From f82bfc64fca9fb46136d7aa26c09d64cde0432d2 Mon Sep 17 00:00:00 2001 From: letouzey Date: Mon, 2 Jun 2008 23:26:13 +0000 Subject: In abstract parts of theories/Numbers, plus/times becomes add/mul, for increased consistency with bignums parts (commit part I: content of files) git-svn-id: svn+ssh://scm.gforge.inria.fr/svn/coq/trunk@11039 85f007b7-540e-0410-9357-904b9bb8a0f7 --- theories/Numbers/NatInt/NZAxioms.v | 20 ++-- theories/Numbers/NatInt/NZPlus.v | 58 +++++----- theories/Numbers/NatInt/NZPlusOrder.v | 98 ++++++++--------- theories/Numbers/NatInt/NZTimes.v | 58 +++++----- theories/Numbers/NatInt/NZTimesOrder.v | 196 ++++++++++++++++----------------- 5 files changed, 215 insertions(+), 215 deletions(-) (limited to 'theories/Numbers/NatInt') diff --git a/theories/Numbers/NatInt/NZAxioms.v b/theories/Numbers/NatInt/NZAxioms.v index ef19069955..516cf5b42b 100644 --- a/theories/Numbers/NatInt/NZAxioms.v +++ b/theories/Numbers/NatInt/NZAxioms.v @@ -19,9 +19,9 @@ Parameter Inline NZeq : NZ -> NZ -> Prop. Parameter Inline NZ0 : NZ. Parameter Inline NZsucc : NZ -> NZ. Parameter Inline NZpred : NZ -> NZ. -Parameter Inline NZplus : NZ -> NZ -> NZ. +Parameter Inline NZadd : NZ -> NZ -> NZ. Parameter Inline NZminus : NZ -> NZ -> NZ. -Parameter Inline NZtimes : NZ -> NZ -> NZ. +Parameter Inline NZmul : NZ -> NZ -> NZ. (* Unary minus (opp) is not defined on natural numbers, so we have it for integers only *) @@ -35,9 +35,9 @@ as NZeq_rel. Add Morphism NZsucc with signature NZeq ==> NZeq as NZsucc_wd. Add Morphism NZpred with signature NZeq ==> NZeq as NZpred_wd. -Add Morphism NZplus with signature NZeq ==> NZeq ==> NZeq as NZplus_wd. +Add Morphism NZadd with signature NZeq ==> NZeq ==> NZeq as NZadd_wd. Add Morphism NZminus with signature NZeq ==> NZeq ==> NZeq as NZminus_wd. -Add Morphism NZtimes with signature NZeq ==> NZeq ==> NZeq as NZtimes_wd. +Add Morphism NZmul with signature NZeq ==> NZeq ==> NZeq as NZmul_wd. Delimit Scope NatIntScope with NatInt. Open Local Scope NatIntScope. @@ -47,9 +47,9 @@ Notation "0" := NZ0 : NatIntScope. Notation S := NZsucc. Notation P := NZpred. Notation "1" := (S 0) : NatIntScope. -Notation "x + y" := (NZplus x y) : NatIntScope. +Notation "x + y" := (NZadd x y) : NatIntScope. Notation "x - y" := (NZminus x y) : NatIntScope. -Notation "x * y" := (NZtimes x y) : NatIntScope. +Notation "x * y" := (NZmul x y) : NatIntScope. Axiom NZpred_succ : forall n : NZ, P (S n) == n. @@ -57,14 +57,14 @@ Axiom NZinduction : forall A : NZ -> Prop, predicate_wd NZeq A -> A 0 -> (forall n : NZ, A n <-> A (S n)) -> forall n : NZ, A n. -Axiom NZplus_0_l : forall n : NZ, 0 + n == n. -Axiom NZplus_succ_l : forall n m : NZ, (S n) + m == S (n + m). +Axiom NZadd_0_l : forall n : NZ, 0 + n == n. +Axiom NZadd_succ_l : forall n m : NZ, (S n) + m == S (n + m). Axiom NZminus_0_r : forall n : NZ, n - 0 == n. Axiom NZminus_succ_r : forall n m : NZ, n - (S m) == P (n - m). -Axiom NZtimes_0_l : forall n : NZ, 0 * n == 0. -Axiom NZtimes_succ_l : forall n m : NZ, S n * m == n * m + m. +Axiom NZmul_0_l : forall n : NZ, 0 * n == 0. +Axiom NZmul_succ_l : forall n m : NZ, S n * m == n * m + m. End NZAxiomsSig. diff --git a/theories/Numbers/NatInt/NZPlus.v b/theories/Numbers/NatInt/NZPlus.v index 673b819ba4..6fb72ed4a9 100644 --- a/theories/Numbers/NatInt/NZPlus.v +++ b/theories/Numbers/NatInt/NZPlus.v @@ -17,69 +17,69 @@ Module NZPlusPropFunct (Import NZAxiomsMod : NZAxiomsSig). Module Export NZBasePropMod := NZBasePropFunct NZAxiomsMod. Open Local Scope NatIntScope. -Theorem NZplus_0_r : forall n : NZ, n + 0 == n. +Theorem NZadd_0_r : forall n : NZ, n + 0 == n. Proof. -NZinduct n. now rewrite NZplus_0_l. -intro. rewrite NZplus_succ_l. now rewrite NZsucc_inj_wd. +NZinduct n. now rewrite NZadd_0_l. +intro. rewrite NZadd_succ_l. now rewrite NZsucc_inj_wd. Qed. -Theorem NZplus_succ_r : forall n m : NZ, n + S m == S (n + m). +Theorem NZadd_succ_r : forall n m : NZ, n + S m == S (n + m). Proof. intros n m; NZinduct n. -now do 2 rewrite NZplus_0_l. -intro. repeat rewrite NZplus_succ_l. now rewrite NZsucc_inj_wd. +now do 2 rewrite NZadd_0_l. +intro. repeat rewrite NZadd_succ_l. now rewrite NZsucc_inj_wd. Qed. -Theorem NZplus_comm : forall n m : NZ, n + m == m + n. +Theorem NZadd_comm : forall n m : NZ, n + m == m + n. Proof. intros n m; NZinduct n. -rewrite NZplus_0_l; now rewrite NZplus_0_r. -intros n. rewrite NZplus_succ_l; rewrite NZplus_succ_r. now rewrite NZsucc_inj_wd. +rewrite NZadd_0_l; now rewrite NZadd_0_r. +intros n. rewrite NZadd_succ_l; rewrite NZadd_succ_r. now rewrite NZsucc_inj_wd. Qed. -Theorem NZplus_1_l : forall