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/Dpow.lean | DividedPowerAlgebra.condτ_rel | [654, 1] | [728, 12] | simp only [hJ'J, Ideal.mem_map_iff_of_surjective g hg] at ha' | case h.right.right
R✝ : Type u
inst✝⁹ : CommRing R✝
A : Type u
inst✝⁸ : CommRing A
R S R' S' : Type u
inst✝⁷ : CommRing R
inst✝⁶ : CommRing S
inst✝⁵ : CommRing R'
inst✝⁴ : CommRing S'
inst✝³ : Algebra A R
inst✝² : Algebra A S
inst✝¹ : Algebra A R'
inst✝ : Algebra A S'
f : R →ₐ[A] R'
hf : Function.Surjective ⇑f
I : Idea... | case h.right.right
R✝ : Type u
inst✝⁹ : CommRing R✝
A : Type u
inst✝⁸ : CommRing A
R S R' S' : Type u
inst✝⁷ : CommRing R
inst✝⁶ : CommRing S
inst✝⁵ : CommRing R'
inst✝⁴ : CommRing S'
inst✝³ : Algebra A R
inst✝² : Algebra A S
inst✝¹ : Algebra A R'
inst✝ : Algebra A S'
f : R →ₐ[A] R'
hf : Function.Surjective ⇑f
I : Idea... | Please generate a tactic in lean4 to solve the state.
STATE:
case h.right.right
R✝ : Type u
inst✝⁹ : CommRing R✝
A : Type u
inst✝⁸ : CommRing A
R S R' S' : Type u
inst✝⁷ : CommRing R
inst✝⁶ : CommRing S
inst✝⁵ : CommRing R'
inst✝⁴ : CommRing S'
inst✝³ : Algebra A R
inst✝² : Algebra A S
inst✝¹ : Algebra A R'
inst✝ : Alg... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/DPAlgebra/Dpow.lean | DividedPowerAlgebra.condτ_rel | [654, 1] | [728, 12] | obtain ⟨a, ha, rfl⟩ := ha' | case h.right.right
R✝ : Type u
inst✝⁹ : CommRing R✝
A : Type u
inst✝⁸ : CommRing A
R S R' S' : Type u
inst✝⁷ : CommRing R
inst✝⁶ : CommRing S
inst✝⁵ : CommRing R'
inst✝⁴ : CommRing S'
inst✝³ : Algebra A R
inst✝² : Algebra A S
inst✝¹ : Algebra A R'
inst✝ : Algebra A S'
f : R →ₐ[A] R'
hf : Function.Surjective ⇑f
I : Idea... | case h.right.right.intro.intro
R✝ : Type u
inst✝⁹ : CommRing R✝
A : Type u
inst✝⁸ : CommRing A
R S R' S' : Type u
inst✝⁷ : CommRing R
inst✝⁶ : CommRing S
inst✝⁵ : CommRing R'
inst✝⁴ : CommRing S'
inst✝³ : Algebra A R
inst✝² : Algebra A S
inst✝¹ : Algebra A R'
inst✝ : Algebra A S'
f : R →ₐ[A] R'
hf : Function.Surjective... | Please generate a tactic in lean4 to solve the state.
STATE:
case h.right.right
R✝ : Type u
inst✝⁹ : CommRing R✝
A : Type u
inst✝⁸ : CommRing A
R S R' S' : Type u
inst✝⁷ : CommRing R
inst✝⁶ : CommRing S
inst✝⁵ : CommRing R'
inst✝⁴ : CommRing S'
inst✝³ : Algebra A R
inst✝² : Algebra A S
inst✝¹ : Algebra A R'
inst✝ : Alg... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/DPAlgebra/Dpow.lean | DividedPowerAlgebra.condτ_rel | [654, 1] | [728, 12] | simp only [i2, AlgHom.coe_toRingHom, Algebra.TensorProduct.includeRight_apply] | case h.right.right.intro.intro
R✝ : Type u
inst✝⁹ : CommRing R✝
A : Type u
inst✝⁸ : CommRing A
R S R' S' : Type u
inst✝⁷ : CommRing R
inst✝⁶ : CommRing S
inst✝⁵ : CommRing R'
inst✝⁴ : CommRing S'
inst✝³ : Algebra A R
inst✝² : Algebra A S
inst✝¹ : Algebra A R'
inst✝ : Algebra A S'
f : R →ₐ[A] R'
hf : Function.Surjective... | case h.right.right.intro.intro
R✝ : Type u
inst✝⁹ : CommRing R✝
A : Type u
inst✝⁸ : CommRing A
R S R' S' : Type u
inst✝⁷ : CommRing R
inst✝⁶ : CommRing S
inst✝⁵ : CommRing R'
inst✝⁴ : CommRing S'
inst✝³ : Algebra A R
inst✝² : Algebra A S
inst✝¹ : Algebra A R'
inst✝ : Algebra A S'
f : R →ₐ[A] R'
hf : Function.Surjective... | Please generate a tactic in lean4 to solve the state.
STATE:
case h.right.right.intro.intro
R✝ : Type u
inst✝⁹ : CommRing R✝
A : Type u
inst✝⁸ : CommRing A
R S R' S' : Type u
inst✝⁷ : CommRing R
inst✝⁶ : CommRing S
inst✝⁵ : CommRing R'
inst✝⁴ : CommRing S'
inst✝³ : Algebra A R
inst✝² : Algebra A S
inst✝¹ : Algebra A R'... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/DPAlgebra/Dpow.lean | DividedPowerAlgebra.condτ_rel | [654, 1] | [728, 12] | suffices ∀ y : S, fg.toRingHom (1 ⊗ₜ[A] y) = 1 ⊗ₜ[A] g y by
rw [← this]
rw [Quotient.OfSurjective.dpow_apply hK s_fg]
have that := hg'.2 n a ha
simp only [AlgHom.coe_toRingHom] at that ; rw [that]
rw [← this]
apply congr_arg
simp only [← Algebra.TensorProduct.includeRight_apply]
exact hK_pd.2.2 n a ha
... | case h.right.right.intro.intro
R✝ : Type u
inst✝⁹ : CommRing R✝
A : Type u
inst✝⁸ : CommRing A
R S R' S' : Type u
inst✝⁷ : CommRing R
inst✝⁶ : CommRing S
inst✝⁵ : CommRing R'
inst✝⁴ : CommRing S'
inst✝³ : Algebra A R
inst✝² : Algebra A S
inst✝¹ : Algebra A R'
inst✝ : Algebra A S'
f : R →ₐ[A] R'
hf : Function.Surjective... | case h.right.right.intro.intro
R✝ : Type u
inst✝⁹ : CommRing R✝
A : Type u
inst✝⁸ : CommRing A
R S R' S' : Type u
inst✝⁷ : CommRing R
inst✝⁶ : CommRing S
inst✝⁵ : CommRing R'
inst✝⁴ : CommRing S'
inst✝³ : Algebra A R
inst✝² : Algebra A S
inst✝¹ : Algebra A R'
inst✝ : Algebra A S'
f : R →ₐ[A] R'
hf : Function.Surjective... | Please generate a tactic in lean4 to solve the state.
STATE:
case h.right.right.intro.intro
R✝ : Type u
inst✝⁹ : CommRing R✝
A : Type u
inst✝⁸ : CommRing A
R S R' S' : Type u
inst✝⁷ : CommRing R
inst✝⁶ : CommRing S
inst✝⁵ : CommRing R'
inst✝⁴ : CommRing S'
inst✝³ : Algebra A R
inst✝² : Algebra A S
inst✝¹ : Algebra A R'... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/DPAlgebra/Dpow.lean | DividedPowerAlgebra.condτ_rel | [654, 1] | [728, 12] | intro x | case h.right.right.intro.intro
R✝ : Type u
inst✝⁹ : CommRing R✝
A : Type u
inst✝⁸ : CommRing A
R S R' S' : Type u
inst✝⁷ : CommRing R
inst✝⁶ : CommRing S
inst✝⁵ : CommRing R'
inst✝⁴ : CommRing S'
inst✝³ : Algebra A R
inst✝² : Algebra A S
inst✝¹ : Algebra A R'
inst✝ : Algebra A S'
f : R →ₐ[A] R'
hf : Function.Surjective... | case h.right.right.intro.intro
R✝ : Type u
inst✝⁹ : CommRing R✝
A : Type u
inst✝⁸ : CommRing A
R S R' S' : Type u
inst✝⁷ : CommRing R
inst✝⁶ : CommRing S
inst✝⁵ : CommRing R'
inst✝⁴ : CommRing S'
inst✝³ : Algebra A R
inst✝² : Algebra A S
inst✝¹ : Algebra A R'
inst✝ : Algebra A S'
f : R →ₐ[A] R'
hf : Function.Surjective... | Please generate a tactic in lean4 to solve the state.
STATE:
case h.right.right.intro.intro
R✝ : Type u
inst✝⁹ : CommRing R✝
A : Type u
inst✝⁸ : CommRing A
R S R' S' : Type u
inst✝⁷ : CommRing R
inst✝⁶ : CommRing S
inst✝⁵ : CommRing R'
inst✝⁴ : CommRing S'
inst✝³ : Algebra A R
inst✝² : Algebra A S
inst✝¹ : Algebra A R'... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/DPAlgebra/Dpow.lean | DividedPowerAlgebra.condτ_rel | [654, 1] | [728, 12] | simp only [AlgHom.toRingHom_eq_coe, RingHom.coe_coe, Algebra.TensorProduct.map_tmul, map_one,
fg] | case h.right.right.intro.intro
R✝ : Type u
inst✝⁹ : CommRing R✝
A : Type u
inst✝⁸ : CommRing A
R S R' S' : Type u
inst✝⁷ : CommRing R
inst✝⁶ : CommRing S
inst✝⁵ : CommRing R'
inst✝⁴ : CommRing S'
inst✝³ : Algebra A R
inst✝² : Algebra A S
inst✝¹ : Algebra A R'
inst✝ : Algebra A S'
f : R →ₐ[A] R'
hf : Function.Surjective... | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case h.right.right.intro.intro
R✝ : Type u
inst✝⁹ : CommRing R✝
A : Type u
inst✝⁸ : CommRing A
R S R' S' : Type u
inst✝⁷ : CommRing R
inst✝⁶ : CommRing S
inst✝⁵ : CommRing R'
inst✝⁴ : CommRing S'
inst✝³ : Algebra A R
inst✝² : Algebra A S
inst✝¹ : Algebra A R'... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/DPAlgebra/Dpow.lean | DividedPowerAlgebra.condτ_rel | [654, 1] | [728, 12] | rw [← this] | R✝ : Type u
inst✝⁹ : CommRing R✝
A : Type u
inst✝⁸ : CommRing A
R S R' S' : Type u
inst✝⁷ : CommRing R
inst✝⁶ : CommRing S
inst✝⁵ : CommRing R'
inst✝⁴ : CommRing S'
inst✝³ : Algebra A R
inst✝² : Algebra A S
inst✝¹ : Algebra A R'
inst✝ : Algebra A S'
f : R →ₐ[A] R'
hf : Function.Surjective ⇑f
I : Ideal R
hI : DividedPow... | R✝ : Type u
inst✝⁹ : CommRing R✝
A : Type u
inst✝⁸ : CommRing A
R S R' S' : Type u
inst✝⁷ : CommRing R
inst✝⁶ : CommRing S
inst✝⁵ : CommRing R'
inst✝⁴ : CommRing S'
inst✝³ : Algebra A R
inst✝² : Algebra A S
inst✝¹ : Algebra A R'
inst✝ : Algebra A S'
f : R →ₐ[A] R'
hf : Function.Surjective ⇑f
I : Ideal R
hI : DividedPow... | Please generate a tactic in lean4 to solve the state.
STATE:
R✝ : Type u
inst✝⁹ : CommRing R✝
A : Type u
inst✝⁸ : CommRing A
R S R' S' : Type u
inst✝⁷ : CommRing R
inst✝⁶ : CommRing S
inst✝⁵ : CommRing R'
inst✝⁴ : CommRing S'
inst✝³ : Algebra A R
inst✝² : Algebra A S
inst✝¹ : Algebra A R'
inst✝ : Algebra A S'
f : R →ₐ[... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/DPAlgebra/Dpow.lean | DividedPowerAlgebra.condτ_rel | [654, 1] | [728, 12] | rw [Quotient.OfSurjective.dpow_apply hK s_fg] | R✝ : Type u
inst✝⁹ : CommRing R✝
A : Type u
inst✝⁸ : CommRing A
R S R' S' : Type u
inst✝⁷ : CommRing R
inst✝⁶ : CommRing S
inst✝⁵ : CommRing R'
inst✝⁴ : CommRing S'
inst✝³ : Algebra A R
inst✝² : Algebra A S
inst✝¹ : Algebra A R'
inst✝ : Algebra A S'
f : R →ₐ[A] R'
hf : Function.Surjective ⇑f
I : Ideal R
hI : DividedPow... | R✝ : Type u
inst✝⁹ : CommRing R✝
A : Type u
inst✝⁸ : CommRing A
R S R' S' : Type u
inst✝⁷ : CommRing R
inst✝⁶ : CommRing S
inst✝⁵ : CommRing R'
inst✝⁴ : CommRing S'
inst✝³ : Algebra A R
inst✝² : Algebra A S
inst✝¹ : Algebra A R'
inst✝ : Algebra A S'
f : R →ₐ[A] R'
hf : Function.Surjective ⇑f
I : Ideal R
hI : DividedPow... | Please generate a tactic in lean4 to solve the state.
STATE:
R✝ : Type u
inst✝⁹ : CommRing R✝
A : Type u
inst✝⁸ : CommRing A
R S R' S' : Type u
inst✝⁷ : CommRing R
inst✝⁶ : CommRing S
inst✝⁵ : CommRing R'
inst✝⁴ : CommRing S'
inst✝³ : Algebra A R
inst✝² : Algebra A S
inst✝¹ : Algebra A R'
inst✝ : Algebra A S'
f : R →ₐ[... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/DPAlgebra/Dpow.lean | DividedPowerAlgebra.condτ_rel | [654, 1] | [728, 12] | have that := hg'.2 n a ha | R✝ : Type u
inst✝⁹ : CommRing R✝
A : Type u
inst✝⁸ : CommRing A
R S R' S' : Type u
inst✝⁷ : CommRing R
inst✝⁶ : CommRing S
inst✝⁵ : CommRing R'
inst✝⁴ : CommRing S'
inst✝³ : Algebra A R
inst✝² : Algebra A S
inst✝¹ : Algebra A R'
inst✝ : Algebra A S'
f : R →ₐ[A] R'
hf : Function.Surjective ⇑f
I : Ideal R
hI : DividedPow... | R✝ : Type u
inst✝⁹ : CommRing R✝
A : Type u
inst✝⁸ : CommRing A
R S R' S' : Type u
inst✝⁷ : CommRing R
inst✝⁶ : CommRing S
inst✝⁵ : CommRing R'
inst✝⁴ : CommRing S'
inst✝³ : Algebra A R
inst✝² : Algebra A S
inst✝¹ : Algebra A R'
inst✝ : Algebra A S'
f : R →ₐ[A] R'
hf : Function.Surjective ⇑f
I : Ideal R
hI : DividedPow... | Please generate a tactic in lean4 to solve the state.
STATE:
R✝ : Type u
inst✝⁹ : CommRing R✝
A : Type u
inst✝⁸ : CommRing A
R S R' S' : Type u
inst✝⁷ : CommRing R
inst✝⁶ : CommRing S
inst✝⁵ : CommRing R'
inst✝⁴ : CommRing S'
inst✝³ : Algebra A R
inst✝² : Algebra A S
inst✝¹ : Algebra A R'
inst✝ : Algebra A S'
f : R →ₐ[... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/DPAlgebra/Dpow.lean | DividedPowerAlgebra.condτ_rel | [654, 1] | [728, 12] | simp only [AlgHom.coe_toRingHom] at that | R✝ : Type u
inst✝⁹ : CommRing R✝
A : Type u
inst✝⁸ : CommRing A
R S R' S' : Type u
inst✝⁷ : CommRing R
inst✝⁶ : CommRing S
inst✝⁵ : CommRing R'
inst✝⁴ : CommRing S'
inst✝³ : Algebra A R
inst✝² : Algebra A S
inst✝¹ : Algebra A R'
inst✝ : Algebra A S'
f : R →ₐ[A] R'
hf : Function.Surjective ⇑f
I : Ideal R
hI : DividedPow... | R✝ : Type u
inst✝⁹ : CommRing R✝
A : Type u
inst✝⁸ : CommRing A
R S R' S' : Type u
inst✝⁷ : CommRing R
inst✝⁶ : CommRing S
inst✝⁵ : CommRing R'
inst✝⁴ : CommRing S'
inst✝³ : Algebra A R
inst✝² : Algebra A S
inst✝¹ : Algebra A R'
inst✝ : Algebra A S'
f : R →ₐ[A] R'
hf : Function.Surjective ⇑f
I : Ideal R
hI : DividedPow... | Please generate a tactic in lean4 to solve the state.
STATE:
R✝ : Type u
inst✝⁹ : CommRing R✝
A : Type u
inst✝⁸ : CommRing A
R S R' S' : Type u
inst✝⁷ : CommRing R
inst✝⁶ : CommRing S
inst✝⁵ : CommRing R'
inst✝⁴ : CommRing S'
inst✝³ : Algebra A R
inst✝² : Algebra A S
inst✝¹ : Algebra A R'
inst✝ : Algebra A S'
f : R →ₐ[... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/DPAlgebra/Dpow.lean | DividedPowerAlgebra.condτ_rel | [654, 1] | [728, 12] | rw [that] | R✝ : Type u
inst✝⁹ : CommRing R✝
A : Type u
inst✝⁸ : CommRing A
R S R' S' : Type u
inst✝⁷ : CommRing R
inst✝⁶ : CommRing S
inst✝⁵ : CommRing R'
inst✝⁴ : CommRing S'
inst✝³ : Algebra A R
inst✝² : Algebra A S
inst✝¹ : Algebra A R'
inst✝ : Algebra A S'
f : R →ₐ[A] R'
hf : Function.Surjective ⇑f
I : Ideal R
hI : DividedPow... | R✝ : Type u
inst✝⁹ : CommRing R✝
A : Type u
inst✝⁸ : CommRing A
R S R' S' : Type u
inst✝⁷ : CommRing R
inst✝⁶ : CommRing S
inst✝⁵ : CommRing R'
inst✝⁴ : CommRing S'
inst✝³ : Algebra A R
inst✝² : Algebra A S
inst✝¹ : Algebra A R'
inst✝ : Algebra A S'
f : R →ₐ[A] R'
hf : Function.Surjective ⇑f
I : Ideal R
hI : DividedPow... | Please generate a tactic in lean4 to solve the state.
