Standard Library | The Coq Proof Assistant

The Coq Proof Assistant

# 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.
Defined.
End TestRing.