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Commit 52e55d5b authored by Jean-Christophe Filliâtre's avatar Jean-Christophe Filliâtre
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[library] renamed permut... symbols in map.MapPermut and array.ArrayPermut, 

as follows:

maps:   permut_sub     -> permut
arrays: map_permut_sub -> permut     (to be consistent with maps)
        permut_sub     -> permut_sub
        permut         -> permut_all
parent 1b81b0ad
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View file @ 52e55d5b
* marks an incompatible change
* [library] fixed inconsistency in libraries map.MapPermut and
array.ArrayPermut; names and meaning of symbols "permut..." have
been modified
version 0.82, December 12, 2013
===============================
......
examples/algo64.mlw
View file @ 52e55d5b
......@@ -50,7 +50,7 @@ module Algo64
end
let qs (a:array int) : unit
ensures { permut (old a) a }
ensures { permut_all (old a) a }
ensures { sorted a }
= if length a > 0 then quicksort a 0 (length a - 1)
......
examples/find.mlw
View file @ 52e55d5b
......@@ -39,18 +39,18 @@ module FIND
let find (a: array int) =
requires { length a = _N+1 }
ensures { found a /\ permut a (old a) }
ensures { found a /\ permut_all a (old a) }
'Init:
let m = ref 1 in let n = ref _N in
while !m < !n do
invariant { m_invariant !m a /\ n_invariant !n a /\
permut a (at a 'Init) /\ 1 <= !m /\ !n <= _N }
permut_all a (at a 'Init) /\ 1 <= !m /\ !n <= _N }
variant { !n - !m }
let r = a[f] in let i = ref !m in let j = ref !n in
while !i <= !j do
invariant { i_invariant !m !n !i r a /\ j_invariant !m !n !j r a /\
m_invariant !m a /\ n_invariant !n a /\ 0 <= !j /\ !i <= _N + 1 /\
termination !i !j !m !n r a /\ permut a (at a 'Init) }
termination !i !j !m !n r a /\ permut_all a (at a 'Init) }
variant { _N + 2 + !j - !i }
'L: while a[!i] < r do
invariant { i_invariant !m !n !i r a /\
......
examples/find/find_WP_FIND_WP_parameter_find_4.v
View file @ 52e55d5b
......@@ -80,48 +80,48 @@ Definition exchange {a:Type} {a_WT:WhyType a} (a1:(@array a a_WT))
((~ (k = i)) -> ((~ (k = j)) -> ((get a1 k) = (get a2 k)))))))).
(* Why3 assumption *)
Definition map_permut_sub {a:Type} {a_WT:WhyType a} (a1:(@array a a_WT))
(a2:(@array a a_WT)) (l:Z) (u:Z): Prop := ((length a1) = (length a2)) /\
Definition permut {a:Type} {a_WT:WhyType a} (a1:(@array a a_WT)) (a2:(@array
a a_WT)) (l:Z) (u:Z): Prop := ((length a1) = (length a2)) /\
(((0%Z <= l)%Z /\ (l <= (length a1))%Z) /\ (((0%Z <= u)%Z /\
(u <= (length a1))%Z) /\ (map.MapPermut.permut_sub (elts a1) (elts a2) l
u))).
(u <= (length a1))%Z) /\ (map.MapPermut.permut (elts a1) (elts a2) l u))).
(* Why3 assumption *)
Definition permut_sub {a:Type} {a_WT:WhyType a} (a1:(@array a a_WT))
(a2:(@array a a_WT)) (l:Z) (u:Z): Prop := (map_eq_sub (elts a1) (elts a2)
0%Z l) /\ ((map_permut_sub a1 a2 l u) /\ (map_eq_sub (elts a1) (elts a2) u
0%Z l) /\ ((permut a1 a2 l u) /\ (map_eq_sub (elts a1) (elts a2) u
(length a1))).
(* Why3 assumption *)
Definition permut {a:Type} {a_WT:WhyType a} (a1:(@array a a_WT)) (a2:(@array
a a_WT)): Prop := ((length a1) = (length a2)) /\ (map.MapPermut.permut_sub
(elts a1) (elts a2) 0%Z (length a1)).
Definition permut_all {a:Type} {a_WT:WhyType a} (a1:(@array a a_WT))
(a2:(@array a a_WT)): Prop := ((length a1) = (length a2)) /\
(map.MapPermut.permut (elts a1) (elts a2) 0%Z (length a1)).
Axiom permut_refl : forall {a:Type} {a_WT:WhyType a}, forall (a1:(@array
a a_WT)), (permut a1 a1).
Axiom permut_all_refl : forall {a:Type} {a_WT:WhyType a}, forall (a1:(@array
a a_WT)), (permut_all a1 a1).
