TPTP Problem File: ITP050^2.p
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%------------------------------------------------------------------------------
% File : ITP050^2 : TPTP v8.2.0. Released v7.5.0.
% Domain : Interactive Theorem Proving
% Problem : Sledgehammer EdmondsKarp_Termination_Abstract problem prob_276__7590636_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_276__7590636_1 [Des21]
% Status : Theorem
% Rating : 0.67 v8.1.0, 0.75 v7.5.0
% Syntax : Number of formulae : 334 ( 71 unt; 45 typ; 0 def)
% Number of atoms : 781 ( 179 equ; 0 cnn)
% Maximal formula atoms : 13 ( 2 avg)
% Number of connectives : 4074 ( 66 ~; 3 |; 36 &;3492 @)
% ( 0 <=>; 477 =>; 0 <=; 0 <~>)
% Maximal formula depth : 27 ( 9 avg)
% Number of types : 3 ( 2 usr)
% Number of type conns : 207 ( 207 >; 0 *; 0 +; 0 <<)
% Number of symbols : 45 ( 43 usr; 5 con; 0-5 aty)
% Number of variables : 1016 ( 52 ^; 914 !; 15 ?;1016 :)
% ( 35 !>; 0 ?*; 0 @-; 0 @+)
% SPC : TH1_THM_EQU_NAR
% Comments : This file was generated by Sledgehammer 2021-02-23 16:16:23.670
%------------------------------------------------------------------------------
% Could-be-implicit typings (5)
thf(ty_t_Product__Type_Oprod,type,
product_prod: $tType > $tType > $tType ).
thf(ty_t_List_Olist,type,
list: $tType > $tType ).
thf(ty_t_Set_Oset,type,
set: $tType > $tType ).
thf(ty_t_Nat_Onat,type,
nat: $tType ).
thf(ty_tf_a,type,
a: $tType ).
% Explicit typings (40)
thf(sy_cl_HOL_Otype,type,
type:
!>[A: $tType] : $o ).
thf(sy_cl_Orderings_Oord,type,
ord:
!>[A: $tType] : $o ).
thf(sy_cl_Orderings_Oorder,type,
order:
!>[A: $tType] : $o ).
thf(sy_cl_Lattices_Olattice,type,
lattice:
!>[A: $tType] : $o ).
thf(sy_cl_Finite__Set_Ofinite,type,
finite_finite:
!>[A: $tType] : $o ).
thf(sy_cl_Orderings_Olinorder,type,
linorder:
!>[A: $tType] : $o ).
thf(sy_cl_Orderings_Opreorder,type,
preorder:
!>[A: $tType] : $o ).
thf(sy_cl_Rings_Olinordered__idom,type,
linordered_idom:
!>[A: $tType] : $o ).
thf(sy_cl_Lattices_Osemilattice__inf,type,
semilattice_inf:
!>[A: $tType] : $o ).
thf(sy_c_EdmondsKarp__Termination__Abstract__Mirabelle__klniualtan_Oek__analysis,type,
edmond2129202899alysis:
!>[A: $tType] : ( ( ( product_prod @ nat @ nat ) > A ) > $o ) ).
thf(sy_c_EdmondsKarp__Termination__Abstract__Mirabelle__klniualtan_Oek__analysis__defs_OspEdges,type,
edmond1803015688pEdges:
!>[A: $tType] : ( ( ( product_prod @ nat @ nat ) > A ) > nat > nat > ( set @ ( product_prod @ nat @ nat ) ) ) ).
thf(sy_c_EdmondsKarp__Termination__Abstract__Mirabelle__klniualtan_Oek__analysis__defs_OuE,type,
edmond259086305sis_uE:
!>[A: $tType] : ( ( ( product_prod @ nat @ nat ) > A ) > ( set @ ( product_prod @ nat @ nat ) ) ) ).
thf(sy_c_Finite__Set_Ocard,type,
finite_card:
!>[B: $tType] : ( ( set @ B ) > nat ) ).
thf(sy_c_Finite__Set_Ofinite,type,
finite_finite2:
!>[A: $tType] : ( ( set @ A ) > $o ) ).
thf(sy_c_Graph_OFinite__Graph,type,
finite_Graph:
!>[A: $tType] : ( ( ( product_prod @ nat @ nat ) > A ) > $o ) ).
thf(sy_c_Graph_OGraph_OE,type,
e:
!>[Capacity: $tType] : ( ( ( product_prod @ nat @ nat ) > Capacity ) > ( set @ ( product_prod @ nat @ nat ) ) ) ).
thf(sy_c_Graph_OGraph_OV,type,
v:
!>[Capacity: $tType] : ( ( ( product_prod @ nat @ nat ) > Capacity ) > ( set @ nat ) ) ).
thf(sy_c_Graph_OGraph_Oadjacent__nodes,type,
adjacent_nodes:
!>[Capacity: $tType] : ( ( ( product_prod @ nat @ nat ) > Capacity ) > nat > ( set @ nat ) ) ).
thf(sy_c_Graph_OGraph_Oincoming,type,
incoming:
!>[Capacity: $tType] : ( ( ( product_prod @ nat @ nat ) > Capacity ) > nat > ( set @ ( product_prod @ nat @ nat ) ) ) ).
thf(sy_c_Graph_OGraph_Oincoming_H,type,
incoming2:
!>[Capacity: $tType] : ( ( ( product_prod @ nat @ nat ) > Capacity ) > ( set @ nat ) > ( set @ ( product_prod @ nat @ nat ) ) ) ).
thf(sy_c_Graph_OGraph_OisPath,type,
isPath:
!>[Capacity: $tType] : ( ( ( product_prod @ nat @ nat ) > Capacity ) > nat > ( list @ ( product_prod @ nat @ nat ) ) > nat > $o ) ).
thf(sy_c_Graph_OGraph_OisShortestPath,type,
isShortestPath:
!>[Capacity: $tType] : ( ( ( product_prod @ nat @ nat ) > Capacity ) > nat > ( list @ ( product_prod @ nat @ nat ) ) > nat > $o ) ).
thf(sy_c_Graph_OGraph_Ooutgoing,type,
outgoing:
!>[Capacity: $tType] : ( ( ( product_prod @ nat @ nat ) > Capacity ) > nat > ( set @ ( product_prod @ nat @ nat ) ) ) ).
thf(sy_c_Graph_OGraph_Ooutgoing_H,type,
outgoing2:
!>[Capacity: $tType] : ( ( ( product_prod @ nat @ nat ) > Capacity ) > ( set @ nat ) > ( set @ ( product_prod @ nat @ nat ) ) ) ).
thf(sy_c_Graph_OGraph_OpathVertices,type,
pathVertices: nat > ( list @ ( product_prod @ nat @ nat ) ) > ( list @ nat ) ).
thf(sy_c_Graph_OGraph_OreachableNodes,type,
reachableNodes:
!>[Capacity: $tType] : ( ( ( product_prod @ nat @ nat ) > Capacity ) > nat > ( set @ nat ) ) ).
thf(sy_c_Lattices_Oinf__class_Oinf,type,
inf_inf:
!>[A: $tType] : ( A > A > A ) ).
thf(sy_c_List_Olist_OCons,type,
cons:
!>[A: $tType] : ( A > ( list @ A ) > ( list @ A ) ) ).
thf(sy_c_List_Olist_ONil,type,
nil:
!>[A: $tType] : ( list @ A ) ).
thf(sy_c_List_Olist_Oset,type,
set2:
!>[A: $tType] : ( ( list @ A ) > ( set @ A ) ) ).
thf(sy_c_Nat_Osize__class_Osize,type,
size_size:
!>[A: $tType] : ( A > nat ) ).
thf(sy_c_Orderings_Oord__class_Oless__eq,type,
ord_less_eq:
!>[A: $tType] : ( A > A > $o ) ).
thf(sy_c_Product__Type_OPair,type,
product_Pair:
!>[A: $tType,B: $tType] : ( A > B > ( product_prod @ A @ B ) ) ).
thf(sy_c_Set_OCollect,type,
collect:
!>[A: $tType] : ( ( A > $o ) > ( set @ A ) ) ).
thf(sy_c_member,type,
member:
!>[A: $tType] : ( A > ( set @ A ) > $o ) ).
thf(sy_v_c,type,
c: ( product_prod @ nat @ nat ) > a ).
thf(sy_v_edges,type,
edges: set @ ( product_prod @ nat @ nat ) ).
thf(sy_v_p,type,
p: list @ ( product_prod @ nat @ nat ) ).
thf(sy_v_s,type,
s: nat ).
thf(sy_v_t,type,
t: nat ).
% Relevant facts (256)
thf(fact_0_SP,axiom,
isShortestPath @ a @ c @ s @ p @ t ).
% SP
thf(fact_1_SP__EDGES,axiom,
ord_less_eq @ ( set @ ( product_prod @ nat @ nat ) ) @ edges @ ( set2 @ ( product_prod @ nat @ nat ) @ p ) ).
% SP_EDGES
thf(fact_2_ek__analysis__axioms,axiom,
edmond2129202899alysis @ a @ c ).
% ek_analysis_axioms
thf(fact_3_incoming_H__edges,axiom,
! [U: set @ nat] : ( ord_less_eq @ ( set @ ( product_prod @ nat @ nat ) ) @ ( incoming2 @ a @ c @ U ) @ ( e @ a @ c ) ) ).
% incoming'_edges
thf(fact_4_outgoing_H__edges,axiom,
! [U: set @ nat] : ( ord_less_eq @ ( set @ ( product_prod @ nat @ nat ) ) @ ( outgoing2 @ a @ c @ U ) @ ( e @ a @ c ) ) ).
% outgoing'_edges
thf(fact_5_E__ss__uE,axiom,
ord_less_eq @ ( set @ ( product_prod @ nat @ nat ) ) @ ( e @ a @ c ) @ ( edmond259086305sis_uE @ a @ c ) ).
% E_ss_uE
thf(fact_6_incoming__edges,axiom,
! [U2: nat] : ( ord_less_eq @ ( set @ ( product_prod @ nat @ nat ) ) @ ( incoming @ a @ c @ U2 ) @ ( e @ a @ c ) ) ).
% incoming_edges
thf(fact_7_outgoing__edges,axiom,
! [U2: nat] : ( ord_less_eq @ ( set @ ( product_prod @ nat @ nat ) ) @ ( outgoing @ a @ c @ U2 ) @ ( e @ a @ c ) ) ).
% outgoing_edges
thf(fact_8_Finite__Graph__axioms,axiom,
finite_Graph @ a @ c ).
% Finite_Graph_axioms
thf(fact_9_subsetI,axiom,
! [A: $tType,A2: set @ A,B2: set @ A] :
( ! [X: A] :
( ( member @ A @ X @ A2 )
=> ( member @ A @ X @ B2 ) )
=> ( ord_less_eq @ ( set @ A ) @ A2 @ B2 ) ) ).
% subsetI
thf(fact_10_subset__antisym,axiom,
! [A: $tType,A2: set @ A,B2: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ A2 @ B2 )
=> ( ( ord_less_eq @ ( set @ A ) @ B2 @ A2 )
=> ( A2 = B2 ) ) ) ).
% subset_antisym
thf(fact_11_order__refl,axiom,
! [A: $tType] :
( ( preorder @ A )
=> ! [X2: A] : ( ord_less_eq @ A @ X2 @ X2 ) ) ).
% order_refl
thf(fact_12_SV,axiom,
member @ nat @ s @ ( v @ a @ c ) ).
% SV
thf(fact_13_spEdges__ss__E,axiom,
ord_less_eq @ ( set @ ( product_prod @ nat @ nat ) ) @ ( edmond1803015688pEdges @ a @ c @ s @ t ) @ ( e @ a @ c ) ).
% spEdges_ss_E
thf(fact_14_isPath__edgeset,axiom,
! [U2: nat,P: list @ ( product_prod @ nat @ nat ),V: nat,E: product_prod @ nat @ nat] :
( ( isPath @ a @ c @ U2 @ P @ V )
=> ( ( member @ ( product_prod @ nat @ nat ) @ E @ ( set2 @ ( product_prod @ nat @ nat ) @ P ) )
=> ( member @ ( product_prod @ nat @ nat ) @ E @ ( e @ a @ c ) ) ) ) ).
% isPath_edgeset
thf(fact_15_shortestPath__is__path,axiom,
! [U2: nat,P: list @ ( product_prod @ nat @ nat ),V: nat] :
( ( isShortestPath @ a @ c @ U2 @ P @ V )
=> ( isPath @ a @ c @ U2 @ P @ V ) ) ).
% shortestPath_is_path
thf(fact_16_adjacent__nodes__ss__V,axiom,
! [U2: nat] : ( ord_less_eq @ ( set @ nat ) @ ( adjacent_nodes @ a @ c @ U2 ) @ ( v @ a @ c ) ) ).
% adjacent_nodes_ss_V
thf(fact_17_ek__analysis__def,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ( ( edmond2129202899alysis @ A )
= ( finite_Graph @ A ) ) ) ).
% ek_analysis_def
thf(fact_18_ek__analysis_Ointro,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [C: ( product_prod @ nat @ nat ) > A] :
( ( finite_Graph @ A @ C )
=> ( edmond2129202899alysis @ A @ C ) ) ) ).
% ek_analysis.intro
thf(fact_19_ek__analysis_Oaxioms,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [C: ( product_prod @ nat @ nat ) > A] :
( ( edmond2129202899alysis @ A @ C )
=> ( finite_Graph @ A @ C ) ) ) ).
% ek_analysis.axioms
thf(fact_20_ek__analysis__defs_OspEdges_Ocong,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ( ( edmond1803015688pEdges @ A )
= ( edmond1803015688pEdges @ A ) ) ) ).
% ek_analysis_defs.spEdges.cong
thf(fact_21_ek__analysis__defs_OuE_Ocong,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ( ( edmond259086305sis_uE @ A )
= ( edmond259086305sis_uE @ A ) ) ) ).
% ek_analysis_defs.uE.cong
thf(fact_22_ek__analysis_OspEdges__ss__E,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [C: ( product_prod @ nat @ nat ) > A,S: nat,T: nat] :
( ( edmond2129202899alysis @ A @ C )
=> ( ord_less_eq @ ( set @ ( product_prod @ nat @ nat ) ) @ ( edmond1803015688pEdges @ A @ C @ S @ T ) @ ( e @ A @ C ) ) ) ) ).
% ek_analysis.spEdges_ss_E
thf(fact_23_ek__analysis_OE__ss__uE,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [C: ( product_prod @ nat @ nat ) > A] :
( ( edmond2129202899alysis @ A @ C )
=> ( ord_less_eq @ ( set @ ( product_prod @ nat @ nat ) ) @ ( e @ A @ C ) @ ( edmond259086305sis_uE @ A @ C ) ) ) ) ).
