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/MvPowerSeries/Basic.lean | MvPowerSeries.coeff_eq_zero_of_constantCoeff_nilpotent | [33, 1] | [70, 22] | simp only [hs_def, mem_filter] at hi | σ : Type u_1
inst✝³ : DecidableEq σ
ι : Type u_2
inst✝² : DecidableEq (ι → σ →₀ ℕ)
α : Type u_3
inst✝¹ : CommSemiring α
inst✝ : DecidableEq (ℕ → σ →₀ ℕ)
f : MvPowerSeries σ α
m : ℕ
hf : (constantCoeff σ α) f ^ m = 0
d : σ →₀ ℕ
n : ℕ
hn : m + degree d ≤ n
k : ℕ →₀ σ →₀ ℕ
hk : k.support ⊆ range n ∧ (range n).sum ⇑k = d
s... | σ : Type u_1
inst✝³ : DecidableEq σ
ι : Type u_2
inst✝² : DecidableEq (ι → σ →₀ ℕ)
α : Type u_3
inst✝¹ : CommSemiring α
inst✝ : DecidableEq (ℕ → σ →₀ ℕ)
f : MvPowerSeries σ α
m : ℕ
hf : (constantCoeff σ α) f ^ m = 0
d : σ →₀ ℕ
n : ℕ
hn : m + degree d ≤ n
k : ℕ →₀ σ →₀ ℕ
hk : k.support ⊆ range n ∧ (range n).sum ⇑k = d
s... | Please generate a tactic in lean4 to solve the state.
STATE:
σ : Type u_1
inst✝³ : DecidableEq σ
ι : Type u_2
inst✝² : DecidableEq (ι → σ →₀ ℕ)
α : Type u_3
inst✝¹ : CommSemiring α
inst✝ : DecidableEq (ℕ → σ →₀ ℕ)
f : MvPowerSeries σ α
m : ℕ
hf : (constantCoeff σ α) f ^ m = 0
d : σ →₀ ℕ
n : ℕ
hn : m + degree d ≤ n
k : ... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/ForMathlib/MvPowerSeries/Basic.lean | MvPowerSeries.coeff_eq_zero_of_constantCoeff_nilpotent | [33, 1] | [70, 22] | rw [hi.2, coeff_zero_eq_constantCoeff] | σ : Type u_1
inst✝³ : DecidableEq σ
ι : Type u_2
inst✝² : DecidableEq (ι → σ →₀ ℕ)
α : Type u_3
inst✝¹ : CommSemiring α
inst✝ : DecidableEq (ℕ → σ →₀ ℕ)
f : MvPowerSeries σ α
m : ℕ
hf : (constantCoeff σ α) f ^ m = 0
d : σ →₀ ℕ
n : ℕ
hn : m + degree d ≤ n
k : ℕ →₀ σ →₀ ℕ
hk : k.support ⊆ range n ∧ (range n).sum ⇑k = d
s... | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
σ : Type u_1
inst✝³ : DecidableEq σ
ι : Type u_2
inst✝² : DecidableEq (ι → σ →₀ ℕ)
α : Type u_3
inst✝¹ : CommSemiring α
inst✝ : DecidableEq (ℕ → σ →₀ ℕ)
f : MvPowerSeries σ α
m : ℕ
hf : (constantCoeff σ α) f ^ m = 0
d : σ →₀ ℕ
n : ℕ
hn : m + degree d ≤ n
k : ... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/ForMathlib/MvPowerSeries/Basic.lean | MvPowerSeries.coeff_eq_zero_of_constantCoeff_nilpotent | [33, 1] | [70, 22] | obtain ⟨m', hm'⟩ := Nat.exists_eq_add_of_le this | σ : Type u_1
inst✝³ : DecidableEq σ
ι : Type u_2
inst✝² : DecidableEq (ι → σ →₀ ℕ)
α : Type u_3
inst✝¹ : CommSemiring α
inst✝ : DecidableEq (ℕ → σ →₀ ℕ)
f : MvPowerSeries σ α
m : ℕ
hf : (constantCoeff σ α) f ^ m = 0
d : σ →₀ ℕ
n : ℕ
hn : m + degree d ≤ n
k : ℕ →₀ σ →₀ ℕ
hk : k.support ⊆ range n ∧ (range n).sum ⇑k = d
s... | case intro
σ : Type u_1
inst✝³ : DecidableEq σ
ι : Type u_2
inst✝² : DecidableEq (ι → σ →₀ ℕ)
α : Type u_3
inst✝¹ : CommSemiring α
inst✝ : DecidableEq (ℕ → σ →₀ ℕ)
f : MvPowerSeries σ α
m : ℕ
hf : (constantCoeff σ α) f ^ m = 0
d : σ →₀ ℕ
n : ℕ
hn : m + degree d ≤ n
k : ℕ →₀ σ →₀ ℕ
hk : k.support ⊆ range n ∧ (range n).s... | Please generate a tactic in lean4 to solve the state.
STATE:
σ : Type u_1
inst✝³ : DecidableEq σ
ι : Type u_2
inst✝² : DecidableEq (ι → σ →₀ ℕ)
α : Type u_3
inst✝¹ : CommSemiring α
inst✝ : DecidableEq (ℕ → σ →₀ ℕ)
f : MvPowerSeries σ α
m : ℕ
hf : (constantCoeff σ α) f ^ m = 0
d : σ →₀ ℕ
n : ℕ
hn : m + degree d ≤ n
k : ... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/ForMathlib/MvPowerSeries/Basic.lean | MvPowerSeries.coeff_eq_zero_of_constantCoeff_nilpotent | [33, 1] | [70, 22] | rw [hm', pow_add, hf, MulZeroClass.zero_mul] | case intro
σ : Type u_1
inst✝³ : DecidableEq σ
ι : Type u_2
inst✝² : DecidableEq (ι → σ →₀ ℕ)
α : Type u_3
inst✝¹ : CommSemiring α
inst✝ : DecidableEq (ℕ → σ →₀ ℕ)
f : MvPowerSeries σ α
m : ℕ
hf : (constantCoeff σ α) f ^ m = 0
d : σ →₀ ℕ
n : ℕ
hn : m + degree d ≤ n
k : ℕ →₀ σ →₀ ℕ
hk : k.support ⊆ range n ∧ (range n).s... | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case intro
σ : Type u_1
inst✝³ : DecidableEq σ
ι : Type u_2
inst✝² : DecidableEq (ι → σ →₀ ℕ)
α : Type u_3
inst✝¹ : CommSemiring α
inst✝ : DecidableEq (ℕ → σ →₀ ℕ)
f : MvPowerSeries σ α
m : ℕ
hf : (constantCoeff σ α) f ^ m = 0
d : σ →₀ ℕ
n : ℕ
hn : m + degree... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/ForMathlib/MvPowerSeries/Basic.lean | MvPowerSeries.coeff_eq_zero_of_constantCoeff_nilpotent | [33, 1] | [70, 22] | intro i hi | σ : Type u_1
inst✝³ : DecidableEq σ
ι : Type u_2
inst✝² : DecidableEq (ι → σ →₀ ℕ)
α : Type u_3
inst✝¹ : CommSemiring α
inst✝ : DecidableEq (ℕ → σ →₀ ℕ)
f : MvPowerSeries σ α
m : ℕ
hf : (constantCoeff σ α) f ^ m = 0
d : σ →₀ ℕ
n : ℕ
hn : m + degree d ≤ n
k : ℕ →₀ σ →₀ ℕ
hk : k.support ⊆ range n ∧ (range n).sum ⇑k = d
s... | σ : Type u_1
inst✝³ : DecidableEq σ
ι : Type u_2
inst✝² : DecidableEq (ι → σ →₀ ℕ)
α : Type u_3
inst✝¹ : CommSemiring α
inst✝ : DecidableEq (ℕ → σ →₀ ℕ)
f : MvPowerSeries σ α
m : ℕ
hf : (constantCoeff σ α) f ^ m = 0
d : σ →₀ ℕ
n : ℕ
hn : m + degree d ≤ n
k : ℕ →₀ σ →₀ ℕ
hk : k.support ⊆ range n ∧ (range n).sum ⇑k = d
s... | Please generate a tactic in lean4 to solve the state.
STATE:
σ : Type u_1
inst✝³ : DecidableEq σ
ι : Type u_2
inst✝² : DecidableEq (ι → σ →₀ ℕ)
α : Type u_3
inst✝¹ : CommSemiring α
inst✝ : DecidableEq (ℕ → σ →₀ ℕ)
f : MvPowerSeries σ α
m : ℕ
hf : (constantCoeff σ α) f ^ m = 0
d : σ →₀ ℕ
n : ℕ
hn : m + degree d ≤ n
k : ... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/ForMathlib/MvPowerSeries/Basic.lean | MvPowerSeries.coeff_eq_zero_of_constantCoeff_nilpotent | [33, 1] | [70, 22] | simp only [hs_def, mem_filter] at hi | σ : Type u_1
inst✝³ : DecidableEq σ
ι : Type u_2
inst✝² : DecidableEq (ι → σ →₀ ℕ)
α : Type u_3
inst✝¹ : CommSemiring α
inst✝ : DecidableEq (ℕ → σ →₀ ℕ)
f : MvPowerSeries σ α
m : ℕ
hf : (constantCoeff σ α) f ^ m = 0
d : σ →₀ ℕ
n : ℕ
hn : m + degree d ≤ n
k : ℕ →₀ σ →₀ ℕ
hk : k.support ⊆ range n ∧ (range n).sum ⇑k = d
s... | σ : Type u_1
inst✝³ : DecidableEq σ
ι : Type u_2
inst✝² : DecidableEq (ι → σ →₀ ℕ)
α : Type u_3
inst✝¹ : CommSemiring α
inst✝ : DecidableEq (ℕ → σ →₀ ℕ)
f : MvPowerSeries σ α
m : ℕ
hf : (constantCoeff σ α) f ^ m = 0
d : σ →₀ ℕ
n : ℕ
hn : m + degree d ≤ n
k : ℕ →₀ σ →₀ ℕ
hk : k.support ⊆ range n ∧ (range n).sum ⇑k = d
s... | Please generate a tactic in lean4 to solve the state.
STATE:
σ : Type u_1
inst✝³ : DecidableEq σ
ι : Type u_2
inst✝² : DecidableEq (ι → σ →₀ ℕ)
α : Type u_3
inst✝¹ : CommSemiring α
inst✝ : DecidableEq (ℕ → σ →₀ ℕ)
f : MvPowerSeries σ α
m : ℕ
hf : (constantCoeff σ α) f ^ m = 0
d : σ →₀ ℕ
n : ℕ
hn : m + degree d ≤ n
k : ... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/ForMathlib/MvPowerSeries/Basic.lean | MvPowerSeries.coeff_eq_zero_of_constantCoeff_nilpotent | [33, 1] | [70, 22] | rw [hi.2, map_zero] | σ : Type u_1
inst✝³ : DecidableEq σ
ι : Type u_2
inst✝² : DecidableEq (ι → σ →₀ ℕ)
α : Type u_3
inst✝¹ : CommSemiring α
inst✝ : DecidableEq (ℕ → σ →₀ ℕ)
f : MvPowerSeries σ α
m : ℕ
hf : (constantCoeff σ α) f ^ m = 0
d : σ →₀ ℕ
n : ℕ
hn : m + degree d ≤ n
k : ℕ →₀ σ →₀ ℕ
hk : k.support ⊆ range n ∧ (range n).sum ⇑k = d
s... | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
σ : Type u_1
inst✝³ : DecidableEq σ
ι : Type u_2
inst✝² : DecidableEq (ι → σ →₀ ℕ)
α : Type u_3
inst✝¹ : CommSemiring α
inst✝ : DecidableEq (ℕ → σ →₀ ℕ)
f : MvPowerSeries σ α
m : ℕ
hf : (constantCoeff σ α) f ^ m = 0
d : σ →₀ ℕ
n : ℕ
hn : m + degree d ≤ n
k : ... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/ForMathlib/MvPowerSeries/Basic.lean | MvPowerSeries.coeff_eq_zero_of_constantCoeff_nilpotent | [33, 1] | [70, 22] | simp only [hs_def, mem_filter, mem_sdiff, mem_range, not_and, and_imp] | case convert_2
σ : Type u_1
inst✝³ : DecidableEq σ
ι : Type u_2
inst✝² : DecidableEq (ι → σ →₀ ℕ)
α : Type u_3
inst✝¹ : CommSemiring α
inst✝ : DecidableEq (ℕ → σ →₀ ℕ)
f : MvPowerSeries σ α
m : ℕ
hf : (constantCoeff σ α) f ^ m = 0
d : σ →₀ ℕ
n : ℕ
hn : m + degree d ≤ n
k : ℕ →₀ σ →₀ ℕ
hk : k.support ⊆ range n ∧ (range ... | case convert_2
σ : Type u_1
inst✝³ : DecidableEq σ
ι : Type u_2
inst✝² : DecidableEq (ι → σ →₀ ℕ)
α : Type u_3
inst✝¹ : CommSemiring α
inst✝ : DecidableEq (ℕ → σ →₀ ℕ)
f : MvPowerSeries σ α
m : ℕ
hf : (constantCoeff σ α) f ^ m = 0
d : σ →₀ ℕ
n : ℕ
hn : m + degree d ≤ n
k : ℕ →₀ σ →₀ ℕ
hk : k.support ⊆ range n ∧ (range ... | Please generate a tactic in lean4 to solve the state.
STATE:
case convert_2
σ : Type u_1
inst✝³ : DecidableEq σ
ι : Type u_2
inst✝² : DecidableEq (ι → σ →₀ ℕ)
α : Type u_3
inst✝¹ : CommSemiring α
inst✝ : DecidableEq (ℕ → σ →₀ ℕ)
f : MvPowerSeries σ α
m : ℕ
hf : (constantCoeff σ α) f ^ m = 0
d : σ →₀ ℕ
n : ℕ
hn : m + de... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/ForMathlib/MvPowerSeries/Basic.lean | MvPowerSeries.coeff_eq_zero_of_constantCoeff_nilpotent | [33, 1] | [70, 22] | intro i hi hi' | case convert_2
σ : Type u_1
inst✝³ : DecidableEq σ
ι : Type u_2
inst✝² : DecidableEq (ι → σ →₀ ℕ)
α : Type u_3
inst✝¹ : CommSemiring α
inst✝ : DecidableEq (ℕ → σ →₀ ℕ)
f : MvPowerSeries σ α
m : ℕ
hf : (constantCoeff σ α) f ^ m = 0
d : σ →₀ ℕ
n : ℕ
hn : m + degree d ≤ n
k : ℕ →₀ σ →₀ ℕ
hk : k.support ⊆ range n ∧ (range ... | case convert_2
σ : Type u_1
inst✝³ : DecidableEq σ
ι : Type u_2
inst✝² : DecidableEq (ι → σ →₀ ℕ)
α : Type u_3
inst✝¹ : CommSemiring α
inst✝ : DecidableEq (ℕ → σ →₀ ℕ)
f : MvPowerSeries σ α
m : ℕ
hf : (constantCoeff σ α) f ^ m = 0
d : σ →₀ ℕ
n : ℕ
hn : m + degree d ≤ n
k : ℕ →₀ σ →₀ ℕ
hk : k.support ⊆ range n ∧ (range ... | Please generate a tactic in lean4 to solve the state.
STATE:
case convert_2
σ : Type u_1
inst✝³ : DecidableEq σ
ι : Type u_2
inst✝² : DecidableEq (ι → σ →₀ ℕ)
α : Type u_3
inst✝¹ : CommSemiring α
inst✝ : DecidableEq (ℕ → σ →₀ ℕ)
f : MvPowerSeries σ α
m : ℕ
hf : (constantCoeff σ α) f ^ m = 0
d : σ →₀ ℕ
n : ℕ
hn : m + de... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/ForMathlib/MvPowerSeries/Basic.lean | MvPowerSeries.coeff_eq_zero_of_constantCoeff_nilpotent | [33, 1] | [70, 22] | rw [← not_lt] | case convert_2
σ : Type u_1
inst✝³ : DecidableEq σ
ι : Type u_2
inst✝² : DecidableEq (ι → σ →₀ ℕ)
α : Type u_3
inst✝¹ : CommSemiring α
inst✝ : DecidableEq (ℕ → σ →₀ ℕ)
f : MvPowerSeries σ α
m : ℕ
hf : (constantCoeff σ α) f ^ m = 0
d : σ →₀ ℕ
n : ℕ
hn : m + degree d ≤ n
k : ℕ →₀ σ →₀ ℕ
hk : k.support ⊆ range n ∧ (range ... | case convert_2
σ : Type u_1
inst✝³ : DecidableEq σ
ι : Type u_2
inst✝² : DecidableEq (ι → σ →₀ ℕ)
α : Type u_3
inst✝¹ : CommSemiring α
inst✝ : DecidableEq (ℕ → σ →₀ ℕ)
f : MvPowerSeries σ α
m : ℕ
hf : (constantCoeff σ α) f ^ m = 0
d : σ →₀ ℕ
n : ℕ
hn : m + degree d ≤ n
k : ℕ →₀ σ →₀ ℕ
hk : k.support ⊆ range n ∧ (range ... | Please generate a tactic in lean4 to solve the state.
