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 | Submodule.mem_sup_iff_exists_add | [77, 1] | [83, 43] | simp only [Submodule.mem_toAddSubmonoid] | R : Type u_1
inst✝² : Semiring R
M : Type u_2
inst✝¹ : AddCommMonoid M
inst✝ : Module R M
M₁ M₂ : Submodule R M
m : M
⊢ (∃ y ∈ M₁.toAddSubmonoid, ∃ z ∈ M₂.toAddSubmonoid, y + z = m) ↔ ∃ m₁ ∈ M₁, ∃ m₂ ∈ M₂, m₁ + m₂ = m | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
R : Type u_1
inst✝² : Semiring R
M : Type u_2
inst✝¹ : AddCommMonoid M
inst✝ : Module R M
M₁ M₂ : Submodule R M
m : M
⊢ (∃ y ∈ M₁.toAddSubmonoid, ∃ z ∈ M₂.toAddSubmonoid, y + z = m) ↔ ∃ m₁ ∈ M₁, ∃ m₂ ∈ M₂, m₁ + m₂ = m
TACTIC:
|
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/DPAlgebra/Dpow.lean | DividedPowerAlgebra.on_dp_algebra_unique | [115, 1] | [122, 74] | apply DividedPowers.dp_uniqueness_self h' h (augIdeal_eq_span R M) | R : Type u
inst✝⁴ : CommRing R
inst✝³ : DecidableEq R
M : Type v
inst✝² : AddCommGroup M
inst✝¹ : DecidableEq M
inst✝ : Module R M
x : M
n : ℕ
h h' : DividedPowers (augIdeal R M)
h1 : ∀ (n : ℕ) (x : M), h.dpow n ((ι R M) x) = dp R n x
h1' : ∀ (n : ℕ) (x : M), h'.dpow n ((ι R M) x) = dp R n x
⊢ h = h' | R : Type u
inst✝⁴ : CommRing R
inst✝³ : DecidableEq R
M : Type v
inst✝² : AddCommGroup M
inst✝¹ : DecidableEq M
inst✝ : Module R M
x : M
n : ℕ
h h' : DividedPowers (augIdeal R M)
h1 : ∀ (n : ℕ) (x : M), h.dpow n ((ι R M) x) = dp R n x
h1' : ∀ (n : ℕ) (x : M), h'.dpow n ((ι R M) x) = dp R n x
⊢ ∀ (n : ℕ), ∀ a ∈ Set.imag... | Please generate a tactic in lean4 to solve the state.
STATE:
R : Type u
inst✝⁴ : CommRing R
inst✝³ : DecidableEq R
M : Type v
inst✝² : AddCommGroup M
inst✝¹ : DecidableEq M
inst✝ : Module R M
x : M
n : ℕ
h h' : DividedPowers (augIdeal R M)
h1 : ∀ (n : ℕ) (x : M), h.dpow n ((ι R M) x) = dp R n x
h1' : ∀ (n : ℕ) (x : M),... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/DPAlgebra/Dpow.lean | DividedPowerAlgebra.on_dp_algebra_unique | [115, 1] | [122, 74] | rintro n f ⟨q, hq : 0 < q, m, _, rfl⟩ | R : Type u
inst✝⁴ : CommRing R
inst✝³ : DecidableEq R
M : Type v
inst✝² : AddCommGroup M
inst✝¹ : DecidableEq M
inst✝ : Module R M
x : M
n : ℕ
h h' : DividedPowers (augIdeal R M)
h1 : ∀ (n : ℕ) (x : M), h.dpow n ((ι R M) x) = dp R n x
h1' : ∀ (n : ℕ) (x : M), h'.dpow n ((ι R M) x) = dp R n x
⊢ ∀ (n : ℕ), ∀ a ∈ Set.imag... | case intro.intro.intro.intro
R : Type u
inst✝⁴ : CommRing R
inst✝³ : DecidableEq R
M : Type v
inst✝² : AddCommGroup M
inst✝¹ : DecidableEq M
inst✝ : Module R M
x : M
n✝ : ℕ
h h' : DividedPowers (augIdeal R M)
h1 : ∀ (n : ℕ) (x : M), h.dpow n ((ι R M) x) = dp R n x
h1' : ∀ (n : ℕ) (x : M), h'.dpow n ((ι R M) x) = dp R n... | Please generate a tactic in lean4 to solve the state.
STATE:
R : Type u
inst✝⁴ : CommRing R
inst✝³ : DecidableEq R
M : Type v
inst✝² : AddCommGroup M
inst✝¹ : DecidableEq M
inst✝ : Module R M
x : M
n : ℕ
h h' : DividedPowers (augIdeal R M)
h1 : ∀ (n : ℕ) (x : M), h.dpow n ((ι R M) x) = dp R n x
h1' : ∀ (n : ℕ) (x : M),... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/DPAlgebra/Dpow.lean | DividedPowerAlgebra.on_dp_algebra_unique | [115, 1] | [122, 74] | nth_rw 1 [← h1' q m] | case intro.intro.intro.intro
R : Type u
inst✝⁴ : CommRing R
inst✝³ : DecidableEq R
M : Type v
inst✝² : AddCommGroup M
inst✝¹ : DecidableEq M
inst✝ : Module R M
x : M
n✝ : ℕ
h h' : DividedPowers (augIdeal R M)
h1 : ∀ (n : ℕ) (x : M), h.dpow n ((ι R M) x) = dp R n x
h1' : ∀ (n : ℕ) (x : M), h'.dpow n ((ι R M) x) = dp R n... | case intro.intro.intro.intro
R : Type u
inst✝⁴ : CommRing R
inst✝³ : DecidableEq R
M : Type v
inst✝² : AddCommGroup M
inst✝¹ : DecidableEq M
inst✝ : Module R M
x : M
n✝ : ℕ
h h' : DividedPowers (augIdeal R M)
h1 : ∀ (n : ℕ) (x : M), h.dpow n ((ι R M) x) = dp R n x
h1' : ∀ (n : ℕ) (x : M), h'.dpow n ((ι R M) x) = dp R n... | Please generate a tactic in lean4 to solve the state.
STATE:
case intro.intro.intro.intro
R : Type u
inst✝⁴ : CommRing R
inst✝³ : DecidableEq R
M : Type v
inst✝² : AddCommGroup M
inst✝¹ : DecidableEq M
inst✝ : Module R M
x : M
n✝ : ℕ
h h' : DividedPowers (augIdeal R M)
h1 : ∀ (n : ℕ) (x : M), h.dpow n ((ι R M) x) = dp ... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/DPAlgebra/Dpow.lean | DividedPowerAlgebra.on_dp_algebra_unique | [115, 1] | [122, 74] | rw [← h1 q m, h.dpow_comp n (ne_of_gt hq) (ι_mem_augIdeal R M m),
h'.dpow_comp n (ne_of_gt hq) (ι_mem_augIdeal R M m), h1 _ m, h1' _ m] | case intro.intro.intro.intro
R : Type u
inst✝⁴ : CommRing R
inst✝³ : DecidableEq R
M : Type v
inst✝² : AddCommGroup M
inst✝¹ : DecidableEq M
inst✝ : Module R M
x : M
n✝ : ℕ
h h' : DividedPowers (augIdeal R M)
h1 : ∀ (n : ℕ) (x : M), h.dpow n ((ι R M) x) = dp R n x
h1' : ∀ (n : ℕ) (x : M), h'.dpow n ((ι R M) x) = dp R n... | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case intro.intro.intro.intro
R : Type u
inst✝⁴ : CommRing R
inst✝³ : DecidableEq R
M : Type v
inst✝² : AddCommGroup M
inst✝¹ : DecidableEq M
inst✝ : Module R M
x : M
n✝ : ℕ
h h' : DividedPowers (augIdeal R M)
h1 : ∀ (n : ℕ) (x : M), h.dpow n ((ι R M) x) = dp ... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/DPAlgebra/Dpow.lean | DividedPowerAlgebra.on_tensorProduct_unique | [177, 1] | [190, 89] | apply eq_of_eq_on_ideal | A R S : Type u
inst✝⁴ : CommRing A
inst✝³ : CommRing R
inst✝² : Algebra A R
inst✝¹ : CommRing S
inst✝ : Algebra A S
I : Ideal R
J : Ideal S
hI : DividedPowers I
hJ : DividedPowers J
hK hK' : DividedPowers (K A I J)
hIK : hI.isDPMorphism hK ↑(i1 A R S)
hIK' : hI.isDPMorphism hK' ↑(i1 A R S)
hJK : hJ.isDPMorphism hK ↑(i2... | case h_eq
A R S : Type u
inst✝⁴ : CommRing A
inst✝³ : CommRing R
inst✝² : Algebra A R
inst✝¹ : CommRing S
inst✝ : Algebra A S
I : Ideal R
J : Ideal S
hI : DividedPowers I
hJ : DividedPowers J
hK hK' : DividedPowers (K A I J)
hIK : hI.isDPMorphism hK ↑(i1 A R S)
hIK' : hI.isDPMorphism hK' ↑(i1 A R S)
hJK : hJ.isDPMorphi... | Please generate a tactic in lean4 to solve the state.
STATE:
A R S : Type u
inst✝⁴ : CommRing A
inst✝³ : CommRing R
inst✝² : Algebra A R
inst✝¹ : CommRing S
inst✝ : Algebra A S
I : Ideal R
J : Ideal S
hI : DividedPowers I
hJ : DividedPowers J
hK hK' : DividedPowers (K A I J)
hIK : hI.isDPMorphism hK ↑(i1 A R S)
hIK' : ... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/DPAlgebra/Dpow.lean | DividedPowerAlgebra.on_tensorProduct_unique | [177, 1] | [190, 89] | intro n x hx | case h_eq
A R S : Type u
inst✝⁴ : CommRing A
inst✝³ : CommRing R
inst✝² : Algebra A R
inst✝¹ : CommRing S
inst✝ : Algebra A S
I : Ideal R
J : Ideal S
hI : DividedPowers I
hJ : DividedPowers J
hK hK' : DividedPowers (K A I J)
hIK : hI.isDPMorphism hK ↑(i1 A R S)
hIK' : hI.isDPMorphism hK' ↑(i1 A R S)
hJK : hJ.isDPMorphi... | case h_eq
A R S : Type u
inst✝⁴ : CommRing A
inst✝³ : CommRing R
inst✝² : Algebra A R
inst✝¹ : CommRing S
inst✝ : Algebra A S
I : Ideal R
J : Ideal S
hI : DividedPowers I
hJ : DividedPowers J
hK hK' : DividedPowers (K A I J)
hIK : hI.isDPMorphism hK ↑(i1 A R S)
hIK' : hI.isDPMorphism hK' ↑(i1 A R S)
hJK : hJ.isDPMorphi... | Please generate a tactic in lean4 to solve the state.
STATE:
case h_eq
A R S : Type u
inst✝⁴ : CommRing A
inst✝³ : CommRing R
inst✝² : Algebra A R
inst✝¹ : CommRing S
inst✝ : Algebra A S
I : Ideal R
J : Ideal S
hI : DividedPowers I
hJ : DividedPowers J
hK hK' : DividedPowers (K A I J)
hIK : hI.isDPMorphism hK ↑(i1 A R ... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/DPAlgebra/Dpow.lean | DividedPowerAlgebra.on_tensorProduct_unique | [177, 1] | [190, 89] | suffices x ∈ dpEqualizer hK hK' by exact ((mem_dpEqualizer_iff _ _).mp this).2 n | case h_eq
A R S : Type u
inst✝⁴ : CommRing A
inst✝³ : CommRing R
inst✝² : Algebra A R
inst✝¹ : CommRing S
inst✝ : Algebra A S
I : Ideal R
J : Ideal S
hI : DividedPowers I
hJ : DividedPowers J
hK hK' : DividedPowers (K A I J)
hIK : hI.isDPMorphism hK ↑(i1 A R S)
hIK' : hI.isDPMorphism hK' ↑(i1 A R S)
hJK : hJ.isDPMorphi... | case h_eq
A R S : Type u
inst✝⁴ : CommRing A
inst✝³ : CommRing R
inst✝² : Algebra A R
inst✝¹ : CommRing S
inst✝ : Algebra A S
I : Ideal R
J : Ideal S
hI : DividedPowers I
hJ : DividedPowers J
hK hK' : DividedPowers (K A I J)
hIK : hI.isDPMorphism hK ↑(i1 A R S)
hIK' : hI.isDPMorphism hK' ↑(i1 A R S)
hJK : hJ.isDPMorphi... | Please generate a tactic in lean4 to solve the state.
STATE:
case h_eq
A R S : Type u
inst✝⁴ : CommRing A
inst✝³ : CommRing R
inst✝² : Algebra A R
inst✝¹ : CommRing S
inst✝ : Algebra A S
I : Ideal R
J : Ideal S
hI : DividedPowers I
hJ : DividedPowers J
hK hK' : DividedPowers (K A I J)
hIK : hI.isDPMorphism hK ↑(i1 A R ... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/DPAlgebra/Dpow.lean | DividedPowerAlgebra.on_tensorProduct_unique | [177, 1] | [190, 89] | suffices h_ss : K A I J ≤ dpEqualizer hK hK' by
exact h_ss hx | case h_eq
A R S : Type u
inst✝⁴ : CommRing A
inst✝³ : CommRing R
inst✝² : Algebra A R
inst✝¹ : CommRing S
inst✝ : Algebra A S
I : Ideal R
J : Ideal S
hI : DividedPowers I
hJ : DividedPowers J
hK hK' : DividedPowers (K A I J)
hIK : hI.isDPMorphism hK ↑(i1 A R S)
hIK' : hI.isDPMorphism hK' ↑(i1 A R S)
hJK : hJ.isDPMorphi... | case h_eq
A R S : Type u
inst✝⁴ : CommRing A
inst✝³ : CommRing R
inst✝² : Algebra A R
inst✝¹ : CommRing S
inst✝ : Algebra A S
I : Ideal R
J : Ideal S
hI : DividedPowers I
hJ : DividedPowers J
hK hK' : DividedPowers (K A I J)
hIK : hI.isDPMorphism hK ↑(i1 A R S)
hIK' : hI.isDPMorphism hK' ↑(i1 A R S)
hJK : hJ.isDPMorphi... | Please generate a tactic in lean4 to solve the state.
STATE:
case h_eq
A R S : Type u
inst✝⁴ : CommRing A
inst✝³ : CommRing R
inst✝² : Algebra A R
inst✝¹ : CommRing S
inst✝ : Algebra A S
I : Ideal R
J : Ideal S
hI : DividedPowers I
hJ : DividedPowers J
hK hK' : DividedPowers (K A I J)
hIK : hI.isDPMorphism hK ↑(i1 A R ... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/DPAlgebra/Dpow.lean | DividedPowerAlgebra.on_tensorProduct_unique | [177, 1] | [190, 89] | dsimp only [K] | case h_eq
A R S : Type u
inst✝⁴ : CommRing A
inst✝³ : CommRing R
inst✝² : Algebra A R
inst✝¹ : CommRing S
inst✝ : Algebra A S
I : Ideal R
J : Ideal S
hI : DividedPowers I
hJ : DividedPowers J
hK hK' : DividedPowers (K A I J)
hIK : hI.isDPMorphism hK ↑(i1 A R S)
hIK' : hI.isDPMorphism hK' ↑(i1 A R S)
hJK : hJ.isDPMorphi... | case h_eq
A R S : Type u
inst✝⁴ : CommRing A
inst✝³ : CommRing R
inst✝² : Algebra A R
inst✝¹ : CommRing S
inst✝ : Algebra A S
I : Ideal R
J : Ideal S
hI : DividedPowers I
hJ : DividedPowers J
hK hK' : DividedPowers (K A I J)
hIK : hI.isDPMorphism hK ↑(i1 A R S)
hIK' : hI.isDPMorphism hK' ↑(i1 A R S)
hJK : hJ.isDPMorphi... | Please generate a tactic in lean4 to solve the state.
STATE:
case h_eq
A R S : Type u
inst✝⁴ : CommRing A
inst✝³ : CommRing R
inst✝² : Algebra A R
inst✝¹ : CommRing S
inst✝ : Algebra A S
I : Ideal R
J : Ideal S
hI : DividedPowers I
hJ : DividedPowers J
hK hK' : DividedPowers (K A I J)
hIK : hI.isDPMorphism hK ↑(i1 A R ... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/DPAlgebra/Dpow.lean | DividedPowerAlgebra.on_tensorProduct_unique | [177, 1] | [190, 89] | rw [sup_le_iff] | case h_eq
A R S : Type u
inst✝⁴ : CommRing A
inst✝³ : CommRing R
inst✝² : Algebra A R
inst✝¹ : CommRing S
inst✝ : Algebra A S
I : Ideal R
J : Ideal S
hI : DividedPowers I
hJ : DividedPowers J
hK hK' : DividedPowers (K A I J)
hIK : hI.isDPMorphism hK ↑(i1 A R S)
hIK' : hI.isDPMorphism hK' ↑(i1 A R S)
hJK : hJ.isDPMorphi... | case h_eq
A R S : Type u
inst✝⁴ : CommRing A
inst✝³ : CommRing R
inst✝² : Algebra A R
inst✝¹ : CommRing S
inst✝ : Algebra A S
I : Ideal R
J : Ideal S
hI : DividedPowers I
hJ : DividedPowers J
hK hK' : DividedPowers (K A I J)
hIK : hI.isDPMorphism hK ↑(i1 A R S)
hIK' : hI.isDPMorphism hK' ↑(i1 A R S)
hJK : hJ.isDPMorphi... | Please generate a tactic in lean4 to solve the state.
STATE:
case h_eq
A R S : Type u
inst✝⁴ : CommRing A
inst✝³ : CommRing R
inst✝² : Algebra A R
inst✝¹ : CommRing S
inst✝ : Algebra A S
I : Ideal R
J : Ideal S
hI : DividedPowers I
hJ : DividedPowers J
hK hK' : DividedPowers (K A I J)
hIK : hI.isDPMorphism hK ↑(i1 A R ... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/DPAlgebra/Dpow.lean | DividedPowerAlgebra.on_tensorProduct_unique | [177, 1] | [190, 89] | constructor | case h_eq
A R S : Type u
inst✝⁴ : CommRing A
inst✝³ : CommRing R
inst✝² : Algebra A R
inst✝¹ : CommRing S
inst✝ : Algebra A S
I : Ideal R
J : Ideal S
hI : DividedPowers I
hJ : DividedPowers J
hK hK' : DividedPowers (K A I J)
hIK : hI.isDPMorphism hK ↑(i1 A R S)
hIK' : hI.isDPMorphism hK' ↑(i1 A R S)
hJK : hJ.isDPMorphi... | case h_eq.left
A R S : Type u
inst✝⁴ : CommRing A
inst✝³ : CommRing R
inst✝² : Algebra A R
inst✝¹ : CommRing S
inst✝ : Algebra A S
I : Ideal R
J : Ideal S
hI : DividedPowers I
hJ : DividedPowers J
hK hK' : DividedPowers (K A I J)
hIK : hI.isDPMorphism hK ↑(i1 A R S)
hIK' : hI.isDPMorphism hK' ↑(i1 A R S)
hJK : hJ.isDPM... | Please generate a tactic in lean4 to solve the state.
