Binary heap update

examples/programs/vacid_0_binary_heaps/elements.why

@@ -12,7 +12,7 @@ function elements (a:array 'a) (i j:int) : bag 'a
 axiom Elements_empty : forall a:array 'a, i j:int.
      i >= j -> (elements a i j) = empty_bag
 
-axiom Elements_union2 : forall a:array 'a, i j :int.
+axiom Elements_add : forall a:array 'a, i j :int.
        i < j ->
        (elements a i j) = (add (A.get a (j-1)) (elements a i (j-1)))
 
@@ -24,7 +24,7 @@ lemma Elements_union : forall a:array 'a, i j k:int.
      i <= j <= k  ->
       (elements a i k) = (union (elements a i j) (elements a j k))
 
-lemma Elements_union1 : forall a:array 'a, i j :int.
+lemma Elements_add1 : forall a:array 'a, i j :int.
       i < j ->
       (elements a i j) = (add (A.get a i) (elements a (i+1) j))
 

examples/programs/vacid_0_binary_heaps/proofs/elements_Elements_Elements_add1_1.v

+(* This file is generated by Why3's Coq driver *)
+(* Beware! Only edit allowed sections below    *)
+Require Import ZArith.
+Require Import Rbase.
+Parameter bag : forall (a:Type), Type.
+
+Parameter nb_occ: forall (a:Type), a -> (bag a) -> Z.
+
+Implicit Arguments nb_occ.
+
+Axiom occ_non_negative : forall (a:Type), forall (b:(bag a)) (x:a),
+  (0%Z <= (nb_occ x b))%Z.
+
+Definition eq_bag (a:Type)(a1:(bag a)) (b:(bag a)): Prop := forall (x:a),
+  ((nb_occ x a1) = (nb_occ x b)).
+Implicit Arguments eq_bag.
+
+Axiom bag_extensionality : forall (a:Type), forall (a1:(bag a)) (b:(bag a)),
+  (eq_bag a1 b) -> (a1 = b).
+
+Parameter empty_bag: forall (a:Type), (bag a).
+
+Set Contextual Implicit.
+Implicit Arguments empty_bag.
+Unset Contextual Implicit.
+
+Axiom occ_empty : forall (a:Type), forall (x:a), ((nb_occ x (empty_bag:(bag
+  a))) = 0%Z).
+
+Axiom is_empty : forall (a:Type), forall (b:(bag a)), (forall (x:a),
+  ((nb_occ x b) = 0%Z)) -> (b = (empty_bag:(bag a))).
+
+Parameter singleton: forall (a:Type), a -> (bag a).
+
+Implicit Arguments singleton.
+
+Axiom occ_singleton : forall (a:Type), forall (x:a) (y:a), ((x = y) /\
+  ((nb_occ y (singleton x)) = 1%Z)) \/ ((~ (x = y)) /\ ((nb_occ y
+  (singleton x)) = 0%Z)).
+
+Axiom occ_singleton_eq : forall (a:Type), forall (x:a) (y:a), (x = y) ->
+  ((nb_occ y (singleton x)) = 1%Z).
+
+Axiom occ_singleton_neq : forall (a:Type), forall (x:a) (y:a), (~ (x = y)) ->
+  ((nb_occ y (singleton x)) = 0%Z).
+
+Parameter union: forall (a:Type), (bag a) -> (bag a) -> (bag a).
+
+Implicit Arguments union.
+
+Axiom occ_union : forall (a:Type), forall (x:a) (a1:(bag a)) (b:(bag a)),