% ek_analysis.E_ss_uE
thf(fact_24_dual__order_Oantisym,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [B3: A,A3: A] :
( ( ord_less_eq @ A @ B3 @ A3 )
=> ( ( ord_less_eq @ A @ A3 @ B3 )
=> ( A3 = B3 ) ) ) ) ).
% dual_order.antisym
thf(fact_25_dual__order_Oeq__iff,axiom,
! [A: $tType] :
( ( order @ A )
=> ( ( ^ [Y: A,Z: A] : Y = Z )
= ( ^ [A4: A,B4: A] :
( ( ord_less_eq @ A @ B4 @ A4 )
& ( ord_less_eq @ A @ A4 @ B4 ) ) ) ) ) ).
% dual_order.eq_iff
thf(fact_26_dual__order_Otrans,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [B3: A,A3: A,C: A] :
( ( ord_less_eq @ A @ B3 @ A3 )
=> ( ( ord_less_eq @ A @ C @ B3 )
=> ( ord_less_eq @ A @ C @ A3 ) ) ) ) ).
% dual_order.trans
thf(fact_27_linorder__wlog,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [P2: A > A > $o,A3: A,B3: A] :
( ! [A5: A,B5: A] :
( ( ord_less_eq @ A @ A5 @ B5 )
=> ( P2 @ A5 @ B5 ) )
=> ( ! [A5: A,B5: A] :
( ( P2 @ B5 @ A5 )
=> ( P2 @ A5 @ B5 ) )
=> ( P2 @ A3 @ B3 ) ) ) ) ).
% linorder_wlog
thf(fact_28_dual__order_Orefl,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [A3: A] : ( ord_less_eq @ A @ A3 @ A3 ) ) ).
% dual_order.refl
thf(fact_29_order__trans,axiom,
! [A: $tType] :
( ( preorder @ A )
=> ! [X2: A,Y2: A,Z2: A] :
( ( ord_less_eq @ A @ X2 @ Y2 )
=> ( ( ord_less_eq @ A @ Y2 @ Z2 )
=> ( ord_less_eq @ A @ X2 @ Z2 ) ) ) ) ).
% order_trans
thf(fact_30_order__class_Oorder_Oantisym,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [A3: A,B3: A] :
( ( ord_less_eq @ A @ A3 @ B3 )
=> ( ( ord_less_eq @ A @ B3 @ A3 )
=> ( A3 = B3 ) ) ) ) ).
% order_class.order.antisym
thf(fact_31_ord__le__eq__trans,axiom,
! [A: $tType] :
( ( ord @ A )
=> ! [A3: A,B3: A,C: A] :
( ( ord_less_eq @ A @ A3 @ B3 )
=> ( ( B3 = C )
=> ( ord_less_eq @ A @ A3 @ C ) ) ) ) ).
% ord_le_eq_trans
thf(fact_32_ord__eq__le__trans,axiom,
! [A: $tType] :
( ( ord @ A )
=> ! [A3: A,B3: A,C: A] :
( ( A3 = B3 )
=> ( ( ord_less_eq @ A @ B3 @ C )
=> ( ord_less_eq @ A @ A3 @ C ) ) ) ) ).
% ord_eq_le_trans
thf(fact_33_order__class_Oorder_Oeq__iff,axiom,
! [A: $tType] :
( ( order @ A )
=> ( ( ^ [Y: A,Z: A] : Y = Z )
= ( ^ [A4: A,B4: A] :
( ( ord_less_eq @ A @ A4 @ B4 )
& ( ord_less_eq @ A @ B4 @ A4 ) ) ) ) ) ).
% order_class.order.eq_iff
thf(fact_34_antisym__conv,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [Y2: A,X2: A] :
( ( ord_less_eq @ A @ Y2 @ X2 )
=> ( ( ord_less_eq @ A @ X2 @ Y2 )
= ( X2 = Y2 ) ) ) ) ).
% antisym_conv
thf(fact_35_le__cases3,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [X2: A,Y2: A,Z2: A] :
( ( ( ord_less_eq @ A @ X2 @ Y2 )
=> ~ ( ord_less_eq @ A @ Y2 @ Z2 ) )
=> ( ( ( ord_less_eq @ A @ Y2 @ X2 )
=> ~ ( ord_less_eq @ A @ X2 @ Z2 ) )
=> ( ( ( ord_less_eq @ A @ X2 @ Z2 )
=> ~ ( ord_less_eq @ A @ Z2 @ Y2 ) )
=> ( ( ( ord_less_eq @ A @ Z2 @ Y2 )
=> ~ ( ord_less_eq @ A @ Y2 @ X2 ) )
=> ( ( ( ord_less_eq @ A @ Y2 @ Z2 )
=> ~ ( ord_less_eq @ A @ Z2 @ X2 ) )
=> ~ ( ( ord_less_eq @ A @ Z2 @ X2 )
=> ~ ( ord_less_eq @ A @ X2 @ Y2 ) ) ) ) ) ) ) ) ).
% le_cases3
thf(fact_36_order_Otrans,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [A3: A,B3: A,C: A] :
( ( ord_less_eq @ A @ A3 @ B3 )
=> ( ( ord_less_eq @ A @ B3 @ C )
=> ( ord_less_eq @ A @ A3 @ C ) ) ) ) ).
% order.trans
thf(fact_37_le__cases,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [X2: A,Y2: A] :
( ~ ( ord_less_eq @ A @ X2 @ Y2 )
=> ( ord_less_eq @ A @ Y2 @ X2 ) ) ) ).
% le_cases
thf(fact_38_eq__refl,axiom,
! [A: $tType] :
( ( preorder @ A )
=> ! [X2: A,Y2: A] :
( ( X2 = Y2 )
=> ( ord_less_eq @ A @ X2 @ Y2 ) ) ) ).
% eq_refl
thf(fact_39_linear,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [X2: A,Y2: A] :
( ( ord_less_eq @ A @ X2 @ Y2 )
| ( ord_less_eq @ A @ Y2 @ X2 ) ) ) ).
% linear
thf(fact_40_antisym,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [X2: A,Y2: A] :
( ( ord_less_eq @ A @ X2 @ Y2 )
=> ( ( ord_less_eq @ A @ Y2 @ X2 )
=> ( X2 = Y2 ) ) ) ) ).
% antisym
thf(fact_41_eq__iff,axiom,
! [A: $tType] :
( ( order @ A )
=> ( ( ^ [Y: A,Z: A] : Y = Z )
= ( ^ [X3: A,Y3: A] :
( ( ord_less_eq @ A @ X3 @ Y3 )
& ( ord_less_eq @ A @ Y3 @ X3 ) ) ) ) ) ).
% eq_iff
thf(fact_42_ord__le__eq__subst,axiom,
! [A: $tType,B: $tType] :
( ( ( ord @ B )
& ( ord @ A ) )
=> ! [A3: A,B3: A,F: A > B,C: B] :
( ( ord_less_eq @ A @ A3 @ B3 )
=> ( ( ( F @ B3 )
= C )
=> ( ! [X: A,Y4: A] :
( ( ord_less_eq @ A @ X @ Y4 )
=> ( ord_less_eq @ B @ ( F @ X ) @ ( F @ Y4 ) ) )
=> ( ord_less_eq @ B @ ( F @ A3 ) @ C ) ) ) ) ) ).
% ord_le_eq_subst
thf(fact_43_ord__eq__le__subst,axiom,
! [A: $tType,B: $tType] :
( ( ( ord @ B )
& ( ord @ A ) )
=> ! [A3: A,F: B > A,B3: B,C: B] :
( ( A3
= ( F @ B3 ) )
=> ( ( ord_less_eq @ B @ B3 @ C )
=> ( ! [X: B,Y4: B] :
( ( ord_less_eq @ B @ X @ Y4 )
=> ( ord_less_eq @ A @ ( F @ X ) @ ( F @ Y4 ) ) )
=> ( ord_less_eq @ A @ A3 @ ( F @ C ) ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_44_order__subst2,axiom,
! [A: $tType,C2: $tType] :
( ( ( order @ C2 )
& ( order @ A ) )
=> ! [A3: A,B3: A,F: A > C2,C: C2] :
( ( ord_less_eq @ A @ A3 @ B3 )
=> ( ( ord_less_eq @ C2 @ ( F @ B3 ) @ C )
=> ( ! [X: A,Y4: A] :
( ( ord_less_eq @ A @ X @ Y4 )
=> ( ord_less_eq @ C2 @ ( F @ X ) @ ( F @ Y4 ) ) )
=> ( ord_less_eq @ C2 @ ( F @ A3 ) @ C ) ) ) ) ) ).
% order_subst2
thf(fact_45_mem__Collect__eq,axiom,
! [A: $tType,A3: A,P2: A > $o] :
( ( member @ A @ A3 @ ( collect @ A @ P2 ) )
= ( P2 @ A3 ) ) ).
% mem_Collect_eq
thf(fact_46_Collect__mem__eq,axiom,
! [A: $tType,A2: set @ A] :
( ( collect @ A
@ ^ [X3: A] : ( member @ A @ X3 @ A2 ) )
= A2 ) ).
% Collect_mem_eq
thf(fact_47_Collect__cong,axiom,
! [A: $tType,P2: A > $o,Q: A > $o] :
( ! [X: A] :
( ( P2 @ X )
= ( Q @ X ) )
=> ( ( collect @ A @ P2 )
= ( collect @ A @ Q ) ) ) ).
% Collect_cong
thf(fact_48_ext,axiom,
! [B: $tType,A: $tType,F: A > B,G: A > B] :
( ! [X: A] :
( ( F @ X )
= ( G @ X ) )
=> ( F = G ) ) ).
% ext
thf(fact_49_order__subst1,axiom,
! [A: $tType,B: $tType] :
( ( ( order @ B )
& ( order @ A ) )
=> ! [A3: A,F: B > A,B3: B,C: B] :
( ( ord_less_eq @ A @ A3 @ ( F @ B3 ) )
=> ( ( ord_less_eq @ B @ B3 @ C )
=> ( ! [X: B,Y4: B] :
( ( ord_less_eq @ B @ X @ Y4 )
=> ( ord_less_eq @ A @ ( F @ X ) @ ( F @ Y4 ) ) )
=> ( ord_less_eq @ A @ A3 @ ( F @ C ) ) ) ) ) ) ).
% order_subst1
thf(fact_50_le__fun__def,axiom,
! [B: $tType,A: $tType] :
( ( ord @ B )
=> ( ( ord_less_eq @ ( A > B ) )
= ( ^ [F2: A > B,G2: A > B] :
! [X3: A] : ( ord_less_eq @ B @ ( F2 @ X3 ) @ ( G2 @ X3 ) ) ) ) ) ).
% le_fun_def
thf(fact_51_le__funI,axiom,
! [B: $tType,A: $tType] :
( ( ord @ B )
=> ! [F: A > B,G: A > B] :
( ! [X: A] : ( ord_less_eq @ B @ ( F @ X ) @ ( G @ X ) )
=> ( ord_less_eq @ ( A > B ) @ F @ G ) ) ) ).
% le_funI
thf(fact_52_le__funE,axiom,
! [B: $tType,A: $tType] :
( ( ord @ B )
=> ! [F: A > B,G: A > B,X2: A] :
( ( ord_less_eq @ ( A > B ) @ F @ G )
=> ( ord_less_eq @ B @ ( F @ X2 ) @ ( G @ X2 ) ) ) ) ).
% le_funE
thf(fact_53_le__funD,axiom,
! [B: $tType,A: $tType] :
( ( ord @ B )
=> ! [F: A > B,G: A > B,X2: A] :
( ( ord_less_eq @ ( A > B ) @ F @ G )
=> ( ord_less_eq @ B @ ( F @ X2 ) @ ( G @ X2 ) ) ) ) ).
% le_funD
thf(fact_54_Collect__mono__iff,axiom,
! [A: $tType,P2: A > $o,Q: A > $o] :
( ( ord_less_eq @ ( set @ A ) @ ( collect @ A @ P2 ) @ ( collect @ A @ Q ) )
= ( ! [X3: A] :
( ( P2 @ X3 )
=> ( Q @ X3 ) ) ) ) ).
% Collect_mono_iff
thf(fact_55_set__eq__subset,axiom,
! [A: $tType] :
( ( ^ [Y: set @ A,Z: set @ A] : Y = Z )
= ( ^ [A6: set @ A,B6: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ A6 @ B6 )
& ( ord_less_eq @ ( set @ A ) @ B6 @ A6 ) ) ) ) ).
% set_eq_subset
thf(fact_56_subset__trans,axiom,
! [A: $tType,A2: set @ A,B2: set @ A,C3: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ A2 @ B2 )
=> ( ( ord_less_eq @ ( set @ A ) @ B2 @ C3 )
=> ( ord_less_eq @ ( set @ A ) @ A2 @ C3 ) ) ) ).
% subset_trans
thf(fact_57_Collect__mono,axiom,
! [A: $tType,P2: A > $o,Q: A > $o] :
( ! [X: A] :
( ( P2 @ X )
=> ( Q @ X ) )
=> ( ord_less_eq @ ( set @ A ) @ ( collect @ A @ P2 ) @ ( collect @ A @ Q ) ) ) ).
% Collect_mono
thf(fact_58_subset__refl,axiom,
! [A: $tType,A2: set @ A] : ( ord_less_eq @ ( set @ A ) @ A2 @ A2 ) ).
% subset_refl
thf(fact_59_subset__iff,axiom,
! [A: $tType] :
( ( ord_less_eq @ ( set @ A ) )
= ( ^ [A6: set @ A,B6: set @ A] :
! [T2: A] :
( ( member @ A @ T2 @ A6 )
=> ( member @ A @ T2 @ B6 ) ) ) ) ).
% subset_iff
thf(fact_60_equalityD2,axiom,
! [A: $tType,A2: set @ A,B2: set @ A] :
( ( A2 = B2 )
=> ( ord_less_eq @ ( set @ A ) @ B2 @ A2 ) ) ).
% equalityD2
thf(fact_61_equalityD1,axiom,
! [A: $tType,A2: set @ A,B2: set @ A] :
( ( A2 = B2 )
=> ( ord_less_eq @ ( set @ A ) @ A2 @ B2 ) ) ).
% equalityD1
thf(fact_62_subset__eq,axiom,
! [A: $tType] :
( ( ord_less_eq @ ( set @ A ) )
= ( ^ [A6: set @ A,B6: set @ A] :
! [X3: A] :
( ( member @ A @ X3 @ A6 )
=> ( member @ A @ X3 @ B6 ) ) ) ) ).