STATE:
case h_eq
A R S : Type u
inst✝⁴ : CommRing A
inst✝³ : CommRing R
inst✝² : Algebra A R
inst✝¹ : CommRing S
inst✝ : Algebra A S
I : Ideal R
J : Ideal S
hI : DividedPowers I
hJ : DividedPowers J
hK hK' : DividedPowers (K A I J)
hIK : hI.isDPMorphism hK ↑(i1 A R ... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/DPAlgebra/Dpow.lean | DividedPowerAlgebra.on_tensorProduct_unique | [177, 1] | [190, 89] | apply le_equalizer_of_dp_morphism hI (i1 A R S).toRingHom le_sup_left hK hK' hIK hIK' | case h_eq.left
A R S : Type u
inst✝⁴ : CommRing A
inst✝³ : CommRing R
inst✝² : Algebra A R
inst✝¹ : CommRing S
inst✝ : Algebra A S
I : Ideal R
J : Ideal S
hI : DividedPowers I
hJ : DividedPowers J
hK hK' : DividedPowers (K A I J)
hIK : hI.isDPMorphism hK ↑(i1 A R S)
hIK' : hI.isDPMorphism hK' ↑(i1 A R S)
hJK : hJ.isDPM... | case h_eq.right
A R S : Type u
inst✝⁴ : CommRing A
inst✝³ : CommRing R
inst✝² : Algebra A R
inst✝¹ : CommRing S
inst✝ : Algebra A S
I : Ideal R
J : Ideal S
hI : DividedPowers I
hJ : DividedPowers J
hK hK' : DividedPowers (K A I J)
hIK : hI.isDPMorphism hK ↑(i1 A R S)
hIK' : hI.isDPMorphism hK' ↑(i1 A R S)
hJK : hJ.isDP... | Please generate a tactic in lean4 to solve the state.
STATE:
case h_eq.left
A R S : Type u
inst✝⁴ : CommRing A
inst✝³ : CommRing R
inst✝² : Algebra A R
inst✝¹ : CommRing S
inst✝ : Algebra A S
I : Ideal R
J : Ideal S
hI : DividedPowers I
hJ : DividedPowers J
hK hK' : DividedPowers (K A I J)
hIK : hI.isDPMorphism hK ↑(i1... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/DPAlgebra/Dpow.lean | DividedPowerAlgebra.on_tensorProduct_unique | [177, 1] | [190, 89] | apply le_equalizer_of_dp_morphism hJ (i2 A R S).toRingHom le_sup_right hK hK' hJK hJK' | case h_eq.right
A R S : Type u
inst✝⁴ : CommRing A
inst✝³ : CommRing R
inst✝² : Algebra A R
inst✝¹ : CommRing S
inst✝ : Algebra A S
I : Ideal R
J : Ideal S
hI : DividedPowers I
hJ : DividedPowers J
hK hK' : DividedPowers (K A I J)
hIK : hI.isDPMorphism hK ↑(i1 A R S)
hIK' : hI.isDPMorphism hK' ↑(i1 A R S)
hJK : hJ.isDP... | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case h_eq.right
A R S : Type u
inst✝⁴ : CommRing A
inst✝³ : CommRing R
inst✝² : Algebra A R
inst✝¹ : CommRing S
inst✝ : Algebra A S
I : Ideal R
J : Ideal S
hI : DividedPowers I
hJ : DividedPowers J
hK hK' : DividedPowers (K A I J)
hIK : hI.isDPMorphism hK ↑(i... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/DPAlgebra/Dpow.lean | DividedPowerAlgebra.on_tensorProduct_unique | [177, 1] | [190, 89] | exact ((mem_dpEqualizer_iff _ _).mp this).2 n | A R S : Type u
inst✝⁴ : CommRing A
inst✝³ : CommRing R
inst✝² : Algebra A R
inst✝¹ : CommRing S
inst✝ : Algebra A S
I : Ideal R
J : Ideal S
hI : DividedPowers I
hJ : DividedPowers J
hK hK' : DividedPowers (K A I J)
hIK : hI.isDPMorphism hK ↑(i1 A R S)
hIK' : hI.isDPMorphism hK' ↑(i1 A R S)
hJK : hJ.isDPMorphism hK ↑(i2... | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
A R S : Type u
inst✝⁴ : CommRing A
inst✝³ : CommRing R
inst✝² : Algebra A R
inst✝¹ : CommRing S
inst✝ : Algebra A S
I : Ideal R
J : Ideal S
hI : DividedPowers I
hJ : DividedPowers J
hK hK' : DividedPowers (K A I J)
hIK : hI.isDPMorphism hK ↑(i1 A R S)
hIK' : ... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/DPAlgebra/Dpow.lean | DividedPowerAlgebra.on_tensorProduct_unique | [177, 1] | [190, 89] | exact h_ss hx | A R S : Type u
inst✝⁴ : CommRing A
inst✝³ : CommRing R
inst✝² : Algebra A R
inst✝¹ : CommRing S
inst✝ : Algebra A S
I : Ideal R
J : Ideal S
hI : DividedPowers I
hJ : DividedPowers J
hK hK' : DividedPowers (K A I J)
hIK : hI.isDPMorphism hK ↑(i1 A R S)
hIK' : hI.isDPMorphism hK' ↑(i1 A R S)
hJK : hJ.isDPMorphism hK ↑(i2... | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
A R S : Type u
inst✝⁴ : CommRing A
inst✝³ : CommRing R
inst✝² : Algebra A R
inst✝¹ : CommRing S
inst✝ : Algebra A S
I : Ideal R
J : Ideal S
hI : DividedPowers I
hJ : DividedPowers J
hK hK' : DividedPowers (K A I J)
hIK : hI.isDPMorphism hK ↑(i1 A R S)
hIK' : ... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/DPAlgebra/Dpow.lean | DividedPowerAlgebra.cond_D_uniqueness | [264, 1] | [293, 40] | constructor | R : Type u
inst✝⁵ : CommRing R
inst✝⁴ : DecidableEq R
M : Type v
inst✝³ : AddCommGroup M
inst✝² : Module R M
h : DividedPowers (augIdeal R M)
hh : ∀ (n : ℕ) (x : M), h.dpow n ((ι R M) x) = dp R n x
S : Type u_1
inst✝¹ : CommRing S
inst✝ : Algebra R S
J : Ideal S
hJ : DividedPowers J
f : M →ₗ[R] S
hf : ∀ (m : M), f m ∈ ... | case left
R : Type u
inst✝⁵ : CommRing R
inst✝⁴ : DecidableEq R
M : Type v
inst✝³ : AddCommGroup M
inst✝² : Module R M
h : DividedPowers (augIdeal R M)
hh : ∀ (n : ℕ) (x : M), h.dpow n ((ι R M) x) = dp R n x
S : Type u_1
inst✝¹ : CommRing S
inst✝ : Algebra R S
J : Ideal S
hJ : DividedPowers J
f : M →ₗ[R] S
hf : ∀ (m : ... | Please generate a tactic in lean4 to solve the state.
STATE:
R : Type u
inst✝⁵ : CommRing R
inst✝⁴ : DecidableEq R
M : Type v
inst✝³ : AddCommGroup M
inst✝² : Module R M
h : DividedPowers (augIdeal R M)
hh : ∀ (n : ℕ) (x : M), h.dpow n ((ι R M) x) = dp R n x
S : Type u_1
inst✝¹ : CommRing S
inst✝ : Algebra R S
J : Idea... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/DPAlgebra/Dpow.lean | DividedPowerAlgebra.cond_D_uniqueness | [264, 1] | [293, 40] | rw [augIdeal_eq_span] | case left
R : Type u
inst✝⁵ : CommRing R
inst✝⁴ : DecidableEq R
M : Type v
inst✝³ : AddCommGroup M
inst✝² : Module R M
h : DividedPowers (augIdeal R M)
hh : ∀ (n : ℕ) (x : M), h.dpow n ((ι R M) x) = dp R n x
S : Type u_1
inst✝¹ : CommRing S
inst✝ : Algebra R S
J : Ideal S
hJ : DividedPowers J
f : M →ₗ[R] S
hf : ∀ (m : ... | case left
R : Type u
inst✝⁵ : CommRing R
inst✝⁴ : DecidableEq R
M : Type v
inst✝³ : AddCommGroup M
inst✝² : Module R M
h : DividedPowers (augIdeal R M)
hh : ∀ (n : ℕ) (x : M), h.dpow n ((ι R M) x) = dp R n x
S : Type u_1
inst✝¹ : CommRing S
inst✝ : Algebra R S
J : Ideal S
hJ : DividedPowers J
f : M →ₗ[R] S
hf : ∀ (m : ... | Please generate a tactic in lean4 to solve the state.
STATE:
case left
R : Type u
inst✝⁵ : CommRing R
inst✝⁴ : DecidableEq R
M : Type v
inst✝³ : AddCommGroup M
inst✝² : Module R M
h : DividedPowers (augIdeal R M)
hh : ∀ (n : ℕ) (x : M), h.dpow n ((ι R M) x) = dp R n x
S : Type u_1
inst✝¹ : CommRing S
inst✝ : Algebra R ... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/DPAlgebra/Dpow.lean | DividedPowerAlgebra.cond_D_uniqueness | [264, 1] | [293, 40] | rw [Ideal.map_span] | case left
R : Type u
inst✝⁵ : CommRing R
inst✝⁴ : DecidableEq R
M : Type v
inst✝³ : AddCommGroup M
inst✝² : Module R M
h : DividedPowers (augIdeal R M)
hh : ∀ (n : ℕ) (x : M), h.dpow n ((ι R M) x) = dp R n x
S : Type u_1
inst✝¹ : CommRing S
inst✝ : Algebra R S
J : Ideal S
hJ : DividedPowers J
f : M →ₗ[R] S
hf : ∀ (m : ... | case left
R : Type u
inst✝⁵ : CommRing R
inst✝⁴ : DecidableEq R
M : Type v
inst✝³ : AddCommGroup M
inst✝² : Module R M
h : DividedPowers (augIdeal R M)
hh : ∀ (n : ℕ) (x : M), h.dpow n ((ι R M) x) = dp R n x
S : Type u_1
inst✝¹ : CommRing S
inst✝ : Algebra R S
J : Ideal S
hJ : DividedPowers J
f : M →ₗ[R] S
hf : ∀ (m : ... | Please generate a tactic in lean4 to solve the state.
STATE:
case left
R : Type u
inst✝⁵ : CommRing R
inst✝⁴ : DecidableEq R
M : Type v
inst✝³ : AddCommGroup M
inst✝² : Module R M
h : DividedPowers (augIdeal R M)
hh : ∀ (n : ℕ) (x : M), h.dpow n ((ι R M) x) = dp R n x
S : Type u_1
inst✝¹ : CommRing S
inst✝ : Algebra R ... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/DPAlgebra/Dpow.lean | DividedPowerAlgebra.cond_D_uniqueness | [264, 1] | [293, 40] | rw [Ideal.span_le] | case left
R : Type u
inst✝⁵ : CommRing R
inst✝⁴ : DecidableEq R
M : Type v
inst✝³ : AddCommGroup M
inst✝² : Module R M
h : DividedPowers (augIdeal R M)
hh : ∀ (n : ℕ) (x : M), h.dpow n ((ι R M) x) = dp R n x
S : Type u_1
inst✝¹ : CommRing S
inst✝ : Algebra R S
J : Ideal S
hJ : DividedPowers J
f : M →ₗ[R] S
hf : ∀ (m : ... | case left
R : Type u
inst✝⁵ : CommRing R
inst✝⁴ : DecidableEq R
M : Type v
inst✝³ : AddCommGroup M
inst✝² : Module R M
h : DividedPowers (augIdeal R M)
hh : ∀ (n : ℕ) (x : M), h.dpow n ((ι R M) x) = dp R n x
S : Type u_1
inst✝¹ : CommRing S
inst✝ : Algebra R S
J : Ideal S
hJ : DividedPowers J
f : M →ₗ[R] S
hf : ∀ (m : ... | Please generate a tactic in lean4 to solve the state.
STATE:
case left
R : Type u
inst✝⁵ : CommRing R
inst✝⁴ : DecidableEq R
M : Type v
inst✝³ : AddCommGroup M
inst✝² : Module R M
h : DividedPowers (augIdeal R M)
hh : ∀ (n : ℕ) (x : M), h.dpow n ((ι R M) x) = dp R n x
S : Type u_1
inst✝¹ : CommRing S
inst✝ : Algebra R ... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/DPAlgebra/Dpow.lean | DividedPowerAlgebra.cond_D_uniqueness | [264, 1] | [293, 40] | intro s | case left
R : Type u
inst✝⁵ : CommRing R
inst✝⁴ : DecidableEq R
M : Type v
inst✝³ : AddCommGroup M
inst✝² : Module R M
h : DividedPowers (augIdeal R M)
hh : ∀ (n : ℕ) (x : M), h.dpow n ((ι R M) x) = dp R n x
S : Type u_1
inst✝¹ : CommRing S
inst✝ : Algebra R S
J : Ideal S
hJ : DividedPowers J
f : M →ₗ[R] S
hf : ∀ (m : ... | case left
R : Type u
inst✝⁵ : CommRing R
inst✝⁴ : DecidableEq R
M : Type v
inst✝³ : AddCommGroup M
inst✝² : Module R M
h : DividedPowers (augIdeal R M)
hh : ∀ (n : ℕ) (x : M), h.dpow n ((ι R M) x) = dp R n x
S : Type u_1
inst✝¹ : CommRing S
inst✝ : Algebra R S
J : Ideal S
hJ : DividedPowers J
f : M →ₗ[R] S
hf : ∀ (m : ... | Please generate a tactic in lean4 to solve the state.
STATE:
case left
R : Type u
inst✝⁵ : CommRing R
inst✝⁴ : DecidableEq R
M : Type v
inst✝³ : AddCommGroup M
inst✝² : Module R M
h : DividedPowers (augIdeal R M)
hh : ∀ (n : ℕ) (x : M), h.dpow n ((ι R M) x) = dp R n x
S : Type u_1
inst✝¹ : CommRing S
inst✝ : Algebra R ... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/DPAlgebra/Dpow.lean | DividedPowerAlgebra.cond_D_uniqueness | [264, 1] | [293, 40] | rintro ⟨a, ⟨n, hn : 0 < n, m, _, rfl⟩, rfl⟩ | case left
R : Type u
inst✝⁵ : CommRing R
inst✝⁴ : DecidableEq R
M : Type v
inst✝³ : AddCommGroup M
inst✝² : Module R M
h : DividedPowers (augIdeal R M)
hh : ∀ (n : ℕ) (x : M), h.dpow n ((ι R M) x) = dp R n x
S : Type u_1
inst✝¹ : CommRing S
inst✝ : Algebra R S
J : Ideal S
hJ : DividedPowers J
f : M →ₗ[R] S
hf : ∀ (m : ... | case left.intro.intro.intro.intro.intro.intro
R : Type u
inst✝⁵ : CommRing R
inst✝⁴ : DecidableEq R
M : Type v
inst✝³ : AddCommGroup M
inst✝² : Module R M
h : DividedPowers (augIdeal R M)
hh : ∀ (n : ℕ) (x : M), h.dpow n ((ι R M) x) = dp R n x
S : Type u_1
inst✝¹ : CommRing S
inst✝ : Algebra R S
J : Ideal S
hJ : Divide... | Please generate a tactic in lean4 to solve the state.
STATE:
case left
R : Type u
inst✝⁵ : CommRing R
inst✝⁴ : DecidableEq R
M : Type v
inst✝³ : AddCommGroup M
inst✝² : Module R M
h : DividedPowers (augIdeal R M)
hh : ∀ (n : ℕ) (x : M), h.dpow n ((ι R M) x) = dp R n x
S : Type u_1
inst✝¹ : CommRing S
inst✝ : Algebra R ... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/DPAlgebra/Dpow.lean | DividedPowerAlgebra.cond_D_uniqueness | [264, 1] | [293, 40] | simp only [AlgHom.coe_toRingHom, SetLike.mem_coe] | case left.intro.intro.intro.intro.intro.intro
R : Type u
inst✝⁵ : CommRing R
inst✝⁴ : DecidableEq R
M : Type v
inst✝³ : AddCommGroup M
inst✝² : Module R M
h : DividedPowers (augIdeal R M)
hh : ∀ (n : ℕ) (x : M), h.dpow n ((ι R M) x) = dp R n x
S : Type u_1
inst✝¹ : CommRing S
inst✝ : Algebra R S
J : Ideal S
hJ : Divide... | case left.intro.intro.intro.intro.intro.intro
R : Type u
inst✝⁵ : CommRing R
inst✝⁴ : DecidableEq R
M : Type v
inst✝³ : AddCommGroup M
inst✝² : Module R M
h : DividedPowers (augIdeal R M)
hh : ∀ (n : ℕ) (x : M), h.dpow n ((ι R M) x) = dp R n x
S : Type u_1
inst✝¹ : CommRing S
inst✝ : Algebra R S
J : Ideal S
hJ : Divide... | Please generate a tactic in lean4 to solve the state.
STATE:
case left.intro.intro.intro.intro.intro.intro
R : Type u
inst✝⁵ : CommRing R
inst✝⁴ : DecidableEq R
M : Type v
inst✝³ : AddCommGroup M
inst✝² : Module R M
h : DividedPowers (augIdeal R M)
hh : ∀ (n : ℕ) (x : M), h.dpow n ((ι R M) x) = dp R n x
S : Type u_1
in... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/DPAlgebra/Dpow.lean | DividedPowerAlgebra.cond_D_uniqueness | [264, 1] | [293, 40] | rw [liftAlgHom_apply_dp] | case left.intro.intro.intro.intro.intro.intro
R : Type u
inst✝⁵ : CommRing R
inst✝⁴ : DecidableEq R
M : Type v
inst✝³ : AddCommGroup M
inst✝² : Module R M
h : DividedPowers (augIdeal R M)
hh : ∀ (n : ℕ) (x : M), h.dpow n ((ι R M) x) = dp R n x
S : Type u_1
inst✝¹ : CommRing S
inst✝ : Algebra R S
J : Ideal S
hJ : Divide... | case left.intro.intro.intro.intro.intro.intro
R : Type u
inst✝⁵ : CommRing R
inst✝⁴ : DecidableEq R
M : Type v
inst✝³ : AddCommGroup M
inst✝² : Module R M
h : DividedPowers (augIdeal R M)
hh : ∀ (n : ℕ) (x : M), h.dpow n ((ι R M) x) = dp R n x
S : Type u_1
inst✝¹ : CommRing S
inst✝ : Algebra R S
J : Ideal S
hJ : Divide... | Please generate a tactic in lean4 to solve the state.
STATE:
case left.intro.intro.intro.intro.intro.intro
R : Type u
inst✝⁵ : CommRing R
inst✝⁴ : DecidableEq R
M : Type v
inst✝³ : AddCommGroup M
inst✝² : Module R M
h : DividedPowers (augIdeal R M)
hh : ∀ (n : ℕ) (x : M), h.dpow n ((ι R M) x) = dp R n x
S : Type u_1
in... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/DPAlgebra/Dpow.lean | DividedPowerAlgebra.cond_D_uniqueness | [264, 1] | [293, 40] | apply hJ.dpow_mem (ne_of_gt hn) (hf m) | case left.intro.intro.intro.intro.intro.intro
R : Type u
inst✝⁵ : CommRing R
inst✝⁴ : DecidableEq R
M : Type v
inst✝³ : AddCommGroup M
inst✝² : Module R M
h : DividedPowers (augIdeal R M)
hh : ∀ (n : ℕ) (x : M), h.dpow n ((ι R M) x) = dp R n x
S : Type u_1
inst✝¹ : CommRing S
inst✝ : Algebra R S
J : Ideal S
hJ : Divide... | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case left.intro.intro.intro.intro.intro.intro
R : Type u
inst✝⁵ : CommRing R
inst✝⁴ : DecidableEq R
M : Type v
inst✝³ : AddCommGroup M
inst✝² : Module R M
h : DividedPowers (augIdeal R M)
hh : ∀ (n : ℕ) (x : M), h.dpow n ((ι R M) x) = dp R n x
S : Type u_1
in... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/DPAlgebra/Dpow.lean | DividedPowerAlgebra.cond_D_uniqueness | [264, 1] | [293, 40] | intro n a ha | case right
R : Type u
inst✝⁵ : CommRing R
inst✝⁴ : DecidableEq R
M : Type v
inst✝³ : AddCommGroup M
inst✝² : Module R M
h : DividedPowers (augIdeal R M)
hh : ∀ (n : ℕ) (x : M), h.dpow n ((ι R M) x) = dp R n x
S : Type u_1
inst✝¹ : CommRing S
inst✝ : Algebra R S
J : Ideal S
hJ : DividedPowers J
f : M →ₗ[R] S
hf : ∀ (m :... | case right
R : Type u
inst✝⁵ : CommRing R
inst✝⁴ : DecidableEq R
M : Type v
inst✝³ : AddCommGroup M
inst✝² : Module R M
h : DividedPowers (augIdeal R M)
hh : ∀ (n : ℕ) (x : M), h.dpow n ((ι R M) x) = dp R n x
S : Type u_1
inst✝¹ : CommRing S
inst✝ : Algebra R S
J : Ideal S
hJ : DividedPowers J
f : M →ₗ[R] S
hf : ∀ (m :... | Please generate a tactic in lean4 to solve the state.
