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/SubDPIdeal.lean | DividedPowers.isSubDPIdeal_sup | [187, 1] | [198, 50] | . apply Ideal.span_mono (Set.subset_union_left _ _)
rw [J.span_eq]; exact hJ.2 n hn a ha | case inl
A : Type u_1
inst✝ : CommSemiring A
I : Ideal A
hI : DividedPowers I
J K : Ideal A
hJ : hI.isSubDPIdeal J
hK : hI.isSubDPIdeal K
n : ℕ
hn : n ≠ 0
a : A
ha : a ∈ ↑J
⊢ hI.dpow n a ∈ Ideal.span (↑J ∪ ↑K)
case inr
A : Type u_1
inst✝ : CommSemiring A
I : Ideal A
hI : DividedPowers I
J K : Ideal A
hJ : hI.isSubDPId... | case inr
A : Type u_1
inst✝ : CommSemiring A
I : Ideal A
hI : DividedPowers I
J K : Ideal A
hJ : hI.isSubDPIdeal J
hK : hI.isSubDPIdeal K
n : ℕ
hn : n ≠ 0
a : A
ha : a ∈ ↑K
⊢ hI.dpow n a ∈ Ideal.span (↑J ∪ ↑K) | Please generate a tactic in lean4 to solve the state.
STATE:
case inl
A : Type u_1
inst✝ : CommSemiring A
I : Ideal A
hI : DividedPowers I
J K : Ideal A
hJ : hI.isSubDPIdeal J
hK : hI.isSubDPIdeal K
n : ℕ
hn : n ≠ 0
a : A
ha : a ∈ ↑J
⊢ hI.dpow n a ∈ Ideal.span (↑J ∪ ↑K)
case inr
A : Type u_1
inst✝ : CommSemiring A
I :... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/SubDPIdeal.lean | DividedPowers.isSubDPIdeal_sup | [187, 1] | [198, 50] | . apply Ideal.span_mono (Set.subset_union_right _ _)
rw [K.span_eq]; exact hK.2 n hn a ha | case inr
A : Type u_1
inst✝ : CommSemiring A
I : Ideal A
hI : DividedPowers I
J K : Ideal A
hJ : hI.isSubDPIdeal J
hK : hI.isSubDPIdeal K
n : ℕ
hn : n ≠ 0
a : A
ha : a ∈ ↑K
⊢ hI.dpow n a ∈ Ideal.span (↑J ∪ ↑K) | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case inr
A : Type u_1
inst✝ : CommSemiring A
I : Ideal A
hI : DividedPowers I
J K : Ideal A
hJ : hI.isSubDPIdeal J
hK : hI.isSubDPIdeal K
n : ℕ
hn : n ≠ 0
a : A
ha : a ∈ ↑K
⊢ hI.dpow n a ∈ Ideal.span (↑J ∪ ↑K)
TACTIC:
|
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/SubDPIdeal.lean | DividedPowers.isSubDPIdeal_sup | [187, 1] | [198, 50] | apply Ideal.span_mono (Set.subset_union_left _ _) | case inl
A : Type u_1
inst✝ : CommSemiring A
I : Ideal A
hI : DividedPowers I
J K : Ideal A
hJ : hI.isSubDPIdeal J
hK : hI.isSubDPIdeal K
n : ℕ
hn : n ≠ 0
a : A
ha : a ∈ ↑J
⊢ hI.dpow n a ∈ Ideal.span (↑J ∪ ↑K) | case inl.a
A : Type u_1
inst✝ : CommSemiring A
I : Ideal A
hI : DividedPowers I
J K : Ideal A
hJ : hI.isSubDPIdeal J
hK : hI.isSubDPIdeal K
n : ℕ
hn : n ≠ 0
a : A
ha : a ∈ ↑J
⊢ hI.dpow n a ∈ Ideal.span ↑J | Please generate a tactic in lean4 to solve the state.
STATE:
case inl
A : Type u_1
inst✝ : CommSemiring A
I : Ideal A
hI : DividedPowers I
J K : Ideal A
hJ : hI.isSubDPIdeal J
hK : hI.isSubDPIdeal K
n : ℕ
hn : n ≠ 0
a : A
ha : a ∈ ↑J
⊢ hI.dpow n a ∈ Ideal.span (↑J ∪ ↑K)
TACTIC:
|
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/SubDPIdeal.lean | DividedPowers.isSubDPIdeal_sup | [187, 1] | [198, 50] | rw [J.span_eq] | case inl.a
A : Type u_1
inst✝ : CommSemiring A
I : Ideal A
hI : DividedPowers I
J K : Ideal A
hJ : hI.isSubDPIdeal J
hK : hI.isSubDPIdeal K
n : ℕ
hn : n ≠ 0
a : A
ha : a ∈ ↑J
⊢ hI.dpow n a ∈ Ideal.span ↑J | case inl.a
A : Type u_1
inst✝ : CommSemiring A
I : Ideal A
hI : DividedPowers I
J K : Ideal A
hJ : hI.isSubDPIdeal J
hK : hI.isSubDPIdeal K
n : ℕ
hn : n ≠ 0
a : A
ha : a ∈ ↑J
⊢ hI.dpow n a ∈ J | Please generate a tactic in lean4 to solve the state.
STATE:
case inl.a
A : Type u_1
inst✝ : CommSemiring A
I : Ideal A
hI : DividedPowers I
J K : Ideal A
hJ : hI.isSubDPIdeal J
hK : hI.isSubDPIdeal K
n : ℕ
hn : n ≠ 0
a : A
ha : a ∈ ↑J
⊢ hI.dpow n a ∈ Ideal.span ↑J
TACTIC:
|
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/SubDPIdeal.lean | DividedPowers.isSubDPIdeal_sup | [187, 1] | [198, 50] | exact hJ.2 n hn a ha | case inl.a
A : Type u_1
inst✝ : CommSemiring A
I : Ideal A
hI : DividedPowers I
J K : Ideal A
hJ : hI.isSubDPIdeal J
hK : hI.isSubDPIdeal K
n : ℕ
hn : n ≠ 0
a : A
ha : a ∈ ↑J
⊢ hI.dpow n a ∈ J | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case inl.a
A : Type u_1
inst✝ : CommSemiring A
I : Ideal A
hI : DividedPowers I
J K : Ideal A
hJ : hI.isSubDPIdeal J
hK : hI.isSubDPIdeal K
n : ℕ
hn : n ≠ 0
a : A
ha : a ∈ ↑J
⊢ hI.dpow n a ∈ J
TACTIC:
|
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/SubDPIdeal.lean | DividedPowers.isSubDPIdeal_sup | [187, 1] | [198, 50] | apply Ideal.span_mono (Set.subset_union_right _ _) | case inr
A : Type u_1
inst✝ : CommSemiring A
I : Ideal A
hI : DividedPowers I
J K : Ideal A
hJ : hI.isSubDPIdeal J
hK : hI.isSubDPIdeal K
n : ℕ
hn : n ≠ 0
a : A
ha : a ∈ ↑K
⊢ hI.dpow n a ∈ Ideal.span (↑J ∪ ↑K) | case inr.a
A : Type u_1
inst✝ : CommSemiring A
I : Ideal A
hI : DividedPowers I
J K : Ideal A
hJ : hI.isSubDPIdeal J
hK : hI.isSubDPIdeal K
n : ℕ
hn : n ≠ 0
a : A
ha : a ∈ ↑K
⊢ hI.dpow n a ∈ Ideal.span ↑K | Please generate a tactic in lean4 to solve the state.
STATE:
case inr
A : Type u_1
inst✝ : CommSemiring A
I : Ideal A
hI : DividedPowers I
J K : Ideal A
hJ : hI.isSubDPIdeal J
hK : hI.isSubDPIdeal K
n : ℕ
hn : n ≠ 0
a : A
ha : a ∈ ↑K
⊢ hI.dpow n a ∈ Ideal.span (↑J ∪ ↑K)
TACTIC:
|
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/SubDPIdeal.lean | DividedPowers.isSubDPIdeal_sup | [187, 1] | [198, 50] | rw [K.span_eq] | case inr.a
A : Type u_1
inst✝ : CommSemiring A
I : Ideal A
hI : DividedPowers I
J K : Ideal A
hJ : hI.isSubDPIdeal J
hK : hI.isSubDPIdeal K
n : ℕ
hn : n ≠ 0
a : A
ha : a ∈ ↑K
⊢ hI.dpow n a ∈ Ideal.span ↑K | case inr.a
A : Type u_1
inst✝ : CommSemiring A
I : Ideal A
hI : DividedPowers I
J K : Ideal A
hJ : hI.isSubDPIdeal J
hK : hI.isSubDPIdeal K
n : ℕ
hn : n ≠ 0
a : A
ha : a ∈ ↑K
⊢ hI.dpow n a ∈ K | Please generate a tactic in lean4 to solve the state.
STATE:
case inr.a
A : Type u_1
inst✝ : CommSemiring A
I : Ideal A
hI : DividedPowers I
J K : Ideal A
hJ : hI.isSubDPIdeal J
hK : hI.isSubDPIdeal K
n : ℕ
hn : n ≠ 0
a : A
ha : a ∈ ↑K
⊢ hI.dpow n a ∈ Ideal.span ↑K
TACTIC:
|
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/SubDPIdeal.lean | DividedPowers.isSubDPIdeal_sup | [187, 1] | [198, 50] | exact hK.2 n hn a ha | case inr.a
A : Type u_1
inst✝ : CommSemiring A
I : Ideal A
hI : DividedPowers I
J K : Ideal A
hJ : hI.isSubDPIdeal J
hK : hI.isSubDPIdeal K
n : ℕ
hn : n ≠ 0
a : A
ha : a ∈ ↑K
⊢ hI.dpow n a ∈ K | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case inr.a
A : Type u_1
inst✝ : CommSemiring A
I : Ideal A
hI : DividedPowers I
J K : Ideal A
hJ : hI.isSubDPIdeal J
hK : hI.isSubDPIdeal K
n : ℕ
hn : n ≠ 0
a : A
ha : a ∈ ↑K
⊢ hI.dpow n a ∈ K
TACTIC:
|
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/SubDPIdeal.lean | DividedPowers.isSubDPIdeal_sup | [187, 1] | [198, 50] | rw [Set.union_subset_iff] | case hS
A : Type u_1
inst✝ : CommSemiring A
I : Ideal A
hI : DividedPowers I
J K : Ideal A
hJ : hI.isSubDPIdeal J
hK : hI.isSubDPIdeal K
⊢ ↑J ∪ ↑K ⊆ ↑I | case hS
A : Type u_1
inst✝ : CommSemiring A
I : Ideal A
hI : DividedPowers I
J K : Ideal A
hJ : hI.isSubDPIdeal J
hK : hI.isSubDPIdeal K
⊢ ↑J ⊆ ↑I ∧ ↑K ⊆ ↑I | Please generate a tactic in lean4 to solve the state.
STATE:
case hS
A : Type u_1
inst✝ : CommSemiring A
I : Ideal A
hI : DividedPowers I
J K : Ideal A
hJ : hI.isSubDPIdeal J
hK : hI.isSubDPIdeal K
⊢ ↑J ∪ ↑K ⊆ ↑I
TACTIC:
|
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/SubDPIdeal.lean | DividedPowers.isSubDPIdeal_sup | [187, 1] | [198, 50] | exact ⟨hJ.1, hK.1⟩ | case hS
A : Type u_1
inst✝ : CommSemiring A
I : Ideal A
hI : DividedPowers I
J K : Ideal A
hJ : hI.isSubDPIdeal J
hK : hI.isSubDPIdeal K
⊢ ↑J ⊆ ↑I ∧ ↑K ⊆ ↑I | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case hS
A : Type u_1
inst✝ : CommSemiring A
I : Ideal A
hI : DividedPowers I
J K : Ideal A
hJ : hI.isSubDPIdeal J
hK : hI.isSubDPIdeal K
⊢ ↑J ⊆ ↑I ∧ ↑K ⊆ ↑I
TACTIC:
|
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/SubDPIdeal.lean | DividedPowers.isSubDPIdeal_iSup | [207, 1] | [218, 28] | rw [Ideal.iSup_eq_span] | A : Type u_2
inst✝ : CommSemiring A
I : Ideal A
hI : DividedPowers I
ι : Type u_1
J : ι → Ideal A
hJ : ∀ (i : ι), hI.isSubDPIdeal (J i)
⊢ hI.isSubDPIdeal (iSup J) | A : Type u_2
inst✝ : CommSemiring A
I : Ideal A
hI : DividedPowers I
ι : Type u_1
J : ι → Ideal A
hJ : ∀ (i : ι), hI.isSubDPIdeal (J i)
⊢ hI.isSubDPIdeal (Ideal.span (⋃ i, ↑(J i))) | Please generate a tactic in lean4 to solve the state.
STATE:
A : Type u_2
inst✝ : CommSemiring A
I : Ideal A
hI : DividedPowers I
ι : Type u_1
J : ι → Ideal A
hJ : ∀ (i : ι), hI.isSubDPIdeal (J i)
⊢ hI.isSubDPIdeal (iSup J)
TACTIC:
|
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/SubDPIdeal.lean | DividedPowers.isSubDPIdeal_iSup | [207, 1] | [218, 28] | rw [span_isSubDPIdeal_iff] | A : Type u_2
inst✝ : CommSemiring A
I : Ideal A
hI : DividedPowers I
ι : Type u_1
J : ι → Ideal A
hJ : ∀ (i : ι), hI.isSubDPIdeal (J i)
⊢ hI.isSubDPIdeal (Ideal.span (⋃ i, ↑(J i))) | A : Type u_2
inst✝ : CommSemiring A
I : Ideal A
hI : DividedPowers I
ι : Type u_1
J : ι → Ideal A
hJ : ∀ (i : ι), hI.isSubDPIdeal (J i)
⊢ ∀ (n : ℕ), n ≠ 0 → ∀ s ∈ ⋃ i, ↑(J i), hI.dpow n s ∈ Ideal.span (⋃ i, ↑(J i))
case hS
A : Type u_2
inst✝ : CommSemiring A
I : Ideal A
hI : DividedPowers I
ι : Type u_1
J : ι → Ideal ... | Please generate a tactic in lean4 to solve the state.
STATE:
A : Type u_2
inst✝ : CommSemiring A
I : Ideal A
hI : DividedPowers I
ι : Type u_1
J : ι → Ideal A
hJ : ∀ (i : ι), hI.isSubDPIdeal (J i)
⊢ hI.isSubDPIdeal (Ideal.span (⋃ i, ↑(J i)))
TACTIC:
|
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/SubDPIdeal.lean | DividedPowers.isSubDPIdeal_iSup | [207, 1] | [218, 28] | rintro n hn a | A : Type u_2
inst✝ : CommSemiring A
I : Ideal A
hI : DividedPowers I
ι : Type u_1
J : ι → Ideal A
hJ : ∀ (i : ι), hI.isSubDPIdeal (J i)
⊢ ∀ (n : ℕ), n ≠ 0 → ∀ s ∈ ⋃ i, ↑(J i), hI.dpow n s ∈ Ideal.span (⋃ i, ↑(J i)) | A : Type u_2
inst✝ : CommSemiring A
I : Ideal A
hI : DividedPowers I
ι : Type u_1
J : ι → Ideal A
hJ : ∀ (i : ι), hI.isSubDPIdeal (J i)
n : ℕ
hn : n ≠ 0
a : A
⊢ a ∈ ⋃ i, ↑(J i) → hI.dpow n a ∈ Ideal.span (⋃ i, ↑(J i)) | Please generate a tactic in lean4 to solve the state.
STATE:
A : Type u_2
inst✝ : CommSemiring A
I : Ideal A
hI : DividedPowers I
ι : Type u_1
J : ι → Ideal A
hJ : ∀ (i : ι), hI.isSubDPIdeal (J i)
⊢ ∀ (n : ℕ), n ≠ 0 → ∀ s ∈ ⋃ i, ↑(J i), hI.dpow n s ∈ Ideal.span (⋃ i, ↑(J i))
TACTIC:
|
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/SubDPIdeal.lean | DividedPowers.isSubDPIdeal_iSup | [207, 1] | [218, 28] | rw [Set.mem_iUnion] | A : Type u_2
inst✝ : CommSemiring A
I : Ideal A
hI : DividedPowers I
ι : Type u_1
J : ι → Ideal A
hJ : ∀ (i : ι), hI.isSubDPIdeal (J i)
n : ℕ
hn : n ≠ 0
a : A
⊢ a ∈ ⋃ i, ↑(J i) → hI.dpow n a ∈ Ideal.span (⋃ i, ↑(J i)) | A : Type u_2
inst✝ : CommSemiring A
I : Ideal A
hI : DividedPowers I
ι : Type u_1
J : ι → Ideal A
hJ : ∀ (i : ι), hI.isSubDPIdeal (J i)
n : ℕ
hn : n ≠ 0
a : A
⊢ (∃ i, a ∈ ↑(J i)) → hI.dpow n a ∈ Ideal.span (⋃ i, ↑(J i)) | Please generate a tactic in lean4 to solve the state.
STATE:
A : Type u_2
inst✝ : CommSemiring A
I : Ideal A
hI : DividedPowers I
ι : Type u_1
J : ι → Ideal A
hJ : ∀ (i : ι), hI.isSubDPIdeal (J i)
n : ℕ
hn : n ≠ 0
a : A
⊢ a ∈ ⋃ i, ↑(J i) → hI.dpow n a ∈ Ideal.span (⋃ i, ↑(J i))
TACTIC:
|
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/SubDPIdeal.lean | DividedPowers.isSubDPIdeal_iSup | [207, 1] | [218, 28] | rintro ⟨i, ha⟩ | A : Type u_2
inst✝ : CommSemiring A
I : Ideal A
hI : DividedPowers I
ι : Type u_1
J : ι → Ideal A
hJ : ∀ (i : ι), hI.isSubDPIdeal (J i)
n : ℕ
hn : n ≠ 0
a : A
⊢ (∃ i, a ∈ ↑(J i)) → hI.dpow n a ∈ Ideal.span (⋃ i, ↑(J i)) | case intro
A : Type u_2
inst✝ : CommSemiring A
I : Ideal A
hI : DividedPowers I
ι : Type u_1
J : ι → Ideal A
hJ : ∀ (i : ι), hI.isSubDPIdeal (J i)
n : ℕ
hn : n ≠ 0
a : A
i : ι
ha : a ∈ ↑(J i)
⊢ hI.dpow n a ∈ Ideal.span (⋃ i, ↑(J i)) | Please generate a tactic in lean4 to solve the state.
