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Diffstat (limited to 'theories/NArith/Pminmax.v')
| -rw-r--r-- | theories/NArith/Pminmax.v | 189 |
1 files changed, 45 insertions, 144 deletions
diff --git a/theories/NArith/Pminmax.v b/theories/NArith/Pminmax.v index afae63f5a..3b3c58841 100644 --- a/theories/NArith/Pminmax.v +++ b/theories/NArith/Pminmax.v @@ -15,211 +15,112 @@ Local Open Scope positive_scope. (** The functions [Pmax] and [Pmin] implement indeed a maximum and a minimum *) -Lemma Pmax_spec : forall x y, - (x<y /\ Pmax x y = y) \/ (y<=x /\ Pmax x y = x). +Lemma Pmax_l : forall x y, y<=x -> Pmax x y = x. Proof. - unfold Plt, Ple, Pmax. intros. - generalize (Pcompare_eq_iff x y). rewrite (ZC4 y x). - destruct ((x ?= y) Eq); simpl; auto; right; intuition; discriminate. + unfold Ple, Pmax. intros x y. + rewrite (ZC4 y x). generalize (Pcompare_eq_iff x y). + destruct ((x ?= y) Eq); intuition. Qed. -Lemma Pmin_spec : forall x y, - (x<y /\ Pmin x y = x) \/ (y<=x /\ Pmin x y = y). +Lemma Pmax_r : forall x y, x<=y -> Pmax x y = y. Proof. - unfold Plt, Ple, Pmin. intros. - generalize (Pcompare_eq_iff x y). rewrite (ZC4 y x). - destruct ((x ?= y) Eq); simpl; auto; right; intuition; discriminate. + unfold Ple, Pmax. intros x y. destruct ((x ?= y) Eq); intuition. Qed. -Module PositiveHasMinMax <: HasMinMax Positive_as_OT. - Definition max := Pmax. - Definition min := Pmin. - Definition max_spec := Pmax_spec. - Definition min_spec := Pmin_spec. -End PositiveHasMinMax. - -(** We obtain hence all the generic properties of max and min. *) - -Module Import NatMinMaxProps := - MinMaxProperties Positive_as_OT PositiveHasMinMax. - - -(** For some generic properties, we can have nicer statements here, - since underlying equality is Leibniz. *) - -Lemma Pmax_case_strong : forall n m (P:positive -> Type), - (m<=n -> P n) -> (n<=m -> P m) -> P (Pmax n m). -Proof. intros; apply max_case_strong; auto. congruence. Defined. - -Lemma Pmax_case : forall n m (P:positive -> Type), - P n -> P m -> P (Pmax n m). -Proof. intros. apply Pmax_case_strong; auto. Defined. - -Lemma Pmax_monotone: forall f, - (Proper (Ple ==> Ple) f) -> - forall x y, Pmax (f x) (f y) = f (Pmax x y). -Proof. intros; apply max_monotone; auto. congruence. Qed. - -Lemma Pmin_case_strong : forall n m (P:positive -> Type), - (n<=m -> P n) -> (m<=n -> P m) -> P (Pmin n m). -Proof. intros; apply min_case_strong; auto. congruence. Defined. - -Lemma Pmin_case : forall n m (P:positive -> Type), - P n -> P m -> P (Pmin n m). -Proof. intros. apply Pmin_case_strong; auto. Defined. - -Lemma Pmin_monotone: forall f, - (Proper (Ple ==> Ple) f) -> - forall x y, Pmin (f x) (f y) = f (Pmin x y). -Proof. intros; apply min_monotone; auto. congruence. Qed. - -Lemma Pmax_min_antimonotone : forall f, - Proper (Ple==>Pge) f -> - forall x y, Pmax (f x) (f y) == f (Pmin x y). +Lemma Pmin_l : forall x y, x<=y -> Pmin x y = x. Proof. - intros f H. apply max_min_antimonotone. congruence. - intros z z' Hz; red. specialize (H _ _ Hz). clear Hz. - unfold Ple, Pge in *. contradict H. rewrite ZC4, H; auto. + unfold Ple, Pmin. intros x y. destruct ((x ?