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/Section02reals/Sheet5.lean | Section2sheet5.tendsTo_add | [31, 1] | [52, 13] | linarith | a b : ℕ → ℝ
t u : ℝ
hb : ∀ (ε : ℝ), 0 < ε → ∃ B, ∀ (n : ℕ), B ≤ n → |b n - u| < ε
ε : ℝ
hε : 0 < ε
X : ℕ
hX : ∀ (n : ℕ), X ≤ n → |a n - t| < ε / 2
⊢ 0 < ε / 2 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
a b : ℕ → ℝ
t u : ℝ
hb : ∀ (ε : ℝ), 0 < ε → ∃ B, ∀ (n : ℕ), B ≤ n → |b n - u| < ε
ε : ℝ
hε : 0 < ε
X : ℕ
hX : ∀ (n : ℕ), X ≤ n → |a n - t| < ε / 2
⊢ 0 < ε / 2
TACTIC:
|
https://github.com/ImperialCollegeLondon/formalising-mathematics-2024.git | b732ed1352e87b4474b0520d1383994e069f8057 | FormalisingMathematics2024/Section02reals/Sheet5.lean | Section2sheet5.tendsTo_sub | [56, 1] | [59, 8] | sorry | a b : ℕ → ℝ
t u : ℝ
ha : TendsTo a t
hb : TendsTo b u
⊢ TendsTo (fun n => a n - b n) (t - u) | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
a b : ℕ → ℝ
t u : ℝ
ha : TendsTo a t
hb : TendsTo b u
⊢ TendsTo (fun n => a n - b n) (t - u)
TACTIC:
|
https://github.com/ImperialCollegeLondon/formalising-mathematics-2024.git | b732ed1352e87b4474b0520d1383994e069f8057 | FormalisingMathematics2024/Section15numberTheory/Sheet3.lean | Section15Sheet3.infinite_setOf_solutions | [45, 1] | [48, 23] | rw [infinite_iff_arb_large] | ⊢ Set.Infinite {n | 5 ∣ 4 * n ^ 2 + 1 ∧ 13 ∣ 4 * n ^ 2 + 1} | ⊢ ∀ (N : ℕ), ∃ n > N, n ∈ {n | 5 ∣ 4 * n ^ 2 + 1 ∧ 13 ∣ 4 * n ^ 2 + 1} | Please generate a tactic in lean4 to solve the state.
STATE:
⊢ Set.Infinite {n | 5 ∣ 4 * n ^ 2 + 1 ∧ 13 ∣ 4 * n ^ 2 + 1}
TACTIC:
|
https://github.com/ImperialCollegeLondon/formalising-mathematics-2024.git | b732ed1352e87b4474b0520d1383994e069f8057 | FormalisingMathematics2024/Section15numberTheory/Sheet3.lean | Section15Sheet3.infinite_setOf_solutions | [45, 1] | [48, 23] | exact arb_large_soln | ⊢ ∀ (N : ℕ), ∃ n > N, n ∈ {n | 5 ∣ 4 * n ^ 2 + 1 ∧ 13 ∣ 4 * n ^ 2 + 1} | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
⊢ ∀ (N : ℕ), ∃ n > N, n ∈ {n | 5 ∣ 4 * n ^ 2 + 1 ∧ 13 ∣ 4 * n ^ 2 + 1}
TACTIC:
|
https://github.com/ImperialCollegeLondon/formalising-mathematics-2024.git | b732ed1352e87b4474b0520d1383994e069f8057 | FormalisingMathematics2024/Solutions/Section10TopologicalSpaces/Sheet1.lean | Section10sheet1Solutions.Real.isOpen_univ | [104, 1] | [107, 11] | intro x hx | X : Type
⊢ IsOpen Set.univ | X : Type
x : ℝ
hx : x ∈ Set.univ
⊢ ∃ δ > 0, ∀ (y : ℝ), x - δ < y ∧ y < x + δ → y ∈ Set.univ | Please generate a tactic in lean4 to solve the state.
STATE:
X : Type
⊢ IsOpen Set.univ
TACTIC:
|
https://github.com/ImperialCollegeLondon/formalising-mathematics-2024.git | b732ed1352e87b4474b0520d1383994e069f8057 | FormalisingMathematics2024/Solutions/Section10TopologicalSpaces/Sheet1.lean | Section10sheet1Solutions.Real.isOpen_univ | [104, 1] | [107, 11] | use 37 | X : Type
x : ℝ
hx : x ∈ Set.univ
⊢ ∃ δ > 0, ∀ (y : ℝ), x - δ < y ∧ y < x + δ → y ∈ Set.univ | case h
X : Type
x : ℝ
hx : x ∈ Set.univ
⊢ 37 > 0 ∧ ∀ (y : ℝ), x - 37 < y ∧ y < x + 37 → y ∈ Set.univ | Please generate a tactic in lean4 to solve the state.
STATE:
X : Type
x : ℝ
hx : x ∈ Set.univ
⊢ ∃ δ > 0, ∀ (y : ℝ), x - δ < y ∧ y < x + δ → y ∈ Set.univ
TACTIC:
|
https://github.com/ImperialCollegeLondon/formalising-mathematics-2024.git | b732ed1352e87b4474b0520d1383994e069f8057 | FormalisingMathematics2024/Solutions/Section10TopologicalSpaces/Sheet1.lean | Section10sheet1Solutions.Real.isOpen_univ | [104, 1] | [107, 11] | norm_num | case h
X : Type
x : ℝ
hx : x ∈ Set.univ
⊢ 37 > 0 ∧ ∀ (y : ℝ), x - 37 < y ∧ y < x + 37 → y ∈ Set.univ | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case h
X : Type
x : ℝ
hx : x ∈ Set.univ
⊢ 37 > 0 ∧ ∀ (y : ℝ), x - 37 < y ∧ y < x + 37 → y ∈ Set.univ
TACTIC:
|
https://github.com/ImperialCollegeLondon/formalising-mathematics-2024.git | b732ed1352e87b4474b0520d1383994e069f8057 | FormalisingMathematics2024/Solutions/Section10TopologicalSpaces/Sheet1.lean | Section10sheet1Solutions.Real.isOpen_inter | [111, 1] | [123, 29] | intro x hx | X : Type
s t : Set ℝ
hs : IsOpen s
ht : IsOpen t
⊢ IsOpen (s ∩ t) | X : Type
s t : Set ℝ
hs : IsOpen s
ht : IsOpen t
x : ℝ
hx : x ∈ s ∩ t
⊢ ∃ δ > 0, ∀ (y : ℝ), x - δ < y ∧ y < x + δ → y ∈ s ∩ t | Please generate a tactic in lean4 to solve the state.
STATE:
X : Type
s t : Set ℝ
hs : IsOpen s
ht : IsOpen t
⊢ IsOpen (s ∩ t)
TACTIC:
|
https://github.com/ImperialCollegeLondon/formalising-mathematics-2024.git | b732ed1352e87b4474b0520d1383994e069f8057 | FormalisingMathematics2024/Solutions/Section10TopologicalSpaces/Sheet1.lean | Section10sheet1Solutions.Real.isOpen_inter | [111, 1] | [123, 29] | obtain ⟨δs, δspos, hs⟩ := hs x (by aesop) | X : Type
s t : Set ℝ
hs : IsOpen s
ht : IsOpen t
x : ℝ
hx : x ∈ s ∩ t
⊢ ∃ δ > 0, ∀ (y : ℝ), x - δ < y ∧ y < x + δ → y ∈ s ∩ t | case intro.intro
X : Type
s t : Set ℝ
hs✝ : IsOpen s
ht : IsOpen t
x : ℝ
hx : x ∈ s ∩ t
δs : ℝ
δspos : δs > 0
hs : ∀ (y : ℝ), x - δs < y ∧ y < x + δs → y ∈ s
⊢ ∃ δ > 0, ∀ (y : ℝ), x - δ < y ∧ y < x + δ → y ∈ s ∩ t | Please generate a tactic in lean4 to solve the state.
STATE:
X : Type
s t : Set ℝ
hs : IsOpen s
ht : IsOpen t
x : ℝ
hx : x ∈ s ∩ t
⊢ ∃ δ > 0, ∀ (y : ℝ), x - δ < y ∧ y < x + δ → y ∈ s ∩ t
TACTIC:
|
https://github.com/ImperialCollegeLondon/formalising-mathematics-2024.git | b732ed1352e87b4474b0520d1383994e069f8057 | FormalisingMathematics2024/Solutions/Section10TopologicalSpaces/Sheet1.lean | Section10sheet1Solutions.Real.isOpen_inter | [111, 1] | [123, 29] | obtain ⟨δt, δtpos, ht⟩ := ht x (by aesop) | case intro.intro
X : Type
s t : Set ℝ
hs✝ : IsOpen s
ht : IsOpen t
x : ℝ
hx : x ∈ s ∩ t
δs : ℝ
δspos : δs > 0
hs : ∀ (y : ℝ), x - δs < y ∧ y < x + δs → y ∈ s
⊢ ∃ δ > 0, ∀ (y : ℝ), x - δ < y ∧ y < x + δ → y ∈ s ∩ t | case intro.intro.intro.intro
X : Type
s t : Set ℝ
hs✝ : IsOpen s
ht✝ : IsOpen t
x : ℝ
hx : x ∈ s ∩ t
δs : ℝ
δspos : δs > 0
hs : ∀ (y : ℝ), x - δs < y ∧ y < x + δs → y ∈ s
δt : ℝ
δtpos : δt > 0
ht : ∀ (y : ℝ), x - δt < y ∧ y < x + δt → y ∈ t
⊢ ∃ δ > 0, ∀ (y : ℝ), x - δ < y ∧ y < x + δ → y ∈ s ∩ t | Please generate a tactic in lean4 to solve the state.
STATE:
case intro.intro
X : Type
s t : Set ℝ
hs✝ : IsOpen s
ht : IsOpen t
x : ℝ
hx : x ∈ s ∩ t
δs : ℝ
δspos : δs > 0
hs : ∀ (y : ℝ), x - δs < y ∧ y < x + δs → y ∈ s
⊢ ∃ δ > 0, ∀ (y : ℝ), x - δ < y ∧ y < x + δ → y ∈ s ∩ t
TACTIC:
|
https://github.com/ImperialCollegeLondon/formalising-mathematics-2024.git | b732ed1352e87b4474b0520d1383994e069f8057 | FormalisingMathematics2024/Solutions/Section10TopologicalSpaces/Sheet1.lean | Section10sheet1Solutions.Real.isOpen_inter | [111, 1] | [123, 29] | use min δs δt, by positivity | case intro.intro.intro.intro
X : Type
s t : Set ℝ
hs✝ : IsOpen s
ht✝ : IsOpen t
x : ℝ
hx : x ∈ s ∩ t
δs : ℝ
δspos : δs > 0
hs : ∀ (y : ℝ), x - δs < y ∧ y < x + δs → y ∈ s
δt : ℝ
δtpos : δt > 0
ht : ∀ (y : ℝ), x - δt < y ∧ y < x + δt → y ∈ t
⊢ ∃ δ > 0, ∀ (y : ℝ), x - δ < y ∧ y < x + δ → y ∈ s ∩ t | case right
X : Type
s t : Set ℝ
hs✝ : IsOpen s
ht✝ : IsOpen t
x : ℝ
hx : x ∈ s ∩ t
δs : ℝ
δspos : δs > 0
hs : ∀ (y : ℝ), x - δs < y ∧ y < x + δs → y ∈ s
δt : ℝ
δtpos : δt > 0
ht : ∀ (y : ℝ), x - δt < y ∧ y < x + δt → y ∈ t
⊢ ∀ (y : ℝ), x - min δs δt < y ∧ y < x + min δs δt → y ∈ s ∩ t | Please generate a tactic in lean4 to solve the state.
STATE:
case intro.intro.intro.intro
X : Type
s t : Set ℝ
hs✝ : IsOpen s
ht✝ : IsOpen t
x : ℝ
hx : x ∈ s ∩ t
δs : ℝ
δspos : δs > 0
hs : ∀ (y : ℝ), x - δs < y ∧ y < x + δs → y ∈ s
δt : ℝ
δtpos : δt > 0
ht : ∀ (y : ℝ), x - δt < y ∧ y < x + δt → y ∈ t
⊢ ∃ δ > 0, ∀ (y : ℝ), x - δ < y ∧ y < x + δ → y ∈ s ∩ t
TACTIC:
|
https://github.com/ImperialCollegeLondon/formalising-mathematics-2024.git | b732ed1352e87b4474b0520d1383994e069f8057 | FormalisingMathematics2024/Solutions/Section10TopologicalSpaces/Sheet1.lean | Section10sheet1Solutions.Real.isOpen_inter | [111, 1] | [123, 29] | rintro y ⟨h1, h2⟩ | case right
X : Type
s t : Set ℝ
hs✝ : IsOpen s
ht✝ : IsOpen t
x : ℝ
hx : x ∈ s ∩ t
δs : ℝ
δspos : δs > 0
hs : ∀ (y : ℝ), x - δs < y ∧ y < x + δs → y ∈ s
δt : ℝ
δtpos : δt > 0
ht : ∀ (y : ℝ), x - δt < y ∧ y < x + δt → y ∈ t
⊢ ∀ (y : ℝ), x - min δs δt < y ∧ y < x + min δs δt → y ∈ s ∩ t | case right.intro
X : Type
s t : Set ℝ
hs✝ : IsOpen s
ht✝ : IsOpen t
x : ℝ
hx : x ∈ s ∩ t
δs : ℝ
δspos : δs > 0
hs : ∀ (y : ℝ), x - δs < y ∧ y < x + δs → y ∈ s
δt : ℝ
δtpos : δt > 0
ht : ∀ (y : ℝ), x - δt < y ∧ y < x + δt → y ∈ t
y : ℝ
h1 : x - min δs δt < y
h2 : y < x + min δs δt
⊢ y ∈ s ∩ t | Please generate a tactic in lean4 to solve the state.
