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/ExponentialModule/Basic.lean | PowerSeries.isExponential_mul | [269, 1] | [277, 87] | repeat
rw [← coe_substAlgHom (substDomain_of_constantCoeff_zero (by simp))] | A : Type u_1
inst✝⁴ : CommRing A
R : Type u_2
inst✝³ : CommRing R
inst✝² : Algebra A R
S : Type u_3
inst✝¹ : CommRing S
inst✝ : Algebra A S
f g : R⟦X⟧
hf : f.IsExponential
hg : g.IsExponential
⊢ subst (MvPowerSeries.X 0 + MvPowerSeries.X 1) (f * g) =
subst (MvPowerSeries.X 0) (f * g) * subst (MvPowerSeries.X 1) (f ... | A : Type u_1
inst✝⁴ : CommRing A
R : Type u_2
inst✝³ : CommRing R
inst✝² : Algebra A R
S : Type u_3
inst✝¹ : CommRing S
inst✝ : Algebra A S
f g : R⟦X⟧
hf : f.IsExponential
hg : g.IsExponential
⊢ (substAlgHom ⋯) (f * g) = (substAlgHom ⋯) (f * g) * (substAlgHom ⋯) (f * g) | Please generate a tactic in lean4 to solve the state.
STATE:
A : Type u_1
inst✝⁴ : CommRing A
R : Type u_2
inst✝³ : CommRing R
inst✝² : Algebra A R
S : Type u_3
inst✝¹ : CommRing S
inst✝ : Algebra A S
f g : R⟦X⟧
hf : f.IsExponential
hg : g.IsExponential
⊢ subst (MvPowerSeries.X 0 + MvPowerSeries.X 1) (f * g) =
subs... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/ExponentialModule/Basic.lean | PowerSeries.isExponential_mul | [269, 1] | [277, 87] | simp only [map_mul, coe_substAlgHom, hf.add_mul, hg.add_mul] | A : Type u_1
inst✝⁴ : CommRing A
R : Type u_2
inst✝³ : CommRing R
inst✝² : Algebra A R
S : Type u_3
inst✝¹ : CommRing S
inst✝ : Algebra A S
f g : R⟦X⟧
hf : f.IsExponential
hg : g.IsExponential
⊢ (substAlgHom ⋯) (f * g) = (substAlgHom ⋯) (f * g) * (substAlgHom ⋯) (f * g) | A : Type u_1
inst✝⁴ : CommRing A
R : Type u_2
inst✝³ : CommRing R
inst✝² : Algebra A R
S : Type u_3
inst✝¹ : CommRing S
inst✝ : Algebra A S
f g : R⟦X⟧
hf : f.IsExponential
hg : g.IsExponential
⊢ subst (MvPowerSeries.X 0) f * subst (MvPowerSeries.X 1) f *
(subst (MvPowerSeries.X 0) g * subst (MvPowerSeries.X 1) g)... | Please generate a tactic in lean4 to solve the state.
STATE:
A : Type u_1
inst✝⁴ : CommRing A
R : Type u_2
inst✝³ : CommRing R
inst✝² : Algebra A R
S : Type u_3
inst✝¹ : CommRing S
inst✝ : Algebra A S
f g : R⟦X⟧
hf : f.IsExponential
hg : g.IsExponential
⊢ (substAlgHom ⋯) (f * g) = (substAlgHom ⋯) (f * g) * (substAlgHom... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/ExponentialModule/Basic.lean | PowerSeries.isExponential_mul | [269, 1] | [277, 87] | ring | A : Type u_1
inst✝⁴ : CommRing A
R : Type u_2
inst✝³ : CommRing R
inst✝² : Algebra A R
S : Type u_3
inst✝¹ : CommRing S
inst✝ : Algebra A S
f g : R⟦X⟧
hf : f.IsExponential
hg : g.IsExponential
⊢ subst (MvPowerSeries.X 0) f * subst (MvPowerSeries.X 1) f *
(subst (MvPowerSeries.X 0) g * subst (MvPowerSeries.X 1) g)... | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
A : Type u_1
inst✝⁴ : CommRing A
R : Type u_2
inst✝³ : CommRing R
inst✝² : Algebra A R
S : Type u_3
inst✝¹ : CommRing S
inst✝ : Algebra A S
f g : R⟦X⟧
hf : f.IsExponential
hg : g.IsExponential
⊢ subst (MvPowerSeries.X 0) f * subst (MvPowerSeries.X 1) f *
... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/ExponentialModule/Basic.lean | PowerSeries.isExponential_mul | [269, 1] | [277, 87] | rw [← coe_substAlgHom (substDomain_of_constantCoeff_zero (by simp))] | A : Type u_1
inst✝⁴ : CommRing A
R : Type u_2
inst✝³ : CommRing R
inst✝² : Algebra A R
S : Type u_3
inst✝¹ : CommRing S
inst✝ : Algebra A S
f g : R⟦X⟧
hf : f.IsExponential
hg : g.IsExponential
⊢ (substAlgHom ⋯) (f * g) = (substAlgHom ⋯) (f * g) * subst (MvPowerSeries.X 1) (f * g) | A : Type u_1
inst✝⁴ : CommRing A
R : Type u_2
inst✝³ : CommRing R
inst✝² : Algebra A R
S : Type u_3
inst✝¹ : CommRing S
inst✝ : Algebra A S
f g : R⟦X⟧
hf : f.IsExponential
hg : g.IsExponential
⊢ (substAlgHom ⋯) (f * g) = (substAlgHom ⋯) (f * g) * (substAlgHom ⋯) (f * g) | Please generate a tactic in lean4 to solve the state.
STATE:
A : Type u_1
inst✝⁴ : CommRing A
R : Type u_2
inst✝³ : CommRing R
inst✝² : Algebra A R
S : Type u_3
inst✝¹ : CommRing S
inst✝ : Algebra A S
f g : R⟦X⟧
hf : f.IsExponential
hg : g.IsExponential
⊢ (substAlgHom ⋯) (f * g) = (substAlgHom ⋯) (f * g) * subst (MvPow... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/ExponentialModule/Basic.lean | PowerSeries.isExponential_mul | [269, 1] | [277, 87] | simp | A : Type u_1
inst✝⁴ : CommRing A
R : Type u_2
inst✝³ : CommRing R
inst✝² : Algebra A R
S : Type u_3
inst✝¹ : CommRing S
inst✝ : Algebra A S
f g : R⟦X⟧
hf : f.IsExponential
hg : g.IsExponential
⊢ (MvPowerSeries.constantCoeff (Fin 2) R) (MvPowerSeries.X 1) = 0 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
A : Type u_1
inst✝⁴ : CommRing A
R : Type u_2
inst✝³ : CommRing R
inst✝² : Algebra A R
S : Type u_3
inst✝¹ : CommRing S
inst✝ : Algebra A S
f g : R⟦X⟧
hf : f.IsExponential
hg : g.IsExponential
⊢ (MvPowerSeries.constantCoeff (Fin 2) R) (MvPowerSeries.X 1) = 0
... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/ExponentialModule/Basic.lean | PowerSeries.isExponential_mul | [269, 1] | [277, 87] | simp only [map_mul, hf.constantCoeff, hg.constantCoeff, mul_one] | A : Type u_1
inst✝⁴ : CommRing A
R : Type u_2
inst✝³ : CommRing R
inst✝² : Algebra A R
S : Type u_3
inst✝¹ : CommRing S
inst✝ : Algebra A S
f g : R⟦X⟧
hf : f.IsExponential
hg : g.IsExponential
⊢ (constantCoeff R) (f * g) = 1 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
A : Type u_1
inst✝⁴ : CommRing A
R : Type u_2
inst✝³ : CommRing R
inst✝² : Algebra A R
S : Type u_3
inst✝¹ : CommRing S
inst✝ : Algebra A S
f g : R⟦X⟧
hf : f.IsExponential
hg : g.IsExponential
⊢ (constantCoeff R) (f * g) = 1
TACTIC:
|
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/ExponentialModule/Basic.lean | PowerSeries.isExponential_npow | [280, 1] | [288, 34] | induction n with
| zero =>
simp only [Nat.zero_eq, pow_zero]
exact isExponential_one
| succ n hn =>
rw [pow_succ]
exact isExponential_mul hn hf | A : Type u_1
inst✝⁴ : CommRing A
R : Type u_2
inst✝³ : CommRing R
inst✝² : Algebra A R
S : Type u_3
inst✝¹ : CommRing S
inst✝ : Algebra A S
f : R⟦X⟧
hf : f.IsExponential
n : ℕ
⊢ (f ^ n).IsExponential | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
A : Type u_1
inst✝⁴ : CommRing A
R : Type u_2
inst✝³ : CommRing R
inst✝² : Algebra A R
S : Type u_3
inst✝¹ : CommRing S
inst✝ : Algebra A S
f : R⟦X⟧
hf : f.IsExponential
n : ℕ
⊢ (f ^ n).IsExponential
TACTIC:
|
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/ExponentialModule/Basic.lean | PowerSeries.isExponential_npow | [280, 1] | [288, 34] | simp only [Nat.zero_eq, pow_zero] | case zero
A : Type u_1
inst✝⁴ : CommRing A
R : Type u_2
inst✝³ : CommRing R
inst✝² : Algebra A R
S : Type u_3
inst✝¹ : CommRing S
inst✝ : Algebra A S
f : R⟦X⟧
hf : f.IsExponential
⊢ (f ^ 0).IsExponential | case zero
A : Type u_1
inst✝⁴ : CommRing A
R : Type u_2
inst✝³ : CommRing R
inst✝² : Algebra A R
S : Type u_3
inst✝¹ : CommRing S
inst✝ : Algebra A S
f : R⟦X⟧
hf : f.IsExponential
⊢ IsExponential 1 | Please generate a tactic in lean4 to solve the state.
STATE:
case zero
A : Type u_1
inst✝⁴ : CommRing A
R : Type u_2
inst✝³ : CommRing R
inst✝² : Algebra A R
S : Type u_3
inst✝¹ : CommRing S
inst✝ : Algebra A S
f : R⟦X⟧
hf : f.IsExponential
⊢ (f ^ 0).IsExponential
TACTIC:
|
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/ExponentialModule/Basic.lean | PowerSeries.isExponential_npow | [280, 1] | [288, 34] | exact isExponential_one | case zero
A : Type u_1
inst✝⁴ : CommRing A
R : Type u_2
inst✝³ : CommRing R
inst✝² : Algebra A R
S : Type u_3
inst✝¹ : CommRing S
inst✝ : Algebra A S
f : R⟦X⟧
hf : f.IsExponential
⊢ IsExponential 1 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case zero
A : Type u_1
inst✝⁴ : CommRing A
R : Type u_2
inst✝³ : CommRing R
inst✝² : Algebra A R
S : Type u_3
inst✝¹ : CommRing S
inst✝ : Algebra A S
f : R⟦X⟧
hf : f.IsExponential
⊢ IsExponential 1
TACTIC:
|
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/ExponentialModule/Basic.lean | PowerSeries.isExponential_npow | [280, 1] | [288, 34] | rw [pow_succ] | case succ
A : Type u_1
inst✝⁴ : CommRing A
R : Type u_2
inst✝³ : CommRing R
inst✝² : Algebra A R
S : Type u_3
inst✝¹ : CommRing S
inst✝ : Algebra A S
f : R⟦X⟧
hf : f.IsExponential
n : ℕ
hn : (f ^ n).IsExponential
⊢ (f ^ (n + 1)).IsExponential | case succ
A : Type u_1
inst✝⁴ : CommRing A
R : Type u_2
inst✝³ : CommRing R
inst✝² : Algebra A R
S : Type u_3
inst✝¹ : CommRing S
inst✝ : Algebra A S
f : R⟦X⟧
hf : f.IsExponential
n : ℕ
hn : (f ^ n).IsExponential
⊢ (f ^ n * f).IsExponential | Please generate a tactic in lean4 to solve the state.
STATE:
case succ
A : Type u_1
inst✝⁴ : CommRing A
R : Type u_2
inst✝³ : CommRing R
inst✝² : Algebra A R
S : Type u_3
inst✝¹ : CommRing S
inst✝ : Algebra A S
f : R⟦X⟧
hf : f.IsExponential
n : ℕ
hn : (f ^ n).IsExponential
⊢ (f ^ (n + 1)).IsExponential
TACTIC:
|
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/ExponentialModule/Basic.lean | PowerSeries.isExponential_npow | [280, 1] | [288, 34] | exact isExponential_mul hn hf | case succ
A : Type u_1
inst✝⁴ : CommRing A
R : Type u_2
inst✝³ : CommRing R
inst✝² : Algebra A R
S : Type u_3
inst✝¹ : CommRing S
inst✝ : Algebra A S
f : R⟦X⟧
hf : f.IsExponential
n : ℕ
hn : (f ^ n).IsExponential
⊢ (f ^ n * f).IsExponential | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case succ
A : Type u_1
inst✝⁴ : CommRing A
R : Type u_2
inst✝³ : CommRing R
inst✝² : Algebra A R
S : Type u_3
inst✝¹ : CommRing S
inst✝ : Algebra A S
f : R⟦X⟧
hf : f.IsExponential
n : ℕ
hn : (f ^ n).IsExponential
⊢ (f ^ n * f).IsExponential
TACTIC:
|
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/ExponentialModule/Basic.lean | PowerSeries.isExponential_scale | [291, 1] | [321, 29] | rw [← coeff_zero_eq_constantCoeff, coeff_scale] | A : Type u_1
inst✝⁴ : CommRing A
R : Type u_2
inst✝³ : CommRing R
inst✝² : Algebra A R
S : Type u_3
inst✝¹ : CommRing S
inst✝ : Algebra A S
a : A
f : R⟦X⟧
hf : f.IsExponential
⊢ (constantCoeff R) (scale a f) = 1 | A : Type u_1
inst✝⁴ : CommRing A
R : Type u_2
inst✝³ : CommRing R
inst✝² : Algebra A R
S : Type u_3
inst✝¹ : CommRing S
inst✝ : Algebra A S
a : A
f : R⟦X⟧
hf : f.IsExponential
⊢ a ^ 0 • (coeff R 0) f = 1 | Please generate a tactic in lean4 to solve the state.
STATE:
A : Type u_1
inst✝⁴ : CommRing A
R : Type u_2
inst✝³ : CommRing R
inst✝² : Algebra A R
S : Type u_3
inst✝¹ : CommRing S
inst✝ : Algebra A S
a : A
f : R⟦X⟧
hf : f.IsExponential
⊢ (constantCoeff R) (scale a f) = 1
TACTIC:
|
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/ExponentialModule/Basic.lean | PowerSeries.isExponential_scale | [291, 1] | [321, 29] | simp only [pow_zero, coeff_zero_eq_constantCoeff, one_smul,
hf.constantCoeff] | A : Type u_1
inst✝⁴ : CommRing A
R : Type u_2
inst✝³ : CommRing R
inst✝² : Algebra A R
S : Type u_3
inst✝¹ : CommRing S
inst✝ : Algebra A S
a : A
f : R⟦X⟧
hf : f.IsExponential
⊢ a ^ 0 • (coeff R 0) f = 1 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
A : Type u_1
inst✝⁴ : CommRing A
R : Type u_2
inst✝³ : CommRing R
inst✝² : Algebra A R
S : Type u_3
inst✝¹ : CommRing S
inst✝ : Algebra A S
a : A
f : R⟦X⟧
hf : f.IsExponential
⊢ a ^ 0 • (coeff R 0) f = 1
TACTIC:
|
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/ExponentialModule/Basic.lean | PowerSeries.isExponential_scale | [291, 1] | [321, 29] | rw [subst_linear_subst_scalar_comm] | A : Type u_1
inst✝⁴ : CommRing A
R : Type u_2
inst✝³ : CommRing R
inst✝² : Algebra A R
S : Type u_3
inst✝¹ : CommRing S
inst✝ : Algebra A S
a : A
f : R⟦X⟧
hf : f.IsExponential
⊢ subst (MvPowerSeries.X 0 + MvPowerSeries.X 1) (scale a f) =
subst (MvPowerSeries.X 0) (scale a f) * subst (MvPowerSeries.X 1) (scale a f) | A : Type u_1
inst✝⁴ : CommRing A
R : Type u_2
inst✝³ : CommRing R
inst✝² : Algebra A R
S : Type u_3
inst✝¹ : CommRing S
inst✝ : Algebra A S
a : A
f : R⟦X⟧
hf : f.IsExponential
⊢ MvPowerSeries.scale (Function.const (Fin 2) a) (subst (MvPowerSeries.X 0 + MvPowerSeries.X 1) f) =
subst (MvPowerSeries.X 0) (scale a f) *... | Please generate a tactic in lean4 to solve the state.
STATE:
A : Type u_1
inst✝⁴ : CommRing A
R : Type u_2
inst✝³ : CommRing R
inst✝² : Algebra A R
S : Type u_3
inst✝¹ : CommRing S
inst✝ : Algebra A S
a : A
f : R⟦X⟧
hf : f.IsExponential
⊢ subst (MvPowerSeries.X 0 + MvPowerSeries.X 1) (scale a f) =
subst (MvPowerSer... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/ExponentialModule/Basic.lean | PowerSeries.isExponential_scale | [291, 1] | [321, 29] | rw [subst_linear_subst_scalar_comm] | A : Type u_1
inst✝⁴ : CommRing A
R : Type u_2
inst✝³ : CommRing R
inst✝² : Algebra A R
S : Type u_3
inst✝¹ : CommRing S
inst✝ : Algebra A S
a : A
f : R⟦X⟧
hf : f.IsExponential
⊢ MvPowerSeries.scale (Function.const (Fin 2) a) (subst (MvPowerSeries.X 0 + MvPowerSeries.X 1) f) =
subst (MvPowerSeries.X 0) (scale a f) *... | A : Type u_1
inst✝⁴ : CommRing A
R : Type u_2
inst✝³ : CommRing R
inst✝² : Algebra A R
S : Type u_3
inst✝¹ : CommRing S
inst✝ : Algebra A S
a : A
f : R⟦X⟧
hf : f.IsExponential
⊢ MvPowerSeries.scale (Function.const (Fin 2) a) (subst (MvPowerSeries.X 0 + MvPowerSeries.X 1) f) =
MvPowerSeries.scale (Function.const (Fi... | Please generate a tactic in lean4 to solve the state.
