diff options
Diffstat (limited to 'theories/Numbers/NatInt/NZDiv.v')
| -rw-r--r-- | theories/Numbers/NatInt/NZDiv.v | 57 |
1 files changed, 25 insertions, 32 deletions
diff --git a/theories/Numbers/NatInt/NZDiv.v b/theories/Numbers/NatInt/NZDiv.v index aaf6bfc22..0807013eb 100644 --- a/theories/Numbers/NatInt/NZDiv.v +++ b/theories/Numbers/NatInt/NZDiv.v @@ -23,26 +23,19 @@ End DivModNotation. Module Type DivMod' (A : Typ) := DivMod A <+ DivModNotation A. -Module Type NZDivCommon (Import A : NZAxiomsSig')(Import B : DivMod' A). +Module Type NZDivSpec (Import A : NZOrdAxiomsSig')(Import B : DivMod' A). Declare Instance div_wd : Proper (eq==>eq==>eq) div. Declare Instance mod_wd : Proper (eq==>eq==>eq) modulo. Axiom div_mod : forall a b, b ~= 0 -> a == b*(a/b) + (a mod b). -End NZDivCommon. + Axiom mod_bound_pos : forall a b, 0<=a -> 0<b -> 0 <= a mod b < b. +End NZDivSpec. (** The different divisions will only differ in the conditions - they impose on [modulo]. For NZ, we only describe behavior - on positive numbers. - - NB: This axiom would also be true for N and Z, but redundant. + they impose on [modulo]. For NZ, we have only described the + behavior on positive numbers. *) -Module Type NZDivSpecific (Import A : NZOrdAxiomsSig')(Import B : DivMod' A). - Axiom mod_bound : forall a b, 0<=a -> 0<b -> 0 <= a mod b < b. -End NZDivSpecific. - -Module Type NZDiv (A : NZOrdAxiomsSig) - := DivMod A <+ NZDivCommon A <+ NZDivSpecific A. - +Module Type NZDiv (A : NZOrdAxiomsSig) := DivMod A <+ NZDivSpec A. Module Type NZDiv' (A : NZOrdAxiomsSig) := NZDiv A <+ DivModNotation A. Module Type NZDivProp @@ -83,7 +76,7 @@ Theorem div_unique: Proof. intros a b q r Ha (Hb,Hr) EQ. destruct (div_mod_unique b q (a/b) r (a mod b)); auto. -apply mod_bound; order. +apply mod_bound_pos; order. rewrite <- div_mod; order. Qed. @@ -93,7 +86,7 @@ Theorem mod_unique: Proof. intros a b q r Ha (Hb,Hr) EQ. destruct (div_mod_unique b q (a/b) r (a mod b)); auto. -apply mod_bound; order. +apply mod_bound_pos; order. rewrite <- div_mod; order. Qed. @@ -193,7 +186,7 @@ Theorem mod_le: forall a b, 0<=a -> 0<b -> a mod b <= a. Proof. intros. destruct (le_gt_cases b a). apply le_trans with b; auto. -apply lt_le_incl. destruct (mod_bound a b); auto. +apply lt_le_incl. destruct (mod_bound_pos a b); auto. rewrite lt_eq_cases; right. apply mod_small; auto. Qed. @@ -215,7 +208,7 @@ Lemma div_str_pos : forall a b, 0<b<=a -> 0 < a/b. Proof. intros a b (Hb,Hab). assert (LE : 0 <= a/b) by (apply div_pos; order). -assert (MOD : a mod b < b) by (destruct (mod_bound a b); order). +assert (MOD : a mod b < b) by (destruct (mod_bound_pos a b); order). rewrite lt_eq_cases in LE; destruct LE as [LT|EQ]; auto. exfalso; revert Hab. rewrite (div_mod a b), <-EQ; nzsimpl; order. @@ -262,7 +255,7 @@ rewrite <- (mul_1_l (a/b)) at 1. rewrite <- mul_lt_mono_pos_r; auto. apply div_str_pos; auto. rewrite <- (add_0_r (b*(a/b))) at 1. -rewrite <- add_le_mono_l. destruct (mod_bound a b); order. +rewrite <- add_le_mono_l. destruct (mod_bound_pos a b); order. Qed. (** [le] is compatible with a positive division. *) @@ -281,8 +274,8 @@ apply lt_le_trans with b; auto. rewrite (div_mod b c) at 1 by order. rewrite <- add_assoc, <- add_le_mono_l. apply le_trans with (c+0). -nzsimpl; destruct (mod_bound b c); order. -rewrite <- add_le_mono_l. destruct (mod_bound a c); order. +nzsimpl; destruct (mod_bound_pos b c); order. +rewrite <- add_le_mono_l. destruct (mod_bound_pos a c); order. Qed. (** The