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/DPAlgebra/PolynomialMap.lean | DividedPowerAlgebra.gamma_mem_grade | [53, 1] | [87, 10] | specialize hx k | case add.a.mk
R : Type u
inst✝⁴ : CommRing R
M : Type u_1
inst✝³ : AddCommGroup M
inst✝² : Module R M
S : Type u_2
inst✝¹ : CommRing S
inst✝ : Algebra R S
x y : S ⊗[R] M
hx : ∀ (n : ℕ), (gamma R M n).toFun S x ∈ LinearMap.range (LinearMap.lTensor S (grade R M n).subtype)
hy : ∀ (n : ℕ), (gamma R M n).toFun S y ∈ Linear... | case add.a.mk
R : Type u
inst✝⁴ : CommRing R
M : Type u_1
inst✝³ : AddCommGroup M
inst✝² : Module R M
S : Type u_2
inst✝¹ : CommRing S
inst✝ : Algebra R S
x y : S ⊗[R] M
hy : ∀ (n : ℕ), (gamma R M n).toFun S y ∈ LinearMap.range (LinearMap.lTensor S (grade R M n).subtype)
n k l : ℕ
hkl : (k, l) ∈ Finset.antidiagonal n
h... | Please generate a tactic in lean4 to solve the state.
STATE:
case add.a.mk
R : Type u
inst✝⁴ : CommRing R
M : Type u_1
inst✝³ : AddCommGroup M
inst✝² : Module R M
S : Type u_2
inst✝¹ : CommRing S
inst✝ : Algebra R S
x y : S ⊗[R] M
hx : ∀ (n : ℕ), (gamma R M n).toFun S x ∈ LinearMap.range (LinearMap.lTensor S (grade R M... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/DPAlgebra/PolynomialMap.lean | DividedPowerAlgebra.gamma_mem_grade | [53, 1] | [87, 10] | specialize hy l | case add.a.mk
R : Type u
inst✝⁴ : CommRing R
M : Type u_1
inst✝³ : AddCommGroup M
inst✝² : Module R M
S : Type u_2
inst✝¹ : CommRing S
inst✝ : Algebra R S
x y : S ⊗[R] M
hy : ∀ (n : ℕ), (gamma R M n).toFun S y ∈ LinearMap.range (LinearMap.lTensor S (grade R M n).subtype)
n k l : ℕ
hkl : (k, l) ∈ Finset.antidiagonal n
h... | case add.a.mk
R : Type u
inst✝⁴ : CommRing R
M : Type u_1
inst✝³ : AddCommGroup M
inst✝² : Module R M
S : Type u_2
inst✝¹ : CommRing S
inst✝ : Algebra R S
x y : S ⊗[R] M
n k l : ℕ
hkl : (k, l) ∈ Finset.antidiagonal n
hx : (gamma R M k).toFun S x ∈ LinearMap.range (LinearMap.lTensor S (grade R M k).subtype)
hy : (gamma ... | Please generate a tactic in lean4 to solve the state.
STATE:
case add.a.mk
R : Type u
inst✝⁴ : CommRing R
M : Type u_1
inst✝³ : AddCommGroup M
inst✝² : Module R M
S : Type u_2
inst✝¹ : CommRing S
inst✝ : Algebra R S
x y : S ⊗[R] M
hy : ∀ (n : ℕ), (gamma R M n).toFun S y ∈ LinearMap.range (LinearMap.lTensor S (grade R M... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/DPAlgebra/PolynomialMap.lean | DividedPowerAlgebra.gamma_mem_grade | [53, 1] | [87, 10] | simp only [gamma_toFun, dpScalarExtensionEquiv, ofAlgHom_symm_apply, LinearMap.mem_range] at hx hy | case add.a.mk
R : Type u
inst✝⁴ : CommRing R
M : Type u_1
inst✝³ : AddCommGroup M
inst✝² : Module R M
S : Type u_2
inst✝¹ : CommRing S
inst✝ : Algebra R S
x y : S ⊗[R] M
n k l : ℕ
hkl : (k, l) ∈ Finset.antidiagonal n
hx : (gamma R M k).toFun S x ∈ LinearMap.range (LinearMap.lTensor S (grade R M k).subtype)
hy : (gamma ... | case add.a.mk
R : Type u
inst✝⁴ : CommRing R
M : Type u_1
inst✝³ : AddCommGroup M
inst✝² : Module R M
S : Type u_2
inst✝¹ : CommRing S
inst✝ : Algebra R S
x y : S ⊗[R] M
n k l : ℕ
hkl : (k, l) ∈ Finset.antidiagonal n
hx : ∃ y, (LinearMap.lTensor S (grade R M k).subtype) y = (dpScalarExtensionInv R S M) (dp S k x)
hy : ... | Please generate a tactic in lean4 to solve the state.
STATE:
case add.a.mk
R : Type u
inst✝⁴ : CommRing R
M : Type u_1
inst✝³ : AddCommGroup M
inst✝² : Module R M
S : Type u_2
inst✝¹ : CommRing S
inst✝ : Algebra R S
x y : S ⊗[R] M
n k l : ℕ
hkl : (k, l) ∈ Finset.antidiagonal n
hx : (gamma R M k).toFun S x ∈ LinearMap.r... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/DPAlgebra/PolynomialMap.lean | DividedPowerAlgebra.gamma_mem_grade | [53, 1] | [87, 10] | obtain ⟨x', hx'⟩ := hx | case add.a.mk
R : Type u
inst✝⁴ : CommRing R
M : Type u_1
inst✝³ : AddCommGroup M
inst✝² : Module R M
S : Type u_2
inst✝¹ : CommRing S
inst✝ : Algebra R S
x y : S ⊗[R] M
n k l : ℕ
hkl : (k, l) ∈ Finset.antidiagonal n
hx : ∃ y, (LinearMap.lTensor S (grade R M k).subtype) y = (dpScalarExtensionInv R S M) (dp S k x)
hy : ... | case add.a.mk.intro
R : Type u
inst✝⁴ : CommRing R
M : Type u_1
inst✝³ : AddCommGroup M
inst✝² : Module R M
S : Type u_2
inst✝¹ : CommRing S
inst✝ : Algebra R S
x y : S ⊗[R] M
n k l : ℕ
hkl : (k, l) ∈ Finset.antidiagonal n
hy : ∃ y_1, (LinearMap.lTensor S (grade R M l).subtype) y_1 = (dpScalarExtensionInv R S M) (dp S ... | Please generate a tactic in lean4 to solve the state.
STATE:
case add.a.mk
R : Type u
inst✝⁴ : CommRing R
M : Type u_1
inst✝³ : AddCommGroup M
inst✝² : Module R M
S : Type u_2
inst✝¹ : CommRing S
inst✝ : Algebra R S
x y : S ⊗[R] M
n k l : ℕ
hkl : (k, l) ∈ Finset.antidiagonal n
hx : ∃ y, (LinearMap.lTensor S (grade R M ... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/DPAlgebra/PolynomialMap.lean | DividedPowerAlgebra.gamma_mem_grade | [53, 1] | [87, 10] | obtain ⟨y', hy'⟩ := hy | case add.a.mk.intro
R : Type u
inst✝⁴ : CommRing R
M : Type u_1
inst✝³ : AddCommGroup M
inst✝² : Module R M
S : Type u_2
inst✝¹ : CommRing S
inst✝ : Algebra R S
x y : S ⊗[R] M
n k l : ℕ
hkl : (k, l) ∈ Finset.antidiagonal n
hy : ∃ y_1, (LinearMap.lTensor S (grade R M l).subtype) y_1 = (dpScalarExtensionInv R S M) (dp S ... | case add.a.mk.intro.intro
R : Type u
inst✝⁴ : CommRing R
M : Type u_1
inst✝³ : AddCommGroup M
inst✝² : Module R M
S : Type u_2
inst✝¹ : CommRing S
inst✝ : Algebra R S
x y : S ⊗[R] M
n k l : ℕ
hkl : (k, l) ∈ Finset.antidiagonal n
x' : S ⊗[R] ↥(grade R M k)
hx' : (LinearMap.lTensor S (grade R M k).subtype) x' = (dpScalar... | Please generate a tactic in lean4 to solve the state.
STATE:
case add.a.mk.intro
R : Type u
inst✝⁴ : CommRing R
M : Type u_1
inst✝³ : AddCommGroup M
inst✝² : Module R M
S : Type u_2
inst✝¹ : CommRing S
inst✝ : Algebra R S
x y : S ⊗[R] M
n k l : ℕ
hkl : (k, l) ∈ Finset.antidiagonal n
hy : ∃ y_1, (LinearMap.lTensor S (gr... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/DPAlgebra/PolynomialMap.lean | DividedPowerAlgebra.gamma_mem_grade | [53, 1] | [87, 10] | simp only [LinearMap.mem_range] | case add.a.mk.intro.intro
R : Type u
inst✝⁴ : CommRing R
M : Type u_1
inst✝³ : AddCommGroup M
inst✝² : Module R M
S : Type u_2
inst✝¹ : CommRing S
inst✝ : Algebra R S
x y : S ⊗[R] M
n k l : ℕ
hkl : (k, l) ∈ Finset.antidiagonal n
x' : S ⊗[R] ↥(grade R M k)
hx' : (LinearMap.lTensor S (grade R M k).subtype) x' = (dpScalar... | case add.a.mk.intro.intro
R : Type u
inst✝⁴ : CommRing R
M : Type u_1
inst✝³ : AddCommGroup M
inst✝² : Module R M
S : Type u_2
inst✝¹ : CommRing S
inst✝ : Algebra R S
x y : S ⊗[R] M
n k l : ℕ
hkl : (k, l) ∈ Finset.antidiagonal n
x' : S ⊗[R] ↥(grade R M k)
hx' : (LinearMap.lTensor S (grade R M k).subtype) x' = (dpScalar... | Please generate a tactic in lean4 to solve the state.
STATE:
case add.a.mk.intro.intro
R : Type u
inst✝⁴ : CommRing R
M : Type u_1
inst✝³ : AddCommGroup M
inst✝² : Module R M
S : Type u_2
inst✝¹ : CommRing S
inst✝ : Algebra R S
x y : S ⊗[R] M
n k l : ℕ
hkl : (k, l) ∈ Finset.antidiagonal n
x' : S ⊗[R] ↥(grade R M k)
hx'... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/DPAlgebra/PolynomialMap.lean | DividedPowerAlgebra.gamma_mem_grade | [53, 1] | [87, 10] | rw [← hx', ← hy'] | case add.a.mk.intro.intro
R : Type u
inst✝⁴ : CommRing R
M : Type u_1
inst✝³ : AddCommGroup M
inst✝² : Module R M
S : Type u_2
inst✝¹ : CommRing S
inst✝ : Algebra R S
x y : S ⊗[R] M
n k l : ℕ
hkl : (k, l) ∈ Finset.antidiagonal n
x' : S ⊗[R] ↥(grade R M k)
hx' : (LinearMap.lTensor S (grade R M k).subtype) x' = (dpScalar... | case add.a.mk.intro.intro
R : Type u
inst✝⁴ : CommRing R
M : Type u_1
inst✝³ : AddCommGroup M
inst✝² : Module R M
S : Type u_2
inst✝¹ : CommRing S
inst✝ : Algebra R S
x y : S ⊗[R] M
n k l : ℕ
hkl : (k, l) ∈ Finset.antidiagonal n
x' : S ⊗[R] ↥(grade R M k)
hx' : (LinearMap.lTensor S (grade R M k).subtype) x' = (dpScalar... | Please generate a tactic in lean4 to solve the state.
STATE:
case add.a.mk.intro.intro
R : Type u
inst✝⁴ : CommRing R
M : Type u_1
inst✝³ : AddCommGroup M
inst✝² : Module R M
S : Type u_2
inst✝¹ : CommRing S
inst✝ : Algebra R S
x y : S ⊗[R] M
n k l : ℕ
hkl : (k, l) ∈ Finset.antidiagonal n
x' : S ⊗[R] ↥(grade R M k)
hx'... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/DPAlgebra/PolynomialMap.lean | DividedPowerAlgebra.gamma_mem_grade | [53, 1] | [87, 10] | sorry | case add.a.mk.intro.intro
R : Type u
inst✝⁴ : CommRing R
M : Type u_1
inst✝³ : AddCommGroup M
inst✝² : Module R M
S : Type u_2
inst✝¹ : CommRing S
inst✝ : Algebra R S
x y : S ⊗[R] M
n k l : ℕ
hkl : (k, l) ∈ Finset.antidiagonal n
x' : S ⊗[R] ↥(grade R M k)
hx' : (LinearMap.lTensor S (grade R M k).subtype) x' = (dpScalar... | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case add.a.mk.intro.intro
R : Type u
inst✝⁴ : CommRing R
M : Type u_1
inst✝³ : AddCommGroup M
inst✝² : Module R M
S : Type u_2
inst✝¹ : CommRing S
inst✝ : Algebra R S
x y : S ⊗[R] M
n k l : ℕ
hkl : (k, l) ∈ Finset.antidiagonal n
x' : S ⊗[R] ↥(grade R M k)
hx'... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/DPAlgebra/PolynomialMap.lean | DividedPowerAlgebra.toModule_free | [133, 1] | [139, 79] | classical
haveI : ∀ i, Module.Free R (grade R M i) := fun i ↦ grade_free R M i
haveI : Module.Free R (DirectSum ℕ fun i ↦ ↥(grade R M i)) := by
apply Module.Free.directSum
apply Module.Free.of_equiv (DirectSum.decomposeLinearEquiv (grade R M)).symm | R : Type u
inst✝³ : CommRing R
M : Type u_1
inst✝² : AddCommGroup M
inst✝¹ : Module R M
inst✝ : Module.Free R M
⊢ Module.Free R (DividedPowerAlgebra R M) | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
R : Type u
inst✝³ : CommRing R
M : Type u_1
inst✝² : AddCommGroup M
inst✝¹ : Module R M
inst✝ : Module.Free R M
⊢ Module.Free R (DividedPowerAlgebra R M)
TACTIC:
|
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/DPAlgebra/PolynomialMap.lean | DividedPowerAlgebra.toModule_free | [133, 1] | [139, 79] | haveI : ∀ i, Module.Free R (grade R M i) := fun i ↦ grade_free R M i | R : Type u
inst✝³ : CommRing R
M : Type u_1
inst✝² : AddCommGroup M
inst✝¹ : Module R M
inst✝ : Module.Free R M
⊢ Module.Free R (DividedPowerAlgebra R M) | R : Type u
inst✝³ : CommRing R
M : Type u_1
inst✝² : AddCommGroup M
inst✝¹ : Module R M
inst✝ : Module.Free R M
this : ∀ (i : ℕ), Module.Free R ↥(grade R M i)
⊢ Module.Free R (DividedPowerAlgebra R M) | Please generate a tactic in lean4 to solve the state.
STATE:
R : Type u
inst✝³ : CommRing R
M : Type u_1
inst✝² : AddCommGroup M
inst✝¹ : Module R M
inst✝ : Module.Free R M
⊢ Module.Free R (DividedPowerAlgebra R M)
TACTIC:
|
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/DPAlgebra/PolynomialMap.lean | DividedPowerAlgebra.toModule_free | [133, 1] | [139, 79] | haveI : Module.Free R (DirectSum ℕ fun i ↦ ↥(grade R M i)) := by
apply Module.Free.directSum | R : Type u
inst✝³ : CommRing R
M : Type u_1
inst✝² : AddCommGroup M
inst✝¹ : Module R M
inst✝ : Module.Free R M
this : ∀ (i : ℕ), Module.Free R ↥(grade R M i)
⊢ Module.Free R (DividedPowerAlgebra R M) | R : Type u
inst✝³ : CommRing R
M : Type u_1
inst✝² : AddCommGroup M
inst✝¹ : Module R M
inst✝ : Module.Free R M
this✝ : ∀ (i : ℕ), Module.Free R ↥(grade R M i)
this : Module.Free R (DirectSum ℕ fun i => ↥(grade R M i))
⊢ Module.Free R (DividedPowerAlgebra R M) | Please generate a tactic in lean4 to solve the state.