STATE:
A : Type u_1
inst✝⁴ : CommRing A
R : Type u_2
inst✝³ : CommRing R
inst✝² : Algebra A R
S : Type u_3
inst✝¹ : CommRing S
inst✝ : Algebra A S
a : A
f : R⟦X⟧
hf : f.IsExponential
⊢ MvPowerSeries.scale (Function.const (Fin 2) a) (subst (MvPowerSeries.X 0 + MvPowe... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/ExponentialModule/Basic.lean | PowerSeries.isExponential_scale | [291, 1] | [321, 29] | rw [subst_linear_subst_scalar_comm] | A : Type u_1
inst✝⁴ : CommRing A
R : Type u_2
inst✝³ : CommRing R
inst✝² : Algebra A R
S : Type u_3
inst✝¹ : CommRing S
inst✝ : Algebra A S
a : A
f : R⟦X⟧
hf : f.IsExponential
⊢ MvPowerSeries.scale (Function.const (Fin 2) a) (subst (MvPowerSeries.X 0 + MvPowerSeries.X 1) f) =
MvPowerSeries.scale (Function.const (Fi... | A : Type u_1
inst✝⁴ : CommRing A
R : Type u_2
inst✝³ : CommRing R
inst✝² : Algebra A R
S : Type u_3
inst✝¹ : CommRing S
inst✝ : Algebra A S
a : A
f : R⟦X⟧
hf : f.IsExponential
⊢ MvPowerSeries.scale (Function.const (Fin 2) a) (subst (MvPowerSeries.X 0 + MvPowerSeries.X 1) f) =
MvPowerSeries.scale (Function.const (Fi... | Please generate a tactic in lean4 to solve the state.
STATE:
A : Type u_1
inst✝⁴ : CommRing A
R : Type u_2
inst✝³ : CommRing R
inst✝² : Algebra A R
S : Type u_3
inst✝¹ : CommRing S
inst✝ : Algebra A S
a : A
f : R⟦X⟧
hf : f.IsExponential
⊢ MvPowerSeries.scale (Function.const (Fin 2) a) (subst (MvPowerSeries.X 0 + MvPowe... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/ExponentialModule/Basic.lean | PowerSeries.isExponential_scale | [291, 1] | [321, 29] | simp only [← MvPowerSeries.coe_scale_algHom, ← map_mul, hf.add_mul] | A : Type u_1
inst✝⁴ : CommRing A
R : Type u_2
inst✝³ : CommRing R
inst✝² : Algebra A R
S : Type u_3
inst✝¹ : CommRing S
inst✝ : Algebra A S
a : A
f : R⟦X⟧
hf : f.IsExponential
⊢ MvPowerSeries.scale (Function.const (Fin 2) a) (subst (MvPowerSeries.X 0 + MvPowerSeries.X 1) f) =
MvPowerSeries.scale (Function.const (Fi... | case hp_lin
A : Type u_1
inst✝⁴ : CommRing A
R : Type u_2
inst✝³ : CommRing R
inst✝² : Algebra A R
S : Type u_3
inst✝¹ : CommRing S
inst✝ : Algebra A S
a : A
f : R⟦X⟧
hf : f.IsExponential
⊢ ∀ (d : Fin 2 →₀ ℕ), (d.sum fun x n => n) ≠ 1 → (MvPowerSeries.coeff R d) (MvPowerSeries.X 1) = 0
case hp_lin
A : Type u_1
inst✝⁴ ... | Please generate a tactic in lean4 to solve the state.
STATE:
A : Type u_1
inst✝⁴ : CommRing A
R : Type u_2
inst✝³ : CommRing R
inst✝² : Algebra A R
S : Type u_3
inst✝¹ : CommRing S
inst✝ : Algebra A S
a : A
f : R⟦X⟧
hf : f.IsExponential
⊢ MvPowerSeries.scale (Function.const (Fin 2) a) (subst (MvPowerSeries.X 0 + MvPowe... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/ExponentialModule/Basic.lean | PowerSeries.isExponential_scale | [291, 1] | [321, 29] | repeat
intro d hd
simp only [Fin.isValue, map_add, MvPowerSeries.coeff_X]
rw [if_neg]
intro hd'
apply hd
rw [hd']
simp only [Fin.isValue, Finsupp.sum_single_index] | case hp_lin
A : Type u_1
inst✝⁴ : CommRing A
R : Type u_2
inst✝³ : CommRing R
inst✝² : Algebra A R
S : Type u_3
inst✝¹ : CommRing S
inst✝ : Algebra A S
a : A
f : R⟦X⟧
hf : f.IsExponential
⊢ ∀ (d : Fin 2 →₀ ℕ), (d.sum fun x n => n) ≠ 1 → (MvPowerSeries.coeff R d) (MvPowerSeries.X 1) = 0
case hp_lin
A : Type u_1
inst✝⁴ ... | case hp_lin
A : Type u_1
inst✝⁴ : CommRing A
R : Type u_2
inst✝³ : CommRing R
inst✝² : Algebra A R
S : Type u_3
inst✝¹ : CommRing S
inst✝ : Algebra A S
a : A
f : R⟦X⟧
hf : f.IsExponential
⊢ ∀ (d : Fin 2 →₀ ℕ), (d.sum fun x n => n) ≠ 1 → (MvPowerSeries.coeff R d) (MvPowerSeries.X 0 + MvPowerSeries.X 1) = 0 | Please generate a tactic in lean4 to solve the state.
STATE:
case hp_lin
A : Type u_1
inst✝⁴ : CommRing A
R : Type u_2
inst✝³ : CommRing R
inst✝² : Algebra A R
S : Type u_3
inst✝¹ : CommRing S
inst✝ : Algebra A S
a : A
f : R⟦X⟧
hf : f.IsExponential
⊢ ∀ (d : Fin 2 →₀ ℕ), (d.sum fun x n => n) ≠ 1 → (MvPowerSeries.coeff R... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/ExponentialModule/Basic.lean | PowerSeries.isExponential_scale | [291, 1] | [321, 29] | intro d hd | case hp_lin
A : Type u_1
inst✝⁴ : CommRing A
R : Type u_2
inst✝³ : CommRing R
inst✝² : Algebra A R
S : Type u_3
inst✝¹ : CommRing S
inst✝ : Algebra A S
a : A
f : R⟦X⟧
hf : f.IsExponential
⊢ ∀ (d : Fin 2 →₀ ℕ), (d.sum fun x n => n) ≠ 1 → (MvPowerSeries.coeff R d) (MvPowerSeries.X 0 + MvPowerSeries.X 1) = 0 | case hp_lin
A : Type u_1
inst✝⁴ : CommRing A
R : Type u_2
inst✝³ : CommRing R
inst✝² : Algebra A R
S : Type u_3
inst✝¹ : CommRing S
inst✝ : Algebra A S
a : A
f : R⟦X⟧
hf : f.IsExponential
d : Fin 2 →₀ ℕ
hd : (d.sum fun x n => n) ≠ 1
⊢ (MvPowerSeries.coeff R d) (MvPowerSeries.X 0 + MvPowerSeries.X 1) = 0 | Please generate a tactic in lean4 to solve the state.
STATE:
case hp_lin
A : Type u_1
inst✝⁴ : CommRing A
R : Type u_2
inst✝³ : CommRing R
inst✝² : Algebra A R
S : Type u_3
inst✝¹ : CommRing S
inst✝ : Algebra A S
a : A
f : R⟦X⟧
hf : f.IsExponential
⊢ ∀ (d : Fin 2 →₀ ℕ), (d.sum fun x n => n) ≠ 1 → (MvPowerSeries.coeff R... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/ExponentialModule/Basic.lean | PowerSeries.isExponential_scale | [291, 1] | [321, 29] | simp only [Fin.isValue, map_add, MvPowerSeries.coeff_X] | case hp_lin
A : Type u_1
inst✝⁴ : CommRing A
R : Type u_2
inst✝³ : CommRing R
inst✝² : Algebra A R
S : Type u_3
inst✝¹ : CommRing S
inst✝ : Algebra A S
a : A
f : R⟦X⟧
hf : f.IsExponential
d : Fin 2 →₀ ℕ
hd : (d.sum fun x n => n) ≠ 1
⊢ (MvPowerSeries.coeff R d) (MvPowerSeries.X 0 + MvPowerSeries.X 1) = 0 | case hp_lin
A : Type u_1
inst✝⁴ : CommRing A
R : Type u_2
inst✝³ : CommRing R
inst✝² : Algebra A R
S : Type u_3
inst✝¹ : CommRing S
inst✝ : Algebra A S
a : A
f : R⟦X⟧
hf : f.IsExponential
d : Fin 2 →₀ ℕ
hd : (d.sum fun x n => n) ≠ 1
⊢ ((if d = Finsupp.single 0 1 then 1 else 0) + if d = Finsupp.single 1 1 then 1 else 0)... | Please generate a tactic in lean4 to solve the state.
STATE:
case hp_lin
A : Type u_1
inst✝⁴ : CommRing A
R : Type u_2
inst✝³ : CommRing R
inst✝² : Algebra A R
S : Type u_3
inst✝¹ : CommRing S
inst✝ : Algebra A S
a : A
f : R⟦X⟧
hf : f.IsExponential
d : Fin 2 →₀ ℕ
hd : (d.sum fun x n => n) ≠ 1
⊢ (MvPowerSeries.coeff R d... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/ExponentialModule/Basic.lean | PowerSeries.isExponential_scale | [291, 1] | [321, 29] | rw [if_neg] | case hp_lin
A : Type u_1
inst✝⁴ : CommRing A
R : Type u_2
inst✝³ : CommRing R
inst✝² : Algebra A R
S : Type u_3
inst✝¹ : CommRing S
inst✝ : Algebra A S
a : A
f : R⟦X⟧
hf : f.IsExponential
d : Fin 2 →₀ ℕ
hd : (d.sum fun x n => n) ≠ 1
⊢ ((if d = Finsupp.single 0 1 then 1 else 0) + if d = Finsupp.single 1 1 then 1 else 0)... | case hp_lin
A : Type u_1
inst✝⁴ : CommRing A
R : Type u_2
inst✝³ : CommRing R
inst✝² : Algebra A R
S : Type u_3
inst✝¹ : CommRing S
inst✝ : Algebra A S
a : A
f : R⟦X⟧
hf : f.IsExponential
d : Fin 2 →₀ ℕ
hd : (d.sum fun x n => n) ≠ 1
⊢ (0 + if d = Finsupp.single 1 1 then 1 else 0) = 0
case hp_lin.hnc
A : Type u_1
inst✝... | Please generate a tactic in lean4 to solve the state.
STATE:
case hp_lin
A : Type u_1
inst✝⁴ : CommRing A
R : Type u_2
inst✝³ : CommRing R
inst✝² : Algebra A R
S : Type u_3
inst✝¹ : CommRing S
inst✝ : Algebra A S
a : A
f : R⟦X⟧
hf : f.IsExponential
d : Fin 2 →₀ ℕ
hd : (d.sum fun x n => n) ≠ 1
⊢ ((if d = Finsupp.single ... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/ExponentialModule/Basic.lean | PowerSeries.isExponential_scale | [291, 1] | [321, 29] | intro hd' | case hp_lin.hnc
A : Type u_1
inst✝⁴ : CommRing A
R : Type u_2
inst✝³ : CommRing R
inst✝² : Algebra A R
S : Type u_3
inst✝¹ : CommRing S
inst✝ : Algebra A S
a : A
f : R⟦X⟧
hf : f.IsExponential
d : Fin 2 →₀ ℕ
hd : (d.sum fun x n => n) ≠ 1
⊢ ¬d = Finsupp.single 0 1
case hp_lin
A : Type u_1
inst✝⁴ : CommRing A
R : Type u_... | case hp_lin.hnc
A : Type u_1
inst✝⁴ : CommRing A
R : Type u_2
inst✝³ : CommRing R
inst✝² : Algebra A R
S : Type u_3
inst✝¹ : CommRing S
inst✝ : Algebra A S
a : A
f : R⟦X⟧
hf : f.IsExponential
d : Fin 2 →₀ ℕ
hd : (d.sum fun x n => n) ≠ 1
hd' : d = Finsupp.single 0 1
⊢ False
case hp_lin
A : Type u_1
inst✝⁴ : CommRing A
... | Please generate a tactic in lean4 to solve the state.
STATE:
case hp_lin.hnc
A : Type u_1
inst✝⁴ : CommRing A
R : Type u_2
inst✝³ : CommRing R
inst✝² : Algebra A R
S : Type u_3
inst✝¹ : CommRing S
inst✝ : Algebra A S
a : A
f : R⟦X⟧
hf : f.IsExponential
d : Fin 2 →₀ ℕ
hd : (d.sum fun x n => n) ≠ 1
⊢ ¬d = Finsupp.single ... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/ExponentialModule/Basic.lean | PowerSeries.isExponential_scale | [291, 1] | [321, 29] | apply hd | case hp_lin.hnc
A : Type u_1
inst✝⁴ : CommRing A
R : Type u_2
inst✝³ : CommRing R
inst✝² : Algebra A R
S : Type u_3
inst✝¹ : CommRing S
inst✝ : Algebra A S
a : A
f : R⟦X⟧
hf : f.IsExponential
d : Fin 2 →₀ ℕ
hd : (d.sum fun x n => n) ≠ 1
hd' : d = Finsupp.single 0 1
⊢ False
case hp_lin
A : Type u_1
inst✝⁴ : CommRing A
... | case hp_lin.hnc
A : Type u_1
inst✝⁴ : CommRing A
R : Type u_2
inst✝³ : CommRing R
inst✝² : Algebra A R
S : Type u_3
inst✝¹ : CommRing S
inst✝ : Algebra A S
a : A
f : R⟦X⟧
hf : f.IsExponential
d : Fin 2 →₀ ℕ
hd : (d.sum fun x n => n) ≠ 1
hd' : d = Finsupp.single 0 1
⊢ (d.sum fun x n => n) = 1
case hp_lin
A : Type u_1
i... | Please generate a tactic in lean4 to solve the state.
STATE:
case hp_lin.hnc
A : Type u_1
inst✝⁴ : CommRing A
R : Type u_2
inst✝³ : CommRing R
inst✝² : Algebra A R
S : Type u_3
inst✝¹ : CommRing S
inst✝ : Algebra A S
a : A
f : R⟦X⟧
hf : f.IsExponential
d : Fin 2 →₀ ℕ
hd : (d.sum fun x n => n) ≠ 1
hd' : d = Finsupp.sing... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/ExponentialModule/Basic.lean | PowerSeries.isExponential_scale | [291, 1] | [321, 29] | rw [hd'] | case hp_lin.hnc
A : Type u_1
inst✝⁴ : CommRing A
R : Type u_2
inst✝³ : CommRing R
inst✝² : Algebra A R
S : Type u_3
inst✝¹ : CommRing S
inst✝ : Algebra A S
a : A
f : R⟦X⟧
hf : f.IsExponential
d : Fin 2 →₀ ℕ
hd : (d.sum fun x n => n) ≠ 1
hd' : d = Finsupp.single 0 1
⊢ (d.sum fun x n => n) = 1
case hp_lin
A : Type u_1
i... | case hp_lin.hnc
A : Type u_1
inst✝⁴ : CommRing A
R : Type u_2
inst✝³ : CommRing R
inst✝² : Algebra A R
S : Type u_3
inst✝¹ : CommRing S
inst✝ : Algebra A S
a : A
f : R⟦X⟧
hf : f.IsExponential
d : Fin 2 →₀ ℕ
hd : (d.sum fun x n => n) ≠ 1
hd' : d = Finsupp.single 0 1
⊢ ((Finsupp.single 0 1).sum fun x n => n) = 1
case hp... | Please generate a tactic in lean4 to solve the state.
STATE:
case hp_lin.hnc
A : Type u_1
inst✝⁴ : CommRing A
R : Type u_2
inst✝³ : CommRing R
inst✝² : Algebra A R
S : Type u_3
inst✝¹ : CommRing S
inst✝ : Algebra A S
a : A
f : R⟦X⟧
hf : f.IsExponential
d : Fin 2 →₀ ℕ
hd : (d.sum fun x n => n) ≠ 1
hd' : d = Finsupp.sing... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/ExponentialModule/Basic.lean | PowerSeries.isExponential_scale | [291, 1] | [321, 29] | simp only [Fin.isValue, Finsupp.sum_single_index] | case hp_lin.hnc
A : Type u_1
inst✝⁴ : CommRing A
R : Type u_2
inst✝³ : CommRing R
inst✝² : Algebra A R
S : Type u_3
inst✝¹ : CommRing S
inst✝ : Algebra A S
a : A
f : R⟦X⟧
hf : f.IsExponential
d : Fin 2 →₀ ℕ
hd : (d.sum fun x n => n) ≠ 1
hd' : d = Finsupp.single 0 1
⊢ ((Finsupp.single 0 1).sum fun x n => n) = 1
case hp... | case hp_lin
A : Type u_1
inst✝⁴ : CommRing A
R : Type u_2
inst✝³ : CommRing R
inst✝² : Algebra A R
S : Type u_3
inst✝¹ : CommRing S
inst✝ : Algebra A S
a : A
f : R⟦X⟧
hf : f.IsExponential
⊢ ∀ (d : Fin 2 →₀ ℕ), (d.sum fun x n => n) ≠ 1 → (MvPowerSeries.coeff R d) (MvPowerSeries.X 0 + MvPowerSeries.X 1) = 0 | Please generate a tactic in lean4 to solve the state.
