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.coeff_add_pow | [64, 1] | [106, 19] | intro hed | case neg.h.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
d : Fin 2 →₀ ℕ
n : ℕ
hmon :
∀ (u v : ℕ),
MvPolynomial.X 0 ^ u * MvPolynomial.X 1 ^ v = (MvPolynomial.monomial (Finsupp.single 0 u + Finsupp.single 1 v)) 1
hd :... | case neg.h.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
d : Fin 2 →₀ ℕ
n : ℕ
hmon :
∀ (u v : ℕ),
MvPolynomial.X 0 ^ u * MvPolynomial.X 1 ^ v = (MvPolynomial.monomial (Finsupp.single 0 u + Finsupp.single 1 v)) 1
hd :... | Please generate a tactic in lean4 to solve the state.
STATE:
case neg.h.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
d : Fin 2 →₀ ℕ
n : ℕ
hmon :
∀ (u v : ℕ),
MvPolynomial.X 0 ^ u * MvPolynomial.X 1 ^ v = (MvPolynomi... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/ExponentialModule/Basic.lean | PowerSeries.coeff_add_pow | [64, 1] | [106, 19] | apply hd | case neg.h.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
d : Fin 2 →₀ ℕ
n : ℕ
hmon :
∀ (u v : ℕ),
MvPolynomial.X 0 ^ u * MvPolynomial.X 1 ^ v = (MvPolynomial.monomial (Finsupp.single 0 u + Finsupp.single 1 v)) 1
hd :... | case neg.h.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
d : Fin 2 →₀ ℕ
n : ℕ
hmon :
∀ (u v : ℕ),
MvPolynomial.X 0 ^ u * MvPolynomial.X 1 ^ v = (MvPolynomial.monomial (Finsupp.single 0 u + Finsupp.single 1 v)) 1
hd :... | Please generate a tactic in lean4 to solve the state.
STATE:
case neg.h.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
d : Fin 2 →₀ ℕ
n : ℕ
hmon :
∀ (u v : ℕ),
MvPolynomial.X 0 ^ u * MvPolynomial.X 1 ^ v = (MvPolynomi... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/ExponentialModule/Basic.lean | PowerSeries.coeff_add_pow | [64, 1] | [106, 19] | rw [← hed, Finset.mem_antidiagonal] | case neg.h.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
d : Fin 2 →₀ ℕ
n : ℕ
hmon :
∀ (u v : ℕ),
MvPolynomial.X 0 ^ u * MvPolynomial.X 1 ^ v = (MvPolynomial.monomial (Finsupp.single 0 u + Finsupp.single 1 v)) 1
hd :... | case neg.h.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
d : Fin 2 →₀ ℕ
n : ℕ
hmon :
∀ (u v : ℕ),
MvPolynomial.X 0 ^ u * MvPolynomial.X 1 ^ v = (MvPolynomial.monomial (Finsupp.single 0 u + Finsupp.single 1 v)) 1
hd :... | Please generate a tactic in lean4 to solve the state.
STATE:
case neg.h.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
d : Fin 2 →₀ ℕ
n : ℕ
hmon :
∀ (u v : ℕ),
MvPolynomial.X 0 ^ u * MvPolynomial.X 1 ^ v = (MvPolynomi... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/ExponentialModule/Basic.lean | PowerSeries.coeff_add_pow | [64, 1] | [106, 19] | simpa using he | case neg.h.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
d : Fin 2 →₀ ℕ
n : ℕ
hmon :
∀ (u v : ℕ),
MvPolynomial.X 0 ^ u * MvPolynomial.X 1 ^ v = (MvPolynomial.monomial (Finsupp.single 0 u + Finsupp.single 1 v)) 1
hd :... | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case neg.h.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
d : Fin 2 →₀ ℕ
n : ℕ
hmon :
∀ (u v : ℕ),
MvPolynomial.X 0 ^ u * MvPolynomial.X 1 ^ v = (MvPolynomi... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/ExponentialModule/Basic.lean | PowerSeries.coeff_subst_single | [108, 1] | [124, 76] | rw [PowerSeries.coeff_subst (PowerSeries.substDomain_of_constantCoeff_zero (by simp))] | A : Type u_1
inst✝⁶ : CommRing A
R : Type u_2
inst✝⁵ : CommRing R
inst✝⁴ : Algebra A R
S : Type u_3
inst✝³ : CommRing S
inst✝² : Algebra A S
σ : Type u_4
inst✝¹ : DecidableEq σ
inst✝ : Finite σ
s : σ
f : R⟦X⟧
e : σ →₀ ℕ
⊢ (MvPowerSeries.coeff R e) (subst (MvPowerSeries.X s) f) = if e = Finsupp.single s (e s) then (coef... | A : Type u_1
inst✝⁶ : CommRing A
R : Type u_2
inst✝⁵ : CommRing R
inst✝⁴ : Algebra A R
S : Type u_3
inst✝³ : CommRing S
inst✝² : Algebra A S
σ : Type u_4
inst✝¹ : DecidableEq σ
inst✝ : Finite σ
s : σ
f : R⟦X⟧
e : σ →₀ ℕ
⊢ ∑ᶠ (d : ℕ), (coeff R d) f • (MvPowerSeries.coeff R e) (MvPowerSeries.X s ^ d) =
if e = Finsupp... | Please generate a tactic in lean4 to solve the state.
STATE:
A : Type u_1
inst✝⁶ : CommRing A
R : Type u_2
inst✝⁵ : CommRing R
inst✝⁴ : Algebra A R
S : Type u_3
inst✝³ : CommRing S
inst✝² : Algebra A S
σ : Type u_4
inst✝¹ : DecidableEq σ
inst✝ : Finite σ
s : σ
f : R⟦X⟧
e : σ →₀ ℕ
⊢ (MvPowerSeries.coeff R e) (subst (MvP... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/ExponentialModule/Basic.lean | PowerSeries.coeff_subst_single | [108, 1] | [124, 76] | rw [finsum_eq_single _ (e s)] | A : Type u_1
inst✝⁶ : CommRing A
R : Type u_2
inst✝⁵ : CommRing R
inst✝⁴ : Algebra A R
S : Type u_3
inst✝³ : CommRing S
inst✝² : Algebra A S
σ : Type u_4
inst✝¹ : DecidableEq σ
inst✝ : Finite σ
s : σ
f : R⟦X⟧
e : σ →₀ ℕ
⊢ ∑ᶠ (d : ℕ), (coeff R d) f • (MvPowerSeries.coeff R e) (MvPowerSeries.X s ^ d) =
if e = Finsupp... | A : Type u_1
inst✝⁶ : CommRing A
R : Type u_2
inst✝⁵ : CommRing R
inst✝⁴ : Algebra A R
S : Type u_3
inst✝³ : CommRing S
inst✝² : Algebra A S
σ : Type u_4
inst✝¹ : DecidableEq σ
inst✝ : Finite σ
s : σ
f : R⟦X⟧
e : σ →₀ ℕ
⊢ (coeff R (e s)) f • (MvPowerSeries.coeff R e) (MvPowerSeries.X s ^ e s) =
if e = Finsupp.singl... | Please generate a tactic in lean4 to solve the state.
STATE:
A : Type u_1
inst✝⁶ : CommRing A
R : Type u_2
inst✝⁵ : CommRing R
inst✝⁴ : Algebra A R
S : Type u_3
inst✝³ : CommRing S
inst✝² : Algebra A S
σ : Type u_4
inst✝¹ : DecidableEq σ
inst✝ : Finite σ
s : σ
f : R⟦X⟧
e : σ →₀ ℕ
⊢ ∑ᶠ (d : ℕ), (coeff R d) f • (MvPowerS... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/ExponentialModule/Basic.lean | PowerSeries.coeff_subst_single | [108, 1] | [124, 76] | simp | A : Type u_1
inst✝⁶ : CommRing A
R : Type u_2
inst✝⁵ : CommRing R
inst✝⁴ : Algebra A R
S : Type u_3
inst✝³ : CommRing S
inst✝² : Algebra A S
σ : Type u_4
inst✝¹ : DecidableEq σ
inst✝ : Finite σ
s : σ
f : R⟦X⟧
e : σ →₀ ℕ
⊢ (MvPowerSeries.constantCoeff σ R) (MvPowerSeries.X s) = 0 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
A : Type u_1
inst✝⁶ : CommRing A
R : Type u_2
inst✝⁵ : CommRing R
inst✝⁴ : Algebra A R
S : Type u_3
inst✝³ : CommRing S
inst✝² : Algebra A S
σ : Type u_4
inst✝¹ : DecidableEq σ
inst✝ : Finite σ
s : σ
f : R⟦X⟧
e : σ →₀ ℕ
⊢ (MvPowerSeries.constantCoeff σ R) (Mv... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/ExponentialModule/Basic.lean | PowerSeries.coeff_subst_single | [108, 1] | [124, 76] | rw [MvPowerSeries.coeff_X_pow] | A : Type u_1
inst✝⁶ : CommRing A
R : Type u_2
inst✝⁵ : CommRing R
inst✝⁴ : Algebra A R
S : Type u_3
inst✝³ : CommRing S
inst✝² : Algebra A S
σ : Type u_4
inst✝¹ : DecidableEq σ
inst✝ : Finite σ
s : σ
f : R⟦X⟧
e : σ →₀ ℕ
⊢ (coeff R (e s)) f • (MvPowerSeries.coeff R e) (MvPowerSeries.X s ^ e s) =
if e = Finsupp.singl... | A : Type u_1
inst✝⁶ : CommRing A
R : Type u_2
inst✝⁵ : CommRing R
inst✝⁴ : Algebra A R
S : Type u_3
inst✝³ : CommRing S
inst✝² : Algebra A S
σ : Type u_4
inst✝¹ : DecidableEq σ
inst✝ : Finite σ
s : σ
f : R⟦X⟧
e : σ →₀ ℕ
⊢ ((coeff R (e s)) f • if e = Finsupp.single s (e s) then 1 else 0) =
if e = Finsupp.single s (e... | Please generate a tactic in lean4 to solve the state.
STATE:
A : Type u_1
inst✝⁶ : CommRing A
R : Type u_2
inst✝⁵ : CommRing R
inst✝⁴ : Algebra A R
S : Type u_3
inst✝³ : CommRing S
inst✝² : Algebra A S
σ : Type u_4
inst✝¹ : DecidableEq σ
inst✝ : Finite σ
s : σ
f : R⟦X⟧
e : σ →₀ ℕ
⊢ (coeff R (e s)) f • (MvPowerSeries.co... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/ExponentialModule/Basic.lean | PowerSeries.coeff_subst_single | [108, 1] | [124, 76] | simp only [Fin.isValue, ↓reduceIte, smul_eq_mul, mul_one] | A : Type u_1
inst✝⁶ : CommRing A
R : Type u_2
inst✝⁵ : CommRing R
inst✝⁴ : Algebra A R
S : Type u_3
inst✝³ : CommRing S
inst✝² : Algebra A S
σ : Type u_4
inst✝¹ : DecidableEq σ
inst✝ : Finite σ
s : σ
f : R⟦X⟧
e : σ →₀ ℕ
⊢ ((coeff R (e s)) f • if e = Finsupp.single s (e s) then 1 else 0) =
if e = Finsupp.single s (e... | A : Type u_1
inst✝⁶ : CommRing A
R : Type u_2
inst✝⁵ : CommRing R
inst✝⁴ : Algebra A R
S : Type u_3
inst✝³ : CommRing S
inst✝² : Algebra A S
σ : Type u_4
inst✝¹ : DecidableEq σ
inst✝ : Finite σ
s : σ
f : R⟦X⟧
e : σ →₀ ℕ
⊢ ((coeff R (e s)) f * if e = Finsupp.single s (e s) then 1 else 0) =
if e = Finsupp.single s (e... | Please generate a tactic in lean4 to solve the state.
STATE:
A : Type u_1
inst✝⁶ : CommRing A
R : Type u_2
inst✝⁵ : CommRing R
inst✝⁴ : Algebra A R
S : Type u_3
inst✝³ : CommRing S
inst✝² : Algebra A S
σ : Type u_4
inst✝¹ : DecidableEq σ
inst✝ : Finite σ
s : σ
f : R⟦X⟧
e : σ →₀ ℕ
⊢ ((coeff R (e s)) f • if e = Finsupp.s... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/ExponentialModule/Basic.lean | PowerSeries.coeff_subst_single | [108, 1] | [124, 76] | split_ifs with he | A : Type u_1
inst✝⁶ : CommRing A
R : Type u_2
inst✝⁵ : CommRing R
inst✝⁴ : Algebra A R
S : Type u_3
inst✝³ : CommRing S
inst✝² : Algebra A S
σ : Type u_4
inst✝¹ : DecidableEq σ
inst✝ : Finite σ
s : σ
f : R⟦X⟧
e : σ →₀ ℕ
⊢ ((coeff R (e s)) f * if e = Finsupp.single s (e s) then 1 else 0) =
if e = Finsupp.single s (e... | case pos
A : Type u_1
inst✝⁶ : CommRing A
R : Type u_2
inst✝⁵ : CommRing R
inst✝⁴ : Algebra A R
S : Type u_3
inst✝³ : CommRing S
inst✝² : Algebra A S
σ : Type u_4
inst✝¹ : DecidableEq σ
inst✝ : Finite σ
s : σ
f : R⟦X⟧
e : σ →₀ ℕ
he : e = Finsupp.single s (e s)
⊢ (coeff R (e s)) f * 1 = (coeff R (e s)) f
case neg
A : T... | Please generate a tactic in lean4 to solve the state.
