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/Basic.lean | DividedPowerAlgebra.surjective_of_supported | [108, 1] | [113, 74] | intro f | R : Type u_1
M : Type u_2
inst✝⁴ : CommSemiring R
inst✝³ : AddCommMonoid M
inst✝² : Module R M
inst✝¹ : DecidableEq R
inst✝ : DecidableEq M
⊢ Surjective ⇑(mk.comp (supported R {nm | 0 < nm.1}).val) | R : Type u_1
M : Type u_2
inst✝⁴ : CommSemiring R
inst✝³ : AddCommMonoid M
inst✝² : Module R M
inst✝¹ : DecidableEq R
inst✝ : DecidableEq M
f : DividedPowerAlgebra R M
⊢ ∃ a, (mk.comp (supported R {nm | 0 < nm.1}).val) a = f | Please generate a tactic in lean4 to solve the state.
STATE:
R : Type u_1
M : Type u_2
inst✝⁴ : CommSemiring R
inst✝³ : AddCommMonoid M
inst✝² : Module R M
inst✝¹ : DecidableEq R
inst✝ : DecidableEq M
⊢ Surjective ⇑(mk.comp (supported R {nm | 0 < nm.1}).val)
TACTIC:
|
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/DPAlgebra/Graded/Basic.lean | DividedPowerAlgebra.surjective_of_supported | [108, 1] | [113, 74] | obtain ⟨p', hp'⟩ := DividedPowerAlgebra.mk_surjective f | R : Type u_1
M : Type u_2
inst✝⁴ : CommSemiring R
inst✝³ : AddCommMonoid M
inst✝² : Module R M
inst✝¹ : DecidableEq R
inst✝ : DecidableEq M
f : DividedPowerAlgebra R M
⊢ ∃ a, (mk.comp (supported R {nm | 0 < nm.1}).val) a = f | case intro
R : Type u_1
M : Type u_2
inst✝⁴ : CommSemiring R
inst✝³ : AddCommMonoid M
inst✝² : Module R M
inst✝¹ : DecidableEq R
inst✝ : DecidableEq M
f : DividedPowerAlgebra R M
p' : MvPolynomial (ℕ × M) R
hp' : mk p' = f
⊢ ∃ a, (mk.comp (supported R {nm | 0 < nm.1}).val) a = f | Please generate a tactic in lean4 to solve the state.
STATE:
R : Type u_1
M : Type u_2
inst✝⁴ : CommSemiring R
inst✝³ : AddCommMonoid M
inst✝² : Module R M
inst✝¹ : DecidableEq R
inst✝ : DecidableEq M
f : DividedPowerAlgebra R M
⊢ ∃ a, (mk.comp (supported R {nm | 0 < nm.1}).val) a = f
TACTIC:
|
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/DPAlgebra/Graded/Basic.lean | DividedPowerAlgebra.surjective_of_supported | [108, 1] | [113, 74] | use toSupported R p' | case intro
R : Type u_1
M : Type u_2
inst✝⁴ : CommSemiring R
inst✝³ : AddCommMonoid M
inst✝² : Module R M
inst✝¹ : DecidableEq R
inst✝ : DecidableEq M
f : DividedPowerAlgebra R M
p' : MvPolynomial (ℕ × M) R
hp' : mk p' = f
⊢ ∃ a, (mk.comp (supported R {nm | 0 < nm.1}).val) a = f | case h
R : Type u_1
M : Type u_2
inst✝⁴ : CommSemiring R
inst✝³ : AddCommMonoid M
inst✝² : Module R M
inst✝¹ : DecidableEq R
inst✝ : DecidableEq M
f : DividedPowerAlgebra R M
p' : MvPolynomial (ℕ × M) R
hp' : mk p' = f
⊢ (mk.comp (supported R {nm | 0 < nm.1}).val) ((toSupported R) p') = f | Please generate a tactic in lean4 to solve the state.
STATE:
case intro
R : Type u_1
M : Type u_2
inst✝⁴ : CommSemiring R
inst✝³ : AddCommMonoid M
inst✝² : Module R M
inst✝¹ : DecidableEq R
inst✝ : DecidableEq M
f : DividedPowerAlgebra R M
p' : MvPolynomial (ℕ × M) R
hp' : mk p' = f
⊢ ∃ a, (mk.comp (supported R {nm | 0... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/DPAlgebra/Graded/Basic.lean | DividedPowerAlgebra.surjective_of_supported | [108, 1] | [113, 74] | rw [← AlgHom.comp_apply, AlgHom.comp_assoc, mk_comp_toSupported, ← hp'] | case h
R : Type u_1
M : Type u_2
inst✝⁴ : CommSemiring R
inst✝³ : AddCommMonoid M
inst✝² : Module R M
inst✝¹ : DecidableEq R
inst✝ : DecidableEq M
f : DividedPowerAlgebra R M
p' : MvPolynomial (ℕ × M) R
hp' : mk p' = f
⊢ (mk.comp (supported R {nm | 0 < nm.1}).val) ((toSupported R) p') = f | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case h
R : Type u_1
M : Type u_2
inst✝⁴ : CommSemiring R
inst✝³ : AddCommMonoid M
inst✝² : Module R M
inst✝¹ : DecidableEq R
inst✝ : DecidableEq M
f : DividedPowerAlgebra R M
p' : MvPolynomial (ℕ × M) R
hp' : mk p' = f
⊢ (mk.comp (supported R {nm | 0 < nm.1})... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/DPAlgebra/Graded/Basic.lean | DividedPowerAlgebra.surjective_of_supported' | [115, 1] | [121, 61] | obtain ⟨p', hpn', hp'⟩ := (mem_grade_iff R M _ _).mpr p.2 | R : Type u_1
M : Type u_2
inst✝⁴ : CommSemiring R
inst✝³ : AddCommMonoid M
inst✝² : Module R M
inst✝¹ : DecidableEq R
inst✝ : DecidableEq M
n : ℕ
p : ↥(grade R M n)
⊢ ∃ q, IsWeightedHomogeneous Prod.fst (↑q) n ∧ mk ↑q = ↑p | case intro.intro
R : Type u_1
M : Type u_2
inst✝⁴ : CommSemiring R
inst✝³ : AddCommMonoid M
inst✝² : Module R M
inst✝¹ : DecidableEq R
inst✝ : DecidableEq M
n : ℕ
p : ↥(grade R M n)
p' : MvPolynomial (ℕ × M) R
hpn' : p' ∈ ↑(weightedHomogeneousSubmodule R Prod.fst n)
hp' : (mkAlgHom R (Rel R M)) p' = ↑p
⊢ ∃ q, IsWeighte... | Please generate a tactic in lean4 to solve the state.
STATE:
R : Type u_1
M : Type u_2
inst✝⁴ : CommSemiring R
inst✝³ : AddCommMonoid M
inst✝² : Module R M
inst✝¹ : DecidableEq R
inst✝ : DecidableEq M
n : ℕ
p : ↥(grade R M n)
⊢ ∃ q, IsWeightedHomogeneous Prod.fst (↑q) n ∧ mk ↑q = ↑p
TACTIC:
|
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/DPAlgebra/Graded/Basic.lean | DividedPowerAlgebra.surjective_of_supported' | [115, 1] | [121, 61] | use toSupported R p' | case intro.intro
R : Type u_1
M : Type u_2
inst✝⁴ : CommSemiring R
inst✝³ : AddCommMonoid M
inst✝² : Module R M
inst✝¹ : DecidableEq R
inst✝ : DecidableEq M
n : ℕ
p : ↥(grade R M n)
p' : MvPolynomial (ℕ × M) R
hpn' : p' ∈ ↑(weightedHomogeneousSubmodule R Prod.fst n)
hp' : (mkAlgHom R (Rel R M)) p' = ↑p
⊢ ∃ q, IsWeighte... | case h
R : Type u_1
M : Type u_2
inst✝⁴ : CommSemiring R
inst✝³ : AddCommMonoid M
inst✝² : Module R M
inst✝¹ : DecidableEq R
inst✝ : DecidableEq M
n : ℕ
p : ↥(grade R M n)
p' : MvPolynomial (ℕ × M) R
hpn' : p' ∈ ↑(weightedHomogeneousSubmodule R Prod.fst n)
hp' : (mkAlgHom R (Rel R M)) p' = ↑p
⊢ IsWeightedHomogeneous Pr... | Please generate a tactic in lean4 to solve the state.
STATE:
case intro.intro
R : Type u_1
M : Type u_2
inst✝⁴ : CommSemiring R
inst✝³ : AddCommMonoid M
inst✝² : Module R M
inst✝¹ : DecidableEq R
inst✝ : DecidableEq M
n : ℕ
p : ↥(grade R M n)
p' : MvPolynomial (ℕ × M) R
hpn' : p' ∈ ↑(weightedHomogeneousSubmodule R Prod... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/DPAlgebra/Graded/Basic.lean | DividedPowerAlgebra.surjective_of_supported' | [115, 1] | [121, 61] | refine ⟨toSupported_isHomogeneous' _ _ _ hpn', ?_⟩ | case h
R : Type u_1
M : Type u_2
inst✝⁴ : CommSemiring R
inst✝³ : AddCommMonoid M
inst✝² : Module R M
inst✝¹ : DecidableEq R
inst✝ : DecidableEq M
n : ℕ
p : ↥(grade R M n)
p' : MvPolynomial (ℕ × M) R
hpn' : p' ∈ ↑(weightedHomogeneousSubmodule R Prod.fst n)
hp' : (mkAlgHom R (Rel R M)) p' = ↑p
⊢ IsWeightedHomogeneous Pr... | case h
R : Type u_1
M : Type u_2
inst✝⁴ : CommSemiring R
inst✝³ : AddCommMonoid M
inst✝² : Module R M
inst✝¹ : DecidableEq R
inst✝ : DecidableEq M
n : ℕ
p : ↥(grade R M n)
p' : MvPolynomial (ℕ × M) R
hpn' : p' ∈ ↑(weightedHomogeneousSubmodule R Prod.fst n)
hp' : (mkAlgHom R (Rel R M)) p' = ↑p
⊢ mk ↑((toSupported R) p')... | Please generate a tactic in lean4 to solve the state.
STATE:
case h
R : Type u_1
M : Type u_2
inst✝⁴ : CommSemiring R
inst✝³ : AddCommMonoid M
inst✝² : Module R M
inst✝¹ : DecidableEq R
inst✝ : DecidableEq M
n : ℕ
p : ↥(grade R M n)
p' : MvPolynomial (ℕ × M) R
hpn' : p' ∈ ↑(weightedHomogeneousSubmodule R Prod.fst n)
hp... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/DPAlgebra/Graded/Basic.lean | DividedPowerAlgebra.surjective_of_supported' | [115, 1] | [121, 61] | erw [DFunLike.congr_fun (mk_comp_toSupported R M) p', hp'] | case h
R : Type u_1
M : Type u_2
inst✝⁴ : CommSemiring R
inst✝³ : AddCommMonoid M
inst✝² : Module R M
inst✝¹ : DecidableEq R
inst✝ : DecidableEq M
n : ℕ
p : ↥(grade R M n)
p' : MvPolynomial (ℕ × M) R
hpn' : p' ∈ ↑(weightedHomogeneousSubmodule R Prod.fst n)
hp' : (mkAlgHom R (Rel R M)) p' = ↑p
⊢ mk ↑((toSupported R) p')... | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case h
R : Type u_1
M : Type u_2
inst✝⁴ : CommSemiring R
inst✝³ : AddCommMonoid M
inst✝² : Module R M
inst✝¹ : DecidableEq R
inst✝ : DecidableEq M
n : ℕ
p : ↥(grade R M n)
p' : MvPolynomial (ℕ × M) R
hpn' : p' ∈ ↑(weightedHomogeneousSubmodule R Prod.fst n)
hp... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/DPAlgebra/Graded/Basic.lean | DividedPowerAlgebra.mem_grade_iff' | [124, 1] | [132, 23] | constructor | R : Type u_1
M : Type u_2
inst✝⁴ : CommSemiring R
inst✝³ : AddCommMonoid M
inst✝² : Module R M
inst✝¹ : DecidableEq R
inst✝ : DecidableEq M
n : ℕ
p : DividedPowerAlgebra R M
⊢ p ∈ grade R M n ↔ ∃ q, IsWeightedHomogeneous Prod.fst (↑q) n ∧ mk ↑q = p | case mp
R : Type u_1
M : Type u_2
inst✝⁴ : CommSemiring R
inst✝³ : AddCommMonoid M
inst✝² : Module R M
inst✝¹ : DecidableEq R
inst✝ : DecidableEq M
n : ℕ
p : DividedPowerAlgebra R M
⊢ p ∈ grade R M n → ∃ q, IsWeightedHomogeneous Prod.fst (↑q) n ∧ mk ↑q = p
case mpr
R : Type u_1
M : Type u_2
inst✝⁴ : CommSemiring R
ins... | Please generate a tactic in lean4 to solve the state.
STATE:
R : Type u_1
M : Type u_2
inst✝⁴ : CommSemiring R
inst✝³ : AddCommMonoid M
inst✝² : Module R M
inst✝¹ : DecidableEq R
inst✝ : DecidableEq M
n : ℕ
p : DividedPowerAlgebra R M
⊢ p ∈ grade R M n ↔ ∃ q, IsWeightedHomogeneous Prod.fst (↑q) n ∧ mk ↑q = p
TACTIC:
|
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/DPAlgebra/Graded/Basic.lean | DividedPowerAlgebra.mem_grade_iff' | [124, 1] | [132, 23] | . intro hp
rw [← Submodule.coe_mk p hp]
apply surjective_of_supported' | case mp
R : Type u_1
M : Type u_2
inst✝⁴ : CommSemiring R
inst✝³ : AddCommMonoid M
inst✝² : Module R M
inst✝¹ : DecidableEq R
inst✝ : DecidableEq M
n : ℕ
p : DividedPowerAlgebra R M
⊢ p ∈ grade R M n → ∃ q, IsWeightedHomogeneous Prod.fst (↑q) n ∧ mk ↑q = p
case mpr
R : Type u_1
M : Type u_2
inst✝⁴ : CommSemiring R
ins... | case mpr
R : Type u_1
M : Type u_2
inst✝⁴ : CommSemiring R
inst✝³ : AddCommMonoid M
inst✝² : Module R M
inst✝¹ : DecidableEq R
inst✝ : DecidableEq M
n : ℕ
p : DividedPowerAlgebra R M
⊢ (∃ q, IsWeightedHomogeneous Prod.fst (↑q) n ∧ mk ↑q = p) → p ∈ grade R M n | Please generate a tactic in lean4 to solve the state.
STATE:
case mp
R : Type u_1
M : Type u_2
inst✝⁴ : CommSemiring R
inst✝³ : AddCommMonoid M
inst✝² : Module R M
inst✝¹ : DecidableEq R
inst✝ : DecidableEq M
n : ℕ
p : DividedPowerAlgebra R M
⊢ p ∈ grade R M n → ∃ q, IsWeightedHomogeneous Prod.fst (↑q) n ∧ mk ↑q = p
c... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/DPAlgebra/Graded/Basic.lean | DividedPowerAlgebra.mem_grade_iff' | [124, 1] | [132, 23] | . rintro ⟨q, hq, rfl⟩
exact ⟨q, hq, rfl⟩ | case mpr
R : Type u_1
M : Type u_2
inst✝⁴ : CommSemiring R
inst✝³ : AddCommMonoid M
inst✝² : Module R M
inst✝¹ : DecidableEq R
inst✝ : DecidableEq M
n : ℕ
p : DividedPowerAlgebra R M
⊢ (∃ q, IsWeightedHomogeneous Prod.fst (↑q) n ∧ mk ↑q = p) → p ∈ grade R M n | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case mpr
R : Type u_1
M : Type u_2
inst✝⁴ : CommSemiring R
inst✝³ : AddCommMonoid M
inst✝² : Module R M
inst✝¹ : DecidableEq R
inst✝ : DecidableEq M
n : ℕ
p : DividedPowerAlgebra R M
⊢ (∃ q, IsWeightedHomogeneous Prod.fst (↑q) n ∧ mk ↑q = p) → p ∈ grade R M n... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/DPAlgebra/Graded/Basic.lean | DividedPowerAlgebra.mem_grade_iff' | [124, 1] | [132, 23] | intro hp | case mp
R : Type u_1
M : Type u_2
inst✝⁴ : CommSemiring R
inst✝³ : AddCommMonoid M
inst✝² : Module R M
inst✝¹ : DecidableEq R
inst✝ : DecidableEq M
n : ℕ
p : DividedPowerAlgebra R M
⊢ p ∈ grade R M n → ∃ q, IsWeightedHomogeneous Prod.fst (↑q) n ∧ mk ↑q = p | case mp
R : Type u_1
M : Type u_2
inst✝⁴ : CommSemiring R
inst✝³ : AddCommMonoid M
inst✝² : Module R M
inst✝¹ : DecidableEq R
inst✝ : DecidableEq M
n : ℕ
p : DividedPowerAlgebra R M
hp : p ∈ grade R M n
⊢ ∃ q, IsWeightedHomogeneous Prod.fst (↑q) n ∧ mk ↑q = p | Please generate a tactic in lean4 to solve the state.