STATE:
case right
X : Type
s t : Set ℝ
hs✝ : IsOpen s
ht✝ : IsOpen t
x : ℝ
hx : x ∈ s ∩ t
δs : ℝ
δspos : δs > 0
hs : ∀ (y : ℝ), x - δs < y ∧ y < x + δs → y ∈ s
δt : ℝ
δtpos : δt > 0
ht : ∀ (y : ℝ), x - δt < y ∧ y < x + δt → y ∈ t
⊢ ∀ (y : ℝ), x - min δs δt < y ∧ y < x + min δs δt → y ∈ s ∩ t
TACTIC:
|
https://github.com/ImperialCollegeLondon/formalising-mathematics-2024.git | b732ed1352e87b4474b0520d1383994e069f8057 | FormalisingMathematics2024/Solutions/Section10TopologicalSpaces/Sheet1.lean | Section10sheet1Solutions.Real.isOpen_inter | [111, 1] | [123, 29] | constructor | case right.intro
X : Type
s t : Set ℝ
hs✝ : IsOpen s
ht✝ : IsOpen t
x : ℝ
hx : x ∈ s ∩ t
δs : ℝ
δspos : δs > 0
hs : ∀ (y : ℝ), x - δs < y ∧ y < x + δs → y ∈ s
δt : ℝ
δtpos : δt > 0
ht : ∀ (y : ℝ), x - δt < y ∧ y < x + δt → y ∈ t
y : ℝ
h1 : x - min δs δt < y
h2 : y < x + min δs δt
⊢ y ∈ s ∩ t | case right.intro.left
X : Type
s t : Set ℝ
hs✝ : IsOpen s
ht✝ : IsOpen t
x : ℝ
hx : x ∈ s ∩ t
δs : ℝ
δspos : δs > 0
hs : ∀ (y : ℝ), x - δs < y ∧ y < x + δs → y ∈ s
δt : ℝ
δtpos : δt > 0
ht : ∀ (y : ℝ), x - δt < y ∧ y < x + δt → y ∈ t
y : ℝ
h1 : x - min δs δt < y
h2 : y < x + min δs δt
⊢ y ∈ s
case right.intro.right
X : Type
s t : Set ℝ
hs✝ : IsOpen s
ht✝ : IsOpen t
x : ℝ
hx : x ∈ s ∩ t
δs : ℝ
δspos : δs > 0
hs : ∀ (y : ℝ), x - δs < y ∧ y < x + δs → y ∈ s
δt : ℝ
δtpos : δt > 0
ht : ∀ (y : ℝ), x - δt < y ∧ y < x + δt → y ∈ t
y : ℝ
h1 : x - min δs δt < y
h2 : y < x + min δs δt
⊢ y ∈ t | Please generate a tactic in lean4 to solve the state.
STATE:
case right.intro
X : Type
s t : Set ℝ
hs✝ : IsOpen s
ht✝ : IsOpen t
x : ℝ
hx : x ∈ s ∩ t
δs : ℝ
δspos : δs > 0
hs : ∀ (y : ℝ), x - δs < y ∧ y < x + δs → y ∈ s
δt : ℝ
δtpos : δt > 0
ht : ∀ (y : ℝ), x - δt < y ∧ y < x + δt → y ∈ t
y : ℝ
h1 : x - min δs δt < y
h2 : y < x + min δs δt
⊢ y ∈ s ∩ t
TACTIC:
|
https://github.com/ImperialCollegeLondon/formalising-mathematics-2024.git | b732ed1352e87b4474b0520d1383994e069f8057 | FormalisingMathematics2024/Solutions/Section10TopologicalSpaces/Sheet1.lean | Section10sheet1Solutions.Real.isOpen_inter | [111, 1] | [123, 29] | aesop | X : Type
s t : Set ℝ
hs : IsOpen s
ht : IsOpen t
x : ℝ
hx : x ∈ s ∩ t
⊢ x ∈ s | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
X : Type
s t : Set ℝ
hs : IsOpen s
ht : IsOpen t
x : ℝ
hx : x ∈ s ∩ t
⊢ x ∈ s
TACTIC:
|
https://github.com/ImperialCollegeLondon/formalising-mathematics-2024.git | b732ed1352e87b4474b0520d1383994e069f8057 | FormalisingMathematics2024/Solutions/Section10TopologicalSpaces/Sheet1.lean | Section10sheet1Solutions.Real.isOpen_inter | [111, 1] | [123, 29] | aesop | X : Type
s t : Set ℝ
hs✝ : IsOpen s
ht : IsOpen t
x : ℝ
hx : x ∈ s ∩ t
δs : ℝ
δspos : δs > 0
hs : ∀ (y : ℝ), x - δs < y ∧ y < x + δs → y ∈ s
⊢ x ∈ t | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
X : Type
s t : Set ℝ
hs✝ : IsOpen s
ht : IsOpen t
x : ℝ
hx : x ∈ s ∩ t
δs : ℝ
δspos : δs > 0
hs : ∀ (y : ℝ), x - δs < y ∧ y < x + δs → y ∈ s
⊢ x ∈ t
TACTIC:
|
https://github.com/ImperialCollegeLondon/formalising-mathematics-2024.git | b732ed1352e87b4474b0520d1383994e069f8057 | FormalisingMathematics2024/Solutions/Section10TopologicalSpaces/Sheet1.lean | Section10sheet1Solutions.Real.isOpen_inter | [111, 1] | [123, 29] | positivity | X : Type
s t : Set ℝ
hs✝ : IsOpen s
ht✝ : IsOpen t
x : ℝ
hx : x ∈ s ∩ t
δs : ℝ
δspos : δs > 0
hs : ∀ (y : ℝ), x - δs < y ∧ y < x + δs → y ∈ s
δt : ℝ
δtpos : δt > 0
ht : ∀ (y : ℝ), x - δt < y ∧ y < x + δt → y ∈ t
⊢ min δs δt > 0 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
X : Type
s t : Set ℝ
hs✝ : IsOpen s
ht✝ : IsOpen t
x : ℝ
hx : x ∈ s ∩ t
δs : ℝ
δspos : δs > 0
hs : ∀ (y : ℝ), x - δs < y ∧ y < x + δs → y ∈ s
δt : ℝ
δtpos : δt > 0
ht : ∀ (y : ℝ), x - δt < y ∧ y < x + δt → y ∈ t
⊢ min δs δt > 0
TACTIC:
|
https://github.com/ImperialCollegeLondon/formalising-mathematics-2024.git | b732ed1352e87b4474b0520d1383994e069f8057 | FormalisingMathematics2024/Solutions/Section10TopologicalSpaces/Sheet1.lean | Section10sheet1Solutions.Real.isOpen_inter | [111, 1] | [123, 29] | apply hs | case right.intro.left
X : Type
s t : Set ℝ
hs✝ : IsOpen s
ht✝ : IsOpen t
x : ℝ
hx : x ∈ s ∩ t
δs : ℝ
δspos : δs > 0
hs : ∀ (y : ℝ), x - δs < y ∧ y < x + δs → y ∈ s
δt : ℝ
δtpos : δt > 0
ht : ∀ (y : ℝ), x - δt < y ∧ y < x + δt → y ∈ t
y : ℝ
h1 : x - min δs δt < y
h2 : y < x + min δs δt
⊢ y ∈ s | case right.intro.left.a
X : Type
s t : Set ℝ
hs✝ : IsOpen s
ht✝ : IsOpen t
x : ℝ
hx : x ∈ s ∩ t
δs : ℝ
δspos : δs > 0
hs : ∀ (y : ℝ), x - δs < y ∧ y < x + δs → y ∈ s
δt : ℝ
δtpos : δt > 0
ht : ∀ (y : ℝ), x - δt < y ∧ y < x + δt → y ∈ t
y : ℝ
h1 : x - min δs δt < y
h2 : y < x + min δs δt
⊢ x - δs < y ∧ y < x + δs | Please generate a tactic in lean4 to solve the state.
STATE:
case right.intro.left
X : Type
s t : Set ℝ
hs✝ : IsOpen s
ht✝ : IsOpen t
x : ℝ
hx : x ∈ s ∩ t
δs : ℝ
δspos : δs > 0
hs : ∀ (y : ℝ), x - δs < y ∧ y < x + δs → y ∈ s
δt : ℝ
δtpos : δt > 0
ht : ∀ (y : ℝ), x - δt < y ∧ y < x + δt → y ∈ t
y : ℝ
h1 : x - min δs δt < y
h2 : y < x + min δs δt
⊢ y ∈ s
TACTIC:
|
https://github.com/ImperialCollegeLondon/formalising-mathematics-2024.git | b732ed1352e87b4474b0520d1383994e069f8057 | FormalisingMathematics2024/Solutions/Section10TopologicalSpaces/Sheet1.lean | Section10sheet1Solutions.Real.isOpen_inter | [111, 1] | [123, 29] | have foo : min δs δt ≤ δs := min_le_left δs δt | case right.intro.left.a
X : Type
s t : Set ℝ
hs✝ : IsOpen s
ht✝ : IsOpen t
x : ℝ
hx : x ∈ s ∩ t
δs : ℝ
δspos : δs > 0
hs : ∀ (y : ℝ), x - δs < y ∧ y < x + δs → y ∈ s
δt : ℝ
δtpos : δt > 0
ht : ∀ (y : ℝ), x - δt < y ∧ y < x + δt → y ∈ t
y : ℝ
h1 : x - min δs δt < y
h2 : y < x + min δs δt
⊢ x - δs < y ∧ y < x + δs | case right.intro.left.a
X : Type
s t : Set ℝ
hs✝ : IsOpen s
ht✝ : IsOpen t
x : ℝ
hx : x ∈ s ∩ t
δs : ℝ
δspos : δs > 0
hs : ∀ (y : ℝ), x - δs < y ∧ y < x + δs → y ∈ s
δt : ℝ
δtpos : δt > 0
ht : ∀ (y : ℝ), x - δt < y ∧ y < x + δt → y ∈ t
y : ℝ
h1 : x - min δs δt < y
h2 : y < x + min δs δt
foo : min δs δt ≤ δs
⊢ x - δs < y ∧ y < x + δs | Please generate a tactic in lean4 to solve the state.
STATE:
case right.intro.left.a
X : Type
s t : Set ℝ
hs✝ : IsOpen s
ht✝ : IsOpen t
x : ℝ
hx : x ∈ s ∩ t
δs : ℝ
δspos : δs > 0
hs : ∀ (y : ℝ), x - δs < y ∧ y < x + δs → y ∈ s
δt : ℝ
δtpos : δt > 0
ht : ∀ (y : ℝ), x - δt < y ∧ y < x + δt → y ∈ t
y : ℝ
h1 : x - min δs δt < y
h2 : y < x + min δs δt
⊢ x - δs < y ∧ y < x + δs
TACTIC:
|
https://github.com/ImperialCollegeLondon/formalising-mathematics-2024.git | b732ed1352e87b4474b0520d1383994e069f8057 | FormalisingMathematics2024/Solutions/Section10TopologicalSpaces/Sheet1.lean | Section10sheet1Solutions.Real.isOpen_inter | [111, 1] | [123, 29] | constructor <;> linarith | case right.intro.left.a
X : Type
s t : Set ℝ
hs✝ : IsOpen s
ht✝ : IsOpen t
x : ℝ
hx : x ∈ s ∩ t
δs : ℝ
δspos : δs > 0
hs : ∀ (y : ℝ), x - δs < y ∧ y < x + δs → y ∈ s
δt : ℝ
δtpos : δt > 0
ht : ∀ (y : ℝ), x - δt < y ∧ y < x + δt → y ∈ t
y : ℝ
h1 : x - min δs δt < y
h2 : y < x + min δs δt
foo : min δs δt ≤ δs
⊢ x - δs < y ∧ y < x + δs | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case right.intro.left.a
X : Type
s t : Set ℝ
hs✝ : IsOpen s
ht✝ : IsOpen t
x : ℝ
hx : x ∈ s ∩ t
δs : ℝ
δspos : δs > 0
hs : ∀ (y : ℝ), x - δs < y ∧ y < x + δs → y ∈ s
δt : ℝ
δtpos : δt > 0
ht : ∀ (y : ℝ), x - δt < y ∧ y < x + δt → y ∈ t
y : ℝ
h1 : x - min δs δt < y
h2 : y < x + min δs δt
foo : min δs δt ≤ δs
⊢ x - δs < y ∧ y < x + δs
TACTIC:
|
https://github.com/ImperialCollegeLondon/formalising-mathematics-2024.git | b732ed1352e87b4474b0520d1383994e069f8057 | FormalisingMathematics2024/Solutions/Section10TopologicalSpaces/Sheet1.lean | Section10sheet1Solutions.Real.isOpen_inter | [111, 1] | [123, 29] | apply ht | case right.intro.right
X : Type
s t : Set ℝ
hs✝ : IsOpen s
ht✝ : IsOpen t
x : ℝ
hx : x ∈ s ∩ t
δs : ℝ
δspos : δs > 0
hs : ∀ (y : ℝ), x - δs < y ∧ y < x + δs → y ∈ s
δt : ℝ
δtpos : δt > 0
ht : ∀ (y : ℝ), x - δt < y ∧ y < x + δt → y ∈ t
y : ℝ
h1 : x - min δs δt < y
h2 : y < x + min δs δt
⊢ y ∈ t | case right.intro.right.a
X : Type
s t : Set ℝ
hs✝ : IsOpen s
ht✝ : IsOpen t
x : ℝ
hx : x ∈ s ∩ t
δs : ℝ
δspos : δs > 0
hs : ∀ (y : ℝ), x - δs < y ∧ y < x + δs → y ∈ s
δt : ℝ
δtpos : δt > 0
ht : ∀ (y : ℝ), x - δt < y ∧ y < x + δt → y ∈ t
y : ℝ
h1 : x - min δs δt < y
h2 : y < x + min δs δt
⊢ x - δt < y ∧ y < x + δt | Please generate a tactic in lean4 to solve the state.
STATE:
case right.intro.right
X : Type
s t : Set ℝ
hs✝ : IsOpen s
ht✝ : IsOpen t
x : ℝ
hx : x ∈ s ∩ t
δs : ℝ
δspos : δs > 0
hs : ∀ (y : ℝ), x - δs < y ∧ y < x + δs → y ∈ s
δt : ℝ
δtpos : δt > 0
ht : ∀ (y : ℝ), x - δt < y ∧ y < x + δt → y ∈ t
y : ℝ
h1 : x - min δs δt < y
h2 : y < x + min δs δt
⊢ y ∈ t
TACTIC:
|
https://github.com/ImperialCollegeLondon/formalising-mathematics-2024.git | b732ed1352e87b4474b0520d1383994e069f8057 | FormalisingMathematics2024/Solutions/Section10TopologicalSpaces/Sheet1.lean | Section10sheet1Solutions.Real.isOpen_inter | [111, 1] | [123, 29] | have foo : min δs δt ≤ δt := min_le_right δs δt | case right.intro.right.a
X : Type
s t : Set ℝ
hs✝ : IsOpen s
ht✝ : IsOpen t
x : ℝ
hx : x ∈ s ∩ t
δs : ℝ
δspos : δs > 0
hs : ∀ (y : ℝ), x - δs < y ∧ y < x + δs → y ∈ s
δt : ℝ
δtpos : δt > 0
ht : ∀ (y : ℝ), x - δt < y ∧ y < x + δt → y ∈ t
y : ℝ
h1 : x - min δs δt < y
h2 : y < x + min δs δt
⊢ x - δt < y ∧ y < x + δt | case right.intro.right.a
X : Type
s t : Set ℝ
hs✝ : IsOpen s
ht✝ : IsOpen t
x : ℝ
hx : x ∈ s ∩ t
δs : ℝ
δspos : δs > 0
hs : ∀ (y : ℝ), x - δs < y ∧ y < x + δs → y ∈ s
δt : ℝ
δtpos : δt > 0
ht : ∀ (y : ℝ), x - δt < y ∧ y < x + δt → y ∈ t
y : ℝ
h1 : x - min δs δt < y
h2 : y < x + min δs δt
foo : min δs δt ≤ δt
⊢ x - δt < y ∧ y < x + δt | Please generate a tactic in lean4 to solve the state.