STATE:
case right
R : Type u
inst✝⁵ : CommRing R
inst✝⁴ : DecidableEq R
M : Type v
inst✝³ : AddCommGroup M
inst✝² : Module R M
h : DividedPowers (augIdeal R M)
hh : ∀ (n : ℕ) (x : M), h.dpow n ((ι R M) x) = dp R n x
S : Type u_1
inst✝¹ : CommRing S
inst✝ : Algebra R... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/DPAlgebra/Dpow.lean | DividedPowerAlgebra.cond_D_uniqueness | [264, 1] | [293, 40] | apply symm | case right
R : Type u
inst✝⁵ : CommRing R
inst✝⁴ : DecidableEq R
M : Type v
inst✝³ : AddCommGroup M
inst✝² : Module R M
h : DividedPowers (augIdeal R M)
hh : ∀ (n : ℕ) (x : M), h.dpow n ((ι R M) x) = dp R n x
S : Type u_1
inst✝¹ : CommRing S
inst✝ : Algebra R S
J : Ideal S
hJ : DividedPowers J
f : M →ₗ[R] S
hf : ∀ (m :... | case right.a
R : Type u
inst✝⁵ : CommRing R
inst✝⁴ : DecidableEq R
M : Type v
inst✝³ : AddCommGroup M
inst✝² : Module R M
h : DividedPowers (augIdeal R M)
hh : ∀ (n : ℕ) (x : M), h.dpow n ((ι R M) x) = dp R n x
S : Type u_1
inst✝¹ : CommRing S
inst✝ : Algebra R S
J : Ideal S
hJ : DividedPowers J
f : M →ₗ[R] S
hf : ∀ (m... | Please generate a tactic in lean4 to solve the state.
STATE:
case right
R : Type u
inst✝⁵ : CommRing R
inst✝⁴ : DecidableEq R
M : Type v
inst✝³ : AddCommGroup M
inst✝² : Module R M
h : DividedPowers (augIdeal R M)
hh : ∀ (n : ℕ) (x : M), h.dpow n ((ι R M) x) = dp R n x
S : Type u_1
inst✝¹ : CommRing S
inst✝ : Algebra R... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/DPAlgebra/Dpow.lean | DividedPowerAlgebra.cond_D_uniqueness | [264, 1] | [293, 40] | rw [(dp_uniqueness h hJ (lift hJ f hf) (augIdeal_eq_span R M) _ _) n a ha] | case right.a
R : Type u
inst✝⁵ : CommRing R
inst✝⁴ : DecidableEq R
M : Type v
inst✝³ : AddCommGroup M
inst✝² : Module R M
h : DividedPowers (augIdeal R M)
hh : ∀ (n : ℕ) (x : M), h.dpow n ((ι R M) x) = dp R n x
S : Type u_1
inst✝¹ : CommRing S
inst✝ : Algebra R S
J : Ideal S
hJ : DividedPowers J
f : M →ₗ[R] S
hf : ∀ (m... | R : Type u
inst✝⁵ : CommRing R
inst✝⁴ : DecidableEq R
M : Type v
inst✝³ : AddCommGroup M
inst✝² : Module R M
h : DividedPowers (augIdeal R M)
hh : ∀ (n : ℕ) (x : M), h.dpow n ((ι R M) x) = dp R n x
S : Type u_1
inst✝¹ : CommRing S
inst✝ : Algebra R S
J : Ideal S
hJ : DividedPowers J
f : M →ₗ[R] S
hf : ∀ (m : M), f m ∈ ... | Please generate a tactic in lean4 to solve the state.
STATE:
case right.a
R : Type u
inst✝⁵ : CommRing R
inst✝⁴ : DecidableEq R
M : Type v
inst✝³ : AddCommGroup M
inst✝² : Module R M
h : DividedPowers (augIdeal R M)
hh : ∀ (n : ℕ) (x : M), h.dpow n ((ι R M) x) = dp R n x
S : Type u_1
inst✝¹ : CommRing S
inst✝ : Algebra... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/DPAlgebra/Dpow.lean | DividedPowerAlgebra.cond_D_uniqueness | [264, 1] | [293, 40] | rintro a ⟨q, hq : 0 < q, m, _, rfl⟩ | R : Type u
inst✝⁵ : CommRing R
inst✝⁴ : DecidableEq R
M : Type v
inst✝³ : AddCommGroup M
inst✝² : Module R M
h : DividedPowers (augIdeal R M)
hh : ∀ (n : ℕ) (x : M), h.dpow n ((ι R M) x) = dp R n x
S : Type u_1
inst✝¹ : CommRing S
inst✝ : Algebra R S
J : Ideal S
hJ : DividedPowers J
f : M →ₗ[R] S
hf : ∀ (m : M), f m ∈ ... | case intro.intro.intro.intro
R : Type u
inst✝⁵ : CommRing R
inst✝⁴ : DecidableEq R
M : Type v
inst✝³ : AddCommGroup M
inst✝² : Module R M
h : DividedPowers (augIdeal R M)
hh : ∀ (n : ℕ) (x : M), h.dpow n ((ι R M) x) = dp R n x
S : Type u_1
inst✝¹ : CommRing S
inst✝ : Algebra R S
J : Ideal S
hJ : DividedPowers J
f : M →... | Please generate a tactic in lean4 to solve the state.
STATE:
R : Type u
inst✝⁵ : CommRing R
inst✝⁴ : DecidableEq R
M : Type v
inst✝³ : AddCommGroup M
inst✝² : Module R M
h : DividedPowers (augIdeal R M)
hh : ∀ (n : ℕ) (x : M), h.dpow n ((ι R M) x) = dp R n x
S : Type u_1
inst✝¹ : CommRing S
inst✝ : Algebra R S
J : Idea... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/DPAlgebra/Dpow.lean | DividedPowerAlgebra.cond_D_uniqueness | [264, 1] | [293, 40] | simp only [AlgHom.coe_toRingHom, liftAlgHom_apply_dp] | case intro.intro.intro.intro
R : Type u
inst✝⁵ : CommRing R
inst✝⁴ : DecidableEq R
M : Type v
inst✝³ : AddCommGroup M
inst✝² : Module R M
h : DividedPowers (augIdeal R M)
hh : ∀ (n : ℕ) (x : M), h.dpow n ((ι R M) x) = dp R n x
S : Type u_1
inst✝¹ : CommRing S
inst✝ : Algebra R S
J : Ideal S
hJ : DividedPowers J
f : M →... | case intro.intro.intro.intro
R : Type u
inst✝⁵ : CommRing R
inst✝⁴ : DecidableEq R
M : Type v
inst✝³ : AddCommGroup M
inst✝² : Module R M
h : DividedPowers (augIdeal R M)
hh : ∀ (n : ℕ) (x : M), h.dpow n ((ι R M) x) = dp R n x
S : Type u_1
inst✝¹ : CommRing S
inst✝ : Algebra R S
J : Ideal S
hJ : DividedPowers J
f : M →... | Please generate a tactic in lean4 to solve the state.
STATE:
case intro.intro.intro.intro
R : Type u
inst✝⁵ : CommRing R
inst✝⁴ : DecidableEq R
M : Type v
inst✝³ : AddCommGroup M
inst✝² : Module R M
h : DividedPowers (augIdeal R M)
hh : ∀ (n : ℕ) (x : M), h.dpow n ((ι R M) x) = dp R n x
S : Type u_1
inst✝¹ : CommRing S... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/DPAlgebra/Dpow.lean | DividedPowerAlgebra.cond_D_uniqueness | [264, 1] | [293, 40] | exact hJ.dpow_mem (ne_of_gt hq) (hf m) | case intro.intro.intro.intro
R : Type u
inst✝⁵ : CommRing R
inst✝⁴ : DecidableEq R
M : Type v
inst✝³ : AddCommGroup M
inst✝² : Module R M
h : DividedPowers (augIdeal R M)
hh : ∀ (n : ℕ) (x : M), h.dpow n ((ι R M) x) = dp R n x
S : Type u_1
inst✝¹ : CommRing S
inst✝ : Algebra R S
J : Ideal S
hJ : DividedPowers J
f : M →... | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case intro.intro.intro.intro
R : Type u
inst✝⁵ : CommRing R
inst✝⁴ : DecidableEq R
M : Type v
inst✝³ : AddCommGroup M
inst✝² : Module R M
h : DividedPowers (augIdeal R M)
hh : ∀ (n : ℕ) (x : M), h.dpow n ((ι R M) x) = dp R n x
S : Type u_1
inst✝¹ : CommRing S... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/DPAlgebra/Dpow.lean | DividedPowerAlgebra.cond_D_uniqueness | [264, 1] | [293, 40] | rintro n a ⟨q, hq : 0 < q, m, _, rfl⟩ | R : Type u
inst✝⁵ : CommRing R
inst✝⁴ : DecidableEq R
M : Type v
inst✝³ : AddCommGroup M
inst✝² : Module R M
h : DividedPowers (augIdeal R M)
hh : ∀ (n : ℕ) (x : M), h.dpow n ((ι R M) x) = dp R n x
S : Type u_1
inst✝¹ : CommRing S
inst✝ : Algebra R S
J : Ideal S
hJ : DividedPowers J
f : M →ₗ[R] S
hf : ∀ (m : M), f m ∈ ... | case intro.intro.intro.intro
R : Type u
inst✝⁵ : CommRing R
inst✝⁴ : DecidableEq R
M : Type v
inst✝³ : AddCommGroup M
inst✝² : Module R M
h : DividedPowers (augIdeal R M)
hh : ∀ (n : ℕ) (x : M), h.dpow n ((ι R M) x) = dp R n x
S : Type u_1
inst✝¹ : CommRing S
inst✝ : Algebra R S
J : Ideal S
hJ : DividedPowers J
f : M →... | Please generate a tactic in lean4 to solve the state.
STATE:
R : Type u
inst✝⁵ : CommRing R
inst✝⁴ : DecidableEq R
M : Type v
inst✝³ : AddCommGroup M
inst✝² : Module R M
h : DividedPowers (augIdeal R M)
hh : ∀ (n : ℕ) (x : M), h.dpow n ((ι R M) x) = dp R n x
S : Type u_1
inst✝¹ : CommRing S
inst✝ : Algebra R S
J : Idea... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/DPAlgebra/Dpow.lean | DividedPowerAlgebra.cond_D_uniqueness | [264, 1] | [293, 40] | simp only [AlgHom.coe_toRingHom, liftAlgHom_apply_dp] | case intro.intro.intro.intro
R : Type u
inst✝⁵ : CommRing R
inst✝⁴ : DecidableEq R
M : Type v
inst✝³ : AddCommGroup M
inst✝² : Module R M
h : DividedPowers (augIdeal R M)
hh : ∀ (n : ℕ) (x : M), h.dpow n ((ι R M) x) = dp R n x
S : Type u_1
inst✝¹ : CommRing S
inst✝ : Algebra R S
J : Ideal S
hJ : DividedPowers J
f : M →... | case intro.intro.intro.intro
R : Type u
inst✝⁵ : CommRing R
inst✝⁴ : DecidableEq R
M : Type v
inst✝³ : AddCommGroup M
inst✝² : Module R M
h : DividedPowers (augIdeal R M)
hh : ∀ (n : ℕ) (x : M), h.dpow n ((ι R M) x) = dp R n x
S : Type u_1
inst✝¹ : CommRing S
inst✝ : Algebra R S
J : Ideal S
hJ : DividedPowers J
f : M →... | Please generate a tactic in lean4 to solve the state.
STATE:
case intro.intro.intro.intro
R : Type u
inst✝⁵ : CommRing R
inst✝⁴ : DecidableEq R
M : Type v
inst✝³ : AddCommGroup M
inst✝² : Module R M
h : DividedPowers (augIdeal R M)
hh : ∀ (n : ℕ) (x : M), h.dpow n ((ι R M) x) = dp R n x
S : Type u_1
inst✝¹ : CommRing S... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/DPAlgebra/Dpow.lean | DividedPowerAlgebra.cond_D_uniqueness | [264, 1] | [293, 40] | rw [hJ.dpow_comp n (ne_of_gt hq) (hf m),← hh q m,
h.dpow_comp n (ne_of_gt hq) (ι_mem_augIdeal R M m), _root_.map_mul, map_natCast] | case intro.intro.intro.intro
R : Type u
inst✝⁵ : CommRing R
inst✝⁴ : DecidableEq R
M : Type v
inst✝³ : AddCommGroup M
inst✝² : Module R M
h : DividedPowers (augIdeal R M)
hh : ∀ (n : ℕ) (x : M), h.dpow n ((ι R M) x) = dp R n x
S : Type u_1
inst✝¹ : CommRing S
inst✝ : Algebra R S
J : Ideal S
hJ : DividedPowers J
f : M →... | case intro.intro.intro.intro
R : Type u
inst✝⁵ : CommRing R
inst✝⁴ : DecidableEq R
M : Type v
inst✝³ : AddCommGroup M
inst✝² : Module R M
h : DividedPowers (augIdeal R M)
hh : ∀ (n : ℕ) (x : M), h.dpow n ((ι R M) x) = dp R n x
S : Type u_1
inst✝¹ : CommRing S
inst✝ : Algebra R S
J : Ideal S
hJ : DividedPowers J
f : M →... | Please generate a tactic in lean4 to solve the state.
STATE:
case intro.intro.intro.intro
R : Type u
inst✝⁵ : CommRing R
inst✝⁴ : DecidableEq R
M : Type v
inst✝³ : AddCommGroup M
inst✝² : Module R M
h : DividedPowers (augIdeal R M)
hh : ∀ (n : ℕ) (x : M), h.dpow n ((ι R M) x) = dp R n x
S : Type u_1
inst✝¹ : CommRing S... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/DPAlgebra/Dpow.lean | DividedPowerAlgebra.cond_D_uniqueness | [264, 1] | [293, 40] | apply congr_arg₂ _ rfl | case intro.intro.intro.intro
R : Type u
inst✝⁵ : CommRing R
inst✝⁴ : DecidableEq R
M : Type v
inst✝³ : AddCommGroup M
inst✝² : Module R M
h : DividedPowers (augIdeal R M)
hh : ∀ (n : ℕ) (x : M), h.dpow n ((ι R M) x) = dp R n x
S : Type u_1
inst✝¹ : CommRing S
inst✝ : Algebra R S
J : Ideal S
hJ : DividedPowers J
f : M →... | case intro.intro.intro.intro
R : Type u
inst✝⁵ : CommRing R
inst✝⁴ : DecidableEq R
M : Type v
inst✝³ : AddCommGroup M
inst✝² : Module R M
h : DividedPowers (augIdeal R M)
hh : ∀ (n : ℕ) (x : M), h.dpow n ((ι R M) x) = dp R n x
S : Type u_1
inst✝¹ : CommRing S
inst✝ : Algebra R S
J : Ideal S
hJ : DividedPowers J
f : M →... | Please generate a tactic in lean4 to solve the state.
STATE:
case intro.intro.intro.intro
R : Type u
inst✝⁵ : CommRing R
inst✝⁴ : DecidableEq R
M : Type v
inst✝³ : AddCommGroup M
inst✝² : Module R M
h : DividedPowers (augIdeal R M)
hh : ∀ (n : ℕ) (x : M), h.dpow n ((ι R M) x) = dp R n x
S : Type u_1
inst✝¹ : CommRing S... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/DPAlgebra/Dpow.lean | DividedPowerAlgebra.cond_D_uniqueness | [264, 1] | [293, 40] | rw [hh] | case intro.intro.intro.intro
R : Type u
inst✝⁵ : CommRing R
inst✝⁴ : DecidableEq R
M : Type v
inst✝³ : AddCommGroup M
inst✝² : Module R M
h : DividedPowers (augIdeal R M)
hh : ∀ (n : ℕ) (x : M), h.dpow n ((ι R M) x) = dp R n x
S : Type u_1
inst✝¹ : CommRing S
inst✝ : Algebra R S
J : Ideal S
hJ : DividedPowers J
f : M →... | case intro.intro.intro.intro
R : Type u
inst✝⁵ : CommRing R
inst✝⁴ : DecidableEq R
M : Type v
inst✝³ : AddCommGroup M
inst✝² : Module R M
h : DividedPowers (augIdeal R M)
hh : ∀ (n : ℕ) (x : M), h.dpow n ((ι R M) x) = dp R n x
S : Type u_1
inst✝¹ : CommRing S
inst✝ : Algebra R S
J : Ideal S
hJ : DividedPowers J
f : M →... | Please generate a tactic in lean4 to solve the state.
STATE:
case intro.intro.intro.intro
R : Type u
inst✝⁵ : CommRing R
inst✝⁴ : DecidableEq R
M : Type v
inst✝³ : AddCommGroup M
inst✝² : Module R M
h : DividedPowers (augIdeal R M)
hh : ∀ (n : ℕ) (x : M), h.dpow n ((ι R M) x) = dp R n x
S : Type u_1
inst✝¹ : CommRing S... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/DPAlgebra/Dpow.lean | DividedPowerAlgebra.cond_D_uniqueness | [264, 1] | [293, 40] | rw [liftAlgHom_apply_dp] | case intro.intro.intro.intro
R : Type u
inst✝⁵ : CommRing R
inst✝⁴ : DecidableEq R
M : Type v
inst✝³ : AddCommGroup M
inst✝² : Module R M
h : DividedPowers (augIdeal R M)
hh : ∀ (n : ℕ) (x : M), h.dpow n ((ι R M) x) = dp R n x
S : Type u_1
inst✝¹ : CommRing S
inst✝ : Algebra R S
J : Ideal S
hJ : DividedPowers J
f : M →... | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case intro.intro.intro.intro
R : Type u
inst✝⁵ : CommRing R
inst✝⁴ : DecidableEq R
M : Type v
inst✝³ : AddCommGroup M
inst✝² : Module R M
h : DividedPowers (augIdeal R M)
hh : ∀ (n : ℕ) (x : M), h.dpow n ((ι R M) x) = dp R n x
S : Type u_1
inst✝¹ : CommRing S... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/DPAlgebra/Dpow.lean | DividedPowerAlgebra.roby4.Algebra.mem_bot_of_subalgebra_iff | [322, 1] | [331, 23] | simp only [Algebra.mem_bot, Set.mem_range, Subtype.exists] | R✝ : Type u
inst✝⁷ : CommRing R✝
A : Type u
inst✝⁶ : CommRing A
inst✝⁵ : DecidableEq A
S✝ : Type u
inst✝⁴ : CommRing S✝
inst✝³ : Algebra A S✝
I : Ideal S✝
hI : DividedPowers I
S₀✝ : Subalgebra A S✝
hIS₀ : IsCompl (Subalgebra.toSubmodule S₀✝) (Submodule.restrictScalars A I)
R : Type u_1
inst✝² : CommRing R
S : Type u_2
... | R✝ : Type u
inst✝⁷ : CommRing R✝
A : Type u
inst✝⁶ : CommRing A
inst✝⁵ : DecidableEq A
S✝ : Type u
inst✝⁴ : CommRing S✝
inst✝³ : Algebra A S✝
I : Ideal S✝
hI : DividedPowers I
S₀✝ : Subalgebra A S✝
hIS₀ : IsCompl (Subalgebra.toSubmodule S₀✝) (Submodule.restrictScalars A I)
R : Type u_1
inst✝² : CommRing R
S : Type u_2
... | Please generate a tactic in lean4 to solve the state.
