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pkuAI4M
/
lean_github

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https://github.com/ImperialCollegeLondon/formalising-mathematics-2024.git
b732ed1352e87b4474b0520d1383994e069f8057
FormalisingMathematics2024/Solutions/Section14UFDsAndPIDsEtc/Sheet3.lean
Ideal.IsPrime.not_one_mem
[114, 1]
[119, 29]
rwa [Ideal.eq_top_iff_one]
R✝ : Type inst✝³ : CommRing R✝ inst✝² : IsDomain R✝ inst✝¹ : UniqueFactorizationMonoid R✝ R : Type inst✝ : CommRing R P : Ideal R hI : IsPrime P h : 1 ∈ P ⊢ P = ⊤
no goals
Please generate a tactic in lean4 to solve the state. STATE: R✝ : Type inst✝³ : CommRing R✝ inst✝² : IsDomain R✝ inst✝¹ : UniqueFactorizationMonoid R✝ R : Type inst✝ : CommRing R P : Ideal R hI : IsPrime P h : 1 ∈ P ⊢ P = ⊤ TACTIC:
https://github.com/ImperialCollegeLondon/formalising-mathematics-2024.git
b732ed1352e87b4474b0520d1383994e069f8057
FormalisingMathematics2024/Solutions/Section14UFDsAndPIDsEtc/Sheet3.lean
Ideal.IsPrime.mem_of_prod_mem
[121, 1]
[133, 51]
refine L.induction_on ?_ ?_
R✝ : Type inst✝³ : CommRing R✝ inst✝² : IsDomain R✝ inst✝¹ : UniqueFactorizationMonoid R✝ R : Type inst✝ : CommRing R P : Ideal R hP : IsPrime P L : Multiset R ⊢ Multiset.prod L ∈ P → ∃ x ∈ L, x ∈ P
case refine_1 R✝ : Type inst✝³ : CommRing R✝ inst✝² : IsDomain R✝ inst✝¹ : UniqueFactorizationMonoid R✝ R : Type inst✝ : CommRing R P : Ideal R hP : IsPrime P L : Multiset R ⊢ Multiset.prod 0 ∈ P → ∃ x ∈ 0, x ∈ P case refine_2 R✝ : Type inst✝³ : CommRing R✝ inst✝² : IsDomain R✝ inst✝¹ : UniqueFactorizationMonoid R✝ R : Type inst✝ : CommRing R P : Ideal R hP : IsPrime P L : Multiset R ⊢ ∀ ⦃a : R⦄ {s : Multiset R}, (Multiset.prod s ∈ P → ∃ x ∈ s, x ∈ P) → Multiset.prod (a ::ₘ s) ∈ P → ∃ x ∈ a ::ₘ s, x ∈ P
Please generate a tactic in lean4 to solve the state. STATE: R✝ : Type inst✝³ : CommRing R✝ inst✝² : IsDomain R✝ inst✝¹ : UniqueFactorizationMonoid R✝ R : Type inst✝ : CommRing R P : Ideal R hP : IsPrime P L : Multiset R ⊢ Multiset.prod L ∈ P → ∃ x ∈ L, x ∈ P TACTIC:
https://github.com/ImperialCollegeLondon/formalising-mathematics-2024.git
b732ed1352e87b4474b0520d1383994e069f8057
FormalisingMathematics2024/Solutions/Section14UFDsAndPIDsEtc/Sheet3.lean
Ideal.IsPrime.mem_of_prod_mem
[121, 1]
[133, 51]
intro h
case refine_1 R✝ : Type inst✝³ : CommRing R✝ inst✝² : IsDomain R✝ inst✝¹ : UniqueFactorizationMonoid R✝ R : Type inst✝ : CommRing R P : Ideal R hP : IsPrime P L : Multiset R ⊢ Multiset.prod 0 ∈ P → ∃ x ∈ 0, x ∈ P
case refine_1 R✝ : Type inst✝³ : CommRing R✝ inst✝² : IsDomain R✝ inst✝¹ : UniqueFactorizationMonoid R✝ R : Type inst✝ : CommRing R P : Ideal R hP : IsPrime P L : Multiset R h : Multiset.prod 0 ∈ P ⊢ ∃ x ∈ 0, x ∈ P
Please generate a tactic in lean4 to solve the state. STATE: case refine_1 R✝ : Type inst✝³ : CommRing R✝ inst✝² : IsDomain R✝ inst✝¹ : UniqueFactorizationMonoid R✝ R : Type inst✝ : CommRing R P : Ideal R hP : IsPrime P L : Multiset R ⊢ Multiset.prod 0 ∈ P → ∃ x ∈ 0, x ∈ P TACTIC:
https://github.com/ImperialCollegeLondon/formalising-mathematics-2024.git
b732ed1352e87b4474b0520d1383994e069f8057
FormalisingMathematics2024/Solutions/Section14UFDsAndPIDsEtc/Sheet3.lean
Ideal.IsPrime.mem_of_prod_mem
[121, 1]
[133, 51]
rw [Multiset.prod_zero] at h
case refine_1 R✝ : Type inst✝³ : CommRing R✝ inst✝² : IsDomain R✝ inst✝¹ : UniqueFactorizationMonoid R✝ R : Type inst✝ : CommRing R P : Ideal R hP : IsPrime P L : Multiset R h : Multiset.prod 0 ∈ P ⊢ ∃ x ∈ 0, x ∈ P
case refine_1 R✝ : Type inst✝³ : CommRing R✝ inst✝² : IsDomain R✝ inst✝¹ : UniqueFactorizationMonoid R✝ R : Type inst✝ : CommRing R P : Ideal R hP : IsPrime P L : Multiset R h : 1 ∈ P ⊢ ∃ x ∈ 0, x ∈ P
Please generate a tactic in lean4 to solve the state. STATE: case refine_1 R✝ : Type inst✝³ : CommRing R✝ inst✝² : IsDomain R✝ inst✝¹ : UniqueFactorizationMonoid R✝ R : Type inst✝ : CommRing R P : Ideal R hP : IsPrime P L : Multiset R h : Multiset.prod 0 ∈ P ⊢ ∃ x ∈ 0, x ∈ P TACTIC:
https://github.com/ImperialCollegeLondon/formalising-mathematics-2024.git
b732ed1352e87b4474b0520d1383994e069f8057
FormalisingMathematics2024/Solutions/Section14UFDsAndPIDsEtc/Sheet3.lean
Ideal.IsPrime.mem_of_prod_mem
[121, 1]
[133, 51]
cases hP.not_one_mem h
case refine_1 R✝ : Type inst✝³ : CommRing R✝ inst✝² : IsDomain R✝ inst✝¹ : UniqueFactorizationMonoid R✝ R : Type inst✝ : CommRing R P : Ideal R hP : IsPrime P L : Multiset R h : 1 ∈ P ⊢ ∃ x ∈ 0, x ∈ P
no goals
Please generate a tactic in lean4 to solve the state. STATE: case refine_1 R✝ : Type inst✝³ : CommRing R✝ inst✝² : IsDomain R✝ inst✝¹ : UniqueFactorizationMonoid R✝ R : Type inst✝ : CommRing R P : Ideal R hP : IsPrime P L : Multiset R h : 1 ∈ P ⊢ ∃ x ∈ 0, x ∈ P TACTIC:
https://github.com/ImperialCollegeLondon/formalising-mathematics-2024.git
b732ed1352e87b4474b0520d1383994e069f8057
FormalisingMathematics2024/Solutions/Section14UFDsAndPIDsEtc/Sheet3.lean
Ideal.IsPrime.mem_of_prod_mem
[121, 1]
[133, 51]
intro a L IH h
case refine_2 R✝ : Type inst✝³ : CommRing R✝ inst✝² : IsDomain R✝ inst✝¹ : UniqueFactorizationMonoid R✝ R : Type inst✝ : CommRing R P : Ideal R hP : IsPrime P L : Multiset R ⊢ ∀ ⦃a : R⦄ {s : Multiset R}, (Multiset.prod s ∈ P → ∃ x ∈ s, x ∈ P) → Multiset.prod (a ::ₘ s) ∈ P → ∃ x ∈ a ::ₘ s, x ∈ P
case refine_2 R✝ : Type inst✝³ : CommRing R✝ inst✝² : IsDomain R✝ inst✝¹ : UniqueFactorizationMonoid R✝ R : Type inst✝ : CommRing R P : Ideal R hP : IsPrime P L✝ : Multiset R a : R L : Multiset R IH : Multiset.prod L ∈ P → ∃ x ∈ L, x ∈ P h : Multiset.prod (a ::ₘ L) ∈ P ⊢ ∃ x ∈ a ::ₘ L, x ∈ P
Please generate a tactic in lean4 to solve the state. STATE: case refine_2 R✝ : Type inst✝³ : CommRing R✝ inst✝² : IsDomain R✝ inst✝¹ : UniqueFactorizationMonoid R✝ R : Type inst✝ : CommRing R P : Ideal R hP : IsPrime P L : Multiset R ⊢ ∀ ⦃a : R⦄ {s : Multiset R}, (Multiset.prod s ∈ P → ∃ x ∈ s, x ∈ P) → Multiset.prod (a ::ₘ s) ∈ P → ∃ x ∈ a ::ₘ s, x ∈ P TACTIC:
https://github.com/ImperialCollegeLondon/formalising-mathematics-2024.git
b732ed1352e87b4474b0520d1383994e069f8057
FormalisingMathematics2024/Solutions/Section14UFDsAndPIDsEtc/Sheet3.lean
Ideal.IsPrime.mem_of_prod_mem
[121, 1]
[133, 51]
simp only [Multiset.prod_cons] at h
case refine_2 R✝ : Type inst✝³ : CommRing R✝ inst✝² : IsDomain R✝ inst✝¹ : UniqueFactorizationMonoid R✝ R : Type inst✝ : CommRing R P : Ideal R hP : IsPrime P L✝ : Multiset R a : R L : Multiset R IH : Multiset.prod L ∈ P → ∃ x ∈ L, x ∈ P h : Multiset.prod (a ::ₘ L) ∈ P ⊢ ∃ x ∈ a ::ₘ L, x ∈ P
case refine_2 R✝ : Type inst✝³ : CommRing R✝ inst✝² : IsDomain R✝ inst✝¹ : UniqueFactorizationMonoid R✝ R : Type inst✝ : CommRing R P : Ideal R hP : IsPrime P L✝ : Multiset R a : R L : Multiset R IH : Multiset.prod L ∈ P → ∃ x ∈ L, x ∈ P h : a * Multiset.prod L ∈ P ⊢ ∃ x ∈ a ::ₘ L, x ∈ P
Please generate a tactic in lean4 to solve the state. STATE: case refine_2 R✝ : Type inst✝³ : CommRing R✝ inst✝² : IsDomain R✝ inst✝¹ : UniqueFactorizationMonoid R✝ R : Type inst✝ : CommRing R P : Ideal R hP : IsPrime P L✝ : Multiset R a : R L : Multiset R IH : Multiset.prod L ∈ P → ∃ x ∈ L, x ∈ P h : Multiset.prod (a ::ₘ L) ∈ P ⊢ ∃ x ∈ a ::ₘ L, x ∈ P TACTIC:
https://github.com/ImperialCollegeLondon/formalising-mathematics-2024.git
b732ed1352e87b4474b0520d1383994e069f8057
FormalisingMathematics2024/Solutions/Section14UFDsAndPIDsEtc/Sheet3.lean
Ideal.IsPrime.mem_of_prod_mem
[121, 1]
[133, 51]
rcases hP.mem_or_mem h with (ha | hL)
case refine_2 R✝ : Type inst✝³ : CommRing R✝ inst✝² : IsDomain R✝ inst✝¹ : UniqueFactorizationMonoid R✝ R : Type inst✝ : CommRing R P : Ideal R hP : IsPrime P L✝ : Multiset R a : R L : Multiset R IH : Multiset.prod L ∈ P → ∃ x ∈ L, x ∈ P h : a * Multiset.prod L ∈ P ⊢ ∃ x ∈ a ::ₘ L, x ∈ P