STATE:
R✝ : Type u
inst✝⁹ : CommRing R✝
A : Type u
inst✝⁸ : CommRing A
R S R' S' : Type u
inst✝⁷ : CommRing R
inst✝⁶ : CommRing S
inst✝⁵ : CommRing R'
inst✝⁴ : CommRing S'
inst✝³ : Algebra A R
inst✝² : Algebra A S
inst✝¹ : Algebra A R'
inst✝ : Algebra A S'
f : R →ₐ[... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/DPAlgebra/Dpow.lean | DividedPowerAlgebra.condτ_rel | [654, 1] | [728, 12] | rw [← this] | R✝ : Type u
inst✝⁹ : CommRing R✝
A : Type u
inst✝⁸ : CommRing A
R S R' S' : Type u
inst✝⁷ : CommRing R
inst✝⁶ : CommRing S
inst✝⁵ : CommRing R'
inst✝⁴ : CommRing S'
inst✝³ : Algebra A R
inst✝² : Algebra A S
inst✝¹ : Algebra A R'
inst✝ : Algebra A S'
f : R →ₐ[A] R'
hf : Function.Surjective ⇑f
I : Ideal R
hI : DividedPow... | R✝ : Type u
inst✝⁹ : CommRing R✝
A : Type u
inst✝⁸ : CommRing A
R S R' S' : Type u
inst✝⁷ : CommRing R
inst✝⁶ : CommRing S
inst✝⁵ : CommRing R'
inst✝⁴ : CommRing S'
inst✝³ : Algebra A R
inst✝² : Algebra A S
inst✝¹ : Algebra A R'
inst✝ : Algebra A S'
f : R →ₐ[A] R'
hf : Function.Surjective ⇑f
I : Ideal R
hI : DividedPow... | Please generate a tactic in lean4 to solve the state.
STATE:
R✝ : Type u
inst✝⁹ : CommRing R✝
A : Type u
inst✝⁸ : CommRing A
R S R' S' : Type u
inst✝⁷ : CommRing R
inst✝⁶ : CommRing S
inst✝⁵ : CommRing R'
inst✝⁴ : CommRing S'
inst✝³ : Algebra A R
inst✝² : Algebra A S
inst✝¹ : Algebra A R'
inst✝ : Algebra A S'
f : R →ₐ[... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/DPAlgebra/Dpow.lean | DividedPowerAlgebra.condτ_rel | [654, 1] | [728, 12] | apply congr_arg | R✝ : Type u
inst✝⁹ : CommRing R✝
A : Type u
inst✝⁸ : CommRing A
R S R' S' : Type u
inst✝⁷ : CommRing R
inst✝⁶ : CommRing S
inst✝⁵ : CommRing R'
inst✝⁴ : CommRing S'
inst✝³ : Algebra A R
inst✝² : Algebra A S
inst✝¹ : Algebra A R'
inst✝ : Algebra A S'
f : R →ₐ[A] R'
hf : Function.Surjective ⇑f
I : Ideal R
hI : DividedPow... | case h
R✝ : Type u
inst✝⁹ : CommRing R✝
A : Type u
inst✝⁸ : CommRing A
R S R' S' : Type u
inst✝⁷ : CommRing R
inst✝⁶ : CommRing S
inst✝⁵ : CommRing R'
inst✝⁴ : CommRing S'
inst✝³ : Algebra A R
inst✝² : Algebra A S
inst✝¹ : Algebra A R'
inst✝ : Algebra A S'
f : R →ₐ[A] R'
hf : Function.Surjective ⇑f
I : Ideal R
hI : Div... | Please generate a tactic in lean4 to solve the state.
STATE:
R✝ : Type u
inst✝⁹ : CommRing R✝
A : Type u
inst✝⁸ : CommRing A
R S R' S' : Type u
inst✝⁷ : CommRing R
inst✝⁶ : CommRing S
inst✝⁵ : CommRing R'
inst✝⁴ : CommRing S'
inst✝³ : Algebra A R
inst✝² : Algebra A S
inst✝¹ : Algebra A R'
inst✝ : Algebra A S'
f : R →ₐ[... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/DPAlgebra/Dpow.lean | DividedPowerAlgebra.condτ_rel | [654, 1] | [728, 12] | simp only [← Algebra.TensorProduct.includeRight_apply] | case h
R✝ : Type u
inst✝⁹ : CommRing R✝
A : Type u
inst✝⁸ : CommRing A
R S R' S' : Type u
inst✝⁷ : CommRing R
inst✝⁶ : CommRing S
inst✝⁵ : CommRing R'
inst✝⁴ : CommRing S'
inst✝³ : Algebra A R
inst✝² : Algebra A S
inst✝¹ : Algebra A R'
inst✝ : Algebra A S'
f : R →ₐ[A] R'
hf : Function.Surjective ⇑f
I : Ideal R
hI : Div... | case h
R✝ : Type u
inst✝⁹ : CommRing R✝
A : Type u
inst✝⁸ : CommRing A
R S R' S' : Type u
inst✝⁷ : CommRing R
inst✝⁶ : CommRing S
inst✝⁵ : CommRing R'
inst✝⁴ : CommRing S'
inst✝³ : Algebra A R
inst✝² : Algebra A S
inst✝¹ : Algebra A R'
inst✝ : Algebra A S'
f : R →ₐ[A] R'
hf : Function.Surjective ⇑f
I : Ideal R
hI : Div... | Please generate a tactic in lean4 to solve the state.
STATE:
case h
R✝ : Type u
inst✝⁹ : CommRing R✝
A : Type u
inst✝⁸ : CommRing A
R S R' S' : Type u
inst✝⁷ : CommRing R
inst✝⁶ : CommRing S
inst✝⁵ : CommRing R'
inst✝⁴ : CommRing S'
inst✝³ : Algebra A R
inst✝² : Algebra A S
inst✝¹ : Algebra A R'
inst✝ : Algebra A S'
f ... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/DPAlgebra/Dpow.lean | DividedPowerAlgebra.condτ_rel | [654, 1] | [728, 12] | exact hK_pd.2.2 n a ha | case h
R✝ : Type u
inst✝⁹ : CommRing R✝
A : Type u
inst✝⁸ : CommRing A
R S R' S' : Type u
inst✝⁷ : CommRing R
inst✝⁶ : CommRing S
inst✝⁵ : CommRing R'
inst✝⁴ : CommRing S'
inst✝³ : Algebra A R
inst✝² : Algebra A S
inst✝¹ : Algebra A R'
inst✝ : Algebra A S'
f : R →ₐ[A] R'
hf : Function.Surjective ⇑f
I : Ideal R
hI : Div... | case ha
R✝ : Type u
inst✝⁹ : CommRing R✝
A : Type u
inst✝⁸ : CommRing A
R S R' S' : Type u
inst✝⁷ : CommRing R
inst✝⁶ : CommRing S
inst✝⁵ : CommRing R'
inst✝⁴ : CommRing S'
inst✝³ : Algebra A R
inst✝² : Algebra A S
inst✝¹ : Algebra A R'
inst✝ : Algebra A S'
f : R →ₐ[A] R'
hf : Function.Surjective ⇑f
I : Ideal R
hI : Di... | Please generate a tactic in lean4 to solve the state.
STATE:
case h
R✝ : Type u
inst✝⁹ : CommRing R✝
A : Type u
inst✝⁸ : CommRing A
R S R' S' : Type u
inst✝⁷ : CommRing R
inst✝⁶ : CommRing S
inst✝⁵ : CommRing R'
inst✝⁴ : CommRing S'
inst✝³ : Algebra A R
inst✝² : Algebra A S
inst✝¹ : Algebra A R'
inst✝ : Algebra A S'
f ... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/DPAlgebra/Dpow.lean | DividedPowerAlgebra.condτ_rel | [654, 1] | [728, 12] | apply Ideal.mem_sup_right | case ha
R✝ : Type u
inst✝⁹ : CommRing R✝
A : Type u
inst✝⁸ : CommRing A
R S R' S' : Type u
inst✝⁷ : CommRing R
inst✝⁶ : CommRing S
inst✝⁵ : CommRing R'
inst✝⁴ : CommRing S'
inst✝³ : Algebra A R
inst✝² : Algebra A S
inst✝¹ : Algebra A R'
inst✝ : Algebra A S'
f : R →ₐ[A] R'
hf : Function.Surjective ⇑f
I : Ideal R
hI : Di... | case ha.a
R✝ : Type u
inst✝⁹ : CommRing R✝
A : Type u
inst✝⁸ : CommRing A
R S R' S' : Type u
inst✝⁷ : CommRing R
inst✝⁶ : CommRing S
inst✝⁵ : CommRing R'
inst✝⁴ : CommRing S'
inst✝³ : Algebra A R
inst✝² : Algebra A S
inst✝¹ : Algebra A R'
inst✝ : Algebra A S'
f : R →ₐ[A] R'
hf : Function.Surjective ⇑f
I : Ideal R
hI : ... | Please generate a tactic in lean4 to solve the state.
STATE:
case ha
R✝ : Type u
inst✝⁹ : CommRing R✝
A : Type u
inst✝⁸ : CommRing A
R S R' S' : Type u
inst✝⁷ : CommRing R
inst✝⁶ : CommRing S
inst✝⁵ : CommRing R'
inst✝⁴ : CommRing S'
inst✝³ : Algebra A R
inst✝² : Algebra A S
inst✝¹ : Algebra A R'
inst✝ : Algebra A S'
f... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/DPAlgebra/Dpow.lean | DividedPowerAlgebra.condτ_rel | [654, 1] | [728, 12] | apply Ideal.mem_map_of_mem _ ha | case ha.a
R✝ : Type u
inst✝⁹ : CommRing R✝
A : Type u
inst✝⁸ : CommRing A
R S R' S' : Type u
inst✝⁷ : CommRing R
inst✝⁶ : CommRing S
inst✝⁵ : CommRing R'
inst✝⁴ : CommRing S'
inst✝³ : Algebra A R
inst✝² : Algebra A S
inst✝¹ : Algebra A R'
inst✝ : Algebra A S'
f : R →ₐ[A] R'
hf : Function.Surjective ⇑f
I : Ideal R
hI : ... | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case ha.a
R✝ : Type u
inst✝⁹ : CommRing R✝
A : Type u
inst✝⁸ : CommRing A
R S R' S' : Type u
inst✝⁷ : CommRing R
inst✝⁶ : CommRing S
inst✝⁵ : CommRing R'
inst✝⁴ : CommRing S'
inst✝³ : Algebra A R
inst✝² : Algebra A S
inst✝¹ : Algebra A R'
inst✝ : Algebra A S'... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/DPAlgebra/Dpow.lean | DividedPowerAlgebra.Ideal.map_coe_toRingHom | [785, 1] | [789, 6] | rfl | R✝ : Type u
inst✝⁵ : CommRing R✝
A : Type u_1
inst✝⁴ : CommRing A
R : Type u_2
S : Type u_3
inst✝³ : CommRing R
inst✝² : CommRing S
inst✝¹ : Algebra A R
inst✝ : Algebra A S
f : R →ₐ[A] S
I : Ideal R
⊢ Ideal.map f I = Ideal.map f.toRingHom I | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
R✝ : Type u
inst✝⁵ : CommRing R✝
A : Type u_1
inst✝⁴ : CommRing A
R : Type u_2
S : Type u_3
inst✝³ : CommRing R
inst✝² : CommRing S
inst✝¹ : Algebra A R
inst✝ : Algebra A S
f : R →ₐ[A] S
I : Ideal R
⊢ Ideal.map f I = Ideal.map f.toRingHom I
TACTIC:
|
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/DPAlgebra/Dpow.lean | DividedPowerAlgebra.condQ_and_condTFree_imply_condT | [792, 1] | [830, 18] | intro R' _ _ I' hI' S' _ _ J' hJ' | R : Type u
inst✝¹ : CommRing R
A : Type u_1
inst✝ : CommRing A
hQ : CondQ A
hT_free : CondTFree A
⊢ CondT A | R : Type u
inst✝⁵ : CommRing R
A : Type u_1
inst✝⁴ : CommRing A
hQ : CondQ A
hT_free : CondTFree A
R' : Type u_1
inst✝³ : CommRing R'
inst✝² : Algebra A R'
I' : Ideal R'
hI' : DividedPowers I'
S' : Type u_1
inst✝¹ : CommRing S'
inst✝ : Algebra A S'
J' : Ideal S'
hJ' : DividedPowers J'
⊢ Condτ A hI' hJ' | Please generate a tactic in lean4 to solve the state.
STATE:
R : Type u
inst✝¹ : CommRing R
A : Type u_1
inst✝ : CommRing A
hQ : CondQ A
hT_free : CondTFree A
⊢ CondT A
TACTIC:
|
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/DPAlgebra/Dpow.lean | DividedPowerAlgebra.condQ_and_condTFree_imply_condT | [792, 1] | [830, 18] | simp only [CondQ] at hQ | R : Type u
inst✝⁵ : CommRing R
A : Type u_1
inst✝⁴ : CommRing A
hQ : CondQ A
hT_free : CondTFree A
R' : Type u_1
inst✝³ : CommRing R'
inst✝² : Algebra A R'
I' : Ideal R'
hI' : DividedPowers I'
S' : Type u_1
inst✝¹ : CommRing S'
inst✝ : Algebra A S'
J' : Ideal S'
hJ' : DividedPowers J'
⊢ Condτ A hI' hJ' | R : Type u
inst✝⁵ : CommRing R
A : Type u_1
inst✝⁴ : CommRing A
hQ :
∀ (R : Type u_1) [inst : CommRing R] [inst_1 : Algebra A R] (I : Ideal R) (hI : DividedPowers I),
∃ T x x_1,
∃ (_ : Module.Free A T), ∃ f J hJ, ∃ (_ : hJ.isDPMorphism hI ↑f), I = Ideal.map f J ∧ Function.Surjective ⇑f
hT_free : CondTFree A... | Please generate a tactic in lean4 to solve the state.
STATE:
R : Type u
inst✝⁵ : CommRing R
A : Type u_1
inst✝⁴ : CommRing A
hQ : CondQ A
hT_free : CondTFree A
R' : Type u_1
inst✝³ : CommRing R'
inst✝² : Algebra A R'
I' : Ideal R'
hI' : DividedPowers I'
S' : Type u_1
inst✝¹ : CommRing S'
inst✝ : Algebra A S'
J' : Ideal... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/DPAlgebra/Dpow.lean | DividedPowerAlgebra.condQ_and_condTFree_imply_condT | [792, 1] | [830, 18] | obtain ⟨R, _, _, hR_free, f, I, hI, hfDP, hfI, hf⟩ := hQ R' I' hI' | R : Type u
inst✝⁵ : CommRing R
A : Type u_1
inst✝⁴ : CommRing A
hQ :
∀ (R : Type u_1) [inst : CommRing R] [inst_1 : Algebra A R] (I : Ideal R) (hI : DividedPowers I),
∃ T x x_1,
∃ (_ : Module.Free A T), ∃ f J hJ, ∃ (_ : hJ.isDPMorphism hI ↑f), I = Ideal.map f J ∧ Function.Surjective ⇑f
hT_free : CondTFree A... | case intro.intro.intro.intro.intro.intro.intro.intro.intro
R✝ : Type u
inst✝⁵ : CommRing R✝
A : Type u_1
inst✝⁴ : CommRing A
hQ :
∀ (R : Type u_1) [inst : CommRing R] [inst_1 : Algebra A R] (I : Ideal R) (hI : DividedPowers I),
∃ T x x_1,
∃ (_ : Module.Free A T), ∃ f J hJ, ∃ (_ : hJ.isDPMorphism hI ↑f), I =... | Please generate a tactic in lean4 to solve the state.
STATE:
R : Type u
inst✝⁵ : CommRing R
A : Type u_1
inst✝⁴ : CommRing A
hQ :
∀ (R : Type u_1) [inst : CommRing R] [inst_1 : Algebra A R] (I : Ideal R) (hI : DividedPowers I),
∃ T x x_1,
∃ (_ : Module.Free A T), ∃ f J hJ, ∃ (_ : hJ.isDPMorphism hI ↑f), I =... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/DPAlgebra/Dpow.lean | DividedPowerAlgebra.condQ_and_condTFree_imply_condT | [792, 1] | [830, 18] | obtain ⟨S, _, _, hS_free, g, J, hJ, hgDP, hgJ, hg⟩ := hQ S' J' hJ' | case intro.intro.intro.intro.intro.intro.intro.intro.intro
R✝ : Type u
inst✝⁵ : CommRing R✝
A : Type u_1
inst✝⁴ : CommRing A
hQ :
∀ (R : Type u_1) [inst : CommRing R] [inst_1 : Algebra A R] (I : Ideal R) (hI : DividedPowers I),
∃ T x x_1,
∃ (_ : Module.Free A T), ∃ f J hJ, ∃ (_ : hJ.isDPMorphism hI ↑f), I =... | case intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro
R✝ : Type u
inst✝⁵ : CommRing R✝
A : Type u_1
inst✝⁴ : CommRing A
hQ :
∀ (R : Type u_1) [inst : CommRing R] [inst_1 : Algebra A R] (I : Ideal R) (hI : DividedPowers I),
∃ T x x_1,
∃ (_ : Module.F... | Please generate a tactic in lean4 to solve the state.
STATE:
case intro.intro.intro.intro.intro.intro.intro.intro.intro
R✝ : Type u
inst✝⁵ : CommRing R✝
A : Type u_1
inst✝⁴ : CommRing A
hQ :
∀ (R : Type u_1) [inst : CommRing R] [inst_1 : Algebra A R] (I : Ideal R) (hI : DividedPowers I),
∃ T x x_1,
∃ (_ : M... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/DPAlgebra/Dpow.lean | DividedPowerAlgebra.condQ_and_condTFree_imply_condT | [792, 1] | [830, 18] | apply condτ_rel A f hf hI hI' hfDP hfI g hg hJ hJ' hgDP hgJ | case intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro
R✝ : Type u
inst✝⁵ : CommRing R✝
A : Type u_1
inst✝⁴ : CommRing A
hQ :
∀ (R : Type u_1) [inst : CommRing R] [inst_1 : Algebra A R] (I : Ideal R) (hI : DividedPowers I),
∃ T x x_1,
∃ (_ : Module.F... | case intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.roby
R✝ : Type u
inst✝⁵ : CommRing R✝
A : Type u_1
inst✝⁴ : CommRing A
hQ :
∀ (R : Type u_1) [inst : CommRing R] [inst_1 : Algebra A R] (I : Ideal R) (hI : DividedPowers I),
∃ T x x_1,
∃ (_ : Mod... | Please generate a tactic in lean4 to solve the state.