+  ((nb_occ x (union a1 b)) = ((nb_occ x a1) + (nb_occ x b))%Z).
+
+Axiom Union_comm : forall (a:Type), forall (a1:(bag a)) (b:(bag a)),
+  ((union a1 b) = (union b a1)).
+
+Axiom Union_identity : forall (a:Type), forall (a1:(bag a)), ((union a1
+  (empty_bag:(bag a))) = a1).
+
+Axiom Union_assoc : forall (a:Type), forall (a1:(bag a)) (b:(bag a)) (c:(bag
+  a)), ((union a1 (union b c)) = (union (union a1 b) c)).
+
+Axiom bag_simpl : forall (a:Type), forall (a1:(bag a)) (b:(bag a)) (c:(bag
+  a)), ((union a1 b) = (union c b)) -> (a1 = c).
+
+Axiom bag_simpl_left : forall (a:Type), forall (a1:(bag a)) (b:(bag a))
+  (c:(bag a)), ((union a1 b) = (union a1 c)) -> (b = c).
+
+Definition add (a:Type)(x:a) (b:(bag a)): (bag a) := (union (singleton x) b).
+Implicit Arguments add.
+
+Axiom occ_add_eq : forall (a:Type), forall (b:(bag a)) (x:a) (y:a),
+  (x = y) -> ((nb_occ x (add x b)) = ((nb_occ x b) + 1%Z)%Z).
+
+Axiom occ_add_neq : forall (a:Type), forall (b:(bag a)) (x:a) (y:a),
+  (~ (x = y)) -> ((nb_occ y (add x b)) = (nb_occ y b)).
+
+Parameter card: forall (a:Type), (bag a) -> Z.
+
+Implicit Arguments card.
+
+Axiom Card_empty : forall (a:Type), ((card (empty_bag:(bag a))) = 0%Z).
+
+Axiom Card_singleton : forall (a:Type), forall (x:a),
+  ((card (singleton x)) = 1%Z).
+
+Axiom Card_union : forall (a:Type), forall (x:(bag a)) (y:(bag a)),
+  ((card (union x y)) = ((card x) + (card y))%Z).
+
+Axiom Card_zero_empty : forall (a:Type), forall (x:(bag a)),
+  ((card x) = 0%Z) -> (x = (empty_bag:(bag a))).
+
+Axiom Max_is_ge : forall (x:Z) (y:Z), (x <= (Zmax x y))%Z /\
+  (y <= (Zmax x y))%Z.
+
+Axiom Max_is_some : forall (x:Z) (y:Z), ((Zmax x y) = x) \/ ((Zmax x y) = y).
+
+Axiom Min_is_le : forall (x:Z) (y:Z), ((Zmin x y) <= x)%Z /\
+  ((Zmin x y) <= y)%Z.
+
+Axiom Min_is_some : forall (x:Z) (y:Z), ((Zmin x y) = x) \/ ((Zmin x y) = y).
+
+Axiom Max_x : forall (x:Z) (y:Z), (y <= x)%Z -> ((Zmax x y) = x).
+
+Axiom Max_y : forall (x:Z) (y:Z), (x <= y)%Z -> ((Zmax x y) = y).
+
+Axiom Min_x : forall (x:Z) (y:Z), (x <= y)%Z -> ((Zmin x y) = x).
+
+Axiom Min_y : forall (x:Z) (y:Z), (y <= x)%Z -> ((Zmin x y) = y).
+
+Axiom Max_sym : forall (x:Z) (y:Z), (y <= x)%Z -> ((Zmax x y) = (Zmax y x)).
+
+Axiom Min_sym : forall (x:Z) (y:Z), (y <= x)%Z -> ((Zmin x y) = (Zmin y x)).
+
+Parameter diff: forall (a:Type), (bag a) -> (bag a) -> (bag a).
+
+Implicit Arguments diff.
+
+Axiom Diff_occ : forall (a:Type), forall (b1:(bag a)) (b2:(bag a)) (x:a),
+  ((nb_occ x (diff b1 b2)) = (Zmax 0%Z ((nb_occ x b1) - (nb_occ x b2))%Z)).