case refine_2.inl R✝ : Type inst✝³ : CommRing R✝ inst✝² : IsDomain R✝ inst✝¹ : UniqueFactorizationMonoid R✝ R : Type inst✝ : CommRing R P : Ideal R hP : IsPrime P L✝ : Multiset R a : R L : Multiset R IH : Multiset.prod L ∈ P → ∃ x ∈ L, x ∈ P h : a * Multiset.prod L ∈ P ha : a ∈ P ⊢ ∃ x ∈ a ::ₘ L, x ∈ P case refine_2.inr R✝ : Type inst✝³ : CommRing R✝ inst✝² : IsDomain R✝ inst✝¹ : UniqueFactorizationMonoid R✝ R : Type inst✝ : CommRing R P : Ideal R hP : IsPrime P L✝ : Multiset R a : R L : Multiset R IH : Multiset.prod L ∈ P → ∃ x ∈ L, x ∈ P h : a * Multiset.prod L ∈ P hL : Multiset.prod L ∈ P ⊢ ∃ x ∈ a ::ₘ L, x ∈ P
Please generate a tactic in lean4 to solve the state. STATE: case refine_2 R✝ : Type inst✝³ : CommRing R✝ inst✝² : IsDomain R✝ inst✝¹ : UniqueFactorizationMonoid R✝ R : Type inst✝ : CommRing R P : Ideal R hP : IsPrime P L✝ : Multiset R a : R L : Multiset R IH : Multiset.prod L ∈ P → ∃ x ∈ L, x ∈ P h : a * Multiset.prod L ∈ P ⊢ ∃ x ∈ a ::ₘ L, x ∈ P TACTIC:
https://github.com/ImperialCollegeLondon/formalising-mathematics-2024.git
b732ed1352e87b4474b0520d1383994e069f8057
FormalisingMathematics2024/Solutions/Section14UFDsAndPIDsEtc/Sheet3.lean
Ideal.IsPrime.mem_of_prod_mem
[121, 1]
[133, 51]
exact ⟨a, by simp, ha⟩
case refine_2.inl R✝ : Type inst✝³ : CommRing R✝ inst✝² : IsDomain R✝ inst✝¹ : UniqueFactorizationMonoid R✝ R : Type inst✝ : CommRing R P : Ideal R hP : IsPrime P L✝ : Multiset R a : R L : Multiset R IH : Multiset.prod L ∈ P → ∃ x ∈ L, x ∈ P h : a * Multiset.prod L ∈ P ha : a ∈ P ⊢ ∃ x ∈ a ::ₘ L, x ∈ P
no goals
Please generate a tactic in lean4 to solve the state. STATE: case refine_2.inl R✝ : Type inst✝³ : CommRing R✝ inst✝² : IsDomain R✝ inst✝¹ : UniqueFactorizationMonoid R✝ R : Type inst✝ : CommRing R P : Ideal R hP : IsPrime P L✝ : Multiset R a : R L : Multiset R IH : Multiset.prod L ∈ P → ∃ x ∈ L, x ∈ P h : a * Multiset.prod L ∈ P ha : a ∈ P ⊢ ∃ x ∈ a ::ₘ L, x ∈ P TACTIC:
https://github.com/ImperialCollegeLondon/formalising-mathematics-2024.git
b732ed1352e87b4474b0520d1383994e069f8057
FormalisingMathematics2024/Solutions/Section14UFDsAndPIDsEtc/Sheet3.lean
Ideal.IsPrime.mem_of_prod_mem
[121, 1]
[133, 51]
simp
R✝ : Type inst✝³ : CommRing R✝ inst✝² : IsDomain R✝ inst✝¹ : UniqueFactorizationMonoid R✝ R : Type inst✝ : CommRing R P : Ideal R hP : IsPrime P L✝ : Multiset R a : R L : Multiset R IH : Multiset.prod L ∈ P → ∃ x ∈ L, x ∈ P h : a * Multiset.prod L ∈ P ha : a ∈ P ⊢ a ∈ a ::ₘ L
no goals
Please generate a tactic in lean4 to solve the state. STATE: R✝ : Type inst✝³ : CommRing R✝ inst✝² : IsDomain R✝ inst✝¹ : UniqueFactorizationMonoid R✝ R : Type inst✝ : CommRing R P : Ideal R hP : IsPrime P L✝ : Multiset R a : R L : Multiset R IH : Multiset.prod L ∈ P → ∃ x ∈ L, x ∈ P h : a * Multiset.prod L ∈ P ha : a ∈ P ⊢ a ∈ a ::ₘ L TACTIC:
https://github.com/ImperialCollegeLondon/formalising-mathematics-2024.git
b732ed1352e87b4474b0520d1383994e069f8057
FormalisingMathematics2024/Solutions/Section14UFDsAndPIDsEtc/Sheet3.lean
Ideal.IsPrime.mem_of_prod_mem
[121, 1]
[133, 51]
obtain ⟨x, hxL, hxP⟩ := IH hL
case refine_2.inr R✝ : Type inst✝³ : CommRing R✝ inst✝² : IsDomain R✝ inst✝¹ : UniqueFactorizationMonoid R✝ R : Type inst✝ : CommRing R P : Ideal R hP : IsPrime P L✝ : Multiset R a : R L : Multiset R IH : Multiset.prod L ∈ P → ∃ x ∈ L, x ∈ P h : a * Multiset.prod L ∈ P hL : Multiset.prod L ∈ P ⊢ ∃ x ∈ a ::ₘ L, x ∈ P
case refine_2.inr.intro.intro R✝ : Type inst✝³ : CommRing R✝ inst✝² : IsDomain R✝ inst✝¹ : UniqueFactorizationMonoid R✝ R : Type inst✝ : CommRing R P : Ideal R hP : IsPrime P L✝ : Multiset R a : R L : Multiset R IH : Multiset.prod L ∈ P → ∃ x ∈ L, x ∈ P h : a * Multiset.prod L ∈ P hL : Multiset.prod L ∈ P x : R hxL : x ∈ L hxP : x ∈ P ⊢ ∃ x ∈ a ::ₘ L, x ∈ P
Please generate a tactic in lean4 to solve the state. STATE: case refine_2.inr R✝ : Type inst✝³ : CommRing R✝ inst✝² : IsDomain R✝ inst✝¹ : UniqueFactorizationMonoid R✝ R : Type inst✝ : CommRing R P : Ideal R hP : IsPrime P L✝ : Multiset R a : R L : Multiset R IH : Multiset.prod L ∈ P → ∃ x ∈ L, x ∈ P h : a * Multiset.prod L ∈ P hL : Multiset.prod L ∈ P ⊢ ∃ x ∈ a ::ₘ L, x ∈ P TACTIC:
https://github.com/ImperialCollegeLondon/formalising-mathematics-2024.git
b732ed1352e87b4474b0520d1383994e069f8057
FormalisingMathematics2024/Solutions/Section14UFDsAndPIDsEtc/Sheet3.lean
Ideal.IsPrime.mem_of_prod_mem
[121, 1]
[133, 51]
exact ⟨x, Multiset.mem_cons_of_mem hxL, hxP⟩
case refine_2.inr.intro.intro R✝ : Type inst✝³ : CommRing R✝ inst✝² : IsDomain R✝ inst✝¹ : UniqueFactorizationMonoid R✝ R : Type inst✝ : CommRing R P : Ideal R hP : IsPrime P L✝ : Multiset R a : R L : Multiset R IH : Multiset.prod L ∈ P → ∃ x ∈ L, x ∈ P h : a * Multiset.prod L ∈ P hL : Multiset.prod L ∈ P x : R hxL : x ∈ L hxP : x ∈ P ⊢ ∃ x ∈ a ::ₘ L, x ∈ P
no goals
Please generate a tactic in lean4 to solve the state. STATE: case refine_2.inr.intro.intro R✝ : Type inst✝³ : CommRing R✝ inst✝² : IsDomain R✝ inst✝¹ : UniqueFactorizationMonoid R✝ R : Type inst✝ : CommRing R P : Ideal R hP : IsPrime P L✝ : Multiset R a : R L : Multiset R IH : Multiset.prod L ∈ P → ∃ x ∈ L, x ∈ P h : a * Multiset.prod L ∈ P hL : Multiset.prod L ∈ P x : R hxL : x ∈ L hxP : x ∈ P ⊢ ∃ x ∈ a ::ₘ L, x ∈ P TACTIC:
https://github.com/ImperialCollegeLondon/formalising-mathematics-2024.git
b732ed1352e87b4474b0520d1383994e069f8057
FormalisingMathematics2024/Solutions/Section14UFDsAndPIDsEtc/Sheet3.lean
Prime.ideal_span_singleton_isPrime
[135, 1]
[139, 19]
rwa [Ideal.span_singleton_prime]
R✝ : Type inst✝³ : CommRing R✝ inst✝² : IsDomain R✝ inst✝¹ : UniqueFactorizationMonoid R✝ R : Type inst✝ : CommRing R p : R hp : Prime p ⊢ Ideal.IsPrime (Ideal.span {p})
R✝ : Type inst✝³ : CommRing R✝ inst✝² : IsDomain R✝ inst✝¹ : UniqueFactorizationMonoid R✝ R : Type inst✝ : CommRing R p : R hp : Prime p ⊢ p ≠ 0
Please generate a tactic in lean4 to solve the state. STATE: R✝ : Type inst✝³ : CommRing R✝ inst✝² : IsDomain R✝ inst✝¹ : UniqueFactorizationMonoid R✝ R : Type inst✝ : CommRing R p : R hp : Prime p ⊢ Ideal.IsPrime (Ideal.span {p}) TACTIC:
https://github.com/ImperialCollegeLondon/formalising-mathematics-2024.git
b732ed1352e87b4474b0520d1383994e069f8057
FormalisingMathematics2024/Solutions/Section14UFDsAndPIDsEtc/Sheet3.lean
Prime.ideal_span_singleton_isPrime
[135, 1]
[139, 19]
exact hp.ne_zero
R✝ : Type inst✝³ : CommRing R✝ inst✝² : IsDomain R✝ inst✝¹ : UniqueFactorizationMonoid R✝ R : Type inst✝ : CommRing R p : R hp : Prime p ⊢ p ≠ 0
no goals
Please generate a tactic in lean4 to solve the state. STATE: R✝ : Type inst✝³ : CommRing R✝ inst✝² : IsDomain R✝ inst✝¹ : UniqueFactorizationMonoid R✝ R : Type inst✝ : CommRing R p : R hp : Prime p ⊢ p ≠ 0 TACTIC:
https://github.com/ImperialCollegeLondon/formalising-mathematics-2024.git
b732ed1352e87b4474b0520d1383994e069f8057
FormalisingMathematics2024/Solutions/Section02reals/Sheet5.lean
Section2sheet5solutions.tendsTo_neg
[16, 1]
[24, 21]
rw [tendsTo_def] at *
a : ℕ → ℝ t : ℝ ha : TendsTo a t ⊢ TendsTo (fun n => -a n) (-t)
a : ℕ → ℝ t : ℝ ha : ∀ (ε : ℝ), 0 < ε → ∃ B, ∀ (n : ℕ), B ≤ n → |a n - t| < ε ⊢ ∀ (ε : ℝ), 0 < ε → ∃ B, ∀ (n : ℕ), B ≤ n → |-a n - -t| < ε
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/Sheet5.lean
Section2sheet5solutions.tendsTo_neg
[16, 1]
[24, 21]
have h : ∀ n, |a n - t| = |-a n - -t| := by intro n rw [abs_sub_comm] congr 1 ring
a : ℕ → ℝ t : ℝ ha : ∀ (ε : ℝ), 0 < ε → ∃ B, ∀ (n : ℕ), B ≤ n → |a n - t| < ε ⊢ ∀ (ε : ℝ), 0 < ε → ∃ B, ∀ (n : ℕ), B ≤ n → |-a n - -t| < ε
a : ℕ → ℝ t : ℝ ha : ∀ (ε : ℝ), 0 < ε → ∃ B, ∀ (n : ℕ), B ≤ n → |a n - t| < ε h : ∀ (n : ℕ), |a n - t| = |-a n - -t| ⊢ ∀ (ε : ℝ), 0 < ε → ∃ B, ∀ (n : ℕ), B ≤ n → |-a n - -t| < ε
Please generate a tactic in lean4 to solve the state. STATE: a : ℕ → ℝ t : ℝ ha : ∀ (ε : ℝ), 0 < ε → ∃ B, ∀ (n : ℕ), B ≤ n → |a n - t| < ε ⊢ ∀ (ε : ℝ), 0 < ε → ∃ B, ∀ (n : ℕ), B ≤ n → |-a n - -t| < ε TACTIC:
https://github.com/ImperialCollegeLondon/formalising-mathematics-2024.git
b732ed1352e87b4474b0520d1383994e069f8057
FormalisingMathematics2024/Solutions/Section02reals/Sheet5.lean
Section2sheet5solutions.tendsTo_neg
[16, 1]
[24, 21]
simpa [h] using ha
a : ℕ → ℝ t : ℝ ha : ∀ (ε : ℝ), 0 < ε → ∃ B, ∀ (n : ℕ), B ≤ n → |a n - t| < ε h : ∀ (n : ℕ), |a n - t| = |-a n - -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 ha : ∀ (ε : ℝ), 0 < ε → ∃ B, ∀ (n : ℕ), B ≤ n → |a n - t| < ε h : ∀ (n : ℕ), |a n - t| = |-a n - -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/Solutions/Section02reals/Sheet5.lean
Section2sheet5solutions.tendsTo_neg
[16, 1]
[24, 21]
intro n
a : ℕ → ℝ t : ℝ ha : ∀ (ε : ℝ), 0 < ε → ∃ B, ∀ (n : ℕ), B ≤ n → |a n - t| < ε ⊢ ∀ (n : ℕ), |a n - t| = |-a n - -t|
a : ℕ → ℝ t : ℝ ha : ∀ (ε : ℝ), 0 < ε → ∃ B, ∀ (n : ℕ), B ≤ n → |a n - t| < ε n : ℕ ⊢ |a n - t| = |-a n - -t|
Please generate a tactic in lean4 to solve the state. STATE: a : ℕ → ℝ t : ℝ ha : ∀ (ε : ℝ), 0 < ε → ∃ B, ∀ (n : ℕ), B ≤ n → |a n - t| < ε ⊢ ∀ (n : ℕ), |a n - t| = |-a n - -t| TACTIC:
https://github.com/ImperialCollegeLondon/formalising-mathematics-2024.git
b732ed1352e87b4474b0520d1383994e069f8057
FormalisingMathematics2024/Solutions/Section02reals/Sheet5.lean
Section2sheet5solutions.tendsTo_neg
[16, 1]
[24, 21]
rw [abs_sub_comm]
a : ℕ → ℝ t : ℝ ha : ∀ (ε : ℝ), 0 < ε → ∃ B, ∀ (n : ℕ), B ≤ n → |a n - t| < ε n : ℕ ⊢ |a n - t| = |-a n - -t|
a : ℕ → ℝ t : ℝ ha : ∀ (ε : ℝ), 0 < ε → ∃ B, ∀ (n : ℕ), B ≤ n → |a n - t| < ε n : ℕ ⊢ |t - a n| = |-a n - -t|
Please generate a tactic in lean4 to solve the state. STATE: a : ℕ → ℝ t : ℝ ha : ∀ (ε : ℝ), 0 < ε → ∃ B, ∀ (n : ℕ), B ≤ n → |a n - t| < ε n : ℕ ⊢ |a n - t| = |-a n - -t| TACTIC:
https://github.com/ImperialCollegeLondon/formalising-mathematics-2024.git
b732ed1352e87b4474b0520d1383994e069f8057
FormalisingMathematics2024/Solutions/Section02reals/Sheet5.lean
Section2sheet5solutions.tendsTo_neg
[16, 1]
[24, 21]
congr 1
a : ℕ → ℝ t : ℝ ha : ∀ (ε : ℝ), 0 < ε → ∃ B, ∀ (n : ℕ), B ≤ n → |a n - t| < ε n : ℕ ⊢ |t - a n| = |-a n - -t|
case e_a a : ℕ → ℝ t : ℝ ha : ∀ (ε : ℝ), 0 < ε → ∃ B, ∀ (n : ℕ), B ≤ n → |a n - t| < ε n : ℕ ⊢ t - a n = -a n - -t
Please generate a tactic in lean4 to solve the state. STATE: a : ℕ → ℝ t : ℝ ha : ∀ (ε : ℝ), 0 < ε → ∃ B, ∀ (n : ℕ), B ≤ n → |a n - t| < ε n : ℕ ⊢ |t - a n| = |-a n - -t| TACTIC:
https://github.com/ImperialCollegeLondon/formalising-mathematics-2024.git
b732ed1352e87b4474b0520d1383994e069f8057
FormalisingMathematics2024/Solutions/Section02reals/Sheet5.lean
Section2sheet5solutions.tendsTo_neg
[16, 1]
[24, 21]
ring
case e_a a : ℕ → ℝ t : ℝ ha : ∀ (ε : ℝ), 0 < ε → ∃ B, ∀ (n : ℕ), B ≤ n → |a n - t| < ε n : ℕ ⊢ t - a n = -a n - -t
no goals
Please generate a tactic in lean4 to solve the state. STATE: case e_a a : ℕ → ℝ t : ℝ ha : ∀ (ε : ℝ), 0 < ε → ∃ B, ∀ (n : ℕ), B ≤ n → |a n - t| < ε n : ℕ ⊢ t - a n = -a n - -t TACTIC:
https://github.com/ImperialCollegeLondon/formalising-mathematics-2024.git
b732ed1352e87b4474b0520d1383994e069f8057
FormalisingMathematics2024/Solutions/Section02reals/Sheet5.lean
Section2sheet5solutions.tendsTo_add
[38, 1]
[59, 13]
rw [tendsTo_def] 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 : ∀ (ε : ℝ), 0 < ε → ∃ B, ∀ (n : ℕ), B ≤ n → |a n - t| < ε hb : ∀ (ε : ℝ), 0 < ε → ∃ B, ∀ (n : ℕ), B ≤ n → |b n - u| < ε ⊢ ∀ (ε : ℝ), 0 < ε → ∃ B, ∀ (n : ℕ), B ≤ n → |a n + b n - (t + u)| < ε
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/Sheet5.lean
Section2sheet5solutions.tendsTo_add
[38, 1]
[59, 13]
intro ε hε
a b : ℕ → ℝ t u : ℝ ha : ∀ (ε : ℝ), 0 < ε → ∃ B, ∀ (n : ℕ), B ≤ n → |a n - t| < ε hb : ∀ (ε : ℝ), 0 < ε → ∃ B, ∀ (n : ℕ), B ≤ n → |b n - u| < ε ⊢ ∀ (ε : ℝ), 0 < ε → ∃ B, ∀ (n : ℕ), B ≤ n → |a n + b n - (t + u)| < ε
a b : ℕ → ℝ t u : ℝ ha : ∀ (ε : ℝ), 0 < ε → ∃ B, ∀ (n : ℕ), B ≤ n → |a n - t| < ε hb : ∀ (ε : ℝ), 0 < ε → ∃ B, ∀ (n : ℕ), B ≤ n → |b n - u| < ε ε : ℝ hε : 0 < ε ⊢ ∃ B, ∀ (n : ℕ), B ≤ n → |a n + b n - (t + u)| < ε
Please generate a tactic in lean4 to solve the state. STATE: a b : ℕ → ℝ t u : ℝ ha : ∀ (ε : ℝ), 0 < ε → ∃ B, ∀ (n : ℕ), B ≤ n → |a n - t| < ε hb : ∀ (ε : ℝ), 0 < ε → ∃ B, ∀ (n : ℕ), B ≤ n → |b n - u| < ε ⊢ ∀ (ε : ℝ), 0 < ε → ∃ B, ∀ (n : ℕ), B ≤ n → |a n + b n - (t + u)| < ε TACTIC:
https://github.com/ImperialCollegeLondon/formalising-mathematics-2024.git
b732ed1352e87b4474b0520d1383994e069f8057
FormalisingMathematics2024/Solutions/Section02reals/Sheet5.lean
Section2sheet5solutions.tendsTo_add
[38, 1]
[59, 13]
specialize ha (ε / 2) (by linarith)
a b : ℕ → ℝ t u : ℝ ha : ∀ (ε : ℝ), 0 < ε → ∃ B, ∀ (n : ℕ), B ≤ n → |a n - t| < ε hb : ∀ (ε : ℝ), 0 < ε → ∃ B, ∀ (n : ℕ), B ≤ n → |b n - u| < ε ε : ℝ hε : 0 < ε ⊢ ∃ B, ∀ (n : ℕ), B ≤ n → |a n + b n - (t + u)| < ε
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 ha : ∃ B, ∀ (n : ℕ), B ≤ n → |a n - t| < ε / 2 ⊢ ∃ B, ∀ (n : ℕ), B ≤ n → |a n + b n - (t + u)| < ε
Please generate a tactic in lean4 to solve the state. STATE: a b : ℕ → ℝ t u : ℝ ha : ∀ (ε : ℝ), 0 < ε → ∃ B, ∀ (n : ℕ), B ≤ n → |a n - t| < ε hb : ∀ (ε : ℝ), 0 < ε → ∃ B, ∀ (n : ℕ), B ≤ n → |b n - u| < ε ε : ℝ hε : 0 < ε ⊢ ∃ B, ∀ (n : ℕ), B ≤ n → |a n + b n - (t + u)| < ε TACTIC:
https://github.com/ImperialCollegeLondon/formalising-mathematics-2024.git
b732ed1352e87b4474b0520d1383994e069f8057
FormalisingMathematics2024/Solutions/Section02reals/Sheet5.lean
Section2sheet5solutions.tendsTo_add
[38, 1]
[59, 13]
cases' ha with X hX
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 ha : ∃ B, ∀ (n : ℕ), B ≤ n → |a n - t| < ε / 2 ⊢ ∃ B, ∀ (n : ℕ), B ≤ n → |a n + b n - (t + u)| < ε
case intro 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 ⊢ ∃ B, ∀ (n : ℕ), B ≤ n → |a n + b n - (t + u)| < ε
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 ha : ∃ B, ∀ (n : ℕ), B ≤ n → |a n - t| < ε / 2 ⊢ ∃ B, ∀ (n : ℕ), B ≤ n → |a n + b n - (t + u)| < ε TACTIC:
https://github.com/ImperialCollegeLondon/formalising-mathematics-2024.git
b732ed1352e87b4474b0520d1383994e069f8057
FormalisingMathematics2024/Solutions/Section02reals/Sheet5.lean
Section2sheet5solutions.tendsTo_add
[38, 1]
[59, 13]
obtain ⟨Y, hY⟩ := hb (ε / 2) (by linarith)
case intro 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 ⊢ ∃ B, ∀ (n : ℕ), B ≤ n → |a n + b n - (t + u)| < ε
case intro.intro 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 Y : ℕ hY : ∀ (n : ℕ), Y ≤ n → |b n - u| < ε / 2 ⊢ ∃ B, ∀ (n : ℕ), B ≤ n → |a n + b n - (t + u)| < ε
Please generate a tactic in lean4 to solve the state. STATE: case intro 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 ⊢ ∃ B, ∀ (n : ℕ), B ≤ n → |a n + b n - (t + u)| < ε TACTIC:
https://github.com/ImperialCollegeLondon/formalising-mathematics-2024.git
b732ed1352e87b4474b0520d1383994e069f8057
FormalisingMathematics2024/Solutions/Section02reals/Sheet5.lean
Section2sheet5solutions.tendsTo_add
[38, 1]
[59, 13]
use max X Y
case intro.intro 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 Y : ℕ hY : ∀ (n : ℕ), Y ≤ n → |b n - u| < ε / 2 ⊢ ∃ B, ∀ (n : ℕ), B ≤ n → |a n + b n - (t + u)| < ε
case h 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 Y : ℕ hY : ∀ (n : ℕ), Y ≤ n → |b n - u| < ε / 2 ⊢ ∀ (n : ℕ), max X Y ≤ n → |a n + b n - (t + u)| < ε
Please generate a tactic in lean4 to solve the state. STATE: case intro.intro 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 Y : ℕ hY : ∀ (n : ℕ), Y ≤ n → |b n - u| < ε / 2 ⊢ ∃ B, ∀ (n : ℕ), B ≤ n → |a n + b n - (t + u)| < ε TACTIC:
https://github.com/ImperialCollegeLondon/formalising-mathematics-2024.git
b732ed1352e87b4474b0520d1383994e069f8057
FormalisingMathematics2024/Solutions/Section02reals/Sheet5.lean
Section2sheet5solutions.tendsTo_add
[38, 1]
[59, 13]
intro n hn