STATE:
R : Type u
inst✝³ : CommRing R
M : Type u_1
inst✝² : AddCommGroup M
inst✝¹ : Module R M
inst✝ : Module.Free R M
this : ∀ (i : ℕ), Module.Free R ↥(grade R M i)
⊢ Module.Free R (DividedPowerAlgebra R M)
TACTIC:
|
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/DPAlgebra/PolynomialMap.lean | DividedPowerAlgebra.toModule_free | [133, 1] | [139, 79] | apply Module.Free.of_equiv (DirectSum.decomposeLinearEquiv (grade R M)).symm | R : Type u
inst✝³ : CommRing R
M : Type u_1
inst✝² : AddCommGroup M
inst✝¹ : Module R M
inst✝ : Module.Free R M
this✝ : ∀ (i : ℕ), Module.Free R ↥(grade R M i)
this : Module.Free R (DirectSum ℕ fun i => ↥(grade R M i))
⊢ Module.Free R (DividedPowerAlgebra R M) | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
R : Type u
inst✝³ : CommRing R
M : Type u_1
inst✝² : AddCommGroup M
inst✝¹ : Module R M
inst✝ : Module.Free R M
this✝ : ∀ (i : ℕ), Module.Free R ↥(grade R M i)
this : Module.Free R (DirectSum ℕ fun i => ↥(grade R M i))
⊢ Module.Free R (DividedPowerAlgebra R M... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/DPAlgebra/PolynomialMap.lean | DividedPowerAlgebra.toModule_free | [133, 1] | [139, 79] | apply Module.Free.directSum | R : Type u
inst✝³ : CommRing R
M : Type u_1
inst✝² : AddCommGroup M
inst✝¹ : Module R M
inst✝ : Module.Free R M
this : ∀ (i : ℕ), Module.Free R ↥(grade R M i)
⊢ Module.Free R (DirectSum ℕ fun i => ↥(grade R M i)) | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
R : Type u
inst✝³ : CommRing R
M : Type u_1
inst✝² : AddCommGroup M
inst✝¹ : Module R M
inst✝ : Module.Free R M
this : ∀ (i : ℕ), Module.Free R ↥(grade R M i)
⊢ Module.Free R (DirectSum ℕ fun i => ↥(grade R M i))
TACTIC:
|
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/RatAlgebra.lean | Nat.isUnitFactorial | [7, 1] | [9, 30] | rw [isUnit_iff_ne_zero, ne_eq, Nat.cast_eq_zero] | n : ℕ
⊢ IsUnit ↑n ! | n : ℕ
⊢ ¬n ! = 0 | Please generate a tactic in lean4 to solve the state.
STATE:
n : ℕ
⊢ IsUnit ↑n !
TACTIC:
|
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/RatAlgebra.lean | Nat.isUnitFactorial | [7, 1] | [9, 30] | apply Nat.factorial_ne_zero | n : ℕ
⊢ ¬n ! = 0 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
n : ℕ
⊢ ¬n ! = 0
TACTIC:
|
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/RatAlgebra.lean | Nat.inv_smul_eq_invCast_mul | [11, 1] | [27, 26] | cases' Nat.eq_zero_or_pos n with hn hn | A : Type u_1
inst✝¹ : CommSemiring A
inst✝ : Algebra ℚ A
n : ℕ
a : A
⊢ Ring.inverse ↑n • a = Ring.inverse ↑n * a | case inl
A : Type u_1
inst✝¹ : CommSemiring A
inst✝ : Algebra ℚ A
n : ℕ
a : A
hn : n = 0
⊢ Ring.inverse ↑n • a = Ring.inverse ↑n * a
case inr
A : Type u_1
inst✝¹ : CommSemiring A
inst✝ : Algebra ℚ A
n : ℕ
a : A
hn : n > 0
⊢ Ring.inverse ↑n • a = Ring.inverse ↑n * a | Please generate a tactic in lean4 to solve the state.
STATE:
A : Type u_1
inst✝¹ : CommSemiring A
inst✝ : Algebra ℚ A
n : ℕ
a : A
⊢ Ring.inverse ↑n • a = Ring.inverse ↑n * a
TACTIC:
|
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/RatAlgebra.lean | Nat.inv_smul_eq_invCast_mul | [11, 1] | [27, 26] | . simp only [hn, Nat.cast_zero, isUnit_zero_iff, not_false_eq_true, Ring.inverse_non_unit, zero_smul,
Ring.inverse_zero, zero_mul] | case inl
A : Type u_1
inst✝¹ : CommSemiring A
inst✝ : Algebra ℚ A
n : ℕ
a : A
hn : n = 0
⊢ Ring.inverse ↑n • a = Ring.inverse ↑n * a
case inr
A : Type u_1
inst✝¹ : CommSemiring A
inst✝ : Algebra ℚ A
n : ℕ
a : A
hn : n > 0
⊢ Ring.inverse ↑n • a = Ring.inverse ↑n * a | case inr
A : Type u_1
inst✝¹ : CommSemiring A
inst✝ : Algebra ℚ A
n : ℕ
a : A
hn : n > 0
⊢ Ring.inverse ↑n • a = Ring.inverse ↑n * a | Please generate a tactic in lean4 to solve the state.
STATE:
case inl
A : Type u_1
inst✝¹ : CommSemiring A
inst✝ : Algebra ℚ A
n : ℕ
a : A
hn : n = 0
⊢ Ring.inverse ↑n • a = Ring.inverse ↑n * a
case inr
A : Type u_1
inst✝¹ : CommSemiring A
inst✝ : Algebra ℚ A
n : ℕ
a : A
hn : n > 0
⊢ Ring.inverse ↑n • a = Ring.inverse... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/RatAlgebra.lean | Nat.inv_smul_eq_invCast_mul | [11, 1] | [27, 26] | . suffices hn' : IsUnit (n : ℚ) by
simp only [Algebra.smul_def, ← map_natCast (algebraMap ℚ A)]
apply symm
rw [Ring.inverse_mul_eq_iff_eq_mul _ _ _ (RingHom.isUnit_map _ hn'), ← mul_assoc]
apply symm
convert @one_mul A _ _
simp only [← map_mul, ← map_one (algebraMap ℚ A)]
apply congr_arg
... | case inr
A : Type u_1
inst✝¹ : CommSemiring A
inst✝ : Algebra ℚ A
n : ℕ
a : A
hn : n > 0
⊢ Ring.inverse ↑n • a = Ring.inverse ↑n * a | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case inr
A : Type u_1
inst✝¹ : CommSemiring A
inst✝ : Algebra ℚ A
n : ℕ
a : A
hn : n > 0
⊢ Ring.inverse ↑n • a = Ring.inverse ↑n * a
TACTIC:
|
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/RatAlgebra.lean | Nat.inv_smul_eq_invCast_mul | [11, 1] | [27, 26] | simp only [hn, Nat.cast_zero, isUnit_zero_iff, not_false_eq_true, Ring.inverse_non_unit, zero_smul,
Ring.inverse_zero, zero_mul] | case inl
A : Type u_1
inst✝¹ : CommSemiring A
inst✝ : Algebra ℚ A
n : ℕ
a : A
hn : n = 0
⊢ Ring.inverse ↑n • a = Ring.inverse ↑n * a | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case inl
A : Type u_1
inst✝¹ : CommSemiring A
inst✝ : Algebra ℚ A
n : ℕ
a : A
hn : n = 0
⊢ Ring.inverse ↑n • a = Ring.inverse ↑n * a
TACTIC:
|
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/RatAlgebra.lean | Nat.inv_smul_eq_invCast_mul | [11, 1] | [27, 26] | suffices hn' : IsUnit (n : ℚ) by
simp only [Algebra.smul_def, ← map_natCast (algebraMap ℚ A)]
apply symm
rw [Ring.inverse_mul_eq_iff_eq_mul _ _ _ (RingHom.isUnit_map _ hn'), ← mul_assoc]
apply symm
convert @one_mul A _ _
simp only [← map_mul, ← map_one (algebraMap ℚ A)]
apply congr_arg
apply symm
rw [... | case inr
A : Type u_1
inst✝¹ : CommSemiring A
inst✝ : Algebra ℚ A
n : ℕ
a : A
hn : n > 0
⊢ Ring.inverse ↑n • a = Ring.inverse ↑n * a | case inr
A : Type u_1
inst✝¹ : CommSemiring A
inst✝ : Algebra ℚ A
n : ℕ
a : A
hn : n > 0
⊢ IsUnit ↑n | Please generate a tactic in lean4 to solve the state.
STATE:
case inr
A : Type u_1
inst✝¹ : CommSemiring A
inst✝ : Algebra ℚ A
n : ℕ
a : A
hn : n > 0
⊢ Ring.inverse ↑n • a = Ring.inverse ↑n * a
TACTIC:
|
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/RatAlgebra.lean | Nat.inv_smul_eq_invCast_mul | [11, 1] | [27, 26] | rw [isUnit_iff_ne_zero, ne_eq, Nat.cast_eq_zero] | case inr
A : Type u_1
inst✝¹ : CommSemiring A
inst✝ : Algebra ℚ A
n : ℕ
a : A
hn : n > 0
⊢ IsUnit ↑n | case inr
A : Type u_1
inst✝¹ : CommSemiring A
inst✝ : Algebra ℚ A
n : ℕ
a : A
hn : n > 0
⊢ ¬n = 0 | Please generate a tactic in lean4 to solve the state.
STATE:
case inr
A : Type u_1
inst✝¹ : CommSemiring A
inst✝ : Algebra ℚ A
n : ℕ
a : A
hn : n > 0
⊢ IsUnit ↑n
TACTIC:
|
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/RatAlgebra.lean | Nat.inv_smul_eq_invCast_mul | [11, 1] | [27, 26] | exact Nat.ne_of_gt hn | case inr
A : Type u_1
inst✝¹ : CommSemiring A
inst✝ : Algebra ℚ A
n : ℕ
a : A
hn : n > 0
⊢ ¬n = 0 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case inr
A : Type u_1
inst✝¹ : CommSemiring A
inst✝ : Algebra ℚ A
n : ℕ
a : A
hn : n > 0
⊢ ¬n = 0
TACTIC:
|
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/RatAlgebra.lean | Nat.inv_smul_eq_invCast_mul | [11, 1] | [27, 26] | simp only [Algebra.smul_def, ← map_natCast (algebraMap ℚ A)] | A : Type u_1
inst✝¹ : CommSemiring A
inst✝ : Algebra ℚ A
n : ℕ
a : A
hn : n > 0
hn' : IsUnit ↑n
⊢ Ring.inverse ↑n • a = Ring.inverse ↑n * a | A : Type u_1
inst✝¹ : CommSemiring A
inst✝ : Algebra ℚ A
n : ℕ
a : A
hn : n > 0
hn' : IsUnit ↑n
⊢ (algebraMap ℚ A) (Ring.inverse ↑n) * a = Ring.inverse ((algebraMap ℚ A) ↑n) * a | Please generate a tactic in lean4 to solve the state.
STATE:
A : Type u_1
inst✝¹ : CommSemiring A
inst✝ : Algebra ℚ A
n : ℕ
a : A
hn : n > 0
hn' : IsUnit ↑n
⊢ Ring.inverse ↑n • a = Ring.inverse ↑n * a
TACTIC:
|
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/RatAlgebra.lean | Nat.inv_smul_eq_invCast_mul | [11, 1] | [27, 26] | apply symm | A : Type u_1
inst✝¹ : CommSemiring A
inst✝ : Algebra ℚ A
n : ℕ
a : A
hn : n > 0
hn' : IsUnit ↑n
⊢ (algebraMap ℚ A) (Ring.inverse ↑n) * a = Ring.inverse ((algebraMap ℚ A) ↑n) * a | case a
A : Type u_1
inst✝¹ : CommSemiring A
inst✝ : Algebra ℚ A
n : ℕ
a : A
hn : n > 0
hn' : IsUnit ↑n
⊢ Ring.inverse ((algebraMap ℚ A) ↑n) * a = (algebraMap ℚ A) (Ring.inverse ↑n) * a | Please generate a tactic in lean4 to solve the state.
STATE:
A : Type u_1
inst✝¹ : CommSemiring A
inst✝ : Algebra ℚ A
n : ℕ
a : A
hn : n > 0
hn' : IsUnit ↑n
⊢ (algebraMap ℚ A) (Ring.inverse ↑n) * a = Ring.inverse ((algebraMap ℚ A) ↑n) * a
TACTIC:
|
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/RatAlgebra.lean | Nat.inv_smul_eq_invCast_mul | [11, 1] | [27, 26] | rw [Ring.inverse_mul_eq_iff_eq_mul _ _ _ (RingHom.isUnit_map _ hn'), ← mul_assoc] | case a
A : Type u_1
inst✝¹ : CommSemiring A
inst✝ : Algebra ℚ A
n : ℕ
a : A
hn : n > 0
hn' : IsUnit ↑n
⊢ Ring.inverse ((algebraMap ℚ A) ↑n) * a = (algebraMap ℚ A) (Ring.inverse ↑n) * a | case a
A : Type u_1
inst✝¹ : CommSemiring A
inst✝ : Algebra ℚ A
n : ℕ
a : A
hn : n > 0
hn' : IsUnit ↑n
⊢ a = (algebraMap ℚ A) ↑n * (algebraMap ℚ A) (Ring.inverse ↑n) * a | Please generate a tactic in lean4 to solve the state.
STATE:
case a
A : Type u_1
inst✝¹ : CommSemiring A
inst✝ : Algebra ℚ A
n : ℕ
a : A
hn : n > 0
hn' : IsUnit ↑n
⊢ Ring.inverse ((algebraMap ℚ A) ↑n) * a = (algebraMap ℚ A) (Ring.inverse ↑n) * a
TACTIC:
|
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/RatAlgebra.lean | Nat.inv_smul_eq_invCast_mul | [11, 1] | [27, 26] | apply symm | case a
A : Type u_1
inst✝¹ : CommSemiring A
inst✝ : Algebra ℚ A
n : ℕ
a : A
hn : n > 0
hn' : IsUnit ↑n
⊢ a = (algebraMap ℚ A) ↑n * (algebraMap ℚ A) (Ring.inverse ↑n) * a | case a.a
A : Type u_1
inst✝¹ : CommSemiring A
inst✝ : Algebra ℚ A
n : ℕ
a : A
hn : n > 0
hn' : IsUnit ↑n
⊢ (algebraMap ℚ A) ↑n * (algebraMap ℚ A) (Ring.inverse ↑n) * a = a | Please generate a tactic in lean4 to solve the state.
STATE:
case a
A : Type u_1
inst✝¹ : CommSemiring A
inst✝ : Algebra ℚ A
n : ℕ
a : A
hn : n > 0
hn' : IsUnit ↑n
⊢ a = (algebraMap ℚ A) ↑n * (algebraMap ℚ A) (Ring.inverse ↑n) * a
TACTIC:
|
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/RatAlgebra.lean | Nat.inv_smul_eq_invCast_mul | [11, 1] | [27, 26] | convert @one_mul A _ _ | case a.a
A : Type u_1
inst✝¹ : CommSemiring A
inst✝ : Algebra ℚ A
n : ℕ
a : A
hn : n > 0
hn' : IsUnit ↑n
⊢ (algebraMap ℚ A) ↑n * (algebraMap ℚ A) (Ring.inverse ↑n) * a = a | case h.e'_2.h.e'_5
A : Type u_1
inst✝¹ : CommSemiring A
inst✝ : Algebra ℚ A
n : ℕ
a : A
hn : n > 0
hn' : IsUnit ↑n
⊢ (algebraMap ℚ A) ↑n * (algebraMap ℚ A) (Ring.inverse ↑n) = 1 | Please generate a tactic in lean4 to solve the state.
STATE:
case a.a
A : Type u_1
inst✝¹ : CommSemiring A
inst✝ : Algebra ℚ A
n : ℕ
a : A
hn : n > 0
hn' : IsUnit ↑n
⊢ (algebraMap ℚ A) ↑n * (algebraMap ℚ A) (Ring.inverse ↑n) * a = a
TACTIC:
|
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/RatAlgebra.lean | Nat.inv_smul_eq_invCast_mul | [11, 1] | [27, 26] | simp only [← map_mul, ← map_one (algebraMap ℚ A)] | case h.e'_2.h.e'_5
A : Type u_1
inst✝¹ : CommSemiring A
inst✝ : Algebra ℚ A
n : ℕ
a : A
hn : n > 0
hn' : IsUnit ↑n
⊢ (algebraMap ℚ A) ↑n * (algebraMap ℚ A) (Ring.inverse ↑n) = 1 | case h.e'_2.h.e'_5
A : Type u_1
inst✝¹ : CommSemiring A
inst✝ : Algebra ℚ A
n : ℕ
a : A
hn : n > 0
hn' : IsUnit ↑n
⊢ (algebraMap ℚ A) (↑n * Ring.inverse ↑n) = (algebraMap ℚ A) 1 | Please generate a tactic in lean4 to solve the state.
