TPTP Problem File: ITP175^2.p
View Solutions
- Solve Problem
%------------------------------------------------------------------------------
% File : ITP175^2 : TPTP v8.2.0. Released v7.5.0.
% Domain : Interactive Theorem Proving
% Problem : Sledgehammer StandardRules problem prob_201__5389132_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_201__5389132_1 [Des21]
% Status : Theorem
% Rating : 0.00 v7.5.0
% Syntax : Number of formulae : 348 ( 122 unt; 69 typ; 0 def)
% Number of atoms : 761 ( 235 equ; 0 cnn)
% Maximal formula atoms : 13 ( 2 avg)
% Number of connectives : 5778 ( 65 ~; 7 |; 39 &;5319 @)
% ( 0 <=>; 348 =>; 0 <=; 0 <~>)
% Maximal formula depth : 28 ( 9 avg)
% Number of types : 4 ( 3 usr)
% Number of type conns : 186 ( 186 >; 0 *; 0 +; 0 <<)
% Number of symbols : 67 ( 66 usr; 5 con; 0-8 aty)
% Number of variables : 1303 ( 50 ^;1150 !; 6 ?;1303 :)
% ( 97 !>; 0 ?*; 0 @-; 0 @+)
% SPC : TH1_THM_EQU_NAR
% Comments : This file was generated by Sledgehammer 2021-02-23 16:22:17.769
%------------------------------------------------------------------------------
% Could-be-implicit typings (8)
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_b,type,
b: $tType ).
thf(ty_tf_a,type,
a: $tType ).
% Explicit typings (61)
thf(sy_cl_HOL_Otype,type,
type:
!>[A: $tType] : $o ).
thf(sy_cl_Orderings_Obot,type,
bot:
!>[A: $tType] : $o ).
thf(sy_cl_Orderings_Oord,type,
ord:
!>[A: $tType] : $o ).
thf(sy_cl_Orderings_Oorder,type,
order:
!>[A: $tType] : $o ).
thf(sy_cl_Orderings_Olinorder,type,
linorder:
!>[A: $tType] : $o ).
thf(sy_cl_Orderings_Opreorder,type,
preorder:
!>[A: $tType] : $o ).
thf(sy_cl_Orderings_Oorder__bot,type,
order_bot:
!>[A: $tType] : $o ).
thf(sy_c_BNF__Greatest__Fixpoint_OrelImage,type,
bNF_Gr1317331620lImage:
!>[B: $tType,A: $tType] : ( ( set @ ( product_prod @ B @ B ) ) > ( B > A ) > ( set @ ( product_prod @ A @ A ) ) ) ).
thf(sy_c_BNF__Greatest__Fixpoint_Ouniv,type,
bNF_Greatest_univ:
!>[B: $tType,A: $tType] : ( ( B > A ) > ( set @ B ) > A ) ).
thf(sy_c_Equiv__Relations_Ocongruent,type,
equiv_congruent:
!>[A: $tType,B: $tType] : ( ( set @ ( product_prod @ A @ A ) ) > ( A > B ) > $o ) ).
thf(sy_c_Equiv__Relations_Ocongruent2,type,
equiv_congruent2:
!>[A: $tType,B: $tType,C: $tType] : ( ( set @ ( product_prod @ A @ A ) ) > ( set @ ( product_prod @ B @ B ) ) > ( A > B > C ) > $o ) ).
thf(sy_c_Equiv__Relations_Oequiv,type,
equiv_equiv:
!>[A: $tType] : ( ( set @ A ) > ( set @ ( product_prod @ A @ A ) ) > $o ) ).
thf(sy_c_Equiv__Relations_Oproj,type,
equiv_proj:
!>[B: $tType,A: $tType] : ( ( set @ ( product_prod @ B @ A ) ) > B > ( set @ A ) ) ).
thf(sy_c_Equiv__Relations_Oquotient,type,
equiv_quotient:
!>[A: $tType] : ( ( set @ A ) > ( set @ ( product_prod @ A @ A ) ) > ( set @ ( set @ A ) ) ) ).
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_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_Oon__triple,type,
on_triple:
!>[B: $tType,C: $tType,A: $tType] : ( ( set @ ( product_prod @ B @ C ) ) > ( set @ ( product_prod @ ( product_prod @ A @ ( product_prod @ B @ B ) ) @ ( product_prod @ A @ ( product_prod @ C @ C ) ) ) ) ) ).
thf(sy_c_LabeledGraphs_Orestrict,type,
restrict:
!>[A: $tType,B: $tType] : ( ( labeled_graph @ A @ B ) > ( labeled_graph @ A @ B ) ) ).
thf(sy_c_MissingRelation_Oidempotent,type,
idempotent:
!>[A: $tType] : ( ( set @ ( product_prod @ A @ A ) ) > $o ) ).
thf(sy_c_Order__Relation_Olinear__order__on,type,
order_1409979114der_on:
!>[A: $tType] : ( ( set @ A ) > ( set @ ( product_prod @ A @ A ) ) > $o ) ).
thf(sy_c_Order__Relation_Opreorder__on,type,
order_preorder_on:
!>[A: $tType] : ( ( set @ A ) > ( set @ ( product_prod @ A @ A ) ) > $o ) ).
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_Ointernal__case__prod,type,
produc2004651681e_prod:
!>[A: $tType,B: $tType,C: $tType] : ( ( A > B > C ) > ( product_prod @ A @ B ) > C ) ).
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_OId__on,type,
id_on:
!>[A: $tType] : ( ( set @ A ) > ( set @ ( product_prod @ A @ A ) ) ) ).
thf(sy_c_Relation_Orefl__on,type,
refl_on:
!>[A: $tType] : ( ( set @ A ) > ( set @ ( product_prod @ A @ A ) ) > $o ) ).
thf(sy_c_Relation_Osym,type,
sym:
!>[A: $tType] : ( ( set @ ( product_prod @ A @ A ) ) > $o ) ).
thf(sy_c_Relation_Ototal__on,type,
total_on:
!>[A: $tType] : ( ( set @ A ) > ( set @ ( product_prod @ A @ A ) ) > $o ) ).
thf(sy_c_Relation_Otrans,type,
trans:
!>[A: $tType] : ( ( 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_Ofin__maintained,type,
fin_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_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__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_Othe__elem,type,
the_elem:
!>[A: $tType] : ( ( set @ A ) > A ) ).
thf(sy_c_StandardRules__Mirabelle__xlrcbcygrg_Ocongruence__rule,type,
standa1068660421e_rule:
!>[L: $tType] : ( L > L > ( product_prod @ ( labeled_graph @ L @ nat ) @ ( labeled_graph @ L @ 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_Oreflexivity__rule,type,
standa245363227y_rule:
!>[L: $tType] : ( L > ( product_prod @ ( labeled_graph @ L @ nat ) @ ( labeled_graph @ L @ nat ) ) ) ).
thf(sy_c_StandardRules__Mirabelle__xlrcbcygrg_Osymmetry__rule,type,
standa1805702094y_rule:
!>[L: $tType] : ( L > ( product_prod @ ( labeled_graph @ L @ nat ) @ ( labeled_graph @ L @ nat ) ) ) ).
thf(sy_c_StandardRules__Mirabelle__xlrcbcygrg_Otransitive__rule,type,
standa2114124375e_rule:
!>[L: $tType] : ( L > ( product_prod @ ( labeled_graph @ L @ nat ) @ ( labeled_graph @ L @ nat ) ) ) ).
thf(sy_c_Wfrec_Osame__fst,type,
same_fst:
!>[A: $tType,B: $tType] : ( ( A > $o ) > ( A > ( set @ ( product_prod @ B @ B ) ) ) > ( set @ ( product_prod @ ( product_prod @ A @ B ) @ ( product_prod @ A @ B ) ) ) ) ).
thf(sy_c_member,type,
member:
!>[A: $tType] : ( A > ( set @ A ) > $o ) ).
thf(sy_v_G,type,
g: labeled_graph @ a @ b ).
thf(sy_v_I,type,
i: a ).
thf(sy_v_l,type,
l: a ).
thf(sy_v_y____,type,
y: b ).
thf(sy_v_z____,type,
z: b ).
% Relevant facts (256)
thf(fact_0__092_060open_062refl__on_A_Ivertices_AG_J_A_IgetRel_AI_AG_J_A_092_060and_062_A_I_092_060forall_062x_Ay_O_A_Ix_M_Ay_J_A_092_060in_062_AgetRel_AI_AG_A_092_060longrightarrow_062_A_Iy_M_Ax_J_A_092_060in_062_AgetRel_AI_AG_J_A_092_060and_062_Atrans_A_IgetRel_AI_AG_J_092_060close_062,axiom,
( ( refl_on @ b @ ( labeled_vertices @ a @ b @ g ) @ ( getRel @ a @ b @ i @ g ) )
& ! [X2: b,Y: b] :
( ( member @ ( product_prod @ b @ b ) @ ( product_Pair @ b @ b @ X2 @ Y ) @ ( getRel @ a @ b @ i @ g ) )
=> ( member @ ( product_prod @ b @ b ) @ ( product_Pair @ b @ b @ Y @ X2 ) @ ( getRel @ a @ b @ i @ g ) ) )
& ( trans @ b @ ( getRel @ a @ b @ i @ g ) ) ) ).
% \<open>refl_on (vertices G) (getRel I G) \<and> (\<forall>x y. (x, y) \<in> getRel I G \<longrightarrow> (y, x) \<in> getRel I G) \<and> trans (getRel I G)\<close>
thf(fact_1_aI,axiom,
member @ ( product_prod @ b @ b ) @ ( product_Pair @ b @ b @ y @ z ) @ ( getRel @ a @ b @ i @ g ) ).
% aI
thf(fact_2_g,axiom,
( g
= ( restrict @ a @ b @ g ) ) ).
% g
thf(fact_3_m,axiom,
maintained @ a @ nat @ b @ ( standa1068660421e_rule @ a @ i @ l ) @ g ).
% m
thf(fact_4_eq,axiom,
equiv_equiv @ b @ ( labeled_vertices @ a @ b @ g ) @ ( getRel @ a @ b @ i @ g ) ).
% eq
thf(fact_5_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_6_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_7_surj__pair,axiom,
! [A: $tType,B: $tType,P: product_prod @ A @ B] :
? [X3: A,Y3: B] :
( P
= ( product_Pair @ A @ B @ X3 @ Y3 ) ) ).
% surj_pair
thf(fact_8_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_9_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_10_prod__cases3,axiom,
! [A: $tType,B: $tType,C: $tType,Y4: product_prod @ A @ ( product_prod @ B @ C )] :
~ ! [A4: A,B4: B,C2: C] :
( Y4
!= ( product_Pair @ A @ ( product_prod @ B @ C ) @ A4 @ ( product_Pair @ B @ C @ B4 @ C2 ) ) ) ).
% prod_cases3
thf(fact_11_prod__cases4,axiom,
! [A: $tType,B: $tType,C: $tType,D: $tType,Y4: product_prod @ A @ ( product_prod @ B @ ( product_prod @ C @ D ) )] :
~ ! [A4: A,B4: B,C2: C,D2: D] :
( Y4
!= ( product_Pair @ A @ ( product_prod @ B @ ( product_prod @ C @ D ) ) @ A4 @ ( product_Pair @ B @ ( product_prod @ C @ D ) @ B4 @ ( product_Pair @ C @ D @ C2 @ D2 ) ) ) ) ).
% prod_cases4
thf(fact_12_prod__cases5,axiom,
! [A: $tType,B: $tType,C: $tType,D: $tType,E: $tType,Y4: product_prod @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D @ E ) ) )] :
~ ! [A4: A,B4: B,C2: C,D2: D,E2: E] :
( Y4
!= ( 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 ) @ C2 @ ( product_Pair @ D @ E @ D2 @ E2 ) ) ) ) ) ).
% prod_cases5
thf(fact_13_prod__cases6,axiom,
! [A: $tType,B: $tType,C: $tType,D: $tType,E: $tType,F: $tType,Y4: product_prod @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D @ ( product_prod @ E @ F ) ) ) )] :
~ ! [A4: A,B4: B,C2: C,D2: D,E2: E,F2: F] :
( Y4
!= ( product_Pair @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D @ ( product_prod @ E @ F ) ) ) ) @ A4 @ ( product_Pair @ B @ ( product_prod @ C @ ( product_prod @ D @ ( product_prod @ E @ F ) ) ) @ B4 @ ( product_Pair @ C @ ( product_prod @ D @ ( product_prod @ E @ F ) ) @ C2 @ ( product_Pair @ D @ ( product_prod @ E @ F ) @ D2 @ ( product_Pair @ E @ F @ E2 @ F2 ) ) ) ) ) ) ).
% prod_cases6
thf(fact_14_prod__cases7,axiom,
! [A: $tType,B: $tType,C: $tType,D: $tType,E: $tType,F: $tType,G: $tType,Y4: product_prod @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D @ ( product_prod @ E @ ( product_prod @ F @ G ) ) ) ) )] :
~ ! [A4: A,B4: B,C2: C,D2: D,E2: E,F2: F,G2: G] :
( Y4
!= ( product_Pair @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D @ ( product_prod @ E @ ( product_prod @ F @ G ) ) ) ) ) @ A4 @ ( product_Pair @ B @ ( product_prod @ C @ ( product_prod @ D @ ( product_prod @ E @ ( product_prod @ F @ G ) ) ) ) @ B4 @ ( product_Pair @ C @ ( product_prod @ D @ ( product_prod @ E @ ( product_prod @ F @ G ) ) ) @ C2 @ ( product_Pair @ D @ ( product_prod @ E @ ( product_prod @ F @ G ) ) @ D2 @ ( product_Pair @ E @ ( product_prod @ F @ G ) @ E2 @ ( product_Pair @ F @ G @ F2 @ G2 ) ) ) ) ) ) ) ).
% prod_cases7
thf(fact_15_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_16_old_Oprod_Oexhaust,axiom,
! [A: $tType,B: $tType,Y4: product_prod @ A @ B] :
~ ! [A4: A,B4: B] :
( Y4
!= ( product_Pair @ A @ B @ A4 @ B4 ) ) ).
