MathClasses.implementations.fast_integers
Require Import
BigZ
interfaces.abstract_algebra interfaces.integers
interfaces.additional_operations fast_naturals.
Require Export
ZType_integers.
Module BigZ_Integers := ZType_Integers BigZ.
Instance inject_fastN_fastZ: Cast bigN bigZ := BigZ.Pos.
Instance: SemiRing_Morphism inject_fastN_fastZ.
Proof. repeat (split; try apply _); intuition. Qed.
Program Instance bigZ_pow: Pow bigZ bigN := λ x n, BigZ.pow x ('n).
Instance: NatPowSpec bigZ bigN _.
Proof.
split; unfold pow, bigZ_pow.
solve_proper.
intro. apply BigZ.pow_0_r.
intros x n.
rewrite rings.preserves_plus, rings.preserves_1, BigZ.add_1_l.
apply BigZ.pow_succ_r.
change (0 ≤ cast bigN bigZ n).
now apply nat_int.to_semiring_nonneg.
Qed.
Instance fastZ_shiftl: ShiftL bigZ bigN := λ x n, BigZ.shiftl x ('n).
Instance: ShiftLSpec bigZ bigN _.
Proof.
apply shiftl_spec_from_nat_pow.
intros. apply BigZ.shiftl_mul_pow2.
change (0 ≤ cast bigN bigZ n).
now apply nat_int.to_semiring_nonneg.
Qed.
BigZ
interfaces.abstract_algebra interfaces.integers
interfaces.additional_operations fast_naturals.
Require Export
ZType_integers.
Module BigZ_Integers := ZType_Integers BigZ.
Instance inject_fastN_fastZ: Cast bigN bigZ := BigZ.Pos.
Instance: SemiRing_Morphism inject_fastN_fastZ.
Proof. repeat (split; try apply _); intuition. Qed.
Program Instance bigZ_pow: Pow bigZ bigN := λ x n, BigZ.pow x ('n).
Instance: NatPowSpec bigZ bigN _.
Proof.
split; unfold pow, bigZ_pow.
solve_proper.
intro. apply BigZ.pow_0_r.
intros x n.
rewrite rings.preserves_plus, rings.preserves_1, BigZ.add_1_l.
apply BigZ.pow_succ_r.
change (0 ≤ cast bigN bigZ n).
now apply nat_int.to_semiring_nonneg.
Qed.
Instance fastZ_shiftl: ShiftL bigZ bigN := λ x n, BigZ.shiftl x ('n).
Instance: ShiftLSpec bigZ bigN _.
Proof.
apply shiftl_spec_from_nat_pow.
intros. apply BigZ.shiftl_mul_pow2.
change (0 ≤ cast bigN bigZ n).
now apply nat_int.to_semiring_nonneg.
Qed.