STATE:
A : Type u_1
inst✝⁶ : CommRing A
R : Type u_2
inst✝⁵ : CommRing R
inst✝⁴ : Algebra A R
S : Type u_3
inst✝³ : CommRing S
inst✝² : Algebra A S
σ : Type u_4
inst✝¹ : DecidableEq σ
inst✝ : Finite σ
s : σ
f : R⟦X⟧
e : σ →₀ ℕ
⊢ ((coeff R (e s)) f * if e = Finsupp.s... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/ExponentialModule/Basic.lean | PowerSeries.coeff_subst_single | [108, 1] | [124, 76] | rw [mul_one] | case pos
A : Type u_1
inst✝⁶ : CommRing A
R : Type u_2
inst✝⁵ : CommRing R
inst✝⁴ : Algebra A R
S : Type u_3
inst✝³ : CommRing S
inst✝² : Algebra A S
σ : Type u_4
inst✝¹ : DecidableEq σ
inst✝ : Finite σ
s : σ
f : R⟦X⟧
e : σ →₀ ℕ
he : e = Finsupp.single s (e s)
⊢ (coeff R (e s)) f * 1 = (coeff R (e s)) f | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case pos
A : Type u_1
inst✝⁶ : CommRing A
R : Type u_2
inst✝⁵ : CommRing R
inst✝⁴ : Algebra A R
S : Type u_3
inst✝³ : CommRing S
inst✝² : Algebra A S
σ : Type u_4
inst✝¹ : DecidableEq σ
inst✝ : Finite σ
s : σ
f : R⟦X⟧
e : σ →₀ ℕ
he : e = Finsupp.single s (e s... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/ExponentialModule/Basic.lean | PowerSeries.coeff_subst_single | [108, 1] | [124, 76] | rw [mul_zero] | case neg
A : Type u_1
inst✝⁶ : CommRing A
R : Type u_2
inst✝⁵ : CommRing R
inst✝⁴ : Algebra A R
S : Type u_3
inst✝³ : CommRing S
inst✝² : Algebra A S
σ : Type u_4
inst✝¹ : DecidableEq σ
inst✝ : Finite σ
s : σ
f : R⟦X⟧
e : σ →₀ ℕ
he : ¬e = Finsupp.single s (e s)
⊢ (coeff R (e s)) f * 0 = 0 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case neg
A : Type u_1
inst✝⁶ : CommRing A
R : Type u_2
inst✝⁵ : CommRing R
inst✝⁴ : Algebra A R
S : Type u_3
inst✝³ : CommRing S
inst✝² : Algebra A S
σ : Type u_4
inst✝¹ : DecidableEq σ
inst✝ : Finite σ
s : σ
f : R⟦X⟧
e : σ →₀ ℕ
he : ¬e = Finsupp.single s (e ... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/ExponentialModule/Basic.lean | PowerSeries.coeff_subst_single | [108, 1] | [124, 76] | intro d hd | A : Type u_1
inst✝⁶ : CommRing A
R : Type u_2
inst✝⁵ : CommRing R
inst✝⁴ : Algebra A R
S : Type u_3
inst✝³ : CommRing S
inst✝² : Algebra A S
σ : Type u_4
inst✝¹ : DecidableEq σ
inst✝ : Finite σ
s : σ
f : R⟦X⟧
e : σ →₀ ℕ
⊢ ∀ (x : ℕ), x ≠ e s → (coeff R x) f • (MvPowerSeries.coeff R e) (MvPowerSeries.X s ^ x) = 0 | A : Type u_1
inst✝⁶ : CommRing A
R : Type u_2
inst✝⁵ : CommRing R
inst✝⁴ : Algebra A R
S : Type u_3
inst✝³ : CommRing S
inst✝² : Algebra A S
σ : Type u_4
inst✝¹ : DecidableEq σ
inst✝ : Finite σ
s : σ
f : R⟦X⟧
e : σ →₀ ℕ
d : ℕ
hd : d ≠ e s
⊢ (coeff R d) f • (MvPowerSeries.coeff R e) (MvPowerSeries.X s ^ d) = 0 | Please generate a tactic in lean4 to solve the state.
STATE:
A : Type u_1
inst✝⁶ : CommRing A
R : Type u_2
inst✝⁵ : CommRing R
inst✝⁴ : Algebra A R
S : Type u_3
inst✝³ : CommRing S
inst✝² : Algebra A S
σ : Type u_4
inst✝¹ : DecidableEq σ
inst✝ : Finite σ
s : σ
f : R⟦X⟧
e : σ →₀ ℕ
⊢ ∀ (x : ℕ), x ≠ e s → (coeff R x) f • ... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/ExponentialModule/Basic.lean | PowerSeries.coeff_subst_single | [108, 1] | [124, 76] | simp only [MvPowerSeries.coeff_X_pow, smul_eq_mul, mul_ite, mul_one, mul_zero, ite_eq_right_iff] | A : Type u_1
inst✝⁶ : CommRing A
R : Type u_2
inst✝⁵ : CommRing R
inst✝⁴ : Algebra A R
S : Type u_3
inst✝³ : CommRing S
inst✝² : Algebra A S
σ : Type u_4
inst✝¹ : DecidableEq σ
inst✝ : Finite σ
s : σ
f : R⟦X⟧
e : σ →₀ ℕ
d : ℕ
hd : d ≠ e s
⊢ (coeff R d) f • (MvPowerSeries.coeff R e) (MvPowerSeries.X s ^ d) = 0 | A : Type u_1
inst✝⁶ : CommRing A
R : Type u_2
inst✝⁵ : CommRing R
inst✝⁴ : Algebra A R
S : Type u_3
inst✝³ : CommRing S
inst✝² : Algebra A S
σ : Type u_4
inst✝¹ : DecidableEq σ
inst✝ : Finite σ
s : σ
f : R⟦X⟧
e : σ →₀ ℕ
d : ℕ
hd : d ≠ e s
⊢ e = Finsupp.single s d → (coeff R d) f = 0 | Please generate a tactic in lean4 to solve the state.
STATE:
A : Type u_1
inst✝⁶ : CommRing A
R : Type u_2
inst✝⁵ : CommRing R
inst✝⁴ : Algebra A R
S : Type u_3
inst✝³ : CommRing S
inst✝² : Algebra A S
σ : Type u_4
inst✝¹ : DecidableEq σ
inst✝ : Finite σ
s : σ
f : R⟦X⟧
e : σ →₀ ℕ
d : ℕ
hd : d ≠ e s
⊢ (coeff R d) f • (M... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/ExponentialModule/Basic.lean | PowerSeries.coeff_subst_single | [108, 1] | [124, 76] | intro hd' | A : Type u_1
inst✝⁶ : CommRing A
R : Type u_2
inst✝⁵ : CommRing R
inst✝⁴ : Algebra A R
S : Type u_3
inst✝³ : CommRing S
inst✝² : Algebra A S
σ : Type u_4
inst✝¹ : DecidableEq σ
inst✝ : Finite σ
s : σ
f : R⟦X⟧
e : σ →₀ ℕ
d : ℕ
hd : d ≠ e s
⊢ e = Finsupp.single s d → (coeff R d) f = 0 | A : Type u_1
inst✝⁶ : CommRing A
R : Type u_2
inst✝⁵ : CommRing R
inst✝⁴ : Algebra A R
S : Type u_3
inst✝³ : CommRing S
inst✝² : Algebra A S
σ : Type u_4
inst✝¹ : DecidableEq σ
inst✝ : Finite σ
s : σ
f : R⟦X⟧
e : σ →₀ ℕ
d : ℕ
hd : d ≠ e s
hd' : e = Finsupp.single s d
⊢ (coeff R d) f = 0 | Please generate a tactic in lean4 to solve the state.
STATE:
A : Type u_1
inst✝⁶ : CommRing A
R : Type u_2
inst✝⁵ : CommRing R
inst✝⁴ : Algebra A R
S : Type u_3
inst✝³ : CommRing S
inst✝² : Algebra A S
σ : Type u_4
inst✝¹ : DecidableEq σ
inst✝ : Finite σ
s : σ
f : R⟦X⟧
e : σ →₀ ℕ
d : ℕ
hd : d ≠ e s
⊢ e = Finsupp.single... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/ExponentialModule/Basic.lean | PowerSeries.coeff_subst_single | [108, 1] | [124, 76] | simp only [hd', Finsupp.single_eq_same, ne_eq, not_true_eq_false] at hd | A : Type u_1
inst✝⁶ : CommRing A
R : Type u_2
inst✝⁵ : CommRing R
inst✝⁴ : Algebra A R
S : Type u_3
inst✝³ : CommRing S
inst✝² : Algebra A S
σ : Type u_4
inst✝¹ : DecidableEq σ
inst✝ : Finite σ
s : σ
f : R⟦X⟧
e : σ →₀ ℕ
d : ℕ
hd : d ≠ e s
hd' : e = Finsupp.single s d
⊢ (coeff R d) f = 0 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
A : Type u_1
inst✝⁶ : CommRing A
R : Type u_2
inst✝⁵ : CommRing R
inst✝⁴ : Algebra A R
S : Type u_3
inst✝³ : CommRing S
inst✝² : Algebra A S
σ : Type u_4
inst✝¹ : DecidableEq σ
inst✝ : Finite σ
s : σ
f : R⟦X⟧
e : σ →₀ ℕ
d : ℕ
hd : d ≠ e s
hd' : e = Finsupp.si... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/ExponentialModule/Basic.lean | PowerSeries.ne_zero_of_mul_ne_zero | [126, 1] | [134, 22] | constructor | A : Type u_1
inst✝⁵ : CommRing A
R : Type u_2
inst✝⁴ : CommRing R
inst✝³ : Algebra A R
S : Type u_3
inst✝² : CommRing S
inst✝¹ : Algebra A S
M : Type u_4
inst✝ : MonoidWithZero M
a b : M
h : a * b ≠ 0
⊢ a ≠ 0 ∧ b ≠ 0 | case left
A : Type u_1
inst✝⁵ : CommRing A
R : Type u_2
inst✝⁴ : CommRing R
inst✝³ : Algebra A R
S : Type u_3
inst✝² : CommRing S
inst✝¹ : Algebra A S
M : Type u_4
inst✝ : MonoidWithZero M
a b : M
h : a * b ≠ 0
⊢ a ≠ 0
case right
A : Type u_1
inst✝⁵ : CommRing A
R : Type u_2
inst✝⁴ : CommRing R
inst✝³ : Algebra A R
S ... | Please generate a tactic in lean4 to solve the state.
STATE:
A : Type u_1
inst✝⁵ : CommRing A
R : Type u_2
inst✝⁴ : CommRing R
inst✝³ : Algebra A R
S : Type u_3
inst✝² : CommRing S
inst✝¹ : Algebra A S
M : Type u_4
inst✝ : MonoidWithZero M
a b : M
h : a * b ≠ 0
⊢ a ≠ 0 ∧ b ≠ 0
TACTIC:
|
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/ExponentialModule/Basic.lean | PowerSeries.ne_zero_of_mul_ne_zero | [126, 1] | [134, 22] | intro ha | case left
A : Type u_1
inst✝⁵ : CommRing A
R : Type u_2
inst✝⁴ : CommRing R
inst✝³ : Algebra A R
S : Type u_3
inst✝² : CommRing S
inst✝¹ : Algebra A S
M : Type u_4
inst✝ : MonoidWithZero M
a b : M
h : a * b ≠ 0
⊢ a ≠ 0 | case left
A : Type u_1
inst✝⁵ : CommRing A
R : Type u_2
inst✝⁴ : CommRing R
inst✝³ : Algebra A R
S : Type u_3
inst✝² : CommRing S
inst✝¹ : Algebra A S
M : Type u_4
inst✝ : MonoidWithZero M
a b : M
h : a * b ≠ 0
ha : a = 0
⊢ False | Please generate a tactic in lean4 to solve the state.
STATE:
case left
A : Type u_1
inst✝⁵ : CommRing A
R : Type u_2
inst✝⁴ : CommRing R
inst✝³ : Algebra A R
S : Type u_3
inst✝² : CommRing S
inst✝¹ : Algebra A S
M : Type u_4
inst✝ : MonoidWithZero M
a b : M
h : a * b ≠ 0
⊢ a ≠ 0
TACTIC:
|
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/ExponentialModule/Basic.lean | PowerSeries.ne_zero_of_mul_ne_zero | [126, 1] | [134, 22] | apply h | case left
A : Type u_1
inst✝⁵ : CommRing A
R : Type u_2
inst✝⁴ : CommRing R
inst✝³ : Algebra A R
S : Type u_3
inst✝² : CommRing S
inst✝¹ : Algebra A S
M : Type u_4
inst✝ : MonoidWithZero M
a b : M
h : a * b ≠ 0
ha : a = 0
⊢ False | case left
A : Type u_1
inst✝⁵ : CommRing A
R : Type u_2
inst✝⁴ : CommRing R
inst✝³ : Algebra A R
S : Type u_3
inst✝² : CommRing S
inst✝¹ : Algebra A S
M : Type u_4
inst✝ : MonoidWithZero M
a b : M
h : a * b ≠ 0
ha : a = 0
⊢ a * b = 0 | Please generate a tactic in lean4 to solve the state.