Axiom exchange_permut : forall {a:Type} {a_WT:WhyType a}, forall (a1:(@array
a a_WT)) (a2:(@array a a_WT)) (i:Z) (j:Z), (exchange a1 a2 i j) -> (permut
a1 a2).
Axiom exchange_permut_all : forall {a:Type} {a_WT:WhyType a},
forall (a1:(@array a a_WT)) (a2:(@array a a_WT)) (i:Z) (j:Z), (exchange a1
a2 i j) -> (permut_all a1 a2).
Axiom permut_sym : forall {a:Type} {a_WT:WhyType a}, forall (a1:(@array
a a_WT)) (a2:(@array a a_WT)), (permut a1 a2) -> (permut a2 a1).
Axiom permut_all_sym : forall {a:Type} {a_WT:WhyType a}, forall (a1:(@array
a a_WT)) (a2:(@array a a_WT)), (permut_all a1 a2) -> (permut_all a2 a1).
Axiom permut_trans : forall {a:Type} {a_WT:WhyType a}, forall (a1:(@array
a a_WT)) (a2:(@array a a_WT)) (a3:(@array a a_WT)), (permut a1 a2) ->
((permut a2 a3) -> (permut a1 a3)).
Axiom permut_all_trans : forall {a:Type} {a_WT:WhyType a}, forall (a1:(@array
a a_WT)) (a2:(@array a a_WT)) (a3:(@array a a_WT)), (permut_all a1 a2) ->
((permut_all a2 a3) -> (permut_all a1 a3)).
Axiom array_eq_permut : forall {a:Type} {a_WT:WhyType a}, forall (a1:(@array
a a_WT)) (a2:(@array a a_WT)), (array_eq a1 a2) -> (permut a1 a2).
Axiom array_eq_permut_all : forall {a:Type} {a_WT:WhyType a},
forall (a1:(@array a a_WT)) (a2:(@array a a_WT)), (array_eq a1 a2) ->
(permut_all a1 a2).
Axiom permut_sub_weakening : forall {a:Type} {a_WT:WhyType a},
forall (a1:(@array a a_WT)) (a2:(@array a a_WT)) (l1:Z) (u1:Z) (l2:Z)
(u2:Z), (permut_sub a1 a2 l1 u1) -> (((0%Z <= l2)%Z /\ (l2 <= l1)%Z) ->
(((u1 <= u2)%Z /\ (u2 <= (length a1))%Z) -> (permut_sub a1 a2 l2 u2))).
Axiom permut_sub_permut : forall {a:Type} {a_WT:WhyType a},
Axiom permut_sub_permut_all : forall {a:Type} {a_WT:WhyType a},
forall (a1:(@array a a_WT)) (a2:(@array a a_WT)) (l:Z) (u:Z), (permut_sub
a1 a2 l u) -> (permut a1 a2).
a1 a2 l u) -> (permut_all a1 a2).
Parameter usN: Z.
......@@ -167,13 +167,13 @@ Import MapPermut.
Theorem WP_parameter_find : forall (a:Z) (a1:(@map.Map.map Z _ Z _)),
let a2 := (mk_array a a1) in (((0%Z <= a)%Z /\ (a = (usN + 1%Z)%Z)) ->
forall (n:Z) (m:Z) (a3:(@map.Map.map Z _ Z _)), let a4 := (mk_array a
a3) in (((m_invariant m a4) /\ ((n_invariant n a4) /\ ((permut a4 a2) /\
((1%Z <= m)%Z /\ (n <= usN)%Z)))) -> ((m < n)%Z -> let o := f in
a3) in (((m_invariant m a4) /\ ((n_invariant n a4) /\ ((permut_all a4
a2) /\ ((1%Z <= m)%Z /\ (n <= usN)%Z)))) -> ((m < n)%Z -> let o := f in
(((0%Z <= a)%Z /\ ((0%Z <= o)%Z /\ (o < a)%Z)) -> let r := (map.Map.get a3
o) in forall (j:Z) (i:Z) (a5:(@map.Map.map Z _ Z _)), let a6 := (mk_array a
a5) in (((i_invariant m n i r a6) /\ ((j_invariant m n j r a6) /\
((m_invariant m a6) /\ ((n_invariant n a6) /\ ((0%Z <= j)%Z /\
((i <= (usN + 1%Z)%Z)%Z /\ ((termination i j m n r a6) /\ (permut a6
((i <= (usN + 1%Z)%Z)%Z /\ ((termination i j m n r a6) /\ (permut_all a6
a2)))))))) -> ((i <= j)%Z -> forall (i1:Z), ((i_invariant m n i1 r a6) /\
(((i <= i1)%Z /\ (i1 <= n)%Z) /\ (termination i1 j m n r a6))) ->
(((0%Z <= a)%Z /\ ((0%Z <= i1)%Z /\ (i1 < a)%Z)) -> ((~ ((map.Map.get a5
......@@ -191,7 +191,8 @@ Theorem WP_parameter_find : forall (a:Z) (a1:(@map.Map.map Z _ Z _)),
(i2 = (i1 + 1%Z)%Z) -> forall (j2:Z), (j2 = (j1 - 1%Z)%Z) -> ((i_invariant
m n i2 r a9) /\ ((j_invariant m n j2 r a9) /\ ((m_invariant m a9) /\
((n_invariant n a9) /\ ((0%Z <= j2)%Z /\ ((i2 <= (usN + 1%Z)%Z)%Z /\
((termination i2 j2 m n r a9) /\ (permut a9 a2)))))))))))))))))))))))))))).