% old.prod.exhaust
thf(fact_17_prod__induct7,axiom,
! [G: $tType,F: $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 @ F @ G ) ) ) ) ) ) > $o,X4: product_prod @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D @ ( product_prod @ E @ ( product_prod @ F @ G ) ) ) ) )] :
( ! [A4: A,B4: B,C2: C,D2: D,E2: E,F2: F,G2: G] : ( P2 @ ( product_Pair @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D @ ( product_prod @ E @ ( product_prod @ F @ G ) ) ) ) ) @ A4 @ ( product_Pair @ B @ ( product_prod @ C @ ( product_prod @ D @ ( product_prod @ E @ ( product_prod @ F @ G ) ) ) ) @ B4 @ ( product_Pair @ C @ ( product_prod @ D @ ( product_prod @ E @ ( product_prod @ F @ G ) ) ) @ C2 @ ( product_Pair @ D @ ( product_prod @ E @ ( product_prod @ F @ G ) ) @ D2 @ ( product_Pair @ E @ ( product_prod @ F @ G ) @ E2 @ ( product_Pair @ F @ G @ F2 @ G2 ) ) ) ) ) ) )
=> ( P2 @ X4 ) ) ).
% prod_induct7
thf(fact_18_prod__induct6,axiom,
! [F: $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 @ F ) ) ) ) ) > $o,X4: product_prod @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D @ ( product_prod @ E @ F ) ) ) )] :
( ! [A4: A,B4: B,C2: C,D2: D,E2: E,F2: F] : ( P2 @ ( product_Pair @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D @ ( product_prod @ E @ F ) ) ) ) @ A4 @ ( product_Pair @ B @ ( product_prod @ C @ ( product_prod @ D @ ( product_prod @ E @ F ) ) ) @ B4 @ ( product_Pair @ C @ ( product_prod @ D @ ( product_prod @ E @ F ) ) @ C2 @ ( product_Pair @ D @ ( product_prod @ E @ F ) @ D2 @ ( product_Pair @ E @ F @ E2 @ F2 ) ) ) ) ) )
=> ( P2 @ X4 ) ) ).
% prod_induct6
thf(fact_19_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,X4: product_prod @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D @ E ) ) )] :
( ! [A4: A,B4: B,C2: 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 ) @ C2 @ ( product_Pair @ D @ E @ D2 @ E2 ) ) ) ) )
=> ( P2 @ X4 ) ) ).
% prod_induct5
thf(fact_20_prod__induct4,axiom,
! [D: $tType,C: $tType,B: $tType,A: $tType,P2: ( product_prod @ A @ ( product_prod @ B @ ( product_prod @ C @ D ) ) ) > $o,X4: product_prod @ A @ ( product_prod @ B @ ( product_prod @ C @ D ) )] :
( ! [A4: A,B4: B,C2: 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 @ C2 @ D2 ) ) ) )
=> ( P2 @ X4 ) ) ).
% prod_induct4
thf(fact_21_prod__induct3,axiom,
! [C: $tType,B: $tType,A: $tType,P2: ( product_prod @ A @ ( product_prod @ B @ C ) ) > $o,X4: product_prod @ A @ ( product_prod @ B @ C )] :
( ! [A4: A,B4: B,C2: C] : ( P2 @ ( product_Pair @ A @ ( product_prod @ B @ C ) @ A4 @ ( product_Pair @ B @ C @ B4 @ C2 ) ) )
=> ( P2 @ X4 ) ) ).
% prod_induct3
thf(fact_22_vertices__restrict,axiom,
! [A: $tType,B: $tType,G3: labeled_graph @ B @ A] :
( ( labeled_vertices @ B @ A @ ( restrict @ B @ A @ G3 ) )
= ( labeled_vertices @ B @ A @ G3 ) ) ).
% vertices_restrict
thf(fact_23_getRel__dom_I1_J,axiom,
! [B: $tType,A: $tType,G3: labeled_graph @ A @ B,A2: B,B2: B,L2: A] :
( ( G3
= ( restrict @ A @ B @ G3 ) )
=> ( ( member @ ( product_prod @ B @ B ) @ ( product_Pair @ B @ B @ A2 @ B2 ) @ ( getRel @ A @ B @ L2 @ G3 ) )
=> ( member @ B @ A2 @ ( labeled_vertices @ A @ B @ G3 ) ) ) ) ).
% getRel_dom(1)
thf(fact_24_getRel__dom_I2_J,axiom,
! [B: $tType,A: $tType,G3: labeled_graph @ A @ B,A2: B,B2: B,L2: A] :
( ( G3
= ( restrict @ A @ B @ G3 ) )
=> ( ( member @ ( product_prod @ B @ B ) @ ( product_Pair @ B @ B @ A2 @ B2 ) @ ( getRel @ A @ B @ L2 @ G3 ) )
=> ( member @ B @ B2 @ ( labeled_vertices @ A @ B @ G3 ) ) ) ) ).
% getRel_dom(2)
thf(fact_25_transitive__rule,axiom,
! [B: $tType,A: $tType,G3: labeled_graph @ A @ B,L2: A] :
( ( G3
= ( restrict @ A @ B @ G3 ) )
=> ( ( maintained @ A @ nat @ B @ ( standa2114124375e_rule @ A @ L2 ) @ G3 )
=> ( trans @ B @ ( getRel @ A @ B @ L2 @ G3 ) ) ) ) ).
% transitive_rule
thf(fact_26_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_27_maintained__refl,axiom,
! [B: $tType,C: $tType,A: $tType,R: labeled_graph @ A @ B,G3: labeled_graph @ A @ C] : ( maintained @ A @ B @ C @ ( product_Pair @ ( labeled_graph @ A @ B ) @ ( labeled_graph @ A @ B ) @ R @ R ) @ G3 ) ).
% maintained_refl
thf(fact_28_reflexivity__rule,axiom,
! [B: $tType,A: $tType,G3: labeled_graph @ A @ B,L2: A] :
( ( G3
= ( restrict @ A @ B @ G3 ) )
=> ( ( maintained @ A @ nat @ B @ ( standa245363227y_rule @ A @ L2 ) @ G3 )
=> ( refl_on @ B @ ( labeled_vertices @ A @ B @ G3 ) @ ( getRel @ A @ B @ L2 @ G3 ) ) ) ) ).
% reflexivity_rule
thf(fact_29_restrict__idemp,axiom,
! [B: $tType,A: $tType,X4: labeled_graph @ A @ B] :
( ( restrict @ A @ B @ ( restrict @ A @ B @ X4 ) )
= ( restrict @ A @ B @ X4 ) ) ).
% restrict_idemp
thf(fact_30_refl__on__domain,axiom,
! [A: $tType,A5: set @ A,R2: set @ ( product_prod @ A @ A ),A2: A,B2: A] :
( ( refl_on @ A @ A5 @ R2 )
=> ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A2 @ B2 ) @ R2 )
=> ( ( member @ A @ A2 @ A5 )
& ( member @ A @ B2 @ A5 ) ) ) ) ).
% refl_on_domain
thf(fact_31_refl__onD2,axiom,
! [A: $tType,A5: set @ A,R2: set @ ( product_prod @ A @ A ),X4: A,Y4: A] :
( ( refl_on @ A @ A5 @ R2 )
=> ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X4 @ Y4 ) @ R2 )
=> ( member @ A @ Y4 @ A5 ) ) ) ).
% refl_onD2
thf(fact_32_transD,axiom,
! [A: $tType,R2: set @ ( product_prod @ A @ A ),X4: A,Y4: A,Z: A] :
( ( trans @ A @ R2 )
=> ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X4 @ Y4 ) @ R2 )
=> ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Y4 @ Z ) @ R2 )
=> ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X4 @ Z ) @ R2 ) ) ) ) ).
% transD
thf(fact_33_transE,axiom,
! [A: $tType,R2: set @ ( product_prod @ A @ A ),X4: A,Y4: A,Z: A] :
( ( trans @ A @ R2 )
=> ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X4 @ Y4 ) @ R2 )
=> ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Y4 @ Z ) @ R2 )
=> ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X4 @ Z ) @ R2 ) ) ) ) ).
% transE
thf(fact_34_transI,axiom,
! [A: $tType,R2: set @ ( product_prod @ A @ A )] :
( ! [X3: A,Y3: A,Z2: A] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X3 @ Y3 ) @ R2 )
=> ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Y3 @ Z2 ) @ R2 )
=> ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X3 @ Z2 ) @ R2 ) ) )
=> ( trans @ A @ R2 ) ) ).
% transI
thf(fact_35_Relation_Otrans__def,axiom,
! [A: $tType] :
( ( trans @ A )
= ( ^ [R3: set @ ( product_prod @ A @ A )] :
! [X5: A,Y5: A,Z3: A] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X5 @ Y5 ) @ R3 )
=> ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Y5 @ Z3 ) @ R3 )
=> ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X5 @ Z3 ) @ R3 ) ) ) ) ) ).
% Relation.trans_def
thf(fact_36_refl__onD,axiom,
! [A: $tType,A5: set @ A,R2: set @ ( product_prod @ A @ A ),A2: A] :
( ( refl_on @ A @ A5 @ R2 )
=> ( ( member @ A @ A2 @ A5 )
=> ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A2 @ A2 ) @ R2 ) ) ) ).
% refl_onD
thf(fact_37_refl__onD1,axiom,
! [A: $tType,A5: set @ A,R2: set @ ( product_prod @ A @ A ),X4: A,Y4: A] :
( ( refl_on @ A @ A5 @ R2 )
=> ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X4 @ Y4 ) @ R2 )
=> ( member @ A @ X4 @ A5 ) ) ) ).
% refl_onD1
thf(fact_38_mA,axiom,
! [X2: product_prod @ ( labeled_graph @ a @ nat ) @ ( labeled_graph @ a @ nat )] :
( ( member @ ( product_prod @ ( labeled_graph @ a @ nat ) @ ( labeled_graph @ a @ nat ) ) @ X2 @ ( insert @ ( product_prod @ ( labeled_graph @ a @ nat ) @ ( labeled_graph @ a @ nat ) ) @ ( standa245363227y_rule @ a @ i ) @ ( insert @ ( product_prod @ ( labeled_graph @ a @ nat ) @ ( labeled_graph @ a @ nat ) ) @ ( standa2114124375e_rule @ a @ i ) @ ( insert @ ( product_prod @ ( labeled_graph @ a @ nat ) @ ( labeled_graph @ a @ nat ) ) @ ( standa1805702094y_rule @ a @ i ) @ ( bot_bot @ ( set @ ( product_prod @ ( labeled_graph @ a @ nat ) @ ( labeled_graph @ a @ nat ) ) ) ) ) ) ) )
=> ( maintained @ a @ nat @ b @ X2 @ g ) ) ).
% mA
thf(fact_39_internal__case__prod__conv,axiom,
! [B: $tType,A: $tType,C: $tType,C3: B > C > A,A2: B,B2: C] :
( ( produc2004651681e_prod @ B @ C @ A @ C3 @ ( product_Pair @ B @ C @ A2 @ B2 ) )
= ( C3 @ A2 @ B2 ) ) ).
% internal_case_prod_conv
thf(fact_40_preorder__on__def,axiom,
! [A: $tType] :
( ( order_preorder_on @ A )
= ( ^ [A6: set @ A,R3: set @ ( product_prod @ A @ A )] :
( ( refl_on @ A @ A6 @ R3 )
& ( trans @ A @ R3 ) ) ) ) ).
% preorder_on_def
thf(fact_41_on__triple,axiom,
! [B: $tType,C: $tType,A: $tType,L1: A,V1: B,V2: B,L22: A,V3: C,V4: C,R: set @ ( product_prod @ B @ C )] :
( ( member @ ( product_prod @ ( product_prod @ A @ ( product_prod @ B @ B ) ) @ ( product_prod @ A @ ( product_prod @ C @ C ) ) ) @ ( product_Pair @ ( product_prod @ A @ ( product_prod @ B @ B ) ) @ ( product_prod @ A @ ( product_prod @ C @ C ) ) @ ( product_Pair @ A @ ( product_prod @ B @ B ) @ L1 @ ( product_Pair @ B @ B @ V1 @ V2 ) ) @ ( product_Pair @ A @ ( product_prod @ C @ C ) @ L22 @ ( product_Pair @ C @ C @ V3 @ V4 ) ) ) @ ( on_triple @ B @ C @ A @ R ) )
= ( ( member @ ( product_prod @ B @ C ) @ ( product_Pair @ B @ C @ V1 @ V3 ) @ R )
& ( member @ ( product_prod @ B @ C ) @ ( product_Pair @ B @ C @ V2 @ V4 ) @ R )
& ( L1 = L22 ) ) ) ).
% on_triple
thf(fact_42_maintained__then__fin__maintained,axiom,
! [B: $tType,C: $tType,A: $tType,R: product_prod @ ( labeled_graph @ A @ B ) @ ( labeled_graph @ A @ B ),G3: labeled_graph @ A @ C] :
( ( maintained @ A @ B @ C @ R @ G3 )
=> ( fin_maintained @ A @ B @ C @ R @ G3 ) ) ).
% maintained_then_fin_maintained
thf(fact_43_refl__trans__impl__idempotent,axiom,
! [A: $tType,A5: set @ A,R2: set @ ( product_prod @ A @ A )] :
( ( refl_on @ A @ A5 @ R2 )
=> ( ( trans @ A @ R2 )
=> ( idempotent @ A @ R2 ) ) ) ).
% refl_trans_impl_idempotent
thf(fact_44_equivalence,axiom,
! [B: $tType,A: $tType,G3: labeled_graph @ A @ B,I: A] :
( ( G3
= ( restrict @ A @ B @ G3 ) )
=> ( ! [X3: product_prod @ ( labeled_graph @ A @ nat ) @ ( labeled_graph @ A @ nat )] :
( ( member @ ( product_prod @ ( labeled_graph @ A @ nat ) @ ( labeled_graph @ A @ nat ) ) @ X3 @ ( insert @ ( product_prod @ ( labeled_graph @ A @ nat ) @ ( labeled_graph @ A @ nat ) ) @ ( standa245363227y_rule @ A @ I ) @ ( insert @ ( product_prod @ ( labeled_graph @ A @ nat ) @ ( labeled_graph @ A @ nat ) ) @ ( standa2114124375e_rule @ A @ I ) @ ( insert @ ( product_prod @ ( labeled_graph @ A @ nat ) @ ( labeled_graph @ A @ nat ) ) @ ( standa1805702094y_rule @ A @ I ) @ ( bot_bot @ ( set @ ( product_prod @ ( labeled_graph @ A @ nat ) @ ( labeled_graph @ A @ nat ) ) ) ) ) ) ) )
=> ( maintained @ A @ nat @ B @ X3 @ G3 ) )
=> ( equiv_equiv @ B @ ( labeled_vertices @ A @ B @ G3 ) @ ( getRel @ A @ B @ I @ G3 ) ) ) ) ).
