 ) ) ).
% insert_Diff
thf(fact_109_Diff__insert,axiom,
! [A: $tType,A3: set @ A,A2: A,B: set @ A] :
( ( minus_minus @ ( set @ A ) @ A3 @ ( insert @ A @ A2 @ B ) )
= ( minus_minus @ ( set @ A ) @ ( minus_minus @ ( set @ A ) @ A3 @ B ) @ ( insert @ A @ A2 @ ( bot_bot @ ( set @ A ) ) ) ) ) ).
% Diff_insert
thf(fact_110_lset__LCons,axiom,
! [A: $tType,X: A,Xs4: coinductive_llist @ A] :
( ( coinductive_lset @ A @ ( coinductive_LCons @ A @ X @ Xs4 ) )
= ( insert @ A @ X @ ( coinductive_lset @ A @ Xs4 ) ) ) ).
% lset_LCons
thf(fact_111_llist_Osimps_I19_J,axiom,
! [A: $tType,X21: A,X22: coinductive_llist @ A] :
( ( coinductive_lset @ A @ ( coinductive_LCons @ A @ X21 @ X22 ) )
= ( insert @ A @ X21 @ ( coinductive_lset @ A @ X22 ) ) ) ).
% llist.simps(19)
thf(fact_112_llist_Oinject,axiom,
! [A: $tType,X21: A,X22: coinductive_llist @ A,Y21: A,Y22: coinductive_llist @ A] :
( ( ( coinductive_LCons @ A @ X21 @ X22 )
= ( coinductive_LCons @ A @ Y21 @ Y22 ) )
= ( ( X21 = Y21 )
& ( X22 = Y22 ) ) ) ).
% llist.inject
thf(fact_113_the__elem__eq,axiom,
! [A: $tType,X: A] :
( ( the_elem @ A @ ( insert @ A @ X @ ( bot_bot @ ( set @ A ) ) ) )
= X ) ).
% the_elem_eq
thf(fact_114_bot__apply,axiom,
! [C4: $tType,D3: $tType] :
( ( bot @ C4 @ ( type2 @ C4 ) )
=> ( ( bot_bot @ ( D3 > C4 ) )
= ( ^ [X2: D3] : ( bot_bot @ C4 ) ) ) ) ).
% bot_apply
thf(fact_115_minus__apply,axiom,
! [B3: $tType,A: $tType] :
( ( minus @ B3 @ ( type2 @ B3 ) )
=> ( ( minus_minus @ ( A > B3 ) )
= ( ^ [A4: A > B3,B4: A > B3,X2: A] : ( minus_minus @ B3 @ ( A4 @ X2 ) @ ( B4 @ X2 ) ) ) ) ) ).
% minus_apply
thf(fact_116_order__refl,axiom,
! [A: $tType] :
( ( preorder @ A @ ( type2 @ A ) )
=> ! [X: A] : ( ord_less_eq @ A @ X @ X ) ) ).
% order_refl
thf(fact_117_is__singletonI,axiom,
! [A: $tType,X: A] : ( is_singleton @ A @ ( insert @ A @ X @ ( bot_bot @ ( set @ A ) ) ) ) ).
% is_singletonI
thf(fact_118_lset__intros_I2_J,axiom,
! [A: $tType,X: A,Xs4: coinductive_llist @ A,X5: A] :
( ( member @ A @ X @ ( coinductive_lset @ A @ Xs4 ) )
=> ( member @ A @ X @ ( coinductive_lset @ A @ ( coinductive_LCons @ A @ X5 @ Xs4 ) ) ) ) ).
% lset_intros(2)
thf(fact_119_lset__intros_I1_J,axiom,
! [A: $tType,X: A,Xs4: coinductive_llist @ A] : ( member @ A @ X @ ( coinductive_lset @ A @ ( coinductive_LCons @ A @ X @ Xs4 ) ) ) ).
% lset_intros(1)
thf(fact_120_bot__set__def,axiom,
! [A: $tType] :
( ( bot_bot @ ( set @ A ) )
= ( collect @ A @ ( bot_bot @ ( A > $o ) ) ) ) ).
% bot_set_def
thf(fact_121_is__singleton__the__elem,axiom,
! [A: $tType] :
( ( is_singleton @ A )
= ( ^ [A4: set @ A] :
( A4
= ( insert @ A @ ( the_elem @ A @ A4 ) @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ).
% is_singleton_the_elem
thf(fact_122_is__singletonI_H,axiom,
! [A: $tType,A3: set @ A] :
( ( A3
!= ( bot_bot @ ( set @ A ) ) )
=> ( ! [X3: A,Y3: A] :
( ( member @ A @ X3 @ A3 )
=> ( ( member @ A @ Y3 @ A3 )
=> ( X3 = Y3 ) ) )
=> ( is_singleton @ A @ A3 ) ) ) ).
% is_singletonI'
thf(fact_123_dual__order_Oantisym,axiom,
! [A: $tType] :
( ( order @ A @ ( type2 @ A ) )
=> ! [B2: A,A2: A] :
( ( ord_less_eq @ A @ B2 @ A2 )
=> ( ( ord_less_eq @ A @ A2 @ B2 )
=> ( A2 = B2 ) ) ) ) ).
% dual_order.antisym
thf(fact_124_dual__order_Otrans,axiom,
! [A: $tType] :
( ( order @ A @ ( type2 @ A ) )
=> ! [B2: A,A2: A,C: A] :
( ( ord_less_eq @ A @ B2 @ A2 )
=> ( ( ord_less_eq @ A @ C @ B2 )
=> ( ord_less_eq @ A @ C @ A2 ) ) ) ) ).
% dual_order.trans
thf(fact_125_linorder__wlog,axiom,
! [A: $tType] :
( ( linorder @ A @ ( type2 @ A ) )
=> ! [P: A > A > $o,A2: A,B2: A] :
( ! [A5: A,B6: A] :
( ( ord_less_eq @ A @ A5 @ B6 )
=> ( P @ A5 @ B6 ) )
=> ( ! [A5: A,B6: A] :
( ( P @ B6 @ A5 )
=> ( P @ A5 @ B6 ) )
=> ( P @ A2 @ B2 ) ) ) ) ).
% linorder_wlog
thf(fact_126_dual__order_Orefl,axiom,
! [A: $tType] :
( ( order @ A @ ( type2 @ A ) )
=> ! [A2: A] : ( ord_less_eq @ A @ A2 @ A2 ) ) ).
% dual_order.refl
thf(fact_127_order__trans,axiom,
! [A: $tType] :
( ( preorder @ A @ ( type2 @ A ) )
=> ! [X: A,Y2: A,Z2: A] :
( ( ord_less_eq @ A @ X @ Y2 )
=> ( ( ord_less_eq @ A @ Y2 @ Z2 )
=> ( ord_less_eq @ A @ X @ Z2 ) ) ) ) ).
% order_trans
thf(fact_128_order__class_Oorder_Oantisym,axiom,
! [A: $tType] :
( ( order @ A @ ( type2 @ A ) )
=> ! [A2: A,B2: A] :
( ( ord_less_eq @ A @ A2 @ B2 )
=> ( ( ord_less_eq @ A @ B2 @ A2 )
=> ( A2 = B2 ) ) ) ) ).
% order_class.order.antisym
thf(fact_129_ord__le__eq__trans,axiom,
! [A: $tType] :
( ( ord @ A @ ( type2 @ A ) )
=> ! [A2: A,B2: A,C: A] :
( ( ord_less_eq @ A @ A2 @ B2 )
=> ( ( B2 = C )
=> ( ord_less_eq @ A @ A2 @ C ) ) ) ) ).
% ord_le_eq_trans
thf(fact_130_ord__eq__le__trans,axiom,
! [A: $tType] :
( ( ord @ A @ ( type2 @ A ) )
=> ! [A2: A,B2: A,C: A] :
( ( A2 = B2 )
=> ( ( ord_less_eq @ A @ B2 @ C )
=> ( ord_less_eq @ A @ A2 @ C ) ) ) ) ).
% ord_eq_le_trans
thf(fact_131_antisym__conv,axiom,
! [A: $tType] :
( ( order @ A @ ( type2 @ A ) )
=> ! [Y2: A,X: A] :
( ( ord_less_eq @ A @ Y2 @ X )
=> ( ( ord_less_eq @ A @ X @ Y2 )
= ( X = Y2 ) ) ) ) ).
