Commit 6dc70dbf authored by MARCHE Claude's avatar MARCHE Claude

LCP: more proofs

parent c833116f
...@@ -78,6 +78,7 @@ that it does so correctly. ...@@ -78,6 +78,7 @@ that it does so correctly.
module LCP "longest common prefix" module LCP "longest common prefix"
use import int.Int use import int.Int
use map.Map
use import array.Array use import array.Array
(* TODO: dire plus precisement que c'est une permutation de 0..b-1 *) (* TODO: dire plus precisement que c'est une permutation de 0..b-1 *)
...@@ -180,11 +181,12 @@ let test1 () = ...@@ -180,11 +181,12 @@ let test1 () =
predicate le (a : array int) (x y:int) = predicate le (a : array int) (x y:int) =
let n = a.length in let n = a.length in
0 <= x < n /\ 0 <= y < n /\
let l = longest_common_prefix a x y in let l = longest_common_prefix a x y in
x+l = n \/ x+l = n \/
(x+l < n /\ y+l < n /\ a[x+l] <= a[y+l]) (x+l < n /\ y+l < n /\ a[x+l] <= a[y+l])
lemma eq_le : lemma le_refl :
forall a:array int, x :int. forall a:array int, x :int.
0 <= x < a.length -> le a x x 0 <= x < a.length -> le a x x
...@@ -213,14 +215,18 @@ let compare (a:array int) (x y:int) : int ...@@ -213,14 +215,18 @@ let compare (a:array int) (x y:int) : int
predicate sorted_sub (a : array int) (data:array int) (l u:int) = predicate sorted_sub (a : array int) (data:array int) (l u:int) =
forall i1 i2 : int. l <= i1 <= i2 < u -> le a data[i1] data[i2] forall i1 i2 : int. l <= i1 <= i2 < u -> le a data[i1] data[i2]
lemma sorted_bounded:
forall a data: array int, l u i:int.
l <= i < u /\ sorted_sub a data l u -> 0 <= data[i] < a.length
lemma sorted_le: lemma sorted_le:
forall a data: array int, l u i x:int. forall a data: array int, l u i x:int.
sorted_sub a data l u /\ le a x data[l] /\ l <= i < u -> l <= i < u /\ sorted_sub a data l u /\ le a x data[l] ->
le a x data[i] le a x data[i]
lemma sorted_ge: lemma sorted_ge:
forall a data: array int, l u i x:int. forall a data: array int, l u i x:int.
sorted_sub a data l u /\ le a data[u] x /\ l <= i < u -> sorted_sub a data l u /\ le a data[u-1] x /\ l <= i < u ->
le a data[i] x le a data[i] x
lemma sorted_sub_sub: lemma sorted_sub_sub:
...@@ -240,6 +246,18 @@ let compare (a:array int) (x y:int) : int ...@@ -240,6 +246,18 @@ let compare (a:array int) (x y:int) : int
le a data[m-1] data[m] -> le a data[m-1] data[m] ->
sorted_sub a data l u sorted_sub a data l u
(*
lemma sorted_sub_set:
forall a data:array int, l u i v:int.
sorted_sub a data l u /\ u <= i ->
sorted_sub a (data[i<-v]) l u
lemma sorted_sub_set2:
forall a data:array int, l u i v:int.