STATE:
case hp_lin.hnc
A : Type u_1
inst✝⁴ : CommRing A
R : Type u_2
inst✝³ : CommRing R
inst✝² : Algebra A R
S : Type u_3
inst✝¹ : CommRing S
inst✝ : Algebra A S
a : A
f : R⟦X⟧
hf : f.IsExponential
d : Fin 2 →₀ ℕ
hd : (d.sum fun x n => n) ≠ 1
hd' : d = Finsupp.sing... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/ExponentialModule/Basic.lean | PowerSeries.isExponential_scale | [291, 1] | [321, 29] | split_ifs with h0 h1 h1 | case hp_lin
A : Type u_1
inst✝⁴ : CommRing A
R : Type u_2
inst✝³ : CommRing R
inst✝² : Algebra A R
S : Type u_3
inst✝¹ : CommRing S
inst✝ : Algebra A S
a : A
f : R⟦X⟧
hf : f.IsExponential
d : Fin 2 →₀ ℕ
hd : (d.sum fun x n => n) ≠ 1
⊢ ((if d = Finsupp.single 0 1 then 1 else 0) + if d = Finsupp.single 1 1 then 1 else 0)... | case pos
A : Type u_1
inst✝⁴ : CommRing A
R : Type u_2
inst✝³ : CommRing R
inst✝² : Algebra A R
S : Type u_3
inst✝¹ : CommRing S
inst✝ : Algebra A S
a : A
f : R⟦X⟧
hf : f.IsExponential
d : Fin 2 →₀ ℕ
hd : (d.sum fun x n => n) ≠ 1
h0 : d = Finsupp.single 0 1
h1 : d = Finsupp.single 1 1
⊢ 1 + 1 = 0
case neg
A : Type u_1... | Please generate a tactic in lean4 to solve the state.
STATE:
case hp_lin
A : Type u_1
inst✝⁴ : CommRing A
R : Type u_2
inst✝³ : CommRing R
inst✝² : Algebra A R
S : Type u_3
inst✝¹ : CommRing S
inst✝ : Algebra A S
a : A
f : R⟦X⟧
hf : f.IsExponential
d : Fin 2 →₀ ℕ
hd : (d.sum fun x n => n) ≠ 1
⊢ ((if d = Finsupp.single ... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/ExponentialModule/Basic.lean | PowerSeries.isExponential_scale | [291, 1] | [321, 29] | rw [h1, Finsupp.single_left_inj (by norm_num)] at h0 | case pos
A : Type u_1
inst✝⁴ : CommRing A
R : Type u_2
inst✝³ : CommRing R
inst✝² : Algebra A R
S : Type u_3
inst✝¹ : CommRing S
inst✝ : Algebra A S
a : A
f : R⟦X⟧
hf : f.IsExponential
d : Fin 2 →₀ ℕ
hd : (d.sum fun x n => n) ≠ 1
h0 : d = Finsupp.single 0 1
h1 : d = Finsupp.single 1 1
⊢ 1 + 1 = 0 | case pos
A : Type u_1
inst✝⁴ : CommRing A
R : Type u_2
inst✝³ : CommRing R
inst✝² : Algebra A R
S : Type u_3
inst✝¹ : CommRing S
inst✝ : Algebra A S
a : A
f : R⟦X⟧
hf : f.IsExponential
d : Fin 2 →₀ ℕ
hd : (d.sum fun x n => n) ≠ 1
h0 : 1 = 0
h1 : d = Finsupp.single 1 1
⊢ 1 + 1 = 0 | Please generate a tactic in lean4 to solve the state.
STATE:
case pos
A : Type u_1
inst✝⁴ : CommRing A
R : Type u_2
inst✝³ : CommRing R
inst✝² : Algebra A R
S : Type u_3
inst✝¹ : CommRing S
inst✝ : Algebra A S
a : A
f : R⟦X⟧
hf : f.IsExponential
d : Fin 2 →₀ ℕ
hd : (d.sum fun x n => n) ≠ 1
h0 : d = Finsupp.single 0 1
h... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/ExponentialModule/Basic.lean | PowerSeries.isExponential_scale | [291, 1] | [321, 29] | exfalso | case pos
A : Type u_1
inst✝⁴ : CommRing A
R : Type u_2
inst✝³ : CommRing R
inst✝² : Algebra A R
S : Type u_3
inst✝¹ : CommRing S
inst✝ : Algebra A S
a : A
f : R⟦X⟧
hf : f.IsExponential
d : Fin 2 →₀ ℕ
hd : (d.sum fun x n => n) ≠ 1
h0 : 1 = 0
h1 : d = Finsupp.single 1 1
⊢ 1 + 1 = 0 | case pos
A : Type u_1
inst✝⁴ : CommRing A
R : Type u_2
inst✝³ : CommRing R
inst✝² : Algebra A R
S : Type u_3
inst✝¹ : CommRing S
inst✝ : Algebra A S
a : A
f : R⟦X⟧
hf : f.IsExponential
d : Fin 2 →₀ ℕ
hd : (d.sum fun x n => n) ≠ 1
h0 : 1 = 0
h1 : d = Finsupp.single 1 1
⊢ False | Please generate a tactic in lean4 to solve the state.
STATE:
case pos
A : Type u_1
inst✝⁴ : CommRing A
R : Type u_2
inst✝³ : CommRing R
inst✝² : Algebra A R
S : Type u_3
inst✝¹ : CommRing S
inst✝ : Algebra A S
a : A
f : R⟦X⟧
hf : f.IsExponential
d : Fin 2 →₀ ℕ
hd : (d.sum fun x n => n) ≠ 1
h0 : 1 = 0
h1 : d = Finsupp.s... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/ExponentialModule/Basic.lean | PowerSeries.isExponential_scale | [291, 1] | [321, 29] | exact one_ne_zero h0 | case pos
A : Type u_1
inst✝⁴ : CommRing A
R : Type u_2
inst✝³ : CommRing R
inst✝² : Algebra A R
S : Type u_3
inst✝¹ : CommRing S
inst✝ : Algebra A S
a : A
f : R⟦X⟧
hf : f.IsExponential
d : Fin 2 →₀ ℕ
hd : (d.sum fun x n => n) ≠ 1
h0 : 1 = 0
h1 : d = Finsupp.single 1 1
⊢ False | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case pos
A : Type u_1
inst✝⁴ : CommRing A
R : Type u_2
inst✝³ : CommRing R
inst✝² : Algebra A R
S : Type u_3
inst✝¹ : CommRing S
inst✝ : Algebra A S
a : A
f : R⟦X⟧
hf : f.IsExponential
d : Fin 2 →₀ ℕ
hd : (d.sum fun x n => n) ≠ 1
h0 : 1 = 0
h1 : d = Finsupp.s... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/ExponentialModule/Basic.lean | PowerSeries.isExponential_scale | [291, 1] | [321, 29] | norm_num | A : Type u_1
inst✝⁴ : CommRing A
R : Type u_2
inst✝³ : CommRing R
inst✝² : Algebra A R
S : Type u_3
inst✝¹ : CommRing S
inst✝ : Algebra A S
a : A
f : R⟦X⟧
hf : f.IsExponential
d : Fin 2 →₀ ℕ
hd : (d.sum fun x n => n) ≠ 1
h0 : Finsupp.single 1 1 = Finsupp.single 0 1
h1 : d = Finsupp.single 1 1
⊢ 1 ≠ 0 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
A : Type u_1
inst✝⁴ : CommRing A
R : Type u_2
inst✝³ : CommRing R
inst✝² : Algebra A R
S : Type u_3
inst✝¹ : CommRing S
inst✝ : Algebra A S
a : A
f : R⟦X⟧
hf : f.IsExponential
d : Fin 2 →₀ ℕ
hd : (d.sum fun x n => n) ≠ 1
h0 : Finsupp.single 1 1 = Finsupp.sing... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/ExponentialModule/Basic.lean | PowerSeries.isExponential_scale | [291, 1] | [321, 29] | exfalso | case neg
A : Type u_1
inst✝⁴ : CommRing A
R : Type u_2
inst✝³ : CommRing R
inst✝² : Algebra A R
S : Type u_3
inst✝¹ : CommRing S
inst✝ : Algebra A S
a : A
f : R⟦X⟧
hf : f.IsExponential
d : Fin 2 →₀ ℕ
hd : (d.sum fun x n => n) ≠ 1
h0 : d = Finsupp.single 0 1
h1 : ¬d = Finsupp.single 1 1
⊢ 1 + 0 = 0 | case neg
A : Type u_1
inst✝⁴ : CommRing A
R : Type u_2
inst✝³ : CommRing R
inst✝² : Algebra A R
S : Type u_3
inst✝¹ : CommRing S
inst✝ : Algebra A S
a : A
f : R⟦X⟧
hf : f.IsExponential
d : Fin 2 →₀ ℕ
hd : (d.sum fun x n => n) ≠ 1
h0 : d = Finsupp.single 0 1
h1 : ¬d = Finsupp.single 1 1
⊢ False | Please generate a tactic in lean4 to solve the state.
STATE:
case neg
A : Type u_1
inst✝⁴ : CommRing A
R : Type u_2
inst✝³ : CommRing R
inst✝² : Algebra A R
S : Type u_3
inst✝¹ : CommRing S
inst✝ : Algebra A S
a : A
f : R⟦X⟧
hf : f.IsExponential
d : Fin 2 →₀ ℕ
hd : (d.sum fun x n => n) ≠ 1
h0 : d = Finsupp.single 0 1
h... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/ExponentialModule/Basic.lean | PowerSeries.isExponential_scale | [291, 1] | [321, 29] | apply hd | case neg
A : Type u_1
inst✝⁴ : CommRing A
R : Type u_2
inst✝³ : CommRing R
inst✝² : Algebra A R
S : Type u_3
inst✝¹ : CommRing S
inst✝ : Algebra A S
a : A
f : R⟦X⟧
hf : f.IsExponential
d : Fin 2 →₀ ℕ
hd : (d.sum fun x n => n) ≠ 1
h0 : d = Finsupp.single 0 1
h1 : ¬d = Finsupp.single 1 1
⊢ False | case neg
A : Type u_1
inst✝⁴ : CommRing A
R : Type u_2
inst✝³ : CommRing R
inst✝² : Algebra A R
S : Type u_3
inst✝¹ : CommRing S
inst✝ : Algebra A S
a : A
f : R⟦X⟧
hf : f.IsExponential
d : Fin 2 →₀ ℕ
hd : (d.sum fun x n => n) ≠ 1
h0 : d = Finsupp.single 0 1
h1 : ¬d = Finsupp.single 1 1
⊢ (d.sum fun x n => n) = 1 | Please generate a tactic in lean4 to solve the state.
STATE:
case neg
A : Type u_1
inst✝⁴ : CommRing A
R : Type u_2
inst✝³ : CommRing R
inst✝² : Algebra A R
S : Type u_3
inst✝¹ : CommRing S
inst✝ : Algebra A S
a : A
f : R⟦X⟧
hf : f.IsExponential
d : Fin 2 →₀ ℕ
hd : (d.sum fun x n => n) ≠ 1
h0 : d = Finsupp.single 0 1
h... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/ExponentialModule/Basic.lean | PowerSeries.isExponential_scale | [291, 1] | [321, 29] | simp only [h0, Fin.isValue, Finsupp.sum_single_index] | case neg
A : Type u_1
inst✝⁴ : CommRing A
R : Type u_2
inst✝³ : CommRing R
inst✝² : Algebra A R
S : Type u_3
inst✝¹ : CommRing S
inst✝ : Algebra A S
a : A
f : R⟦X⟧
hf : f.IsExponential
d : Fin 2 →₀ ℕ
hd : (d.sum fun x n => n) ≠ 1
h0 : d = Finsupp.single 0 1
h1 : ¬d = Finsupp.single 1 1
⊢ (d.sum fun x n => n) = 1 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case neg
A : Type u_1
inst✝⁴ : CommRing A
R : Type u_2
inst✝³ : CommRing R
inst✝² : Algebra A R
S : Type u_3
inst✝¹ : CommRing S
inst✝ : Algebra A S
a : A
f : R⟦X⟧
hf : f.IsExponential
d : Fin 2 →₀ ℕ
hd : (d.sum fun x n => n) ≠ 1
h0 : d = Finsupp.single 0 1
h... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/ExponentialModule/Basic.lean | PowerSeries.isExponential_scale | [291, 1] | [321, 29] | exfalso | case pos
A : Type u_1
inst✝⁴ : CommRing A
R : Type u_2
inst✝³ : CommRing R
inst✝² : Algebra A R
S : Type u_3
inst✝¹ : CommRing S
inst✝ : Algebra A S
a : A
f : R⟦X⟧
hf : f.IsExponential
d : Fin 2 →₀ ℕ
hd : (d.sum fun x n => n) ≠ 1
h0 : ¬d = Finsupp.single 0 1
h1 : d = Finsupp.single 1 1
⊢ 0 + 1 = 0 | case pos
A : Type u_1
inst✝⁴ : CommRing A
R : Type u_2
inst✝³ : CommRing R
inst✝² : Algebra A R
S : Type u_3
inst✝¹ : CommRing S
inst✝ : Algebra A S
a : A
f : R⟦X⟧
hf : f.IsExponential
d : Fin 2 →₀ ℕ
hd : (d.sum fun x n => n) ≠ 1
h0 : ¬d = Finsupp.single 0 1
h1 : d = Finsupp.single 1 1
⊢ False | Please generate a tactic in lean4 to solve the state.
STATE:
case pos
A : Type u_1
inst✝⁴ : CommRing A
R : Type u_2
inst✝³ : CommRing R
inst✝² : Algebra A R
S : Type u_3
inst✝¹ : CommRing S
inst✝ : Algebra A S
a : A
f : R⟦X⟧
hf : f.IsExponential
d : Fin 2 →₀ ℕ
hd : (d.sum fun x n => n) ≠ 1
h0 : ¬d = Finsupp.single 0 1
... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/ExponentialModule/Basic.lean | PowerSeries.isExponential_scale | [291, 1] | [321, 29] | apply hd | case pos
A : Type u_1
inst✝⁴ : CommRing A
R : Type u_2
inst✝³ : CommRing R
inst✝² : Algebra A R
S : Type u_3
inst✝¹ : CommRing S
inst✝ : Algebra A S
a : A
f : R⟦X⟧
hf : f.IsExponential
d : Fin 2 →₀ ℕ
hd : (d.sum fun x n => n) ≠ 1
h0 : ¬d = Finsupp.single 0 1
h1 : d = Finsupp.single 1 1
⊢ False | case pos
A : Type u_1
inst✝⁴ : CommRing A
R : Type u_2
inst✝³ : CommRing R
inst✝² : Algebra A R
S : Type u_3
inst✝¹ : CommRing S
inst✝ : Algebra A S
a : A
f : R⟦X⟧
hf : f.IsExponential
d : Fin 2 →₀ ℕ
hd : (d.sum fun x n => n) ≠ 1
h0 : ¬d = Finsupp.single 0 1
h1 : d = Finsupp.single 1 1
⊢ (d.sum fun x n => n) = 1 | Please generate a tactic in lean4 to solve the state.
STATE:
case pos
A : Type u_1
inst✝⁴ : CommRing A
R : Type u_2
inst✝³ : CommRing R
inst✝² : Algebra A R
S : Type u_3
inst✝¹ : CommRing S
inst✝ : Algebra A S
a : A
f : R⟦X⟧
hf : f.IsExponential
d : Fin 2 →₀ ℕ
hd : (d.sum fun x n => n) ≠ 1
h0 : ¬d = Finsupp.single 0 1
... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/ExponentialModule/Basic.lean | PowerSeries.isExponential_scale | [291, 1] | [321, 29] | simp only [h1, Fin.isValue, Finsupp.sum_single_index] | case pos
A : Type u_1
inst✝⁴ : CommRing A
R : Type u_2
inst✝³ : CommRing R
inst✝² : Algebra A R
S : Type u_3
inst✝¹ : CommRing S
inst✝ : Algebra A S
a : A
f : R⟦X⟧
hf : f.IsExponential
d : Fin 2 →₀ ℕ
hd : (d.sum fun x n => n) ≠ 1
h0 : ¬d = Finsupp.single 0 1
h1 : d = Finsupp.single 1 1
⊢ (d.sum fun x n => n) = 1 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case pos
A : Type u_1
inst✝⁴ : CommRing A
R : Type u_2
inst✝³ : CommRing R
inst✝² : Algebra A R
S : Type u_3
inst✝¹ : CommRing S
inst✝ : Algebra A S
a : A
f : R⟦X⟧
hf : f.IsExponential
d : Fin 2 →₀ ℕ
hd : (d.sum fun x n => n) ≠ 1
h0 : ¬d = Finsupp.single 0 1
... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/ExponentialModule/Basic.lean | PowerSeries.isExponential_scale | [291, 1] | [321, 29] | simp only [add_zero] | case neg
A : Type u_1
inst✝⁴ : CommRing A
R : Type u_2
inst✝³ : CommRing R
inst✝² : Algebra A R
S : Type u_3
inst✝¹ : CommRing S
inst✝ : Algebra A S
a : A
f : R⟦X⟧
hf : f.IsExponential
d : Fin 2 →₀ ℕ
hd : (d.sum fun x n => n) ≠ 1
h0 : ¬d = Finsupp.single 0 1
h1 : ¬d = Finsupp.single 1 1
⊢ 0 + 0 = 0 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case neg
A : Type u_1
inst✝⁴ : CommRing A
R : Type u_2
inst✝³ : CommRing R
inst✝² : Algebra A R
S : Type u_3
inst✝¹ : CommRing S
inst✝ : Algebra A S
a : A
f : R⟦X⟧
hf : f.IsExponential
d : Fin 2 →₀ ℕ
hd : (d.sum fun x n => n) ≠ 1
h0 : ¬d = Finsupp.single 0 1
... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/ExponentialModule/Basic.lean | PowerSeries.isExponential_scale_add | [323, 1] | [350, 69] | let a : Fin 2 → PowerSeries R
| 0 => r • X
| 1 => s • X | A : Type u_1
inst✝⁴ : CommRing A
R : Type u_2
inst✝³ : CommRing R
inst✝² : Algebra A R
S : Type u_3
inst✝¹ : CommRing S
inst✝ : Algebra A S
r s : A
f : R⟦X⟧
hf : f.IsExponential
⊢ scale (r + s) f = scale r f * scale s f | A : Type u_1
inst✝⁴ : CommRing A
R : Type u_2
inst✝³ : CommRing R
inst✝² : Algebra A R
S : Type u_3
inst✝¹ : CommRing S
inst✝ : Algebra A S
r s : A
f : R⟦X⟧
hf : f.IsExponential
a : Fin 2 → R⟦X⟧ :=
fun x =>
match x with
| 0 => r • X
| 1 => s • X
⊢ scale (r + s) f = scale r f * scale s f | Please generate a tactic in lean4 to solve the state.