% antisym_conv
thf(fact_132_le__cases3,axiom,
! [A: $tType] :
( ( linorder @ A @ ( type2 @ A ) )
=> ! [X: A,Y2: A,Z2: A] :
( ( ( ord_less_eq @ A @ X @ Y2 )
=> ~ ( ord_less_eq @ A @ Y2 @ Z2 ) )
=> ( ( ( ord_less_eq @ A @ Y2 @ X )
=> ~ ( ord_less_eq @ A @ X @ Z2 ) )
=> ( ( ( ord_less_eq @ A @ X @ Z2 )
=> ~ ( ord_less_eq @ A @ Z2 @ Y2 ) )
=> ( ( ( ord_less_eq @ A @ Z2 @ Y2 )
=> ~ ( ord_less_eq @ A @ Y2 @ X ) )
=> ( ( ( ord_less_eq @ A @ Y2 @ Z2 )
=> ~ ( ord_less_eq @ A @ Z2 @ X ) )
=> ~ ( ( ord_less_eq @ A @ Z2 @ X )
=> ~ ( ord_less_eq @ A @ X @ Y2 ) ) ) ) ) ) ) ) ).
% le_cases3
thf(fact_133_order_Otrans,axiom,
! [A: $tType] :
( ( order @ A @ ( type2 @ A ) )
=> ! [A2: A,B2: A,C: A] :
( ( ord_less_eq @ A @ A2 @ B2 )
=> ( ( ord_less_eq @ A @ B2 @ C )
=> ( ord_less_eq @ A @ A2 @ C ) ) ) ) ).
% order.trans
thf(fact_134_le__cases,axiom,
! [A: $tType] :
( ( linorder @ A @ ( type2 @ A ) )
=> ! [X: A,Y2: A] :
( ~ ( ord_less_eq @ A @ X @ Y2 )
=> ( ord_less_eq @ A @ Y2 @ X ) ) ) ).
% le_cases
thf(fact_135_wlog__linorder__le,axiom,
! [A: $tType] :
( ( linorder @ A @ ( type2 @ A ) )
=> ! [P: A > A > $o,B2: A,A2: A] :
( ! [A5: A,B6: A] :
( ( ord_less_eq @ A @ A5 @ B6 )
=> ( P @ A5 @ B6 ) )
=> ( ( ( P @ B2 @ A2 )
=> ( P @ A2 @ B2 ) )
=> ( P @ A2 @ B2 ) ) ) ) ).
% wlog_linorder_le
thf(fact_136_eq__refl,axiom,
! [A: $tType] :
( ( preorder @ A @ ( type2 @ A ) )
=> ! [X: A,Y2: A] :
( ( X = Y2 )
=> ( ord_less_eq @ A @ X @ Y2 ) ) ) ).
% eq_refl
thf(fact_137_linear,axiom,
! [A: $tType] :
( ( linorder @ A @ ( type2 @ A ) )
=> ! [X: A,Y2: A] :
( ( ord_less_eq @ A @ X @ Y2 )
| ( ord_less_eq @ A @ Y2 @ X ) ) ) ).
% linear
thf(fact_138_antisym,axiom,
! [A: $tType] :
( ( order @ A @ ( type2 @ A ) )
=> ! [X: A,Y2: A] :
( ( ord_less_eq @ A @ X @ Y2 )
=> ( ( ord_less_eq @ A @ Y2 @ X )
=> ( X = Y2 ) ) ) ) ).
% antisym
thf(fact_139_eq__iff,axiom,
! [A: $tType] :
( ( order @ A @ ( type2 @ A ) )
=> ( ( ^ [Y: A,Z: A] : Y = Z )
= ( ^ [X2: A,Y4: A] :
( ( ord_less_eq @ A @ X2 @ Y4 )
& ( ord_less_eq @ A @ Y4 @ X2 ) ) ) ) ) ).
% eq_iff
thf(fact_140_ord__le__eq__subst,axiom,
! [A: $tType,B3: $tType] :
( ( ( ord @ B3 @ ( type2 @ B3 ) )
& ( ord @ A @ ( type2 @ A ) ) )
=> ! [A2: A,B2: A,F: A > B3,C: B3] :
( ( ord_less_eq @ A @ A2 @ B2 )
=> ( ( ( F @ B2 )
= C )
=> ( ! [X3: A,Y3: A] :
( ( ord_less_eq @ A @ X3 @ Y3 )
=> ( ord_less_eq @ B3 @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq @ B3 @ ( F @ A2 ) @ C ) ) ) ) ) ).
% ord_le_eq_subst
thf(fact_141_ord__eq__le__subst,axiom,
! [A: $tType,B3: $tType] :
( ( ( ord @ B3 @ ( type2 @ B3 ) )
& ( ord @ A @ ( type2 @ A ) ) )
=> ! [A2: A,F: B3 > A,B2: B3,C: B3] :
( ( A2
= ( F @ B2 ) )
=> ( ( ord_less_eq @ B3 @ B2 @ C )
=> ( ! [X3: B3,Y3: B3] :
( ( ord_less_eq @ B3 @ X3 @ Y3 )
=> ( ord_less_eq @ A @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq @ A @ A2 @ ( F @ C ) ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_142_order__subst2,axiom,
! [A: $tType,C4: $tType] :
( ( ( order @ C4 @ ( type2 @ C4 ) )
& ( order @ A @ ( type2 @ A ) ) )
=> ! [A2: A,B2: A,F: A > C4,C: C4] :
( ( ord_less_eq @ A @ A2 @ B2 )
=> ( ( ord_less_eq @ C4 @ ( F @ B2 ) @ C )
=> ( ! [X3: A,Y3: A] :
( ( ord_less_eq @ A @ X3 @ Y3 )
=> ( ord_less_eq @ C4 @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq @ C4 @ ( F @ A2 ) @ C ) ) ) ) ) ).
% order_subst2
thf(fact_143_order__subst1,axiom,
! [A: $tType,B3: $tType] :
( ( ( order @ B3 @ ( type2 @ B3 ) )
& ( order @ A @ ( type2 @ A ) ) )
=> ! [A2: A,F: B3 > A,B2: B3,C: B3] :
( ( ord_less_eq @ A @ A2 @ ( F @ B2 ) )
=> ( ( ord_less_eq @ B3 @ B2 @ C )
=> ( ! [X3: B3,Y3: B3] :
( ( ord_less_eq @ B3 @ X3 @ Y3 )
=> ( ord_less_eq @ A @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq @ A @ A2 @ ( F @ C ) ) ) ) ) ) ).
% order_subst1
thf(fact_144_le__fun__def,axiom,
! [B3: $tType,A: $tType] :
( ( ord @ B3 @ ( type2 @ B3 ) )
=> ( ( ord_less_eq @ ( A > B3 ) )
= ( ^ [F2: A > B3,G2: A > B3] :
! [X2: A] : ( ord_less_eq @ B3 @ ( F2 @ X2 ) @ ( G2 @ X2 ) ) ) ) ) ).
% le_fun_def
thf(fact_145_le__funI,axiom,
! [B3: $tType,A: $tType] :
( ( ord @ B3 @ ( type2 @ B3 ) )
=> ! [F: A > B3,G: A > B3] :
( ! [X3: A] : ( ord_less_eq @ B3 @ ( F @ X3 ) @ ( G @ X3 ) )
=> ( ord_less_eq @ ( A > B3 ) @ F @ G ) ) ) ).
% le_funI
thf(fact_146_le__funE,axiom,
! [B3: $tType,A: $tType] :
( ( ord @ B3 @ ( type2 @ B3 ) )
=> ! [F: A > B3,G: A > B3,X: A] :
( ( ord_less_eq @ ( A > B3 ) @ F @ G )
=> ( ord_less_eq @ B3 @ ( F @ X ) @ ( G @ X ) ) ) ) ).
% le_funE
thf(fact_147_le__funD,axiom,
! [B3: $tType,A: $tType] :
( ( ord @ B3 @ ( type2 @ B3 ) )
=> ! [F: A > B3,G: A > B3,X: A] :
( ( ord_less_eq @ ( A > B3 ) @ F @ G )
=> ( ord_less_eq @ B3 @ ( F @ X ) @ ( G @ X ) ) ) ) ).