n : NZ, 1 + n == S n. +Theorem NZadd_1_l : forall n : NZ, 1 + n == S n. Proof. -intro n; rewrite NZplus_succ_l; now rewrite NZplus_0_l. +intro n; rewrite NZadd_succ_l; now rewrite NZadd_0_l. Qed. -Theorem NZplus_1_r : forall n : NZ, n + 1 == S n. +Theorem NZadd_1_r : forall n : NZ, n + 1 == S n. Proof. -intro n; rewrite NZplus_comm; apply NZplus_1_l. +intro n; rewrite NZadd_comm; apply NZadd_1_l. Qed. -Theorem NZplus_assoc : forall n m p : NZ, n + (m + p) == (n + m) + p. +Theorem NZadd_assoc : forall n m p : NZ, n + (m + p) == (n + m) + p. Proof. intros n m p; NZinduct n. -now do 2 rewrite NZplus_0_l. -intro. do 3 rewrite NZplus_succ_l. now rewrite NZsucc_inj_wd. +now do 2 rewrite NZadd_0_l. +intro. do 3 rewrite NZadd_succ_l. now rewrite NZsucc_inj_wd. Qed. -Theorem NZplus_shuffle1 : forall n m p q : NZ, (n + m) + (p + q) == (n + p) + (m + q). +Theorem NZadd_shuffle1 : forall n m p q : NZ, (n + m) + (p + q) == (n + p) + (m + q). Proof. intros n m p q. -rewrite <- (NZplus_assoc n m (p + q)). rewrite (NZplus_comm m (p + q)). -rewrite <- (NZplus_assoc p q m). rewrite (NZplus_assoc n p (q + m)). -now rewrite (NZplus_comm q m). +rewrite <- (NZadd_assoc n m (p + q)). rewrite (NZadd_comm m (p + q)). +rewrite <- (NZadd_assoc p q m). rewrite (NZadd_assoc n p (q + m)). +now rewrite (NZadd_comm q m). Qed. -Theorem NZplus_shuffle2 : forall n m p q : NZ, (n + m) + (p + q) == (n + q) + (m + p). +Theorem NZadd_shuffle2 : forall n m p q : NZ, (n + m) + (p + q) == (n + q) + (m + p). Proof. intros n m p q. -rewrite <- (NZplus_assoc n m (p + q)). rewrite (NZplus_assoc m p q). -rewrite (NZplus_comm (m + p) q). now rewrite <- (NZplus_assoc n q (m + p)). +rewrite <- (NZadd_assoc n m (p + q)). rewrite (NZadd_assoc m p q). +rewrite (NZadd_comm (m + p) q). now rewrite <- (NZadd_assoc n q (m + p)). Qed. -Theorem NZplus_cancel_l : forall n m p : NZ, p + n == p + m <-> n == m. +Theorem NZadd_cancel_l : forall n m p : NZ, p + n == p + m <-> n == m. Proof. intros n m p; NZinduct p. -now do 2 rewrite NZplus_0_l. -intro p. do 2 rewrite NZplus_succ_l. now rewrite NZsucc_inj_wd. +now do 2 rewrite NZadd_0_l. +intro p. do 2 rewrite NZadd_succ_l. now rewrite NZsucc_inj_wd. Qed. -Theorem NZplus_cancel_r : forall n m p : NZ, n + p == m + p <-> n == m. +Theorem NZadd_cancel_r : forall n m p : NZ, n + p == m + p <-> n == m. Proof. -intros n m p. rewrite (NZplus_comm n p); rewrite (NZplus_comm m p). -apply NZplus_cancel_l. +intros n m p. rewrite (NZadd_comm n p); rewrite (NZadd_comm m p). +apply NZadd_cancel_l. Qed. Theorem NZminus_1_r : forall n : NZ, n - 1 == P n. diff --git a/theories/Numbers/NatInt/NZPlusOrder.v b/theories/Numbers/NatInt/NZPlusOrder.v index 45148bc20c..00d178c0d9 100644 --- a/theories/Numbers/NatInt/NZPlusOrder.v +++ b/theories/Numbers/NatInt/NZPlusOrder.v @@ -17,149 +17,149 @@ Module NZPlusOrderPropFunct (Import NZOrdAxiomsMod : NZOrdAxiomsSig). Module Export NZOrderPropMod := NZOrderPropFunct NZOrdAxiomsMod. Open Local Scope NatIntScope. -Theorem NZplus_lt_mono_l : forall n m p : NZ, n < m <-> p + n < p + m. +Theorem NZadd_lt_mono_l : forall n m p : NZ, n < m <-> p + n < p + m. Proof. intros n m p; NZinduct p. -now do 2 rewrite NZplus_0_l. -intro p. do 2 rewrite NZplus_succ_l. now rewrite <- NZsucc_lt_mono. +now do 2 rewrite NZadd_0_l. +intro p. do 2 rewrite NZadd_succ_l. now rewrite <- NZsucc_lt_mono. Qed. -Theorem NZplus_lt_mono_r : forall n m p : NZ, n < m <-> n + p < m + p. +Theorem NZadd_lt_mono_r : forall n m p : NZ, n < m <-> n + p < m + p. Proof. intros n m p. -rewrite (NZplus_comm n p); rewrite (NZplus_comm m p); apply NZplus_lt_mono_l. +rewrite (NZadd_comm n p); rewrite (NZadd_comm m p); apply NZadd_lt_mono_l. Qed. -Theorem NZplus_lt_mono : forall n m p q : NZ, n < m -> p < q -> n + p < m + q. +Theorem NZadd_lt_mono : forall n m p q : NZ, n < m -> p < q -> n + p < m + q. Proof. intros n m p q H1 H2. apply NZlt_trans with (m + p); -[now apply -> NZplus_lt_mono_r | now apply -> NZplus_lt_mono_l]. +[now apply -> NZadd_lt_mono_r | now apply -> NZadd_lt_mono_l]. Qed. -Theorem NZplus_le_mono_l : forall n m p : NZ, n <= m <-> p + n <= p + m. +Theorem NZadd_le_mono_l : forall n m p : NZ, n <= m <-> p + n <= p + m. Proof. intros n m p; NZinduct p. -now do 2 rewrite NZplus_0_l. -intro p. do 2 rewrite NZplus_succ_l. now rewrite <- NZsucc_le_mono. +now do 2 rewrite NZadd_0_l. +intro p. do 2 rewrite NZadd_succ_l. now rewrite <- NZsucc_le_mono. Qed. -Theorem NZplus_le_mono_r : forall n m p : NZ, n <= m <-> n + p <= m + p. +Theorem NZadd_le_mono_r : forall n m p : NZ, n <= m <-> n + p <= m + p. Proof. intros n m p. -rewrite (NZplus_comm