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/RingTheory/MvPowerSeries/Trunc.lean | MvPowerSeries.trunc_one' | [58, 1] | [73, 31] | exact orderBot.proof_1 n | case neg
σ : Type u_1
R : Type u_2
inst✝ : CommSemiring R
n✝ n : σ →₀ ℕ
H : ¬0 ≤ n
⊢ 0 ≤ n | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case neg
σ : Type u_1
R : Type u_2
inst✝ : CommSemiring R
n✝ n : σ →₀ ℕ
H : ¬0 ≤ n
⊢ 0 ≤ n
TACTIC:
|
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/ForMathlib/RingTheory/MvPowerSeries/Trunc.lean | MvPowerSeries.trunc_c | [76, 1] | [81, 56] | classical
rw [coeff_trunc', coeff_C, MvPolynomial.coeff_C]
split_ifs with H <;> first |rfl|try simp_all
exfalso; apply H; subst m; exact orderBot.proof_1 n | σ : Type u_1
R : Type u_2
inst✝ : CommSemiring R
n✝ n : σ →₀ ℕ
a : R
m : σ →₀ ℕ
⊢ MvPolynomial.coeff m ((trunc' R n) ((C σ R) a)) = MvPolynomial.coeff m (MvPolynomial.C a) | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
σ : Type u_1
R : Type u_2
inst✝ : CommSemiring R
n✝ n : σ →₀ ℕ
a : R
m : σ →₀ ℕ
⊢ MvPolynomial.coeff m ((trunc' R n) ((C σ R) a)) = MvPolynomial.coeff m (MvPolynomial.C a)
TACTIC:
|
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/ForMathlib/RingTheory/MvPowerSeries/Trunc.lean | MvPowerSeries.trunc_c | [76, 1] | [81, 56] | rw [coeff_trunc', coeff_C, MvPolynomial.coeff_C] | σ : Type u_1
R : Type u_2
inst✝ : CommSemiring R
n✝ n : σ →₀ ℕ
a : R
m : σ →₀ ℕ
⊢ MvPolynomial.coeff m ((trunc' R n) ((C σ R) a)) = MvPolynomial.coeff m (MvPolynomial.C a) | σ : Type u_1
R : Type u_2
inst✝ : CommSemiring R
n✝ n : σ →₀ ℕ
a : R
m : σ →₀ ℕ
⊢ (if m ≤ n then if m = 0 then a else 0 else 0) = if 0 = m then a else 0 | Please generate a tactic in lean4 to solve the state.
STATE:
σ : Type u_1
R : Type u_2
inst✝ : CommSemiring R
n✝ n : σ →₀ ℕ
a : R
m : σ →₀ ℕ
⊢ MvPolynomial.coeff m ((trunc' R n) ((C σ R) a)) = MvPolynomial.coeff m (MvPolynomial.C a)
TACTIC:
|
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/ForMathlib/RingTheory/MvPowerSeries/Trunc.lean | MvPowerSeries.trunc_c | [76, 1] | [81, 56] | split_ifs with H <;> first |rfl|try simp_all | σ : Type u_1
R : Type u_2
inst✝ : CommSemiring R
n✝ n : σ →₀ ℕ
a : R
m : σ →₀ ℕ
⊢ (if m ≤ n then if m = 0 then a else 0 else 0) = if 0 = m then a else 0 | case pos
σ : Type u_1
R : Type u_2
inst✝ : CommSemiring R
n✝ n : σ →₀ ℕ
a : R
m : σ →₀ ℕ
H : ¬m ≤ n
h✝ : 0 = m
⊢ 0 = a | Please generate a tactic in lean4 to solve the state.
STATE:
σ : Type u_1
R : Type u_2
inst✝ : CommSemiring R
n✝ n : σ →₀ ℕ
a : R
m : σ →₀ ℕ
⊢ (if m ≤ n then if m = 0 then a else 0 else 0) = if 0 = m then a else 0
TACTIC:
|
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/ForMathlib/RingTheory/MvPowerSeries/Trunc.lean | MvPowerSeries.trunc_c | [76, 1] | [81, 56] | exfalso | case pos
σ : Type u_1
R : Type u_2
inst✝ : CommSemiring R
n✝ n : σ →₀ ℕ
a : R
m : σ →₀ ℕ
H : ¬m ≤ n
h✝ : 0 = m
⊢ 0 = a | case pos
σ : Type u_1
R : Type u_2
inst✝ : CommSemiring R
n✝ n : σ →₀ ℕ
a : R
m : σ →₀ ℕ
H : ¬m ≤ n
h✝ : 0 = m
⊢ False | Please generate a tactic in lean4 to solve the state.
STATE:
case pos
σ : Type u_1
R : Type u_2
inst✝ : CommSemiring R
n✝ n : σ →₀ ℕ
a : R
m : σ →₀ ℕ
H : ¬m ≤ n
h✝ : 0 = m
⊢ 0 = a
TACTIC:
|
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/ForMathlib/RingTheory/MvPowerSeries/Trunc.lean | MvPowerSeries.trunc_c | [76, 1] | [81, 56] | apply H | case pos
σ : Type u_1
R : Type u_2
inst✝ : CommSemiring R
n✝ n : σ →₀ ℕ
a : R
m : σ →₀ ℕ
H : ¬m ≤ n
h✝ : 0 = m
⊢ False | case pos
σ : Type u_1
R : Type u_2
inst✝ : CommSemiring R
n✝ n : σ →₀ ℕ
a : R
m : σ →₀ ℕ
H : ¬m ≤ n
h✝ : 0 = m
⊢ m ≤ n | Please generate a tactic in lean4 to solve the state.
STATE:
case pos
σ : Type u_1
R : Type u_2
inst✝ : CommSemiring R
n✝ n : σ →₀ ℕ
a : R
m : σ →₀ ℕ
H : ¬m ≤ n
h✝ : 0 = m
⊢ False
TACTIC:
|
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/ForMathlib/RingTheory/MvPowerSeries/Trunc.lean | MvPowerSeries.trunc_c | [76, 1] | [81, 56] | subst m | case pos
σ : Type u_1
R : Type u_2
inst✝ : CommSemiring R
n✝ n : σ →₀ ℕ
a : R
m : σ →₀ ℕ
H : ¬m ≤ n
h✝ : 0 = m
⊢ m ≤ n | case pos
σ : Type u_1
R : Type u_2
inst✝ : CommSemiring R
n✝ n : σ →₀ ℕ
a : R
H : ¬0 ≤ n
⊢ 0 ≤ n | Please generate a tactic in lean4 to solve the state.
STATE:
case pos
σ : Type u_1
R : Type u_2
inst✝ : CommSemiring R
n✝ n : σ →₀ ℕ
a : R
m : σ →₀ ℕ
H : ¬m ≤ n
h✝ : 0 = m
⊢ m ≤ n
TACTIC:
|
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/ForMathlib/RingTheory/MvPowerSeries/Trunc.lean | MvPowerSeries.trunc_c | [76, 1] | [81, 56] | exact orderBot.proof_1 n | case pos
σ : Type u_1
R : Type u_2
inst✝ : CommSemiring R
n✝ n : σ →₀ ℕ
a : R
H : ¬0 ≤ n
⊢ 0 ≤ n | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case pos
σ : Type u_1
R : Type u_2
inst✝ : CommSemiring R
n✝ n : σ →₀ ℕ
a : R
H : ¬0 ≤ n
⊢ 0 ≤ n
TACTIC:
|
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/ForMathlib/RingTheory/MvPowerSeries/Trunc.lean | MvPowerSeries.trunc_c | [76, 1] | [81, 56] | rfl | case neg
σ : Type u_1
R : Type u_2
inst✝ : CommSemiring R
n✝ n : σ →₀ ℕ
a : R
m : σ →₀ ℕ
H : ¬m ≤ n
h✝ : ¬0 = m
⊢ 0 = 0 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case neg
σ : Type u_1
R : Type u_2
inst✝ : CommSemiring R
n✝ n : σ →₀ ℕ
a : R
m : σ →₀ ℕ
H : ¬m ≤ n
h✝ : ¬0 = m
⊢ 0 = 0
TACTIC:
|
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/ForMathlib/RingTheory/MvPowerSeries/Trunc.lean | MvPowerSeries.trunc_c | [76, 1] | [81, 56] | try simp_all | case pos
σ : Type u_1
R : Type u_2
inst✝ : CommSemiring R
n✝ n : σ →₀ ℕ
a : R
m : σ →₀ ℕ
H : m ≤ n
h✝¹ : ¬m = 0
h✝ : 0 = m
⊢ 0 = a | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case pos
σ : Type u_1
R : Type u_2
inst✝ : CommSemiring R
n✝ n : σ →₀ ℕ
a : R
m : σ →₀ ℕ
H : m ≤ n
h✝¹ : ¬m = 0
h✝ : 0 = m
⊢ 0 = a
TACTIC:
|
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/ForMathlib/RingTheory/MvPowerSeries/Trunc.lean | MvPowerSeries.trunc_c | [76, 1] | [81, 56] | simp_all | case pos
σ : Type u_1
R : Type u_2
inst✝ : CommSemiring R
n✝ n : σ →₀ ℕ
a : R
m : σ →₀ ℕ
H : m ≤ n
h✝¹ : ¬m = 0
h✝ : 0 = m
⊢ 0 = a | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case pos
σ : Type u_1
R : Type u_2
inst✝ : CommSemiring R
n✝ n : σ →₀ ℕ
a : R
m : σ →₀ ℕ
H : m ≤ n
h✝¹ : ¬m = 0
h✝ : 0 = m
⊢ 0 = a
TACTIC:
|
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/ForMathlib/RingTheory/MvPowerSeries/Trunc.lean | MvPowerSeries.coeff_mul_trunc' | [83, 1] | [95, 53] | classical
simp only [MvPowerSeries.coeff_mul, MvPolynomial.coeff_mul]
apply Finset.sum_congr rfl
rintro ⟨i, j⟩ hij
simp only [mem_antidiagonal] at hij
rw [← hij] at h
simp only
apply congr_arg₂
rw [coeff_trunc', if_pos (le_trans le_self_add h)]
rw [coeff_trunc', if_pos (le_trans le_add_self h)] | σ : Type u_1
R : Type u_2
inst✝ : CommSemiring R
n✝ n : σ →₀ ℕ
f g : MvPowerSeries σ R
m : σ →₀ ℕ
h : m ≤ n
⊢ MvPolynomial.coeff m ((trunc' R n) f * (trunc' R n) g) = (coeff R m) (f * g) | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
σ : Type u_1
R : Type u_2
inst✝ : CommSemiring R
n✝ n : σ →₀ ℕ
f g : MvPowerSeries σ R
m : σ →₀ ℕ
h : m ≤ n
⊢ MvPolynomial.coeff m ((trunc' R n) f * (trunc' R n) g) = (coeff R m) (f * g)
TACTIC:
|
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/ForMathlib/RingTheory/MvPowerSeries/Trunc.lean | MvPowerSeries.coeff_mul_trunc' | [83, 1] | [95, 53] | simp only [MvPowerSeries.coeff_mul, MvPolynomial.coeff_mul] | σ : Type u_1
R : Type u_2
inst✝ : CommSemiring R
n✝ n : σ →₀ ℕ
f g : MvPowerSeries σ R
m : σ →₀ ℕ
h : m ≤ n
⊢ MvPolynomial.coeff m ((trunc' R n) f * (trunc' R n) g) = (coeff R m) (f * g) | σ : Type u_1
R : Type u_2
inst✝ : CommSemiring R
n✝ n : σ →₀ ℕ
f g : MvPowerSeries σ R
m : σ →₀ ℕ
h : m ≤ n
⊢ ∑ x ∈ antidiagonal m, MvPolynomial.coeff x.1 ((trunc' R n) f) * MvPolynomial.coeff x.2 ((trunc' R n) g) =
∑ p ∈ antidiagonal m, (coeff R p.1) f * (coeff R p.2) g | Please generate a tactic in lean4 to solve the state.
STATE:
σ : Type u_1
R : Type u_2
inst✝ : CommSemiring R
n✝ n : σ →₀ ℕ
f g : MvPowerSeries σ R
m : σ →₀ ℕ
h : m ≤ n
⊢ MvPolynomial.coeff m ((trunc' R n) f * (trunc' R n) g) = (coeff R m) (f * g)
TACTIC:
|
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/ForMathlib/RingTheory/MvPowerSeries/Trunc.lean | MvPowerSeries.coeff_mul_trunc' | [83, 1] | [95, 53] | apply Finset.sum_congr rfl | σ : Type u_1
R : Type u_2
inst✝ : CommSemiring R
n✝ n : σ →₀ ℕ
f g : MvPowerSeries σ R
m : σ →₀ ℕ
h : m ≤ n
⊢ ∑ x ∈ antidiagonal m, MvPolynomial.coeff x.1 ((trunc' R n) f) * MvPolynomial.coeff x.2 ((trunc' R n) g) =
∑ p ∈ antidiagonal m, (coeff R p.1) f * (coeff R p.2) g | σ : Type u_1
R : Type u_2
inst✝ : CommSemiring R
n✝ n : σ →₀ ℕ
f g : MvPowerSeries σ R
m : σ →₀ ℕ
h : m ≤ n
⊢ ∀ x ∈ antidiagonal m,
MvPolynomial.coeff x.1 ((trunc' R n) f) * MvPolynomial.coeff x.2 ((trunc' R n) g) =
(coeff R x.1) f * (coeff R x.2) g | Please generate a tactic in lean4 to solve the state.
STATE:
σ : Type u_1
R : Type u_2
inst✝ : CommSemiring R
n✝ n : σ →₀ ℕ
f g : MvPowerSeries σ R
m : σ →₀ ℕ
h : m ≤ n
⊢ ∑ x ∈ antidiagonal m, MvPolynomial.coeff x.1 ((trunc' R n) f) * MvPolynomial.coeff x.2 ((trunc' R n) g) =
∑ p ∈ antidiagonal m, (coeff R p.1) f *... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/ForMathlib/RingTheory/MvPowerSeries/Trunc.lean | MvPowerSeries.coeff_mul_trunc' | [83, 1] | [95, 53] | rintro ⟨i, j⟩ hij | σ : Type u_1
R : Type u_2
inst✝ : CommSemiring R
n✝ n : σ →₀ ℕ
f g : MvPowerSeries σ R
m : σ →₀ ℕ
h : m ≤ n
⊢ ∀ x ∈ antidiagonal m,
MvPolynomial.coeff x.1 ((trunc' R n) f) * MvPolynomial.coeff x.2 ((trunc' R n) g) =
(coeff R x.1) f * (coeff R x.2) g | case mk
σ : Type u_1
R : Type u_2
inst✝ : CommSemiring R
n✝ n : σ →₀ ℕ
f g : MvPowerSeries σ R
m : σ →₀ ℕ
h : m ≤ n
i j : σ →₀ ℕ
hij : (i, j) ∈ antidiagonal m
⊢ MvPolynomial.coeff (i, j).1 ((trunc' R n) f) * MvPolynomial.coeff (i, j).2 ((trunc' R n) g) =
(coeff R (i, j).1) f * (coeff R (i, j).2) g | Please generate a tactic in lean4 to solve the state.
STATE:
σ : Type u_1
R : Type u_2
inst✝ : CommSemiring R
n✝ n : σ →₀ ℕ
f g : MvPowerSeries σ R
m : σ →₀ ℕ
h : m ≤ n
⊢ ∀ x ∈ antidiagonal m,
MvPolynomial.coeff x.1 ((trunc' R n) f) * MvPolynomial.coeff x.2 ((trunc' R n) g) =
(coeff R x.1) f * (coeff R x.2) g... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/ForMathlib/RingTheory/MvPowerSeries/Trunc.lean | MvPowerSeries.coeff_mul_trunc' | [83, 1] | [95, 53] | simp only [mem_antidiagonal] at hij | case mk
σ : Type u_1
R : Type u_2
inst✝ : CommSemiring R
n✝ n : σ →₀ ℕ
f g : MvPowerSeries σ R
m : σ →₀ ℕ
h : m ≤ n
i j : σ →₀ ℕ
hij : (i, j) ∈ antidiagonal m
⊢ MvPolynomial.coeff (i, j).1 ((trunc' R n) f) * MvPolynomial.coeff (i, j).2 ((trunc' R n) g) =
(coeff R (i, j).1) f * (coeff R (i, j).2) g | case mk
σ : Type u_1
R : Type u_2
inst✝ : CommSemiring R
n✝ n : σ →₀ ℕ
f g : MvPowerSeries σ R
m : σ →₀ ℕ
h : m ≤ n
i j : σ →₀ ℕ
hij : i + j = m
⊢ MvPolynomial.coeff (i, j).1 ((trunc' R n) f) * MvPolynomial.coeff (i, j).2 ((trunc' R n) g) =
(coeff R (i, j).1) f * (coeff R (i, j).2) g | Please generate a tactic in lean4 to solve the state.