case h 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 Y : ℕ hY : ∀ (n : ℕ), Y ≤ n → |b n - u| < ε / 2 ⊢ ∀ (n : ℕ), max X Y ≤ n → |a n + b n - (t + u)| < ε
case h 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 Y : ℕ hY : ∀ (n : ℕ), Y ≤ n → |b n - u| < ε / 2 n : ℕ hn : max X Y ≤ n ⊢ |a n + b n - (t + u)| < ε
Please generate a tactic in lean4 to solve the state. STATE: case h 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 Y : ℕ hY : ∀ (n : ℕ), Y ≤ n → |b n - u| < ε / 2 ⊢ ∀ (n : ℕ), max X Y ≤ n → |a n + b n - (t + u)| < ε TACTIC:
https://github.com/ImperialCollegeLondon/formalising-mathematics-2024.git
b732ed1352e87b4474b0520d1383994e069f8057
FormalisingMathematics2024/Solutions/Section02reals/Sheet5.lean
Section2sheet5solutions.tendsTo_add
[38, 1]
[59, 13]
rw [max_le_iff] at hn
case h 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 Y : ℕ hY : ∀ (n : ℕ), Y ≤ n → |b n - u| < ε / 2 n : ℕ hn : max X Y ≤ n ⊢ |a n + b n - (t + u)| < ε
case h 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 Y : ℕ hY : ∀ (n : ℕ), Y ≤ n → |b n - u| < ε / 2 n : ℕ hn : X ≤ n ∧ Y ≤ n ⊢ |a n + b n - (t + u)| < ε
Please generate a tactic in lean4 to solve the state. STATE: case h 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 Y : ℕ hY : ∀ (n : ℕ), Y ≤ n → |b n - u| < ε / 2 n : ℕ hn : max X Y ≤ n ⊢ |a n + b n - (t + u)| < ε TACTIC:
https://github.com/ImperialCollegeLondon/formalising-mathematics-2024.git
b732ed1352e87b4474b0520d1383994e069f8057
FormalisingMathematics2024/Solutions/Section02reals/Sheet5.lean
Section2sheet5solutions.tendsTo_add
[38, 1]
[59, 13]
specialize hX n hn.1
case h 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 Y : ℕ hY : ∀ (n : ℕ), Y ≤ n → |b n - u| < ε / 2 n : ℕ hn : X ≤ n ∧ Y ≤ n ⊢ |a n + b n - (t + u)| < ε
case h 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 Y : ℕ hY : ∀ (n : ℕ), Y ≤ n → |b n - u| < ε / 2 n : ℕ hn : X ≤ n ∧ Y ≤ n hX : |a n - t| < ε / 2 ⊢ |a n + b n - (t + u)| < ε
Please generate a tactic in lean4 to solve the state. STATE: case h 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 Y : ℕ hY : ∀ (n : ℕ), Y ≤ n → |b n - u| < ε / 2 n : ℕ hn : X ≤ n ∧ Y ≤ n ⊢ |a n + b n - (t + u)| < ε TACTIC:
https://github.com/ImperialCollegeLondon/formalising-mathematics-2024.git
b732ed1352e87b4474b0520d1383994e069f8057
FormalisingMathematics2024/Solutions/Section02reals/Sheet5.lean
Section2sheet5solutions.tendsTo_add
[38, 1]
[59, 13]
specialize hY n hn.2
case h 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 Y : ℕ hY : ∀ (n : ℕ), Y ≤ n → |b n - u| < ε / 2 n : ℕ hn : X ≤ n ∧ Y ≤ n hX : |a n - t| < ε / 2 ⊢ |a n + b n - (t + u)| < ε
case h 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 Y n : ℕ hn : X ≤ n ∧ Y ≤ n hX : |a n - t| < ε / 2 hY : |b n - u| < ε / 2 ⊢ |a n + b n - (t + u)| < ε
Please generate a tactic in lean4 to solve the state. STATE: case h 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 Y : ℕ hY : ∀ (n : ℕ), Y ≤ n → |b n - u| < ε / 2 n : ℕ hn : X ≤ n ∧ Y ≤ n hX : |a n - t| < ε / 2 ⊢ |a n + b n - (t + u)| < ε TACTIC:
https://github.com/ImperialCollegeLondon/formalising-mathematics-2024.git
b732ed1352e87b4474b0520d1383994e069f8057
FormalisingMathematics2024/Solutions/Section02reals/Sheet5.lean
Section2sheet5solutions.tendsTo_add
[38, 1]
[59, 13]
rw [abs_lt] at *
case h 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 Y n : ℕ hn : X ≤ n ∧ Y ≤ n hX : |a n - t| < ε / 2 hY : |b n - u| < ε / 2 ⊢ |a n + b n - (t + u)| < ε
case h 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 Y n : ℕ hn : X ≤ n ∧ Y ≤ n hX : -(ε / 2) < a n - t ∧ a n - t < ε / 2 hY : -(ε / 2) < b n - u ∧ b n - u < ε / 2 ⊢ -ε < a n + b n - (t + u) ∧ a n + b n - (t + u) < ε
Please generate a tactic in lean4 to solve the state. STATE: case h 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 Y n : ℕ hn : X ≤ n ∧ Y ≤ n hX : |a n - t| < ε / 2 hY : |b n - u| < ε / 2 ⊢ |a n + b n - (t + u)| < ε TACTIC:
https://github.com/ImperialCollegeLondon/formalising-mathematics-2024.git
b732ed1352e87b4474b0520d1383994e069f8057
FormalisingMathematics2024/Solutions/Section02reals/Sheet5.lean
Section2sheet5solutions.tendsTo_add
[38, 1]
[59, 13]
constructor <;>linarith
case h 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 Y n : ℕ hn : X ≤ n ∧ Y ≤ n hX : -(ε / 2) < a n - t ∧ a n - t < ε / 2 hY : -(ε / 2) < b n - u ∧ b n - u < ε / 2 ⊢ -ε < a n + b n - (t + u) ∧ a n + b n - (t + u) < ε
no goals
Please generate a tactic in lean4 to solve the state. STATE: case h 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 Y n : ℕ hn : X ≤ n ∧ Y ≤ n hX : -(ε / 2) < a n - t ∧ a n - t < ε / 2 hY : -(ε / 2) < b n - u ∧ b n - u < ε / 2 ⊢ -ε < a n + b n - (t + u) ∧ a n + b n - (t + u) < ε TACTIC:
https://github.com/ImperialCollegeLondon/formalising-mathematics-2024.git
b732ed1352e87b4474b0520d1383994e069f8057
FormalisingMathematics2024/Solutions/Section02reals/Sheet5.lean
Section2sheet5solutions.tendsTo_add
[38, 1]
[59, 13]
linarith
a b : ℕ → ℝ t u : ℝ ha : ∀ (ε : ℝ), 0 < ε → ∃ B, ∀ (n : ℕ), B ≤ n → |a n - t| < ε hb : ∀ (ε : ℝ), 0 < ε → ∃ B, ∀ (n : ℕ), B ≤ n → |b n - u| < ε ε : ℝ hε : 0 < ε ⊢ 0 < ε / 2
no goals
Please generate a tactic in lean4 to solve the state. STATE: a b : ℕ → ℝ t u : ℝ ha : ∀ (ε : ℝ), 0 < ε → ∃ B, ∀ (n : ℕ), B ≤ n → |a n - t| < ε hb : ∀ (ε : ℝ), 0 < ε → ∃ B, ∀ (n : ℕ), B ≤ n → |b n - u| < ε ε : ℝ hε : 0 < ε ⊢ 0 < ε / 2 TACTIC:
https://github.com/ImperialCollegeLondon/formalising-mathematics-2024.git
b732ed1352e87b4474b0520d1383994e069f8057
FormalisingMathematics2024/Solutions/Section02reals/Sheet5.lean
Section2sheet5solutions.tendsTo_add
[38, 1]
[59, 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/Solutions/Section02reals/Sheet5.lean
Section2sheet5solutions.tendsTo_sub
[63, 1]
[65, 63]
simpa [sub_eq_add_neg] using tendsTo_add ha (tendsTo_neg hb)
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/Solutions/Section15numberTheory/Sheet3.lean
Section15Sheet3Solutions.divides_of_cong_four
[29, 1]
[35, 9]
constructor
t : ℕ ⊢ 5 ∣ 4 * (65 * t + 4) ^ 2 + 1 ∧ 13 ∣ 4 * (65 * t + 4) ^ 2 + 1
case left t : ℕ ⊢ 5 ∣ 4 * (65 * t + 4) ^ 2 + 1 case right t : ℕ ⊢ 13 ∣ 4 * (65 * t + 4) ^ 2 + 1
Please generate a tactic in lean4 to solve the state. STATE: t : ℕ ⊢ 5 ∣ 4 * (65 * t + 4) ^ 2 + 1 ∧ 13 ∣ 4 * (65 * t + 4) ^ 2 + 1 TACTIC:
https://github.com/ImperialCollegeLondon/formalising-mathematics-2024.git
b732ed1352e87b4474b0520d1383994e069f8057
FormalisingMathematics2024/Solutions/Section15numberTheory/Sheet3.lean
Section15Sheet3Solutions.divides_of_cong_four
[29, 1]
[35, 9]
use 3380 * t ^ 2 + 416 * t + 13
case left t : ℕ ⊢ 5 ∣ 4 * (65 * t + 4) ^ 2 + 1
case h t : ℕ ⊢ 4 * (65 * t + 4) ^ 2 + 1 = 5 * (3380 * t ^ 2 + 416 * t + 13)
Please generate a tactic in lean4 to solve the state. STATE: case left t : ℕ ⊢ 5 ∣ 4 * (65 * t + 4) ^ 2 + 1 TACTIC:
https://github.com/ImperialCollegeLondon/formalising-mathematics-2024.git
b732ed1352e87b4474b0520d1383994e069f8057
FormalisingMathematics2024/Solutions/Section15numberTheory/Sheet3.lean
Section15Sheet3Solutions.divides_of_cong_four
[29, 1]
[35, 9]
ring
case h t : ℕ ⊢ 4 * (65 * t + 4) ^ 2 + 1 = 5 * (3380 * t ^ 2 + 416 * t + 13)
no goals
Please generate a tactic in lean4 to solve the state. STATE: case h t : ℕ ⊢ 4 * (65 * t + 4) ^ 2 + 1 = 5 * (3380 * t ^ 2 + 416 * t + 13) TACTIC:
https://github.com/ImperialCollegeLondon/formalising-mathematics-2024.git
b732ed1352e87b4474b0520d1383994e069f8057
FormalisingMathematics2024/Solutions/Section15numberTheory/Sheet3.lean
Section15Sheet3Solutions.divides_of_cong_four
[29, 1]
[35, 9]
use 1300 * t ^ 2 + 160 * t + 5
case right t : ℕ ⊢ 13 ∣ 4 * (65 * t + 4) ^ 2 + 1
case h t : ℕ ⊢ 4 * (65 * t + 4) ^ 2 + 1 = 13 * (1300 * t ^ 2 + 160 * t + 5)
Please generate a tactic in lean4 to solve the state. STATE: case right t : ℕ ⊢ 13 ∣ 4 * (65 * t + 4) ^ 2 + 1 TACTIC:
https://github.com/ImperialCollegeLondon/formalising-mathematics-2024.git
b732ed1352e87b4474b0520d1383994e069f8057
FormalisingMathematics2024/Solutions/Section15numberTheory/Sheet3.lean
Section15Sheet3Solutions.divides_of_cong_four
[29, 1]
[35, 9]
ring
case h t : ℕ ⊢ 4 * (65 * t + 4) ^ 2 + 1 = 13 * (1300 * t ^ 2 + 160 * t + 5)
no goals
Please generate a tactic in lean4 to solve the state. STATE: case h t : ℕ ⊢ 4 * (65 * t + 4) ^ 2 + 1 = 13 * (1300 * t ^ 2 + 160 * t + 5) TACTIC:
https://github.com/ImperialCollegeLondon/formalising-mathematics-2024.git
b732ed1352e87b4474b0520d1383994e069f8057
FormalisingMathematics2024/Solutions/Section15numberTheory/Sheet3.lean
Section15Sheet3Solutions.arb_large_soln
[38, 1]
[45, 31]
intro N
⊢ ∀ (N : ℕ), ∃ n > N, 5 ∣ 4 * n ^ 2 + 1 ∧ 13 ∣ 4 * n ^ 2 + 1
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: ⊢ ∀ (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/Section15numberTheory/Sheet3.lean
Section15Sheet3Solutions.arb_large_soln
[38, 1]
[45, 31]
use 65 * N + 4
N : ℕ ⊢ ∃ n > N, 5 ∣ 4 * n ^ 2 + 1 ∧ 13 ∣ 4 * n ^ 2 + 1
case h N : ℕ ⊢ 65 * N + 4 > N ∧ 5 ∣ 4 * (65 * N + 4) ^ 2 + 1 ∧ 13 ∣ 4 * (65 * N + 4) ^ 2 + 1
Please generate a tactic in lean4 to solve the state. STATE: 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/Section15numberTheory/Sheet3.lean
Section15Sheet3Solutions.arb_large_soln
[38, 1]
[45, 31]
constructor
case h N : ℕ ⊢ 65 * N + 4 > N ∧ 5 ∣ 4 * (65 * N + 4) ^ 2 + 1 ∧ 13 ∣ 4 * (65 * N + 4) ^ 2 + 1
case h.left N : ℕ ⊢ 65 * N + 4 > N case h.right N : ℕ ⊢ 5 ∣ 4 * (65 * N + 4) ^ 2 + 1 ∧ 13 ∣ 4 * (65 * N + 4) ^ 2 + 1
Please generate a tactic in lean4 to solve the state. STATE: case h N : ℕ ⊢ 65 * N + 4 > N ∧ 5 ∣ 4 * (65 * N + 4) ^ 2 + 1 ∧ 13 ∣ 4 * (65 * N + 4) ^ 2 + 1 TACTIC:
https://github.com/ImperialCollegeLondon/formalising-mathematics-2024.git
b732ed1352e87b4474b0520d1383994e069f8057
FormalisingMathematics2024/Solutions/Section15numberTheory/Sheet3.lean
Section15Sheet3Solutions.arb_large_soln
[38, 1]
[45, 31]
linarith
case h.left N : ℕ ⊢ 65 * N + 4 > N
no goals
Please generate a tactic in lean4 to solve the state. STATE: case h.left N : ℕ ⊢ 65 * N + 4 > N TACTIC:
https://github.com/ImperialCollegeLondon/formalising-mathematics-2024.git
b732ed1352e87b4474b0520d1383994e069f8057
FormalisingMathematics2024/Solutions/Section15numberTheory/Sheet3.lean
Section15Sheet3Solutions.arb_large_soln
[38, 1]
[45, 31]
apply divides_of_cong_four
case h.right N : ℕ ⊢ 5 ∣ 4 * (65 * N + 4) ^ 2 + 1 ∧ 13 ∣ 4 * (65 * N + 4) ^ 2 + 1
no goals
Please generate a tactic in lean4 to solve the state. STATE: case h.right N : ℕ ⊢ 5 ∣ 4 * (65 * N + 4) ^ 2 + 1 ∧ 13 ∣ 4 * (65 * N + 4) ^ 2 + 1 TACTIC:
https://github.com/ImperialCollegeLondon/formalising-mathematics-2024.git
b732ed1352e87b4474b0520d1383994e069f8057
FormalisingMathematics2024/Solutions/Section15numberTheory/Sheet3.lean
Section15Sheet3Solutions.infinite_iff_arb_large
[51, 1]
[77, 13]
constructor
S : Set ℕ ⊢ Set.Infinite S ↔ ∀ (N : ℕ), ∃ n > N, n ∈ S
case mp S : Set ℕ ⊢ Set.Infinite S → ∀ (N : ℕ), ∃ n > N, n ∈ S case mpr S : Set ℕ ⊢ (∀ (N : ℕ), ∃ n > N, n ∈ S) → Set.Infinite S
Please generate a tactic in lean4 to solve the state. STATE: S : Set ℕ ⊢ Set.Infinite S ↔ ∀ (N : ℕ), ∃ n > N, n ∈ S TACTIC:
https://github.com/ImperialCollegeLondon/formalising-mathematics-2024.git
b732ed1352e87b4474b0520d1383994e069f8057
FormalisingMathematics2024/Solutions/Section15numberTheory/Sheet3.lean
Section15Sheet3Solutions.infinite_iff_arb_large
[51, 1]
[77, 13]
intro h n
case mp S : Set ℕ ⊢ Set.Infinite S → ∀ (N : ℕ), ∃ n > N, n ∈ S
case mp S : Set ℕ h : Set.Infinite S n : ℕ ⊢ ∃ n_1 > n, n_1 ∈ S
Please generate a tactic in lean4 to solve the state. STATE: case mp S : Set ℕ ⊢ Set.Infinite S → ∀ (N : ℕ), ∃ n > N, n ∈ S TACTIC:
https://github.com/ImperialCollegeLondon/formalising-mathematics-2024.git
b732ed1352e87b4474b0520d1383994e069f8057
FormalisingMathematics2024/Solutions/Section15numberTheory/Sheet3.lean
Section15Sheet3Solutions.infinite_iff_arb_large
[51, 1]
[77, 13]
have h2 := Set.Infinite.exists_not_mem_finset h (Finset.range (n + 1))
case mp S : Set ℕ h : Set.Infinite S n : ℕ ⊢ ∃ n_1 > n, n_1 ∈ S
case mp S : Set ℕ h : Set.Infinite S n : ℕ h2 : ∃ a ∈ S, a ∉ Finset.range (n + 1) ⊢ ∃ n_1 > n, n_1 ∈ S
Please generate a tactic in lean4 to solve the state. STATE: case mp S : Set ℕ h : Set.Infinite S n : ℕ ⊢ ∃ n_1 > n, n_1 ∈ S TACTIC:
https://github.com/ImperialCollegeLondon/formalising-mathematics-2024.git
b732ed1352e87b4474b0520d1383994e069f8057
FormalisingMathematics2024/Solutions/Section15numberTheory/Sheet3.lean
Section15Sheet3Solutions.infinite_iff_arb_large
[51, 1]
[77, 13]
rcases h2 with ⟨m, hm, h3⟩
case mp S : Set ℕ h : Set.Infinite S n : ℕ h2 : ∃ a ∈ S, a ∉ Finset.range (n + 1) ⊢ ∃ n_1 > n, n_1 ∈ S
case mp.intro.intro S : Set ℕ h : Set.Infinite S n m : ℕ hm : m ∈ S h3 : m ∉ Finset.range (n + 1) ⊢ ∃ n_1 > n, n_1 ∈ S
Please generate a tactic in lean4 to solve the state. STATE: case mp S : Set ℕ h : Set.Infinite S n : ℕ h2 : ∃ a ∈ S, a ∉ Finset.range (n + 1) ⊢ ∃ n_1 > n, n_1 ∈ S TACTIC:
https://github.com/ImperialCollegeLondon/formalising-mathematics-2024.git
b732ed1352e87b4474b0520d1383994e069f8057
FormalisingMathematics2024/Solutions/Section15numberTheory/Sheet3.lean
Section15Sheet3Solutions.infinite_iff_arb_large
[51, 1]
[77, 13]
use m
case mp.intro.intro S : Set ℕ h : Set.Infinite S n m : ℕ hm : m ∈ S h3 : m ∉ Finset.range (n + 1) ⊢ ∃ n_1 > n, n_1 ∈ S
case h S : Set ℕ h : Set.Infinite S n m : ℕ hm : m ∈ S h3 : m ∉ Finset.range (n + 1) ⊢ m > n ∧ m ∈ S
Please generate a tactic in lean4 to solve the state. STATE: case mp.intro.intro S : Set ℕ h : Set.Infinite S n m : ℕ hm : m ∈ S h3 : m ∉ Finset.range (n + 1) ⊢ ∃ n_1 > n, n_1 ∈ S TACTIC:
https://github.com/ImperialCollegeLondon/formalising-mathematics-2024.git
b732ed1352e87b4474b0520d1383994e069f8057
FormalisingMathematics2024/Solutions/Section15numberTheory/Sheet3.lean
Section15Sheet3Solutions.infinite_iff_arb_large
[51, 1]
[77, 13]
refine' ⟨_, hm⟩
case h S : Set ℕ h : Set.Infinite S n m : ℕ hm : m ∈ S h3 : m ∉ Finset.range (n + 1) ⊢ m > n ∧ m ∈ S
case h S : Set ℕ h : Set.Infinite S n m : ℕ hm : m ∈ S h3 : m ∉ Finset.range (n + 1) ⊢ m > n
Please generate a tactic in lean4 to solve the state. STATE: case h S : Set ℕ h : Set.Infinite S n m : ℕ hm : m ∈ S h3 : m ∉ Finset.range (n + 1) ⊢ m > n ∧ m ∈ S TACTIC:
https://github.com/ImperialCollegeLondon/formalising-mathematics-2024.git
b732ed1352e87b4474b0520d1383994e069f8057
FormalisingMathematics2024/Solutions/Section15numberTheory/Sheet3.lean
Section15Sheet3Solutions.infinite_iff_arb_large
[51, 1]
[77, 13]
contrapose! h3
case h S : Set ℕ h : Set.Infinite S n m : ℕ hm : m ∈ S h3 : m ∉ Finset.range (n + 1) ⊢ m > n
case h S : Set ℕ h : Set.Infinite S n m : ℕ hm : m ∈ S h3 : m ≤ n ⊢ m ∈ Finset.range (n + 1)
Please generate a tactic in lean4 to solve the state. STATE: case h S : Set ℕ h : Set.Infinite S n m : ℕ hm : m ∈ S h3 : m ∉ Finset.range (n + 1) ⊢ m > n TACTIC:
https://github.com/ImperialCollegeLondon/formalising-mathematics-2024.git
b732ed1352e87b4474b0520d1383994e069f8057
FormalisingMathematics2024/Solutions/Section15numberTheory/Sheet3.lean
Section15Sheet3Solutions.infinite_iff_arb_large
[51, 1]
[77, 13]
exact Finset.mem_range_succ_iff.mpr h3
case h S : Set ℕ h : Set.Infinite S n m : ℕ hm : m ∈ S h3 : m ≤ n ⊢ m ∈ Finset.range (n + 1)
no goals
Please generate a tactic in lean4 to solve the state. STATE: case h S : Set ℕ h : Set.Infinite S n m : ℕ hm : m ∈ S h3 : m ≤ n ⊢ m ∈ Finset.range (n + 1) TACTIC:
https://github.com/ImperialCollegeLondon/formalising-mathematics-2024.git
b732ed1352e87b4474b0520d1383994e069f8057
FormalisingMathematics2024/Solutions/Section15numberTheory/Sheet3.lean
Section15Sheet3Solutions.infinite_iff_arb_large
[51, 1]
[77, 13]
contrapose!