% equivalence
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
@ ^ [X5: A] : ( member @ A @ X5 @ A5 ) )
= A5 ) ).
% Collect_mem_eq
thf(fact_47_Collect__cong,axiom,
! [A: $tType,P2: A > $o,Q: A > $o] :
( ! [X3: A] :
( ( P2 @ X3 )
= ( Q @ X3 ) )
=> ( ( collect @ A @ P2 )
= ( collect @ A @ Q ) ) ) ).
% Collect_cong
thf(fact_48_ext,axiom,
! [B: $tType,A: $tType,F3: A > B,G4: A > B] :
( ! [X3: A] :
( ( F3 @ X3 )
= ( G4 @ X3 ) )
=> ( F3 = G4 ) ) ).
% ext
thf(fact_49_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_50_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_51_refl__on__singleton,axiom,
! [A: $tType,X4: A] : ( refl_on @ A @ ( insert @ A @ X4 @ ( bot_bot @ ( set @ A ) ) ) @ ( insert @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X4 @ X4 ) @ ( bot_bot @ ( set @ ( product_prod @ A @ A ) ) ) ) ) ).
% refl_on_singleton
thf(fact_52_preorder__on__empty,axiom,
! [A: $tType] : ( order_preorder_on @ A @ ( bot_bot @ ( set @ A ) ) @ ( bot_bot @ ( set @ ( product_prod @ A @ A ) ) ) ) ).
% preorder_on_empty
thf(fact_53_refl__on__empty,axiom,
! [A: $tType] : ( refl_on @ A @ ( bot_bot @ ( set @ A ) ) @ ( bot_bot @ ( set @ ( product_prod @ A @ A ) ) ) ) ).
% refl_on_empty
thf(fact_54_trans__singleton,axiom,
! [A: $tType,A2: A] : ( trans @ A @ ( insert @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A2 @ A2 ) @ ( bot_bot @ ( set @ ( product_prod @ A @ A ) ) ) ) ) ).
% trans_singleton
thf(fact_55_idempotent__impl__trans,axiom,
! [A: $tType,R2: set @ ( product_prod @ A @ A )] :
( ( idempotent @ A @ R2 )
=> ( trans @ A @ R2 ) ) ).
% idempotent_impl_trans
thf(fact_56_trans__empty,axiom,
! [A: $tType] : ( trans @ A @ ( bot_bot @ ( set @ ( product_prod @ A @ A ) ) ) ) ).
% trans_empty
thf(fact_57_on__tripleD_I1_J,axiom,
! [B: $tType,C: $tType,A: $tType,L1: A,V1: B,V2: B,L22: A,V3: C,V4: C,R: set @ ( product_prod @ B @ C )] :
( ( member @ ( product_prod @ ( product_prod @ A @ ( product_prod @ B @ B ) ) @ ( product_prod @ A @ ( product_prod @ C @ C ) ) ) @ ( product_Pair @ ( product_prod @ A @ ( product_prod @ B @ B ) ) @ ( product_prod @ A @ ( product_prod @ C @ C ) ) @ ( product_Pair @ A @ ( product_prod @ B @ B ) @ L1 @ ( product_Pair @ B @ B @ V1 @ V2 ) ) @ ( product_Pair @ A @ ( product_prod @ C @ C ) @ L22 @ ( product_Pair @ C @ C @ V3 @ V4 ) ) ) @ ( on_triple @ B @ C @ A @ R ) )
=> ( L22 = L1 ) ) ).
% on_tripleD(1)
thf(fact_58_on__tripleD_I2_J,axiom,
! [A: $tType,C: $tType,B: $tType,L1: A,V1: B,V2: B,L22: A,V3: C,V4: C,R: set @ ( product_prod @ B @ C )] :
( ( member @ ( product_prod @ ( product_prod @ A @ ( product_prod @ B @ B ) ) @ ( product_prod @ A @ ( product_prod @ C @ C ) ) ) @ ( product_Pair @ ( product_prod @ A @ ( product_prod @ B @ B ) ) @ ( product_prod @ A @ ( product_prod @ C @ C ) ) @ ( product_Pair @ A @ ( product_prod @ B @ B ) @ L1 @ ( product_Pair @ B @ B @ V1 @ V2 ) ) @ ( product_Pair @ A @ ( product_prod @ C @ C ) @ L22 @ ( product_Pair @ C @ C @ V3 @ V4 ) ) ) @ ( on_triple @ B @ C @ A @ R ) )
=> ( member @ ( product_prod @ B @ C ) @ ( product_Pair @ B @ C @ V1 @ V3 ) @ R ) ) ).
% on_tripleD(2)
thf(fact_59_on__tripleD_I3_J,axiom,
! [A: $tType,C: $tType,B: $tType,L1: A,V1: B,V2: B,L22: A,V3: C,V4: C,R: set @ ( product_prod @ B @ C )] :
( ( member @ ( product_prod @ ( product_prod @ A @ ( product_prod @ B @ B ) ) @ ( product_prod @ A @ ( product_prod @ C @ C ) ) ) @ ( product_Pair @ ( product_prod @ A @ ( product_prod @ B @ B ) ) @ ( product_prod @ A @ ( product_prod @ C @ C ) ) @ ( product_Pair @ A @ ( product_prod @ B @ B ) @ L1 @ ( product_Pair @ B @ B @ V1 @ V2 ) ) @ ( product_Pair @ A @ ( product_prod @ C @ C ) @ L22 @ ( product_Pair @ C @ C @ V3 @ V4 ) ) ) @ ( on_triple @ B @ C @ A @ R ) )
=> ( member @ ( product_prod @ B @ C ) @ ( product_Pair @ B @ C @ V2 @ V4 ) @ R ) ) ).
% on_tripleD(3)
thf(fact_60_singletonI,axiom,
! [A: $tType,A2: A] : ( member @ A @ A2 @ ( insert @ A @ A2 @ ( bot_bot @ ( set @ A ) ) ) ) ).
% singletonI
thf(fact_61_nonempty__rule,axiom,
! [A: $tType,B: $tType,G3: labeled_graph @ A @ B] :
( ( G3
= ( restrict @ A @ B @ G3 ) )
=> ( ( maintained @ A @ nat @ B @ ( standa244753842y_rule @ A ) @ G3 )
= ( ( labeled_vertices @ A @ B @ G3 )
!= ( bot_bot @ ( set @ B ) ) ) ) ) ).
% nonempty_rule
thf(fact_62_insertCI,axiom,
! [A: $tType,A2: A,B5: set @ A,B2: A] :
( ( ~ ( member @ A @ A2 @ B5 )
=> ( A2 = B2 ) )
=> ( member @ A @ A2 @ ( insert @ A @ B2 @ B5 ) ) ) ).
% insertCI
thf(fact_63_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_64_insert__absorb2,axiom,
! [A: $tType,X4: A,A5: set @ A] :
( ( insert @ A @ X4 @ ( insert @ A @ X4 @ A5 ) )
= ( insert @ A @ X4 @ A5 ) ) ).
% insert_absorb2
thf(fact_65_empty__iff,axiom,
! [A: $tType,C3: A] :
~ ( member @ A @ C3 @ ( bot_bot @ ( set @ A ) ) ) ).
% empty_iff
thf(fact_66_all__not__in__conv,axiom,
! [A: $tType,A5: set @ A] :
( ( ! [X5: A] :
~ ( member @ A @ X5 @ A5 ) )
= ( A5
= ( bot_bot @ ( set @ A ) ) ) ) ).
% all_not_in_conv
thf(fact_67_Collect__empty__eq,axiom,
! [A: $tType,P2: A > $o] :
( ( ( collect @ A @ P2 )
= ( bot_bot @ ( set @ A ) ) )
= ( ! [X5: A] :
~ ( P2 @ X5 ) ) ) ).
% Collect_empty_eq
thf(fact_68_empty__Collect__eq,axiom,
! [A: $tType,P2: A > $o] :
( ( ( bot_bot @ ( set @ A ) )
= ( collect @ A @ P2 ) )
= ( ! [X5: A] :
~ ( P2 @ X5 ) ) ) ).
% empty_Collect_eq
thf(fact_69_ex__in__conv,axiom,
! [A: $tType,A5: set @ A] :
( ( ? [X5: A] : ( member @ A @ X5 @ A5 ) )
= ( A5
!= ( bot_bot @ ( set @ A ) ) ) ) ).
% ex_in_conv
thf(fact_70_equals0I,axiom,
! [A: $tType,A5: set @ A] :
( ! [Y3: A] :
~ ( member @ A @ Y3 @ A5 )
=> ( A5
= ( bot_bot @ ( set @ A ) ) ) ) ).
% equals0I
thf(fact_71_equals0D,axiom,
! [A: $tType,A5: set @ A,A2: A] :
( ( A5
= ( bot_bot @ ( set @ A ) ) )
=> ~ ( member @ A @ A2 @ A5 ) ) ).
% equals0D
thf(fact_72_emptyE,axiom,
! [A: $tType,A2: A] :
~ ( member @ A @ A2 @ ( bot_bot @ ( set @ A ) ) ) ).
% emptyE
thf(fact_73_mk__disjoint__insert,axiom,
! [A: $tType,A2: A,A5: set @ A] :
( ( member @ A @ A2 @ A5 )
=> ? [B6: set @ A] :
( ( A5
= ( insert @ A @ A2 @ B6 ) )
& ~ ( member @ A @ A2 @ B6 ) ) ) ).
% mk_disjoint_insert
thf(fact_74_insert__commute,axiom,
! [A: $tType,X4: A,Y4: A,A5: set @ A] :
( ( insert @ A @ X4 @ ( insert @ A @ Y4 @ A5 ) )
= ( insert @ A @ Y4 @ ( insert @ A @ X4 @ A5 ) ) ) ).
% insert_commute
thf(fact_75_insert__eq__iff,axiom,
! [A: $tType,A2: A,A5: set @ A,B2: A,B5: set @ A] :
( ~ ( member @ A @ A2 @ A5 )
=> ( ~ ( member @ A @ B2 @ B5 )
=> ( ( ( insert @ A @ A2 @ A5 )
= ( insert @ A @ B2 @ B5 ) )
= ( ( ( A2 = B2 )
=> ( A5 = B5 ) )
& ( ( A2 != B2 )
=> ? [C4: set @ A] :
( ( A5
= ( insert @ A @ B2 @ C4 ) )
& ~ ( member @ A @ B2 @ C4 )
& ( B5
= ( insert @ A @ A2 @ C4 ) )
& ~ ( member @ A @ A2 @ C4 ) ) ) ) ) ) ) ).
% insert_eq_iff
thf(fact_76_insert__absorb,axiom,
! [A: $tType,A2: A,A5: set @ A] :
( ( member @ A @ A2 @ A5 )
=> ( ( insert @ A @ A2 @ A5 )
= A5 ) ) ).
% insert_absorb
thf(fact_77_insert__ident,axiom,
! [A: $tType,X4: A,A5: set @ A,B5: set @ A] :
( ~ ( member @ A @ X4 @ A5 )
=> ( ~ ( member @ A @ X4 @ B5 )
=> ( ( ( insert @ A @ X4 @ A5 )
= ( insert @ A @ X4 @ B5 ) )
= ( A5 = B5 ) ) ) ) ).
% insert_ident
thf(fact_78_Set_Oset__insert,axiom,
! [A: $tType,X4: A,A5: set @ A] :
( ( member @ A @ X4 @ A5 )
=> ~ ! [B6: set @ A] :
( ( A5
= ( insert @ A @ X4 @ B6 ) )
=> ( member @ A @ X4 @ B6 ) ) ) ).
% Set.set_insert
thf(fact_79_insertI2,axiom,
! [A: $tType,A2: A,B5: set @ A,B2: A] :
( ( member @ A @ A2 @ B5 )
=> ( member @ A @ A2 @ ( insert @ A @ B2 @ B5 ) ) ) ).
% insertI2
thf(fact_80_insertI1,axiom,
! [A: $tType,A2: A,B5: set @ A] : ( member @ A @ A2 @ ( insert @ A @ A2 @ B5 ) ) ).
% insertI1
thf(fact_81_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_82_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_83_insert__not__empty,axiom,
! [A: $tType,A2: A,A5: set @ A] :
( ( insert @ A @ A2 @ A5 )
!= ( bot_bot @ ( set @ A ) ) ) ).
% insert_not_empty
thf(fact_84_doubleton__eq__iff,axiom,
! [A: $tType,A2: A,B2: A,C3: A,D3: A] :
( ( ( insert @ A @ A2 @ ( insert @ A @ B2 @ ( bot_bot @ ( set @ A ) ) ) )
= ( insert @ A @ C3 @ ( insert @ A @ D3 @ ( bot_bot @ ( set @ A ) ) ) ) )
= ( ( ( A2 = C3 )
& ( B2 = D3 ) )
| ( ( A2 = D3 )
& ( B2 = C3 ) ) ) ) ).
% doubleton_eq_iff
thf(fact_85_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_86_singletonD,axiom,
! [A: $tType,B2: A,A2: A] :
( ( member @ A @ B2 @ ( insert @ A @ A2 @ ( bot_bot @ ( set @ A ) ) ) )
=> ( B2 = A2 ) ) ).
% singletonD
thf(fact_87_the__elem__eq,axiom,
! [A: $tType,X4: A] :
( ( the_elem @ A @ ( insert @ A @ X4 @ ( bot_bot @ ( set @ A ) ) ) )
= X4 ) ).
