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add Nele's topological sort to examples directory
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| class Kahn | ||
| { | ||
| //graph[i][j]: there is an edge from Node i to Node j | ||
| // `G` is a graph in from of an adjacency matrix if `G` is an `n*n` matrix. | ||
| pure boolean isAdjacencyMatrix(int n, seq<seq<boolean>> G) = |G| == n && (\forall seq<boolean> e; e in G; {:|e|:} == n); | ||
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| pure boolean distinct(seq<int> s) = | ||
| (\forall int i = 0 .. |s|-1; | ||
| (\forall int j = 0 .. |s|; s[i] == s[j] ==> i == j)); | ||
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| requires(distinct(xs)); | ||
| requires(xs.contains(n)); | ||
| ensures (\result < |xs| && \result >= 0); | ||
| ensures (xs[\result] == n); | ||
| pure int getPosition(seq<int> xs, int n) = | ||
| xs[0] == n ? 0 : 1 + getPosition(xs[1..], n); | ||
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| requires(distinct(xs)); | ||
| requires(xs.contains(n)); | ||
| ensures(|\result| == |xs|-1); | ||
| pure seq<int> removeValue(seq<int> xs, int n) = | ||
| xs[..getPosition(xs, n)] + xs[getPosition(xs, n)+1..]; | ||
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| inline pure int last(seq<int> xs) = xs[|xs| - 1]; | ||
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| yields seq<int> topResult; | ||
| context_everywhere N >= 0; | ||
| context_everywhere isAdjacencyMatrix(N, graph); | ||
| context_everywhere indegree != null; | ||
| context_everywhere indegree.length == N; | ||
| context_everywhere Perm(indegree[*], write); | ||
| requires (\forall int j; 0 <= j && j < N; {:indegree[j]:} == seq<int> { }); //indegree list is all zero at the beginning | ||
| ensures (\forall int j; 0 <= j && j < |topResult|; {:topResult[j]:} >= 0 && {:topResult[j]:} < N); | ||
| ensures distinct(topResult); | ||
| ensures (\forall int j; 0 <= j && j < |topResult|; (\forall int i; 0 <= i && i < N; graph[i][topResult[j]] ==> topResult[..j].contains(i))); //TopologySort!! | ||
| boolean topologySort(int N, seq<seq<boolean>> graph, seq<int>[] indegree) | ||
| { | ||
| topResult = seq<int> { }; | ||
| seq<int> queue = seq<int> { }; | ||
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| loop_invariant 0 <= toIx && toIx <= N; | ||
| loop_invariant (\forall int j; 0 <= j && j < N; (\forall int i; 0 <= i && i < |indegree[j]|; {:indegree[j][i]:} >= 0 && {:indegree[j][i]:} < N)); | ||
| loop_invariant (\forall int j; 0 <= j && j < N; distinct({:indegree[j]:})); | ||
| loop_invariant (\forall int j; 0 <= j && j < toIx; (\forall int i; 0 <= i && i < |indegree[j]|; graph[{:indegree[j][i]:}][j])); | ||
| loop_invariant (\forall int i; 0 <= i && i < toIx; (\forall int j; 0 <= j && j < N; {:graph[j][i]:} ==> indegree[i].contains(j))); | ||
| loop_invariant (\forall int j; toIx <= j && j < N; {:indegree[j]:} == seq<int> { }); | ||
| for (int toIx = 0; toIx < N; toIx++) | ||
| { | ||
| seq<seq<int>> temp = \values(indegree, 0, N); | ||
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| loop_invariant 0 <= fromIx && fromIx <= N; | ||
| loop_invariant 0 <= toIx && toIx < N; | ||
| loop_invariant (\forall int j; 0 <= j && j < toIx; {:indegree[j]:} == temp[j]); | ||
| loop_invariant (\forall int j; toIx < j && j < N; {:indegree[j]:} == temp[j]); | ||