STATE:
case mk
σ : Type u_1
R : Type u_2
inst✝ : CommSemiring R
n✝ n : σ →₀ ℕ
f g : MvPowerSeries σ R
m : σ →₀ ℕ
h : m ≤ n
i j : σ →₀ ℕ
hij : (i, j) ∈ antidiagonal m
⊢ MvPolynomial.coeff (i, j).1 ((trunc' R n) f) * MvPolynomial.coeff (i, j).2 ((trunc' R n) g) =
... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/ForMathlib/RingTheory/MvPowerSeries/Trunc.lean | MvPowerSeries.coeff_mul_trunc' | [83, 1] | [95, 53] | rw [← hij] at h | case mk
σ : Type u_1
R : Type u_2
inst✝ : CommSemiring R
n✝ n : σ →₀ ℕ
f g : MvPowerSeries σ R
m : σ →₀ ℕ
h : m ≤ n
i j : σ →₀ ℕ
hij : i + j = m
⊢ MvPolynomial.coeff (i, j).1 ((trunc' R n) f) * MvPolynomial.coeff (i, j).2 ((trunc' R n) g) =
(coeff R (i, j).1) f * (coeff R (i, j).2) g | case mk
σ : Type u_1
R : Type u_2
inst✝ : CommSemiring R
n✝ n : σ →₀ ℕ
f g : MvPowerSeries σ R
m i j : σ →₀ ℕ
h : i + j ≤ n
hij : i + j = m
⊢ MvPolynomial.coeff (i, j).1 ((trunc' R n) f) * MvPolynomial.coeff (i, j).2 ((trunc' R n) g) =
(coeff R (i, j).1) f * (coeff R (i, j).2) g | Please generate a tactic in lean4 to solve the state.
STATE:
case mk
σ : Type u_1
R : Type u_2
inst✝ : CommSemiring R
n✝ n : σ →₀ ℕ
f g : MvPowerSeries σ R
m : σ →₀ ℕ
h : m ≤ n
i j : σ →₀ ℕ
hij : i + j = m
⊢ MvPolynomial.coeff (i, j).1 ((trunc' R n) f) * MvPolynomial.coeff (i, j).2 ((trunc' R n) g) =
(coeff R (i, j... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/ForMathlib/RingTheory/MvPowerSeries/Trunc.lean | MvPowerSeries.coeff_mul_trunc' | [83, 1] | [95, 53] | simp only | case mk
σ : Type u_1
R : Type u_2
inst✝ : CommSemiring R
n✝ n : σ →₀ ℕ
f g : MvPowerSeries σ R
m i j : σ →₀ ℕ
h : i + j ≤ n
hij : i + j = m
⊢ MvPolynomial.coeff (i, j).1 ((trunc' R n) f) * MvPolynomial.coeff (i, j).2 ((trunc' R n) g) =
(coeff R (i, j).1) f * (coeff R (i, j).2) g | case mk
σ : Type u_1
R : Type u_2
inst✝ : CommSemiring R
n✝ n : σ →₀ ℕ
f g : MvPowerSeries σ R
m i j : σ →₀ ℕ
h : i + j ≤ n
hij : i + j = m
⊢ MvPolynomial.coeff i ((trunc' R n) f) * MvPolynomial.coeff j ((trunc' R n) g) = (coeff R i) f * (coeff R j) g | Please generate a tactic in lean4 to solve the state.
STATE:
case mk
σ : Type u_1
R : Type u_2
inst✝ : CommSemiring R
n✝ n : σ →₀ ℕ
f g : MvPowerSeries σ R
m i j : σ →₀ ℕ
h : i + j ≤ n
hij : i + j = m
⊢ MvPolynomial.coeff (i, j).1 ((trunc' R n) f) * MvPolynomial.coeff (i, j).2 ((trunc' R n) g) =
(coeff R (i, j).1) ... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/ForMathlib/RingTheory/MvPowerSeries/Trunc.lean | MvPowerSeries.coeff_mul_trunc' | [83, 1] | [95, 53] | apply congr_arg₂ | case mk
σ : Type u_1
R : Type u_2
inst✝ : CommSemiring R
n✝ n : σ →₀ ℕ
f g : MvPowerSeries σ R
m i j : σ →₀ ℕ
h : i + j ≤ n
hij : i + j = m
⊢ MvPolynomial.coeff i ((trunc' R n) f) * MvPolynomial.coeff j ((trunc' R n) g) = (coeff R i) f * (coeff R j) g | case mk.hx
σ : Type u_1
R : Type u_2
inst✝ : CommSemiring R
n✝ n : σ →₀ ℕ
f g : MvPowerSeries σ R
m i j : σ →₀ ℕ
h : i + j ≤ n
hij : i + j = m
⊢ MvPolynomial.coeff i ((trunc' R n) f) = (coeff R i) f
case mk.hy
σ : Type u_1
R : Type u_2
inst✝ : CommSemiring R
n✝ n : σ →₀ ℕ
f g : MvPowerSeries σ R
m i j : σ →₀ ℕ
h : i +... | Please generate a tactic in lean4 to solve the state.
STATE:
case mk
σ : Type u_1
R : Type u_2
inst✝ : CommSemiring R
n✝ n : σ →₀ ℕ
f g : MvPowerSeries σ R
m i j : σ →₀ ℕ
h : i + j ≤ n
hij : i + j = m
⊢ MvPolynomial.coeff i ((trunc' R n) f) * MvPolynomial.coeff j ((trunc' R n) g) = (coeff R i) f * (coeff R j) g
TACTIC:... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/ForMathlib/RingTheory/MvPowerSeries/Trunc.lean | MvPowerSeries.coeff_mul_trunc' | [83, 1] | [95, 53] | rw [coeff_trunc', if_pos (le_trans le_self_add h)] | case mk.hx
σ : Type u_1
R : Type u_2
inst✝ : CommSemiring R
n✝ n : σ →₀ ℕ
f g : MvPowerSeries σ R
m i j : σ →₀ ℕ
h : i + j ≤ n
hij : i + j = m
⊢ MvPolynomial.coeff i ((trunc' R n) f) = (coeff R i) f
case mk.hy
σ : Type u_1
R : Type u_2
inst✝ : CommSemiring R
n✝ n : σ →₀ ℕ
f g : MvPowerSeries σ R
m i j : σ →₀ ℕ
h : i +... | case mk.hy
σ : Type u_1
R : Type u_2
inst✝ : CommSemiring R
n✝ n : σ →₀ ℕ
f g : MvPowerSeries σ R
m i j : σ →₀ ℕ
h : i + j ≤ n
hij : i + j = m
⊢ MvPolynomial.coeff j ((trunc' R n) g) = (coeff R j) g | Please generate a tactic in lean4 to solve the state.
STATE:
case mk.hx
σ : Type u_1
R : Type u_2
inst✝ : CommSemiring R
n✝ n : σ →₀ ℕ
f g : MvPowerSeries σ R
m i j : σ →₀ ℕ
h : i + j ≤ n
hij : i + j = m
⊢ MvPolynomial.coeff i ((trunc' R n) f) = (coeff R i) f
case mk.hy
σ : Type u_1
R : Type u_2
inst✝ : CommSemiring R... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/ForMathlib/RingTheory/MvPowerSeries/Trunc.lean | MvPowerSeries.coeff_mul_trunc' | [83, 1] | [95, 53] | rw [coeff_trunc', if_pos (le_trans le_add_self h)] | case mk.hy
σ : Type u_1
R : Type u_2
inst✝ : CommSemiring R
n✝ n : σ →₀ ℕ
f g : MvPowerSeries σ R
m i j : σ →₀ ℕ
h : i + j ≤ n
hij : i + j = m
⊢ MvPolynomial.coeff j ((trunc' R n) g) = (coeff R j) g | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case mk.hy
σ : Type u_1
R : Type u_2
inst✝ : CommSemiring R
n✝ n : σ →₀ ℕ
f g : MvPowerSeries σ R
m i j : σ →₀ ℕ
h : i + j ≤ n
hij : i + j = m
⊢ MvPolynomial.coeff j ((trunc' R n) g) = (coeff R j) g
TACTIC:
|
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/ForMathlib/RingTheory/MvPowerSeries/Trunc.lean | MvPowerSeries.continuous_trunc' | [107, 1] | [115, 27] | rw [continuous_induced_rng] | σ : Type u_1
R : Type u_2
inst✝³ : CommSemiring R
n✝ : σ →₀ ℕ
inst✝² inst✝¹ : TopologicalSpace R
inst✝ : TopologicalSemiring R
n : σ →₀ ℕ
⊢ Continuous ⇑(trunc' R n) | σ : Type u_1
R : Type u_2
inst✝³ : CommSemiring R
n✝ : σ →₀ ℕ
inst✝² inst✝¹ : TopologicalSpace R
inst✝ : TopologicalSemiring R
n : σ →₀ ℕ
⊢ Continuous (MvPolynomial.toMvPowerSeries ∘ ⇑(trunc' R n)) | Please generate a tactic in lean4 to solve the state.
STATE:
σ : Type u_1
R : Type u_2
inst✝³ : CommSemiring R
n✝ : σ →₀ ℕ
inst✝² inst✝¹ : TopologicalSpace R
inst✝ : TopologicalSemiring R
n : σ →₀ ℕ
⊢ Continuous ⇑(trunc' R n)
TACTIC:
|
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/ForMathlib/RingTheory/MvPowerSeries/Trunc.lean | MvPowerSeries.continuous_trunc' | [107, 1] | [115, 27] | apply continuous_pi | σ : Type u_1
R : Type u_2
inst✝³ : CommSemiring R
n✝ : σ →₀ ℕ
inst✝² inst✝¹ : TopologicalSpace R
inst✝ : TopologicalSemiring R
n : σ →₀ ℕ
⊢ Continuous (MvPolynomial.toMvPowerSeries ∘ ⇑(trunc' R n)) | case h
σ : Type u_1
R : Type u_2
inst✝³ : CommSemiring R
n✝ : σ →₀ ℕ
inst✝² inst✝¹ : TopologicalSpace R
inst✝ : TopologicalSemiring R
n : σ →₀ ℕ
⊢ ∀ (i : σ →₀ ℕ), Continuous fun a => (MvPolynomial.toMvPowerSeries ∘ ⇑(trunc' R n)) a i | Please generate a tactic in lean4 to solve the state.
STATE:
σ : Type u_1
R : Type u_2
inst✝³ : CommSemiring R
n✝ : σ →₀ ℕ
inst✝² inst✝¹ : TopologicalSpace R
inst✝ : TopologicalSemiring R
n : σ →₀ ℕ
⊢ Continuous (MvPolynomial.toMvPowerSeries ∘ ⇑(trunc' R n))
TACTIC:
|
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/ForMathlib/RingTheory/MvPowerSeries/Trunc.lean | MvPowerSeries.continuous_trunc' | [107, 1] | [115, 27] | intro m | case h
σ : Type u_1
R : Type u_2
inst✝³ : CommSemiring R
n✝ : σ →₀ ℕ
inst✝² inst✝¹ : TopologicalSpace R
inst✝ : TopologicalSemiring R
n : σ →₀ ℕ
⊢ ∀ (i : σ →₀ ℕ), Continuous fun a => (MvPolynomial.toMvPowerSeries ∘ ⇑(trunc' R n)) a i | case h
σ : Type u_1
R : Type u_2
inst✝³ : CommSemiring R
n✝ : σ →₀ ℕ
inst✝² inst✝¹ : TopologicalSpace R
inst✝ : TopologicalSemiring R
n m : σ →₀ ℕ
⊢ Continuous fun a => (MvPolynomial.toMvPowerSeries ∘ ⇑(trunc' R n)) a m | Please generate a tactic in lean4 to solve the state.
STATE:
case h
σ : Type u_1
R : Type u_2
inst✝³ : CommSemiring R
n✝ : σ →₀ ℕ
inst✝² inst✝¹ : TopologicalSpace R
inst✝ : TopologicalSemiring R
n : σ →₀ ℕ
⊢ ∀ (i : σ →₀ ℕ), Continuous fun a => (MvPolynomial.toMvPowerSeries ∘ ⇑(trunc' R n)) a i
TACTIC:
|
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/ForMathlib/RingTheory/MvPowerSeries/Trunc.lean | MvPowerSeries.continuous_trunc' | [107, 1] | [115, 27] | simp only [Function.comp_apply, MvPolynomial.coe_def, coeff_trunc'] | case h
σ : Type u_1
R : Type u_2
inst✝³ : CommSemiring R
n✝ : σ →₀ ℕ
inst✝² inst✝¹ : TopologicalSpace R
inst✝ : TopologicalSemiring R
n m : σ →₀ ℕ
⊢ Continuous fun a => (MvPolynomial.toMvPowerSeries ∘ ⇑(trunc' R n)) a m | case h
σ : Type u_1
R : Type u_2
inst✝³ : CommSemiring R
n✝ : σ →₀ ℕ
inst✝² inst✝¹ : TopologicalSpace R
inst✝ : TopologicalSemiring R
n m : σ →₀ ℕ
⊢ Continuous fun a => if m ≤ n then (coeff R m) a else 0 | Please generate a tactic in lean4 to solve the state.
STATE:
case h
σ : Type u_1
R : Type u_2
inst✝³ : CommSemiring R
n✝ : σ →₀ ℕ
inst✝² inst✝¹ : TopologicalSpace R
inst✝ : TopologicalSemiring R
n m : σ →₀ ℕ
⊢ Continuous fun a => (MvPolynomial.toMvPowerSeries ∘ ⇑(trunc' R n)) a m
TACTIC:
|
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/ForMathlib/RingTheory/MvPowerSeries/Trunc.lean | MvPowerSeries.continuous_trunc' | [107, 1] | [115, 27] | split_ifs with h | case h
σ : Type u_1
R : Type u_2
inst✝³ : CommSemiring R
n✝ : σ →₀ ℕ
inst✝² inst✝¹ : TopologicalSpace R
inst✝ : TopologicalSemiring R
n m : σ →₀ ℕ
⊢ Continuous fun a => if m ≤ n then (coeff R m) a else 0 | case pos
σ : Type u_1
R : Type u_2
inst✝³ : CommSemiring R
n✝ : σ →₀ ℕ
inst✝² inst✝¹ : TopologicalSpace R
inst✝ : TopologicalSemiring R
n m : σ →₀ ℕ
h : m ≤ n
⊢ Continuous fun a => (coeff R m) a
case neg
σ : Type u_1
R : Type u_2
inst✝³ : CommSemiring R
n✝ : σ →₀ ℕ
inst✝² inst✝¹ : TopologicalSpace R
inst✝ : Topologica... | Please generate a tactic in lean4 to solve the state.
STATE:
case h
σ : Type u_1
R : Type u_2
inst✝³ : CommSemiring R
n✝ : σ →₀ ℕ
inst✝² inst✝¹ : TopologicalSpace R
inst✝ : TopologicalSemiring R
n m : σ →₀ ℕ
⊢ Continuous fun a => if m ≤ n then (coeff R m) a else 0
TACTIC:
|
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/ForMathlib/RingTheory/MvPowerSeries/Trunc.lean | MvPowerSeries.continuous_trunc' | [107, 1] | [115, 27] | exact continuous_apply m | case pos
σ : Type u_1
R : Type u_2
inst✝³ : CommSemiring R
n✝ : σ →₀ ℕ
inst✝² inst✝¹ : TopologicalSpace R
inst✝ : TopologicalSemiring R
n m : σ →₀ ℕ
h : m ≤ n
⊢ Continuous fun a => (coeff R m) a | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case pos
σ : Type u_1
R : Type u_2
inst✝³ : CommSemiring R
n✝ : σ →₀ ℕ
inst✝² inst✝¹ : TopologicalSpace R
inst✝ : TopologicalSemiring R
n m : σ →₀ ℕ
h : m ≤ n
⊢ Continuous fun a => (coeff R m) a
TACTIC:
|
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/ForMathlib/RingTheory/MvPowerSeries/Trunc.lean | MvPowerSeries.continuous_trunc' | [107, 1] | [115, 27] | exact continuous_const | case neg
σ : Type u_1
R : Type u_2
inst✝³ : CommSemiring R
n✝ : σ →₀ ℕ
inst✝² inst✝¹ : TopologicalSpace R
inst✝ : TopologicalSemiring R
n m : σ →₀ ℕ
h : ¬m ≤ n
⊢ Continuous fun a => 0 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case neg
σ : Type u_1
R : Type u_2
inst✝³ : CommSemiring R
n✝ : σ →₀ ℕ
inst✝² inst✝¹ : TopologicalSpace R
inst✝ : TopologicalSemiring R
n m : σ →₀ ℕ
h : ¬m ≤ n
⊢ Continuous fun a => 0
TACTIC:
|
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/DPMorphism.lean | DividedPowers.isDPMorphism.comp | [33, 1] | [47, 36] | intro hg hf | A : Type u_1
B : Type u_2
C : Type u_3
inst✝² : CommSemiring A
inst✝¹ : CommSemiring B
inst✝ : CommSemiring C
I : Ideal A
J : Ideal B
K : Ideal C
hI : DividedPowers I
hJ : DividedPowers J
hK : DividedPowers K
f : A →+* B
g : B →+* C
h : A →+* C
hcomp : g.comp f = h
⊢ hJ.isDPMorphism hK g → hI.isDPMorphism hJ f → hI.isD... | A : Type u_1
B : Type u_2
C : Type u_3
inst✝² : CommSemiring A
inst✝¹ : CommSemiring B
inst✝ : CommSemiring C
I : Ideal A
J : Ideal B
K : Ideal C
hI : DividedPowers I
hJ : DividedPowers J
hK : DividedPowers K
f : A →+* B
g : B →+* C
h : A →+* C
hcomp : g.comp f = h
hg : hJ.isDPMorphism hK g
hf : hI.isDPMorphism hJ f
⊢ ... | Please generate a tactic in lean4 to solve the state.