% le_funD
thf(fact_148_fun__diff__def,axiom,
! [B3: $tType,A: $tType] :
( ( minus @ B3 @ ( type2 @ B3 ) )
=> ( ( minus_minus @ ( A > B3 ) )
= ( ^ [A4: A > B3,B4: A > B3,X2: A] : ( minus_minus @ B3 @ ( A4 @ X2 ) @ ( B4 @ X2 ) ) ) ) ) ).
% fun_diff_def
thf(fact_149_bot__fun__def,axiom,
! [B3: $tType,A: $tType] :
( ( bot @ B3 @ ( type2 @ B3 ) )
=> ( ( bot_bot @ ( A > B3 ) )
= ( ^ [X2: A] : ( bot_bot @ B3 ) ) ) ) ).
% bot_fun_def
thf(fact_150_is__singletonE,axiom,
! [A: $tType,A3: set @ A] :
( ( is_singleton @ A @ A3 )
=> ~ ! [X3: A] :
( A3
!= ( insert @ A @ X3 @ ( bot_bot @ ( set @ A ) ) ) ) ) ).
% is_singletonE
thf(fact_151_is__singleton__def,axiom,
! [A: $tType] :
( ( is_singleton @ A )
= ( ^ [A4: set @ A] :
? [X2: A] :
( A4
= ( insert @ A @ X2 @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ).
% is_singleton_def
thf(fact_152_bot_Oextremum,axiom,
! [A: $tType] :
( ( order_bot @ A @ ( type2 @ A ) )
=> ! [A2: A] : ( ord_less_eq @ A @ ( bot_bot @ A ) @ A2 ) ) ).
% bot.extremum
thf(fact_153_bot_Oextremum__unique,axiom,
! [A: $tType] :
( ( order_bot @ A @ ( type2 @ A ) )
=> ! [A2: A] :
( ( ord_less_eq @ A @ A2 @ ( bot_bot @ A ) )
= ( A2
= ( bot_bot @ A ) ) ) ) ).
% bot.extremum_unique
thf(fact_154_bot_Oextremum__uniqueI,axiom,
! [A: $tType] :
( ( order_bot @ A @ ( type2 @ A ) )
=> ! [A2: A] :
( ( ord_less_eq @ A @ A2 @ ( bot_bot @ A ) )
=> ( A2
= ( bot_bot @ A ) ) ) ) ).
% bot.extremum_uniqueI
thf(fact_155_llist_Oset__induct,axiom,
! [A: $tType,X: A,A2: coinductive_llist @ A,P: A > ( coinductive_llist @ A ) > $o] :
( ( member @ A @ X @ ( coinductive_lset @ A @ A2 ) )
=> ( ! [Z1: A,Z22: coinductive_llist @ A] : ( P @ Z1 @ ( coinductive_LCons @ A @ Z1 @ Z22 ) )
=> ( ! [Z1: A,Z22: coinductive_llist @ A,Xa: A] :
( ( member @ A @ Xa @ ( coinductive_lset @ A @ Z22 ) )
=> ( ( P @ Xa @ Z22 )
=> ( P @ Xa @ ( coinductive_LCons @ A @ Z1 @ Z22 ) ) ) )
=> ( P @ X @ A2 ) ) ) ) ).
% llist.set_induct
thf(fact_156_llist_Oset__cases,axiom,
! [A: $tType,E: A,A2: coinductive_llist @ A] :
( ( member @ A @ E @ ( coinductive_lset @ A @ A2 ) )
=> ( ! [Z22: coinductive_llist @ A] :
( A2
!= ( coinductive_LCons @ A @ E @ Z22 ) )
=> ~ ! [Z1: A,Z22: coinductive_llist @ A] :
( ( A2
= ( coinductive_LCons @ A @ Z1 @ Z22 ) )
=> ~ ( member @ A @ E @ ( coinductive_lset @ A @ Z22 ) ) ) ) ) ).
% llist.set_cases
thf(fact_157_lset__induct_H,axiom,
! [A: $tType,X: A,Xs4: coinductive_llist @ A,P: ( coinductive_llist @ A ) > $o] :
( ( member @ A @ X @ ( coinductive_lset @ A @ Xs4 ) )
=> ( ! [Xs3: coinductive_llist @ A] : ( P @ ( coinductive_LCons @ A @ X @ Xs3 ) )
=> ( ! [X6: A,Xs3: coinductive_llist @ A] :
( ( member @ A @ X @ ( coinductive_lset @ A @ Xs3 ) )
=> ( ( P @ Xs3 )
=> ( P @ ( coinductive_LCons @ A @ X6 @ Xs3 ) ) ) )
=> ( P @ Xs4 ) ) ) ) ).
% lset_induct'
thf(fact_158_lset__induct,axiom,
! [A: $tType,X: A,Xs4: coinductive_llist @ A,P: ( coinductive_llist @ A ) > $o] :
( ( member @ A @ X @ ( coinductive_lset @ A @ Xs4 ) )
=> ( ! [Xs3: coinductive_llist @ A] : ( P @ ( coinductive_LCons @ A @ X @ Xs3 ) )
=> ( ! [X6: A,Xs3: coinductive_llist @ A] :
( ( member @ A @ X @ ( coinductive_lset @ A @ Xs3 ) )
=> ( ( X != X6 )
=> ( ( P @ Xs3 )
=> ( P @ ( coinductive_LCons @ A @ X6 @ Xs3 ) ) ) ) )
=> ( P @ Xs4 ) ) ) ) ).
% lset_induct
thf(fact_159_lset__cases,axiom,
! [A: $tType,X: A,Xs4: coinductive_llist @ A] :
( ( member @ A @ X @ ( coinductive_lset @ A @ Xs4 ) )
=> ( ! [Xs: coinductive_llist @ A] :
( Xs4
!= ( coinductive_LCons @ A @ X @ Xs ) )
=> ~ ! [X6: A,Xs: coinductive_llist @ A] :
( ( Xs4
= ( coinductive_LCons @ A @ X6 @ Xs ) )
=> ~ ( member @ A @ X @ ( coinductive_lset @ A @ Xs ) ) ) ) ) ).
% lset_cases
thf(fact_160_llist_Oset__intros_I1_J,axiom,
! [A: $tType,A1: A,A22: coinductive_llist @ A] : ( member @ A @ A1 @ ( coinductive_lset @ A @ ( coinductive_LCons @ A @ A1 @ A22 ) ) ) ).
% llist.set_intros(1)
thf(fact_161_llist_Oset__intros_I2_J,axiom,
! [A: $tType,X: A,A22: coinductive_llist @ A,A1: A] :
( ( member @ A @ X @ ( coinductive_lset @ A @ A22 ) )
=> ( member @ A @ X @ ( coinductive_lset @ A @ ( coinductive_LCons @ A @ A1 @ A22 ) ) ) ) ).
% llist.set_intros(2)
thf(fact_162_insert__subsetI,axiom,
! [A: $tType,X: A,A3: set @ A,X4: set @ A] :
( ( member @ A @ X @ A3 )
=> ( ( ord_less_eq @ ( set @ A ) @ X4 @ A3 )
=> ( ord_less_eq @ ( set @ A ) @ ( insert @ A @ X @ X4 ) @ A3 ) ) ) ).
% insert_subsetI
thf(fact_163_subset__emptyI,axiom,
! [A: $tType,A3: set @ A] :
( ! [X3: A] :
~ ( member @ A @ X3 @ A3 )
=> ( ord_less_eq @ ( set @ A ) @ A3 @ ( bot_bot @ ( set @ A ) ) ) ) ).
% subset_emptyI
thf(fact_164_diff__mono,axiom,
! [A: $tType] :
( ( ordered_ab_group_add @ A @ ( type2 @ A ) )
=> ! [A2: A,B2: A,D2: A,C: A] :
( ( ord_less_eq @ A @ A2 @ B2 )
=> ( ( ord_less_eq @ A @ D2 @ C )
=> ( ord_less_eq @ A @ ( minus_minus @ A @ A2 @ C ) @ ( minus_minus @ A @ B2 @ D2 ) ) ) ) ) ).
% diff_mono
thf(fact_165_diff__left__mono,axiom,
! [A: $tType] :
( ( ordered_ab_group_add @ A @ ( type2 @ A ) )
=> ! [B2: A,A2: A,C: A] :
( ( ord_less_eq @ A @ B2 @ A2 )
=> ( ord_less_eq @ A @ ( minus_minus @ A @ C @ A2 ) @ ( minus_minus @ A @ C @ B2 ) ) ) ) ).