STATE:
case mp
R : Type u_1
M : Type u_2
inst✝⁴ : CommSemiring R
inst✝³ : AddCommMonoid M
inst✝² : Module R M
inst✝¹ : DecidableEq R
inst✝ : DecidableEq M
n : ℕ
p : DividedPowerAlgebra R M
⊢ p ∈ grade R M n → ∃ q, IsWeightedHomogeneous Prod.fst (↑q) n ∧ mk ↑q = p
TA... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/DPAlgebra/Graded/Basic.lean | DividedPowerAlgebra.mem_grade_iff' | [124, 1] | [132, 23] | rw [← Submodule.coe_mk p hp] | case mp
R : Type u_1
M : Type u_2
inst✝⁴ : CommSemiring R
inst✝³ : AddCommMonoid M
inst✝² : Module R M
inst✝¹ : DecidableEq R
inst✝ : DecidableEq M
n : ℕ
p : DividedPowerAlgebra R M
hp : p ∈ grade R M n
⊢ ∃ q, IsWeightedHomogeneous Prod.fst (↑q) n ∧ mk ↑q = p | case mp
R : Type u_1
M : Type u_2
inst✝⁴ : CommSemiring R
inst✝³ : AddCommMonoid M
inst✝² : Module R M
inst✝¹ : DecidableEq R
inst✝ : DecidableEq M
n : ℕ
p : DividedPowerAlgebra R M
hp : p ∈ grade R M n
⊢ ∃ q, IsWeightedHomogeneous Prod.fst (↑q) n ∧ mk ↑q = ↑⟨p, hp⟩ | Please generate a tactic in lean4 to solve the state.
STATE:
case mp
R : Type u_1
M : Type u_2
inst✝⁴ : CommSemiring R
inst✝³ : AddCommMonoid M
inst✝² : Module R M
inst✝¹ : DecidableEq R
inst✝ : DecidableEq M
n : ℕ
p : DividedPowerAlgebra R M
hp : p ∈ grade R M n
⊢ ∃ q, IsWeightedHomogeneous Prod.fst (↑q) n ∧ mk ↑q = p... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/DPAlgebra/Graded/Basic.lean | DividedPowerAlgebra.mem_grade_iff' | [124, 1] | [132, 23] | apply surjective_of_supported' | case mp
R : Type u_1
M : Type u_2
inst✝⁴ : CommSemiring R
inst✝³ : AddCommMonoid M
inst✝² : Module R M
inst✝¹ : DecidableEq R
inst✝ : DecidableEq M
n : ℕ
p : DividedPowerAlgebra R M
hp : p ∈ grade R M n
⊢ ∃ q, IsWeightedHomogeneous Prod.fst (↑q) n ∧ mk ↑q = ↑⟨p, hp⟩ | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case mp
R : Type u_1
M : Type u_2
inst✝⁴ : CommSemiring R
inst✝³ : AddCommMonoid M
inst✝² : Module R M
inst✝¹ : DecidableEq R
inst✝ : DecidableEq M
n : ℕ
p : DividedPowerAlgebra R M
hp : p ∈ grade R M n
⊢ ∃ q, IsWeightedHomogeneous Prod.fst (↑q) n ∧ mk ↑q = ↑... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/DPAlgebra/Graded/Basic.lean | DividedPowerAlgebra.mem_grade_iff' | [124, 1] | [132, 23] | rintro ⟨q, hq, rfl⟩ | case mpr
R : Type u_1
M : Type u_2
inst✝⁴ : CommSemiring R
inst✝³ : AddCommMonoid M
inst✝² : Module R M
inst✝¹ : DecidableEq R
inst✝ : DecidableEq M
n : ℕ
p : DividedPowerAlgebra R M
⊢ (∃ q, IsWeightedHomogeneous Prod.fst (↑q) n ∧ mk ↑q = p) → p ∈ grade R M n | case mpr.intro.intro
R : Type u_1
M : Type u_2
inst✝⁴ : CommSemiring R
inst✝³ : AddCommMonoid M
inst✝² : Module R M
inst✝¹ : DecidableEq R
inst✝ : DecidableEq M
n : ℕ
q : ↥(supported R {nm | 0 < nm.1})
hq : IsWeightedHomogeneous Prod.fst (↑q) n
⊢ mk ↑q ∈ grade R M n | Please generate a tactic in lean4 to solve the state.
STATE:
case mpr
R : Type u_1
M : Type u_2
inst✝⁴ : CommSemiring R
inst✝³ : AddCommMonoid M
inst✝² : Module R M
inst✝¹ : DecidableEq R
inst✝ : DecidableEq M
n : ℕ
p : DividedPowerAlgebra R M
⊢ (∃ q, IsWeightedHomogeneous Prod.fst (↑q) n ∧ mk ↑q = p) → p ∈ grade R M n... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/DPAlgebra/Graded/Basic.lean | DividedPowerAlgebra.mem_grade_iff' | [124, 1] | [132, 23] | exact ⟨q, hq, rfl⟩ | case mpr.intro.intro
R : Type u_1
M : Type u_2
inst✝⁴ : CommSemiring R
inst✝³ : AddCommMonoid M
inst✝² : Module R M
inst✝¹ : DecidableEq R
inst✝ : DecidableEq M
n : ℕ
q : ↥(supported R {nm | 0 < nm.1})
hq : IsWeightedHomogeneous Prod.fst (↑q) n
⊢ mk ↑q ∈ grade R M n | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case mpr.intro.intro
R : Type u_1
M : Type u_2
inst✝⁴ : CommSemiring R
inst✝³ : AddCommMonoid M
inst✝² : Module R M
inst✝¹ : DecidableEq R
inst✝ : DecidableEq M
n : ℕ
q : ↥(supported R {nm | 0 < nm.1})
hq : IsWeightedHomogeneous Prod.fst (↑q) n
⊢ mk ↑q ∈ grad... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/DPAlgebra/Graded/Basic.lean | DividedPowerAlgebra.ι_comp_lift | [162, 9] | [168, 27] | ext m | R : Type u_1
M : Type u_2
inst✝⁶ : CommSemiring R
inst✝⁵ : AddCommMonoid M
inst✝⁴ : Module R M
inst✝³ : DecidableEq R
inst✝² : DecidableEq M
A : Type u_3
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).toLinearMap ∘ₗ ι R M = φ | case h
R : Type u_1
M : Type u_2
inst✝⁶ : CommSemiring R
inst✝⁵ : AddCommMonoid M
inst✝⁴ : Module R M
inst✝³ : DecidableEq R
inst✝² : DecidableEq M
A : Type u_3
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
m : M
⊢ ((lift hI φ hφ).toLinearMap ∘ₗ ι R M) m ... | Please generate a tactic in lean4 to solve the state.
STATE:
R : Type u_1
M : Type u_2
inst✝⁶ : CommSemiring R
inst✝⁵ : AddCommMonoid M
inst✝⁴ : Module R M
inst✝³ : DecidableEq R
inst✝² : DecidableEq M
A : Type u_3
inst✝¹ : CommRing A
inst✝ : Algebra R A
I : Ideal A
hI : DividedPowers I
φ : M →ₗ[R] A
hφ : ∀ (m : M), φ ... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/DPAlgebra/Graded/Basic.lean | DividedPowerAlgebra.ι_comp_lift | [162, 9] | [168, 27] | simp only [LinearMap.coe_comp, Function.comp_apply, AlgHom.toLinearMap_apply, ι_def,
liftAlgHom_apply_dp] | case h
R : Type u_1
M : Type u_2
inst✝⁶ : CommSemiring R
inst✝⁵ : AddCommMonoid M
inst✝⁴ : Module R M
inst✝³ : DecidableEq R
inst✝² : DecidableEq M
A : Type u_3
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
m : M
⊢ ((lift hI φ hφ).toLinearMap ∘ₗ ι R M) m ... | case h
R : Type u_1
M : Type u_2
inst✝⁶ : CommSemiring R
inst✝⁵ : AddCommMonoid M
inst✝⁴ : Module R M
inst✝³ : DecidableEq R
inst✝² : DecidableEq M
A : Type u_3
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
m : M
⊢ hI.dpow 1 (φ m) = φ m | Please generate a tactic in lean4 to solve the state.
STATE:
case h
R : Type u_1
M : Type u_2
inst✝⁶ : CommSemiring R
inst✝⁵ : AddCommMonoid M
inst✝⁴ : Module R M
inst✝³ : DecidableEq R
inst✝² : DecidableEq M
A : Type u_3
inst✝¹ : CommRing A
inst✝ : Algebra R A
I : Ideal A
hI : DividedPowers I
φ : M →ₗ[R] A
hφ : ∀ (m :... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/DPAlgebra/Graded/Basic.lean | DividedPowerAlgebra.ι_comp_lift | [162, 9] | [168, 27] | exact hI.dpow_one (hφ m) | case h
R : Type u_1
M : Type u_2
inst✝⁶ : CommSemiring R
inst✝⁵ : AddCommMonoid M
inst✝⁴ : Module R M
inst✝³ : DecidableEq R
inst✝² : DecidableEq M
A : Type u_3
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
m : M
⊢ hI.dpow 1 (φ m) = φ m | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case h
R : Type u_1
M : Type u_2
inst✝⁶ : CommSemiring R
inst✝⁵ : AddCommMonoid M
inst✝⁴ : Module R M
inst✝³ : DecidableEq R
inst✝² : DecidableEq M
A : Type u_3
inst✝¹ : CommRing A
inst✝ : Algebra R A
I : Ideal A
hI : DividedPowers I
φ : M →ₗ[R] A
hφ : ∀ (m :... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/DPAlgebra/Graded/Basic.lean | DividedPowerAlgebra.lift_ι_apply | [172, 9] | [176, 6] | conv_rhs => rw [← ι_comp_lift R hI hφ] | R : Type u_1
M : Type u_2
inst✝⁶ : CommSemiring R
inst✝⁵ : AddCommMonoid M
inst✝⁴ : Module R M
inst✝³ : DecidableEq R
inst✝² : DecidableEq M
A : Type u_3
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
x : M
⊢ (lift hI φ hφ) ((ι R M) x) = φ x | R : Type u_1
M : Type u_2
inst✝⁶ : CommSemiring R
inst✝⁵ : AddCommMonoid M
inst✝⁴ : Module R M
inst✝³ : DecidableEq R
inst✝² : DecidableEq M
A : Type u_3
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
x : M
⊢ (lift hI φ hφ) ((ι R M) x) = ((lift hI φ hφ).to... | Please generate a tactic in lean4 to solve the state.
STATE:
R : Type u_1
M : Type u_2
inst✝⁶ : CommSemiring R
inst✝⁵ : AddCommMonoid M
inst✝⁴ : Module R M
inst✝³ : DecidableEq R
inst✝² : DecidableEq M
A : Type u_3
inst✝¹ : CommRing A
inst✝ : Algebra R A
I : Ideal A
hI : DividedPowers I
φ : M →ₗ[R] A
hφ : ∀ (m : M), φ ... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/DPAlgebra/Graded/Basic.lean | DividedPowerAlgebra.lift_ι_apply | [172, 9] | [176, 6] | rfl | R : Type u_1
M : Type u_2
inst✝⁶ : CommSemiring R
inst✝⁵ : AddCommMonoid M
inst✝⁴ : Module R M
inst✝³ : DecidableEq R
inst✝² : DecidableEq M
A : Type u_3
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
x : M
⊢ (lift hI φ hφ) ((ι R M) x) = ((lift hI φ hφ).to... | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
R : Type u_1
M : Type u_2
inst✝⁶ : CommSemiring R
inst✝⁵ : AddCommMonoid M
inst✝⁴ : Module R M
inst✝³ : DecidableEq R
inst✝² : DecidableEq M
A : Type u_3
inst✝¹ : CommRing A
inst✝ : Algebra R A
I : Ideal A
hI : DividedPowers I
φ : M →ₗ[R] A
hφ : ∀ (m : M), φ ... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/DPAlgebra/Graded/Basic.lean | DividedPowerAlgebra.liftAux_isHomogeneous | [188, 1] | [206, 62] | intro i a | R : Type u_1
M : Type u_2
inst✝¹¹ : CommSemiring R
inst✝¹⁰ : AddCommMonoid M
inst✝⁹ : Module R M
inst✝⁸ : DecidableEq R
inst✝⁷ : DecidableEq M
S : Type u_3
inst✝⁶ : CommSemiring S
inst✝⁵ : Algebra R S
A : Type u_4
inst✝⁴ : CommSemiring A
inst✝³ : Algebra R A
inst✝² : Algebra S A
inst✝¹ : IsScalarTower R S A
𝒜 : ℕ → Su... | R : Type u_1
M : Type u_2
inst✝¹¹ : CommSemiring R
inst✝¹⁰ : AddCommMonoid M
inst✝⁹ : Module R M
inst✝⁸ : DecidableEq R
inst✝⁷ : DecidableEq M
S : Type u_3
inst✝⁶ : CommSemiring S
inst✝⁵ : Algebra R S
A : Type u_4
inst✝⁴ : CommSemiring A
inst✝³ : Algebra R A
inst✝² : Algebra S A
inst✝¹ : IsScalarTower R S A
𝒜 : ℕ → Su... | Please generate a tactic in lean4 to solve the state.
STATE:
R : Type u_1
M : Type u_2
inst✝¹¹ : CommSemiring R
inst✝¹⁰ : AddCommMonoid M
inst✝⁹ : Module R M
inst✝⁸ : DecidableEq R
inst✝⁷ : DecidableEq M
S : Type u_3
inst✝⁶ : CommSemiring S
inst✝⁵ : Algebra R S
A : Type u_4
inst✝⁴ : CommSemiring A
inst✝³ : Algebra R A
... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/DPAlgebra/Graded/Basic.lean | DividedPowerAlgebra.liftAux_isHomogeneous | [188, 1] | [206, 62] | simp only [mem_grade_iff] | R : Type u_1
M : Type u_2
inst✝¹¹ : CommSemiring R
inst✝¹⁰ : AddCommMonoid M
inst✝⁹ : Module R M
inst✝⁸ : DecidableEq R
inst✝⁷ : DecidableEq M
S : Type u_3
inst✝⁶ : CommSemiring S
inst✝⁵ : Algebra R S
A : Type u_4
inst✝⁴ : CommSemiring A
inst✝³ : Algebra R A
inst✝² : Algebra S A
inst✝¹ : IsScalarTower R S A
𝒜 : ℕ → Su... | R : Type u_1
M : Type u_2
inst✝¹¹ : CommSemiring R
inst✝¹⁰ : AddCommMonoid M
inst✝⁹ : Module R M
inst✝⁸ : DecidableEq R
inst✝⁷ : DecidableEq M
S : Type u_3
inst✝⁶ : CommSemiring S
inst✝⁵ : Algebra R S
A : Type u_4
inst✝⁴ : CommSemiring A
inst✝³ : Algebra R A
inst✝² : Algebra S A
inst✝¹ : IsScalarTower R S A
𝒜 : ℕ → Su... | Please generate a tactic in lean4 to solve the state.
STATE:
R : Type u_1
M : Type u_2
inst✝¹¹ : CommSemiring R
inst✝¹⁰ : AddCommMonoid M
inst✝⁹ : Module R M
inst✝⁸ : DecidableEq R
inst✝⁷ : DecidableEq M
S : Type u_3
inst✝⁶ : CommSemiring S
inst✝⁵ : Algebra R S
A : Type u_4
inst✝⁴ : CommSemiring A
inst✝³ : Algebra R A
... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/DPAlgebra/Graded/Basic.lean | DividedPowerAlgebra.liftAux_isHomogeneous | [188, 1] | [206, 62] | rintro ⟨p, hp, rfl⟩ | R : Type u_1
M : Type u_2
inst✝¹¹ : CommSemiring R
inst✝¹⁰ : AddCommMonoid M
inst✝⁹ : Module R M
inst✝⁸ : DecidableEq R
inst✝⁷ : DecidableEq M
S : Type u_3
inst✝⁶ : CommSemiring S
inst✝⁵ : Algebra R S
A : Type u_4
inst✝⁴ : CommSemiring A
inst✝³ : Algebra R A
inst✝² : Algebra S A
inst✝¹ : IsScalarTower R S A
𝒜 : ℕ → Su... | case intro.intro
R : Type u_1
M : Type u_2
inst✝¹¹ : CommSemiring R
inst✝¹⁰ : AddCommMonoid M
inst✝⁹ : Module R M
inst✝⁸ : DecidableEq R
inst✝⁷ : DecidableEq M
S : Type u_3
inst✝⁶ : CommSemiring S
inst✝⁵ : Algebra R S
A : Type u_4
inst✝⁴ : CommSemiring A
inst✝³ : Algebra R A
inst✝² : Algebra S A
inst✝¹ : IsScalarTower ... | Please generate a tactic in lean4 to solve the state.
STATE:
R : Type u_1
M : Type u_2
inst✝¹¹ : CommSemiring R
inst✝¹⁰ : AddCommMonoid M
inst✝⁹ : Module R M
inst✝⁸ : DecidableEq R
inst✝⁷ : DecidableEq M
S : Type u_3
inst✝⁶ : CommSemiring S
inst✝⁵ : Algebra R S
A : Type u_4
inst✝⁴ : CommSemiring A
inst✝³ : Algebra R A
... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/DPAlgebra/Graded/Basic.lean | DividedPowerAlgebra.liftAux_isHomogeneous | [188, 1] | [206, 62] | rw [lift'AlgHom_apply, p.as_sum, aeval_sum] | case intro.intro
R : Type u_1
M : Type u_2
inst✝¹¹ : CommSemiring R
inst✝¹⁰ : AddCommMonoid M
inst✝⁹ : Module R M
inst✝⁸ : DecidableEq R
inst✝⁷ : DecidableEq M
S : Type u_3
inst✝⁶ : CommSemiring S
inst✝⁵ : Algebra R S
A : Type u_4
inst✝⁴ : CommSemiring A
inst✝³ : Algebra R A
inst✝² : Algebra S A
inst✝¹ : IsScalarTower ... | case intro.intro
R : Type u_1
M : Type u_2
inst✝¹¹ : CommSemiring R
inst✝¹⁰ : AddCommMonoid M
inst✝⁹ : Module R M
inst✝⁸ : DecidableEq R
inst✝⁷ : DecidableEq M
S : Type u_3
inst✝⁶ : CommSemiring S
inst✝⁵ : Algebra R S
A : Type u_4
inst✝⁴ : CommSemiring A
inst✝³ : Algebra R A
inst✝² : Algebra S A
inst✝¹ : IsScalarTower ... | Please generate a tactic in lean4 to solve the state.