n p); rewrite (NZplus_comm m p); apply NZplus_le_mono_l. +rewrite (NZadd_comm n p); rewrite (NZadd_comm m p); apply NZadd_le_mono_l. Qed. -Theorem NZplus_le_mono : forall n m p q : NZ, n <= m -> p <= q -> n + p <= m + q. +Theorem NZadd_le_mono : forall n m p q : NZ, n <= m -> p <= q -> n + p <= m + q. Proof. intros n m p q H1 H2. apply NZle_trans with (m + p); -[now apply -> NZplus_le_mono_r | now apply -> NZplus_le_mono_l]. +[now apply -> NZadd_le_mono_r | now apply -> NZadd_le_mono_l]. Qed. -Theorem NZplus_lt_le_mono : forall n m p q : NZ, n < m -> p <= q -> n + p < m + q. +Theorem NZadd_lt_le_mono : forall n m p q : NZ, n < m -> p <= q -> n + p < m + q. Proof. intros n m p q H1 H2. apply NZlt_le_trans with (m + p); -[now apply -> NZplus_lt_mono_r | now apply -> NZplus_le_mono_l]. +[now apply -> NZadd_lt_mono_r | now apply -> NZadd_le_mono_l]. Qed. -Theorem NZplus_le_lt_mono : forall n m p q : NZ, n <= m -> p < q -> n + p < m + q. +Theorem NZadd_le_lt_mono : forall n m p q : NZ, n <= m -> p < q -> n + p < m + q. Proof. intros n m p q H1 H2. apply NZle_lt_trans with (m + p); -[now apply -> NZplus_le_mono_r | now apply -> NZplus_lt_mono_l]. +[now apply -> NZadd_le_mono_r | now apply -> NZadd_lt_mono_l]. Qed. -Theorem NZplus_pos_pos : forall n m : NZ, 0 < n -> 0 < m -> 0 < n + m. +Theorem NZadd_pos_pos : forall n m : NZ, 0 < n -> 0 < m -> 0 < n + m. Proof. -intros n m H1 H2. rewrite <- (NZplus_0_l 0). now apply NZplus_lt_mono. +intros n m H1 H2. rewrite <- (NZadd_0_l 0). now apply NZadd_lt_mono. Qed. -Theorem NZplus_pos_nonneg : forall n m : NZ, 0 < n -> 0 <= m -> 0 < n + m. +Theorem NZadd_pos_nonneg : forall n m : NZ, 0 < n -> 0 <= m -> 0 < n + m. Proof. -intros n m H1 H2. rewrite <- (NZplus_0_l 0). now apply NZplus_lt_le_mono. +intros n m H1 H2. rewrite <- (NZadd_0_l 0). now apply NZadd_lt_le_mono. Qed. -Theorem NZplus_nonneg_pos : forall n m : NZ, 0 <= n -> 0 < m -> 0 < n + m. +Theorem NZadd_nonneg_pos : forall n m : NZ, 0 <= n -> 0 < m -> 0 < n + m. Proof. -intros n m H1 H2. rewrite <- (NZplus_0_l 0). now apply NZplus_le_lt_mono. +intros n m H1 H2. rewrite <- (NZadd_0_l 0). now apply NZadd_le_lt_mono. Qed. -Theorem NZplus_nonneg_nonneg : forall n m : NZ, 0 <= n -> 0 <= m -> 0 <= n + m. +Theorem NZadd_nonneg_nonneg : forall n m : NZ, 0 <= n -> 0 <= m -> 0 <= n + m. Proof. -intros n m H1 H2. rewrite <- (NZplus_0_l 0). now apply NZplus_le_mono. +intros n m H1 H2. rewrite <- (NZadd_0_l 0). now apply NZadd_le_mono. Qed. -Theorem NZlt_plus_pos_l : forall n m : NZ, 0 < n -> m < n + m. +Theorem NZlt_add_pos_l : forall n m : NZ, 0 < n -> m < n + m. Proof. -intros n m H. apply -> (NZplus_lt_mono_r 0 n m) in H. -now rewrite NZplus_0_l in H. +intros n m H. apply -> (NZadd_lt_mono_r 0 n m) in H. +now rewrite NZadd_0_l in H. Qed. -Theorem NZlt_plus_pos_r : forall n m : NZ, 0 < n -> m < m + n. +Theorem NZlt_add_pos_r : forall n m : NZ, 0 < n -> m < m + n. Proof. -intros; rewrite NZplus_comm; now apply NZlt_plus_pos_l. +intros; rewrite NZadd_comm; now apply NZlt_add_pos_l. Qed. -Theorem NZle_lt_plus_lt : forall n m p q : NZ, n <= m -> p + m < q + n -> p < q. +Theorem NZle_lt_add_lt : forall n m p q : NZ, n <= m -> p + m < q + n -> p < q. Proof. intros n m p q H1 H2. destruct (NZle_gt_cases q p); [| assumption]. -pose proof (NZplus_le_mono q p n m H H1) as H3. apply <- NZnle_gt in H2. +pose proof (NZadd_le_mono q p n m H H1) as H3. apply <- NZnle_gt in H2. false_hyp H3 H2. Qed. -Theorem NZlt_le_plus_lt : forall n m p q : NZ, n < m -> p + m <= q + n -> p < q. +Theorem NZlt_le_add_lt : forall n m p q : NZ, n < m -> p + m <= q + n -> p < q. Proof. intros n m p q H1 H2. destruct (NZle_gt_cases q p); [| assumption]. -pose proof (NZplus_le_lt_mono q p n m H H1) as H3. apply <- NZnle_gt in H3. +pose proof (NZadd_le_lt_mono q p n m H H1) as H3. apply <- NZnle_gt in H3. false_hyp H2 H3. Qed. -Theorem NZle_le_plus_le : forall n m p q : NZ, n <= m -> p + m <= q + n -> p <= q. +Theorem NZle_le_add_le : forall n m p q : NZ, n <= m -> p + m <= q + n -> p <= q. Proof. intros n m p q H1 H2. destruct (NZle_gt_cases p q); [assumption |]. -pose proof (NZplus_lt_le_mono q p n m H H1) as H3. apply <- NZnle_gt in H3. +pose proof (NZadd_lt_le_mono q p n m H H1) as H3. apply <- NZnle_gt in H3. false_hyp H2 H3. Qed. -Theorem NZplus_lt_cases : forall n m p q : NZ, n + m < p + q -> n < p \/ m < q. +Theorem NZadd_lt_cases : forall n m p q : NZ, n + m < p + q -> n < p \/ m < q. Proof. intros n m p q H; destruct (NZle_gt_cases p n) as [H1 | H1]. destruct (NZle_gt_cases q m) as [H2 | H2]. -pose proof (NZplus_le_mono p n q m H1 H2) as H3. apply -> NZle_ngt in