STATE:
case left
A : Type u_1
inst✝⁵ : CommRing A
R : Type u_2
inst✝⁴ : CommRing R
inst✝³ : Algebra A R
S : Type u_3
inst✝² : CommRing S
inst✝¹ : Algebra A S
M : Type u_4
inst✝ : MonoidWithZero M
a b : M
h : a * b ≠ 0
ha : a = 0
⊢ False
TACTIC:
|
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/ExponentialModule/Basic.lean | PowerSeries.ne_zero_of_mul_ne_zero | [126, 1] | [134, 22] | rw [ha, zero_mul] | case left
A : Type u_1
inst✝⁵ : CommRing A
R : Type u_2
inst✝⁴ : CommRing R
inst✝³ : Algebra A R
S : Type u_3
inst✝² : CommRing S
inst✝¹ : Algebra A S
M : Type u_4
inst✝ : MonoidWithZero M
a b : M
h : a * b ≠ 0
ha : a = 0
⊢ a * b = 0 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case left
A : Type u_1
inst✝⁵ : CommRing A
R : Type u_2
inst✝⁴ : CommRing R
inst✝³ : Algebra A R
S : Type u_3
inst✝² : CommRing S
inst✝¹ : Algebra A S
M : Type u_4
inst✝ : MonoidWithZero M
a b : M
h : a * b ≠ 0
ha : a = 0
⊢ a * b = 0
TACTIC:
|
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/ExponentialModule/Basic.lean | PowerSeries.ne_zero_of_mul_ne_zero | [126, 1] | [134, 22] | intro hb | case right
A : Type u_1
inst✝⁵ : CommRing A
R : Type u_2
inst✝⁴ : CommRing R
inst✝³ : Algebra A R
S : Type u_3
inst✝² : CommRing S
inst✝¹ : Algebra A S
M : Type u_4
inst✝ : MonoidWithZero M
a b : M
h : a * b ≠ 0
⊢ b ≠ 0 | case right
A : Type u_1
inst✝⁵ : CommRing A
R : Type u_2
inst✝⁴ : CommRing R
inst✝³ : Algebra A R
S : Type u_3
inst✝² : CommRing S
inst✝¹ : Algebra A S
M : Type u_4
inst✝ : MonoidWithZero M
a b : M
h : a * b ≠ 0
hb : b = 0
⊢ False | Please generate a tactic in lean4 to solve the state.
STATE:
case right
A : Type u_1
inst✝⁵ : CommRing A
R : Type u_2
inst✝⁴ : CommRing R
inst✝³ : Algebra A R
S : Type u_3
inst✝² : CommRing S
inst✝¹ : Algebra A S
M : Type u_4
inst✝ : MonoidWithZero M
a b : M
h : a * b ≠ 0
⊢ b ≠ 0
TACTIC:
|
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/ExponentialModule/Basic.lean | PowerSeries.ne_zero_of_mul_ne_zero | [126, 1] | [134, 22] | apply h | case right
A : Type u_1
inst✝⁵ : CommRing A
R : Type u_2
inst✝⁴ : CommRing R
inst✝³ : Algebra A R
S : Type u_3
inst✝² : CommRing S
inst✝¹ : Algebra A S
M : Type u_4
inst✝ : MonoidWithZero M
a b : M
h : a * b ≠ 0
hb : b = 0
⊢ False | case right
A : Type u_1
inst✝⁵ : CommRing A
R : Type u_2
inst✝⁴ : CommRing R
inst✝³ : Algebra A R
S : Type u_3
inst✝² : CommRing S
inst✝¹ : Algebra A S
M : Type u_4
inst✝ : MonoidWithZero M
a b : M
h : a * b ≠ 0
hb : b = 0
⊢ a * b = 0 | Please generate a tactic in lean4 to solve the state.
STATE:
case right
A : Type u_1
inst✝⁵ : CommRing A
R : Type u_2
inst✝⁴ : CommRing R
inst✝³ : Algebra A R
S : Type u_3
inst✝² : CommRing S
inst✝¹ : Algebra A S
M : Type u_4
inst✝ : MonoidWithZero M
a b : M
h : a * b ≠ 0
hb : b = 0
⊢ False
TACTIC:
|
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/ExponentialModule/Basic.lean | PowerSeries.ne_zero_of_mul_ne_zero | [126, 1] | [134, 22] | rw [hb, mul_zero] | case right
A : Type u_1
inst✝⁵ : CommRing A
R : Type u_2
inst✝⁴ : CommRing R
inst✝³ : Algebra A R
S : Type u_3
inst✝² : CommRing S
inst✝¹ : Algebra A S
M : Type u_4
inst✝ : MonoidWithZero M
a b : M
h : a * b ≠ 0
hb : b = 0
⊢ a * b = 0 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case right
A : Type u_1
inst✝⁵ : CommRing A
R : Type u_2
inst✝⁴ : CommRing R
inst✝³ : Algebra A R
S : Type u_3
inst✝² : CommRing S
inst✝¹ : Algebra A S
M : Type u_4
inst✝ : MonoidWithZero M
a b : M
h : a * b ≠ 0
hb : b = 0
⊢ a * b = 0
TACTIC:
|
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/ExponentialModule/Basic.lean | PowerSeries.forall_congr_curry | [151, 1] | [166, 12] | 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
α : Type u_4
p : (Fin 2 → α) → Prop
q : α → α → Prop
hpq : ∀ (e : Fin 2 → α), p e ↔ q (e 0) (e 1)
⊢ (∀ (e : Fin 2 → α), p e) ↔ ∀ (u v : α), q u v | case mp
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
α : Type u_4
p : (Fin 2 → α) → Prop
q : α → α → Prop
hpq : ∀ (e : Fin 2 → α), p e ↔ q (e 0) (e 1)
⊢ (∀ (e : Fin 2 → α), p e) → ∀ (u v : α), q u v
case mpr
A : Type u_1
ins... | 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
α : Type u_4
p : (Fin 2 → α) → Prop
q : α → α → Prop
hpq : ∀ (e : Fin 2 → α), p e ↔ q (e 0) (e 1)
⊢ (∀ (e : Fin 2 → α),... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/ExponentialModule/Basic.lean | PowerSeries.forall_congr_curry | [151, 1] | [166, 12] | intro H u v | case mp
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
α : Type u_4
p : (Fin 2 → α) → Prop
q : α → α → Prop
hpq : ∀ (e : Fin 2 → α), p e ↔ q (e 0) (e 1)
⊢ (∀ (e : Fin 2 → α), p e) → ∀ (u v : α), q u v | case mp
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
α : Type u_4
p : (Fin 2 → α) → Prop
q : α → α → Prop
hpq : ∀ (e : Fin 2 → α), p e ↔ q (e 0) (e 1)
H : ∀ (e : Fin 2 → α), p e
u v : α
⊢ q u v | Please generate a tactic in lean4 to solve the state.
STATE:
case mp
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
α : Type u_4
p : (Fin 2 → α) → Prop
q : α → α → Prop
hpq : ∀ (e : Fin 2 → α), p e ↔ q (e 0) (e 1)
⊢ (∀ (e : Fin... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/ExponentialModule/Basic.lean | PowerSeries.forall_congr_curry | [151, 1] | [166, 12] | set e : Fin 2 → α := fun
| 0 => u
| 1 => v | case mp
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
α : Type u_4
p : (Fin 2 → α) → Prop
q : α → α → Prop
hpq : ∀ (e : Fin 2 → α), p e ↔ q (e 0) (e 1)
H : ∀ (e : Fin 2 → α), p e
u v : α
⊢ q u v | case mp
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
α : Type u_4
p : (Fin 2 → α) → Prop
q : α → α → Prop
hpq : ∀ (e : Fin 2 → α), p e ↔ q (e 0) (e 1)
H : ∀ (e : Fin 2 → α), p e
u v : α
e : Fin 2 → α :=
fun x =>
match x... | Please generate a tactic in lean4 to solve the state.
STATE:
case mp
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
α : Type u_4
p : (Fin 2 → α) → Prop
q : α → α → Prop
hpq : ∀ (e : Fin 2 → α), p e ↔ q (e 0) (e 1)
H : ∀ (e : Fi... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/ExponentialModule/Basic.lean | PowerSeries.forall_congr_curry | [151, 1] | [166, 12] | specialize hpq e | case mp
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
α : Type u_4
p : (Fin 2 → α) → Prop
q : α → α → Prop
hpq : ∀ (e : Fin 2 → α), p e ↔ q (e 0) (e 1)
H : ∀ (e : Fin 2 → α), p e
u v : α
e : Fin 2 → α :=
fun x =>
match x... | case mp
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
α : Type u_4
p : (Fin 2 → α) → Prop
q : α → α → Prop
H : ∀ (e : Fin 2 → α), p e
u v : α
e : Fin 2 → α :=
fun x =>
match x with
| 0 => u
| 1 => v
hpq : p e ↔ q... | Please generate a tactic in lean4 to solve the state.
STATE:
case mp
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
α : Type u_4
p : (Fin 2 → α) → Prop
q : α → α → Prop
hpq : ∀ (e : Fin 2 → α), p e ↔ q (e 0) (e 1)
H : ∀ (e : Fi... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/ExponentialModule/Basic.lean | PowerSeries.forall_congr_curry | [151, 1] | [166, 12] | simp [e] at hpq | case mp
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
α : Type u_4
p : (Fin 2 → α) → Prop
q : α → α → Prop
H : ∀ (e : Fin 2 → α), p e
u v : α
e : Fin 2 → α :=
fun x =>
match x with
| 0 => u
| 1 => v
hpq : p e ↔ q... | case mp
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
α : Type u_4
p : (Fin 2 → α) → Prop
q : α → α → Prop
H : ∀ (e : Fin 2 → α), p e
u v : α
e : Fin 2 → α :=
fun x =>
match x with
| 0 => u
| 1 => v
hpq :
(p fu... | Please generate a tactic in lean4 to solve the state.
STATE:
case mp
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
α : Type u_4
p : (Fin 2 → α) → Prop
q : α → α → Prop
H : ∀ (e : Fin 2 → α), p e
u v : α
e : Fin 2 → α :=
fun ... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/ExponentialModule/Basic.lean | PowerSeries.forall_congr_curry | [151, 1] | [166, 12] | rw [← hpq] | case mp
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
α : Type u_4
p : (Fin 2 → α) → Prop
q : α → α → Prop
H : ∀ (e : Fin 2 → α), p e
u v : α
e : Fin 2 → α :=
fun x =>
match x with
| 0 => u
| 1 => v
hpq :
(p fu... | case mp
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
α : Type u_4
p : (Fin 2 → α) → Prop
q : α → α → Prop
H : ∀ (e : Fin 2 → α), p e
u v : α
e : Fin 2 → α :=
fun x =>
match x with
| 0 => u
| 1 => v
hpq :
(p fu... | Please generate a tactic in lean4 to solve the state.
STATE:
case mp
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
α : Type u_4
p : (Fin 2 → α) → Prop
q : α → α → Prop
H : ∀ (e : Fin 2 → α), p e
u v : α
e : Fin 2 → α :=
fun ... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/ExponentialModule/Basic.lean | PowerSeries.forall_congr_curry | [151, 1] | [166, 12] | apply H | case mp
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
α : Type u_4
p : (Fin 2 → α) → Prop
q : α → α → Prop
H : ∀ (e : Fin 2 → α), p e
u v : α
e : Fin 2 → α :=
fun x =>
match x with
| 0 => u
| 1 => v
hpq :
(p fu... | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case mp
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
α : Type u_4
p : (Fin 2 → α) → Prop
q : α → α → Prop
H : ∀ (e : Fin 2 → α), p e
u v : α
e : Fin 2 → α :=
fun ... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/ExponentialModule/Basic.lean | PowerSeries.forall_congr_curry | [151, 1] | [166, 12] | intro H e | case mpr
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
α : Type u_4
p : (Fin 2 → α) → Prop
q : α → α → Prop
hpq : ∀ (e : Fin 2 → α), p e ↔ q (e 0) (e 1)
⊢ (∀ (u v : α), q u v) → ∀ (e : Fin 2 → α), p e | case mpr
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
α : Type u_4
p : (Fin 2 → α) → Prop
q : α → α → Prop
hpq : ∀ (e : Fin 2 → α), p e ↔ q (e 0) (e 1)
H : ∀ (u v : α), q u v
e : Fin 2 → α
⊢ p e | Please generate a tactic in lean4 to solve the state.
STATE:
case mpr
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
α : Type u_4
p : (Fin 2 → α) → Prop
q : α → α → Prop
hpq : ∀ (e : Fin 2 → α), p e ↔ q (e 0) (e 1)
⊢ (∀ (u v : ... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/ExponentialModule/Basic.lean | PowerSeries.forall_congr_curry | [151, 1] | [166, 12] | rw [hpq] | case mpr
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
α : Type u_4
p : (Fin 2 → α) → Prop
q : α → α → Prop
hpq : ∀ (e : Fin 2 → α), p e ↔ q (e 0) (e 1)
H : ∀ (u v : α), q u v
e : Fin 2 → α
⊢ p e | case mpr
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
α : Type u_4
p : (Fin 2 → α) → Prop
q : α → α → Prop
hpq : ∀ (e : Fin 2 → α), p e ↔ q (e 0) (e 1)
H : ∀ (u v : α), q u v
e : Fin 2 → α
⊢ q (e 0) (e 1) | Please generate a tactic in lean4 to solve the state.
