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(************************************************************************)
(* v * The Coq Proof Assistant / The Coq Development Team *)
(* <O___,, * INRIA - CNRS - LIX - LRI - PPS - Copyright 1999-2011 *)
(* \VV/ **************************************************************)
(* // * This file is distributed under the terms of the *)
(* * GNU Lesser General Public License Version 2.1 *)
(************************************************************************)
(* *)
(* Micromega: A reflexive tactic using the Positivstellensatz *)
(* *)
(* Frédéric Besson (Irisa/Inria) 2006-2008 *)
(* *)
(************************************************************************)
Require Import ZArith.
Require Import Coq.Arith.Max.
Require Import List.
Set Implicit Arguments.
(* I have addded a Leaf constructor to the varmap data structure (/plugins/ring/Quote.v)
-- this is harmless and spares a lot of Empty.
This means smaller proof-terms.
BTW, by dropping the polymorphism, I get small (yet noticeable) speed-up.
*)
Section S.
Variable D :Type.
Definition Env := positive -> D.
Definition jump (j:positive) (e:Env) := fun x => e (Pplus x j).
Definition nth (n:positive) (e : Env ) := e n.
Definition hd (x:D) (e: Env) := nth xH e.
Definition tail (e: Env) := jump xH e.
Lemma psucc : forall p, (match p with
| xI y' => xO (Psucc y')
| xO y' => xI y'
| 1%positive => 2%positive
end) = (p+1)%positive.
Proof.
destruct p.
auto with zarith.
rewrite xI_succ_xO.
auto with zarith.
reflexivity.
Qed.
Lemma jump_Pplus : forall i j l,
forall x, jump (i + j) l x = jump i (jump j l) x.
Proof.
unfold jump.
intros.
rewrite Pplus_assoc.
reflexivity.
Qed.
Lemma jump_simpl : forall p l,
forall x, jump p l x =
match p with
| xH => tail l x
| xO p => jump p (jump p l) x
| xI p => jump p (jump p (tail l)) x
end.
Proof.
destruct p ; unfold tail ; intros ; repeat rewrite <- jump_Pplus.
(* xI p = p + p + 1 *)
rewrite xI_succ_xO.
rewrite Pplus_diag.
rewrite <- Pplus_one_succ_r.
reflexivity.
(* xO p = p + p *)
rewrite Pplus_diag.
reflexivity.
reflexivity.
Qed.
Ltac jump_s :=
repeat
match goal with
| |- context [jump xH ?e] => rewrite (jump_simpl xH)
| |- context [jump (xO ?p) ?e] => rewrite (jump_simpl (xO p))
| |- context [jump (xI ?p) ?e] => rewrite (jump_simpl (xI p))
end.
Lemma jump_tl : forall j l, forall x, tail (jump j l) x = jump j (tail l) x.
Proof.
unfold tail.
intros.
repeat rewrite <- jump_Pplus.
rewrite Pplus_comm.
reflexivity.
Qed.
Lemma jump_Psucc : forall j l,
forall x, (jump (Psucc j) l x) = (jump 1 (jump j l) x).
Proof.
intros.
rewrite <- jump_Pplus.
rewrite Pplus_one_succ_r.
rewrite Pplus_comm.
reflexivity.
Qed.
Lemma jump_Pdouble_minus_one : forall i l,
forall x, (jump (Pdouble_minus_one i) (tail l)) x = (jump i (jump i l)) x.
Proof.
unfold tail.
intros.
repeat rewrite <- jump_Pplus.
rewrite <- Pplus_one_succ_r.
rewrite Psucc_o_double_minus_one_eq_xO.
rewrite Pplus_diag.
reflexivity.
Qed.
Lemma jump_x0_tail : forall p l, forall x, jump (xO p) (tail l) x = jump (xI p) l x.
Proof.
intros.
unfold jump.
unfold tail.
unfold jump.
rewrite <- Pplus_assoc.
simpl.
reflexivity.
Qed.
Lemma nth_spec : forall p l x,
nth p l =
match p with
| xH => hd x l
| xO p => nth p (jump p l)
| xI p => nth p (jump p (tail l))
end.
Proof.
unfold nth.
destruct p.
intros.
unfold jump, tail.
unfold jump.
rewrite Pplus_diag.
rewrite xI_succ_xO.
simpl.
reflexivity.
unfold jump.
rewrite Pplus_diag.
reflexivity.
unfold hd.
unfold nth.
reflexivity.
Qed.
Lemma nth_jump : forall p l x, nth p (tail l) = hd x (jump p l).
Proof.
unfold tail.
unfold hd.
unfold jump.
unfold nth.
intros.
rewrite Pplus_comm.
reflexivity.
Qed.
Lemma nth_Pdouble_minus_one :
forall p l, nth (Pdouble_minus_one p) (tail l) = nth p (jump p l).
Proof.
intros.
unfold tail.
unfold nth, jump.
rewrite Pplus_diag.
rewrite <- Psucc_o_double_minus_one_eq_xO.
rewrite Pplus_one_succ_r.
reflexivity.
Qed.
End S.
|
