TPTP Problem File: ITP180^2.p
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%------------------------------------------------------------------------------
% File : ITP180^2 : TPTP v8.2.0. Released v7.5.0.
% Domain : Interactive Theorem Proving
% Problem : Sledgehammer StandardRules problem prob_651__5393624_1
% Version : Especial.
% English :
% Refs : [BH+15] Blanchette et al. (2015), Mining the Archive of Formal
% : [Des21] Desharnais (2021), Email to Geoff Sutcliffe
% Source : [Des21]
% Names : StandardRules/prob_651__5393624_1 [Des21]
% Status : Theorem
% Rating : 0.33 v8.1.0, 0.75 v7.5.0
% Syntax : Number of formulae : 343 ( 144 unt; 64 typ; 0 def)
% Number of atoms : 623 ( 280 equ; 0 cnn)
% Maximal formula atoms : 6 ( 2 avg)
% Number of connectives : 5623 ( 63 ~; 11 |; 40 &;5289 @)
% ( 0 <=>; 220 =>; 0 <=; 0 <~>)
% Maximal formula depth : 28 ( 8 avg)
% Number of types : 4 ( 3 usr)
% Number of type conns : 173 ( 173 >; 0 *; 0 +; 0 <<)
% Number of symbols : 62 ( 61 usr; 6 con; 0-8 aty)
% Number of variables : 1204 ( 46 ^;1064 !; 12 ?;1204 :)
% ( 82 !>; 0 ?*; 0 @-; 0 @+)
% SPC : TH1_THM_EQU_NAR
% Comments : This file was generated by Sledgehammer 2021-02-23 16:24:22.396
%------------------------------------------------------------------------------
% Could-be-implicit typings (9)
thf(ty_t_LabeledGraphSemantics_OStandard__Constant,type,
standard_Constant: $tType > $tType ).
thf(ty_t_LabeledGraphSemantics_Oallegorical__term,type,
allegorical_term: $tType > $tType ).
thf(ty_t_LabeledGraphs_Olabeled__graph,type,
labeled_graph: $tType > $tType > $tType ).
thf(ty_t_Product__Type_Oprod,type,
product_prod: $tType > $tType > $tType ).
thf(ty_t_Set_Oset,type,
set: $tType > $tType ).
thf(ty_t_Nat_Onat,type,
nat: $tType ).
thf(ty_t_itself,type,
itself: $tType > $tType ).
thf(ty_tf_V_H,type,
v: $tType ).
thf(ty_tf_V,type,
v2: $tType ).
% Explicit typings (55)
thf(sy_cl_Lattices_Obounded__lattice,type,
bounded_lattice:
!>[A: $tType] : $o ).
thf(sy_cl_HOL_Otype,type,
type:
!>[A: $tType] : $o ).
thf(sy_cl_Orderings_Obot,type,
bot:
!>[A: $tType] : $o ).
thf(sy_cl_Lattices_Olattice,type,
lattice:
!>[A: $tType] : $o ).
thf(sy_cl_Orderings_Opreorder,type,
preorder:
!>[A: $tType] : $o ).
thf(sy_cl_Lattices_Osemilattice__sup,type,
semilattice_sup:
!>[A: $tType] : $o ).
thf(sy_cl_Lattices_Obounded__semilattice__sup__bot,type,
bounde1808546759up_bot:
!>[A: $tType] : $o ).
thf(sy_c_BNF__Def_OGr,type,
bNF_Gr:
!>[A: $tType,B: $tType] : ( ( set @ A ) > ( A > B ) > ( set @ ( product_prod @ A @ B ) ) ) ).
thf(sy_c_Fun_Ocomp,type,
comp:
!>[B: $tType,C: $tType,A: $tType] : ( ( B > C ) > ( A > B ) > A > C ) ).
thf(sy_c_LabeledGraphSemantics_OStandard__Constant_OS__Bot,type,
standard_S_Bot:
!>[V: $tType] : ( standard_Constant @ V ) ).
thf(sy_c_LabeledGraphSemantics_OStandard__Constant_OS__Const,type,
standard_S_Const:
!>[V: $tType] : ( V > ( standard_Constant @ V ) ) ).
thf(sy_c_LabeledGraphSemantics_OStandard__Constant_OS__Top,type,
standard_S_Top:
!>[V: $tType] : ( standard_Constant @ V ) ).
thf(sy_c_LabeledGraphSemantics_OgetRel,type,
getRel:
!>[B: $tType,A: $tType] : ( B > ( labeled_graph @ B @ A ) > ( set @ ( product_prod @ A @ A ) ) ) ).
thf(sy_c_LabeledGraphSemantics_Osemantics,type,
semantics:
!>[A: $tType,B: $tType] : ( ( labeled_graph @ A @ B ) > ( allegorical_term @ A ) > ( set @ ( product_prod @ B @ B ) ) ) ).
thf(sy_c_LabeledGraphs_Oedge__preserving,type,
edge_preserving:
!>[A: $tType,B: $tType,C: $tType] : ( ( set @ ( product_prod @ A @ B ) ) > ( set @ ( product_prod @ C @ ( product_prod @ A @ A ) ) ) > ( set @ ( product_prod @ C @ ( product_prod @ B @ B ) ) ) > $o ) ).
thf(sy_c_LabeledGraphs_Ograph__homomorphism,type,
graph_homomorphism:
!>[A: $tType,B: $tType,C: $tType] : ( ( labeled_graph @ A @ B ) > ( labeled_graph @ A @ C ) > ( set @ ( product_prod @ B @ C ) ) > $o ) ).
thf(sy_c_LabeledGraphs_Ograph__union,type,
graph_union:
!>[A: $tType,B: $tType] : ( ( labeled_graph @ A @ B ) > ( labeled_graph @ A @ B ) > ( labeled_graph @ A @ B ) ) ).
thf(sy_c_LabeledGraphs_Olabeled__graph_OLG,type,
labeled_LG:
!>[L: $tType,V: $tType] : ( ( set @ ( product_prod @ L @ ( product_prod @ V @ V ) ) ) > ( set @ V ) > ( labeled_graph @ L @ V ) ) ).
thf(sy_c_LabeledGraphs_Olabeled__graph_Ocase__labeled__graph,type,
labele1974067554_graph:
!>[L: $tType,V: $tType,A: $tType] : ( ( ( set @ ( product_prod @ L @ ( product_prod @ V @ V ) ) ) > ( set @ V ) > A ) > ( labeled_graph @ L @ V ) > A ) ).
thf(sy_c_LabeledGraphs_Olabeled__graph_Oedges,type,
labeled_edges:
!>[L: $tType,V: $tType] : ( ( labeled_graph @ L @ V ) > ( set @ ( product_prod @ L @ ( product_prod @ V @ V ) ) ) ) ).
thf(sy_c_LabeledGraphs_Olabeled__graph_Overtices,type,
labeled_vertices:
!>[L: $tType,V: $tType] : ( ( labeled_graph @ L @ V ) > ( set @ V ) ) ).
thf(sy_c_LabeledGraphs_Omap__graph,type,
map_graph:
!>[C: $tType,B: $tType,A: $tType] : ( ( set @ ( product_prod @ C @ B ) ) > ( labeled_graph @ A @ C ) > ( labeled_graph @ A @ B ) ) ).
thf(sy_c_LabeledGraphs_Orestrict,type,
restrict:
!>[A: $tType,B: $tType] : ( ( labeled_graph @ A @ B ) > ( labeled_graph @ A @ B ) ) ).
thf(sy_c_Lattices_Osup__class_Osup,type,
sup_sup:
!>[A: $tType] : ( A > A > A ) ).
thf(sy_c_Orderings_Obot__class_Obot,type,
bot_bot:
!>[A: $tType] : A ).
thf(sy_c_Orderings_Oord__class_Oless__eq,type,
ord_less_eq:
!>[A: $tType] : ( A > A > $o ) ).
thf(sy_c_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_Relation_OField,type,
field:
!>[A: $tType] : ( ( set @ ( product_prod @ A @ A ) ) > ( set @ A ) ) ).
thf(sy_c_Relation_OId__on,type,
id_on:
!>[A: $tType] : ( ( set @ A ) > ( set @ ( product_prod @ A @ A ) ) ) ).
thf(sy_c_Relation_Ototal__on,type,
total_on:
!>[A: $tType] : ( ( set @ A ) > ( set @ ( product_prod @ A @ A ) ) > $o ) ).
thf(sy_c_RulesAndChains_Oagree__on,type,
agree_on:
!>[A: $tType,B: $tType,C: $tType] : ( ( labeled_graph @ A @ B ) > ( set @ ( product_prod @ B @ C ) ) > ( set @ ( product_prod @ B @ C ) ) > $o ) ).
thf(sy_c_RulesAndChains_Oconsequence__graph,type,
consequence_graph:
!>[A: $tType,B: $tType,C: $tType] : ( ( set @ ( product_prod @ ( labeled_graph @ A @ B ) @ ( labeled_graph @ A @ B ) ) ) > ( labeled_graph @ A @ C ) > $o ) ).
thf(sy_c_RulesAndChains_Oextensible,type,
extensible:
!>[L: $tType,X: $tType,V: $tType] : ( ( product_prod @ ( labeled_graph @ L @ X ) @ ( labeled_graph @ L @ X ) ) > ( labeled_graph @ L @ V ) > ( set @ ( product_prod @ X @ V ) ) > $o ) ).
thf(sy_c_RulesAndChains_Oleast,type,
least:
!>[X: $tType,L: $tType,V: $tType,C: $tType] : ( ( itself @ X ) > ( set @ ( product_prod @ ( labeled_graph @ L @ V ) @ ( labeled_graph @ L @ V ) ) ) > ( labeled_graph @ L @ C ) > ( labeled_graph @ L @ C ) > $o ) ).
thf(sy_c_RulesAndChains_Oleast__consequence__graph,type,
least_559130134_graph:
!>[X: $tType,L: $tType,V: $tType,C: $tType] : ( ( itself @ X ) > ( set @ ( product_prod @ ( labeled_graph @ L @ V ) @ ( labeled_graph @ L @ V ) ) ) > ( labeled_graph @ L @ C ) > ( labeled_graph @ L @ C ) > $o ) ).
thf(sy_c_RulesAndChains_Omaintained,type,
maintained:
!>[L: $tType,X: $tType,V: $tType] : ( ( product_prod @ ( labeled_graph @ L @ X ) @ ( labeled_graph @ L @ X ) ) > ( labeled_graph @ L @ V ) > $o ) ).
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__singleton,type,
is_singleton:
!>[A: $tType] : ( ( set @ A ) > $o ) ).
thf(sy_c_Set_Othe__elem,type,
the_elem:
!>[A: $tType] : ( ( set @ A ) > A ) ).
thf(sy_c_StandardRules__Mirabelle__xlrcbcygrg_Oconstant__rules,type,
standa1138209853_rules:
!>[A: $tType] : ( ( set @ A ) > ( set @ ( product_prod @ ( labeled_graph @ ( standard_Constant @ A ) @ nat ) @ ( labeled_graph @ ( standard_Constant @ A ) @ nat ) ) ) ) ).
thf(sy_c_StandardRules__Mirabelle__xlrcbcygrg_Oidentity__rules,type,
standa2002409347_rules:
!>[A: $tType] : ( ( set @ ( standard_Constant @ A ) ) > ( set @ ( product_prod @ ( labeled_graph @ ( standard_Constant @ A ) @ nat ) @ ( labeled_graph @ ( standard_Constant @ A ) @ nat ) ) ) ) ).
thf(sy_c_StandardRules__Mirabelle__xlrcbcygrg_Ononempty__rule,type,
standa244753842y_rule:
!>[L: $tType] : ( product_prod @ ( labeled_graph @ L @ nat ) @ ( labeled_graph @ L @ nat ) ) ).
thf(sy_c_StandardRules__Mirabelle__xlrcbcygrg_Ostandard__rules,type,
standa438229444_rules:
!>[A: $tType] : ( ( set @ A ) > ( set @ ( standard_Constant @ A ) ) > ( set @ ( product_prod @ ( labeled_graph @ ( standard_Constant @ A ) @ nat ) @ ( labeled_graph @ ( standard_Constant @ A ) @ nat ) ) ) ) ).
thf(sy_c_StandardRules__Mirabelle__xlrcbcygrg_Otop__rule,type,
standa400804411p_rule:
!>[L: $tType] : ( L > ( product_prod @ ( labeled_graph @ L @ nat ) @ ( labeled_graph @ L @ nat ) ) ) ).
thf(sy_c_member,type,
member:
!>[A: $tType] : ( A > ( set @ A ) > $o ) ).
thf(sy_v_C,type,
c: set @ v2 ).
thf(sy_v_G_H,type,
g: labeled_graph @ ( standard_Constant @ v2 ) @ v ).
thf(sy_v_L,type,
l: set @ ( standard_Constant @ v2 ) ).
thf(sy_v_h____,type,
h: v > v ).
thf(sy_v_m____,type,
m: v2 > v ).
thf(sy_v_x____,type,
x: v ).
thf(sy_v_xa____,type,
xa: v2 ).
thf(sy_v_y____,type,
y: v2 ).
% Relevant facts (256)
thf(fact_0_x,axiom,
member @ v @ x @ ( labeled_vertices @ ( standard_Constant @ v2 ) @ v @ g ) ).
% x
thf(fact_1__092_060open_062_092_060And_062thesis_O_A_I_092_060And_062x_O_Ax_A_092_060in_062_Avertices_AG_H_A_092_060Longrightarrow_062_Athesis_J_A_092_060Longrightarrow_062_Athesis_092_060close_062,axiom,
~ ! [X2: v] :
~ ( member @ v @ X2 @ ( labeled_vertices @ ( standard_Constant @ v2 ) @ v @ g ) ) ).
% \<open>\<And>thesis. (\<And>x. x \<in> vertices G' \<Longrightarrow> thesis) \<Longrightarrow> thesis\<close>
thf(fact_2__C1_C_I3_J,axiom,
( ( m @ xa )
= ( m @ y ) ) ).
% "1"(3)
thf(fact_3__C1_C_I1_J,axiom,
member @ v2 @ xa @ c ).
% "1"(1)
thf(fact_4__C1_C_I2_J,axiom,
member @ v2 @ y @ c ).
% "1"(2)
thf(fact_5__092_060open_062_Im_Ax_M_Am_Ax_J_A_092_060in_062_AgetRel_A_IS__Const_Ax_J_A_Imap__graph__fn_AG_H_Ah_J_092_060close_062,axiom,
member @ ( product_prod @ v @ v ) @ ( product_Pair @ v @ v @ ( m @ xa ) @ ( m @ xa ) ) @ ( getRel @ ( standard_Constant @ v2 ) @ v @ ( standard_S_Const @ v2 @ xa ) @ ( map_graph @ v @ v @ ( standard_Constant @ v2 ) @ ( bNF_Gr @ v @ v @ ( labeled_vertices @ ( standard_Constant @ v2 ) @ v @ g ) @ h ) @ g ) ) ).
% \<open>(m x, m x) \<in> getRel (S_Const x) (map_graph_fn G' h)\<close>
thf(fact_6__092_060open_062_Im_Ax_M_Am_Ax_J_A_092_060in_062_AgetRel_A_IS__Const_Ay_J_A_Imap__graph__fn_AG_H_Ah_J_092_060close_062,axiom,
member @ ( product_prod @ v @ v ) @ ( product_Pair @ v @ v @ ( m @ xa ) @ ( m @ xa ) ) @ ( getRel @ ( standard_Constant @ v2 ) @ v @ ( standard_S_Const @ v2 @ y ) @ ( map_graph @ v @ v @ ( standard_Constant @ v2 ) @ ( bNF_Gr @ v @ v @ ( labeled_vertices @ ( standard_Constant @ v2 ) @ v @ g ) @ h ) @ g ) ) ).
% \<open>(m x, m x) \<in> getRel (S_Const y) (map_graph_fn G' h)\<close>
thf(fact_7_gr_I1_J,axiom,
( g
= ( restrict @ ( standard_Constant @ v2 ) @ v @ g ) ) ).
% gr(1)
thf(fact_8_h_I3_J,axiom,
graph_homomorphism @ ( standard_Constant @ v2 ) @ v @ v @ ( map_graph @ v @ v @ ( standard_Constant @ v2 ) @ ( bNF_Gr @ v @ v @ ( labeled_vertices @ ( standard_Constant @ v2 ) @ v @ g ) @ h ) @ g ) @ g @ ( id_on @ v @ ( labeled_vertices @ ( standard_Constant @ v2 ) @ v @ ( map_graph @ v @ v @ ( standard_Constant @ v2 ) @ ( bNF_Gr @ v @ v @ ( labeled_vertices @ ( standard_Constant @ v2 ) @ v @ g ) @ h ) @ g ) ) ) ).
% h(3)
thf(fact_9_Standard__Constant_Oinject,axiom,
! [V: $tType,X4: V,Y4: V] :
( ( ( standard_S_Const @ V @ X4 )
= ( standard_S_Const @ V @ Y4 ) )
= ( X4 = Y4 ) ) ).
% Standard_Constant.inject
thf(fact_10_prod_Oinject,axiom,
! [A: $tType,B: $tType,X1: A,X22: B,Y1: A,Y2: B] :
( ( ( product_Pair @ A @ B @ X1 @ X22 )
= ( product_Pair @ A @ B @ Y1 @ Y2 ) )
= ( ( X1 = Y1 )
& ( X22 = Y2 ) ) ) ).
% prod.inject
thf(fact_11_old_Oprod_Oinject,axiom,
! [A: $tType,B: $tType,A2: A,B2: B,A3: A,B3: B] :
( ( ( product_Pair @ A @ B @ A2 @ B2 )
= ( product_Pair @ A @ B @ A3 @ B3 ) )
= ( ( A2 = A3 )
& ( B2 = B3 ) ) ) ).