sorted_sub a data l u /\ u <= i ->
sorted_sub a { length = a.length; elts = Map.set data.elts i v } l u
*)
predicate sorted (a : array int) (data:array int) = predicate sorted (a : array int) (data:array int) =
sorted_sub a data 0 data.length sorted_sub a data 0 data.length
...@@ -267,11 +285,14 @@ let compare (a:array int) (x y:int) : int ...@@ -267,11 +285,14 @@ let compare (a:array int) (x y:int) : int
assert { le a data[!j] data[!j-1] }; assert { le a data[!j] data[!j-1] };
let b = !j - 1 in let b = !j - 1 in
let t = data[!j] in let t = data[!j] in
assert { sorted_sub a data 0 (!j-1) };
data[!j] <- data[b]; data[!j] <- data[b];
assert { sorted_sub a data 0 (!j-1) };
data[b] <- t; data[b] <- t;
assert { le a data[!j-1] data[!j] }; assert { le a data[!j-1] data[!j] };
assert { forall k:int. !j <= k <=i -> le a data[!j] data[k] }; assert { forall k:int. !j <= k <=i -> le a data[!j] data[k] };
assert { exchange (at data 'L) data (!j-1) !j }; assert { exchange (at data 'L) data (!j-1) !j };
assert { sorted_sub a data 0 (!j-1) };
decr j decr j
done; done;
assert { !j > 0 -> le a data[!j-1] data[!j] } assert { !j > 0 -> le a data[!j-1] data[!j] }
......
(* This file is generated by Why3's Coq driver *)
(* Beware! Only edit allowed sections below *)
Require Import BuiltIn.
Require BuiltIn.
Require int.Int.
(* Why3 assumption *)
Definition unit := unit.
Axiom map : forall (a:Type) {a_WT:WhyType a} (b:Type) {b_WT:WhyType b}, Type.
Parameter map_WhyType : forall (a:Type) {a_WT:WhyType a}
(b:Type) {b_WT:WhyType b}, WhyType (map a b).
Existing Instance map_WhyType.
Parameter get: forall {a:Type} {a_WT:WhyType a} {b:Type} {b_WT:WhyType b},
(map a b) -> a -> b.
Parameter set: forall {a:Type} {a_WT:WhyType a} {b:Type} {b_WT:WhyType b},
(map a b) -> a -> b -> (map a b).
Axiom Select_eq : forall {a:Type} {a_WT:WhyType a} {b:Type} {b_WT:WhyType b},
forall (m:(map a b)), forall (a1:a) (a2:a), forall (b1:b), (a1 = a2) ->
((get (set m a1 b1) a2) = b1).
Axiom Select_neq : forall {a:Type} {a_WT:WhyType a}
{b:Type} {b_WT:WhyType b}, forall (m:(map a b)), forall (a1:a) (a2:a),
forall (b1:b), (~ (a1 = a2)) -> ((get (set m a1 b1) a2) = (get m a2)).
Parameter const: forall {a:Type} {a_WT:WhyType a} {b:Type} {b_WT:WhyType b},
b -> (map a b).
Axiom Const : forall {a:Type} {a_WT:WhyType a} {b:Type} {b_WT:WhyType b},
forall (b1:b) (a1:a), ((get (const b1:(map a b)) a1) = b1).
(* Why3 assumption *)
Inductive array (a:Type) {a_WT:WhyType a} :=
| mk_array : Z -> (map Z a) -> array a.
Axiom array_WhyType : forall (a:Type) {a_WT:WhyType a}, WhyType (array a).
Existing Instance array_WhyType.
Implicit Arguments mk_array [[a] [a_WT]].
(* Why3 assumption *)
Definition elts {a:Type} {a_WT:WhyType a}(v:(array a)): (map Z a) :=
match v with
| (mk_array x x1) => x1
end.
(* Why3 assumption *)
Definition length {a:Type} {a_WT:WhyType a}(v:(array a)): Z :=
match v with
| (mk_array x x1) => x
end.
(* Why3 assumption *)
Definition get1 {a:Type} {a_WT:WhyType a}(a1:(array a)) (i:Z): a :=
(get (elts a1) i).
(* Why3 assumption *)
Definition set1 {a:Type} {a_WT:WhyType a}(a1:(array a)) (i:Z) (v:a): (array
a) := (mk_array (length a1) (set (elts a1) i v)).
(* Why3 assumption *)
Definition make {a:Type} {a_WT:WhyType a}(n:Z) (v:a): (array a) :=
(mk_array n (const v:(map Z a))).
(* Why3 assumption *)
Definition array_bounded(a:(array Z)) (b:Z): Prop := forall (i:Z),
((0%Z <= i)%Z /\ (i < (length a))%Z) -> ((0%Z <= (get1 a i))%Z /\ ((get1 a
i) < b)%Z).
