Library Coq.Numbers.Cyclic.Int31.Ring31
This library has been deprecated since Coq version 8.10.
Int31 numbers defines Z/(2^31)Z, and can hence be equipped
with a ring structure and a ring tactic
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
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.