% diff_left_mono
thf(fact_166_diff__right__commute,axiom,
! [A: $tType] :
( ( cancel146912293up_add @ A @ ( type2 @ A ) )
=> ! [A2: A,C: A,B2: A] :
( ( minus_minus @ A @ ( minus_minus @ A @ A2 @ C ) @ B2 )
= ( minus_minus @ A @ ( minus_minus @ A @ A2 @ B2 ) @ C ) ) ) ).
% diff_right_commute
thf(fact_167_diff__eq__diff__eq,axiom,
! [A: $tType] :
( ( group_add @ A @ ( type2 @ A ) )
=> ! [A2: A,B2: A,C: A,D2: A] :
( ( ( minus_minus @ A @ A2 @ B2 )
= ( minus_minus @ A @ C @ D2 ) )
=> ( ( A2 = B2 )
= ( C = D2 ) ) ) ) ).
% diff_eq_diff_eq
thf(fact_168_diff__eq__diff__less__eq,axiom,
! [A: $tType] :
( ( ordered_ab_group_add @ A @ ( type2 @ A ) )
=> ! [A2: A,B2: A,C: A,D2: A] :
( ( ( minus_minus @ A @ A2 @ B2 )
= ( minus_minus @ A @ C @ D2 ) )
=> ( ( ord_less_eq @ A @ A2 @ B2 )
= ( ord_less_eq @ A @ C @ D2 ) ) ) ) ).
% diff_eq_diff_less_eq
thf(fact_169_diff__right__mono,axiom,
! [A: $tType] :
( ( ordered_ab_group_add @ A @ ( type2 @ A ) )
=> ! [A2: A,B2: A,C: A] :
( ( ord_less_eq @ A @ A2 @ B2 )
=> ( ord_less_eq @ A @ ( minus_minus @ A @ A2 @ C ) @ ( minus_minus @ A @ B2 @ C ) ) ) ) ).
% diff_right_mono
thf(fact_170_bot__empty__eq,axiom,
! [A: $tType] :
( ( bot_bot @ ( A > $o ) )
= ( ^ [X2: A] : ( member @ A @ X2 @ ( bot_bot @ ( set @ A ) ) ) ) ) ).
% bot_empty_eq
thf(fact_171_Collect__empty__eq__bot,axiom,
! [A: $tType,P: A > $o] :
( ( ( collect @ A @ P )
= ( bot_bot @ ( set @ A ) ) )
= ( P
= ( bot_bot @ ( A > $o ) ) ) ) ).
% Collect_empty_eq_bot
thf(fact_172_remove__def,axiom,
! [A: $tType] :
( ( remove @ A )
= ( ^ [X2: A,A4: set @ A] : ( minus_minus @ ( set @ A ) @ A4 @ ( insert @ A @ X2 @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ).
% remove_def
thf(fact_173_member__remove,axiom,
! [A: $tType,X: A,Y2: A,A3: set @ A] :
( ( member @ A @ X @ ( remove @ A @ Y2 @ A3 ) )
= ( ( member @ A @ X @ A3 )
& ( X != Y2 ) ) ) ).
% member_remove
thf(fact_174_lset__code,axiom,
! [A: $tType] :
( ( coinductive_lset @ A )
= ( coinductive_gen_lset @ A @ ( bot_bot @ ( set @ A ) ) ) ) ).
% lset_code
thf(fact_175_gen__lset__code_I2_J,axiom,
! [A: $tType,A3: set @ A,X: A,Xs4: coinductive_llist @ A] :
( ( coinductive_gen_lset @ A @ A3 @ ( coinductive_LCons @ A @ X @ Xs4 ) )
= ( coinductive_gen_lset @ A @ ( insert @ A @ X @ A3 ) @ Xs4 ) ) ).
% gen_lset_code(2)
thf(fact_176_Set_Ois__empty__def,axiom,
! [A: $tType] :
( ( is_empty @ A )
= ( ^ [A4: set @ A] :
( A4
= ( bot_bot @ ( set @ A ) ) ) ) ) ).
% Set.is_empty_def
thf(fact_177_psubset__insert__iff,axiom,
! [A: $tType,A3: set @ A,X: A,B: set @ A] :
( ( ord_less @ ( set @ A ) @ A3 @ ( insert @ A @ X @ B ) )
= ( ( ( member @ A @ X @ B )
=> ( ord_less @ ( set @ A ) @ A3 @ B ) )
& ( ~ ( member @ A @ X @ B )
=> ( ( ( member @ A @ X @ A3 )
=> ( ord_less @ ( set @ A ) @ ( minus_minus @ ( set @ A ) @ A3 @ ( insert @ A @ X @ ( bot_bot @ ( set @ A ) ) ) ) @ B ) )
& ( ~ ( member @ A @ X @ A3 )
=> ( ord_less_eq @ ( set @ A ) @ A3 @ B ) ) ) ) ) ) ).
% psubset_insert_iff
thf(fact_178_pairwise__singleton,axiom,
! [A: $tType,P: A > A > $o,A3: A] : ( pairwise @ A @ P @ ( insert @ A @ A3 @ ( bot_bot @ ( set @ A ) ) ) ) ).
% pairwise_singleton
thf(fact_179_subset__Compl__singleton,axiom,
! [A: $tType,A3: set @ A,B2: A] :
( ( ord_less_eq @ ( set @ A ) @ A3 @ ( uminus_uminus @ ( set @ A ) @ ( insert @ A @ B2 @ ( bot_bot @ ( set @ A ) ) ) ) )
= ( ~ ( member @ A @ B2 @ A3 ) ) ) ).
% subset_Compl_singleton
thf(fact_180_add_Oinverse__inverse,axiom,
! [A: $tType] :
( ( group_add @ A @ ( type2 @ A ) )
=> ! [A2: A] :
( ( uminus_uminus @ A @ ( uminus_uminus @ A @ A2 ) )
= A2 ) ) ).
% add.inverse_inverse
thf(fact_181_neg__equal__iff__equal,axiom,
! [A: $tType] :
( ( group_add @ A @ ( type2 @ A ) )
=> ! [A2: A,B2: A] :
( ( ( uminus_uminus @ A @ A2 )
= ( uminus_uminus @ A @ B2 ) )
= ( A2 = B2 ) ) ) ).
% neg_equal_iff_equal
thf(fact_182_uminus__apply,axiom,
! [B3: $tType,A: $tType] :
( ( uminus @ B3 @ ( type2 @ B3 ) )
=> ( ( uminus_uminus @ ( A > B3 ) )
= ( ^ [A4: A > B3,X2: A] : ( uminus_uminus @ B3 @ ( A4 @ X2 ) ) ) ) ) ).
% uminus_apply
thf(fact_183_double__compl,axiom,
! [A: $tType] :
( ( boolean_algebra @ A @ ( type2 @ A ) )
=> ! [X: A] :
( ( uminus_uminus @ A @ ( uminus_uminus @ A @ X ) )
= X ) ) ).
% double_compl
thf(fact_184_compl__eq__compl__iff,axiom,
! [A: $tType] :
( ( boolean_algebra @ A @ ( type2 @ A ) )
=> ! [X: A,Y2: A] :
( ( ( uminus_uminus @ A @ X )
= ( uminus_uminus @ A @ Y2 ) )
= ( X = Y2 ) ) ) ).
% compl_eq_compl_iff
thf(fact_185_ComplI,axiom,
! [A: $tType,C: A,A3: set @ A] :
( ~ ( member @ A @ C @ A3 )
=> ( member @ A @ C @ ( uminus_uminus @ ( set @ A ) @ A3 ) ) ) ).
% ComplI
thf(fact_186_Compl__iff,axiom,
! [A: $tType,C: A,A3: set @ A] :
( ( member @ A @ C @ ( uminus_uminus @ ( set @ A ) @ A3 ) )
= ( ~ ( member @ A @ C @ A3 ) ) ) ).
% Compl_iff
thf(fact_187_Compl__eq__Compl__iff,axiom,
! [A: $tType,A3: set @ A,B: set @ A] :
( ( ( uminus_uminus @ ( set @ A ) @ A3 )
= ( uminus_uminus @ ( set @ A ) @ B ) )
= ( A3 = B ) ) ).