STATE:
A : Type u_1
B : Type u_2
C : Type u_3
inst✝² : CommSemiring A
inst✝¹ : CommSemiring B
inst✝ : CommSemiring C
I : Ideal A
J : Ideal B
K : Ideal C
hI : DividedPowers I
hJ : DividedPowers J
hK : DividedPowers K
f : A →+* B
g : B →+* C
h : A →+* C
hcomp : g.comp... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/DPMorphism.lean | DividedPowers.isDPMorphism.comp | [33, 1] | [47, 36] | rw [← hcomp] | A : Type u_1
B : Type u_2
C : Type u_3
inst✝² : CommSemiring A
inst✝¹ : CommSemiring B
inst✝ : CommSemiring C
I : Ideal A
J : Ideal B
K : Ideal C
hI : DividedPowers I
hJ : DividedPowers J
hK : DividedPowers K
f : A →+* B
g : B →+* C
h : A →+* C
hcomp : g.comp f = h
hg : hJ.isDPMorphism hK g
hf : hI.isDPMorphism hJ f
⊢ ... | A : Type u_1
B : Type u_2
C : Type u_3
inst✝² : CommSemiring A
inst✝¹ : CommSemiring B
inst✝ : CommSemiring C
I : Ideal A
J : Ideal B
K : Ideal C
hI : DividedPowers I
hJ : DividedPowers J
hK : DividedPowers K
f : A →+* B
g : B →+* C
h : A →+* C
hcomp : g.comp f = h
hg : hJ.isDPMorphism hK g
hf : hI.isDPMorphism hJ f
⊢ ... | Please generate a tactic in lean4 to solve the state.
STATE:
A : Type u_1
B : Type u_2
C : Type u_3
inst✝² : CommSemiring A
inst✝¹ : CommSemiring B
inst✝ : CommSemiring C
I : Ideal A
J : Ideal B
K : Ideal C
hI : DividedPowers I
hJ : DividedPowers J
hK : DividedPowers K
f : A →+* B
g : B →+* C
h : A →+* C
hcomp : g.comp... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/DPMorphism.lean | DividedPowers.isDPMorphism.comp | [33, 1] | [47, 36] | constructor | A : Type u_1
B : Type u_2
C : Type u_3
inst✝² : CommSemiring A
inst✝¹ : CommSemiring B
inst✝ : CommSemiring C
I : Ideal A
J : Ideal B
K : Ideal C
hI : DividedPowers I
hJ : DividedPowers J
hK : DividedPowers K
f : A →+* B
g : B →+* C
h : A →+* C
hcomp : g.comp f = h
hg : hJ.isDPMorphism hK g
hf : hI.isDPMorphism hJ f
⊢ ... | case left
A : Type u_1
B : Type u_2
C : Type u_3
inst✝² : CommSemiring A
inst✝¹ : CommSemiring B
inst✝ : CommSemiring C
I : Ideal A
J : Ideal B
K : Ideal C
hI : DividedPowers I
hJ : DividedPowers J
hK : DividedPowers K
f : A →+* B
g : B →+* C
h : A →+* C
hcomp : g.comp f = h
hg : hJ.isDPMorphism hK g
hf : hI.isDPMorphi... | Please generate a tactic in lean4 to solve the state.
STATE:
A : Type u_1
B : Type u_2
C : Type u_3
inst✝² : CommSemiring A
inst✝¹ : CommSemiring B
inst✝ : CommSemiring C
I : Ideal A
J : Ideal B
K : Ideal C
hI : DividedPowers I
hJ : DividedPowers J
hK : DividedPowers K
f : A →+* B
g : B →+* C
h : A →+* C
hcomp : g.comp... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/DPMorphism.lean | DividedPowers.isDPMorphism.comp | [33, 1] | [47, 36] | apply le_trans _ hg.1 | case left
A : Type u_1
B : Type u_2
C : Type u_3
inst✝² : CommSemiring A
inst✝¹ : CommSemiring B
inst✝ : CommSemiring C
I : Ideal A
J : Ideal B
K : Ideal C
hI : DividedPowers I
hJ : DividedPowers J
hK : DividedPowers K
f : A →+* B
g : B →+* C
h : A →+* C
hcomp : g.comp f = h
hg : hJ.isDPMorphism hK g
hf : hI.isDPMorphi... | A : Type u_1
B : Type u_2
C : Type u_3
inst✝² : CommSemiring A
inst✝¹ : CommSemiring B
inst✝ : CommSemiring C
I : Ideal A
J : Ideal B
K : Ideal C
hI : DividedPowers I
hJ : DividedPowers J
hK : DividedPowers K
f : A →+* B
g : B →+* C
h : A →+* C
hcomp : g.comp f = h
hg : hJ.isDPMorphism hK g
hf : hI.isDPMorphism hJ f
⊢ ... | Please generate a tactic in lean4 to solve the state.
STATE:
case left
A : Type u_1
B : Type u_2
C : Type u_3
inst✝² : CommSemiring A
inst✝¹ : CommSemiring B
inst✝ : CommSemiring C
I : Ideal A
J : Ideal B
K : Ideal C
hI : DividedPowers I
hJ : DividedPowers J
hK : DividedPowers K
f : A →+* B
g : B →+* C
h : A →+* C
hcom... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/DPMorphism.lean | DividedPowers.isDPMorphism.comp | [33, 1] | [47, 36] | rw [← Ideal.map_map] | A : Type u_1
B : Type u_2
C : Type u_3
inst✝² : CommSemiring A
inst✝¹ : CommSemiring B
inst✝ : CommSemiring C
I : Ideal A
J : Ideal B
K : Ideal C
hI : DividedPowers I
hJ : DividedPowers J
hK : DividedPowers K
f : A →+* B
g : B →+* C
h : A →+* C
hcomp : g.comp f = h
hg : hJ.isDPMorphism hK g
hf : hI.isDPMorphism hJ f
⊢ ... | A : Type u_1
B : Type u_2
C : Type u_3
inst✝² : CommSemiring A
inst✝¹ : CommSemiring B
inst✝ : CommSemiring C
I : Ideal A
J : Ideal B
K : Ideal C
hI : DividedPowers I
hJ : DividedPowers J
hK : DividedPowers K
f : A →+* B
g : B →+* C
h : A →+* C
hcomp : g.comp f = h
hg : hJ.isDPMorphism hK g
hf : hI.isDPMorphism hJ f
⊢ ... | Please generate a tactic in lean4 to solve the state.
STATE:
A : Type u_1
B : Type u_2
C : Type u_3
inst✝² : CommSemiring A
inst✝¹ : CommSemiring B
inst✝ : CommSemiring C
I : Ideal A
J : Ideal B
K : Ideal C
hI : DividedPowers I
hJ : DividedPowers J
hK : DividedPowers K
f : A →+* B
g : B →+* C
h : A →+* C
hcomp : g.comp... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/DPMorphism.lean | DividedPowers.isDPMorphism.comp | [33, 1] | [47, 36] | exact Ideal.map_mono hf.1 | A : Type u_1
B : Type u_2
C : Type u_3
inst✝² : CommSemiring A
inst✝¹ : CommSemiring B
inst✝ : CommSemiring C
I : Ideal A
J : Ideal B
K : Ideal C
hI : DividedPowers I
hJ : DividedPowers J
hK : DividedPowers K
f : A →+* B
g : B →+* C
h : A →+* C
hcomp : g.comp f = h
hg : hJ.isDPMorphism hK g
hf : hI.isDPMorphism hJ f
⊢ ... | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
A : Type u_1
B : Type u_2
C : Type u_3
inst✝² : CommSemiring A
inst✝¹ : CommSemiring B
inst✝ : CommSemiring C
I : Ideal A
J : Ideal B
K : Ideal C
hI : DividedPowers I
hJ : DividedPowers J
hK : DividedPowers K
f : A →+* B
g : B →+* C
h : A →+* C
hcomp : g.comp... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/DPMorphism.lean | DividedPowers.isDPMorphism.comp | [33, 1] | [47, 36] | intro n a ha | case right
A : Type u_1
B : Type u_2
C : Type u_3
inst✝² : CommSemiring A
inst✝¹ : CommSemiring B
inst✝ : CommSemiring C
I : Ideal A
J : Ideal B
K : Ideal C
hI : DividedPowers I
hJ : DividedPowers J
hK : DividedPowers K
f : A →+* B
g : B →+* C
h : A →+* C
hcomp : g.comp f = h
hg : hJ.isDPMorphism hK g
hf : hI.isDPMorph... | case right
A : Type u_1
B : Type u_2
C : Type u_3
inst✝² : CommSemiring A
inst✝¹ : CommSemiring B
inst✝ : CommSemiring C
I : Ideal A
J : Ideal B
K : Ideal C
hI : DividedPowers I
hJ : DividedPowers J
hK : DividedPowers K
f : A →+* B
g : B →+* C
h : A →+* C
hcomp : g.comp f = h
hg : hJ.isDPMorphism hK g
hf : hI.isDPMorph... | Please generate a tactic in lean4 to solve the state.
STATE:
case right
A : Type u_1
B : Type u_2
C : Type u_3
inst✝² : CommSemiring A
inst✝¹ : CommSemiring B
inst✝ : CommSemiring C
I : Ideal A
J : Ideal B
K : Ideal C
hI : DividedPowers I
hJ : DividedPowers J
hK : DividedPowers K
f : A →+* B
g : B →+* C
h : A →+* C
hco... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/DPMorphism.lean | DividedPowers.isDPMorphism.comp | [33, 1] | [47, 36] | simp only [RingHom.coe_comp, Function.comp_apply] | case right
A : Type u_1
B : Type u_2
C : Type u_3
inst✝² : CommSemiring A
inst✝¹ : CommSemiring B
inst✝ : CommSemiring C
I : Ideal A
J : Ideal B
K : Ideal C
hI : DividedPowers I
hJ : DividedPowers J
hK : DividedPowers K
f : A →+* B
g : B →+* C
h : A →+* C
hcomp : g.comp f = h
hg : hJ.isDPMorphism hK g
hf : hI.isDPMorph... | case right
A : Type u_1
B : Type u_2
C : Type u_3
inst✝² : CommSemiring A
inst✝¹ : CommSemiring B
inst✝ : CommSemiring C
I : Ideal A
J : Ideal B
K : Ideal C
hI : DividedPowers I
hJ : DividedPowers J
hK : DividedPowers K
f : A →+* B
g : B →+* C
h : A →+* C
hcomp : g.comp f = h
hg : hJ.isDPMorphism hK g
hf : hI.isDPMorph... | Please generate a tactic in lean4 to solve the state.
STATE:
case right
A : Type u_1
B : Type u_2
C : Type u_3
inst✝² : CommSemiring A
inst✝¹ : CommSemiring B
inst✝ : CommSemiring C
I : Ideal A
J : Ideal B
K : Ideal C
hI : DividedPowers I
hJ : DividedPowers J
hK : DividedPowers K
f : A →+* B
g : B →+* C
h : A →+* C
hco... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/DPMorphism.lean | DividedPowers.isDPMorphism.comp | [33, 1] | [47, 36] | rw [← hf.2 n a ha] | case right
A : Type u_1
B : Type u_2
C : Type u_3
inst✝² : CommSemiring A
inst✝¹ : CommSemiring B
inst✝ : CommSemiring C
I : Ideal A
J : Ideal B
K : Ideal C
hI : DividedPowers I
hJ : DividedPowers J
hK : DividedPowers K
f : A →+* B
g : B →+* C
h : A →+* C
hcomp : g.comp f = h
hg : hJ.isDPMorphism hK g
hf : hI.isDPMorph... | case right
A : Type u_1
B : Type u_2
C : Type u_3
inst✝² : CommSemiring A
inst✝¹ : CommSemiring B
inst✝ : CommSemiring C
I : Ideal A
J : Ideal B
K : Ideal C
hI : DividedPowers I
hJ : DividedPowers J
hK : DividedPowers K
f : A →+* B
g : B →+* C
h : A →+* C
hcomp : g.comp f = h
hg : hJ.isDPMorphism hK g
hf : hI.isDPMorph... | Please generate a tactic in lean4 to solve the state.
STATE:
case right
A : Type u_1
B : Type u_2
C : Type u_3
inst✝² : CommSemiring A
inst✝¹ : CommSemiring B
inst✝ : CommSemiring C
I : Ideal A
J : Ideal B
K : Ideal C
hI : DividedPowers I
hJ : DividedPowers J
hK : DividedPowers K
f : A →+* B
g : B →+* C
h : A →+* C
hco... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/DPMorphism.lean | DividedPowers.isDPMorphism.comp | [33, 1] | [47, 36] | rw [hg.2] | case right
A : Type u_1
B : Type u_2
C : Type u_3
inst✝² : CommSemiring A
inst✝¹ : CommSemiring B
inst✝ : CommSemiring C
I : Ideal A
J : Ideal B
K : Ideal C
hI : DividedPowers I
hJ : DividedPowers J
hK : DividedPowers K
f : A →+* B
g : B →+* C
h : A →+* C
hcomp : g.comp f = h
hg : hJ.isDPMorphism hK g
hf : hI.isDPMorph... | case right.a
A : Type u_1
B : Type u_2
C : Type u_3
inst✝² : CommSemiring A
inst✝¹ : CommSemiring B
inst✝ : CommSemiring C
I : Ideal A
J : Ideal B
K : Ideal C
hI : DividedPowers I
hJ : DividedPowers J
hK : DividedPowers K
f : A →+* B
g : B →+* C
h : A →+* C
hcomp : g.comp f = h
hg : hJ.isDPMorphism hK g
hf : hI.isDPMor... | Please generate a tactic in lean4 to solve the state.
STATE:
case right
A : Type u_1
B : Type u_2
C : Type u_3
inst✝² : CommSemiring A
inst✝¹ : CommSemiring B
inst✝ : CommSemiring C
I : Ideal A
J : Ideal B
K : Ideal C
hI : DividedPowers I
hJ : DividedPowers J
hK : DividedPowers K
f : A →+* B
g : B →+* C
h : A →+* C
hco... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/DPMorphism.lean | DividedPowers.isDPMorphism.comp | [33, 1] | [47, 36] | apply hf.1 | case right.a
A : Type u_1
B : Type u_2
C : Type u_3
inst✝² : CommSemiring A
inst✝¹ : CommSemiring B
inst✝ : CommSemiring C
I : Ideal A
J : Ideal B
K : Ideal C
hI : DividedPowers I
hJ : DividedPowers J
hK : DividedPowers K
f : A →+* B
g : B →+* C
h : A →+* C
hcomp : g.comp f = h
hg : hJ.isDPMorphism hK g
hf : hI.isDPMor... | case right.a.a
A : Type u_1
B : Type u_2
C : Type u_3
inst✝² : CommSemiring A
inst✝¹ : CommSemiring B
inst✝ : CommSemiring C
I : Ideal A
J : Ideal B
K : Ideal C
hI : DividedPowers I
hJ : DividedPowers J
hK : DividedPowers K
f : A →+* B
g : B →+* C
h : A →+* C
hcomp : g.comp f = h
hg : hJ.isDPMorphism hK g
hf : hI.isDPM... | Please generate a tactic in lean4 to solve the state.
