Commit 4b9ec76c authored by MARCHE Claude's avatar MARCHE Claude

LCP: only one lemma left: antisymmetry of le

parent 5774a161
......@@ -119,7 +119,14 @@ axiom lcp_spec:
*)
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))
is_common_prefix a x y l /\
forall m:int. l < m -> not (is_common_prefix a x y m)
(* for proving lcp_sym and le_trans lemmas, and compare function in the absurd case *)
lemma longest_common_prefix_succ:
forall a:array int, x y l:int.
is_common_prefix a x y l /\ not (is_common_prefix a x y (l+1)) ->
is_longest_common_prefix a x y l
use import ref.Refint
......@@ -190,13 +197,14 @@ let test1 () =
check {x = 1}
predicate le (a : array int) (x y:int) =
predicate lt (a : array int) (x y:int) =
let n = a.length in
0 <= x <= n /\ 0 <= y <= n /\
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]))
(y+l < n /\ (x+l = n \/ a[x+l] < a[y+l]))
predicate le (a : array int) (x y:int) = x = y \/ lt a x y
lemma le_refl :
forall a:array int, x :int.
0 <= x <= a.length -> le a x x
......
(* This file is generated by Why3's Coq driver *)
(* This file is generated by Why3's Coq 8.4 driver *)
(* Beware! Only edit allowed sections below *)
Require Import BuiltIn.
Require BuiltIn.
Require int.Int.
Require int.MinMax.
Require map.Map.
Require map.MapPermut.
(* 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).
Definition unit := unit.
(* Why3 assumption *)
Definition injective(a:(map Z Z)) (n:Z): Prop := forall (i:Z) (j:Z),
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)) -> ~ ((get a i) = (get a j)))).
((~ (i = j)) -> ~ ((map.Map.get a i) = (map.Map.get a j)))).
(* Why3 assumption *)
Definition surjective(a:(map Z Z)) (n:Z): Prop := forall (i:Z),
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) /\
((get a j) = i).
((map.Map.get a j) = i).
(* Why3 assumption *)
Definition range(a:(map Z Z)) (n:Z): Prop := forall (i:Z), ((0%Z <= i)%Z /\
(i < n)%Z) -> ((0%Z <= (get a i))%Z /\ ((get a i) < n)%Z).
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 Z Z)) (n:Z), (injective a n) ->
((range a n) -> (surjective a n)).
Axiom injective_surjective : forall (a:(map.Map.map Z Z)) (n:Z), (injective a
n) -> ((range a n) -> (surjective a n)).
(* Why3 assumption *)
Inductive array (a:Type) {a_WT:WhyType a} :=
| mk_array : Z -> (map Z a) -> array a.
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 Z a) :=
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 :=
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)).
Definition get {a:Type} {a_WT:WhyType a} (a1:(array a)) (i:Z): a :=
(map.Map.get (elts a1) i).
(* Why3 assumption *)
Definition make {a:Type} {a_WT:WhyType a}(n:Z) (v:a): (array a) :=
(mk_array n (const v:(map Z a))).
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 permutation(a:(array Z)): Prop := (range (elts a) (length a)) /\
(injective (elts a) (length a)).
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 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)).
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))).
(* 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
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 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))
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).
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
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)).
......@@ -150,15 +125,20 @@ 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)).
(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
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
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)).
......@@ -169,50 +149,20 @@ Axiom array_eq_sub_permut : forall {a:Type} {a_WT:WhyType a},
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_permutation : forall (a1:(array Z)) (a2:(array Z)), ((permut a1
a2) /\ (permutation a1)) -> (permutation a2).
(* Why3 assumption *)
Definition is_common_prefix(a:(array Z)) (x:Z) (y:Z) (l:Z): Prop :=
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).
(i < l)%Z) -> ((get a (x + i)%Z) = (get a (y + i)%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)).
(l:Z), ((0%Z <= l)%Z /\ (((x + l)%Z < (length a))%Z /\
(((y + l)%Z < (length a))%Z /\ ~ ((get a (x + l)%Z) = (get a
(y + l)%Z))))) -> ~ (is_common_prefix a x y (l + 1%Z)%Z).
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).
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 is_longest_common_prefix (a:(array Z)) (x:Z) (y:Z) (l:Z): Prop :=
(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} :=
......@@ -222,16 +172,34 @@ 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 :=
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) (l:Z), (((0%Z <= x)%Z /\
(x <= (length a))%Z) /\ ((0%Z <= y)%Z /\ (y <= (length a))%Z)) ->
((is_longest_common_prefix a x y l) -> (is_common_prefix a x y l)).