% old.prod.inject
thf(fact_12_m,axiom,
! [X3: v2] :
( ( member @ v2 @ X3 @ c )
=> ( member @ ( product_prod @ v @ v ) @ ( product_Pair @ v @ v @ ( m @ X3 ) @ ( m @ X3 ) ) @ ( getRel @ ( standard_Constant @ v2 ) @ v @ ( standard_S_Const @ v2 @ X3 ) @ ( map_graph @ v @ v @ ( standard_Constant @ v2 ) @ ( bNF_Gr @ v @ v @ ( labeled_vertices @ ( standard_Constant @ v2 ) @ v @ g ) @ h ) @ g ) ) ) ) ).
% m
thf(fact_13_top__nonempty,axiom,
member @ ( product_prod @ v @ v ) @ ( product_Pair @ v @ v @ x @ x ) @ ( getRel @ ( standard_Constant @ v2 ) @ v @ ( standard_S_Top @ v2 ) @ g ) ).
% top_nonempty
thf(fact_14_cf,axiom,
( ( getRel @ ( standard_Constant @ v2 ) @ v @ ( standard_S_Bot @ v2 ) @ g )
= ( bot_bot @ ( set @ ( product_prod @ v @ v ) ) ) ) ).
% cf
thf(fact_15__092_060open_062_092_060And_062c_O_Ac_A_092_060in_062_AC_A_092_060Longrightarrow_062_A_092_060exists_062v_O_A_Iv_M_Av_J_A_092_060in_062_AgetRel_A_IS__Const_Ac_J_A_Imap__graph__fn_AG_H_Ah_J_092_060close_062,axiom,
! [C2: v2] :
( ( member @ v2 @ C2 @ c )
=> ? [V2: v] : ( member @ ( product_prod @ v @ v ) @ ( product_Pair @ v @ v @ V2 @ V2 ) @ ( getRel @ ( standard_Constant @ v2 ) @ v @ ( standard_S_Const @ v2 @ C2 ) @ ( map_graph @ v @ v @ ( standard_Constant @ v2 ) @ ( bNF_Gr @ v @ v @ ( labeled_vertices @ ( standard_Constant @ v2 ) @ v @ g ) @ h ) @ g ) ) ) ) ).
% \<open>\<And>c. c \<in> C \<Longrightarrow> \<exists>v. (v, v) \<in> getRel (S_Const c) (map_graph_fn G' h)\<close>
thf(fact_16__092_060open_062_092_060forall_062c_O_A_092_060exists_062v_O_Ac_A_092_060in_062_AC_A_092_060longrightarrow_062_A_Iv_M_Av_J_A_092_060in_062_AgetRel_A_IS__Const_Ac_J_A_Imap__graph__fn_AG_H_Ah_J_092_060close_062,axiom,
! [C3: v2] :
? [V2: v] :
( ( member @ v2 @ C3 @ c )
=> ( member @ ( product_prod @ v @ v ) @ ( product_Pair @ v @ v @ V2 @ V2 ) @ ( getRel @ ( standard_Constant @ v2 ) @ v @ ( standard_S_Const @ v2 @ C3 ) @ ( map_graph @ v @ v @ ( standard_Constant @ v2 ) @ ( bNF_Gr @ v @ v @ ( labeled_vertices @ ( standard_Constant @ v2 ) @ v @ g ) @ h ) @ g ) ) ) ) ).
% \<open>\<forall>c. \<exists>v. c \<in> C \<longrightarrow> (v, v) \<in> getRel (S_Const c) (map_graph_fn G' h)\<close>
thf(fact_17__092_060open_062_092_060exists_062f_O_A_092_060forall_062x_O_Ax_A_092_060in_062_AC_A_092_060longrightarrow_062_A_If_Ax_M_Af_Ax_J_A_092_060in_062_AgetRel_A_IS__Const_Ax_J_A_Imap__graph__fn_AG_H_Ah_J_092_060close_062,axiom,
? [F: v2 > v] :
! [X5: v2] :
( ( member @ v2 @ X5 @ c )
=> ( member @ ( product_prod @ v @ v ) @ ( product_Pair @ v @ v @ ( F @ X5 ) @ ( F @ X5 ) ) @ ( getRel @ ( standard_Constant @ v2 ) @ v @ ( standard_S_Const @ v2 @ X5 ) @ ( map_graph @ v @ v @ ( standard_Constant @ v2 ) @ ( bNF_Gr @ v @ v @ ( labeled_vertices @ ( standard_Constant @ v2 ) @ v @ g ) @ h ) @ g ) ) ) ) ).
% \<open>\<exists>f. \<forall>x. x \<in> C \<longrightarrow> (f x, f x) \<in> getRel (S_Const x) (map_graph_fn G' h)\<close>
thf(fact_18__092_060open_062_092_060And_062thesis_O_A_I_092_060And_062m_O_A_I_092_060And_062x_O_Ax_A_092_060in_062_AC_A_092_060Longrightarrow_062_A_Im_Ax_M_Am_Ax_J_A_092_060in_062_AgetRel_A_IS__Const_Ax_J_A_Imap__graph__fn_AG_H_Ah_J_J_A_092_060Longrightarrow_062_Athesis_J_A_092_060Longrightarrow_062_Athesis_092_060close_062,axiom,
~ ! [M: v2 > v] :
~ ! [X5: v2] :
( ( member @ v2 @ X5 @ c )
=> ( member @ ( product_prod @ v @ v ) @ ( product_Pair @ v @ v @ ( M @ X5 ) @ ( M @ X5 ) ) @ ( getRel @ ( standard_Constant @ v2 ) @ v @ ( standard_S_Const @ v2 @ X5 ) @ ( map_graph @ v @ v @ ( standard_Constant @ v2 ) @ ( bNF_Gr @ v @ v @ ( labeled_vertices @ ( standard_Constant @ v2 ) @ v @ g ) @ h ) @ g ) ) ) ) ).
% \<open>\<And>thesis. (\<And>m. (\<And>x. x \<in> C \<Longrightarrow> (m x, m x) \<in> getRel (S_Const x) (map_graph_fn G' h)) \<Longrightarrow> thesis) \<Longrightarrow> thesis\<close>
thf(fact_19_getRel__homR,axiom,
! [C: $tType,A: $tType,B: $tType,Y: A,Z: A,L2: B,G: labeled_graph @ B @ A,U: C,F2: set @ ( product_prod @ A @ C ),V3: C] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Y @ Z ) @ ( getRel @ B @ A @ L2 @ G ) )
=> ( ( member @ ( product_prod @ A @ C ) @ ( product_Pair @ A @ C @ Y @ U ) @ F2 )
=> ( ( member @ ( product_prod @ A @ C ) @ ( product_Pair @ A @ C @ Z @ V3 ) @ F2 )
=> ( member @ ( product_prod @ C @ C ) @ ( product_Pair @ C @ C @ U @ V3 ) @ ( getRel @ B @ C @ L2 @ ( map_graph @ A @ C @ B @ F2 @ G ) ) ) ) ) ) ).
% getRel_homR
thf(fact_20_surj__pair,axiom,
! [A: $tType,B: $tType,P: product_prod @ A @ B] :
? [X2: A,Y3: B] :
( P
= ( product_Pair @ A @ B @ X2 @ Y3 ) ) ).
% surj_pair
thf(fact_21_h_I1_J,axiom,
( ( comp @ v @ v @ v @ h @ h )
= h ) ).
% h(1)
thf(fact_22_getRel__map__fn,axiom,
! [C: $tType,A: $tType,B: $tType,A22: A,G: labeled_graph @ B @ A,B22: A,L2: B,F2: A > C,A2: C,B2: C] :
( ( member @ A @ A22 @ ( labeled_vertices @ B @ A @ G ) )
=> ( ( member @ A @ B22 @ ( labeled_vertices @ B @ A @ G ) )
=> ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A22 @ B22 ) @ ( getRel @ B @ A @ L2 @ G ) )
=> ( ( ( F2 @ A22 )
= A2 )
=> ( ( ( F2 @ B22 )
= B2 )
=> ( member @ ( product_prod @ C @ C ) @ ( product_Pair @ C @ C @ A2 @ B2 ) @ ( getRel @ B @ C @ L2 @ ( map_graph @ A @ C @ B @ ( bNF_Gr @ A @ C @ ( labeled_vertices @ B @ A @ G ) @ F2 ) @ G ) ) ) ) ) ) ) ) ).
% getRel_map_fn
thf(fact_23_getRel__hom,axiom,
! [C: $tType,A: $tType,B: $tType,Y: A,Z: A,L2: B,G: labeled_graph @ B @ A,F2: A > C] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Y @ Z ) @ ( getRel @ B @ A @ L2 @ G ) )
=> ( ( member @ A @ Y @ ( labeled_vertices @ B @ A @ G ) )
=> ( ( member @ A @ Z @ ( labeled_vertices @ B @ A @ G ) )
=> ( member @ ( product_prod @ C @ C ) @ ( product_Pair @ C @ C @ ( F2 @ Y ) @ ( F2 @ Z ) ) @ ( getRel @ B @ C @ L2 @ ( map_graph @ A @ C @ B @ ( bNF_Gr @ A @ C @ ( labeled_vertices @ B @ A @ G ) @ F2 ) @ G ) ) ) ) ) ) ).
% getRel_hom
thf(fact_24_Standard__Constant_Odistinct_I1_J,axiom,
! [V: $tType] :
( ( standard_S_Top @ V )
!= ( standard_S_Bot @ V ) ) ).
% Standard_Constant.distinct(1)
thf(fact_25_Standard__Constant_Odistinct_I5_J,axiom,
! [V: $tType,X4: V] :
( ( standard_S_Top @ V )
!= ( standard_S_Const @ V @ X4 ) ) ).
% Standard_Constant.distinct(5)
thf(fact_26_Standard__Constant_Odistinct_I9_J,axiom,
! [V: $tType,X4: V] :
( ( standard_S_Bot @ V )
!= ( standard_S_Const @ V @ X4 ) ) ).
% Standard_Constant.distinct(9)
thf(fact_27_getRel__dom_I1_J,axiom,
! [B: $tType,A: $tType,G: labeled_graph @ A @ B,A2: B,B2: B,L2: A] :
( ( G
= ( restrict @ A @ B @ G ) )
=> ( ( member @ ( product_prod @ B @ B ) @ ( product_Pair @ B @ B @ A2 @ B2 ) @ ( getRel @ A @ B @ L2 @ G ) )
=> ( member @ B @ A2 @ ( labeled_vertices @ A @ B @ G ) ) ) ) ).
% getRel_dom(1)
thf(fact_28_getRel__dom_I2_J,axiom,
! [B: $tType,A: $tType,G: labeled_graph @ A @ B,A2: B,B2: B,L2: A] :
( ( G
= ( restrict @ A @ B @ G ) )
=> ( ( member @ ( product_prod @ B @ B ) @ ( product_Pair @ B @ B @ A2 @ B2 ) @ ( getRel @ A @ B @ L2 @ G ) )
=> ( member @ B @ B2 @ ( labeled_vertices @ A @ B @ G ) ) ) ) ).
% getRel_dom(2)
thf(fact_29_getRel__subgraph,axiom,
! [A: $tType,B: $tType,Y: A,Z: A,L2: B,G: labeled_graph @ B @ A,G2: labeled_graph @ B @ A] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Y @ Z ) @ ( getRel @ B @ A @ L2 @ G ) )
=> ( ( graph_homomorphism @ B @ A @ A @ G @ G2 @ ( id_on @ A @ ( labeled_vertices @ B @ A @ G ) ) )
=> ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Y @ Z ) @ ( getRel @ B @ A @ L2 @ G2 ) ) ) ) ).
% getRel_subgraph
thf(fact_30_old_Oprod_Oinducts,axiom,
! [B: $tType,A: $tType,P2: ( product_prod @ A @ B ) > $o,Prod: product_prod @ A @ B] :
( ! [A4: A,B4: B] : ( P2 @ ( product_Pair @ A @ B @ A4 @ B4 ) )
=> ( P2 @ Prod ) ) ).
% old.prod.inducts
thf(fact_31_old_Oprod_Oexhaust,axiom,
! [A: $tType,B: $tType,Y: product_prod @ A @ B] :
~ ! [A4: A,B4: B] :
( Y
!= ( product_Pair @ A @ B @ A4 @ B4 ) ) ).
% old.prod.exhaust
thf(fact_32_prod__induct7,axiom,
! [G3: $tType,F3: $tType,E: $tType,D: $tType,C: $tType,B: $tType,A: $tType,P2: ( product_prod @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D @ ( product_prod @ E @ ( product_prod @ F3 @ G3 ) ) ) ) ) ) > $o,X6: product_prod @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D @ ( product_prod @ E @ ( product_prod @ F3 @ G3 ) ) ) ) )] :
( ! [A4: A,B4: B,C4: C,D2: D,E2: E,F: F3,G4: G3] : ( P2 @ ( product_Pair @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D @ ( product_prod @ E @ ( product_prod @ F3 @ G3 ) ) ) ) ) @ A4 @ ( product_Pair @ B @ ( product_prod @ C @ ( product_prod @ D @ ( product_prod @ E @ ( product_prod @ F3 @ G3 ) ) ) ) @ B4 @ ( product_Pair @ C @ ( product_prod @ D @ ( product_prod @ E @ ( product_prod @ F3 @ G3 ) ) ) @ C4 @ ( product_Pair @ D @ ( product_prod @ E @ ( product_prod @ F3 @ G3 ) ) @ D2 @ ( product_Pair @ E @ ( product_prod @ F3 @ G3 ) @ E2 @ ( product_Pair @ F3 @ G3 @ F @ G4 ) ) ) ) ) ) )
=> ( P2 @ X6 ) ) ).
% prod_induct7
thf(fact_33_prod__induct6,axiom,
! [F3: $tType,E: $tType,D: $tType,C: $tType,B: $tType,A: $tType,P2: ( product_prod @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D @ ( product_prod @ E @ F3 ) ) ) ) ) > $o,X6: product_prod @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D @ ( product_prod @ E @ F3 ) ) ) )] :
( ! [A4: A,B4: B,C4: C,D2: D,E2: E,F: F3] : ( P2 @ ( product_Pair @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D @ ( product_prod @ E @ F3 ) ) ) ) @ A4 @ ( product_Pair @ B @ ( product_prod @ C @ ( product_prod @ D @ ( product_prod @ E @ F3 ) ) ) @ B4 @ ( product_Pair @ C @ ( product_prod @ D @ ( product_prod @ E @ F3 ) ) @ C4 @ ( product_Pair @ D @ ( product_prod @ E @ F3 ) @ D2 @ ( product_Pair @ E @ F3 @ E2 @ F ) ) ) ) ) )
=> ( P2 @ X6 ) ) ).
% prod_induct6
thf(fact_34_prod__induct5,axiom,
! [E: $tType,D: $tType,C: $tType,B: $tType,A: $tType,P2: ( product_prod @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D @ E ) ) ) ) > $o,X6: product_prod @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D @ E ) ) )] :
( ! [A4: A,B4: B,C4: C,D2: D,E2: E] : ( P2 @ ( product_Pair @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D @ E ) ) ) @ A4 @ ( product_Pair @ B @ ( product_prod @ C @ ( product_prod @ D @ E ) ) @ B4 @ ( product_Pair @ C @ ( product_prod @ D @ E ) @ C4 @ ( product_Pair @ D @ E @ D2 @ E2 ) ) ) ) )
=> ( P2 @ X6 ) ) ).
% prod_induct5
thf(fact_35_prod__induct4,axiom,
! [D: $tType,C: $tType,B: $tType,A: $tType,P2: ( product_prod @ A @ ( product_prod @ B @ ( product_prod @ C @ D ) ) ) > $o,X6: product_prod @ A @ ( product_prod @ B @ ( product_prod @ C @ D ) )] :
( ! [A4: A,B4: B,C4: C,D2: D] : ( P2 @ ( product_Pair @ A @ ( product_prod @ B @ ( product_prod @ C @ D ) ) @ A4 @ ( product_Pair @ B @ ( product_prod @ C @ D ) @ B4 @ ( product_Pair @ C @ D @ C4 @ D2 ) ) ) )
=> ( P2 @ X6 ) ) ).
% prod_induct4
thf(fact_36_prod__induct3,axiom,
! [C: $tType,B: $tType,A: $tType,P2: ( product_prod @ A @ ( product_prod @ B @ C ) ) > $o,X6: product_prod @ A @ ( product_prod @ B @ C )] :
( ! [A4: A,B4: B,C4: C] : ( P2 @ ( product_Pair @ A @ ( product_prod @ B @ C ) @ A4 @ ( product_Pair @ B @ C @ B4 @ C4 ) ) )
=> ( P2 @ X6 ) ) ).
% prod_induct3
thf(fact_37_prod__cases7,axiom,
! [A: $tType,B: $tType,C: $tType,D: $tType,E: $tType,F3: $tType,G3: $tType,Y: product_prod @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D @ ( product_prod @ E @ ( product_prod @ F3 @ G3 ) ) ) ) )] :
~ ! [A4: A,B4: B,C4: C,D2: D,E2: E,F: F3,G4: G3] :
( Y
!= ( product_Pair @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D @ ( product_prod @ E @ ( product_prod @ F3 @ G3 ) ) ) ) ) @ A4 @ ( product_Pair @ B @ ( product_prod @ C @ ( product_prod @ D @ ( product_prod @ E @ ( product_prod @ F3 @ G3 ) ) ) ) @ B4 @ ( product_Pair @ C @ ( product_prod @ D @ ( product_prod @ E @ ( product_prod @ F3 @ G3 ) ) ) @ C4 @ ( product_Pair @ D @ ( product_prod @ E @ ( product_prod @ F3 @ G3 ) ) @ D2 @ ( product_Pair @ E @ ( product_prod @ F3 @ G3 ) @ E2 @ ( product_Pair @ F3 @ G3 @ F @ G4 ) ) ) ) ) ) ) ).