STATE:
case right.a
A : Type u_1
B : Type u_2
C : Type u_3
inst✝² : CommSemiring A
inst✝¹ : CommSemiring B
inst✝ : CommSemiring C
I : Ideal A
J : Ideal B
K : Ideal C
hI : DividedPowers I
hJ : DividedPowers J
hK : DividedPowers K
f : A →+* B
g : B →+* C
h : A →+* C
h... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/DPMorphism.lean | DividedPowers.isDPMorphism.comp | [33, 1] | [47, 36] | exact Ideal.mem_map_of_mem f ha | case right.a.a
A : Type u_1
B : Type u_2
C : Type u_3
inst✝² : CommSemiring A
inst✝¹ : CommSemiring B
inst✝ : CommSemiring C
I : Ideal A
J : Ideal B
K : Ideal C
hI : DividedPowers I
hJ : DividedPowers J
hK : DividedPowers K
f : A →+* B
g : B →+* C
h : A →+* C
hcomp : g.comp f = h
hg : hJ.isDPMorphism hK g
hf : hI.isDPM... | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case right.a.a
A : Type u_1
B : Type u_2
C : Type u_3
inst✝² : CommSemiring A
inst✝¹ : CommSemiring B
inst✝ : CommSemiring C
I : Ideal A
J : Ideal B
K : Ideal C
hI : DividedPowers I
hJ : DividedPowers J
hK : DividedPowers K
f : A →+* B
g : B →+* C
h : A →+* C... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/DPMorphism.lean | DividedPowers.isDPMorphism_on_span | [169, 1] | [180, 18] | suffices h : I.map f ≤ J by
apply And.intro h
let dp_f := dpMorphismFromGens hI hJ hS h hdp
intro n a ha
rw [← dpMorphismFromGens_coe hI hJ hS h hdp, dp_f.dpow_comp n a ha] | A : Type u_1
B : Type u_2
inst✝¹ : CommSemiring A
inst✝ : CommSemiring B
I : Ideal A
J : Ideal B
hI : DividedPowers I
hJ : DividedPowers J
f : A →+* B
S : Set A
hS : I = Ideal.span S
hS' : ∀ s ∈ S, f s ∈ J
hdp : ∀ (n : ℕ), ∀ a ∈ S, f (hI.dpow n a) = hJ.dpow n (f a)
⊢ hI.isDPMorphism hJ f | A : Type u_1
B : Type u_2
inst✝¹ : CommSemiring A
inst✝ : CommSemiring B
I : Ideal A
J : Ideal B
hI : DividedPowers I
hJ : DividedPowers J
f : A →+* B
S : Set A
hS : I = Ideal.span S
hS' : ∀ s ∈ S, f s ∈ J
hdp : ∀ (n : ℕ), ∀ a ∈ S, f (hI.dpow n a) = hJ.dpow n (f a)
⊢ Ideal.map f I ≤ J | Please generate a tactic in lean4 to solve the state.
STATE:
A : Type u_1
B : Type u_2
inst✝¹ : CommSemiring A
inst✝ : CommSemiring B
I : Ideal A
J : Ideal B
hI : DividedPowers I
hJ : DividedPowers J
f : A →+* B
S : Set A
hS : I = Ideal.span S
hS' : ∀ s ∈ S, f s ∈ J
hdp : ∀ (n : ℕ), ∀ a ∈ S, f (hI.dpow n a) = hJ.dpow n... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/DPMorphism.lean | DividedPowers.isDPMorphism_on_span | [169, 1] | [180, 18] | rw [hS, Ideal.map_span, Ideal.span_le] | A : Type u_1
B : Type u_2
inst✝¹ : CommSemiring A
inst✝ : CommSemiring B
I : Ideal A
J : Ideal B
hI : DividedPowers I
hJ : DividedPowers J
f : A →+* B
S : Set A
hS : I = Ideal.span S
hS' : ∀ s ∈ S, f s ∈ J
hdp : ∀ (n : ℕ), ∀ a ∈ S, f (hI.dpow n a) = hJ.dpow n (f a)
⊢ Ideal.map f I ≤ J | A : Type u_1
B : Type u_2
inst✝¹ : CommSemiring A
inst✝ : CommSemiring B
I : Ideal A
J : Ideal B
hI : DividedPowers I
hJ : DividedPowers J
f : A →+* B
S : Set A
hS : I = Ideal.span S
hS' : ∀ s ∈ S, f s ∈ J
hdp : ∀ (n : ℕ), ∀ a ∈ S, f (hI.dpow n a) = hJ.dpow n (f a)
⊢ ⇑f '' S ⊆ ↑J | Please generate a tactic in lean4 to solve the state.
STATE:
A : Type u_1
B : Type u_2
inst✝¹ : CommSemiring A
inst✝ : CommSemiring B
I : Ideal A
J : Ideal B
hI : DividedPowers I
hJ : DividedPowers J
f : A →+* B
S : Set A
hS : I = Ideal.span S
hS' : ∀ s ∈ S, f s ∈ J
hdp : ∀ (n : ℕ), ∀ a ∈ S, f (hI.dpow n a) = hJ.dpow n... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/DPMorphism.lean | DividedPowers.isDPMorphism_on_span | [169, 1] | [180, 18] | rintro b ⟨a, has, rfl⟩ | A : Type u_1
B : Type u_2
inst✝¹ : CommSemiring A
inst✝ : CommSemiring B
I : Ideal A
J : Ideal B
hI : DividedPowers I
hJ : DividedPowers J
f : A →+* B
S : Set A
hS : I = Ideal.span S
hS' : ∀ s ∈ S, f s ∈ J
hdp : ∀ (n : ℕ), ∀ a ∈ S, f (hI.dpow n a) = hJ.dpow n (f a)
⊢ ⇑f '' S ⊆ ↑J | case intro.intro
A : Type u_1
B : Type u_2
inst✝¹ : CommSemiring A
inst✝ : CommSemiring B
I : Ideal A
J : Ideal B
hI : DividedPowers I
hJ : DividedPowers J
f : A →+* B
S : Set A
hS : I = Ideal.span S
hS' : ∀ s ∈ S, f s ∈ J
hdp : ∀ (n : ℕ), ∀ a ∈ S, f (hI.dpow n a) = hJ.dpow n (f a)
a : A
has : a ∈ S
⊢ f a ∈ ↑J | Please generate a tactic in lean4 to solve the state.
STATE:
A : Type u_1
B : Type u_2
inst✝¹ : CommSemiring A
inst✝ : CommSemiring B
I : Ideal A
J : Ideal B
hI : DividedPowers I
hJ : DividedPowers J
f : A →+* B
S : Set A
hS : I = Ideal.span S
hS' : ∀ s ∈ S, f s ∈ J
hdp : ∀ (n : ℕ), ∀ a ∈ S, f (hI.dpow n a) = hJ.dpow n... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/DPMorphism.lean | DividedPowers.isDPMorphism_on_span | [169, 1] | [180, 18] | exact hS' a has | case intro.intro
A : Type u_1
B : Type u_2
inst✝¹ : CommSemiring A
inst✝ : CommSemiring B
I : Ideal A
J : Ideal B
hI : DividedPowers I
hJ : DividedPowers J
f : A →+* B
S : Set A
hS : I = Ideal.span S
hS' : ∀ s ∈ S, f s ∈ J
hdp : ∀ (n : ℕ), ∀ a ∈ S, f (hI.dpow n a) = hJ.dpow n (f a)
a : A
has : a ∈ S
⊢ f a ∈ ↑J | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case intro.intro
A : Type u_1
B : Type u_2
inst✝¹ : CommSemiring A
inst✝ : CommSemiring B
I : Ideal A
J : Ideal B
hI : DividedPowers I
hJ : DividedPowers J
f : A →+* B
S : Set A
hS : I = Ideal.span S
hS' : ∀ s ∈ S, f s ∈ J
hdp : ∀ (n : ℕ), ∀ a ∈ S, f (hI.dpow... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/DPMorphism.lean | DividedPowers.isDPMorphism_on_span | [169, 1] | [180, 18] | apply And.intro h | A : Type u_1
B : Type u_2
inst✝¹ : CommSemiring A
inst✝ : CommSemiring B
I : Ideal A
J : Ideal B
hI : DividedPowers I
hJ : DividedPowers J
f : A →+* B
S : Set A
hS : I = Ideal.span S
hS' : ∀ s ∈ S, f s ∈ J
hdp : ∀ (n : ℕ), ∀ a ∈ S, f (hI.dpow n a) = hJ.dpow n (f a)
h : Ideal.map f I ≤ J
⊢ hI.isDPMorphism hJ f | A : Type u_1
B : Type u_2
inst✝¹ : CommSemiring A
inst✝ : CommSemiring B
I : Ideal A
J : Ideal B
hI : DividedPowers I
hJ : DividedPowers J
f : A →+* B
S : Set A
hS : I = Ideal.span S
hS' : ∀ s ∈ S, f s ∈ J
hdp : ∀ (n : ℕ), ∀ a ∈ S, f (hI.dpow n a) = hJ.dpow n (f a)
h : Ideal.map f I ≤ J
⊢ ∀ (n : ℕ), ∀ a ∈ I, hJ.dpow n ... | Please generate a tactic in lean4 to solve the state.
STATE:
A : Type u_1
B : Type u_2
inst✝¹ : CommSemiring A
inst✝ : CommSemiring B
I : Ideal A
J : Ideal B
hI : DividedPowers I
hJ : DividedPowers J
f : A →+* B
S : Set A
hS : I = Ideal.span S
hS' : ∀ s ∈ S, f s ∈ J
hdp : ∀ (n : ℕ), ∀ a ∈ S, f (hI.dpow n a) = hJ.dpow n... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/DPMorphism.lean | DividedPowers.isDPMorphism_on_span | [169, 1] | [180, 18] | let dp_f := dpMorphismFromGens hI hJ hS h hdp | A : Type u_1
B : Type u_2
inst✝¹ : CommSemiring A
inst✝ : CommSemiring B
I : Ideal A
J : Ideal B
hI : DividedPowers I
hJ : DividedPowers J
f : A →+* B
S : Set A
hS : I = Ideal.span S
hS' : ∀ s ∈ S, f s ∈ J
hdp : ∀ (n : ℕ), ∀ a ∈ S, f (hI.dpow n a) = hJ.dpow n (f a)
h : Ideal.map f I ≤ J
⊢ ∀ (n : ℕ), ∀ a ∈ I, hJ.dpow n ... | A : Type u_1
B : Type u_2
inst✝¹ : CommSemiring A
inst✝ : CommSemiring B
I : Ideal A
J : Ideal B
hI : DividedPowers I
hJ : DividedPowers J
f : A →+* B
S : Set A
hS : I = Ideal.span S
hS' : ∀ s ∈ S, f s ∈ J
hdp : ∀ (n : ℕ), ∀ a ∈ S, f (hI.dpow n a) = hJ.dpow n (f a)
h : Ideal.map f I ≤ J
dp_f : hI.dpMorphism hJ := hI.dp... | Please generate a tactic in lean4 to solve the state.
STATE:
A : Type u_1
B : Type u_2
inst✝¹ : CommSemiring A
inst✝ : CommSemiring B
I : Ideal A
J : Ideal B
hI : DividedPowers I
hJ : DividedPowers J
f : A →+* B
S : Set A
hS : I = Ideal.span S
hS' : ∀ s ∈ S, f s ∈ J
hdp : ∀ (n : ℕ), ∀ a ∈ S, f (hI.dpow n a) = hJ.dpow n... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/DPMorphism.lean | DividedPowers.isDPMorphism_on_span | [169, 1] | [180, 18] | intro n a ha | A : Type u_1
B : Type u_2
inst✝¹ : CommSemiring A
inst✝ : CommSemiring B
I : Ideal A
J : Ideal B
hI : DividedPowers I
hJ : DividedPowers J
f : A →+* B
S : Set A
hS : I = Ideal.span S
hS' : ∀ s ∈ S, f s ∈ J
hdp : ∀ (n : ℕ), ∀ a ∈ S, f (hI.dpow n a) = hJ.dpow n (f a)
h : Ideal.map f I ≤ J
dp_f : hI.dpMorphism hJ := hI.dp... | A : Type u_1
B : Type u_2
inst✝¹ : CommSemiring A
inst✝ : CommSemiring B
I : Ideal A
J : Ideal B
hI : DividedPowers I
hJ : DividedPowers J
f : A →+* B
S : Set A
hS : I = Ideal.span S
hS' : ∀ s ∈ S, f s ∈ J
hdp : ∀ (n : ℕ), ∀ a ∈ S, f (hI.dpow n a) = hJ.dpow n (f a)
h : Ideal.map f I ≤ J
dp_f : hI.dpMorphism hJ := hI.dp... | Please generate a tactic in lean4 to solve the state.
STATE:
A : Type u_1
B : Type u_2
inst✝¹ : CommSemiring A
inst✝ : CommSemiring B
I : Ideal A
J : Ideal B
hI : DividedPowers I
hJ : DividedPowers J
f : A →+* B
S : Set A
hS : I = Ideal.span S
hS' : ∀ s ∈ S, f s ∈ J
hdp : ∀ (n : ℕ), ∀ a ∈ S, f (hI.dpow n a) = hJ.dpow n... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/DPMorphism.lean | DividedPowers.isDPMorphism_on_span | [169, 1] | [180, 18] | rw [← dpMorphismFromGens_coe hI hJ hS h hdp, dp_f.dpow_comp n a ha] | A : Type u_1
B : Type u_2
inst✝¹ : CommSemiring A
inst✝ : CommSemiring B
I : Ideal A
J : Ideal B
hI : DividedPowers I
hJ : DividedPowers J
f : A →+* B
S : Set A
hS : I = Ideal.span S
hS' : ∀ s ∈ S, f s ∈ J
hdp : ∀ (n : ℕ), ∀ a ∈ S, f (hI.dpow n a) = hJ.dpow n (f a)
h : Ideal.map f I ≤ J
dp_f : hI.dpMorphism hJ := hI.dp... | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
A : Type u_1
B : Type u_2
inst✝¹ : CommSemiring A
inst✝ : CommSemiring B
I : Ideal A
J : Ideal B
hI : DividedPowers I
hJ : DividedPowers J
f : A →+* B
S : Set A
hS : I = Ideal.span S
hS' : ∀ s ∈ S, f s ∈ J
hdp : ∀ (n : ℕ), ∀ a ∈ S, f (hI.dpow n a) = hJ.dpow n... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/DPMorphism.lean | DividedPowers.isDPMorphism.of_comp | [190, 1] | [201, 19] | intro hf hh | A : Type u_1
B : Type u_2
C : Type u_3
inst✝² : CommSemiring A
inst✝¹ : CommSemiring B
inst✝ : CommSemiring C
I : Ideal A
J : Ideal B
K : Ideal C
hI : DividedPowers I
hJ : DividedPowers J
hK : DividedPowers K
f : A →+* B
g : B →+* C
h : A →+* C
hcomp : g.comp f = h
sf : J = Ideal.map f I
⊢ hI.isDPMorphism hJ f → hI.isD... | A : Type u_1
B : Type u_2
C : Type u_3
inst✝² : CommSemiring A
inst✝¹ : CommSemiring B
inst✝ : CommSemiring C
I : Ideal A
J : Ideal B
K : Ideal C
hI : DividedPowers I
hJ : DividedPowers J
hK : DividedPowers K
f : A →+* B
g : B →+* C
h : A →+* C
hcomp : g.comp f = h
sf : J = Ideal.map f I
hf : hI.isDPMorphism hJ f
hh : ... | Please generate a tactic in lean4 to solve the state.
STATE:
A : Type u_1
B : Type u_2
C : Type u_3
inst✝² : CommSemiring A
inst✝¹ : CommSemiring B
inst✝ : CommSemiring C
I : Ideal A
J : Ideal B
K : Ideal C
hI : DividedPowers I
hJ : DividedPowers J
hK : DividedPowers K
f : A →+* B
g : B →+* C
h : A →+* C
hcomp : g.comp... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/DPMorphism.lean | DividedPowers.isDPMorphism.of_comp | [190, 1] | [201, 19] | apply isDPMorphism_on_span | A : Type u_1
B : Type u_2
C : Type u_3
inst✝² : CommSemiring A
inst✝¹ : CommSemiring B
inst✝ : CommSemiring C
I : Ideal A
J : Ideal B
K : Ideal C
hI : DividedPowers I
hJ : DividedPowers J
hK : DividedPowers K
f : A →+* B
g : B →+* C
h : A →+* C
hcomp : g.comp f = h
sf : J = Ideal.map f I
hf : hI.isDPMorphism hJ f
hh : ... | case hS
A : Type u_1
B : Type u_2
C : Type u_3
inst✝² : CommSemiring A
inst✝¹ : CommSemiring B
inst✝ : CommSemiring C
I : Ideal A
J : Ideal B
K : Ideal C
hI : DividedPowers I
hJ : DividedPowers J
hK : DividedPowers K
f : A →+* B
g : B →+* C
h : A →+* C
hcomp : g.comp f = h
sf : J = Ideal.map f I
hf : hI.isDPMorphism hJ... | Please generate a tactic in lean4 to solve the state.