= y) Eq); intuition. Qed. -Lemma Pmin_max_antimonotone : forall f, - Proper (Ple==>Pge) f -> - forall x y, Pmin (f x) (f y) == f (Pmax x y). +Lemma Pmin_r : forall x y, y<=x -> Pmin x y = y. Proof. - intros f H. apply min_max_antimonotone. congruence. - intros z z' Hz; red. specialize (H _ _ Hz). clear Hz. - unfold Ple, Pge in *. contradict H. rewrite ZC4, H; auto. + unfold Ple, Pmin. intros x y. + rewrite (ZC4 y x). generalize (Pcompare_eq_iff x y). + destruct ((x ?= y) Eq); intuition. Qed. -(** For the other generic properties, we make aliases, - since otherwise SearchAbout misses some of them - (bad interaction with an Include). - See GenericMinMax (or SearchAbout) for the statements. *) - -Definition Pmax_spec_le := max_spec_le. -Definition Pmax_dec := max_dec. -Definition Pmax_unicity := max_unicity. -Definition Pmax_unicity_ext := max_unicity_ext. -Definition Pmax_id := max_id. -Notation Pmax_idempotent := Pmax_id (only parsing). -Definition Pmax_assoc := max_assoc. -Definition Pmax_comm := max_comm. -Definition Pmax_l := max_l. -Definition Pmax_r := max_r. -Definition Ple_max_l := le_max_l. -Definition Ple_max_r := le_max_r. -Definition Pmax_le := max_le. -Definition Pmax_le_iff := max_le_iff. -Definition Pmax_lt_iff := max_lt_iff. -Definition Pmax_lub_l := max_lub_l. -Definition Pmax_lub_r := max_lub_r. -Definition Pmax_lub := max_lub. -Definition Pmax_lub_iff := max_lub_iff. -Definition Pmax_lub_lt := max_lub_lt. -Definition Pmax_lub_lt_iff := max_lub_lt_iff. -Definition Pmax_le_compat_l := max_le_compat_l. -Definition Pmax_le_compat_r := max_le_compat_r. -Definition Pmax_le_compat := max_le_compat. - -Definition Pmin_spec_le := min_spec_le. -Definition Pmin_dec := min_dec. -Definition Pmin_unicity := min_unicity. -Definition Pmin_unicity_ext := min_unicity_ext. -Definition Pmin_id := min_id. -Notation Pmin_idempotent := Pmin_id (only parsing). -Definition Pmin_assoc := min_assoc. -Definition Pmin_comm := min_comm. -Definition Pmin_l := min_l. -Definition Pmin_r := min_r. -Definition Ple_min_l := le_min_l. -Definition Ple_min_r := le_min_r. -Definition Pmin_le := min_le. -Definition Pmin_le_iff := min_le_iff. -Definition Pmin_lt_iff := min_lt_iff. -Definition Pmin_glb_l := min_glb_l. -Definition Pmin_glb_r := min_glb_r. -Definition Pmin_glb := min_glb. -Definition Pmin_glb_iff := min_glb_iff. -Definition Pmin_glb_lt := min_glb_lt. -Definition Pmin_glb_lt_iff := min_glb_lt_iff. -Definition Pmin_le_compat_l := min_le_compat_l. -Definition Pmin_le_compat_r := min_le_compat_r. -Definition Pmin_le_compat := min_le_compat. - -Definition Pmin_max_absorption := min_max_absorption. -Definition Pmax_min_absorption := max_min_absorption. -Definition Pmax_min_distr := max_min_distr. -Definition Pmin_max_distr := min_max_distr. -Definition Pmax_min_modular := max_min_modular. -Definition Pmin_max_modular := min_max_modular. -Definition Pmax_min_disassoc := max_min_disassoc. +Module PositiveHasMinMax <: HasMinMax Positive_as_OT. + Definition max := Pmax. + Definition min := Pmin. + Definition max_l := Pmax_l. + Definition max_r := Pmax_r. + Definition min_l := Pmin_l. + Definition min_r := Pmin_r. +End PositiveHasMinMax. + +Module P. +(** We obtain hence all the generic properties of max and min. *) + +Include UsualMinMaxProperties Positive_as_OT PositiveHasMinMax. (** * Properties specific to the [positive] domain *) (** Simplifications *) -Lemma Pmax_1_l : forall n, Pmax 1 n = n. +Lemma max_1_l : forall n, Pmax 1 n = n. Proof. intros. unfold Pmax. rewrite ZC4. generalize (Pcompare_1 n). destruct (n ?