% prod_cases7
thf(fact_38_prod__cases6,axiom,
! [A: $tType,B: $tType,C: $tType,D: $tType,E: $tType,F3: $tType,Y: product_prod @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D @ ( product_prod @ E @ F3 ) ) ) )] :
~ ! [A4: A,B4: B,C4: C,D2: D,E2: E,F: F3] :
( Y
!= ( product_Pair @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D @ ( product_prod @ E @ F3 ) ) ) ) @ A4 @ ( product_Pair @ B @ ( product_prod @ C @ ( product_prod @ D @ ( product_prod @ E @ F3 ) ) ) @ B4 @ ( product_Pair @ C @ ( product_prod @ D @ ( product_prod @ E @ F3 ) ) @ C4 @ ( product_Pair @ D @ ( product_prod @ E @ F3 ) @ D2 @ ( product_Pair @ E @ F3 @ E2 @ F ) ) ) ) ) ) ).
% prod_cases6
thf(fact_39_prod__cases5,axiom,
! [A: $tType,B: $tType,C: $tType,D: $tType,E: $tType,Y: product_prod @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D @ E ) ) )] :
~ ! [A4: A,B4: B,C4: C,D2: D,E2: E] :
( Y
!= ( product_Pair @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D @ E ) ) ) @ A4 @ ( product_Pair @ B @ ( product_prod @ C @ ( product_prod @ D @ E ) ) @ B4 @ ( product_Pair @ C @ ( product_prod @ D @ E ) @ C4 @ ( product_Pair @ D @ E @ D2 @ E2 ) ) ) ) ) ).
% prod_cases5
thf(fact_40_prod__cases4,axiom,
! [A: $tType,B: $tType,C: $tType,D: $tType,Y: product_prod @ A @ ( product_prod @ B @ ( product_prod @ C @ D ) )] :
~ ! [A4: A,B4: B,C4: C,D2: D] :
( Y
!= ( product_Pair @ A @ ( product_prod @ B @ ( product_prod @ C @ D ) ) @ A4 @ ( product_Pair @ B @ ( product_prod @ C @ D ) @ B4 @ ( product_Pair @ C @ D @ C4 @ D2 ) ) ) ) ).
% prod_cases4
thf(fact_41_prod__cases3,axiom,
! [A: $tType,B: $tType,C: $tType,Y: product_prod @ A @ ( product_prod @ B @ C )] :
~ ! [A4: A,B4: B,C4: C] :
( Y
!= ( product_Pair @ A @ ( product_prod @ B @ C ) @ A4 @ ( product_Pair @ B @ C @ B4 @ C4 ) ) ) ).
% prod_cases3
thf(fact_42_Pair__inject,axiom,
! [A: $tType,B: $tType,A2: A,B2: B,A3: A,B3: B] :
( ( ( product_Pair @ A @ B @ A2 @ B2 )
= ( product_Pair @ A @ B @ A3 @ B3 ) )
=> ~ ( ( A2 = A3 )
=> ( B2 != B3 ) ) ) ).
% Pair_inject
thf(fact_43_prod__cases,axiom,
! [B: $tType,A: $tType,P2: ( product_prod @ A @ B ) > $o,P: product_prod @ A @ B] :
( ! [A4: A,B4: B] : ( P2 @ ( product_Pair @ A @ B @ A4 @ B4 ) )
=> ( P2 @ P ) ) ).
% prod_cases
thf(fact_44_mg,axiom,
! [B5: $tType,R: product_prod @ ( labeled_graph @ ( standard_Constant @ v2 ) @ B5 ) @ ( labeled_graph @ ( standard_Constant @ v2 ) @ B5 )] :
( ( maintained @ ( standard_Constant @ v2 ) @ B5 @ v @ R @ g )
=> ( maintained @ ( standard_Constant @ v2 ) @ B5 @ v @ R @ ( map_graph @ v @ v @ ( standard_Constant @ v2 ) @ ( bNF_Gr @ v @ v @ ( labeled_vertices @ ( standard_Constant @ v2 ) @ v @ g ) @ h ) @ g ) ) ) ).
% mg
thf(fact_45_mem__Collect__eq,axiom,
! [A: $tType,A2: A,P2: A > $o] :
( ( member @ A @ A2 @ ( collect @ A @ P2 ) )
= ( P2 @ A2 ) ) ).
% mem_Collect_eq
thf(fact_46_Collect__mem__eq,axiom,
! [A: $tType,A5: set @ A] :
( ( collect @ A
@ ^ [X7: A] : ( member @ A @ X7 @ A5 ) )
= A5 ) ).
% Collect_mem_eq
thf(fact_47_Collect__cong,axiom,
! [A: $tType,P2: A > $o,Q: A > $o] :
( ! [X2: A] :
( ( P2 @ X2 )
= ( Q @ X2 ) )
=> ( ( collect @ A @ P2 )
= ( collect @ A @ Q ) ) ) ).
% Collect_cong
thf(fact_48_ext,axiom,
! [B: $tType,A: $tType,F2: A > B,G5: A > B] :
( ! [X2: A] :
( ( F2 @ X2 )
= ( G5 @ X2 ) )
=> ( F2 = G5 ) ) ).
% ext
thf(fact_49_graph__homo,axiom,
! [A: $tType,C: $tType,B: $tType,G: labeled_graph @ A @ B,F2: B > C] :
( ( G
= ( restrict @ A @ B @ G ) )
=> ( graph_homomorphism @ A @ B @ C @ G @ ( map_graph @ B @ C @ A @ ( bNF_Gr @ B @ C @ ( labeled_vertices @ A @ B @ G ) @ F2 ) @ G ) @ ( bNF_Gr @ B @ C @ ( labeled_vertices @ A @ B @ G ) @ F2 ) ) ) ).
% graph_homo
thf(fact_50_map__graph__fn__id_I2_J,axiom,
! [B: $tType,A: $tType,X8: labeled_graph @ A @ B] :
( ( map_graph @ B @ B @ A @ ( id_on @ B @ ( labeled_vertices @ A @ B @ X8 ) ) @ X8 )
= ( restrict @ A @ B @ X8 ) ) ).
% map_graph_fn_id(2)
thf(fact_51_map__graph__fn__graphI,axiom,
! [B: $tType,C: $tType,A: $tType,G: labeled_graph @ A @ C,F2: C > B] :
( ( map_graph @ C @ B @ A @ ( bNF_Gr @ C @ B @ ( labeled_vertices @ A @ C @ G ) @ F2 ) @ G )
= ( restrict @ A @ B @ ( map_graph @ C @ B @ A @ ( bNF_Gr @ C @ B @ ( labeled_vertices @ A @ C @ G ) @ F2 ) @ G ) ) ) ).
% map_graph_fn_graphI
thf(fact_52_map__graph__preserves__subgraph,axiom,
! [C: $tType,B: $tType,A: $tType,A5: labeled_graph @ A @ B,B6: labeled_graph @ A @ B,F2: set @ ( product_prod @ B @ C )] :
( ( graph_homomorphism @ A @ B @ B @ A5 @ B6 @ ( id_on @ B @ ( labeled_vertices @ A @ B @ A5 ) ) )
=> ( graph_homomorphism @ A @ C @ C @ ( map_graph @ B @ C @ A @ F2 @ A5 ) @ ( map_graph @ B @ C @ A @ F2 @ B6 ) @ ( id_on @ C @ ( labeled_vertices @ A @ C @ ( map_graph @ B @ C @ A @ F2 @ A5 ) ) ) ) ) ).
% map_graph_preserves_subgraph
thf(fact_53_subgraph__refl,axiom,
! [B: $tType,A: $tType,G: labeled_graph @ A @ B] :
( ( graph_homomorphism @ A @ B @ B @ G @ G @ ( id_on @ B @ ( labeled_vertices @ A @ B @ G ) ) )
= ( G
= ( restrict @ A @ B @ G ) ) ) ).
% subgraph_refl
thf(fact_54_subgraph__restrict,axiom,
! [B: $tType,A: $tType,G: labeled_graph @ A @ B] :
( ( graph_homomorphism @ A @ B @ B @ G @ ( restrict @ A @ B @ G ) @ ( id_on @ B @ ( labeled_vertices @ A @ B @ G ) ) )
= ( G
= ( restrict @ A @ B @ G ) ) ) ).
% subgraph_restrict
thf(fact_55_graph__homomorphism__Id,axiom,
! [B: $tType,A: $tType,A2: labeled_graph @ A @ B] : ( graph_homomorphism @ A @ B @ B @ ( restrict @ A @ B @ A2 ) @ ( restrict @ A @ B @ A2 ) @ ( id_on @ B @ ( labeled_vertices @ A @ B @ A2 ) ) ) ).
% graph_homomorphism_Id
thf(fact_56_Id__on__empty,axiom,
! [A: $tType] :
( ( id_on @ A @ ( bot_bot @ ( set @ A ) ) )
= ( bot_bot @ ( set @ ( product_prod @ A @ A ) ) ) ) ).
% Id_on_empty
thf(fact_57_Gr__empty,axiom,
! [B: $tType,A: $tType,F2: A > B] :
( ( bNF_Gr @ A @ B @ ( bot_bot @ ( set @ A ) ) @ F2 )
= ( bot_bot @ ( set @ ( product_prod @ A @ B ) ) ) ) ).
% Gr_empty
thf(fact_58_restrict__idemp,axiom,
! [B: $tType,A: $tType,X6: labeled_graph @ A @ B] :
( ( restrict @ A @ B @ ( restrict @ A @ B @ X6 ) )
= ( restrict @ A @ B @ X6 ) ) ).
% restrict_idemp
thf(fact_59_in__Gr,axiom,
! [A: $tType,B: $tType,X6: A,Y: B,A5: set @ A,F2: A > B] :
( ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X6 @ Y ) @ ( bNF_Gr @ A @ B @ A5 @ F2 ) )
= ( ( member @ A @ X6 @ A5 )
& ( ( F2 @ X6 )
= Y ) ) ) ).
% in_Gr
thf(fact_60_Gr__not__in,axiom,
! [B: $tType,A: $tType,X6: A,F4: set @ A,F2: A > B,Y: B] :
( ( ~ ( member @ A @ X6 @ F4 )
| ( ( F2 @ X6 )
!= Y ) )
=> ~ ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X6 @ Y ) @ ( bNF_Gr @ A @ B @ F4 @ F2 ) ) ) ).
% Gr_not_in
thf(fact_61_Id__onI,axiom,
! [A: $tType,A2: A,A5: set @ A] :
( ( member @ A @ A2 @ A5 )
=> ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A2 @ A2 ) @ ( id_on @ A @ A5 ) ) ) ).
% Id_onI
thf(fact_62_vertices__restrict,axiom,
! [A: $tType,B: $tType,G: labeled_graph @ B @ A] :
( ( labeled_vertices @ B @ A @ ( restrict @ B @ A @ G ) )
= ( labeled_vertices @ B @ A @ G ) ) ).
% vertices_restrict
thf(fact_63_map__graph__preserves__restricted,axiom,
! [C: $tType,B: $tType,A: $tType,G: labeled_graph @ A @ B,F2: set @ ( product_prod @ B @ C )] :
( ( G
= ( restrict @ A @ B @ G ) )
=> ( ( map_graph @ B @ C @ A @ F2 @ G )
= ( restrict @ A @ C @ ( map_graph @ B @ C @ A @ F2 @ G ) ) ) ) ).
% map_graph_preserves_restricted
thf(fact_64_map__graph__fn__comp,axiom,
! [B: $tType,D: $tType,C: $tType,A: $tType,G: labeled_graph @ A @ C,F2: D > B,G5: C > D] :
( ( map_graph @ C @ B @ A @ ( bNF_Gr @ C @ B @ ( labeled_vertices @ A @ C @ G ) @ ( comp @ D @ B @ C @ F2 @ G5 ) ) @ G )
= ( map_graph @ D @ B @ A @ ( bNF_Gr @ D @ B @ ( labeled_vertices @ A @ D @ ( map_graph @ C @ D @ A @ ( bNF_Gr @ C @ D @ ( labeled_vertices @ A @ C @ G ) @ G5 ) @ G ) ) @ F2 ) @ ( map_graph @ C @ D @ A @ ( bNF_Gr @ C @ D @ ( labeled_vertices @ A @ C @ G ) @ G5 ) @ G ) ) ) ).
% map_graph_fn_comp
thf(fact_65_assms_I1_J,axiom,
! [X5: product_prod @ ( labeled_graph @ ( standard_Constant @ v2 ) @ nat ) @ ( labeled_graph @ ( standard_Constant @ v2 ) @ nat )] :
( ( member @ ( product_prod @ ( labeled_graph @ ( standard_Constant @ v2 ) @ nat ) @ ( labeled_graph @ ( standard_Constant @ v2 ) @ nat ) ) @ X5 @ ( standa438229444_rules @ v2 @ c @ l ) )
=> ( maintained @ ( standard_Constant @ v2 ) @ nat @ v @ X5 @ g ) ) ).
% assms(1)
thf(fact_66_ne,axiom,
maintained @ ( standard_Constant @ v2 ) @ nat @ v @ ( standa244753842y_rule @ ( standard_Constant @ v2 ) ) @ g ).
% ne
thf(fact_67__092_060open_062maintained_Anonempty__rule_AG_H_A_061_A_Ivertices_AG_H_A_092_060noteq_062_A_123_125_J_092_060close_062,axiom,
( ( maintained @ ( standard_Constant @ v2 ) @ nat @ v @ ( standa244753842y_rule @ ( standard_Constant @ v2 ) ) @ g )
= ( ( labeled_vertices @ ( standard_Constant @ v2 ) @ v @ g )
!= ( bot_bot @ ( set @ v ) ) ) ) ).
% \<open>maintained nonempty_rule G' = (vertices G' \<noteq> {})\<close>
thf(fact_68_maintained__preserved__by__isomorphism,axiom,
! [C: $tType,D: $tType,A: $tType,B: $tType,G: labeled_graph @ B @ A,F2: C > A,G5: A > C,R: product_prod @ ( labeled_graph @ B @ D ) @ ( labeled_graph @ B @ D )] :
( ! [X2: A] :
( ( member @ A @ X2 @ ( labeled_vertices @ B @ A @ G ) )
=> ( ( comp @ C @ A @ A @ F2 @ G5 @ X2 )
= X2 ) )
=> ( ( G
= ( restrict @ B @ A @ G ) )
=> ( ( maintained @ B @ D @ C @ R @ ( map_graph @ A @ C @ B @ ( bNF_Gr @ A @ C @ ( labeled_vertices @ B @ A @ G ) @ G5 ) @ G ) )
=> ( maintained @ B @ D @ A @ R @ G ) ) ) ) ).
% maintained_preserved_by_isomorphism
thf(fact_69_idemp__embedding__maintained__preserved,axiom,
! [C: $tType,B: $tType,A: $tType,G: labeled_graph @ A @ B,F2: B > B,R: product_prod @ ( labeled_graph @ A @ C ) @ ( labeled_graph @ A @ C )] :
( ( graph_homomorphism @ A @ B @ B @ ( map_graph @ B @ B @ A @ ( bNF_Gr @ B @ B @ ( labeled_vertices @ A @ B @ G ) @ F2 ) @ G ) @ G @ ( id_on @ B @ ( labeled_vertices @ A @ B @ ( map_graph @ B @ B @ A @ ( bNF_Gr @ B @ B @ ( labeled_vertices @ A @ B @ G ) @ F2 ) @ G ) ) ) )
=> ( ! [X2: B] :
( ( member @ B @ X2 @ ( labeled_vertices @ A @ B @ G ) )
=> ( ( comp @ B @ B @ B @ F2 @ F2 @ X2 )
= ( F2 @ X2 ) ) )
=> ( ( maintained @ A @ C @ B @ R @ G )
=> ( maintained @ A @ C @ B @ R @ ( map_graph @ B @ B @ A @ ( bNF_Gr @ B @ B @ ( labeled_vertices @ A @ B @ G ) @ F2 ) @ G ) ) ) ) ) ).
% idemp_embedding_maintained_preserved
thf(fact_70_Id__onE,axiom,
! [A: $tType,C5: product_prod @ A @ A,A5: set @ A] :
( ( member @ ( product_prod @ A @ A ) @ C5 @ ( id_on @ A @ A5 ) )
=> ~ ! [X2: A] :
( ( member @ A @ X2 @ A5 )
=> ( C5
!= ( product_Pair @ A @ A @ X2 @ X2 ) ) ) ) ).
% Id_onE
thf(fact_71_Id__on__eqI,axiom,
! [A: $tType,A2: A,B2: A,A5: set @ A] :
( ( A2 = B2 )
=> ( ( member @ A @ A2 @ A5 )
=> ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A2 @ B2 ) @ ( id_on @ A @ A5 ) ) ) ) ).
% Id_on_eqI
thf(fact_72_Id__on__iff,axiom,
! [A: $tType,X6: A,Y: A,A5: set @ A] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X6 @ Y ) @ ( id_on @ A @ A5 ) )
= ( ( X6 = Y )
& ( member @ A @ X6 @ A5 ) ) ) ).
% Id_on_iff
thf(fact_73_subgraph__preserves__hom,axiom,
! [A: $tType,B: $tType,C: $tType,A5: labeled_graph @ A @ B,B6: labeled_graph @ A @ B,X8: labeled_graph @ A @ C,H: set @ ( product_prod @ C @ B )] :
( ( graph_homomorphism @ A @ B @ B @ A5 @ B6 @ ( id_on @ B @ ( labeled_vertices @ A @ B @ A5 ) ) )
=> ( ( graph_homomorphism @ A @ C @ B @ X8 @ A5 @ H )
=> ( graph_homomorphism @ A @ C @ B @ X8 @ B6 @ H ) ) ) ).