STATE:
A : Type u_1
B : Type u_2
C : Type u_3
inst✝² : CommSemiring A
inst✝¹ : CommSemiring B
inst✝ : CommSemiring C
I : Ideal A
J : Ideal B
K : Ideal C
hI : DividedPowers I
hJ : DividedPowers J
hK : DividedPowers K
f : A →+* B
g : B →+* C
h : A →+* C
hcomp : g.comp... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/DPMorphism.lean | DividedPowers.isDPMorphism.of_comp | [190, 1] | [201, 19] | exact sf | case hS
A : Type u_1
B : Type u_2
C : Type u_3
inst✝² : CommSemiring A
inst✝¹ : CommSemiring B
inst✝ : CommSemiring C
I : Ideal A
J : Ideal B
K : Ideal C
hI : DividedPowers I
hJ : DividedPowers J
hK : DividedPowers K
f : A →+* B
g : B →+* C
h : A →+* C
hcomp : g.comp f = h
sf : J = Ideal.map f I
hf : hI.isDPMorphism hJ... | case hS'
A : Type u_1
B : Type u_2
C : Type u_3
inst✝² : CommSemiring A
inst✝¹ : CommSemiring B
inst✝ : CommSemiring C
I : Ideal A
J : Ideal B
K : Ideal C
hI : DividedPowers I
hJ : DividedPowers J
hK : DividedPowers K
f : A →+* B
g : B →+* C
h : A →+* C
hcomp : g.comp f = h
sf : J = Ideal.map f I
hf : hI.isDPMorphism h... | Please generate a tactic in lean4 to solve the state.
STATE:
case hS
A : Type u_1
B : Type u_2
C : Type u_3
inst✝² : CommSemiring A
inst✝¹ : CommSemiring B
inst✝ : CommSemiring C
I : Ideal A
J : Ideal B
K : Ideal C
hI : DividedPowers I
hJ : DividedPowers J
hK : DividedPowers K
f : A →+* B
g : B →+* C
h : A →+* C
hcomp ... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/DPMorphism.lean | DividedPowers.isDPMorphism.of_comp | [190, 1] | [201, 19] | rintro b ⟨a, ha, rfl⟩ | case hS'
A : Type u_1
B : Type u_2
C : Type u_3
inst✝² : CommSemiring A
inst✝¹ : CommSemiring B
inst✝ : CommSemiring C
I : Ideal A
J : Ideal B
K : Ideal C
hI : DividedPowers I
hJ : DividedPowers J
hK : DividedPowers K
f : A →+* B
g : B →+* C
h : A →+* C
hcomp : g.comp f = h
sf : J = Ideal.map f I
hf : hI.isDPMorphism h... | case hS'.intro.intro
A : Type u_1
B : Type u_2
C : Type u_3
inst✝² : CommSemiring A
inst✝¹ : CommSemiring B
inst✝ : CommSemiring C
I : Ideal A
J : Ideal B
K : Ideal C
hI : DividedPowers I
hJ : DividedPowers J
hK : DividedPowers K
f : A →+* B
g : B →+* C
h : A →+* C
hcomp : g.comp f = h
sf : J = Ideal.map f I
hf : hI.is... | Please generate a tactic in lean4 to solve the state.
STATE:
case hS'
A : Type u_1
B : Type u_2
C : Type u_3
inst✝² : CommSemiring A
inst✝¹ : CommSemiring B
inst✝ : CommSemiring C
I : Ideal A
J : Ideal B
K : Ideal C
hI : DividedPowers I
hJ : DividedPowers J
hK : DividedPowers K
f : A →+* B
g : B →+* C
h : A →+* C
hcomp... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/DPMorphism.lean | DividedPowers.isDPMorphism.of_comp | [190, 1] | [201, 19] | rw [← RingHom.comp_apply] | case hS'.intro.intro
A : Type u_1
B : Type u_2
C : Type u_3
inst✝² : CommSemiring A
inst✝¹ : CommSemiring B
inst✝ : CommSemiring C
I : Ideal A
J : Ideal B
K : Ideal C
hI : DividedPowers I
hJ : DividedPowers J
hK : DividedPowers K
f : A →+* B
g : B →+* C
h : A →+* C
hcomp : g.comp f = h
sf : J = Ideal.map f I
hf : hI.is... | case hS'.intro.intro
A : Type u_1
B : Type u_2
C : Type u_3
inst✝² : CommSemiring A
inst✝¹ : CommSemiring B
inst✝ : CommSemiring C
I : Ideal A
J : Ideal B
K : Ideal C
hI : DividedPowers I
hJ : DividedPowers J
hK : DividedPowers K
f : A →+* B
g : B →+* C
h : A →+* C
hcomp : g.comp f = h
sf : J = Ideal.map f I
hf : hI.is... | Please generate a tactic in lean4 to solve the state.
STATE:
case hS'.intro.intro
A : Type u_1
B : Type u_2
C : Type u_3
inst✝² : CommSemiring A
inst✝¹ : CommSemiring B
inst✝ : CommSemiring C
I : Ideal A
J : Ideal B
K : Ideal C
hI : DividedPowers I
hJ : DividedPowers J
hK : DividedPowers K
f : A →+* B
g : B →+* C
h : A... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/DPMorphism.lean | DividedPowers.isDPMorphism.of_comp | [190, 1] | [201, 19] | rw [hcomp] | case hS'.intro.intro
A : Type u_1
B : Type u_2
C : Type u_3
inst✝² : CommSemiring A
inst✝¹ : CommSemiring B
inst✝ : CommSemiring C
I : Ideal A
J : Ideal B
K : Ideal C
hI : DividedPowers I
hJ : DividedPowers J
hK : DividedPowers K
f : A →+* B
g : B →+* C
h : A →+* C
hcomp : g.comp f = h
sf : J = Ideal.map f I
hf : hI.is... | case hS'.intro.intro
A : Type u_1
B : Type u_2
C : Type u_3
inst✝² : CommSemiring A
inst✝¹ : CommSemiring B
inst✝ : CommSemiring C
I : Ideal A
J : Ideal B
K : Ideal C
hI : DividedPowers I
hJ : DividedPowers J
hK : DividedPowers K
f : A →+* B
g : B →+* C
h : A →+* C
hcomp : g.comp f = h
sf : J = Ideal.map f I
hf : hI.is... | Please generate a tactic in lean4 to solve the state.
STATE:
case hS'.intro.intro
A : Type u_1
B : Type u_2
C : Type u_3
inst✝² : CommSemiring A
inst✝¹ : CommSemiring B
inst✝ : CommSemiring C
I : Ideal A
J : Ideal B
K : Ideal C
hI : DividedPowers I
hJ : DividedPowers J
hK : DividedPowers K
f : A →+* B
g : B →+* C
h : A... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/DPMorphism.lean | DividedPowers.isDPMorphism.of_comp | [190, 1] | [201, 19] | apply hh.1 | case hS'.intro.intro
A : Type u_1
B : Type u_2
C : Type u_3
inst✝² : CommSemiring A
inst✝¹ : CommSemiring B
inst✝ : CommSemiring C
I : Ideal A
J : Ideal B
K : Ideal C
hI : DividedPowers I
hJ : DividedPowers J
hK : DividedPowers K
f : A →+* B
g : B →+* C
h : A →+* C
hcomp : g.comp f = h
sf : J = Ideal.map f I
hf : hI.is... | case hS'.intro.intro.a
A : Type u_1
B : Type u_2
C : Type u_3
inst✝² : CommSemiring A
inst✝¹ : CommSemiring B
inst✝ : CommSemiring C
I : Ideal A
J : Ideal B
K : Ideal C
hI : DividedPowers I
hJ : DividedPowers J
hK : DividedPowers K
f : A →+* B
g : B →+* C
h : A →+* C
hcomp : g.comp f = h
sf : J = Ideal.map f I
hf : hI.... | Please generate a tactic in lean4 to solve the state.
STATE:
case hS'.intro.intro
A : Type u_1
B : Type u_2
C : Type u_3
inst✝² : CommSemiring A
inst✝¹ : CommSemiring B
inst✝ : CommSemiring C
I : Ideal A
J : Ideal B
K : Ideal C
hI : DividedPowers I
hJ : DividedPowers J
hK : DividedPowers K
f : A →+* B
g : B →+* C
h : A... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/DPMorphism.lean | DividedPowers.isDPMorphism.of_comp | [190, 1] | [201, 19] | apply Ideal.mem_map_of_mem | case hS'.intro.intro.a
A : Type u_1
B : Type u_2
C : Type u_3
inst✝² : CommSemiring A
inst✝¹ : CommSemiring B
inst✝ : CommSemiring C
I : Ideal A
J : Ideal B
K : Ideal C
hI : DividedPowers I
hJ : DividedPowers J
hK : DividedPowers K
f : A →+* B
g : B →+* C
h : A →+* C
hcomp : g.comp f = h
sf : J = Ideal.map f I
hf : hI.... | case hS'.intro.intro.a.h
A : Type u_1
B : Type u_2
C : Type u_3
inst✝² : CommSemiring A
inst✝¹ : CommSemiring B
inst✝ : CommSemiring C
I : Ideal A
J : Ideal B
K : Ideal C
hI : DividedPowers I
hJ : DividedPowers J
hK : DividedPowers K
f : A →+* B
g : B →+* C
h : A →+* C
hcomp : g.comp f = h
sf : J = Ideal.map f I
hf : h... | Please generate a tactic in lean4 to solve the state.
STATE:
case hS'.intro.intro.a
A : Type u_1
B : Type u_2
C : Type u_3
inst✝² : CommSemiring A
inst✝¹ : CommSemiring B
inst✝ : CommSemiring C
I : Ideal A
J : Ideal B
K : Ideal C
hI : DividedPowers I
hJ : DividedPowers J
hK : DividedPowers K
f : A →+* B
g : B →+* C
h :... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/DPMorphism.lean | DividedPowers.isDPMorphism.of_comp | [190, 1] | [201, 19] | exact ha | case hS'.intro.intro.a.h
A : Type u_1
B : Type u_2
C : Type u_3
inst✝² : CommSemiring A
inst✝¹ : CommSemiring B
inst✝ : CommSemiring C
I : Ideal A
J : Ideal B
K : Ideal C
hI : DividedPowers I
hJ : DividedPowers J
hK : DividedPowers K
f : A →+* B
g : B →+* C
h : A →+* C
hcomp : g.comp f = h
sf : J = Ideal.map f I
hf : h... | case hdp
A : Type u_1
B : Type u_2
C : Type u_3
inst✝² : CommSemiring A
inst✝¹ : CommSemiring B
inst✝ : CommSemiring C
I : Ideal A
J : Ideal B
K : Ideal C
hI : DividedPowers I
hJ : DividedPowers J
hK : DividedPowers K
f : A →+* B
g : B →+* C
h : A →+* C
hcomp : g.comp f = h
sf : J = Ideal.map f I
hf : hI.isDPMorphism h... | Please generate a tactic in lean4 to solve the state.
STATE:
case hS'.intro.intro.a.h
A : Type u_1
B : Type u_2
C : Type u_3
inst✝² : CommSemiring A
inst✝¹ : CommSemiring B
inst✝ : CommSemiring C
I : Ideal A
J : Ideal B
K : Ideal C
hI : DividedPowers I
hJ : DividedPowers J
hK : DividedPowers K
f : A →+* B
g : B →+* C
h... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/DPMorphism.lean | DividedPowers.isDPMorphism.of_comp | [190, 1] | [201, 19] | rintro n b ⟨a, ha, rfl⟩ | case hdp
A : Type u_1
B : Type u_2
C : Type u_3
inst✝² : CommSemiring A
inst✝¹ : CommSemiring B
inst✝ : CommSemiring C
I : Ideal A
J : Ideal B
K : Ideal C
hI : DividedPowers I
hJ : DividedPowers J
hK : DividedPowers K
f : A →+* B
g : B →+* C
h : A →+* C
hcomp : g.comp f = h
sf : J = Ideal.map f I
hf : hI.isDPMorphism h... | case hdp.intro.intro
A : Type u_1
B : Type u_2
C : Type u_3
inst✝² : CommSemiring A
inst✝¹ : CommSemiring B
inst✝ : CommSemiring C
I : Ideal A
J : Ideal B
K : Ideal C
hI : DividedPowers I
hJ : DividedPowers J
hK : DividedPowers K
f : A →+* B
g : B →+* C
h : A →+* C
hcomp : g.comp f = h
sf : J = Ideal.map f I
hf : hI.is... | Please generate a tactic in lean4 to solve the state.
STATE:
case hdp
A : Type u_1
B : Type u_2
C : Type u_3
inst✝² : CommSemiring A
inst✝¹ : CommSemiring B
inst✝ : CommSemiring C
I : Ideal A
J : Ideal B
K : Ideal C
hI : DividedPowers I
hJ : DividedPowers J
hK : DividedPowers K
f : A →+* B
g : B →+* C
h : A →+* C
hcomp... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/DPMorphism.lean | DividedPowers.isDPMorphism.of_comp | [190, 1] | [201, 19] | rw [← RingHom.comp_apply, hcomp, hh.2 n a ha, ← hcomp, RingHom.comp_apply] | case hdp.intro.intro
A : Type u_1
B : Type u_2
C : Type u_3
inst✝² : CommSemiring A
inst✝¹ : CommSemiring B
inst✝ : CommSemiring C
I : Ideal A
J : Ideal B
K : Ideal C
hI : DividedPowers I
hJ : DividedPowers J
hK : DividedPowers K
f : A →+* B
g : B →+* C
h : A →+* C
hcomp : g.comp f = h
sf : J = Ideal.map f I
hf : hI.is... | case hdp.intro.intro
A : Type u_1
B : Type u_2
C : Type u_3
inst✝² : CommSemiring A
inst✝¹ : CommSemiring B
inst✝ : CommSemiring C
I : Ideal A
J : Ideal B
K : Ideal C
hI : DividedPowers I
hJ : DividedPowers J
hK : DividedPowers K
f : A →+* B
g : B →+* C
h : A →+* C
hcomp : g.comp f = h
sf : J = Ideal.map f I
hf : hI.is... | Please generate a tactic in lean4 to solve the state.
STATE:
case hdp.intro.intro
A : Type u_1
B : Type u_2
C : Type u_3
inst✝² : CommSemiring A
inst✝¹ : CommSemiring B
inst✝ : CommSemiring C
I : Ideal A
J : Ideal B
K : Ideal C
hI : DividedPowers I
hJ : DividedPowers J
hK : DividedPowers K
f : A →+* B
g : B →+* C
h : A... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/DPMorphism.lean | DividedPowers.isDPMorphism.of_comp | [190, 1] | [201, 19] | rw [hf.2 n a ha] | case hdp.intro.intro
A : Type u_1
B : Type u_2
C : Type u_3
inst✝² : CommSemiring A
inst✝¹ : CommSemiring B
inst✝ : CommSemiring C
I : Ideal A
J : Ideal B
K : Ideal C
hI : DividedPowers I
hJ : DividedPowers J
hK : DividedPowers K
f : A →+* B
g : B →+* C
h : A →+* C
hcomp : g.comp f = h
sf : J = Ideal.map f I
hf : hI.is... | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case hdp.intro.intro
A : Type u_1
B : Type u_2
C : Type u_3
inst✝² : CommSemiring A
inst✝¹ : CommSemiring B
inst✝ : CommSemiring C
I : Ideal A
J : Ideal B
K : Ideal C
hI : DividedPowers I
hJ : DividedPowers J
hK : DividedPowers K
f : A →+* B
g : B →+* C
h : A... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/DPMorphism.lean | DividedPowers.dp_uniqueness_self | [206, 1] | [216, 43] | ext n a | A : Type u_1
inst✝ : CommSemiring A
I : Ideal A
hI hI' : DividedPowers I
S : Set A
hS : I = Ideal.span S
hdp : ∀ (n : ℕ), ∀ a ∈ S, hI.dpow n a = hI'.dpow n a
⊢ hI' = hI | case dpow.h.h
A : Type u_1
inst✝ : CommSemiring A
I : Ideal A
hI hI' : DividedPowers I
S : Set A
hS : I = Ideal.span S
hdp : ∀ (n : ℕ), ∀ a ∈ S, hI.dpow n a = hI'.dpow n a
n : ℕ
a : A
⊢ hI'.dpow n a = hI.dpow n a | Please generate a tactic in lean4 to solve the state.
STATE:
A : Type u_1
inst✝ : CommSemiring A
I : Ideal A
hI hI' : DividedPowers I
S : Set A
hS : I = Ideal.span S
hdp : ∀ (n : ℕ), ∀ a ∈ S, hI.dpow n a = hI'.dpow n a
⊢ hI' = hI
TACTIC:
|
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/DPMorphism.lean | DividedPowers.dp_uniqueness_self | [206, 1] | [216, 43] | by_cases ha : a ∈ I | case dpow.h.h
A : Type u_1
inst✝ : CommSemiring A
I : Ideal A
hI hI' : DividedPowers I
S : Set A
hS : I = Ideal.span S
hdp : ∀ (n : ℕ), ∀ a ∈ S, hI.dpow n a = hI'.dpow n a
n : ℕ
a : A
⊢ hI'.dpow n a = hI.dpow n a | case pos
A : Type u_1
inst✝ : CommSemiring A
I : Ideal A
hI hI' : DividedPowers I
S : Set A
hS : I = Ideal.span S
hdp : ∀ (n : ℕ), ∀ a ∈ S, hI.dpow n a = hI'.dpow n a
n : ℕ
a : A
ha : a ∈ I
⊢ hI'.dpow n a = hI.dpow n a
case neg
A : Type u_1
inst✝ : CommSemiring A
I : Ideal A
hI hI' : DividedPowers I
S : Set A
hS : I =... | Please generate a tactic in lean4 to solve the state.
