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/ImperialCollegeLondon/formalising-mathematics-2024.git | b732ed1352e87b4474b0520d1383994e069f8057 | FormalisingMathematics2024/Solutions/Section02reals/Sheet6.lean | Section2sheet6Solutions.tendsTo_unique | [135, 1] | [150, 11] | by_contra h | a : ℕ → ℝ
s t : ℝ
hs : TendsTo a s
ht : TendsTo a t
⊢ s = t | a : ℕ → ℝ
s t : ℝ
hs : TendsTo a s
ht : TendsTo a t
h : ¬s = t
⊢ False | Please generate a tactic in lean4 to solve the state.
STATE:
a : ℕ → ℝ
s t : ℝ
hs : TendsTo a s
ht : TendsTo a t
⊢ s = t
TACTIC:
|
https://github.com/ImperialCollegeLondon/formalising-mathematics-2024.git | b732ed1352e87b4474b0520d1383994e069f8057 | FormalisingMathematics2024/Solutions/Section02reals/Sheet6.lean | Section2sheet6Solutions.tendsTo_unique | [135, 1] | [150, 11] | wlog h2 : s < t | a : ℕ → ℝ
s t : ℝ
hs : TendsTo a s
ht : TendsTo a t
h : ¬s = t
⊢ False | case inr
a : ℕ → ℝ
s t : ℝ
hs : TendsTo a s
ht : TendsTo a t
h : ¬s = t
this : ∀ (a : ℕ → ℝ) (s t : ℝ), TendsTo a s → TendsTo a t → ¬s = t → s < t → False
h2 : ¬s < t
⊢ False
a : ℕ → ℝ
s t : ℝ
hs : TendsTo a s
ht : TendsTo a t
h : ¬s = t
h2 : s < t
⊢ False | Please generate a tactic in lean4 to solve the state.
STATE:
a : ℕ → ℝ
s t : ℝ
hs : TendsTo a s
ht : TendsTo a t
h : ¬s = t
⊢ False
TACTIC:
|
https://github.com/ImperialCollegeLondon/formalising-mathematics-2024.git | b732ed1352e87b4474b0520d1383994e069f8057 | FormalisingMathematics2024/Solutions/Section02reals/Sheet6.lean | Section2sheet6Solutions.tendsTo_unique | [135, 1] | [150, 11] | set ε := t - s with hε | a : ℕ → ℝ
s t : ℝ
hs : TendsTo a s
ht : TendsTo a t
h : ¬s = t
h2 : s < t
⊢ False | a : ℕ → ℝ
s t : ℝ
hs : TendsTo a s
ht : TendsTo a t
h : ¬s = t
h2 : s < t
ε : ℝ := t - s
hε : ε = t - s
⊢ False | Please generate a tactic in lean4 to solve the state.
STATE:
a : ℕ → ℝ
s t : ℝ
hs : TendsTo a s
ht : TendsTo a t
h : ¬s = t
h2 : s < t
⊢ False
TACTIC:
|
https://github.com/ImperialCollegeLondon/formalising-mathematics-2024.git | b732ed1352e87b4474b0520d1383994e069f8057 | FormalisingMathematics2024/Solutions/Section02reals/Sheet6.lean | Section2sheet6Solutions.tendsTo_unique | [135, 1] | [150, 11] | have hε : 0 < ε := sub_pos.mpr h2 | a : ℕ → ℝ
s t : ℝ
hs : TendsTo a s
ht : TendsTo a t
h : ¬s = t
h2 : s < t
ε : ℝ := t - s
hε : ε = t - s
⊢ False | a : ℕ → ℝ
s t : ℝ
hs : TendsTo a s
ht : TendsTo a t
h : ¬s = t
h2 : s < t
ε : ℝ := t - s
hε✝ : ε = t - s
hε : 0 < ε
⊢ False | Please generate a tactic in lean4 to solve the state.
STATE:
a : ℕ → ℝ
s t : ℝ
hs : TendsTo a s
ht : TendsTo a t
h : ¬s = t
h2 : s < t
ε : ℝ := t - s
hε : ε = t - s
⊢ False
TACTIC:
|
https://github.com/ImperialCollegeLondon/formalising-mathematics-2024.git | b732ed1352e87b4474b0520d1383994e069f8057 | FormalisingMathematics2024/Solutions/Section02reals/Sheet6.lean | Section2sheet6Solutions.tendsTo_unique | [135, 1] | [150, 11] | obtain ⟨X, hX⟩ := hs (ε / 2) (by linarith) | a : ℕ → ℝ
s t : ℝ
hs : TendsTo a s
ht : TendsTo a t
h : ¬s = t
h2 : s < t
ε : ℝ := t - s
hε✝ : ε = t - s
hε : 0 < ε
⊢ False | case intro
a : ℕ → ℝ
s t : ℝ
hs : TendsTo a s
ht : TendsTo a t
h : ¬s = t
h2 : s < t
ε : ℝ := t - s
hε✝ : ε = t - s
hε : 0 < ε
X : ℕ
hX : ∀ (n : ℕ), X ≤ n → |a n - s| < ε / 2
⊢ False | Please generate a tactic in lean4 to solve the state.
STATE:
a : ℕ → ℝ
s t : ℝ
hs : TendsTo a s
ht : TendsTo a t
h : ¬s = t
h2 : s < t
ε : ℝ := t - s
hε✝ : ε = t - s
hε : 0 < ε
⊢ False
TACTIC:
|
https://github.com/ImperialCollegeLondon/formalising-mathematics-2024.git | b732ed1352e87b4474b0520d1383994e069f8057 | FormalisingMathematics2024/Solutions/Section02reals/Sheet6.lean | Section2sheet6Solutions.tendsTo_unique | [135, 1] | [150, 11] | obtain ⟨Y, hY⟩ := ht (ε / 2) (by linarith) | case intro
a : ℕ → ℝ
s t : ℝ
hs : TendsTo a s
ht : TendsTo a t
h : ¬s = t
h2 : s < t
ε : ℝ := t - s
hε✝ : ε = t - s
hε : 0 < ε
X : ℕ
hX : ∀ (n : ℕ), X ≤ n → |a n - s| < ε / 2
⊢ False | case intro.intro
a : ℕ → ℝ
s t : ℝ
hs : TendsTo a s
ht : TendsTo a t
h : ¬s = t
h2 : s < t
ε : ℝ := t - s
hε✝ : ε = t - s
hε : 0 < ε
X : ℕ
hX : ∀ (n : ℕ), X ≤ n → |a n - s| < ε / 2
Y : ℕ
hY : ∀ (n : ℕ), Y ≤ n → |a n - t| < ε / 2
⊢ False | Please generate a tactic in lean4 to solve the state.
STATE:
case intro
a : ℕ → ℝ
s t : ℝ
hs : TendsTo a s
ht : TendsTo a t
h : ¬s = t
h2 : s < t
ε : ℝ := t - s
hε✝ : ε = t - s
hε : 0 < ε
X : ℕ
hX : ∀ (n : ℕ), X ≤ n → |a n - s| < ε / 2
⊢ False
TACTIC:
|
https://github.com/ImperialCollegeLondon/formalising-mathematics-2024.git | b732ed1352e87b4474b0520d1383994e069f8057 | FormalisingMathematics2024/Solutions/Section02reals/Sheet6.lean | Section2sheet6Solutions.tendsTo_unique | [135, 1] | [150, 11] | specialize hX (max X Y) (le_max_left X Y) | case intro.intro
a : ℕ → ℝ
s t : ℝ
hs : TendsTo a s
ht : TendsTo a t
h : ¬s = t
h2 : s < t
ε : ℝ := t - s
hε✝ : ε = t - s
hε : 0 < ε
X : ℕ
hX : ∀ (n : ℕ), X ≤ n → |a n - s| < ε / 2
Y : ℕ
hY : ∀ (n : ℕ), Y ≤ n → |a n - t| < ε / 2
⊢ False | case intro.intro
a : ℕ → ℝ
s t : ℝ
hs : TendsTo a s
ht : TendsTo a t
h : ¬s = t
h2 : s < t
ε : ℝ := t - s
hε✝ : ε = t - s
hε : 0 < ε
X Y : ℕ
hY : ∀ (n : ℕ), Y ≤ n → |a n - t| < ε / 2
hX : |a (max X Y) - s| < ε / 2
⊢ False | Please generate a tactic in lean4 to solve the state.
STATE:
case intro.intro
a : ℕ → ℝ
s t : ℝ
hs : TendsTo a s
ht : TendsTo a t
h : ¬s = t
h2 : s < t
ε : ℝ := t - s
hε✝ : ε = t - s
hε : 0 < ε
X : ℕ
hX : ∀ (n : ℕ), X ≤ n → |a n - s| < ε / 2
Y : ℕ
hY : ∀ (n : ℕ), Y ≤ n → |a n - t| < ε / 2
⊢ False
TACTIC:
|
https://github.com/ImperialCollegeLondon/formalising-mathematics-2024.git | b732ed1352e87b4474b0520d1383994e069f8057 | FormalisingMathematics2024/Solutions/Section02reals/Sheet6.lean | Section2sheet6Solutions.tendsTo_unique | [135, 1] | [150, 11] | specialize hY (max X Y) (le_max_right X Y) | case intro.intro
a : ℕ → ℝ
s t : ℝ
hs : TendsTo a s
ht : TendsTo a t
h : ¬s = t
h2 : s < t
ε : ℝ := t - s
hε✝ : ε = t - s
hε : 0 < ε
X Y : ℕ
hY : ∀ (n : ℕ), Y ≤ n → |a n - t| < ε / 2
hX : |a (max X Y) - s| < ε / 2
⊢ False | case intro.intro
a : ℕ → ℝ
s t : ℝ
hs : TendsTo a s
ht : TendsTo a t
h : ¬s = t
h2 : s < t
ε : ℝ := t - s
hε✝ : ε = t - s
hε : 0 < ε
X Y : ℕ
hX : |a (max X Y) - s| < ε / 2
hY : |a (max X Y) - t| < ε / 2
⊢ False | Please generate a tactic in lean4 to solve the state.
STATE:
case intro.intro
a : ℕ → ℝ
s t : ℝ
hs : TendsTo a s
ht : TendsTo a t
h : ¬s = t
h2 : s < t
ε : ℝ := t - s
hε✝ : ε = t - s
hε : 0 < ε
X Y : ℕ
hY : ∀ (n : ℕ), Y ≤ n → |a n - t| < ε / 2
hX : |a (max X Y) - s| < ε / 2
⊢ False
TACTIC:
|
https://github.com/ImperialCollegeLondon/formalising-mathematics-2024.git | b732ed1352e87b4474b0520d1383994e069f8057 | FormalisingMathematics2024/Solutions/Section02reals/Sheet6.lean | Section2sheet6Solutions.tendsTo_unique | [135, 1] | [150, 11] | rw [abs_lt] at hX hY | case intro.intro
a : ℕ → ℝ
s t : ℝ
hs : TendsTo a s
ht : TendsTo a t
h : ¬s = t
h2 : s < t
ε : ℝ := t - s
hε✝ : ε = t - s
hε : 0 < ε
X Y : ℕ
hX : |a (max X Y) - s| < ε / 2
hY : |a (max X Y) - t| < ε / 2
⊢ False | case intro.intro
a : ℕ → ℝ
s t : ℝ
hs : TendsTo a s
ht : TendsTo a t
h : ¬s = t
h2 : s < t
ε : ℝ := t - s
hε✝ : ε = t - s
hε : 0 < ε
X Y : ℕ
hX : -(ε / 2) < a (max X Y) - s ∧ a (max X Y) - s < ε / 2
hY : -(ε / 2) < a (max X Y) - t ∧ a (max X Y) - t < ε / 2
⊢ False | Please generate a tactic in lean4 to solve the state.
STATE:
case intro.intro
a : ℕ → ℝ
s t : ℝ
hs : TendsTo a s
ht : TendsTo a t
h : ¬s = t
h2 : s < t
ε : ℝ := t - s
hε✝ : ε = t - s
hε : 0 < ε
X Y : ℕ
hX : |a (max X Y) - s| < ε / 2
hY : |a (max X Y) - t| < ε / 2
⊢ False
TACTIC:
|
https://github.com/ImperialCollegeLondon/formalising-mathematics-2024.git | b732ed1352e87b4474b0520d1383994e069f8057 | FormalisingMathematics2024/Solutions/Section02reals/Sheet6.lean | Section2sheet6Solutions.tendsTo_unique | [135, 1] | [150, 11] | linarith | case intro.intro
a : ℕ → ℝ
s t : ℝ
hs : TendsTo a s
ht : TendsTo a t
h : ¬s = t
h2 : s < t
ε : ℝ := t - s
hε✝ : ε = t - s
hε : 0 < ε
X Y : ℕ
hX : -(ε / 2) < a (max X Y) - s ∧ a (max X Y) - s < ε / 2
hY : -(ε / 2) < a (max X Y) - t ∧ a (max X Y) - t < ε / 2
⊢ False | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case intro.intro
a : ℕ → ℝ
s t : ℝ
hs : TendsTo a s
ht : TendsTo a t
h : ¬s = t
h2 : s < t
ε : ℝ := t - s
hε✝ : ε = t - s
hε : 0 < ε
X Y : ℕ
hX : -(ε / 2) < a (max X Y) - s ∧ a (max X Y) - s < ε / 2
hY : -(ε / 2) < a (max X Y) - t ∧ a (max X Y) - t < ε / 2
⊢ False
TACTIC:
|
https://github.com/ImperialCollegeLondon/formalising-mathematics-2024.git | b732ed1352e87b4474b0520d1383994e069f8057 | FormalisingMathematics2024/Solutions/Section02reals/Sheet6.lean | Section2sheet6Solutions.tendsTo_unique | [135, 1] | [150, 11] | cases' Ne.lt_or_lt h with h3 h3 | case inr
a : ℕ → ℝ
s t : ℝ
hs : TendsTo a s
ht : TendsTo a t
h : ¬s = t
this : ∀ (a : ℕ → ℝ) (s t : ℝ), TendsTo a s → TendsTo a t → ¬s = t → s < t → False
h2 : ¬s < t
⊢ False | case inr.inl
a : ℕ → ℝ
s t : ℝ
hs : TendsTo a s
ht : TendsTo a t
h : ¬s = t
this : ∀ (a : ℕ → ℝ) (s t : ℝ), TendsTo a s → TendsTo a t → ¬s = t → s < t → False
h2 : ¬s < t
h3 : s < t
⊢ False
case inr.inr
a : ℕ → ℝ
s t : ℝ
hs : TendsTo a s
ht : TendsTo a t
h : ¬s = t
this : ∀ (a : ℕ → ℝ) (s t : ℝ), TendsTo a s → TendsTo a t → ¬s = t → s < t → False
h2 : ¬s < t
h3 : t < s
⊢ False | Please generate a tactic in lean4 to solve the state.
