LCP: proof of LRS function in progress

examples/programs/verifythis_fm2012_lcp.mlw

@@ -89,7 +89,7 @@ predicate permutation (a:array int) =
 use import array.ArrayPermut
 
 lemma permut_permutation :
-  forall a1 a2:array int, n:int.
+  forall a1 a2:array int.
   permut a1 a2 /\ permutation a1 -> permutation a2
 
 
@@ -136,6 +136,11 @@ lemma lcp_refl :
   forall a:array int, x:int.
      0 <= x <= a.length -> longest_common_prefix a x x = a.length - x
 
+lemma lcp_sym :
+  forall a:array int, x y:int.
+     0 <= x <= a.length /\ 0 <= y <= a.length -> 
+       longest_common_prefix a x y = longest_common_prefix a y x
+
 
 use import ref.Refint
 
@@ -303,6 +308,16 @@ let compare (a:array int) (x y:int) : int
    done
 
 
+(* additional properties relating le and longest_common_prefix, needed
+   for LRS
+
+*)
+
+  lemma lcp_le_le:
+    forall a:array int, x y z:int.
+     le a x y /\ le a y z ->
+     longest_common_prefix a x z <= longest_common_prefix a y z
+
 end
 
 
@@ -392,6 +407,7 @@ module LRS "longest repeated substring"
   use import int.Int
   use import ref.Ref
   use import array.Array
+  use map.MapInjection
   use LCP
   use SuffixArray
 
@@ -402,14 +418,15 @@ module LRS "longest repeated substring"
       requires { a.length > 0 }
       ensures { 0 <= !solLength <= a.length }
       ensures { 0 <= !solStart <= a.length }
-      ensures { forall x y:int.
-                  0 <= x < a.length /\ 0 <= y <= a.length /\ x <> y ->
-                  !solLength >= LCP.longest_common_prefix a x y }
       ensures { exists y:int.
                   0 <= y <= a.length /\ !solStart <> y /\
                   !solLength = LCP.longest_common_prefix a !solStart y }
+      ensures { forall x y:int.
+                  0 <= x < y < a.length ->
+                  !solLength >= LCP.longest_common_prefix a x y }
     =
       let sa = SuffixArray.create a in
+      assert { LCP.permutation sa.SuffixArray.suffixes };
       solStart := 0;
       solLength := 0;
       for i=1 to a.length - 1 do
@@ -418,12 +435,40 @@ module LRS "longest repeated substring"
         invariant { exists y:int.
                   0 <= y <= a.length /\ !solStart <> y /\
                   !solLength = LCP.longest_common_prefix a !solStart y }
+        invariant { forall j k:int.
+                  0 <= j < k < i ->
+                  !solLength >= LCP.longest_common_prefix a
+                    sa.SuffixArray.suffixes[j] sa.SuffixArray.suffixes[k] }
          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
+                             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[j]
+                             sa.SuffixArray.suffixes[k] };
          if l > !solLength then begin
             solStart := SuffixArray.select sa i;
             solLength := l
          end
-      done
+      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 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] }
 
   let test () =
     let arr = Array.make 4 0 in

examples/programs/verifythis_fm2012_lcp/verifythis_fm2012_lcp_LRS_WP_parameter_lrs_2.v

@@ -77,9 +77,25 @@ 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).
+Definition injective(a:(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)))).
+
+(* Why3 assumption *)
+Definition surjective(a:(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).
+
+(* 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).
+
+Axiom injective_surjective : forall (a:(map Z Z)) (n:Z), (injective a n) ->
+  ((range a n) -> (surjective a n)).
+
+(* Why3 assumption *)
+Definition permutation(a:(array Z)): Prop := (range (elts a) (length a)) /\
+  (injective (elts a) (length a)).
 