STATE:
case h.e'_2.h.e'_5
A : Type u_1
inst✝¹ : CommSemiring A
inst✝ : Algebra ℚ A
n : ℕ
a : A
hn : n > 0
hn' : IsUnit ↑n
⊢ (algebraMap ℚ A) ↑n * (algebraMap ℚ A) (Ring.inverse ↑n) = 1
TACTIC:
|
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/RatAlgebra.lean | Nat.inv_smul_eq_invCast_mul | [11, 1] | [27, 26] | apply congr_arg | case h.e'_2.h.e'_5
A : Type u_1
inst✝¹ : CommSemiring A
inst✝ : Algebra ℚ A
n : ℕ
a : A
hn : n > 0
hn' : IsUnit ↑n
⊢ (algebraMap ℚ A) (↑n * Ring.inverse ↑n) = (algebraMap ℚ A) 1 | case h.e'_2.h.e'_5.h
A : Type u_1
inst✝¹ : CommSemiring A
inst✝ : Algebra ℚ A
n : ℕ
a : A
hn : n > 0
hn' : IsUnit ↑n
⊢ ↑n * Ring.inverse ↑n = 1 | Please generate a tactic in lean4 to solve the state.
STATE:
case h.e'_2.h.e'_5
A : Type u_1
inst✝¹ : CommSemiring A
inst✝ : Algebra ℚ A
n : ℕ
a : A
hn : n > 0
hn' : IsUnit ↑n
⊢ (algebraMap ℚ A) (↑n * Ring.inverse ↑n) = (algebraMap ℚ A) 1
TACTIC:
|
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/RatAlgebra.lean | Nat.inv_smul_eq_invCast_mul | [11, 1] | [27, 26] | apply symm | case h.e'_2.h.e'_5.h
A : Type u_1
inst✝¹ : CommSemiring A
inst✝ : Algebra ℚ A
n : ℕ
a : A
hn : n > 0
hn' : IsUnit ↑n
⊢ ↑n * Ring.inverse ↑n = 1 | case h.e'_2.h.e'_5.h.a
A : Type u_1
inst✝¹ : CommSemiring A
inst✝ : Algebra ℚ A
n : ℕ
a : A
hn : n > 0
hn' : IsUnit ↑n
⊢ 1 = ↑n * Ring.inverse ↑n | Please generate a tactic in lean4 to solve the state.
STATE:
case h.e'_2.h.e'_5.h
A : Type u_1
inst✝¹ : CommSemiring A
inst✝ : Algebra ℚ A
n : ℕ
a : A
hn : n > 0
hn' : IsUnit ↑n
⊢ ↑n * Ring.inverse ↑n = 1
TACTIC:
|
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/RatAlgebra.lean | Nat.inv_smul_eq_invCast_mul | [11, 1] | [27, 26] | rw [Ring.eq_mul_inverse_iff_mul_eq _ _ _ hn', one_mul] | case h.e'_2.h.e'_5.h.a
A : Type u_1
inst✝¹ : CommSemiring A
inst✝ : Algebra ℚ A
n : ℕ
a : A
hn : n > 0
hn' : IsUnit ↑n
⊢ 1 = ↑n * Ring.inverse ↑n | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case h.e'_2.h.e'_5.h.a
A : Type u_1
inst✝¹ : CommSemiring A
inst✝ : Algebra ℚ A
n : ℕ
a : A
hn : n > 0
hn' : IsUnit ↑n
⊢ 1 = ↑n * Ring.inverse ↑n
TACTIC:
|
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/RatAlgebra.lean | DividedPowers.OfInvertibleFactorial.dpow_eq_of_mem | [41, 1] | [43, 36] | simp only [dpow] | A : Type u_1
inst✝ : CommSemiring A
I✝ I : Ideal A
m : ℕ
x : A
hx : x ∈ I
⊢ dpow I m x = Ring.inverse ↑m ! * x ^ m | A : Type u_1
inst✝ : CommSemiring A
I✝ I : Ideal A
m : ℕ
x : A
hx : x ∈ I
⊢ (if x ∈ I then Ring.inverse ↑m ! * x ^ m else 0) = Ring.inverse ↑m ! * x ^ m | Please generate a tactic in lean4 to solve the state.
STATE:
A : Type u_1
inst✝ : CommSemiring A
I✝ I : Ideal A
m : ℕ
x : A
hx : x ∈ I
⊢ dpow I m x = Ring.inverse ↑m ! * x ^ m
TACTIC:
|
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/RatAlgebra.lean | DividedPowers.OfInvertibleFactorial.dpow_eq_of_mem | [41, 1] | [43, 36] | rw [if_pos hx] | A : Type u_1
inst✝ : CommSemiring A
I✝ I : Ideal A
m : ℕ
x : A
hx : x ∈ I
⊢ (if x ∈ I then Ring.inverse ↑m ! * x ^ m else 0) = Ring.inverse ↑m ! * x ^ m | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
A : Type u_1
inst✝ : CommSemiring A
I✝ I : Ideal A
m : ℕ
x : A
hx : x ∈ I
⊢ (if x ∈ I then Ring.inverse ↑m ! * x ^ m else 0) = Ring.inverse ↑m ! * x ^ m
TACTIC:
|
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/RatAlgebra.lean | DividedPowers.OfInvertibleFactorial.dpow_eq_of_not_mem | [46, 1] | [47, 36] | simp only [dpow] | A : Type u_1
inst✝ : CommSemiring A
I✝ I : Ideal A
m : ℕ
x : A
hx : x ∉ I
⊢ dpow I m x = 0 | A : Type u_1
inst✝ : CommSemiring A
I✝ I : Ideal A
m : ℕ
x : A
hx : x ∉ I
⊢ (if x ∈ I then Ring.inverse ↑m ! * x ^ m else 0) = 0 | Please generate a tactic in lean4 to solve the state.
STATE:
A : Type u_1
inst✝ : CommSemiring A
I✝ I : Ideal A
m : ℕ
x : A
hx : x ∉ I
⊢ dpow I m x = 0
TACTIC:
|
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/RatAlgebra.lean | DividedPowers.OfInvertibleFactorial.dpow_eq_of_not_mem | [46, 1] | [47, 36] | rw [if_neg hx] | A : Type u_1
inst✝ : CommSemiring A
I✝ I : Ideal A
m : ℕ
x : A
hx : x ∉ I
⊢ (if x ∈ I then Ring.inverse ↑m ! * x ^ m else 0) = 0 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
A : Type u_1
inst✝ : CommSemiring A
I✝ I : Ideal A
m : ℕ
x : A
hx : x ∉ I
⊢ (if x ∈ I then Ring.inverse ↑m ! * x ^ m else 0) = 0
TACTIC:
|
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/RatAlgebra.lean | DividedPowers.OfInvertibleFactorial.dpow_null | [50, 1] | [51, 36] | simp only [dpow] | A : Type u_1
inst✝ : CommSemiring A
I : Ideal A
m : ℕ
x : A
hx : x ∉ I
⊢ dpow I m x = 0 | A : Type u_1
inst✝ : CommSemiring A
I : Ideal A
m : ℕ
x : A
hx : x ∉ I
⊢ (if x ∈ I then Ring.inverse ↑m ! * x ^ m else 0) = 0 | Please generate a tactic in lean4 to solve the state.
STATE:
A : Type u_1
inst✝ : CommSemiring A
I : Ideal A
m : ℕ
x : A
hx : x ∉ I
⊢ dpow I m x = 0
TACTIC:
|
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/RatAlgebra.lean | DividedPowers.OfInvertibleFactorial.dpow_null | [50, 1] | [51, 36] | rw [if_neg hx] | A : Type u_1
inst✝ : CommSemiring A
I : Ideal A
m : ℕ
x : A
hx : x ∉ I
⊢ (if x ∈ I then Ring.inverse ↑m ! * x ^ m else 0) = 0 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
A : Type u_1
inst✝ : CommSemiring A
I : Ideal A
m : ℕ
x : A
hx : x ∉ I
⊢ (if x ∈ I then Ring.inverse ↑m ! * x ^ m else 0) = 0
TACTIC:
|
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/RatAlgebra.lean | DividedPowers.OfInvertibleFactorial.dpow_zero | [54, 1] | [55, 100] | simp only [dpow, factorial_zero, cast_one, Ring.inverse_one, _root_.pow_zero, mul_one, if_pos hx] | A : Type u_1
inst✝ : CommSemiring A
I : Ideal A
x : A
hx : x ∈ I
⊢ dpow I 0 x = 1 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
A : Type u_1
inst✝ : CommSemiring A
I : Ideal A
x : A
hx : x ∈ I
⊢ dpow I 0 x = 1
TACTIC:
|
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/RatAlgebra.lean | DividedPowers.OfInvertibleFactorial.dpow_one | [58, 1] | [59, 96] | rw [dpow_eq_of_mem 1 hx, pow_one, Nat.factorial_one, Nat.cast_one, Ring.inverse_one, one_mul] | A : Type u_1
inst✝ : CommSemiring A
I : Ideal A
x : A
hx : x ∈ I
⊢ dpow I 1 x = x | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
A : Type u_1
inst✝ : CommSemiring A
I : Ideal A
x : A
hx : x ∈ I
⊢ dpow I 1 x = x
TACTIC:
|
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/RatAlgebra.lean | DividedPowers.OfInvertibleFactorial.dpow_mem | [62, 1] | [65, 85] | rw [dpow_eq_of_mem m hx] | A : Type u_1
inst✝ : CommSemiring A
I : Ideal A
m : ℕ
hm : m ≠ 0
x : A
hx : x ∈ I
⊢ dpow I m x ∈ I | A : Type u_1
inst✝ : CommSemiring A
I : Ideal A
m : ℕ
hm : m ≠ 0
x : A
hx : x ∈ I
⊢ Ring.inverse ↑m ! * x ^ m ∈ I | Please generate a tactic in lean4 to solve the state.
STATE:
A : Type u_1
inst✝ : CommSemiring A
I : Ideal A
m : ℕ
hm : m ≠ 0
x : A
hx : x ∈ I
⊢ dpow I m x ∈ I
TACTIC:
|
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/RatAlgebra.lean | DividedPowers.OfInvertibleFactorial.dpow_mem | [62, 1] | [65, 85] | exact Ideal.mul_mem_left I _ (Ideal.pow_mem_of_mem I hx _ (Nat.pos_of_ne_zero hm)) | A : Type u_1
inst✝ : CommSemiring A
I : Ideal A
m : ℕ
hm : m ≠ 0
x : A
hx : x ∈ I
⊢ Ring.inverse ↑m ! * x ^ m ∈ I | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
A : Type u_1
inst✝ : CommSemiring A
I : Ideal A
m : ℕ
hm : m ≠ 0
x : A
hx : x ∈ I
⊢ Ring.inverse ↑m ! * x ^ m ∈ I
TACTIC:
|
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/RatAlgebra.lean | DividedPowers.OfInvertibleFactorial.foo | [68, 1] | [80, 34] | rw [Finset.mem_antidiagonal] at hk | A : Type u_1
inst✝ : CommSemiring A
I : Ideal A
m : ℕ
k : ℕ × ℕ
hm : IsUnit ↑m !
hk : k ∈ Finset.antidiagonal m
⊢ ↑(m.choose k.1) = ↑m ! * Ring.inverse ↑k.1! * Ring.inverse ↑k.2! | A : Type u_1
inst✝ : CommSemiring A
I : Ideal A
m : ℕ
k : ℕ × ℕ
hm : IsUnit ↑m !
hk : k.1 + k.2 = m
⊢ ↑(m.choose k.1) = ↑m ! * Ring.inverse ↑k.1! * Ring.inverse ↑k.2! | Please generate a tactic in lean4 to solve the state.
STATE:
A : Type u_1
inst✝ : CommSemiring A
I : Ideal A
m : ℕ
k : ℕ × ℕ
hm : IsUnit ↑m !
hk : k ∈ Finset.antidiagonal m
⊢ ↑(m.choose k.1) = ↑m ! * Ring.inverse ↑k.1! * Ring.inverse ↑k.2!
TACTIC:
|
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/RatAlgebra.lean | DividedPowers.OfInvertibleFactorial.foo | [68, 1] | [80, 34] | rw [Ring.eq_mul_inverse_iff_mul_eq] | A : Type u_1
inst✝ : CommSemiring A
I : Ideal A
m : ℕ
k : ℕ × ℕ
hm : IsUnit ↑m !
hk : k.1 + k.2 = m
⊢ ↑(m.choose k.1) = ↑m ! * Ring.inverse ↑k.1! * Ring.inverse ↑k.2! | A : Type u_1
inst✝ : CommSemiring A
I : Ideal A
m : ℕ
k : ℕ × ℕ
hm : IsUnit ↑m !
hk : k.1 + k.2 = m
⊢ ↑(m.choose k.1) * ↑k.2! = ↑m ! * Ring.inverse ↑k.1!
case h
A : Type u_1
inst✝ : CommSemiring A
I : Ideal A
m : ℕ
k : ℕ × ℕ
hm : IsUnit ↑m !
hk : k.1 + k.2 = m
⊢ IsUnit ↑k.2! | Please generate a tactic in lean4 to solve the state.
STATE:
A : Type u_1
inst✝ : CommSemiring A
I : Ideal A
m : ℕ
k : ℕ × ℕ
hm : IsUnit ↑m !
hk : k.1 + k.2 = m
⊢ ↑(m.choose k.1) = ↑m ! * Ring.inverse ↑k.1! * Ring.inverse ↑k.2!
TACTIC:
|
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/RatAlgebra.lean | DividedPowers.OfInvertibleFactorial.foo | [68, 1] | [80, 34] | rw [Ring.eq_mul_inverse_iff_mul_eq] | A : Type u_1
inst✝ : CommSemiring A
I : Ideal A
m : ℕ
k : ℕ × ℕ
hm : IsUnit ↑m !
hk : k.1 + k.2 = m
⊢ ↑(m.choose k.1) * ↑k.2! = ↑m ! * Ring.inverse ↑k.1!
case h
A : Type u_1
inst✝ : CommSemiring A
I : Ideal A
m : ℕ
k : ℕ × ℕ
hm : IsUnit ↑m !
hk : k.1 + k.2 = m
⊢ IsUnit ↑k.2! | A : Type u_1
inst✝ : CommSemiring A
I : Ideal A
m : ℕ
k : ℕ × ℕ
hm : IsUnit ↑m !
hk : k.1 + k.2 = m
⊢ ↑(m.choose k.1) * ↑k.2! * ↑k.1! = ↑m !
case h
A : Type u_1
inst✝ : CommSemiring A
I : Ideal A
m : ℕ
k : ℕ × ℕ
hm : IsUnit ↑m !
hk : k.1 + k.2 = m
⊢ IsUnit ↑k.1!
case h
A : Type u_1
inst✝ : CommSemiring A
I : Ideal A
... | Please generate a tactic in lean4 to solve the state.
STATE:
A : Type u_1
inst✝ : CommSemiring A
I : Ideal A
m : ℕ
k : ℕ × ℕ
hm : IsUnit ↑m !
hk : k.1 + k.2 = m
⊢ ↑(m.choose k.1) * ↑k.2! = ↑m ! * Ring.inverse ↑k.1!
case h
A : Type u_1
inst✝ : CommSemiring A
I : Ideal A
m : ℕ
k : ℕ × ℕ
hm : IsUnit ↑m !
hk : k.1 + k.2 =... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/RatAlgebra.lean | DividedPowers.OfInvertibleFactorial.foo | [68, 1] | [80, 34] | rw [← hk, ← Nat.cast_mul, ← Nat.cast_mul, add_comm, Nat.add_choose_mul_factorial_mul_factorial] | A : Type u_1
inst✝ : CommSemiring A
I : Ideal A
m : ℕ
k : ℕ × ℕ
hm : IsUnit ↑m !
hk : k.1 + k.2 = m
⊢ ↑(m.choose k.1) * ↑k.2! * ↑k.1! = ↑m ! | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
A : Type u_1
inst✝ : CommSemiring A
I : Ideal A
m : ℕ
k : ℕ × ℕ
hm : IsUnit ↑m !
hk : k.1 + k.2 = m
⊢ ↑(m.choose k.1) * ↑k.2! * ↑k.1! = ↑m !
TACTIC:
|
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/RatAlgebra.lean | DividedPowers.OfInvertibleFactorial.foo | [68, 1] | [80, 34] | apply factorial_isUnit' hm | case h
A : Type u_1
inst✝ : CommSemiring A
I : Ideal A
m : ℕ
k : ℕ × ℕ
hm : IsUnit ↑m !
hk : k.1 + k.2 = m
⊢ IsUnit ↑k.1! | case h
A : Type u_1
inst✝ : CommSemiring A
I : Ideal A
m : ℕ
k : ℕ × ℕ
hm : IsUnit ↑m !
hk : k.1 + k.2 = m
⊢ k.1 ≤ m | Please generate a tactic in lean4 to solve the state.