% subset_eq
thf(fact_63_equalityE,axiom,
! [A: $tType,A2: set @ A,B2: set @ A] :
( ( A2 = B2 )
=> ~ ( ( ord_less_eq @ ( set @ A ) @ A2 @ B2 )
=> ~ ( ord_less_eq @ ( set @ A ) @ B2 @ A2 ) ) ) ).
% equalityE
thf(fact_64_subsetD,axiom,
! [A: $tType,A2: set @ A,B2: set @ A,C: A] :
( ( ord_less_eq @ ( set @ A ) @ A2 @ B2 )
=> ( ( member @ A @ C @ A2 )
=> ( member @ A @ C @ B2 ) ) ) ).
% subsetD
thf(fact_65_in__mono,axiom,
! [A: $tType,A2: set @ A,B2: set @ A,X2: A] :
( ( ord_less_eq @ ( set @ A ) @ A2 @ B2 )
=> ( ( member @ A @ X2 @ A2 )
=> ( member @ A @ X2 @ B2 ) ) ) ).
% in_mono
thf(fact_66_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_67_transfer__path,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [P: list @ ( product_prod @ nat @ nat ),C4: ( product_prod @ nat @ nat ) > A,U2: nat,V: nat] :
( ( ord_less_eq @ ( set @ ( product_prod @ nat @ nat ) ) @ ( inf_inf @ ( set @ ( product_prod @ nat @ nat ) ) @ ( set2 @ ( product_prod @ nat @ nat ) @ P ) @ ( e @ a @ c ) ) @ ( e @ A @ C4 ) )
=> ( ( isPath @ a @ c @ U2 @ P @ V )
=> ( isPath @ A @ C4 @ U2 @ P @ V ) ) ) ) ).
% transfer_path
thf(fact_68_card__spEdges__le,axiom,
ord_less_eq @ nat @ ( finite_card @ ( product_prod @ nat @ nat ) @ ( edmond1803015688pEdges @ a @ c @ s @ t ) ) @ ( finite_card @ ( product_prod @ nat @ nat ) @ ( edmond259086305sis_uE @ a @ c ) ) ).
% card_spEdges_le
thf(fact_69_Finite__Graph__EI,axiom,
( ( finite_finite2 @ ( product_prod @ nat @ nat ) @ ( e @ a @ c ) )
=> ( finite_Graph @ a @ c ) ) ).
% Finite_Graph_EI
thf(fact_70_Graph_Oincoming_H__edges,axiom,
! [Capacity: $tType] :
( ( linordered_idom @ Capacity )
=> ! [C: ( product_prod @ nat @ nat ) > Capacity,U: set @ nat] : ( ord_less_eq @ ( set @ ( product_prod @ nat @ nat ) ) @ ( incoming2 @ Capacity @ C @ U ) @ ( e @ Capacity @ C ) ) ) ).
% Graph.incoming'_edges
thf(fact_71_Graph_Ooutgoing_H__edges,axiom,
! [Capacity: $tType] :
( ( linordered_idom @ Capacity )
=> ! [C: ( product_prod @ nat @ nat ) > Capacity,U: set @ nat] : ( ord_less_eq @ ( set @ ( product_prod @ nat @ nat ) ) @ ( outgoing2 @ Capacity @ C @ U ) @ ( e @ Capacity @ C ) ) ) ).
% Graph.outgoing'_edges
thf(fact_72_Graph_Oincoming__edges,axiom,
! [Capacity: $tType] :
( ( linordered_idom @ Capacity )
=> ! [C: ( product_prod @ nat @ nat ) > Capacity,U2: nat] : ( ord_less_eq @ ( set @ ( product_prod @ nat @ nat ) ) @ ( incoming @ Capacity @ C @ U2 ) @ ( e @ Capacity @ C ) ) ) ).
% Graph.incoming_edges
thf(fact_73_Graph_Ooutgoing__edges,axiom,
! [Capacity: $tType] :
( ( linordered_idom @ Capacity )
=> ! [C: ( product_prod @ nat @ nat ) > Capacity,U2: nat] : ( ord_less_eq @ ( set @ ( product_prod @ nat @ nat ) ) @ ( outgoing @ Capacity @ C @ U2 ) @ ( e @ Capacity @ C ) ) ) ).
% Graph.outgoing_edges
thf(fact_74_Graph_OisPath__edgeset,axiom,
! [Capacity: $tType] :
( ( linordered_idom @ Capacity )
=> ! [C: ( product_prod @ nat @ nat ) > Capacity,U2: nat,P: list @ ( product_prod @ nat @ nat ),V: nat,E: product_prod @ nat @ nat] :
( ( isPath @ Capacity @ C @ U2 @ P @ V )
=> ( ( member @ ( product_prod @ nat @ nat ) @ E @ ( set2 @ ( product_prod @ nat @ nat ) @ P ) )
=> ( member @ ( product_prod @ nat @ nat ) @ E @ ( e @ Capacity @ C ) ) ) ) ) ).
% Graph.isPath_edgeset
thf(fact_75_isPath__ex__edge2,axiom,
! [U2: nat,P: list @ ( product_prod @ nat @ nat ),V: nat,U1: nat,V1: nat] :
( ( isPath @ a @ c @ U2 @ P @ V )
=> ( ( member @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ U1 @ V1 ) @ ( set2 @ ( product_prod @ nat @ nat ) @ P ) )
=> ( ( V1 != V )
=> ? [V2: nat] : ( member @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ V1 @ V2 ) @ ( set2 @ ( product_prod @ nat @ nat ) @ P ) ) ) ) ) ).
% isPath_ex_edge2
thf(fact_76_IntI,axiom,
! [A: $tType,C: A,A2: set @ A,B2: set @ A] :
( ( member @ A @ C @ A2 )
=> ( ( member @ A @ C @ B2 )
=> ( member @ A @ C @ ( inf_inf @ ( set @ A ) @ A2 @ B2 ) ) ) ) ).
% IntI
thf(fact_77_Int__iff,axiom,
! [A: $tType,C: A,A2: set @ A,B2: set @ A] :
( ( member @ A @ C @ ( inf_inf @ ( set @ A ) @ A2 @ B2 ) )
= ( ( member @ A @ C @ A2 )
& ( member @ A @ C @ B2 ) ) ) ).
% Int_iff
thf(fact_78_isPath__ex__edge1,axiom,
! [U2: nat,P: list @ ( product_prod @ nat @ nat ),V: nat,U1: nat,V1: nat] :
( ( isPath @ a @ c @ U2 @ P @ V )
=> ( ( member @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ U1 @ V1 ) @ ( set2 @ ( product_prod @ nat @ nat ) @ P ) )
=> ( ( U1 != U2 )
=> ? [U22: nat] : ( member @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ U22 @ U1 ) @ ( set2 @ ( product_prod @ nat @ nat ) @ P ) ) ) ) ) ).
% isPath_ex_edge1
thf(fact_79_reachableNodes__append__edge,axiom,
! [U2: nat,S: nat,V: nat] :
( ( member @ nat @ U2 @ ( reachableNodes @ a @ c @ S ) )
=> ( ( member @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ U2 @ V ) @ ( e @ a @ c ) )
=> ( member @ nat @ V @ ( reachableNodes @ a @ c @ S ) ) ) ) ).
% reachableNodes_append_edge
thf(fact_80_Int__subset__iff,axiom,
! [A: $tType,C3: set @ A,A2: set @ A,B2: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ C3 @ ( inf_inf @ ( set @ A ) @ A2 @ B2 ) )
= ( ( ord_less_eq @ ( set @ A ) @ C3 @ A2 )
& ( ord_less_eq @ ( set @ A ) @ C3 @ B2 ) ) ) ).
% Int_subset_iff
thf(fact_81_finite__E,axiom,
finite_finite2 @ ( product_prod @ nat @ nat ) @ ( e @ a @ c ) ).
% finite_E
thf(fact_82_finite__uE,axiom,
finite_finite2 @ ( product_prod @ nat @ nat ) @ ( edmond259086305sis_uE @ a @ c ) ).
% finite_uE
thf(fact_83_finite__spEdges,axiom,
finite_finite2 @ ( product_prod @ nat @ nat ) @ ( edmond1803015688pEdges @ a @ c @ s @ t ) ).
% finite_spEdges
thf(fact_84_IntE,axiom,
! [A: $tType,C: A,A2: set @ A,B2: set @ A] :
( ( member @ A @ C @ ( inf_inf @ ( set @ A ) @ A2 @ B2 ) )
=> ~ ( ( member @ A @ C @ A2 )
=> ~ ( member @ A @ C @ B2 ) ) ) ).
% IntE
thf(fact_85_IntD1,axiom,
! [A: $tType,C: A,A2: set @ A,B2: set @ A] :
( ( member @ A @ C @ ( inf_inf @ ( set @ A ) @ A2 @ B2 ) )
=> ( member @ A @ C @ A2 ) ) ).
% IntD1
thf(fact_86_IntD2,axiom,
! [A: $tType,C: A,A2: set @ A,B2: set @ A] :
( ( member @ A @ C @ ( inf_inf @ ( set @ A ) @ A2 @ B2 ) )
=> ( member @ A @ C @ B2 ) ) ).
% IntD2
thf(fact_87_Int__assoc,axiom,
! [A: $tType,A2: set @ A,B2: set @ A,C3: set @ A] :
( ( inf_inf @ ( set @ A ) @ ( inf_inf @ ( set @ A ) @ A2 @ B2 ) @ C3 )
= ( inf_inf @ ( set @ A ) @ A2 @ ( inf_inf @ ( set @ A ) @ B2 @ C3 ) ) ) ).
% Int_assoc
thf(fact_88_Int__absorb,axiom,
! [A: $tType,A2: set @ A] :
( ( inf_inf @ ( set @ A ) @ A2 @ A2 )
= A2 ) ).
% Int_absorb
thf(fact_89_Int__commute,axiom,
! [A: $tType] :
( ( inf_inf @ ( set @ A ) )
= ( ^ [A6: set @ A,B6: set @ A] : ( inf_inf @ ( set @ A ) @ B6 @ A6 ) ) ) ).
% Int_commute
thf(fact_90_Int__left__absorb,axiom,
! [A: $tType,A2: set @ A,B2: set @ A] :
( ( inf_inf @ ( set @ A ) @ A2 @ ( inf_inf @ ( set @ A ) @ A2 @ B2 ) )
= ( inf_inf @ ( set @ A ) @ A2 @ B2 ) ) ).
% Int_left_absorb
thf(fact_91_Int__left__commute,axiom,
! [A: $tType,A2: set @ A,B2: set @ A,C3: set @ A] :
( ( inf_inf @ ( set @ A ) @ A2 @ ( inf_inf @ ( set @ A ) @ B2 @ C3 ) )
= ( inf_inf @ ( set @ A ) @ B2 @ ( inf_inf @ ( set @ A ) @ A2 @ C3 ) ) ) ).
% Int_left_commute
thf(fact_92_Graph_OreachableNodes__append__edge,axiom,
! [Capacity: $tType] :
( ( linordered_idom @ Capacity )
=> ! [U2: nat,C: ( product_prod @ nat @ nat ) > Capacity,S: nat,V: nat] :
( ( member @ nat @ U2 @ ( reachableNodes @ Capacity @ C @ S ) )
=> ( ( member @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ U2 @ V ) @ ( e @ Capacity @ C ) )
=> ( member @ nat @ V @ ( reachableNodes @ Capacity @ C @ S ) ) ) ) ) ).
% Graph.reachableNodes_append_edge
thf(fact_93_Graph_OreachableNodes_Ocong,axiom,
! [Capacity: $tType] :
( ( linordered_idom @ Capacity )
=> ( ( reachableNodes @ Capacity )
= ( reachableNodes @ Capacity ) ) ) ).
% Graph.reachableNodes.cong
thf(fact_94_Graph_Oadjacent__nodes_Ocong,axiom,
! [Capacity: $tType] :
( ( linordered_idom @ Capacity )
=> ( ( adjacent_nodes @ Capacity )
= ( adjacent_nodes @ Capacity ) ) ) ).
% Graph.adjacent_nodes.cong
thf(fact_95_Graph_Oreachable__ss__V,axiom,
! [Capacity: $tType] :
( ( linordered_idom @ Capacity )
=> ! [S: nat,C: ( product_prod @ nat @ nat ) > Capacity] :
( ( member @ nat @ S @ ( v @ Capacity @ C ) )
=> ( ord_less_eq @ ( set @ nat ) @ ( reachableNodes @ Capacity @ C @ S ) @ ( v @ Capacity @ C ) ) ) ) ).
% Graph.reachable_ss_V
thf(fact_96_Int__mono,axiom,
! [A: $tType,A2: set @ A,C3: set @ A,B2: set @ A,D: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ A2 @ C3 )
=> ( ( ord_less_eq @ ( set @ A ) @ B2 @ D )
=> ( ord_less_eq @ ( set @ A ) @ ( inf_inf @ ( set @ A ) @ A2 @ B2 ) @ ( inf_inf @ ( set @ A ) @ C3 @ D ) ) ) ) ).
% Int_mono
thf(fact_97_Int__lower1,axiom,
! [A: $tType,A2: set @ A,B2: set @ A] : ( ord_less_eq @ ( set @ A ) @ ( inf_inf @ ( set @ A ) @ A2 @ B2 ) @ A2 ) ).
% Int_lower1
thf(fact_98_Int__lower2,axiom,
! [A: $tType,A2: set @ A,B2: set @ A] : ( ord_less_eq @ ( set @ A ) @ ( inf_inf @ ( set @ A ) @ A2 @ B2 ) @ B2 ) ).
% Int_lower2
thf(fact_99_Int__absorb1,axiom,
! [A: $tType,B2: set @ A,A2: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ B2 @ A2 )
=> ( ( inf_inf @ ( set @ A ) @ A2 @ B2 )
= B2 ) ) ).
% Int_absorb1
thf(fact_100_Int__absorb2,axiom,
! [A: $tType,A2: set @ A,B2: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ A2 @ B2 )
=> ( ( inf_inf @ ( set @ A ) @ A2 @ B2 )
= A2 ) ) ).
% Int_absorb2
thf(fact_101_Int__greatest,axiom,
! [A: $tType,C3: set @ A,A2: set @ A,B2: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ C3 @ A2 )
=> ( ( ord_less_eq @ ( set @ A ) @ C3 @ B2 )
=> ( ord_less_eq @ ( set @ A ) @ C3 @ ( inf_inf @ ( set @ A ) @ A2 @ B2 ) ) ) ) ).