STATE:
R✝ : Type u
inst✝⁷ : CommRing R✝
A : Type u
inst✝⁶ : CommRing A
inst✝⁵ : DecidableEq A
S✝ : Type u
inst✝⁴ : CommRing S✝
inst✝³ : Algebra A S✝
I : Ideal S✝
hI : DividedPowers I
S₀✝ : Subalgebra A S✝
hIS₀ : IsCompl (Subalgebra.toSubmodule S₀✝) (Submodule.restri... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/DPAlgebra/Dpow.lean | DividedPowerAlgebra.roby4.Algebra.mem_bot_of_subalgebra_iff | [322, 1] | [331, 23] | constructor | R✝ : Type u
inst✝⁷ : CommRing R✝
A : Type u
inst✝⁶ : CommRing A
inst✝⁵ : DecidableEq A
S✝ : Type u
inst✝⁴ : CommRing S✝
inst✝³ : Algebra A S✝
I : Ideal S✝
hI : DividedPowers I
S₀✝ : Subalgebra A S✝
hIS₀ : IsCompl (Subalgebra.toSubmodule S₀✝) (Submodule.restrictScalars A I)
R : Type u_1
inst✝² : CommRing R
S : Type u_2
... | case mp
R✝ : Type u
inst✝⁷ : CommRing R✝
A : Type u
inst✝⁶ : CommRing A
inst✝⁵ : DecidableEq A
S✝ : Type u
inst✝⁴ : CommRing S✝
inst✝³ : Algebra A S✝
I : Ideal S✝
hI : DividedPowers I
S₀✝ : Subalgebra A S✝
hIS₀ : IsCompl (Subalgebra.toSubmodule S₀✝) (Submodule.restrictScalars A I)
R : Type u_1
inst✝² : CommRing R
S : T... | Please generate a tactic in lean4 to solve the state.
STATE:
R✝ : Type u
inst✝⁷ : CommRing R✝
A : Type u
inst✝⁶ : CommRing A
inst✝⁵ : DecidableEq A
S✝ : Type u
inst✝⁴ : CommRing S✝
inst✝³ : Algebra A S✝
I : Ideal S✝
hI : DividedPowers I
S₀✝ : Subalgebra A S✝
hIS₀ : IsCompl (Subalgebra.toSubmodule S₀✝) (Submodule.restri... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/DPAlgebra/Dpow.lean | DividedPowerAlgebra.roby4.Algebra.mem_bot_of_subalgebra_iff | [322, 1] | [331, 23] | rintro ⟨s, hs, rfl⟩ | case mp
R✝ : Type u
inst✝⁷ : CommRing R✝
A : Type u
inst✝⁶ : CommRing A
inst✝⁵ : DecidableEq A
S✝ : Type u
inst✝⁴ : CommRing S✝
inst✝³ : Algebra A S✝
I : Ideal S✝
hI : DividedPowers I
S₀✝ : Subalgebra A S✝
hIS₀ : IsCompl (Subalgebra.toSubmodule S₀✝) (Submodule.restrictScalars A I)
R : Type u_1
inst✝² : CommRing R
S : T... | case mp.intro.intro
R✝ : Type u
inst✝⁷ : CommRing R✝
A : Type u
inst✝⁶ : CommRing A
inst✝⁵ : DecidableEq A
S✝ : Type u
inst✝⁴ : CommRing S✝
inst✝³ : Algebra A S✝
I : Ideal S✝
hI : DividedPowers I
S₀✝ : Subalgebra A S✝
hIS₀ : IsCompl (Subalgebra.toSubmodule S₀✝) (Submodule.restrictScalars A I)
R : Type u_1
inst✝² : Comm... | Please generate a tactic in lean4 to solve the state.
STATE:
case mp
R✝ : Type u
inst✝⁷ : CommRing R✝
A : Type u
inst✝⁶ : CommRing A
inst✝⁵ : DecidableEq A
S✝ : Type u
inst✝⁴ : CommRing S✝
inst✝³ : Algebra A S✝
I : Ideal S✝
hI : DividedPowers I
S₀✝ : Subalgebra A S✝
hIS₀ : IsCompl (Subalgebra.toSubmodule S₀✝) (Submodul... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/DPAlgebra/Dpow.lean | DividedPowerAlgebra.roby4.Algebra.mem_bot_of_subalgebra_iff | [322, 1] | [331, 23] | exact hs | case mp.intro.intro
R✝ : Type u
inst✝⁷ : CommRing R✝
A : Type u
inst✝⁶ : CommRing A
inst✝⁵ : DecidableEq A
S✝ : Type u
inst✝⁴ : CommRing S✝
inst✝³ : Algebra A S✝
I : Ideal S✝
hI : DividedPowers I
S₀✝ : Subalgebra A S✝
hIS₀ : IsCompl (Subalgebra.toSubmodule S₀✝) (Submodule.restrictScalars A I)
R : Type u_1
inst✝² : Comm... | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case mp.intro.intro
R✝ : Type u
inst✝⁷ : CommRing R✝
A : Type u
inst✝⁶ : CommRing A
inst✝⁵ : DecidableEq A
S✝ : Type u
inst✝⁴ : CommRing S✝
inst✝³ : Algebra A S✝
I : Ideal S✝
hI : DividedPowers I
S₀✝ : Subalgebra A S✝
hIS₀ : IsCompl (Subalgebra.toSubmodule S₀... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/DPAlgebra/Dpow.lean | DividedPowerAlgebra.roby4.Algebra.mem_bot_of_subalgebra_iff | [322, 1] | [331, 23] | intro hs | case mpr
R✝ : Type u
inst✝⁷ : CommRing R✝
A : Type u
inst✝⁶ : CommRing A
inst✝⁵ : DecidableEq A
S✝ : Type u
inst✝⁴ : CommRing S✝
inst✝³ : Algebra A S✝
I : Ideal S✝
hI : DividedPowers I
S₀✝ : Subalgebra A S✝
hIS₀ : IsCompl (Subalgebra.toSubmodule S₀✝) (Submodule.restrictScalars A I)
R : Type u_1
inst✝² : CommRing R
S : ... | case mpr
R✝ : Type u
inst✝⁷ : CommRing R✝
A : Type u
inst✝⁶ : CommRing A
inst✝⁵ : DecidableEq A
S✝ : Type u
inst✝⁴ : CommRing S✝
inst✝³ : Algebra A S✝
I : Ideal S✝
hI : DividedPowers I
S₀✝ : Subalgebra A S✝
hIS₀ : IsCompl (Subalgebra.toSubmodule S₀✝) (Submodule.restrictScalars A I)
R : Type u_1
inst✝² : CommRing R
S : ... | Please generate a tactic in lean4 to solve the state.
STATE:
case mpr
R✝ : Type u
inst✝⁷ : CommRing R✝
A : Type u
inst✝⁶ : CommRing A
inst✝⁵ : DecidableEq A
S✝ : Type u
inst✝⁴ : CommRing S✝
inst✝³ : Algebra A S✝
I : Ideal S✝
hI : DividedPowers I
S₀✝ : Subalgebra A S✝
hIS₀ : IsCompl (Subalgebra.toSubmodule S₀✝) (Submodu... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/DPAlgebra/Dpow.lean | DividedPowerAlgebra.roby4.Algebra.mem_bot_of_subalgebra_iff | [322, 1] | [331, 23] | exact ⟨s, hs, rfl⟩ | case mpr
R✝ : Type u
inst✝⁷ : CommRing R✝
A : Type u
inst✝⁶ : CommRing A
inst✝⁵ : DecidableEq A
S✝ : Type u
inst✝⁴ : CommRing S✝
inst✝³ : Algebra A S✝
I : Ideal S✝
hI : DividedPowers I
S₀✝ : Subalgebra A S✝
hIS₀ : IsCompl (Subalgebra.toSubmodule S₀✝) (Submodule.restrictScalars A I)
R : Type u_1
inst✝² : CommRing R
S : ... | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case mpr
R✝ : Type u
inst✝⁷ : CommRing R✝
A : Type u
inst✝⁶ : CommRing A
inst✝⁵ : DecidableEq A
S✝ : Type u
inst✝⁴ : CommRing S✝
inst✝³ : Algebra A S✝
I : Ideal S✝
hI : DividedPowers I
S₀✝ : Subalgebra A S✝
hIS₀ : IsCompl (Subalgebra.toSubmodule S₀✝) (Submodu... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/DPAlgebra/Dpow.lean | DividedPowerAlgebra.roby4.isAugmentation_iff_isCompl | [333, 1] | [346, 44] | unfold IsAugmentation | R : Type u
inst✝⁴ : CommRing R
A : Type u
inst✝³ : CommRing A
inst✝² : DecidableEq A
S : Type u
inst✝¹ : CommRing S
inst✝ : Algebra A S
I : Ideal S
hI : DividedPowers I
S₀ : Subalgebra A S
hIS₀ : IsCompl (Subalgebra.toSubmodule S₀) (Submodule.restrictScalars A I)
⊢ IsAugmentation (↥S₀) I ↔ IsCompl (Subalgebra.toSubmodu... | R : Type u
inst✝⁴ : CommRing R
A : Type u
inst✝³ : CommRing A
inst✝² : DecidableEq A
S : Type u
inst✝¹ : CommRing S
inst✝ : Algebra A S
I : Ideal S
hI : DividedPowers I
S₀ : Subalgebra A S
hIS₀ : IsCompl (Subalgebra.toSubmodule S₀) (Submodule.restrictScalars A I)
⊢ IsCompl (Subalgebra.toSubmodule ⊥) (Submodule.restrict... | Please generate a tactic in lean4 to solve the state.
STATE:
R : Type u
inst✝⁴ : CommRing R
A : Type u
inst✝³ : CommRing A
inst✝² : DecidableEq A
S : Type u
inst✝¹ : CommRing S
inst✝ : Algebra A S
I : Ideal S
hI : DividedPowers I
S₀ : Subalgebra A S
hIS₀ : IsCompl (Subalgebra.toSubmodule S₀) (Submodule.restrictScalars ... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/DPAlgebra/Dpow.lean | DividedPowerAlgebra.roby4.isAugmentation_iff_isCompl | [333, 1] | [346, 44] | rw [← Submodule.isCompl_restrictScalars_iff A] | R : Type u
inst✝⁴ : CommRing R
A : Type u
inst✝³ : CommRing A
inst✝² : DecidableEq A
S : Type u
inst✝¹ : CommRing S
inst✝ : Algebra A S
I : Ideal S
hI : DividedPowers I
S₀ : Subalgebra A S
hIS₀ : IsCompl (Subalgebra.toSubmodule S₀) (Submodule.restrictScalars A I)
⊢ IsCompl (Subalgebra.toSubmodule ⊥) (Submodule.restrict... | R : Type u
inst✝⁴ : CommRing R
A : Type u
inst✝³ : CommRing A
inst✝² : DecidableEq A
S : Type u
inst✝¹ : CommRing S
inst✝ : Algebra A S
I : Ideal S
hI : DividedPowers I
S₀ : Subalgebra A S
hIS₀ : IsCompl (Subalgebra.toSubmodule S₀) (Submodule.restrictScalars A I)
⊢ IsCompl (Submodule.restrictScalars A (Subalgebra.toSub... | Please generate a tactic in lean4 to solve the state.
STATE:
R : Type u
inst✝⁴ : CommRing R
A : Type u
inst✝³ : CommRing A
inst✝² : DecidableEq A
S : Type u
inst✝¹ : CommRing S
inst✝ : Algebra A S
I : Ideal S
hI : DividedPowers I
S₀ : Subalgebra A S
hIS₀ : IsCompl (Subalgebra.toSubmodule S₀) (Submodule.restrictScalars ... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/DPAlgebra/Dpow.lean | DividedPowerAlgebra.roby4.isAugmentation_iff_isCompl | [333, 1] | [346, 44] | suffices Submodule.restrictScalars A (Submodule.restrictScalars S₀ I) =
Submodule.restrictScalars A I by
rw [this]
suffices Submodule.restrictScalars A _ = Subalgebra.toSubmodule S₀ by
rw [this]
ext x
simp only [Submodule.restrictScalars_mem, Subalgebra.mem_toSubmodule]
apply Algebra.mem_bot_of_subalgeb... | R : Type u
inst✝⁴ : CommRing R
A : Type u
inst✝³ : CommRing A
inst✝² : DecidableEq A
S : Type u
inst✝¹ : CommRing S
inst✝ : Algebra A S
I : Ideal S
hI : DividedPowers I
S₀ : Subalgebra A S
hIS₀ : IsCompl (Subalgebra.toSubmodule S₀) (Submodule.restrictScalars A I)
⊢ IsCompl (Submodule.restrictScalars A (Subalgebra.toSub... | R : Type u
inst✝⁴ : CommRing R
A : Type u
inst✝³ : CommRing A
inst✝² : DecidableEq A
S : Type u
inst✝¹ : CommRing S
inst✝ : Algebra A S
I : Ideal S
hI : DividedPowers I
S₀ : Subalgebra A S
hIS₀ : IsCompl (Subalgebra.toSubmodule S₀) (Submodule.restrictScalars A I)
⊢ Submodule.restrictScalars A (Submodule.restrictScalars... | Please generate a tactic in lean4 to solve the state.
STATE:
R : Type u
inst✝⁴ : CommRing R
A : Type u
inst✝³ : CommRing A
inst✝² : DecidableEq A
S : Type u
inst✝¹ : CommRing S
inst✝ : Algebra A S
I : Ideal S
hI : DividedPowers I
S₀ : Subalgebra A S
hIS₀ : IsCompl (Subalgebra.toSubmodule S₀) (Submodule.restrictScalars ... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/DPAlgebra/Dpow.lean | DividedPowerAlgebra.roby4.isAugmentation_iff_isCompl | [333, 1] | [346, 44] | ext x | R : Type u
inst✝⁴ : CommRing R
A : Type u
inst✝³ : CommRing A
inst✝² : DecidableEq A
S : Type u
inst✝¹ : CommRing S
inst✝ : Algebra A S
I : Ideal S
hI : DividedPowers I
S₀ : Subalgebra A S
hIS₀ : IsCompl (Subalgebra.toSubmodule S₀) (Submodule.restrictScalars A I)
⊢ Submodule.restrictScalars A (Submodule.restrictScalars... | case h
R : Type u
inst✝⁴ : CommRing R
A : Type u
inst✝³ : CommRing A
inst✝² : DecidableEq A
S : Type u
inst✝¹ : CommRing S
inst✝ : Algebra A S
I : Ideal S
hI : DividedPowers I
S₀ : Subalgebra A S
hIS₀ : IsCompl (Subalgebra.toSubmodule S₀) (Submodule.restrictScalars A I)
x : S
⊢ x ∈ Submodule.restrictScalars A (Submodul... | Please generate a tactic in lean4 to solve the state.
STATE:
R : Type u
inst✝⁴ : CommRing R
A : Type u
inst✝³ : CommRing A
inst✝² : DecidableEq A
S : Type u
inst✝¹ : CommRing S
inst✝ : Algebra A S
I : Ideal S
hI : DividedPowers I
S₀ : Subalgebra A S
hIS₀ : IsCompl (Subalgebra.toSubmodule S₀) (Submodule.restrictScalars ... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/DPAlgebra/Dpow.lean | DividedPowerAlgebra.roby4.isAugmentation_iff_isCompl | [333, 1] | [346, 44] | simp only [Submodule.restrictScalars_mem] | case h
R : Type u
inst✝⁴ : CommRing R
A : Type u
inst✝³ : CommRing A
inst✝² : DecidableEq A
S : Type u
inst✝¹ : CommRing S
inst✝ : Algebra A S
I : Ideal S
hI : DividedPowers I
S₀ : Subalgebra A S
hIS₀ : IsCompl (Subalgebra.toSubmodule S₀) (Submodule.restrictScalars A I)
x : S
⊢ x ∈ Submodule.restrictScalars A (Submodul... | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case h
R : Type u
inst✝⁴ : CommRing R
A : Type u
inst✝³ : CommRing A
inst✝² : DecidableEq A
S : Type u
inst✝¹ : CommRing S
inst✝ : Algebra A S
I : Ideal S
hI : DividedPowers I
S₀ : Subalgebra A S
hIS₀ : IsCompl (Subalgebra.toSubmodule S₀) (Submodule.restrictS... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/DPAlgebra/Dpow.lean | DividedPowerAlgebra.roby4.isAugmentation_iff_isCompl | [333, 1] | [346, 44] | rw [this] | R : Type u
inst✝⁴ : CommRing R
A : Type u
inst✝³ : CommRing A
inst✝² : DecidableEq A
S : Type u
inst✝¹ : CommRing S
inst✝ : Algebra A S
I : Ideal S
hI : DividedPowers I
S₀ : Subalgebra A S
hIS₀ : IsCompl (Subalgebra.toSubmodule S₀) (Submodule.restrictScalars A I)
this : Submodule.restrictScalars A (Submodule.restrictSc... | R : Type u
inst✝⁴ : CommRing R
A : Type u
inst✝³ : CommRing A
inst✝² : DecidableEq A
S : Type u
inst✝¹ : CommRing S
inst✝ : Algebra A S
I : Ideal S
hI : DividedPowers I
S₀ : Subalgebra A S
hIS₀ : IsCompl (Subalgebra.toSubmodule S₀) (Submodule.restrictScalars A I)
this : Submodule.restrictScalars A (Submodule.restrictSc... | Please generate a tactic in lean4 to solve the state.
STATE:
R : Type u
inst✝⁴ : CommRing R
A : Type u
inst✝³ : CommRing A
inst✝² : DecidableEq A
S : Type u
inst✝¹ : CommRing S
inst✝ : Algebra A S
I : Ideal S
hI : DividedPowers I
S₀ : Subalgebra A S
hIS₀ : IsCompl (Subalgebra.toSubmodule S₀) (Submodule.restrictScalars ... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/DPAlgebra/Dpow.lean | DividedPowerAlgebra.roby4.isAugmentation_iff_isCompl | [333, 1] | [346, 44] | suffices Submodule.restrictScalars A _ = Subalgebra.toSubmodule S₀ by
rw [this] | R : Type u
inst✝⁴ : CommRing R
A : Type u
inst✝³ : CommRing A
inst✝² : DecidableEq A
S : Type u
inst✝¹ : CommRing S
inst✝ : Algebra A S
I : Ideal S
hI : DividedPowers I
S₀ : Subalgebra A S
hIS₀ : IsCompl (Subalgebra.toSubmodule S₀) (Submodule.restrictScalars A I)
this : Submodule.restrictScalars A (Submodule.restrictSc... | R : Type u
inst✝⁴ : CommRing R
A : Type u
inst✝³ : CommRing A
inst✝² : DecidableEq A
S : Type u
inst✝¹ : CommRing S
inst✝ : Algebra A S
I : Ideal S
hI : DividedPowers I
S₀ : Subalgebra A S
hIS₀ : IsCompl (Subalgebra.toSubmodule S₀) (Submodule.restrictScalars A I)
this : Submodule.restrictScalars A (Submodule.restrictSc... | Please generate a tactic in lean4 to solve the state.