Axiom lcp_eq : 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)) ->
((is_longest_common_prefix a x y l) -> forall (i:Z), ((0%Z <= i)%Z /\
(i < l)%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) -> (is_longest_common_prefix a x x
((length a) - x)%Z).
Axiom lcp_sym : 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)) ->
((is_longest_common_prefix a x y l) -> (is_longest_common_prefix a y x l)).
(* 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))))).
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) /\
exists l:Z, (is_common_prefix a x y l) /\ (((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).
......@@ -246,52 +214,33 @@ Axiom le_asym : forall (a:(array Z)) (x:Z) (y:Z), (((0%Z <= x)%Z /\
y))) -> (le a y x).
(* 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_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)).
Axiom sorted_ge : forall (a:(array Z)) (data:(array Z)) (l:Z) (u:Z) (i:Z)
(x:Z), ((sorted_sub a data l u) /\ ((le a (get1 data (u - 1%Z)%Z) x) /\
((l <= i)%Z /\ (i < u)%Z))) -> (le a (get1 data i) x).
Axiom sorted_sub_sub : forall (a:(array Z)) (data:(array Z)) (l:Z) (u:Z)
(l':Z) (u':Z), ((l <= l')%Z /\ (u' <= u)%Z) -> ((sorted_sub a data l u) ->
(sorted_sub a data l' u')).
Axiom sorted_sub_add : forall (a:(array Z)) (data:(array Z)) (l:Z) (u:Z),
((sorted_sub a data (l + 1%Z)%Z u) /\ (le a (get1 data l) (get1 data
(l + 1%Z)%Z))) -> (sorted_sub a data l u).
Definition permutation (m:(map.Map.map Z Z)) (u:Z): Prop := (range m u) /\
(injective m u).
Axiom sorted_sub_concat : forall (a:(array Z)) (data:(array Z)) (l:Z) (m:Z)
(u:Z), (((l <= m)%Z /\ (m <= u)%Z) /\ ((sorted_sub a data l m) /\
((sorted_sub a data m u) /\ (le a (get1 data (m - 1%Z)%Z) (get1 data
m))))) -> (sorted_sub a data l u).
Axiom sorted_sub_set : forall (a:(array Z)) (data:(array Z)) (l:Z) (u:Z)
(i:Z) (v:Z), ((sorted_sub a data l u) /\ (u <= i)%Z) -> (sorted_sub a
(set1 data i v) l u).
Axiom sorted_sub_set2 : forall (a:(array Z)) (data:(array Z)) (l:Z) (u:Z)
(i:Z) (v:Z), ((sorted_sub a data l u) /\ (u <= i)%Z) -> (sorted_sub a
(mk_array (length a) (set (elts data) i v)) l u).
(* 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 data
0%Z (length data)).
Definition sorted (a:(array Z)) (data:(array Z)): Prop := (sorted_sub a
(elts data) 0%Z (length data)).
Axiom lcp_le_le_min : forall (a:(array Z)) (x:Z) (y:Z) (z:Z), ((le a x y) /\
(le a y z)) -> ((longest_common_prefix a x
z) = (Zmin (longest_common_prefix a x y) (longest_common_prefix a y z))).
Axiom permut_permutation : forall (a1:(array Z)) (a2:(array Z)), (permut a1
a2) -> ((permutation (elts a1) (length a1)) -> (permutation (elts a2)
(length a2))).
Axiom lcp_le_le_aux : forall (a:(array Z)) (x:Z) (y:Z) (z:Z) (l:Z), ((le a x
y) /\ (le a y z)) -> ((is_common_prefix a x z l) -> (is_common_prefix a y z
l)).
(* Why3 goal *)
Theorem lcp_le_le : forall (a:(array Z)) (x:Z) (y:Z) (z:Z), ((le a x y) /\
(le a y z)) -> ((longest_common_prefix a x z) <= (longest_common_prefix a y
z))%Z.
intros a x y z (h1,h2).
Theorem lcp_le_le : forall (a:(array Z)) (x:Z) (y:Z) (z:Z) (l:Z) (m:Z), ((le
a x y) /\ (le a y z)) -> (((is_longest_common_prefix a x z l) /\
(is_longest_common_prefix a y z m)) -> (l <= m)%Z).
unfold is_longest_common_prefix.
intros a x y z l m (h1,h2) ((h3,_),(_,h4)).
rewrite lcp_le_le_min with (y:=y); auto.
apply Zle_min_r.
Qed.
......
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