url stringclasses 147
values | commit stringclasses 147
values | file_path stringlengths 7 101 | full_name stringlengths 1 94 | start stringlengths 6 10 | end stringlengths 6 11 | tactic stringlengths 1 11.2k | state_before stringlengths 3 2.09M | state_after stringlengths 6 2.09M | input stringlengths 73 2.09M |
|---|---|---|---|---|---|---|---|---|---|
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/DPAlgebra/Graded/GradeZero.lean | DividedPowerAlgebra.coeff_zero_of_mem_augIdeal | [174, 1] | [196, 34] | rw [map_sum, Finset.sum_eq_single 0] | R : Type u
M : Type v
inst✝⁴ : CommRing R
inst✝³ : AddCommMonoid M
inst✝² : Module R M
inst✝¹ : DecidableEq R
inst✝ : DecidableEq M
f : MvPolynomial (ℕ × M) R
hf : f ∈ supported R {nm | 0 < nm.1}
hf0 : (algebraMapInv R M) (mk f) = 0
⊢ (aeval fun nm => if 0 < nm.1 then 0 else 1) (∑ v ∈ f.support, (monomial v) (coeff v f... | R : Type u
M : Type v
inst✝⁴ : CommRing R
inst✝³ : AddCommMonoid M
inst✝² : Module R M
inst✝¹ : DecidableEq R
inst✝ : DecidableEq M
f : MvPolynomial (ℕ × M) R
hf : f ∈ supported R {nm | 0 < nm.1}
hf0 : (algebraMapInv R M) (mk f) = 0
⊢ (aeval fun nm => if 0 < nm.1 then 0 else 1) ((monomial 0) (coeff 0 f)) = coeff 0 f
c... | Please generate a tactic in lean4 to solve the state.
STATE:
R : Type u
M : Type v
inst✝⁴ : CommRing R
inst✝³ : AddCommMonoid M
inst✝² : Module R M
inst✝¹ : DecidableEq R
inst✝ : DecidableEq M
f : MvPolynomial (ℕ × M) R
hf : f ∈ supported R {nm | 0 < nm.1}
hf0 : (algebraMapInv R M) (mk f) = 0
⊢ (aeval fun nm => if 0 < ... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/DPAlgebra/Graded/GradeZero.lean | DividedPowerAlgebra.coeff_zero_of_mem_augIdeal | [174, 1] | [196, 34] | . simp only [monomial_zero', aeval_C, Algebra.id.map_eq_id, RingHom.id_apply] | R : Type u
M : Type v
inst✝⁴ : CommRing R
inst✝³ : AddCommMonoid M
inst✝² : Module R M
inst✝¹ : DecidableEq R
inst✝ : DecidableEq M
f : MvPolynomial (ℕ × M) R
hf : f ∈ supported R {nm | 0 < nm.1}
hf0 : (algebraMapInv R M) (mk f) = 0
⊢ (aeval fun nm => if 0 < nm.1 then 0 else 1) ((monomial 0) (coeff 0 f)) = coeff 0 f
c... | case h₀
R : Type u
M : Type v
inst✝⁴ : CommRing R
inst✝³ : AddCommMonoid M
inst✝² : Module R M
inst✝¹ : DecidableEq R
inst✝ : DecidableEq M
f : MvPolynomial (ℕ × M) R
hf : f ∈ supported R {nm | 0 < nm.1}
hf0 : (algebraMapInv R M) (mk f) = 0
⊢ ∀ b ∈ f.support, b ≠ 0 → (aeval fun nm => if 0 < nm.1 then 0 else 1) ((monomi... | Please generate a tactic in lean4 to solve the state.
STATE:
R : Type u
M : Type v
inst✝⁴ : CommRing R
inst✝³ : AddCommMonoid M
inst✝² : Module R M
inst✝¹ : DecidableEq R
inst✝ : DecidableEq M
f : MvPolynomial (ℕ × M) R
hf : f ∈ supported R {nm | 0 < nm.1}
hf0 : (algebraMapInv R M) (mk f) = 0
⊢ (aeval fun nm => if 0 < ... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/DPAlgebra/Graded/GradeZero.lean | DividedPowerAlgebra.coeff_zero_of_mem_augIdeal | [174, 1] | [196, 34] | simp only [monomial_zero', aeval_C, Algebra.id.map_eq_id, RingHom.id_apply] | R : Type u
M : Type v
inst✝⁴ : CommRing R
inst✝³ : AddCommMonoid M
inst✝² : Module R M
inst✝¹ : DecidableEq R
inst✝ : DecidableEq M
f : MvPolynomial (ℕ × M) R
hf : f ∈ supported R {nm | 0 < nm.1}
hf0 : (algebraMapInv R M) (mk f) = 0
⊢ (aeval fun nm => if 0 < nm.1 then 0 else 1) ((monomial 0) (coeff 0 f)) = coeff 0 f | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
R : Type u
M : Type v
inst✝⁴ : CommRing R
inst✝³ : AddCommMonoid M
inst✝² : Module R M
inst✝¹ : DecidableEq R
inst✝ : DecidableEq M
f : MvPolynomial (ℕ × M) R
hf : f ∈ supported R {nm | 0 < nm.1}
hf0 : (algebraMapInv R M) (mk f) = 0
⊢ (aeval fun nm => if 0 < ... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/DPAlgebra/Graded/GradeZero.lean | DividedPowerAlgebra.coeff_zero_of_mem_augIdeal | [174, 1] | [196, 34] | intro b hb hb0 | case h₀
R : Type u
M : Type v
inst✝⁴ : CommRing R
inst✝³ : AddCommMonoid M
inst✝² : Module R M
inst✝¹ : DecidableEq R
inst✝ : DecidableEq M
f : MvPolynomial (ℕ × M) R
hf : f ∈ supported R {nm | 0 < nm.1}
hf0 : (algebraMapInv R M) (mk f) = 0
⊢ ∀ b ∈ f.support, b ≠ 0 → (aeval fun nm => if 0 < nm.1 then 0 else 1) ((monomi... | case h₀
R : Type u
M : Type v
inst✝⁴ : CommRing R
inst✝³ : AddCommMonoid M
inst✝² : Module R M
inst✝¹ : DecidableEq R
inst✝ : DecidableEq M
f : MvPolynomial (ℕ × M) R
hf : f ∈ supported R {nm | 0 < nm.1}
hf0 : (algebraMapInv R M) (mk f) = 0
b : ℕ × M →₀ ℕ
hb : b ∈ f.support
hb0 : b ≠ 0
⊢ (aeval fun nm => if 0 < nm.1 th... | Please generate a tactic in lean4 to solve the state.
STATE:
case h₀
R : Type u
M : Type v
inst✝⁴ : CommRing R
inst✝³ : AddCommMonoid M
inst✝² : Module R M
inst✝¹ : DecidableEq R
inst✝ : DecidableEq M
f : MvPolynomial (ℕ × M) R
hf : f ∈ supported R {nm | 0 < nm.1}
hf0 : (algebraMapInv R M) (mk f) = 0
⊢ ∀ b ∈ f.support,... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/DPAlgebra/Graded/GradeZero.lean | DividedPowerAlgebra.coeff_zero_of_mem_augIdeal | [174, 1] | [196, 34] | rw [aeval_monomial, Algebra.id.map_eq_id, RingHom.id_apply] | case h₀
R : Type u
M : Type v
inst✝⁴ : CommRing R
inst✝³ : AddCommMonoid M
inst✝² : Module R M
inst✝¹ : DecidableEq R
inst✝ : DecidableEq M
f : MvPolynomial (ℕ × M) R
hf : f ∈ supported R {nm | 0 < nm.1}
hf0 : (algebraMapInv R M) (mk f) = 0
b : ℕ × M →₀ ℕ
hb : b ∈ f.support
hb0 : b ≠ 0
⊢ (aeval fun nm => if 0 < nm.1 th... | case h₀
R : Type u
M : Type v
inst✝⁴ : CommRing R
inst✝³ : AddCommMonoid M
inst✝² : Module R M
inst✝¹ : DecidableEq R
inst✝ : DecidableEq M
f : MvPolynomial (ℕ × M) R
hf : f ∈ supported R {nm | 0 < nm.1}
hf0 : (algebraMapInv R M) (mk f) = 0
b : ℕ × M →₀ ℕ
hb : b ∈ f.support
hb0 : b ≠ 0
⊢ (coeff b f * b.prod fun i k => ... | Please generate a tactic in lean4 to solve the state.
STATE:
case h₀
R : Type u
M : Type v
inst✝⁴ : CommRing R
inst✝³ : AddCommMonoid M
inst✝² : Module R M
inst✝¹ : DecidableEq R
inst✝ : DecidableEq M
f : MvPolynomial (ℕ × M) R
hf : f ∈ supported R {nm | 0 < nm.1}
hf0 : (algebraMapInv R M) (mk f) = 0
b : ℕ × M →₀ ℕ
hb ... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/DPAlgebra/Graded/GradeZero.lean | DividedPowerAlgebra.coeff_zero_of_mem_augIdeal | [174, 1] | [196, 34] | convert mul_zero (coeff b f) | case h₀
R : Type u
M : Type v
inst✝⁴ : CommRing R
inst✝³ : AddCommMonoid M
inst✝² : Module R M
inst✝¹ : DecidableEq R
inst✝ : DecidableEq M
f : MvPolynomial (ℕ × M) R
hf : f ∈ supported R {nm | 0 < nm.1}
hf0 : (algebraMapInv R M) (mk f) = 0
b : ℕ × M →₀ ℕ
hb : b ∈ f.support
hb0 : b ≠ 0
⊢ (coeff b f * b.prod fun i k => ... | case h.e'_2.h.e'_6
R : Type u
M : Type v
inst✝⁴ : CommRing R
inst✝³ : AddCommMonoid M
inst✝² : Module R M
inst✝¹ : DecidableEq R
inst✝ : DecidableEq M
f : MvPolynomial (ℕ × M) R
hf : f ∈ supported R {nm | 0 < nm.1}
hf0 : (algebraMapInv R M) (mk f) = 0
b : ℕ × M →₀ ℕ
hb : b ∈ f.support
hb0 : b ≠ 0
⊢ (b.prod fun i k => (... | Please generate a tactic in lean4 to solve the state.
STATE:
case h₀
R : Type u
M : Type v
inst✝⁴ : CommRing R
inst✝³ : AddCommMonoid M
inst✝² : Module R M
inst✝¹ : DecidableEq R
inst✝ : DecidableEq M
f : MvPolynomial (ℕ × M) R
hf : f ∈ supported R {nm | 0 < nm.1}
hf0 : (algebraMapInv R M) (mk f) = 0
b : ℕ × M →₀ ℕ
hb ... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/DPAlgebra/Graded/GradeZero.lean | DividedPowerAlgebra.coeff_zero_of_mem_augIdeal | [174, 1] | [196, 34] | obtain ⟨i, hi⟩ := Finsupp.support_nonempty_iff.mpr hb0 | case h.e'_2.h.e'_6
R : Type u
M : Type v
inst✝⁴ : CommRing R
inst✝³ : AddCommMonoid M
inst✝² : Module R M
inst✝¹ : DecidableEq R
inst✝ : DecidableEq M
f : MvPolynomial (ℕ × M) R
hf : f ∈ supported R {nm | 0 < nm.1}
hf0 : (algebraMapInv R M) (mk f) = 0
b : ℕ × M →₀ ℕ
hb : b ∈ f.support
hb0 : b ≠ 0
⊢ (b.prod fun i k => (... | case h.e'_2.h.e'_6.intro
R : Type u
M : Type v
inst✝⁴ : CommRing R
inst✝³ : AddCommMonoid M
inst✝² : Module R M
inst✝¹ : DecidableEq R
inst✝ : DecidableEq M
f : MvPolynomial (ℕ × M) R
hf : f ∈ supported R {nm | 0 < nm.1}
hf0 : (algebraMapInv R M) (mk f) = 0
b : ℕ × M →₀ ℕ
hb : b ∈ f.support
hb0 : b ≠ 0
i : ℕ × M
hi : i... | Please generate a tactic in lean4 to solve the state.
STATE:
case h.e'_2.h.e'_6
R : Type u
M : Type v
inst✝⁴ : CommRing R
inst✝³ : AddCommMonoid M
inst✝² : Module R M
inst✝¹ : DecidableEq R
inst✝ : DecidableEq M
f : MvPolynomial (ℕ × M) R
hf : f ∈ supported R {nm | 0 < nm.1}
hf0 : (algebraMapInv R M) (mk f) = 0
b : ℕ ×... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/DPAlgebra/Graded/GradeZero.lean | DividedPowerAlgebra.coeff_zero_of_mem_augIdeal | [174, 1] | [196, 34] | rw [Finsupp.prod] | case h.e'_2.h.e'_6.intro
R : Type u
M : Type v
inst✝⁴ : CommRing R
inst✝³ : AddCommMonoid M
inst✝² : Module R M
inst✝¹ : DecidableEq R
inst✝ : DecidableEq M
f : MvPolynomial (ℕ × M) R
hf : f ∈ supported R {nm | 0 < nm.1}
hf0 : (algebraMapInv R M) (mk f) = 0
b : ℕ × M →₀ ℕ
hb : b ∈ f.support
hb0 : b ≠ 0
i : ℕ × M
hi : i... | case h.e'_2.h.e'_6.intro
R : Type u
M : Type v
inst✝⁴ : CommRing R
inst✝³ : AddCommMonoid M
inst✝² : Module R M
inst✝¹ : DecidableEq R
inst✝ : DecidableEq M
f : MvPolynomial (ℕ × M) R
hf : f ∈ supported R {nm | 0 < nm.1}
hf0 : (algebraMapInv R M) (mk f) = 0
b : ℕ × M →₀ ℕ
hb : b ∈ f.support
hb0 : b ≠ 0
i : ℕ × M
hi : i... | Please generate a tactic in lean4 to solve the state.
STATE:
case h.e'_2.h.e'_6.intro
R : Type u
M : Type v
inst✝⁴ : CommRing R
inst✝³ : AddCommMonoid M
inst✝² : Module R M
inst✝¹ : DecidableEq R
inst✝ : DecidableEq M
f : MvPolynomial (ℕ × M) R
hf : f ∈ supported R {nm | 0 < nm.1}
hf0 : (algebraMapInv R M) (mk f) = 0
b... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/DPAlgebra/Graded/GradeZero.lean | DividedPowerAlgebra.coeff_zero_of_mem_augIdeal | [174, 1] | [196, 34] | apply Finset.prod_eq_zero hi | case h.e'_2.h.e'_6.intro
R : Type u
M : Type v
inst✝⁴ : CommRing R
inst✝³ : AddCommMonoid M
inst✝² : Module R M
inst✝¹ : DecidableEq R
inst✝ : DecidableEq M
f : MvPolynomial (ℕ × M) R
hf : f ∈ supported R {nm | 0 < nm.1}
hf0 : (algebraMapInv R M) (mk f) = 0
b : ℕ × M →₀ ℕ
hb : b ∈ f.support
hb0 : b ≠ 0
i : ℕ × M
hi : i... | case h.e'_2.h.e'_6.intro
R : Type u
M : Type v
inst✝⁴ : CommRing R
inst✝³ : AddCommMonoid M
inst✝² : Module R M
inst✝¹ : DecidableEq R
inst✝ : DecidableEq M
f : MvPolynomial (ℕ × M) R
hf : f ∈ supported R {nm | 0 < nm.1}
hf0 : (algebraMapInv R M) (mk f) = 0
b : ℕ × M →₀ ℕ
hb : b ∈ f.support
hb0 : b ≠ 0
i : ℕ × M
hi : i... | Please generate a tactic in lean4 to solve the state.
STATE:
case h.e'_2.h.e'_6.intro
R : Type u
M : Type v
inst✝⁴ : CommRing R
inst✝³ : AddCommMonoid M
inst✝² : Module R M
inst✝¹ : DecidableEq R
inst✝ : DecidableEq M
f : MvPolynomial (ℕ × M) R
hf : f ∈ supported R {nm | 0 < nm.1}
hf0 : (algebraMapInv R M) (mk f) = 0
b... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/DPAlgebra/Graded/GradeZero.lean | DividedPowerAlgebra.coeff_zero_of_mem_augIdeal | [174, 1] | [196, 34] | have hi' : 0 < i.fst := by
apply mem_supported.mp hf
rw [Finset.mem_coe, mem_vars]
exact ⟨b, ⟨hb, hi⟩⟩ | case h.e'_2.h.e'_6.intro
R : Type u
M : Type v
inst✝⁴ : CommRing R
inst✝³ : AddCommMonoid M
inst✝² : Module R M
inst✝¹ : DecidableEq R
inst✝ : DecidableEq M
f : MvPolynomial (ℕ × M) R
hf : f ∈ supported R {nm | 0 < nm.1}
hf0 : (algebraMapInv R M) (mk f) = 0
b : ℕ × M →₀ ℕ
hb : b ∈ f.support
hb0 : b ≠ 0
i : ℕ × M
hi : i... | case h.e'_2.h.e'_6.intro
R : Type u
M : Type v
inst✝⁴ : CommRing R
inst✝³ : AddCommMonoid M
inst✝² : Module R M
inst✝¹ : DecidableEq R
inst✝ : DecidableEq M
f : MvPolynomial (ℕ × M) R
hf : f ∈ supported R {nm | 0 < nm.1}
hf0 : (algebraMapInv R M) (mk f) = 0
b : ℕ × M →₀ ℕ
hb : b ∈ f.support
hb0 : b ≠ 0
i : ℕ × M
hi : i... | Please generate a tactic in lean4 to solve the state.