STATE:
case dpow.h.h
A : Type u_1
inst✝ : CommSemiring A
I : Ideal A
hI hI' : DividedPowers I
S : Set A
hS : I = Ideal.span S
hdp : ∀ (n : ℕ), ∀ a ∈ S, hI.dpow n a = hI'.dpow n a
n : ℕ
a : A
⊢ hI'.dpow n a = hI.dpow n a
TACTIC:
|
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/DPMorphism.lean | DividedPowers.dp_uniqueness_self | [206, 1] | [216, 43] | . refine' hI.dp_uniqueness hI' (RingHom.id A) hS _ _ n a ha
. intro s hs
simp only [RingHom.id_apply, hS]
exact Ideal.subset_span hs
. simpa only [RingHom.id_apply] using hdp | case pos
A : Type u_1
inst✝ : CommSemiring A
I : Ideal A
hI hI' : DividedPowers I
S : Set A
hS : I = Ideal.span S
hdp : ∀ (n : ℕ), ∀ a ∈ S, hI.dpow n a = hI'.dpow n a
n : ℕ
a : A
ha : a ∈ I
⊢ hI'.dpow n a = hI.dpow n a
case neg
A : Type u_1
inst✝ : CommSemiring A
I : Ideal A
hI hI' : DividedPowers I
S : Set A
hS : I =... | case neg
A : Type u_1
inst✝ : CommSemiring A
I : Ideal A
hI hI' : DividedPowers I
S : Set A
hS : I = Ideal.span S
hdp : ∀ (n : ℕ), ∀ a ∈ S, hI.dpow n a = hI'.dpow n a
n : ℕ
a : A
ha : a ∉ I
⊢ hI'.dpow n a = hI.dpow n a | Please generate a tactic in lean4 to solve the state.
STATE:
case pos
A : Type u_1
inst✝ : CommSemiring A
I : Ideal A
hI hI' : DividedPowers I
S : Set A
hS : I = Ideal.span S
hdp : ∀ (n : ℕ), ∀ a ∈ S, hI.dpow n a = hI'.dpow n a
n : ℕ
a : A
ha : a ∈ I
⊢ hI'.dpow n a = hI.dpow n a
case neg
A : Type u_1
inst✝ : CommSemir... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/DPMorphism.lean | DividedPowers.dp_uniqueness_self | [206, 1] | [216, 43] | refine' hI.dp_uniqueness hI' (RingHom.id A) hS _ _ n a ha | case pos
A : Type u_1
inst✝ : CommSemiring A
I : Ideal A
hI hI' : DividedPowers I
S : Set A
hS : I = Ideal.span S
hdp : ∀ (n : ℕ), ∀ a ∈ S, hI.dpow n a = hI'.dpow n a
n : ℕ
a : A
ha : a ∈ I
⊢ hI'.dpow n a = hI.dpow n a | case pos.refine'_1
A : Type u_1
inst✝ : CommSemiring A
I : Ideal A
hI hI' : DividedPowers I
S : Set A
hS : I = Ideal.span S
hdp : ∀ (n : ℕ), ∀ a ∈ S, hI.dpow n a = hI'.dpow n a
n : ℕ
a : A
ha : a ∈ I
⊢ ∀ s ∈ S, (RingHom.id A) s ∈ I
case pos.refine'_2
A : Type u_1
inst✝ : CommSemiring A
I : Ideal A
hI hI' : DividedPowe... | Please generate a tactic in lean4 to solve the state.
STATE:
case pos
A : Type u_1
inst✝ : CommSemiring A
I : Ideal A
hI hI' : DividedPowers I
S : Set A
hS : I = Ideal.span S
hdp : ∀ (n : ℕ), ∀ a ∈ S, hI.dpow n a = hI'.dpow n a
n : ℕ
a : A
ha : a ∈ I
⊢ hI'.dpow n a = hI.dpow n a
TACTIC:
|
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/DPMorphism.lean | DividedPowers.dp_uniqueness_self | [206, 1] | [216, 43] | . intro s hs
simp only [RingHom.id_apply, hS]
exact Ideal.subset_span hs | case pos.refine'_1
A : Type u_1
inst✝ : CommSemiring A
I : Ideal A
hI hI' : DividedPowers I
S : Set A
hS : I = Ideal.span S
hdp : ∀ (n : ℕ), ∀ a ∈ S, hI.dpow n a = hI'.dpow n a
n : ℕ
a : A
ha : a ∈ I
⊢ ∀ s ∈ S, (RingHom.id A) s ∈ I
case pos.refine'_2
A : Type u_1
inst✝ : CommSemiring A
I : Ideal A
hI hI' : DividedPowe... | case pos.refine'_2
A : Type u_1
inst✝ : CommSemiring A
I : Ideal A
hI hI' : DividedPowers I
S : Set A
hS : I = Ideal.span S
hdp : ∀ (n : ℕ), ∀ a ∈ S, hI.dpow n a = hI'.dpow n a
n : ℕ
a : A
ha : a ∈ I
⊢ ∀ (n : ℕ), ∀ a ∈ S, (RingHom.id A) (hI.dpow n a) = hI'.dpow n ((RingHom.id A) a) | Please generate a tactic in lean4 to solve the state.
STATE:
case pos.refine'_1
A : Type u_1
inst✝ : CommSemiring A
I : Ideal A
hI hI' : DividedPowers I
S : Set A
hS : I = Ideal.span S
hdp : ∀ (n : ℕ), ∀ a ∈ S, hI.dpow n a = hI'.dpow n a
n : ℕ
a : A
ha : a ∈ I
⊢ ∀ s ∈ S, (RingHom.id A) s ∈ I
case pos.refine'_2
A : Typ... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/DPMorphism.lean | DividedPowers.dp_uniqueness_self | [206, 1] | [216, 43] | . simpa only [RingHom.id_apply] using hdp | case pos.refine'_2
A : Type u_1
inst✝ : CommSemiring A
I : Ideal A
hI hI' : DividedPowers I
S : Set A
hS : I = Ideal.span S
hdp : ∀ (n : ℕ), ∀ a ∈ S, hI.dpow n a = hI'.dpow n a
n : ℕ
a : A
ha : a ∈ I
⊢ ∀ (n : ℕ), ∀ a ∈ S, (RingHom.id A) (hI.dpow n a) = hI'.dpow n ((RingHom.id A) a) | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case pos.refine'_2
A : Type u_1
inst✝ : CommSemiring A
I : Ideal A
hI hI' : DividedPowers I
S : Set A
hS : I = Ideal.span S
hdp : ∀ (n : ℕ), ∀ a ∈ S, hI.dpow n a = hI'.dpow n a
n : ℕ
a : A
ha : a ∈ I
⊢ ∀ (n : ℕ), ∀ a ∈ S, (RingHom.id A) (hI.dpow n a) = hI'.dp... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/DPMorphism.lean | DividedPowers.dp_uniqueness_self | [206, 1] | [216, 43] | intro s hs | case pos.refine'_1
A : Type u_1
inst✝ : CommSemiring A
I : Ideal A
hI hI' : DividedPowers I
S : Set A
hS : I = Ideal.span S
hdp : ∀ (n : ℕ), ∀ a ∈ S, hI.dpow n a = hI'.dpow n a
n : ℕ
a : A
ha : a ∈ I
⊢ ∀ s ∈ S, (RingHom.id A) s ∈ I | case pos.refine'_1
A : Type u_1
inst✝ : CommSemiring A
I : Ideal A
hI hI' : DividedPowers I
S : Set A
hS : I = Ideal.span S
hdp : ∀ (n : ℕ), ∀ a ∈ S, hI.dpow n a = hI'.dpow n a
n : ℕ
a : A
ha : a ∈ I
s : A
hs : s ∈ S
⊢ (RingHom.id A) s ∈ I | Please generate a tactic in lean4 to solve the state.
STATE:
case pos.refine'_1
A : Type u_1
inst✝ : CommSemiring A
I : Ideal A
hI hI' : DividedPowers I
S : Set A
hS : I = Ideal.span S
hdp : ∀ (n : ℕ), ∀ a ∈ S, hI.dpow n a = hI'.dpow n a
n : ℕ
a : A
ha : a ∈ I
⊢ ∀ s ∈ S, (RingHom.id A) s ∈ I
TACTIC:
|
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/DPMorphism.lean | DividedPowers.dp_uniqueness_self | [206, 1] | [216, 43] | simp only [RingHom.id_apply, hS] | case pos.refine'_1
A : Type u_1
inst✝ : CommSemiring A
I : Ideal A
hI hI' : DividedPowers I
S : Set A
hS : I = Ideal.span S
hdp : ∀ (n : ℕ), ∀ a ∈ S, hI.dpow n a = hI'.dpow n a
n : ℕ
a : A
ha : a ∈ I
s : A
hs : s ∈ S
⊢ (RingHom.id A) s ∈ I | case pos.refine'_1
A : Type u_1
inst✝ : CommSemiring A
I : Ideal A
hI hI' : DividedPowers I
S : Set A
hS : I = Ideal.span S
hdp : ∀ (n : ℕ), ∀ a ∈ S, hI.dpow n a = hI'.dpow n a
n : ℕ
a : A
ha : a ∈ I
s : A
hs : s ∈ S
⊢ s ∈ Ideal.span S | Please generate a tactic in lean4 to solve the state.
STATE:
case pos.refine'_1
A : Type u_1
inst✝ : CommSemiring A
I : Ideal A
hI hI' : DividedPowers I
S : Set A
hS : I = Ideal.span S
hdp : ∀ (n : ℕ), ∀ a ∈ S, hI.dpow n a = hI'.dpow n a
n : ℕ
a : A
ha : a ∈ I
s : A
hs : s ∈ S
⊢ (RingHom.id A) s ∈ I
TACTIC:
|
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/DPMorphism.lean | DividedPowers.dp_uniqueness_self | [206, 1] | [216, 43] | exact Ideal.subset_span hs | case pos.refine'_1
A : Type u_1
inst✝ : CommSemiring A
I : Ideal A
hI hI' : DividedPowers I
S : Set A
hS : I = Ideal.span S
hdp : ∀ (n : ℕ), ∀ a ∈ S, hI.dpow n a = hI'.dpow n a
n : ℕ
a : A
ha : a ∈ I
s : A
hs : s ∈ S
⊢ s ∈ Ideal.span S | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case pos.refine'_1
A : Type u_1
inst✝ : CommSemiring A
I : Ideal A
hI hI' : DividedPowers I
S : Set A
hS : I = Ideal.span S
hdp : ∀ (n : ℕ), ∀ a ∈ S, hI.dpow n a = hI'.dpow n a
n : ℕ
a : A
ha : a ∈ I
s : A
hs : s ∈ S
⊢ s ∈ Ideal.span S
TACTIC:
|
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/DPMorphism.lean | DividedPowers.dp_uniqueness_self | [206, 1] | [216, 43] | simpa only [RingHom.id_apply] using hdp | case pos.refine'_2
A : Type u_1
inst✝ : CommSemiring A
I : Ideal A
hI hI' : DividedPowers I
S : Set A
hS : I = Ideal.span S
hdp : ∀ (n : ℕ), ∀ a ∈ S, hI.dpow n a = hI'.dpow n a
n : ℕ
a : A
ha : a ∈ I
⊢ ∀ (n : ℕ), ∀ a ∈ S, (RingHom.id A) (hI.dpow n a) = hI'.dpow n ((RingHom.id A) a) | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case pos.refine'_2
A : Type u_1
inst✝ : CommSemiring A
I : Ideal A
hI hI' : DividedPowers I
S : Set A
hS : I = Ideal.span S
hdp : ∀ (n : ℕ), ∀ a ∈ S, hI.dpow n a = hI'.dpow n a
n : ℕ
a : A
ha : a ∈ I
⊢ ∀ (n : ℕ), ∀ a ∈ S, (RingHom.id A) (hI.dpow n a) = hI'.dp... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/DPMorphism.lean | DividedPowers.dp_uniqueness_self | [206, 1] | [216, 43] | rw [hI.dpow_null ha, hI'.dpow_null ha] | case neg
A : Type u_1
inst✝ : CommSemiring A
I : Ideal A
hI hI' : DividedPowers I
S : Set A
hS : I = Ideal.span S
hdp : ∀ (n : ℕ), ∀ a ∈ S, hI.dpow n a = hI'.dpow n a
n : ℕ
a : A
ha : a ∉ I
⊢ hI'.dpow n a = hI.dpow n a | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case neg
A : Type u_1
inst✝ : CommSemiring A
I : Ideal A
hI hI' : DividedPowers I
S : Set A
hS : I = Ideal.span S
hdp : ∀ (n : ℕ), ∀ a ∈ S, hI.dpow n a = hI'.dpow n a
n : ℕ
a : A
ha : a ∉ I
⊢ hI'.dpow n a = hI.dpow n a
TACTIC:
|
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/PolynomialMap/Homogeneous.lean | Polynomial.coeff_baseChange_apply | [73, 1] | [83, 100] | rw [baseChange, AlgHom.coe_mk, coe_eval₂RingHom] | R : Type u_1
inst✝⁴ : CommSemiring R
S : Type u_2
inst✝³ : CommSemiring S
inst✝² : Algebra R S
S' : Type u_3
inst✝¹ : CommSemiring S'
inst✝ : Algebra R S'
φ : S →ₐ[R] S'
f : S[X]
p : ℕ
⊢ ((baseChange φ) f).coeff p = φ (f.coeff p) | R : Type u_1
inst✝⁴ : CommSemiring R
S : Type u_2
inst✝³ : CommSemiring S
inst✝² : Algebra R S
S' : Type u_3
inst✝¹ : CommSemiring S'
inst✝ : Algebra R S'
φ : S →ₐ[R] S'
f : S[X]
p : ℕ
⊢ (eval₂ (C.comp ↑φ) X f).coeff p = φ (f.coeff p) | Please generate a tactic in lean4 to solve the state.