STATE:
A : Type u_2
inst✝ : CommSemiring A
I : Ideal A
hI : DividedPowers I
ι : Type u_1
J : ι → Ideal A
hJ : ∀ (i : ι), hI.isSubDPIdeal (J i)
n : ℕ
hn : n ≠ 0
a : A
⊢ (∃ i, a ∈ ↑(J i)) → hI.dpow n a ∈ Ideal.span (⋃ i, ↑(J i))
TACTIC:
|
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/SubDPIdeal.lean | DividedPowers.isSubDPIdeal_iSup | [207, 1] | [218, 28] | apply Ideal.span_mono (Set.subset_iUnion _ i) | case intro
A : Type u_2
inst✝ : CommSemiring A
I : Ideal A
hI : DividedPowers I
ι : Type u_1
J : ι → Ideal A
hJ : ∀ (i : ι), hI.isSubDPIdeal (J i)
n : ℕ
hn : n ≠ 0
a : A
i : ι
ha : a ∈ ↑(J i)
⊢ hI.dpow n a ∈ Ideal.span (⋃ i, ↑(J i)) | case intro.a
A : Type u_2
inst✝ : CommSemiring A
I : Ideal A
hI : DividedPowers I
ι : Type u_1
J : ι → Ideal A
hJ : ∀ (i : ι), hI.isSubDPIdeal (J i)
n : ℕ
hn : n ≠ 0
a : A
i : ι
ha : a ∈ ↑(J i)
⊢ hI.dpow n a ∈ Ideal.span ↑(J i) | Please generate a tactic in lean4 to solve the state.
STATE:
case intro
A : Type u_2
inst✝ : CommSemiring A
I : Ideal A
hI : DividedPowers I
ι : Type u_1
J : ι → Ideal A
hJ : ∀ (i : ι), hI.isSubDPIdeal (J i)
n : ℕ
hn : n ≠ 0
a : A
i : ι
ha : a ∈ ↑(J i)
⊢ hI.dpow n a ∈ Ideal.span (⋃ i, ↑(J i))
TACTIC:
|
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/SubDPIdeal.lean | DividedPowers.isSubDPIdeal_iSup | [207, 1] | [218, 28] | rw [Ideal.span_eq] | case intro.a
A : Type u_2
inst✝ : CommSemiring A
I : Ideal A
hI : DividedPowers I
ι : Type u_1
J : ι → Ideal A
hJ : ∀ (i : ι), hI.isSubDPIdeal (J i)
n : ℕ
hn : n ≠ 0
a : A
i : ι
ha : a ∈ ↑(J i)
⊢ hI.dpow n a ∈ Ideal.span ↑(J i) | case intro.a
A : Type u_2
inst✝ : CommSemiring A
I : Ideal A
hI : DividedPowers I
ι : Type u_1
J : ι → Ideal A
hJ : ∀ (i : ι), hI.isSubDPIdeal (J i)
n : ℕ
hn : n ≠ 0
a : A
i : ι
ha : a ∈ ↑(J i)
⊢ hI.dpow n a ∈ J i | Please generate a tactic in lean4 to solve the state.
STATE:
case intro.a
A : Type u_2
inst✝ : CommSemiring A
I : Ideal A
hI : DividedPowers I
ι : Type u_1
J : ι → Ideal A
hJ : ∀ (i : ι), hI.isSubDPIdeal (J i)
n : ℕ
hn : n ≠ 0
a : A
i : ι
ha : a ∈ ↑(J i)
⊢ hI.dpow n a ∈ Ideal.span ↑(J i)
TACTIC:
|
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/SubDPIdeal.lean | DividedPowers.isSubDPIdeal_iSup | [207, 1] | [218, 28] | exact (hJ i).2 n hn a ha | case intro.a
A : Type u_2
inst✝ : CommSemiring A
I : Ideal A
hI : DividedPowers I
ι : Type u_1
J : ι → Ideal A
hJ : ∀ (i : ι), hI.isSubDPIdeal (J i)
n : ℕ
hn : n ≠ 0
a : A
i : ι
ha : a ∈ ↑(J i)
⊢ hI.dpow n a ∈ J i | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case intro.a
A : Type u_2
inst✝ : CommSemiring A
I : Ideal A
hI : DividedPowers I
ι : Type u_1
J : ι → Ideal A
hJ : ∀ (i : ι), hI.isSubDPIdeal (J i)
n : ℕ
hn : n ≠ 0
a : A
i : ι
ha : a ∈ ↑(J i)
⊢ hI.dpow n a ∈ J i
TACTIC:
|
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/SubDPIdeal.lean | DividedPowers.isSubDPIdeal_iSup | [207, 1] | [218, 28] | rw [Set.iUnion_subset_iff] | case hS
A : Type u_2
inst✝ : CommSemiring A
I : Ideal A
hI : DividedPowers I
ι : Type u_1
J : ι → Ideal A
hJ : ∀ (i : ι), hI.isSubDPIdeal (J i)
⊢ ⋃ i, ↑(J i) ⊆ ↑I | case hS
A : Type u_2
inst✝ : CommSemiring A
I : Ideal A
hI : DividedPowers I
ι : Type u_1
J : ι → Ideal A
hJ : ∀ (i : ι), hI.isSubDPIdeal (J i)
⊢ ∀ (i : ι), ↑(J i) ⊆ ↑I | Please generate a tactic in lean4 to solve the state.
STATE:
case hS
A : Type u_2
inst✝ : CommSemiring A
I : Ideal A
hI : DividedPowers I
ι : Type u_1
J : ι → Ideal A
hJ : ∀ (i : ι), hI.isSubDPIdeal (J i)
⊢ ⋃ i, ↑(J i) ⊆ ↑I
TACTIC:
|
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/SubDPIdeal.lean | DividedPowers.isSubDPIdeal_iSup | [207, 1] | [218, 28] | intro i | case hS
A : Type u_2
inst✝ : CommSemiring A
I : Ideal A
hI : DividedPowers I
ι : Type u_1
J : ι → Ideal A
hJ : ∀ (i : ι), hI.isSubDPIdeal (J i)
⊢ ∀ (i : ι), ↑(J i) ⊆ ↑I | case hS
A : Type u_2
inst✝ : CommSemiring A
I : Ideal A
hI : DividedPowers I
ι : Type u_1
J : ι → Ideal A
hJ : ∀ (i : ι), hI.isSubDPIdeal (J i)
i : ι
⊢ ↑(J i) ⊆ ↑I | Please generate a tactic in lean4 to solve the state.
STATE:
case hS
A : Type u_2
inst✝ : CommSemiring A
I : Ideal A
hI : DividedPowers I
ι : Type u_1
J : ι → Ideal A
hJ : ∀ (i : ι), hI.isSubDPIdeal (J i)
⊢ ∀ (i : ι), ↑(J i) ⊆ ↑I
TACTIC:
|
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/SubDPIdeal.lean | DividedPowers.isSubDPIdeal_iSup | [207, 1] | [218, 28] | exact (hJ i).1 | case hS
A : Type u_2
inst✝ : CommSemiring A
I : Ideal A
hI : DividedPowers I
ι : Type u_1
J : ι → Ideal A
hJ : ∀ (i : ι), hI.isSubDPIdeal (J i)
i : ι
⊢ ↑(J i) ⊆ ↑I | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case hS
A : Type u_2
inst✝ : CommSemiring A
I : Ideal A
hI : DividedPowers I
ι : Type u_1
J : ι → Ideal A
hJ : ∀ (i : ι), hI.isSubDPIdeal (J i)
i : ι
⊢ ↑(J i) ⊆ ↑I
TACTIC:
|
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/SubDPIdeal.lean | DividedPowers.isSubDPIdeal_map | [221, 1] | [232, 48] | simp only [Ideal.map] | A : Type u_2
inst✝¹ : CommSemiring A
I : Ideal A
hI : DividedPowers I
B : Type u_1
inst✝ : CommSemiring B
J : Ideal B
hJ : DividedPowers J
f : A →+* B
hf : hI.isDPMorphism hJ f
K : Ideal A
hK : hI.isSubDPIdeal K
⊢ hJ.isSubDPIdeal (Ideal.map f K) | A : Type u_2
inst✝¹ : CommSemiring A
I : Ideal A
hI : DividedPowers I
B : Type u_1
inst✝ : CommSemiring B
J : Ideal B
hJ : DividedPowers J
f : A →+* B
hf : hI.isDPMorphism hJ f
K : Ideal A
hK : hI.isSubDPIdeal K
⊢ hJ.isSubDPIdeal (Ideal.span (⇑f '' ↑K)) | Please generate a tactic in lean4 to solve the state.
STATE:
A : Type u_2
inst✝¹ : CommSemiring A
I : Ideal A
hI : DividedPowers I
B : Type u_1
inst✝ : CommSemiring B
J : Ideal B
hJ : DividedPowers J
f : A →+* B
hf : hI.isDPMorphism hJ f
K : Ideal A
hK : hI.isSubDPIdeal K
⊢ hJ.isSubDPIdeal (Ideal.map f K)
TACTIC:
|
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/SubDPIdeal.lean | DividedPowers.isSubDPIdeal_map | [221, 1] | [232, 48] | rw [span_isSubDPIdeal_iff] | A : Type u_2
inst✝¹ : CommSemiring A
I : Ideal A
hI : DividedPowers I
B : Type u_1
inst✝ : CommSemiring B
J : Ideal B
hJ : DividedPowers J
f : A →+* B
hf : hI.isDPMorphism hJ f
K : Ideal A
hK : hI.isSubDPIdeal K
⊢ hJ.isSubDPIdeal (Ideal.span (⇑f '' ↑K)) | A : Type u_2
inst✝¹ : CommSemiring A
I : Ideal A
hI : DividedPowers I
B : Type u_1
inst✝ : CommSemiring B
J : Ideal B
hJ : DividedPowers J
f : A →+* B
hf : hI.isDPMorphism hJ f
K : Ideal A
hK : hI.isSubDPIdeal K
⊢ ∀ (n : ℕ), n ≠ 0 → ∀ s ∈ ⇑f '' ↑K, hJ.dpow n s ∈ Ideal.span (⇑f '' ↑K)
case hS
A : Type u_2
inst✝¹ : Comm... | Please generate a tactic in lean4 to solve the state.
STATE:
A : Type u_2
inst✝¹ : CommSemiring A
I : Ideal A
hI : DividedPowers I
B : Type u_1
inst✝ : CommSemiring B
J : Ideal B
hJ : DividedPowers J
f : A →+* B
hf : hI.isDPMorphism hJ f
K : Ideal A
hK : hI.isSubDPIdeal K
⊢ hJ.isSubDPIdeal (Ideal.span (⇑f '' ↑K))
TACTI... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/SubDPIdeal.lean | DividedPowers.isSubDPIdeal_map | [221, 1] | [232, 48] | rintro n hn y ⟨x, hx, rfl⟩ | A : Type u_2
inst✝¹ : CommSemiring A
I : Ideal A
hI : DividedPowers I
B : Type u_1
inst✝ : CommSemiring B
J : Ideal B
hJ : DividedPowers J
f : A →+* B
hf : hI.isDPMorphism hJ f
K : Ideal A
hK : hI.isSubDPIdeal K
⊢ ∀ (n : ℕ), n ≠ 0 → ∀ s ∈ ⇑f '' ↑K, hJ.dpow n s ∈ Ideal.span (⇑f '' ↑K)
case hS
A : Type u_2
inst✝¹ : Comm... | case intro.intro
A : Type u_2
inst✝¹ : CommSemiring A
I : Ideal A
hI : DividedPowers I
B : Type u_1
inst✝ : CommSemiring B
J : Ideal B
hJ : DividedPowers J
f : A →+* B
hf : hI.isDPMorphism hJ f
K : Ideal A
hK : hI.isSubDPIdeal K
n : ℕ
hn : n ≠ 0
x : A
hx : x ∈ ↑K
⊢ hJ.dpow n (f x) ∈ Ideal.span (⇑f '' ↑K)
case hS
A : T... | Please generate a tactic in lean4 to solve the state.
STATE:
A : Type u_2
inst✝¹ : CommSemiring A
I : Ideal A
hI : DividedPowers I
B : Type u_1
inst✝ : CommSemiring B
J : Ideal B
hJ : DividedPowers J
f : A →+* B
hf : hI.isDPMorphism hJ f
K : Ideal A
hK : hI.isSubDPIdeal K
⊢ ∀ (n : ℕ), n ≠ 0 → ∀ s ∈ ⇑f '' ↑K, hJ.dpow n ... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/SubDPIdeal.lean | DividedPowers.isSubDPIdeal_map | [221, 1] | [232, 48] | rw [hf.2 n x (hK.1 hx)] | case intro.intro
A : Type u_2
inst✝¹ : CommSemiring A
I : Ideal A
hI : DividedPowers I
B : Type u_1
inst✝ : CommSemiring B
J : Ideal B
hJ : DividedPowers J
f : A →+* B
hf : hI.isDPMorphism hJ f
K : Ideal A
hK : hI.isSubDPIdeal K
n : ℕ
hn : n ≠ 0
x : A
hx : x ∈ ↑K
⊢ hJ.dpow n (f x) ∈ Ideal.span (⇑f '' ↑K)
case hS
A : T... | case intro.intro
A : Type u_2
inst✝¹ : CommSemiring A
I : Ideal A
hI : DividedPowers I
B : Type u_1
inst✝ : CommSemiring B
J : Ideal B
hJ : DividedPowers J
f : A →+* B
hf : hI.isDPMorphism hJ f
K : Ideal A
hK : hI.isSubDPIdeal K
n : ℕ
hn : n ≠ 0
x : A
hx : x ∈ ↑K
⊢ f (hI.dpow n x) ∈ Ideal.span (⇑f '' ↑K)
case hS
A : T... | Please generate a tactic in lean4 to solve the state.
STATE:
case intro.intro
A : Type u_2
inst✝¹ : CommSemiring A
I : Ideal A
hI : DividedPowers I
B : Type u_1
inst✝ : CommSemiring B
J : Ideal B
hJ : DividedPowers J
f : A →+* B
hf : hI.isDPMorphism hJ f
K : Ideal A
hK : hI.isSubDPIdeal K
n : ℕ
hn : n ≠ 0
x : A
hx : x ... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/SubDPIdeal.lean | DividedPowers.isSubDPIdeal_map | [221, 1] | [232, 48] | apply Ideal.mem_map_of_mem | case intro.intro
A : Type u_2
inst✝¹ : CommSemiring A
I : Ideal A
hI : DividedPowers I
B : Type u_1
inst✝ : CommSemiring B
J : Ideal B
hJ : DividedPowers J
f : A →+* B
hf : hI.isDPMorphism hJ f
K : Ideal A
hK : hI.isSubDPIdeal K
n : ℕ
hn : n ≠ 0
x : A
hx : x ∈ ↑K
⊢ f (hI.dpow n x) ∈ Ideal.span (⇑f '' ↑K)
case hS
A : T... | case intro.intro.h
A : Type u_2
inst✝¹ : CommSemiring A
I : Ideal A
hI : DividedPowers I
B : Type u_1
inst✝ : CommSemiring B
J : Ideal B
hJ : DividedPowers J
f : A →+* B
hf : hI.isDPMorphism hJ f
K : Ideal A
hK : hI.isSubDPIdeal K
n : ℕ
hn : n ≠ 0
x : A
hx : x ∈ ↑K
⊢ hI.dpow n x ∈ K
case hS
A : Type u_2
inst✝¹ : CommS... | Please generate a tactic in lean4 to solve the state.
STATE:
case intro.intro
A : Type u_2
inst✝¹ : CommSemiring A
I : Ideal A
hI : DividedPowers I
B : Type u_1
inst✝ : CommSemiring B
J : Ideal B
hJ : DividedPowers J
f : A →+* B
hf : hI.isDPMorphism hJ f
K : Ideal A
hK : hI.isSubDPIdeal K
n : ℕ
hn : n ≠ 0
x : A
hx : x ... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/SubDPIdeal.lean | DividedPowers.isSubDPIdeal_map | [221, 1] | [232, 48] | exact hK.2 n hn x hx | case intro.intro.h
A : Type u_2
inst✝¹ : CommSemiring A
I : Ideal A
hI : DividedPowers I
B : Type u_1
inst✝ : CommSemiring B
J : Ideal B
hJ : DividedPowers J
f : A →+* B
hf : hI.isDPMorphism hJ f
K : Ideal A
hK : hI.isSubDPIdeal K
n : ℕ
hn : n ≠ 0
x : A
hx : x ∈ ↑K
⊢ hI.dpow n x ∈ K
case hS
A : Type u_2
inst✝¹ : CommS... | case hS
A : Type u_2
inst✝¹ : CommSemiring A
I : Ideal A
hI : DividedPowers I
B : Type u_1
inst✝ : CommSemiring B
J : Ideal B
hJ : DividedPowers J
f : A →+* B
hf : hI.isDPMorphism hJ f
K : Ideal A
hK : hI.isSubDPIdeal K
⊢ ⇑f '' ↑K ⊆ ↑J | Please generate a tactic in lean4 to solve the state.