H3. +pose proof (NZadd_le_mono p n q m H1 H2) as H3. apply -> NZle_ngt in H3. false_hyp H H3. now right. now left. Qed. -Theorem NZplus_le_cases : forall n m p q : NZ, n + m <= p + q -> n <= p \/ m <= q. +Theorem NZadd_le_cases : forall n m p q : NZ, n + m <= p + q -> n <= p \/ m <= q. Proof. intros n m p q H. destruct (NZle_gt_cases n p) as [H1 | H1]. now left. destruct (NZle_gt_cases m q) as [H2 | H2]. now right. -assert (H3 : p + q < n + m) by now apply NZplus_lt_mono. +assert (H3 : p + q < n + m) by now apply NZadd_lt_mono. apply -> NZle_ngt in H. false_hyp H3 H. Qed. -Theorem NZplus_neg_cases : forall n m : NZ, n + m < 0 -> n < 0 \/ m < 0. +Theorem NZadd_neg_cases : forall n m : NZ, n + m < 0 -> n < 0 \/ m < 0. Proof. -intros n m H; apply NZplus_lt_cases; now rewrite NZplus_0_l. +intros n m H; apply NZadd_lt_cases; now rewrite NZadd_0_l. Qed. -Theorem NZplus_pos_cases : forall n m : NZ, 0 < n + m -> 0 < n \/ 0 < m. +Theorem NZadd_pos_cases : forall n m : NZ, 0 < n + m -> 0 < n \/ 0 < m. Proof. -intros n m H; apply NZplus_lt_cases; now rewrite NZplus_0_l. +intros n m H; apply NZadd_lt_cases; now rewrite NZadd_0_l. Qed. -Theorem NZplus_nonpos_cases : forall n m : NZ, n + m <= 0 -> n <= 0 \/ m <= 0. +Theorem NZadd_nonpos_cases : forall n m : NZ, n + m <= 0 -> n <= 0 \/ m <= 0. Proof. -intros n m H; apply NZplus_le_cases; now rewrite NZplus_0_l. +intros n m H; apply NZadd_le_cases; now rewrite NZadd_0_l. Qed. -Theorem NZplus_nonneg_cases : forall n m : NZ, 0 <= n + m -> 0 <= n \/ 0 <= m. +Theorem NZadd_nonneg_cases : forall n m : NZ, 0 <= n + m -> 0 <= n \/ 0 <= m. Proof. -intros n m H; apply NZplus_le_cases; now rewrite NZplus_0_l. +intros n m H; apply NZadd_le_cases; now rewrite NZadd_0_l. Qed. End NZPlusOrderPropFunct. diff --git a/theories/Numbers/NatInt/NZTimes.v b/theories/Numbers/NatInt/NZTimes.v index 7503ddce22..9f93e0a1bf 100644 --- a/theories/Numbers/NatInt/NZTimes.v +++ b/theories/Numbers/NatInt/NZTimes.v @@ -17,63 +17,63 @@ Module NZTimesPropFunct (Import NZAxiomsMod : NZAxiomsSig). Module Export NZPlusPropMod := NZPlusPropFunct NZAxiomsMod. Open Local Scope NatIntScope. -Theorem NZtimes_0_r : forall n : NZ, n * 0 == 0. +Theorem NZmul_0_r : forall n : NZ, n * 0 == 0. Proof. NZinduct n. -now rewrite NZtimes_0_l. -intro. rewrite NZtimes_succ_l. now rewrite NZplus_0_r. +now rewrite NZmul_0_l. +intro. rewrite NZmul_succ_l. now rewrite NZadd_0_r. Qed. -Theorem NZtimes_succ_r : forall n m : NZ, n * (S m) == n * m + n. +Theorem NZmul_succ_r : forall n m : NZ, n * (S m) == n * m + n. Proof. intros n m; NZinduct n. -do 2 rewrite NZtimes_0_l; now rewrite NZplus_0_l. -intro n. do 2 rewrite NZtimes_succ_l. do 2 rewrite NZplus_succ_r. -rewrite NZsucc_inj_wd. rewrite <- (NZplus_assoc (n * m) m n). -rewrite (NZplus_comm m n). rewrite NZplus_assoc. -now rewrite NZplus_cancel_r. +do 2 rewrite NZmul_0_l; now rewrite NZadd_0_l. +intro n. do 2 rewrite NZmul_succ_l. do 2 rewrite NZadd_succ_r. +rewrite NZsucc_inj_wd. rewrite <- (NZadd_assoc (n * m) m n). +rewrite (NZadd_comm m n). rewrite NZadd_assoc. +now rewrite NZadd_cancel_r. Qed. -Theorem NZtimes_comm : forall n m : NZ, n * m == m * n. +Theorem NZmul_comm : forall n m : NZ, n * m == m * n. Proof. intros n m; NZinduct n. -rewrite NZtimes_0_l; now rewrite NZtimes_0_r. -intro. rewrite NZtimes_succ_l; rewrite NZtimes_succ_r. now rewrite NZplus_cancel_r. +rewrite NZmul_0_l; now rewrite NZmul_0_r. +intro. rewrite NZmul_succ_l; rewrite NZmul_succ_r. now rewrite NZadd_cancel_r. Qed. -Theorem NZtimes_plus_distr_r : forall n m p : NZ, (n + m) * p == n * p + m * p. +Theorem NZmul_add_distr_r : forall n m p : NZ, (n + m) * p == n * p + m * p. Proof. intros n m p; NZinduct n. -rewrite NZtimes_0_l. now do 2 rewrite NZplus_0_l. -intro n. rewrite NZplus_succ_l. do 2 rewrite NZtimes_succ_l. -rewrite <- (NZplus_assoc (n * p) p (m * p)). -rewrite (NZplus_comm p (m * p)). rewrite (NZplus_assoc (n * p) (m * p) p). -now rewrite NZplus_cancel_r. +rewrite NZmul_0_l. now do 2 rewrite NZadd_0_l. +intro n. rewrite NZadd_succ_l. do 2 rewrite NZmul_succ_l. +rewrite <- (NZadd_assoc (n * p) p (m * p)). +rewrite (NZadd_comm p (m * p)). rewrite (NZadd_assoc (n * p) (m * p) p). +now rewrite NZadd_cancel_r. Qed. -Theorem NZtimes_plus_distr_l : forall n m p : NZ, n * (m + p) == n * m + n * p. +Theorem NZmul_add_distr_l : forall n m p : NZ, n * (m + p) == n * m + n * p. Proof. intros n m p. -rewrite (NZtimes_comm n (m + p)). rewrite (NZtimes_comm n m). -rewrite (NZtimes_comm n p). apply NZtimes_plus_distr_r. +rewrite (NZmul_comm n (m + p)). rewrite (NZmul_comm n m). +rewrite (NZmul_comm n p). apply