GLIB_004: Proof of Dijkstra's Shortest Path Algorithm & Prim's Minimum Spanning Tree Algorithm

:: Proof of Dijkstra's Shortest Path Algorithm & Prim's Minimum Spanning Tree Algorithm
:: by Gilbert Lee and Piotr Rudnicki
::
:: Received February 22, 2005
:: Copyright (c) 2005-2021 Association of Mizar Users

Lm1: for F being Function
for x, y being set holds dom (F +* (x .--> y)) = (dom F) \/ {x}

proof end;

Lm2: for F being Function
for x, y, z being set st z in dom (F +* (x .--> y)) & not z in dom F holds
x = z

proof end;

theorem Th1: :: GLIB_004:1

for f, g being Function holds support (f +* g) c= (support f) \/ (support g)

proof end;

theorem Th2: :: GLIB_004:2

for f being Function
for x, y being object holds support (f +* (x .--> y)) c= (support f) \/ {x}

proof end;

theorem Th3: :: GLIB_004:3

for A, B being set
for b being Rbag of A
for b1 being Rbag of B
for b2 being Rbag of A \ B st b = b1 +* b2 holds
Sum b = (Sum b1) + (Sum b2)

proof end;

theorem Th4: :: GLIB_004:4

for X, x being set
for b being Rbag of X st dom b = {x} holds
Sum b = b . x

proof end;

theorem Th5: :: GLIB_004:5

for A being set
for b1, b2 being Rbag of A st ( for x being set st x in A holds
b1 . x <= b2 . x ) holds
Sum b1 <= Sum b2

proof end;

theorem :: GLIB_004:6

for A being set
for b1, b2 being Rbag of A st ( for x being set st x in A holds
b1 . x = b2 . x ) holds
Sum b1 = Sum b2

proof end;

theorem :: GLIB_004:7

for A1, A2 being set
for b1 being Rbag of A1
for b2 being Rbag of A2 st b1 = b2 holds
Sum b1 = Sum b2

proof end;

theorem Th8: :: GLIB_004:8

for X, x being set
for b being Rbag of X
for y being Real st b = (EmptyBag X) +* (x .--> y) holds
Sum b = y

proof end;

theorem :: GLIB_004:9

for X, x being set
for b1, b2 being Rbag of X
for y being Real st b2 = b1 +* (x .--> y) holds
Sum b2 = ((Sum b1) + y) - (b1 . x)

proof end;

definition

let G1 be real-weighted WGraph;
let G2 be WSubgraph of G1;
let v be set ;

pred G2 is_mincost_DTree_rooted_at v means :: GLIB_004:def 1
( G2 is Tree-like & ( for x being Vertex of G2 ex W2 being DPath of G2 st
( W2 is_Walk_from v,x & ( for W1 being DPath of G1 st W1 is_Walk_from v,x holds
W2 .cost() <= W1 .cost() ) ) ) );

end;

:: deftheorem defines is_mincost_DTree_rooted_at GLIB_004:def 1 :

definition

let G be real-weighted WGraph;
let W be DPath of G;
let x, y be set ;

pred W is_mincost_DPath_from x,y means :: GLIB_004:def 2
( W is_Walk_from x,y & ( for W2 being DPath of G st W2 is_Walk_from x,y holds
W .cost() <= W2 .cost() ) );

end;

:: deftheorem defines is_mincost_DPath_from GLIB_004:def 2 :

definition

let G be _finite real-weighted WGraph;
let x, y be set ;

func G .min_DPath_cost (x,y) -> Real means :Def3: :: GLIB_004:def 3
ex W being DPath of G st
( W is_mincost_DPath_from x,y & it = W .cost() ) if ex W being DWalk of G st W is_Walk_from x,y
otherwise it = 0 ;

existence

proof end;

uniqueness

proof end;

consistency ;

end;

:: deftheorem Def3 defines .min_DPath_cost GLIB_004:def 3 :

definition end;

:: deftheorem defines WGraphSelectors GLIB_004:def 4 :

Lm3: for G being WGraph holds WGraphSelectors c= dom G

proof end;

registration

let G be WGraph;

cluster G | WGraphSelectors -> [Graph-like] [Weighted] ;

coherence

proof end;

end;