% Int_greatest
thf(fact_102_Int__Collect__mono,axiom,
! [A: $tType,A2: set @ A,B2: set @ A,P2: A > $o,Q: A > $o] :
( ( ord_less_eq @ ( set @ A ) @ A2 @ B2 )
=> ( ! [X: A] :
( ( member @ A @ X @ A2 )
=> ( ( P2 @ X )
=> ( Q @ X ) ) )
=> ( ord_less_eq @ ( set @ A ) @ ( inf_inf @ ( set @ A ) @ A2 @ ( collect @ A @ P2 ) ) @ ( inf_inf @ ( set @ A ) @ B2 @ ( collect @ A @ Q ) ) ) ) ) ).
% Int_Collect_mono
thf(fact_103_Graph_Oadjacent__nodes__ss__V,axiom,
! [Capacity: $tType] :
( ( linordered_idom @ Capacity )
=> ! [C: ( product_prod @ nat @ nat ) > Capacity,U2: nat] : ( ord_less_eq @ ( set @ nat ) @ ( adjacent_nodes @ Capacity @ C @ U2 ) @ ( v @ Capacity @ C ) ) ) ).
% Graph.adjacent_nodes_ss_V
thf(fact_104_Graph_OisPath__ex__edge2,axiom,
! [Capacity: $tType] :
( ( linordered_idom @ Capacity )
=> ! [C: ( product_prod @ nat @ nat ) > Capacity,U2: nat,P: list @ ( product_prod @ nat @ nat ),V: nat,U1: nat,V1: nat] :
( ( isPath @ Capacity @ C @ U2 @ P @ V )
=> ( ( member @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ U1 @ V1 ) @ ( set2 @ ( product_prod @ nat @ nat ) @ P ) )
=> ( ( V1 != V )
=> ? [V2: nat] : ( member @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ V1 @ V2 ) @ ( set2 @ ( product_prod @ nat @ nat ) @ P ) ) ) ) ) ) ).
% Graph.isPath_ex_edge2
thf(fact_105_Graph_OisPath__ex__edge1,axiom,
! [Capacity: $tType] :
( ( linordered_idom @ Capacity )
=> ! [C: ( product_prod @ nat @ nat ) > Capacity,U2: nat,P: list @ ( product_prod @ nat @ nat ),V: nat,U1: nat,V1: nat] :
( ( isPath @ Capacity @ C @ U2 @ P @ V )
=> ( ( member @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ U1 @ V1 ) @ ( set2 @ ( product_prod @ nat @ nat ) @ P ) )
=> ( ( U1 != U2 )
=> ? [U22: nat] : ( member @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ U22 @ U1 ) @ ( set2 @ ( product_prod @ nat @ nat ) @ P ) ) ) ) ) ) ).
% Graph.isPath_ex_edge1
thf(fact_106_Graph_OFinite__Graph__EI,axiom,
! [Capacity: $tType] :
( ( linordered_idom @ Capacity )
=> ! [C: ( product_prod @ nat @ nat ) > Capacity] :
( ( finite_finite2 @ ( product_prod @ nat @ nat ) @ ( e @ Capacity @ C ) )
=> ( finite_Graph @ Capacity @ C ) ) ) ).
% Graph.Finite_Graph_EI
thf(fact_107_Finite__Graph_Ofinite__E,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [C: ( product_prod @ nat @ nat ) > A] :
( ( finite_Graph @ A @ C )
=> ( finite_finite2 @ ( product_prod @ nat @ nat ) @ ( e @ A @ C ) ) ) ) ).
% Finite_Graph.finite_E
thf(fact_108_ek__analysis_Ofinite__spEdges,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [C: ( product_prod @ nat @ nat ) > A,S: nat,T: nat] :
( ( edmond2129202899alysis @ A @ C )
=> ( finite_finite2 @ ( product_prod @ nat @ nat ) @ ( edmond1803015688pEdges @ A @ C @ S @ T ) ) ) ) ).
% ek_analysis.finite_spEdges
thf(fact_109_ek__analysis_Ofinite__uE,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [C: ( product_prod @ nat @ nat ) > A] :
( ( edmond2129202899alysis @ A @ C )
=> ( finite_finite2 @ ( product_prod @ nat @ nat ) @ ( edmond259086305sis_uE @ A @ C ) ) ) ) ).
% ek_analysis.finite_uE
thf(fact_110_Graph_Otransfer__path,axiom,
! [Capacity: $tType,A: $tType] :
( ( ( linordered_idom @ A )
& ( linordered_idom @ Capacity ) )
=> ! [P: list @ ( product_prod @ nat @ nat ),C: ( product_prod @ nat @ nat ) > Capacity,C4: ( product_prod @ nat @ nat ) > A,U2: nat,V: nat] :
( ( ord_less_eq @ ( set @ ( product_prod @ nat @ nat ) ) @ ( inf_inf @ ( set @ ( product_prod @ nat @ nat ) ) @ ( set2 @ ( product_prod @ nat @ nat ) @ P ) @ ( e @ Capacity @ C ) ) @ ( e @ A @ C4 ) )
=> ( ( isPath @ Capacity @ C @ U2 @ P @ V )
=> ( isPath @ A @ C4 @ U2 @ P @ V ) ) ) ) ).
% Graph.transfer_path
thf(fact_111_Graph_OE_Ocong,axiom,
! [Capacity: $tType] :
( ( linordered_idom @ Capacity )
=> ( ( e @ Capacity )
= ( e @ Capacity ) ) ) ).
% Graph.E.cong
thf(fact_112_Graph_OisPath_Ocong,axiom,
! [Capacity: $tType] :
( ( linordered_idom @ Capacity )
=> ( ( isPath @ Capacity )
= ( isPath @ Capacity ) ) ) ).
% Graph.isPath.cong
thf(fact_113_Graph_OV_Ocong,axiom,
! [Capacity: $tType] :
( ( linordered_idom @ Capacity )
=> ( ( v @ Capacity )
= ( v @ Capacity ) ) ) ).
% Graph.V.cong
thf(fact_114_ek__analysis_Ocard__spEdges__le,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [C: ( product_prod @ nat @ nat ) > A,S: nat,T: nat] :
( ( edmond2129202899alysis @ A @ C )
=> ( ord_less_eq @ nat @ ( finite_card @ ( product_prod @ nat @ nat ) @ ( edmond1803015688pEdges @ A @ C @ S @ T ) ) @ ( finite_card @ ( product_prod @ nat @ nat ) @ ( edmond259086305sis_uE @ A @ C ) ) ) ) ) ).
% ek_analysis.card_spEdges_le
thf(fact_115_Graph_OisShortestPath_Ocong,axiom,
! [Capacity: $tType] :
( ( linordered_idom @ Capacity )
=> ( ( isShortestPath @ Capacity )
= ( isShortestPath @ Capacity ) ) ) ).
% Graph.isShortestPath.cong
thf(fact_116_Graph_Ooutgoing_Ocong,axiom,
! [Capacity: $tType] :
( ( linordered_idom @ Capacity )
=> ( ( outgoing @ Capacity )
= ( outgoing @ Capacity ) ) ) ).
% Graph.outgoing.cong
thf(fact_117_Graph_Oincoming_Ocong,axiom,
! [Capacity: $tType] :
( ( linordered_idom @ Capacity )
=> ( ( incoming @ Capacity )
= ( incoming @ Capacity ) ) ) ).
% Graph.incoming.cong
thf(fact_118_Graph_Ooutgoing_H_Ocong,axiom,
! [Capacity: $tType] :
( ( linordered_idom @ Capacity )
=> ( ( outgoing2 @ Capacity )
= ( outgoing2 @ Capacity ) ) ) ).
% Graph.outgoing'.cong
thf(fact_119_Graph_Oincoming_H_Ocong,axiom,
! [Capacity: $tType] :
( ( linordered_idom @ Capacity )
=> ( ( incoming2 @ Capacity )
= ( incoming2 @ Capacity ) ) ) ).
% Graph.incoming'.cong
thf(fact_120_Graph_OshortestPath__is__path,axiom,
! [Capacity: $tType] :
( ( linordered_idom @ Capacity )
=> ! [C: ( product_prod @ nat @ nat ) > Capacity,U2: nat,P: list @ ( product_prod @ nat @ nat ),V: nat] :
( ( isShortestPath @ Capacity @ C @ U2 @ P @ V )
=> ( isPath @ Capacity @ C @ U2 @ P @ V ) ) ) ).
% Graph.shortestPath_is_path
thf(fact_121_finite__Int,axiom,
! [A: $tType,F3: set @ A,G3: set @ A] :
( ( ( finite_finite2 @ A @ F3 )
| ( finite_finite2 @ A @ G3 ) )
=> ( finite_finite2 @ A @ ( inf_inf @ ( set @ A ) @ F3 @ G3 ) ) ) ).
% finite_Int
thf(fact_122_List_Ofinite__set,axiom,
! [A: $tType,Xs: list @ A] : ( finite_finite2 @ A @ ( set2 @ A @ Xs ) ) ).
% List.finite_set
thf(fact_123_le__inf__iff,axiom,
! [A: $tType] :
( ( semilattice_inf @ A )
=> ! [X2: A,Y2: A,Z2: A] :
( ( ord_less_eq @ A @ X2 @ ( inf_inf @ A @ Y2 @ Z2 ) )
= ( ( ord_less_eq @ A @ X2 @ Y2 )
& ( ord_less_eq @ A @ X2 @ Z2 ) ) ) ) ).
% le_inf_iff
thf(fact_124_inf_Obounded__iff,axiom,
! [A: $tType] :
( ( semilattice_inf @ A )
=> ! [A3: A,B3: A,C: A] :
( ( ord_less_eq @ A @ A3 @ ( inf_inf @ A @ B3 @ C ) )
= ( ( ord_less_eq @ A @ A3 @ B3 )
& ( ord_less_eq @ A @ A3 @ C ) ) ) ) ).
% inf.bounded_iff
thf(fact_125_card__mono,axiom,
! [A: $tType,B2: set @ A,A2: set @ A] :
( ( finite_finite2 @ A @ B2 )
=> ( ( ord_less_eq @ ( set @ A ) @ A2 @ B2 )
=> ( ord_less_eq @ nat @ ( finite_card @ A @ A2 ) @ ( finite_card @ A @ B2 ) ) ) ) ).
% card_mono
thf(fact_126_card__seteq,axiom,
! [A: $tType,B2: set @ A,A2: set @ A] :
( ( finite_finite2 @ A @ B2 )
=> ( ( ord_less_eq @ ( set @ A ) @ A2 @ B2 )
=> ( ( ord_less_eq @ nat @ ( finite_card @ A @ B2 ) @ ( finite_card @ A @ A2 ) )
=> ( A2 = B2 ) ) ) ) ).
% card_seteq
thf(fact_127_finite__code,axiom,
! [A: $tType] :
( ( finite_finite @ A )
=> ( ( finite_finite2 @ A )
= ( ^ [A6: set @ A] : $true ) ) ) ).
% finite_code
thf(fact_128_inf__apply,axiom,
! [B: $tType,A: $tType] :
( ( semilattice_inf @ B )
=> ( ( inf_inf @ ( A > B ) )
= ( ^ [F2: A > B,G2: A > B,X3: A] : ( inf_inf @ B @ ( F2 @ X3 ) @ ( G2 @ X3 ) ) ) ) ) ).
% inf_apply
thf(fact_129_inf_Oidem,axiom,
! [A: $tType] :
( ( semilattice_inf @ A )
=> ! [A3: A] :
( ( inf_inf @ A @ A3 @ A3 )
= A3 ) ) ).
% inf.idem
thf(fact_130_inf__idem,axiom,
! [A: $tType] :
( ( semilattice_inf @ A )
=> ! [X2: A] :
( ( inf_inf @ A @ X2 @ X2 )
= X2 ) ) ).
% inf_idem
thf(fact_131_inf_Oleft__idem,axiom,
! [A: $tType] :
( ( semilattice_inf @ A )
=> ! [A3: A,B3: A] :
( ( inf_inf @ A @ A3 @ ( inf_inf @ A @ A3 @ B3 ) )
= ( inf_inf @ A @ A3 @ B3 ) ) ) ).
% inf.left_idem
thf(fact_132_inf__left__idem,axiom,
! [A: $tType] :
( ( semilattice_inf @ A )
=> ! [X2: A,Y2: A] :
( ( inf_inf @ A @ X2 @ ( inf_inf @ A @ X2 @ Y2 ) )
= ( inf_inf @ A @ X2 @ Y2 ) ) ) ).
% inf_left_idem
thf(fact_133_inf_Oright__idem,axiom,
! [A: $tType] :
( ( semilattice_inf @ A )
=> ! [A3: A,B3: A] :
( ( inf_inf @ A @ ( inf_inf @ A @ A3 @ B3 ) @ B3 )
= ( inf_inf @ A @ A3 @ B3 ) ) ) ).
% inf.right_idem
thf(fact_134_inf__right__idem,axiom,
! [A: $tType] :
( ( semilattice_inf @ A )
=> ! [X2: A,Y2: A] :
( ( inf_inf @ A @ ( inf_inf @ A @ X2 @ Y2 ) @ Y2 )
= ( inf_inf @ A @ X2 @ Y2 ) ) ) ).
% inf_right_idem
thf(fact_135_Efin__imp__Vfin,axiom,
( ( finite_finite2 @ ( product_prod @ nat @ nat ) @ ( e @ a @ c ) )
=> ( finite_finite2 @ nat @ ( v @ a @ c ) ) ) ).
% Efin_imp_Vfin
thf(fact_136_finite__V,axiom,
finite_finite2 @ nat @ ( v @ a @ c ) ).
% finite_V
thf(fact_137_adjacent__nodes__finite,axiom,
! [U2: nat] : ( finite_finite2 @ nat @ ( adjacent_nodes @ a @ c @ U2 ) ) ).
% adjacent_nodes_finite
thf(fact_138_Vfin__imp__Efin,axiom,
( ( finite_finite2 @ nat @ ( v @ a @ c ) )
=> ( finite_finite2 @ ( product_prod @ nat @ nat ) @ ( e @ a @ c ) ) ) ).
% Vfin_imp_Efin
thf(fact_139_finite__incoming,axiom,
! [U2: nat] :
( ( finite_finite2 @ nat @ ( v @ a @ c ) )
=> ( finite_finite2 @ ( product_prod @ nat @ nat ) @ ( incoming @ a @ c @ U2 ) ) ) ).
% finite_incoming
thf(fact_140_finite__outgoing,axiom,
! [U2: nat] :
( ( finite_finite2 @ nat @ ( v @ a @ c ) )
=> ( finite_finite2 @ ( product_prod @ nat @ nat ) @ ( outgoing @ a @ c @ U2 ) ) ) ).
% finite_outgoing
thf(fact_141_finite__incoming_H,axiom,
! [U: set @ nat] :
( ( finite_finite2 @ nat @ ( v @ a @ c ) )
=> ( finite_finite2 @ ( product_prod @ nat @ nat ) @ ( incoming2 @ a @ c @ U ) ) ) ).
