TPTP Problem File: ITP049^1.p
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%------------------------------------------------------------------------------
% File : ITP049^1 : TPTP v8.2.0. Released v7.5.0.
% Domain : Interactive Theorem Proving
% Problem : Sledgehammer EdmondsKarp_Termination_Abstract problem prob_229__7588066_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 : EdmondsKarp_Termination_Abstract/prob_229__7588066_1 [Des21]
% Status : Theorem
% Rating : 0.30 v8.2.0, 0.15 v8.1.0, 0.18 v7.5.0
% Syntax : Number of formulae : 435 ( 102 unt; 80 typ; 0 def)
% Number of atoms : 1099 ( 200 equ; 0 cnn)
% Maximal formula atoms : 12 ( 3 avg)
% Number of connectives : 3642 ( 106 ~; 10 |; 89 &;2864 @)
% ( 0 <=>; 573 =>; 0 <=; 0 <~>)
% Maximal formula depth : 22 ( 8 avg)
% Number of types : 16 ( 15 usr)
% Number of type conns : 343 ( 343 >; 0 *; 0 +; 0 <<)
% Number of symbols : 66 ( 65 usr; 2 con; 0-4 aty)
% Number of variables : 1227 ( 124 ^;1026 !; 77 ?;1227 :)
% SPC : TH0_THM_EQU_NAR
% Comments : This file was generated by Sledgehammer 2021-02-23 15:30:52.673
%------------------------------------------------------------------------------
% Could-be-implicit typings (15)
thf(ty_n_t__Set__Oset_It__Product____Type__Oprod_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_Mt__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J_J,type,
set_Pr1490359111at_nat: $tType ).
thf(ty_n_t__Product____Type__Oprod_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_Mt__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J,type,
produc842455143at_nat: $tType ).
thf(ty_n_t__Set__Oset_It__Product____Type__Oprod_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_Mt__Nat__Onat_J_J,type,
set_Pr1287749686at_nat: $tType ).
thf(ty_n_t__Set__Oset_It__Product____Type__Oprod_It__Nat__Onat_Mt__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J_J,type,
set_Pr1746169692at_nat: $tType ).
thf(ty_n_t__Product____Type__Oprod_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_Mt__Nat__Onat_J,type,
produc1271302400at_nat: $tType ).
thf(ty_n_t__Product____Type__Oprod_It__Nat__Onat_Mt__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J,type,
produc1695820582at_nat: $tType ).
thf(ty_n_t__Set__Oset_It__Set__Oset_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J_J,type,
set_se1612935105at_nat: $tType ).
thf(ty_n_t__List__Olist_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J,type,
list_P559422087at_nat: $tType ).
thf(ty_n_t__Set__Oset_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J,type,
set_Pr1986765409at_nat: $tType ).
thf(ty_n_t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J,type,
product_prod_nat_nat: $tType ).
thf(ty_n_t__Set__Oset_It__Set__Oset_It__Nat__Onat_J_J,type,
set_set_nat: $tType ).
thf(ty_n_t__List__Olist_It__Nat__Onat_J,type,
list_nat: $tType ).
thf(ty_n_t__Set__Oset_It__Nat__Onat_J,type,
set_nat: $tType ).
thf(ty_n_t__Nat__Onat,type,
nat: $tType ).
thf(ty_n_tf__a,type,
a: $tType ).
% Explicit typings (65)
thf(sy_c_EdmondsKarp__Termination__Abstract__Mirabelle__pndqeoznpl_Oek__analysis_001tf__a,type,
edmond1517640972ysis_a: ( product_prod_nat_nat > a ) > $o ).
thf(sy_c_EdmondsKarp__Termination__Abstract__Mirabelle__pndqeoznpl_Oek__analysis__defs_OspEdges_001tf__a,type,
edmond475474835dges_a: ( product_prod_nat_nat > a ) > nat > nat > set_Pr1986765409at_nat ).
thf(sy_c_Finite__Set_Ofinite_001t__Nat__Onat,type,
finite_finite_nat: set_nat > $o ).
thf(sy_c_Finite__Set_Ofinite_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J,type,
finite772653738at_nat: set_Pr1986765409at_nat > $o ).
thf(sy_c_Finite__Set_Ofinite_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J,type,
finite277291581at_nat: set_Pr1746169692at_nat > $o ).
thf(sy_c_Finite__Set_Ofinite_001t__Product____Type__Oprod_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_Mt__Nat__Onat_J,type,
finite2000257047at_nat: set_Pr1287749686at_nat > $o ).
thf(sy_c_Finite__Set_Ofinite_001t__Product____Type__Oprod_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_Mt__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J,type,
finite48957584at_nat: set_Pr1490359111at_nat > $o ).
thf(sy_c_Finite__Set_Ofinite_001t__Set__Oset_It__Nat__Onat_J,type,
finite2012248349et_nat: set_set_nat > $o ).
thf(sy_c_Finite__Set_Ofinite_001t__Set__Oset_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J,type,
finite1457549322at_nat: set_se1612935105at_nat > $o ).
thf(sy_c_Graph_OFinite__Graph_001tf__a,type,
finite_Graph_a: ( product_prod_nat_nat > a ) > $o ).
thf(sy_c_Graph_OGraph_OE_001tf__a,type,
e_a: ( product_prod_nat_nat > a ) > set_Pr1986765409at_nat ).
thf(sy_c_Graph_OGraph_OV_001tf__a,type,
v_a: ( product_prod_nat_nat > a ) > set_nat ).
thf(sy_c_Graph_OGraph_Oadjacent__nodes_001tf__a,type,
adjacent_nodes_a: ( product_prod_nat_nat > a ) > nat > set_nat ).
thf(sy_c_Graph_OGraph_Oconnected_001tf__a,type,
connected_a: ( product_prod_nat_nat > a ) > nat > nat > $o ).
thf(sy_c_Graph_OGraph_Oincoming_001tf__a,type,
incoming_a: ( product_prod_nat_nat > a ) > nat > set_Pr1986765409at_nat ).
thf(sy_c_Graph_OGraph_Oincoming_H_001tf__a,type,
incoming_a2: ( product_prod_nat_nat > a ) > set_nat > set_Pr1986765409at_nat ).
thf(sy_c_Graph_OGraph_OisPath_001tf__a,type,
isPath_a: ( product_prod_nat_nat > a ) > nat > list_P559422087at_nat > nat > $o ).
thf(sy_c_Graph_OGraph_OisShortestPath_001tf__a,type,
isShortestPath_a: ( product_prod_nat_nat > a ) > nat > list_P559422087at_nat > nat > $o ).
thf(sy_c_Graph_OGraph_OisSimplePath_001tf__a,type,
isSimplePath_a: ( product_prod_nat_nat > a ) > nat > list_P559422087at_nat > nat > $o ).
thf(sy_c_Graph_OGraph_Ooutgoing_001tf__a,type,
outgoing_a: ( product_prod_nat_nat > a ) > nat > set_Pr1986765409at_nat ).
thf(sy_c_Graph_OGraph_Ooutgoing_H_001tf__a,type,
outgoing_a2: ( product_prod_nat_nat > a ) > set_nat > set_Pr1986765409at_nat ).
thf(sy_c_Graph_OGraph_OreachableNodes_001tf__a,type,
reachableNodes_a: ( product_prod_nat_nat > a ) > nat > set_nat ).
thf(sy_c_List_Olist_Oset_001t__Nat__Onat,type,
set_nat2: list_nat > set_nat ).
thf(sy_c_List_Olist_Oset_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J,type,
set_Pr2131844118at_nat: list_P559422087at_nat > set_Pr1986765409at_nat ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001_062_It__Nat__Onat_M_062_It__Nat__Onat_M_Eo_J_J,type,
ord_le1578155910_nat_o: ( nat > nat > $o ) > ( nat > nat > $o ) > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001_062_It__Nat__Onat_M_Eo_J,type,
ord_less_eq_nat_o: ( nat > $o ) > ( nat > $o ) > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001_062_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_M_Eo_J,type,
ord_le1039616028_nat_o: ( product_prod_nat_nat > $o ) > ( product_prod_nat_nat > $o ) > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Nat__Onat,type,
ord_less_eq_nat: nat > nat > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_It__Nat__Onat_J,type,
ord_less_eq_set_nat: set_nat > set_nat > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J,type,
ord_le841296385at_nat: set_Pr1986765409at_nat > set_Pr1986765409at_nat > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_It__Product____Type__Oprod_It__Nat__Onat_Mt__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J_J,type,
ord_le148299708at_nat: set_Pr1746169692at_nat > set_Pr1746169692at_nat > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_It__Product____Type__Oprod_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_Mt__Nat__Onat_J_J,type,
ord_le1837363350at_nat: set_Pr1287749686at_nat > set_Pr1287749686at_nat > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_It__Product____Type__Oprod_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_Mt__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J_J,type,
ord_le465611495at_nat: set_Pr1490359111at_nat > set_Pr1490359111at_nat > $o ).
thf(sy_c_Product__Type_OPair_001t__Nat__Onat_001t__Nat__Onat,type,
product_Pair_nat_nat: nat > nat > product_prod_nat_nat ).
thf(sy_c_Product__Type_OPair_001t__Nat__Onat_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J,type,
produc1933845336at_nat: nat > product_prod_nat_nat > produc1695820582at_nat ).
thf(sy_c_Product__Type_OPair_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_001t__Nat__Onat,type,
produc947540346at_nat: product_prod_nat_nat > nat > produc1271302400at_nat ).
thf(sy_c_Product__Type_OPair_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J,type,
produc1168807639at_nat: product_prod_nat_nat > product_prod_nat_nat > produc842455143at_nat ).
thf(sy_c_Product__Type_OSigma_001t__Nat__Onat_001t__Nat__Onat,type,
produc45129834at_nat: set_nat > ( nat > set_nat ) > set_Pr1986765409at_nat ).
thf(sy_c_Product__Type_OSigma_001t__Nat__Onat_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J,type,
produc894163943at_nat: set_nat > ( nat > set_Pr1986765409at_nat ) > set_Pr1746169692at_nat ).
thf(sy_c_Product__Type_OSigma_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_001t__Nat__Onat,type,
produc2055342601at_nat: set_Pr1986765409at_nat > ( product_prod_nat_nat > set_nat ) > set_Pr1287749686at_nat ).
thf(sy_c_Product__Type_OSigma_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J,type,
produc2056081288at_nat: set_Pr1986765409at_nat > ( product_prod_nat_nat > set_Pr1986765409at_nat ) > set_Pr1490359111at_nat ).
thf(sy_c_Product__Type_Oprod_Oswap_001t__Nat__Onat_001t__Nat__Onat,type,
product_swap_nat_nat: product_prod_nat_nat > product_prod_nat_nat ).
thf(sy_c_Relation_ODomain_001t__Nat__Onat_001t__Nat__Onat,type,
domain_nat_nat: set_Pr1986765409at_nat > set_nat ).
thf(sy_c_Relation_ODomain_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J,type,
domain37934156at_nat: set_Pr1490359111at_nat > set_Pr1986765409at_nat ).
thf(sy_c_Relation_OImage_001t__Nat__Onat_001t__Nat__Onat,type,
image_nat_nat: set_Pr1986765409at_nat > set_nat > set_nat ).
thf(sy_c_Relation_OImage_001t__Nat__Onat_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J,type,
image_981520924at_nat: set_Pr1746169692at_nat > set_nat > set_Pr1986765409at_nat ).
thf(sy_c_Relation_OImage_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_001t__Nat__Onat,type,
image_2142699582at_nat: set_Pr1287749686at_nat > set_Pr1986765409at_nat > set_nat ).
thf(sy_c_Relation_OImage_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J,type,
image_1655366419at_nat: set_Pr1490359111at_nat > set_Pr1986765409at_nat > set_Pr1986765409at_nat ).
thf(sy_c_Relation_ORange_001t__Nat__Onat_001t__Nat__Onat,type,
range_nat_nat: set_Pr1986765409at_nat > set_nat ).
thf(sy_c_Set_OCollect_001t__Nat__Onat,type,
collect_nat: ( nat > $o ) > set_nat ).
thf(sy_c_Set_OCollect_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J,type,
collec7649004at_nat: ( product_prod_nat_nat > $o ) > set_Pr1986765409at_nat ).
thf(sy_c_Set_OCollect_001t__Set__Oset_It__Nat__Onat_J,type,
collect_set_nat: ( set_nat > $o ) > set_set_nat ).
thf(sy_c_Set_OCollect_001t__Set__Oset_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J,type,
collec1606769740at_nat: ( set_Pr1986765409at_nat > $o ) > set_se1612935105at_nat ).
thf(sy_c_Transitive__Closure_Ortrancl_001t__Nat__Onat,type,
transi1645084429cl_nat: set_Pr1986765409at_nat > set_Pr1986765409at_nat ).
thf(sy_c_Transitive__Closure_Ortrancl_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J,type,
transi1484096900at_nat: set_Pr1490359111at_nat > set_Pr1490359111at_nat ).
thf(sy_c_member_001t__Nat__Onat,type,
member_nat: nat > set_nat > $o ).
thf(sy_c_member_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J,type,
member701585322at_nat: product_prod_nat_nat > set_Pr1986765409at_nat > $o ).
thf(sy_c_member_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J,type,
member1293241661at_nat: produc1695820582at_nat > set_Pr1746169692at_nat > $o ).
thf(sy_c_member_001t__Product____Type__Oprod_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_Mt__Nat__Onat_J,type,
member868723479at_nat: produc1271302400at_nat > set_Pr1287749686at_nat > $o ).
thf(sy_c_member_001t__Product____Type__Oprod_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_Mt__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J,type,
member2027625872at_nat: produc842455143at_nat > set_Pr1490359111at_nat > $o ).
thf(sy_c_member_001t__Set__Oset_It__Nat__Onat_J,type,
member_set_nat: set_nat > set_set_nat > $o ).
thf(sy_c_member_001t__Set__Oset_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J,type,
member298845450at_nat: set_Pr1986765409at_nat > set_se1612935105at_nat > $o ).
thf(sy_v_c,type,
c: product_prod_nat_nat > a ).
thf(sy_v_s,type,
s: nat ).
thf(sy_v_t,type,
t: nat ).
% Relevant facts (354)
thf(fact_0_ek__analysis__axioms,axiom,
edmond1517640972ysis_a @ c ).
% ek_analysis_axioms
thf(fact_1__092_060open_062finite_AE_A_092_060Longrightarrow_062_Afinite_AspEdges_092_060close_062,axiom,
( ( finite772653738at_nat @ ( e_a @ c ) )
=> ( finite772653738at_nat @ ( edmond475474835dges_a @ c @ s @ t ) ) ) ).
% \<open>finite E \<Longrightarrow> finite spEdges\<close>
thf(fact_2_ek__analysis__defs_OspEdges_Ocong,axiom,
edmond475474835dges_a = edmond475474835dges_a ).
% ek_analysis_defs.spEdges.cong
thf(fact_3_finite__E,axiom,
finite772653738at_nat @ ( e_a @ c ) ).
% finite_E
thf(fact_4_spEdges__ss__E,axiom,
ord_le841296385at_nat @ ( edmond475474835dges_a @ c @ s @ t ) @ ( e_a @ c ) ).
% spEdges_ss_E
thf(fact_5_Finite__Graph__axioms,axiom,
finite_Graph_a @ c ).
% Finite_Graph_axioms
thf(fact_6_Finite__Graph__EI,axiom,
( ( finite772653738at_nat @ ( e_a @ c ) )
=> ( finite_Graph_a @ c ) ) ).
% Finite_Graph_EI
thf(fact_7_connected__refl,axiom,
! [V: nat] : ( connected_a @ c @ V @ V ) ).
% connected_refl
thf(fact_8_spEdges__def,axiom,
( ( edmond475474835dges_a @ c @ s @ t )
= ( collec7649004at_nat
@ ^ [E: product_prod_nat_nat] :
? [P: list_P559422087at_nat] :
( ( member701585322at_nat @ E @ ( set_Pr2131844118at_nat @ P ) )
& ( isShortestPath_a @ c @ s @ P @ t ) ) ) ) ).
% spEdges_def
thf(fact_9_finite__Range,axiom,
! [R: set_Pr1986765409at_nat] :
( ( finite772653738at_nat @ R )
=> ( finite_finite_nat @ ( range_nat_nat @ R ) ) ) ).
% finite_Range
thf(fact_10_finite__Domain,axiom,
! [R: set_Pr1986765409at_nat] :
( ( finite772653738at_nat @ R )
=> ( finite_finite_nat @ ( domain_nat_nat @ R ) ) ) ).
% finite_Domain
thf(fact_11_Efin__imp__Vfin,axiom,
( ( finite772653738at_nat @ ( e_a @ c ) )
=> ( finite_finite_nat @ ( v_a @ c ) ) ) ).
% Efin_imp_Vfin
thf(fact_12_connected__inV__iff,axiom,
! [U: nat,V: nat] :
( ( connected_a @ c @ U @ V )
=> ( ( member_nat @ V @ ( v_a @ c ) )
= ( member_nat @ U @ ( v_a @ c ) ) ) ) ).
% connected_inV_iff
thf(fact_13_obtain__shortest__path,axiom,
! [U: nat,V: nat] :
( ( connected_a @ c @ U @ V )
=> ~ ! [P2: list_P559422087at_nat] :
~ ( isShortestPath_a @ c @ U @ P2 @ V ) ) ).
% obtain_shortest_path
thf(fact_14_finite__Collect__conjI,axiom,
! [P3: product_prod_nat_nat > $o,Q: product_prod_nat_nat > $o] :
( ( ( finite772653738at_nat @ ( collec7649004at_nat @ P3 ) )
| ( finite772653738at_nat @ ( collec7649004at_nat @ Q ) ) )
=> ( finite772653738at_nat
@ ( collec7649004at_nat
@ ^ [X: product_prod_nat_nat] :
( ( P3 @ X )
& ( Q @ X ) ) ) ) ) ).
% finite_Collect_conjI
thf(fact_15_finite__Collect__conjI,axiom,
! [P3: nat > $o,Q: nat > $o] :
( ( ( finite_finite_nat @ ( collect_nat @ P3 ) )
| ( finite_finite_nat @ ( collect_nat @ Q ) ) )
=> ( finite_finite_nat
@ ( collect_nat
@ ^ [X: nat] :
( ( P3 @ X )
& ( Q @ X ) ) ) ) ) ).
% finite_Collect_conjI
thf(fact_16_finite__Collect__disjI,axiom,
! [P3: product_prod_nat_nat > $o,Q: product_prod_nat_nat > $o] :
( ( finite772653738at_nat
@ ( collec7649004at_nat
@ ^ [X: product_prod_nat_nat] :
( ( P3 @ X )
| ( Q @ X ) ) ) )
= ( ( finite772653738at_nat @ ( collec7649004at_nat @ P3 ) )
& ( finite772653738at_nat @ ( collec7649004at_nat @ Q ) ) ) ) ).
% finite_Collect_disjI
thf(fact_17_finite__Collect__disjI,axiom,
! [P3: nat > $o,Q: nat > $o] :
( ( finite_finite_nat
@ ( collect_nat
@ ^ [X: nat] :
( ( P3 @ X )
| ( Q @ X ) ) ) )
= ( ( finite_finite_nat @ ( collect_nat @ P3 ) )
& ( finite_finite_nat @ ( collect_nat @ Q ) ) ) ) ).
% finite_Collect_disjI
thf(fact_18_finite__Collect__bounded__ex,axiom,
! [P3: product_prod_nat_nat > $o,Q: product_prod_nat_nat > product_prod_nat_nat > $o] :
( ( finite772653738at_nat @ ( collec7649004at_nat @ P3 ) )
=> ( ( finite772653738at_nat
@ ( collec7649004at_nat
@ ^ [X: product_prod_nat_nat] :
? [Y: product_prod_nat_nat] :
( ( P3 @ Y )
& ( Q @ X @ Y ) ) ) )
= ( ! [Y: product_prod_nat_nat] :
( ( P3 @ Y )
=> ( finite772653738at_nat
@ ( collec7649004at_nat
@ ^ [X: product_prod_nat_nat] : ( Q @ X @ Y ) ) ) ) ) ) ) ).
% finite_Collect_bounded_ex
thf(fact_19_finite__Collect__bounded__ex,axiom,
! [P3: product_prod_nat_nat > $o,Q: nat > product_prod_nat_nat > $o] :
( ( finite772653738at_nat @ ( collec7649004at_nat @ P3 ) )
=> ( ( finite_finite_nat
@ ( collect_nat
@ ^ [X: nat] :
? [Y: product_prod_nat_nat] :
( ( P3 @ Y )
& ( Q @ X @ Y ) ) ) )
= ( ! [Y: product_prod_nat_nat] :
( ( P3 @ Y )
=> ( finite_finite_nat
@ ( collect_nat
@ ^ [X: nat] : ( Q @ X @ Y ) ) ) ) ) ) ) ).
% finite_Collect_bounded_ex
thf(fact_20_finite__Collect__bounded__ex,axiom,
! [P3: nat > $o,Q: product_prod_nat_nat > nat > $o] :
( ( finite_finite_nat @ ( collect_nat @ P3 ) )
=> ( ( finite772653738at_nat
@ ( collec7649004at_nat
@ ^ [X: product_prod_nat_nat] :
? [Y: nat] :
( ( P3 @ Y )
& ( Q @ X @ Y ) ) ) )
= ( ! [Y: nat] :
( ( P3 @ Y )
=> ( finite772653738at_nat
@ ( collec7649004at_nat
@ ^ [X: product_prod_nat_nat] : ( Q @ X @ Y ) ) ) ) ) ) ) ).
% finite_Collect_bounded_ex
thf(fact_21_finite__Collect__bounded__ex,axiom,
! [P3: nat > $o,Q: nat > nat > $o] :
( ( finite_finite_nat @ ( collect_nat @ P3 ) )
=> ( ( finite_finite_nat
@ ( collect_nat
@ ^ [X: nat] :
? [Y: nat] :
( ( P3 @ Y )
& ( Q @ X @ Y ) ) ) )
= ( ! [Y: nat] :
( ( P3 @ Y )
=> ( finite_finite_nat
@ ( collect_nat
@ ^ [X: nat] : ( Q @ X @ Y ) ) ) ) ) ) ) ).
% finite_Collect_bounded_ex
thf(fact_22_finite__Collect__subsets,axiom,
! [A: set_Pr1986765409at_nat] :
( ( finite772653738at_nat @ A )
=> ( finite1457549322at_nat
@ ( collec1606769740at_nat
@ ^ [B: set_Pr1986765409at_nat] : ( ord_le841296385at_nat @ B @ A ) ) ) ) ).
% finite_Collect_subsets
thf(fact_23_finite__Collect__subsets,axiom,
! [A: set_nat] :
( ( finite_finite_nat @ A )
=> ( finite2012248349et_nat
@ ( collect_set_nat
@ ^ [B: set_nat] : ( ord_less_eq_set_nat @ B @ A ) ) ) ) ).
% finite_Collect_subsets
thf(fact_24_finite__V,axiom,
finite_finite_nat @ ( v_a @ c ) ).
% finite_V
thf(fact_25_Vfin__imp__Efin,axiom,
( ( finite_finite_nat @ ( v_a @ c ) )
=> ( finite772653738at_nat @ ( e_a @ c ) ) ) ).
% Vfin_imp_Efin
thf(fact_26_Range__mono,axiom,
! [R: set_Pr1986765409at_nat,S: set_Pr1986765409at_nat] :
( ( ord_le841296385at_nat @ R @ S )
=> ( ord_less_eq_set_nat @ ( range_nat_nat @ R ) @ ( range_nat_nat @ S ) ) ) ).
% Range_mono
thf(fact_27_Domain__mono,axiom,
! [R: set_Pr1986765409at_nat,S: set_Pr1986765409at_nat] :
( ( ord_le841296385at_nat @ R @ S )
=> ( ord_less_eq_set_nat @ ( domain_nat_nat @ R ) @ ( domain_nat_nat @ S ) ) ) ).
% Domain_mono
thf(fact_28_finite__image__set,axiom,
! [P3: product_prod_nat_nat > $o,F: product_prod_nat_nat > product_prod_nat_nat] :
( ( finite772653738at_nat @ ( collec7649004at_nat @ P3 ) )
=> ( finite772653738at_nat
@ ( collec7649004at_nat
@ ^ [Uu: product_prod_nat_nat] :
? [X: product_prod_nat_nat] :
( ( Uu
= ( F @ X ) )
& ( P3 @ X ) ) ) ) ) ).
% finite_image_set
thf(fact_29_finite__image__set,axiom,
! [P3: product_prod_nat_nat > $o,F: product_prod_nat_nat > nat] :
( ( finite772653738at_nat @ ( collec7649004at_nat @ P3 ) )
=> ( finite_finite_nat
@ ( collect_nat
@ ^ [Uu: nat] :
? [X: product_prod_nat_nat] :
( ( Uu
= ( F @ X ) )
& ( P3 @ X ) ) ) ) ) ).
% finite_image_set
thf(fact_30_finite__image__set,axiom,
! [P3: nat > $o,F: nat > product_prod_nat_nat] :
( ( finite_finite_nat @ ( collect_nat @ P3 ) )
=> ( finite772653738at_nat
@ ( collec7649004at_nat
@ ^ [Uu: product_prod_nat_nat] :
? [X: nat] :
( ( Uu
= ( F @ X ) )
& ( P3 @ X ) ) ) ) ) ).
% finite_image_set
thf(fact_31_finite__image__set,axiom,
! [P3: nat > $o,F: nat > nat] :
( ( finite_finite_nat @ ( collect_nat @ P3 ) )
=> ( finite_finite_nat
@ ( collect_nat
@ ^ [Uu: nat] :
? [X: nat] :
( ( Uu
= ( F @ X ) )
& ( P3 @ X ) ) ) ) ) ).
% finite_image_set
thf(fact_32_finite__image__set2,axiom,
! [P3: product_prod_nat_nat > $o,Q: product_prod_nat_nat > $o,F: product_prod_nat_nat > product_prod_nat_nat > product_prod_nat_nat] :
( ( finite772653738at_nat @ ( collec7649004at_nat @ P3 ) )
=> ( ( finite772653738at_nat @ ( collec7649004at_nat @ Q ) )
=> ( finite772653738at_nat
@ ( collec7649004at_nat
@ ^ [Uu: product_prod_nat_nat] :
? [X: product_prod_nat_nat,Y: product_prod_nat_nat] :
( ( Uu
= ( F @ X @ Y ) )
& ( P3 @ X )
& ( Q @ Y ) ) ) ) ) ) ).
% finite_image_set2
thf(fact_33_finite__image__set2,axiom,
! [P3: product_prod_nat_nat > $o,Q: product_prod_nat_nat > $o,F: product_prod_nat_nat > product_prod_nat_nat > nat] :
( ( finite772653738at_nat @ ( collec7649004at_nat @ P3 ) )
=> ( ( finite772653738at_nat @ ( collec7649004at_nat @ Q ) )
=> ( finite_finite_nat
@ ( collect_nat
@ ^ [Uu: nat] :
? [X: product_prod_nat_nat,Y: product_prod_nat_nat] :
( ( Uu
= ( F @ X @ Y ) )
& ( P3 @ X )
& ( Q @ Y ) ) ) ) ) ) ).
% finite_image_set2
thf(fact_34_finite__image__set2,axiom,
! [P3: product_prod_nat_nat > $o,Q: nat > $o,F: product_prod_nat_nat > nat > product_prod_nat_nat] :
( ( finite772653738at_nat @ ( collec7649004at_nat @ P3 ) )
=> ( ( finite_finite_nat @ ( collect_nat @ Q ) )
=> ( finite772653738at_nat
@ ( collec7649004at_nat
@ ^ [Uu: product_prod_nat_nat] :
? [X: product_prod_nat_nat,Y: nat] :
( ( Uu
= ( F @ X @ Y ) )
& ( P3 @ X )
& ( Q @ Y ) ) ) ) ) ) ).
% finite_image_set2
thf(fact_35_finite__image__set2,axiom,
! [P3: product_prod_nat_nat > $o,Q: nat > $o,F: product_prod_nat_nat > nat > nat] :
( ( finite772653738at_nat @ ( collec7649004at_nat @ P3 ) )
=> ( ( finite_finite_nat @ ( collect_nat @ Q ) )
=> ( finite_finite_nat
@ ( collect_nat
@ ^ [Uu: nat] :
? [X: product_prod_nat_nat,Y: nat] :
( ( Uu
= ( F @ X @ Y ) )
& ( P3 @ X )
& ( Q @ Y ) ) ) ) ) ) ).
% finite_image_set2
thf(fact_36_finite__image__set2,axiom,
! [P3: nat > $o,Q: product_prod_nat_nat > $o,F: nat > product_prod_nat_nat > product_prod_nat_nat] :
( ( finite_finite_nat @ ( collect_nat @ P3 ) )
=> ( ( finite772653738at_nat @ ( collec7649004at_nat @ Q ) )
=> ( finite772653738at_nat
@ ( collec7649004at_nat
@ ^ [Uu: product_prod_nat_nat] :
? [X: nat,Y: product_prod_nat_nat] :
( ( Uu
= ( F @ X @ Y ) )
& ( P3 @ X )
& ( Q @ Y ) ) ) ) ) ) ).
% finite_image_set2
thf(fact_37_finite__image__set2,axiom,
! [P3: nat > $o,Q: product_prod_nat_nat > $o,F: nat > product_prod_nat_nat > nat] :
( ( finite_finite_nat @ ( collect_nat @ P3 ) )
=> ( ( finite772653738at_nat @ ( collec7649004at_nat @ Q ) )
=> ( finite_finite_nat
@ ( collect_nat
@ ^ [Uu: nat] :
? [X: nat,Y: product_prod_nat_nat] :
( ( Uu
= ( F @ X @ Y ) )
& ( P3 @ X )
& ( Q @ Y ) ) ) ) ) ) ).
% finite_image_set2
thf(fact_38_finite__image__set2,axiom,
! [P3: nat > $o,Q: nat > $o,F: nat > nat > product_prod_nat_nat] :
( ( finite_finite_nat @ ( collect_nat @ P3 ) )
=> ( ( finite_finite_nat @ ( collect_nat @ Q ) )
=> ( finite772653738at_nat
@ ( collec7649004at_nat
@ ^ [Uu: product_prod_nat_nat] :
? [X: nat,Y: nat] :
( ( Uu
= ( F @ X @ Y ) )
& ( P3 @ X )
& ( Q @ Y ) ) ) ) ) ) ).
% finite_image_set2
thf(fact_39_finite__image__set2,axiom,
! [P3: nat > $o,Q: nat > $o,F: nat > nat > nat] :
( ( finite_finite_nat @ ( collect_nat @ P3 ) )
=> ( ( finite_finite_nat @ ( collect_nat @ Q ) )
=> ( finite_finite_nat
@ ( collect_nat
@ ^ [Uu: nat] :
? [X: nat,Y: nat] :
( ( Uu
= ( F @ X @ Y ) )
& ( P3 @ X )
& ( Q @ Y ) ) ) ) ) ) ).
% finite_image_set2
thf(fact_40_not__finite__existsD,axiom,
! [P3: product_prod_nat_nat > $o] :
( ~ ( finite772653738at_nat @ ( collec7649004at_nat @ P3 ) )
=> ? [X_1: product_prod_nat_nat] : ( P3 @ X_1 ) ) ).
% not_finite_existsD
thf(fact_41_not__finite__existsD,axiom,
! [P3: nat > $o] :
( ~ ( finite_finite_nat @ ( collect_nat @ P3 ) )
=> ? [X_1: nat] : ( P3 @ X_1 ) ) ).
% not_finite_existsD
thf(fact_42_pigeonhole__infinite__rel,axiom,
! [A: set_Pr1986765409at_nat,B2: set_Pr1986765409at_nat,R2: product_prod_nat_nat > product_prod_nat_nat > $o] :
( ~ ( finite772653738at_nat @ A )
=> ( ( finite772653738at_nat @ B2 )
=> ( ! [X2: product_prod_nat_nat] :
( ( member701585322at_nat @ X2 @ A )
=> ? [Xa: product_prod_nat_nat] :
( ( member701585322at_nat @ Xa @ B2 )
& ( R2 @ X2 @ Xa ) ) )
=> ? [X2: product_prod_nat_nat] :
( ( member701585322at_nat @ X2 @ B2 )
& ~ ( finite772653738at_nat
@ ( collec7649004at_nat
@ ^ [A2: product_prod_nat_nat] :
( ( member701585322at_nat @ A2 @ A )
& ( R2 @ A2 @ X2 ) ) ) ) ) ) ) ) ).
% pigeonhole_infinite_rel
thf(fact_43_pigeonhole__infinite__rel,axiom,
! [A: set_Pr1986765409at_nat,B2: set_nat,R2: product_prod_nat_nat > nat > $o] :
( ~ ( finite772653738at_nat @ A )
=> ( ( finite_finite_nat @ B2 )
=> ( ! [X2: product_prod_nat_nat] :
( ( member701585322at_nat @ X2 @ A )
=> ? [Xa: nat] :
( ( member_nat @ Xa @ B2 )
& ( R2 @ X2 @ Xa ) ) )
=> ? [X2: nat] :
( ( member_nat @ X2 @ B2 )
& ~ ( finite772653738at_nat
@ ( collec7649004at_nat
@ ^ [A2: product_prod_nat_nat] :
( ( member701585322at_nat @ A2 @ A )
& ( R2 @ A2 @ X2 ) ) ) ) ) ) ) ) ).
% pigeonhole_infinite_rel
thf(fact_44_pigeonhole__infinite__rel,axiom,
! [A: set_nat,B2: set_Pr1986765409at_nat,R2: nat > product_prod_nat_nat > $o] :
( ~ ( finite_finite_nat @ A )
=> ( ( finite772653738at_nat @ B2 )
=> ( ! [X2: nat] :
( ( member_nat @ X2 @ A )
=> ? [Xa: product_prod_nat_nat] :
( ( member701585322at_nat @ Xa @ B2 )
& ( R2 @ X2 @ Xa ) ) )
=> ? [X2: product_prod_nat_nat] :
( ( member701585322at_nat @ X2 @ B2 )
& ~ ( finite_finite_nat
@ ( collect_nat
@ ^ [A2: nat] :
( ( member_nat @ A2 @ A )
& ( R2 @ A2 @ X2 ) ) ) ) ) ) ) ) ).
% pigeonhole_infinite_rel
thf(fact_45_pigeonhole__infinite__rel,axiom,
! [A: set_nat,B2: set_nat,R2: nat > nat > $o] :
( ~ ( finite_finite_nat @ A )
=> ( ( finite_finite_nat @ B2 )
=> ( ! [X2: nat] :
( ( member_nat @ X2 @ A )
=> ? [Xa: nat] :
( ( member_nat @ Xa @ B2 )
& ( R2 @ X2 @ Xa ) ) )
=> ? [X2: nat] :
( ( member_nat @ X2 @ B2 )
& ~ ( finite_finite_nat
@ ( collect_nat
@ ^ [A2: nat] :
( ( member_nat @ A2 @ A )
& ( R2 @ A2 @ X2 ) ) ) ) ) ) ) ) ).
% pigeonhole_infinite_rel
thf(fact_46_ek__analysis__def,axiom,
edmond1517640972ysis_a = finite_Graph_a ).
% ek_analysis_def
thf(fact_47_ek__analysis_Ointro,axiom,
! [C: product_prod_nat_nat > a] :
( ( finite_Graph_a @ C )
=> ( edmond1517640972ysis_a @ C ) ) ).
% ek_analysis.intro
thf(fact_48_ek__analysis_Oaxioms,axiom,
! [C: product_prod_nat_nat > a] :
( ( edmond1517640972ysis_a @ C )
=> ( finite_Graph_a @ C ) ) ).
% ek_analysis.axioms
thf(fact_49_ek__analysis__defs_OspEdges__def,axiom,
( edmond475474835dges_a
= ( ^ [C2: product_prod_nat_nat > a,S2: nat,T: nat] :
( collec7649004at_nat
@ ^ [E: product_prod_nat_nat] :
? [P: list_P559422087at_nat] :
( ( member701585322at_nat @ E @ ( set_Pr2131844118at_nat @ P ) )
& ( isShortestPath_a @ C2 @ S2 @ P @ T ) ) ) ) ) ).
% ek_analysis_defs.spEdges_def
thf(fact_50_finite__has__minimal2,axiom,
! [A: set_se1612935105at_nat,A3: set_Pr1986765409at_nat] :
( ( finite1457549322at_nat @ A )
=> ( ( member298845450at_nat @ A3 @ A )
=> ? [X2: set_Pr1986765409at_nat] :
( ( member298845450at_nat @ X2 @ A )
& ( ord_le841296385at_nat @ X2 @ A3 )
& ! [Xa: set_Pr1986765409at_nat] :
( ( member298845450at_nat @ Xa @ A )
=> ( ( ord_le841296385at_nat @ Xa @ X2 )
=> ( X2 = Xa ) ) ) ) ) ) ).
% finite_has_minimal2
thf(fact_51_finite__has__minimal2,axiom,
! [A: set_set_nat,A3: set_nat] :
( ( finite2012248349et_nat @ A )
=> ( ( member_set_nat @ A3 @ A )
=> ? [X2: set_nat] :
( ( member_set_nat @ X2 @ A )
& ( ord_less_eq_set_nat @ X2 @ A3 )
& ! [Xa: set_nat] :
( ( member_set_nat @ Xa @ A )
=> ( ( ord_less_eq_set_nat @ Xa @ X2 )
=> ( X2 = Xa ) ) ) ) ) ) ).
% finite_has_minimal2
thf(fact_52_finite__has__minimal2,axiom,
! [A: set_nat,A3: nat] :
( ( finite_finite_nat @ A )
=> ( ( member_nat @ A3 @ A )
=> ? [X2: nat] :
( ( member_nat @ X2 @ A )
& ( ord_less_eq_nat @ X2 @ A3 )
& ! [Xa: nat] :
( ( member_nat @ Xa @ A )
=> ( ( ord_less_eq_nat @ Xa @ X2 )
=> ( X2 = Xa ) ) ) ) ) ) ).
% finite_has_minimal2
thf(fact_53_finite__has__maximal2,axiom,
! [A: set_se1612935105at_nat,A3: set_Pr1986765409at_nat] :
( ( finite1457549322at_nat @ A )
=> ( ( member298845450at_nat @ A3 @ A )
=> ? [X2: set_Pr1986765409at_nat] :
( ( member298845450at_nat @ X2 @ A )
& ( ord_le841296385at_nat @ A3 @ X2 )
& ! [Xa: set_Pr1986765409at_nat] :
( ( member298845450at_nat @ Xa @ A )
=> ( ( ord_le841296385at_nat @ X2 @ Xa )
=> ( X2 = Xa ) ) ) ) ) ) ).
% finite_has_maximal2
thf(fact_54_finite__has__maximal2,axiom,
! [A: set_set_nat,A3: set_nat] :
( ( finite2012248349et_nat @ A )
=> ( ( member_set_nat @ A3 @ A )
=> ? [X2: set_nat] :
( ( member_set_nat @ X2 @ A )
& ( ord_less_eq_set_nat @ A3 @ X2 )
& ! [Xa: set_nat] :
( ( member_set_nat @ Xa @ A )
=> ( ( ord_less_eq_set_nat @ X2 @ Xa )
=> ( X2 = Xa ) ) ) ) ) ) ).
% finite_has_maximal2
thf(fact_55_finite__has__maximal2,axiom,
! [A: set_nat,A3: nat] :
( ( finite_finite_nat @ A )
=> ( ( member_nat @ A3 @ A )
=> ? [X2: nat] :
( ( member_nat @ X2 @ A )
& ( ord_less_eq_nat @ A3 @ X2 )
& ! [Xa: nat] :
( ( member_nat @ Xa @ A )
=> ( ( ord_less_eq_nat @ X2 @ Xa )
=> ( X2 = Xa ) ) ) ) ) ) ).
% finite_has_maximal2
thf(fact_56_rev__finite__subset,axiom,
! [B2: set_Pr1986765409at_nat,A: set_Pr1986765409at_nat] :
( ( finite772653738at_nat @ B2 )
=> ( ( ord_le841296385at_nat @ A @ B2 )
=> ( finite772653738at_nat @ A ) ) ) ).
% rev_finite_subset
thf(fact_57_rev__finite__subset,axiom,
! [B2: set_nat,A: set_nat] :
( ( finite_finite_nat @ B2 )
=> ( ( ord_less_eq_set_nat @ A @ B2 )
=> ( finite_finite_nat @ A ) ) ) ).
% rev_finite_subset
thf(fact_58_infinite__super,axiom,
! [S3: set_Pr1986765409at_nat,T2: set_Pr1986765409at_nat] :
( ( ord_le841296385at_nat @ S3 @ T2 )
=> ( ~ ( finite772653738at_nat @ S3 )
=> ~ ( finite772653738at_nat @ T2 ) ) ) ).
% infinite_super
thf(fact_59_infinite__super,axiom,
! [S3: set_nat,T2: set_nat] :
( ( ord_less_eq_set_nat @ S3 @ T2 )
=> ( ~ ( finite_finite_nat @ S3 )
=> ~ ( finite_finite_nat @ T2 ) ) ) ).
% infinite_super
thf(fact_60_finite__subset,axiom,
! [A: set_Pr1986765409at_nat,B2: set_Pr1986765409at_nat] :
( ( ord_le841296385at_nat @ A @ B2 )
=> ( ( finite772653738at_nat @ B2 )
=> ( finite772653738at_nat @ A ) ) ) ).
% finite_subset
thf(fact_61_finite__subset,axiom,
! [A: set_nat,B2: set_nat] :
( ( ord_less_eq_set_nat @ A @ B2 )
=> ( ( finite_finite_nat @ B2 )
=> ( finite_finite_nat @ A ) ) ) ).
% finite_subset
thf(fact_62_ek__analysis_OspEdges__ss__E,axiom,
! [C: product_prod_nat_nat > a,S: nat,T3: nat] :
( ( edmond1517640972ysis_a @ C )
=> ( ord_le841296385at_nat @ ( edmond475474835dges_a @ C @ S @ T3 ) @ ( e_a @ C ) ) ) ).
% ek_analysis.spEdges_ss_E
thf(fact_63_incoming_H__edges,axiom,
! [U2: set_nat] : ( ord_le841296385at_nat @ ( incoming_a2 @ c @ U2 ) @ ( e_a @ c ) ) ).
% incoming'_edges
thf(fact_64_outgoing_H__edges,axiom,
! [U2: set_nat] : ( ord_le841296385at_nat @ ( outgoing_a2 @ c @ U2 ) @ ( e_a @ c ) ) ).
% outgoing'_edges
thf(fact_65_reachableNodes__def,axiom,
! [U: nat] :
( ( reachableNodes_a @ c @ U )
= ( collect_nat @ ( connected_a @ c @ U ) ) ) ).
% reachableNodes_def
thf(fact_66_List_Ofinite__set,axiom,
! [Xs: list_P559422087at_nat] : ( finite772653738at_nat @ ( set_Pr2131844118at_nat @ Xs ) ) ).
% List.finite_set
thf(fact_67_List_Ofinite__set,axiom,
! [Xs: list_nat] : ( finite_finite_nat @ ( set_nat2 @ Xs ) ) ).
% List.finite_set
thf(fact_68_Graph_OEfin__imp__Vfin,axiom,
! [C: product_prod_nat_nat > a] :
( ( finite772653738at_nat @ ( e_a @ C ) )
=> ( finite_finite_nat @ ( v_a @ C ) ) ) ).
% Graph.Efin_imp_Vfin
thf(fact_69_Graph_OVfin__imp__Efin,axiom,
! [C: product_prod_nat_nat > a] :
( ( finite_finite_nat @ ( v_a @ C ) )
=> ( finite772653738at_nat @ ( e_a @ C ) ) ) ).
% Graph.Vfin_imp_Efin
thf(fact_70_mem__Collect__eq,axiom,
! [A3: product_prod_nat_nat,P3: product_prod_nat_nat > $o] :
( ( member701585322at_nat @ A3 @ ( collec7649004at_nat @ P3 ) )
= ( P3 @ A3 ) ) ).
% mem_Collect_eq
thf(fact_71_mem__Collect__eq,axiom,
! [A3: nat,P3: nat > $o] :
( ( member_nat @ A3 @ ( collect_nat @ P3 ) )
= ( P3 @ A3 ) ) ).
% mem_Collect_eq
thf(fact_72_Collect__mem__eq,axiom,
! [A: set_Pr1986765409at_nat] :
( ( collec7649004at_nat
@ ^ [X: product_prod_nat_nat] : ( member701585322at_nat @ X @ A ) )
= A ) ).
% Collect_mem_eq
thf(fact_73_Collect__mem__eq,axiom,
! [A: set_nat] :
( ( collect_nat
@ ^ [X: nat] : ( member_nat @ X @ A ) )
= A ) ).
% Collect_mem_eq
thf(fact_74_Collect__cong,axiom,
! [P3: product_prod_nat_nat > $o,Q: product_prod_nat_nat > $o] :
( ! [X2: product_prod_nat_nat] :
( ( P3 @ X2 )
= ( Q @ X2 ) )
=> ( ( collec7649004at_nat @ P3 )
= ( collec7649004at_nat @ Q ) ) ) ).
% Collect_cong
thf(fact_75_Collect__cong,axiom,
! [P3: nat > $o,Q: nat > $o] :
( ! [X2: nat] :
( ( P3 @ X2 )
= ( Q @ X2 ) )
=> ( ( collect_nat @ P3 )
= ( collect_nat @ Q ) ) ) ).
% Collect_cong
thf(fact_76_isShortestPath__level__edge_I1_J,axiom,
! [S: nat,P4: list_P559422087at_nat,T3: nat,U: nat,V: nat] :
( ( isShortestPath_a @ c @ S @ P4 @ T3 )
=> ( ( member701585322at_nat @ ( product_Pair_nat_nat @ U @ V ) @ ( set_Pr2131844118at_nat @ P4 ) )
=> ( connected_a @ c @ S @ U ) ) ) ).
% isShortestPath_level_edge(1)
thf(fact_77_isShortestPath__level__edge_I2_J,axiom,
! [S: nat,P4: list_P559422087at_nat,T3: nat,U: nat,V: nat] :
( ( isShortestPath_a @ c @ S @ P4 @ T3 )
=> ( ( member701585322at_nat @ ( product_Pair_nat_nat @ U @ V ) @ ( set_Pr2131844118at_nat @ P4 ) )
=> ( connected_a @ c @ U @ V ) ) ) ).
% isShortestPath_level_edge(2)
thf(fact_78_isShortestPath__level__edge_I3_J,axiom,
! [S: nat,P4: list_P559422087at_nat,T3: nat,U: nat,V: nat] :
( ( isShortestPath_a @ c @ S @ P4 @ T3 )
=> ( ( member701585322at_nat @ ( product_Pair_nat_nat @ U @ V ) @ ( set_Pr2131844118at_nat @ P4 ) )
=> ( connected_a @ c @ V @ T3 ) ) ) ).
% isShortestPath_level_edge(3)
thf(fact_79_incoming__edges,axiom,
! [U: nat] : ( ord_le841296385at_nat @ ( incoming_a @ c @ U ) @ ( e_a @ c ) ) ).
% incoming_edges
thf(fact_80_reachable__ss__V,axiom,
! [S: nat] :
( ( member_nat @ S @ ( v_a @ c ) )
=> ( ord_less_eq_set_nat @ ( reachableNodes_a @ c @ S ) @ ( v_a @ c ) ) ) ).
% reachable_ss_V
thf(fact_81_connected__append__edge,axiom,
! [U: nat,V: nat,W: nat] :
( ( connected_a @ c @ U @ V )
=> ( ( member701585322at_nat @ ( product_Pair_nat_nat @ V @ W ) @ ( e_a @ c ) )
=> ( connected_a @ c @ U @ W ) ) ) ).
% connected_append_edge
thf(fact_82_reachableNodes__append__edge,axiom,
! [U: nat,S: nat,V: nat] :
( ( member_nat @ U @ ( reachableNodes_a @ c @ S ) )
=> ( ( member701585322at_nat @ ( product_Pair_nat_nat @ U @ V ) @ ( e_a @ c ) )
=> ( member_nat @ V @ ( reachableNodes_a @ c @ S ) ) ) ) ).
% reachableNodes_append_edge
thf(fact_83_V__def,axiom,
( ( v_a @ c )
= ( collect_nat
@ ^ [U3: nat] :
? [V2: nat] :
( ( member701585322at_nat @ ( product_Pair_nat_nat @ U3 @ V2 ) @ ( e_a @ c ) )
| ( member701585322at_nat @ ( product_Pair_nat_nat @ V2 @ U3 ) @ ( e_a @ c ) ) ) ) ) ).
% V_def
thf(fact_84_incoming__def,axiom,
! [V: nat] :
( ( incoming_a @ c @ V )
= ( collec7649004at_nat
@ ^ [Uu: product_prod_nat_nat] :
? [U3: nat] :
( ( Uu
= ( product_Pair_nat_nat @ U3 @ V ) )
& ( member701585322at_nat @ ( product_Pair_nat_nat @ U3 @ V ) @ ( e_a @ c ) ) ) ) ) ).
% incoming_def
thf(fact_85_outgoing_H__def,axiom,
! [K: set_nat] :
( ( outgoing_a2 @ c @ K )
= ( collec7649004at_nat
@ ^ [Uu: product_prod_nat_nat] :
? [U3: nat,V2: nat] :
( ( Uu
= ( product_Pair_nat_nat @ V2 @ U3 ) )
& ~ ( member_nat @ U3 @ K )
& ( member_nat @ V2 @ K )
& ( member701585322at_nat @ ( product_Pair_nat_nat @ V2 @ U3 ) @ ( e_a @ c ) ) ) ) ) ).
% outgoing'_def
thf(fact_86_incoming_H__def,axiom,
! [K: set_nat] :
( ( incoming_a2 @ c @ K )
= ( collec7649004at_nat
@ ^ [Uu: product_prod_nat_nat] :
? [U3: nat,V2: nat] :
( ( Uu
= ( product_Pair_nat_nat @ U3 @ V2 ) )
& ~ ( member_nat @ U3 @ K )
& ( member_nat @ V2 @ K )
& ( member701585322at_nat @ ( product_Pair_nat_nat @ U3 @ V2 ) @ ( e_a @ c ) ) ) ) ) ).
% incoming'_def
thf(fact_87_finite__Collect__le__nat,axiom,
! [K: nat] :
( finite_finite_nat
@ ( collect_nat
@ ^ [N: nat] : ( ord_less_eq_nat @ N @ K ) ) ) ).
% finite_Collect_le_nat
thf(fact_88_finite__incoming,axiom,
! [U: nat] :
( ( finite_finite_nat @ ( v_a @ c ) )
=> ( finite772653738at_nat @ ( incoming_a @ c @ U ) ) ) ).
% finite_incoming
thf(fact_89_finite__outgoing_H,axiom,
! [U2: set_nat] :
( ( finite_finite_nat @ ( v_a @ c ) )
=> ( finite772653738at_nat @ ( outgoing_a2 @ c @ U2 ) ) ) ).
% finite_outgoing'
thf(fact_90_finite__incoming_H,axiom,
! [U2: set_nat] :
( ( finite_finite_nat @ ( v_a @ c ) )
=> ( finite772653738at_nat @ ( incoming_a2 @ c @ U2 ) ) ) ).
% finite_incoming'
thf(fact_91_adjacent__nodes__ss__V,axiom,
! [U: nat] : ( ord_less_eq_set_nat @ ( adjacent_nodes_a @ c @ U ) @ ( v_a @ c ) ) ).
% adjacent_nodes_ss_V
thf(fact_92_Graph_Oincoming__def,axiom,
( incoming_a
= ( ^ [C2: product_prod_nat_nat > a,V2: nat] :
( collec7649004at_nat
@ ^ [Uu: product_prod_nat_nat] :
? [U3: nat] :
( ( Uu
= ( product_Pair_nat_nat @ U3 @ V2 ) )
& ( member701585322at_nat @ ( product_Pair_nat_nat @ U3 @ V2 ) @ ( e_a @ C2 ) ) ) ) ) ) ).
% Graph.incoming_def
thf(fact_93_Graph_Oincoming_H__def,axiom,
( incoming_a2
= ( ^ [C2: product_prod_nat_nat > a,K2: set_nat] :
( collec7649004at_nat
@ ^ [Uu: product_prod_nat_nat] :
? [U3: nat,V2: nat] :
( ( Uu
= ( product_Pair_nat_nat @ U3 @ V2 ) )
& ~ ( member_nat @ U3 @ K2 )
& ( member_nat @ V2 @ K2 )
& ( member701585322at_nat @ ( product_Pair_nat_nat @ U3 @ V2 ) @ ( e_a @ C2 ) ) ) ) ) ) ).
% Graph.incoming'_def
thf(fact_94_Graph_Ooutgoing_H__def,axiom,
( outgoing_a2
= ( ^ [C2: product_prod_nat_nat > a,K2: set_nat] :
( collec7649004at_nat
@ ^ [Uu: product_prod_nat_nat] :
? [U3: nat,V2: nat] :
( ( Uu
= ( product_Pair_nat_nat @ V2 @ U3 ) )
& ~ ( member_nat @ U3 @ K2 )
& ( member_nat @ V2 @ K2 )
& ( member701585322at_nat @ ( product_Pair_nat_nat @ V2 @ U3 ) @ ( e_a @ C2 ) ) ) ) ) ) ).
% Graph.outgoing'_def
thf(fact_95_Graph_Oreachable__ss__V,axiom,
! [S: nat,C: product_prod_nat_nat > a] :
( ( member_nat @ S @ ( v_a @ C ) )
=> ( ord_less_eq_set_nat @ ( reachableNodes_a @ C @ S ) @ ( v_a @ C ) ) ) ).
% Graph.reachable_ss_V
thf(fact_96_Graph_OreachableNodes__append__edge,axiom,
! [U: nat,C: product_prod_nat_nat > a,S: nat,V: nat] :
( ( member_nat @ U @ ( reachableNodes_a @ C @ S ) )
=> ( ( member701585322at_nat @ ( product_Pair_nat_nat @ U @ V ) @ ( e_a @ C ) )
=> ( member_nat @ V @ ( reachableNodes_a @ C @ S ) ) ) ) ).
% Graph.reachableNodes_append_edge
thf(fact_97_Graph_OreachableNodes_Ocong,axiom,
reachableNodes_a = reachableNodes_a ).
% Graph.reachableNodes.cong
thf(fact_98_Graph_Ooutgoing_H_Ocong,axiom,
outgoing_a2 = outgoing_a2 ).
% Graph.outgoing'.cong
thf(fact_99_Graph_Oincoming_H_Ocong,axiom,
incoming_a2 = incoming_a2 ).
% Graph.incoming'.cong
thf(fact_100_Graph_Oincoming_Ocong,axiom,
incoming_a = incoming_a ).
% Graph.incoming.cong
thf(fact_101_pred__equals__eq2,axiom,
! [R2: set_Pr1986765409at_nat,S3: set_Pr1986765409at_nat] :
( ( ( ^ [X: nat,Y: nat] : ( member701585322at_nat @ ( product_Pair_nat_nat @ X @ Y ) @ R2 ) )
= ( ^ [X: nat,Y: nat] : ( member701585322at_nat @ ( product_Pair_nat_nat @ X @ Y ) @ S3 ) ) )
= ( R2 = S3 ) ) ).
% pred_equals_eq2
thf(fact_102_pred__subset__eq2,axiom,
! [R2: set_Pr1986765409at_nat,S3: set_Pr1986765409at_nat] :
( ( ord_le1578155910_nat_o
@ ^ [X: nat,Y: nat] : ( member701585322at_nat @ ( product_Pair_nat_nat @ X @ Y ) @ R2 )
@ ^ [X: nat,Y: nat] : ( member701585322at_nat @ ( product_Pair_nat_nat @ X @ Y ) @ S3 ) )
= ( ord_le841296385at_nat @ R2 @ S3 ) ) ).
% pred_subset_eq2
thf(fact_103_Graph_OV__def,axiom,
( v_a
= ( ^ [C2: product_prod_nat_nat > a] :
( collect_nat
@ ^ [U3: nat] :
? [V2: nat] :
( ( member701585322at_nat @ ( product_Pair_nat_nat @ U3 @ V2 ) @ ( e_a @ C2 ) )
| ( member701585322at_nat @ ( product_Pair_nat_nat @ V2 @ U3 ) @ ( e_a @ C2 ) ) ) ) ) ) ).
% Graph.V_def
thf(fact_104_subrelI,axiom,
! [R: set_Pr1986765409at_nat,S: set_Pr1986765409at_nat] :
( ! [X2: nat,Y2: nat] :
( ( member701585322at_nat @ ( product_Pair_nat_nat @ X2 @ Y2 ) @ R )
=> ( member701585322at_nat @ ( product_Pair_nat_nat @ X2 @ Y2 ) @ S ) )
=> ( ord_le841296385at_nat @ R @ S ) ) ).
% subrelI
thf(fact_105_Domain_Oinducts,axiom,
! [X3: nat,R: set_Pr1986765409at_nat,P3: nat > $o] :
( ( member_nat @ X3 @ ( domain_nat_nat @ R ) )
=> ( ! [A4: nat,B3: nat] :
( ( member701585322at_nat @ ( product_Pair_nat_nat @ A4 @ B3 ) @ R )
=> ( P3 @ A4 ) )
=> ( P3 @ X3 ) ) ) ).
% Domain.inducts
thf(fact_106_Domain_ODomainI,axiom,
! [A3: nat,B4: nat,R: set_Pr1986765409at_nat] :
( ( member701585322at_nat @ ( product_Pair_nat_nat @ A3 @ B4 ) @ R )
=> ( member_nat @ A3 @ ( domain_nat_nat @ R ) ) ) ).
% Domain.DomainI
thf(fact_107_Domain_Osimps,axiom,
! [A3: nat,R: set_Pr1986765409at_nat] :
( ( member_nat @ A3 @ ( domain_nat_nat @ R ) )
= ( ? [A2: nat,B5: nat] :
( ( A3 = A2 )
& ( member701585322at_nat @ ( product_Pair_nat_nat @ A2 @ B5 ) @ R ) ) ) ) ).
% Domain.simps
thf(fact_108_Domain_Ocases,axiom,
! [A3: nat,R: set_Pr1986765409at_nat] :
( ( member_nat @ A3 @ ( domain_nat_nat @ R ) )
=> ~ ! [B3: nat] :
~ ( member701585322at_nat @ ( product_Pair_nat_nat @ A3 @ B3 ) @ R ) ) ).
% Domain.cases
thf(fact_109_Domain__iff,axiom,
! [A3: nat,R: set_Pr1986765409at_nat] :
( ( member_nat @ A3 @ ( domain_nat_nat @ R ) )
= ( ? [Y: nat] : ( member701585322at_nat @ ( product_Pair_nat_nat @ A3 @ Y ) @ R ) ) ) ).
% Domain_iff
thf(fact_110_DomainE,axiom,
! [A3: nat,R: set_Pr1986765409at_nat] :
( ( member_nat @ A3 @ ( domain_nat_nat @ R ) )
=> ~ ! [B3: nat] :
~ ( member701585322at_nat @ ( product_Pair_nat_nat @ A3 @ B3 ) @ R ) ) ).
% DomainE
thf(fact_111_Range_Oinducts,axiom,
! [X3: nat,R: set_Pr1986765409at_nat,P3: nat > $o] :
( ( member_nat @ X3 @ ( range_nat_nat @ R ) )
=> ( ! [A4: nat,B3: nat] :
( ( member701585322at_nat @ ( product_Pair_nat_nat @ A4 @ B3 ) @ R )
=> ( P3 @ B3 ) )
=> ( P3 @ X3 ) ) ) ).
% Range.inducts
thf(fact_112_Range_Ointros,axiom,
! [A3: nat,B4: nat,R: set_Pr1986765409at_nat] :
( ( member701585322at_nat @ ( product_Pair_nat_nat @ A3 @ B4 ) @ R )
=> ( member_nat @ B4 @ ( range_nat_nat @ R ) ) ) ).
% Range.intros
thf(fact_113_Range_Osimps,axiom,
! [A3: nat,R: set_Pr1986765409at_nat] :
( ( member_nat @ A3 @ ( range_nat_nat @ R ) )
= ( ? [A2: nat,B5: nat] :
( ( A3 = B5 )
& ( member701585322at_nat @ ( product_Pair_nat_nat @ A2 @ B5 ) @ R ) ) ) ) ).
% Range.simps
thf(fact_114_Range_Ocases,axiom,
! [A3: nat,R: set_Pr1986765409at_nat] :
( ( member_nat @ A3 @ ( range_nat_nat @ R ) )
=> ~ ! [A4: nat] :
~ ( member701585322at_nat @ ( product_Pair_nat_nat @ A4 @ A3 ) @ R ) ) ).
% Range.cases
thf(fact_115_Range__iff,axiom,
! [A3: nat,R: set_Pr1986765409at_nat] :
( ( member_nat @ A3 @ ( range_nat_nat @ R ) )
= ( ? [Y: nat] : ( member701585322at_nat @ ( product_Pair_nat_nat @ Y @ A3 ) @ R ) ) ) ).
% Range_iff
thf(fact_116_RangeE,axiom,
! [B4: nat,R: set_Pr1986765409at_nat] :
( ( member_nat @ B4 @ ( range_nat_nat @ R ) )
=> ~ ! [A4: nat] :
~ ( member701585322at_nat @ ( product_Pair_nat_nat @ A4 @ B4 ) @ R ) ) ).
% RangeE
thf(fact_117_Graph_OreachableNodes__def,axiom,
( reachableNodes_a
= ( ^ [C2: product_prod_nat_nat > a,U3: nat] : ( collect_nat @ ( connected_a @ C2 @ U3 ) ) ) ) ).
% Graph.reachableNodes_def
thf(fact_118_pred__subset__eq,axiom,
! [R2: set_Pr1986765409at_nat,S3: set_Pr1986765409at_nat] :
( ( ord_le1039616028_nat_o
@ ^ [X: product_prod_nat_nat] : ( member701585322at_nat @ X @ R2 )
@ ^ [X: product_prod_nat_nat] : ( member701585322at_nat @ X @ S3 ) )
= ( ord_le841296385at_nat @ R2 @ S3 ) ) ).
% pred_subset_eq
thf(fact_119_pred__subset__eq,axiom,
! [R2: set_nat,S3: set_nat] :
( ( ord_less_eq_nat_o
@ ^ [X: nat] : ( member_nat @ X @ R2 )
@ ^ [X: nat] : ( member_nat @ X @ S3 ) )
= ( ord_less_eq_set_nat @ R2 @ S3 ) ) ).
% pred_subset_eq
thf(fact_120_Graph_Oconnected__append__edge,axiom,
! [C: product_prod_nat_nat > a,U: nat,V: nat,W: nat] :
( ( connected_a @ C @ U @ V )
=> ( ( member701585322at_nat @ ( product_Pair_nat_nat @ V @ W ) @ ( e_a @ C ) )
=> ( connected_a @ C @ U @ W ) ) ) ).
% Graph.connected_append_edge
thf(fact_121_Graph_Oincoming__edges,axiom,
! [C: product_prod_nat_nat > a,U: nat] : ( ord_le841296385at_nat @ ( incoming_a @ C @ U ) @ ( e_a @ C ) ) ).
% Graph.incoming_edges
thf(fact_122_Graph_OisShortestPath__level__edge_I1_J,axiom,
! [C: product_prod_nat_nat > a,S: nat,P4: list_P559422087at_nat,T3: nat,U: nat,V: nat] :
( ( isShortestPath_a @ C @ S @ P4 @ T3 )
=> ( ( member701585322at_nat @ ( product_Pair_nat_nat @ U @ V ) @ ( set_Pr2131844118at_nat @ P4 ) )
=> ( connected_a @ C @ S @ U ) ) ) ).
% Graph.isShortestPath_level_edge(1)
thf(fact_123_Graph_OisShortestPath__level__edge_I2_J,axiom,
! [C: product_prod_nat_nat > a,S: nat,P4: list_P559422087at_nat,T3: nat,U: nat,V: nat] :
( ( isShortestPath_a @ C @ S @ P4 @ T3 )
=> ( ( member701585322at_nat @ ( product_Pair_nat_nat @ U @ V ) @ ( set_Pr2131844118at_nat @ P4 ) )
=> ( connected_a @ C @ U @ V ) ) ) ).
% Graph.isShortestPath_level_edge(2)
thf(fact_124_Graph_OisShortestPath__level__edge_I3_J,axiom,
! [C: product_prod_nat_nat > a,S: nat,P4: list_P559422087at_nat,T3: nat,U: nat,V: nat] :
( ( isShortestPath_a @ C @ S @ P4 @ T3 )
=> ( ( member701585322at_nat @ ( product_Pair_nat_nat @ U @ V ) @ ( set_Pr2131844118at_nat @ P4 ) )
=> ( connected_a @ C @ V @ T3 ) ) ) ).
% Graph.isShortestPath_level_edge(3)
thf(fact_125_Graph_Ooutgoing_H__edges,axiom,
! [C: product_prod_nat_nat > a,U2: set_nat] : ( ord_le841296385at_nat @ ( outgoing_a2 @ C @ U2 ) @ ( e_a @ C ) ) ).
% Graph.outgoing'_edges
thf(fact_126_Graph_Oincoming_H__edges,axiom,
! [C: product_prod_nat_nat > a,U2: set_nat] : ( ord_le841296385at_nat @ ( incoming_a2 @ C @ U2 ) @ ( e_a @ C ) ) ).
% Graph.incoming'_edges
thf(fact_127_Domain__unfold,axiom,
( domain_nat_nat
= ( ^ [R3: set_Pr1986765409at_nat] :
( collect_nat
@ ^ [X: nat] :
? [Y: nat] : ( member701585322at_nat @ ( product_Pair_nat_nat @ X @ Y ) @ R3 ) ) ) ) ).
% Domain_unfold
thf(fact_128_Graph_Ofinite__incoming,axiom,
! [C: product_prod_nat_nat > a,U: nat] :
( ( finite_finite_nat @ ( v_a @ C ) )
=> ( finite772653738at_nat @ ( incoming_a @ C @ U ) ) ) ).
% Graph.finite_incoming
thf(fact_129_Graph_OE_Ocong,axiom,
e_a = e_a ).
% Graph.E.cong
thf(fact_130_Graph_OV_Ocong,axiom,
v_a = v_a ).
% Graph.V.cong
thf(fact_131_Graph_Oconnected__refl,axiom,
! [C: product_prod_nat_nat > a,V: nat] : ( connected_a @ C @ V @ V ) ).
% Graph.connected_refl
thf(fact_132_Graph_Oconnected_Ocong,axiom,
connected_a = connected_a ).
% Graph.connected.cong
thf(fact_133_Graph_Ofinite__outgoing_H,axiom,
! [C: product_prod_nat_nat > a,U2: set_nat] :
( ( finite_finite_nat @ ( v_a @ C ) )
=> ( finite772653738at_nat @ ( outgoing_a2 @ C @ U2 ) ) ) ).
% Graph.finite_outgoing'
thf(fact_134_Graph_OisShortestPath_Ocong,axiom,
isShortestPath_a = isShortestPath_a ).
% Graph.isShortestPath.cong
thf(fact_135_Graph_Ofinite__incoming_H,axiom,
! [C: product_prod_nat_nat > a,U2: set_nat] :
( ( finite_finite_nat @ ( v_a @ C ) )
=> ( finite772653738at_nat @ ( incoming_a2 @ C @ U2 ) ) ) ).
% Graph.finite_incoming'
thf(fact_136_finite__list,axiom,
! [A: set_Pr1986765409at_nat] :
( ( finite772653738at_nat @ A )
=> ? [Xs2: list_P559422087at_nat] :
( ( set_Pr2131844118at_nat @ Xs2 )
= A ) ) ).
% finite_list
thf(fact_137_finite__list,axiom,
! [A: set_nat] :
( ( finite_finite_nat @ A )
=> ? [Xs2: list_nat] :
( ( set_nat2 @ Xs2 )
= A ) ) ).
% finite_list
thf(fact_138_subset__code_I1_J,axiom,
! [Xs: list_P559422087at_nat,B2: set_Pr1986765409at_nat] :
( ( ord_le841296385at_nat @ ( set_Pr2131844118at_nat @ Xs ) @ B2 )
= ( ! [X: product_prod_nat_nat] :
( ( member701585322at_nat @ X @ ( set_Pr2131844118at_nat @ Xs ) )
=> ( member701585322at_nat @ X @ B2 ) ) ) ) ).
% subset_code(1)
thf(fact_139_subset__code_I1_J,axiom,
! [Xs: list_nat,B2: set_nat] :
( ( ord_less_eq_set_nat @ ( set_nat2 @ Xs ) @ B2 )
= ( ! [X: nat] :
( ( member_nat @ X @ ( set_nat2 @ Xs ) )
=> ( member_nat @ X @ B2 ) ) ) ) ).
% subset_code(1)
thf(fact_140_Graph_Oconnected__inV__iff,axiom,
! [C: product_prod_nat_nat > a,U: nat,V: nat] :
( ( connected_a @ C @ U @ V )
=> ( ( member_nat @ V @ ( v_a @ C ) )
= ( member_nat @ U @ ( v_a @ C ) ) ) ) ).
% Graph.connected_inV_iff
thf(fact_141_Graph_Oobtain__shortest__path,axiom,
! [C: product_prod_nat_nat > a,U: nat,V: nat] :
( ( connected_a @ C @ U @ V )
=> ~ ! [P2: list_P559422087at_nat] :
~ ( isShortestPath_a @ C @ U @ P2 @ V ) ) ).
% Graph.obtain_shortest_path
thf(fact_142_Finite__Graph_Ofinite__V,axiom,
! [C: product_prod_nat_nat > a] :
( ( finite_Graph_a @ C )
=> ( finite_finite_nat @ ( v_a @ C ) ) ) ).
% Finite_Graph.finite_V
thf(fact_143_Finite__Graph_Ointro,axiom,
! [C: product_prod_nat_nat > a] :
( ( finite_finite_nat @ ( v_a @ C ) )
=> ( finite_Graph_a @ C ) ) ).
% Finite_Graph.intro
thf(fact_144_Finite__Graph__def,axiom,
( finite_Graph_a
= ( ^ [C2: product_prod_nat_nat > a] : ( finite_finite_nat @ ( v_a @ C2 ) ) ) ) ).
% Finite_Graph_def
thf(fact_145_Finite__Graph_Ofinite__E,axiom,
! [C: product_prod_nat_nat > a] :
( ( finite_Graph_a @ C )
=> ( finite772653738at_nat @ ( e_a @ C ) ) ) ).
% Finite_Graph.finite_E
thf(fact_146_Graph_OFinite__Graph__EI,axiom,
! [C: product_prod_nat_nat > a] :
( ( finite772653738at_nat @ ( e_a @ C ) )
=> ( finite_Graph_a @ C ) ) ).
% Graph.Finite_Graph_EI
thf(fact_147_outgoing__def,axiom,
! [V: nat] :
( ( outgoing_a @ c @ V )
= ( collec7649004at_nat
@ ^ [Uu: product_prod_nat_nat] :
? [U3: nat] :
( ( Uu
= ( product_Pair_nat_nat @ V @ U3 ) )
& ( member701585322at_nat @ ( product_Pair_nat_nat @ V @ U3 ) @ ( e_a @ c ) ) ) ) ) ).
% outgoing_def
thf(fact_148_outgoing__edges,axiom,
! [U: nat] : ( ord_le841296385at_nat @ ( outgoing_a @ c @ U ) @ ( e_a @ c ) ) ).
% outgoing_edges
thf(fact_149_E__ss__VxV,axiom,
( ord_le841296385at_nat @ ( e_a @ c )
@ ( produc45129834at_nat @ ( v_a @ c )
@ ^ [Uu: nat] : ( v_a @ c ) ) ) ).
% E_ss_VxV
thf(fact_150_connected__edgeRtc,axiom,
! [U: nat,V: nat] :
( ( connected_a @ c @ U @ V )
= ( member701585322at_nat @ ( product_Pair_nat_nat @ U @ V ) @ ( transi1645084429cl_nat @ ( e_a @ c ) ) ) ) ).
% connected_edgeRtc
thf(fact_151_isSPath__no__selfloop,axiom,
! [U: nat,P4: list_P559422087at_nat,V: nat,U1: nat] :
( ( isSimplePath_a @ c @ U @ P4 @ V )
=> ~ ( member701585322at_nat @ ( product_Pair_nat_nat @ U1 @ U1 ) @ ( set_Pr2131844118at_nat @ P4 ) ) ) ).
% isSPath_no_selfloop
thf(fact_152_isSPath__sg__incoming,axiom,
! [U: nat,P4: list_P559422087at_nat,V: nat,U1: nat,V1: nat,U22: nat] :
( ( isSimplePath_a @ c @ U @ P4 @ V )
=> ( ( member701585322at_nat @ ( product_Pair_nat_nat @ U1 @ V1 ) @ ( set_Pr2131844118at_nat @ P4 ) )
=> ( ( U1 != U22 )
=> ~ ( member701585322at_nat @ ( product_Pair_nat_nat @ U22 @ V1 ) @ ( set_Pr2131844118at_nat @ P4 ) ) ) ) ) ).
% isSPath_sg_incoming
thf(fact_153_isSPath__sg__outgoing,axiom,
! [U: nat,P4: list_P559422087at_nat,V: nat,U1: nat,V1: nat,V22: nat] :
( ( isSimplePath_a @ c @ U @ P4 @ V )
=> ( ( member701585322at_nat @ ( product_Pair_nat_nat @ U1 @ V1 ) @ ( set_Pr2131844118at_nat @ P4 ) )
=> ( ( V1 != V22 )
=> ~ ( member701585322at_nat @ ( product_Pair_nat_nat @ U1 @ V22 ) @ ( set_Pr2131844118at_nat @ P4 ) ) ) ) ) ).
% isSPath_sg_outgoing
thf(fact_154_reachableNodes__E__closed,axiom,
! [S: nat] : ( ord_less_eq_set_nat @ ( image_nat_nat @ ( e_a @ c ) @ ( reachableNodes_a @ c @ S ) ) @ ( reachableNodes_a @ c @ S ) ) ).
% reachableNodes_E_closed
thf(fact_155_shortestPath__is__simple,axiom,
! [S: nat,P4: list_P559422087at_nat,T3: nat] :
( ( isShortestPath_a @ c @ S @ P4 @ T3 )
=> ( isSimplePath_a @ c @ S @ P4 @ T3 ) ) ).
% shortestPath_is_simple
thf(fact_156_ImageI,axiom,
! [A3: product_prod_nat_nat,B4: product_prod_nat_nat,R: set_Pr1490359111at_nat,A: set_Pr1986765409at_nat] :
( ( member2027625872at_nat @ ( produc1168807639at_nat @ A3 @ B4 ) @ R )
=> ( ( member701585322at_nat @ A3 @ A )
=> ( member701585322at_nat @ B4 @ ( image_1655366419at_nat @ R @ A ) ) ) ) ).
% ImageI
thf(fact_157_ImageI,axiom,
! [A3: product_prod_nat_nat,B4: nat,R: set_Pr1287749686at_nat,A: set_Pr1986765409at_nat] :
( ( member868723479at_nat @ ( produc947540346at_nat @ A3 @ B4 ) @ R )
=> ( ( member701585322at_nat @ A3 @ A )
=> ( member_nat @ B4 @ ( image_2142699582at_nat @ R @ A ) ) ) ) ).
% ImageI
thf(fact_158_ImageI,axiom,
! [A3: nat,B4: product_prod_nat_nat,R: set_Pr1746169692at_nat,A: set_nat] :
( ( member1293241661at_nat @ ( produc1933845336at_nat @ A3 @ B4 ) @ R )
=> ( ( member_nat @ A3 @ A )
=> ( member701585322at_nat @ B4 @ ( image_981520924at_nat @ R @ A ) ) ) ) ).
% ImageI
thf(fact_159_ImageI,axiom,
! [A3: nat,B4: nat,R: set_Pr1986765409at_nat,A: set_nat] :
( ( member701585322at_nat @ ( product_Pair_nat_nat @ A3 @ B4 ) @ R )
=> ( ( member_nat @ A3 @ A )
=> ( member_nat @ B4 @ ( image_nat_nat @ R @ A ) ) ) ) ).
% ImageI
thf(fact_160_finite__SigmaI,axiom,
! [A: set_Pr1986765409at_nat,B2: product_prod_nat_nat > set_Pr1986765409at_nat] :
( ( finite772653738at_nat @ A )
=> ( ! [A4: product_prod_nat_nat] :
( ( member701585322at_nat @ A4 @ A )
=> ( finite772653738at_nat @ ( B2 @ A4 ) ) )
=> ( finite48957584at_nat @ ( produc2056081288at_nat @ A @ B2 ) ) ) ) ).
% finite_SigmaI
thf(fact_161_finite__SigmaI,axiom,
! [A: set_Pr1986765409at_nat,B2: product_prod_nat_nat > set_nat] :
( ( finite772653738at_nat @ A )
=> ( ! [A4: product_prod_nat_nat] :
( ( member701585322at_nat @ A4 @ A )
=> ( finite_finite_nat @ ( B2 @ A4 ) ) )
=> ( finite2000257047at_nat @ ( produc2055342601at_nat @ A @ B2 ) ) ) ) ).
% finite_SigmaI
thf(fact_162_finite__SigmaI,axiom,
! [A: set_nat,B2: nat > set_Pr1986765409at_nat] :
( ( finite_finite_nat @ A )
=> ( ! [A4: nat] :
( ( member_nat @ A4 @ A )
=> ( finite772653738at_nat @ ( B2 @ A4 ) ) )
=> ( finite277291581at_nat @ ( produc894163943at_nat @ A @ B2 ) ) ) ) ).
% finite_SigmaI
thf(fact_163_finite__SigmaI,axiom,
! [A: set_nat,B2: nat > set_nat] :
( ( finite_finite_nat @ A )
=> ( ! [A4: nat] :
( ( member_nat @ A4 @ A )
=> ( finite_finite_nat @ ( B2 @ A4 ) ) )
=> ( finite772653738at_nat @ ( produc45129834at_nat @ A @ B2 ) ) ) ) ).
% finite_SigmaI
thf(fact_164_adjacent__nodes__finite,axiom,
! [U: nat] : ( finite_finite_nat @ ( adjacent_nodes_a @ c @ U ) ) ).
% adjacent_nodes_finite
thf(fact_165_finite__outgoing,axiom,
! [U: nat] :
( ( finite_finite_nat @ ( v_a @ c ) )
=> ( finite772653738at_nat @ ( outgoing_a @ c @ U ) ) ) ).
% finite_outgoing
thf(fact_166_Graph_Oadjacent__nodes_Ocong,axiom,
adjacent_nodes_a = adjacent_nodes_a ).
% Graph.adjacent_nodes.cong
thf(fact_167_Graph_OisSimplePath_Ocong,axiom,
isSimplePath_a = isSimplePath_a ).
% Graph.isSimplePath.cong
thf(fact_168_Graph_Ooutgoing_Ocong,axiom,
outgoing_a = outgoing_a ).
% Graph.outgoing.cong
thf(fact_169_rev__ImageI,axiom,
! [A3: product_prod_nat_nat,A: set_Pr1986765409at_nat,B4: product_prod_nat_nat,R: set_Pr1490359111at_nat] :
( ( member701585322at_nat @ A3 @ A )
=> ( ( member2027625872at_nat @ ( produc1168807639at_nat @ A3 @ B4 ) @ R )
=> ( member701585322at_nat @ B4 @ ( image_1655366419at_nat @ R @ A ) ) ) ) ).
% rev_ImageI
thf(fact_170_rev__ImageI,axiom,
! [A3: product_prod_nat_nat,A: set_Pr1986765409at_nat,B4: nat,R: set_Pr1287749686at_nat] :
( ( member701585322at_nat @ A3 @ A )
=> ( ( member868723479at_nat @ ( produc947540346at_nat @ A3 @ B4 ) @ R )
=> ( member_nat @ B4 @ ( image_2142699582at_nat @ R @ A ) ) ) ) ).
% rev_ImageI
thf(fact_171_rev__ImageI,axiom,
! [A3: nat,A: set_nat,B4: product_prod_nat_nat,R: set_Pr1746169692at_nat] :
( ( member_nat @ A3 @ A )
=> ( ( member1293241661at_nat @ ( produc1933845336at_nat @ A3 @ B4 ) @ R )
=> ( member701585322at_nat @ B4 @ ( image_981520924at_nat @ R @ A ) ) ) ) ).
% rev_ImageI
thf(fact_172_rev__ImageI,axiom,
! [A3: nat,A: set_nat,B4: nat,R: set_Pr1986765409at_nat] :
( ( member_nat @ A3 @ A )
=> ( ( member701585322at_nat @ ( product_Pair_nat_nat @ A3 @ B4 ) @ R )
=> ( member_nat @ B4 @ ( image_nat_nat @ R @ A ) ) ) ) ).
% rev_ImageI
thf(fact_173_Image__iff,axiom,
! [B4: nat,R: set_Pr1986765409at_nat,A: set_nat] :
( ( member_nat @ B4 @ ( image_nat_nat @ R @ A ) )
= ( ? [X: nat] :
( ( member_nat @ X @ A )
& ( member701585322at_nat @ ( product_Pair_nat_nat @ X @ B4 ) @ R ) ) ) ) ).
% Image_iff
thf(fact_174_ImageE,axiom,
! [B4: product_prod_nat_nat,R: set_Pr1490359111at_nat,A: set_Pr1986765409at_nat] :
( ( member701585322at_nat @ B4 @ ( image_1655366419at_nat @ R @ A ) )
=> ~ ! [X2: product_prod_nat_nat] :
( ( member2027625872at_nat @ ( produc1168807639at_nat @ X2 @ B4 ) @ R )
=> ~ ( member701585322at_nat @ X2 @ A ) ) ) ).
% ImageE
thf(fact_175_ImageE,axiom,
! [B4: product_prod_nat_nat,R: set_Pr1746169692at_nat,A: set_nat] :
( ( member701585322at_nat @ B4 @ ( image_981520924at_nat @ R @ A ) )
=> ~ ! [X2: nat] :
( ( member1293241661at_nat @ ( produc1933845336at_nat @ X2 @ B4 ) @ R )
=> ~ ( member_nat @ X2 @ A ) ) ) ).
% ImageE
thf(fact_176_ImageE,axiom,
! [B4: nat,R: set_Pr1287749686at_nat,A: set_Pr1986765409at_nat] :
( ( member_nat @ B4 @ ( image_2142699582at_nat @ R @ A ) )
=> ~ ! [X2: product_prod_nat_nat] :
( ( member868723479at_nat @ ( produc947540346at_nat @ X2 @ B4 ) @ R )
=> ~ ( member701585322at_nat @ X2 @ A ) ) ) ).
% ImageE
thf(fact_177_ImageE,axiom,
! [B4: nat,R: set_Pr1986765409at_nat,A: set_nat] :
( ( member_nat @ B4 @ ( image_nat_nat @ R @ A ) )
=> ~ ! [X2: nat] :
( ( member701585322at_nat @ ( product_Pair_nat_nat @ X2 @ B4 ) @ R )
=> ~ ( member_nat @ X2 @ A ) ) ) ).
% ImageE
thf(fact_178_Image__subset,axiom,
! [R: set_Pr1986765409at_nat,A: set_nat,B2: set_nat,C3: set_nat] :
( ( ord_le841296385at_nat @ R
@ ( produc45129834at_nat @ A
@ ^ [Uu: nat] : B2 ) )
=> ( ord_less_eq_set_nat @ ( image_nat_nat @ R @ C3 ) @ B2 ) ) ).
% Image_subset
thf(fact_179_Graph_OshortestPath__is__simple,axiom,
! [C: product_prod_nat_nat > a,S: nat,P4: list_P559422087at_nat,T3: nat] :
( ( isShortestPath_a @ C @ S @ P4 @ T3 )
=> ( isSimplePath_a @ C @ S @ P4 @ T3 ) ) ).
% Graph.shortestPath_is_simple
thf(fact_180_Image__mono,axiom,
! [R4: set_Pr1490359111at_nat,R: set_Pr1490359111at_nat,A5: set_Pr1986765409at_nat,A: set_Pr1986765409at_nat] :
( ( ord_le465611495at_nat @ R4 @ R )
=> ( ( ord_le841296385at_nat @ A5 @ A )
=> ( ord_le841296385at_nat @ ( image_1655366419at_nat @ R4 @ A5 ) @ ( image_1655366419at_nat @ R @ A ) ) ) ) ).
% Image_mono
thf(fact_181_Image__mono,axiom,
! [R4: set_Pr1287749686at_nat,R: set_Pr1287749686at_nat,A5: set_Pr1986765409at_nat,A: set_Pr1986765409at_nat] :
( ( ord_le1837363350at_nat @ R4 @ R )
=> ( ( ord_le841296385at_nat @ A5 @ A )
=> ( ord_less_eq_set_nat @ ( image_2142699582at_nat @ R4 @ A5 ) @ ( image_2142699582at_nat @ R @ A ) ) ) ) ).
% Image_mono
thf(fact_182_Image__mono,axiom,
! [R4: set_Pr1746169692at_nat,R: set_Pr1746169692at_nat,A5: set_nat,A: set_nat] :
( ( ord_le148299708at_nat @ R4 @ R )
=> ( ( ord_less_eq_set_nat @ A5 @ A )
=> ( ord_le841296385at_nat @ ( image_981520924at_nat @ R4 @ A5 ) @ ( image_981520924at_nat @ R @ A ) ) ) ) ).
% Image_mono
thf(fact_183_Image__mono,axiom,
! [R4: set_Pr1986765409at_nat,R: set_Pr1986765409at_nat,A5: set_nat,A: set_nat] :
( ( ord_le841296385at_nat @ R4 @ R )
=> ( ( ord_less_eq_set_nat @ A5 @ A )
=> ( ord_less_eq_set_nat @ ( image_nat_nat @ R4 @ A5 ) @ ( image_nat_nat @ R @ A ) ) ) ) ).
% Image_mono
thf(fact_184_Relation_Ofinite__Image,axiom,
! [R2: set_Pr1986765409at_nat,A: set_nat] :
( ( finite772653738at_nat @ R2 )
=> ( finite_finite_nat @ ( image_nat_nat @ R2 @ A ) ) ) ).
% Relation.finite_Image
thf(fact_185_Graph_OreachableNodes__E__closed,axiom,
! [C: product_prod_nat_nat > a,S: nat] : ( ord_less_eq_set_nat @ ( image_nat_nat @ ( e_a @ C ) @ ( reachableNodes_a @ C @ S ) ) @ ( reachableNodes_a @ C @ S ) ) ).
% Graph.reachableNodes_E_closed
thf(fact_186_finite__cartesian__product,axiom,
! [A: set_Pr1986765409at_nat,B2: set_Pr1986765409at_nat] :
( ( finite772653738at_nat @ A )
=> ( ( finite772653738at_nat @ B2 )
=> ( finite48957584at_nat
@ ( produc2056081288at_nat @ A
@ ^ [Uu: product_prod_nat_nat] : B2 ) ) ) ) ).
% finite_cartesian_product
thf(fact_187_finite__cartesian__product,axiom,
! [A: set_Pr1986765409at_nat,B2: set_nat] :
( ( finite772653738at_nat @ A )
=> ( ( finite_finite_nat @ B2 )
=> ( finite2000257047at_nat
@ ( produc2055342601at_nat @ A
@ ^ [Uu: product_prod_nat_nat] : B2 ) ) ) ) ).
% finite_cartesian_product
thf(fact_188_finite__cartesian__product,axiom,
! [A: set_nat,B2: set_Pr1986765409at_nat] :
( ( finite_finite_nat @ A )
=> ( ( finite772653738at_nat @ B2 )
=> ( finite277291581at_nat
@ ( produc894163943at_nat @ A
@ ^ [Uu: nat] : B2 ) ) ) ) ).
% finite_cartesian_product
thf(fact_189_finite__cartesian__product,axiom,
! [A: set_nat,B2: set_nat] :
( ( finite_finite_nat @ A )
=> ( ( finite_finite_nat @ B2 )
=> ( finite772653738at_nat
@ ( produc45129834at_nat @ A
@ ^ [Uu: nat] : B2 ) ) ) ) ).
% finite_cartesian_product
thf(fact_190_Graph_OisSPath__sg__outgoing,axiom,
! [C: product_prod_nat_nat > a,U: nat,P4: list_P559422087at_nat,V: nat,U1: nat,V1: nat,V22: nat] :
( ( isSimplePath_a @ C @ U @ P4 @ V )
=> ( ( member701585322at_nat @ ( product_Pair_nat_nat @ U1 @ V1 ) @ ( set_Pr2131844118at_nat @ P4 ) )
=> ( ( V1 != V22 )
=> ~ ( member701585322at_nat @ ( product_Pair_nat_nat @ U1 @ V22 ) @ ( set_Pr2131844118at_nat @ P4 ) ) ) ) ) ).
% Graph.isSPath_sg_outgoing
thf(fact_191_Graph_OisSPath__sg__incoming,axiom,
! [C: product_prod_nat_nat > a,U: nat,P4: list_P559422087at_nat,V: nat,U1: nat,V1: nat,U22: nat] :
( ( isSimplePath_a @ C @ U @ P4 @ V )
=> ( ( member701585322at_nat @ ( product_Pair_nat_nat @ U1 @ V1 ) @ ( set_Pr2131844118at_nat @ P4 ) )
=> ( ( U1 != U22 )
=> ~ ( member701585322at_nat @ ( product_Pair_nat_nat @ U22 @ V1 ) @ ( set_Pr2131844118at_nat @ P4 ) ) ) ) ) ).
% Graph.isSPath_sg_incoming
thf(fact_192_Graph_OisSPath__no__selfloop,axiom,
! [C: product_prod_nat_nat > a,U: nat,P4: list_P559422087at_nat,V: nat,U1: nat] :
( ( isSimplePath_a @ C @ U @ P4 @ V )
=> ~ ( member701585322at_nat @ ( product_Pair_nat_nat @ U1 @ U1 ) @ ( set_Pr2131844118at_nat @ P4 ) ) ) ).
% Graph.isSPath_no_selfloop
thf(fact_193_Graph_Oadjacent__nodes__ss__V,axiom,
! [C: product_prod_nat_nat > a,U: nat] : ( ord_less_eq_set_nat @ ( adjacent_nodes_a @ C @ U ) @ ( v_a @ C ) ) ).
% Graph.adjacent_nodes_ss_V
thf(fact_194_Graph_Ooutgoing__edges,axiom,
! [C: product_prod_nat_nat > a,U: nat] : ( ord_le841296385at_nat @ ( outgoing_a @ C @ U ) @ ( e_a @ C ) ) ).
% Graph.outgoing_edges
thf(fact_195_Graph_Ooutgoing__def,axiom,
( outgoing_a
= ( ^ [C2: product_prod_nat_nat > a,V2: nat] :
( collec7649004at_nat
@ ^ [Uu: product_prod_nat_nat] :
? [U3: nat] :
( ( Uu
= ( product_Pair_nat_nat @ V2 @ U3 ) )
& ( member701585322at_nat @ ( product_Pair_nat_nat @ V2 @ U3 ) @ ( e_a @ C2 ) ) ) ) ) ) ).
% Graph.outgoing_def
thf(fact_196_Graph_Oconnected__edgeRtc,axiom,
( connected_a
= ( ^ [C2: product_prod_nat_nat > a,U3: nat,V2: nat] : ( member701585322at_nat @ ( product_Pair_nat_nat @ U3 @ V2 ) @ ( transi1645084429cl_nat @ ( e_a @ C2 ) ) ) ) ) ).
% Graph.connected_edgeRtc
thf(fact_197_Graph_OE__ss__VxV,axiom,
! [C: product_prod_nat_nat > a] :
( ord_le841296385at_nat @ ( e_a @ C )
@ ( produc45129834at_nat @ ( v_a @ C )
@ ^ [Uu: nat] : ( v_a @ C ) ) ) ).
% Graph.E_ss_VxV
thf(fact_198_Graph_Ofinite__outgoing,axiom,
! [C: product_prod_nat_nat > a,U: nat] :
( ( finite_finite_nat @ ( v_a @ C ) )
=> ( finite772653738at_nat @ ( outgoing_a @ C @ U ) ) ) ).
% Graph.finite_outgoing
thf(fact_199_Finite__Graph_Oadjacent__nodes__finite,axiom,
! [C: product_prod_nat_nat > a,U: nat] :
( ( finite_Graph_a @ C )
=> ( finite_finite_nat @ ( adjacent_nodes_a @ C @ U ) ) ) ).
% Finite_Graph.adjacent_nodes_finite
thf(fact_200_isSPath__nt__parallel,axiom,
! [S: nat,P4: list_P559422087at_nat,T3: nat,E2: product_prod_nat_nat] :
( ( isSimplePath_a @ c @ S @ P4 @ T3 )
=> ( ( member701585322at_nat @ E2 @ ( set_Pr2131844118at_nat @ P4 ) )
=> ~ ( member701585322at_nat @ ( product_swap_nat_nat @ E2 ) @ ( set_Pr2131844118at_nat @ P4 ) ) ) ) ).
% isSPath_nt_parallel
thf(fact_201_mem__Sigma__iff,axiom,
! [A3: product_prod_nat_nat,B4: product_prod_nat_nat,A: set_Pr1986765409at_nat,B2: product_prod_nat_nat > set_Pr1986765409at_nat] :
( ( member2027625872at_nat @ ( produc1168807639at_nat @ A3 @ B4 ) @ ( produc2056081288at_nat @ A @ B2 ) )
= ( ( member701585322at_nat @ A3 @ A )
& ( member701585322at_nat @ B4 @ ( B2 @ A3 ) ) ) ) ).
% mem_Sigma_iff
thf(fact_202_mem__Sigma__iff,axiom,
! [A3: product_prod_nat_nat,B4: nat,A: set_Pr1986765409at_nat,B2: product_prod_nat_nat > set_nat] :
( ( member868723479at_nat @ ( produc947540346at_nat @ A3 @ B4 ) @ ( produc2055342601at_nat @ A @ B2 ) )
= ( ( member701585322at_nat @ A3 @ A )
& ( member_nat @ B4 @ ( B2 @ A3 ) ) ) ) ).
% mem_Sigma_iff
thf(fact_203_mem__Sigma__iff,axiom,
! [A3: nat,B4: product_prod_nat_nat,A: set_nat,B2: nat > set_Pr1986765409at_nat] :
( ( member1293241661at_nat @ ( produc1933845336at_nat @ A3 @ B4 ) @ ( produc894163943at_nat @ A @ B2 ) )
= ( ( member_nat @ A3 @ A )
& ( member701585322at_nat @ B4 @ ( B2 @ A3 ) ) ) ) ).
% mem_Sigma_iff
thf(fact_204_mem__Sigma__iff,axiom,
! [A3: nat,B4: nat,A: set_nat,B2: nat > set_nat] :
( ( member701585322at_nat @ ( product_Pair_nat_nat @ A3 @ B4 ) @ ( produc45129834at_nat @ A @ B2 ) )
= ( ( member_nat @ A3 @ A )
& ( member_nat @ B4 @ ( B2 @ A3 ) ) ) ) ).
% mem_Sigma_iff
thf(fact_205_SigmaI,axiom,
! [A3: product_prod_nat_nat,A: set_Pr1986765409at_nat,B4: product_prod_nat_nat,B2: product_prod_nat_nat > set_Pr1986765409at_nat] :
( ( member701585322at_nat @ A3 @ A )
=> ( ( member701585322at_nat @ B4 @ ( B2 @ A3 ) )
=> ( member2027625872at_nat @ ( produc1168807639at_nat @ A3 @ B4 ) @ ( produc2056081288at_nat @ A @ B2 ) ) ) ) ).
% SigmaI
thf(fact_206_SigmaI,axiom,
! [A3: product_prod_nat_nat,A: set_Pr1986765409at_nat,B4: nat,B2: product_prod_nat_nat > set_nat] :
( ( member701585322at_nat @ A3 @ A )
=> ( ( member_nat @ B4 @ ( B2 @ A3 ) )
=> ( member868723479at_nat @ ( produc947540346at_nat @ A3 @ B4 ) @ ( produc2055342601at_nat @ A @ B2 ) ) ) ) ).
% SigmaI
thf(fact_207_SigmaI,axiom,
! [A3: nat,A: set_nat,B4: product_prod_nat_nat,B2: nat > set_Pr1986765409at_nat] :
( ( member_nat @ A3 @ A )
=> ( ( member701585322at_nat @ B4 @ ( B2 @ A3 ) )
=> ( member1293241661at_nat @ ( produc1933845336at_nat @ A3 @ B4 ) @ ( produc894163943at_nat @ A @ B2 ) ) ) ) ).
% SigmaI
thf(fact_208_SigmaI,axiom,
! [A3: nat,A: set_nat,B4: nat,B2: nat > set_nat] :
( ( member_nat @ A3 @ A )
=> ( ( member_nat @ B4 @ ( B2 @ A3 ) )
=> ( member701585322at_nat @ ( product_Pair_nat_nat @ A3 @ B4 ) @ ( produc45129834at_nat @ A @ B2 ) ) ) ) ).
% SigmaI
thf(fact_209_trancl__subset__Sigma__aux,axiom,
! [A3: product_prod_nat_nat,B4: product_prod_nat_nat,R: set_Pr1490359111at_nat,A: set_Pr1986765409at_nat] :
( ( member2027625872at_nat @ ( produc1168807639at_nat @ A3 @ B4 ) @ ( transi1484096900at_nat @ R ) )
=> ( ( ord_le465611495at_nat @ R
@ ( produc2056081288at_nat @ A
@ ^ [Uu: product_prod_nat_nat] : A ) )
=> ( ( A3 = B4 )
| ( member701585322at_nat @ A3 @ A ) ) ) ) ).
% trancl_subset_Sigma_aux
thf(fact_210_trancl__subset__Sigma__aux,axiom,
! [A3: nat,B4: nat,R: set_Pr1986765409at_nat,A: set_nat] :
( ( member701585322at_nat @ ( product_Pair_nat_nat @ A3 @ B4 ) @ ( transi1645084429cl_nat @ R ) )
=> ( ( ord_le841296385at_nat @ R
@ ( produc45129834at_nat @ A
@ ^ [Uu: nat] : A ) )
=> ( ( A3 = B4 )
| ( member_nat @ A3 @ A ) ) ) ) ).
% trancl_subset_Sigma_aux
thf(fact_211_old_Oprod_Oinject,axiom,
! [A3: nat,B4: nat,A6: nat,B6: nat] :
( ( ( product_Pair_nat_nat @ A3 @ B4 )
= ( product_Pair_nat_nat @ A6 @ B6 ) )
= ( ( A3 = A6 )
& ( B4 = B6 ) ) ) ).
% old.prod.inject
thf(fact_212_prod_Oinject,axiom,
! [X1: nat,X22: nat,Y1: nat,Y22: nat] :
( ( ( product_Pair_nat_nat @ X1 @ X22 )
= ( product_Pair_nat_nat @ Y1 @ Y22 ) )
= ( ( X1 = Y1 )
& ( X22 = Y22 ) ) ) ).
% prod.inject
thf(fact_213_r__into__rtrancl,axiom,
! [P4: product_prod_nat_nat,R: set_Pr1986765409at_nat] :
( ( member701585322at_nat @ P4 @ R )
=> ( member701585322at_nat @ P4 @ ( transi1645084429cl_nat @ R ) ) ) ).
% r_into_rtrancl
thf(fact_214_rtrancl__idemp,axiom,
! [R: set_Pr1986765409at_nat] :
( ( transi1645084429cl_nat @ ( transi1645084429cl_nat @ R ) )
= ( transi1645084429cl_nat @ R ) ) ).
% rtrancl_idemp
thf(fact_215_swap__swap,axiom,
! [P4: product_prod_nat_nat] :
( ( product_swap_nat_nat @ ( product_swap_nat_nat @ P4 ) )
= P4 ) ).
% swap_swap
thf(fact_216_swap__simp,axiom,
! [X3: nat,Y3: nat] :
( ( product_swap_nat_nat @ ( product_Pair_nat_nat @ X3 @ Y3 ) )
= ( product_Pair_nat_nat @ Y3 @ X3 ) ) ).
% swap_simp
thf(fact_217_converse__rtrancl__induct2,axiom,
! [Ax: nat,Ay: nat,Bx: nat,By: nat,R: set_Pr1490359111at_nat,P3: nat > nat > $o] :
( ( member2027625872at_nat @ ( produc1168807639at_nat @ ( product_Pair_nat_nat @ Ax @ Ay ) @ ( product_Pair_nat_nat @ Bx @ By ) ) @ ( transi1484096900at_nat @ R ) )
=> ( ( P3 @ Bx @ By )
=> ( ! [A4: nat,B3: nat,Aa: nat,Ba: nat] :
( ( member2027625872at_nat @ ( produc1168807639at_nat @ ( product_Pair_nat_nat @ A4 @ B3 ) @ ( product_Pair_nat_nat @ Aa @ Ba ) ) @ R )
=> ( ( member2027625872at_nat @ ( produc1168807639at_nat @ ( product_Pair_nat_nat @ Aa @ Ba ) @ ( product_Pair_nat_nat @ Bx @ By ) ) @ ( transi1484096900at_nat @ R ) )
=> ( ( P3 @ Aa @ Ba )
=> ( P3 @ A4 @ B3 ) ) ) )
=> ( P3 @ Ax @ Ay ) ) ) ) ).
% converse_rtrancl_induct2
thf(fact_218_converse__rtranclE2,axiom,
! [Xa2: nat,Xb: nat,Za: nat,Zb: nat,R: set_Pr1490359111at_nat] :
( ( member2027625872at_nat @ ( produc1168807639at_nat @ ( product_Pair_nat_nat @ Xa2 @ Xb ) @ ( product_Pair_nat_nat @ Za @ Zb ) ) @ ( transi1484096900at_nat @ R ) )
=> ( ( ( product_Pair_nat_nat @ Xa2 @ Xb )
!= ( product_Pair_nat_nat @ Za @ Zb ) )
=> ~ ! [A4: nat,B3: nat] :
( ( member2027625872at_nat @ ( produc1168807639at_nat @ ( product_Pair_nat_nat @ Xa2 @ Xb ) @ ( product_Pair_nat_nat @ A4 @ B3 ) ) @ R )
=> ~ ( member2027625872at_nat @ ( produc1168807639at_nat @ ( product_Pair_nat_nat @ A4 @ B3 ) @ ( product_Pair_nat_nat @ Za @ Zb ) ) @ ( transi1484096900at_nat @ R ) ) ) ) ) ).
% converse_rtranclE2
thf(fact_219_rtrancl__induct2,axiom,
! [Ax: nat,Ay: nat,Bx: nat,By: nat,R: set_Pr1490359111at_nat,P3: nat > nat > $o] :
( ( member2027625872at_nat @ ( produc1168807639at_nat @ ( product_Pair_nat_nat @ Ax @ Ay ) @ ( product_Pair_nat_nat @ Bx @ By ) ) @ ( transi1484096900at_nat @ R ) )
=> ( ( P3 @ Ax @ Ay )
=> ( ! [A4: nat,B3: nat,Aa: nat,Ba: nat] :
( ( member2027625872at_nat @ ( produc1168807639at_nat @ ( product_Pair_nat_nat @ Ax @ Ay ) @ ( product_Pair_nat_nat @ A4 @ B3 ) ) @ ( transi1484096900at_nat @ R ) )
=> ( ( member2027625872at_nat @ ( produc1168807639at_nat @ ( product_Pair_nat_nat @ A4 @ B3 ) @ ( product_Pair_nat_nat @ Aa @ Ba ) ) @ R )
=> ( ( P3 @ A4 @ B3 )
=> ( P3 @ Aa @ Ba ) ) ) )
=> ( P3 @ Bx @ By ) ) ) ) ).
% rtrancl_induct2
thf(fact_220_old_Oprod_Oinducts,axiom,
! [P3: product_prod_nat_nat > $o,Prod: product_prod_nat_nat] :
( ! [A4: nat,B3: nat] : ( P3 @ ( product_Pair_nat_nat @ A4 @ B3 ) )
=> ( P3 @ Prod ) ) ).
% old.prod.inducts
thf(fact_221_old_Oprod_Oexhaust,axiom,
! [Y3: product_prod_nat_nat] :
~ ! [A4: nat,B3: nat] :
( Y3
!= ( product_Pair_nat_nat @ A4 @ B3 ) ) ).
% old.prod.exhaust
thf(fact_222_Pair__inject,axiom,
! [A3: nat,B4: nat,A6: nat,B6: nat] :
( ( ( product_Pair_nat_nat @ A3 @ B4 )
= ( product_Pair_nat_nat @ A6 @ B6 ) )
=> ~ ( ( A3 = A6 )
=> ( B4 != B6 ) ) ) ).
% Pair_inject
thf(fact_223_prod__cases,axiom,
! [P3: product_prod_nat_nat > $o,P4: product_prod_nat_nat] :
( ! [A4: nat,B3: nat] : ( P3 @ ( product_Pair_nat_nat @ A4 @ B3 ) )
=> ( P3 @ P4 ) ) ).
% prod_cases
thf(fact_224_surj__pair,axiom,
! [P4: product_prod_nat_nat] :
? [X2: nat,Y2: nat] :
( P4
= ( product_Pair_nat_nat @ X2 @ Y2 ) ) ).
% surj_pair
thf(fact_225_Graph_OisSPath__nt__parallel,axiom,
! [C: product_prod_nat_nat > a,S: nat,P4: list_P559422087at_nat,T3: nat,E2: product_prod_nat_nat] :
( ( isSimplePath_a @ C @ S @ P4 @ T3 )
=> ( ( member701585322at_nat @ E2 @ ( set_Pr2131844118at_nat @ P4 ) )
=> ~ ( member701585322at_nat @ ( product_swap_nat_nat @ E2 ) @ ( set_Pr2131844118at_nat @ P4 ) ) ) ) ).
% Graph.isSPath_nt_parallel
thf(fact_226_Sigma__cong,axiom,
! [A: set_nat,B2: set_nat,C3: nat > set_nat,D: nat > set_nat] :
( ( A = B2 )
=> ( ! [X2: nat] :
( ( member_nat @ X2 @ B2 )
=> ( ( C3 @ X2 )
= ( D @ X2 ) ) )
=> ( ( produc45129834at_nat @ A @ C3 )
= ( produc45129834at_nat @ B2 @ D ) ) ) ) ).
% Sigma_cong
thf(fact_227_Times__eq__cancel2,axiom,
! [X3: nat,C3: set_nat,A: set_nat,B2: set_nat] :
( ( member_nat @ X3 @ C3 )
=> ( ( ( produc45129834at_nat @ A
@ ^ [Uu: nat] : C3 )
= ( produc45129834at_nat @ B2
@ ^ [Uu: nat] : C3 ) )
= ( A = B2 ) ) ) ).
% Times_eq_cancel2
thf(fact_228_SigmaE,axiom,
! [C: produc842455143at_nat,A: set_Pr1986765409at_nat,B2: product_prod_nat_nat > set_Pr1986765409at_nat] :
( ( member2027625872at_nat @ C @ ( produc2056081288at_nat @ A @ B2 ) )
=> ~ ! [X2: product_prod_nat_nat] :
( ( member701585322at_nat @ X2 @ A )
=> ! [Y2: product_prod_nat_nat] :
( ( member701585322at_nat @ Y2 @ ( B2 @ X2 ) )
=> ( C
!= ( produc1168807639at_nat @ X2 @ Y2 ) ) ) ) ) ).
% SigmaE
thf(fact_229_SigmaE,axiom,
! [C: produc1271302400at_nat,A: set_Pr1986765409at_nat,B2: product_prod_nat_nat > set_nat] :
( ( member868723479at_nat @ C @ ( produc2055342601at_nat @ A @ B2 ) )
=> ~ ! [X2: product_prod_nat_nat] :
( ( member701585322at_nat @ X2 @ A )
=> ! [Y2: nat] :
( ( member_nat @ Y2 @ ( B2 @ X2 ) )
=> ( C
!= ( produc947540346at_nat @ X2 @ Y2 ) ) ) ) ) ).
% SigmaE
thf(fact_230_SigmaE,axiom,
! [C: produc1695820582at_nat,A: set_nat,B2: nat > set_Pr1986765409at_nat] :
( ( member1293241661at_nat @ C @ ( produc894163943at_nat @ A @ B2 ) )
=> ~ ! [X2: nat] :
( ( member_nat @ X2 @ A )
=> ! [Y2: product_prod_nat_nat] :
( ( member701585322at_nat @ Y2 @ ( B2 @ X2 ) )
=> ( C
!= ( produc1933845336at_nat @ X2 @ Y2 ) ) ) ) ) ).
% SigmaE
thf(fact_231_SigmaE,axiom,
! [C: product_prod_nat_nat,A: set_nat,B2: nat > set_nat] :
( ( member701585322at_nat @ C @ ( produc45129834at_nat @ A @ B2 ) )
=> ~ ! [X2: nat] :
( ( member_nat @ X2 @ A )
=> ! [Y2: nat] :
( ( member_nat @ Y2 @ ( B2 @ X2 ) )
=> ( C
!= ( product_Pair_nat_nat @ X2 @ Y2 ) ) ) ) ) ).
% SigmaE
thf(fact_232_SigmaD1,axiom,
! [A3: nat,B4: nat,A: set_nat,B2: nat > set_nat] :
( ( member701585322at_nat @ ( product_Pair_nat_nat @ A3 @ B4 ) @ ( produc45129834at_nat @ A @ B2 ) )
=> ( member_nat @ A3 @ A ) ) ).
% SigmaD1
thf(fact_233_SigmaD2,axiom,
! [A3: nat,B4: nat,A: set_nat,B2: nat > set_nat] :
( ( member701585322at_nat @ ( product_Pair_nat_nat @ A3 @ B4 ) @ ( produc45129834at_nat @ A @ B2 ) )
=> ( member_nat @ B4 @ ( B2 @ A3 ) ) ) ).
% SigmaD2
thf(fact_234_SigmaE2,axiom,
! [A3: product_prod_nat_nat,B4: product_prod_nat_nat,A: set_Pr1986765409at_nat,B2: product_prod_nat_nat > set_Pr1986765409at_nat] :
( ( member2027625872at_nat @ ( produc1168807639at_nat @ A3 @ B4 ) @ ( produc2056081288at_nat @ A @ B2 ) )
=> ~ ( ( member701585322at_nat @ A3 @ A )
=> ~ ( member701585322at_nat @ B4 @ ( B2 @ A3 ) ) ) ) ).
% SigmaE2
thf(fact_235_SigmaE2,axiom,
! [A3: product_prod_nat_nat,B4: nat,A: set_Pr1986765409at_nat,B2: product_prod_nat_nat > set_nat] :
( ( member868723479at_nat @ ( produc947540346at_nat @ A3 @ B4 ) @ ( produc2055342601at_nat @ A @ B2 ) )
=> ~ ( ( member701585322at_nat @ A3 @ A )
=> ~ ( member_nat @ B4 @ ( B2 @ A3 ) ) ) ) ).
% SigmaE2
thf(fact_236_SigmaE2,axiom,
! [A3: nat,B4: product_prod_nat_nat,A: set_nat,B2: nat > set_Pr1986765409at_nat] :
( ( member1293241661at_nat @ ( produc1933845336at_nat @ A3 @ B4 ) @ ( produc894163943at_nat @ A @ B2 ) )
=> ~ ( ( member_nat @ A3 @ A )
=> ~ ( member701585322at_nat @ B4 @ ( B2 @ A3 ) ) ) ) ).
% SigmaE2
thf(fact_237_SigmaE2,axiom,
! [A3: nat,B4: nat,A: set_nat,B2: nat > set_nat] :
( ( member701585322at_nat @ ( product_Pair_nat_nat @ A3 @ B4 ) @ ( produc45129834at_nat @ A @ B2 ) )
=> ~ ( ( member_nat @ A3 @ A )
=> ~ ( member_nat @ B4 @ ( B2 @ A3 ) ) ) ) ).
% SigmaE2
thf(fact_238_rtranclE,axiom,
! [A3: nat,B4: nat,R: set_Pr1986765409at_nat] :
( ( member701585322at_nat @ ( product_Pair_nat_nat @ A3 @ B4 ) @ ( transi1645084429cl_nat @ R ) )
=> ( ( A3 != B4 )
=> ~ ! [Y2: nat] :
( ( member701585322at_nat @ ( product_Pair_nat_nat @ A3 @ Y2 ) @ ( transi1645084429cl_nat @ R ) )
=> ~ ( member701585322at_nat @ ( product_Pair_nat_nat @ Y2 @ B4 ) @ R ) ) ) ) ).
% rtranclE
thf(fact_239_rtrancl_Ocases,axiom,
! [A1: nat,A22: nat,R: set_Pr1986765409at_nat] :
( ( member701585322at_nat @ ( product_Pair_nat_nat @ A1 @ A22 ) @ ( transi1645084429cl_nat @ R ) )
=> ( ( A22 != A1 )
=> ~ ! [B3: nat] :
( ( member701585322at_nat @ ( product_Pair_nat_nat @ A1 @ B3 ) @ ( transi1645084429cl_nat @ R ) )
=> ~ ( member701585322at_nat @ ( product_Pair_nat_nat @ B3 @ A22 ) @ R ) ) ) ) ).
% rtrancl.cases
thf(fact_240_rtrancl_Osimps,axiom,
! [A1: nat,A22: nat,R: set_Pr1986765409at_nat] :
( ( member701585322at_nat @ ( product_Pair_nat_nat @ A1 @ A22 ) @ ( transi1645084429cl_nat @ R ) )
= ( ? [A2: nat] :
( ( A1 = A2 )
& ( A22 = A2 ) )
| ? [A2: nat,B5: nat,C2: nat] :
( ( A1 = A2 )
& ( A22 = C2 )
& ( member701585322at_nat @ ( product_Pair_nat_nat @ A2 @ B5 ) @ ( transi1645084429cl_nat @ R ) )
& ( member701585322at_nat @ ( product_Pair_nat_nat @ B5 @ C2 ) @ R ) ) ) ) ).
% rtrancl.simps
thf(fact_241_rtrancl__trans,axiom,
! [X3: nat,Y3: nat,R: set_Pr1986765409at_nat,Z: nat] :
( ( member701585322at_nat @ ( product_Pair_nat_nat @ X3 @ Y3 ) @ ( transi1645084429cl_nat @ R ) )
=> ( ( member701585322at_nat @ ( product_Pair_nat_nat @ Y3 @ Z ) @ ( transi1645084429cl_nat @ R ) )
=> ( member701585322at_nat @ ( product_Pair_nat_nat @ X3 @ Z ) @ ( transi1645084429cl_nat @ R ) ) ) ) ).
% rtrancl_trans
thf(fact_242_rtrancl__induct,axiom,
! [A3: nat,B4: nat,R: set_Pr1986765409at_nat,P3: nat > $o] :
( ( member701585322at_nat @ ( product_Pair_nat_nat @ A3 @ B4 ) @ ( transi1645084429cl_nat @ R ) )
=> ( ( P3 @ A3 )
=> ( ! [Y2: nat,Z2: nat] :
( ( member701585322at_nat @ ( product_Pair_nat_nat @ A3 @ Y2 ) @ ( transi1645084429cl_nat @ R ) )
=> ( ( member701585322at_nat @ ( product_Pair_nat_nat @ Y2 @ Z2 ) @ R )
=> ( ( P3 @ Y2 )
=> ( P3 @ Z2 ) ) ) )
=> ( P3 @ B4 ) ) ) ) ).
% rtrancl_induct
thf(fact_243_rtrancl_Oinducts,axiom,
! [X1: nat,X22: nat,R: set_Pr1986765409at_nat,P3: nat > nat > $o] :
( ( member701585322at_nat @ ( product_Pair_nat_nat @ X1 @ X22 ) @ ( transi1645084429cl_nat @ R ) )
=> ( ! [A4: nat] : ( P3 @ A4 @ A4 )
=> ( ! [A4: nat,B3: nat,C4: nat] :
( ( member701585322at_nat @ ( product_Pair_nat_nat @ A4 @ B3 ) @ ( transi1645084429cl_nat @ R ) )
=> ( ( P3 @ A4 @ B3 )
=> ( ( member701585322at_nat @ ( product_Pair_nat_nat @ B3 @ C4 ) @ R )
=> ( P3 @ A4 @ C4 ) ) ) )
=> ( P3 @ X1 @ X22 ) ) ) ) ).
% rtrancl.inducts
thf(fact_244_converse__rtranclE,axiom,
! [X3: nat,Z: nat,R: set_Pr1986765409at_nat] :
( ( member701585322at_nat @ ( product_Pair_nat_nat @ X3 @ Z ) @ ( transi1645084429cl_nat @ R ) )
=> ( ( X3 != Z )
=> ~ ! [Y2: nat] :
( ( member701585322at_nat @ ( product_Pair_nat_nat @ X3 @ Y2 ) @ R )
=> ~ ( member701585322at_nat @ ( product_Pair_nat_nat @ Y2 @ Z ) @ ( transi1645084429cl_nat @ R ) ) ) ) ) ).
% converse_rtranclE
thf(fact_245_rtrancl_Ortrancl__refl,axiom,
! [A3: nat,R: set_Pr1986765409at_nat] : ( member701585322at_nat @ ( product_Pair_nat_nat @ A3 @ A3 ) @ ( transi1645084429cl_nat @ R ) ) ).
% rtrancl.rtrancl_refl
thf(fact_246_converse__rtrancl__induct,axiom,
! [A3: nat,B4: nat,R: set_Pr1986765409at_nat,P3: nat > $o] :
( ( member701585322at_nat @ ( product_Pair_nat_nat @ A3 @ B4 ) @ ( transi1645084429cl_nat @ R ) )
=> ( ( P3 @ B4 )
=> ( ! [Y2: nat,Z2: nat] :
( ( member701585322at_nat @ ( product_Pair_nat_nat @ Y2 @ Z2 ) @ R )
=> ( ( member701585322at_nat @ ( product_Pair_nat_nat @ Z2 @ B4 ) @ ( transi1645084429cl_nat @ R ) )
=> ( ( P3 @ Z2 )
=> ( P3 @ Y2 ) ) ) )
=> ( P3 @ A3 ) ) ) ) ).
% converse_rtrancl_induct
thf(fact_247_rtrancl_Ortrancl__into__rtrancl,axiom,
! [A3: nat,B4: nat,R: set_Pr1986765409at_nat,C: nat] :
( ( member701585322at_nat @ ( product_Pair_nat_nat @ A3 @ B4 ) @ ( transi1645084429cl_nat @ R ) )
=> ( ( member701585322at_nat @ ( product_Pair_nat_nat @ B4 @ C ) @ R )
=> ( member701585322at_nat @ ( product_Pair_nat_nat @ A3 @ C ) @ ( transi1645084429cl_nat @ R ) ) ) ) ).
% rtrancl.rtrancl_into_rtrancl
thf(fact_248_converse__rtrancl__into__rtrancl,axiom,
! [A3: nat,B4: nat,R: set_Pr1986765409at_nat,C: nat] :
( ( member701585322at_nat @ ( product_Pair_nat_nat @ A3 @ B4 ) @ R )
=> ( ( member701585322at_nat @ ( product_Pair_nat_nat @ B4 @ C ) @ ( transi1645084429cl_nat @ R ) )
=> ( member701585322at_nat @ ( product_Pair_nat_nat @ A3 @ C ) @ ( transi1645084429cl_nat @ R ) ) ) ) ).
% converse_rtrancl_into_rtrancl
thf(fact_249_rtrancl__mono,axiom,
! [R: set_Pr1986765409at_nat,S: set_Pr1986765409at_nat] :
( ( ord_le841296385at_nat @ R @ S )
=> ( ord_le841296385at_nat @ ( transi1645084429cl_nat @ R ) @ ( transi1645084429cl_nat @ S ) ) ) ).
% rtrancl_mono
thf(fact_250_rtrancl__subset,axiom,
! [R2: set_Pr1986765409at_nat,S3: set_Pr1986765409at_nat] :
( ( ord_le841296385at_nat @ R2 @ S3 )
=> ( ( ord_le841296385at_nat @ S3 @ ( transi1645084429cl_nat @ R2 ) )
=> ( ( transi1645084429cl_nat @ S3 )
= ( transi1645084429cl_nat @ R2 ) ) ) ) ).
% rtrancl_subset
thf(fact_251_rtrancl__subset__rtrancl,axiom,
! [R: set_Pr1986765409at_nat,S: set_Pr1986765409at_nat] :
( ( ord_le841296385at_nat @ R @ ( transi1645084429cl_nat @ S ) )
=> ( ord_le841296385at_nat @ ( transi1645084429cl_nat @ R ) @ ( transi1645084429cl_nat @ S ) ) ) ).
% rtrancl_subset_rtrancl
thf(fact_252_Sigma__mono,axiom,
! [A: set_Pr1986765409at_nat,C3: set_Pr1986765409at_nat,B2: product_prod_nat_nat > set_Pr1986765409at_nat,D: product_prod_nat_nat > set_Pr1986765409at_nat] :
( ( ord_le841296385at_nat @ A @ C3 )
=> ( ! [X2: product_prod_nat_nat] :
( ( member701585322at_nat @ X2 @ A )
=> ( ord_le841296385at_nat @ ( B2 @ X2 ) @ ( D @ X2 ) ) )
=> ( ord_le465611495at_nat @ ( produc2056081288at_nat @ A @ B2 ) @ ( produc2056081288at_nat @ C3 @ D ) ) ) ) ).
% Sigma_mono
thf(fact_253_Sigma__mono,axiom,
! [A: set_Pr1986765409at_nat,C3: set_Pr1986765409at_nat,B2: product_prod_nat_nat > set_nat,D: product_prod_nat_nat > set_nat] :
( ( ord_le841296385at_nat @ A @ C3 )
=> ( ! [X2: product_prod_nat_nat] :
( ( member701585322at_nat @ X2 @ A )
=> ( ord_less_eq_set_nat @ ( B2 @ X2 ) @ ( D @ X2 ) ) )
=> ( ord_le1837363350at_nat @ ( produc2055342601at_nat @ A @ B2 ) @ ( produc2055342601at_nat @ C3 @ D ) ) ) ) ).
% Sigma_mono
thf(fact_254_Sigma__mono,axiom,
! [A: set_nat,C3: set_nat,B2: nat > set_Pr1986765409at_nat,D: nat > set_Pr1986765409at_nat] :
( ( ord_less_eq_set_nat @ A @ C3 )
=> ( ! [X2: nat] :
( ( member_nat @ X2 @ A )
=> ( ord_le841296385at_nat @ ( B2 @ X2 ) @ ( D @ X2 ) ) )
=> ( ord_le148299708at_nat @ ( produc894163943at_nat @ A @ B2 ) @ ( produc894163943at_nat @ C3 @ D ) ) ) ) ).
% Sigma_mono
thf(fact_255_Sigma__mono,axiom,
! [A: set_nat,C3: set_nat,B2: nat > set_nat,D: nat > set_nat] :
( ( ord_less_eq_set_nat @ A @ C3 )
=> ( ! [X2: nat] :
( ( member_nat @ X2 @ A )
=> ( ord_less_eq_set_nat @ ( B2 @ X2 ) @ ( D @ X2 ) ) )
=> ( ord_le841296385at_nat @ ( produc45129834at_nat @ A @ B2 ) @ ( produc45129834at_nat @ C3 @ D ) ) ) ) ).
% Sigma_mono
thf(fact_256_Image__closed__trancl,axiom,
! [R: set_Pr1490359111at_nat,X4: set_Pr1986765409at_nat] :
( ( ord_le841296385at_nat @ ( image_1655366419at_nat @ R @ X4 ) @ X4 )
=> ( ( image_1655366419at_nat @ ( transi1484096900at_nat @ R ) @ X4 )
= X4 ) ) ).
% Image_closed_trancl
thf(fact_257_Image__closed__trancl,axiom,
! [R: set_Pr1986765409at_nat,X4: set_nat] :
( ( ord_less_eq_set_nat @ ( image_nat_nat @ R @ X4 ) @ X4 )
=> ( ( image_nat_nat @ ( transi1645084429cl_nat @ R ) @ X4 )
= X4 ) ) ).
% Image_closed_trancl
thf(fact_258_Not__Domain__rtrancl,axiom,
! [X3: product_prod_nat_nat,R2: set_Pr1490359111at_nat,Y3: product_prod_nat_nat] :
( ~ ( member701585322at_nat @ X3 @ ( domain37934156at_nat @ R2 ) )
=> ( ( member2027625872at_nat @ ( produc1168807639at_nat @ X3 @ Y3 ) @ ( transi1484096900at_nat @ R2 ) )
= ( X3 = Y3 ) ) ) ).
% Not_Domain_rtrancl
thf(fact_259_Not__Domain__rtrancl,axiom,
! [X3: nat,R2: set_Pr1986765409at_nat,Y3: nat] :
( ~ ( member_nat @ X3 @ ( domain_nat_nat @ R2 ) )
=> ( ( member701585322at_nat @ ( product_Pair_nat_nat @ X3 @ Y3 ) @ ( transi1645084429cl_nat @ R2 ) )
= ( X3 = Y3 ) ) ) ).
% Not_Domain_rtrancl
thf(fact_260_Times__subset__cancel2,axiom,
! [X3: product_prod_nat_nat,C3: set_Pr1986765409at_nat,A: set_Pr1986765409at_nat,B2: set_Pr1986765409at_nat] :
( ( member701585322at_nat @ X3 @ C3 )
=> ( ( ord_le465611495at_nat
@ ( produc2056081288at_nat @ A
@ ^ [Uu: product_prod_nat_nat] : C3 )
@ ( produc2056081288at_nat @ B2
@ ^ [Uu: product_prod_nat_nat] : C3 ) )
= ( ord_le841296385at_nat @ A @ B2 ) ) ) ).
% Times_subset_cancel2
thf(fact_261_Times__subset__cancel2,axiom,
! [X3: nat,C3: set_nat,A: set_Pr1986765409at_nat,B2: set_Pr1986765409at_nat] :
( ( member_nat @ X3 @ C3 )
=> ( ( ord_le1837363350at_nat
@ ( produc2055342601at_nat @ A
@ ^ [Uu: product_prod_nat_nat] : C3 )
@ ( produc2055342601at_nat @ B2
@ ^ [Uu: product_prod_nat_nat] : C3 ) )
= ( ord_le841296385at_nat @ A @ B2 ) ) ) ).
% Times_subset_cancel2
thf(fact_262_Times__subset__cancel2,axiom,
! [X3: product_prod_nat_nat,C3: set_Pr1986765409at_nat,A: set_nat,B2: set_nat] :
( ( member701585322at_nat @ X3 @ C3 )
=> ( ( ord_le148299708at_nat
@ ( produc894163943at_nat @ A
@ ^ [Uu: nat] : C3 )
@ ( produc894163943at_nat @ B2
@ ^ [Uu: nat] : C3 ) )
= ( ord_less_eq_set_nat @ A @ B2 ) ) ) ).
% Times_subset_cancel2
thf(fact_263_Times__subset__cancel2,axiom,
! [X3: nat,C3: set_nat,A: set_nat,B2: set_nat] :
( ( member_nat @ X3 @ C3 )
=> ( ( ord_le841296385at_nat
@ ( produc45129834at_nat @ A
@ ^ [Uu: nat] : C3 )
@ ( produc45129834at_nat @ B2
@ ^ [Uu: nat] : C3 ) )
= ( ord_less_eq_set_nat @ A @ B2 ) ) ) ).
% Times_subset_cancel2
thf(fact_264_finite__rtrancl__Image,axiom,
! [R2: set_Pr1490359111at_nat,A: set_Pr1986765409at_nat] :
( ( finite48957584at_nat @ R2 )
=> ( ( finite772653738at_nat @ A )
=> ( finite772653738at_nat @ ( image_1655366419at_nat @ ( transi1484096900at_nat @ R2 ) @ A ) ) ) ) ).
% finite_rtrancl_Image
thf(fact_265_finite__rtrancl__Image,axiom,
! [R2: set_Pr1986765409at_nat,A: set_nat] :
( ( finite772653738at_nat @ R2 )
=> ( ( finite_finite_nat @ A )
=> ( finite_finite_nat @ ( image_nat_nat @ ( transi1645084429cl_nat @ R2 ) @ A ) ) ) ) ).
% finite_rtrancl_Image
thf(fact_266_infinite__cartesian__product,axiom,
! [A: set_Pr1986765409at_nat,B2: set_Pr1986765409at_nat] :
( ~ ( finite772653738at_nat @ A )
=> ( ~ ( finite772653738at_nat @ B2 )
=> ~ ( finite48957584at_nat
@ ( produc2056081288at_nat @ A
@ ^ [Uu: product_prod_nat_nat] : B2 ) ) ) ) ).
% infinite_cartesian_product
thf(fact_267_infinite__cartesian__product,axiom,
! [A: set_Pr1986765409at_nat,B2: set_nat] :
( ~ ( finite772653738at_nat @ A )
=> ( ~ ( finite_finite_nat @ B2 )
=> ~ ( finite2000257047at_nat
@ ( produc2055342601at_nat @ A
@ ^ [Uu: product_prod_nat_nat] : B2 ) ) ) ) ).
% infinite_cartesian_product
thf(fact_268_infinite__cartesian__product,axiom,
! [A: set_nat,B2: set_Pr1986765409at_nat] :
( ~ ( finite_finite_nat @ A )
=> ( ~ ( finite772653738at_nat @ B2 )
=> ~ ( finite277291581at_nat
@ ( produc894163943at_nat @ A
@ ^ [Uu: nat] : B2 ) ) ) ) ).
% infinite_cartesian_product
thf(fact_269_infinite__cartesian__product,axiom,
! [A: set_nat,B2: set_nat] :
( ~ ( finite_finite_nat @ A )
=> ( ~ ( finite_finite_nat @ B2 )
=> ~ ( finite772653738at_nat
@ ( produc45129834at_nat @ A
@ ^ [Uu: nat] : B2 ) ) ) ) ).
% infinite_cartesian_product
thf(fact_270_order__refl,axiom,
! [X3: set_Pr1986765409at_nat] : ( ord_le841296385at_nat @ X3 @ X3 ) ).
% order_refl
thf(fact_271_order__refl,axiom,
! [X3: set_nat] : ( ord_less_eq_set_nat @ X3 @ X3 ) ).
% order_refl
thf(fact_272_order__refl,axiom,
! [X3: nat] : ( ord_less_eq_nat @ X3 @ X3 ) ).
% order_refl
thf(fact_273_order__subst1,axiom,
! [A3: set_Pr1986765409at_nat,F: set_Pr1986765409at_nat > set_Pr1986765409at_nat,B4: set_Pr1986765409at_nat,C: set_Pr1986765409at_nat] :
( ( ord_le841296385at_nat @ A3 @ ( F @ B4 ) )
=> ( ( ord_le841296385at_nat @ B4 @ C )
=> ( ! [X2: set_Pr1986765409at_nat,Y2: set_Pr1986765409at_nat] :
( ( ord_le841296385at_nat @ X2 @ Y2 )
=> ( ord_le841296385at_nat @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_le841296385at_nat @ A3 @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_274_order__subst1,axiom,
! [A3: set_Pr1986765409at_nat,F: set_nat > set_Pr1986765409at_nat,B4: set_nat,C: set_nat] :
( ( ord_le841296385at_nat @ A3 @ ( F @ B4 ) )
=> ( ( ord_less_eq_set_nat @ B4 @ C )
=> ( ! [X2: set_nat,Y2: set_nat] :
( ( ord_less_eq_set_nat @ X2 @ Y2 )
=> ( ord_le841296385at_nat @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_le841296385at_nat @ A3 @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_275_order__subst1,axiom,
! [A3: set_Pr1986765409at_nat,F: nat > set_Pr1986765409at_nat,B4: nat,C: nat] :
( ( ord_le841296385at_nat @ A3 @ ( F @ B4 ) )
=> ( ( ord_less_eq_nat @ B4 @ C )
=> ( ! [X2: nat,Y2: nat] :
( ( ord_less_eq_nat @ X2 @ Y2 )
=> ( ord_le841296385at_nat @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_le841296385at_nat @ A3 @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_276_order__subst1,axiom,
! [A3: set_nat,F: set_Pr1986765409at_nat > set_nat,B4: set_Pr1986765409at_nat,C: set_Pr1986765409at_nat] :
( ( ord_less_eq_set_nat @ A3 @ ( F @ B4 ) )
=> ( ( ord_le841296385at_nat @ B4 @ C )
=> ( ! [X2: set_Pr1986765409at_nat,Y2: set_Pr1986765409at_nat] :
( ( ord_le841296385at_nat @ X2 @ Y2 )
=> ( ord_less_eq_set_nat @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_set_nat @ A3 @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_277_order__subst1,axiom,
! [A3: set_nat,F: set_nat > set_nat,B4: set_nat,C: set_nat] :
( ( ord_less_eq_set_nat @ A3 @ ( F @ B4 ) )
=> ( ( ord_less_eq_set_nat @ B4 @ C )
=> ( ! [X2: set_nat,Y2: set_nat] :
( ( ord_less_eq_set_nat @ X2 @ Y2 )
=> ( ord_less_eq_set_nat @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_set_nat @ A3 @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_278_order__subst1,axiom,
! [A3: set_nat,F: nat > set_nat,B4: nat,C: nat] :
( ( ord_less_eq_set_nat @ A3 @ ( F @ B4 ) )
=> ( ( ord_less_eq_nat @ B4 @ C )
=> ( ! [X2: nat,Y2: nat] :
( ( ord_less_eq_nat @ X2 @ Y2 )
=> ( ord_less_eq_set_nat @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_set_nat @ A3 @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_279_order__subst1,axiom,
! [A3: nat,F: set_Pr1986765409at_nat > nat,B4: set_Pr1986765409at_nat,C: set_Pr1986765409at_nat] :
( ( ord_less_eq_nat @ A3 @ ( F @ B4 ) )
=> ( ( ord_le841296385at_nat @ B4 @ C )
=> ( ! [X2: set_Pr1986765409at_nat,Y2: set_Pr1986765409at_nat] :
( ( ord_le841296385at_nat @ X2 @ Y2 )
=> ( ord_less_eq_nat @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_nat @ A3 @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_280_order__subst1,axiom,
! [A3: nat,F: set_nat > nat,B4: set_nat,C: set_nat] :
( ( ord_less_eq_nat @ A3 @ ( F @ B4 ) )
=> ( ( ord_less_eq_set_nat @ B4 @ C )
=> ( ! [X2: set_nat,Y2: set_nat] :
( ( ord_less_eq_set_nat @ X2 @ Y2 )
=> ( ord_less_eq_nat @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_nat @ A3 @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_281_order__subst1,axiom,
! [A3: nat,F: nat > nat,B4: nat,C: nat] :
( ( ord_less_eq_nat @ A3 @ ( F @ B4 ) )
=> ( ( ord_less_eq_nat @ B4 @ C )
=> ( ! [X2: nat,Y2: nat] :
( ( ord_less_eq_nat @ X2 @ Y2 )
=> ( ord_less_eq_nat @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_nat @ A3 @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_282_order__subst2,axiom,
! [A3: set_Pr1986765409at_nat,B4: set_Pr1986765409at_nat,F: set_Pr1986765409at_nat > set_Pr1986765409at_nat,C: set_Pr1986765409at_nat] :
( ( ord_le841296385at_nat @ A3 @ B4 )
=> ( ( ord_le841296385at_nat @ ( F @ B4 ) @ C )
=> ( ! [X2: set_Pr1986765409at_nat,Y2: set_Pr1986765409at_nat] :
( ( ord_le841296385at_nat @ X2 @ Y2 )
=> ( ord_le841296385at_nat @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_le841296385at_nat @ ( F @ A3 ) @ C ) ) ) ) ).
% order_subst2
thf(fact_283_order__subst2,axiom,
! [A3: set_Pr1986765409at_nat,B4: set_Pr1986765409at_nat,F: set_Pr1986765409at_nat > set_nat,C: set_nat] :
( ( ord_le841296385at_nat @ A3 @ B4 )
=> ( ( ord_less_eq_set_nat @ ( F @ B4 ) @ C )
=> ( ! [X2: set_Pr1986765409at_nat,Y2: set_Pr1986765409at_nat] :
( ( ord_le841296385at_nat @ X2 @ Y2 )
=> ( ord_less_eq_set_nat @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_set_nat @ ( F @ A3 ) @ C ) ) ) ) ).
% order_subst2
thf(fact_284_order__subst2,axiom,
! [A3: set_Pr1986765409at_nat,B4: set_Pr1986765409at_nat,F: set_Pr1986765409at_nat > nat,C: nat] :
( ( ord_le841296385at_nat @ A3 @ B4 )
=> ( ( ord_less_eq_nat @ ( F @ B4 ) @ C )
=> ( ! [X2: set_Pr1986765409at_nat,Y2: set_Pr1986765409at_nat] :
( ( ord_le841296385at_nat @ X2 @ Y2 )
=> ( ord_less_eq_nat @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_nat @ ( F @ A3 ) @ C ) ) ) ) ).
% order_subst2
thf(fact_285_order__subst2,axiom,
! [A3: set_nat,B4: set_nat,F: set_nat > set_Pr1986765409at_nat,C: set_Pr1986765409at_nat] :
( ( ord_less_eq_set_nat @ A3 @ B4 )
=> ( ( ord_le841296385at_nat @ ( F @ B4 ) @ C )
=> ( ! [X2: set_nat,Y2: set_nat] :
( ( ord_less_eq_set_nat @ X2 @ Y2 )
=> ( ord_le841296385at_nat @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_le841296385at_nat @ ( F @ A3 ) @ C ) ) ) ) ).
% order_subst2
thf(fact_286_order__subst2,axiom,
! [A3: set_nat,B4: set_nat,F: set_nat > set_nat,C: set_nat] :
( ( ord_less_eq_set_nat @ A3 @ B4 )
=> ( ( ord_less_eq_set_nat @ ( F @ B4 ) @ C )
=> ( ! [X2: set_nat,Y2: set_nat] :
( ( ord_less_eq_set_nat @ X2 @ Y2 )
=> ( ord_less_eq_set_nat @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_set_nat @ ( F @ A3 ) @ C ) ) ) ) ).
% order_subst2
thf(fact_287_order__subst2,axiom,
! [A3: set_nat,B4: set_nat,F: set_nat > nat,C: nat] :
( ( ord_less_eq_set_nat @ A3 @ B4 )
=> ( ( ord_less_eq_nat @ ( F @ B4 ) @ C )
=> ( ! [X2: set_nat,Y2: set_nat] :
( ( ord_less_eq_set_nat @ X2 @ Y2 )
=> ( ord_less_eq_nat @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_nat @ ( F @ A3 ) @ C ) ) ) ) ).
% order_subst2
thf(fact_288_order__subst2,axiom,
! [A3: nat,B4: nat,F: nat > set_Pr1986765409at_nat,C: set_Pr1986765409at_nat] :
( ( ord_less_eq_nat @ A3 @ B4 )
=> ( ( ord_le841296385at_nat @ ( F @ B4 ) @ C )
=> ( ! [X2: nat,Y2: nat] :
( ( ord_less_eq_nat @ X2 @ Y2 )
=> ( ord_le841296385at_nat @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_le841296385at_nat @ ( F @ A3 ) @ C ) ) ) ) ).
% order_subst2
thf(fact_289_order__subst2,axiom,
! [A3: nat,B4: nat,F: nat > set_nat,C: set_nat] :
( ( ord_less_eq_nat @ A3 @ B4 )
=> ( ( ord_less_eq_set_nat @ ( F @ B4 ) @ C )
=> ( ! [X2: nat,Y2: nat] :
( ( ord_less_eq_nat @ X2 @ Y2 )
=> ( ord_less_eq_set_nat @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_set_nat @ ( F @ A3 ) @ C ) ) ) ) ).
% order_subst2
thf(fact_290_order__subst2,axiom,
! [A3: nat,B4: nat,F: nat > nat,C: nat] :
( ( ord_less_eq_nat @ A3 @ B4 )
=> ( ( ord_less_eq_nat @ ( F @ B4 ) @ C )
=> ( ! [X2: nat,Y2: nat] :
( ( ord_less_eq_nat @ X2 @ Y2 )
=> ( ord_less_eq_nat @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_nat @ ( F @ A3 ) @ C ) ) ) ) ).