STATE:
case h
A : Type u_1
inst✝ : CommSemiring A
I : Ideal A
m : ℕ
k : ℕ × ℕ
hm : IsUnit ↑m !
hk : k.1 + k.2 = m
⊢ IsUnit ↑k.1!
TACTIC:
|
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/RatAlgebra.lean | DividedPowers.OfInvertibleFactorial.foo | [68, 1] | [80, 34] | rw [← hk] | case h
A : Type u_1
inst✝ : CommSemiring A
I : Ideal A
m : ℕ
k : ℕ × ℕ
hm : IsUnit ↑m !
hk : k.1 + k.2 = m
⊢ k.1 ≤ m | case h
A : Type u_1
inst✝ : CommSemiring A
I : Ideal A
m : ℕ
k : ℕ × ℕ
hm : IsUnit ↑m !
hk : k.1 + k.2 = m
⊢ k.1 ≤ k.1 + k.2 | Please generate a tactic in lean4 to solve the state.
STATE:
case h
A : Type u_1
inst✝ : CommSemiring A
I : Ideal A
m : ℕ
k : ℕ × ℕ
hm : IsUnit ↑m !
hk : k.1 + k.2 = m
⊢ k.1 ≤ m
TACTIC:
|
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/RatAlgebra.lean | DividedPowers.OfInvertibleFactorial.foo | [68, 1] | [80, 34] | exact Nat.le_add_right k.1 k.2 | case h
A : Type u_1
inst✝ : CommSemiring A
I : Ideal A
m : ℕ
k : ℕ × ℕ
hm : IsUnit ↑m !
hk : k.1 + k.2 = m
⊢ k.1 ≤ k.1 + k.2 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case h
A : Type u_1
inst✝ : CommSemiring A
I : Ideal A
m : ℕ
k : ℕ × ℕ
hm : IsUnit ↑m !
hk : k.1 + k.2 = m
⊢ k.1 ≤ k.1 + k.2
TACTIC:
|
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/RatAlgebra.lean | DividedPowers.OfInvertibleFactorial.foo | [68, 1] | [80, 34] | apply factorial_isUnit' hm | case h
A : Type u_1
inst✝ : CommSemiring A
I : Ideal A
m : ℕ
k : ℕ × ℕ
hm : IsUnit ↑m !
hk : k.1 + k.2 = m
⊢ IsUnit ↑k.2! | case h
A : Type u_1
inst✝ : CommSemiring A
I : Ideal A
m : ℕ
k : ℕ × ℕ
hm : IsUnit ↑m !
hk : k.1 + k.2 = m
⊢ k.2 ≤ m | Please generate a tactic in lean4 to solve the state.
STATE:
case h
A : Type u_1
inst✝ : CommSemiring A
I : Ideal A
m : ℕ
k : ℕ × ℕ
hm : IsUnit ↑m !
hk : k.1 + k.2 = m
⊢ IsUnit ↑k.2!
TACTIC:
|
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/RatAlgebra.lean | DividedPowers.OfInvertibleFactorial.foo | [68, 1] | [80, 34] | rw [← hk] | case h
A : Type u_1
inst✝ : CommSemiring A
I : Ideal A
m : ℕ
k : ℕ × ℕ
hm : IsUnit ↑m !
hk : k.1 + k.2 = m
⊢ k.2 ≤ m | case h
A : Type u_1
inst✝ : CommSemiring A
I : Ideal A
m : ℕ
k : ℕ × ℕ
hm : IsUnit ↑m !
hk : k.1 + k.2 = m
⊢ k.2 ≤ k.1 + k.2 | Please generate a tactic in lean4 to solve the state.
STATE:
case h
A : Type u_1
inst✝ : CommSemiring A
I : Ideal A
m : ℕ
k : ℕ × ℕ
hm : IsUnit ↑m !
hk : k.1 + k.2 = m
⊢ k.2 ≤ m
TACTIC:
|
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/RatAlgebra.lean | DividedPowers.OfInvertibleFactorial.foo | [68, 1] | [80, 34] | exact Nat.le_add_left k.2 k.1 | case h
A : Type u_1
inst✝ : CommSemiring A
I : Ideal A
m : ℕ
k : ℕ × ℕ
hm : IsUnit ↑m !
hk : k.1 + k.2 = m
⊢ k.2 ≤ k.1 + k.2 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case h
A : Type u_1
inst✝ : CommSemiring A
I : Ideal A
m : ℕ
k : ℕ × ℕ
hm : IsUnit ↑m !
hk : k.1 + k.2 = m
⊢ k.2 ≤ k.1 + k.2
TACTIC:
|
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/RatAlgebra.lean | DividedPowers.OfInvertibleFactorial.dpow_add_dif_pos | [82, 1] | [97, 65] | rw [dpow_eq_of_mem m (Ideal.add_mem I hx hy)] | A : Type u_1
inst✝ : CommSemiring A
I : Ideal A
n : ℕ
hn_fac : IsUnit ↑(n - 1)!
m : ℕ
hmn : m < n
x y : A
hx : x ∈ I
hy : y ∈ I
⊢ dpow I m (x + y) = ∑ k ∈ Finset.antidiagonal m, dpow I k.1 x * dpow I k.2 y | A : Type u_1
inst✝ : CommSemiring A
I : Ideal A
n : ℕ
hn_fac : IsUnit ↑(n - 1)!
m : ℕ
hmn : m < n
x y : A
hx : x ∈ I
hy : y ∈ I
⊢ Ring.inverse ↑m ! * (x + y) ^ m = ∑ k ∈ Finset.antidiagonal m, dpow I k.1 x * dpow I k.2 y | Please generate a tactic in lean4 to solve the state.
STATE:
A : Type u_1
inst✝ : CommSemiring A
I : Ideal A
n : ℕ
hn_fac : IsUnit ↑(n - 1)!
m : ℕ
hmn : m < n
x y : A
hx : x ∈ I
hy : y ∈ I
⊢ dpow I m (x + y) = ∑ k ∈ Finset.antidiagonal m, dpow I k.1 x * dpow I k.2 y
TACTIC:
|
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/RatAlgebra.lean | DividedPowers.OfInvertibleFactorial.dpow_add_dif_pos | [82, 1] | [97, 65] | simp only [dpow] | A : Type u_1
inst✝ : CommSemiring A
I : Ideal A
n : ℕ
hn_fac : IsUnit ↑(n - 1)!
m : ℕ
hmn : m < n
x y : A
hx : x ∈ I
hy : y ∈ I
⊢ Ring.inverse ↑m ! * (x + y) ^ m = ∑ k ∈ Finset.antidiagonal m, dpow I k.1 x * dpow I k.2 y | A : Type u_1
inst✝ : CommSemiring A
I : Ideal A
n : ℕ
hn_fac : IsUnit ↑(n - 1)!
m : ℕ
hmn : m < n
x y : A
hx : x ∈ I
hy : y ∈ I
⊢ Ring.inverse ↑m ! * (x + y) ^ m =
∑ x_1 ∈ Finset.antidiagonal m,
(if x ∈ I then Ring.inverse ↑x_1.1! * x ^ x_1.1 else 0) * if y ∈ I then Ring.inverse ↑x_1.2! * y ^ x_1.2 else 0 | Please generate a tactic in lean4 to solve the state.
STATE:
A : Type u_1
inst✝ : CommSemiring A
I : Ideal A
n : ℕ
hn_fac : IsUnit ↑(n - 1)!
m : ℕ
hmn : m < n
x y : A
hx : x ∈ I
hy : y ∈ I
⊢ Ring.inverse ↑m ! * (x + y) ^ m = ∑ k ∈ Finset.antidiagonal m, dpow I k.1 x * dpow I k.2 y
TACTIC:
|
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/RatAlgebra.lean | DividedPowers.OfInvertibleFactorial.dpow_add_dif_pos | [82, 1] | [97, 65] | rw [Ring.inverse_mul_eq_iff_eq_mul _ _ _ (factorial_isUnit hn_fac hmn), Finset.mul_sum, Commute.add_pow' (Commute.all _ _)] | A : Type u_1
inst✝ : CommSemiring A
I : Ideal A
n : ℕ
hn_fac : IsUnit ↑(n - 1)!
m : ℕ
hmn : m < n
x y : A
hx : x ∈ I
hy : y ∈ I
⊢ Ring.inverse ↑m ! * (x + y) ^ m =
∑ x_1 ∈ Finset.antidiagonal m,
(if x ∈ I then Ring.inverse ↑x_1.1! * x ^ x_1.1 else 0) * if y ∈ I then Ring.inverse ↑x_1.2! * y ^ x_1.2 else 0 | A : Type u_1
inst✝ : CommSemiring A
I : Ideal A
n : ℕ
hn_fac : IsUnit ↑(n - 1)!
m : ℕ
hmn : m < n
x y : A
hx : x ∈ I
hy : y ∈ I
⊢ ∑ m_1 ∈ Finset.antidiagonal m, m.choose m_1.1 • (x ^ m_1.1 * y ^ m_1.2) =
∑ i ∈ Finset.antidiagonal m,
↑m ! * ((if x ∈ I then Ring.inverse ↑i.1! * x ^ i.1 else 0) * if y ∈ I then R... | Please generate a tactic in lean4 to solve the state.
STATE:
A : Type u_1
inst✝ : CommSemiring A
I : Ideal A
n : ℕ
hn_fac : IsUnit ↑(n - 1)!
m : ℕ
hmn : m < n
x y : A
hx : x ∈ I
hy : y ∈ I
⊢ Ring.inverse ↑m ! * (x + y) ^ m =
∑ x_1 ∈ Finset.antidiagonal m,
(if x ∈ I then Ring.inverse ↑x_1.1! * x ^ x_1.1 else 0... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/RatAlgebra.lean | DividedPowers.OfInvertibleFactorial.dpow_add_dif_pos | [82, 1] | [97, 65] | apply Finset.sum_congr rfl | A : Type u_1
inst✝ : CommSemiring A
I : Ideal A
n : ℕ
hn_fac : IsUnit ↑(n - 1)!
m : ℕ
hmn : m < n
x y : A
hx : x ∈ I
hy : y ∈ I
⊢ ∑ m_1 ∈ Finset.antidiagonal m, m.choose m_1.1 • (x ^ m_1.1 * y ^ m_1.2) =
∑ i ∈ Finset.antidiagonal m,
↑m ! * ((if x ∈ I then Ring.inverse ↑i.1! * x ^ i.1 else 0) * if y ∈ I then R... | A : Type u_1
inst✝ : CommSemiring A
I : Ideal A
n : ℕ
hn_fac : IsUnit ↑(n - 1)!
m : ℕ
hmn : m < n
x y : A
hx : x ∈ I
hy : y ∈ I
⊢ ∀ x_1 ∈ Finset.antidiagonal m,
m.choose x_1.1 • (x ^ x_1.1 * y ^ x_1.2) =
↑m ! *
((if x ∈ I then Ring.inverse ↑x_1.1! * x ^ x_1.1 else 0) *
if y ∈ I then Ring.inv... | Please generate a tactic in lean4 to solve the state.
STATE:
A : Type u_1
inst✝ : CommSemiring A
I : Ideal A
n : ℕ
hn_fac : IsUnit ↑(n - 1)!
m : ℕ
hmn : m < n
x y : A
hx : x ∈ I
hy : y ∈ I
⊢ ∑ m_1 ∈ Finset.antidiagonal m, m.choose m_1.1 • (x ^ m_1.1 * y ^ m_1.2) =
∑ i ∈ Finset.antidiagonal m,
↑m ! * ((if x ∈ ... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/RatAlgebra.lean | DividedPowers.OfInvertibleFactorial.dpow_add_dif_pos | [82, 1] | [97, 65] | intro k hk | A : Type u_1
inst✝ : CommSemiring A
I : Ideal A
n : ℕ
hn_fac : IsUnit ↑(n - 1)!
m : ℕ
hmn : m < n
x y : A
hx : x ∈ I
hy : y ∈ I
⊢ ∀ x_1 ∈ Finset.antidiagonal m,
m.choose x_1.1 • (x ^ x_1.1 * y ^ x_1.2) =
↑m ! *
((if x ∈ I then Ring.inverse ↑x_1.1! * x ^ x_1.1 else 0) *
if y ∈ I then Ring.inv... | A : Type u_1
inst✝ : CommSemiring A
I : Ideal A
n : ℕ
hn_fac : IsUnit ↑(n - 1)!
m : ℕ
hmn : m < n
x y : A
hx : x ∈ I
hy : y ∈ I
k : ℕ × ℕ
hk : k ∈ Finset.antidiagonal m
⊢ m.choose k.1 • (x ^ k.1 * y ^ k.2) =
↑m ! * ((if x ∈ I then Ring.inverse ↑k.1! * x ^ k.1 else 0) * if y ∈ I then Ring.inverse ↑k.2! * y ^ k.2 els... | Please generate a tactic in lean4 to solve the state.
STATE:
A : Type u_1
inst✝ : CommSemiring A
I : Ideal A
n : ℕ
hn_fac : IsUnit ↑(n - 1)!
m : ℕ
hmn : m < n
x y : A
hx : x ∈ I
hy : y ∈ I
⊢ ∀ x_1 ∈ Finset.antidiagonal m,
m.choose x_1.1 • (x ^ x_1.1 * y ^ x_1.2) =
↑m ! *
((if x ∈ I then Ring.inverse ↑... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/RatAlgebra.lean | DividedPowers.OfInvertibleFactorial.dpow_add_dif_pos | [82, 1] | [97, 65] | rw [if_pos hx, if_pos hy] | A : Type u_1
inst✝ : CommSemiring A
I : Ideal A
n : ℕ
hn_fac : IsUnit ↑(n - 1)!
m : ℕ
hmn : m < n
x y : A
hx : x ∈ I
hy : y ∈ I
k : ℕ × ℕ
hk : k ∈ Finset.antidiagonal m
⊢ m.choose k.1 • (x ^ k.1 * y ^ k.2) =
↑m ! * ((if x ∈ I then Ring.inverse ↑k.1! * x ^ k.1 else 0) * if y ∈ I then Ring.inverse ↑k.2! * y ^ k.2 els... | A : Type u_1
inst✝ : CommSemiring A
I : Ideal A
n : ℕ
hn_fac : IsUnit ↑(n - 1)!
m : ℕ
hmn : m < n
x y : A
hx : x ∈ I
hy : y ∈ I
k : ℕ × ℕ
hk : k ∈ Finset.antidiagonal m
⊢ m.choose k.1 • (x ^ k.1 * y ^ k.2) = ↑m ! * (Ring.inverse ↑k.1! * x ^ k.1 * (Ring.inverse ↑k.2! * y ^ k.2)) | Please generate a tactic in lean4 to solve the state.
STATE:
A : Type u_1
inst✝ : CommSemiring A
I : Ideal A
n : ℕ
hn_fac : IsUnit ↑(n - 1)!
m : ℕ
hmn : m < n
x y : A
hx : x ∈ I
hy : y ∈ I
k : ℕ × ℕ
hk : k ∈ Finset.antidiagonal m
⊢ m.choose k.1 • (x ^ k.1 * y ^ k.2) =
↑m ! * ((if x ∈ I then Ring.inverse ↑k.1! * x ^... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/RatAlgebra.lean | DividedPowers.OfInvertibleFactorial.dpow_add_dif_pos | [82, 1] | [97, 65] | ring_nf | A : Type u_1
inst✝ : CommSemiring A
I : Ideal A
n : ℕ
hn_fac : IsUnit ↑(n - 1)!
m : ℕ
hmn : m < n
x y : A
hx : x ∈ I
hy : y ∈ I
k : ℕ × ℕ
hk : k ∈ Finset.antidiagonal m
⊢ m.choose k.1 • (x ^ k.1 * y ^ k.2) = ↑m ! * (Ring.inverse ↑k.1! * x ^ k.1 * (Ring.inverse ↑k.2! * y ^ k.2)) | A : Type u_1
inst✝ : CommSemiring A
I : Ideal A
n : ℕ
hn_fac : IsUnit ↑(n - 1)!
m : ℕ
hmn : m < n
x y : A
hx : x ∈ I
hy : y ∈ I
k : ℕ × ℕ
hk : k ∈ Finset.antidiagonal m
⊢ x ^ k.1 * y ^ k.2 * ↑(m.choose k.1) = x ^ k.1 * y ^ k.2 * ↑m ! * Ring.inverse ↑k.1! * Ring.inverse ↑k.2! | Please generate a tactic in lean4 to solve the state.
