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Add morphism instances for N.to_nat and N.of_nat #48

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Jul 11, 2023
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Add morphism instances for N.to_nat and N.of_nat #48

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32 changes: 30 additions & 2 deletions theories/ssrZ.v
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Expand Up @@ -95,19 +95,47 @@ HB.instance Definition _ := GRing.isSemiAdditive.Build N nat nat_of_bin
nat_of_bin_is_semi_additive.

Fact bin_of_nat_is_multiplicative : multiplicative bin_of_nat.
Proof. by split => // m n; rewrite /GRing.mul /=; lia. Qed.
Proof. by split=> // m n; rewrite /GRing.mul /=; lia. Qed.

#[export]
HB.instance Definition _ := GRing.isMultiplicative.Build nat N bin_of_nat
bin_of_nat_is_multiplicative.

Fact nat_of_bin_is_multiplicative : multiplicative nat_of_bin.
Proof. exact: can2_rmorphism bin_of_natK nat_of_binK. Qed.
Proof. by split=> // m n; rewrite /GRing.mul /=; lia. Qed.

#[export]
HB.instance Definition _ := GRing.isMultiplicative.Build N nat nat_of_bin
nat_of_bin_is_multiplicative.

Fact N_of_nat_is_semi_additive : semi_additive N.of_nat.
Proof. by split=> //= m n; rewrite /GRing.add /=; lia. Qed.

#[export]
HB.instance Definition _ := GRing.isSemiAdditive.Build nat N N.of_nat
N_of_nat_is_semi_additive.

Fact N_to_nat_is_semi_additive : semi_additive N.to_nat.
Proof. by split=> //= m n; rewrite /GRing.add /=; lia. Qed.

#[export]
HB.instance Definition _ := GRing.isSemiAdditive.Build N nat N.to_nat
N_to_nat_is_semi_additive.

Fact N_of_nat_is_multiplicative : multiplicative N.of_nat.
Proof. by split=> // m n; rewrite /GRing.mul /=; lia. Qed.

#[export]
HB.instance Definition _ := GRing.isMultiplicative.Build nat N N.of_nat
N_of_nat_is_multiplicative.

Fact N_to_nat_is_multiplicative : multiplicative N.to_nat.
Proof. by split=> // m n; rewrite /GRing.mul /=; lia. Qed.

#[export]
HB.instance Definition _ := GRing.isMultiplicative.Build N nat N.to_nat
N_to_nat_is_multiplicative.

Implicit Types (m n : Z).

Fact eqZP : Equality.axiom Z.eqb.
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