% subgraph_preserves_hom
thf(fact_74_subgraph__trans,axiom,
! [B: $tType,A: $tType,G_1: labeled_graph @ A @ B,G_2: labeled_graph @ A @ B,G_3: labeled_graph @ A @ B] :
( ( graph_homomorphism @ A @ B @ B @ G_1 @ G_2 @ ( id_on @ B @ ( labeled_vertices @ A @ B @ G_1 ) ) )
=> ( ( graph_homomorphism @ A @ B @ B @ G_2 @ G_3 @ ( id_on @ B @ ( labeled_vertices @ A @ B @ G_2 ) ) )
=> ( graph_homomorphism @ A @ B @ B @ G_1 @ G_3 @ ( id_on @ B @ ( labeled_vertices @ A @ B @ G_1 ) ) ) ) ) ).
% subgraph_trans
thf(fact_75_map__graph__fn__eqI,axiom,
! [C: $tType,A: $tType,B: $tType,G: labeled_graph @ B @ A,F2: A > C,G5: A > C] :
( ! [X2: A] :
( ( member @ A @ X2 @ ( labeled_vertices @ B @ A @ G ) )
=> ( ( F2 @ X2 )
= ( G5 @ X2 ) ) )
=> ( ( map_graph @ A @ C @ B @ ( bNF_Gr @ A @ C @ ( labeled_vertices @ B @ A @ G ) @ F2 ) @ G )
= ( map_graph @ A @ C @ B @ ( bNF_Gr @ A @ C @ ( labeled_vertices @ B @ A @ G ) @ G5 ) @ G ) ) ) ).
% map_graph_fn_eqI
thf(fact_76_tr,axiom,
maintained @ ( standard_Constant @ v2 ) @ nat @ v @ ( standa400804411p_rule @ ( standard_Constant @ v2 ) @ ( standard_S_Top @ v2 ) ) @ g ).
% tr
thf(fact_77_h_I4_J,axiom,
! [L2: standard_Constant @ v2,X6: v,Y: v] :
( ( member @ ( product_prod @ ( standard_Constant @ v2 ) @ ( product_prod @ v @ v ) ) @ ( product_Pair @ ( standard_Constant @ v2 ) @ ( product_prod @ v @ v ) @ L2 @ ( product_Pair @ v @ v @ X6 @ Y ) ) @ ( labeled_edges @ ( standard_Constant @ v2 ) @ v @ g ) )
= ( member @ ( product_prod @ ( standard_Constant @ v2 ) @ ( product_prod @ v @ v ) ) @ ( product_Pair @ ( standard_Constant @ v2 ) @ ( product_prod @ v @ v ) @ L2 @ ( product_Pair @ v @ v @ ( h @ X6 ) @ ( h @ Y ) ) ) @ ( labeled_edges @ ( standard_Constant @ v2 ) @ v @ g ) ) ) ).
% h(4)
thf(fact_78_old_Oprod_Orec,axiom,
! [A: $tType,T: $tType,B: $tType,F1: A > B > T,A2: A,B2: B] :
( ( product_rec_prod @ A @ B @ T @ F1 @ ( product_Pair @ A @ B @ A2 @ B2 ) )
= ( F1 @ A2 @ B2 ) ) ).
% old.prod.rec
thf(fact_79_maintained__refl,axiom,
! [B: $tType,C: $tType,A: $tType,R2: labeled_graph @ A @ B,G: labeled_graph @ A @ C] : ( maintained @ A @ B @ C @ ( product_Pair @ ( labeled_graph @ A @ B ) @ ( labeled_graph @ A @ B ) @ R2 @ R2 ) @ G ) ).
% maintained_refl
thf(fact_80_comp__apply,axiom,
! [C: $tType,A: $tType,B: $tType] :
( ( comp @ B @ A @ C )
= ( ^ [F5: B > A,G6: C > B,X7: C] : ( F5 @ ( G6 @ X7 ) ) ) ) ).
% comp_apply
thf(fact_81_empty__Collect__eq,axiom,
! [A: $tType,P2: A > $o] :
( ( ( bot_bot @ ( set @ A ) )
= ( collect @ A @ P2 ) )
= ( ! [X7: A] :
~ ( P2 @ X7 ) ) ) ).
% empty_Collect_eq
thf(fact_82_Collect__empty__eq,axiom,
! [A: $tType,P2: A > $o] :
( ( ( collect @ A @ P2 )
= ( bot_bot @ ( set @ A ) ) )
= ( ! [X7: A] :
~ ( P2 @ X7 ) ) ) ).
% Collect_empty_eq
thf(fact_83_empty__iff,axiom,
! [A: $tType,C5: A] :
~ ( member @ A @ C5 @ ( bot_bot @ ( set @ A ) ) ) ).
% empty_iff
thf(fact_84_all__not__in__conv,axiom,
! [A: $tType,A5: set @ A] :
( ( ! [X7: A] :
~ ( member @ A @ X7 @ A5 ) )
= ( A5
= ( bot_bot @ ( set @ A ) ) ) ) ).
% all_not_in_conv
thf(fact_85_nonempty__rule,axiom,
! [A: $tType,B: $tType,G: labeled_graph @ A @ B] :
( ( G
= ( restrict @ A @ B @ G ) )
=> ( ( maintained @ A @ nat @ B @ ( standa244753842y_rule @ A ) @ G )
= ( ( labeled_vertices @ A @ B @ G )
!= ( bot_bot @ ( set @ B ) ) ) ) ) ).
% nonempty_rule
thf(fact_86__092_060open_062maintainedA_A_Iidentity__rules_AL_J_AG_H_092_060close_062,axiom,
! [X5: product_prod @ ( labeled_graph @ ( standard_Constant @ v2 ) @ nat ) @ ( labeled_graph @ ( standard_Constant @ v2 ) @ nat )] :
( ( member @ ( product_prod @ ( labeled_graph @ ( standard_Constant @ v2 ) @ nat ) @ ( labeled_graph @ ( standard_Constant @ v2 ) @ nat ) ) @ X5 @ ( standa2002409347_rules @ v2 @ l ) )
=> ( maintained @ ( standard_Constant @ v2 ) @ nat @ v @ X5 @ g ) ) ).
% \<open>maintainedA (identity_rules L) G'\<close>
thf(fact_87_labeled__graph_Oexpand,axiom,
! [V: $tType,L: $tType,Labeled_graph: labeled_graph @ L @ V,Labeled_graph2: labeled_graph @ L @ V] :
( ( ( ( labeled_edges @ L @ V @ Labeled_graph )
= ( labeled_edges @ L @ V @ Labeled_graph2 ) )
& ( ( labeled_vertices @ L @ V @ Labeled_graph )
= ( labeled_vertices @ L @ V @ Labeled_graph2 ) ) )
=> ( Labeled_graph = Labeled_graph2 ) ) ).
% labeled_graph.expand
thf(fact_88_emptyE,axiom,
! [A: $tType,A2: A] :
~ ( member @ A @ A2 @ ( bot_bot @ ( set @ A ) ) ) ).
% emptyE
thf(fact_89_equals0D,axiom,
! [A: $tType,A5: set @ A,A2: A] :
( ( A5
= ( bot_bot @ ( set @ A ) ) )
=> ~ ( member @ A @ A2 @ A5 ) ) ).
% equals0D
thf(fact_90_equals0I,axiom,
! [A: $tType,A5: set @ A] :
( ! [Y3: A] :
~ ( member @ A @ Y3 @ A5 )
=> ( A5
= ( bot_bot @ ( set @ A ) ) ) ) ).
% equals0I
thf(fact_91_ex__in__conv,axiom,
! [A: $tType,A5: set @ A] :
( ( ? [X7: A] : ( member @ A @ X7 @ A5 ) )
= ( A5
!= ( bot_bot @ ( set @ A ) ) ) ) ).
% ex_in_conv
thf(fact_92_comp__def,axiom,
! [A: $tType,C: $tType,B: $tType] :
( ( comp @ B @ C @ A )
= ( ^ [F5: B > C,G6: A > B,X7: A] : ( F5 @ ( G6 @ X7 ) ) ) ) ).
% comp_def
thf(fact_93_comp__assoc,axiom,
! [B: $tType,D: $tType,C: $tType,A: $tType,F2: D > B,G5: C > D,H: A > C] :
( ( comp @ C @ B @ A @ ( comp @ D @ B @ C @ F2 @ G5 ) @ H )
= ( comp @ D @ B @ A @ F2 @ ( comp @ C @ D @ A @ G5 @ H ) ) ) ).
% comp_assoc
thf(fact_94_comp__eq__dest,axiom,
! [C: $tType,B: $tType,D: $tType,A: $tType,A2: C > B,B2: A > C,C5: D > B,D3: A > D,V3: A] :
( ( ( comp @ C @ B @ A @ A2 @ B2 )
= ( comp @ D @ B @ A @ C5 @ D3 ) )
=> ( ( A2 @ ( B2 @ V3 ) )
= ( C5 @ ( D3 @ V3 ) ) ) ) ).
% comp_eq_dest
thf(fact_95_comp__eq__elim,axiom,
! [C: $tType,B: $tType,D: $tType,A: $tType,A2: C > B,B2: A > C,C5: D > B,D3: A > D] :
( ( ( comp @ C @ B @ A @ A2 @ B2 )
= ( comp @ D @ B @ A @ C5 @ D3 ) )
=> ! [V4: A] :
( ( A2 @ ( B2 @ V4 ) )
= ( C5 @ ( D3 @ V4 ) ) ) ) ).
% comp_eq_elim
thf(fact_96_comp__eq__dest__lhs,axiom,
! [C: $tType,B: $tType,A: $tType,A2: C > B,B2: A > C,C5: A > B,V3: A] :
( ( ( comp @ C @ B @ A @ A2 @ B2 )
= C5 )
=> ( ( A2 @ ( B2 @ V3 ) )
= ( C5 @ V3 ) ) ) ).
% comp_eq_dest_lhs
thf(fact_97_edge__preserving__on__graphI,axiom,
! [B: $tType,C: $tType,A: $tType,X8: labeled_graph @ A @ B,F2: B > C,Y5: set @ ( product_prod @ A @ ( product_prod @ C @ C ) )] :
( ! [L3: A,X2: B,Y3: B] :
( ( member @ ( product_prod @ A @ ( product_prod @ B @ B ) ) @ ( product_Pair @ A @ ( product_prod @ B @ B ) @ L3 @ ( product_Pair @ B @ B @ X2 @ Y3 ) ) @ ( labeled_edges @ A @ B @ X8 ) )
=> ( ( member @ B @ X2 @ ( labeled_vertices @ A @ B @ X8 ) )
=> ( ( member @ B @ Y3 @ ( labeled_vertices @ A @ B @ X8 ) )
=> ( member @ ( product_prod @ A @ ( product_prod @ C @ C ) ) @ ( product_Pair @ A @ ( product_prod @ C @ C ) @ L3 @ ( product_Pair @ C @ C @ ( F2 @ X2 ) @ ( F2 @ Y3 ) ) ) @ Y5 ) ) ) )
=> ( edge_preserving @ B @ C @ A @ ( bNF_Gr @ B @ C @ ( labeled_vertices @ A @ B @ X8 ) @ F2 ) @ ( labeled_edges @ A @ B @ X8 ) @ Y5 ) ) ).
% edge_preserving_on_graphI
thf(fact_98_bot__apply,axiom,
! [C: $tType,D: $tType] :
( ( bot @ C )
=> ( ( bot_bot @ ( D > C ) )
= ( ^ [X7: D] : ( bot_bot @ C ) ) ) ) ).
% bot_apply
thf(fact_99_least__consequence__graphI,axiom,
! [X: $tType,A: $tType,C: $tType,L: $tType,Rs: set @ ( product_prod @ ( labeled_graph @ L @ A ) @ ( labeled_graph @ L @ A ) ),G: labeled_graph @ L @ C,S: labeled_graph @ L @ C,T2: itself @ X] :
( ( consequence_graph @ L @ A @ C @ Rs @ G )
=> ( ( graph_homomorphism @ L @ C @ C @ S @ G @ ( id_on @ C @ ( labeled_vertices @ L @ C @ S ) ) )
=> ( ! [C6: labeled_graph @ L @ X] :
( ( consequence_graph @ L @ A @ X @ Rs @ C6 )
=> ( maintained @ L @ C @ X @ ( product_Pair @ ( labeled_graph @ L @ C ) @ ( labeled_graph @ L @ C ) @ S @ G ) @ C6 ) )
=> ( least_559130134_graph @ X @ L @ A @ C @ T2 @ Rs @ S @ G ) ) ) ) ).
% least_consequence_graphI
thf(fact_100_GrD2,axiom,
! [A: $tType,B: $tType,X6: A,Fx: B,A5: set @ A,F2: A > B] :
( ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X6 @ Fx ) @ ( bNF_Gr @ A @ B @ A5 @ F2 ) )
=> ( ( F2 @ X6 )
= Fx ) ) ).
% GrD2
thf(fact_101_GrD1,axiom,
! [B: $tType,A: $tType,X6: A,Fx: B,A5: set @ A,F2: A > B] :
( ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X6 @ Fx ) @ ( bNF_Gr @ A @ B @ A5 @ F2 ) )
=> ( member @ A @ X6 @ A5 ) ) ).
% GrD1
thf(fact_102_leastI,axiom,
! [X: $tType,A: $tType,C: $tType,L: $tType,S: labeled_graph @ L @ C,G: labeled_graph @ L @ C,Rs: set @ ( product_prod @ ( labeled_graph @ L @ A ) @ ( labeled_graph @ L @ A ) ),T2: itself @ X] :
( ( graph_homomorphism @ L @ C @ C @ S @ G @ ( id_on @ C @ ( labeled_vertices @ L @ C @ S ) ) )
=> ( ! [C6: labeled_graph @ L @ X] :
( ( consequence_graph @ L @ A @ X @ Rs @ C6 )
=> ( maintained @ L @ C @ X @ ( product_Pair @ ( labeled_graph @ L @ C ) @ ( labeled_graph @ L @ C ) @ S @ G ) @ C6 ) )
=> ( least @ X @ L @ A @ C @ T2 @ Rs @ S @ G ) ) ) ).
% leastI
thf(fact_103_edge__preserving__Id,axiom,
! [A: $tType,B: $tType,Y: set @ A,X6: set @ ( product_prod @ B @ ( product_prod @ A @ A ) )] : ( edge_preserving @ A @ A @ B @ ( id_on @ A @ Y ) @ X6 @ X6 ) ).
% edge_preserving_Id
thf(fact_104_map__graph__edge__preserving,axiom,
! [B: $tType,A: $tType,C: $tType,F2: set @ ( product_prod @ A @ B ),G: labeled_graph @ C @ A] : ( edge_preserving @ A @ B @ C @ F2 @ ( labeled_edges @ C @ A @ G ) @ ( labeled_edges @ C @ B @ ( map_graph @ A @ B @ C @ F2 @ G ) ) ) ).
% map_graph_edge_preserving
thf(fact_105_least__consequence__graph__def,axiom,
! [L: $tType,V: $tType,C: $tType,X: $tType] :
( ( least_559130134_graph @ X @ L @ V @ C )
= ( ^ [T3: itself @ X,Rs2: set @ ( product_prod @ ( labeled_graph @ L @ V ) @ ( labeled_graph @ L @ V ) ),S2: labeled_graph @ L @ C,G7: labeled_graph @ L @ C] :
( ( consequence_graph @ L @ V @ C @ Rs2 @ G7 )
& ( least @ X @ L @ V @ C @ T3 @ Rs2 @ S2 @ G7 ) ) ) ) ).
% least_consequence_graph_def
thf(fact_106_bot__empty__eq,axiom,
! [A: $tType] :
( ( bot_bot @ ( A > $o ) )
= ( ^ [X7: A] : ( member @ A @ X7 @ ( bot_bot @ ( set @ A ) ) ) ) ) ).
% bot_empty_eq
thf(fact_107_bot__set__def,axiom,
! [A: $tType] :
( ( bot_bot @ ( set @ A ) )
= ( collect @ A @ ( bot_bot @ ( A > $o ) ) ) ) ).
% bot_set_def
thf(fact_108_consequence__graphD_I3_J,axiom,
! [B: $tType,C: $tType,A: $tType,Rs: set @ ( product_prod @ ( labeled_graph @ A @ B ) @ ( labeled_graph @ A @ B ) ),G: labeled_graph @ A @ C] :
( ( consequence_graph @ A @ B @ C @ Rs @ G )
=> ( G
= ( restrict @ A @ C @ G ) ) ) ).
% consequence_graphD(3)
thf(fact_109_consequence__graphD_I1_J,axiom,
! [B: $tType,C: $tType,A: $tType,Rs: set @ ( product_prod @ ( labeled_graph @ A @ B ) @ ( labeled_graph @ A @ B ) ),G: labeled_graph @ A @ C,R2: product_prod @ ( labeled_graph @ A @ B ) @ ( labeled_graph @ A @ B )] :
( ( consequence_graph @ A @ B @ C @ Rs @ G )
=> ( ( member @ ( product_prod @ ( labeled_graph @ A @ B ) @ ( labeled_graph @ A @ B ) ) @ R2 @ Rs )
=> ( maintained @ A @ B @ C @ R2 @ G ) ) ) ).