STATE:
case intro.intro.h
A : Type u_2
inst✝¹ : CommSemiring A
I : Ideal A
hI : DividedPowers I
B : Type u_1
inst✝ : CommSemiring B
J : Ideal B
hJ : DividedPowers J
f : A →+* B
hf : hI.isDPMorphism hJ f
K : Ideal A
hK : hI.isSubDPIdeal K
n : ℕ
hn : n ≠ 0
x : A
hx : ... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/SubDPIdeal.lean | DividedPowers.isSubDPIdeal_map | [221, 1] | [232, 48] | rintro y ⟨x, hx, rfl⟩ | case hS
A : Type u_2
inst✝¹ : CommSemiring A
I : Ideal A
hI : DividedPowers I
B : Type u_1
inst✝ : CommSemiring B
J : Ideal B
hJ : DividedPowers J
f : A →+* B
hf : hI.isDPMorphism hJ f
K : Ideal A
hK : hI.isSubDPIdeal K
⊢ ⇑f '' ↑K ⊆ ↑J | case hS.intro.intro
A : Type u_2
inst✝¹ : CommSemiring A
I : Ideal A
hI : DividedPowers I
B : Type u_1
inst✝ : CommSemiring B
J : Ideal B
hJ : DividedPowers J
f : A →+* B
hf : hI.isDPMorphism hJ f
K : Ideal A
hK : hI.isSubDPIdeal K
x : A
hx : x ∈ ↑K
⊢ f x ∈ ↑J | Please generate a tactic in lean4 to solve the state.
STATE:
case hS
A : Type u_2
inst✝¹ : CommSemiring A
I : Ideal A
hI : DividedPowers I
B : Type u_1
inst✝ : CommSemiring B
J : Ideal B
hJ : DividedPowers J
f : A →+* B
hf : hI.isDPMorphism hJ f
K : Ideal A
hK : hI.isSubDPIdeal K
⊢ ⇑f '' ↑K ⊆ ↑J
TACTIC:
|
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/SubDPIdeal.lean | DividedPowers.isSubDPIdeal_map | [221, 1] | [232, 48] | exact hf.1 (Ideal.mem_map_of_mem f (hK.1 hx)) | case hS.intro.intro
A : Type u_2
inst✝¹ : CommSemiring A
I : Ideal A
hI : DividedPowers I
B : Type u_1
inst✝ : CommSemiring B
J : Ideal B
hJ : DividedPowers J
f : A →+* B
hf : hI.isDPMorphism hJ f
K : Ideal A
hK : hI.isSubDPIdeal K
x : A
hx : x ∈ ↑K
⊢ f x ∈ ↑J | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case hS.intro.intro
A : Type u_2
inst✝¹ : CommSemiring A
I : Ideal A
hI : DividedPowers I
B : Type u_1
inst✝ : CommSemiring B
J : Ideal B
hJ : DividedPowers J
f : A →+* B
hf : hI.isDPMorphism hJ f
K : Ideal A
hK : hI.isSubDPIdeal K
x : A
hx : x ∈ ↑K
⊢ f x ∈ ↑... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/SubDPIdeal.lean | DividedPowers.SubDPIdeal.le_generatedDpow | [499, 1] | [500, 73] | rw [hI.dpow_one (hS hx)] | A : Type u_1
inst✝ : CommSemiring A
I : Ideal A
hI : DividedPowers I
S : Set A
hS : S ⊆ ↑I
x : A
hx : x ∈ S
⊢ x = hI.dpow 1 x | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
A : Type u_1
inst✝ : CommSemiring A
I : Ideal A
hI : DividedPowers I
S : Set A
hS : S ⊆ ↑I
x : A
hx : x ∈ S
⊢ x = hI.dpow 1 x
TACTIC:
|
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/SubDPIdeal.lean | DividedPowers.SubDPIdeal.generated_dpow_le | [503, 1] | [509, 35] | rw [Ideal.span_le] | A : Type u_1
inst✝ : CommSemiring A
I : Ideal A
hI : DividedPowers I
S : Set A
J : hI.SubDPIdeal
hSJ : S ⊆ ↑J.carrier
⊢ Ideal.span {y | ∃ n, ∃ (_ : n ≠ 0), ∃ x, ∃ (_ : x ∈ S), y = hI.dpow n x} ≤ J.carrier | A : Type u_1
inst✝ : CommSemiring A
I : Ideal A
hI : DividedPowers I
S : Set A
J : hI.SubDPIdeal
hSJ : S ⊆ ↑J.carrier
⊢ {y | ∃ n, ∃ (_ : n ≠ 0), ∃ x, ∃ (_ : x ∈ S), y = hI.dpow n x} ⊆ ↑J.carrier | Please generate a tactic in lean4 to solve the state.
STATE:
A : Type u_1
inst✝ : CommSemiring A
I : Ideal A
hI : DividedPowers I
S : Set A
J : hI.SubDPIdeal
hSJ : S ⊆ ↑J.carrier
⊢ Ideal.span {y | ∃ n, ∃ (_ : n ≠ 0), ∃ x, ∃ (_ : x ∈ S), y = hI.dpow n x} ≤ J.carrier
TACTIC:
|
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/SubDPIdeal.lean | DividedPowers.SubDPIdeal.generated_dpow_le | [503, 1] | [509, 35] | rintro y ⟨n, hn, x, hx, hxy⟩ | A : Type u_1
inst✝ : CommSemiring A
I : Ideal A
hI : DividedPowers I
S : Set A
J : hI.SubDPIdeal
hSJ : S ⊆ ↑J.carrier
⊢ {y | ∃ n, ∃ (_ : n ≠ 0), ∃ x, ∃ (_ : x ∈ S), y = hI.dpow n x} ⊆ ↑J.carrier | case intro.intro.intro.intro
A : Type u_1
inst✝ : CommSemiring A
I : Ideal A
hI : DividedPowers I
S : Set A
J : hI.SubDPIdeal
hSJ : S ⊆ ↑J.carrier
y : A
n : ℕ
hn : n ≠ 0
x : A
hx : x ∈ S
hxy : y = hI.dpow n x
⊢ y ∈ ↑J.carrier | Please generate a tactic in lean4 to solve the state.
STATE:
A : Type u_1
inst✝ : CommSemiring A
I : Ideal A
hI : DividedPowers I
S : Set A
J : hI.SubDPIdeal
hSJ : S ⊆ ↑J.carrier
⊢ {y | ∃ n, ∃ (_ : n ≠ 0), ∃ x, ∃ (_ : x ∈ S), y = hI.dpow n x} ⊆ ↑J.carrier
TACTIC:
|
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/SubDPIdeal.lean | DividedPowers.SubDPIdeal.generated_dpow_le | [503, 1] | [509, 35] | rw [hxy] | case intro.intro.intro.intro
A : Type u_1
inst✝ : CommSemiring A
I : Ideal A
hI : DividedPowers I
S : Set A
J : hI.SubDPIdeal
hSJ : S ⊆ ↑J.carrier
y : A
n : ℕ
hn : n ≠ 0
x : A
hx : x ∈ S
hxy : y = hI.dpow n x
⊢ y ∈ ↑J.carrier | case intro.intro.intro.intro
A : Type u_1
inst✝ : CommSemiring A
I : Ideal A
hI : DividedPowers I
S : Set A
J : hI.SubDPIdeal
hSJ : S ⊆ ↑J.carrier
y : A
n : ℕ
hn : n ≠ 0
x : A
hx : x ∈ S
hxy : y = hI.dpow n x
⊢ hI.dpow n x ∈ ↑J.carrier | Please generate a tactic in lean4 to solve the state.
STATE:
case intro.intro.intro.intro
A : Type u_1
inst✝ : CommSemiring A
I : Ideal A
hI : DividedPowers I
S : Set A
J : hI.SubDPIdeal
hSJ : S ⊆ ↑J.carrier
y : A
n : ℕ
hn : n ≠ 0
x : A
hx : x ∈ S
hxy : y = hI.dpow n x
⊢ y ∈ ↑J.carrier
TACTIC:
|
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/SubDPIdeal.lean | DividedPowers.SubDPIdeal.generated_dpow_le | [503, 1] | [509, 35] | exact J.dpow_mem n hn x (hSJ hx) | case intro.intro.intro.intro
A : Type u_1
inst✝ : CommSemiring A
I : Ideal A
hI : DividedPowers I
S : Set A
J : hI.SubDPIdeal
hSJ : S ⊆ ↑J.carrier
y : A
n : ℕ
hn : n ≠ 0
x : A
hx : x ∈ S
hxy : y = hI.dpow n x
⊢ hI.dpow n x ∈ ↑J.carrier | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case intro.intro.intro.intro
A : Type u_1
inst✝ : CommSemiring A
I : Ideal A
hI : DividedPowers I
S : Set A
J : hI.SubDPIdeal
hSJ : S ⊆ ↑J.carrier
y : A
n : ℕ
hn : n ≠ 0
x : A
hx : x ∈ S
hxy : y = hI.dpow n x
⊢ hI.dpow n x ∈ ↑J.carrier
TACTIC:
|
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/SubDPIdeal.lean | DividedPowers.SubDPIdeal.generated_carrier_eq | [512, 1] | [531, 16] | simp only [generated, sInf_carrier_def] | A : Type u_1
inst✝ : CommSemiring A
I : Ideal A
hI : DividedPowers I
S : Set A
hS : S ⊆ ↑I
⊢ (generated hI S).carrier = Ideal.span {y | ∃ n, ∃ (_ : n ≠ 0), ∃ x, ∃ (_ : x ∈ S), y = hI.dpow n x} | A : Type u_1
inst✝ : CommSemiring A
I : Ideal A
hI : DividedPowers I
S : Set A
hS : S ⊆ ↑I
⊢ ⨅ s ∈ insert ⊤ {J | S ⊆ ↑J.carrier}, s.carrier =
Ideal.span {y | ∃ n, ∃ (_ : n ≠ 0), ∃ x, ∃ (_ : x ∈ S), y = hI.dpow n x} | Please generate a tactic in lean4 to solve the state.
STATE:
A : Type u_1
inst✝ : CommSemiring A
I : Ideal A
hI : DividedPowers I
S : Set A
hS : S ⊆ ↑I
⊢ (generated hI S).carrier = Ideal.span {y | ∃ n, ∃ (_ : n ≠ 0), ∃ x, ∃ (_ : x ∈ S), y = hI.dpow n x}
TACTIC:
|
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/SubDPIdeal.lean | DividedPowers.SubDPIdeal.generated_carrier_eq | [512, 1] | [531, 16] | apply le_antisymm | A : Type u_1
inst✝ : CommSemiring A
I : Ideal A
hI : DividedPowers I
S : Set A
hS : S ⊆ ↑I
⊢ ⨅ s ∈ insert ⊤ {J | S ⊆ ↑J.carrier}, s.carrier =
Ideal.span {y | ∃ n, ∃ (_ : n ≠ 0), ∃ x, ∃ (_ : x ∈ S), y = hI.dpow n x} | case a
A : Type u_1
inst✝ : CommSemiring A
I : Ideal A
hI : DividedPowers I
S : Set A
hS : S ⊆ ↑I
⊢ ⨅ s ∈ insert ⊤ {J | S ⊆ ↑J.carrier}, s.carrier ≤
Ideal.span {y | ∃ n, ∃ (_ : n ≠ 0), ∃ x, ∃ (_ : x ∈ S), y = hI.dpow n x}
case a
A : Type u_1
inst✝ : CommSemiring A
I : Ideal A
hI : DividedPowers I
S : Set A
hS : S ... | Please generate a tactic in lean4 to solve the state.
STATE:
A : Type u_1
inst✝ : CommSemiring A
I : Ideal A
hI : DividedPowers I
S : Set A
hS : S ⊆ ↑I
⊢ ⨅ s ∈ insert ⊤ {J | S ⊆ ↑J.carrier}, s.carrier =
Ideal.span {y | ∃ n, ∃ (_ : n ≠ 0), ∃ x, ∃ (_ : x ∈ S), y = hI.dpow n x}
TACTIC:
|
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/SubDPIdeal.lean | DividedPowers.SubDPIdeal.generated_carrier_eq | [512, 1] | [531, 16] | have h : generatedDpow hI hS ∈ insert ⊤ {J : hI.SubDPIdeal | S ⊆ ↑J.carrier} :=
by
apply Set.mem_insert_of_mem
simp only [Set.mem_setOf_eq, generatedDpow_carrier]
exact le_generatedDpow hI hS | case a
A : Type u_1
inst✝ : CommSemiring A
I : Ideal A
hI : DividedPowers I
S : Set A
hS : S ⊆ ↑I
⊢ ⨅ s ∈ insert ⊤ {J | S ⊆ ↑J.carrier}, s.carrier ≤
Ideal.span {y | ∃ n, ∃ (_ : n ≠ 0), ∃ x, ∃ (_ : x ∈ S), y = hI.dpow n x} | case a
A : Type u_1
inst✝ : CommSemiring A
I : Ideal A
hI : DividedPowers I
S : Set A
hS : S ⊆ ↑I
h : generatedDpow hI hS ∈ insert ⊤ {J | S ⊆ ↑J.carrier}
⊢ ⨅ s ∈ insert ⊤ {J | S ⊆ ↑J.carrier}, s.carrier ≤
Ideal.span {y | ∃ n, ∃ (_ : n ≠ 0), ∃ x, ∃ (_ : x ∈ S), y = hI.dpow n x} | Please generate a tactic in lean4 to solve the state.
STATE:
case a
A : Type u_1
inst✝ : CommSemiring A
I : Ideal A
hI : DividedPowers I
S : Set A
hS : S ⊆ ↑I
⊢ ⨅ s ∈ insert ⊤ {J | S ⊆ ↑J.carrier}, s.carrier ≤
Ideal.span {y | ∃ n, ∃ (_ : n ≠ 0), ∃ x, ∃ (_ : x ∈ S), y = hI.dpow n x}
TACTIC:
|
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/SubDPIdeal.lean | DividedPowers.SubDPIdeal.generated_carrier_eq | [512, 1] | [531, 16] | refine' sInf_le_of_le ⟨generatedDpow hI hS, _⟩ (le_refl _) | case a
A : Type u_1
inst✝ : CommSemiring A
I : Ideal A
hI : DividedPowers I
S : Set A
hS : S ⊆ ↑I
h : generatedDpow hI hS ∈ insert ⊤ {J | S ⊆ ↑J.carrier}
⊢ ⨅ s ∈ insert ⊤ {J | S ⊆ ↑J.carrier}, s.carrier ≤
Ideal.span {y | ∃ n, ∃ (_ : n ≠ 0), ∃ x, ∃ (_ : x ∈ S), y = hI.dpow n x} | case a
A : Type u_1
inst✝ : CommSemiring A
I : Ideal A
hI : DividedPowers I
S : Set A
hS : S ⊆ ↑I
h : generatedDpow hI hS ∈ insert ⊤ {J | S ⊆ ↑J.carrier}
⊢ (fun s => ⨅ (_ : s ∈ insert ⊤ {J | S ⊆ ↑J.carrier}), s.carrier) (generatedDpow hI hS) =
Ideal.span {y | ∃ n, ∃ (_ : n ≠ 0), ∃ x, ∃ (_ : x ∈ S), y = hI.dpow n x} | Please generate a tactic in lean4 to solve the state.
STATE:
case a
A : Type u_1
inst✝ : CommSemiring A
I : Ideal A
hI : DividedPowers I
S : Set A
hS : S ⊆ ↑I
h : generatedDpow hI hS ∈ insert ⊤ {J | S ⊆ ↑J.carrier}
⊢ ⨅ s ∈ insert ⊤ {J | S ⊆ ↑J.carrier}, s.carrier ≤
Ideal.span {y | ∃ n, ∃ (_ : n ≠ 0), ∃ x, ∃ (_ : x ... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/SubDPIdeal.lean | DividedPowers.SubDPIdeal.generated_carrier_eq | [512, 1] | [531, 16] | simp only [h, ciInf_pos] | case a
A : Type u_1
inst✝ : CommSemiring A
I : Ideal A
hI : DividedPowers I
S : Set A
hS : S ⊆ ↑I
h : generatedDpow hI hS ∈ insert ⊤ {J | S ⊆ ↑J.carrier}
⊢ (fun s => ⨅ (_ : s ∈ insert ⊤ {J | S ⊆ ↑J.carrier}), s.carrier) (generatedDpow hI hS) =
Ideal.span {y | ∃ n, ∃ (_ : n ≠ 0), ∃ x, ∃ (_ : x ∈ S), y = hI.dpow n x} | case a
A : Type u_1
inst✝ : CommSemiring A
I : Ideal A
hI : DividedPowers I
S : Set A
hS : S ⊆ ↑I
h : generatedDpow hI hS ∈ insert ⊤ {J | S ⊆ ↑J.carrier}
⊢ (generatedDpow hI hS).carrier = Ideal.span {y | ∃ n, ∃ (_ : n ≠ 0), ∃ x, ∃ (_ : x ∈ S), y = hI.dpow n x} | Please generate a tactic in lean4 to solve the state.
STATE:
case a
A : Type u_1
inst✝ : CommSemiring A
I : Ideal A
hI : DividedPowers I
S : Set A
hS : S ⊆ ↑I
h : generatedDpow hI hS ∈ insert ⊤ {J | S ⊆ ↑J.carrier}
⊢ (fun s => ⨅ (_ : s ∈ insert ⊤ {J | S ⊆ ↑J.carrier}), s.carrier) (generatedDpow hI hS) =
Ideal.span ... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/SubDPIdeal.lean | DividedPowers.SubDPIdeal.generated_carrier_eq | [512, 1] | [531, 16] | rfl | case a
A : Type u_1
inst✝ : CommSemiring A
I : Ideal A
hI : DividedPowers I
S : Set A
hS : S ⊆ ↑I
h : generatedDpow hI hS ∈ insert ⊤ {J | S ⊆ ↑J.carrier}
⊢ (generatedDpow hI hS).carrier = Ideal.span {y | ∃ n, ∃ (_ : n ≠ 0), ∃ x, ∃ (_ : x ∈ S), y = hI.dpow n x} | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case a
A : Type u_1
inst✝ : CommSemiring A
I : Ideal A
hI : DividedPowers I
S : Set A
hS : S ⊆ ↑I
h : generatedDpow hI hS ∈ insert ⊤ {J | S ⊆ ↑J.carrier}
⊢ (generatedDpow hI hS).carrier = Ideal.span {y | ∃ n, ∃ (_ : n ≠ 0), ∃ x, ∃ (_ : x ∈ S), y = hI.dpow n x... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/SubDPIdeal.lean | DividedPowers.SubDPIdeal.generated_carrier_eq | [512, 1] | [531, 16] | apply Set.mem_insert_of_mem | A : Type u_1
inst✝ : CommSemiring A
I : Ideal A
hI : DividedPowers I
S : Set A
hS : S ⊆ ↑I
⊢ generatedDpow hI hS ∈ insert ⊤ {J | S ⊆ ↑J.carrier} | case a
A : Type u_1
inst✝ : CommSemiring A
I : Ideal A
hI : DividedPowers I
S : Set A
hS : S ⊆ ↑I
⊢ generatedDpow hI hS ∈ {J | S ⊆ ↑J.carrier} | Please generate a tactic in lean4 to solve the state.
