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/ForMathlib/WeightedHomogeneous.lean | MvPolynomial.weightedHomogeneousComponent_of_weightedHomogeneous_ne | [604, 1] | [615, 71] | by_cases zero_coeff : coeff x p = 0 | case a
R : Type u_2
M : Type u_1
inst✝² : CommSemiring R
σ : Type u_3
inst✝¹ : AddCommMonoid M
w : σ → M
n✝ : M
φ ψ : MvPolynomial σ R
inst✝ : DecidableEq M
m n : M
p : MvPolynomial σ R
hp : IsWeightedHomogeneous w p m
hn : n ≠ m
x : σ →₀ ℕ
⊢ (if (weightedDegree w) x = n then coeff x p else 0) = coeff x 0 | case pos
R : Type u_2
M : Type u_1
inst✝² : CommSemiring R
σ : Type u_3
inst✝¹ : AddCommMonoid M
w : σ → M
n✝ : M
φ ψ : MvPolynomial σ R
inst✝ : DecidableEq M
m n : M
p : MvPolynomial σ R
hp : IsWeightedHomogeneous w p m
hn : n ≠ m
x : σ →₀ ℕ
zero_coeff : coeff x p = 0
⊢ (if (weightedDegree w) x = n then coeff x p else... | Please generate a tactic in lean4 to solve the state.
STATE:
case a
R : Type u_2
M : Type u_1
inst✝² : CommSemiring R
σ : Type u_3
inst✝¹ : AddCommMonoid M
w : σ → M
n✝ : M
φ ψ : MvPolynomial σ R
inst✝ : DecidableEq M
m n : M
p : MvPolynomial σ R
hp : IsWeightedHomogeneous w p m
hn : n ≠ m
x : σ →₀ ℕ
⊢ (if (weightedDeg... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/ForMathlib/WeightedHomogeneous.lean | MvPolynomial.weightedHomogeneousComponent_of_weightedHomogeneous_ne | [604, 1] | [615, 71] | split_ifs | case pos
R : Type u_2
M : Type u_1
inst✝² : CommSemiring R
σ : Type u_3
inst✝¹ : AddCommMonoid M
w : σ → M
n✝ : M
φ ψ : MvPolynomial σ R
inst✝ : DecidableEq M
m n : M
p : MvPolynomial σ R
hp : IsWeightedHomogeneous w p m
hn : n ≠ m
x : σ →₀ ℕ
zero_coeff : coeff x p = 0
⊢ (if (weightedDegree w) x = n then coeff x p else... | case pos
R : Type u_2
M : Type u_1
inst✝² : CommSemiring R
σ : Type u_3
inst✝¹ : AddCommMonoid M
w : σ → M
n✝ : M
φ ψ : MvPolynomial σ R
inst✝ : DecidableEq M
m n : M
p : MvPolynomial σ R
hp : IsWeightedHomogeneous w p m
hn : n ≠ m
x : σ →₀ ℕ
zero_coeff : coeff x p = 0
h✝ : (weightedDegree w) x = n
⊢ coeff x p = coeff ... | Please generate a tactic in lean4 to solve the state.
STATE:
case pos
R : Type u_2
M : Type u_1
inst✝² : CommSemiring R
σ : Type u_3
inst✝¹ : AddCommMonoid M
w : σ → M
n✝ : M
φ ψ : MvPolynomial σ R
inst✝ : DecidableEq M
m n : M
p : MvPolynomial σ R
hp : IsWeightedHomogeneous w p m
hn : n ≠ m
x : σ →₀ ℕ
zero_coeff : coe... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/ForMathlib/WeightedHomogeneous.lean | MvPolynomial.weightedHomogeneousComponent_of_weightedHomogeneous_ne | [604, 1] | [615, 71] | rw [zero_coeff] | case pos
R : Type u_2
M : Type u_1
inst✝² : CommSemiring R
σ : Type u_3
inst✝¹ : AddCommMonoid M
w : σ → M
n✝ : M
φ ψ : MvPolynomial σ R
inst✝ : DecidableEq M
m n : M
p : MvPolynomial σ R
hp : IsWeightedHomogeneous w p m
hn : n ≠ m
x : σ →₀ ℕ
zero_coeff : coeff x p = 0
h✝ : (weightedDegree w) x = n
⊢ coeff x p = coeff ... | case pos
R : Type u_2
M : Type u_1
inst✝² : CommSemiring R
σ : Type u_3
inst✝¹ : AddCommMonoid M
w : σ → M
n✝ : M
φ ψ : MvPolynomial σ R
inst✝ : DecidableEq M
m n : M
p : MvPolynomial σ R
hp : IsWeightedHomogeneous w p m
hn : n ≠ m
x : σ →₀ ℕ
zero_coeff : coeff x p = 0
h✝ : (weightedDegree w) x = n
⊢ 0 = coeff x 0
cas... | Please generate a tactic in lean4 to solve the state.
STATE:
case pos
R : Type u_2
M : Type u_1
inst✝² : CommSemiring R
σ : Type u_3
inst✝¹ : AddCommMonoid M
w : σ → M
n✝ : M
φ ψ : MvPolynomial σ R
inst✝ : DecidableEq M
m n : M
p : MvPolynomial σ R
hp : IsWeightedHomogeneous w p m
hn : n ≠ m
x : σ →₀ ℕ
zero_coeff : coe... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/ForMathlib/WeightedHomogeneous.lean | MvPolynomial.weightedHomogeneousComponent_of_weightedHomogeneous_ne | [604, 1] | [615, 71] | rw [coeff_zero] | case pos
R : Type u_2
M : Type u_1
inst✝² : CommSemiring R
σ : Type u_3
inst✝¹ : AddCommMonoid M
w : σ → M
n✝ : M
φ ψ : MvPolynomial σ R
inst✝ : DecidableEq M
m n : M
p : MvPolynomial σ R
hp : IsWeightedHomogeneous w p m
hn : n ≠ m
x : σ →₀ ℕ
zero_coeff : coeff x p = 0
h✝ : (weightedDegree w) x = n
⊢ 0 = coeff x 0
cas... | case neg
R : Type u_2
M : Type u_1
inst✝² : CommSemiring R
σ : Type u_3
inst✝¹ : AddCommMonoid M
w : σ → M
n✝ : M
φ ψ : MvPolynomial σ R
inst✝ : DecidableEq M
m n : M
p : MvPolynomial σ R
hp : IsWeightedHomogeneous w p m
hn : n ≠ m
x : σ →₀ ℕ
zero_coeff : coeff x p = 0
h✝ : ¬(weightedDegree w) x = n
⊢ 0 = coeff x 0 | Please generate a tactic in lean4 to solve the state.
STATE:
case pos
R : Type u_2
M : Type u_1
inst✝² : CommSemiring R
σ : Type u_3
inst✝¹ : AddCommMonoid M
w : σ → M
n✝ : M
φ ψ : MvPolynomial σ R
inst✝ : DecidableEq M
m n : M
p : MvPolynomial σ R
hp : IsWeightedHomogeneous w p m
hn : n ≠ m
x : σ →₀ ℕ
zero_coeff : coe... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/ForMathlib/WeightedHomogeneous.lean | MvPolynomial.weightedHomogeneousComponent_of_weightedHomogeneous_ne | [604, 1] | [615, 71] | rw [coeff_zero] | case neg
R : Type u_2
M : Type u_1
inst✝² : CommSemiring R
σ : Type u_3
inst✝¹ : AddCommMonoid M
w : σ → M
n✝ : M
φ ψ : MvPolynomial σ R
inst✝ : DecidableEq M
m n : M
p : MvPolynomial σ R
hp : IsWeightedHomogeneous w p m
hn : n ≠ m
x : σ →₀ ℕ
zero_coeff : coeff x p = 0
h✝ : ¬(weightedDegree w) x = n
⊢ 0 = coeff x 0 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case neg
R : Type u_2
M : Type u_1
inst✝² : CommSemiring R
σ : Type u_3
inst✝¹ : AddCommMonoid M
w : σ → M
n✝ : M
φ ψ : MvPolynomial σ R
inst✝ : DecidableEq M
m n : M
p : MvPolynomial σ R
hp : IsWeightedHomogeneous w p m
hn : n ≠ m
x : σ →₀ ℕ
zero_coeff : coe... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/ForMathlib/WeightedHomogeneous.lean | MvPolynomial.weightedHomogeneousComponent_of_weightedHomogeneous_ne | [604, 1] | [615, 71] | rw [if_neg] | case neg
R : Type u_2
M : Type u_1
inst✝² : CommSemiring R
σ : Type u_3
inst✝¹ : AddCommMonoid M
w : σ → M
n✝ : M
φ ψ : MvPolynomial σ R
inst✝ : DecidableEq M
m n : M
p : MvPolynomial σ R
hp : IsWeightedHomogeneous w p m
hn : n ≠ m
x : σ →₀ ℕ
zero_coeff : ¬coeff x p = 0
⊢ (if (weightedDegree w) x = n then coeff x p els... | case neg
R : Type u_2
M : Type u_1
inst✝² : CommSemiring R
σ : Type u_3
inst✝¹ : AddCommMonoid M
w : σ → M
n✝ : M
φ ψ : MvPolynomial σ R
inst✝ : DecidableEq M
m n : M
p : MvPolynomial σ R
hp : IsWeightedHomogeneous w p m
hn : n ≠ m
x : σ →₀ ℕ
zero_coeff : ¬coeff x p = 0
⊢ 0 = coeff x 0
case neg.hnc
R : Type u_2
M : Ty... | Please generate a tactic in lean4 to solve the state.
STATE:
case neg
R : Type u_2
M : Type u_1
inst✝² : CommSemiring R
σ : Type u_3
inst✝¹ : AddCommMonoid M
w : σ → M
n✝ : M
φ ψ : MvPolynomial σ R
inst✝ : DecidableEq M
m n : M
p : MvPolynomial σ R
hp : IsWeightedHomogeneous w p m
hn : n ≠ m
x : σ →₀ ℕ
zero_coeff : ¬co... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/ForMathlib/WeightedHomogeneous.lean | MvPolynomial.weightedHomogeneousComponent_of_weightedHomogeneous_ne | [604, 1] | [615, 71] | rw [coeff_zero] | case neg
R : Type u_2
M : Type u_1
inst✝² : CommSemiring R
σ : Type u_3
inst✝¹ : AddCommMonoid M
w : σ → M
n✝ : M
φ ψ : MvPolynomial σ R
inst✝ : DecidableEq M
m n : M
p : MvPolynomial σ R
hp : IsWeightedHomogeneous w p m
hn : n ≠ m
x : σ →₀ ℕ
zero_coeff : ¬coeff x p = 0
⊢ 0 = coeff x 0
case neg.hnc
R : Type u_2
M : Ty... | case neg.hnc
R : Type u_2
M : Type u_1
inst✝² : CommSemiring R
σ : Type u_3
inst✝¹ : AddCommMonoid M
w : σ → M
n✝ : M
φ ψ : MvPolynomial σ R
inst✝ : DecidableEq M
m n : M
p : MvPolynomial σ R
hp : IsWeightedHomogeneous w p m
hn : n ≠ m
x : σ →₀ ℕ
zero_coeff : ¬coeff x p = 0
⊢ ¬(weightedDegree w) x = n | Please generate a tactic in lean4 to solve the state.
STATE:
case neg
R : Type u_2
M : Type u_1
inst✝² : CommSemiring R
σ : Type u_3
inst✝¹ : AddCommMonoid M
w : σ → M
n✝ : M
φ ψ : MvPolynomial σ R
inst✝ : DecidableEq M
m n : M
p : MvPolynomial σ R
hp : IsWeightedHomogeneous w p m
hn : n ≠ m
x : σ →₀ ℕ
zero_coeff : ¬co... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/ForMathlib/WeightedHomogeneous.lean | MvPolynomial.weightedHomogeneousComponent_of_weightedHomogeneous_ne | [604, 1] | [615, 71] | rw [hp zero_coeff] | case neg.hnc
R : Type u_2
M : Type u_1
inst✝² : CommSemiring R
σ : Type u_3
inst✝¹ : AddCommMonoid M
w : σ → M
n✝ : M
φ ψ : MvPolynomial σ R
inst✝ : DecidableEq M
m n : M
p : MvPolynomial σ R
hp : IsWeightedHomogeneous w p m
hn : n ≠ m
x : σ →₀ ℕ
zero_coeff : ¬coeff x p = 0
⊢ ¬(weightedDegree w) x = n | case neg.hnc
R : Type u_2
M : Type u_1
inst✝² : CommSemiring R
σ : Type u_3
inst✝¹ : AddCommMonoid M
w : σ → M
n✝ : M
φ ψ : MvPolynomial σ R
inst✝ : DecidableEq M
m n : M
p : MvPolynomial σ R
hp : IsWeightedHomogeneous w p m
hn : n ≠ m
x : σ →₀ ℕ
zero_coeff : ¬coeff x p = 0
⊢ ¬m = n | Please generate a tactic in lean4 to solve the state.
STATE:
case neg.hnc
R : Type u_2
M : Type u_1
inst✝² : CommSemiring R
σ : Type u_3
inst✝¹ : AddCommMonoid M
w : σ → M
n✝ : M
φ ψ : MvPolynomial σ R
inst✝ : DecidableEq M
m n : M
p : MvPolynomial σ R
hp : IsWeightedHomogeneous w p m
hn : n ≠ m
x : σ →₀ ℕ
zero_coeff :... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/ForMathlib/WeightedHomogeneous.lean | MvPolynomial.weightedHomogeneousComponent_of_weightedHomogeneous_ne | [604, 1] | [615, 71] | exact Ne.symm hn | case neg.hnc
R : Type u_2
M : Type u_1
inst✝² : CommSemiring R
σ : Type u_3
inst✝¹ : AddCommMonoid M
w : σ → M
n✝ : M
φ ψ : MvPolynomial σ R
inst✝ : DecidableEq M
m n : M
p : MvPolynomial σ R
hp : IsWeightedHomogeneous w p m
hn : n ≠ m
x : σ →₀ ℕ
zero_coeff : ¬coeff x p = 0
⊢ ¬m = n | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case neg.hnc
R : Type u_2
M : Type u_1
inst✝² : CommSemiring R
σ : Type u_3
inst✝¹ : AddCommMonoid M
w : σ → M
n✝ : M
φ ψ : MvPolynomial σ R
inst✝ : DecidableEq M
m n : M
p : MvPolynomial σ R
hp : IsWeightedHomogeneous w p m
hn : n ≠ m
x : σ →₀ ℕ
zero_coeff :... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/ForMathlib/WeightedHomogeneous.lean | MvPolynomial.DirectSum.coeLinearMap_eq_support_sum | [638, 1] | [643, 49] | rw [DirectSum.coeLinearMap_eq_dfinsupp_sum] | R : Type u_2
M : Type u_3
inst✝⁴ : CommSemiring R
σ : Type u_1
inst✝³ : AddCommMonoid M
w : σ → M
n : M
φ ψ : MvPolynomial σ R
inst✝² : DecidableEq σ
inst✝¹ : DecidableEq R
inst✝ : DecidableEq M
x : DirectSum M fun i => ↥(weightedHomogeneousSubmodule R w i)
⊢ (DirectSum.coeLinearMap fun i => weightedHomogeneousSubmodul... | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
R : Type u_2
M : Type u_3
inst✝⁴ : CommSemiring R
σ : Type u_1
inst✝³ : AddCommMonoid M
w : σ → M
n : M
φ ψ : MvPolynomial σ R
inst✝² : DecidableEq σ
inst✝¹ : DecidableEq R
inst✝ : DecidableEq M
x : DirectSum M fun i => ↥(weightedHomogeneousSubmodule R w i)
⊢... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/ForMathlib/WeightedHomogeneous.lean | MvPolynomial.DirectSum.coeLinearMap_eq_finsum | [658, 1] | [666, 43] | rw [DirectSum.coeLinearMap_eq_support_sum, DFinsupp.sum] | R : Type u_2
M : Type u_3
inst✝⁴ : CommSemiring R
σ : Type u_1
inst✝³ : AddCommMonoid M
w : σ → M
n : M
φ ψ : MvPolynomial σ R
inst✝² : DecidableEq σ
inst✝¹ : DecidableEq R
inst✝ : DecidableEq M
x : DirectSum M fun i => ↥(weightedHomogeneousSubmodule R w i)
⊢ (DirectSum.coeLinearMap fun i => weightedHomogeneousSubmodul... | R : Type u_2
M : Type u_3
inst✝⁴ : CommSemiring R
σ : Type u_1
inst✝³ : AddCommMonoid M
w : σ → M
n : M
φ ψ : MvPolynomial σ R
inst✝² : DecidableEq σ
inst✝¹ : DecidableEq R
inst✝ : DecidableEq M
x : DirectSum M fun i => ↥(weightedHomogeneousSubmodule R w i)
⊢ ∑ i ∈ DFinsupp.support x, ↑(x i) = ∑ᶠ (m : M), ↑(x m) | Please generate a tactic in lean4 to solve the state.
STATE:
R : Type u_2
M : Type u_3
inst✝⁴ : CommSemiring R
σ : Type u_1
inst✝³ : AddCommMonoid M
w : σ → M
n : M
φ ψ : MvPolynomial σ R
inst✝² : DecidableEq σ
inst✝¹ : DecidableEq R
inst✝ : DecidableEq M
x : DirectSum M fun i => ↥(weightedHomogeneousSubmodule R w i)
⊢... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/ForMathlib/WeightedHomogeneous.lean | MvPolynomial.DirectSum.coeLinearMap_eq_finsum | [658, 1] | [666, 43] | rw [finsum_eq_sum_of_support_subset] | R : Type u_2
M : Type u_3
inst✝⁴ : CommSemiring R
σ : Type u_1
inst✝³ : AddCommMonoid M
w : σ → M
n : M
φ ψ : MvPolynomial σ R
inst✝² : DecidableEq σ
inst✝¹ : DecidableEq R
inst✝ : DecidableEq M
x : DirectSum M fun i => ↥(weightedHomogeneousSubmodule R w i)
⊢ ∑ i ∈ DFinsupp.support x, ↑(x i) = ∑ᶠ (m : M), ↑(x m) | case h
R : Type u_2
M : Type u_3
inst✝⁴ : CommSemiring R
σ : Type u_1
inst✝³ : AddCommMonoid M
w : σ → M
n : M
φ ψ : MvPolynomial σ R
inst✝² : DecidableEq σ
inst✝¹ : DecidableEq R
inst✝ : DecidableEq M
x : DirectSum M fun i => ↥(weightedHomogeneousSubmodule R w i)
⊢ (Function.support fun m => ↑(x m)) ⊆ ↑(DFinsupp.suppo... | Please generate a tactic in lean4 to solve the state.
