Added the directory theories/Numbers where axiomatizations and implementations (unary, binary, etc.) of different number classes (natural, integer, rational, real, complex, etc.) will be stored.Currently there are axiomatized natural numbers with two implementations and axiomatized integers. Modified Makefile accordingly but dod not include the new files in THEORIESVO yet.

git-svn-id: svn+ssh://scm.gforge.inria.fr/svn/coq/trunk@9916 85f007b7-540e-0410-9357-904b9bb8a0f7

1 files changed, 221 insertions, 0 deletions

diff --git a/theories/Numbers/Natural/Peano/NPeano.v b/theories/Numbers/Natural/Peano/NPeano.v
new file mode 100644
index 000000000..403521e7c
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+++ b/theories/Numbers/Natural/Peano/NPeano.v
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+Require Import NDepRec.
+Require Import NPlus.
+Require Import NTimes.
+Require Import NLt.
+Require Import NPlusLt.
+Require Import NTimesLt.
+Require Import NMiscFunct.
+
+Module PeanoDomain <: DomainEqSignature.
+
+Definition N := nat.
+Definition E := (@eq nat).
+Definition e := eq_nat_bool.
+
+Theorem E_equiv_e : forall x y : N, E x y <-> e x y.
+Proof.
+unfold E, e; intros x y; split; intro H;
+[rewrite H; apply eq_nat_bool_refl |
+now apply eq_nat_bool_implies_eq].
+Qed.
+
+Definition E_equiv : equiv N E := eq_equiv N.
+
+Add Relation N E
+ reflexivity proved by (proj1 E_equiv)
+ symmetry proved by (proj2 (proj2 E_equiv))
+ transitivity proved by (proj1 (proj2 E_equiv))
+as E_rel.
+
+End PeanoDomain.
+
+Module PeanoNat <: NatSignature.
+
+Module Export DomainModule := PeanoDomain.
+
+Definition O := 0.
+Definition S := S.
+
+Add Morphism S with signature E ==> E as S_wd.
+Proof.
+congruence.
+Qed.
+
+Theorem induction :
+  forall P : nat -> Prop, pred_wd (@eq nat) P ->
+    P 0 -> (forall n, P n -> P (S n)) -> forall n, P n.
+Proof.
+intros P W Base Step n; elim n; assumption.
+Qed.
+
+Definition recursion := fun A : Set => nat_rec (fun _ => A).
+Implicit Arguments recursion [A].
+
+Theorem recursion_wd :
+forall (A : Set) (EA : relation A),
+  forall a a' : A, EA a a' ->
+    forall f f' : N -> A -> A, eq_fun2 E EA EA f f' ->
+      forall x x' : N, x = x' ->
+        EA (recursion a f x) (recursion a' f' x').
+Proof.
+unfold fun2_wd, E.
+intros A EA a a' Eaa' f f' Eff'.
+induction x as [| n IH]; intros x' H; rewrite <- H; simpl.
+assumption.
+apply Eff'; [reflexivity | now apply IH].
+Qed.
+
+Theorem recursion_0 :
+  forall (A : Set) (a : A) (f : N -> A -> A), recursion a f O = a.
+Proof.
+reflexivity.
+Qed.
+
+Theorem recursion_S :
+forall (A : Set) (EA : relation A) (a : A) (f : N -> A -> A),
+  EA a a -> fun2_wd E EA EA f ->
+    forall n : N, EA (recursion a f (S n)) (f n (recursion a f n)).
+Proof.
+intros A EA a f EAaa f_wd. unfold fun2_wd, E in *.
+induction n; simpl; now apply f_wd.
+Qed.
+
+End PeanoNat.
+
+Module PeanoDepRec <: DepRecSignature.
+
+Module Export DomainModule := PeanoDomain.
+Module Export NatModule <: NatSignature := PeanoNat.
+
+Definition dep_recursion := nat_rec.
+
+Theorem dep_recursion_0 :
+  forall (A : N -> Set) (a : A 0) (f : forall n, A n -> A (S n)),