STATE:
A : Type u_1
inst✝⁴ : CommRing A
R : Type u_2
inst✝³ : CommRing R
inst✝² : Algebra A R
S : Type u_3
inst✝¹ : CommRing S
inst✝ : Algebra A S
r s : A
f : R⟦X⟧
hf : f.IsExponential
⊢ scale (r + s) f = scale r f * scale s f
TACTIC:
|
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/ExponentialModule/Basic.lean | PowerSeries.isExponential_scale_add | [323, 1] | [350, 69] | have ha : MvPowerSeries.SubstDomain a := by
apply MvPowerSeries.substDomain_of_constantCoeff_zero
intro i
simp only [X, a]
match i with
| 0 =>
rw [MvPowerSeries.constantCoeff_smul, MvPowerSeries.constantCoeff_X, smul_zero]
| 1 =>
rw [MvPowerSeries.constantCoeff_smul, MvPowerSeries.constantCoeff_X, s... | A : Type u_1
inst✝⁴ : CommRing A
R : Type u_2
inst✝³ : CommRing R
inst✝² : Algebra A R
S : Type u_3
inst✝¹ : CommRing S
inst✝ : Algebra A S
r s : A
f : R⟦X⟧
hf : f.IsExponential
a : Fin 2 → R⟦X⟧ :=
fun x =>
match x with
| 0 => r • X
| 1 => s • X
⊢ scale (r + s) f = scale r f * scale s f | A : Type u_1
inst✝⁴ : CommRing A
R : Type u_2
inst✝³ : CommRing R
inst✝² : Algebra A R
S : Type u_3
inst✝¹ : CommRing S
inst✝ : Algebra A S
r s : A
f : R⟦X⟧
hf : f.IsExponential
a : Fin 2 → R⟦X⟧ :=
fun x =>
match x with
| 0 => r • X
| 1 => s • X
ha : MvPowerSeries.SubstDomain a
⊢ scale (r + s) f = scale r... | Please generate a tactic in lean4 to solve the state.
STATE:
A : Type u_1
inst✝⁴ : CommRing A
R : Type u_2
inst✝³ : CommRing R
inst✝² : Algebra A R
S : Type u_3
inst✝¹ : CommRing S
inst✝ : Algebra A S
r s : A
f : R⟦X⟧
hf : f.IsExponential
a : Fin 2 → R⟦X⟧ :=
fun x =>
match x with
| 0 => r • X
| 1 => s • X... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/ExponentialModule/Basic.lean | PowerSeries.isExponential_scale_add | [323, 1] | [350, 69] | have hf' := congr_arg (MvPowerSeries.subst a) hf.add_mul | A : Type u_1
inst✝⁴ : CommRing A
R : Type u_2
inst✝³ : CommRing R
inst✝² : Algebra A R
S : Type u_3
inst✝¹ : CommRing S
inst✝ : Algebra A S
r s : A
f : R⟦X⟧
hf : f.IsExponential
a : Fin 2 → R⟦X⟧ :=
fun x =>
match x with
| 0 => r • X
| 1 => s • X
ha : MvPowerSeries.SubstDomain a
⊢ scale (r + s) f = scale r... | A : Type u_1
inst✝⁴ : CommRing A
R : Type u_2
inst✝³ : CommRing R
inst✝² : Algebra A R
S : Type u_3
inst✝¹ : CommRing S
inst✝ : Algebra A S
r s : A
f : R⟦X⟧
hf : f.IsExponential
a : Fin 2 → R⟦X⟧ :=
fun x =>
match x with
| 0 => r • X
| 1 => s • X
ha : MvPowerSeries.SubstDomain a
hf' :
MvPowerSeries.subst... | Please generate a tactic in lean4 to solve the state.
STATE:
A : Type u_1
inst✝⁴ : CommRing A
R : Type u_2
inst✝³ : CommRing R
inst✝² : Algebra A R
S : Type u_3
inst✝¹ : CommRing S
inst✝ : Algebra A S
r s : A
f : R⟦X⟧
hf : f.IsExponential
a : Fin 2 → R⟦X⟧ :=
fun x =>
match x with
| 0 => r • X
| 1 => s • X... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/ExponentialModule/Basic.lean | PowerSeries.isExponential_scale_add | [323, 1] | [350, 69] | simp only [PowerSeries.subst] at hf' | A : Type u_1
inst✝⁴ : CommRing A
R : Type u_2
inst✝³ : CommRing R
inst✝² : Algebra A R
S : Type u_3
inst✝¹ : CommRing S
inst✝ : Algebra A S
r s : A
f : R⟦X⟧
hf : f.IsExponential
a : Fin 2 → R⟦X⟧ :=
fun x =>
match x with
| 0 => r • X
| 1 => s • X
ha : MvPowerSeries.SubstDomain a
hf' :
MvPowerSeries.subst... | A : Type u_1
inst✝⁴ : CommRing A
R : Type u_2
inst✝³ : CommRing R
inst✝² : Algebra A R
S : Type u_3
inst✝¹ : CommRing S
inst✝ : Algebra A S
r s : A
f : R⟦X⟧
hf : f.IsExponential
a : Fin 2 → R⟦X⟧ :=
fun x =>
match x with
| 0 => r • X
| 1 => s • X
ha : MvPowerSeries.SubstDomain a
hf' :
MvPowerSeries.subst... | Please generate a tactic in lean4 to solve the state.
STATE:
A : Type u_1
inst✝⁴ : CommRing A
R : Type u_2
inst✝³ : CommRing R
inst✝² : Algebra A R
S : Type u_3
inst✝¹ : CommRing S
inst✝ : Algebra A S
r s : A
f : R⟦X⟧
hf : f.IsExponential
a : Fin 2 → R⟦X⟧ :=
fun x =>
match x with
| 0 => r • X
| 1 => s • X... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/ExponentialModule/Basic.lean | PowerSeries.isExponential_scale_add | [323, 1] | [350, 69] | rw [← MvPowerSeries.coe_substAlgHom ha] at hf' | A : Type u_1
inst✝⁴ : CommRing A
R : Type u_2
inst✝³ : CommRing R
inst✝² : Algebra A R
S : Type u_3
inst✝¹ : CommRing S
inst✝ : Algebra A S
r s : A
f : R⟦X⟧
hf : f.IsExponential
a : Fin 2 → R⟦X⟧ :=
fun x =>
match x with
| 0 => r • X
| 1 => s • X
ha : MvPowerSeries.SubstDomain a
hf' :
MvPowerSeries.subst... | A : Type u_1
inst✝⁴ : CommRing A
R : Type u_2
inst✝³ : CommRing R
inst✝² : Algebra A R
S : Type u_3
inst✝¹ : CommRing S
inst✝ : Algebra A S
r s : A
f : R⟦X⟧
hf : f.IsExponential
a : Fin 2 → R⟦X⟧ :=
fun x =>
match x with
| 0 => r • X
| 1 => s • X
ha : MvPowerSeries.SubstDomain a
hf' :
(MvPowerSeries.subs... | Please generate a tactic in lean4 to solve the state.
STATE:
A : Type u_1
inst✝⁴ : CommRing A
R : Type u_2
inst✝³ : CommRing R
inst✝² : Algebra A R
S : Type u_3
inst✝¹ : CommRing S
inst✝ : Algebra A S
r s : A
f : R⟦X⟧
hf : f.IsExponential
a : Fin 2 → R⟦X⟧ :=
fun x =>
match x with
| 0 => r • X
| 1 => s • X... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/ExponentialModule/Basic.lean | PowerSeries.isExponential_scale_add | [323, 1] | [350, 69] | rw [← MvPowerSeries.coe_substAlgHom (MvPowerSeries.substDomain_of_constantCoeff_zero (by simp))] at hf' | A : Type u_1
inst✝⁴ : CommRing A
R : Type u_2
inst✝³ : CommRing R
inst✝² : Algebra A R
S : Type u_3
inst✝¹ : CommRing S
inst✝ : Algebra A S
r s : A
f : R⟦X⟧
hf : f.IsExponential
a : Fin 2 → R⟦X⟧ :=
fun x =>
match x with
| 0 => r • X
| 1 => s • X
ha : MvPowerSeries.SubstDomain a
hf' :
(MvPowerSeries.subs... | A : Type u_1
inst✝⁴ : CommRing A
R : Type u_2
inst✝³ : CommRing R
inst✝² : Algebra A R
S : Type u_3
inst✝¹ : CommRing S
inst✝ : Algebra A S
r s : A
f : R⟦X⟧
hf : f.IsExponential
a : Fin 2 → R⟦X⟧ :=
fun x =>
match x with
| 0 => r • X
| 1 => s • X
ha : MvPowerSeries.SubstDomain a
hf' :
(MvPowerSeries.subs... | Please generate a tactic in lean4 to solve the state.
STATE:
A : Type u_1
inst✝⁴ : CommRing A
R : Type u_2
inst✝³ : CommRing R
inst✝² : Algebra A R
S : Type u_3
inst✝¹ : CommRing S
inst✝ : Algebra A S
r s : A
f : R⟦X⟧
hf : f.IsExponential
a : Fin 2 → R⟦X⟧ :=
fun x =>
match x with
| 0 => r • X
| 1 => s • X... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/ExponentialModule/Basic.lean | PowerSeries.isExponential_scale_add | [323, 1] | [350, 69] | rw [← MvPowerSeries.coe_substAlgHom (MvPowerSeries.substDomain_of_constantCoeff_zero (by simp))] at hf' | A : Type u_1
inst✝⁴ : CommRing A
R : Type u_2
inst✝³ : CommRing R
inst✝² : Algebra A R
S : Type u_3
inst✝¹ : CommRing S
inst✝ : Algebra A S
r s : A
f : R⟦X⟧
hf : f.IsExponential
a : Fin 2 → R⟦X⟧ :=
fun x =>
match x with
| 0 => r • X
| 1 => s • X
ha : MvPowerSeries.SubstDomain a
hf' :
(MvPowerSeries.subs... | A : Type u_1
inst✝⁴ : CommRing A
R : Type u_2
inst✝³ : CommRing R
inst✝² : Algebra A R
S : Type u_3
inst✝¹ : CommRing S
inst✝ : Algebra A S
r s : A
f : R⟦X⟧
hf : f.IsExponential
a : Fin 2 → R⟦X⟧ :=
fun x =>
match x with
| 0 => r • X
| 1 => s • X
ha : MvPowerSeries.SubstDomain a
hf' :
(MvPowerSeries.subs... | Please generate a tactic in lean4 to solve the state.
STATE:
A : Type u_1
inst✝⁴ : CommRing A
R : Type u_2
inst✝³ : CommRing R
inst✝² : Algebra A R
S : Type u_3
inst✝¹ : CommRing S
inst✝ : Algebra A S
r s : A
f : R⟦X⟧
hf : f.IsExponential
a : Fin 2 → R⟦X⟧ :=
fun x =>
match x with
| 0 => r • X
| 1 => s • X... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/ExponentialModule/Basic.lean | PowerSeries.isExponential_scale_add | [323, 1] | [350, 69] | rw [← MvPowerSeries.coe_substAlgHom (MvPowerSeries.substDomain_of_constantCoeff_zero (by simp))] at hf' | A : Type u_1
inst✝⁴ : CommRing A
R : Type u_2
inst✝³ : CommRing R
inst✝² : Algebra A R
S : Type u_3
inst✝¹ : CommRing S
inst✝ : Algebra A S
r s : A
f : R⟦X⟧
hf : f.IsExponential
a : Fin 2 → R⟦X⟧ :=
fun x =>
match x with
| 0 => r • X
| 1 => s • X
ha : MvPowerSeries.SubstDomain a
hf' :
(MvPowerSeries.subs... | A : Type u_1
inst✝⁴ : CommRing A
R : Type u_2
inst✝³ : CommRing R
inst✝² : Algebra A R
S : Type u_3
inst✝¹ : CommRing S
inst✝ : Algebra A S
r s : A
f : R⟦X⟧
hf : f.IsExponential
a : Fin 2 → R⟦X⟧ :=
fun x =>
match x with
| 0 => r • X
| 1 => s • X
ha : MvPowerSeries.SubstDomain a
hf' :
(MvPowerSeries.subs... | Please generate a tactic in lean4 to solve the state.
STATE:
A : Type u_1
inst✝⁴ : CommRing A
R : Type u_2
inst✝³ : CommRing R
inst✝² : Algebra A R
S : Type u_3
inst✝¹ : CommRing S
inst✝ : Algebra A S
r s : A
f : R⟦X⟧
hf : f.IsExponential
a : Fin 2 → R⟦X⟧ :=
fun x =>
match x with
| 0 => r • X
| 1 => s • X... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/ExponentialModule/Basic.lean | PowerSeries.isExponential_scale_add | [323, 1] | [350, 69] | simp only [MvPowerSeries.substAlgHom_comp_substAlgHom_apply, map_mul] at hf' | A : Type u_1
inst✝⁴ : CommRing A
R : Type u_2
inst✝³ : CommRing R
inst✝² : Algebra A R
S : Type u_3
inst✝¹ : CommRing S
inst✝ : Algebra A S
r s : A
f : R⟦X⟧
hf : f.IsExponential
a : Fin 2 → R⟦X⟧ :=
fun x =>
match x with
| 0 => r • X
| 1 => s • X
ha : MvPowerSeries.SubstDomain a
hf' :
(MvPowerSeries.subs... | A : Type u_1
inst✝⁴ : CommRing A
R : Type u_2
inst✝³ : CommRing R
inst✝² : Algebra A R
S : Type u_3
inst✝¹ : CommRing S
inst✝ : Algebra A S
r s : A
f : R⟦X⟧
hf : f.IsExponential
a : Fin 2 → R⟦X⟧ :=
fun x =>
match x with
| 0 => r • X
| 1 => s • X
ha : MvPowerSeries.SubstDomain a
hf' : (MvPowerSeries.substA... | Please generate a tactic in lean4 to solve the state.
STATE:
A : Type u_1
inst✝⁴ : CommRing A
R : Type u_2
inst✝³ : CommRing R
inst✝² : Algebra A R
S : Type u_3
inst✝¹ : CommRing S
inst✝ : Algebra A S
r s : A
f : R⟦X⟧
hf : f.IsExponential
a : Fin 2 → R⟦X⟧ :=
fun x =>
match x with
| 0 => r • X
| 1 => s • X... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/ExponentialModule/Basic.lean | PowerSeries.isExponential_scale_add | [323, 1] | [350, 69] | simp only [MvPowerSeries.coe_substAlgHom] at hf' | A : Type u_1
inst✝⁴ : CommRing A
R : Type u_2
inst✝³ : CommRing R
inst✝² : Algebra A R
S : Type u_3
inst✝¹ : CommRing S
inst✝ : Algebra A S
r s : A
f : R⟦X⟧
hf : f.IsExponential
a : Fin 2 → R⟦X⟧ :=
fun x =>
match x with
| 0 => r • X
| 1 => s • X
ha : MvPowerSeries.SubstDomain a
hf' : (MvPowerSeries.substA... | A : Type u_1
inst✝⁴ : CommRing A
R : Type u_2
inst✝³ : CommRing R
inst✝² : Algebra A R
S : Type u_3
inst✝¹ : CommRing S
inst✝ : Algebra A S
r s : A
f : R⟦X⟧
hf : f.IsExponential
a : Fin 2 → R⟦X⟧ :=
fun x =>
match x with
| 0 => r • X
| 1 => s • X
ha : MvPowerSeries.SubstDomain a
hf' :
MvPowerSeries.subst... | Please generate a tactic in lean4 to solve the state.
STATE:
A : Type u_1
inst✝⁴ : CommRing A
R : Type u_2
inst✝³ : CommRing R
inst✝² : Algebra A R
S : Type u_3
inst✝¹ : CommRing S
inst✝ : Algebra A S
r s : A
f : R⟦X⟧
hf : f.IsExponential
a : Fin 2 → R⟦X⟧ :=
fun x =>
match x with
| 0 => r • X
| 1 => s • X... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/ExponentialModule/Basic.lean | PowerSeries.isExponential_scale_add | [323, 1] | [350, 69] | simp only [scale_eq_subst, subst] | A : Type u_1
inst✝⁴ : CommRing A
R : Type u_2
inst✝³ : CommRing R
inst✝² : Algebra A R
S : Type u_3
inst✝¹ : CommRing S
inst✝ : Algebra A S
r s : A
f : R⟦X⟧
hf : f.IsExponential
a : Fin 2 → R⟦X⟧ :=
fun x =>
match x with
| 0 => r • X
| 1 => s • X
ha : MvPowerSeries.SubstDomain a
hf' :
MvPowerSeries.subst... | A : Type u_1
inst✝⁴ : CommRing A
R : Type u_2
inst✝³ : CommRing R
inst✝² : Algebra A R
S : Type u_3
inst✝¹ : CommRing S
inst✝ : Algebra A S
r s : A
f : R⟦X⟧
hf : f.IsExponential
a : Fin 2 → R⟦X⟧ :=
fun x =>
match x with
| 0 => r • X
| 1 => s • X
ha : MvPowerSeries.SubstDomain a
hf' :
MvPowerSeries.subst... | Please generate a tactic in lean4 to solve the state.