STATE:
case right.intro.right.a
X : Type
s t : Set ℝ
hs✝ : IsOpen s
ht✝ : IsOpen t
x : ℝ
hx : x ∈ s ∩ t
δs : ℝ
δspos : δs > 0
hs : ∀ (y : ℝ), x - δs < y ∧ y < x + δs → y ∈ s
δt : ℝ
δtpos : δt > 0
ht : ∀ (y : ℝ), x - δt < y ∧ y < x + δt → y ∈ t
y : ℝ
h1 : x - min δs δt < y
h2 : y < x + min δs δt
⊢ x - δt < y ∧ y < x + δt
TACTIC:
|
https://github.com/ImperialCollegeLondon/formalising-mathematics-2024.git | b732ed1352e87b4474b0520d1383994e069f8057 | FormalisingMathematics2024/Solutions/Section10TopologicalSpaces/Sheet1.lean | Section10sheet1Solutions.Real.isOpen_inter | [111, 1] | [123, 29] | constructor <;> linarith | case right.intro.right.a
X : Type
s t : Set ℝ
hs✝ : IsOpen s
ht✝ : IsOpen t
x : ℝ
hx : x ∈ s ∩ t
δs : ℝ
δspos : δs > 0
hs : ∀ (y : ℝ), x - δs < y ∧ y < x + δs → y ∈ s
δt : ℝ
δtpos : δt > 0
ht : ∀ (y : ℝ), x - δt < y ∧ y < x + δt → y ∈ t
y : ℝ
h1 : x - min δs δt < y
h2 : y < x + min δs δt
foo : min δs δt ≤ δt
⊢ x - δt < y ∧ y < x + δt | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case right.intro.right.a
X : Type
s t : Set ℝ
hs✝ : IsOpen s
ht✝ : IsOpen t
x : ℝ
hx : x ∈ s ∩ t
δs : ℝ
δspos : δs > 0
hs : ∀ (y : ℝ), x - δs < y ∧ y < x + δs → y ∈ s
δt : ℝ
δtpos : δt > 0
ht : ∀ (y : ℝ), x - δt < y ∧ y < x + δt → y ∈ t
y : ℝ
h1 : x - min δs δt < y
h2 : y < x + min δs δt
foo : min δs δt ≤ δt
⊢ x - δt < y ∧ y < x + δt
TACTIC:
|
https://github.com/ImperialCollegeLondon/formalising-mathematics-2024.git | b732ed1352e87b4474b0520d1383994e069f8057 | FormalisingMathematics2024/Solutions/Section10TopologicalSpaces/Sheet1.lean | Section10sheet1Solutions.Real.isOpen_sUnion | [125, 1] | [132, 13] | intro x hx | X : Type
F : Set (Set ℝ)
hF : ∀ s ∈ F, IsOpen s
⊢ IsOpen (⋃₀ F) | X : Type
F : Set (Set ℝ)
hF : ∀ s ∈ F, IsOpen s
x : ℝ
hx : x ∈ ⋃₀ F
⊢ ∃ δ > 0, ∀ (y : ℝ), x - δ < y ∧ y < x + δ → y ∈ ⋃₀ F | Please generate a tactic in lean4 to solve the state.
STATE:
X : Type
F : Set (Set ℝ)
hF : ∀ s ∈ F, IsOpen s
⊢ IsOpen (⋃₀ F)
TACTIC:
|
https://github.com/ImperialCollegeLondon/formalising-mathematics-2024.git | b732ed1352e87b4474b0520d1383994e069f8057 | FormalisingMathematics2024/Solutions/Section10TopologicalSpaces/Sheet1.lean | Section10sheet1Solutions.Real.isOpen_sUnion | [125, 1] | [132, 13] | simp_rw [Set.mem_sUnion] at hx ⊢ | X : Type
F : Set (Set ℝ)
hF : ∀ s ∈ F, IsOpen s
x : ℝ
hx : x ∈ ⋃₀ F
⊢ ∃ δ > 0, ∀ (y : ℝ), x - δ < y ∧ y < x + δ → y ∈ ⋃₀ F | X : Type
F : Set (Set ℝ)
hF : ∀ s ∈ F, IsOpen s
x : ℝ
hx : ∃ t ∈ F, x ∈ t
⊢ ∃ δ > 0, ∀ (y : ℝ), x - δ < y ∧ y < x + δ → ∃ t ∈ F, y ∈ t | Please generate a tactic in lean4 to solve the state.
STATE:
X : Type
F : Set (Set ℝ)
hF : ∀ s ∈ F, IsOpen s
x : ℝ
hx : x ∈ ⋃₀ F
⊢ ∃ δ > 0, ∀ (y : ℝ), x - δ < y ∧ y < x + δ → y ∈ ⋃₀ F
TACTIC:
|
https://github.com/ImperialCollegeLondon/formalising-mathematics-2024.git | b732ed1352e87b4474b0520d1383994e069f8057 | FormalisingMathematics2024/Solutions/Section10TopologicalSpaces/Sheet1.lean | Section10sheet1Solutions.Real.isOpen_sUnion | [125, 1] | [132, 13] | rcases hx with ⟨t, htF, hxt⟩ | X : Type
F : Set (Set ℝ)
hF : ∀ s ∈ F, IsOpen s
x : ℝ
hx : ∃ t ∈ F, x ∈ t
⊢ ∃ δ > 0, ∀ (y : ℝ), x - δ < y ∧ y < x + δ → ∃ t ∈ F, y ∈ t | case intro.intro
X : Type
F : Set (Set ℝ)
hF : ∀ s ∈ F, IsOpen s
x : ℝ
t : Set ℝ
htF : t ∈ F
hxt : x ∈ t
⊢ ∃ δ > 0, ∀ (y : ℝ), x - δ < y ∧ y < x + δ → ∃ t ∈ F, y ∈ t | Please generate a tactic in lean4 to solve the state.
STATE:
X : Type
F : Set (Set ℝ)
hF : ∀ s ∈ F, IsOpen s
x : ℝ
hx : ∃ t ∈ F, x ∈ t
⊢ ∃ δ > 0, ∀ (y : ℝ), x - δ < y ∧ y < x + δ → ∃ t ∈ F, y ∈ t
TACTIC:
|
https://github.com/ImperialCollegeLondon/formalising-mathematics-2024.git | b732ed1352e87b4474b0520d1383994e069f8057 | FormalisingMathematics2024/Solutions/Section10TopologicalSpaces/Sheet1.lean | Section10sheet1Solutions.Real.isOpen_sUnion | [125, 1] | [132, 13] | obtain ⟨δ, hδpos, h⟩ := hF t htF x hxt | case intro.intro
X : Type
F : Set (Set ℝ)
hF : ∀ s ∈ F, IsOpen s
x : ℝ
t : Set ℝ
htF : t ∈ F
hxt : x ∈ t
⊢ ∃ δ > 0, ∀ (y : ℝ), x - δ < y ∧ y < x + δ → ∃ t ∈ F, y ∈ t | case intro.intro.intro.intro
X : Type
F : Set (Set ℝ)
hF : ∀ s ∈ F, IsOpen s
x : ℝ
t : Set ℝ
htF : t ∈ F
hxt : x ∈ t
δ : ℝ
hδpos : δ > 0
h : ∀ (y : ℝ), x - δ < y ∧ y < x + δ → y ∈ t
⊢ ∃ δ > 0, ∀ (y : ℝ), x - δ < y ∧ y < x + δ → ∃ t ∈ F, y ∈ t | Please generate a tactic in lean4 to solve the state.
STATE:
case intro.intro
X : Type
F : Set (Set ℝ)
hF : ∀ s ∈ F, IsOpen s
x : ℝ
t : Set ℝ
htF : t ∈ F
hxt : x ∈ t
⊢ ∃ δ > 0, ∀ (y : ℝ), x - δ < y ∧ y < x + δ → ∃ t ∈ F, y ∈ t
TACTIC:
|
https://github.com/ImperialCollegeLondon/formalising-mathematics-2024.git | b732ed1352e87b4474b0520d1383994e069f8057 | FormalisingMathematics2024/Solutions/Section10TopologicalSpaces/Sheet1.lean | Section10sheet1Solutions.Real.isOpen_sUnion | [125, 1] | [132, 13] | use δ, hδpos | case intro.intro.intro.intro
X : Type
F : Set (Set ℝ)
hF : ∀ s ∈ F, IsOpen s
x : ℝ
t : Set ℝ
htF : t ∈ F
hxt : x ∈ t
δ : ℝ
hδpos : δ > 0
h : ∀ (y : ℝ), x - δ < y ∧ y < x + δ → y ∈ t
⊢ ∃ δ > 0, ∀ (y : ℝ), x - δ < y ∧ y < x + δ → ∃ t ∈ F, y ∈ t | case right
X : Type
F : Set (Set ℝ)
hF : ∀ s ∈ F, IsOpen s
x : ℝ
t : Set ℝ
htF : t ∈ F
hxt : x ∈ t
δ : ℝ
hδpos : δ > 0
h : ∀ (y : ℝ), x - δ < y ∧ y < x + δ → y ∈ t
⊢ ∀ (y : ℝ), x - δ < y ∧ y < x + δ → ∃ t ∈ F, y ∈ t | Please generate a tactic in lean4 to solve the state.
STATE:
case intro.intro.intro.intro
X : Type
F : Set (Set ℝ)
hF : ∀ s ∈ F, IsOpen s
x : ℝ
t : Set ℝ
htF : t ∈ F
hxt : x ∈ t
δ : ℝ
hδpos : δ > 0
h : ∀ (y : ℝ), x - δ < y ∧ y < x + δ → y ∈ t
⊢ ∃ δ > 0, ∀ (y : ℝ), x - δ < y ∧ y < x + δ → ∃ t ∈ F, y ∈ t
TACTIC:
|
https://github.com/ImperialCollegeLondon/formalising-mathematics-2024.git | b732ed1352e87b4474b0520d1383994e069f8057 | FormalisingMathematics2024/Solutions/Section10TopologicalSpaces/Sheet1.lean | Section10sheet1Solutions.Real.isOpen_sUnion | [125, 1] | [132, 13] | peel h with h1 y hyt | case right
X : Type
F : Set (Set ℝ)
hF : ∀ s ∈ F, IsOpen s
x : ℝ
t : Set ℝ
htF : t ∈ F
hxt : x ∈ t
δ : ℝ
hδpos : δ > 0
h : ∀ (y : ℝ), x - δ < y ∧ y < x + δ → y ∈ t
⊢ ∀ (y : ℝ), x - δ < y ∧ y < x + δ → ∃ t ∈ F, y ∈ t | case right.h.h
X : Type
F : Set (Set ℝ)
hF : ∀ s ∈ F, IsOpen s
x : ℝ
t : Set ℝ
htF : t ∈ F
hxt : x ∈ t
δ : ℝ
hδpos : δ > 0
h : ∀ (y : ℝ), x - δ < y ∧ y < x + δ → y ∈ t
y : ℝ
hyt : x - δ < y ∧ y < x + δ
h1 : y ∈ t
⊢ ∃ t ∈ F, y ∈ t | Please generate a tactic in lean4 to solve the state.
STATE:
case right
X : Type
F : Set (Set ℝ)
hF : ∀ s ∈ F, IsOpen s
x : ℝ
t : Set ℝ
htF : t ∈ F
hxt : x ∈ t
δ : ℝ
hδpos : δ > 0
h : ∀ (y : ℝ), x - δ < y ∧ y < x + δ → y ∈ t
⊢ ∀ (y : ℝ), x - δ < y ∧ y < x + δ → ∃ t ∈ F, y ∈ t
TACTIC:
|
https://github.com/ImperialCollegeLondon/formalising-mathematics-2024.git | b732ed1352e87b4474b0520d1383994e069f8057 | FormalisingMathematics2024/Solutions/Section10TopologicalSpaces/Sheet1.lean | Section10sheet1Solutions.Real.isOpen_sUnion | [125, 1] | [132, 13] | use t, htF | case right.h.h
X : Type
F : Set (Set ℝ)
hF : ∀ s ∈ F, IsOpen s
x : ℝ
t : Set ℝ
htF : t ∈ F
hxt : x ∈ t
δ : ℝ
hδpos : δ > 0
h : ∀ (y : ℝ), x - δ < y ∧ y < x + δ → y ∈ t
y : ℝ
hyt : x - δ < y ∧ y < x + δ
h1 : y ∈ t
⊢ ∃ t ∈ F, y ∈ t | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case right.h.h
X : Type
F : Set (Set ℝ)
hF : ∀ s ∈ F, IsOpen s
x : ℝ
t : Set ℝ
htF : t ∈ F
hxt : x ∈ t
δ : ℝ
hδpos : δ > 0
h : ∀ (y : ℝ), x - δ < y ∧ y < x + δ → y ∈ t
y : ℝ
hyt : x - δ < y ∧ y < x + δ
h1 : y ∈ t
⊢ ∃ t ∈ F, y ∈ t
TACTIC:
|
https://github.com/ImperialCollegeLondon/formalising-mathematics-2024.git | b732ed1352e87b4474b0520d1383994e069f8057 | FormalisingMathematics2024/Section02reals/Sheet3.lean | Section2sheet3.tendsTo_def | [60, 1] | [62, 6] | rfl | a : ℕ → ℝ
t : ℝ
⊢ TendsTo a t ↔ ∀ (ε : ℝ), 0 < ε → ∃ B, ∀ (n : ℕ), B ≤ n → |a n - t| < ε | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
a : ℕ → ℝ
t : ℝ
⊢ TendsTo a t ↔ ∀ (ε : ℝ), 0 < ε → ∃ B, ∀ (n : ℕ), B ≤ n → |a n - t| < ε
TACTIC:
|
https://github.com/ImperialCollegeLondon/formalising-mathematics-2024.git | b732ed1352e87b4474b0520d1383994e069f8057 | FormalisingMathematics2024/Section02reals/Sheet3.lean | Section2sheet3.tendsTo_thirtyseven | [77, 1] | [84, 11] | rw [tendsTo_def] | ⊢ TendsTo (fun n => 37) 37 | ⊢ ∀ (ε : ℝ), 0 < ε → ∃ B, ∀ (n : ℕ), B ≤ n → |37 - 37| < ε | Please generate a tactic in lean4 to solve the state.
STATE:
⊢ TendsTo (fun n => 37) 37
TACTIC:
|
https://github.com/ImperialCollegeLondon/formalising-mathematics-2024.git | b732ed1352e87b4474b0520d1383994e069f8057 | FormalisingMathematics2024/Section02reals/Sheet3.lean | Section2sheet3.tendsTo_thirtyseven | [77, 1] | [84, 11] | intro ε hε | ⊢ ∀ (ε : ℝ), 0 < ε → ∃ B, ∀ (n : ℕ), B ≤ n → |37 - 37| < ε | ε : ℝ
hε : 0 < ε
⊢ ∃ B, ∀ (n : ℕ), B ≤ n → |37 - 37| < ε | Please generate a tactic in lean4 to solve the state.
STATE:
⊢ ∀ (ε : ℝ), 0 < ε → ∃ B, ∀ (n : ℕ), B ≤ n → |37 - 37| < ε
TACTIC:
|
https://github.com/ImperialCollegeLondon/formalising-mathematics-2024.git | b732ed1352e87b4474b0520d1383994e069f8057 | FormalisingMathematics2024/Section02reals/Sheet3.lean | Section2sheet3.tendsTo_thirtyseven | [77, 1] | [84, 11] | use 100 | ε : ℝ
hε : 0 < ε
⊢ ∃ B, ∀ (n : ℕ), B ≤ n → |37 - 37| < ε | case h
ε : ℝ
hε : 0 < ε
⊢ ∀ (n : ℕ), 100 ≤ n → |37 - 37| < ε | Please generate a tactic in lean4 to solve the state.