STATE:
case intro.intro
R : Type u_1
M : Type u_2
inst✝¹¹ : CommSemiring R
inst✝¹⁰ : AddCommMonoid M
inst✝⁹ : Module R M
inst✝⁸ : DecidableEq R
inst✝⁷ : DecidableEq M
S : Type u_3
inst✝⁶ : CommSemiring S
inst✝⁵ : Algebra R S
A : Type u_4
inst✝⁴ : CommSemiring A
inst... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/DPAlgebra/Graded/Basic.lean | DividedPowerAlgebra.liftAux_isHomogeneous | [188, 1] | [206, 62] | apply _root_.sum_mem | case intro.intro
R : Type u_1
M : Type u_2
inst✝¹¹ : CommSemiring R
inst✝¹⁰ : AddCommMonoid M
inst✝⁹ : Module R M
inst✝⁸ : DecidableEq R
inst✝⁷ : DecidableEq M
S : Type u_3
inst✝⁶ : CommSemiring S
inst✝⁵ : Algebra R S
A : Type u_4
inst✝⁴ : CommSemiring A
inst✝³ : Algebra R A
inst✝² : Algebra S A
inst✝¹ : IsScalarTower ... | case intro.intro.h
R : Type u_1
M : Type u_2
inst✝¹¹ : CommSemiring R
inst✝¹⁰ : AddCommMonoid M
inst✝⁹ : Module R M
inst✝⁸ : DecidableEq R
inst✝⁷ : DecidableEq M
S : Type u_3
inst✝⁶ : CommSemiring S
inst✝⁵ : Algebra R S
A : Type u_4
inst✝⁴ : CommSemiring A
inst✝³ : Algebra R A
inst✝² : Algebra S A
inst✝¹ : IsScalarTowe... | Please generate a tactic in lean4 to solve the state.
STATE:
case intro.intro
R : Type u_1
M : Type u_2
inst✝¹¹ : CommSemiring R
inst✝¹⁰ : AddCommMonoid M
inst✝⁹ : Module R M
inst✝⁸ : DecidableEq R
inst✝⁷ : DecidableEq M
S : Type u_3
inst✝⁶ : CommSemiring S
inst✝⁵ : Algebra R S
A : Type u_4
inst✝⁴ : CommSemiring A
inst... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/DPAlgebra/Graded/Basic.lean | DividedPowerAlgebra.liftAux_isHomogeneous | [188, 1] | [206, 62] | intro c hc | case intro.intro.h
R : Type u_1
M : Type u_2
inst✝¹¹ : CommSemiring R
inst✝¹⁰ : AddCommMonoid M
inst✝⁹ : Module R M
inst✝⁸ : DecidableEq R
inst✝⁷ : DecidableEq M
S : Type u_3
inst✝⁶ : CommSemiring S
inst✝⁵ : Algebra R S
A : Type u_4
inst✝⁴ : CommSemiring A
inst✝³ : Algebra R A
inst✝² : Algebra S A
inst✝¹ : IsScalarTowe... | case intro.intro.h
R : Type u_1
M : Type u_2
inst✝¹¹ : CommSemiring R
inst✝¹⁰ : AddCommMonoid M
inst✝⁹ : Module R M
inst✝⁸ : DecidableEq R
inst✝⁷ : DecidableEq M
S : Type u_3
inst✝⁶ : CommSemiring S
inst✝⁵ : Algebra R S
A : Type u_4
inst✝⁴ : CommSemiring A
inst✝³ : Algebra R A
inst✝² : Algebra S A
inst✝¹ : IsScalarTowe... | Please generate a tactic in lean4 to solve the state.
STATE:
case intro.intro.h
R : Type u_1
M : Type u_2
inst✝¹¹ : CommSemiring R
inst✝¹⁰ : AddCommMonoid M
inst✝⁹ : Module R M
inst✝⁸ : DecidableEq R
inst✝⁷ : DecidableEq M
S : Type u_3
inst✝⁶ : CommSemiring S
inst✝⁵ : Algebra R S
A : Type u_4
inst✝⁴ : CommSemiring A
in... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/DPAlgebra/Graded/Basic.lean | DividedPowerAlgebra.liftAux_isHomogeneous | [188, 1] | [206, 62] | rw [aeval_monomial, ← smul_eq_mul, algebraMap_smul A, algebra_compatible_smul S (coeff c p)] | case intro.intro.h
R : Type u_1
M : Type u_2
inst✝¹¹ : CommSemiring R
inst✝¹⁰ : AddCommMonoid M
inst✝⁹ : Module R M
inst✝⁸ : DecidableEq R
inst✝⁷ : DecidableEq M
S : Type u_3
inst✝⁶ : CommSemiring S
inst✝⁵ : Algebra R S
A : Type u_4
inst✝⁴ : CommSemiring A
inst✝³ : Algebra R A
inst✝² : Algebra S A
inst✝¹ : IsScalarTowe... | case intro.intro.h
R : Type u_1
M : Type u_2
inst✝¹¹ : CommSemiring R
inst✝¹⁰ : AddCommMonoid M
inst✝⁹ : Module R M
inst✝⁸ : DecidableEq R
inst✝⁷ : DecidableEq M
S : Type u_3
inst✝⁶ : CommSemiring S
inst✝⁵ : Algebra R S
A : Type u_4
inst✝⁴ : CommSemiring A
inst✝³ : Algebra R A
inst✝² : Algebra S A
inst✝¹ : IsScalarTowe... | Please generate a tactic in lean4 to solve the state.
STATE:
case intro.intro.h
R : Type u_1
M : Type u_2
inst✝¹¹ : CommSemiring R
inst✝¹⁰ : AddCommMonoid M
inst✝⁹ : Module R M
inst✝⁸ : DecidableEq R
inst✝⁷ : DecidableEq M
S : Type u_3
inst✝⁶ : CommSemiring S
inst✝⁵ : Algebra R S
A : Type u_4
inst✝⁴ : CommSemiring A
in... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/DPAlgebra/Graded/Basic.lean | DividedPowerAlgebra.liftAux_isHomogeneous | [188, 1] | [206, 62] | apply Submodule.smul_mem | case intro.intro.h
R : Type u_1
M : Type u_2
inst✝¹¹ : CommSemiring R
inst✝¹⁰ : AddCommMonoid M
inst✝⁹ : Module R M
inst✝⁸ : DecidableEq R
inst✝⁷ : DecidableEq M
S : Type u_3
inst✝⁶ : CommSemiring S
inst✝⁵ : Algebra R S
A : Type u_4
inst✝⁴ : CommSemiring A
inst✝³ : Algebra R A
inst✝² : Algebra S A
inst✝¹ : IsScalarTowe... | case intro.intro.h.h
R : Type u_1
M : Type u_2
inst✝¹¹ : CommSemiring R
inst✝¹⁰ : AddCommMonoid M
inst✝⁹ : Module R M
inst✝⁸ : DecidableEq R
inst✝⁷ : DecidableEq M
S : Type u_3
inst✝⁶ : CommSemiring S
inst✝⁵ : Algebra R S
A : Type u_4
inst✝⁴ : CommSemiring A
inst✝³ : Algebra R A
inst✝² : Algebra S A
inst✝¹ : IsScalarTo... | Please generate a tactic in lean4 to solve the state.
STATE:
case intro.intro.h
R : Type u_1
M : Type u_2
inst✝¹¹ : CommSemiring R
inst✝¹⁰ : AddCommMonoid M
inst✝⁹ : Module R M
inst✝⁸ : DecidableEq R
inst✝⁷ : DecidableEq M
S : Type u_3
inst✝⁶ : CommSemiring S
inst✝⁵ : Algebra R S
A : Type u_4
inst✝⁴ : CommSemiring A
in... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/DPAlgebra/Graded/Basic.lean | DividedPowerAlgebra.liftAux_isHomogeneous | [188, 1] | [206, 62] | rw [← hp (mem_support_iff.mp hc)] | case intro.intro.h.h
R : Type u_1
M : Type u_2
inst✝¹¹ : CommSemiring R
inst✝¹⁰ : AddCommMonoid M
inst✝⁹ : Module R M
inst✝⁸ : DecidableEq R
inst✝⁷ : DecidableEq M
S : Type u_3
inst✝⁶ : CommSemiring S
inst✝⁵ : Algebra R S
A : Type u_4
inst✝⁴ : CommSemiring A
inst✝³ : Algebra R A
inst✝² : Algebra S A
inst✝¹ : IsScalarTo... | case intro.intro.h.h
R : Type u_1
M : Type u_2
inst✝¹¹ : CommSemiring R
inst✝¹⁰ : AddCommMonoid M
inst✝⁹ : Module R M
inst✝⁸ : DecidableEq R
inst✝⁷ : DecidableEq M
S : Type u_3
inst✝⁶ : CommSemiring S
inst✝⁵ : Algebra R S
A : Type u_4
inst✝⁴ : CommSemiring A
inst✝³ : Algebra R A
inst✝² : Algebra S A
inst✝¹ : IsScalarTo... | Please generate a tactic in lean4 to solve the state.
STATE:
case intro.intro.h.h
R : Type u_1
M : Type u_2
inst✝¹¹ : CommSemiring R
inst✝¹⁰ : AddCommMonoid M
inst✝⁹ : Module R M
inst✝⁸ : DecidableEq R
inst✝⁷ : DecidableEq M
S : Type u_3
inst✝⁶ : CommSemiring S
inst✝⁵ : Algebra R S
A : Type u_4
inst✝⁴ : CommSemiring A
... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/DPAlgebra/Graded/Basic.lean | DividedPowerAlgebra.liftAux_isHomogeneous | [188, 1] | [206, 62] | exact Finsupp.prod.mem_grade _ _ _ _ fun ⟨n, m⟩ _ => hf n m | case intro.intro.h.h
R : Type u_1
M : Type u_2
inst✝¹¹ : CommSemiring R
inst✝¹⁰ : AddCommMonoid M
inst✝⁹ : Module R M
inst✝⁸ : DecidableEq R
inst✝⁷ : DecidableEq M
S : Type u_3
inst✝⁶ : CommSemiring S
inst✝⁵ : Algebra R S
A : Type u_4
inst✝⁴ : CommSemiring A
inst✝³ : Algebra R A
inst✝² : Algebra S A
inst✝¹ : IsScalarTo... | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case intro.intro.h.h
R : Type u_1
M : Type u_2
inst✝¹¹ : CommSemiring R
inst✝¹⁰ : AddCommMonoid M
inst✝⁹ : Module R M
inst✝⁸ : DecidableEq R
inst✝⁷ : DecidableEq M
S : Type u_3
inst✝⁶ : CommSemiring S
inst✝⁵ : Algebra R S
A : Type u_4
inst✝⁴ : CommSemiring A
... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/DPAlgebra/Graded/Basic.lean | DividedPowerAlgebra.lift_isHomogeneous | [212, 1] | [218, 82] | apply liftAux_isHomogeneous | R : Type u_1
M : Type u_2
inst✝⁹ : CommSemiring R
inst✝⁸ : AddCommMonoid M
inst✝⁷ : Module R M
inst✝⁶ : DecidableEq R
inst✝⁵ : DecidableEq M
S : Type u_3
inst✝⁴ : CommSemiring S
inst✝³ : Algebra R S
A : Type u_4
inst✝² : CommSemiring A
inst✝¹ : Algebra R A
𝒜 : ℕ → Submodule R A
inst✝ : GradedAlgebra 𝒜
I : Ideal A
hI ... | case hf
R : Type u_1
M : Type u_2
inst✝⁹ : CommSemiring R
inst✝⁸ : AddCommMonoid M
inst✝⁷ : Module R M
inst✝⁶ : DecidableEq R
inst✝⁵ : DecidableEq M
S : Type u_3
inst✝⁴ : CommSemiring S
inst✝³ : Algebra R S
A : Type u_4
inst✝² : CommSemiring A
inst✝¹ : Algebra R A
𝒜 : ℕ → Submodule R A
inst✝ : GradedAlgebra 𝒜
I : Ide... | Please generate a tactic in lean4 to solve the state.
STATE:
R : Type u_1
M : Type u_2
inst✝⁹ : CommSemiring R
inst✝⁸ : AddCommMonoid M
inst✝⁷ : Module R M
inst✝⁶ : DecidableEq R
inst✝⁵ : DecidableEq M
S : Type u_3
inst✝⁴ : CommSemiring S
inst✝³ : Algebra R S
A : Type u_4
inst✝² : CommSemiring A
inst✝¹ : Algebra R A
𝒜... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/DPAlgebra/Graded/Basic.lean | DividedPowerAlgebra.lift_isHomogeneous | [212, 1] | [218, 82] | intro n m | case hf
R : Type u_1
M : Type u_2
inst✝⁹ : CommSemiring R
inst✝⁸ : AddCommMonoid M
inst✝⁷ : Module R M
inst✝⁶ : DecidableEq R
inst✝⁵ : DecidableEq M
S : Type u_3
inst✝⁴ : CommSemiring S
inst✝³ : Algebra R S
A : Type u_4
inst✝² : CommSemiring A
inst✝¹ : Algebra R A
𝒜 : ℕ → Submodule R A
inst✝ : GradedAlgebra 𝒜
I : Ide... | case hf
R : Type u_1
M : Type u_2
inst✝⁹ : CommSemiring R
inst✝⁸ : AddCommMonoid M
inst✝⁷ : Module R M
inst✝⁶ : DecidableEq R
inst✝⁵ : DecidableEq M
S : Type u_3
inst✝⁴ : CommSemiring S
inst✝³ : Algebra R S
A : Type u_4
inst✝² : CommSemiring A
inst✝¹ : Algebra R A
𝒜 : ℕ → Submodule R A
inst✝ : GradedAlgebra 𝒜
I : Ide... | Please generate a tactic in lean4 to solve the state.
STATE:
case hf
R : Type u_1
M : Type u_2
inst✝⁹ : CommSemiring R
inst✝⁸ : AddCommMonoid M
inst✝⁷ : Module R M
inst✝⁶ : DecidableEq R
inst✝⁵ : DecidableEq M
S : Type u_3
inst✝⁴ : CommSemiring S
inst✝³ : Algebra R S
A : Type u_4
inst✝² : CommSemiring A
inst✝¹ : Algebr... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/DPAlgebra/Graded/Basic.lean | DividedPowerAlgebra.lift_isHomogeneous | [212, 1] | [218, 82] | simpa only [Algebra.id.smul_eq_mul, mul_one] using hI' (φ m) (hφ m) 1 (hφ' m) n | case hf
R : Type u_1
M : Type u_2
inst✝⁹ : CommSemiring R
inst✝⁸ : AddCommMonoid M
inst✝⁷ : Module R M
inst✝⁶ : DecidableEq R
inst✝⁵ : DecidableEq M
S : Type u_3
inst✝⁴ : CommSemiring S
inst✝³ : Algebra R S
A : Type u_4
inst✝² : CommSemiring A
inst✝¹ : Algebra R A
𝒜 : ℕ → Submodule R A
inst✝ : GradedAlgebra 𝒜
I : Ide... | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case hf
R : Type u_1
M : Type u_2
inst✝⁹ : CommSemiring R
inst✝⁸ : AddCommMonoid M
inst✝⁷ : Module R M
inst✝⁶ : DecidableEq R
inst✝⁵ : DecidableEq M
S : Type u_3
inst✝⁴ : CommSemiring S
inst✝³ : Algebra R S
A : Type u_4
inst✝² : CommSemiring A
inst✝¹ : Algebr... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/DPAlgebra/Graded/Basic.lean | DividedPowerAlgebra.proj'_zero_one | [234, 1] | [235, 72] | rw [proj', proj, LinearMap.coe_mk, AddHom.coe_mk, decompose_one] | R : Type u_1
M : Type u_2
inst✝¹² : CommSemiring R
inst✝¹¹ : AddCommMonoid M
inst✝¹⁰ : Module R M
inst✝⁹ : DecidableEq R
inst✝⁸ : DecidableEq M
S : Type u_3
inst✝⁷ : CommSemiring S
inst✝⁶ : Algebra R S
N : Type u_4
inst✝⁵ : AddCommMonoid N
inst✝⁴ : DecidableEq S
inst✝³ : DecidableEq N
inst✝² : Module R N
inst✝¹ : Modul... | R : Type u_1
M : Type u_2
inst✝¹² : CommSemiring R
inst✝¹¹ : AddCommMonoid M
inst✝¹⁰ : Module R M
inst✝⁹ : DecidableEq R
inst✝⁸ : DecidableEq M
S : Type u_3
inst✝⁷ : CommSemiring S
inst✝⁶ : Algebra R S
N : Type u_4
inst✝⁵ : AddCommMonoid N
inst✝⁴ : DecidableEq S
inst✝³ : DecidableEq N
inst✝² : Module R N
inst✝¹ : Modul... | Please generate a tactic in lean4 to solve the state.