Lm4: for G being WGraph holds
( G == G | WGraphSelectors & the_Weight_of G = the_Weight_of (G | WGraphSelectors) )

proof end;

Lm5: for G being WGraph holds dom (G | WGraphSelectors) = WGraphSelectors

proof end;

definition

let G be WGraph;

func G .allWSubgraphs() -> non empty set means :Def5: :: GLIB_004:def 5
for x being set holds
( x in it iff ex G2 being WSubgraph of G st
( x = G2 & dom G2 = WGraphSelectors ) );

existence

proof end;

uniqueness

proof end;

end;

:: deftheorem Def5 defines .allWSubgraphs() GLIB_004:def 5 :

registration

let G be _finite WGraph;

cluster G .allWSubgraphs() -> non empty finite ;

coherence

proof end;

end;

definition

let G be WGraph;
let X be non empty Subset of (G .allWSubgraphs());
:: original: Element
redefine mode Element of X -> WSubgraph of G;
coherence

proof end;

end;

definition

let G be _finite real-weighted WGraph;

func G .cost() -> Real equals :: GLIB_004:def 6
Sum (the_Weight_of G);

coherence ;

end;

:: deftheorem defines .cost() GLIB_004:def 6 :

theorem Th10: :: GLIB_004:10

for G1 being _finite real-weighted WGraph
for e being set
for G2 being [Weighted] weight-inheriting removeEdge of G1,e st e in the_Edges_of G1 holds
G1 .cost() = (G2 .cost()) + ((the_Weight_of G1) . e)

proof end;

theorem Th11: :: GLIB_004:11

for G being _finite real-weighted WGraph
for V1 being non empty Subset of (the_Vertices_of G)
for E1 being Subset of (G .edgesBetween V1)
for G1 being inducedWSubgraph of G,V1,E1
for e being set
for G2 being inducedWSubgraph of G,V1,E1 \/ {e} st not e in E1 & e in G .edgesBetween V1 holds
(G1 .cost()) + ((the_Weight_of G) . e) = G2 .cost()

proof end;

theorem Th12: :: GLIB_004:12

for G being _finite nonnegative-weighted WGraph
for W being DPath of G
for x, y being set
for m, n being Element of NAT st W is_mincost_DPath_from x,y holds
W .cut (m,n) is_mincost_DPath_from (W .cut (m,n)) .first() ,(W .cut (m,n)) .last()

proof end;

theorem :: GLIB_004:13

for G being _finite real-weighted WGraph
for W1, W2 being DPath of G
for x, y being set st W1 is_mincost_DPath_from x,y & W2 is_mincost_DPath_from x,y holds
W1 .cost() = W2 .cost()

proof end;

theorem Th14: :: GLIB_004:14

for G being _finite real-weighted WGraph
for W being DPath of G
for x, y being set st W is_mincost_DPath_from x,y holds
G .min_DPath_cost (x,y) = W .cost() by Def3;

definition

let G be _Graph;
mode DIJK:Labeling of G is Element of [:(PFuncs ((the_Vertices_of G),REAL)),(bool (the_Edges_of G)):];

end;

definition

let X1, X3 be set ;
let X2 be non empty set ;
let x be Element of [:(PFuncs (X1,X3)),X2:];
:: original: `1
redefine func x `1 -> Element of PFuncs (X1,X3);
coherence by MCART_1:10;

end;

registration

let G be _finite _Graph;
let L be DIJK:Labeling of G;

cluster L `1 -> finite for set ;

coherence

proof end;

cluster L `2 -> finite for set ;

coherence ;

end;

definition

let G be real-weighted WGraph;
let L be DIJK:Labeling of G;

func DIJK:NextBestEdges L -> Subset of (the_Edges_of G) means :Def7: :: GLIB_004:def 7
for e1 being set holds
( e1 in it iff ( e1 DSJoins dom (L `1),(the_Vertices_of G) \ (dom (L `1)),G & ( for e2 being set st e2 DSJoins dom (L `1),(the_Vertices_of G) \ (dom (L `1)),G holds
((L `1) . ((the_Source_of G) . e1)) + ((the_Weight_of G) . e1) <= ((L `1) . ((the_Source_of G) . e2)) + ((the_Weight_of G) . e2) ) ) );

existence

proof end;

uniqueness

proof end;

end;