STATE:
A : Type u_1
inst✝⁴ : CommRing A
R : Type u_2
inst✝³ : CommRing R
inst✝² : Algebra A R
S : Type u_3
inst✝¹ : CommRing S
inst✝ : Algebra A S
r s : A
f : R⟦X⟧
hf : f.IsExponential
a : Fin 2 → R⟦X⟧ :=
fun x =>
match x with
| 0 => r • X
| 1 => s • X... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/ExponentialModule/Basic.lean | PowerSeries.isExponential_scale_add | [323, 1] | [350, 69] | convert hf' | A : Type u_1
inst✝⁴ : CommRing A
R : Type u_2
inst✝³ : CommRing R
inst✝² : Algebra A R
S : Type u_3
inst✝¹ : CommRing S
inst✝ : Algebra A S
r s : A
f : R⟦X⟧
hf : f.IsExponential
a : Fin 2 → R⟦X⟧ :=
fun x =>
match x with
| 0 => r • X
| 1 => s • X
ha : MvPowerSeries.SubstDomain a
hf' :
MvPowerSeries.subst... | case h.e'_2.h.h.e'_8.h
A : Type u_1
inst✝⁴ : CommRing A
R : Type u_2
inst✝³ : CommRing R
inst✝² : Algebra A R
S : Type u_3
inst✝¹ : CommRing S
inst✝ : Algebra A S
r s : A
f : R⟦X⟧
hf : f.IsExponential
a : Fin 2 → R⟦X⟧ :=
fun x =>
match x with
| 0 => r • X
| 1 => s • X
ha : MvPowerSeries.SubstDomain a
hf' ... | Please generate a tactic in lean4 to solve the state.
STATE:
A : Type u_1
inst✝⁴ : CommRing A
R : Type u_2
inst✝³ : CommRing R
inst✝² : Algebra A R
S : Type u_3
inst✝¹ : CommRing S
inst✝ : Algebra A S
r s : A
f : R⟦X⟧
hf : f.IsExponential
a : Fin 2 → R⟦X⟧ :=
fun x =>
match x with
| 0 => r • X
| 1 => s • X... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/ExponentialModule/Basic.lean | PowerSeries.isExponential_scale_add | [323, 1] | [350, 69] | repeat
simp only [← MvPolynomial.coe_X, ← MvPolynomial.coe_add,
MvPowerSeries.subst_coe ha]
simp only [Fin.isValue, map_add, MvPolynomial.aeval_X, add_smul] | case h.e'_2.h.h.e'_8.h
A : Type u_1
inst✝⁴ : CommRing A
R : Type u_2
inst✝³ : CommRing R
inst✝² : Algebra A R
S : Type u_3
inst✝¹ : CommRing S
inst✝ : Algebra A S
r s : A
f : R⟦X⟧
hf : f.IsExponential
a : Fin 2 → R⟦X⟧ :=
fun x =>
match x with
| 0 => r • X
| 1 => s • X
ha : MvPowerSeries.SubstDomain a
hf' ... | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case h.e'_2.h.h.e'_8.h
A : Type u_1
inst✝⁴ : CommRing A
R : Type u_2
inst✝³ : CommRing R
inst✝² : Algebra A R
S : Type u_3
inst✝¹ : CommRing S
inst✝ : Algebra A S
r s : A
f : R⟦X⟧
hf : f.IsExponential
a : Fin 2 → R⟦X⟧ :=
fun x =>
match x with
| 0 =>... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/ExponentialModule/Basic.lean | PowerSeries.isExponential_scale_add | [323, 1] | [350, 69] | apply MvPowerSeries.substDomain_of_constantCoeff_zero | A : Type u_1
inst✝⁴ : CommRing A
R : Type u_2
inst✝³ : CommRing R
inst✝² : Algebra A R
S : Type u_3
inst✝¹ : CommRing S
inst✝ : Algebra A S
r s : A
f : R⟦X⟧
hf : f.IsExponential
a : Fin 2 → R⟦X⟧ :=
fun x =>
match x with
| 0 => r • X
| 1 => s • X
⊢ MvPowerSeries.SubstDomain a | case ha
A : Type u_1
inst✝⁴ : CommRing A
R : Type u_2
inst✝³ : CommRing R
inst✝² : Algebra A R
S : Type u_3
inst✝¹ : CommRing S
inst✝ : Algebra A S
r s : A
f : R⟦X⟧
hf : f.IsExponential
a : Fin 2 → R⟦X⟧ :=
fun x =>
match x with
| 0 => r • X
| 1 => s • X
⊢ ∀ (s : Fin 2), (MvPowerSeries.constantCoeff Unit R... | Please generate a tactic in lean4 to solve the state.
STATE:
A : Type u_1
inst✝⁴ : CommRing A
R : Type u_2
inst✝³ : CommRing R
inst✝² : Algebra A R
S : Type u_3
inst✝¹ : CommRing S
inst✝ : Algebra A S
r s : A
f : R⟦X⟧
hf : f.IsExponential
a : Fin 2 → R⟦X⟧ :=
fun x =>
match x with
| 0 => r • X
| 1 => s • X... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/ExponentialModule/Basic.lean | PowerSeries.isExponential_scale_add | [323, 1] | [350, 69] | intro i | case ha
A : Type u_1
inst✝⁴ : CommRing A
R : Type u_2
inst✝³ : CommRing R
inst✝² : Algebra A R
S : Type u_3
inst✝¹ : CommRing S
inst✝ : Algebra A S
r s : A
f : R⟦X⟧
hf : f.IsExponential
a : Fin 2 → R⟦X⟧ :=
fun x =>
match x with
| 0 => r • X
| 1 => s • X
⊢ ∀ (s : Fin 2), (MvPowerSeries.constantCoeff Unit R... | case ha
A : Type u_1
inst✝⁴ : CommRing A
R : Type u_2
inst✝³ : CommRing R
inst✝² : Algebra A R
S : Type u_3
inst✝¹ : CommRing S
inst✝ : Algebra A S
r s : A
f : R⟦X⟧
hf : f.IsExponential
a : Fin 2 → R⟦X⟧ :=
fun x =>
match x with
| 0 => r • X
| 1 => s • X
i : Fin 2
⊢ (MvPowerSeries.constantCoeff Unit R) (a ... | Please generate a tactic in lean4 to solve the state.
STATE:
case ha
A : Type u_1
inst✝⁴ : CommRing A
R : Type u_2
inst✝³ : CommRing R
inst✝² : Algebra A R
S : Type u_3
inst✝¹ : CommRing S
inst✝ : Algebra A S
r s : A
f : R⟦X⟧
hf : f.IsExponential
a : Fin 2 → R⟦X⟧ :=
fun x =>
match x with
| 0 => r • X
| 1 ... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/ExponentialModule/Basic.lean | PowerSeries.isExponential_scale_add | [323, 1] | [350, 69] | simp only [X, a] | case ha
A : Type u_1
inst✝⁴ : CommRing A
R : Type u_2
inst✝³ : CommRing R
inst✝² : Algebra A R
S : Type u_3
inst✝¹ : CommRing S
inst✝ : Algebra A S
r s : A
f : R⟦X⟧
hf : f.IsExponential
a : Fin 2 → R⟦X⟧ :=
fun x =>
match x with
| 0 => r • X
| 1 => s • X
i : Fin 2
⊢ (MvPowerSeries.constantCoeff Unit R) (a ... | case ha
A : Type u_1
inst✝⁴ : CommRing A
R : Type u_2
inst✝³ : CommRing R
inst✝² : Algebra A R
S : Type u_3
inst✝¹ : CommRing S
inst✝ : Algebra A S
r s : A
f : R⟦X⟧
hf : f.IsExponential
a : Fin 2 → R⟦X⟧ :=
fun x =>
match x with
| 0 => r • X
| 1 => s • X
i : Fin 2
⊢ (MvPowerSeries.constantCoeff Unit R)
... | Please generate a tactic in lean4 to solve the state.
STATE:
case ha
A : Type u_1
inst✝⁴ : CommRing A
R : Type u_2
inst✝³ : CommRing R
inst✝² : Algebra A R
S : Type u_3
inst✝¹ : CommRing S
inst✝ : Algebra A S
r s : A
f : R⟦X⟧
hf : f.IsExponential
a : Fin 2 → R⟦X⟧ :=
fun x =>
match x with
| 0 => r • X
| 1 ... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/ExponentialModule/Basic.lean | PowerSeries.isExponential_scale_add | [323, 1] | [350, 69] | match i with
| 0 =>
rw [MvPowerSeries.constantCoeff_smul, MvPowerSeries.constantCoeff_X, smul_zero]
| 1 =>
rw [MvPowerSeries.constantCoeff_smul, MvPowerSeries.constantCoeff_X, smul_zero] | case ha
A : Type u_1
inst✝⁴ : CommRing A
R : Type u_2
inst✝³ : CommRing R
inst✝² : Algebra A R
S : Type u_3
inst✝¹ : CommRing S
inst✝ : Algebra A S
r s : A
f : R⟦X⟧
hf : f.IsExponential
a : Fin 2 → R⟦X⟧ :=
fun x =>
match x with
| 0 => r • X
| 1 => s • X
i : Fin 2
⊢ (MvPowerSeries.constantCoeff Unit R)
... | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case ha
A : Type u_1
inst✝⁴ : CommRing A
R : Type u_2
inst✝³ : CommRing R
inst✝² : Algebra A R
S : Type u_3
inst✝¹ : CommRing S
inst✝ : Algebra A S
r s : A
f : R⟦X⟧
hf : f.IsExponential
a : Fin 2 → R⟦X⟧ :=
fun x =>
match x with
| 0 => r • X
| 1 ... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/ExponentialModule/Basic.lean | PowerSeries.isExponential_scale_add | [323, 1] | [350, 69] | rw [MvPowerSeries.constantCoeff_smul, MvPowerSeries.constantCoeff_X, smul_zero] | A : Type u_1
inst✝⁴ : CommRing A
R : Type u_2
inst✝³ : CommRing R
inst✝² : Algebra A R
S : Type u_3
inst✝¹ : CommRing S
inst✝ : Algebra A S
r s : A
f : R⟦X⟧
hf : f.IsExponential
a : Fin 2 → R⟦X⟧ :=
fun x =>
match x with
| 0 => r • X
| 1 => s • X
i : Fin 2
⊢ (MvPowerSeries.constantCoeff Unit R)
(matc... | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
A : Type u_1
inst✝⁴ : CommRing A
R : Type u_2
inst✝³ : CommRing R
inst✝² : Algebra A R
S : Type u_3
inst✝¹ : CommRing S
inst✝ : Algebra A S
r s : A
f : R⟦X⟧
hf : f.IsExponential
a : Fin 2 → R⟦X⟧ :=
fun x =>
match x with
| 0 => r • X
| 1 => s • X... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/ExponentialModule/Basic.lean | PowerSeries.isExponential_scale_add | [323, 1] | [350, 69] | rw [MvPowerSeries.constantCoeff_smul, MvPowerSeries.constantCoeff_X, smul_zero] | A : Type u_1
inst✝⁴ : CommRing A
R : Type u_2
inst✝³ : CommRing R
inst✝² : Algebra A R
S : Type u_3
inst✝¹ : CommRing S
inst✝ : Algebra A S
r s : A
f : R⟦X⟧
hf : f.IsExponential
a : Fin 2 → R⟦X⟧ :=
fun x =>
match x with
| 0 => r • X
| 1 => s • X
i : Fin 2
⊢ (MvPowerSeries.constantCoeff Unit R)
(matc... | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
A : Type u_1
inst✝⁴ : CommRing A
R : Type u_2
inst✝³ : CommRing R
inst✝² : Algebra A R
S : Type u_3
inst✝¹ : CommRing S
inst✝ : Algebra A S
r s : A
f : R⟦X⟧
hf : f.IsExponential
a : Fin 2 → R⟦X⟧ :=
fun x =>
match x with
| 0 => r • X
| 1 => s • X... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/ExponentialModule/Basic.lean | PowerSeries.isExponential_scale_add | [323, 1] | [350, 69] | simp | A : Type u_1
inst✝⁴ : CommRing A
R : Type u_2
inst✝³ : CommRing R
inst✝² : Algebra A R
S : Type u_3
inst✝¹ : CommRing S
inst✝ : Algebra A S
r s : A
f : R⟦X⟧
hf : f.IsExponential
a : Fin 2 → R⟦X⟧ :=
fun x =>
match x with
| 0 => r • X
| 1 => s • X
ha : MvPowerSeries.SubstDomain a
hf' :
(MvPowerSeries.subs... | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
A : Type u_1
inst✝⁴ : CommRing A
R : Type u_2
inst✝³ : CommRing R
inst✝² : Algebra A R
S : Type u_3
inst✝¹ : CommRing S
inst✝ : Algebra A S
r s : A
f : R⟦X⟧
hf : f.IsExponential
a : Fin 2 → R⟦X⟧ :=
fun x =>
match x with
| 0 => r • X
| 1 => s • X... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/ExponentialModule/Basic.lean | PowerSeries.isExponential_scale_add | [323, 1] | [350, 69] | simp | A : Type u_1
inst✝⁴ : CommRing A
R : Type u_2
inst✝³ : CommRing R
inst✝² : Algebra A R
S : Type u_3
inst✝¹ : CommRing S
inst✝ : Algebra A S
r s : A
f : R⟦X⟧
hf : f.IsExponential
a : Fin 2 → R⟦X⟧ :=
fun x =>
match x with
| 0 => r • X
| 1 => s • X
ha : MvPowerSeries.SubstDomain a
hf' :
(MvPowerSeries.subs... | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
A : Type u_1
inst✝⁴ : CommRing A
R : Type u_2
inst✝³ : CommRing R
inst✝² : Algebra A R
S : Type u_3
inst✝¹ : CommRing S
inst✝ : Algebra A S
r s : A
f : R⟦X⟧
hf : f.IsExponential
a : Fin 2 → R⟦X⟧ :=
fun x =>
match x with
| 0 => r • X
| 1 => s • X... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/ExponentialModule/Basic.lean | PowerSeries.isExponential_scale_add | [323, 1] | [350, 69] | simp | A : Type u_1
inst✝⁴ : CommRing A
R : Type u_2
inst✝³ : CommRing R
inst✝² : Algebra A R
S : Type u_3
inst✝¹ : CommRing S
inst✝ : Algebra A S
r s : A
f : R⟦X⟧
hf : f.IsExponential
a : Fin 2 → R⟦X⟧ :=
fun x =>
match x with
| 0 => r • X
| 1 => s • X
ha : MvPowerSeries.SubstDomain a
hf' :
(MvPowerSeries.subs... | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
A : Type u_1
inst✝⁴ : CommRing A
R : Type u_2
inst✝³ : CommRing R
inst✝² : Algebra A R
S : Type u_3
inst✝¹ : CommRing S
inst✝ : Algebra A S
r s : A
f : R⟦X⟧
hf : f.IsExponential
a : Fin 2 → R⟦X⟧ :=
fun x =>
match x with
| 0 => r • X
| 1 => s • X... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/ExponentialModule/Basic.lean | PowerSeries.isExponential_scale_add | [323, 1] | [350, 69] | simp only [← MvPolynomial.coe_X, ← MvPolynomial.coe_add,
MvPowerSeries.subst_coe ha] | case h.e'_3.h.h.e'_6.h.h.e'_8.h
A : Type u_1
inst✝⁴ : CommRing A
R : Type u_2
inst✝³ : CommRing R
inst✝² : Algebra A R
S : Type u_3
inst✝¹ : CommRing S
inst✝ : Algebra A S
r s : A
f : R⟦X⟧
hf : f.IsExponential
a : Fin 2 → R⟦X⟧ :=
fun x =>
match x with
| 0 => r • X
| 1 => s • X
ha : MvPowerSeries.SubstDoma... | case h.e'_3.h.h.e'_6.h.h.e'_8.h
A : Type u_1
inst✝⁴ : CommRing A
R : Type u_2
inst✝³ : CommRing R
inst✝² : Algebra A R
S : Type u_3
inst✝¹ : CommRing S
inst✝ : Algebra A S
r s : A
f : R⟦X⟧
hf : f.IsExponential
a : Fin 2 → R⟦X⟧ :=
fun x =>
match x with
| 0 => r • X
| 1 => s • X
ha : MvPowerSeries.SubstDoma... | Please generate a tactic in lean4 to solve the state.
STATE:
case h.e'_3.h.h.e'_6.h.h.e'_8.h
A : Type u_1
inst✝⁴ : CommRing A
R : Type u_2
inst✝³ : CommRing R
inst✝² : Algebra A R
S : Type u_3
inst✝¹ : CommRing S
inst✝ : Algebra A S
r s : A
f : R⟦X⟧
hf : f.IsExponential
a : Fin 2 → R⟦X⟧ :=
fun x =>
match x with
... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/ExponentialModule/Basic.lean | PowerSeries.isExponential_scale_add | [323, 1] | [350, 69] | simp only [Fin.isValue, map_add, MvPolynomial.aeval_X, add_smul] | case h.e'_3.h.h.e'_6.h.h.e'_8.h
A : Type u_1
inst✝⁴ : CommRing A
R : Type u_2
inst✝³ : CommRing R
inst✝² : Algebra A R
S : Type u_3
inst✝¹ : CommRing S
inst✝ : Algebra A S
r s : A
f : R⟦X⟧
hf : f.IsExponential
a : Fin 2 → R⟦X⟧ :=
fun x =>
match x with
| 0 => r • X
| 1 => s • X
ha : MvPowerSeries.SubstDoma... | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case h.e'_3.h.h.e'_6.h.h.e'_8.h
A : Type u_1
inst✝⁴ : CommRing A
R : Type u_2
inst✝³ : CommRing R
inst✝² : Algebra A R
S : Type u_3
inst✝¹ : CommRing S
inst✝ : Algebra A S
r s : A
f : R⟦X⟧
hf : f.IsExponential
a : Fin 2 → R⟦X⟧ :=
fun x =>
match x with
... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/ExponentialModule/Basic.lean | PowerSeries.isExponential_self_mul_neg_eq_one | [355, 1] | [359, 75] | convert (isExponential_scale_add (1 : A) (-1 : A) hf).symm | A : Type u_1
inst✝⁴ : CommRing A
R : Type u_2
inst✝³ : CommRing R
inst✝² : Algebra A R
S : Type u_3
inst✝¹ : CommRing S
inst✝ : Algebra A S
f : R⟦X⟧
hf : f.IsExponential
⊢ f * scale (-1) f = 1 | case h.e'_2.h.e'_5
A : Type u_1
inst✝⁴ : CommRing A
R : Type u_2
inst✝³ : CommRing R
inst✝² : Algebra A R
S : Type u_3
inst✝¹ : CommRing S
inst✝ : Algebra A S
f : R⟦X⟧
hf : f.IsExponential
⊢ f = scale 1 f
case h.e'_3
A : Type u_1
inst✝⁴ : CommRing A
R : Type u_2
inst✝³ : CommRing R
inst✝² : Algebra A R
S : Type u_3
in... | Please generate a tactic in lean4 to solve the state.