STATE:
case convert_2
σ : Type u_1
inst✝³ : DecidableEq σ
ι : Type u_2
inst✝² : DecidableEq (ι → σ →₀ ℕ)
α : Type u_3
inst✝¹ : CommSemiring α
inst✝ : DecidableEq (ℕ → σ →₀ ℕ)
f : MvPowerSeries σ α
m : ℕ
hf : (constantCoeff σ α) f ^ m = 0
d : σ →₀ ℕ
n : ℕ
hn : m + de... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/ForMathlib/MvPowerSeries/Basic.lean | MvPowerSeries.coeff_eq_zero_of_constantCoeff_nilpotent | [33, 1] | [70, 22] | intro h | case convert_2
σ : Type u_1
inst✝³ : DecidableEq σ
ι : Type u_2
inst✝² : DecidableEq (ι → σ →₀ ℕ)
α : Type u_3
inst✝¹ : CommSemiring α
inst✝ : DecidableEq (ℕ → σ →₀ ℕ)
f : MvPowerSeries σ α
m : ℕ
hf : (constantCoeff σ α) f ^ m = 0
d : σ →₀ ℕ
n : ℕ
hn : m + degree d ≤ n
k : ℕ →₀ σ →₀ ℕ
hk : k.support ⊆ range n ∧ (range ... | case convert_2
σ : Type u_1
inst✝³ : DecidableEq σ
ι : Type u_2
inst✝² : DecidableEq (ι → σ →₀ ℕ)
α : Type u_3
inst✝¹ : CommSemiring α
inst✝ : DecidableEq (ℕ → σ →₀ ℕ)
f : MvPowerSeries σ α
m : ℕ
hf : (constantCoeff σ α) f ^ m = 0
d : σ →₀ ℕ
n : ℕ
hn : m + degree d ≤ n
k : ℕ →₀ σ →₀ ℕ
hk : k.support ⊆ range n ∧ (range ... | Please generate a tactic in lean4 to solve the state.
STATE:
case convert_2
σ : Type u_1
inst✝³ : DecidableEq σ
ι : Type u_2
inst✝² : DecidableEq (ι → σ →₀ ℕ)
α : Type u_3
inst✝¹ : CommSemiring α
inst✝ : DecidableEq (ℕ → σ →₀ ℕ)
f : MvPowerSeries σ α
m : ℕ
hf : (constantCoeff σ α) f ^ m = 0
d : σ →₀ ℕ
n : ℕ
hn : m + de... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/ForMathlib/MvPowerSeries/Basic.lean | MvPowerSeries.coeff_eq_zero_of_constantCoeff_nilpotent | [33, 1] | [70, 22] | apply hi' hi | case convert_2
σ : Type u_1
inst✝³ : DecidableEq σ
ι : Type u_2
inst✝² : DecidableEq (ι → σ →₀ ℕ)
α : Type u_3
inst✝¹ : CommSemiring α
inst✝ : DecidableEq (ℕ → σ →₀ ℕ)
f : MvPowerSeries σ α
m : ℕ
hf : (constantCoeff σ α) f ^ m = 0
d : σ →₀ ℕ
n : ℕ
hn : m + degree d ≤ n
k : ℕ →₀ σ →₀ ℕ
hk : k.support ⊆ range n ∧ (range ... | case convert_2
σ : Type u_1
inst✝³ : DecidableEq σ
ι : Type u_2
inst✝² : DecidableEq (ι → σ →₀ ℕ)
α : Type u_3
inst✝¹ : CommSemiring α
inst✝ : DecidableEq (ℕ → σ →₀ ℕ)
f : MvPowerSeries σ α
m : ℕ
hf : (constantCoeff σ α) f ^ m = 0
d : σ →₀ ℕ
n : ℕ
hn : m + degree d ≤ n
k : ℕ →₀ σ →₀ ℕ
hk : k.support ⊆ range n ∧ (range ... | Please generate a tactic in lean4 to solve the state.
STATE:
case convert_2
σ : Type u_1
inst✝³ : DecidableEq σ
ι : Type u_2
inst✝² : DecidableEq (ι → σ →₀ ℕ)
α : Type u_3
inst✝¹ : CommSemiring α
inst✝ : DecidableEq (ℕ → σ →₀ ℕ)
f : MvPowerSeries σ α
m : ℕ
hf : (constantCoeff σ α) f ^ m = 0
d : σ →₀ ℕ
n : ℕ
hn : m + de... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/ForMathlib/MvPowerSeries/Basic.lean | MvPowerSeries.coeff_eq_zero_of_constantCoeff_nilpotent | [33, 1] | [70, 22] | simpa only [Nat.lt_one_iff, degree_eq_zero_iff] using h | case convert_2
σ : Type u_1
inst✝³ : DecidableEq σ
ι : Type u_2
inst✝² : DecidableEq (ι → σ →₀ ℕ)
α : Type u_3
inst✝¹ : CommSemiring α
inst✝ : DecidableEq (ℕ → σ →₀ ℕ)
f : MvPowerSeries σ α
m : ℕ
hf : (constantCoeff σ α) f ^ m = 0
d : σ →₀ ℕ
n : ℕ
hn : m + degree d ≤ n
k : ℕ →₀ σ →₀ ℕ
hk : k.support ⊆ range n ∧ (range ... | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case convert_2
σ : Type u_1
inst✝³ : DecidableEq σ
ι : Type u_2
inst✝² : DecidableEq (ι → σ →₀ ℕ)
α : Type u_3
inst✝¹ : CommSemiring α
inst✝ : DecidableEq (ℕ → σ →₀ ℕ)
f : MvPowerSeries σ α
m : ℕ
hf : (constantCoeff σ α) f ^ m = 0
d : σ →₀ ℕ
n : ℕ
hn : m + de... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/ForMathlib/AlgebraLemmas.lean | factorial_isUnit | [26, 1] | [31, 33] | apply isUnit_of_dvd_unit _ hn_fac | A : Type u_1
inst✝ : CommSemiring A
I : Ideal A
n : ℕ
hn_fac : IsUnit ↑(n - 1)!
m : ℕ
hmn : m < n
⊢ IsUnit ↑m ! | A : Type u_1
inst✝ : CommSemiring A
I : Ideal A
n : ℕ
hn_fac : IsUnit ↑(n - 1)!
m : ℕ
hmn : m < n
⊢ ↑m ! ∣ ↑(n - 1)! | Please generate a tactic in lean4 to solve the state.
STATE:
A : Type u_1
inst✝ : CommSemiring A
I : Ideal A
n : ℕ
hn_fac : IsUnit ↑(n - 1)!
m : ℕ
hmn : m < n
⊢ IsUnit ↑m !
TACTIC:
|
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/ForMathlib/AlgebraLemmas.lean | factorial_isUnit | [26, 1] | [31, 33] | apply Nat.cast_dvd_cast | A : Type u_1
inst✝ : CommSemiring A
I : Ideal A
n : ℕ
hn_fac : IsUnit ↑(n - 1)!
m : ℕ
hmn : m < n
⊢ ↑m ! ∣ ↑(n - 1)! | case h
A : Type u_1
inst✝ : CommSemiring A
I : Ideal A
n : ℕ
hn_fac : IsUnit ↑(n - 1)!
m : ℕ
hmn : m < n
⊢ m ! ∣ (n - 1)! | Please generate a tactic in lean4 to solve the state.
STATE:
A : Type u_1
inst✝ : CommSemiring A
I : Ideal A
n : ℕ
hn_fac : IsUnit ↑(n - 1)!
m : ℕ
hmn : m < n
⊢ ↑m ! ∣ ↑(n - 1)!
TACTIC:
|
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/ForMathlib/AlgebraLemmas.lean | factorial_isUnit | [26, 1] | [31, 33] | apply Nat.factorial_dvd_factorial | case h
A : Type u_1
inst✝ : CommSemiring A
I : Ideal A
n : ℕ
hn_fac : IsUnit ↑(n - 1)!
m : ℕ
hmn : m < n
⊢ m ! ∣ (n - 1)! | case h.h
A : Type u_1
inst✝ : CommSemiring A
I : Ideal A
n : ℕ
hn_fac : IsUnit ↑(n - 1)!
m : ℕ
hmn : m < n
⊢ m ≤ n - 1 | Please generate a tactic in lean4 to solve the state.
STATE:
case h
A : Type u_1
inst✝ : CommSemiring A
I : Ideal A
n : ℕ
hn_fac : IsUnit ↑(n - 1)!
m : ℕ
hmn : m < n
⊢ m ! ∣ (n - 1)!
TACTIC:
|
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/ForMathlib/AlgebraLemmas.lean | factorial_isUnit | [26, 1] | [31, 33] | exact Nat.le_sub_one_of_lt hmn | case h.h
A : Type u_1
inst✝ : CommSemiring A
I : Ideal A
n : ℕ
hn_fac : IsUnit ↑(n - 1)!
m : ℕ
hmn : m < n
⊢ m ≤ n - 1 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case h.h
A : Type u_1
inst✝ : CommSemiring A
I : Ideal A
n : ℕ
hn_fac : IsUnit ↑(n - 1)!
m : ℕ
hmn : m < n
⊢ m ≤ n - 1
TACTIC:
|
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/ForMathlib/AlgebraLemmas.lean | factorial_isUnit' | [34, 1] | [39, 12] | apply isUnit_of_dvd_unit _ hn_fac | A : Type u_1
inst✝ : CommSemiring A
I : Ideal A
n : ℕ
hn_fac : IsUnit ↑n !
m : ℕ
hmn : m ≤ n
⊢ IsUnit ↑m ! | A : Type u_1
inst✝ : CommSemiring A
I : Ideal A
n : ℕ
hn_fac : IsUnit ↑n !
m : ℕ
hmn : m ≤ n
⊢ ↑m ! ∣ ↑n ! | Please generate a tactic in lean4 to solve the state.
STATE:
A : Type u_1
inst✝ : CommSemiring A
I : Ideal A
n : ℕ
hn_fac : IsUnit ↑n !
m : ℕ
hmn : m ≤ n
⊢ IsUnit ↑m !
TACTIC:
|
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/ForMathlib/AlgebraLemmas.lean | factorial_isUnit' | [34, 1] | [39, 12] | apply Nat.cast_dvd_cast | A : Type u_1
inst✝ : CommSemiring A
I : Ideal A
n : ℕ
hn_fac : IsUnit ↑n !
m : ℕ
hmn : m ≤ n
⊢ ↑m ! ∣ ↑n ! | case h
A : Type u_1
inst✝ : CommSemiring A
I : Ideal A
n : ℕ
hn_fac : IsUnit ↑n !
m : ℕ
hmn : m ≤ n
⊢ m ! ∣ n ! | Please generate a tactic in lean4 to solve the state.
STATE:
A : Type u_1
inst✝ : CommSemiring A
I : Ideal A
n : ℕ
hn_fac : IsUnit ↑n !
m : ℕ
hmn : m ≤ n
⊢ ↑m ! ∣ ↑n !
TACTIC:
|
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/ForMathlib/AlgebraLemmas.lean | factorial_isUnit' | [34, 1] | [39, 12] | apply Nat.factorial_dvd_factorial | case h
A : Type u_1
inst✝ : CommSemiring A
I : Ideal A
n : ℕ
hn_fac : IsUnit ↑n !
m : ℕ
hmn : m ≤ n
⊢ m ! ∣ n ! | case h.h
A : Type u_1
inst✝ : CommSemiring A
I : Ideal A
n : ℕ
hn_fac : IsUnit ↑n !
m : ℕ
hmn : m ≤ n
⊢ m ≤ n | Please generate a tactic in lean4 to solve the state.
STATE:
case h
A : Type u_1
inst✝ : CommSemiring A
I : Ideal A
n : ℕ
hn_fac : IsUnit ↑n !
m : ℕ
hmn : m ≤ n
⊢ m ! ∣ n !
TACTIC:
|
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/ForMathlib/AlgebraLemmas.lean | factorial_isUnit' | [34, 1] | [39, 12] | exact hmn | case h.h
A : Type u_1
inst✝ : CommSemiring A
I : Ideal A
n : ℕ
hn_fac : IsUnit ↑n !
m : ℕ
hmn : m ≤ n
⊢ m ≤ n | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case h.h
A : Type u_1
inst✝ : CommSemiring A
I : Ideal A
n : ℕ
hn_fac : IsUnit ↑n !
m : ℕ
hmn : m ≤ n
⊢ m ≤ n
TACTIC:
|
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/ForMathlib/AlgebraLemmas.lean | Factorial.isUnit | [41, 1] | [47, 32] | rw [← map_natCast (algebraMap ℚ A)] | A : Type u_1
inst✝¹ : CommSemiring A
I : Ideal A
inst✝ : Algebra ℚ A
n : ℕ
⊢ IsUnit ↑n ! | A : Type u_1
inst✝¹ : CommSemiring A
I : Ideal A
inst✝ : Algebra ℚ A
n : ℕ
⊢ IsUnit ((algebraMap ℚ A) ↑n !) | Please generate a tactic in lean4 to solve the state.
STATE:
A : Type u_1
inst✝¹ : CommSemiring A
I : Ideal A
inst✝ : Algebra ℚ A
n : ℕ
⊢ IsUnit ↑n !
TACTIC:
|
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/ForMathlib/AlgebraLemmas.lean | Factorial.isUnit | [41, 1] | [47, 32] | apply IsUnit.map | A : Type u_1
inst✝¹ : CommSemiring A
I : Ideal A
inst✝ : Algebra ℚ A
n : ℕ
⊢ IsUnit ((algebraMap ℚ A) ↑n !) | case h
A : Type u_1
inst✝¹ : CommSemiring A
I : Ideal A
inst✝ : Algebra ℚ A
n : ℕ
⊢ IsUnit ↑n ! | Please generate a tactic in lean4 to solve the state.
STATE:
A : Type u_1
inst✝¹ : CommSemiring A
I : Ideal A
inst✝ : Algebra ℚ A
n : ℕ
⊢ IsUnit ((algebraMap ℚ A) ↑n !)
TACTIC:
|
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/ForMathlib/AlgebraLemmas.lean | Factorial.isUnit | [41, 1] | [47, 32] | rw [isUnit_iff_ne_zero] | case h
A : Type u_1
inst✝¹ : CommSemiring A
I : Ideal A
inst✝ : Algebra ℚ A
n : ℕ
⊢ IsUnit ↑n ! | case h
A : Type u_1
inst✝¹ : CommSemiring A
I : Ideal A
inst✝ : Algebra ℚ A
n : ℕ
⊢ ↑n ! ≠ 0 | Please generate a tactic in lean4 to solve the state.
STATE:
case h
A : Type u_1
inst✝¹ : CommSemiring A
I : Ideal A
inst✝ : Algebra ℚ A
n : ℕ
⊢ IsUnit ↑n !
TACTIC:
|
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/ForMathlib/AlgebraLemmas.lean | Factorial.isUnit | [41, 1] | [47, 32] | simp only [ne_eq, Nat.cast_eq_zero] | case h
A : Type u_1
inst✝¹ : CommSemiring A
I : Ideal A
inst✝ : Algebra ℚ A
n : ℕ
⊢ ↑n ! ≠ 0 | case h
A : Type u_1
inst✝¹ : CommSemiring A
I : Ideal A
inst✝ : Algebra ℚ A
n : ℕ
⊢ ¬n ! = 0 | Please generate a tactic in lean4 to solve the state.
STATE:
case h
A : Type u_1
inst✝¹ : CommSemiring A
I : Ideal A
inst✝ : Algebra ℚ A
n : ℕ
⊢ ↑n ! ≠ 0
TACTIC:
|
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/ForMathlib/AlgebraLemmas.lean | Factorial.isUnit | [41, 1] | [47, 32] | exact Nat.factorial_ne_zero n | case h
A : Type u_1
inst✝¹ : CommSemiring A
I : Ideal A
inst✝ : Algebra ℚ A
n : ℕ
⊢ ¬n ! = 0 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case h
A : Type u_1
inst✝¹ : CommSemiring A
I : Ideal A
inst✝ : Algebra ℚ A
n : ℕ
⊢ ¬n ! = 0
TACTIC:
|
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/ForMathlib/AlgebraLemmas.lean | Ring.inverse_pow_mul_eq_iff_eq_mul | [56, 1] | [58, 80] | rw [Ring.inverse_pow, Ring.inverse_mul_eq_iff_eq_mul _ _ _ (IsUnit.pow _ ha)] | M₀ : Type u_1
inst✝ : CommMonoidWithZero M₀
a b c : M₀
ha : IsUnit a
k : ℕ
⊢ inverse a ^ k * b = c ↔ b = a ^ k * c | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
M₀ : Type u_1
inst✝ : CommMonoidWithZero M₀
a b c : M₀
ha : IsUnit a
k : ℕ
⊢ inverse a ^ k * b = c ↔ b = a ^ k * c
TACTIC:
|
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/ForMathlib/AlgebraLemmas.lean | Ideal.mem_pow_eq_zero | [65, 1] | [73, 47] | have hxn : x ^ n = 0 := by
rw [Ideal.zero_eq_bot] at hnI
rw [← Ideal.mem_bot, ← hnI]
exact Ideal.pow_mem_pow hx n | A : Type u_1
inst✝ : CommSemiring A
I : Ideal A
n m : ℕ
hnI : I ^ n = 0
hmn : n ≤ m
x : A
hx : x ∈ I
⊢ x ^ m = 0 | A : Type u_1
inst✝ : CommSemiring A
I : Ideal A
n m : ℕ
hnI : I ^ n = 0
hmn : n ≤ m
x : A
hx : x ∈ I
hxn : x ^ n = 0
⊢ x ^ m = 0 | Please generate a tactic in lean4 to solve the state.
