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|
(************************************************************************)
(* v * The Coq Proof Assistant / The Coq Development Team *)
(* <O___,, * INRIA - CNRS - LIX - LRI - PPS - Copyright 1999-2010 *)
(* \VV/ **************************************************************)
(* // * This file is distributed under the terms of the *)
(* * GNU Lesser General Public License Version 2.1 *)
(************************************************************************)
Require Import ZArith Nnat ZAxioms ZDivFloor ZSig.
(** * The interface [ZSig.ZType] implies the interface [ZAxiomsSig]
It also provides [sgn], [abs], [div], [mod]
*)
Module ZTypeIsZAxioms (Import Z : ZType').
Hint Rewrite
spec_0 spec_1 spec_2 spec_add spec_sub spec_pred spec_succ
spec_mul spec_opp spec_of_Z spec_div spec_modulo spec_sqrt
spec_compare spec_eq_bool spec_max spec_min spec_abs spec_sgn
spec_pow spec_log2 spec_even spec_odd
: zsimpl.
Ltac zsimpl := autorewrite with zsimpl.
Ltac zcongruence := repeat red; intros; zsimpl; congruence.
Ltac zify := unfold eq, lt, le in *; zsimpl.
Instance eq_equiv : Equivalence eq.
Proof. unfold eq. firstorder. Qed.
Local Obligation Tactic := zcongruence.
Program Instance succ_wd : Proper (eq ==> eq) succ.
Program Instance pred_wd : Proper (eq ==> eq) pred.
Program Instance add_wd : Proper (eq ==> eq ==> eq) add.
Program Instance sub_wd : Proper (eq ==> eq ==> eq) sub.
Program Instance mul_wd : Proper (eq ==> eq ==> eq) mul.
Theorem pred_succ : forall n, pred (succ n) == n.
Proof.
intros. zify. auto with zarith.
Qed.
Theorem one_succ : 1 == succ 0.
Proof.
now zify.
Qed.
Theorem two_succ : 2 == succ 1.
Proof.
now zify.
Qed.
Section Induction.
Variable A : Z.t -> Prop.
Hypothesis A_wd : Proper (eq==>iff) A.
Hypothesis A0 : A 0.
Hypothesis AS : forall n, A n <-> A (succ n).
Let B (z : Z) := A (of_Z z).
Lemma B0 : B 0.
Proof.
unfold B; simpl.
rewrite <- (A_wd 0); auto.
zify. auto.
Qed.
Lemma BS : forall z : Z, B z -> B (z + 1).
Proof.
intros z H.
unfold B in *. apply -> AS in H.
setoid_replace (of_Z (z + 1)) with (succ (of_Z z)); auto.
zify. auto.
Qed.
Lemma BP : forall z : Z, B z -> B (z - 1).
Proof.
intros z H.
unfold B in *. rewrite AS.
setoid_replace (succ (of_Z (z - 1))) with (of_Z z); auto.
zify. auto with zarith.
Qed.
Lemma B_holds : forall z : Z, B z.
Proof.
intros; destruct (Z_lt_le_dec 0 z).
apply natlike_ind; auto with zarith.
apply B0.
intros; apply BS; auto.
replace z with (-(-z))%Z in * by (auto with zarith).
remember (-z)%Z as z'.
pattern z'; apply natlike_ind.
apply B0.
intros; rewrite Zopp_succ; unfold Zpred; apply BP; auto.
subst z'; auto with zarith.
Qed.
Theorem bi_induction : forall n, A n.
Proof.
intro n. setoid_replace n with (of_Z (to_Z n)).
apply B_holds.
zify. auto.
Qed.
End Induction.
Theorem add_0_l : forall n, 0 + n == n.
Proof.
intros. zify. auto with zarith.
Qed.
Theorem add_succ_l : forall n m, (succ n) + m == succ (n + m).
Proof.
intros. zify. auto with zarith.
Qed.
Theorem sub_0_r : forall n, n - 0 == n.
Proof.
intros. zify. auto with zarith.
Qed.
Theorem sub_succ_r : forall n m, n - (succ m) == pred (n - m).
Proof.
intros. zify. auto with zarith.
Qed.
Theorem mul_0_l : forall n, 0 * n == 0.
Proof.
intros. zify. auto with zarith.
Qed.
Theorem mul_succ_l : forall n m, (succ n) * m == n * m + m.
Proof.
intros. zify. ring.
Qed.
(** Order *)
Lemma compare_spec : forall x y, CompSpec eq lt x y (compare x y).
Proof.
intros. zify. destruct (Zcompare_spec [x] [y]); auto.
Qed.
Definition eqb := eq_bool.
Lemma eqb_eq : forall x y, eq_bool x y = true <-> x == y.
Proof.
intros. zify. symmetry. apply Zeq_is_eq_bool.
Qed.
Instance compare_wd : Proper (eq ==> eq ==> Logic.eq) compare.
Proof.
intros x x' Hx y y' Hy. rewrite 2 spec_compare, Hx, Hy; intuition.
Qed.
Instance lt_wd : Proper (eq ==> eq ==> iff) lt.
Proof.
intros x x' Hx y y' Hy; unfold lt; rewrite Hx, Hy; intuition.
Qed.
Theorem lt_eq_cases : forall n m, n <= m <-> n < m \/ n == m.
Proof.
intros. zify. omega.
Qed.
Theorem lt_irrefl : forall n, ~ n < n.
Proof.
intros. zify. omega.
Qed.
Theorem lt_succ_r : forall n m, n < (succ m) <-> n <= m.