(* Why3 assumption *)
Definition map_eq_sub {a:Type} {a_WT:WhyType a}(a1:(map Z a)) (a2:(map Z a))
(l:Z) (u:Z): Prop := forall (i:Z), ((l <= i)%Z /\ (i < u)%Z) -> ((get a1
i) = (get a2 i)).
(* Why3 assumption *)
Definition exchange {a:Type} {a_WT:WhyType a}(a1:(map Z a)) (a2:(map Z a))
(i:Z) (j:Z): Prop := ((get a1 i) = (get a2 j)) /\ (((get a2 i) = (get a1
j)) /\ forall (k:Z), ((~ (k = i)) /\ ~ (k = j)) -> ((get a1 k) = (get a2
k))).
Axiom exchange_set : forall {a:Type} {a_WT:WhyType a}, forall (a1:(map Z a)),
forall (i:Z) (j:Z), (exchange a1 (set (set a1 i (get a1 j)) j (get a1 i)) i
j).
(* Why3 assumption *)
Inductive permut_sub{a:Type} {a_WT:WhyType a} : (map Z a) -> (map Z a) -> Z
-> Z -> Prop :=
| permut_refl : forall (a1:(map Z a)) (a2:(map Z a)), forall (l:Z) (u:Z),
(map_eq_sub a1 a2 l u) -> (permut_sub a1 a2 l u)
| permut_sym : forall (a1:(map Z a)) (a2:(map Z a)), forall (l:Z) (u:Z),
(permut_sub a1 a2 l u) -> (permut_sub a2 a1 l u)
| permut_trans : forall (a1:(map Z a)) (a2:(map Z a)) (a3:(map Z a)),
forall (l:Z) (u:Z), (permut_sub a1 a2 l u) -> ((permut_sub a2 a3 l
u) -> (permut_sub a1 a3 l u))
| permut_exchange : forall (a1:(map Z a)) (a2:(map Z a)), forall (l:Z)
(u:Z) (i:Z) (j:Z), ((l <= i)%Z /\ (i < u)%Z) -> (((l <= j)%Z /\
(j < u)%Z) -> ((exchange a1 a2 i j) -> (permut_sub a1 a2 l u))).
Axiom permut_weakening : forall {a:Type} {a_WT:WhyType a}, forall (a1:(map Z
a)) (a2:(map Z a)), forall (l1:Z) (r1:Z) (l2:Z) (r2:Z), (((l1 <= l2)%Z /\
(l2 <= r2)%Z) /\ (r2 <= r1)%Z) -> ((permut_sub a1 a2 l2 r2) ->
(permut_sub a1 a2 l1 r1)).
Axiom permut_eq : forall {a:Type} {a_WT:WhyType a}, forall (a1:(map Z a))
(a2:(map Z a)), forall (l:Z) (u:Z), (permut_sub a1 a2 l u) -> forall (i:Z),
((i < l)%Z \/ (u <= i)%Z) -> ((get a2 i) = (get a1 i)).
Axiom permut_exists : forall {a:Type} {a_WT:WhyType a}, forall (a1:(map Z a))
(a2:(map Z a)), forall (l:Z) (u:Z), (permut_sub a1 a2 l u) -> forall (i:Z),
((l <= i)%Z /\ (i < u)%Z) -> exists j:Z, ((l <= j)%Z /\ (j < u)%Z) /\
((get a2 i) = (get a1 j)).
(* Why3 assumption *)
Definition exchange1 {a:Type} {a_WT:WhyType a}(a1:(array a)) (a2:(array a))
(i:Z) (j:Z): Prop := (exchange (elts a1) (elts a2) i j).
(* Why3 assumption *)
Definition permut_sub1 {a:Type} {a_WT:WhyType a}(a1:(array a)) (a2:(array a))
(l:Z) (u:Z): Prop := (permut_sub (elts a1) (elts a2) l u).