STATE:
A : Type u_1
inst✝ : CommSemiring A
I : Ideal A
n m : ℕ
hnI : I ^ n = 0
hmn : n ≤ m
x : A
hx : x ∈ I
⊢ x ^ m = 0
TACTIC:
|
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/ForMathlib/AlgebraLemmas.lean | Ideal.mem_pow_eq_zero | [65, 1] | [73, 47] | obtain ⟨c, hc⟩ := Nat.exists_eq_add_of_le hmn | A : Type u_1
inst✝ : CommSemiring A
I : Ideal A
n m : ℕ
hnI : I ^ n = 0
hmn : n ≤ m
x : A
hx : x ∈ I
hxn : x ^ n = 0
⊢ x ^ m = 0 | case intro
A : Type u_1
inst✝ : CommSemiring A
I : Ideal A
n m : ℕ
hnI : I ^ n = 0
hmn : n ≤ m
x : A
hx : x ∈ I
hxn : x ^ n = 0
c : ℕ
hc : m = n + c
⊢ x ^ m = 0 | Please generate a tactic in lean4 to solve the state.
STATE:
A : Type u_1
inst✝ : CommSemiring A
I : Ideal A
n m : ℕ
hnI : I ^ n = 0
hmn : n ≤ m
x : A
hx : x ∈ I
hxn : x ^ n = 0
⊢ x ^ m = 0
TACTIC:
|
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/ForMathlib/AlgebraLemmas.lean | Ideal.mem_pow_eq_zero | [65, 1] | [73, 47] | rw [hc, pow_add, hxn, MulZeroClass.zero_mul] | case intro
A : Type u_1
inst✝ : CommSemiring A
I : Ideal A
n m : ℕ
hnI : I ^ n = 0
hmn : n ≤ m
x : A
hx : x ∈ I
hxn : x ^ n = 0
c : ℕ
hc : m = n + c
⊢ x ^ m = 0 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case intro
A : Type u_1
inst✝ : CommSemiring A
I : Ideal A
n m : ℕ
hnI : I ^ n = 0
hmn : n ≤ m
x : A
hx : x ∈ I
hxn : x ^ n = 0
c : ℕ
hc : m = n + c
⊢ x ^ m = 0
TACTIC:
|
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/ForMathlib/AlgebraLemmas.lean | Ideal.mem_pow_eq_zero | [65, 1] | [73, 47] | rw [Ideal.zero_eq_bot] at hnI | A : Type u_1
inst✝ : CommSemiring A
I : Ideal A
n m : ℕ
hnI : I ^ n = 0
hmn : n ≤ m
x : A
hx : x ∈ I
⊢ x ^ n = 0 | A : Type u_1
inst✝ : CommSemiring A
I : Ideal A
n m : ℕ
hnI : I ^ n = ⊥
hmn : n ≤ m
x : A
hx : x ∈ I
⊢ x ^ n = 0 | Please generate a tactic in lean4 to solve the state.
STATE:
A : Type u_1
inst✝ : CommSemiring A
I : Ideal A
n m : ℕ
hnI : I ^ n = 0
hmn : n ≤ m
x : A
hx : x ∈ I
⊢ x ^ n = 0
TACTIC:
|
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/ForMathlib/AlgebraLemmas.lean | Ideal.mem_pow_eq_zero | [65, 1] | [73, 47] | rw [← Ideal.mem_bot, ← hnI] | A : Type u_1
inst✝ : CommSemiring A
I : Ideal A
n m : ℕ
hnI : I ^ n = ⊥
hmn : n ≤ m
x : A
hx : x ∈ I
⊢ x ^ n = 0 | A : Type u_1
inst✝ : CommSemiring A
I : Ideal A
n m : ℕ
hnI : I ^ n = ⊥
hmn : n ≤ m
x : A
hx : x ∈ I
⊢ x ^ n ∈ I ^ n | Please generate a tactic in lean4 to solve the state.
STATE:
A : Type u_1
inst✝ : CommSemiring A
I : Ideal A
n m : ℕ
hnI : I ^ n = ⊥
hmn : n ≤ m
x : A
hx : x ∈ I
⊢ x ^ n = 0
TACTIC:
|
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/ForMathlib/AlgebraLemmas.lean | Ideal.mem_pow_eq_zero | [65, 1] | [73, 47] | exact Ideal.pow_mem_pow hx n | A : Type u_1
inst✝ : CommSemiring A
I : Ideal A
n m : ℕ
hnI : I ^ n = ⊥
hmn : n ≤ m
x : A
hx : x ∈ I
⊢ x ^ n ∈ I ^ n | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
A : Type u_1
inst✝ : CommSemiring A
I : Ideal A
n m : ℕ
hnI : I ^ n = ⊥
hmn : n ≤ m
x : A
hx : x ∈ I
⊢ x ^ n ∈ I ^ n
TACTIC:
|
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/DPAlgebra/BaseChange.lean | Algebra.TensorProduct.coe_map_id₁ | [43, 1] | [46, 100] | rfl | R : Type u_1
inst✝⁶ : CommSemiring R
S : Type u_2
S' : Type u_3
inst✝⁵ : Semiring S
inst✝⁴ : Semiring S'
inst✝³ : Algebra R S
inst✝² : Algebra R S'
φ : S →ₐ[R] S'
A : Type u_4
inst✝¹ : Semiring A
inst✝ : Algebra R A
⊢ ⇑(map (AlgHom.id R A) φ) = ⇑(LinearMap.lTensor A φ.toLinearMap) | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
R : Type u_1
inst✝⁶ : CommSemiring R
S : Type u_2
S' : Type u_3
inst✝⁵ : Semiring S
inst✝⁴ : Semiring S'
inst✝³ : Algebra R S
inst✝² : Algebra R S'
φ : S →ₐ[R] S'
A : Type u_4
inst✝¹ : Semiring A
inst✝ : Algebra R A
⊢ ⇑(map (AlgHom.id R A) φ) = ⇑(LinearMap.lT... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/DPAlgebra/BaseChange.lean | Algebra.TensorProduct.coe_map_id₂ | [48, 1] | [51, 100] | rfl | R : Type u_1
inst✝⁶ : CommSemiring R
S : Type u_2
S' : Type u_3
inst✝⁵ : Semiring S
inst✝⁴ : Semiring S'
inst✝³ : Algebra R S
inst✝² : Algebra R S'
φ : S →ₐ[R] S'
A : Type u_4
inst✝¹ : Semiring A
inst✝ : Algebra R A
⊢ ⇑(map φ (AlgHom.id R A)) = ⇑(LinearMap.rTensor A φ.toLinearMap) | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
R : Type u_1
inst✝⁶ : CommSemiring R
S : Type u_2
S' : Type u_3
inst✝⁵ : Semiring S
inst✝⁴ : Semiring S'
inst✝³ : Algebra R S
inst✝² : Algebra R S'
φ : S →ₐ[R] S'
A : Type u_4
inst✝¹ : Semiring A
inst✝ : Algebra R A
⊢ ⇑(map φ (AlgHom.id R A)) = ⇑(LinearMap.rT... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/DPAlgebra/BaseChange.lean | MvPolynomial.coeff_baseChange_apply | [106, 1] | [124, 29] | rw [baseChange, AlgHom.coe_mk, coe_eval₂RingHom] | R : Type u_1
inst✝⁴ : CommSemiring R
S : Type u_4
inst✝³ : CommSemiring S
inst✝² : Algebra R S
S' : Type u_2
inst✝¹ : CommSemiring S'
inst✝ : Algebra R S'
σ : Type u_3
m : σ →₀ ℕ
φ : S →ₐ[R] S'
f : MvPolynomial σ S
⊢ coeff m ((baseChange φ) f) = φ (coeff m f) | R : Type u_1
inst✝⁴ : CommSemiring R
S : Type u_4
inst✝³ : CommSemiring S
inst✝² : Algebra R S
S' : Type u_2
inst✝¹ : CommSemiring S'
inst✝ : Algebra R S'
σ : Type u_3
m : σ →₀ ℕ
φ : S →ₐ[R] S'
f : MvPolynomial σ S
⊢ coeff m (eval₂ (C.comp ↑φ) X f) = φ (coeff m f) | Please generate a tactic in lean4 to solve the state.
STATE:
R : Type u_1
inst✝⁴ : CommSemiring R
S : Type u_4
inst✝³ : CommSemiring S
inst✝² : Algebra R S
S' : Type u_2
inst✝¹ : CommSemiring S'
inst✝ : Algebra R S'
σ : Type u_3
m : σ →₀ ℕ
φ : S →ₐ[R] S'
f : MvPolynomial σ S
⊢ coeff m ((baseChange φ) f) = φ (coeff m f)... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/DPAlgebra/BaseChange.lean | MvPolynomial.coeff_baseChange_apply | [106, 1] | [124, 29] | simp only [eval₂_C, RingHom.coe_comp, RingHom.coe_coe, Function.comp_apply, coeff_C] | case h_C
R : Type u_1
inst✝⁴ : CommSemiring R
S : Type u_4
inst✝³ : CommSemiring S
inst✝² : Algebra R S
S' : Type u_2
inst✝¹ : CommSemiring S'
inst✝ : Algebra R S'
σ : Type u_3
φ : S →ₐ[R] S'
r : S
m : σ →₀ ℕ
⊢ coeff m (eval₂ (C.comp ↑φ) X (C r)) = φ (coeff m (C r)) | case h_C
R : Type u_1
inst✝⁴ : CommSemiring R
S : Type u_4
inst✝³ : CommSemiring S
inst✝² : Algebra R S
S' : Type u_2
inst✝¹ : CommSemiring S'
inst✝ : Algebra R S'
σ : Type u_3
φ : S →ₐ[R] S'
r : S
m : σ →₀ ℕ
⊢ (if 0 = m then φ r else 0) = φ (if 0 = m then r else 0) | Please generate a tactic in lean4 to solve the state.
STATE:
case h_C
R : Type u_1
inst✝⁴ : CommSemiring R
S : Type u_4
inst✝³ : CommSemiring S
inst✝² : Algebra R S
S' : Type u_2
inst✝¹ : CommSemiring S'
inst✝ : Algebra R S'
σ : Type u_3
φ : S →ₐ[R] S'
r : S
m : σ →₀ ℕ
⊢ coeff m (eval₂ (C.comp ↑φ) X (C r)) = φ (coeff m... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/DPAlgebra/BaseChange.lean | MvPolynomial.coeff_baseChange_apply | [106, 1] | [124, 29] | split_ifs | case h_C
R : Type u_1
inst✝⁴ : CommSemiring R
S : Type u_4
inst✝³ : CommSemiring S
inst✝² : Algebra R S
S' : Type u_2
inst✝¹ : CommSemiring S'
inst✝ : Algebra R S'
σ : Type u_3
φ : S →ₐ[R] S'
r : S
m : σ →₀ ℕ
⊢ (if 0 = m then φ r else 0) = φ (if 0 = m then r else 0) | case pos
R : Type u_1
inst✝⁴ : CommSemiring R
S : Type u_4
inst✝³ : CommSemiring S
inst✝² : Algebra R S
S' : Type u_2
inst✝¹ : CommSemiring S'
inst✝ : Algebra R S'
σ : Type u_3
φ : S →ₐ[R] S'
r : S
m : σ →₀ ℕ
h✝ : 0 = m
⊢ φ r = φ r
case neg
R : Type u_1
inst✝⁴ : CommSemiring R
S : Type u_4
inst✝³ : CommSemiring S
inst... | Please generate a tactic in lean4 to solve the state.
STATE:
case h_C
R : Type u_1
inst✝⁴ : CommSemiring R
S : Type u_4
inst✝³ : CommSemiring S
inst✝² : Algebra R S
S' : Type u_2
inst✝¹ : CommSemiring S'
inst✝ : Algebra R S'
σ : Type u_3
φ : S →ₐ[R] S'
r : S
m : σ →₀ ℕ
⊢ (if 0 = m then φ r else 0) = φ (if 0 = m then r ... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/DPAlgebra/BaseChange.lean | MvPolynomial.coeff_baseChange_apply | [106, 1] | [124, 29] | rfl | case pos
R : Type u_1
inst✝⁴ : CommSemiring R
S : Type u_4
inst✝³ : CommSemiring S
inst✝² : Algebra R S
S' : Type u_2
inst✝¹ : CommSemiring S'
inst✝ : Algebra R S'
σ : Type u_3
φ : S →ₐ[R] S'
r : S
m : σ →₀ ℕ
h✝ : 0 = m
⊢ φ r = φ r | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case pos
R : Type u_1
inst✝⁴ : CommSemiring R
S : Type u_4
inst✝³ : CommSemiring S
inst✝² : Algebra R S
S' : Type u_2
inst✝¹ : CommSemiring S'
inst✝ : Algebra R S'
σ : Type u_3
φ : S →ₐ[R] S'
r : S
m : σ →₀ ℕ
h✝ : 0 = m
⊢ φ r = φ r
TACTIC:
|
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/DPAlgebra/BaseChange.lean | MvPolynomial.coeff_baseChange_apply | [106, 1] | [124, 29] | rw [map_zero] | case neg
R : Type u_1
inst✝⁴ : CommSemiring R
S : Type u_4
inst✝³ : CommSemiring S
inst✝² : Algebra R S
S' : Type u_2
inst✝¹ : CommSemiring S'
inst✝ : Algebra R S'
σ : Type u_3
φ : S →ₐ[R] S'
r : S
m : σ →₀ ℕ
h✝ : ¬0 = m
⊢ 0 = φ 0 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case neg
R : Type u_1
inst✝⁴ : CommSemiring R
S : Type u_4
inst✝³ : CommSemiring S
inst✝² : Algebra R S
S' : Type u_2
inst✝¹ : CommSemiring S'
inst✝ : Algebra R S'
σ : Type u_3
φ : S →ₐ[R] S'
r : S
m : σ →₀ ℕ
h✝ : ¬0 = m
⊢ 0 = φ 0
TACTIC:
|
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/DPAlgebra/BaseChange.lean | MvPolynomial.coeff_baseChange_apply | [106, 1] | [124, 29] | simp only [eval₂_add, coeff_add, hf, hg, map_add] | case h_add
R : Type u_1
inst✝⁴ : CommSemiring R
S : Type u_4
inst✝³ : CommSemiring S
inst✝² : Algebra R S
S' : Type u_2
inst✝¹ : CommSemiring S'
inst✝ : Algebra R S'
σ : Type u_3
φ : S →ₐ[R] S'
f g : MvPolynomial σ S
hf : ∀ (m : σ →₀ ℕ), coeff m (eval₂ (C.comp ↑φ) X f) = φ (coeff m f)
hg : ∀ (m : σ →₀ ℕ), coeff m (eval... | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case h_add
R : Type u_1
inst✝⁴ : CommSemiring R
S : Type u_4
inst✝³ : CommSemiring S
inst✝² : Algebra R S
S' : Type u_2
inst✝¹ : CommSemiring S'
inst✝ : Algebra R S'
σ : Type u_3
φ : S →ₐ[R] S'
f g : MvPolynomial σ S
hf : ∀ (m : σ →₀ ℕ), coeff m (eval₂ (C.com... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/DPAlgebra/BaseChange.lean | MvPolynomial.coeff_baseChange_apply | [106, 1] | [124, 29] | simp only [eval₂_mul, eval₂_X, coeff_mul, map_sum, _root_.map_mul] | case h_X
R : Type u_1
inst✝⁴ : CommSemiring R
S : Type u_4
inst✝³ : CommSemiring S
inst✝² : Algebra R S
S' : Type u_2
inst✝¹ : CommSemiring S'
inst✝ : Algebra R S'
σ : Type u_3
φ : S →ₐ[R] S'
p : MvPolynomial σ S
s : σ
h : ∀ (m : σ →₀ ℕ), coeff m (eval₂ (C.comp ↑φ) X p) = φ (coeff m p)
m : σ →₀ ℕ
⊢ coeff m (eval₂ (C.co... | case h_X
R : Type u_1
inst✝⁴ : CommSemiring R
S : Type u_4
inst✝³ : CommSemiring S
inst✝² : Algebra R S
S' : Type u_2
inst✝¹ : CommSemiring S'
inst✝ : Algebra R S'
σ : Type u_3
φ : S →ₐ[R] S'
p : MvPolynomial σ S
s : σ
h : ∀ (m : σ →₀ ℕ), coeff m (eval₂ (C.comp ↑φ) X p) = φ (coeff m p)
m : σ →₀ ℕ
⊢ ∑ x ∈ Finset.antidia... | Please generate a tactic in lean4 to solve the state.
STATE:
case h_X
R : Type u_1
inst✝⁴ : CommSemiring R
S : Type u_4
inst✝³ : CommSemiring S
inst✝² : Algebra R S
S' : Type u_2
inst✝¹ : CommSemiring S'
inst✝ : Algebra R S'
σ : Type u_3
φ : S →ₐ[R] S'
p : MvPolynomial σ S
s : σ
h : ∀ (m : σ →₀ ℕ), coeff m (eval₂ (C.co... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/DPAlgebra/BaseChange.lean | MvPolynomial.coeff_baseChange_apply | [106, 1] | [124, 29] | apply Finset.sum_congr rfl | case h_X
R : Type u_1
inst✝⁴ : CommSemiring R
S : Type u_4
inst✝³ : CommSemiring S
inst✝² : Algebra R S
S' : Type u_2
inst✝¹ : CommSemiring S'
inst✝ : Algebra R S'
σ : Type u_3
φ : S →ₐ[R] S'
p : MvPolynomial σ S
s : σ
h : ∀ (m : σ →₀ ℕ), coeff m (eval₂ (C.comp ↑φ) X p) = φ (coeff m p)
m : σ →₀ ℕ
⊢ ∑ x ∈ Finset.antidia... | case h_X
R : Type u_1
inst✝⁴ : CommSemiring R
S : Type u_4
inst✝³ : CommSemiring S
inst✝² : Algebra R S
S' : Type u_2
inst✝¹ : CommSemiring S'
inst✝ : Algebra R S'
σ : Type u_3
φ : S →ₐ[R] S'
p : MvPolynomial σ S
s : σ
h : ∀ (m : σ →₀ ℕ), coeff m (eval₂ (C.comp ↑φ) X p) = φ (coeff m p)
m : σ →₀ ℕ
⊢ ∀ x ∈ Finset.antidia... | Please generate a tactic in lean4 to solve the state.