((termination i2 j2 m n r a9) /\ (permut_all a9
a2)))))))))))))))))))))))))))).
(* Why3 intros a a1 a2 (h1,h2) n m a3 a4 (h3,(h4,(h5,(h6,h7)))) h8 o
(h9,(h10,h11)) r j i a5 a6
(h12,(h13,(h14,(h15,(h16,(h17,(h18,h19))))))) h20 i1
......@@ -336,8 +337,8 @@ intuition.
red; simpl.
split. trivial.
apply permut_trans with a5.
assert (permut (mk_array a a8) (mk_array a a5)).
apply exchange_permut with i1 j1; auto.
assert (permut_all (mk_array a a8) (mk_array a a5)).
apply exchange_permut_all with i1 j1; auto.
destruct H54; intuition.
red in H19; intuition.
Qed.
......
examples/find/why3session.xml
View file @ 52e55d5b
This diff is collapsed.
examples/flag.mlw
View file @ 52e55d5b
......@@ -28,7 +28,7 @@ module Flag
monochrome a 0 b Blue /\
monochrome a b r White /\
monochrome a r (length a) Red }
ensures { permut (old a) a }
ensures { permut_all (old a) a }
=
let b = ref 0 in
let i = ref 0 in
......@@ -39,7 +39,7 @@ module Flag
invariant { monochrome a 0 !b Blue }
invariant { monochrome a !b !i White }
invariant { monochrome a !r (length a) Red }
invariant { permut (at a 'Init) a }
invariant { permut_all (at a 'Init) a }
variant { !r - !i }
match a[!i] with
| Blue ->
......
examples/flag/why3session.xml
View file @ 52e55d5b
This diff is collapsed.
examples/insertion_sort.mlw
View file @ 52e55d5b
......@@ -11,16 +11,16 @@ module InsertionSort
use import array.ArrayEq
let insertion_sort (a: array int) =
ensures { sorted a /\ permut (old a) a }
ensures { sorted a /\ permut_all (old a) a }
'L:
for i = 1 to length a - 1 do
(* a[0..i[ is sorted; now insert a[i] *)
invariant { sorted_sub a 0 i /\ permut (at a 'L) a }
invariant { sorted_sub a 0 i /\ permut_all (at a 'L) a }
let v = a[i] in
let j = ref i in
while !j > 0 && a[!j - 1] > v do
invariant { 0 <= !j <= i }
invariant { permut (at a 'L) a[!j <- v] }
invariant { permut_all (at a 'L) a[!j <- v] }
invariant { forall k1 k2: int.
0 <= k1 <= k2 <= i -> k1 <> !j -> k2 <> !j -> a[k1] <= a[k2] }
invariant { forall k: int. !j+1 <= k <= i -> v < a[k] }
......@@ -86,16 +86,16 @@ module InsertionSortGen
M.sorted_sub a.elts 0 a.length
let insertion_sort (a: array elt) =
ensures { sorted a /\ permut (old a) a }
ensures { sorted a /\ permut_all (old a) a }
'L:
for i = 1 to length a - 1 do
(* a[0..i[ is sorted; now insert a[i] *)
invariant { sorted_sub a 0 i /\ permut (at a 'L) a }
invariant { sorted_sub a 0 i /\ permut_all (at a 'L) a }
let v = a[i] in
let j = ref i in
while !j > 0 && not (le a[!j - 1] v) do
invariant { 0 <= !j <= i }
invariant { permut (at a 'L) a[!j <- v] }
invariant { permut_all (at a 'L) a[!j <- v] }
invariant { forall k1 k2: int.