Proof.
intros. zify. omega.
Qed.
Theorem min_l : forall n m, n <= m -> min n m == n.
Proof.
intros n m. zify. omega with *.
Qed.
Theorem min_r : forall n m, m <= n -> min n m == m.
Proof.
intros n m. zify. omega with *.
Qed.
Theorem max_l : forall n m, m <= n -> max n m == n.
Proof.
intros n m. zify. omega with *.
Qed.
Theorem max_r : forall n m, n <= m -> max n m == m.
Proof.
intros n m. zify. omega with *.
Qed.
(** Part specific to integers, not natural numbers *)
Theorem succ_pred : forall n, succ (pred n) == n.
Proof.
intros. zify. auto with zarith.
Qed.
(** Opp *)
Program Instance opp_wd : Proper (eq ==> eq) opp.
Theorem opp_0 : - 0 == 0.
Proof.
intros. zify. auto with zarith.
Qed.
Theorem opp_succ : forall n, - (succ n) == pred (- n).
Proof.
intros. zify. auto with zarith.
Qed.
(** Abs / Sgn *)
Theorem abs_eq : forall n, 0 <= n -> abs n == n.
Proof.
intros n. zify. omega with *.
Qed.
Theorem abs_neq : forall n, n <= 0 -> abs n == -n.
Proof.
intros n. zify. omega with *.
Qed.
Theorem sgn_null : forall n, n==0 -> sgn n == 0.
Proof.
intros n. zify. omega with *.
Qed.
Theorem sgn_pos : forall n, 0<n -> sgn n == 1.
Proof.
intros n. zify. omega with *.
Qed.
Theorem sgn_neg : forall n, n<0 -> sgn n == opp 1.
Proof.
intros n. zify. omega with *.
Qed.
(** Power *)
Program Instance pow_wd : Proper (eq==>eq==>eq) pow.
Lemma pow_0_r : forall a, a^0 == 1.
Proof.
intros. now zify.
Qed.
Lemma pow_succ_r : forall a b, 0<=b -> a^(succ b) == a * a^b.
Proof.
intros a b. zify. intros Hb.
rewrite Zpower_exp; auto with zarith.
simpl. unfold Zpower_pos; simpl. ring.
Qed.
Lemma pow_neg_r : forall a b, b<0 -> a^b == 0.
Proof.
intros a b. zify. intros Hb.
destruct [b]; reflexivity || discriminate.
Qed.
Lemma pow_pow_N : forall a b, 0<=b -> a^b == pow_N a (Zabs_N (to_Z b)).
Proof.
intros a b. zify. intros Hb.
now rewrite spec_pow_N, Z_of_N_abs, Zabs_eq.
Qed.
Lemma pow_pos_N : forall a p, pow_pos a p == pow_N a (Npos p).
Proof.
intros a b. red. now rewrite spec_pow_N, spec_pow_pos.
Qed.
(** Sqrt *)
Lemma sqrt_spec : forall n, 0<=n ->
(sqrt n)*(sqrt n) <= n /\ n < (succ (sqrt n))*(succ (sqrt n)).
Proof.
intros n. zify. apply Zsqrt_spec.
Qed.
Lemma sqrt_neg : forall n, n<0 -> sqrt n == 0.
Proof.
intros n. zify. apply Zsqrt_neg.
Qed.
(** Log2 *)
Lemma log2_spec : forall n, 0<n ->
2^(log2 n) <= n /\ n < 2^(succ (log2 n)).
Proof.
intros n. zify. apply Zlog2_spec.
Qed.
Lemma log2_nonpos : forall n, n<=0 -> log2 n == 0.
Proof.
intros n. zify. apply Zlog2_nonpos.
Qed.
(** Even / Odd *)
Definition Even n := exists m, n == 2*m.
Definition Odd n := exists m, n == 2*m+1.
Lemma even_spec : forall n, even n = true <-> Even n.
Proof.
intros n. unfold Even. zify.
rewrite Zeven_bool_iff, Zeven_ex_iff.
split; intros (m,Hm).
exists (of_Z m). now zify.
exists [m]. revert Hm. now zify.
Qed.
Lemma odd_spec : forall n, odd n = true <-> Odd n.
Proof.
intros n. unfold Odd. zify.
rewrite Zodd_bool_iff, Zodd_ex_iff.
split; intros (m,Hm).
exists (of_Z m). now zify.
exists [m]. revert Hm. now zify.
Qed.
(** Div / Mod *)
Program Instance div_wd : Proper (eq==>eq==>eq) div.
Program Instance mod_wd : Proper (eq==>eq==>eq) modulo.
Theorem div_mod : forall a b, ~b==0 -> a == b*(div a b) + (modulo a b).
Proof.
intros a b. zify. intros. apply Z_div_mod_eq_full; auto.
Qed.
Theorem mod_pos_bound :
forall a b, 0 < b -> 0 <= modulo a b /\ modulo a b < b.
Proof.
intros a b. zify. intros. apply Z_mod_lt; auto with zarith.
Qed.
Theorem mod_neg_bound :
forall a b, b < 0 -> b < modulo a b /\ modulo a b <= 0.
Proof.
intros a b. zify. intros. apply Z_mod_neg; auto with zarith.
Qed.
End ZTypeIsZAxioms.
Module ZType_ZAxioms (Z : ZType)
<: ZAxiomsSig <: ZDivSig <: HasCompare Z <: HasEqBool Z <: HasMinMax Z
:= Z <+ ZTypeIsZAxioms.
|