STATE:
case intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro
R✝ : Type u
inst✝⁵ : CommRing R✝
A : Type u_1
inst✝⁴ : CommRing A
hQ :
∀ (R : Type u_1) [inst : CommRing R] [inst_1 : Algebra A R] (I : Ideal R)... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/DPAlgebra/Dpow.lean | DividedPowerAlgebra.condQ_and_condTFree_imply_condT | [792, 1] | [830, 18] | rw [Algebra.TensorProduct.map_ker _ _ hf hg] | case intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.roby
R✝ : Type u
inst✝⁵ : CommRing R✝
A : Type u_1
inst✝⁴ : CommRing A
hQ :
∀ (R : Type u_1) [inst : CommRing R] [inst_1 : Algebra A R] (I : Ideal R) (hI : DividedPowers I),
∃ T x x_1,
∃ (_ : Mod... | case intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.roby
R✝ : Type u
inst✝⁵ : CommRing R✝
A : Type u_1
inst✝⁴ : CommRing A
hQ :
∀ (R : Type u_1) [inst : CommRing R] [inst_1 : Algebra A R] (I : Ideal R) (hI : DividedPowers I),
∃ T x x_1,
∃ (_ : Mod... | Please generate a tactic in lean4 to solve the state.
STATE:
case intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.roby
R✝ : Type u
inst✝⁵ : CommRing R✝
A : Type u_1
inst✝⁴ : CommRing A
hQ :
∀ (R : Type u_1) [inst : CommRing R] [inst_1 : Algebra A R] (I : Ide... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/DPAlgebra/Dpow.lean | DividedPowerAlgebra.condQ_and_condTFree_imply_condT | [792, 1] | [830, 18] | sorry | case intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.roby
R✝ : Type u
inst✝⁵ : CommRing R✝
A : Type u_1
inst✝⁴ : CommRing A
hQ :
∀ (R : Type u_1) [inst : CommRing R] [inst_1 : Algebra A R] (I : Ideal R) (hI : DividedPowers I),
∃ T x x_1,
∃ (_ : Mod... | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.roby
R✝ : Type u
inst✝⁵ : CommRing R✝
A : Type u_1
inst✝⁴ : CommRing A
hQ :
∀ (R : Type u_1) [inst : CommRing R] [inst_1 : Algebra A R] (I : Ide... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/DPAlgebra/Dpow.lean | DividedPowerAlgebra.condQ_and_condTFree_imply_condT | [792, 1] | [830, 18] | apply hT_free | case intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.hRS
R✝ : Type u
inst✝⁵ : CommRing R✝
A : Type u_1
inst✝⁴ : CommRing A
hQ :
∀ (R : Type u_1) [inst : CommRing R] [inst_1 : Algebra A R] (I : Ideal R) (hI : DividedPowers I),
∃ T x x_1,
∃ (_ : Modu... | case intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.hRS.x
R✝ : Type u
inst✝⁵ : CommRing R✝
A : Type u_1
inst✝⁴ : CommRing A
hQ :
∀ (R : Type u_1) [inst : CommRing R] [inst_1 : Algebra A R] (I : Ideal R) (hI : DividedPowers I),
∃ T x x_1,
∃ (_ : Mo... | Please generate a tactic in lean4 to solve the state.
STATE:
case intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.hRS
R✝ : Type u
inst✝⁵ : CommRing R✝
A : Type u_1
inst✝⁴ : CommRing A
hQ :
∀ (R : Type u_1) [inst : CommRing R] [inst_1 : Algebra A R] (I : Idea... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/DPAlgebra/Dpow.lean | DividedPowerAlgebra.condQ_and_condTFree_imply_condT | [792, 1] | [830, 18] | exact hR_free | case intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.hRS.x
R✝ : Type u
inst✝⁵ : CommRing R✝
A : Type u_1
inst✝⁴ : CommRing A
hQ :
∀ (R : Type u_1) [inst : CommRing R] [inst_1 : Algebra A R] (I : Ideal R) (hI : DividedPowers I),
∃ T x x_1,
∃ (_ : Mo... | case intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.hRS.x
R✝ : Type u
inst✝⁵ : CommRing R✝
A : Type u_1
inst✝⁴ : CommRing A
hQ :
∀ (R : Type u_1) [inst : CommRing R] [inst_1 : Algebra A R] (I : Ideal R) (hI : DividedPowers I),
∃ T x x_1,
∃ (_ : Mo... | Please generate a tactic in lean4 to solve the state.
STATE:
case intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.hRS.x
R✝ : Type u
inst✝⁵ : CommRing R✝
A : Type u_1
inst✝⁴ : CommRing A
hQ :
∀ (R : Type u_1) [inst : CommRing R] [inst_1 : Algebra A R] (I : Id... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/DPAlgebra/Dpow.lean | DividedPowerAlgebra.condQ_and_condTFree_imply_condT | [792, 1] | [830, 18] | exact hS_free | case intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.hRS.x
R✝ : Type u
inst✝⁵ : CommRing R✝
A : Type u_1
inst✝⁴ : CommRing A
hQ :
∀ (R : Type u_1) [inst : CommRing R] [inst_1 : Algebra A R] (I : Ideal R) (hI : DividedPowers I),
∃ T x x_1,
∃ (_ : Mo... | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.hRS.x
R✝ : Type u
inst✝⁵ : CommRing R✝
A : Type u_1
inst✝⁴ : CommRing A
hQ :
∀ (R : Type u_1) [inst : CommRing R] [inst_1 : Algebra A R] (I : Id... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/DPAlgebra/Dpow.lean | DividedPowerAlgebra.dp_comp | [924, 1] | [927, 84] | erw [← (roby_D A M).choose_spec, dpow_comp _ m hn (ι_mem_augIdeal A M x), dpow_ι] | A : Type u
inst✝³ : CommRing A
inst✝² : DecidableEq A
M : Type u
inst✝¹ : AddCommGroup M
inst✝ : Module A M
x : M
n m : ℕ
hn : n ≠ 0
⊢ (dividedPowers' A M).dpow m (dp A n x) = ↑(mchoose m n) * dp A (m * n) x | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
A : Type u
inst✝³ : CommRing A
inst✝² : DecidableEq A
M : Type u
inst✝¹ : AddCommGroup M
inst✝ : Module A M
x : M
n m : ℕ
hn : n ≠ 0
⊢ (dividedPowers' A M).dpow m (dp A n x) = ↑(mchoose m n) * dp A (m * n) x
TACTIC:
|
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/DPAlgebra/Dpow.lean | DividedPowerAlgebra.roby_theorem_2 | [930, 1] | [937, 14] | apply cond_D_uniqueness | R : Type u
inst✝⁵ : CommRing R
inst✝⁴ : DecidableEq R
M : Type u
inst✝³ : AddCommGroup M
inst✝² : Module R M
A : Type u
inst✝¹ : CommRing A
inst✝ : Algebra R A
I : Ideal A
hI : DividedPowers I
φ : M →ₗ[R] A
hφ : ∀ (m : M), φ m ∈ I
⊢ (dividedPowers' R M).isDPMorphism hI ↑(lift hI φ hφ) | case hh
R : Type u
inst✝⁵ : CommRing R
inst✝⁴ : DecidableEq R
M : Type u
inst✝³ : AddCommGroup M
inst✝² : Module R M
A : Type u
inst✝¹ : CommRing A
inst✝ : Algebra R A
I : Ideal A
hI : DividedPowers I
φ : M →ₗ[R] A
hφ : ∀ (m : M), φ m ∈ I
⊢ ∀ (n : ℕ) (x : M), (dividedPowers' R M).dpow n ((ι R M) x) = dp R n x | Please generate a tactic in lean4 to solve the state.
STATE:
R : Type u
inst✝⁵ : CommRing R
inst✝⁴ : DecidableEq R
M : Type u
inst✝³ : AddCommGroup M
inst✝² : Module R M
A : Type u
inst✝¹ : CommRing A
inst✝ : Algebra R A
I : Ideal A
hI : DividedPowers I
φ : M →ₗ[R] A
hφ : ∀ (m : M), φ m ∈ I
⊢ (dividedPowers' R M).isDPM... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/DPAlgebra/Dpow.lean | DividedPowerAlgebra.roby_theorem_2 | [930, 1] | [937, 14] | intro m n | case hh
R : Type u
inst✝⁵ : CommRing R
inst✝⁴ : DecidableEq R
M : Type u
inst✝³ : AddCommGroup M
inst✝² : Module R M
A : Type u
inst✝¹ : CommRing A
inst✝ : Algebra R A
I : Ideal A
hI : DividedPowers I
φ : M →ₗ[R] A
hφ : ∀ (m : M), φ m ∈ I
⊢ ∀ (n : ℕ) (x : M), (dividedPowers' R M).dpow n ((ι R M) x) = dp R n x | case hh
R : Type u
inst✝⁵ : CommRing R
inst✝⁴ : DecidableEq R
M : Type u
inst✝³ : AddCommGroup M
inst✝² : Module R M
A : Type u
inst✝¹ : CommRing A
inst✝ : Algebra R A
I : Ideal A
hI : DividedPowers I
φ : M →ₗ[R] A
hφ : ∀ (m : M), φ m ∈ I
m : ℕ
n : M
⊢ (dividedPowers' R M).dpow m ((ι R M) n) = dp R m n | Please generate a tactic in lean4 to solve the state.
STATE:
case hh
R : Type u
inst✝⁵ : CommRing R
inst✝⁴ : DecidableEq R
M : Type u
inst✝³ : AddCommGroup M
inst✝² : Module R M
A : Type u
inst✝¹ : CommRing A
inst✝ : Algebra R A
I : Ideal A
hI : DividedPowers I
φ : M →ₗ[R] A
hφ : ∀ (m : M), φ m ∈ I
⊢ ∀ (n : ℕ) (x : M),... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/DPAlgebra/Dpow.lean | DividedPowerAlgebra.roby_theorem_2 | [930, 1] | [937, 14] | rw [dpow_ι] | case hh
R : Type u
inst✝⁵ : CommRing R
inst✝⁴ : DecidableEq R
M : Type u
inst✝³ : AddCommGroup M
inst✝² : Module R M
A : Type u
inst✝¹ : CommRing A
inst✝ : Algebra R A
I : Ideal A
hI : DividedPowers I
φ : M →ₗ[R] A
hφ : ∀ (m : M), φ m ∈ I
m : ℕ
n : M
⊢ (dividedPowers' R M).dpow m ((ι R M) n) = dp R m n | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case hh
R : Type u
inst✝⁵ : CommRing R
inst✝⁴ : DecidableEq R
M : Type u
inst✝³ : AddCommGroup M
inst✝² : Module R M
A : Type u
inst✝¹ : CommRing A
inst✝ : Algebra R A
I : Ideal A
hI : DividedPowers I
φ : M →ₗ[R] A
hφ : ∀ (m : M), φ m ∈ I
m : ℕ
n : M
⊢ (divid... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/DPAlgebra/Dpow.lean | DividedPowerAlgebra.lift'_eq_dp_lift | [941, 1] | [961, 59] | have hφ : ∀ m, ((ι S N).restrictScalars R).comp f m ∈ augIdeal S N := by
intro m
simp only [LinearMap.coe_comp, LinearMap.coe_restrictScalars,
Function.comp_apply, ι_mem_augIdeal S N (f m)] | R : Type u
inst✝⁹ : CommRing R
M : Type v
inst✝⁸ : AddCommGroup M
inst✝⁷ : Module R M
S : Type w
inst✝⁶ : CommRing S
inst✝⁵ : DecidableEq S
inst✝⁴ : Algebra R S
N : Type w
inst✝³ : AddCommGroup N
inst✝² : Module R N
inst✝¹ : Module S N
inst✝ : IsScalarTower R S N
f : M →ₗ[R] N
⊢ ∃ (hφ : ∀ (m : M), (↑R (ι S N) ∘ₗ f) m ∈... | R : Type u
inst✝⁹ : CommRing R
M : Type v
inst✝⁸ : AddCommGroup M
inst✝⁷ : Module R M
S : Type w
inst✝⁶ : CommRing S
inst✝⁵ : DecidableEq S
inst✝⁴ : Algebra R S
N : Type w
inst✝³ : AddCommGroup N
inst✝² : Module R N
inst✝¹ : Module S N
inst✝ : IsScalarTower R S N
f : M →ₗ[R] N
hφ : ∀ (m : M), (↑R (ι S N) ∘ₗ f) m ∈ augI... | Please generate a tactic in lean4 to solve the state.
STATE:
R : Type u
inst✝⁹ : CommRing R
M : Type v
inst✝⁸ : AddCommGroup M
inst✝⁷ : Module R M
S : Type w
inst✝⁶ : CommRing S
inst✝⁵ : DecidableEq S
inst✝⁴ : Algebra R S
N : Type w
inst✝³ : AddCommGroup N
inst✝² : Module R N
inst✝¹ : Module S N
inst✝ : IsScalarTower R... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/DPAlgebra/Dpow.lean | DividedPowerAlgebra.lift'_eq_dp_lift | [941, 1] | [961, 59] | use hφ | R : Type u
inst✝⁹ : CommRing R
M : Type v
inst✝⁸ : AddCommGroup M
inst✝⁷ : Module R M
S : Type w
inst✝⁶ : CommRing S
inst✝⁵ : DecidableEq S
inst✝⁴ : Algebra R S
N : Type w
inst✝³ : AddCommGroup N
inst✝² : Module R N
inst✝¹ : Module S N
inst✝ : IsScalarTower R S N
f : M →ₗ[R] N
hφ : ∀ (m : M), (↑R (ι S N) ∘ₗ f) m ∈ augI... | case h
R : Type u
inst✝⁹ : CommRing R
M : Type v
inst✝⁸ : AddCommGroup M
inst✝⁷ : Module R M
S : Type w
inst✝⁶ : CommRing S
inst✝⁵ : DecidableEq S
inst✝⁴ : Algebra R S
N : Type w
inst✝³ : AddCommGroup N
inst✝² : Module R N
inst✝¹ : Module S N
inst✝ : IsScalarTower R S N
f : M →ₗ[R] N
hφ : ∀ (m : M), (↑R (ι S N) ∘ₗ f) m... | Please generate a tactic in lean4 to solve the state.
STATE:
R : Type u
inst✝⁹ : CommRing R
M : Type v
inst✝⁸ : AddCommGroup M
inst✝⁷ : Module R M
S : Type w
inst✝⁶ : CommRing S
inst✝⁵ : DecidableEq S
inst✝⁴ : Algebra R S
N : Type w
inst✝³ : AddCommGroup N
inst✝² : Module R N
inst✝¹ : Module S N
inst✝ : IsScalarTower R... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/DPAlgebra/Dpow.lean | DividedPowerAlgebra.lift'_eq_dp_lift | [941, 1] | [961, 59] | apply DividedPowerAlgebra.ext | case h
R : Type u
inst✝⁹ : CommRing R
M : Type v
inst✝⁸ : AddCommGroup M
inst✝⁷ : Module R M
S : Type w
inst✝⁶ : CommRing S
inst✝⁵ : DecidableEq S
inst✝⁴ : Algebra R S
N : Type w
inst✝³ : AddCommGroup N
inst✝² : Module R N
inst✝¹ : Module S N
inst✝ : IsScalarTower R S N
f : M →ₗ[R] N
hφ : ∀ (m : M), (↑R (ι S N) ∘ₗ f) m... | case h.h
R : Type u
inst✝⁹ : CommRing R
M : Type v
inst✝⁸ : AddCommGroup M
inst✝⁷ : Module R M
S : Type w
inst✝⁶ : CommRing S
inst✝⁵ : DecidableEq S
inst✝⁴ : Algebra R S
N : Type w
inst✝³ : AddCommGroup N
inst✝² : Module R N
inst✝¹ : Module S N
inst✝ : IsScalarTower R S N
f : M →ₗ[R] N
hφ : ∀ (m : M), (↑R (ι S N) ∘ₗ f)... | Please generate a tactic in lean4 to solve the state.
STATE:
case h
R : Type u
inst✝⁹ : CommRing R
M : Type v
inst✝⁸ : AddCommGroup M
inst✝⁷ : Module R M
S : Type w
inst✝⁶ : CommRing S
inst✝⁵ : DecidableEq S
inst✝⁴ : Algebra R S
N : Type w
inst✝³ : AddCommGroup N
inst✝² : Module R N
inst✝¹ : Module S N
inst✝ : IsScalar... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/DPAlgebra/Dpow.lean | DividedPowerAlgebra.lift'_eq_dp_lift | [941, 1] | [961, 59] | intro n m | case h.h
R : Type u
inst✝⁹ : CommRing R
M : Type v
inst✝⁸ : AddCommGroup M
inst✝⁷ : Module R M
S : Type w
inst✝⁶ : CommRing S
inst✝⁵ : DecidableEq S
inst✝⁴ : Algebra R S
N : Type w
inst✝³ : AddCommGroup N
inst✝² : Module R N
inst✝¹ : Module S N
inst✝ : IsScalarTower R S N
f : M →ₗ[R] N
hφ : ∀ (m : M), (↑R (ι S N) ∘ₗ f)... | case h.h
R : Type u
inst✝⁹ : CommRing R
M : Type v
inst✝⁸ : AddCommGroup M
inst✝⁷ : Module R M
S : Type w
inst✝⁶ : CommRing S
inst✝⁵ : DecidableEq S
inst✝⁴ : Algebra R S
N : Type w
inst✝³ : AddCommGroup N
inst✝² : Module R N
inst✝¹ : Module S N
inst✝ : IsScalarTower R S N
f : M →ₗ[R] N
hφ : ∀ (m : M), (↑R (ι S N) ∘ₗ f)... | Please generate a tactic in lean4 to solve the state.