STATE:
R : Type u_1
inst✝⁴ : CommSemiring R
S : Type u_2
inst✝³ : CommSemiring S
inst✝² : Algebra R S
S' : Type u_3
inst✝¹ : CommSemiring S'
inst✝ : Algebra R S'
φ : S →ₐ[R] S'
f : S[X]
p : ℕ
⊢ ((baseChange φ) f).coeff p = φ (f.coeff p)
TACTIC:
|
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/PolynomialMap/Homogeneous.lean | Polynomial.coeff_baseChange_apply | [73, 1] | [83, 100] | induction f using Polynomial.induction_on with
| h_C r =>
simp only [eval₂_C, RingHom.coe_comp, RingHom.coe_coe, Function.comp_apply, coeff_C,
apply_ite φ, map_zero]
| h_add f g hf hg => simp only [eval₂_add, coeff_add, hf, hg, map_add]
| h_monomial n r _ =>
simp only [eval₂_mul, eval₂_C, RingHom.coe_comp, Ring... | R : Type u_1
inst✝⁴ : CommSemiring R
S : Type u_2
inst✝³ : CommSemiring S
inst✝² : Algebra R S
S' : Type u_3
inst✝¹ : CommSemiring S'
inst✝ : Algebra R S'
φ : S →ₐ[R] S'
f : S[X]
p : ℕ
⊢ (eval₂ (C.comp ↑φ) X f).coeff p = φ (f.coeff p) | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
R : Type u_1
inst✝⁴ : CommSemiring R
S : Type u_2
inst✝³ : CommSemiring S
inst✝² : Algebra R S
S' : Type u_3
inst✝¹ : CommSemiring S'
inst✝ : Algebra R S'
φ : S →ₐ[R] S'
f : S[X]
p : ℕ
⊢ (eval₂ (C.comp ↑φ) X f).coeff p = φ (f.coeff p)
TACTIC:
|
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/PolynomialMap/Homogeneous.lean | Polynomial.coeff_baseChange_apply | [73, 1] | [83, 100] | simp only [eval₂_C, RingHom.coe_comp, RingHom.coe_coe, Function.comp_apply, coeff_C,
apply_ite φ, map_zero] | case h_C
R : Type u_1
inst✝⁴ : CommSemiring R
S : Type u_2
inst✝³ : CommSemiring S
inst✝² : Algebra R S
S' : Type u_3
inst✝¹ : CommSemiring S'
inst✝ : Algebra R S'
φ : S →ₐ[R] S'
p : ℕ
r : S
⊢ (eval₂ (C.comp ↑φ) X (C r)).coeff p = φ ((C r).coeff p) | no goals | 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_2
inst✝³ : CommSemiring S
inst✝² : Algebra R S
S' : Type u_3
inst✝¹ : CommSemiring S'
inst✝ : Algebra R S'
φ : S →ₐ[R] S'
p : ℕ
r : S
⊢ (eval₂ (C.comp ↑φ) X (C r)).coeff p = φ ((C r).coeff p)
TACTIC:
|
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/PolynomialMap/Homogeneous.lean | Polynomial.coeff_baseChange_apply | [73, 1] | [83, 100] | simp only [eval₂_add, coeff_add, hf, hg, map_add] | case h_add
R : Type u_1
inst✝⁴ : CommSemiring R
S : Type u_2
inst✝³ : CommSemiring S
inst✝² : Algebra R S
S' : Type u_3
inst✝¹ : CommSemiring S'
inst✝ : Algebra R S'
φ : S →ₐ[R] S'
p : ℕ
f g : S[X]
hf : (eval₂ (C.comp ↑φ) X f).coeff p = φ (f.coeff p)
hg : (eval₂ (C.comp ↑φ) X g).coeff p = φ (g.coeff p)
⊢ (eval₂ (C.comp... | 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_2
inst✝³ : CommSemiring S
inst✝² : Algebra R S
S' : Type u_3
inst✝¹ : CommSemiring S'
inst✝ : Algebra R S'
φ : S →ₐ[R] S'
p : ℕ
f g : S[X]
hf : (eval₂ (C.comp ↑φ) X f).coeff p = φ (f.coeff p)
hg : (ev... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/PolynomialMap/Homogeneous.lean | Polynomial.coeff_baseChange_apply | [73, 1] | [83, 100] | simp only [eval₂_mul, eval₂_C, RingHom.coe_comp, RingHom.coe_coe, Function.comp_apply,
eval₂_X_pow, coeff_C_mul, _root_.map_mul, coeff_X_pow, apply_ite φ, _root_.map_one, map_zero] | case h_monomial
R : Type u_1
inst✝⁴ : CommSemiring R
S : Type u_2
inst✝³ : CommSemiring S
inst✝² : Algebra R S
S' : Type u_3
inst✝¹ : CommSemiring S'
inst✝ : Algebra R S'
φ : S →ₐ[R] S'
p n : ℕ
r : S
a✝ : (eval₂ (C.comp ↑φ) X (C r * X ^ n)).coeff p = φ ((C r * X ^ n).coeff p)
⊢ (eval₂ (C.comp ↑φ) X (C r * X ^ (n + 1)))... | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case h_monomial
R : Type u_1
inst✝⁴ : CommSemiring R
S : Type u_2
inst✝³ : CommSemiring S
inst✝² : Algebra R S
S' : Type u_3
inst✝¹ : CommSemiring S'
inst✝ : Algebra R S'
φ : S →ₐ[R] S'
p n : ℕ
r : S
a✝ : (eval₂ (C.comp ↑φ) X (C r * X ^ n)).coeff p = φ ((C r ... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/PolynomialMap/Homogeneous.lean | Polynomial.lcoeff_comp_baseChange_eq | [85, 1] | [90, 54] | ext f | R : Type u_1
inst✝⁴ : CommSemiring R
S : Type u_2
inst✝³ : CommSemiring S
inst✝² : Algebra R S
S' : Type u_3
inst✝¹ : CommSemiring S'
inst✝ : Algebra R S'
φ : S →ₐ[R] S'
p : ℕ
⊢ φ.toLinearMap ∘ₗ ↑R (lcoeff S p) = ↑R (lcoeff S' p) ∘ₗ (baseChange φ).toLinearMap | case h
R : Type u_1
inst✝⁴ : CommSemiring R
S : Type u_2
inst✝³ : CommSemiring S
inst✝² : Algebra R S
S' : Type u_3
inst✝¹ : CommSemiring S'
inst✝ : Algebra R S'
φ : S →ₐ[R] S'
p : ℕ
f : S[X]
⊢ (φ.toLinearMap ∘ₗ ↑R (lcoeff S p)) f = (↑R (lcoeff S' p) ∘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_2
inst✝³ : CommSemiring S
inst✝² : Algebra R S
S' : Type u_3
inst✝¹ : CommSemiring S'
inst✝ : Algebra R S'
φ : S →ₐ[R] S'
p : ℕ
⊢ φ.toLinearMap ∘ₗ ↑R (lcoeff S p) = ↑R (lcoeff S' p) ∘ₗ (baseChange φ).toLinearMap... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/PolynomialMap/Homogeneous.lean | Polynomial.lcoeff_comp_baseChange_eq | [85, 1] | [90, 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_2
inst✝³ : CommSemiring S
inst✝² : Algebra R S
S' : Type u_3
inst✝¹ : CommSemiring S'
inst✝ : Algebra R S'
φ : S →ₐ[R] S'
p : ℕ
f : S[X]
⊢ (φ.toLinearMap ∘ₗ ↑R (lcoeff S p)) f = (↑R (lcoeff S' p) ∘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_2
inst✝³ : CommSemiring S
inst✝² : Algebra R S
S' : Type u_3
inst✝¹ : CommSemiring S'
inst✝ : Algebra R S'
φ : S →ₐ[R] S'
p : ℕ
f : S[X]
⊢ (φ.toLinearMap ∘ₗ ↑R (lcoeff S p)) f = (↑R (lcoeff S' p) ∘ₗ (base... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/PolynomialMap/Homogeneous.lean | Polynomial.baseChange_monomial | [92, 1] | [95, 67] | simp only [baseChange, AlgHom.coe_mk, coe_eval₂RingHom, eval₂_monomial, RingHom.coe_comp,
RingHom.coe_coe, Function.comp_apply, C_mul_X_pow_eq_monomial] | R : Type u_1
inst✝⁴ : CommSemiring R
S : Type u_2
inst✝³ : CommSemiring S
inst✝² : Algebra R S
S' : Type u_3
inst✝¹ : CommSemiring S'
inst✝ : Algebra R S'
φ : S →ₐ[R] S'
n : ℕ
a : S
⊢ (baseChange φ) ((monomial n) a) = (monomial n) (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_2
inst✝³ : CommSemiring S
inst✝² : Algebra R S
S' : Type u_3
inst✝¹ : CommSemiring S'
inst✝ : Algebra R S'
φ : S →ₐ[R] S'
n : ℕ
a : S
⊢ (baseChange φ) ((monomial n) a) = (monomial n) (φ a)
TACTIC:
|
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/PolynomialMap/Homogeneous.lean | TensorProduct.exists_Finsupp | [108, 1] | [118, 88] | classical
induction μ using TensorProduct.induction_on with
| zero => use 0; simp
| tmul s m => use single s m; simp
| add x y hx hy =>
obtain ⟨sx, rfl⟩ := hx
obtain ⟨sy, rfl⟩ := hy
use sx + sy
rw [sum_add_index (fun _ ↦ by simp) (fun _ _ _ _ ↦ by rw [TensorProduct.tmul_add])] | R : Type u_1
inst✝⁴ : CommSemiring R
M : Type u_2
inst✝³ : AddCommMonoid M
inst✝² : Module R M
S : Type u_3
inst✝¹ : CommSemiring S
inst✝ : Algebra R S
μ : S ⊗[R] M
⊢ ∃ m, μ = m.sum fun s x => s ⊗ₜ[R] x | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
R : Type u_1
inst✝⁴ : CommSemiring R
M : Type u_2
inst✝³ : AddCommMonoid M
inst✝² : Module R M
S : Type u_3
inst✝¹ : CommSemiring S
inst✝ : Algebra R S
μ : S ⊗[R] M
⊢ ∃ m, μ = m.sum fun s x => s ⊗ₜ[R] x
TACTIC:
|
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/PolynomialMap/Homogeneous.lean | TensorProduct.exists_Finsupp | [108, 1] | [118, 88] | induction μ using TensorProduct.induction_on with
| zero => use 0; simp
| tmul s m => use single s m; simp
| add x y hx hy =>
obtain ⟨sx, rfl⟩ := hx
obtain ⟨sy, rfl⟩ := hy
use sx + sy
rw [sum_add_index (fun _ ↦ by simp) (fun _ _ _ _ ↦ by rw [TensorProduct.tmul_add])] | R : Type u_1
inst✝⁴ : CommSemiring R
M : Type u_2
inst✝³ : AddCommMonoid M
inst✝² : Module R M
S : Type u_3
inst✝¹ : CommSemiring S
inst✝ : Algebra R S
μ : S ⊗[R] M
⊢ ∃ m, μ = m.sum fun s x => s ⊗ₜ[R] x | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
R : Type u_1
inst✝⁴ : CommSemiring R
M : Type u_2
inst✝³ : AddCommMonoid M
inst✝² : Module R M
S : Type u_3
inst✝¹ : CommSemiring S
inst✝ : Algebra R S
μ : S ⊗[R] M
⊢ ∃ m, μ = m.sum fun s x => s ⊗ₜ[R] x
TACTIC:
|
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/PolynomialMap/Homogeneous.lean | TensorProduct.exists_Finsupp | [108, 1] | [118, 88] | use 0 | case zero
R : Type u_1
inst✝⁴ : CommSemiring R
M : Type u_2
inst✝³ : AddCommMonoid M
inst✝² : Module R M
S : Type u_3
inst✝¹ : CommSemiring S
inst✝ : Algebra R S
⊢ ∃ m, 0 = m.sum fun s x => s ⊗ₜ[R] x | case h
R : Type u_1
inst✝⁴ : CommSemiring R
M : Type u_2
inst✝³ : AddCommMonoid M
inst✝² : Module R M
S : Type u_3
inst✝¹ : CommSemiring S
inst✝ : Algebra R S
⊢ 0 = sum 0 fun s x => s ⊗ₜ[R] x | Please generate a tactic in lean4 to solve the state.
STATE:
case zero
R : Type u_1
inst✝⁴ : CommSemiring R
M : Type u_2
inst✝³ : AddCommMonoid M
inst✝² : Module R M
S : Type u_3
inst✝¹ : CommSemiring S
inst✝ : Algebra R S
⊢ ∃ m, 0 = m.sum fun s x => s ⊗ₜ[R] x
TACTIC:
|
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/PolynomialMap/Homogeneous.lean | TensorProduct.exists_Finsupp | [108, 1] | [118, 88] | simp | case h
R : Type u_1
inst✝⁴ : CommSemiring R
M : Type u_2
inst✝³ : AddCommMonoid M
inst✝² : Module R M
S : Type u_3
inst✝¹ : CommSemiring S
inst✝ : Algebra R S
⊢ 0 = sum 0 fun s x => s ⊗ₜ[R] x | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case h
R : Type u_1
inst✝⁴ : CommSemiring R
M : Type u_2
inst✝³ : AddCommMonoid M
inst✝² : Module R M
S : Type u_3
inst✝¹ : CommSemiring S
inst✝ : Algebra R S
⊢ 0 = sum 0 fun s x => s ⊗ₜ[R] x
TACTIC:
|
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/PolynomialMap/Homogeneous.lean | TensorProduct.exists_Finsupp | [108, 1] | [118, 88] | use single s m | case tmul
R : Type u_1
inst✝⁴ : CommSemiring R
M : Type u_2
inst✝³ : AddCommMonoid M
inst✝² : Module R M
S : Type u_3
inst✝¹ : CommSemiring S
inst✝ : Algebra R S
s : S
m : M
⊢ ∃ m_1, s ⊗ₜ[R] m = m_1.sum fun s x => s ⊗ₜ[R] x | case h
R : Type u_1
inst✝⁴ : CommSemiring R
M : Type u_2
inst✝³ : AddCommMonoid M
inst✝² : Module R M
S : Type u_3
inst✝¹ : CommSemiring S
inst✝ : Algebra R S
s : S
m : M
⊢ s ⊗ₜ[R] m = (Finsupp.single s m).sum fun s x => s ⊗ₜ[R] x | Please generate a tactic in lean4 to solve the state.
STATE:
case tmul
R : Type u_1
inst✝⁴ : CommSemiring R
M : Type u_2
inst✝³ : AddCommMonoid M
inst✝² : Module R M
S : Type u_3
inst✝¹ : CommSemiring S
inst✝ : Algebra R S
s : S
m : M
⊢ ∃ m_1, s ⊗ₜ[R] m = m_1.sum fun s x => s ⊗ₜ[R] x
TACTIC:
|
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/PolynomialMap/Homogeneous.lean | TensorProduct.exists_Finsupp | [108, 1] | [118, 88] | simp | case h
R : Type u_1
inst✝⁴ : CommSemiring R
M : Type u_2
inst✝³ : AddCommMonoid M
inst✝² : Module R M
S : Type u_3
inst✝¹ : CommSemiring S
inst✝ : Algebra R S
s : S
m : M
⊢ s ⊗ₜ[R] m = (Finsupp.single s m).sum fun s x => s ⊗ₜ[R] x | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case h
R : Type u_1
inst✝⁴ : CommSemiring R
M : Type u_2
inst✝³ : AddCommMonoid M
inst✝² : Module R M
S : Type u_3
inst✝¹ : CommSemiring S
inst✝ : Algebra R S
s : S
m : M
⊢ s ⊗ₜ[R] m = (Finsupp.single s m).sum fun s x => s ⊗ₜ[R] x
TACTIC:
|
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/PolynomialMap/Homogeneous.lean | TensorProduct.exists_Finsupp | [108, 1] | [118, 88] | obtain ⟨sx, rfl⟩ := hx | case add
R : Type u_1
inst✝⁴ : CommSemiring R
M : Type u_2
inst✝³ : AddCommMonoid M
inst✝² : Module R M
S : Type u_3
inst✝¹ : CommSemiring S
inst✝ : Algebra R S
x y : S ⊗[R] M
hx : ∃ m, x = m.sum fun s x => s ⊗ₜ[R] x
hy : ∃ m, y = m.sum fun s x => s ⊗ₜ[R] x
⊢ ∃ m, x + y = m.sum fun s x => s ⊗ₜ[R] x | case add.intro
R : Type u_1
inst✝⁴ : CommSemiring R
M : Type u_2
inst✝³ : AddCommMonoid M
inst✝² : Module R M
S : Type u_3
inst✝¹ : CommSemiring S
inst✝ : Algebra R S
y : S ⊗[R] M
hy : ∃ m, y = m.sum fun s x => s ⊗ₜ[R] x
sx : S →₀ M
⊢ ∃ m, (sx.sum fun s x => s ⊗ₜ[R] x) + y = m.sum fun s x => s ⊗ₜ[R] x | Please generate a tactic in lean4 to solve the state.
STATE:
case add
R : Type u_1
inst✝⁴ : CommSemiring R
M : Type u_2
inst✝³ : AddCommMonoid M
inst✝² : Module R M
S : Type u_3
inst✝¹ : CommSemiring S
inst✝ : Algebra R S
x y : S ⊗[R] M
hx : ∃ m, x = m.sum fun s x => s ⊗ₜ[R] x
hy : ∃ m, y = m.sum fun s x => s ⊗ₜ[R] x
⊢... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/PolynomialMap/Homogeneous.lean | TensorProduct.exists_Finsupp | [108, 1] | [118, 88] | obtain ⟨sy, rfl⟩ := hy | case add.intro
R : Type u_1
inst✝⁴ : CommSemiring R
M : Type u_2
inst✝³ : AddCommMonoid M
inst✝² : Module R M
S : Type u_3
inst✝¹ : CommSemiring S
inst✝ : Algebra R S
y : S ⊗[R] M
hy : ∃ m, y = m.sum fun s x => s ⊗ₜ[R] x
sx : S →₀ M
⊢ ∃ m, (sx.sum fun s x => s ⊗ₜ[R] x) + y = m.sum fun s x => s ⊗ₜ[R] x | case add.intro.intro
R : Type u_1
inst✝⁴ : CommSemiring R
M : Type u_2
inst✝³ : AddCommMonoid M
inst✝² : Module R M
S : Type u_3
inst✝¹ : CommSemiring S
inst✝ : Algebra R S
sx sy : S →₀ M
⊢ ∃ m, ((sx.sum fun s x => s ⊗ₜ[R] x) + sy.sum fun s x => s ⊗ₜ[R] x) = m.sum fun s x => s ⊗ₜ[R] x | Please generate a tactic in lean4 to solve the state.
STATE:
case add.intro
R : Type u_1
inst✝⁴ : CommSemiring R
M : Type u_2
inst✝³ : AddCommMonoid M
inst✝² : Module R M
S : Type u_3
inst✝¹ : CommSemiring S
inst✝ : Algebra R S
y : S ⊗[R] M
hy : ∃ m, y = m.sum fun s x => s ⊗ₜ[R] x
sx : S →₀ M
⊢ ∃ m, (sx.sum fun s x => ... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/PolynomialMap/Homogeneous.lean | TensorProduct.exists_Finsupp | [108, 1] | [118, 88] | use sx + sy | case add.intro.intro
R : Type u_1
inst✝⁴ : CommSemiring R
M : Type u_2
inst✝³ : AddCommMonoid M
inst✝² : Module R M
S : Type u_3
inst✝¹ : CommSemiring S
inst✝ : Algebra R S
sx sy : S →₀ M
⊢ ∃ m, ((sx.sum fun s x => s ⊗ₜ[R] x) + sy.sum fun s x => s ⊗ₜ[R] x) = m.sum fun s x => s ⊗ₜ[R] x | case h
R : Type u_1
inst✝⁴ : CommSemiring R
M : Type u_2
inst✝³ : AddCommMonoid M
inst✝² : Module R M
S : Type u_3
inst✝¹ : CommSemiring S
inst✝ : Algebra R S
sx sy : S →₀ M
⊢ ((sx.sum fun s x => s ⊗ₜ[R] x) + sy.sum fun s x => s ⊗ₜ[R] x) = (sx + sy).sum fun s x => s ⊗ₜ[R] x | Please generate a tactic in lean4 to solve the state.