% Compl_eq_Compl_iff
thf(fact_188_neg__le__iff__le,axiom,
! [A: $tType] :
( ( ordered_ab_group_add @ A @ ( type2 @ A ) )
=> ! [B2: A,A2: A] :
( ( ord_less_eq @ A @ ( uminus_uminus @ A @ B2 ) @ ( uminus_uminus @ A @ A2 ) )
= ( ord_less_eq @ A @ A2 @ B2 ) ) ) ).
% neg_le_iff_le
thf(fact_189_compl__le__compl__iff,axiom,
! [A: $tType] :
( ( boolean_algebra @ A @ ( type2 @ A ) )
=> ! [X: A,Y2: A] :
( ( ord_less_eq @ A @ ( uminus_uminus @ A @ X ) @ ( uminus_uminus @ A @ Y2 ) )
= ( ord_less_eq @ A @ Y2 @ X ) ) ) ).
% compl_le_compl_iff
thf(fact_190_neg__less__iff__less,axiom,
! [A: $tType] :
( ( ordered_ab_group_add @ A @ ( type2 @ A ) )
=> ! [B2: A,A2: A] :
( ( ord_less @ A @ ( uminus_uminus @ A @ B2 ) @ ( uminus_uminus @ A @ A2 ) )
= ( ord_less @ A @ A2 @ B2 ) ) ) ).
% neg_less_iff_less
thf(fact_191_minus__diff__eq,axiom,
! [A: $tType] :
( ( group_add @ A @ ( type2 @ A ) )
=> ! [A2: A,B2: A] :
( ( uminus_uminus @ A @ ( minus_minus @ A @ A2 @ B2 ) )
= ( minus_minus @ A @ B2 @ A2 ) ) ) ).
% minus_diff_eq
thf(fact_192_psubsetI,axiom,
! [A: $tType,A3: set @ A,B: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ A3 @ B )
=> ( ( A3 != B )
=> ( ord_less @ ( set @ A ) @ A3 @ B ) ) ) ).
% psubsetI
thf(fact_193_Compl__anti__mono,axiom,
! [A: $tType,A3: set @ A,B: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ A3 @ B )
=> ( ord_less_eq @ ( set @ A ) @ ( uminus_uminus @ ( set @ A ) @ B ) @ ( uminus_uminus @ ( set @ A ) @ A3 ) ) ) ).
% Compl_anti_mono
thf(fact_194_Compl__subset__Compl__iff,axiom,
! [A: $tType,A3: set @ A,B: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ ( uminus_uminus @ ( set @ A ) @ A3 ) @ ( uminus_uminus @ ( set @ A ) @ B ) )
= ( ord_less_eq @ ( set @ A ) @ B @ A3 ) ) ).
% Compl_subset_Compl_iff
thf(fact_195_le__imp__neg__le,axiom,
! [A: $tType] :
( ( ordered_ab_group_add @ A @ ( type2 @ A ) )
=> ! [A2: A,B2: A] :
( ( ord_less_eq @ A @ A2 @ B2 )
=> ( ord_less_eq @ A @ ( uminus_uminus @ A @ B2 ) @ ( uminus_uminus @ A @ A2 ) ) ) ) ).
% le_imp_neg_le
thf(fact_196_minus__le__iff,axiom,
! [A: $tType] :
( ( ordered_ab_group_add @ A @ ( type2 @ A ) )
=> ! [A2: A,B2: A] :
( ( ord_less_eq @ A @ ( uminus_uminus @ A @ A2 ) @ B2 )
= ( ord_less_eq @ A @ ( uminus_uminus @ A @ B2 ) @ A2 ) ) ) ).
% minus_le_iff
thf(fact_197_le__minus__iff,axiom,
! [A: $tType] :
( ( ordered_ab_group_add @ A @ ( type2 @ A ) )
=> ! [A2: A,B2: A] :
( ( ord_less_eq @ A @ A2 @ ( uminus_uminus @ A @ B2 ) )
= ( ord_less_eq @ A @ B2 @ ( uminus_uminus @ A @ A2 ) ) ) ) ).
% le_minus_iff
thf(fact_198_compl__le__swap2,axiom,
! [A: $tType] :
( ( boolean_algebra @ A @ ( type2 @ A ) )
=> ! [Y2: A,X: A] :
( ( ord_less_eq @ A @ ( uminus_uminus @ A @ Y2 ) @ X )
=> ( ord_less_eq @ A @ ( uminus_uminus @ A @ X ) @ Y2 ) ) ) ).
% compl_le_swap2
thf(fact_199_compl__le__swap1,axiom,
! [A: $tType] :
( ( boolean_algebra @ A @ ( type2 @ A ) )
=> ! [Y2: A,X: A] :
( ( ord_less_eq @ A @ Y2 @ ( uminus_uminus @ A @ X ) )
=> ( ord_less_eq @ A @ X @ ( uminus_uminus @ A @ Y2 ) ) ) ) ).
% compl_le_swap1
thf(fact_200_compl__mono,axiom,
! [A: $tType] :
( ( boolean_algebra @ A @ ( type2 @ A ) )
=> ! [X: A,Y2: A] :
( ( ord_less_eq @ A @ X @ Y2 )
=> ( ord_less_eq @ A @ ( uminus_uminus @ A @ Y2 ) @ ( uminus_uminus @ A @ X ) ) ) ) ).
% compl_mono
thf(fact_201_order_Onot__eq__order__implies__strict,axiom,
! [A: $tType] :
( ( order @ A @ ( type2 @ A ) )
=> ! [A2: A,B2: A] :
( ( A2 != B2 )
=> ( ( ord_less_eq @ A @ A2 @ B2 )
=> ( ord_less @ A @ A2 @ B2 ) ) ) ) ).
% order.not_eq_order_implies_strict
thf(fact_202_dual__order_Ostrict__implies__order,axiom,
! [A: $tType] :
( ( order @ A @ ( type2 @ A ) )
=> ! [B2: A,A2: A] :
( ( ord_less @ A @ B2 @ A2 )
=> ( ord_less_eq @ A @ B2 @ A2 ) ) ) ).
% dual_order.strict_implies_order
thf(fact_203_dual__order_Ostrict__iff__order,axiom,
! [A: $tType] :
( ( order @ A @ ( type2 @ A ) )
=> ( ( ord_less @ A )
= ( ^ [B7: A,A6: A] :
( ( ord_less_eq @ A @ B7 @ A6 )
& ( A6 != B7 ) ) ) ) ) ).
% dual_order.strict_iff_order
thf(fact_204_dual__order_Oorder__iff__strict,axiom,
! [A: $tType] :
( ( order @ A @ ( type2 @ A ) )
=> ( ( ord_less_eq @ A )
= ( ^ [B7: A,A6: A] :
( ( ord_less @ A @ B7 @ A6 )
| ( A6 = B7 ) ) ) ) ) ).
% dual_order.order_iff_strict
thf(fact_205_order_Ostrict__implies__order,axiom,
! [A: $tType] :
( ( order @ A @ ( type2 @ A ) )
=> ! [A2: A,B2: A] :
( ( ord_less @ A @ A2 @ B2 )
=> ( ord_less_eq @ A @ A2 @ B2 ) ) ) ).
% order.strict_implies_order
thf(fact_206_dense__le__bounded,axiom,
! [A: $tType] :
( ( dense_linorder @ A @ ( type2 @ A ) )
=> ! [X: A,Y2: A,Z2: A] :
( ( ord_less @ A @ X @ Y2 )
=> ( ! [W: A] :
( ( ord_less @ A @ X @ W )
=> ( ( ord_less @ A @ W @ Y2 )
=> ( ord_less_eq @ A @ W @ Z2 ) ) )
=> ( ord_less_eq @ A @ Y2 @ Z2 ) ) ) ) ).
% dense_le_bounded
thf(fact_207_dense__ge__bounded,axiom,
! [A: $tType] :
( ( dense_linorder @ A @ ( type2 @ A ) )
=> ! [Z2: A,X: A,Y2: A] :
( ( ord_less @ A @ Z2 @ X )
=> ( ! [W: A] :
( ( ord_less @ A @ Z2 @ W )
=> ( ( ord_less @ A @ W @ X )
=> ( ord_less_eq @ A @ Y2 @ W ) ) )
=> ( ord_less_eq @ A @ Y2 @ Z2 ) ) ) ) ).
% dense_ge_bounded
thf(fact_208_dual__order_Ostrict__trans2,axiom,
! [A: $tType] :
( ( order @ A @ ( type2 @ A ) )
=> ! [B2: A,A2: A,C: A] :
( ( ord_less @ A @ B2 @ A2 )
=> ( ( ord_less_eq @ A @ C @ B2 )
=> ( ord_less @ A @ C @ A2 ) ) ) ) ).
% dual_order.strict_trans2
thf(fact_209_dual__order_Ostrict__trans1,axiom,
! [A: $tType] :
( ( order @ A @ ( type2 @ A ) )
=> ! [B2: A,A2: A,C: A] :
( ( ord_less_eq @ A @ B2 @ A2 )
=> ( ( ord_less @ A @ C @ B2 )
=> ( ord_less @ A @ C @ A2 ) ) ) ) ).
% dual_order.strict_trans1
thf(fact_210_order_Ostrict__iff__order,axiom,
! [A: $tType] :
( ( order @ A @ ( type2 @ A ) )
=> ( ( ord_less @ A )
= ( ^ [A6: A,B7: A] :
( ( ord_less_eq @ A @ A6 @ B7 )
& ( A6 != B7 ) ) ) ) ) ).
% order.strict_iff_order
thf(fact_211_order_Oorder__iff__strict,axiom,
! [A: $tType] :
( ( order @ A @ ( type2 @ A ) )
=> ( ( ord_less_eq @ A )
= ( ^ [A6: A,B7: A] :
( ( ord_less @ A @ A6 @ B7 )
| ( A6 = B7 ) ) ) ) ) ).
% order.order_iff_strict
thf(fact_212_order_Ostrict__trans2,axiom,
! [A: $tType] :
( ( order @ A @ ( type2 @ A ) )
=> ! [A2: A,B2: A,C: A] :
( ( ord_less @ A @ A2 @ B2 )
=> ( ( ord_less_eq @ A @ B2 @ C )
=> ( ord_less @ A @ A2 @ C ) ) ) ) ).
% order.strict_trans2
thf(fact_213_order_Ostrict__trans1,axiom,
! [A: $tType] :
( ( order @ A @ ( type2 @ A ) )
=> ! [A2: A,B2: A,C: A] :
( ( ord_less_eq @ A @ A2 @ B2 )
=> ( ( ord_less @ A @ B2 @ C )
=> ( ord_less @ A @ A2 @ C ) ) ) ) ).
% order.strict_trans1
thf(fact_214_not__le__imp__less,axiom,
! [A: $tType] :
( ( linorder @ A @ ( type2 @ A ) )
=> ! [Y2: A,X: A] :
( ~ ( ord_less_eq @ A @ Y2 @ X )
=> ( ord_less @ A @ X @ Y2 ) ) ) ).
% not_le_imp_less
thf(fact_215_less__le__not__le,axiom,
! [A: $tType] :
( ( preorder @ A @ ( type2 @ A ) )
=> ( ( ord_less @ A )
= ( ^ [X2: A,Y4: A] :
( ( ord_less_eq @ A @ X2 @ Y4 )
& ~ ( ord_less_eq @ A @ Y4 @ X2 ) ) ) ) ) ).
% less_le_not_le
thf(fact_216_le__imp__less__or__eq,axiom,
! [A: $tType] :
( ( order @ A @ ( type2 @ A ) )
=> ! [X: A,Y2: A] :
( ( ord_less_eq @ A @ X @ Y2 )
=> ( ( ord_less @ A @ X @ Y2 )
| ( X = Y2 ) ) ) ) ).
% le_imp_less_or_eq
thf(fact_217_le__less__linear,axiom,
! [A: $tType] :
( ( linorder @ A @ ( type2 @ A ) )
=> ! [X: A,Y2: A] :
( ( ord_less_eq @ A @ X @ Y2 )
| ( ord_less @ A @ Y2 @ X ) ) ) ).
% le_less_linear
thf(fact_218_dense__le,axiom,
! [A: $tType] :
( ( dense_linorder @ A @ ( type2 @ A ) )
=> ! [Y2: A,Z2: A] :
( ! [X3: A] :
( ( ord_less @ A @ X3 @ Y2 )
=> ( ord_less_eq @ A @ X3 @ Z2 ) )
=> ( ord_less_eq @ A @ Y2 @ Z2 ) ) ) ).
% dense_le
thf(fact_219_dense__ge,axiom,
! [A: $tType] :
( ( dense_linorder @ A @ ( type2 @ A ) )
=> ! [Z2: A,Y2: A] :
( ! [X3: A] :
( ( ord_less @ A @ Z2 @ X3 )
=> ( ord_less_eq @ A @ Y2 @ X3 ) )
=> ( ord_less_eq @ A @ Y2 @ Z2 ) ) ) ).
% dense_ge
thf(fact_220_less__le__trans,axiom,
! [A: $tType] :
( ( preorder @ A @ ( type2 @ A ) )
=> ! [X: A,Y2: A,Z2: A] :
( ( ord_less @ A @ X @ Y2 )
=> ( ( ord_less_eq @ A @ Y2 @ Z2 )
=> ( ord_less @ A @ X @ Z2 ) ) ) ) ).
% less_le_trans
thf(fact_221_le__less__trans,axiom,
! [A: $tType] :
( ( preorder @ A @ ( type2 @ A ) )
=> ! [X: A,Y2: A,Z2: A] :
( ( ord_less_eq @ A @ X @ Y2 )
=> ( ( ord_less @ A @ Y2 @ Z2 )
=> ( ord_less @ A @ X @ Z2 ) ) ) ) ).
% le_less_trans
thf(fact_222_antisym__conv2,axiom,
! [A: $tType] :
( ( linorder @ A @ ( type2 @ A ) )
=> ! [X: A,Y2: A] :
( ( ord_less_eq @ A @ X @ Y2 )
=> ( ( ~ ( ord_less @ A @ X @ Y2 ) )
= ( X = Y2 ) ) ) ) ).
% antisym_conv2
thf(fact_223_antisym__conv1,axiom,
! [A: $tType] :
( ( linorder @ A @ ( type2 @ A ) )
=> ! [X: A,Y2: A] :
( ~ ( ord_less @ A @ X @ Y2 )
=> ( ( ord_less_eq @ A @ X @ Y2 )
= ( X = Y2 ) ) ) ) ).
% antisym_conv1
thf(fact_224_less__imp__le,axiom,
! [A: $tType] :
( ( preorder @ A @ ( type2 @ A ) )
=> ! [X: A,Y2: A] :
( ( ord_less @ A @ X @ Y2 )
=> ( ord_less_eq @ A @ X @ Y2 ) ) ) ).
% less_imp_le
thf(fact_225_le__neq__trans,axiom,
! [A: $tType] :
( ( order @ A @ ( type2 @ A ) )
=> ! [A2: A,B2: A] :
( ( ord_less_eq @ A @ A2 @ B2 )
=> ( ( A2 != B2 )
=> ( ord_less @ A @ A2 @ B2 ) ) ) ) ).
% le_neq_trans
thf(fact_226_not__less,axiom,
! [A: $tType] :
( ( linorder @ A @ ( type2 @ A ) )
=> ! [X: A,Y2: A] :
( ( ~ ( ord_less @ A @ X @ Y2 ) )
= ( ord_less_eq @ A @ Y2 @ X ) ) ) ).
% not_less
thf(fact_227_not__le,axiom,
! [A: $tType] :
( ( linorder @ A @ ( type2 @ A ) )
=> ! [X: A,Y2: A] :
( ( ~ ( ord_less_eq @ A @ X @ Y2 ) )
= ( ord_less @ A @ Y2 @ X ) ) ) ).
% not_le
thf(fact_228_order__less__le__subst2,axiom,
! [A: $tType,C4: $tType] :
( ( ( order @ C4 @ ( type2 @ C4 ) )
& ( order @ A @ ( type2 @ A ) ) )
=> ! [A2: A,B2: A,F: A > C4,C: C4] :
( ( ord_less @ A @ A2 @ B2 )
=> ( ( ord_less_eq @ C4 @ ( F @ B2 ) @ C )
=> ( ! [X3: A,Y3: A] :
( ( ord_less @ A @ X3 @ Y3 )
=> ( ord_less @ C4 @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less @ C4 @ ( F @ A2 ) @ C ) ) ) ) ) ).
