heap_Heap_Is_heap_relation_1.v 4.64 KB
 MARCHE Claude committed Sep 20, 2011 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 124 125 126 127 128 129 130 131 132 133 134 135 ``````(* This file is generated by Why3's Coq driver *) (* Beware! Only edit allowed sections below *) Require Import ZArith. Require Import Rbase. Require Import ZOdiv. Axiom Abs_le : forall (x:Z) (y:Z), ((Zabs x) <= y)%Z <-> (((-y)%Z <= x)%Z /\ (x <= y)%Z). Definition left(i:Z): Z := ((2%Z * i)%Z + 1%Z)%Z. Definition right(i:Z): Z := ((2%Z * i)%Z + 2%Z)%Z. Definition parent(i:Z): Z := (ZOdiv (i - 1%Z)%Z 2%Z). Axiom Parent_inf : forall (i:Z), (0%Z < i)%Z -> ((parent i) < i)%Z. Axiom Left_sup : forall (i:Z), (0%Z <= i)%Z -> (i < (left i))%Z. Axiom Right_sup : forall (i:Z), (0%Z <= i)%Z -> (i < (right i))%Z. Axiom Parent_right : forall (i:Z), (0%Z <= i)%Z -> ((parent (right i)) = i). Axiom Parent_left : forall (i:Z), (0%Z <= i)%Z -> ((parent (left i)) = i). Axiom Inf_parent : forall (i:Z) (j:Z), ((0%Z < j)%Z /\ (j <= (right i))%Z) -> ((parent j) <= i)%Z. Axiom Child_parent : forall (i:Z), (0%Z < i)%Z -> ((i = (left (parent i))) \/ (i = (right (parent i)))). Axiom Parent_pos : forall (j:Z), (0%Z < j)%Z -> (0%Z <= (parent j))%Z. Definition parentChild(i:Z) (j:Z): Prop := ((0%Z <= i)%Z /\ (i < j)%Z) -> ((j = (left i)) \/ (j = (right i))). 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 map1 := (map Z Z). Definition logic_heap := ((map Z Z)* Z)%type. Definition is_heap_array(a:(map Z Z)) (idx:Z) (sz:Z): Prop := (0%Z <= idx)%Z -> forall (i:Z) (j:Z), (((idx <= i)%Z /\ (i < j)%Z) /\ (j < sz)%Z) -> ((parentChild i j) -> ((get a i) <= (get a j))%Z). Definition is_heap(h:((map Z Z)* Z)%type): Prop := match h with | (a, sz) => (0%Z <= sz)%Z /\ (is_heap_array a 0%Z sz) end. Axiom Is_heap_when_no_element : forall (a:(map Z Z)) (idx:Z) (n:Z), ((0%Z <= n)%Z /\ (n <= idx)%Z) -> (is_heap_array a idx n). Axiom Is_heap_sub : forall (a:(map Z Z)) (i:Z) (n:Z), (is_heap_array a i n) -> forall (j:Z), ((i <= j)%Z /\ (j <= n)%Z) -> (is_heap_array a i j). Axiom Is_heap_sub2 : forall (a:(map Z Z)) (n:Z), (is_heap_array a 0%Z n) -> forall (j:Z), ((0%Z <= j)%Z /\ (j <= n)%Z) -> (is_heap_array a j n). Axiom Is_heap_when_node_modified : forall (a:(map Z Z)) (n:Z) (e:Z) (idx:Z) (i:Z), ((0%Z <= i)%Z /\ (i < n)%Z) -> ((is_heap_array a idx n) -> (((0%Z < i)%Z -> ((get a (parent i)) <= e)%Z) -> ((((left i) < n)%Z -> (e <= (get a (left i)))%Z) -> ((((right i) < n)%Z -> (e <= (get a (right i)))%Z) -> (is_heap_array (set a i e) idx n))))). Axiom Is_heap_add_last : forall (a:(map Z Z)) (n:Z) (e:Z), (0%Z < n)%Z -> (((is_heap_array a 0%Z n) /\ ((get a (parent n)) <= e)%Z) -> (is_heap_array (set a n e) 0%Z (n + 1%Z)%Z)). Axiom Parent_inf_el : forall (a:(map Z Z)) (n:Z), (is_heap_array a 0%Z n) -> forall (j:Z), ((0%Z < j)%Z /\ (j < n)%Z) -> ((get a (parent j)) <= (get a j))%Z. Axiom Left_sup_el : forall (a:(map Z Z)) (n:Z), (is_heap_array a 0%Z n) -> forall (j:Z), ((0%Z <= j)%Z /\ (j < n)%Z) -> (((left j) < n)%Z -> ((get a j) <= (get a (left j)))%Z). Axiom Right_sup_el : forall (a:(map Z Z)) (n:Z), (is_heap_array a 0%Z n) -> forall (j:Z), ((0%Z <= j)%Z /\ (j < n)%Z) -> (((right j) < n)%Z -> ((get a j) <= (get a (right j)))%Z). (* YOU MAY EDIT THE CONTEXT BELOW *) (* DO NOT EDIT BELOW *) Theorem Is_heap_relation : forall (a:(map Z Z)) (n:Z), (0%Z < n)%Z -> ((is_heap_array a 0%Z n) -> forall (j:Z), (0%Z <= j)%Z -> ((j < n)%Z -> ((get a 0%Z) <= (get a j))%Z)). (* YOU MAY EDIT THE PROOF BELOW *) intros a n H H_heap. intros j H1. apply Zlt_lower_bound_ind with (z:=0) (P:= fun j => j < n -> get a 0 <= get a j); auto with zarith. intros x Hind H_x_pos H_x_lt_n. assert (h: x=0 \/ 0 < x) by omega. destruct h. (*case x = 0*) subst; auto with zarith. (*case x > 0*) apply Zle_trans with (get a (parent x)). apply Hind. split. apply Parent_pos; auto with zarith. apply Parent_inf; auto with zarith. apply Zlt_trans with x; auto. apply Parent_inf; auto with zarith. apply Parent_inf_el with (n := n); auto with zarith. Qed. `````` Asma Tafat-Bouzid committed Oct 11, 2011 136 ``(* DO NOT EDIT BELOW *)``