% the_elem_eq
thf(fact_88_bot__apply,axiom,
! [C: $tType,D: $tType] :
( ( bot @ C )
=> ( ( bot_bot @ ( D > C ) )
= ( ^ [X5: D] : ( bot_bot @ C ) ) ) ) ).
% bot_apply
thf(fact_89_graph__single,axiom,
! [A: $tType,B: $tType,A2: A,B2: B,C3: 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 @ C3 ) ) @ ( bot_bot @ ( set @ ( product_prod @ A @ ( product_prod @ B @ B ) ) ) ) ) @ ( insert @ B @ B2 @ ( insert @ B @ C3 @ ( 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 @ C3 ) ) @ ( bot_bot @ ( set @ ( product_prod @ A @ ( product_prod @ B @ B ) ) ) ) ) @ ( insert @ B @ B2 @ ( insert @ B @ C3 @ ( bot_bot @ ( set @ B ) ) ) ) ) ) ) ).
% graph_single
thf(fact_90_is__singletonI,axiom,
! [A: $tType,X4: A] : ( is_singleton @ A @ ( insert @ A @ X4 @ ( bot_bot @ ( set @ A ) ) ) ) ).
% is_singletonI
thf(fact_91_linear__order__on__singleton,axiom,
! [A: $tType,X4: A] : ( order_1409979114der_on @ A @ ( insert @ A @ X4 @ ( bot_bot @ ( set @ A ) ) ) @ ( insert @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X4 @ X4 ) @ ( bot_bot @ ( set @ ( product_prod @ A @ A ) ) ) ) ) ).
% linear_order_on_singleton
thf(fact_92_total__on__singleton,axiom,
! [A: $tType,X4: A] : ( total_on @ A @ ( insert @ A @ X4 @ ( bot_bot @ ( set @ A ) ) ) @ ( insert @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X4 @ X4 ) @ ( bot_bot @ ( set @ ( product_prod @ A @ A ) ) ) ) ) ).
% total_on_singleton
thf(fact_93_symmetry__rule,axiom,
! [B: $tType,A: $tType,G3: labeled_graph @ A @ B,L2: A] :
( ( G3
= ( restrict @ A @ B @ G3 ) )
=> ( ( maintained @ A @ nat @ B @ ( standa1805702094y_rule @ A @ L2 ) @ G3 )
=> ( sym @ B @ ( getRel @ A @ B @ L2 @ G3 ) ) ) ) ).
% symmetry_rule
thf(fact_94_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_95_graph__empty__e,axiom,
! [A: $tType,B: $tType,V5: set @ B] :
( ( labeled_LG @ A @ B @ ( bot_bot @ ( set @ ( product_prod @ A @ ( product_prod @ B @ B ) ) ) ) @ V5 )
= ( restrict @ A @ B @ ( labeled_LG @ A @ B @ ( bot_bot @ ( set @ ( product_prod @ A @ ( product_prod @ B @ B ) ) ) ) @ V5 ) ) ) ).
% graph_empty_e
thf(fact_96_labeled__graph_Oexhaust,axiom,
! [L: $tType,V: $tType,Y4: labeled_graph @ L @ V] :
~ ! [X12: set @ ( product_prod @ L @ ( product_prod @ V @ V ) ),X23: set @ V] :
( Y4
!= ( labeled_LG @ L @ V @ X12 @ X23 ) ) ).
% labeled_graph.exhaust
thf(fact_97_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_98_bot__empty__eq,axiom,
! [A: $tType] :
( ( bot_bot @ ( A > $o ) )
= ( ^ [X5: A] : ( member @ A @ X5 @ ( bot_bot @ ( set @ A ) ) ) ) ) ).
% bot_empty_eq
thf(fact_99_bot__set__def,axiom,
! [A: $tType] :
( ( bot_bot @ ( set @ A ) )
= ( collect @ A @ ( bot_bot @ ( A > $o ) ) ) ) ).
% bot_set_def
thf(fact_100_sym__def,axiom,
! [A: $tType] :
( ( sym @ A )
= ( ^ [R3: set @ ( product_prod @ A @ A )] :
! [X5: A,Y5: A] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X5 @ Y5 ) @ R3 )
=> ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Y5 @ X5 ) @ R3 ) ) ) ) ).
% sym_def
thf(fact_101_symI,axiom,
! [A: $tType,R2: set @ ( product_prod @ A @ A )] :
( ! [A4: A,B4: A] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A4 @ B4 ) @ R2 )
=> ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ B4 @ A4 ) @ R2 ) )
=> ( sym @ A @ R2 ) ) ).
% symI
thf(fact_102_symE,axiom,
! [A: $tType,R2: set @ ( product_prod @ A @ A ),B2: A,A2: A] :
( ( sym @ A @ R2 )
=> ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ B2 @ A2 ) @ R2 )
=> ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A2 @ B2 ) @ R2 ) ) ) ).
% symE
thf(fact_103_symD,axiom,
! [A: $tType,R2: set @ ( product_prod @ A @ A ),B2: A,A2: A] :
( ( sym @ A @ R2 )
=> ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ B2 @ A2 ) @ R2 )
=> ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A2 @ B2 ) @ R2 ) ) ) ).
% symD
thf(fact_104_total__on__def,axiom,
! [A: $tType] :
( ( total_on @ A )
= ( ^ [A6: set @ A,R3: set @ ( product_prod @ A @ A )] :
! [X5: A] :
( ( member @ A @ X5 @ A6 )
=> ! [Y5: A] :
( ( member @ A @ Y5 @ A6 )
=> ( ( X5 != Y5 )
=> ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X5 @ Y5 ) @ R3 )
| ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Y5 @ X5 ) @ R3 ) ) ) ) ) ) ) ).
% total_on_def
thf(fact_105_total__onI,axiom,
! [A: $tType,A5: set @ A,R2: set @ ( product_prod @ A @ A )] :
( ! [X3: A,Y3: A] :
( ( member @ A @ X3 @ A5 )
=> ( ( member @ A @ Y3 @ A5 )
=> ( ( X3 != Y3 )
=> ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X3 @ Y3 ) @ R2 )
| ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Y3 @ X3 ) @ R2 ) ) ) ) )
=> ( total_on @ A @ A5 @ R2 ) ) ).
% total_onI
thf(fact_106_total__on__empty,axiom,
! [A: $tType,R2: set @ ( product_prod @ A @ A )] : ( total_on @ A @ ( bot_bot @ ( set @ A ) ) @ R2 ) ).
% total_on_empty
thf(fact_107_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_108_is__singleton__the__elem,axiom,
! [A: $tType] :
( ( is_singleton @ A )
= ( ^ [A6: set @ A] :
( A6
= ( insert @ A @ ( the_elem @ A @ A6 ) @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ).
% is_singleton_the_elem
thf(fact_109_is__singletonI_H,axiom,
! [A: $tType,A5: set @ A] :
( ( A5
!= ( bot_bot @ ( set @ A ) ) )
=> ( ! [X3: A,Y3: A] :
( ( member @ A @ X3 @ A5 )
=> ( ( member @ A @ Y3 @ A5 )
=> ( X3 = Y3 ) ) )
=> ( is_singleton @ A @ A5 ) ) ) ).
% is_singletonI'
thf(fact_110_lnear__order__on__empty,axiom,
! [A: $tType] : ( order_1409979114der_on @ A @ ( bot_bot @ ( set @ A ) ) @ ( bot_bot @ ( set @ ( product_prod @ A @ A ) ) ) ) ).
% lnear_order_on_empty
thf(fact_111_bot__fun__def,axiom,
! [B: $tType,A: $tType] :
( ( bot @ B )
=> ( ( bot_bot @ ( A > B ) )
= ( ^ [X5: A] : ( bot_bot @ B ) ) ) ) ).
% bot_fun_def
thf(fact_112_is__singletonE,axiom,
! [A: $tType,A5: set @ A] :
( ( is_singleton @ A @ A5 )
=> ~ ! [X3: A] :
( A5
!= ( insert @ A @ X3 @ ( bot_bot @ ( set @ A ) ) ) ) ) ).
% is_singletonE
thf(fact_113_is__singleton__def,axiom,
! [A: $tType] :
( ( is_singleton @ A )
= ( ^ [A6: set @ A] :
? [X5: A] :
( A6
= ( insert @ A @ X5 @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ).
% is_singleton_def
thf(fact_114_equivE,axiom,
! [A: $tType,A5: set @ A,R2: set @ ( product_prod @ A @ A )] :
( ( equiv_equiv @ A @ A5 @ R2 )
=> ~ ( ( refl_on @ A @ A5 @ R2 )
=> ( ( sym @ A @ R2 )
=> ~ ( trans @ A @ R2 ) ) ) ) ).
% equivE
thf(fact_115_equivI,axiom,
! [A: $tType,A5: set @ A,R2: set @ ( product_prod @ A @ A )] :
( ( refl_on @ A @ A5 @ R2 )
=> ( ( sym @ A @ R2 )
=> ( ( trans @ A @ R2 )
=> ( equiv_equiv @ A @ A5 @ R2 ) ) ) ) ).
% equivI
thf(fact_116_equiv__def,axiom,
! [A: $tType] :
( ( equiv_equiv @ A )
= ( ^ [A6: set @ A,R3: set @ ( product_prod @ A @ A )] :
( ( refl_on @ A @ A6 @ R3 )
& ( sym @ A @ R3 )
& ( trans @ A @ R3 ) ) ) ) ).
% equiv_def
thf(fact_117_agree__on__empty,axiom,
! [A: $tType,C: $tType,B: $tType,F3: set @ ( product_prod @ B @ C ),G4: 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 ) ) ) @ F3 @ G4 ) ).
% agree_on_empty
thf(fact_118_graph__homomorphism__empty,axiom,
! [B: $tType,C: $tType,A: $tType,G3: labeled_graph @ A @ C,F3: 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 ) ) ) @ G3 @ F3 )
= ( ( F3
= ( bot_bot @ ( set @ ( product_prod @ B @ C ) ) ) )
& ( G3
= ( restrict @ A @ C @ G3 ) ) ) ) ).
% graph_homomorphism_empty
thf(fact_119_agree__on__comm,axiom,
! [C: $tType,B: $tType,A: $tType] :
( ( agree_on @ A @ B @ C )
= ( ^ [X6: labeled_graph @ A @ B,F4: set @ ( product_prod @ B @ C ),G5: set @ ( product_prod @ B @ C )] : ( agree_on @ A @ B @ C @ X6 @ G5 @ F4 ) ) ) ).
% agree_on_comm
thf(fact_120_agree__on__refl,axiom,
! [A: $tType,C: $tType,B: $tType,R: labeled_graph @ A @ B,F3: set @ ( product_prod @ B @ C )] : ( agree_on @ A @ B @ C @ R @ F3 @ F3 ) ).
% agree_on_refl
thf(fact_121_agree__on__trans,axiom,
! [A: $tType,C: $tType,B: $tType,X7: labeled_graph @ A @ B,F3: set @ ( product_prod @ B @ C ),G4: set @ ( product_prod @ B @ C ),H: set @ ( product_prod @ B @ C )] :
( ( agree_on @ A @ B @ C @ X7 @ F3 @ G4 )
=> ( ( agree_on @ A @ B @ C @ X7 @ G4 @ H )
=> ( agree_on @ A @ B @ C @ X7 @ F3 @ H ) ) ) ).
% agree_on_trans
thf(fact_122_graph__homomorphism__semantics,axiom,
! [B: $tType,C: $tType,A: $tType,A5: labeled_graph @ A @ B,B5: labeled_graph @ A @ C,F3: set @ ( product_prod @ B @ C ),A2: B,B2: B,E3: allegorical_term @ A,A3: C,B3: C] :
( ( graph_homomorphism @ A @ B @ C @ A5 @ B5 @ F3 )
=> ( ( 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 ) @ F3 )
=> ( ( member @ ( product_prod @ B @ C ) @ ( product_Pair @ B @ C @ B2 @ B3 ) @ F3 )
=> ( member @ ( product_prod @ C @ C ) @ ( product_Pair @ C @ C @ A3 @ B3 ) @ ( semantics @ A @ C @ B5 @ E3 ) ) ) ) ) ) ).
% graph_homomorphism_semantics
thf(fact_123_graph__homomorphism__nonempty,axiom,
! [B: $tType,A: $tType,C: $tType,A5: labeled_graph @ A @ B,B5: labeled_graph @ A @ C,F3: set @ ( product_prod @ B @ C ),E3: allegorical_term @ A] :
( ( graph_homomorphism @ A @ B @ C @ A5 @ B5 @ F3 )
=> ( ( ( semantics @ A @ B @ A5 @ E3 )
!= ( bot_bot @ ( set @ ( product_prod @ B @ B ) ) ) )
=> ( ( semantics @ A @ C @ B5 @ E3 )
!= ( bot_bot @ ( set @ ( product_prod @ C @ C ) ) ) ) ) ) ).
% graph_homomorphism_nonempty
thf(fact_124_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_125_extensibleI,axiom,
! [A: $tType,C: $tType,B: $tType,R22: labeled_graph @ A @ B,G3: labeled_graph @ A @ C,G4: set @ ( product_prod @ B @ C ),R1: labeled_graph @ A @ B,F3: set @ ( product_prod @ B @ C )] :
( ( graph_homomorphism @ A @ B @ C @ R22 @ G3 @ G4 )
=> ( ( agree_on @ A @ B @ C @ R1 @ F3 @ G4 )
=> ( extensible @ A @ B @ C @ ( product_Pair @ ( labeled_graph @ A @ B ) @ ( labeled_graph @ A @ B ) @ R1 @ R22 ) @ G3 @ F3 ) ) ) ).
% extensibleI
thf(fact_126_maintainedI,axiom,
! [A: $tType,B: $tType,C: $tType,A5: labeled_graph @ C @ A,G3: labeled_graph @ C @ B,B5: labeled_graph @ C @ A] :
( ! [F2: set @ ( product_prod @ A @ B )] :
( ( graph_homomorphism @ C @ A @ B @ A5 @ G3 @ F2 )
=> ( extensible @ C @ A @ B @ ( product_Pair @ ( labeled_graph @ C @ A ) @ ( labeled_graph @ C @ A ) @ A5 @ B5 ) @ G3 @ F2 ) )
=> ( maintained @ C @ A @ B @ ( product_Pair @ ( labeled_graph @ C @ A ) @ ( labeled_graph @ C @ A ) @ A5 @ B5 ) @ G3 ) ) ).