(* Why3 assumption *)
Definition permut {a:Type} {a_WT:WhyType a}(a1:(array a)) (a2:(array
a)): Prop := ((length a1) = (length a2)) /\ (permut_sub (elts a1) (elts a2)
0%Z (length a1)).
Axiom exchange_permut : forall {a:Type} {a_WT:WhyType a}, forall (a1:(array
a)) (a2:(array a)) (i:Z) (j:Z), (exchange1 a1 a2 i j) ->
(((length a1) = (length a2)) -> (((0%Z <= i)%Z /\ (i < (length a1))%Z) ->
(((0%Z <= j)%Z /\ (j < (length a1))%Z) -> (permut a1 a2)))).
Axiom permut_sym1 : forall {a:Type} {a_WT:WhyType a}, forall (a1:(array a))
(a2:(array a)), (permut a1 a2) -> (permut a2 a1).
Axiom permut_trans1 : forall {a:Type} {a_WT:WhyType a}, forall (a1:(array a))
(a2:(array a)) (a3:(array a)), (permut a1 a2) -> ((permut a2 a3) ->
(permut a1 a3)).
(* Why3 assumption *)
Definition array_eq_sub {a:Type} {a_WT:WhyType a}(a1:(array a)) (a2:(array
a)) (l:Z) (u:Z): Prop := (map_eq_sub (elts a1) (elts a2) l u).
(* Why3 assumption *)
Definition array_eq {a:Type} {a_WT:WhyType a}(a1:(array a)) (a2:(array
a)): Prop := ((length a1) = (length a2)) /\ (array_eq_sub a1 a2 0%Z
(length a1)).
Axiom array_eq_sub_permut : forall {a:Type} {a_WT:WhyType a},
forall (a1:(array a)) (a2:(array a)) (l:Z) (u:Z), (array_eq_sub a1 a2 l
u) -> (permut_sub1 a1 a2 l u).
Axiom array_eq_permut : forall {a:Type} {a_WT:WhyType a}, forall (a1:(array
a)) (a2:(array a)), (array_eq a1 a2) -> (permut a1 a2).
Axiom permut_bounded : forall (a1:(array Z)) (a2:(array Z)) (n:Z),
((permut a1 a2) /\ (array_bounded a1 n)) -> (array_bounded a2 n).
(* Why3 assumption *)
Definition is_common_prefix(a:(array Z)) (x:Z) (y:Z) (l:Z): Prop :=
(0%Z <= l)%Z /\ (((x + l)%Z <= (length a))%Z /\
(((y + l)%Z <= (length a))%Z /\ forall (i:Z), ((0%Z <= i)%Z /\
(i < l)%Z) -> ((get1 a (x + i)%Z) = (get1 a (y + i)%Z)))).
Axiom common_prefix_eq : forall (a:(array Z)) (x:Z), ((0%Z <= x)%Z /\
(x < (length a))%Z) -> (is_common_prefix a x x ((length a) - x)%Z).
Axiom common_prefix_eq2 : forall (a:(array Z)) (x:Z), ((0%Z <= x)%Z /\
(x < (length a))%Z) -> ~ (is_common_prefix a x x
(((length a) - x)%Z + 1%Z)%Z).
Axiom not_common_prefix_if_last_different : forall (a:(array Z)) (x:Z) (y:Z)
(l:Z), ((0%Z < l)%Z /\ (((x + l)%Z < (length a))%Z /\
(((y + l)%Z < (length a))%Z /\ ~ ((get1 a (x + (l - 1%Z)%Z)%Z) = (get1 a
(y + (l - 1%Z)%Z)%Z))))) -> ~ (is_common_prefix a x y l).
Parameter longest_common_prefix: (array Z) -> Z -> Z -> Z.
Axiom lcp_spec : forall (a:(array Z)) (x:Z) (y:Z) (l:Z), (((0%Z <= x)%Z /\
(x < (length a))%Z) /\ ((0%Z <= y)%Z /\ (y < (length a))%Z)) ->
((l = (longest_common_prefix a x y)) <-> ((is_common_prefix a x y l) /\
~ (is_common_prefix a x y (l + 1%Z)%Z))).