+
+Axiom Diff_empty_right : forall (a:Type), forall (b:(bag a)), ((diff b
+  (empty_bag:(bag a))) = b).
+
+Axiom Diff_empty_left : forall (a:Type), forall (b:(bag a)),
+  ((diff (empty_bag:(bag a)) b) = (empty_bag:(bag a))).
+
+Axiom Diff_add : forall (a:Type), forall (b:(bag a)) (x:a), ((diff (add x b)
+  (singleton x)) = b).
+
+Axiom Diff_comm : forall (a:Type), forall (b:(bag a)) (b1:(bag a)) (b2:(bag
+  a)), ((diff (diff b b1) b2) = (diff (diff b b2) b1)).
+
+Axiom Add_diff : forall (a:Type), forall (b:(bag a)) (x:a), (0%Z <  (nb_occ x
+  b))%Z -> ((add x (diff b (singleton x))) = b).
+
+Parameter map : forall (a:Type) (b:Type), Type.
+
+Parameter get: forall (a:Type) (b:Type), (map a b) -> a -> b.
+
+Implicit Arguments get.
+
+Parameter set: forall (a:Type) (b:Type), (map a b) -> a -> b -> (map a b).
+
+Implicit Arguments set.
+
+Axiom Select_eq : forall (a:Type) (b:Type), 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) (b:Type), 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 (b:Type) (a:Type), b -> (map a b).
+
+Set Contextual Implicit.
+Implicit Arguments const.
+Unset Contextual Implicit.
+
+Axiom Const : forall (b:Type) (a:Type), forall (b1:b) (a1:a), ((get (const(
+  b1):(map a b)) a1) = b1).
+
+Definition array (a:Type) := (map Z a).
+
+Parameter elements: forall (a:Type), (map Z a) -> Z -> Z -> (bag a).
+
+Implicit Arguments elements.
+
+Axiom Elements_empty : forall (a:Type), forall (a1:(map Z a)) (i:Z) (j:Z),
+  (j <= i)%Z -> ((elements a1 i j) = (empty_bag:(bag a))).
+
+Axiom Elements_add : forall (a:Type), forall (a1:(map Z a)) (i:Z) (j:Z),
+  (i <  j)%Z -> ((elements a1 i j) = (add (get a1 (j - 1%Z)%Z) (elements a1 i
+  (j - 1%Z)%Z))).
+
+Axiom Elements_singleton : forall (a:Type), forall (a1:(map Z a)) (i:Z)
+  (j:Z), (j = (i + 1%Z)%Z) -> ((elements a1 i j) = (singleton (get a1 i))).
+
+Axiom Elements_union : forall (a:Type), forall (a1:(map Z a)) (i:Z) (j:Z)
+  (k:Z), ((i <= j)%Z /\ (j <= k)%Z) -> ((elements a1 i
+  k) = (union (elements a1 i j) (elements a1 j k))).
+
+(* YOU MAY EDIT THE CONTEXT BELOW *)
+
+(* DO NOT EDIT BELOW *)
+
+Theorem Elements_add1 : forall (a:Type), forall (a1:(map Z a)) (i:Z) (j:Z),
+  (i <  j)%Z -> ((elements a1 i j) = (add (get a1 i) (elements a1 (i + 1%Z)%Z
+  j))).
+(* YOU MAY EDIT THE PROOF BELOW *)
+intros X a i j Hij.
+unfold add.
+pattern (elements a i j); 
+  rewrite Elements_union with (j:= (i+1)%Z); 
+  auto with zarith.
+rewrite Elements_singleton; auto with zarith.
+Qed.
+(* DO NOT EDIT BELOW *)
+
+

