Commit 82b3fb69 authored by MARCHE Claude's avatar MARCHE Claude

LCP: removed the axiom, but some proofs need to be completed

parent ff035059
......@@ -99,6 +99,7 @@ lemma not_common_prefix_if_last_different:
0 < l /\ x+l < a.length /\ y+l < a.length /\ a[x+(l-1)] <> a[y+(l-1)] ->
not is_common_prefix a x y l
(*
function longest_common_prefix (a:array int) (x y:int) :int
axiom lcp_spec:
......@@ -106,45 +107,51 @@ axiom lcp_spec:
0 <= x <= a.length /\ 0 <= y <= a.length ->
(l = longest_common_prefix a x y <->
is_common_prefix a x y l /\ not is_common_prefix a x y (l+1))
*)
predicate is_longest_common_prefix (a:array int) (x y:int) (l:int) =
is_common_prefix a x y l /\ not (is_common_prefix a x y (l+1))
use import ref.Refint
let lcp (a:array int) (x y:int) : int
requires { 0 <= x <= a.length }
requires { 0 <= y <= a.length }
ensures { result = longest_common_prefix a x y }
ensures { is_longest_common_prefix a x y result }
= let n = a.length in
let l = ref 0 in
while x + !l < n && y + !l < n && a[x + !l] = a[y + !l] do
invariant { is_common_prefix a x y !l }
incr l
done;
(*
assert { is_common_prefix a x y !l };
assert { x + !l < n /\ y + !l < n -> a[x + !l] <> a[y + !l] };
assert { not is_common_prefix a x y (!l+1) };
*)
!l
lemma lcp_is_cp :
forall a:array int, x y:int.
forall a:array int, x y l:int.
0 <= x <= a.length /\ 0 <= y <= a.length ->
let l = longest_common_prefix a x y in
is_longest_common_prefix a x y l ->
is_common_prefix a x y l
lemma lcp_eq :
forall a:array int, x y:int.
forall a:array int, x y l:int.
0 <= x <= a.length /\ 0 <= y <= a.length ->
let l = longest_common_prefix a x y in
is_longest_common_prefix a x y l ->
forall i:int. 0 <= i < l -> a[x+i] = a[y+i]
lemma lcp_refl :
forall a:array int, x:int.
0 <= x <= a.length -> longest_common_prefix a x x = a.length - x
0 <= x <= a.length -> is_longest_common_prefix a x x (a.length - x)
lemma lcp_sym :
forall a:array int, x y:int.
forall a:array int, x y l:int.
0 <= x <= a.length /\ 0 <= y <= a.length ->
longest_common_prefix a x y = longest_common_prefix a y x
is_longest_common_prefix a x y l -> is_longest_common_prefix a y x l
let test1 () =
let arr = Array.make 4 0 in
......@@ -175,9 +182,9 @@ let test1 () =
predicate le (a : array int) (x y:int) =
let n = a.length in
0 <= x <= n /\ 0 <= y <= n /\
let l = longest_common_prefix a x y in
x+l = n \/
(x+l < n /\ y+l < n /\ a[x+l] <= a[y+l])
exists l:int. is_common_prefix a x y l /\
(x+l = n \/
(x+l < n /\ y+l < n /\ a[x+l] <= a[y+l]))
lemma le_refl :
forall a:array int, x :int.
......@@ -213,13 +220,9 @@ let compare (a:array int) (x y:int) : int
use map.MapPermut
use map.MapInjection
predicate map_permutation (m:Map.map int int) (u : int) =
predicate permutation (m:Map.map int int) (u : int) =
MapInjection.range m u /\ MapInjection.injective m u
predicate permutation (a:array int) =
map_permutation a.elts a.length
predicate sorted_sub (a:array int) (data:Map.map int int) (l u:int) =
forall i1 i2 : int. l <= i1 <= i2 < u ->
le a (Map.get data i1) (Map.get data i2)
......@@ -272,20 +275,27 @@ lemma map_permut_permutation :
lemma permut_permutation :
forall a1 a2:array int.