% consequence_graphD(1)
thf(fact_110_edge__preserving__atomic,axiom,
! [A: $tType,B: $tType,C: $tType,H1: set @ ( product_prod @ A @ B ),E1: set @ ( product_prod @ C @ ( product_prod @ A @ A ) ),E22: set @ ( product_prod @ C @ ( product_prod @ B @ B ) ),V1: A,V12: B,V22: A,V23: B,K: C] :
( ( edge_preserving @ A @ B @ C @ H1 @ E1 @ E22 )
=> ( ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ V1 @ V12 ) @ H1 )
=> ( ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ V22 @ V23 ) @ H1 )
=> ( ( member @ ( product_prod @ C @ ( product_prod @ A @ A ) ) @ ( product_Pair @ C @ ( product_prod @ A @ A ) @ K @ ( product_Pair @ A @ A @ V1 @ V22 ) ) @ E1 )
=> ( member @ ( product_prod @ C @ ( product_prod @ B @ B ) ) @ ( product_Pair @ C @ ( product_prod @ B @ B ) @ K @ ( product_Pair @ B @ B @ V12 @ V23 ) ) @ E22 ) ) ) ) ) ).
% edge_preserving_atomic
thf(fact_111_bot__fun__def,axiom,
! [B: $tType,A: $tType] :
( ( bot @ B )
=> ( ( bot_bot @ ( A > B ) )
= ( ^ [X7: A] : ( bot_bot @ B ) ) ) ) ).
% bot_fun_def
thf(fact_112_least__def,axiom,
! [L: $tType,V: $tType,C: $tType,X: $tType] :
( ( least @ X @ L @ V @ C )
= ( ^ [Uu: itself @ X,Rs2: set @ ( product_prod @ ( labeled_graph @ L @ V ) @ ( labeled_graph @ L @ V ) ),S2: labeled_graph @ L @ C,G7: labeled_graph @ L @ C] :
( ( graph_homomorphism @ L @ C @ C @ S2 @ G7 @ ( id_on @ C @ ( labeled_vertices @ L @ C @ S2 ) ) )
& ! [C7: labeled_graph @ L @ X] :
( ( consequence_graph @ L @ V @ X @ Rs2 @ C7 )
=> ( maintained @ L @ C @ X @ ( product_Pair @ ( labeled_graph @ L @ C ) @ ( labeled_graph @ L @ C ) @ S2 @ G7 ) @ C7 ) ) ) ) ) ).
% least_def
thf(fact_113_mnt,axiom,
! [X5: product_prod @ ( labeled_graph @ ( standard_Constant @ v2 ) @ nat ) @ ( labeled_graph @ ( standard_Constant @ v2 ) @ nat )] :
( ( member @ ( product_prod @ ( labeled_graph @ ( standard_Constant @ v2 ) @ nat ) @ ( labeled_graph @ ( standard_Constant @ v2 ) @ nat ) ) @ X5 @ ( sup_sup @ ( set @ ( product_prod @ ( labeled_graph @ ( standard_Constant @ v2 ) @ nat ) @ ( labeled_graph @ ( standard_Constant @ v2 ) @ nat ) ) ) @ ( sup_sup @ ( set @ ( product_prod @ ( labeled_graph @ ( standard_Constant @ v2 ) @ nat ) @ ( labeled_graph @ ( standard_Constant @ v2 ) @ nat ) ) ) @ ( standa1138209853_rules @ v2 @ c ) @ ( standa2002409347_rules @ v2 @ l ) ) @ ( insert @ ( product_prod @ ( labeled_graph @ ( standard_Constant @ v2 ) @ nat ) @ ( labeled_graph @ ( standard_Constant @ v2 ) @ nat ) ) @ ( standa400804411p_rule @ ( standard_Constant @ v2 ) @ ( standard_S_Top @ v2 ) ) @ ( insert @ ( product_prod @ ( labeled_graph @ ( standard_Constant @ v2 ) @ nat ) @ ( labeled_graph @ ( standard_Constant @ v2 ) @ nat ) ) @ ( standa244753842y_rule @ ( standard_Constant @ v2 ) ) @ ( bot_bot @ ( set @ ( product_prod @ ( labeled_graph @ ( standard_Constant @ v2 ) @ nat ) @ ( labeled_graph @ ( standard_Constant @ v2 ) @ nat ) ) ) ) ) ) ) )
=> ( maintained @ ( standard_Constant @ v2 ) @ nat @ v @ X5 @ g ) ) ).
% mnt
thf(fact_114_map__graph__in,axiom,
! [B: $tType,C: $tType,A: $tType,G: labeled_graph @ A @ B,A2: B,B2: B,E3: allegorical_term @ A,F2: B > C] :
( ( G
= ( restrict @ A @ B @ G ) )
=> ( ( member @ ( product_prod @ B @ B ) @ ( product_Pair @ B @ B @ A2 @ B2 ) @ ( semantics @ A @ B @ G @ E3 ) )
=> ( member @ ( product_prod @ C @ C ) @ ( product_Pair @ C @ C @ ( F2 @ A2 ) @ ( F2 @ B2 ) ) @ ( semantics @ A @ C @ ( map_graph @ B @ C @ A @ ( bNF_Gr @ B @ C @ ( labeled_vertices @ A @ B @ G ) @ F2 ) @ G ) @ E3 ) ) ) ) ).
% map_graph_in
thf(fact_115_maintainedI,axiom,
! [A: $tType,B: $tType,C: $tType,A5: labeled_graph @ C @ A,G: labeled_graph @ C @ B,B6: labeled_graph @ C @ A] :
( ! [F: set @ ( product_prod @ A @ B )] :
( ( graph_homomorphism @ C @ A @ B @ A5 @ G @ F )
=> ( extensible @ C @ A @ B @ ( product_Pair @ ( labeled_graph @ C @ A ) @ ( labeled_graph @ C @ A ) @ A5 @ B6 ) @ G @ F ) )
=> ( maintained @ C @ A @ B @ ( product_Pair @ ( labeled_graph @ C @ A ) @ ( labeled_graph @ C @ A ) @ A5 @ B6 ) @ G ) ) ).
% maintainedI
thf(fact_116_graph__homomorphism__empty,axiom,
! [B: $tType,C: $tType,A: $tType,G: labeled_graph @ A @ C,F2: set @ ( product_prod @ B @ C )] :
( ( graph_homomorphism @ A @ B @ C @ ( labeled_LG @ A @ B @ ( bot_bot @ ( set @ ( product_prod @ A @ ( product_prod @ B @ B ) ) ) ) @ ( bot_bot @ ( set @ B ) ) ) @ G @ F2 )
= ( ( F2
= ( bot_bot @ ( set @ ( product_prod @ B @ C ) ) ) )
& ( G
= ( restrict @ A @ C @ G ) ) ) ) ).
% graph_homomorphism_empty
thf(fact_117_subgraph__def,axiom,
! [B: $tType,A: $tType,G_1: labeled_graph @ A @ B,G_2: labeled_graph @ A @ B] :
( ( graph_homomorphism @ A @ B @ B @ G_1 @ G_2 @ ( id_on @ B @ ( labeled_vertices @ A @ B @ G_1 ) ) )
= ( ( G_1
= ( restrict @ A @ B @ G_1 ) )
& ( G_2
= ( restrict @ A @ B @ G_2 ) )
& ( ( graph_union @ A @ B @ G_1 @ G_2 )
= G_2 ) ) ) ).
% subgraph_def
thf(fact_118_insert__absorb2,axiom,
! [A: $tType,X6: A,A5: set @ A] :
( ( insert @ A @ X6 @ ( insert @ A @ X6 @ A5 ) )
= ( insert @ A @ X6 @ A5 ) ) ).
% insert_absorb2
thf(fact_119_insert__iff,axiom,
! [A: $tType,A2: A,B2: A,A5: set @ A] :
( ( member @ A @ A2 @ ( insert @ A @ B2 @ A5 ) )
= ( ( A2 = B2 )
| ( member @ A @ A2 @ A5 ) ) ) ).
% insert_iff
thf(fact_120_insertCI,axiom,
! [A: $tType,A2: A,B6: set @ A,B2: A] :
( ( ~ ( member @ A @ A2 @ B6 )
=> ( A2 = B2 ) )
=> ( member @ A @ A2 @ ( insert @ A @ B2 @ B6 ) ) ) ).
% insertCI
thf(fact_121_Un__iff,axiom,
! [A: $tType,C5: A,A5: set @ A,B6: set @ A] :
( ( member @ A @ C5 @ ( sup_sup @ ( set @ A ) @ A5 @ B6 ) )
= ( ( member @ A @ C5 @ A5 )
| ( member @ A @ C5 @ B6 ) ) ) ).
% Un_iff
thf(fact_122_UnCI,axiom,
! [A: $tType,C5: A,B6: set @ A,A5: set @ A] :
( ( ~ ( member @ A @ C5 @ B6 )
=> ( member @ A @ C5 @ A5 ) )
=> ( member @ A @ C5 @ ( sup_sup @ ( set @ A ) @ A5 @ B6 ) ) ) ).
% UnCI
thf(fact_123_labeled__graph_Oinject,axiom,
! [L: $tType,V: $tType,X1: set @ ( product_prod @ L @ ( product_prod @ V @ V ) ),X22: set @ V,Y1: set @ ( product_prod @ L @ ( product_prod @ V @ V ) ),Y2: set @ V] :
( ( ( labeled_LG @ L @ V @ X1 @ X22 )
= ( labeled_LG @ L @ V @ Y1 @ Y2 ) )
= ( ( X1 = Y1 )
& ( X22 = Y2 ) ) ) ).
% labeled_graph.inject
thf(fact_124_graph__union__idemp_I1_J,axiom,
! [B: $tType,A: $tType,A5: labeled_graph @ A @ B] :
( ( graph_union @ A @ B @ A5 @ A5 )
= A5 ) ).
% graph_union_idemp(1)
thf(fact_125_graph__union__idemp_I2_J,axiom,
! [B: $tType,A: $tType,A5: labeled_graph @ A @ B,B6: labeled_graph @ A @ B] :
( ( graph_union @ A @ B @ A5 @ ( graph_union @ A @ B @ A5 @ B6 ) )
= ( graph_union @ A @ B @ A5 @ B6 ) ) ).
% graph_union_idemp(2)
thf(fact_126_graph__union__idemp_I3_J,axiom,
! [B: $tType,A: $tType,A5: labeled_graph @ A @ B,B6: labeled_graph @ A @ B] :
( ( graph_union @ A @ B @ A5 @ ( graph_union @ A @ B @ B6 @ A5 ) )
= ( graph_union @ A @ B @ B6 @ A5 ) ) ).
% graph_union_idemp(3)
thf(fact_127_singletonI,axiom,
! [A: $tType,A2: A] : ( member @ A @ A2 @ ( insert @ A @ A2 @ ( bot_bot @ ( set @ A ) ) ) ) ).
% singletonI
thf(fact_128_Un__empty,axiom,
! [A: $tType,A5: set @ A,B6: set @ A] :
( ( ( sup_sup @ ( set @ A ) @ A5 @ B6 )
= ( bot_bot @ ( set @ A ) ) )
= ( ( A5
= ( bot_bot @ ( set @ A ) ) )
& ( B6
= ( bot_bot @ ( set @ A ) ) ) ) ) ).
% Un_empty
thf(fact_129_Un__insert__right,axiom,
! [A: $tType,A5: set @ A,A2: A,B6: set @ A] :
( ( sup_sup @ ( set @ A ) @ A5 @ ( insert @ A @ A2 @ B6 ) )
= ( insert @ A @ A2 @ ( sup_sup @ ( set @ A ) @ A5 @ B6 ) ) ) ).
% Un_insert_right
thf(fact_130_Un__insert__left,axiom,
! [A: $tType,A2: A,B6: set @ A,C8: set @ A] :
( ( sup_sup @ ( set @ A ) @ ( insert @ A @ A2 @ B6 ) @ C8 )
= ( insert @ A @ A2 @ ( sup_sup @ ( set @ A ) @ B6 @ C8 ) ) ) ).
% Un_insert_left
thf(fact_131_graph__union__preserves__restrict,axiom,
! [B: $tType,A: $tType,G_1: labeled_graph @ A @ B,G_2: labeled_graph @ A @ B] :
( ( G_1
= ( restrict @ A @ B @ G_1 ) )
=> ( ( G_2
= ( restrict @ A @ B @ G_2 ) )
=> ( ( graph_union @ A @ B @ G_1 @ G_2 )
= ( restrict @ A @ B @ ( graph_union @ A @ B @ G_1 @ G_2 ) ) ) ) ) ).
% graph_union_preserves_restrict
thf(fact_132_graph__union__vertices,axiom,
! [A: $tType,B: $tType,G_1: labeled_graph @ B @ A,G_2: labeled_graph @ B @ A] :
( ( labeled_vertices @ B @ A @ ( graph_union @ B @ A @ G_1 @ G_2 ) )
= ( sup_sup @ ( set @ A ) @ ( labeled_vertices @ B @ A @ G_1 ) @ ( labeled_vertices @ B @ A @ G_2 ) ) ) ).
% graph_union_vertices
thf(fact_133_labeled__graph_Ocollapse,axiom,
! [V: $tType,L: $tType,Labeled_graph: labeled_graph @ L @ V] :
( ( labeled_LG @ L @ V @ ( labeled_edges @ L @ V @ Labeled_graph ) @ ( labeled_vertices @ L @ V @ Labeled_graph ) )
= Labeled_graph ) ).
% labeled_graph.collapse
thf(fact_134_extensible__refl,axiom,
! [A: $tType,C: $tType,B: $tType,R2: labeled_graph @ A @ B,G: labeled_graph @ A @ C,F2: set @ ( product_prod @ B @ C )] :
( ( graph_homomorphism @ A @ B @ C @ R2 @ G @ F2 )
=> ( extensible @ A @ B @ C @ ( product_Pair @ ( labeled_graph @ A @ B ) @ ( labeled_graph @ A @ B ) @ R2 @ R2 ) @ G @ F2 ) ) ).
% extensible_refl
thf(fact_135_graph__empty__e,axiom,
! [A: $tType,B: $tType,V3: set @ B] :
( ( labeled_LG @ A @ B @ ( bot_bot @ ( set @ ( product_prod @ A @ ( product_prod @ B @ B ) ) ) ) @ V3 )
= ( restrict @ A @ B @ ( labeled_LG @ A @ B @ ( bot_bot @ ( set @ ( product_prod @ A @ ( product_prod @ B @ B ) ) ) ) @ V3 ) ) ) ).
% graph_empty_e
thf(fact_136_constant__rules__empty,axiom,
! [A: $tType] :
( ( standa1138209853_rules @ A @ ( bot_bot @ ( set @ A ) ) )
= ( bot_bot @ ( set @ ( product_prod @ ( labeled_graph @ ( standard_Constant @ A ) @ nat ) @ ( labeled_graph @ ( standard_Constant @ A ) @ nat ) ) ) ) ) ).
% constant_rules_empty
thf(fact_137_Gr__insert,axiom,
! [B: $tType,A: $tType,X6: A,F4: set @ A,F2: A > B] :
( ( bNF_Gr @ A @ B @ ( insert @ A @ X6 @ F4 ) @ F2 )
= ( insert @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X6 @ ( F2 @ X6 ) ) @ ( bNF_Gr @ A @ B @ F4 @ F2 ) ) ) ).
% Gr_insert
thf(fact_138_extensible__refl__concr,axiom,
! [A: $tType,C: $tType,B: $tType,E_1: set @ ( product_prod @ A @ ( product_prod @ B @ B ) ),V3: set @ B,G: labeled_graph @ A @ C,F2: set @ ( product_prod @ B @ C ),E_2: set @ ( product_prod @ A @ ( product_prod @ B @ B ) )] :
( ( graph_homomorphism @ A @ B @ C @ ( labeled_LG @ A @ B @ E_1 @ V3 ) @ G @ F2 )
=> ( ( extensible @ A @ B @ C @ ( product_Pair @ ( labeled_graph @ A @ B ) @ ( labeled_graph @ A @ B ) @ ( labeled_LG @ A @ B @ E_1 @ V3 ) @ ( labeled_LG @ A @ B @ E_2 @ V3 ) ) @ G @ F2 )
= ( graph_homomorphism @ A @ B @ C @ ( labeled_LG @ A @ B @ E_2 @ V3 ) @ G @ F2 ) ) ) ).
% extensible_refl_concr
thf(fact_139_graph__single,axiom,
! [A: $tType,B: $tType,A2: A,B2: B,C5: B] :
( ( labeled_LG @ A @ B @ ( insert @ ( product_prod @ A @ ( product_prod @ B @ B ) ) @ ( product_Pair @ A @ ( product_prod @ B @ B ) @ A2 @ ( product_Pair @ B @ B @ B2 @ C5 ) ) @ ( bot_bot @ ( set @ ( product_prod @ A @ ( product_prod @ B @ B ) ) ) ) ) @ ( insert @ B @ B2 @ ( insert @ B @ C5 @ ( bot_bot @ ( set @ B ) ) ) ) )
= ( restrict @ A @ B @ ( labeled_LG @ A @ B @ ( insert @ ( product_prod @ A @ ( product_prod @ B @ B ) ) @ ( product_Pair @ A @ ( product_prod @ B @ B ) @ A2 @ ( product_Pair @ B @ B @ B2 @ C5 ) ) @ ( bot_bot @ ( set @ ( product_prod @ A @ ( product_prod @ B @ B ) ) ) ) ) @ ( insert @ B @ B2 @ ( insert @ B @ C5 @ ( bot_bot @ ( set @ B ) ) ) ) ) ) ) ).
% graph_single
thf(fact_140_insert__is__Un,axiom,
! [A: $tType] :
( ( insert @ A )
= ( ^ [A6: A] : ( sup_sup @ ( set @ A ) @ ( insert @ A @ A6 @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ).
% insert_is_Un
thf(fact_141_Un__singleton__iff,axiom,
! [A: $tType,A5: set @ A,B6: set @ A,X6: A] :
( ( ( sup_sup @ ( set @ A ) @ A5 @ B6 )
= ( insert @ A @ X6 @ ( bot_bot @ ( set @ A ) ) ) )
= ( ( ( A5
= ( bot_bot @ ( set @ A ) ) )
& ( B6
= ( insert @ A @ X6 @ ( bot_bot @ ( set @ A ) ) ) ) )
| ( ( A5
= ( insert @ A @ X6 @ ( bot_bot @ ( set @ A ) ) ) )
& ( B6
= ( bot_bot @ ( set @ A ) ) ) )
| ( ( A5
= ( insert @ A @ X6 @ ( bot_bot @ ( set @ A ) ) ) )
& ( B6
= ( insert @ A @ X6 @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ) ).
