Commit 87fe8329 by Martin Clochard

### (WIP) continued dyck words & Warshall algorithm

parent 75821528
 ... ... @@ -15,16 +15,16 @@ theory Dyck | Dyck_ind: forall w1 w2. dyck w1 -> dyck w2 -> dyck (Cons L (w1 ++ Cons R w2)) lemma dyck_concat: forall w1 w2. dyck w1 -> dyck w2 -> dyck (w1 ++ w2) (* the first letter, if any, must be L *) lemma dyck_word_first: forall w: word. dyck w -> match w with Nil -> true | Cons c _ -> c = L end (*(* Concatenation of dyck words is a dyck word. *) lemma dyck_concat : forall w1 w2:word. dyck w1 -> dyck w2 -> dyck (w1 ++ w2) lemma dyck_decomp: forall w1 w2: word. dyck (w1 ++ w2) -> dyck w1 -> dyck w2 forall w1 w2: word. dyck (w1 ++ w2) -> dyck w1 -> dyck w2*) end ... ... @@ -32,29 +32,53 @@ module Check use import Dyck use import list.Length use import ref.Ref exception Failure let rec is_dyck_rec (w: word) : word ensures { exists p: word. dyck p && w = p ++ result && match result with Cons L _ -> false | _ -> true end } raises { Failure -> forall p s: word. w = p ++ s -> not (dyck p) } (* A fall of a word is a decomposition p ++ s with p a dyck word and s a word not starting by L. *) predicate fall (p s:word) = dyck p /\ match s with Cons L _ -> false | _ -> true end let rec lemma same_prefix (p s s2:word) : unit requires { p ++ s = p ++ s2 } ensures { s = s2 } variant { p } = match p with Nil -> () | Cons _ q -> same_prefix q s s2 end (* Compute the fall decomposition, if it exists. As a side-effect, prove its unicity. *) let rec is_dyck_rec (ghost p:ref word) (w: word) : word ensures { w = !p ++ result && fall !p result && (forall p2 s: word. w = p2 ++ s /\ fall p2 s -> p2 = !p && s = result) } writes { p } raises { Failure -> (forall p s:word. w = p ++ s -> not fall p s) } variant { length w } = match w with | Cons L w -> match is_dyck_rec w with | Cons R w -> is_dyck_rec w | _ -> raise Failure end | _ -> w | Cons L w0 -> let ghost p0 = ref Nil in match is_dyck_rec p0 w0 with | Cons _ w1 -> assert { forall p s p1 p2:word. dyck p /\ w = p ++ s /\ dyck p1 /\ dyck p2 /\ p = Cons L (p1 ++ Cons R p2) -> w0 = p1 ++ (Cons R (p2 ++ s)) && p1 = !p0 && w1 = p2 ++ s }; let ghost p1 = ref Nil in let w = is_dyck_rec p1 w1 in p := Cons L (!p0 ++ Cons R !p1); w | _ -> raise Failure end | _ -> p := Nil; w end let is_dyck (w: word) : bool ensures { result <-> dyck w } = try is_dyck_rec w = Nil with Failure -> false end try match is_dyck_rec (ref Nil) w with | Nil -> true | _ -> false end with Failure -> false end end
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 ... ... @@ -20,8 +20,12 @@ module WarshallAlgorithm 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 <= k2 <= k1 -> forall m i j k1 k2. 0 <= k1 <= k2 -> path m i j k1 -> path m i j k2 lemma decomposition: forall m i j k. 0 <= k /\ 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 } ... ... @@ -37,12 +41,15 @@ module WarshallAlgorithm for i = 0 to n - 1 do invariant { forall x y. 0 <= x < n -> 0 <= y < n -> get t (x,y) <-> (path m x y k || x < i && path m x y (k+1)) } 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) <-> (path m x y k || x < i && path m x y (k+1) || x = i && y < j && path m x y (k+1)) } 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 ... ...
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