STATE:
case h.e'_2.h.e'_6.intro
R : Type u
M : Type v
inst✝⁴ : CommRing R
inst✝³ : AddCommMonoid M
inst✝² : Module R M
inst✝¹ : DecidableEq R
inst✝ : DecidableEq M
f : MvPolynomial (ℕ × M) R
hf : f ∈ supported R {nm | 0 < nm.1}
hf0 : (algebraMapInv R M) (mk f) = 0
b... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/DPAlgebra/Graded/GradeZero.lean | DividedPowerAlgebra.coeff_zero_of_mem_augIdeal | [174, 1] | [196, 34] | rw [if_pos hi'] | case h.e'_2.h.e'_6.intro
R : Type u
M : Type v
inst✝⁴ : CommRing R
inst✝³ : AddCommMonoid M
inst✝² : Module R M
inst✝¹ : DecidableEq R
inst✝ : DecidableEq M
f : MvPolynomial (ℕ × M) R
hf : f ∈ supported R {nm | 0 < nm.1}
hf0 : (algebraMapInv R M) (mk f) = 0
b : ℕ × M →₀ ℕ
hb : b ∈ f.support
hb0 : b ≠ 0
i : ℕ × M
hi : i... | case h.e'_2.h.e'_6.intro
R : Type u
M : Type v
inst✝⁴ : CommRing R
inst✝³ : AddCommMonoid M
inst✝² : Module R M
inst✝¹ : DecidableEq R
inst✝ : DecidableEq M
f : MvPolynomial (ℕ × M) R
hf : f ∈ supported R {nm | 0 < nm.1}
hf0 : (algebraMapInv R M) (mk f) = 0
b : ℕ × M →₀ ℕ
hb : b ∈ f.support
hb0 : b ≠ 0
i : ℕ × M
hi : i... | Please generate a tactic in lean4 to solve the state.
STATE:
case h.e'_2.h.e'_6.intro
R : Type u
M : Type v
inst✝⁴ : CommRing R
inst✝³ : AddCommMonoid M
inst✝² : Module R M
inst✝¹ : DecidableEq R
inst✝ : DecidableEq M
f : MvPolynomial (ℕ × M) R
hf : f ∈ supported R {nm | 0 < nm.1}
hf0 : (algebraMapInv R M) (mk f) = 0
b... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/DPAlgebra/Graded/GradeZero.lean | DividedPowerAlgebra.coeff_zero_of_mem_augIdeal | [174, 1] | [196, 34] | exact zero_pow (Finsupp.mem_support_iff.mp hi) | case h.e'_2.h.e'_6.intro
R : Type u
M : Type v
inst✝⁴ : CommRing R
inst✝³ : AddCommMonoid M
inst✝² : Module R M
inst✝¹ : DecidableEq R
inst✝ : DecidableEq M
f : MvPolynomial (ℕ × M) R
hf : f ∈ supported R {nm | 0 < nm.1}
hf0 : (algebraMapInv R M) (mk f) = 0
b : ℕ × M →₀ ℕ
hb : b ∈ f.support
hb0 : b ≠ 0
i : ℕ × M
hi : i... | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case h.e'_2.h.e'_6.intro
R : Type u
M : Type v
inst✝⁴ : CommRing R
inst✝³ : AddCommMonoid M
inst✝² : Module R M
inst✝¹ : DecidableEq R
inst✝ : DecidableEq M
f : MvPolynomial (ℕ × M) R
hf : f ∈ supported R {nm | 0 < nm.1}
hf0 : (algebraMapInv R M) (mk f) = 0
b... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/DPAlgebra/Graded/GradeZero.lean | DividedPowerAlgebra.coeff_zero_of_mem_augIdeal | [174, 1] | [196, 34] | apply mem_supported.mp hf | R : Type u
M : Type v
inst✝⁴ : CommRing R
inst✝³ : AddCommMonoid M
inst✝² : Module R M
inst✝¹ : DecidableEq R
inst✝ : DecidableEq M
f : MvPolynomial (ℕ × M) R
hf : f ∈ supported R {nm | 0 < nm.1}
hf0 : (algebraMapInv R M) (mk f) = 0
b : ℕ × M →₀ ℕ
hb : b ∈ f.support
hb0 : b ≠ 0
i : ℕ × M
hi : i ∈ b.support
⊢ 0 < i.1 | case a
R : Type u
M : Type v
inst✝⁴ : CommRing R
inst✝³ : AddCommMonoid M
inst✝² : Module R M
inst✝¹ : DecidableEq R
inst✝ : DecidableEq M
f : MvPolynomial (ℕ × M) R
hf : f ∈ supported R {nm | 0 < nm.1}
hf0 : (algebraMapInv R M) (mk f) = 0
b : ℕ × M →₀ ℕ
hb : b ∈ f.support
hb0 : b ≠ 0
i : ℕ × M
hi : i ∈ b.support
⊢ i ∈... | Please generate a tactic in lean4 to solve the state.
STATE:
R : Type u
M : Type v
inst✝⁴ : CommRing R
inst✝³ : AddCommMonoid M
inst✝² : Module R M
inst✝¹ : DecidableEq R
inst✝ : DecidableEq M
f : MvPolynomial (ℕ × M) R
hf : f ∈ supported R {nm | 0 < nm.1}
hf0 : (algebraMapInv R M) (mk f) = 0
b : ℕ × M →₀ ℕ
hb : b ∈ f.... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/DPAlgebra/Graded/GradeZero.lean | DividedPowerAlgebra.coeff_zero_of_mem_augIdeal | [174, 1] | [196, 34] | rw [Finset.mem_coe, mem_vars] | case a
R : Type u
M : Type v
inst✝⁴ : CommRing R
inst✝³ : AddCommMonoid M
inst✝² : Module R M
inst✝¹ : DecidableEq R
inst✝ : DecidableEq M
f : MvPolynomial (ℕ × M) R
hf : f ∈ supported R {nm | 0 < nm.1}
hf0 : (algebraMapInv R M) (mk f) = 0
b : ℕ × M →₀ ℕ
hb : b ∈ f.support
hb0 : b ≠ 0
i : ℕ × M
hi : i ∈ b.support
⊢ i ∈... | case a
R : Type u
M : Type v
inst✝⁴ : CommRing R
inst✝³ : AddCommMonoid M
inst✝² : Module R M
inst✝¹ : DecidableEq R
inst✝ : DecidableEq M
f : MvPolynomial (ℕ × M) R
hf : f ∈ supported R {nm | 0 < nm.1}
hf0 : (algebraMapInv R M) (mk f) = 0
b : ℕ × M →₀ ℕ
hb : b ∈ f.support
hb0 : b ≠ 0
i : ℕ × M
hi : i ∈ b.support
⊢ ∃ d... | Please generate a tactic in lean4 to solve the state.
STATE:
case a
R : Type u
M : Type v
inst✝⁴ : CommRing R
inst✝³ : AddCommMonoid M
inst✝² : Module R M
inst✝¹ : DecidableEq R
inst✝ : DecidableEq M
f : MvPolynomial (ℕ × M) R
hf : f ∈ supported R {nm | 0 < nm.1}
hf0 : (algebraMapInv R M) (mk f) = 0
b : ℕ × M →₀ ℕ
hb :... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/DPAlgebra/Graded/GradeZero.lean | DividedPowerAlgebra.coeff_zero_of_mem_augIdeal | [174, 1] | [196, 34] | exact ⟨b, ⟨hb, hi⟩⟩ | case a
R : Type u
M : Type v
inst✝⁴ : CommRing R
inst✝³ : AddCommMonoid M
inst✝² : Module R M
inst✝¹ : DecidableEq R
inst✝ : DecidableEq M
f : MvPolynomial (ℕ × M) R
hf : f ∈ supported R {nm | 0 < nm.1}
hf0 : (algebraMapInv R M) (mk f) = 0
b : ℕ × M →₀ ℕ
hb : b ∈ f.support
hb0 : b ≠ 0
i : ℕ × M
hi : i ∈ b.support
⊢ ∃ d... | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case a
R : Type u
M : Type v
inst✝⁴ : CommRing R
inst✝³ : AddCommMonoid M
inst✝² : Module R M
inst✝¹ : DecidableEq R
inst✝ : DecidableEq M
f : MvPolynomial (ℕ × M) R
hf : f ∈ supported R {nm | 0 < nm.1}
hf0 : (algebraMapInv R M) (mk f) = 0
b : ℕ × M →₀ ℕ
hb :... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/DPAlgebra/Graded/GradeZero.lean | DividedPowerAlgebra.coeff_zero_of_mem_augIdeal | [174, 1] | [196, 34] | intro hf' | case h₁
R : Type u
M : Type v
inst✝⁴ : CommRing R
inst✝³ : AddCommMonoid M
inst✝² : Module R M
inst✝¹ : DecidableEq R
inst✝ : DecidableEq M
f : MvPolynomial (ℕ × M) R
hf : f ∈ supported R {nm | 0 < nm.1}
hf0 : (algebraMapInv R M) (mk f) = 0
⊢ 0 ∉ f.support → (aeval fun nm => if 0 < nm.1 then 0 else 1) ((monomial 0) (co... | case h₁
R : Type u
M : Type v
inst✝⁴ : CommRing R
inst✝³ : AddCommMonoid M
inst✝² : Module R M
inst✝¹ : DecidableEq R
inst✝ : DecidableEq M
f : MvPolynomial (ℕ × M) R
hf : f ∈ supported R {nm | 0 < nm.1}
hf0 : (algebraMapInv R M) (mk f) = 0
hf' : 0 ∉ f.support
⊢ (aeval fun nm => if 0 < nm.1 then 0 else 1) ((monomial 0)... | Please generate a tactic in lean4 to solve the state.
STATE:
case h₁
R : Type u
M : Type v
inst✝⁴ : CommRing R
inst✝³ : AddCommMonoid M
inst✝² : Module R M
inst✝¹ : DecidableEq R
inst✝ : DecidableEq M
f : MvPolynomial (ℕ × M) R
hf : f ∈ supported R {nm | 0 < nm.1}
hf0 : (algebraMapInv R M) (mk f) = 0
⊢ 0 ∉ f.support → ... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/DPAlgebra/Graded/GradeZero.lean | DividedPowerAlgebra.coeff_zero_of_mem_augIdeal | [174, 1] | [196, 34] | rw [monomial_zero', aeval_C, Algebra.id.map_eq_id, RingHom.id_apply, ←
not_mem_support_iff.mp hf'] | case h₁
R : Type u
M : Type v
inst✝⁴ : CommRing R
inst✝³ : AddCommMonoid M
inst✝² : Module R M
inst✝¹ : DecidableEq R
inst✝ : DecidableEq M
f : MvPolynomial (ℕ × M) R
hf : f ∈ supported R {nm | 0 < nm.1}
hf0 : (algebraMapInv R M) (mk f) = 0
hf' : 0 ∉ f.support
⊢ (aeval fun nm => if 0 < nm.1 then 0 else 1) ((monomial 0)... | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case h₁
R : Type u
M : Type v
inst✝⁴ : CommRing R
inst✝³ : AddCommMonoid M
inst✝² : Module R M
inst✝¹ : DecidableEq R
inst✝ : DecidableEq M
f : MvPolynomial (ℕ × M) R
hf : f ∈ supported R {nm | 0 < nm.1}
hf0 : (algebraMapInv R M) (mk f) = 0
hf' : 0 ∉ f.suppor... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/DPAlgebra/Graded/GradeZero.lean | DividedPowerAlgebra.augIdeal_eq_span | [204, 1] | [246, 54] | apply le_antisymm | R : Type u
M : Type v
inst✝⁴ : CommRing R
inst✝³ : AddCommMonoid M
inst✝² : Module R M
inst✝¹ : DecidableEq R
inst✝ : DecidableEq M
⊢ augIdeal R M = span (Set.image2 (dp R) {n | 0 < n} Set.univ) | case a
R : Type u
M : Type v
inst✝⁴ : CommRing R
inst✝³ : AddCommMonoid M
inst✝² : Module R M
inst✝¹ : DecidableEq R
inst✝ : DecidableEq M
⊢ augIdeal R M ≤ span (Set.image2 (dp R) {n | 0 < n} Set.univ)
case a
R : Type u
M : Type v
inst✝⁴ : CommRing R
inst✝³ : AddCommMonoid M
inst✝² : Module R M
inst✝¹ : DecidableEq R
... | Please generate a tactic in lean4 to solve the state.
STATE:
R : Type u
M : Type v
inst✝⁴ : CommRing R
inst✝³ : AddCommMonoid M
inst✝² : Module R M
inst✝¹ : DecidableEq R
inst✝ : DecidableEq M
⊢ augIdeal R M = span (Set.image2 (dp R) {n | 0 < n} Set.univ)
TACTIC:
|
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/DPAlgebra/Graded/GradeZero.lean | DividedPowerAlgebra.augIdeal_eq_span | [204, 1] | [246, 54] | intro f0 hf0 | case a
R : Type u
M : Type v
inst✝⁴ : CommRing R
inst✝³ : AddCommMonoid M
inst✝² : Module R M
inst✝¹ : DecidableEq R
inst✝ : DecidableEq M
⊢ augIdeal R M ≤ span (Set.image2 (dp R) {n | 0 < n} Set.univ) | case a
R : Type u
M : Type v
inst✝⁴ : CommRing R
inst✝³ : AddCommMonoid M
inst✝² : Module R M
inst✝¹ : DecidableEq R
inst✝ : DecidableEq M
f0 : DividedPowerAlgebra R M
hf0 : f0 ∈ augIdeal R M
⊢ f0 ∈ span (Set.image2 (dp R) {n | 0 < n} Set.univ) | Please generate a tactic in lean4 to solve the state.
STATE:
case a
R : Type u
M : Type v
inst✝⁴ : CommRing R
inst✝³ : AddCommMonoid M
inst✝² : Module R M
inst✝¹ : DecidableEq R
inst✝ : DecidableEq M
⊢ augIdeal R M ≤ span (Set.image2 (dp R) {n | 0 < n} Set.univ)
TACTIC:
|
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/DPAlgebra/Graded/GradeZero.lean | DividedPowerAlgebra.augIdeal_eq_span | [204, 1] | [246, 54] | obtain ⟨⟨f, hf⟩, rfl⟩ := surjective_of_supported R M f0 | case a
R : Type u
M : Type v
inst✝⁴ : CommRing R
inst✝³ : AddCommMonoid M
inst✝² : Module R M
inst✝¹ : DecidableEq R
inst✝ : DecidableEq M
f0 : DividedPowerAlgebra R M
hf0 : f0 ∈ augIdeal R M
⊢ f0 ∈ span (Set.image2 (dp R) {n | 0 < n} Set.univ) | case a.intro.mk
R : Type u
M : Type v
inst✝⁴ : CommRing R
inst✝³ : AddCommMonoid M
inst✝² : Module R M
inst✝¹ : DecidableEq R
inst✝ : DecidableEq M
f : MvPolynomial (ℕ × M) R
hf : f ∈ supported R {nm | 0 < nm.1}
hf0 : (mk.comp (supported R {nm | 0 < nm.1}).val) ⟨f, hf⟩ ∈ augIdeal R M
⊢ (mk.comp (supported R {nm | 0 < n... | Please generate a tactic in lean4 to solve the state.
STATE:
case a
R : Type u
M : Type v
inst✝⁴ : CommRing R
inst✝³ : AddCommMonoid M
inst✝² : Module R M
inst✝¹ : DecidableEq R
inst✝ : DecidableEq M
f0 : DividedPowerAlgebra R M
hf0 : f0 ∈ augIdeal R M
⊢ f0 ∈ span (Set.image2 (dp R) {n | 0 < n} Set.univ)
TACTIC:
|
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/DPAlgebra/Graded/GradeZero.lean | DividedPowerAlgebra.augIdeal_eq_span | [204, 1] | [246, 54] | have hf0' : coeff 0 f = 0 := coeff_zero_of_mem_augIdeal R M hf hf0 | case a.intro.mk
R : Type u
M : Type v
inst✝⁴ : CommRing R
inst✝³ : AddCommMonoid M
inst✝² : Module R M
inst✝¹ : DecidableEq R
inst✝ : DecidableEq M
f : MvPolynomial (ℕ × M) R
hf : f ∈ supported R {nm | 0 < nm.1}
hf0 : (mk.comp (supported R {nm | 0 < nm.1}).val) ⟨f, hf⟩ ∈ augIdeal R M
⊢ (mk.comp (supported R {nm | 0 < n... | case a.intro.mk
R : Type u
M : Type v
inst✝⁴ : CommRing R
inst✝³ : AddCommMonoid M
inst✝² : Module R M
inst✝¹ : DecidableEq R
inst✝ : DecidableEq M
f : MvPolynomial (ℕ × M) R
hf : f ∈ supported R {nm | 0 < nm.1}
hf0 : (mk.comp (supported R {nm | 0 < nm.1}).val) ⟨f, hf⟩ ∈ augIdeal R M
hf0' : coeff 0 f = 0
⊢ (mk.comp (su... | Please generate a tactic in lean4 to solve the state.
STATE:
case a.intro.mk
R : Type u
M : Type v
inst✝⁴ : CommRing R
inst✝³ : AddCommMonoid M
inst✝² : Module R M
inst✝¹ : DecidableEq R
inst✝ : DecidableEq M
f : MvPolynomial (ℕ × M) R
hf : f ∈ supported R {nm | 0 < nm.1}
hf0 : (mk.comp (supported R {nm | 0 < nm.1}).va... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/DPAlgebra/Graded/GradeZero.lean | DividedPowerAlgebra.augIdeal_eq_span | [204, 1] | [246, 54] | simp only [AlgHom.coe_comp, mkₐ_eq_mk, Subalgebra.coe_val, Function.comp_apply] at hf0 ⊢ | case a.intro.mk
R : Type u
M : Type v
inst✝⁴ : CommRing R
inst✝³ : AddCommMonoid M
inst✝² : Module R M
inst✝¹ : DecidableEq R
inst✝ : DecidableEq M
f : MvPolynomial (ℕ × M) R
hf : f ∈ supported R {nm | 0 < nm.1}
hf0 : (mk.comp (supported R {nm | 0 < nm.1}).val) ⟨f, hf⟩ ∈ augIdeal R M
hf0' : coeff 0 f = 0
⊢ (mk.comp (su... | case a.intro.mk
R : Type u
M : Type v
inst✝⁴ : CommRing R
inst✝³ : AddCommMonoid M
inst✝² : Module R M
inst✝¹ : DecidableEq R
inst✝ : DecidableEq M
f : MvPolynomial (ℕ × M) R
hf : f ∈ supported R {nm | 0 < nm.1}
hf0 : mk f ∈ augIdeal R M
hf0' : coeff 0 f = 0
⊢ mk f ∈ span (Set.image2 (dp R) {n | 0 < n} Set.univ) | Please generate a tactic in lean4 to solve the state.