STATE:
R : Type u_1
M : Type u_2
inst✝¹² : CommSemiring R
inst✝¹¹ : AddCommMonoid M
inst✝¹⁰ : Module R M
inst✝⁹ : DecidableEq R
inst✝⁸ : DecidableEq M
S : Type u_3
inst✝⁷ : CommSemiring S
inst✝⁶ : Algebra R S
N : Type u_4
inst✝⁵ : AddCommMonoid N
inst✝⁴ : DecidableE... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/DPAlgebra/Graded/Basic.lean | DividedPowerAlgebra.proj'_zero_one | [234, 1] | [235, 72] | rfl | R : Type u_1
M : Type u_2
inst✝¹² : CommSemiring R
inst✝¹¹ : AddCommMonoid M
inst✝¹⁰ : Module R M
inst✝⁹ : DecidableEq R
inst✝⁸ : DecidableEq M
S : Type u_3
inst✝⁷ : CommSemiring S
inst✝⁶ : Algebra R S
N : Type u_4
inst✝⁵ : AddCommMonoid N
inst✝⁴ : DecidableEq S
inst✝³ : DecidableEq N
inst✝² : Module R N
inst✝¹ : Modul... | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
R : Type u_1
M : Type u_2
inst✝¹² : CommSemiring R
inst✝¹¹ : AddCommMonoid M
inst✝¹⁰ : Module R M
inst✝⁹ : DecidableEq R
inst✝⁸ : DecidableEq M
S : Type u_3
inst✝⁷ : CommSemiring S
inst✝⁶ : Algebra R S
N : Type u_4
inst✝⁵ : AddCommMonoid N
inst✝⁴ : DecidableE... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/DPAlgebra/Graded/Basic.lean | DividedPowerAlgebra.proj'_zero_mul | [237, 1] | [239, 62] | simp only [proj', ← projZeroRingHom'_apply, _root_.map_mul] | R : Type u_1
M : Type u_2
inst✝¹² : CommSemiring R
inst✝¹¹ : AddCommMonoid M
inst✝¹⁰ : Module R M
inst✝⁹ : DecidableEq R
inst✝⁸ : DecidableEq M
S : Type u_3
inst✝⁷ : CommSemiring S
inst✝⁶ : Algebra R S
N : Type u_4
inst✝⁵ : AddCommMonoid N
inst✝⁴ : DecidableEq S
inst✝³ : DecidableEq N
inst✝² : Module R N
inst✝¹ : Modul... | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
R : Type u_1
M : Type u_2
inst✝¹² : CommSemiring R
inst✝¹¹ : AddCommMonoid M
inst✝¹⁰ : Module R M
inst✝⁹ : DecidableEq R
inst✝⁸ : DecidableEq M
S : Type u_3
inst✝⁷ : CommSemiring S
inst✝⁶ : Algebra R S
N : Type u_4
inst✝⁵ : AddCommMonoid N
inst✝⁴ : DecidableE... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | unused_files/Ideal.lean | Ideal.image_eq_map_of_surjective | [6, 1] | [13, 45] | symm | A : Type u_1
B : Type u_2
inst✝¹ : Semiring A
inst✝ : Semiring B
f : A →+* B
I : Ideal A
hf : Function.Surjective ⇑f
⊢ ⇑f '' ↑I = ↑(map f I) | A : Type u_1
B : Type u_2
inst✝¹ : Semiring A
inst✝ : Semiring B
f : A →+* B
I : Ideal A
hf : Function.Surjective ⇑f
⊢ ↑(map f I) = ⇑f '' ↑I | Please generate a tactic in lean4 to solve the state.
STATE:
A : Type u_1
B : Type u_2
inst✝¹ : Semiring A
inst✝ : Semiring B
f : A →+* B
I : Ideal A
hf : Function.Surjective ⇑f
⊢ ⇑f '' ↑I = ↑(map f I)
TACTIC:
|
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | unused_files/Ideal.lean | Ideal.image_eq_map_of_surjective | [6, 1] | [13, 45] | ext x | A : Type u_1
B : Type u_2
inst✝¹ : Semiring A
inst✝ : Semiring B
f : A →+* B
I : Ideal A
hf : Function.Surjective ⇑f
⊢ ↑(map f I) = ⇑f '' ↑I | case h
A : Type u_1
B : Type u_2
inst✝¹ : Semiring A
inst✝ : Semiring B
f : A →+* B
I : Ideal A
hf : Function.Surjective ⇑f
x : B
⊢ x ∈ ↑(map f I) ↔ x ∈ ⇑f '' ↑I | Please generate a tactic in lean4 to solve the state.
STATE:
A : Type u_1
B : Type u_2
inst✝¹ : Semiring A
inst✝ : Semiring B
f : A →+* B
I : Ideal A
hf : Function.Surjective ⇑f
⊢ ↑(map f I) = ⇑f '' ↑I
TACTIC:
|
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | unused_files/Ideal.lean | Ideal.image_eq_map_of_surjective | [6, 1] | [13, 45] | simp only [Set.mem_image, SetLike.mem_coe] | case h
A : Type u_1
B : Type u_2
inst✝¹ : Semiring A
inst✝ : Semiring B
f : A →+* B
I : Ideal A
hf : Function.Surjective ⇑f
x : B
⊢ x ∈ ↑(map f I) ↔ x ∈ ⇑f '' ↑I | case h
A : Type u_1
B : Type u_2
inst✝¹ : Semiring A
inst✝ : Semiring B
f : A →+* B
I : Ideal A
hf : Function.Surjective ⇑f
x : B
⊢ x ∈ map f I ↔ ∃ x_1 ∈ I, f x_1 = x | Please generate a tactic in lean4 to solve the state.
STATE:
case h
A : Type u_1
B : Type u_2
inst✝¹ : Semiring A
inst✝ : Semiring B
f : A →+* B
I : Ideal A
hf : Function.Surjective ⇑f
x : B
⊢ x ∈ ↑(map f I) ↔ x ∈ ⇑f '' ↑I
TACTIC:
|
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | unused_files/Ideal.lean | Ideal.image_eq_map_of_surjective | [6, 1] | [13, 45] | apply Ideal.mem_map_iff_of_surjective _ hf | case h
A : Type u_1
B : Type u_2
inst✝¹ : Semiring A
inst✝ : Semiring B
f : A →+* B
I : Ideal A
hf : Function.Surjective ⇑f
x : B
⊢ x ∈ map f I ↔ ∃ x_1 ∈ I, f x_1 = x | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case h
A : Type u_1
B : Type u_2
inst✝¹ : Semiring A
inst✝ : Semiring B
f : A →+* B
I : Ideal A
hf : Function.Surjective ⇑f
x : B
⊢ x ∈ map f I ↔ ∃ x_1 ∈ I, f x_1 = x
TACTIC:
|
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/DPAlgebra/Graded/GradeZero.lean | DividedPowerAlgebra.algebraMapInv_eq | [26, 1] | [38, 51] | rw [← AlgHom.comp_apply] | R : Type u
M : Type v
inst✝⁴ : CommSemiring R
inst✝³ : AddCommMonoid M
inst✝² : Module R M
inst✝¹ : DecidableEq R
inst✝ : DecidableEq M
f : MvPolynomial (ℕ × M) R
⊢ (algebraMapInv R M) (mk f) = (aeval fun nm => if 0 < nm.1 then 0 else 1) f | R : Type u
M : Type v
inst✝⁴ : CommSemiring R
inst✝³ : AddCommMonoid M
inst✝² : Module R M
inst✝¹ : DecidableEq R
inst✝ : DecidableEq M
f : MvPolynomial (ℕ × M) R
⊢ ((algebraMapInv R M).comp mk) f = (aeval fun nm => if 0 < nm.1 then 0 else 1) f | Please generate a tactic in lean4 to solve the state.
STATE:
R : Type u
M : Type v
inst✝⁴ : CommSemiring R
inst✝³ : AddCommMonoid M
inst✝² : Module R M
inst✝¹ : DecidableEq R
inst✝ : DecidableEq M
f : MvPolynomial (ℕ × M) R
⊢ (algebraMapInv R M) (mk f) = (aeval fun nm => if 0 < nm.1 then 0 else 1) f
TACTIC:
|
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/DPAlgebra/Graded/GradeZero.lean | DividedPowerAlgebra.algebraMapInv_eq | [26, 1] | [38, 51] | apply AlgHom.congr_fun | R : Type u
M : Type v
inst✝⁴ : CommSemiring R
inst✝³ : AddCommMonoid M
inst✝² : Module R M
inst✝¹ : DecidableEq R
inst✝ : DecidableEq M
f : MvPolynomial (ℕ × M) R
⊢ ((algebraMapInv R M).comp mk) f = (aeval fun nm => if 0 < nm.1 then 0 else 1) f | case H
R : Type u
M : Type v
inst✝⁴ : CommSemiring R
inst✝³ : AddCommMonoid M
inst✝² : Module R M
inst✝¹ : DecidableEq R
inst✝ : DecidableEq M
f : MvPolynomial (ℕ × M) R
⊢ (algebraMapInv R M).comp mk = aeval fun nm => if 0 < nm.1 then 0 else 1 | Please generate a tactic in lean4 to solve the state.
STATE:
R : Type u
M : Type v
inst✝⁴ : CommSemiring R
inst✝³ : AddCommMonoid M
inst✝² : Module R M
inst✝¹ : DecidableEq R
inst✝ : DecidableEq M
f : MvPolynomial (ℕ × M) R
⊢ ((algebraMapInv R M).comp mk) f = (aeval fun nm => if 0 < nm.1 then 0 else 1) f
TACTIC:
|
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/DPAlgebra/Graded/GradeZero.lean | DividedPowerAlgebra.algebraMapInv_eq | [26, 1] | [38, 51] | ext ⟨n, m⟩ | case H
R : Type u
M : Type v
inst✝⁴ : CommSemiring R
inst✝³ : AddCommMonoid M
inst✝² : Module R M
inst✝¹ : DecidableEq R
inst✝ : DecidableEq M
f : MvPolynomial (ℕ × M) R
⊢ (algebraMapInv R M).comp mk = aeval fun nm => if 0 < nm.1 then 0 else 1 | case H.hf.mk
R : Type u
M : Type v
inst✝⁴ : CommSemiring R
inst✝³ : AddCommMonoid M
inst✝² : Module R M
inst✝¹ : DecidableEq R
inst✝ : DecidableEq M
f : MvPolynomial (ℕ × M) R
n : ℕ
m : M
⊢ ((algebraMapInv R M).comp mk) (X (n, m)) = (aeval fun nm => if 0 < nm.1 then 0 else 1) (X (n, m)) | Please generate a tactic in lean4 to solve the state.
STATE:
case H
R : Type u
M : Type v
inst✝⁴ : CommSemiring R
inst✝³ : AddCommMonoid M
inst✝² : Module R M
inst✝¹ : DecidableEq R
inst✝ : DecidableEq M
f : MvPolynomial (ℕ × M) R
⊢ (algebraMapInv R M).comp mk = aeval fun nm => if 0 < nm.1 then 0 else 1
TACTIC:
|
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/DPAlgebra/Graded/GradeZero.lean | DividedPowerAlgebra.algebraMapInv_eq | [26, 1] | [38, 51] | simp only [algebraMapInv, AlgHom.comp_apply, aeval_X] | case H.hf.mk
R : Type u
M : Type v
inst✝⁴ : CommSemiring R
inst✝³ : AddCommMonoid M
inst✝² : Module R M
inst✝¹ : DecidableEq R
inst✝ : DecidableEq M
f : MvPolynomial (ℕ × M) R
n : ℕ
m : M
⊢ ((algebraMapInv R M).comp mk) (X (n, m)) = (aeval fun nm => if 0 < nm.1 then 0 else 1) (X (n, m)) | case H.hf.mk
R : Type u
M : Type v
inst✝⁴ : CommSemiring R
inst✝³ : AddCommMonoid M
inst✝² : Module R M
inst✝¹ : DecidableEq R
inst✝ : DecidableEq M
f : MvPolynomial (ℕ × M) R
n : ℕ
m : M
⊢ (lift (dividedPowersBot R) 0 ⋯) (mk (X (n, m))) = if 0 < n then 0 else 1 | Please generate a tactic in lean4 to solve the state.
STATE:
case H.hf.mk
R : Type u
M : Type v
inst✝⁴ : CommSemiring R
inst✝³ : AddCommMonoid M
inst✝² : Module R M
inst✝¹ : DecidableEq R
inst✝ : DecidableEq M
f : MvPolynomial (ℕ × M) R
n : ℕ
m : M
⊢ ((algebraMapInv R M).comp mk) (X (n, m)) = (aeval fun nm => if 0 < nm... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/DPAlgebra/Graded/GradeZero.lean | DividedPowerAlgebra.algebraMapInv_eq | [26, 1] | [38, 51] | by_cases hn : 0 < n | case H.hf.mk
R : Type u
M : Type v
inst✝⁴ : CommSemiring R
inst✝³ : AddCommMonoid M
inst✝² : Module R M
inst✝¹ : DecidableEq R
inst✝ : DecidableEq M
f : MvPolynomial (ℕ × M) R
n : ℕ
m : M
⊢ (lift (dividedPowersBot R) 0 ⋯) (mk (X (n, m))) = if 0 < n then 0 else 1 | case pos
R : Type u
M : Type v
inst✝⁴ : CommSemiring R
inst✝³ : AddCommMonoid M
inst✝² : Module R M
inst✝¹ : DecidableEq R
inst✝ : DecidableEq M
f : MvPolynomial (ℕ × M) R
n : ℕ
m : M
hn : 0 < n
⊢ (lift (dividedPowersBot R) 0 ⋯) (mk (X (n, m))) = if 0 < n then 0 else 1
case neg
R : Type u
M : Type v
inst✝⁴ : CommSemir... | Please generate a tactic in lean4 to solve the state.
STATE:
case H.hf.mk
R : Type u
M : Type v
inst✝⁴ : CommSemiring R
inst✝³ : AddCommMonoid M
inst✝² : Module R M
inst✝¹ : DecidableEq R
inst✝ : DecidableEq M
f : MvPolynomial (ℕ × M) R
n : ℕ
m : M
⊢ (lift (dividedPowersBot R) 0 ⋯) (mk (X (n, m))) = if 0 < n then 0 els... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/DPAlgebra/Graded/GradeZero.lean | DividedPowerAlgebra.algebraMapInv_eq | [26, 1] | [38, 51] | simp only [if_pos hn, liftAlgHom_apply, LinearMap.zero_apply, aeval_X] | case pos
R : Type u
M : Type v
inst✝⁴ : CommSemiring R
inst✝³ : AddCommMonoid M
inst✝² : Module R M
inst✝¹ : DecidableEq R
inst✝ : DecidableEq M
f : MvPolynomial (ℕ × M) R
n : ℕ
m : M
hn : 0 < n
⊢ (lift (dividedPowersBot R) 0 ⋯) (mk (X (n, m))) = if 0 < n then 0 else 1 | case pos
R : Type u
M : Type v
inst✝⁴ : CommSemiring R
inst✝³ : AddCommMonoid M
inst✝² : Module R M
inst✝¹ : DecidableEq R
inst✝ : DecidableEq M
f : MvPolynomial (ℕ × M) R
n : ℕ
m : M
hn : 0 < n
⊢ (dividedPowersBot R).dpow n 0 = 0 | Please generate a tactic in lean4 to solve the state.
STATE:
case pos
R : Type u
M : Type v
inst✝⁴ : CommSemiring R
inst✝³ : AddCommMonoid M
inst✝² : Module R M
inst✝¹ : DecidableEq R
inst✝ : DecidableEq M
f : MvPolynomial (ℕ × M) R
n : ℕ
m : M
hn : 0 < n
⊢ (lift (dividedPowersBot R) 0 ⋯) (mk (X (n, m))) = if 0 < n the... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/DPAlgebra/Graded/GradeZero.lean | DividedPowerAlgebra.algebraMapInv_eq | [26, 1] | [38, 51] | rw [DividedPowers.dpow_eval_zero _ (ne_of_gt hn)] | case pos
R : Type u
M : Type v
inst✝⁴ : CommSemiring R
inst✝³ : AddCommMonoid M
inst✝² : Module R M
inst✝¹ : DecidableEq R
inst✝ : DecidableEq M
f : MvPolynomial (ℕ × M) R
n : ℕ
m : M
hn : 0 < n
⊢ (dividedPowersBot R).dpow n 0 = 0 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case pos
R : Type u
M : Type v
inst✝⁴ : CommSemiring R
inst✝³ : AddCommMonoid M
inst✝² : Module R M
inst✝¹ : DecidableEq R
inst✝ : DecidableEq M
f : MvPolynomial (ℕ × M) R
n : ℕ
m : M
hn : 0 < n
⊢ (dividedPowersBot R).dpow n 0 = 0
TACTIC:
|
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/DPAlgebra/Graded/GradeZero.lean | DividedPowerAlgebra.algebraMapInv_eq | [26, 1] | [38, 51] | rw [if_neg hn] | case neg
R : Type u
M : Type v
inst✝⁴ : CommSemiring R
inst✝³ : AddCommMonoid M
inst✝² : Module R M
inst✝¹ : DecidableEq R
inst✝ : DecidableEq M
f : MvPolynomial (ℕ × M) R
n : ℕ
m : M
hn : ¬0 < n
⊢ (lift (dividedPowersBot R) 0 ⋯) (mk (X (n, m))) = if 0 < n then 0 else 1 | case neg
R : Type u
M : Type v
inst✝⁴ : CommSemiring R
inst✝³ : AddCommMonoid M
inst✝² : Module R M
inst✝¹ : DecidableEq R
inst✝ : DecidableEq M
f : MvPolynomial (ℕ × M) R
n : ℕ
m : M
hn : ¬0 < n
⊢ (lift (dividedPowersBot R) 0 ⋯) (mk (X (n, m))) = 1 | Please generate a tactic in lean4 to solve the state.
STATE:
case neg
R : Type u
M : Type v
inst✝⁴ : CommSemiring R
inst✝³ : AddCommMonoid M
inst✝² : Module R M
inst✝¹ : DecidableEq R
inst✝ : DecidableEq M
f : MvPolynomial (ℕ × M) R
n : ℕ
m : M
hn : ¬0 < n
⊢ (lift (dividedPowersBot R) 0 ⋯) (mk (X (n, m))) = if 0 < n th... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/DPAlgebra/Graded/GradeZero.lean | DividedPowerAlgebra.algebraMapInv_eq | [26, 1] | [38, 51] | rw [not_lt, le_zero_iff] at hn | case neg
R : Type u
M : Type v
inst✝⁴ : CommSemiring R
inst✝³ : AddCommMonoid M
inst✝² : Module R M
inst✝¹ : DecidableEq R
inst✝ : DecidableEq M
f : MvPolynomial (ℕ × M) R
n : ℕ
m : M
hn : ¬0 < n
⊢ (lift (dividedPowersBot R) 0 ⋯) (mk (X (n, m))) = 1 | case neg
R : Type u
M : Type v
inst✝⁴ : CommSemiring R
inst✝³ : AddCommMonoid M
inst✝² : Module R M
inst✝¹ : DecidableEq R
inst✝ : DecidableEq M
f : MvPolynomial (ℕ × M) R
n : ℕ
m : M
hn : n = 0
⊢ (lift (dividedPowersBot R) 0 ⋯) (mk (X (n, m))) = 1 | Please generate a tactic in lean4 to solve the state.