STATE:
R : Type u_2
M : Type u_3
inst✝⁴ : CommSemiring R
σ : Type u_1
inst✝³ : AddCommMonoid M
w : σ → M
n : M
φ ψ : MvPolynomial σ R
inst✝² : DecidableEq σ
inst✝¹ : DecidableEq R
inst✝ : DecidableEq M
x : DirectSum M fun i => ↥(weightedHomogeneousSubmodule R w i)
⊢... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/ForMathlib/WeightedHomogeneous.lean | MvPolynomial.DirectSum.coeLinearMap_eq_finsum | [658, 1] | [666, 43] | apply DirectSum.support_subset_submodule | case h
R : Type u_2
M : Type u_3
inst✝⁴ : CommSemiring R
σ : Type u_1
inst✝³ : AddCommMonoid M
w : σ → M
n : M
φ ψ : MvPolynomial σ R
inst✝² : DecidableEq σ
inst✝¹ : DecidableEq R
inst✝ : DecidableEq M
x : DirectSum M fun i => ↥(weightedHomogeneousSubmodule R w i)
⊢ (Function.support fun m => ↑(x m)) ⊆ ↑(DFinsupp.suppo... | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case h
R : Type u_2
M : Type u_3
inst✝⁴ : CommSemiring R
σ : Type u_1
inst✝³ : AddCommMonoid M
w : σ → M
n : M
φ ψ : MvPolynomial σ R
inst✝² : DecidableEq σ
inst✝¹ : DecidableEq R
inst✝ : DecidableEq M
x : DirectSum M fun i => ↥(weightedHomogeneousSubmodule R... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/ForMathlib/WeightedHomogeneous.lean | MvPolynomial.weightedHomogeneousComponent_weightedHomogeneousPolynomial' | [679, 1] | [685, 16] | rw [weightedHomogeneousComponent_weighted_homogeneous_polynomial m m _ x.prop,
if_pos rfl] | R : Type u_2
M : Type u_3
inst✝⁴ : CommSemiring R
σ : Type u_1
inst✝³ : AddCommMonoid M
w : σ → M
n : M
φ ψ : MvPolynomial σ R
inst✝² : DecidableEq σ
inst✝¹ : DecidableEq R
inst✝ : DecidableEq M
m : M
x : ↥(weightedHomogeneousSubmodule R w m)
⊢ (weightedHomogeneousComponent w m) ↑x = ↑x | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
R : Type u_2
M : Type u_3
inst✝⁴ : CommSemiring R
σ : Type u_1
inst✝³ : AddCommMonoid M
w : σ → M
n : M
φ ψ : MvPolynomial σ R
inst✝² : DecidableEq σ
inst✝¹ : DecidableEq R
inst✝ : DecidableEq M
m : M
x : ↥(weightedHomogeneousSubmodule R w m)
⊢ (weightedHomog... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/ForMathlib/WeightedHomogeneous.lean | MvPolynomial.weightedHomogeneousComponent_directSum | [689, 1] | [706, 52] | rw [DirectSum.coeLinearMap_eq_dfinsupp_sum, DFinsupp.sum, map_sum] | R : Type u_2
M : Type u_3
inst✝⁴ : CommSemiring R
σ : Type u_1
inst✝³ : AddCommMonoid M
w : σ → M
n : M
φ ψ : MvPolynomial σ R
inst✝² : DecidableEq σ
inst✝¹ : DecidableEq R
inst✝ : DecidableEq M
x : DirectSum M fun i => ↥(weightedHomogeneousSubmodule R w i)
m : M
⊢ (weightedHomogeneousComponent w m) ((DirectSum.coeLine... | R : Type u_2
M : Type u_3
inst✝⁴ : CommSemiring R
σ : Type u_1
inst✝³ : AddCommMonoid M
w : σ → M
n : M
φ ψ : MvPolynomial σ R
inst✝² : DecidableEq σ
inst✝¹ : DecidableEq R
inst✝ : DecidableEq M
x : DirectSum M fun i => ↥(weightedHomogeneousSubmodule R w i)
m : M
⊢ ∑ x_1 ∈ DFinsupp.support x, (weightedHomogeneousCompon... | Please generate a tactic in lean4 to solve the state.
STATE:
R : Type u_2
M : Type u_3
inst✝⁴ : CommSemiring R
σ : Type u_1
inst✝³ : AddCommMonoid M
w : σ → M
n : M
φ ψ : MvPolynomial σ R
inst✝² : DecidableEq σ
inst✝¹ : DecidableEq R
inst✝ : DecidableEq M
x : DirectSum M fun i => ↥(weightedHomogeneousSubmodule R w i)
m... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/ForMathlib/WeightedHomogeneous.lean | MvPolynomial.weightedHomogeneousComponent_directSum | [689, 1] | [706, 52] | convert @Finset.sum_eq_single M (MvPolynomial σ R) _ (DFinsupp.support x) _ m _ _ | R : Type u_2
M : Type u_3
inst✝⁴ : CommSemiring R
σ : Type u_1
inst✝³ : AddCommMonoid M
w : σ → M
n : M
φ ψ : MvPolynomial σ R
inst✝² : DecidableEq σ
inst✝¹ : DecidableEq R
inst✝ : DecidableEq M
x : DirectSum M fun i => ↥(weightedHomogeneousSubmodule R w i)
m : M
⊢ ∑ x_1 ∈ DFinsupp.support x, (weightedHomogeneousCompon... | case h.e'_3
R : Type u_2
M : Type u_3
inst✝⁴ : CommSemiring R
σ : Type u_1
inst✝³ : AddCommMonoid M
w : σ → M
n : M
φ ψ : MvPolynomial σ R
inst✝² : DecidableEq σ
inst✝¹ : DecidableEq R
inst✝ : DecidableEq M
x : DirectSum M fun i => ↥(weightedHomogeneousSubmodule R w i)
m : M
⊢ ↑(x m) = (weightedHomogeneousComponent w m... | Please generate a tactic in lean4 to solve the state.
STATE:
R : Type u_2
M : Type u_3
inst✝⁴ : CommSemiring R
σ : Type u_1
inst✝³ : AddCommMonoid M
w : σ → M
n : M
φ ψ : MvPolynomial σ R
inst✝² : DecidableEq σ
inst✝¹ : DecidableEq R
inst✝ : DecidableEq M
x : DirectSum M fun i => ↥(weightedHomogeneousSubmodule R w i)
m... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/ForMathlib/WeightedHomogeneous.lean | MvPolynomial.weightedHomogeneousComponent_directSum | [689, 1] | [706, 52] | rw [weightedHomogeneousComponent_of_weighted_homogeneous_polynomial_same] | case h.e'_3
R : Type u_2
M : Type u_3
inst✝⁴ : CommSemiring R
σ : Type u_1
inst✝³ : AddCommMonoid M
w : σ → M
n : M
φ ψ : MvPolynomial σ R
inst✝² : DecidableEq σ
inst✝¹ : DecidableEq R
inst✝ : DecidableEq M
x : DirectSum M fun i => ↥(weightedHomogeneousSubmodule R w i)
m : M
⊢ ↑(x m) = (weightedHomogeneousComponent w m... | case h.e'_3.hp
R : Type u_2
M : Type u_3
inst✝⁴ : CommSemiring R
σ : Type u_1
inst✝³ : AddCommMonoid M
w : σ → M
n : M
φ ψ : MvPolynomial σ R
inst✝² : DecidableEq σ
inst✝¹ : DecidableEq R
inst✝ : DecidableEq M
x : DirectSum M fun i => ↥(weightedHomogeneousSubmodule R w i)
m : M
⊢ IsWeightedHomogeneous w (↑(x m)) m | Please generate a tactic in lean4 to solve the state.
STATE:
case h.e'_3
R : Type u_2
M : Type u_3
inst✝⁴ : CommSemiring R
σ : Type u_1
inst✝³ : AddCommMonoid M
w : σ → M
n : M
φ ψ : MvPolynomial σ R
inst✝² : DecidableEq σ
inst✝¹ : DecidableEq R
inst✝ : DecidableEq M
x : DirectSum M fun i => ↥(weightedHomogeneousSubmod... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/ForMathlib/WeightedHomogeneous.lean | MvPolynomial.weightedHomogeneousComponent_directSum | [689, 1] | [706, 52] | rw [← mem_weightedHomogeneousSubmodule] | case h.e'_3.hp
R : Type u_2
M : Type u_3
inst✝⁴ : CommSemiring R
σ : Type u_1
inst✝³ : AddCommMonoid M
w : σ → M
n : M
φ ψ : MvPolynomial σ R
inst✝² : DecidableEq σ
inst✝¹ : DecidableEq R
inst✝ : DecidableEq M
x : DirectSum M fun i => ↥(weightedHomogeneousSubmodule R w i)
m : M
⊢ IsWeightedHomogeneous w (↑(x m)) m | case h.e'_3.hp
R : Type u_2
M : Type u_3
inst✝⁴ : CommSemiring R
σ : Type u_1
inst✝³ : AddCommMonoid M
w : σ → M
n : M
φ ψ : MvPolynomial σ R
inst✝² : DecidableEq σ
inst✝¹ : DecidableEq R
inst✝ : DecidableEq M
x : DirectSum M fun i => ↥(weightedHomogeneousSubmodule R w i)
m : M
⊢ ↑(x m) ∈ weightedHomogeneousSubmodule R... | Please generate a tactic in lean4 to solve the state.
STATE:
case h.e'_3.hp
R : Type u_2
M : Type u_3
inst✝⁴ : CommSemiring R
σ : Type u_1
inst✝³ : AddCommMonoid M
w : σ → M
n : M
φ ψ : MvPolynomial σ R
inst✝² : DecidableEq σ
inst✝¹ : DecidableEq R
inst✝ : DecidableEq M
x : DirectSum M fun i => ↥(weightedHomogeneousSub... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/ForMathlib/WeightedHomogeneous.lean | MvPolynomial.weightedHomogeneousComponent_directSum | [689, 1] | [706, 52] | exact (x m).prop | case h.e'_3.hp
R : Type u_2
M : Type u_3
inst✝⁴ : CommSemiring R
σ : Type u_1
inst✝³ : AddCommMonoid M
w : σ → M
n : M
φ ψ : MvPolynomial σ R
inst✝² : DecidableEq σ
inst✝¹ : DecidableEq R
inst✝ : DecidableEq M
x : DirectSum M fun i => ↥(weightedHomogeneousSubmodule R w i)
m : M
⊢ ↑(x m) ∈ weightedHomogeneousSubmodule R... | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case h.e'_3.hp
R : Type u_2
M : Type u_3
inst✝⁴ : CommSemiring R
σ : Type u_1
inst✝³ : AddCommMonoid M
w : σ → M
n : M
φ ψ : MvPolynomial σ R
inst✝² : DecidableEq σ
inst✝¹ : DecidableEq R
inst✝ : DecidableEq M
x : DirectSum M fun i => ↥(weightedHomogeneousSub... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/ForMathlib/WeightedHomogeneous.lean | MvPolynomial.weightedHomogeneousComponent_directSum | [689, 1] | [706, 52] | intro n _ hmn | case convert_2
R : Type u_2
M : Type u_3
inst✝⁴ : CommSemiring R
σ : Type u_1
inst✝³ : AddCommMonoid M
w : σ → M
n : M
φ ψ : MvPolynomial σ R
inst✝² : DecidableEq σ
inst✝¹ : DecidableEq R
inst✝ : DecidableEq M
x : DirectSum M fun i => ↥(weightedHomogeneousSubmodule R w i)
m : M
⊢ ∀ b ∈ DFinsupp.support x, b ≠ m → (weig... | case convert_2
R : Type u_2
M : Type u_3
inst✝⁴ : CommSemiring R
σ : Type u_1
inst✝³ : AddCommMonoid M
w : σ → M
n✝ : M
φ ψ : MvPolynomial σ R
inst✝² : DecidableEq σ
inst✝¹ : DecidableEq R
inst✝ : DecidableEq M
x : DirectSum M fun i => ↥(weightedHomogeneousSubmodule R w i)
m n : M
a✝ : n ∈ DFinsupp.support x
hmn : n ≠ ... | Please generate a tactic in lean4 to solve the state.
STATE:
case convert_2
R : Type u_2
M : Type u_3
inst✝⁴ : CommSemiring R
σ : Type u_1
inst✝³ : AddCommMonoid M
w : σ → M
n : M
φ ψ : MvPolynomial σ R
inst✝² : DecidableEq σ
inst✝¹ : DecidableEq R
inst✝ : DecidableEq M
x : DirectSum M fun i => ↥(weightedHomogeneousSub... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/ForMathlib/WeightedHomogeneous.lean | MvPolynomial.weightedHomogeneousComponent_directSum | [689, 1] | [706, 52] | rw [weightedHomogeneousComponent_of_weightedHomogeneous_ne n m] | case convert_2
R : Type u_2
M : Type u_3
inst✝⁴ : CommSemiring R
σ : Type u_1
inst✝³ : AddCommMonoid M
w : σ → M
n✝ : M
φ ψ : MvPolynomial σ R
inst✝² : DecidableEq σ
inst✝¹ : DecidableEq R
inst✝ : DecidableEq M
x : DirectSum M fun i => ↥(weightedHomogeneousSubmodule R w i)
m n : M
a✝ : n ∈ DFinsupp.support x
hmn : n ≠ ... | case convert_2.hp
R : Type u_2
M : Type u_3
inst✝⁴ : CommSemiring R
σ : Type u_1
inst✝³ : AddCommMonoid M
w : σ → M
n✝ : M
φ ψ : MvPolynomial σ R
inst✝² : DecidableEq σ
inst✝¹ : DecidableEq R
inst✝ : DecidableEq M
x : DirectSum M fun i => ↥(weightedHomogeneousSubmodule R w i)
m n : M
a✝ : n ∈ DFinsupp.support x
hmn : n... | Please generate a tactic in lean4 to solve the state.
STATE:
case convert_2
R : Type u_2
M : Type u_3
inst✝⁴ : CommSemiring R
σ : Type u_1
inst✝³ : AddCommMonoid M
w : σ → M
n✝ : M
φ ψ : MvPolynomial σ R
inst✝² : DecidableEq σ
inst✝¹ : DecidableEq R
inst✝ : DecidableEq M
x : DirectSum M fun i => ↥(weightedHomogeneousSu... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/ForMathlib/WeightedHomogeneous.lean | MvPolynomial.weightedHomogeneousComponent_directSum | [689, 1] | [706, 52] | rw [← mem_weightedHomogeneousSubmodule] | case convert_2.hp
R : Type u_2
M : Type u_3
inst✝⁴ : CommSemiring R
σ : Type u_1
inst✝³ : AddCommMonoid M
w : σ → M
n✝ : M
φ ψ : MvPolynomial σ R
inst✝² : DecidableEq σ
inst✝¹ : DecidableEq R
inst✝ : DecidableEq M
x : DirectSum M fun i => ↥(weightedHomogeneousSubmodule R w i)
m n : M
a✝ : n ∈ DFinsupp.support x
hmn : n... | case convert_2.hp
R : Type u_2
M : Type u_3
inst✝⁴ : CommSemiring R
σ : Type u_1
inst✝³ : AddCommMonoid M
w : σ → M
n✝ : M
φ ψ : MvPolynomial σ R
inst✝² : DecidableEq σ
inst✝¹ : DecidableEq R
inst✝ : DecidableEq M
x : DirectSum M fun i => ↥(weightedHomogeneousSubmodule R w i)
m n : M
a✝ : n ∈ DFinsupp.support x
hmn : n... | Please generate a tactic in lean4 to solve the state.
STATE:
case convert_2.hp
R : Type u_2
M : Type u_3
inst✝⁴ : CommSemiring R
σ : Type u_1
inst✝³ : AddCommMonoid M
w : σ → M
n✝ : M
φ ψ : MvPolynomial σ R
inst✝² : DecidableEq σ
inst✝¹ : DecidableEq R
inst✝ : DecidableEq M
x : DirectSum M fun i => ↥(weightedHomogeneou... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/ForMathlib/WeightedHomogeneous.lean | MvPolynomial.weightedHomogeneousComponent_directSum | [689, 1] | [706, 52] | exact (x n).prop | case convert_2.hp
R : Type u_2
M : Type u_3
inst✝⁴ : CommSemiring R
σ : Type u_1
inst✝³ : AddCommMonoid M
w : σ → M
n✝ : M
φ ψ : MvPolynomial σ R
inst✝² : DecidableEq σ
inst✝¹ : DecidableEq R
inst✝ : DecidableEq M
x : DirectSum M fun i => ↥(weightedHomogeneousSubmodule R w i)
m n : M
a✝ : n ∈ DFinsupp.support x
hmn : n... | case convert_2.a
R : Type u_2
M : Type u_3
inst✝⁴ : CommSemiring R
σ : Type u_1
inst✝³ : AddCommMonoid M
w : σ → M
n✝ : M
φ ψ : MvPolynomial σ R
inst✝² : DecidableEq σ
inst✝¹ : DecidableEq R
inst✝ : DecidableEq M
x : DirectSum M fun i => ↥(weightedHomogeneousSubmodule R w i)
m n : M
a✝ : n ∈ DFinsupp.support x
hmn : n ... | Please generate a tactic in lean4 to solve the state.