% maintainedI
thf(fact_127_extensible__refl__concr,axiom,
! [A: $tType,C: $tType,B: $tType,E_1: set @ ( product_prod @ A @ ( product_prod @ B @ B ) ),V5: set @ B,G3: labeled_graph @ A @ C,F3: 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 @ V5 ) @ G3 @ F3 )
=> ( ( extensible @ A @ B @ C @ ( product_Pair @ ( labeled_graph @ A @ B ) @ ( labeled_graph @ A @ B ) @ ( labeled_LG @ A @ B @ E_1 @ V5 ) @ ( labeled_LG @ A @ B @ E_2 @ V5 ) ) @ G3 @ F3 )
= ( graph_homomorphism @ A @ B @ C @ ( labeled_LG @ A @ B @ E_2 @ V5 ) @ G3 @ F3 ) ) ) ).
% extensible_refl_concr
thf(fact_128_extensible__refl,axiom,
! [A: $tType,C: $tType,B: $tType,R: labeled_graph @ A @ B,G3: labeled_graph @ A @ C,F3: set @ ( product_prod @ B @ C )] :
( ( graph_homomorphism @ A @ B @ C @ R @ G3 @ F3 )
=> ( extensible @ A @ B @ C @ ( product_Pair @ ( labeled_graph @ A @ B ) @ ( labeled_graph @ A @ B ) @ R @ R ) @ G3 @ F3 ) ) ).
% extensible_refl
thf(fact_129_maintainedD,axiom,
! [A: $tType,C: $tType,B: $tType,A5: labeled_graph @ A @ B,B5: labeled_graph @ A @ B,G3: labeled_graph @ A @ C,F3: set @ ( product_prod @ B @ C )] :
( ( maintained @ A @ B @ C @ ( product_Pair @ ( labeled_graph @ A @ B ) @ ( labeled_graph @ A @ B ) @ A5 @ B5 ) @ G3 )
=> ( ( graph_homomorphism @ A @ B @ C @ A5 @ G3 @ F3 )
=> ( extensible @ A @ B @ C @ ( product_Pair @ ( labeled_graph @ A @ B ) @ ( labeled_graph @ A @ B ) @ A5 @ B5 ) @ G3 @ F3 ) ) ) ).
% maintainedD
thf(fact_130_Set_Ois__empty__def,axiom,
! [A: $tType] :
( ( is_empty @ A )
= ( ^ [A6: set @ A] :
( A6
= ( bot_bot @ ( set @ A ) ) ) ) ) ).
% Set.is_empty_def
thf(fact_131_congruent2I,axiom,
! [C: $tType,B: $tType,A: $tType,A1: set @ A,R12: set @ ( product_prod @ A @ A ),A22: set @ B,R23: set @ ( product_prod @ B @ B ),F3: A > B > C] :
( ( equiv_equiv @ A @ A1 @ R12 )
=> ( ( equiv_equiv @ B @ A22 @ R23 )
=> ( ! [Y3: A,Z2: A,W: B] :
( ( member @ B @ W @ A22 )
=> ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Y3 @ Z2 ) @ R12 )
=> ( ( F3 @ Y3 @ W )
= ( F3 @ Z2 @ W ) ) ) )
=> ( ! [Y3: B,Z2: B,W: A] :
( ( member @ A @ W @ A1 )
=> ( ( member @ ( product_prod @ B @ B ) @ ( product_Pair @ B @ B @ Y3 @ Z2 ) @ R23 )
=> ( ( F3 @ W @ Y3 )
= ( F3 @ W @ Z2 ) ) ) )
=> ( equiv_congruent2 @ A @ B @ C @ R12 @ R23 @ F3 ) ) ) ) ) ).
% congruent2I
thf(fact_132_congruent2__commuteI,axiom,
! [B: $tType,A: $tType,A5: set @ A,R2: set @ ( product_prod @ A @ A ),F3: A > A > B] :
( ( equiv_equiv @ A @ A5 @ R2 )
=> ( ! [Y3: A,Z2: A] :
( ( member @ A @ Y3 @ A5 )
=> ( ( member @ A @ Z2 @ A5 )
=> ( ( F3 @ Y3 @ Z2 )
= ( F3 @ Z2 @ Y3 ) ) ) )
=> ( ! [Y3: A,Z2: A,W: A] :
( ( member @ A @ W @ A5 )
=> ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Y3 @ Z2 ) @ R2 )
=> ( ( F3 @ W @ Y3 )
= ( F3 @ W @ Z2 ) ) ) )
=> ( equiv_congruent2 @ A @ A @ B @ R2 @ R2 @ F3 ) ) ) ) ).
% congruent2_commuteI
thf(fact_133_eq__equiv__class__iff2,axiom,
! [A: $tType,A5: set @ A,R2: set @ ( product_prod @ A @ A ),X4: A,Y4: A] :
( ( equiv_equiv @ A @ A5 @ R2 )
=> ( ( member @ A @ X4 @ A5 )
=> ( ( member @ A @ Y4 @ A5 )
=> ( ( ( equiv_quotient @ A @ ( insert @ A @ X4 @ ( bot_bot @ ( set @ A ) ) ) @ R2 )
= ( equiv_quotient @ A @ ( insert @ A @ Y4 @ ( bot_bot @ ( set @ A ) ) ) @ R2 ) )
= ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X4 @ Y4 ) @ R2 ) ) ) ) ) ).
% eq_equiv_class_iff2
thf(fact_134_quotient__is__empty2,axiom,
! [A: $tType,A5: set @ A,R2: set @ ( product_prod @ A @ A )] :
( ( ( bot_bot @ ( set @ ( set @ A ) ) )
= ( equiv_quotient @ A @ A5 @ R2 ) )
= ( A5
= ( bot_bot @ ( set @ A ) ) ) ) ).
% quotient_is_empty2
thf(fact_135_quotient__is__empty,axiom,
! [A: $tType,A5: set @ A,R2: set @ ( product_prod @ A @ A )] :
( ( ( equiv_quotient @ A @ A5 @ R2 )
= ( bot_bot @ ( set @ ( set @ A ) ) ) )
= ( A5
= ( bot_bot @ ( set @ A ) ) ) ) ).
% quotient_is_empty
thf(fact_136_quotient__empty,axiom,
! [A: $tType,R2: set @ ( product_prod @ A @ A )] :
( ( equiv_quotient @ A @ ( bot_bot @ ( set @ A ) ) @ R2 )
= ( bot_bot @ ( set @ ( set @ A ) ) ) ) ).
% quotient_empty
thf(fact_137_in__quotient__imp__closed,axiom,
! [A: $tType,A5: set @ A,R2: set @ ( product_prod @ A @ A ),X7: set @ A,X4: A,Y4: A] :
( ( equiv_equiv @ A @ A5 @ R2 )
=> ( ( member @ ( set @ A ) @ X7 @ ( equiv_quotient @ A @ A5 @ R2 ) )
=> ( ( member @ A @ X4 @ X7 )
=> ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X4 @ Y4 ) @ R2 )
=> ( member @ A @ Y4 @ X7 ) ) ) ) ) ).
% in_quotient_imp_closed
thf(fact_138_quotient__eq__iff,axiom,
! [A: $tType,A5: set @ A,R2: set @ ( product_prod @ A @ A ),X7: set @ A,Y6: set @ A,X4: A,Y4: A] :
( ( equiv_equiv @ A @ A5 @ R2 )
=> ( ( member @ ( set @ A ) @ X7 @ ( equiv_quotient @ A @ A5 @ R2 ) )
=> ( ( member @ ( set @ A ) @ Y6 @ ( equiv_quotient @ A @ A5 @ R2 ) )
=> ( ( member @ A @ X4 @ X7 )
=> ( ( member @ A @ Y4 @ Y6 )
=> ( ( X7 = Y6 )
= ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X4 @ Y4 ) @ R2 ) ) ) ) ) ) ) ).
% quotient_eq_iff
thf(fact_139_quotient__eqI,axiom,
! [A: $tType,A5: set @ A,R2: set @ ( product_prod @ A @ A ),X7: set @ A,Y6: set @ A,X4: A,Y4: A] :
( ( equiv_equiv @ A @ A5 @ R2 )
=> ( ( member @ ( set @ A ) @ X7 @ ( equiv_quotient @ A @ A5 @ R2 ) )
=> ( ( member @ ( set @ A ) @ Y6 @ ( equiv_quotient @ A @ A5 @ R2 ) )
=> ( ( member @ A @ X4 @ X7 )
=> ( ( member @ A @ Y4 @ Y6 )
=> ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X4 @ Y4 ) @ R2 )
=> ( X7 = Y6 ) ) ) ) ) ) ) ).
% quotient_eqI
thf(fact_140_in__quotient__imp__non__empty,axiom,
! [A: $tType,A5: set @ A,R2: set @ ( product_prod @ A @ A ),X7: set @ A] :
( ( equiv_equiv @ A @ A5 @ R2 )
=> ( ( member @ ( set @ A ) @ X7 @ ( equiv_quotient @ A @ A5 @ R2 ) )
=> ( X7
!= ( bot_bot @ ( set @ A ) ) ) ) ) ).
% in_quotient_imp_non_empty
thf(fact_141_congruent2I_H,axiom,
! [C: $tType,B: $tType,A: $tType,R12: set @ ( product_prod @ A @ A ),R23: set @ ( product_prod @ B @ B ),F3: A > B > C] :
( ! [Y12: A,Z1: A,Y22: B,Z22: B] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Y12 @ Z1 ) @ R12 )
=> ( ( member @ ( product_prod @ B @ B ) @ ( product_Pair @ B @ B @ Y22 @ Z22 ) @ R23 )
=> ( ( F3 @ Y12 @ Y22 )
= ( F3 @ Z1 @ Z22 ) ) ) )
=> ( equiv_congruent2 @ A @ B @ C @ R12 @ R23 @ F3 ) ) ).
% congruent2I'
thf(fact_142_congruent2D,axiom,
! [A: $tType,C: $tType,B: $tType,R12: set @ ( product_prod @ A @ A ),R23: set @ ( product_prod @ B @ B ),F3: A > B > C,Y1: A,Z12: A,Y2: B,Z23: B] :
( ( equiv_congruent2 @ A @ B @ C @ R12 @ R23 @ F3 )
=> ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Y1 @ Z12 ) @ R12 )
=> ( ( member @ ( product_prod @ B @ B ) @ ( product_Pair @ B @ B @ Y2 @ Z23 ) @ R23 )
=> ( ( F3 @ Y1 @ Y2 )
= ( F3 @ Z12 @ Z23 ) ) ) ) ) ).
% congruent2D
thf(fact_143_congruent2__implies__congruent,axiom,
! [B: $tType,C: $tType,A: $tType,A5: set @ A,R12: set @ ( product_prod @ A @ A ),R23: set @ ( product_prod @ B @ B ),F3: A > B > C,A2: A] :
( ( equiv_equiv @ A @ A5 @ R12 )
=> ( ( equiv_congruent2 @ A @ B @ C @ R12 @ R23 @ F3 )
=> ( ( member @ A @ A2 @ A5 )
=> ( equiv_congruent @ B @ C @ R23 @ ( F3 @ A2 ) ) ) ) ) ).
% congruent2_implies_congruent
thf(fact_144_in__quotient__imp__in__rel,axiom,
! [A: $tType,A5: set @ A,R2: set @ ( product_prod @ A @ A ),X7: set @ A,X4: A,Y4: A] :
( ( equiv_equiv @ A @ A5 @ R2 )
=> ( ( member @ ( set @ A ) @ X7 @ ( equiv_quotient @ A @ A5 @ R2 ) )
=> ( ( ord_less_eq @ ( set @ A ) @ ( insert @ A @ X4 @ ( insert @ A @ Y4 @ ( bot_bot @ ( set @ A ) ) ) ) @ X7 )
=> ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X4 @ Y4 ) @ R2 ) ) ) ) ).
% in_quotient_imp_in_rel
thf(fact_145_same__fstI,axiom,
! [B: $tType,A: $tType,P2: A > $o,X4: A,Y7: B,Y4: B,R: A > ( set @ ( product_prod @ B @ B ) )] :
( ( P2 @ X4 )
=> ( ( member @ ( product_prod @ B @ B ) @ ( product_Pair @ B @ B @ Y7 @ Y4 ) @ ( R @ X4 ) )
=> ( member @ ( product_prod @ ( product_prod @ A @ B ) @ ( product_prod @ A @ B ) ) @ ( product_Pair @ ( product_prod @ A @ B ) @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X4 @ Y7 ) @ ( product_Pair @ A @ B @ X4 @ Y4 ) ) @ ( same_fst @ A @ B @ P2 @ R ) ) ) ) ).
% same_fstI
thf(fact_146_getRel__subgraph,axiom,
! [A: $tType,B: $tType,Y4: A,Z: A,L2: B,G3: labeled_graph @ B @ A,G6: labeled_graph @ B @ A] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Y4 @ Z ) @ ( getRel @ B @ A @ L2 @ G3 ) )
=> ( ( graph_homomorphism @ B @ A @ A @ G3 @ G6 @ ( id_on @ A @ ( labeled_vertices @ B @ A @ G3 ) ) )
=> ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Y4 @ Z ) @ ( getRel @ B @ A @ L2 @ G6 ) ) ) ) ).
% getRel_subgraph
thf(fact_147_order__refl,axiom,
! [A: $tType] :
( ( preorder @ A )
=> ! [X4: A] : ( ord_less_eq @ A @ X4 @ X4 ) ) ).
% order_refl
thf(fact_148_subset__antisym,axiom,
! [A: $tType,A5: set @ A,B5: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ A5 @ B5 )
=> ( ( ord_less_eq @ ( set @ A ) @ B5 @ A5 )
=> ( A5 = B5 ) ) ) ).
% subset_antisym
thf(fact_149_subsetI,axiom,
! [A: $tType,A5: set @ A,B5: set @ A] :
( ! [X3: A] :
( ( member @ A @ X3 @ A5 )
=> ( member @ A @ X3 @ B5 ) )
=> ( ord_less_eq @ ( set @ A ) @ A5 @ B5 ) ) ).
% subsetI
thf(fact_150_empty__subsetI,axiom,
! [A: $tType,A5: set @ A] : ( ord_less_eq @ ( set @ A ) @ ( bot_bot @ ( set @ A ) ) @ A5 ) ).
% empty_subsetI
thf(fact_151_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_152_insert__subset,axiom,
! [A: $tType,X4: A,A5: set @ A,B5: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ ( insert @ A @ X4 @ A5 ) @ B5 )
= ( ( member @ A @ X4 @ B5 )
& ( ord_less_eq @ ( set @ A ) @ A5 @ B5 ) ) ) ).
% insert_subset
thf(fact_153_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_154_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_155_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_156_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_157_subgraph__refl,axiom,
! [B: $tType,A: $tType,G3: labeled_graph @ A @ B] :
( ( graph_homomorphism @ A @ B @ B @ G3 @ G3 @ ( id_on @ B @ ( labeled_vertices @ A @ B @ G3 ) ) )
= ( G3
= ( restrict @ A @ B @ G3 ) ) ) ).
% subgraph_refl
thf(fact_158_subgraph__restrict,axiom,
! [B: $tType,A: $tType,G3: labeled_graph @ A @ B] :
( ( graph_homomorphism @ A @ B @ B @ G3 @ ( restrict @ A @ B @ G3 ) @ ( id_on @ B @ ( labeled_vertices @ A @ B @ G3 ) ) )
= ( G3
= ( restrict @ A @ B @ G3 ) ) ) ).
% subgraph_restrict
thf(fact_159_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_160_subgraph__subset_I1_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 @ B ) @ ( labeled_vertices @ A @ B @ G_1 ) @ ( labeled_vertices @ A @ B @ G_2 ) ) ) ).
% subgraph_subset(1)
thf(fact_161_Id__onE,axiom,
! [A: $tType,C3: product_prod @ A @ A,A5: set @ A] :
( ( member @ ( product_prod @ A @ A ) @ C3 @ ( id_on @ A @ A5 ) )
=> ~ ! [X3: A] :
( ( member @ A @ X3 @ A5 )
=> ( C3
!= ( product_Pair @ A @ A @ X3 @ X3 ) ) ) ) ).
% Id_onE
thf(fact_162_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_163_Id__on__iff,axiom,
! [A: $tType,X4: A,Y4: A,A5: set @ A] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X4 @ Y4 ) @ ( id_on @ A @ A5 ) )
= ( ( X4 = Y4 )
& ( member @ A @ X4 @ A5 ) ) ) ).
% Id_on_iff
thf(fact_164_bot_Oextremum__uniqueI,axiom,
! [A: $tType] :
( ( order_bot @ A )
=> ! [A2: A] :
( ( ord_less_eq @ A @ A2 @ ( bot_bot @ A ) )
=> ( A2
= ( bot_bot @ A ) ) ) ) ).
% bot.extremum_uniqueI
thf(fact_165_bot_Oextremum__unique,axiom,
! [A: $tType] :
( ( order_bot @ A )
=> ! [A2: A] :
( ( ord_less_eq @ A @ A2 @ ( bot_bot @ A ) )
= ( A2
= ( bot_bot @ A ) ) ) ) ).
% bot.extremum_unique
thf(fact_166_bot_Oextremum,axiom,
! [A: $tType] :
( ( order_bot @ A )
=> ! [A2: A] : ( ord_less_eq @ A @ ( bot_bot @ A ) @ A2 ) ) ).
% bot.extremum
thf(fact_167_subrelI,axiom,
! [B: $tType,A: $tType,R2: set @ ( product_prod @ A @ B ),S: set @ ( product_prod @ A @ B )] :
( ! [X3: A,Y3: B] :
( ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X3 @ Y3 ) @ R2 )
=> ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X3 @ Y3 ) @ S ) )
=> ( ord_less_eq @ ( set @ ( product_prod @ A @ B ) ) @ R2 @ S ) ) ).
% subrelI
thf(fact_168_insert__mono,axiom,
! [A: $tType,C5: set @ A,D4: set @ A,A2: A] :
( ( ord_less_eq @ ( set @ A ) @ C5 @ D4 )
=> ( ord_less_eq @ ( set @ A ) @ ( insert @ A @ A2 @ C5 ) @ ( insert @ A @ A2 @ D4 ) ) ) ).
% insert_mono
thf(fact_169_subset__insert,axiom,
! [A: $tType,X4: A,A5: set @ A,B5: set @ A] :
( ~ ( member @ A @ X4 @ A5 )
=> ( ( ord_less_eq @ ( set @ A ) @ A5 @ ( insert @ A @ X4 @ B5 ) )
= ( ord_less_eq @ ( set @ A ) @ A5 @ B5 ) ) ) ).
% subset_insert
thf(fact_170_subset__insertI,axiom,
! [A: $tType,B5: set @ A,A2: A] : ( ord_less_eq @ ( set @ A ) @ B5 @ ( insert @ A @ A2 @ B5 ) ) ).
% subset_insertI
thf(fact_171_subset__insertI2,axiom,
! [A: $tType,A5: set @ A,B5: set @ A,B2: A] :
( ( ord_less_eq @ ( set @ A ) @ A5 @ B5 )
=> ( ord_less_eq @ ( set @ A ) @ A5 @ ( insert @ A @ B2 @ B5 ) ) ) ).
% subset_insertI2
thf(fact_172_trans__Id__on,axiom,
! [A: $tType,A5: set @ A] : ( trans @ A @ ( id_on @ A @ A5 ) ) ).
% trans_Id_on
thf(fact_173_refl__on__Id__on,axiom,
! [A: $tType,A5: set @ A] : ( refl_on @ A @ A5 @ ( id_on @ A @ A5 ) ) ).
% refl_on_Id_on
thf(fact_174_Collect__mono__iff,axiom,
! [A: $tType,P2: A > $o,Q: A > $o] :
( ( ord_less_eq @ ( set @ A ) @ ( collect @ A @ P2 ) @ ( collect @ A @ Q ) )
= ( ! [X5: A] :
( ( P2 @ X5 )
=> ( Q @ X5 ) ) ) ) ).
% Collect_mono_iff
thf(fact_175_set__eq__subset,axiom,
! [A: $tType] :
( ( ^ [Y8: set @ A,Z4: set @ A] : Y8 = Z4 )
= ( ^ [A6: set @ A,B7: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ A6 @ B7 )
& ( ord_less_eq @ ( set @ A ) @ B7 @ A6 ) ) ) ) ).
% set_eq_subset
thf(fact_176_subset__trans,axiom,
! [A: $tType,A5: set @ A,B5: set @ A,C5: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ A5 @ B5 )
=> ( ( ord_less_eq @ ( set @ A ) @ B5 @ C5 )
=> ( ord_less_eq @ ( set @ A ) @ A5 @ C5 ) ) ) ).
% subset_trans
thf(fact_177_Collect__mono,axiom,
! [A: $tType,P2: A > $o,Q: A > $o] :
( ! [X3: A] :
( ( P2 @ X3 )
=> ( Q @ X3 ) )
=> ( ord_less_eq @ ( set @ A ) @ ( collect @ A @ P2 ) @ ( collect @ A @ Q ) ) ) ).
% Collect_mono
thf(fact_178_subset__refl,axiom,
! [A: $tType,A5: set @ A] : ( ord_less_eq @ ( set @ A ) @ A5 @ A5 ) ).
% subset_refl
thf(fact_179_subset__iff,axiom,
! [A: $tType] :
( ( ord_less_eq @ ( set @ A ) )
= ( ^ [A6: set @ A,B7: set @ A] :
! [T2: A] :
( ( member @ A @ T2 @ A6 )
=> ( member @ A @ T2 @ B7 ) ) ) ) ).
% subset_iff
thf(fact_180_equalityD2,axiom,
! [A: $tType,A5: set @ A,B5: set @ A] :
( ( A5 = B5 )
=> ( ord_less_eq @ ( set @ A ) @ B5 @ A5 ) ) ).
% equalityD2
thf(fact_181_equalityD1,axiom,
! [A: $tType,A5: set @ A,B5: set @ A] :
( ( A5 = B5 )
=> ( ord_less_eq @ ( set @ A ) @ A5 @ B5 ) ) ).
% equalityD1
thf(fact_182_subset__eq,axiom,
! [A: $tType] :
( ( ord_less_eq @ ( set @ A ) )
= ( ^ [A6: set @ A,B7: set @ A] :
! [X5: A] :
( ( member @ A @ X5 @ A6 )
=> ( member @ A @ X5 @ B7 ) ) ) ) ).
% subset_eq
thf(fact_183_equalityE,axiom,
! [A: $tType,A5: set @ A,B5: set @ A] :
( ( A5 = B5 )
=> ~ ( ( ord_less_eq @ ( set @ A ) @ A5 @ B5 )
=> ~ ( ord_less_eq @ ( set @ A ) @ B5 @ A5 ) ) ) ).
% equalityE
thf(fact_184_subsetD,axiom,
! [A: $tType,A5: set @ A,B5: set @ A,C3: A] :
( ( ord_less_eq @ ( set @ A ) @ A5 @ B5 )
=> ( ( member @ A @ C3 @ A5 )
=> ( member @ A @ C3 @ B5 ) ) ) ).
% subsetD
thf(fact_185_in__mono,axiom,
! [A: $tType,A5: set @ A,B5: set @ A,X4: A] :
( ( ord_less_eq @ ( set @ A ) @ A5 @ B5 )
=> ( ( member @ A @ X4 @ A5 )
=> ( member @ A @ X4 @ B5 ) ) ) ).
% in_mono
thf(fact_186_dual__order_Oantisym,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [B2: A,A2: A] :
( ( ord_less_eq @ A @ B2 @ A2 )
=> ( ( ord_less_eq @ A @ A2 @ B2 )
=> ( A2 = B2 ) ) ) ) ).
% dual_order.antisym
thf(fact_187_dual__order_Oeq__iff,axiom,
! [A: $tType] :
( ( order @ A )
=> ( ( ^ [Y8: A,Z4: A] : Y8 = Z4 )
= ( ^ [A7: A,B8: A] :
( ( ord_less_eq @ A @ B8 @ A7 )
& ( ord_less_eq @ A @ A7 @ B8 ) ) ) ) ) ).
% dual_order.eq_iff
thf(fact_188_dual__order_Otrans,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [B2: A,A2: A,C3: A] :
( ( ord_less_eq @ A @ B2 @ A2 )
=> ( ( ord_less_eq @ A @ C3 @ B2 )
=> ( ord_less_eq @ A @ C3 @ A2 ) ) ) ) ).
% dual_order.trans
thf(fact_189_linorder__wlog,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [P2: A > A > $o,A2: A,B2: A] :
( ! [A4: A,B4: A] :
( ( ord_less_eq @ A @ A4 @ B4 )
=> ( P2 @ A4 @ B4 ) )
=> ( ! [A4: A,B4: A] :
( ( P2 @ B4 @ A4 )
=> ( P2 @ A4 @ B4 ) )
=> ( P2 @ A2 @ B2 ) ) ) ) ).
% linorder_wlog
thf(fact_190_dual__order_Orefl,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [A2: A] : ( ord_less_eq @ A @ A2 @ A2 ) ) ).
% dual_order.refl
thf(fact_191_order__trans,axiom,
! [A: $tType] :
( ( preorder @ A )
=> ! [X4: A,Y4: A,Z: A] :
( ( ord_less_eq @ A @ X4 @ Y4 )
=> ( ( ord_less_eq @ A @ Y4 @ Z )
=> ( ord_less_eq @ A @ X4 @ Z ) ) ) ) ).
% order_trans
thf(fact_192_order__class_Oorder_Oantisym,axiom,
! [A: $tType] :
( ( order @ 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_193_ord__le__eq__trans,axiom,
! [A: $tType] :
( ( ord @ A )
=> ! [A2: A,B2: A,C3: A] :
( ( ord_less_eq @ A @ A2 @ B2 )
=> ( ( B2 = C3 )
=> ( ord_less_eq @ A @ A2 @ C3 ) ) ) ) ).
% ord_le_eq_trans
thf(fact_194_ord__eq__le__trans,axiom,
! [A: $tType] :
( ( ord @ A )
=> ! [A2: A,B2: A,C3: A] :
( ( A2 = B2 )
=> ( ( ord_less_eq @ A @ B2 @ C3 )
=> ( ord_less_eq @ A @ A2 @ C3 ) ) ) ) ).
% ord_eq_le_trans
thf(fact_195_order__class_Oorder_Oeq__iff,axiom,
! [A: $tType] :
( ( order @ A )
=> ( ( ^ [Y8: A,Z4: A] : Y8 = Z4 )
= ( ^ [A7: A,B8: A] :
( ( ord_less_eq @ A @ A7 @ B8 )
& ( ord_less_eq @ A @ B8 @ A7 ) ) ) ) ) ).
% order_class.order.eq_iff
thf(fact_196_antisym__conv,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [Y4: A,X4: A] :
( ( ord_less_eq @ A @ Y4 @ X4 )
=> ( ( ord_less_eq @ A @ X4 @ Y4 )
= ( X4 = Y4 ) ) ) ) ).
% antisym_conv
thf(fact_197_le__cases3,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [X4: A,Y4: A,Z: A] :
( ( ( ord_less_eq @ A @ X4 @ Y4 )
=> ~ ( ord_less_eq @ A @ Y4 @ Z ) )
=> ( ( ( ord_less_eq @ A @ Y4 @ X4 )
=> ~ ( ord_less_eq @ A @ X4 @ Z ) )
=> ( ( ( ord_less_eq @ A @ X4 @ Z )
=> ~ ( ord_less_eq @ A @ Z @ Y4 ) )
=> ( ( ( ord_less_eq @ A @ Z @ Y4 )
=> ~ ( ord_less_eq @ A @ Y4 @ X4 ) )
=> ( ( ( ord_less_eq @ A @ Y4 @ Z )
=> ~ ( ord_less_eq @ A @ Z @ X4 ) )
=> ~ ( ( ord_less_eq @ A @ Z @ X4 )
=> ~ ( ord_less_eq @ A @ X4 @ Y4 ) ) ) ) ) ) ) ) ).