STATE:
case add.intro.intro
R : Type u_1
inst✝⁴ : CommSemiring R
M : Type u_2
inst✝³ : AddCommMonoid M
inst✝² : Module R M
S : Type u_3
inst✝¹ : CommSemiring S
inst✝ : Algebra R S
sx sy : S →₀ M
⊢ ∃ m, ((sx.sum fun s x => s ⊗ₜ[R] x) + sy.sum fun s x => s ⊗ₜ[R] x) = ... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/PolynomialMap/Homogeneous.lean | TensorProduct.exists_Finsupp | [108, 1] | [118, 88] | rw [sum_add_index (fun _ ↦ by simp) (fun _ _ _ _ ↦ by rw [TensorProduct.tmul_add])] | case h
R : Type u_1
inst✝⁴ : CommSemiring R
M : Type u_2
inst✝³ : AddCommMonoid M
inst✝² : Module R M
S : Type u_3
inst✝¹ : CommSemiring S
inst✝ : Algebra R S
sx sy : S →₀ M
⊢ ((sx.sum fun s x => s ⊗ₜ[R] x) + sy.sum fun s x => s ⊗ₜ[R] x) = (sx + sy).sum fun s x => s ⊗ₜ[R] x | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case h
R : Type u_1
inst✝⁴ : CommSemiring R
M : Type u_2
inst✝³ : AddCommMonoid M
inst✝² : Module R M
S : Type u_3
inst✝¹ : CommSemiring S
inst✝ : Algebra R S
sx sy : S →₀ M
⊢ ((sx.sum fun s x => s ⊗ₜ[R] x) + sy.sum fun s x => s ⊗ₜ[R] x) = (sx + sy).sum fun s... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/PolynomialMap/Homogeneous.lean | TensorProduct.exists_Finsupp | [108, 1] | [118, 88] | simp | R : Type u_1
inst✝⁴ : CommSemiring R
M : Type u_2
inst✝³ : AddCommMonoid M
inst✝² : Module R M
S : Type u_3
inst✝¹ : CommSemiring S
inst✝ : Algebra R S
sx sy : S →₀ M
x✝ : S
⊢ x✝ ∈ sx.support ∪ sy.support → x✝ ⊗ₜ[R] 0 = 0 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
R : Type u_1
inst✝⁴ : CommSemiring R
M : Type u_2
inst✝³ : AddCommMonoid M
inst✝² : Module R M
S : Type u_3
inst✝¹ : CommSemiring S
inst✝ : Algebra R S
sx sy : S →₀ M
x✝ : S
⊢ x✝ ∈ sx.support ∪ sy.support → x✝ ⊗ₜ[R] 0 = 0
TACTIC:
|
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/PolynomialMap/Homogeneous.lean | TensorProduct.exists_Finsupp | [108, 1] | [118, 88] | rw [TensorProduct.tmul_add] | R : Type u_1
inst✝⁴ : CommSemiring R
M : Type u_2
inst✝³ : AddCommMonoid M
inst✝² : Module R M
S : Type u_3
inst✝¹ : CommSemiring S
inst✝ : Algebra R S
sx sy : S →₀ M
x✝³ : S
x✝² : x✝³ ∈ sx.support ∪ sy.support
x✝¹ x✝ : M
⊢ x✝³ ⊗ₜ[R] (x✝¹ + x✝) = x✝³ ⊗ₜ[R] x✝¹ + x✝³ ⊗ₜ[R] x✝ | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
R : Type u_1
inst✝⁴ : CommSemiring R
M : Type u_2
inst✝³ : AddCommMonoid M
inst✝² : Module R M
S : Type u_3
inst✝¹ : CommSemiring S
inst✝ : Algebra R S
sx sy : S →₀ M
x✝³ : S
x✝² : x✝³ ∈ sx.support ∪ sy.support
x✝¹ x✝ : M
⊢ x✝³ ⊗ₜ[R] (x✝¹ + x✝) = x✝³ ⊗ₜ[R] x✝... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/PolynomialMap/Homogeneous.lean | TensorProduct.exists_Fin | [121, 1] | [128, 36] | obtain ⟨m, rfl⟩ := TensorProduct.exists_Finsupp S sm | R : Type u_1
inst✝⁴ : CommSemiring R
M : Type u_2
inst✝³ : AddCommMonoid M
inst✝² : Module R M
S : Type u_3
inst✝¹ : CommRing S
inst✝ : Algebra R S
sm : S ⊗[R] M
⊢ ∃ n s m, sm = ∑ i : Fin n, s i ⊗ₜ[R] m i | case intro
R : Type u_1
inst✝⁴ : CommSemiring R
M : Type u_2
inst✝³ : AddCommMonoid M
inst✝² : Module R M
S : Type u_3
inst✝¹ : CommRing S
inst✝ : Algebra R S
m : S →₀ M
⊢ ∃ n s m_1, (m.sum fun s x => s ⊗ₜ[R] x) = ∑ i : Fin n, s i ⊗ₜ[R] m_1 i | Please generate a tactic in lean4 to solve the state.
STATE:
R : Type u_1
inst✝⁴ : CommSemiring R
M : Type u_2
inst✝³ : AddCommMonoid M
inst✝² : Module R M
S : Type u_3
inst✝¹ : CommRing S
inst✝ : Algebra R S
sm : S ⊗[R] M
⊢ ∃ n s m, sm = ∑ i : Fin n, s i ⊗ₜ[R] m i
TACTIC:
|
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/PolynomialMap/Homogeneous.lean | TensorProduct.exists_Fin | [121, 1] | [128, 36] | let e : m.support ≃ Fin (m.support.card) := Finset.equivFin _ | case intro
R : Type u_1
inst✝⁴ : CommSemiring R
M : Type u_2
inst✝³ : AddCommMonoid M
inst✝² : Module R M
S : Type u_3
inst✝¹ : CommRing S
inst✝ : Algebra R S
m : S →₀ M
⊢ ∃ n s m_1, (m.sum fun s x => s ⊗ₜ[R] x) = ∑ i : Fin n, s i ⊗ₜ[R] m_1 i | case intro
R : Type u_1
inst✝⁴ : CommSemiring R
M : Type u_2
inst✝³ : AddCommMonoid M
inst✝² : Module R M
S : Type u_3
inst✝¹ : CommRing S
inst✝ : Algebra R S
m : S →₀ M
e : { x // x ∈ m.support } ≃ Fin m.support.card := m.support.equivFin
⊢ ∃ n s m_1, (m.sum fun s x => s ⊗ₜ[R] x) = ∑ i : Fin n, s i ⊗ₜ[R] m_1 i | Please generate a tactic in lean4 to solve the state.
STATE:
case intro
R : Type u_1
inst✝⁴ : CommSemiring R
M : Type u_2
inst✝³ : AddCommMonoid M
inst✝² : Module R M
S : Type u_3
inst✝¹ : CommRing S
inst✝ : Algebra R S
m : S →₀ M
⊢ ∃ n s m_1, (m.sum fun s x => s ⊗ₜ[R] x) = ∑ i : Fin n, s i ⊗ₜ[R] m_1 i
TACTIC:
|
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/PolynomialMap/Homogeneous.lean | TensorProduct.exists_Fin | [121, 1] | [128, 36] | use m.support.card, fun i ↦ e.symm i, fun i ↦ m (e.symm i) | case intro
R : Type u_1
inst✝⁴ : CommSemiring R
M : Type u_2
inst✝³ : AddCommMonoid M
inst✝² : Module R M
S : Type u_3
inst✝¹ : CommRing S
inst✝ : Algebra R S
m : S →₀ M
e : { x // x ∈ m.support } ≃ Fin m.support.card := m.support.equivFin
⊢ ∃ n s m_1, (m.sum fun s x => s ⊗ₜ[R] x) = ∑ i : Fin n, s i ⊗ₜ[R] m_1 i | case h
R : Type u_1
inst✝⁴ : CommSemiring R
M : Type u_2
inst✝³ : AddCommMonoid M
inst✝² : Module R M
S : Type u_3
inst✝¹ : CommRing S
inst✝ : Algebra R S
m : S →₀ M
e : { x // x ∈ m.support } ≃ Fin m.support.card := m.support.equivFin
⊢ (m.sum fun s x => s ⊗ₜ[R] x) = ∑ i : Fin m.support.card, ↑(e.symm i) ⊗ₜ[R] m ↑(e.s... | Please generate a tactic in lean4 to solve the state.
STATE:
case intro
R : Type u_1
inst✝⁴ : CommSemiring R
M : Type u_2
inst✝³ : AddCommMonoid M
inst✝² : Module R M
S : Type u_3
inst✝¹ : CommRing S
inst✝ : Algebra R S
m : S →₀ M
e : { x // x ∈ m.support } ≃ Fin m.support.card := m.support.equivFin
⊢ ∃ n s m_1, (m.sum... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/PolynomialMap/Homogeneous.lean | TensorProduct.exists_Fin | [121, 1] | [128, 36] | rw [sum, ← Finset.sum_attach] | case h
R : Type u_1
inst✝⁴ : CommSemiring R
M : Type u_2
inst✝³ : AddCommMonoid M
inst✝² : Module R M
S : Type u_3
inst✝¹ : CommRing S
inst✝ : Algebra R S
m : S →₀ M
e : { x // x ∈ m.support } ≃ Fin m.support.card := m.support.equivFin
⊢ (m.sum fun s x => s ⊗ₜ[R] x) = ∑ i : Fin m.support.card, ↑(e.symm i) ⊗ₜ[R] m ↑(e.s... | case h
R : Type u_1
inst✝⁴ : CommSemiring R
M : Type u_2
inst✝³ : AddCommMonoid M
inst✝² : Module R M
S : Type u_3
inst✝¹ : CommRing S
inst✝ : Algebra R S
m : S →₀ M
e : { x // x ∈ m.support } ≃ Fin m.support.card := m.support.equivFin
⊢ ∑ x ∈ m.support.attach, ↑x ⊗ₜ[R] m ↑x = ∑ i : Fin m.support.card, ↑(e.symm i) ⊗ₜ[R... | Please generate a tactic in lean4 to solve the state.
STATE:
case h
R : Type u_1
inst✝⁴ : CommSemiring R
M : Type u_2
inst✝³ : AddCommMonoid M
inst✝² : Module R M
S : Type u_3
inst✝¹ : CommRing S
inst✝ : Algebra R S
m : S →₀ M
e : { x // x ∈ m.support } ≃ Fin m.support.card := m.support.equivFin
⊢ (m.sum fun s x => s ⊗... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/PolynomialMap/Homogeneous.lean | TensorProduct.exists_Fin | [121, 1] | [128, 36] | apply Finset.sum_equiv e <;> simp | case h
R : Type u_1
inst✝⁴ : CommSemiring R
M : Type u_2
inst✝³ : AddCommMonoid M
inst✝² : Module R M
S : Type u_3
inst✝¹ : CommRing S
inst✝ : Algebra R S
m : S →₀ M
e : { x // x ∈ m.support } ≃ Fin m.support.card := m.support.equivFin
⊢ ∑ x ∈ m.support.attach, ↑x ⊗ₜ[R] m ↑x = ∑ i : Fin m.support.card, ↑(e.symm i) ⊗ₜ[R... | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case h
R : Type u_1
inst✝⁴ : CommSemiring R
M : Type u_2
inst✝³ : AddCommMonoid M
inst✝² : Module R M
S : Type u_3
inst✝¹ : CommRing S
inst✝ : Algebra R S
m : S →₀ M
e : { x // x ∈ m.support } ≃ Fin m.support.card := m.support.equivFin
⊢ ∑ x ∈ m.support.attac... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/PolynomialMap/Homogeneous.lean | TensorProduct.smul_rTensor | [130, 1] | [138, 38] | induction m using TensorProduct.induction_on with
| zero => simp
| tmul s' m =>
simp only [rTensor_tmul, AlgHom.toLinearMap_apply, smul_tmul', smul_eq_mul, _root_.map_mul]
| add m m' hm hm' => simp [hm, hm'] | R✝ : Type u_1
inst✝⁹ : CommSemiring R✝
M✝ : Type u_2
inst✝⁸ : AddCommMonoid M✝
inst✝⁷ : Module R✝ M✝
R : Type u_3
inst✝⁶ : CommSemiring R
M : Type u_4
inst✝⁵ : AddCommMonoid M
inst✝⁴ : Module R M
S : Type u_5
inst✝³ : Semiring S
inst✝² : Algebra R S
T : Type u_6
inst✝¹ : Semiring T
inst✝ : Algebra R T
φ : S →ₐ[R] T
s :... | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
R✝ : Type u_1
inst✝⁹ : CommSemiring R✝
M✝ : Type u_2
inst✝⁸ : AddCommMonoid M✝
inst✝⁷ : Module R✝ M✝
R : Type u_3
inst✝⁶ : CommSemiring R
M : Type u_4
inst✝⁵ : AddCommMonoid M
inst✝⁴ : Module R M
S : Type u_5
inst✝³ : Semiring S
inst✝² : Algebra R S
T : Type ... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/PolynomialMap/Homogeneous.lean | TensorProduct.smul_rTensor | [130, 1] | [138, 38] | simp | case zero
R✝ : Type u_1
inst✝⁹ : CommSemiring R✝
M✝ : Type u_2
inst✝⁸ : AddCommMonoid M✝
inst✝⁷ : Module R✝ M✝
R : Type u_3
inst✝⁶ : CommSemiring R
M : Type u_4
inst✝⁵ : AddCommMonoid M
inst✝⁴ : Module R M
S : Type u_5
inst✝³ : Semiring S
inst✝² : Algebra R S
T : Type u_6
inst✝¹ : Semiring T
inst✝ : Algebra R T
φ : S →... | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case zero
R✝ : Type u_1
inst✝⁹ : CommSemiring R✝
M✝ : Type u_2
inst✝⁸ : AddCommMonoid M✝
inst✝⁷ : Module R✝ M✝
R : Type u_3
inst✝⁶ : CommSemiring R
M : Type u_4
inst✝⁵ : AddCommMonoid M
inst✝⁴ : Module R M
S : Type u_5
inst✝³ : Semiring S
inst✝² : Algebra R S... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/PolynomialMap/Homogeneous.lean | TensorProduct.smul_rTensor | [130, 1] | [138, 38] | simp only [rTensor_tmul, AlgHom.toLinearMap_apply, smul_tmul', smul_eq_mul, _root_.map_mul] | case tmul
R✝ : Type u_1
inst✝⁹ : CommSemiring R✝
M✝ : Type u_2
inst✝⁸ : AddCommMonoid M✝
inst✝⁷ : Module R✝ M✝
R : Type u_3
inst✝⁶ : CommSemiring R
M : Type u_4
inst✝⁵ : AddCommMonoid M
inst✝⁴ : Module R M
S : Type u_5
inst✝³ : Semiring S
inst✝² : Algebra R S
T : Type u_6
inst✝¹ : Semiring T
inst✝ : Algebra R T
φ : S →... | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case tmul
R✝ : Type u_1
inst✝⁹ : CommSemiring R✝
M✝ : Type u_2
inst✝⁸ : AddCommMonoid M✝
inst✝⁷ : Module R✝ M✝
R : Type u_3
inst✝⁶ : CommSemiring R
M : Type u_4
inst✝⁵ : AddCommMonoid M
inst✝⁴ : Module R M
S : Type u_5
inst✝³ : Semiring S
inst✝² : Algebra R S... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/PolynomialMap/Homogeneous.lean | TensorProduct.smul_rTensor | [130, 1] | [138, 38] | simp [hm, hm'] | case add
R✝ : Type u_1
inst✝⁹ : CommSemiring R✝
M✝ : Type u_2
inst✝⁸ : AddCommMonoid M✝
inst✝⁷ : Module R✝ M✝
R : Type u_3
inst✝⁶ : CommSemiring R
M : Type u_4
inst✝⁵ : AddCommMonoid M
inst✝⁴ : Module R M
S : Type u_5
inst✝³ : Semiring S
inst✝² : Algebra R S
T : Type u_6
inst✝¹ : Semiring T
inst✝ : Algebra R T
φ : S →ₐ... | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case add
R✝ : Type u_1
inst✝⁹ : CommSemiring R✝
M✝ : Type u_2
inst✝⁸ : AddCommMonoid M✝
inst✝⁷ : Module R✝ M✝
R : Type u_3
inst✝⁶ : CommSemiring R
M : Type u_4
inst✝⁵ : AddCommMonoid M
inst✝⁴ : Module R M
S : Type u_5
inst✝³ : Semiring S
inst✝² : Algebra R S
... |
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