case mpr S : Set ℕ ⊢ (∀ (N : ℕ), ∃ n > N, n ∈ S) → Set.Infinite S
case mpr S : Set ℕ ⊢ ¬Set.Infinite S → ∃ N, ∀ n > N, n ∉ S
Please generate a tactic in lean4 to solve the state. STATE: case mpr S : Set ℕ ⊢ (∀ (N : ℕ), ∃ n > N, n ∈ S) → Set.Infinite S TACTIC:
https://github.com/ImperialCollegeLondon/formalising-mathematics-2024.git
b732ed1352e87b4474b0520d1383994e069f8057
FormalisingMathematics2024/Solutions/Section15numberTheory/Sheet3.lean
Section15Sheet3Solutions.infinite_iff_arb_large
[51, 1]
[77, 13]
intro h
case mpr S : Set ℕ ⊢ ¬Set.Infinite S → ∃ N, ∀ n > N, n ∉ S
case mpr S : Set ℕ h : ¬Set.Infinite S ⊢ ∃ N, ∀ n > N, n ∉ S
Please generate a tactic in lean4 to solve the state. STATE: case mpr S : Set ℕ ⊢ ¬Set.Infinite S → ∃ N, ∀ n > N, n ∉ S TACTIC:
https://github.com/ImperialCollegeLondon/formalising-mathematics-2024.git
b732ed1352e87b4474b0520d1383994e069f8057
FormalisingMathematics2024/Solutions/Section15numberTheory/Sheet3.lean
Section15Sheet3Solutions.infinite_iff_arb_large
[51, 1]
[77, 13]
rw [Set.not_infinite] at h
case mpr S : Set ℕ h : ¬Set.Infinite S ⊢ ∃ N, ∀ n > N, n ∉ S
case mpr S : Set ℕ h : Set.Finite S ⊢ ∃ N, ∀ n > N, n ∉ S
Please generate a tactic in lean4 to solve the state. STATE: case mpr S : Set ℕ h : ¬Set.Infinite S ⊢ ∃ N, ∀ n > N, n ∉ S TACTIC:
https://github.com/ImperialCollegeLondon/formalising-mathematics-2024.git
b732ed1352e87b4474b0520d1383994e069f8057
FormalisingMathematics2024/Solutions/Section15numberTheory/Sheet3.lean
Section15Sheet3Solutions.infinite_iff_arb_large
[51, 1]
[77, 13]
let S2 : Finset ℕ := Set.Finite.toFinset h
case mpr S : Set ℕ h : Set.Finite S ⊢ ∃ N, ∀ n > N, n ∉ S
case mpr S : Set ℕ h : Set.Finite S S2 : Finset ℕ := Set.Finite.toFinset h ⊢ ∃ N, ∀ n > N, n ∉ S
Please generate a tactic in lean4 to solve the state. STATE: case mpr S : Set ℕ h : Set.Finite S ⊢ ∃ N, ∀ n > N, n ∉ S TACTIC:
https://github.com/ImperialCollegeLondon/formalising-mathematics-2024.git
b732ed1352e87b4474b0520d1383994e069f8057
FormalisingMathematics2024/Solutions/Section15numberTheory/Sheet3.lean
Section15Sheet3Solutions.infinite_iff_arb_large
[51, 1]
[77, 13]
have h2 : ∃ B, ∀ n ∈ S2, n ≤ B := by use Finset.sup S2 id intros _ H apply Finset.le_sup H
case mpr S : Set ℕ h : Set.Finite S S2 : Finset ℕ := Set.Finite.toFinset h ⊢ ∃ N, ∀ n > N, n ∉ S
case mpr S : Set ℕ h : Set.Finite S S2 : Finset ℕ := Set.Finite.toFinset h h2 : ∃ B, ∀ n ∈ S2, n ≤ B ⊢ ∃ N, ∀ n > N, n ∉ S
Please generate a tactic in lean4 to solve the state. STATE: case mpr S : Set ℕ h : Set.Finite S S2 : Finset ℕ := Set.Finite.toFinset h ⊢ ∃ N, ∀ n > N, n ∉ S TACTIC:
https://github.com/ImperialCollegeLondon/formalising-mathematics-2024.git
b732ed1352e87b4474b0520d1383994e069f8057
FormalisingMathematics2024/Solutions/Section15numberTheory/Sheet3.lean
Section15Sheet3Solutions.infinite_iff_arb_large
[51, 1]
[77, 13]
cases' h2 with N hN
case mpr S : Set ℕ h : Set.Finite S S2 : Finset ℕ := Set.Finite.toFinset h h2 : ∃ B, ∀ n ∈ S2, n ≤ B ⊢ ∃ N, ∀ n > N, n ∉ S
case mpr.intro S : Set ℕ h : Set.Finite S S2 : Finset ℕ := Set.Finite.toFinset h N : ℕ hN : ∀ n ∈ S2, n ≤ N ⊢ ∃ N, ∀ n > N, n ∉ S
Please generate a tactic in lean4 to solve the state. STATE: case mpr S : Set ℕ h : Set.Finite S S2 : Finset ℕ := Set.Finite.toFinset h h2 : ∃ B, ∀ n ∈ S2, n ≤ B ⊢ ∃ N, ∀ n > N, n ∉ S TACTIC:
https://github.com/ImperialCollegeLondon/formalising-mathematics-2024.git
b732ed1352e87b4474b0520d1383994e069f8057
FormalisingMathematics2024/Solutions/Section15numberTheory/Sheet3.lean
Section15Sheet3Solutions.infinite_iff_arb_large
[51, 1]
[77, 13]
use N
case mpr.intro S : Set ℕ h : Set.Finite S S2 : Finset ℕ := Set.Finite.toFinset h N : ℕ hN : ∀ n ∈ S2, n ≤ N ⊢ ∃ N, ∀ n > N, n ∉ S
case h S : Set ℕ h : Set.Finite S S2 : Finset ℕ := Set.Finite.toFinset h N : ℕ hN : ∀ n ∈ S2, n ≤ N ⊢ ∀ n > N, n ∉ S
Please generate a tactic in lean4 to solve the state. STATE: case mpr.intro S : Set ℕ h : Set.Finite S S2 : Finset ℕ := Set.Finite.toFinset h N : ℕ hN : ∀ n ∈ S2, n ≤ N ⊢ ∃ N, ∀ n > N, n ∉ S TACTIC:
https://github.com/ImperialCollegeLondon/formalising-mathematics-2024.git
b732ed1352e87b4474b0520d1383994e069f8057
FormalisingMathematics2024/Solutions/Section15numberTheory/Sheet3.lean
Section15Sheet3Solutions.infinite_iff_arb_large
[51, 1]
[77, 13]
have h3 : ∀ n : ℕ, n ∈ S ↔ n ∈ S2
case h S : Set ℕ h : Set.Finite S S2 : Finset ℕ := Set.Finite.toFinset h N : ℕ hN : ∀ n ∈ S2, n ≤ N ⊢ ∀ n > N, n ∉ S
case h3 S : Set ℕ h : Set.Finite S S2 : Finset ℕ := Set.Finite.toFinset h N : ℕ hN : ∀ n ∈ S2, n ≤ N ⊢ ∀ (n : ℕ), n ∈ S ↔ n ∈ S2 case h S : Set ℕ h : Set.Finite S S2 : Finset ℕ := Set.Finite.toFinset h N : ℕ hN : ∀ n ∈ S2, n ≤ N h3 : ∀ (n : ℕ), n ∈ S ↔ n ∈ S2 ⊢ ∀ n > N, n ∉ S
Please generate a tactic in lean4 to solve the state. STATE: case h S : Set ℕ h : Set.Finite S S2 : Finset ℕ := Set.Finite.toFinset h N : ℕ hN : ∀ n ∈ S2, n ≤ N ⊢ ∀ n > N, n ∉ S TACTIC:
https://github.com/ImperialCollegeLondon/formalising-mathematics-2024.git
b732ed1352e87b4474b0520d1383994e069f8057
FormalisingMathematics2024/Solutions/Section15numberTheory/Sheet3.lean
Section15Sheet3Solutions.infinite_iff_arb_large
[51, 1]
[77, 13]
intro n
case h3 S : Set ℕ h : Set.Finite S S2 : Finset ℕ := Set.Finite.toFinset h N : ℕ hN : ∀ n ∈ S2, n ≤ N ⊢ ∀ (n : ℕ), n ∈ S ↔ n ∈ S2 case h S : Set ℕ h : Set.Finite S S2 : Finset ℕ := Set.Finite.toFinset h N : ℕ hN : ∀ n ∈ S2, n ≤ N h3 : ∀ (n : ℕ), n ∈ S ↔ n ∈ S2 ⊢ ∀ n > N, n ∉ S
case h3 S : Set ℕ h : Set.Finite S S2 : Finset ℕ := Set.Finite.toFinset h N : ℕ hN : ∀ n ∈ S2, n ≤ N n : ℕ ⊢ n ∈ S ↔ n ∈ S2 case h S : Set ℕ h : Set.Finite S S2 : Finset ℕ := Set.Finite.toFinset h N : ℕ hN : ∀ n ∈ S2, n ≤ N h3 : ∀ (n : ℕ), n ∈ S ↔ n ∈ S2 ⊢ ∀ n > N, n ∉ S
Please generate a tactic in lean4 to solve the state. STATE: case h3 S : Set ℕ h : Set.Finite S S2 : Finset ℕ := Set.Finite.toFinset h N : ℕ hN : ∀ n ∈ S2, n ≤ N ⊢ ∀ (n : ℕ), n ∈ S ↔ n ∈ S2 case h S : Set ℕ h : Set.Finite S S2 : Finset ℕ := Set.Finite.toFinset h N : ℕ hN : ∀ n ∈ S2, n ≤ N h3 : ∀ (n : ℕ), n ∈ S ↔ n ∈ S2 ⊢ ∀ n > N, n ∉ S TACTIC:
https://github.com/ImperialCollegeLondon/formalising-mathematics-2024.git
b732ed1352e87b4474b0520d1383994e069f8057
FormalisingMathematics2024/Solutions/Section15numberTheory/Sheet3.lean
Section15Sheet3Solutions.infinite_iff_arb_large
[51, 1]
[77, 13]
exact (Set.Finite.mem_toFinset h).symm
case h3 S : Set ℕ h : Set.Finite S S2 : Finset ℕ := Set.Finite.toFinset h N : ℕ hN : ∀ n ∈ S2, n ≤ N n : ℕ ⊢ n ∈ S ↔ n ∈ S2 case h S : Set ℕ h : Set.Finite S S2 : Finset ℕ := Set.Finite.toFinset h N : ℕ hN : ∀ n ∈ S2, n ≤ N h3 : ∀ (n : ℕ), n ∈ S ↔ n ∈ S2 ⊢ ∀ n > N, n ∉ S
case h S : Set ℕ h : Set.Finite S S2 : Finset ℕ := Set.Finite.toFinset h N : ℕ hN : ∀ n ∈ S2, n ≤ N h3 : ∀ (n : ℕ), n ∈ S ↔ n ∈ S2 ⊢ ∀ n > N, n ∉ S
Please generate a tactic in lean4 to solve the state. STATE: case h3 S : Set ℕ h : Set.Finite S S2 : Finset ℕ := Set.Finite.toFinset h N : ℕ hN : ∀ n ∈ S2, n ≤ N n : ℕ ⊢ n ∈ S ↔ n ∈ S2 case h S : Set ℕ h : Set.Finite S S2 : Finset ℕ := Set.Finite.toFinset h N : ℕ hN : ∀ n ∈ S2, n ≤ N h3 : ∀ (n : ℕ), n ∈ S ↔ n ∈ S2 ⊢ ∀ n > N, n ∉ S TACTIC:
https://github.com/ImperialCollegeLondon/formalising-mathematics-2024.git
b732ed1352e87b4474b0520d1383994e069f8057
FormalisingMathematics2024/Solutions/Section15numberTheory/Sheet3.lean
Section15Sheet3Solutions.infinite_iff_arb_large
[51, 1]
[77, 13]
intro n hn h4
case h S : Set ℕ h : Set.Finite S S2 : Finset ℕ := Set.Finite.toFinset h N : ℕ hN : ∀ n ∈ S2, n ≤ N h3 : ∀ (n : ℕ), n ∈ S ↔ n ∈ S2 ⊢ ∀ n > N, n ∉ S
case h S : Set ℕ h : Set.Finite S S2 : Finset ℕ := Set.Finite.toFinset h N : ℕ hN : ∀ n ∈ S2, n ≤ N h3 : ∀ (n : ℕ), n ∈ S ↔ n ∈ S2 n : ℕ hn : n > N h4 : n ∈ S ⊢ False
Please generate a tactic in lean4 to solve the state. STATE: case h S : Set ℕ h : Set.Finite S S2 : Finset ℕ := Set.Finite.toFinset h N : ℕ hN : ∀ n ∈ S2, n ≤ N h3 : ∀ (n : ℕ), n ∈ S ↔ n ∈ S2 ⊢ ∀ n > N, n ∉ S TACTIC:
https://github.com/ImperialCollegeLondon/formalising-mathematics-2024.git
b732ed1352e87b4474b0520d1383994e069f8057
FormalisingMathematics2024/Solutions/Section15numberTheory/Sheet3.lean
Section15Sheet3Solutions.infinite_iff_arb_large
[51, 1]
[77, 13]
rw [h3] at h4
case h S : Set ℕ h : Set.Finite S S2 : Finset ℕ := Set.Finite.toFinset h N : ℕ hN : ∀ n ∈ S2, n ≤ N h3 : ∀ (n : ℕ), n ∈ S ↔ n ∈ S2 n : ℕ hn : n > N h4 : n ∈ S ⊢ False
case h S : Set ℕ h : Set.Finite S S2 : Finset ℕ := Set.Finite.toFinset h N : ℕ hN : ∀ n ∈ S2, n ≤ N h3 : ∀ (n : ℕ), n ∈ S ↔ n ∈ S2 n : ℕ hn : n > N h4 : n ∈ S2 ⊢ False
Please generate a tactic in lean4 to solve the state. STATE: case h S : Set ℕ h : Set.Finite S S2 : Finset ℕ := Set.Finite.toFinset h N : ℕ hN : ∀ n ∈ S2, n ≤ N h3 : ∀ (n : ℕ), n ∈ S ↔ n ∈ S2 n : ℕ hn : n > N h4 : n ∈ S ⊢ False TACTIC:
https://github.com/ImperialCollegeLondon/formalising-mathematics-2024.git
b732ed1352e87b4474b0520d1383994e069f8057
FormalisingMathematics2024/Solutions/Section15numberTheory/Sheet3.lean
Section15Sheet3Solutions.infinite_iff_arb_large
[51, 1]
[77, 13]
specialize hN n h4
case h S : Set ℕ h : Set.Finite S S2 : Finset ℕ := Set.Finite.toFinset h N : ℕ hN : ∀ n ∈ S2, n ≤ N h3 : ∀ (n : ℕ), n ∈ S ↔ n ∈ S2 n : ℕ hn : n > N h4 : n ∈ S2 ⊢ False