% order_less_le_subst2
thf(fact_229_order__less__le__subst1,axiom,
! [A: $tType,B3: $tType] :
( ( ( order @ B3 @ ( type2 @ B3 ) )
& ( order @ A @ ( type2 @ A ) ) )
=> ! [A2: A,F: B3 > A,B2: B3,C: B3] :
( ( ord_less @ A @ A2 @ ( F @ B2 ) )
=> ( ( ord_less_eq @ B3 @ B2 @ C )
=> ( ! [X3: B3,Y3: B3] :
( ( ord_less_eq @ B3 @ X3 @ Y3 )
=> ( ord_less_eq @ A @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less @ A @ A2 @ ( F @ C ) ) ) ) ) ) ).
% order_less_le_subst1
thf(fact_230_order__le__less__subst2,axiom,
! [A: $tType,C4: $tType] :
( ( ( order @ C4 @ ( type2 @ C4 ) )
& ( order @ A @ ( type2 @ A ) ) )
=> ! [A2: A,B2: A,F: A > C4,C: C4] :
( ( ord_less_eq @ A @ A2 @ B2 )
=> ( ( ord_less @ C4 @ ( F @ B2 ) @ C )
=> ( ! [X3: A,Y3: A] :
( ( ord_less_eq @ A @ X3 @ Y3 )
=> ( ord_less_eq @ C4 @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less @ C4 @ ( F @ A2 ) @ C ) ) ) ) ) ).
% order_le_less_subst2
thf(fact_231_order__le__less__subst1,axiom,
! [A: $tType,B3: $tType] :
( ( ( order @ B3 @ ( type2 @ B3 ) )
& ( order @ A @ ( type2 @ A ) ) )
=> ! [A2: A,F: B3 > A,B2: B3,C: B3] :
( ( ord_less_eq @ A @ A2 @ ( F @ B2 ) )
=> ( ( ord_less @ B3 @ B2 @ C )
=> ( ! [X3: B3,Y3: B3] :
( ( ord_less @ B3 @ X3 @ Y3 )
=> ( ord_less @ A @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less @ A @ A2 @ ( F @ C ) ) ) ) ) ) ).
% order_le_less_subst1
thf(fact_232_less__le,axiom,
! [A: $tType] :
( ( order @ A @ ( type2 @ A ) )
=> ( ( ord_less @ A )
= ( ^ [X2: A,Y4: A] :
( ( ord_less_eq @ A @ X2 @ Y4 )
& ( X2 != Y4 ) ) ) ) ) ).
% less_le
thf(fact_233_le__less,axiom,
! [A: $tType] :
( ( order @ A @ ( type2 @ A ) )
=> ( ( ord_less_eq @ A )
= ( ^ [X2: A,Y4: A] :
( ( ord_less @ A @ X2 @ Y4 )
| ( X2 = Y4 ) ) ) ) ) ).
% le_less
thf(fact_234_leI,axiom,
! [A: $tType] :
( ( linorder @ A @ ( type2 @ A ) )
=> ! [X: A,Y2: A] :
( ~ ( ord_less @ A @ X @ Y2 )
=> ( ord_less_eq @ A @ Y2 @ X ) ) ) ).
% leI
thf(fact_235_leD,axiom,
! [A: $tType] :
( ( linorder @ A @ ( type2 @ A ) )
=> ! [Y2: A,X: A] :
( ( ord_less_eq @ A @ Y2 @ X )
=> ~ ( ord_less @ A @ X @ Y2 ) ) ) ).
% leD
thf(fact_236_diff__strict__mono,axiom,
! [A: $tType] :
( ( ordered_ab_group_add @ A @ ( type2 @ A ) )
=> ! [A2: A,B2: A,D2: A,C: A] :
( ( ord_less @ A @ A2 @ B2 )
=> ( ( ord_less @ A @ D2 @ C )
=> ( ord_less @ A @ ( minus_minus @ A @ A2 @ C ) @ ( minus_minus @ A @ B2 @ D2 ) ) ) ) ) ).
% diff_strict_mono
thf(fact_237_diff__eq__diff__less,axiom,
! [A: $tType] :
( ( ordered_ab_group_add @ A @ ( type2 @ A ) )
=> ! [A2: A,B2: A,C: A,D2: A] :
( ( ( minus_minus @ A @ A2 @ B2 )
= ( minus_minus @ A @ C @ D2 ) )
=> ( ( ord_less @ A @ A2 @ B2 )
= ( ord_less @ A @ C @ D2 ) ) ) ) ).
% diff_eq_diff_less
thf(fact_238_diff__strict__left__mono,axiom,
! [A: $tType] :
( ( ordered_ab_group_add @ A @ ( type2 @ A ) )
=> ! [B2: A,A2: A,C: A] :
( ( ord_less @ A @ B2 @ A2 )
=> ( ord_less @ A @ ( minus_minus @ A @ C @ A2 ) @ ( minus_minus @ A @ C @ B2 ) ) ) ) ).
% diff_strict_left_mono
thf(fact_239_diff__strict__right__mono,axiom,
! [A: $tType] :
( ( ordered_ab_group_add @ A @ ( type2 @ A ) )
=> ! [A2: A,B2: A,C: A] :
( ( ord_less @ A @ A2 @ B2 )
=> ( ord_less @ A @ ( minus_minus @ A @ A2 @ C ) @ ( minus_minus @ A @ B2 @ C ) ) ) ) ).
% diff_strict_right_mono
thf(fact_240_bot_Oextremum__strict,axiom,
! [A: $tType] :
( ( order_bot @ A @ ( type2 @ A ) )
=> ! [A2: A] :
~ ( ord_less @ A @ A2 @ ( bot_bot @ A ) ) ) ).
% bot.extremum_strict
thf(fact_241_bot_Onot__eq__extremum,axiom,
! [A: $tType] :
( ( order_bot @ A @ ( type2 @ A ) )
=> ! [A2: A] :
( ( A2
!= ( bot_bot @ A ) )
= ( ord_less @ A @ ( bot_bot @ A ) @ A2 ) ) ) ).
% bot.not_eq_extremum
thf(fact_242_not__psubset__empty,axiom,
! [A: $tType,A3: set @ A] :
~ ( ord_less @ ( set @ A ) @ A3 @ ( bot_bot @ ( set @ A ) ) ) ).
% not_psubset_empty
thf(fact_243_subset__iff__psubset__eq,axiom,
! [A: $tType] :
( ( ord_less_eq @ ( set @ A ) )
= ( ^ [A4: set @ A,B4: set @ A] :
( ( ord_less @ ( set @ A ) @ A4 @ B4 )
| ( A4 = B4 ) ) ) ) ).
% subset_iff_psubset_eq
thf(fact_244_subset__psubset__trans,axiom,
! [A: $tType,A3: set @ A,B: set @ A,C2: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ A3 @ B )
=> ( ( ord_less @ ( set @ A ) @ B @ C2 )
=> ( ord_less @ ( set @ A ) @ A3 @ C2 ) ) ) ).
% subset_psubset_trans
thf(fact_245_subset__not__subset__eq,axiom,
! [A: $tType] :
( ( ord_less @ ( set @ A ) )
= ( ^ [A4: set @ A,B4: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ A4 @ B4 )
& ~ ( ord_less_eq @ ( set @ A ) @ B4 @ A4 ) ) ) ) ).
% subset_not_subset_eq
thf(fact_246_psubset__subset__trans,axiom,
! [A: $tType,A3: set @ A,B: set @ A,C2: set @ A] :
( ( ord_less @ ( set @ A ) @ A3 @ B )
=> ( ( ord_less_eq @ ( set @ A ) @ B @ C2 )
=> ( ord_less @ ( set @ A ) @ A3 @ C2 ) ) ) ).
% psubset_subset_trans
thf(fact_247_psubset__imp__subset,axiom,
! [A: $tType,A3: set @ A,B: set @ A] :
( ( ord_less @ ( set @ A ) @ A3 @ B )
=> ( ord_less_eq @ ( set @ A ) @ A3 @ B ) ) ).
% psubset_imp_subset
thf(fact_248_psubset__eq,axiom,
! [A: $tType] :
( ( ord_less @ ( set @ A ) )
= ( ^ [A4: set @ A,B4: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ A4 @ B4 )
& ( A4 != B4 ) ) ) ) ).