permut a1 a2 -> permutation a1 -> permutation a2
permut a1 a2 -> permutation a1.elts a1.length -> permutation a2.elts a2.length
use import int.MinMax
(*
lemma lcp_le_le_min:
forall a:array int, x y z:int [le a x y, le a y z].
le a x y /\ le a y z ->
longest_common_prefix a x z =
min (longest_common_prefix a x y) (longest_common_prefix a y z)
*)
lemma lcp_le_le_aux:
forall a:array int, x y z l:int.
le a x y /\ le a y z ->
is_common_prefix a x z l -> is_common_prefix a y z l
lemma lcp_le_le:
forall a:array int, x y z:int.
forall a:array int, x y z l m :int.
le a x y /\ le a y z ->
longest_common_prefix a x z <= longest_common_prefix a y z
is_longest_common_prefix a x z l /\ is_longest_common_prefix a y z m
-> l <= m
end
......@@ -309,7 +319,7 @@ use map.MapInjection
predicate inv(s:suffixArray) =
s.values.length = s.suffixes.length /\
permutation s.suffixes /\
permutation s.suffixes.elts s.suffixes.length /\
sorted s.values s.suffixes
let select (s:suffixArray) (i:int) : int
......@@ -332,8 +342,7 @@ let create (a:array int) : suffixArray
let lcp (s:suffixArray) (i:int) : int
requires { inv s }
requires { 0 < i < s.values.length }
ensures { result =
longest_common_prefix s.values s.suffixes[i-1] s.suffixes[i] }
ensures { is_longest_common_prefix s.values s.suffixes[i-1] s.suffixes[i] result }
=
LCP.lcp s.values s.suffixes[i] s.suffixes[i-1]
......@@ -392,13 +401,15 @@ module LRS "longest repeated substring"
ensures { 0 <= !solLength <= a.length }
ensures { 0 <= !solStart <= a.length }
ensures { 0 <= !solStart2 <= a.length /\ !solStart <> !solStart2 /\
!solLength = LCP.longest_common_prefix a !solStart !solStart2 }
ensures { forall x y:int.
0 <= x < y < a.length ->
!solLength >= LCP.longest_common_prefix a x y }
LCP.is_longest_common_prefix a !solStart !solStart2 !solLength }
ensures { forall x y l:int.
0 <= x < y < a.length /\
LCP.is_longest_common_prefix a x y l ->
!solLength >= l }
=
let sa = SuffixArray.create a in
assert { LCP.permutation sa.SuffixArray.suffixes };
assert { LCP.permutation sa.SuffixArray.suffixes.elts
sa.SuffixArray.suffixes.length };
solStart := 0;
solLength := 0;
solStart2 := a.length;
......@@ -407,24 +418,26 @@ module LRS "longest repeated substring"
invariant { 0 <= !solStart <= a.length }
invariant {
0 <= !solStart2 <= a.length /\ !solStart <> !solStart2 /\
!solLength = LCP.longest_common_prefix a !solStart !solStart2 }
invariant { forall j k:int.
0 <= j < k < i ->
!solLength >= LCP.longest_common_prefix a
sa.SuffixArray.suffixes[j] sa.SuffixArray.suffixes[k] }
LCP.is_longest_common_prefix a !solStart !solStart2 !solLength }
invariant { forall j k l:int.
