Library Coq.Numbers.Cyclic.Int31.Ring31


Int31 numbers defines Z/(2^31)Z, and can hence be equipped

with a ring structure and a ring tactic

Require Import Int31 Cyclic31 CyclicAxioms.

Local Open Scope int31_scope.

Detection of constants

Local Open Scope list_scope.

Ltac isInt31cst_lst l :=
 match l with
 | nil => constr:(true)
 | ?t::?l => match t with
               | D1 => isInt31cst_lst l
               | D0 => isInt31cst_lst l
               | _ => constr:(false)
             end
 | _ => constr:(false)
 end.

Ltac isInt31cst t :=
 match t with
 | I31 ?i0 ?i1 ?i2 ?i3 ?i4 ?i5 ?i6 ?i7 ?i8 ?i9 ?i10
       ?i11 ?i12 ?i13 ?i14 ?i15 ?i16 ?i17 ?i18 ?i19 ?i20
       ?i21 ?i22 ?i23 ?i24 ?i25 ?i26 ?i27 ?i28 ?i29 ?i30 =>
    let l :=
      constr:(i0::i1::i2::i3::i4::i5::i6::i7::i8::i9::i10
      ::i11::i12::i13::i14::i15::i16::i17::i18::i19::i20
      ::i21::i22::i23::i24::i25::i26::i27::i28::i29::i30::nil)
    in isInt31cst_lst l
 | Int31.On => constr:(true)
 | Int31.In => constr:(true)
 | Int31.Tn => constr:(true)
 | Int31.Twon => constr:(true)
 | _ => constr:(false)
 end.

Ltac Int31cst t :=
 match isInt31cst t with
 | true => constr:(t)
 | false => constr:(NotConstant)
 end.

The generic ring structure inferred from the Cyclic structure

Module Int31ring := CyclicRing Int31Cyclic.

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

Lemma Int31_canonic : forall x y, phi x = phi 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 Int31Ring : ring_theory 0 1 add31 mul31 sub31 opp31 Logic.eq.

Lemma eqb31_eq : forall x y, eqb31 x y = true <-> x=y.

Lemma eqb31_correct : forall x y, eqb31 x y = true -> x=y.

Add Ring Int31Ring : Int31Ring
 (decidable eqb31_correct,
  constants [Int31cst]).

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