diff options
Diffstat (limited to 'theories/Numbers/Natural/Abstract/NSub.v')
| -rw-r--r-- | theories/Numbers/Natural/Abstract/NSub.v | 196 |
1 files changed, 159 insertions, 37 deletions
diff --git a/theories/Numbers/Natural/Abstract/NSub.v b/theories/Numbers/Natural/Abstract/NSub.v index f67689dd..35d3b8aa 100644 --- a/theories/Numbers/Natural/Abstract/NSub.v +++ b/theories/Numbers/Natural/Abstract/NSub.v @@ -8,49 +8,33 @@ (* Evgeny Makarov, INRIA, 2007 *) (************************************************************************) -(*i $Id: NSub.v 11040 2008-06-03 00:04:16Z letouzey $ i*) +(*i $Id$ i*) Require Export NMulOrder. -Module NSubPropFunct (Import NAxiomsMod : NAxiomsSig). -Module Export NMulOrderPropMod := NMulOrderPropFunct NAxiomsMod. -Open Local Scope NatScope. +Module Type NSubPropFunct (Import N : NAxiomsSig'). +Include NMulOrderPropFunct N. -Theorem sub_wd : - forall n1 n2 : N, n1 == n2 -> forall m1 m2 : N, m1 == m2 -> n1 - m1 == n2 - m2. -Proof NZsub_wd. - -Theorem sub_0_r : forall n : N, n - 0 == n. -Proof NZsub_0_r. - -Theorem sub_succ_r : forall n m : N, n - (S m) == P (n - m). -Proof NZsub_succ_r. - -Theorem sub_1_r : forall n : N, n - 1 == P n. -Proof. -intro n; rewrite sub_succ_r; now rewrite sub_0_r. -Qed. - -Theorem sub_0_l : forall n : N, 0 - n == 0. +Theorem sub_0_l : forall n, 0 - n == 0. Proof. induct n. apply sub_0_r. intros n IH; rewrite sub_succ_r; rewrite IH. now apply pred_0. Qed. -Theorem sub_succ : forall n m : N, S n - S m == n - m. +Theorem sub_succ : forall n m, S n - S m == n - m. Proof. intro n; induct m. rewrite sub_succ_r. do 2 rewrite sub_0_r. now rewrite pred_succ. intros m IH. rewrite sub_succ_r. rewrite IH. now rewrite sub_succ_r. Qed. -Theorem sub_diag : forall n : N, n - n == 0. +Theorem sub_diag : forall n, n - n == 0. Proof. induct n. apply sub_0_r. intros n IH; rewrite sub_succ; now rewrite IH. Qed. -Theorem sub_gt : forall n m : N, n > m -> n - m ~= 0. +Theorem sub_gt : forall n m, n > m -> n - m ~= 0. Proof. intros n m H; elim H using lt_ind_rel; clear n m H. solve_relation_wd. @@ -58,7 +42,7 @@ intro; rewrite sub_0_r; apply neq_succ_0. intros; now rewrite sub_succ. Qed. -Theorem add_sub_assoc : forall n m p : N, p <= m -> n + (m - p) == (n + m) - p. +Theorem add_sub_assoc : forall n m p, p <= m -> n + (m - p) == (n + m) - p. Proof. intros n m p; induct p. intro; now do 2 rewrite sub_0_r. @@ -68,32 +52,32 @@ rewrite add_pred_r by (apply sub_gt; now apply -> le_succ_l). reflexivity. Qed. -Theorem sub_succ_l : forall n m : N, n <= m -> S m - n == S (m - n). +Theorem sub_succ_l : forall n m, n <= m -> S m - n == S (m - n). Proof. intros n m H. rewrite <- (add_1_l m). rewrite <- (add_1_l (m - n)). symmetry; now apply add_sub_assoc. Qed. -Theorem add_sub : forall n m : N, (n + m) - m == n. +Theorem add_sub : forall n m, (n + m) - m == n. Proof. intros n m. rewrite <- add_sub_assoc by (apply le_refl). rewrite sub_diag; now rewrite add_0_r. Qed. -Theorem sub_add : forall n m : N, n <= m -> (m - n) + n == m. +Theorem sub_add : forall n m, n <= m -> (m - n) + n == m. Proof. intros n m H. rewrite add_comm. rewrite add_sub_assoc by assumption. rewrite add_comm. apply add_sub. Qed. -Theorem add_sub_eq_l : forall n m p : N, m + p == n -> n - m == p. +Theorem add_sub_eq_l : forall n m p, m + p == n -> n - m == p. Proof. intros n m p H. symmetry. assert (H1 : m + p - m == n - m) by now rewrite H. rewrite add_comm in H1. now rewrite add_sub in H1. Qed. -Theorem