STATE:
A : Type u_1
inst✝ : CommSemiring A
I : Ideal A
hI : DividedPowers I
S : Set A
hS : S ⊆ ↑I
⊢ generatedDpow hI hS ∈ insert ⊤ {J | S ⊆ ↑J.carrier}
TACTIC:
|
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/SubDPIdeal.lean | DividedPowers.SubDPIdeal.generated_carrier_eq | [512, 1] | [531, 16] | simp only [Set.mem_setOf_eq, generatedDpow_carrier] | case a
A : Type u_1
inst✝ : CommSemiring A
I : Ideal A
hI : DividedPowers I
S : Set A
hS : S ⊆ ↑I
⊢ generatedDpow hI hS ∈ {J | S ⊆ ↑J.carrier} | case a
A : Type u_1
inst✝ : CommSemiring A
I : Ideal A
hI : DividedPowers I
S : Set A
hS : S ⊆ ↑I
⊢ S ⊆ ↑(Ideal.span {y | ∃ n, ∃ (_ : n ≠ 0), ∃ x, ∃ (_ : x ∈ S), y = hI.dpow n x}) | Please generate a tactic in lean4 to solve the state.
STATE:
case a
A : Type u_1
inst✝ : CommSemiring A
I : Ideal A
hI : DividedPowers I
S : Set A
hS : S ⊆ ↑I
⊢ generatedDpow hI hS ∈ {J | S ⊆ ↑J.carrier}
TACTIC:
|
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/SubDPIdeal.lean | DividedPowers.SubDPIdeal.generated_carrier_eq | [512, 1] | [531, 16] | exact le_generatedDpow hI hS | case a
A : Type u_1
inst✝ : CommSemiring A
I : Ideal A
hI : DividedPowers I
S : Set A
hS : S ⊆ ↑I
⊢ S ⊆ ↑(Ideal.span {y | ∃ n, ∃ (_ : n ≠ 0), ∃ x, ∃ (_ : x ∈ S), y = hI.dpow n x}) | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case a
A : Type u_1
inst✝ : CommSemiring A
I : Ideal A
hI : DividedPowers I
S : Set A
hS : S ⊆ ↑I
⊢ S ⊆ ↑(Ideal.span {y | ∃ n, ∃ (_ : n ≠ 0), ∃ x, ∃ (_ : x ∈ S), y = hI.dpow n x})
TACTIC:
|
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/SubDPIdeal.lean | DividedPowers.SubDPIdeal.generated_carrier_eq | [512, 1] | [531, 16] | rw [le_iInf₂_iff] | case a
A : Type u_1
inst✝ : CommSemiring A
I : Ideal A
hI : DividedPowers I
S : Set A
hS : S ⊆ ↑I
⊢ Ideal.span {y | ∃ n, ∃ (_ : n ≠ 0), ∃ x, ∃ (_ : x ∈ S), y = hI.dpow n x} ≤
⨅ s ∈ insert ⊤ {J | S ⊆ ↑J.carrier}, s.carrier | case a
A : Type u_1
inst✝ : CommSemiring A
I : Ideal A
hI : DividedPowers I
S : Set A
hS : S ⊆ ↑I
⊢ ∀ i ∈ insert ⊤ {J | S ⊆ ↑J.carrier},
Ideal.span {y | ∃ n, ∃ (_ : n ≠ 0), ∃ x, ∃ (_ : x ∈ S), y = hI.dpow n x} ≤ i.carrier | Please generate a tactic in lean4 to solve the state.
STATE:
case a
A : Type u_1
inst✝ : CommSemiring A
I : Ideal A
hI : DividedPowers I
S : Set A
hS : S ⊆ ↑I
⊢ Ideal.span {y | ∃ n, ∃ (_ : n ≠ 0), ∃ x, ∃ (_ : x ∈ S), y = hI.dpow n x} ≤
⨅ s ∈ insert ⊤ {J | S ⊆ ↑J.carrier}, s.carrier
TACTIC:
|
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/SubDPIdeal.lean | DividedPowers.SubDPIdeal.generated_carrier_eq | [512, 1] | [531, 16] | rintro J hJ | case a
A : Type u_1
inst✝ : CommSemiring A
I : Ideal A
hI : DividedPowers I
S : Set A
hS : S ⊆ ↑I
⊢ ∀ i ∈ insert ⊤ {J | S ⊆ ↑J.carrier},
Ideal.span {y | ∃ n, ∃ (_ : n ≠ 0), ∃ x, ∃ (_ : x ∈ S), y = hI.dpow n x} ≤ i.carrier | case a
A : Type u_1
inst✝ : CommSemiring A
I : Ideal A
hI : DividedPowers I
S : Set A
hS : S ⊆ ↑I
J : hI.SubDPIdeal
hJ : J ∈ insert ⊤ {J | S ⊆ ↑J.carrier}
⊢ Ideal.span {y | ∃ n, ∃ (_ : n ≠ 0), ∃ x, ∃ (_ : x ∈ S), y = hI.dpow n x} ≤ J.carrier | Please generate a tactic in lean4 to solve the state.
STATE:
case a
A : Type u_1
inst✝ : CommSemiring A
I : Ideal A
hI : DividedPowers I
S : Set A
hS : S ⊆ ↑I
⊢ ∀ i ∈ insert ⊤ {J | S ⊆ ↑J.carrier},
Ideal.span {y | ∃ n, ∃ (_ : n ≠ 0), ∃ x, ∃ (_ : x ∈ S), y = hI.dpow n x} ≤ i.carrier
TACTIC:
|
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/SubDPIdeal.lean | DividedPowers.SubDPIdeal.generated_carrier_eq | [512, 1] | [531, 16] | refine' generated_dpow_le hI S J _ | case a
A : Type u_1
inst✝ : CommSemiring A
I : Ideal A
hI : DividedPowers I
S : Set A
hS : S ⊆ ↑I
J : hI.SubDPIdeal
hJ : J ∈ insert ⊤ {J | S ⊆ ↑J.carrier}
⊢ Ideal.span {y | ∃ n, ∃ (_ : n ≠ 0), ∃ x, ∃ (_ : x ∈ S), y = hI.dpow n x} ≤ J.carrier | case a
A : Type u_1
inst✝ : CommSemiring A
I : Ideal A
hI : DividedPowers I
S : Set A
hS : S ⊆ ↑I
J : hI.SubDPIdeal
hJ : J ∈ insert ⊤ {J | S ⊆ ↑J.carrier}
⊢ S ⊆ ↑J.carrier | Please generate a tactic in lean4 to solve the state.
STATE:
case a
A : Type u_1
inst✝ : CommSemiring A
I : Ideal A
hI : DividedPowers I
S : Set A
hS : S ⊆ ↑I
J : hI.SubDPIdeal
hJ : J ∈ insert ⊤ {J | S ⊆ ↑J.carrier}
⊢ Ideal.span {y | ∃ n, ∃ (_ : n ≠ 0), ∃ x, ∃ (_ : x ∈ S), y = hI.dpow n x} ≤ J.carrier
TACTIC:
|
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/SubDPIdeal.lean | DividedPowers.SubDPIdeal.generated_carrier_eq | [512, 1] | [531, 16] | simp only [Set.mem_setOf_eq, Set.mem_insert_iff] at hJ | case a
A : Type u_1
inst✝ : CommSemiring A
I : Ideal A
hI : DividedPowers I
S : Set A
hS : S ⊆ ↑I
J : hI.SubDPIdeal
hJ : J ∈ insert ⊤ {J | S ⊆ ↑J.carrier}
⊢ S ⊆ ↑J.carrier | case a
A : Type u_1
inst✝ : CommSemiring A
I : Ideal A
hI : DividedPowers I
S : Set A
hS : S ⊆ ↑I
J : hI.SubDPIdeal
hJ : J = ⊤ ∨ S ⊆ ↑J.carrier
⊢ S ⊆ ↑J.carrier | Please generate a tactic in lean4 to solve the state.
STATE:
case a
A : Type u_1
inst✝ : CommSemiring A
I : Ideal A
hI : DividedPowers I
S : Set A
hS : S ⊆ ↑I
J : hI.SubDPIdeal
hJ : J ∈ insert ⊤ {J | S ⊆ ↑J.carrier}
⊢ S ⊆ ↑J.carrier
TACTIC:
|
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/SubDPIdeal.lean | DividedPowers.SubDPIdeal.generated_carrier_eq | [512, 1] | [531, 16] | cases' hJ with hJI hJS | case a
A : Type u_1
inst✝ : CommSemiring A
I : Ideal A
hI : DividedPowers I
S : Set A
hS : S ⊆ ↑I
J : hI.SubDPIdeal
hJ : J = ⊤ ∨ S ⊆ ↑J.carrier
⊢ S ⊆ ↑J.carrier | case a.inl
A : Type u_1
inst✝ : CommSemiring A
I : Ideal A
hI : DividedPowers I
S : Set A
hS : S ⊆ ↑I
J : hI.SubDPIdeal
hJI : J = ⊤
⊢ S ⊆ ↑J.carrier
case a.inr
A : Type u_1
inst✝ : CommSemiring A
I : Ideal A
hI : DividedPowers I
S : Set A
hS : S ⊆ ↑I
J : hI.SubDPIdeal
hJS : S ⊆ ↑J.carrier
⊢ S ⊆ ↑J.carrier | Please generate a tactic in lean4 to solve the state.
STATE:
case a
A : Type u_1
inst✝ : CommSemiring A
I : Ideal A
hI : DividedPowers I
S : Set A
hS : S ⊆ ↑I
J : hI.SubDPIdeal
hJ : J = ⊤ ∨ S ⊆ ↑J.carrier
⊢ S ⊆ ↑J.carrier
TACTIC:
|
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/SubDPIdeal.lean | DividedPowers.SubDPIdeal.generated_carrier_eq | [512, 1] | [531, 16] | rw [hJI] | case a.inl
A : Type u_1
inst✝ : CommSemiring A
I : Ideal A
hI : DividedPowers I
S : Set A
hS : S ⊆ ↑I
J : hI.SubDPIdeal
hJI : J = ⊤
⊢ S ⊆ ↑J.carrier | case a.inl
A : Type u_1
inst✝ : CommSemiring A
I : Ideal A
hI : DividedPowers I
S : Set A
hS : S ⊆ ↑I
J : hI.SubDPIdeal
hJI : J = ⊤
⊢ S ⊆ ↑⊤.carrier | Please generate a tactic in lean4 to solve the state.
STATE:
case a.inl
A : Type u_1
inst✝ : CommSemiring A
I : Ideal A
hI : DividedPowers I
S : Set A
hS : S ⊆ ↑I
J : hI.SubDPIdeal
hJI : J = ⊤
⊢ S ⊆ ↑J.carrier
TACTIC:
|
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/SubDPIdeal.lean | DividedPowers.SubDPIdeal.generated_carrier_eq | [512, 1] | [531, 16] | exact hS | case a.inl
A : Type u_1
inst✝ : CommSemiring A
I : Ideal A
hI : DividedPowers I
S : Set A
hS : S ⊆ ↑I
J : hI.SubDPIdeal
hJI : J = ⊤
⊢ S ⊆ ↑⊤.carrier | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case a.inl
A : Type u_1
inst✝ : CommSemiring A
I : Ideal A
hI : DividedPowers I
S : Set A
hS : S ⊆ ↑I
J : hI.SubDPIdeal
hJI : J = ⊤
⊢ S ⊆ ↑⊤.carrier
TACTIC:
|
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/SubDPIdeal.lean | DividedPowers.SubDPIdeal.generated_carrier_eq | [512, 1] | [531, 16] | exact hJS | case a.inr
A : Type u_1
inst✝ : CommSemiring A
I : Ideal A
hI : DividedPowers I
S : Set A
hS : S ⊆ ↑I
J : hI.SubDPIdeal
hJS : S ⊆ ↑J.carrier
⊢ S ⊆ ↑J.carrier | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case a.inr
A : Type u_1
inst✝ : CommSemiring A
I : Ideal A
hI : DividedPowers I
S : Set A
hS : S ⊆ ↑I
J : hI.SubDPIdeal
hJS : S ⊆ ↑J.carrier
⊢ S ⊆ ↑J.carrier
TACTIC:
|
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/SubDPIdeal.lean | DividedPowers.isSubDPIdeal_ker | [545, 1] | [552, 20] | rw [isSubDPIdeal_inf_iff] | A : Type u_2
inst✝¹ : CommRing A
I : Ideal A
hI : DividedPowers I
B : Type u_1
inst✝ : CommRing B
J : Ideal B
hJ : DividedPowers J
f : A →+* B
hf : hI.isDPMorphism hJ f
⊢ hI.isSubDPIdeal (RingHom.ker f ⊓ I) | A : Type u_2
inst✝¹ : CommRing A
I : Ideal A
hI : DividedPowers I
B : Type u_1
inst✝ : CommRing B
J : Ideal B
hJ : DividedPowers J
f : A →+* B
hf : hI.isDPMorphism hJ f
⊢ ∀ (n : ℕ) (a b : A), a ∈ I → b ∈ I → a - b ∈ RingHom.ker f → hI.dpow n a - hI.dpow n b ∈ RingHom.ker f | Please generate a tactic in lean4 to solve the state.
STATE:
A : Type u_2
inst✝¹ : CommRing A
I : Ideal A
hI : DividedPowers I
B : Type u_1
inst✝ : CommRing B
J : Ideal B
hJ : DividedPowers J
f : A →+* B
hf : hI.isDPMorphism hJ f
⊢ hI.isSubDPIdeal (RingHom.ker f ⊓ I)
TACTIC:
|
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/SubDPIdeal.lean | DividedPowers.isSubDPIdeal_ker | [545, 1] | [552, 20] | simp only [isDPMorphism] at hf | A : Type u_2
inst✝¹ : CommRing A
I : Ideal A
hI : DividedPowers I
B : Type u_1
inst✝ : CommRing B
J : Ideal B
hJ : DividedPowers J
f : A →+* B
hf : hI.isDPMorphism hJ f
⊢ ∀ (n : ℕ) (a b : A), a ∈ I → b ∈ I → a - b ∈ RingHom.ker f → hI.dpow n a - hI.dpow n b ∈ RingHom.ker f | A : Type u_2
inst✝¹ : CommRing A
I : Ideal A
hI : DividedPowers I
B : Type u_1
inst✝ : CommRing B
J : Ideal B
hJ : DividedPowers J
f : A →+* B
hf : Ideal.map f I ≤ J ∧ ∀ (n : ℕ), ∀ a ∈ I, hJ.dpow n (f a) = f (hI.dpow n a)
⊢ ∀ (n : ℕ) (a b : A), a ∈ I → b ∈ I → a - b ∈ RingHom.ker f → hI.dpow n a - hI.dpow n b ∈ RingHom... | Please generate a tactic in lean4 to solve the state.
STATE:
A : Type u_2
inst✝¹ : CommRing A
I : Ideal A
hI : DividedPowers I
B : Type u_1
inst✝ : CommRing B
J : Ideal B
hJ : DividedPowers J
f : A →+* B
hf : hI.isDPMorphism hJ f
⊢ ∀ (n : ℕ) (a b : A), a ∈ I → b ∈ I → a - b ∈ RingHom.ker f → hI.dpow n a - hI.dpow n b ∈... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/SubDPIdeal.lean | DividedPowers.isSubDPIdeal_ker | [545, 1] | [552, 20] | intro n a b ha hb | A : Type u_2
inst✝¹ : CommRing A
I : Ideal A
hI : DividedPowers I
B : Type u_1
inst✝ : CommRing B
J : Ideal B
hJ : DividedPowers J
f : A →+* B
hf : Ideal.map f I ≤ J ∧ ∀ (n : ℕ), ∀ a ∈ I, hJ.dpow n (f a) = f (hI.dpow n a)
⊢ ∀ (n : ℕ) (a b : A), a ∈ I → b ∈ I → a - b ∈ RingHom.ker f → hI.dpow n a - hI.dpow n b ∈ RingHom... | A : Type u_2
inst✝¹ : CommRing A
I : Ideal A
hI : DividedPowers I
B : Type u_1
inst✝ : CommRing B
J : Ideal B
hJ : DividedPowers J
f : A →+* B
hf : Ideal.map f I ≤ J ∧ ∀ (n : ℕ), ∀ a ∈ I, hJ.dpow n (f a) = f (hI.dpow n a)
n : ℕ
a b : A
ha : a ∈ I
hb : b ∈ I
⊢ a - b ∈ RingHom.ker f → hI.dpow n a - hI.dpow n b ∈ RingHom.... | Please generate a tactic in lean4 to solve the state.