STATE:
R : Type u
inst✝⁴ : CommRing R
A : Type u
inst✝³ : CommRing A
inst✝² : DecidableEq A
S : Type u
inst✝¹ : CommRing S
inst✝ : Algebra A S
I : Ideal S
hI : DividedPowers I
S₀ : Subalgebra A S
hIS₀ : IsCompl (Subalgebra.toSubmodule S₀) (Submodule.restrictScalars ... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/DPAlgebra/Dpow.lean | DividedPowerAlgebra.roby4.isAugmentation_iff_isCompl | [333, 1] | [346, 44] | ext x | R : Type u
inst✝⁴ : CommRing R
A : Type u
inst✝³ : CommRing A
inst✝² : DecidableEq A
S : Type u
inst✝¹ : CommRing S
inst✝ : Algebra A S
I : Ideal S
hI : DividedPowers I
S₀ : Subalgebra A S
hIS₀ : IsCompl (Subalgebra.toSubmodule S₀) (Submodule.restrictScalars A I)
this : Submodule.restrictScalars A (Submodule.restrictSc... | case h
R : Type u
inst✝⁴ : CommRing R
A : Type u
inst✝³ : CommRing A
inst✝² : DecidableEq A
S : Type u
inst✝¹ : CommRing S
inst✝ : Algebra A S
I : Ideal S
hI : DividedPowers I
S₀ : Subalgebra A S
hIS₀ : IsCompl (Subalgebra.toSubmodule S₀) (Submodule.restrictScalars A I)
this : Submodule.restrictScalars A (Submodule.res... | Please generate a tactic in lean4 to solve the state.
STATE:
R : Type u
inst✝⁴ : CommRing R
A : Type u
inst✝³ : CommRing A
inst✝² : DecidableEq A
S : Type u
inst✝¹ : CommRing S
inst✝ : Algebra A S
I : Ideal S
hI : DividedPowers I
S₀ : Subalgebra A S
hIS₀ : IsCompl (Subalgebra.toSubmodule S₀) (Submodule.restrictScalars ... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/DPAlgebra/Dpow.lean | DividedPowerAlgebra.roby4.isAugmentation_iff_isCompl | [333, 1] | [346, 44] | simp only [Submodule.restrictScalars_mem, Subalgebra.mem_toSubmodule] | case h
R : Type u
inst✝⁴ : CommRing R
A : Type u
inst✝³ : CommRing A
inst✝² : DecidableEq A
S : Type u
inst✝¹ : CommRing S
inst✝ : Algebra A S
I : Ideal S
hI : DividedPowers I
S₀ : Subalgebra A S
hIS₀ : IsCompl (Subalgebra.toSubmodule S₀) (Submodule.restrictScalars A I)
this : Submodule.restrictScalars A (Submodule.res... | case h
R : Type u
inst✝⁴ : CommRing R
A : Type u
inst✝³ : CommRing A
inst✝² : DecidableEq A
S : Type u
inst✝¹ : CommRing S
inst✝ : Algebra A S
I : Ideal S
hI : DividedPowers I
S₀ : Subalgebra A S
hIS₀ : IsCompl (Subalgebra.toSubmodule S₀) (Submodule.restrictScalars A I)
this : Submodule.restrictScalars A (Submodule.res... | Please generate a tactic in lean4 to solve the state.
STATE:
case h
R : Type u
inst✝⁴ : CommRing R
A : Type u
inst✝³ : CommRing A
inst✝² : DecidableEq A
S : Type u
inst✝¹ : CommRing S
inst✝ : Algebra A S
I : Ideal S
hI : DividedPowers I
S₀ : Subalgebra A S
hIS₀ : IsCompl (Subalgebra.toSubmodule S₀) (Submodule.restrictS... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/DPAlgebra/Dpow.lean | DividedPowerAlgebra.roby4.isAugmentation_iff_isCompl | [333, 1] | [346, 44] | apply Algebra.mem_bot_of_subalgebra_iff | case h
R : Type u
inst✝⁴ : CommRing R
A : Type u
inst✝³ : CommRing A
inst✝² : DecidableEq A
S : Type u
inst✝¹ : CommRing S
inst✝ : Algebra A S
I : Ideal S
hI : DividedPowers I
S₀ : Subalgebra A S
hIS₀ : IsCompl (Subalgebra.toSubmodule S₀) (Submodule.restrictScalars A I)
this : Submodule.restrictScalars A (Submodule.res... | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case h
R : Type u
inst✝⁴ : CommRing R
A : Type u
inst✝³ : CommRing A
inst✝² : DecidableEq A
S : Type u
inst✝¹ : CommRing S
inst✝ : Algebra A S
I : Ideal S
hI : DividedPowers I
S₀ : Subalgebra A S
hIS₀ : IsCompl (Subalgebra.toSubmodule S₀) (Submodule.restrictS... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/DPAlgebra/Dpow.lean | DividedPowerAlgebra.roby4.isAugmentation_iff_isCompl | [333, 1] | [346, 44] | rw [this] | R : Type u
inst✝⁴ : CommRing R
A : Type u
inst✝³ : CommRing A
inst✝² : DecidableEq A
S : Type u
inst✝¹ : CommRing S
inst✝ : Algebra A S
I : Ideal S
hI : DividedPowers I
S₀ : Subalgebra A S
hIS₀ : IsCompl (Subalgebra.toSubmodule S₀) (Submodule.restrictScalars A I)
this✝ : Submodule.restrictScalars A (Submodule.restrictS... | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
R : Type u
inst✝⁴ : CommRing R
A : Type u
inst✝³ : CommRing A
inst✝² : DecidableEq A
S : Type u
inst✝¹ : CommRing S
inst✝ : Algebra A S
I : Ideal S
hI : DividedPowers I
S₀ : Subalgebra A S
hIS₀ : IsCompl (Subalgebra.toSubmodule S₀) (Submodule.restrictScalars ... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/DPAlgebra/Dpow.lean | DividedPowerAlgebra.roby4.f_mem_I | [369, 1] | [375, 78] | suffices LinearMap.range (f A I) ≤ Submodule.restrictScalars A I by
apply this
simp only [LinearMap.mem_range, exists_apply_eq_apply] | R : Type u
inst✝⁴ : CommRing R
A : Type u
inst✝³ : CommRing A
inst✝² : DecidableEq A
S : Type u
inst✝¹ : CommRing S
inst✝ : Algebra A S
I : Ideal S
hI : DividedPowers I
S₀ : Subalgebra A S
hIS₀ : IsCompl (Subalgebra.toSubmodule S₀) (Submodule.restrictScalars A I)
p : ↥I →₀ A
⊢ (f A I) p ∈ I | R : Type u
inst✝⁴ : CommRing R
A : Type u
inst✝³ : CommRing A
inst✝² : DecidableEq A
S : Type u
inst✝¹ : CommRing S
inst✝ : Algebra A S
I : Ideal S
hI : DividedPowers I
S₀ : Subalgebra A S
hIS₀ : IsCompl (Subalgebra.toSubmodule S₀) (Submodule.restrictScalars A I)
p : ↥I →₀ A
⊢ LinearMap.range (f A I) ≤ Submodule.restri... | Please generate a tactic in lean4 to solve the state.
STATE:
R : Type u
inst✝⁴ : CommRing R
A : Type u
inst✝³ : CommRing A
inst✝² : DecidableEq A
S : Type u
inst✝¹ : CommRing S
inst✝ : Algebra A S
I : Ideal S
hI : DividedPowers I
S₀ : Subalgebra A S
hIS₀ : IsCompl (Subalgebra.toSubmodule S₀) (Submodule.restrictScalars ... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/DPAlgebra/Dpow.lean | DividedPowerAlgebra.roby4.f_mem_I | [369, 1] | [375, 78] | simp only [f, Basis.constr_range, Submodule.span_le] | R : Type u
inst✝⁴ : CommRing R
A : Type u
inst✝³ : CommRing A
inst✝² : DecidableEq A
S : Type u
inst✝¹ : CommRing S
inst✝ : Algebra A S
I : Ideal S
hI : DividedPowers I
S₀ : Subalgebra A S
hIS₀ : IsCompl (Subalgebra.toSubmodule S₀) (Submodule.restrictScalars A I)
p : ↥I →₀ A
⊢ LinearMap.range (f A I) ≤ Submodule.restri... | R : Type u
inst✝⁴ : CommRing R
A : Type u
inst✝³ : CommRing A
inst✝² : DecidableEq A
S : Type u
inst✝¹ : CommRing S
inst✝ : Algebra A S
I : Ideal S
hI : DividedPowers I
S₀ : Subalgebra A S
hIS₀ : IsCompl (Subalgebra.toSubmodule S₀) (Submodule.restrictScalars A I)
p : ↥I →₀ A
⊢ (Set.range fun i => ↑i) ⊆ ↑(Submodule.rest... | Please generate a tactic in lean4 to solve the state.
STATE:
R : Type u
inst✝⁴ : CommRing R
A : Type u
inst✝³ : CommRing A
inst✝² : DecidableEq A
S : Type u
inst✝¹ : CommRing S
inst✝ : Algebra A S
I : Ideal S
hI : DividedPowers I
S₀ : Subalgebra A S
hIS₀ : IsCompl (Subalgebra.toSubmodule S₀) (Submodule.restrictScalars ... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/DPAlgebra/Dpow.lean | DividedPowerAlgebra.roby4.f_mem_I | [369, 1] | [375, 78] | rintro _ ⟨i, rfl⟩ | R : Type u
inst✝⁴ : CommRing R
A : Type u
inst✝³ : CommRing A
inst✝² : DecidableEq A
S : Type u
inst✝¹ : CommRing S
inst✝ : Algebra A S
I : Ideal S
hI : DividedPowers I
S₀ : Subalgebra A S
hIS₀ : IsCompl (Subalgebra.toSubmodule S₀) (Submodule.restrictScalars A I)
p : ↥I →₀ A
⊢ (Set.range fun i => ↑i) ⊆ ↑(Submodule.rest... | case intro
R : Type u
inst✝⁴ : CommRing R
A : Type u
inst✝³ : CommRing A
inst✝² : DecidableEq A
S : Type u
inst✝¹ : CommRing S
inst✝ : Algebra A S
I : Ideal S
hI : DividedPowers I
S₀ : Subalgebra A S
hIS₀ : IsCompl (Subalgebra.toSubmodule S₀) (Submodule.restrictScalars A I)
p : ↥I →₀ A
i : ↥I
⊢ (fun i => ↑i) i ∈ ↑(Subm... | Please generate a tactic in lean4 to solve the state.
STATE:
R : Type u
inst✝⁴ : CommRing R
A : Type u
inst✝³ : CommRing A
inst✝² : DecidableEq A
S : Type u
inst✝¹ : CommRing S
inst✝ : Algebra A S
I : Ideal S
hI : DividedPowers I
S₀ : Subalgebra A S
hIS₀ : IsCompl (Subalgebra.toSubmodule S₀) (Submodule.restrictScalars ... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/DPAlgebra/Dpow.lean | DividedPowerAlgebra.roby4.f_mem_I | [369, 1] | [375, 78] | simp only [Submodule.coe_restrictScalars, SetLike.mem_coe, SetLike.coe_mem] | case intro
R : Type u
inst✝⁴ : CommRing R
A : Type u
inst✝³ : CommRing A
inst✝² : DecidableEq A
S : Type u
inst✝¹ : CommRing S
inst✝ : Algebra A S
I : Ideal S
hI : DividedPowers I
S₀ : Subalgebra A S
hIS₀ : IsCompl (Subalgebra.toSubmodule S₀) (Submodule.restrictScalars A I)
p : ↥I →₀ A
i : ↥I
⊢ (fun i => ↑i) i ∈ ↑(Subm... | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case intro
R : Type u
inst✝⁴ : CommRing R
A : Type u
inst✝³ : CommRing A
inst✝² : DecidableEq A
S : Type u
inst✝¹ : CommRing S
inst✝ : Algebra A S
I : Ideal S
hI : DividedPowers I
S₀ : Subalgebra A S
hIS₀ : IsCompl (Subalgebra.toSubmodule S₀) (Submodule.restr... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/DPAlgebra/Dpow.lean | DividedPowerAlgebra.roby4.f_mem_I | [369, 1] | [375, 78] | apply this | R : Type u
inst✝⁴ : CommRing R
A : Type u
inst✝³ : CommRing A
inst✝² : DecidableEq A
S : Type u
inst✝¹ : CommRing S
inst✝ : Algebra A S
I : Ideal S
hI : DividedPowers I
S₀ : Subalgebra A S
hIS₀ : IsCompl (Subalgebra.toSubmodule S₀) (Submodule.restrictScalars A I)
p : ↥I →₀ A
this : LinearMap.range (f A I) ≤ Submodule.r... | case a
R : Type u
inst✝⁴ : CommRing R
A : Type u
inst✝³ : CommRing A
inst✝² : DecidableEq A
S : Type u
inst✝¹ : CommRing S
inst✝ : Algebra A S
I : Ideal S
hI : DividedPowers I
S₀ : Subalgebra A S
hIS₀ : IsCompl (Subalgebra.toSubmodule S₀) (Submodule.restrictScalars A I)
p : ↥I →₀ A
this : LinearMap.range (f A I) ≤ Subm... | Please generate a tactic in lean4 to solve the state.
STATE:
R : Type u
inst✝⁴ : CommRing R
A : Type u
inst✝³ : CommRing A
inst✝² : DecidableEq A
S : Type u
inst✝¹ : CommRing S
inst✝ : Algebra A S
I : Ideal S
hI : DividedPowers I
S₀ : Subalgebra A S
hIS₀ : IsCompl (Subalgebra.toSubmodule S₀) (Submodule.restrictScalars ... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/DPAlgebra/Dpow.lean | DividedPowerAlgebra.roby4.f_mem_I | [369, 1] | [375, 78] | simp only [LinearMap.mem_range, exists_apply_eq_apply] | case a
R : Type u
inst✝⁴ : CommRing R
A : Type u
inst✝³ : CommRing A
inst✝² : DecidableEq A
S : Type u
inst✝¹ : CommRing S
inst✝ : Algebra A S
I : Ideal S
hI : DividedPowers I
S₀ : Subalgebra A S
hIS₀ : IsCompl (Subalgebra.toSubmodule S₀) (Submodule.restrictScalars A I)
p : ↥I →₀ A
this : LinearMap.range (f A I) ≤ Subm... | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case a
R : Type u
inst✝⁴ : CommRing R
A : Type u
inst✝³ : CommRing A
inst✝² : DecidableEq A
S : Type u
inst✝¹ : CommRing S
inst✝ : Algebra A S
I : Ideal S
hI : DividedPowers I
S₀ : Subalgebra A S
hIS₀ : IsCompl (Subalgebra.toSubmodule S₀) (Submodule.restrictS... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/DPAlgebra/Dpow.lean | DividedPowerAlgebra.roby4.condτ | [391, 1] | [394, 17] | apply condTFree | R : Type u
inst✝⁴ : CommRing R
A : Type u
inst✝³ : CommRing A
inst✝² : DecidableEq A
S : Type u
inst✝¹ : CommRing S
inst✝ : Algebra A S
I : Ideal S
hI : DividedPowers I
S₀ : Subalgebra A S
hIS₀ : IsCompl (Subalgebra.toSubmodule S₀) (Submodule.restrictScalars A I)
condTFree : CondTFree A
condD : CondD A
hM : DividedPowe... | case x
R : Type u
inst✝⁴ : CommRing R
A : Type u
inst✝³ : CommRing A
inst✝² : DecidableEq A
S : Type u
inst✝¹ : CommRing S
inst✝ : Algebra A S
I : Ideal S
hI : DividedPowers I
S₀ : Subalgebra A S
hIS₀ : IsCompl (Subalgebra.toSubmodule S₀) (Submodule.restrictScalars A I)
condTFree : CondTFree A
condD : CondD A
hM : Divi... | Please generate a tactic in lean4 to solve the state.
STATE:
R : Type u
inst✝⁴ : CommRing R
A : Type u
inst✝³ : CommRing A
inst✝² : DecidableEq A
S : Type u
inst✝¹ : CommRing S
inst✝ : Algebra A S
I : Ideal S
hI : DividedPowers I
S₀ : Subalgebra A S
hIS₀ : IsCompl (Subalgebra.toSubmodule S₀) (Submodule.restrictScalars ... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/DPAlgebra/Dpow.lean | DividedPowerAlgebra.roby4.condτ | [391, 1] | [394, 17] | infer_instance | case x
R : Type u
inst✝⁴ : CommRing R
A : Type u
inst✝³ : CommRing A
inst✝² : DecidableEq A
S : Type u
inst✝¹ : CommRing S
inst✝ : Algebra A S
I : Ideal S
hI : DividedPowers I
S₀ : Subalgebra A S
hIS₀ : IsCompl (Subalgebra.toSubmodule S₀) (Submodule.restrictScalars A I)
condTFree : CondTFree A
condD : CondD A
hM : Divi... | case x
R : Type u
inst✝⁴ : CommRing R
A : Type u
inst✝³ : CommRing A
inst✝² : DecidableEq A
S : Type u
inst✝¹ : CommRing S
inst✝ : Algebra A S
I : Ideal S
hI : DividedPowers I
S₀ : Subalgebra A S
hIS₀ : IsCompl (Subalgebra.toSubmodule S₀) (Submodule.restrictScalars A I)
condTFree : CondTFree A
condD : CondD A
hM : Divi... | Please generate a tactic in lean4 to solve the state.
STATE:
case x
R : Type u
inst✝⁴ : CommRing R
A : Type u
inst✝³ : CommRing A
inst✝² : DecidableEq A
S : Type u
inst✝¹ : CommRing S
inst✝ : Algebra A S
I : Ideal S
hI : DividedPowers I
S₀ : Subalgebra A S
hIS₀ : IsCompl (Subalgebra.toSubmodule S₀) (Submodule.restrictS... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/DPAlgebra/Dpow.lean | DividedPowerAlgebra.roby4.condτ | [391, 1] | [394, 17] | infer_instance | case x
R : Type u
inst✝⁴ : CommRing R
A : Type u
inst✝³ : CommRing A
inst✝² : DecidableEq A
S : Type u
inst✝¹ : CommRing S
inst✝ : Algebra A S
I : Ideal S
hI : DividedPowers I
S₀ : Subalgebra A S
hIS₀ : IsCompl (Subalgebra.toSubmodule S₀) (Submodule.restrictScalars A I)
condTFree : CondTFree A
condD : CondD A
hM : Divi... | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case x
R : Type u
inst✝⁴ : CommRing R
A : Type u
inst✝³ : CommRing A
inst✝² : DecidableEq A
S : Type u
inst✝¹ : CommRing S
inst✝ : Algebra A S
I : Ideal S
hI : DividedPowers I
S₀ : Subalgebra A S
hIS₀ : IsCompl (Subalgebra.toSubmodule S₀) (Submodule.restrictS... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/DPAlgebra/Dpow.lean | DividedPowerAlgebra.roby4.Ψ_eq | [425, 1] | [427, 37] | simp [Ψ, Φ, f, Basis.constr_apply] | R : Type u
inst✝⁴ : CommRing R
A : Type u
inst✝³ : CommRing A
inst✝² : DecidableEq A
S : Type u
inst✝¹ : CommRing S
inst✝ : Algebra A S
I : Ideal S
hI : DividedPowers I
S₀ : Subalgebra A S
hIS₀ : IsCompl (Subalgebra.toSubmodule S₀) (Submodule.restrictScalars A I)
condTFree : CondTFree A
condD : CondD A
hM : DividedPowe... | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
R : Type u
inst✝⁴ : CommRing R
A : Type u
inst✝³ : CommRing A
inst✝² : DecidableEq A
S : Type u
inst✝¹ : CommRing S
inst✝ : Algebra A S
I : Ideal S
hI : DividedPowers I
S₀ : Subalgebra A S
hIS₀ : IsCompl (Subalgebra.toSubmodule S₀) (Submodule.restrictScalars ... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/DPAlgebra/Dpow.lean | DividedPowerAlgebra.roby4.Ψ_surjective | [429, 1] | [445, 42] | rw [← Algebra.range_top_iff_surjective _, eq_top_iff] | R : Type u
inst✝⁴ : CommRing R
A : Type u
inst✝³ : CommRing A
inst✝² : DecidableEq A
S : Type u
inst✝¹ : CommRing S
inst✝ : Algebra A S
I : Ideal S
hI : DividedPowers I
S₀ : Subalgebra A S
hIS₀ : IsCompl (Subalgebra.toSubmodule S₀) (Submodule.restrictScalars A I)
condTFree : CondTFree A
condD : CondD A
hM : DividedPowe... | R : Type u
inst✝⁴ : CommRing R
A : Type u
inst✝³ : CommRing A
inst✝² : DecidableEq A
S : Type u
inst✝¹ : CommRing S
inst✝ : Algebra A S
I : Ideal S
hI : DividedPowers I
S₀ : Subalgebra A S
hIS₀ : IsCompl (Subalgebra.toSubmodule S₀) (Submodule.restrictScalars A I)
condTFree : CondTFree A
condD : CondD A
hM : DividedPowe... | Please generate a tactic in lean4 to solve the state.