examples/programs/vacid_0_binary_heaps/proofs/elements_Elements_Elements_set_inside_1.v

@@ -4,7 +4,7 @@ Require Import ZArith.
 Require Import Rbase.
 Parameter bag : forall (a:Type), Type.
 
-Parameter nb_occ: forall (a:Type), a -> (bag a)  -> Z.
+Parameter nb_occ: forall (a:Type), a -> (bag a) -> Z.
 
 Implicit Arguments nb_occ.
 
@@ -18,7 +18,7 @@ Implicit Arguments eq_bag.
 Axiom bag_extensionality : forall (a:Type), forall (a1:(bag a)) (b:(bag a)),
   (eq_bag a1 b) -> (a1 = b).
 
-Parameter empty_bag: forall (a:Type),  (bag a).
+Parameter empty_bag: forall (a:Type), (bag a).
 
 Set Contextual Implicit.
 Implicit Arguments empty_bag.
@@ -30,7 +30,7 @@ Axiom occ_empty : forall (a:Type), forall (x:a), ((nb_occ x (empty_bag:(bag
 Axiom is_empty : forall (a:Type), forall (b:(bag a)), (forall (x:a),
   ((nb_occ x b) = 0%Z)) -> (b = (empty_bag:(bag a))).
 
-Parameter singleton: forall (a:Type), a  -> (bag a).
+Parameter singleton: forall (a:Type), a -> (bag a).
 
 Implicit Arguments singleton.
 
@@ -44,7 +44,7 @@ Axiom occ_singleton_eq : forall (a:Type), forall (x:a) (y:a), (x = y) ->
 Axiom occ_singleton_neq : forall (a:Type), forall (x:a) (y:a), (~ (x = y)) ->
   ((nb_occ y (singleton x)) = 0%Z).
 
-Parameter union: forall (a:Type), (bag a) -> (bag a)  -> (bag a).
+Parameter union: forall (a:Type), (bag a) -> (bag a) -> (bag a).
 
 Implicit Arguments union.
 
@@ -75,7 +75,7 @@ Axiom occ_add_eq : forall (a:Type), forall (b:(bag a)) (x:a) (y:a),
 Axiom occ_add_neq : forall (a:Type), forall (b:(bag a)) (x:a) (y:a),
   (~ (x = y)) -> ((nb_occ y (add x b)) = (nb_occ y b)).
 
-Parameter card: forall (a:Type), (bag a)  -> Z.
+Parameter card: forall (a:Type), (bag a) -> Z.
 
 Implicit Arguments card.
 
@@ -112,7 +112,7 @@ Axiom Max_sym : forall (x:Z) (y:Z), (y <= x)%Z -> ((Zmax x y) = (Zmax y x)).
 
 Axiom Min_sym : forall (x:Z) (y:Z), (y <= x)%Z -> ((Zmin x y) = (Zmin y x)).
 
-Parameter diff: forall (a:Type), (bag a) -> (bag a)  -> (bag a).
+Parameter diff: forall (a:Type), (bag a) -> (bag a) -> (bag a).
 
 Implicit Arguments diff.
 
@@ -136,11 +136,11 @@ Axiom Add_diff : forall (a:Type), forall (b:(bag a)) (x:a), (0%Z <  (nb_occ x
 
 Parameter map : forall (a:Type) (b:Type), Type.
 
-Parameter get: forall (a:Type) (b:Type), (map a b) -> a  -> b.
+Parameter get: forall (a:Type) (b:Type), (map a b) -> a -> b.
 
 Implicit Arguments get.
 
-Parameter set: forall (a:Type) (b:Type), (map a b) -> a -> b  -> (map a b).
+Parameter set: forall (a:Type) (b:Type), (map a b) -> a -> b -> (map a b).
 
 Implicit Arguments set.
 
@@ -152,7 +152,7 @@ Axiom Select_neq : forall (a:Type) (b:Type), 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 (b:Type) (a:Type), b  -> (map a b).
+Parameter const: forall (b:Type) (a:Type), b -> (map a b).
 
 Set Contextual Implicit.
 Implicit Arguments const.
@@ -163,13 +163,17 @@ Axiom Const : forall (b:Type) (a:Type), forall (b1:b) (a1:a), ((get (const(
 
 Definition array (a:Type) := (map Z a).
 
-Parameter elements: forall (a:Type), (map Z a) -> Z -> Z  -> (bag a).
+Parameter elements: forall (a:Type), (map Z a) -> Z -> Z -> (bag a).
 
 Implicit Arguments elements.
 
 Axiom Elements_empty : forall (a:Type), forall (a1:(map Z a)) (i:Z) (j:Z),
   (j <= i)%Z -> ((elements a1 i j) = (empty_bag:(bag a))).
 
+Axiom Elements_add : forall (a:Type), forall (a1:(map Z a)) (i:Z) (j:Z),
+  (i <  j)%Z -> ((elements a1 i j) = (add (get a1 (j - 1%Z)%Z) (elements a1 i
+  (j - 1%Z)%Z))).
+
 Axiom Elements_singleton : forall (a:Type), forall (a1:(map Z a)) (i:Z)
   (j:Z), (j = (i + 1%Z)%Z) -> ((elements a1 i j) = (singleton (get a1 i))).
 
@@ -177,14 +181,10 @@ Axiom Elements_union : forall (a:Type), forall (a1:(map Z a)) (i:Z) (j:Z)
   (k:Z), ((i <= j)%Z /\ (j <= k)%Z) -> ((elements a1 i
   k) = (union (elements a1 i j) (elements a1 j k))).
 
-Axiom Elements_union1 : forall (a:Type), forall (a1:(map Z a)) (i:Z) (j:Z),
+Axiom Elements_add1 : forall (a:Type), forall (a1:(map Z a)) (i:Z) (j:Z),
   (i <  j)%Z -> ((elements a1 i j) = (add (get a1 i) (elements a1 (i + 1%Z)%Z
   j))).
 
-Axiom Elements_union2 : forall (a:Type), forall (a1:(map Z a)) (i:Z) (j:Z),
-  (i <  j)%Z -> ((elements a1 i j) = (add (get a1 (j - 1%Z)%Z) (elements a1 i
-  (j - 1%Z)%Z))).
-
 Axiom Elements_remove_last : forall (a:Type), forall (a1:(map Z a)) (i:Z)
   (j:Z), (i <  (j - 1%Z)%Z)%Z -> ((elements a1 i
   (j - 1%Z)%Z) = (diff (elements a1 i j) (singleton (get a1 (j - 1%Z)%Z)))).
@@ -214,11 +214,11 @@ unfold add; rewrite Union_identity.
 unfold add in H; rewrite <- H; clear H.
 rewrite Elements_union with (j:=j); auto with zarith.
 rewrite Elements_set_outside; auto with zarith.
-pattern (elements (set a j e) j n); rewrite Elements_union1; auto with zarith.
+pattern (elements (set a j e) j n); rewrite Elements_add1; auto with zarith.
 rewrite Select_eq; auto.
 rewrite Elements_set_outside; auto with zarith.
 rewrite Elements_union with (i:=i) (j:=j) (k:=n); auto with zarith.
-pattern (elements a j n); rewrite Elements_union1; auto with zarith.
+pattern (elements a j n); rewrite Elements_add1; auto with zarith.
 (* AC1 equality *)
 rewrite Union_identity.
 rewrite Union_assoc.

examples/programs/vacid_0_binary_heaps/proofs/elements_Elements_Elements_set_outside_1.v

@@ -2,131 +2,196 @@
 (* Beware! Only edit allowed sections below    *)
 Require Import ZArith.
 Require Import Rbase.
-Parameter map : forall (a:Type) (b:Type), Type.
+Parameter bag : forall (a:Type), Type.
 
-Parameter get: forall (a:Type) (b:Type), (map a b) -> a  -> b.
+Parameter nb_occ: forall (a:Type), a -> (bag a) -> Z.
 
-Implicit Arguments get.
+Implicit Arguments nb_occ.
 
-Parameter set: forall (a:Type) (b:Type), (map a b) -> a -> b  -> (map a b).
+Axiom occ_non_negative : forall (a:Type), forall (b:(bag a)) (x:a),
+  (0%Z <= (nb_occ x b))%Z.
 
-Implicit Arguments set.
-
-Axiom Select_eq : forall (a:Type) (b:Type), forall (m:(map a b)),
-  forall (a1:a) (a2:a), forall (b1:b), (a1 = a2) -> ((get (set m a1 b1)
-  a2) = b1).
+Definition eq_bag (a:Type)(a1:(bag a)) (b:(bag a)): Prop := forall (x:a),
+  ((nb_occ x a1) = (nb_occ x b)).
+Implicit Arguments eq_bag.
 
-Axiom Select_neq : forall (a:Type) (b:Type), forall (m:(map a b)),
-  forall (a1:a) (a2:a), forall (b1:b), (~ (a1 = a2)) -> ((get (set m a1 b1)
-  a2) = (get m a2)).
+Axiom bag_extensionality : forall (a:Type), forall (a1:(bag a)) (b:(bag a)),
+  (eq_bag a1 b) -> (a1 = b).
 
-Parameter const: forall (b:Type) (a:Type), b  -> (map a b).
+Parameter empty_bag: forall (a:Type), (bag a).
 
 Set Contextual Implicit.
-Implicit Arguments const.
+Implicit Arguments empty_bag.
 Unset Contextual Implicit.
 