STATE:
case inr
a : ℕ → ℝ
s t : ℝ
hs : TendsTo a s
ht : TendsTo a t
h : ¬s = t
this : ∀ (a : ℕ → ℝ) (s t : ℝ), TendsTo a s → TendsTo a t → ¬s = t → s < t → False
h2 : ¬s < t
⊢ False
TACTIC:
|
https://github.com/ImperialCollegeLondon/formalising-mathematics-2024.git | b732ed1352e87b4474b0520d1383994e069f8057 | FormalisingMathematics2024/Solutions/Section02reals/Sheet6.lean | Section2sheet6Solutions.tendsTo_unique | [135, 1] | [150, 11] | contradiction | case inr.inl
a : ℕ → ℝ
s t : ℝ
hs : TendsTo a s
ht : TendsTo a t
h : ¬s = t
this : ∀ (a : ℕ → ℝ) (s t : ℝ), TendsTo a s → TendsTo a t → ¬s = t → s < t → False
h2 : ¬s < t
h3 : s < t
⊢ False | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case inr.inl
a : ℕ → ℝ
s t : ℝ
hs : TendsTo a s
ht : TendsTo a t
h : ¬s = t
this : ∀ (a : ℕ → ℝ) (s t : ℝ), TendsTo a s → TendsTo a t → ¬s = t → s < t → False
h2 : ¬s < t
h3 : s < t
⊢ False
TACTIC:
|
https://github.com/ImperialCollegeLondon/formalising-mathematics-2024.git | b732ed1352e87b4474b0520d1383994e069f8057 | FormalisingMathematics2024/Solutions/Section02reals/Sheet6.lean | Section2sheet6Solutions.tendsTo_unique | [135, 1] | [150, 11] | apply this _ _ _ ht hs _ h3 | case inr.inr
a : ℕ → ℝ
s t : ℝ
hs : TendsTo a s
ht : TendsTo a t
h : ¬s = t
this : ∀ (a : ℕ → ℝ) (s t : ℝ), TendsTo a s → TendsTo a t → ¬s = t → s < t → False
h2 : ¬s < t
h3 : t < s
⊢ False | a : ℕ → ℝ
s t : ℝ
hs : TendsTo a s
ht : TendsTo a t
h : ¬s = t
this : ∀ (a : ℕ → ℝ) (s t : ℝ), TendsTo a s → TendsTo a t → ¬s = t → s < t → False
h2 : ¬s < t
h3 : t < s
⊢ ¬t = s | Please generate a tactic in lean4 to solve the state.
STATE:
case inr.inr
a : ℕ → ℝ
s t : ℝ
hs : TendsTo a s
ht : TendsTo a t
h : ¬s = t
this : ∀ (a : ℕ → ℝ) (s t : ℝ), TendsTo a s → TendsTo a t → ¬s = t → s < t → False
h2 : ¬s < t
h3 : t < s
⊢ False
TACTIC:
|
https://github.com/ImperialCollegeLondon/formalising-mathematics-2024.git | b732ed1352e87b4474b0520d1383994e069f8057 | FormalisingMathematics2024/Solutions/Section02reals/Sheet6.lean | Section2sheet6Solutions.tendsTo_unique | [135, 1] | [150, 11] | exact ne_comm.mp h | a : ℕ → ℝ
s t : ℝ
hs : TendsTo a s
ht : TendsTo a t
h : ¬s = t
this : ∀ (a : ℕ → ℝ) (s t : ℝ), TendsTo a s → TendsTo a t → ¬s = t → s < t → False
h2 : ¬s < t
h3 : t < s
⊢ ¬t = s | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
a : ℕ → ℝ
s t : ℝ
hs : TendsTo a s
ht : TendsTo a t
h : ¬s = t
this : ∀ (a : ℕ → ℝ) (s t : ℝ), TendsTo a s → TendsTo a t → ¬s = t → s < t → False
h2 : ¬s < t
h3 : t < s
⊢ ¬t = s
TACTIC:
|
https://github.com/ImperialCollegeLondon/formalising-mathematics-2024.git | b732ed1352e87b4474b0520d1383994e069f8057 | FormalisingMathematics2024/Solutions/Section02reals/Sheet6.lean | Section2sheet6Solutions.tendsTo_unique | [135, 1] | [150, 11] | linarith | a : ℕ → ℝ
s t : ℝ
hs : TendsTo a s
ht : TendsTo a t
h : ¬s = t
h2 : s < t
ε : ℝ := t - s
hε✝ : ε = t - s
hε : 0 < ε
⊢ ε / 2 > 0 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
a : ℕ → ℝ
s t : ℝ
hs : TendsTo a s
ht : TendsTo a t
h : ¬s = t
h2 : s < t
ε : ℝ := t - s
hε✝ : ε = t - s
hε : 0 < ε
⊢ ε / 2 > 0
TACTIC:
|
https://github.com/ImperialCollegeLondon/formalising-mathematics-2024.git | b732ed1352e87b4474b0520d1383994e069f8057 | FormalisingMathematics2024/Solutions/Section02reals/Sheet6.lean | Section2sheet6Solutions.tendsTo_unique | [135, 1] | [150, 11] | linarith | a : ℕ → ℝ
s t : ℝ
hs : TendsTo a s
ht : TendsTo a t
h : ¬s = t
h2 : s < t
ε : ℝ := t - s
hε✝ : ε = t - s
hε : 0 < ε
X : ℕ
hX : ∀ (n : ℕ), X ≤ n → |a n - s| < ε / 2
⊢ ε / 2 > 0 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
a : ℕ → ℝ
s t : ℝ
hs : TendsTo a s
ht : TendsTo a t
h : ¬s = t
h2 : s < t
ε : ℝ := t - s
hε✝ : ε = t - s
hε : 0 < ε
X : ℕ
hX : ∀ (n : ℕ), X ≤ n → |a n - s| < ε / 2
⊢ ε / 2 > 0
TACTIC:
|
https://github.com/ImperialCollegeLondon/formalising-mathematics-2024.git | b732ed1352e87b4474b0520d1383994e069f8057 | FormalisingMathematics2024/Solutions/Section02reals/Sheet6.lean | Section2sheet6Solutions.eq_zero_of_abs_lt_eps | [155, 1] | [162, 11] | by_contra h2 | r : ℝ
h : ∀ ε > 0, |r| < ε
⊢ r = 0 | r : ℝ
h : ∀ ε > 0, |r| < ε
h2 : ¬r = 0
⊢ False | Please generate a tactic in lean4 to solve the state.
STATE:
r : ℝ
h : ∀ ε > 0, |r| < ε
⊢ r = 0
TACTIC:
|
https://github.com/ImperialCollegeLondon/formalising-mathematics-2024.git | b732ed1352e87b4474b0520d1383994e069f8057 | FormalisingMathematics2024/Solutions/Section02reals/Sheet6.lean | Section2sheet6Solutions.eq_zero_of_abs_lt_eps | [155, 1] | [162, 11] | specialize h |r| (abs_pos.mpr h2) | r : ℝ
h : ∀ ε > 0, |r| < ε
h2 : ¬r = 0
⊢ False | r : ℝ
h2 : ¬r = 0
h : |r| < |r|
⊢ False | Please generate a tactic in lean4 to solve the state.
STATE:
r : ℝ
h : ∀ ε > 0, |r| < ε
h2 : ¬r = 0
⊢ False
TACTIC:
|
https://github.com/ImperialCollegeLondon/formalising-mathematics-2024.git | b732ed1352e87b4474b0520d1383994e069f8057 | FormalisingMathematics2024/Solutions/Section02reals/Sheet6.lean | Section2sheet6Solutions.eq_zero_of_abs_lt_eps | [155, 1] | [162, 11] | linarith | r : ℝ
h2 : ¬r = 0
h : |r| < |r|
⊢ False | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
r : ℝ
h2 : ¬r = 0
h : |r| < |r|
⊢ False
TACTIC:
|
https://github.com/ImperialCollegeLondon/formalising-mathematics-2024.git | b732ed1352e87b4474b0520d1383994e069f8057 | FormalisingMathematics2024/Solutions/Section02reals/Sheet6.lean | Section2sheet6Solutions.tendsTo_unique' | [165, 1] | [178, 17] | have h := tendsTo_sub hs ht | a : ℕ → ℝ
s t : ℝ
hs : TendsTo a s
ht : TendsTo a t
⊢ s = t | a : ℕ → ℝ
s t : ℝ
hs : TendsTo a s
ht : TendsTo a t
h : TendsTo (fun n => a n - a n) (s - t)
⊢ s = t | Please generate a tactic in lean4 to solve the state.
STATE:
a : ℕ → ℝ
s t : ℝ
hs : TendsTo a s
ht : TendsTo a t
⊢ s = t
TACTIC:
|
https://github.com/ImperialCollegeLondon/formalising-mathematics-2024.git | b732ed1352e87b4474b0520d1383994e069f8057 | FormalisingMathematics2024/Solutions/Section02reals/Sheet6.lean | Section2sheet6Solutions.tendsTo_unique' | [165, 1] | [178, 17] | suffices ∀ ε > 0, |t - s| < ε by linarith [eq_zero_of_abs_lt_eps this] | a : ℕ → ℝ
s t : ℝ
hs : TendsTo a s
ht : TendsTo a t
h : TendsTo (fun n => a n - a n) (s - t)
⊢ s = t | a : ℕ → ℝ
s t : ℝ
hs : TendsTo a s
ht : TendsTo a t
h : TendsTo (fun n => a n - a n) (s - t)
⊢ ∀ ε > 0, |t - s| < ε | Please generate a tactic in lean4 to solve the state.
STATE:
a : ℕ → ℝ
s t : ℝ
hs : TendsTo a s
ht : TendsTo a t
h : TendsTo (fun n => a n - a n) (s - t)
⊢ s = t
TACTIC:
|
https://github.com/ImperialCollegeLondon/formalising-mathematics-2024.git | b732ed1352e87b4474b0520d1383994e069f8057 | FormalisingMathematics2024/Solutions/Section02reals/Sheet6.lean | Section2sheet6Solutions.tendsTo_unique' | [165, 1] | [178, 17] | intro ε hε | a : ℕ → ℝ
s t : ℝ
hs : TendsTo a s
ht : TendsTo a t
h : TendsTo (fun n => a n - a n) (s - t)
⊢ ∀ ε > 0, |t - s| < ε | a : ℕ → ℝ
s t : ℝ
hs : TendsTo a s
ht : TendsTo a t
h : TendsTo (fun n => a n - a n) (s - t)
ε : ℝ
hε : ε > 0
⊢ |t - s| < ε | Please generate a tactic in lean4 to solve the state.
STATE:
a : ℕ → ℝ
s t : ℝ
hs : TendsTo a s
ht : TendsTo a t
h : TendsTo (fun n => a n - a n) (s - t)
⊢ ∀ ε > 0, |t - s| < ε
TACTIC:
|
https://github.com/ImperialCollegeLondon/formalising-mathematics-2024.git | b732ed1352e87b4474b0520d1383994e069f8057 | FormalisingMathematics2024/Solutions/Section02reals/Sheet6.lean | Section2sheet6Solutions.tendsTo_unique' | [165, 1] | [178, 17] | obtain ⟨X, hX⟩ := h ε hε | a : ℕ → ℝ
s t : ℝ
hs : TendsTo a s
ht : TendsTo a t
h : TendsTo (fun n => a n - a n) (s - t)
ε : ℝ
hε : ε > 0
⊢ |t - s| < ε | case intro
a : ℕ → ℝ
s t : ℝ
hs : TendsTo a s
ht : TendsTo a t
h : TendsTo (fun n => a n - a n) (s - t)
ε : ℝ
hε : ε > 0
X : ℕ
hX : ∀ (n : ℕ), X ≤ n → |(fun n => a n - a n) n - (s - t)| < ε
⊢ |t - s| < ε | Please generate a tactic in lean4 to solve the state.
STATE:
a : ℕ → ℝ
s t : ℝ
hs : TendsTo a s
ht : TendsTo a t
h : TendsTo (fun n => a n - a n) (s - t)
ε : ℝ
hε : ε > 0
⊢ |t - s| < ε
TACTIC:
|
https://github.com/ImperialCollegeLondon/formalising-mathematics-2024.git | b732ed1352e87b4474b0520d1383994e069f8057 | FormalisingMathematics2024/Solutions/Section02reals/Sheet6.lean | Section2sheet6Solutions.tendsTo_unique' | [165, 1] | [178, 17] | specialize hX X (by rfl) | case intro
a : ℕ → ℝ
s t : ℝ
hs : TendsTo a s
ht : TendsTo a t
h : TendsTo (fun n => a n - a n) (s - t)
ε : ℝ
hε : ε > 0
X : ℕ
hX : ∀ (n : ℕ), X ≤ n → |(fun n => a n - a n) n - (s - t)| < ε
⊢ |t - s| < ε | case intro
a : ℕ → ℝ
s t : ℝ
hs : TendsTo a s
ht : TendsTo a t
h : TendsTo (fun n => a n - a n) (s - t)
ε : ℝ
hε : ε > 0
X : ℕ
hX : |(fun n => a n - a n) X - (s - t)| < ε
⊢ |t - s| < ε | Please generate a tactic in lean4 to solve the state.