STATE:
A : Type u_1
inst✝ : CommSemiring A
I : Ideal A
n : ℕ
hn_fac : IsUnit ↑(n - 1)!
m : ℕ
hmn : m < n
x y : A
hx : x ∈ I
hy : y ∈ I
k : ℕ × ℕ
hk : k ∈ Finset.antidiagonal m
⊢ m.choose k.1 • (x ^ k.1 * y ^ k.2) = ↑m ! * (Ring.inverse ↑k.1! * x ^ k.1 * (Ring.invers... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/RatAlgebra.lean | DividedPowers.OfInvertibleFactorial.dpow_add_dif_pos | [82, 1] | [97, 65] | simp only [mul_assoc] | A : Type u_1
inst✝ : CommSemiring A
I : Ideal A
n : ℕ
hn_fac : IsUnit ↑(n - 1)!
m : ℕ
hmn : m < n
x y : A
hx : x ∈ I
hy : y ∈ I
k : ℕ × ℕ
hk : k ∈ Finset.antidiagonal m
⊢ x ^ k.1 * y ^ k.2 * ↑(m.choose k.1) = x ^ k.1 * y ^ k.2 * ↑m ! * Ring.inverse ↑k.1! * Ring.inverse ↑k.2! | A : Type u_1
inst✝ : CommSemiring A
I : Ideal A
n : ℕ
hn_fac : IsUnit ↑(n - 1)!
m : ℕ
hmn : m < n
x y : A
hx : x ∈ I
hy : y ∈ I
k : ℕ × ℕ
hk : k ∈ Finset.antidiagonal m
⊢ x ^ k.1 * (y ^ k.2 * ↑(m.choose k.1)) = x ^ k.1 * (y ^ k.2 * (↑m ! * (Ring.inverse ↑k.1! * Ring.inverse ↑k.2!))) | Please generate a tactic in lean4 to solve the state.
STATE:
A : Type u_1
inst✝ : CommSemiring A
I : Ideal A
n : ℕ
hn_fac : IsUnit ↑(n - 1)!
m : ℕ
hmn : m < n
x y : A
hx : x ∈ I
hy : y ∈ I
k : ℕ × ℕ
hk : k ∈ Finset.antidiagonal m
⊢ x ^ k.1 * y ^ k.2 * ↑(m.choose k.1) = x ^ k.1 * y ^ k.2 * ↑m ! * Ring.inverse ↑k.1! * Ri... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/RatAlgebra.lean | DividedPowers.OfInvertibleFactorial.dpow_add_dif_pos | [82, 1] | [97, 65] | apply congr_arg₂ _ rfl | A : Type u_1
inst✝ : CommSemiring A
I : Ideal A
n : ℕ
hn_fac : IsUnit ↑(n - 1)!
m : ℕ
hmn : m < n
x y : A
hx : x ∈ I
hy : y ∈ I
k : ℕ × ℕ
hk : k ∈ Finset.antidiagonal m
⊢ x ^ k.1 * (y ^ k.2 * ↑(m.choose k.1)) = x ^ k.1 * (y ^ k.2 * (↑m ! * (Ring.inverse ↑k.1! * Ring.inverse ↑k.2!))) | A : Type u_1
inst✝ : CommSemiring A
I : Ideal A
n : ℕ
hn_fac : IsUnit ↑(n - 1)!
m : ℕ
hmn : m < n
x y : A
hx : x ∈ I
hy : y ∈ I
k : ℕ × ℕ
hk : k ∈ Finset.antidiagonal m
⊢ y ^ k.2 * ↑(m.choose k.1) = y ^ k.2 * (↑m ! * (Ring.inverse ↑k.1! * Ring.inverse ↑k.2!)) | Please generate a tactic in lean4 to solve the state.
STATE:
A : Type u_1
inst✝ : CommSemiring A
I : Ideal A
n : ℕ
hn_fac : IsUnit ↑(n - 1)!
m : ℕ
hmn : m < n
x y : A
hx : x ∈ I
hy : y ∈ I
k : ℕ × ℕ
hk : k ∈ Finset.antidiagonal m
⊢ x ^ k.1 * (y ^ k.2 * ↑(m.choose k.1)) = x ^ k.1 * (y ^ k.2 * (↑m ! * (Ring.inverse ↑k.1!... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/RatAlgebra.lean | DividedPowers.OfInvertibleFactorial.dpow_add_dif_pos | [82, 1] | [97, 65] | apply congr_arg₂ _ rfl | A : Type u_1
inst✝ : CommSemiring A
I : Ideal A
n : ℕ
hn_fac : IsUnit ↑(n - 1)!
m : ℕ
hmn : m < n
x y : A
hx : x ∈ I
hy : y ∈ I
k : ℕ × ℕ
hk : k ∈ Finset.antidiagonal m
⊢ y ^ k.2 * ↑(m.choose k.1) = y ^ k.2 * (↑m ! * (Ring.inverse ↑k.1! * Ring.inverse ↑k.2!)) | A : Type u_1
inst✝ : CommSemiring A
I : Ideal A
n : ℕ
hn_fac : IsUnit ↑(n - 1)!
m : ℕ
hmn : m < n
x y : A
hx : x ∈ I
hy : y ∈ I
k : ℕ × ℕ
hk : k ∈ Finset.antidiagonal m
⊢ ↑(m.choose k.1) = ↑m ! * (Ring.inverse ↑k.1! * Ring.inverse ↑k.2!) | Please generate a tactic in lean4 to solve the state.
STATE:
A : Type u_1
inst✝ : CommSemiring A
I : Ideal A
n : ℕ
hn_fac : IsUnit ↑(n - 1)!
m : ℕ
hmn : m < n
x y : A
hx : x ∈ I
hy : y ∈ I
k : ℕ × ℕ
hk : k ∈ Finset.antidiagonal m
⊢ y ^ k.2 * ↑(m.choose k.1) = y ^ k.2 * (↑m ! * (Ring.inverse ↑k.1! * Ring.inverse ↑k.2!))... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/RatAlgebra.lean | DividedPowers.OfInvertibleFactorial.dpow_add_dif_pos | [82, 1] | [97, 65] | rw [← mul_assoc] | A : Type u_1
inst✝ : CommSemiring A
I : Ideal A
n : ℕ
hn_fac : IsUnit ↑(n - 1)!
m : ℕ
hmn : m < n
x y : A
hx : x ∈ I
hy : y ∈ I
k : ℕ × ℕ
hk : k ∈ Finset.antidiagonal m
⊢ ↑(m.choose k.1) = ↑m ! * (Ring.inverse ↑k.1! * Ring.inverse ↑k.2!) | A : Type u_1
inst✝ : CommSemiring A
I : Ideal A
n : ℕ
hn_fac : IsUnit ↑(n - 1)!
m : ℕ
hmn : m < n
x y : A
hx : x ∈ I
hy : y ∈ I
k : ℕ × ℕ
hk : k ∈ Finset.antidiagonal m
⊢ ↑(m.choose k.1) = ↑m ! * Ring.inverse ↑k.1! * Ring.inverse ↑k.2! | Please generate a tactic in lean4 to solve the state.
STATE:
A : Type u_1
inst✝ : CommSemiring A
I : Ideal A
n : ℕ
hn_fac : IsUnit ↑(n - 1)!
m : ℕ
hmn : m < n
x y : A
hx : x ∈ I
hy : y ∈ I
k : ℕ × ℕ
hk : k ∈ Finset.antidiagonal m
⊢ ↑(m.choose k.1) = ↑m ! * (Ring.inverse ↑k.1! * Ring.inverse ↑k.2!)
TACTIC:
|
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/RatAlgebra.lean | DividedPowers.OfInvertibleFactorial.dpow_add_dif_pos | [82, 1] | [97, 65] | exact foo (factorial_isUnit' hn_fac (le_sub_one_of_lt hmn)) hk | A : Type u_1
inst✝ : CommSemiring A
I : Ideal A
n : ℕ
hn_fac : IsUnit ↑(n - 1)!
m : ℕ
hmn : m < n
x y : A
hx : x ∈ I
hy : y ∈ I
k : ℕ × ℕ
hk : k ∈ Finset.antidiagonal m
⊢ ↑(m.choose k.1) = ↑m ! * Ring.inverse ↑k.1! * Ring.inverse ↑k.2! | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
A : Type u_1
inst✝ : CommSemiring A
I : Ideal A
n : ℕ
hn_fac : IsUnit ↑(n - 1)!
m : ℕ
hmn : m < n
x y : A
hx : x ∈ I
hy : y ∈ I
k : ℕ × ℕ
hk : k ∈ Finset.antidiagonal m
⊢ ↑(m.choose k.1) = ↑m ! * Ring.inverse ↑k.1! * Ring.inverse ↑k.2!
TACTIC:
|
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/RatAlgebra.lean | DividedPowers.OfInvertibleFactorial.dpow_add | [100, 1] | [121, 78] | by_cases hmn : m < n | A : Type u_1
inst✝ : CommSemiring A
I : Ideal A
n : ℕ
hn_fac : IsUnit ↑(n - 1)!
hnI : I ^ n = 0
m : ℕ
x : A
hx : x ∈ I
y : A
hy : y ∈ I
⊢ dpow I m (x + y) = ∑ k ∈ Finset.antidiagonal m, dpow I k.1 x * dpow I k.2 y | case pos
A : Type u_1
inst✝ : CommSemiring A
I : Ideal A
n : ℕ
hn_fac : IsUnit ↑(n - 1)!
hnI : I ^ n = 0
m : ℕ
x : A
hx : x ∈ I
y : A
hy : y ∈ I
hmn : m < n
⊢ dpow I m (x + y) = ∑ k ∈ Finset.antidiagonal m, dpow I k.1 x * dpow I k.2 y
case neg
A : Type u_1
inst✝ : CommSemiring A
I : Ideal A
n : ℕ
hn_fac : IsUnit ↑(n -... | Please generate a tactic in lean4 to solve the state.
STATE:
A : Type u_1
inst✝ : CommSemiring A
I : Ideal A
n : ℕ
hn_fac : IsUnit ↑(n - 1)!
hnI : I ^ n = 0
m : ℕ
x : A
hx : x ∈ I
y : A
hy : y ∈ I
⊢ dpow I m (x + y) = ∑ k ∈ Finset.antidiagonal m, dpow I k.1 x * dpow I k.2 y
TACTIC:
|
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/RatAlgebra.lean | DividedPowers.OfInvertibleFactorial.dpow_add | [100, 1] | [121, 78] | exact dpow_add_dif_pos hn_fac hmn hx hy | case pos
A : Type u_1
inst✝ : CommSemiring A
I : Ideal A
n : ℕ
hn_fac : IsUnit ↑(n - 1)!
hnI : I ^ n = 0
m : ℕ
x : A
hx : x ∈ I
y : A
hy : y ∈ I
hmn : m < n
⊢ dpow I m (x + y) = ∑ k ∈ Finset.antidiagonal m, dpow I k.1 x * dpow I k.2 y | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case pos
A : Type u_1
inst✝ : CommSemiring A
I : Ideal A
n : ℕ
hn_fac : IsUnit ↑(n - 1)!
hnI : I ^ n = 0
m : ℕ
x : A
hx : x ∈ I
y : A
hy : y ∈ I
hmn : m < n
⊢ dpow I m (x + y) = ∑ k ∈ Finset.antidiagonal m, dpow I k.1 x * dpow I k.2 y
TACTIC:
|
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/RatAlgebra.lean | DividedPowers.OfInvertibleFactorial.dpow_add | [100, 1] | [121, 78] | have h_sub : I ^ m ≤ I ^ n := Ideal.pow_le_pow_right (not_lt.mp hmn) | case neg
A : Type u_1
inst✝ : CommSemiring A
I : Ideal A
n : ℕ
hn_fac : IsUnit ↑(n - 1)!
hnI : I ^ n = 0
m : ℕ
x : A
hx : x ∈ I
y : A
hy : y ∈ I
hmn : ¬m < n
⊢ dpow I m (x + y) = ∑ k ∈ Finset.antidiagonal m, dpow I k.1 x * dpow I k.2 y | case neg
A : Type u_1
inst✝ : CommSemiring A
I : Ideal A
n : ℕ
hn_fac : IsUnit ↑(n - 1)!
hnI : I ^ n = 0
m : ℕ
x : A
hx : x ∈ I
y : A
hy : y ∈ I
hmn : ¬m < n
h_sub : I ^ m ≤ I ^ n
⊢ dpow I m (x + y) = ∑ k ∈ Finset.antidiagonal m, dpow I k.1 x * dpow I k.2 y | Please generate a tactic in lean4 to solve the state.
STATE:
case neg
A : Type u_1
inst✝ : CommSemiring A
I : Ideal A
n : ℕ
hn_fac : IsUnit ↑(n - 1)!
hnI : I ^ n = 0
m : ℕ
x : A
hx : x ∈ I
y : A
hy : y ∈ I
hmn : ¬m < n
⊢ dpow I m (x + y) = ∑ k ∈ Finset.antidiagonal m, dpow I k.1 x * dpow I k.2 y
TACTIC:
|
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/RatAlgebra.lean | DividedPowers.OfInvertibleFactorial.dpow_add | [100, 1] | [121, 78] | rw [dpow_eq_of_mem m (Ideal.add_mem I hx hy)] | case neg
A : Type u_1
inst✝ : CommSemiring A
I : Ideal A
n : ℕ
hn_fac : IsUnit ↑(n - 1)!
hnI : I ^ n = 0
m : ℕ
x : A
hx : x ∈ I
y : A
hy : y ∈ I
hmn : ¬m < n
h_sub : I ^ m ≤ I ^ n
⊢ dpow I m (x + y) = ∑ k ∈ Finset.antidiagonal m, dpow I k.1 x * dpow I k.2 y | case neg
A : Type u_1
inst✝ : CommSemiring A
I : Ideal A
n : ℕ
hn_fac : IsUnit ↑(n - 1)!
hnI : I ^ n = 0
m : ℕ
x : A
hx : x ∈ I
y : A
hy : y ∈ I
hmn : ¬m < n
h_sub : I ^ m ≤ I ^ n
⊢ Ring.inverse ↑m ! * (x + y) ^ m = ∑ k ∈ Finset.antidiagonal m, dpow I k.1 x * dpow I k.2 y | Please generate a tactic in lean4 to solve the state.
STATE:
case neg
A : Type u_1
inst✝ : CommSemiring A
I : Ideal A
n : ℕ
hn_fac : IsUnit ↑(n - 1)!
hnI : I ^ n = 0
m : ℕ
x : A
hx : x ∈ I
y : A
hy : y ∈ I
hmn : ¬m < n
h_sub : I ^ m ≤ I ^ n
⊢ dpow I m (x + y) = ∑ k ∈ Finset.antidiagonal m, dpow I k.1 x * dpow I k.2 y
T... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/RatAlgebra.lean | DividedPowers.OfInvertibleFactorial.dpow_add | [100, 1] | [121, 78] | simp only [dpow] | case neg
A : Type u_1
inst✝ : CommSemiring A
I : Ideal A
n : ℕ
hn_fac : IsUnit ↑(n - 1)!
hnI : I ^ n = 0
m : ℕ
x : A
hx : x ∈ I
y : A
hy : y ∈ I
hmn : ¬m < n
h_sub : I ^ m ≤ I ^ n
⊢ Ring.inverse ↑m ! * (x + y) ^ m = ∑ k ∈ Finset.antidiagonal m, dpow I k.1 x * dpow I k.2 y | case neg
A : Type u_1
inst✝ : CommSemiring A
I : Ideal A
n : ℕ
hn_fac : IsUnit ↑(n - 1)!
hnI : I ^ n = 0
m : ℕ
x : A
hx : x ∈ I
y : A
hy : y ∈ I
hmn : ¬m < n
h_sub : I ^ m ≤ I ^ n
⊢ Ring.inverse ↑m ! * (x + y) ^ m =
∑ x_1 ∈ Finset.antidiagonal m,
(if x ∈ I then Ring.inverse ↑x_1.1! * x ^ x_1.1 else 0) * if y ... | Please generate a tactic in lean4 to solve the state.
