I'm trying to import Library Coq.Structures.OrdersFacts as usual with:
Require Import Coq.Structures.OrdersFacts
Then I try to use of the lemmas there with either:
apply CompareFacts.compare_nlt_iff. or apply compare_nlt_iff.
But none work ... what am I missing?
CompareFacts is a Module Type, not a Module. You can see that if you do
Require Import Coq.Structures.OrdersFacts.
Print OrdersFacts.CompareFacts.
Find a Module of this type and apply its Lemmas instead.
EDIT:
I meant that to use the lemmas on i.e. nat, you need a module that shows that nat is a DecStrOrder' (and Nat from PeanoNat is such a module), and also one that specializes CompareFacts for nat .
Perhaps an example is more useful.
Require Import Coq.Structures.OrdersFacts.
Module mymodule (O:DecStrOrder') (T: CompareFacts O).
Import T.
Import O.
Check compare_eq_iff. (* from CompareFacts *)
(* a theorem about terms of type O.t *)
Lemma lem1 a b c: (a ?= b) = Eq -> b == c -> c == a.
intros.
rewrite compare_eq_iff in H. (* here we use the lemma *)
rewrite H.
rewrite H0.
apply eq_equiv.
Qed.
End mymodule.
(* the above module functor can be specialised for i.e. nat *)
Require Import PeanoNat.
Print CompareFacts.
Module M : CompareFacts Nat.
Definition compare_eq_iff := Nat.compare_eq_iff.
Definition compare_eq := Nat.compare_eq.
Definition compare_lt_iff := Nat.compare_lt_iff.
Definition compare_gt_iff := Nat.compare_gt_iff.
Definition compare_nlt_iff := Nat.compare_nlt_iff.
Definition compare_ngt_iff := Nat.compare_ngt_iff.
Definition compare_refl := Nat.compare_refl.
Definition compare_compat: Proper (eq==>eq==>eq) Nat.compare.
intros x y Hxy a b Hab; now subst. Defined.
Definition compare_antisym := Nat.compare_antisym.
End M.
Module natmodule := mymodule Nat M.
Check natmodule.lem1.