STATE:
case intro
a : ℕ → ℝ
s t : ℝ
hs : TendsTo a s
ht : TendsTo a t
h : TendsTo (fun n => a n - a n) (s - t)
ε : ℝ
hε : ε > 0
X : ℕ
hX : ∀ (n : ℕ), X ≤ n → |(fun n => a n - a n) n - (s - t)| < ε
⊢ |t - s| < ε
TACTIC:
|
https://github.com/ImperialCollegeLondon/formalising-mathematics-2024.git | b732ed1352e87b4474b0520d1383994e069f8057 | FormalisingMathematics2024/Solutions/Section02reals/Sheet6.lean | Section2sheet6Solutions.tendsTo_unique' | [165, 1] | [178, 17] | simpa using hX | case intro
a : ℕ → ℝ
s t : ℝ
hs : TendsTo a s
ht : TendsTo a t
h : TendsTo (fun n => a n - a n) (s - t)
ε : ℝ
hε : ε > 0
X : ℕ
hX : |(fun n => a n - a n) X - (s - t)| < ε
⊢ |t - s| < ε | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case intro
a : ℕ → ℝ
s t : ℝ
hs : TendsTo a s
ht : TendsTo a t
h : TendsTo (fun n => a n - a n) (s - t)
ε : ℝ
hε : ε > 0
X : ℕ
hX : |(fun n => a n - a n) X - (s - t)| < ε
⊢ |t - s| < ε
TACTIC:
|
https://github.com/ImperialCollegeLondon/formalising-mathematics-2024.git | b732ed1352e87b4474b0520d1383994e069f8057 | FormalisingMathematics2024/Solutions/Section02reals/Sheet6.lean | Section2sheet6Solutions.tendsTo_unique' | [165, 1] | [178, 17] | linarith [eq_zero_of_abs_lt_eps this] | a : ℕ → ℝ
s t : ℝ
hs : TendsTo a s
ht : TendsTo a t
h : TendsTo (fun n => a n - a n) (s - t)
this : ∀ ε > 0, |t - s| < ε
⊢ s = t | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
a : ℕ → ℝ
s t : ℝ
hs : TendsTo a s
ht : TendsTo a t
h : TendsTo (fun n => a n - a n) (s - t)
this : ∀ ε > 0, |t - s| < ε
⊢ s = t
TACTIC:
|
https://github.com/ImperialCollegeLondon/formalising-mathematics-2024.git | b732ed1352e87b4474b0520d1383994e069f8057 | FormalisingMathematics2024/Solutions/Section02reals/Sheet6.lean | Section2sheet6Solutions.tendsTo_unique' | [165, 1] | [178, 17] | rfl | a : ℕ → ℝ
s t : ℝ
hs : TendsTo a s
ht : TendsTo a t
h : TendsTo (fun n => a n - a n) (s - t)
ε : ℝ
hε : ε > 0
X : ℕ
hX : ∀ (n : ℕ), X ≤ n → |(fun n => a n - a n) n - (s - t)| < ε
⊢ X ≤ X | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
a : ℕ → ℝ
s t : ℝ
hs : TendsTo a s
ht : TendsTo a t
h : TendsTo (fun n => a n - a n) (s - t)
ε : ℝ
hε : ε > 0
X : ℕ
hX : ∀ (n : ℕ), X ≤ n → |(fun n => a n - a n) n - (s - t)| < ε
⊢ X ≤ X
TACTIC:
|
https://github.com/ImperialCollegeLondon/formalising-mathematics-2024.git | b732ed1352e87b4474b0520d1383994e069f8057 | FormalisingMathematics2024/Section07subgroupsAndHomomorphisms/Sheet2.lean | Secion7Sheet2.comp_id | [68, 1] | [69, 8] | sorry | G H : Type
inst✝² : Group G
inst✝¹ : Group H
φ : G →* H
a : G
K : Type
inst✝ : Group K
ψ : H →* K
⊢ MonoidHom.comp φ (MonoidHom.id G) = φ | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
G H : Type
inst✝² : Group G
inst✝¹ : Group H
φ : G →* H
a : G
K : Type
inst✝ : Group K
ψ : H →* K
⊢ MonoidHom.comp φ (MonoidHom.id G) = φ
TACTIC:
|
https://github.com/ImperialCollegeLondon/formalising-mathematics-2024.git | b732ed1352e87b4474b0520d1383994e069f8057 | FormalisingMathematics2024/Section07subgroupsAndHomomorphisms/Sheet2.lean | Secion7Sheet2.id_comp | [71, 1] | [72, 8] | sorry | G H : Type
inst✝² : Group G
inst✝¹ : Group H
φ : G →* H
a : G
K : Type
inst✝ : Group K
ψ : H →* K
⊢ MonoidHom.comp (MonoidHom.id H) φ = φ | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
G H : Type
inst✝² : Group G
inst✝¹ : Group H
φ : G →* H
a : G
K : Type
inst✝ : Group K
ψ : H →* K
⊢ MonoidHom.comp (MonoidHom.id H) φ = φ
TACTIC:
|
https://github.com/ImperialCollegeLondon/formalising-mathematics-2024.git | b732ed1352e87b4474b0520d1383994e069f8057 | FormalisingMathematics2024/Section07subgroupsAndHomomorphisms/Sheet2.lean | Secion7Sheet2.comp_assoc | [74, 1] | [76, 8] | sorry | G H : Type
inst✝³ : Group G
inst✝² : Group H
φ : G →* H
a : G
K : Type
inst✝¹ : Group K
ψ : H →* K
L : Type
inst✝ : Group L
ρ : K →* L
⊢ MonoidHom.comp (MonoidHom.comp ρ ψ) φ = MonoidHom.comp ρ (MonoidHom.comp ψ φ) | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
G H : Type
inst✝³ : Group G
inst✝² : Group H
φ : G →* H
a : G
K : Type
inst✝¹ : Group K
ψ : H →* K
L : Type
inst✝ : Group L
ρ : K →* L
⊢ MonoidHom.comp (MonoidHom.comp ρ ψ) φ = MonoidHom.comp ρ (MonoidHom.comp ψ φ)
TACTIC:
|
https://github.com/ImperialCollegeLondon/formalising-mathematics-2024.git | b732ed1352e87b4474b0520d1383994e069f8057 | FormalisingMathematics2024/Section04sets/Sheet4.lean | Section4sheet4.mem_def | [38, 1] | [40, 6] | rfl | X : Type
P : X → Prop
a : X
⊢ a ∈ {x | P x} ↔ P a | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
X : Type
P : X → Prop
a : X
⊢ a ∈ {x | P x} ↔ P a
TACTIC:
|
https://github.com/ImperialCollegeLondon/formalising-mathematics-2024.git | b732ed1352e87b4474b0520d1383994e069f8057 | FormalisingMathematics2024/Solutions/Section16commutativeAlgebra/Sheet2.lean | Section16Sheet2solutions.lemma1 | [37, 1] | [54, 22] | rw [Set.nonempty_iff_ne_empty] | A : Type
inst✝¹ : CommRing A
inst✝ : DecidableEq A
I : Ideal A
hInonfg : ¬Ideal.FG I
n : ℕ
g : (m : ℕ) → m < n → ↥I
⊢ Set.Nonempty (↑I \ ↑(Ideal.span ↑(Finset.image (fun m => ↑(g ↑m (_ : ↑m < n))) Finset.univ))) | A : Type
inst✝¹ : CommRing A
inst✝ : DecidableEq A
I : Ideal A
hInonfg : ¬Ideal.FG I
n : ℕ
g : (m : ℕ) → m < n → ↥I
⊢ ↑I \ ↑(Ideal.span ↑(Finset.image (fun m => ↑(g ↑m (_ : ↑m < n))) Finset.univ)) ≠ ∅ | Please generate a tactic in lean4 to solve the state.
STATE:
A : Type
inst✝¹ : CommRing A
inst✝ : DecidableEq A
I : Ideal A
hInonfg : ¬Ideal.FG I
n : ℕ
g : (m : ℕ) → m < n → ↥I
⊢ Set.Nonempty (↑I \ ↑(Ideal.span ↑(Finset.image (fun m => ↑(g ↑m (_ : ↑m < n))) Finset.univ)))
TACTIC:
|
https://github.com/ImperialCollegeLondon/formalising-mathematics-2024.git | b732ed1352e87b4474b0520d1383994e069f8057 | FormalisingMathematics2024/Solutions/Section16commutativeAlgebra/Sheet2.lean | Section16Sheet2solutions.lemma1 | [37, 1] | [54, 22] | intro h | A : Type
inst✝¹ : CommRing A
inst✝ : DecidableEq A
I : Ideal A
hInonfg : ¬Ideal.FG I
n : ℕ
g : (m : ℕ) → m < n → ↥I
⊢ ↑I \ ↑(Ideal.span ↑(Finset.image (fun m => ↑(g ↑m (_ : ↑m < n))) Finset.univ)) ≠ ∅ | A : Type
inst✝¹ : CommRing A
inst✝ : DecidableEq A
I : Ideal A
hInonfg : ¬Ideal.FG I
n : ℕ
g : (m : ℕ) → m < n → ↥I
h : ↑I \ ↑(Ideal.span ↑(Finset.image (fun m => ↑(g ↑m (_ : ↑m < n))) Finset.univ)) = ∅
⊢ False | Please generate a tactic in lean4 to solve the state.
STATE:
A : Type
inst✝¹ : CommRing A
inst✝ : DecidableEq A
I : Ideal A
hInonfg : ¬Ideal.FG I
n : ℕ
g : (m : ℕ) → m < n → ↥I
⊢ ↑I \ ↑(Ideal.span ↑(Finset.image (fun m => ↑(g ↑m (_ : ↑m < n))) Finset.univ)) ≠ ∅
TACTIC:
|
https://github.com/ImperialCollegeLondon/formalising-mathematics-2024.git | b732ed1352e87b4474b0520d1383994e069f8057 | FormalisingMathematics2024/Solutions/Section16commutativeAlgebra/Sheet2.lean | Section16Sheet2solutions.lemma1 | [37, 1] | [54, 22] | rw [Set.diff_eq_empty] at h | A : Type
inst✝¹ : CommRing A
inst✝ : DecidableEq A
I : Ideal A
hInonfg : ¬Ideal.FG I
n : ℕ
g : (m : ℕ) → m < n → ↥I
h : ↑I \ ↑(Ideal.span ↑(Finset.image (fun m => ↑(g ↑m (_ : ↑m < n))) Finset.univ)) = ∅
⊢ False | A : Type
inst✝¹ : CommRing A
inst✝ : DecidableEq A
I : Ideal A
hInonfg : ¬Ideal.FG I
n : ℕ
g : (m : ℕ) → m < n → ↥I
h : ↑I ⊆ ↑(Ideal.span ↑(Finset.image (fun m => ↑(g ↑m (_ : ↑m < n))) Finset.univ))
⊢ False | Please generate a tactic in lean4 to solve the state.
STATE:
A : Type
inst✝¹ : CommRing A
inst✝ : DecidableEq A
I : Ideal A
hInonfg : ¬Ideal.FG I
n : ℕ
g : (m : ℕ) → m < n → ↥I
h : ↑I \ ↑(Ideal.span ↑(Finset.image (fun m => ↑(g ↑m (_ : ↑m < n))) Finset.univ)) = ∅
⊢ False
TACTIC:
|
https://github.com/ImperialCollegeLondon/formalising-mathematics-2024.git | b732ed1352e87b4474b0520d1383994e069f8057 | FormalisingMathematics2024/Solutions/Section16commutativeAlgebra/Sheet2.lean | Section16Sheet2solutions.lemma1 | [37, 1] | [54, 22] | apply hInonfg | A : Type
inst✝¹ : CommRing A
inst✝ : DecidableEq A
I : Ideal A
hInonfg : ¬Ideal.FG I
n : ℕ
g : (m : ℕ) → m < n → ↥I
h : ↑I ⊆ ↑(Ideal.span ↑(Finset.image (fun m => ↑(g ↑m (_ : ↑m < n))) Finset.univ))
⊢ False | A : Type
inst✝¹ : CommRing A
inst✝ : DecidableEq A
I : Ideal A
hInonfg : ¬Ideal.FG I
n : ℕ
g : (m : ℕ) → m < n → ↥I
h : ↑I ⊆ ↑(Ideal.span ↑(Finset.image (fun m => ↑(g ↑m (_ : ↑m < n))) Finset.univ))
⊢ Ideal.FG I | Please generate a tactic in lean4 to solve the state.
STATE:
A : Type
inst✝¹ : CommRing A
inst✝ : DecidableEq A
I : Ideal A
hInonfg : ¬Ideal.FG I
n : ℕ
g : (m : ℕ) → m < n → ↥I
h : ↑I ⊆ ↑(Ideal.span ↑(Finset.image (fun m => ↑(g ↑m (_ : ↑m < n))) Finset.univ))
⊢ False
TACTIC:
|
https://github.com/ImperialCollegeLondon/formalising-mathematics-2024.git | b732ed1352e87b4474b0520d1383994e069f8057 | FormalisingMathematics2024/Solutions/Section16commutativeAlgebra/Sheet2.lean | Section16Sheet2solutions.lemma1 | [37, 1] | [54, 22] | refine' ⟨Finset.image (fun m : Fin n => (g m.1 m.2).1) Finset.univ, _⟩ | A : Type
inst✝¹ : CommRing A
inst✝ : DecidableEq A
I : Ideal A
hInonfg : ¬Ideal.FG I
n : ℕ
g : (m : ℕ) → m < n → ↥I
h : ↑I ⊆ ↑(Ideal.span ↑(Finset.image (fun m => ↑(g ↑m (_ : ↑m < n))) Finset.univ))
⊢ Ideal.FG I | A : Type
inst✝¹ : CommRing A
inst✝ : DecidableEq A
I : Ideal A
hInonfg : ¬Ideal.FG I
n : ℕ
g : (m : ℕ) → m < n → ↥I
h : ↑I ⊆ ↑(Ideal.span ↑(Finset.image (fun m => ↑(g ↑m (_ : ↑m < n))) Finset.univ))
⊢ Ideal.span ↑(Finset.image (fun m => ↑(g ↑m (_ : ↑m < n))) Finset.univ) = I | Please generate a tactic in lean4 to solve the state.
STATE:
A : Type
inst✝¹ : CommRing A
inst✝ : DecidableEq A
I : Ideal A
hInonfg : ¬Ideal.FG I
n : ℕ
g : (m : ℕ) → m < n → ↥I
h : ↑I ⊆ ↑(Ideal.span ↑(Finset.image (fun m => ↑(g ↑m (_ : ↑m < n))) Finset.univ))
⊢ Ideal.FG I
TACTIC:
|
https://github.com/ImperialCollegeLondon/formalising-mathematics-2024.git | b732ed1352e87b4474b0520d1383994e069f8057 | FormalisingMathematics2024/Solutions/Section16commutativeAlgebra/Sheet2.lean | Section16Sheet2solutions.lemma1 | [37, 1] | [54, 22] | refine' le_antisymm _ h | A : Type
inst✝¹ : CommRing A
inst✝ : DecidableEq A
I : Ideal A
hInonfg : ¬Ideal.FG I
n : ℕ
g : (m : ℕ) → m < n → ↥I
h : ↑I ⊆ ↑(Ideal.span ↑(Finset.image (fun m => ↑(g ↑m (_ : ↑m < n))) Finset.univ))
⊢ Ideal.span ↑(Finset.image (fun m => ↑(g ↑m (_ : ↑m < n))) Finset.univ) = I | A : Type
inst✝¹ : CommRing A
inst✝ : DecidableEq A
I : Ideal A
hInonfg : ¬Ideal.FG I
n : ℕ
g : (m : ℕ) → m < n → ↥I
h : ↑I ⊆ ↑(Ideal.span ↑(Finset.image (fun m => ↑(g ↑m (_ : ↑m < n))) Finset.univ))
⊢ Ideal.span ↑(Finset.image (fun m => ↑(g ↑m (_ : ↑m < n))) Finset.univ) ≤ I | Please generate a tactic in lean4 to solve the state.