:: deftheorem Def7 defines DIJK:NextBestEdges GLIB_004:def 7 :

definition

let G be real-weighted WGraph;
let L be DIJK:Labeling of G;

func DIJK:Step L -> DIJK:Labeling of G equals :Def8: :: GLIB_004:def 8
L if DIJK:NextBestEdges L = {}
otherwise [((L `1) +* (((the_Target_of G) . the Element of DIJK:NextBestEdges L) .--> (((L `1) . ((the_Source_of G) . the Element of DIJK:NextBestEdges L)) + ((the_Weight_of G) . the Element of DIJK:NextBestEdges L)))),((L `2) \/ { the Element of DIJK:NextBestEdges L})];

coherence

proof end;

consistency ;

end;

:: deftheorem Def8 defines DIJK:Step GLIB_004:def 8 :

definition

let G be real-weighted WGraph;
let src be Vertex of G;

func DIJK:Init src -> DIJK:Labeling of G equals :: GLIB_004:def 9
[(src .--> 0),{}];

coherence

proof end;

end;

:: deftheorem defines DIJK:Init GLIB_004:def 9 :

definition

let G be real-weighted WGraph;

mode DIJK:LabelingSeq of G -> ManySortedSet of NAT means :Def10: :: GLIB_004:def 10
for n being Nat holds it . n is DIJK:Labeling of G;

existence

proof end;

end;

:: deftheorem Def10 defines DIJK:LabelingSeq GLIB_004:def 10 :

definition

let G be real-weighted WGraph;
let S be DIJK:LabelingSeq of G;
let n be Nat;
:: original: .
redefine func S . n -> DIJK:Labeling of G;
coherence by Def10;

end;

definition

let G be real-weighted WGraph;
let src be Vertex of G;

func DIJK:CompSeq src -> DIJK:LabelingSeq of G means :Def11: :: GLIB_004:def 11
( it . 0 = DIJK:Init src & ( for n being Nat holds it . (n + 1) = DIJK:Step (it . n) ) );

existence

proof end;

uniqueness

proof end;

end;

:: deftheorem Def11 defines DIJK:CompSeq GLIB_004:def 11 :

definition

let G be real-weighted WGraph;
let src be Vertex of G;

func DIJK:SSSP (G,src) -> DIJK:Labeling of G equals :: GLIB_004:def 12
(DIJK:CompSeq src) .Result() ;

coherence

proof end;

end;

:: deftheorem defines DIJK:SSSP GLIB_004:def 12 :

theorem Th15: :: GLIB_004:15

for G being _finite real-weighted WGraph
for L being DIJK:Labeling of G holds
( ( card (dom ((DIJK:Step L) `1)) = card (dom (L `1)) implies DIJK:NextBestEdges L = {} ) & ( DIJK:NextBestEdges L = {} implies card (dom ((DIJK:Step L) `1)) = card (dom (L `1)) ) & ( card (dom ((DIJK:Step L) `1)) = (card (dom (L `1))) + 1 implies DIJK:NextBestEdges L <> {} ) & ( DIJK:NextBestEdges L <> {} implies card (dom ((DIJK:Step L) `1)) = (card (dom (L `1))) + 1 ) )

proof end;

theorem Th16: :: GLIB_004:16

for G being real-weighted WGraph
for L being DIJK:Labeling of G holds
( dom (L `1) c= dom ((DIJK:Step L) `1) & L `2 c= (DIJK:Step L) `2 )

proof end;

theorem :: GLIB_004:17

for G being real-weighted WGraph
for src being Vertex of G holds dom ((DIJK:Init src) `1) = {src} ;

theorem Th18: :: GLIB_004:18

for G being real-weighted WGraph
for src being Vertex of G
for i, j being Nat st i <= j holds
( dom (((DIJK:CompSeq src) . i) `1) c= dom (((DIJK:CompSeq src) . j) `1) & ((DIJK:CompSeq src) . i) `2 c= ((DIJK:CompSeq src) . j) `2 )

proof end;

theorem Th19: :: GLIB_004:19

for G being _finite real-weighted WGraph
for src being Vertex of G
for n being Nat holds dom (((DIJK:CompSeq src) . n) `1) c= G .reachableDFrom src