STATE:
case a.intro.mk
R : Type u
M : Type v
inst✝⁴ : CommRing R
inst✝³ : AddCommMonoid M
inst✝² : Module R M
inst✝¹ : DecidableEq R
inst✝ : DecidableEq M
f : MvPolynomial (ℕ × M) R
hf : f ∈ supported R {nm | 0 < nm.1}
hf0 : (mk.comp (supported R {nm | 0 < nm.1}).va... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/DPAlgebra/Graded/GradeZero.lean | DividedPowerAlgebra.augIdeal_eq_span | [204, 1] | [246, 54] | rw [f.as_sum, map_sum] | case a.intro.mk
R : Type u
M : Type v
inst✝⁴ : CommRing R
inst✝³ : AddCommMonoid M
inst✝² : Module R M
inst✝¹ : DecidableEq R
inst✝ : DecidableEq M
f : MvPolynomial (ℕ × M) R
hf : f ∈ supported R {nm | 0 < nm.1}
hf0 : mk f ∈ augIdeal R M
hf0' : coeff 0 f = 0
⊢ mk f ∈ span (Set.image2 (dp R) {n | 0 < n} Set.univ) | case a.intro.mk
R : Type u
M : Type v
inst✝⁴ : CommRing R
inst✝³ : AddCommMonoid M
inst✝² : Module R M
inst✝¹ : DecidableEq R
inst✝ : DecidableEq M
f : MvPolynomial (ℕ × M) R
hf : f ∈ supported R {nm | 0 < nm.1}
hf0 : mk f ∈ augIdeal R M
hf0' : coeff 0 f = 0
⊢ ∑ x ∈ f.support, mk ((monomial x) (coeff x f)) ∈ span (Set.... | Please generate a tactic in lean4 to solve the state.
STATE:
case a.intro.mk
R : Type u
M : Type v
inst✝⁴ : CommRing R
inst✝³ : AddCommMonoid M
inst✝² : Module R M
inst✝¹ : DecidableEq R
inst✝ : DecidableEq M
f : MvPolynomial (ℕ × M) R
hf : f ∈ supported R {nm | 0 < nm.1}
hf0 : mk f ∈ augIdeal R M
hf0' : coeff 0 f = 0
... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/DPAlgebra/Graded/GradeZero.lean | DividedPowerAlgebra.augIdeal_eq_span | [204, 1] | [246, 54] | refine' Ideal.sum_mem _ _ | case a.intro.mk
R : Type u
M : Type v
inst✝⁴ : CommRing R
inst✝³ : AddCommMonoid M
inst✝² : Module R M
inst✝¹ : DecidableEq R
inst✝ : DecidableEq M
f : MvPolynomial (ℕ × M) R
hf : f ∈ supported R {nm | 0 < nm.1}
hf0 : mk f ∈ augIdeal R M
hf0' : coeff 0 f = 0
⊢ ∑ x ∈ f.support, mk ((monomial x) (coeff x f)) ∈ span (Set.... | case a.intro.mk
R : Type u
M : Type v
inst✝⁴ : CommRing R
inst✝³ : AddCommMonoid M
inst✝² : Module R M
inst✝¹ : DecidableEq R
inst✝ : DecidableEq M
f : MvPolynomial (ℕ × M) R
hf : f ∈ supported R {nm | 0 < nm.1}
hf0 : mk f ∈ augIdeal R M
hf0' : coeff 0 f = 0
⊢ ∀ c ∈ f.support, mk ((monomial c) (coeff c f)) ∈ span (Set.... | Please generate a tactic in lean4 to solve the state.
STATE:
case a.intro.mk
R : Type u
M : Type v
inst✝⁴ : CommRing R
inst✝³ : AddCommMonoid M
inst✝² : Module R M
inst✝¹ : DecidableEq R
inst✝ : DecidableEq M
f : MvPolynomial (ℕ × M) R
hf : f ∈ supported R {nm | 0 < nm.1}
hf0 : mk f ∈ augIdeal R M
hf0' : coeff 0 f = 0
... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/DPAlgebra/Graded/GradeZero.lean | DividedPowerAlgebra.augIdeal_eq_span | [204, 1] | [246, 54] | intro c hc | case a.intro.mk
R : Type u
M : Type v
inst✝⁴ : CommRing R
inst✝³ : AddCommMonoid M
inst✝² : Module R M
inst✝¹ : DecidableEq R
inst✝ : DecidableEq M
f : MvPolynomial (ℕ × M) R
hf : f ∈ supported R {nm | 0 < nm.1}
hf0 : mk f ∈ augIdeal R M
hf0' : coeff 0 f = 0
⊢ ∀ c ∈ f.support, mk ((monomial c) (coeff c f)) ∈ span (Set.... | case a.intro.mk
R : Type u
M : Type v
inst✝⁴ : CommRing R
inst✝³ : AddCommMonoid M
inst✝² : Module R M
inst✝¹ : DecidableEq R
inst✝ : DecidableEq M
f : MvPolynomial (ℕ × M) R
hf : f ∈ supported R {nm | 0 < nm.1}
hf0 : mk f ∈ augIdeal R M
hf0' : coeff 0 f = 0
c : ℕ × M →₀ ℕ
hc : c ∈ f.support
⊢ mk ((monomial c) (coeff c... | Please generate a tactic in lean4 to solve the state.
STATE:
case a.intro.mk
R : Type u
M : Type v
inst✝⁴ : CommRing R
inst✝³ : AddCommMonoid M
inst✝² : Module R M
inst✝¹ : DecidableEq R
inst✝ : DecidableEq M
f : MvPolynomial (ℕ × M) R
hf : f ∈ supported R {nm | 0 < nm.1}
hf0 : mk f ∈ augIdeal R M
hf0' : coeff 0 f = 0
... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/DPAlgebra/Graded/GradeZero.lean | DividedPowerAlgebra.augIdeal_eq_span | [204, 1] | [246, 54] | rw [monomial_eq, Finsupp.prod] | case a.intro.mk
R : Type u
M : Type v
inst✝⁴ : CommRing R
inst✝³ : AddCommMonoid M
inst✝² : Module R M
inst✝¹ : DecidableEq R
inst✝ : DecidableEq M
f : MvPolynomial (ℕ × M) R
hf : f ∈ supported R {nm | 0 < nm.1}
hf0 : mk f ∈ augIdeal R M
hf0' : coeff 0 f = 0
c : ℕ × M →₀ ℕ
hc : c ∈ f.support
⊢ mk ((monomial c) (coeff c... | case a.intro.mk
R : Type u
M : Type v
inst✝⁴ : CommRing R
inst✝³ : AddCommMonoid M
inst✝² : Module R M
inst✝¹ : DecidableEq R
inst✝ : DecidableEq M
f : MvPolynomial (ℕ × M) R
hf : f ∈ supported R {nm | 0 < nm.1}
hf0 : mk f ∈ augIdeal R M
hf0' : coeff 0 f = 0
c : ℕ × M →₀ ℕ
hc : c ∈ f.support
⊢ mk (C (coeff c f) * ∏ a ∈... | Please generate a tactic in lean4 to solve the state.
STATE:
case a.intro.mk
R : Type u
M : Type v
inst✝⁴ : CommRing R
inst✝³ : AddCommMonoid M
inst✝² : Module R M
inst✝¹ : DecidableEq R
inst✝ : DecidableEq M
f : MvPolynomial (ℕ × M) R
hf : f ∈ supported R {nm | 0 < nm.1}
hf0 : mk f ∈ augIdeal R M
hf0' : coeff 0 f = 0
... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/DPAlgebra/Graded/GradeZero.lean | DividedPowerAlgebra.augIdeal_eq_span | [204, 1] | [246, 54] | simp only [_root_.map_mul] | case a.intro.mk
R : Type u
M : Type v
inst✝⁴ : CommRing R
inst✝³ : AddCommMonoid M
inst✝² : Module R M
inst✝¹ : DecidableEq R
inst✝ : DecidableEq M
f : MvPolynomial (ℕ × M) R
hf : f ∈ supported R {nm | 0 < nm.1}
hf0 : mk f ∈ augIdeal R M
hf0' : coeff 0 f = 0
c : ℕ × M →₀ ℕ
hc : c ∈ f.support
⊢ mk (C (coeff c f) * ∏ a ∈... | case a.intro.mk
R : Type u
M : Type v
inst✝⁴ : CommRing R
inst✝³ : AddCommMonoid M
inst✝² : Module R M
inst✝¹ : DecidableEq R
inst✝ : DecidableEq M
f : MvPolynomial (ℕ × M) R
hf : f ∈ supported R {nm | 0 < nm.1}
hf0 : mk f ∈ augIdeal R M
hf0' : coeff 0 f = 0
c : ℕ × M →₀ ℕ
hc : c ∈ f.support
⊢ mk (C (coeff c f)) * mk (... | Please generate a tactic in lean4 to solve the state.
STATE:
case a.intro.mk
R : Type u
M : Type v
inst✝⁴ : CommRing R
inst✝³ : AddCommMonoid M
inst✝² : Module R M
inst✝¹ : DecidableEq R
inst✝ : DecidableEq M
f : MvPolynomial (ℕ × M) R
hf : f ∈ supported R {nm | 0 < nm.1}
hf0 : mk f ∈ augIdeal R M
hf0' : coeff 0 f = 0
... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/DPAlgebra/Graded/GradeZero.lean | DividedPowerAlgebra.augIdeal_eq_span | [204, 1] | [246, 54] | refine' mul_mem_left _ _ _ | case a.intro.mk
R : Type u
M : Type v
inst✝⁴ : CommRing R
inst✝³ : AddCommMonoid M
inst✝² : Module R M
inst✝¹ : DecidableEq R
inst✝ : DecidableEq M
f : MvPolynomial (ℕ × M) R
hf : f ∈ supported R {nm | 0 < nm.1}
hf0 : mk f ∈ augIdeal R M
hf0' : coeff 0 f = 0
c : ℕ × M →₀ ℕ
hc : c ∈ f.support
⊢ mk (C (coeff c f)) * mk (... | case a.intro.mk
R : Type u
M : Type v
inst✝⁴ : CommRing R
inst✝³ : AddCommMonoid M
inst✝² : Module R M
inst✝¹ : DecidableEq R
inst✝ : DecidableEq M
f : MvPolynomial (ℕ × M) R
hf : f ∈ supported R {nm | 0 < nm.1}
hf0 : mk f ∈ augIdeal R M
hf0' : coeff 0 f = 0
c : ℕ × M →₀ ℕ
hc : c ∈ f.support
⊢ mk (∏ a ∈ c.support, X a ... | Please generate a tactic in lean4 to solve the state.
STATE:
case a.intro.mk
R : Type u
M : Type v
inst✝⁴ : CommRing R
inst✝³ : AddCommMonoid M
inst✝² : Module R M
inst✝¹ : DecidableEq R
inst✝ : DecidableEq M
f : MvPolynomial (ℕ × M) R
hf : f ∈ supported R {nm | 0 < nm.1}
hf0 : mk f ∈ augIdeal R M
hf0' : coeff 0 f = 0
... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/DPAlgebra/Graded/GradeZero.lean | DividedPowerAlgebra.augIdeal_eq_span | [204, 1] | [246, 54] | by_cases hc0 : c.support.Nonempty | R : Type u
M : Type v
inst✝⁴ : CommRing R
inst✝³ : AddCommMonoid M
inst✝² : Module R M
inst✝¹ : DecidableEq R
inst✝ : DecidableEq M
f : MvPolynomial (ℕ × M) R
hf : f ∈ supported R {nm | 0 < nm.1}
hf0 : mk f ∈ augIdeal R M
hf0' : coeff 0 f = 0
c : ℕ × M →₀ ℕ
hc : c ∈ f.support
supp_ss : ↑c.support ⊆ {nm | 0 < nm.1}
⊢ mk... | case pos
R : Type u
M : Type v
inst✝⁴ : CommRing R
inst✝³ : AddCommMonoid M
inst✝² : Module R M
inst✝¹ : DecidableEq R
inst✝ : DecidableEq M
f : MvPolynomial (ℕ × M) R
hf : f ∈ supported R {nm | 0 < nm.1}
hf0 : mk f ∈ augIdeal R M
hf0' : coeff 0 f = 0
c : ℕ × M →₀ ℕ
hc : c ∈ f.support
supp_ss : ↑c.support ⊆ {nm | 0 < n... | Please generate a tactic in lean4 to solve the state.
STATE:
R : Type u
M : Type v
inst✝⁴ : CommRing R
inst✝³ : AddCommMonoid M
inst✝² : Module R M
inst✝¹ : DecidableEq R
inst✝ : DecidableEq M
f : MvPolynomial (ℕ × M) R
hf : f ∈ supported R {nm | 0 < nm.1}
hf0 : mk f ∈ augIdeal R M
hf0' : coeff 0 f = 0
c : ℕ × M →₀ ℕ
h... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/DPAlgebra/Graded/GradeZero.lean | DividedPowerAlgebra.augIdeal_eq_span | [204, 1] | [246, 54] | obtain ⟨⟨n, m⟩, hnm⟩ := hc0 | case pos
R : Type u
M : Type v
inst✝⁴ : CommRing R
inst✝³ : AddCommMonoid M
inst✝² : Module R M
inst✝¹ : DecidableEq R
inst✝ : DecidableEq M
f : MvPolynomial (ℕ × M) R
hf : f ∈ supported R {nm | 0 < nm.1}
hf0 : mk f ∈ augIdeal R M
hf0' : coeff 0 f = 0
c : ℕ × M →₀ ℕ
hc : c ∈ f.support
supp_ss : ↑c.support ⊆ {nm | 0 < n... | case pos.intro.mk
R : Type u
M : Type v
inst✝⁴ : CommRing R
inst✝³ : AddCommMonoid M
inst✝² : Module R M
inst✝¹ : DecidableEq R
inst✝ : DecidableEq M
f : MvPolynomial (ℕ × M) R
hf : f ∈ supported R {nm | 0 < nm.1}
hf0 : mk f ∈ augIdeal R M
hf0' : coeff 0 f = 0
c : ℕ × M →₀ ℕ
hc : c ∈ f.support
supp_ss : ↑c.support ⊆ {n... | Please generate a tactic in lean4 to solve the state.
STATE:
case pos
R : Type u
M : Type v
inst✝⁴ : CommRing R
inst✝³ : AddCommMonoid M
inst✝² : Module R M
inst✝¹ : DecidableEq R
inst✝ : DecidableEq M
f : MvPolynomial (ℕ × M) R
hf : f ∈ supported R {nm | 0 < nm.1}
hf0 : mk f ∈ augIdeal R M
hf0' : coeff 0 f = 0
c : ℕ ×... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/DPAlgebra/Graded/GradeZero.lean | DividedPowerAlgebra.augIdeal_eq_span | [204, 1] | [246, 54] | rw [Finset.prod_eq_mul_prod_diff_singleton hnm] | case pos.intro.mk
R : Type u
M : Type v
inst✝⁴ : CommRing R
inst✝³ : AddCommMonoid M
inst✝² : Module R M
inst✝¹ : DecidableEq R
inst✝ : DecidableEq M
f : MvPolynomial (ℕ × M) R
hf : f ∈ supported R {nm | 0 < nm.1}
hf0 : mk f ∈ augIdeal R M
hf0' : coeff 0 f = 0
c : ℕ × M →₀ ℕ
hc : c ∈ f.support
supp_ss : ↑c.support ⊆ {n... | case pos.intro.mk
R : Type u
M : Type v
inst✝⁴ : CommRing R
inst✝³ : AddCommMonoid M
inst✝² : Module R M
inst✝¹ : DecidableEq R
inst✝ : DecidableEq M
f : MvPolynomial (ℕ × M) R
hf : f ∈ supported R {nm | 0 < nm.1}
hf0 : mk f ∈ augIdeal R M
hf0' : coeff 0 f = 0
c : ℕ × M →₀ ℕ
hc : c ∈ f.support
supp_ss : ↑c.support ⊆ {n... | Please generate a tactic in lean4 to solve the state.
STATE:
case pos.intro.mk
R : Type u
M : Type v
inst✝⁴ : CommRing R
inst✝³ : AddCommMonoid M
inst✝² : Module R M
inst✝¹ : DecidableEq R
inst✝ : DecidableEq M
f : MvPolynomial (ℕ × M) R
hf : f ∈ supported R {nm | 0 < nm.1}
hf0 : mk f ∈ augIdeal R M
hf0' : coeff 0 f = ... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/DPAlgebra/Graded/GradeZero.lean | DividedPowerAlgebra.augIdeal_eq_span | [204, 1] | [246, 54] | simp only [_root_.map_mul, map_pow] | case pos.intro.mk
R : Type u
M : Type v
inst✝⁴ : CommRing R
inst✝³ : AddCommMonoid M
inst✝² : Module R M
inst✝¹ : DecidableEq R
inst✝ : DecidableEq M
f : MvPolynomial (ℕ × M) R
hf : f ∈ supported R {nm | 0 < nm.1}
hf0 : mk f ∈ augIdeal R M
hf0' : coeff 0 f = 0
c : ℕ × M →₀ ℕ
hc : c ∈ f.support
supp_ss : ↑c.support ⊆ {n... | case pos.intro.mk
R : Type u
M : Type v
inst✝⁴ : CommRing R
inst✝³ : AddCommMonoid M
inst✝² : Module R M
inst✝¹ : DecidableEq R
inst✝ : DecidableEq M
f : MvPolynomial (ℕ × M) R
hf : f ∈ supported R {nm | 0 < nm.1}
hf0 : mk f ∈ augIdeal R M
hf0' : coeff 0 f = 0
c : ℕ × M →₀ ℕ
hc : c ∈ f.support
supp_ss : ↑c.support ⊆ {n... | Please generate a tactic in lean4 to solve the state.