STATE:
A : Type
inst✝¹ : CommRing A
inst✝ : DecidableEq A
I : Ideal A
hInonfg : ¬Ideal.FG I
n : ℕ
g : (m : ℕ) → m < n → ↥I
h : ↑I ⊆ ↑(Ideal.span ↑(Finset.image (fun m => ↑(g ↑m (_ : ↑m < n))) Finset.univ))
⊢ Ideal.span ↑(Finset.image (fun m => ↑(g ↑m (_ : ↑m < n))) Finset.univ) = I
TACTIC:
|
https://github.com/ImperialCollegeLondon/formalising-mathematics-2024.git | b732ed1352e87b4474b0520d1383994e069f8057 | FormalisingMathematics2024/Solutions/Section16commutativeAlgebra/Sheet2.lean | Section16Sheet2solutions.lemma1 | [37, 1] | [54, 22] | rw [Ideal.span_le] | A : Type
inst✝¹ : CommRing A
inst✝ : DecidableEq A
I : Ideal A
hInonfg : ¬Ideal.FG I
n : ℕ
g : (m : ℕ) → m < n → ↥I
h : ↑I ⊆ ↑(Ideal.span ↑(Finset.image (fun m => ↑(g ↑m (_ : ↑m < n))) Finset.univ))
⊢ Ideal.span ↑(Finset.image (fun m => ↑(g ↑m (_ : ↑m < n))) Finset.univ) ≤ I | A : Type
inst✝¹ : CommRing A
inst✝ : DecidableEq A
I : Ideal A
hInonfg : ¬Ideal.FG I
n : ℕ
g : (m : ℕ) → m < n → ↥I
h : ↑I ⊆ ↑(Ideal.span ↑(Finset.image (fun m => ↑(g ↑m (_ : ↑m < n))) Finset.univ))
⊢ ↑(Finset.image (fun m => ↑(g ↑m (_ : ↑m < n))) Finset.univ) ⊆ ↑I | Please generate a tactic in lean4 to solve the state.
STATE:
A : Type
inst✝¹ : CommRing A
inst✝ : DecidableEq A
I : Ideal A
hInonfg : ¬Ideal.FG I
n : ℕ
g : (m : ℕ) → m < n → ↥I
h : ↑I ⊆ ↑(Ideal.span ↑(Finset.image (fun m => ↑(g ↑m (_ : ↑m < n))) Finset.univ))
⊢ Ideal.span ↑(Finset.image (fun m => ↑(g ↑m (_ : ↑m < n))) Finset.univ) ≤ I
TACTIC:
|
https://github.com/ImperialCollegeLondon/formalising-mathematics-2024.git | b732ed1352e87b4474b0520d1383994e069f8057 | FormalisingMathematics2024/Solutions/Section16commutativeAlgebra/Sheet2.lean | Section16Sheet2solutions.lemma1 | [37, 1] | [54, 22] | intro a ha | A : Type
inst✝¹ : CommRing A
inst✝ : DecidableEq A
I : Ideal A
hInonfg : ¬Ideal.FG I
n : ℕ
g : (m : ℕ) → m < n → ↥I
h : ↑I ⊆ ↑(Ideal.span ↑(Finset.image (fun m => ↑(g ↑m (_ : ↑m < n))) Finset.univ))
⊢ ↑(Finset.image (fun m => ↑(g ↑m (_ : ↑m < n))) Finset.univ) ⊆ ↑I | A : Type
inst✝¹ : CommRing A
inst✝ : DecidableEq A
I : Ideal A
hInonfg : ¬Ideal.FG I
n : ℕ
g : (m : ℕ) → m < n → ↥I
h : ↑I ⊆ ↑(Ideal.span ↑(Finset.image (fun m => ↑(g ↑m (_ : ↑m < n))) Finset.univ))
a : A
ha : a ∈ ↑(Finset.image (fun m => ↑(g ↑m (_ : ↑m < n))) Finset.univ)
⊢ a ∈ ↑I | Please generate a tactic in lean4 to solve the state.
STATE:
A : Type
inst✝¹ : CommRing A
inst✝ : DecidableEq A
I : Ideal A
hInonfg : ¬Ideal.FG I
n : ℕ
g : (m : ℕ) → m < n → ↥I
h : ↑I ⊆ ↑(Ideal.span ↑(Finset.image (fun m => ↑(g ↑m (_ : ↑m < n))) Finset.univ))
⊢ ↑(Finset.image (fun m => ↑(g ↑m (_ : ↑m < n))) Finset.univ) ⊆ ↑I
TACTIC:
|
https://github.com/ImperialCollegeLondon/formalising-mathematics-2024.git | b732ed1352e87b4474b0520d1383994e069f8057 | FormalisingMathematics2024/Solutions/Section16commutativeAlgebra/Sheet2.lean | Section16Sheet2solutions.lemma1 | [37, 1] | [54, 22] | simp only [Finset.coe_image, Finset.coe_univ, Set.image_univ,
Set.mem_range] at ha | A : Type
inst✝¹ : CommRing A
inst✝ : DecidableEq A
I : Ideal A
hInonfg : ¬Ideal.FG I
n : ℕ
g : (m : ℕ) → m < n → ↥I
h : ↑I ⊆ ↑(Ideal.span ↑(Finset.image (fun m => ↑(g ↑m (_ : ↑m < n))) Finset.univ))
a : A
ha : a ∈ ↑(Finset.image (fun m => ↑(g ↑m (_ : ↑m < n))) Finset.univ)
⊢ a ∈ ↑I | A : Type
inst✝¹ : CommRing A
inst✝ : DecidableEq A
I : Ideal A
hInonfg : ¬Ideal.FG I
n : ℕ
g : (m : ℕ) → m < n → ↥I
h : ↑I ⊆ ↑(Ideal.span ↑(Finset.image (fun m => ↑(g ↑m (_ : ↑m < n))) Finset.univ))
a : A
ha : ∃ y, ↑(g ↑y (_ : ↑y < n)) = a
⊢ a ∈ ↑I | Please generate a tactic in lean4 to solve the state.
STATE:
A : Type
inst✝¹ : CommRing A
inst✝ : DecidableEq A
I : Ideal A
hInonfg : ¬Ideal.FG I
n : ℕ
g : (m : ℕ) → m < n → ↥I
h : ↑I ⊆ ↑(Ideal.span ↑(Finset.image (fun m => ↑(g ↑m (_ : ↑m < n))) Finset.univ))
a : A
ha : a ∈ ↑(Finset.image (fun m => ↑(g ↑m (_ : ↑m < n))) Finset.univ)
⊢ a ∈ ↑I
TACTIC:
|
https://github.com/ImperialCollegeLondon/formalising-mathematics-2024.git | b732ed1352e87b4474b0520d1383994e069f8057 | FormalisingMathematics2024/Solutions/Section16commutativeAlgebra/Sheet2.lean | Section16Sheet2solutions.lemma1 | [37, 1] | [54, 22] | rcases ha with ⟨y, rfl⟩ | A : Type
inst✝¹ : CommRing A
inst✝ : DecidableEq A
I : Ideal A
hInonfg : ¬Ideal.FG I
n : ℕ
g : (m : ℕ) → m < n → ↥I
h : ↑I ⊆ ↑(Ideal.span ↑(Finset.image (fun m => ↑(g ↑m (_ : ↑m < n))) Finset.univ))
a : A
ha : ∃ y, ↑(g ↑y (_ : ↑y < n)) = a
⊢ a ∈ ↑I | case intro
A : Type
inst✝¹ : CommRing A
inst✝ : DecidableEq A
I : Ideal A
hInonfg : ¬Ideal.FG I
n : ℕ
g : (m : ℕ) → m < n → ↥I
h : ↑I ⊆ ↑(Ideal.span ↑(Finset.image (fun m => ↑(g ↑m (_ : ↑m < n))) Finset.univ))
y : Fin n
⊢ ↑(g ↑y (_ : ↑y < n)) ∈ ↑I | Please generate a tactic in lean4 to solve the state.
STATE:
A : Type
inst✝¹ : CommRing A
inst✝ : DecidableEq A
I : Ideal A
hInonfg : ¬Ideal.FG I
n : ℕ
g : (m : ℕ) → m < n → ↥I
h : ↑I ⊆ ↑(Ideal.span ↑(Finset.image (fun m => ↑(g ↑m (_ : ↑m < n))) Finset.univ))
a : A
ha : ∃ y, ↑(g ↑y (_ : ↑y < n)) = a
⊢ a ∈ ↑I
TACTIC:
|
https://github.com/ImperialCollegeLondon/formalising-mathematics-2024.git | b732ed1352e87b4474b0520d1383994e069f8057 | FormalisingMathematics2024/Solutions/Section16commutativeAlgebra/Sheet2.lean | Section16Sheet2solutions.lemma1 | [37, 1] | [54, 22] | exact (g y.1 y.2).2 | case intro
A : Type
inst✝¹ : CommRing A
inst✝ : DecidableEq A
I : Ideal A
hInonfg : ¬Ideal.FG I
n : ℕ
g : (m : ℕ) → m < n → ↥I
h : ↑I ⊆ ↑(Ideal.span ↑(Finset.image (fun m => ↑(g ↑m (_ : ↑m < n))) Finset.univ))
y : Fin n
⊢ ↑(g ↑y (_ : ↑y < n)) ∈ ↑I | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case intro
A : Type
inst✝¹ : CommRing A
inst✝ : DecidableEq A
I : Ideal A
hInonfg : ¬Ideal.FG I
n : ℕ
g : (m : ℕ) → m < n → ↥I
h : ↑I ⊆ ↑(Ideal.span ↑(Finset.image (fun m => ↑(g ↑m (_ : ↑m < n))) Finset.univ))
y : Fin n
⊢ ↑(g ↑y (_ : ↑y < n)) ∈ ↑I
TACTIC:
|
https://github.com/ImperialCollegeLondon/formalising-mathematics-2024.git | b732ed1352e87b4474b0520d1383994e069f8057 | FormalisingMathematics2024/Solutions/Section16commutativeAlgebra/Sheet2.lean | Section16Sheet2solutions.lemma1' | [56, 1] | [70, 13] | set S : Set A := _ | A : Type
inst✝¹ : CommRing A
inst✝ : DecidableEq A
I : Ideal A
hInonfg : ¬Ideal.FG I
n : ℕ
g : (m : ℕ) → m < n → ↥I
⊢ Set.Nonempty {x | ↑x ∉ Ideal.span ↑(Finset.image (fun m => ↑(g ↑m (_ : ↑m < n))) Finset.univ)} | A : Type
inst✝¹ : CommRing A
inst✝ : DecidableEq A
I : Ideal A
hInonfg : ¬Ideal.FG I
n : ℕ
g : (m : ℕ) → m < n → ↥I
S : Set A := ?m.3713
⊢ Set.Nonempty {x | ↑x ∉ Ideal.span ↑(Finset.image (fun m => ↑(g ↑m (_ : ↑m < n))) Finset.univ)} | Please generate a tactic in lean4 to solve the state.
STATE:
A : Type
inst✝¹ : CommRing A
inst✝ : DecidableEq A
I : Ideal A
hInonfg : ¬Ideal.FG I
n : ℕ
g : (m : ℕ) → m < n → ↥I
⊢ Set.Nonempty {x | ↑x ∉ Ideal.span ↑(Finset.image (fun m => ↑(g ↑m (_ : ↑m < n))) Finset.univ)}
TACTIC:
|
https://github.com/ImperialCollegeLondon/formalising-mathematics-2024.git | b732ed1352e87b4474b0520d1383994e069f8057 | FormalisingMathematics2024/Solutions/Section16commutativeAlgebra/Sheet2.lean | Section16Sheet2solutions.lemma1' | [56, 1] | [70, 13] | have ne1 : Set.Nonempty S := lemma1 hInonfg n g | A : Type
inst✝¹ : CommRing A
inst✝ : DecidableEq A
I : Ideal A
hInonfg : ¬Ideal.FG I
n : ℕ
g : (m : ℕ) → m < n → ↥I
S : Set A := ?m.3713
⊢ Set.Nonempty {x | ↑x ∉ Ideal.span ↑(Finset.image (fun m => ↑(g ↑m (_ : ↑m < n))) Finset.univ)} | A : Type
inst✝¹ : CommRing A
inst✝ : DecidableEq A
I : Ideal A
hInonfg : ¬Ideal.FG I
n : ℕ
g : (m : ℕ) → m < n → ↥I
S : Set A := ↑I \ ↑(Ideal.span ↑(Finset.image (fun m => ↑(g ↑m (_ : ↑m < n))) Finset.univ))
ne1 : Set.Nonempty S
⊢ Set.Nonempty {x | ↑x ∉ Ideal.span ↑(Finset.image (fun m => ↑(g ↑m (_ : ↑m < n))) Finset.univ)} | Please generate a tactic in lean4 to solve the state.
STATE:
A : Type
inst✝¹ : CommRing A
inst✝ : DecidableEq A
I : Ideal A
hInonfg : ¬Ideal.FG I
n : ℕ
g : (m : ℕ) → m < n → ↥I
S : Set A := ?m.3713
⊢ Set.Nonempty {x | ↑x ∉ Ideal.span ↑(Finset.image (fun m => ↑(g ↑m (_ : ↑m < n))) Finset.univ)}
TACTIC:
|
https://github.com/ImperialCollegeLondon/formalising-mathematics-2024.git | b732ed1352e87b4474b0520d1383994e069f8057 | FormalisingMathematics2024/Solutions/Section16commutativeAlgebra/Sheet2.lean | Section16Sheet2solutions.lemma1' | [56, 1] | [70, 13] | choose x hx using ne1 | A : Type
inst✝¹ : CommRing A
inst✝ : DecidableEq A
I : Ideal A
hInonfg : ¬Ideal.FG I
n : ℕ
g : (m : ℕ) → m < n → ↥I
S : Set A := ↑I \ ↑(Ideal.span ↑(Finset.image (fun m => ↑(g ↑m (_ : ↑m < n))) Finset.univ))
ne1 : Set.Nonempty S
⊢ Set.Nonempty {x | ↑x ∉ Ideal.span ↑(Finset.image (fun m => ↑(g ↑m (_ : ↑m < n))) Finset.univ)} | A : Type
inst✝¹ : CommRing A
inst✝ : DecidableEq A
I : Ideal A
hInonfg : ¬Ideal.FG I
n : ℕ
g : (m : ℕ) → m < n → ↥I
S : Set A := ↑I \ ↑(Ideal.span ↑(Finset.image (fun m => ↑(g ↑m (_ : ↑m < n))) Finset.univ))
x : A
hx : x ∈ S
⊢ Set.Nonempty {x | ↑x ∉ Ideal.span ↑(Finset.image (fun m => ↑(g ↑m (_ : ↑m < n))) Finset.univ)} | Please generate a tactic in lean4 to solve the state.