case h S : Set ℕ h : Set.Finite S S2 : Finset ℕ := Set.Finite.toFinset h N : ℕ h3 : ∀ (n : ℕ), n ∈ S ↔ n ∈ S2 n : ℕ hn : n > N h4 : n ∈ S2 hN : n ≤ N ⊢ False
Please generate a tactic in lean4 to solve the state. STATE: case h S : Set ℕ h : Set.Finite S S2 : Finset ℕ := Set.Finite.toFinset h N : ℕ hN : ∀ n ∈ S2, n ≤ N h3 : ∀ (n : ℕ), n ∈ S ↔ n ∈ S2 n : ℕ hn : n > N h4 : n ∈ S2 ⊢ False TACTIC:
https://github.com/ImperialCollegeLondon/formalising-mathematics-2024.git
b732ed1352e87b4474b0520d1383994e069f8057
FormalisingMathematics2024/Solutions/Section15numberTheory/Sheet3.lean
Section15Sheet3Solutions.infinite_iff_arb_large
[51, 1]
[77, 13]
linarith
case h S : Set ℕ h : Set.Finite S S2 : Finset ℕ := Set.Finite.toFinset h N : ℕ h3 : ∀ (n : ℕ), n ∈ S ↔ n ∈ S2 n : ℕ hn : n > N h4 : n ∈ S2 hN : n ≤ N ⊢ False
no goals
Please generate a tactic in lean4 to solve the state. STATE: case h S : Set ℕ h : Set.Finite S S2 : Finset ℕ := Set.Finite.toFinset h N : ℕ h3 : ∀ (n : ℕ), n ∈ S ↔ n ∈ S2 n : ℕ hn : n > N h4 : n ∈ S2 hN : n ≤ N ⊢ False TACTIC:
https://github.com/ImperialCollegeLondon/formalising-mathematics-2024.git
b732ed1352e87b4474b0520d1383994e069f8057
FormalisingMathematics2024/Solutions/Section15numberTheory/Sheet3.lean
Section15Sheet3Solutions.infinite_iff_arb_large
[51, 1]
[77, 13]
use Finset.sup S2 id
S : Set ℕ h : Set.Finite S S2 : Finset ℕ := Set.Finite.toFinset h ⊢ ∃ B, ∀ n ∈ S2, n ≤ B
case h S : Set ℕ h : Set.Finite S S2 : Finset ℕ := Set.Finite.toFinset h ⊢ ∀ n ∈ S2, n ≤ Finset.sup S2 id
Please generate a tactic in lean4 to solve the state. STATE: S : Set ℕ h : Set.Finite S S2 : Finset ℕ := Set.Finite.toFinset h ⊢ ∃ B, ∀ n ∈ S2, n ≤ B TACTIC:
https://github.com/ImperialCollegeLondon/formalising-mathematics-2024.git
b732ed1352e87b4474b0520d1383994e069f8057
FormalisingMathematics2024/Solutions/Section15numberTheory/Sheet3.lean
Section15Sheet3Solutions.infinite_iff_arb_large
[51, 1]
[77, 13]
intros _ H
case h S : Set ℕ h : Set.Finite S S2 : Finset ℕ := Set.Finite.toFinset h ⊢ ∀ n ∈ S2, n ≤ Finset.sup S2 id
case h S : Set ℕ h : Set.Finite S S2 : Finset ℕ := Set.Finite.toFinset h n✝ : ℕ H : n✝ ∈ S2 ⊢ n✝ ≤ Finset.sup S2 id
Please generate a tactic in lean4 to solve the state. STATE: case h S : Set ℕ h : Set.Finite S S2 : Finset ℕ := Set.Finite.toFinset h ⊢ ∀ n ∈ S2, n ≤ Finset.sup S2 id TACTIC:
https://github.com/ImperialCollegeLondon/formalising-mathematics-2024.git
b732ed1352e87b4474b0520d1383994e069f8057
FormalisingMathematics2024/Solutions/Section15numberTheory/Sheet3.lean
Section15Sheet3Solutions.infinite_iff_arb_large
[51, 1]
[77, 13]
apply Finset.le_sup H
case h S : Set ℕ h : Set.Finite S S2 : Finset ℕ := Set.Finite.toFinset h n✝ : ℕ H : n✝ ∈ S2 ⊢ n✝ ≤ Finset.sup S2 id
no goals
Please generate a tactic in lean4 to solve the state. STATE: case h S : Set ℕ h : Set.Finite S S2 : Finset ℕ := Set.Finite.toFinset h n✝ : ℕ H : n✝ ∈ S2 ⊢ n✝ ≤ Finset.sup S2 id TACTIC:
https://github.com/ImperialCollegeLondon/formalising-mathematics-2024.git
b732ed1352e87b4474b0520d1383994e069f8057
FormalisingMathematics2024/Solutions/Section15numberTheory/Sheet3.lean
Section15Sheet3Solutions.infinite_setOf_solutions
[81, 1]
[84, 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/Solutions/Section15numberTheory/Sheet3.lean
Section15Sheet3Solutions.infinite_setOf_solutions
[81, 1]
[84, 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/Section04sets/Sheet1.lean
Section4sheet1.subset_def
[57, 1]
[59, 6]
rfl
X : Type A B C D : Set X ⊢ A ⊆ B ↔ ∀ x ∈ A, x ∈ B
no goals
Please generate a tactic in lean4 to solve the state. STATE: X : Type A B C D : Set X ⊢ A ⊆ B ↔ ∀ x ∈ A, x ∈ B TACTIC:
https://github.com/ImperialCollegeLondon/formalising-mathematics-2024.git
b732ed1352e87b4474b0520d1383994e069f8057
FormalisingMathematics2024/Section04sets/Sheet1.lean
Section4sheet1.mem_union_iff
[64, 1]
[65, 6]
rfl
X : Type A B C D : Set X x : X ⊢ x ∈ A ∪ B ↔ x ∈ A ∨ x ∈ B
no goals
Please generate a tactic in lean4 to solve the state. STATE: X : Type A B C D : Set X x : X ⊢ x ∈ A ∪ B ↔ x ∈ A ∨ x ∈ B TACTIC:
https://github.com/ImperialCollegeLondon/formalising-mathematics-2024.git
b732ed1352e87b4474b0520d1383994e069f8057
FormalisingMathematics2024/Section05groups/Sheet2.lean
Section5sheet2.WeakGroup.mul_left_cancel
[56, 1]
[56, 64]
sorry
G : Type inst✝ : WeakGroup G a b c : G h : a * b = a * c ⊢ b = c
no goals
Please generate a tactic in lean4 to solve the state. STATE: G : Type inst✝ : WeakGroup G a b c : G h : a * b = a * c ⊢ b = c TACTIC:
https://github.com/ImperialCollegeLondon/formalising-mathematics-2024.git
b732ed1352e87b4474b0520d1383994e069f8057
FormalisingMathematics2024/Section05groups/Sheet2.lean
Section5sheet2.WeakGroup.mul_eq_of_eq_inv_mul
[58, 1]
[58, 71]
sorry
G : Type inst✝ : WeakGroup G a b c : G h : b = a⁻¹ * c ⊢ a * b = c
no goals
Please generate a tactic in lean4 to solve the state. STATE: G : Type inst✝ : WeakGroup G a b c : G h : b = a⁻¹ * c ⊢ a * b = c TACTIC:
https://github.com/ImperialCollegeLondon/formalising-mathematics-2024.git
b732ed1352e87b4474b0520d1383994e069f8057
FormalisingMathematics2024/Section05groups/Sheet2.lean
Section5sheet2.WeakGroup.mul_one
[60, 1]
[60, 48]
sorry
G : Type inst✝ : WeakGroup G a✝ b c a : G ⊢ a * 1 = a
no goals
Please generate a tactic in lean4 to solve the state. STATE: G : Type inst✝ : WeakGroup G a✝ b c a : G ⊢ a * 1 = a TACTIC:
https://github.com/ImperialCollegeLondon/formalising-mathematics-2024.git
b732ed1352e87b4474b0520d1383994e069f8057
FormalisingMathematics2024/Section05groups/Sheet2.lean
Section5sheet2.WeakGroup.mul_inv_self
[62, 1]
[62, 55]
sorry
G : Type inst✝ : WeakGroup G a✝ b c a : G ⊢ a * a⁻¹ = 1
no goals
Please generate a tactic in lean4 to solve the state. STATE: G : Type inst✝ : WeakGroup G a✝ b c a : G ⊢ a * a⁻¹ = 1 TACTIC:
https://github.com/ImperialCollegeLondon/formalising-mathematics-2024.git
b732ed1352e87b4474b0520d1383994e069f8057
FormalisingMathematics2024/Solutions/Section03functions/Sheet3.lean
Section3sheet1solutions.Yb_ne_Yc
[50, 1]
[53, 10]
intro h
⊢ Y.b ≠ Y.c
h : Y.b = Y.c ⊢ False
Please generate a tactic in lean4 to solve the state. STATE: ⊢ Y.b ≠ Y.c TACTIC:
https://github.com/ImperialCollegeLondon/formalising-mathematics-2024.git
b732ed1352e87b4474b0520d1383994e069f8057
FormalisingMathematics2024/Solutions/Section03functions/Sheet3.lean
Section3sheet1solutions.Yb_ne_Yc
[50, 1]
[53, 10]
cases h
h : Y.b = Y.c ⊢ False
no goals
Please generate a tactic in lean4 to solve the state. STATE: h : Y.b = Y.c ⊢ False TACTIC:
https://github.com/ImperialCollegeLondon/formalising-mathematics-2024.git
b732ed1352e87b4474b0520d1383994e069f8057
FormalisingMathematics2024/Solutions/Section03functions/Sheet3.lean
Section3sheet1solutions.gYb_eq_gYc
[56, 1]
[58, 6]
rfl
⊢ g Y.b = g Y.c
no goals
Please generate a tactic in lean4 to solve the state. STATE: ⊢ g Y.b = g Y.c TACTIC:
https://github.com/ImperialCollegeLondon/formalising-mathematics-2024.git
b732ed1352e87b4474b0520d1383994e069f8057
FormalisingMathematics2024/Solutions/Section03functions/Sheet3.lean
Section3sheet1solutions.gf_injective
[62, 1]
[66, 6]
rintro ⟨_⟩ ⟨_⟩ _
⊢ Injective (g ∘ f)
case a.a a✝ : (g ∘ f) X.a = (g ∘ f) X.a ⊢ X.a = X.a
Please generate a tactic in lean4 to solve the state. STATE: ⊢ Injective (g ∘ f) TACTIC:
https://github.com/ImperialCollegeLondon/formalising-mathematics-2024.git
b732ed1352e87b4474b0520d1383994e069f8057
FormalisingMathematics2024/Solutions/Section03functions/Sheet3.lean
Section3sheet1solutions.gf_injective
[62, 1]
[66, 6]
rfl
case a.a a✝ : (g ∘ f) X.a = (g ∘ f) X.a ⊢ X.a = X.a
no goals
Please generate a tactic in lean4 to solve the state. STATE: case a.a a✝ : (g ∘ f) X.a = (g ∘ f) X.a ⊢ X.a = X.a TACTIC:
https://github.com/ImperialCollegeLondon/formalising-mathematics-2024.git
b732ed1352e87b4474b0520d1383994e069f8057
FormalisingMathematics2024/Solutions/Section03functions/Sheet3.lean
Section3sheet1solutions.gf_surjective
[78, 1]
[80, 10]
intro z
⊢ Surjective (g ∘ f)
z : Z ⊢ ∃ a, (g ∘ f) a = z
Please generate a tactic in lean4 to solve the state. STATE: ⊢ Surjective (g ∘ f) TACTIC:
https://github.com/ImperialCollegeLondon/formalising-mathematics-2024.git
b732ed1352e87b4474b0520d1383994e069f8057
FormalisingMathematics2024/Solutions/Section03functions/Sheet3.lean
Section3sheet1solutions.gf_surjective
[78, 1]
[80, 10]
use X.a
z : Z ⊢ ∃ a, (g ∘ f) a = z
no goals
Please generate a tactic in lean4 to solve the state. STATE: z : Z ⊢ ∃ a, (g ∘ f) a = z TACTIC:
https://github.com/ImperialCollegeLondon/formalising-mathematics-2024.git
b732ed1352e87b4474b0520d1383994e069f8057
FormalisingMathematics2024/Section02reals/Sheet5.lean
Section2sheet5.tendsTo_neg
[16, 1]
[17, 8]
sorry
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/Section02reals/Sheet5.lean
Section2sheet5.tendsTo_add
[31, 1]
[52, 13]
rw [tendsTo_def] 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 : ∀ (ε : ℝ), 0 < ε → ∃ B, ∀ (n : ℕ), B ≤ n → |a n - t| < ε hb : ∀ (ε : ℝ), 0 < ε → ∃ B, ∀ (n : ℕ), B ≤ n → |b n - u| < ε ⊢ ∀ (ε : ℝ), 0 < ε → ∃ B, ∀ (n : ℕ), B ≤ n → |a n + b n - (t + u)| < ε
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/Section02reals/Sheet5.lean