STATE:
case pos.intro.mk
R : Type u
M : Type v
inst✝⁴ : CommRing R
inst✝³ : AddCommMonoid M
inst✝² : Module R M
inst✝¹ : DecidableEq R
inst✝ : DecidableEq M
f : MvPolynomial (ℕ × M) R
hf : f ∈ supported R {nm | 0 < nm.1}
hf0 : mk f ∈ augIdeal R M
hf0' : coeff 0 f = ... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/DPAlgebra/Graded/GradeZero.lean | DividedPowerAlgebra.augIdeal_eq_span | [204, 1] | [246, 54] | apply
mul_mem_right _ _
(pow_mem_of_mem _ _ _ (Nat.pos_of_ne_zero (Finsupp.mem_support_iff.mp hnm))) | case pos.intro.mk
R : Type u
M : Type v
inst✝⁴ : CommRing R
inst✝³ : AddCommMonoid M
inst✝² : Module R M
inst✝¹ : DecidableEq R
inst✝ : DecidableEq M
f : MvPolynomial (ℕ × M) R
hf : f ∈ supported R {nm | 0 < nm.1}
hf0 : mk f ∈ augIdeal R M
hf0' : coeff 0 f = 0
c : ℕ × M →₀ ℕ
hc : c ∈ f.support
supp_ss : ↑c.support ⊆ {n... | R : Type u
M : Type v
inst✝⁴ : CommRing R
inst✝³ : AddCommMonoid M
inst✝² : Module R M
inst✝¹ : DecidableEq R
inst✝ : DecidableEq M
f : MvPolynomial (ℕ × M) R
hf : f ∈ supported R {nm | 0 < nm.1}
hf0 : mk f ∈ augIdeal R M
hf0' : coeff 0 f = 0
c : ℕ × M →₀ ℕ
hc : c ∈ f.support
supp_ss : ↑c.support ⊆ {nm | 0 < nm.1}
n : ... | Please generate a tactic in lean4 to solve the state.
STATE:
case pos.intro.mk
R : Type u
M : Type v
inst✝⁴ : CommRing R
inst✝³ : AddCommMonoid M
inst✝² : Module R M
inst✝¹ : DecidableEq R
inst✝ : DecidableEq M
f : MvPolynomial (ℕ × M) R
hf : f ∈ supported R {nm | 0 < nm.1}
hf0 : mk f ∈ augIdeal R M
hf0' : coeff 0 f = ... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/DPAlgebra/Graded/GradeZero.lean | DividedPowerAlgebra.augIdeal_eq_span | [204, 1] | [246, 54] | refine subset_span ⟨n,
by simpa only [Set.mem_setOf_eq] using supp_ss hnm,
m, trivial , rfl⟩ | R : Type u
M : Type v
inst✝⁴ : CommRing R
inst✝³ : AddCommMonoid M
inst✝² : Module R M
inst✝¹ : DecidableEq R
inst✝ : DecidableEq M
f : MvPolynomial (ℕ × M) R
hf : f ∈ supported R {nm | 0 < nm.1}
hf0 : mk f ∈ augIdeal R M
hf0' : coeff 0 f = 0
c : ℕ × M →₀ ℕ
hc : c ∈ f.support
supp_ss : ↑c.support ⊆ {nm | 0 < nm.1}
n : ... | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
R : Type u
M : Type v
inst✝⁴ : CommRing R
inst✝³ : AddCommMonoid M
inst✝² : Module R M
inst✝¹ : DecidableEq R
inst✝ : DecidableEq M
f : MvPolynomial (ℕ × M) R
hf : f ∈ supported R {nm | 0 < nm.1}
hf0 : mk f ∈ augIdeal R M
hf0' : coeff 0 f = 0
c : ℕ × M →₀ ℕ
h... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/DPAlgebra/Graded/GradeZero.lean | DividedPowerAlgebra.augIdeal_eq_span | [204, 1] | [246, 54] | simpa only [Set.mem_setOf_eq] using supp_ss hnm | R : Type u
M : Type v
inst✝⁴ : CommRing R
inst✝³ : AddCommMonoid M
inst✝² : Module R M
inst✝¹ : DecidableEq R
inst✝ : DecidableEq M
f : MvPolynomial (ℕ × M) R
hf : f ∈ supported R {nm | 0 < nm.1}
hf0 : mk f ∈ augIdeal R M
hf0' : coeff 0 f = 0
c : ℕ × M →₀ ℕ
hc : c ∈ f.support
supp_ss : ↑c.support ⊆ {nm | 0 < nm.1}
n : ... | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
R : Type u
M : Type v
inst✝⁴ : CommRing R
inst✝³ : AddCommMonoid M
inst✝² : Module R M
inst✝¹ : DecidableEq R
inst✝ : DecidableEq M
f : MvPolynomial (ℕ × M) R
hf : f ∈ supported R {nm | 0 < nm.1}
hf0 : mk f ∈ augIdeal R M
hf0' : coeff 0 f = 0
c : ℕ × M →₀ ℕ
h... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/DPAlgebra/Graded/GradeZero.lean | DividedPowerAlgebra.augIdeal_eq_span | [204, 1] | [246, 54] | rw [not_nonempty_iff_eq_empty, Finsupp.support_eq_empty] at hc0 | case neg
R : Type u
M : Type v
inst✝⁴ : CommRing R
inst✝³ : AddCommMonoid M
inst✝² : Module R M
inst✝¹ : DecidableEq R
inst✝ : DecidableEq M
f : MvPolynomial (ℕ × M) R
hf : f ∈ supported R {nm | 0 < nm.1}
hf0 : mk f ∈ augIdeal R M
hf0' : coeff 0 f = 0
c : ℕ × M →₀ ℕ
hc : c ∈ f.support
supp_ss : ↑c.support ⊆ {nm | 0 < n... | case neg
R : Type u
M : Type v
inst✝⁴ : CommRing R
inst✝³ : AddCommMonoid M
inst✝² : Module R M
inst✝¹ : DecidableEq R
inst✝ : DecidableEq M
f : MvPolynomial (ℕ × M) R
hf : f ∈ supported R {nm | 0 < nm.1}
hf0 : mk f ∈ augIdeal R M
hf0' : coeff 0 f = 0
c : ℕ × M →₀ ℕ
hc : c ∈ f.support
supp_ss : ↑c.support ⊆ {nm | 0 < n... | Please generate a tactic in lean4 to solve the state.
STATE:
case neg
R : Type u
M : Type v
inst✝⁴ : CommRing R
inst✝³ : AddCommMonoid M
inst✝² : Module R M
inst✝¹ : DecidableEq R
inst✝ : DecidableEq M
f : MvPolynomial (ℕ × M) R
hf : f ∈ supported R {nm | 0 < nm.1}
hf0 : mk f ∈ augIdeal R M
hf0' : coeff 0 f = 0
c : ℕ ×... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/DPAlgebra/Graded/GradeZero.lean | DividedPowerAlgebra.augIdeal_eq_span | [204, 1] | [246, 54] | rw [hc0] at hc | case neg
R : Type u
M : Type v
inst✝⁴ : CommRing R
inst✝³ : AddCommMonoid M
inst✝² : Module R M
inst✝¹ : DecidableEq R
inst✝ : DecidableEq M
f : MvPolynomial (ℕ × M) R
hf : f ∈ supported R {nm | 0 < nm.1}
hf0 : mk f ∈ augIdeal R M
hf0' : coeff 0 f = 0
c : ℕ × M →₀ ℕ
hc : c ∈ f.support
supp_ss : ↑c.support ⊆ {nm | 0 < n... | case neg
R : Type u
M : Type v
inst✝⁴ : CommRing R
inst✝³ : AddCommMonoid M
inst✝² : Module R M
inst✝¹ : DecidableEq R
inst✝ : DecidableEq M
f : MvPolynomial (ℕ × M) R
hf : f ∈ supported R {nm | 0 < nm.1}
hf0 : mk f ∈ augIdeal R M
hf0' : coeff 0 f = 0
c : ℕ × M →₀ ℕ
hc : 0 ∈ f.support
supp_ss : ↑c.support ⊆ {nm | 0 < n... | Please generate a tactic in lean4 to solve the state.
STATE:
case neg
R : Type u
M : Type v
inst✝⁴ : CommRing R
inst✝³ : AddCommMonoid M
inst✝² : Module R M
inst✝¹ : DecidableEq R
inst✝ : DecidableEq M
f : MvPolynomial (ℕ × M) R
hf : f ∈ supported R {nm | 0 < nm.1}
hf0 : mk f ∈ augIdeal R M
hf0' : coeff 0 f = 0
c : ℕ ×... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/DPAlgebra/Graded/GradeZero.lean | DividedPowerAlgebra.augIdeal_eq_span | [204, 1] | [246, 54] | exact absurd hf0' (mem_support_iff.mp hc) | case neg
R : Type u
M : Type v
inst✝⁴ : CommRing R
inst✝³ : AddCommMonoid M
inst✝² : Module R M
inst✝¹ : DecidableEq R
inst✝ : DecidableEq M
f : MvPolynomial (ℕ × M) R
hf : f ∈ supported R {nm | 0 < nm.1}
hf0 : mk f ∈ augIdeal R M
hf0' : coeff 0 f = 0
c : ℕ × M →₀ ℕ
hc : 0 ∈ f.support
supp_ss : ↑c.support ⊆ {nm | 0 < n... | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case neg
R : Type u
M : Type v
inst✝⁴ : CommRing R
inst✝³ : AddCommMonoid M
inst✝² : Module R M
inst✝¹ : DecidableEq R
inst✝ : DecidableEq M
f : MvPolynomial (ℕ × M) R
hf : f ∈ supported R {nm | 0 < nm.1}
hf0 : mk f ∈ augIdeal R M
hf0' : coeff 0 f = 0
c : ℕ ×... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/DPAlgebra/Graded/GradeZero.lean | DividedPowerAlgebra.augIdeal_eq_span | [204, 1] | [246, 54] | intro nm hnm | case a.intro.mk
R : Type u
M : Type v
inst✝⁴ : CommRing R
inst✝³ : AddCommMonoid M
inst✝² : Module R M
inst✝¹ : DecidableEq R
inst✝ : DecidableEq M
f : MvPolynomial (ℕ × M) R
hf : f ∈ supported R {nm | 0 < nm.1}
hf0 : mk f ∈ augIdeal R M
hf0' : coeff 0 f = 0
c : ℕ × M →₀ ℕ
hc : c ∈ f.support
⊢ ↑c.support ⊆ {nm | 0 < nm... | case a.intro.mk
R : Type u
M : Type v
inst✝⁴ : CommRing R
inst✝³ : AddCommMonoid M
inst✝² : Module R M
inst✝¹ : DecidableEq R
inst✝ : DecidableEq M
f : MvPolynomial (ℕ × M) R
hf : f ∈ supported R {nm | 0 < nm.1}
hf0 : mk f ∈ augIdeal R M
hf0' : coeff 0 f = 0
c : ℕ × M →₀ ℕ
hc : c ∈ f.support
nm : ℕ × M
hnm : nm ∈ ↑c.su... | Please generate a tactic in lean4 to solve the state.
STATE:
case a.intro.mk
R : Type u
M : Type v
inst✝⁴ : CommRing R
inst✝³ : AddCommMonoid M
inst✝² : Module R M
inst✝¹ : DecidableEq R
inst✝ : DecidableEq M
f : MvPolynomial (ℕ × M) R
hf : f ∈ supported R {nm | 0 < nm.1}
hf0 : mk f ∈ augIdeal R M
hf0' : coeff 0 f = 0
... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/DPAlgebra/Graded/GradeZero.lean | DividedPowerAlgebra.augIdeal_eq_span | [204, 1] | [246, 54] | apply mem_supported.mp hf | case a.intro.mk
R : Type u
M : Type v
inst✝⁴ : CommRing R
inst✝³ : AddCommMonoid M
inst✝² : Module R M
inst✝¹ : DecidableEq R
inst✝ : DecidableEq M
f : MvPolynomial (ℕ × M) R
hf : f ∈ supported R {nm | 0 < nm.1}
hf0 : mk f ∈ augIdeal R M
hf0' : coeff 0 f = 0
c : ℕ × M →₀ ℕ
hc : c ∈ f.support
nm : ℕ × M
hnm : nm ∈ ↑c.su... | case a.intro.mk.a
R : Type u
M : Type v
inst✝⁴ : CommRing R
inst✝³ : AddCommMonoid M
inst✝² : Module R M
inst✝¹ : DecidableEq R
inst✝ : DecidableEq M
f : MvPolynomial (ℕ × M) R
hf : f ∈ supported R {nm | 0 < nm.1}
hf0 : mk f ∈ augIdeal R M
hf0' : coeff 0 f = 0
c : ℕ × M →₀ ℕ
hc : c ∈ f.support
nm : ℕ × M
hnm : nm ∈ ↑c.... | Please generate a tactic in lean4 to solve the state.
STATE:
case a.intro.mk
R : Type u
M : Type v
inst✝⁴ : CommRing R
inst✝³ : AddCommMonoid M
inst✝² : Module R M
inst✝¹ : DecidableEq R
inst✝ : DecidableEq M
f : MvPolynomial (ℕ × M) R
hf : f ∈ supported R {nm | 0 < nm.1}
hf0 : mk f ∈ augIdeal R M
hf0' : coeff 0 f = 0
... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/DPAlgebra/Graded/GradeZero.lean | DividedPowerAlgebra.augIdeal_eq_span | [204, 1] | [246, 54] | simp only [mem_vars, mem_coe, mem_support_iff, ne_eq, Finsupp.mem_support_iff, exists_prop] | case a.intro.mk.a
R : Type u
M : Type v
inst✝⁴ : CommRing R
inst✝³ : AddCommMonoid M
inst✝² : Module R M
inst✝¹ : DecidableEq R
inst✝ : DecidableEq M
f : MvPolynomial (ℕ × M) R
hf : f ∈ supported R {nm | 0 < nm.1}
hf0 : mk f ∈ augIdeal R M
hf0' : coeff 0 f = 0
c : ℕ × M →₀ ℕ
hc : c ∈ f.support
nm : ℕ × M
hnm : nm ∈ ↑c.... | case a.intro.mk.a
R : Type u
M : Type v
inst✝⁴ : CommRing R
inst✝³ : AddCommMonoid M
inst✝² : Module R M
inst✝¹ : DecidableEq R
inst✝ : DecidableEq M
f : MvPolynomial (ℕ × M) R
hf : f ∈ supported R {nm | 0 < nm.1}
hf0 : mk f ∈ augIdeal R M
hf0' : coeff 0 f = 0
c : ℕ × M →₀ ℕ
hc : c ∈ f.support
nm : ℕ × M
hnm : nm ∈ ↑c.... | Please generate a tactic in lean4 to solve the state.
STATE:
case a.intro.mk.a
R : Type u
M : Type v
inst✝⁴ : CommRing R
inst✝³ : AddCommMonoid M
inst✝² : Module R M
inst✝¹ : DecidableEq R
inst✝ : DecidableEq M
f : MvPolynomial (ℕ × M) R
hf : f ∈ supported R {nm | 0 < nm.1}
hf0 : mk f ∈ augIdeal R M
hf0' : coeff 0 f = ... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/DPAlgebra/Graded/GradeZero.lean | DividedPowerAlgebra.augIdeal_eq_span | [204, 1] | [246, 54] | rw [mem_coe, Finsupp.mem_support_iff] at hnm | case a.intro.mk.a
R : Type u
M : Type v
inst✝⁴ : CommRing R
inst✝³ : AddCommMonoid M
inst✝² : Module R M
inst✝¹ : DecidableEq R
inst✝ : DecidableEq M
f : MvPolynomial (ℕ × M) R
hf : f ∈ supported R {nm | 0 < nm.1}
hf0 : mk f ∈ augIdeal R M
hf0' : coeff 0 f = 0
c : ℕ × M →₀ ℕ
hc : c ∈ f.support
nm : ℕ × M
hnm : nm ∈ ↑c.... | case a.intro.mk.a
R : Type u
M : Type v
inst✝⁴ : CommRing R
inst✝³ : AddCommMonoid M
inst✝² : Module R M
inst✝¹ : DecidableEq R
inst✝ : DecidableEq M
f : MvPolynomial (ℕ × M) R
hf : f ∈ supported R {nm | 0 < nm.1}
hf0 : mk f ∈ augIdeal R M
hf0' : coeff 0 f = 0
c : ℕ × M →₀ ℕ
hc : c ∈ f.support
nm : ℕ × M
hnm : c nm ≠ 0... | Please generate a tactic in lean4 to solve the state.
STATE:
case a.intro.mk.a
R : Type u
M : Type v
inst✝⁴ : CommRing R
inst✝³ : AddCommMonoid M
inst✝² : Module R M
inst✝¹ : DecidableEq R
inst✝ : DecidableEq M
f : MvPolynomial (ℕ × M) R
hf : f ∈ supported R {nm | 0 < nm.1}
hf0 : mk f ∈ augIdeal R M
hf0' : coeff 0 f = ... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/DPAlgebra/Graded/GradeZero.lean | DividedPowerAlgebra.augIdeal_eq_span | [204, 1] | [246, 54] | exact ⟨c, ⟨mem_support_iff.mp hc, hnm⟩⟩ | case a.intro.mk.a
R : Type u
M : Type v
inst✝⁴ : CommRing R
inst✝³ : AddCommMonoid M
inst✝² : Module R M
inst✝¹ : DecidableEq R
inst✝ : DecidableEq M
f : MvPolynomial (ℕ × M) R
hf : f ∈ supported R {nm | 0 < nm.1}
hf0 : mk f ∈ augIdeal R M
hf0' : coeff 0 f = 0
c : ℕ × M →₀ ℕ
hc : c ∈ f.support
nm : ℕ × M
hnm : c nm ≠ 0... | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case a.intro.mk.a
R : Type u
M : Type v
inst✝⁴ : CommRing R
inst✝³ : AddCommMonoid M
inst✝² : Module R M
inst✝¹ : DecidableEq R
inst✝ : DecidableEq M
f : MvPolynomial (ℕ × M) R
hf : f ∈ supported R {nm | 0 < nm.1}
hf0 : mk f ∈ augIdeal R M
hf0' : coeff 0 f = ... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/DPAlgebra/Graded/GradeZero.lean | DividedPowerAlgebra.augIdeal_eq_span | [204, 1] | [246, 54] | rw [span_le] | case a
R : Type u
M : Type v
inst✝⁴ : CommRing R
inst✝³ : AddCommMonoid M
inst✝² : Module R M
inst✝¹ : DecidableEq R
inst✝ : DecidableEq M
⊢ span (Set.image2 (dp R) {n | 0 < n} Set.univ) ≤ augIdeal R M | case a
R : Type u
M : Type v
inst✝⁴ : CommRing R
inst✝³ : AddCommMonoid M
inst✝² : Module R M
inst✝¹ : DecidableEq R
inst✝ : DecidableEq M
⊢ Set.image2 (dp R) {n | 0 < n} Set.univ ⊆ ↑(augIdeal R M) | Please generate a tactic in lean4 to solve the state.
STATE:
case a
R : Type u
M : Type v
inst✝⁴ : CommRing R
inst✝³ : AddCommMonoid M
inst✝² : Module R M
inst✝¹ : DecidableEq R
inst✝ : DecidableEq M
⊢ span (Set.image2 (dp R) {n | 0 < n} Set.univ) ≤ augIdeal R M
TACTIC:
|
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/DPAlgebra/Graded/GradeZero.lean | DividedPowerAlgebra.augIdeal_eq_span | [204, 1] | [246, 54] | intro f | case a
R : Type u
M : Type v
inst✝⁴ : CommRing R
inst✝³ : AddCommMonoid M
inst✝² : Module R M
inst✝¹ : DecidableEq R
inst✝ : DecidableEq M
⊢ Set.image2 (dp R) {n | 0 < n} Set.univ ⊆ ↑(augIdeal R M) | case a
R : Type u
M : Type v
inst✝⁴ : CommRing R
inst✝³ : AddCommMonoid M
inst✝² : Module R M
inst✝¹ : DecidableEq R
inst✝ : DecidableEq M
f : DividedPowerAlgebra R M
⊢ f ∈ Set.image2 (dp R) {n | 0 < n} Set.univ → f ∈ ↑(augIdeal R M) | Please generate a tactic in lean4 to solve the state.
STATE:
case a
R : Type u
M : Type v
inst✝⁴ : CommRing R
inst✝³ : AddCommMonoid M
inst✝² : Module R M
inst✝¹ : DecidableEq R
inst✝ : DecidableEq M
⊢ Set.image2 (dp R) {n | 0 < n} Set.univ ⊆ ↑(augIdeal R M)
TACTIC:
|
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/DPAlgebra/Graded/GradeZero.lean | DividedPowerAlgebra.augIdeal_eq_span | [204, 1] | [246, 54] | simp only [Set.mem_image2, Set.mem_setOf_eq, Set.mem_univ, true_and_iff, exists_and_left,
SetLike.mem_coe, forall_exists_index, and_imp] | case a
R : Type u
M : Type v
inst✝⁴ : CommRing R
inst✝³ : AddCommMonoid M
inst✝² : Module R M
inst✝¹ : DecidableEq R
inst✝ : DecidableEq M
f : DividedPowerAlgebra R M
⊢ f ∈ Set.image2 (dp R) {n | 0 < n} Set.univ → f ∈ ↑(augIdeal R M) | case a
R : Type u
M : Type v
inst✝⁴ : CommRing R
inst✝³ : AddCommMonoid M
inst✝² : Module R M
inst✝¹ : DecidableEq R
inst✝ : DecidableEq M
f : DividedPowerAlgebra R M
⊢ ∀ (x : ℕ), 0 < x → ∀ (x_1 : M), dp R x x_1 = f → f ∈ augIdeal R M | Please generate a tactic in lean4 to solve the state.
STATE:
case a
R : Type u
M : Type v
inst✝⁴ : CommRing R
inst✝³ : AddCommMonoid M
inst✝² : Module R M
inst✝¹ : DecidableEq R
inst✝ : DecidableEq M
f : DividedPowerAlgebra R M
⊢ f ∈ Set.image2 (dp R) {n | 0 < n} Set.univ → f ∈ ↑(augIdeal R M)
TACTIC:
|
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/DPAlgebra/Graded/GradeZero.lean | DividedPowerAlgebra.augIdeal_eq_span | [204, 1] | [246, 54] | intro n hn m hf | case a
R : Type u
M : Type v
inst✝⁴ : CommRing R
inst✝³ : AddCommMonoid M
inst✝² : Module R M
inst✝¹ : DecidableEq R
inst✝ : DecidableEq M
f : DividedPowerAlgebra R M
⊢ ∀ (x : ℕ), 0 < x → ∀ (x_1 : M), dp R x x_1 = f → f ∈ augIdeal R M | case a
R : Type u
M : Type v
inst✝⁴ : CommRing R
inst✝³ : AddCommMonoid M
inst✝² : Module R M
inst✝¹ : DecidableEq R
inst✝ : DecidableEq M
f : DividedPowerAlgebra R M
n : ℕ
hn : 0 < n
m : M
hf : dp R n m = f
⊢ f ∈ augIdeal R M | Please generate a tactic in lean4 to solve the state.
STATE:
case a
R : Type u
M : Type v
inst✝⁴ : CommRing R
inst✝³ : AddCommMonoid M
inst✝² : Module R M
inst✝¹ : DecidableEq R
inst✝ : DecidableEq M
f : DividedPowerAlgebra R M
⊢ ∀ (x : ℕ), 0 < x → ∀ (x_1 : M), dp R x x_1 = f → f ∈ augIdeal R M
TACTIC:
|
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/DPAlgebra/Graded/GradeZero.lean | DividedPowerAlgebra.augIdeal_eq_span | [204, 1] | [246, 54] | rw [← hf, mem_augIdeal_iff, algebraMapInv, liftAlgHom_apply_dp] | case a
R : Type u
M : Type v
inst✝⁴ : CommRing R
inst✝³ : AddCommMonoid M
inst✝² : Module R M
inst✝¹ : DecidableEq R
inst✝ : DecidableEq M
f : DividedPowerAlgebra R M
n : ℕ
hn : 0 < n
m : M
hf : dp R n m = f
⊢ f ∈ augIdeal R M | case a
R : Type u
M : Type v
inst✝⁴ : CommRing R
inst✝³ : AddCommMonoid M
inst✝² : Module R M
inst✝¹ : DecidableEq R
inst✝ : DecidableEq M
f : DividedPowerAlgebra R M
n : ℕ
hn : 0 < n
m : M
hf : dp R n m = f
⊢ (dividedPowersBot R).dpow n (0 m) = 0 | Please generate a tactic in lean4 to solve the state.
STATE:
case a
R : Type u
M : Type v
inst✝⁴ : CommRing R
inst✝³ : AddCommMonoid M
inst✝² : Module R M
inst✝¹ : DecidableEq R
inst✝ : DecidableEq M
f : DividedPowerAlgebra R M
n : ℕ
hn : 0 < n
m : M
hf : dp R n m = f
⊢ f ∈ augIdeal R M
TACTIC:
|
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/DPAlgebra/Graded/GradeZero.lean | DividedPowerAlgebra.augIdeal_eq_span | [204, 1] | [246, 54] | simp_rw [LinearMap.zero_apply] | case a
R : Type u
M : Type v
inst✝⁴ : CommRing R
inst✝³ : AddCommMonoid M
inst✝² : Module R M
inst✝¹ : DecidableEq R
inst✝ : DecidableEq M
f : DividedPowerAlgebra R M
n : ℕ
hn : 0 < n
m : M
hf : dp R n m = f
⊢ (dividedPowersBot R).dpow n (0 m) = 0 | case a
R : Type u
M : Type v
inst✝⁴ : CommRing R
inst✝³ : AddCommMonoid M
inst✝² : Module R M
inst✝¹ : DecidableEq R
inst✝ : DecidableEq M
f : DividedPowerAlgebra R M
n : ℕ
hn : 0 < n
m : M
hf : dp R n m = f
⊢ (dividedPowersBot R).dpow n 0 = 0 | Please generate a tactic in lean4 to solve the state.
STATE:
case a
R : Type u
M : Type v
inst✝⁴ : CommRing R
inst✝³ : AddCommMonoid M
inst✝² : Module R M
inst✝¹ : DecidableEq R
inst✝ : DecidableEq M
f : DividedPowerAlgebra R M
n : ℕ
hn : 0 < n
m : M
hf : dp R n m = f
⊢ (dividedPowersBot R).dpow n (0 m) = 0
TACTIC:
|
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/DPAlgebra/Graded/GradeZero.lean | DividedPowerAlgebra.augIdeal_eq_span | [204, 1] | [246, 54] | rw [DividedPowers.dpow_eval_zero _ (ne_of_gt hn)] | case a
R : Type u
M : Type v
inst✝⁴ : CommRing R
inst✝³ : AddCommMonoid M
inst✝² : Module R M
inst✝¹ : DecidableEq R
inst✝ : DecidableEq M
f : DividedPowerAlgebra R M
n : ℕ
hn : 0 < n
m : M
hf : dp R n m = f
⊢ (dividedPowersBot R).dpow n 0 = 0 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case a
R : Type u
M : Type v
inst✝⁴ : CommRing R
inst✝³ : AddCommMonoid M
inst✝² : Module R M
inst✝¹ : DecidableEq R
inst✝ : DecidableEq M
f : DividedPowerAlgebra R M
n : ℕ
hn : 0 < n
m : M
hf : dp R n m = f
⊢ (dividedPowersBot R).dpow n 0 = 0
TACTIC:
|
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/DPAlgebra/Graded/GradeZero.lean | DividedPowerAlgebra.right_inv' | [248, 1] | [251, 37] | rw [proj'_zero_comp_algebraMap] | R : Type u
M : Type v
inst✝⁴ : CommRing R
inst✝³ : AddCommMonoid M
inst✝² : Module R M
inst✝¹ : DecidableEq R
inst✝ : DecidableEq M
x : R
⊢ (algebraMapInv R M) ↑((⇑(proj' R M 0) ∘ ⇑(algebraMap R (DividedPowerAlgebra R M))) x) = x | R : Type u
M : Type v
inst✝⁴ : CommRing R
inst✝³ : AddCommMonoid M
inst✝² : Module R M
inst✝¹ : DecidableEq R
inst✝ : DecidableEq M
x : R
⊢ (algebraMapInv R M) ((algebraMap R (DividedPowerAlgebra R M)) x) = x | Please generate a tactic in lean4 to solve the state.
STATE:
R : Type u
M : Type v
inst✝⁴ : CommRing R
inst✝³ : AddCommMonoid M
inst✝² : Module R M
inst✝¹ : DecidableEq R
inst✝ : DecidableEq M
x : R
⊢ (algebraMapInv R M) ↑((⇑(proj' R M 0) ∘ ⇑(algebraMap R (DividedPowerAlgebra R M))) x) = x
TACTIC:
|
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/DPAlgebra/Graded/GradeZero.lean | DividedPowerAlgebra.right_inv' | [248, 1] | [251, 37] | exact algebraMap_leftInverse R M x | R : Type u
M : Type v
inst✝⁴ : CommRing R
inst✝³ : AddCommMonoid M
inst✝² : Module R M
inst✝¹ : DecidableEq R
inst✝ : DecidableEq M
x : R
⊢ (algebraMapInv R M) ((algebraMap R (DividedPowerAlgebra R M)) x) = x | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
R : Type u
M : Type v
inst✝⁴ : CommRing R
inst✝³ : AddCommMonoid M
inst✝² : Module R M
inst✝¹ : DecidableEq R
inst✝ : DecidableEq M
x : R
⊢ (algebraMapInv R M) ((algebraMap R (DividedPowerAlgebra R M)) x) = x
TACTIC:
|
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/DPAlgebra/Graded/GradeZero.lean | DividedPowerAlgebra.left_inv' | [253, 1] | [258, 83] | ext | R : Type u
M : Type v
inst✝⁴ : CommRing R
inst✝³ : AddCommMonoid M
inst✝² : Module R M
inst✝¹ : DecidableEq R
inst✝ : DecidableEq M
x : ↥(grade R M 0)
⊢ (⇑(proj' R M 0) ∘ ⇑(algebraMap R (DividedPowerAlgebra R M))) ((algebraMapInv R M) ↑x) = x | case a
R : Type u
M : Type v
inst✝⁴ : CommRing R
inst✝³ : AddCommMonoid M
inst✝² : Module R M
inst✝¹ : DecidableEq R
inst✝ : DecidableEq M
x : ↥(grade R M 0)
⊢ ↑((⇑(proj' R M 0) ∘ ⇑(algebraMap R (DividedPowerAlgebra R M))) ((algebraMapInv R M) ↑x)) = ↑x | Please generate a tactic in lean4 to solve the state.
STATE:
R : Type u
M : Type v
inst✝⁴ : CommRing R
inst✝³ : AddCommMonoid M
inst✝² : Module R M
inst✝¹ : DecidableEq R
inst✝ : DecidableEq M
x : ↥(grade R M 0)
⊢ (⇑(proj' R M 0) ∘ ⇑(algebraMap R (DividedPowerAlgebra R M))) ((algebraMapInv R M) ↑x) = x
TACTIC:
|
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/DPAlgebra/Graded/GradeZero.lean | DividedPowerAlgebra.left_inv' | [253, 1] | [258, 83] | simp only [proj', proj, LinearMap.coe_mk, AddHom.coe_mk, Function.comp_apply] | case a
R : Type u
M : Type v
inst✝⁴ : CommRing R
inst✝³ : AddCommMonoid M
inst✝² : Module R M
inst✝¹ : DecidableEq R
inst✝ : DecidableEq M
x : ↥(grade R M 0)
⊢ ↑((⇑(proj' R M 0) ∘ ⇑(algebraMap R (DividedPowerAlgebra R M))) ((algebraMapInv R M) ↑x)) = ↑x | case a
R : Type u
M : Type v
inst✝⁴ : CommRing R
inst✝³ : AddCommMonoid M
inst✝² : Module R M
inst✝¹ : DecidableEq R
inst✝ : DecidableEq M
x : ↥(grade R M 0)
⊢ ↑(((DirectSum.decompose (grade R M)) ((algebraMap R (DividedPowerAlgebra R M)) ((algebraMapInv R M) ↑x))) 0) = ↑x | Please generate a tactic in lean4 to solve the state.
STATE:
case a
R : Type u
M : Type v
inst✝⁴ : CommRing R
inst✝³ : AddCommMonoid M
inst✝² : Module R M
inst✝¹ : DecidableEq R
inst✝ : DecidableEq M
x : ↥(grade R M 0)
⊢ ↑((⇑(proj' R M 0) ∘ ⇑(algebraMap R (DividedPowerAlgebra R M))) ((algebraMapInv R M) ↑x)) = ↑x
TACTI... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/DPAlgebra/Graded/GradeZero.lean | DividedPowerAlgebra.left_inv' | [253, 1] | [258, 83] | conv_rhs => rw [← DirectSum.decompose_of_mem_same _ x.2] | case a
R : Type u
M : Type v
inst✝⁴ : CommRing R
inst✝³ : AddCommMonoid M
inst✝² : Module R M
inst✝¹ : DecidableEq R
inst✝ : DecidableEq M
x : ↥(grade R M 0)
⊢ ↑(((DirectSum.decompose (grade R M)) ((algebraMap R (DividedPowerAlgebra R M)) ((algebraMapInv R M) ↑x))) 0) = ↑x | case a
R : Type u
M : Type v
inst✝⁴ : CommRing R
inst✝³ : AddCommMonoid M
inst✝² : Module R M
inst✝¹ : DecidableEq R
inst✝ : DecidableEq M
x : ↥(grade R M 0)
⊢ ↑(((DirectSum.decompose (grade R M)) ((algebraMap R (DividedPowerAlgebra R M)) ((algebraMapInv R M) ↑x))) 0) =
↑(((DirectSum.decompose (grade R M)) ↑x) 0) | Please generate a tactic in lean4 to solve the state.
