diff options
Diffstat (limited to 'theories/Numbers/Natural/Abstract/NAdd.v')
| -rw-r--r-- | theories/Numbers/Natural/Abstract/NAdd.v | 109 |
1 files changed, 17 insertions, 92 deletions
diff --git a/theories/Numbers/Natural/Abstract/NAdd.v b/theories/Numbers/Natural/Abstract/NAdd.v index 91ae5b70..9f0b54a6 100644 --- a/theories/Numbers/Natural/Abstract/NAdd.v +++ b/theories/Numbers/Natural/Abstract/NAdd.v @@ -8,74 +8,30 @@ (* Evgeny Makarov, INRIA, 2007 *) (************************************************************************) -(*i $Id: NAdd.v 11674 2008-12-12 19:48:40Z letouzey $ i*) +(*i $Id$ i*) Require Export NBase. -Module NAddPropFunct (Import NAxiomsMod : NAxiomsSig). -Module Export NBasePropMod := NBasePropFunct NAxiomsMod. +Module NAddPropFunct (Import N : NAxiomsSig'). +Include NBasePropFunct N. -Open Local Scope NatScope. +(** For theorems about [add] that are both valid for [N] and [Z], see [NZAdd] *) +(** Now comes theorems valid for natural numbers but not for Z *) -Theorem add_wd : - forall n1 n2 : N, n1 == n2 -> forall m1 m2 : N, m1 == m2 -> n1 + m1 == n2 + m2. -Proof NZadd_wd. - -Theorem add_0_l : forall n : N, 0 + n == n. -Proof NZadd_0_l. - -Theorem add_succ_l : forall n m : N, (S n) + m == S (n + m). -Proof NZadd_succ_l. - -(** Theorems that are valid for both natural numbers and integers *) - -Theorem add_0_r : forall n : N, n + 0 == n. -Proof NZadd_0_r. - -Theorem add_succ_r : forall n m : N, n + S m == S (n + m). -Proof NZadd_succ_r. - -Theorem add_comm : forall n m : N, n + m == m + n. -Proof NZadd_comm. - -Theorem add_assoc : forall n m p : N, n + (m + p) == (n + m) + p. -Proof NZadd_assoc. - -Theorem add_shuffle1 : forall n m p q : N, (n + m) + (p + q) == (n + p) + (m + q). -Proof NZadd_shuffle1. - -Theorem add_shuffle2 : forall n m p q : N, (n + m) + (p + q) == (n + q) + (m + p). -Proof NZadd_shuffle2. - -Theorem add_1_l : forall n : N, 1 + n == S n. -Proof NZadd_1_l. - -Theorem add_1_r : forall n : N, n + 1 == S n. -Proof NZadd_1_r. - -Theorem add_cancel_l : forall n m p : N, p + n == p + m <-> n == m. -Proof NZadd_cancel_l. - -Theorem add_cancel_r : forall n m p : N, n + p == m + p <-> n == m. -Proof NZadd_cancel_r. - -(* Theorems that are valid for natural numbers but cannot be proved for Z *) - -Theorem eq_add_0 : forall n m : N, n + m == 0 <-> n == 0 /\ m == 0. +Theorem eq_add_0 : forall n m, n + m == 0 <-> n == 0 /\ m == 0. Proof. intros n m; induct n. -(* The next command does not work with the axiom add_0_l from NAddSig *) -rewrite add_0_l. intuition reflexivity. -intros n IH. rewrite add_succ_l. -setoid_replace (S (n + m) == 0) with False using relation iff by +nzsimpl; intuition. +intros n IH. nzsimpl. +setoid_replace (S (n + m) == 0) with False by (apply -> neg_false; apply neq_succ_0). -setoid_replace (S n == 0) with False using relation iff by +setoid_replace (S n == 0) with False by (apply -> neg_false; apply neq_succ_0). tauto. Qed. Theorem eq_add_succ : - forall n m : N, (exists p : N, n + m == S p) <-> - (exists n' : N, n == S n') \/ (exists m' : N, m == S m'). + forall n m, (exists p, n + m == S p) <-> + (exists n', n == S n') \/ (exists m', m == S m'). Proof. intros n m; cases n. split; intro H. @@ -88,11 +44,11 @@ left; now exists n. exists (n + m); now rewrite add_succ_l. Qed. -Theorem eq_add_1 : forall n m : N, +Theorem eq_add_1 : forall n m, n + m == 1 -> n == 1 /\ m == 0 \/ n == 0 /\ m == 1. Proof. intros n m H. -assert (H1 : exists p : N, n + m == S p) by now exists 0. +assert (H1 : exists p, n + m == S p) by now exists 0. apply -> eq_add_succ in H1. destruct H1 as [[n' H1] | [m' H1]]. left. rewrite H1 in H; rewrite add_succ_l in H; apply succ_inj in H. apply -> eq_add_0 in H. destruct H as [H2 H3]; rewrite H2 in H1; now split. @@ -100,7 +56,7 @@ right. rewrite H1 in H; rewrite add_succ_r in H; apply succ_inj in H. apply -> eq_add_0 in H. destruct H as [H2 H3]; rewrite H3 in H1; now split. Qed. -Theorem succ_add_discr : forall n m : N, m ~= S (n + m). +Theorem succ_add_discr : forall n m, m ~= S (n + m). Proof. intro n; induct m. apply neq_sym. apply neq_succ_0. @@ -108,49 +64,18 @@ intros m IH H. apply succ_inj in H. rewrite add_succ_r in H. unfold not in IH; now apply IH. Qed. -Theorem add_pred_l : forall n m : N, n ~= 0 -> P n + m == P (n + m). +Theorem add_pred_l : forall n m, n ~= 0 -> P n + m == P (n + m). Proof. intros n m; cases n. intro H; now elim H. intros n IH; rewrite add_succ_l; now do 2 rewrite pred_succ. Qed. -Theorem add_pred_r : forall n m : N, m ~= 0 -> n + P m == P (n + m). +Theorem add_pred_r : forall n m, m ~= 0 -> n + P m == P (n + m). Proof. intros n m H; rewrite (add_comm n (P m)); rewrite (add_comm n m); now apply add_pred_l. Qed. -(* One could define n <= m as exists p : N, p + n == m. Then we have -dichotomy: - -forall n m : N, n <= m \/ m <= n, - -i.e., - -forall n m : N, (exists p : N, p + n == m) \/ (exists p : N, p + m == n) (1) - -We will need (1) in the proof of induction principle for integers -constructed as pairs of natural numbers. The formula (1) can be proved -using properties of order and truncated subtraction. Thus, p would be -m - n or n - m and (1) would hold by theorem sub_add from Sub.v -depending on whether n <= m or m <= n. However, in proving induction -for integers constructed from natural numbers we do not need to -require implementations of order and sub; it is enough to prove (1) -here. *) - -Theorem add_dichotomy : - forall n m : N, (exists p : N, p + n == m) \/ (exists p : N, p + 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; rewrite 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. - End NAddPropFunct. |