STATE:
case neg
R : Type u
M : Type v
inst✝⁴ : CommSemiring R
inst✝³ : AddCommMonoid M
inst✝² : Module R M
inst✝¹ : DecidableEq R
inst✝ : DecidableEq M
f : MvPolynomial (ℕ × M) R
n : ℕ
m : M
hn : ¬0 < n
⊢ (lift (dividedPowersBot R) 0 ⋯) (mk (X (n, m))) = 1
TACTIC:
|
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/DPAlgebra/Graded/GradeZero.lean | DividedPowerAlgebra.algebraMapInv_eq | [26, 1] | [38, 51] | simp only [hn, liftAlgHom_apply, LinearMap.zero_apply, aeval_X,
DividedPowers.dpow_zero _ (mem_bot.mpr rfl)] | case neg
R : Type u
M : Type v
inst✝⁴ : CommSemiring R
inst✝³ : AddCommMonoid M
inst✝² : Module R M
inst✝¹ : DecidableEq R
inst✝ : DecidableEq M
f : MvPolynomial (ℕ × M) R
n : ℕ
m : M
hn : n = 0
⊢ (lift (dividedPowersBot R) 0 ⋯) (mk (X (n, m))) = 1 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case neg
R : Type u
M : Type v
inst✝⁴ : CommSemiring R
inst✝³ : AddCommMonoid M
inst✝² : Module R M
inst✝¹ : DecidableEq R
inst✝ : DecidableEq M
f : MvPolynomial (ℕ × M) R
n : ℕ
m : M
hn : n = 0
⊢ (lift (dividedPowersBot R) 0 ⋯) (mk (X (n, m))) = 1
TACTIC:
|
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/DPAlgebra/Graded/GradeZero.lean | DividedPowerAlgebra.proj'_zero_comp_algebraMap | [40, 1] | [45, 6] | simp only [proj', proj, LinearMap.coe_mk, AddHom.coe_mk, Function.comp_apply,
Algebra.algebraMap_eq_smul_one, decompose_smul, decompose_one] | R : Type u
M : Type v
inst✝⁴ : CommSemiring R
inst✝³ : AddCommMonoid M
inst✝² : Module R M
inst✝¹ : DecidableEq R
inst✝ : DecidableEq M
x : R
⊢ ↑((⇑(proj' R M 0) ∘ ⇑(algebraMap R (DividedPowerAlgebra R M))) x) = (algebraMap R (DividedPowerAlgebra R M)) x | R : Type u
M : Type v
inst✝⁴ : CommSemiring R
inst✝³ : AddCommMonoid M
inst✝² : Module R M
inst✝¹ : DecidableEq R
inst✝ : DecidableEq M
x : R
⊢ ↑((x • 1) 0) = x • 1 | Please generate a tactic in lean4 to solve the state.
STATE:
R : Type u
M : Type v
inst✝⁴ : CommSemiring R
inst✝³ : AddCommMonoid M
inst✝² : Module R M
inst✝¹ : DecidableEq R
inst✝ : DecidableEq M
x : R
⊢ ↑((⇑(proj' R M 0) ∘ ⇑(algebraMap R (DividedPowerAlgebra R M))) x) = (algebraMap R (DividedPowerAlgebra R M)) x
TACT... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/DPAlgebra/Graded/GradeZero.lean | DividedPowerAlgebra.proj'_zero_comp_algebraMap | [40, 1] | [45, 6] | rfl | R : Type u
M : Type v
inst✝⁴ : CommSemiring R
inst✝³ : AddCommMonoid M
inst✝² : Module R M
inst✝¹ : DecidableEq R
inst✝ : DecidableEq M
x : R
⊢ ↑((x • 1) 0) = x • 1 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
R : Type u
M : Type v
inst✝⁴ : CommSemiring R
inst✝³ : AddCommMonoid M
inst✝² : Module R M
inst✝¹ : DecidableEq R
inst✝ : DecidableEq M
x : R
⊢ ↑((x • 1) 0) = x • 1
TACTIC:
|
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/DPAlgebra/Graded/GradeZero.lean | DividedPowerAlgebra.algebraMap_leftInverse | [47, 1] | [49, 70] | simp only [AlgHom.commutes, Algebra.id.map_eq_id, RingHom.id_apply] | R : Type u
M : Type v
inst✝⁴ : CommSemiring R
inst✝³ : AddCommMonoid M
inst✝² : Module R M
inst✝¹ : DecidableEq R
inst✝ : DecidableEq M
m : R
⊢ (algebraMapInv R M) ((algebraMap R (DividedPowerAlgebra R M)) m) = m | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
R : Type u
M : Type v
inst✝⁴ : CommSemiring R
inst✝³ : AddCommMonoid M
inst✝² : Module R M
inst✝¹ : DecidableEq R
inst✝ : DecidableEq M
m : R
⊢ (algebraMapInv R M) ((algebraMap R (DividedPowerAlgebra R M)) m) = m
TACTIC:
|
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/DPAlgebra/Graded/GradeZero.lean | MvPolynomial.mkₐ_eq_aeval | [64, 1] | [67, 33] | ext d | R : Type u
M : Type v
inst✝⁵ : CommSemiring R
inst✝⁴ : AddCommMonoid M
inst✝³ : Module R M
inst✝² : DecidableEq R
inst✝¹ : DecidableEq M
C : Type u_1
inst✝ : CommRing C
D : Type u_2
I : Ideal (MvPolynomial D C)
⊢ mkₐ C I = aeval fun d => (Ideal.Quotient.mk I) (X d) | case hf
R : Type u
M : Type v
inst✝⁵ : CommSemiring R
inst✝⁴ : AddCommMonoid M
inst✝³ : Module R M
inst✝² : DecidableEq R
inst✝¹ : DecidableEq M
C : Type u_1
inst✝ : CommRing C
D : Type u_2
I : Ideal (MvPolynomial D C)
d : D
⊢ (mkₐ C I) (X d) = (aeval fun d => (Ideal.Quotient.mk I) (X d)) (X d) | Please generate a tactic in lean4 to solve the state.
STATE:
R : Type u
M : Type v
inst✝⁵ : CommSemiring R
inst✝⁴ : AddCommMonoid M
inst✝³ : Module R M
inst✝² : DecidableEq R
inst✝¹ : DecidableEq M
C : Type u_1
inst✝ : CommRing C
D : Type u_2
I : Ideal (MvPolynomial D C)
⊢ mkₐ C I = aeval fun d => (Ideal.Quotient.mk I)... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/DPAlgebra/Graded/GradeZero.lean | MvPolynomial.mkₐ_eq_aeval | [64, 1] | [67, 33] | simp only [mkₐ_eq_mk, aeval_X] | case hf
R : Type u
M : Type v
inst✝⁵ : CommSemiring R
inst✝⁴ : AddCommMonoid M
inst✝³ : Module R M
inst✝² : DecidableEq R
inst✝¹ : DecidableEq M
C : Type u_1
inst✝ : CommRing C
D : Type u_2
I : Ideal (MvPolynomial D C)
d : D
⊢ (mkₐ C I) (X d) = (aeval fun d => (Ideal.Quotient.mk I) (X d)) (X d) | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case hf
R : Type u
M : Type v
inst✝⁵ : CommSemiring R
inst✝⁴ : AddCommMonoid M
inst✝³ : Module R M
inst✝² : DecidableEq R
inst✝¹ : DecidableEq M
C : Type u_1
inst✝ : CommRing C
D : Type u_2
I : Ideal (MvPolynomial D C)
d : D
⊢ (mkₐ C I) (X d) = (aeval fun d =... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/DPAlgebra/Graded/GradeZero.lean | MvPolynomial.mk_eq_eval₂ | [70, 1] | [75, 6] | ext d | R : Type u
M : Type v
inst✝⁵ : CommSemiring R
inst✝⁴ : AddCommMonoid M
inst✝³ : Module R M
inst✝² : DecidableEq R
inst✝¹ : DecidableEq M
C : Type u_1
inst✝ : CommRing C
D : Type u_2
I : Ideal (MvPolynomial D C)
⊢ (↑↑(Ideal.Quotient.mk I)).toFun = eval₂ (algebraMap C (MvPolynomial D C ⧸ I)) fun d => (Ideal.Quotient.mk I... | case h
R : Type u
M : Type v
inst✝⁵ : CommSemiring R
inst✝⁴ : AddCommMonoid M
inst✝³ : Module R M
inst✝² : DecidableEq R
inst✝¹ : DecidableEq M
C : Type u_1
inst✝ : CommRing C
D : Type u_2
I : Ideal (MvPolynomial D C)
d : MvPolynomial D C
⊢ (↑↑(Ideal.Quotient.mk I)).toFun d =
eval₂ (algebraMap C (MvPolynomial D C ⧸... | Please generate a tactic in lean4 to solve the state.
STATE:
R : Type u
M : Type v
inst✝⁵ : CommSemiring R
inst✝⁴ : AddCommMonoid M
inst✝³ : Module R M
inst✝² : DecidableEq R
inst✝¹ : DecidableEq M
C : Type u_1
inst✝ : CommRing C
D : Type u_2
I : Ideal (MvPolynomial D C)
⊢ (↑↑(Ideal.Quotient.mk I)).toFun = eval₂ (algeb... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/DPAlgebra/Graded/GradeZero.lean | MvPolynomial.mk_eq_eval₂ | [70, 1] | [75, 6] | simp_rw [RingHom.toFun_eq_coe, ← mkₐ_eq_mk C, mkₐ_eq_aeval, aeval_X] | case h
R : Type u
M : Type v
inst✝⁵ : CommSemiring R
inst✝⁴ : AddCommMonoid M
inst✝³ : Module R M
inst✝² : DecidableEq R
inst✝¹ : DecidableEq M
C : Type u_1
inst✝ : CommRing C
D : Type u_2
I : Ideal (MvPolynomial D C)
d : MvPolynomial D C
⊢ (↑↑(Ideal.Quotient.mk I)).toFun d =
eval₂ (algebraMap C (MvPolynomial D C ⧸... | case h
R : Type u
M : Type v
inst✝⁵ : CommSemiring R
inst✝⁴ : AddCommMonoid M
inst✝³ : Module R M
inst✝² : DecidableEq R
inst✝¹ : DecidableEq M
C : Type u_1
inst✝ : CommRing C
D : Type u_2
I : Ideal (MvPolynomial D C)
d : MvPolynomial D C
⊢ (aeval fun d => (Ideal.Quotient.mk I) (X d)) d =
eval₂ (algebraMap C (MvPol... | Please generate a tactic in lean4 to solve the state.
STATE:
case h
R : Type u
M : Type v
inst✝⁵ : CommSemiring R
inst✝⁴ : AddCommMonoid M
inst✝³ : Module R M
inst✝² : DecidableEq R
inst✝¹ : DecidableEq M
C : Type u_1
inst✝ : CommRing C
D : Type u_2
I : Ideal (MvPolynomial D C)
d : MvPolynomial D C
⊢ (↑↑(Ideal.Quotient... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/DPAlgebra/Graded/GradeZero.lean | MvPolynomial.mk_eq_eval₂ | [70, 1] | [75, 6] | rfl | case h
R : Type u
M : Type v
inst✝⁵ : CommSemiring R
inst✝⁴ : AddCommMonoid M
inst✝³ : Module R M
inst✝² : DecidableEq R
inst✝¹ : DecidableEq M
C : Type u_1
inst✝ : CommRing C
D : Type u_2
I : Ideal (MvPolynomial D C)
d : MvPolynomial D C
⊢ (aeval fun d => (Ideal.Quotient.mk I) (X d)) d =
eval₂ (algebraMap C (MvPol... | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case h
R : Type u
M : Type v
inst✝⁵ : CommSemiring R
inst✝⁴ : AddCommMonoid M
inst✝³ : Module R M
inst✝² : DecidableEq R
inst✝¹ : DecidableEq M
C : Type u_1
inst✝ : CommRing C
D : Type u_2
I : Ideal (MvPolynomial D C)
d : MvPolynomial D C
⊢ (aeval fun d => (I... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/DPAlgebra/Graded/GradeZero.lean | DividedPowerAlgebra.algebraMap_right_inv_of_degree_zero | [77, 1] | [105, 26] | obtain ⟨p, hp0, hpx⟩ := (mem_grade_iff' _ _ _).mp x.2 | R : Type u
M : Type v
inst✝⁴ : CommSemiring R
inst✝³ : AddCommMonoid M
inst✝² : Module R M
inst✝¹ : DecidableEq R
inst✝ : DecidableEq M
x : ↥(grade R M 0)
⊢ (algebraMap R (DividedPowerAlgebra R M)) ((algebraMapInv R M) ↑x) = ↑x | case intro.intro
R : Type u
M : Type v
inst✝⁴ : CommSemiring R
inst✝³ : AddCommMonoid M
inst✝² : Module R M
inst✝¹ : DecidableEq R
inst✝ : DecidableEq M
x : ↥(grade R M 0)
p : ↥(supported R {nm | 0 < nm.1})
hp0 : IsWeightedHomogeneous Prod.fst (↑p) 0
hpx : mk ↑p = ↑x
⊢ (algebraMap R (DividedPowerAlgebra R M)) ((algebra... | Please generate a tactic in lean4 to solve the state.
STATE:
R : Type u
M : Type v
inst✝⁴ : CommSemiring R
inst✝³ : AddCommMonoid M
inst✝² : Module R M
inst✝¹ : DecidableEq R
inst✝ : DecidableEq M
x : ↥(grade 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.algebraMap_right_inv_of_degree_zero | [77, 1] | [105, 26] | suffices ∃ (a : R), p.val = C a by
obtain ⟨a, ha⟩ := this
simp only [← hpx, ha, mk_C, AlgHom.commutes, Algebra.id.map_eq_id, RingHom.id_apply] | case intro.intro
R : Type u
M : Type v
inst✝⁴ : CommSemiring R
inst✝³ : AddCommMonoid M
inst✝² : Module R M
inst✝¹ : DecidableEq R
inst✝ : DecidableEq M
x : ↥(grade R M 0)
p : ↥(supported R {nm | 0 < nm.1})
hp0 : IsWeightedHomogeneous Prod.fst (↑p) 0
hpx : mk ↑p = ↑x
⊢ (algebraMap R (DividedPowerAlgebra R M)) ((algebra... | case intro.intro
R : Type u
M : Type v
inst✝⁴ : CommSemiring R
inst✝³ : AddCommMonoid M
inst✝² : Module R M
inst✝¹ : DecidableEq R
inst✝ : DecidableEq M
x : ↥(grade R M 0)
p : ↥(supported R {nm | 0 < nm.1})
hp0 : IsWeightedHomogeneous Prod.fst (↑p) 0
hpx : mk ↑p = ↑x
⊢ ∃ a, ↑p = C a | Please generate a tactic in lean4 to solve the state.
STATE:
case intro.intro
R : Type u
M : Type v
inst✝⁴ : CommSemiring R
inst✝³ : AddCommMonoid M
inst✝² : Module R M
inst✝¹ : DecidableEq R
inst✝ : DecidableEq M
x : ↥(grade R M 0)
p : ↥(supported R {nm | 0 < nm.1})
hp0 : IsWeightedHomogeneous Prod.fst (↑p) 0
hpx : mk... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/DPAlgebra/Graded/GradeZero.lean | DividedPowerAlgebra.algebraMap_right_inv_of_degree_zero | [77, 1] | [105, 26] | use constantCoeff p.val | case intro.intro
R : Type u
M : Type v
inst✝⁴ : CommSemiring R
inst✝³ : AddCommMonoid M
inst✝² : Module R M
inst✝¹ : DecidableEq R
inst✝ : DecidableEq M
x : ↥(grade R M 0)
p : ↥(supported R {nm | 0 < nm.1})
hp0 : IsWeightedHomogeneous Prod.fst (↑p) 0
hpx : mk ↑p = ↑x
⊢ ∃ a, ↑p = C a | case h
R : Type u
M : Type v
inst✝⁴ : CommSemiring R
inst✝³ : AddCommMonoid M
inst✝² : Module R M
inst✝¹ : DecidableEq R
inst✝ : DecidableEq M
x : ↥(grade R M 0)
p : ↥(supported R {nm | 0 < nm.1})
hp0 : IsWeightedHomogeneous Prod.fst (↑p) 0
hpx : mk ↑p = ↑x
⊢ ↑p = C (constantCoeff ↑p) | Please generate a tactic in lean4 to solve the state.