proof end;

theorem Th20: :: GLIB_004:20

for G being _finite real-weighted WGraph
for src being Vertex of G
for n being Nat holds
( DIJK:NextBestEdges ((DIJK:CompSeq src) . n) = {} iff dom (((DIJK:CompSeq src) . n) `1) = G .reachableDFrom src )

proof end;

theorem Th21: :: GLIB_004:21

for G being _finite real-weighted WGraph
for s being Vertex of G
for n being Nat holds card (dom (((DIJK:CompSeq s) . n) `1)) = min ((n + 1),(card (G .reachableDFrom s)))

proof end;

theorem Th22: :: GLIB_004:22

for G being _finite real-weighted WGraph
for src being Vertex of G
for n being Nat holds ((DIJK:CompSeq src) . n) `2 c= G .edgesBetween (dom (((DIJK:CompSeq src) . n) `1))

proof end;

theorem Th23: :: GLIB_004:23

for G being _finite nonnegative-weighted WGraph
for s being Vertex of G
for n being Nat
for G2 being inducedWSubgraph of G, dom (((DIJK:CompSeq s) . n) `1),((DIJK:CompSeq s) . n) `2 holds
( G2 is_mincost_DTree_rooted_at s & ( for v being Vertex of G st v in dom (((DIJK:CompSeq s) . n) `1) holds
G .min_DPath_cost (s,v) = (((DIJK:CompSeq s) . n) `1) . v ) )

proof end;

theorem Th24: :: GLIB_004:24

for G being _finite real-weighted WGraph
for s being Vertex of G holds DIJK:CompSeq s is halting

proof end;

registration

let G be _finite real-weighted WGraph;
let src be Vertex of G;

cluster DIJK:CompSeq src -> halting ;

coherence by Th24;

end;

theorem Th25: :: GLIB_004:25

for G being _finite real-weighted WGraph
for s being Vertex of G holds ((DIJK:CompSeq s) .Lifespan()) + 1 = card (G .reachableDFrom s)

proof end;

theorem Th26: :: GLIB_004:26

for G being _finite real-weighted WGraph
for s being Vertex of G holds dom ((DIJK:SSSP (G,s)) `1) = G .reachableDFrom s

proof end;

Dijkstra's shortest path algorithm

theorem :: GLIB_004:27

for G being _finite nonnegative-weighted WGraph
for s being Vertex of G
for G2 being inducedWSubgraph of G, dom ((DIJK:SSSP (G,s)) `1),(DIJK:SSSP (G,s)) `2 holds
( G2 is_mincost_DTree_rooted_at s & ( for v being Vertex of G st v in G .reachableDFrom s holds
( v in the_Vertices_of G2 & G .min_DPath_cost (s,v) = ((DIJK:SSSP (G,s)) `1) . v ) ) )

proof end;

definition

let G be _Graph;
mode PRIM:Labeling of G is Element of [:(bool (the_Vertices_of G)),(bool (the_Edges_of G)):];

end;

registration

let G be _finite _Graph;
let L be PRIM:Labeling of G;

cluster L `1 -> finite for set ;

coherence ;

cluster L `2 -> finite for set ;

coherence ;

end;

definition

let G be real-weighted WGraph;
let L be PRIM:Labeling of G;

func PRIM:NextBestEdges L -> Subset of (the_Edges_of G) means :Def13: :: GLIB_004:def 13
for e1 being set holds
( e1 in it iff ( e1 SJoins L `1 ,(the_Vertices_of G) \ (L `1),G & ( for e2 being set st e2 SJoins L `1 ,(the_Vertices_of G) \ (L `1),G holds
(the_Weight_of G) . e1 <= (the_Weight_of G) . e2 ) ) );

existence

proof end;

uniqueness

proof end;

end;

:: deftheorem Def13 defines PRIM:NextBestEdges GLIB_004:def 13 :

definition

let G be real-weighted WGraph;

func PRIM:Init G -> PRIM:Labeling of G equals :: GLIB_004:def 14
[{ the Element of the_Vertices_of G},{}];

coherence

proof end;

end;