STATE:
R : Type u
inst✝⁴ : CommRing R
A : Type u
inst✝³ : CommRing A
inst✝² : DecidableEq A
S : Type u
inst✝¹ : CommRing S
inst✝ : Algebra A S
I : Ideal S
hI : DividedPowers I
S₀ : Subalgebra A S
hIS₀ : IsCompl (Subalgebra.toSubmodule S₀) (Submodule.restrictScalars ... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/DPAlgebra/Dpow.lean | DividedPowerAlgebra.roby4.Ψ_surjective | [429, 1] | [445, 42] | intro s _ | R : Type u
inst✝⁴ : CommRing R
A : Type u
inst✝³ : CommRing A
inst✝² : DecidableEq A
S : Type u
inst✝¹ : CommRing S
inst✝ : Algebra A S
I : Ideal S
hI : DividedPowers I
S₀ : Subalgebra A S
hIS₀ : IsCompl (Subalgebra.toSubmodule S₀) (Submodule.restrictScalars A I)
condTFree : CondTFree A
condD : CondD A
hM : DividedPowe... | R : Type u
inst✝⁴ : CommRing R
A : Type u
inst✝³ : CommRing A
inst✝² : DecidableEq A
S : Type u
inst✝¹ : CommRing S
inst✝ : Algebra A S
I : Ideal S
hI : DividedPowers I
S₀ : Subalgebra A S
hIS₀ : IsCompl (Subalgebra.toSubmodule S₀) (Submodule.restrictScalars A I)
condTFree : CondTFree A
condD : CondD A
hM : DividedPowe... | Please generate a tactic in lean4 to solve the state.
STATE:
R : Type u
inst✝⁴ : CommRing R
A : Type u
inst✝³ : CommRing A
inst✝² : DecidableEq A
S : Type u
inst✝¹ : CommRing S
inst✝ : Algebra A S
I : Ideal S
hI : DividedPowers I
S₀ : Subalgebra A S
hIS₀ : IsCompl (Subalgebra.toSubmodule S₀) (Submodule.restrictScalars ... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/DPAlgebra/Dpow.lean | DividedPowerAlgebra.roby4.Ψ_surjective | [429, 1] | [445, 42] | obtain ⟨s₀, hs₀, s₁, hs₁, rfl⟩ := Submodule.exists_add_eq_of_codisjoint (hIS₀.codisjoint) s | R : Type u
inst✝⁴ : CommRing R
A : Type u
inst✝³ : CommRing A
inst✝² : DecidableEq A
S : Type u
inst✝¹ : CommRing S
inst✝ : Algebra A S
I : Ideal S
hI : DividedPowers I
S₀ : Subalgebra A S
hIS₀ : IsCompl (Subalgebra.toSubmodule S₀) (Submodule.restrictScalars A I)
condTFree : CondTFree A
condD : CondD A
hM : DividedPowe... | case intro.intro.intro.intro
R : Type u
inst✝⁴ : CommRing R
A : Type u
inst✝³ : CommRing A
inst✝² : DecidableEq A
S : Type u
inst✝¹ : CommRing S
inst✝ : Algebra A S
I : Ideal S
hI : DividedPowers I
S₀ : Subalgebra A S
hIS₀ : IsCompl (Subalgebra.toSubmodule S₀) (Submodule.restrictScalars A I)
condTFree : CondTFree A
con... | Please generate a tactic in lean4 to solve the state.
STATE:
R : Type u
inst✝⁴ : CommRing R
A : Type u
inst✝³ : CommRing A
inst✝² : DecidableEq A
S : Type u
inst✝¹ : CommRing S
inst✝ : Algebra A S
I : Ideal S
hI : DividedPowers I
S₀ : Subalgebra A S
hIS₀ : IsCompl (Subalgebra.toSubmodule S₀) (Submodule.restrictScalars ... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/DPAlgebra/Dpow.lean | DividedPowerAlgebra.roby4.Ψ_surjective | [429, 1] | [445, 42] | apply Subalgebra.add_mem | case intro.intro.intro.intro
R : Type u
inst✝⁴ : CommRing R
A : Type u
inst✝³ : CommRing A
inst✝² : DecidableEq A
S : Type u
inst✝¹ : CommRing S
inst✝ : Algebra A S
I : Ideal S
hI : DividedPowers I
S₀ : Subalgebra A S
hIS₀ : IsCompl (Subalgebra.toSubmodule S₀) (Submodule.restrictScalars A I)
condTFree : CondTFree A
con... | case intro.intro.intro.intro.hx
R : Type u
inst✝⁴ : CommRing R
A : Type u
inst✝³ : CommRing A
inst✝² : DecidableEq A
S : Type u
inst✝¹ : CommRing S
inst✝ : Algebra A S
I : Ideal S
hI : DividedPowers I
S₀ : Subalgebra A S
hIS₀ : IsCompl (Subalgebra.toSubmodule S₀) (Submodule.restrictScalars A I)
condTFree : CondTFree A
... | Please generate a tactic in lean4 to solve the state.
STATE:
case intro.intro.intro.intro
R : Type u
inst✝⁴ : CommRing R
A : Type u
inst✝³ : CommRing A
inst✝² : DecidableEq A
S : Type u
inst✝¹ : CommRing S
inst✝ : Algebra A S
I : Ideal S
hI : DividedPowers I
S₀ : Subalgebra A S
hIS₀ : IsCompl (Subalgebra.toSubmodule S₀... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/DPAlgebra/Dpow.lean | DividedPowerAlgebra.roby4.Ψ_surjective | [429, 1] | [445, 42] | use (X ⟨s₀, hs₀⟩) ⊗ₜ[A] 1 | case intro.intro.intro.intro.hx
R : Type u
inst✝⁴ : CommRing R
A : Type u
inst✝³ : CommRing A
inst✝² : DecidableEq A
S : Type u
inst✝¹ : CommRing S
inst✝ : Algebra A S
I : Ideal S
hI : DividedPowers I
S₀ : Subalgebra A S
hIS₀ : IsCompl (Subalgebra.toSubmodule S₀) (Submodule.restrictScalars A I)
condTFree : CondTFree A
... | case h
R : Type u
inst✝⁴ : CommRing R
A : Type u
inst✝³ : CommRing A
inst✝² : DecidableEq A
S : Type u
inst✝¹ : CommRing S
inst✝ : Algebra A S
I : Ideal S
hI : DividedPowers I
S₀ : Subalgebra A S
hIS₀ : IsCompl (Subalgebra.toSubmodule S₀) (Submodule.restrictScalars A I)
condTFree : CondTFree A
condD : CondD A
hM : Divi... | Please generate a tactic in lean4 to solve the state.
STATE:
case intro.intro.intro.intro.hx
R : Type u
inst✝⁴ : CommRing R
A : Type u
inst✝³ : CommRing A
inst✝² : DecidableEq A
S : Type u
inst✝¹ : CommRing S
inst✝ : Algebra A S
I : Ideal S
hI : DividedPowers I
S₀ : Subalgebra A S
hIS₀ : IsCompl (Subalgebra.toSubmodule... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/DPAlgebra/Dpow.lean | DividedPowerAlgebra.roby4.Ψ_surjective | [429, 1] | [445, 42] | simp only [Ψ, AlgHom.toRingHom_eq_coe, RingHom.coe_coe,
Algebra.TensorProduct.productMap_apply_tmul, AlgHom.coe_comp, Subalgebra.coe_val,
IsScalarTower.coe_toAlgHom', algebraMap_eq, Function.comp_apply, aeval_X, id_eq, map_one,
mul_one] | case h
R : Type u
inst✝⁴ : CommRing R
A : Type u
inst✝³ : CommRing A
inst✝² : DecidableEq A
S : Type u
inst✝¹ : CommRing S
inst✝ : Algebra A S
I : Ideal S
hI : DividedPowers I
S₀ : Subalgebra A S
hIS₀ : IsCompl (Subalgebra.toSubmodule S₀) (Submodule.restrictScalars A I)
condTFree : CondTFree A
condD : CondD A
hM : Divi... | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case h
R : Type u
inst✝⁴ : CommRing R
A : Type u
inst✝³ : CommRing A
inst✝² : DecidableEq A
S : Type u
inst✝¹ : CommRing S
inst✝ : Algebra A S
I : Ideal S
hI : DividedPowers I
S₀ : Subalgebra A S
hIS₀ : IsCompl (Subalgebra.toSubmodule S₀) (Submodule.restrictS... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/DPAlgebra/Dpow.lean | DividedPowerAlgebra.roby4.Ψ_surjective | [429, 1] | [445, 42] | use 1 ⊗ₜ[A] (ι A _ (Finsupp.single ⟨s₁, hs₁⟩ 1)) | case intro.intro.intro.intro.hy
R : Type u
inst✝⁴ : CommRing R
A : Type u
inst✝³ : CommRing A
inst✝² : DecidableEq A
S : Type u
inst✝¹ : CommRing S
inst✝ : Algebra A S
I : Ideal S
hI : DividedPowers I
S₀ : Subalgebra A S
hIS₀ : IsCompl (Subalgebra.toSubmodule S₀) (Submodule.restrictScalars A I)
condTFree : CondTFree A
... | case h
R : Type u
inst✝⁴ : CommRing R
A : Type u
inst✝³ : CommRing A
inst✝² : DecidableEq A
S : Type u
inst✝¹ : CommRing S
inst✝ : Algebra A S
I : Ideal S
hI : DividedPowers I
S₀ : Subalgebra A S
hIS₀ : IsCompl (Subalgebra.toSubmodule S₀) (Submodule.restrictScalars A I)
condTFree : CondTFree A
condD : CondD A
hM : Divi... | Please generate a tactic in lean4 to solve the state.
STATE:
case intro.intro.intro.intro.hy
R : Type u
inst✝⁴ : CommRing R
A : Type u
inst✝³ : CommRing A
inst✝² : DecidableEq A
S : Type u
inst✝¹ : CommRing S
inst✝ : Algebra A S
I : Ideal S
hI : DividedPowers I
S₀ : Subalgebra A S
hIS₀ : IsCompl (Subalgebra.toSubmodule... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/DPAlgebra/Dpow.lean | DividedPowerAlgebra.roby4.Ψ_surjective | [429, 1] | [445, 42] | simp only [Ψ, Φ, f, AlgHom.toRingHom_eq_coe, RingHom.coe_coe,
Algebra.TensorProduct.productMap_apply_tmul, map_one, lift_ι_apply, one_mul] | case h
R : Type u
inst✝⁴ : CommRing R
A : Type u
inst✝³ : CommRing A
inst✝² : DecidableEq A
S : Type u
inst✝¹ : CommRing S
inst✝ : Algebra A S
I : Ideal S
hI : DividedPowers I
S₀ : Subalgebra A S
hIS₀ : IsCompl (Subalgebra.toSubmodule S₀) (Submodule.restrictScalars A I)
condTFree : CondTFree A
condD : CondD A
hM : Divi... | case h
R : Type u
inst✝⁴ : CommRing R
A : Type u
inst✝³ : CommRing A
inst✝² : DecidableEq A
S : Type u
inst✝¹ : CommRing S
inst✝ : Algebra A S
I : Ideal S
hI : DividedPowers I
S₀ : Subalgebra A S
hIS₀ : IsCompl (Subalgebra.toSubmodule S₀) (Submodule.restrictScalars A I)
condTFree : CondTFree A
condD : CondD A
hM : Divi... | Please generate a tactic in lean4 to solve the state.
STATE:
case h
R : Type u
inst✝⁴ : CommRing R
A : Type u
inst✝³ : CommRing A
inst✝² : DecidableEq A
S : Type u
inst✝¹ : CommRing S
inst✝ : Algebra A S
I : Ideal S
hI : DividedPowers I
S₀ : Subalgebra A S
hIS₀ : IsCompl (Subalgebra.toSubmodule S₀) (Submodule.restrictS... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/DPAlgebra/Dpow.lean | DividedPowerAlgebra.roby4.Ψ_surjective | [429, 1] | [445, 42] | simp only [Basis.constr_apply, Finsupp.basisSingleOne_repr, LinearEquiv.refl_apply, zero_smul,
Finsupp.sum_single_index, one_smul] | case h
R : Type u
inst✝⁴ : CommRing R
A : Type u
inst✝³ : CommRing A
inst✝² : DecidableEq A
S : Type u
inst✝¹ : CommRing S
inst✝ : Algebra A S
I : Ideal S
hI : DividedPowers I
S₀ : Subalgebra A S
hIS₀ : IsCompl (Subalgebra.toSubmodule S₀) (Submodule.restrictScalars A I)
condTFree : CondTFree A
condD : CondD A
hM : Divi... | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case h
R : Type u
inst✝⁴ : CommRing R
A : Type u
inst✝³ : CommRing A
inst✝² : DecidableEq A
S : Type u
inst✝¹ : CommRing S
inst✝ : Algebra A S
I : Ideal S
hI : DividedPowers I
S₀ : Subalgebra A S
hIS₀ : IsCompl (Subalgebra.toSubmodule S₀) (Submodule.restrictS... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/DPAlgebra/Dpow.lean | DividedPowerAlgebra.roby4.Ψ_map_eq | [447, 1] | [462, 73] | ext x | R : Type u
inst✝⁴ : CommRing R
A : Type u
inst✝³ : CommRing A
inst✝² : DecidableEq A
S : Type u
inst✝¹ : CommRing S
inst✝ : Algebra A S
I : Ideal S
hI : DividedPowers I
S₀ : Subalgebra A S
hIS₀ : IsCompl (Subalgebra.toSubmodule S₀) (Submodule.restrictScalars A I)
condTFree : CondTFree A
condD : CondD A
hM : DividedPowe... | case h
R : Type u
inst✝⁴ : CommRing R
A : Type u
inst✝³ : CommRing A
inst✝² : DecidableEq A
S : Type u
inst✝¹ : CommRing S
inst✝ : Algebra A S
I : Ideal S
hI : DividedPowers I
S₀ : Subalgebra A S
hIS₀ : IsCompl (Subalgebra.toSubmodule S₀) (Submodule.restrictScalars A I)
condTFree : CondTFree A
condD : CondD A
hM : Divi... | Please generate a tactic in lean4 to solve the state.
STATE:
R : Type u
inst✝⁴ : CommRing R
A : Type u
inst✝³ : CommRing A
inst✝² : DecidableEq A
S : Type u
inst✝¹ : CommRing S
inst✝ : Algebra A S
I : Ideal S
hI : DividedPowers I
S₀ : Subalgebra A S
hIS₀ : IsCompl (Subalgebra.toSubmodule S₀) (Submodule.restrictScalars ... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/DPAlgebra/Dpow.lean | DividedPowerAlgebra.roby4.Ψ_map_eq | [447, 1] | [462, 73] | simp only [Subalgebra.mem_map, Subalgebra.mem_restrictScalars] | case h
R : Type u
inst✝⁴ : CommRing R
A : Type u
inst✝³ : CommRing A
inst✝² : DecidableEq A
S : Type u
inst✝¹ : CommRing S
inst✝ : Algebra A S
I : Ideal S
hI : DividedPowers I
S₀ : Subalgebra A S
hIS₀ : IsCompl (Subalgebra.toSubmodule S₀) (Submodule.restrictScalars A I)
condTFree : CondTFree A
condD : CondD A
hM : Divi... | case h
R : Type u
inst✝⁴ : CommRing R
A : Type u
inst✝³ : CommRing A
inst✝² : DecidableEq A
S : Type u
inst✝¹ : CommRing S
inst✝ : Algebra A S
I : Ideal S
hI : DividedPowers I
S₀ : Subalgebra A S
hIS₀ : IsCompl (Subalgebra.toSubmodule S₀) (Submodule.restrictScalars A I)
condTFree : CondTFree A
condD : CondD A
hM : Divi... | Please generate a tactic in lean4 to solve the state.
STATE:
case h
R : Type u
inst✝⁴ : CommRing R
A : Type u
inst✝³ : CommRing A
inst✝² : DecidableEq A
S : Type u
inst✝¹ : CommRing S
inst✝ : Algebra A S
I : Ideal S
hI : DividedPowers I
S₀ : Subalgebra A S
hIS₀ : IsCompl (Subalgebra.toSubmodule S₀) (Submodule.restrictS... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/DPAlgebra/Dpow.lean | DividedPowerAlgebra.roby4.Ψ_map_eq | [447, 1] | [462, 73] | simp only [Algebra.mem_bot, Set.mem_range, Algebra.TensorProduct.algebraMap_apply,
Algebra.id.map_eq_id, RingHom.id_apply, exists_exists_eq_and] | case h
R : Type u
inst✝⁴ : CommRing R
A : Type u
inst✝³ : CommRing A
inst✝² : DecidableEq A
S : Type u
inst✝¹ : CommRing S
inst✝ : Algebra A S
I : Ideal S
hI : DividedPowers I
S₀ : Subalgebra A S
hIS₀ : IsCompl (Subalgebra.toSubmodule S₀) (Submodule.restrictScalars A I)
condTFree : CondTFree A
condD : CondD A
hM : Divi... | case h
R : Type u
inst✝⁴ : CommRing R
A : Type u
inst✝³ : CommRing A
inst✝² : DecidableEq A
S : Type u
inst✝¹ : CommRing S
inst✝ : Algebra A S
I : Ideal S
hI : DividedPowers I
S₀ : Subalgebra A S
hIS₀ : IsCompl (Subalgebra.toSubmodule S₀) (Submodule.restrictScalars A I)
condTFree : CondTFree A
condD : CondD A
hM : Divi... | Please generate a tactic in lean4 to solve the state.
STATE:
case h
R : Type u
inst✝⁴ : CommRing R
A : Type u
inst✝³ : CommRing A
inst✝² : DecidableEq A
S : Type u
inst✝¹ : CommRing S
inst✝ : Algebra A S
I : Ideal S
hI : DividedPowers I
S₀ : Subalgebra A S
hIS₀ : IsCompl (Subalgebra.toSubmodule S₀) (Submodule.restrictS... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/DPAlgebra/Dpow.lean | DividedPowerAlgebra.roby4.Ψ_map_eq | [447, 1] | [462, 73] | constructor | case h
R : Type u
inst✝⁴ : CommRing R
A : Type u
inst✝³ : CommRing A
inst✝² : DecidableEq A
S : Type u
inst✝¹ : CommRing S
inst✝ : Algebra A S
I : Ideal S
hI : DividedPowers I
S₀ : Subalgebra A S
hIS₀ : IsCompl (Subalgebra.toSubmodule S₀) (Submodule.restrictScalars A I)
condTFree : CondTFree A
condD : CondD A
hM : Divi... | case h.mp
R : Type u
inst✝⁴ : CommRing R
A : Type u
inst✝³ : CommRing A
inst✝² : DecidableEq A
S : Type u
inst✝¹ : CommRing S
inst✝ : Algebra A S
I : Ideal S
hI : DividedPowers I
S₀ : Subalgebra A S
hIS₀ : IsCompl (Subalgebra.toSubmodule S₀) (Submodule.restrictScalars A I)
condTFree : CondTFree A
condD : CondD A
hM : D... | Please generate a tactic in lean4 to solve the state.