STATE:
case h_X
R : Type u_1
inst✝⁴ : CommSemiring R
S : Type u_4
inst✝³ : CommSemiring S
inst✝² : Algebra R S
S' : Type u_2
inst✝¹ : CommSemiring S'
inst✝ : Algebra R S'
σ : Type u_3
φ : S →ₐ[R] S'
p : MvPolynomial σ S
s : σ
h : ∀ (m : σ →₀ ℕ), coeff m (eval₂ (C.co... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/DPAlgebra/BaseChange.lean | MvPolynomial.coeff_baseChange_apply | [106, 1] | [124, 29] | intro x _ | case h_X
R : Type u_1
inst✝⁴ : CommSemiring R
S : Type u_4
inst✝³ : CommSemiring S
inst✝² : Algebra R S
S' : Type u_2
inst✝¹ : CommSemiring S'
inst✝ : Algebra R S'
σ : Type u_3
φ : S →ₐ[R] S'
p : MvPolynomial σ S
s : σ
h : ∀ (m : σ →₀ ℕ), coeff m (eval₂ (C.comp ↑φ) X p) = φ (coeff m p)
m : σ →₀ ℕ
⊢ ∀ x ∈ Finset.antidia... | case h_X
R : Type u_1
inst✝⁴ : CommSemiring R
S : Type u_4
inst✝³ : CommSemiring S
inst✝² : Algebra R S
S' : Type u_2
inst✝¹ : CommSemiring S'
inst✝ : Algebra R S'
σ : Type u_3
φ : S →ₐ[R] S'
p : MvPolynomial σ S
s : σ
h : ∀ (m : σ →₀ ℕ), coeff m (eval₂ (C.comp ↑φ) X p) = φ (coeff m p)
m : σ →₀ ℕ
x : (σ →₀ ℕ) × (σ →₀ ℕ... | Please generate a tactic in lean4 to solve the state.
STATE:
case h_X
R : Type u_1
inst✝⁴ : CommSemiring R
S : Type u_4
inst✝³ : CommSemiring S
inst✝² : Algebra R S
S' : Type u_2
inst✝¹ : CommSemiring S'
inst✝ : Algebra R S'
σ : Type u_3
φ : S →ₐ[R] S'
p : MvPolynomial σ S
s : σ
h : ∀ (m : σ →₀ ℕ), coeff m (eval₂ (C.co... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/DPAlgebra/BaseChange.lean | MvPolynomial.coeff_baseChange_apply | [106, 1] | [124, 29] | simp only [h x.1, coeff_X'] | case h_X
R : Type u_1
inst✝⁴ : CommSemiring R
S : Type u_4
inst✝³ : CommSemiring S
inst✝² : Algebra R S
S' : Type u_2
inst✝¹ : CommSemiring S'
inst✝ : Algebra R S'
σ : Type u_3
φ : S →ₐ[R] S'
p : MvPolynomial σ S
s : σ
h : ∀ (m : σ →₀ ℕ), coeff m (eval₂ (C.comp ↑φ) X p) = φ (coeff m p)
m : σ →₀ ℕ
x : (σ →₀ ℕ) × (σ →₀ ℕ... | case h_X
R : Type u_1
inst✝⁴ : CommSemiring R
S : Type u_4
inst✝³ : CommSemiring S
inst✝² : Algebra R S
S' : Type u_2
inst✝¹ : CommSemiring S'
inst✝ : Algebra R S'
σ : Type u_3
φ : S →ₐ[R] S'
p : MvPolynomial σ S
s : σ
h : ∀ (m : σ →₀ ℕ), coeff m (eval₂ (C.comp ↑φ) X p) = φ (coeff m p)
m : σ →₀ ℕ
x : (σ →₀ ℕ) × (σ →₀ ℕ... | Please generate a tactic in lean4 to solve the state.
STATE:
case h_X
R : Type u_1
inst✝⁴ : CommSemiring R
S : Type u_4
inst✝³ : CommSemiring S
inst✝² : Algebra R S
S' : Type u_2
inst✝¹ : CommSemiring S'
inst✝ : Algebra R S'
σ : Type u_3
φ : S →ₐ[R] S'
p : MvPolynomial σ S
s : σ
h : ∀ (m : σ →₀ ℕ), coeff m (eval₂ (C.co... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/DPAlgebra/BaseChange.lean | MvPolynomial.coeff_baseChange_apply | [106, 1] | [124, 29] | split_ifs | case h_X
R : Type u_1
inst✝⁴ : CommSemiring R
S : Type u_4
inst✝³ : CommSemiring S
inst✝² : Algebra R S
S' : Type u_2
inst✝¹ : CommSemiring S'
inst✝ : Algebra R S'
σ : Type u_3
φ : S →ₐ[R] S'
p : MvPolynomial σ S
s : σ
h : ∀ (m : σ →₀ ℕ), coeff m (eval₂ (C.comp ↑φ) X p) = φ (coeff m p)
m : σ →₀ ℕ
x : (σ →₀ ℕ) × (σ →₀ ℕ... | case pos
R : Type u_1
inst✝⁴ : CommSemiring R
S : Type u_4
inst✝³ : CommSemiring S
inst✝² : Algebra R S
S' : Type u_2
inst✝¹ : CommSemiring S'
inst✝ : Algebra R S'
σ : Type u_3
φ : S →ₐ[R] S'
p : MvPolynomial σ S
s : σ
h : ∀ (m : σ →₀ ℕ), coeff m (eval₂ (C.comp ↑φ) X p) = φ (coeff m p)
m : σ →₀ ℕ
x : (σ →₀ ℕ) × (σ →₀ ℕ... | Please generate a tactic in lean4 to solve the state.
STATE:
case h_X
R : Type u_1
inst✝⁴ : CommSemiring R
S : Type u_4
inst✝³ : CommSemiring S
inst✝² : Algebra R S
S' : Type u_2
inst✝¹ : CommSemiring S'
inst✝ : Algebra R S'
σ : Type u_3
φ : S →ₐ[R] S'
p : MvPolynomial σ S
s : σ
h : ∀ (m : σ →₀ ℕ), coeff m (eval₂ (C.co... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/DPAlgebra/BaseChange.lean | MvPolynomial.coeff_baseChange_apply | [106, 1] | [124, 29] | rw [_root_.map_one] | case pos
R : Type u_1
inst✝⁴ : CommSemiring R
S : Type u_4
inst✝³ : CommSemiring S
inst✝² : Algebra R S
S' : Type u_2
inst✝¹ : CommSemiring S'
inst✝ : Algebra R S'
σ : Type u_3
φ : S →ₐ[R] S'
p : MvPolynomial σ S
s : σ
h : ∀ (m : σ →₀ ℕ), coeff m (eval₂ (C.comp ↑φ) X p) = φ (coeff m p)
m : σ →₀ ℕ
x : (σ →₀ ℕ) × (σ →₀ ℕ... | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case pos
R : Type u_1
inst✝⁴ : CommSemiring R
S : Type u_4
inst✝³ : CommSemiring S
inst✝² : Algebra R S
S' : Type u_2
inst✝¹ : CommSemiring S'
inst✝ : Algebra R S'
σ : Type u_3
φ : S →ₐ[R] S'
p : MvPolynomial σ S
s : σ
h : ∀ (m : σ →₀ ℕ), coeff m (eval₂ (C.co... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/DPAlgebra/BaseChange.lean | MvPolynomial.coeff_baseChange_apply | [106, 1] | [124, 29] | rw [_root_.map_zero] | case neg
R : Type u_1
inst✝⁴ : CommSemiring R
S : Type u_4
inst✝³ : CommSemiring S
inst✝² : Algebra R S
S' : Type u_2
inst✝¹ : CommSemiring S'
inst✝ : Algebra R S'
σ : Type u_3
φ : S →ₐ[R] S'
p : MvPolynomial σ S
s : σ
h : ∀ (m : σ →₀ ℕ), coeff m (eval₂ (C.comp ↑φ) X p) = φ (coeff m p)
m : σ →₀ ℕ
x : (σ →₀ ℕ) × (σ →₀ ℕ... | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case neg
R : Type u_1
inst✝⁴ : CommSemiring R
S : Type u_4
inst✝³ : CommSemiring S
inst✝² : Algebra R S
S' : Type u_2
inst✝¹ : CommSemiring S'
inst✝ : Algebra R S'
σ : Type u_3
φ : S →ₐ[R] S'
p : MvPolynomial σ S
s : σ
h : ∀ (m : σ →₀ ℕ), coeff m (eval₂ (C.co... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/DPAlgebra/BaseChange.lean | MvPolynomial.lcoeff_comp_baseChange_eq | [126, 1] | [131, 54] | ext f | R : Type u_1
inst✝⁴ : CommSemiring R
S : Type u_4
inst✝³ : CommSemiring S
inst✝² : Algebra R S
S' : Type u_2
inst✝¹ : CommSemiring S'
inst✝ : Algebra R S'
σ : Type u_3
φ : S →ₐ[R] S'
m : σ →₀ ℕ
⊢ φ.toLinearMap ∘ₗ ↑R (lcoeff S m) = ↑R (lcoeff S' m) ∘ₗ (baseChange φ).toLinearMap | case h
R : Type u_1
inst✝⁴ : CommSemiring R
S : Type u_4
inst✝³ : CommSemiring S
inst✝² : Algebra R S
S' : Type u_2
inst✝¹ : CommSemiring S'
inst✝ : Algebra R S'
σ : Type u_3
φ : S →ₐ[R] S'
m : σ →₀ ℕ
f : MvPolynomial σ S
⊢ (φ.toLinearMap ∘ₗ ↑R (lcoeff S m)) f = (↑R (lcoeff S' m) ∘ₗ (baseChange φ).toLinearMap) f | Please generate a tactic in lean4 to solve the state.
STATE:
R : Type u_1
inst✝⁴ : CommSemiring R
S : Type u_4
inst✝³ : CommSemiring S
inst✝² : Algebra R S
S' : Type u_2
inst✝¹ : CommSemiring S'
inst✝ : Algebra R S'
σ : Type u_3
φ : S →ₐ[R] S'
m : σ →₀ ℕ
⊢ φ.toLinearMap ∘ₗ ↑R (lcoeff S m) = ↑R (lcoeff S' m) ∘ₗ (baseCha... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/DPAlgebra/BaseChange.lean | MvPolynomial.lcoeff_comp_baseChange_eq | [126, 1] | [131, 54] | simp only [LinearMap.coe_comp, LinearMap.coe_restrictScalars, Function.comp_apply, lcoeff_apply,
AlgHom.toLinearMap_apply, coeff_baseChange_apply] | case h
R : Type u_1
inst✝⁴ : CommSemiring R
S : Type u_4
inst✝³ : CommSemiring S
inst✝² : Algebra R S
S' : Type u_2
inst✝¹ : CommSemiring S'
inst✝ : Algebra R S'
σ : Type u_3
φ : S →ₐ[R] S'
m : σ →₀ ℕ
f : MvPolynomial σ S
⊢ (φ.toLinearMap ∘ₗ ↑R (lcoeff S m)) f = (↑R (lcoeff S' m) ∘ₗ (baseChange φ).toLinearMap) f | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case h
R : Type u_1
inst✝⁴ : CommSemiring R
S : Type u_4
inst✝³ : CommSemiring S
inst✝² : Algebra R S
S' : Type u_2
inst✝¹ : CommSemiring S'
inst✝ : Algebra R S'
σ : Type u_3
φ : S →ₐ[R] S'
m : σ →₀ ℕ
f : MvPolynomial σ S
⊢ (φ.toLinearMap ∘ₗ ↑R (lcoeff S m)) ... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/DPAlgebra/BaseChange.lean | MvPolynomial.baseChange_monomial | [133, 1] | [137, 19] | simp only [baseChange, coe_mk, coe_eval₂RingHom, eval₂_monomial, RingHom.coe_comp,
RingHom.coe_coe, Function.comp_apply] | R : Type u_1
inst✝⁴ : CommSemiring R
S : Type u_4
inst✝³ : CommSemiring S
inst✝² : Algebra R S
S' : Type u_2
inst✝¹ : CommSemiring S'
inst✝ : Algebra R S'
σ : Type u_3
φ : S →ₐ[R] S'
m : σ →₀ ℕ
a : S
⊢ (baseChange φ) ((monomial m) a) = (monomial m) (φ a) | R : Type u_1
inst✝⁴ : CommSemiring R
S : Type u_4
inst✝³ : CommSemiring S
inst✝² : Algebra R S
S' : Type u_2
inst✝¹ : CommSemiring S'
inst✝ : Algebra R S'
σ : Type u_3
φ : S →ₐ[R] S'
m : σ →₀ ℕ
a : S
⊢ (C (φ a) * m.prod fun n e => X n ^ e) = (monomial m) (φ a) | Please generate a tactic in lean4 to solve the state.
STATE:
R : Type u_1
inst✝⁴ : CommSemiring R
S : Type u_4
inst✝³ : CommSemiring S
inst✝² : Algebra R S
S' : Type u_2
inst✝¹ : CommSemiring S'
inst✝ : Algebra R S'
σ : Type u_3
φ : S →ₐ[R] S'
m : σ →₀ ℕ
a : S
⊢ (baseChange φ) ((monomial m) a) = (monomial m) (φ a)
TACT... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/DPAlgebra/BaseChange.lean | MvPolynomial.baseChange_monomial | [133, 1] | [137, 19] | rw [monomial_eq] | R : Type u_1
inst✝⁴ : CommSemiring R
S : Type u_4
inst✝³ : CommSemiring S
inst✝² : Algebra R S
S' : Type u_2
inst✝¹ : CommSemiring S'
inst✝ : Algebra R S'
σ : Type u_3
φ : S →ₐ[R] S'
m : σ →₀ ℕ
a : S
⊢ (C (φ a) * m.prod fun n e => X n ^ e) = (monomial m) (φ a) | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
R : Type u_1
inst✝⁴ : CommSemiring R
S : Type u_4
inst✝³ : CommSemiring S
inst✝² : Algebra R S
S' : Type u_2
inst✝¹ : CommSemiring S'
inst✝ : Algebra R S'
σ : Type u_3
φ : S →ₐ[R] S'
m : σ →₀ ℕ
a : S
⊢ (C (φ a) * m.prod fun n e => X n ^ e) = (monomial m) (φ a... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/DPAlgebra/BaseChange.lean | DividedPowerAlgebra.AlgHom.baseChange_tmul | [154, 1] | [159, 65] | simp only [baseChange, toRingHom_eq_coe, coe_mk, RingHom.coe_coe, productMap_apply_tmul,
IsScalarTower.coe_toAlgHom', ← smul_eq_mul, algebraMap_smul] | R : Type u_1
A : Type u_2
B : Type u_3
C : Type u_4
inst✝⁸ : CommSemiring R
inst✝⁷ : CommSemiring A
inst✝⁶ : Algebra R A
inst✝⁵ : CommSemiring B
inst✝⁴ : Algebra R B
inst✝³ : CommSemiring C
inst✝² : Algebra R C
inst✝¹ : Algebra A C
inst✝ : IsScalarTower R A C
φ : B →ₐ[R] C
a : A
b : B
⊢ (baseChange φ) (a ⊗ₜ[R] b) = a •... | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
R : Type u_1
A : Type u_2
B : Type u_3
C : Type u_4
inst✝⁸ : CommSemiring R
inst✝⁷ : CommSemiring A
inst✝⁶ : Algebra R A
inst✝⁵ : CommSemiring B
inst✝⁴ : Algebra R B
inst✝³ : CommSemiring C
inst✝² : Algebra R C
inst✝¹ : Algebra A C
inst✝ : IsScalarTower R A C... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/DPAlgebra/BaseChange.lean | DividedPowerAlgebra.dpScalarExtension_apply_dp | [197, 1] | [200, 26] | rw [dpScalarExtension, AlgHom.baseChange_tmul, lift'AlgHom_apply_dp, ← dp_smul, smul_tmul',
smul_eq_mul, mul_one] | A : Type u_1
inst✝⁴ : CommSemiring A
R : Type u_2
inst✝³ : CommSemiring R
inst✝² : Algebra A R
M : Type u_3
inst✝¹ : AddCommMonoid M
inst✝ : Module A M
r : R
n : ℕ
m : M
⊢ (dpScalarExtension A R M) ((r ^ n) ⊗ₜ[A] dp A n m) = dp R n (r ⊗ₜ[A] m) | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
A : Type u_1
inst✝⁴ : CommSemiring A
R : Type u_2
inst✝³ : CommSemiring R
inst✝² : Algebra A R
M : Type u_3
inst✝¹ : AddCommMonoid M
inst✝ : Module A M
r : R
n : ℕ
m : M
⊢ (dpScalarExtension A R M) ((r ^ n) ⊗ₜ[A] dp A n m) = dp R n (r ⊗ₜ[A] m)
TACTIC:
|
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/DPAlgebra/BaseChange.lean | DividedPowerAlgebra.dpScalarExtension_apply_one_dp | [202, 1] | [204, 56] | rw [← one_pow n, dpScalarExtension_apply_dp, one_pow] | A : Type u_1
inst✝⁴ : CommSemiring A
R : Type u_2
inst✝³ : CommSemiring R
inst✝² : Algebra A R
M : Type u_3
inst✝¹ : AddCommMonoid M
inst✝ : Module A M
n : ℕ
m : M
⊢ (dpScalarExtension A R M) (1 ⊗ₜ[A] dp A n m) = dp R n (1 ⊗ₜ[A] m) | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
A : Type u_1
inst✝⁴ : CommSemiring A
R : Type u_2
inst✝³ : CommSemiring R
inst✝² : Algebra A R
M : Type u_3
inst✝¹ : AddCommMonoid M
inst✝ : Module A M
n : ℕ
m : M
⊢ (dpScalarExtension A R M) (1 ⊗ₜ[A] dp A n m) = dp R n (1 ⊗ₜ[A] m)
TACTIC:
|
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/DPAlgebra/BaseChange.lean | DividedPowerAlgebra.dpScalarExtension_tmul | [206, 1] | [208, 66] | simp only [dpScalarExtension, AlgHom.baseChange_tmul, one_smul] | A : Type u_1
inst✝⁴ : CommSemiring A
R : Type u_2
inst✝³ : CommSemiring R
inst✝² : Algebra A R
M : Type u_3
inst✝¹ : AddCommMonoid M
inst✝ : Module A M
r : R
m : DividedPowerAlgebra A M
⊢ (dpScalarExtension A R M) (r ⊗ₜ[A] m) = r • (dpScalarExtension A R M) (1 ⊗ₜ[A] m) | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
A : Type u_1
inst✝⁴ : CommSemiring A
R : Type u_2
inst✝³ : CommSemiring R
inst✝² : Algebra A R
M : Type u_3
inst✝¹ : AddCommMonoid M
inst✝ : Module A M
r : R
m : DividedPowerAlgebra A M
⊢ (dpScalarExtension A R M) (r ⊗ₜ[A] m) = r • (dpScalarExtension A R M) (... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/DPAlgebra/BaseChange.lean | DividedPowerAlgebra.dpScalarExtension_unique | [211, 1] | [229, 49] | apply AlgHom.ext | A : Type u_1
inst✝⁴ : CommSemiring A
R : Type u_2
inst✝³ : CommSemiring R
inst✝² : Algebra A R
M : Type u_3
inst✝¹ : AddCommMonoid M
inst✝ : Module A M
φ : R ⊗[A] DividedPowerAlgebra A M →ₐ[R] DividedPowerAlgebra R (R ⊗[A] M)
hφ : ∀ (n : ℕ) (m : M), φ (1 ⊗ₜ[A] dp A n m) = dp R n (1 ⊗ₜ[A] m)
⊢ φ = dpScalarExtension A R ... | case H
A : Type u_1
inst✝⁴ : CommSemiring A
R : Type u_2
inst✝³ : CommSemiring R
inst✝² : Algebra A R
M : Type u_3
inst✝¹ : AddCommMonoid M
inst✝ : Module A M
φ : R ⊗[A] DividedPowerAlgebra A M →ₐ[R] DividedPowerAlgebra R (R ⊗[A] M)
hφ : ∀ (n : ℕ) (m : M), φ (1 ⊗ₜ[A] dp A n m) = dp R n (1 ⊗ₜ[A] m)
⊢ ∀ (x : R ⊗[A] Divid... | Please generate a tactic in lean4 to solve the state.