= 1); intuition. Qed. -Lemma Pmax_1_r : forall n, Pmax n 1 = n. -Proof. intros. rewrite max_comm. apply Pmax_1_l. Qed. +Lemma max_1_r : forall n, Pmax n 1 = n. +Proof. intros. rewrite P.max_comm. apply max_1_l. Qed. -Lemma Pmin_1_l : forall n, Pmin 1 n = 1. +Lemma min_1_l : forall n, Pmin 1 n = 1. Proof. intros. unfold Pmin. rewrite ZC4. generalize (Pcompare_1 n). destruct (n ?= 1); intuition. Qed. -Lemma Pmin_1_r : forall n, Pmin n 1 = 1. -Proof. intros. rewrite min_comm. apply Pmin_1_l. Qed. +Lemma min_1_r : forall n, Pmin n 1 = 1. +Proof. intros. rewrite P.min_comm. apply min_1_l. Qed. (** Compatibilities (consequences of monotonicity) *) -Lemma Psucc_max_distr : +Lemma succ_max_distr : forall n m, Psucc (Pmax n m) = Pmax (Psucc n) (Psucc m). Proof. - intros. symmetry. apply Pmax_monotone. + intros. symmetry. apply max_monotone. intros x x'. unfold Ple. rewrite 2 nat_of_P_compare_morphism, 2 nat_of_P_succ_morphism. simpl; auto. Qed. -Lemma Psucc_min_distr : forall n m, Psucc (Pmin n m) = Pmin (Psucc n) (Psucc m). +Lemma succ_min_distr : forall n m, Psucc (Pmin n m) = Pmin (Psucc n) (Psucc m). Proof. - intros. symmetry. apply Pmin_monotone. + intros. symmetry. apply min_monotone. intros x x'. unfold Ple. rewrite 2 nat_of_P_compare_morphism, 2 nat_of_P_succ_morphism. simpl; auto. Qed. -Lemma Pplus_max_distr_l : forall n m p, Pmax (p + n) (p + m) = p + Pmax n m. +Lemma plus_max_distr_l : forall n m p, Pmax (p + n) (p + m) = p + Pmax n m. Proof. - intros. apply Pmax_monotone. + intros. apply max_monotone. intros x x'. unfold Ple. rewrite 2 nat_of_P_compare_morphism, 2 nat_of_P_plus_morphism. rewrite <- 2 Compare_dec.nat_compare_le. auto with arith. Qed. -Lemma Pplus_max_distr_r : forall n m p, Pmax (n + p) (m + p) = Pmax n m + p. +Lemma plus_max_distr_r : forall n m p, Pmax (n + p) (m + p) = Pmax n m + p. Proof. intros. rewrite (Pplus_comm n p), (Pplus_comm m p), (Pplus_comm _ p). - apply Pplus_max_distr_l. + apply plus_max_distr_l. Qed. -Lemma Pplus_min_distr_l : forall n m p, Pmin (p + n) (p + m) = p + Pmin n m. +Lemma plus_min_distr_l : forall n m p, Pmin (p + n) (p + m) = p + Pmin n m. Proof. - intros. apply Pmin_monotone. + intros. apply min_monotone. intros x x'. unfold Ple. rewrite 2 nat_of_P_compare_morphism, 2 nat_of_P_plus_morphism. rewrite <- 2 Compare_dec.nat_compare_le. auto with arith. Qed. -Lemma Pplus_min_distr_r : forall n m p, Pmin (n + p) (m + p) = Pmin n m + p. +Lemma plus_min_distr_r : forall n m p, Pmin (n + p) (m + p) = Pmin n m + p. Proof. intros. rewrite (Pplus_comm n p), (Pplus_comm m p), (Pplus_comm _ p). - apply Pplus_min_distr_l. + apply plus_min_distr_l. Qed. + +End P.
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