-Axiom Const : forall (b:Type) (a:Type), forall (b1:b) (a1:a), ((get (const(
-  b1):(map a b)) a1) = b1).
-
-Definition bag (a:Type) := (map a Z).
-
-Axiom occ_non_negative : forall (a:Type), forall (b:(map a Z)) (x:a),
-  (0%Z <= (get b x))%Z.
-
-Definition eq_bag (a:Type)(a1:(map a Z)) (b:(map a Z)): Prop := forall (x:a),
-  ((get a1 x) = (get b x)).
-Implicit Arguments eq_bag.
-
-Axiom bag_extensionality : forall (a:Type), forall (a1:(map a Z)) (b:(map a
-  Z)), (eq_bag a1 b) -> (a1 = b).
+Axiom occ_empty : forall (a:Type), forall (x:a), ((nb_occ x (empty_bag:(bag
+  a))) = 0%Z).
 
-Parameter empty_bag: forall (a:Type),  (map a Z).
+Axiom is_empty : forall (a:Type), forall (b:(bag a)), (forall (x:a),
+  ((nb_occ x b) = 0%Z)) -> (b = (empty_bag:(bag a))).
 
-Set Contextual Implicit.
-Implicit Arguments empty_bag.
-Unset Contextual Implicit.
+Parameter singleton: forall (a:Type), a -> (bag a).
 
-Axiom occ_empty : forall (a:Type), forall (x:a), ((get (empty_bag:(map a Z))
-  x) = 0%Z).
+Implicit Arguments singleton.
 
-Axiom is_empty : forall (a:Type), forall (b:(map a Z)), (forall (x:a),
-  ((get b x) = 0%Z)) -> (b = (empty_bag:(map a Z))).
+Axiom occ_singleton : forall (a:Type), forall (x:a) (y:a), ((x = y) /\
+  ((nb_occ y (singleton x)) = 1%Z)) \/ ((~ (x = y)) /\ ((nb_occ y
+  (singleton x)) = 0%Z)).
 
 Axiom occ_singleton_eq : forall (a:Type), forall (x:a) (y:a), (x = y) ->
-  ((get (set (empty_bag:(map a Z)) x 1%Z) y) = 1%Z).
+  ((nb_occ y (singleton x)) = 1%Z).
 
 Axiom occ_singleton_neq : forall (a:Type), forall (x:a) (y:a), (~ (x = y)) ->
-  ((get (set (empty_bag:(map a Z)) x 1%Z) y) = 0%Z).
+  ((nb_occ y (singleton x)) = 0%Z).
 
-Parameter union: forall (a:Type), (map a Z) -> (map a Z)  -> (map a Z).
+Parameter union: forall (a:Type), (bag a) -> (bag a) -> (bag a).
 
 Implicit Arguments union.
 
-Axiom occ_union : forall (a:Type), forall (x:a) (a1:(map a Z)) (b:(map a Z)),
-  ((get (union a1 b) x) = ((get a1 x) + (get b x))%Z).
+Axiom occ_union : forall (a:Type), forall (x:a) (a1:(bag a)) (b:(bag a)),
+  ((nb_occ x (union a1 b)) = ((nb_occ x a1) + (nb_occ x b))%Z).
 
-Axiom Union_comm : forall (a:Type), forall (a1:(map a Z)) (b:(map a Z)),
+Axiom Union_comm : forall (a:Type), forall (a1:(bag a)) (b:(bag a)),
   ((union a1 b) = (union b a1)).
 
-Axiom Union_identity : forall (a:Type), forall (a1:(map a Z)), ((union a1
-  (empty_bag:(map a Z))) = a1).
+Axiom Union_identity : forall (a:Type), forall (a1:(bag a)), ((union a1
+  (empty_bag:(bag a))) = a1).
 
-Axiom Union_assoc : forall (a:Type), forall (a1:(map a Z)) (b:(map a Z))
-  (c:(map a Z)), ((union a1 (union b c)) = (union (union a1 b) c)).
+Axiom Union_assoc : forall (a:Type), forall (a1:(bag a)) (b:(bag a)) (c:(bag
+  a)), ((union a1 (union b c)) = (union (union a1 b) c)).
 