STATE:
A : Type u_1
inst✝⁴ : CommSemiring A
R : Type u_2
inst✝³ : CommSemiring R
inst✝² : Algebra A R
M : Type u_3
inst✝¹ : AddCommMonoid M
inst✝ : Module A M
φ : R ⊗[A] DividedPowerAlgebra A M →ₐ[R] DividedPowerAlgebra R (R ⊗[A] M)
hφ : ∀ (n : ℕ) (m : M), φ (1 ⊗ₜ[A... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/DPAlgebra/BaseChange.lean | DividedPowerAlgebra.dpScalarExtension_unique | [211, 1] | [229, 49] | intro x | case H
A : Type u_1
inst✝⁴ : CommSemiring A
R : Type u_2
inst✝³ : CommSemiring R
inst✝² : Algebra A R
M : Type u_3
inst✝¹ : AddCommMonoid M
inst✝ : Module A M
φ : R ⊗[A] DividedPowerAlgebra A M →ₐ[R] DividedPowerAlgebra R (R ⊗[A] M)
hφ : ∀ (n : ℕ) (m : M), φ (1 ⊗ₜ[A] dp A n m) = dp R n (1 ⊗ₜ[A] m)
⊢ ∀ (x : R ⊗[A] Divid... | case H
A : Type u_1
inst✝⁴ : CommSemiring A
R : Type u_2
inst✝³ : CommSemiring R
inst✝² : Algebra A R
M : Type u_3
inst✝¹ : AddCommMonoid M
inst✝ : Module A M
φ : R ⊗[A] DividedPowerAlgebra A M →ₐ[R] DividedPowerAlgebra R (R ⊗[A] M)
hφ : ∀ (n : ℕ) (m : M), φ (1 ⊗ₜ[A] dp A n m) = dp R n (1 ⊗ₜ[A] m)
x : R ⊗[A] DividedPow... | Please generate a tactic in lean4 to solve the state.
STATE:
case H
A : Type u_1
inst✝⁴ : CommSemiring A
R : Type u_2
inst✝³ : CommSemiring R
inst✝² : Algebra A R
M : Type u_3
inst✝¹ : AddCommMonoid M
inst✝ : Module A M
φ : R ⊗[A] DividedPowerAlgebra A M →ₐ[R] DividedPowerAlgebra R (R ⊗[A] M)
hφ : ∀ (n : ℕ) (m : M), φ ... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/DPAlgebra/BaseChange.lean | DividedPowerAlgebra.dpScalarExtension_unique | [211, 1] | [229, 49] | induction x using TensorProduct.induction_on with
| zero => simp only [map_zero]
| tmul s x =>
induction x using DividedPowerAlgebra.induction_on with
| h_C a =>
rw [mk_C, Algebra.algebraMap_eq_smul_one, tmul_smul, smul_tmul', ← mul_one s, ← smul_eq_mul,
← smul_assoc, ← smul_tmul', map_smul,map_smul, ← Al... | case H
A : Type u_1
inst✝⁴ : CommSemiring A
R : Type u_2
inst✝³ : CommSemiring R
inst✝² : Algebra A R
M : Type u_3
inst✝¹ : AddCommMonoid M
inst✝ : Module A M
φ : R ⊗[A] DividedPowerAlgebra A M →ₐ[R] DividedPowerAlgebra R (R ⊗[A] M)
hφ : ∀ (n : ℕ) (m : M), φ (1 ⊗ₜ[A] dp A n m) = dp R n (1 ⊗ₜ[A] m)
x : R ⊗[A] DividedPow... | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case H
A : Type u_1
inst✝⁴ : CommSemiring A
R : Type u_2
inst✝³ : CommSemiring R
inst✝² : Algebra A R
M : Type u_3
inst✝¹ : AddCommMonoid M
inst✝ : Module A M
φ : R ⊗[A] DividedPowerAlgebra A M →ₐ[R] DividedPowerAlgebra R (R ⊗[A] M)
hφ : ∀ (n : ℕ) (m : M), φ ... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/DPAlgebra/BaseChange.lean | DividedPowerAlgebra.dpScalarExtension_unique | [211, 1] | [229, 49] | simp only [map_zero] | case H.zero
A : Type u_1
inst✝⁴ : CommSemiring A
R : Type u_2
inst✝³ : CommSemiring R
inst✝² : Algebra A R
M : Type u_3
inst✝¹ : AddCommMonoid M
inst✝ : Module A M
φ : R ⊗[A] DividedPowerAlgebra A M →ₐ[R] DividedPowerAlgebra R (R ⊗[A] M)
hφ : ∀ (n : ℕ) (m : M), φ (1 ⊗ₜ[A] dp A n m) = dp R n (1 ⊗ₜ[A] m)
⊢ φ 0 = (dpScala... | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case H.zero
A : Type u_1
inst✝⁴ : CommSemiring A
R : Type u_2
inst✝³ : CommSemiring R
inst✝² : Algebra A R
M : Type u_3
inst✝¹ : AddCommMonoid M
inst✝ : Module A M
φ : R ⊗[A] DividedPowerAlgebra A M →ₐ[R] DividedPowerAlgebra R (R ⊗[A] M)
hφ : ∀ (n : ℕ) (m : M... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/DPAlgebra/BaseChange.lean | DividedPowerAlgebra.dpScalarExtension_unique | [211, 1] | [229, 49] | induction x using DividedPowerAlgebra.induction_on with
| h_C a =>
rw [mk_C, Algebra.algebraMap_eq_smul_one, tmul_smul, smul_tmul', ← mul_one s, ← smul_eq_mul,
← smul_assoc, ← smul_tmul', map_smul,map_smul, ← Algebra.TensorProduct.one_def,
_root_.map_one, _root_.map_one]
| h_add f g hf hg => simp only [tmul_a... | case H.tmul
A : Type u_1
inst✝⁴ : CommSemiring A
R : Type u_2
inst✝³ : CommSemiring R
inst✝² : Algebra A R
M : Type u_3
inst✝¹ : AddCommMonoid M
inst✝ : Module A M
φ : R ⊗[A] DividedPowerAlgebra A M →ₐ[R] DividedPowerAlgebra R (R ⊗[A] M)
hφ : ∀ (n : ℕ) (m : M), φ (1 ⊗ₜ[A] dp A n m) = dp R n (1 ⊗ₜ[A] m)
s : R
x : Divide... | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case H.tmul
A : Type u_1
inst✝⁴ : CommSemiring A
R : Type u_2
inst✝³ : CommSemiring R
inst✝² : Algebra A R
M : Type u_3
inst✝¹ : AddCommMonoid M
inst✝ : Module A M
φ : R ⊗[A] DividedPowerAlgebra A M →ₐ[R] DividedPowerAlgebra R (R ⊗[A] M)
hφ : ∀ (n : ℕ) (m : M... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/DPAlgebra/BaseChange.lean | DividedPowerAlgebra.dpScalarExtension_unique | [211, 1] | [229, 49] | rw [mk_C, Algebra.algebraMap_eq_smul_one, tmul_smul, smul_tmul', ← mul_one s, ← smul_eq_mul,
← smul_assoc, ← smul_tmul', map_smul,map_smul, ← Algebra.TensorProduct.one_def,
_root_.map_one, _root_.map_one] | case H.tmul.h_C
A : Type u_1
inst✝⁴ : CommSemiring A
R : Type u_2
inst✝³ : CommSemiring R
inst✝² : Algebra A R
M : Type u_3
inst✝¹ : AddCommMonoid M
inst✝ : Module A M
φ : R ⊗[A] DividedPowerAlgebra A M →ₐ[R] DividedPowerAlgebra R (R ⊗[A] M)
hφ : ∀ (n : ℕ) (m : M), φ (1 ⊗ₜ[A] dp A n m) = dp R n (1 ⊗ₜ[A] m)
s : R
a : A
... | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case H.tmul.h_C
A : Type u_1
inst✝⁴ : CommSemiring A
R : Type u_2
inst✝³ : CommSemiring R
inst✝² : Algebra A R
M : Type u_3
inst✝¹ : AddCommMonoid M
inst✝ : Module A M
φ : R ⊗[A] DividedPowerAlgebra A M →ₐ[R] DividedPowerAlgebra R (R ⊗[A] M)
hφ : ∀ (n : ℕ) (m... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/DPAlgebra/BaseChange.lean | DividedPowerAlgebra.dpScalarExtension_unique | [211, 1] | [229, 49] | simp only [tmul_add, map_add, hf, hg] | case H.tmul.h_add
A : Type u_1
inst✝⁴ : CommSemiring A
R : Type u_2
inst✝³ : CommSemiring R
inst✝² : Algebra A R
M : Type u_3
inst✝¹ : AddCommMonoid M
inst✝ : Module A M
φ : R ⊗[A] DividedPowerAlgebra A M →ₐ[R] DividedPowerAlgebra R (R ⊗[A] M)
hφ : ∀ (n : ℕ) (m : M), φ (1 ⊗ₜ[A] dp A n m) = dp R n (1 ⊗ₜ[A] m)
s : R
f g ... | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case H.tmul.h_add
A : Type u_1
inst✝⁴ : CommSemiring A
R : Type u_2
inst✝³ : CommSemiring R
inst✝² : Algebra A R
M : Type u_3
inst✝¹ : AddCommMonoid M
inst✝ : Module A M
φ : R ⊗[A] DividedPowerAlgebra A M →ₐ[R] DividedPowerAlgebra R (R ⊗[A] M)
hφ : ∀ (n : ℕ) ... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/DPAlgebra/BaseChange.lean | DividedPowerAlgebra.dpScalarExtension_unique | [211, 1] | [229, 49] | rw [← mul_one s, ← tmul_mul_tmul, _root_.map_mul, _root_.map_mul, hf, hφ n m,
dpScalarExtension_apply_one_dp] | case H.tmul.h_dp
A : Type u_1
inst✝⁴ : CommSemiring A
R : Type u_2
inst✝³ : CommSemiring R
inst✝² : Algebra A R
M : Type u_3
inst✝¹ : AddCommMonoid M
inst✝ : Module A M
φ : R ⊗[A] DividedPowerAlgebra A M →ₐ[R] DividedPowerAlgebra R (R ⊗[A] M)
hφ : ∀ (n : ℕ) (m : M), φ (1 ⊗ₜ[A] dp A n m) = dp R n (1 ⊗ₜ[A] m)
s : R
f : D... | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case H.tmul.h_dp
A : Type u_1
inst✝⁴ : CommSemiring A
R : Type u_2
inst✝³ : CommSemiring R
inst✝² : Algebra A R
M : Type u_3
inst✝¹ : AddCommMonoid M
inst✝ : Module A M
φ : R ⊗[A] DividedPowerAlgebra A M →ₐ[R] DividedPowerAlgebra R (R ⊗[A] M)
hφ : ∀ (n : ℕ) (... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/DPAlgebra/BaseChange.lean | DividedPowerAlgebra.dpScalarExtension_unique | [211, 1] | [229, 49] | simp only [map_add, hx, hy] | case H.add
A : Type u_1
inst✝⁴ : CommSemiring A
R : Type u_2
inst✝³ : CommSemiring R
inst✝² : Algebra A R
M : Type u_3
inst✝¹ : AddCommMonoid M
inst✝ : Module A M
φ : R ⊗[A] DividedPowerAlgebra A M →ₐ[R] DividedPowerAlgebra R (R ⊗[A] M)
hφ : ∀ (n : ℕ) (m : M), φ (1 ⊗ₜ[A] dp A n m) = dp R n (1 ⊗ₜ[A] m)
x y : R ⊗[A] Divi... | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case H.add
A : Type u_1
inst✝⁴ : CommSemiring A
R : Type u_2
inst✝³ : CommSemiring R
inst✝² : Algebra A R
M : Type u_3
inst✝¹ : AddCommMonoid M
inst✝ : Module A M
φ : R ⊗[A] DividedPowerAlgebra A M →ₐ[R] DividedPowerAlgebra R (R ⊗[A] M)
hφ : ∀ (n : ℕ) (m : M)... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/DPAlgebra/BaseChange.lean | DividedPowerAlgebra.dpScalarExtensionInv_apply_dp | [247, 1] | [255, 74] | rw [dpScalarExtensionInv, dividedPowerAlgebra_exponentialModule_equiv_symm_apply] | R : Type u_1
inst✝⁴ : CommRing R
S : Type u_2
inst✝³ : CommRing S
inst✝² : Algebra R S
M : Type u_3
inst✝¹ : AddCommMonoid M
inst✝ : Module R M
n : ℕ
s : S
m : M
⊢ (dpScalarExtensionInv R S M) (dp S n (s ⊗ₜ[R] m)) = (s ^ n) ⊗ₜ[R] dp R n m | R : Type u_1
inst✝⁴ : CommRing R
S : Type u_2
inst✝³ : CommRing S
inst✝² : Algebra R S
M : Type u_3
inst✝¹ : AddCommMonoid M
inst✝ : Module R M
n : ℕ
s : S
m : M
⊢ (PowerSeries.coeff (S ⊗[R] DividedPowerAlgebra R M) n) ↑((dpScalarExtensionExp R S M) (s ⊗ₜ[R] m)) =
(s ^ n) ⊗ₜ[R] dp R n m | Please generate a tactic in lean4 to solve the state.
STATE:
R : Type u_1
inst✝⁴ : CommRing R
S : Type u_2
inst✝³ : CommRing S
inst✝² : Algebra R S
M : Type u_3
inst✝¹ : AddCommMonoid M
inst✝ : Module R M
n : ℕ
s : S
m : M
⊢ (dpScalarExtensionInv R S M) (dp S n (s ⊗ₜ[R] m)) = (s ^ n) ⊗ₜ[R] dp R n m
TACTIC:
|
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/DPAlgebra/BaseChange.lean | DividedPowerAlgebra.dpScalarExtensionInv_apply_dp | [247, 1] | [255, 74] | simp only [dpScalarExtensionExp, LinearMap.baseChangeEquiv, LinearEquiv.coe_symm_mk,
AlgebraTensorModule.lift_apply, lift.tmul, LinearMap.coe_restrictScalars, LinearMap.flip_apply,
LinearMap.lsmul_apply, LinearMap.smul_apply, LinearMap.coe_comp, Function.comp_apply] | R : Type u_1
inst✝⁴ : CommRing R
S : Type u_2
inst✝³ : CommRing S
inst✝² : Algebra R S
M : Type u_3
inst✝¹ : AddCommMonoid M
inst✝ : Module R M
n : ℕ
s : S
m : M
⊢ (PowerSeries.coeff (S ⊗[R] DividedPowerAlgebra R M) n) ↑((dpScalarExtensionExp R S M) (s ⊗ₜ[R] m)) =
(s ^ n) ⊗ₜ[R] dp R n m | R : Type u_1
inst✝⁴ : CommRing R
S : Type u_2
inst✝³ : CommRing S
inst✝² : Algebra R S
M : Type u_3
inst✝¹ : AddCommMonoid M
inst✝ : Module R M
n : ℕ
s : S
m : M
⊢ (PowerSeries.coeff (S ⊗[R] DividedPowerAlgebra R M) n)
↑(s • (ExponentialModule.linearMap Algebra.TensorProduct.includeRight) ((exp_LinearMap R M) m))... | Please generate a tactic in lean4 to solve the state.