% Un_singleton_iff
thf(fact_142_singleton__Un__iff,axiom,
! [A: $tType,X6: A,A5: set @ A,B6: set @ A] :
( ( ( insert @ A @ X6 @ ( bot_bot @ ( set @ A ) ) )
= ( sup_sup @ ( set @ A ) @ A5 @ B6 ) )
= ( ( ( A5
= ( bot_bot @ ( set @ A ) ) )
& ( B6
= ( insert @ A @ X6 @ ( bot_bot @ ( set @ A ) ) ) ) )
| ( ( A5
= ( insert @ A @ X6 @ ( bot_bot @ ( set @ A ) ) ) )
& ( B6
= ( bot_bot @ ( set @ A ) ) ) )
| ( ( A5
= ( insert @ A @ X6 @ ( bot_bot @ ( set @ A ) ) ) )
& ( B6
= ( insert @ A @ X6 @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ) ).
% singleton_Un_iff
thf(fact_143_graph__union__def,axiom,
! [B: $tType,A: $tType] :
( ( graph_union @ A @ B )
= ( ^ [G_12: labeled_graph @ A @ B,G_22: labeled_graph @ A @ B] : ( labeled_LG @ A @ B @ ( sup_sup @ ( set @ ( product_prod @ A @ ( product_prod @ B @ B ) ) ) @ ( labeled_edges @ A @ B @ G_12 ) @ ( labeled_edges @ A @ B @ G_22 ) ) @ ( sup_sup @ ( set @ B ) @ ( labeled_vertices @ A @ B @ G_12 ) @ ( labeled_vertices @ A @ B @ G_22 ) ) ) ) ) ).
% graph_union_def
thf(fact_144_Un__empty__left,axiom,
! [A: $tType,B6: set @ A] :
( ( sup_sup @ ( set @ A ) @ ( bot_bot @ ( set @ A ) ) @ B6 )
= B6 ) ).
% Un_empty_left
thf(fact_145_Un__empty__right,axiom,
! [A: $tType,A5: set @ A] :
( ( sup_sup @ ( set @ A ) @ A5 @ ( bot_bot @ ( set @ A ) ) )
= A5 ) ).
% Un_empty_right
thf(fact_146_singletonD,axiom,
! [A: $tType,B2: A,A2: A] :
( ( member @ A @ B2 @ ( insert @ A @ A2 @ ( bot_bot @ ( set @ A ) ) ) )
=> ( B2 = A2 ) ) ).
% singletonD
thf(fact_147_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_148_doubleton__eq__iff,axiom,
! [A: $tType,A2: A,B2: A,C5: A,D3: A] :
( ( ( insert @ A @ A2 @ ( insert @ A @ B2 @ ( bot_bot @ ( set @ A ) ) ) )
= ( insert @ A @ C5 @ ( insert @ A @ D3 @ ( bot_bot @ ( set @ A ) ) ) ) )
= ( ( ( A2 = C5 )
& ( B2 = D3 ) )
| ( ( A2 = D3 )
& ( B2 = C5 ) ) ) ) ).
% doubleton_eq_iff
thf(fact_149_insert__not__empty,axiom,
! [A: $tType,A2: A,A5: set @ A] :
( ( insert @ A @ A2 @ A5 )
!= ( bot_bot @ ( set @ A ) ) ) ).
% insert_not_empty
thf(fact_150_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_151_labeled__graph_Oexhaust,axiom,
! [L: $tType,V: $tType,Y: labeled_graph @ L @ V] :
~ ! [X12: set @ ( product_prod @ L @ ( product_prod @ V @ V ) ),X23: set @ V] :
( Y
!= ( labeled_LG @ L @ V @ X12 @ X23 ) ) ).
% labeled_graph.exhaust
thf(fact_152_labeled__graph_Oinduct,axiom,
! [V: $tType,L: $tType,P2: ( labeled_graph @ L @ V ) > $o,Labeled_graph: labeled_graph @ L @ V] :
( ! [X1a: set @ ( product_prod @ L @ ( product_prod @ V @ V ) ),X2a: set @ V] : ( P2 @ ( labeled_LG @ L @ V @ X1a @ X2a ) )
=> ( P2 @ Labeled_graph ) ) ).
% labeled_graph.induct
thf(fact_153_mk__disjoint__insert,axiom,
! [A: $tType,A2: A,A5: set @ A] :
( ( member @ A @ A2 @ A5 )
=> ? [B7: set @ A] :
( ( A5
= ( insert @ A @ A2 @ B7 ) )
& ~ ( member @ A @ A2 @ B7 ) ) ) ).
% mk_disjoint_insert
thf(fact_154_Un__left__commute,axiom,
! [A: $tType,A5: set @ A,B6: set @ A,C8: set @ A] :
( ( sup_sup @ ( set @ A ) @ A5 @ ( sup_sup @ ( set @ A ) @ B6 @ C8 ) )
= ( sup_sup @ ( set @ A ) @ B6 @ ( sup_sup @ ( set @ A ) @ A5 @ C8 ) ) ) ).
% Un_left_commute
thf(fact_155_insert__commute,axiom,
! [A: $tType,X6: A,Y: A,A5: set @ A] :
( ( insert @ A @ X6 @ ( insert @ A @ Y @ A5 ) )
= ( insert @ A @ Y @ ( insert @ A @ X6 @ A5 ) ) ) ).
% insert_commute
thf(fact_156_Un__left__absorb,axiom,
! [A: $tType,A5: set @ A,B6: set @ A] :
( ( sup_sup @ ( set @ A ) @ A5 @ ( sup_sup @ ( set @ A ) @ A5 @ B6 ) )
= ( sup_sup @ ( set @ A ) @ A5 @ B6 ) ) ).
% Un_left_absorb
thf(fact_157_insert__eq__iff,axiom,
! [A: $tType,A2: A,A5: set @ A,B2: A,B6: set @ A] :
( ~ ( member @ A @ A2 @ A5 )
=> ( ~ ( member @ A @ B2 @ B6 )
=> ( ( ( insert @ A @ A2 @ A5 )
= ( insert @ A @ B2 @ B6 ) )
= ( ( ( A2 = B2 )
=> ( A5 = B6 ) )
& ( ( A2 != B2 )
=> ? [C7: set @ A] :
( ( A5
= ( insert @ A @ B2 @ C7 ) )
& ~ ( member @ A @ B2 @ C7 )
& ( B6
= ( insert @ A @ A2 @ C7 ) )
& ~ ( member @ A @ A2 @ C7 ) ) ) ) ) ) ) ).
% insert_eq_iff
thf(fact_158_insert__absorb,axiom,
! [A: $tType,A2: A,A5: set @ A] :
( ( member @ A @ A2 @ A5 )
=> ( ( insert @ A @ A2 @ A5 )
= A5 ) ) ).
% insert_absorb
thf(fact_159_insert__ident,axiom,
! [A: $tType,X6: A,A5: set @ A,B6: set @ A] :
( ~ ( member @ A @ X6 @ A5 )
=> ( ~ ( member @ A @ X6 @ B6 )
=> ( ( ( insert @ A @ X6 @ A5 )
= ( insert @ A @ X6 @ B6 ) )
= ( A5 = B6 ) ) ) ) ).
% insert_ident
thf(fact_160_Set_Oset__insert,axiom,
! [A: $tType,X6: A,A5: set @ A] :
( ( member @ A @ X6 @ A5 )
=> ~ ! [B7: set @ A] :
( ( A5
= ( insert @ A @ X6 @ B7 ) )
=> ( member @ A @ X6 @ B7 ) ) ) ).
% Set.set_insert
thf(fact_161_Un__commute,axiom,
! [A: $tType] :
( ( sup_sup @ ( set @ A ) )
= ( ^ [A7: set @ A,B8: set @ A] : ( sup_sup @ ( set @ A ) @ B8 @ A7 ) ) ) ).
% Un_commute
thf(fact_162_Un__absorb,axiom,
! [A: $tType,A5: set @ A] :
( ( sup_sup @ ( set @ A ) @ A5 @ A5 )
= A5 ) ).
% Un_absorb
thf(fact_163_insertI2,axiom,
! [A: $tType,A2: A,B6: set @ A,B2: A] :
( ( member @ A @ A2 @ B6 )
=> ( member @ A @ A2 @ ( insert @ A @ B2 @ B6 ) ) ) ).
% insertI2
thf(fact_164_insertI1,axiom,
! [A: $tType,A2: A,B6: set @ A] : ( member @ A @ A2 @ ( insert @ A @ A2 @ B6 ) ) ).
% insertI1
thf(fact_165_Un__assoc,axiom,
! [A: $tType,A5: set @ A,B6: set @ A,C8: set @ A] :
( ( sup_sup @ ( set @ A ) @ ( sup_sup @ ( set @ A ) @ A5 @ B6 ) @ C8 )
= ( sup_sup @ ( set @ A ) @ A5 @ ( sup_sup @ ( set @ A ) @ B6 @ C8 ) ) ) ).
% Un_assoc
thf(fact_166_insertE,axiom,
! [A: $tType,A2: A,B2: A,A5: set @ A] :
( ( member @ A @ A2 @ ( insert @ A @ B2 @ A5 ) )
=> ( ( A2 != B2 )
=> ( member @ A @ A2 @ A5 ) ) ) ).
% insertE
thf(fact_167_ball__Un,axiom,
! [A: $tType,A5: set @ A,B6: set @ A,P2: A > $o] :
( ( ! [X7: A] :
( ( member @ A @ X7 @ ( sup_sup @ ( set @ A ) @ A5 @ B6 ) )
=> ( P2 @ X7 ) ) )
= ( ! [X7: A] :
( ( member @ A @ X7 @ A5 )
=> ( P2 @ X7 ) )
& ! [X7: A] :
( ( member @ A @ X7 @ B6 )
=> ( P2 @ X7 ) ) ) ) ).
% ball_Un
thf(fact_168_bex__Un,axiom,
! [A: $tType,A5: set @ A,B6: set @ A,P2: A > $o] :
( ( ? [X7: A] :
( ( member @ A @ X7 @ ( sup_sup @ ( set @ A ) @ A5 @ B6 ) )
& ( P2 @ X7 ) ) )
= ( ? [X7: A] :
( ( member @ A @ X7 @ A5 )
& ( P2 @ X7 ) )
| ? [X7: A] :
( ( member @ A @ X7 @ B6 )
& ( P2 @ X7 ) ) ) ) ).
% bex_Un
thf(fact_169_UnI2,axiom,
! [A: $tType,C5: A,B6: set @ A,A5: set @ A] :
( ( member @ A @ C5 @ B6 )
=> ( member @ A @ C5 @ ( sup_sup @ ( set @ A ) @ A5 @ B6 ) ) ) ).
% UnI2
thf(fact_170_UnI1,axiom,
! [A: $tType,C5: A,A5: set @ A,B6: set @ A] :
( ( member @ A @ C5 @ A5 )
=> ( member @ A @ C5 @ ( sup_sup @ ( set @ A ) @ A5 @ B6 ) ) ) ).
% UnI1
thf(fact_171_UnE,axiom,
! [A: $tType,C5: A,A5: set @ A,B6: set @ A] :
( ( member @ A @ C5 @ ( sup_sup @ ( set @ A ) @ A5 @ B6 ) )
=> ( ~ ( member @ A @ C5 @ A5 )
=> ( member @ A @ C5 @ B6 ) ) ) ).
% UnE
thf(fact_172_labeled__graph_Osel_I2_J,axiom,
! [L: $tType,V: $tType,X1: set @ ( product_prod @ L @ ( product_prod @ V @ V ) ),X22: set @ V] :
( ( labeled_vertices @ L @ V @ ( labeled_LG @ L @ V @ X1 @ X22 ) )
= X22 ) ).
% labeled_graph.sel(2)
thf(fact_173_labeled__graph_Osel_I1_J,axiom,
! [V: $tType,L: $tType,X1: set @ ( product_prod @ L @ ( product_prod @ V @ V ) ),X22: set @ V] :
( ( labeled_edges @ L @ V @ ( labeled_LG @ L @ V @ X1 @ X22 ) )
= X1 ) ).
% labeled_graph.sel(1)
thf(fact_174_graph__homomorphism__semantics,axiom,
! [B: $tType,C: $tType,A: $tType,A5: labeled_graph @ A @ B,B6: labeled_graph @ A @ C,F2: set @ ( product_prod @ B @ C ),A2: B,B2: B,E3: allegorical_term @ A,A3: C,B3: C] :
( ( graph_homomorphism @ A @ B @ C @ A5 @ B6 @ F2 )
=> ( ( member @ ( product_prod @ B @ B ) @ ( product_Pair @ B @ B @ A2 @ B2 ) @ ( semantics @ A @ B @ A5 @ E3 ) )
=> ( ( member @ ( product_prod @ B @ C ) @ ( product_Pair @ B @ C @ A2 @ A3 ) @ F2 )
=> ( ( member @ ( product_prod @ B @ C ) @ ( product_Pair @ B @ C @ B2 @ B3 ) @ F2 )
=> ( member @ ( product_prod @ C @ C ) @ ( product_Pair @ C @ C @ A3 @ B3 ) @ ( semantics @ A @ C @ B6 @ E3 ) ) ) ) ) ) ).
% graph_homomorphism_semantics
thf(fact_175_graph__homomorphism__nonempty,axiom,
! [B: $tType,A: $tType,C: $tType,A5: labeled_graph @ A @ B,B6: labeled_graph @ A @ C,F2: set @ ( product_prod @ B @ C ),E3: allegorical_term @ A] :
( ( graph_homomorphism @ A @ B @ C @ A5 @ B6 @ F2 )
=> ( ( ( semantics @ A @ B @ A5 @ E3 )
!= ( bot_bot @ ( set @ ( product_prod @ B @ B ) ) ) )
=> ( ( semantics @ A @ C @ B6 @ E3 )
!= ( bot_bot @ ( set @ ( product_prod @ C @ C ) ) ) ) ) ) ).
% graph_homomorphism_nonempty
thf(fact_176_labeled__graph_Oexhaust__sel,axiom,
! [V: $tType,L: $tType,Labeled_graph: labeled_graph @ L @ V] :
( Labeled_graph
= ( labeled_LG @ L @ V @ ( labeled_edges @ L @ V @ Labeled_graph ) @ ( labeled_vertices @ L @ V @ Labeled_graph ) ) ) ).
% labeled_graph.exhaust_sel
thf(fact_177_standard__rules__def,axiom,
! [A: $tType] :
( ( standa438229444_rules @ A )
= ( ^ [C7: set @ A,L4: set @ ( standard_Constant @ A )] : ( sup_sup @ ( set @ ( product_prod @ ( labeled_graph @ ( standard_Constant @ A ) @ nat ) @ ( labeled_graph @ ( standard_Constant @ A ) @ nat ) ) ) @ ( sup_sup @ ( set @ ( product_prod @ ( labeled_graph @ ( standard_Constant @ A ) @ nat ) @ ( labeled_graph @ ( standard_Constant @ A ) @ nat ) ) ) @ ( standa1138209853_rules @ A @ C7 ) @ ( standa2002409347_rules @ A @ L4 ) ) @ ( insert @ ( product_prod @ ( labeled_graph @ ( standard_Constant @ A ) @ nat ) @ ( labeled_graph @ ( standard_Constant @ A ) @ nat ) ) @ ( standa400804411p_rule @ ( standard_Constant @ A ) @ ( standard_S_Top @ A ) ) @ ( insert @ ( product_prod @ ( labeled_graph @ ( standard_Constant @ A ) @ nat ) @ ( labeled_graph @ ( standard_Constant @ A ) @ nat ) ) @ ( standa244753842y_rule @ ( standard_Constant @ A ) ) @ ( bot_bot @ ( set @ ( product_prod @ ( labeled_graph @ ( standard_Constant @ A ) @ nat ) @ ( labeled_graph @ ( standard_Constant @ A ) @ nat ) ) ) ) ) ) ) ) ) ).
% standard_rules_def
thf(fact_178_semantics__in__vertices_I2_J,axiom,
! [B: $tType,A: $tType,A5: labeled_graph @ A @ B,A2: B,B2: B,E3: allegorical_term @ A] :
( ( A5
= ( restrict @ A @ B @ A5 ) )
=> ( ( member @ ( product_prod @ B @ B ) @ ( product_Pair @ B @ B @ A2 @ B2 ) @ ( semantics @ A @ B @ A5 @ E3 ) )
=> ( member @ B @ B2 @ ( labeled_vertices @ A @ B @ A5 ) ) ) ) ).
% semantics_in_vertices(2)
thf(fact_179_semantics__in__vertices_I1_J,axiom,
! [B: $tType,A: $tType,A5: labeled_graph @ A @ B,A2: B,B2: B,E3: allegorical_term @ A] :
( ( A5
= ( restrict @ A @ B @ A5 ) )
=> ( ( member @ ( product_prod @ B @ B ) @ ( product_Pair @ B @ B @ A2 @ B2 ) @ ( semantics @ A @ B @ A5 @ E3 ) )
=> ( member @ B @ A2 @ ( labeled_vertices @ A @ B @ A5 ) ) ) ) ).