0 <= k1 <= k2 <= i -> k1 <> !j -> k2 <> !j -> le a[k1] a[k2] }
invariant { forall k: int. !j+1 <= k <= i -> le v a[k] }
......
examples/insertion_sort/insertion_sort_InsertionSortGen_WP_parameter_insertion_sort_1.v
View file @ 52e55d5b
......@@ -80,50 +80,48 @@ Definition exchange {a:Type} {a_WT:WhyType a} (a1:(@array a a_WT))
((~ (k = i)) -> ((~ (k = j)) -> ((get a1 k) = (get a2 k)))))))).
(* Why3 assumption *)
Definition map_permut_sub {a:Type} {a_WT:WhyType a} (a1:(@array a a_WT))
(a2:(@array a a_WT)) (l:Z) (u:Z): Prop := ((length a1) = (length a2)) /\
Definition permut {a:Type} {a_WT:WhyType a} (a1:(@array a a_WT)) (a2:(@array
a a_WT)) (l:Z) (u:Z): Prop := ((length a1) = (length a2)) /\
(((0%Z <= l)%Z /\ (l <= (length a1))%Z) /\ (((0%Z <= u)%Z /\
(u <= (length a1))%Z) /\ (map.MapPermut.permut_sub (elts a1) (elts a2) l
u))).
(u <= (length a1))%Z) /\ (map.MapPermut.permut (elts a1) (elts a2) l u))).
(* Why3 assumption *)
Definition permut_sub {a:Type} {a_WT:WhyType a} (a1:(@array a a_WT))
(a2:(@array a a_WT)) (l:Z) (u:Z): Prop := ((length a1) = (length a2)) /\
(((0%Z <= l)%Z /\ (l <= (length a1))%Z) /\ (((0%Z <= u)%Z /\
(u <= (length a1))%Z) /\ ((map_eq_sub (elts a1) (elts a2) 0%Z l) /\
((map.MapPermut.permut_sub (elts a1) (elts a2) l u) /\ (map_eq_sub
(elts a2) (elts a2) u (length a1)))))).
(a2:(@array a a_WT)) (l:Z) (u:Z): Prop := (map_eq_sub (elts a1) (elts a2)
0%Z l) /\ ((permut a1 a2 l u) /\ (map_eq_sub (elts a1) (elts a2) u
(length a1))).
(* Why3 assumption *)
Definition permut {a:Type} {a_WT:WhyType a} (a1:(@array a a_WT)) (a2:(@array
a a_WT)): Prop := ((length a1) = (length a2)) /\ (map.MapPermut.permut_sub
(elts a1) (elts a2) 0%Z (length a1)).
Definition permut_all {a:Type} {a_WT:WhyType a} (a1:(@array a a_WT))
(a2:(@array a a_WT)): Prop := ((length a1) = (length a2)) /\
(map.MapPermut.permut (elts a1) (elts a2) 0%Z (length a1)).
Axiom permut_refl : forall {a:Type} {a_WT:WhyType a}, forall (a1:(@array
a a_WT)), (permut a1 a1).
Axiom permut_all_refl : forall {a:Type} {a_WT:WhyType a}, forall (a1:(@array
a a_WT)), (permut_all a1 a1).
Axiom exchange_permut : forall {a:Type} {a_WT:WhyType a}, forall (a1:(@array
a a_WT)) (a2:(@array a a_WT)) (i:Z) (j:Z), (exchange a1 a2 i j) -> (permut
a1 a2).
Axiom exchange_permut_all : forall {a:Type} {a_WT:WhyType a},
forall (a1:(@array a a_WT)) (a2:(@array a a_WT)) (i:Z) (j:Z), (exchange a1
a2 i j) -> (permut_all a1 a2).
Axiom permut_sym : forall {a:Type} {a_WT:WhyType a}, forall (a1:(@array
a a_WT)) (a2:(@array a a_WT)), (permut a1 a2) -> (permut a2 a1).
Axiom permut_all_sym : forall {a:Type} {a_WT:WhyType a}, forall (a1:(@array
a a_WT)) (a2:(@array a a_WT)), (permut_all a1 a2) -> (permut_all a2 a1).
Axiom permut_trans : forall {a:Type} {a_WT:WhyType a}, forall (a1:(@array
a a_WT)) (a2:(@array a a_WT)) (a3:(@array a a_WT)), (permut a1 a2) ->
((permut a2 a3) -> (permut a1 a3)).
Axiom permut_all_trans : forall {a:Type} {a_WT:WhyType a}, forall (a1:(@array
a a_WT)) (a2:(@array a a_WT)) (a3:(@array a a_WT)), (permut_all a1 a2) ->
((permut_all a2 a3) -> (permut_all a1 a3)).