STATE:
A : Type u_2
inst✝¹ : CommRing A
I : Ideal A
hI : DividedPowers I
B : Type u_1
inst✝ : CommRing B
J : Ideal B
hJ : DividedPowers J
f : A →+* B
hf : Ideal.map f I ≤ J ∧ ∀ (n : ℕ), ∀ a ∈ I, hJ.dpow n (f a) = f (hI.dpow n a)
⊢ ∀ (n : ℕ) (a b : A), a ∈ I → b ∈ I ... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/SubDPIdeal.lean | DividedPowers.isSubDPIdeal_ker | [545, 1] | [552, 20] | simp only [RingHom.sub_mem_ker_iff] | A : Type u_2
inst✝¹ : CommRing A
I : Ideal A
hI : DividedPowers I
B : Type u_1
inst✝ : CommRing B
J : Ideal B
hJ : DividedPowers J
f : A →+* B
hf : Ideal.map f I ≤ J ∧ ∀ (n : ℕ), ∀ a ∈ I, hJ.dpow n (f a) = f (hI.dpow n a)
n : ℕ
a b : A
ha : a ∈ I
hb : b ∈ I
⊢ a - b ∈ RingHom.ker f → hI.dpow n a - hI.dpow n b ∈ RingHom.... | A : Type u_2
inst✝¹ : CommRing A
I : Ideal A
hI : DividedPowers I
B : Type u_1
inst✝ : CommRing B
J : Ideal B
hJ : DividedPowers J
f : A →+* B
hf : Ideal.map f I ≤ J ∧ ∀ (n : ℕ), ∀ a ∈ I, hJ.dpow n (f a) = f (hI.dpow n a)
n : ℕ
a b : A
ha : a ∈ I
hb : b ∈ I
⊢ f a = f b → f (hI.dpow n a) = f (hI.dpow n b) | Please generate a tactic in lean4 to solve the state.
STATE:
A : Type u_2
inst✝¹ : CommRing A
I : Ideal A
hI : DividedPowers I
B : Type u_1
inst✝ : CommRing B
J : Ideal B
hJ : DividedPowers J
f : A →+* B
hf : Ideal.map f I ≤ J ∧ ∀ (n : ℕ), ∀ a ∈ I, hJ.dpow n (f a) = f (hI.dpow n a)
n : ℕ
a b : A
ha : a ∈ I
hb : b ∈ I
⊢... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/SubDPIdeal.lean | DividedPowers.isSubDPIdeal_ker | [545, 1] | [552, 20] | rw [← hf.2 n a ha, ← hf.2 n b hb] | A : Type u_2
inst✝¹ : CommRing A
I : Ideal A
hI : DividedPowers I
B : Type u_1
inst✝ : CommRing B
J : Ideal B
hJ : DividedPowers J
f : A →+* B
hf : Ideal.map f I ≤ J ∧ ∀ (n : ℕ), ∀ a ∈ I, hJ.dpow n (f a) = f (hI.dpow n a)
n : ℕ
a b : A
ha : a ∈ I
hb : b ∈ I
⊢ f a = f b → f (hI.dpow n a) = f (hI.dpow n b) | A : Type u_2
inst✝¹ : CommRing A
I : Ideal A
hI : DividedPowers I
B : Type u_1
inst✝ : CommRing B
J : Ideal B
hJ : DividedPowers J
f : A →+* B
hf : Ideal.map f I ≤ J ∧ ∀ (n : ℕ), ∀ a ∈ I, hJ.dpow n (f a) = f (hI.dpow n a)
n : ℕ
a b : A
ha : a ∈ I
hb : b ∈ I
⊢ f a = f b → hJ.dpow n (f a) = hJ.dpow n (f b) | Please generate a tactic in lean4 to solve the state.
STATE:
A : Type u_2
inst✝¹ : CommRing A
I : Ideal A
hI : DividedPowers I
B : Type u_1
inst✝ : CommRing B
J : Ideal B
hJ : DividedPowers J
f : A →+* B
hf : Ideal.map f I ≤ J ∧ ∀ (n : ℕ), ∀ a ∈ I, hJ.dpow n (f a) = f (hI.dpow n a)
n : ℕ
a b : A
ha : a ∈ I
hb : b ∈ I
⊢... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/SubDPIdeal.lean | DividedPowers.isSubDPIdeal_ker | [545, 1] | [552, 20] | exact congr_arg _ | A : Type u_2
inst✝¹ : CommRing A
I : Ideal A
hI : DividedPowers I
B : Type u_1
inst✝ : CommRing B
J : Ideal B
hJ : DividedPowers J
f : A →+* B
hf : Ideal.map f I ≤ J ∧ ∀ (n : ℕ), ∀ a ∈ I, hJ.dpow n (f a) = f (hI.dpow n a)
n : ℕ
a b : A
ha : a ∈ I
hb : b ∈ I
⊢ f a = f b → hJ.dpow n (f a) = hJ.dpow n (f b) | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
A : Type u_2
inst✝¹ : CommRing A
I : Ideal A
hI : DividedPowers I
B : Type u_1
inst✝ : CommRing B
J : Ideal B
hJ : DividedPowers J
f : A →+* B
hf : Ideal.map f I ≤ J ∧ ∀ (n : ℕ), ∀ a ∈ I, hJ.dpow n (f a) = f (hI.dpow n a)
n : ℕ
a b : A
ha : a ∈ I
hb : b ∈ I
⊢... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/SubDPIdeal.lean | DividedPowers.mem_dpEqualizer_iff | [601, 1] | [603, 107] | simp only [dpEqualizer, Submodule.mem_mk, AddSubmonoid.mem_mk, AddSubsemigroup.mem_mk, Set.mem_setOf_eq] | A✝ : Type ?u.85550
inst✝¹ : CommSemiring A✝
I✝ : Ideal A✝
hI✝ : DividedPowers I✝
A : Type u_1
inst✝ : CommSemiring A
I : Ideal A
hI hI' : DividedPowers I
x : A
⊢ x ∈ hI.dpEqualizer hI' ↔ x ∈ I ∧ ∀ (n : ℕ), hI.dpow n x = hI'.dpow n x | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
A✝ : Type ?u.85550
inst✝¹ : CommSemiring A✝
I✝ : Ideal A✝
hI✝ : DividedPowers I✝
A : Type u_1
inst✝ : CommSemiring A
I : Ideal A
hI hI' : DividedPowers I
x : A
⊢ x ∈ hI.dpEqualizer hI' ↔ x ∈ I ∧ ∀ (n : ℕ), hI.dpow n x = hI'.dpow n x
TACTIC:
|
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/SubDPIdeal.lean | DividedPowers.dpEqualizer_is_dp_ideal_left | [607, 1] | [619, 69] | apply isSubDPIdeal.mk | A✝ : Type ?u.86677
inst✝¹ : CommSemiring A✝
I✝ : Ideal A✝
hI✝ : DividedPowers I✝
A : Type u_1
inst✝ : CommSemiring A
I : Ideal A
hI hI' : DividedPowers I
⊢ hI.isSubDPIdeal (hI.dpEqualizer hI') | case isSubIdeal
A✝ : Type ?u.86677
inst✝¹ : CommSemiring A✝
I✝ : Ideal A✝
hI✝ : DividedPowers I✝
A : Type u_1
inst✝ : CommSemiring A
I : Ideal A
hI hI' : DividedPowers I
⊢ hI.dpEqualizer hI' ≤ I
case dpow_mem
A✝ : Type ?u.86677
inst✝¹ : CommSemiring A✝
I✝ : Ideal A✝
hI✝ : DividedPowers I✝
A : Type u_1
inst✝ : CommSemi... | Please generate a tactic in lean4 to solve the state.
STATE:
A✝ : Type ?u.86677
inst✝¹ : CommSemiring A✝
I✝ : Ideal A✝
hI✝ : DividedPowers I✝
A : Type u_1
inst✝ : CommSemiring A
I : Ideal A
hI hI' : DividedPowers I
⊢ hI.isSubDPIdeal (hI.dpEqualizer hI')
TACTIC:
|
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/SubDPIdeal.lean | DividedPowers.dpEqualizer_is_dp_ideal_left | [607, 1] | [619, 69] | intro x hx | case isSubIdeal
A✝ : Type ?u.86677
inst✝¹ : CommSemiring A✝
I✝ : Ideal A✝
hI✝ : DividedPowers I✝
A : Type u_1
inst✝ : CommSemiring A
I : Ideal A
hI hI' : DividedPowers I
⊢ hI.dpEqualizer hI' ≤ I | case isSubIdeal
A✝ : Type ?u.86677
inst✝¹ : CommSemiring A✝
I✝ : Ideal A✝
hI✝ : DividedPowers I✝
A : Type u_1
inst✝ : CommSemiring A
I : Ideal A
hI hI' : DividedPowers I
x : A
hx : x ∈ hI.dpEqualizer hI'
⊢ x ∈ I | Please generate a tactic in lean4 to solve the state.
STATE:
case isSubIdeal
A✝ : Type ?u.86677
inst✝¹ : CommSemiring A✝
I✝ : Ideal A✝
hI✝ : DividedPowers I✝
A : Type u_1
inst✝ : CommSemiring A
I : Ideal A
hI hI' : DividedPowers I
⊢ hI.dpEqualizer hI' ≤ I
TACTIC:
|
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/SubDPIdeal.lean | DividedPowers.dpEqualizer_is_dp_ideal_left | [607, 1] | [619, 69] | rw [mem_dpEqualizer_iff] at hx | case isSubIdeal
A✝ : Type ?u.86677
inst✝¹ : CommSemiring A✝
I✝ : Ideal A✝
hI✝ : DividedPowers I✝
A : Type u_1
inst✝ : CommSemiring A
I : Ideal A
hI hI' : DividedPowers I
x : A
hx : x ∈ hI.dpEqualizer hI'
⊢ x ∈ I | case isSubIdeal
A✝ : Type ?u.86677
inst✝¹ : CommSemiring A✝
I✝ : Ideal A✝
hI✝ : DividedPowers I✝
A : Type u_1
inst✝ : CommSemiring A
I : Ideal A
hI hI' : DividedPowers I
x : A
hx : x ∈ I ∧ ∀ (n : ℕ), hI.dpow n x = hI'.dpow n x
⊢ x ∈ I | Please generate a tactic in lean4 to solve the state.
STATE:
case isSubIdeal
A✝ : Type ?u.86677
inst✝¹ : CommSemiring A✝
I✝ : Ideal A✝
hI✝ : DividedPowers I✝
A : Type u_1
inst✝ : CommSemiring A
I : Ideal A
hI hI' : DividedPowers I
x : A
hx : x ∈ hI.dpEqualizer hI'
⊢ x ∈ I
TACTIC:
|
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/SubDPIdeal.lean | DividedPowers.dpEqualizer_is_dp_ideal_left | [607, 1] | [619, 69] | exact hx.1 | case isSubIdeal
A✝ : Type ?u.86677
inst✝¹ : CommSemiring A✝
I✝ : Ideal A✝
hI✝ : DividedPowers I✝
A : Type u_1
inst✝ : CommSemiring A
I : Ideal A
hI hI' : DividedPowers I
x : A
hx : x ∈ I ∧ ∀ (n : ℕ), hI.dpow n x = hI'.dpow n x
⊢ x ∈ I | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case isSubIdeal
A✝ : Type ?u.86677
inst✝¹ : CommSemiring A✝
I✝ : Ideal A✝
hI✝ : DividedPowers I✝
A : Type u_1
inst✝ : CommSemiring A
I : Ideal A
hI hI' : DividedPowers I
x : A
hx : x ∈ I ∧ ∀ (n : ℕ), hI.dpow n x = hI'.dpow n x
⊢ x ∈ I
TACTIC:
|
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/SubDPIdeal.lean | DividedPowers.dpEqualizer_is_dp_ideal_left | [607, 1] | [619, 69] | intro n hn x hx | case dpow_mem
A✝ : Type ?u.86677
inst✝¹ : CommSemiring A✝
I✝ : Ideal A✝
hI✝ : DividedPowers I✝
A : Type u_1
inst✝ : CommSemiring A
I : Ideal A
hI hI' : DividedPowers I
⊢ ∀ (n : ℕ), n ≠ 0 → ∀ j ∈ hI.dpEqualizer hI', hI.dpow n j ∈ hI.dpEqualizer hI' | case dpow_mem
A✝ : Type ?u.86677
inst✝¹ : CommSemiring A✝
I✝ : Ideal A✝
hI✝ : DividedPowers I✝
A : Type u_1
inst✝ : CommSemiring A
I : Ideal A
hI hI' : DividedPowers I
n : ℕ
hn : n ≠ 0
x : A
hx : x ∈ hI.dpEqualizer hI'
⊢ hI.dpow n x ∈ hI.dpEqualizer hI' | Please generate a tactic in lean4 to solve the state.
STATE:
case dpow_mem
A✝ : Type ?u.86677
inst✝¹ : CommSemiring A✝
I✝ : Ideal A✝
hI✝ : DividedPowers I✝
A : Type u_1
inst✝ : CommSemiring A
I : Ideal A
hI hI' : DividedPowers I
⊢ ∀ (n : ℕ), n ≠ 0 → ∀ j ∈ hI.dpEqualizer hI', hI.dpow n j ∈ hI.dpEqualizer hI'
TACTIC:
|
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/SubDPIdeal.lean | DividedPowers.dpEqualizer_is_dp_ideal_left | [607, 1] | [619, 69] | rw [mem_dpEqualizer_iff] at hx ⊢ | case dpow_mem
A✝ : Type ?u.86677
inst✝¹ : CommSemiring A✝
I✝ : Ideal A✝
hI✝ : DividedPowers I✝
A : Type u_1
inst✝ : CommSemiring A
I : Ideal A
hI hI' : DividedPowers I
n : ℕ
hn : n ≠ 0
x : A
hx : x ∈ hI.dpEqualizer hI'
⊢ hI.dpow n x ∈ hI.dpEqualizer hI' | case dpow_mem
A✝ : Type ?u.86677
inst✝¹ : CommSemiring A✝
I✝ : Ideal A✝
hI✝ : DividedPowers I✝
A : Type u_1
inst✝ : CommSemiring A
I : Ideal A
hI hI' : DividedPowers I
n : ℕ
hn : n ≠ 0
x : A
hx : x ∈ I ∧ ∀ (n : ℕ), hI.dpow n x = hI'.dpow n x
⊢ hI.dpow n x ∈ I ∧ ∀ (n_1 : ℕ), hI.dpow n_1 (hI.dpow n x) = hI'.dpow n_1 (hI.... | Please generate a tactic in lean4 to solve the state.
STATE:
case dpow_mem
A✝ : Type ?u.86677
inst✝¹ : CommSemiring A✝
I✝ : Ideal A✝
hI✝ : DividedPowers I✝
A : Type u_1
inst✝ : CommSemiring A
I : Ideal A
hI hI' : DividedPowers I
n : ℕ
hn : n ≠ 0
x : A
hx : x ∈ hI.dpEqualizer hI'
⊢ hI.dpow n x ∈ hI.dpEqualizer hI'
TACTI... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/SubDPIdeal.lean | DividedPowers.dpEqualizer_is_dp_ideal_left | [607, 1] | [619, 69] | constructor | case dpow_mem
A✝ : Type ?u.86677
inst✝¹ : CommSemiring A✝
I✝ : Ideal A✝
hI✝ : DividedPowers I✝
A : Type u_1
inst✝ : CommSemiring A
I : Ideal A
hI hI' : DividedPowers I
n : ℕ
hn : n ≠ 0
x : A
hx : x ∈ I ∧ ∀ (n : ℕ), hI.dpow n x = hI'.dpow n x
⊢ hI.dpow n x ∈ I ∧ ∀ (n_1 : ℕ), hI.dpow n_1 (hI.dpow n x) = hI'.dpow n_1 (hI.... | case dpow_mem.left
A✝ : Type ?u.86677
inst✝¹ : CommSemiring A✝
I✝ : Ideal A✝
hI✝ : DividedPowers I✝
A : Type u_1
inst✝ : CommSemiring A
I : Ideal A
hI hI' : DividedPowers I
n : ℕ
hn : n ≠ 0
x : A
hx : x ∈ I ∧ ∀ (n : ℕ), hI.dpow n x = hI'.dpow n x
⊢ hI.dpow n x ∈ I
case dpow_mem.right
A✝ : Type ?u.86677
inst✝¹ : CommSe... | Please generate a tactic in lean4 to solve the state.
STATE:
case dpow_mem
A✝ : Type ?u.86677
inst✝¹ : CommSemiring A✝
I✝ : Ideal A✝
hI✝ : DividedPowers I✝
A : Type u_1
inst✝ : CommSemiring A
I : Ideal A
hI hI' : DividedPowers I
n : ℕ
hn : n ≠ 0
x : A
hx : x ∈ I ∧ ∀ (n : ℕ), hI.dpow n x = hI'.dpow n x
⊢ hI.dpow n x ∈ I... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/SubDPIdeal.lean | DividedPowers.dpEqualizer_is_dp_ideal_left | [607, 1] | [619, 69] | apply hI.dpow_mem hn hx.1 | case dpow_mem.left
A✝ : Type ?u.86677
inst✝¹ : CommSemiring A✝
I✝ : Ideal A✝
hI✝ : DividedPowers I✝
A : Type u_1
inst✝ : CommSemiring A
I : Ideal A
hI hI' : DividedPowers I
n : ℕ
hn : n ≠ 0
x : A
hx : x ∈ I ∧ ∀ (n : ℕ), hI.dpow n x = hI'.dpow n x
⊢ hI.dpow n x ∈ I
case dpow_mem.right
A✝ : Type ?u.86677
inst✝¹ : CommSe... | case dpow_mem.right
A✝ : Type ?u.86677
inst✝¹ : CommSemiring A✝
I✝ : Ideal A✝
hI✝ : DividedPowers I✝
A : Type u_1
inst✝ : CommSemiring A
I : Ideal A
hI hI' : DividedPowers I
n : ℕ
hn : n ≠ 0
x : A
hx : x ∈ I ∧ ∀ (n : ℕ), hI.dpow n x = hI'.dpow n x
⊢ ∀ (n_1 : ℕ), hI.dpow n_1 (hI.dpow n x) = hI'.dpow n_1 (hI.dpow n x) | Please generate a tactic in lean4 to solve the state.