% finite_incoming'
thf(fact_142_finite__outgoing_H,axiom,
! [U: set @ nat] :
( ( finite_finite2 @ nat @ ( v @ a @ c ) )
=> ( finite_finite2 @ ( product_prod @ nat @ nat ) @ ( outgoing2 @ a @ c @ U ) ) ) ).
% finite_outgoing'
thf(fact_143_Finite__Graph__def,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ( ( finite_Graph @ A )
= ( ^ [C5: ( product_prod @ nat @ nat ) > A] : ( finite_finite2 @ nat @ ( v @ A @ C5 ) ) ) ) ) ).
% Finite_Graph_def
thf(fact_144_Finite__Graph_Ointro,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [C: ( product_prod @ nat @ nat ) > A] :
( ( finite_finite2 @ nat @ ( v @ A @ C ) )
=> ( finite_Graph @ A @ C ) ) ) ).
% Finite_Graph.intro
thf(fact_145_Finite__Graph_Ofinite__V,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [C: ( product_prod @ nat @ nat ) > A] :
( ( finite_Graph @ A @ C )
=> ( finite_finite2 @ nat @ ( v @ A @ C ) ) ) ) ).
% Finite_Graph.finite_V
thf(fact_146_Finite__Graph_Oadjacent__nodes__finite,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [C: ( product_prod @ nat @ nat ) > A,U2: nat] :
( ( finite_Graph @ A @ C )
=> ( finite_finite2 @ nat @ ( adjacent_nodes @ A @ C @ U2 ) ) ) ) ).
% Finite_Graph.adjacent_nodes_finite
thf(fact_147_finite__set__choice,axiom,
! [B: $tType,A: $tType,A2: set @ A,P2: A > B > $o] :
( ( finite_finite2 @ A @ A2 )
=> ( ! [X: A] :
( ( member @ A @ X @ A2 )
=> ? [X_1: B] : ( P2 @ X @ X_1 ) )
=> ? [F4: A > B] :
! [X4: A] :
( ( member @ A @ X4 @ A2 )
=> ( P2 @ X4 @ ( F4 @ X4 ) ) ) ) ) ).
% finite_set_choice
thf(fact_148_finite,axiom,
! [A: $tType] :
( ( finite_finite @ A )
=> ! [A2: set @ A] : ( finite_finite2 @ A @ A2 ) ) ).
% finite
thf(fact_149_inf__sup__aci_I4_J,axiom,
! [A: $tType] :
( ( lattice @ A )
=> ! [X2: A,Y2: A] :
( ( inf_inf @ A @ X2 @ ( inf_inf @ A @ X2 @ Y2 ) )
= ( inf_inf @ A @ X2 @ Y2 ) ) ) ).
% inf_sup_aci(4)
thf(fact_150_inf__sup__aci_I3_J,axiom,
! [A: $tType] :
( ( lattice @ A )
=> ! [X2: A,Y2: A,Z2: A] :
( ( inf_inf @ A @ X2 @ ( inf_inf @ A @ Y2 @ Z2 ) )
= ( inf_inf @ A @ Y2 @ ( inf_inf @ A @ X2 @ Z2 ) ) ) ) ).
% inf_sup_aci(3)
thf(fact_151_inf__sup__aci_I2_J,axiom,
! [A: $tType] :
( ( lattice @ A )
=> ! [X2: A,Y2: A,Z2: A] :
( ( inf_inf @ A @ ( inf_inf @ A @ X2 @ Y2 ) @ Z2 )
= ( inf_inf @ A @ X2 @ ( inf_inf @ A @ Y2 @ Z2 ) ) ) ) ).
% inf_sup_aci(2)
thf(fact_152_inf__sup__aci_I1_J,axiom,
! [A: $tType] :
( ( lattice @ A )
=> ( ( inf_inf @ A )
= ( ^ [X3: A,Y3: A] : ( inf_inf @ A @ Y3 @ X3 ) ) ) ) ).
% inf_sup_aci(1)
thf(fact_153_inf__fun__def,axiom,
! [B: $tType,A: $tType] :
( ( semilattice_inf @ B )
=> ( ( inf_inf @ ( A > B ) )
= ( ^ [F2: A > B,G2: A > B,X3: A] : ( inf_inf @ B @ ( F2 @ X3 ) @ ( G2 @ X3 ) ) ) ) ) ).
% inf_fun_def
thf(fact_154_boolean__algebra__cancel_Oinf1,axiom,
! [A: $tType] :
( ( semilattice_inf @ A )
=> ! [A2: A,K: A,A3: A,B3: A] :
( ( A2
= ( inf_inf @ A @ K @ A3 ) )
=> ( ( inf_inf @ A @ A2 @ B3 )
= ( inf_inf @ A @ K @ ( inf_inf @ A @ A3 @ B3 ) ) ) ) ) ).
% boolean_algebra_cancel.inf1
thf(fact_155_boolean__algebra__cancel_Oinf2,axiom,
! [A: $tType] :
( ( semilattice_inf @ A )
=> ! [B2: A,K: A,B3: A,A3: A] :
( ( B2
= ( inf_inf @ A @ K @ B3 ) )
=> ( ( inf_inf @ A @ A3 @ B2 )
= ( inf_inf @ A @ K @ ( inf_inf @ A @ A3 @ B3 ) ) ) ) ) ).
% boolean_algebra_cancel.inf2
thf(fact_156_inf_Oassoc,axiom,
! [A: $tType] :
( ( semilattice_inf @ A )
=> ! [A3: A,B3: A,C: A] :
( ( inf_inf @ A @ ( inf_inf @ A @ A3 @ B3 ) @ C )
= ( inf_inf @ A @ A3 @ ( inf_inf @ A @ B3 @ C ) ) ) ) ).
% inf.assoc
thf(fact_157_inf__assoc,axiom,
! [A: $tType] :
( ( semilattice_inf @ A )
=> ! [X2: A,Y2: A,Z2: A] :
( ( inf_inf @ A @ ( inf_inf @ A @ X2 @ Y2 ) @ Z2 )
= ( inf_inf @ A @ X2 @ ( inf_inf @ A @ Y2 @ Z2 ) ) ) ) ).
% inf_assoc
thf(fact_158_inf_Ocommute,axiom,
! [A: $tType] :
( ( semilattice_inf @ A )
=> ( ( inf_inf @ A )
= ( ^ [A4: A,B4: A] : ( inf_inf @ A @ B4 @ A4 ) ) ) ) ).
% inf.commute
thf(fact_159_inf__commute,axiom,
! [A: $tType] :
( ( semilattice_inf @ A )
=> ( ( inf_inf @ A )
= ( ^ [X3: A,Y3: A] : ( inf_inf @ A @ Y3 @ X3 ) ) ) ) ).
% inf_commute
thf(fact_160_inf_Oleft__commute,axiom,
! [A: $tType] :
( ( semilattice_inf @ A )
=> ! [B3: A,A3: A,C: A] :
( ( inf_inf @ A @ B3 @ ( inf_inf @ A @ A3 @ C ) )
= ( inf_inf @ A @ A3 @ ( inf_inf @ A @ B3 @ C ) ) ) ) ).
% inf.left_commute
thf(fact_161_inf__left__commute,axiom,
! [A: $tType] :
( ( semilattice_inf @ A )
=> ! [X2: A,Y2: A,Z2: A] :
( ( inf_inf @ A @ X2 @ ( inf_inf @ A @ Y2 @ Z2 ) )
= ( inf_inf @ A @ Y2 @ ( inf_inf @ A @ X2 @ Z2 ) ) ) ) ).
% inf_left_commute
thf(fact_162_Graph_OEfin__imp__Vfin,axiom,
! [Capacity: $tType] :
( ( linordered_idom @ Capacity )
=> ! [C: ( product_prod @ nat @ nat ) > Capacity] :
( ( finite_finite2 @ ( product_prod @ nat @ nat ) @ ( e @ Capacity @ C ) )
=> ( finite_finite2 @ nat @ ( v @ Capacity @ C ) ) ) ) ).
% Graph.Efin_imp_Vfin
thf(fact_163_Graph_OVfin__imp__Efin,axiom,
! [Capacity: $tType] :
( ( linordered_idom @ Capacity )
=> ! [C: ( product_prod @ nat @ nat ) > Capacity] :
( ( finite_finite2 @ nat @ ( v @ Capacity @ C ) )
=> ( finite_finite2 @ ( product_prod @ nat @ nat ) @ ( e @ Capacity @ C ) ) ) ) ).
% Graph.Vfin_imp_Efin
thf(fact_164_Graph_Ofinite__outgoing,axiom,
! [Capacity: $tType] :
( ( linordered_idom @ Capacity )
=> ! [C: ( product_prod @ nat @ nat ) > Capacity,U2: nat] :
( ( finite_finite2 @ nat @ ( v @ Capacity @ C ) )
=> ( finite_finite2 @ ( product_prod @ nat @ nat ) @ ( outgoing @ Capacity @ C @ U2 ) ) ) ) ).
% Graph.finite_outgoing
thf(fact_165_Graph_Ofinite__incoming,axiom,
! [Capacity: $tType] :
( ( linordered_idom @ Capacity )
=> ! [C: ( product_prod @ nat @ nat ) > Capacity,U2: nat] :
( ( finite_finite2 @ nat @ ( v @ Capacity @ C ) )
=> ( finite_finite2 @ ( product_prod @ nat @ nat ) @ ( incoming @ Capacity @ C @ U2 ) ) ) ) ).
% Graph.finite_incoming
thf(fact_166_Graph_Ofinite__outgoing_H,axiom,
! [Capacity: $tType] :
( ( linordered_idom @ Capacity )
=> ! [C: ( product_prod @ nat @ nat ) > Capacity,U: set @ nat] :
( ( finite_finite2 @ nat @ ( v @ Capacity @ C ) )
=> ( finite_finite2 @ ( product_prod @ nat @ nat ) @ ( outgoing2 @ Capacity @ C @ U ) ) ) ) ).
% Graph.finite_outgoing'
thf(fact_167_Graph_Ofinite__incoming_H,axiom,
! [Capacity: $tType] :
( ( linordered_idom @ Capacity )
=> ! [C: ( product_prod @ nat @ nat ) > Capacity,U: set @ nat] :
( ( finite_finite2 @ nat @ ( v @ Capacity @ C ) )
=> ( finite_finite2 @ ( product_prod @ nat @ nat ) @ ( incoming2 @ Capacity @ C @ U ) ) ) ) ).
% Graph.finite_incoming'
thf(fact_168_finite__has__minimal2,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [A2: set @ A,A3: A] :
( ( finite_finite2 @ A @ A2 )
=> ( ( member @ A @ A3 @ A2 )
=> ? [X: A] :
( ( member @ A @ X @ A2 )
& ( ord_less_eq @ A @ X @ A3 )
& ! [Xa: A] :
( ( member @ A @ Xa @ A2 )
=> ( ( ord_less_eq @ A @ Xa @ X )
=> ( X = Xa ) ) ) ) ) ) ) ).
% finite_has_minimal2
thf(fact_169_finite__has__maximal2,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [A2: set @ A,A3: A] :
( ( finite_finite2 @ A @ A2 )
=> ( ( member @ A @ A3 @ A2 )
=> ? [X: A] :
( ( member @ A @ X @ A2 )
& ( ord_less_eq @ A @ A3 @ X )
& ! [Xa: A] :
( ( member @ A @ Xa @ A2 )
=> ( ( ord_less_eq @ A @ X @ Xa )
=> ( X = Xa ) ) ) ) ) ) ) ).
% finite_has_maximal2
thf(fact_170_inf_OcoboundedI2,axiom,
! [A: $tType] :
( ( semilattice_inf @ A )
=> ! [B3: A,C: A,A3: A] :
( ( ord_less_eq @ A @ B3 @ C )
=> ( ord_less_eq @ A @ ( inf_inf @ A @ A3 @ B3 ) @ C ) ) ) ).
% inf.coboundedI2
thf(fact_171_inf_OcoboundedI1,axiom,
! [A: $tType] :
( ( semilattice_inf @ A )
=> ! [A3: A,C: A,B3: A] :
( ( ord_less_eq @ A @ A3 @ C )
=> ( ord_less_eq @ A @ ( inf_inf @ A @ A3 @ B3 ) @ C ) ) ) ).
% inf.coboundedI1
thf(fact_172_inf_Oabsorb__iff2,axiom,
! [A: $tType] :
( ( semilattice_inf @ A )
=> ( ( ord_less_eq @ A )
= ( ^ [B4: A,A4: A] :
( ( inf_inf @ A @ A4 @ B4 )
= B4 ) ) ) ) ).
% inf.absorb_iff2
thf(fact_173_inf_Oabsorb__iff1,axiom,
! [A: $tType] :
( ( semilattice_inf @ A )
=> ( ( ord_less_eq @ A )
= ( ^ [A4: A,B4: A] :
( ( inf_inf @ A @ A4 @ B4 )
= A4 ) ) ) ) ).
% inf.absorb_iff1
thf(fact_174_inf_Ocobounded2,axiom,
! [A: $tType] :
( ( semilattice_inf @ A )
=> ! [A3: A,B3: A] : ( ord_less_eq @ A @ ( inf_inf @ A @ A3 @ B3 ) @ B3 ) ) ).
% inf.cobounded2
thf(fact_175_inf_Ocobounded1,axiom,
! [A: $tType] :
( ( semilattice_inf @ A )
=> ! [A3: A,B3: A] : ( ord_less_eq @ A @ ( inf_inf @ A @ A3 @ B3 ) @ A3 ) ) ).
% inf.cobounded1
thf(fact_176_inf_Oorder__iff,axiom,
! [A: $tType] :
( ( semilattice_inf @ A )
=> ( ( ord_less_eq @ A )
= ( ^ [A4: A,B4: A] :
( A4
= ( inf_inf @ A @ A4 @ B4 ) ) ) ) ) ).
% inf.order_iff
thf(fact_177_inf__greatest,axiom,
! [A: $tType] :
( ( semilattice_inf @ A )
=> ! [X2: A,Y2: A,Z2: A] :
( ( ord_less_eq @ A @ X2 @ Y2 )
=> ( ( ord_less_eq @ A @ X2 @ Z2 )
=> ( ord_less_eq @ A @ X2 @ ( inf_inf @ A @ Y2 @ Z2 ) ) ) ) ) ).
% inf_greatest
thf(fact_178_inf_OboundedI,axiom,
! [A: $tType] :
( ( semilattice_inf @ A )
=> ! [A3: A,B3: A,C: A] :
( ( ord_less_eq @ A @ A3 @ B3 )
=> ( ( ord_less_eq @ A @ A3 @ C )
=> ( ord_less_eq @ A @ A3 @ ( inf_inf @ A @ B3 @ C ) ) ) ) ) ).