| loop_invariant (\forall int j; 0 <= j && j < |indegree[toIx]|; {:indegree[toIx][j]:} >= 0 && {:indegree[toIx][j]:} < N); | ||
| loop_invariant (\forall int j; 0 <= j && j < |indegree[toIx]|; {:indegree[toIx][j]:} < fromIx); | ||
| loop_invariant (distinct(indegree[toIx])); | ||
| loop_invariant (\forall int j; 0 <= j && j < fromIx; {:graph[j][toIx]:} ==> indegree[toIx].contains(j)); | ||
| loop_invariant (\forall int i; 0 <= i && i < |indegree[toIx]|; {:graph[indegree[toIx][i]][toIx]:}); | ||
| for (int fromIx = 0; fromIx < N; fromIx++) | ||
| { | ||
| if (graph[fromIx][toIx] == true) | ||
| { | ||
| indegree[toIx] = indegree[toIx] + seq<int> {fromIx}; | ||
| } | ||
| } | ||
| } | ||
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| loop_invariant 0 <= toIndex && toIndex <= N; | ||
| loop_invariant (\forall int j; 0 <= j && j < N; (\forall int i; 0 <= i && i < |indegree[j]|; {:indegree[j][i]:} >= 0 && {:indegree[j][i]:} < N)); | ||
| loop_invariant (\forall int j; 0 <= j && j < N; distinct({:indegree[j]:})); | ||
| loop_invariant (\forall int j; 0 <= j && j < |queue|; {:queue[j]:} >= 0 && {:queue[j]:} < N); | ||
| loop_invariant (\forall int j; 0 <= j && j < |queue|; |indegree[{:queue[j]:}]| == 0); | ||
| loop_invariant (\forall int j; 0 <= j && j < toIndex; {:|indegree[j]|:} == 0 ==> queue.contains(j)); | ||
| loop_invariant (\forall int j; 0 <= j && j < |queue|; {:queue[j]:} < toIndex); | ||
| loop_invariant (distinct(queue)); | ||
| loop_invariant (\forall int j; 0 <= j && j < N; (\forall int i; 0 <= i && i < |indegree[j]|; {:graph[indegree[j][i]][j]:})); | ||
| loop_invariant (\forall int j; 0 <= j && j < N; (\forall int i; 0 <= i && i < N; {:graph[i][j]:} ==> indegree[j].contains(i))); | ||
| for (int toIndex = 0; toIndex < N; toIndex++) | ||
| { | ||
| if (|indegree[toIndex]| == 0) | ||
| { | ||
| queue = queue + seq<int> { toIndex }; | ||
| } | ||
| } | ||
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| loop_invariant (\forall int j; 0 <= j && j < N; (\forall int i; 0 <= i && i < |indegree[j]|; {:indegree[j][i]:} >= 0 && indegree[j][i] < N)); | ||
| loop_invariant (\forall int j; 0 <= j && j < N; distinct({:indegree[j]:})); | ||
| loop_invariant (\forall int j; 0 <= j && j < |queue|; {:queue[j]:} >= 0 && {:queue[j]:} < N); | ||
| loop_invariant (\forall int j; 0 <= j && j < |topResult|; {:topResult[j]:} >= 0 && {:topResult[j]:} < N); | ||
| loop_invariant (\forall int j; 0 <= j && j < |queue|; {:|indegree[queue[j]]|:} == 0); | ||
| loop_invariant (\forall int j; 0 <= j && j < |topResult|; {:|indegree[topResult[j]]|:} == 0); | ||
| loop_invariant distinct(topResult); | ||
| loop_invariant distinct(queue); | ||
| loop_invariant distinct(topResult + queue); | ||
| loop_invariant (\forall int j; 0 <= j && j < |topResult|; !{:queue.contains(topResult[j]):}); | ||
| loop_invariant (\forall int j; 0 <= j && j < N; {:|indegree[j]|:} > 0 ==> !topResult.contains(j) && !queue.contains(j)); | ||
| loop_invariant (\forall int j; 0 <= j && j < N; (\forall int i; 0 <= i && i < |indegree[j]|; {:graph[indegree[j][i]][j]:})); | ||
| loop_invariant (\forall int j; 0 <= j && j < |queue|; (\forall int i; 0 <= i && i < N; graph[i][queue[j]] ==> topResult.contains(i))); | ||