+    dep_recursion A a f 0 = a.
+Proof.
+reflexivity.
+Qed.
+
+Theorem dep_recursion_S :
+  forall (A : N -> Set) (a : A 0) (f : forall n, A n -> A (S n)) (n : N),
+    dep_recursion A a f (S n) = f n (dep_recursion A a f n).
+Proof.
+reflexivity.
+Qed.
+
+End PeanoDepRec.
+
+Module PeanoPlus <: PlusSignature.
+
+Module Export NatModule := PeanoNat.
+
+Definition plus := plus.
+
+Add Morphism plus with signature E ==> E ==> E as plus_wd.
+Proof.
+unfold E; congruence.
+Qed.
+
+Theorem plus_0_n : forall n, 0 + n = n.
+Proof.
+reflexivity.
+Qed.
+
+Theorem plus_Sn_m : forall n m, (S n) + m = S (n + m).
+Proof.
+reflexivity.
+Qed.
+
+End PeanoPlus.
+
+Module PeanoTimes <: TimesSignature.
+Module Export PlusModule := PeanoPlus.
+
+Definition times := mult.
+
+Add Morphism times with signature E ==> E ==> E as times_wd.
+Proof.
+unfold E; congruence.
+Qed.
+
+Theorem times_0_n : forall n, 0 * n = 0.
+Proof.
+auto.
+Qed.
+
+Theorem times_Sn_m : forall n m, (S n) * m = m + n * m.
+Proof.
+auto.
+Qed.
+
+End PeanoTimes.
+
+(* Some checks:
+Check times_eq_1 : forall n m, n * m = 1 -> n = 1 /\ m = 1.
+Eval compute in times_eq_0_dec 0 5.
+Eval compute in times_eq_0_dec 5 0. *)
+
+Module PeanoLt <: LtSignature.
+Module Export NatModule := PeanoNat.
+
+Definition lt := lt_bool.
+
+Add Morphism lt with signature E ==> E ==> eq_bool as lt_wd.
+Proof.
+unfold E, eq_bool; congruence.
+Qed.
+
+Theorem lt_0 : forall x, ~ (lt x 0).
+Proof.
+exact lt_bool_0.
+Qed.
+
+Theorem lt_S : forall x y, lt x (S y) <-> lt x y \/ x = y.
+Proof.
+exact lt_bool_S.
+Qed.
+
+End PeanoLt.
+
+(* Obtaining properties for +, *, <, and their combinations *)
+
+Module Export PeanoPlusProperties := PlusProperties PeanoPlus.
+Module Export PeanoTimesProperties := TimesProperties PeanoTimes.
+Module Export PeanoLtProperties := LtProperties PeanoLt.
+Module Export PeanoPlusLtProperties := PlusLtProperties PeanoPlus PeanoLt.
+Module Export PeanoTimesLtProperties := TimesLtProperties PeanoTimes PeanoLt.
+Module Export PeanoDepRecTimesProperties :=
+  DepRecTimesProperties PeanoDepRec PeanoTimes.
+
+Module MiscFunctModule := MiscFunctFunctor PeanoNat.
+
+(*Eval compute in MiscFunctModule.lt 6 5.*)
+
+(*Set Printing All.*)
+(*Check plus_comm.
+Goal forall x y : nat, x + y = y + x.
+intros x y.
+rewrite plus_comm. reflexivity. (* does now work -- but the next line does *)
+apply plus_comm.*)
+
+(*Opaque plus.
+Eval compute in (forall n m : N, E m (PeanoPlus.Nat.S (PeanoPlus.plus n m)) -> False).
+
+Eval compute in (plus_eq_1_dec 1 1).
+Opaque plus_eq_1_dec.
+Check plus_eq_1.
+Eval compute in (forall m n : N,
+       E (PeanoPlus.plus m n) (PeanoPlus.Nat.S PeanoPlus.Nat.O) ->
+       (plus_eq_1_dec m n = true ->
+        E m PeanoPlus.Nat.O /\ E n (PeanoPlus.Nat.S PeanoPlus.Nat.O)) /\
+       (plus_eq_1_dec m n = false ->
+        E m (PeanoPlus.Nat.S PeanoPlus.Nat.O) /\ E n PeanoPlus.Nat.O)).*)
+
+(*Require Import rec_ex.
+
+Module Import PeanoRecursionExamples := RecursionExamples PeanoNat.
+
+Eval compute in mult 3 15.
+Eval compute in e 100 100.
+Eval compute in log 8.
+Eval compute in half 0.*)