Axiom array_eq_permut : forall {a:Type} {a_WT:WhyType a}, forall (a1:(@array
a a_WT)) (a2:(@array a a_WT)), (array_eq a1 a2) -> (permut a1 a2).
Axiom array_eq_permut_all : forall {a:Type} {a_WT:WhyType a},
forall (a1:(@array a a_WT)) (a2:(@array a a_WT)), (array_eq a1 a2) ->
(permut_all a1 a2).
Axiom permut_sub_weakening : forall {a:Type} {a_WT:WhyType a},
forall (a1:(@array a a_WT)) (a2:(@array a a_WT)) (l1:Z) (u1:Z) (l2:Z)
(u2:Z), (permut_sub a1 a2 l1 u1) -> (((0%Z <= l2)%Z /\ (l2 <= l1)%Z) ->
(((u1 <= u2)%Z /\ (u2 <= (length a1))%Z) -> (permut_sub a1 a2 l2 u2))).
Axiom permut_sub_permut : forall {a:Type} {a_WT:WhyType a},
Axiom permut_sub_permut_all : forall {a:Type} {a_WT:WhyType a},
forall (a1:(@array a a_WT)) (a2:(@array a a_WT)) (l:Z) (u:Z), (permut_sub
a1 a2 l u) -> (permut a1 a2).
a1 a2 l u) -> (permut_all a1 a2).
Axiom elt : Type.
Parameter elt_WhyType : WhyType elt.
......@@ -161,21 +159,22 @@ Theorem WP_parameter_insertion_sort : forall (a:Z) (a1:(@map.Map.map Z _
elt elt_WhyType)), let a2 := (mk_array a a1) in ((0%Z <= a)%Z -> let o :=
(a - 1%Z)%Z in ((1%Z <= o)%Z -> forall (a3:(@map.Map.map Z _
elt elt_WhyType)), forall (i:Z), ((1%Z <= i)%Z /\ (i <= o)%Z) ->
(((sorted_sub a3 0%Z i) /\ (permut a2 (mk_array a a3))) ->
(((sorted_sub a3 0%Z i) /\ (permut_all a2 (mk_array a a3))) ->
(((0%Z <= a)%Z /\ ((0%Z <= i)%Z /\ (i < a)%Z)) -> let v := (map.Map.get a3
i) in forall (j:Z) (a4:(@map.Map.map Z _ elt elt_WhyType)),
(((0%Z <= j)%Z /\ (j <= i)%Z) /\ ((permut a2 (mk_array a (map.Map.set a4 j
v))) /\ ((forall (k1:Z) (k2:Z), ((0%Z <= k1)%Z /\ ((k1 <= k2)%Z /\
(k2 <= i)%Z)) -> ((~ (k1 = j)) -> ((~ (k2 = j)) -> (le (map.Map.get a4 k1)
(map.Map.get a4 k2))))) /\ forall (k:Z), (((j + 1%Z)%Z <= k)%Z /\
(k <= i)%Z) -> (le v (map.Map.get a4 k))))) -> ((0%Z < j)%Z -> let o1 :=
(j - 1%Z)%Z in (((0%Z <= a)%Z /\ ((0%Z <= o1)%Z /\ (o1 < a)%Z)) -> ((~ (le
(map.Map.get a4 o1) v)) -> let o2 := (j - 1%Z)%Z in (((0%Z <= o2)%Z /\
(o2 < a)%Z) -> (((0%Z <= j)%Z /\ (j < a)%Z) -> forall (a5:(@map.Map.map Z _
(((0%Z <= j)%Z /\ (j <= i)%Z) /\ ((permut_all a2 (mk_array a
(map.Map.set a4 j v))) /\ ((forall (k1:Z) (k2:Z), ((0%Z <= k1)%Z /\
((k1 <= k2)%Z /\ (k2 <= i)%Z)) -> ((~ (k1 = j)) -> ((~ (k2 = j)) -> (le
(map.Map.get a4 k1) (map.Map.get a4 k2))))) /\ forall (k:Z),
(((j + 1%Z)%Z <= k)%Z /\ (k <= i)%Z) -> (le v (map.Map.get a4 k))))) ->
((0%Z < j)%Z -> let o1 := (j - 1%Z)%Z in (((0%Z <= a)%Z /\
((0%Z <= o1)%Z /\ (o1 < a)%Z)) -> ((~ (le (map.Map.get a4 o1) v)) ->
let o2 := (j - 1%Z)%Z in (((0%Z <= o2)%Z /\ (o2 < a)%Z) ->
(((0%Z <= j)%Z /\ (j < a)%Z) -> forall (a5:(@map.Map.map Z _
elt elt_WhyType)), ((0%Z <= a)%Z /\ (a5 = (map.Map.set a4 j (map.Map.get a4
o2)))) -> ((exchange (mk_array a (map.Map.set a4 j v)) (mk_array a
(map.Map.set a5 (j - 1%Z)%Z v)) (j - 1%Z)%Z j) -> forall (j1:Z),
(j1 = (j - 1%Z)%Z) -> (permut a2 (mk_array a (map.Map.set a5 j1
(j1 = (j - 1%Z)%Z) -> (permut_all a2 (mk_array a (map.Map.set a5 j1
v))))))))))))).