:: deftheorem defines PRIM:Init GLIB_004:def 14 :

definition

let G be real-weighted WGraph;
let L be PRIM:Labeling of G;

func PRIM:Step L -> PRIM:Labeling of G equals :Def15: :: GLIB_004:def 15
L if PRIM:NextBestEdges L = {}
[((L `1) \/ {((the_Target_of G) . the Element of PRIM:NextBestEdges L)}),((L `2) \/ { the Element of PRIM:NextBestEdges L})] if ( PRIM:NextBestEdges L <> {} & (the_Source_of G) . the Element of PRIM:NextBestEdges L in L `1 )
otherwise [((L `1) \/ {((the_Source_of G) . the Element of PRIM:NextBestEdges L)}),((L `2) \/ { the Element of PRIM:NextBestEdges L})];

coherence

proof end;

consistency ;

end;

:: deftheorem Def15 defines PRIM:Step GLIB_004:def 15 :

definition

let G be real-weighted WGraph;

mode PRIM:LabelingSeq of G -> ManySortedSet of NAT means :Def16: :: GLIB_004:def 16
for n being Nat holds it . n is PRIM:Labeling of G;

existence

proof end;

end;

:: deftheorem Def16 defines PRIM:LabelingSeq GLIB_004:def 16 :

definition

let G be real-weighted WGraph;
let S be PRIM:LabelingSeq of G;
let n be Nat;
:: original: .
redefine func S . n -> PRIM:Labeling of G;
coherence by Def16;

end;

definition

let G be real-weighted WGraph;

func PRIM:CompSeq G -> PRIM:LabelingSeq of G means :Def17: :: GLIB_004:def 17
( it . 0 = PRIM:Init G & ( for n being Nat holds it . (n + 1) = PRIM:Step (it . n) ) );

existence

proof end;

uniqueness

proof end;

end;

:: deftheorem Def17 defines PRIM:CompSeq GLIB_004:def 17 :

definition

let G be real-weighted WGraph;

func PRIM:MST G -> PRIM:Labeling of G equals :: GLIB_004:def 18
(PRIM:CompSeq G) .Result() ;

coherence

proof end;

end;

:: deftheorem defines PRIM:MST GLIB_004:def 18 :

theorem Th28: :: GLIB_004:28

for G being real-weighted WGraph
for L being PRIM:Labeling of G st PRIM:NextBestEdges L <> {} holds
ex v being Vertex of G st
( not v in L `1 & PRIM:Step L = [((L `1) \/ {v}),((L `2) \/ { the Element of PRIM:NextBestEdges L})] )

proof end;

theorem Th29: :: GLIB_004:29

for G being real-weighted WGraph
for L being PRIM:Labeling of G holds
( L `1 c= (PRIM:Step L) `1 & L `2 c= (PRIM:Step L) `2 )

proof end;

theorem Th30: :: GLIB_004:30

for G being _finite real-weighted WGraph
for n being Nat holds
( ((PRIM:CompSeq G) . n) `1 is non empty Subset of (the_Vertices_of G) & ((PRIM:CompSeq G) . n) `2 c= G .edgesBetween (((PRIM:CompSeq G) . n) `1) )

proof end;

theorem Th31: :: GLIB_004:31

for G1 being _finite real-weighted WGraph
for n being Nat
for G2 being inducedSubgraph of G1,((PRIM:CompSeq G1) . n) `1 ,((PRIM:CompSeq G1) . n) `2 holds G2 is connected

proof end;

theorem Th32: :: GLIB_004:32

for G1 being _finite real-weighted WGraph
for n being Nat
for G2 being inducedSubgraph of G1,(((PRIM:CompSeq G1) . n) `1) holds G2 is connected

proof end;

registration

let G1 be _finite real-weighted WGraph;
let n be Nat;

cluster -> connected for inducedSubgraph of G1,((PRIM:CompSeq G1) . n) `1 ,G1 .edgesBetween (((PRIM:CompSeq G1) . n) `1);

coherence by Th32;

end;

registration

let G1 be _finite real-weighted WGraph;
let n be Nat;

cluster -> connected for inducedSubgraph of G1,((PRIM:CompSeq G1) . n) `1 ,((PRIM:CompSeq G1) . n) `2 ;

coherence by Th31;