0 <= j < k < i /\
LCP.is_longest_common_prefix a
sa.SuffixArray.suffixes[j] sa.SuffixArray.suffixes[k] l ->
!solLength >= l }
let l = SuffixArray.lcp sa i in
assert { forall j:int. 0 <= j < i ->
LCP.le a sa.SuffixArray.suffixes[j]
sa.SuffixArray.suffixes[i] };
assert { forall j:int. 0 <= j < i ->
LCP.longest_common_prefix a
assert { forall j m:int. 0 <= j < i /\
LCP.is_longest_common_prefix a
sa.SuffixArray.suffixes[j]
sa.SuffixArray.suffixes[i]
<= l };
assert { forall j k:int. 0 <= j < k < i-1 ->
!solLength >= LCP.longest_common_prefix a
sa.SuffixArray.suffixes[i] m ->
m <= l };
assert { forall j k m:int. 0 <= j < k < i-1 /\
LCP.is_longest_common_prefix a
sa.SuffixArray.suffixes[j]
sa.SuffixArray.suffixes[k] };
sa.SuffixArray.suffixes[k] m -> !solLength >= m};
if l > !solLength then begin
solStart := SuffixArray.select sa i;
solStart2 := SuffixArray.select sa (i-1);
......@@ -433,16 +446,18 @@ module LRS "longest repeated substring"
done;
assert { let s = sa.SuffixArray.suffixes in
MapInjection.surjective s.elts s.length };
assert { forall j k:int.
0 <= j < a.length /\ 0 <= k < a.length /\ j <> k ->
!solLength >= LCP.longest_common_prefix a
sa.SuffixArray.suffixes[j] sa.SuffixArray.suffixes[k] };
assert { forall j k l:int.
0 <= j < a.length /\ 0 <= k < a.length /\ j <> k /\
LCP.is_longest_common_prefix a
sa.SuffixArray.suffixes[j] sa.SuffixArray.suffixes[k] l ->
!solLength >= l};
assert { forall x y:int.
0 <= x < y < a.length ->
exists j k : int.
0 <= j < a.length /\ 0 <= k < a.length /\ j <> k /\
x = sa.SuffixArray.suffixes[j] /\
y = sa.SuffixArray.suffixes[k] }
y = sa.SuffixArray.suffixes[k] };
()
(*
let test () =
......
(* This file is generated by Why3's Coq driver *)
(* Beware! Only edit allowed sections below *)
Require Import BuiltIn.
Require BuiltIn.
Require int.Int.
Require map.Map.
Require map.MapPermut.
(* Why3 assumption *)
Definition unit := unit.
(* Why3 assumption *)
Inductive array (a:Type) {a_WT:WhyType a} :=
| mk_array : Z -> (map.Map.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.Map.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 get {a:Type} {a_WT:WhyType a}(a1:(array a)) (i:Z): a :=
(map.Map.get (elts a1) i).
(* Why3 assumption *)
Definition set {a:Type} {a_WT:WhyType a}(a1:(array a)) (i:Z) (v:a): (array
a) := (mk_array (length a1) (map.Map.set (elts a1) i v)).
(* Why3 assumption *)
Definition make {a:Type} {a_WT:WhyType a}(n:Z) (v:a): (array a) :=
(mk_array n (map.Map.const v:(map.Map.map Z a))).
(* Why3 assumption *)
Definition exchange {a:Type} {a_WT:WhyType a}(a1:(map.Map.map Z a))
(a2:(map.Map.map Z a)) (i:Z) (j:Z): Prop := ((map.Map.get a1
i) = (map.Map.get a2 j)) /\ (((map.Map.get a2 i) = (map.Map.get a1 j)) /\
forall (k:Z), ((~ (k = i)) /\ ~ (k = j)) -> ((map.Map.get a1
k) = (map.Map.get a2 k))).
Axiom exchange_set : forall {a:Type} {a_WT:WhyType a},
forall (a1:(map.Map.map Z a)), forall (i:Z) (j:Z), (exchange a1
(map.Map.set (map.Map.set a1 i (map.Map.get a1 j)) j (map.Map.get a1 i)) i
j).
(* Why3 assumption *)
Inductive permut_sub{a:Type} {a_WT:WhyType a} : (map.Map.map Z a)
-> (map.Map.map Z a) -> Z -> Z -> Prop :=
| permut_refl : forall (a1:(map.Map.map Z a)), forall (l:Z) (u:Z),
(permut_sub a1 a1 l u)
| permut_sym : forall (a1:(map.Map.map Z a)) (a2:(map.Map.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.Map.map Z a)) (a2:(map.Map.map Z a))
(a3:(map.Map.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.Map.map Z a)) (a2:(map.Map.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.Map.map Z a)) (a2:(map.Map.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.Map.map Z
a)) (a2:(map.Map.map Z a)), forall (l:Z) (u:Z), (permut_sub a1 a2 l u) ->
forall (i:Z), ((i < l)%Z \/ (u <= i)%Z) -> ((map.Map.get a2
i) = (map.Map.get a1 i)).