STATE:
case convert_2.hp
R : Type u_2
M : Type u_3
inst✝⁴ : CommSemiring R
σ : Type u_1
inst✝³ : AddCommMonoid M
w : σ → M
n✝ : M
φ ψ : MvPolynomial σ R
inst✝² : DecidableEq σ
inst✝¹ : DecidableEq R
inst✝ : DecidableEq M
x : DirectSum M fun i => ↥(weightedHomogeneou... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/ForMathlib/WeightedHomogeneous.lean | MvPolynomial.weightedHomogeneousComponent_directSum | [689, 1] | [706, 52] | exact Ne.symm hmn | case convert_2.a
R : Type u_2
M : Type u_3
inst✝⁴ : CommSemiring R
σ : Type u_1
inst✝³ : AddCommMonoid M
w : σ → M
n✝ : M
φ ψ : MvPolynomial σ R
inst✝² : DecidableEq σ
inst✝¹ : DecidableEq R
inst✝ : DecidableEq M
x : DirectSum M fun i => ↥(weightedHomogeneousSubmodule R w i)
m n : M
a✝ : n ∈ DFinsupp.support x
hmn : n ... | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case convert_2.a
R : Type u_2
M : Type u_3
inst✝⁴ : CommSemiring R
σ : Type u_1
inst✝³ : AddCommMonoid M
w : σ → M
n✝ : M
φ ψ : MvPolynomial σ R
inst✝² : DecidableEq σ
inst✝¹ : DecidableEq R
inst✝ : DecidableEq M
x : DirectSum M fun i => ↥(weightedHomogeneous... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/ForMathlib/WeightedHomogeneous.lean | MvPolynomial.weightedHomogeneousComponent_directSum | [689, 1] | [706, 52] | rw [DFinsupp.not_mem_support_iff] | case convert_3
R : Type u_2
M : Type u_3
inst✝⁴ : CommSemiring R
σ : Type u_1
inst✝³ : AddCommMonoid M
w : σ → M
n : M
φ ψ : MvPolynomial σ R
inst✝² : DecidableEq σ
inst✝¹ : DecidableEq R
inst✝ : DecidableEq M
x : DirectSum M fun i => ↥(weightedHomogeneousSubmodule R w i)
m : M
⊢ m ∉ DFinsupp.support x → (weightedHomog... | case convert_3
R : Type u_2
M : Type u_3
inst✝⁴ : CommSemiring R
σ : Type u_1
inst✝³ : AddCommMonoid M
w : σ → M
n : M
φ ψ : MvPolynomial σ R
inst✝² : DecidableEq σ
inst✝¹ : DecidableEq R
inst✝ : DecidableEq M
x : DirectSum M fun i => ↥(weightedHomogeneousSubmodule R w i)
m : M
⊢ x m = 0 → (weightedHomogeneousComponent... | Please generate a tactic in lean4 to solve the state.
STATE:
case convert_3
R : Type u_2
M : Type u_3
inst✝⁴ : CommSemiring R
σ : Type u_1
inst✝³ : AddCommMonoid M
w : σ → M
n : M
φ ψ : MvPolynomial σ R
inst✝² : DecidableEq σ
inst✝¹ : DecidableEq R
inst✝ : DecidableEq M
x : DirectSum M fun i => ↥(weightedHomogeneousSub... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/ForMathlib/WeightedHomogeneous.lean | MvPolynomial.weightedHomogeneousComponent_directSum | [689, 1] | [706, 52] | intro hm | case convert_3
R : Type u_2
M : Type u_3
inst✝⁴ : CommSemiring R
σ : Type u_1
inst✝³ : AddCommMonoid M
w : σ → M
n : M
φ ψ : MvPolynomial σ R
inst✝² : DecidableEq σ
inst✝¹ : DecidableEq R
inst✝ : DecidableEq M
x : DirectSum M fun i => ↥(weightedHomogeneousSubmodule R w i)
m : M
⊢ x m = 0 → (weightedHomogeneousComponent... | case convert_3
R : Type u_2
M : Type u_3
inst✝⁴ : CommSemiring R
σ : Type u_1
inst✝³ : AddCommMonoid M
w : σ → M
n : M
φ ψ : MvPolynomial σ R
inst✝² : DecidableEq σ
inst✝¹ : DecidableEq R
inst✝ : DecidableEq M
x : DirectSum M fun i => ↥(weightedHomogeneousSubmodule R w i)
m : M
hm : x m = 0
⊢ (weightedHomogeneousCompon... | Please generate a tactic in lean4 to solve the state.
STATE:
case convert_3
R : Type u_2
M : Type u_3
inst✝⁴ : CommSemiring R
σ : Type u_1
inst✝³ : AddCommMonoid M
w : σ → M
n : M
φ ψ : MvPolynomial σ R
inst✝² : DecidableEq σ
inst✝¹ : DecidableEq R
inst✝ : DecidableEq M
x : DirectSum M fun i => ↥(weightedHomogeneousSub... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/ForMathlib/WeightedHomogeneous.lean | MvPolynomial.weightedHomogeneousComponent_directSum | [689, 1] | [706, 52] | rw [hm, Submodule.coe_zero, map_zero] | case convert_3
R : Type u_2
M : Type u_3
inst✝⁴ : CommSemiring R
σ : Type u_1
inst✝³ : AddCommMonoid M
w : σ → M
n : M
φ ψ : MvPolynomial σ R
inst✝² : DecidableEq σ
inst✝¹ : DecidableEq R
inst✝ : DecidableEq M
x : DirectSum M fun i => ↥(weightedHomogeneousSubmodule R w i)
m : M
hm : x m = 0
⊢ (weightedHomogeneousCompon... | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case convert_3
R : Type u_2
M : Type u_3
inst✝⁴ : CommSemiring R
σ : Type u_1
inst✝³ : AddCommMonoid M
w : σ → M
n : M
φ ψ : MvPolynomial σ R
inst✝² : DecidableEq σ
inst✝¹ : DecidableEq R
inst✝ : DecidableEq M
x : DirectSum M fun i => ↥(weightedHomogeneousSub... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/ForMathlib/WeightedHomogeneous.lean | MvPolynomial.nonTorsionWeight_of_nonZero | [725, 1] | [730, 25] | intro n x | R : Type ?u.38955
M : Type u_1
inst✝² : CommSemiring R
σ : Type u_2
inst✝¹ : CanonicallyOrderedAddCommMonoid M
w : σ → M
φ : MvPolynomial σ R
inst✝ : NoZeroSMulDivisors ℕ M
hw : ∀ (i : σ), w i ≠ 0
⊢ NonTorsionWeight w | R : Type ?u.38955
M : Type u_1
inst✝² : CommSemiring R
σ : Type u_2
inst✝¹ : CanonicallyOrderedAddCommMonoid M
w : σ → M
φ : MvPolynomial σ R
inst✝ : NoZeroSMulDivisors ℕ M
hw : ∀ (i : σ), w i ≠ 0
n : ℕ
x : σ
⊢ n • w x = 0 → n = 0 | Please generate a tactic in lean4 to solve the state.
STATE:
R : Type ?u.38955
M : Type u_1
inst✝² : CommSemiring R
σ : Type u_2
inst✝¹ : CanonicallyOrderedAddCommMonoid M
w : σ → M
φ : MvPolynomial σ R
inst✝ : NoZeroSMulDivisors ℕ M
hw : ∀ (i : σ), w i ≠ 0
⊢ NonTorsionWeight w
TACTIC:
|
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/ForMathlib/WeightedHomogeneous.lean | MvPolynomial.nonTorsionWeight_of_nonZero | [725, 1] | [730, 25] | rw [smul_eq_zero] | R : Type ?u.38955
M : Type u_1
inst✝² : CommSemiring R
σ : Type u_2
inst✝¹ : CanonicallyOrderedAddCommMonoid M
w : σ → M
φ : MvPolynomial σ R
inst✝ : NoZeroSMulDivisors ℕ M
hw : ∀ (i : σ), w i ≠ 0
n : ℕ
x : σ
⊢ n • w x = 0 → n = 0 | R : Type ?u.38955
M : Type u_1
inst✝² : CommSemiring R
σ : Type u_2
inst✝¹ : CanonicallyOrderedAddCommMonoid M
w : σ → M
φ : MvPolynomial σ R
inst✝ : NoZeroSMulDivisors ℕ M
hw : ∀ (i : σ), w i ≠ 0
n : ℕ
x : σ
⊢ n = 0 ∨ w x = 0 → n = 0 | Please generate a tactic in lean4 to solve the state.
STATE:
R : Type ?u.38955
M : Type u_1
inst✝² : CommSemiring R
σ : Type u_2
inst✝¹ : CanonicallyOrderedAddCommMonoid M
w : σ → M
φ : MvPolynomial σ R
inst✝ : NoZeroSMulDivisors ℕ M
hw : ∀ (i : σ), w i ≠ 0
n : ℕ
x : σ
⊢ n • w x = 0 → n = 0
TACTIC:
|
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/ForMathlib/WeightedHomogeneous.lean | MvPolynomial.nonTorsionWeight_of_nonZero | [725, 1] | [730, 25] | intro hnx | R : Type ?u.38955
M : Type u_1
inst✝² : CommSemiring R
σ : Type u_2
inst✝¹ : CanonicallyOrderedAddCommMonoid M
w : σ → M
φ : MvPolynomial σ R
inst✝ : NoZeroSMulDivisors ℕ M
hw : ∀ (i : σ), w i ≠ 0
n : ℕ
x : σ
⊢ n = 0 ∨ w x = 0 → n = 0 | R : Type ?u.38955
M : Type u_1
inst✝² : CommSemiring R
σ : Type u_2
inst✝¹ : CanonicallyOrderedAddCommMonoid M
w : σ → M
φ : MvPolynomial σ R
inst✝ : NoZeroSMulDivisors ℕ M
hw : ∀ (i : σ), w i ≠ 0
n : ℕ
x : σ
hnx : n = 0 ∨ w x = 0
⊢ n = 0 | Please generate a tactic in lean4 to solve the state.
STATE:
R : Type ?u.38955
M : Type u_1
inst✝² : CommSemiring R
σ : Type u_2
inst✝¹ : CanonicallyOrderedAddCommMonoid M
w : σ → M
φ : MvPolynomial σ R
inst✝ : NoZeroSMulDivisors ℕ M
hw : ∀ (i : σ), w i ≠ 0
n : ℕ
x : σ
⊢ n = 0 ∨ w x = 0 → n = 0
TACTIC:
|
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/ForMathlib/WeightedHomogeneous.lean | MvPolynomial.nonTorsionWeight_of_nonZero | [725, 1] | [730, 25] | cases' hnx with hn hx | R : Type ?u.38955
M : Type u_1
inst✝² : CommSemiring R
σ : Type u_2
inst✝¹ : CanonicallyOrderedAddCommMonoid M
w : σ → M
φ : MvPolynomial σ R
inst✝ : NoZeroSMulDivisors ℕ M
hw : ∀ (i : σ), w i ≠ 0
n : ℕ
x : σ
hnx : n = 0 ∨ w x = 0
⊢ n = 0 | case inl
R : Type ?u.38955
M : Type u_1
inst✝² : CommSemiring R
σ : Type u_2
inst✝¹ : CanonicallyOrderedAddCommMonoid M
w : σ → M
φ : MvPolynomial σ R
inst✝ : NoZeroSMulDivisors ℕ M
hw : ∀ (i : σ), w i ≠ 0
n : ℕ
x : σ
hn : n = 0
⊢ n = 0
case inr
R : Type ?u.38955
M : Type u_1
inst✝² : CommSemiring R
σ : Type u_2
inst✝... | Please generate a tactic in lean4 to solve the state.
STATE:
R : Type ?u.38955
M : Type u_1
inst✝² : CommSemiring R
σ : Type u_2
inst✝¹ : CanonicallyOrderedAddCommMonoid M
w : σ → M
φ : MvPolynomial σ R
inst✝ : NoZeroSMulDivisors ℕ M
hw : ∀ (i : σ), w i ≠ 0
n : ℕ
x : σ
hnx : n = 0 ∨ w x = 0
⊢ n = 0
TACTIC:
|
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/ForMathlib/WeightedHomogeneous.lean | MvPolynomial.nonTorsionWeight_of_nonZero | [725, 1] | [730, 25] | exact hn | case inl
R : Type ?u.38955
M : Type u_1
inst✝² : CommSemiring R
σ : Type u_2
inst✝¹ : CanonicallyOrderedAddCommMonoid M
w : σ → M
φ : MvPolynomial σ R
inst✝ : NoZeroSMulDivisors ℕ M
hw : ∀ (i : σ), w i ≠ 0
n : ℕ
x : σ
hn : n = 0
⊢ n = 0
case inr
R : Type ?u.38955
M : Type u_1
inst✝² : CommSemiring R
σ : Type u_2
inst✝... | case inr
R : Type ?u.38955
M : Type u_1
inst✝² : CommSemiring R
σ : Type u_2
inst✝¹ : CanonicallyOrderedAddCommMonoid M
w : σ → M
φ : MvPolynomial σ R
inst✝ : NoZeroSMulDivisors ℕ M
hw : ∀ (i : σ), w i ≠ 0
n : ℕ
x : σ
hx : w x = 0
⊢ n = 0 | Please generate a tactic in lean4 to solve the state.
STATE:
case inl
R : Type ?u.38955
M : Type u_1
inst✝² : CommSemiring R
σ : Type u_2
inst✝¹ : CanonicallyOrderedAddCommMonoid M
w : σ → M
φ : MvPolynomial σ R
inst✝ : NoZeroSMulDivisors ℕ M
hw : ∀ (i : σ), w i ≠ 0
n : ℕ
x : σ
hn : n = 0
⊢ n = 0
case inr
R : Type ?u.... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/ForMathlib/WeightedHomogeneous.lean | MvPolynomial.nonTorsionWeight_of_nonZero | [725, 1] | [730, 25] | exfalso | case inr
R : Type ?u.38955
M : Type u_1
inst✝² : CommSemiring R
σ : Type u_2
inst✝¹ : CanonicallyOrderedAddCommMonoid M
w : σ → M
φ : MvPolynomial σ R
inst✝ : NoZeroSMulDivisors ℕ M
hw : ∀ (i : σ), w i ≠ 0
n : ℕ
x : σ
hx : w x = 0
⊢ n = 0 | case inr
R : Type ?u.38955
M : Type u_1
inst✝² : CommSemiring R
σ : Type u_2
inst✝¹ : CanonicallyOrderedAddCommMonoid M
w : σ → M
φ : MvPolynomial σ R
inst✝ : NoZeroSMulDivisors ℕ M
hw : ∀ (i : σ), w i ≠ 0
n : ℕ
x : σ
hx : w x = 0
⊢ False | Please generate a tactic in lean4 to solve the state.
STATE:
case inr
R : Type ?u.38955
M : Type u_1
inst✝² : CommSemiring R
σ : Type u_2
inst✝¹ : CanonicallyOrderedAddCommMonoid M
w : σ → M
φ : MvPolynomial σ R
inst✝ : NoZeroSMulDivisors ℕ M
hw : ∀ (i : σ), w i ≠ 0
n : ℕ
x : σ
hx : w x = 0
⊢ n = 0
TACTIC:
|
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/ForMathlib/WeightedHomogeneous.lean | MvPolynomial.nonTorsionWeight_of_nonZero | [725, 1] | [730, 25] | exact hw x hx | case inr
R : Type ?u.38955
M : Type u_1
inst✝² : CommSemiring R
σ : Type u_2
inst✝¹ : CanonicallyOrderedAddCommMonoid M
w : σ → M
φ : MvPolynomial σ R
inst✝ : NoZeroSMulDivisors ℕ M
hw : ∀ (i : σ), w i ≠ 0
n : ℕ
x : σ
hx : w x = 0
⊢ False | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case inr
R : Type ?u.38955
M : Type u_1
inst✝² : CommSemiring R
σ : Type u_2
inst✝¹ : CanonicallyOrderedAddCommMonoid M
w : σ → M
φ : MvPolynomial σ R
inst✝ : NoZeroSMulDivisors ℕ M
hw : ∀ (i : σ), w i ≠ 0
n : ℕ
x : σ
hx : w x = 0
⊢ False
TACTIC:
|
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/ForMathlib/WeightedHomogeneous.lean | MvPolynomial.decompose'_aux | [824, 9] | [834, 11] | apply weightedHomogeneousComponent_eq_zero' | R : Type u_2
M : Type u_1
inst✝² : CommSemiring R
σ : Type u_3
w : σ → M
inst✝¹ : AddCommMonoid M
inst✝ : DecidableEq M
φ : MvPolynomial σ R
i : M
hi : i ∉ Finset.image (⇑(weightedDegree w)) φ.support
⊢ (weightedHomogeneousComponent w i) φ = 0 | case h
R : Type u_2
M : Type u_1
inst✝² : CommSemiring R
σ : Type u_3
w : σ → M
inst✝¹ : AddCommMonoid M
inst✝ : DecidableEq M
φ : MvPolynomial σ R
i : M
hi : i ∉ Finset.image (⇑(weightedDegree w)) φ.support
⊢ ∀ d ∈ φ.support, (weightedDegree w) d ≠ i | Please generate a tactic in lean4 to solve the state.
STATE:
R : Type u_2
M : Type u_1
inst✝² : CommSemiring R
σ : Type u_3
w : σ → M
inst✝¹ : AddCommMonoid M
inst✝ : DecidableEq M
φ : MvPolynomial σ R
i : M
hi : i ∉ Finset.image (⇑(weightedDegree w)) φ.support
⊢ (weightedHomogeneousComponent w i) φ = 0
TACTIC:
|
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/ForMathlib/WeightedHomogeneous.lean | MvPolynomial.decompose'_aux | [824, 9] | [834, 11] | simp only [Finset.mem_image, mem_support_iff, ne_eq, exists_prop, not_exists, not_and] at hi | case h
R : Type u_2
M : Type u_1
inst✝² : CommSemiring R
σ : Type u_3
w : σ → M
inst✝¹ : AddCommMonoid M
inst✝ : DecidableEq M
φ : MvPolynomial σ R
i : M
hi : i ∉ Finset.image (⇑(weightedDegree w)) φ.support
⊢ ∀ d ∈ φ.support, (weightedDegree w) d ≠ i | case h
R : Type u_2
M : Type u_1
inst✝² : CommSemiring R
σ : Type u_3
w : σ → M
inst✝¹ : AddCommMonoid M
inst✝ : DecidableEq M
φ : MvPolynomial σ R
i : M
hi : ∀ (x : σ →₀ ℕ), ¬coeff x φ = 0 → ¬(weightedDegree w) x = i
⊢ ∀ d ∈ φ.support, (weightedDegree w) d ≠ i | Please generate a tactic in lean4 to solve the state.
