[coq] Add example of HB mathcomp eqtype

coq/third-party/hb-eqtype.v

@@ -0,0 +1,51 @@
+From HB Require Import structures.
+From mathcomp Require Import all_ssreflect.
+
+Inductive t: Type :=
+| Zero: t
+| One: t
+| Add: t -> t -> t.
+
+Fixpoint eqb (a b: t): bool :=
+  match a, b with
+  | Zero, Zero => true
+  | One, One => true
+  | Add x1 y1, Add x2 y2 => eqb x1 x2 && eqb y1 y2
+  | _, _ => false
+  end.
+
+Section EqLemmas.
+  Lemma lm_eqb1 (a: t): a = a -> eqb a a.
+  Proof.
+    elim: a => //= a IHa b IHb H.
+    by rewrite IHa // IHb.
+  Qed.
+
+  Lemma lm_eqb2r (a b: t): a = b -> eqb a b.
+  Proof.
+    move <-.
+    by apply: lm_eqb1.
+  Qed.
+
+  Lemma lm_eqb2l (a b: t): eqb a b -> a = b.
+  Proof.
+    elim: a b => [||a IHa b IHb]; case => // x y /andP [H1 H2].
+    by rewrite (IHa x H1) (IHb y H2).
+  Qed.
+
+  Theorem eqP (a b: t): reflect (a=b) (eqb a b).
+  Proof.
+    apply (iffP idP).
+    - apply: lm_eqb2l.
+    - apply: lm_eqb2r.
+  Qed.
+End EqLemmas.
+
+HB.instance Definition _ := hasDecEq.Build t eqP.
+
+HB.about t.
+(*
+HB: t is canonically equipped with structures:
+    - eqtype.Equality
+      (from "(stdin)", line 2)
+*)