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[Merged by Bors] - feat(data/zmod/basic): Explicitly state computable right_inverses instead of just surjectivity #5797

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[Merged by Bors] - feat(data/zmod/basic): Explicitly state computable right_inverses instead of just surjectivity #5797

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19 changes: 14 additions & 5 deletions src/data/zmod/basic.lean
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Original file line number Diff line number Diff line change
Expand Up @@ -214,8 +214,11 @@ begin
rw [val, fin.ext_iff, fin.coe_coe_eq_self] }
end

lemma nat_cast_right_inverse [fact (0 < n)] : function.right_inverse val (coe : ℕ → zmod n) :=
nat_cast_zmod_val

lemma nat_cast_zmod_surjective [fact (0 < n)] : function.surjective (coe : ℕ → zmod n) :=
function.right_inverse.surjective nat_cast_zmod_val
nat_cast_right_inverse.surjective

/-- So-named because the outer coercion is `int.cast` into `zmod`. For `int.cast` into an arbitrary
ring, see `zmod.int_cast_cast`. -/
Expand All @@ -226,8 +229,11 @@ begin
{ rw [coe_coe, int.nat_cast_eq_coe_nat, int.cast_coe_nat, fin.coe_coe_eq_self] }
end

lemma int_cast_right_inverse : function.right_inverse (coe : zmod n → ℤ) (coe : ℤ → zmod n) :=
int_cast_zmod_cast

lemma int_cast_surjective : function.surjective (coe : ℤ → zmod n) :=
function.right_inverse.surjective int_cast_zmod_cast
int_cast_right_inverse.surjective

@[norm_cast]
lemma cast_id : ∀ n (i : zmod n), ↑i = i
Expand Down Expand Up @@ -818,9 +824,12 @@ instance subsingleton_ring_equiv [semiring R] : subsingleton (zmod n ≃+* R) :=
f k = k :=
by { cases n; simp }

lemma ring_hom_surjective [ring R] (f : R →+* (zmod n)) :
function.surjective f :=
function.right_inverse.surjective (ring_hom_map_cast f)
lemma ring_hom_right_inverse [ring R] (f : R →+* (zmod n)) :
function.right_inverse (coe : zmod n → R) f :=
ring_hom_map_cast f

lemma ring_hom_surjective [ring R] (f : R →+* (zmod n)) : function.surjective f :=
(ring_hom_right_inverse f).surjective

lemma ring_hom_eq_of_ker_eq [comm_ring R] (f g : R →+* (zmod n))
(h : f.ker = g.ker) : f = g :=
Expand Down