% inf.boundedI
thf(fact_179_inf_OboundedE,axiom,
! [A: $tType] :
( ( semilattice_inf @ A )
=> ! [A3: A,B3: A,C: A] :
( ( ord_less_eq @ A @ A3 @ ( inf_inf @ A @ B3 @ C ) )
=> ~ ( ( ord_less_eq @ A @ A3 @ B3 )
=> ~ ( ord_less_eq @ A @ A3 @ C ) ) ) ) ).
% inf.boundedE
thf(fact_180_inf__absorb2,axiom,
! [A: $tType] :
( ( semilattice_inf @ A )
=> ! [Y2: A,X2: A] :
( ( ord_less_eq @ A @ Y2 @ X2 )
=> ( ( inf_inf @ A @ X2 @ Y2 )
= Y2 ) ) ) ).
% inf_absorb2
thf(fact_181_inf__absorb1,axiom,
! [A: $tType] :
( ( semilattice_inf @ A )
=> ! [X2: A,Y2: A] :
( ( ord_less_eq @ A @ X2 @ Y2 )
=> ( ( inf_inf @ A @ X2 @ Y2 )
= X2 ) ) ) ).
% inf_absorb1
thf(fact_182_inf_Oabsorb2,axiom,
! [A: $tType] :
( ( semilattice_inf @ A )
=> ! [B3: A,A3: A] :
( ( ord_less_eq @ A @ B3 @ A3 )
=> ( ( inf_inf @ A @ A3 @ B3 )
= B3 ) ) ) ).
% inf.absorb2
thf(fact_183_inf_Oabsorb1,axiom,
! [A: $tType] :
( ( semilattice_inf @ A )
=> ! [A3: A,B3: A] :
( ( ord_less_eq @ A @ A3 @ B3 )
=> ( ( inf_inf @ A @ A3 @ B3 )
= A3 ) ) ) ).
% inf.absorb1
thf(fact_184_le__iff__inf,axiom,
! [A: $tType] :
( ( semilattice_inf @ A )
=> ( ( ord_less_eq @ A )
= ( ^ [X3: A,Y3: A] :
( ( inf_inf @ A @ X3 @ Y3 )
= X3 ) ) ) ) ).
% le_iff_inf
thf(fact_185_inf__unique,axiom,
! [A: $tType] :
( ( semilattice_inf @ A )
=> ! [F: A > A > A,X2: A,Y2: A] :
( ! [X: A,Y4: A] : ( ord_less_eq @ A @ ( F @ X @ Y4 ) @ X )
=> ( ! [X: A,Y4: A] : ( ord_less_eq @ A @ ( F @ X @ Y4 ) @ Y4 )
=> ( ! [X: A,Y4: A,Z3: A] :
( ( ord_less_eq @ A @ X @ Y4 )
=> ( ( ord_less_eq @ A @ X @ Z3 )
=> ( ord_less_eq @ A @ X @ ( F @ Y4 @ Z3 ) ) ) )
=> ( ( inf_inf @ A @ X2 @ Y2 )
= ( F @ X2 @ Y2 ) ) ) ) ) ) ).
% inf_unique
thf(fact_186_inf_OorderI,axiom,
! [A: $tType] :
( ( semilattice_inf @ A )
=> ! [A3: A,B3: A] :
( ( A3
= ( inf_inf @ A @ A3 @ B3 ) )
=> ( ord_less_eq @ A @ A3 @ B3 ) ) ) ).
% inf.orderI
thf(fact_187_inf_OorderE,axiom,
! [A: $tType] :
( ( semilattice_inf @ A )
=> ! [A3: A,B3: A] :
( ( ord_less_eq @ A @ A3 @ B3 )
=> ( A3
= ( inf_inf @ A @ A3 @ B3 ) ) ) ) ).
% inf.orderE
thf(fact_188_le__infI2,axiom,
! [A: $tType] :
( ( semilattice_inf @ A )
=> ! [B3: A,X2: A,A3: A] :
( ( ord_less_eq @ A @ B3 @ X2 )
=> ( ord_less_eq @ A @ ( inf_inf @ A @ A3 @ B3 ) @ X2 ) ) ) ).
% le_infI2
thf(fact_189_le__infI1,axiom,
! [A: $tType] :
( ( semilattice_inf @ A )
=> ! [A3: A,X2: A,B3: A] :
( ( ord_less_eq @ A @ A3 @ X2 )
=> ( ord_less_eq @ A @ ( inf_inf @ A @ A3 @ B3 ) @ X2 ) ) ) ).
% le_infI1
thf(fact_190_inf__mono,axiom,
! [A: $tType] :
( ( semilattice_inf @ A )
=> ! [A3: A,C: A,B3: A,D2: A] :
( ( ord_less_eq @ A @ A3 @ C )
=> ( ( ord_less_eq @ A @ B3 @ D2 )
=> ( ord_less_eq @ A @ ( inf_inf @ A @ A3 @ B3 ) @ ( inf_inf @ A @ C @ D2 ) ) ) ) ) ).
% inf_mono
thf(fact_191_le__infI,axiom,
! [A: $tType] :
( ( semilattice_inf @ A )
=> ! [X2: A,A3: A,B3: A] :
( ( ord_less_eq @ A @ X2 @ A3 )
=> ( ( ord_less_eq @ A @ X2 @ B3 )
=> ( ord_less_eq @ A @ X2 @ ( inf_inf @ A @ A3 @ B3 ) ) ) ) ) ).
% le_infI
thf(fact_192_le__infE,axiom,
! [A: $tType] :
( ( semilattice_inf @ A )
=> ! [X2: A,A3: A,B3: A] :
( ( ord_less_eq @ A @ X2 @ ( inf_inf @ A @ A3 @ B3 ) )
=> ~ ( ( ord_less_eq @ A @ X2 @ A3 )
=> ~ ( ord_less_eq @ A @ X2 @ B3 ) ) ) ) ).
% le_infE
thf(fact_193_inf__le2,axiom,
! [A: $tType] :
( ( semilattice_inf @ A )
=> ! [X2: A,Y2: A] : ( ord_less_eq @ A @ ( inf_inf @ A @ X2 @ Y2 ) @ Y2 ) ) ).
% inf_le2
thf(fact_194_inf__le1,axiom,
! [A: $tType] :
( ( semilattice_inf @ A )
=> ! [X2: A,Y2: A] : ( ord_less_eq @ A @ ( inf_inf @ A @ X2 @ Y2 ) @ X2 ) ) ).
% inf_le1
thf(fact_195_inf__sup__ord_I1_J,axiom,
! [A: $tType] :
( ( lattice @ A )
=> ! [X2: A,Y2: A] : ( ord_less_eq @ A @ ( inf_inf @ A @ X2 @ Y2 ) @ X2 ) ) ).
% inf_sup_ord(1)
thf(fact_196_inf__sup__ord_I2_J,axiom,
! [A: $tType] :
( ( lattice @ A )
=> ! [X2: A,Y2: A] : ( ord_less_eq @ A @ ( inf_inf @ A @ X2 @ Y2 ) @ Y2 ) ) ).
% inf_sup_ord(2)
thf(fact_197_rev__finite__subset,axiom,
! [A: $tType,B2: set @ A,A2: set @ A] :
( ( finite_finite2 @ A @ B2 )
=> ( ( ord_less_eq @ ( set @ A ) @ A2 @ B2 )
=> ( finite_finite2 @ A @ A2 ) ) ) ).
% rev_finite_subset
thf(fact_198_infinite__super,axiom,
! [A: $tType,S2: set @ A,T3: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ S2 @ T3 )
=> ( ~ ( finite_finite2 @ A @ S2 )
=> ~ ( finite_finite2 @ A @ T3 ) ) ) ).
% infinite_super
thf(fact_199_finite__subset,axiom,
! [A: $tType,A2: set @ A,B2: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ A2 @ B2 )
=> ( ( finite_finite2 @ A @ B2 )
=> ( finite_finite2 @ A @ A2 ) ) ) ).
% finite_subset
thf(fact_200_finite__list,axiom,
! [A: $tType,A2: set @ A] :
( ( finite_finite2 @ A @ A2 )
=> ? [Xs2: list @ A] :
( ( set2 @ A @ Xs2 )
= A2 ) ) ).
% finite_list
thf(fact_201_subset__code_I1_J,axiom,
! [A: $tType,Xs: list @ A,B2: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ ( set2 @ A @ Xs ) @ B2 )
= ( ! [X3: A] :
( ( member @ A @ X3 @ ( set2 @ A @ Xs ) )
=> ( member @ A @ X3 @ B2 ) ) ) ) ).
% subset_code(1)
thf(fact_202_infinite__arbitrarily__large,axiom,
! [A: $tType,A2: set @ A,N: nat] :
( ~ ( finite_finite2 @ A @ A2 )
=> ? [B7: set @ A] :
( ( finite_finite2 @ A @ B7 )
& ( ( finite_card @ A @ B7 )
= N )
& ( ord_less_eq @ ( set @ A ) @ B7 @ A2 ) ) ) ).
% infinite_arbitrarily_large
thf(fact_203_card__subset__eq,axiom,
! [A: $tType,B2: set @ A,A2: set @ A] :
( ( finite_finite2 @ A @ B2 )
=> ( ( ord_less_eq @ ( set @ A ) @ A2 @ B2 )
=> ( ( ( finite_card @ A @ A2 )
= ( finite_card @ A @ B2 ) )
=> ( A2 = B2 ) ) ) ) ).
% card_subset_eq
thf(fact_204_finite__if__finite__subsets__card__bdd,axiom,
! [A: $tType,F3: set @ A,C3: nat] :
( ! [G4: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ G4 @ F3 )
=> ( ( finite_finite2 @ A @ G4 )
=> ( ord_less_eq @ nat @ ( finite_card @ A @ G4 ) @ C3 ) ) )
=> ( ( finite_finite2 @ A @ F3 )
& ( ord_less_eq @ nat @ ( finite_card @ A @ F3 ) @ C3 ) ) ) ).
% finite_if_finite_subsets_card_bdd
thf(fact_205_pathVertices__edgeset,axiom,
! [U2: nat,P: list @ ( product_prod @ nat @ nat ),V: nat] :
( ( member @ nat @ U2 @ ( v @ a @ c ) )
=> ( ( isPath @ a @ c @ U2 @ P @ V )
=> ( ord_less_eq @ ( set @ nat ) @ ( set2 @ nat @ ( pathVertices @ U2 @ P ) ) @ ( v @ a @ c ) ) ) ) ).
% pathVertices_edgeset
thf(fact_206_obtain__subset__with__card__n,axiom,
! [A: $tType,N: nat,S2: set @ A] :
( ( ord_less_eq @ nat @ N @ ( finite_card @ A @ S2 ) )
=> ~ ! [T4: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ T4 @ S2 )
=> ( ( ( finite_card @ A @ T4 )
= N )
=> ~ ( finite_finite2 @ A @ T4 ) ) ) ) ).
% obtain_subset_with_card_n
thf(fact_207_isPath_Osimps_I2_J,axiom,
! [U2: nat,X2: nat,Y2: nat,P: list @ ( product_prod @ nat @ nat ),V: nat] :
( ( isPath @ a @ c @ U2 @ ( cons @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ X2 @ Y2 ) @ P ) @ V )
= ( ( U2 = X2 )
& ( member @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ X2 @ Y2 ) @ ( e @ a @ c ) )
& ( isPath @ a @ c @ Y2 @ P @ V ) ) ) ).
% isPath.simps(2)
thf(fact_208_isShortestPath__def,axiom,
! [U2: nat,P: list @ ( product_prod @ nat @ nat ),V: nat] :
( ( isShortestPath @ a @ c @ U2 @ P @ V )
= ( ( isPath @ a @ c @ U2 @ P @ V )
& ! [P3: list @ ( product_prod @ nat @ nat )] :
( ( isPath @ a @ c @ U2 @ P3 @ V )
=> ( ord_less_eq @ nat @ ( size_size @ ( list @ ( product_prod @ nat @ nat ) ) @ P ) @ ( size_size @ ( list @ ( product_prod @ nat @ nat ) ) @ P3 ) ) ) ) ) ).
% isShortestPath_def
thf(fact_209_list_Oinject,axiom,
! [A: $tType,X21: A,X22: list @ A,Y21: A,Y22: list @ A] :
( ( ( cons @ A @ X21 @ X22 )
= ( cons @ A @ Y21 @ Y22 ) )
= ( ( X21 = Y21 )
& ( X22 = Y22 ) ) ) ).
% list.inject
thf(fact_210_impossible__Cons,axiom,
! [A: $tType,Xs: list @ A,Ys: list @ A,X2: A] :
( ( ord_less_eq @ nat @ ( size_size @ ( list @ A ) @ Xs ) @ ( size_size @ ( list @ A ) @ Ys ) )
=> ( Xs
!= ( cons @ A @ X2 @ Ys ) ) ) ).
% impossible_Cons
thf(fact_211_list_Oset__cases,axiom,
! [A: $tType,E: A,A3: list @ A] :
( ( member @ A @ E @ ( set2 @ A @ A3 ) )
=> ( ! [Z22: list @ A] :
( A3
!= ( cons @ A @ E @ Z22 ) )
=> ~ ! [Z1: A,Z22: list @ A] :
( ( A3
= ( cons @ A @ Z1 @ Z22 ) )
=> ~ ( member @ A @ E @ ( set2 @ A @ Z22 ) ) ) ) ) ).
% list.set_cases
thf(fact_212_set__ConsD,axiom,
! [A: $tType,Y2: A,X2: A,Xs: list @ A] :
( ( member @ A @ Y2 @ ( set2 @ A @ ( cons @ A @ X2 @ Xs ) ) )
=> ( ( Y2 = X2 )
| ( member @ A @ Y2 @ ( set2 @ A @ Xs ) ) ) ) ).
% set_ConsD
thf(fact_213_list_Oset__intros_I1_J,axiom,
! [A: $tType,X21: A,X22: list @ A] : ( member @ A @ X21 @ ( set2 @ A @ ( cons @ A @ X21 @ X22 ) ) ) ).
% list.set_intros(1)
thf(fact_214_list_Oset__intros_I2_J,axiom,
! [A: $tType,Y2: A,X22: list @ A,X21: A] :
( ( member @ A @ Y2 @ ( set2 @ A @ X22 ) )
=> ( member @ A @ Y2 @ ( set2 @ A @ ( cons @ A @ X21 @ X22 ) ) ) ) ).