STATE:
case intro.intro
R : Type u
M : Type v
inst✝⁴ : CommSemiring R
inst✝³ : AddCommMonoid M
inst✝² : Module R M
inst✝¹ : DecidableEq R
inst✝ : DecidableEq M
x : ↥(grade R M 0)
p : ↥(supported R {nm | 0 < nm.1})
hp0 : IsWeightedHomogeneous Prod.fst (↑p) 0
hpx : mk... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/DPAlgebra/Graded/GradeZero.lean | DividedPowerAlgebra.algebraMap_right_inv_of_degree_zero | [77, 1] | [105, 26] | ext exp | case h
R : Type u
M : Type v
inst✝⁴ : CommSemiring R
inst✝³ : AddCommMonoid M
inst✝² : Module R M
inst✝¹ : DecidableEq R
inst✝ : DecidableEq M
x : ↥(grade R M 0)
p : ↥(supported R {nm | 0 < nm.1})
hp0 : IsWeightedHomogeneous Prod.fst (↑p) 0
hpx : mk ↑p = ↑x
⊢ ↑p = C (constantCoeff ↑p) | case h.a
R : Type u
M : Type v
inst✝⁴ : CommSemiring R
inst✝³ : AddCommMonoid M
inst✝² : Module R M
inst✝¹ : DecidableEq R
inst✝ : DecidableEq M
x : ↥(grade R M 0)
p : ↥(supported R {nm | 0 < nm.1})
hp0 : IsWeightedHomogeneous Prod.fst (↑p) 0
hpx : mk ↑p = ↑x
exp : ℕ × M →₀ ℕ
⊢ coeff exp ↑p = coeff exp (C (constantCoef... | Please generate a tactic in lean4 to solve the state.
STATE:
case h
R : Type u
M : Type v
inst✝⁴ : CommSemiring R
inst✝³ : AddCommMonoid M
inst✝² : Module R M
inst✝¹ : DecidableEq R
inst✝ : DecidableEq M
x : ↥(grade R M 0)
p : ↥(supported R {nm | 0 < nm.1})
hp0 : IsWeightedHomogeneous Prod.fst (↑p) 0
hpx : mk ↑p = ↑x
⊢... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/DPAlgebra/Graded/GradeZero.lean | DividedPowerAlgebra.algebraMap_right_inv_of_degree_zero | [77, 1] | [105, 26] | simp only [coeff_C] | case h.a
R : Type u
M : Type v
inst✝⁴ : CommSemiring R
inst✝³ : AddCommMonoid M
inst✝² : Module R M
inst✝¹ : DecidableEq R
inst✝ : DecidableEq M
x : ↥(grade R M 0)
p : ↥(supported R {nm | 0 < nm.1})
hp0 : IsWeightedHomogeneous Prod.fst (↑p) 0
hpx : mk ↑p = ↑x
exp : ℕ × M →₀ ℕ
⊢ coeff exp ↑p = coeff exp (C (constantCoef... | case h.a
R : Type u
M : Type v
inst✝⁴ : CommSemiring R
inst✝³ : AddCommMonoid M
inst✝² : Module R M
inst✝¹ : DecidableEq R
inst✝ : DecidableEq M
x : ↥(grade R M 0)
p : ↥(supported R {nm | 0 < nm.1})
hp0 : IsWeightedHomogeneous Prod.fst (↑p) 0
hpx : mk ↑p = ↑x
exp : ℕ × M →₀ ℕ
⊢ coeff exp ↑p = if 0 = exp then constantCo... | Please generate a tactic in lean4 to solve the state.
STATE:
case h.a
R : Type u
M : Type v
inst✝⁴ : CommSemiring R
inst✝³ : AddCommMonoid M
inst✝² : Module R M
inst✝¹ : DecidableEq R
inst✝ : DecidableEq M
x : ↥(grade R M 0)
p : ↥(supported R {nm | 0 < nm.1})
hp0 : IsWeightedHomogeneous Prod.fst (↑p) 0
hpx : mk ↑p = ↑x... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/DPAlgebra/Graded/GradeZero.lean | DividedPowerAlgebra.algebraMap_right_inv_of_degree_zero | [77, 1] | [105, 26] | split_ifs with hexp | case h.a
R : Type u
M : Type v
inst✝⁴ : CommSemiring R
inst✝³ : AddCommMonoid M
inst✝² : Module R M
inst✝¹ : DecidableEq R
inst✝ : DecidableEq M
x : ↥(grade R M 0)
p : ↥(supported R {nm | 0 < nm.1})
hp0 : IsWeightedHomogeneous Prod.fst (↑p) 0
hpx : mk ↑p = ↑x
exp : ℕ × M →₀ ℕ
⊢ coeff exp ↑p = if 0 = exp then constantCo... | case pos
R : Type u
M : Type v
inst✝⁴ : CommSemiring R
inst✝³ : AddCommMonoid M
inst✝² : Module R M
inst✝¹ : DecidableEq R
inst✝ : DecidableEq M
x : ↥(grade R M 0)
p : ↥(supported R {nm | 0 < nm.1})
hp0 : IsWeightedHomogeneous Prod.fst (↑p) 0
hpx : mk ↑p = ↑x
exp : ℕ × M →₀ ℕ
hexp : 0 = exp
⊢ coeff exp ↑p = constantCoe... | Please generate a tactic in lean4 to solve the state.
STATE:
case h.a
R : Type u
M : Type v
inst✝⁴ : CommSemiring R
inst✝³ : AddCommMonoid M
inst✝² : Module R M
inst✝¹ : DecidableEq R
inst✝ : DecidableEq M
x : ↥(grade R M 0)
p : ↥(supported R {nm | 0 < nm.1})
hp0 : IsWeightedHomogeneous Prod.fst (↑p) 0
hpx : mk ↑p = ↑x... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/DPAlgebra/Graded/GradeZero.lean | DividedPowerAlgebra.algebraMap_right_inv_of_degree_zero | [77, 1] | [105, 26] | . rw [← hexp, constantCoeff_eq] | case pos
R : Type u
M : Type v
inst✝⁴ : CommSemiring R
inst✝³ : AddCommMonoid M
inst✝² : Module R M
inst✝¹ : DecidableEq R
inst✝ : DecidableEq M
x : ↥(grade R M 0)
p : ↥(supported R {nm | 0 < nm.1})
hp0 : IsWeightedHomogeneous Prod.fst (↑p) 0
hpx : mk ↑p = ↑x
exp : ℕ × M →₀ ℕ
hexp : 0 = exp
⊢ coeff exp ↑p = constantCoe... | case neg
R : Type u
M : Type v
inst✝⁴ : CommSemiring R
inst✝³ : AddCommMonoid M
inst✝² : Module R M
inst✝¹ : DecidableEq R
inst✝ : DecidableEq M
x : ↥(grade R M 0)
p : ↥(supported R {nm | 0 < nm.1})
hp0 : IsWeightedHomogeneous Prod.fst (↑p) 0
hpx : mk ↑p = ↑x
exp : ℕ × M →₀ ℕ
hexp : ¬0 = exp
⊢ coeff exp ↑p = 0 | Please generate a tactic in lean4 to solve the state.
STATE:
case pos
R : Type u
M : Type v
inst✝⁴ : CommSemiring R
inst✝³ : AddCommMonoid M
inst✝² : Module R M
inst✝¹ : DecidableEq R
inst✝ : DecidableEq M
x : ↥(grade R M 0)
p : ↥(supported R {nm | 0 < nm.1})
hp0 : IsWeightedHomogeneous Prod.fst (↑p) 0
hpx : mk ↑p = ↑x... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/DPAlgebra/Graded/GradeZero.lean | DividedPowerAlgebra.algebraMap_right_inv_of_degree_zero | [77, 1] | [105, 26] | . simp only [IsWeightedHomogeneous] at hp0
by_contra h
rw [eq_comm, ← Finsupp.support_eq_empty] at hexp
obtain ⟨nm, hnm⟩ := nonempty_of_ne_empty hexp
specialize hp0 h
simp only [weightedDegree, LinearMap.toAddMonoidHom_coe, Finsupp.total_apply, Finsupp.sum,
sum_eq_zero_iff] at hp0
specialize hp0 nm hnm... | case neg
R : Type u
M : Type v
inst✝⁴ : CommSemiring R
inst✝³ : AddCommMonoid M
inst✝² : Module R M
inst✝¹ : DecidableEq R
inst✝ : DecidableEq M
x : ↥(grade R M 0)
p : ↥(supported R {nm | 0 < nm.1})
hp0 : IsWeightedHomogeneous Prod.fst (↑p) 0
hpx : mk ↑p = ↑x
exp : ℕ × M →₀ ℕ
hexp : ¬0 = exp
⊢ coeff exp ↑p = 0 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case neg
R : Type u
M : Type v
inst✝⁴ : CommSemiring R
inst✝³ : AddCommMonoid M
inst✝² : Module R M
inst✝¹ : DecidableEq R
inst✝ : DecidableEq M
x : ↥(grade R M 0)
p : ↥(supported R {nm | 0 < nm.1})
hp0 : IsWeightedHomogeneous Prod.fst (↑p) 0
hpx : mk ↑p = ↑x... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/DPAlgebra/Graded/GradeZero.lean | DividedPowerAlgebra.algebraMap_right_inv_of_degree_zero | [77, 1] | [105, 26] | obtain ⟨a, ha⟩ := this | R : Type u
M : Type v
inst✝⁴ : CommSemiring R
inst✝³ : AddCommMonoid M
inst✝² : Module R M
inst✝¹ : DecidableEq R
inst✝ : DecidableEq M
x : ↥(grade R M 0)
p : ↥(supported R {nm | 0 < nm.1})
hp0 : IsWeightedHomogeneous Prod.fst (↑p) 0
hpx : mk ↑p = ↑x
this : ∃ a, ↑p = C a
⊢ (algebraMap R (DividedPowerAlgebra R M)) ((alg... | case intro
R : Type u
M : Type v
inst✝⁴ : CommSemiring R
inst✝³ : AddCommMonoid M
inst✝² : Module R M
inst✝¹ : DecidableEq R
inst✝ : DecidableEq M
x : ↥(grade R M 0)
p : ↥(supported R {nm | 0 < nm.1})
hp0 : IsWeightedHomogeneous Prod.fst (↑p) 0
hpx : mk ↑p = ↑x
a : R
ha : ↑p = C a
⊢ (algebraMap R (DividedPowerAlgebra R... | Please generate a tactic in lean4 to solve the state.
STATE:
R : Type u
M : Type v
inst✝⁴ : CommSemiring R
inst✝³ : AddCommMonoid M
inst✝² : Module R M
inst✝¹ : DecidableEq R
inst✝ : DecidableEq M
x : ↥(grade R M 0)
p : ↥(supported R {nm | 0 < nm.1})
hp0 : IsWeightedHomogeneous Prod.fst (↑p) 0
hpx : mk ↑p = ↑x
this : ∃... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/DPAlgebra/Graded/GradeZero.lean | DividedPowerAlgebra.algebraMap_right_inv_of_degree_zero | [77, 1] | [105, 26] | simp only [← hpx, ha, mk_C, AlgHom.commutes, Algebra.id.map_eq_id, RingHom.id_apply] | case intro
R : Type u
M : Type v
inst✝⁴ : CommSemiring R
inst✝³ : AddCommMonoid M
inst✝² : Module R M
inst✝¹ : DecidableEq R
inst✝ : DecidableEq M
x : ↥(grade R M 0)
p : ↥(supported R {nm | 0 < nm.1})
hp0 : IsWeightedHomogeneous Prod.fst (↑p) 0
hpx : mk ↑p = ↑x
a : R
ha : ↑p = C a
⊢ (algebraMap R (DividedPowerAlgebra R... | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case intro
R : Type u
M : Type v
inst✝⁴ : CommSemiring R
inst✝³ : AddCommMonoid M
inst✝² : Module R M
inst✝¹ : DecidableEq R
inst✝ : DecidableEq M
x : ↥(grade R M 0)
p : ↥(supported R {nm | 0 < nm.1})
hp0 : IsWeightedHomogeneous Prod.fst (↑p) 0
hpx : mk ↑p = ... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/DPAlgebra/Graded/GradeZero.lean | DividedPowerAlgebra.algebraMap_right_inv_of_degree_zero | [77, 1] | [105, 26] | rw [← hexp, constantCoeff_eq] | case pos
R : Type u
M : Type v
inst✝⁴ : CommSemiring R
inst✝³ : AddCommMonoid M
inst✝² : Module R M
inst✝¹ : DecidableEq R
inst✝ : DecidableEq M
x : ↥(grade R M 0)
p : ↥(supported R {nm | 0 < nm.1})
hp0 : IsWeightedHomogeneous Prod.fst (↑p) 0
hpx : mk ↑p = ↑x
exp : ℕ × M →₀ ℕ
hexp : 0 = exp
⊢ coeff exp ↑p = constantCoe... | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case pos
R : Type u
M : Type v
inst✝⁴ : CommSemiring R
inst✝³ : AddCommMonoid M
inst✝² : Module R M
inst✝¹ : DecidableEq R
inst✝ : DecidableEq M
x : ↥(grade R M 0)
p : ↥(supported R {nm | 0 < nm.1})
hp0 : IsWeightedHomogeneous Prod.fst (↑p) 0
hpx : mk ↑p = ↑x... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/DPAlgebra/Graded/GradeZero.lean | DividedPowerAlgebra.algebraMap_right_inv_of_degree_zero | [77, 1] | [105, 26] | simp only [IsWeightedHomogeneous] at hp0 | case neg
R : Type u
M : Type v
inst✝⁴ : CommSemiring R
inst✝³ : AddCommMonoid M
inst✝² : Module R M
inst✝¹ : DecidableEq R
inst✝ : DecidableEq M
x : ↥(grade R M 0)
p : ↥(supported R {nm | 0 < nm.1})
hp0 : IsWeightedHomogeneous Prod.fst (↑p) 0
hpx : mk ↑p = ↑x
exp : ℕ × M →₀ ℕ
hexp : ¬0 = exp
⊢ coeff exp ↑p = 0 | case neg
R : Type u
M : Type v
inst✝⁴ : CommSemiring R
inst✝³ : AddCommMonoid M
inst✝² : Module R M
inst✝¹ : DecidableEq R
inst✝ : DecidableEq M
x : ↥(grade R M 0)
p : ↥(supported R {nm | 0 < nm.1})
hp0 : ∀ ⦃d : ℕ × M →₀ ℕ⦄, coeff d ↑p ≠ 0 → (weightedDegree Prod.fst) d = 0
hpx : mk ↑p = ↑x
exp : ℕ × M →₀ ℕ
hexp : ¬0 = ... | Please generate a tactic in lean4 to solve the state.
STATE:
case neg
R : Type u
M : Type v
inst✝⁴ : CommSemiring R
inst✝³ : AddCommMonoid M
inst✝² : Module R M
inst✝¹ : DecidableEq R
inst✝ : DecidableEq M
x : ↥(grade R M 0)
p : ↥(supported R {nm | 0 < nm.1})
hp0 : IsWeightedHomogeneous Prod.fst (↑p) 0
hpx : mk ↑p = ↑x... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/DPAlgebra/Graded/GradeZero.lean | DividedPowerAlgebra.algebraMap_right_inv_of_degree_zero | [77, 1] | [105, 26] | by_contra h | case neg
R : Type u
M : Type v
inst✝⁴ : CommSemiring R
inst✝³ : AddCommMonoid M
inst✝² : Module R M
inst✝¹ : DecidableEq R
inst✝ : DecidableEq M
x : ↥(grade R M 0)
p : ↥(supported R {nm | 0 < nm.1})
hp0 : ∀ ⦃d : ℕ × M →₀ ℕ⦄, coeff d ↑p ≠ 0 → (weightedDegree Prod.fst) d = 0
hpx : mk ↑p = ↑x
exp : ℕ × M →₀ ℕ
hexp : ¬0 = ... | case neg
R : Type u
M : Type v
inst✝⁴ : CommSemiring R
inst✝³ : AddCommMonoid M
inst✝² : Module R M
inst✝¹ : DecidableEq R
inst✝ : DecidableEq M
x : ↥(grade R M 0)
p : ↥(supported R {nm | 0 < nm.1})
hp0 : ∀ ⦃d : ℕ × M →₀ ℕ⦄, coeff d ↑p ≠ 0 → (weightedDegree Prod.fst) d = 0
hpx : mk ↑p = ↑x
exp : ℕ × M →₀ ℕ
hexp : ¬0 = ... | Please generate a tactic in lean4 to solve the state.
STATE:
case neg
R : Type u
M : Type v
inst✝⁴ : CommSemiring R
inst✝³ : AddCommMonoid M
inst✝² : Module R M
inst✝¹ : DecidableEq R
inst✝ : DecidableEq M
x : ↥(grade R M 0)
p : ↥(supported R {nm | 0 < nm.1})
hp0 : ∀ ⦃d : ℕ × M →₀ ℕ⦄, coeff d ↑p ≠ 0 → (weightedDegree P... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/DPAlgebra/Graded/GradeZero.lean | DividedPowerAlgebra.algebraMap_right_inv_of_degree_zero | [77, 1] | [105, 26] | rw [eq_comm, ← Finsupp.support_eq_empty] at hexp | case neg
R : Type u
M : Type v
inst✝⁴ : CommSemiring R
inst✝³ : AddCommMonoid M
inst✝² : Module R M
inst✝¹ : DecidableEq R
inst✝ : DecidableEq M
x : ↥(grade R M 0)
p : ↥(supported R {nm | 0 < nm.1})
hp0 : ∀ ⦃d : ℕ × M →₀ ℕ⦄, coeff d ↑p ≠ 0 → (weightedDegree Prod.fst) d = 0
hpx : mk ↑p = ↑x
exp : ℕ × M →₀ ℕ
hexp : ¬0 = ... | case neg
R : Type u
M : Type v
inst✝⁴ : CommSemiring R
inst✝³ : AddCommMonoid M
inst✝² : Module R M
inst✝¹ : DecidableEq R
inst✝ : DecidableEq M
x : ↥(grade R M 0)
p : ↥(supported R {nm | 0 < nm.1})
hp0 : ∀ ⦃d : ℕ × M →₀ ℕ⦄, coeff d ↑p ≠ 0 → (weightedDegree Prod.fst) d = 0
hpx : mk ↑p = ↑x
exp : ℕ × M →₀ ℕ
hexp : ¬exp.... | Please generate a tactic in lean4 to solve the state.
