(** Warshall algorithm Computes the transitive closure of a graph implemented as a Boolean matrix. Authors: Martin Clochard (École Normale Supérieure) Jean-Christophe Filliâtre (CNRS) *) module WarshallAlgorithm use import int.Int use import matrix.Matrix (* path m i j k = there is a path from i to j, using only vertices smaller than k *) inductive path (matrix bool) int int int = | Path_empty: forall m: matrix bool, i j k: int. get m i j -> path m i j k | Path_cons: forall m: matrix bool, i x j k: int. 0 <= x < k -> path m i x k -> path m x j k -> path m i j k lemma weakening: forall m i j k1 k2. 0 <= k1 <= k2 -> path m i j k1 -> path m i j k2 lemma decomposition: forall m k. 0 <= k -> forall i j. path m i j (k+1) -> (path m i j k \/ (path m i k k /\ path m k j k)) let transitive_closure (m: matrix bool) : matrix bool requires { m.rows = m.columns } ensures { let n = m.rows in forall x y: int. 0 <= x < n -> 0 <= y < n -> get result x y <-> path m x y n } = let t = copy m in let n = m.rows in for k = 0 to n - 1 do invariant { forall x y. 0 <= x < n -> 0 <= y < n -> get t x y <-> path m x y k } for i = 0 to n - 1 do invariant { forall x y. 0 <= x < n -> 0 <= y < n -> get t x y <-> if x < i then path m x y (k+1) else path m x y k } for j = 0 to n - 1 do invariant { forall x y. 0 <= x < n -> 0 <= y < n -> get t x y <-> if x < i \/ (x = i /\ y < j) then path m x y (k+1) else path m x y k } set t i j (get t i j || get t i k && get t k j) done done done; t end