STATE:
case h.h
R : Type u
inst✝⁹ : CommRing R
M : Type v
inst✝⁸ : AddCommGroup M
inst✝⁷ : Module R M
S : Type w
inst✝⁶ : CommRing S
inst✝⁵ : DecidableEq S
inst✝⁴ : Algebra R S
N : Type w
inst✝³ : AddCommGroup N
inst✝² : Module R N
inst✝¹ : Module S N
inst✝ : IsScal... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/DPAlgebra/Dpow.lean | DividedPowerAlgebra.lift'_eq_dp_lift | [941, 1] | [961, 59] | simp only [liftAlgHom_apply_dp, LinearMap.coe_comp, LinearMap.coe_restrictScalars,
Function.comp_apply] | case h.h
R : Type u
inst✝⁹ : CommRing R
M : Type v
inst✝⁸ : AddCommGroup M
inst✝⁷ : Module R M
S : Type w
inst✝⁶ : CommRing S
inst✝⁵ : DecidableEq S
inst✝⁴ : Algebra R S
N : Type w
inst✝³ : AddCommGroup N
inst✝² : Module R N
inst✝¹ : Module S N
inst✝ : IsScalarTower R S N
f : M →ₗ[R] N
hφ : ∀ (m : M), (↑R (ι S N) ∘ₗ f)... | case h.h
R : Type u
inst✝⁹ : CommRing R
M : Type v
inst✝⁸ : AddCommGroup M
inst✝⁷ : Module R M
S : Type w
inst✝⁶ : CommRing S
inst✝⁵ : DecidableEq S
inst✝⁴ : Algebra R S
N : Type w
inst✝³ : AddCommGroup N
inst✝² : Module R N
inst✝¹ : Module S N
inst✝ : IsScalarTower R S N
f : M →ₗ[R] N
hφ : ∀ (m : M), (↑R (ι S N) ∘ₗ f)... | Please generate a tactic in lean4 to solve the state.
STATE:
case h.h
R : Type u
inst✝⁹ : CommRing R
M : Type v
inst✝⁸ : AddCommGroup M
inst✝⁷ : Module R M
S : Type w
inst✝⁶ : CommRing S
inst✝⁵ : DecidableEq S
inst✝⁴ : Algebra R S
N : Type w
inst✝³ : AddCommGroup N
inst✝² : Module R N
inst✝¹ : Module S N
inst✝ : IsScal... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/DPAlgebra/Dpow.lean | DividedPowerAlgebra.lift'_eq_dp_lift | [941, 1] | [961, 59] | simp only [LinearMap.liftAlgHom_dp] | case h.h
R : Type u
inst✝⁹ : CommRing R
M : Type v
inst✝⁸ : AddCommGroup M
inst✝⁷ : Module R M
S : Type w
inst✝⁶ : CommRing S
inst✝⁵ : DecidableEq S
inst✝⁴ : Algebra R S
N : Type w
inst✝³ : AddCommGroup N
inst✝² : Module R N
inst✝¹ : Module S N
inst✝ : IsScalarTower R S N
f : M →ₗ[R] N
hφ : ∀ (m : M), (↑R (ι S N) ∘ₗ f)... | case h.h
R : Type u
inst✝⁹ : CommRing R
M : Type v
inst✝⁸ : AddCommGroup M
inst✝⁷ : Module R M
S : Type w
inst✝⁶ : CommRing S
inst✝⁵ : DecidableEq S
inst✝⁴ : Algebra R S
N : Type w
inst✝³ : AddCommGroup N
inst✝² : Module R N
inst✝¹ : Module S N
inst✝ : IsScalarTower R S N
f : M →ₗ[R] N
hφ : ∀ (m : M), (↑R (ι S N) ∘ₗ f)... | Please generate a tactic in lean4 to solve the state.
STATE:
case h.h
R : Type u
inst✝⁹ : CommRing R
M : Type v
inst✝⁸ : AddCommGroup M
inst✝⁷ : Module R M
S : Type w
inst✝⁶ : CommRing S
inst✝⁵ : DecidableEq S
inst✝⁴ : Algebra R S
N : Type w
inst✝³ : AddCommGroup N
inst✝² : Module R N
inst✝¹ : Module S N
inst✝ : IsScal... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/DPAlgebra/Dpow.lean | DividedPowerAlgebra.lift'_eq_dp_lift | [941, 1] | [961, 59] | simp only [ι, LinearMap.coe_mk, AddHom.coe_mk] | case h.h
R : Type u
inst✝⁹ : CommRing R
M : Type v
inst✝⁸ : AddCommGroup M
inst✝⁷ : Module R M
S : Type w
inst✝⁶ : CommRing S
inst✝⁵ : DecidableEq S
inst✝⁴ : Algebra R S
N : Type w
inst✝³ : AddCommGroup N
inst✝² : Module R N
inst✝¹ : Module S N
inst✝ : IsScalarTower R S N
f : M →ₗ[R] N
hφ : ∀ (m : M), (↑R (ι S N) ∘ₗ f)... | case h.h
R : Type u
inst✝⁹ : CommRing R
M : Type v
inst✝⁸ : AddCommGroup M
inst✝⁷ : Module R M
S : Type w
inst✝⁶ : CommRing S
inst✝⁵ : DecidableEq S
inst✝⁴ : Algebra R S
N : Type w
inst✝³ : AddCommGroup N
inst✝² : Module R N
inst✝¹ : Module S N
inst✝ : IsScalarTower R S N
f : M →ₗ[R] N
hφ : ∀ (m : M), (↑R (ι S N) ∘ₗ f)... | Please generate a tactic in lean4 to solve the state.
STATE:
case h.h
R : Type u
inst✝⁹ : CommRing R
M : Type v
inst✝⁸ : AddCommGroup M
inst✝⁷ : Module R M
S : Type w
inst✝⁶ : CommRing S
inst✝⁵ : DecidableEq S
inst✝⁴ : Algebra R S
N : Type w
inst✝³ : AddCommGroup N
inst✝² : Module R N
inst✝¹ : Module S N
inst✝ : IsScal... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/DPAlgebra/Dpow.lean | DividedPowerAlgebra.lift'_eq_dp_lift | [941, 1] | [961, 59] | rw [dp_comp _ _ _ _ Nat.one_ne_zero] | case h.h
R : Type u
inst✝⁹ : CommRing R
M : Type v
inst✝⁸ : AddCommGroup M
inst✝⁷ : Module R M
S : Type w
inst✝⁶ : CommRing S
inst✝⁵ : DecidableEq S
inst✝⁴ : Algebra R S
N : Type w
inst✝³ : AddCommGroup N
inst✝² : Module R N
inst✝¹ : Module S N
inst✝ : IsScalarTower R S N
f : M →ₗ[R] N
hφ : ∀ (m : M), (↑R (ι S N) ∘ₗ f)... | case h.h
R : Type u
inst✝⁹ : CommRing R
M : Type v
inst✝⁸ : AddCommGroup M
inst✝⁷ : Module R M
S : Type w
inst✝⁶ : CommRing S
inst✝⁵ : DecidableEq S
inst✝⁴ : Algebra R S
N : Type w
inst✝³ : AddCommGroup N
inst✝² : Module R N
inst✝¹ : Module S N
inst✝ : IsScalarTower R S N
f : M →ₗ[R] N
hφ : ∀ (m : M), (↑R (ι S N) ∘ₗ f)... | Please generate a tactic in lean4 to solve the state.
STATE:
case h.h
R : Type u
inst✝⁹ : CommRing R
M : Type v
inst✝⁸ : AddCommGroup M
inst✝⁷ : Module R M
S : Type w
inst✝⁶ : CommRing S
inst✝⁵ : DecidableEq S
inst✝⁴ : Algebra R S
N : Type w
inst✝³ : AddCommGroup N
inst✝² : Module R N
inst✝¹ : Module S N
inst✝ : IsScal... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/DPAlgebra/Dpow.lean | DividedPowerAlgebra.lift'_eq_dp_lift | [941, 1] | [961, 59] | simp only [mchoose_one', Nat.cast_one, mul_one, one_mul] | case h.h
R : Type u
inst✝⁹ : CommRing R
M : Type v
inst✝⁸ : AddCommGroup M
inst✝⁷ : Module R M
S : Type w
inst✝⁶ : CommRing S
inst✝⁵ : DecidableEq S
inst✝⁴ : Algebra R S
N : Type w
inst✝³ : AddCommGroup N
inst✝² : Module R N
inst✝¹ : Module S N
inst✝ : IsScalarTower R S N
f : M →ₗ[R] N
hφ : ∀ (m : M), (↑R (ι S N) ∘ₗ f)... | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case h.h
R : Type u
inst✝⁹ : CommRing R
M : Type v
inst✝⁸ : AddCommGroup M
inst✝⁷ : Module R M
S : Type w
inst✝⁶ : CommRing S
inst✝⁵ : DecidableEq S
inst✝⁴ : Algebra R S
N : Type w
inst✝³ : AddCommGroup N
inst✝² : Module R N
inst✝¹ : Module S N
inst✝ : IsScal... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/DPAlgebra/Dpow.lean | DividedPowerAlgebra.lift'_eq_dp_lift | [941, 1] | [961, 59] | intro m | R : Type u
inst✝⁹ : CommRing R
M : Type v
inst✝⁸ : AddCommGroup M
inst✝⁷ : Module R M
S : Type w
inst✝⁶ : CommRing S
inst✝⁵ : DecidableEq S
inst✝⁴ : Algebra R S
N : Type w
inst✝³ : AddCommGroup N
inst✝² : Module R N
inst✝¹ : Module S N
inst✝ : IsScalarTower R S N
f : M →ₗ[R] N
⊢ ∀ (m : M), (↑R (ι S N) ∘ₗ f) m ∈ augIdea... | R : Type u
inst✝⁹ : CommRing R
M : Type v
inst✝⁸ : AddCommGroup M
inst✝⁷ : Module R M
S : Type w
inst✝⁶ : CommRing S
inst✝⁵ : DecidableEq S
inst✝⁴ : Algebra R S
N : Type w
inst✝³ : AddCommGroup N
inst✝² : Module R N
inst✝¹ : Module S N
inst✝ : IsScalarTower R S N
f : M →ₗ[R] N
m : M
⊢ (↑R (ι S N) ∘ₗ f) m ∈ augIdeal S N | Please generate a tactic in lean4 to solve the state.
STATE:
R : Type u
inst✝⁹ : CommRing R
M : Type v
inst✝⁸ : AddCommGroup M
inst✝⁷ : Module R M
S : Type w
inst✝⁶ : CommRing S
inst✝⁵ : DecidableEq S
inst✝⁴ : Algebra R S
N : Type w
inst✝³ : AddCommGroup N
inst✝² : Module R N
inst✝¹ : Module S N
inst✝ : IsScalarTower R... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/DPAlgebra/Dpow.lean | DividedPowerAlgebra.lift'_eq_dp_lift | [941, 1] | [961, 59] | simp only [LinearMap.coe_comp, LinearMap.coe_restrictScalars,
Function.comp_apply, ι_mem_augIdeal S N (f m)] | R : Type u
inst✝⁹ : CommRing R
M : Type v
inst✝⁸ : AddCommGroup M
inst✝⁷ : Module R M
S : Type w
inst✝⁶ : CommRing S
inst✝⁵ : DecidableEq S
inst✝⁴ : Algebra R S
N : Type w
inst✝³ : AddCommGroup N
inst✝² : Module R N
inst✝¹ : Module S N
inst✝ : IsScalarTower R S N
f : M →ₗ[R] N
m : M
⊢ (↑R (ι S N) ∘ₗ f) m ∈ augIdeal S N | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
R : Type u
inst✝⁹ : CommRing R
M : Type v
inst✝⁸ : AddCommGroup M
inst✝⁷ : Module R M
S : Type w
inst✝⁶ : CommRing S
inst✝⁵ : DecidableEq S
inst✝⁴ : Algebra R S
N : Type w
inst✝³ : AddCommGroup N
inst✝² : Module R N
inst✝¹ : Module S N
inst✝ : IsScalarTower R... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/DPAlgebra/Dpow.lean | DividedPowerAlgebra.roby_prop_8 | [963, 1] | [970, 53] | obtain ⟨hφ, phφ'⟩ := lift'_eq_dp_lift R S f | R : Type u
inst✝¹⁰ : DecidableEq R
inst✝⁹ : CommRing R
M : Type u
inst✝⁸ : AddCommGroup M
inst✝⁷ : Module R M
S : Type u
inst✝⁶ : DecidableEq S
inst✝⁵ : CommRing S
inst✝⁴ : Algebra R S
N : Type u
inst✝³ : AddCommGroup N
inst✝² : Module R N
inst✝¹ : Module S N
inst✝ : IsScalarTower R S N
f : M →ₗ[R] N
⊢ (dividedPowers' ... | case intro
R : Type u
inst✝¹⁰ : DecidableEq R
inst✝⁹ : CommRing R
M : Type u
inst✝⁸ : AddCommGroup M
inst✝⁷ : Module R M
S : Type u
inst✝⁶ : DecidableEq S
inst✝⁵ : CommRing S
inst✝⁴ : Algebra R S
N : Type u
inst✝³ : AddCommGroup N
inst✝² : Module R N
inst✝¹ : Module S N
inst✝ : IsScalarTower R S N
f : M →ₗ[R] N
hφ : ∀ ... | Please generate a tactic in lean4 to solve the state.
STATE:
R : Type u
inst✝¹⁰ : DecidableEq R
inst✝⁹ : CommRing R
M : Type u
inst✝⁸ : AddCommGroup M
inst✝⁷ : Module R M
S : Type u
inst✝⁶ : DecidableEq S
inst✝⁵ : CommRing S
inst✝⁴ : Algebra R S
N : Type u
inst✝³ : AddCommGroup N
inst✝² : Module R N
inst✝¹ : Module S N... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/DPAlgebra/Dpow.lean | DividedPowerAlgebra.roby_prop_8 | [963, 1] | [970, 53] | convert roby_theorem_2 R M (dividedPowers' S N) hφ | case intro
R : Type u
inst✝¹⁰ : DecidableEq R
inst✝⁹ : CommRing R
M : Type u
inst✝⁸ : AddCommGroup M
inst✝⁷ : Module R M
S : Type u
inst✝⁶ : DecidableEq S
inst✝⁵ : CommRing S
inst✝⁴ : Algebra R S
N : Type u
inst✝³ : AddCommGroup N
inst✝² : Module R N
inst✝¹ : Module S N
inst✝ : IsScalarTower R S N
f : M →ₗ[R] N
hφ : ∀ ... | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case intro
R : Type u
inst✝¹⁰ : DecidableEq R
inst✝⁹ : CommRing R
M : Type u
inst✝⁸ : AddCommGroup M
inst✝⁷ : Module R M
S : Type u
inst✝⁶ : DecidableEq S
inst✝⁵ : CommRing S
inst✝⁴ : Algebra R S
N : Type u
inst✝³ : AddCommGroup N
inst✝² : Module R N
inst✝¹ :... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/Basic.lean | DividedPowers.add_dpowExp | [161, 1] | [165, 54] | ext n | A : Type u_1
inst✝ : CommSemiring A
I : Ideal A
hI : DividedPowers I
a b : A
ha : a ∈ I
hb : b ∈ I
⊢ hI.dpowExp (a + b) = hI.dpowExp a * hI.dpowExp b | case h
A : Type u_1
inst✝ : CommSemiring A
I : Ideal A
hI : DividedPowers I
a b : A
ha : a ∈ I
hb : b ∈ I
n : ℕ
⊢ (PowerSeries.coeff A n) (hI.dpowExp (a + b)) = (PowerSeries.coeff A n) (hI.dpowExp a * hI.dpowExp b) | Please generate a tactic in lean4 to solve the state.
STATE:
A : Type u_1
inst✝ : CommSemiring A
I : Ideal A
hI : DividedPowers I
a b : A
ha : a ∈ I
hb : b ∈ I
⊢ hI.dpowExp (a + b) = hI.dpowExp a * hI.dpowExp b
TACTIC:
|
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/Basic.lean | DividedPowers.add_dpowExp | [161, 1] | [165, 54] | simp only [dpowExp, PowerSeries.coeff_mk, PowerSeries.coeff_mul, hI.dpow_add n ha hb,
Finset.Nat.sum_antidiagonal_eq_sum_range_succ_mk] | case h
A : Type u_1
inst✝ : CommSemiring A
I : Ideal A
hI : DividedPowers I
a b : A
ha : a ∈ I
hb : b ∈ I
n : ℕ
⊢ (PowerSeries.coeff A n) (hI.dpowExp (a + b)) = (PowerSeries.coeff A n) (hI.dpowExp a * hI.dpowExp b) | 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
hI : DividedPowers I
a b : A
ha : a ∈ I
hb : b ∈ I
n : ℕ
⊢ (PowerSeries.coeff A n) (hI.dpowExp (a + b)) = (PowerSeries.coeff A n) (hI.dpowExp a * hI.dpowExp b)
TACTIC:
|
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/Basic.lean | DividedPowers.eq_of_eq_on_ideal | [168, 1] | [173, 43] | ext n x | A : Type u_1
inst✝ : CommSemiring A
I : Ideal A
hI hI' : DividedPowers I
h_eq : ∀ (n : ℕ) {x : A}, x ∈ I → hI.dpow n x = hI'.dpow n x
⊢ hI = hI' | case dpow.h.h
A : Type u_1
inst✝ : CommSemiring A
I : Ideal A
hI hI' : DividedPowers I
h_eq : ∀ (n : ℕ) {x : A}, x ∈ I → hI.dpow n x = hI'.dpow n x
n : ℕ
x : A
⊢ hI.dpow n x = hI'.dpow n x | Please generate a tactic in lean4 to solve the state.
STATE:
A : Type u_1
inst✝ : CommSemiring A
I : Ideal A
hI hI' : DividedPowers I
h_eq : ∀ (n : ℕ) {x : A}, x ∈ I → hI.dpow n x = hI'.dpow n x
⊢ hI = hI'
TACTIC:
|
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/Basic.lean | DividedPowers.eq_of_eq_on_ideal | [168, 1] | [173, 43] | by_cases hx : x ∈ I | case dpow.h.h
A : Type u_1
inst✝ : CommSemiring A
I : Ideal A
hI hI' : DividedPowers I
h_eq : ∀ (n : ℕ) {x : A}, x ∈ I → hI.dpow n x = hI'.dpow n x
n : ℕ
x : A
⊢ hI.dpow n x = hI'.dpow n x | case pos
A : Type u_1
inst✝ : CommSemiring A
I : Ideal A
hI hI' : DividedPowers I
h_eq : ∀ (n : ℕ) {x : A}, x ∈ I → hI.dpow n x = hI'.dpow n x
n : ℕ
x : A
hx : x ∈ I
⊢ hI.dpow n x = hI'.dpow n x
case neg
A : Type u_1
inst✝ : CommSemiring A
I : Ideal A
hI hI' : DividedPowers I
h_eq : ∀ (n : ℕ) {x : A}, x ∈ I → hI.dpow ... | Please generate a tactic in lean4 to solve the state.