STATE:
R : Type u_1
inst✝⁴ : CommRing R
S : Type u_2
inst✝³ : CommRing S
inst✝² : Algebra R S
M : Type u_3
inst✝¹ : AddCommMonoid M
inst✝ : Module R M
n : ℕ
s : S
m : M
⊢ (PowerSeries.coeff (S ⊗[R] DividedPowerAlgebra R M) n) ↑((dpScalarExtensionExp R S M) (s ⊗ₜ[R] ... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/DPAlgebra/BaseChange.lean | DividedPowerAlgebra.dpScalarExtensionInv_apply_dp | [247, 1] | [255, 74] | rw [ExponentialModule.coe_smul, PowerSeries.coeff_scale, ExponentialModule.coeff_linearMap,
Algebra.TensorProduct.includeRight, coe_mk, RingHom.coe_mk, MonoidHom.coe_mk,
OneHom.coe_mk, coeff_exp_LinearMap, smul_tmul', smul_eq_mul, mul_one] | R : Type u_1
inst✝⁴ : CommRing R
S : Type u_2
inst✝³ : CommRing S
inst✝² : Algebra R S
M : Type u_3
inst✝¹ : AddCommMonoid M
inst✝ : Module R M
n : ℕ
s : S
m : M
⊢ (PowerSeries.coeff (S ⊗[R] DividedPowerAlgebra R M) n)
↑(s • (ExponentialModule.linearMap Algebra.TensorProduct.includeRight) ((exp_LinearMap R M) m))... | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
R : Type u_1
inst✝⁴ : CommRing R
S : Type u_2
inst✝³ : CommRing S
inst✝² : Algebra R S
M : Type u_3
inst✝¹ : AddCommMonoid M
inst✝ : Module R M
n : ℕ
s : S
m : M
⊢ (PowerSeries.coeff (S ⊗[R] DividedPowerAlgebra R M) n)
↑(s • (ExponentialModule.linearMap... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/DPAlgebra/BaseChange.lean | DividedPowerAlgebra.dpScalarExtensionInv_unique | [258, 1] | [282, 28] | apply AlgHom.ext | R : Type u_1
inst✝⁴ : CommRing R
S : Type u_2
inst✝³ : CommRing S
inst✝² : Algebra R S
M : Type u_3
inst✝¹ : AddCommMonoid M
inst✝ : Module R M
φ : DividedPowerAlgebra S (S ⊗[R] M) →ₐ[S] S ⊗[R] DividedPowerAlgebra R M
hφ : ∀ (n : ℕ) (s : S) (m : M), φ (dp S n (s ⊗ₜ[R] m)) = (s ^ n) ⊗ₜ[R] dp R n m
⊢ φ = dpScalarExtensio... | case H
R : Type u_1
inst✝⁴ : CommRing R
S : Type u_2
inst✝³ : CommRing S
inst✝² : Algebra R S
M : Type u_3
inst✝¹ : AddCommMonoid M
inst✝ : Module R M
φ : DividedPowerAlgebra S (S ⊗[R] M) →ₐ[S] S ⊗[R] DividedPowerAlgebra R M
hφ : ∀ (n : ℕ) (s : S) (m : M), φ (dp S n (s ⊗ₜ[R] m)) = (s ^ n) ⊗ₜ[R] dp R n m
⊢ ∀ (x : Divide... | Please generate a tactic in lean4 to solve the state.
STATE:
R : Type u_1
inst✝⁴ : CommRing R
S : Type u_2
inst✝³ : CommRing S
inst✝² : Algebra R S
M : Type u_3
inst✝¹ : AddCommMonoid M
inst✝ : Module R M
φ : DividedPowerAlgebra S (S ⊗[R] M) →ₐ[S] S ⊗[R] DividedPowerAlgebra R M
hφ : ∀ (n : ℕ) (s : S) (m : M), φ (dp S n... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/DPAlgebra/BaseChange.lean | DividedPowerAlgebra.dpScalarExtensionInv_unique | [258, 1] | [282, 28] | intro x | case H
R : Type u_1
inst✝⁴ : CommRing R
S : Type u_2
inst✝³ : CommRing S
inst✝² : Algebra R S
M : Type u_3
inst✝¹ : AddCommMonoid M
inst✝ : Module R M
φ : DividedPowerAlgebra S (S ⊗[R] M) →ₐ[S] S ⊗[R] DividedPowerAlgebra R M
hφ : ∀ (n : ℕ) (s : S) (m : M), φ (dp S n (s ⊗ₜ[R] m)) = (s ^ n) ⊗ₜ[R] dp R n m
⊢ ∀ (x : Divide... | case H
R : Type u_1
inst✝⁴ : CommRing R
S : Type u_2
inst✝³ : CommRing S
inst✝² : Algebra R S
M : Type u_3
inst✝¹ : AddCommMonoid M
inst✝ : Module R M
φ : DividedPowerAlgebra S (S ⊗[R] M) →ₐ[S] S ⊗[R] DividedPowerAlgebra R M
hφ : ∀ (n : ℕ) (s : S) (m : M), φ (dp S n (s ⊗ₜ[R] m)) = (s ^ n) ⊗ₜ[R] dp R n m
x : DividedPowe... | Please generate a tactic in lean4 to solve the state.
STATE:
case H
R : Type u_1
inst✝⁴ : CommRing R
S : Type u_2
inst✝³ : CommRing S
inst✝² : Algebra R S
M : Type u_3
inst✝¹ : AddCommMonoid M
inst✝ : Module R M
φ : DividedPowerAlgebra S (S ⊗[R] M) →ₐ[S] S ⊗[R] DividedPowerAlgebra R M
hφ : ∀ (n : ℕ) (s : S) (m : M), φ ... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/DPAlgebra/BaseChange.lean | DividedPowerAlgebra.dpScalarExtensionInv_unique | [258, 1] | [282, 28] | rw [mk_C, Algebra.algebraMap_eq_smul_one, map_smul, _root_.map_one, map_smul, _root_.map_one] | case H.h_C
R : Type u_1
inst✝⁴ : CommRing R
S : Type u_2
inst✝³ : CommRing S
inst✝² : Algebra R S
M : Type u_3
inst✝¹ : AddCommMonoid M
inst✝ : Module R M
φ : DividedPowerAlgebra S (S ⊗[R] M) →ₐ[S] S ⊗[R] DividedPowerAlgebra R M
hφ : ∀ (n : ℕ) (s : S) (m : M), φ (dp S n (s ⊗ₜ[R] m)) = (s ^ n) ⊗ₜ[R] dp R n m
a : S
⊢ φ (... | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case H.h_C
R : Type u_1
inst✝⁴ : CommRing R
S : Type u_2
inst✝³ : CommRing S
inst✝² : Algebra R S
M : Type u_3
inst✝¹ : AddCommMonoid M
inst✝ : Module R M
φ : DividedPowerAlgebra S (S ⊗[R] M) →ₐ[S] S ⊗[R] DividedPowerAlgebra R M
hφ : ∀ (n : ℕ) (s : S) (m : M)... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/DPAlgebra/BaseChange.lean | DividedPowerAlgebra.dpScalarExtensionInv_unique | [258, 1] | [282, 28] | simp only [map_add, hf, hg] | case H.h_add
R : Type u_1
inst✝⁴ : CommRing R
S : Type u_2
inst✝³ : CommRing S
inst✝² : Algebra R S
M : Type u_3
inst✝¹ : AddCommMonoid M
inst✝ : Module R M
φ : DividedPowerAlgebra S (S ⊗[R] M) →ₐ[S] S ⊗[R] DividedPowerAlgebra R M
hφ : ∀ (n : ℕ) (s : S) (m : M), φ (dp S n (s ⊗ₜ[R] m)) = (s ^ n) ⊗ₜ[R] dp R n m
f g : Div... | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case H.h_add
R : Type u_1
inst✝⁴ : CommRing R
S : Type u_2
inst✝³ : CommRing S
inst✝² : Algebra R S
M : Type u_3
inst✝¹ : AddCommMonoid M
inst✝ : Module R M
φ : DividedPowerAlgebra S (S ⊗[R] M) →ₐ[S] S ⊗[R] DividedPowerAlgebra R M
hφ : ∀ (n : ℕ) (s : S) (m : ... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/DPAlgebra/BaseChange.lean | DividedPowerAlgebra.dpScalarExtensionInv_unique | [258, 1] | [282, 28] | suffices h_eq : φ (dp S n sm) = (dpScalarExtensionInv R S M) (dp S n sm) by
rw [_root_.map_mul, _root_.map_mul, hf, h_eq] | case H.h_dp
R : Type u_1
inst✝⁴ : CommRing R
S : Type u_2
inst✝³ : CommRing S
inst✝² : Algebra R S
M : Type u_3
inst✝¹ : AddCommMonoid M
inst✝ : Module R M
φ : DividedPowerAlgebra S (S ⊗[R] M) →ₐ[S] S ⊗[R] DividedPowerAlgebra R M
hφ : ∀ (n : ℕ) (s : S) (m : M), φ (dp S n (s ⊗ₜ[R] m)) = (s ^ n) ⊗ₜ[R] dp R n m
f : Divide... | case H.h_dp
R : Type u_1
inst✝⁴ : CommRing R
S : Type u_2
inst✝³ : CommRing S
inst✝² : Algebra R S
M : Type u_3
inst✝¹ : AddCommMonoid M
inst✝ : Module R M
φ : DividedPowerAlgebra S (S ⊗[R] M) →ₐ[S] S ⊗[R] DividedPowerAlgebra R M
hφ : ∀ (n : ℕ) (s : S) (m : M), φ (dp S n (s ⊗ₜ[R] m)) = (s ^ n) ⊗ₜ[R] dp R n m
f : Divide... | Please generate a tactic in lean4 to solve the state.
STATE:
case H.h_dp
R : Type u_1
inst✝⁴ : CommRing R
S : Type u_2
inst✝³ : CommRing S
inst✝² : Algebra R S
M : Type u_3
inst✝¹ : AddCommMonoid M
inst✝ : Module R M
φ : DividedPowerAlgebra S (S ⊗[R] M) →ₐ[S] S ⊗[R] DividedPowerAlgebra R M
hφ : ∀ (n : ℕ) (s : S) (m : M... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/DPAlgebra/BaseChange.lean | DividedPowerAlgebra.dpScalarExtensionInv_unique | [258, 1] | [282, 28] | rw [_root_.map_mul, _root_.map_mul, hf, h_eq] | R : Type u_1
inst✝⁴ : CommRing R
S : Type u_2
inst✝³ : CommRing S
inst✝² : Algebra R S
M : Type u_3
inst✝¹ : AddCommMonoid M
inst✝ : Module R M
φ : DividedPowerAlgebra S (S ⊗[R] M) →ₐ[S] S ⊗[R] DividedPowerAlgebra R M
hφ : ∀ (n : ℕ) (s : S) (m : M), φ (dp S n (s ⊗ₜ[R] m)) = (s ^ n) ⊗ₜ[R] dp R n m
f : DividedPowerAlgebr... | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
R : Type u_1
inst✝⁴ : CommRing R
S : Type u_2
inst✝³ : CommRing S
inst✝² : Algebra R S
M : Type u_3
inst✝¹ : AddCommMonoid M
inst✝ : Module R M
φ : DividedPowerAlgebra S (S ⊗[R] M) →ₐ[S] S ⊗[R] DividedPowerAlgebra R M
hφ : ∀ (n : ℕ) (s : S) (m : M), φ (dp S n... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/DPAlgebra/BaseChange.lean | DividedPowerAlgebra.dpScalarExtensionInv_unique | [258, 1] | [282, 28] | rw [dp_null] | case H.h_dp.zero
R : Type u_1
inst✝⁴ : CommRing R
S : Type u_2
inst✝³ : CommRing S
inst✝² : Algebra R S
M : Type u_3
inst✝¹ : AddCommMonoid M
inst✝ : Module R M
φ : DividedPowerAlgebra S (S ⊗[R] M) →ₐ[S] S ⊗[R] DividedPowerAlgebra R M
hφ : ∀ (n : ℕ) (s : S) (m : M), φ (dp S n (s ⊗ₜ[R] m)) = (s ^ n) ⊗ₜ[R] dp R n m
f : D... | case H.h_dp.zero
R : Type u_1
inst✝⁴ : CommRing R
S : Type u_2
inst✝³ : CommRing S
inst✝² : Algebra R S
M : Type u_3
inst✝¹ : AddCommMonoid M
inst✝ : Module R M
φ : DividedPowerAlgebra S (S ⊗[R] M) →ₐ[S] S ⊗[R] DividedPowerAlgebra R M
hφ : ∀ (n : ℕ) (s : S) (m : M), φ (dp S n (s ⊗ₜ[R] m)) = (s ^ n) ⊗ₜ[R] dp R n m
f : D... | Please generate a tactic in lean4 to solve the state.
STATE:
case H.h_dp.zero
R : Type u_1
inst✝⁴ : CommRing R
S : Type u_2
inst✝³ : CommRing S
inst✝² : Algebra R S
M : Type u_3
inst✝¹ : AddCommMonoid M
inst✝ : Module R M
φ : DividedPowerAlgebra S (S ⊗[R] M) →ₐ[S] S ⊗[R] DividedPowerAlgebra R M
hφ : ∀ (n : ℕ) (s : S) (... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/DPAlgebra/BaseChange.lean | DividedPowerAlgebra.dpScalarExtensionInv_unique | [258, 1] | [282, 28] | split_ifs | case H.h_dp.zero
R : Type u_1
inst✝⁴ : CommRing R
S : Type u_2
inst✝³ : CommRing S
inst✝² : Algebra R S
M : Type u_3
inst✝¹ : AddCommMonoid M
inst✝ : Module R M
φ : DividedPowerAlgebra S (S ⊗[R] M) →ₐ[S] S ⊗[R] DividedPowerAlgebra R M
hφ : ∀ (n : ℕ) (s : S) (m : M), φ (dp S n (s ⊗ₜ[R] m)) = (s ^ n) ⊗ₜ[R] dp R n m
f : D... | case pos
R : Type u_1
inst✝⁴ : CommRing R
S : Type u_2
inst✝³ : CommRing S
inst✝² : Algebra R S
M : Type u_3
inst✝¹ : AddCommMonoid M
inst✝ : Module R M
φ : DividedPowerAlgebra S (S ⊗[R] M) →ₐ[S] S ⊗[R] DividedPowerAlgebra R M
hφ : ∀ (n : ℕ) (s : S) (m : M), φ (dp S n (s ⊗ₜ[R] m)) = (s ^ n) ⊗ₜ[R] dp R n m
f : DividedPo... | Please generate a tactic in lean4 to solve the state.