STATE:
case neg
A : Type u_1
inst✝ : CommSemiring A
I : Ideal A
n : ℕ
hn_fac : IsUnit ↑(n - 1)!
hnI : I ^ n = 0
m : ℕ
x : A
hx : x ∈ I
y : A
hy : y ∈ I
hmn : ¬m < n
h_sub : I ^ m ≤ I ^ n
⊢ Ring.inverse ↑m ! * (x + y) ^ m = ∑ k ∈ Finset.antidiagonal m, dpow I k.1 x *... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/RatAlgebra.lean | DividedPowers.OfInvertibleFactorial.dpow_add | [100, 1] | [121, 78] | have hxy : (x + y) ^ m = 0 :=
by
rw [← Ideal.mem_bot, ← Ideal.zero_eq_bot, ← hnI]
apply Set.mem_of_subset_of_mem h_sub (Ideal.pow_mem_pow (Ideal.add_mem I hx hy) m) | case neg
A : Type u_1
inst✝ : CommSemiring A
I : Ideal A
n : ℕ
hn_fac : IsUnit ↑(n - 1)!
hnI : I ^ n = 0
m : ℕ
x : A
hx : x ∈ I
y : A
hy : y ∈ I
hmn : ¬m < n
h_sub : I ^ m ≤ I ^ n
⊢ Ring.inverse ↑m ! * (x + y) ^ m =
∑ x_1 ∈ Finset.antidiagonal m,
(if x ∈ I then Ring.inverse ↑x_1.1! * x ^ x_1.1 else 0) * if y ... | case neg
A : Type u_1
inst✝ : CommSemiring A
I : Ideal A
n : ℕ
hn_fac : IsUnit ↑(n - 1)!
hnI : I ^ n = 0
m : ℕ
x : A
hx : x ∈ I
y : A
hy : y ∈ I
hmn : ¬m < n
h_sub : I ^ m ≤ I ^ n
hxy : (x + y) ^ m = 0
⊢ Ring.inverse ↑m ! * (x + y) ^ m =
∑ x_1 ∈ Finset.antidiagonal m,
(if x ∈ I then Ring.inverse ↑x_1.1! * x ^... | Please generate a tactic in lean4 to solve the state.
STATE:
case neg
A : Type u_1
inst✝ : CommSemiring A
I : Ideal A
n : ℕ
hn_fac : IsUnit ↑(n - 1)!
hnI : I ^ n = 0
m : ℕ
x : A
hx : x ∈ I
y : A
hy : y ∈ I
hmn : ¬m < n
h_sub : I ^ m ≤ I ^ n
⊢ Ring.inverse ↑m ! * (x + y) ^ m =
∑ x_1 ∈ Finset.antidiagonal m,
(i... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/RatAlgebra.lean | DividedPowers.OfInvertibleFactorial.dpow_add | [100, 1] | [121, 78] | rw [hxy, MulZeroClass.mul_zero, eq_comm] | case neg
A : Type u_1
inst✝ : CommSemiring A
I : Ideal A
n : ℕ
hn_fac : IsUnit ↑(n - 1)!
hnI : I ^ n = 0
m : ℕ
x : A
hx : x ∈ I
y : A
hy : y ∈ I
hmn : ¬m < n
h_sub : I ^ m ≤ I ^ n
hxy : (x + y) ^ m = 0
⊢ Ring.inverse ↑m ! * (x + y) ^ m =
∑ x_1 ∈ Finset.antidiagonal m,
(if x ∈ I then Ring.inverse ↑x_1.1! * x ^... | case neg
A : Type u_1
inst✝ : CommSemiring A
I : Ideal A
n : ℕ
hn_fac : IsUnit ↑(n - 1)!
hnI : I ^ n = 0
m : ℕ
x : A
hx : x ∈ I
y : A
hy : y ∈ I
hmn : ¬m < n
h_sub : I ^ m ≤ I ^ n
hxy : (x + y) ^ m = 0
⊢ (∑ x_1 ∈ Finset.antidiagonal m,
(if x ∈ I then Ring.inverse ↑x_1.1! * x ^ x_1.1 else 0) * if y ∈ I then Ring.i... | Please generate a tactic in lean4 to solve the state.
STATE:
case neg
A : Type u_1
inst✝ : CommSemiring A
I : Ideal A
n : ℕ
hn_fac : IsUnit ↑(n - 1)!
hnI : I ^ n = 0
m : ℕ
x : A
hx : x ∈ I
y : A
hy : y ∈ I
hmn : ¬m < n
h_sub : I ^ m ≤ I ^ n
hxy : (x + y) ^ m = 0
⊢ Ring.inverse ↑m ! * (x + y) ^ m =
∑ x_1 ∈ Finset.an... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/RatAlgebra.lean | DividedPowers.OfInvertibleFactorial.dpow_add | [100, 1] | [121, 78] | apply Finset.sum_eq_zero | case neg
A : Type u_1
inst✝ : CommSemiring A
I : Ideal A
n : ℕ
hn_fac : IsUnit ↑(n - 1)!
hnI : I ^ n = 0
m : ℕ
x : A
hx : x ∈ I
y : A
hy : y ∈ I
hmn : ¬m < n
h_sub : I ^ m ≤ I ^ n
hxy : (x + y) ^ m = 0
⊢ (∑ x_1 ∈ Finset.antidiagonal m,
(if x ∈ I then Ring.inverse ↑x_1.1! * x ^ x_1.1 else 0) * if y ∈ I then Ring.i... | case neg.h
A : Type u_1
inst✝ : CommSemiring A
I : Ideal A
n : ℕ
hn_fac : IsUnit ↑(n - 1)!
hnI : I ^ n = 0
m : ℕ
x : A
hx : x ∈ I
y : A
hy : y ∈ I
hmn : ¬m < n
h_sub : I ^ m ≤ I ^ n
hxy : (x + y) ^ m = 0
⊢ ∀ x_1 ∈ Finset.antidiagonal m,
((if x ∈ I then Ring.inverse ↑x_1.1! * x ^ x_1.1 else 0) * if y ∈ I then Ring.i... | Please generate a tactic in lean4 to solve the state.
STATE:
case neg
A : Type u_1
inst✝ : CommSemiring A
I : Ideal A
n : ℕ
hn_fac : IsUnit ↑(n - 1)!
hnI : I ^ n = 0
m : ℕ
x : A
hx : x ∈ I
y : A
hy : y ∈ I
hmn : ¬m < n
h_sub : I ^ m ≤ I ^ n
hxy : (x + y) ^ m = 0
⊢ (∑ x_1 ∈ Finset.antidiagonal m,
(if x ∈ I then Ri... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/RatAlgebra.lean | DividedPowers.OfInvertibleFactorial.dpow_add | [100, 1] | [121, 78] | intro k hk | case neg.h
A : Type u_1
inst✝ : CommSemiring A
I : Ideal A
n : ℕ
hn_fac : IsUnit ↑(n - 1)!
hnI : I ^ n = 0
m : ℕ
x : A
hx : x ∈ I
y : A
hy : y ∈ I
hmn : ¬m < n
h_sub : I ^ m ≤ I ^ n
hxy : (x + y) ^ m = 0
⊢ ∀ x_1 ∈ Finset.antidiagonal m,
((if x ∈ I then Ring.inverse ↑x_1.1! * x ^ x_1.1 else 0) * if y ∈ I then Ring.i... | case neg.h
A : Type u_1
inst✝ : CommSemiring A
I : Ideal A
n : ℕ
hn_fac : IsUnit ↑(n - 1)!
hnI : I ^ n = 0
m : ℕ
x : A
hx : x ∈ I
y : A
hy : y ∈ I
hmn : ¬m < n
h_sub : I ^ m ≤ I ^ n
hxy : (x + y) ^ m = 0
k : ℕ × ℕ
hk : k ∈ Finset.antidiagonal m
⊢ ((if x ∈ I then Ring.inverse ↑k.1! * x ^ k.1 else 0) * if y ∈ I then Ring... | Please generate a tactic in lean4 to solve the state.
STATE:
case neg.h
A : Type u_1
inst✝ : CommSemiring A
I : Ideal A
n : ℕ
hn_fac : IsUnit ↑(n - 1)!
hnI : I ^ n = 0
m : ℕ
x : A
hx : x ∈ I
y : A
hy : y ∈ I
hmn : ¬m < n
h_sub : I ^ m ≤ I ^ n
hxy : (x + y) ^ m = 0
⊢ ∀ x_1 ∈ Finset.antidiagonal m,
((if x ∈ I then Ri... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/RatAlgebra.lean | DividedPowers.OfInvertibleFactorial.dpow_add | [100, 1] | [121, 78] | rw [if_pos hx, if_pos hy, mul_assoc, mul_comm (x ^ k.1), mul_assoc, ← mul_assoc] | case neg.h
A : Type u_1
inst✝ : CommSemiring A
I : Ideal A
n : ℕ
hn_fac : IsUnit ↑(n - 1)!
hnI : I ^ n = 0
m : ℕ
x : A
hx : x ∈ I
y : A
hy : y ∈ I
hmn : ¬m < n
h_sub : I ^ m ≤ I ^ n
hxy : (x + y) ^ m = 0
k : ℕ × ℕ
hk : k ∈ Finset.antidiagonal m
⊢ ((if x ∈ I then Ring.inverse ↑k.1! * x ^ k.1 else 0) * if y ∈ I then Ring... | case neg.h
A : Type u_1
inst✝ : CommSemiring A
I : Ideal A
n : ℕ
hn_fac : IsUnit ↑(n - 1)!
hnI : I ^ n = 0
m : ℕ
x : A
hx : x ∈ I
y : A
hy : y ∈ I
hmn : ¬m < n
h_sub : I ^ m ≤ I ^ n
hxy : (x + y) ^ m = 0
k : ℕ × ℕ
hk : k ∈ Finset.antidiagonal m
⊢ Ring.inverse ↑k.1! * Ring.inverse ↑k.2! * (y ^ k.2 * x ^ k.1) = 0 | Please generate a tactic in lean4 to solve the state.
STATE:
case neg.h
A : Type u_1
inst✝ : CommSemiring A
I : Ideal A
n : ℕ
hn_fac : IsUnit ↑(n - 1)!
hnI : I ^ n = 0
m : ℕ
x : A
hx : x ∈ I
y : A
hy : y ∈ I
hmn : ¬m < n
h_sub : I ^ m ≤ I ^ n
hxy : (x + y) ^ m = 0
k : ℕ × ℕ
hk : k ∈ Finset.antidiagonal m
⊢ ((if x ∈ I t... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/RatAlgebra.lean | DividedPowers.OfInvertibleFactorial.dpow_add | [100, 1] | [121, 78] | apply mul_eq_zero_of_right | case neg.h
A : Type u_1
inst✝ : CommSemiring A
I : Ideal A
n : ℕ
hn_fac : IsUnit ↑(n - 1)!
hnI : I ^ n = 0
m : ℕ
x : A
hx : x ∈ I
y : A
hy : y ∈ I
hmn : ¬m < n
h_sub : I ^ m ≤ I ^ n
hxy : (x + y) ^ m = 0
k : ℕ × ℕ
hk : k ∈ Finset.antidiagonal m
⊢ Ring.inverse ↑k.1! * Ring.inverse ↑k.2! * (y ^ k.2 * x ^ k.1) = 0 | case neg.h.h
A : Type u_1
inst✝ : CommSemiring A
I : Ideal A
n : ℕ
hn_fac : IsUnit ↑(n - 1)!
hnI : I ^ n = 0
m : ℕ
x : A
hx : x ∈ I
y : A
hy : y ∈ I
hmn : ¬m < n
h_sub : I ^ m ≤ I ^ n
hxy : (x + y) ^ m = 0
k : ℕ × ℕ
hk : k ∈ Finset.antidiagonal m
⊢ y ^ k.2 * x ^ k.1 = 0 | Please generate a tactic in lean4 to solve the state.
STATE:
case neg.h
A : Type u_1
inst✝ : CommSemiring A
I : Ideal A
n : ℕ
hn_fac : IsUnit ↑(n - 1)!
hnI : I ^ n = 0
m : ℕ
x : A
hx : x ∈ I
y : A
hy : y ∈ I
hmn : ¬m < n
h_sub : I ^ m ≤ I ^ n
hxy : (x + y) ^ m = 0
k : ℕ × ℕ
hk : k ∈ Finset.antidiagonal m
⊢ Ring.inverse... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/RatAlgebra.lean | DividedPowers.OfInvertibleFactorial.dpow_add | [100, 1] | [121, 78] | rw [← Ideal.mem_bot, ← Ideal.zero_eq_bot, ← hnI] | case neg.h.h
A : Type u_1
inst✝ : CommSemiring A
I : Ideal A
n : ℕ
hn_fac : IsUnit ↑(n - 1)!
hnI : I ^ n = 0
m : ℕ
x : A
hx : x ∈ I
y : A
hy : y ∈ I
hmn : ¬m < n
h_sub : I ^ m ≤ I ^ n
hxy : (x + y) ^ m = 0
k : ℕ × ℕ
hk : k ∈ Finset.antidiagonal m
⊢ y ^ k.2 * x ^ k.1 = 0 | case neg.h.h
A : Type u_1
inst✝ : CommSemiring A
I : Ideal A
n : ℕ
hn_fac : IsUnit ↑(n - 1)!
hnI : I ^ n = 0
m : ℕ
x : A
hx : x ∈ I
y : A
hy : y ∈ I
hmn : ¬m < n
h_sub : I ^ m ≤ I ^ n
hxy : (x + y) ^ m = 0
k : ℕ × ℕ
hk : k ∈ Finset.antidiagonal m
⊢ y ^ k.2 * x ^ k.1 ∈ I ^ n | Please generate a tactic in lean4 to solve the state.
STATE:
case neg.h.h
A : Type u_1
inst✝ : CommSemiring A
I : Ideal A
n : ℕ
hn_fac : IsUnit ↑(n - 1)!
hnI : I ^ n = 0
m : ℕ
x : A
hx : x ∈ I
y : A
hy : y ∈ I
hmn : ¬m < n
h_sub : I ^ m ≤ I ^ n
hxy : (x + y) ^ m = 0
k : ℕ × ℕ
hk : k ∈ Finset.antidiagonal m
⊢ y ^ k.2 * ... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/RatAlgebra.lean | DividedPowers.OfInvertibleFactorial.dpow_add | [100, 1] | [121, 78] | apply Set.mem_of_subset_of_mem h_sub | case neg.h.h
A : Type u_1
inst✝ : CommSemiring A
I : Ideal A
n : ℕ
hn_fac : IsUnit ↑(n - 1)!
hnI : I ^ n = 0
m : ℕ
x : A
hx : x ∈ I
y : A
hy : y ∈ I
hmn : ¬m < n
h_sub : I ^ m ≤ I ^ n
hxy : (x + y) ^ m = 0
k : ℕ × ℕ
hk : k ∈ Finset.antidiagonal m
⊢ y ^ k.2 * x ^ k.1 ∈ I ^ n | case neg.h.h
A : Type u_1
inst✝ : CommSemiring A
I : Ideal A
n : ℕ
hn_fac : IsUnit ↑(n - 1)!
hnI : I ^ n = 0
m : ℕ
x : A
hx : x ∈ I
y : A
hy : y ∈ I
hmn : ¬m < n
h_sub : I ^ m ≤ I ^ n
hxy : (x + y) ^ m = 0
k : ℕ × ℕ
hk : k ∈ Finset.antidiagonal m
⊢ y ^ k.2 * x ^ k.1 ∈ ↑(I ^ m) | Please generate a tactic in lean4 to solve the state.
STATE:
case neg.h.h
A : Type u_1
inst✝ : CommSemiring A
I : Ideal A
n : ℕ
hn_fac : IsUnit ↑(n - 1)!
hnI : I ^ n = 0
m : ℕ
x : A
hx : x ∈ I
y : A
hy : y ∈ I
hmn : ¬m < n
h_sub : I ^ m ≤ I ^ n
hxy : (x + y) ^ m = 0
k : ℕ × ℕ
hk : k ∈ Finset.antidiagonal m
⊢ y ^ k.2 * ... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/RatAlgebra.lean | DividedPowers.OfInvertibleFactorial.dpow_add | [100, 1] | [121, 78] | rw [Finset.mem_antidiagonal] at hk | case neg.h.h
A : Type u_1
inst✝ : CommSemiring A
I : Ideal A
n : ℕ
hn_fac : IsUnit ↑(n - 1)!
hnI : I ^ n = 0
m : ℕ
x : A
hx : x ∈ I
y : A
hy : y ∈ I
hmn : ¬m < n
h_sub : I ^ m ≤ I ^ n
hxy : (x + y) ^ m = 0
k : ℕ × ℕ
hk : k ∈ Finset.antidiagonal m
⊢ y ^ k.2 * x ^ k.1 ∈ ↑(I ^ m) | case neg.h.h
A : Type u_1
inst✝ : CommSemiring A
I : Ideal A
n : ℕ
hn_fac : IsUnit ↑(n - 1)!
hnI : I ^ n = 0
m : ℕ
x : A
hx : x ∈ I
y : A
hy : y ∈ I
hmn : ¬m < n
h_sub : I ^ m ≤ I ^ n
hxy : (x + y) ^ m = 0
k : ℕ × ℕ
hk : k.1 + k.2 = m
⊢ y ^ k.2 * x ^ k.1 ∈ ↑(I ^ m) | Please generate a tactic in lean4 to solve the state.