following two properties could be used as specification of div *) @@ -292,7 +285,7 @@ Proof. intros. rewrite (add_le_mono_r _ _ (a mod b)), <- div_mod by order. rewrite <- (add_0_r a) at 1. -rewrite <- add_le_mono_l. destruct (mod_bound a b); order. +rewrite <- add_le_mono_l. destruct (mod_bound_pos a b); order. Qed. Lemma mul_succ_div_gt : forall a b, 0<=a -> 0<b -> a < b*(S (a/b)). @@ -301,7 +294,7 @@ intros. rewrite (div_mod a b) at 1 by order. rewrite (mul_succ_r). rewrite <- add_lt_mono_l. -destruct (mod_bound a b); auto. +destruct (mod_bound_pos a b); auto. Qed. @@ -358,7 +351,7 @@ Proof. apply mul_le_mono_nonneg_r; try order. apply div_pos; order. rewrite <- (add_0_r (r*(p/r))) at 1. - rewrite <- add_le_mono_l. destruct (mod_bound p r); order. + rewrite <- add_le_mono_l. destruct (mod_bound_pos p r); order. Qed. @@ -370,7 +363,7 @@ Proof. intros. symmetry. apply mod_unique with (a/c+b); auto. - apply mod_bound; auto. + apply mod_bound_pos; auto. rewrite mul_add_distr_l, add_shuffle0, <- div_mod by order. now rewrite mul_comm. Qed. @@ -403,8 +396,8 @@ Proof. apply div_unique with ((a mod b)*c). apply mul_nonneg_nonneg; order. split. - apply mul_nonneg_nonneg; destruct (mod_bound a b); order. - rewrite <- mul_lt_mono_pos_r; auto. destruct (mod_bound a b); auto. + apply mul_nonneg_nonneg; destruct (mod_bound_pos a b); order. + rewrite <- mul_lt_mono_pos_r; auto. destruct (mod_bound_pos a b); auto. rewrite (div_mod a b) at 1 by order. rewrite mul_add_distr_r. rewrite add_cancel_r. @@ -440,7 +433,7 @@ Qed. Theorem mod_mod: forall a n, 0<=a -> 0<n -> (a mod n) mod n == a mod n. Proof. - intros. destruct (mod_bound a n); auto. now rewrite mod_small_iff. + intros. destruct (mod_bound_pos a n); auto. now rewrite mod_small_iff. Qed. Lemma mul_mod_idemp_l : forall a b n, 0<=a -> 0<=b -> 0<n -> @@ -453,7 +446,7 @@ Proof. rewrite mul_add_distr_l, mul_assoc. intros. rewrite mod_add; auto. now rewrite mul_comm. - apply mul_nonneg_nonneg; destruct (mod_bound a n); auto. + apply mul_nonneg_nonneg; destruct (mod_bound_pos a n); auto. Qed. Lemma mul_mod_idemp_r : forall a b n, 0<=a -> 0<=b -> 0<n -> @@ -466,7 +459,7 @@ Theorem mul_mod: forall a b n, 0<=a -> 0<=b -> 0<n -> (a * b) mod n == ((a mod n) * (b mod n)) mod n. Proof. intros. rewrite mul_mod_idemp_l, mul_mod_idemp_r; trivial. reflexivity. - now destruct (mod_bound b n). + now destruct (mod_bound_pos b n). Qed. Lemma add_mod_idemp_l : forall a b n, 0<=a -> 0<=b -> 0<n -> @@ -477,7 +470,7 @@ Proof. rewrite (div_mod a n) at 1 2 by order. rewrite <- add_assoc, add_comm, mul_comm. intros. rewrite mod_add; trivial. reflexivity. - apply add_nonneg_nonneg; auto. destruct (mod_bound a n); auto. + apply add_nonneg_nonneg; auto. destruct (mod_bound_pos a n); auto. Qed. Lemma add_mod_idemp_r : forall a b n, 0<=a -> 0<=b -> 0<n -> @@ -490,7 +483,7 @@ Theorem add_mod: forall a b n, 0<=a -> 0<=b -> 0<n -> (a+b) mod n == (a mod n + b mod n) mod n. Proof. intros. rewrite add_mod_idemp_l, add_mod_idemp_r; trivial. reflexivity. - now destruct (mod_bound b n). + now destruct (mod_bound_pos b n). Qed. Lemma div_div : forall a b c, 0<=a -> 0<b -> 0<c -> @@ -499,7 +492,7 @@ Proof. intros a b c Ha Hb Hc. apply div_unique with (b*((a/b) mod c) + a mod b); trivial. (* begin 0<= ... <b*c *) - destruct (mod_bound (a/b) c), (mod_bound a b); auto using div_pos. + destruct (mod_bound_pos (a/b) c), (mod_bound_pos a b); auto using div_pos. split. apply add_nonneg_nonneg; auto. apply mul_nonneg_nonneg; order. |