NZmul_add_distr_r. Qed. -Theorem NZtimes_assoc : forall n m p : NZ, n * (m * p) == (n * m) * p. +Theorem NZmul_assoc : forall n m p : NZ, n * (m * p) == (n * m) * p. Proof. intros n m p; NZinduct n. -now do 3 rewrite NZtimes_0_l. -intro n. do 2 rewrite NZtimes_succ_l. rewrite NZtimes_plus_distr_r. -now rewrite NZplus_cancel_r. +now do 3 rewrite NZmul_0_l. +intro n. do 2 rewrite NZmul_succ_l. rewrite NZmul_add_distr_r. +now rewrite NZadd_cancel_r. Qed. -Theorem NZtimes_1_l : forall n : NZ, 1 * n == n. +Theorem NZmul_1_l : forall n : NZ, 1 * n == n. Proof. -intro n. rewrite NZtimes_succ_l; rewrite NZtimes_0_l. now rewrite NZplus_0_l. +intro n. rewrite NZmul_succ_l; rewrite NZmul_0_l. now rewrite NZadd_0_l. Qed. -Theorem NZtimes_1_r : forall n : NZ, n * 1 == n. +Theorem NZmul_1_r : forall n : NZ, n * 1 == n. Proof. -intro n; rewrite NZtimes_comm; apply NZtimes_1_l. +intro n; rewrite NZmul_comm; apply NZmul_1_l. Qed. End NZTimesPropFunct. diff --git a/theories/Numbers/NatInt/NZTimesOrder.v b/theories/Numbers/NatInt/NZTimesOrder.v index b48acc598d..ebb2a9b5d0 100644 --- a/theories/Numbers/NatInt/NZTimesOrder.v +++ b/theories/Numbers/NatInt/NZTimesOrder.v @@ -17,263 +17,263 @@ Module NZTimesOrderPropFunct (Import NZOrdAxiomsMod : NZOrdAxiomsSig). Module Export NZPlusOrderPropMod := NZPlusOrderPropFunct NZOrdAxiomsMod. Open Local Scope NatIntScope. -Theorem NZtimes_lt_pred : +Theorem NZmul_lt_pred : forall p q n m : NZ, S p == q -> (p * n < p * m <-> q * n + m < q * m + n). Proof. -intros p q n m H. rewrite <- H. do 2 rewrite NZtimes_succ_l. -rewrite <- (NZplus_assoc (p * n) n m). -rewrite <- (NZplus_assoc (p * m) m n). -rewrite (NZplus_comm n m). now rewrite <- NZplus_lt_mono_r. +intros p q n m H. rewrite <- H. do 2 rewrite NZmul_succ_l. +rewrite <- (NZadd_assoc (p * n) n m). +rewrite <- (NZadd_assoc (p * m) m n). +rewrite (NZadd_comm n m). now rewrite <- NZadd_lt_mono_r. Qed. -Theorem NZtimes_lt_mono_pos_l : forall p n m : NZ, 0 < p -> (n < m <-> p * n < p * m). +Theorem NZmul_lt_mono_pos_l : forall p n m : NZ, 0 < p -> (n < m <-> p * n < p * m). Proof. NZord_induct p. intros n m H; false_hyp H NZlt_irrefl. -intros p H IH n m H1. do 2 rewrite NZtimes_succ_l. +intros p H IH n m H1. do 2 rewrite NZmul_succ_l. le_elim H. assert (LR : forall n m : NZ, n < m -> p * n + n < p * m + m). -intros n1 m1 H2. apply NZplus_lt_mono; [now apply -> IH | assumption]. +intros n1 m1 H2. apply NZadd_lt_mono; [now apply -> IH | assumption]. split; [apply LR |]. intro H2. apply -> NZlt_dne; intro H3. apply <- NZle_ngt in H3. le_elim H3. apply NZlt_asymm in H2. apply H2. now apply LR. rewrite H3 in H2; false_hyp H2 NZlt_irrefl. -rewrite <- H; do 2 rewrite NZtimes_0_l; now do 2 rewrite NZplus_0_l. +rewrite <- H; do 2 rewrite NZmul_0_l; now do 2 rewrite NZadd_0_l. intros p H1 _ n m H2. apply NZlt_asymm in H1. false_hyp H2 H1. Qed. -Theorem NZtimes_lt_mono_pos_r : forall p n m : NZ, 0 < p -> (n < m <-> n * p < m * p). +Theorem NZmul_lt_mono_pos_r : forall p n m : NZ, 0 < p -> (n < m <-> n * p < m * p). Proof. intros p n m. -rewrite (NZtimes_comm n p); rewrite (NZtimes_comm m p). now apply NZtimes_lt_mono_pos_l. +rewrite (NZmul_comm n p); rewrite (NZmul_comm m p). now apply NZmul_lt_mono_pos_l. Qed. -Theorem NZtimes_lt_mono_neg_l : forall p n m : NZ, p < 0 -> (n < m <-> p * m < p * n). +Theorem NZmul_lt_mono_neg_l : forall p n m : NZ, p < 0 -> (n < m <-> p * m < p * n). Proof. NZord_induct p. intros n m H; false_hyp H NZlt_irrefl. intros p H1 _ n m H2. apply NZlt_succ_l in H2. apply <- NZnle_gt in H2. false_hyp H1 H2. intros p H IH n m H1. apply <- NZle_succ_l in H. le_elim H. assert (LR : forall n m : NZ, n < m -> p * m < p * n). -intros n1 m1 H2. apply (NZle_lt_plus_lt n1 m1). -now apply NZlt_le_incl. do 2 rewrite <- NZtimes_succ_l. now apply -> IH. +intros n1 m1 H2. apply (NZle_lt_add_lt n1 m1). +now apply NZlt_le_incl. do 2 rewrite <- NZmul_succ_l. now apply -> IH. split; [apply LR |]. intro H2. apply -> NZlt_dne; intro H3. apply <- NZle_ngt in H3. le_elim H3. apply NZlt_asymm in H2. apply H2. now apply LR. rewrite H3 in H2; false_hyp H2 NZlt_irrefl. -rewrite (NZtimes_lt_pred p (S p)) by reflexivity. -rewrite H; do 2 rewrite NZtimes_0_l; now do 2 rewrite NZplus_0_l. +rewrite (NZmul_lt_pred p (S p)) by reflexivity. +rewrite H; do 2 rewrite NZmul_0_l; now do 2 rewrite NZadd_0_l. Qed. -Theorem NZtimes_lt_mono_neg_r : forall p n m : NZ, p < 0 -> (n < m <-> m * p < n * p). +Theorem NZmul_lt_mono_neg_r : forall p n m : NZ, p < 0 -> (n < m <-> m * p < n * p). Proof. intros p n m. -rewrite (NZtimes_comm n p); rewrite (NZtimes_comm m p). now apply