% list.set_intros(2)
thf(fact_215_not__Cons__self2,axiom,
! [A: $tType,X2: A,Xs: list @ A] :
( ( cons @ A @ X2 @ Xs )
!= Xs ) ).
% not_Cons_self2
thf(fact_216_Ex__list__of__length,axiom,
! [A: $tType,N: nat] :
? [Xs2: list @ A] :
( ( size_size @ ( list @ A ) @ Xs2 )
= N ) ).
% Ex_list_of_length
thf(fact_217_neq__if__length__neq,axiom,
! [A: $tType,Xs: list @ A,Ys: list @ A] :
( ( ( size_size @ ( list @ A ) @ Xs )
!= ( size_size @ ( list @ A ) @ Ys ) )
=> ( Xs != Ys ) ) ).
% neq_if_length_neq
thf(fact_218_set__subset__Cons,axiom,
! [A: $tType,Xs: list @ A,X2: A] : ( ord_less_eq @ ( set @ A ) @ ( set2 @ A @ Xs ) @ ( set2 @ A @ ( cons @ A @ X2 @ Xs ) ) ) ).
% set_subset_Cons
thf(fact_219_card__length,axiom,
! [A: $tType,Xs: list @ A] : ( ord_less_eq @ nat @ ( finite_card @ A @ ( set2 @ A @ Xs ) ) @ ( size_size @ ( list @ A ) @ Xs ) ) ).
% card_length
thf(fact_220_Graph_OisPath_Osimps_I2_J,axiom,
! [Capacity: $tType] :
( ( linordered_idom @ Capacity )
=> ! [C: ( product_prod @ nat @ nat ) > Capacity,U2: nat,X2: nat,Y2: nat,P: list @ ( product_prod @ nat @ nat ),V: nat] :
( ( isPath @ Capacity @ C @ U2 @ ( cons @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ X2 @ Y2 ) @ P ) @ V )
= ( ( U2 = X2 )
& ( member @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ X2 @ Y2 ) @ ( e @ Capacity @ C ) )
& ( isPath @ Capacity @ C @ Y2 @ P @ V ) ) ) ) ).
% Graph.isPath.simps(2)
thf(fact_221_Graph_OisShortestPath__def,axiom,
! [Capacity: $tType] :
( ( linordered_idom @ Capacity )
=> ( ( isShortestPath @ Capacity )
= ( ^ [C5: ( product_prod @ nat @ nat ) > Capacity,U3: nat,P4: list @ ( product_prod @ nat @ nat ),V3: nat] :
( ( isPath @ Capacity @ C5 @ U3 @ P4 @ V3 )
& ! [P3: list @ ( product_prod @ nat @ nat )] :
( ( isPath @ Capacity @ C5 @ U3 @ P3 @ V3 )
=> ( ord_less_eq @ nat @ ( size_size @ ( list @ ( product_prod @ nat @ nat ) ) @ P4 ) @ ( size_size @ ( list @ ( product_prod @ nat @ nat ) ) @ P3 ) ) ) ) ) ) ) ).
% Graph.isShortestPath_def
thf(fact_222_bounded__Max__nat,axiom,
! [P2: nat > $o,X2: nat,M: nat] :
( ( P2 @ X2 )
=> ( ! [X: nat] :
( ( P2 @ X )
=> ( ord_less_eq @ nat @ X @ M ) )
=> ~ ! [M2: nat] :
( ( P2 @ M2 )
=> ~ ! [X4: nat] :
( ( P2 @ X4 )
=> ( ord_less_eq @ nat @ X4 @ M2 ) ) ) ) ) ).
% bounded_Max_nat
thf(fact_223_finite__nat__set__iff__bounded__le,axiom,
( ( finite_finite2 @ nat )
= ( ^ [N2: set @ nat] :
? [M3: nat] :
! [X3: nat] :
( ( member @ nat @ X3 @ N2 )
=> ( ord_less_eq @ nat @ X3 @ M3 ) ) ) ) ).
% finite_nat_set_iff_bounded_le
thf(fact_224_Graph_OpathVertices__edgeset,axiom,
! [Capacity: $tType] :
( ( linordered_idom @ Capacity )
=> ! [U2: nat,C: ( product_prod @ nat @ nat ) > Capacity,P: list @ ( product_prod @ nat @ nat ),V: nat] :
( ( member @ nat @ U2 @ ( v @ Capacity @ C ) )
=> ( ( isPath @ Capacity @ C @ U2 @ P @ V )
=> ( ord_less_eq @ ( set @ nat ) @ ( set2 @ nat @ ( pathVertices @ U2 @ P ) ) @ ( v @ Capacity @ C ) ) ) ) ) ).
% Graph.pathVertices_edgeset
thf(fact_225_card__le__if__inj__on__rel,axiom,
! [B: $tType,A: $tType,B2: set @ A,A2: set @ B,R: B > A > $o] :
( ( finite_finite2 @ A @ B2 )
=> ( ! [A5: B] :
( ( member @ B @ A5 @ A2 )
=> ? [B8: A] :
( ( member @ A @ B8 @ B2 )
& ( R @ A5 @ B8 ) ) )
=> ( ! [A1: B,A22: B,B5: A] :
( ( member @ B @ A1 @ A2 )
=> ( ( member @ B @ A22 @ A2 )
=> ( ( member @ A @ B5 @ B2 )
=> ( ( R @ A1 @ B5 )
=> ( ( R @ A22 @ B5 )
=> ( A1 = A22 ) ) ) ) ) )
=> ( ord_less_eq @ nat @ ( finite_card @ B @ A2 ) @ ( finite_card @ A @ B2 ) ) ) ) ) ).
% card_le_if_inj_on_rel
thf(fact_226_isPath_Oelims_I3_J,axiom,
! [X2: nat,Xa2: list @ ( product_prod @ nat @ nat ),Xb: nat] :
( ~ ( isPath @ a @ c @ X2 @ Xa2 @ Xb )
=> ( ( ( Xa2
= ( nil @ ( product_prod @ nat @ nat ) ) )
=> ( X2 = Xb ) )
=> ~ ! [X: nat,Y4: nat,P5: list @ ( product_prod @ nat @ nat )] :
( ( Xa2
= ( cons @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ X @ Y4 ) @ P5 ) )
=> ( ( X2 = X )
& ( member @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ X @ Y4 ) @ ( e @ a @ c ) )
& ( isPath @ a @ c @ Y4 @ P5 @ Xb ) ) ) ) ) ).
% isPath.elims(3)
thf(fact_227_isPath_Oelims_I2_J,axiom,
! [X2: nat,Xa2: list @ ( product_prod @ nat @ nat ),Xb: nat] :
( ( isPath @ a @ c @ X2 @ Xa2 @ Xb )
=> ( ( ( Xa2
= ( nil @ ( product_prod @ nat @ nat ) ) )
=> ( X2 != Xb ) )
=> ~ ! [X: nat,Y4: nat,P5: list @ ( product_prod @ nat @ nat )] :
( ( Xa2
= ( cons @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ X @ Y4 ) @ P5 ) )
=> ~ ( ( X2 = X )
& ( member @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ X @ Y4 ) @ ( e @ a @ c ) )
& ( isPath @ a @ c @ Y4 @ P5 @ Xb ) ) ) ) ) ).
% isPath.elims(2)
thf(fact_228_isPath_Osimps_I1_J,axiom,
! [U2: nat,V: nat] :
( ( isPath @ a @ c @ U2 @ ( nil @ ( product_prod @ nat @ nat ) ) @ V )
= ( U2 = V ) ) ).
% isPath.simps(1)
thf(fact_229_isPath__fwd__cases,axiom,
! [S: nat,P: list @ ( product_prod @ nat @ nat ),T: nat] :
( ( isPath @ a @ c @ S @ P @ T )
=> ( ( ( P
= ( nil @ ( product_prod @ nat @ nat ) ) )
=> ( T != S ) )
=> ~ ! [P6: list @ ( product_prod @ nat @ nat ),U4: nat] :
( ( P
= ( cons @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ S @ U4 ) @ P6 ) )
=> ( ( member @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ S @ U4 ) @ ( e @ a @ c ) )
=> ~ ( isPath @ a @ c @ U4 @ P6 @ T ) ) ) ) ) ).
% isPath_fwd_cases
thf(fact_230_isPath_Oelims_I1_J,axiom,
! [X2: nat,Xa2: list @ ( product_prod @ nat @ nat ),Xb: nat,Y2: $o] :
( ( ( isPath @ a @ c @ X2 @ Xa2 @ Xb )
= Y2 )
=> ( ( ( Xa2
= ( nil @ ( product_prod @ nat @ nat ) ) )
=> ( Y2
= ( X2 != Xb ) ) )
=> ~ ! [X: nat,Y4: nat,P5: list @ ( product_prod @ nat @ nat )] :
( ( Xa2
= ( cons @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ X @ Y4 ) @ P5 ) )
=> ( Y2
= ( ~ ( ( X2 = X )
& ( member @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ X @ Y4 ) @ ( e @ a @ c ) )
& ( isPath @ a @ c @ Y4 @ P5 @ Xb ) ) ) ) ) ) ) ).
% isPath.elims(1)
thf(fact_231_list_Odistinct_I1_J,axiom,
! [A: $tType,X21: A,X22: list @ A] :
( ( nil @ A )
!= ( cons @ A @ X21 @ X22 ) ) ).
% list.distinct(1)
thf(fact_232_list_OdiscI,axiom,
! [A: $tType,List: list @ A,X21: A,X22: list @ A] :
( ( List
= ( cons @ A @ X21 @ X22 ) )
=> ( List
!= ( nil @ A ) ) ) ).
% list.discI
thf(fact_233_list_Oexhaust,axiom,
! [A: $tType,Y2: list @ A] :
( ( Y2
!= ( nil @ A ) )
=> ~ ! [X212: A,X222: list @ A] :
( Y2
!= ( cons @ A @ X212 @ X222 ) ) ) ).
% list.exhaust
thf(fact_234_list_Oinducts,axiom,
! [A: $tType,P2: ( list @ A ) > $o,List: list @ A] :
( ( P2 @ ( nil @ A ) )
=> ( ! [X1: A,X23: list @ A] :
( ( P2 @ X23 )
=> ( P2 @ ( cons @ A @ X1 @ X23 ) ) )
=> ( P2 @ List ) ) ) ).
% list.inducts
thf(fact_235_list__induct2,axiom,
! [A: $tType,B: $tType,Xs: list @ A,Ys: list @ B,P2: ( list @ A ) > ( list @ B ) > $o] :
( ( ( size_size @ ( list @ A ) @ Xs )
= ( size_size @ ( list @ B ) @ Ys ) )
=> ( ( P2 @ ( nil @ A ) @ ( nil @ B ) )
=> ( ! [X: A,Xs2: list @ A,Y4: B,Ys2: list @ B] :
( ( ( size_size @ ( list @ A ) @ Xs2 )
= ( size_size @ ( list @ B ) @ Ys2 ) )
=> ( ( P2 @ Xs2 @ Ys2 )
=> ( P2 @ ( cons @ A @ X @ Xs2 ) @ ( cons @ B @ Y4 @ Ys2 ) ) ) )
=> ( P2 @ Xs @ Ys ) ) ) ) ).
% list_induct2
thf(fact_236_list__induct3,axiom,
! [B: $tType,A: $tType,C2: $tType,Xs: list @ A,Ys: list @ B,Zs: list @ C2,P2: ( list @ A ) > ( list @ B ) > ( list @ C2 ) > $o] :
( ( ( size_size @ ( list @ A ) @ Xs )
= ( size_size @ ( list @ B ) @ Ys ) )
=> ( ( ( size_size @ ( list @ B ) @ Ys )
= ( size_size @ ( list @ C2 ) @ Zs ) )
=> ( ( P2 @ ( nil @ A ) @ ( nil @ B ) @ ( nil @ C2 ) )
=> ( ! [X: A,Xs2: list @ A,Y4: B,Ys2: list @ B,Z3: C2,Zs2: list @ C2] :
( ( ( size_size @ ( list @ A ) @ Xs2 )
= ( size_size @ ( list @ B ) @ Ys2 ) )
=> ( ( ( size_size @ ( list @ B ) @ Ys2 )
= ( size_size @ ( list @ C2 ) @ Zs2 ) )
=> ( ( P2 @ Xs2 @ Ys2 @ Zs2 )
=> ( P2 @ ( cons @ A @ X @ Xs2 ) @ ( cons @ B @ Y4 @ Ys2 ) @ ( cons @ C2 @ Z3 @ Zs2 ) ) ) ) )
=> ( P2 @ Xs @ Ys @ Zs ) ) ) ) ) ).
% list_induct3
thf(fact_237_neq__Nil__conv,axiom,
! [A: $tType,Xs: list @ A] :
( ( Xs
!= ( nil @ A ) )
= ( ? [Y3: A,Ys3: list @ A] :
( Xs
= ( cons @ A @ Y3 @ Ys3 ) ) ) ) ).
% neq_Nil_conv
thf(fact_238_list__induct2_H,axiom,
! [A: $tType,B: $tType,P2: ( list @ A ) > ( list @ B ) > $o,Xs: list @ A,Ys: list @ B] :
( ( P2 @ ( nil @ A ) @ ( nil @ B ) )
=> ( ! [X: A,Xs2: list @ A] : ( P2 @ ( cons @ A @ X @ Xs2 ) @ ( nil @ B ) )
=> ( ! [Y4: B,Ys2: list @ B] : ( P2 @ ( nil @ A ) @ ( cons @ B @ Y4 @ Ys2 ) )
=> ( ! [X: A,Xs2: list @ A,Y4: B,Ys2: list @ B] :
( ( P2 @ Xs2 @ Ys2 )
=> ( P2 @ ( cons @ A @ X @ Xs2 ) @ ( cons @ B @ Y4 @ Ys2 ) ) )
=> ( P2 @ Xs @ Ys ) ) ) ) ) ).
% list_induct2'
thf(fact_239_splice_Oinduct,axiom,
! [A: $tType,P2: ( list @ A ) > ( list @ A ) > $o,A0: list @ A,A12: list @ A] :
( ! [X_12: list @ A] : ( P2 @ ( nil @ A ) @ X_12 )
=> ( ! [X: A,Xs2: list @ A,Ys2: list @ A] :
( ( P2 @ Ys2 @ Xs2 )
=> ( P2 @ ( cons @ A @ X @ Xs2 ) @ Ys2 ) )
=> ( P2 @ A0 @ A12 ) ) ) ).
