Skip to content

feat(RingTheory/MvPolynomial/MonomialOrder): remove unnecessary hypothesis #38323

New issue

Have a question about this project? Sign up for a free GitHub account to open an issue and contact its maintainers and the community.

Sign up for GitHub

By clicking “Sign up for GitHub”, you agree to our terms of service and privacy statement. We’ll occasionally send you account related emails.

Already on GitHub? Sign in to your account

Open
NoahW314 wants to merge 1 commit into
base: master
Choose a base branch
from
+13 −5
Open

feat(RingTheory/MvPolynomial/MonomialOrder): remove unnecessary hypothesis #38323

Changes from all commits
Commits
Show all changes
1 commit
Select commit
File filter

Filter by extension

Filter by extension
Conversations
Failed to load comments.
Loading
Jump to
Jump to file
Failed to load files.
Loading
Diff view
Diff view
18 changes: 13 additions & 5 deletions Mathlib/RingTheory/MvPolynomial/MonomialOrder.lean
View file
Open in desktop
Original file line number Diff line number Diff line change
Expand Up @@ -461,13 +461,17 @@ theorem degree_mul_of_right_mem_nonZeroDivisors {f g : MvPolynomial σ R}
add_comm (m.degree f) (m.degree g) ▸ mul_comm f g ▸ degree_mul_of_left_mem_nonZeroDivisors hg hf

theorem leadingCoeff_mul_of_left_mem_nonZeroDivisors {f g : MvPolynomial σ R}
(hf : m.leadingCoeff f ∈ nonZeroDivisors _) (hg : g ≠ 0) :
(hf : m.leadingCoeff f ∈ nonZeroDivisors _) :
m.leadingCoeff (f * g) = m.leadingCoeff f * m.leadingCoeff g := by
by_cases hg : g = 0
· simp [hg]
simp only [leadingCoeff, degree_mul_of_left_mem_nonZeroDivisors hf hg, coeff_mul_of_degree_add]

theorem leadingCoeff_mul_of_right_mem_nonZeroDivisors {f g : MvPolynomial σ R}
(hf : f ≠ 0) (hg : m.leadingCoeff g ∈ nonZeroDivisors _) :
(hg : m.leadingCoeff g ∈ nonZeroDivisors _) :
m.leadingCoeff (f * g) = m.leadingCoeff f * m.leadingCoeff g := by
by_cases hf : f = 0
· simp [hf]
simp only [leadingCoeff, degree_mul_of_right_mem_nonZeroDivisors hf hg, coeff_mul_of_degree_add]

/-- Monomial degree of product -/
Expand All @@ -481,8 +485,10 @@ theorem degree_mul_of_isRegular_left {f g : MvPolynomial σ R}

/-- Multiplicativity of leading coefficients -/
theorem leadingCoeff_mul_of_isRegular_left {f g : MvPolynomial σ R}
(hf : IsRegular (m.leadingCoeff f)) (hg : g ≠ 0) :
(hf : IsRegular (m.leadingCoeff f)) :
m.leadingCoeff (f * g) = m.leadingCoeff f * m.leadingCoeff g := by
by_cases hg : g = 0
· simp [hg]
simp only [leadingCoeff, degree_mul_of_isRegular_left hf hg, coeff_mul_of_degree_add]

/-- Monomial degree of product -/
Expand All @@ -493,8 +499,10 @@ theorem degree_mul_of_isRegular_right {f g : MvPolynomial σ R}

/-- Multiplicativity of leading coefficients -/
theorem leadingCoeff_mul_of_isRegular_right {f g : MvPolynomial σ R}
(hf : f ≠ 0) (hg : IsRegular (m.leadingCoeff g)) :
(hg : IsRegular (m.leadingCoeff g)) :
m.leadingCoeff (f * g) = m.leadingCoeff f * m.leadingCoeff g := by
by_cases hf : f = 0
· simp [hf]
simp only [leadingCoeff, degree_mul_of_isRegular_right hf hg, coeff_mul_of_degree_add]

theorem Monic.mul {f g : MvPolynomial σ R} (hf : m.Monic f) (hg : m.Monic g) :
Expand Down Expand Up @@ -829,7 +837,7 @@ lemma mem_nonZeroDivisors_of_leadingCoeff_mem_nonZeroDivisors
intro g
rw [← not_imp_not, ← m.leadingCoeff_eq_zero_iff (f := f * g)]
intro h
rwa [m.leadingCoeff_mul_of_left_mem_nonZeroDivisors hf h,
rwa [m.leadingCoeff_mul_of_left_mem_nonZeroDivisors hf,
mul_left_mem_nonZeroDivisors_eq_zero_iff hf, m.leadingCoeff_eq_zero_iff]

end Semiring
Expand Down
Loading