add_sub_eq_r : forall n m p : N, m + p == n -> n - p == m. +Theorem add_sub_eq_r : forall n m p, m + p == n -> n - p == m. Proof. intros n m p H; rewrite add_comm in H; now apply add_sub_eq_l. Qed. @@ -101,7 +85,7 @@ Qed. (* This could be proved by adding m to both sides. Then the proof would use add_sub_assoc and sub_0_le, which is proven below. *) -Theorem add_sub_eq_nz : forall n m p : N, p ~= 0 -> n - m == p -> m + p == n. +Theorem add_sub_eq_nz : forall n m p, p ~= 0 -> n - m == p -> m + p == n. Proof. intros n m p H; double_induct n m. intros m H1; rewrite sub_0_l in H1. symmetry in H1; false_hyp H1 H. @@ -110,14 +94,14 @@ intros n m IH H1. rewrite sub_succ in H1. apply IH in H1. rewrite add_succ_l; now rewrite H1. Qed. -Theorem sub_add_distr : forall n m p : N, n - (m + p) == (n - m) - p. +Theorem sub_add_distr : forall n m p, n - (m + p) == (n - m) - p. Proof. intros n m; induct p. rewrite add_0_r; now rewrite sub_0_r. intros p IH. rewrite add_succ_r; do 2 rewrite sub_succ_r. now rewrite IH. Qed. -Theorem add_sub_swap : forall n m p : N, p <= n -> n + m - p == n - p + m. +Theorem add_sub_swap : forall n m p, p <= n -> n + m - p == n - p + m. Proof. intros n m p H. rewrite (add_comm n m). @@ -127,7 +111,7 @@ Qed. (** Sub and order *) -Theorem le_sub_l : forall n m : N, n - m <= n. +Theorem le_sub_l : forall n m, n - m <= n. Proof. intro n; induct m. rewrite sub_0_r; now apply eq_le_incl. @@ -135,7 +119,7 @@ intros m IH. rewrite sub_succ_r. apply le_trans with (n - m); [apply le_pred_l | assumption]. Qed. -Theorem sub_0_le : forall n m : N, n - m == 0 <-> n <= m. +Theorem sub_0_le : forall n m, n - m == 0 <-> n <= m. Proof. double_induct n m. intro m; split; intro; [apply le_0_l | apply sub_0_l]. @@ -144,9 +128,86 @@ intro m; rewrite sub_0_r; split; intro H; intros n m H. rewrite <- succ_le_mono. now rewrite sub_succ. Qed. +Theorem sub_add_le : forall n m, n <= n - m + m. +Proof. +intros. +destruct (le_ge_cases n m) as [LE|GE]. +rewrite <- sub_0_le in LE. rewrite LE; nzsimpl. +now rewrite <- sub_0_le. +rewrite sub_add by assumption. apply le_refl. +Qed. + +Theorem le_sub_le_add_r : forall n m p, + n - p <= m <-> n <= m + p. +Proof. +intros n m p. +split; intros LE. +rewrite (add_le_mono_r _ _ p) in LE. +apply le_trans with (n-p+p); auto using sub_add_le. +destruct (le_ge_cases n p) as [LE'|GE]. +rewrite <- sub_0_le in LE'. rewrite LE'. apply le_0_l. +rewrite (add_le_mono_r _ _ p). now rewrite sub_add. +Qed. + +Theorem le_sub_le_add_l : forall n m p, n - m <= p <-> n <= m + p. +Proof. +intros n m p. rewrite add_comm; apply le_sub_le_add_r. +Qed. + +Theorem lt_sub_lt_add_r : forall n m p, + n - p < m -> n < m + p. +Proof. +intros n m p LT. +rewrite (add_lt_mono_r _ _ p) in LT. +apply le_lt_trans with (n-p+p); auto using sub_add_le. +Qed. + +(** Unfortunately, we do not have [n < m + p -> n - p < m]. + For instance [1<0+2] but not [1-2<0]. *) + +Theorem lt_sub_lt_add_l : forall n m p, n - m < p -> n < m + p. +Proof. +intros n m p. rewrite add_comm; apply lt_sub_lt_add_r. +Qed. + +Theorem le_add_le_sub_r : forall n m p, n + p <= m -> n <= m - p. +Proof. +intros n m p LE. +apply (add_le_mono_r _ _ p). +rewrite sub_add. assumption. +apply le_trans with (n+p); trivial. +rewrite <- (add_0_l p) at 1. rewrite <- add_le_mono_r. apply le_0_l. +Qed. + +(** Unfortunately, we do not have [n <= m - p -> n + p <= m]. + For instance [0<=1-2] but not [2+0<=1]. *) + +Theorem le_add_le_sub_l : forall n m p, n + p <= m -> p <= m - n. +Proof. +intros n m p. rewrite add_comm; apply le_add_le_sub_r. +Qed. + +Theorem lt_add_lt_sub_r : forall n m p, n + p < m <-> n < m - p. +Proof. +intros n m p. +destruct (le_ge_cases p m) as [LE|GE]. +rewrite <- (sub_add p m) at 1 by assumption. +now rewrite <- add_lt_mono_r. +assert (GE' := GE). rewrite <- sub_0_le in GE'; rewrite GE'. +split; intros LT. +elim (lt_irrefl m). apply le_lt_trans with (n+p); trivial. + rewrite <- (add_0_l m). apply add_le_mono. apply le_0_l. assumption. +now elim (nlt_0_r n). +Qed. + +Theorem lt_add_lt_sub_l : forall n m p, n + p < m <-> p < m - n. +Proof. +intros n m p. rewrite add_comm; apply lt_add_lt_sub_r. +Qed. + (** Sub and mul *) -Theorem mul_pred_r : forall n m : N, n * (P m) == n * m - n. +Theorem mul_pred_r : forall n m, n * (P m) == n * m - n. Proof. intros n m; cases m. now rewrite pred_0, mul_0_r, sub_0_l. @@ -155,7 +216,7 @@ now rewrite sub_diag, add_0_r. now apply eq_le_incl. Qed. -Theorem mul_sub_distr_r : forall n m p : N, (n - m) * p == n * p - m * p. +Theorem mul_sub_distr_r : forall n m p, (n - m) * p == n * p - m * p. Proof. intros n m p; induct n. now rewrite sub_0_l, mul_0_l, sub_0_l. @@ -170,11 +231,72 @@ setoid_replace ((S n * p) - m * p) with 0 by (apply <- sub_0_le; now apply mul_l apply mul_0_l. Qed. -Theorem mul_sub_distr_l : forall n m p : N, p * (n - m) == p * n - p * m. +Theorem mul_sub_distr_l : forall n m p, p * (n - m) == p * n - p * m. Proof. intros n m p; rewrite (mul_comm p (n - m)), (mul_comm p n), (mul_comm p m). apply mul_sub_distr_r. Qed. +(** Alternative definitions of [<=] and [<] based on [+] *) + +Definition le_alt n m := exists p, p + n == m. +Definition lt_alt n m := exists p, S p + n == m. + +Lemma le_equiv : forall n m, le_alt n m <-> n <= m. +Proof. +split. +intros (p,H). rewrite <- H, add_comm. apply le_add_r. +intro H. exists (m-n). now apply sub_add. +Qed. + +Lemma lt_equiv : forall n m, lt_alt n m <-> n < m. +Proof. +split. +intros (p,H). rewrite <- H, add_succ_l, lt_succ_r, add_comm. apply le_add_r. +intro H. exists (m-S n). rewrite add_succ_l, <- add_succ_r. +apply sub_add. now rewrite le_succ_l. +Qed. + +Instance le_alt_wd : Proper (eq==>eq==>iff) le_alt. +Proof. + intros x x' Hx y y' Hy; unfold le_alt. + setoid_rewrite Hx. setoid_rewrite Hy. auto with *. +Qed. + +Instance lt_alt_wd : Proper (eq==>eq==>iff) lt_alt. +Proof. + intros x x' Hx y y' Hy; unfold lt_alt. + setoid_rewrite Hx. setoid_rewrite Hy. auto with *. +Qed. + +(** With these alternative definition, the dichotomy: + +[forall n m, n <= m \/ m <= n] + +becomes: + +[forall n m, (exists p, p + n == m) \/ (exists p, p + m == n)] + +We will need this in the proof of induction principle for integers +constructed as pairs of natural numbers. This formula can be proved +from know properties of [<=]. However, it can also be done directly. *) + +Theorem le_alt_dichotomy : forall n m, le_alt n m \/ le_alt m n. +Proof. +intros n m; induct n. +left; exists m; apply add_0_r. +intros n IH. +destruct IH as [[p H] | [p H]]. +destruct (zero_or_succ p) as [H1 | [p' H1]]; rewrite H1 in H. +rewrite add_0_l in H. right; exists (S 0); rewrite H, add_succ_l; + now rewrite add_0_l. +left; exists p'; rewrite add_succ_r; now rewrite add_succ_l in H. +right; exists (S p). rewrite add_succ_l; now rewrite H. +Qed. + +Theorem add_dichotomy : + forall n m, (exists p, p + n == m) \/ (exists p, p + m == n). +Proof. exact le_alt_dichotomy. Qed. + End NSubPropFunct. |