STATE:
A : Type u_1
inst✝⁴ : CommRing A
R : Type u_2
inst✝³ : CommRing R
inst✝² : Algebra A R
S : Type u_3
inst✝¹ : CommRing S
inst✝ : Algebra A S
f : R⟦X⟧
hf : f.IsExponential
⊢ f * scale (-1) f = 1
TACTIC:
|
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/ExponentialModule/Basic.lean | PowerSeries.isExponential_self_mul_neg_eq_one | [355, 1] | [359, 75] | rw [scale_one, id_eq] | case h.e'_2.h.e'_5
A : Type u_1
inst✝⁴ : CommRing A
R : Type u_2
inst✝³ : CommRing R
inst✝² : Algebra A R
S : Type u_3
inst✝¹ : CommRing S
inst✝ : Algebra A S
f : R⟦X⟧
hf : f.IsExponential
⊢ f = scale 1 f | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case h.e'_2.h.e'_5
A : Type u_1
inst✝⁴ : CommRing A
R : Type u_2
inst✝³ : CommRing R
inst✝² : Algebra A R
S : Type u_3
inst✝¹ : CommRing S
inst✝ : Algebra A S
f : R⟦X⟧
hf : f.IsExponential
⊢ f = scale 1 f
TACTIC:
|
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/ExponentialModule/Basic.lean | PowerSeries.isExponential_self_mul_neg_eq_one | [355, 1] | [359, 75] | simp only [add_right_neg, scale_zero_apply, hf.constantCoeff, map_one] | case h.e'_3
A : Type u_1
inst✝⁴ : CommRing A
R : Type u_2
inst✝³ : CommRing R
inst✝² : Algebra A R
S : Type u_3
inst✝¹ : CommRing S
inst✝ : Algebra A S
f : R⟦X⟧
hf : f.IsExponential
⊢ 1 = scale (1 + -1) f | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case h.e'_3
A : Type u_1
inst✝⁴ : CommRing A
R : Type u_2
inst✝³ : CommRing R
inst✝² : Algebra A R
S : Type u_3
inst✝¹ : CommRing S
inst✝ : Algebra A S
f : R⟦X⟧
hf : f.IsExponential
⊢ 1 = scale (1 + -1) f
TACTIC:
|
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/ExponentialModule/Basic.lean | PowerSeries.isExponential_neg_mul_self_eq_one | [361, 1] | [363, 54] | rw [mul_comm, isExponential_self_mul_neg_eq_one hf] | A : Type u_1
inst✝⁴ : CommRing A
R : Type u_2
inst✝³ : CommRing R
inst✝² : Algebra A R
S : Type u_3
inst✝¹ : CommRing S
inst✝ : Algebra A S
f : R⟦X⟧
hf : f.IsExponential
⊢ scale (-1) f * f = 1 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
A : Type u_1
inst✝⁴ : CommRing A
R : Type u_2
inst✝³ : CommRing R
inst✝² : Algebra A R
S : Type u_3
inst✝¹ : CommRing S
inst✝ : Algebra A S
f : R⟦X⟧
hf : f.IsExponential
⊢ scale (-1) f * f = 1
TACTIC:
|
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/ExponentialModule/Basic.lean | PowerSeries.isExponential_map | [365, 1] | [373, 37] | rw [isExponential_iff] | A : Type u_1
inst✝⁴ : CommRing A
R : Type u_2
inst✝³ : CommRing R
inst✝² : Algebra A R
S : Type u_3
inst✝¹ : CommRing S
inst✝ : Algebra A S
φ : R →+* S
f : R⟦X⟧
hf : f.IsExponential
⊢ ((map φ) f).IsExponential | A : Type u_1
inst✝⁴ : CommRing A
R : Type u_2
inst✝³ : CommRing R
inst✝² : Algebra A R
S : Type u_3
inst✝¹ : CommRing S
inst✝ : Algebra A S
φ : R →+* S
f : R⟦X⟧
hf : f.IsExponential
⊢ (constantCoeff S) ((map φ) f) = 1 ∧
∀ (p q : ℕ), ↑((p + q).choose p) * (coeff S (p + q)) ((map φ) f) = (coeff S p) ((map φ) f) * (co... | Please generate a tactic in lean4 to solve the state.
STATE:
A : Type u_1
inst✝⁴ : CommRing A
R : Type u_2
inst✝³ : CommRing R
inst✝² : Algebra A R
S : Type u_3
inst✝¹ : CommRing S
inst✝ : Algebra A S
φ : R →+* S
f : R⟦X⟧
hf : f.IsExponential
⊢ ((map φ) f).IsExponential
TACTIC:
|
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/ExponentialModule/Basic.lean | PowerSeries.isExponential_map | [365, 1] | [373, 37] | constructor | A : Type u_1
inst✝⁴ : CommRing A
R : Type u_2
inst✝³ : CommRing R
inst✝² : Algebra A R
S : Type u_3
inst✝¹ : CommRing S
inst✝ : Algebra A S
φ : R →+* S
f : R⟦X⟧
hf : f.IsExponential
⊢ (constantCoeff S) ((map φ) f) = 1 ∧
∀ (p q : ℕ), ↑((p + q).choose p) * (coeff S (p + q)) ((map φ) f) = (coeff S p) ((map φ) f) * (co... | case left
A : Type u_1
inst✝⁴ : CommRing A
R : Type u_2
inst✝³ : CommRing R
inst✝² : Algebra A R
S : Type u_3
inst✝¹ : CommRing S
inst✝ : Algebra A S
φ : R →+* S
f : R⟦X⟧
hf : f.IsExponential
⊢ (constantCoeff S) ((map φ) f) = 1
case right
A : Type u_1
inst✝⁴ : CommRing A
R : Type u_2
inst✝³ : CommRing R
inst✝² : Algeb... | Please generate a tactic in lean4 to solve the state.
STATE:
A : Type u_1
inst✝⁴ : CommRing A
R : Type u_2
inst✝³ : CommRing R
inst✝² : Algebra A R
S : Type u_3
inst✝¹ : CommRing S
inst✝ : Algebra A S
φ : R →+* S
f : R⟦X⟧
hf : f.IsExponential
⊢ (constantCoeff S) ((map φ) f) = 1 ∧
∀ (p q : ℕ), ↑((p + q).choose p) * ... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/ExponentialModule/Basic.lean | PowerSeries.isExponential_map | [365, 1] | [373, 37] | rw [← coeff_zero_eq_constantCoeff_apply, coeff_map,
coeff_zero_eq_constantCoeff, hf.constantCoeff, map_one] | case left
A : Type u_1
inst✝⁴ : CommRing A
R : Type u_2
inst✝³ : CommRing R
inst✝² : Algebra A R
S : Type u_3
inst✝¹ : CommRing S
inst✝ : Algebra A S
φ : R →+* S
f : R⟦X⟧
hf : f.IsExponential
⊢ (constantCoeff S) ((map φ) f) = 1 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case left
A : Type u_1
inst✝⁴ : CommRing A
R : Type u_2
inst✝³ : CommRing R
inst✝² : Algebra A R
S : Type u_3
inst✝¹ : CommRing S
inst✝ : Algebra A S
φ : R →+* S
f : R⟦X⟧
hf : f.IsExponential
⊢ (constantCoeff S) ((map φ) f) = 1
TACTIC:
|
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/ExponentialModule/Basic.lean | PowerSeries.isExponential_map | [365, 1] | [373, 37] | intro p q | case right
A : Type u_1
inst✝⁴ : CommRing A
R : Type u_2
inst✝³ : CommRing R
inst✝² : Algebra A R
S : Type u_3
inst✝¹ : CommRing S
inst✝ : Algebra A S
φ : R →+* S
f : R⟦X⟧
hf : f.IsExponential
⊢ ∀ (p q : ℕ), ↑((p + q).choose p) * (coeff S (p + q)) ((map φ) f) = (coeff S p) ((map φ) f) * (coeff S q) ((map φ) f) | case right
A : Type u_1
inst✝⁴ : CommRing A
R : Type u_2
inst✝³ : CommRing R
inst✝² : Algebra A R
S : Type u_3
inst✝¹ : CommRing S
inst✝ : Algebra A S
φ : R →+* S
f : R⟦X⟧
hf : f.IsExponential
p q : ℕ
⊢ ↑((p + q).choose p) * (coeff S (p + q)) ((map φ) f) = (coeff S p) ((map φ) f) * (coeff S q) ((map φ) f) | Please generate a tactic in lean4 to solve the state.
STATE:
case right
A : Type u_1
inst✝⁴ : CommRing A
R : Type u_2
inst✝³ : CommRing R
inst✝² : Algebra A R
S : Type u_3
inst✝¹ : CommRing S
inst✝ : Algebra A S
φ : R →+* S
f : R⟦X⟧
hf : f.IsExponential
⊢ ∀ (p q : ℕ), ↑((p + q).choose p) * (coeff S (p + q)) ((map φ) f)... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/ExponentialModule/Basic.lean | PowerSeries.isExponential_map | [365, 1] | [373, 37] | simp only [coeff_map, ← map_mul, ← ((isExponential_iff f).mp hf).2 p q] | case right
A : Type u_1
inst✝⁴ : CommRing A
R : Type u_2
inst✝³ : CommRing R
inst✝² : Algebra A R
S : Type u_3
inst✝¹ : CommRing S
inst✝ : Algebra A S
φ : R →+* S
f : R⟦X⟧
hf : f.IsExponential
p q : ℕ
⊢ ↑((p + q).choose p) * (coeff S (p + q)) ((map φ) f) = (coeff S p) ((map φ) f) * (coeff S q) ((map φ) f) | case right
A : Type u_1
inst✝⁴ : CommRing A
R : Type u_2
inst✝³ : CommRing R
inst✝² : Algebra A R
S : Type u_3
inst✝¹ : CommRing S
inst✝ : Algebra A S
φ : R →+* S
f : R⟦X⟧
hf : f.IsExponential
p q : ℕ
⊢ ↑((p + q).choose p) * φ ((coeff R (p + q)) f) = φ (↑((p + q).choose p) * (coeff R (p + q)) f) | Please generate a tactic in lean4 to solve the state.
STATE:
case right
A : Type u_1
inst✝⁴ : CommRing A
R : Type u_2
inst✝³ : CommRing R
inst✝² : Algebra A R
S : Type u_3
inst✝¹ : CommRing S
inst✝ : Algebra A S
φ : R →+* S
f : R⟦X⟧
hf : f.IsExponential
p q : ℕ
⊢ ↑((p + q).choose p) * (coeff S (p + q)) ((map φ) f) = (c... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/ExponentialModule/Basic.lean | PowerSeries.isExponential_map | [365, 1] | [373, 37] | simp only [map_mul, map_natCast] | case right
A : Type u_1
inst✝⁴ : CommRing A
R : Type u_2
inst✝³ : CommRing R
inst✝² : Algebra A R
S : Type u_3
inst✝¹ : CommRing S
inst✝ : Algebra A S
φ : R →+* S
f : R⟦X⟧
hf : f.IsExponential
p q : ℕ
⊢ ↑((p + q).choose p) * φ ((coeff R (p + q)) f) = φ (↑((p + q).choose p) * (coeff R (p + q)) f) | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case right
A : Type u_1
inst✝⁴ : CommRing A
R : Type u_2
inst✝³ : CommRing R
inst✝² : Algebra A R
S : Type u_3
inst✝¹ : CommRing S
inst✝ : Algebra A S
φ : R →+* S
f : R⟦X⟧
hf : f.IsExponential
p q : ℕ
⊢ ↑((p + q).choose p) * φ ((coeff R (p + q)) f) = φ (↑((p ... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/ExponentialModule/Basic.lean | ExponentialModule.coe_injective | [440, 1] | [442, 91] | simp only [toPowerSeries, EmbeddingLike.apply_eq_iff_eq, SetLike.coe_eq_coe, imp_self] | A✝ : Type u_1
R✝ : Type u_2
inst✝⁵ : CommRing A✝
inst✝⁴ : CommRing R✝
inst✝³ : Algebra A✝ R✝
A : Type u_3
R : Type u_4
inst✝² : CommRing A
inst✝¹ : CommRing R
inst✝ : Algebra A R
f g : ↥(ExponentialModule R)
⊢ ↑f = ↑g → f = g | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
A✝ : Type u_1
R✝ : Type u_2
inst✝⁵ : CommRing A✝
inst✝⁴ : CommRing R✝
inst✝³ : Algebra A✝ R✝
A : Type u_3
R : Type u_4
inst✝² : CommRing A
inst✝¹ : CommRing R
inst✝ : Algebra A R
f g : ↥(ExponentialModule R)
⊢ ↑f = ↑g → f = g
TACTIC:
|
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/ExponentialModule/Basic.lean | ExponentialModule.coe_add | [488, 1] | [489, 61] | simp only [toPowerSeries, AddSubmonoid.coe_add, toMul_add] | A✝ : Type u_1
R✝ : Type u_2
inst✝⁵ : CommRing A✝
inst✝⁴ : CommRing R✝
inst✝³ : Algebra A✝ R✝
A : Type u_3
R : Type u_4
inst✝² : CommRing A
inst✝¹ : CommRing R
inst✝ : Algebra A R
f g : ↥(ExponentialModule R)
⊢ ↑(f + g) = ↑f * ↑g | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
A✝ : Type u_1
R✝ : Type u_2
inst✝⁵ : CommRing A✝
inst✝⁴ : CommRing R✝
inst✝³ : Algebra A✝ R✝
A : Type u_3
R : Type u_4
inst✝² : CommRing A
inst✝¹ : CommRing R
inst✝ : Algebra A R
f g : ↥(ExponentialModule R)
⊢ ↑(f + g) = ↑f * ↑g
TACTIC:
|
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/DPAlgebra/PolynomialMap.lean | DividedPowerAlgebra.gamma_toFun | [39, 1] | [43, 65] | obtain ⟨k, ψ, p, rfl⟩ := PolynomialMap.exists_lift m | R : Type u
inst✝⁴ : CommRing R
M : Type u_1
inst✝³ : AddCommGroup M
inst✝² : Module R M
n : ℕ
S : Type u_2
inst✝¹ : CommRing S
inst✝ : Algebra R S
m : S ⊗[R] M
⊢ (gamma R M n).toFun S m = (dpScalarExtensionEquiv R S M).symm (dp S n m) | case intro.intro.intro
R : Type u
inst✝⁴ : CommRing R
M : Type u_1
inst✝³ : AddCommGroup M
inst✝² : Module R M
n : ℕ
S : Type u_2
inst✝¹ : CommRing S
inst✝ : Algebra R S
k : ℕ
ψ : MvPolynomial (Fin k) R →ₐ[R] S
p : MvPolynomial (Fin k) R ⊗[R] M
⊢ (gamma R M n).toFun S ((LinearMap.rTensor M ψ.toLinearMap) p) =
(dpSc... | Please generate a tactic in lean4 to solve the state.
STATE:
R : Type u
inst✝⁴ : CommRing R
M : Type u_1
inst✝³ : AddCommGroup M
inst✝² : Module R M
n : ℕ
S : Type u_2
inst✝¹ : CommRing S
inst✝ : Algebra R S
m : S ⊗[R] M
⊢ (gamma R M n).toFun S m = (dpScalarExtensionEquiv R S M).symm (dp S n m)
TACTIC:
|
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/DPAlgebra/PolynomialMap.lean | DividedPowerAlgebra.gamma_toFun | [39, 1] | [43, 65] | rw [← (gamma R M n).isCompat_apply, PolynomialMap.toFun_eq_toFun'] | case intro.intro.intro
R : Type u
inst✝⁴ : CommRing R
M : Type u_1
inst✝³ : AddCommGroup M
inst✝² : Module R M
n : ℕ
S : Type u_2
inst✝¹ : CommRing S
inst✝ : Algebra R S
k : ℕ
ψ : MvPolynomial (Fin k) R →ₐ[R] S
p : MvPolynomial (Fin k) R ⊗[R] M
⊢ (gamma R M n).toFun S ((LinearMap.rTensor M ψ.toLinearMap) p) =
(dpSc... | case intro.intro.intro
R : Type u
inst✝⁴ : CommRing R
M : Type u_1
inst✝³ : AddCommGroup M
inst✝² : Module R M
n : ℕ
S : Type u_2
inst✝¹ : CommRing S
inst✝ : Algebra R S
k : ℕ
ψ : MvPolynomial (Fin k) R →ₐ[R] S
p : MvPolynomial (Fin k) R ⊗[R] M
⊢ (LinearMap.rTensor (DividedPowerAlgebra R M) ψ.toLinearMap) ((gamma R M n... | Please generate a tactic in lean4 to solve the state.