% semantics_in_vertices(1)
thf(fact_180_subgraph__semantics,axiom,
! [B: $tType,A: $tType,A5: labeled_graph @ A @ B,B6: labeled_graph @ A @ B,A2: B,B2: B,E3: allegorical_term @ A] :
( ( graph_homomorphism @ A @ B @ B @ A5 @ B6 @ ( id_on @ B @ ( labeled_vertices @ A @ B @ A5 ) ) )
=> ( ( member @ ( product_prod @ B @ B ) @ ( product_Pair @ B @ B @ A2 @ B2 ) @ ( semantics @ A @ B @ A5 @ E3 ) )
=> ( member @ ( product_prod @ B @ B ) @ ( product_Pair @ B @ B @ A2 @ B2 ) @ ( semantics @ A @ B @ B6 @ E3 ) ) ) ) ).
% subgraph_semantics
thf(fact_181_maintainedD,axiom,
! [A: $tType,C: $tType,B: $tType,A5: labeled_graph @ A @ B,B6: labeled_graph @ A @ B,G: labeled_graph @ A @ C,F2: set @ ( product_prod @ B @ C )] :
( ( maintained @ A @ B @ C @ ( product_Pair @ ( labeled_graph @ A @ B ) @ ( labeled_graph @ A @ B ) @ A5 @ B6 ) @ G )
=> ( ( graph_homomorphism @ A @ B @ C @ A5 @ G @ F2 )
=> ( extensible @ A @ B @ C @ ( product_Pair @ ( labeled_graph @ A @ B ) @ ( labeled_graph @ A @ B ) @ A5 @ B6 ) @ G @ F2 ) ) ) ).
% maintainedD
thf(fact_182_sup__bot_Oright__neutral,axiom,
! [A: $tType] :
( ( bounde1808546759up_bot @ A )
=> ! [A2: A] :
( ( sup_sup @ A @ A2 @ ( bot_bot @ A ) )
= A2 ) ) ).
% sup_bot.right_neutral
thf(fact_183_sup__bot_Oneutr__eq__iff,axiom,
! [A: $tType] :
( ( bounde1808546759up_bot @ A )
=> ! [A2: A,B2: A] :
( ( ( bot_bot @ A )
= ( sup_sup @ A @ A2 @ B2 ) )
= ( ( A2
= ( bot_bot @ A ) )
& ( B2
= ( bot_bot @ A ) ) ) ) ) ).
% sup_bot.neutr_eq_iff
thf(fact_184_sup__bot_Oleft__neutral,axiom,
! [A: $tType] :
( ( bounde1808546759up_bot @ A )
=> ! [A2: A] :
( ( sup_sup @ A @ ( bot_bot @ A ) @ A2 )
= A2 ) ) ).
% sup_bot.left_neutral
thf(fact_185_sup__bot_Oeq__neutr__iff,axiom,
! [A: $tType] :
( ( bounde1808546759up_bot @ A )
=> ! [A2: A,B2: A] :
( ( ( sup_sup @ A @ A2 @ B2 )
= ( bot_bot @ A ) )
= ( ( A2
= ( bot_bot @ A ) )
& ( B2
= ( bot_bot @ A ) ) ) ) ) ).
% sup_bot.eq_neutr_iff
thf(fact_186_sup_Oright__idem,axiom,
! [A: $tType] :
( ( semilattice_sup @ A )
=> ! [A2: A,B2: A] :
( ( sup_sup @ A @ ( sup_sup @ A @ A2 @ B2 ) @ B2 )
= ( sup_sup @ A @ A2 @ B2 ) ) ) ).
% sup.right_idem
thf(fact_187_sup__left__idem,axiom,
! [A: $tType] :
( ( semilattice_sup @ A )
=> ! [X6: A,Y: A] :
( ( sup_sup @ A @ X6 @ ( sup_sup @ A @ X6 @ Y ) )
= ( sup_sup @ A @ X6 @ Y ) ) ) ).
% sup_left_idem
thf(fact_188_sup_Oleft__idem,axiom,
! [A: $tType] :
( ( semilattice_sup @ A )
=> ! [A2: A,B2: A] :
( ( sup_sup @ A @ A2 @ ( sup_sup @ A @ A2 @ B2 ) )
= ( sup_sup @ A @ A2 @ B2 ) ) ) ).
% sup.left_idem
thf(fact_189_sup__idem,axiom,
! [A: $tType] :
( ( semilattice_sup @ A )
=> ! [X6: A] :
( ( sup_sup @ A @ X6 @ X6 )
= X6 ) ) ).
% sup_idem
thf(fact_190_sup_Oidem,axiom,
! [A: $tType] :
( ( semilattice_sup @ A )
=> ! [A2: A] :
( ( sup_sup @ A @ A2 @ A2 )
= A2 ) ) ).
% sup.idem
thf(fact_191_sup__apply,axiom,
! [B: $tType,A: $tType] :
( ( semilattice_sup @ B )
=> ( ( sup_sup @ ( A > B ) )
= ( ^ [F5: A > B,G6: A > B,X7: A] : ( sup_sup @ B @ ( F5 @ X7 ) @ ( G6 @ X7 ) ) ) ) ) ).
% sup_apply
thf(fact_192_sup__bot__left,axiom,
! [A: $tType] :
( ( bounde1808546759up_bot @ A )
=> ! [X6: A] :
( ( sup_sup @ A @ ( bot_bot @ A ) @ X6 )
= X6 ) ) ).
% sup_bot_left
thf(fact_193_sup__bot__right,axiom,
! [A: $tType] :
( ( bounde1808546759up_bot @ A )
=> ! [X6: A] :
( ( sup_sup @ A @ X6 @ ( bot_bot @ A ) )
= X6 ) ) ).
% sup_bot_right
thf(fact_194_bot__eq__sup__iff,axiom,
! [A: $tType] :
( ( bounde1808546759up_bot @ A )
=> ! [X6: A,Y: A] :
( ( ( bot_bot @ A )
= ( sup_sup @ A @ X6 @ Y ) )
= ( ( X6
= ( bot_bot @ A ) )
& ( Y
= ( bot_bot @ A ) ) ) ) ) ).
% bot_eq_sup_iff
thf(fact_195_sup__eq__bot__iff,axiom,
! [A: $tType] :
( ( bounde1808546759up_bot @ A )
=> ! [X6: A,Y: A] :
( ( ( sup_sup @ A @ X6 @ Y )
= ( bot_bot @ A ) )
= ( ( X6
= ( bot_bot @ A ) )
& ( Y
= ( bot_bot @ A ) ) ) ) ) ).
% sup_eq_bot_iff
thf(fact_196_edge__preserving__unionI,axiom,
! [A: $tType,B: $tType,C: $tType,F2: set @ ( product_prod @ A @ B ),A5: set @ ( product_prod @ C @ ( product_prod @ A @ A ) ),G: set @ ( product_prod @ C @ ( product_prod @ B @ B ) ),B6: set @ ( product_prod @ C @ ( product_prod @ A @ A ) )] :
( ( edge_preserving @ A @ B @ C @ F2 @ A5 @ G )
=> ( ( edge_preserving @ A @ B @ C @ F2 @ B6 @ G )
=> ( edge_preserving @ A @ B @ C @ F2 @ ( sup_sup @ ( set @ ( product_prod @ C @ ( product_prod @ A @ A ) ) ) @ A5 @ B6 ) @ G ) ) ) ).
% edge_preserving_unionI
thf(fact_197_graph__union__edges,axiom,
! [B: $tType,A: $tType,G_1: labeled_graph @ A @ B,G_2: labeled_graph @ A @ B] :
( ( labeled_edges @ A @ B @ ( graph_union @ A @ B @ G_1 @ G_2 ) )
= ( sup_sup @ ( set @ ( product_prod @ A @ ( product_prod @ B @ B ) ) ) @ ( labeled_edges @ A @ B @ G_1 ) @ ( labeled_edges @ A @ B @ G_2 ) ) ) ).
% graph_union_edges
thf(fact_198_sup__left__commute,axiom,
! [A: $tType] :
( ( semilattice_sup @ A )
=> ! [X6: A,Y: A,Z: A] :
( ( sup_sup @ A @ X6 @ ( sup_sup @ A @ Y @ Z ) )
= ( sup_sup @ A @ Y @ ( sup_sup @ A @ X6 @ Z ) ) ) ) ).
% sup_left_commute
thf(fact_199_sup_Oleft__commute,axiom,
! [A: $tType] :
( ( semilattice_sup @ A )
=> ! [B2: A,A2: A,C5: A] :
( ( sup_sup @ A @ B2 @ ( sup_sup @ A @ A2 @ C5 ) )
= ( sup_sup @ A @ A2 @ ( sup_sup @ A @ B2 @ C5 ) ) ) ) ).
% sup.left_commute
thf(fact_200_sup__commute,axiom,
! [A: $tType] :
( ( semilattice_sup @ A )
=> ( ( sup_sup @ A )
= ( ^ [X7: A,Y6: A] : ( sup_sup @ A @ Y6 @ X7 ) ) ) ) ).
% sup_commute
thf(fact_201_sup_Ocommute,axiom,
! [A: $tType] :
( ( semilattice_sup @ A )
=> ( ( sup_sup @ A )
= ( ^ [A6: A,B9: A] : ( sup_sup @ A @ B9 @ A6 ) ) ) ) ).
% sup.commute
thf(fact_202_sup__assoc,axiom,
! [A: $tType] :
( ( semilattice_sup @ A )
=> ! [X6: A,Y: A,Z: A] :
( ( sup_sup @ A @ ( sup_sup @ A @ X6 @ Y ) @ Z )
= ( sup_sup @ A @ X6 @ ( sup_sup @ A @ Y @ Z ) ) ) ) ).
% sup_assoc
thf(fact_203_sup_Oassoc,axiom,
! [A: $tType] :
( ( semilattice_sup @ A )
=> ! [A2: A,B2: A,C5: A] :
( ( sup_sup @ A @ ( sup_sup @ A @ A2 @ B2 ) @ C5 )
= ( sup_sup @ A @ A2 @ ( sup_sup @ A @ B2 @ C5 ) ) ) ) ).
% sup.assoc
thf(fact_204_boolean__algebra__cancel_Osup2,axiom,
! [A: $tType] :
( ( semilattice_sup @ A )
=> ! [B6: A,K: A,B2: A,A2: A] :
( ( B6
= ( sup_sup @ A @ K @ B2 ) )
=> ( ( sup_sup @ A @ A2 @ B6 )
= ( sup_sup @ A @ K @ ( sup_sup @ A @ A2 @ B2 ) ) ) ) ) ).
% boolean_algebra_cancel.sup2
thf(fact_205_boolean__algebra__cancel_Osup1,axiom,
! [A: $tType] :
( ( semilattice_sup @ A )
=> ! [A5: A,K: A,A2: A,B2: A] :
( ( A5
= ( sup_sup @ A @ K @ A2 ) )
=> ( ( sup_sup @ A @ A5 @ B2 )
= ( sup_sup @ A @ K @ ( sup_sup @ A @ A2 @ B2 ) ) ) ) ) ).
% boolean_algebra_cancel.sup1
thf(fact_206_sup__fun__def,axiom,
! [B: $tType,A: $tType] :
( ( semilattice_sup @ B )
=> ( ( sup_sup @ ( A > B ) )
= ( ^ [F5: A > B,G6: A > B,X7: A] : ( sup_sup @ B @ ( F5 @ X7 ) @ ( G6 @ X7 ) ) ) ) ) ).
% sup_fun_def
thf(fact_207_inf__sup__aci_I5_J,axiom,
! [A: $tType] :
( ( lattice @ A )
=> ( ( sup_sup @ A )
= ( ^ [X7: A,Y6: A] : ( sup_sup @ A @ Y6 @ X7 ) ) ) ) ).
% inf_sup_aci(5)
thf(fact_208_inf__sup__aci_I6_J,axiom,
! [A: $tType] :
( ( lattice @ A )
=> ! [X6: A,Y: A,Z: A] :
( ( sup_sup @ A @ ( sup_sup @ A @ X6 @ Y ) @ Z )
= ( sup_sup @ A @ X6 @ ( sup_sup @ A @ Y @ Z ) ) ) ) ).
% inf_sup_aci(6)
thf(fact_209_inf__sup__aci_I7_J,axiom,
! [A: $tType] :
( ( lattice @ A )
=> ! [X6: A,Y: A,Z: A] :
( ( sup_sup @ A @ X6 @ ( sup_sup @ A @ Y @ Z ) )
= ( sup_sup @ A @ Y @ ( sup_sup @ A @ X6 @ Z ) ) ) ) ).
% inf_sup_aci(7)
thf(fact_210_inf__sup__aci_I8_J,axiom,
! [A: $tType] :
( ( lattice @ A )
=> ! [X6: A,Y: A] :
( ( sup_sup @ A @ X6 @ ( sup_sup @ A @ X6 @ Y ) )
= ( sup_sup @ A @ X6 @ Y ) ) ) ).
% inf_sup_aci(8)
thf(fact_211_the__elem__eq,axiom,
! [A: $tType,X6: A] :
( ( the_elem @ A @ ( insert @ A @ X6 @ ( bot_bot @ ( set @ A ) ) ) )
= X6 ) ).
% the_elem_eq
thf(fact_212_Collect__empty__eq__bot,axiom,
! [A: $tType,P2: A > $o] :
( ( ( collect @ A @ P2 )
= ( bot_bot @ ( set @ A ) ) )
= ( P2
= ( bot_bot @ ( A > $o ) ) ) ) ).
% Collect_empty_eq_bot
thf(fact_213_is__singletonI,axiom,
! [A: $tType,X6: A] : ( is_singleton @ A @ ( insert @ A @ X6 @ ( bot_bot @ ( set @ A ) ) ) ) ).
% is_singletonI
thf(fact_214_is__singleton__the__elem,axiom,
! [A: $tType] :
( ( is_singleton @ A )
= ( ^ [A7: set @ A] :
( A7
= ( insert @ A @ ( the_elem @ A @ A7 ) @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ).
% is_singleton_the_elem
thf(fact_215_is__singletonI_H,axiom,
! [A: $tType,A5: set @ A] :
( ( A5
!= ( bot_bot @ ( set @ A ) ) )
=> ( ! [X2: A,Y3: A] :
( ( member @ A @ X2 @ A5 )
=> ( ( member @ A @ Y3 @ A5 )
=> ( X2 = Y3 ) ) )
=> ( is_singleton @ A @ A5 ) ) ) ).
% is_singletonI'
thf(fact_216_is__singletonE,axiom,
! [A: $tType,A5: set @ A] :
( ( is_singleton @ A @ A5 )
=> ~ ! [X2: A] :
( A5
!= ( insert @ A @ X2 @ ( bot_bot @ ( set @ A ) ) ) ) ) ).
% is_singletonE
thf(fact_217_is__singleton__def,axiom,
! [A: $tType] :
( ( is_singleton @ A )
= ( ^ [A7: set @ A] :
? [X7: A] :
( A7
= ( insert @ A @ X7 @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ).
% is_singleton_def
thf(fact_218_labeled__graph_Osplit__sel,axiom,
! [A: $tType,V: $tType,L: $tType,P2: A > $o,F2: ( set @ ( product_prod @ L @ ( product_prod @ V @ V ) ) ) > ( set @ V ) > A,Labeled_graph: labeled_graph @ L @ V] :
( ( P2 @ ( labele1974067554_graph @ L @ V @ A @ F2 @ Labeled_graph ) )
= ( ( Labeled_graph
= ( labeled_LG @ L @ V @ ( labeled_edges @ L @ V @ Labeled_graph ) @ ( labeled_vertices @ L @ V @ Labeled_graph ) ) )
=> ( P2 @ ( F2 @ ( labeled_edges @ L @ V @ Labeled_graph ) @ ( labeled_vertices @ L @ V @ Labeled_graph ) ) ) ) ) ).
% labeled_graph.split_sel
thf(fact_219_labeled__graph_Osplit__sel__asm,axiom,
! [A: $tType,V: $tType,L: $tType,P2: A > $o,F2: ( set @ ( product_prod @ L @ ( product_prod @ V @ V ) ) ) > ( set @ V ) > A,Labeled_graph: labeled_graph @ L @ V] :
( ( P2 @ ( labele1974067554_graph @ L @ V @ A @ F2 @ Labeled_graph ) )
= ( ~ ( ( Labeled_graph
= ( labeled_LG @ L @ V @ ( labeled_edges @ L @ V @ Labeled_graph ) @ ( labeled_vertices @ L @ V @ Labeled_graph ) ) )
& ~ ( P2 @ ( F2 @ ( labeled_edges @ L @ V @ Labeled_graph ) @ ( labeled_vertices @ L @ V @ Labeled_graph ) ) ) ) ) ) ).
% labeled_graph.split_sel_asm
thf(fact_220_labeled__graph_Ocase,axiom,
! [L: $tType,A: $tType,V: $tType,F2: ( set @ ( product_prod @ L @ ( product_prod @ V @ V ) ) ) > ( set @ V ) > A,X1: set @ ( product_prod @ L @ ( product_prod @ V @ V ) ),X22: set @ V] :
( ( labele1974067554_graph @ L @ V @ A @ F2 @ ( labeled_LG @ L @ V @ X1 @ X22 ) )
= ( F2 @ X1 @ X22 ) ) ).
% labeled_graph.case
thf(fact_221_labeled__graph_Ocase__eq__if,axiom,
! [A: $tType,V: $tType,L: $tType] :
( ( labele1974067554_graph @ L @ V @ A )
= ( ^ [F5: ( set @ ( product_prod @ L @ ( product_prod @ V @ V ) ) ) > ( set @ V ) > A,Labeled_graph3: labeled_graph @ L @ V] : ( F5 @ ( labeled_edges @ L @ V @ Labeled_graph3 ) @ ( labeled_vertices @ L @ V @ Labeled_graph3 ) ) ) ) ).