STATE:
case h
R : Type u_2
M : Type u_1
inst✝² : CommSemiring R
σ : Type u_3
w : σ → M
inst✝¹ : AddCommMonoid M
inst✝ : DecidableEq M
φ : MvPolynomial σ R
i : M
hi : i ∉ Finset.image (⇑(weightedDegree w)) φ.support
⊢ ∀ d ∈ φ.support, (weightedDegree w) d ≠ i
TACTIC:... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/ForMathlib/WeightedHomogeneous.lean | MvPolynomial.decompose'_aux | [824, 9] | [834, 11] | intro m hm | case h
R : Type u_2
M : Type u_1
inst✝² : CommSemiring R
σ : Type u_3
w : σ → M
inst✝¹ : AddCommMonoid M
inst✝ : DecidableEq M
φ : MvPolynomial σ R
i : M
hi : ∀ (x : σ →₀ ℕ), ¬coeff x φ = 0 → ¬(weightedDegree w) x = i
⊢ ∀ d ∈ φ.support, (weightedDegree w) d ≠ i | case h
R : Type u_2
M : Type u_1
inst✝² : CommSemiring R
σ : Type u_3
w : σ → M
inst✝¹ : AddCommMonoid M
inst✝ : DecidableEq M
φ : MvPolynomial σ R
i : M
hi : ∀ (x : σ →₀ ℕ), ¬coeff x φ = 0 → ¬(weightedDegree w) x = i
m : σ →₀ ℕ
hm : m ∈ φ.support
⊢ (weightedDegree w) m ≠ i | Please generate a tactic in lean4 to solve the state.
STATE:
case h
R : Type u_2
M : Type u_1
inst✝² : CommSemiring R
σ : Type u_3
w : σ → M
inst✝¹ : AddCommMonoid M
inst✝ : DecidableEq M
φ : MvPolynomial σ R
i : M
hi : ∀ (x : σ →₀ ℕ), ¬coeff x φ = 0 → ¬(weightedDegree w) x = i
⊢ ∀ d ∈ φ.support, (weightedDegree w) d ≠... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/ForMathlib/WeightedHomogeneous.lean | MvPolynomial.decompose'_aux | [824, 9] | [834, 11] | apply hi m | case h
R : Type u_2
M : Type u_1
inst✝² : CommSemiring R
σ : Type u_3
w : σ → M
inst✝¹ : AddCommMonoid M
inst✝ : DecidableEq M
φ : MvPolynomial σ R
i : M
hi : ∀ (x : σ →₀ ℕ), ¬coeff x φ = 0 → ¬(weightedDegree w) x = i
m : σ →₀ ℕ
hm : m ∈ φ.support
⊢ (weightedDegree w) m ≠ i | case h
R : Type u_2
M : Type u_1
inst✝² : CommSemiring R
σ : Type u_3
w : σ → M
inst✝¹ : AddCommMonoid M
inst✝ : DecidableEq M
φ : MvPolynomial σ R
i : M
hi : ∀ (x : σ →₀ ℕ), ¬coeff x φ = 0 → ¬(weightedDegree w) x = i
m : σ →₀ ℕ
hm : m ∈ φ.support
⊢ ¬coeff m φ = 0 | Please generate a tactic in lean4 to solve the state.
STATE:
case h
R : Type u_2
M : Type u_1
inst✝² : CommSemiring R
σ : Type u_3
w : σ → M
inst✝¹ : AddCommMonoid M
inst✝ : DecidableEq M
φ : MvPolynomial σ R
i : M
hi : ∀ (x : σ →₀ ℕ), ¬coeff x φ = 0 → ¬(weightedDegree w) x = i
m : σ →₀ ℕ
hm : m ∈ φ.support
⊢ (weighted... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/ForMathlib/WeightedHomogeneous.lean | MvPolynomial.decompose'_aux | [824, 9] | [834, 11] | rw [mem_support_iff] at hm | case h
R : Type u_2
M : Type u_1
inst✝² : CommSemiring R
σ : Type u_3
w : σ → M
inst✝¹ : AddCommMonoid M
inst✝ : DecidableEq M
φ : MvPolynomial σ R
i : M
hi : ∀ (x : σ →₀ ℕ), ¬coeff x φ = 0 → ¬(weightedDegree w) x = i
m : σ →₀ ℕ
hm : m ∈ φ.support
⊢ ¬coeff m φ = 0 | case h
R : Type u_2
M : Type u_1
inst✝² : CommSemiring R
σ : Type u_3
w : σ → M
inst✝¹ : AddCommMonoid M
inst✝ : DecidableEq M
φ : MvPolynomial σ R
i : M
hi : ∀ (x : σ →₀ ℕ), ¬coeff x φ = 0 → ¬(weightedDegree w) x = i
m : σ →₀ ℕ
hm : coeff m φ ≠ 0
⊢ ¬coeff m φ = 0 | Please generate a tactic in lean4 to solve the state.
STATE:
case h
R : Type u_2
M : Type u_1
inst✝² : CommSemiring R
σ : Type u_3
w : σ → M
inst✝¹ : AddCommMonoid M
inst✝ : DecidableEq M
φ : MvPolynomial σ R
i : M
hi : ∀ (x : σ →₀ ℕ), ¬coeff x φ = 0 → ¬(weightedDegree w) x = i
m : σ →₀ ℕ
hm : m ∈ φ.support
⊢ ¬coeff m ... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/ForMathlib/WeightedHomogeneous.lean | MvPolynomial.decompose'_aux | [824, 9] | [834, 11] | exact hm | case h
R : Type u_2
M : Type u_1
inst✝² : CommSemiring R
σ : Type u_3
w : σ → M
inst✝¹ : AddCommMonoid M
inst✝ : DecidableEq M
φ : MvPolynomial σ R
i : M
hi : ∀ (x : σ →₀ ℕ), ¬coeff x φ = 0 → ¬(weightedDegree w) x = i
m : σ →₀ ℕ
hm : coeff m φ ≠ 0
⊢ ¬coeff m φ = 0 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case h
R : Type u_2
M : Type u_1
inst✝² : CommSemiring R
σ : Type u_3
w : σ → M
inst✝¹ : AddCommMonoid M
inst✝ : DecidableEq M
φ : MvPolynomial σ R
i : M
hi : ∀ (x : σ →₀ ℕ), ¬coeff x φ = 0 → ¬(weightedDegree w) x = i
m : σ →₀ ℕ
hm : coeff m φ ≠ 0
⊢ ¬coeff m ... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/ForMathlib/WeightedHomogeneous.lean | MvPolynomial.decompose'_toFun_apply | [843, 9] | [849, 85] | rw [decompose'_toFun] | R : Type u_2
M : Type u_1
inst✝² : CommSemiring R
σ : Type u_3
w : σ → M
inst✝¹ : AddCommMonoid M
inst✝ : DecidableEq M
φ : MvPolynomial σ R
m : M
⊢ ↑((MvPolynomial.decompose'_toFun R w φ) m) = (weightedHomogeneousComponent w m) φ | R : Type u_2
M : Type u_1
inst✝² : CommSemiring R
σ : Type u_3
w : σ → M
inst✝¹ : AddCommMonoid M
inst✝ : DecidableEq M
φ : MvPolynomial σ R
m : M
⊢ ↑(((DirectSum.mk (fun i => ↥(weightedHomogeneousSubmodule R w i)) (Finset.image (⇑(weightedDegree w)) φ.support))
fun m => ⟨(weightedHomogeneousComponent w ↑m) φ... | Please generate a tactic in lean4 to solve the state.
STATE:
R : Type u_2
M : Type u_1
inst✝² : CommSemiring R
σ : Type u_3
w : σ → M
inst✝¹ : AddCommMonoid M
inst✝ : DecidableEq M
φ : MvPolynomial σ R
m : M
⊢ ↑((MvPolynomial.decompose'_toFun R w φ) m) = (weightedHomogeneousComponent w m) φ
TACTIC:
|
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/ForMathlib/WeightedHomogeneous.lean | MvPolynomial.decompose'_toFun_apply | [843, 9] | [849, 85] | by_cases hm : m ∈ Finset.image (weightedDegree w) φ.support | R : Type u_2
M : Type u_1
inst✝² : CommSemiring R
σ : Type u_3
w : σ → M
inst✝¹ : AddCommMonoid M
inst✝ : DecidableEq M
φ : MvPolynomial σ R
m : M
⊢ ↑(((DirectSum.mk (fun i => ↥(weightedHomogeneousSubmodule R w i)) (Finset.image (⇑(weightedDegree w)) φ.support))
fun m => ⟨(weightedHomogeneousComponent w ↑m) φ... | case pos
R : Type u_2
M : Type u_1
inst✝² : CommSemiring R
σ : Type u_3
w : σ → M
inst✝¹ : AddCommMonoid M
inst✝ : DecidableEq M
φ : MvPolynomial σ R
m : M
hm : m ∈ Finset.image (⇑(weightedDegree w)) φ.support
⊢ ↑(((DirectSum.mk (fun i => ↥(weightedHomogeneousSubmodule R w i)) (Finset.image (⇑(weightedDegree w)) φ.supp... | Please generate a tactic in lean4 to solve the state.
STATE:
R : Type u_2
M : Type u_1
inst✝² : CommSemiring R
σ : Type u_3
w : σ → M
inst✝¹ : AddCommMonoid M
inst✝ : DecidableEq M
φ : MvPolynomial σ R
m : M
⊢ ↑(((DirectSum.mk (fun i => ↥(weightedHomogeneousSubmodule R w i)) (Finset.image (⇑(weightedDegree w)) φ.suppor... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/ForMathlib/WeightedHomogeneous.lean | MvPolynomial.decompose'_toFun_apply | [843, 9] | [849, 85] | simp only [DirectSum.mk_apply_of_mem hm, Subtype.coe_mk] | case pos
R : Type u_2
M : Type u_1
inst✝² : CommSemiring R
σ : Type u_3
w : σ → M
inst✝¹ : AddCommMonoid M
inst✝ : DecidableEq M
φ : MvPolynomial σ R
m : M
hm : m ∈ Finset.image (⇑(weightedDegree w)) φ.support
⊢ ↑(((DirectSum.mk (fun i => ↥(weightedHomogeneousSubmodule R w i)) (Finset.image (⇑(weightedDegree w)) φ.supp... | case neg
R : Type u_2
M : Type u_1
inst✝² : CommSemiring R
σ : Type u_3
w : σ → M
inst✝¹ : AddCommMonoid M
inst✝ : DecidableEq M
φ : MvPolynomial σ R
m : M
hm : m ∉ Finset.image (⇑(weightedDegree w)) φ.support
⊢ ↑(((DirectSum.mk (fun i => ↥(weightedHomogeneousSubmodule R w i)) (Finset.image (⇑(weightedDegree w)) φ.supp... | Please generate a tactic in lean4 to solve the state.
STATE:
case pos
R : Type u_2
M : Type u_1
inst✝² : CommSemiring R
σ : Type u_3
w : σ → M
inst✝¹ : AddCommMonoid M
inst✝ : DecidableEq M
φ : MvPolynomial σ R
m : M
hm : m ∈ Finset.image (⇑(weightedDegree w)) φ.support
⊢ ↑(((DirectSum.mk (fun i => ↥(weightedHomogeneou... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/ForMathlib/WeightedHomogeneous.lean | MvPolynomial.decompose'_toFun_apply | [843, 9] | [849, 85] | rw [DirectSum.mk_apply_of_not_mem hm, Submodule.coe_zero, decompose'_aux w φ m hm] | case neg
R : Type u_2
M : Type u_1
inst✝² : CommSemiring R
σ : Type u_3
w : σ → M
inst✝¹ : AddCommMonoid M
inst✝ : DecidableEq M
φ : MvPolynomial σ R
m : M
hm : m ∉ Finset.image (⇑(weightedDegree w)) φ.support
⊢ ↑(((DirectSum.mk (fun i => ↥(weightedHomogeneousSubmodule R w i)) (Finset.image (⇑(weightedDegree w)) φ.supp... | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case neg
R : Type u_2
M : Type u_1
inst✝² : CommSemiring R
σ : Type u_3
w : σ → M
inst✝¹ : AddCommMonoid M
inst✝ : DecidableEq M
φ : MvPolynomial σ R
m : M
hm : m ∉ Finset.image (⇑(weightedDegree w)) φ.support
⊢ ↑(((DirectSum.mk (fun i => ↥(weightedHomogeneou... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/SubDPIdeal.lean | Submodule.iSup_eq_span' | [13, 1] | [16, 53] | simp_rw [← Submodule.iSup_span, Submodule.span_eq] | R : Type u_1
M : Type u_2
inst✝² : Semiring R
inst✝¹ : AddCommMonoid M
inst✝ : Module R M
ι : Sort u_3
p : ι → Submodule R M
h : ι → Prop
⊢ ⨆ i, ⨆ (_ : h i), p i = span R (⋃ i, ⋃ (_ : h i), ↑(p i)) | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
R : Type u_1
M : Type u_2
inst✝² : Semiring R
inst✝¹ : AddCommMonoid M
inst✝ : Module R M
ι : Sort u_3
p : ι → Submodule R M
h : ι → Prop
⊢ ⨆ i, ⨆ (_ : h i), p i = span R (⋃ i, ⋃ (_ : h i), ↑(p i))
TACTIC:
|
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/SubDPIdeal.lean | Subideal.bot_def | [41, 1] | [41, 81] | rw [mk_bot] | A : Type u_1
inst✝ : CommSemiring A
I : Ideal A
⊢ ⟨⊥, ⋯⟩ = ⊥ | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
A : Type u_1
inst✝ : CommSemiring A
I : Ideal A
⊢ ⟨⊥, ⋯⟩ = ⊥
TACTIC:
|
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/SubDPIdeal.lean | Subideal.inf_def | [44, 1] | [48, 59] | ext x | A : Type u_1
inst✝ : CommSemiring A
I : Ideal A
J J' : { J // J ≤ I }
⊢ J ⊓ J' = ⟨↑J ⊓ ↑J', ⋯⟩ | case a.h
A : Type u_1
inst✝ : CommSemiring A
I : Ideal A
J J' : { J // J ≤ I }
x : A
⊢ x ∈ ↑(J ⊓ J') ↔ x ∈ ↑⟨↑J ⊓ ↑J', ⋯⟩ | Please generate a tactic in lean4 to solve the state.
STATE:
A : Type u_1
inst✝ : CommSemiring A
I : Ideal A
J J' : { J // J ≤ I }
⊢ J ⊓ J' = ⟨↑J ⊓ ↑J', ⋯⟩
TACTIC:
|
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/SubDPIdeal.lean | Subideal.inf_def | [44, 1] | [48, 59] | exact ⟨fun ⟨h, _⟩ => h, fun h => ⟨h, J.property h.left⟩⟩ | case a.h
A : Type u_1
inst✝ : CommSemiring A
I : Ideal A
J J' : { J // J ≤ I }
x : A
⊢ x ∈ ↑(J ⊓ J') ↔ x ∈ ↑⟨↑J ⊓ ↑J', ⋯⟩ | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case a.h
A : Type u_1
inst✝ : CommSemiring A
I : Ideal A
J J' : { J // J ≤ I }
x : A
⊢ x ∈ ↑(J ⊓ J') ↔ x ∈ ↑⟨↑J ⊓ ↑J', ⋯⟩
TACTIC:
|
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/SubDPIdeal.lean | Subideal.sInf_def | [54, 1] | [58, 6] | ext x | A : Type u_1
inst✝ : CommSemiring A
I : Ideal A
S : Set { J // J ≤ I }
⊢ sInf S = ⟨sInf (val '' S) ⊓ I, ⋯⟩ | case a.h
A : Type u_1
inst✝ : CommSemiring A
I : Ideal A
S : Set { J // J ≤ I }
x : A
⊢ x ∈ ↑(sInf S) ↔ x ∈ ↑⟨sInf (val '' S) ⊓ I, ⋯⟩ | Please generate a tactic in lean4 to solve the state.
STATE:
A : Type u_1
inst✝ : CommSemiring A
I : Ideal A
S : Set { J // J ≤ I }
⊢ sInf S = ⟨sInf (val '' S) ⊓ I, ⋯⟩
TACTIC:
|
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/SubDPIdeal.lean | Subideal.sInf_def | [54, 1] | [58, 6] | rfl | case a.h
A : Type u_1
inst✝ : CommSemiring A
I : Ideal A
S : Set { J // J ≤ I }
x : A
⊢ x ∈ ↑(sInf S) ↔ x ∈ ↑⟨sInf (val '' S) ⊓ I, ⋯⟩ | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case a.h
A : Type u_1
inst✝ : CommSemiring A
I : Ideal A
S : Set { J // J ≤ I }
x : A
⊢ x ∈ ↑(sInf S) ↔ x ∈ ↑⟨sInf (val '' S) ⊓ I, ⋯⟩
TACTIC:
|
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/SubDPIdeal.lean | Subideal.sup_def | [61, 1] | [68, 24] | ext x | A : Type u_1
inst✝ : CommSemiring A
I : Ideal A
J J' : { J // J ≤ I }
⊢ J ⊔ J' = ⟨sInf {B | ↑J ≤ B ∧ ↑J' ≤ B}, ⋯⟩ | case a.h
A : Type u_1
inst✝ : CommSemiring A
I : Ideal A
J J' : { J // J ≤ I }
x : A
⊢ x ∈ ↑(J ⊔ J') ↔ x ∈ ↑⟨sInf {B | ↑J ≤ B ∧ ↑J' ≤ B}, ⋯⟩ | Please generate a tactic in lean4 to solve the state.
STATE:
A : Type u_1
inst✝ : CommSemiring A
I : Ideal A
J J' : { J // J ≤ I }
⊢ J ⊔ J' = ⟨sInf {B | ↑J ≤ B ∧ ↑J' ≤ B}, ⋯⟩
TACTIC:
|
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/SubDPIdeal.lean | Subideal.sup_def | [61, 1] | [68, 24] | refine' ⟨fun ⟨h, _⟩ => h, fun h => ⟨h, _⟩⟩ | case a.h
A : Type u_1
inst✝ : CommSemiring A
I : Ideal A
J J' : { J // J ≤ I }
x : A
⊢ x ∈ ↑(J ⊔ J') ↔ x ∈ ↑⟨sInf {B | ↑J ≤ B ∧ ↑J' ≤ B}, ⋯⟩ | case a.h
A : Type u_1
inst✝ : CommSemiring A
I : Ideal A
J J' : { J // J ≤ I }
x : A
h : x ∈ ↑⟨sInf {B | ↑J ≤ B ∧ ↑J' ≤ B}, ⋯⟩
⊢ x ∈ ↑I | Please generate a tactic in lean4 to solve the state.