STATE:
case a
R : Type u
M : Type v
inst✝⁴ : CommRing R
inst✝³ : AddCommMonoid M
inst✝² : Module R M
inst✝¹ : DecidableEq R
inst✝ : DecidableEq M
x : ↥(grade R M 0)
⊢ ↑(((DirectSum.decompose (grade R M)) ((algebraMap R (DividedPowerAlgebra R M)) ((algebraMapInv R M)... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/DPAlgebra/Graded/GradeZero.lean | DividedPowerAlgebra.left_inv' | [253, 1] | [258, 83] | simp only [algebraMap_right_inv_of_degree_zero R M x, decompose_coe, of_eq_same] | case a
R : Type u
M : Type v
inst✝⁴ : CommRing R
inst✝³ : AddCommMonoid M
inst✝² : Module R M
inst✝¹ : DecidableEq R
inst✝ : DecidableEq M
x : ↥(grade R M 0)
⊢ ↑(((DirectSum.decompose (grade R M)) ((algebraMap R (DividedPowerAlgebra R M)) ((algebraMapInv R M) ↑x))) 0) =
↑(((DirectSum.decompose (grade R M)) ↑x) 0) | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case a
R : Type u
M : Type v
inst✝⁴ : CommRing R
inst✝³ : AddCommMonoid M
inst✝² : Module R M
inst✝¹ : DecidableEq R
inst✝ : DecidableEq M
x : ↥(grade R M 0)
⊢ ↑(((DirectSum.decompose (grade R M)) ((algebraMap R (DividedPowerAlgebra R M)) ((algebraMapInv R M)... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/DPAlgebra/Graded/GradeZero.lean | DividedPowerAlgebra.lift_augIdeal_le | [260, 1] | [266, 42] | simp only [augIdeal_eq_span, Ideal.map_span, Ideal.span_le, SetLike.mem_coe] | R : Type u
M : Type v
inst✝⁶ : CommRing R
inst✝⁵ : AddCommMonoid M
inst✝⁴ : Module R M
inst✝³ : DecidableEq R
inst✝² : DecidableEq M
A : Type u_1
inst✝¹ : CommRing A
inst✝ : Algebra R A
I : Ideal A
hI : DividedPowers I
φ : M →ₗ[R] A
hφ : ∀ (m : M), φ m ∈ I
⊢ Ideal.map (lift hI φ hφ) (augIdeal R M) ≤ I | R : Type u
M : Type v
inst✝⁶ : CommRing R
inst✝⁵ : AddCommMonoid M
inst✝⁴ : Module R M
inst✝³ : DecidableEq R
inst✝² : DecidableEq M
A : Type u_1
inst✝¹ : CommRing A
inst✝ : Algebra R A
I : Ideal A
hI : DividedPowers I
φ : M →ₗ[R] A
hφ : ∀ (m : M), φ m ∈ I
⊢ ⇑(lift hI φ hφ) '' Set.image2 (dp R) {n | 0 < n} Set.univ ⊆ ↑... | Please generate a tactic in lean4 to solve the state.
STATE:
R : Type u
M : Type v
inst✝⁶ : CommRing R
inst✝⁵ : AddCommMonoid M
inst✝⁴ : Module R M
inst✝³ : DecidableEq R
inst✝² : DecidableEq M
A : Type u_1
inst✝¹ : CommRing A
inst✝ : Algebra R A
I : Ideal A
hI : DividedPowers I
φ : M →ₗ[R] A
hφ : ∀ (m : M), φ m ∈ I
⊢ ... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/DPAlgebra/Graded/GradeZero.lean | DividedPowerAlgebra.lift_augIdeal_le | [260, 1] | [266, 42] | rintro y ⟨x, ⟨n, hn, m, _, rfl⟩, rfl⟩ | R : Type u
M : Type v
inst✝⁶ : CommRing R
inst✝⁵ : AddCommMonoid M
inst✝⁴ : Module R M
inst✝³ : DecidableEq R
inst✝² : DecidableEq M
A : Type u_1
inst✝¹ : CommRing A
inst✝ : Algebra R A
I : Ideal A
hI : DividedPowers I
φ : M →ₗ[R] A
hφ : ∀ (m : M), φ m ∈ I
⊢ ⇑(lift hI φ hφ) '' Set.image2 (dp R) {n | 0 < n} Set.univ ⊆ ↑... | case intro.intro.intro.intro.intro.intro
R : Type u
M : Type v
inst✝⁶ : CommRing R
inst✝⁵ : AddCommMonoid M
inst✝⁴ : Module R M
inst✝³ : DecidableEq R
inst✝² : DecidableEq M
A : Type u_1
inst✝¹ : CommRing A
inst✝ : Algebra R A
I : Ideal A
hI : DividedPowers I
φ : M →ₗ[R] A
hφ : ∀ (m : M), φ m ∈ I
n : ℕ
hn : n ∈ {n | 0 ... | Please generate a tactic in lean4 to solve the state.
STATE:
R : Type u
M : Type v
inst✝⁶ : CommRing R
inst✝⁵ : AddCommMonoid M
inst✝⁴ : Module R M
inst✝³ : DecidableEq R
inst✝² : DecidableEq M
A : Type u_1
inst✝¹ : CommRing A
inst✝ : Algebra R A
I : Ideal A
hI : DividedPowers I
φ : M →ₗ[R] A
hφ : ∀ (m : M), φ m ∈ I
⊢ ... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/DPAlgebra/Graded/GradeZero.lean | DividedPowerAlgebra.lift_augIdeal_le | [260, 1] | [266, 42] | simp only [liftAlgHom_apply_dp] | case intro.intro.intro.intro.intro.intro
R : Type u
M : Type v
inst✝⁶ : CommRing R
inst✝⁵ : AddCommMonoid M
inst✝⁴ : Module R M
inst✝³ : DecidableEq R
inst✝² : DecidableEq M
A : Type u_1
inst✝¹ : CommRing A
inst✝ : Algebra R A
I : Ideal A
hI : DividedPowers I
φ : M →ₗ[R] A
hφ : ∀ (m : M), φ m ∈ I
n : ℕ
hn : n ∈ {n | 0 ... | case intro.intro.intro.intro.intro.intro
R : Type u
M : Type v
inst✝⁶ : CommRing R
inst✝⁵ : AddCommMonoid M
inst✝⁴ : Module R M
inst✝³ : DecidableEq R
inst✝² : DecidableEq M
A : Type u_1
inst✝¹ : CommRing A
inst✝ : Algebra R A
I : Ideal A
hI : DividedPowers I
φ : M →ₗ[R] A
hφ : ∀ (m : M), φ m ∈ I
n : ℕ
hn : n ∈ {n | 0 ... | Please generate a tactic in lean4 to solve the state.
STATE:
case intro.intro.intro.intro.intro.intro
R : Type u
M : Type v
inst✝⁶ : CommRing R
inst✝⁵ : AddCommMonoid M
inst✝⁴ : Module R M
inst✝³ : DecidableEq R
inst✝² : DecidableEq M
A : Type u_1
inst✝¹ : CommRing A
inst✝ : Algebra R A
I : Ideal A
hI : DividedPowers I... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/DPAlgebra/Graded/GradeZero.lean | DividedPowerAlgebra.lift_augIdeal_le | [260, 1] | [266, 42] | refine hI.dpow_mem (ne_of_gt hn) (hφ m) | case intro.intro.intro.intro.intro.intro
R : Type u
M : Type v
inst✝⁶ : CommRing R
inst✝⁵ : AddCommMonoid M
inst✝⁴ : Module R M
inst✝³ : DecidableEq R
inst✝² : DecidableEq M
A : Type u_1
inst✝¹ : CommRing A
inst✝ : Algebra R A
I : Ideal A
hI : DividedPowers I
φ : M →ₗ[R] A
hφ : ∀ (m : M), φ m ∈ I
n : ℕ
hn : n ∈ {n | 0 ... | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case intro.intro.intro.intro.intro.intro
R : Type u
M : Type v
inst✝⁶ : CommRing R
inst✝⁵ : AddCommMonoid M
inst✝⁴ : Module R M
inst✝³ : DecidableEq R
inst✝² : DecidableEq M
A : Type u_1
inst✝¹ : CommRing A
inst✝ : Algebra R A
I : Ideal A
hI : DividedPowers I... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/DPAlgebra/Graded/GradeZero.lean | DividedPowerAlgebra.algebraMap_mem_grade_zero | [295, 1] | [302, 14] | rw [mem_grade_iff] | R : Type u
M : Type v
inst✝⁴ : CommRing R
inst✝³ : AddCommMonoid M
inst✝² : Module R M
inst✝¹ : DecidableEq R
inst✝ : DecidableEq M
r : R
⊢ (algebraMap R (DividedPowerAlgebra R M)) r ∈ grade R M 0 | R : Type u
M : Type v
inst✝⁴ : CommRing R
inst✝³ : AddCommMonoid M
inst✝² : Module R M
inst✝¹ : DecidableEq R
inst✝ : DecidableEq M
r : R
⊢ ∃ p ∈ weightedHomogeneousSubmodule R Prod.fst 0, mk p = (algebraMap R (DividedPowerAlgebra R M)) r | Please generate a tactic in lean4 to solve the state.
STATE:
R : Type u
M : Type v
inst✝⁴ : CommRing R
inst✝³ : AddCommMonoid M
inst✝² : Module R M
inst✝¹ : DecidableEq R
inst✝ : DecidableEq M
r : R
⊢ (algebraMap R (DividedPowerAlgebra R M)) r ∈ grade R M 0
TACTIC:
|
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/DPAlgebra/Graded/GradeZero.lean | DividedPowerAlgebra.algebraMap_mem_grade_zero | [295, 1] | [302, 14] | use C r | R : Type u
M : Type v
inst✝⁴ : CommRing R
inst✝³ : AddCommMonoid M
inst✝² : Module R M
inst✝¹ : DecidableEq R
inst✝ : DecidableEq M
r : R
⊢ ∃ p ∈ weightedHomogeneousSubmodule R Prod.fst 0, mk p = (algebraMap R (DividedPowerAlgebra R M)) r | case h
R : Type u
M : Type v
inst✝⁴ : CommRing R
inst✝³ : AddCommMonoid M
inst✝² : Module R M
inst✝¹ : DecidableEq R
inst✝ : DecidableEq M
r : R
⊢ C r ∈ weightedHomogeneousSubmodule R Prod.fst 0 ∧ mk (C r) = (algebraMap R (DividedPowerAlgebra R M)) r | Please generate a tactic in lean4 to solve the state.
STATE:
R : Type u
M : Type v
inst✝⁴ : CommRing R
inst✝³ : AddCommMonoid M
inst✝² : Module R M
inst✝¹ : DecidableEq R
inst✝ : DecidableEq M
r : R
⊢ ∃ p ∈ weightedHomogeneousSubmodule R Prod.fst 0, mk p = (algebraMap R (DividedPowerAlgebra R M)) r
TACTIC:
|
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/DPAlgebra/Graded/GradeZero.lean | DividedPowerAlgebra.algebraMap_mem_grade_zero | [295, 1] | [302, 14] | constructor | case h
R : Type u
M : Type v
inst✝⁴ : CommRing R
inst✝³ : AddCommMonoid M
inst✝² : Module R M
inst✝¹ : DecidableEq R
inst✝ : DecidableEq M
r : R
⊢ C r ∈ weightedHomogeneousSubmodule R Prod.fst 0 ∧ mk (C r) = (algebraMap R (DividedPowerAlgebra R M)) r | case h.left
R : Type u
M : Type v
inst✝⁴ : CommRing R
inst✝³ : AddCommMonoid M
inst✝² : Module R M
inst✝¹ : DecidableEq R
inst✝ : DecidableEq M
r : R
⊢ C r ∈ weightedHomogeneousSubmodule R Prod.fst 0
case h.right
R : Type u
M : Type v
inst✝⁴ : CommRing R
inst✝³ : AddCommMonoid M
inst✝² : Module R M
inst✝¹ : DecidableE... | Please generate a tactic in lean4 to solve the state.
STATE:
case h
R : Type u
M : Type v
inst✝⁴ : CommRing R
inst✝³ : AddCommMonoid M
inst✝² : Module R M
inst✝¹ : DecidableEq R
inst✝ : DecidableEq M
r : R
⊢ C r ∈ weightedHomogeneousSubmodule R Prod.fst 0 ∧ mk (C r) = (algebraMap R (DividedPowerAlgebra R M)) r
TACTIC:
|
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/DPAlgebra/Graded/GradeZero.lean | DividedPowerAlgebra.algebraMap_mem_grade_zero | [295, 1] | [302, 14] | simp only [mem_weightedHomogeneousSubmodule] | case h.left
R : Type u
M : Type v
inst✝⁴ : CommRing R
inst✝³ : AddCommMonoid M
inst✝² : Module R M
inst✝¹ : DecidableEq R
inst✝ : DecidableEq M
r : R
⊢ C r ∈ weightedHomogeneousSubmodule R Prod.fst 0 | case h.left
R : Type u
M : Type v
inst✝⁴ : CommRing R
inst✝³ : AddCommMonoid M
inst✝² : Module R M
inst✝¹ : DecidableEq R
inst✝ : DecidableEq M
r : R
⊢ IsWeightedHomogeneous Prod.fst (C r) 0 | Please generate a tactic in lean4 to solve the state.
STATE:
case h.left
R : Type u
M : Type v
inst✝⁴ : CommRing R
inst✝³ : AddCommMonoid M
inst✝² : Module R M
inst✝¹ : DecidableEq R
inst✝ : DecidableEq M
r : R
⊢ C r ∈ weightedHomogeneousSubmodule R Prod.fst 0
TACTIC:
|
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/DPAlgebra/Graded/GradeZero.lean | DividedPowerAlgebra.algebraMap_mem_grade_zero | [295, 1] | [302, 14] | exact isWeightedHomogeneous_C Prod.fst r | case h.left
R : Type u
M : Type v
inst✝⁴ : CommRing R
inst✝³ : AddCommMonoid M
inst✝² : Module R M
inst✝¹ : DecidableEq R
inst✝ : DecidableEq M
r : R
⊢ IsWeightedHomogeneous Prod.fst (C r) 0 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case h.left
R : Type u
M : Type v
inst✝⁴ : CommRing R
inst✝³ : AddCommMonoid M
inst✝² : Module R M
inst✝¹ : DecidableEq R
inst✝ : DecidableEq M
r : R
⊢ IsWeightedHomogeneous Prod.fst (C r) 0
TACTIC:
|
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/DPAlgebra/Graded/GradeZero.lean | DividedPowerAlgebra.algebraMap_mem_grade_zero | [295, 1] | [302, 14] | rw [mk_C] | case h.right
R : Type u
M : Type v
inst✝⁴ : CommRing R
inst✝³ : AddCommMonoid M
inst✝² : Module R M
inst✝¹ : DecidableEq R
inst✝ : DecidableEq M
r : R
⊢ mk (C r) = (algebraMap R (DividedPowerAlgebra R M)) r | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case h.right
R : Type u
M : Type v
inst✝⁴ : CommRing R
inst✝³ : AddCommMonoid M
inst✝² : Module R M
inst✝¹ : DecidableEq R
inst✝ : DecidableEq M
r : R
⊢ mk (C r) = (algebraMap R (DividedPowerAlgebra R M)) r
TACTIC:
|
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/DPAlgebra/Graded/GradeZero.lean | DividedPowerAlgebra.grade0Subalgebra_eq_bot | [313, 1] | [318, 63] | rw [eq_bot_iff] | R : Type u
M : Type v
inst✝⁴ : CommRing R
inst✝³ : AddCommMonoid M
inst✝² : Module R M
inst✝¹ : DecidableEq R
inst✝ : DecidableEq M
⊢ grade0Subalgebra R M = ⊥ | R : Type u
M : Type v
inst✝⁴ : CommRing R
inst✝³ : AddCommMonoid M
inst✝² : Module R M
inst✝¹ : DecidableEq R
inst✝ : DecidableEq M
⊢ grade0Subalgebra R M ≤ ⊥ | Please generate a tactic in lean4 to solve the state.
STATE:
R : Type u
M : Type v
inst✝⁴ : CommRing R
inst✝³ : AddCommMonoid M
inst✝² : Module R M
inst✝¹ : DecidableEq R
inst✝ : DecidableEq M
⊢ grade0Subalgebra R M = ⊥
TACTIC:
|
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/DPAlgebra/Graded/GradeZero.lean | DividedPowerAlgebra.grade0Subalgebra_eq_bot | [313, 1] | [318, 63] | intro p hp | R : Type u
M : Type v
inst✝⁴ : CommRing R
inst✝³ : AddCommMonoid M
inst✝² : Module R M
inst✝¹ : DecidableEq R
inst✝ : DecidableEq M
⊢ grade0Subalgebra R M ≤ ⊥ | R : Type u
M : Type v
inst✝⁴ : CommRing R
inst✝³ : AddCommMonoid M
inst✝² : Module R M
inst✝¹ : DecidableEq R
inst✝ : DecidableEq M
p : DividedPowerAlgebra R M
hp : p ∈ grade0Subalgebra R M
⊢ p ∈ ⊥ | Please generate a tactic in lean4 to solve the state.
STATE:
R : Type u
M : Type v
inst✝⁴ : CommRing R
inst✝³ : AddCommMonoid M
inst✝² : Module R M
inst✝¹ : DecidableEq R
inst✝ : DecidableEq M
⊢ grade0Subalgebra R M ≤ ⊥
TACTIC:
|
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/DPAlgebra/Graded/GradeZero.lean | DividedPowerAlgebra.grade0Subalgebra_eq_bot | [313, 1] | [318, 63] | rw [Algebra.mem_bot] | R : Type u
M : Type v
inst✝⁴ : CommRing R
inst✝³ : AddCommMonoid M
inst✝² : Module R M
inst✝¹ : DecidableEq R
inst✝ : DecidableEq M
p : DividedPowerAlgebra R M
hp : p ∈ grade0Subalgebra R M
⊢ p ∈ ⊥ | R : Type u
M : Type v
inst✝⁴ : CommRing R
inst✝³ : AddCommMonoid M
inst✝² : Module R M
inst✝¹ : DecidableEq R
inst✝ : DecidableEq M
p : DividedPowerAlgebra R M
hp : p ∈ grade0Subalgebra R M
⊢ p ∈ Set.range ⇑(algebraMap R (DividedPowerAlgebra R M)) | Please generate a tactic in lean4 to solve the state.
