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Diffstat (limited to 'theories/Arith/Le.v')
| -rwxr-xr-x | theories/Arith/Le.v | 106 |
1 files changed, 53 insertions, 53 deletions
diff --git a/theories/Arith/Le.v b/theories/Arith/Le.v index c80689836e..d311046650 100755 --- a/theories/Arith/Le.v +++ b/theories/Arith/Le.v @@ -9,114 +9,114 @@ (*i $Id$ i*) (** Order on natural numbers *) -V7only [Import nat_scope.]. Open Local Scope nat_scope. -Implicit Variables Type m,n,p:nat. +Implicit Types m n p : nat. (** Reflexivity *) -Theorem le_refl : (n:nat)(le n n). +Theorem le_refl : forall n, n <= n. Proof. -Exact le_n. +exact le_n. Qed. (** Transitivity *) -Theorem le_trans : (n,m,p:nat)(le n m)->(le m p)->(le n p). +Theorem le_trans : forall n m p, n <= m -> m <= p -> n <= p. Proof. - NewInduction 2; Auto. + induction 2; auto. Qed. -Hints Resolve le_trans : arith v62. +Hint Resolve le_trans: arith v62. (** Order, successor and predecessor *) -Theorem le_n_S : (n,m:nat)(le n m)->(le (S n) (S m)). +Theorem le_n_S : forall n m, n <= m -> S n <= S m. Proof. - NewInduction 1; Auto. + induction 1; auto. Qed. -Theorem le_n_Sn : (n:nat)(le n (S n)). +Theorem le_n_Sn : forall n, n <= S n. Proof. - Auto. + auto. Qed. -Theorem le_O_n : (n:nat)(le O n). +Theorem le_O_n : forall n, 0 <= n. Proof. - NewInduction n ; Auto. + induction n; auto. Qed. -Hints Resolve le_n_S le_n_Sn le_O_n le_n_S : arith v62. +Hint Resolve le_n_S le_n_Sn le_O_n le_n_S: arith v62. -Theorem le_pred_n : (n:nat)(le (pred n) n). +Theorem le_pred_n : forall n, pred n <= n. Proof. -NewInduction n ; Auto with arith. +induction n; auto with arith. Qed. -Hints Resolve le_pred_n : arith v62. +Hint Resolve le_pred_n: arith v62. -Theorem le_trans_S : (n,m:nat)(le (S n) m)->(le n m). +Theorem le_Sn_le : forall n m, S n <= m -> n <= m. Proof. -Intros n m H ; Apply le_trans with (S n); Auto with arith. +intros n m H; apply le_trans with (S n); auto with arith. Qed. -Hints Immediate le_trans_S : arith v62. +Hint Immediate le_Sn_le: arith v62. -Theorem le_S_n : (n,m:nat)(le (S n) (S m))->(le n m). +Theorem le_S_n : forall n m, S n <= S m -> n <= m. Proof. -Intros n m H ; Change (le (pred (S n)) (pred (S m))). -Elim H ; Simpl ; Auto with arith. +intros n m H; change (pred (S n) <= pred (S m)) in |- *. +elim H; simpl in |- *; auto with arith. Qed. -Hints Immediate le_S_n : arith v62. +Hint Immediate le_S_n: arith v62. -Theorem le_pred : (n,m:nat)(le n m)->(le (pred n) (pred m)). +Theorem le_pred : forall n m, n <= m -> pred n <= pred m. Proof. -NewInduction n as [|n IHn]. Simpl. Auto with arith. -NewDestruct m as [|m]. Simpl. Intro H. Inversion H. -Simpl. Auto with arith. +induction n as [| n IHn]. simpl in |- *. auto with arith. +destruct m as [| m]. simpl in |- *. intro H. inversion H. +simpl in |- *. auto with arith. Qed. (** Comparison to 0 *) -Theorem le_Sn_O : (n:nat)~(le (S n) O). +Theorem le_Sn_O : forall n, ~ S n <= 0. Proof. -Red ; Intros n H. -Change (IsSucc O) ; Elim H ; Simpl ; Auto with arith. +red in |- *; intros n H. +change (IsSucc 0) in |- *; elim H; simpl in |- *; auto with arith. Qed. -Hints Resolve le_Sn_O : arith v62. +Hint Resolve le_Sn_O: arith v62. -Theorem le_n_O_eq : (n:nat)(le n O)->(O=n). +Theorem le_n_O_eq : forall n, n <= 0 -> 0 = n. Proof. -NewInduction n; Auto with arith. -Intro; Contradiction le_Sn_O with n. +induction n; auto with arith. +intro; contradiction le_Sn_O with n. Qed. -Hints Immediate le_n_O_eq : arith v62. +Hint Immediate le_n_O_eq: arith v62. (** Negative properties *) -Theorem le_Sn_n : (n:nat)~(le (S n) n). +Theorem le_Sn_n : forall n, ~ S n <= n. Proof. -NewInduction n; Auto with arith. +induction n; auto with arith. Qed. -Hints Resolve le_Sn_n : arith v62. +Hint Resolve le_Sn_n: arith v62. (** Antisymmetry *) -Theorem le_antisym : (n,m:nat)(le n m)->(le m n)->(n=m). +Theorem le_antisym : forall n m, n <= m -> m <= n -> n = m. Proof. -Intros n m h ; NewDestruct h as [|m0]; Auto with arith. -Intros H1. -Absurd (le (S m0) m0) ; Auto with arith. -Apply le_trans with n ; Auto with arith. +intros n m h; destruct h as [| m0 H]; auto with arith. +intros H1. +absurd (S m0 <= m0); auto with arith. +apply le_trans with n; auto with arith. Qed. -Hints Immediate le_antisym : arith v62. +Hint Immediate le_antisym: arith v62. (** A different elimination principle for the order on natural numbers *) -Lemma le_elim_rel : (P:nat->nat->Prop) - ((p:nat)(P O p))-> - ((p,q:nat)(le p q)->(P p q)->(P (S p) (S q)))-> - (n,m:nat)(le n m)->(P n m). +Lemma le_elim_rel : + forall P:nat -> nat -> Prop, + (forall p, P 0 p) -> + (forall p (q:nat), p <= q -> P p q -> P (S p) (S q)) -> + forall n m, n <= m -> P n m. Proof. -NewInduction n; Auto with arith. -Intros m Le. -Elim Le; Auto with arith. -Qed. +induction n; auto with arith. +intros m Le. +elim Le; auto with arith. +Qed.
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