STATE:
case dpow_mem.left
A✝ : Type ?u.86677
inst✝¹ : CommSemiring A✝
I✝ : Ideal A✝
hI✝ : DividedPowers I✝
A : Type u_1
inst✝ : CommSemiring A
I : Ideal A
hI hI' : DividedPowers I
n : ℕ
hn : n ≠ 0
x : A
hx : x ∈ I ∧ ∀ (n : ℕ), hI.dpow n x = hI'.dpow n x
⊢ hI.dpow n ... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/SubDPIdeal.lean | DividedPowers.dpEqualizer_is_dp_ideal_left | [607, 1] | [619, 69] | intro m | case dpow_mem.right
A✝ : Type ?u.86677
inst✝¹ : CommSemiring A✝
I✝ : Ideal A✝
hI✝ : DividedPowers I✝
A : Type u_1
inst✝ : CommSemiring A
I : Ideal A
hI hI' : DividedPowers I
n : ℕ
hn : n ≠ 0
x : A
hx : x ∈ I ∧ ∀ (n : ℕ), hI.dpow n x = hI'.dpow n x
⊢ ∀ (n_1 : ℕ), hI.dpow n_1 (hI.dpow n x) = hI'.dpow n_1 (hI.dpow n x) | case dpow_mem.right
A✝ : Type ?u.86677
inst✝¹ : CommSemiring A✝
I✝ : Ideal A✝
hI✝ : DividedPowers I✝
A : Type u_1
inst✝ : CommSemiring A
I : Ideal A
hI hI' : DividedPowers I
n : ℕ
hn : n ≠ 0
x : A
hx : x ∈ I ∧ ∀ (n : ℕ), hI.dpow n x = hI'.dpow n x
m : ℕ
⊢ hI.dpow m (hI.dpow n x) = hI'.dpow m (hI.dpow n x) | Please generate a tactic in lean4 to solve the state.
STATE:
case dpow_mem.right
A✝ : Type ?u.86677
inst✝¹ : CommSemiring A✝
I✝ : Ideal A✝
hI✝ : DividedPowers I✝
A : Type u_1
inst✝ : CommSemiring A
I : Ideal A
hI hI' : DividedPowers I
n : ℕ
hn : n ≠ 0
x : A
hx : x ∈ I ∧ ∀ (n : ℕ), hI.dpow n x = hI'.dpow n x
⊢ ∀ (n_1 : ... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/SubDPIdeal.lean | DividedPowers.dpEqualizer_is_dp_ideal_left | [607, 1] | [619, 69] | rw [hI.dpow_comp m hn hx.1, hx.2, hx.2, hI'.dpow_comp m hn hx.1] | case dpow_mem.right
A✝ : Type ?u.86677
inst✝¹ : CommSemiring A✝
I✝ : Ideal A✝
hI✝ : DividedPowers I✝
A : Type u_1
inst✝ : CommSemiring A
I : Ideal A
hI hI' : DividedPowers I
n : ℕ
hn : n ≠ 0
x : A
hx : x ∈ I ∧ ∀ (n : ℕ), hI.dpow n x = hI'.dpow n x
m : ℕ
⊢ hI.dpow m (hI.dpow n x) = hI'.dpow m (hI.dpow n x) | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case dpow_mem.right
A✝ : Type ?u.86677
inst✝¹ : CommSemiring A✝
I✝ : Ideal A✝
hI✝ : DividedPowers I✝
A : Type u_1
inst✝ : CommSemiring A
I : Ideal A
hI hI' : DividedPowers I
n : ℕ
hn : n ≠ 0
x : A
hx : x ∈ I ∧ ∀ (n : ℕ), hI.dpow n x = hI'.dpow n x
m : ℕ
⊢ hI.... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/SubDPIdeal.lean | DividedPowers.dpEqualizer_is_dp_ideal_right | [622, 1] | [634, 77] | apply isSubDPIdeal.mk | A✝ : Type ?u.87809
inst✝¹ : CommSemiring A✝
I✝ : Ideal A✝
hI✝ : DividedPowers I✝
A : Type u_1
inst✝ : CommSemiring A
I : Ideal A
hI hI' : DividedPowers I
⊢ hI'.isSubDPIdeal (hI.dpEqualizer hI') | case isSubIdeal
A✝ : Type ?u.87809
inst✝¹ : CommSemiring A✝
I✝ : Ideal A✝
hI✝ : DividedPowers I✝
A : Type u_1
inst✝ : CommSemiring A
I : Ideal A
hI hI' : DividedPowers I
⊢ hI.dpEqualizer hI' ≤ I
case dpow_mem
A✝ : Type ?u.87809
inst✝¹ : CommSemiring A✝
I✝ : Ideal A✝
hI✝ : DividedPowers I✝
A : Type u_1
inst✝ : CommSemi... | Please generate a tactic in lean4 to solve the state.
STATE:
A✝ : Type ?u.87809
inst✝¹ : CommSemiring A✝
I✝ : Ideal A✝
hI✝ : DividedPowers I✝
A : Type u_1
inst✝ : CommSemiring A
I : Ideal A
hI hI' : DividedPowers I
⊢ hI'.isSubDPIdeal (hI.dpEqualizer hI')
TACTIC:
|
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/SubDPIdeal.lean | DividedPowers.dpEqualizer_is_dp_ideal_right | [622, 1] | [634, 77] | intro x hx | case isSubIdeal
A✝ : Type ?u.87809
inst✝¹ : CommSemiring A✝
I✝ : Ideal A✝
hI✝ : DividedPowers I✝
A : Type u_1
inst✝ : CommSemiring A
I : Ideal A
hI hI' : DividedPowers I
⊢ hI.dpEqualizer hI' ≤ I | case isSubIdeal
A✝ : Type ?u.87809
inst✝¹ : CommSemiring A✝
I✝ : Ideal A✝
hI✝ : DividedPowers I✝
A : Type u_1
inst✝ : CommSemiring A
I : Ideal A
hI hI' : DividedPowers I
x : A
hx : x ∈ hI.dpEqualizer hI'
⊢ x ∈ I | Please generate a tactic in lean4 to solve the state.
STATE:
case isSubIdeal
A✝ : Type ?u.87809
inst✝¹ : CommSemiring A✝
I✝ : Ideal A✝
hI✝ : DividedPowers I✝
A : Type u_1
inst✝ : CommSemiring A
I : Ideal A
hI hI' : DividedPowers I
⊢ hI.dpEqualizer hI' ≤ I
TACTIC:
|
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/SubDPIdeal.lean | DividedPowers.dpEqualizer_is_dp_ideal_right | [622, 1] | [634, 77] | rw [mem_dpEqualizer_iff] at hx | case isSubIdeal
A✝ : Type ?u.87809
inst✝¹ : CommSemiring A✝
I✝ : Ideal A✝
hI✝ : DividedPowers I✝
A : Type u_1
inst✝ : CommSemiring A
I : Ideal A
hI hI' : DividedPowers I
x : A
hx : x ∈ hI.dpEqualizer hI'
⊢ x ∈ I | case isSubIdeal
A✝ : Type ?u.87809
inst✝¹ : CommSemiring A✝
I✝ : Ideal A✝
hI✝ : DividedPowers I✝
A : Type u_1
inst✝ : CommSemiring A
I : Ideal A
hI hI' : DividedPowers I
x : A
hx : x ∈ I ∧ ∀ (n : ℕ), hI.dpow n x = hI'.dpow n x
⊢ x ∈ I | Please generate a tactic in lean4 to solve the state.
STATE:
case isSubIdeal
A✝ : Type ?u.87809
inst✝¹ : CommSemiring A✝
I✝ : Ideal A✝
hI✝ : DividedPowers I✝
A : Type u_1
inst✝ : CommSemiring A
I : Ideal A
hI hI' : DividedPowers I
x : A
hx : x ∈ hI.dpEqualizer hI'
⊢ x ∈ I
TACTIC:
|
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/SubDPIdeal.lean | DividedPowers.dpEqualizer_is_dp_ideal_right | [622, 1] | [634, 77] | exact hx.1 | case isSubIdeal
A✝ : Type ?u.87809
inst✝¹ : CommSemiring A✝
I✝ : Ideal A✝
hI✝ : DividedPowers I✝
A : Type u_1
inst✝ : CommSemiring A
I : Ideal A
hI hI' : DividedPowers I
x : A
hx : x ∈ I ∧ ∀ (n : ℕ), hI.dpow n x = hI'.dpow n x
⊢ x ∈ I | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case isSubIdeal
A✝ : Type ?u.87809
inst✝¹ : CommSemiring A✝
I✝ : Ideal A✝
hI✝ : DividedPowers I✝
A : Type u_1
inst✝ : CommSemiring A
I : Ideal A
hI hI' : DividedPowers I
x : A
hx : x ∈ I ∧ ∀ (n : ℕ), hI.dpow n x = hI'.dpow n x
⊢ x ∈ I
TACTIC:
|
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/SubDPIdeal.lean | DividedPowers.dpEqualizer_is_dp_ideal_right | [622, 1] | [634, 77] | intro n hn x hx | case dpow_mem
A✝ : Type ?u.87809
inst✝¹ : CommSemiring A✝
I✝ : Ideal A✝
hI✝ : DividedPowers I✝
A : Type u_1
inst✝ : CommSemiring A
I : Ideal A
hI hI' : DividedPowers I
⊢ ∀ (n : ℕ), n ≠ 0 → ∀ j ∈ hI.dpEqualizer hI', hI'.dpow n j ∈ hI.dpEqualizer hI' | case dpow_mem
A✝ : Type ?u.87809
inst✝¹ : CommSemiring A✝
I✝ : Ideal A✝
hI✝ : DividedPowers I✝
A : Type u_1
inst✝ : CommSemiring A
I : Ideal A
hI hI' : DividedPowers I
n : ℕ
hn : n ≠ 0
x : A
hx : x ∈ hI.dpEqualizer hI'
⊢ hI'.dpow n x ∈ hI.dpEqualizer hI' | Please generate a tactic in lean4 to solve the state.
STATE:
case dpow_mem
A✝ : Type ?u.87809
inst✝¹ : CommSemiring A✝
I✝ : Ideal A✝
hI✝ : DividedPowers I✝
A : Type u_1
inst✝ : CommSemiring A
I : Ideal A
hI hI' : DividedPowers I
⊢ ∀ (n : ℕ), n ≠ 0 → ∀ j ∈ hI.dpEqualizer hI', hI'.dpow n j ∈ hI.dpEqualizer hI'
TACTIC:
|
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/SubDPIdeal.lean | DividedPowers.dpEqualizer_is_dp_ideal_right | [622, 1] | [634, 77] | rw [mem_dpEqualizer_iff] at hx ⊢ | case dpow_mem
A✝ : Type ?u.87809
inst✝¹ : CommSemiring A✝
I✝ : Ideal A✝
hI✝ : DividedPowers I✝
A : Type u_1
inst✝ : CommSemiring A
I : Ideal A
hI hI' : DividedPowers I
n : ℕ
hn : n ≠ 0
x : A
hx : x ∈ hI.dpEqualizer hI'
⊢ hI'.dpow n x ∈ hI.dpEqualizer hI' | case dpow_mem
A✝ : Type ?u.87809
inst✝¹ : CommSemiring A✝
I✝ : Ideal A✝
hI✝ : DividedPowers I✝
A : Type u_1
inst✝ : CommSemiring A
I : Ideal A
hI hI' : DividedPowers I
n : ℕ
hn : n ≠ 0
x : A
hx : x ∈ I ∧ ∀ (n : ℕ), hI.dpow n x = hI'.dpow n x
⊢ hI'.dpow n x ∈ I ∧ ∀ (n_1 : ℕ), hI.dpow n_1 (hI'.dpow n x) = hI'.dpow n_1 (h... | Please generate a tactic in lean4 to solve the state.
STATE:
case dpow_mem
A✝ : Type ?u.87809
inst✝¹ : CommSemiring A✝
I✝ : Ideal A✝
hI✝ : DividedPowers I✝
A : Type u_1
inst✝ : CommSemiring A
I : Ideal A
hI hI' : DividedPowers I
n : ℕ
hn : n ≠ 0
x : A
hx : x ∈ hI.dpEqualizer hI'
⊢ hI'.dpow n x ∈ hI.dpEqualizer hI'
TACT... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/SubDPIdeal.lean | DividedPowers.dpEqualizer_is_dp_ideal_right | [622, 1] | [634, 77] | constructor | case dpow_mem
A✝ : Type ?u.87809
inst✝¹ : CommSemiring A✝
I✝ : Ideal A✝
hI✝ : DividedPowers I✝
A : Type u_1
inst✝ : CommSemiring A
I : Ideal A
hI hI' : DividedPowers I
n : ℕ
hn : n ≠ 0
x : A
hx : x ∈ I ∧ ∀ (n : ℕ), hI.dpow n x = hI'.dpow n x
⊢ hI'.dpow n x ∈ I ∧ ∀ (n_1 : ℕ), hI.dpow n_1 (hI'.dpow n x) = hI'.dpow n_1 (h... | case dpow_mem.left
A✝ : Type ?u.87809
inst✝¹ : CommSemiring A✝
I✝ : Ideal A✝
hI✝ : DividedPowers I✝
A : Type u_1
inst✝ : CommSemiring A
I : Ideal A
hI hI' : DividedPowers I
n : ℕ
hn : n ≠ 0
x : A
hx : x ∈ I ∧ ∀ (n : ℕ), hI.dpow n x = hI'.dpow n x
⊢ hI'.dpow n x ∈ I
case dpow_mem.right
A✝ : Type ?u.87809
inst✝¹ : CommS... | Please generate a tactic in lean4 to solve the state.
STATE:
case dpow_mem
A✝ : Type ?u.87809
inst✝¹ : CommSemiring A✝
I✝ : Ideal A✝
hI✝ : DividedPowers I✝
A : Type u_1
inst✝ : CommSemiring A
I : Ideal A
hI hI' : DividedPowers I
n : ℕ
hn : n ≠ 0
x : A
hx : x ∈ I ∧ ∀ (n : ℕ), hI.dpow n x = hI'.dpow n x
⊢ hI'.dpow n x ∈ ... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/SubDPIdeal.lean | DividedPowers.dpEqualizer_is_dp_ideal_right | [622, 1] | [634, 77] | apply hI'.dpow_mem hn hx.1 | case dpow_mem.left
A✝ : Type ?u.87809
inst✝¹ : CommSemiring A✝
I✝ : Ideal A✝
hI✝ : DividedPowers I✝
A : Type u_1
inst✝ : CommSemiring A
I : Ideal A
hI hI' : DividedPowers I
n : ℕ
hn : n ≠ 0
x : A
hx : x ∈ I ∧ ∀ (n : ℕ), hI.dpow n x = hI'.dpow n x
⊢ hI'.dpow n x ∈ I
case dpow_mem.right
A✝ : Type ?u.87809
inst✝¹ : CommS... | case dpow_mem.right
A✝ : Type ?u.87809
inst✝¹ : CommSemiring A✝
I✝ : Ideal A✝
hI✝ : DividedPowers I✝
A : Type u_1
inst✝ : CommSemiring A
I : Ideal A
hI hI' : DividedPowers I
n : ℕ
hn : n ≠ 0
x : A
hx : x ∈ I ∧ ∀ (n : ℕ), hI.dpow n x = hI'.dpow n x
⊢ ∀ (n_1 : ℕ), hI.dpow n_1 (hI'.dpow n x) = hI'.dpow n_1 (hI'.dpow n x) | Please generate a tactic in lean4 to solve the state.
STATE:
case dpow_mem.left
A✝ : Type ?u.87809
inst✝¹ : CommSemiring A✝
I✝ : Ideal A✝
hI✝ : DividedPowers I✝
A : Type u_1
inst✝ : CommSemiring A
I : Ideal A
hI hI' : DividedPowers I
n : ℕ
hn : n ≠ 0
x : A
hx : x ∈ I ∧ ∀ (n : ℕ), hI.dpow n x = hI'.dpow n x
⊢ hI'.dpow n... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/SubDPIdeal.lean | DividedPowers.dpEqualizer_is_dp_ideal_right | [622, 1] | [634, 77] | intro m | case dpow_mem.right
A✝ : Type ?u.87809
inst✝¹ : CommSemiring A✝
I✝ : Ideal A✝
hI✝ : DividedPowers I✝
A : Type u_1
inst✝ : CommSemiring A
I : Ideal A
hI hI' : DividedPowers I
n : ℕ
hn : n ≠ 0
x : A
hx : x ∈ I ∧ ∀ (n : ℕ), hI.dpow n x = hI'.dpow n x
⊢ ∀ (n_1 : ℕ), hI.dpow n_1 (hI'.dpow n x) = hI'.dpow n_1 (hI'.dpow n x) | case dpow_mem.right
A✝ : Type ?u.87809
inst✝¹ : CommSemiring A✝
I✝ : Ideal A✝
hI✝ : DividedPowers I✝
A : Type u_1
inst✝ : CommSemiring A
I : Ideal A
hI hI' : DividedPowers I
n : ℕ
hn : n ≠ 0
x : A
hx : x ∈ I ∧ ∀ (n : ℕ), hI.dpow n x = hI'.dpow n x
m : ℕ
⊢ hI.dpow m (hI'.dpow n x) = hI'.dpow m (hI'.dpow n x) | Please generate a tactic in lean4 to solve the state.