STATE:
ε : ℝ
hε : 0 < ε
⊢ ∃ B, ∀ (n : ℕ), B ≤ n → |37 - 37| < ε
TACTIC:
|
https://github.com/ImperialCollegeLondon/formalising-mathematics-2024.git | b732ed1352e87b4474b0520d1383994e069f8057 | FormalisingMathematics2024/Section02reals/Sheet3.lean | Section2sheet3.tendsTo_thirtyseven | [77, 1] | [84, 11] | intro n hn | case h
ε : ℝ
hε : 0 < ε
⊢ ∀ (n : ℕ), 100 ≤ n → |37 - 37| < ε | case h
ε : ℝ
hε : 0 < ε
n : ℕ
hn : 100 ≤ n
⊢ |37 - 37| < ε | Please generate a tactic in lean4 to solve the state.
STATE:
case h
ε : ℝ
hε : 0 < ε
⊢ ∀ (n : ℕ), 100 ≤ n → |37 - 37| < ε
TACTIC:
|
https://github.com/ImperialCollegeLondon/formalising-mathematics-2024.git | b732ed1352e87b4474b0520d1383994e069f8057 | FormalisingMathematics2024/Section02reals/Sheet3.lean | Section2sheet3.tendsTo_thirtyseven | [77, 1] | [84, 11] | norm_num | case h
ε : ℝ
hε : 0 < ε
n : ℕ
hn : 100 ≤ n
⊢ |37 - 37| < ε | case h
ε : ℝ
hε : 0 < ε
n : ℕ
hn : 100 ≤ n
⊢ 0 < ε | Please generate a tactic in lean4 to solve the state.
STATE:
case h
ε : ℝ
hε : 0 < ε
n : ℕ
hn : 100 ≤ n
⊢ |37 - 37| < ε
TACTIC:
|
https://github.com/ImperialCollegeLondon/formalising-mathematics-2024.git | b732ed1352e87b4474b0520d1383994e069f8057 | FormalisingMathematics2024/Section02reals/Sheet3.lean | Section2sheet3.tendsTo_thirtyseven | [77, 1] | [84, 11] | exact hε | case h
ε : ℝ
hε : 0 < ε
n : ℕ
hn : 100 ≤ n
⊢ 0 < ε | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case h
ε : ℝ
hε : 0 < ε
n : ℕ
hn : 100 ≤ n
⊢ 0 < ε
TACTIC:
|
https://github.com/ImperialCollegeLondon/formalising-mathematics-2024.git | b732ed1352e87b4474b0520d1383994e069f8057 | FormalisingMathematics2024/Section02reals/Sheet3.lean | Section2sheet3.tendsTo_const | [87, 1] | [95, 11] | intro ε hε | c : ℝ
⊢ TendsTo (fun n => c) c | c ε : ℝ
hε : ε > 0
⊢ ∃ B, ∀ (n : ℕ), B ≤ n → |(fun n => c) n - c| < ε | Please generate a tactic in lean4 to solve the state.
STATE:
c : ℝ
⊢ TendsTo (fun n => c) c
TACTIC:
|
https://github.com/ImperialCollegeLondon/formalising-mathematics-2024.git | b732ed1352e87b4474b0520d1383994e069f8057 | FormalisingMathematics2024/Section02reals/Sheet3.lean | Section2sheet3.tendsTo_const | [87, 1] | [95, 11] | dsimp only | c ε : ℝ
hε : ε > 0
⊢ ∃ B, ∀ (n : ℕ), B ≤ n → |(fun n => c) n - c| < ε | c ε : ℝ
hε : ε > 0
⊢ ∃ B, ∀ (n : ℕ), B ≤ n → |c - c| < ε | Please generate a tactic in lean4 to solve the state.
STATE:
c ε : ℝ
hε : ε > 0
⊢ ∃ B, ∀ (n : ℕ), B ≤ n → |(fun n => c) n - c| < ε
TACTIC:
|
https://github.com/ImperialCollegeLondon/formalising-mathematics-2024.git | b732ed1352e87b4474b0520d1383994e069f8057 | FormalisingMathematics2024/Section02reals/Sheet3.lean | Section2sheet3.tendsTo_const | [87, 1] | [95, 11] | use 37 | c ε : ℝ
hε : ε > 0
⊢ ∃ B, ∀ (n : ℕ), B ≤ n → |c - c| < ε | case h
c ε : ℝ
hε : ε > 0
⊢ ∀ (n : ℕ), 37 ≤ n → |c - c| < ε | Please generate a tactic in lean4 to solve the state.
STATE:
c ε : ℝ
hε : ε > 0
⊢ ∃ B, ∀ (n : ℕ), B ≤ n → |c - c| < ε
TACTIC:
|
https://github.com/ImperialCollegeLondon/formalising-mathematics-2024.git | b732ed1352e87b4474b0520d1383994e069f8057 | FormalisingMathematics2024/Section02reals/Sheet3.lean | Section2sheet3.tendsTo_const | [87, 1] | [95, 11] | intro n hn | case h
c ε : ℝ
hε : ε > 0
⊢ ∀ (n : ℕ), 37 ≤ n → |c - c| < ε | case h
c ε : ℝ
hε : ε > 0
n : ℕ
hn : 37 ≤ n
⊢ |c - c| < ε | Please generate a tactic in lean4 to solve the state.
STATE:
case h
c ε : ℝ
hε : ε > 0
⊢ ∀ (n : ℕ), 37 ≤ n → |c - c| < ε
TACTIC:
|
https://github.com/ImperialCollegeLondon/formalising-mathematics-2024.git | b732ed1352e87b4474b0520d1383994e069f8057 | FormalisingMathematics2024/Section02reals/Sheet3.lean | Section2sheet3.tendsTo_const | [87, 1] | [95, 11] | ring_nf | case h
c ε : ℝ
hε : ε > 0
n : ℕ
hn : 37 ≤ n
⊢ |c - c| < ε | case h
c ε : ℝ
hε : ε > 0
n : ℕ
hn : 37 ≤ n
⊢ |0| < ε | Please generate a tactic in lean4 to solve the state.
STATE:
case h
c ε : ℝ
hε : ε > 0
n : ℕ
hn : 37 ≤ n
⊢ |c - c| < ε
TACTIC:
|
https://github.com/ImperialCollegeLondon/formalising-mathematics-2024.git | b732ed1352e87b4474b0520d1383994e069f8057 | FormalisingMathematics2024/Section02reals/Sheet3.lean | Section2sheet3.tendsTo_const | [87, 1] | [95, 11] | norm_num | case h
c ε : ℝ
hε : ε > 0
n : ℕ
hn : 37 ≤ n
⊢ |0| < ε | case h
c ε : ℝ
hε : ε > 0
n : ℕ
hn : 37 ≤ n
⊢ 0 < ε | Please generate a tactic in lean4 to solve the state.
STATE:
case h
c ε : ℝ
hε : ε > 0
n : ℕ
hn : 37 ≤ n
⊢ |0| < ε
TACTIC:
|
https://github.com/ImperialCollegeLondon/formalising-mathematics-2024.git | b732ed1352e87b4474b0520d1383994e069f8057 | FormalisingMathematics2024/Section02reals/Sheet3.lean | Section2sheet3.tendsTo_const | [87, 1] | [95, 11] | exact hε | case h
c ε : ℝ
hε : ε > 0
n : ℕ
hn : 37 ≤ n
⊢ 0 < ε | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case h
c ε : ℝ
hε : ε > 0
n : ℕ
hn : 37 ≤ n
⊢ 0 < ε
TACTIC:
|
https://github.com/ImperialCollegeLondon/formalising-mathematics-2024.git | b732ed1352e87b4474b0520d1383994e069f8057 | FormalisingMathematics2024/Section02reals/Sheet3.lean | Section2sheet3.tendsTo_add_const | [98, 1] | [107, 8] | sorry | a : ℕ → ℝ
t c : ℝ
h : TendsTo a t
⊢ TendsTo (fun n => a n + c) (t + c) | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
a : ℕ → ℝ
t c : ℝ
h : TendsTo a t
⊢ TendsTo (fun n => a n + c) (t + c)
TACTIC:
|
https://github.com/ImperialCollegeLondon/formalising-mathematics-2024.git | b732ed1352e87b4474b0520d1383994e069f8057 | FormalisingMathematics2024/Solutions/Section02reals/Sheet6.lean | Section2sheet6Solutions.tendsTo_thirtyseven_mul | [31, 1] | [39, 54] | intro ε hε | a : ℕ → ℝ
t : ℝ
h : TendsTo a t
⊢ TendsTo (fun n => 37 * a n) (37 * t) | a : ℕ → ℝ
t : ℝ
h : TendsTo a t
ε : ℝ
hε : ε > 0
⊢ ∃ B, ∀ (n : ℕ), B ≤ n → |(fun n => 37 * a n) n - 37 * t| < ε | Please generate a tactic in lean4 to solve the state.
STATE:
a : ℕ → ℝ
t : ℝ
h : TendsTo a t
⊢ TendsTo (fun n => 37 * a n) (37 * t)
TACTIC:
|
https://github.com/ImperialCollegeLondon/formalising-mathematics-2024.git | b732ed1352e87b4474b0520d1383994e069f8057 | FormalisingMathematics2024/Solutions/Section02reals/Sheet6.lean | Section2sheet6Solutions.tendsTo_thirtyseven_mul | [31, 1] | [39, 54] | obtain ⟨X, hX⟩ := h (ε / 37) (by linarith) | a : ℕ → ℝ
t : ℝ
h : TendsTo a t
ε : ℝ
hε : ε > 0
⊢ ∃ B, ∀ (n : ℕ), B ≤ n → |(fun n => 37 * a n) n - 37 * t| < ε | case intro
a : ℕ → ℝ
t : ℝ
h : TendsTo a t
ε : ℝ
hε : ε > 0
X : ℕ
hX : ∀ (n : ℕ), X ≤ n → |a n - t| < ε / 37
⊢ ∃ B, ∀ (n : ℕ), B ≤ n → |(fun n => 37 * a n) n - 37 * t| < ε | Please generate a tactic in lean4 to solve the state.
STATE:
a : ℕ → ℝ
t : ℝ
h : TendsTo a t
ε : ℝ
hε : ε > 0
⊢ ∃ B, ∀ (n : ℕ), B ≤ n → |(fun n => 37 * a n) n - 37 * t| < ε
TACTIC:
|
https://github.com/ImperialCollegeLondon/formalising-mathematics-2024.git | b732ed1352e87b4474b0520d1383994e069f8057 | FormalisingMathematics2024/Solutions/Section02reals/Sheet6.lean | Section2sheet6Solutions.tendsTo_thirtyseven_mul | [31, 1] | [39, 54] | use X | case intro
a : ℕ → ℝ
t : ℝ
h : TendsTo a t
ε : ℝ
hε : ε > 0
X : ℕ
hX : ∀ (n : ℕ), X ≤ n → |a n - t| < ε / 37
⊢ ∃ B, ∀ (n : ℕ), B ≤ n → |(fun n => 37 * a n) n - 37 * t| < ε | case h
a : ℕ → ℝ
t : ℝ
h : TendsTo a t
ε : ℝ
hε : ε > 0
X : ℕ
hX : ∀ (n : ℕ), X ≤ n → |a n - t| < ε / 37
⊢ ∀ (n : ℕ), X ≤ n → |(fun n => 37 * a n) n - 37 * t| < ε | Please generate a tactic in lean4 to solve the state.
STATE:
case intro
a : ℕ → ℝ
t : ℝ
h : TendsTo a t
ε : ℝ
hε : ε > 0
X : ℕ
hX : ∀ (n : ℕ), X ≤ n → |a n - t| < ε / 37
⊢ ∃ B, ∀ (n : ℕ), B ≤ n → |(fun n => 37 * a n) n - 37 * t| < ε
TACTIC:
|
https://github.com/ImperialCollegeLondon/formalising-mathematics-2024.git | b732ed1352e87b4474b0520d1383994e069f8057 | FormalisingMathematics2024/Solutions/Section02reals/Sheet6.lean | Section2sheet6Solutions.tendsTo_thirtyseven_mul | [31, 1] | [39, 54] | intro n hn | case h
a : ℕ → ℝ
t : ℝ
h : TendsTo a t
ε : ℝ
hε : ε > 0
X : ℕ
hX : ∀ (n : ℕ), X ≤ n → |a n - t| < ε / 37
⊢ ∀ (n : ℕ), X ≤ n → |(fun n => 37 * a n) n - 37 * t| < ε | case h
a : ℕ → ℝ
t : ℝ
h : TendsTo a t
ε : ℝ
hε : ε > 0
X : ℕ
hX : ∀ (n : ℕ), X ≤ n → |a n - t| < ε / 37
n : ℕ
hn : X ≤ n
⊢ |(fun n => 37 * a n) n - 37 * t| < ε | Please generate a tactic in lean4 to solve the state.
STATE:
case h
a : ℕ → ℝ
t : ℝ
h : TendsTo a t
ε : ℝ
hε : ε > 0
X : ℕ
hX : ∀ (n : ℕ), X ≤ n → |a n - t| < ε / 37
⊢ ∀ (n : ℕ), X ≤ n → |(fun n => 37 * a n) n - 37 * t| < ε
TACTIC:
|
https://github.com/ImperialCollegeLondon/formalising-mathematics-2024.git | b732ed1352e87b4474b0520d1383994e069f8057 | FormalisingMathematics2024/Solutions/Section02reals/Sheet6.lean | Section2sheet6Solutions.tendsTo_thirtyseven_mul | [31, 1] | [39, 54] | specialize hX n hn | case h
a : ℕ → ℝ
t : ℝ
h : TendsTo a t
ε : ℝ
hε : ε > 0
X : ℕ
hX : ∀ (n : ℕ), X ≤ n → |a n - t| < ε / 37
n : ℕ
hn : X ≤ n
⊢ |(fun n => 37 * a n) n - 37 * t| < ε | case h
a : ℕ → ℝ
t : ℝ
h : TendsTo a t
ε : ℝ
hε : ε > 0
X n : ℕ
hn : X ≤ n
hX : |a n - t| < ε / 37
⊢ |(fun n => 37 * a n) n - 37 * t| < ε | Please generate a tactic in lean4 to solve the state.
STATE:
case h
a : ℕ → ℝ
t : ℝ
h : TendsTo a t
ε : ℝ
hε : ε > 0
X : ℕ
hX : ∀ (n : ℕ), X ≤ n → |a n - t| < ε / 37
n : ℕ
hn : X ≤ n
⊢ |(fun n => 37 * a n) n - 37 * t| < ε
TACTIC:
|
https://github.com/ImperialCollegeLondon/formalising-mathematics-2024.git | b732ed1352e87b4474b0520d1383994e069f8057 | FormalisingMathematics2024/Solutions/Section02reals/Sheet6.lean | Section2sheet6Solutions.tendsTo_thirtyseven_mul | [31, 1] | [39, 54] | simp | case h
a : ℕ → ℝ
t : ℝ
h : TendsTo a t
ε : ℝ
hε : ε > 0
X n : ℕ
hn : X ≤ n
hX : |a n - t| < ε / 37
⊢ |(fun n => 37 * a n) n - 37 * t| < ε | case h
a : ℕ → ℝ
t : ℝ
h : TendsTo a t
ε : ℝ
hε : ε > 0
X n : ℕ
hn : X ≤ n
hX : |a n - t| < ε / 37
⊢ |37 * a n - 37 * t| < ε | Please generate a tactic in lean4 to solve the state.
STATE:
case h
a : ℕ → ℝ
t : ℝ
h : TendsTo a t
ε : ℝ
hε : ε > 0
X n : ℕ
hn : X ≤ n
hX : |a n - t| < ε / 37
⊢ |(fun n => 37 * a n) n - 37 * t| < ε
TACTIC:
|
https://github.com/ImperialCollegeLondon/formalising-mathematics-2024.git | b732ed1352e87b4474b0520d1383994e069f8057 | FormalisingMathematics2024/Solutions/Section02reals/Sheet6.lean | Section2sheet6Solutions.tendsTo_thirtyseven_mul | [31, 1] | [39, 54] | rw [← mul_sub, abs_mul, abs_of_nonneg] <;> linarith | case h
a : ℕ → ℝ
t : ℝ
h : TendsTo a t
ε : ℝ
hε : ε > 0
X n : ℕ
hn : X ≤ n
hX : |a n - t| < ε / 37
⊢ |37 * a n - 37 * t| < ε | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case h
a : ℕ → ℝ
t : ℝ
h : TendsTo a t
ε : ℝ
hε : ε > 0
X n : ℕ
hn : X ≤ n
hX : |a n - t| < ε / 37
⊢ |37 * a n - 37 * t| < ε
TACTIC:
|
https://github.com/ImperialCollegeLondon/formalising-mathematics-2024.git | b732ed1352e87b4474b0520d1383994e069f8057 | FormalisingMathematics2024/Solutions/Section02reals/Sheet6.lean | Section2sheet6Solutions.tendsTo_thirtyseven_mul | [31, 1] | [39, 54] | linarith | a : ℕ → ℝ
t : ℝ
h : TendsTo a t
ε : ℝ
hε : ε > 0
⊢ ε / 37 > 0 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
a : ℕ → ℝ
t : ℝ
h : TendsTo a t
ε : ℝ
hε : ε > 0
⊢ ε / 37 > 0
TACTIC:
|
https://github.com/ImperialCollegeLondon/formalising-mathematics-2024.git | b732ed1352e87b4474b0520d1383994e069f8057 | FormalisingMathematics2024/Solutions/Section02reals/Sheet6.lean | Section2sheet6Solutions.tendsTo_pos_const_mul | [43, 1] | [52, 31] | intro ε hε | a : ℕ → ℝ
t : ℝ
h : TendsTo a t
c : ℝ
hc : 0 < c
⊢ TendsTo (fun n => c * a n) (c * t) | a : ℕ → ℝ
t : ℝ
h : TendsTo a t
c : ℝ
hc : 0 < c
ε : ℝ
hε : ε > 0
⊢ ∃ B, ∀ (n : ℕ), B ≤ n → |(fun n => c * a n) n - c * t| < ε | Please generate a tactic in lean4 to solve the state.
STATE:
a : ℕ → ℝ
t : ℝ
h : TendsTo a t
c : ℝ
hc : 0 < c
⊢ TendsTo (fun n => c * a n) (c * t)
TACTIC:
|
https://github.com/ImperialCollegeLondon/formalising-mathematics-2024.git | b732ed1352e87b4474b0520d1383994e069f8057 | FormalisingMathematics2024/Solutions/Section02reals/Sheet6.lean | Section2sheet6Solutions.tendsTo_pos_const_mul | [43, 1] | [52, 31] | obtain ⟨X, hX⟩ := h (ε / c) (div_pos hε hc) | a : ℕ → ℝ
t : ℝ
h : TendsTo a t
c : ℝ
hc : 0 < c
ε : ℝ
hε : ε > 0
⊢ ∃ B, ∀ (n : ℕ), B ≤ n → |(fun n => c * a n) n - c * t| < ε | case intro
a : ℕ → ℝ
t : ℝ
h : TendsTo a t
c : ℝ
hc : 0 < c
ε : ℝ
hε : ε > 0
X : ℕ
hX : ∀ (n : ℕ), X ≤ n → |a n - t| < ε / c
⊢ ∃ B, ∀ (n : ℕ), B ≤ n → |(fun n => c * a n) n - c * t| < ε | Please generate a tactic in lean4 to solve the state.
STATE:
a : ℕ → ℝ
t : ℝ
h : TendsTo a t
c : ℝ
hc : 0 < c
ε : ℝ
hε : ε > 0
⊢ ∃ B, ∀ (n : ℕ), B ≤ n → |(fun n => c * a n) n - c * t| < ε
TACTIC:
|
https://github.com/ImperialCollegeLondon/formalising-mathematics-2024.git | b732ed1352e87b4474b0520d1383994e069f8057 | FormalisingMathematics2024/Solutions/Section02reals/Sheet6.lean | Section2sheet6Solutions.tendsTo_pos_const_mul | [43, 1] | [52, 31] | use X | case intro
a : ℕ → ℝ
t : ℝ
h : TendsTo a t
c : ℝ
hc : 0 < c
ε : ℝ
hε : ε > 0
X : ℕ
hX : ∀ (n : ℕ), X ≤ n → |a n - t| < ε / c
⊢ ∃ B, ∀ (n : ℕ), B ≤ n → |(fun n => c * a n) n - c * t| < ε | case h
a : ℕ → ℝ
t : ℝ
h : TendsTo a t
c : ℝ
hc : 0 < c
ε : ℝ
hε : ε > 0
X : ℕ
hX : ∀ (n : ℕ), X ≤ n → |a n - t| < ε / c
⊢ ∀ (n : ℕ), X ≤ n → |(fun n => c * a n) n - c * t| < ε | Please generate a tactic in lean4 to solve the state.
STATE:
case intro
a : ℕ → ℝ
t : ℝ
h : TendsTo a t
c : ℝ
hc : 0 < c
ε : ℝ
hε : ε > 0
X : ℕ
hX : ∀ (n : ℕ), X ≤ n → |a n - t| < ε / c
⊢ ∃ B, ∀ (n : ℕ), B ≤ n → |(fun n => c * a n) n - c * t| < ε
TACTIC:
|
https://github.com/ImperialCollegeLondon/formalising-mathematics-2024.git | b732ed1352e87b4474b0520d1383994e069f8057 | FormalisingMathematics2024/Solutions/Section02reals/Sheet6.lean | Section2sheet6Solutions.tendsTo_pos_const_mul | [43, 1] | [52, 31] | intro n hn | case h
a : ℕ → ℝ
t : ℝ
h : TendsTo a t
c : ℝ
hc : 0 < c
ε : ℝ
hε : ε > 0
X : ℕ
hX : ∀ (n : ℕ), X ≤ n → |a n - t| < ε / c
⊢ ∀ (n : ℕ), X ≤ n → |(fun n => c * a n) n - c * t| < ε | case h
a : ℕ → ℝ
t : ℝ
h : TendsTo a t
c : ℝ
hc : 0 < c
ε : ℝ
hε : ε > 0
X : ℕ
hX : ∀ (n : ℕ), X ≤ n → |a n - t| < ε / c
n : ℕ
hn : X ≤ n
⊢ |(fun n => c * a n) n - c * t| < ε | Please generate a tactic in lean4 to solve the state.
STATE:
case h
a : ℕ → ℝ
t : ℝ
h : TendsTo a t
c : ℝ
hc : 0 < c
ε : ℝ
hε : ε > 0
X : ℕ
hX : ∀ (n : ℕ), X ≤ n → |a n - t| < ε / c
⊢ ∀ (n : ℕ), X ≤ n → |(fun n => c * a n) n - c * t| < ε
TACTIC:
|
https://github.com/ImperialCollegeLondon/formalising-mathematics-2024.git | b732ed1352e87b4474b0520d1383994e069f8057 | FormalisingMathematics2024/Solutions/Section02reals/Sheet6.lean | Section2sheet6Solutions.tendsTo_pos_const_mul | [43, 1] | [52, 31] | specialize hX n hn | case h
a : ℕ → ℝ
t : ℝ
h : TendsTo a t
c : ℝ
hc : 0 < c
ε : ℝ
hε : ε > 0
X : ℕ
hX : ∀ (n : ℕ), X ≤ n → |a n - t| < ε / c
n : ℕ
hn : X ≤ n
⊢ |(fun n => c * a n) n - c * t| < ε | case h
a : ℕ → ℝ
t : ℝ
h : TendsTo a t
c : ℝ
hc : 0 < c
ε : ℝ
hε : ε > 0
X n : ℕ
hn : X ≤ n
hX : |a n - t| < ε / c
⊢ |(fun n => c * a n) n - c * t| < ε | Please generate a tactic in lean4 to solve the state.
STATE:
case h
a : ℕ → ℝ
t : ℝ
h : TendsTo a t
c : ℝ
hc : 0 < c
ε : ℝ
hε : ε > 0
X : ℕ
hX : ∀ (n : ℕ), X ≤ n → |a n - t| < ε / c
n : ℕ
hn : X ≤ n
⊢ |(fun n => c * a n) n - c * t| < ε
TACTIC:
|
https://github.com/ImperialCollegeLondon/formalising-mathematics-2024.git | b732ed1352e87b4474b0520d1383994e069f8057 | FormalisingMathematics2024/Solutions/Section02reals/Sheet6.lean | Section2sheet6Solutions.tendsTo_pos_const_mul | [43, 1] | [52, 31] | simp | case h
a : ℕ → ℝ
t : ℝ
h : TendsTo a t
c : ℝ
hc : 0 < c
ε : ℝ
hε : ε > 0
X n : ℕ
hn : X ≤ n
hX : |a n - t| < ε / c
⊢ |(fun n => c * a n) n - c * t| < ε | case h
a : ℕ → ℝ
t : ℝ
h : TendsTo a t
c : ℝ
hc : 0 < c
ε : ℝ
hε : ε > 0
X n : ℕ
hn : X ≤ n
hX : |a n - t| < ε / c
⊢ |c * a n - c * t| < ε | Please generate a tactic in lean4 to solve the state.
STATE:
case h
a : ℕ → ℝ
t : ℝ
h : TendsTo a t
c : ℝ
hc : 0 < c
ε : ℝ
hε : ε > 0
X n : ℕ
hn : X ≤ n
hX : |a n - t| < ε / c
⊢ |(fun n => c * a n) n - c * t| < ε
TACTIC:
|
https://github.com/ImperialCollegeLondon/formalising-mathematics-2024.git | b732ed1352e87b4474b0520d1383994e069f8057 | FormalisingMathematics2024/Solutions/Section02reals/Sheet6.lean | Section2sheet6Solutions.tendsTo_pos_const_mul | [43, 1] | [52, 31] | rw [← mul_sub, abs_mul, abs_of_pos hc] | case h
a : ℕ → ℝ
t : ℝ
h : TendsTo a t
c : ℝ
hc : 0 < c
ε : ℝ
hε : ε > 0
X n : ℕ
hn : X ≤ n
hX : |a n - t| < ε / c
⊢ |c * a n - c * t| < ε | case h
a : ℕ → ℝ
t : ℝ
h : TendsTo a t
c : ℝ
hc : 0 < c
ε : ℝ
hε : ε > 0
X n : ℕ
hn : X ≤ n
hX : |a n - t| < ε / c
⊢ c * |a n - t| < ε | Please generate a tactic in lean4 to solve the state.
STATE:
case h
a : ℕ → ℝ
t : ℝ
h : TendsTo a t
c : ℝ
hc : 0 < c
ε : ℝ
hε : ε > 0
X n : ℕ
hn : X ≤ n
hX : |a n - t| < ε / c
⊢ |c * a n - c * t| < ε
TACTIC:
|
https://github.com/ImperialCollegeLondon/formalising-mathematics-2024.git | b732ed1352e87b4474b0520d1383994e069f8057 | FormalisingMathematics2024/Solutions/Section02reals/Sheet6.lean | Section2sheet6Solutions.tendsTo_pos_const_mul | [43, 1] | [52, 31] | exact (lt_div_iff' hc).mp hX | case h
a : ℕ → ℝ
t : ℝ
h : TendsTo a t
c : ℝ
hc : 0 < c
ε : ℝ
hε : ε > 0
X n : ℕ
hn : X ≤ n
hX : |a n - t| < ε / c
⊢ c * |a n - t| < ε | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case h
a : ℕ → ℝ
t : ℝ
h : TendsTo a t
c : ℝ
hc : 0 < c
ε : ℝ
hε : ε > 0
X n : ℕ
hn : X ≤ n
hX : |a n - t| < ε / c
⊢ c * |a n - t| < ε
TACTIC:
|
https://github.com/ImperialCollegeLondon/formalising-mathematics-2024.git | b732ed1352e87b4474b0520d1383994e069f8057 | FormalisingMathematics2024/Solutions/Section02reals/Sheet6.lean | Section2sheet6Solutions.tendsTo_neg_const_mul | [56, 1] | [67, 32] | have hc' : 0 < -c := neg_pos.mpr hc | a : ℕ → ℝ
t : ℝ
h : TendsTo a t
c : ℝ
hc : c < 0
⊢ TendsTo (fun n => c * a n) (c * t) | a : ℕ → ℝ
t : ℝ
h : TendsTo a t
c : ℝ
hc : c < 0
hc' : 0 < -c
⊢ TendsTo (fun n => c * a n) (c * t) | Please generate a tactic in lean4 to solve the state.
STATE:
a : ℕ → ℝ
t : ℝ
h : TendsTo a t
c : ℝ
hc : c < 0
⊢ TendsTo (fun n => c * a n) (c * t)
TACTIC:
|
https://github.com/ImperialCollegeLondon/formalising-mathematics-2024.git | b732ed1352e87b4474b0520d1383994e069f8057 | FormalisingMathematics2024/Solutions/Section02reals/Sheet6.lean | Section2sheet6Solutions.tendsTo_neg_const_mul | [56, 1] | [67, 32] | intro ε hε | a : ℕ → ℝ
t : ℝ
h : TendsTo a t
c : ℝ
hc : c < 0
hc' : 0 < -c
⊢ TendsTo (fun n => c * a n) (c * t) | a : ℕ → ℝ
t : ℝ
h : TendsTo a t
c : ℝ
hc : c < 0
hc' : 0 < -c
ε : ℝ
hε : ε > 0
⊢ ∃ B, ∀ (n : ℕ), B ≤ n → |(fun n => c * a n) n - c * t| < ε | Please generate a tactic in lean4 to solve the state.