% le_cases3
thf(fact_198_order_Otrans,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [A2: A,B2: A,C3: A] :
( ( ord_less_eq @ A @ A2 @ B2 )
=> ( ( ord_less_eq @ A @ B2 @ C3 )
=> ( ord_less_eq @ A @ A2 @ C3 ) ) ) ) ).
% order.trans
thf(fact_199_le__cases,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [X4: A,Y4: A] :
( ~ ( ord_less_eq @ A @ X4 @ Y4 )
=> ( ord_less_eq @ A @ Y4 @ X4 ) ) ) ).
% le_cases
thf(fact_200_eq__refl,axiom,
! [A: $tType] :
( ( preorder @ A )
=> ! [X4: A,Y4: A] :
( ( X4 = Y4 )
=> ( ord_less_eq @ A @ X4 @ Y4 ) ) ) ).
% eq_refl
thf(fact_201_linear,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [X4: A,Y4: A] :
( ( ord_less_eq @ A @ X4 @ Y4 )
| ( ord_less_eq @ A @ Y4 @ X4 ) ) ) ).
% linear
thf(fact_202_antisym,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [X4: A,Y4: A] :
( ( ord_less_eq @ A @ X4 @ Y4 )
=> ( ( ord_less_eq @ A @ Y4 @ X4 )
=> ( X4 = Y4 ) ) ) ) ).
% antisym
thf(fact_203_eq__iff,axiom,
! [A: $tType] :
( ( order @ A )
=> ( ( ^ [Y8: A,Z4: A] : Y8 = Z4 )
= ( ^ [X5: A,Y5: A] :
( ( ord_less_eq @ A @ X5 @ Y5 )
& ( ord_less_eq @ A @ Y5 @ X5 ) ) ) ) ) ).
% eq_iff
thf(fact_204_ord__le__eq__subst,axiom,
! [A: $tType,B: $tType] :
( ( ( ord @ B )
& ( ord @ A ) )
=> ! [A2: A,B2: A,F3: A > B,C3: B] :
( ( ord_less_eq @ A @ A2 @ B2 )
=> ( ( ( F3 @ B2 )
= C3 )
=> ( ! [X3: A,Y3: A] :
( ( ord_less_eq @ A @ X3 @ Y3 )
=> ( ord_less_eq @ B @ ( F3 @ X3 ) @ ( F3 @ Y3 ) ) )
=> ( ord_less_eq @ B @ ( F3 @ A2 ) @ C3 ) ) ) ) ) ).
% ord_le_eq_subst
thf(fact_205_ord__eq__le__subst,axiom,
! [A: $tType,B: $tType] :
( ( ( ord @ B )
& ( ord @ A ) )
=> ! [A2: A,F3: B > A,B2: B,C3: B] :
( ( A2
= ( F3 @ B2 ) )
=> ( ( ord_less_eq @ B @ B2 @ C3 )
=> ( ! [X3: B,Y3: B] :
( ( ord_less_eq @ B @ X3 @ Y3 )
=> ( ord_less_eq @ A @ ( F3 @ X3 ) @ ( F3 @ Y3 ) ) )
=> ( ord_less_eq @ A @ A2 @ ( F3 @ C3 ) ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_206_order__subst2,axiom,
! [A: $tType,C: $tType] :
( ( ( order @ C )
& ( order @ A ) )
=> ! [A2: A,B2: A,F3: A > C,C3: C] :
( ( ord_less_eq @ A @ A2 @ B2 )
=> ( ( ord_less_eq @ C @ ( F3 @ B2 ) @ C3 )
=> ( ! [X3: A,Y3: A] :
( ( ord_less_eq @ A @ X3 @ Y3 )
=> ( ord_less_eq @ C @ ( F3 @ X3 ) @ ( F3 @ Y3 ) ) )
=> ( ord_less_eq @ C @ ( F3 @ A2 ) @ C3 ) ) ) ) ) ).
% order_subst2
thf(fact_207_order__subst1,axiom,
! [A: $tType,B: $tType] :
( ( ( order @ B )
& ( order @ A ) )
=> ! [A2: A,F3: B > A,B2: B,C3: B] :
( ( ord_less_eq @ A @ A2 @ ( F3 @ B2 ) )
=> ( ( ord_less_eq @ B @ B2 @ C3 )
=> ( ! [X3: B,Y3: B] :
( ( ord_less_eq @ B @ X3 @ Y3 )
=> ( ord_less_eq @ A @ ( F3 @ X3 ) @ ( F3 @ Y3 ) ) )
=> ( ord_less_eq @ A @ A2 @ ( F3 @ C3 ) ) ) ) ) ) ).
% order_subst1
thf(fact_208_le__fun__def,axiom,
! [B: $tType,A: $tType] :
( ( ord @ B )
=> ( ( ord_less_eq @ ( A > B ) )
= ( ^ [F4: A > B,G5: A > B] :
! [X5: A] : ( ord_less_eq @ B @ ( F4 @ X5 ) @ ( G5 @ X5 ) ) ) ) ) ).
% le_fun_def
thf(fact_209_le__funI,axiom,
! [B: $tType,A: $tType] :
( ( ord @ B )
=> ! [F3: A > B,G4: A > B] :
( ! [X3: A] : ( ord_less_eq @ B @ ( F3 @ X3 ) @ ( G4 @ X3 ) )
=> ( ord_less_eq @ ( A > B ) @ F3 @ G4 ) ) ) ).
% le_funI
thf(fact_210_le__funE,axiom,
! [B: $tType,A: $tType] :
( ( ord @ B )
=> ! [F3: A > B,G4: A > B,X4: A] :
( ( ord_less_eq @ ( A > B ) @ F3 @ G4 )
=> ( ord_less_eq @ B @ ( F3 @ X4 ) @ ( G4 @ X4 ) ) ) ) ).
% le_funE
thf(fact_211_le__funD,axiom,
! [B: $tType,A: $tType] :
( ( ord @ B )
=> ! [F3: A > B,G4: A > B,X4: A] :
( ( ord_less_eq @ ( A > B ) @ F3 @ G4 )
=> ( ord_less_eq @ B @ ( F3 @ X4 ) @ ( G4 @ X4 ) ) ) ) ).
% le_funD
thf(fact_212_sym__Id__on,axiom,
! [A: $tType,A5: set @ A] : ( sym @ A @ ( id_on @ A @ A5 ) ) ).
% sym_Id_on
thf(fact_213_subset__singletonD,axiom,
! [A: $tType,A5: set @ A,X4: A] :
( ( ord_less_eq @ ( set @ A ) @ A5 @ ( insert @ A @ X4 @ ( bot_bot @ ( set @ A ) ) ) )
=> ( ( A5
= ( bot_bot @ ( set @ A ) ) )
| ( A5
= ( insert @ A @ X4 @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ).
% subset_singletonD
thf(fact_214_subset__singleton__iff,axiom,
! [A: $tType,X7: set @ A,A2: A] :
( ( ord_less_eq @ ( set @ A ) @ X7 @ ( insert @ A @ A2 @ ( bot_bot @ ( set @ A ) ) ) )
= ( ( X7
= ( bot_bot @ ( set @ A ) ) )
| ( X7
= ( insert @ A @ A2 @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ).
% subset_singleton_iff
thf(fact_215_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_216_subgraph__preserves__hom,axiom,
! [A: $tType,B: $tType,C: $tType,A5: labeled_graph @ A @ B,B5: labeled_graph @ A @ B,X7: labeled_graph @ A @ C,H: set @ ( product_prod @ C @ B )] :
( ( graph_homomorphism @ A @ B @ B @ A5 @ B5 @ ( id_on @ B @ ( labeled_vertices @ A @ B @ A5 ) ) )
=> ( ( graph_homomorphism @ A @ C @ B @ X7 @ A5 @ H )
=> ( graph_homomorphism @ A @ C @ B @ X7 @ B5 @ H ) ) ) ).
% subgraph_preserves_hom
thf(fact_217_congruentD,axiom,
! [B: $tType,A: $tType,R2: set @ ( product_prod @ A @ A ),F3: A > B,Y4: A,Z: A] :
( ( equiv_congruent @ A @ B @ R2 @ F3 )
=> ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Y4 @ Z ) @ R2 )
=> ( ( F3 @ Y4 )
= ( F3 @ Z ) ) ) ) ).
% congruentD
thf(fact_218_congruentI,axiom,
! [B: $tType,A: $tType,R2: set @ ( product_prod @ A @ A ),F3: A > B] :
( ! [Y3: A,Z2: A] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Y3 @ Z2 ) @ R2 )
=> ( ( F3 @ Y3 )
= ( F3 @ Z2 ) ) )
=> ( equiv_congruent @ A @ B @ R2 @ F3 ) ) ).
% congruentI
thf(fact_219_in__quotient__imp__subset,axiom,
! [A: $tType,A5: set @ A,R2: set @ ( product_prod @ A @ A ),X7: set @ A] :
( ( equiv_equiv @ A @ A5 @ R2 )
=> ( ( member @ ( set @ A ) @ X7 @ ( equiv_quotient @ A @ A5 @ R2 ) )
=> ( ord_less_eq @ ( set @ A ) @ X7 @ A5 ) ) ) ).
% in_quotient_imp_subset
thf(fact_220_agree__on__subg__compose,axiom,
! [A: $tType,C: $tType,B: $tType,R: labeled_graph @ A @ B,G4: set @ ( product_prod @ B @ C ),H: set @ ( product_prod @ B @ C ),F5: labeled_graph @ A @ B,F3: set @ ( product_prod @ B @ C )] :
( ( agree_on @ A @ B @ C @ R @ G4 @ H )
=> ( ( agree_on @ A @ B @ C @ F5 @ F3 @ G4 )
=> ( ( graph_homomorphism @ A @ B @ B @ F5 @ R @ ( id_on @ B @ ( labeled_vertices @ A @ B @ F5 ) ) )
=> ( agree_on @ A @ B @ C @ F5 @ F3 @ H ) ) ) ) ).
% agree_on_subg_compose
thf(fact_221_maintainedD2,axiom,
! [A: $tType,C: $tType,B: $tType,A5: labeled_graph @ A @ B,B5: labeled_graph @ A @ B,G3: labeled_graph @ A @ C,F3: set @ ( product_prod @ B @ C )] :
( ( maintained @ A @ B @ C @ ( product_Pair @ ( labeled_graph @ A @ B ) @ ( labeled_graph @ A @ B ) @ A5 @ B5 ) @ G3 )
=> ( ( graph_homomorphism @ A @ B @ C @ A5 @ G3 @ F3 )
=> ~ ! [G2: set @ ( product_prod @ B @ C )] :
( ( graph_homomorphism @ A @ B @ C @ B5 @ G3 @ G2 )
=> ~ ( ord_less_eq @ ( set @ ( product_prod @ B @ C ) ) @ F3 @ G2 ) ) ) ) ).
% maintainedD2
thf(fact_222_subgraph__semantics,axiom,
! [B: $tType,A: $tType,A5: labeled_graph @ A @ B,B5: labeled_graph @ A @ B,A2: B,B2: B,E3: allegorical_term @ A] :
( ( graph_homomorphism @ A @ B @ B @ A5 @ B5 @ ( 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 @ B5 @ E3 ) ) ) ) ).
% subgraph_semantics
thf(fact_223_proj__iff,axiom,
! [A: $tType,A5: set @ A,R2: set @ ( product_prod @ A @ A ),X4: A,Y4: A] :
( ( equiv_equiv @ A @ A5 @ R2 )
=> ( ( ord_less_eq @ ( set @ A ) @ ( insert @ A @ X4 @ ( insert @ A @ Y4 @ ( bot_bot @ ( set @ A ) ) ) ) @ A5 )
=> ( ( ( equiv_proj @ A @ A @ R2 @ X4 )
= ( equiv_proj @ A @ A @ R2 @ Y4 ) )
= ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X4 @ Y4 ) @ R2 ) ) ) ) ).
% proj_iff
thf(fact_224_least__consequence__graphI,axiom,
! [X: $tType,A: $tType,C: $tType,L: $tType,Rs: set @ ( product_prod @ ( labeled_graph @ L @ A ) @ ( labeled_graph @ L @ A ) ),G3: labeled_graph @ L @ C,S2: labeled_graph @ L @ C,T3: itself @ X] :
( ( consequence_graph @ L @ A @ C @ Rs @ G3 )
=> ( ( graph_homomorphism @ L @ C @ C @ S2 @ G3 @ ( id_on @ C @ ( labeled_vertices @ L @ C @ S2 ) ) )
=> ( ! [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 ) @ S2 @ G3 ) @ C6 ) )
=> ( least_559130134_graph @ X @ L @ A @ C @ T3 @ Rs @ S2 @ G3 ) ) ) ) ).
% least_consequence_graphI
thf(fact_225_leastI,axiom,
! [X: $tType,A: $tType,C: $tType,L: $tType,S2: labeled_graph @ L @ C,G3: labeled_graph @ L @ C,Rs: set @ ( product_prod @ ( labeled_graph @ L @ A ) @ ( labeled_graph @ L @ A ) ),T3: itself @ X] :
( ( graph_homomorphism @ L @ C @ C @ S2 @ G3 @ ( id_on @ C @ ( labeled_vertices @ L @ C @ S2 ) ) )
=> ( ! [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 ) @ S2 @ G3 ) @ C6 ) )
=> ( least @ X @ L @ A @ C @ T3 @ Rs @ S2 @ G3 ) ) ) ).
% leastI
thf(fact_226_least__consequence__graph__def,axiom,
! [L: $tType,V: $tType,C: $tType,X: $tType] :
( ( least_559130134_graph @ X @ L @ V @ C )
= ( ^ [T2: itself @ X,Rs2: set @ ( product_prod @ ( labeled_graph @ L @ V ) @ ( labeled_graph @ L @ V ) ),S3: labeled_graph @ L @ C,G7: labeled_graph @ L @ C] :
( ( consequence_graph @ L @ V @ C @ Rs2 @ G7 )
& ( least @ X @ L @ V @ C @ T2 @ Rs2 @ S3 @ G7 ) ) ) ) ).