NZtimes_lt_mono_neg_l. +rewrite (NZmul_comm n p); rewrite (NZmul_comm m p). now apply NZmul_lt_mono_neg_l. Qed. -Theorem NZtimes_le_mono_nonneg_l : forall n m p : NZ, 0 <= p -> n <= m -> p * n <= p * m. +Theorem NZmul_le_mono_nonneg_l : forall n m p : NZ, 0 <= p -> n <= m -> p * n <= p * m. Proof. intros n m p H1 H2. le_elim H1. -le_elim H2. apply NZlt_le_incl. now apply -> NZtimes_lt_mono_pos_l. +le_elim H2. apply NZlt_le_incl. now apply -> NZmul_lt_mono_pos_l. apply NZeq_le_incl; now rewrite H2. -apply NZeq_le_incl; rewrite <- H1; now do 2 rewrite NZtimes_0_l. +apply NZeq_le_incl; rewrite <- H1; now do 2 rewrite NZmul_0_l. Qed. -Theorem NZtimes_le_mono_nonpos_l : forall n m p : NZ, p <= 0 -> n <= m -> p * m <= p * n. +Theorem NZmul_le_mono_nonpos_l : forall n m p : NZ, p <= 0 -> n <= m -> p * m <= p * n. Proof. intros n m p H1 H2. le_elim H1. -le_elim H2. apply NZlt_le_incl. now apply -> NZtimes_lt_mono_neg_l. +le_elim H2. apply NZlt_le_incl. now apply -> NZmul_lt_mono_neg_l. apply NZeq_le_incl; now rewrite H2. -apply NZeq_le_incl; rewrite H1; now do 2 rewrite NZtimes_0_l. +apply NZeq_le_incl; rewrite H1; now do 2 rewrite NZmul_0_l. Qed. -Theorem NZtimes_le_mono_nonneg_r : forall n m p : NZ, 0 <= p -> n <= m -> n * p <= m * p. +Theorem NZmul_le_mono_nonneg_r : forall n m p : NZ, 0 <= p -> n <= m -> n * p <= m * p. Proof. -intros n m p H1 H2; rewrite (NZtimes_comm n p); rewrite (NZtimes_comm m p); -now apply NZtimes_le_mono_nonneg_l. +intros n m p H1 H2; rewrite (NZmul_comm n p); rewrite (NZmul_comm m p); +now apply NZmul_le_mono_nonneg_l. Qed. -Theorem NZtimes_le_mono_nonpos_r : forall n m p : NZ, p <= 0 -> n <= m -> m * p <= n * p. +Theorem NZmul_le_mono_nonpos_r : forall n m p : NZ, p <= 0 -> n <= m -> m * p <= n * p. Proof. -intros n m p H1 H2; rewrite (NZtimes_comm n p); rewrite (NZtimes_comm m p); -now apply NZtimes_le_mono_nonpos_l. +intros n m p H1 H2; rewrite (NZmul_comm n p); rewrite (NZmul_comm m p); +now apply NZmul_le_mono_nonpos_l. Qed. -Theorem NZtimes_cancel_l : forall n m p : NZ, p ~= 0 -> (p * n == p * m <-> n == m). +Theorem NZmul_cancel_l : forall n m p : NZ, p ~= 0 -> (p * n == p * m <-> n == m). Proof. intros n m p H; split; intro H1. destruct (NZlt_trichotomy p 0) as [H2 | [H2 | H2]]. apply -> NZeq_dne; intro H3. apply -> NZlt_gt_cases in H3. destruct H3 as [H3 | H3]. -assert (H4 : p * m < p * n); [now apply -> NZtimes_lt_mono_neg_l |]. +assert (H4 : p * m < p * n); [now apply -> NZmul_lt_mono_neg_l |]. rewrite H1 in H4; false_hyp H4 NZlt_irrefl. -assert (H4 : p * n < p * m); [now apply -> NZtimes_lt_mono_neg_l |]. +assert (H4 : p * n < p * m); [now apply -> NZmul_lt_mono_neg_l |]. rewrite H1 in H4; false_hyp H4 NZlt_irrefl. false_hyp H2 H. apply -> NZeq_dne; intro H3. apply -> NZlt_gt_cases in H3. destruct H3 as [H3 | H3]. -assert (H4 : p * n < p * m) by (now apply -> NZtimes_lt_mono_pos_l). +assert (H4 : p * n < p * m) by (now apply -> NZmul_lt_mono_pos_l). rewrite H1 in H4; false_hyp H4 NZlt_irrefl. -assert (H4 : p * m < p * n) by (now apply -> NZtimes_lt_mono_pos_l). +assert (H4 : p * m < p * n) by (now apply -> NZmul_lt_mono_pos_l). rewrite H1 in H4; false_hyp H4 NZlt_irrefl. now rewrite H1. Qed. -Theorem NZtimes_cancel_r : forall n m p : NZ, p ~= 0 -> (n * p == m * p <-> n == m). +Theorem NZmul_cancel_r : forall n m p : NZ, p ~= 0 -> (n * p == m * p <-> n == m). Proof. -intros n m p. rewrite (NZtimes_comm n p), (NZtimes_comm m p); apply NZtimes_cancel_l. +intros n m p. rewrite (NZmul_comm n p), (NZmul_comm m p); apply NZmul_cancel_l. Qed. -Theorem NZtimes_id_l : forall n m : NZ, m ~= 0 -> (n * m == m <-> n == 1). +Theorem NZmul_id_l : forall n m : NZ, m ~= 0 -> (n * m == m <-> n == 1). Proof. intros n m H. -stepl (n * m == 1 * m) by now rewrite NZtimes_1_l. now apply NZtimes_cancel_r. +stepl (n * m == 1 * m) by now rewrite NZmul_1_l. now apply NZmul_cancel_r. Qed. -Theorem NZtimes_id_r : forall n m : NZ, n ~= 0 -> (n * m == n <-> m == 1). +Theorem NZmul_id_r : forall n m : NZ, n ~= 0 -> (n * m == n <-> m == 1). Proof. -intros n m; rewrite NZtimes_comm; apply NZtimes_id_l. +intros n m; rewrite NZmul_comm; apply NZmul_id_l. Qed. -Theorem NZtimes_le_mono_pos_l : forall n m p : NZ, 0 < p -> (n <= m <-> p * n <= p * m). +Theorem NZmul_le_mono_pos_l : forall n m p : NZ, 0 < p -> (n <= m <-> p * n <= p * m). Proof. intros n m p H; do 2 rewrite NZlt_eq_cases. -rewrite (NZtimes_lt_mono_pos_l p n m) by assumption. -now rewrite -> (NZtimes_cancel_l n m p) by +rewrite (NZmul_lt_mono_pos_l p n m) by assumption. +now rewrite -> (NZmul_cancel_l n m p) by (intro H1; rewrite H1 in H; false_hyp H