STATE:
case intro.intro.intro
R : Type u
inst✝⁴ : CommRing R
M : Type u_1
inst✝³ : AddCommGroup M
inst✝² : Module R M
n : ℕ
S : Type u_2
inst✝¹ : CommRing S
inst✝ : Algebra R S
k : ℕ
ψ : MvPolynomial (Fin k) R →ₐ[R] S
p : MvPolynomial (Fin k) R ⊗[R] M
⊢ (gamma R M n... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/DPAlgebra/PolynomialMap.lean | DividedPowerAlgebra.gamma_toFun | [39, 1] | [43, 65] | simp only [gamma, rTensor_comp_dpScalarExtensionEquiv_symm_eq] | case intro.intro.intro
R : Type u
inst✝⁴ : CommRing R
M : Type u_1
inst✝³ : AddCommGroup M
inst✝² : Module R M
n : ℕ
S : Type u_2
inst✝¹ : CommRing S
inst✝ : Algebra R S
k : ℕ
ψ : MvPolynomial (Fin k) R →ₐ[R] S
p : MvPolynomial (Fin k) R ⊗[R] M
⊢ (LinearMap.rTensor (DividedPowerAlgebra R M) ψ.toLinearMap) ((gamma R M n... | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case intro.intro.intro
R : Type u
inst✝⁴ : CommRing R
M : Type u_1
inst✝³ : AddCommGroup M
inst✝² : Module R M
n : ℕ
S : Type u_2
inst✝¹ : CommRing S
inst✝ : Algebra R S
k : ℕ
ψ : MvPolynomial (Fin k) R →ₐ[R] S
p : MvPolynomial (Fin k) R ⊗[R] M
⊢ (LinearMap.r... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/DPAlgebra/PolynomialMap.lean | DividedPowerAlgebra.isHomogeneousOfDegree_gamma | [45, 1] | [51, 15] | intro S _ _ r sm | R : Type u
inst✝² : CommRing R
M : Type u_1
inst✝¹ : AddCommGroup M
inst✝ : Module R M
n : ℕ
⊢ PolynomialMap.IsHomogeneousOfDegree n (gamma R M n) | R : Type u
inst✝⁴ : CommRing R
M : Type u_1
inst✝³ : AddCommGroup M
inst✝² : Module R M
n : ℕ
S : Type u
inst✝¹ : CommRing S
inst✝ : Algebra R S
r : S
sm : S ⊗[R] M
⊢ (gamma R M n).toFun' S (r • sm) = r ^ n • (gamma R M n).toFun' S sm | Please generate a tactic in lean4 to solve the state.
STATE:
R : Type u
inst✝² : CommRing R
M : Type u_1
inst✝¹ : AddCommGroup M
inst✝ : Module R M
n : ℕ
⊢ PolynomialMap.IsHomogeneousOfDegree n (gamma R M n)
TACTIC:
|
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/DPAlgebra/PolynomialMap.lean | DividedPowerAlgebra.isHomogeneousOfDegree_gamma | [45, 1] | [51, 15] | simp only [gamma] | R : Type u
inst✝⁴ : CommRing R
M : Type u_1
inst✝³ : AddCommGroup M
inst✝² : Module R M
n : ℕ
S : Type u
inst✝¹ : CommRing S
inst✝ : Algebra R S
r : S
sm : S ⊗[R] M
⊢ (gamma R M n).toFun' S (r • sm) = r ^ n • (gamma R M n).toFun' S sm | R : Type u
inst✝⁴ : CommRing R
M : Type u_1
inst✝³ : AddCommGroup M
inst✝² : Module R M
n : ℕ
S : Type u
inst✝¹ : CommRing S
inst✝ : Algebra R S
r : S
sm : S ⊗[R] M
⊢ (dpScalarExtensionEquiv R S M).symm (dp S n (r • sm)) = r ^ n • (dpScalarExtensionEquiv R S M).symm (dp S n sm) | Please generate a tactic in lean4 to solve the state.
STATE:
R : Type u
inst✝⁴ : CommRing R
M : Type u_1
inst✝³ : AddCommGroup M
inst✝² : Module R M
n : ℕ
S : Type u
inst✝¹ : CommRing S
inst✝ : Algebra R S
r : S
sm : S ⊗[R] M
⊢ (gamma R M n).toFun' S (r • sm) = r ^ n • (gamma R M n).toFun' S sm
TACTIC:
|
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/DPAlgebra/PolynomialMap.lean | DividedPowerAlgebra.isHomogeneousOfDegree_gamma | [45, 1] | [51, 15] | apply (dpScalarExtensionEquiv R S M).injective | R : Type u
inst✝⁴ : CommRing R
M : Type u_1
inst✝³ : AddCommGroup M
inst✝² : Module R M
n : ℕ
S : Type u
inst✝¹ : CommRing S
inst✝ : Algebra R S
r : S
sm : S ⊗[R] M
⊢ (dpScalarExtensionEquiv R S M).symm (dp S n (r • sm)) = r ^ n • (dpScalarExtensionEquiv R S M).symm (dp S n sm) | case a
R : Type u
inst✝⁴ : CommRing R
M : Type u_1
inst✝³ : AddCommGroup M
inst✝² : Module R M
n : ℕ
S : Type u
inst✝¹ : CommRing S
inst✝ : Algebra R S
r : S
sm : S ⊗[R] M
⊢ (dpScalarExtensionEquiv R S M) ((dpScalarExtensionEquiv R S M).symm (dp S n (r • sm))) =
(dpScalarExtensionEquiv R S M) (r ^ n • (dpScalarExte... | Please generate a tactic in lean4 to solve the state.
STATE:
R : Type u
inst✝⁴ : CommRing R
M : Type u_1
inst✝³ : AddCommGroup M
inst✝² : Module R M
n : ℕ
S : Type u
inst✝¹ : CommRing S
inst✝ : Algebra R S
r : S
sm : S ⊗[R] M
⊢ (dpScalarExtensionEquiv R S M).symm (dp S n (r • sm)) = r ^ n • (dpScalarExtensionEquiv R S ... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/DPAlgebra/PolynomialMap.lean | DividedPowerAlgebra.isHomogeneousOfDegree_gamma | [45, 1] | [51, 15] | simp only [apply_symm_apply, LinearMapClass.map_smul] | case a
R : Type u
inst✝⁴ : CommRing R
M : Type u_1
inst✝³ : AddCommGroup M
inst✝² : Module R M
n : ℕ
S : Type u
inst✝¹ : CommRing S
inst✝ : Algebra R S
r : S
sm : S ⊗[R] M
⊢ (dpScalarExtensionEquiv R S M) ((dpScalarExtensionEquiv R S M).symm (dp S n (r • sm))) =
(dpScalarExtensionEquiv R S M) (r ^ n • (dpScalarExte... | case a
R : Type u
inst✝⁴ : CommRing R
M : Type u_1
inst✝³ : AddCommGroup M
inst✝² : Module R M
n : ℕ
S : Type u
inst✝¹ : CommRing S
inst✝ : Algebra R S
r : S
sm : S ⊗[R] M
⊢ dp S n (r • sm) = r ^ n • dp S n sm | Please generate a tactic in lean4 to solve the state.
STATE:
case a
R : Type u
inst✝⁴ : CommRing R
M : Type u_1
inst✝³ : AddCommGroup M
inst✝² : Module R M
n : ℕ
S : Type u
inst✝¹ : CommRing S
inst✝ : Algebra R S
r : S
sm : S ⊗[R] M
⊢ (dpScalarExtensionEquiv R S M) ((dpScalarExtensionEquiv R S M).symm (dp S n (r • sm))... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/DPAlgebra/PolynomialMap.lean | DividedPowerAlgebra.isHomogeneousOfDegree_gamma | [45, 1] | [51, 15] | rw [dp_smul] | case a
R : Type u
inst✝⁴ : CommRing R
M : Type u_1
inst✝³ : AddCommGroup M
inst✝² : Module R M
n : ℕ
S : Type u
inst✝¹ : CommRing S
inst✝ : Algebra R S
r : S
sm : S ⊗[R] M
⊢ dp S n (r • sm) = r ^ n • dp S n sm | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case a
R : Type u
inst✝⁴ : CommRing R
M : Type u_1
inst✝³ : AddCommGroup M
inst✝² : Module R M
n : ℕ
S : Type u
inst✝¹ : CommRing S
inst✝ : Algebra R S
r : S
sm : S ⊗[R] M
⊢ dp S n (r • sm) = r ^ n • dp S n sm
TACTIC:
|
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/DPAlgebra/PolynomialMap.lean | DividedPowerAlgebra.gamma_mem_grade | [53, 1] | [87, 10] | simp only [gamma_toFun, dp_null] | case zero
R : Type u
inst✝⁴ : CommRing R
M : Type u_1
inst✝³ : AddCommGroup M
inst✝² : Module R M
S : Type u_2
inst✝¹ : CommRing S
inst✝ : Algebra R S
n : ℕ
⊢ (gamma R M n).toFun S 0 ∈ LinearMap.range (LinearMap.lTensor S (grade R M n).subtype) | case zero
R : Type u
inst✝⁴ : CommRing R
M : Type u_1
inst✝³ : AddCommGroup M
inst✝² : Module R M
S : Type u_2
inst✝¹ : CommRing S
inst✝ : Algebra R S
n : ℕ
⊢ (dpScalarExtensionEquiv R S M).symm (if n = 0 then 1 else 0) ∈
LinearMap.range (LinearMap.lTensor S (grade R M n).subtype) | Please generate a tactic in lean4 to solve the state.
STATE:
case zero
R : Type u
inst✝⁴ : CommRing R
M : Type u_1
inst✝³ : AddCommGroup M
inst✝² : Module R M
S : Type u_2
inst✝¹ : CommRing S
inst✝ : Algebra R S
n : ℕ
⊢ (gamma R M n).toFun S 0 ∈ LinearMap.range (LinearMap.lTensor S (grade R M n).subtype)
TACTIC:
|
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/DPAlgebra/PolynomialMap.lean | DividedPowerAlgebra.gamma_mem_grade | [53, 1] | [87, 10] | split_ifs with h | case zero
R : Type u
inst✝⁴ : CommRing R
M : Type u_1
inst✝³ : AddCommGroup M
inst✝² : Module R M
S : Type u_2
inst✝¹ : CommRing S
inst✝ : Algebra R S
n : ℕ
⊢ (dpScalarExtensionEquiv R S M).symm (if n = 0 then 1 else 0) ∈
LinearMap.range (LinearMap.lTensor S (grade R M n).subtype) | case pos
R : Type u
inst✝⁴ : CommRing R
M : Type u_1
inst✝³ : AddCommGroup M
inst✝² : Module R M
S : Type u_2
inst✝¹ : CommRing S
inst✝ : Algebra R S
n : ℕ
h : n = 0
⊢ (dpScalarExtensionEquiv R S M).symm 1 ∈ LinearMap.range (LinearMap.lTensor S (grade R M n).subtype)
case neg
R : Type u
inst✝⁴ : CommRing R
M : Type u_... | Please generate a tactic in lean4 to solve the state.
STATE:
case zero
R : Type u
inst✝⁴ : CommRing R
M : Type u_1
inst✝³ : AddCommGroup M
inst✝² : Module R M
S : Type u_2
inst✝¹ : CommRing S
inst✝ : Algebra R S
n : ℕ
⊢ (dpScalarExtensionEquiv R S M).symm (if n = 0 then 1 else 0) ∈
LinearMap.range (LinearMap.lTenso... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/DPAlgebra/PolynomialMap.lean | DividedPowerAlgebra.gamma_mem_grade | [53, 1] | [87, 10] | rw [AlgEquiv.map_one, h] | case pos
R : Type u
inst✝⁴ : CommRing R
M : Type u_1
inst✝³ : AddCommGroup M
inst✝² : Module R M
S : Type u_2
inst✝¹ : CommRing S
inst✝ : Algebra R S
n : ℕ
h : n = 0
⊢ (dpScalarExtensionEquiv R S M).symm 1 ∈ LinearMap.range (LinearMap.lTensor S (grade R M n).subtype) | case pos
R : Type u
inst✝⁴ : CommRing R
M : Type u_1
inst✝³ : AddCommGroup M
inst✝² : Module R M
S : Type u_2
inst✝¹ : CommRing S
inst✝ : Algebra R S
n : ℕ
h : n = 0
⊢ 1 ∈ LinearMap.range (LinearMap.lTensor S (grade R M 0).subtype) | Please generate a tactic in lean4 to solve the state.
STATE:
case pos
R : Type u
inst✝⁴ : CommRing R
M : Type u_1
inst✝³ : AddCommGroup M
inst✝² : Module R M
S : Type u_2
inst✝¹ : CommRing S
inst✝ : Algebra R S
n : ℕ
h : n = 0
⊢ (dpScalarExtensionEquiv R S M).symm 1 ∈ LinearMap.range (LinearMap.lTensor S (grade R M n).... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/DPAlgebra/PolynomialMap.lean | DividedPowerAlgebra.gamma_mem_grade | [53, 1] | [87, 10] | simp only [LinearMap.mem_range] | case pos
R : Type u
inst✝⁴ : CommRing R
M : Type u_1
inst✝³ : AddCommGroup M
inst✝² : Module R M
S : Type u_2
inst✝¹ : CommRing S
inst✝ : Algebra R S
n : ℕ
h : n = 0
⊢ 1 ∈ LinearMap.range (LinearMap.lTensor S (grade R M 0).subtype) | case pos
R : Type u
inst✝⁴ : CommRing R
M : Type u_1
inst✝³ : AddCommGroup M
inst✝² : Module R M
S : Type u_2
inst✝¹ : CommRing S
inst✝ : Algebra R S
n : ℕ
h : n = 0
⊢ ∃ y, (LinearMap.lTensor S (grade R M 0).subtype) y = 1 | Please generate a tactic in lean4 to solve the state.
STATE:
case pos
R : Type u
inst✝⁴ : CommRing R
M : Type u_1
inst✝³ : AddCommGroup M
inst✝² : Module R M
S : Type u_2
inst✝¹ : CommRing S
inst✝ : Algebra R S
n : ℕ
h : n = 0
⊢ 1 ∈ LinearMap.range (LinearMap.lTensor S (grade R M 0).subtype)
TACTIC:
|
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/DPAlgebra/PolynomialMap.lean | DividedPowerAlgebra.gamma_mem_grade | [53, 1] | [87, 10] | use (1 : S) ⊗ₜ[R] ⟨(1 : DividedPowerAlgebra R M), one_mem R M⟩ | case pos
R : Type u
inst✝⁴ : CommRing R
M : Type u_1
inst✝³ : AddCommGroup M
inst✝² : Module R M
S : Type u_2
inst✝¹ : CommRing S
inst✝ : Algebra R S
n : ℕ
h : n = 0
⊢ ∃ y, (LinearMap.lTensor S (grade R M 0).subtype) y = 1 | case h
R : Type u
inst✝⁴ : CommRing R
M : Type u_1
inst✝³ : AddCommGroup M
inst✝² : Module R M
S : Type u_2
inst✝¹ : CommRing S
inst✝ : Algebra R S
n : ℕ
h : n = 0
⊢ (LinearMap.lTensor S (grade R M 0).subtype) (1 ⊗ₜ[R] ⟨1, ⋯⟩) = 1 | Please generate a tactic in lean4 to solve the state.
STATE:
case pos
R : Type u
inst✝⁴ : CommRing R
M : Type u_1
inst✝³ : AddCommGroup M
inst✝² : Module R M
S : Type u_2
inst✝¹ : CommRing S
inst✝ : Algebra R S
n : ℕ
h : n = 0
⊢ ∃ y, (LinearMap.lTensor S (grade R M 0).subtype) y = 1
TACTIC:
|
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/DPAlgebra/PolynomialMap.lean | DividedPowerAlgebra.gamma_mem_grade | [53, 1] | [87, 10] | simp only [LinearMap.lTensor_tmul, Submodule.coeSubtype] | case h
R : Type u
inst✝⁴ : CommRing R
M : Type u_1
inst✝³ : AddCommGroup M
inst✝² : Module R M
S : Type u_2
inst✝¹ : CommRing S
inst✝ : Algebra R S
n : ℕ
h : n = 0
⊢ (LinearMap.lTensor S (grade R M 0).subtype) (1 ⊗ₜ[R] ⟨1, ⋯⟩) = 1 | case h
R : Type u
inst✝⁴ : CommRing R
M : Type u_1
inst✝³ : AddCommGroup M
inst✝² : Module R M
S : Type u_2
inst✝¹ : CommRing S
inst✝ : Algebra R S
n : ℕ
h : n = 0
⊢ 1 ⊗ₜ[R] 1 = 1 | Please generate a tactic in lean4 to solve the state.
STATE:
case h
R : Type u
inst✝⁴ : CommRing R
M : Type u_1
inst✝³ : AddCommGroup M
inst✝² : Module R M
S : Type u_2
inst✝¹ : CommRing S
inst✝ : Algebra R S
n : ℕ
h : n = 0
⊢ (LinearMap.lTensor S (grade R M 0).subtype) (1 ⊗ₜ[R] ⟨1, ⋯⟩) = 1
TACTIC:
|
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/DPAlgebra/PolynomialMap.lean | DividedPowerAlgebra.gamma_mem_grade | [53, 1] | [87, 10] | rw [Algebra.TensorProduct.one_def] | case h
R : Type u
inst✝⁴ : CommRing R
M : Type u_1
inst✝³ : AddCommGroup M
inst✝² : Module R M
S : Type u_2
inst✝¹ : CommRing S
inst✝ : Algebra R S
n : ℕ
h : n = 0
⊢ 1 ⊗ₜ[R] 1 = 1 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case h
R : Type u
inst✝⁴ : CommRing R
M : Type u_1
inst✝³ : AddCommGroup M
inst✝² : Module R M
S : Type u_2
inst✝¹ : CommRing S
inst✝ : Algebra R S
n : ℕ
h : n = 0
⊢ 1 ⊗ₜ[R] 1 = 1
TACTIC:
|
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/DPAlgebra/PolynomialMap.lean | DividedPowerAlgebra.gamma_mem_grade | [53, 1] | [87, 10] | simp only [map_zero, zero_mem] | case neg
R : Type u
inst✝⁴ : CommRing R
M : Type u_1
inst✝³ : AddCommGroup M
inst✝² : Module R M
S : Type u_2
inst✝¹ : CommRing S
inst✝ : Algebra R S
n : ℕ
h : ¬n = 0
⊢ (dpScalarExtensionEquiv R S M).symm 0 ∈ LinearMap.range (LinearMap.lTensor S (grade R M n).subtype) | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case neg
R : Type u
inst✝⁴ : CommRing R
M : Type u_1
inst✝³ : AddCommGroup M
inst✝² : Module R M
S : Type u_2
inst✝¹ : CommRing S
inst✝ : Algebra R S
n : ℕ
h : ¬n = 0
⊢ (dpScalarExtensionEquiv R S M).symm 0 ∈ LinearMap.range (LinearMap.lTensor S (grade R M n)... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/DPAlgebra/PolynomialMap.lean | DividedPowerAlgebra.gamma_mem_grade | [53, 1] | [87, 10] | simp only [gamma_toFun, dpScalarExtensionEquiv] | case tmul
R : Type u
inst✝⁴ : CommRing R
M : Type u_1
inst✝³ : AddCommGroup M
inst✝² : Module R M
S : Type u_2
inst✝¹ : CommRing S
inst✝ : Algebra R S
s : S
m : M
n : ℕ
⊢ (gamma R M n).toFun S (s ⊗ₜ[R] m) ∈ LinearMap.range (LinearMap.lTensor S (grade R M n).subtype) | case tmul
R : Type u
inst✝⁴ : CommRing R
M : Type u_1
inst✝³ : AddCommGroup M
inst✝² : Module R M
S : Type u_2
inst✝¹ : CommRing S
inst✝ : Algebra R S
s : S
m : M
n : ℕ
⊢ (ofAlgHom (dpScalarExtension R S M) (dpScalarExtensionInv R S M) ⋯ ⋯).symm (dp S n (s ⊗ₜ[R] m)) ∈
LinearMap.range (LinearMap.lTensor S (grade R M... | Please generate a tactic in lean4 to solve the state.