end;

theorem Th33: :: GLIB_004:33

for G being _finite real-weighted WGraph
for n being Nat holds ((PRIM:CompSeq G) . n) `1 c= G .reachableFrom the Element of the_Vertices_of G

proof end;

theorem Th34: :: GLIB_004:34

for G being _finite real-weighted WGraph
for i, j being Nat st i <= j holds
( ((PRIM:CompSeq G) . i) `1 c= ((PRIM:CompSeq G) . j) `1 & ((PRIM:CompSeq G) . i) `2 c= ((PRIM:CompSeq G) . j) `2 )

proof end;

theorem Th35: :: GLIB_004:35

for G being _finite real-weighted WGraph
for n being Nat holds
( PRIM:NextBestEdges ((PRIM:CompSeq G) . n) = {} iff ((PRIM:CompSeq G) . n) `1 = G .reachableFrom the Element of the_Vertices_of G )

proof end;

theorem Th36: :: GLIB_004:36

for G being _finite real-weighted WGraph
for n being Nat holds card (((PRIM:CompSeq G) . n) `1) = min ((n + 1),(card (G .reachableFrom the Element of the_Vertices_of G)))

proof end;

theorem Th37: :: GLIB_004:37

for G being _finite real-weighted WGraph holds
( PRIM:CompSeq G is halting & ((PRIM:CompSeq G) .Lifespan()) + 1 = card (G .reachableFrom the Element of the_Vertices_of G) )

proof end;

theorem Th38: :: GLIB_004:38

for G1 being _finite real-weighted WGraph
for n being Nat
for G2 being inducedSubgraph of G1,((PRIM:CompSeq G1) . n) `1 ,((PRIM:CompSeq G1) . n) `2 holds G2 is Tree-like

proof end;

theorem Th39: :: GLIB_004:39

for G being _finite connected real-weighted WGraph holds (PRIM:MST G) `1 = the_Vertices_of G

proof end;

registration

let G be _finite connected real-weighted WGraph;

cluster Relation-like NAT -defined Function-like finite finite-support [Graph-like] _finite spanning Tree-like [Weighted] weight-inheriting real-weighted for Subgraph of G;

existence

proof end;

end;

definition

let G1 be _finite connected real-weighted WGraph;
let G2 be spanning Tree-like WSubgraph of G1;

attr G2 is min-cost means :Def19: :: GLIB_004:def 19
for G3 being spanning Tree-like WSubgraph of G1 holds G2 .cost() <= G3 .cost() ;

end;

:: deftheorem Def19 defines min-cost GLIB_004:def 19 :

registration

let G1 be _finite connected real-weighted WGraph;

cluster Relation-like NAT -defined Function-like finite finite-support [Graph-like] _finite loopless non-multi non-Dmulti simple Dsimple spanning connected acyclic Tree-like [Weighted] weight-inheriting real-weighted min-cost for Subgraph of G1;

existence

proof end;

end;

definition

let G be _finite connected real-weighted WGraph;
mode minimumSpanningTree of G is spanning Tree-like min-cost WSubgraph of G;

end;

theorem :: GLIB_004:40

for G1, G2 being _finite connected real-weighted WGraph
for G3 being WSubgraph of G1 st G3 is minimumSpanningTree of G1 & G1 == G2 & the_Weight_of G1 = the_Weight_of G2 holds
G3 is minimumSpanningTree of G2

proof end;

theorem Th41: :: GLIB_004:41

for G being _finite connected real-weighted WGraph
for G1 being minimumSpanningTree of G
for G2 being WGraph st G1 == G2 & the_Weight_of G1 = the_Weight_of G2 holds
G2 is minimumSpanningTree of G

proof end;

theorem Th42: :: GLIB_004:42

for G being _finite connected real-weighted WGraph
for n being Nat holds ((PRIM:CompSeq G) . n) `2 c= (PRIM:MST G) `2

proof end;

Prim's Minimum Spanning Tree Algorithm

theorem :: GLIB_004:43

for G1 being _finite connected real-weighted WGraph
for G2 being inducedWSubgraph of G1,(PRIM:MST G1) `1 ,(PRIM:MST G1) `2 holds G2 is minimumSpanningTree of G1

proof end;