[coq] add number notation example

coq/mathcomp/mixins-hello.v

@@ -0,0 +1,57 @@
+From mathcomp Require Import all_ssreflect all_algebra.
+
+(* Example from Chapter 8 of mathcomp book *)
+
+Inductive dir: predArgType :=
+| South | North | East | West.
+
+(* dir to an existing ordinal ?? *)
+Definition d2o (d: dir): 'I_4 :=
+  match d with
+  | South => inord 0
+  | North => inord 1
+  | East => inord 2
+  | West => inord 3
+  end.
+
+Definition o2d (o: 'I_4): option dir :=
+  match val o with
+  | 0 => Some South
+  | 1 => Some North
+  | 2 => Some East
+  | 3 => Some West
+  | _ => None
+  end.
+
+(* d2o and o2d cancel out *)
+Lemma pcan_do4: pcancel d2o o2d.
+Proof.
+  by case; rewrite /o2d /= inordK.
+Qed.
+
+Compute (North \in dir).
+(* = true : bool *)
+
+Fail Check (North != South).  (* needs EqType *)
+Fail Check (#| dir | == 4).   (* needs FinType *)
+
+(* Give structure of ordinals to [dir] *)
+Definition dir_eqMixin := PcanEqMixin pcan_do4.
+Canonical dir_eqType := EqType dir dir_eqMixin.
+Check (North != South).
+(* North != South : bool *)
+
+(* [CountType] implies [ChoiceType] by the way.
+   Here it's explicitly given though *)
+Definition dir_choiceMixin := PcanChoiceMixin pcan_do4.
+Canonical dir_choiceType := ChoiceType dir dir_choiceMixin.
+
+Definition dir_countMixin := PcanCountMixin pcan_do4.
+Canonical dir_countType := CountType dir dir_countMixin.
+
+Definition dir_finMixin := PcanFinMixin pcan_do4.
+Canonical dir_finType := FinType dir dir_finMixin.
+Check (#| dir | == 4).
+(* #|dir| == 4 : bool *)
+
+Check (North != South) && (North \in dir) && (#| dir | == 4).

coq/number-notation.v

@@ -0,0 +1,60 @@
+(* From https://coq.inria.fr/refman/user-extensions/syntax-extensions.html#number-notations *)
+
+Inductive radix2: Set :=
+| x0: radix2
+| x2p0: radix2 -> radix2
+| x2p1: radix2 -> radix2.
+
+Print Decimal.
+
+Definition of_uint_dec (u : Decimal.uint) : option radix2 :=
+  let fix f u := match u with
+    | Decimal.Nil => Some x0
+    | Decimal.D0 u =>
+        match f u with
+        | Some u => Some (x2p0 u)
+        | None => None
+        end
+    | Decimal.D1 u =>
+        match f u with
+        | Some u => Some (x2p1 u)
+        | None => None
+        end
+    | _ => None end in
+  f (Decimal.rev u).
+
+Definition of_uint (u : Number.uint) : option radix2 :=
+  match u with
+  | Number.UIntDecimal u => of_uint_dec u
+  | Number.UIntHexadecimal _ => None
+  end.
+
+(* Printing stuff *)
+Definition to_uint_dec (x : radix2) : Decimal.uint :=
+  let fix f x :=
+    match x with
+    | x0 => Decimal.Nil
+    | x2p0 x => Decimal.D0 (f x)
+    | x2p1 x => Decimal.D1 (f x)
+    end in
+  Decimal.rev (f x).
+Definition to_uint (x : radix2) : Number.uint := Number.UIntDecimal (to_uint_dec x).
+
+Declare Scope radix2_scope.
+Delimit Scope radix2_scope with r2.
+Number Notation radix2 of_uint to_uint : radix2_scope.
+
+Check 01%r2.
+(* 01%r2 : radix2 *)
+
+Fail Check 02%r2.
+(*
+The command has indeed failed with message:
+Cannot interpret this number as a value of type radix2
+*)
+
+Check (x2p1 x0).
+(* 1%r2 : radix2 *)
+
+Check (x2p1 (x2p0 x0)).
+(* 01%r2 : radix2 *)