STATE:
case dpow_mem.right
A✝ : Type ?u.87809
inst✝¹ : CommSemiring A✝
I✝ : Ideal A✝
hI✝ : DividedPowers I✝
A : Type u_1
inst✝ : CommSemiring A
I : Ideal A
hI hI' : DividedPowers I
n : ℕ
hn : n ≠ 0
x : A
hx : x ∈ I ∧ ∀ (n : ℕ), hI.dpow n x = hI'.dpow n x
⊢ ∀ (n_1 : ... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/SubDPIdeal.lean | DividedPowers.dpEqualizer_is_dp_ideal_right | [622, 1] | [634, 77] | rw [← hx.2, hI.dpow_comp m hn hx.1, hx.2, hx.2, hI'.dpow_comp m hn hx.1] | case dpow_mem.right
A✝ : Type ?u.87809
inst✝¹ : CommSemiring A✝
I✝ : Ideal A✝
hI✝ : DividedPowers I✝
A : Type u_1
inst✝ : CommSemiring A
I : Ideal A
hI hI' : DividedPowers I
n : ℕ
hn : n ≠ 0
x : A
hx : x ∈ I ∧ ∀ (n : ℕ), hI.dpow n x = hI'.dpow n x
m : ℕ
⊢ hI.dpow m (hI'.dpow n x) = hI'.dpow m (hI'.dpow n x) | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case dpow_mem.right
A✝ : Type ?u.87809
inst✝¹ : CommSemiring A✝
I✝ : Ideal A✝
hI✝ : DividedPowers I✝
A : Type u_1
inst✝ : CommSemiring A
I : Ideal A
hI hI' : DividedPowers I
n : ℕ
hn : n ≠ 0
x : A
hx : x ∈ I ∧ ∀ (n : ℕ), hI.dpow n x = hI'.dpow n x
m : ℕ
⊢ hI.... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/SubDPIdeal.lean | DividedPowers.le_equalizer_of_dp_morphism | [638, 1] | [650, 35] | rw [Ideal.map] | A✝ : Type ?u.88948
inst✝² : CommSemiring A✝
I✝ : Ideal A✝
hI✝ : DividedPowers I✝
A : Type u_1
inst✝¹ : CommSemiring A
I : Ideal A
hI : DividedPowers I
B : Type u_2
inst✝ : CommSemiring B
f : A →+* B
K : Ideal B
hI_le_K : Ideal.map f I ≤ K
hK hK' : DividedPowers K
hIK : hI.isDPMorphism hK f
hIK' : hI.isDPMorphism hK' f
... | A✝ : Type ?u.88948
inst✝² : CommSemiring A✝
I✝ : Ideal A✝
hI✝ : DividedPowers I✝
A : Type u_1
inst✝¹ : CommSemiring A
I : Ideal A
hI : DividedPowers I
B : Type u_2
inst✝ : CommSemiring B
f : A →+* B
K : Ideal B
hI_le_K : Ideal.map f I ≤ K
hK hK' : DividedPowers K
hIK : hI.isDPMorphism hK f
hIK' : hI.isDPMorphism hK' f
... | Please generate a tactic in lean4 to solve the state.
STATE:
A✝ : Type ?u.88948
inst✝² : CommSemiring A✝
I✝ : Ideal A✝
hI✝ : DividedPowers I✝
A : Type u_1
inst✝¹ : CommSemiring A
I : Ideal A
hI : DividedPowers I
B : Type u_2
inst✝ : CommSemiring B
f : A →+* B
K : Ideal B
hI_le_K : Ideal.map f I ≤ K
hK hK' : DividedPowe... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/SubDPIdeal.lean | DividedPowers.le_equalizer_of_dp_morphism | [638, 1] | [650, 35] | rw [Ideal.span_le] | A✝ : Type ?u.88948
inst✝² : CommSemiring A✝
I✝ : Ideal A✝
hI✝ : DividedPowers I✝
A : Type u_1
inst✝¹ : CommSemiring A
I : Ideal A
hI : DividedPowers I
B : Type u_2
inst✝ : CommSemiring B
f : A →+* B
K : Ideal B
hI_le_K : Ideal.map f I ≤ K
hK hK' : DividedPowers K
hIK : hI.isDPMorphism hK f
hIK' : hI.isDPMorphism hK' f
... | A✝ : Type ?u.88948
inst✝² : CommSemiring A✝
I✝ : Ideal A✝
hI✝ : DividedPowers I✝
A : Type u_1
inst✝¹ : CommSemiring A
I : Ideal A
hI : DividedPowers I
B : Type u_2
inst✝ : CommSemiring B
f : A →+* B
K : Ideal B
hI_le_K : Ideal.map f I ≤ K
hK hK' : DividedPowers K
hIK : hI.isDPMorphism hK f
hIK' : hI.isDPMorphism hK' f
... | Please generate a tactic in lean4 to solve the state.
STATE:
A✝ : Type ?u.88948
inst✝² : CommSemiring A✝
I✝ : Ideal A✝
hI✝ : DividedPowers I✝
A : Type u_1
inst✝¹ : CommSemiring A
I : Ideal A
hI : DividedPowers I
B : Type u_2
inst✝ : CommSemiring B
f : A →+* B
K : Ideal B
hI_le_K : Ideal.map f I ≤ K
hK hK' : DividedPowe... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/SubDPIdeal.lean | DividedPowers.le_equalizer_of_dp_morphism | [638, 1] | [650, 35] | rintro b ⟨a, ha, rfl⟩ | A✝ : Type ?u.88948
inst✝² : CommSemiring A✝
I✝ : Ideal A✝
hI✝ : DividedPowers I✝
A : Type u_1
inst✝¹ : CommSemiring A
I : Ideal A
hI : DividedPowers I
B : Type u_2
inst✝ : CommSemiring B
f : A →+* B
K : Ideal B
hI_le_K : Ideal.map f I ≤ K
hK hK' : DividedPowers K
hIK : hI.isDPMorphism hK f
hIK' : hI.isDPMorphism hK' f
... | case intro.intro
A✝ : Type ?u.88948
inst✝² : CommSemiring A✝
I✝ : Ideal A✝
hI✝ : DividedPowers I✝
A : Type u_1
inst✝¹ : CommSemiring A
I : Ideal A
hI : DividedPowers I
B : Type u_2
inst✝ : CommSemiring B
f : A →+* B
K : Ideal B
hI_le_K : Ideal.map f I ≤ K
hK hK' : DividedPowers K
hIK : hI.isDPMorphism hK f
hIK' : hI.is... | Please generate a tactic in lean4 to solve the state.
STATE:
A✝ : Type ?u.88948
inst✝² : CommSemiring A✝
I✝ : Ideal A✝
hI✝ : DividedPowers I✝
A : Type u_1
inst✝¹ : CommSemiring A
I : Ideal A
hI : DividedPowers I
B : Type u_2
inst✝ : CommSemiring B
f : A →+* B
K : Ideal B
hI_le_K : Ideal.map f I ≤ K
hK hK' : DividedPowe... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/SubDPIdeal.lean | DividedPowers.le_equalizer_of_dp_morphism | [638, 1] | [650, 35] | simp only [SetLike.mem_coe] at ha ⊢ | case intro.intro
A✝ : Type ?u.88948
inst✝² : CommSemiring A✝
I✝ : Ideal A✝
hI✝ : DividedPowers I✝
A : Type u_1
inst✝¹ : CommSemiring A
I : Ideal A
hI : DividedPowers I
B : Type u_2
inst✝ : CommSemiring B
f : A →+* B
K : Ideal B
hI_le_K : Ideal.map f I ≤ K
hK hK' : DividedPowers K
hIK : hI.isDPMorphism hK f
hIK' : hI.is... | case intro.intro
A✝ : Type ?u.88948
inst✝² : CommSemiring A✝
I✝ : Ideal A✝
hI✝ : DividedPowers I✝
A : Type u_1
inst✝¹ : CommSemiring A
I : Ideal A
hI : DividedPowers I
B : Type u_2
inst✝ : CommSemiring B
f : A →+* B
K : Ideal B
hI_le_K : Ideal.map f I ≤ K
hK hK' : DividedPowers K
hIK : hI.isDPMorphism hK f
hIK' : hI.is... | Please generate a tactic in lean4 to solve the state.
STATE:
case intro.intro
A✝ : Type ?u.88948
inst✝² : CommSemiring A✝
I✝ : Ideal A✝
hI✝ : DividedPowers I✝
A : Type u_1
inst✝¹ : CommSemiring A
I : Ideal A
hI : DividedPowers I
B : Type u_2
inst✝ : CommSemiring B
f : A →+* B
K : Ideal B
hI_le_K : Ideal.map f I ≤ K
hK ... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/SubDPIdeal.lean | DividedPowers.le_equalizer_of_dp_morphism | [638, 1] | [650, 35] | rw [mem_dpEqualizer_iff] | case intro.intro
A✝ : Type ?u.88948
inst✝² : CommSemiring A✝
I✝ : Ideal A✝
hI✝ : DividedPowers I✝
A : Type u_1
inst✝¹ : CommSemiring A
I : Ideal A
hI : DividedPowers I
B : Type u_2
inst✝ : CommSemiring B
f : A →+* B
K : Ideal B
hI_le_K : Ideal.map f I ≤ K
hK hK' : DividedPowers K
hIK : hI.isDPMorphism hK f
hIK' : hI.is... | case intro.intro
A✝ : Type ?u.88948
inst✝² : CommSemiring A✝
I✝ : Ideal A✝
hI✝ : DividedPowers I✝
A : Type u_1
inst✝¹ : CommSemiring A
I : Ideal A
hI : DividedPowers I
B : Type u_2
inst✝ : CommSemiring B
f : A →+* B
K : Ideal B
hI_le_K : Ideal.map f I ≤ K
hK hK' : DividedPowers K
hIK : hI.isDPMorphism hK f
hIK' : hI.is... | Please generate a tactic in lean4 to solve the state.
STATE:
case intro.intro
A✝ : Type ?u.88948
inst✝² : CommSemiring A✝
I✝ : Ideal A✝
hI✝ : DividedPowers I✝
A : Type u_1
inst✝¹ : CommSemiring A
I : Ideal A
hI : DividedPowers I
B : Type u_2
inst✝ : CommSemiring B
f : A →+* B
K : Ideal B
hI_le_K : Ideal.map f I ≤ K
hK ... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/SubDPIdeal.lean | DividedPowers.le_equalizer_of_dp_morphism | [638, 1] | [650, 35] | constructor | case intro.intro
A✝ : Type ?u.88948
inst✝² : CommSemiring A✝
I✝ : Ideal A✝
hI✝ : DividedPowers I✝
A : Type u_1
inst✝¹ : CommSemiring A
I : Ideal A
hI : DividedPowers I
B : Type u_2
inst✝ : CommSemiring B
f : A →+* B
K : Ideal B
hI_le_K : Ideal.map f I ≤ K
hK hK' : DividedPowers K
hIK : hI.isDPMorphism hK f
hIK' : hI.is... | case intro.intro.left
A✝ : Type ?u.88948
inst✝² : CommSemiring A✝
I✝ : Ideal A✝
hI✝ : DividedPowers I✝
A : Type u_1
inst✝¹ : CommSemiring A
I : Ideal A
hI : DividedPowers I
B : Type u_2
inst✝ : CommSemiring B
f : A →+* B
K : Ideal B
hI_le_K : Ideal.map f I ≤ K
hK hK' : DividedPowers K
hIK : hI.isDPMorphism hK f
hIK' : ... | Please generate a tactic in lean4 to solve the state.
STATE:
case intro.intro
A✝ : Type ?u.88948
inst✝² : CommSemiring A✝
I✝ : Ideal A✝
hI✝ : DividedPowers I✝
A : Type u_1
inst✝¹ : CommSemiring A
I : Ideal A
hI : DividedPowers I
B : Type u_2
inst✝ : CommSemiring B
f : A →+* B
K : Ideal B
hI_le_K : Ideal.map f I ≤ K
hK ... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/SubDPIdeal.lean | DividedPowers.le_equalizer_of_dp_morphism | [638, 1] | [650, 35] | apply hI_le_K | case intro.intro.left
A✝ : Type ?u.88948
inst✝² : CommSemiring A✝
I✝ : Ideal A✝
hI✝ : DividedPowers I✝
A : Type u_1
inst✝¹ : CommSemiring A
I : Ideal A
hI : DividedPowers I
B : Type u_2
inst✝ : CommSemiring B
f : A →+* B
K : Ideal B
hI_le_K : Ideal.map f I ≤ K
hK hK' : DividedPowers K
hIK : hI.isDPMorphism hK f
hIK' : ... | case intro.intro.left.a
A✝ : Type ?u.88948
inst✝² : CommSemiring A✝
I✝ : Ideal A✝
hI✝ : DividedPowers I✝
A : Type u_1
inst✝¹ : CommSemiring A
I : Ideal A
hI : DividedPowers I
B : Type u_2
inst✝ : CommSemiring B
f : A →+* B
K : Ideal B
hI_le_K : Ideal.map f I ≤ K
hK hK' : DividedPowers K
hIK : hI.isDPMorphism hK f
hIK' ... | Please generate a tactic in lean4 to solve the state.
STATE:
case intro.intro.left
A✝ : Type ?u.88948
inst✝² : CommSemiring A✝
I✝ : Ideal A✝
hI✝ : DividedPowers I✝
A : Type u_1
inst✝¹ : CommSemiring A
I : Ideal A
hI : DividedPowers I
B : Type u_2
inst✝ : CommSemiring B
f : A →+* B
K : Ideal B
hI_le_K : Ideal.map f I ≤ ... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/SubDPIdeal.lean | DividedPowers.le_equalizer_of_dp_morphism | [638, 1] | [650, 35] | exact Ideal.mem_map_of_mem f ha | case intro.intro.left.a
A✝ : Type ?u.88948
inst✝² : CommSemiring A✝
I✝ : Ideal A✝
hI✝ : DividedPowers I✝
A : Type u_1
inst✝¹ : CommSemiring A
I : Ideal A
hI : DividedPowers I
B : Type u_2
inst✝ : CommSemiring B
f : A →+* B
K : Ideal B
hI_le_K : Ideal.map f I ≤ K
hK hK' : DividedPowers K
hIK : hI.isDPMorphism hK f
hIK' ... | case intro.intro.right
A✝ : Type ?u.88948
inst✝² : CommSemiring A✝
I✝ : Ideal A✝
hI✝ : DividedPowers I✝
A : Type u_1
inst✝¹ : CommSemiring A
I : Ideal A
hI : DividedPowers I
B : Type u_2
inst✝ : CommSemiring B
f : A →+* B
K : Ideal B
hI_le_K : Ideal.map f I ≤ K
hK hK' : DividedPowers K
hIK : hI.isDPMorphism hK f
hIK' :... | Please generate a tactic in lean4 to solve the state.
STATE:
case intro.intro.left.a
A✝ : Type ?u.88948
inst✝² : CommSemiring A✝
I✝ : Ideal A✝
hI✝ : DividedPowers I✝
A : Type u_1
inst✝¹ : CommSemiring A
I : Ideal A
hI : DividedPowers I
B : Type u_2
inst✝ : CommSemiring B
f : A →+* B
K : Ideal B
hI_le_K : Ideal.map f I ... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/SubDPIdeal.lean | DividedPowers.le_equalizer_of_dp_morphism | [638, 1] | [650, 35] | simp only [isDPMorphism, Ideal.map_id, RingHom.id_apply] at hIK hIK' | case intro.intro.right
A✝ : Type ?u.88948
inst✝² : CommSemiring A✝
I✝ : Ideal A✝
hI✝ : DividedPowers I✝
A : Type u_1
inst✝¹ : CommSemiring A
I : Ideal A
hI : DividedPowers I
B : Type u_2
inst✝ : CommSemiring B
f : A →+* B
K : Ideal B
hI_le_K : Ideal.map f I ≤ K
hK hK' : DividedPowers K
hIK : hI.isDPMorphism hK f
hIK' :... | case intro.intro.right
A✝ : Type ?u.88948
inst✝² : CommSemiring A✝
I✝ : Ideal A✝
hI✝ : DividedPowers I✝
A : Type u_1
inst✝¹ : CommSemiring A
I : Ideal A
hI : DividedPowers I
B : Type u_2
inst✝ : CommSemiring B
f : A →+* B
K : Ideal B
hI_le_K : Ideal.map f I ≤ K
hK hK' : DividedPowers K
hIK : Ideal.map f I ≤ K ∧ ∀ (n : ... | Please generate a tactic in lean4 to solve the state.
STATE:
case intro.intro.right
A✝ : Type ?u.88948
inst✝² : CommSemiring A✝
I✝ : Ideal A✝
hI✝ : DividedPowers I✝
A : Type u_1
inst✝¹ : CommSemiring A
I : Ideal A
hI : DividedPowers I
B : Type u_2
inst✝ : CommSemiring B
f : A →+* B
K : Ideal B
hI_le_K : Ideal.map f I ≤... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/SubDPIdeal.lean | DividedPowers.le_equalizer_of_dp_morphism | [638, 1] | [650, 35] | intro n | case intro.intro.right
A✝ : Type ?u.88948
inst✝² : CommSemiring A✝
I✝ : Ideal A✝
hI✝ : DividedPowers I✝
A : Type u_1
inst✝¹ : CommSemiring A
I : Ideal A
hI : DividedPowers I
B : Type u_2
inst✝ : CommSemiring B
f : A →+* B
K : Ideal B
hI_le_K : Ideal.map f I ≤ K
hK hK' : DividedPowers K
hIK : Ideal.map f I ≤ K ∧ ∀ (n : ... | case intro.intro.right
A✝ : Type ?u.88948
inst✝² : CommSemiring A✝
I✝ : Ideal A✝
hI✝ : DividedPowers I✝
A : Type u_1
inst✝¹ : CommSemiring A
I : Ideal A
hI : DividedPowers I
B : Type u_2
inst✝ : CommSemiring B
f : A →+* B
K : Ideal B
hI_le_K : Ideal.map f I ≤ K
hK hK' : DividedPowers K
hIK : Ideal.map f I ≤ K ∧ ∀ (n : ... | Please generate a tactic in lean4 to solve the state.