STATE:
case dpow.h.h
A : Type u_1
inst✝ : CommSemiring A
I : Ideal A
hI hI' : DividedPowers I
h_eq : ∀ (n : ℕ) {x : A}, x ∈ I → hI.dpow n x = hI'.dpow n x
n : ℕ
x : A
⊢ hI.dpow n x = hI'.dpow n x
TACTIC:
|
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/Basic.lean | DividedPowers.eq_of_eq_on_ideal | [168, 1] | [173, 43] | exact h_eq n hx | case pos
A : Type u_1
inst✝ : CommSemiring A
I : Ideal A
hI hI' : DividedPowers I
h_eq : ∀ (n : ℕ) {x : A}, x ∈ I → hI.dpow n x = hI'.dpow n x
n : ℕ
x : A
hx : x ∈ I
⊢ hI.dpow n x = hI'.dpow n 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
hI hI' : DividedPowers I
h_eq : ∀ (n : ℕ) {x : A}, x ∈ I → hI.dpow n x = hI'.dpow n x
n : ℕ
x : A
hx : x ∈ I
⊢ hI.dpow n x = hI'.dpow n x
TACTIC:
|
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/Basic.lean | DividedPowers.eq_of_eq_on_ideal | [168, 1] | [173, 43] | rw [hI.dpow_null hx, hI'.dpow_null hx] | case neg
A : Type u_1
inst✝ : CommSemiring A
I : Ideal A
hI hI' : DividedPowers I
h_eq : ∀ (n : ℕ) {x : A}, x ∈ I → hI.dpow n x = hI'.dpow n x
n : ℕ
x : A
hx : x ∉ I
⊢ hI.dpow n x = hI'.dpow n 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
hI hI' : DividedPowers I
h_eq : ∀ (n : ℕ) {x : A}, x ∈ I → hI.dpow n x = hI'.dpow n x
n : ℕ
x : A
hx : x ∉ I
⊢ hI.dpow n x = hI'.dpow n x
TACTIC:
|
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/Basic.lean | DividedPowers.dpow_smul' | [194, 1] | [196, 44] | simp only [smul_eq_mul, hI.dpow_smul, hx] | A : Type u_1
inst✝ : CommSemiring A
I : Ideal A
hI : DividedPowers I
n : ℕ
a x : A
hx : x ∈ I
⊢ hI.dpow n (a • x) = a ^ n • hI.dpow n x | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
A : Type u_1
inst✝ : CommSemiring A
I : Ideal A
hI : DividedPowers I
n : ℕ
a x : A
hx : x ∈ I
⊢ hI.dpow n (a • x) = a ^ n • hI.dpow n x
TACTIC:
|
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/Basic.lean | DividedPowers.dpow_mul_right | [199, 1] | [201, 45] | rw [mul_comm, hI.dpow_smul n ha, mul_comm] | A : Type u_1
inst✝ : CommSemiring A
I : Ideal A
hI : DividedPowers I
n : ℕ
a : A
ha : a ∈ I
x : A
⊢ hI.dpow n (a * x) = hI.dpow n a * x ^ n | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
A : Type u_1
inst✝ : CommSemiring A
I : Ideal A
hI : DividedPowers I
n : ℕ
a : A
ha : a ∈ I
x : A
⊢ hI.dpow n (a * x) = hI.dpow n a * x ^ n
TACTIC:
|
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/Basic.lean | DividedPowers.dpow_smul_right | [204, 1] | [206, 56] | rw [smul_eq_mul, hI.dpow_mul_right n ha, smul_eq_mul] | A : Type u_1
inst✝ : CommSemiring A
I : Ideal A
hI : DividedPowers I
n : ℕ
a : A
ha : a ∈ I
x : A
⊢ hI.dpow n (a • x) = hI.dpow n a • x ^ n | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
A : Type u_1
inst✝ : CommSemiring A
I : Ideal A
hI : DividedPowers I
n : ℕ
a : A
ha : a ∈ I
x : A
⊢ hI.dpow n (a • x) = hI.dpow n a • x ^ n
TACTIC:
|
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/Basic.lean | DividedPowers.factorial_mul_dpow_eq_pow | [209, 1] | [217, 82] | induction n with
| zero => rw [Nat.factorial_zero, Nat.cast_one, one_mul, pow_zero, hI.dpow_zero hx]
| add_one n ih =>
rw [Nat.factorial_succ, mul_comm (n + 1)]
nth_rewrite 1 [← (n + 1).choose_one_right]
rw [← Nat.choose_symm_add, Nat.cast_mul, mul_assoc,
← hI.dpow_mul n 1 hx, ← mul_assoc, ih, hI.dpow_one hx,... | A : Type u_1
inst✝ : CommSemiring A
I : Ideal A
hI : DividedPowers I
n : ℕ
x : A
hx : x ∈ I
⊢ ↑n.factorial * hI.dpow n x = x ^ n | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
A : Type u_1
inst✝ : CommSemiring A
I : Ideal A
hI : DividedPowers I
n : ℕ
x : A
hx : x ∈ I
⊢ ↑n.factorial * hI.dpow n x = x ^ n
TACTIC:
|
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/Basic.lean | DividedPowers.factorial_mul_dpow_eq_pow | [209, 1] | [217, 82] | rw [Nat.factorial_zero, Nat.cast_one, one_mul, pow_zero, hI.dpow_zero hx] | case zero
A : Type u_1
inst✝ : CommSemiring A
I : Ideal A
hI : DividedPowers I
x : A
hx : x ∈ I
⊢ ↑(Nat.factorial 0) * hI.dpow 0 x = x ^ 0 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case zero
A : Type u_1
inst✝ : CommSemiring A
I : Ideal A
hI : DividedPowers I
x : A
hx : x ∈ I
⊢ ↑(Nat.factorial 0) * hI.dpow 0 x = x ^ 0
TACTIC:
|
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/Basic.lean | DividedPowers.factorial_mul_dpow_eq_pow | [209, 1] | [217, 82] | rw [Nat.factorial_succ, mul_comm (n + 1)] | case add_one
A : Type u_1
inst✝ : CommSemiring A
I : Ideal A
hI : DividedPowers I
x : A
hx : x ∈ I
n : ℕ
ih : ↑n.factorial * hI.dpow n x = x ^ n
⊢ ↑(n + 1).factorial * hI.dpow (n + 1) x = x ^ (n + 1) | case add_one
A : Type u_1
inst✝ : CommSemiring A
I : Ideal A
hI : DividedPowers I
x : A
hx : x ∈ I
n : ℕ
ih : ↑n.factorial * hI.dpow n x = x ^ n
⊢ ↑(n.factorial * (n + 1)) * hI.dpow (n + 1) x = x ^ (n + 1) | Please generate a tactic in lean4 to solve the state.
STATE:
case add_one
A : Type u_1
inst✝ : CommSemiring A
I : Ideal A
hI : DividedPowers I
x : A
hx : x ∈ I
n : ℕ
ih : ↑n.factorial * hI.dpow n x = x ^ n
⊢ ↑(n + 1).factorial * hI.dpow (n + 1) x = x ^ (n + 1)
TACTIC:
|
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/Basic.lean | DividedPowers.factorial_mul_dpow_eq_pow | [209, 1] | [217, 82] | nth_rewrite 1 [← (n + 1).choose_one_right] | case add_one
A : Type u_1
inst✝ : CommSemiring A
I : Ideal A
hI : DividedPowers I
x : A
hx : x ∈ I
n : ℕ
ih : ↑n.factorial * hI.dpow n x = x ^ n
⊢ ↑(n.factorial * (n + 1)) * hI.dpow (n + 1) x = x ^ (n + 1) | case add_one
A : Type u_1
inst✝ : CommSemiring A
I : Ideal A
hI : DividedPowers I
x : A
hx : x ∈ I
n : ℕ
ih : ↑n.factorial * hI.dpow n x = x ^ n
⊢ ↑(n.factorial * (n + 1).choose 1) * hI.dpow (n + 1) x = x ^ (n + 1) | Please generate a tactic in lean4 to solve the state.
STATE:
case add_one
A : Type u_1
inst✝ : CommSemiring A
I : Ideal A
hI : DividedPowers I
x : A
hx : x ∈ I
n : ℕ
ih : ↑n.factorial * hI.dpow n x = x ^ n
⊢ ↑(n.factorial * (n + 1)) * hI.dpow (n + 1) x = x ^ (n + 1)
TACTIC:
|
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/Basic.lean | DividedPowers.factorial_mul_dpow_eq_pow | [209, 1] | [217, 82] | rw [← Nat.choose_symm_add, Nat.cast_mul, mul_assoc,
← hI.dpow_mul n 1 hx, ← mul_assoc, ih, hI.dpow_one hx, pow_succ', mul_comm] | case add_one
A : Type u_1
inst✝ : CommSemiring A
I : Ideal A
hI : DividedPowers I
x : A
hx : x ∈ I
n : ℕ
ih : ↑n.factorial * hI.dpow n x = x ^ n
⊢ ↑(n.factorial * (n + 1).choose 1) * hI.dpow (n + 1) x = x ^ (n + 1) | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case add_one
A : Type u_1
inst✝ : CommSemiring A
I : Ideal A
hI : DividedPowers I
x : A
hx : x ∈ I
n : ℕ
ih : ↑n.factorial * hI.dpow n x = x ^ n
⊢ ↑(n.factorial * (n + 1).choose 1) * hI.dpow (n + 1) x = x ^ (n + 1)
TACTIC:
|
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/Basic.lean | DividedPowers.dpow_eval_zero | [220, 1] | [222, 37] | rw [← MulZeroClass.mul_zero (0 : A), hI.dpow_smul n I.zero_mem,
zero_pow hn, zero_mul, zero_mul] | A : Type u_1
inst✝ : CommSemiring A
I : Ideal A
hI : DividedPowers I
n : ℕ
hn : n ≠ 0
⊢ hI.dpow n 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
hI : DividedPowers I
n : ℕ
hn : n ≠ 0
⊢ hI.dpow n 0 = 0
TACTIC:
|
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/Basic.lean | DividedPowers.nilpotent_of_mem_dpIdeal | [226, 1] | [233, 73] | have h_fac : (n.factorial : A) * hI.dpow n x =
n • ((n - 1).factorial : A) * hI.dpow n x := by
rw [nsmul_eq_mul, ← Nat.cast_mul, Nat.mul_factorial_pred (Nat.pos_of_ne_zero hn)] | A : Type u_1
inst✝ : CommSemiring A
I : Ideal A
hI✝ hI : DividedPowers I
n : ℕ
hn : n ≠ 0
hnI : ∀ {y : A}, y ∈ I → n • y = 0
x : A
hx : x ∈ I
⊢ x ^ n = 0 | A : Type u_1
inst✝ : CommSemiring A
I : Ideal A
hI✝ hI : DividedPowers I
n : ℕ
hn : n ≠ 0
hnI : ∀ {y : A}, y ∈ I → n • y = 0
x : A
hx : x ∈ I
h_fac : ↑n.factorial * hI.dpow n x = n • ↑(n - 1).factorial * hI.dpow n x
⊢ x ^ n = 0 | Please generate a tactic in lean4 to solve the state.
STATE:
A : Type u_1
inst✝ : CommSemiring A
I : Ideal A
hI✝ hI : DividedPowers I
n : ℕ
hn : n ≠ 0
hnI : ∀ {y : A}, y ∈ I → n • y = 0
x : A
hx : x ∈ I
⊢ x ^ n = 0
TACTIC:
|
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/Basic.lean | DividedPowers.nilpotent_of_mem_dpIdeal | [226, 1] | [233, 73] | rw [← factorial_mul_dpow_eq_pow hI _ _ hx, h_fac, smul_mul_assoc] | A : Type u_1
inst✝ : CommSemiring A
I : Ideal A
hI✝ hI : DividedPowers I
n : ℕ
hn : n ≠ 0
hnI : ∀ {y : A}, y ∈ I → n • y = 0
x : A
hx : x ∈ I
h_fac : ↑n.factorial * hI.dpow n x = n • ↑(n - 1).factorial * hI.dpow n x
⊢ x ^ n = 0 | A : Type u_1
inst✝ : CommSemiring A
I : Ideal A
hI✝ hI : DividedPowers I
n : ℕ
hn : n ≠ 0
hnI : ∀ {y : A}, y ∈ I → n • y = 0
x : A
hx : x ∈ I
h_fac : ↑n.factorial * hI.dpow n x = n • ↑(n - 1).factorial * hI.dpow n x
⊢ n • (↑(n - 1).factorial * hI.dpow n x) = 0 | Please generate a tactic in lean4 to solve the state.
STATE:
A : Type u_1
inst✝ : CommSemiring A
I : Ideal A
hI✝ hI : DividedPowers I
n : ℕ
hn : n ≠ 0
hnI : ∀ {y : A}, y ∈ I → n • y = 0
x : A
hx : x ∈ I
h_fac : ↑n.factorial * hI.dpow n x = n • ↑(n - 1).factorial * hI.dpow n x
⊢ x ^ n = 0
TACTIC:
|
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/Basic.lean | DividedPowers.nilpotent_of_mem_dpIdeal | [226, 1] | [233, 73] | exact hnI (I.mul_mem_left ((n - 1).factorial : A) (hI.dpow_mem hn hx)) | A : Type u_1
inst✝ : CommSemiring A
I : Ideal A
hI✝ hI : DividedPowers I
n : ℕ
hn : n ≠ 0
hnI : ∀ {y : A}, y ∈ I → n • y = 0
x : A
hx : x ∈ I
h_fac : ↑n.factorial * hI.dpow n x = n • ↑(n - 1).factorial * hI.dpow n x
⊢ n • (↑(n - 1).factorial * hI.dpow n x) = 0 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
A : Type u_1
inst✝ : CommSemiring A
I : Ideal A
hI✝ hI : DividedPowers I
n : ℕ
hn : n ≠ 0
hnI : ∀ {y : A}, y ∈ I → n • y = 0
x : A
hx : x ∈ I
h_fac : ↑n.factorial * hI.dpow n x = n • ↑(n - 1).factorial * hI.dpow n x
⊢ n • (↑(n - 1).factorial * hI.dpow n x) = ... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/Basic.lean | DividedPowers.nilpotent_of_mem_dpIdeal | [226, 1] | [233, 73] | rw [nsmul_eq_mul, ← Nat.cast_mul, Nat.mul_factorial_pred (Nat.pos_of_ne_zero hn)] | A : Type u_1
inst✝ : CommSemiring A
I : Ideal A
hI✝ hI : DividedPowers I
n : ℕ
hn : n ≠ 0
hnI : ∀ {y : A}, y ∈ I → n • y = 0
x : A
hx : x ∈ I
⊢ ↑n.factorial * hI.dpow n x = n • ↑(n - 1).factorial * hI.dpow n x | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
A : Type u_1
inst✝ : CommSemiring A
I : Ideal A
hI✝ hI : DividedPowers I
n : ℕ
hn : n ≠ 0
hnI : ∀ {y : A}, y ∈ I → n • y = 0
x : A
hx : x ∈ I
⊢ ↑n.factorial * hI.dpow n x = n • ↑(n - 1).factorial * hI.dpow n x
TACTIC:
|
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/Basic.lean | DividedPowers.coincide_on_smul | [240, 1] | [252, 18] | induction ha using Submodule.smul_induction_on' generalizing n with
| smul a ha b hb =>
rw [Algebra.id.smul_eq_mul, hJ.dpow_smul n hb, mul_comm a b, hI.dpow_smul n ha, ←
hJ.factorial_mul_dpow_eq_pow n b hb, ← hI.factorial_mul_dpow_eq_pow n a ha]
ring
| add x hx y hy hx' hy' =>
rw [hI.dpow_add n (Ideal.mul_le_... | A : Type u_1
inst✝ : CommSemiring A
I : Ideal A
hI : DividedPowers I
J : Ideal A
hJ : DividedPowers J
n : ℕ
a : A
ha : a ∈ I • J
⊢ hI.dpow n a = hJ.dpow n a | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
A : Type u_1
inst✝ : CommSemiring A
I : Ideal A
hI : DividedPowers I
J : Ideal A
hJ : DividedPowers J
n : ℕ
a : A
ha : a ∈ I • J
⊢ hI.dpow n a = hJ.dpow n a
TACTIC:
|
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/Basic.lean | DividedPowers.coincide_on_smul | [240, 1] | [252, 18] | rw [Algebra.id.smul_eq_mul, hJ.dpow_smul n hb, mul_comm a b, hI.dpow_smul n ha, ←
hJ.factorial_mul_dpow_eq_pow n b hb, ← hI.factorial_mul_dpow_eq_pow n a ha] | case smul
A : Type u_1
inst✝ : CommSemiring A
I : Ideal A
hI : DividedPowers I
J : Ideal A
hJ : DividedPowers J
a✝ a : A
ha : a ∈ I
b : A
hb : b ∈ J
n : ℕ
⊢ hI.dpow n (a • b) = hJ.dpow n (a • b) | case smul
A : Type u_1
inst✝ : CommSemiring A
I : Ideal A
hI : DividedPowers I
J : Ideal A
hJ : DividedPowers J
a✝ a : A
ha : a ∈ I
b : A
hb : b ∈ J
n : ℕ
⊢ ↑n.factorial * hJ.dpow n b * hI.dpow n a = ↑n.factorial * hI.dpow n a * hJ.dpow n b | Please generate a tactic in lean4 to solve the state.