STATE:
R : Type u
M : Type v
inst✝⁴ : CommRing R
inst✝³ : AddCommMonoid M
inst✝² : Module R M
inst✝¹ : DecidableEq R
inst✝ : DecidableEq M
p : DividedPowerAlgebra R M
hp : p ∈ grade0Subalgebra R M
⊢ p ∈ ⊥
TACTIC:
|
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/DPAlgebra/Graded/GradeZero.lean | DividedPowerAlgebra.grade0Subalgebra_eq_bot | [313, 1] | [318, 63] | convert Set.mem_range_self ((algebraMapInv R M) p) | R : Type u
M : Type v
inst✝⁴ : CommRing R
inst✝³ : AddCommMonoid M
inst✝² : Module R M
inst✝¹ : DecidableEq R
inst✝ : DecidableEq M
p : DividedPowerAlgebra R M
hp : p ∈ grade0Subalgebra R M
⊢ p ∈ Set.range ⇑(algebraMap R (DividedPowerAlgebra R M)) | case h.e'_4
R : Type u
M : Type v
inst✝⁴ : CommRing R
inst✝³ : AddCommMonoid M
inst✝² : Module R M
inst✝¹ : DecidableEq R
inst✝ : DecidableEq M
p : DividedPowerAlgebra R M
hp : p ∈ grade0Subalgebra R M
⊢ p = (algebraMap R (DividedPowerAlgebra R M)) ((algebraMapInv R M) p) | Please generate a tactic in lean4 to solve the state.
STATE:
R : Type u
M : Type v
inst✝⁴ : CommRing R
inst✝³ : AddCommMonoid M
inst✝² : Module R M
inst✝¹ : DecidableEq R
inst✝ : DecidableEq M
p : DividedPowerAlgebra R M
hp : p ∈ grade0Subalgebra R M
⊢ p ∈ Set.range ⇑(algebraMap R (DividedPowerAlgebra R M))
TACTIC:
|
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/DPAlgebra/Graded/GradeZero.lean | DividedPowerAlgebra.grade0Subalgebra_eq_bot | [313, 1] | [318, 63] | exact (algebraMap_right_inv_of_degree_zero R M ⟨p, hp⟩).symm | case h.e'_4
R : Type u
M : Type v
inst✝⁴ : CommRing R
inst✝³ : AddCommMonoid M
inst✝² : Module R M
inst✝¹ : DecidableEq R
inst✝ : DecidableEq M
p : DividedPowerAlgebra R M
hp : p ∈ grade0Subalgebra R M
⊢ p = (algebraMap R (DividedPowerAlgebra R M)) ((algebraMapInv R M) p) | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case h.e'_4
R : Type u
M : Type v
inst✝⁴ : CommRing R
inst✝³ : AddCommMonoid M
inst✝² : Module R M
inst✝¹ : DecidableEq R
inst✝ : DecidableEq M
p : DividedPowerAlgebra R M
hp : p ∈ grade0Subalgebra R M
⊢ p = (algebraMap R (DividedPowerAlgebra R M)) ((algebraM... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/DPAlgebra/Graded/GradeZero.lean | DividedPowerAlgebra.isCompl_augIdeal | [320, 1] | [338, 34] | apply IsCompl.mk | R : Type u
M : Type v
inst✝⁴ : CommRing R
inst✝³ : AddCommMonoid M
inst✝² : Module R M
inst✝¹ : DecidableEq R
inst✝ : DecidableEq M
⊢ IsAugmentation R (augIdeal R M) | case disjoint
R : Type u
M : Type v
inst✝⁴ : CommRing R
inst✝³ : AddCommMonoid M
inst✝² : Module R M
inst✝¹ : DecidableEq R
inst✝ : DecidableEq M
⊢ Disjoint (Subalgebra.toSubmodule ⊥) (Submodule.restrictScalars R (augIdeal R M))
case codisjoint
R : Type u
M : Type v
inst✝⁴ : CommRing R
inst✝³ : AddCommMonoid M
inst✝² ... | Please generate a tactic in lean4 to solve the state.
STATE:
R : Type u
M : Type v
inst✝⁴ : CommRing R
inst✝³ : AddCommMonoid M
inst✝² : Module R M
inst✝¹ : DecidableEq R
inst✝ : DecidableEq M
⊢ IsAugmentation R (augIdeal R M)
TACTIC:
|
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/DPAlgebra/Graded/GradeZero.lean | DividedPowerAlgebra.isCompl_augIdeal | [320, 1] | [338, 34] | rw [Submodule.disjoint_def] | case disjoint
R : Type u
M : Type v
inst✝⁴ : CommRing R
inst✝³ : AddCommMonoid M
inst✝² : Module R M
inst✝¹ : DecidableEq R
inst✝ : DecidableEq M
⊢ Disjoint (Subalgebra.toSubmodule ⊥) (Submodule.restrictScalars R (augIdeal R M)) | case disjoint
R : Type u
M : Type v
inst✝⁴ : CommRing R
inst✝³ : AddCommMonoid M
inst✝² : Module R M
inst✝¹ : DecidableEq R
inst✝ : DecidableEq M
⊢ ∀ x ∈ Subalgebra.toSubmodule ⊥, x ∈ Submodule.restrictScalars R (augIdeal R M) → x = 0 | Please generate a tactic in lean4 to solve the state.
STATE:
case disjoint
R : Type u
M : Type v
inst✝⁴ : CommRing R
inst✝³ : AddCommMonoid M
inst✝² : Module R M
inst✝¹ : DecidableEq R
inst✝ : DecidableEq M
⊢ Disjoint (Subalgebra.toSubmodule ⊥) (Submodule.restrictScalars R (augIdeal R M))
TACTIC:
|
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/DPAlgebra/Graded/GradeZero.lean | DividedPowerAlgebra.isCompl_augIdeal | [320, 1] | [338, 34] | intro x | case disjoint
R : Type u
M : Type v
inst✝⁴ : CommRing R
inst✝³ : AddCommMonoid M
inst✝² : Module R M
inst✝¹ : DecidableEq R
inst✝ : DecidableEq M
⊢ ∀ x ∈ Subalgebra.toSubmodule ⊥, x ∈ Submodule.restrictScalars R (augIdeal R M) → x = 0 | case disjoint
R : Type u
M : Type v
inst✝⁴ : CommRing R
inst✝³ : AddCommMonoid M
inst✝² : Module R M
inst✝¹ : DecidableEq R
inst✝ : DecidableEq M
x : DividedPowerAlgebra R M
⊢ x ∈ Subalgebra.toSubmodule ⊥ → x ∈ Submodule.restrictScalars R (augIdeal R M) → x = 0 | Please generate a tactic in lean4 to solve the state.
STATE:
case disjoint
R : Type u
M : Type v
inst✝⁴ : CommRing R
inst✝³ : AddCommMonoid M
inst✝² : Module R M
inst✝¹ : DecidableEq R
inst✝ : DecidableEq M
⊢ ∀ x ∈ Subalgebra.toSubmodule ⊥, x ∈ Submodule.restrictScalars R (augIdeal R M) → x = 0
TACTIC:
|
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/DPAlgebra/Graded/GradeZero.lean | DividedPowerAlgebra.isCompl_augIdeal | [320, 1] | [338, 34] | simp only [Subalgebra.mem_toSubmodule, Algebra.mem_bot] | case disjoint
R : Type u
M : Type v
inst✝⁴ : CommRing R
inst✝³ : AddCommMonoid M
inst✝² : Module R M
inst✝¹ : DecidableEq R
inst✝ : DecidableEq M
x : DividedPowerAlgebra R M
⊢ x ∈ Subalgebra.toSubmodule ⊥ → x ∈ Submodule.restrictScalars R (augIdeal R M) → x = 0 | case disjoint
R : Type u
M : Type v
inst✝⁴ : CommRing R
inst✝³ : AddCommMonoid M
inst✝² : Module R M
inst✝¹ : DecidableEq R
inst✝ : DecidableEq M
x : DividedPowerAlgebra R M
⊢ x ∈ Set.range ⇑(algebraMap R (DividedPowerAlgebra R M)) → x ∈ Submodule.restrictScalars R (augIdeal R M) → x = 0 | Please generate a tactic in lean4 to solve the state.
STATE:
case disjoint
R : Type u
M : Type v
inst✝⁴ : CommRing R
inst✝³ : AddCommMonoid M
inst✝² : Module R M
inst✝¹ : DecidableEq R
inst✝ : DecidableEq M
x : DividedPowerAlgebra R M
⊢ x ∈ Subalgebra.toSubmodule ⊥ → x ∈ Submodule.restrictScalars R (augIdeal R M) → x =... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/DPAlgebra/Graded/GradeZero.lean | DividedPowerAlgebra.isCompl_augIdeal | [320, 1] | [338, 34] | rintro ⟨r, rfl⟩ | case disjoint
R : Type u
M : Type v
inst✝⁴ : CommRing R
inst✝³ : AddCommMonoid M
inst✝² : Module R M
inst✝¹ : DecidableEq R
inst✝ : DecidableEq M
x : DividedPowerAlgebra R M
⊢ x ∈ Set.range ⇑(algebraMap R (DividedPowerAlgebra R M)) → x ∈ Submodule.restrictScalars R (augIdeal R M) → x = 0 | case disjoint.intro
R : Type u
M : Type v
inst✝⁴ : CommRing R
inst✝³ : AddCommMonoid M
inst✝² : Module R M
inst✝¹ : DecidableEq R
inst✝ : DecidableEq M
r : R
⊢ (algebraMap R (DividedPowerAlgebra R M)) r ∈ Submodule.restrictScalars R (augIdeal R M) →
(algebraMap R (DividedPowerAlgebra R M)) r = 0 | Please generate a tactic in lean4 to solve the state.
STATE:
case disjoint
R : Type u
M : Type v
inst✝⁴ : CommRing R
inst✝³ : AddCommMonoid M
inst✝² : Module R M
inst✝¹ : DecidableEq R
inst✝ : DecidableEq M
x : DividedPowerAlgebra R M
⊢ x ∈ Set.range ⇑(algebraMap R (DividedPowerAlgebra R M)) → x ∈ Submodule.restrictSca... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/DPAlgebra/Graded/GradeZero.lean | DividedPowerAlgebra.isCompl_augIdeal | [320, 1] | [338, 34] | simp only [Submodule.restrictScalars_mem, mem_augIdeal_iff, AlgHom.commutes,
Algebra.id.map_eq_id, RingHom.id_apply] | case disjoint.intro
R : Type u
M : Type v
inst✝⁴ : CommRing R
inst✝³ : AddCommMonoid M
inst✝² : Module R M
inst✝¹ : DecidableEq R
inst✝ : DecidableEq M
r : R
⊢ (algebraMap R (DividedPowerAlgebra R M)) r ∈ Submodule.restrictScalars R (augIdeal R M) →
(algebraMap R (DividedPowerAlgebra R M)) r = 0 | case disjoint.intro
R : Type u
M : Type v
inst✝⁴ : CommRing R
inst✝³ : AddCommMonoid M
inst✝² : Module R M
inst✝¹ : DecidableEq R
inst✝ : DecidableEq M
r : R
⊢ r = 0 → (algebraMap R (DividedPowerAlgebra R M)) r = 0 | Please generate a tactic in lean4 to solve the state.
STATE:
case disjoint.intro
R : Type u
M : Type v
inst✝⁴ : CommRing R
inst✝³ : AddCommMonoid M
inst✝² : Module R M
inst✝¹ : DecidableEq R
inst✝ : DecidableEq M
r : R
⊢ (algebraMap R (DividedPowerAlgebra R M)) r ∈ Submodule.restrictScalars R (augIdeal R M) →
(alge... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/DPAlgebra/Graded/GradeZero.lean | DividedPowerAlgebra.isCompl_augIdeal | [320, 1] | [338, 34] | intro hr | case disjoint.intro
R : Type u
M : Type v
inst✝⁴ : CommRing R
inst✝³ : AddCommMonoid M
inst✝² : Module R M
inst✝¹ : DecidableEq R
inst✝ : DecidableEq M
r : R
⊢ r = 0 → (algebraMap R (DividedPowerAlgebra R M)) r = 0 | case disjoint.intro
R : Type u
M : Type v
inst✝⁴ : CommRing R
inst✝³ : AddCommMonoid M
inst✝² : Module R M
inst✝¹ : DecidableEq R
inst✝ : DecidableEq M
r : R
hr : r = 0
⊢ (algebraMap R (DividedPowerAlgebra R M)) r = 0 | Please generate a tactic in lean4 to solve the state.
STATE:
case disjoint.intro
R : Type u
M : Type v
inst✝⁴ : CommRing R
inst✝³ : AddCommMonoid M
inst✝² : Module R M
inst✝¹ : DecidableEq R
inst✝ : DecidableEq M
r : R
⊢ r = 0 → (algebraMap R (DividedPowerAlgebra R M)) r = 0
TACTIC:
|
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/DPAlgebra/Graded/GradeZero.lean | DividedPowerAlgebra.isCompl_augIdeal | [320, 1] | [338, 34] | rw [hr, map_zero] | case disjoint.intro
R : Type u
M : Type v
inst✝⁴ : CommRing R
inst✝³ : AddCommMonoid M
inst✝² : Module R M
inst✝¹ : DecidableEq R
inst✝ : DecidableEq M
r : R
hr : r = 0
⊢ (algebraMap R (DividedPowerAlgebra R M)) r = 0 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case disjoint.intro
R : Type u
M : Type v
inst✝⁴ : CommRing R
inst✝³ : AddCommMonoid M
inst✝² : Module R M
inst✝¹ : DecidableEq R
inst✝ : DecidableEq M
r : R
hr : r = 0
⊢ (algebraMap R (DividedPowerAlgebra R M)) r = 0
TACTIC:
|
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/DPAlgebra/Graded/GradeZero.lean | DividedPowerAlgebra.isCompl_augIdeal | [320, 1] | [338, 34] | rw [codisjoint_iff, eq_top_iff] | case codisjoint
R : Type u
M : Type v
inst✝⁴ : CommRing R
inst✝³ : AddCommMonoid M
inst✝² : Module R M
inst✝¹ : DecidableEq R
inst✝ : DecidableEq M
⊢ Codisjoint (Subalgebra.toSubmodule ⊥) (Submodule.restrictScalars R (augIdeal R M)) | case codisjoint
R : Type u
M : Type v
inst✝⁴ : CommRing R
inst✝³ : AddCommMonoid M
inst✝² : Module R M
inst✝¹ : DecidableEq R
inst✝ : DecidableEq M
⊢ ⊤ ≤ Subalgebra.toSubmodule ⊥ ⊔ Submodule.restrictScalars R (augIdeal R M) | Please generate a tactic in lean4 to solve the state.
STATE:
case codisjoint
R : Type u
M : Type v
inst✝⁴ : CommRing R
inst✝³ : AddCommMonoid M
inst✝² : Module R M
inst✝¹ : DecidableEq R
inst✝ : DecidableEq M
⊢ Codisjoint (Subalgebra.toSubmodule ⊥) (Submodule.restrictScalars R (augIdeal R M))
TACTIC:
|
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/DPAlgebra/Graded/GradeZero.lean | DividedPowerAlgebra.isCompl_augIdeal | [320, 1] | [338, 34] | intro p _ | case codisjoint
R : Type u
M : Type v
inst✝⁴ : CommRing R
inst✝³ : AddCommMonoid M
inst✝² : Module R M
inst✝¹ : DecidableEq R
inst✝ : DecidableEq M
⊢ ⊤ ≤ Subalgebra.toSubmodule ⊥ ⊔ Submodule.restrictScalars R (augIdeal R M) | case codisjoint
R : Type u
M : Type v
inst✝⁴ : CommRing R
inst✝³ : AddCommMonoid M
inst✝² : Module R M
inst✝¹ : DecidableEq R
inst✝ : DecidableEq M
p : DividedPowerAlgebra R M
a✝ : p ∈ ⊤
⊢ p ∈ Subalgebra.toSubmodule ⊥ ⊔ Submodule.restrictScalars R (augIdeal R M) | Please generate a tactic in lean4 to solve the state.
STATE:
case codisjoint
R : Type u
M : Type v
inst✝⁴ : CommRing R
inst✝³ : AddCommMonoid M
inst✝² : Module R M
inst✝¹ : DecidableEq R
inst✝ : DecidableEq M
⊢ ⊤ ≤ Subalgebra.toSubmodule ⊥ ⊔ Submodule.restrictScalars R (augIdeal R M)
TACTIC:
|
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/DPAlgebra/Graded/GradeZero.lean | DividedPowerAlgebra.isCompl_augIdeal | [320, 1] | [338, 34] | simp only [Submodule.mem_sup, Subalgebra.mem_toSubmodule, Submodule.restrictScalars_mem] | case codisjoint
R : Type u
M : Type v
inst✝⁴ : CommRing R
inst✝³ : AddCommMonoid M
inst✝² : Module R M
inst✝¹ : DecidableEq R
inst✝ : DecidableEq M
p : DividedPowerAlgebra R M
a✝ : p ∈ ⊤
⊢ p ∈ Subalgebra.toSubmodule ⊥ ⊔ Submodule.restrictScalars R (augIdeal R M) | case codisjoint
R : Type u
M : Type v
inst✝⁴ : CommRing R
inst✝³ : AddCommMonoid M
inst✝² : Module R M
inst✝¹ : DecidableEq R
inst✝ : DecidableEq M
p : DividedPowerAlgebra R M
a✝ : p ∈ ⊤
⊢ ∃ y ∈ ⊥, ∃ z ∈ augIdeal R M, y + z = p | Please generate a tactic in lean4 to solve the state.