STATE:
case intro.intro.right
A✝ : Type ?u.88948
inst✝² : CommSemiring A✝
I✝ : Ideal A✝
hI✝ : DividedPowers I✝
A : Type u_1
inst✝¹ : CommSemiring A
I : Ideal A
hI : DividedPowers I
B : Type u_2
inst✝ : CommSemiring B
f : A →+* B
K : Ideal B
hI_le_K : Ideal.map f I ≤... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/SubDPIdeal.lean | DividedPowers.le_equalizer_of_dp_morphism | [638, 1] | [650, 35] | rw [hIK.2 n a ha, hIK'.2 n a ha] | case intro.intro.right
A✝ : Type ?u.88948
inst✝² : CommSemiring A✝
I✝ : Ideal A✝
hI✝ : DividedPowers I✝
A : Type u_1
inst✝¹ : CommSemiring A
I : Ideal A
hI : DividedPowers I
B : Type u_2
inst✝ : CommSemiring B
f : A →+* B
K : Ideal B
hI_le_K : Ideal.map f I ≤ K
hK hK' : DividedPowers K
hIK : Ideal.map f I ≤ K ∧ ∀ (n : ... | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case intro.intro.right
A✝ : Type ?u.88948
inst✝² : CommSemiring A✝
I✝ : Ideal A✝
hI✝ : DividedPowers I✝
A : Type u_1
inst✝¹ : CommSemiring A
I : Ideal A
hI : DividedPowers I
B : Type u_2
inst✝ : CommSemiring B
f : A →+* B
K : Ideal B
hI_le_K : Ideal.map f I ≤... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/SubDPIdeal.lean | DividedPowers.Quotient.OfSurjective.dpow_apply' | [703, 1] | [713, 32] | classical
simp only [dpow, Function.extend_def]
have h : ∃ (a_1 : I), f ↑a_1 = f a := by use ⟨a, ha⟩
rw [dif_pos h]
rw [← sub_eq_zero, ← map_sub, ← RingHom.mem_ker]
rw [isSubDPIdeal_inf_iff] at hIf
apply hIf n _ a _ ha
rw [RingHom.mem_ker, map_sub, sub_eq_zero]
rw [h.choose_spec]
simp only [Submodule.coe_mem] | A : Type u_1
inst✝¹ : CommRing A
I : Ideal A
hI : DividedPowers I
B : Type u_2
inst✝ : CommRing B
f : A →+* B
hf : Function.Surjective ⇑f
hIf : hI.isSubDPIdeal (RingHom.ker f ⊓ I)
n : ℕ
a : A
ha : a ∈ I
⊢ dpow hI f n (f a) = f (hI.dpow n a) | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
A : Type u_1
inst✝¹ : CommRing A
I : Ideal A
hI : DividedPowers I
B : Type u_2
inst✝ : CommRing B
f : A →+* B
hf : Function.Surjective ⇑f
hIf : hI.isSubDPIdeal (RingHom.ker f ⊓ I)
n : ℕ
a : A
ha : a ∈ I
⊢ dpow hI f n (f a) = f (hI.dpow n a)
TACTIC:
|
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/SubDPIdeal.lean | DividedPowers.Quotient.OfSurjective.dpow_apply' | [703, 1] | [713, 32] | simp only [dpow, Function.extend_def] | A : Type u_1
inst✝¹ : CommRing A
I : Ideal A
hI : DividedPowers I
B : Type u_2
inst✝ : CommRing B
f : A →+* B
hf : Function.Surjective ⇑f
hIf : hI.isSubDPIdeal (RingHom.ker f ⊓ I)
n : ℕ
a : A
ha : a ∈ I
⊢ dpow hI f n (f a) = f (hI.dpow n a) | A : Type u_1
inst✝¹ : CommRing A
I : Ideal A
hI : DividedPowers I
B : Type u_2
inst✝ : CommRing B
f : A →+* B
hf : Function.Surjective ⇑f
hIf : hI.isSubDPIdeal (RingHom.ker f ⊓ I)
n : ℕ
a : A
ha : a ∈ I
⊢ (if h : ∃ a_1, f ↑a_1 = f a then f (hI.dpow n ↑(Classical.choose ⋯)) else OfNat.ofNat 0 (f a)) = f (hI.dpow n a) | Please generate a tactic in lean4 to solve the state.
STATE:
A : Type u_1
inst✝¹ : CommRing A
I : Ideal A
hI : DividedPowers I
B : Type u_2
inst✝ : CommRing B
f : A →+* B
hf : Function.Surjective ⇑f
hIf : hI.isSubDPIdeal (RingHom.ker f ⊓ I)
n : ℕ
a : A
ha : a ∈ I
⊢ dpow hI f n (f a) = f (hI.dpow n a)
TACTIC:
|
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/SubDPIdeal.lean | DividedPowers.Quotient.OfSurjective.dpow_apply' | [703, 1] | [713, 32] | have h : ∃ (a_1 : I), f ↑a_1 = f a := by use ⟨a, ha⟩ | A : Type u_1
inst✝¹ : CommRing A
I : Ideal A
hI : DividedPowers I
B : Type u_2
inst✝ : CommRing B
f : A →+* B
hf : Function.Surjective ⇑f
hIf : hI.isSubDPIdeal (RingHom.ker f ⊓ I)
n : ℕ
a : A
ha : a ∈ I
⊢ (if h : ∃ a_1, f ↑a_1 = f a then f (hI.dpow n ↑(Classical.choose ⋯)) else OfNat.ofNat 0 (f a)) = f (hI.dpow n a) | A : Type u_1
inst✝¹ : CommRing A
I : Ideal A
hI : DividedPowers I
B : Type u_2
inst✝ : CommRing B
f : A →+* B
hf : Function.Surjective ⇑f
hIf : hI.isSubDPIdeal (RingHom.ker f ⊓ I)
n : ℕ
a : A
ha : a ∈ I
h : ∃ a_1, f ↑a_1 = f a
⊢ (if h : ∃ a_1, f ↑a_1 = f a then f (hI.dpow n ↑(Classical.choose ⋯)) else OfNat.ofNat 0 (f ... | Please generate a tactic in lean4 to solve the state.
STATE:
A : Type u_1
inst✝¹ : CommRing A
I : Ideal A
hI : DividedPowers I
B : Type u_2
inst✝ : CommRing B
f : A →+* B
hf : Function.Surjective ⇑f
hIf : hI.isSubDPIdeal (RingHom.ker f ⊓ I)
n : ℕ
a : A
ha : a ∈ I
⊢ (if h : ∃ a_1, f ↑a_1 = f a then f (hI.dpow n ↑(Classi... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/SubDPIdeal.lean | DividedPowers.Quotient.OfSurjective.dpow_apply' | [703, 1] | [713, 32] | rw [dif_pos h] | A : Type u_1
inst✝¹ : CommRing A
I : Ideal A
hI : DividedPowers I
B : Type u_2
inst✝ : CommRing B
f : A →+* B
hf : Function.Surjective ⇑f
hIf : hI.isSubDPIdeal (RingHom.ker f ⊓ I)
n : ℕ
a : A
ha : a ∈ I
h : ∃ a_1, f ↑a_1 = f a
⊢ (if h : ∃ a_1, f ↑a_1 = f a then f (hI.dpow n ↑(Classical.choose ⋯)) else OfNat.ofNat 0 (f ... | A : Type u_1
inst✝¹ : CommRing A
I : Ideal A
hI : DividedPowers I
B : Type u_2
inst✝ : CommRing B
f : A →+* B
hf : Function.Surjective ⇑f
hIf : hI.isSubDPIdeal (RingHom.ker f ⊓ I)
n : ℕ
a : A
ha : a ∈ I
h : ∃ a_1, f ↑a_1 = f a
⊢ f (hI.dpow n ↑(Classical.choose ⋯)) = f (hI.dpow n a) | Please generate a tactic in lean4 to solve the state.
STATE:
A : Type u_1
inst✝¹ : CommRing A
I : Ideal A
hI : DividedPowers I
B : Type u_2
inst✝ : CommRing B
f : A →+* B
hf : Function.Surjective ⇑f
hIf : hI.isSubDPIdeal (RingHom.ker f ⊓ I)
n : ℕ
a : A
ha : a ∈ I
h : ∃ a_1, f ↑a_1 = f a
⊢ (if h : ∃ a_1, f ↑a_1 = f a th... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/SubDPIdeal.lean | DividedPowers.Quotient.OfSurjective.dpow_apply' | [703, 1] | [713, 32] | rw [← sub_eq_zero, ← map_sub, ← RingHom.mem_ker] | A : Type u_1
inst✝¹ : CommRing A
I : Ideal A
hI : DividedPowers I
B : Type u_2
inst✝ : CommRing B
f : A →+* B
hf : Function.Surjective ⇑f
hIf : hI.isSubDPIdeal (RingHom.ker f ⊓ I)
n : ℕ
a : A
ha : a ∈ I
h : ∃ a_1, f ↑a_1 = f a
⊢ f (hI.dpow n ↑(Classical.choose ⋯)) = f (hI.dpow n a) | A : Type u_1
inst✝¹ : CommRing A
I : Ideal A
hI : DividedPowers I
B : Type u_2
inst✝ : CommRing B
f : A →+* B
hf : Function.Surjective ⇑f
hIf : hI.isSubDPIdeal (RingHom.ker f ⊓ I)
n : ℕ
a : A
ha : a ∈ I
h : ∃ a_1, f ↑a_1 = f a
⊢ hI.dpow n ↑(Classical.choose ⋯) - hI.dpow n a ∈ RingHom.ker f | Please generate a tactic in lean4 to solve the state.
STATE:
A : Type u_1
inst✝¹ : CommRing A
I : Ideal A
hI : DividedPowers I
B : Type u_2
inst✝ : CommRing B
f : A →+* B
hf : Function.Surjective ⇑f
hIf : hI.isSubDPIdeal (RingHom.ker f ⊓ I)
n : ℕ
a : A
ha : a ∈ I
h : ∃ a_1, f ↑a_1 = f a
⊢ f (hI.dpow n ↑(Classical.choos... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/SubDPIdeal.lean | DividedPowers.Quotient.OfSurjective.dpow_apply' | [703, 1] | [713, 32] | rw [isSubDPIdeal_inf_iff] at hIf | A : Type u_1
inst✝¹ : CommRing A
I : Ideal A
hI : DividedPowers I
B : Type u_2
inst✝ : CommRing B
f : A →+* B
hf : Function.Surjective ⇑f
hIf : hI.isSubDPIdeal (RingHom.ker f ⊓ I)
n : ℕ
a : A
ha : a ∈ I
h : ∃ a_1, f ↑a_1 = f a
⊢ hI.dpow n ↑(Classical.choose ⋯) - hI.dpow n a ∈ RingHom.ker f | A : Type u_1
inst✝¹ : CommRing A
I : Ideal A
hI : DividedPowers I
B : Type u_2
inst✝ : CommRing B
f : A →+* B
hf : Function.Surjective ⇑f
hIf : ∀ (n : ℕ) (a b : A), a ∈ I → b ∈ I → a - b ∈ RingHom.ker f → hI.dpow n a - hI.dpow n b ∈ RingHom.ker f
n : ℕ
a : A
ha : a ∈ I
h : ∃ a_1, f ↑a_1 = f a
⊢ hI.dpow n ↑(Classical.ch... | Please generate a tactic in lean4 to solve the state.
STATE:
A : Type u_1
inst✝¹ : CommRing A
I : Ideal A
hI : DividedPowers I
B : Type u_2
inst✝ : CommRing B
f : A →+* B
hf : Function.Surjective ⇑f
hIf : hI.isSubDPIdeal (RingHom.ker f ⊓ I)
n : ℕ
a : A
ha : a ∈ I
h : ∃ a_1, f ↑a_1 = f a
⊢ hI.dpow n ↑(Classical.choose ⋯... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/SubDPIdeal.lean | DividedPowers.Quotient.OfSurjective.dpow_apply' | [703, 1] | [713, 32] | apply hIf n _ a _ ha | A : Type u_1
inst✝¹ : CommRing A
I : Ideal A
hI : DividedPowers I
B : Type u_2
inst✝ : CommRing B
f : A →+* B
hf : Function.Surjective ⇑f
hIf : ∀ (n : ℕ) (a b : A), a ∈ I → b ∈ I → a - b ∈ RingHom.ker f → hI.dpow n a - hI.dpow n b ∈ RingHom.ker f
n : ℕ
a : A
ha : a ∈ I
h : ∃ a_1, f ↑a_1 = f a
⊢ hI.dpow n ↑(Classical.ch... | A : Type u_1
inst✝¹ : CommRing A
I : Ideal A
hI : DividedPowers I
B : Type u_2
inst✝ : CommRing B
f : A →+* B
hf : Function.Surjective ⇑f
hIf : ∀ (n : ℕ) (a b : A), a ∈ I → b ∈ I → a - b ∈ RingHom.ker f → hI.dpow n a - hI.dpow n b ∈ RingHom.ker f
n : ℕ
a : A
ha : a ∈ I
h : ∃ a_1, f ↑a_1 = f a
⊢ ↑(Classical.choose ⋯) - ... | Please generate a tactic in lean4 to solve the state.
STATE:
A : Type u_1
inst✝¹ : CommRing A
I : Ideal A
hI : DividedPowers I
B : Type u_2
inst✝ : CommRing B
f : A →+* B
hf : Function.Surjective ⇑f
hIf : ∀ (n : ℕ) (a b : A), a ∈ I → b ∈ I → a - b ∈ RingHom.ker f → hI.dpow n a - hI.dpow n b ∈ RingHom.ker f
n : ℕ
a : A
... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/SubDPIdeal.lean | DividedPowers.Quotient.OfSurjective.dpow_apply' | [703, 1] | [713, 32] | rw [RingHom.mem_ker, map_sub, sub_eq_zero] | A : Type u_1
inst✝¹ : CommRing A
I : Ideal A
hI : DividedPowers I
B : Type u_2
inst✝ : CommRing B
f : A →+* B
hf : Function.Surjective ⇑f
hIf : ∀ (n : ℕ) (a b : A), a ∈ I → b ∈ I → a - b ∈ RingHom.ker f → hI.dpow n a - hI.dpow n b ∈ RingHom.ker f
n : ℕ
a : A
ha : a ∈ I
h : ∃ a_1, f ↑a_1 = f a
⊢ ↑(Classical.choose ⋯) - ... | A : Type u_1
inst✝¹ : CommRing A
I : Ideal A
hI : DividedPowers I
B : Type u_2
inst✝ : CommRing B
f : A →+* B
hf : Function.Surjective ⇑f
hIf : ∀ (n : ℕ) (a b : A), a ∈ I → b ∈ I → a - b ∈ RingHom.ker f → hI.dpow n a - hI.dpow n b ∈ RingHom.ker f
n : ℕ
a : A
ha : a ∈ I
h : ∃ a_1, f ↑a_1 = f a
⊢ f ↑(Classical.choose ⋯) ... | Please generate a tactic in lean4 to solve the state.
STATE:
A : Type u_1
inst✝¹ : CommRing A
I : Ideal A
hI : DividedPowers I
B : Type u_2
inst✝ : CommRing B
f : A →+* B
hf : Function.Surjective ⇑f
hIf : ∀ (n : ℕ) (a b : A), a ∈ I → b ∈ I → a - b ∈ RingHom.ker f → hI.dpow n a - hI.dpow n b ∈ RingHom.ker f
n : ℕ
a : A
... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/SubDPIdeal.lean | DividedPowers.Quotient.OfSurjective.dpow_apply' | [703, 1] | [713, 32] | rw [h.choose_spec] | A : Type u_1
inst✝¹ : CommRing A
I : Ideal A
hI : DividedPowers I
B : Type u_2
inst✝ : CommRing B
f : A →+* B
hf : Function.Surjective ⇑f
hIf : ∀ (n : ℕ) (a b : A), a ∈ I → b ∈ I → a - b ∈ RingHom.ker f → hI.dpow n a - hI.dpow n b ∈ RingHom.ker f
n : ℕ
a : A
ha : a ∈ I
h : ∃ a_1, f ↑a_1 = f a
⊢ f ↑(Classical.choose ⋯) ... | A : Type u_1
inst✝¹ : CommRing A
I : Ideal A
hI : DividedPowers I
B : Type u_2
inst✝ : CommRing B
f : A →+* B
hf : Function.Surjective ⇑f
hIf : ∀ (n : ℕ) (a b : A), a ∈ I → b ∈ I → a - b ∈ RingHom.ker f → hI.dpow n a - hI.dpow n b ∈ RingHom.ker f
n : ℕ
a : A
ha : a ∈ I
h : ∃ a_1, f ↑a_1 = f a
⊢ ↑(Classical.choose ⋯) ∈ ... | Please generate a tactic in lean4 to solve the state.
STATE:
A : Type u_1
inst✝¹ : CommRing A
I : Ideal A
hI : DividedPowers I
B : Type u_2
inst✝ : CommRing B
f : A →+* B
hf : Function.Surjective ⇑f
hIf : ∀ (n : ℕ) (a b : A), a ∈ I → b ∈ I → a - b ∈ RingHom.ker f → hI.dpow n a - hI.dpow n b ∈ RingHom.ker f
n : ℕ
a : A
... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/SubDPIdeal.lean | DividedPowers.Quotient.OfSurjective.dpow_apply' | [703, 1] | [713, 32] | simp only [Submodule.coe_mem] | A : Type u_1
inst✝¹ : CommRing A
I : Ideal A
hI : DividedPowers I
B : Type u_2
inst✝ : CommRing B
f : A →+* B
hf : Function.Surjective ⇑f
hIf : ∀ (n : ℕ) (a b : A), a ∈ I → b ∈ I → a - b ∈ RingHom.ker f → hI.dpow n a - hI.dpow n b ∈ RingHom.ker f
n : ℕ
a : A
ha : a ∈ I
h : ∃ a_1, f ↑a_1 = f a
⊢ ↑(Classical.choose ⋯) ∈ ... | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
A : Type u_1
inst✝¹ : CommRing A
I : Ideal A
hI : DividedPowers I
B : Type u_2
inst✝ : CommRing B
f : A →+* B
hf : Function.Surjective ⇑f
hIf : ∀ (n : ℕ) (a b : A), a ∈ I → b ∈ I → a - b ∈ RingHom.ker f → hI.dpow n a - hI.dpow n b ∈ RingHom.ker f
n : ℕ
a : A
... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/SubDPIdeal.lean | DividedPowers.Quotient.OfSurjective.dpow_apply' | [703, 1] | [713, 32] | use ⟨a, ha⟩ | A : Type u_1
inst✝¹ : CommRing A
I : Ideal A
hI : DividedPowers I
B : Type u_2
inst✝ : CommRing B
f : A →+* B
hf : Function.Surjective ⇑f
hIf : hI.isSubDPIdeal (RingHom.ker f ⊓ I)
n : ℕ
a : A
ha : a ∈ I
⊢ ∃ a_1, f ↑a_1 = f a | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
A : Type u_1
inst✝¹ : CommRing A
I : Ideal A
hI : DividedPowers I
B : Type u_2
inst✝ : CommRing B
f : A →+* B
hf : Function.Surjective ⇑f
hIf : hI.isSubDPIdeal (RingHom.ker f ⊓ I)
n : ℕ
a : A
ha : a ∈ I
⊢ ∃ a_1, f ↑a_1 = f a
TACTIC:
|
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