STATE:
case smul
A : Type u_1
inst✝ : CommSemiring A
I : Ideal A
hI : DividedPowers I
J : Ideal A
hJ : DividedPowers J
a✝ a : A
ha : a ∈ I
b : A
hb : b ∈ J
n : ℕ
⊢ hI.dpow n (a • b) = hJ.dpow n (a • b)
TACTIC:
|
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/Basic.lean | DividedPowers.coincide_on_smul | [240, 1] | [252, 18] | ring | case smul
A : Type u_1
inst✝ : CommSemiring A
I : Ideal A
hI : DividedPowers I
J : Ideal A
hJ : DividedPowers J
a✝ a : A
ha : a ∈ I
b : A
hb : b ∈ J
n : ℕ
⊢ ↑n.factorial * hJ.dpow n b * hI.dpow n a = ↑n.factorial * hI.dpow n a * hJ.dpow n b | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case smul
A : Type u_1
inst✝ : CommSemiring A
I : Ideal A
hI : DividedPowers I
J : Ideal A
hJ : DividedPowers J
a✝ a : A
ha : a ∈ I
b : A
hb : b ∈ J
n : ℕ
⊢ ↑n.factorial * hJ.dpow n b * hI.dpow n a = ↑n.factorial * hI.dpow n a * hJ.dpow n b
TACTIC:
|
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/Basic.lean | DividedPowers.coincide_on_smul | [240, 1] | [252, 18] | rw [hI.dpow_add n (Ideal.mul_le_right hx) (Ideal.mul_le_right hy),
hJ.dpow_add n (Ideal.mul_le_left hx) (Ideal.mul_le_left hy)] | case add
A : Type u_1
inst✝ : CommSemiring A
I : Ideal A
hI : DividedPowers I
J : Ideal A
hJ : DividedPowers J
a x : A
hx : x ∈ I • J
y : A
hy : y ∈ I • J
hx' : ∀ {n : ℕ}, hI.dpow n x = hJ.dpow n x
hy' : ∀ {n : ℕ}, hI.dpow n y = hJ.dpow n y
n : ℕ
⊢ hI.dpow n (x + y) = hJ.dpow n (x + y) | case add
A : Type u_1
inst✝ : CommSemiring A
I : Ideal A
hI : DividedPowers I
J : Ideal A
hJ : DividedPowers J
a x : A
hx : x ∈ I • J
y : A
hy : y ∈ I • J
hx' : ∀ {n : ℕ}, hI.dpow n x = hJ.dpow n x
hy' : ∀ {n : ℕ}, hI.dpow n y = hJ.dpow n y
n : ℕ
⊢ ∑ k ∈ Finset.antidiagonal n, hI.dpow k.1 x * hI.dpow k.2 y =
∑ k ∈ ... | Please generate a tactic in lean4 to solve the state.
STATE:
case add
A : Type u_1
inst✝ : CommSemiring A
I : Ideal A
hI : DividedPowers I
J : Ideal A
hJ : DividedPowers J
a x : A
hx : x ∈ I • J
y : A
hy : y ∈ I • J
hx' : ∀ {n : ℕ}, hI.dpow n x = hJ.dpow n x
hy' : ∀ {n : ℕ}, hI.dpow n y = hJ.dpow n y
n : ℕ
⊢ hI.dpow n ... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/Basic.lean | DividedPowers.coincide_on_smul | [240, 1] | [252, 18] | apply Finset.sum_congr rfl | case add
A : Type u_1
inst✝ : CommSemiring A
I : Ideal A
hI : DividedPowers I
J : Ideal A
hJ : DividedPowers J
a x : A
hx : x ∈ I • J
y : A
hy : y ∈ I • J
hx' : ∀ {n : ℕ}, hI.dpow n x = hJ.dpow n x
hy' : ∀ {n : ℕ}, hI.dpow n y = hJ.dpow n y
n : ℕ
⊢ ∑ k ∈ Finset.antidiagonal n, hI.dpow k.1 x * hI.dpow k.2 y =
∑ k ∈ ... | case add
A : Type u_1
inst✝ : CommSemiring A
I : Ideal A
hI : DividedPowers I
J : Ideal A
hJ : DividedPowers J
a x : A
hx : x ∈ I • J
y : A
hy : y ∈ I • J
hx' : ∀ {n : ℕ}, hI.dpow n x = hJ.dpow n x
hy' : ∀ {n : ℕ}, hI.dpow n y = hJ.dpow n y
n : ℕ
⊢ ∀ x_1 ∈ Finset.antidiagonal n, hI.dpow x_1.1 x * hI.dpow x_1.2 y = hJ.d... | Please generate a tactic in lean4 to solve the state.
STATE:
case add
A : Type u_1
inst✝ : CommSemiring A
I : Ideal A
hI : DividedPowers I
J : Ideal A
hJ : DividedPowers J
a x : A
hx : x ∈ I • J
y : A
hy : y ∈ I • J
hx' : ∀ {n : ℕ}, hI.dpow n x = hJ.dpow n x
hy' : ∀ {n : ℕ}, hI.dpow n y = hJ.dpow n y
n : ℕ
⊢ ∑ k ∈ Fins... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/Basic.lean | DividedPowers.coincide_on_smul | [240, 1] | [252, 18] | intro k _ | case add
A : Type u_1
inst✝ : CommSemiring A
I : Ideal A
hI : DividedPowers I
J : Ideal A
hJ : DividedPowers J
a x : A
hx : x ∈ I • J
y : A
hy : y ∈ I • J
hx' : ∀ {n : ℕ}, hI.dpow n x = hJ.dpow n x
hy' : ∀ {n : ℕ}, hI.dpow n y = hJ.dpow n y
n : ℕ
⊢ ∀ x_1 ∈ Finset.antidiagonal n, hI.dpow x_1.1 x * hI.dpow x_1.2 y = hJ.d... | case add
A : Type u_1
inst✝ : CommSemiring A
I : Ideal A
hI : DividedPowers I
J : Ideal A
hJ : DividedPowers J
a x : A
hx : x ∈ I • J
y : A
hy : y ∈ I • J
hx' : ∀ {n : ℕ}, hI.dpow n x = hJ.dpow n x
hy' : ∀ {n : ℕ}, hI.dpow n y = hJ.dpow n y
n : ℕ
k : ℕ × ℕ
a✝ : k ∈ Finset.antidiagonal n
⊢ hI.dpow k.1 x * hI.dpow k.2 y ... | Please generate a tactic in lean4 to solve the state.
STATE:
case add
A : Type u_1
inst✝ : CommSemiring A
I : Ideal A
hI : DividedPowers I
J : Ideal A
hJ : DividedPowers J
a x : A
hx : x ∈ I • J
y : A
hy : y ∈ I • J
hx' : ∀ {n : ℕ}, hI.dpow n x = hJ.dpow n x
hy' : ∀ {n : ℕ}, hI.dpow n y = hJ.dpow n y
n : ℕ
⊢ ∀ x_1 ∈ Fi... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/Basic.lean | DividedPowers.coincide_on_smul | [240, 1] | [252, 18] | rw [hx', hy'] | case add
A : Type u_1
inst✝ : CommSemiring A
I : Ideal A
hI : DividedPowers I
J : Ideal A
hJ : DividedPowers J
a x : A
hx : x ∈ I • J
y : A
hy : y ∈ I • J
hx' : ∀ {n : ℕ}, hI.dpow n x = hJ.dpow n x
hy' : ∀ {n : ℕ}, hI.dpow n y = hJ.dpow n y
n : ℕ
k : ℕ × ℕ
a✝ : k ∈ Finset.antidiagonal n
⊢ hI.dpow k.1 x * hI.dpow k.2 y ... | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case add
A : Type u_1
inst✝ : CommSemiring A
I : Ideal A
hI : DividedPowers I
J : Ideal A
hJ : DividedPowers J
a x : A
hx : x ∈ I • J
y : A
hy : y ∈ I • J
hx' : ∀ {n : ℕ}, hI.dpow n x = hJ.dpow n x
hy' : ∀ {n : ℕ}, hI.dpow n y = hJ.dpow n y
n : ℕ
k : ℕ × ℕ
a✝... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/Basic.lean | DividedPowers.mul_dpow | [259, 1] | [270, 81] | classical
induction s using Finset.induction with
| empty =>
simp only [prod_empty, Nat.multinomial_empty, Nat.cast_one, sum_empty, one_mul]
rw [hI.dpow_zero ha]
| insert hi hrec =>
rw [Finset.prod_insert hi, hrec, ← mul_assoc, mul_comm (hI.dpow (n _) a),
mul_assoc, dpow_mul _ _ _ ha, ← Finset.sum_insert hi, ... | A : Type u_2
inst✝ : CommSemiring A
I : Ideal A
hI : DividedPowers I
ι : Type u_1
s : Finset ι
n : ι → ℕ
a : A
ha : a ∈ I
⊢ ∏ i ∈ s, hI.dpow (n i) a = ↑(Nat.multinomial s n) * hI.dpow (s.sum n) a | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
A : Type u_2
inst✝ : CommSemiring A
I : Ideal A
hI : DividedPowers I
ι : Type u_1
s : Finset ι
n : ι → ℕ
a : A
ha : a ∈ I
⊢ ∏ i ∈ s, hI.dpow (n i) a = ↑(Nat.multinomial s n) * hI.dpow (s.sum n) a
TACTIC:
|
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/Basic.lean | DividedPowers.mul_dpow | [259, 1] | [270, 81] | induction s using Finset.induction with
| empty =>
simp only [prod_empty, Nat.multinomial_empty, Nat.cast_one, sum_empty, one_mul]
rw [hI.dpow_zero ha]
| insert hi hrec =>
rw [Finset.prod_insert hi, hrec, ← mul_assoc, mul_comm (hI.dpow (n _) a),
mul_assoc, dpow_mul _ _ _ ha, ← Finset.sum_insert hi, ← mul_asso... | A : Type u_2
inst✝ : CommSemiring A
I : Ideal A
hI : DividedPowers I
ι : Type u_1
s : Finset ι
n : ι → ℕ
a : A
ha : a ∈ I
⊢ ∏ i ∈ s, hI.dpow (n i) a = ↑(Nat.multinomial s n) * hI.dpow (s.sum n) a | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
A : Type u_2
inst✝ : CommSemiring A
I : Ideal A
hI : DividedPowers I
ι : Type u_1
s : Finset ι
n : ι → ℕ
a : A
ha : a ∈ I
⊢ ∏ i ∈ s, hI.dpow (n i) a = ↑(Nat.multinomial s n) * hI.dpow (s.sum n) a
TACTIC:
|
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/Basic.lean | DividedPowers.mul_dpow | [259, 1] | [270, 81] | simp only [prod_empty, Nat.multinomial_empty, Nat.cast_one, sum_empty, one_mul] | case empty
A : Type u_2
inst✝ : CommSemiring A
I : Ideal A
hI : DividedPowers I
ι : Type u_1
n : ι → ℕ
a : A
ha : a ∈ I
⊢ ∏ i ∈ ∅, hI.dpow (n i) a = ↑(Nat.multinomial ∅ n) * hI.dpow (∅.sum n) a | case empty
A : Type u_2
inst✝ : CommSemiring A
I : Ideal A
hI : DividedPowers I
ι : Type u_1
n : ι → ℕ
a : A
ha : a ∈ I
⊢ 1 = hI.dpow 0 a | Please generate a tactic in lean4 to solve the state.
STATE:
case empty
A : Type u_2
inst✝ : CommSemiring A
I : Ideal A
hI : DividedPowers I
ι : Type u_1
n : ι → ℕ
a : A
ha : a ∈ I
⊢ ∏ i ∈ ∅, hI.dpow (n i) a = ↑(Nat.multinomial ∅ n) * hI.dpow (∅.sum n) a
TACTIC:
|
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/Basic.lean | DividedPowers.mul_dpow | [259, 1] | [270, 81] | rw [hI.dpow_zero ha] | case empty
A : Type u_2
inst✝ : CommSemiring A
I : Ideal A
hI : DividedPowers I
ι : Type u_1
n : ι → ℕ
a : A
ha : a ∈ I
⊢ 1 = hI.dpow 0 a | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case empty
A : Type u_2
inst✝ : CommSemiring A
I : Ideal A
hI : DividedPowers I
ι : Type u_1
n : ι → ℕ
a : A
ha : a ∈ I
⊢ 1 = hI.dpow 0 a
TACTIC:
|
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/Basic.lean | DividedPowers.mul_dpow | [259, 1] | [270, 81] | rw [Finset.prod_insert hi, hrec, ← mul_assoc, mul_comm (hI.dpow (n _) a),
mul_assoc, dpow_mul _ _ _ ha, ← Finset.sum_insert hi, ← mul_assoc] | case insert
A : Type u_2
inst✝ : CommSemiring A
I : Ideal A
hI : DividedPowers I
ι : Type u_1
n : ι → ℕ
a : A
ha : a ∈ I
a✝ : ι
s✝ : Finset ι
hi : a✝ ∉ s✝
hrec : ∏ i ∈ s✝, hI.dpow (n i) a = ↑(Nat.multinomial s✝ n) * hI.dpow (s✝.sum n) a
⊢ ∏ i ∈ insert a✝ s✝, hI.dpow (n i) a = ↑(Nat.multinomial (insert a✝ s✝) n) * hI.dp... | case insert
A : Type u_2
inst✝ : CommSemiring A
I : Ideal A
hI : DividedPowers I
ι : Type u_1
n : ι → ℕ
a : A
ha : a ∈ I
a✝ : ι
s✝ : Finset ι
hi : a✝ ∉ s✝
hrec : ∏ i ∈ s✝, hI.dpow (n i) a = ↑(Nat.multinomial s✝ n) * hI.dpow (s✝.sum n) a
⊢ ↑(Nat.multinomial s✝ n) * ↑((∑ x ∈ insert a✝ s✝, n x).choose (n a✝)) * hI.dpow (∑... | Please generate a tactic in lean4 to solve the state.
STATE:
case insert
A : Type u_2
inst✝ : CommSemiring A
I : Ideal A
hI : DividedPowers I
ι : Type u_1
n : ι → ℕ
a : A
ha : a ∈ I
a✝ : ι
s✝ : Finset ι
hi : a✝ ∉ s✝
hrec : ∏ i ∈ s✝, hI.dpow (n i) a = ↑(Nat.multinomial s✝ n) * hI.dpow (s✝.sum n) a
⊢ ∏ i ∈ insert a✝ s✝, ... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/Basic.lean | DividedPowers.mul_dpow | [259, 1] | [270, 81] | apply congr_arg₂ _ _ rfl | case insert
A : Type u_2
inst✝ : CommSemiring A
I : Ideal A
hI : DividedPowers I
ι : Type u_1
n : ι → ℕ
a : A
ha : a ∈ I
a✝ : ι
s✝ : Finset ι
hi : a✝ ∉ s✝
hrec : ∏ i ∈ s✝, hI.dpow (n i) a = ↑(Nat.multinomial s✝ n) * hI.dpow (s✝.sum n) a
⊢ ↑(Nat.multinomial s✝ n) * ↑((∑ x ∈ insert a✝ s✝, n x).choose (n a✝)) * hI.dpow (∑... | A : Type u_2
inst✝ : CommSemiring A
I : Ideal A
hI : DividedPowers I
ι : Type u_1
n : ι → ℕ
a : A
ha : a ∈ I
a✝ : ι
s✝ : Finset ι
hi : a✝ ∉ s✝
hrec : ∏ i ∈ s✝, hI.dpow (n i) a = ↑(Nat.multinomial s✝ n) * hI.dpow (s✝.sum n) a
⊢ ↑(Nat.multinomial s✝ n) * ↑((∑ x ∈ insert a✝ s✝, n x).choose (n a✝)) = ↑(Nat.multinomial (ins... | Please generate a tactic in lean4 to solve the state.
STATE:
case insert
A : Type u_2
inst✝ : CommSemiring A
I : Ideal A
hI : DividedPowers I
ι : Type u_1
n : ι → ℕ
a : A
ha : a ∈ I
a✝ : ι
s✝ : Finset ι
hi : a✝ ∉ s✝
hrec : ∏ i ∈ s✝, hI.dpow (n i) a = ↑(Nat.multinomial s✝ n) * hI.dpow (s✝.sum n) a
⊢ ↑(Nat.multinomial s✝... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/Basic.lean | DividedPowers.mul_dpow | [259, 1] | [270, 81] | rw [Nat.multinomial_insert hi, mul_comm, Nat.cast_mul, Finset.sum_insert hi] | A : Type u_2
inst✝ : CommSemiring A
I : Ideal A
hI : DividedPowers I
ι : Type u_1
n : ι → ℕ
a : A
ha : a ∈ I
a✝ : ι
s✝ : Finset ι
hi : a✝ ∉ s✝
hrec : ∏ i ∈ s✝, hI.dpow (n i) a = ↑(Nat.multinomial s✝ n) * hI.dpow (s✝.sum n) a
⊢ ↑(Nat.multinomial s✝ n) * ↑((∑ x ∈ insert a✝ s✝, n x).choose (n a✝)) = ↑(Nat.multinomial (ins... | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
A : Type u_2
inst✝ : CommSemiring A
I : Ideal A
hI : DividedPowers I
ι : Type u_1
n : ι → ℕ
a : A
ha : a ∈ I
a✝ : ι
s✝ : Finset ι
hi : a✝ ∉ s✝
hrec : ∏ i ∈ s✝, hI.dpow (n i) a = ↑(Nat.multinomial s✝ n) * hI.dpow (s✝.sum n) a
⊢ ↑(Nat.multinomial s✝ n) * ↑((∑ x... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/Basic.lean | DividedPowers.dpow_sum_aux | [284, 1] | [345, 27] | simp only [Finset.Nat.sum_antidiagonal_eq_sum_range_succ_mk] at dpow_add | A : Type u_1
inst✝¹ : CommSemiring A
I : Ideal A
hI : DividedPowers I
dpow : ℕ → A → A
dpow_zero : ∀ {x : A}, x ∈ I → dpow 0 x = 1
dpow_add : ∀ (n : ℕ) {x y : A}, x ∈ I → y ∈ I → dpow n (x + y) = ∑ k ∈ antidiagonal n, dpow k.1 x * dpow k.2 y
dpow_eval_zero : ∀ {n : ℕ}, n ≠ 0 → dpow n 0 = 0
ι : Type u_2
inst✝ : Decidabl... | A : Type u_1
inst✝¹ : CommSemiring A
I : Ideal A
hI : DividedPowers I
dpow : ℕ → A → A
dpow_zero : ∀ {x : A}, x ∈ I → dpow 0 x = 1
dpow_eval_zero : ∀ {n : ℕ}, n ≠ 0 → dpow n 0 = 0
ι : Type u_2
inst✝ : DecidableEq ι
s : Finset ι
x : ι → A
hx : ∀ i ∈ s, x i ∈ I
n : ℕ
dpow_add : ∀ (n : ℕ) {x y : A}, x ∈ I → y ∈ I → dpow n... | Please generate a tactic in lean4 to solve the state.