STATE:
a : ℕ → ℝ
t : ℝ
h : TendsTo a t
c : ℝ
hc : c < 0
hc' : 0 < -c
⊢ TendsTo (fun n => c * a n) (c * t)
TACTIC:
|
https://github.com/ImperialCollegeLondon/formalising-mathematics-2024.git | b732ed1352e87b4474b0520d1383994e069f8057 | FormalisingMathematics2024/Solutions/Section02reals/Sheet6.lean | Section2sheet6Solutions.tendsTo_neg_const_mul | [56, 1] | [67, 32] | obtain ⟨X, hX⟩ := h (ε / -c) (div_pos hε hc') | a : ℕ → ℝ
t : ℝ
h : TendsTo a t
c : ℝ
hc : c < 0
hc' : 0 < -c
ε : ℝ
hε : ε > 0
⊢ ∃ B, ∀ (n : ℕ), B ≤ n → |(fun n => c * a n) n - c * t| < ε | case intro
a : ℕ → ℝ
t : ℝ
h : TendsTo a t
c : ℝ
hc : c < 0
hc' : 0 < -c
ε : ℝ
hε : ε > 0
X : ℕ
hX : ∀ (n : ℕ), X ≤ n → |a n - t| < ε / -c
⊢ ∃ B, ∀ (n : ℕ), B ≤ n → |(fun n => c * a n) n - c * t| < ε | Please generate a tactic in lean4 to solve the state.
STATE:
a : ℕ → ℝ
t : ℝ
h : TendsTo a t
c : ℝ
hc : c < 0
hc' : 0 < -c
ε : ℝ
hε : ε > 0
⊢ ∃ B, ∀ (n : ℕ), B ≤ n → |(fun n => c * a n) n - c * t| < ε
TACTIC:
|
https://github.com/ImperialCollegeLondon/formalising-mathematics-2024.git | b732ed1352e87b4474b0520d1383994e069f8057 | FormalisingMathematics2024/Solutions/Section02reals/Sheet6.lean | Section2sheet6Solutions.tendsTo_neg_const_mul | [56, 1] | [67, 32] | use X | case intro
a : ℕ → ℝ
t : ℝ
h : TendsTo a t
c : ℝ
hc : c < 0
hc' : 0 < -c
ε : ℝ
hε : ε > 0
X : ℕ
hX : ∀ (n : ℕ), X ≤ n → |a n - t| < ε / -c
⊢ ∃ B, ∀ (n : ℕ), B ≤ n → |(fun n => c * a n) n - c * t| < ε | case h
a : ℕ → ℝ
t : ℝ
h : TendsTo a t
c : ℝ
hc : c < 0
hc' : 0 < -c
ε : ℝ
hε : ε > 0
X : ℕ
hX : ∀ (n : ℕ), X ≤ n → |a n - t| < ε / -c
⊢ ∀ (n : ℕ), X ≤ n → |(fun n => c * a n) n - c * t| < ε | Please generate a tactic in lean4 to solve the state.
STATE:
case intro
a : ℕ → ℝ
t : ℝ
h : TendsTo a t
c : ℝ
hc : c < 0
hc' : 0 < -c
ε : ℝ
hε : ε > 0
X : ℕ
hX : ∀ (n : ℕ), X ≤ n → |a n - t| < ε / -c
⊢ ∃ B, ∀ (n : ℕ), B ≤ n → |(fun n => c * a n) n - c * t| < ε
TACTIC:
|
https://github.com/ImperialCollegeLondon/formalising-mathematics-2024.git | b732ed1352e87b4474b0520d1383994e069f8057 | FormalisingMathematics2024/Solutions/Section02reals/Sheet6.lean | Section2sheet6Solutions.tendsTo_neg_const_mul | [56, 1] | [67, 32] | intro n hn | case h
a : ℕ → ℝ
t : ℝ
h : TendsTo a t
c : ℝ
hc : c < 0
hc' : 0 < -c
ε : ℝ
hε : ε > 0
X : ℕ
hX : ∀ (n : ℕ), X ≤ n → |a n - t| < ε / -c
⊢ ∀ (n : ℕ), X ≤ n → |(fun n => c * a n) n - c * t| < ε | case h
a : ℕ → ℝ
t : ℝ
h : TendsTo a t
c : ℝ
hc : c < 0
hc' : 0 < -c
ε : ℝ
hε : ε > 0
X : ℕ
hX : ∀ (n : ℕ), X ≤ n → |a n - t| < ε / -c
n : ℕ
hn : X ≤ n
⊢ |(fun n => c * a n) n - c * t| < ε | Please generate a tactic in lean4 to solve the state.
STATE:
case h
a : ℕ → ℝ
t : ℝ
h : TendsTo a t
c : ℝ
hc : c < 0
hc' : 0 < -c
ε : ℝ
hε : ε > 0
X : ℕ
hX : ∀ (n : ℕ), X ≤ n → |a n - t| < ε / -c
⊢ ∀ (n : ℕ), X ≤ n → |(fun n => c * a n) n - c * t| < ε
TACTIC:
|
https://github.com/ImperialCollegeLondon/formalising-mathematics-2024.git | b732ed1352e87b4474b0520d1383994e069f8057 | FormalisingMathematics2024/Solutions/Section02reals/Sheet6.lean | Section2sheet6Solutions.tendsTo_neg_const_mul | [56, 1] | [67, 32] | specialize hX n hn | case h
a : ℕ → ℝ
t : ℝ
h : TendsTo a t
c : ℝ
hc : c < 0
hc' : 0 < -c
ε : ℝ
hε : ε > 0
X : ℕ
hX : ∀ (n : ℕ), X ≤ n → |a n - t| < ε / -c
n : ℕ
hn : X ≤ n
⊢ |(fun n => c * a n) n - c * t| < ε | case h
a : ℕ → ℝ
t : ℝ
h : TendsTo a t
c : ℝ
hc : c < 0
hc' : 0 < -c
ε : ℝ
hε : ε > 0
X n : ℕ
hn : X ≤ n
hX : |a n - t| < ε / -c
⊢ |(fun n => c * a n) n - c * t| < ε | Please generate a tactic in lean4 to solve the state.
STATE:
case h
a : ℕ → ℝ
t : ℝ
h : TendsTo a t
c : ℝ
hc : c < 0
hc' : 0 < -c
ε : ℝ
hε : ε > 0
X : ℕ
hX : ∀ (n : ℕ), X ≤ n → |a n - t| < ε / -c
n : ℕ
hn : X ≤ n
⊢ |(fun n => c * a n) n - c * t| < ε
TACTIC:
|
https://github.com/ImperialCollegeLondon/formalising-mathematics-2024.git | b732ed1352e87b4474b0520d1383994e069f8057 | FormalisingMathematics2024/Solutions/Section02reals/Sheet6.lean | Section2sheet6Solutions.tendsTo_neg_const_mul | [56, 1] | [67, 32] | simp | case h
a : ℕ → ℝ
t : ℝ
h : TendsTo a t
c : ℝ
hc : c < 0
hc' : 0 < -c
ε : ℝ
hε : ε > 0
X n : ℕ
hn : X ≤ n
hX : |a n - t| < ε / -c
⊢ |(fun n => c * a n) n - c * t| < ε | case h
a : ℕ → ℝ
t : ℝ
h : TendsTo a t
c : ℝ
hc : c < 0
hc' : 0 < -c
ε : ℝ
hε : ε > 0
X n : ℕ
hn : X ≤ n
hX : |a n - t| < ε / -c
⊢ |c * a n - c * t| < ε | Please generate a tactic in lean4 to solve the state.
STATE:
case h
a : ℕ → ℝ
t : ℝ
h : TendsTo a t
c : ℝ
hc : c < 0
hc' : 0 < -c
ε : ℝ
hε : ε > 0
X n : ℕ
hn : X ≤ n
hX : |a n - t| < ε / -c
⊢ |(fun n => c * a n) n - c * t| < ε
TACTIC:
|
https://github.com/ImperialCollegeLondon/formalising-mathematics-2024.git | b732ed1352e87b4474b0520d1383994e069f8057 | FormalisingMathematics2024/Solutions/Section02reals/Sheet6.lean | Section2sheet6Solutions.tendsTo_neg_const_mul | [56, 1] | [67, 32] | rw [← mul_sub, abs_mul, abs_of_neg hc] | case h
a : ℕ → ℝ
t : ℝ
h : TendsTo a t
c : ℝ
hc : c < 0
hc' : 0 < -c
ε : ℝ
hε : ε > 0
X n : ℕ
hn : X ≤ n
hX : |a n - t| < ε / -c
⊢ |c * a n - c * t| < ε | case h
a : ℕ → ℝ
t : ℝ
h : TendsTo a t
c : ℝ
hc : c < 0
hc' : 0 < -c
ε : ℝ
hε : ε > 0
X n : ℕ
hn : X ≤ n
hX : |a n - t| < ε / -c
⊢ -c * |a n - t| < ε | Please generate a tactic in lean4 to solve the state.
STATE:
case h
a : ℕ → ℝ
t : ℝ
h : TendsTo a t
c : ℝ
hc : c < 0
hc' : 0 < -c
ε : ℝ
hε : ε > 0
X n : ℕ
hn : X ≤ n
hX : |a n - t| < ε / -c
⊢ |c * a n - c * t| < ε
TACTIC:
|
https://github.com/ImperialCollegeLondon/formalising-mathematics-2024.git | b732ed1352e87b4474b0520d1383994e069f8057 | FormalisingMathematics2024/Solutions/Section02reals/Sheet6.lean | Section2sheet6Solutions.tendsTo_neg_const_mul | [56, 1] | [67, 32] | exact (lt_div_iff' hc').mp hX | case h
a : ℕ → ℝ
t : ℝ
h : TendsTo a t
c : ℝ
hc : c < 0
hc' : 0 < -c
ε : ℝ
hε : ε > 0
X n : ℕ
hn : X ≤ n
hX : |a n - t| < ε / -c
⊢ -c * |a n - t| < ε | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case h
a : ℕ → ℝ
t : ℝ
h : TendsTo a t
c : ℝ
hc : c < 0
hc' : 0 < -c
ε : ℝ
hε : ε > 0
X n : ℕ
hn : X ≤ n
hX : |a n - t| < ε / -c
⊢ -c * |a n - t| < ε
TACTIC:
|
https://github.com/ImperialCollegeLondon/formalising-mathematics-2024.git | b732ed1352e87b4474b0520d1383994e069f8057 | FormalisingMathematics2024/Solutions/Section02reals/Sheet6.lean | Section2sheet6Solutions.tendsTo_const_mul | [71, 1] | [77, 37] | obtain hc | rfl | hc := lt_trichotomy 0 c | a : ℕ → ℝ
t c : ℝ
h : TendsTo a t
⊢ TendsTo (fun n => c * a n) (c * t) | case inl
a : ℕ → ℝ
t c : ℝ
h : TendsTo a t
hc : 0 < c
⊢ TendsTo (fun n => c * a n) (c * t)
case inr.inl
a : ℕ → ℝ
t : ℝ
h : TendsTo a t
⊢ TendsTo (fun n => 0 * a n) (0 * t)
case inr.inr
a : ℕ → ℝ
t c : ℝ
h : TendsTo a t
hc : c < 0
⊢ TendsTo (fun n => c * a n) (c * t) | Please generate a tactic in lean4 to solve the state.
STATE:
a : ℕ → ℝ
t c : ℝ
h : TendsTo a t
⊢ TendsTo (fun n => c * a n) (c * t)
TACTIC:
|
https://github.com/ImperialCollegeLondon/formalising-mathematics-2024.git | b732ed1352e87b4474b0520d1383994e069f8057 | FormalisingMathematics2024/Solutions/Section02reals/Sheet6.lean | Section2sheet6Solutions.tendsTo_const_mul | [71, 1] | [77, 37] | exact tendsTo_pos_const_mul h hc | case inl
a : ℕ → ℝ
t c : ℝ
h : TendsTo a t
hc : 0 < c
⊢ TendsTo (fun n => c * a n) (c * t) | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case inl
a : ℕ → ℝ
t c : ℝ
h : TendsTo a t
hc : 0 < c
⊢ TendsTo (fun n => c * a n) (c * t)
TACTIC:
|
https://github.com/ImperialCollegeLondon/formalising-mathematics-2024.git | b732ed1352e87b4474b0520d1383994e069f8057 | FormalisingMathematics2024/Solutions/Section02reals/Sheet6.lean | Section2sheet6Solutions.tendsTo_const_mul | [71, 1] | [77, 37] | simpa using tendsTo_const 0 | case inr.inl
a : ℕ → ℝ
t : ℝ
h : TendsTo a t
⊢ TendsTo (fun n => 0 * a n) (0 * t) | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case inr.inl
a : ℕ → ℝ
t : ℝ
h : TendsTo a t
⊢ TendsTo (fun n => 0 * a n) (0 * t)
TACTIC:
|
https://github.com/ImperialCollegeLondon/formalising-mathematics-2024.git | b732ed1352e87b4474b0520d1383994e069f8057 | FormalisingMathematics2024/Solutions/Section02reals/Sheet6.lean | Section2sheet6Solutions.tendsTo_const_mul | [71, 1] | [77, 37] | exact tendsTo_neg_const_mul h hc | case inr.inr
a : ℕ → ℝ
t c : ℝ
h : TendsTo a t
hc : c < 0
⊢ TendsTo (fun n => c * a n) (c * t) | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case inr.inr
a : ℕ → ℝ
t c : ℝ
h : TendsTo a t
hc : c < 0
⊢ TendsTo (fun n => c * a n) (c * t)
TACTIC:
|
https://github.com/ImperialCollegeLondon/formalising-mathematics-2024.git | b732ed1352e87b4474b0520d1383994e069f8057 | FormalisingMathematics2024/Solutions/Section02reals/Sheet6.lean | Section2sheet6Solutions.tendsTo_mul_const | [81, 1] | [82, 89] | simpa [mul_comm] using tendsTo_const_mul c h | a : ℕ → ℝ
t c : ℝ
h : TendsTo a t
⊢ TendsTo (fun n => a n * c) (t * c) | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
a : ℕ → ℝ
t c : ℝ
h : TendsTo a t
⊢ TendsTo (fun n => a n * c) (t * c)
TACTIC:
|
https://github.com/ImperialCollegeLondon/formalising-mathematics-2024.git | b732ed1352e87b4474b0520d1383994e069f8057 | FormalisingMathematics2024/Solutions/Section02reals/Sheet6.lean | Section2sheet6Solutions.tendsTo_neg' | [86, 1] | [87, 40] | simpa using tendsTo_const_mul (-1) ha | a : ℕ → ℝ
t : ℝ
ha : TendsTo a t
⊢ TendsTo (fun n => -a n) (-t) | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
a : ℕ → ℝ
t : ℝ
ha : TendsTo a t
⊢ TendsTo (fun n => -a n) (-t)
TACTIC:
|
https://github.com/ImperialCollegeLondon/formalising-mathematics-2024.git | b732ed1352e87b4474b0520d1383994e069f8057 | FormalisingMathematics2024/Solutions/Section02reals/Sheet6.lean | Section2sheet6Solutions.tendsTo_of_tendsTo_sub | [91, 1] | [92, 79] | simpa using tendsTo_add h1 h2 | a b : ℕ → ℝ
t u : ℝ
h1 : TendsTo (fun n => a n - b n) t
h2 : TendsTo b u
⊢ TendsTo a (t + u) | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
a b : ℕ → ℝ
t u : ℝ
h1 : TendsTo (fun n => a n - b n) t
h2 : TendsTo b u
⊢ TendsTo a (t + u)
TACTIC:
|
https://github.com/ImperialCollegeLondon/formalising-mathematics-2024.git | b732ed1352e87b4474b0520d1383994e069f8057 | FormalisingMathematics2024/Solutions/Section02reals/Sheet6.lean | Section2sheet6Solutions.tendsTo_sub_lim_iff | [95, 1] | [101, 48] | constructor | a : ℕ → ℝ
t : ℝ
⊢ TendsTo a t ↔ TendsTo (fun n => a n - t) 0 | case mp
a : ℕ → ℝ
t : ℝ
⊢ TendsTo a t → TendsTo (fun n => a n - t) 0
case mpr
a : ℕ → ℝ
t : ℝ
⊢ TendsTo (fun n => a n - t) 0 → TendsTo a t | Please generate a tactic in lean4 to solve the state.
STATE:
a : ℕ → ℝ
t : ℝ
⊢ TendsTo a t ↔ TendsTo (fun n => a n - t) 0
TACTIC:
|
https://github.com/ImperialCollegeLondon/formalising-mathematics-2024.git | b732ed1352e87b4474b0520d1383994e069f8057 | FormalisingMathematics2024/Solutions/Section02reals/Sheet6.lean | Section2sheet6Solutions.tendsTo_sub_lim_iff | [95, 1] | [101, 48] | intro h | case mp
a : ℕ → ℝ
t : ℝ
⊢ TendsTo a t → TendsTo (fun n => a n - t) 0 | case mp
a : ℕ → ℝ
t : ℝ
h : TendsTo a t
⊢ TendsTo (fun n => a n - t) 0 | Please generate a tactic in lean4 to solve the state.
STATE:
case mp
a : ℕ → ℝ
t : ℝ
⊢ TendsTo a t → TendsTo (fun n => a n - t) 0
TACTIC:
|
https://github.com/ImperialCollegeLondon/formalising-mathematics-2024.git | b732ed1352e87b4474b0520d1383994e069f8057 | FormalisingMathematics2024/Solutions/Section02reals/Sheet6.lean | Section2sheet6Solutions.tendsTo_sub_lim_iff | [95, 1] | [101, 48] | simpa using tendsTo_sub h (tendsTo_const t) | case mp
a : ℕ → ℝ
t : ℝ
h : TendsTo a t
⊢ TendsTo (fun n => a n - t) 0 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case mp
a : ℕ → ℝ
t : ℝ
h : TendsTo a t
⊢ TendsTo (fun n => a n - t) 0
TACTIC:
|
https://github.com/ImperialCollegeLondon/formalising-mathematics-2024.git | b732ed1352e87b4474b0520d1383994e069f8057 | FormalisingMathematics2024/Solutions/Section02reals/Sheet6.lean | Section2sheet6Solutions.tendsTo_sub_lim_iff | [95, 1] | [101, 48] | intro h | case mpr
a : ℕ → ℝ
t : ℝ
⊢ TendsTo (fun n => a n - t) 0 → TendsTo a t | case mpr
a : ℕ → ℝ
t : ℝ
h : TendsTo (fun n => a n - t) 0
⊢ TendsTo a t | Please generate a tactic in lean4 to solve the state.
STATE:
case mpr
a : ℕ → ℝ
t : ℝ
⊢ TendsTo (fun n => a n - t) 0 → TendsTo a t
TACTIC:
|
https://github.com/ImperialCollegeLondon/formalising-mathematics-2024.git | b732ed1352e87b4474b0520d1383994e069f8057 | FormalisingMathematics2024/Solutions/Section02reals/Sheet6.lean | Section2sheet6Solutions.tendsTo_sub_lim_iff | [95, 1] | [101, 48] | simpa using tendsTo_add h (tendsTo_const t) | case mpr
a : ℕ → ℝ
t : ℝ
h : TendsTo (fun n => a n - t) 0
⊢ TendsTo a t | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case mpr
a : ℕ → ℝ
t : ℝ
h : TendsTo (fun n => a n - t) 0
⊢ TendsTo a t
TACTIC:
|
https://github.com/ImperialCollegeLondon/formalising-mathematics-2024.git | b732ed1352e87b4474b0520d1383994e069f8057 | FormalisingMathematics2024/Solutions/Section02reals/Sheet6.lean | Section2sheet6Solutions.tendsTo_zero_mul_tendsTo_zero | [105, 1] | [114, 43] | intro ε hε | a b : ℕ → ℝ
ha : TendsTo a 0
hb : TendsTo b 0
⊢ TendsTo (fun n => a n * b n) 0 | a b : ℕ → ℝ
ha : TendsTo a 0
hb : TendsTo b 0
ε : ℝ
hε : ε > 0
⊢ ∃ B, ∀ (n : ℕ), B ≤ n → |(fun n => a n * b n) n - 0| < ε | Please generate a tactic in lean4 to solve the state.
STATE:
a b : ℕ → ℝ
ha : TendsTo a 0
hb : TendsTo b 0
⊢ TendsTo (fun n => a n * b n) 0
TACTIC:
|
https://github.com/ImperialCollegeLondon/formalising-mathematics-2024.git | b732ed1352e87b4474b0520d1383994e069f8057 | FormalisingMathematics2024/Solutions/Section02reals/Sheet6.lean | Section2sheet6Solutions.tendsTo_zero_mul_tendsTo_zero | [105, 1] | [114, 43] | obtain ⟨X, hX⟩ := ha ε hε | a b : ℕ → ℝ
ha : TendsTo a 0
hb : TendsTo b 0
ε : ℝ
hε : ε > 0
⊢ ∃ B, ∀ (n : ℕ), B ≤ n → |(fun n => a n * b n) n - 0| < ε | case intro
a b : ℕ → ℝ
ha : TendsTo a 0
hb : TendsTo b 0
ε : ℝ
hε : ε > 0
X : ℕ
hX : ∀ (n : ℕ), X ≤ n → |a n - 0| < ε
⊢ ∃ B, ∀ (n : ℕ), B ≤ n → |(fun n => a n * b n) n - 0| < ε | Please generate a tactic in lean4 to solve the state.
STATE:
a b : ℕ → ℝ
ha : TendsTo a 0
hb : TendsTo b 0
ε : ℝ
hε : ε > 0
⊢ ∃ B, ∀ (n : ℕ), B ≤ n → |(fun n => a n * b n) n - 0| < ε
TACTIC:
|
https://github.com/ImperialCollegeLondon/formalising-mathematics-2024.git | b732ed1352e87b4474b0520d1383994e069f8057 | FormalisingMathematics2024/Solutions/Section02reals/Sheet6.lean | Section2sheet6Solutions.tendsTo_zero_mul_tendsTo_zero | [105, 1] | [114, 43] | obtain ⟨Y, hY⟩ := hb 1 zero_lt_one | case intro
a b : ℕ → ℝ
ha : TendsTo a 0
hb : TendsTo b 0
ε : ℝ
hε : ε > 0
X : ℕ
hX : ∀ (n : ℕ), X ≤ n → |a n - 0| < ε
⊢ ∃ B, ∀ (n : ℕ), B ≤ n → |(fun n => a n * b n) n - 0| < ε | case intro.intro
a b : ℕ → ℝ
ha : TendsTo a 0
hb : TendsTo b 0
ε : ℝ
hε : ε > 0
X : ℕ
hX : ∀ (n : ℕ), X ≤ n → |a n - 0| < ε
Y : ℕ
hY : ∀ (n : ℕ), Y ≤ n → |b n - 0| < 1
⊢ ∃ B, ∀ (n : ℕ), B ≤ n → |(fun n => a n * b n) n - 0| < ε | Please generate a tactic in lean4 to solve the state.
STATE:
case intro
a b : ℕ → ℝ
ha : TendsTo a 0
hb : TendsTo b 0
ε : ℝ
hε : ε > 0
X : ℕ
hX : ∀ (n : ℕ), X ≤ n → |a n - 0| < ε
⊢ ∃ B, ∀ (n : ℕ), B ≤ n → |(fun n => a n * b n) n - 0| < ε
TACTIC:
|
https://github.com/ImperialCollegeLondon/formalising-mathematics-2024.git | b732ed1352e87b4474b0520d1383994e069f8057 | FormalisingMathematics2024/Solutions/Section02reals/Sheet6.lean | Section2sheet6Solutions.tendsTo_zero_mul_tendsTo_zero | [105, 1] | [114, 43] | use max X Y | case intro.intro
a b : ℕ → ℝ
ha : TendsTo a 0
hb : TendsTo b 0
ε : ℝ
hε : ε > 0
X : ℕ
hX : ∀ (n : ℕ), X ≤ n → |a n - 0| < ε
Y : ℕ
hY : ∀ (n : ℕ), Y ≤ n → |b n - 0| < 1
⊢ ∃ B, ∀ (n : ℕ), B ≤ n → |(fun n => a n * b n) n - 0| < ε | case h
a b : ℕ → ℝ
ha : TendsTo a 0
hb : TendsTo b 0
ε : ℝ
hε : ε > 0
X : ℕ
hX : ∀ (n : ℕ), X ≤ n → |a n - 0| < ε
Y : ℕ
hY : ∀ (n : ℕ), Y ≤ n → |b n - 0| < 1
⊢ ∀ (n : ℕ), max X Y ≤ n → |(fun n => a n * b n) n - 0| < ε | Please generate a tactic in lean4 to solve the state.
STATE:
case intro.intro
a b : ℕ → ℝ
ha : TendsTo a 0
hb : TendsTo b 0
ε : ℝ
hε : ε > 0
X : ℕ
hX : ∀ (n : ℕ), X ≤ n → |a n - 0| < ε
Y : ℕ
hY : ∀ (n : ℕ), Y ≤ n → |b n - 0| < 1
⊢ ∃ B, ∀ (n : ℕ), B ≤ n → |(fun n => a n * b n) n - 0| < ε
TACTIC:
|
https://github.com/ImperialCollegeLondon/formalising-mathematics-2024.git | b732ed1352e87b4474b0520d1383994e069f8057 | FormalisingMathematics2024/Solutions/Section02reals/Sheet6.lean | Section2sheet6Solutions.tendsTo_zero_mul_tendsTo_zero | [105, 1] | [114, 43] | intro n hn | case h
a b : ℕ → ℝ
ha : TendsTo a 0
hb : TendsTo b 0
ε : ℝ
hε : ε > 0
X : ℕ
hX : ∀ (n : ℕ), X ≤ n → |a n - 0| < ε
Y : ℕ
hY : ∀ (n : ℕ), Y ≤ n → |b n - 0| < 1
⊢ ∀ (n : ℕ), max X Y ≤ n → |(fun n => a n * b n) n - 0| < ε | case h
a b : ℕ → ℝ
ha : TendsTo a 0
hb : TendsTo b 0
ε : ℝ
hε : ε > 0
X : ℕ
hX : ∀ (n : ℕ), X ≤ n → |a n - 0| < ε
Y : ℕ
hY : ∀ (n : ℕ), Y ≤ n → |b n - 0| < 1
n : ℕ
hn : max X Y ≤ n
⊢ |(fun n => a n * b n) n - 0| < ε | Please generate a tactic in lean4 to solve the state.
STATE:
case h
a b : ℕ → ℝ
ha : TendsTo a 0
hb : TendsTo b 0
ε : ℝ
hε : ε > 0
X : ℕ
hX : ∀ (n : ℕ), X ≤ n → |a n - 0| < ε
Y : ℕ
hY : ∀ (n : ℕ), Y ≤ n → |b n - 0| < 1
⊢ ∀ (n : ℕ), max X Y ≤ n → |(fun n => a n * b n) n - 0| < ε
TACTIC:
|
https://github.com/ImperialCollegeLondon/formalising-mathematics-2024.git | b732ed1352e87b4474b0520d1383994e069f8057 | FormalisingMathematics2024/Solutions/Section02reals/Sheet6.lean | Section2sheet6Solutions.tendsTo_zero_mul_tendsTo_zero | [105, 1] | [114, 43] | specialize hX n (le_of_max_le_left hn) | case h
a b : ℕ → ℝ
ha : TendsTo a 0
hb : TendsTo b 0
ε : ℝ
hε : ε > 0
X : ℕ
hX : ∀ (n : ℕ), X ≤ n → |a n - 0| < ε
Y : ℕ
hY : ∀ (n : ℕ), Y ≤ n → |b n - 0| < 1
n : ℕ
hn : max X Y ≤ n
⊢ |(fun n => a n * b n) n - 0| < ε | case h
a b : ℕ → ℝ
ha : TendsTo a 0
hb : TendsTo b 0
ε : ℝ
hε : ε > 0
X Y : ℕ
hY : ∀ (n : ℕ), Y ≤ n → |b n - 0| < 1
n : ℕ
hn : max X Y ≤ n
hX : |a n - 0| < ε
⊢ |(fun n => a n * b n) n - 0| < ε | Please generate a tactic in lean4 to solve the state.
STATE:
case h
a b : ℕ → ℝ
ha : TendsTo a 0
hb : TendsTo b 0
ε : ℝ
hε : ε > 0
X : ℕ
hX : ∀ (n : ℕ), X ≤ n → |a n - 0| < ε
Y : ℕ
hY : ∀ (n : ℕ), Y ≤ n → |b n - 0| < 1
n : ℕ
hn : max X Y ≤ n
⊢ |(fun n => a n * b n) n - 0| < ε
TACTIC:
|
https://github.com/ImperialCollegeLondon/formalising-mathematics-2024.git | b732ed1352e87b4474b0520d1383994e069f8057 | FormalisingMathematics2024/Solutions/Section02reals/Sheet6.lean | Section2sheet6Solutions.tendsTo_zero_mul_tendsTo_zero | [105, 1] | [114, 43] | specialize hY n (le_of_max_le_right hn) | case h
a b : ℕ → ℝ
ha : TendsTo a 0
hb : TendsTo b 0
ε : ℝ
hε : ε > 0
X Y : ℕ
hY : ∀ (n : ℕ), Y ≤ n → |b n - 0| < 1
n : ℕ
hn : max X Y ≤ n
hX : |a n - 0| < ε
⊢ |(fun n => a n * b n) n - 0| < ε | case h
a b : ℕ → ℝ
ha : TendsTo a 0
hb : TendsTo b 0
ε : ℝ
hε : ε > 0
X Y n : ℕ
hn : max X Y ≤ n
hX : |a n - 0| < ε
hY : |b n - 0| < 1
⊢ |(fun n => a n * b n) n - 0| < ε | Please generate a tactic in lean4 to solve the state.