STATE:
case mpr
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
α : Type u_4
p : (Fin 2 → α) → Prop
q : α → α → Prop
hpq : ∀ (e : Fin 2 → α), p e ↔ q (e 0) (e 1)
H : ∀ (u v :... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/ExponentialModule/Basic.lean | PowerSeries.forall_congr_curry | [151, 1] | [166, 12] | apply H | case mpr
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
α : Type u_4
p : (Fin 2 → α) → Prop
q : α → α → Prop
hpq : ∀ (e : Fin 2 → α), p e ↔ q (e 0) (e 1)
H : ∀ (u v : α), q u v
e : Fin 2 → α
⊢ q (e 0) (e 1) | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case mpr
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
α : Type u_4
p : (Fin 2 → α) → Prop
q : α → α → Prop
hpq : ∀ (e : Fin 2 → α), p e ↔ q (e 0) (e 1)
H : ∀ (u v :... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/ExponentialModule/Basic.lean | PowerSeries.forall_congr_curry₀ | [168, 1] | [183, 12] | constructor | A : Type u_1
inst✝⁵ : CommRing A
R : Type u_2
inst✝⁴ : CommRing R
inst✝³ : Algebra A R
S : Type u_3
inst✝² : CommRing S
inst✝¹ : Algebra A S
α : Type u_4
inst✝ : Zero α
p : (Fin 2 →₀ α) → Prop
q : α → α → Prop
hpq : ∀ (e : Fin 2 →₀ α), p e ↔ q (e 0) (e 1)
⊢ (∀ (e : Fin 2 →₀ α), p e) ↔ ∀ (u v : α), q u v | case mp
A : Type u_1
inst✝⁵ : CommRing A
R : Type u_2
inst✝⁴ : CommRing R
inst✝³ : Algebra A R
S : Type u_3
inst✝² : CommRing S
inst✝¹ : Algebra A S
α : Type u_4
inst✝ : Zero α
p : (Fin 2 →₀ α) → Prop
q : α → α → Prop
hpq : ∀ (e : Fin 2 →₀ α), p e ↔ q (e 0) (e 1)
⊢ (∀ (e : Fin 2 →₀ α), p e) → ∀ (u v : α), q u v
case m... | Please generate a tactic in lean4 to solve the state.
STATE:
A : Type u_1
inst✝⁵ : CommRing A
R : Type u_2
inst✝⁴ : CommRing R
inst✝³ : Algebra A R
S : Type u_3
inst✝² : CommRing S
inst✝¹ : Algebra A S
α : Type u_4
inst✝ : Zero α
p : (Fin 2 →₀ α) → Prop
q : α → α → Prop
hpq : ∀ (e : Fin 2 →₀ α), p e ↔ q (e 0) (e 1)
⊢ (... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/ExponentialModule/Basic.lean | PowerSeries.forall_congr_curry₀ | [168, 1] | [183, 12] | intro H u v | case mp
A : Type u_1
inst✝⁵ : CommRing A
R : Type u_2
inst✝⁴ : CommRing R
inst✝³ : Algebra A R
S : Type u_3
inst✝² : CommRing S
inst✝¹ : Algebra A S
α : Type u_4
inst✝ : Zero α
p : (Fin 2 →₀ α) → Prop
q : α → α → Prop
hpq : ∀ (e : Fin 2 →₀ α), p e ↔ q (e 0) (e 1)
⊢ (∀ (e : Fin 2 →₀ α), p e) → ∀ (u v : α), q u v | case mp
A : Type u_1
inst✝⁵ : CommRing A
R : Type u_2
inst✝⁴ : CommRing R
inst✝³ : Algebra A R
S : Type u_3
inst✝² : CommRing S
inst✝¹ : Algebra A S
α : Type u_4
inst✝ : Zero α
p : (Fin 2 →₀ α) → Prop
q : α → α → Prop
hpq : ∀ (e : Fin 2 →₀ α), p e ↔ q (e 0) (e 1)
H : ∀ (e : Fin 2 →₀ α), p e
u v : α
⊢ q u v | Please generate a tactic in lean4 to solve the state.
STATE:
case mp
A : Type u_1
inst✝⁵ : CommRing A
R : Type u_2
inst✝⁴ : CommRing R
inst✝³ : Algebra A R
S : Type u_3
inst✝² : CommRing S
inst✝¹ : Algebra A S
α : Type u_4
inst✝ : Zero α
p : (Fin 2 →₀ α) → Prop
q : α → α → Prop
hpq : ∀ (e : Fin 2 →₀ α), p e ↔ q (e 0) (... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/ExponentialModule/Basic.lean | PowerSeries.forall_congr_curry₀ | [168, 1] | [183, 12] | set e : Fin 2 → α := fun
| 0 => u
| 1 => v | case mp
A : Type u_1
inst✝⁵ : CommRing A
R : Type u_2
inst✝⁴ : CommRing R
inst✝³ : Algebra A R
S : Type u_3
inst✝² : CommRing S
inst✝¹ : Algebra A S
α : Type u_4
inst✝ : Zero α
p : (Fin 2 →₀ α) → Prop
q : α → α → Prop
hpq : ∀ (e : Fin 2 →₀ α), p e ↔ q (e 0) (e 1)
H : ∀ (e : Fin 2 →₀ α), p e
u v : α
⊢ q u v | case mp
A : Type u_1
inst✝⁵ : CommRing A
R : Type u_2
inst✝⁴ : CommRing R
inst✝³ : Algebra A R
S : Type u_3
inst✝² : CommRing S
inst✝¹ : Algebra A S
α : Type u_4
inst✝ : Zero α
p : (Fin 2 →₀ α) → Prop
q : α → α → Prop
hpq : ∀ (e : Fin 2 →₀ α), p e ↔ q (e 0) (e 1)
H : ∀ (e : Fin 2 →₀ α), p e
u v : α
e : Fin 2 → α :=
f... | Please generate a tactic in lean4 to solve the state.
STATE:
case mp
A : Type u_1
inst✝⁵ : CommRing A
R : Type u_2
inst✝⁴ : CommRing R
inst✝³ : Algebra A R
S : Type u_3
inst✝² : CommRing S
inst✝¹ : Algebra A S
α : Type u_4
inst✝ : Zero α
p : (Fin 2 →₀ α) → Prop
q : α → α → Prop
hpq : ∀ (e : Fin 2 →₀ α), p e ↔ q (e 0) (... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/ExponentialModule/Basic.lean | PowerSeries.forall_congr_curry₀ | [168, 1] | [183, 12] | specialize hpq (Finsupp.equivFunOnFinite.symm e) | case mp
A : Type u_1
inst✝⁵ : CommRing A
R : Type u_2
inst✝⁴ : CommRing R
inst✝³ : Algebra A R
S : Type u_3
inst✝² : CommRing S
inst✝¹ : Algebra A S
α : Type u_4
inst✝ : Zero α
p : (Fin 2 →₀ α) → Prop
q : α → α → Prop
hpq : ∀ (e : Fin 2 →₀ α), p e ↔ q (e 0) (e 1)
H : ∀ (e : Fin 2 →₀ α), p e
u v : α
e : Fin 2 → α :=
f... | case mp
A : Type u_1
inst✝⁵ : CommRing A
R : Type u_2
inst✝⁴ : CommRing R
inst✝³ : Algebra A R
S : Type u_3
inst✝² : CommRing S
inst✝¹ : Algebra A S
α : Type u_4
inst✝ : Zero α
p : (Fin 2 →₀ α) → Prop
q : α → α → Prop
H : ∀ (e : Fin 2 →₀ α), p e
u v : α
e : Fin 2 → α :=
fun x =>
match x with
| 0 => u
| 1 ... | Please generate a tactic in lean4 to solve the state.
STATE:
case mp
A : Type u_1
inst✝⁵ : CommRing A
R : Type u_2
inst✝⁴ : CommRing R
inst✝³ : Algebra A R
S : Type u_3
inst✝² : CommRing S
inst✝¹ : Algebra A S
α : Type u_4
inst✝ : Zero α
p : (Fin 2 →₀ α) → Prop
q : α → α → Prop
hpq : ∀ (e : Fin 2 →₀ α), p e ↔ q (e 0) (... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/ExponentialModule/Basic.lean | PowerSeries.forall_congr_curry₀ | [168, 1] | [183, 12] | simp [e] at hpq | case mp
A : Type u_1
inst✝⁵ : CommRing A
R : Type u_2
inst✝⁴ : CommRing R
inst✝³ : Algebra A R
S : Type u_3
inst✝² : CommRing S
inst✝¹ : Algebra A S
α : Type u_4
inst✝ : Zero α
p : (Fin 2 →₀ α) → Prop
q : α → α → Prop
H : ∀ (e : Fin 2 →₀ α), p e
u v : α
e : Fin 2 → α :=
fun x =>
match x with
| 0 => u
| 1 ... | case mp
A : Type u_1
inst✝⁵ : CommRing A
R : Type u_2
inst✝⁴ : CommRing R
inst✝³ : Algebra A R
S : Type u_3
inst✝² : CommRing S
inst✝¹ : Algebra A S
α : Type u_4
inst✝ : Zero α
p : (Fin 2 →₀ α) → Prop
q : α → α → Prop
H : ∀ (e : Fin 2 →₀ α), p e
u v : α
e : Fin 2 → α :=
fun x =>
match x with
| 0 => u
| 1 ... | Please generate a tactic in lean4 to solve the state.
STATE:
case mp
A : Type u_1
inst✝⁵ : CommRing A
R : Type u_2
inst✝⁴ : CommRing R
inst✝³ : Algebra A R
S : Type u_3
inst✝² : CommRing S
inst✝¹ : Algebra A S
α : Type u_4
inst✝ : Zero α
p : (Fin 2 →₀ α) → Prop
q : α → α → Prop
H : ∀ (e : Fin 2 →₀ α), p e
u v : α
e : F... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/ExponentialModule/Basic.lean | PowerSeries.forall_congr_curry₀ | [168, 1] | [183, 12] | rw [← hpq] | case mp
A : Type u_1
inst✝⁵ : CommRing A
R : Type u_2
inst✝⁴ : CommRing R
inst✝³ : Algebra A R
S : Type u_3
inst✝² : CommRing S
inst✝¹ : Algebra A S
α : Type u_4
inst✝ : Zero α
p : (Fin 2 →₀ α) → Prop
q : α → α → Prop
H : ∀ (e : Fin 2 →₀ α), p e
u v : α
e : Fin 2 → α :=
fun x =>
match x with
| 0 => u
| 1 ... | case mp
A : Type u_1
inst✝⁵ : CommRing A
R : Type u_2
inst✝⁴ : CommRing R
inst✝³ : Algebra A R
S : Type u_3
inst✝² : CommRing S
inst✝¹ : Algebra A S
α : Type u_4
inst✝ : Zero α
p : (Fin 2 →₀ α) → Prop
q : α → α → Prop
H : ∀ (e : Fin 2 →₀ α), p e
u v : α
e : Fin 2 → α :=
fun x =>
match x with
| 0 => u
| 1 ... | Please generate a tactic in lean4 to solve the state.
STATE:
case mp
A : Type u_1
inst✝⁵ : CommRing A
R : Type u_2
inst✝⁴ : CommRing R
inst✝³ : Algebra A R
S : Type u_3
inst✝² : CommRing S
inst✝¹ : Algebra A S
α : Type u_4
inst✝ : Zero α
p : (Fin 2 →₀ α) → Prop
q : α → α → Prop
H : ∀ (e : Fin 2 →₀ α), p e
u v : α
e : F... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/ExponentialModule/Basic.lean | PowerSeries.forall_congr_curry₀ | [168, 1] | [183, 12] | apply H | case mp
A : Type u_1
inst✝⁵ : CommRing A
R : Type u_2
inst✝⁴ : CommRing R
inst✝³ : Algebra A R
S : Type u_3
inst✝² : CommRing S
inst✝¹ : Algebra A S
α : Type u_4
inst✝ : Zero α
p : (Fin 2 →₀ α) → Prop
q : α → α → Prop
H : ∀ (e : Fin 2 →₀ α), p e
u v : α
e : Fin 2 → α :=
fun x =>
match x with
| 0 => u
| 1 ... | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case mp
A : Type u_1
inst✝⁵ : CommRing A
R : Type u_2
inst✝⁴ : CommRing R
inst✝³ : Algebra A R
S : Type u_3
inst✝² : CommRing S
inst✝¹ : Algebra A S
α : Type u_4
inst✝ : Zero α
p : (Fin 2 →₀ α) → Prop
q : α → α → Prop
H : ∀ (e : Fin 2 →₀ α), p e
u v : α
e : F... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/ExponentialModule/Basic.lean | PowerSeries.forall_congr_curry₀ | [168, 1] | [183, 12] | intro H e | case mpr
A : Type u_1
inst✝⁵ : CommRing A
R : Type u_2
inst✝⁴ : CommRing R
inst✝³ : Algebra A R
S : Type u_3
inst✝² : CommRing S
inst✝¹ : Algebra A S
α : Type u_4
inst✝ : Zero α
p : (Fin 2 →₀ α) → Prop
q : α → α → Prop
hpq : ∀ (e : Fin 2 →₀ α), p e ↔ q (e 0) (e 1)
⊢ (∀ (u v : α), q u v) → ∀ (e : Fin 2 →₀ α), p e | case mpr
A : Type u_1
inst✝⁵ : CommRing A
R : Type u_2
inst✝⁴ : CommRing R
inst✝³ : Algebra A R
S : Type u_3
inst✝² : CommRing S
inst✝¹ : Algebra A S
α : Type u_4
inst✝ : Zero α
p : (Fin 2 →₀ α) → Prop
q : α → α → Prop
hpq : ∀ (e : Fin 2 →₀ α), p e ↔ q (e 0) (e 1)
H : ∀ (u v : α), q u v
e : Fin 2 →₀ α
⊢ p e | Please generate a tactic in lean4 to solve the state.
