diff options
Diffstat (limited to 'theories/Numbers/Natural/SpecViaZ/NSigNAxioms.v')
-rw-r--r-- | theories/Numbers/Natural/SpecViaZ/NSigNAxioms.v | 111 |
1 files changed, 35 insertions, 76 deletions
diff --git a/theories/Numbers/Natural/SpecViaZ/NSigNAxioms.v b/theories/Numbers/Natural/SpecViaZ/NSigNAxioms.v index 919701879..2b199858f 100644 --- a/theories/Numbers/Natural/SpecViaZ/NSigNAxioms.v +++ b/theories/Numbers/Natural/SpecViaZ/NSigNAxioms.v @@ -17,59 +17,36 @@ Require Import NSig. Module NSig_NAxioms (N:NType) <: NAxiomsSig. -Delimit Scope IntScope with Int. -Bind Scope IntScope with N.t. -Local Open Scope IntScope. -Notation "[ x ]" := (N.to_Z x) : IntScope. -Infix "==" := N.eq (at level 70) : IntScope. -Notation "0" := N.zero : IntScope. -Infix "+" := N.add : IntScope. -Infix "-" := N.sub : IntScope. -Infix "*" := N.mul : IntScope. - -Hint Rewrite N.spec_0 N.spec_succ N.spec_add N.spec_mul : int. -Ltac isimpl := autorewrite with int. +Delimit Scope NumScope with Num. +Bind Scope NumScope with N.t. +Local Open Scope NumScope. +Notation "[ x ]" := (N.to_Z x) : NumScope. +Infix "==" := N.eq (at level 70) : NumScope. +Notation "0" := N.zero : NumScope. +Infix "+" := N.add : NumScope. +Infix "-" := N.sub : NumScope. +Infix "*" := N.mul : NumScope. + +Hint Rewrite + N.spec_0 N.spec_succ N.spec_add N.spec_mul N.spec_pred N.spec_sub : num. +Ltac nsimpl := autorewrite with num. +Ltac ncongruence := unfold N.eq; repeat red; intros; nsimpl; congruence. + +Obligation Tactic := ncongruence. Instance eq_equiv : Equivalence N.eq. -Instance succ_wd : Proper (N.eq==>N.eq) N.succ. -Proof. -unfold N.eq; repeat red; intros; isimpl; f_equal; auto. -Qed. - -Instance pred_wd : Proper (N.eq==>N.eq) N.pred. -Proof. -unfold N.eq; repeat red; intros. -generalize (N.spec_pos y) (N.spec_pos x) (N.spec_eq_bool x 0). -destruct N.eq_bool; rewrite N.spec_0; intros. -rewrite 2 N.spec_pred0; congruence. -rewrite 2 N.spec_pred; f_equal; auto; try omega. -Qed. - -Instance add_wd : Proper (N.eq==>N.eq==>N.eq) N.add. -Proof. -unfold N.eq; repeat red; intros; isimpl; f_equal; auto. -Qed. - -Instance sub_wd : Proper (N.eq==>N.eq==>N.eq) N.sub. -Proof. -unfold N.eq; intros x x' Hx y y' Hy. -destruct (Z_lt_le_dec [x] [y]). -rewrite 2 N.spec_sub0; f_equal; congruence. -rewrite 2 N.spec_sub; f_equal; congruence. -Qed. - -Instance mul_wd : Proper (N.eq==>N.eq==>N.eq) N.mul. -Proof. -unfold N.eq; repeat red; intros; isimpl; f_equal; auto. -Qed. +Program Instance succ_wd : Proper (N.eq==>N.eq) N.succ. +Program Instance pred_wd : Proper (N.eq==>N.eq) N.pred. +Program Instance add_wd : Proper (N.eq==>N.eq==>N.eq) N.add. +Program Instance sub_wd : Proper (N.eq==>N.eq==>N.eq) N.sub. +Program Instance mul_wd : Proper (N.eq==>N.eq==>N.eq) N.mul. Theorem pred_succ : forall n, N.pred (N.succ n) == n. Proof. unfold N.eq; repeat red; intros. rewrite N.spec_pred; rewrite