STATE:
case H.h_dp.zero
R : Type u_1
inst✝⁴ : CommRing R
S : Type u_2
inst✝³ : CommRing S
inst✝² : Algebra R S
M : Type u_3
inst✝¹ : AddCommMonoid M
inst✝ : Module R M
φ : DividedPowerAlgebra S (S ⊗[R] M) →ₐ[S] S ⊗[R] DividedPowerAlgebra R M
hφ : ∀ (n : ℕ) (s : S) (... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/DPAlgebra/BaseChange.lean | DividedPowerAlgebra.dpScalarExtensionInv_unique | [258, 1] | [282, 28] | simp only [_root_.map_one] | case pos
R : Type u_1
inst✝⁴ : CommRing R
S : Type u_2
inst✝³ : CommRing S
inst✝² : Algebra R S
M : Type u_3
inst✝¹ : AddCommMonoid M
inst✝ : Module R M
φ : DividedPowerAlgebra S (S ⊗[R] M) →ₐ[S] S ⊗[R] DividedPowerAlgebra R M
hφ : ∀ (n : ℕ) (s : S) (m : M), φ (dp S n (s ⊗ₜ[R] m)) = (s ^ n) ⊗ₜ[R] dp R n m
f : DividedPo... | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case pos
R : Type u_1
inst✝⁴ : CommRing R
S : Type u_2
inst✝³ : CommRing S
inst✝² : Algebra R S
M : Type u_3
inst✝¹ : AddCommMonoid M
inst✝ : Module R M
φ : DividedPowerAlgebra S (S ⊗[R] M) →ₐ[S] S ⊗[R] DividedPowerAlgebra R M
hφ : ∀ (n : ℕ) (s : S) (m : M), ... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/DPAlgebra/BaseChange.lean | DividedPowerAlgebra.dpScalarExtensionInv_unique | [258, 1] | [282, 28] | simp only [_root_.map_zero] | case neg
R : Type u_1
inst✝⁴ : CommRing R
S : Type u_2
inst✝³ : CommRing S
inst✝² : Algebra R S
M : Type u_3
inst✝¹ : AddCommMonoid M
inst✝ : Module R M
φ : DividedPowerAlgebra S (S ⊗[R] M) →ₐ[S] S ⊗[R] DividedPowerAlgebra R M
hφ : ∀ (n : ℕ) (s : S) (m : M), φ (dp S n (s ⊗ₜ[R] m)) = (s ^ n) ⊗ₜ[R] dp R n m
f : DividedPo... | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case neg
R : Type u_1
inst✝⁴ : CommRing R
S : Type u_2
inst✝³ : CommRing S
inst✝² : Algebra R S
M : Type u_3
inst✝¹ : AddCommMonoid M
inst✝ : Module R M
φ : DividedPowerAlgebra S (S ⊗[R] M) →ₐ[S] S ⊗[R] DividedPowerAlgebra R M
hφ : ∀ (n : ℕ) (s : S) (m : M), ... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/DPAlgebra/BaseChange.lean | DividedPowerAlgebra.dpScalarExtensionInv_unique | [258, 1] | [282, 28] | rw [hφ, dpScalarExtensionInv_apply_dp] | case H.h_dp.tmul
R : Type u_1
inst✝⁴ : CommRing R
S : Type u_2
inst✝³ : CommRing S
inst✝² : Algebra R S
M : Type u_3
inst✝¹ : AddCommMonoid M
inst✝ : Module R M
φ : DividedPowerAlgebra S (S ⊗[R] M) →ₐ[S] S ⊗[R] DividedPowerAlgebra R M
hφ : ∀ (n : ℕ) (s : S) (m : M), φ (dp S n (s ⊗ₜ[R] m)) = (s ^ n) ⊗ₜ[R] dp R n m
f : D... | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case H.h_dp.tmul
R : Type u_1
inst✝⁴ : CommRing R
S : Type u_2
inst✝³ : CommRing S
inst✝² : Algebra R S
M : Type u_3
inst✝¹ : AddCommMonoid M
inst✝ : Module R M
φ : DividedPowerAlgebra S (S ⊗[R] M) →ₐ[S] S ⊗[R] DividedPowerAlgebra R M
hφ : ∀ (n : ℕ) (s : S) (... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/DPAlgebra/BaseChange.lean | DividedPowerAlgebra.dpScalarExtensionInv_unique | [258, 1] | [282, 28] | simp only [dp_add, map_sum, _root_.map_mul] | case H.h_dp.add
R : Type u_1
inst✝⁴ : CommRing R
S : Type u_2
inst✝³ : CommRing S
inst✝² : Algebra R S
M : Type u_3
inst✝¹ : AddCommMonoid M
inst✝ : Module R M
φ : DividedPowerAlgebra S (S ⊗[R] M) →ₐ[S] S ⊗[R] DividedPowerAlgebra R M
hφ : ∀ (n : ℕ) (s : S) (m : M), φ (dp S n (s ⊗ₜ[R] m)) = (s ^ n) ⊗ₜ[R] dp R n m
f : Di... | case H.h_dp.add
R : Type u_1
inst✝⁴ : CommRing R
S : Type u_2
inst✝³ : CommRing S
inst✝² : Algebra R S
M : Type u_3
inst✝¹ : AddCommMonoid M
inst✝ : Module R M
φ : DividedPowerAlgebra S (S ⊗[R] M) →ₐ[S] S ⊗[R] DividedPowerAlgebra R M
hφ : ∀ (n : ℕ) (s : S) (m : M), φ (dp S n (s ⊗ₜ[R] m)) = (s ^ n) ⊗ₜ[R] dp R n m
f : Di... | Please generate a tactic in lean4 to solve the state.
STATE:
case H.h_dp.add
R : Type u_1
inst✝⁴ : CommRing R
S : Type u_2
inst✝³ : CommRing S
inst✝² : Algebra R S
M : Type u_3
inst✝¹ : AddCommMonoid M
inst✝ : Module R M
φ : DividedPowerAlgebra S (S ⊗[R] M) →ₐ[S] S ⊗[R] DividedPowerAlgebra R M
hφ : ∀ (n : ℕ) (s : S) (m... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/DPAlgebra/BaseChange.lean | DividedPowerAlgebra.dpScalarExtensionInv_unique | [258, 1] | [282, 28] | apply Finset.sum_congr rfl | case H.h_dp.add
R : Type u_1
inst✝⁴ : CommRing R
S : Type u_2
inst✝³ : CommRing S
inst✝² : Algebra R S
M : Type u_3
inst✝¹ : AddCommMonoid M
inst✝ : Module R M
φ : DividedPowerAlgebra S (S ⊗[R] M) →ₐ[S] S ⊗[R] DividedPowerAlgebra R M
hφ : ∀ (n : ℕ) (s : S) (m : M), φ (dp S n (s ⊗ₜ[R] m)) = (s ^ n) ⊗ₜ[R] dp R n m
f : Di... | case H.h_dp.add
R : Type u_1
inst✝⁴ : CommRing R
S : Type u_2
inst✝³ : CommRing S
inst✝² : Algebra R S
M : Type u_3
inst✝¹ : AddCommMonoid M
inst✝ : Module R M
φ : DividedPowerAlgebra S (S ⊗[R] M) →ₐ[S] S ⊗[R] DividedPowerAlgebra R M
hφ : ∀ (n : ℕ) (s : S) (m : M), φ (dp S n (s ⊗ₜ[R] m)) = (s ^ n) ⊗ₜ[R] dp R n m
f : Di... | Please generate a tactic in lean4 to solve the state.
STATE:
case H.h_dp.add
R : Type u_1
inst✝⁴ : CommRing R
S : Type u_2
inst✝³ : CommRing S
inst✝² : Algebra R S
M : Type u_3
inst✝¹ : AddCommMonoid M
inst✝ : Module R M
φ : DividedPowerAlgebra S (S ⊗[R] M) →ₐ[S] S ⊗[R] DividedPowerAlgebra R M
hφ : ∀ (n : ℕ) (s : S) (m... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/DPAlgebra/BaseChange.lean | DividedPowerAlgebra.dpScalarExtensionInv_unique | [258, 1] | [282, 28] | intro nn _hnn | case H.h_dp.add
R : Type u_1
inst✝⁴ : CommRing R
S : Type u_2
inst✝³ : CommRing S
inst✝² : Algebra R S
M : Type u_3
inst✝¹ : AddCommMonoid M
inst✝ : Module R M
φ : DividedPowerAlgebra S (S ⊗[R] M) →ₐ[S] S ⊗[R] DividedPowerAlgebra R M
hφ : ∀ (n : ℕ) (s : S) (m : M), φ (dp S n (s ⊗ₜ[R] m)) = (s ^ n) ⊗ₜ[R] dp R n m
f : Di... | case H.h_dp.add
R : Type u_1
inst✝⁴ : CommRing R
S : Type u_2
inst✝³ : CommRing S
inst✝² : Algebra R S
M : Type u_3
inst✝¹ : AddCommMonoid M
inst✝ : Module R M
φ : DividedPowerAlgebra S (S ⊗[R] M) →ₐ[S] S ⊗[R] DividedPowerAlgebra R M
hφ : ∀ (n : ℕ) (s : S) (m : M), φ (dp S n (s ⊗ₜ[R] m)) = (s ^ n) ⊗ₜ[R] dp R n m
f : Di... | Please generate a tactic in lean4 to solve the state.
STATE:
case H.h_dp.add
R : Type u_1
inst✝⁴ : CommRing R
S : Type u_2
inst✝³ : CommRing S
inst✝² : Algebra R S
M : Type u_3
inst✝¹ : AddCommMonoid M
inst✝ : Module R M
φ : DividedPowerAlgebra S (S ⊗[R] M) →ₐ[S] S ⊗[R] DividedPowerAlgebra R M
hφ : ∀ (n : ℕ) (s : S) (m... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/DPAlgebra/BaseChange.lean | DividedPowerAlgebra.dpScalarExtensionInv_unique | [258, 1] | [282, 28] | rw [hx nn.1, hy nn.2] | case H.h_dp.add
R : Type u_1
inst✝⁴ : CommRing R
S : Type u_2
inst✝³ : CommRing S
inst✝² : Algebra R S
M : Type u_3
inst✝¹ : AddCommMonoid M
inst✝ : Module R M
φ : DividedPowerAlgebra S (S ⊗[R] M) →ₐ[S] S ⊗[R] DividedPowerAlgebra R M
hφ : ∀ (n : ℕ) (s : S) (m : M), φ (dp S n (s ⊗ₜ[R] m)) = (s ^ n) ⊗ₜ[R] dp R n m
f : Di... | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case H.h_dp.add
R : Type u_1
inst✝⁴ : CommRing R
S : Type u_2
inst✝³ : CommRing S
inst✝² : Algebra R S
M : Type u_3
inst✝¹ : AddCommMonoid M
inst✝ : Module R M
φ : DividedPowerAlgebra S (S ⊗[R] M) →ₐ[S] S ⊗[R] DividedPowerAlgebra R M
hφ : ∀ (n : ℕ) (s : S) (m... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/DPAlgebra/BaseChange.lean | DividedPowerAlgebra.coe_dpScalarExtensionEquiv | [319, 1] | [321, 6] | rfl | R : Type u_1
inst✝⁴ : CommRing R
S : Type u_2
inst✝³ : CommRing S
inst✝² : Algebra R S
M : Type u_3
inst✝¹ : AddCommMonoid M
inst✝ : Module R M
⊢ ⇑(dpScalarExtensionEquiv R S M) = ⇑(dpScalarExtension R S M) | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
R : Type u_1
inst✝⁴ : CommRing R
S : Type u_2
inst✝³ : CommRing S
inst✝² : Algebra R S
M : Type u_3
inst✝¹ : AddCommMonoid M
inst✝ : Module R M
⊢ ⇑(dpScalarExtensionEquiv R S M) = ⇑(dpScalarExtension R S M)
TACTIC:
|
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/DPAlgebra/BaseChange.lean | DividedPowerAlgebra.coe_dpScalarExtensionEquiv_symm | [323, 1] | [325, 6] | rfl | R : Type u_1
inst✝⁴ : CommRing R
S : Type u_2
inst✝³ : CommRing S
inst✝² : Algebra R S
M : Type u_3
inst✝¹ : AddCommMonoid M
inst✝ : Module R M
⊢ ⇑(dpScalarExtensionEquiv R S M).symm = ⇑(dpScalarExtensionInv R S M) | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
R : Type u_1
inst✝⁴ : CommRing R
S : Type u_2
inst✝³ : CommRing S
inst✝² : Algebra R S
M : Type u_3
inst✝¹ : AddCommMonoid M
inst✝ : Module R M
⊢ ⇑(dpScalarExtensionEquiv R S M).symm = ⇑(dpScalarExtensionInv R S M)
TACTIC:
|
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/DPAlgebra/BaseChange.lean | DividedPowerAlgebra.rTensor_comp_dpScalarExtensionEquiv_symm_eq | [332, 1] | [347, 16] | rw [← coe_map_id₂] | R : Type u_1
inst✝⁸ : CommRing R
S✝ : Type u_2
inst✝⁷ : CommRing S✝
inst✝⁶ : Algebra R S✝
M : Type u_3
inst✝⁵ : AddCommMonoid M
inst✝⁴ : Module R M
S : Type u_4
inst✝³ : CommRing S
inst✝² : Algebra R S
S' : Type u_5
inst✝¹ : CommRing S'
inst✝ : Algebra R S'
φ : S →ₐ[R] S'
n : ℕ
m : S ⊗[R] M
⊢ (LinearMap.rTensor (Divide... | R : Type u_1
inst✝⁸ : CommRing R
S✝ : Type u_2
inst✝⁷ : CommRing S✝
inst✝⁶ : Algebra R S✝
M : Type u_3
inst✝⁵ : AddCommMonoid M
inst✝⁴ : Module R M
S : Type u_4
inst✝³ : CommRing S
inst✝² : Algebra R S
S' : Type u_5
inst✝¹ : CommRing S'
inst✝ : Algebra R S'
φ : S →ₐ[R] S'
n : ℕ
m : S ⊗[R] M
⊢ (Algebra.TensorProduct.map... | Please generate a tactic in lean4 to solve the state.
STATE:
R : Type u_1
inst✝⁸ : CommRing R
S✝ : Type u_2
inst✝⁷ : CommRing S✝
inst✝⁶ : Algebra R S✝
M : Type u_3
inst✝⁵ : AddCommMonoid M
inst✝⁴ : Module R M
S : Type u_4
inst✝³ : CommRing S
inst✝² : Algebra R S
S' : Type u_5
inst✝¹ : CommRing S'
inst✝ : Algebra R S'
φ... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/DPAlgebra/BaseChange.lean | DividedPowerAlgebra.rTensor_comp_dpScalarExtensionEquiv_symm_eq | [332, 1] | [347, 16] | induction m using TensorProduct.induction_on generalizing n with
| zero => simp only [Function.comp_apply, dp_null, RingHom.map_ite_one_zero, map_zero]
| tmul s m =>
simp only [coe_dpScalarExtensionEquiv_symm, dpScalarExtensionInv_apply_dp, toLinearMap_apply,
Algebra.TensorProduct.map_tmul, map_pow, coe_id, id_eq... | R : Type u_1
inst✝⁸ : CommRing R
S✝ : Type u_2
inst✝⁷ : CommRing S✝
inst✝⁶ : Algebra R S✝
M : Type u_3
inst✝⁵ : AddCommMonoid M
inst✝⁴ : Module R M
S : Type u_4
inst✝³ : CommRing S
inst✝² : Algebra R S
S' : Type u_5
inst✝¹ : CommRing S'
inst✝ : Algebra R S'
φ : S →ₐ[R] S'
n : ℕ
m : S ⊗[R] M
⊢ (Algebra.TensorProduct.map... | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
R : Type u_1
inst✝⁸ : CommRing R
S✝ : Type u_2
inst✝⁷ : CommRing S✝
inst✝⁶ : Algebra R S✝
M : Type u_3
inst✝⁵ : AddCommMonoid M
inst✝⁴ : Module R M
S : Type u_4
inst✝³ : CommRing S
inst✝² : Algebra R S
S' : Type u_5
inst✝¹ : CommRing S'
inst✝ : Algebra R S'
φ... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/DPAlgebra/BaseChange.lean | DividedPowerAlgebra.rTensor_comp_dpScalarExtensionEquiv_symm_eq | [332, 1] | [347, 16] | simp only [Function.comp_apply, dp_null, RingHom.map_ite_one_zero, map_zero] | case zero
R : Type u_1
inst✝⁸ : CommRing R
S✝ : Type u_2
inst✝⁷ : CommRing S✝
inst✝⁶ : Algebra R S✝
M : Type u_3
inst✝⁵ : AddCommMonoid M
inst✝⁴ : Module R M
S : Type u_4
inst✝³ : CommRing S
inst✝² : Algebra R S
S' : Type u_5
inst✝¹ : CommRing S'
inst✝ : Algebra R S'
φ : S →ₐ[R] S'
n : ℕ
⊢ (Algebra.TensorProduct.map φ ... | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case zero
R : Type u_1
inst✝⁸ : CommRing R
S✝ : Type u_2
inst✝⁷ : CommRing S✝
inst✝⁶ : Algebra R S✝
M : Type u_3
inst✝⁵ : AddCommMonoid M
inst✝⁴ : Module R M
S : Type u_4
inst✝³ : CommRing S
inst✝² : Algebra R S
S' : Type u_5
inst✝¹ : CommRing S'
inst✝ : Alge... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/DPAlgebra/BaseChange.lean | DividedPowerAlgebra.rTensor_comp_dpScalarExtensionEquiv_symm_eq | [332, 1] | [347, 16] | simp only [coe_dpScalarExtensionEquiv_symm, dpScalarExtensionInv_apply_dp, toLinearMap_apply,
Algebra.TensorProduct.map_tmul, map_pow, coe_id, id_eq, LinearMap.rTensor_tmul ] | case tmul
R : Type u_1
inst✝⁸ : CommRing R
S✝ : Type u_2
inst✝⁷ : CommRing S✝
inst✝⁶ : Algebra R S✝
M : Type u_3
inst✝⁵ : AddCommMonoid M
inst✝⁴ : Module R M
S : Type u_4
inst✝³ : CommRing S
inst✝² : Algebra R S
S' : Type u_5
inst✝¹ : CommRing S'
inst✝ : Algebra R S'
φ : S →ₐ[R] S'
s : S
m : M
n : ℕ
⊢ (Algebra.TensorPr... | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case tmul
R : Type u_1
inst✝⁸ : CommRing R
S✝ : Type u_2
inst✝⁷ : CommRing S✝
inst✝⁶ : Algebra R S✝
M : Type u_3
inst✝⁵ : AddCommMonoid M
inst✝⁴ : Module R M
S : Type u_4
inst✝³ : CommRing S
inst✝² : Algebra R S
S' : Type u_5
inst✝¹ : CommRing S'
inst✝ : Alge... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/DPAlgebra/BaseChange.lean | DividedPowerAlgebra.rTensor_comp_dpScalarExtensionEquiv_symm_eq | [332, 1] | [347, 16] | simp only [dp_add, _root_.map_sum, _root_.map_mul, map_add] | case add
R : Type u_1
inst✝⁸ : CommRing R
S✝ : Type u_2
inst✝⁷ : CommRing S✝
inst✝⁶ : Algebra R S✝
M : Type u_3
inst✝⁵ : AddCommMonoid M
inst✝⁴ : Module R M
S : Type u_4
inst✝³ : CommRing S
inst✝² : Algebra R S
S' : Type u_5
inst✝¹ : CommRing S'
inst✝ : Algebra R S'
φ : S →ₐ[R] S'
x y : S ⊗[R] M
hx :
∀ (n : ℕ),
(... | case add
R : Type u_1
inst✝⁸ : CommRing R
S✝ : Type u_2
inst✝⁷ : CommRing S✝
inst✝⁶ : Algebra R S✝
M : Type u_3
inst✝⁵ : AddCommMonoid M
inst✝⁴ : Module R M
S : Type u_4
inst✝³ : CommRing S
inst✝² : Algebra R S
S' : Type u_5
inst✝¹ : CommRing S'
inst✝ : Algebra R S'
φ : S →ₐ[R] S'
x y : S ⊗[R] M
hx :
∀ (n : ℕ),
(... | Please generate a tactic in lean4 to solve the state.