STATE:
case h
R : Type u
inst✝⁴ : CommRing R
A : Type u
inst✝³ : CommRing A
inst✝² : DecidableEq A
S : Type u
inst✝¹ : CommRing S
inst✝ : Algebra A S
I : Ideal S
hI : DividedPowers I
S₀ : Subalgebra A S
hIS₀ : IsCompl (Subalgebra.toSubmodule S₀) (Submodule.restrictS... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/DPAlgebra/Dpow.lean | DividedPowerAlgebra.roby4.Ψ_map_eq | [447, 1] | [462, 73] | rintro ⟨p, rfl⟩ | case h.mp
R : Type u
inst✝⁴ : CommRing R
A : Type u
inst✝³ : CommRing A
inst✝² : DecidableEq A
S : Type u
inst✝¹ : CommRing S
inst✝ : Algebra A S
I : Ideal S
hI : DividedPowers I
S₀ : Subalgebra A S
hIS₀ : IsCompl (Subalgebra.toSubmodule S₀) (Submodule.restrictScalars A I)
condTFree : CondTFree A
condD : CondD A
hM : D... | case h.mp.intro
R : Type u
inst✝⁴ : CommRing R
A : Type u
inst✝³ : CommRing A
inst✝² : DecidableEq A
S : Type u
inst✝¹ : CommRing S
inst✝ : Algebra A S
I : Ideal S
hI : DividedPowers I
S₀ : Subalgebra A S
hIS₀ : IsCompl (Subalgebra.toSubmodule S₀) (Submodule.restrictScalars A I)
condTFree : CondTFree A
condD : CondD A
... | Please generate a tactic in lean4 to solve the state.
STATE:
case h.mp
R : Type u
inst✝⁴ : CommRing R
A : Type u
inst✝³ : CommRing A
inst✝² : DecidableEq A
S : Type u
inst✝¹ : CommRing S
inst✝ : Algebra A S
I : Ideal S
hI : DividedPowers I
S₀ : Subalgebra A S
hIS₀ : IsCompl (Subalgebra.toSubmodule S₀) (Submodule.restri... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/DPAlgebra/Dpow.lean | DividedPowerAlgebra.roby4.Ψ_map_eq | [447, 1] | [462, 73] | simp only [Ψ, Algebra.TensorProduct.productMap_apply_tmul, AlgHom.coe_comp, Subalgebra.coe_val,
IsScalarTower.coe_toAlgHom', Function.comp_apply, map_one, mul_one, SetLike.coe_mem] | case h.mp.intro
R : Type u
inst✝⁴ : CommRing R
A : Type u
inst✝³ : CommRing A
inst✝² : DecidableEq A
S : Type u
inst✝¹ : CommRing S
inst✝ : Algebra A S
I : Ideal S
hI : DividedPowers I
S₀ : Subalgebra A S
hIS₀ : IsCompl (Subalgebra.toSubmodule S₀) (Submodule.restrictScalars A I)
condTFree : CondTFree A
condD : CondD A
... | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case h.mp.intro
R : Type u
inst✝⁴ : CommRing R
A : Type u
inst✝³ : CommRing A
inst✝² : DecidableEq A
S : Type u
inst✝¹ : CommRing S
inst✝ : Algebra A S
I : Ideal S
hI : DividedPowers I
S₀ : Subalgebra A S
hIS₀ : IsCompl (Subalgebra.toSubmodule S₀) (Submodule.... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/DPAlgebra/Dpow.lean | DividedPowerAlgebra.roby4.Ψ_map_eq | [447, 1] | [462, 73] | intro hx | case h.mpr
R : Type u
inst✝⁴ : CommRing R
A : Type u
inst✝³ : CommRing A
inst✝² : DecidableEq A
S : Type u
inst✝¹ : CommRing S
inst✝ : Algebra A S
I : Ideal S
hI : DividedPowers I
S₀ : Subalgebra A S
hIS₀ : IsCompl (Subalgebra.toSubmodule S₀) (Submodule.restrictScalars A I)
condTFree : CondTFree A
condD : CondD A
hM : ... | case h.mpr
R : Type u
inst✝⁴ : CommRing R
A : Type u
inst✝³ : CommRing A
inst✝² : DecidableEq A
S : Type u
inst✝¹ : CommRing S
inst✝ : Algebra A S
I : Ideal S
hI : DividedPowers I
S₀ : Subalgebra A S
hIS₀ : IsCompl (Subalgebra.toSubmodule S₀) (Submodule.restrictScalars A I)
condTFree : CondTFree A
condD : CondD A
hM : ... | Please generate a tactic in lean4 to solve the state.
STATE:
case h.mpr
R : Type u
inst✝⁴ : CommRing R
A : Type u
inst✝³ : CommRing A
inst✝² : DecidableEq A
S : Type u
inst✝¹ : CommRing S
inst✝ : Algebra A S
I : Ideal S
hI : DividedPowers I
S₀ : Subalgebra A S
hIS₀ : IsCompl (Subalgebra.toSubmodule S₀) (Submodule.restr... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/DPAlgebra/Dpow.lean | DividedPowerAlgebra.roby4.Ψ_map_eq | [447, 1] | [462, 73] | use MvPolynomial.X ⟨x, hx⟩ | case h.mpr
R : Type u
inst✝⁴ : CommRing R
A : Type u
inst✝³ : CommRing A
inst✝² : DecidableEq A
S : Type u
inst✝¹ : CommRing S
inst✝ : Algebra A S
I : Ideal S
hI : DividedPowers I
S₀ : Subalgebra A S
hIS₀ : IsCompl (Subalgebra.toSubmodule S₀) (Submodule.restrictScalars A I)
condTFree : CondTFree A
condD : CondD A
hM : ... | case h
R : Type u
inst✝⁴ : CommRing R
A : Type u
inst✝³ : CommRing A
inst✝² : DecidableEq A
S : Type u
inst✝¹ : CommRing S
inst✝ : Algebra A S
I : Ideal S
hI : DividedPowers I
S₀ : Subalgebra A S
hIS₀ : IsCompl (Subalgebra.toSubmodule S₀) (Submodule.restrictScalars A I)
condTFree : CondTFree A
condD : CondD A
hM : Divi... | Please generate a tactic in lean4 to solve the state.
STATE:
case h.mpr
R : Type u
inst✝⁴ : CommRing R
A : Type u
inst✝³ : CommRing A
inst✝² : DecidableEq A
S : Type u
inst✝¹ : CommRing S
inst✝ : Algebra A S
I : Ideal S
hI : DividedPowers I
S₀ : Subalgebra A S
hIS₀ : IsCompl (Subalgebra.toSubmodule S₀) (Submodule.restr... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/DPAlgebra/Dpow.lean | DividedPowerAlgebra.roby4.Ψ_map_eq | [447, 1] | [462, 73] | simp only [Ψ, Algebra.TensorProduct.productMap_apply_tmul, AlgHom.coe_comp, Subalgebra.coe_val,
IsScalarTower.coe_toAlgHom', Function.comp_apply, map_one, mul_one] | case h
R : Type u
inst✝⁴ : CommRing R
A : Type u
inst✝³ : CommRing A
inst✝² : DecidableEq A
S : Type u
inst✝¹ : CommRing S
inst✝ : Algebra A S
I : Ideal S
hI : DividedPowers I
S₀ : Subalgebra A S
hIS₀ : IsCompl (Subalgebra.toSubmodule S₀) (Submodule.restrictScalars A I)
condTFree : CondTFree A
condD : CondD A
hM : Divi... | case h
R : Type u
inst✝⁴ : CommRing R
A : Type u
inst✝³ : CommRing A
inst✝² : DecidableEq A
S : Type u
inst✝¹ : CommRing S
inst✝ : Algebra A S
I : Ideal S
hI : DividedPowers I
S₀ : Subalgebra A S
hIS₀ : IsCompl (Subalgebra.toSubmodule S₀) (Submodule.restrictScalars A I)
condTFree : CondTFree A
condD : CondD A
hM : Divi... | Please generate a tactic in lean4 to solve the state.
STATE:
case h
R : Type u
inst✝⁴ : CommRing R
A : Type u
inst✝³ : CommRing A
inst✝² : DecidableEq A
S : Type u
inst✝¹ : CommRing S
inst✝ : Algebra A S
I : Ideal S
hI : DividedPowers I
S₀ : Subalgebra A S
hIS₀ : IsCompl (Subalgebra.toSubmodule S₀) (Submodule.restrictS... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/DPAlgebra/Dpow.lean | DividedPowerAlgebra.roby4.Ψ_map_eq | [447, 1] | [462, 73] | rw [algebraMap_eq] | case h
R : Type u
inst✝⁴ : CommRing R
A : Type u
inst✝³ : CommRing A
inst✝² : DecidableEq A
S : Type u
inst✝¹ : CommRing S
inst✝ : Algebra A S
I : Ideal S
hI : DividedPowers I
S₀ : Subalgebra A S
hIS₀ : IsCompl (Subalgebra.toSubmodule S₀) (Submodule.restrictScalars A I)
condTFree : CondTFree A
condD : CondD A
hM : Divi... | case h
R : Type u
inst✝⁴ : CommRing R
A : Type u
inst✝³ : CommRing A
inst✝² : DecidableEq A
S : Type u
inst✝¹ : CommRing S
inst✝ : Algebra A S
I : Ideal S
hI : DividedPowers I
S₀ : Subalgebra A S
hIS₀ : IsCompl (Subalgebra.toSubmodule S₀) (Submodule.restrictScalars A I)
condTFree : CondTFree A
condD : CondD A
hM : Divi... | Please generate a tactic in lean4 to solve the state.
STATE:
case h
R : Type u
inst✝⁴ : CommRing R
A : Type u
inst✝³ : CommRing A
inst✝² : DecidableEq A
S : Type u
inst✝¹ : CommRing S
inst✝ : Algebra A S
I : Ideal S
hI : DividedPowers I
S₀ : Subalgebra A S
hIS₀ : IsCompl (Subalgebra.toSubmodule S₀) (Submodule.restrictS... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/DPAlgebra/Dpow.lean | DividedPowerAlgebra.roby4.Ψ_map_eq | [447, 1] | [462, 73] | simp only [AlgHom.toRingHom_eq_coe, RingHom.coe_coe, aeval_X, id_eq] | case h
R : Type u
inst✝⁴ : CommRing R
A : Type u
inst✝³ : CommRing A
inst✝² : DecidableEq A
S : Type u
inst✝¹ : CommRing S
inst✝ : Algebra A S
I : Ideal S
hI : DividedPowers I
S₀ : Subalgebra A S
hIS₀ : IsCompl (Subalgebra.toSubmodule S₀) (Submodule.restrictScalars A I)
condTFree : CondTFree A
condD : CondD A
hM : Divi... | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case h
R : Type u
inst✝⁴ : CommRing R
A : Type u
inst✝³ : CommRing A
inst✝² : DecidableEq A
S : Type u
inst✝¹ : CommRing S
inst✝ : Algebra A S
I : Ideal S
hI : DividedPowers I
S₀ : Subalgebra A S
hIS₀ : IsCompl (Subalgebra.toSubmodule S₀) (Submodule.restrictS... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/DPAlgebra/Dpow.lean | DividedPowerAlgebra.roby4.K_eq_span | [465, 1] | [470, 73] | simp [K, i1, i2] | R : Type u
inst✝⁴ : CommRing R
A : Type u
inst✝³ : CommRing A
inst✝² : DecidableEq A
S : Type u
inst✝¹ : CommRing S
inst✝ : Algebra A S
I : Ideal S
hI : DividedPowers I
S₀ : Subalgebra A S
hIS₀ : IsCompl (Subalgebra.toSubmodule S₀) (Submodule.restrictScalars A I)
condTFree : CondTFree A
condD : CondD A
hM : DividedPowe... | R : Type u
inst✝⁴ : CommRing R
A : Type u
inst✝³ : CommRing A
inst✝² : DecidableEq A
S : Type u
inst✝¹ : CommRing S
inst✝ : Algebra A S
I : Ideal S
hI : DividedPowers I
S₀ : Subalgebra A S
hIS₀ : IsCompl (Subalgebra.toSubmodule S₀) (Submodule.restrictScalars A I)
condTFree : CondTFree A
condD : CondD A
hM : DividedPowe... | Please generate a tactic in lean4 to solve the state.
STATE:
R : Type u
inst✝⁴ : CommRing R
A : Type u
inst✝³ : CommRing A
inst✝² : DecidableEq A
S : Type u
inst✝¹ : CommRing S
inst✝ : Algebra A S
I : Ideal S
hI : DividedPowers I
S₀ : Subalgebra A S
hIS₀ : IsCompl (Subalgebra.toSubmodule S₀) (Submodule.restrictScalars ... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/DPAlgebra/Dpow.lean | DividedPowerAlgebra.roby4.K_eq_span | [465, 1] | [470, 73] | rw [augIdeal_eq_span, Ideal.map_span] | R : Type u
inst✝⁴ : CommRing R
A : Type u
inst✝³ : CommRing A
inst✝² : DecidableEq A
S : Type u
inst✝¹ : CommRing S
inst✝ : Algebra A S
I : Ideal S
hI : DividedPowers I
S₀ : Subalgebra A S
hIS₀ : IsCompl (Subalgebra.toSubmodule S₀) (Submodule.restrictScalars A I)
condTFree : CondTFree A
condD : CondD A
hM : DividedPowe... | R : Type u
inst✝⁴ : CommRing R
A : Type u
inst✝³ : CommRing A
inst✝² : DecidableEq A
S : Type u
inst✝¹ : CommRing S
inst✝ : Algebra A S
I : Ideal S
hI : DividedPowers I
S₀ : Subalgebra A S
hIS₀ : IsCompl (Subalgebra.toSubmodule S₀) (Submodule.restrictScalars A I)
condTFree : CondTFree A
condD : CondD A
hM : DividedPowe... | Please generate a tactic in lean4 to solve the state.
STATE:
R : Type u
inst✝⁴ : CommRing R
A : Type u
inst✝³ : CommRing A
inst✝² : DecidableEq A
S : Type u
inst✝¹ : CommRing S
inst✝ : Algebra A S
I : Ideal S
hI : DividedPowers I
S₀ : Subalgebra A S
hIS₀ : IsCompl (Subalgebra.toSubmodule S₀) (Submodule.restrictScalars ... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/DPAlgebra/Dpow.lean | DividedPowerAlgebra.roby4.K_eq_span | [465, 1] | [470, 73] | simp only [Algebra.TensorProduct.includeRight_apply, Set.image_image2] | R : Type u
inst✝⁴ : CommRing R
A : Type u
inst✝³ : CommRing A
inst✝² : DecidableEq A
S : Type u
inst✝¹ : CommRing S
inst✝ : Algebra A S
I : Ideal S
hI : DividedPowers I
S₀ : Subalgebra A S
hIS₀ : IsCompl (Subalgebra.toSubmodule S₀) (Submodule.restrictScalars A I)
condTFree : CondTFree A
condD : CondD A
hM : DividedPowe... | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
R : Type u
inst✝⁴ : CommRing R
A : Type u
inst✝³ : CommRing A
inst✝² : DecidableEq A
S : Type u
inst✝¹ : CommRing S
inst✝ : Algebra A S
I : Ideal S
hI : DividedPowers I
S₀ : Subalgebra A S
hIS₀ : IsCompl (Subalgebra.toSubmodule S₀) (Submodule.restrictScalars ... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/DPAlgebra/Dpow.lean | DividedPowerAlgebra.T_free_and_D_to_QSplit | [506, 1] | [595, 17] | intro S _ _ I hI S₀ hIS₀ | R : Type u
inst✝⁴ : CommRing R
A : Type u
inst✝³ : CommRing A
inst✝² : DecidableEq A
S : Type u
inst✝¹ : CommRing S
inst✝ : Algebra A S
I : Ideal S
hI : DividedPowers I
S₀ : Subalgebra A S
hIS₀ : IsCompl (Subalgebra.toSubmodule S₀) (Submodule.restrictScalars A I)
condTFree : CondTFree A
condD : CondD A
hM : DividedPowe... | R : Type u
inst✝⁶ : CommRing R
A : Type u
inst✝⁵ : CommRing A
inst✝⁴ : DecidableEq A
S✝ : Type u
inst✝³ : CommRing S✝
inst✝² : Algebra A S✝
I✝ : Ideal S✝
hI✝ : DividedPowers I✝
S₀✝ : Subalgebra A S✝
hIS₀✝ : IsCompl (Subalgebra.toSubmodule S₀✝) (Submodule.restrictScalars A I✝)
condTFree : CondTFree A
condD : CondD A
hM ... | Please generate a tactic in lean4 to solve the state.
STATE:
R : Type u
inst✝⁴ : CommRing R
A : Type u
inst✝³ : CommRing A
inst✝² : DecidableEq A
S : Type u
inst✝¹ : CommRing S
inst✝ : Algebra A S
I : Ideal S
hI : DividedPowers I
S₀ : Subalgebra A S
hIS₀ : IsCompl (Subalgebra.toSubmodule S₀) (Submodule.restrictScalars ... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/DPAlgebra/Dpow.lean | DividedPowerAlgebra.T_free_and_D_to_QSplit | [506, 1] | [595, 17] | let M := I →₀ A | R : Type u
inst✝⁶ : CommRing R
A : Type u
inst✝⁵ : CommRing A
inst✝⁴ : DecidableEq A
S✝ : Type u
inst✝³ : CommRing S✝
inst✝² : Algebra A S✝
I✝ : Ideal S✝
hI✝ : DividedPowers I✝
S₀✝ : Subalgebra A S✝
hIS₀✝ : IsCompl (Subalgebra.toSubmodule S₀✝) (Submodule.restrictScalars A I✝)
condTFree : CondTFree A
condD : CondD A
hM ... | R : Type u
inst✝⁶ : CommRing R
A : Type u
inst✝⁵ : CommRing A
inst✝⁴ : DecidableEq A
S✝ : Type u
inst✝³ : CommRing S✝
inst✝² : Algebra A S✝
I✝ : Ideal S✝
hI✝ : DividedPowers I✝
S₀✝ : Subalgebra A S✝
hIS₀✝ : IsCompl (Subalgebra.toSubmodule S₀✝) (Submodule.restrictScalars A I✝)
condTFree : CondTFree A
condD : CondD A
hM ... | Please generate a tactic in lean4 to solve the state.
