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123 lines
3.2 KiB
Coq
123 lines
3.2 KiB
Coq
(* -*- coding:utf-8 -*- *)
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Require Import List Ascii.
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Require Import Pow MultiByte ListUtil.
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Open Scope char_scope.
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(** MsgPackで使うオブジェクトの定義 *)
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Inductive object :=
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| Bool (_ : bool)
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| Nil
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| PFixnum (_ : ascii8)
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| NFixnum (_ : ascii8)
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| Uint8 (_ : ascii8)
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| Uint16 (_ : ascii16)
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| Uint32 (_ : ascii32)
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| Uint64 (_ : ascii64)
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| Int8 (_ : ascii8)
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| Int16 (_ : ascii16)
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| Int32 (_ : ascii32)
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| Int64 (_ : ascii64)
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| Float (_ : ascii32)
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| Double (_ : ascii64)
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| FixRaw (_ : list ascii8)
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| Raw16 (_ : list ascii8)
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| Raw32 (_ : list ascii8)
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| FixArray ( _ : list object)
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| Array16 ( _ : list object)
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| Array32 ( _ : list object)
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| FixMap ( _ : list (object * object)%type)
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| Map16 ( _ : list (object * object)%type)
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| Map32 ( _ : list (object * object)%type).
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(** 妥当なオブジェクトの定義 *)
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Inductive Valid : object -> Prop :=
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| VBool : forall b,
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Valid (Bool b)
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| VPFixNum : forall n,
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nat_of_ascii8 n < 128 -> Valid (PFixnum n)
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| VNFixNum : forall n,
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(* 負の数を導入したくないので、補数表現を使う *)
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223 < nat_of_ascii8 n /\ nat_of_ascii8 n < 256 -> Valid (NFixnum n)
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| VUint8 : forall c, Valid (Uint8 c)
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| VUint16 : forall c, Valid (Uint16 c)
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| VUint32 : forall c, Valid (Uint32 c)
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| VUint64 : forall c, Valid (Uint64 c)
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| VInt8 : forall c, Valid (Int8 c)
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| VInt16 : forall c, Valid (Int16 c)
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| VInt32 : forall c, Valid (Int32 c)
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| VInt64 : forall c, Valid (Int64 c)
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| VFloat : forall c, Valid (Float c)
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| VDouble : forall c, Valid (Double c)
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| VFixRaw : forall xs,
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length xs < pow 5 -> Valid (FixRaw xs)
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| VRaw16 : forall xs,
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length xs < pow 16 -> Valid (Raw16 xs)
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| VRaw32 : forall xs,
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length xs < pow 32 -> Valid (Raw32 xs)
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| VFixArrayNil :
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Valid (FixArray [])
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| VFixArrayCons : forall x xs,
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Valid x ->
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Valid (FixArray xs) ->
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length (x::xs) < pow 4 ->
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Valid (FixArray (x::xs))
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| VArray16Nil :
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Valid (Array16 [])
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| VArray16Cons: forall x xs,
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Valid x ->
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Valid (Array16 xs) ->
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length (x::xs) < pow 16 ->
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Valid (Array16 (x::xs))
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| VArray32Nil :
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Valid (Array32 [])
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| VArray32Cons : forall x xs,
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Valid x ->
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Valid (Array32 xs) ->
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length (x::xs) < pow 32 ->
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Valid (Array32 (x::xs))
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| VFixMapNil:
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Valid (FixMap [])
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| VFixMapCons : forall k v xs,
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Valid k ->
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Valid v ->
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Valid (FixMap xs) ->
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length ((k,v)::xs) < pow 4 ->
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Valid (FixMap ((k,v)::xs))
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| VMap16Nil :
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Valid (Map16 [])
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| VMap16Cons : forall k v xs,
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Valid k ->
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Valid v ->
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Valid (Map16 xs) ->
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length ((k,v)::xs) < pow 16 ->
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Valid (Map16 ((k,v)::xs))
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| VMap32Nil :
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Valid (Map32 [])
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| VMap32Cons : forall k v xs,
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Valid k ->
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Valid v ->
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Valid (Map32 xs) ->
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length ((k,v)::xs) < pow 32 ->
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Valid (Map32 ((k,v)::xs)).
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Lemma varray16_inv1: forall x xs,
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Valid (Array16 (x::xs)) ->
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("000", "000") <> ascii16_of_nat (length (x :: xs)).
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Proof.
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intros.
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apply ascii16_not_O.
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split; [ apply length_lt_O | inversion H; auto ].
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Qed.
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Lemma varray16_inv2 : forall A (x y : A) xs ys,
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pow 16 > length (x :: xs) ->
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pow 16 > length (y :: ys) ->
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ascii16_of_nat (length (x :: xs)) = ascii16_of_nat (length (y :: ys)) ->
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ascii16_of_nat (length xs) = ascii16_of_nat (length ys).
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Proof.
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intros.
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apply ascii16_of_nat_eq in H1; auto.
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Qed.
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