% least_consequence_graph_def
thf(fact_227_consequence__graphD_I3_J,axiom,
! [B: $tType,C: $tType,A: $tType,Rs: set @ ( product_prod @ ( labeled_graph @ A @ B ) @ ( labeled_graph @ A @ B ) ),G3: labeled_graph @ A @ C] :
( ( consequence_graph @ A @ B @ C @ Rs @ G3 )
=> ( G3
= ( restrict @ A @ C @ G3 ) ) ) ).
% consequence_graphD(3)
thf(fact_228_consequence__graphD_I1_J,axiom,
! [B: $tType,C: $tType,A: $tType,Rs: set @ ( product_prod @ ( labeled_graph @ A @ B ) @ ( labeled_graph @ A @ B ) ),G3: labeled_graph @ A @ C,R: product_prod @ ( labeled_graph @ A @ B ) @ ( labeled_graph @ A @ B )] :
( ( consequence_graph @ A @ B @ C @ Rs @ G3 )
=> ( ( member @ ( product_prod @ ( labeled_graph @ A @ B ) @ ( labeled_graph @ A @ B ) ) @ R @ Rs )
=> ( maintained @ A @ B @ C @ R @ G3 ) ) ) ).
% consequence_graphD(1)
thf(fact_229_proj__in__iff,axiom,
! [A: $tType,A5: set @ A,R2: set @ ( product_prod @ A @ A ),X4: A] :
( ( equiv_equiv @ A @ A5 @ R2 )
=> ( ( member @ ( set @ A ) @ ( equiv_proj @ A @ A @ R2 @ X4 ) @ ( equiv_quotient @ A @ A5 @ R2 ) )
= ( member @ A @ X4 @ A5 ) ) ) ).
% proj_in_iff
thf(fact_230_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 ) ),S3: labeled_graph @ L @ C,G7: labeled_graph @ L @ C] :
( ( graph_homomorphism @ L @ C @ C @ S3 @ G7 @ ( id_on @ C @ ( labeled_vertices @ L @ C @ S3 ) ) )
& ! [C4: labeled_graph @ L @ X] :
( ( consequence_graph @ L @ V @ X @ Rs2 @ C4 )
=> ( maintained @ L @ C @ X @ ( product_Pair @ ( labeled_graph @ L @ C ) @ ( labeled_graph @ L @ C ) @ S3 @ G7 ) @ C4 ) ) ) ) ) ).
% least_def
thf(fact_231_less__by__empty,axiom,
! [A: $tType,A5: set @ ( product_prod @ A @ A ),B5: set @ ( product_prod @ A @ A )] :
( ( A5
= ( bot_bot @ ( set @ ( product_prod @ A @ A ) ) ) )
=> ( ord_less_eq @ ( set @ ( product_prod @ A @ A ) ) @ A5 @ B5 ) ) ).
% less_by_empty
thf(fact_232_insert__subsetI,axiom,
! [A: $tType,X4: A,A5: set @ A,X7: set @ A] :
( ( member @ A @ X4 @ A5 )
=> ( ( ord_less_eq @ ( set @ A ) @ X7 @ A5 )
=> ( ord_less_eq @ ( set @ A ) @ ( insert @ A @ X4 @ X7 ) @ A5 ) ) ) ).
% insert_subsetI
thf(fact_233_ssubst__Pair__rhs,axiom,
! [B: $tType,A: $tType,R2: A,S: B,R: set @ ( product_prod @ A @ B ),S4: B] :
( ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ R2 @ S ) @ R )
=> ( ( S4 = S )
=> ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ R2 @ S4 ) @ R ) ) ) ).
% ssubst_Pair_rhs
thf(fact_234_subset__emptyI,axiom,
! [A: $tType,A5: set @ A] :
( ! [X3: A] :
~ ( member @ A @ X3 @ A5 )
=> ( ord_less_eq @ ( set @ A ) @ A5 @ ( bot_bot @ ( set @ A ) ) ) ) ).
% subset_emptyI
thf(fact_235_univ__commute,axiom,
! [B: $tType,A: $tType,A5: set @ A,R2: set @ ( product_prod @ A @ A ),F3: A > B,X4: A] :
( ( equiv_equiv @ A @ A5 @ R2 )
=> ( ( equiv_congruent @ A @ B @ R2 @ F3 )
=> ( ( member @ A @ X4 @ A5 )
=> ( ( bNF_Greatest_univ @ A @ B @ F3 @ ( equiv_proj @ A @ A @ R2 @ X4 ) )
= ( F3 @ X4 ) ) ) ) ) ).
% univ_commute
thf(fact_236_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_237_graph__union__idemp_I3_J,axiom,
! [B: $tType,A: $tType,A5: labeled_graph @ A @ B,B5: labeled_graph @ A @ B] :
( ( graph_union @ A @ B @ A5 @ ( graph_union @ A @ B @ B5 @ A5 ) )
= ( graph_union @ A @ B @ B5 @ A5 ) ) ).
% graph_union_idemp(3)
thf(fact_238_graph__union__idemp_I2_J,axiom,
! [B: $tType,A: $tType,A5: labeled_graph @ A @ B,B5: labeled_graph @ A @ B] :
( ( graph_union @ A @ B @ A5 @ ( graph_union @ A @ B @ A5 @ B5 ) )
= ( graph_union @ A @ B @ A5 @ B5 ) ) ).
% graph_union_idemp(2)
thf(fact_239_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_240_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_241_univ__preserves,axiom,
! [A: $tType,B: $tType,A5: set @ A,R2: set @ ( product_prod @ A @ A ),F3: A > B,B5: set @ B] :
( ( equiv_equiv @ A @ A5 @ R2 )
=> ( ( equiv_congruent @ A @ B @ R2 @ F3 )
=> ( ! [X3: A] :
( ( member @ A @ X3 @ A5 )
=> ( member @ B @ ( F3 @ X3 ) @ B5 ) )
=> ! [X2: set @ A] :
( ( member @ ( set @ A ) @ X2 @ ( equiv_quotient @ A @ A5 @ R2 ) )
=> ( member @ B @ ( bNF_Greatest_univ @ A @ B @ F3 @ X2 ) @ B5 ) ) ) ) ) ).
% univ_preserves
thf(fact_242_relImage__proj,axiom,
! [A: $tType,A5: set @ A,R: set @ ( product_prod @ A @ A )] :
( ( equiv_equiv @ A @ A5 @ R )
=> ( ord_less_eq @ ( set @ ( product_prod @ ( set @ A ) @ ( set @ A ) ) ) @ ( bNF_Gr1317331620lImage @ A @ ( set @ A ) @ R @ ( equiv_proj @ A @ A @ R ) ) @ ( id_on @ ( set @ A ) @ ( equiv_quotient @ A @ A5 @ R ) ) ) ) ).
% relImage_proj
thf(fact_243_subgraph__def2,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_homomorphism @ A @ B @ B @ G_1 @ G_2 @ ( id_on @ B @ ( labeled_vertices @ A @ B @ G_1 ) ) )
= ( ( ord_less_eq @ ( set @ B ) @ ( labeled_vertices @ A @ B @ G_1 ) @ ( labeled_vertices @ A @ B @ G_2 ) )
& ( ord_less_eq @ ( set @ ( product_prod @ A @ ( product_prod @ B @ B ) ) ) @ ( labeled_edges @ A @ B @ G_1 ) @ ( labeled_edges @ A @ B @ G_2 ) ) ) ) ) ) ).
% subgraph_def2
thf(fact_244_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_245_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_246_restrict__subsD,axiom,
! [B: $tType,A: $tType,G3: labeled_graph @ A @ B] :
( ( ord_less_eq @ ( set @ ( product_prod @ A @ ( product_prod @ B @ B ) ) ) @ ( labeled_edges @ A @ B @ G3 ) @ ( labeled_edges @ A @ B @ ( restrict @ A @ B @ G3 ) ) )
=> ( G3
= ( restrict @ A @ B @ G3 ) ) ) ).
% restrict_subsD
thf(fact_247_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_248_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_249_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_250_graph__union__iff,axiom,
! [B: $tType,A: $tType,G_1: labeled_graph @ A @ B,G_2: labeled_graph @ A @ B] :
( ( ( graph_union @ A @ B @ G_1 @ G_2 )
= G_2 )
= ( ( 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_iff
thf(fact_251_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_252_labeled__graph_Osplit__sel__asm,axiom,
! [A: $tType,V: $tType,L: $tType,P2: A > $o,F3: ( set @ ( product_prod @ L @ ( product_prod @ V @ V ) ) ) > ( set @ V ) > A,Labeled_graph: labeled_graph @ L @ V] :
( ( P2 @ ( labele1974067554_graph @ L @ V @ A @ F3 @ Labeled_graph ) )
= ( ~ ( ( Labeled_graph
= ( labeled_LG @ L @ V @ ( labeled_edges @ L @ V @ Labeled_graph ) @ ( labeled_vertices @ L @ V @ Labeled_graph ) ) )
& ~ ( P2 @ ( F3 @ ( labeled_edges @ L @ V @ Labeled_graph ) @ ( labeled_vertices @ L @ V @ Labeled_graph ) ) ) ) ) ) ).
% labeled_graph.split_sel_asm
thf(fact_253_labeled__graph_Osplit__sel,axiom,
! [A: $tType,V: $tType,L: $tType,P2: A > $o,F3: ( set @ ( product_prod @ L @ ( product_prod @ V @ V ) ) ) > ( set @ V ) > A,Labeled_graph: labeled_graph @ L @ V] :
( ( P2 @ ( labele1974067554_graph @ L @ V @ A @ F3 @ Labeled_graph ) )
= ( ( Labeled_graph
= ( labeled_LG @ L @ V @ ( labeled_edges @ L @ V @ Labeled_graph ) @ ( labeled_vertices @ L @ V @ Labeled_graph ) ) )
=> ( P2 @ ( F3 @ ( labeled_edges @ L @ V @ Labeled_graph ) @ ( labeled_vertices @ L @ V @ Labeled_graph ) ) ) ) ) ).
% labeled_graph.split_sel
thf(fact_254_labeled__graph_Ocase,axiom,
! [L: $tType,A: $tType,V: $tType,F3: ( 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 @ F3 @ ( labeled_LG @ L @ V @ X1 @ X22 ) )
= ( F3 @ X1 @ X22 ) ) ).
% labeled_graph.case
thf(fact_255_labeled__graph_Ocase__eq__if,axiom,
! [A: $tType,V: $tType,L: $tType] :
( ( labele1974067554_graph @ L @ V @ A )
= ( ^ [F4: ( set @ ( product_prod @ L @ ( product_prod @ V @ V ) ) ) > ( set @ V ) > A,Labeled_graph3: labeled_graph @ L @ V] : ( F4 @ ( labeled_edges @ L @ V @ Labeled_graph3 ) @ ( labeled_vertices @ L @ V @ Labeled_graph3 ) ) ) ) ).
% labeled_graph.case_eq_if
% Type constructors (22)
thf(tcon_fun___Orderings_Oorder__bot,axiom,
! [A8: $tType,A9: $tType] :
( ( order_bot @ A9 )
=> ( order_bot @ ( A8 > A9 ) ) ) ).
thf(tcon_fun___Orderings_Opreorder,axiom,
! [A8: $tType,A9: $tType] :
( ( preorder @ A9 )
=> ( preorder @ ( A8 > A9 ) ) ) ).
thf(tcon_fun___Orderings_Oorder,axiom,
! [A8: $tType,A9: $tType] :
( ( order @ A9 )
=> ( order @ ( A8 > A9 ) ) ) ).
thf(tcon_fun___Orderings_Oord,axiom,
! [A8: $tType,A9: $tType] :
( ( ord @ A9 )
=> ( ord @ ( A8 > A9 ) ) ) ).
thf(tcon_fun___Orderings_Obot,axiom,
! [A8: $tType,A9: $tType] :
( ( bot @ A9 )
=> ( bot @ ( A8 > A9 ) ) ) ).
thf(tcon_Nat_Onat___Orderings_Oorder__bot_1,axiom,
order_bot @ nat ).
thf(tcon_Nat_Onat___Orderings_Opreorder_2,axiom,
preorder @ nat ).
thf(tcon_Nat_Onat___Orderings_Olinorder,axiom,
linorder @ nat ).
thf(tcon_Nat_Onat___Orderings_Oorder_3,axiom,
order @ nat ).
thf(tcon_Nat_Onat___Orderings_Oord_4,axiom,
ord @ nat ).
thf(tcon_Nat_Onat___Orderings_Obot_5,axiom,
bot @ nat ).
thf(tcon_Set_Oset___Orderings_Oorder__bot_6,axiom,
! [A8: $tType] : ( order_bot @ ( set @ A8 ) ) ).
thf(tcon_Set_Oset___Orderings_Opreorder_7,axiom,
! [A8: $tType] : ( preorder @ ( set @ A8 ) ) ).
thf(tcon_Set_Oset___Orderings_Oorder_8,axiom,
! [A8: $tType] : ( order @ ( set @ A8 ) ) ).
thf(tcon_Set_Oset___Orderings_Oord_9,axiom,
! [A8: $tType] : ( ord @ ( set @ A8 ) ) ).
thf(tcon_Set_Oset___Orderings_Obot_10,axiom,
! [A8: $tType] : ( bot @ ( set @ A8 ) ) ).
thf(tcon_HOL_Obool___Orderings_Oorder__bot_11,axiom,
order_bot @ $o ).
thf(tcon_HOL_Obool___Orderings_Opreorder_12,axiom,
preorder @ $o ).
thf(tcon_HOL_Obool___Orderings_Olinorder_13,axiom,
linorder @ $o ).
thf(tcon_HOL_Obool___Orderings_Oorder_14,axiom,
order @ $o ).
thf(tcon_HOL_Obool___Orderings_Oord_15,axiom,
ord @ $o ).
thf(tcon_HOL_Obool___Orderings_Obot_16,axiom,
bot @ $o ).
% Conjectures (1)
thf(conj_0,conjecture,
member @ ( product_prod @ b @ b ) @ ( product_Pair @ b @ b @ z @ y ) @ ( getRel @ a @ b @ i @ g ) ).
%------------------------------------------------------------------------------