(* Why3 intros a a1 a2 h1 o h2 a3 i (h3,h4) (h5,h6) (h7,(h8,h9)) v j a4
((h10,h11),(h12,(h13,h14))) h15 o1 (h16,(h17,h18)) h19 o2 (h20,h21)
......@@ -183,7 +182,7 @@ Theorem WP_parameter_insertion_sort : forall (a:Z) (a1:(@map.Map.map Z _
intros a a1 a2 h1 o h2 a3 i (h3,h4) (h5,h6) (h7,(h8,h9)) v j a4
((h10,h11),(h12,(h13,h14))) h15 o1 (h16,(h17,h18)) h19 o2 (h20,h21)
(h22,h23) a5 (h24,h25) h26 j1 h27.
unfold permut in *.
unfold permut_all in *.
simpl; split; auto.
simpl in h12.
destruct h12 as (h12a & h12b).
......
examples/insertion_sort/insertion_sort_WP_InsertionSort_WP_parameter_insertion_sort_1.v
View file @ 52e55d5b
......@@ -93,50 +93,48 @@ Definition exchange {a:Type} {a_WT:WhyType a} (a1:(@array a a_WT))
((~ (k = i)) -> ((~ (k = j)) -> ((get a1 k) = (get a2 k)))))))).
(* Why3 assumption *)
Definition map_permut_sub {a:Type} {a_WT:WhyType a} (a1:(@array a a_WT))
(a2:(@array a a_WT)) (l:Z) (u:Z): Prop := ((length a1) = (length a2)) /\
Definition permut {a:Type} {a_WT:WhyType a} (a1:(@array a a_WT)) (a2:(@array
a a_WT)) (l:Z) (u:Z): Prop := ((length a1) = (length a2)) /\
(((0%Z <= l)%Z /\ (l <= (length a1))%Z) /\ (((0%Z <= u)%Z /\
(u <= (length a1))%Z) /\ (map.MapPermut.permut_sub (elts a1) (elts a2) l
u))).
(u <= (length a1))%Z) /\ (map.MapPermut.permut (elts a1) (elts a2) l u))).
(* Why3 assumption *)
Definition permut_sub {a:Type} {a_WT:WhyType a} (a1:(@array a a_WT))
(a2:(@array a a_WT)) (l:Z) (u:Z): Prop := ((length a1) = (length a2)) /\
(((0%Z <= l)%Z /\ (l <= (length a1))%Z) /\ (((0%Z <= u)%Z /\
(u <= (length a1))%Z) /\ ((map_eq_sub (elts a1) (elts a2) 0%Z l) /\
((map.MapPermut.permut_sub (elts a1) (elts a2) l u) /\ (map_eq_sub
(elts a2) (elts a2) u (length a1)))))).
(a2:(@array a a_WT)) (l:Z) (u:Z): Prop := (map_eq_sub (elts a1) (elts a2)
0%Z l) /\ ((permut a1 a2 l u) /\ (map_eq_sub (elts a1) (elts a2) u
(length a1))).
(* Why3 assumption *)
Definition permut {a:Type} {a_WT:WhyType a} (a1:(@array a a_WT)) (a2:(@array
a a_WT)): Prop := ((length a1) = (length a2)) /\ (map.MapPermut.permut_sub
(elts a1) (elts a2) 0%Z (length a1)).
Definition permut_all {a:Type} {a_WT:WhyType a} (a1:(@array a a_WT))
(a2:(@array a a_WT)): Prop := ((length a1) = (length a2)) /\
(map.MapPermut.permut (elts a1) (elts a2) 0%Z (length a1)).
Axiom permut_refl : forall {a:Type} {a_WT:WhyType a}, forall (a1:(@array
a a_WT)), (permut a1 a1).
Axiom permut_all_refl : forall {a:Type} {a_WT:WhyType a}, forall (a1:(@array
a a_WT)), (permut_all a1 a1).
Axiom exchange_permut : forall {a:Type} {a_WT:WhyType a}, forall (a1:(@array
a a_WT)) (a2:(@array a a_WT)) (i:Z) (j:Z), (exchange a1 a2 i j) -> (permut
a1 a2).