STATE:
case neg.h.h
A : Type u_1
inst✝ : CommSemiring A
I : Ideal A
n : ℕ
hn_fac : IsUnit ↑(n - 1)!
hnI : I ^ n = 0
m : ℕ
x : A
hx : x ∈ I
y : A
hy : y ∈ I
hmn : ¬m < n
h_sub : I ^ m ≤ I ^ n
hxy : (x + y) ^ m = 0
k : ℕ × ℕ
hk : k ∈ Finset.antidiagonal m
⊢ y ^ k.2 * ... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/RatAlgebra.lean | DividedPowers.OfInvertibleFactorial.dpow_add | [100, 1] | [121, 78] | rw [← hk, add_comm, pow_add] | case neg.h.h
A : Type u_1
inst✝ : CommSemiring A
I : Ideal A
n : ℕ
hn_fac : IsUnit ↑(n - 1)!
hnI : I ^ n = 0
m : ℕ
x : A
hx : x ∈ I
y : A
hy : y ∈ I
hmn : ¬m < n
h_sub : I ^ m ≤ I ^ n
hxy : (x + y) ^ m = 0
k : ℕ × ℕ
hk : k.1 + k.2 = m
⊢ y ^ k.2 * x ^ k.1 ∈ ↑(I ^ m) | case neg.h.h
A : Type u_1
inst✝ : CommSemiring A
I : Ideal A
n : ℕ
hn_fac : IsUnit ↑(n - 1)!
hnI : I ^ n = 0
m : ℕ
x : A
hx : x ∈ I
y : A
hy : y ∈ I
hmn : ¬m < n
h_sub : I ^ m ≤ I ^ n
hxy : (x + y) ^ m = 0
k : ℕ × ℕ
hk : k.1 + k.2 = m
⊢ y ^ k.2 * x ^ k.1 ∈ ↑(I ^ k.2 * I ^ k.1) | Please generate a tactic in lean4 to solve the state.
STATE:
case neg.h.h
A : Type u_1
inst✝ : CommSemiring A
I : Ideal A
n : ℕ
hn_fac : IsUnit ↑(n - 1)!
hnI : I ^ n = 0
m : ℕ
x : A
hx : x ∈ I
y : A
hy : y ∈ I
hmn : ¬m < n
h_sub : I ^ m ≤ I ^ n
hxy : (x + y) ^ m = 0
k : ℕ × ℕ
hk : k.1 + k.2 = m
⊢ y ^ k.2 * x ^ k.1 ∈ ↑(... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/RatAlgebra.lean | DividedPowers.OfInvertibleFactorial.dpow_add | [100, 1] | [121, 78] | exact Ideal.mul_mem_mul (Ideal.pow_mem_pow hy _) (Ideal.pow_mem_pow hx _) | case neg.h.h
A : Type u_1
inst✝ : CommSemiring A
I : Ideal A
n : ℕ
hn_fac : IsUnit ↑(n - 1)!
hnI : I ^ n = 0
m : ℕ
x : A
hx : x ∈ I
y : A
hy : y ∈ I
hmn : ¬m < n
h_sub : I ^ m ≤ I ^ n
hxy : (x + y) ^ m = 0
k : ℕ × ℕ
hk : k.1 + k.2 = m
⊢ y ^ k.2 * x ^ k.1 ∈ ↑(I ^ k.2 * I ^ k.1) | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case neg.h.h
A : Type u_1
inst✝ : CommSemiring A
I : Ideal A
n : ℕ
hn_fac : IsUnit ↑(n - 1)!
hnI : I ^ n = 0
m : ℕ
x : A
hx : x ∈ I
y : A
hy : y ∈ I
hmn : ¬m < n
h_sub : I ^ m ≤ I ^ n
hxy : (x + y) ^ m = 0
k : ℕ × ℕ
hk : k.1 + k.2 = m
⊢ y ^ k.2 * x ^ k.1 ∈ ↑(... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/RatAlgebra.lean | DividedPowers.OfInvertibleFactorial.dpow_add | [100, 1] | [121, 78] | rw [← Ideal.mem_bot, ← Ideal.zero_eq_bot, ← hnI] | A : Type u_1
inst✝ : CommSemiring A
I : Ideal A
n : ℕ
hn_fac : IsUnit ↑(n - 1)!
hnI : I ^ n = 0
m : ℕ
x : A
hx : x ∈ I
y : A
hy : y ∈ I
hmn : ¬m < n
h_sub : I ^ m ≤ I ^ n
⊢ (x + y) ^ m = 0 | A : Type u_1
inst✝ : CommSemiring A
I : Ideal A
n : ℕ
hn_fac : IsUnit ↑(n - 1)!
hnI : I ^ n = 0
m : ℕ
x : A
hx : x ∈ I
y : A
hy : y ∈ I
hmn : ¬m < n
h_sub : I ^ m ≤ I ^ n
⊢ (x + y) ^ m ∈ I ^ n | Please generate a tactic in lean4 to solve the state.
STATE:
A : Type u_1
inst✝ : CommSemiring A
I : Ideal A
n : ℕ
hn_fac : IsUnit ↑(n - 1)!
hnI : I ^ n = 0
m : ℕ
x : A
hx : x ∈ I
y : A
hy : y ∈ I
hmn : ¬m < n
h_sub : I ^ m ≤ I ^ n
⊢ (x + y) ^ m = 0
TACTIC:
|
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/RatAlgebra.lean | DividedPowers.OfInvertibleFactorial.dpow_add | [100, 1] | [121, 78] | apply Set.mem_of_subset_of_mem h_sub (Ideal.pow_mem_pow (Ideal.add_mem I hx hy) m) | A : Type u_1
inst✝ : CommSemiring A
I : Ideal A
n : ℕ
hn_fac : IsUnit ↑(n - 1)!
hnI : I ^ n = 0
m : ℕ
x : A
hx : x ∈ I
y : A
hy : y ∈ I
hmn : ¬m < n
h_sub : I ^ m ≤ I ^ n
⊢ (x + y) ^ m ∈ I ^ n | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
A : Type u_1
inst✝ : CommSemiring A
I : Ideal A
n : ℕ
hn_fac : IsUnit ↑(n - 1)!
hnI : I ^ n = 0
m : ℕ
x : A
hx : x ∈ I
y : A
hy : y ∈ I
hmn : ¬m < n
h_sub : I ^ m ≤ I ^ n
⊢ (x + y) ^ m ∈ I ^ n
TACTIC:
|
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/RatAlgebra.lean | DividedPowers.OfInvertibleFactorial.dpow_smul | [124, 1] | [126, 57] | rw [dpow_eq_of_mem m (Ideal.mul_mem_left I _ hx), dpow_eq_of_mem m hx,
mul_pow, ← mul_assoc, mul_comm _ (a ^ m), mul_assoc] | A : Type u_1
inst✝ : CommSemiring A
I : Ideal A
m : ℕ
a x : A
hx : x ∈ I
⊢ dpow I m (a * x) = a ^ m * dpow I m x | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
A : Type u_1
inst✝ : CommSemiring A
I : Ideal A
m : ℕ
a x : A
hx : x ∈ I
⊢ dpow I m (a * x) = a ^ m * dpow I m x
TACTIC:
|
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/RatAlgebra.lean | DividedPowers.OfInvertibleFactorial.dpow_mul_dif_pos | [129, 1] | [143, 101] | have hm : m < n := lt_of_le_of_lt le_self_add hkm | A : Type u_1
inst✝ : CommSemiring A
I : Ideal A
n : ℕ
hn_fac : IsUnit ↑(n - 1)!
m k : ℕ
hkm : m + k < n
x : A
hx : x ∈ I
⊢ dpow I m x * dpow I k x = ↑((m + k).choose m) * dpow I (m + k) x | A : Type u_1
inst✝ : CommSemiring A
I : Ideal A
n : ℕ
hn_fac : IsUnit ↑(n - 1)!
m k : ℕ
hkm : m + k < n
x : A
hx : x ∈ I
hm : m < n
⊢ dpow I m x * dpow I k x = ↑((m + k).choose m) * dpow I (m + k) x | Please generate a tactic in lean4 to solve the state.
STATE:
A : Type u_1
inst✝ : CommSemiring A
I : Ideal A
n : ℕ
hn_fac : IsUnit ↑(n - 1)!
m k : ℕ
hkm : m + k < n
x : A
hx : x ∈ I
⊢ dpow I m x * dpow I k x = ↑((m + k).choose m) * dpow I (m + k) x
TACTIC:
|
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/RatAlgebra.lean | DividedPowers.OfInvertibleFactorial.dpow_mul_dif_pos | [129, 1] | [143, 101] | have hk : k < n := lt_of_le_of_lt le_add_self hkm | A : Type u_1
inst✝ : CommSemiring A
I : Ideal A
n : ℕ
hn_fac : IsUnit ↑(n - 1)!
m k : ℕ
hkm : m + k < n
x : A
hx : x ∈ I
hm : m < n
⊢ dpow I m x * dpow I k x = ↑((m + k).choose m) * dpow I (m + k) x | A : Type u_1
inst✝ : CommSemiring A
I : Ideal A
n : ℕ
hn_fac : IsUnit ↑(n - 1)!
m k : ℕ
hkm : m + k < n
x : A
hx : x ∈ I
hm : m < n
hk : k < n
⊢ dpow I m x * dpow I k x = ↑((m + k).choose m) * dpow I (m + k) x | Please generate a tactic in lean4 to solve the state.
STATE:
A : Type u_1
inst✝ : CommSemiring A
I : Ideal A
n : ℕ
hn_fac : IsUnit ↑(n - 1)!
m k : ℕ
hkm : m + k < n
x : A
hx : x ∈ I
hm : m < n
⊢ dpow I m x * dpow I k x = ↑((m + k).choose m) * dpow I (m + k) x
TACTIC:
|
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/RatAlgebra.lean | DividedPowers.OfInvertibleFactorial.dpow_mul_dif_pos | [129, 1] | [143, 101] | rw [dpow_eq_of_mem _ hx, dpow_eq_of_mem _ hx, dpow_eq_of_mem _ hx,
mul_assoc, ← mul_assoc (x ^ m), mul_comm (x ^ m), mul_assoc _ (x ^ m),
← pow_add, ← mul_assoc, ← mul_assoc] | A : Type u_1
inst✝ : CommSemiring A
I : Ideal A
n : ℕ
hn_fac : IsUnit ↑(n - 1)!
m k : ℕ
hkm : m + k < n
x : A
hx : x ∈ I
hm : m < n
hk : k < n
⊢ dpow I m x * dpow I k x = ↑((m + k).choose m) * dpow I (m + k) x | A : Type u_1
inst✝ : CommSemiring A
I : Ideal A
n : ℕ
hn_fac : IsUnit ↑(n - 1)!
m k : ℕ
hkm : m + k < n
x : A
hx : x ∈ I
hm : m < n
hk : k < n
⊢ Ring.inverse ↑m ! * Ring.inverse ↑k ! * x ^ (m + k) = ↑((m + k).choose m) * Ring.inverse ↑(m + k)! * x ^ (m + k) | Please generate a tactic in lean4 to solve the state.
STATE:
A : Type u_1
inst✝ : CommSemiring A
I : Ideal A
n : ℕ
hn_fac : IsUnit ↑(n - 1)!
m k : ℕ
hkm : m + k < n
x : A
hx : x ∈ I
hm : m < n
hk : k < n
⊢ dpow I m x * dpow I k x = ↑((m + k).choose m) * dpow I (m + k) x
TACTIC:
|
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/RatAlgebra.lean | DividedPowers.OfInvertibleFactorial.dpow_mul_dif_pos | [129, 1] | [143, 101] | apply congr_arg₂ _ _ rfl | A : Type u_1
inst✝ : CommSemiring A
I : Ideal A
n : ℕ
hn_fac : IsUnit ↑(n - 1)!
m k : ℕ
hkm : m + k < n
x : A
hx : x ∈ I
hm : m < n
hk : k < n
⊢ Ring.inverse ↑m ! * Ring.inverse ↑k ! * x ^ (m + k) = ↑((m + k).choose m) * Ring.inverse ↑(m + k)! * x ^ (m + k) | A : Type u_1
inst✝ : CommSemiring A
I : Ideal A
n : ℕ
hn_fac : IsUnit ↑(n - 1)!
m k : ℕ
hkm : m + k < n
x : A
hx : x ∈ I
hm : m < n
hk : k < n
⊢ Ring.inverse ↑m ! * Ring.inverse ↑k ! = ↑((m + k).choose m) * Ring.inverse ↑(m + k)! | Please generate a tactic in lean4 to solve the state.
STATE:
A : Type u_1
inst✝ : CommSemiring A
I : Ideal A
n : ℕ
hn_fac : IsUnit ↑(n - 1)!
m k : ℕ
hkm : m + k < n
x : A
hx : x ∈ I
hm : m < n
hk : k < n
⊢ Ring.inverse ↑m ! * Ring.inverse ↑k ! * x ^ (m + k) = ↑((m + k).choose m) * Ring.inverse ↑(m + k)! * x ^ (m + k)
T... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/RatAlgebra.lean | DividedPowers.OfInvertibleFactorial.dpow_mul_dif_pos | [129, 1] | [143, 101] | rw [Ring.eq_mul_inverse_iff_mul_eq _ _ _ (factorial_isUnit hn_fac hkm),
mul_assoc,
Ring.inverse_mul_eq_iff_eq_mul _ _ _ (factorial_isUnit hn_fac hm),
Ring.inverse_mul_eq_iff_eq_mul _ _ _ (factorial_isUnit hn_fac hk)] | A : Type u_1
inst✝ : CommSemiring A
I : Ideal A
n : ℕ
hn_fac : IsUnit ↑(n - 1)!
m k : ℕ
hkm : m + k < n
x : A
hx : x ∈ I
hm : m < n
hk : k < n
⊢ Ring.inverse ↑m ! * Ring.inverse ↑k ! = ↑((m + k).choose m) * Ring.inverse ↑(m + k)! | A : Type u_1
inst✝ : CommSemiring A
I : Ideal A
n : ℕ
hn_fac : IsUnit ↑(n - 1)!
m k : ℕ
hkm : m + k < n
x : A
hx : x ∈ I
hm : m < n
hk : k < n
⊢ ↑(m + k)! = ↑k ! * (↑m ! * ↑((m + k).choose m)) | Please generate a tactic in lean4 to solve the state.
STATE:
A : Type u_1
inst✝ : CommSemiring A
I : Ideal A
n : ℕ
hn_fac : IsUnit ↑(n - 1)!
m k : ℕ
hkm : m + k < n
x : A
hx : x ∈ I
hm : m < n
hk : k < n
⊢ Ring.inverse ↑m ! * Ring.inverse ↑k ! = ↑((m + k).choose m) * Ring.inverse ↑(m + k)!
TACTIC:
|
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/RatAlgebra.lean | DividedPowers.OfInvertibleFactorial.dpow_mul_dif_pos | [129, 1] | [143, 101] | norm_cast | A : Type u_1
inst✝ : CommSemiring A
I : Ideal A
n : ℕ
hn_fac : IsUnit ↑(n - 1)!
m k : ℕ
hkm : m + k < n
x : A
hx : x ∈ I
hm : m < n
hk : k < n
⊢ ↑(m + k)! = ↑k ! * (↑m ! * ↑((m + k).choose m)) | A : Type u_1
inst✝ : CommSemiring A
I : Ideal A
n : ℕ
hn_fac : IsUnit ↑(n - 1)!
m k : ℕ
hkm : m + k < n
x : A
hx : x ∈ I
hm : m < n
hk : k < n
⊢ ↑(m + k)! = ↑(k ! * (m ! * (m + k).choose m)) | Please generate a tactic in lean4 to solve the state.
STATE:
A : Type u_1
inst✝ : CommSemiring A
I : Ideal A
n : ℕ
hn_fac : IsUnit ↑(n - 1)!
m k : ℕ
hkm : m + k < n
x : A
hx : x ∈ I
hm : m < n
hk : k < n
⊢ ↑(m + k)! = ↑k ! * (↑m ! * ↑((m + k).choose m))
TACTIC:
|
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/RatAlgebra.lean | DividedPowers.OfInvertibleFactorial.dpow_mul_dif_pos | [129, 1] | [143, 101] | apply congr_arg | A : Type u_1
inst✝ : CommSemiring A
I : Ideal A
n : ℕ
hn_fac : IsUnit ↑(n - 1)!
m k : ℕ
hkm : m + k < n
x : A
hx : x ∈ I
hm : m < n
hk : k < n
⊢ ↑(m + k)! = ↑(k ! * (m ! * (m + k).choose m)) | case h
A : Type u_1
inst✝ : CommSemiring A
I : Ideal A
n : ℕ
hn_fac : IsUnit ↑(n - 1)!
m k : ℕ
hkm : m + k < n
x : A
hx : x ∈ I
hm : m < n
hk : k < n
⊢ (m + k)! = k ! * (m ! * (m + k).choose m) | Please generate a tactic in lean4 to solve the state.