STATE:
case mpr
A : Type u_1
inst✝⁵ : CommRing A
R : Type u_2
inst✝⁴ : CommRing R
inst✝³ : Algebra A R
S : Type u_3
inst✝² : CommRing S
inst✝¹ : Algebra A S
α : Type u_4
inst✝ : Zero α
p : (Fin 2 →₀ α) → Prop
q : α → α → Prop
hpq : ∀ (e : Fin 2 →₀ α), p e ↔ q (e 0) ... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/ExponentialModule/Basic.lean | PowerSeries.forall_congr_curry₀ | [168, 1] | [183, 12] | rw [hpq] | case mpr
A : Type u_1
inst✝⁵ : CommRing A
R : Type u_2
inst✝⁴ : CommRing R
inst✝³ : Algebra A R
S : Type u_3
inst✝² : CommRing S
inst✝¹ : Algebra A S
α : Type u_4
inst✝ : Zero α
p : (Fin 2 →₀ α) → Prop
q : α → α → Prop
hpq : ∀ (e : Fin 2 →₀ α), p e ↔ q (e 0) (e 1)
H : ∀ (u v : α), q u v
e : Fin 2 →₀ α
⊢ p e | case mpr
A : Type u_1
inst✝⁵ : CommRing A
R : Type u_2
inst✝⁴ : CommRing R
inst✝³ : Algebra A R
S : Type u_3
inst✝² : CommRing S
inst✝¹ : Algebra A S
α : Type u_4
inst✝ : Zero α
p : (Fin 2 →₀ α) → Prop
q : α → α → Prop
hpq : ∀ (e : Fin 2 →₀ α), p e ↔ q (e 0) (e 1)
H : ∀ (u v : α), q u v
e : Fin 2 →₀ α
⊢ q (e 0) (e 1) | Please generate a tactic in lean4 to solve the state.
STATE:
case mpr
A : Type u_1
inst✝⁵ : CommRing A
R : Type u_2
inst✝⁴ : CommRing R
inst✝³ : Algebra A R
S : Type u_3
inst✝² : CommRing S
inst✝¹ : Algebra A S
α : Type u_4
inst✝ : Zero α
p : (Fin 2 →₀ α) → Prop
q : α → α → Prop
hpq : ∀ (e : Fin 2 →₀ α), p e ↔ q (e 0) ... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/ExponentialModule/Basic.lean | PowerSeries.forall_congr_curry₀ | [168, 1] | [183, 12] | apply H | case mpr
A : Type u_1
inst✝⁵ : CommRing A
R : Type u_2
inst✝⁴ : CommRing R
inst✝³ : Algebra A R
S : Type u_3
inst✝² : CommRing S
inst✝¹ : Algebra A S
α : Type u_4
inst✝ : Zero α
p : (Fin 2 →₀ α) → Prop
q : α → α → Prop
hpq : ∀ (e : Fin 2 →₀ α), p e ↔ q (e 0) (e 1)
H : ∀ (u v : α), q u v
e : Fin 2 →₀ α
⊢ q (e 0) (e 1) | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case mpr
A : Type u_1
inst✝⁵ : CommRing A
R : Type u_2
inst✝⁴ : CommRing R
inst✝³ : Algebra A R
S : Type u_3
inst✝² : CommRing S
inst✝¹ : Algebra A S
α : Type u_4
inst✝ : Zero α
p : (Fin 2 →₀ α) → Prop
q : α → α → Prop
hpq : ∀ (e : Fin 2 →₀ α), p e ↔ q (e 0) ... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/ExponentialModule/Basic.lean | PowerSeries.isExponential_add_mul_iff | [188, 1] | [243, 15] | rw [MvPowerSeries.ext_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
f : R⟦X⟧
he :
∀ (e : Fin 2 →₀ ℕ),
(MvPowerSeries.coeff R e) (subst (MvPowerSeries.X 0 + MvPowerSeries.X 1) f) =
↑((e 0 + e 1).choose (e 0)) * (coeff R (e 0 + e 1)) f
he... | 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⟧
he :
∀ (e : Fin 2 →₀ ℕ),
(MvPowerSeries.coeff R e) (subst (MvPowerSeries.X 0 + MvPowerSeries.X 1) f) =
↑((e 0 + e 1).choose (e 0)) * (coeff R (e 0 + e 1)) f
he... | 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⟧
he :
∀ (e : Fin 2 →₀ ℕ),
(MvPowerSeries.coeff R e) (subst (MvPowerSeries.X 0 + MvPowerSeries.X 1) f) =
... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/ExponentialModule/Basic.lean | PowerSeries.isExponential_add_mul_iff | [188, 1] | [243, 15] | convert forall_congr_curry₀ _ | 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⟧
he :
∀ (e : Fin 2 →₀ ℕ),
(MvPowerSeries.coeff R e) (subst (MvPowerSeries.X 0 + MvPowerSeries.X 1) f) =
↑((e 0 + e 1).choose (e 0)) * (coeff R (e 0 + e 1)) f
he... | case convert_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⟧
he :
∀ (e : Fin 2 →₀ ℕ),
(MvPowerSeries.coeff R e) (subst (MvPowerSeries.X 0 + MvPowerSeries.X 1) f) =
↑((e 0 + e 1).choose (e 0)) * (coeff R (e... | 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⟧
he :
∀ (e : Fin 2 →₀ ℕ),
(MvPowerSeries.coeff R e) (subst (MvPowerSeries.X 0 + MvPowerSeries.X 1) f) =
... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/ExponentialModule/Basic.lean | PowerSeries.isExponential_add_mul_iff | [188, 1] | [243, 15] | intro e | case convert_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⟧
he :
∀ (e : Fin 2 →₀ ℕ),
(MvPowerSeries.coeff R e) (subst (MvPowerSeries.X 0 + MvPowerSeries.X 1) f) =
↑((e 0 + e 1).choose (e 0)) * (coeff R (e... | case convert_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⟧
he :
∀ (e : Fin 2 →₀ ℕ),
(MvPowerSeries.coeff R e) (subst (MvPowerSeries.X 0 + MvPowerSeries.X 1) f) =
↑((e 0 + e 1).choose (e 0)) * (coeff R (e... | Please generate a tactic in lean4 to solve the state.
STATE:
case convert_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⟧
he :
∀ (e : Fin 2 →₀ ℕ),
(MvPowerSeries.coeff R e) (subst (MvPowerSeries.X 0 + MvPowerSer... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/ExponentialModule/Basic.lean | PowerSeries.isExponential_add_mul_iff | [188, 1] | [243, 15] | rw [he, he'] | case convert_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⟧
he :
∀ (e : Fin 2 →₀ ℕ),
(MvPowerSeries.coeff R e) (subst (MvPowerSeries.X 0 + MvPowerSeries.X 1) f) =
↑((e 0 + e 1).choose (e 0)) * (coeff R (e... | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case convert_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⟧
he :
∀ (e : Fin 2 →₀ ℕ),
(MvPowerSeries.coeff R e) (subst (MvPowerSeries.X 0 + MvPowerSer... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/ExponentialModule/Basic.lean | PowerSeries.isExponential_add_mul_iff | [188, 1] | [243, 15] | intro e | 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⟧
⊢ ∀ (e : Fin 2 →₀ ℕ),
(MvPowerSeries.coeff R e) (subst (MvPowerSeries.X 0 + MvPowerSeries.X 1) f) =
↑((e 0 + e 1).choose (e 0)) * (coeff R (e 0 + e 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 : R⟦X⟧
e : Fin 2 →₀ ℕ
⊢ (MvPowerSeries.coeff R e) (subst (MvPowerSeries.X 0 + MvPowerSeries.X 1) f) =
↑((e 0 + e 1).choose (e 0)) * (coeff R (e 0 + e 1)) 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
f : R⟦X⟧
⊢ ∀ (e : Fin 2 →₀ ℕ),
(MvPowerSeries.coeff R e) (subst (MvPowerSeries.X 0 + MvPowerSeries.X 1) f) =
... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/ExponentialModule/Basic.lean | PowerSeries.isExponential_add_mul_iff | [188, 1] | [243, 15] | rw [PowerSeries.subst, MvPowerSeries.coeff_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
f : R⟦X⟧
e : Fin 2 →₀ ℕ
⊢ (MvPowerSeries.coeff R e) (subst (MvPowerSeries.X 0 + MvPowerSeries.X 1) f) =
↑((e 0 + e 1).choose (e 0)) * (coeff R (e 0 + e 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 : R⟦X⟧
e : Fin 2 →₀ ℕ
⊢ ∑ᶠ (d : Unit →₀ ℕ),
(MvPowerSeries.coeff R d) f •
(MvPowerSeries.coeff R e) (d.prod fun s e => (MvPowerSeries.X 0 + MvPowerSeries.X 1) ^ e) ... | 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⟧
e : Fin 2 →₀ ℕ
⊢ (MvPowerSeries.coeff R e) (subst (MvPowerSeries.X 0 + MvPowerSeries.X 1) f) =
↑((e 0 + e ... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/ExponentialModule/Basic.lean | PowerSeries.isExponential_add_mul_iff | [188, 1] | [243, 15] | simp only [Fin.isValue, Finsupp.prod_pow, Finset.univ_unique,
PUnit.default_eq_unit, Finset.prod_singleton, smul_eq_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 : R⟦X⟧
e : Fin 2 →₀ ℕ
⊢ ∑ᶠ (d : Unit →₀ ℕ),
(MvPowerSeries.coeff R d) f •
(MvPowerSeries.coeff R e) (d.prod fun s e => (MvPowerSeries.X 0 + MvPowerSeries.X 1) ^ e) ... | 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⟧
e : Fin 2 →₀ ℕ
⊢ ∑ᶠ (d : Unit →₀ ℕ),
(MvPowerSeries.coeff R d) f * (MvPowerSeries.coeff R e) ((MvPowerSeries.X 0 + MvPowerSeries.X 1) ^ d PUnit.unit) =
↑((e 0 + ... | 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⟧
e : Fin 2 →₀ ℕ
⊢ ∑ᶠ (d : Unit →₀ ℕ),
(MvPowerSeries.coeff R d) f •
(MvPowerSeries.coeff R e) (d.... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/ExponentialModule/Basic.lean | PowerSeries.isExponential_add_mul_iff | [188, 1] | [243, 15] | simp only [← MvPolynomial.coe_X, ← MvPolynomial.coe_add, ← MvPolynomial.coe_pow, MvPolynomial.coeff_coe] | 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⟧
e : Fin 2 →₀ ℕ
⊢ ∑ᶠ (d : Unit →₀ ℕ),
(MvPowerSeries.coeff R d) f * (MvPowerSeries.coeff R e) ((MvPowerSeries.X 0 + MvPowerSeries.X 1) ^ d PUnit.unit) =
↑((e 0 + ... | 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⟧
e : Fin 2 →₀ ℕ
⊢ ∑ᶠ (d : Unit →₀ ℕ),
(MvPowerSeries.coeff R d) f * MvPolynomial.coeff e ((MvPolynomial.X 0 + MvPolynomial.X 1) ^ d PUnit.unit) =
↑((e 0 + e 1).ch... | 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⟧
e : Fin 2 →₀ ℕ
⊢ ∑ᶠ (d : Unit →₀ ℕ),
(MvPowerSeries.coeff R d) f * (MvPowerSeries.coeff R e) ((MvPowerSe... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/ExponentialModule/Basic.lean | PowerSeries.isExponential_add_mul_iff | [188, 1] | [243, 15] | rw [finsum_eq_single _ (Finsupp.single () (e 0 + e 1)), mul_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
f : R⟦X⟧
e : Fin 2 →₀ ℕ
⊢ ∑ᶠ (d : Unit →₀ ℕ),
(MvPowerSeries.coeff R d) f * MvPolynomial.coeff e ((MvPolynomial.X 0 + MvPolynomial.X 1) ^ d PUnit.unit) =
↑((e 0 + e 1).ch... | 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⟧
e : Fin 2 →₀ ℕ
⊢ MvPolynomial.coeff e ((MvPolynomial.X 0 + MvPolynomial.X 1) ^ (Finsupp.single () (e 0 + e 1)) PUnit.unit) *
(MvPowerSeries.coeff R (Finsupp.single (... | 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⟧
e : Fin 2 →₀ ℕ
⊢ ∑ᶠ (d : Unit →₀ ℕ),
(MvPowerSeries.coeff R d) f * MvPolynomial.coeff e ((MvPolynomial.X... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/ExponentialModule/Basic.lean | PowerSeries.isExponential_add_mul_iff | [188, 1] | [243, 15] | apply congr_arg₂ | 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⟧
e : Fin 2 →₀ ℕ
⊢ MvPolynomial.coeff e ((MvPolynomial.X 0 + MvPolynomial.X 1) ^ (Finsupp.single () (e 0 + e 1)) PUnit.unit) *
(MvPowerSeries.coeff R (Finsupp.single (... | case hx
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⟧
e : Fin 2 →₀ ℕ
⊢ MvPolynomial.coeff e ((MvPolynomial.X 0 + MvPolynomial.X 1) ^ (Finsupp.single () (e 0 + e 1)) PUnit.unit) =
↑((e 0 + e 1).choose (e 0))
case ... | 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⟧
e : Fin 2 →₀ ℕ
⊢ MvPolynomial.coeff e ((MvPolynomial.X 0 + MvPolynomial.X 1) ^ (Finsupp.single () (e 0 + e 1))... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/ExponentialModule/Basic.lean | PowerSeries.isExponential_add_mul_iff | [188, 1] | [243, 15] | simp only [Finsupp.single_add, Finsupp.coe_add, Pi.add_apply, Finsupp.single_eq_same] | case hx
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⟧
e : Fin 2 →₀ ℕ
⊢ MvPolynomial.coeff e ((MvPolynomial.X 0 + MvPolynomial.X 1) ^ (Finsupp.single () (e 0 + e 1)) PUnit.unit) =
↑((e 0 + e 1).choose (e 0))
case ... | case hx
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⟧
e : Fin 2 →₀ ℕ
⊢ MvPolynomial.coeff e ((MvPolynomial.X 0 + MvPolynomial.X 1) ^ (e 0 + e 1)) = ↑((e 0 + e 1).choose (e 0))
case hy
A : Type u_1
inst✝⁴ : CommRing A... | Please generate a tactic in lean4 to solve the state.
STATE:
case hx
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⟧
e : Fin 2 →₀ ℕ
⊢ MvPolynomial.coeff e ((MvPolynomial.X 0 + MvPolynomial.X 1) ^ (Finsupp.single () (e 0... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/ExponentialModule/Basic.lean | PowerSeries.isExponential_add_mul_iff | [188, 1] | [243, 15] | simp only [Fin.isValue, coeff_add_pow e _, Finset.mem_antidiagonal, ↓reduceIte, coeff] | case hx
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⟧
e : Fin 2 →₀ ℕ
⊢ MvPolynomial.coeff e ((MvPolynomial.X 0 + MvPolynomial.X 1) ^ (e 0 + e 1)) = ↑((e 0 + e 1).choose (e 0))
case hy
A : Type u_1
inst✝⁴ : CommRing A... | case hy
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⟧
e : Fin 2 →₀ ℕ
⊢ (MvPowerSeries.coeff R (Finsupp.single () (e 0 + e 1))) f = (coeff R (e 0 + e 1)) f | Please generate a tactic in lean4 to solve the state.
STATE:
case hx
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⟧
e : Fin 2 →₀ ℕ
⊢ MvPolynomial.coeff e ((MvPolynomial.X 0 + MvPolynomial.X 1) ^ (e 0 + e 1)) = ↑((e 0 +... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/ExponentialModule/Basic.lean | PowerSeries.isExponential_add_mul_iff | [188, 1] | [243, 15] | rfl | case hy
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⟧
e : Fin 2 →₀ ℕ
⊢ (MvPowerSeries.coeff R (Finsupp.single () (e 0 + e 1))) f = (coeff R (e 0 + e 1)) f | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case hy
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⟧
e : Fin 2 →₀ ℕ
⊢ (MvPowerSeries.coeff R (Finsupp.single () (e 0 + e 1))) f = (coeff R (e 0 + e 1)) f
T... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/ExponentialModule/Basic.lean | PowerSeries.isExponential_add_mul_iff | [188, 1] | [243, 15] | intro d hd' | 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⟧
e : Fin 2 →₀ ℕ
⊢ ∀ (x : Unit →₀ ℕ),
x ≠ Finsupp.single () (e 0 + e 1) →
(MvPowerSeries.coeff R x) f * MvPolynomial.coeff e ((MvPolynomial.X 0 + MvPolynomial.X 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
f : R⟦X⟧
e : Fin 2 →₀ ℕ
d : Unit →₀ ℕ
hd' : d ≠ Finsupp.single () (e 0 + e 1)
⊢ (MvPowerSeries.coeff R d) f * MvPolynomial.coeff e ((MvPolynomial.X 0 + MvPolynomial.X 1) ^ d PUnit.... | 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⟧
e : Fin 2 →₀ ℕ
⊢ ∀ (x : Unit →₀ ℕ),
x ≠ Finsupp.single () (e 0 + e 1) →
(MvPowerSeries.coeff R x) f ... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/ExponentialModule/Basic.lean | PowerSeries.isExponential_add_mul_iff | [188, 1] | [243, 15] | simp [coeff_add_pow] | 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⟧
e : Fin 2 →₀ ℕ
d : Unit →₀ ℕ
hd' : d ≠ Finsupp.single () (e 0 + e 1)
⊢ (MvPowerSeries.coeff R d) f * MvPolynomial.coeff e ((MvPolynomial.X 0 + MvPolynomial.X 1) ^ d PUnit.... | 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⟧
e : Fin 2 →₀ ℕ
d : Unit →₀ ℕ
hd' : d ≠ Finsupp.single () (e 0 + e 1)
⊢ e 0 + e 1 = d PUnit.unit → (MvPowerSeries.coeff R d) f * ↑((d PUnit.unit).choose (e 0)) = 0 | 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⟧
e : Fin 2 →₀ ℕ
d : Unit →₀ ℕ
hd' : d ≠ Finsupp.single () (e 0 + e 1)
⊢ (MvPowerSeries.coeff R d) f * MvPolynom... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/ExponentialModule/Basic.lean | PowerSeries.isExponential_add_mul_iff | [188, 1] | [243, 15] | intro hd | 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⟧
e : Fin 2 →₀ ℕ
d : Unit →₀ ℕ
hd' : d ≠ Finsupp.single () (e 0 + e 1)
⊢ e 0 + e 1 = d PUnit.unit → (MvPowerSeries.coeff R d) f * ↑((d PUnit.unit).choose (e 0)) = 0 | 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⟧
e : Fin 2 →₀ ℕ
d : Unit →₀ ℕ
hd' : d ≠ Finsupp.single () (e 0 + e 1)
hd : e 0 + e 1 = d PUnit.unit
⊢ (MvPowerSeries.coeff R d) f * ↑((d PUnit.unit).choose (e 0)) = 0 | 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⟧
e : Fin 2 →₀ ℕ
d : Unit →₀ ℕ
hd' : d ≠ Finsupp.single () (e 0 + e 1)
⊢ e 0 + e 1 = d PUnit.unit → (MvPowerSeri... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/ExponentialModule/Basic.lean | PowerSeries.isExponential_add_mul_iff | [188, 1] | [243, 15] | exfalso | 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⟧
e : Fin 2 →₀ ℕ
d : Unit →₀ ℕ
hd' : d ≠ Finsupp.single () (e 0 + e 1)
hd : e 0 + e 1 = d PUnit.unit
⊢ (MvPowerSeries.coeff R d) f * ↑((d PUnit.unit).choose (e 0)) = 0 | 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⟧
e : Fin 2 →₀ ℕ
d : Unit →₀ ℕ
hd' : d ≠ Finsupp.single () (e 0 + e 1)
hd : e 0 + e 1 = d PUnit.unit
⊢ False | 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⟧
e : Fin 2 →₀ ℕ
d : Unit →₀ ℕ
hd' : d ≠ Finsupp.single () (e 0 + e 1)
hd : e 0 + e 1 = d PUnit.unit
⊢ (MvPowerS... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/ExponentialModule/Basic.lean | PowerSeries.isExponential_add_mul_iff | [188, 1] | [243, 15] | apply hd' | 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⟧
e : Fin 2 →₀ ℕ
d : Unit →₀ ℕ
hd' : d ≠ Finsupp.single () (e 0 + e 1)
hd : e 0 + e 1 = d PUnit.unit
⊢ False | 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⟧
e : Fin 2 →₀ ℕ
d : Unit →₀ ℕ
hd' : d ≠ Finsupp.single () (e 0 + e 1)
hd : e 0 + e 1 = d PUnit.unit
⊢ d = Finsupp.single () (e 0 + e 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
f : R⟦X⟧
e : Fin 2 →₀ ℕ
d : Unit →₀ ℕ
hd' : d ≠ Finsupp.single () (e 0 + e 1)
hd : e 0 + e 1 = d PUnit.unit
⊢ False
TAC... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/ExponentialModule/Basic.lean | PowerSeries.isExponential_add_mul_iff | [188, 1] | [243, 15] | ext | 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⟧
e : Fin 2 →₀ ℕ
d : Unit →₀ ℕ
hd' : d ≠ Finsupp.single () (e 0 + e 1)
hd : e 0 + e 1 = d PUnit.unit
⊢ d = Finsupp.single () (e 0 + e 1) | case 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
f : R⟦X⟧
e : Fin 2 →₀ ℕ
d : Unit →₀ ℕ
hd' : d ≠ Finsupp.single () (e 0 + e 1)
hd : e 0 + e 1 = d PUnit.unit
⊢ d default = (Finsupp.single () (e 0 + e 1)) default | 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⟧
e : Fin 2 →₀ ℕ
d : Unit →₀ ℕ
hd' : d ≠ Finsupp.single () (e 0 + e 1)
hd : e 0 + e 1 = d PUnit.unit
⊢ d = Finsu... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/ExponentialModule/Basic.lean | PowerSeries.isExponential_add_mul_iff | [188, 1] | [243, 15] | simp only [PUnit.default_eq_unit, hd, Finsupp.single_eq_same] | case 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
f : R⟦X⟧
e : Fin 2 →₀ ℕ
d : Unit →₀ ℕ
hd' : d ≠ Finsupp.single () (e 0 + e 1)
hd : e 0 + e 1 = d PUnit.unit
⊢ d default = (Finsupp.single () (e 0 + e 1)) default | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case 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
f : R⟦X⟧
e : Fin 2 →₀ ℕ
d : Unit →₀ ℕ
hd' : d ≠ Finsupp.single () (e 0 + e 1)
hd : e 0 + e 1 = d PUnit.unit
⊢ d ... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/ExponentialModule/Basic.lean | PowerSeries.isExponential_add_mul_iff | [188, 1] | [243, 15] | exact MvPowerSeries.substDomain_of_constantCoeff_zero
(fun _ ↦ 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 : R⟦X⟧
e : Fin 2 →₀ ℕ
⊢ MvPowerSeries.SubstDomain fun x => MvPowerSeries.X 0 + MvPowerSeries.X 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⟧
e : Fin 2 →₀ ℕ
⊢ MvPowerSeries.SubstDomain fun x => MvPowerSeries.X 0 + MvPowerSeries.X 1
TACTIC:
|
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/ExponentialModule/Basic.lean | PowerSeries.isExponential_add_mul_iff | [188, 1] | [243, 15] | 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 : R⟦X⟧
e : Fin 2 →₀ ℕ
x✝ : Unit
⊢ (MvPowerSeries.constantCoeff (Fin 2) R) (MvPowerSeries.X 0 + 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 : R⟦X⟧
e : Fin 2 →₀ ℕ
x✝ : Unit
⊢ (MvPowerSeries.constantCoeff (Fin 2) R) (MvPowerSeries.X 0 + MvPowerSeries.X 1) = 0... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/ExponentialModule/Basic.lean | PowerSeries.isExponential_add_mul_iff | [188, 1] | [243, 15] | intro e | 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⟧
he :
∀ (e : Fin 2 →₀ ℕ),
(MvPowerSeries.coeff R e) (subst (MvPowerSeries.X 0 + MvPowerSeries.X 1) f) =
↑((e 0 + e 1).choose (e 0)) * (coeff R (e 0 + e 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 : R⟦X⟧
he :
∀ (e : Fin 2 →₀ ℕ),
(MvPowerSeries.coeff R e) (subst (MvPowerSeries.X 0 + MvPowerSeries.X 1) f) =
↑((e 0 + e 1).choose (e 0)) * (coeff R (e 0 + e 1)) f
e ... | 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⟧
he :
∀ (e : Fin 2 →₀ ℕ),
(MvPowerSeries.coeff R e) (subst (MvPowerSeries.X 0 + MvPowerSeries.X 1) f) =
... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/ExponentialModule/Basic.lean | PowerSeries.isExponential_add_mul_iff | [188, 1] | [243, 15] | rw [MvPowerSeries.coeff_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 : R⟦X⟧
he :
∀ (e : Fin 2 →₀ ℕ),
(MvPowerSeries.coeff R e) (subst (MvPowerSeries.X 0 + MvPowerSeries.X 1) f) =
↑((e 0 + e 1).choose (e 0)) * (coeff R (e 0 + e 1)) f
e ... | 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⟧
he :
∀ (e : Fin 2 →₀ ℕ),
(MvPowerSeries.coeff R e) (subst (MvPowerSeries.X 0 + MvPowerSeries.X 1) f) =
↑((e 0 + e 1).choose (e 0)) * (coeff R (e 0 + e 1)) f
e ... | 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⟧
he :
∀ (e : Fin 2 →₀ ℕ),
(MvPowerSeries.coeff R e) (subst (MvPowerSeries.X 0 + MvPowerSeries.X 1) f) =
... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/ExponentialModule/Basic.lean | PowerSeries.isExponential_add_mul_iff | [188, 1] | [243, 15] | rw [Finset.sum_eq_single (Finsupp.single 0 (e 0), Finsupp.single 1 (e 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
f : R⟦X⟧
he :
∀ (e : Fin 2 →₀ ℕ),
(MvPowerSeries.coeff R e) (subst (MvPowerSeries.X 0 + MvPowerSeries.X 1) f) =
↑((e 0 + e 1).choose (e 0)) * (coeff R (e 0 + e 1)) f
e ... | 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⟧
he :
∀ (e : Fin 2 →₀ ℕ),
(MvPowerSeries.coeff R e) (subst (MvPowerSeries.X 0 + MvPowerSeries.X 1) f) =
↑((e 0 + e 1).choose (e 0)) * (coeff R (e 0 + e 1)) f
e ... | 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⟧
he :
∀ (e : Fin 2 →₀ ℕ),
(MvPowerSeries.coeff R e) (subst (MvPowerSeries.X 0 + MvPowerSeries.X 1) f) =
... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/ExponentialModule/Basic.lean | PowerSeries.isExponential_add_mul_iff | [188, 1] | [243, 15] | apply congr_arg₂ | 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⟧
he :
∀ (e : Fin 2 →₀ ℕ),
(MvPowerSeries.coeff R e) (subst (MvPowerSeries.X 0 + MvPowerSeries.X 1) f) =
↑((e 0 + e 1).choose (e 0)) * (coeff R (e 0 + e 1)) f
e ... | case hx
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⟧
he :
∀ (e : Fin 2 →₀ ℕ),
(MvPowerSeries.coeff R e) (subst (MvPowerSeries.X 0 + MvPowerSeries.X 1) f) =
↑((e 0 + e 1).choose (e 0)) * (coeff R (e 0 + e ... | 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⟧
he :
∀ (e : Fin 2 →₀ ℕ),
(MvPowerSeries.coeff R e) (subst (MvPowerSeries.X 0 + MvPowerSeries.X 1) f) =
... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/ExponentialModule/Basic.lean | PowerSeries.isExponential_add_mul_iff | [188, 1] | [243, 15] | simp only [coeff_subst_single, Finsupp.single_eq_same, if_pos] | case hx
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⟧
he :
∀ (e : Fin 2 →₀ ℕ),
(MvPowerSeries.coeff R e) (subst (MvPowerSeries.X 0 + MvPowerSeries.X 1) f) =
↑((e 0 + e 1).choose (e 0)) * (coeff R (e 0 + e ... | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case hx
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⟧
he :
∀ (e : Fin 2 →₀ ℕ),
(MvPowerSeries.coeff R e) (subst (MvPowerSeries.X 0 + MvPowerSeries.X 1... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/ExponentialModule/Basic.lean | PowerSeries.isExponential_add_mul_iff | [188, 1] | [243, 15] | simp only [coeff_subst_single, Finsupp.single_eq_same, if_pos] | case hy
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⟧
he :
∀ (e : Fin 2 →₀ ℕ),
(MvPowerSeries.coeff R e) (subst (MvPowerSeries.X 0 + MvPowerSeries.X 1) f) =
↑((e 0 + e 1).choose (e 0)) * (coeff R (e 0 + e ... | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case hy
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⟧
he :
∀ (e : Fin 2 →₀ ℕ),
(MvPowerSeries.coeff R e) (subst (MvPowerSeries.X 0 + MvPowerSeries.X 1... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/ExponentialModule/Basic.lean | PowerSeries.isExponential_add_mul_iff | [188, 1] | [243, 15] | intro b hb hb' | case h₀
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⟧
he :
∀ (e : Fin 2 →₀ ℕ),
(MvPowerSeries.coeff R e) (subst (MvPowerSeries.X 0 + MvPowerSeries.X 1) f) =
↑((e 0 + e 1).choose (e 0)) * (coeff R (e 0 + e ... | case h₀
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⟧
he :
∀ (e : Fin 2 →₀ ℕ),
(MvPowerSeries.coeff R e) (subst (MvPowerSeries.X 0 + MvPowerSeries.X 1) f) =
↑((e 0 + e 1).choose (e 0)) * (coeff R (e 0 + e ... | Please generate a tactic in lean4 to solve the state.
STATE:
case h₀
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⟧
he :
∀ (e : Fin 2 →₀ ℕ),
(MvPowerSeries.coeff R e) (subst (MvPowerSeries.X 0 + MvPowerSeries.X 1... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/ExponentialModule/Basic.lean | PowerSeries.isExponential_add_mul_iff | [188, 1] | [243, 15] | rw [Finset.mem_antidiagonal] at hb | case h₀
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⟧
he :
∀ (e : Fin 2 →₀ ℕ),
(MvPowerSeries.coeff R e) (subst (MvPowerSeries.X 0 + MvPowerSeries.X 1) f) =
↑((e 0 + e 1).choose (e 0)) * (coeff R (e 0 + e ... | case h₀
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⟧
he :
∀ (e : Fin 2 →₀ ℕ),
(MvPowerSeries.coeff R e) (subst (MvPowerSeries.X 0 + MvPowerSeries.X 1) f) =
↑((e 0 + e 1).choose (e 0)) * (coeff R (e 0 + e ... | Please generate a tactic in lean4 to solve the state.
STATE:
case h₀
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⟧
he :
∀ (e : Fin 2 →₀ ℕ),
(MvPowerSeries.coeff R e) (subst (MvPowerSeries.X 0 + MvPowerSeries.X 1... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/ExponentialModule/Basic.lean | PowerSeries.isExponential_add_mul_iff | [188, 1] | [243, 15] | by_contra hmul_ne_zero | case h₀
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⟧
he :
∀ (e : Fin 2 →₀ ℕ),
(MvPowerSeries.coeff R e) (subst (MvPowerSeries.X 0 + MvPowerSeries.X 1) f) =
↑((e 0 + e 1).choose (e 0)) * (coeff R (e 0 + e ... | case h₀
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⟧
he :
∀ (e : Fin 2 →₀ ℕ),
(MvPowerSeries.coeff R e) (subst (MvPowerSeries.X 0 + MvPowerSeries.X 1) f) =
↑((e 0 + e 1).choose (e 0)) * (coeff R (e 0 + e ... | Please generate a tactic in lean4 to solve the state.
STATE:
case h₀
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⟧
he :
∀ (e : Fin 2 →₀ ℕ),
(MvPowerSeries.coeff R e) (subst (MvPowerSeries.X 0 + MvPowerSeries.X 1... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/ExponentialModule/Basic.lean | PowerSeries.isExponential_add_mul_iff | [188, 1] | [243, 15] | rcases ne_zero_of_mul_ne_zero hmul_ne_zero with ⟨h0, h1⟩ | case h₀
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⟧
he :
∀ (e : Fin 2 →₀ ℕ),
(MvPowerSeries.coeff R e) (subst (MvPowerSeries.X 0 + MvPowerSeries.X 1) f) =
↑((e 0 + e 1).choose (e 0)) * (coeff R (e 0 + e ... | case h₀.intro
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⟧
he :
∀ (e : Fin 2 →₀ ℕ),
(MvPowerSeries.coeff R e) (subst (MvPowerSeries.X 0 + MvPowerSeries.X 1) f) =
↑((e 0 + e 1).choose (e 0)) * (coeff R (e ... | Please generate a tactic in lean4 to solve the state.
STATE:
case h₀
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⟧
he :
∀ (e : Fin 2 →₀ ℕ),
(MvPowerSeries.coeff R e) (subst (MvPowerSeries.X 0 + MvPowerSeries.X 1... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/ExponentialModule/Basic.lean | PowerSeries.isExponential_add_mul_iff | [188, 1] | [243, 15] | simp only [Fin.isValue, coeff_subst_single, ne_eq, ite_eq_right_iff,
not_forall, exists_prop] at h0 h1 | case h₀.intro
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⟧
he :
∀ (e : Fin 2 →₀ ℕ),
(MvPowerSeries.coeff R e) (subst (MvPowerSeries.X 0 + MvPowerSeries.X 1) f) =
↑((e 0 + e 1).choose (e 0)) * (coeff R (e ... | case h₀.intro
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⟧
he :
∀ (e : Fin 2 →₀ ℕ),
(MvPowerSeries.coeff R e) (subst (MvPowerSeries.X 0 + MvPowerSeries.X 1) f) =
↑((e 0 + e 1).choose (e 0)) * (coeff R (e ... | Please generate a tactic in lean4 to solve the state.
STATE:
case h₀.intro
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⟧
he :
∀ (e : Fin 2 →₀ ℕ),
(MvPowerSeries.coeff R e) (subst (MvPowerSeries.X 0 + MvPowerSeri... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/ExponentialModule/Basic.lean | PowerSeries.isExponential_add_mul_iff | [188, 1] | [243, 15] | apply hb' | case h₀.intro
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⟧
he :
∀ (e : Fin 2 →₀ ℕ),
(MvPowerSeries.coeff R e) (subst (MvPowerSeries.X 0 + MvPowerSeries.X 1) f) =
↑((e 0 + e 1).choose (e 0)) * (coeff R (e ... | case h₀.intro
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⟧
he :
∀ (e : Fin 2 →₀ ℕ),
(MvPowerSeries.coeff R e) (subst (MvPowerSeries.X 0 + MvPowerSeries.X 1) f) =
↑((e 0 + e 1).choose (e 0)) * (coeff R (e ... | Please generate a tactic in lean4 to solve the state.
STATE:
case h₀.intro
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⟧
he :
∀ (e : Fin 2 →₀ ℕ),
(MvPowerSeries.coeff R e) (subst (MvPowerSeries.X 0 + MvPowerSeri... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/ExponentialModule/Basic.lean | PowerSeries.isExponential_add_mul_iff | [188, 1] | [243, 15] | rw [Prod.ext_iff, ← hb, h0.1, h1.1] | case h₀.intro
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⟧
he :
∀ (e : Fin 2 →₀ ℕ),
(MvPowerSeries.coeff R e) (subst (MvPowerSeries.X 0 + MvPowerSeries.X 1) f) =
↑((e 0 + e 1).choose (e 0)) * (coeff R (e ... | case h₀.intro
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⟧
he :
∀ (e : Fin 2 →₀ ℕ),
(MvPowerSeries.coeff R e) (subst (MvPowerSeries.X 0 + MvPowerSeries.X 1) f) =
↑((e 0 + e 1).choose (e 0)) * (coeff R (e ... | Please generate a tactic in lean4 to solve the state.
STATE:
case h₀.intro
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⟧
he :
∀ (e : Fin 2 →₀ ℕ),
(MvPowerSeries.coeff R e) (subst (MvPowerSeries.X 0 + MvPowerSeri... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/ExponentialModule/Basic.lean | PowerSeries.isExponential_add_mul_iff | [188, 1] | [243, 15] | simp | case h₀.intro
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⟧
he :
∀ (e : Fin 2 →₀ ℕ),
(MvPowerSeries.coeff R e) (subst (MvPowerSeries.X 0 + MvPowerSeries.X 1) f) =
↑((e 0 + e 1).choose (e 0)) * (coeff R (e ... | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case h₀.intro
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⟧
he :
∀ (e : Fin 2 →₀ ℕ),
(MvPowerSeries.coeff R e) (subst (MvPowerSeries.X 0 + MvPowerSeri... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/ExponentialModule/Basic.lean | PowerSeries.isExponential_add_mul_iff | [188, 1] | [243, 15] | intro he | case h₁
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⟧
he :
∀ (e : Fin 2 →₀ ℕ),
(MvPowerSeries.coeff R e) (subst (MvPowerSeries.X 0 + MvPowerSeries.X 1) f) =
↑((e 0 + e 1).choose (e 0)) * (coeff R (e 0 + e ... | case h₁
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⟧
he✝ :
∀ (e : Fin 2 →₀ ℕ),
(MvPowerSeries.coeff R e) (subst (MvPowerSeries.X 0 + MvPowerSeries.X 1) f) =
↑((e 0 + e 1).choose (e 0)) * (coeff R (e 0 + e... | Please generate a tactic in lean4 to solve the state.
STATE:
case h₁
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⟧
he :
∀ (e : Fin 2 →₀ ℕ),
(MvPowerSeries.coeff R e) (subst (MvPowerSeries.X 0 + MvPowerSeries.X 1... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/ExponentialModule/Basic.lean | PowerSeries.isExponential_add_mul_iff | [188, 1] | [243, 15] | exfalso | case h₁
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⟧
he✝ :
∀ (e : Fin 2 →₀ ℕ),
(MvPowerSeries.coeff R e) (subst (MvPowerSeries.X 0 + MvPowerSeries.X 1) f) =
↑((e 0 + e 1).choose (e 0)) * (coeff R (e 0 + e... | case h₁
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⟧
he✝ :
∀ (e : Fin 2 →₀ ℕ),
(MvPowerSeries.coeff R e) (subst (MvPowerSeries.X 0 + MvPowerSeries.X 1) f) =
↑((e 0 + e 1).choose (e 0)) * (coeff R (e 0 + e... | Please generate a tactic in lean4 to solve the state.