Axiom lcp_is_cp : forall (a:(array Z)) (x:Z) (y:Z), (((0%Z <= x)%Z /\
(x < (length a))%Z) /\ ((0%Z <= y)%Z /\ (y < (length a))%Z)) ->
(is_common_prefix a x y (longest_common_prefix a x y)).
Axiom lcp_eq : forall (a:(array Z)) (x:Z) (y:Z), (((0%Z <= x)%Z /\
(x < (length a))%Z) /\ ((0%Z <= y)%Z /\ (y < (length a))%Z)) ->
forall (i:Z), ((0%Z <= i)%Z /\ (i < (longest_common_prefix a x y))%Z) ->
((get1 a (x + i)%Z) = (get1 a (y + i)%Z)).
Axiom lcp_refl : forall (a:(array Z)) (x:Z), ((0%Z <= x)%Z /\
(x < (length a))%Z) -> ((longest_common_prefix a x
x) = ((length a) - x)%Z).
(* Why3 assumption *)
Inductive ref (a:Type) {a_WT:WhyType a} :=
| mk_ref : a -> ref a.
Axiom ref_WhyType : forall (a:Type) {a_WT:WhyType a}, WhyType (ref a).
Existing Instance ref_WhyType.
Implicit Arguments mk_ref [[a] [a_WT]].
(* Why3 assumption *)
Definition contents {a:Type} {a_WT:WhyType a}(v:(ref a)): a :=
match v with
| (mk_ref x) => x
end.
(* Why3 assumption *)
Definition le(a:(array Z)) (x:Z) (y:Z): Prop := let n := (length a) in
(((0%Z <= x)%Z /\ (x < n)%Z) /\ (((0%Z <= y)%Z /\ (y < n)%Z) /\ let l :=
(longest_common_prefix a x y) in (((x + l)%Z = n) \/ (((x + l)%Z < n)%Z /\
(((y + l)%Z < n)%Z /\ ((get1 a (x + l)%Z) <= (get1 a (y + l)%Z))%Z))))).
Axiom le_refl : forall (a:(array Z)) (x:Z), ((0%Z <= x)%Z /\
(x < (length a))%Z) -> (le a x x).
Axiom le_trans : forall (a:(array Z)) (x:Z) (y:Z) (z:Z), (((0%Z <= x)%Z /\
(x < (length a))%Z) /\ (((0%Z <= y)%Z /\ (y < (length a))%Z) /\
(((0%Z <= z)%Z /\ (z < (length a))%Z) /\ ((le a x y) /\ (le a y z))))) ->
(le a x z).
(* Why3 assumption *)
Definition sorted_sub(a:(array Z)) (data:(array Z)) (l:Z) (u:Z): Prop :=
forall (i1:Z) (i2:Z), (((l <= i1)%Z /\ (i1 <= i2)%Z) /\ (i2 < u)%Z) ->
(le a (get1 data i1) (get1 data i2)).
Axiom sorted_bounded : forall (a:(array Z)) (data:(array Z)) (l:Z) (u:Z)
(i:Z), (((l <= i)%Z /\ (i < u)%Z) /\ (sorted_sub a data l u)) ->
((0%Z <= (get1 data i))%Z /\ ((get1 data i) < (length a))%Z).
Require Import Why3.
Ltac ae := why3 "alt-ergo" timelimit 3.
(* Why3 goal *)
Theorem sorted_le : forall (a:(array Z)) (data:(array Z)) (l:Z) (u:Z) (i:Z)
(x:Z), (((l <= i)%Z /\ (i < u)%Z) /\ ((sorted_sub a data l u) /\ (le a x
(get1 data l)))) -> (le a x (get1 data i)).
intros a data l u i x ((h1,h2),(h3,h4)).
apply le_trans with (get1 data l).
ae.
Qed.
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