STATE:
case neg
R : Type u
M : Type v
inst✝⁴ : CommSemiring R
inst✝³ : AddCommMonoid M
inst✝² : Module R M
inst✝¹ : DecidableEq R
inst✝ : DecidableEq M
x : ↥(grade R M 0)
p : ↥(supported R {nm | 0 < nm.1})
hp0 : ∀ ⦃d : ℕ × M →₀ ℕ⦄, coeff d ↑p ≠ 0 → (weightedDegree P... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/DPAlgebra/Graded/GradeZero.lean | DividedPowerAlgebra.algebraMap_right_inv_of_degree_zero | [77, 1] | [105, 26] | obtain ⟨nm, hnm⟩ := nonempty_of_ne_empty hexp | case neg
R : Type u
M : Type v
inst✝⁴ : CommSemiring R
inst✝³ : AddCommMonoid M
inst✝² : Module R M
inst✝¹ : DecidableEq R
inst✝ : DecidableEq M
x : ↥(grade R M 0)
p : ↥(supported R {nm | 0 < nm.1})
hp0 : ∀ ⦃d : ℕ × M →₀ ℕ⦄, coeff d ↑p ≠ 0 → (weightedDegree Prod.fst) d = 0
hpx : mk ↑p = ↑x
exp : ℕ × M →₀ ℕ
hexp : ¬exp.... | case neg.intro
R : Type u
M : Type v
inst✝⁴ : CommSemiring R
inst✝³ : AddCommMonoid M
inst✝² : Module R M
inst✝¹ : DecidableEq R
inst✝ : DecidableEq M
x : ↥(grade R M 0)
p : ↥(supported R {nm | 0 < nm.1})
hp0 : ∀ ⦃d : ℕ × M →₀ ℕ⦄, coeff d ↑p ≠ 0 → (weightedDegree Prod.fst) d = 0
hpx : mk ↑p = ↑x
exp : ℕ × M →₀ ℕ
hexp :... | Please generate a tactic in lean4 to solve the state.
STATE:
case neg
R : Type u
M : Type v
inst✝⁴ : CommSemiring R
inst✝³ : AddCommMonoid M
inst✝² : Module R M
inst✝¹ : DecidableEq R
inst✝ : DecidableEq M
x : ↥(grade R M 0)
p : ↥(supported R {nm | 0 < nm.1})
hp0 : ∀ ⦃d : ℕ × M →₀ ℕ⦄, coeff d ↑p ≠ 0 → (weightedDegree P... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/DPAlgebra/Graded/GradeZero.lean | DividedPowerAlgebra.algebraMap_right_inv_of_degree_zero | [77, 1] | [105, 26] | specialize hp0 h | case neg.intro
R : Type u
M : Type v
inst✝⁴ : CommSemiring R
inst✝³ : AddCommMonoid M
inst✝² : Module R M
inst✝¹ : DecidableEq R
inst✝ : DecidableEq M
x : ↥(grade R M 0)
p : ↥(supported R {nm | 0 < nm.1})
hp0 : ∀ ⦃d : ℕ × M →₀ ℕ⦄, coeff d ↑p ≠ 0 → (weightedDegree Prod.fst) d = 0
hpx : mk ↑p = ↑x
exp : ℕ × M →₀ ℕ
hexp :... | case neg.intro
R : Type u
M : Type v
inst✝⁴ : CommSemiring R
inst✝³ : AddCommMonoid M
inst✝² : Module R M
inst✝¹ : DecidableEq R
inst✝ : DecidableEq M
x : ↥(grade R M 0)
p : ↥(supported R {nm | 0 < nm.1})
hpx : mk ↑p = ↑x
exp : ℕ × M →₀ ℕ
hexp : ¬exp.support = ∅
h : ¬coeff exp ↑p = 0
nm : ℕ × M
hnm : nm ∈ exp.support
h... | Please generate a tactic in lean4 to solve the state.
STATE:
case neg.intro
R : Type u
M : Type v
inst✝⁴ : CommSemiring R
inst✝³ : AddCommMonoid M
inst✝² : Module R M
inst✝¹ : DecidableEq R
inst✝ : DecidableEq M
x : ↥(grade R M 0)
p : ↥(supported R {nm | 0 < nm.1})
hp0 : ∀ ⦃d : ℕ × M →₀ ℕ⦄, coeff d ↑p ≠ 0 → (weightedDe... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/DPAlgebra/Graded/GradeZero.lean | DividedPowerAlgebra.algebraMap_right_inv_of_degree_zero | [77, 1] | [105, 26] | simp only [weightedDegree, LinearMap.toAddMonoidHom_coe, Finsupp.total_apply, Finsupp.sum,
sum_eq_zero_iff] at hp0 | case neg.intro
R : Type u
M : Type v
inst✝⁴ : CommSemiring R
inst✝³ : AddCommMonoid M
inst✝² : Module R M
inst✝¹ : DecidableEq R
inst✝ : DecidableEq M
x : ↥(grade R M 0)
p : ↥(supported R {nm | 0 < nm.1})
hpx : mk ↑p = ↑x
exp : ℕ × M →₀ ℕ
hexp : ¬exp.support = ∅
h : ¬coeff exp ↑p = 0
nm : ℕ × M
hnm : nm ∈ exp.support
h... | case neg.intro
R : Type u
M : Type v
inst✝⁴ : CommSemiring R
inst✝³ : AddCommMonoid M
inst✝² : Module R M
inst✝¹ : DecidableEq R
inst✝ : DecidableEq M
x : ↥(grade R M 0)
p : ↥(supported R {nm | 0 < nm.1})
hpx : mk ↑p = ↑x
exp : ℕ × M →₀ ℕ
hexp : ¬exp.support = ∅
h : ¬coeff exp ↑p = 0
nm : ℕ × M
hnm : nm ∈ exp.support
h... | Please generate a tactic in lean4 to solve the state.
STATE:
case neg.intro
R : Type u
M : Type v
inst✝⁴ : CommSemiring R
inst✝³ : AddCommMonoid M
inst✝² : Module R M
inst✝¹ : DecidableEq R
inst✝ : DecidableEq M
x : ↥(grade R M 0)
p : ↥(supported R {nm | 0 < nm.1})
hpx : mk ↑p = ↑x
exp : ℕ × M →₀ ℕ
hexp : ¬exp.support ... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/DPAlgebra/Graded/GradeZero.lean | DividedPowerAlgebra.algebraMap_right_inv_of_degree_zero | [77, 1] | [105, 26] | specialize hp0 nm hnm | case neg.intro
R : Type u
M : Type v
inst✝⁴ : CommSemiring R
inst✝³ : AddCommMonoid M
inst✝² : Module R M
inst✝¹ : DecidableEq R
inst✝ : DecidableEq M
x : ↥(grade R M 0)
p : ↥(supported R {nm | 0 < nm.1})
hpx : mk ↑p = ↑x
exp : ℕ × M →₀ ℕ
hexp : ¬exp.support = ∅
h : ¬coeff exp ↑p = 0
nm : ℕ × M
hnm : nm ∈ exp.support
h... | case neg.intro
R : Type u
M : Type v
inst✝⁴ : CommSemiring R
inst✝³ : AddCommMonoid M
inst✝² : Module R M
inst✝¹ : DecidableEq R
inst✝ : DecidableEq M
x : ↥(grade R M 0)
p : ↥(supported R {nm | 0 < nm.1})
hpx : mk ↑p = ↑x
exp : ℕ × M →₀ ℕ
hexp : ¬exp.support = ∅
h : ¬coeff exp ↑p = 0
nm : ℕ × M
hnm : nm ∈ exp.support
h... | Please generate a tactic in lean4 to solve the state.
STATE:
case neg.intro
R : Type u
M : Type v
inst✝⁴ : CommSemiring R
inst✝³ : AddCommMonoid M
inst✝² : Module R M
inst✝¹ : DecidableEq R
inst✝ : DecidableEq M
x : ↥(grade R M 0)
p : ↥(supported R {nm | 0 < nm.1})
hpx : mk ↑p = ↑x
exp : ℕ × M →₀ ℕ
hexp : ¬exp.support ... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/DPAlgebra/Graded/GradeZero.lean | DividedPowerAlgebra.algebraMap_right_inv_of_degree_zero | [77, 1] | [105, 26] | simp only [smul_eq_mul, mul_eq_zero] at hp0 | case neg.intro
R : Type u
M : Type v
inst✝⁴ : CommSemiring R
inst✝³ : AddCommMonoid M
inst✝² : Module R M
inst✝¹ : DecidableEq R
inst✝ : DecidableEq M
x : ↥(grade R M 0)
p : ↥(supported R {nm | 0 < nm.1})
hpx : mk ↑p = ↑x
exp : ℕ × M →₀ ℕ
hexp : ¬exp.support = ∅
h : ¬coeff exp ↑p = 0
nm : ℕ × M
hnm : nm ∈ exp.support
h... | case neg.intro
R : Type u
M : Type v
inst✝⁴ : CommSemiring R
inst✝³ : AddCommMonoid M
inst✝² : Module R M
inst✝¹ : DecidableEq R
inst✝ : DecidableEq M
x : ↥(grade R M 0)
p : ↥(supported R {nm | 0 < nm.1})
hpx : mk ↑p = ↑x
exp : ℕ × M →₀ ℕ
hexp : ¬exp.support = ∅
h : ¬coeff exp ↑p = 0
nm : ℕ × M
hnm : nm ∈ exp.support
h... | Please generate a tactic in lean4 to solve the state.
STATE:
case neg.intro
R : Type u
M : Type v
inst✝⁴ : CommSemiring R
inst✝³ : AddCommMonoid M
inst✝² : Module R M
inst✝¹ : DecidableEq R
inst✝ : DecidableEq M
x : ↥(grade R M 0)
p : ↥(supported R {nm | 0 < nm.1})
hpx : mk ↑p = ↑x
exp : ℕ × M →₀ ℕ
hexp : ¬exp.support ... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/DPAlgebra/Graded/GradeZero.lean | DividedPowerAlgebra.algebraMap_right_inv_of_degree_zero | [77, 1] | [105, 26] | cases' hp0 with hnm0 hnm0 | case neg.intro
R : Type u
M : Type v
inst✝⁴ : CommSemiring R
inst✝³ : AddCommMonoid M
inst✝² : Module R M
inst✝¹ : DecidableEq R
inst✝ : DecidableEq M
x : ↥(grade R M 0)
p : ↥(supported R {nm | 0 < nm.1})
hpx : mk ↑p = ↑x
exp : ℕ × M →₀ ℕ
hexp : ¬exp.support = ∅
h : ¬coeff exp ↑p = 0
nm : ℕ × M
hnm : nm ∈ exp.support
h... | case neg.intro.inl
R : Type u
M : Type v
inst✝⁴ : CommSemiring R
inst✝³ : AddCommMonoid M
inst✝² : Module R M
inst✝¹ : DecidableEq R
inst✝ : DecidableEq M
x : ↥(grade R M 0)
p : ↥(supported R {nm | 0 < nm.1})
hpx : mk ↑p = ↑x
exp : ℕ × M →₀ ℕ
hexp : ¬exp.support = ∅
h : ¬coeff exp ↑p = 0
nm : ℕ × M
hnm : nm ∈ exp.suppo... | Please generate a tactic in lean4 to solve the state.
STATE:
case neg.intro
R : Type u
M : Type v
inst✝⁴ : CommSemiring R
inst✝³ : AddCommMonoid M
inst✝² : Module R M
inst✝¹ : DecidableEq R
inst✝ : DecidableEq M
x : ↥(grade R M 0)
p : ↥(supported R {nm | 0 < nm.1})
hpx : mk ↑p = ↑x
exp : ℕ × M →₀ ℕ
hexp : ¬exp.support ... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/DPAlgebra/Graded/GradeZero.lean | DividedPowerAlgebra.algebraMap_right_inv_of_degree_zero | [77, 1] | [105, 26] | . simp only [Finsupp.mem_support_iff] at hnm
exact hnm hnm0 | case neg.intro.inl
R : Type u
M : Type v
inst✝⁴ : CommSemiring R
inst✝³ : AddCommMonoid M
inst✝² : Module R M
inst✝¹ : DecidableEq R
inst✝ : DecidableEq M
x : ↥(grade R M 0)
p : ↥(supported R {nm | 0 < nm.1})
hpx : mk ↑p = ↑x
exp : ℕ × M →₀ ℕ
hexp : ¬exp.support = ∅
h : ¬coeff exp ↑p = 0
nm : ℕ × M
hnm : nm ∈ exp.suppo... | case neg.intro.inr
R : Type u
M : Type v
inst✝⁴ : CommSemiring R
inst✝³ : AddCommMonoid M
inst✝² : Module R M
inst✝¹ : DecidableEq R
inst✝ : DecidableEq M
x : ↥(grade R M 0)
p : ↥(supported R {nm | 0 < nm.1})
hpx : mk ↑p = ↑x
exp : ℕ × M →₀ ℕ
hexp : ¬exp.support = ∅
h : ¬coeff exp ↑p = 0
nm : ℕ × M
hnm : nm ∈ exp.suppo... | Please generate a tactic in lean4 to solve the state.
STATE:
case neg.intro.inl
R : Type u
M : Type v
inst✝⁴ : CommSemiring R
inst✝³ : AddCommMonoid M
inst✝² : Module R M
inst✝¹ : DecidableEq R
inst✝ : DecidableEq M
x : ↥(grade R M 0)
p : ↥(supported R {nm | 0 < nm.1})
hpx : mk ↑p = ↑x
exp : ℕ × M →₀ ℕ
hexp : ¬exp.supp... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/DPAlgebra/Graded/GradeZero.lean | DividedPowerAlgebra.algebraMap_right_inv_of_degree_zero | [77, 1] | [105, 26] | . apply lt_irrefl 0
nth_rewrite 2 [← hnm0]
apply MvPolynomial.mem_supported.mp p.prop
simp only [mem_coe, mem_vars, Finsupp.mem_support_iff, ne_eq, mem_support_iff, exists_prop]
simp only [Finsupp.mem_support_iff] at hnm
exact ⟨exp, h, hnm⟩ | case neg.intro.inr
R : Type u
M : Type v
inst✝⁴ : CommSemiring R
inst✝³ : AddCommMonoid M
inst✝² : Module R M
inst✝¹ : DecidableEq R
inst✝ : DecidableEq M
x : ↥(grade R M 0)
p : ↥(supported R {nm | 0 < nm.1})
hpx : mk ↑p = ↑x
exp : ℕ × M →₀ ℕ
hexp : ¬exp.support = ∅
h : ¬coeff exp ↑p = 0
nm : ℕ × M
hnm : nm ∈ exp.suppo... | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case neg.intro.inr
R : Type u
M : Type v
inst✝⁴ : CommSemiring R
inst✝³ : AddCommMonoid M
inst✝² : Module R M
inst✝¹ : DecidableEq R
inst✝ : DecidableEq M
x : ↥(grade R M 0)
p : ↥(supported R {nm | 0 < nm.1})
hpx : mk ↑p = ↑x
exp : ℕ × M →₀ ℕ
hexp : ¬exp.supp... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/DPAlgebra/Graded/GradeZero.lean | DividedPowerAlgebra.algebraMap_right_inv_of_degree_zero | [77, 1] | [105, 26] | simp only [Finsupp.mem_support_iff] at hnm | case neg.intro.inl
R : Type u
M : Type v
inst✝⁴ : CommSemiring R
inst✝³ : AddCommMonoid M
inst✝² : Module R M
inst✝¹ : DecidableEq R
inst✝ : DecidableEq M
x : ↥(grade R M 0)
p : ↥(supported R {nm | 0 < nm.1})
hpx : mk ↑p = ↑x
exp : ℕ × M →₀ ℕ
hexp : ¬exp.support = ∅
h : ¬coeff exp ↑p = 0
nm : ℕ × M
hnm : nm ∈ exp.suppo... | case neg.intro.inl
R : Type u
M : Type v
inst✝⁴ : CommSemiring R
inst✝³ : AddCommMonoid M
inst✝² : Module R M
inst✝¹ : DecidableEq R
inst✝ : DecidableEq M
x : ↥(grade R M 0)
p : ↥(supported R {nm | 0 < nm.1})
hpx : mk ↑p = ↑x
exp : ℕ × M →₀ ℕ
hexp : ¬exp.support = ∅
h : ¬coeff exp ↑p = 0
nm : ℕ × M
hnm0 : exp nm = 0
hn... | Please generate a tactic in lean4 to solve the state.
STATE:
case neg.intro.inl
R : Type u
M : Type v
inst✝⁴ : CommSemiring R
inst✝³ : AddCommMonoid M
inst✝² : Module R M
inst✝¹ : DecidableEq R
inst✝ : DecidableEq M
x : ↥(grade R M 0)
p : ↥(supported R {nm | 0 < nm.1})
hpx : mk ↑p = ↑x
exp : ℕ × M →₀ ℕ
hexp : ¬exp.supp... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/DPAlgebra/Graded/GradeZero.lean | DividedPowerAlgebra.algebraMap_right_inv_of_degree_zero | [77, 1] | [105, 26] | exact hnm hnm0 | case neg.intro.inl
R : Type u
M : Type v
inst✝⁴ : CommSemiring R
inst✝³ : AddCommMonoid M
inst✝² : Module R M
inst✝¹ : DecidableEq R
inst✝ : DecidableEq M
x : ↥(grade R M 0)
p : ↥(supported R {nm | 0 < nm.1})
hpx : mk ↑p = ↑x
exp : ℕ × M →₀ ℕ
hexp : ¬exp.support = ∅
h : ¬coeff exp ↑p = 0
nm : ℕ × M
hnm0 : exp nm = 0
hn... | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case neg.intro.inl
R : Type u
M : Type v
inst✝⁴ : CommSemiring R
inst✝³ : AddCommMonoid M
inst✝² : Module R M
inst✝¹ : DecidableEq R
inst✝ : DecidableEq M
x : ↥(grade R M 0)
p : ↥(supported R {nm | 0 < nm.1})
hpx : mk ↑p = ↑x
exp : ℕ × M →₀ ℕ
hexp : ¬exp.supp... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/DPAlgebra/Graded/GradeZero.lean | DividedPowerAlgebra.algebraMap_right_inv_of_degree_zero | [77, 1] | [105, 26] | apply lt_irrefl 0 | case neg.intro.inr
R : Type u
M : Type v
inst✝⁴ : CommSemiring R
inst✝³ : AddCommMonoid M
inst✝² : Module R M
inst✝¹ : DecidableEq R
inst✝ : DecidableEq M
x : ↥(grade R M 0)
p : ↥(supported R {nm | 0 < nm.1})
hpx : mk ↑p = ↑x
exp : ℕ × M →₀ ℕ
hexp : ¬exp.support = ∅
h : ¬coeff exp ↑p = 0
nm : ℕ × M
hnm : nm ∈ exp.suppo... | case neg.intro.inr
R : Type u
M : Type v
inst✝⁴ : CommSemiring R
inst✝³ : AddCommMonoid M
inst✝² : Module R M
inst✝¹ : DecidableEq R
inst✝ : DecidableEq M
x : ↥(grade R M 0)
p : ↥(supported R {nm | 0 < nm.1})
hpx : mk ↑p = ↑x
exp : ℕ × M →₀ ℕ
hexp : ¬exp.support = ∅
h : ¬coeff exp ↑p = 0
nm : ℕ × M
hnm : nm ∈ exp.suppo... | Please generate a tactic in lean4 to solve the state.