STATE:
case a.h
A : Type u_1
inst✝ : CommSemiring A
I : Ideal A
J J' : { J // J ≤ I }
x : A
⊢ x ∈ ↑(J ⊔ J') ↔ x ∈ ↑⟨sInf {B | ↑J ≤ B ∧ ↑J' ≤ B}, ⋯⟩
TACTIC:
|
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/SubDPIdeal.lean | Subideal.sup_def | [61, 1] | [68, 24] | rw [coe_mk, Submodule.mem_sInf] at h | case a.h
A : Type u_1
inst✝ : CommSemiring A
I : Ideal A
J J' : { J // J ≤ I }
x : A
h : x ∈ ↑⟨sInf {B | ↑J ≤ B ∧ ↑J' ≤ B}, ⋯⟩
⊢ x ∈ ↑I | case a.h
A : Type u_1
inst✝ : CommSemiring A
I : Ideal A
J J' : { J // J ≤ I }
x : A
h : ∀ p ∈ {B | ↑J ≤ B ∧ ↑J' ≤ B}, x ∈ p
⊢ x ∈ ↑I | Please generate a tactic in lean4 to solve the state.
STATE:
case a.h
A : Type u_1
inst✝ : CommSemiring A
I : Ideal A
J J' : { J // J ≤ I }
x : A
h : x ∈ ↑⟨sInf {B | ↑J ≤ B ∧ ↑J' ≤ B}, ⋯⟩
⊢ x ∈ ↑I
TACTIC:
|
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/SubDPIdeal.lean | Subideal.sup_def | [61, 1] | [68, 24] | exact h I ⟨J.2, J'.2⟩ | case a.h
A : Type u_1
inst✝ : CommSemiring A
I : Ideal A
J J' : { J // J ≤ I }
x : A
h : ∀ p ∈ {B | ↑J ≤ B ∧ ↑J' ≤ B}, x ∈ p
⊢ x ∈ ↑I | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case a.h
A : Type u_1
inst✝ : CommSemiring A
I : Ideal A
J J' : { J // J ≤ I }
x : A
h : ∀ p ∈ {B | ↑J ≤ B ∧ ↑J' ≤ B}, x ∈ p
⊢ x ∈ ↑I
TACTIC:
|
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/SubDPIdeal.lean | Subideal.sSup_def | [71, 1] | [74, 6] | ext x | A : Type u_1
inst✝ : CommSemiring A
I : Ideal A
S : Set { J // J ≤ I }
⊢ sSup S = ⟨sSup (val '' S) ⊓ I, ⋯⟩ | case a.h
A : Type u_1
inst✝ : CommSemiring A
I : Ideal A
S : Set { J // J ≤ I }
x : A
⊢ x ∈ ↑(sSup S) ↔ x ∈ ↑⟨sSup (val '' S) ⊓ I, ⋯⟩ | Please generate a tactic in lean4 to solve the state.
STATE:
A : Type u_1
inst✝ : CommSemiring A
I : Ideal A
S : Set { J // J ≤ I }
⊢ sSup S = ⟨sSup (val '' S) ⊓ I, ⋯⟩
TACTIC:
|
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/SubDPIdeal.lean | Subideal.sSup_def | [71, 1] | [74, 6] | rfl | case a.h
A : Type u_1
inst✝ : CommSemiring A
I : Ideal A
S : Set { J // J ≤ I }
x : A
⊢ x ∈ ↑(sSup S) ↔ x ∈ ↑⟨sSup (val '' S) ⊓ I, ⋯⟩ | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case a.h
A : Type u_1
inst✝ : CommSemiring A
I : Ideal A
S : Set { J // J ≤ I }
x : A
⊢ x ∈ ↑(sSup S) ↔ x ∈ ↑⟨sSup (val '' S) ⊓ I, ⋯⟩
TACTIC:
|
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/SubDPIdeal.lean | DividedPowers.isSubDPIdeal_inf_iff | [118, 1] | [139, 71] | refine' ⟨fun hIJ n a b ha hb hab => _, fun hIJ => _⟩ | A✝ : Type ?u.19361
inst✝¹ : CommSemiring A✝
I✝ : Ideal A✝
hI✝ : DividedPowers I✝
A : Type u_1
inst✝ : CommRing A
I : Ideal A
hI : DividedPowers I
J : Ideal A
⊢ hI.isSubDPIdeal (J ⊓ I) ↔ ∀ (n : ℕ) (a b : A), a ∈ I → b ∈ I → a - b ∈ J → hI.dpow n a - hI.dpow n b ∈ J | case refine'_1
A✝ : Type ?u.19361
inst✝¹ : CommSemiring A✝
I✝ : Ideal A✝
hI✝ : DividedPowers I✝
A : Type u_1
inst✝ : CommRing A
I : Ideal A
hI : DividedPowers I
J : Ideal A
hIJ : hI.isSubDPIdeal (J ⊓ I)
n : ℕ
a b : A
ha : a ∈ I
hb : b ∈ I
hab : a - b ∈ J
⊢ hI.dpow n a - hI.dpow n b ∈ J
case refine'_2
A✝ : Type ?u.1936... | Please generate a tactic in lean4 to solve the state.
STATE:
A✝ : Type ?u.19361
inst✝¹ : CommSemiring A✝
I✝ : Ideal A✝
hI✝ : DividedPowers I✝
A : Type u_1
inst✝ : CommRing A
I : Ideal A
hI : DividedPowers I
J : Ideal A
⊢ hI.isSubDPIdeal (J ⊓ I) ↔ ∀ (n : ℕ) (a b : A), a ∈ I → b ∈ I → a - b ∈ J → hI.dpow n a - hI.dpow n ... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/SubDPIdeal.lean | DividedPowers.isSubDPIdeal_inf_iff | [118, 1] | [139, 71] | have hab' : a - b ∈ I := I.sub_mem ha hb | case refine'_1
A✝ : Type ?u.19361
inst✝¹ : CommSemiring A✝
I✝ : Ideal A✝
hI✝ : DividedPowers I✝
A : Type u_1
inst✝ : CommRing A
I : Ideal A
hI : DividedPowers I
J : Ideal A
hIJ : hI.isSubDPIdeal (J ⊓ I)
n : ℕ
a b : A
ha : a ∈ I
hb : b ∈ I
hab : a - b ∈ J
⊢ hI.dpow n a - hI.dpow n b ∈ J | case refine'_1
A✝ : Type ?u.19361
inst✝¹ : CommSemiring A✝
I✝ : Ideal A✝
hI✝ : DividedPowers I✝
A : Type u_1
inst✝ : CommRing A
I : Ideal A
hI : DividedPowers I
J : Ideal A
hIJ : hI.isSubDPIdeal (J ⊓ I)
n : ℕ
a b : A
ha : a ∈ I
hb : b ∈ I
hab : a - b ∈ J
hab' : a - b ∈ I
⊢ hI.dpow n a - hI.dpow n b ∈ J | Please generate a tactic in lean4 to solve the state.
STATE:
case refine'_1
A✝ : Type ?u.19361
inst✝¹ : CommSemiring A✝
I✝ : Ideal A✝
hI✝ : DividedPowers I✝
A : Type u_1
inst✝ : CommRing A
I : Ideal A
hI : DividedPowers I
J : Ideal A
hIJ : hI.isSubDPIdeal (J ⊓ I)
n : ℕ
a b : A
ha : a ∈ I
hb : b ∈ I
hab : a - b ∈ J
⊢ hI... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/SubDPIdeal.lean | DividedPowers.isSubDPIdeal_inf_iff | [118, 1] | [139, 71] | rw [← add_sub_cancel b a, hI.dpow_add' n hb hab', Finset.range_succ,
Finset.sum_insert Finset.not_mem_range_self, tsub_self, hI.dpow_zero hab', mul_one,
add_sub_cancel_left] | case refine'_1
A✝ : Type ?u.19361
inst✝¹ : CommSemiring A✝
I✝ : Ideal A✝
hI✝ : DividedPowers I✝
A : Type u_1
inst✝ : CommRing A
I : Ideal A
hI : DividedPowers I
J : Ideal A
hIJ : hI.isSubDPIdeal (J ⊓ I)
n : ℕ
a b : A
ha : a ∈ I
hb : b ∈ I
hab : a - b ∈ J
hab' : a - b ∈ I
⊢ hI.dpow n a - hI.dpow n b ∈ J | case refine'_1
A✝ : Type ?u.19361
inst✝¹ : CommSemiring A✝
I✝ : Ideal A✝
hI✝ : DividedPowers I✝
A : Type u_1
inst✝ : CommRing A
I : Ideal A
hI : DividedPowers I
J : Ideal A
hIJ : hI.isSubDPIdeal (J ⊓ I)
n : ℕ
a b : A
ha : a ∈ I
hb : b ∈ I
hab : a - b ∈ J
hab' : a - b ∈ I
⊢ ∑ x ∈ Finset.range n, hI.dpow x b * hI.dpow (n... | Please generate a tactic in lean4 to solve the state.
STATE:
case refine'_1
A✝ : Type ?u.19361
inst✝¹ : CommSemiring A✝
I✝ : Ideal A✝
hI✝ : DividedPowers I✝
A : Type u_1
inst✝ : CommRing A
I : Ideal A
hI : DividedPowers I
J : Ideal A
hIJ : hI.isSubDPIdeal (J ⊓ I)
n : ℕ
a b : A
ha : a ∈ I
hb : b ∈ I
hab : a - b ∈ J
hab'... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/SubDPIdeal.lean | DividedPowers.isSubDPIdeal_inf_iff | [118, 1] | [139, 71] | apply Ideal.sum_mem | case refine'_1
A✝ : Type ?u.19361
inst✝¹ : CommSemiring A✝
I✝ : Ideal A✝
hI✝ : DividedPowers I✝
A : Type u_1
inst✝ : CommRing A
I : Ideal A
hI : DividedPowers I
J : Ideal A
hIJ : hI.isSubDPIdeal (J ⊓ I)
n : ℕ
a b : A
ha : a ∈ I
hb : b ∈ I
hab : a - b ∈ J
hab' : a - b ∈ I
⊢ ∑ x ∈ Finset.range n, hI.dpow x b * hI.dpow (n... | case refine'_1.a
A✝ : Type ?u.19361
inst✝¹ : CommSemiring A✝
I✝ : Ideal A✝
hI✝ : DividedPowers I✝
A : Type u_1
inst✝ : CommRing A
I : Ideal A
hI : DividedPowers I
J : Ideal A
hIJ : hI.isSubDPIdeal (J ⊓ I)
n : ℕ
a b : A
ha : a ∈ I
hb : b ∈ I
hab : a - b ∈ J
hab' : a - b ∈ I
⊢ ∀ c ∈ Finset.range n, hI.dpow c b * hI.dpow ... | Please generate a tactic in lean4 to solve the state.
STATE:
case refine'_1
A✝ : Type ?u.19361
inst✝¹ : CommSemiring A✝
I✝ : Ideal A✝
hI✝ : DividedPowers I✝
A : Type u_1
inst✝ : CommRing A
I : Ideal A
hI : DividedPowers I
J : Ideal A
hIJ : hI.isSubDPIdeal (J ⊓ I)
n : ℕ
a b : A
ha : a ∈ I
hb : b ∈ I
hab : a - b ∈ J
hab'... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/SubDPIdeal.lean | DividedPowers.isSubDPIdeal_inf_iff | [118, 1] | [139, 71] | intro i hi | case refine'_1.a
A✝ : Type ?u.19361
inst✝¹ : CommSemiring A✝
I✝ : Ideal A✝
hI✝ : DividedPowers I✝
A : Type u_1
inst✝ : CommRing A
I : Ideal A
hI : DividedPowers I
J : Ideal A
hIJ : hI.isSubDPIdeal (J ⊓ I)
n : ℕ
a b : A
ha : a ∈ I
hb : b ∈ I
hab : a - b ∈ J
hab' : a - b ∈ I
⊢ ∀ c ∈ Finset.range n, hI.dpow c b * hI.dpow ... | case refine'_1.a
A✝ : Type ?u.19361
inst✝¹ : CommSemiring A✝
I✝ : Ideal A✝
hI✝ : DividedPowers I✝
A : Type u_1
inst✝ : CommRing A
I : Ideal A
hI : DividedPowers I
J : Ideal A
hIJ : hI.isSubDPIdeal (J ⊓ I)
n : ℕ
a b : A
ha : a ∈ I
hb : b ∈ I
hab : a - b ∈ J
hab' : a - b ∈ I
i : ℕ
hi : i ∈ Finset.range n
⊢ hI.dpow i b * ... | Please generate a tactic in lean4 to solve the state.
STATE:
case refine'_1.a
A✝ : Type ?u.19361
inst✝¹ : CommSemiring A✝
I✝ : Ideal A✝
hI✝ : DividedPowers I✝
A : Type u_1
inst✝ : CommRing A
I : Ideal A
hI : DividedPowers I
J : Ideal A
hIJ : hI.isSubDPIdeal (J ⊓ I)
n : ℕ
a b : A
ha : a ∈ I
hb : b ∈ I
hab : a - b ∈ J
ha... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/SubDPIdeal.lean | DividedPowers.isSubDPIdeal_inf_iff | [118, 1] | [139, 71] | apply SemilatticeInf.inf_le_left J I | case refine'_1.a
A✝ : Type ?u.19361
inst✝¹ : CommSemiring A✝
I✝ : Ideal A✝
hI✝ : DividedPowers I✝
A : Type u_1
inst✝ : CommRing A
I : Ideal A
hI : DividedPowers I
J : Ideal A
hIJ : hI.isSubDPIdeal (J ⊓ I)
n : ℕ
a b : A
ha : a ∈ I
hb : b ∈ I
hab : a - b ∈ J
hab' : a - b ∈ I
i : ℕ
hi : i ∈ Finset.range n
⊢ hI.dpow i b * ... | case refine'_1.a.a
A✝ : Type ?u.19361
inst✝¹ : CommSemiring A✝
I✝ : Ideal A✝
hI✝ : DividedPowers I✝
A : Type u_1
inst✝ : CommRing A
I : Ideal A
hI : DividedPowers I
J : Ideal A
hIJ : hI.isSubDPIdeal (J ⊓ I)
n : ℕ
a b : A
ha : a ∈ I
hb : b ∈ I
hab : a - b ∈ J
hab' : a - b ∈ I
i : ℕ
hi : i ∈ Finset.range n
⊢ hI.dpow i b ... | Please generate a tactic in lean4 to solve the state.
STATE:
case refine'_1.a
A✝ : Type ?u.19361
inst✝¹ : CommSemiring A✝
I✝ : Ideal A✝
hI✝ : DividedPowers I✝
A : Type u_1
inst✝ : CommRing A
I : Ideal A
hI : DividedPowers I
J : Ideal A
hIJ : hI.isSubDPIdeal (J ⊓ I)
n : ℕ
a b : A
ha : a ∈ I
hb : b ∈ I
hab : a - b ∈ J
ha... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/SubDPIdeal.lean | DividedPowers.isSubDPIdeal_inf_iff | [118, 1] | [139, 71] | exact
(J ⊓ I).smul_mem _
(hIJ.dpow_mem (n - i) (ne_of_gt (Nat.sub_pos_of_lt (Finset.mem_range.mp hi))) _
⟨hab, hab'⟩) | case refine'_1.a.a
A✝ : Type ?u.19361
inst✝¹ : CommSemiring A✝
I✝ : Ideal A✝
hI✝ : DividedPowers I✝
A : Type u_1
inst✝ : CommRing A
I : Ideal A
hI : DividedPowers I
J : Ideal A
hIJ : hI.isSubDPIdeal (J ⊓ I)
n : ℕ
a b : A
ha : a ∈ I
hb : b ∈ I
hab : a - b ∈ J
hab' : a - b ∈ I
i : ℕ
hi : i ∈ Finset.range n
⊢ hI.dpow i b ... | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case refine'_1.a.a
A✝ : Type ?u.19361
inst✝¹ : CommSemiring A✝
I✝ : Ideal A✝
hI✝ : DividedPowers I✝
A : Type u_1
inst✝ : CommRing A
I : Ideal A
hI : DividedPowers I
J : Ideal A
hIJ : hI.isSubDPIdeal (J ⊓ I)
n : ℕ
a b : A
ha : a ∈ I
hb : b ∈ I
hab : a - b ∈ J
... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/SubDPIdeal.lean | DividedPowers.isSubDPIdeal_inf_iff | [118, 1] | [139, 71] | refine' ⟨SemilatticeInf.inf_le_right J I, fun n hn a ha => ⟨_, hI.dpow_mem hn ha.right⟩⟩ | case refine'_2
A✝ : Type ?u.19361
inst✝¹ : CommSemiring A✝
I✝ : Ideal A✝
hI✝ : DividedPowers I✝
A : Type u_1
inst✝ : CommRing A
I : Ideal A
hI : DividedPowers I
J : Ideal A
hIJ : ∀ (n : ℕ) (a b : A), a ∈ I → b ∈ I → a - b ∈ J → hI.dpow n a - hI.dpow n b ∈ J
⊢ hI.isSubDPIdeal (J ⊓ I) | case refine'_2
A✝ : Type ?u.19361
inst✝¹ : CommSemiring A✝
I✝ : Ideal A✝
hI✝ : DividedPowers I✝
A : Type u_1
inst✝ : CommRing A
I : Ideal A
hI : DividedPowers I
J : Ideal A
hIJ : ∀ (n : ℕ) (a b : A), a ∈ I → b ∈ I → a - b ∈ J → hI.dpow n a - hI.dpow n b ∈ J
n : ℕ
hn : n ≠ 0
a : A
ha : a ∈ J ⊓ I
⊢ hI.dpow n a ∈ ↑J | Please generate a tactic in lean4 to solve the state.