% psubset_eq
thf(fact_249_psubsetE,axiom,
! [A: $tType,A3: set @ A,B: set @ A] :
( ( ord_less @ ( set @ A ) @ A3 @ B )
=> ~ ( ( ord_less_eq @ ( set @ A ) @ A3 @ B )
=> ( ord_less_eq @ ( set @ A ) @ B @ A3 ) ) ) ).
% psubsetE
thf(fact_250_equation__minus__iff,axiom,
! [A: $tType] :
( ( group_add @ A @ ( type2 @ A ) )
=> ! [A2: A,B2: A] :
( ( A2
= ( uminus_uminus @ A @ B2 ) )
= ( B2
= ( uminus_uminus @ A @ A2 ) ) ) ) ).
% equation_minus_iff
thf(fact_251_minus__equation__iff,axiom,
! [A: $tType] :
( ( group_add @ A @ ( type2 @ A ) )
=> ! [A2: A,B2: A] :
( ( ( uminus_uminus @ A @ A2 )
= B2 )
= ( ( uminus_uminus @ A @ B2 )
= A2 ) ) ) ).
% minus_equation_iff
thf(fact_252_less__minus__iff,axiom,
! [A: $tType] :
( ( ordered_ab_group_add @ A @ ( type2 @ A ) )
=> ! [A2: A,B2: A] :
( ( ord_less @ A @ A2 @ ( uminus_uminus @ A @ B2 ) )
= ( ord_less @ A @ B2 @ ( uminus_uminus @ A @ A2 ) ) ) ) ).
% less_minus_iff
thf(fact_253_minus__less__iff,axiom,
! [A: $tType] :
( ( ordered_ab_group_add @ A @ ( type2 @ A ) )
=> ! [A2: A,B2: A] :
( ( ord_less @ A @ ( uminus_uminus @ A @ A2 ) @ B2 )
= ( ord_less @ A @ ( uminus_uminus @ A @ B2 ) @ A2 ) ) ) ).
% minus_less_iff
thf(fact_254_ComplD,axiom,
! [A: $tType,C: A,A3: set @ A] :
( ( member @ A @ C @ ( uminus_uminus @ ( set @ A ) @ A3 ) )
=> ~ ( member @ A @ C @ A3 ) ) ).
% ComplD
thf(fact_255_psubsetD,axiom,
! [A: $tType,A3: set @ A,B: set @ A,C: A] :
( ( ord_less @ ( set @ A ) @ A3 @ B )
=> ( ( member @ A @ C @ A3 )
=> ( member @ A @ C @ B ) ) ) ).
% psubsetD
%----Type constructors (25)
thf(tcon_fun___Lattices_Oboolean__algebra,axiom,
! [A7: $tType,A8: $tType] :
( ( boolean_algebra @ A8 @ ( type2 @ A8 ) )
=> ( boolean_algebra @ ( A7 > A8 ) @ ( type2 @ ( A7 > A8 ) ) ) ) ).
thf(tcon_fun___Orderings_Oorder__bot,axiom,
! [A7: $tType,A8: $tType] :
( ( order_bot @ A8 @ ( type2 @ A8 ) )
=> ( order_bot @ ( A7 > A8 ) @ ( type2 @ ( A7 > A8 ) ) ) ) ).
thf(tcon_fun___Orderings_Opreorder,axiom,
! [A7: $tType,A8: $tType] :
( ( preorder @ A8 @ ( type2 @ A8 ) )
=> ( preorder @ ( A7 > A8 ) @ ( type2 @ ( A7 > A8 ) ) ) ) ).
thf(tcon_fun___Orderings_Oorder,axiom,
! [A7: $tType,A8: $tType] :
( ( order @ A8 @ ( type2 @ A8 ) )
=> ( order @ ( A7 > A8 ) @ ( type2 @ ( A7 > A8 ) ) ) ) ).
thf(tcon_fun___Orderings_Oord,axiom,
! [A7: $tType,A8: $tType] :
( ( ord @ A8 @ ( type2 @ A8 ) )
=> ( ord @ ( A7 > A8 ) @ ( type2 @ ( A7 > A8 ) ) ) ) ).
thf(tcon_fun___Orderings_Obot,axiom,
! [A7: $tType,A8: $tType] :
( ( bot @ A8 @ ( type2 @ A8 ) )
=> ( bot @ ( A7 > A8 ) @ ( type2 @ ( A7 > A8 ) ) ) ) ).
thf(tcon_fun___Groups_Ouminus,axiom,
! [A7: $tType,A8: $tType] :
( ( uminus @ A8 @ ( type2 @ A8 ) )
=> ( uminus @ ( A7 > A8 ) @ ( type2 @ ( A7 > A8 ) ) ) ) ).
thf(tcon_fun___Groups_Ominus,axiom,
! [A7: $tType,A8: $tType] :
( ( minus @ A8 @ ( type2 @ A8 ) )
=> ( minus @ ( A7 > A8 ) @ ( type2 @ ( A7 > A8 ) ) ) ) ).
thf(tcon_Set_Oset___Lattices_Oboolean__algebra_1,axiom,
! [A7: $tType] : ( boolean_algebra @ ( set @ A7 ) @ ( type2 @ ( set @ A7 ) ) ) ).
thf(tcon_Set_Oset___Orderings_Oorder__bot_2,axiom,
! [A7: $tType] : ( order_bot @ ( set @ A7 ) @ ( type2 @ ( set @ A7 ) ) ) ).
thf(tcon_Set_Oset___Orderings_Opreorder_3,axiom,
! [A7: $tType] : ( preorder @ ( set @ A7 ) @ ( type2 @ ( set @ A7 ) ) ) ).
thf(tcon_Set_Oset___Orderings_Oorder_4,axiom,
! [A7: $tType] : ( order @ ( set @ A7 ) @ ( type2 @ ( set @ A7 ) ) ) ).
thf(tcon_Set_Oset___Orderings_Oord_5,axiom,
! [A7: $tType] : ( ord @ ( set @ A7 ) @ ( type2 @ ( set @ A7 ) ) ) ).
thf(tcon_Set_Oset___Orderings_Obot_6,axiom,
! [A7: $tType] : ( bot @ ( set @ A7 ) @ ( type2 @ ( set @ A7 ) ) ) ).
thf(tcon_Set_Oset___Groups_Ouminus_7,axiom,
! [A7: $tType] : ( uminus @ ( set @ A7 ) @ ( type2 @ ( set @ A7 ) ) ) ).
thf(tcon_Set_Oset___Groups_Ominus_8,axiom,
! [A7: $tType] : ( minus @ ( set @ A7 ) @ ( type2 @ ( set @ A7 ) ) ) ).
thf(tcon_HOL_Obool___Lattices_Oboolean__algebra_9,axiom,
boolean_algebra @ $o @ ( type2 @ $o ) ).
thf(tcon_HOL_Obool___Orderings_Oorder__bot_10,axiom,
order_bot @ $o @ ( type2 @ $o ) ).
thf(tcon_HOL_Obool___Orderings_Opreorder_11,axiom,
preorder @ $o @ ( type2 @ $o ) ).
thf(tcon_HOL_Obool___Orderings_Olinorder,axiom,
linorder @ $o @ ( type2 @ $o ) ).
thf(tcon_HOL_Obool___Orderings_Oorder_12,axiom,
order @ $o @ ( type2 @ $o ) ).
thf(tcon_HOL_Obool___Orderings_Oord_13,axiom,
ord @ $o @ ( type2 @ $o ) ).
thf(tcon_HOL_Obool___Orderings_Obot_14,axiom,
bot @ $o @ ( type2 @ $o ) ).
thf(tcon_HOL_Obool___Groups_Ouminus_15,axiom,
uminus @ $o @ ( type2 @ $o ) ).
thf(tcon_HOL_Obool___Groups_Ominus_16,axiom,
minus @ $o @ ( type2 @ $o ) ).
%----Conjectures (1)
thf(conj_0,conjecture,
member @ a @ x @ ( insert @ a @ n2 @ ( koenig317145564le_via @ a @ graph @ ( minus_minus @ ( set @ a ) @ ns @ ( insert @ a @ n2 @ ( bot_bot @ ( set @ a ) ) ) ) @ n2 ) ) ).
%------------------------------------------------------------------------------