STATE:
A : Type u_1
inst✝¹ : CommSemiring A
I : Ideal A
hI : DividedPowers I
dpow : ℕ → A → A
dpow_zero : ∀ {x : A}, x ∈ I → dpow 0 x = 1
dpow_add : ∀ (n : ℕ) {x y : A}, x ∈ I → y ∈ I → dpow n (x + y) = ∑ k ∈ antidiagonal n, dpow k.1 x * dpow k.2 y
dpow_eval_zero : ... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/Basic.lean | DividedPowers.dpow_sum_aux | [284, 1] | [345, 27] | simp only [sum_empty, prod_empty, sum_const, nsmul_eq_mul, mul_one] | case empty
A : Type u_1
inst✝¹ : CommSemiring A
I : Ideal A
hI : DividedPowers I
dpow : ℕ → A → A
dpow_zero : ∀ {x : A}, x ∈ I → dpow 0 x = 1
dpow_eval_zero : ∀ {n : ℕ}, n ≠ 0 → dpow n 0 = 0
ι : Type u_2
inst✝ : DecidableEq ι
x : ι → A
dpow_add : ∀ (n : ℕ) {x y : A}, x ∈ I → y ∈ I → dpow n (x + y) = ∑ x_3 ∈ range n.suc... | case empty
A : Type u_1
inst✝¹ : CommSemiring A
I : Ideal A
hI : DividedPowers I
dpow : ℕ → A → A
dpow_zero : ∀ {x : A}, x ∈ I → dpow 0 x = 1
dpow_eval_zero : ∀ {n : ℕ}, n ≠ 0 → dpow n 0 = 0
ι : Type u_2
inst✝ : DecidableEq ι
x : ι → A
dpow_add : ∀ (n : ℕ) {x y : A}, x ∈ I → y ∈ I → dpow n (x + y) = ∑ x_3 ∈ range n.suc... | Please generate a tactic in lean4 to solve the state.
STATE:
case empty
A : Type u_1
inst✝¹ : CommSemiring A
I : Ideal A
hI : DividedPowers I
dpow : ℕ → A → A
dpow_zero : ∀ {x : A}, x ∈ I → dpow 0 x = 1
dpow_eval_zero : ∀ {n : ℕ}, n ≠ 0 → dpow n 0 = 0
ι : Type u_2
inst✝ : DecidableEq ι
x : ι → A
dpow_add : ∀ (n : ℕ) {x... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/Basic.lean | DividedPowers.dpow_sum_aux | [284, 1] | [345, 27] | by_cases hn : n = 0 | case empty
A : Type u_1
inst✝¹ : CommSemiring A
I : Ideal A
hI : DividedPowers I
dpow : ℕ → A → A
dpow_zero : ∀ {x : A}, x ∈ I → dpow 0 x = 1
dpow_eval_zero : ∀ {n : ℕ}, n ≠ 0 → dpow n 0 = 0
ι : Type u_2
inst✝ : DecidableEq ι
x : ι → A
dpow_add : ∀ (n : ℕ) {x y : A}, x ∈ I → y ∈ I → dpow n (x + y) = ∑ x_3 ∈ range n.suc... | case pos
A : Type u_1
inst✝¹ : CommSemiring A
I : Ideal A
hI : DividedPowers I
dpow : ℕ → A → A
dpow_zero : ∀ {x : A}, x ∈ I → dpow 0 x = 1
dpow_eval_zero : ∀ {n : ℕ}, n ≠ 0 → dpow n 0 = 0
ι : Type u_2
inst✝ : DecidableEq ι
x : ι → A
dpow_add : ∀ (n : ℕ) {x y : A}, x ∈ I → y ∈ I → dpow n (x + y) = ∑ x_3 ∈ range n.succ,... | Please generate a tactic in lean4 to solve the state.
STATE:
case empty
A : Type u_1
inst✝¹ : CommSemiring A
I : Ideal A
hI : DividedPowers I
dpow : ℕ → A → A
dpow_zero : ∀ {x : A}, x ∈ I → dpow 0 x = 1
dpow_eval_zero : ∀ {n : ℕ}, n ≠ 0 → dpow n 0 = 0
ι : Type u_2
inst✝ : DecidableEq ι
x : ι → A
dpow_add : ∀ (n : ℕ) {x... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/Basic.lean | DividedPowers.dpow_sum_aux | [284, 1] | [345, 27] | rw [hn] | case pos
A : Type u_1
inst✝¹ : CommSemiring A
I : Ideal A
hI : DividedPowers I
dpow : ℕ → A → A
dpow_zero : ∀ {x : A}, x ∈ I → dpow 0 x = 1
dpow_eval_zero : ∀ {n : ℕ}, n ≠ 0 → dpow n 0 = 0
ι : Type u_2
inst✝ : DecidableEq ι
x : ι → A
dpow_add : ∀ (n : ℕ) {x y : A}, x ∈ I → y ∈ I → dpow n (x + y) = ∑ x_3 ∈ range n.succ,... | case pos
A : Type u_1
inst✝¹ : CommSemiring A
I : Ideal A
hI : DividedPowers I
dpow : ℕ → A → A
dpow_zero : ∀ {x : A}, x ∈ I → dpow 0 x = 1
dpow_eval_zero : ∀ {n : ℕ}, n ≠ 0 → dpow n 0 = 0
ι : Type u_2
inst✝ : DecidableEq ι
x : ι → A
dpow_add : ∀ (n : ℕ) {x y : A}, x ∈ I → y ∈ I → dpow n (x + y) = ∑ x_3 ∈ range n.succ,... | Please generate a tactic in lean4 to solve the state.
STATE:
case pos
A : Type u_1
inst✝¹ : CommSemiring A
I : Ideal A
hI : DividedPowers I
dpow : ℕ → A → A
dpow_zero : ∀ {x : A}, x ∈ I → dpow 0 x = 1
dpow_eval_zero : ∀ {n : ℕ}, n ≠ 0 → dpow n 0 = 0
ι : Type u_2
inst✝ : DecidableEq ι
x : ι → A
dpow_add : ∀ (n : ℕ) {x y... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/Basic.lean | DividedPowers.dpow_sum_aux | [284, 1] | [345, 27] | rw [dpow_zero I.zero_mem] | case pos
A : Type u_1
inst✝¹ : CommSemiring A
I : Ideal A
hI : DividedPowers I
dpow : ℕ → A → A
dpow_zero : ∀ {x : A}, x ∈ I → dpow 0 x = 1
dpow_eval_zero : ∀ {n : ℕ}, n ≠ 0 → dpow n 0 = 0
ι : Type u_2
inst✝ : DecidableEq ι
x : ι → A
dpow_add : ∀ (n : ℕ) {x y : A}, x ∈ I → y ∈ I → dpow n (x + y) = ∑ x_3 ∈ range n.succ,... | case pos
A : Type u_1
inst✝¹ : CommSemiring A
I : Ideal A
hI : DividedPowers I
dpow : ℕ → A → A
dpow_zero : ∀ {x : A}, x ∈ I → dpow 0 x = 1
dpow_eval_zero : ∀ {n : ℕ}, n ≠ 0 → dpow n 0 = 0
ι : Type u_2
inst✝ : DecidableEq ι
x : ι → A
dpow_add : ∀ (n : ℕ) {x y : A}, x ∈ I → y ∈ I → dpow n (x + y) = ∑ x_3 ∈ range n.succ,... | Please generate a tactic in lean4 to solve the state.
STATE:
case pos
A : Type u_1
inst✝¹ : CommSemiring A
I : Ideal A
hI : DividedPowers I
dpow : ℕ → A → A
dpow_zero : ∀ {x : A}, x ∈ I → dpow 0 x = 1
dpow_eval_zero : ∀ {n : ℕ}, n ≠ 0 → dpow n 0 = 0
ι : Type u_2
inst✝ : DecidableEq ι
x : ι → A
dpow_add : ∀ (n : ℕ) {x y... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/Basic.lean | DividedPowers.dpow_sum_aux | [284, 1] | [345, 27] | simp only [sym_zero, card_singleton, Nat.cast_one] | case pos
A : Type u_1
inst✝¹ : CommSemiring A
I : Ideal A
hI : DividedPowers I
dpow : ℕ → A → A
dpow_zero : ∀ {x : A}, x ∈ I → dpow 0 x = 1
dpow_eval_zero : ∀ {n : ℕ}, n ≠ 0 → dpow n 0 = 0
ι : Type u_2
inst✝ : DecidableEq ι
x : ι → A
dpow_add : ∀ (n : ℕ) {x y : A}, x ∈ I → y ∈ I → dpow n (x + y) = ∑ x_3 ∈ range n.succ,... | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case pos
A : Type u_1
inst✝¹ : CommSemiring A
I : Ideal A
hI : DividedPowers I
dpow : ℕ → A → A
dpow_zero : ∀ {x : A}, x ∈ I → dpow 0 x = 1
dpow_eval_zero : ∀ {n : ℕ}, n ≠ 0 → dpow n 0 = 0
ι : Type u_2
inst✝ : DecidableEq ι
x : ι → A
dpow_add : ∀ (n : ℕ) {x y... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/Basic.lean | DividedPowers.dpow_sum_aux | [284, 1] | [345, 27] | rw [dpow_eval_zero hn] | case neg
A : Type u_1
inst✝¹ : CommSemiring A
I : Ideal A
hI : DividedPowers I
dpow : ℕ → A → A
dpow_zero : ∀ {x : A}, x ∈ I → dpow 0 x = 1
dpow_eval_zero : ∀ {n : ℕ}, n ≠ 0 → dpow n 0 = 0
ι : Type u_2
inst✝ : DecidableEq ι
x : ι → A
dpow_add : ∀ (n : ℕ) {x y : A}, x ∈ I → y ∈ I → dpow n (x + y) = ∑ x_3 ∈ range n.succ,... | case neg
A : Type u_1
inst✝¹ : CommSemiring A
I : Ideal A
hI : DividedPowers I
dpow : ℕ → A → A
dpow_zero : ∀ {x : A}, x ∈ I → dpow 0 x = 1
dpow_eval_zero : ∀ {n : ℕ}, n ≠ 0 → dpow n 0 = 0
ι : Type u_2
inst✝ : DecidableEq ι
x : ι → A
dpow_add : ∀ (n : ℕ) {x y : A}, x ∈ I → y ∈ I → dpow n (x + y) = ∑ x_3 ∈ range n.succ,... | Please generate a tactic in lean4 to solve the state.
STATE:
case neg
A : Type u_1
inst✝¹ : CommSemiring A
I : Ideal A
hI : DividedPowers I
dpow : ℕ → A → A
dpow_zero : ∀ {x : A}, x ∈ I → dpow 0 x = 1
dpow_eval_zero : ∀ {n : ℕ}, n ≠ 0 → dpow n 0 = 0
ι : Type u_2
inst✝ : DecidableEq ι
x : ι → A
dpow_add : ∀ (n : ℕ) {x y... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/Basic.lean | DividedPowers.dpow_sum_aux | [284, 1] | [345, 27] | apply symm | case neg
A : Type u_1
inst✝¹ : CommSemiring A
I : Ideal A
hI : DividedPowers I
dpow : ℕ → A → A
dpow_zero : ∀ {x : A}, x ∈ I → dpow 0 x = 1
dpow_eval_zero : ∀ {n : ℕ}, n ≠ 0 → dpow n 0 = 0
ι : Type u_2
inst✝ : DecidableEq ι
x : ι → A
dpow_add : ∀ (n : ℕ) {x y : A}, x ∈ I → y ∈ I → dpow n (x + y) = ∑ x_3 ∈ range n.succ,... | case neg.a
A : Type u_1
inst✝¹ : CommSemiring A
I : Ideal A
hI : DividedPowers I
dpow : ℕ → A → A
dpow_zero : ∀ {x : A}, x ∈ I → dpow 0 x = 1
dpow_eval_zero : ∀ {n : ℕ}, n ≠ 0 → dpow n 0 = 0
ι : Type u_2
inst✝ : DecidableEq ι
x : ι → A
dpow_add : ∀ (n : ℕ) {x y : A}, x ∈ I → y ∈ I → dpow n (x + y) = ∑ x_3 ∈ range n.suc... | Please generate a tactic in lean4 to solve the state.
STATE:
case neg
A : Type u_1
inst✝¹ : CommSemiring A
I : Ideal A
hI : DividedPowers I
dpow : ℕ → A → A
dpow_zero : ∀ {x : A}, x ∈ I → dpow 0 x = 1
dpow_eval_zero : ∀ {n : ℕ}, n ≠ 0 → dpow n 0 = 0
ι : Type u_2
inst✝ : DecidableEq ι
x : ι → A
dpow_add : ∀ (n : ℕ) {x y... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/Basic.lean | DividedPowers.dpow_sum_aux | [284, 1] | [345, 27] | rw [← Nat.cast_zero] | case neg.a
A : Type u_1
inst✝¹ : CommSemiring A
I : Ideal A
hI : DividedPowers I
dpow : ℕ → A → A
dpow_zero : ∀ {x : A}, x ∈ I → dpow 0 x = 1
dpow_eval_zero : ∀ {n : ℕ}, n ≠ 0 → dpow n 0 = 0
ι : Type u_2
inst✝ : DecidableEq ι
x : ι → A
dpow_add : ∀ (n : ℕ) {x y : A}, x ∈ I → y ∈ I → dpow n (x + y) = ∑ x_3 ∈ range n.suc... | case neg.a
A : Type u_1
inst✝¹ : CommSemiring A
I : Ideal A
hI : DividedPowers I
dpow : ℕ → A → A
dpow_zero : ∀ {x : A}, x ∈ I → dpow 0 x = 1
dpow_eval_zero : ∀ {n : ℕ}, n ≠ 0 → dpow n 0 = 0
ι : Type u_2
inst✝ : DecidableEq ι
x : ι → A
dpow_add : ∀ (n : ℕ) {x y : A}, x ∈ I → y ∈ I → dpow n (x + y) = ∑ x_3 ∈ range n.suc... | Please generate a tactic in lean4 to solve the state.
STATE:
case neg.a
A : Type u_1
inst✝¹ : CommSemiring A
I : Ideal A
hI : DividedPowers I
dpow : ℕ → A → A
dpow_zero : ∀ {x : A}, x ∈ I → dpow 0 x = 1
dpow_eval_zero : ∀ {n : ℕ}, n ≠ 0 → dpow n 0 = 0
ι : Type u_2
inst✝ : DecidableEq ι
x : ι → A
dpow_add : ∀ (n : ℕ) {x... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/Basic.lean | DividedPowers.dpow_sum_aux | [284, 1] | [345, 27] | apply congr_arg | case neg.a
A : Type u_1
inst✝¹ : CommSemiring A
I : Ideal A
hI : DividedPowers I
dpow : ℕ → A → A
dpow_zero : ∀ {x : A}, x ∈ I → dpow 0 x = 1
dpow_eval_zero : ∀ {n : ℕ}, n ≠ 0 → dpow n 0 = 0
ι : Type u_2
inst✝ : DecidableEq ι
x : ι → A
dpow_add : ∀ (n : ℕ) {x y : A}, x ∈ I → y ∈ I → dpow n (x + y) = ∑ x_3 ∈ range n.suc... | case neg.a.h
A : Type u_1
inst✝¹ : CommSemiring A
I : Ideal A
hI : DividedPowers I
dpow : ℕ → A → A
dpow_zero : ∀ {x : A}, x ∈ I → dpow 0 x = 1
dpow_eval_zero : ∀ {n : ℕ}, n ≠ 0 → dpow n 0 = 0
ι : Type u_2
inst✝ : DecidableEq ι
x : ι → A
dpow_add : ∀ (n : ℕ) {x y : A}, x ∈ I → y ∈ I → dpow n (x + y) = ∑ x_3 ∈ range n.s... | Please generate a tactic in lean4 to solve the state.
STATE:
case neg.a
A : Type u_1
inst✝¹ : CommSemiring A
I : Ideal A
hI : DividedPowers I
dpow : ℕ → A → A
dpow_zero : ∀ {x : A}, x ∈ I → dpow 0 x = 1
dpow_eval_zero : ∀ {n : ℕ}, n ≠ 0 → dpow n 0 = 0
ι : Type u_2
inst✝ : DecidableEq ι
x : ι → A
dpow_add : ∀ (n : ℕ) {x... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/Basic.lean | DividedPowers.dpow_sum_aux | [284, 1] | [345, 27] | rw [card_eq_zero] | case neg.a.h
A : Type u_1
inst✝¹ : CommSemiring A
I : Ideal A
hI : DividedPowers I
dpow : ℕ → A → A
dpow_zero : ∀ {x : A}, x ∈ I → dpow 0 x = 1
dpow_eval_zero : ∀ {n : ℕ}, n ≠ 0 → dpow n 0 = 0
ι : Type u_2
inst✝ : DecidableEq ι
x : ι → A
dpow_add : ∀ (n : ℕ) {x y : A}, x ∈ I → y ∈ I → dpow n (x + y) = ∑ x_3 ∈ range n.s... | case neg.a.h
A : Type u_1
inst✝¹ : CommSemiring A
I : Ideal A
hI : DividedPowers I
dpow : ℕ → A → A
dpow_zero : ∀ {x : A}, x ∈ I → dpow 0 x = 1
dpow_eval_zero : ∀ {n : ℕ}, n ≠ 0 → dpow n 0 = 0
ι : Type u_2
inst✝ : DecidableEq ι
x : ι → A
dpow_add : ∀ (n : ℕ) {x y : A}, x ∈ I → y ∈ I → dpow n (x + y) = ∑ x_3 ∈ range n.s... | Please generate a tactic in lean4 to solve the state.
STATE:
case neg.a.h
A : Type u_1
inst✝¹ : CommSemiring A
I : Ideal A
hI : DividedPowers I
dpow : ℕ → A → A
dpow_zero : ∀ {x : A}, x ∈ I → dpow 0 x = 1
dpow_eval_zero : ∀ {n : ℕ}, n ≠ 0 → dpow n 0 = 0
ι : Type u_2
inst✝ : DecidableEq ι
x : ι → A
dpow_add : ∀ (n : ℕ) ... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/Basic.lean | DividedPowers.dpow_sum_aux | [284, 1] | [345, 27] | rw [sym_eq_empty] | case neg.a.h
A : Type u_1
inst✝¹ : CommSemiring A
I : Ideal A
hI : DividedPowers I
dpow : ℕ → A → A
dpow_zero : ∀ {x : A}, x ∈ I → dpow 0 x = 1
dpow_eval_zero : ∀ {n : ℕ}, n ≠ 0 → dpow n 0 = 0
ι : Type u_2
inst✝ : DecidableEq ι
x : ι → A
dpow_add : ∀ (n : ℕ) {x y : A}, x ∈ I → y ∈ I → dpow n (x + y) = ∑ x_3 ∈ range n.s... | case neg.a.h
A : Type u_1
inst✝¹ : CommSemiring A
I : Ideal A
hI : DividedPowers I
dpow : ℕ → A → A
dpow_zero : ∀ {x : A}, x ∈ I → dpow 0 x = 1
dpow_eval_zero : ∀ {n : ℕ}, n ≠ 0 → dpow n 0 = 0
ι : Type u_2
inst✝ : DecidableEq ι
x : ι → A
dpow_add : ∀ (n : ℕ) {x y : A}, x ∈ I → y ∈ I → dpow n (x + y) = ∑ x_3 ∈ range n.s... | Please generate a tactic in lean4 to solve the state.