STATE:
case refine'_2
A✝ : Type ?u.19361
inst✝¹ : CommSemiring A✝
I✝ : Ideal A✝
hI✝ : DividedPowers I✝
A : Type u_1
inst✝ : CommRing A
I : Ideal A
hI : DividedPowers I
J : Ideal A
hIJ : ∀ (n : ℕ) (a b : A), a ∈ I → b ∈ I → a - b ∈ J → hI.dpow n a - hI.dpow n b ∈ J
⊢... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/SubDPIdeal.lean | DividedPowers.isSubDPIdeal_inf_iff | [118, 1] | [139, 71] | rw [← sub_zero (hI.dpow n a), ← hI.dpow_eval_zero hn] | case refine'_2
A✝ : Type ?u.19361
inst✝¹ : CommSemiring A✝
I✝ : Ideal A✝
hI✝ : DividedPowers I✝
A : Type u_1
inst✝ : CommRing A
I : Ideal A
hI : DividedPowers I
J : Ideal A
hIJ : ∀ (n : ℕ) (a b : A), a ∈ I → b ∈ I → a - b ∈ J → hI.dpow n a - hI.dpow n b ∈ J
n : ℕ
hn : n ≠ 0
a : A
ha : a ∈ J ⊓ I
⊢ hI.dpow n a ∈ ↑J | case refine'_2
A✝ : Type ?u.19361
inst✝¹ : CommSemiring A✝
I✝ : Ideal A✝
hI✝ : DividedPowers I✝
A : Type u_1
inst✝ : CommRing A
I : Ideal A
hI : DividedPowers I
J : Ideal A
hIJ : ∀ (n : ℕ) (a b : A), a ∈ I → b ∈ I → a - b ∈ J → hI.dpow n a - hI.dpow n b ∈ J
n : ℕ
hn : n ≠ 0
a : A
ha : a ∈ J ⊓ I
⊢ hI.dpow n a - hI.dpow ... | Please generate a tactic in lean4 to solve the state.
STATE:
case refine'_2
A✝ : Type ?u.19361
inst✝¹ : CommSemiring A✝
I✝ : Ideal A✝
hI✝ : DividedPowers I✝
A : Type u_1
inst✝ : CommRing A
I : Ideal A
hI : DividedPowers I
J : Ideal A
hIJ : ∀ (n : ℕ) (a b : A), a ∈ I → b ∈ I → a - b ∈ J → hI.dpow n a - hI.dpow n b ∈ J
n... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/SubDPIdeal.lean | DividedPowers.isSubDPIdeal_inf_iff | [118, 1] | [139, 71] | exact hIJ n a 0 ha.right I.zero_mem (J.sub_mem ha.left J.zero_mem) | case refine'_2
A✝ : Type ?u.19361
inst✝¹ : CommSemiring A✝
I✝ : Ideal A✝
hI✝ : DividedPowers I✝
A : Type u_1
inst✝ : CommRing A
I : Ideal A
hI : DividedPowers I
J : Ideal A
hIJ : ∀ (n : ℕ) (a b : A), a ∈ I → b ∈ I → a - b ∈ J → hI.dpow n a - hI.dpow n b ∈ J
n : ℕ
hn : n ≠ 0
a : A
ha : a ∈ J ⊓ I
⊢ hI.dpow n a - hI.dpow ... | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case refine'_2
A✝ : Type ?u.19361
inst✝¹ : CommSemiring A✝
I✝ : Ideal A✝
hI✝ : DividedPowers I✝
A : Type u_1
inst✝ : CommRing A
I : Ideal A
hI : DividedPowers I
J : Ideal A
hIJ : ∀ (n : ℕ) (a b : A), a ∈ I → b ∈ I → a - b ∈ J → hI.dpow n a - hI.dpow n b ∈ J
n... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/SubDPIdeal.lean | DividedPowers.span_isSubDPIdeal_iff | [143, 1] | [175, 64] | constructor | A : Type u_1
inst✝ : CommSemiring A
I : Ideal A
hI : DividedPowers I
S : Set A
hS : S ⊆ ↑I
⊢ hI.isSubDPIdeal (Ideal.span S) ↔ ∀ (n : ℕ), n ≠ 0 → ∀ s ∈ S, hI.dpow n s ∈ Ideal.span S | case mp
A : Type u_1
inst✝ : CommSemiring A
I : Ideal A
hI : DividedPowers I
S : Set A
hS : S ⊆ ↑I
⊢ hI.isSubDPIdeal (Ideal.span S) → ∀ (n : ℕ), n ≠ 0 → ∀ s ∈ S, hI.dpow n s ∈ Ideal.span S
case mpr
A : Type u_1
inst✝ : CommSemiring A
I : Ideal A
hI : DividedPowers I
S : Set A
hS : S ⊆ ↑I
⊢ (∀ (n : ℕ), n ≠ 0 → ∀ s ∈ S,... | Please generate a tactic in lean4 to solve the state.
STATE:
A : Type u_1
inst✝ : CommSemiring A
I : Ideal A
hI : DividedPowers I
S : Set A
hS : S ⊆ ↑I
⊢ hI.isSubDPIdeal (Ideal.span S) ↔ ∀ (n : ℕ), n ≠ 0 → ∀ s ∈ S, hI.dpow n s ∈ Ideal.span S
TACTIC:
|
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/SubDPIdeal.lean | DividedPowers.span_isSubDPIdeal_iff | [143, 1] | [175, 64] | intro hhI h hn s hs | case mp
A : Type u_1
inst✝ : CommSemiring A
I : Ideal A
hI : DividedPowers I
S : Set A
hS : S ⊆ ↑I
⊢ hI.isSubDPIdeal (Ideal.span S) → ∀ (n : ℕ), n ≠ 0 → ∀ s ∈ S, hI.dpow n s ∈ Ideal.span S | case mp
A : Type u_1
inst✝ : CommSemiring A
I : Ideal A
hI : DividedPowers I
S : Set A
hS : S ⊆ ↑I
hhI : hI.isSubDPIdeal (Ideal.span S)
h : ℕ
hn : h ≠ 0
s : A
hs : s ∈ S
⊢ hI.dpow h s ∈ Ideal.span S | Please generate a tactic in lean4 to solve the state.
STATE:
case mp
A : Type u_1
inst✝ : CommSemiring A
I : Ideal A
hI : DividedPowers I
S : Set A
hS : S ⊆ ↑I
⊢ hI.isSubDPIdeal (Ideal.span S) → ∀ (n : ℕ), n ≠ 0 → ∀ s ∈ S, hI.dpow n s ∈ Ideal.span S
TACTIC:
|
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/SubDPIdeal.lean | DividedPowers.span_isSubDPIdeal_iff | [143, 1] | [175, 64] | apply hhI.dpow_mem h hn s (Ideal.subset_span hs) | case mp
A : Type u_1
inst✝ : CommSemiring A
I : Ideal A
hI : DividedPowers I
S : Set A
hS : S ⊆ ↑I
hhI : hI.isSubDPIdeal (Ideal.span S)
h : ℕ
hn : h ≠ 0
s : A
hs : s ∈ S
⊢ hI.dpow h s ∈ Ideal.span S | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case mp
A : Type u_1
inst✝ : CommSemiring A
I : Ideal A
hI : DividedPowers I
S : Set A
hS : S ⊆ ↑I
hhI : hI.isSubDPIdeal (Ideal.span S)
h : ℕ
hn : h ≠ 0
s : A
hs : s ∈ S
⊢ hI.dpow h s ∈ Ideal.span S
TACTIC:
|
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/SubDPIdeal.lean | DividedPowers.span_isSubDPIdeal_iff | [143, 1] | [175, 64] | intro hhI | case mpr
A : Type u_1
inst✝ : CommSemiring A
I : Ideal A
hI : DividedPowers I
S : Set A
hS : S ⊆ ↑I
⊢ (∀ (n : ℕ), n ≠ 0 → ∀ s ∈ S, hI.dpow n s ∈ Ideal.span S) → hI.isSubDPIdeal (Ideal.span S) | case mpr
A : Type u_1
inst✝ : CommSemiring A
I : Ideal A
hI : DividedPowers I
S : Set A
hS : S ⊆ ↑I
hhI : ∀ (n : ℕ), n ≠ 0 → ∀ s ∈ S, hI.dpow n s ∈ Ideal.span S
⊢ hI.isSubDPIdeal (Ideal.span S) | Please generate a tactic in lean4 to solve the state.
STATE:
case mpr
A : Type u_1
inst✝ : CommSemiring A
I : Ideal A
hI : DividedPowers I
S : Set A
hS : S ⊆ ↑I
⊢ (∀ (n : ℕ), n ≠ 0 → ∀ s ∈ S, hI.dpow n s ∈ Ideal.span S) → hI.isSubDPIdeal (Ideal.span S)
TACTIC:
|
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/SubDPIdeal.lean | DividedPowers.span_isSubDPIdeal_iff | [143, 1] | [175, 64] | have hSI := Ideal.span_le.mpr hS | case mpr
A : Type u_1
inst✝ : CommSemiring A
I : Ideal A
hI : DividedPowers I
S : Set A
hS : S ⊆ ↑I
hhI : ∀ (n : ℕ), n ≠ 0 → ∀ s ∈ S, hI.dpow n s ∈ Ideal.span S
⊢ hI.isSubDPIdeal (Ideal.span S) | case mpr
A : Type u_1
inst✝ : CommSemiring A
I : Ideal A
hI : DividedPowers I
S : Set A
hS : S ⊆ ↑I
hhI : ∀ (n : ℕ), n ≠ 0 → ∀ s ∈ S, hI.dpow n s ∈ Ideal.span S
hSI : Ideal.span S ≤ I
⊢ hI.isSubDPIdeal (Ideal.span S) | Please generate a tactic in lean4 to solve the state.
STATE:
case mpr
A : Type u_1
inst✝ : CommSemiring A
I : Ideal A
hI : DividedPowers I
S : Set A
hS : S ⊆ ↑I
hhI : ∀ (n : ℕ), n ≠ 0 → ∀ s ∈ S, hI.dpow n s ∈ Ideal.span S
⊢ hI.isSubDPIdeal (Ideal.span S)
TACTIC:
|
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/SubDPIdeal.lean | DividedPowers.span_isSubDPIdeal_iff | [143, 1] | [175, 64] | apply isSubDPIdeal.mk hSI | case mpr
A : Type u_1
inst✝ : CommSemiring A
I : Ideal A
hI : DividedPowers I
S : Set A
hS : S ⊆ ↑I
hhI : ∀ (n : ℕ), n ≠ 0 → ∀ s ∈ S, hI.dpow n s ∈ Ideal.span S
hSI : Ideal.span S ≤ I
⊢ hI.isSubDPIdeal (Ideal.span S) | case mpr
A : Type u_1
inst✝ : CommSemiring A
I : Ideal A
hI : DividedPowers I
S : Set A
hS : S ⊆ ↑I
hhI : ∀ (n : ℕ), n ≠ 0 → ∀ s ∈ S, hI.dpow n s ∈ Ideal.span S
hSI : Ideal.span S ≤ I
⊢ ∀ (n : ℕ), n ≠ 0 → ∀ j ∈ Ideal.span S, hI.dpow n j ∈ Ideal.span S | Please generate a tactic in lean4 to solve the state.
STATE:
case mpr
A : Type u_1
inst✝ : CommSemiring A
I : Ideal A
hI : DividedPowers I
S : Set A
hS : S ⊆ ↑I
hhI : ∀ (n : ℕ), n ≠ 0 → ∀ s ∈ S, hI.dpow n s ∈ Ideal.span S
hSI : Ideal.span S ≤ I
⊢ hI.isSubDPIdeal (Ideal.span S)
TACTIC:
|
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/SubDPIdeal.lean | DividedPowers.span_isSubDPIdeal_iff | [143, 1] | [175, 64] | intro n hn z hz | case mpr
A : Type u_1
inst✝ : CommSemiring A
I : Ideal A
hI : DividedPowers I
S : Set A
hS : S ⊆ ↑I
hhI : ∀ (n : ℕ), n ≠ 0 → ∀ s ∈ S, hI.dpow n s ∈ Ideal.span S
hSI : Ideal.span S ≤ I
⊢ ∀ (n : ℕ), n ≠ 0 → ∀ j ∈ Ideal.span S, hI.dpow n j ∈ Ideal.span S | case mpr
A : Type u_1
inst✝ : CommSemiring A
I : Ideal A
hI : DividedPowers I
S : Set A
hS : S ⊆ ↑I
hhI : ∀ (n : ℕ), n ≠ 0 → ∀ s ∈ S, hI.dpow n s ∈ Ideal.span S
hSI : Ideal.span S ≤ I
n : ℕ
hn : n ≠ 0
z : A
hz : z ∈ Ideal.span S
⊢ hI.dpow n z ∈ Ideal.span S | Please generate a tactic in lean4 to solve the state.
STATE:
case mpr
A : Type u_1
inst✝ : CommSemiring A
I : Ideal A
hI : DividedPowers I
S : Set A
hS : S ⊆ ↑I
hhI : ∀ (n : ℕ), n ≠ 0 → ∀ s ∈ S, hI.dpow n s ∈ Ideal.span S
hSI : Ideal.span S ≤ I
⊢ ∀ (n : ℕ), n ≠ 0 → ∀ j ∈ Ideal.span S, hI.dpow n j ∈ Ideal.span S
TACTIC:... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/SubDPIdeal.lean | DividedPowers.span_isSubDPIdeal_iff | [143, 1] | [175, 64] | revert n | case mpr
A : Type u_1
inst✝ : CommSemiring A
I : Ideal A
hI : DividedPowers I
S : Set A
hS : S ⊆ ↑I
hhI : ∀ (n : ℕ), n ≠ 0 → ∀ s ∈ S, hI.dpow n s ∈ Ideal.span S
hSI : Ideal.span S ≤ I
n : ℕ
hn : n ≠ 0
z : A
hz : z ∈ Ideal.span S
⊢ hI.dpow n z ∈ Ideal.span S | case mpr
A : Type u_1
inst✝ : CommSemiring A
I : Ideal A
hI : DividedPowers I
S : Set A
hS : S ⊆ ↑I
hhI : ∀ (n : ℕ), n ≠ 0 → ∀ s ∈ S, hI.dpow n s ∈ Ideal.span S
hSI : Ideal.span S ≤ I
z : A
hz : z ∈ Ideal.span S
⊢ ∀ (n : ℕ), n ≠ 0 → hI.dpow n z ∈ Ideal.span S | Please generate a tactic in lean4 to solve the state.
STATE:
case mpr
A : Type u_1
inst✝ : CommSemiring A
I : Ideal A
hI : DividedPowers I
S : Set A
hS : S ⊆ ↑I
hhI : ∀ (n : ℕ), n ≠ 0 → ∀ s ∈ S, hI.dpow n s ∈ Ideal.span S
hSI : Ideal.span S ≤ I
n : ℕ
hn : n ≠ 0
z : A
hz : z ∈ Ideal.span S
⊢ hI.dpow n z ∈ Ideal.span S
T... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/SubDPIdeal.lean | DividedPowers.span_isSubDPIdeal_iff | [143, 1] | [175, 64] | refine' Submodule.span_induction' _ _ _ _ hz | case mpr
A : Type u_1
inst✝ : CommSemiring A
I : Ideal A
hI : DividedPowers I
S : Set A
hS : S ⊆ ↑I
hhI : ∀ (n : ℕ), n ≠ 0 → ∀ s ∈ S, hI.dpow n s ∈ Ideal.span S
hSI : Ideal.span S ≤ I
z : A
hz : z ∈ Ideal.span S
⊢ ∀ (n : ℕ), n ≠ 0 → hI.dpow n z ∈ Ideal.span S | case mpr.refine'_1
A : Type u_1
inst✝ : CommSemiring A
I : Ideal A
hI : DividedPowers I
S : Set A
hS : S ⊆ ↑I
hhI : ∀ (n : ℕ), n ≠ 0 → ∀ s ∈ S, hI.dpow n s ∈ Ideal.span S
hSI : Ideal.span S ≤ I
z : A
hz : z ∈ Ideal.span S
⊢ ∀ x ∈ S, ∀ (n : ℕ), n ≠ 0 → hI.dpow n x ∈ Ideal.span S
case mpr.refine'_2
A : Type u_1
inst✝ : ... | Please generate a tactic in lean4 to solve the state.
STATE:
case mpr
A : Type u_1
inst✝ : CommSemiring A
I : Ideal A
hI : DividedPowers I
S : Set A
hS : S ⊆ ↑I
hhI : ∀ (n : ℕ), n ≠ 0 → ∀ s ∈ S, hI.dpow n s ∈ Ideal.span S
hSI : Ideal.span S ≤ I
z : A
hz : z ∈ Ideal.span S
⊢ ∀ (n : ℕ), n ≠ 0 → hI.dpow n z ∈ Ideal.span S... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/SubDPIdeal.lean | DividedPowers.span_isSubDPIdeal_iff | [143, 1] | [175, 64] | intro s hs n hn | case mpr.refine'_1
A : Type u_1
inst✝ : CommSemiring A
I : Ideal A
hI : DividedPowers I
S : Set A
hS : S ⊆ ↑I
hhI : ∀ (n : ℕ), n ≠ 0 → ∀ s ∈ S, hI.dpow n s ∈ Ideal.span S
hSI : Ideal.span S ≤ I
z : A
hz : z ∈ Ideal.span S
⊢ ∀ x ∈ S, ∀ (n : ℕ), n ≠ 0 → hI.dpow n x ∈ Ideal.span S | case mpr.refine'_1
A : Type u_1
inst✝ : CommSemiring A
I : Ideal A
hI : DividedPowers I
S : Set A
hS : S ⊆ ↑I
hhI : ∀ (n : ℕ), n ≠ 0 → ∀ s ∈ S, hI.dpow n s ∈ Ideal.span S
hSI : Ideal.span S ≤ I
z : A
hz : z ∈ Ideal.span S
s : A
hs : s ∈ S
n : ℕ
hn : n ≠ 0
⊢ hI.dpow n s ∈ Ideal.span S | Please generate a tactic in lean4 to solve the state.
STATE:
case mpr.refine'_1
A : Type u_1
inst✝ : CommSemiring A
I : Ideal A
hI : DividedPowers I
S : Set A
hS : S ⊆ ↑I
hhI : ∀ (n : ℕ), n ≠ 0 → ∀ s ∈ S, hI.dpow n s ∈ Ideal.span S
hSI : Ideal.span S ≤ I
z : A
hz : z ∈ Ideal.span S
⊢ ∀ x ∈ S, ∀ (n : ℕ), n ≠ 0 → hI.dpow... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/SubDPIdeal.lean | DividedPowers.span_isSubDPIdeal_iff | [143, 1] | [175, 64] | exact hhI n hn s hs | case mpr.refine'_1
A : Type u_1
inst✝ : CommSemiring A
I : Ideal A
hI : DividedPowers I
S : Set A
hS : S ⊆ ↑I
hhI : ∀ (n : ℕ), n ≠ 0 → ∀ s ∈ S, hI.dpow n s ∈ Ideal.span S
hSI : Ideal.span S ≤ I
z : A
hz : z ∈ Ideal.span S
s : A
hs : s ∈ S
n : ℕ
hn : n ≠ 0
⊢ hI.dpow n s ∈ Ideal.span S | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case mpr.refine'_1
A : Type u_1
inst✝ : CommSemiring A
I : Ideal A
hI : DividedPowers I
S : Set A
hS : S ⊆ ↑I
hhI : ∀ (n : ℕ), n ≠ 0 → ∀ s ∈ S, hI.dpow n s ∈ Ideal.span S
hSI : Ideal.span S ≤ I
z : A
hz : z ∈ Ideal.span S
s : A
hs : s ∈ S
n : ℕ
hn : n ≠ 0
⊢ h... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/SubDPIdeal.lean | DividedPowers.span_isSubDPIdeal_iff | [143, 1] | [175, 64] | intro n hn | case mpr.refine'_2
A : Type u_1
inst✝ : CommSemiring A
I : Ideal A
hI : DividedPowers I
S : Set A
hS : S ⊆ ↑I
hhI : ∀ (n : ℕ), n ≠ 0 → ∀ s ∈ S, hI.dpow n s ∈ Ideal.span S
hSI : Ideal.span S ≤ I
z : A
hz : z ∈ Ideal.span S
⊢ ∀ (n : ℕ), n ≠ 0 → hI.dpow n 0 ∈ Ideal.span S | case mpr.refine'_2
A : Type u_1
inst✝ : CommSemiring A
I : Ideal A
hI : DividedPowers I
S : Set A
hS : S ⊆ ↑I
hhI : ∀ (n : ℕ), n ≠ 0 → ∀ s ∈ S, hI.dpow n s ∈ Ideal.span S
hSI : Ideal.span S ≤ I
z : A
hz : z ∈ Ideal.span S
n : ℕ
hn : n ≠ 0
⊢ hI.dpow n 0 ∈ Ideal.span S | Please generate a tactic in lean4 to solve the state.
STATE:
case mpr.refine'_2
A : Type u_1
inst✝ : CommSemiring A
I : Ideal A
hI : DividedPowers I
S : Set A
hS : S ⊆ ↑I
hhI : ∀ (n : ℕ), n ≠ 0 → ∀ s ∈ S, hI.dpow n s ∈ Ideal.span S
hSI : Ideal.span S ≤ I
z : A
hz : z ∈ Ideal.span S
⊢ ∀ (n : ℕ), n ≠ 0 → hI.dpow n 0 ∈ Id... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/SubDPIdeal.lean | DividedPowers.span_isSubDPIdeal_iff | [143, 1] | [175, 64] | rw [hI.dpow_eval_zero hn] | case mpr.refine'_2
A : Type u_1
inst✝ : CommSemiring A
I : Ideal A
hI : DividedPowers I
S : Set A
hS : S ⊆ ↑I
hhI : ∀ (n : ℕ), n ≠ 0 → ∀ s ∈ S, hI.dpow n s ∈ Ideal.span S
hSI : Ideal.span S ≤ I
z : A
hz : z ∈ Ideal.span S
n : ℕ
hn : n ≠ 0
⊢ hI.dpow n 0 ∈ Ideal.span S | case mpr.refine'_2
A : Type u_1
inst✝ : CommSemiring A
I : Ideal A
hI : DividedPowers I
S : Set A
hS : S ⊆ ↑I
hhI : ∀ (n : ℕ), n ≠ 0 → ∀ s ∈ S, hI.dpow n s ∈ Ideal.span S
hSI : Ideal.span S ≤ I
z : A
hz : z ∈ Ideal.span S
n : ℕ
hn : n ≠ 0
⊢ 0 ∈ Ideal.span S | Please generate a tactic in lean4 to solve the state.
STATE:
case mpr.refine'_2
A : Type u_1
inst✝ : CommSemiring A
I : Ideal A
hI : DividedPowers I
S : Set A
hS : S ⊆ ↑I
hhI : ∀ (n : ℕ), n ≠ 0 → ∀ s ∈ S, hI.dpow n s ∈ Ideal.span S
hSI : Ideal.span S ≤ I
z : A
hz : z ∈ Ideal.span S
n : ℕ
hn : n ≠ 0
⊢ hI.dpow n 0 ∈ Idea... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/SubDPIdeal.lean | DividedPowers.span_isSubDPIdeal_iff | [143, 1] | [175, 64] | apply Ideal.zero_mem _ | case mpr.refine'_2
A : Type u_1
inst✝ : CommSemiring A
I : Ideal A
hI : DividedPowers I
S : Set A
hS : S ⊆ ↑I
hhI : ∀ (n : ℕ), n ≠ 0 → ∀ s ∈ S, hI.dpow n s ∈ Ideal.span S
hSI : Ideal.span S ≤ I
z : A
hz : z ∈ Ideal.span S
n : ℕ
hn : n ≠ 0
⊢ 0 ∈ Ideal.span S | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case mpr.refine'_2
A : Type u_1
inst✝ : CommSemiring A
I : Ideal A
hI : DividedPowers I
S : Set A
hS : S ⊆ ↑I
hhI : ∀ (n : ℕ), n ≠ 0 → ∀ s ∈ S, hI.dpow n s ∈ Ideal.span S
hSI : Ideal.span S ≤ I
z : A
hz : z ∈ Ideal.span S
n : ℕ
hn : n ≠ 0
⊢ 0 ∈ Ideal.span S
T... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/SubDPIdeal.lean | DividedPowers.span_isSubDPIdeal_iff | [143, 1] | [175, 64] | rintro x hxI y hyI hx hy n hn | case mpr.refine'_3
A : Type u_1
inst✝ : CommSemiring A
I : Ideal A
hI : DividedPowers I
S : Set A
hS : S ⊆ ↑I
hhI : ∀ (n : ℕ), n ≠ 0 → ∀ s ∈ S, hI.dpow n s ∈ Ideal.span S
hSI : Ideal.span S ≤ I
z : A
hz : z ∈ Ideal.span S
⊢ ∀ x ∈ Submodule.span A S,
∀ y ∈ Submodule.span A S,
(∀ (n : ℕ), n ≠ 0 → hI.dpow n x ∈ ... | case mpr.refine'_3
A : Type u_1
inst✝ : CommSemiring A
I : Ideal A
hI : DividedPowers I
S : Set A
hS : S ⊆ ↑I
hhI : ∀ (n : ℕ), n ≠ 0 → ∀ s ∈ S, hI.dpow n s ∈ Ideal.span S
hSI : Ideal.span S ≤ I
z : A
hz : z ∈ Ideal.span S
x : A
hxI : x ∈ Submodule.span A S
y : A
hyI : y ∈ Submodule.span A S
hx : ∀ (n : ℕ), n ≠ 0 → hI.d... | Please generate a tactic in lean4 to solve the state.
STATE:
case mpr.refine'_3
A : Type u_1
inst✝ : CommSemiring A
I : Ideal A
hI : DividedPowers I
S : Set A
hS : S ⊆ ↑I
hhI : ∀ (n : ℕ), n ≠ 0 → ∀ s ∈ S, hI.dpow n s ∈ Ideal.span S
hSI : Ideal.span S ≤ I
z : A
hz : z ∈ Ideal.span S
⊢ ∀ x ∈ Submodule.span A S,
∀ y ∈... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/SubDPIdeal.lean | DividedPowers.span_isSubDPIdeal_iff | [143, 1] | [175, 64] | rw [hI.dpow_add' n (hSI hxI) (hSI hyI)] | case mpr.refine'_3
A : Type u_1
inst✝ : CommSemiring A
I : Ideal A
hI : DividedPowers I
S : Set A
hS : S ⊆ ↑I
hhI : ∀ (n : ℕ), n ≠ 0 → ∀ s ∈ S, hI.dpow n s ∈ Ideal.span S
hSI : Ideal.span S ≤ I
z : A
hz : z ∈ Ideal.span S
x : A
hxI : x ∈ Submodule.span A S
y : A
hyI : y ∈ Submodule.span A S
hx : ∀ (n : ℕ), n ≠ 0 → hI.d... | case mpr.refine'_3
A : Type u_1
inst✝ : CommSemiring A
I : Ideal A
hI : DividedPowers I
S : Set A
hS : S ⊆ ↑I
hhI : ∀ (n : ℕ), n ≠ 0 → ∀ s ∈ S, hI.dpow n s ∈ Ideal.span S
hSI : Ideal.span S ≤ I
z : A
hz : z ∈ Ideal.span S
x : A
hxI : x ∈ Submodule.span A S
y : A
hyI : y ∈ Submodule.span A S
hx : ∀ (n : ℕ), n ≠ 0 → hI.d... | Please generate a tactic in lean4 to solve the state.
STATE:
case mpr.refine'_3
A : Type u_1
inst✝ : CommSemiring A
I : Ideal A
hI : DividedPowers I
S : Set A
hS : S ⊆ ↑I
hhI : ∀ (n : ℕ), n ≠ 0 → ∀ s ∈ S, hI.dpow n s ∈ Ideal.span S
hSI : Ideal.span S ≤ I
z : A
hz : z ∈ Ideal.span S
x : A
hxI : x ∈ Submodule.span A S
y ... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/SubDPIdeal.lean | DividedPowers.span_isSubDPIdeal_iff | [143, 1] | [175, 64] | apply Submodule.sum_mem (Ideal.span S) | case mpr.refine'_3
A : Type u_1
inst✝ : CommSemiring A
I : Ideal A
hI : DividedPowers I
S : Set A
hS : S ⊆ ↑I
hhI : ∀ (n : ℕ), n ≠ 0 → ∀ s ∈ S, hI.dpow n s ∈ Ideal.span S
hSI : Ideal.span S ≤ I
z : A
hz : z ∈ Ideal.span S
x : A
hxI : x ∈ Submodule.span A S
y : A
hyI : y ∈ Submodule.span A S
hx : ∀ (n : ℕ), n ≠ 0 → hI.d... | case mpr.refine'_3
A : Type u_1
inst✝ : CommSemiring A
I : Ideal A
hI : DividedPowers I
S : Set A
hS : S ⊆ ↑I
hhI : ∀ (n : ℕ), n ≠ 0 → ∀ s ∈ S, hI.dpow n s ∈ Ideal.span S
hSI : Ideal.span S ≤ I
z : A
hz : z ∈ Ideal.span S
x : A
hxI : x ∈ Submodule.span A S
y : A
hyI : y ∈ Submodule.span A S
hx : ∀ (n : ℕ), n ≠ 0 → hI.d... | Please generate a tactic in lean4 to solve the state.
STATE:
case mpr.refine'_3
A : Type u_1
inst✝ : CommSemiring A
I : Ideal A
hI : DividedPowers I
S : Set A
hS : S ⊆ ↑I
hhI : ∀ (n : ℕ), n ≠ 0 → ∀ s ∈ S, hI.dpow n s ∈ Ideal.span S
hSI : Ideal.span S ≤ I
z : A
hz : z ∈ Ideal.span S
x : A
hxI : x ∈ Submodule.span A S
y ... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/SubDPIdeal.lean | DividedPowers.span_isSubDPIdeal_iff | [143, 1] | [175, 64] | intro m _ | case mpr.refine'_3
A : Type u_1
inst✝ : CommSemiring A
I : Ideal A
hI : DividedPowers I
S : Set A
hS : S ⊆ ↑I
hhI : ∀ (n : ℕ), n ≠ 0 → ∀ s ∈ S, hI.dpow n s ∈ Ideal.span S
hSI : Ideal.span S ≤ I
z : A
hz : z ∈ Ideal.span S
x : A
hxI : x ∈ Submodule.span A S
y : A
hyI : y ∈ Submodule.span A S
hx : ∀ (n : ℕ), n ≠ 0 → hI.d... | case mpr.refine'_3
A : Type u_1
inst✝ : CommSemiring A
I : Ideal A
hI : DividedPowers I
S : Set A
hS : S ⊆ ↑I
hhI : ∀ (n : ℕ), n ≠ 0 → ∀ s ∈ S, hI.dpow n s ∈ Ideal.span S
hSI : Ideal.span S ≤ I
z : A
hz : z ∈ Ideal.span S
x : A
hxI : x ∈ Submodule.span A S
y : A
hyI : y ∈ Submodule.span A S
hx : ∀ (n : ℕ), n ≠ 0 → hI.d... | Please generate a tactic in lean4 to solve the state.
STATE:
case mpr.refine'_3
A : Type u_1
inst✝ : CommSemiring A
I : Ideal A
hI : DividedPowers I
S : Set A
hS : S ⊆ ↑I
hhI : ∀ (n : ℕ), n ≠ 0 → ∀ s ∈ S, hI.dpow n s ∈ Ideal.span S
hSI : Ideal.span S ≤ I
z : A
hz : z ∈ Ideal.span S
x : A
hxI : x ∈ Submodule.span A S
y ... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/SubDPIdeal.lean | DividedPowers.span_isSubDPIdeal_iff | [143, 1] | [175, 64] | by_cases hm0 : m = 0 | case mpr.refine'_3
A : Type u_1
inst✝ : CommSemiring A
I : Ideal A
hI : DividedPowers I
S : Set A
hS : S ⊆ ↑I
hhI : ∀ (n : ℕ), n ≠ 0 → ∀ s ∈ S, hI.dpow n s ∈ Ideal.span S
hSI : Ideal.span S ≤ I
z : A
hz : z ∈ Ideal.span S
x : A
hxI : x ∈ Submodule.span A S
y : A
hyI : y ∈ Submodule.span A S
hx : ∀ (n : ℕ), n ≠ 0 → hI.d... | case pos
A : Type u_1
inst✝ : CommSemiring A
I : Ideal A
hI : DividedPowers I
S : Set A
hS : S ⊆ ↑I
hhI : ∀ (n : ℕ), n ≠ 0 → ∀ s ∈ S, hI.dpow n s ∈ Ideal.span S
hSI : Ideal.span S ≤ I
z : A
hz : z ∈ Ideal.span S
x : A
hxI : x ∈ Submodule.span A S
y : A
hyI : y ∈ Submodule.span A S
hx : ∀ (n : ℕ), n ≠ 0 → hI.dpow n x ∈ ... | Please generate a tactic in lean4 to solve the state.
STATE:
case mpr.refine'_3
A : Type u_1
inst✝ : CommSemiring A
I : Ideal A
hI : DividedPowers I
S : Set A
hS : S ⊆ ↑I
hhI : ∀ (n : ℕ), n ≠ 0 → ∀ s ∈ S, hI.dpow n s ∈ Ideal.span S
hSI : Ideal.span S ≤ I
z : A
hz : z ∈ Ideal.span S
x : A
hxI : x ∈ Submodule.span A S
y ... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/SubDPIdeal.lean | DividedPowers.span_isSubDPIdeal_iff | [143, 1] | [175, 64] | rw [hm0] | case pos
A : Type u_1
inst✝ : CommSemiring A
I : Ideal A
hI : DividedPowers I
S : Set A
hS : S ⊆ ↑I
hhI : ∀ (n : ℕ), n ≠ 0 → ∀ s ∈ S, hI.dpow n s ∈ Ideal.span S
hSI : Ideal.span S ≤ I
z : A
hz : z ∈ Ideal.span S
x : A
hxI : x ∈ Submodule.span A S
y : A
hyI : y ∈ Submodule.span A S
hx : ∀ (n : ℕ), n ≠ 0 → hI.dpow n x ∈ ... | case pos
A : Type u_1
inst✝ : CommSemiring A
I : Ideal A
hI : DividedPowers I
S : Set A
hS : S ⊆ ↑I
hhI : ∀ (n : ℕ), n ≠ 0 → ∀ s ∈ S, hI.dpow n s ∈ Ideal.span S
hSI : Ideal.span S ≤ I
z : A
hz : z ∈ Ideal.span S
x : A
hxI : x ∈ Submodule.span A S
y : A
hyI : y ∈ Submodule.span A S
hx : ∀ (n : ℕ), n ≠ 0 → hI.dpow n x ∈ ... | Please generate a tactic in lean4 to solve the state.