STATE:
A : Type
inst✝¹ : CommRing A
inst✝ : DecidableEq A
I : Ideal A
hInonfg : ¬Ideal.FG I
n : ℕ
g : (m : ℕ) → m < n → ↥I
S : Set A := ↑I \ ↑(Ideal.span ↑(Finset.image (fun m => ↑(g ↑m (_ : ↑m < n))) Finset.univ))
ne1 : Set.Nonempty S
⊢ Set.Nonempty {x | ↑x ∉ Ideal.span ↑(Finset.image (fun m => ↑(g ↑m (_ : ↑m < n))) Finset.univ)}
TACTIC:
|
https://github.com/ImperialCollegeLondon/formalising-mathematics-2024.git | b732ed1352e87b4474b0520d1383994e069f8057 | FormalisingMathematics2024/Solutions/Section16commutativeAlgebra/Sheet2.lean | Section16Sheet2solutions.lemma1' | [56, 1] | [70, 13] | refine ⟨⟨x, hx.1⟩, ?_⟩ | A : Type
inst✝¹ : CommRing A
inst✝ : DecidableEq A
I : Ideal A
hInonfg : ¬Ideal.FG I
n : ℕ
g : (m : ℕ) → m < n → ↥I
S : Set A := ↑I \ ↑(Ideal.span ↑(Finset.image (fun m => ↑(g ↑m (_ : ↑m < n))) Finset.univ))
x : A
hx : x ∈ S
⊢ Set.Nonempty {x | ↑x ∉ Ideal.span ↑(Finset.image (fun m => ↑(g ↑m (_ : ↑m < n))) Finset.univ)} | A : Type
inst✝¹ : CommRing A
inst✝ : DecidableEq A
I : Ideal A
hInonfg : ¬Ideal.FG I
n : ℕ
g : (m : ℕ) → m < n → ↥I
S : Set A := ↑I \ ↑(Ideal.span ↑(Finset.image (fun m => ↑(g ↑m (_ : ↑m < n))) Finset.univ))
x : A
hx : x ∈ S
⊢ { val := x, property := (_ : x ∈ ↑I) } ∈
{x | ↑x ∉ Ideal.span ↑(Finset.image (fun m => ↑(g ↑m (_ : ↑m < n))) Finset.univ)} | Please generate a tactic in lean4 to solve the state.
STATE:
A : Type
inst✝¹ : CommRing A
inst✝ : DecidableEq A
I : Ideal A
hInonfg : ¬Ideal.FG I
n : ℕ
g : (m : ℕ) → m < n → ↥I
S : Set A := ↑I \ ↑(Ideal.span ↑(Finset.image (fun m => ↑(g ↑m (_ : ↑m < n))) Finset.univ))
x : A
hx : x ∈ S
⊢ Set.Nonempty {x | ↑x ∉ Ideal.span ↑(Finset.image (fun m => ↑(g ↑m (_ : ↑m < n))) Finset.univ)}
TACTIC:
|
https://github.com/ImperialCollegeLondon/formalising-mathematics-2024.git | b732ed1352e87b4474b0520d1383994e069f8057 | FormalisingMathematics2024/Solutions/Section16commutativeAlgebra/Sheet2.lean | Section16Sheet2solutions.lemma1' | [56, 1] | [70, 13] | simp only [Finset.coe_image, Finset.coe_univ, Set.image_univ, Set.mem_diff,
SetLike.mem_coe, Set.mem_setOf_eq] at hx ⊢ | A : Type
inst✝¹ : CommRing A
inst✝ : DecidableEq A
I : Ideal A
hInonfg : ¬Ideal.FG I
n : ℕ
g : (m : ℕ) → m < n → ↥I
S : Set A := ↑I \ ↑(Ideal.span ↑(Finset.image (fun m => ↑(g ↑m (_ : ↑m < n))) Finset.univ))
x : A
hx : x ∈ S
⊢ { val := x, property := (_ : x ∈ ↑I) } ∈
{x | ↑x ∉ Ideal.span ↑(Finset.image (fun m => ↑(g ↑m (_ : ↑m < n))) Finset.univ)} | A : Type
inst✝¹ : CommRing A
inst✝ : DecidableEq A
I : Ideal A
hInonfg : ¬Ideal.FG I
n : ℕ
g : (m : ℕ) → m < n → ↥I
S : Set A := ↑I \ ↑(Ideal.span ↑(Finset.image (fun m => ↑(g ↑m (_ : ↑m < n))) Finset.univ))
x : A
hx : x ∈ I ∧ x ∉ Ideal.span (Set.range fun m => ↑(g ↑m (_ : ↑m < n)))
⊢ x ∉ Ideal.span (Set.range fun m => ↑(g ↑m (_ : ↑m < n))) | Please generate a tactic in lean4 to solve the state.
STATE:
A : Type
inst✝¹ : CommRing A
inst✝ : DecidableEq A
I : Ideal A
hInonfg : ¬Ideal.FG I
n : ℕ
g : (m : ℕ) → m < n → ↥I
S : Set A := ↑I \ ↑(Ideal.span ↑(Finset.image (fun m => ↑(g ↑m (_ : ↑m < n))) Finset.univ))
x : A
hx : x ∈ S
⊢ { val := x, property := (_ : x ∈ ↑I) } ∈
{x | ↑x ∉ Ideal.span ↑(Finset.image (fun m => ↑(g ↑m (_ : ↑m < n))) Finset.univ)}
TACTIC:
|
https://github.com/ImperialCollegeLondon/formalising-mathematics-2024.git | b732ed1352e87b4474b0520d1383994e069f8057 | FormalisingMathematics2024/Solutions/Section16commutativeAlgebra/Sheet2.lean | Section16Sheet2solutions.lemma1' | [56, 1] | [70, 13] | exact hx.2 | A : Type
inst✝¹ : CommRing A
inst✝ : DecidableEq A
I : Ideal A
hInonfg : ¬Ideal.FG I
n : ℕ
g : (m : ℕ) → m < n → ↥I
S : Set A := ↑I \ ↑(Ideal.span ↑(Finset.image (fun m => ↑(g ↑m (_ : ↑m < n))) Finset.univ))
x : A
hx : x ∈ I ∧ x ∉ Ideal.span (Set.range fun m => ↑(g ↑m (_ : ↑m < n)))
⊢ x ∉ Ideal.span (Set.range fun m => ↑(g ↑m (_ : ↑m < n))) | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
A : Type
inst✝¹ : CommRing A
inst✝ : DecidableEq A
I : Ideal A
hInonfg : ¬Ideal.FG I
n : ℕ
g : (m : ℕ) → m < n → ↥I
S : Set A := ↑I \ ↑(Ideal.span ↑(Finset.image (fun m => ↑(g ↑m (_ : ↑m < n))) Finset.univ))
x : A
hx : x ∈ I ∧ x ∉ Ideal.span (Set.range fun m => ↑(g ↑m (_ : ↑m < n)))
⊢ x ∉ Ideal.span (Set.range fun m => ↑(g ↑m (_ : ↑m < n)))
TACTIC:
|
https://github.com/ImperialCollegeLondon/formalising-mathematics-2024.git | b732ed1352e87b4474b0520d1383994e069f8057 | FormalisingMathematics2024/Solutions/Section07subgroupsAndHomomorphisms/Sheet1.lean | conjugate.one_mem | [122, 1] | [126, 10] | use 1 | G : Type
inst✝ : Group G
a b : G
H K : Subgroup G
x y z : G
⊢ 1 ∈ {a | ∃ h ∈ H, a = x * h * x⁻¹} | case h
G : Type
inst✝ : Group G
a b : G
H K : Subgroup G
x y z : G
⊢ 1 ∈ H ∧ 1 = x * 1 * x⁻¹ | Please generate a tactic in lean4 to solve the state.
STATE:
G : Type
inst✝ : Group G
a b : G
H K : Subgroup G
x y z : G
⊢ 1 ∈ {a | ∃ h ∈ H, a = x * h * x⁻¹}
TACTIC:
|
https://github.com/ImperialCollegeLondon/formalising-mathematics-2024.git | b732ed1352e87b4474b0520d1383994e069f8057 | FormalisingMathematics2024/Solutions/Section07subgroupsAndHomomorphisms/Sheet1.lean | conjugate.one_mem | [122, 1] | [126, 10] | constructor | case h
G : Type
inst✝ : Group G
a b : G
H K : Subgroup G
x y z : G
⊢ 1 ∈ H ∧ 1 = x * 1 * x⁻¹ | case h.left
G : Type
inst✝ : Group G
a b : G
H K : Subgroup G
x y z : G
⊢ 1 ∈ H
case h.right
G : Type
inst✝ : Group G
a b : G
H K : Subgroup G
x y z : G
⊢ 1 = x * 1 * x⁻¹ | Please generate a tactic in lean4 to solve the state.
STATE:
case h
G : Type
inst✝ : Group G
a b : G
H K : Subgroup G
x y z : G
⊢ 1 ∈ H ∧ 1 = x * 1 * x⁻¹
TACTIC:
|
https://github.com/ImperialCollegeLondon/formalising-mathematics-2024.git | b732ed1352e87b4474b0520d1383994e069f8057 | FormalisingMathematics2024/Solutions/Section07subgroupsAndHomomorphisms/Sheet1.lean | conjugate.one_mem | [122, 1] | [126, 10] | exact H.one_mem | case h.left
G : Type
inst✝ : Group G
a b : G
H K : Subgroup G
x y z : G
⊢ 1 ∈ H | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case h.left
G : Type
inst✝ : Group G
a b : G
H K : Subgroup G
x y z : G
⊢ 1 ∈ H
TACTIC:
|
https://github.com/ImperialCollegeLondon/formalising-mathematics-2024.git | b732ed1352e87b4474b0520d1383994e069f8057 | FormalisingMathematics2024/Solutions/Section07subgroupsAndHomomorphisms/Sheet1.lean | conjugate.one_mem | [122, 1] | [126, 10] | group | case h.right
G : Type
inst✝ : Group G
a b : G
H K : Subgroup G
x y z : G
⊢ 1 = x * 1 * x⁻¹ | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case h.right
G : Type
inst✝ : Group G
a b : G
H K : Subgroup G
x y z : G
⊢ 1 = x * 1 * x⁻¹
TACTIC:
|
https://github.com/ImperialCollegeLondon/formalising-mathematics-2024.git | b732ed1352e87b4474b0520d1383994e069f8057 | FormalisingMathematics2024/Solutions/Section07subgroupsAndHomomorphisms/Sheet1.lean | conjugate.inv_mem | [128, 1] | [132, 8] | rcases hy with ⟨g, hg, rfl⟩ | G : Type
inst✝ : Group G
a b : G
H K : Subgroup G
x y z : G
hy : y ∈ {a | ∃ h ∈ H, a = x * h * x⁻¹}
⊢ y⁻¹ ∈ {a | ∃ h ∈ H, a = x * h * x⁻¹} | case intro.intro
G : Type
inst✝ : Group G
a b : G
H K : Subgroup G
x z g : G
hg : g ∈ H
⊢ (x * g * x⁻¹)⁻¹ ∈ {a | ∃ h ∈ H, a = x * h * x⁻¹} | Please generate a tactic in lean4 to solve the state.
STATE:
G : Type
inst✝ : Group G
a b : G
H K : Subgroup G
x y z : G
hy : y ∈ {a | ∃ h ∈ H, a = x * h * x⁻¹}
⊢ y⁻¹ ∈ {a | ∃ h ∈ H, a = x * h * x⁻¹}
TACTIC:
|
https://github.com/ImperialCollegeLondon/formalising-mathematics-2024.git | b732ed1352e87b4474b0520d1383994e069f8057 | FormalisingMathematics2024/Solutions/Section07subgroupsAndHomomorphisms/Sheet1.lean | conjugate.inv_mem | [128, 1] | [132, 8] | use g⁻¹, inv_mem hg | case intro.intro
G : Type
inst✝ : Group G
a b : G
H K : Subgroup G
x z g : G
hg : g ∈ H
⊢ (x * g * x⁻¹)⁻¹ ∈ {a | ∃ h ∈ H, a = x * h * x⁻¹} | case right
G : Type
inst✝ : Group G
a b : G
H K : Subgroup G
x z g : G
hg : g ∈ H
⊢ (x * g * x⁻¹)⁻¹ = x * g⁻¹ * x⁻¹ | Please generate a tactic in lean4 to solve the state.
STATE:
case intro.intro
G : Type
inst✝ : Group G
a b : G
H K : Subgroup G
x z g : G
hg : g ∈ H
⊢ (x * g * x⁻¹)⁻¹ ∈ {a | ∃ h ∈ H, a = x * h * x⁻¹}
TACTIC:
|
https://github.com/ImperialCollegeLondon/formalising-mathematics-2024.git | b732ed1352e87b4474b0520d1383994e069f8057 | FormalisingMathematics2024/Solutions/Section07subgroupsAndHomomorphisms/Sheet1.lean | conjugate.inv_mem | [128, 1] | [132, 8] | group | case right
G : Type
inst✝ : Group G
a b : G
H K : Subgroup G
x z g : G
hg : g ∈ H
⊢ (x * g * x⁻¹)⁻¹ = x * g⁻¹ * x⁻¹ | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case right
G : Type
inst✝ : Group G
a b : G
H K : Subgroup G
x z g : G
hg : g ∈ H
⊢ (x * g * x⁻¹)⁻¹ = x * g⁻¹ * x⁻¹
TACTIC:
|
https://github.com/ImperialCollegeLondon/formalising-mathematics-2024.git | b732ed1352e87b4474b0520d1383994e069f8057 | FormalisingMathematics2024/Solutions/Section07subgroupsAndHomomorphisms/Sheet1.lean | conjugate.mul_mem | [134, 1] | [140, 8] | rcases hy with ⟨g, hg, rfl⟩ | G : Type
inst✝ : Group G
a b : G
H K : Subgroup G
x y z : G
hy : y ∈ {a | ∃ h ∈ H, a = x * h * x⁻¹}
hz : z ∈ {a | ∃ h ∈ H, a = x * h * x⁻¹}
⊢ y * z ∈ {a | ∃ h ∈ H, a = x * h * x⁻¹} | case intro.intro
G : Type
inst✝ : Group G
a b : G
H K : Subgroup G
x z : G
hz : z ∈ {a | ∃ h ∈ H, a = x * h * x⁻¹}
g : G
hg : g ∈ H
⊢ x * g * x⁻¹ * z ∈ {a | ∃ h ∈ H, a = x * h * x⁻¹} | Please generate a tactic in lean4 to solve the state.
STATE:
G : Type
inst✝ : Group G
a b : G
H K : Subgroup G
x y z : G
hy : y ∈ {a | ∃ h ∈ H, a = x * h * x⁻¹}
hz : z ∈ {a | ∃ h ∈ H, a = x * h * x⁻¹}
⊢ y * z ∈ {a | ∃ h ∈ H, a = x * h * x⁻¹}
TACTIC:
|
https://github.com/ImperialCollegeLondon/formalising-mathematics-2024.git | b732ed1352e87b4474b0520d1383994e069f8057 | FormalisingMathematics2024/Solutions/Section07subgroupsAndHomomorphisms/Sheet1.lean | conjugate.mul_mem | [134, 1] | [140, 8] | rcases hz with ⟨k, hk, rfl⟩ | case intro.intro
G : Type
inst✝ : Group G
a b : G
H K : Subgroup G
x z : G
hz : z ∈ {a | ∃ h ∈ H, a = x * h * x⁻¹}
g : G
hg : g ∈ H
⊢ x * g * x⁻¹ * z ∈ {a | ∃ h ∈ H, a = x * h * x⁻¹} | case intro.intro.intro.intro
G : Type
inst✝ : Group G
a b : G
H K : Subgroup G
x g : G
hg : g ∈ H
k : G
hk : k ∈ H
⊢ x * g * x⁻¹ * (x * k * x⁻¹) ∈ {a | ∃ h ∈ H, a = x * h * x⁻¹} | Please generate a tactic in lean4 to solve the state.
