Module Coq.Arith.Le

Order on natural numbers

Theorem le_n_S : (n,m:nat)(le n m)->(le (S n) (S m)).
Proof.
  NewInduction 1; Auto.
Qed.

Theorem le_trans : (n,m,p:nat)(le n m)->(le m p)->(le n p).
Proof.
  NewInduction 2; Auto.
Qed.

Theorem le_n_Sn : (n:nat)(le n (S n)).
Proof.
  Auto.
Qed.

Theorem le_O_n : (n:nat)(le O n).
Proof.
  NewInduction n ; Auto.
Qed.

Hints Resolve le_n_S le_n_Sn le_O_n le_n_S le_trans : arith v62.

Theorem le_pred_n : (n:nat)(le (pred n) n).
Proof.
NewInduction n ; Auto with arith.
Qed.
Hints Resolve le_pred_n : arith v62.

Theorem le_trans_S : (n,m:nat)(le (S n) m)->(le n m).
Proof.
Intros n m H ; Apply le_trans with (S n) ; Auto with arith.
Qed.
Hints Immediate le_trans_S : arith v62.

Theorem le_S_n : (n,m:nat)(le (S n) (S m))->(le n m).
Proof.
Intros n m H ; Change (le (pred (S n)) (pred (S m))).
Elim H ; Simpl ; Auto with arith.
Qed.
Hints Immediate le_S_n : arith v62.

Negative properties

Theorem le_Sn_O : (n:nat)~(le (S n) O).
Proof.
Red ; Intros n H.
Change (IsSucc O) ; Elim H ; Simpl ; Auto with arith.
Qed.
Hints Resolve le_Sn_O : arith v62.

Theorem le_Sn_n : (n:nat)~(le (S n) n).
Proof.
NewInduction n; Auto with arith.
Qed.
Hints Resolve le_Sn_n : arith v62.

Theorem le_antisym : (n,m:nat)(le n m)->(le m n)->(n=m).
Proof.
Intros n m h ; Elim h ; Auto with arith.
Intros m0 H H0 H1.
Absurd (le (S m0) m0) ; Auto with arith.
Apply le_trans with n ; Auto with arith.
Qed.
Hints Immediate le_antisym : arith v62.

Theorem le_n_O_eq : (n:nat)(le n O)->(O=n).
Proof.
Auto with arith.
Qed.
Hints Immediate le_n_O_eq : 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).
Proof.
NewInduction n; Auto with arith.
Intros m Le.
Elim Le; Auto with arith.
Qed.


Index