NZlt_irrefl). Qed. -Theorem NZtimes_le_mono_pos_r : forall n m p : NZ, 0 < p -> (n <= m <-> n * p <= m * p). +Theorem NZmul_le_mono_pos_r : forall n m p : NZ, 0 < p -> (n <= m <-> n * p <= m * p). Proof. -intros n m p. rewrite (NZtimes_comm n p); rewrite (NZtimes_comm m p); -apply NZtimes_le_mono_pos_l. +intros n m p. rewrite (NZmul_comm n p); rewrite (NZmul_comm m p); +apply NZmul_le_mono_pos_l. Qed. -Theorem NZtimes_le_mono_neg_l : forall n m p : NZ, p < 0 -> (n <= m <-> p * m <= p * n). +Theorem NZmul_le_mono_neg_l : forall n m p : NZ, p < 0 -> (n <= m <-> p * m <= p * n). Proof. intros n m p H; do 2 rewrite NZlt_eq_cases. -rewrite (NZtimes_lt_mono_neg_l p n m); [| assumption]. -rewrite -> (NZtimes_cancel_l m n p) by (intro H1; rewrite H1 in H; false_hyp H NZlt_irrefl). +rewrite (NZmul_lt_mono_neg_l p n m); [| assumption]. +rewrite -> (NZmul_cancel_l m n p) by (intro H1; rewrite H1 in H; false_hyp H NZlt_irrefl). now setoid_replace (n == m) with (m == n) using relation iff by (split; now intro). Qed. -Theorem NZtimes_le_mono_neg_r : forall n m p : NZ, p < 0 -> (n <= m <-> m * p <= n * p). +Theorem NZmul_le_mono_neg_r : forall n m p : NZ, p < 0 -> (n <= m <-> m * p <= n * p). Proof. -intros n m p. rewrite (NZtimes_comm n p); rewrite (NZtimes_comm m p); -apply NZtimes_le_mono_neg_l. +intros n m p. rewrite (NZmul_comm n p); rewrite (NZmul_comm m p); +apply NZmul_le_mono_neg_l. Qed. -Theorem NZtimes_lt_mono_nonneg : +Theorem NZmul_lt_mono_nonneg : forall n m p q : NZ, 0 <= n -> n < m -> 0 <= p -> p < q -> n * p < m * q. Proof. intros n m p q H1 H2 H3 H4. apply NZle_lt_trans with (m * p). -apply NZtimes_le_mono_nonneg_r; [assumption | now apply NZlt_le_incl]. -apply -> NZtimes_lt_mono_pos_l; [assumption | now apply NZle_lt_trans with n]. +apply NZmul_le_mono_nonneg_r; [assumption | now apply NZlt_le_incl]. +apply -> NZmul_lt_mono_pos_l; [assumption | now apply NZle_lt_trans with n]. Qed. (* There are still many variants of the theorem above. One can assume 0 < n or 0 < p or n <= m or p <= q. *) -Theorem NZtimes_le_mono_nonneg : +Theorem NZmul_le_mono_nonneg : forall n m p q : NZ, 0 <= n -> n <= m -> 0 <= p -> p <= q -> n * p <= m * q. Proof. intros n m p q H1 H2 H3 H4. le_elim H2; le_elim H4. -apply NZlt_le_incl; now apply NZtimes_lt_mono_nonneg. -rewrite <- H4; apply NZtimes_le_mono_nonneg_r; [assumption | now apply NZlt_le_incl]. -rewrite <- H2; apply NZtimes_le_mono_nonneg_l; [assumption | now apply NZlt_le_incl]. +apply NZlt_le_incl; now apply NZmul_lt_mono_nonneg. +rewrite <- H4; apply NZmul_le_mono_nonneg_r; [assumption | now apply NZlt_le_incl]. +rewrite <- H2; apply NZmul_le_mono_nonneg_l; [assumption | now apply NZlt_le_incl]. rewrite H2; rewrite H4; now apply NZeq_le_incl. Qed. -Theorem NZtimes_pos_pos : forall n m : NZ, 0 < n -> 0 < m -> 0 < n * m. +Theorem NZmul_pos_pos : forall n m : NZ, 0 < n -> 0 < m -> 0 < n * m. Proof. intros n m H1 H2. -rewrite <- (NZtimes_0_l m). now apply -> NZtimes_lt_mono_pos_r. +rewrite <- (NZmul_0_l m). now apply -> NZmul_lt_mono_pos_r. Qed. -Theorem NZtimes_neg_neg : forall n m : NZ, n < 0 -> m < 0 -> 0 < n * m. +Theorem NZmul_neg_neg : forall n m : NZ, n < 0 -> m < 0 -> 0 < n * m. Proof. intros n m H1 H2. -rewrite <- (NZtimes_0_l m). now apply -> NZtimes_lt_mono_neg_r. +rewrite <- (NZmul_0_l m). now apply -> NZmul_lt_mono_neg_r. Qed. -Theorem NZtimes_pos_neg : forall n m : NZ, 0 < n -> m < 0 -> n * m < 0. +Theorem NZmul_pos_neg : forall n m : NZ, 0 < n -> m < 0 -> n * m < 0. Proof. intros n m H1 H2. -rewrite <- (NZtimes_0_l m). now apply -> NZtimes_lt_mono_neg_r. +rewrite <- (NZmul_0_l m). now apply -> NZmul_lt_mono_neg_r. Qed. -Theorem NZtimes_neg_pos : forall n m : NZ, n < 0 -> 0 < m -> n * m < 0. +Theorem NZmul_neg_pos : forall n m : NZ, n < 0 -> 0 < m -> n * m < 0. Proof. -intros; rewrite NZtimes_comm; now apply NZtimes_pos_neg. +intros; rewrite NZmul_comm; now apply NZmul_pos_neg. Qed. -Theorem NZlt_1_times_pos : forall n m : NZ, 1 < n -> 0 < m -> 1 < n * m. +Theorem NZlt_1_mul_pos : forall n m : NZ, 1 < n -> 0 < m -> 1 < n * m. Proof. -intros n m H1 H2. apply -> (NZtimes_lt_mono_pos_r m) in H1. -rewrite NZtimes_1_l in H1. now apply NZlt_1_l with m. +intros n m H1 H2. apply -> (NZmul_lt_mono_pos_r m) in H1. +rewrite NZmul_1_l in H1. now apply NZlt_1_l with m. assumption. Qed. -Theorem NZeq_times_0 : forall n m : NZ, n * m == 0 <-> n == 0 \/ m == 0. +Theorem NZeq_mul_0 : forall n m : NZ, n * m == 0 <-> n == 0 \/ m == 0. Proof. intros n m; split. intro H; destruct (NZlt_trichotomy n 0) as [H1 | [H1 | H1]]; destruct (NZlt_trichotomy m 0) as [H2 | [H2 | H2]]; try (now right); try (now