STATE:
case intro.intro
G : Type
inst✝ : Group G
a b : G
H K : Subgroup G
x z : G
hz : z ∈ {a | ∃ h ∈ H, a = x * h * x⁻¹}
g : G
hg : g ∈ H
⊢ x * g * x⁻¹ * z ∈ {a | ∃ h ∈ H, a = x * h * x⁻¹}
TACTIC:
|
https://github.com/ImperialCollegeLondon/formalising-mathematics-2024.git | b732ed1352e87b4474b0520d1383994e069f8057 | FormalisingMathematics2024/Solutions/Section07subgroupsAndHomomorphisms/Sheet1.lean | conjugate.mul_mem | [134, 1] | [140, 8] | use g * k, mul_mem hg hk | case intro.intro.intro.intro
G : Type
inst✝ : Group G
a b : G
H K : Subgroup G
x g : G
hg : g ∈ H
k : G
hk : k ∈ H
⊢ x * g * x⁻¹ * (x * k * x⁻¹) ∈ {a | ∃ h ∈ H, a = x * h * x⁻¹} | case right
G : Type
inst✝ : Group G
a b : G
H K : Subgroup G
x g : G
hg : g ∈ H
k : G
hk : k ∈ H
⊢ x * g * x⁻¹ * (x * k * x⁻¹) = x * (g * k) * x⁻¹ | Please generate a tactic in lean4 to solve the state.
STATE:
case intro.intro.intro.intro
G : Type
inst✝ : Group G
a b : G
H K : Subgroup G
x g : G
hg : g ∈ H
k : G
hk : k ∈ H
⊢ x * g * x⁻¹ * (x * k * x⁻¹) ∈ {a | ∃ h ∈ H, a = x * h * x⁻¹}
TACTIC:
|
https://github.com/ImperialCollegeLondon/formalising-mathematics-2024.git | b732ed1352e87b4474b0520d1383994e069f8057 | FormalisingMathematics2024/Solutions/Section07subgroupsAndHomomorphisms/Sheet1.lean | conjugate.mul_mem | [134, 1] | [140, 8] | group | case right
G : Type
inst✝ : Group G
a b : G
H K : Subgroup G
x g : G
hg : g ∈ H
k : G
hk : k ∈ H
⊢ x * g * x⁻¹ * (x * k * x⁻¹) = x * (g * k) * x⁻¹ | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case right
G : Type
inst✝ : Group G
a b : G
H K : Subgroup G
x g : G
hg : g ∈ H
k : G
hk : k ∈ H
⊢ x * g * x⁻¹ * (x * k * x⁻¹) = x * (g * k) * x⁻¹
TACTIC:
|
https://github.com/ImperialCollegeLondon/formalising-mathematics-2024.git | b732ed1352e87b4474b0520d1383994e069f8057 | FormalisingMathematics2024/Solutions/Section07subgroupsAndHomomorphisms/Sheet1.lean | mem_conjugate_iff | [175, 1] | [177, 6] | rfl | G : Type
inst✝ : Group G
a b : G
H K : Subgroup G
x y z : G
⊢ a ∈ conjugate H x ↔ ∃ h ∈ H, a = x * h * x⁻¹ | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
G : Type
inst✝ : Group G
a b : G
H K : Subgroup G
x y z : G
⊢ a ∈ conjugate H x ↔ ∃ h ∈ H, a = x * h * x⁻¹
TACTIC:
|
https://github.com/ImperialCollegeLondon/formalising-mathematics-2024.git | b732ed1352e87b4474b0520d1383994e069f8057 | FormalisingMathematics2024/Solutions/Section07subgroupsAndHomomorphisms/Sheet1.lean | conjugate_mono | [179, 1] | [185, 23] | intro g | G : Type
inst✝ : Group G
a b : G
H✝ K✝ : Subgroup G
x y z : G
H K : Subgroup G
h : H ≤ K
⊢ conjugate H x ≤ conjugate K x | G : Type
inst✝ : Group G
a b : G
H✝ K✝ : Subgroup G
x y z : G
H K : Subgroup G
h : H ≤ K
g : G
⊢ g ∈ conjugate H x → g ∈ conjugate K x | Please generate a tactic in lean4 to solve the state.
STATE:
G : Type
inst✝ : Group G
a b : G
H✝ K✝ : Subgroup G
x y z : G
H K : Subgroup G
h : H ≤ K
⊢ conjugate H x ≤ conjugate K x
TACTIC:
|
https://github.com/ImperialCollegeLondon/formalising-mathematics-2024.git | b732ed1352e87b4474b0520d1383994e069f8057 | FormalisingMathematics2024/Solutions/Section07subgroupsAndHomomorphisms/Sheet1.lean | conjugate_mono | [179, 1] | [185, 23] | rintro ⟨t, ht, rfl⟩ | G : Type
inst✝ : Group G
a b : G
H✝ K✝ : Subgroup G
x y z : G
H K : Subgroup G
h : H ≤ K
g : G
⊢ g ∈ conjugate H x → g ∈ conjugate K x | case intro.intro
G : Type
inst✝ : Group G
a b : G
H✝ K✝ : Subgroup G
x y z : G
H K : Subgroup G
h : H ≤ K
t : G
ht : t ∈ H
⊢ x * t * x⁻¹ ∈ conjugate K x | Please generate a tactic in lean4 to solve the state.
STATE:
G : Type
inst✝ : Group G
a b : G
H✝ K✝ : Subgroup G
x y z : G
H K : Subgroup G
h : H ≤ K
g : G
⊢ g ∈ conjugate H x → g ∈ conjugate K x
TACTIC:
|
https://github.com/ImperialCollegeLondon/formalising-mathematics-2024.git | b732ed1352e87b4474b0520d1383994e069f8057 | FormalisingMathematics2024/Solutions/Section07subgroupsAndHomomorphisms/Sheet1.lean | conjugate_mono | [179, 1] | [185, 23] | exact ⟨t, h ht, rfl⟩ | case intro.intro
G : Type
inst✝ : Group G
a b : G
H✝ K✝ : Subgroup G
x y z : G
H K : Subgroup G
h : H ≤ K
t : G
ht : t ∈ H
⊢ x * t * x⁻¹ ∈ conjugate K x | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case intro.intro
G : Type
inst✝ : Group G
a b : G
H✝ K✝ : Subgroup G
x y z : G
H K : Subgroup G
h : H ≤ K
t : G
ht : t ∈ H
⊢ x * t * x⁻¹ ∈ conjugate K x
TACTIC:
|
https://github.com/ImperialCollegeLondon/formalising-mathematics-2024.git | b732ed1352e87b4474b0520d1383994e069f8057 | FormalisingMathematics2024/Solutions/Section07subgroupsAndHomomorphisms/Sheet1.lean | conjugate_bot | [187, 1] | [203, 12] | ext a | G : Type
inst✝ : Group G
a b : G
H K : Subgroup G
x y z : G
⊢ conjugate ⊥ x = ⊥ | case h
G : Type
inst✝ : Group G
a✝ b : G
H K : Subgroup G
x y z a : G
⊢ a ∈ conjugate ⊥ x ↔ a ∈ ⊥ | Please generate a tactic in lean4 to solve the state.
STATE:
G : Type
inst✝ : Group G
a b : G
H K : Subgroup G
x y z : G
⊢ conjugate ⊥ x = ⊥
TACTIC:
|
https://github.com/ImperialCollegeLondon/formalising-mathematics-2024.git | b732ed1352e87b4474b0520d1383994e069f8057 | FormalisingMathematics2024/Solutions/Section07subgroupsAndHomomorphisms/Sheet1.lean | conjugate_bot | [187, 1] | [203, 12] | rw [mem_conjugate_iff] | case h
G : Type
inst✝ : Group G
a✝ b : G
H K : Subgroup G
x y z a : G
⊢ a ∈ conjugate ⊥ x ↔ a ∈ ⊥ | case h
G : Type
inst✝ : Group G
a✝ b : G
H K : Subgroup G
x y z a : G
⊢ (∃ h ∈ ⊥, a = x * h * x⁻¹) ↔ a ∈ ⊥ | Please generate a tactic in lean4 to solve the state.
STATE:
case h
G : Type
inst✝ : Group G
a✝ b : G
H K : Subgroup G
x y z a : G
⊢ a ∈ conjugate ⊥ x ↔ a ∈ ⊥
TACTIC:
|
https://github.com/ImperialCollegeLondon/formalising-mathematics-2024.git | b732ed1352e87b4474b0520d1383994e069f8057 | FormalisingMathematics2024/Solutions/Section07subgroupsAndHomomorphisms/Sheet1.lean | conjugate_bot | [187, 1] | [203, 12] | rw [Subgroup.mem_bot] | case h
G : Type
inst✝ : Group G
a✝ b : G
H K : Subgroup G
x y z a : G
⊢ (∃ h ∈ ⊥, a = x * h * x⁻¹) ↔ a ∈ ⊥ | case h
G : Type
inst✝ : Group G
a✝ b : G
H K : Subgroup G
x y z a : G
⊢ (∃ h ∈ ⊥, a = x * h * x⁻¹) ↔ a = 1 | Please generate a tactic in lean4 to solve the state.
STATE:
case h
G : Type
inst✝ : Group G
a✝ b : G
H K : Subgroup G
x y z a : G
⊢ (∃ h ∈ ⊥, a = x * h * x⁻¹) ↔ a ∈ ⊥
TACTIC:
|
https://github.com/ImperialCollegeLondon/formalising-mathematics-2024.git | b732ed1352e87b4474b0520d1383994e069f8057 | FormalisingMathematics2024/Solutions/Section07subgroupsAndHomomorphisms/Sheet1.lean | conjugate_bot | [187, 1] | [203, 12] | constructor | case h
G : Type
inst✝ : Group G
a✝ b : G
H K : Subgroup G
x y z a : G
⊢ (∃ h ∈ ⊥, a = x * h * x⁻¹) ↔ a = 1 | case h.mp
G : Type
inst✝ : Group G
a✝ b : G
H K : Subgroup G
x y z a : G
⊢ (∃ h ∈ ⊥, a = x * h * x⁻¹) → a = 1
case h.mpr
G : Type
inst✝ : Group G
a✝ b : G
H K : Subgroup G
x y z a : G
⊢ a = 1 → ∃ h ∈ ⊥, a = x * h * x⁻¹ | Please generate a tactic in lean4 to solve the state.
STATE:
case h
G : Type
inst✝ : Group G
a✝ b : G
H K : Subgroup G
x y z a : G
⊢ (∃ h ∈ ⊥, a = x * h * x⁻¹) ↔ a = 1
TACTIC:
|
https://github.com/ImperialCollegeLondon/formalising-mathematics-2024.git | b732ed1352e87b4474b0520d1383994e069f8057 | FormalisingMathematics2024/Solutions/Section07subgroupsAndHomomorphisms/Sheet1.lean | conjugate_bot | [187, 1] | [203, 12] | rintro ⟨b, hb, rfl⟩ | case h.mp
G : Type
inst✝ : Group G
a✝ b : G
H K : Subgroup G
x y z a : G
⊢ (∃ h ∈ ⊥, a = x * h * x⁻¹) → a = 1 | case h.mp.intro.intro
G : Type
inst✝ : Group G
a b✝ : G
H K : Subgroup G
x y z b : G
hb : b ∈ ⊥
⊢ x * b * x⁻¹ = 1 | Please generate a tactic in lean4 to solve the state.
STATE:
case h.mp
G : Type
inst✝ : Group G
a✝ b : G
H K : Subgroup G
x y z a : G
⊢ (∃ h ∈ ⊥, a = x * h * x⁻¹) → a = 1
TACTIC:
|
https://github.com/ImperialCollegeLondon/formalising-mathematics-2024.git | b732ed1352e87b4474b0520d1383994e069f8057 | FormalisingMathematics2024/Solutions/Section07subgroupsAndHomomorphisms/Sheet1.lean | conjugate_bot | [187, 1] | [203, 12] | rw [Subgroup.mem_bot] at hb | case h.mp.intro.intro
G : Type
inst✝ : Group G
a b✝ : G
H K : Subgroup G
x y z b : G
hb : b ∈ ⊥
⊢ x * b * x⁻¹ = 1 | case h.mp.intro.intro
G : Type
inst✝ : Group G
a b✝ : G
H K : Subgroup G
x y z b : G
hb : b = 1
⊢ x * b * x⁻¹ = 1 | Please generate a tactic in lean4 to solve the state.
STATE:
case h.mp.intro.intro
G : Type
inst✝ : Group G
a b✝ : G
H K : Subgroup G
x y z b : G
hb : b ∈ ⊥
⊢ x * b * x⁻¹ = 1
TACTIC:
|
https://github.com/ImperialCollegeLondon/formalising-mathematics-2024.git | b732ed1352e87b4474b0520d1383994e069f8057 | FormalisingMathematics2024/Solutions/Section07subgroupsAndHomomorphisms/Sheet1.lean | conjugate_bot | [187, 1] | [203, 12] | rw [hb] | case h.mp.intro.intro
G : Type
inst✝ : Group G
a b✝ : G
H K : Subgroup G
x y z b : G
hb : b = 1
⊢ x * b * x⁻¹ = 1 | case h.mp.intro.intro
G : Type
inst✝ : Group G
a b✝ : G
H K : Subgroup G
x y z b : G
hb : b = 1
⊢ x * 1 * x⁻¹ = 1 | Please generate a tactic in lean4 to solve the state.
STATE:
case h.mp.intro.intro
G : Type
inst✝ : Group G
a b✝ : G
H K : Subgroup G
x y z b : G
hb : b = 1
⊢ x * b * x⁻¹ = 1
TACTIC:
|
https://github.com/ImperialCollegeLondon/formalising-mathematics-2024.git | b732ed1352e87b4474b0520d1383994e069f8057 | FormalisingMathematics2024/Solutions/Section07subgroupsAndHomomorphisms/Sheet1.lean | conjugate_bot | [187, 1] | [203, 12] | group | case h.mp.intro.intro
G : Type
inst✝ : Group G
a b✝ : G
H K : Subgroup G
x y z b : G
hb : b = 1
⊢ x * 1 * x⁻¹ = 1 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case h.mp.intro.intro
G : Type
inst✝ : Group G
a b✝ : G
H K : Subgroup G
x y z b : G
hb : b = 1
⊢ x * 1 * x⁻¹ = 1
TACTIC:
|
https://github.com/ImperialCollegeLondon/formalising-mathematics-2024.git | b732ed1352e87b4474b0520d1383994e069f8057 | FormalisingMathematics2024/Solutions/Section07subgroupsAndHomomorphisms/Sheet1.lean | conjugate_bot | [187, 1] | [203, 12] | rintro rfl | case h.mpr
G : Type
inst✝ : Group G
a✝ b : G
H K : Subgroup G
x y z a : G
⊢ a = 1 → ∃ h ∈ ⊥, a = x * h * x⁻¹ | case h.mpr
G : Type
inst✝ : Group G
a b : G
H K : Subgroup G
x y z : G
⊢ ∃ h ∈ ⊥, 1 = x * h * x⁻¹ | Please generate a tactic in lean4 to solve the state.
STATE:
case h.mpr
G : Type
inst✝ : Group G
a✝ b : G
H K : Subgroup G
x y z a : G
⊢ a = 1 → ∃ h ∈ ⊥, a = x * h * x⁻¹
TACTIC:
|
https://github.com/ImperialCollegeLondon/formalising-mathematics-2024.git | b732ed1352e87b4474b0520d1383994e069f8057 | FormalisingMathematics2024/Solutions/Section07subgroupsAndHomomorphisms/Sheet1.lean | conjugate_bot | [187, 1] | [203, 12] | use 1 | case h.mpr
G : Type
inst✝ : Group G
a b : G
H K : Subgroup G
x y z : G
⊢ ∃ h ∈ ⊥, 1 = x * h * x⁻¹ | case h
G : Type
inst✝ : Group G
a b : G
H K : Subgroup G
x y z : G
⊢ 1 ∈ ⊥ ∧ 1 = x * 1 * x⁻¹ | Please generate a tactic in lean4 to solve the state.
STATE:
case h.mpr
G : Type
inst✝ : Group G
a b : G
H K : Subgroup G
x y z : G
⊢ ∃ h ∈ ⊥, 1 = x * h * x⁻¹
TACTIC:
|
https://github.com/ImperialCollegeLondon/formalising-mathematics-2024.git | b732ed1352e87b4474b0520d1383994e069f8057 | FormalisingMathematics2024/Solutions/Section07subgroupsAndHomomorphisms/Sheet1.lean | conjugate_bot | [187, 1] | [203, 12] | constructor | case h
G : Type
inst✝ : Group G
a b : G
H K : Subgroup G
x y z : G
⊢ 1 ∈ ⊥ ∧ 1 = x * 1 * x⁻¹ | case h.left
G : Type
inst✝ : Group G
a b : G
H K : Subgroup G
x y z : G
⊢ 1 ∈ ⊥
case h.right
G : Type
inst✝ : Group G
a b : G
H K : Subgroup G
x y z : G
⊢ 1 = x * 1 * x⁻¹ | Please generate a tactic in lean4 to solve the state.
STATE:
case h
G : Type
inst✝ : Group G
a b : G
H K : Subgroup G
x y z : G
⊢ 1 ∈ ⊥ ∧ 1 = x * 1 * x⁻¹
TACTIC:
|
https://github.com/ImperialCollegeLondon/formalising-mathematics-2024.git | b732ed1352e87b4474b0520d1383994e069f8057 | FormalisingMathematics2024/Solutions/Section07subgroupsAndHomomorphisms/Sheet1.lean | conjugate_bot | [187, 1] | [203, 12] | rw [Subgroup.mem_bot] | case h.left
G : Type
inst✝ : Group G
a b : G
H K : Subgroup G
x y z : G
⊢ 1 ∈ ⊥ | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case h.left
G : Type
inst✝ : Group G
a b : G
H K : Subgroup G
x y z : G
⊢ 1 ∈ ⊥
TACTIC:
|
https://github.com/ImperialCollegeLondon/formalising-mathematics-2024.git | b732ed1352e87b4474b0520d1383994e069f8057 | FormalisingMathematics2024/Solutions/Section07subgroupsAndHomomorphisms/Sheet1.lean | conjugate_bot | [187, 1] | [203, 12] | group | case h.right
G : Type
inst✝ : Group G
a b : G
H K : Subgroup G
x y z : G
⊢ 1 = x * 1 * x⁻¹ | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case h.right
G : Type
inst✝ : Group G
a b : G
H K : Subgroup G
x y z : G
⊢ 1 = x * 1 * x⁻¹
TACTIC:
|
https://github.com/ImperialCollegeLondon/formalising-mathematics-2024.git | b732ed1352e87b4474b0520d1383994e069f8057 | FormalisingMathematics2024/Solutions/Section07subgroupsAndHomomorphisms/Sheet1.lean | conjugate_top | [205, 1] | [215, 12] | ext a | G : Type
inst✝ : Group G
a b : G
H K : Subgroup G
x y z : G
⊢ conjugate ⊤ x = ⊤ | case h
G : Type
inst✝ : Group G
a✝ b : G
H K : Subgroup G
x y z a : G
⊢ a ∈ conjugate ⊤ x ↔ a ∈ ⊤ | Please generate a tactic in lean4 to solve the state.
STATE:
G : Type
inst✝ : Group G
a b : G
H K : Subgroup G
x y z : G
⊢ conjugate ⊤ x = ⊤
TACTIC:
|
https://github.com/ImperialCollegeLondon/formalising-mathematics-2024.git | b732ed1352e87b4474b0520d1383994e069f8057 | FormalisingMathematics2024/Solutions/Section07subgroupsAndHomomorphisms/Sheet1.lean | conjugate_top | [205, 1] | [215, 12] | rw [mem_conjugate_iff] | case h
G : Type
inst✝ : Group G
a✝ b : G
H K : Subgroup G
x y z a : G
⊢ a ∈ conjugate ⊤ x ↔ a ∈ ⊤ | case h
G : Type
inst✝ : Group G
a✝ b : G
H K : Subgroup G
x y z a : G
⊢ (∃ h ∈ ⊤, a = x * h * x⁻¹) ↔ a ∈ ⊤ | Please generate a tactic in lean4 to solve the state.
STATE:
case h
G : Type
inst✝ : Group G
a✝ b : G
H K : Subgroup G
x y z a : G
⊢ a ∈ conjugate ⊤ x ↔ a ∈ ⊤
TACTIC:
|
https://github.com/ImperialCollegeLondon/formalising-mathematics-2024.git | b732ed1352e87b4474b0520d1383994e069f8057 | FormalisingMathematics2024/Solutions/Section07subgroupsAndHomomorphisms/Sheet1.lean | conjugate_top | [205, 1] | [215, 12] | constructor | case h
G : Type
inst✝ : Group G
a✝ b : G
H K : Subgroup G
x y z a : G
⊢ (∃ h ∈ ⊤, a = x * h * x⁻¹) ↔ a ∈ ⊤ | case h.mp
G : Type
inst✝ : Group G
a✝ b : G
H K : Subgroup G
x y z a : G
⊢ (∃ h ∈ ⊤, a = x * h * x⁻¹) → a ∈ ⊤
case h.mpr
G : Type
inst✝ : Group G
a✝ b : G
H K : Subgroup G
x y z a : G
⊢ a ∈ ⊤ → ∃ h ∈ ⊤, a = x * h * x⁻¹ | Please generate a tactic in lean4 to solve the state.
STATE:
case h
G : Type
inst✝ : Group G
a✝ b : G
H K : Subgroup G
x y z a : G
⊢ (∃ h ∈ ⊤, a = x * h * x⁻¹) ↔ a ∈ ⊤
TACTIC:
|
https://github.com/ImperialCollegeLondon/formalising-mathematics-2024.git | b732ed1352e87b4474b0520d1383994e069f8057 | FormalisingMathematics2024/Solutions/Section07subgroupsAndHomomorphisms/Sheet1.lean | conjugate_top | [205, 1] | [215, 12] | intro h | case h.mp
G : Type
inst✝ : Group G
a✝ b : G
H K : Subgroup G
x y z a : G
⊢ (∃ h ∈ ⊤, a = x * h * x⁻¹) → a ∈ ⊤ | case h.mp
G : Type
inst✝ : Group G
a✝ b : G
H K : Subgroup G
x y z a : G
h : ∃ h ∈ ⊤, a = x * h * x⁻¹
⊢ a ∈ ⊤ | Please generate a tactic in lean4 to solve the state.
STATE:
case h.mp
G : Type
inst✝ : Group G
a✝ b : G
H K : Subgroup G
x y z a : G
⊢ (∃ h ∈ ⊤, a = x * h * x⁻¹) → a ∈ ⊤
TACTIC:
|
https://github.com/ImperialCollegeLondon/formalising-mathematics-2024.git | b732ed1352e87b4474b0520d1383994e069f8057 | FormalisingMathematics2024/Solutions/Section07subgroupsAndHomomorphisms/Sheet1.lean | conjugate_top | [205, 1] | [215, 12] | exact Subgroup.mem_top a | case h.mp
G : Type
inst✝ : Group G
a✝ b : G
H K : Subgroup G
x y z a : G
h : ∃ h ∈ ⊤, a = x * h * x⁻¹
⊢ a ∈ ⊤ | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case h.mp
G : Type
inst✝ : Group G
a✝ b : G
H K : Subgroup G
x y z a : G
h : ∃ h ∈ ⊤, a = x * h * x⁻¹
⊢ a ∈ ⊤
TACTIC:
|
https://github.com/ImperialCollegeLondon/formalising-mathematics-2024.git | b732ed1352e87b4474b0520d1383994e069f8057 | FormalisingMathematics2024/Solutions/Section07subgroupsAndHomomorphisms/Sheet1.lean | conjugate_top | [205, 1] | [215, 12] | intro h | case h.mpr
G : Type
inst✝ : Group G
a✝ b : G
H K : Subgroup G
x y z a : G
⊢ a ∈ ⊤ → ∃ h ∈ ⊤, a = x * h * x⁻¹ | case h.mpr
G : Type
inst✝ : Group G
a✝ b : G
H K : Subgroup G
x y z a : G
h : a ∈ ⊤
⊢ ∃ h ∈ ⊤, a = x * h * x⁻¹ | Please generate a tactic in lean4 to solve the state.
STATE:
case h.mpr
G : Type
inst✝ : Group G
a✝ b : G
H K : Subgroup G
x y z a : G
⊢ a ∈ ⊤ → ∃ h ∈ ⊤, a = x * h * x⁻¹
TACTIC:
|
https://github.com/ImperialCollegeLondon/formalising-mathematics-2024.git | b732ed1352e87b4474b0520d1383994e069f8057 | FormalisingMathematics2024/Solutions/Section07subgroupsAndHomomorphisms/Sheet1.lean | conjugate_top | [205, 1] | [215, 12] | use x⁻¹ * a * x | case h.mpr
G : Type
inst✝ : Group G
a✝ b : G
H K : Subgroup G
x y z a : G
h : a ∈ ⊤
⊢ ∃ h ∈ ⊤, a = x * h * x⁻¹ | case h
G : Type
inst✝ : Group G
a✝ b : G
H K : Subgroup G
x y z a : G
h : a ∈ ⊤
⊢ x⁻¹ * a * x ∈ ⊤ ∧ a = x * (x⁻¹ * a * x) * x⁻¹ | Please generate a tactic in lean4 to solve the state.
STATE:
case h.mpr
G : Type
inst✝ : Group G
a✝ b : G
H K : Subgroup G
x y z a : G
h : a ∈ ⊤
⊢ ∃ h ∈ ⊤, a = x * h * x⁻¹
TACTIC:
|
https://github.com/ImperialCollegeLondon/formalising-mathematics-2024.git | b732ed1352e87b4474b0520d1383994e069f8057 | FormalisingMathematics2024/Solutions/Section07subgroupsAndHomomorphisms/Sheet1.lean | conjugate_top | [205, 1] | [215, 12] | constructor | case h
G : Type
inst✝ : Group G
a✝ b : G
H K : Subgroup G
x y z a : G
h : a ∈ ⊤
⊢ x⁻¹ * a * x ∈ ⊤ ∧ a = x * (x⁻¹ * a * x) * x⁻¹ | case h.left
G : Type
inst✝ : Group G
a✝ b : G
H K : Subgroup G
x y z a : G
h : a ∈ ⊤
⊢ x⁻¹ * a * x ∈ ⊤
case h.right
G : Type
inst✝ : Group G
a✝ b : G
H K : Subgroup G
x y z a : G
h : a ∈ ⊤
⊢ a = x * (x⁻¹ * a * x) * x⁻¹ | Please generate a tactic in lean4 to solve the state.
STATE:
case h
G : Type
inst✝ : Group G
a✝ b : G
H K : Subgroup G
x y z a : G
h : a ∈ ⊤
⊢ x⁻¹ * a * x ∈ ⊤ ∧ a = x * (x⁻¹ * a * x) * x⁻¹
TACTIC:
|
https://github.com/ImperialCollegeLondon/formalising-mathematics-2024.git | b732ed1352e87b4474b0520d1383994e069f8057 | FormalisingMathematics2024/Solutions/Section07subgroupsAndHomomorphisms/Sheet1.lean | conjugate_top | [205, 1] | [215, 12] | apply Subgroup.mem_top | case h.left
G : Type
inst✝ : Group G
a✝ b : G
H K : Subgroup G
x y z a : G
h : a ∈ ⊤
⊢ x⁻¹ * a * x ∈ ⊤ | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case h.left
G : Type
inst✝ : Group G
a✝ b : G
H K : Subgroup G
x y z a : G
h : a ∈ ⊤
⊢ x⁻¹ * a * x ∈ ⊤
TACTIC:
|
https://github.com/ImperialCollegeLondon/formalising-mathematics-2024.git | b732ed1352e87b4474b0520d1383994e069f8057 | FormalisingMathematics2024/Solutions/Section07subgroupsAndHomomorphisms/Sheet1.lean | conjugate_top | [205, 1] | [215, 12] | group | case h.right
G : Type
inst✝ : Group G
a✝ b : G
H K : Subgroup G
x y z a : G
h : a ∈ ⊤
⊢ a = x * (x⁻¹ * a * x) * x⁻¹ | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case h.right
G : Type
inst✝ : Group G
a✝ b : G
H K : Subgroup G
x y z a : G
h : a ∈ ⊤
⊢ a = x * (x⁻¹ * a * x) * x⁻¹
TACTIC:
|
https://github.com/ImperialCollegeLondon/formalising-mathematics-2024.git | b732ed1352e87b4474b0520d1383994e069f8057 | FormalisingMathematics2024/Solutions/Section07subgroupsAndHomomorphisms/Sheet1.lean | conjugate_eq_of_abelian | [217, 1] | [229, 42] | ext a | G : Type
inst✝ : Group G
a b : G
H K : Subgroup G
x y z : G
habelian : ∀ (a b : G), a * b = b * a
⊢ conjugate H x = H | case h
G : Type
inst✝ : Group G
a✝ b : G
H K : Subgroup G
x y z : G
habelian : ∀ (a b : G), a * b = b * a
a : G
⊢ a ∈ conjugate H x ↔ a ∈ H | Please generate a tactic in lean4 to solve the state.