N.spec_succ. -omega. -generalize (N.spec_pos n); omega. +generalize (N.spec_pos n); omega with *. Qed. Definition N_of_Z z := N.of_N (Zabs_N z). @@ -118,52 +95,38 @@ End Induction. Theorem add_0_l : forall n, 0 + n == n. Proof. -intros; red; isimpl; auto with zarith. +intros; red; nsimpl; auto with zarith. Qed. Theorem add_succ_l : forall n m, (N.succ n) + m == N.succ (n + m). Proof. -intros; red; isimpl; auto with zarith. +intros; red; nsimpl; auto with zarith. Qed. Theorem sub_0_r : forall n, n - 0 == n. Proof. -intros; red; rewrite N.spec_sub; rewrite N.spec_0; auto with zarith. -apply N.spec_pos. +intros; red; nsimpl. generalize (N.spec_pos n); omega with *. Qed. Theorem sub_succ_r : forall n m, n - (N.succ m) == N.pred (n - m). Proof. -intros; red. -destruct (Z_lt_le_dec [n] [N.succ m]) as [H|H]. -rewrite N.spec_sub0; auto. -rewrite N.spec_succ in H. -rewrite N.spec_pred0; auto. -destruct (Z_eq_dec [n] [m]). -rewrite N.spec_sub; auto with zarith. -rewrite N.spec_sub0; auto with zarith. - -rewrite N.spec_sub, N.spec_succ in *; auto. -rewrite N.spec_pred, N.spec_sub; auto with zarith. -rewrite N.spec_sub; auto with zarith. +intros; red; nsimpl. omega with *. Qed. Theorem mul_0_l : forall n, 0 * n == 0. Proof. -intros; red. -rewrite N.spec_mul, N.spec_0; auto with zarith. +intros; red; nsimpl; auto with zarith. Qed. Theorem mul_succ_l : forall n m, (N.succ n) * m == n * m + m. Proof. -intros; red. -rewrite N.spec_add, 2 N.spec_mul, N.spec_succ; ring. +intros; red; nsimpl. ring. Qed. (** Order *) -Infix "<=" := N.le : IntScope. -Infix "<" := N.lt : IntScope. +Infix "<=" := N.le : NumScope. +Infix "<" := N.lt : NumScope. Lemma spec_compare_alt : forall x y, N.compare x y = ([x] ?= [y])%Z. Proof. @@ -223,33 +186,29 @@ Qed. Theorem min_l : forall n m, n <= m -> N.min n m == n. Proof. -intros n m; unfold N.eq; rewrite spec_le, spec_min. -generalize (Zmin_spec [n] [m]); omega. +intros n m; red; rewrite spec_le, spec_min; omega with *. Qed. Theorem min_r : forall n m, m <= n -> N.min n m == m. Proof. -intros n m; unfold N.eq; rewrite spec_le, spec_min. -generalize (Zmin_spec [n] [m]); omega. +intros n m; red; rewrite spec_le, spec_min; omega with *. Qed. Theorem max_l : forall n m, m <= n -> N.max n m == n. Proof. -intros n m; unfold N.eq; rewrite spec_le, spec_max. -generalize (Zmax_spec [n] [m]); omega. +intros n m; red; rewrite spec_le, spec_max; omega with *. Qed. Theorem max_r : forall n m, n <= m -> N.max n m == m. Proof. -intros n m; unfold N.eq; rewrite spec_le, spec_max. -generalize (Zmax_spec [n] [m]); omega. +intros n m; red; rewrite spec_le, spec_max; omega with *. Qed. (** Properties specific to natural numbers, not integers. *) Theorem pred_0 : N.pred 0 == 0. Proof. -red; rewrite N.spec_pred0; rewrite N.spec_0; auto. +red; nsimpl; auto. Qed. Definition recursion (A : Type) (a : A) (f : N.t -> A -> A) (n : N.t) := |