STATE:
R : Type u
inst✝⁶ : CommRing R
A : Type u
inst✝⁵ : CommRing A
inst✝⁴ : DecidableEq A
S✝ : Type u
inst✝³ : CommRing S✝
inst✝² : Algebra A S✝
I✝ : Ideal S✝
hI✝ : DividedPowers I✝
S₀✝ : Subalgebra A S✝
hIS₀✝ : IsCompl (Subalgebra.toSubmodule S₀✝) (Submodule.rest... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/DPAlgebra/Dpow.lean | DividedPowerAlgebra.T_free_and_D_to_QSplit | [506, 1] | [595, 17] | let R := MvPolynomial S₀ A | R : Type u
inst✝⁶ : CommRing R
A : Type u
inst✝⁵ : CommRing A
inst✝⁴ : DecidableEq A
S✝ : Type u
inst✝³ : CommRing S✝
inst✝² : Algebra A S✝
I✝ : Ideal S✝
hI✝ : DividedPowers I✝
S₀✝ : Subalgebra A S✝
hIS₀✝ : IsCompl (Subalgebra.toSubmodule S₀✝) (Submodule.restrictScalars A I✝)
condTFree : CondTFree A
condD : CondD A
hM ... | R✝ : Type u
inst✝⁶ : CommRing R✝
A : Type u
inst✝⁵ : CommRing A
inst✝⁴ : DecidableEq A
S✝ : Type u
inst✝³ : CommRing S✝
inst✝² : Algebra A S✝
I✝ : Ideal S✝
hI✝ : DividedPowers I✝
S₀✝ : Subalgebra A S✝
hIS₀✝ : IsCompl (Subalgebra.toSubmodule S₀✝) (Submodule.restrictScalars A I✝)
condTFree : CondTFree A
condD : CondD A
h... | Please generate a tactic in lean4 to solve the state.
STATE:
R : Type u
inst✝⁶ : CommRing R
A : Type u
inst✝⁵ : CommRing A
inst✝⁴ : DecidableEq A
S✝ : Type u
inst✝³ : CommRing S✝
inst✝² : Algebra A S✝
I✝ : Ideal S✝
hI✝ : DividedPowers I✝
S₀✝ : Subalgebra A S✝
hIS₀✝ : IsCompl (Subalgebra.toSubmodule S₀✝) (Submodule.rest... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/DPAlgebra/Dpow.lean | DividedPowerAlgebra.T_free_and_D_to_QSplit | [506, 1] | [595, 17] | let D := DividedPowerAlgebra A M | R✝ : Type u
inst✝⁶ : CommRing R✝
A : Type u
inst✝⁵ : CommRing A
inst✝⁴ : DecidableEq A
S✝ : Type u
inst✝³ : CommRing S✝
inst✝² : Algebra A S✝
I✝ : Ideal S✝
hI✝ : DividedPowers I✝
S₀✝ : Subalgebra A S✝
hIS₀✝ : IsCompl (Subalgebra.toSubmodule S₀✝) (Submodule.restrictScalars A I✝)
condTFree : CondTFree A
condD : CondD A
h... | R✝ : Type u
inst✝⁶ : CommRing R✝
A : Type u
inst✝⁵ : CommRing A
inst✝⁴ : DecidableEq A
S✝ : Type u
inst✝³ : CommRing S✝
inst✝² : Algebra A S✝
I✝ : Ideal S✝
hI✝ : DividedPowers I✝
S₀✝ : Subalgebra A S✝
hIS₀✝ : IsCompl (Subalgebra.toSubmodule S₀✝) (Submodule.restrictScalars A I✝)
condTFree : CondTFree A
condD : CondD A
h... | Please generate a tactic in lean4 to solve the state.
STATE:
R✝ : Type u
inst✝⁶ : CommRing R✝
A : Type u
inst✝⁵ : CommRing A
inst✝⁴ : DecidableEq A
S✝ : Type u
inst✝³ : CommRing S✝
inst✝² : Algebra A S✝
I✝ : Ideal S✝
hI✝ : DividedPowers I✝
S₀✝ : Subalgebra A S✝
hIS₀✝ : IsCompl (Subalgebra.toSubmodule S₀✝) (Submodule.re... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/DPAlgebra/Dpow.lean | DividedPowerAlgebra.T_free_and_D_to_QSplit | [506, 1] | [595, 17] | obtain ⟨hM, hM_eq⟩ := condD M | R✝ : Type u
inst✝⁶ : CommRing R✝
A : Type u
inst✝⁵ : CommRing A
inst✝⁴ : DecidableEq A
S✝ : Type u
inst✝³ : CommRing S✝
inst✝² : Algebra A S✝
I✝ : Ideal S✝
hI✝ : DividedPowers I✝
S₀✝ : Subalgebra A S✝
hIS₀✝ : IsCompl (Subalgebra.toSubmodule S₀✝) (Submodule.restrictScalars A I✝)
condTFree : CondTFree A
condD : CondD A
h... | case intro
R✝ : Type u
inst✝⁶ : CommRing R✝
A : Type u
inst✝⁵ : CommRing A
inst✝⁴ : DecidableEq A
S✝ : Type u
inst✝³ : CommRing S✝
inst✝² : Algebra A S✝
I✝ : Ideal S✝
hI✝ : DividedPowers I✝
S₀✝ : Subalgebra A S✝
hIS₀✝ : IsCompl (Subalgebra.toSubmodule S₀✝) (Submodule.restrictScalars A I✝)
condTFree : CondTFree A
condD ... | Please generate a tactic in lean4 to solve the state.
STATE:
R✝ : Type u
inst✝⁶ : CommRing R✝
A : Type u
inst✝⁵ : CommRing A
inst✝⁴ : DecidableEq A
S✝ : Type u
inst✝³ : CommRing S✝
inst✝² : Algebra A S✝
I✝ : Ideal S✝
hI✝ : DividedPowers I✝
S₀✝ : Subalgebra A S✝
hIS₀✝ : IsCompl (Subalgebra.toSubmodule S₀✝) (Submodule.re... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/DPAlgebra/Dpow.lean | DividedPowerAlgebra.T_free_and_D_to_QSplit | [506, 1] | [595, 17] | haveI hdpM_free : Module.Free A D := DividedPowerAlgebra.toModule_free A M | case intro
R✝ : Type u
inst✝⁶ : CommRing R✝
A : Type u
inst✝⁵ : CommRing A
inst✝⁴ : DecidableEq A
S✝ : Type u
inst✝³ : CommRing S✝
inst✝² : Algebra A S✝
I✝ : Ideal S✝
hI✝ : DividedPowers I✝
S₀✝ : Subalgebra A S✝
hIS₀✝ : IsCompl (Subalgebra.toSubmodule S₀✝) (Submodule.restrictScalars A I✝)
condTFree : CondTFree A
condD ... | case intro
R✝ : Type u
inst✝⁶ : CommRing R✝
A : Type u
inst✝⁵ : CommRing A
inst✝⁴ : DecidableEq A
S✝ : Type u
inst✝³ : CommRing S✝
inst✝² : Algebra A S✝
I✝ : Ideal S✝
hI✝ : DividedPowers I✝
S₀✝ : Subalgebra A S✝
hIS₀✝ : IsCompl (Subalgebra.toSubmodule S₀✝) (Submodule.restrictScalars A I✝)
condTFree : CondTFree A
condD ... | Please generate a tactic in lean4 to solve the state.
STATE:
case intro
R✝ : Type u
inst✝⁶ : CommRing R✝
A : Type u
inst✝⁵ : CommRing A
inst✝⁴ : DecidableEq A
S✝ : Type u
inst✝³ : CommRing S✝
inst✝² : Algebra A S✝
I✝ : Ideal S✝
hI✝ : DividedPowers I✝
S₀✝ : Subalgebra A S✝
hIS₀✝ : IsCompl (Subalgebra.toSubmodule S₀✝) (S... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/DPAlgebra/Dpow.lean | DividedPowerAlgebra.T_free_and_D_to_QSplit | [506, 1] | [595, 17] | haveI hR_free : Module.Free A R :=
Module.Free.of_basis (MvPolynomial.basisMonomials _ _) | case intro
R✝ : Type u
inst✝⁶ : CommRing R✝
A : Type u
inst✝⁵ : CommRing A
inst✝⁴ : DecidableEq A
S✝ : Type u
inst✝³ : CommRing S✝
inst✝² : Algebra A S✝
I✝ : Ideal S✝
hI✝ : DividedPowers I✝
S₀✝ : Subalgebra A S✝
hIS₀✝ : IsCompl (Subalgebra.toSubmodule S₀✝) (Submodule.restrictScalars A I✝)
condTFree : CondTFree A
condD ... | case intro
R✝ : Type u
inst✝⁶ : CommRing R✝
A : Type u
inst✝⁵ : CommRing A
inst✝⁴ : DecidableEq A
S✝ : Type u
inst✝³ : CommRing S✝
inst✝² : Algebra A S✝
I✝ : Ideal S✝
hI✝ : DividedPowers I✝
S₀✝ : Subalgebra A S✝
hIS₀✝ : IsCompl (Subalgebra.toSubmodule S₀✝) (Submodule.restrictScalars A I✝)
condTFree : CondTFree A
condD ... | Please generate a tactic in lean4 to solve the state.
STATE:
case intro
R✝ : Type u
inst✝⁶ : CommRing R✝
A : Type u
inst✝⁵ : CommRing A
inst✝⁴ : DecidableEq A
S✝ : Type u
inst✝³ : CommRing S✝
inst✝² : Algebra A S✝
I✝ : Ideal S✝
hI✝ : DividedPowers I✝
S₀✝ : Subalgebra A S✝
hIS₀✝ : IsCompl (Subalgebra.toSubmodule S₀✝) (S... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/DPAlgebra/Dpow.lean | DividedPowerAlgebra.T_free_and_D_to_QSplit | [506, 1] | [595, 17] | let T := R ⊗[A] D | case intro
R✝ : Type u
inst✝⁶ : CommRing R✝
A : Type u
inst✝⁵ : CommRing A
inst✝⁴ : DecidableEq A
S✝ : Type u
inst✝³ : CommRing S✝
inst✝² : Algebra A S✝
I✝ : Ideal S✝
hI✝ : DividedPowers I✝
S₀✝ : Subalgebra A S✝
hIS₀✝ : IsCompl (Subalgebra.toSubmodule S₀✝) (Submodule.restrictScalars A I✝)
condTFree : CondTFree A
condD ... | case intro
R✝ : Type u
inst✝⁶ : CommRing R✝
A : Type u
inst✝⁵ : CommRing A
inst✝⁴ : DecidableEq A
S✝ : Type u
inst✝³ : CommRing S✝
inst✝² : Algebra A S✝
I✝ : Ideal S✝
hI✝ : DividedPowers I✝
S₀✝ : Subalgebra A S✝
hIS₀✝ : IsCompl (Subalgebra.toSubmodule S₀✝) (Submodule.restrictScalars A I✝)
condTFree : CondTFree A
condD ... | Please generate a tactic in lean4 to solve the state.
STATE:
case intro
R✝ : Type u
inst✝⁶ : CommRing R✝
A : Type u
inst✝⁵ : CommRing A
inst✝⁴ : DecidableEq A
S✝ : Type u
inst✝³ : CommRing S✝
inst✝² : Algebra A S✝
I✝ : Ideal S✝
hI✝ : DividedPowers I✝
S₀✝ : Subalgebra A S✝
hIS₀✝ : IsCompl (Subalgebra.toSubmodule S₀✝) (S... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/DPAlgebra/Dpow.lean | DividedPowerAlgebra.T_free_and_D_to_QSplit | [506, 1] | [595, 17] | use T, by infer_instance, by infer_instance | case intro
R✝ : Type u
inst✝⁶ : CommRing R✝
A : Type u
inst✝⁵ : CommRing A
inst✝⁴ : DecidableEq A
S✝ : Type u
inst✝³ : CommRing S✝
inst✝² : Algebra A S✝
I✝ : Ideal S✝
hI✝ : DividedPowers I✝
S₀✝ : Subalgebra A S✝
hIS₀✝ : IsCompl (Subalgebra.toSubmodule S₀✝) (Submodule.restrictScalars A I✝)
condTFree : CondTFree A
condD ... | case h
R✝ : Type u
inst✝⁶ : CommRing R✝
A : Type u
inst✝⁵ : CommRing A
inst✝⁴ : DecidableEq A
S✝ : Type u
inst✝³ : CommRing S✝
inst✝² : Algebra A S✝
I✝ : Ideal S✝
hI✝ : DividedPowers I✝
S₀✝ : Subalgebra A S✝
hIS₀✝ : IsCompl (Subalgebra.toSubmodule S₀✝) (Submodule.restrictScalars A I✝)
condTFree : CondTFree A
condD : Co... | Please generate a tactic in lean4 to solve the state.
STATE:
case intro
R✝ : Type u
inst✝⁶ : CommRing R✝
A : Type u
inst✝⁵ : CommRing A
inst✝⁴ : DecidableEq A
S✝ : Type u
inst✝³ : CommRing S✝
inst✝² : Algebra A S✝
I✝ : Ideal S✝
hI✝ : DividedPowers I✝
S₀✝ : Subalgebra A S✝
hIS₀✝ : IsCompl (Subalgebra.toSubmodule S₀✝) (S... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/DPAlgebra/Dpow.lean | DividedPowerAlgebra.T_free_and_D_to_QSplit | [506, 1] | [595, 17] | let idK : Ideal T := K A ⊥ (augIdeal A M) | case h
R✝ : Type u
inst✝⁶ : CommRing R✝
A : Type u
inst✝⁵ : CommRing A
inst✝⁴ : DecidableEq A
S✝ : Type u
inst✝³ : CommRing S✝
inst✝² : Algebra A S✝
I✝ : Ideal S✝
hI✝ : DividedPowers I✝
S₀✝ : Subalgebra A S✝
hIS₀✝ : IsCompl (Subalgebra.toSubmodule S₀✝) (Submodule.restrictScalars A I✝)
condTFree : CondTFree A
condD : Co... | case h
R✝ : Type u
inst✝⁶ : CommRing R✝
A : Type u
inst✝⁵ : CommRing A
inst✝⁴ : DecidableEq A
S✝ : Type u
inst✝³ : CommRing S✝
inst✝² : Algebra A S✝
I✝ : Ideal S✝
hI✝ : DividedPowers I✝
S₀✝ : Subalgebra A S✝
hIS₀✝ : IsCompl (Subalgebra.toSubmodule S₀✝) (Submodule.restrictScalars A I✝)
condTFree : CondTFree A
condD : Co... | Please generate a tactic in lean4 to solve the state.
STATE:
case h
R✝ : Type u
inst✝⁶ : CommRing R✝
A : Type u
inst✝⁵ : CommRing A
inst✝⁴ : DecidableEq A
S✝ : Type u
inst✝³ : CommRing S✝
inst✝² : Algebra A S✝
I✝ : Ideal S✝
hI✝ : DividedPowers I✝
S₀✝ : Subalgebra A S✝
hIS₀✝ : IsCompl (Subalgebra.toSubmodule S₀✝) (Submo... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/DPAlgebra/Dpow.lean | DividedPowerAlgebra.T_free_and_D_to_QSplit | [506, 1] | [595, 17] | have hidK : idK = Ideal.map Algebra.TensorProduct.includeRight (augIdeal A M) := by
simp only [K, Ideal.map_bot, i2, ge_iff_le, bot_le, sup_of_le_right, idK] | case h
R✝ : Type u
inst✝⁶ : CommRing R✝
A : Type u
inst✝⁵ : CommRing A
inst✝⁴ : DecidableEq A
S✝ : Type u
inst✝³ : CommRing S✝
inst✝² : Algebra A S✝
I✝ : Ideal S✝
hI✝ : DividedPowers I✝
S₀✝ : Subalgebra A S✝
hIS₀✝ : IsCompl (Subalgebra.toSubmodule S₀✝) (Submodule.restrictScalars A I✝)
condTFree : CondTFree A
condD : Co... | case h
R✝ : Type u
inst✝⁶ : CommRing R✝
A : Type u
inst✝⁵ : CommRing A
inst✝⁴ : DecidableEq A
S✝ : Type u
inst✝³ : CommRing S✝
inst✝² : Algebra A S✝
I✝ : Ideal S✝
hI✝ : DividedPowers I✝
S₀✝ : Subalgebra A S✝
hIS₀✝ : IsCompl (Subalgebra.toSubmodule S₀✝) (Submodule.restrictScalars A I✝)
condTFree : CondTFree A
condD : Co... | Please generate a tactic in lean4 to solve the state.
STATE:
case h
R✝ : Type u
inst✝⁶ : CommRing R✝
A : Type u
inst✝⁵ : CommRing A
inst✝⁴ : DecidableEq A
S✝ : Type u
inst✝³ : CommRing S✝
inst✝² : Algebra A S✝
I✝ : Ideal S✝
hI✝ : DividedPowers I✝
S₀✝ : Subalgebra A S✝
hIS₀✝ : IsCompl (Subalgebra.toSubmodule S₀✝) (Submo... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/DPAlgebra/Dpow.lean | DividedPowerAlgebra.T_free_and_D_to_QSplit | [506, 1] | [595, 17] | use idK | case h
R✝ : Type u
inst✝⁶ : CommRing R✝
A : Type u
inst✝⁵ : CommRing A
inst✝⁴ : DecidableEq A
S✝ : Type u
inst✝³ : CommRing S✝
inst✝² : Algebra A S✝
I✝ : Ideal S✝
hI✝ : DividedPowers I✝
S₀✝ : Subalgebra A S✝
hIS₀✝ : IsCompl (Subalgebra.toSubmodule S₀✝) (Submodule.restrictScalars A I✝)
condTFree : CondTFree A
condD : Co... | case h
R✝ : Type u
inst✝⁶ : CommRing R✝
A : Type u
inst✝⁵ : CommRing A
inst✝⁴ : DecidableEq A
S✝ : Type u
inst✝³ : CommRing S✝
inst✝² : Algebra A S✝
I✝ : Ideal S✝
hI✝ : DividedPowers I✝
S₀✝ : Subalgebra A S✝
hIS₀✝ : IsCompl (Subalgebra.toSubmodule S₀✝) (Submodule.restrictScalars A I✝)
condTFree : CondTFree A
condD : Co... | Please generate a tactic in lean4 to solve the state.
STATE:
case h
R✝ : Type u
inst✝⁶ : CommRing R✝
A : Type u
inst✝⁵ : CommRing A
inst✝⁴ : DecidableEq A
S✝ : Type u
inst✝³ : CommRing S✝
inst✝² : Algebra A S✝
I✝ : Ideal S✝
hI✝ : DividedPowers I✝
S₀✝ : Subalgebra A S✝
hIS₀✝ : IsCompl (Subalgebra.toSubmodule S₀✝) (Submo... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/DPAlgebra/Dpow.lean | DividedPowerAlgebra.T_free_and_D_to_QSplit | [506, 1] | [595, 17] | let hK : DividedPowers idK := (condτ A S I S₀ condTFree hM).choose | case h
R✝ : Type u
inst✝⁶ : CommRing R✝
A : Type u
inst✝⁵ : CommRing A
inst✝⁴ : DecidableEq A
S✝ : Type u
inst✝³ : CommRing S✝
inst✝² : Algebra A S✝
I✝ : Ideal S✝
hI✝ : DividedPowers I✝
S₀✝ : Subalgebra A S✝
hIS₀✝ : IsCompl (Subalgebra.toSubmodule S₀✝) (Submodule.restrictScalars A I✝)
condTFree : CondTFree A
condD : Co... | case h
R✝ : Type u
inst✝⁶ : CommRing R✝
A : Type u
inst✝⁵ : CommRing A
inst✝⁴ : DecidableEq A
S✝ : Type u
inst✝³ : CommRing S✝
inst✝² : Algebra A S✝
I✝ : Ideal S✝
hI✝ : DividedPowers I✝
S₀✝ : Subalgebra A S✝
hIS₀✝ : IsCompl (Subalgebra.toSubmodule S₀✝) (Submodule.restrictScalars A I✝)
condTFree : CondTFree A
condD : Co... | Please generate a tactic in lean4 to solve the state.
STATE:
case h
R✝ : Type u
inst✝⁶ : CommRing R✝
A : Type u
inst✝⁵ : CommRing A
inst✝⁴ : DecidableEq A
S✝ : Type u
inst✝³ : CommRing S✝
inst✝² : Algebra A S✝
I✝ : Ideal S✝
hI✝ : DividedPowers I✝
S₀✝ : Subalgebra A S✝
hIS₀✝ : IsCompl (Subalgebra.toSubmodule S₀✝) (Submo... |
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