left). -elimtype False; now apply (NZlt_neq 0 (n * m)); [apply NZtimes_neg_neg |]. -elimtype False; now apply (NZlt_neq (n * m) 0); [apply NZtimes_neg_pos |]. -elimtype False; now apply (NZlt_neq (n * m) 0); [apply NZtimes_pos_neg |]. -elimtype False; now apply (NZlt_neq 0 (n * m)); [apply NZtimes_pos_pos |]. -intros [H | H]. now rewrite H, NZtimes_0_l. now rewrite H, NZtimes_0_r. +elimtype False; now apply (NZlt_neq 0 (n * m)); [apply NZmul_neg_neg |]. +elimtype False; now apply (NZlt_neq (n * m) 0); [apply NZmul_neg_pos |]. +elimtype False; now apply (NZlt_neq (n * m) 0); [apply NZmul_pos_neg |]. +elimtype False; now apply (NZlt_neq 0 (n * m)); [apply NZmul_pos_pos |]. +intros [H | H]. now rewrite H, NZmul_0_l. now rewrite H, NZmul_0_r. Qed. -Theorem NZneq_times_0 : forall n m : NZ, n ~= 0 /\ m ~= 0 <-> n * m ~= 0. +Theorem NZneq_mul_0 : forall n m : NZ, n ~= 0 /\ m ~= 0 <-> n * m ~= 0. Proof. intros n m; split; intro H. -intro H1; apply -> NZeq_times_0 in H1. tauto. +intro H1; apply -> NZeq_mul_0 in H1. tauto. split; intro H1; rewrite H1 in H; -(rewrite NZtimes_0_l in H || rewrite NZtimes_0_r in H); now apply H. +(rewrite NZmul_0_l in H || rewrite NZmul_0_r in H); now apply H. Qed. Theorem NZeq_square_0 : forall n : NZ, n * n == 0 <-> n == 0. Proof. -intro n; rewrite NZeq_times_0; tauto. +intro n; rewrite NZeq_mul_0; tauto. Qed. -Theorem NZeq_times_0_l : forall n m : NZ, n * m == 0 -> m ~= 0 -> n == 0. +Theorem NZeq_mul_0_l : forall n m : NZ, n * m == 0 -> m ~= 0 -> n == 0. Proof. -intros n m H1 H2. apply -> NZeq_times_0 in H1. destruct H1 as [H1 | H1]. +intros n m H1 H2. apply -> NZeq_mul_0 in H1. destruct H1 as [H1 | H1]. assumption. false_hyp H1 H2. Qed. -Theorem NZeq_times_0_r : forall n m : NZ, n * m == 0 -> n ~= 0 -> m == 0. +Theorem NZeq_mul_0_r : forall n m : NZ, n * m == 0 -> n ~= 0 -> m == 0. Proof. -intros n m H1 H2; apply -> NZeq_times_0 in H1. destruct H1 as [H1 | H1]. +intros n m H1 H2; apply -> NZeq_mul_0 in H1. destruct H1 as [H1 | H1]. false_hyp H1 H2. assumption. Qed. -Theorem NZlt_0_times : forall n m : NZ, 0 < n * m <-> (0 < n /\ 0 < m) \/ (m < 0 /\ n < 0). +Theorem NZlt_0_mul : forall n m : NZ, 0 < n * m <-> (0 < n /\ 0 < m) \/ (m < 0 /\ n < 0). Proof. intros n m; split; [intro H | intros [[H1 H2] | [H1 H2]]]. destruct (NZlt_trichotomy n 0) as [H1 | [H1 | H1]]; -[| rewrite H1 in H; rewrite NZtimes_0_l in H; false_hyp H NZlt_irrefl |]; +[| rewrite H1 in H; rewrite NZmul_0_l in H; false_hyp H NZlt_irrefl |]; (destruct (NZlt_trichotomy m 0) as [H2 | [H2 | H2]]; -[| rewrite H2 in H; rewrite NZtimes_0_r in H; false_hyp H NZlt_irrefl |]); +[| rewrite H2 in H; rewrite NZmul_0_r in H; false_hyp H NZlt_irrefl |]); try (left; now split); try (right; now split). -assert (H3 : n * m < 0) by now apply NZtimes_neg_pos. +assert (H3 : n * m < 0) by now apply NZmul_neg_pos. elimtype False; now apply (NZlt_asymm (n * m) 0). -assert (H3 : n * m < 0) by now apply NZtimes_pos_neg. +assert (H3 : n * m < 0) by now apply NZmul_pos_neg. elimtype False; now apply (NZlt_asymm (n * m) 0). -now apply NZtimes_pos_pos. now apply NZtimes_neg_neg. +now apply NZmul_pos_pos. now apply NZmul_neg_neg. Qed. Theorem NZsquare_lt_mono_nonneg : forall n m : NZ, 0 <= n -> n < m -> n * n < m * m. Proof. -intros n m H1 H2. now apply NZtimes_lt_mono_nonneg. +intros n m H1 H2. now apply NZmul_lt_mono_nonneg. Qed. Theorem NZsquare_le_mono_nonneg : forall n m : NZ, 0 <= n -> n <= m -> n * n <= m * m. Proof. -intros n m H1 H2. now apply NZtimes_le_mono_nonneg. +intros n m H1 H2. now apply NZmul_le_mono_nonneg. Qed. (* The converse theorems require nonnegativity (or nonpositivity) of the @@ -297,14 +297,14 @@ assumption. assert (F : m * m < n * n) by now apply NZsquare_lt_mono_nonneg. apply -> NZlt_nge in F. false_hyp H2 F. Qed. -Theorem NZtimes_2_mono_l : forall n m : NZ, n < m -> 1 + (1 + 1) * n < (1 + 1) * m. +Theorem NZmul_2_mono_l : forall n m : NZ, n < m -> 1 + (1 + 1) * n < (1 + 1) * m. Proof. intros n m H. apply <- NZle_succ_l in H. -apply -> (NZtimes_le_mono_pos_l (S n) m (1 + 1)) in H. -repeat rewrite NZtimes_plus_distr_r in *; repeat rewrite NZtimes_1_l in *. -repeat rewrite NZplus_succ_r in *. repeat rewrite NZplus_succ_l in *. rewrite NZplus_0_l. +apply -> (NZmul_le_mono_pos_l (S n) m (1 + 1)) in H. +repeat rewrite NZmul_add_distr_r in *; repeat rewrite NZmul_1_l in *. +repeat rewrite NZadd_succ_r in *. repeat rewrite NZadd_succ_l in *. rewrite NZadd_0_l. now apply -> NZle_succ_l. -apply NZplus_pos_pos; now apply NZlt_succ_diag_r. +apply NZadd_pos_pos; now apply NZlt_succ_diag_r. Qed. End NZTimesOrderPropFunct. -- cgit v1.2.3