% labeled_graph.case_eq_if
thf(fact_222_extensibleI,axiom,
! [A: $tType,C: $tType,B: $tType,R22: labeled_graph @ A @ B,G: labeled_graph @ A @ C,G5: set @ ( product_prod @ B @ C ),R1: labeled_graph @ A @ B,F2: set @ ( product_prod @ B @ C )] :
( ( graph_homomorphism @ A @ B @ C @ R22 @ G @ G5 )
=> ( ( agree_on @ A @ B @ C @ R1 @ F2 @ G5 )
=> ( extensible @ A @ B @ C @ ( product_Pair @ ( labeled_graph @ A @ B ) @ ( labeled_graph @ A @ B ) @ R1 @ R22 ) @ G @ F2 ) ) ) ).
% extensibleI
thf(fact_223_total__on__singleton,axiom,
! [A: $tType,X6: A] : ( total_on @ A @ ( insert @ A @ X6 @ ( bot_bot @ ( set @ A ) ) ) @ ( insert @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X6 @ X6 ) @ ( bot_bot @ ( set @ ( product_prod @ A @ A ) ) ) ) ) ).
% total_on_singleton
thf(fact_224_agree__on__refl,axiom,
! [A: $tType,C: $tType,B: $tType,R2: labeled_graph @ A @ B,F2: set @ ( product_prod @ B @ C )] : ( agree_on @ A @ B @ C @ R2 @ F2 @ F2 ) ).
% agree_on_refl
thf(fact_225_agree__on__comm,axiom,
! [C: $tType,B: $tType,A: $tType] :
( ( agree_on @ A @ B @ C )
= ( ^ [X9: labeled_graph @ A @ B,F5: set @ ( product_prod @ B @ C ),G6: set @ ( product_prod @ B @ C )] : ( agree_on @ A @ B @ C @ X9 @ G6 @ F5 ) ) ) ).
% agree_on_comm
thf(fact_226_agree__on__empty,axiom,
! [A: $tType,C: $tType,B: $tType,F2: set @ ( product_prod @ B @ C ),G5: set @ ( product_prod @ B @ C )] : ( agree_on @ A @ B @ C @ ( labeled_LG @ A @ B @ ( bot_bot @ ( set @ ( product_prod @ A @ ( product_prod @ B @ B ) ) ) ) @ ( bot_bot @ ( set @ B ) ) ) @ F2 @ G5 ) ).
% agree_on_empty
thf(fact_227_total__onI,axiom,
! [A: $tType,A5: set @ A,R: set @ ( product_prod @ A @ A )] :
( ! [X2: A,Y3: A] :
( ( member @ A @ X2 @ A5 )
=> ( ( member @ A @ Y3 @ A5 )
=> ( ( X2 != Y3 )
=> ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X2 @ Y3 ) @ R )
| ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Y3 @ X2 ) @ R ) ) ) ) )
=> ( total_on @ A @ A5 @ R ) ) ).
% total_onI
thf(fact_228_total__on__def,axiom,
! [A: $tType] :
( ( total_on @ A )
= ( ^ [A7: set @ A,R3: set @ ( product_prod @ A @ A )] :
! [X7: A] :
( ( member @ A @ X7 @ A7 )
=> ! [Y6: A] :
( ( member @ A @ Y6 @ A7 )
=> ( ( X7 != Y6 )
=> ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X7 @ Y6 ) @ R3 )
| ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Y6 @ X7 ) @ R3 ) ) ) ) ) ) ) ).
% total_on_def
thf(fact_229_total__on__empty,axiom,
! [A: $tType,R: set @ ( product_prod @ A @ A )] : ( total_on @ A @ ( bot_bot @ ( set @ A ) ) @ R ) ).
% total_on_empty
thf(fact_230_agree__on__trans,axiom,
! [A: $tType,C: $tType,B: $tType,X8: labeled_graph @ A @ B,F2: set @ ( product_prod @ B @ C ),G5: set @ ( product_prod @ B @ C ),H: set @ ( product_prod @ B @ C )] :
( ( agree_on @ A @ B @ C @ X8 @ F2 @ G5 )
=> ( ( agree_on @ A @ B @ C @ X8 @ G5 @ H )
=> ( agree_on @ A @ B @ C @ X8 @ F2 @ H ) ) ) ).
% agree_on_trans
thf(fact_231_agree__on__subg__compose,axiom,
! [A: $tType,C: $tType,B: $tType,R2: labeled_graph @ A @ B,G5: set @ ( product_prod @ B @ C ),H: set @ ( product_prod @ B @ C ),F4: labeled_graph @ A @ B,F2: set @ ( product_prod @ B @ C )] :
( ( agree_on @ A @ B @ C @ R2 @ G5 @ H )
=> ( ( agree_on @ A @ B @ C @ F4 @ F2 @ G5 )
=> ( ( graph_homomorphism @ A @ B @ B @ F4 @ R2 @ ( id_on @ B @ ( labeled_vertices @ A @ B @ F4 ) ) )
=> ( agree_on @ A @ B @ C @ F4 @ F2 @ H ) ) ) ) ).
% agree_on_subg_compose
thf(fact_232_Field__insert,axiom,
! [A: $tType,A2: A,B2: A,R: set @ ( product_prod @ A @ A )] :
( ( field @ A @ ( insert @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A2 @ B2 ) @ R ) )
= ( sup_sup @ ( set @ A ) @ ( insert @ A @ A2 @ ( insert @ A @ B2 @ ( bot_bot @ ( set @ A ) ) ) ) @ ( field @ A @ R ) ) ) ).
% Field_insert
thf(fact_233_subgraph__subset_I2_J,axiom,
! [B: $tType,A: $tType,G_1: labeled_graph @ A @ B,G_2: labeled_graph @ A @ B] :
( ( graph_homomorphism @ A @ B @ B @ G_1 @ G_2 @ ( id_on @ B @ ( labeled_vertices @ A @ B @ G_1 ) ) )
=> ( ord_less_eq @ ( set @ ( product_prod @ A @ ( product_prod @ B @ B ) ) ) @ ( labeled_edges @ A @ B @ ( restrict @ A @ B @ G_1 ) ) @ ( labeled_edges @ A @ B @ G_2 ) ) ) ).
% subgraph_subset(2)
thf(fact_234_order__refl,axiom,
! [A: $tType] :
( ( preorder @ A )
=> ! [X6: A] : ( ord_less_eq @ A @ X6 @ X6 ) ) ).
% order_refl
thf(fact_235_subsetI,axiom,
! [A: $tType,A5: set @ A,B6: set @ A] :
( ! [X2: A] :
( ( member @ A @ X2 @ A5 )
=> ( member @ A @ X2 @ B6 ) )
=> ( ord_less_eq @ ( set @ A ) @ A5 @ B6 ) ) ).
% subsetI
thf(fact_236_subset__antisym,axiom,
! [A: $tType,A5: set @ A,B6: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ A5 @ B6 )
=> ( ( ord_less_eq @ ( set @ A ) @ B6 @ A5 )
=> ( A5 = B6 ) ) ) ).
% subset_antisym
thf(fact_237_le__sup__iff,axiom,
! [A: $tType] :
( ( semilattice_sup @ A )
=> ! [X6: A,Y: A,Z: A] :
( ( ord_less_eq @ A @ ( sup_sup @ A @ X6 @ Y ) @ Z )
= ( ( ord_less_eq @ A @ X6 @ Z )
& ( ord_less_eq @ A @ Y @ Z ) ) ) ) ).
% le_sup_iff
thf(fact_238_sup_Obounded__iff,axiom,
! [A: $tType] :
( ( semilattice_sup @ A )
=> ! [B2: A,C5: A,A2: A] :
( ( ord_less_eq @ A @ ( sup_sup @ A @ B2 @ C5 ) @ A2 )
= ( ( ord_less_eq @ A @ B2 @ A2 )
& ( ord_less_eq @ A @ C5 @ A2 ) ) ) ) ).
% sup.bounded_iff
thf(fact_239_empty__subsetI,axiom,
! [A: $tType,A5: set @ A] : ( ord_less_eq @ ( set @ A ) @ ( bot_bot @ ( set @ A ) ) @ A5 ) ).
% empty_subsetI
thf(fact_240_subset__empty,axiom,
! [A: $tType,A5: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ A5 @ ( bot_bot @ ( set @ A ) ) )
= ( A5
= ( bot_bot @ ( set @ A ) ) ) ) ).
% subset_empty
thf(fact_241_insert__subset,axiom,
! [A: $tType,X6: A,A5: set @ A,B6: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ ( insert @ A @ X6 @ A5 ) @ B6 )
= ( ( member @ A @ X6 @ B6 )
& ( ord_less_eq @ ( set @ A ) @ A5 @ B6 ) ) ) ).
% insert_subset
thf(fact_242_Un__subset__iff,axiom,
! [A: $tType,A5: set @ A,B6: set @ A,C8: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ ( sup_sup @ ( set @ A ) @ A5 @ B6 ) @ C8 )
= ( ( ord_less_eq @ ( set @ A ) @ A5 @ C8 )
& ( ord_less_eq @ ( set @ A ) @ B6 @ C8 ) ) ) ).
% Un_subset_iff
thf(fact_243_singleton__insert__inj__eq,axiom,
! [A: $tType,B2: A,A2: A,A5: set @ A] :
( ( ( insert @ A @ B2 @ ( bot_bot @ ( set @ A ) ) )
= ( insert @ A @ A2 @ A5 ) )
= ( ( A2 = B2 )
& ( ord_less_eq @ ( set @ A ) @ A5 @ ( insert @ A @ B2 @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ).
% singleton_insert_inj_eq
thf(fact_244_singleton__insert__inj__eq_H,axiom,
! [A: $tType,A2: A,A5: set @ A,B2: A] :
( ( ( insert @ A @ A2 @ A5 )
= ( insert @ A @ B2 @ ( bot_bot @ ( set @ A ) ) ) )
= ( ( A2 = B2 )
& ( ord_less_eq @ ( set @ A ) @ A5 @ ( insert @ A @ B2 @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ).
% singleton_insert_inj_eq'
thf(fact_245_Field__empty,axiom,
! [A: $tType] :
( ( field @ A @ ( bot_bot @ ( set @ ( product_prod @ A @ A ) ) ) )
= ( bot_bot @ ( set @ A ) ) ) ).
% Field_empty
thf(fact_246_Field__Un,axiom,
! [A: $tType,R: set @ ( product_prod @ A @ A ),S3: set @ ( product_prod @ A @ A )] :
( ( field @ A @ ( sup_sup @ ( set @ ( product_prod @ A @ A ) ) @ R @ S3 ) )
= ( sup_sup @ ( set @ A ) @ ( field @ A @ R ) @ ( field @ A @ S3 ) ) ) ).
% Field_Un
thf(fact_247_graph__unionI,axiom,
! [B: $tType,A: $tType,G_1: labeled_graph @ A @ B,G_2: labeled_graph @ A @ B] :
( ( ord_less_eq @ ( set @ ( product_prod @ A @ ( product_prod @ B @ B ) ) ) @ ( labeled_edges @ A @ B @ G_1 ) @ ( labeled_edges @ A @ B @ G_2 ) )
=> ( ( ord_less_eq @ ( set @ B ) @ ( labeled_vertices @ A @ B @ G_1 ) @ ( labeled_vertices @ A @ B @ G_2 ) )
=> ( ( graph_union @ A @ B @ G_1 @ G_2 )
= G_2 ) ) ) ).
% graph_unionI
thf(fact_248_inf__sup__ord_I4_J,axiom,
! [A: $tType] :
( ( lattice @ A )
=> ! [Y: A,X6: A] : ( ord_less_eq @ A @ Y @ ( sup_sup @ A @ X6 @ Y ) ) ) ).
% inf_sup_ord(4)
thf(fact_249_inf__sup__ord_I3_J,axiom,
! [A: $tType] :
( ( lattice @ A )
=> ! [X6: A,Y: A] : ( ord_less_eq @ A @ X6 @ ( sup_sup @ A @ X6 @ Y ) ) ) ).
% inf_sup_ord(3)
thf(fact_250_le__supE,axiom,
! [A: $tType] :
( ( semilattice_sup @ A )
=> ! [A2: A,B2: A,X6: A] :
( ( ord_less_eq @ A @ ( sup_sup @ A @ A2 @ B2 ) @ X6 )
=> ~ ( ( ord_less_eq @ A @ A2 @ X6 )
=> ~ ( ord_less_eq @ A @ B2 @ X6 ) ) ) ) ).
% le_supE
thf(fact_251_le__supI,axiom,
! [A: $tType] :
( ( semilattice_sup @ A )
=> ! [A2: A,X6: A,B2: A] :
( ( ord_less_eq @ A @ A2 @ X6 )
=> ( ( ord_less_eq @ A @ B2 @ X6 )
=> ( ord_less_eq @ A @ ( sup_sup @ A @ A2 @ B2 ) @ X6 ) ) ) ) ).
% le_supI
thf(fact_252_sup__ge1,axiom,
! [A: $tType] :
( ( semilattice_sup @ A )
=> ! [X6: A,Y: A] : ( ord_less_eq @ A @ X6 @ ( sup_sup @ A @ X6 @ Y ) ) ) ).
% sup_ge1
thf(fact_253_sup__ge2,axiom,
! [A: $tType] :
( ( semilattice_sup @ A )
=> ! [Y: A,X6: A] : ( ord_less_eq @ A @ Y @ ( sup_sup @ A @ X6 @ Y ) ) ) ).
% sup_ge2
thf(fact_254_le__supI1,axiom,
! [A: $tType] :
( ( semilattice_sup @ A )
=> ! [X6: A,A2: A,B2: A] :
( ( ord_less_eq @ A @ X6 @ A2 )
=> ( ord_less_eq @ A @ X6 @ ( sup_sup @ A @ A2 @ B2 ) ) ) ) ).
% le_supI1
thf(fact_255_le__supI2,axiom,
! [A: $tType] :
( ( semilattice_sup @ A )
=> ! [X6: A,B2: A,A2: A] :
( ( ord_less_eq @ A @ X6 @ B2 )
=> ( ord_less_eq @ A @ X6 @ ( sup_sup @ A @ A2 @ B2 ) ) ) ) ).
% le_supI2
% Type constructors (22)
thf(tcon_HOL_Obool___Lattices_Obounded__lattice,axiom,
bounded_lattice @ $o ).
thf(tcon_Set_Oset___Lattices_Obounded__lattice_1,axiom,
! [A8: $tType] : ( bounded_lattice @ ( set @ A8 ) ) ).
thf(tcon_fun___Lattices_Obounded__lattice_2,axiom,
! [A8: $tType,A9: $tType] :
( ( bounded_lattice @ A9 )
=> ( bounded_lattice @ ( A8 > A9 ) ) ) ).
thf(tcon_fun___Lattices_Obounded__semilattice__sup__bot,axiom,
! [A8: $tType,A9: $tType] :
( ( bounded_lattice @ A9 )
=> ( bounde1808546759up_bot @ ( A8 > A9 ) ) ) ).
thf(tcon_fun___Lattices_Osemilattice__sup,axiom,
! [A8: $tType,A9: $tType] :
( ( semilattice_sup @ A9 )
=> ( semilattice_sup @ ( A8 > A9 ) ) ) ).
thf(tcon_fun___Orderings_Opreorder,axiom,
! [A8: $tType,A9: $tType] :
( ( preorder @ A9 )
=> ( preorder @ ( A8 > A9 ) ) ) ).
thf(tcon_fun___Lattices_Olattice,axiom,
! [A8: $tType,A9: $tType] :
( ( lattice @ A9 )
=> ( lattice @ ( A8 > A9 ) ) ) ).
thf(tcon_fun___Orderings_Obot,axiom,
! [A8: $tType,A9: $tType] :
( ( bot @ A9 )
=> ( bot @ ( A8 > A9 ) ) ) ).
thf(tcon_Nat_Onat___Lattices_Osemilattice__sup_3,axiom,
semilattice_sup @ nat ).
thf(tcon_Nat_Onat___Orderings_Opreorder_4,axiom,
preorder @ nat ).
thf(tcon_Nat_Onat___Lattices_Olattice_5,axiom,
lattice @ nat ).
thf(tcon_Nat_Onat___Orderings_Obot_6,axiom,
bot @ nat ).
thf(tcon_Set_Oset___Lattices_Obounded__semilattice__sup__bot_7,axiom,
! [A8: $tType] : ( bounde1808546759up_bot @ ( set @ A8 ) ) ).
thf(tcon_Set_Oset___Lattices_Osemilattice__sup_8,axiom,
! [A8: $tType] : ( semilattice_sup @ ( set @ A8 ) ) ).
thf(tcon_Set_Oset___Orderings_Opreorder_9,axiom,
! [A8: $tType] : ( preorder @ ( set @ A8 ) ) ).
thf(tcon_Set_Oset___Lattices_Olattice_10,axiom,
! [A8: $tType] : ( lattice @ ( set @ A8 ) ) ).
thf(tcon_Set_Oset___Orderings_Obot_11,axiom,
! [A8: $tType] : ( bot @ ( set @ A8 ) ) ).
thf(tcon_HOL_Obool___Lattices_Obounded__semilattice__sup__bot_12,axiom,
bounde1808546759up_bot @ $o ).
thf(tcon_HOL_Obool___Lattices_Osemilattice__sup_13,axiom,
semilattice_sup @ $o ).
thf(tcon_HOL_Obool___Orderings_Opreorder_14,axiom,
preorder @ $o ).
thf(tcon_HOL_Obool___Lattices_Olattice_15,axiom,
lattice @ $o ).
thf(tcon_HOL_Obool___Orderings_Obot_16,axiom,
bot @ $o ).
% Conjectures (1)
thf(conj_0,conjecture,
member @ ( product_prod @ v @ v ) @ ( product_Pair @ v @ v @ ( m @ xa ) @ ( m @ xa ) ) @ ( getRel @ ( standard_Constant @ v2 ) @ v @ ( standard_S_Const @ v2 @ y ) @ g ) ).
%------------------------------------------------------------------------------