STATE:
case h
a b : ℕ → ℝ
ha : TendsTo a 0
hb : TendsTo b 0
ε : ℝ
hε : ε > 0
X Y : ℕ
hY : ∀ (n : ℕ), Y ≤ n → |b n - 0| < 1
n : ℕ
hn : max X Y ≤ n
hX : |a n - 0| < ε
⊢ |(fun n => a n * b n) n - 0| < ε
TACTIC:
|
https://github.com/ImperialCollegeLondon/formalising-mathematics-2024.git | b732ed1352e87b4474b0520d1383994e069f8057 | FormalisingMathematics2024/Solutions/Section02reals/Sheet6.lean | Section2sheet6Solutions.tendsTo_zero_mul_tendsTo_zero | [105, 1] | [114, 43] | simpa [abs_mul] using mul_lt_mul'' hX hY | case h
a b : ℕ → ℝ
ha : TendsTo a 0
hb : TendsTo b 0
ε : ℝ
hε : ε > 0
X Y n : ℕ
hn : max X Y ≤ n
hX : |a n - 0| < ε
hY : |b n - 0| < 1
⊢ |(fun n => a n * b n) n - 0| < ε | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case h
a b : ℕ → ℝ
ha : TendsTo a 0
hb : TendsTo b 0
ε : ℝ
hε : ε > 0
X Y n : ℕ
hn : max X Y ≤ n
hX : |a n - 0| < ε
hY : |b n - 0| < 1
⊢ |(fun n => a n * b n) n - 0| < ε
TACTIC:
|
https://github.com/ImperialCollegeLondon/formalising-mathematics-2024.git | b732ed1352e87b4474b0520d1383994e069f8057 | FormalisingMathematics2024/Solutions/Section02reals/Sheet6.lean | Section2sheet6Solutions.tendsTo_mul | [118, 1] | [131, 33] | rw [tendsTo_sub_lim_iff] at * | a b : ℕ → ℝ
t u : ℝ
ha : TendsTo a t
hb : TendsTo b u
⊢ TendsTo (fun n => a n * b n) (t * u) | a b : ℕ → ℝ
t u : ℝ
ha : TendsTo (fun n => a n - t) 0
hb : TendsTo (fun n => b n - u) 0
⊢ TendsTo (fun n => a n * b n - t * u) 0 | Please generate a tactic in lean4 to solve the state.
STATE:
a b : ℕ → ℝ
t u : ℝ
ha : TendsTo a t
hb : TendsTo b u
⊢ TendsTo (fun n => a n * b n) (t * u)
TACTIC:
|
https://github.com/ImperialCollegeLondon/formalising-mathematics-2024.git | b732ed1352e87b4474b0520d1383994e069f8057 | FormalisingMathematics2024/Solutions/Section02reals/Sheet6.lean | Section2sheet6Solutions.tendsTo_mul | [118, 1] | [131, 33] | have h : ∀ n, a n * b n - t * u = (a n - t) * (b n - u) + t * (b n - u) + (a n - t) * u := by
intro n; ring | a b : ℕ → ℝ
t u : ℝ
ha : TendsTo (fun n => a n - t) 0
hb : TendsTo (fun n => b n - u) 0
⊢ TendsTo (fun n => a n * b n - t * u) 0 | a b : ℕ → ℝ
t u : ℝ
ha : TendsTo (fun n => a n - t) 0
hb : TendsTo (fun n => b n - u) 0
h : ∀ (n : ℕ), a n * b n - t * u = (a n - t) * (b n - u) + t * (b n - u) + (a n - t) * u
⊢ TendsTo (fun n => a n * b n - t * u) 0 | Please generate a tactic in lean4 to solve the state.
STATE:
a b : ℕ → ℝ
t u : ℝ
ha : TendsTo (fun n => a n - t) 0
hb : TendsTo (fun n => b n - u) 0
⊢ TendsTo (fun n => a n * b n - t * u) 0
TACTIC:
|
https://github.com/ImperialCollegeLondon/formalising-mathematics-2024.git | b732ed1352e87b4474b0520d1383994e069f8057 | FormalisingMathematics2024/Solutions/Section02reals/Sheet6.lean | Section2sheet6Solutions.tendsTo_mul | [118, 1] | [131, 33] | simp [h] | a b : ℕ → ℝ
t u : ℝ
ha : TendsTo (fun n => a n - t) 0
hb : TendsTo (fun n => b n - u) 0
h : ∀ (n : ℕ), a n * b n - t * u = (a n - t) * (b n - u) + t * (b n - u) + (a n - t) * u
⊢ TendsTo (fun n => a n * b n - t * u) 0 | a b : ℕ → ℝ
t u : ℝ
ha : TendsTo (fun n => a n - t) 0
hb : TendsTo (fun n => b n - u) 0
h : ∀ (n : ℕ), a n * b n - t * u = (a n - t) * (b n - u) + t * (b n - u) + (a n - t) * u
⊢ TendsTo (fun n => (a n - t) * (b n - u) + t * (b n - u) + (a n - t) * u) 0 | Please generate a tactic in lean4 to solve the state.
STATE:
a b : ℕ → ℝ
t u : ℝ
ha : TendsTo (fun n => a n - t) 0
hb : TendsTo (fun n => b n - u) 0
h : ∀ (n : ℕ), a n * b n - t * u = (a n - t) * (b n - u) + t * (b n - u) + (a n - t) * u
⊢ TendsTo (fun n => a n * b n - t * u) 0
TACTIC:
|
https://github.com/ImperialCollegeLondon/formalising-mathematics-2024.git | b732ed1352e87b4474b0520d1383994e069f8057 | FormalisingMathematics2024/Solutions/Section02reals/Sheet6.lean | Section2sheet6Solutions.tendsTo_mul | [118, 1] | [131, 33] | rw [show (0 : ℝ) = 0 + t * 0 + 0 * u by simp] | a b : ℕ → ℝ
t u : ℝ
ha : TendsTo (fun n => a n - t) 0
hb : TendsTo (fun n => b n - u) 0
h : ∀ (n : ℕ), a n * b n - t * u = (a n - t) * (b n - u) + t * (b n - u) + (a n - t) * u
⊢ TendsTo (fun n => (a n - t) * (b n - u) + t * (b n - u) + (a n - t) * u) 0 | a b : ℕ → ℝ
t u : ℝ
ha : TendsTo (fun n => a n - t) 0
hb : TendsTo (fun n => b n - u) 0
h : ∀ (n : ℕ), a n * b n - t * u = (a n - t) * (b n - u) + t * (b n - u) + (a n - t) * u
⊢ TendsTo (fun n => (a n - t) * (b n - u) + t * (b n - u) + (a n - t) * u) (0 + t * 0 + 0 * u) | Please generate a tactic in lean4 to solve the state.
STATE:
a b : ℕ → ℝ
t u : ℝ
ha : TendsTo (fun n => a n - t) 0
hb : TendsTo (fun n => b n - u) 0
h : ∀ (n : ℕ), a n * b n - t * u = (a n - t) * (b n - u) + t * (b n - u) + (a n - t) * u
⊢ TendsTo (fun n => (a n - t) * (b n - u) + t * (b n - u) + (a n - t) * u) 0
TACTIC:
|
https://github.com/ImperialCollegeLondon/formalising-mathematics-2024.git | b732ed1352e87b4474b0520d1383994e069f8057 | FormalisingMathematics2024/Solutions/Section02reals/Sheet6.lean | Section2sheet6Solutions.tendsTo_mul | [118, 1] | [131, 33] | refine' tendsTo_add (tendsTo_add _ _) _ | a b : ℕ → ℝ
t u : ℝ
ha : TendsTo (fun n => a n - t) 0
hb : TendsTo (fun n => b n - u) 0
h : ∀ (n : ℕ), a n * b n - t * u = (a n - t) * (b n - u) + t * (b n - u) + (a n - t) * u
⊢ TendsTo (fun n => (a n - t) * (b n - u) + t * (b n - u) + (a n - t) * u) (0 + t * 0 + 0 * u) | case refine'_1
a b : ℕ → ℝ
t u : ℝ
ha : TendsTo (fun n => a n - t) 0
hb : TendsTo (fun n => b n - u) 0
h : ∀ (n : ℕ), a n * b n - t * u = (a n - t) * (b n - u) + t * (b n - u) + (a n - t) * u
⊢ TendsTo (fun n => (a n - t) * (b n - u)) 0
case refine'_2
a b : ℕ → ℝ
t u : ℝ
ha : TendsTo (fun n => a n - t) 0
hb : TendsTo (fun n => b n - u) 0
h : ∀ (n : ℕ), a n * b n - t * u = (a n - t) * (b n - u) + t * (b n - u) + (a n - t) * u
⊢ TendsTo (fun n => t * (b n - u)) (t * 0)
case refine'_3
a b : ℕ → ℝ
t u : ℝ
ha : TendsTo (fun n => a n - t) 0
hb : TendsTo (fun n => b n - u) 0
h : ∀ (n : ℕ), a n * b n - t * u = (a n - t) * (b n - u) + t * (b n - u) + (a n - t) * u
⊢ TendsTo (fun n => (a n - t) * u) (0 * u) | Please generate a tactic in lean4 to solve the state.
STATE:
a b : ℕ → ℝ
t u : ℝ
ha : TendsTo (fun n => a n - t) 0
hb : TendsTo (fun n => b n - u) 0
h : ∀ (n : ℕ), a n * b n - t * u = (a n - t) * (b n - u) + t * (b n - u) + (a n - t) * u
⊢ TendsTo (fun n => (a n - t) * (b n - u) + t * (b n - u) + (a n - t) * u) (0 + t * 0 + 0 * u)
TACTIC:
|
https://github.com/ImperialCollegeLondon/formalising-mathematics-2024.git | b732ed1352e87b4474b0520d1383994e069f8057 | FormalisingMathematics2024/Solutions/Section02reals/Sheet6.lean | Section2sheet6Solutions.tendsTo_mul | [118, 1] | [131, 33] | intro n | a b : ℕ → ℝ
t u : ℝ
ha : TendsTo (fun n => a n - t) 0
hb : TendsTo (fun n => b n - u) 0
⊢ ∀ (n : ℕ), a n * b n - t * u = (a n - t) * (b n - u) + t * (b n - u) + (a n - t) * u | a b : ℕ → ℝ
t u : ℝ
ha : TendsTo (fun n => a n - t) 0
hb : TendsTo (fun n => b n - u) 0
n : ℕ
⊢ a n * b n - t * u = (a n - t) * (b n - u) + t * (b n - u) + (a n - t) * u | Please generate a tactic in lean4 to solve the state.
STATE:
a b : ℕ → ℝ
t u : ℝ
ha : TendsTo (fun n => a n - t) 0
hb : TendsTo (fun n => b n - u) 0
⊢ ∀ (n : ℕ), a n * b n - t * u = (a n - t) * (b n - u) + t * (b n - u) + (a n - t) * u
TACTIC:
|
https://github.com/ImperialCollegeLondon/formalising-mathematics-2024.git | b732ed1352e87b4474b0520d1383994e069f8057 | FormalisingMathematics2024/Solutions/Section02reals/Sheet6.lean | Section2sheet6Solutions.tendsTo_mul | [118, 1] | [131, 33] | ring | a b : ℕ → ℝ
t u : ℝ
ha : TendsTo (fun n => a n - t) 0
hb : TendsTo (fun n => b n - u) 0
n : ℕ
⊢ a n * b n - t * u = (a n - t) * (b n - u) + t * (b n - u) + (a n - t) * u | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
a b : ℕ → ℝ
t u : ℝ
ha : TendsTo (fun n => a n - t) 0
hb : TendsTo (fun n => b n - u) 0
n : ℕ
⊢ a n * b n - t * u = (a n - t) * (b n - u) + t * (b n - u) + (a n - t) * u
TACTIC:
|
https://github.com/ImperialCollegeLondon/formalising-mathematics-2024.git | b732ed1352e87b4474b0520d1383994e069f8057 | FormalisingMathematics2024/Solutions/Section02reals/Sheet6.lean | Section2sheet6Solutions.tendsTo_mul | [118, 1] | [131, 33] | simp | a b : ℕ → ℝ
t u : ℝ
ha : TendsTo (fun n => a n - t) 0
hb : TendsTo (fun n => b n - u) 0
h : ∀ (n : ℕ), a n * b n - t * u = (a n - t) * (b n - u) + t * (b n - u) + (a n - t) * u
⊢ 0 = 0 + t * 0 + 0 * u | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
a b : ℕ → ℝ
t u : ℝ
ha : TendsTo (fun n => a n - t) 0
hb : TendsTo (fun n => b n - u) 0
h : ∀ (n : ℕ), a n * b n - t * u = (a n - t) * (b n - u) + t * (b n - u) + (a n - t) * u
⊢ 0 = 0 + t * 0 + 0 * u
TACTIC:
|
https://github.com/ImperialCollegeLondon/formalising-mathematics-2024.git | b732ed1352e87b4474b0520d1383994e069f8057 | FormalisingMathematics2024/Solutions/Section02reals/Sheet6.lean | Section2sheet6Solutions.tendsTo_mul | [118, 1] | [131, 33] | exact tendsTo_zero_mul_tendsTo_zero ha hb | case refine'_1
a b : ℕ → ℝ
t u : ℝ
ha : TendsTo (fun n => a n - t) 0
hb : TendsTo (fun n => b n - u) 0
h : ∀ (n : ℕ), a n * b n - t * u = (a n - t) * (b n - u) + t * (b n - u) + (a n - t) * u
⊢ TendsTo (fun n => (a n - t) * (b n - u)) 0 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case refine'_1
a b : ℕ → ℝ
t u : ℝ
ha : TendsTo (fun n => a n - t) 0
hb : TendsTo (fun n => b n - u) 0
h : ∀ (n : ℕ), a n * b n - t * u = (a n - t) * (b n - u) + t * (b n - u) + (a n - t) * u
⊢ TendsTo (fun n => (a n - t) * (b n - u)) 0
TACTIC:
|
https://github.com/ImperialCollegeLondon/formalising-mathematics-2024.git | b732ed1352e87b4474b0520d1383994e069f8057 | FormalisingMathematics2024/Solutions/Section02reals/Sheet6.lean | Section2sheet6Solutions.tendsTo_mul | [118, 1] | [131, 33] | exact tendsTo_const_mul t hb | case refine'_2
a b : ℕ → ℝ
t u : ℝ
ha : TendsTo (fun n => a n - t) 0
hb : TendsTo (fun n => b n - u) 0
h : ∀ (n : ℕ), a n * b n - t * u = (a n - t) * (b n - u) + t * (b n - u) + (a n - t) * u
⊢ TendsTo (fun n => t * (b n - u)) (t * 0) | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case refine'_2
a b : ℕ → ℝ
t u : ℝ
ha : TendsTo (fun n => a n - t) 0
hb : TendsTo (fun n => b n - u) 0
h : ∀ (n : ℕ), a n * b n - t * u = (a n - t) * (b n - u) + t * (b n - u) + (a n - t) * u
⊢ TendsTo (fun n => t * (b n - u)) (t * 0)
TACTIC:
|
https://github.com/ImperialCollegeLondon/formalising-mathematics-2024.git | b732ed1352e87b4474b0520d1383994e069f8057 | FormalisingMathematics2024/Solutions/Section02reals/Sheet6.lean | Section2sheet6Solutions.tendsTo_mul | [118, 1] | [131, 33] | exact tendsTo_mul_const u ha | case refine'_3
a b : ℕ → ℝ
t u : ℝ
ha : TendsTo (fun n => a n - t) 0
hb : TendsTo (fun n => b n - u) 0
h : ∀ (n : ℕ), a n * b n - t * u = (a n - t) * (b n - u) + t * (b n - u) + (a n - t) * u
⊢ TendsTo (fun n => (a n - t) * u) (0 * u) | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case refine'_3
a b : ℕ → ℝ
t u : ℝ
ha : TendsTo (fun n => a n - t) 0
hb : TendsTo (fun n => b n - u) 0
h : ∀ (n : ℕ), a n * b n - t * u = (a n - t) * (b n - u) + t * (b n - u) + (a n - t) * u
⊢ TendsTo (fun n => (a n - t) * u) (0 * u)
TACTIC:
|
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