STATE:
case add
R : Type u_1
inst✝⁸ : CommRing R
S✝ : Type u_2
inst✝⁷ : CommRing S✝
inst✝⁶ : Algebra R S✝
M : Type u_3
inst✝⁵ : AddCommMonoid M
inst✝⁴ : Module R M
S : Type u_4
inst✝³ : CommRing S
inst✝² : Algebra R S
S' : Type u_5
inst✝¹ : CommRing S'
inst✝ : Algeb... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/DPAlgebra/BaseChange.lean | DividedPowerAlgebra.rTensor_comp_dpScalarExtensionEquiv_symm_eq | [332, 1] | [347, 16] | apply Finset.sum_congr rfl | case add
R : Type u_1
inst✝⁸ : CommRing R
S✝ : Type u_2
inst✝⁷ : CommRing S✝
inst✝⁶ : Algebra R S✝
M : Type u_3
inst✝⁵ : AddCommMonoid M
inst✝⁴ : Module R M
S : Type u_4
inst✝³ : CommRing S
inst✝² : Algebra R S
S' : Type u_5
inst✝¹ : CommRing S'
inst✝ : Algebra R S'
φ : S →ₐ[R] S'
x y : S ⊗[R] M
hx :
∀ (n : ℕ),
(... | case add
R : Type u_1
inst✝⁸ : CommRing R
S✝ : Type u_2
inst✝⁷ : CommRing S✝
inst✝⁶ : Algebra R S✝
M : Type u_3
inst✝⁵ : AddCommMonoid M
inst✝⁴ : Module R M
S : Type u_4
inst✝³ : CommRing S
inst✝² : Algebra R S
S' : Type u_5
inst✝¹ : CommRing S'
inst✝ : Algebra R S'
φ : S →ₐ[R] S'
x y : S ⊗[R] M
hx :
∀ (n : ℕ),
(... | Please generate a tactic in lean4 to solve the state.
STATE:
case add
R : Type u_1
inst✝⁸ : CommRing R
S✝ : Type u_2
inst✝⁷ : CommRing S✝
inst✝⁶ : Algebra R S✝
M : Type u_3
inst✝⁵ : AddCommMonoid M
inst✝⁴ : Module R M
S : Type u_4
inst✝³ : CommRing S
inst✝² : Algebra R S
S' : Type u_5
inst✝¹ : CommRing S'
inst✝ : Algeb... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/DPAlgebra/BaseChange.lean | DividedPowerAlgebra.rTensor_comp_dpScalarExtensionEquiv_symm_eq | [332, 1] | [347, 16] | rintro ⟨k, l⟩ _ | case add
R : Type u_1
inst✝⁸ : CommRing R
S✝ : Type u_2
inst✝⁷ : CommRing S✝
inst✝⁶ : Algebra R S✝
M : Type u_3
inst✝⁵ : AddCommMonoid M
inst✝⁴ : Module R M
S : Type u_4
inst✝³ : CommRing S
inst✝² : Algebra R S
S' : Type u_5
inst✝¹ : CommRing S'
inst✝ : Algebra R S'
φ : S →ₐ[R] S'
x y : S ⊗[R] M
hx :
∀ (n : ℕ),
(... | case add.mk
R : Type u_1
inst✝⁸ : CommRing R
S✝ : Type u_2
inst✝⁷ : CommRing S✝
inst✝⁶ : Algebra R S✝
M : Type u_3
inst✝⁵ : AddCommMonoid M
inst✝⁴ : Module R M
S : Type u_4
inst✝³ : CommRing S
inst✝² : Algebra R S
S' : Type u_5
inst✝¹ : CommRing S'
inst✝ : Algebra R S'
φ : S →ₐ[R] S'
x y : S ⊗[R] M
hx :
∀ (n : ℕ),
... | Please generate a tactic in lean4 to solve the state.
STATE:
case add
R : Type u_1
inst✝⁸ : CommRing R
S✝ : Type u_2
inst✝⁷ : CommRing S✝
inst✝⁶ : Algebra R S✝
M : Type u_3
inst✝⁵ : AddCommMonoid M
inst✝⁴ : Module R M
S : Type u_4
inst✝³ : CommRing S
inst✝² : Algebra R S
S' : Type u_5
inst✝¹ : CommRing S'
inst✝ : Algeb... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/DPAlgebra/BaseChange.lean | DividedPowerAlgebra.rTensor_comp_dpScalarExtensionEquiv_symm_eq | [332, 1] | [347, 16] | rw [hx, hy] | case add.mk
R : Type u_1
inst✝⁸ : CommRing R
S✝ : Type u_2
inst✝⁷ : CommRing S✝
inst✝⁶ : Algebra R S✝
M : Type u_3
inst✝⁵ : AddCommMonoid M
inst✝⁴ : Module R M
S : Type u_4
inst✝³ : CommRing S
inst✝² : Algebra R S
S' : Type u_5
inst✝¹ : CommRing S'
inst✝ : Algebra R S'
φ : S →ₐ[R] S'
x y : S ⊗[R] M
hx :
∀ (n : ℕ),
... | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case add.mk
R : Type u_1
inst✝⁸ : CommRing R
S✝ : Type u_2
inst✝⁷ : CommRing S✝
inst✝⁶ : Algebra R S✝
M : Type u_3
inst✝⁵ : AddCommMonoid M
inst✝⁴ : Module R M
S : Type u_4
inst✝³ : CommRing S
inst✝² : Algebra R S
S' : Type u_5
inst✝¹ : CommRing S'
inst✝ : Al... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/DPAlgebra/BaseChange.lean | DividedPowerAlgebra.dpScalarExtension_mem_grade | [356, 1] | [420, 98] | rw [mem_grade_iff] | R : Type u_1
inst✝⁴ : CommRing R
S : Type u_2
inst✝³ : CommRing S
inst✝² : Algebra R S
M : Type u_3
inst✝¹ : AddCommMonoid M
inst✝ : Module R M
hinj : Function.Injective ⇑(algebraMap R S)
a : DividedPowerAlgebra R M
n : ℕ
ha : a ∈ grade R M n
s : S
⊢ (dpScalarExtension R S M) (s ⊗ₜ[R] a) ∈ grade S (S ⊗[R] M) n | R : Type u_1
inst✝⁴ : CommRing R
S : Type u_2
inst✝³ : CommRing S
inst✝² : Algebra R S
M : Type u_3
inst✝¹ : AddCommMonoid M
inst✝ : Module R M
hinj : Function.Injective ⇑(algebraMap R S)
a : DividedPowerAlgebra R M
n : ℕ
ha : a ∈ grade R M n
s : S
⊢ ∃ p ∈ weightedHomogeneousSubmodule S Prod.fst n, mk p = (dpScalarExte... | Please generate a tactic in lean4 to solve the state.
STATE:
R : Type u_1
inst✝⁴ : CommRing R
S : Type u_2
inst✝³ : CommRing S
inst✝² : Algebra R S
M : Type u_3
inst✝¹ : AddCommMonoid M
inst✝ : Module R M
hinj : Function.Injective ⇑(algebraMap R S)
a : DividedPowerAlgebra R M
n : ℕ
ha : a ∈ grade R M n
s : S
⊢ (dpScala... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/DPAlgebra/BaseChange.lean | DividedPowerAlgebra.dpScalarExtension_mem_grade | [356, 1] | [420, 98] | set f : R →ₐ[R] S := { (algebraMap R S) with
commutes' := fun r => rfl } with hf | R : Type u_1
inst✝⁴ : CommRing R
S : Type u_2
inst✝³ : CommRing S
inst✝² : Algebra R S
M : Type u_3
inst✝¹ : AddCommMonoid M
inst✝ : Module R M
hinj : Function.Injective ⇑(algebraMap R S)
a : DividedPowerAlgebra R M
n : ℕ
ha : a ∈ grade R M n
s : S
⊢ ∃ p ∈ weightedHomogeneousSubmodule S Prod.fst n, mk p = (dpScalarExte... | R : Type u_1
inst✝⁴ : CommRing R
S : Type u_2
inst✝³ : CommRing S
inst✝² : Algebra R S
M : Type u_3
inst✝¹ : AddCommMonoid M
inst✝ : Module R M
hinj : Function.Injective ⇑(algebraMap R S)
a : DividedPowerAlgebra R M
n : ℕ
ha : a ∈ grade R M n
s : S
f : R →ₐ[R] S :=
let __src := algebraMap R S;
{ toRingHom := __src,... | Please generate a tactic in lean4 to solve the state.
STATE:
R : Type u_1
inst✝⁴ : CommRing R
S : Type u_2
inst✝³ : CommRing S
inst✝² : Algebra R S
M : Type u_3
inst✝¹ : AddCommMonoid M
inst✝ : Module R M
hinj : Function.Injective ⇑(algebraMap R S)
a : DividedPowerAlgebra R M
n : ℕ
ha : a ∈ grade R M n
s : S
⊢ ∃ p ∈ we... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/DPAlgebra/BaseChange.lean | DividedPowerAlgebra.dpScalarExtension_mem_grade | [356, 1] | [420, 98] | obtain ⟨p, hpn, hpa⟩ := ha | R : Type u_1
inst✝⁴ : CommRing R
S : Type u_2
inst✝³ : CommRing S
inst✝² : Algebra R S
M : Type u_3
inst✝¹ : AddCommMonoid M
inst✝ : Module R M
hinj : Function.Injective ⇑(algebraMap R S)
a : DividedPowerAlgebra R M
n : ℕ
ha : a ∈ grade R M n
s : S
f : R →ₐ[R] S :=
let __src := algebraMap R S;
{ toRingHom := __src,... | case intro.intro
R : Type u_1
inst✝⁴ : CommRing R
S : Type u_2
inst✝³ : CommRing S
inst✝² : Algebra R S
M : Type u_3
inst✝¹ : AddCommMonoid M
inst✝ : Module R M
hinj : Function.Injective ⇑(algebraMap R S)
a : DividedPowerAlgebra R M
n : ℕ
s : S
f : R →ₐ[R] S :=
let __src := algebraMap R S;
{ toRingHom := __src, com... | Please generate a tactic in lean4 to solve the state.
STATE:
R : Type u_1
inst✝⁴ : CommRing R
S : Type u_2
inst✝³ : CommRing S
inst✝² : Algebra R S
M : Type u_3
inst✝¹ : AddCommMonoid M
inst✝ : Module R M
hinj : Function.Injective ⇑(algebraMap R S)
a : DividedPowerAlgebra R M
n : ℕ
ha : a ∈ grade R M n
s : S
f : R →ₐ[R... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/DPAlgebra/BaseChange.lean | DividedPowerAlgebra.dpScalarExtension_mem_grade | [356, 1] | [420, 98] | set p' : MvPolynomial (ℕ × M) S := MvPolynomial.baseChange f p with hp' | case intro.intro
R : Type u_1
inst✝⁴ : CommRing R
S : Type u_2
inst✝³ : CommRing S
inst✝² : Algebra R S
M : Type u_3
inst✝¹ : AddCommMonoid M
inst✝ : Module R M
hinj : Function.Injective ⇑(algebraMap R S)
a : DividedPowerAlgebra R M
n : ℕ
s : S
f : R →ₐ[R] S :=
let __src := algebraMap R S;
{ toRingHom := __src, com... | case intro.intro
R : Type u_1
inst✝⁴ : CommRing R
S : Type u_2
inst✝³ : CommRing S
inst✝² : Algebra R S
M : Type u_3
inst✝¹ : AddCommMonoid M
inst✝ : Module R M
hinj : Function.Injective ⇑(algebraMap R S)
a : DividedPowerAlgebra R M
n : ℕ
s : S
f : R →ₐ[R] S :=
let __src := algebraMap R S;
{ toRingHom := __src, com... | Please generate a tactic in lean4 to solve the state.
STATE:
case intro.intro
R : Type u_1
inst✝⁴ : CommRing R
S : Type u_2
inst✝³ : CommRing S
inst✝² : Algebra R S
M : Type u_3
inst✝¹ : AddCommMonoid M
inst✝ : Module R M
hinj : Function.Injective ⇑(algebraMap R S)
a : DividedPowerAlgebra R M
n : ℕ
s : S
f : R →ₐ[R] S ... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/DPAlgebra/BaseChange.lean | DividedPowerAlgebra.dpScalarExtension_mem_grade | [356, 1] | [420, 98] | use s • rename (Prod.map id (fun m => 1 ⊗ₜ m)) p' | case intro.intro
R : Type u_1
inst✝⁴ : CommRing R
S : Type u_2
inst✝³ : CommRing S
inst✝² : Algebra R S
M : Type u_3
inst✝¹ : AddCommMonoid M
inst✝ : Module R M
hinj : Function.Injective ⇑(algebraMap R S)
a : DividedPowerAlgebra R M
n : ℕ
s : S
f : R →ₐ[R] S :=
let __src := algebraMap R S;
{ toRingHom := __src, com... | case h
R : Type u_1
inst✝⁴ : CommRing R
S : Type u_2
inst✝³ : CommRing S
inst✝² : Algebra R S
M : Type u_3
inst✝¹ : AddCommMonoid M
inst✝ : Module R M
hinj : Function.Injective ⇑(algebraMap R S)
a : DividedPowerAlgebra R M
n : ℕ
s : S
f : R →ₐ[R] S :=
let __src := algebraMap R S;
{ toRingHom := __src, commutes' := ... | Please generate a tactic in lean4 to solve the state.
STATE:
case intro.intro
R : Type u_1
inst✝⁴ : CommRing R
S : Type u_2
inst✝³ : CommRing S
inst✝² : Algebra R S
M : Type u_3
inst✝¹ : AddCommMonoid M
inst✝ : Module R M
hinj : Function.Injective ⇑(algebraMap R S)
a : DividedPowerAlgebra R M
n : ℕ
s : S
f : R →ₐ[R] S ... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/DPAlgebra/BaseChange.lean | DividedPowerAlgebra.dpScalarExtension_mem_grade | [356, 1] | [420, 98] | constructor | case h
R : Type u_1
inst✝⁴ : CommRing R
S : Type u_2
inst✝³ : CommRing S
inst✝² : Algebra R S
M : Type u_3
inst✝¹ : AddCommMonoid M
inst✝ : Module R M
hinj : Function.Injective ⇑(algebraMap R S)
a : DividedPowerAlgebra R M
n : ℕ
s : S
f : R →ₐ[R] S :=
let __src := algebraMap R S;
{ toRingHom := __src, commutes' := ... | case h.left
R : Type u_1
inst✝⁴ : CommRing R
S : Type u_2
inst✝³ : CommRing S
inst✝² : Algebra R S
M : Type u_3
inst✝¹ : AddCommMonoid M
inst✝ : Module R M
hinj : Function.Injective ⇑(algebraMap R S)
a : DividedPowerAlgebra R M
n : ℕ
s : S
f : R →ₐ[R] S :=
let __src := algebraMap R S;
{ toRingHom := __src, commutes... | Please generate a tactic in lean4 to solve the state.
STATE:
case h
R : Type u_1
inst✝⁴ : CommRing R
S : Type u_2
inst✝³ : CommRing S
inst✝² : Algebra R S
M : Type u_3
inst✝¹ : AddCommMonoid M
inst✝ : Module R M
hinj : Function.Injective ⇑(algebraMap R S)
a : DividedPowerAlgebra R M
n : ℕ
s : S
f : R →ₐ[R] S :=
let _... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/DPAlgebra/BaseChange.lean | DividedPowerAlgebra.dpScalarExtension_mem_grade | [356, 1] | [420, 98] | simp only [SetLike.mem_coe, mem_weightedHomogeneousSubmodule, IsWeightedHomogeneous] at hpn ⊢ | case h.left
R : Type u_1
inst✝⁴ : CommRing R
S : Type u_2
inst✝³ : CommRing S
inst✝² : Algebra R S
M : Type u_3
inst✝¹ : AddCommMonoid M
inst✝ : Module R M
hinj : Function.Injective ⇑(algebraMap R S)
a : DividedPowerAlgebra R M
n : ℕ
s : S
f : R →ₐ[R] S :=
let __src := algebraMap R S;
{ toRingHom := __src, commutes... | case h.left
R : Type u_1
inst✝⁴ : CommRing R
S : Type u_2
inst✝³ : CommRing S
inst✝² : Algebra R S
M : Type u_3
inst✝¹ : AddCommMonoid M
inst✝ : Module R M
hinj : Function.Injective ⇑(algebraMap R S)
a : DividedPowerAlgebra R M
n : ℕ
s : S
f : R →ₐ[R] S :=
let __src := algebraMap R S;
{ toRingHom := __src, commutes... | Please generate a tactic in lean4 to solve the state.
STATE:
case h.left
R : Type u_1
inst✝⁴ : CommRing R
S : Type u_2
inst✝³ : CommRing S
inst✝² : Algebra R S
M : Type u_3
inst✝¹ : AddCommMonoid M
inst✝ : Module R M
hinj : Function.Injective ⇑(algebraMap R S)
a : DividedPowerAlgebra R M
n : ℕ
s : S
f : R →ₐ[R] S :=
... |
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