| loop_invariant (\forall int j; 0 <= j && j < N; (|indegree[j]| == 0 ==> (\forall int i; 0 <= i && i < N; (graph[i][j] ==> topResult.contains(i))))); | ||
| loop_invariant (\forall int j; 0 <= j && j < N; (\forall int i; 0 <= i && i < N; (graph[i][j] && !indegree[j].contains(i)) ==> topResult.contains(i))); | ||
| loop_invariant (\forall int j; 0 <= j && j < |topResult|; (\forall int i; 0 <= i && i < N; graph[i][topResult[j]] ==> topResult[..j].contains(i))); | ||
| while (|queue| > 0) | ||
| { | ||
| seq<seq<int>> temp2 = \values(indegree, 0, N); | ||
| seq<int> temppppQueue = queue; | ||
| int currentNode = queue[0]; | ||
| queue = queue[1..]; | ||
| // the assert below was not needed in VerCors v2.0.0-beta.1, but became necessary in 2.0.0 | ||
| assert (\let seq<int> concated = topResult+temppppQueue; currentNode == concated[|topResult|] && queue == concated[|topResult|+1..]); | ||
| topResult = topResult + seq<int> { currentNode }; | ||
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| loop_invariant 0 <= toIndex && toIndex <= N; | ||
| loop_invariant 0 <= currentNode && currentNode < N; | ||
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| loop_invariant (\forall int j; 0 <= j && j < N; (\forall int i; 0 <= i && i < |indegree[j]|; {:indegree[j][i]:} >= 0 && indegree[j][i] < N)); | ||
| loop_invariant (\forall int j; 0 <= j && j < N; distinct({:indegree[j]:})); | ||
| loop_invariant (\forall int j; toIndex <= j && j < N; temp2[j] == {:indegree[j]:}); | ||
| loop_invariant (\forall int j; 0 <= j && j < |topResult|; {:topResult[j]:} >= 0 && {:topResult[j]:} < N); | ||
| loop_invariant (\forall int j; 0 <= j && j < |queue|; {:queue[j]:} >= 0 && {:queue[j]:} < N); | ||
| loop_invariant (\forall int j; 0 <= j && j < |topResult|; {:|indegree[topResult[j]]|:} == 0); | ||
| loop_invariant (\forall int j; 0 <= j && j < |queue|; {:|indegree[queue[j]]|:} == 0); | ||
| loop_invariant distinct(topResult); | ||
| loop_invariant distinct(queue); | ||
| loop_invariant distinct(topResult + queue); | ||
| loop_invariant (\forall int j; 0 <= j && j < |queue|; !topResult.contains({:queue[j]:})); | ||
| loop_invariant (\forall int j; 0 <= j && j < N; {:|indegree[j]|:} > 0 ==> (!topResult.contains(j) && !queue.contains(j))); | ||
| loop_invariant (\forall int j; 0 <= j && j < N; (\forall int i; 0 <= i && i < |indegree[j]|; {:graph[indegree[j][i]][j]:})); | ||
| loop_invariant (\forall int j; 0 <= j && j < |queue|; (\forall int i; 0 <= i && i < N; graph[i][queue[j]] ==> topResult.contains(i))); | ||
| loop_invariant (\forall int j; 0 <= j && j < N; (|indegree[j]| == 0 ==> (\forall int i; 0 <= i && i < N; (graph[i][j] ==> topResult.contains(i))))); | ||
| loop_invariant (\forall int j; 0 <= j && j < N; (\forall int i; 0 <= i && i < N; (graph[i][j] && !indegree[j].contains(i)) ==> topResult.contains(i))); | ||
| loop_invariant (\forall int j; 0 <= j && j < |topResult|; (\forall int i; 0 <= i && i < N; graph[i][topResult[j]] ==> topResult[..j].contains(i))); | ||
| for (int toIndex = 0; toIndex < N; toIndex++) | ||
| { | ||
| if (graph[currentNode][toIndex] == true){ | ||
| indegree[toIndex] = removeValue(indegree[toIndex], currentNode); | ||
| if (|indegree[toIndex]| == 0){ | ||
| queue = queue + seq<int> { toIndex }; | ||
| } | ||
| } | ||
| } | ||
| } | ||
| return |topResult| == N; | ||
| } | ||
| } |
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