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(* -*- coding: utf-8 -*- *)
(************************************************************************)
(* v * The Coq Proof Assistant / The Coq Development Team *)
(* <O___,, * CNRS-Ecole Polytechnique-INRIA Futurs-Universite Paris Sud *)
(* \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 MakeVarMap.
Variable A : Type.
Variable default : A.
Inductive t : Type :=
| Empty : t
| Leaf : A -> t
| Node : t -> A -> t -> t .
Fixpoint find (vm : t ) (p:positive) {struct vm} : A :=
match vm with
| Empty => default
| Leaf i => i
| Node l e r => match p with
| xH => e
| xO p => find l p
| xI p => find r p
end
end.
(* an off_map (a map with offset) offers the same functionalites as /plugins/setoid_ring/BinList.v - it is used in EnvRing.v *)
(*
Definition off_map := (option positive *t )%type.
Definition jump (j:positive) (l:off_map ) :=
let (o,m) := l in
match o with
| None => (Some j,m)
| Some j0 => (Some (j+j0)%positive,m)
end.
Definition nth (n:positive) (l: off_map ) :=
let (o,m) := l in
let idx := match o with
| None => n
| Some i => i + n
end%positive in
find idx m.
Definition hd (l:off_map) := nth xH l.
Definition tail (l:off_map ) := jump xH l.
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,
(jump (i + j) l) = (jump i (jump j l)).
Proof.
unfold jump.
destruct l.
destruct o.
rewrite Pplus_assoc.
reflexivity.
reflexivity.
Qed.
Lemma jump_simpl : forall p l,
jump p l =
match p with
| xH => tail l
| xO p => jump p (jump p l)
| xI p => jump p (jump p (tail l))
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, tail (jump j l) = jump j (tail l).
Proof.
unfold tail.
intros.
repeat rewrite <- jump_Pplus.
rewrite Pplus_comm.
reflexivity.
Qed.
Lemma jump_Psucc : forall j l,
(jump (Psucc j) l) = (jump 1 (jump j l)).
Proof.
intros.
rewrite <- jump_Pplus.
rewrite Pplus_one_succ_r.
rewrite Pplus_comm.
reflexivity.
Qed.
Lemma jump_Pdouble_minus_one : forall i l,
(jump (Pdouble_minus_one i) (tail l)) = (jump i (jump i l)).
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, jump (xO p) (tail l) = jump (xI p) l.
Proof.
intros.
jump_s.
repeat rewrite <- jump_Pplus.
reflexivity.
Qed.
Lemma nth_spec : forall p l,
nth p l =
match p with
| xH => hd l
| xO p => nth p (jump p l)
| xI p => nth p (jump p (tail l))
end.
Proof.
unfold nth.
destruct l.
destruct o.
simpl.
rewrite psucc.
destruct p.
replace (p0 + xI p)%positive with ((p + (p0 + 1) + p))%positive.
reflexivity.
rewrite xI_succ_xO.
rewrite Pplus_one_succ_r.
rewrite <- Pplus_diag.
rewrite Pplus_comm.
symmetry.
rewrite (Pplus_comm p0).
rewrite <- Pplus_assoc.
rewrite (Pplus_comm 1)%positive.
rewrite <- Pplus_assoc.
reflexivity.
(**)
replace ((p0 + xO p))%positive with (p + p0 + p)%positive.
reflexivity.
rewrite <- Pplus_diag.
rewrite <- Pplus_assoc.
rewrite Pplus_comm.
rewrite Pplus_assoc.
reflexivity.
reflexivity.
simpl.
destruct p.
rewrite xI_succ_xO.
rewrite Pplus_one_succ_r.
rewrite <- Pplus_diag.
symmetry.
rewrite Pplus_comm.
rewrite Pplus_assoc.
reflexivity.
rewrite Pplus_diag.
reflexivity.
reflexivity.
Qed.
Lemma nth_jump : forall p l, nth p (tail l) = hd (jump p l).
Proof.
destruct l.
unfold tail.
unfold hd.
unfold jump.
unfold nth.
destruct o.
symmetry.
rewrite Pplus_comm.
rewrite <- Pplus_assoc.
rewrite (Pplus_comm p0).
reflexivity.
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.
destruct l.
unfold tail.
unfold nth, jump.
destruct o.
rewrite ((Pplus_comm p)).
rewrite <- (Pplus_assoc p0).
rewrite Pplus_diag.
rewrite <- Psucc_o_double_minus_one_eq_xO.
rewrite Pplus_one_succ_r.
rewrite (Pplus_comm (Pdouble_minus_one p)).
rewrite Pplus_assoc.
rewrite (Pplus_comm p0).
reflexivity.
rewrite <- Pplus_one_succ_l.
rewrite Psucc_o_double_minus_one_eq_xO.
rewrite Pplus_diag.
reflexivity.
Qed.
*)
End MakeVarMap.
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