STATE:
case neg.a.h
A : Type u_1
inst✝¹ : CommSemiring A
I : Ideal A
hI : DividedPowers I
dpow : ℕ → A → A
dpow_zero : ∀ {x : A}, x ∈ I → dpow 0 x = 1
dpow_eval_zero : ∀ {n : ℕ}, n ≠ 0 → dpow n 0 = 0
ι : Type u_2
inst✝ : DecidableEq ι
x : ι → A
dpow_add : ∀ (n : ℕ) ... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/Basic.lean | DividedPowers.dpow_sum_aux | [284, 1] | [345, 27] | exact ⟨hn, rfl⟩ | case neg.a.h
A : Type u_1
inst✝¹ : CommSemiring A
I : Ideal A
hI : DividedPowers I
dpow : ℕ → A → A
dpow_zero : ∀ {x : A}, x ∈ I → dpow 0 x = 1
dpow_eval_zero : ∀ {n : ℕ}, n ≠ 0 → dpow n 0 = 0
ι : Type u_2
inst✝ : DecidableEq ι
x : ι → A
dpow_add : ∀ (n : ℕ) {x y : A}, x ∈ I → y ∈ I → dpow n (x + y) = ∑ x_3 ∈ range n.s... | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case neg.a.h
A : Type u_1
inst✝¹ : CommSemiring A
I : Ideal A
hI : DividedPowers I
dpow : ℕ → A → A
dpow_zero : ∀ {x : A}, x ∈ I → dpow 0 x = 1
dpow_eval_zero : ∀ {n : ℕ}, n ≠ 0 → dpow n 0 = 0
ι : Type u_2
inst✝ : DecidableEq ι
x : ι → A
dpow_add : ∀ (n : ℕ) ... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/Basic.lean | DividedPowers.dpow_sum_aux | [284, 1] | [345, 27] | have hx' : ∀ i, i ∈ s → x i ∈ I := fun i hi => hx i (Finset.mem_insert_of_mem hi) | case insert
A : Type u_1
inst✝¹ : CommSemiring A
I : Ideal A
hI : DividedPowers I
dpow : ℕ → A → A
dpow_zero : ∀ {x : A}, x ∈ I → dpow 0 x = 1
dpow_eval_zero : ∀ {n : ℕ}, n ≠ 0 → dpow n 0 = 0
ι : Type u_2
inst✝ : DecidableEq ι
x : ι → A
dpow_add : ∀ (n : ℕ) {x y : A}, x ∈ I → y ∈ I → dpow n (x + y) = ∑ x_3 ∈ range n.su... | case insert
A : Type u_1
inst✝¹ : CommSemiring A
I : Ideal A
hI : DividedPowers I
dpow : ℕ → A → A
dpow_zero : ∀ {x : A}, x ∈ I → dpow 0 x = 1
dpow_eval_zero : ∀ {n : ℕ}, n ≠ 0 → dpow n 0 = 0
ι : Type u_2
inst✝ : DecidableEq ι
x : ι → A
dpow_add : ∀ (n : ℕ) {x y : A}, x ∈ I → y ∈ I → dpow n (x + y) = ∑ x_3 ∈ range n.su... | Please generate a tactic in lean4 to solve the state.
STATE:
case insert
A : Type u_1
inst✝¹ : CommSemiring A
I : Ideal A
hI : DividedPowers I
dpow : ℕ → A → A
dpow_zero : ∀ {x : A}, x ∈ I → dpow 0 x = 1
dpow_eval_zero : ∀ {n : ℕ}, n ≠ 0 → dpow n 0 = 0
ι : Type u_2
inst✝ : DecidableEq ι
x : ι → A
dpow_add : ∀ (n : ℕ) {... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/Basic.lean | DividedPowers.dpow_sum_aux | [284, 1] | [345, 27] | simp_rw [sum_insert ha,
dpow_add n (hx a (Finset.mem_insert_self a s)) (I.sum_mem fun i => hx' i),
sum_range, ih hx', mul_sum, sum_sigma'] | case insert
A : Type u_1
inst✝¹ : CommSemiring A
I : Ideal A
hI : DividedPowers I
dpow : ℕ → A → A
dpow_zero : ∀ {x : A}, x ∈ I → dpow 0 x = 1
dpow_eval_zero : ∀ {n : ℕ}, n ≠ 0 → dpow n 0 = 0
ι : Type u_2
inst✝ : DecidableEq ι
x : ι → A
dpow_add : ∀ (n : ℕ) {x y : A}, x ∈ I → y ∈ I → dpow n (x + y) = ∑ x_3 ∈ range n.su... | case insert
A : Type u_1
inst✝¹ : CommSemiring A
I : Ideal A
hI : DividedPowers I
dpow : ℕ → A → A
dpow_zero : ∀ {x : A}, x ∈ I → dpow 0 x = 1
dpow_eval_zero : ∀ {n : ℕ}, n ≠ 0 → dpow n 0 = 0
ι : Type u_2
inst✝ : DecidableEq ι
x : ι → A
dpow_add : ∀ (n : ℕ) {x y : A}, x ∈ I → y ∈ I → dpow n (x + y) = ∑ x_3 ∈ range n.su... | Please generate a tactic in lean4 to solve the state.
STATE:
case insert
A : Type u_1
inst✝¹ : CommSemiring A
I : Ideal A
hI : DividedPowers I
dpow : ℕ → A → A
dpow_zero : ∀ {x : A}, x ∈ I → dpow 0 x = 1
dpow_eval_zero : ∀ {n : ℕ}, n ≠ 0 → dpow n 0 = 0
ι : Type u_2
inst✝ : DecidableEq ι
x : ι → A
dpow_add : ∀ (n : ℕ) {... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/Basic.lean | DividedPowers.dpow_sum_aux | [284, 1] | [345, 27] | apply symm | case insert
A : Type u_1
inst✝¹ : CommSemiring A
I : Ideal A
hI : DividedPowers I
dpow : ℕ → A → A
dpow_zero : ∀ {x : A}, x ∈ I → dpow 0 x = 1
dpow_eval_zero : ∀ {n : ℕ}, n ≠ 0 → dpow n 0 = 0
ι : Type u_2
inst✝ : DecidableEq ι
x : ι → A
dpow_add : ∀ (n : ℕ) {x y : A}, x ∈ I → y ∈ I → dpow n (x + y) = ∑ x_3 ∈ range n.su... | case insert.a
A : Type u_1
inst✝¹ : CommSemiring A
I : Ideal A
hI : DividedPowers I
dpow : ℕ → A → A
dpow_zero : ∀ {x : A}, x ∈ I → dpow 0 x = 1
dpow_eval_zero : ∀ {n : ℕ}, n ≠ 0 → dpow n 0 = 0
ι : Type u_2
inst✝ : DecidableEq ι
x : ι → A
dpow_add : ∀ (n : ℕ) {x y : A}, x ∈ I → y ∈ I → dpow n (x + y) = ∑ x_3 ∈ range n.... | Please generate a tactic in lean4 to solve the state.
STATE:
case insert
A : Type u_1
inst✝¹ : CommSemiring A
I : Ideal A
hI : DividedPowers I
dpow : ℕ → A → A
dpow_zero : ∀ {x : A}, x ∈ I → dpow 0 x = 1
dpow_eval_zero : ∀ {n : ℕ}, n ≠ 0 → dpow n 0 = 0
ι : Type u_2
inst✝ : DecidableEq ι
x : ι → A
dpow_add : ∀ (n : ℕ) {... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/Basic.lean | DividedPowers.dpow_sum_aux | [284, 1] | [345, 27] | apply sum_bij'
(fun m _ => Sym.filterNe a m)
(fun m _ => m.2.fill a m.1)
(fun m hm => Finset.mem_sigma.2 ⟨mem_univ _, _⟩)
(fun m hm => by
rw [mem_sym_iff]
intro i hi
rw [Sym.mem_fill_iff] at hi
cases hi with
| inl hi =>
rw [hi.2]
exact mem_insert_self a s
| inr hi =>
si... | case insert.a
A : Type u_1
inst✝¹ : CommSemiring A
I : Ideal A
hI : DividedPowers I
dpow : ℕ → A → A
dpow_zero : ∀ {x : A}, x ∈ I → dpow 0 x = 1
dpow_eval_zero : ∀ {n : ℕ}, n ≠ 0 → dpow n 0 = 0
ι : Type u_2
inst✝ : DecidableEq ι
x : ι → A
dpow_add : ∀ (n : ℕ) {x y : A}, x ∈ I → y ∈ I → dpow n (x + y) = ∑ x_3 ∈ range n.... | case insert.a.right_inv
A : Type u_1
inst✝¹ : CommSemiring A
I : Ideal A
hI : DividedPowers I
dpow : ℕ → A → A
dpow_zero : ∀ {x : A}, x ∈ I → dpow 0 x = 1
dpow_eval_zero : ∀ {n : ℕ}, n ≠ 0 → dpow n 0 = 0
ι : Type u_2
inst✝ : DecidableEq ι
x : ι → A
dpow_add : ∀ (n : ℕ) {x y : A}, x ∈ I → y ∈ I → dpow n (x + y) = ∑ x_3 ... | Please generate a tactic in lean4 to solve the state.
STATE:
case insert.a
A : Type u_1
inst✝¹ : CommSemiring A
I : Ideal A
hI : DividedPowers I
dpow : ℕ → A → A
dpow_zero : ∀ {x : A}, x ∈ I → dpow 0 x = 1
dpow_eval_zero : ∀ {n : ℕ}, n ≠ 0 → dpow n 0 = 0
ι : Type u_2
inst✝ : DecidableEq ι
x : ι → A
dpow_add : ∀ (n : ℕ)... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/Basic.lean | DividedPowers.dpow_sum_aux | [284, 1] | [345, 27] | rw [mem_sym_iff] | A : Type u_1
inst✝¹ : CommSemiring A
I : Ideal A
hI : DividedPowers I
dpow : ℕ → A → A
dpow_zero : ∀ {x : A}, x ∈ I → dpow 0 x = 1
dpow_eval_zero : ∀ {n : ℕ}, n ≠ 0 → dpow n 0 = 0
ι : Type u_2
inst✝ : DecidableEq ι
x : ι → A
dpow_add : ∀ (n : ℕ) {x y : A}, x ∈ I → y ∈ I → dpow n (x + y) = ∑ x_3 ∈ range n.succ, dpow x_3... | A : Type u_1
inst✝¹ : CommSemiring A
I : Ideal A
hI : DividedPowers I
dpow : ℕ → A → A
dpow_zero : ∀ {x : A}, x ∈ I → dpow 0 x = 1
dpow_eval_zero : ∀ {n : ℕ}, n ≠ 0 → dpow n 0 = 0
ι : Type u_2
inst✝ : DecidableEq ι
x : ι → A
dpow_add : ∀ (n : ℕ) {x y : A}, x ∈ I → y ∈ I → dpow n (x + y) = ∑ x_3 ∈ range n.succ, dpow x_3... | Please generate a tactic in lean4 to solve the state.
STATE:
A : Type u_1
inst✝¹ : CommSemiring A
I : Ideal A
hI : DividedPowers I
dpow : ℕ → A → A
dpow_zero : ∀ {x : A}, x ∈ I → dpow 0 x = 1
dpow_eval_zero : ∀ {n : ℕ}, n ≠ 0 → dpow n 0 = 0
ι : Type u_2
inst✝ : DecidableEq ι
x : ι → A
dpow_add : ∀ (n : ℕ) {x y : A}, x ... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/Basic.lean | DividedPowers.dpow_sum_aux | [284, 1] | [345, 27] | intro i hi | A : Type u_1
inst✝¹ : CommSemiring A
I : Ideal A
hI : DividedPowers I
dpow : ℕ → A → A
dpow_zero : ∀ {x : A}, x ∈ I → dpow 0 x = 1
dpow_eval_zero : ∀ {n : ℕ}, n ≠ 0 → dpow n 0 = 0
ι : Type u_2
inst✝ : DecidableEq ι
x : ι → A
dpow_add : ∀ (n : ℕ) {x y : A}, x ∈ I → y ∈ I → dpow n (x + y) = ∑ x_3 ∈ range n.succ, dpow x_3... | A : Type u_1
inst✝¹ : CommSemiring A
I : Ideal A
hI : DividedPowers I
dpow : ℕ → A → A
dpow_zero : ∀ {x : A}, x ∈ I → dpow 0 x = 1
dpow_eval_zero : ∀ {n : ℕ}, n ≠ 0 → dpow n 0 = 0
ι : Type u_2
inst✝ : DecidableEq ι
x : ι → A
dpow_add : ∀ (n : ℕ) {x y : A}, x ∈ I → y ∈ I → dpow n (x + y) = ∑ x_3 ∈ range n.succ, dpow x_3... | Please generate a tactic in lean4 to solve the state.
STATE:
A : Type u_1
inst✝¹ : CommSemiring A
I : Ideal A
hI : DividedPowers I
dpow : ℕ → A → A
dpow_zero : ∀ {x : A}, x ∈ I → dpow 0 x = 1
dpow_eval_zero : ∀ {n : ℕ}, n ≠ 0 → dpow n 0 = 0
ι : Type u_2
inst✝ : DecidableEq ι
x : ι → A
dpow_add : ∀ (n : ℕ) {x y : A}, x ... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/Basic.lean | DividedPowers.dpow_sum_aux | [284, 1] | [345, 27] | rw [Sym.mem_fill_iff] at hi | A : Type u_1
inst✝¹ : CommSemiring A
I : Ideal A
hI : DividedPowers I
dpow : ℕ → A → A
dpow_zero : ∀ {x : A}, x ∈ I → dpow 0 x = 1
dpow_eval_zero : ∀ {n : ℕ}, n ≠ 0 → dpow n 0 = 0
ι : Type u_2
inst✝ : DecidableEq ι
x : ι → A
dpow_add : ∀ (n : ℕ) {x y : A}, x ∈ I → y ∈ I → dpow n (x + y) = ∑ x_3 ∈ range n.succ, dpow x_3... | A : Type u_1
inst✝¹ : CommSemiring A
I : Ideal A
hI : DividedPowers I
dpow : ℕ → A → A
dpow_zero : ∀ {x : A}, x ∈ I → dpow 0 x = 1
dpow_eval_zero : ∀ {n : ℕ}, n ≠ 0 → dpow n 0 = 0
ι : Type u_2
inst✝ : DecidableEq ι
x : ι → A
dpow_add : ∀ (n : ℕ) {x y : A}, x ∈ I → y ∈ I → dpow n (x + y) = ∑ x_3 ∈ range n.succ, dpow x_3... | Please generate a tactic in lean4 to solve the state.
STATE:
A : Type u_1
inst✝¹ : CommSemiring A
I : Ideal A
hI : DividedPowers I
dpow : ℕ → A → A
dpow_zero : ∀ {x : A}, x ∈ I → dpow 0 x = 1
dpow_eval_zero : ∀ {n : ℕ}, n ≠ 0 → dpow n 0 = 0
ι : Type u_2
inst✝ : DecidableEq ι
x : ι → A
dpow_add : ∀ (n : ℕ) {x y : A}, x ... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/Basic.lean | DividedPowers.dpow_sum_aux | [284, 1] | [345, 27] | cases hi with
| inl hi =>
rw [hi.2]
exact mem_insert_self a s
| inr hi =>
simp only [mem_sigma, mem_univ, mem_sym_iff, true_and] at hm
exact mem_insert_of_mem (hm i hi) | A : Type u_1
inst✝¹ : CommSemiring A
I : Ideal A
hI : DividedPowers I
dpow : ℕ → A → A
dpow_zero : ∀ {x : A}, x ∈ I → dpow 0 x = 1
dpow_eval_zero : ∀ {n : ℕ}, n ≠ 0 → dpow n 0 = 0
ι : Type u_2
inst✝ : DecidableEq ι
x : ι → A
dpow_add : ∀ (n : ℕ) {x y : A}, x ∈ I → y ∈ I → dpow n (x + y) = ∑ x_3 ∈ range n.succ, dpow x_3... | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
A : Type u_1
inst✝¹ : CommSemiring A
I : Ideal A
hI : DividedPowers I
dpow : ℕ → A → A
dpow_zero : ∀ {x : A}, x ∈ I → dpow 0 x = 1
dpow_eval_zero : ∀ {n : ℕ}, n ≠ 0 → dpow n 0 = 0
ι : Type u_2
inst✝ : DecidableEq ι
x : ι → A
dpow_add : ∀ (n : ℕ) {x y : A}, x ... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/Basic.lean | DividedPowers.dpow_sum_aux | [284, 1] | [345, 27] | rw [hi.2] | case inl
A : Type u_1
inst✝¹ : CommSemiring A
I : Ideal A
hI : DividedPowers I
dpow : ℕ → A → A
dpow_zero : ∀ {x : A}, x ∈ I → dpow 0 x = 1
dpow_eval_zero : ∀ {n : ℕ}, n ≠ 0 → dpow n 0 = 0
ι : Type u_2
inst✝ : DecidableEq ι
x : ι → A
dpow_add : ∀ (n : ℕ) {x y : A}, x ∈ I → y ∈ I → dpow n (x + y) = ∑ x_3 ∈ range n.succ,... | case inl
A : Type u_1
inst✝¹ : CommSemiring A
I : Ideal A
hI : DividedPowers I
dpow : ℕ → A → A
dpow_zero : ∀ {x : A}, x ∈ I → dpow 0 x = 1
dpow_eval_zero : ∀ {n : ℕ}, n ≠ 0 → dpow n 0 = 0
ι : Type u_2
inst✝ : DecidableEq ι
x : ι → A
dpow_add : ∀ (n : ℕ) {x y : A}, x ∈ I → y ∈ I → dpow n (x + y) = ∑ x_3 ∈ range n.succ,... | Please generate a tactic in lean4 to solve the state.
STATE:
case inl
A : Type u_1
inst✝¹ : CommSemiring A
I : Ideal A
hI : DividedPowers I
dpow : ℕ → A → A
dpow_zero : ∀ {x : A}, x ∈ I → dpow 0 x = 1
dpow_eval_zero : ∀ {n : ℕ}, n ≠ 0 → dpow n 0 = 0
ι : Type u_2
inst✝ : DecidableEq ι
x : ι → A
dpow_add : ∀ (n : ℕ) {x y... |
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