STATE:
case neg.intro.inr
R : Type u
M : Type v
inst✝⁴ : CommSemiring R
inst✝³ : AddCommMonoid M
inst✝² : Module R M
inst✝¹ : DecidableEq R
inst✝ : DecidableEq M
x : ↥(grade R M 0)
p : ↥(supported R {nm | 0 < nm.1})
hpx : mk ↑p = ↑x
exp : ℕ × M →₀ ℕ
hexp : ¬exp.supp... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/DPAlgebra/Graded/GradeZero.lean | DividedPowerAlgebra.algebraMap_right_inv_of_degree_zero | [77, 1] | [105, 26] | nth_rewrite 2 [← hnm0] | case neg.intro.inr
R : Type u
M : Type v
inst✝⁴ : CommSemiring R
inst✝³ : AddCommMonoid M
inst✝² : Module R M
inst✝¹ : DecidableEq R
inst✝ : DecidableEq M
x : ↥(grade R M 0)
p : ↥(supported R {nm | 0 < nm.1})
hpx : mk ↑p = ↑x
exp : ℕ × M →₀ ℕ
hexp : ¬exp.support = ∅
h : ¬coeff exp ↑p = 0
nm : ℕ × M
hnm : nm ∈ exp.suppo... | case neg.intro.inr
R : Type u
M : Type v
inst✝⁴ : CommSemiring R
inst✝³ : AddCommMonoid M
inst✝² : Module R M
inst✝¹ : DecidableEq R
inst✝ : DecidableEq M
x : ↥(grade R M 0)
p : ↥(supported R {nm | 0 < nm.1})
hpx : mk ↑p = ↑x
exp : ℕ × M →₀ ℕ
hexp : ¬exp.support = ∅
h : ¬coeff exp ↑p = 0
nm : ℕ × M
hnm : nm ∈ exp.suppo... | Please generate a tactic in lean4 to solve the state.
STATE:
case neg.intro.inr
R : Type u
M : Type v
inst✝⁴ : CommSemiring R
inst✝³ : AddCommMonoid M
inst✝² : Module R M
inst✝¹ : DecidableEq R
inst✝ : DecidableEq M
x : ↥(grade R M 0)
p : ↥(supported R {nm | 0 < nm.1})
hpx : mk ↑p = ↑x
exp : ℕ × M →₀ ℕ
hexp : ¬exp.supp... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/DPAlgebra/Graded/GradeZero.lean | DividedPowerAlgebra.algebraMap_right_inv_of_degree_zero | [77, 1] | [105, 26] | apply MvPolynomial.mem_supported.mp p.prop | case neg.intro.inr
R : Type u
M : Type v
inst✝⁴ : CommSemiring R
inst✝³ : AddCommMonoid M
inst✝² : Module R M
inst✝¹ : DecidableEq R
inst✝ : DecidableEq M
x : ↥(grade R M 0)
p : ↥(supported R {nm | 0 < nm.1})
hpx : mk ↑p = ↑x
exp : ℕ × M →₀ ℕ
hexp : ¬exp.support = ∅
h : ¬coeff exp ↑p = 0
nm : ℕ × M
hnm : nm ∈ exp.suppo... | case neg.intro.inr.a
R : Type u
M : Type v
inst✝⁴ : CommSemiring R
inst✝³ : AddCommMonoid M
inst✝² : Module R M
inst✝¹ : DecidableEq R
inst✝ : DecidableEq M
x : ↥(grade R M 0)
p : ↥(supported R {nm | 0 < nm.1})
hpx : mk ↑p = ↑x
exp : ℕ × M →₀ ℕ
hexp : ¬exp.support = ∅
h : ¬coeff exp ↑p = 0
nm : ℕ × M
hnm : nm ∈ exp.sup... | Please generate a tactic in lean4 to solve the state.
STATE:
case neg.intro.inr
R : Type u
M : Type v
inst✝⁴ : CommSemiring R
inst✝³ : AddCommMonoid M
inst✝² : Module R M
inst✝¹ : DecidableEq R
inst✝ : DecidableEq M
x : ↥(grade R M 0)
p : ↥(supported R {nm | 0 < nm.1})
hpx : mk ↑p = ↑x
exp : ℕ × M →₀ ℕ
hexp : ¬exp.supp... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/DPAlgebra/Graded/GradeZero.lean | DividedPowerAlgebra.algebraMap_right_inv_of_degree_zero | [77, 1] | [105, 26] | simp only [mem_coe, mem_vars, Finsupp.mem_support_iff, ne_eq, mem_support_iff, exists_prop] | case neg.intro.inr.a
R : Type u
M : Type v
inst✝⁴ : CommSemiring R
inst✝³ : AddCommMonoid M
inst✝² : Module R M
inst✝¹ : DecidableEq R
inst✝ : DecidableEq M
x : ↥(grade R M 0)
p : ↥(supported R {nm | 0 < nm.1})
hpx : mk ↑p = ↑x
exp : ℕ × M →₀ ℕ
hexp : ¬exp.support = ∅
h : ¬coeff exp ↑p = 0
nm : ℕ × M
hnm : nm ∈ exp.sup... | case neg.intro.inr.a
R : Type u
M : Type v
inst✝⁴ : CommSemiring R
inst✝³ : AddCommMonoid M
inst✝² : Module R M
inst✝¹ : DecidableEq R
inst✝ : DecidableEq M
x : ↥(grade R M 0)
p : ↥(supported R {nm | 0 < nm.1})
hpx : mk ↑p = ↑x
exp : ℕ × M →₀ ℕ
hexp : ¬exp.support = ∅
h : ¬coeff exp ↑p = 0
nm : ℕ × M
hnm : nm ∈ exp.sup... | Please generate a tactic in lean4 to solve the state.
STATE:
case neg.intro.inr.a
R : Type u
M : Type v
inst✝⁴ : CommSemiring R
inst✝³ : AddCommMonoid M
inst✝² : Module R M
inst✝¹ : DecidableEq R
inst✝ : DecidableEq M
x : ↥(grade R M 0)
p : ↥(supported R {nm | 0 < nm.1})
hpx : mk ↑p = ↑x
exp : ℕ × M →₀ ℕ
hexp : ¬exp.su... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/DPAlgebra/Graded/GradeZero.lean | DividedPowerAlgebra.algebraMap_right_inv_of_degree_zero | [77, 1] | [105, 26] | simp only [Finsupp.mem_support_iff] at hnm | case neg.intro.inr.a
R : Type u
M : Type v
inst✝⁴ : CommSemiring R
inst✝³ : AddCommMonoid M
inst✝² : Module R M
inst✝¹ : DecidableEq R
inst✝ : DecidableEq M
x : ↥(grade R M 0)
p : ↥(supported R {nm | 0 < nm.1})
hpx : mk ↑p = ↑x
exp : ℕ × M →₀ ℕ
hexp : ¬exp.support = ∅
h : ¬coeff exp ↑p = 0
nm : ℕ × M
hnm : nm ∈ exp.sup... | case neg.intro.inr.a
R : Type u
M : Type v
inst✝⁴ : CommSemiring R
inst✝³ : AddCommMonoid M
inst✝² : Module R M
inst✝¹ : DecidableEq R
inst✝ : DecidableEq M
x : ↥(grade R M 0)
p : ↥(supported R {nm | 0 < nm.1})
hpx : mk ↑p = ↑x
exp : ℕ × M →₀ ℕ
hexp : ¬exp.support = ∅
h : ¬coeff exp ↑p = 0
nm : ℕ × M
hnm0 : nm.1 = 0
hn... | Please generate a tactic in lean4 to solve the state.
STATE:
case neg.intro.inr.a
R : Type u
M : Type v
inst✝⁴ : CommSemiring R
inst✝³ : AddCommMonoid M
inst✝² : Module R M
inst✝¹ : DecidableEq R
inst✝ : DecidableEq M
x : ↥(grade R M 0)
p : ↥(supported R {nm | 0 < nm.1})
hpx : mk ↑p = ↑x
exp : ℕ × M →₀ ℕ
hexp : ¬exp.su... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/DPAlgebra/Graded/GradeZero.lean | DividedPowerAlgebra.algebraMap_right_inv_of_degree_zero | [77, 1] | [105, 26] | exact ⟨exp, h, hnm⟩ | case neg.intro.inr.a
R : Type u
M : Type v
inst✝⁴ : CommSemiring R
inst✝³ : AddCommMonoid M
inst✝² : Module R M
inst✝¹ : DecidableEq R
inst✝ : DecidableEq M
x : ↥(grade R M 0)
p : ↥(supported R {nm | 0 < nm.1})
hpx : mk ↑p = ↑x
exp : ℕ × M →₀ ℕ
hexp : ¬exp.support = ∅
h : ¬coeff exp ↑p = 0
nm : ℕ × M
hnm0 : nm.1 = 0
hn... | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case neg.intro.inr.a
R : Type u
M : Type v
inst✝⁴ : CommSemiring R
inst✝³ : AddCommMonoid M
inst✝² : Module R M
inst✝¹ : DecidableEq R
inst✝ : DecidableEq M
x : ↥(grade R M 0)
p : ↥(supported R {nm | 0 < nm.1})
hpx : mk ↑p = ↑x
exp : ℕ × M →₀ ℕ
hexp : ¬exp.su... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/DPAlgebra/Graded/GradeZero.lean | DividedPowerAlgebra.mem_augIdeal_iff | [122, 1] | [124, 33] | rw [augIdeal, RingHom.mem_ker] | 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 ∈ augIdeal R M ↔ (algebraMapInv R M) f = 0 | 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 : DividedPowerAlgebra R M
⊢ f ∈ augIdeal R M ↔ (algebraMapInv R M) f = 0
TACTIC:
|
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/DPAlgebra/Graded/GradeZero.lean | DividedPowerAlgebra.dp_mem_augIdeal_iff | [127, 1] | [130, 63] | rw [mem_augIdeal_iff, dp, algebraMapInv_eq, aeval_X] | R : Type u
M : Type v
inst✝⁵ : CommRing R
inst✝⁴ : AddCommMonoid M
inst✝³ : Module R M
inst✝² : DecidableEq R
inst✝¹ : DecidableEq M
inst✝ : Nontrivial R
n : ℕ
m : M
⊢ dp R n m ∈ augIdeal R M ↔ 0 < n | R : Type u
M : Type v
inst✝⁵ : CommRing R
inst✝⁴ : AddCommMonoid M
inst✝³ : Module R M
inst✝² : DecidableEq R
inst✝¹ : DecidableEq M
inst✝ : Nontrivial R
n : ℕ
m : M
⊢ (if 0 < (n, m).1 then 0 else 1) = 0 ↔ 0 < n | Please generate a tactic in lean4 to solve the state.
STATE:
R : Type u
M : Type v
inst✝⁵ : CommRing R
inst✝⁴ : AddCommMonoid M
inst✝³ : Module R M
inst✝² : DecidableEq R
inst✝¹ : DecidableEq M
inst✝ : Nontrivial R
n : ℕ
m : M
⊢ dp R n m ∈ augIdeal R M ↔ 0 < n
TACTIC:
|
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/DPAlgebra/Graded/GradeZero.lean | DividedPowerAlgebra.dp_mem_augIdeal_iff | [127, 1] | [130, 63] | simp only [ite_eq_left_iff, not_not, one_ne_zero, imp_false] | R : Type u
M : Type v
inst✝⁵ : CommRing R
inst✝⁴ : AddCommMonoid M
inst✝³ : Module R M
inst✝² : DecidableEq R
inst✝¹ : DecidableEq M
inst✝ : Nontrivial R
n : ℕ
m : M
⊢ (if 0 < (n, m).1 then 0 else 1) = 0 ↔ 0 < n | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
R : Type u
M : Type v
inst✝⁵ : CommRing R
inst✝⁴ : AddCommMonoid M
inst✝³ : Module R M
inst✝² : DecidableEq R
inst✝¹ : DecidableEq M
inst✝ : Nontrivial R
n : ℕ
m : M
⊢ (if 0 < (n, m).1 then 0 else 1) = 0 ↔ 0 < n
TACTIC:
|
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/DPAlgebra/Graded/GradeZero.lean | DividedPowerAlgebra.dp_mem_augIdeal | [133, 1] | [135, 66] | rw [mem_augIdeal_iff, dp, algebraMapInv_eq, aeval_X, if_pos hn] | 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
n : ℕ
hn : 0 < n
m : M
⊢ dp R n m ∈ augIdeal R M | 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
n : ℕ
hn : 0 < n
m : M
⊢ dp R n m ∈ augIdeal R M
TACTIC:
|
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/DPAlgebra/Graded/GradeZero.lean | DividedPowerAlgebra.ι_mem_augIdeal | [138, 1] | [139, 92] | simp only [mem_augIdeal_iff, ι_def, dp, algebraMapInv_eq, aeval_X, zero_lt_one, ite_true] | 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
m : M
⊢ (ι R M) m ∈ augIdeal R M | 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
m : M
⊢ (ι R M) m ∈ augIdeal R M
TACTIC:
|
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/DPAlgebra/Graded/GradeZero.lean | DividedPowerAlgebra.augIdeal_isAugmentationIdeal' | [164, 1] | [169, 32] | dsimp only [algebraMap_comp_kerLiftAlg] | 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 : DividedPowerAlgebra R M ⧸ augIdeal R M
⊢ (Ideal.Quotient.mk (augIdeal R M)) ((algebraMap_comp_kerLiftAlg R M) r) = 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 : DividedPowerAlgebra R M ⧸ augIdeal R M
⊢ (Ideal.Quotient.mk (augIdeal R M))
(((algebraMap R (DividedPowerAlgebra R M)).comp (kerLiftAlg_algebraMapInv R M).toRingHom) r) =
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 : DividedPowerAlgebra R M ⧸ augIdeal R M
⊢ (Ideal.Quotient.mk (augIdeal R M)) ((algebraMap_comp_kerLiftAlg R M) r) = r
TACTIC... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/DPAlgebra/Graded/GradeZero.lean | DividedPowerAlgebra.augIdeal_isAugmentationIdeal' | [164, 1] | [169, 32] | rw [RingHom.coe_comp, Function.comp_apply, Ideal.Quotient.mk_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
r : DividedPowerAlgebra R M ⧸ augIdeal R M
⊢ (Ideal.Quotient.mk (augIdeal R M))
(((algebraMap R (DividedPowerAlgebra R M)).comp (kerLiftAlg_algebraMapInv R M).toRingHom) r) =
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 : DividedPowerAlgebra R M ⧸ augIdeal R M
⊢ (algebraMap R (DividedPowerAlgebra R M ⧸ augIdeal R M)) ((kerLiftAlg_algebraMapInv R M).toRingHom r) = 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 : DividedPowerAlgebra R M ⧸ augIdeal R M
⊢ (Ideal.Quotient.mk (augIdeal R M))
(((algebraMap R (DividedPowerAlgebra R M)... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/DPAlgebra/Graded/GradeZero.lean | DividedPowerAlgebra.augIdeal_isAugmentationIdeal' | [164, 1] | [169, 32] | apply kerLiftAlg_rightInverse | 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 : DividedPowerAlgebra R M ⧸ augIdeal R M
⊢ (algebraMap R (DividedPowerAlgebra R M ⧸ augIdeal R M)) ((kerLiftAlg_algebraMapInv R M).toRingHom r) = r | 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
r : DividedPowerAlgebra R M ⧸ augIdeal R M
⊢ (algebraMap R (DividedPowerAlgebra R M ⧸ augIdeal R M)) ((kerLiftAlg_algebraMapInv... |
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 [augIdeal, AlgHom.toRingHom_eq_coe, RingHom.mem_ker, RingHom.coe_coe] at hf0 | 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
⊢ coeff 0 f = 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
⊢ coeff 0 f = 0 | 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
⊢ coeff 0 f = 0
TACTIC:
|
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/DPAlgebra/Graded/GradeZero.lean | DividedPowerAlgebra.coeff_zero_of_mem_augIdeal | [174, 1] | [196, 34] | rw [← hf0, algebraMapInv_eq R M, eq_comm] | 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
⊢ coeff 0 f = 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) f = coeff 0 f | 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
⊢ coeff 0 f = 0
TACTIC:
|
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/DPAlgebra/Graded/GradeZero.lean | DividedPowerAlgebra.coeff_zero_of_mem_augIdeal | [174, 1] | [196, 34] | conv_lhs => rw [f.as_sum] | 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) f = coeff 0 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) (∑ v ∈ f.support, (monomial v) (coeff v f... | 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 < ... |
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