STATE:
case codisjoint
R : Type u
M : Type v
inst✝⁴ : CommRing R
inst✝³ : AddCommMonoid M
inst✝² : Module R M
inst✝¹ : DecidableEq R
inst✝ : DecidableEq M
p : DividedPowerAlgebra R M
a✝ : p ∈ ⊤
⊢ p ∈ Subalgebra.toSubmodule ⊥ ⊔ Submodule.restrictScalars R (augIdeal R... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/DPAlgebra/Graded/GradeZero.lean | DividedPowerAlgebra.isCompl_augIdeal | [320, 1] | [338, 34] | refine ⟨algebraMap R _ (algebraMapInv R M p), ?_, _, ?_, add_sub_cancel _ p⟩ | case codisjoint
R : Type u
M : Type v
inst✝⁴ : CommRing R
inst✝³ : AddCommMonoid M
inst✝² : Module R M
inst✝¹ : DecidableEq R
inst✝ : DecidableEq M
p : DividedPowerAlgebra R M
a✝ : p ∈ ⊤
⊢ ∃ y ∈ ⊥, ∃ z ∈ augIdeal R M, y + z = p | case codisjoint.refine_1
R : Type u
M : Type v
inst✝⁴ : CommRing R
inst✝³ : AddCommMonoid M
inst✝² : Module R M
inst✝¹ : DecidableEq R
inst✝ : DecidableEq M
p : DividedPowerAlgebra R M
a✝ : p ∈ ⊤
⊢ (algebraMap R (DividedPowerAlgebra R M)) ((algebraMapInv R M) p) ∈ ⊥
case codisjoint.refine_2
R : Type u
M : Type v
inst✝... | Please generate a tactic in lean4 to solve the state.
STATE:
case codisjoint
R : Type u
M : Type v
inst✝⁴ : CommRing R
inst✝³ : AddCommMonoid M
inst✝² : Module R M
inst✝¹ : DecidableEq R
inst✝ : DecidableEq M
p : DividedPowerAlgebra R M
a✝ : p ∈ ⊤
⊢ ∃ y ∈ ⊥, ∃ z ∈ augIdeal R M, y + z = p
TACTIC:
|
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/DPAlgebra/Graded/GradeZero.lean | DividedPowerAlgebra.isCompl_augIdeal | [320, 1] | [338, 34] | rw [Algebra.mem_bot] | case codisjoint.refine_1
R : Type u
M : Type v
inst✝⁴ : CommRing R
inst✝³ : AddCommMonoid M
inst✝² : Module R M
inst✝¹ : DecidableEq R
inst✝ : DecidableEq M
p : DividedPowerAlgebra R M
a✝ : p ∈ ⊤
⊢ (algebraMap R (DividedPowerAlgebra R M)) ((algebraMapInv R M) p) ∈ ⊥ | case codisjoint.refine_1
R : Type u
M : Type v
inst✝⁴ : CommRing R
inst✝³ : AddCommMonoid M
inst✝² : Module R M
inst✝¹ : DecidableEq R
inst✝ : DecidableEq M
p : DividedPowerAlgebra R M
a✝ : p ∈ ⊤
⊢ (algebraMap R (DividedPowerAlgebra R M)) ((algebraMapInv R M) p) ∈ Set.range ⇑(algebraMap R (DividedPowerAlgebra R M)) | Please generate a tactic in lean4 to solve the state.
STATE:
case codisjoint.refine_1
R : Type u
M : Type v
inst✝⁴ : CommRing R
inst✝³ : AddCommMonoid M
inst✝² : Module R M
inst✝¹ : DecidableEq R
inst✝ : DecidableEq M
p : DividedPowerAlgebra R M
a✝ : p ∈ ⊤
⊢ (algebraMap R (DividedPowerAlgebra R M)) ((algebraMapInv R M)... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/DPAlgebra/Graded/GradeZero.lean | DividedPowerAlgebra.isCompl_augIdeal | [320, 1] | [338, 34] | exact Set.mem_range_self ((algebraMapInv R M) p) | case codisjoint.refine_1
R : Type u
M : Type v
inst✝⁴ : CommRing R
inst✝³ : AddCommMonoid M
inst✝² : Module R M
inst✝¹ : DecidableEq R
inst✝ : DecidableEq M
p : DividedPowerAlgebra R M
a✝ : p ∈ ⊤
⊢ (algebraMap R (DividedPowerAlgebra R M)) ((algebraMapInv R M) p) ∈ Set.range ⇑(algebraMap R (DividedPowerAlgebra R M)) | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case codisjoint.refine_1
R : Type u
M : Type v
inst✝⁴ : CommRing R
inst✝³ : AddCommMonoid M
inst✝² : Module R M
inst✝¹ : DecidableEq R
inst✝ : DecidableEq M
p : DividedPowerAlgebra R M
a✝ : p ∈ ⊤
⊢ (algebraMap R (DividedPowerAlgebra R M)) ((algebraMapInv R M)... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/DPAlgebra/Graded/GradeZero.lean | DividedPowerAlgebra.isCompl_augIdeal | [320, 1] | [338, 34] | simp only [mem_augIdeal_iff, map_sub, AlgHom.commutes, Algebra.id.map_eq_id,
RingHom.id_apply, sub_self] | case codisjoint.refine_2
R : Type u
M : Type v
inst✝⁴ : CommRing R
inst✝³ : AddCommMonoid M
inst✝² : Module R M
inst✝¹ : DecidableEq R
inst✝ : DecidableEq M
p : DividedPowerAlgebra R M
a✝ : p ∈ ⊤
⊢ p - (algebraMap R (DividedPowerAlgebra R M)) ((algebraMapInv R M) p) ∈ augIdeal R M | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case codisjoint.refine_2
R : Type u
M : Type v
inst✝⁴ : CommRing R
inst✝³ : AddCommMonoid M
inst✝² : Module R M
inst✝¹ : DecidableEq R
inst✝ : DecidableEq M
p : DividedPowerAlgebra R M
a✝ : p ∈ ⊤
⊢ p - (algebraMap R (DividedPowerAlgebra R M)) ((algebraMapInv ... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/ForMathlib/RingTheory/MvPowerSeries/Trunc.lean | MvPowerSeries.coeff_truncFun' | [27, 1] | [30, 43] | classical
simp [truncFun', MvPolynomial.coeff_sum] | σ : Type u_1
R : Type u_2
inst✝ : CommSemiring R
n m : σ →₀ ℕ
φ : MvPowerSeries σ R
⊢ MvPolynomial.coeff m (truncFun' n φ) = if m ≤ n then (coeff R m) φ else 0 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
σ : Type u_1
R : Type u_2
inst✝ : CommSemiring R
n m : σ →₀ ℕ
φ : MvPowerSeries σ R
⊢ MvPolynomial.coeff m (truncFun' n φ) = if m ≤ n then (coeff R m) φ else 0
TACTIC:
|
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/ForMathlib/RingTheory/MvPowerSeries/Trunc.lean | MvPowerSeries.coeff_truncFun' | [27, 1] | [30, 43] | simp [truncFun', MvPolynomial.coeff_sum] | σ : Type u_1
R : Type u_2
inst✝ : CommSemiring R
n m : σ →₀ ℕ
φ : MvPowerSeries σ R
⊢ MvPolynomial.coeff m (truncFun' n φ) = if m ≤ n then (coeff R m) φ else 0 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
σ : Type u_1
R : Type u_2
inst✝ : CommSemiring R
n m : σ →₀ ℕ
φ : MvPowerSeries σ R
⊢ MvPolynomial.coeff m (truncFun' n φ) = if m ≤ n then (coeff R m) φ else 0
TACTIC:
|
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/ForMathlib/RingTheory/MvPowerSeries/Trunc.lean | MvPowerSeries.trunc_one' | [58, 1] | [73, 31] | rw [coeff_trunc', coeff_one] | σ : Type u_1
R : Type u_2
inst✝ : CommSemiring R
n✝ n m : σ →₀ ℕ
⊢ MvPolynomial.coeff m ((trunc' R n) 1) = MvPolynomial.coeff m 1 | σ : Type u_1
R : Type u_2
inst✝ : CommSemiring R
n✝ n m : σ →₀ ℕ
⊢ (if m ≤ n then if m = 0 then 1 else 0 else 0) = MvPolynomial.coeff m 1 | Please generate a tactic in lean4 to solve the state.
STATE:
σ : Type u_1
R : Type u_2
inst✝ : CommSemiring R
n✝ n m : σ →₀ ℕ
⊢ MvPolynomial.coeff m ((trunc' R n) 1) = MvPolynomial.coeff m 1
TACTIC:
|
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/ForMathlib/RingTheory/MvPowerSeries/Trunc.lean | MvPowerSeries.trunc_one' | [58, 1] | [73, 31] | split_ifs with H H' | σ : Type u_1
R : Type u_2
inst✝ : CommSemiring R
n✝ n m : σ →₀ ℕ
⊢ (if m ≤ n then if m = 0 then 1 else 0 else 0) = MvPolynomial.coeff m 1 | case pos
σ : Type u_1
R : Type u_2
inst✝ : CommSemiring R
n✝ n m : σ →₀ ℕ
H : m ≤ n
H' : m = 0
⊢ 1 = MvPolynomial.coeff m 1
case neg
σ : Type u_1
R : Type u_2
inst✝ : CommSemiring R
n✝ n m : σ →₀ ℕ
H : m ≤ n
H' : ¬m = 0
⊢ 0 = MvPolynomial.coeff m 1
case neg
σ : Type u_1
R : Type u_2
inst✝ : CommSemiring R
n✝ n m : σ ... | Please generate a tactic in lean4 to solve the state.
STATE:
σ : Type u_1
R : Type u_2
inst✝ : CommSemiring R
n✝ n m : σ →₀ ℕ
⊢ (if m ≤ n then if m = 0 then 1 else 0 else 0) = MvPolynomial.coeff m 1
TACTIC:
|
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/ForMathlib/RingTheory/MvPowerSeries/Trunc.lean | MvPowerSeries.trunc_one' | [58, 1] | [73, 31] | subst m | case pos
σ : Type u_1
R : Type u_2
inst✝ : CommSemiring R
n✝ n m : σ →₀ ℕ
H : m ≤ n
H' : m = 0
⊢ 1 = MvPolynomial.coeff m 1 | case pos
σ : Type u_1
R : Type u_2
inst✝ : CommSemiring R
n✝ n : σ →₀ ℕ
H : 0 ≤ n
⊢ 1 = MvPolynomial.coeff 0 1 | Please generate a tactic in lean4 to solve the state.
STATE:
case pos
σ : Type u_1
R : Type u_2
inst✝ : CommSemiring R
n✝ n m : σ →₀ ℕ
H : m ≤ n
H' : m = 0
⊢ 1 = MvPolynomial.coeff m 1
TACTIC:
|
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/ForMathlib/RingTheory/MvPowerSeries/Trunc.lean | MvPowerSeries.trunc_one' | [58, 1] | [73, 31] | simp | case pos
σ : Type u_1
R : Type u_2
inst✝ : CommSemiring R
n✝ n : σ →₀ ℕ
H : 0 ≤ n
⊢ 1 = MvPolynomial.coeff 0 1 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case pos
σ : Type u_1
R : Type u_2
inst✝ : CommSemiring R
n✝ n : σ →₀ ℕ
H : 0 ≤ n
⊢ 1 = MvPolynomial.coeff 0 1
TACTIC:
|
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/ForMathlib/RingTheory/MvPowerSeries/Trunc.lean | MvPowerSeries.trunc_one' | [58, 1] | [73, 31] | symm | case neg
σ : Type u_1
R : Type u_2
inst✝ : CommSemiring R
n✝ n m : σ →₀ ℕ
H : m ≤ n
H' : ¬m = 0
⊢ 0 = MvPolynomial.coeff m 1 | case neg
σ : Type u_1
R : Type u_2
inst✝ : CommSemiring R
n✝ n m : σ →₀ ℕ
H : m ≤ n
H' : ¬m = 0
⊢ MvPolynomial.coeff m 1 = 0 | Please generate a tactic in lean4 to solve the state.
STATE:
case neg
σ : Type u_1
R : Type u_2
inst✝ : CommSemiring R
n✝ n m : σ →₀ ℕ
H : m ≤ n
H' : ¬m = 0
⊢ 0 = MvPolynomial.coeff m 1
TACTIC:
|
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/ForMathlib/RingTheory/MvPowerSeries/Trunc.lean | MvPowerSeries.trunc_one' | [58, 1] | [73, 31] | rw [MvPolynomial.coeff_one] | case neg
σ : Type u_1
R : Type u_2
inst✝ : CommSemiring R
n✝ n m : σ →₀ ℕ
H : m ≤ n
H' : ¬m = 0
⊢ MvPolynomial.coeff m 1 = 0 | case neg
σ : Type u_1
R : Type u_2
inst✝ : CommSemiring R
n✝ n m : σ →₀ ℕ
H : m ≤ n
H' : ¬m = 0
⊢ (if 0 = m then 1 else 0) = 0 | Please generate a tactic in lean4 to solve the state.
STATE:
case neg
σ : Type u_1
R : Type u_2
inst✝ : CommSemiring R
n✝ n m : σ →₀ ℕ
H : m ≤ n
H' : ¬m = 0
⊢ MvPolynomial.coeff m 1 = 0
TACTIC:
|
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/ForMathlib/RingTheory/MvPowerSeries/Trunc.lean | MvPowerSeries.trunc_one' | [58, 1] | [73, 31] | exact if_neg (Ne.symm H') | case neg
σ : Type u_1
R : Type u_2
inst✝ : CommSemiring R
n✝ n m : σ →₀ ℕ
H : m ≤ n
H' : ¬m = 0
⊢ (if 0 = m then 1 else 0) = 0 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case neg
σ : Type u_1
R : Type u_2
inst✝ : CommSemiring R
n✝ n m : σ →₀ ℕ
H : m ≤ n
H' : ¬m = 0
⊢ (if 0 = m then 1 else 0) = 0
TACTIC:
|
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/ForMathlib/RingTheory/MvPowerSeries/Trunc.lean | MvPowerSeries.trunc_one' | [58, 1] | [73, 31] | symm | case neg
σ : Type u_1
R : Type u_2
inst✝ : CommSemiring R
n✝ n m : σ →₀ ℕ
H : ¬m ≤ n
⊢ 0 = MvPolynomial.coeff m 1 | case neg
σ : Type u_1
R : Type u_2
inst✝ : CommSemiring R
n✝ n m : σ →₀ ℕ
H : ¬m ≤ n
⊢ MvPolynomial.coeff m 1 = 0 | Please generate a tactic in lean4 to solve the state.
STATE:
case neg
σ : Type u_1
R : Type u_2
inst✝ : CommSemiring R
n✝ n m : σ →₀ ℕ
H : ¬m ≤ n
⊢ 0 = MvPolynomial.coeff m 1
TACTIC:
|
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/ForMathlib/RingTheory/MvPowerSeries/Trunc.lean | MvPowerSeries.trunc_one' | [58, 1] | [73, 31] | rw [MvPolynomial.coeff_one] | case neg
σ : Type u_1
R : Type u_2
inst✝ : CommSemiring R
n✝ n m : σ →₀ ℕ
H : ¬m ≤ n
⊢ MvPolynomial.coeff m 1 = 0 | case neg
σ : Type u_1
R : Type u_2
inst✝ : CommSemiring R
n✝ n m : σ →₀ ℕ
H : ¬m ≤ n
⊢ (if 0 = m then 1 else 0) = 0 | Please generate a tactic in lean4 to solve the state.
STATE:
case neg
σ : Type u_1
R : Type u_2
inst✝ : CommSemiring R
n✝ n m : σ →₀ ℕ
H : ¬m ≤ n
⊢ MvPolynomial.coeff m 1 = 0
TACTIC:
|
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/ForMathlib/RingTheory/MvPowerSeries/Trunc.lean | MvPowerSeries.trunc_one' | [58, 1] | [73, 31] | refine' if_neg _ | case neg
σ : Type u_1
R : Type u_2
inst✝ : CommSemiring R
n✝ n m : σ →₀ ℕ
H : ¬m ≤ n
⊢ (if 0 = m then 1 else 0) = 0 | case neg
σ : Type u_1
R : Type u_2
inst✝ : CommSemiring R
n✝ n m : σ →₀ ℕ
H : ¬m ≤ n
⊢ ¬0 = m | Please generate a tactic in lean4 to solve the state.
STATE:
case neg
σ : Type u_1
R : Type u_2
inst✝ : CommSemiring R
n✝ n m : σ →₀ ℕ
H : ¬m ≤ n
⊢ (if 0 = m then 1 else 0) = 0
TACTIC:
|
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/ForMathlib/RingTheory/MvPowerSeries/Trunc.lean | MvPowerSeries.trunc_one' | [58, 1] | [73, 31] | rintro rfl | case neg
σ : Type u_1
R : Type u_2
inst✝ : CommSemiring R
n✝ n m : σ →₀ ℕ
H : ¬m ≤ n
⊢ ¬0 = m | case neg
σ : Type u_1
R : Type u_2
inst✝ : CommSemiring R
n✝ n : σ →₀ ℕ
H : ¬0 ≤ n
⊢ False | Please generate a tactic in lean4 to solve the state.
STATE:
case neg
σ : Type u_1
R : Type u_2
inst✝ : CommSemiring R
n✝ n m : σ →₀ ℕ
H : ¬m ≤ n
⊢ ¬0 = m
TACTIC:
|
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/ForMathlib/RingTheory/MvPowerSeries/Trunc.lean | MvPowerSeries.trunc_one' | [58, 1] | [73, 31] | apply H | case neg
σ : Type u_1
R : Type u_2
inst✝ : CommSemiring R
n✝ n : σ →₀ ℕ
H : ¬0 ≤ n
⊢ False | case neg
σ : Type u_1
R : Type u_2
inst✝ : CommSemiring R
n✝ n : σ →₀ ℕ
H : ¬0 ≤ n
⊢ 0 ≤ n | Please generate a tactic in lean4 to solve the state.
STATE:
case neg
σ : Type u_1
R : Type u_2
inst✝ : CommSemiring R
n✝ n : σ →₀ ℕ
H : ¬0 ≤ n
⊢ False
TACTIC:
|
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