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(* 02110-1301 USA                                                     *)
(* Contribution to the Coq Library   V6.3 (July 1999)                    *)
(****************************************************************)
(*           The Calculus of Inductive Constructions            *)
(*                       COQ v5.10                              *)
(*                                                              *)
(* Laurent Arditi.  Laboratoire I3S. CNRS ura 1376.             *)
(* Universite de Nice - Sophia Antipolis                        *)
(* arditi@unice.fr, http://wwwi3s.unice.fr/~arditi/lolo.html    *)
(*                                                              *)
(* date: november 1995                                          *)
(* file: Fill_spec.v                                            *)
(* contents: proof of a memory block instruction: Fills cx words*)
(* in memory starting at address di with value al.              *)
(* Implementation level                                         *)
(****************************************************************)
Require Export Fill_defs.
(* Implementation:  cx: register a_size di: register a_size al: register d_size mem: memory of d_size bit words, addresses of a_size bits ad: register a_size da: register d_size  ad <- di da <- al while not(cx=0) do begin mem[ad] <- da ad <- ad+1 cx <- cx-1 end di <- ad *)
(****************************************************************)
(* Implementation *)
(* seq1 *)
Definition di1 (di0 cx0 al0 : BV) (mem0 : Memo) (ad0 da0 : BV) := di0.
Definition cx1 (di0 cx0 al0 : BV) (mem0 : Memo) (ad0 da0 : BV) := cx0.
Definition al1 (di0 cx0 al0 : BV) (mem0 : Memo) (ad0 da0 : BV) := al0.
Definition mem1 (di0 cx0 al0 : BV) (mem0 : Memo) (ad0 da0 : BV) := mem0.
Definition ad1 (di0 cx0 al0 : BV) (mem0 : Memo) (ad0 da0 : BV) := di0.
Definition da1 (di0 cx0 al0 : BV) (mem0 : Memo) (ad0 da0 : BV) := al0.
(* seq2 *)
Fixpoint di2 (st : nat) : BV -> BV -> BV -> Memo -> BV -> BV -> BV :=
fun (di0 cx0 al0 : BV) (mem0 : Memo) (ad0 da0 : BV) =>
match st return BV with
| O => di0
| S t =>
match IsNull (cx2 t di0 cx0 al0 mem0 ad0 da0) return BV with
| true => di2 t di0 cx0 al0 mem0 ad0 da0
| false => di2 t di0 cx0 al0 mem0 ad0 da0
end
end
with cx2 (st : nat) : BV -> BV -> BV -> Memo -> BV -> BV -> BV :=
fun (di0 cx0 al0 : BV) (mem0 : Memo) (ad0 da0 : BV) =>
match st return BV with
| O => cx0
| S t =>
match IsNull (cx2 t di0 cx0 al0 mem0 ad0 da0) return BV with
| true => cx2 t di0 cx0 al0 mem0 ad0 da0
| false => BV_decrement (cx2 t di0 cx0 al0 mem0 ad0 da0)
end
end
with al2 (st : nat) : BV -> BV -> BV -> Memo -> BV -> BV -> BV :=
fun (di0 cx0 al0 : BV) (mem0 : Memo) (ad0 da0 : BV) =>
match st return BV with
| O => al0
| S t =>
match IsNull (cx2 t di0 cx0 al0 mem0 ad0 da0) return BV with
| true => al2 t di0 cx0 al0 mem0 ad0 da0
| false => al2 t di0 cx0 al0 mem0 ad0 da0
end
end
with mem2 (st : nat) : BV -> BV -> BV -> Memo -> BV -> BV -> Memo :=
fun (di0 cx0 al0 : BV) (mem0 : Memo) (ad0 da0 : BV) =>
match st return Memo with
| O => mem0
| S t =>
match IsNull (cx2 t di0 cx0 al0 mem0 ad0 da0) return Memo with
| true => mem2 t di0 cx0 al0 mem0 ad0 da0
| false =>
MemoWrite (mem2 t di0 cx0 al0 mem0 ad0 da0)
(BV_to_nat (ad2 t di0 cx0 al0 mem0 ad0 da0))
(da2 t di0 cx0 al0 mem0 ad0 da0)
end
end
with ad2 (st : nat) : BV -> BV -> BV -> Memo -> BV -> BV -> BV :=
fun (di0 cx0 al0 : BV) (mem0 : Memo) (ad0 da0 : BV) =>
match st return BV with
| O => ad0
| S t =>
match IsNull (cx2 t di0 cx0 al0 mem0 ad0 da0) return BV with
| true => ad2 t di0 cx0 al0 mem0 ad0 da0
| false => BV_increment (ad2 t di0 cx0 al0 mem0 ad0 da0)
end
end
with da2 (st : nat) : BV -> BV -> BV -> Memo -> BV -> BV -> BV :=
fun (di0 cx0 al0 : BV) (mem0 : Memo) (ad0 da0 : BV) =>
match st return BV with
| O => da0
| S t =>
match IsNull (cx2 t di0 cx0 al0 mem0 ad0 da0) return BV with
| true => da2 t di0 cx0 al0 mem0 ad0 da0
| false => da2 t di0 cx0 al0 mem0 ad0 da0
end
end.
auto.
Qed.
simple induction t. simpl in |- *; trivial.
intros. rewrite di2_t. elim (IsNull (cx2 n di0 cx0 al0 mem0 ad0 da0)). apply H.
apply H.
Qed.

Lemma cx2_t :
forall (t : nat) (di0 cx0 al0 : BV) (mem0 : Memo) (ad0 da0 : BV),
cx2 (S t) di0 cx0 al0 mem0 ad0 da0 =
match IsNull (cx2 t di0 cx0 al0 mem0 ad0 da0) return BV with
| true => cx2 t di0 cx0 al0 mem0 ad0 da0
| false => BV_decrement (cx2 t di0 cx0 al0 mem0 ad0 da0)
end.