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Commit 4b9ec76c by 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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