Axiom exchange_permut_all : forall {a:Type} {a_WT:WhyType a},
forall (a1:(@array a a_WT)) (a2:(@array a a_WT)) (i:Z) (j:Z), (exchange a1
a2 i j) -> (permut_all a1 a2).
Axiom permut_sym : forall {a:Type} {a_WT:WhyType a}, forall (a1:(@array
a a_WT)) (a2:(@array a a_WT)), (permut a1 a2) -> (permut a2 a1).
Axiom permut_all_sym : forall {a:Type} {a_WT:WhyType a}, forall (a1:(@array
a a_WT)) (a2:(@array a a_WT)), (permut_all a1 a2) -> (permut_all a2 a1).
Axiom permut_trans : forall {a:Type} {a_WT:WhyType a}, forall (a1:(@array
a a_WT)) (a2:(@array a a_WT)) (a3:(@array a a_WT)), (permut a1 a2) ->
((permut a2 a3) -> (permut a1 a3)).
Axiom permut_all_trans : forall {a:Type} {a_WT:WhyType a}, forall (a1:(@array
a a_WT)) (a2:(@array a a_WT)) (a3:(@array a a_WT)), (permut_all a1 a2) ->
((permut_all a2 a3) -> (permut_all a1 a3)).
Axiom array_eq_permut : forall {a:Type} {a_WT:WhyType a}, forall (a1:(@array
a a_WT)) (a2:(@array a a_WT)), (array_eq a1 a2) -> (permut a1 a2).
Axiom array_eq_permut_all : forall {a:Type} {a_WT:WhyType a},
forall (a1:(@array a a_WT)) (a2:(@array a a_WT)), (array_eq a1 a2) ->
(permut_all a1 a2).
Axiom permut_sub_weakening : forall {a:Type} {a_WT:WhyType a},
forall (a1:(@array a a_WT)) (a2:(@array a a_WT)) (l1:Z) (u1:Z) (l2:Z)
(u2:Z), (permut_sub a1 a2 l1 u1) -> (((0%Z <= l2)%Z /\ (l2 <= l1)%Z) ->
(((u1 <= u2)%Z /\ (u2 <= (length a1))%Z) -> (permut_sub a1 a2 l2 u2))).
Axiom permut_sub_permut : forall {a:Type} {a_WT:WhyType a},
Axiom permut_sub_permut_all : forall {a:Type} {a_WT:WhyType a},
forall (a1:(@array a a_WT)) (a2:(@array a a_WT)) (l:Z) (u:Z), (permut_sub
a1 a2 l u) -> (permut a1 a2).
a1 a2 l u) -> (permut_all a1 a2).
Import MapPermut.
......@@ -145,10 +143,10 @@ Theorem WP_parameter_insertion_sort : forall (a:Z) (a1:(@map.Map.map Z _
Z _)), let a2 := (mk_array a a1) in ((0%Z <= a)%Z -> let o :=
(a - 1%Z)%Z in ((1%Z <= o)%Z -> forall (a3:(@map.Map.map Z _ Z _)),
forall (i:Z), ((1%Z <= i)%Z /\ (i <= o)%Z) -> (((sorted_sub a3 0%Z i) /\
(permut a2 (mk_array a a3))) -> (((0%Z <= a)%Z /\ ((0%Z <= i)%Z /\
(permut_all a2 (mk_array a a3))) -> (((0%Z <= a)%Z /\ ((0%Z <= i)%Z /\
(i < a)%Z)) -> let v := (map.Map.get a3 i) in forall (j:Z)
(a4:(@map.Map.map Z _ Z _)), (((0%Z <= j)%Z /\ (j <= i)%Z) /\ ((permut a2
(mk_array a (map.Map.set a4 j v))) /\ ((forall (k1:Z) (k2:Z),
(a4:(@map.Map.map Z _ Z _)), (((0%Z <= j)%Z /\ (j <= i)%Z) /\ ((permut_all
a2 (mk_array a (map.Map.set a4 j v))) /\ ((forall (k1:Z) (k2:Z),
((0%Z <= k1)%Z /\ ((k1 <= k2)%Z /\ (k2 <= i)%Z)) -> ((~ (k1 = j)) ->
((~ (k2 = j)) -> ((map.Map.get a4 k1) <= (map.Map.get a4 k2))%Z))) /\
forall (k:Z), (((j + 1%Z)%Z <= k)%Z /\ (k <= i)%Z) -> (v < (map.Map.get a4
......@@ -158,7 +156,7 @@ Theorem WP_parameter_insertion_sort : forall (a:Z) (a1:(@map.Map.map Z _
(j < a)%Z) -> forall (a5:(@map.Map.map Z _ Z _)), ((0%Z <= a)%Z /\
(a5 = (map.Map.set a4 j (map.Map.get a4 o2)))) -> ((exchange (mk_array a
(map.Map.set a4 j v)) (mk_array a (map.Map.set a5 (j - 1%Z)%Z v))
(j - 1%Z)%Z j) -> forall (j1:Z), (j1 = (j - 1%Z)%Z) -> (permut a2
(j - 1%Z)%Z j) -> forall (j1:Z), (j1 = (j - 1%Z)%Z) -> (permut_all a2
(mk_array a (map.Map.set a5 j1 v))))))))))))).