 (* Why3 assumption *)
 Definition map_eq_sub {a:Type} {a_WT:WhyType a}(a1:(map Z a)) (a2:(map Z a))
@@ -165,8 +181,8 @@ 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_bounded : forall (a1:(array Z)) (a2:(array Z)) (n:Z),
-  ((permut a1 a2) /\ (array_bounded a1 n)) -> (array_bounded a2 n).
+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 :=
@@ -206,6 +222,10 @@ 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 :=
@@ -262,6 +282,10 @@ Axiom sorted_sub_set2 : forall (a:(array Z)) (data:(array Z)) (l:Z) (u:Z)
 Definition sorted(a:(array Z)) (data:(array Z)): Prop := (sorted_sub a data
   0%Z (length data)).
 
+Axiom 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.
+
 (* Why3 assumption *)
 Inductive suffixArray  :=
   | mk_suffixArray : (array Z) -> (array Z) -> suffixArray .
@@ -283,19 +307,27 @@ Definition values(v:suffixArray): (array Z) :=
 (* Why3 assumption *)
 Definition inv(s:suffixArray): Prop :=
   ((length (values s)) = (length (suffixes s))) /\
-  (array_bounded (suffixes s) (length (values s))).
+  (permutation (suffixes s)).
 
 Require Import Why3.
 
 (* Why3 goal *)
-Theorem WP_parameter_lrs : forall (a:Z), forall (a1:(map Z Z)),
-  (0%Z < a)%Z -> ((0%Z <= a)%Z -> forall (sa:Z) (sa1:(map Z Z)) (sa2:Z)
-  (sa3:(map Z Z)), ((sa = a) /\ (inv (mk_suffixArray (mk_array sa sa1)
-  (mk_array sa2 sa3)))) -> forall (solStart:Z), (solStart = 0%Z) ->
-  forall (solLength:Z), (solLength = 0%Z) -> (((a - 1%Z)%Z < 1%Z)%Z ->
-  exists y:Z, ((0%Z <= y)%Z /\ (y <= a)%Z) /\ ((~ (solStart = y)) /\
-  (solLength = (longest_common_prefix (mk_array a a1) solStart y))))).
-intros a a1 h1 h2 sa sa1 sa2 sa3 (h3,h4) solStart h5 solLength h6 h7.
+Theorem WP_parameter_lrs : forall (a:Z), forall (a1:(map Z Z)), let a2 :=
+  (mk_array a a1) in ((0%Z < a)%Z -> ((0%Z <= a)%Z -> forall (sa:Z) (sa1:(map
+  Z Z)) (sa2:Z) (sa3:(map Z Z)), (((sa = a) /\ (sa1 = a1)) /\
+  (inv (mk_suffixArray (mk_array sa sa1) (mk_array sa2 sa3)))) ->
+  ((permutation (mk_array sa2 sa3)) -> forall (solStart:Z),
+  (solStart = 0%Z) -> forall (solLength:Z), (solLength = 0%Z) ->
+  (((a - 1%Z)%Z < 1%Z)%Z -> ((surjective sa3 sa2) -> ((forall (j:Z) (k:Z),
+  (((0%Z <= j)%Z /\ (j < a)%Z) /\ (((0%Z <= k)%Z /\ (k < a)%Z) /\
+  ~ (j = k))) -> ((longest_common_prefix a2 (get sa3 j) (get sa3
+  k)) <= solLength)%Z) -> ((forall (x:Z) (y:Z), (((0%Z <= x)%Z /\
+  (x < y)%Z) /\ (y < a)%Z) -> exists j:Z, exists k:Z, ((0%Z <= j)%Z /\
+  (j < a)%Z) /\ (((0%Z <= k)%Z /\ (k < a)%Z) /\ ((~ (j = k)) /\
+  ((x = (get sa3 j)) /\ (y = (get sa3 k)))))) -> exists y:Z, ((0%Z <= y)%Z /\
+  (y <= a)%Z) /\ ((~ (solStart = y)) /\
+  (solLength = (longest_common_prefix a2 solStart y)))))))))).
+intro a; intros.
 exists a.
 why3 "alt-ergo".
 Qed.