% order_subst2
thf(fact_291_ord__eq__le__subst,axiom,
! [A3: set_Pr1986765409at_nat,F: set_Pr1986765409at_nat > set_Pr1986765409at_nat,B4: set_Pr1986765409at_nat,C: set_Pr1986765409at_nat] :
( ( A3
= ( F @ B4 ) )
=> ( ( ord_le841296385at_nat @ B4 @ C )
=> ( ! [X2: set_Pr1986765409at_nat,Y2: set_Pr1986765409at_nat] :
( ( ord_le841296385at_nat @ X2 @ Y2 )
=> ( ord_le841296385at_nat @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_le841296385at_nat @ A3 @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_292_ord__eq__le__subst,axiom,
! [A3: set_nat,F: set_Pr1986765409at_nat > set_nat,B4: set_Pr1986765409at_nat,C: set_Pr1986765409at_nat] :
( ( A3
= ( F @ B4 ) )
=> ( ( ord_le841296385at_nat @ B4 @ C )
=> ( ! [X2: set_Pr1986765409at_nat,Y2: set_Pr1986765409at_nat] :
( ( ord_le841296385at_nat @ X2 @ Y2 )
=> ( ord_less_eq_set_nat @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_set_nat @ A3 @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_293_ord__eq__le__subst,axiom,
! [A3: nat,F: set_Pr1986765409at_nat > nat,B4: set_Pr1986765409at_nat,C: set_Pr1986765409at_nat] :
( ( A3
= ( F @ B4 ) )
=> ( ( ord_le841296385at_nat @ B4 @ C )
=> ( ! [X2: set_Pr1986765409at_nat,Y2: set_Pr1986765409at_nat] :
( ( ord_le841296385at_nat @ X2 @ Y2 )
=> ( ord_less_eq_nat @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_nat @ A3 @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_294_ord__eq__le__subst,axiom,
! [A3: set_Pr1986765409at_nat,F: set_nat > set_Pr1986765409at_nat,B4: set_nat,C: set_nat] :
( ( A3
= ( F @ B4 ) )
=> ( ( ord_less_eq_set_nat @ B4 @ C )
=> ( ! [X2: set_nat,Y2: set_nat] :
( ( ord_less_eq_set_nat @ X2 @ Y2 )
=> ( ord_le841296385at_nat @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_le841296385at_nat @ A3 @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_295_ord__eq__le__subst,axiom,
! [A3: set_nat,F: set_nat > set_nat,B4: set_nat,C: set_nat] :
( ( A3
= ( F @ B4 ) )
=> ( ( ord_less_eq_set_nat @ B4 @ C )
=> ( ! [X2: set_nat,Y2: set_nat] :
( ( ord_less_eq_set_nat @ X2 @ Y2 )
=> ( ord_less_eq_set_nat @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_set_nat @ A3 @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_296_ord__eq__le__subst,axiom,
! [A3: nat,F: set_nat > nat,B4: set_nat,C: set_nat] :
( ( A3
= ( F @ B4 ) )
=> ( ( ord_less_eq_set_nat @ B4 @ C )
=> ( ! [X2: set_nat,Y2: set_nat] :
( ( ord_less_eq_set_nat @ X2 @ Y2 )
=> ( ord_less_eq_nat @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_nat @ A3 @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_297_ord__eq__le__subst,axiom,
! [A3: set_Pr1986765409at_nat,F: nat > set_Pr1986765409at_nat,B4: nat,C: nat] :
( ( A3
= ( F @ B4 ) )
=> ( ( ord_less_eq_nat @ B4 @ C )
=> ( ! [X2: nat,Y2: nat] :
( ( ord_less_eq_nat @ X2 @ Y2 )
=> ( ord_le841296385at_nat @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_le841296385at_nat @ A3 @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_298_ord__eq__le__subst,axiom,
! [A3: set_nat,F: nat > set_nat,B4: nat,C: nat] :
( ( A3
= ( F @ B4 ) )
=> ( ( ord_less_eq_nat @ B4 @ C )
=> ( ! [X2: nat,Y2: nat] :
( ( ord_less_eq_nat @ X2 @ Y2 )
=> ( ord_less_eq_set_nat @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_set_nat @ A3 @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_299_ord__eq__le__subst,axiom,
! [A3: nat,F: nat > nat,B4: nat,C: nat] :
( ( A3
= ( F @ B4 ) )
=> ( ( ord_less_eq_nat @ B4 @ C )
=> ( ! [X2: nat,Y2: nat] :
( ( ord_less_eq_nat @ X2 @ Y2 )
=> ( ord_less_eq_nat @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_nat @ A3 @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_300_ord__le__eq__subst,axiom,
! [A3: set_Pr1986765409at_nat,B4: set_Pr1986765409at_nat,F: set_Pr1986765409at_nat > set_Pr1986765409at_nat,C: set_Pr1986765409at_nat] :
( ( ord_le841296385at_nat @ A3 @ B4 )
=> ( ( ( F @ B4 )
= C )
=> ( ! [X2: set_Pr1986765409at_nat,Y2: set_Pr1986765409at_nat] :
( ( ord_le841296385at_nat @ X2 @ Y2 )
=> ( ord_le841296385at_nat @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_le841296385at_nat @ ( F @ A3 ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_301_ord__le__eq__subst,axiom,
! [A3: set_Pr1986765409at_nat,B4: set_Pr1986765409at_nat,F: set_Pr1986765409at_nat > set_nat,C: set_nat] :
( ( ord_le841296385at_nat @ A3 @ B4 )
=> ( ( ( F @ B4 )
= C )
=> ( ! [X2: set_Pr1986765409at_nat,Y2: set_Pr1986765409at_nat] :
( ( ord_le841296385at_nat @ X2 @ Y2 )
=> ( ord_less_eq_set_nat @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_set_nat @ ( F @ A3 ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_302_ord__le__eq__subst,axiom,
! [A3: set_Pr1986765409at_nat,B4: set_Pr1986765409at_nat,F: set_Pr1986765409at_nat > nat,C: nat] :
( ( ord_le841296385at_nat @ A3 @ B4 )
=> ( ( ( F @ B4 )
= C )
=> ( ! [X2: set_Pr1986765409at_nat,Y2: set_Pr1986765409at_nat] :
( ( ord_le841296385at_nat @ X2 @ Y2 )
=> ( ord_less_eq_nat @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_nat @ ( F @ A3 ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_303_ord__le__eq__subst,axiom,
! [A3: set_nat,B4: set_nat,F: set_nat > set_Pr1986765409at_nat,C: set_Pr1986765409at_nat] :
( ( ord_less_eq_set_nat @ A3 @ B4 )
=> ( ( ( F @ B4 )
= C )
=> ( ! [X2: set_nat,Y2: set_nat] :
( ( ord_less_eq_set_nat @ X2 @ Y2 )
=> ( ord_le841296385at_nat @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_le841296385at_nat @ ( F @ A3 ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_304_ord__le__eq__subst,axiom,
! [A3: set_nat,B4: set_nat,F: set_nat > set_nat,C: set_nat] :
( ( ord_less_eq_set_nat @ A3 @ B4 )
=> ( ( ( F @ B4 )
= C )
=> ( ! [X2: set_nat,Y2: set_nat] :
( ( ord_less_eq_set_nat @ X2 @ Y2 )
=> ( ord_less_eq_set_nat @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_set_nat @ ( F @ A3 ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_305_ord__le__eq__subst,axiom,
! [A3: set_nat,B4: set_nat,F: set_nat > nat,C: nat] :
( ( ord_less_eq_set_nat @ A3 @ B4 )
=> ( ( ( F @ B4 )
= C )
=> ( ! [X2: set_nat,Y2: set_nat] :
( ( ord_less_eq_set_nat @ X2 @ Y2 )
=> ( ord_less_eq_nat @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_nat @ ( F @ A3 ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_306_ord__le__eq__subst,axiom,
! [A3: nat,B4: nat,F: nat > set_Pr1986765409at_nat,C: set_Pr1986765409at_nat] :
( ( ord_less_eq_nat @ A3 @ B4 )
=> ( ( ( F @ B4 )
= C )
=> ( ! [X2: nat,Y2: nat] :
( ( ord_less_eq_nat @ X2 @ Y2 )
=> ( ord_le841296385at_nat @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_le841296385at_nat @ ( F @ A3 ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_307_ord__le__eq__subst,axiom,
! [A3: nat,B4: nat,F: nat > set_nat,C: set_nat] :
( ( ord_less_eq_nat @ A3 @ B4 )
=> ( ( ( F @ B4 )
= C )
=> ( ! [X2: nat,Y2: nat] :
( ( ord_less_eq_nat @ X2 @ Y2 )
=> ( ord_less_eq_set_nat @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_set_nat @ ( F @ A3 ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_308_ord__le__eq__subst,axiom,
! [A3: nat,B4: nat,F: nat > nat,C: nat] :
( ( ord_less_eq_nat @ A3 @ B4 )
=> ( ( ( F @ B4 )
= C )
=> ( ! [X2: nat,Y2: nat] :
( ( ord_less_eq_nat @ X2 @ Y2 )
=> ( ord_less_eq_nat @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_nat @ ( F @ A3 ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_309_eq__iff,axiom,
( ( ^ [Y4: set_Pr1986765409at_nat,Z3: set_Pr1986765409at_nat] : Y4 = Z3 )
= ( ^ [X: set_Pr1986765409at_nat,Y: set_Pr1986765409at_nat] :
( ( ord_le841296385at_nat @ X @ Y )
& ( ord_le841296385at_nat @ Y @ X ) ) ) ) ).
% eq_iff
thf(fact_310_eq__iff,axiom,
( ( ^ [Y4: set_nat,Z3: set_nat] : Y4 = Z3 )
= ( ^ [X: set_nat,Y: set_nat] :
( ( ord_less_eq_set_nat @ X @ Y )
& ( ord_less_eq_set_nat @ Y @ X ) ) ) ) ).
% eq_iff
thf(fact_311_eq__iff,axiom,
( ( ^ [Y4: nat,Z3: nat] : Y4 = Z3 )
= ( ^ [X: nat,Y: nat] :
( ( ord_less_eq_nat @ X @ Y )
& ( ord_less_eq_nat @ Y @ X ) ) ) ) ).
% eq_iff
thf(fact_312_antisym,axiom,
! [X3: set_Pr1986765409at_nat,Y3: set_Pr1986765409at_nat] :
( ( ord_le841296385at_nat @ X3 @ Y3 )
=> ( ( ord_le841296385at_nat @ Y3 @ X3 )
=> ( X3 = Y3 ) ) ) ).
% antisym
thf(fact_313_antisym,axiom,
! [X3: set_nat,Y3: set_nat] :
( ( ord_less_eq_set_nat @ X3 @ Y3 )
=> ( ( ord_less_eq_set_nat @ Y3 @ X3 )
=> ( X3 = Y3 ) ) ) ).
% antisym
thf(fact_314_antisym,axiom,
! [X3: nat,Y3: nat] :
( ( ord_less_eq_nat @ X3 @ Y3 )
=> ( ( ord_less_eq_nat @ Y3 @ X3 )
=> ( X3 = Y3 ) ) ) ).
% antisym
thf(fact_315_linear,axiom,
! [X3: nat,Y3: nat] :
( ( ord_less_eq_nat @ X3 @ Y3 )
| ( ord_less_eq_nat @ Y3 @ X3 ) ) ).
% linear
thf(fact_316_eq__refl,axiom,
! [X3: set_Pr1986765409at_nat,Y3: set_Pr1986765409at_nat] :
( ( X3 = Y3 )
=> ( ord_le841296385at_nat @ X3 @ Y3 ) ) ).
% eq_refl
thf(fact_317_eq__refl,axiom,
! [X3: set_nat,Y3: set_nat] :
( ( X3 = Y3 )
=> ( ord_less_eq_set_nat @ X3 @ Y3 ) ) ).
% eq_refl
thf(fact_318_eq__refl,axiom,
! [X3: nat,Y3: nat] :
( ( X3 = Y3 )
=> ( ord_less_eq_nat @ X3 @ Y3 ) ) ).
% eq_refl
thf(fact_319_le__cases,axiom,
! [X3: nat,Y3: nat] :
( ~ ( ord_less_eq_nat @ X3 @ Y3 )
=> ( ord_less_eq_nat @ Y3 @ X3 ) ) ).
% le_cases
thf(fact_320_order_Otrans,axiom,
! [A3: set_Pr1986765409at_nat,B4: set_Pr1986765409at_nat,C: set_Pr1986765409at_nat] :
( ( ord_le841296385at_nat @ A3 @ B4 )
=> ( ( ord_le841296385at_nat @ B4 @ C )
=> ( ord_le841296385at_nat @ A3 @ C ) ) ) ).
% order.trans
thf(fact_321_order_Otrans,axiom,
! [A3: set_nat,B4: set_nat,C: set_nat] :
( ( ord_less_eq_set_nat @ A3 @ B4 )
=> ( ( ord_less_eq_set_nat @ B4 @ C )
=> ( ord_less_eq_set_nat @ A3 @ C ) ) ) ).
% order.trans
thf(fact_322_order_Otrans,axiom,
! [A3: nat,B4: nat,C: nat] :
( ( ord_less_eq_nat @ A3 @ B4 )
=> ( ( ord_less_eq_nat @ B4 @ C )
=> ( ord_less_eq_nat @ A3 @ C ) ) ) ).
% order.trans
thf(fact_323_le__cases3,axiom,
! [X3: nat,Y3: nat,Z: nat] :
( ( ( ord_less_eq_nat @ X3 @ Y3 )
=> ~ ( ord_less_eq_nat @ Y3 @ Z ) )
=> ( ( ( ord_less_eq_nat @ Y3 @ X3 )
=> ~ ( ord_less_eq_nat @ X3 @ Z ) )
=> ( ( ( ord_less_eq_nat @ X3 @ Z )
=> ~ ( ord_less_eq_nat @ Z @ Y3 ) )
=> ( ( ( ord_less_eq_nat @ Z @ Y3 )
=> ~ ( ord_less_eq_nat @ Y3 @ X3 ) )
=> ( ( ( ord_less_eq_nat @ Y3 @ Z )
=> ~ ( ord_less_eq_nat @ Z @ X3 ) )
=> ~ ( ( ord_less_eq_nat @ Z @ X3 )
=> ~ ( ord_less_eq_nat @ X3 @ Y3 ) ) ) ) ) ) ) ).
% le_cases3
thf(fact_324_antisym__conv,axiom,
! [Y3: set_Pr1986765409at_nat,X3: set_Pr1986765409at_nat] :
( ( ord_le841296385at_nat @ Y3 @ X3 )
=> ( ( ord_le841296385at_nat @ X3 @ Y3 )
= ( X3 = Y3 ) ) ) ).
% antisym_conv
thf(fact_325_antisym__conv,axiom,
! [Y3: set_nat,X3: set_nat] :
( ( ord_less_eq_set_nat @ Y3 @ X3 )
=> ( ( ord_less_eq_set_nat @ X3 @ Y3 )
= ( X3 = Y3 ) ) ) ).
% antisym_conv
thf(fact_326_antisym__conv,axiom,
! [Y3: nat,X3: nat] :
( ( ord_less_eq_nat @ Y3 @ X3 )
=> ( ( ord_less_eq_nat @ X3 @ Y3 )
= ( X3 = Y3 ) ) ) ).
% antisym_conv
thf(fact_327_order__class_Oorder_Oeq__iff,axiom,
( ( ^ [Y4: set_Pr1986765409at_nat,Z3: set_Pr1986765409at_nat] : Y4 = Z3 )
= ( ^ [A2: set_Pr1986765409at_nat,B5: set_Pr1986765409at_nat] :
( ( ord_le841296385at_nat @ A2 @ B5 )
& ( ord_le841296385at_nat @ B5 @ A2 ) ) ) ) ).
% order_class.order.eq_iff
thf(fact_328_order__class_Oorder_Oeq__iff,axiom,
( ( ^ [Y4: set_nat,Z3: set_nat] : Y4 = Z3 )
= ( ^ [A2: set_nat,B5: set_nat] :
( ( ord_less_eq_set_nat @ A2 @ B5 )
& ( ord_less_eq_set_nat @ B5 @ A2 ) ) ) ) ).
% order_class.order.eq_iff
thf(fact_329_order__class_Oorder_Oeq__iff,axiom,
( ( ^ [Y4: nat,Z3: nat] : Y4 = Z3 )
= ( ^ [A2: nat,B5: nat] :
( ( ord_less_eq_nat @ A2 @ B5 )
& ( ord_less_eq_nat @ B5 @ A2 ) ) ) ) ).
% order_class.order.eq_iff
thf(fact_330_ord__eq__le__trans,axiom,
! [A3: set_Pr1986765409at_nat,B4: set_Pr1986765409at_nat,C: set_Pr1986765409at_nat] :
( ( A3 = B4 )
=> ( ( ord_le841296385at_nat @ B4 @ C )
=> ( ord_le841296385at_nat @ A3 @ C ) ) ) ).
% ord_eq_le_trans
thf(fact_331_ord__eq__le__trans,axiom,
! [A3: set_nat,B4: set_nat,C: set_nat] :
( ( A3 = B4 )
=> ( ( ord_less_eq_set_nat @ B4 @ C )
=> ( ord_less_eq_set_nat @ A3 @ C ) ) ) ).
% ord_eq_le_trans
thf(fact_332_ord__eq__le__trans,axiom,
! [A3: nat,B4: nat,C: nat] :
( ( A3 = B4 )
=> ( ( ord_less_eq_nat @ B4 @ C )
=> ( ord_less_eq_nat @ A3 @ C ) ) ) ).
% ord_eq_le_trans
thf(fact_333_ord__le__eq__trans,axiom,
! [A3: set_Pr1986765409at_nat,B4: set_Pr1986765409at_nat,C: set_Pr1986765409at_nat] :
( ( ord_le841296385at_nat @ A3 @ B4 )
=> ( ( B4 = C )
=> ( ord_le841296385at_nat @ A3 @ C ) ) ) ).
% ord_le_eq_trans
thf(fact_334_ord__le__eq__trans,axiom,
! [A3: set_nat,B4: set_nat,C: set_nat] :
( ( ord_less_eq_set_nat @ A3 @ B4 )
=> ( ( B4 = C )
=> ( ord_less_eq_set_nat @ A3 @ C ) ) ) ).
% ord_le_eq_trans
thf(fact_335_ord__le__eq__trans,axiom,
! [A3: nat,B4: nat,C: nat] :
( ( ord_less_eq_nat @ A3 @ B4 )
=> ( ( B4 = C )
=> ( ord_less_eq_nat @ A3 @ C ) ) ) ).
% ord_le_eq_trans
thf(fact_336_order__class_Oorder_Oantisym,axiom,
! [A3: set_Pr1986765409at_nat,B4: set_Pr1986765409at_nat] :
( ( ord_le841296385at_nat @ A3 @ B4 )
=> ( ( ord_le841296385at_nat @ B4 @ A3 )
=> ( A3 = B4 ) ) ) ).
% order_class.order.antisym
thf(fact_337_order__class_Oorder_Oantisym,axiom,
! [A3: set_nat,B4: set_nat] :
( ( ord_less_eq_set_nat @ A3 @ B4 )
=> ( ( ord_less_eq_set_nat @ B4 @ A3 )
=> ( A3 = B4 ) ) ) ).
% order_class.order.antisym
thf(fact_338_order__class_Oorder_Oantisym,axiom,
! [A3: nat,B4: nat] :
( ( ord_less_eq_nat @ A3 @ B4 )
=> ( ( ord_less_eq_nat @ B4 @ A3 )
=> ( A3 = B4 ) ) ) ).
% order_class.order.antisym
thf(fact_339_order__trans,axiom,
! [X3: set_Pr1986765409at_nat,Y3: set_Pr1986765409at_nat,Z: set_Pr1986765409at_nat] :
( ( ord_le841296385at_nat @ X3 @ Y3 )
=> ( ( ord_le841296385at_nat @ Y3 @ Z )
=> ( ord_le841296385at_nat @ X3 @ Z ) ) ) ).
% order_trans
thf(fact_340_order__trans,axiom,
! [X3: set_nat,Y3: set_nat,Z: set_nat] :
( ( ord_less_eq_set_nat @ X3 @ Y3 )
=> ( ( ord_less_eq_set_nat @ Y3 @ Z )
=> ( ord_less_eq_set_nat @ X3 @ Z ) ) ) ).
% order_trans
thf(fact_341_order__trans,axiom,
! [X3: nat,Y3: nat,Z: nat] :
( ( ord_less_eq_nat @ X3 @ Y3 )
=> ( ( ord_less_eq_nat @ Y3 @ Z )
=> ( ord_less_eq_nat @ X3 @ Z ) ) ) ).
% order_trans
thf(fact_342_dual__order_Orefl,axiom,
! [A3: set_Pr1986765409at_nat] : ( ord_le841296385at_nat @ A3 @ A3 ) ).
% dual_order.refl
thf(fact_343_dual__order_Orefl,axiom,
! [A3: set_nat] : ( ord_less_eq_set_nat @ A3 @ A3 ) ).
% dual_order.refl
thf(fact_344_dual__order_Orefl,axiom,
! [A3: nat] : ( ord_less_eq_nat @ A3 @ A3 ) ).
% dual_order.refl
thf(fact_345_linorder__wlog,axiom,
! [P3: nat > nat > $o,A3: nat,B4: nat] :
( ! [A4: nat,B3: nat] :
( ( ord_less_eq_nat @ A4 @ B3 )
=> ( P3 @ A4 @ B3 ) )
=> ( ! [A4: nat,B3: nat] :
( ( P3 @ B3 @ A4 )
=> ( P3 @ A4 @ B3 ) )
=> ( P3 @ A3 @ B4 ) ) ) ).
% linorder_wlog
thf(fact_346_rtc__isPath,axiom,
! [U: nat,V: nat] :
( ( member701585322at_nat @ ( product_Pair_nat_nat @ U @ V ) @ ( transi1645084429cl_nat @ ( e_a @ c ) ) )
=> ? [P2: list_P559422087at_nat] : ( isPath_a @ c @ U @ P2 @ V ) ) ).
% rtc_isPath
thf(fact_347_isSPath__pathLE,axiom,
! [S: nat,P4: list_P559422087at_nat,T3: nat] :
( ( isPath_a @ c @ S @ P4 @ T3 )
=> ? [P5: list_P559422087at_nat] : ( isSimplePath_a @ c @ S @ P5 @ T3 ) ) ).
% isSPath_pathLE
thf(fact_348_connected__def,axiom,
! [U: nat,V: nat] :
( ( connected_a @ c @ U @ V )
= ( ? [P: list_P559422087at_nat] : ( isPath_a @ c @ U @ P @ V ) ) ) ).
% connected_def
thf(fact_349_shortestPath__is__path,axiom,
! [U: nat,P4: list_P559422087at_nat,V: nat] :
( ( isShortestPath_a @ c @ U @ P4 @ V )
=> ( isPath_a @ c @ U @ P4 @ V ) ) ).
% shortestPath_is_path
thf(fact_350_isPath__ex__edge1,axiom,
! [U: nat,P4: list_P559422087at_nat,V: nat,U1: nat,V1: nat] :
( ( isPath_a @ c @ U @ P4 @ V )
=> ( ( member701585322at_nat @ ( product_Pair_nat_nat @ U1 @ V1 ) @ ( set_Pr2131844118at_nat @ P4 ) )
=> ( ( U1 != U )
=> ? [U23: nat] : ( member701585322at_nat @ ( product_Pair_nat_nat @ U23 @ U1 ) @ ( set_Pr2131844118at_nat @ P4 ) ) ) ) ) ).
% isPath_ex_edge1
thf(fact_351_isPath__ex__edge2,axiom,
! [U: nat,P4: list_P559422087at_nat,V: nat,U1: nat,V1: nat] :
( ( isPath_a @ c @ U @ P4 @ V )
=> ( ( member701585322at_nat @ ( product_Pair_nat_nat @ U1 @ V1 ) @ ( set_Pr2131844118at_nat @ P4 ) )
=> ( ( V1 != V )
=> ? [V23: nat] : ( member701585322at_nat @ ( product_Pair_nat_nat @ V1 @ V23 ) @ ( set_Pr2131844118at_nat @ P4 ) ) ) ) ) ).
% isPath_ex_edge2
thf(fact_352_isPath__edgeset,axiom,
! [U: nat,P4: list_P559422087at_nat,V: nat,E2: product_prod_nat_nat] :
( ( isPath_a @ c @ U @ P4 @ V )
=> ( ( member701585322at_nat @ E2 @ ( set_Pr2131844118at_nat @ P4 ) )
=> ( member701585322at_nat @ E2 @ ( e_a @ c ) ) ) ) ).
% isPath_edgeset
thf(fact_353_isPath__rtc,axiom,
! [U: nat,P4: list_P559422087at_nat,V: nat] :
( ( isPath_a @ c @ U @ P4 @ V )
=> ( member701585322at_nat @ ( product_Pair_nat_nat @ U @ V ) @ ( transi1645084429cl_nat @ ( e_a @ c ) ) ) ) ).
% isPath_rtc
% Conjectures (1)
thf(conj_0,conjecture,
finite772653738at_nat @ ( edmond475474835dges_a @ c @ s @ t ) ).
%------------------------------------------------------------------------------