STATE:
A : Type u_1
inst✝ : CommSemiring A
I : Ideal A
n : ℕ
hn_fac : IsUnit ↑(n - 1)!
m k : ℕ
hkm : m + k < n
x : A
hx : x ∈ I
hm : m < n
hk : k < n
⊢ ↑(m + k)! = ↑(k ! * (m ! * (m + k).choose m))
TACTIC:
|
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/RatAlgebra.lean | DividedPowers.OfInvertibleFactorial.dpow_mul_dif_pos | [129, 1] | [143, 101] | rw [← Nat.add_choose_mul_factorial_mul_factorial, mul_comm, mul_comm _ (m !), Nat.choose_symm_add] | case h
A : Type u_1
inst✝ : CommSemiring A
I : Ideal A
n : ℕ
hn_fac : IsUnit ↑(n - 1)!
m k : ℕ
hkm : m + k < n
x : A
hx : x ∈ I
hm : m < n
hk : k < n
⊢ (m + k)! = k ! * (m ! * (m + k).choose m) | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case h
A : Type u_1
inst✝ : CommSemiring A
I : Ideal A
n : ℕ
hn_fac : IsUnit ↑(n - 1)!
m k : ℕ
hkm : m + k < n
x : A
hx : x ∈ I
hm : m < n
hk : k < n
⊢ (m + k)! = k ! * (m ! * (m + k).choose m)
TACTIC:
|
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/RatAlgebra.lean | DividedPowers.OfInvertibleFactorial.dpow_mul | [146, 1] | [153, 98] | by_cases hkm : m + k < n | A : Type u_1
inst✝ : CommSemiring A
I : Ideal A
n : ℕ
hn_fac : IsUnit ↑(n - 1)!
hnI : I ^ n = 0
m k : ℕ
x : A
hx : x ∈ I
⊢ dpow I m x * dpow I k x = ↑((m + k).choose m) * dpow I (m + k) x | case pos
A : Type u_1
inst✝ : CommSemiring A
I : Ideal A
n : ℕ
hn_fac : IsUnit ↑(n - 1)!
hnI : I ^ n = 0
m k : ℕ
x : A
hx : x ∈ I
hkm : m + k < n
⊢ dpow I m x * dpow I k x = ↑((m + k).choose m) * dpow I (m + k) x
case neg
A : Type u_1
inst✝ : CommSemiring A
I : Ideal A
n : ℕ
hn_fac : IsUnit ↑(n - 1)!
hnI : I ^ n = 0
m... | Please generate a tactic in lean4 to solve the state.
STATE:
A : Type u_1
inst✝ : CommSemiring A
I : Ideal A
n : ℕ
hn_fac : IsUnit ↑(n - 1)!
hnI : I ^ n = 0
m k : ℕ
x : A
hx : x ∈ I
⊢ dpow I m x * dpow I k x = ↑((m + k).choose m) * dpow I (m + k) x
TACTIC:
|
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/RatAlgebra.lean | DividedPowers.OfInvertibleFactorial.dpow_mul | [146, 1] | [153, 98] | exact dpow_mul_dif_pos hn_fac hkm hx | case pos
A : Type u_1
inst✝ : CommSemiring A
I : Ideal A
n : ℕ
hn_fac : IsUnit ↑(n - 1)!
hnI : I ^ n = 0
m k : ℕ
x : A
hx : x ∈ I
hkm : m + k < n
⊢ dpow I m x * dpow I k x = ↑((m + k).choose m) * dpow I (m + k) x | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case pos
A : Type u_1
inst✝ : CommSemiring A
I : Ideal A
n : ℕ
hn_fac : IsUnit ↑(n - 1)!
hnI : I ^ n = 0
m k : ℕ
x : A
hx : x ∈ I
hkm : m + k < n
⊢ dpow I m x * dpow I k x = ↑((m + k).choose m) * dpow I (m + k) x
TACTIC:
|
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/RatAlgebra.lean | DividedPowers.OfInvertibleFactorial.dpow_mul | [146, 1] | [153, 98] | have hxmk : x ^ (m + k) = 0 := Ideal.mem_pow_eq_zero n (m + k) hnI (not_lt.mp hkm) hx | case neg
A : Type u_1
inst✝ : CommSemiring A
I : Ideal A
n : ℕ
hn_fac : IsUnit ↑(n - 1)!
hnI : I ^ n = 0
m k : ℕ
x : A
hx : x ∈ I
hkm : ¬m + k < n
⊢ dpow I m x * dpow I k x = ↑((m + k).choose m) * dpow I (m + k) x | case neg
A : Type u_1
inst✝ : CommSemiring A
I : Ideal A
n : ℕ
hn_fac : IsUnit ↑(n - 1)!
hnI : I ^ n = 0
m k : ℕ
x : A
hx : x ∈ I
hkm : ¬m + k < n
hxmk : x ^ (m + k) = 0
⊢ dpow I m x * dpow I k x = ↑((m + k).choose m) * dpow I (m + k) x | Please generate a tactic in lean4 to solve the state.
STATE:
case neg
A : Type u_1
inst✝ : CommSemiring A
I : Ideal A
n : ℕ
hn_fac : IsUnit ↑(n - 1)!
hnI : I ^ n = 0
m k : ℕ
x : A
hx : x ∈ I
hkm : ¬m + k < n
⊢ dpow I m x * dpow I k x = ↑((m + k).choose m) * dpow I (m + k) x
TACTIC:
|
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/RatAlgebra.lean | DividedPowers.OfInvertibleFactorial.dpow_mul | [146, 1] | [153, 98] | rw [dpow_eq_of_mem m hx, dpow_eq_of_mem k hx, dpow_eq_of_mem (m + k) hx,
mul_assoc, ← mul_assoc (x ^ m), mul_comm (x ^ m), mul_assoc _ (x ^ m), ← pow_add, hxmk,
MulZeroClass.mul_zero, MulZeroClass.mul_zero, MulZeroClass.mul_zero, MulZeroClass.mul_zero] | case neg
A : Type u_1
inst✝ : CommSemiring A
I : Ideal A
n : ℕ
hn_fac : IsUnit ↑(n - 1)!
hnI : I ^ n = 0
m k : ℕ
x : A
hx : x ∈ I
hkm : ¬m + k < n
hxmk : x ^ (m + k) = 0
⊢ dpow I m x * dpow I k x = ↑((m + k).choose m) * dpow I (m + k) x | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case neg
A : Type u_1
inst✝ : CommSemiring A
I : Ideal A
n : ℕ
hn_fac : IsUnit ↑(n - 1)!
hnI : I ^ n = 0
m k : ℕ
x : A
hx : x ∈ I
hkm : ¬m + k < n
hxmk : x ^ (m + k) = 0
⊢ dpow I m x * dpow I k x = ↑((m + k).choose m) * dpow I (m + k) x
TACTIC:
|
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/RatAlgebra.lean | DividedPowers.OfInvertibleFactorial.dpow_comp_dif_pos | [156, 1] | [172, 39] | have hmn : m < n := lt_of_le_of_lt (Nat.le_mul_of_pos_right _ (Nat.pos_of_ne_zero hk)) hkm | A : Type u_1
inst✝ : CommSemiring A
I : Ideal A
n : ℕ
hn_fac : IsUnit ↑(n - 1)!
m k : ℕ
hk : k ≠ 0
hkm : m * k < n
x : A
hx : x ∈ I
⊢ dpow I m (dpow I k x) = ↑(mchoose m k) * dpow I (m * k) x | A : Type u_1
inst✝ : CommSemiring A
I : Ideal A
n : ℕ
hn_fac : IsUnit ↑(n - 1)!
m k : ℕ
hk : k ≠ 0
hkm : m * k < n
x : A
hx : x ∈ I
hmn : m < n
⊢ dpow I m (dpow I k x) = ↑(mchoose m k) * dpow I (m * k) x | Please generate a tactic in lean4 to solve the state.
STATE:
A : Type u_1
inst✝ : CommSemiring A
I : Ideal A
n : ℕ
hn_fac : IsUnit ↑(n - 1)!
m k : ℕ
hk : k ≠ 0
hkm : m * k < n
x : A
hx : x ∈ I
⊢ dpow I m (dpow I k x) = ↑(mchoose m k) * dpow I (m * k) x
TACTIC:
|
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/RatAlgebra.lean | DividedPowers.OfInvertibleFactorial.dpow_comp_dif_pos | [156, 1] | [172, 39] | rw [dpow_eq_of_mem (m * k) hx, dpow_eq_of_mem _ (dpow_mem hk hx)] | A : Type u_1
inst✝ : CommSemiring A
I : Ideal A
n : ℕ
hn_fac : IsUnit ↑(n - 1)!
m k : ℕ
hk : k ≠ 0
hkm : m * k < n
x : A
hx : x ∈ I
hmn : m < n
⊢ dpow I m (dpow I k x) = ↑(mchoose m k) * dpow I (m * k) x | A : Type u_1
inst✝ : CommSemiring A
I : Ideal A
n : ℕ
hn_fac : IsUnit ↑(n - 1)!
m k : ℕ
hk : k ≠ 0
hkm : m * k < n
x : A
hx : x ∈ I
hmn : m < n
⊢ Ring.inverse ↑m ! * dpow I k x ^ m = ↑(mchoose m k) * (Ring.inverse ↑(m * k)! * x ^ (m * k)) | Please generate a tactic in lean4 to solve the state.
STATE:
A : Type u_1
inst✝ : CommSemiring A
I : Ideal A
n : ℕ
hn_fac : IsUnit ↑(n - 1)!
m k : ℕ
hk : k ≠ 0
hkm : m * k < n
x : A
hx : x ∈ I
hmn : m < n
⊢ dpow I m (dpow I k x) = ↑(mchoose m k) * dpow I (m * k) x
TACTIC:
|
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/RatAlgebra.lean | DividedPowers.OfInvertibleFactorial.dpow_comp_dif_pos | [156, 1] | [172, 39] | by_cases hm0 : m = 0 | A : Type u_1
inst✝ : CommSemiring A
I : Ideal A
n : ℕ
hn_fac : IsUnit ↑(n - 1)!
m k : ℕ
hk : k ≠ 0
hkm : m * k < n
x : A
hx : x ∈ I
hmn : m < n
⊢ Ring.inverse ↑m ! * dpow I k x ^ m = ↑(mchoose m k) * (Ring.inverse ↑(m * k)! * x ^ (m * k)) | case pos
A : Type u_1
inst✝ : CommSemiring A
I : Ideal A
n : ℕ
hn_fac : IsUnit ↑(n - 1)!
m k : ℕ
hk : k ≠ 0
hkm : m * k < n
x : A
hx : x ∈ I
hmn : m < n
hm0 : m = 0
⊢ Ring.inverse ↑m ! * dpow I k x ^ m = ↑(mchoose m k) * (Ring.inverse ↑(m * k)! * x ^ (m * k))
case neg
A : Type u_1
inst✝ : CommSemiring A
I : Ideal A
n ... | Please generate a tactic in lean4 to solve the state.
STATE:
A : Type u_1
inst✝ : CommSemiring A
I : Ideal A
n : ℕ
hn_fac : IsUnit ↑(n - 1)!
m k : ℕ
hk : k ≠ 0
hkm : m * k < n
x : A
hx : x ∈ I
hmn : m < n
⊢ Ring.inverse ↑m ! * dpow I k x ^ m = ↑(mchoose m k) * (Ring.inverse ↑(m * k)! * x ^ (m * k))
TACTIC:
|
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/RatAlgebra.lean | DividedPowers.OfInvertibleFactorial.dpow_comp_dif_pos | [156, 1] | [172, 39] | simp only [hm0, MulZeroClass.zero_mul, _root_.pow_zero, mul_one, mchoose_zero, Nat.cast_one, one_mul] | case pos
A : Type u_1
inst✝ : CommSemiring A
I : Ideal A
n : ℕ
hn_fac : IsUnit ↑(n - 1)!
m k : ℕ
hk : k ≠ 0
hkm : m * k < n
x : A
hx : x ∈ I
hmn : m < n
hm0 : m = 0
⊢ Ring.inverse ↑m ! * dpow I k x ^ m = ↑(mchoose m k) * (Ring.inverse ↑(m * k)! * x ^ (m * k)) | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case pos
A : Type u_1
inst✝ : CommSemiring A
I : Ideal A
n : ℕ
hn_fac : IsUnit ↑(n - 1)!
m k : ℕ
hk : k ≠ 0
hkm : m * k < n
x : A
hx : x ∈ I
hmn : m < n
hm0 : m = 0
⊢ Ring.inverse ↑m ! * dpow I k x ^ m = ↑(mchoose m k) * (Ring.inverse ↑(m * k)! * x ^ (m * k))... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/RatAlgebra.lean | DividedPowers.OfInvertibleFactorial.dpow_comp_dif_pos | [156, 1] | [172, 39] | have hkn : k < n := lt_of_le_of_lt (Nat.le_mul_of_pos_left _ (Nat.pos_of_ne_zero hm0)) hkm | case neg
A : Type u_1
inst✝ : CommSemiring A
I : Ideal A
n : ℕ
hn_fac : IsUnit ↑(n - 1)!
m k : ℕ
hk : k ≠ 0
hkm : m * k < n
x : A
hx : x ∈ I
hmn : m < n
hm0 : ¬m = 0
⊢ Ring.inverse ↑m ! * dpow I k x ^ m = ↑(mchoose m k) * (Ring.inverse ↑(m * k)! * x ^ (m * k)) | case neg
A : Type u_1
inst✝ : CommSemiring A
I : Ideal A
n : ℕ
hn_fac : IsUnit ↑(n - 1)!
m k : ℕ
hk : k ≠ 0
hkm : m * k < n
x : A
hx : x ∈ I
hmn : m < n
hm0 : ¬m = 0
hkn : k < n
⊢ Ring.inverse ↑m ! * dpow I k x ^ m = ↑(mchoose m k) * (Ring.inverse ↑(m * k)! * x ^ (m * k)) | Please generate a tactic in lean4 to solve the state.
STATE:
case neg
A : Type u_1
inst✝ : CommSemiring A
I : Ideal A
n : ℕ
hn_fac : IsUnit ↑(n - 1)!
m k : ℕ
hk : k ≠ 0
hkm : m * k < n
x : A
hx : x ∈ I
hmn : m < n
hm0 : ¬m = 0
⊢ Ring.inverse ↑m ! * dpow I k x ^ m = ↑(mchoose m k) * (Ring.inverse ↑(m * k)! * x ^ (m * k)... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/RatAlgebra.lean | DividedPowers.OfInvertibleFactorial.dpow_comp_dif_pos | [156, 1] | [172, 39] | rw [dpow_eq_of_mem _ hx] | case neg
A : Type u_1
inst✝ : CommSemiring A
I : Ideal A
n : ℕ
hn_fac : IsUnit ↑(n - 1)!
m k : ℕ
hk : k ≠ 0
hkm : m * k < n
x : A
hx : x ∈ I
hmn : m < n
hm0 : ¬m = 0
hkn : k < n
⊢ Ring.inverse ↑m ! * dpow I k x ^ m = ↑(mchoose m k) * (Ring.inverse ↑(m * k)! * x ^ (m * k)) | case neg
A : Type u_1
inst✝ : CommSemiring A
I : Ideal A
n : ℕ
hn_fac : IsUnit ↑(n - 1)!
m k : ℕ
hk : k ≠ 0
hkm : m * k < n
x : A
hx : x ∈ I
hmn : m < n
hm0 : ¬m = 0
hkn : k < n
⊢ Ring.inverse ↑m ! * (Ring.inverse ↑k ! * x ^ k) ^ m = ↑(mchoose m k) * (Ring.inverse ↑(m * k)! * x ^ (m * k)) | Please generate a tactic in lean4 to solve the state.
STATE:
case neg
A : Type u_1
inst✝ : CommSemiring A
I : Ideal A
n : ℕ
hn_fac : IsUnit ↑(n - 1)!
m k : ℕ
hk : k ≠ 0
hkm : m * k < n
x : A
hx : x ∈ I
hmn : m < n
hm0 : ¬m = 0
hkn : k < n
⊢ Ring.inverse ↑m ! * dpow I k x ^ m = ↑(mchoose m k) * (Ring.inverse ↑(m * k)! *... |
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