Library Coq.Numbers.Cyclic.Int63.Ring63


Uint63 numbers defines Z/(2^63)Z, and can hence be equipped

with a ring structure and a ring tactic

Require Import Cyclic63 CyclicAxioms.

Local Open Scope uint63_scope.

Detection of constants

Ltac isUint63cst t :=
  match eval lazy delta [add] in (t + 1)%uint63 with
  | add _ _ => constr:(false)
  | _ => constr:(true)
  end.

Ltac Uint63cst t :=
  match eval lazy delta [add] in (t + 1)%uint63 with
  | add _ _ => constr:(NotConstant)
  | _ => constr:(t)
  end.

The generic ring structure inferred from the Cyclic structure

Module Uint63ring := CyclicRing Uint63Cyclic.

Unlike in the generic CyclicRing, we can use Leibniz here.

Lemma Uint63_canonic : forall x y, to_Z x = to_Z y -> x = y.

Lemma ring_theory_switch_eq :
 forall A (R R':A->A->Prop) zero one add mul sub opp,
  (forall x y : A, R x y -> R' x y) ->
  ring_theory zero one add mul sub opp R ->
  ring_theory zero one add mul sub opp R'.

Lemma Uint63Ring : ring_theory 0 1 add mul sub opp Logic.eq.

Lemma eq31_correct : forall x y, eqb x y = true -> x=y.

Add Ring Uint63Ring : Uint63Ring
 (decidable eq31_correct,
  constants [Uint63cst]).

Section TestRing.
Let test : forall x y, 1 + x*y + x*x + 1 = 1*1 + 1 + y*x + 1*x*x.
Qed.
End TestRing.