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Diffstat (limited to 'theories/Arith/Lt.v')
| -rwxr-xr-x | theories/Arith/Lt.v | 145 |
1 files changed, 72 insertions, 73 deletions
diff --git a/theories/Arith/Lt.v b/theories/Arith/Lt.v index 8c80e64c25..425087ea56 100755 --- a/theories/Arith/Lt.v +++ b/theories/Arith/Lt.v @@ -8,169 +8,168 @@ (*i $Id$ i*) -Require Le. -V7only [Import nat_scope.]. +Require Import Le. Open Local Scope nat_scope. -Implicit Variables Type m,n,p:nat. +Implicit Types m n p : nat. (** Irreflexivity *) -Theorem lt_n_n : (n:nat)~(lt n n). +Theorem lt_irrefl : forall n, ~ n < n. Proof le_Sn_n. -Hints Resolve lt_n_n : arith v62. +Hint Resolve lt_irrefl: arith v62. (** Relationship between [le] and [lt] *) -Theorem lt_le_S : (n,p:nat)(lt n p)->(le (S n) p). +Theorem lt_le_S : forall n m, n < m -> S n <= m. Proof. -Auto with arith. +auto with arith. Qed. -Hints Immediate lt_le_S : arith v62. +Hint Immediate lt_le_S: arith v62. -Theorem lt_n_Sm_le : (n,m:nat)(lt n (S m))->(le n m). +Theorem lt_n_Sm_le : forall n m, n < S m -> n <= m. Proof. -Auto with arith. +auto with arith. Qed. -Hints Immediate lt_n_Sm_le : arith v62. +Hint Immediate lt_n_Sm_le: arith v62. -Theorem le_lt_n_Sm : (n,m:nat)(le n m)->(lt n (S m)). +Theorem le_lt_n_Sm : forall n m, n <= m -> n < S m. Proof. -Auto with arith. +auto with arith. Qed. -Hints Immediate le_lt_n_Sm : arith v62. +Hint Immediate le_lt_n_Sm: arith v62. -Theorem le_not_lt : (n,m:nat)(le n m) -> ~(lt m n). +Theorem le_not_lt : forall n m, n <= m -> ~ m < n. Proof. -NewInduction 1; Auto with arith. +induction 1; auto with arith. Qed. -Theorem lt_not_le : (n,m:nat)(lt n m) -> ~(le m n). +Theorem lt_not_le : forall n m, n < m -> ~ m <= n. Proof. -Red; Intros n m Lt Le; Exact (le_not_lt m n Le Lt). +red in |- *; intros n m Lt Le; exact (le_not_lt m n Le Lt). Qed. -Hints Immediate le_not_lt lt_not_le : arith v62. +Hint Immediate le_not_lt lt_not_le: arith v62. (** Asymmetry *) -Theorem lt_not_sym : (n,m:nat)(lt n m) -> ~(lt m n). +Theorem lt_asym : forall n m, n < m -> ~ m < n. Proof. -NewInduction 1; Auto with arith. +induction 1; auto with arith. Qed. (** Order and successor *) -Theorem lt_n_Sn : (n:nat)(lt n (S n)). +Theorem lt_n_Sn : forall n, n < S n. Proof. -Auto with arith. +auto with arith. Qed. -Hints Resolve lt_n_Sn : arith v62. +Hint Resolve lt_n_Sn: arith v62. -Theorem lt_S : (n,m:nat)(lt n m)->(lt n (S m)). +Theorem lt_S : forall n m, n < m -> n < S m. Proof. -Auto with arith. +auto with arith. Qed. -Hints Resolve lt_S : arith v62. +Hint Resolve lt_S: arith v62. -Theorem lt_n_S : (n,m:nat)(lt n m)->(lt (S n) (S m)). +Theorem lt_n_S : forall n m, n < m -> S n < S m. Proof. -Auto with arith. +auto with arith. Qed. -Hints Resolve lt_n_S : arith v62. +Hint Resolve lt_n_S: arith v62. -Theorem lt_S_n : (n,m:nat)(lt (S n) (S m))->(lt n m). +Theorem lt_S_n : forall n m, S n < S m -> n < m. Proof. -Auto with arith. +auto with arith. Qed. -Hints Immediate lt_S_n : arith v62. +Hint Immediate lt_S_n: arith v62. -Theorem lt_O_Sn : (n:nat)(lt O (S n)). +Theorem lt_O_Sn : forall n, 0 < S n. Proof. -Auto with arith. +auto with arith. Qed. -Hints Resolve lt_O_Sn : arith v62. +Hint Resolve lt_O_Sn: arith v62. -Theorem lt_n_O : (n:nat)~(lt n O). +Theorem lt_n_O : forall n, ~ n < 0. Proof le_Sn_O. -Hints Resolve lt_n_O : arith v62. +Hint Resolve lt_n_O: arith v62. (** Predecessor *) -Lemma S_pred : (n,m:nat)(lt m n)->n=(S (pred n)). +Lemma S_pred : forall n m, m < n -> n = S (pred n). Proof. -NewInduction 1; Auto with arith. +induction 1; auto with arith. Qed. -Lemma lt_pred : (n,p:nat)(lt (S n) p)->(lt n (pred p)). +Lemma lt_pred : forall n m, S n < m -> n < pred m. Proof. -NewInduction 1; Simpl; Auto with arith. +induction 1; simpl in |- *; auto with arith. Qed. -Hints Immediate lt_pred : arith v62. +Hint Immediate lt_pred: arith v62. -Lemma lt_pred_n_n : (n:nat)(lt O n)->(lt (pred n) n). -NewDestruct 1; Simpl; Auto with arith. +Lemma lt_pred_n_n : forall n, 0 < n -> pred n < n. +destruct 1; simpl in |- *; auto with arith. Qed. -Hints Resolve lt_pred_n_n : arith v62. +Hint Resolve lt_pred_n_n: arith v62. (** Transitivity properties *) -Theorem lt_trans : (n,m,p:nat)(lt n m)->(lt m p)->(lt n p). +Theorem lt_trans : forall n m p, n < m -> m < p -> n < p. Proof. -NewInduction 2; Auto with arith. +induction 2; auto with arith. Qed. -Theorem lt_le_trans : (n,m,p:nat)(lt n m)->(le m p)->(lt n p). +Theorem lt_le_trans : forall n m p, n < m -> m <= p -> n < p. Proof. -NewInduction 2; Auto with arith. +induction 2; auto with arith. Qed. -Theorem le_lt_trans : (n,m,p:nat)(le n m)->(lt m p)->(lt n p). +Theorem le_lt_trans : forall n m p, n <= m -> m < p -> n < p. Proof. -NewInduction 2; Auto with arith. +induction 2; auto with arith. Qed. -Hints Resolve lt_trans lt_le_trans le_lt_trans : arith v62. +Hint Resolve lt_trans lt_le_trans le_lt_trans: arith v62. (** Large = strict or equal *) -Theorem le_lt_or_eq : (n,m:nat)(le n m)->((lt n m) \/ n=m). +Theorem le_lt_or_eq : forall n m, n <= m -> n < m \/ n = m. Proof. -NewInduction 1; Auto with arith. +induction 1; auto with arith. Qed. -Theorem lt_le_weak : (n,m:nat)(lt n m)->(le n m). +Theorem lt_le_weak : forall n m, n < m -> n <= m. Proof. -Auto with arith. +auto with arith. Qed. -Hints Immediate lt_le_weak : arith v62. +Hint Immediate lt_le_weak: arith v62. (** Dichotomy *) -Theorem le_or_lt : (n,m:nat)((le n m)\/(lt m n)). +Theorem le_or_lt : forall n m, n <= m \/ m < n. Proof. -Intros n m; Pattern n m; Apply nat_double_ind; Auto with arith. -NewInduction 1; Auto with arith. +intros n m; pattern n, m in |- *; apply nat_double_ind; auto with arith. +induction 1; auto with arith. Qed. -Theorem nat_total_order: (m,n: nat) ~ m = n -> (lt m n) \/ (lt n m). +Theorem nat_total_order : forall n m, n <> m -> n < m \/ m < n. Proof. -Intros m n diff. -Elim (le_or_lt n m); [Intro H'0 | Auto with arith]. -Elim (le_lt_or_eq n m); Auto with arith. -Intro H'; Elim diff; Auto with arith. +intros m n diff. +elim (le_or_lt n m); [ intro H'0 | auto with arith ]. +elim (le_lt_or_eq n m); auto with arith. +intro H'; elim diff; auto with arith. Qed. (** Comparison to 0 *) -Theorem neq_O_lt : (n:nat)(~O=n)->(lt O n). +Theorem neq_O_lt : forall n, 0 <> n -> 0 < n. Proof. -NewInduction n; Auto with arith. -Intros; Absurd O=O; Trivial with arith. +induction n; auto with arith. +intros; absurd (0 = 0); trivial with arith. Qed. -Hints Immediate neq_O_lt : arith v62. +Hint Immediate neq_O_lt: arith v62. -Theorem lt_O_neq : (n:nat)(lt O n)->(~O=n). +Theorem lt_O_neq : forall n, 0 < n -> 0 <> n. Proof. -NewInduction 1; Auto with arith. +induction 1; auto with arith. Qed. -Hints Immediate lt_O_neq : arith v62. +Hint Immediate lt_O_neq: arith v62.
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