(* Why3 intros a a1 a2 h1 o h2 a3 i (h3,h4) (h5,h6) (h7,(h8,h9)) v j a4
((h10,h11),(h12,(h13,h14))) h15 o1 (h16,(h17,h18)) h19 o2 (h20,h21)
......@@ -167,7 +165,7 @@ intros a a1 a2 h1 o h2 a3 i (h3,h4) (h5,h6) (h7,(h8,h9)) v j a4
((h10,h11),(h12,(h13,h14))) h15 o1 (h16,(h17,h18)) h19 o2 (h20,h21)
(h22,h23) a5 (h24,h25) h26 j1 h27.
intuition.
unfold permut.
unfold permut_all.
split.
simpl.
auto.
......@@ -175,10 +173,10 @@ subst a5.
simpl.
apply permut_trans with (elts (set (mk_array a a4) j (Map.get a3 i))); auto.
subst j1.
unfold permut in h12.
unfold permut_all in h12.
intuition.
generalize (exchange_permut _ _ _ _ h26).
unfold permut; simpl; intuition.
generalize (exchange_permut_all _ _ _ _ h26).
unfold permut_all; simpl; intuition.
subst j1; assumption.
Qed.
examples/insertion_sort/why3session.xml
View file @ 52e55d5b
This diff is collapsed.
examples/insertion_sort_naive.mlw
View file @ 52e55d5b
......@@ -17,16 +17,16 @@ module InsertionSortNaive
let sort (a:array int)
ensures { sorted a }
ensures { permut (old a) a }
ensures { permut_all (old a) a }
=
'Init:
for i = 0 to a.length - 1 do
invariant { permut (at a 'Init) a }
invariant { permut_all (at a 'Init) a }
invariant { sorted_sub a 0 i }
let j = ref i in
while !j > 0 && a[!j-1] > a[!j] do
invariant { 0 <= !j <= i }
invariant { permut (at a 'Init) a }
invariant { permut_all (at a 'Init) a }
invariant { sorted_sub a 0 !j }
invariant { sorted_sub a !j (i+1) }
invariant { forall k1 k2:int. 0 <= k1 < !j /\ !j+1 <= k2 <= i ->
......@@ -70,16 +70,16 @@ module InsertionSortNaiveGen
let sort (a:array elt)
ensures { sorted a }
ensures { permut (old a) a }
ensures { permut_all (old a) a }
=
'Init:
for i = 0 to a.length - 1 do
invariant { permut (at a 'Init) a }
invariant { permut_all (at a 'Init) a }
invariant { sorted_sub a 0 i }
let j = ref i in
while !j > 0 && not (le a[!j-1] a[!j]) do
invariant { 0 <= !j <= i }
invariant { permut (at a 'Init) a }
invariant { permut_all (at a 'Init) a }
invariant { sorted_sub a 0 !j }
invariant { sorted_sub a !j (i+1) }
invariant { forall k1 k2:int. 0 <= k1 < !j /\ !j+1 <= k2 <= i ->
......@@ -130,16 +130,16 @@ module InsertionSortParam
let sort (p:param) (a:array elt)
ensures { sorted p a }
ensures { permut (old a) a }
ensures { permut_all (old a) a }
=
'Init:
for i = 0 to a.length - 1 do
invariant { permut (at a 'Init) a }
invariant { permut_all (at a 'Init) a }
invariant { sorted_sub p a.elts 0 i }
let j = ref i in
while !j > 0 && not (le p a[!j-1] a[!j]) do
invariant { 0 <= !j <= i }
invariant { permut (at a 'Init) a }
invariant { permut_all (at a 'Init) a }
invariant { sorted_sub p a.elts 0 !j }
invariant { sorted_sub p a.elts !j (i+1) }
invariant { forall k1 k2:int. 0 <= k1 < !j /\ !j+1 <= k2 <= i ->
......@@ -191,16 +191,16 @@ module InsertionSortParamBad
let sort (p:param) (a:array elt)
ensures { sorted p a }
ensures { permut (old a) a }
ensures { permut_all (old a) a }