STATE:
G : Type
inst✝ : Group G
a b : G
H K : Subgroup G
x y z : G
habelian : ∀ (a b : G), a * b = b * a
⊢ conjugate H x = H
TACTIC:
|
https://github.com/ImperialCollegeLondon/formalising-mathematics-2024.git | b732ed1352e87b4474b0520d1383994e069f8057 | FormalisingMathematics2024/Solutions/Section07subgroupsAndHomomorphisms/Sheet1.lean | conjugate_eq_of_abelian | [217, 1] | [229, 42] | rw [mem_conjugate_iff] | case h
G : Type
inst✝ : Group G
a✝ b : G
H K : Subgroup G
x y z : G
habelian : ∀ (a b : G), a * b = b * a
a : G
⊢ a ∈ conjugate H x ↔ a ∈ H | case h
G : Type
inst✝ : Group G
a✝ b : G
H K : Subgroup G
x y z : G
habelian : ∀ (a b : G), a * b = b * a
a : G
⊢ (∃ h ∈ H, a = x * h * x⁻¹) ↔ a ∈ H | Please generate a tactic in lean4 to solve the state.
STATE:
case h
G : Type
inst✝ : Group G
a✝ b : G
H K : Subgroup G
x y z : G
habelian : ∀ (a b : G), a * b = b * a
a : G
⊢ a ∈ conjugate H x ↔ a ∈ H
TACTIC:
|
https://github.com/ImperialCollegeLondon/formalising-mathematics-2024.git | b732ed1352e87b4474b0520d1383994e069f8057 | FormalisingMathematics2024/Solutions/Section07subgroupsAndHomomorphisms/Sheet1.lean | conjugate_eq_of_abelian | [217, 1] | [229, 42] | constructor | case h
G : Type
inst✝ : Group G
a✝ b : G
H K : Subgroup G
x y z : G
habelian : ∀ (a b : G), a * b = b * a
a : G
⊢ (∃ h ∈ H, a = x * h * x⁻¹) ↔ a ∈ H | case h.mp
G : Type
inst✝ : Group G
a✝ b : G
H K : Subgroup G
x y z : G
habelian : ∀ (a b : G), a * b = b * a
a : G
⊢ (∃ h ∈ H, a = x * h * x⁻¹) → a ∈ H
case h.mpr
G : Type
inst✝ : Group G
a✝ b : G
H K : Subgroup G
x y z : G
habelian : ∀ (a b : G), a * b = b * a
a : G
⊢ a ∈ H → ∃ h ∈ H, a = x * h * x⁻¹ | Please generate a tactic in lean4 to solve the state.
STATE:
case h
G : Type
inst✝ : Group G
a✝ b : G
H K : Subgroup G
x y z : G
habelian : ∀ (a b : G), a * b = b * a
a : G
⊢ (∃ h ∈ H, a = x * h * x⁻¹) ↔ a ∈ H
TACTIC:
|
https://github.com/ImperialCollegeLondon/formalising-mathematics-2024.git | b732ed1352e87b4474b0520d1383994e069f8057 | FormalisingMathematics2024/Solutions/Section07subgroupsAndHomomorphisms/Sheet1.lean | conjugate_eq_of_abelian | [217, 1] | [229, 42] | rintro ⟨b, hb, rfl⟩ | case h.mp
G : Type
inst✝ : Group G
a✝ b : G
H K : Subgroup G
x y z : G
habelian : ∀ (a b : G), a * b = b * a
a : G
⊢ (∃ h ∈ H, a = x * h * x⁻¹) → a ∈ H | case h.mp.intro.intro
G : Type
inst✝ : Group G
a b✝ : G
H K : Subgroup G
x y z : G
habelian : ∀ (a b : G), a * b = b * a
b : G
hb : b ∈ H
⊢ x * b * x⁻¹ ∈ H | Please generate a tactic in lean4 to solve the state.
STATE:
case h.mp
G : Type
inst✝ : Group G
a✝ b : G
H K : Subgroup G
x y z : G
habelian : ∀ (a b : G), a * b = b * a
a : G
⊢ (∃ h ∈ H, a = x * h * x⁻¹) → a ∈ H
TACTIC:
|
https://github.com/ImperialCollegeLondon/formalising-mathematics-2024.git | b732ed1352e87b4474b0520d1383994e069f8057 | FormalisingMathematics2024/Solutions/Section07subgroupsAndHomomorphisms/Sheet1.lean | conjugate_eq_of_abelian | [217, 1] | [229, 42] | rw [habelian] | case h.mp.intro.intro
G : Type
inst✝ : Group G
a b✝ : G
H K : Subgroup G
x y z : G
habelian : ∀ (a b : G), a * b = b * a
b : G
hb : b ∈ H
⊢ x * b * x⁻¹ ∈ H | case h.mp.intro.intro
G : Type
inst✝ : Group G
a b✝ : G
H K : Subgroup G
x y z : G
habelian : ∀ (a b : G), a * b = b * a
b : G
hb : b ∈ H
⊢ x⁻¹ * (x * b) ∈ H | Please generate a tactic in lean4 to solve the state.
STATE:
case h.mp.intro.intro
G : Type
inst✝ : Group G
a b✝ : G
H K : Subgroup G
x y z : G
habelian : ∀ (a b : G), a * b = b * a
b : G
hb : b ∈ H
⊢ x * b * x⁻¹ ∈ H
TACTIC:
|
https://github.com/ImperialCollegeLondon/formalising-mathematics-2024.git | b732ed1352e87b4474b0520d1383994e069f8057 | FormalisingMathematics2024/Solutions/Section07subgroupsAndHomomorphisms/Sheet1.lean | conjugate_eq_of_abelian | [217, 1] | [229, 42] | rwa [show x⁻¹ * (x * b) = b by group] | case h.mp.intro.intro
G : Type
inst✝ : Group G
a b✝ : G
H K : Subgroup G
x y z : G
habelian : ∀ (a b : G), a * b = b * a
b : G
hb : b ∈ H
⊢ x⁻¹ * (x * b) ∈ H | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case h.mp.intro.intro
G : Type
inst✝ : Group G
a b✝ : G
H K : Subgroup G
x y z : G
habelian : ∀ (a b : G), a * b = b * a
b : G
hb : b ∈ H
⊢ x⁻¹ * (x * b) ∈ H
TACTIC:
|
https://github.com/ImperialCollegeLondon/formalising-mathematics-2024.git | b732ed1352e87b4474b0520d1383994e069f8057 | FormalisingMathematics2024/Solutions/Section07subgroupsAndHomomorphisms/Sheet1.lean | conjugate_eq_of_abelian | [217, 1] | [229, 42] | group | G : Type
inst✝ : Group G
a b✝ : G
H K : Subgroup G
x y z : G
habelian : ∀ (a b : G), a * b = b * a
b : G
hb : b ∈ H
⊢ x⁻¹ * (x * b) = b | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
G : Type
inst✝ : Group G
a b✝ : G
H K : Subgroup G
x y z : G
habelian : ∀ (a b : G), a * b = b * a
b : G
hb : b ∈ H
⊢ x⁻¹ * (x * b) = b
TACTIC:
|
https://github.com/ImperialCollegeLondon/formalising-mathematics-2024.git | b732ed1352e87b4474b0520d1383994e069f8057 | FormalisingMathematics2024/Solutions/Section07subgroupsAndHomomorphisms/Sheet1.lean | conjugate_eq_of_abelian | [217, 1] | [229, 42] | intro ha | case h.mpr
G : Type
inst✝ : Group G
a✝ b : G
H K : Subgroup G
x y z : G
habelian : ∀ (a b : G), a * b = b * a
a : G
⊢ a ∈ H → ∃ h ∈ H, a = x * h * x⁻¹ | case h.mpr
G : Type
inst✝ : Group G
a✝ b : G
H K : Subgroup G
x y z : G
habelian : ∀ (a b : G), a * b = b * a
a : G
ha : a ∈ H
⊢ ∃ h ∈ H, a = x * h * x⁻¹ | Please generate a tactic in lean4 to solve the state.
STATE:
case h.mpr
G : Type
inst✝ : Group G
a✝ b : G
H K : Subgroup G
x y z : G
habelian : ∀ (a b : G), a * b = b * a
a : G
⊢ a ∈ H → ∃ h ∈ H, a = x * h * x⁻¹
TACTIC:
|
https://github.com/ImperialCollegeLondon/formalising-mathematics-2024.git | b732ed1352e87b4474b0520d1383994e069f8057 | FormalisingMathematics2024/Solutions/Section07subgroupsAndHomomorphisms/Sheet1.lean | conjugate_eq_of_abelian | [217, 1] | [229, 42] | use x⁻¹ * a * x | case h.mpr
G : Type
inst✝ : Group G
a✝ b : G
H K : Subgroup G
x y z : G
habelian : ∀ (a b : G), a * b = b * a
a : G
ha : a ∈ H
⊢ ∃ h ∈ H, a = x * h * x⁻¹ | case h
G : Type
inst✝ : Group G
a✝ b : G
H K : Subgroup G
x y z : G
habelian : ∀ (a b : G), a * b = b * a
a : G
ha : a ∈ H
⊢ x⁻¹ * a * x ∈ H ∧ a = x * (x⁻¹ * a * x) * x⁻¹ | Please generate a tactic in lean4 to solve the state.
STATE:
case h.mpr
G : Type
inst✝ : Group G
a✝ b : G
H K : Subgroup G
x y z : G
habelian : ∀ (a b : G), a * b = b * a
a : G
ha : a ∈ H
⊢ ∃ h ∈ H, a = x * h * x⁻¹
TACTIC:
|
https://github.com/ImperialCollegeLondon/formalising-mathematics-2024.git | b732ed1352e87b4474b0520d1383994e069f8057 | FormalisingMathematics2024/Solutions/Section07subgroupsAndHomomorphisms/Sheet1.lean | conjugate_eq_of_abelian | [217, 1] | [229, 42] | refine' ⟨_, by group⟩ | case h
G : Type
inst✝ : Group G
a✝ b : G
H K : Subgroup G
x y z : G
habelian : ∀ (a b : G), a * b = b * a
a : G
ha : a ∈ H
⊢ x⁻¹ * a * x ∈ H ∧ a = x * (x⁻¹ * a * x) * x⁻¹ | case h
G : Type
inst✝ : Group G
a✝ b : G
H K : Subgroup G
x y z : G
habelian : ∀ (a b : G), a * b = b * a
a : G
ha : a ∈ H
⊢ x⁻¹ * a * x ∈ H | Please generate a tactic in lean4 to solve the state.
STATE:
case h
G : Type
inst✝ : Group G
a✝ b : G
H K : Subgroup G
x y z : G
habelian : ∀ (a b : G), a * b = b * a
a : G
ha : a ∈ H
⊢ x⁻¹ * a * x ∈ H ∧ a = x * (x⁻¹ * a * x) * x⁻¹
TACTIC:
|
https://github.com/ImperialCollegeLondon/formalising-mathematics-2024.git | b732ed1352e87b4474b0520d1383994e069f8057 | FormalisingMathematics2024/Solutions/Section07subgroupsAndHomomorphisms/Sheet1.lean | conjugate_eq_of_abelian | [217, 1] | [229, 42] | rw [habelian] | case h
G : Type
inst✝ : Group G
a✝ b : G
H K : Subgroup G
x y z : G
habelian : ∀ (a b : G), a * b = b * a
a : G
ha : a ∈ H
⊢ x⁻¹ * a * x ∈ H | case h
G : Type
inst✝ : Group G
a✝ b : G
H K : Subgroup G
x y z : G
habelian : ∀ (a b : G), a * b = b * a
a : G
ha : a ∈ H
⊢ x * (x⁻¹ * a) ∈ H | Please generate a tactic in lean4 to solve the state.
STATE:
case h
G : Type
inst✝ : Group G
a✝ b : G
H K : Subgroup G
x y z : G
habelian : ∀ (a b : G), a * b = b * a
a : G
ha : a ∈ H
⊢ x⁻¹ * a * x ∈ H
TACTIC:
|
https://github.com/ImperialCollegeLondon/formalising-mathematics-2024.git | b732ed1352e87b4474b0520d1383994e069f8057 | FormalisingMathematics2024/Solutions/Section07subgroupsAndHomomorphisms/Sheet1.lean | conjugate_eq_of_abelian | [217, 1] | [229, 42] | rwa [show x * (x⁻¹ * a) = a by group] | case h
G : Type
inst✝ : Group G
a✝ b : G
H K : Subgroup G
x y z : G
habelian : ∀ (a b : G), a * b = b * a
a : G
ha : a ∈ H
⊢ x * (x⁻¹ * a) ∈ H | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case h
G : Type
inst✝ : Group G
a✝ b : G
H K : Subgroup G
x y z : G
habelian : ∀ (a b : G), a * b = b * a
a : G
ha : a ∈ H
⊢ x * (x⁻¹ * a) ∈ H
TACTIC:
|
https://github.com/ImperialCollegeLondon/formalising-mathematics-2024.git | b732ed1352e87b4474b0520d1383994e069f8057 | FormalisingMathematics2024/Solutions/Section07subgroupsAndHomomorphisms/Sheet1.lean | conjugate_eq_of_abelian | [217, 1] | [229, 42] | group | G : Type
inst✝ : Group G
a✝ b : G
H K : Subgroup G
x y z : G
habelian : ∀ (a b : G), a * b = b * a
a : G
ha : a ∈ H
⊢ a = x * (x⁻¹ * a * x) * x⁻¹ | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
G : Type
inst✝ : Group G
a✝ b : G
H K : Subgroup G
x y z : G
habelian : ∀ (a b : G), a * b = b * a
a : G
ha : a ∈ H
⊢ a = x * (x⁻¹ * a * x) * x⁻¹
TACTIC:
|
https://github.com/ImperialCollegeLondon/formalising-mathematics-2024.git | b732ed1352e87b4474b0520d1383994e069f8057 | FormalisingMathematics2024/Solutions/Section07subgroupsAndHomomorphisms/Sheet1.lean | conjugate_eq_of_abelian | [217, 1] | [229, 42] | group | G : Type
inst✝ : Group G
a✝ b : G
H K : Subgroup G
x y z : G
habelian : ∀ (a b : G), a * b = b * a
a : G
ha : a ∈ H
⊢ x * (x⁻¹ * a) = a | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
G : Type
inst✝ : Group G
a✝ b : G
H K : Subgroup G
x y z : G
habelian : ∀ (a b : G), a * b = b * a
a : G
ha : a ∈ H
⊢ x * (x⁻¹ * a) = a
TACTIC:
|
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