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author | 2009-11-03 08:24:06 +0000 | |
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committer | 2009-11-03 08:24:06 +0000 | |
commit | 4f0ad99adb04e7f2888e75f2a10e8c916dde179b (patch) | |
tree | 4b52d7436fe06f4b2babfd5bfed84762440e7de7 /theories/NArith/Pminmax.v | |
parent | 4e68924f48d3f6d5ffdf1cd394b590b5a6e15ea1 (diff) |
OrderedType implementation for various numerical datatypes + min/max structures
- A richer OrderedTypeFull interface : OrderedType + predicate "le"
- Implementations {Nat,N,P,Z,Q}OrderedType.v, also providing "order" tactics
- By the way: as suggested by S. Lescuyer, specification of compare is
now inductive
- GenericMinMax: axiomatisation + properties of min and max out of
OrderedTypeFull structures.
- MinMax.v, {Z,P,N,Q}minmax.v are specialization of GenericMinMax,
with also some domain-specific results, and compatibility layer
with already existing results.
- Some ML code of plugins had to be adapted, otherwise wrong "eq",
"lt" or simimlar constants were found by functions like coq_constant.
- Beware of the aliasing problems: for instance eq:=@eq t instead of
eq:=@eq M.t in Make_UDT made (r)omega stopped working (Z_as_OT.t
instead of Z in statement of Zmax_spec).
- Some Morphism declaration are now ambiguous: switch to new syntax
anyway.
- Misc adaptations of FSets/MSets
- Classes/RelationPairs.v: from two relations over A and B, we
inspect relations over A*B and their properties in terms of classes.
git-svn-id: svn+ssh://scm.gforge.inria.fr/svn/coq/trunk@12461 85f007b7-540e-0410-9357-904b9bb8a0f7
Diffstat (limited to 'theories/NArith/Pminmax.v')
-rw-r--r-- | theories/NArith/Pminmax.v | 225 |
1 files changed, 225 insertions, 0 deletions
diff --git a/theories/NArith/Pminmax.v b/theories/NArith/Pminmax.v new file mode 100644 index 000000000..18008d18d --- /dev/null +++ b/theories/NArith/Pminmax.v @@ -0,0 +1,225 @@ +(************************************************************************) +(* 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 *) +(************************************************************************) + +Require Import OrderedType2 BinPos Pnat POrderedType GenericMinMax. + +(** * Maximum and Minimum of two positive numbers *) + +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). +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. +Qed. + +Lemma Pmin_spec : forall x y, + (x<y /\ Pmin x y = x) \/ (y<=x /\ Pmin 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. +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), + (m<=n -> P m) -> (n<=m -> P n) -> 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). +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. +Qed. + +Lemma Pmin_max_antimonotone : forall f, + Proper (Ple==>Pge) f -> + forall x y, Pmin (f x) (f y) == f (Pmax x 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. +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. + + +(** * Properties specific to the [positive] domain *) + +(** Simplifications *) + +Lemma Pmax_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 Pmin_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. + +(** Compatibilities (consequences of monotonicity) *) + +Lemma Psucc_max_distr : + forall n m, Psucc (Pmax n m) = Pmax (Psucc n) (Psucc m). +Proof. + intros. symmetry. apply Pmax_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). +Proof. + intros. symmetry. apply Pmin_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. +Proof. + intros. apply Pmax_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. +Proof. + intros. rewrite (Pplus_comm n p), (Pplus_comm m p), (Pplus_comm _ p). + apply Pplus_max_distr_l. +Qed. + +Lemma Pplus_min_distr_l : forall n m p, Pmin (p + n) (p + m) = p + Pmin n m. +Proof. + intros. apply Pmin_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. +Proof. + intros. rewrite (Pplus_comm n p), (Pplus_comm m p), (Pplus_comm _ p). + apply Pplus_min_distr_l. +Qed. |