STATE:
case pos
A : Type u_1
inst✝ : CommSemiring A
I : Ideal A
hI : DividedPowers I
S : Set A
hS : S ⊆ ↑I
hhI : ∀ (n : ℕ), n ≠ 0 → ∀ s ∈ S, hI.dpow n s ∈ Ideal.span S
hSI : Ideal.span S ≤ I
z : A
hz : z ∈ Ideal.span S
x : A
hxI : x ∈ Submodule.span A S
y : A
hyI : ... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/SubDPIdeal.lean | DividedPowers.span_isSubDPIdeal_iff | [143, 1] | [175, 64] | exact Ideal.mul_mem_left (Ideal.span S) _ (hy n hn) | case pos
A : Type u_1
inst✝ : CommSemiring A
I : Ideal A
hI : DividedPowers I
S : Set A
hS : S ⊆ ↑I
hhI : ∀ (n : ℕ), n ≠ 0 → ∀ s ∈ S, hI.dpow n s ∈ Ideal.span S
hSI : Ideal.span S ≤ I
z : A
hz : z ∈ Ideal.span S
x : A
hxI : x ∈ Submodule.span A S
y : A
hyI : y ∈ Submodule.span A S
hx : ∀ (n : ℕ), n ≠ 0 → hI.dpow n x ∈ ... | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case pos
A : Type u_1
inst✝ : CommSemiring A
I : Ideal A
hI : DividedPowers I
S : Set A
hS : S ⊆ ↑I
hhI : ∀ (n : ℕ), n ≠ 0 → ∀ s ∈ S, hI.dpow n s ∈ Ideal.span S
hSI : Ideal.span S ≤ I
z : A
hz : z ∈ Ideal.span S
x : A
hxI : x ∈ Submodule.span A S
y : A
hyI : ... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/SubDPIdeal.lean | DividedPowers.span_isSubDPIdeal_iff | [143, 1] | [175, 64] | exact Ideal.mul_mem_right _ (Ideal.span S) (hx m hm0) | case neg
A : Type u_1
inst✝ : CommSemiring A
I : Ideal A
hI : DividedPowers I
S : Set A
hS : S ⊆ ↑I
hhI : ∀ (n : ℕ), n ≠ 0 → ∀ s ∈ S, hI.dpow n s ∈ Ideal.span S
hSI : Ideal.span S ≤ I
z : A
hz : z ∈ Ideal.span S
x : A
hxI : x ∈ Submodule.span A S
y : A
hyI : y ∈ Submodule.span A S
hx : ∀ (n : ℕ), n ≠ 0 → hI.dpow n x ∈ ... | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case neg
A : Type u_1
inst✝ : CommSemiring A
I : Ideal A
hI : DividedPowers I
S : Set A
hS : S ⊆ ↑I
hhI : ∀ (n : ℕ), n ≠ 0 → ∀ s ∈ S, hI.dpow n s ∈ Ideal.span S
hSI : Ideal.span S ≤ I
z : A
hz : z ∈ Ideal.span S
x : A
hxI : x ∈ Submodule.span A S
y : A
hyI : ... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/SubDPIdeal.lean | DividedPowers.span_isSubDPIdeal_iff | [143, 1] | [175, 64] | intro a x hxI hx n hn | case mpr.refine'_4
A : Type u_1
inst✝ : CommSemiring A
I : Ideal A
hI : DividedPowers I
S : Set A
hS : S ⊆ ↑I
hhI : ∀ (n : ℕ), n ≠ 0 → ∀ s ∈ S, hI.dpow n s ∈ Ideal.span S
hSI : Ideal.span S ≤ I
z : A
hz : z ∈ Ideal.span S
⊢ ∀ (a x : A),
x ∈ Submodule.span A S →
(∀ (n : ℕ), n ≠ 0 → hI.dpow n x ∈ Ideal.span S) ... | case mpr.refine'_4
A : Type u_1
inst✝ : CommSemiring A
I : Ideal A
hI : DividedPowers I
S : Set A
hS : S ⊆ ↑I
hhI : ∀ (n : ℕ), n ≠ 0 → ∀ s ∈ S, hI.dpow n s ∈ Ideal.span S
hSI : Ideal.span S ≤ I
z : A
hz : z ∈ Ideal.span S
a x : A
hxI : x ∈ Submodule.span A S
hx : ∀ (n : ℕ), n ≠ 0 → hI.dpow n x ∈ Ideal.span S
n : ℕ
hn :... | Please generate a tactic in lean4 to solve the state.
STATE:
case mpr.refine'_4
A : Type u_1
inst✝ : CommSemiring A
I : Ideal A
hI : DividedPowers I
S : Set A
hS : S ⊆ ↑I
hhI : ∀ (n : ℕ), n ≠ 0 → ∀ s ∈ S, hI.dpow n s ∈ Ideal.span S
hSI : Ideal.span S ≤ I
z : A
hz : z ∈ Ideal.span S
⊢ ∀ (a x : A),
x ∈ Submodule.span... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/SubDPIdeal.lean | DividedPowers.span_isSubDPIdeal_iff | [143, 1] | [175, 64] | simp only [Algebra.id.smul_eq_mul] | case mpr.refine'_4
A : Type u_1
inst✝ : CommSemiring A
I : Ideal A
hI : DividedPowers I
S : Set A
hS : S ⊆ ↑I
hhI : ∀ (n : ℕ), n ≠ 0 → ∀ s ∈ S, hI.dpow n s ∈ Ideal.span S
hSI : Ideal.span S ≤ I
z : A
hz : z ∈ Ideal.span S
a x : A
hxI : x ∈ Submodule.span A S
hx : ∀ (n : ℕ), n ≠ 0 → hI.dpow n x ∈ Ideal.span S
n : ℕ
hn :... | case mpr.refine'_4
A : Type u_1
inst✝ : CommSemiring A
I : Ideal A
hI : DividedPowers I
S : Set A
hS : S ⊆ ↑I
hhI : ∀ (n : ℕ), n ≠ 0 → ∀ s ∈ S, hI.dpow n s ∈ Ideal.span S
hSI : Ideal.span S ≤ I
z : A
hz : z ∈ Ideal.span S
a x : A
hxI : x ∈ Submodule.span A S
hx : ∀ (n : ℕ), n ≠ 0 → hI.dpow n x ∈ Ideal.span S
n : ℕ
hn :... | Please generate a tactic in lean4 to solve the state.
STATE:
case mpr.refine'_4
A : Type u_1
inst✝ : CommSemiring A
I : Ideal A
hI : DividedPowers I
S : Set A
hS : S ⊆ ↑I
hhI : ∀ (n : ℕ), n ≠ 0 → ∀ s ∈ S, hI.dpow n s ∈ Ideal.span S
hSI : Ideal.span S ≤ I
z : A
hz : z ∈ Ideal.span S
a x : A
hxI : x ∈ Submodule.span A S
... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/SubDPIdeal.lean | DividedPowers.span_isSubDPIdeal_iff | [143, 1] | [175, 64] | rw [hI.dpow_smul n (hSI hxI)] | case mpr.refine'_4
A : Type u_1
inst✝ : CommSemiring A
I : Ideal A
hI : DividedPowers I
S : Set A
hS : S ⊆ ↑I
hhI : ∀ (n : ℕ), n ≠ 0 → ∀ s ∈ S, hI.dpow n s ∈ Ideal.span S
hSI : Ideal.span S ≤ I
z : A
hz : z ∈ Ideal.span S
a x : A
hxI : x ∈ Submodule.span A S
hx : ∀ (n : ℕ), n ≠ 0 → hI.dpow n x ∈ Ideal.span S
n : ℕ
hn :... | case mpr.refine'_4
A : Type u_1
inst✝ : CommSemiring A
I : Ideal A
hI : DividedPowers I
S : Set A
hS : S ⊆ ↑I
hhI : ∀ (n : ℕ), n ≠ 0 → ∀ s ∈ S, hI.dpow n s ∈ Ideal.span S
hSI : Ideal.span S ≤ I
z : A
hz : z ∈ Ideal.span S
a x : A
hxI : x ∈ Submodule.span A S
hx : ∀ (n : ℕ), n ≠ 0 → hI.dpow n x ∈ Ideal.span S
n : ℕ
hn :... | Please generate a tactic in lean4 to solve the state.
STATE:
case mpr.refine'_4
A : Type u_1
inst✝ : CommSemiring A
I : Ideal A
hI : DividedPowers I
S : Set A
hS : S ⊆ ↑I
hhI : ∀ (n : ℕ), n ≠ 0 → ∀ s ∈ S, hI.dpow n s ∈ Ideal.span S
hSI : Ideal.span S ≤ I
z : A
hz : z ∈ Ideal.span S
a x : A
hxI : x ∈ Submodule.span A S
... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/SubDPIdeal.lean | DividedPowers.span_isSubDPIdeal_iff | [143, 1] | [175, 64] | exact Ideal.mul_mem_left (Ideal.span S) (a ^ n) (hx n hn) | case mpr.refine'_4
A : Type u_1
inst✝ : CommSemiring A
I : Ideal A
hI : DividedPowers I
S : Set A
hS : S ⊆ ↑I
hhI : ∀ (n : ℕ), n ≠ 0 → ∀ s ∈ S, hI.dpow n s ∈ Ideal.span S
hSI : Ideal.span S ≤ I
z : A
hz : z ∈ Ideal.span S
a x : A
hxI : x ∈ Submodule.span A S
hx : ∀ (n : ℕ), n ≠ 0 → hI.dpow n x ∈ Ideal.span S
n : ℕ
hn :... | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case mpr.refine'_4
A : Type u_1
inst✝ : CommSemiring A
I : Ideal A
hI : DividedPowers I
S : Set A
hS : S ⊆ ↑I
hhI : ∀ (n : ℕ), n ≠ 0 → ∀ s ∈ S, hI.dpow n s ∈ Ideal.span S
hSI : Ideal.span S ≤ I
z : A
hz : z ∈ Ideal.span S
a x : A
hxI : x ∈ Submodule.span A S
... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/SubDPIdeal.lean | DividedPowers.generated_dpow_isSubIdeal | [178, 1] | [184, 31] | rw [Ideal.span_le] | A : Type u_1
inst✝ : CommSemiring A
I : Ideal A
hI : DividedPowers I
S : Set A
hS : S ⊆ ↑I
⊢ Ideal.span {y | ∃ n, ∃ (_ : n ≠ 0), ∃ x, ∃ (_ : x ∈ S), y = hI.dpow n x} ≤ I | A : Type u_1
inst✝ : CommSemiring A
I : Ideal A
hI : DividedPowers I
S : Set A
hS : S ⊆ ↑I
⊢ {y | ∃ n, ∃ (_ : n ≠ 0), ∃ x, ∃ (_ : x ∈ S), y = hI.dpow n x} ⊆ ↑I | Please generate a tactic in lean4 to solve the state.
STATE:
A : Type u_1
inst✝ : CommSemiring A
I : Ideal A
hI : DividedPowers I
S : Set A
hS : S ⊆ ↑I
⊢ Ideal.span {y | ∃ n, ∃ (_ : n ≠ 0), ∃ x, ∃ (_ : x ∈ S), y = hI.dpow n x} ≤ I
TACTIC:
|
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/SubDPIdeal.lean | DividedPowers.generated_dpow_isSubIdeal | [178, 1] | [184, 31] | rintro y ⟨n, hn, x, hx, hxy⟩ | A : Type u_1
inst✝ : CommSemiring A
I : Ideal A
hI : DividedPowers I
S : Set A
hS : S ⊆ ↑I
⊢ {y | ∃ n, ∃ (_ : n ≠ 0), ∃ x, ∃ (_ : x ∈ S), y = hI.dpow n x} ⊆ ↑I | case intro.intro.intro.intro
A : Type u_1
inst✝ : CommSemiring A
I : Ideal A
hI : DividedPowers I
S : Set A
hS : S ⊆ ↑I
y : A
n : ℕ
hn : n ≠ 0
x : A
hx : x ∈ S
hxy : y = hI.dpow n x
⊢ y ∈ ↑I | Please generate a tactic in lean4 to solve the state.
STATE:
A : Type u_1
inst✝ : CommSemiring A
I : Ideal A
hI : DividedPowers I
S : Set A
hS : S ⊆ ↑I
⊢ {y | ∃ n, ∃ (_ : n ≠ 0), ∃ x, ∃ (_ : x ∈ S), y = hI.dpow n x} ⊆ ↑I
TACTIC:
|
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/SubDPIdeal.lean | DividedPowers.generated_dpow_isSubIdeal | [178, 1] | [184, 31] | rw [hxy] | case intro.intro.intro.intro
A : Type u_1
inst✝ : CommSemiring A
I : Ideal A
hI : DividedPowers I
S : Set A
hS : S ⊆ ↑I
y : A
n : ℕ
hn : n ≠ 0
x : A
hx : x ∈ S
hxy : y = hI.dpow n x
⊢ y ∈ ↑I | case intro.intro.intro.intro
A : Type u_1
inst✝ : CommSemiring A
I : Ideal A
hI : DividedPowers I
S : Set A
hS : S ⊆ ↑I
y : A
n : ℕ
hn : n ≠ 0
x : A
hx : x ∈ S
hxy : y = hI.dpow n x
⊢ hI.dpow n x ∈ ↑I | Please generate a tactic in lean4 to solve the state.
STATE:
case intro.intro.intro.intro
A : Type u_1
inst✝ : CommSemiring A
I : Ideal A
hI : DividedPowers I
S : Set A
hS : S ⊆ ↑I
y : A
n : ℕ
hn : n ≠ 0
x : A
hx : x ∈ S
hxy : y = hI.dpow n x
⊢ y ∈ ↑I
TACTIC:
|
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/SubDPIdeal.lean | DividedPowers.generated_dpow_isSubIdeal | [178, 1] | [184, 31] | exact hI.dpow_mem hn (hS hx) | case intro.intro.intro.intro
A : Type u_1
inst✝ : CommSemiring A
I : Ideal A
hI : DividedPowers I
S : Set A
hS : S ⊆ ↑I
y : A
n : ℕ
hn : n ≠ 0
x : A
hx : x ∈ S
hxy : y = hI.dpow n x
⊢ hI.dpow n x ∈ ↑I | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case intro.intro.intro.intro
A : Type u_1
inst✝ : CommSemiring A
I : Ideal A
hI : DividedPowers I
S : Set A
hS : S ⊆ ↑I
y : A
n : ℕ
hn : n ≠ 0
x : A
hx : x ∈ S
hxy : y = hI.dpow n x
⊢ hI.dpow n x ∈ ↑I
TACTIC:
|
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/SubDPIdeal.lean | DividedPowers.isSubDPIdeal_sup | [187, 1] | [198, 50] | rw [← J.span_eq, ← K.span_eq, ← Ideal.span_union, span_isSubDPIdeal_iff] | A : Type u_1
inst✝ : CommSemiring A
I : Ideal A
hI : DividedPowers I
J K : Ideal A
hJ : hI.isSubDPIdeal J
hK : hI.isSubDPIdeal K
⊢ hI.isSubDPIdeal (J ⊔ K) | A : Type u_1
inst✝ : CommSemiring A
I : Ideal A
hI : DividedPowers I
J K : Ideal A
hJ : hI.isSubDPIdeal J
hK : hI.isSubDPIdeal K
⊢ ∀ (n : ℕ), n ≠ 0 → ∀ s ∈ ↑J ∪ ↑K, hI.dpow n s ∈ Ideal.span (↑J ∪ ↑K)
case hS
A : Type u_1
inst✝ : CommSemiring A
I : Ideal A
hI : DividedPowers I
J K : Ideal A
hJ : hI.isSubDPIdeal J
hK : ... | Please generate a tactic in lean4 to solve the state.
STATE:
A : Type u_1
inst✝ : CommSemiring A
I : Ideal A
hI : DividedPowers I
J K : Ideal A
hJ : hI.isSubDPIdeal J
hK : hI.isSubDPIdeal K
⊢ hI.isSubDPIdeal (J ⊔ K)
TACTIC:
|
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/SubDPIdeal.lean | DividedPowers.isSubDPIdeal_sup | [187, 1] | [198, 50] | . rw [Set.union_subset_iff]; exact ⟨hJ.1, hK.1⟩ | case hS
A : Type u_1
inst✝ : CommSemiring A
I : Ideal A
hI : DividedPowers I
J K : Ideal A
hJ : hI.isSubDPIdeal J
hK : hI.isSubDPIdeal K
⊢ ↑J ∪ ↑K ⊆ ↑I | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case hS
A : Type u_1
inst✝ : CommSemiring A
I : Ideal A
hI : DividedPowers I
J K : Ideal A
hJ : hI.isSubDPIdeal J
hK : hI.isSubDPIdeal K
⊢ ↑J ∪ ↑K ⊆ ↑I
TACTIC:
|
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/SubDPIdeal.lean | DividedPowers.isSubDPIdeal_sup | [187, 1] | [198, 50] | intro n hn a ha | A : Type u_1
inst✝ : CommSemiring A
I : Ideal A
hI : DividedPowers I
J K : Ideal A
hJ : hI.isSubDPIdeal J
hK : hI.isSubDPIdeal K
⊢ ∀ (n : ℕ), n ≠ 0 → ∀ s ∈ ↑J ∪ ↑K, hI.dpow n s ∈ Ideal.span (↑J ∪ ↑K) | A : Type u_1
inst✝ : CommSemiring A
I : Ideal A
hI : DividedPowers I
J K : Ideal A
hJ : hI.isSubDPIdeal J
hK : hI.isSubDPIdeal K
n : ℕ
hn : n ≠ 0
a : A
ha : a ∈ ↑J ∪ ↑K
⊢ hI.dpow n a ∈ Ideal.span (↑J ∪ ↑K) | Please generate a tactic in lean4 to solve the state.
STATE:
A : Type u_1
inst✝ : CommSemiring A
I : Ideal A
hI : DividedPowers I
J K : Ideal A
hJ : hI.isSubDPIdeal J
hK : hI.isSubDPIdeal K
⊢ ∀ (n : ℕ), n ≠ 0 → ∀ s ∈ ↑J ∪ ↑K, hI.dpow n s ∈ Ideal.span (↑J ∪ ↑K)
TACTIC:
|
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/SubDPIdeal.lean | DividedPowers.isSubDPIdeal_sup | [187, 1] | [198, 50] | cases' ha with ha ha | A : Type u_1
inst✝ : CommSemiring A
I : Ideal A
hI : DividedPowers I
J K : Ideal A
hJ : hI.isSubDPIdeal J
hK : hI.isSubDPIdeal K
n : ℕ
hn : n ≠ 0
a : A
ha : a ∈ ↑J ∪ ↑K
⊢ hI.dpow n a ∈ Ideal.span (↑J ∪ ↑K) | case inl
A : Type u_1
inst✝ : CommSemiring A
I : Ideal A
hI : DividedPowers I
J K : Ideal A
hJ : hI.isSubDPIdeal J
hK : hI.isSubDPIdeal K
n : ℕ
hn : n ≠ 0
a : A
ha : a ∈ ↑J
⊢ hI.dpow n a ∈ Ideal.span (↑J ∪ ↑K)
case inr
A : Type u_1
inst✝ : CommSemiring A
I : Ideal A
hI : DividedPowers I
J K : Ideal A
hJ : hI.isSubDPId... | Please generate a tactic in lean4 to solve the state.
STATE:
A : Type u_1
inst✝ : CommSemiring A
I : Ideal A
hI : DividedPowers I
J K : Ideal A
hJ : hI.isSubDPIdeal J
hK : hI.isSubDPIdeal K
n : ℕ
hn : n ≠ 0
a : A
ha : a ∈ ↑J ∪ ↑K
⊢ hI.dpow n a ∈ Ideal.span (↑J ∪ ↑K)
TACTIC:
|
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