% splice.induct
thf(fact_240_induct__list012,axiom,
! [A: $tType,P2: ( list @ A ) > $o,Xs: list @ A] :
( ( P2 @ ( nil @ A ) )
=> ( ! [X: A] : ( P2 @ ( cons @ A @ X @ ( nil @ A ) ) )
=> ( ! [X: A,Y4: A,Zs2: list @ A] :
( ( P2 @ Zs2 )
=> ( ( P2 @ ( cons @ A @ Y4 @ Zs2 ) )
=> ( P2 @ ( cons @ A @ X @ ( cons @ A @ Y4 @ Zs2 ) ) ) ) )
=> ( P2 @ Xs ) ) ) ) ).
% induct_list012
thf(fact_241_min__list_Ocases,axiom,
! [A: $tType] :
( ( ord @ A )
=> ! [X2: list @ A] :
( ! [X: A,Xs2: list @ A] :
( X2
!= ( cons @ A @ X @ Xs2 ) )
=> ( X2
= ( nil @ A ) ) ) ) ).
% min_list.cases
thf(fact_242_min__list_Oinduct,axiom,
! [A: $tType] :
( ( ord @ A )
=> ! [P2: ( list @ A ) > $o,A0: list @ A] :
( ! [X: A,Xs2: list @ A] :
( ! [X213: A,X223: list @ A] :
( ( Xs2
= ( cons @ A @ X213 @ X223 ) )
=> ( P2 @ Xs2 ) )
=> ( P2 @ ( cons @ A @ X @ Xs2 ) ) )
=> ( ( P2 @ ( nil @ A ) )
=> ( P2 @ A0 ) ) ) ) ).
% min_list.induct
thf(fact_243_shuffles_Oinduct,axiom,
! [A: $tType,P2: ( list @ A ) > ( list @ A ) > $o,A0: list @ A,A12: list @ A] :
( ! [X_12: list @ A] : ( P2 @ ( nil @ A ) @ X_12 )
=> ( ! [Xs2: list @ A] : ( P2 @ Xs2 @ ( nil @ A ) )
=> ( ! [X: A,Xs2: list @ A,Y4: A,Ys2: list @ A] :
( ( P2 @ Xs2 @ ( cons @ A @ Y4 @ Ys2 ) )
=> ( ( P2 @ ( cons @ A @ X @ Xs2 ) @ Ys2 )
=> ( P2 @ ( cons @ A @ X @ Xs2 ) @ ( cons @ A @ Y4 @ Ys2 ) ) ) )
=> ( P2 @ A0 @ A12 ) ) ) ) ).
% shuffles.induct
thf(fact_244_transpose_Ocases,axiom,
! [A: $tType,X2: list @ ( list @ A )] :
( ( X2
!= ( nil @ ( list @ A ) ) )
=> ( ! [Xss: list @ ( list @ A )] :
( X2
!= ( cons @ ( list @ A ) @ ( nil @ A ) @ Xss ) )
=> ~ ! [X: A,Xs2: list @ A,Xss: list @ ( list @ A )] :
( X2
!= ( cons @ ( list @ A ) @ ( cons @ A @ X @ Xs2 ) @ Xss ) ) ) ) ).
% transpose.cases
thf(fact_245_remdups__adj_Ocases,axiom,
! [A: $tType,X2: list @ A] :
( ( X2
!= ( nil @ A ) )
=> ( ! [X: A] :
( X2
!= ( cons @ A @ X @ ( nil @ A ) ) )
=> ~ ! [X: A,Y4: A,Xs2: list @ A] :
( X2
!= ( cons @ A @ X @ ( cons @ A @ Y4 @ Xs2 ) ) ) ) ) ).
% remdups_adj.cases
thf(fact_246_sorted__wrt_Oinduct,axiom,
! [A: $tType,P2: ( A > A > $o ) > ( list @ A ) > $o,A0: A > A > $o,A12: list @ A] :
( ! [P7: A > A > $o] : ( P2 @ P7 @ ( nil @ A ) )
=> ( ! [P7: A > A > $o,X: A,Ys2: list @ A] :
( ( P2 @ P7 @ Ys2 )
=> ( P2 @ P7 @ ( cons @ A @ X @ Ys2 ) ) )
=> ( P2 @ A0 @ A12 ) ) ) ).
% sorted_wrt.induct
thf(fact_247_remdups__adj_Oinduct,axiom,
! [A: $tType,P2: ( list @ A ) > $o,A0: list @ A] :
( ( P2 @ ( nil @ A ) )
=> ( ! [X: A] : ( P2 @ ( cons @ A @ X @ ( nil @ A ) ) )
=> ( ! [X: A,Y4: A,Xs2: list @ A] :
( ( ( X = Y4 )
=> ( P2 @ ( cons @ A @ X @ Xs2 ) ) )
=> ( ( ( X != Y4 )
=> ( P2 @ ( cons @ A @ Y4 @ Xs2 ) ) )
=> ( P2 @ ( cons @ A @ X @ ( cons @ A @ Y4 @ Xs2 ) ) ) ) )
=> ( P2 @ A0 ) ) ) ) ).
% remdups_adj.induct
thf(fact_248_arg__min__list_Oinduct,axiom,
! [B: $tType,A: $tType] :
( ( linorder @ B )
=> ! [P2: ( A > B ) > ( list @ A ) > $o,A0: A > B,A12: list @ A] :
( ! [F4: A > B,X: A] : ( P2 @ F4 @ ( cons @ A @ X @ ( nil @ A ) ) )
=> ( ! [F4: A > B,X: A,Y4: A,Zs2: list @ A] :
( ( P2 @ F4 @ ( cons @ A @ Y4 @ Zs2 ) )
=> ( P2 @ F4 @ ( cons @ A @ X @ ( cons @ A @ Y4 @ Zs2 ) ) ) )
=> ( ! [A5: A > B] : ( P2 @ A5 @ ( nil @ A ) )
=> ( P2 @ A0 @ A12 ) ) ) ) ) ).
% arg_min_list.induct
thf(fact_249_successively_Oinduct,axiom,
! [A: $tType,P2: ( A > A > $o ) > ( list @ A ) > $o,A0: A > A > $o,A12: list @ A] :
( ! [P7: A > A > $o] : ( P2 @ P7 @ ( nil @ A ) )
=> ( ! [P7: A > A > $o,X: A] : ( P2 @ P7 @ ( cons @ A @ X @ ( nil @ A ) ) )
=> ( ! [P7: A > A > $o,X: A,Y4: A,Xs2: list @ A] :
( ( P2 @ P7 @ ( cons @ A @ Y4 @ Xs2 ) )
=> ( P2 @ P7 @ ( cons @ A @ X @ ( cons @ A @ Y4 @ Xs2 ) ) ) )
=> ( P2 @ A0 @ A12 ) ) ) ) ).
% successively.induct
thf(fact_250_list__nonempty__induct,axiom,
! [A: $tType,Xs: list @ A,P2: ( list @ A ) > $o] :
( ( Xs
!= ( nil @ A ) )
=> ( ! [X: A] : ( P2 @ ( cons @ A @ X @ ( nil @ A ) ) )
=> ( ! [X: A,Xs2: list @ A] :
( ( Xs2
!= ( nil @ A ) )
=> ( ( P2 @ Xs2 )
=> ( P2 @ ( cons @ A @ X @ Xs2 ) ) ) )
=> ( P2 @ Xs ) ) ) ) ).
% list_nonempty_induct
thf(fact_251_map__tailrec__rev_Oinduct,axiom,
! [A: $tType,B: $tType,P2: ( A > B ) > ( list @ A ) > ( list @ B ) > $o,A0: A > B,A12: list @ A,A23: list @ B] :
( ! [F4: A > B,X_12: list @ B] : ( P2 @ F4 @ ( nil @ A ) @ X_12 )
=> ( ! [F4: A > B,A5: A,As: list @ A,Bs: list @ B] :
( ( P2 @ F4 @ As @ ( cons @ B @ ( F4 @ A5 ) @ Bs ) )
=> ( P2 @ F4 @ ( cons @ A @ A5 @ As ) @ Bs ) )
=> ( P2 @ A0 @ A12 @ A23 ) ) ) ).
% map_tailrec_rev.induct
thf(fact_252_strict__sorted_Ocases,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [X2: list @ A] :
( ( X2
!= ( nil @ A ) )
=> ~ ! [X: A,Ys2: list @ A] :
( X2
!= ( cons @ A @ X @ Ys2 ) ) ) ) ).
% strict_sorted.cases
thf(fact_253_strict__sorted_Oinduct,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [P2: ( list @ A ) > $o,A0: list @ A] :
( ( P2 @ ( nil @ A ) )
=> ( ! [X: A,Ys2: list @ A] :
( ( P2 @ Ys2 )
=> ( P2 @ ( cons @ A @ X @ Ys2 ) ) )
=> ( P2 @ A0 ) ) ) ) ).
% strict_sorted.induct
thf(fact_254_Graph_OisPath_Oinduct,axiom,
! [P2: nat > ( list @ ( product_prod @ nat @ nat ) ) > nat > $o,A0: nat,A12: list @ ( product_prod @ nat @ nat ),A23: nat] :
( ! [U4: nat,X_12: nat] : ( P2 @ U4 @ ( nil @ ( product_prod @ nat @ nat ) ) @ X_12 )
=> ( ! [U4: nat,X: nat,Y4: nat,P5: list @ ( product_prod @ nat @ nat ),V4: nat] :
( ( P2 @ Y4 @ P5 @ V4 )
=> ( P2 @ U4 @ ( cons @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ X @ Y4 ) @ P5 ) @ V4 ) )
=> ( P2 @ A0 @ A12 @ A23 ) ) ) ).
% Graph.isPath.induct
thf(fact_255_Graph_OisPath_Osimps_I1_J,axiom,
! [Capacity: $tType] :
( ( linordered_idom @ Capacity )
=> ! [C: ( product_prod @ nat @ nat ) > Capacity,U2: nat,V: nat] :
( ( isPath @ Capacity @ C @ U2 @ ( nil @ ( product_prod @ nat @ nat ) ) @ V )
= ( U2 = V ) ) ) ).
% Graph.isPath.simps(1)
% Subclasses (5)
thf(subcl_Rings_Olinordered__idom___HOL_Otype,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ( type @ A ) ) ).
thf(subcl_Rings_Olinordered__idom___Orderings_Oord,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ( ord @ A ) ) ).
thf(subcl_Rings_Olinordered__idom___Orderings_Oorder,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ( order @ A ) ) ).
thf(subcl_Rings_Olinordered__idom___Orderings_Olinorder,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ( linorder @ A ) ) ).
thf(subcl_Rings_Olinordered__idom___Orderings_Opreorder,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ( preorder @ A ) ) ).
% Type constructors (26)
thf(tcon_fun___Lattices_Osemilattice__inf,axiom,
! [A7: $tType,A8: $tType] :
( ( semilattice_inf @ A8 )
=> ( semilattice_inf @ ( A7 > A8 ) ) ) ).
thf(tcon_fun___Orderings_Opreorder,axiom,
! [A7: $tType,A8: $tType] :
( ( preorder @ A8 )
=> ( preorder @ ( A7 > A8 ) ) ) ).
thf(tcon_fun___Finite__Set_Ofinite,axiom,
! [A7: $tType,A8: $tType] :
( ( ( finite_finite @ A7 )
& ( finite_finite @ A8 ) )
=> ( finite_finite @ ( A7 > A8 ) ) ) ).
thf(tcon_fun___Lattices_Olattice,axiom,
! [A7: $tType,A8: $tType] :
( ( lattice @ A8 )
=> ( lattice @ ( A7 > A8 ) ) ) ).
thf(tcon_fun___Orderings_Oorder,axiom,
! [A7: $tType,A8: $tType] :
( ( order @ A8 )
=> ( order @ ( A7 > A8 ) ) ) ).
thf(tcon_fun___Orderings_Oord,axiom,
! [A7: $tType,A8: $tType] :
( ( ord @ A8 )
=> ( ord @ ( A7 > A8 ) ) ) ).
thf(tcon_Nat_Onat___Lattices_Osemilattice__inf_1,axiom,
semilattice_inf @ nat ).
thf(tcon_Nat_Onat___Orderings_Opreorder_2,axiom,
preorder @ nat ).
thf(tcon_Nat_Onat___Orderings_Olinorder,axiom,
linorder @ nat ).
thf(tcon_Nat_Onat___Lattices_Olattice_3,axiom,
lattice @ nat ).
thf(tcon_Nat_Onat___Orderings_Oorder_4,axiom,
order @ nat ).
thf(tcon_Nat_Onat___Orderings_Oord_5,axiom,
ord @ nat ).
thf(tcon_Set_Oset___Lattices_Osemilattice__inf_6,axiom,
! [A7: $tType] : ( semilattice_inf @ ( set @ A7 ) ) ).
thf(tcon_Set_Oset___Orderings_Opreorder_7,axiom,
! [A7: $tType] : ( preorder @ ( set @ A7 ) ) ).
thf(tcon_Set_Oset___Finite__Set_Ofinite_8,axiom,
! [A7: $tType] :
( ( finite_finite @ A7 )
=> ( finite_finite @ ( set @ A7 ) ) ) ).
thf(tcon_Set_Oset___Lattices_Olattice_9,axiom,
! [A7: $tType] : ( lattice @ ( set @ A7 ) ) ).
thf(tcon_Set_Oset___Orderings_Oorder_10,axiom,
! [A7: $tType] : ( order @ ( set @ A7 ) ) ).
thf(tcon_Set_Oset___Orderings_Oord_11,axiom,
! [A7: $tType] : ( ord @ ( set @ A7 ) ) ).
thf(tcon_HOL_Obool___Lattices_Osemilattice__inf_12,axiom,
semilattice_inf @ $o ).
thf(tcon_HOL_Obool___Orderings_Opreorder_13,axiom,
preorder @ $o ).
thf(tcon_HOL_Obool___Orderings_Olinorder_14,axiom,
linorder @ $o ).
thf(tcon_HOL_Obool___Finite__Set_Ofinite_15,axiom,
finite_finite @ $o ).
thf(tcon_HOL_Obool___Lattices_Olattice_16,axiom,
lattice @ $o ).
thf(tcon_HOL_Obool___Orderings_Oorder_17,axiom,
order @ $o ).
thf(tcon_HOL_Obool___Orderings_Oord_18,axiom,
ord @ $o ).
thf(tcon_Product__Type_Oprod___Finite__Set_Ofinite_19,axiom,
! [A7: $tType,A8: $tType] :
( ( ( finite_finite @ A7 )
& ( finite_finite @ A8 ) )
=> ( finite_finite @ ( product_prod @ A7 @ A8 ) ) ) ).
% Free types (1)
thf(tfree_0,hypothesis,
linordered_idom @ a ).
% Conjectures (1)
thf(conj_0,conjecture,
ord_less_eq @ ( set @ ( product_prod @ nat @ nat ) ) @ edges @ ( e @ a @ c ) ).
%------------------------------------------------------------------------------