Axiom permut_exists : forall {a:Type} {a_WT:WhyType a},
forall (a1:(map.Map.map Z a)) (a2:(map.Map.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) /\ ((map.Map.get a2
i) = (map.Map.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 map_eq_sub {a:Type} {a_WT:WhyType a}(a1:(map.Map.map Z a))
(a2:(map.Map.map Z a)) (l:Z) (u:Z): Prop := forall (i:Z), ((l <= i)%Z /\
(i < u)%Z) -> ((map.Map.get a1 i) = (map.Map.get a2 i)).
(* 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).
(* 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) -> ((get a (x + i)%Z) = (get 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 /\ ~ ((get a (x + (l - 1%Z)%Z)%Z) = (get 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))).
(* 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.
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) ->
((get a (x + i)%Z) = (get 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).
Axiom lcp_sym : forall (a:(array Z)) (x:Z) (y:Z), (((0%Z <= x)%Z /\
(x <= (length a))%Z) /\ ((0%Z <= y)%Z /\ (y <= (length a))%Z)) ->
((longest_common_prefix a x y) = (longest_common_prefix a y x)).
(* 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 /\ ((get a (x + l)%Z) <= (get 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).
Axiom le_asym : forall (a:(array Z)) (x:Z) (y:Z), (((0%Z <= x)%Z /\
(x <= (length a))%Z) /\ (((0%Z <= y)%Z /\ (y <= (length a))%Z) /\ ~ (le a x
y))) -> (le a y x).
(* Why3 assumption *)
Definition injective(a:(map.Map.map Z Z)) (n:Z): Prop := forall (i:Z) (j:Z),
((0%Z <= i)%Z /\ (i < n)%Z) -> (((0%Z <= j)%Z /\ (j < n)%Z) ->
((~ (i = j)) -> ~ ((map.Map.get a i) = (map.Map.get a j)))).
(* Why3 assumption *)
Definition surjective(a:(map.Map.map Z Z)) (n:Z): Prop := forall (i:Z),
((0%Z <= i)%Z /\ (i < n)%Z) -> exists j:Z, ((0%Z <= j)%Z /\ (j < n)%Z) /\
((map.Map.get a j) = i).
(* Why3 assumption *)
Definition range(a:(map.Map.map Z Z)) (n:Z): Prop := forall (i:Z),
((0%Z <= i)%Z /\ (i < n)%Z) -> ((0%Z <= (map.Map.get a i))%Z /\
((map.Map.get a i) < n)%Z).
Axiom injective_surjective : forall (a:(map.Map.map Z Z)) (n:Z), (injective a
n) -> ((range a n) -> (surjective a n)).
(* Why3 assumption *)
Definition map_permutation(m:(map.Map.map Z Z)) (u:Z): Prop := (range m u) /\
(injective m u).
(* Why3 assumption *)
Definition permutation(a:(array Z)): Prop := (map_permutation (elts a)
(length a)).
(* Why3 assumption *)
Definition sorted_sub(a:(array Z)) (data:(map.Map.map Z Z)) (l:Z)
(u:Z): Prop := forall (i1:Z) (i2:Z), (((l <= i1)%Z /\ (i1 <= i2)%Z) /\
(i2 < u)%Z) -> (le a (map.Map.get data i1) (map.Map.get data i2)).
(* Why3 assumption *)
Definition sorted(a:(array Z)) (data:(array Z)): Prop := (sorted_sub a
(elts data) 0%Z (length data)).
(* Why3 goal *)
Theorem permut_permutation : forall (a1:(array Z)) (a2:(array Z)), (permut a1
a2) -> ((permutation a1) -> (permutation a2)).
intros a1 a2 h1 h2.
Qed.
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