Section2sheet5.tendsTo_add
[31, 1]
[52, 13]
intro ε hε
a b : ℕ → ℝ t u : ℝ ha : ∀ (ε : ℝ), 0 < ε → ∃ B, ∀ (n : ℕ), B ≤ n → |a n - t| < ε hb : ∀ (ε : ℝ), 0 < ε → ∃ B, ∀ (n : ℕ), B ≤ n → |b n - u| < ε ⊢ ∀ (ε : ℝ), 0 < ε → ∃ B, ∀ (n : ℕ), B ≤ n → |a n + b n - (t + u)| < ε
a b : ℕ → ℝ t u : ℝ ha : ∀ (ε : ℝ), 0 < ε → ∃ B, ∀ (n : ℕ), B ≤ n → |a n - t| < ε hb : ∀ (ε : ℝ), 0 < ε → ∃ B, ∀ (n : ℕ), B ≤ n → |b n - u| < ε ε : ℝ hε : 0 < ε ⊢ ∃ B, ∀ (n : ℕ), B ≤ n → |a n + b n - (t + u)| < ε
Please generate a tactic in lean4 to solve the state. STATE: a b : ℕ → ℝ t u : ℝ ha : ∀ (ε : ℝ), 0 < ε → ∃ B, ∀ (n : ℕ), B ≤ n → |a n - t| < ε hb : ∀ (ε : ℝ), 0 < ε → ∃ B, ∀ (n : ℕ), B ≤ n → |b n - u| < ε ⊢ ∀ (ε : ℝ), 0 < ε → ∃ B, ∀ (n : ℕ), B ≤ n → |a n + b n - (t + u)| < ε TACTIC:
https://github.com/ImperialCollegeLondon/formalising-mathematics-2024.git
b732ed1352e87b4474b0520d1383994e069f8057
FormalisingMathematics2024/Section02reals/Sheet5.lean
Section2sheet5.tendsTo_add
[31, 1]
[52, 13]
specialize ha (ε / 2) (by linarith)
a b : ℕ → ℝ t u : ℝ ha : ∀ (ε : ℝ), 0 < ε → ∃ B, ∀ (n : ℕ), B ≤ n → |a n - t| < ε hb : ∀ (ε : ℝ), 0 < ε → ∃ B, ∀ (n : ℕ), B ≤ n → |b n - u| < ε ε : ℝ hε : 0 < ε ⊢ ∃ B, ∀ (n : ℕ), B ≤ n → |a n + b n - (t + u)| < ε
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 ha : ∃ B, ∀ (n : ℕ), B ≤ n → |a n - t| < ε / 2 ⊢ ∃ B, ∀ (n : ℕ), B ≤ n → |a n + b n - (t + u)| < ε
Please generate a tactic in lean4 to solve the state. STATE: a b : ℕ → ℝ t u : ℝ ha : ∀ (ε : ℝ), 0 < ε → ∃ B, ∀ (n : ℕ), B ≤ n → |a n - t| < ε hb : ∀ (ε : ℝ), 0 < ε → ∃ B, ∀ (n : ℕ), B ≤ n → |b n - u| < ε ε : ℝ hε : 0 < ε ⊢ ∃ B, ∀ (n : ℕ), B ≤ n → |a n + b n - (t + u)| < ε TACTIC:
https://github.com/ImperialCollegeLondon/formalising-mathematics-2024.git
b732ed1352e87b4474b0520d1383994e069f8057
FormalisingMathematics2024/Section02reals/Sheet5.lean
Section2sheet5.tendsTo_add
[31, 1]
[52, 13]
cases' ha with X hX
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 ha : ∃ B, ∀ (n : ℕ), B ≤ n → |a n - t| < ε / 2 ⊢ ∃ B, ∀ (n : ℕ), B ≤ n → |a n + b n - (t + u)| < ε
case intro 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 ⊢ ∃ B, ∀ (n : ℕ), B ≤ n → |a n + b n - (t + u)| < ε
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 ha : ∃ B, ∀ (n : ℕ), B ≤ n → |a n - t| < ε / 2 ⊢ ∃ B, ∀ (n : ℕ), B ≤ n → |a n + b n - (t + u)| < ε TACTIC:
https://github.com/ImperialCollegeLondon/formalising-mathematics-2024.git
b732ed1352e87b4474b0520d1383994e069f8057
FormalisingMathematics2024/Section02reals/Sheet5.lean
Section2sheet5.tendsTo_add
[31, 1]
[52, 13]
obtain ⟨Y, hY⟩ := hb (ε / 2) (by linarith)
case intro 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 ⊢ ∃ B, ∀ (n : ℕ), B ≤ n → |a n + b n - (t + u)| < ε
case intro.intro 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 Y : ℕ hY : ∀ (n : ℕ), Y ≤ n → |b n - u| < ε / 2 ⊢ ∃ B, ∀ (n : ℕ), B ≤ n → |a n + b n - (t + u)| < ε
Please generate a tactic in lean4 to solve the state. STATE: case intro 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 ⊢ ∃ B, ∀ (n : ℕ), B ≤ n → |a n + b n - (t + u)| < ε TACTIC:
https://github.com/ImperialCollegeLondon/formalising-mathematics-2024.git
b732ed1352e87b4474b0520d1383994e069f8057
FormalisingMathematics2024/Section02reals/Sheet5.lean
Section2sheet5.tendsTo_add
[31, 1]
[52, 13]
use max X Y
case intro.intro 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 Y : ℕ hY : ∀ (n : ℕ), Y ≤ n → |b n - u| < ε / 2 ⊢ ∃ B, ∀ (n : ℕ), B ≤ n → |a n + b n - (t + u)| < ε
case h 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 Y : ℕ hY : ∀ (n : ℕ), Y ≤ n → |b n - u| < ε / 2 ⊢ ∀ (n : ℕ), max X Y ≤ n → |a n + b n - (t + u)| < ε
Please generate a tactic in lean4 to solve the state. STATE: case intro.intro 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 Y : ℕ hY : ∀ (n : ℕ), Y ≤ n → |b n - u| < ε / 2 ⊢ ∃ B, ∀ (n : ℕ), B ≤ n → |a n + b n - (t + u)| < ε TACTIC:
https://github.com/ImperialCollegeLondon/formalising-mathematics-2024.git
b732ed1352e87b4474b0520d1383994e069f8057
FormalisingMathematics2024/Section02reals/Sheet5.lean
Section2sheet5.tendsTo_add
[31, 1]
[52, 13]
intro n hn
case h 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 Y : ℕ hY : ∀ (n : ℕ), Y ≤ n → |b n - u| < ε / 2 ⊢ ∀ (n : ℕ), max X Y ≤ n → |a n + b n - (t + u)| < ε
case h 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 Y : ℕ hY : ∀ (n : ℕ), Y ≤ n → |b n - u| < ε / 2 n : ℕ hn : max X Y ≤ n ⊢ |a n + b n - (t + u)| < ε
Please generate a tactic in lean4 to solve the state. STATE: case h 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 Y : ℕ hY : ∀ (n : ℕ), Y ≤ n → |b n - u| < ε / 2 ⊢ ∀ (n : ℕ), max X Y ≤ n → |a n + b n - (t + u)| < ε TACTIC:
https://github.com/ImperialCollegeLondon/formalising-mathematics-2024.git
b732ed1352e87b4474b0520d1383994e069f8057
FormalisingMathematics2024/Section02reals/Sheet5.lean
Section2sheet5.tendsTo_add
[31, 1]
[52, 13]
rw [max_le_iff] at hn
case h 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 Y : ℕ hY : ∀ (n : ℕ), Y ≤ n → |b n - u| < ε / 2 n : ℕ hn : max X Y ≤ n ⊢ |a n + b n - (t + u)| < ε
case h 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 Y : ℕ hY : ∀ (n : ℕ), Y ≤ n → |b n - u| < ε / 2 n : ℕ hn : X ≤ n ∧ Y ≤ n ⊢ |a n + b n - (t + u)| < ε
Please generate a tactic in lean4 to solve the state. STATE: case h 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 Y : ℕ hY : ∀ (n : ℕ), Y ≤ n → |b n - u| < ε / 2 n : ℕ hn : max X Y ≤ n ⊢ |a n + b n - (t + u)| < ε TACTIC:
https://github.com/ImperialCollegeLondon/formalising-mathematics-2024.git
b732ed1352e87b4474b0520d1383994e069f8057
FormalisingMathematics2024/Section02reals/Sheet5.lean
Section2sheet5.tendsTo_add
[31, 1]
[52, 13]
specialize hX n hn.1
case h 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 Y : ℕ hY : ∀ (n : ℕ), Y ≤ n → |b n - u| < ε / 2 n : ℕ hn : X ≤ n ∧ Y ≤ n ⊢ |a n + b n - (t + u)| < ε
case h 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 Y : ℕ hY : ∀ (n : ℕ), Y ≤ n → |b n - u| < ε / 2 n : ℕ hn : X ≤ n ∧ Y ≤ n hX : |a n - t| < ε / 2 ⊢ |a n + b n - (t + u)| < ε
Please generate a tactic in lean4 to solve the state. STATE: case h 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 Y : ℕ hY : ∀ (n : ℕ), Y ≤ n → |b n - u| < ε / 2 n : ℕ hn : X ≤ n ∧ Y ≤ n ⊢ |a n + b n - (t + u)| < ε TACTIC:
https://github.com/ImperialCollegeLondon/formalising-mathematics-2024.git
b732ed1352e87b4474b0520d1383994e069f8057
FormalisingMathematics2024/Section02reals/Sheet5.lean
Section2sheet5.tendsTo_add
[31, 1]
[52, 13]
specialize hY n hn.2
case h 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 Y : ℕ hY : ∀ (n : ℕ), Y ≤ n → |b n - u| < ε / 2 n : ℕ hn : X ≤ n ∧ Y ≤ n hX : |a n - t| < ε / 2 ⊢ |a n + b n - (t + u)| < ε
case h 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 Y n : ℕ hn : X ≤ n ∧ Y ≤ n hX : |a n - t| < ε / 2 hY : |b n - u| < ε / 2 ⊢ |a n + b n - (t + u)| < ε
Please generate a tactic in lean4 to solve the state. STATE: case h 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 Y : ℕ hY : ∀ (n : ℕ), Y ≤ n → |b n - u| < ε / 2 n : ℕ hn : X ≤ n ∧ Y ≤ n hX : |a n - t| < ε / 2 ⊢ |a n + b n - (t + u)| < ε TACTIC:
https://github.com/ImperialCollegeLondon/formalising-mathematics-2024.git
b732ed1352e87b4474b0520d1383994e069f8057
FormalisingMathematics2024/Section02reals/Sheet5.lean
Section2sheet5.tendsTo_add
[31, 1]
[52, 13]
rw [abs_lt] at *
case h 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 Y n : ℕ hn : X ≤ n ∧ Y ≤ n hX : |a n - t| < ε / 2 hY : |b n - u| < ε / 2 ⊢ |a n + b n - (t + u)| < ε
case h 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 Y n : ℕ hn : X ≤ n ∧ Y ≤ n hX : -(ε / 2) < a n - t ∧ a n - t < ε / 2 hY : -(ε / 2) < b n - u ∧ b n - u < ε / 2 ⊢ -ε < a n + b n - (t + u) ∧ a n + b n - (t + u) < ε
Please generate a tactic in lean4 to solve the state. STATE: case h 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 Y n : ℕ hn : X ≤ n ∧ Y ≤ n hX : |a n - t| < ε / 2 hY : |b n - u| < ε / 2 ⊢ |a n + b n - (t + u)| < ε TACTIC:
https://github.com/ImperialCollegeLondon/formalising-mathematics-2024.git
b732ed1352e87b4474b0520d1383994e069f8057
FormalisingMathematics2024/Section02reals/Sheet5.lean
Section2sheet5.tendsTo_add
[31, 1]
[52, 13]
constructor <;>linarith
case h 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 Y n : ℕ hn : X ≤ n ∧ Y ≤ n hX : -(ε / 2) < a n - t ∧ a n - t < ε / 2 hY : -(ε / 2) < b n - u ∧ b n - u < ε / 2 ⊢ -ε < a n + b n - (t + u) ∧ a n + b n - (t + u) < ε
no goals
Please generate a tactic in lean4 to solve the state. STATE: case h 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 Y n : ℕ hn : X ≤ n ∧ Y ≤ n hX : -(ε / 2) < a n - t ∧ a n - t < ε / 2 hY : -(ε / 2) < b n - u ∧ b n - u < ε / 2 ⊢ -ε < a n + b n - (t + u) ∧ a n + b n - (t + u) < ε TACTIC:
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 ha : ∀ (ε : ℝ), 0 < ε → ∃ B, ∀ (n : ℕ), B ≤ n → |a n - t| < ε hb : ∀ (ε : ℝ), 0 < ε → ∃ B, ∀ (n : ℕ), B ≤ n → |b n - u| < ε ε : ℝ hε : 0 < ε ⊢ 0 < ε / 2
no goals
Please generate a tactic in lean4 to solve the state. STATE: a b : ℕ → ℝ t u : ℝ ha : ∀ (ε : ℝ), 0 < ε → ∃ B, ∀ (n : ℕ), B ≤ n → |a n - t| < ε hb : ∀ (ε : ℝ), 0 < ε → ∃ B, ∀ (n : ℕ), B ≤ n → |b n - u| < ε ε : ℝ hε : 0 < ε ⊢ 0 < ε / 2 TACTIC: