# Library Coq.Arith.Compare_dec

Require Import Le Lt Gt Decidable PeanoNat.

Local Open Scope nat_scope.

Implicit Types m n x y : nat.

Definition zerop n : {n = 0} + {0 < n}.

Definition lt_eq_lt_dec n m : {n < m} + {n = m} + {m < n}.

Definition gt_eq_gt_dec n m : {m > n} + {n = m} + {n > m}.

Definition le_lt_dec n m : {n <= m} + {m < n}.

Definition le_le_S_dec n m : {n <= m} + {S m <= n}.

Definition le_ge_dec n m : {n <= m} + {n >= m}.

Definition le_gt_dec n m : {n <= m} + {n > m}.

Definition le_lt_eq_dec n m : n <= m -> {n < m} + {n = m}.

Theorem le_dec n m : {n <= m} + {~ n <= m}.

Theorem lt_dec n m : {n < m} + {~ n < m}.

Theorem gt_dec n m : {n > m} + {~ n > m}.

Theorem ge_dec n m : {n >= m} + {~ n >= m}.

Proofs of decidability

Theorem dec_le n m : decidable (n <= m).

Theorem dec_lt n m : decidable (n < m).

Theorem dec_gt n m : decidable (n > m).

Theorem dec_ge n m : decidable (n >= m).

Theorem not_eq n m : n <> m -> n < m \/ m < n.

Theorem not_le n m : ~ n <= m -> n > m.

Theorem not_gt n m : ~ n > m -> n <= m.

Theorem not_ge n m : ~ n >= m -> n < m.

Theorem not_lt n m : ~ n < m -> n >= m.

A ternary comparison function in the spirit of Z.compare. See now Nat.compare and its properties. In scope nat_scope, the notation for Nat.compare is "?="

Notation nat_compare := Nat.compare (compat "8.7").

Notation nat_compare_spec := Nat.compare_spec (compat "8.7").
Notation nat_compare_eq_iff := Nat.compare_eq_iff (compat "8.7").
Notation nat_compare_S := Nat.compare_succ (only parsing).

Lemma nat_compare_lt n m : n<m <-> (n ?= m) = Lt.

Lemma nat_compare_gt n m : n>m <-> (n ?= m) = Gt.

Lemma nat_compare_le n m : n<=m <-> (n ?= m) <> Gt.

Lemma nat_compare_ge n m : n>=m <-> (n ?= m) <> Lt.

Some projections of the above equivalences.

Lemma nat_compare_eq n m : (n ?= m) = Eq -> n = m.

Lemma nat_compare_Lt_lt n m : (n ?= m) = Lt -> n<m.

Lemma nat_compare_Gt_gt n m : (n ?= m) = Gt -> n>m.

A previous definition of nat_compare in terms of lt_eq_lt_dec. The new version avoids the creation of proof parts.

Definition nat_compare_alt (n m:nat) :=
match lt_eq_lt_dec n m with
| inleft (left _) => Lt
| inleft (right _) => Eq
| inright _ => Gt
end.

Lemma nat_compare_equiv n m : (n ?= m) = nat_compare_alt n m.

A boolean version of le over nat. See now Nat.leb and its properties. In scope nat_scope, the notation for Nat.leb is "<=?"

Notation leb := Nat.leb (only parsing).

Notation leb_iff := Nat.leb_le (only parsing).

Lemma leb_iff_conv m n : (n <=? m) = false <-> m < n.

Lemma leb_correct m n : m <= n -> (m <=? n) = true.

Lemma leb_complete m n : (m <=? n) = true -> m <= n.

Lemma leb_correct_conv m n : m < n -> (n <=? m) = false.

Lemma leb_complete_conv m n : (n <=? m) = false -> m < n.

Lemma leb_compare n m : (n <=? m) = true <-> (n ?= m) <> Gt.