STATE:
case h₁
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⟧
he✝ :
∀ (e : Fin 2 →₀ ℕ),
(MvPowerSeries.coeff R e) (subst (MvPowerSeries.X 0 + MvPowerSeries.X ... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/ExponentialModule/Basic.lean | PowerSeries.isExponential_add_mul_iff | [188, 1] | [243, 15] | apply he | case h₁
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⟧
he✝ :
∀ (e : Fin 2 →₀ ℕ),
(MvPowerSeries.coeff R e) (subst (MvPowerSeries.X 0 + MvPowerSeries.X 1) f) =
↑((e 0 + e 1).choose (e 0)) * (coeff R (e 0 + e... | case h₁
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⟧
he✝ :
∀ (e : Fin 2 →₀ ℕ),
(MvPowerSeries.coeff R e) (subst (MvPowerSeries.X 0 + MvPowerSeries.X 1) f) =
↑((e 0 + e 1).choose (e 0)) * (coeff R (e 0 + e... | Please generate a tactic in lean4 to solve the state.
STATE:
case h₁
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⟧
he✝ :
∀ (e : Fin 2 →₀ ℕ),
(MvPowerSeries.coeff R e) (subst (MvPowerSeries.X 0 + MvPowerSeries.X ... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/ExponentialModule/Basic.lean | PowerSeries.isExponential_add_mul_iff | [188, 1] | [243, 15] | simp only [Finset.mem_antidiagonal] | case h₁
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⟧
he✝ :
∀ (e : Fin 2 →₀ ℕ),
(MvPowerSeries.coeff R e) (subst (MvPowerSeries.X 0 + MvPowerSeries.X 1) f) =
↑((e 0 + e 1).choose (e 0)) * (coeff R (e 0 + e... | case h₁
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⟧
he✝ :
∀ (e : Fin 2 →₀ ℕ),
(MvPowerSeries.coeff R e) (subst (MvPowerSeries.X 0 + MvPowerSeries.X 1) f) =
↑((e 0 + e 1).choose (e 0)) * (coeff R (e 0 + e... | Please generate a tactic in lean4 to solve the state.
STATE:
case h₁
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⟧
he✝ :
∀ (e : Fin 2 →₀ ℕ),
(MvPowerSeries.coeff R e) (subst (MvPowerSeries.X 0 + MvPowerSeries.X ... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/ExponentialModule/Basic.lean | PowerSeries.isExponential_add_mul_iff | [188, 1] | [243, 15] | ext i | case h₁
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⟧
he✝ :
∀ (e : Fin 2 →₀ ℕ),
(MvPowerSeries.coeff R e) (subst (MvPowerSeries.X 0 + MvPowerSeries.X 1) f) =
↑((e 0 + e 1).choose (e 0)) * (coeff R (e 0 + e... | case h₁.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
f : R⟦X⟧
he✝ :
∀ (e : Fin 2 →₀ ℕ),
(MvPowerSeries.coeff R e) (subst (MvPowerSeries.X 0 + MvPowerSeries.X 1) f) =
↑((e 0 + e 1).choose (e 0)) * (coeff R (e 0 +... | Please generate a tactic in lean4 to solve the state.
STATE:
case h₁
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⟧
he✝ :
∀ (e : Fin 2 →₀ ℕ),
(MvPowerSeries.coeff R e) (subst (MvPowerSeries.X 0 + MvPowerSeries.X ... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/ExponentialModule/Basic.lean | PowerSeries.isExponential_add_mul_iff | [188, 1] | [243, 15] | match i with
| 0 => simp
| 1 => simp | case h₁.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
f : R⟦X⟧
he✝ :
∀ (e : Fin 2 →₀ ℕ),
(MvPowerSeries.coeff R e) (subst (MvPowerSeries.X 0 + MvPowerSeries.X 1) f) =
↑((e 0 + e 1).choose (e 0)) * (coeff R (e 0 +... | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case h₁.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
f : R⟦X⟧
he✝ :
∀ (e : Fin 2 →₀ ℕ),
(MvPowerSeries.coeff R e) (subst (MvPowerSeries.X 0 + MvPowerSeries.... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/ExponentialModule/Basic.lean | PowerSeries.isExponential_add_mul_iff | [188, 1] | [243, 15] | 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 : R⟦X⟧
he✝ :
∀ (e : Fin 2 →₀ ℕ),
(MvPowerSeries.coeff R e) (subst (MvPowerSeries.X 0 + MvPowerSeries.X 1) f) =
↑((e 0 + e 1).choose (e 0)) * (coeff R (e 0 + e 1)) f
e... | 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⟧
he✝ :
∀ (e : Fin 2 →₀ ℕ),
(MvPowerSeries.coeff R e) (subst (MvPowerSeries.X 0 + MvPowerSeries.X 1) f) =
... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/ExponentialModule/Basic.lean | PowerSeries.isExponential_add_mul_iff | [188, 1] | [243, 15] | 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 : R⟦X⟧
he✝ :
∀ (e : Fin 2 →₀ ℕ),
(MvPowerSeries.coeff R e) (subst (MvPowerSeries.X 0 + MvPowerSeries.X 1) f) =
↑((e 0 + e 1).choose (e 0)) * (coeff R (e 0 + e 1)) f
e... | 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⟧
he✝ :
∀ (e : Fin 2 →₀ ℕ),
(MvPowerSeries.coeff R e) (subst (MvPowerSeries.X 0 + MvPowerSeries.X 1) f) =
... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/ExponentialModule/Basic.lean | PowerSeries.isExponential_iff | [248, 1] | [256, 24] | rw [← isExponential_add_mul_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
f : R⟦X⟧
⊢ f.IsExponential ↔
(constantCoeff R) f = 1 ∧ ∀ (p q : ℕ), ↑((p + q).choose p) * (coeff R (p + q)) f = (coeff R p) f * (coeff R q) 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 : R⟦X⟧
⊢ f.IsExponential ↔
(constantCoeff R) f = 1 ∧
subst (MvPowerSeries.X 0 + MvPowerSeries.X 1) f = subst (MvPowerSeries.X 0) f * subst (MvPowerSeries.X 1) 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
f : R⟦X⟧
⊢ f.IsExponential ↔
(constantCoeff R) f = 1 ∧ ∀ (p q : ℕ), ↑((p + q).choose p) * (coeff R (p + q)) f = (co... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/ExponentialModule/Basic.lean | PowerSeries.isExponential_iff | [248, 1] | [256, 24] | 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
f : R⟦X⟧
⊢ f.IsExponential ↔
(constantCoeff R) f = 1 ∧
subst (MvPowerSeries.X 0 + MvPowerSeries.X 1) f = subst (MvPowerSeries.X 0) f * subst (MvPowerSeries.X 1) f | case mp
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⟧
⊢ f.IsExponential →
(constantCoeff R) f = 1 ∧
subst (MvPowerSeries.X 0 + MvPowerSeries.X 1) f = subst (MvPowerSeries.X 0) f * subst (MvPowerSeries.X 1) 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
f : R⟦X⟧
⊢ f.IsExponential ↔
(constantCoeff R) f = 1 ∧
subst (MvPowerSeries.X 0 + MvPowerSeries.X 1) f = subs... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/ExponentialModule/Basic.lean | PowerSeries.isExponential_iff | [248, 1] | [256, 24] | exact fun hf ↦ ⟨hf.constantCoeff, hf.add_mul⟩ | case mp
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⟧
⊢ f.IsExponential →
(constantCoeff R) f = 1 ∧
subst (MvPowerSeries.X 0 + MvPowerSeries.X 1) f = subst (MvPowerSeries.X 0) f * subst (MvPowerSeries.X 1) f | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case mp
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⟧
⊢ f.IsExponential →
(constantCoeff R) f = 1 ∧
subst (MvPowerSeries.X 0 + MvPowerSeries.X 1) ... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/ExponentialModule/Basic.lean | PowerSeries.isExponential_iff | [248, 1] | [256, 24] | exact fun hf ↦ {
constantCoeff := hf.1
add_mul := hf.2 } | case mpr
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⟧
⊢ (constantCoeff R) f = 1 ∧
subst (MvPowerSeries.X 0 + MvPowerSeries.X 1) f = subst (MvPowerSeries.X 0) f * subst (MvPowerSeries.X 1) f →
f.IsExponentia... | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case mpr
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⟧
⊢ (constantCoeff R) f = 1 ∧
subst (MvPowerSeries.X 0 + MvPowerSeries.X 1) f = subst (MvPowerSer... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/ExponentialModule/Basic.lean | PowerSeries.isExponential_one | [259, 1] | [266, 42] | rw [← Polynomial.coe_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
⊢ subst (MvPowerSeries.X 0 + MvPowerSeries.X 1) 1 = subst (MvPowerSeries.X 0) 1 * subst (MvPowerSeries.X 1) 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
⊢ subst (MvPowerSeries.X 0 + MvPowerSeries.X 1) ↑1 = subst (MvPowerSeries.X 0) ↑1 * subst (MvPowerSeries.X 1) ↑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
⊢ subst (MvPowerSeries.X 0 + MvPowerSeries.X 1) 1 = subst (MvPowerSeries.X 0) 1 * subst (MvPowerSeries.X 1) 1
TACTIC:
|
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/ExponentialModule/Basic.lean | PowerSeries.isExponential_one | [259, 1] | [266, 42] | rw [subst_coe (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
⊢ subst (MvPowerSeries.X 0 + MvPowerSeries.X 1) ↑1 = subst (MvPowerSeries.X 0) ↑1 * subst (MvPowerSeries.X 1) ↑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
⊢ (Polynomial.aeval (MvPowerSeries.X 0 + MvPowerSeries.X 1)) 1 =
subst (MvPowerSeries.X 0) ↑1 * subst (MvPowerSeries.X 1) ↑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
⊢ subst (MvPowerSeries.X 0 + MvPowerSeries.X 1) ↑1 = subst (MvPowerSeries.X 0) ↑1 * subst (MvPowerSeries.X 1) ↑1
TACTIC... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/ExponentialModule/Basic.lean | PowerSeries.isExponential_one | [259, 1] | [266, 42] | rw [subst_coe (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
⊢ (Polynomial.aeval (MvPowerSeries.X 0 + MvPowerSeries.X 1)) 1 =
subst (MvPowerSeries.X 0) ↑1 * subst (MvPowerSeries.X 1) ↑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
⊢ (Polynomial.aeval (MvPowerSeries.X 0 + MvPowerSeries.X 1)) 1 =
(Polynomial.aeval (MvPowerSeries.X 0)) 1 * subst (MvPowerSeries.X 1) ↑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
⊢ (Polynomial.aeval (MvPowerSeries.X 0 + MvPowerSeries.X 1)) 1 =
subst (MvPowerSeries.X 0) ↑1 * subst (MvPowerSerie... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/ExponentialModule/Basic.lean | PowerSeries.isExponential_one | [259, 1] | [266, 42] | rw [subst_coe (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
⊢ (Polynomial.aeval (MvPowerSeries.X 0 + MvPowerSeries.X 1)) 1 =
(Polynomial.aeval (MvPowerSeries.X 0)) 1 * subst (MvPowerSeries.X 1) ↑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
⊢ (Polynomial.aeval (MvPowerSeries.X 0 + MvPowerSeries.X 1)) 1 =
(Polynomial.aeval (MvPowerSeries.X 0)) 1 * (Polynomial.aeval (MvPowerSeries.X 1)) 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
⊢ (Polynomial.aeval (MvPowerSeries.X 0 + MvPowerSeries.X 1)) 1 =
(Polynomial.aeval (MvPowerSeries.X 0)) 1 * subst (... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/ExponentialModule/Basic.lean | PowerSeries.isExponential_one | [259, 1] | [266, 42] | simp only [map_one, 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
⊢ (Polynomial.aeval (MvPowerSeries.X 0 + MvPowerSeries.X 1)) 1 =
(Polynomial.aeval (MvPowerSeries.X 0)) 1 * (Polynomial.aeval (MvPowerSeries.X 1)) 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
⊢ (Polynomial.aeval (MvPowerSeries.X 0 + MvPowerSeries.X 1)) 1 =
(Polynomial.aeval (MvPowerSeries.X 0)) 1 * (Polyno... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/ExponentialModule/Basic.lean | PowerSeries.isExponential_one | [259, 1] | [266, 42] | 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
⊢ (MvPowerSeries.constantCoeff (Fin 2) R) (MvPowerSeries.X 0 + 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
⊢ (MvPowerSeries.constantCoeff (Fin 2) R) (MvPowerSeries.X 0 + MvPowerSeries.X 1) = 0
TACTIC:
|
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/ExponentialModule/Basic.lean | PowerSeries.isExponential_one | [259, 1] | [266, 42] | 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
⊢ (MvPowerSeries.constantCoeff (Fin 2) R) (MvPowerSeries.X 0) = 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
⊢ (MvPowerSeries.constantCoeff (Fin 2) R) (MvPowerSeries.X 0) = 0
TACTIC:
|
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/ExponentialModule/Basic.lean | PowerSeries.isExponential_one | [259, 1] | [266, 42] | 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
⊢ (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
⊢ (MvPowerSeries.constantCoeff (Fin 2) R) (MvPowerSeries.X 1) = 0
TACTIC:
|
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/ExponentialModule/Basic.lean | PowerSeries.isExponential_one | [259, 1] | [266, 42] | simp only [map_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
⊢ (constantCoeff R) 1 = 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
⊢ (constantCoeff R) 1 = 1
TACTIC:
|
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