-Axiom bag_simpl : forall (a:Type), forall (a1:(map a Z)) (b:(map a Z))
-  (c:(map a Z)), ((union a1 b) = (union c b)) -> (a1 = c).
+Axiom bag_simpl : forall (a:Type), forall (a1:(bag a)) (b:(bag a)) (c:(bag
+  a)), ((union a1 b) = (union c b)) -> (a1 = c).
 
-Definition add (a:Type)(x:a) (b:(map a Z)): (map a Z) :=
-  (union (set (empty_bag:(map a Z)) x 1%Z) b).
+Axiom bag_simpl_left : forall (a:Type), forall (a1:(bag a)) (b:(bag a))
+  (c:(bag a)), ((union a1 b) = (union a1 c)) -> (b = c).
+
+Definition add (a:Type)(x:a) (b:(bag a)): (bag a) := (union (singleton x) b).
 Implicit Arguments add.
 
-Axiom occ_add_eq : forall (a:Type), forall (b:(map a Z)) (x:a) (y:a),
-  (x = y) -> ((get (add x b) x) = ((get b x) + 1%Z)%Z).
+Axiom occ_add_eq : forall (a:Type), forall (b:(bag a)) (x:a) (y:a),
+  (x = y) -> ((nb_occ x (add x b)) = ((nb_occ x b) + 1%Z)%Z).
 
-Axiom occ_add_neq : forall (a:Type), forall (b:(map a Z)) (x:a) (y:a),
-  (~ (x = y)) -> ((get (add x b) y) = (get b y)).
+Axiom occ_add_neq : forall (a:Type), forall (b:(bag a)) (x:a) (y:a),
+  (~ (x = y)) -> ((nb_occ y (add x b)) = (nb_occ y b)).
 
-Parameter card: forall (a:Type), (map a Z)  -> Z.
+Parameter card: forall (a:Type), (bag a) -> Z.
 
 Implicit Arguments card.
 
-Axiom Card_empty : forall (a:Type), ((card (empty_bag:(map a Z))) = 0%Z).
+Axiom Card_empty : forall (a:Type), ((card (empty_bag:(bag a))) = 0%Z).
 
 Axiom Card_singleton : forall (a:Type), forall (x:a),
-  ((card (set (empty_bag:(map a Z)) x 1%Z)) = 1%Z).
+  ((card (singleton x)) = 1%Z).
 
-Axiom Card_union : forall (a:Type), forall (x:(map a Z)) (y:(map a Z)),
+Axiom Card_union : forall (a:Type), forall (x:(bag a)) (y:(bag a)),
   ((card (union x y)) = ((card x) + (card y))%Z).
 
-Axiom Card_zero_empty : forall (a:Type), forall (x:(map a Z)),
-  ((card x) = 0%Z) -> (x = (empty_bag:(map a Z))).
+Axiom Card_zero_empty : forall (a:Type), forall (x:(bag a)),
+  ((card x) = 0%Z) -> (x = (empty_bag:(bag a))).
+
+Axiom Max_is_ge : forall (x:Z) (y:Z), (x <= (Zmax x y))%Z /\
+  (y <= (Zmax x y))%Z.
+
+Axiom Max_is_some : forall (x:Z) (y:Z), ((Zmax x y) = x) \/ ((Zmax x y) = y).
+
+Axiom Min_is_le : forall (x:Z) (y:Z), ((Zmin x y) <= x)%Z /\
+  ((Zmin x y) <= y)%Z.
+
+Axiom Min_is_some : forall (x:Z) (y:Z), ((Zmin x y) = x) \/ ((Zmin x y) = y).
+
+Axiom Max_x : forall (x:Z) (y:Z), (y <= x)%Z -> ((Zmax x y) = x).
+
+Axiom Max_y : forall (x:Z) (y:Z), (x <= y)%Z -> ((Zmax x y) = y).
+
+Axiom Min_x : forall (x:Z) (y:Z), (x <= y)%Z -> ((Zmin x y) = x).
+
+Axiom Min_y : forall (x:Z) (y:Z), (y <= x)%Z -> ((Zmin x y) = y).
+