STATE:
case tmul
R : Type u
inst✝⁴ : CommRing R
M : Type u_1
inst✝³ : AddCommGroup M
inst✝² : Module R M
S : Type u_2
inst✝¹ : CommRing S
inst✝ : Algebra R S
s : S
m : M
n : ℕ
⊢ (gamma R M n).toFun S (s ⊗ₜ[R] m) ∈ LinearMap.range (LinearMap.lTensor S (grade R M n).s... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/DPAlgebra/PolynomialMap.lean | DividedPowerAlgebra.gamma_mem_grade | [53, 1] | [87, 10] | simp only [ofAlgHom_symm_apply] | case tmul
R : Type u
inst✝⁴ : CommRing R
M : Type u_1
inst✝³ : AddCommGroup M
inst✝² : Module R M
S : Type u_2
inst✝¹ : CommRing S
inst✝ : Algebra R S
s : S
m : M
n : ℕ
⊢ (ofAlgHom (dpScalarExtension R S M) (dpScalarExtensionInv R S M) ⋯ ⋯).symm (dp S n (s ⊗ₜ[R] m)) ∈
LinearMap.range (LinearMap.lTensor S (grade R M... | case tmul
R : Type u
inst✝⁴ : CommRing R
M : Type u_1
inst✝³ : AddCommGroup M
inst✝² : Module R M
S : Type u_2
inst✝¹ : CommRing S
inst✝ : Algebra R S
s : S
m : M
n : ℕ
⊢ (dpScalarExtensionInv R S M) (dp S n (s ⊗ₜ[R] m)) ∈ LinearMap.range (LinearMap.lTensor S (grade R M n).subtype) | Please generate a tactic in lean4 to solve the state.
STATE:
case tmul
R : Type u
inst✝⁴ : CommRing R
M : Type u_1
inst✝³ : AddCommGroup M
inst✝² : Module R M
S : Type u_2
inst✝¹ : CommRing S
inst✝ : Algebra R S
s : S
m : M
n : ℕ
⊢ (ofAlgHom (dpScalarExtension R S M) (dpScalarExtensionInv R S M) ⋯ ⋯).symm (dp S n (s ⊗ₜ... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/DPAlgebra/PolynomialMap.lean | DividedPowerAlgebra.gamma_mem_grade | [53, 1] | [87, 10] | rw [dpScalarExtensionInv_apply_dp] | case tmul
R : Type u
inst✝⁴ : CommRing R
M : Type u_1
inst✝³ : AddCommGroup M
inst✝² : Module R M
S : Type u_2
inst✝¹ : CommRing S
inst✝ : Algebra R S
s : S
m : M
n : ℕ
⊢ (dpScalarExtensionInv R S M) (dp S n (s ⊗ₜ[R] m)) ∈ LinearMap.range (LinearMap.lTensor S (grade R M n).subtype) | case tmul
R : Type u
inst✝⁴ : CommRing R
M : Type u_1
inst✝³ : AddCommGroup M
inst✝² : Module R M
S : Type u_2
inst✝¹ : CommRing S
inst✝ : Algebra R S
s : S
m : M
n : ℕ
⊢ (s ^ n) ⊗ₜ[R] dp R n m ∈ LinearMap.range (LinearMap.lTensor S (grade R M n).subtype) | Please generate a tactic in lean4 to solve the state.
STATE:
case tmul
R : Type u
inst✝⁴ : CommRing R
M : Type u_1
inst✝³ : AddCommGroup M
inst✝² : Module R M
S : Type u_2
inst✝¹ : CommRing S
inst✝ : Algebra R S
s : S
m : M
n : ℕ
⊢ (dpScalarExtensionInv R S M) (dp S n (s ⊗ₜ[R] m)) ∈ LinearMap.range (LinearMap.lTensor S... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/DPAlgebra/PolynomialMap.lean | DividedPowerAlgebra.gamma_mem_grade | [53, 1] | [87, 10] | simp only [LinearMap.mem_range] | case tmul
R : Type u
inst✝⁴ : CommRing R
M : Type u_1
inst✝³ : AddCommGroup M
inst✝² : Module R M
S : Type u_2
inst✝¹ : CommRing S
inst✝ : Algebra R S
s : S
m : M
n : ℕ
⊢ (s ^ n) ⊗ₜ[R] dp R n m ∈ LinearMap.range (LinearMap.lTensor S (grade R M n).subtype) | case tmul
R : Type u
inst✝⁴ : CommRing R
M : Type u_1
inst✝³ : AddCommGroup M
inst✝² : Module R M
S : Type u_2
inst✝¹ : CommRing S
inst✝ : Algebra R S
s : S
m : M
n : ℕ
⊢ ∃ y, (LinearMap.lTensor S (grade R M n).subtype) y = (s ^ n) ⊗ₜ[R] dp R n m | Please generate a tactic in lean4 to solve the state.
STATE:
case tmul
R : Type u
inst✝⁴ : CommRing R
M : Type u_1
inst✝³ : AddCommGroup M
inst✝² : Module R M
S : Type u_2
inst✝¹ : CommRing S
inst✝ : Algebra R S
s : S
m : M
n : ℕ
⊢ (s ^ n) ⊗ₜ[R] dp R n m ∈ LinearMap.range (LinearMap.lTensor S (grade R M n).subtype)
TAC... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/DPAlgebra/PolynomialMap.lean | DividedPowerAlgebra.gamma_mem_grade | [53, 1] | [87, 10] | use (s ^ n) ⊗ₜ[R] ⟨dp R n m, dp_mem_grade R M n m⟩ | case tmul
R : Type u
inst✝⁴ : CommRing R
M : Type u_1
inst✝³ : AddCommGroup M
inst✝² : Module R M
S : Type u_2
inst✝¹ : CommRing S
inst✝ : Algebra R S
s : S
m : M
n : ℕ
⊢ ∃ y, (LinearMap.lTensor S (grade R M n).subtype) y = (s ^ n) ⊗ₜ[R] dp R n m | case h
R : Type u
inst✝⁴ : CommRing R
M : Type u_1
inst✝³ : AddCommGroup M
inst✝² : Module R M
S : Type u_2
inst✝¹ : CommRing S
inst✝ : Algebra R S
s : S
m : M
n : ℕ
⊢ (LinearMap.lTensor S (grade R M n).subtype) ((s ^ n) ⊗ₜ[R] ⟨dp R n m, ⋯⟩) = (s ^ n) ⊗ₜ[R] dp R n m | Please generate a tactic in lean4 to solve the state.
STATE:
case tmul
R : Type u
inst✝⁴ : CommRing R
M : Type u_1
inst✝³ : AddCommGroup M
inst✝² : Module R M
S : Type u_2
inst✝¹ : CommRing S
inst✝ : Algebra R S
s : S
m : M
n : ℕ
⊢ ∃ y, (LinearMap.lTensor S (grade R M n).subtype) y = (s ^ n) ⊗ₜ[R] dp R n m
TACTIC:
|
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/DPAlgebra/PolynomialMap.lean | DividedPowerAlgebra.gamma_mem_grade | [53, 1] | [87, 10] | simp only [LinearMap.lTensor_tmul, Submodule.coeSubtype] | case h
R : Type u
inst✝⁴ : CommRing R
M : Type u_1
inst✝³ : AddCommGroup M
inst✝² : Module R M
S : Type u_2
inst✝¹ : CommRing S
inst✝ : Algebra R S
s : S
m : M
n : ℕ
⊢ (LinearMap.lTensor S (grade R M n).subtype) ((s ^ n) ⊗ₜ[R] ⟨dp R n m, ⋯⟩) = (s ^ n) ⊗ₜ[R] dp R n m | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case h
R : Type u
inst✝⁴ : CommRing R
M : Type u_1
inst✝³ : AddCommGroup M
inst✝² : Module R M
S : Type u_2
inst✝¹ : CommRing S
inst✝ : Algebra R S
s : S
m : M
n : ℕ
⊢ (LinearMap.lTensor S (grade R M n).subtype) ((s ^ n) ⊗ₜ[R] ⟨dp R n m, ⋯⟩) = (s ^ n) ⊗ₜ[R] d... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/DPAlgebra/PolynomialMap.lean | DividedPowerAlgebra.gamma_mem_grade | [53, 1] | [87, 10] | simp only [gamma_toFun, dpScalarExtensionEquiv, ofAlgHom_symm_apply] | case add
R : Type u
inst✝⁴ : CommRing R
M : Type u_1
inst✝³ : AddCommGroup M
inst✝² : Module R M
S : Type u_2
inst✝¹ : CommRing S
inst✝ : Algebra R S
x y : S ⊗[R] M
hx : ∀ (n : ℕ), (gamma R M n).toFun S x ∈ LinearMap.range (LinearMap.lTensor S (grade R M n).subtype)
hy : ∀ (n : ℕ), (gamma R M n).toFun S y ∈ LinearMap.r... | case add
R : Type u
inst✝⁴ : CommRing R
M : Type u_1
inst✝³ : AddCommGroup M
inst✝² : Module R M
S : Type u_2
inst✝¹ : CommRing S
inst✝ : Algebra R S
x y : S ⊗[R] M
hx : ∀ (n : ℕ), (gamma R M n).toFun S x ∈ LinearMap.range (LinearMap.lTensor S (grade R M n).subtype)
hy : ∀ (n : ℕ), (gamma R M n).toFun S y ∈ LinearMap.r... | Please generate a tactic in lean4 to solve the state.
STATE:
case add
R : Type u
inst✝⁴ : CommRing R
M : Type u_1
inst✝³ : AddCommGroup M
inst✝² : Module R M
S : Type u_2
inst✝¹ : CommRing S
inst✝ : Algebra R S
x y : S ⊗[R] M
hx : ∀ (n : ℕ), (gamma R M n).toFun S x ∈ LinearMap.range (LinearMap.lTensor S (grade R M n).s... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/DPAlgebra/PolynomialMap.lean | DividedPowerAlgebra.gamma_mem_grade | [53, 1] | [87, 10] | simp only [dp_add, _root_.map_sum] | case add
R : Type u
inst✝⁴ : CommRing R
M : Type u_1
inst✝³ : AddCommGroup M
inst✝² : Module R M
S : Type u_2
inst✝¹ : CommRing S
inst✝ : Algebra R S
x y : S ⊗[R] M
hx : ∀ (n : ℕ), (gamma R M n).toFun S x ∈ LinearMap.range (LinearMap.lTensor S (grade R M n).subtype)
hy : ∀ (n : ℕ), (gamma R M n).toFun S y ∈ LinearMap.r... | case add
R : Type u
inst✝⁴ : CommRing R
M : Type u_1
inst✝³ : AddCommGroup M
inst✝² : Module R M
S : Type u_2
inst✝¹ : CommRing S
inst✝ : Algebra R S
x y : S ⊗[R] M
hx : ∀ (n : ℕ), (gamma R M n).toFun S x ∈ LinearMap.range (LinearMap.lTensor S (grade R M n).subtype)
hy : ∀ (n : ℕ), (gamma R M n).toFun S y ∈ LinearMap.r... | Please generate a tactic in lean4 to solve the state.
STATE:
case add
R : Type u
inst✝⁴ : CommRing R
M : Type u_1
inst✝³ : AddCommGroup M
inst✝² : Module R M
S : Type u_2
inst✝¹ : CommRing S
inst✝ : Algebra R S
x y : S ⊗[R] M
hx : ∀ (n : ℕ), (gamma R M n).toFun S x ∈ LinearMap.range (LinearMap.lTensor S (grade R M n).s... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/DPAlgebra/PolynomialMap.lean | DividedPowerAlgebra.gamma_mem_grade | [53, 1] | [87, 10] | apply Submodule.sum_mem | case add
R : Type u
inst✝⁴ : CommRing R
M : Type u_1
inst✝³ : AddCommGroup M
inst✝² : Module R M
S : Type u_2
inst✝¹ : CommRing S
inst✝ : Algebra R S
x y : S ⊗[R] M
hx : ∀ (n : ℕ), (gamma R M n).toFun S x ∈ LinearMap.range (LinearMap.lTensor S (grade R M n).subtype)
hy : ∀ (n : ℕ), (gamma R M n).toFun S y ∈ LinearMap.r... | case add.a
R : Type u
inst✝⁴ : CommRing R
M : Type u_1
inst✝³ : AddCommGroup M
inst✝² : Module R M
S : Type u_2
inst✝¹ : CommRing S
inst✝ : Algebra R S
x y : S ⊗[R] M
hx : ∀ (n : ℕ), (gamma R M n).toFun S x ∈ LinearMap.range (LinearMap.lTensor S (grade R M n).subtype)
hy : ∀ (n : ℕ), (gamma R M n).toFun S y ∈ LinearMap... | Please generate a tactic in lean4 to solve the state.
STATE:
case add
R : Type u
inst✝⁴ : CommRing R
M : Type u_1
inst✝³ : AddCommGroup M
inst✝² : Module R M
S : Type u_2
inst✝¹ : CommRing S
inst✝ : Algebra R S
x y : S ⊗[R] M
hx : ∀ (n : ℕ), (gamma R M n).toFun S x ∈ LinearMap.range (LinearMap.lTensor S (grade R M n).s... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/DPAlgebra/PolynomialMap.lean | DividedPowerAlgebra.gamma_mem_grade | [53, 1] | [87, 10] | rintro ⟨k, l⟩ hkl | case add.a
R : Type u
inst✝⁴ : CommRing R
M : Type u_1
inst✝³ : AddCommGroup M
inst✝² : Module R M
S : Type u_2
inst✝¹ : CommRing S
inst✝ : Algebra R S
x y : S ⊗[R] M
hx : ∀ (n : ℕ), (gamma R M n).toFun S x ∈ LinearMap.range (LinearMap.lTensor S (grade R M n).subtype)
hy : ∀ (n : ℕ), (gamma R M n).toFun S y ∈ LinearMap... | case add.a.mk
R : Type u
inst✝⁴ : CommRing R
M : Type u_1
inst✝³ : AddCommGroup M
inst✝² : Module R M
S : Type u_2
inst✝¹ : CommRing S
inst✝ : Algebra R S
x y : S ⊗[R] M
hx : ∀ (n : ℕ), (gamma R M n).toFun S x ∈ LinearMap.range (LinearMap.lTensor S (grade R M n).subtype)
hy : ∀ (n : ℕ), (gamma R M n).toFun S y ∈ Linear... | Please generate a tactic in lean4 to solve the state.
STATE:
case add.a
R : Type u
inst✝⁴ : CommRing R
M : Type u_1
inst✝³ : AddCommGroup M
inst✝² : Module R M
S : Type u_2
inst✝¹ : CommRing S
inst✝ : Algebra R S
x y : S ⊗[R] M
hx : ∀ (n : ℕ), (gamma R M n).toFun S x ∈ LinearMap.range (LinearMap.lTensor S (grade R M n)... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/DPAlgebra/PolynomialMap.lean | DividedPowerAlgebra.gamma_mem_grade | [53, 1] | [87, 10] | simp only [_root_.map_mul] | case add.a.mk
R : Type u
inst✝⁴ : CommRing R
M : Type u_1
inst✝³ : AddCommGroup M
inst✝² : Module R M
S : Type u_2
inst✝¹ : CommRing S
inst✝ : Algebra R S
x y : S ⊗[R] M
hx : ∀ (n : ℕ), (gamma R M n).toFun S x ∈ LinearMap.range (LinearMap.lTensor S (grade R M n).subtype)
hy : ∀ (n : ℕ), (gamma R M n).toFun S y ∈ Linear... | case add.a.mk
R : Type u
inst✝⁴ : CommRing R
M : Type u_1
inst✝³ : AddCommGroup M
inst✝² : Module R M
S : Type u_2
inst✝¹ : CommRing S
inst✝ : Algebra R S
x y : S ⊗[R] M
hx : ∀ (n : ℕ), (gamma R M n).toFun S x ∈ LinearMap.range (LinearMap.lTensor S (grade R M n).subtype)
hy : ∀ (n : ℕ), (gamma R M n).toFun S y ∈ Linear... | Please generate a tactic in lean4 to solve the state.
STATE:
case add.a.mk
R : Type u
inst✝⁴ : CommRing R
M : Type u_1
inst✝³ : AddCommGroup M
inst✝² : Module R M
S : Type u_2
inst✝¹ : CommRing S
inst✝ : Algebra R S
x y : S ⊗[R] M
hx : ∀ (n : ℕ), (gamma R M n).toFun S x ∈ LinearMap.range (LinearMap.lTensor S (grade R M... |
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