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/fpvandoorn/LeanCourse23.git | 7b0a3cf61b802764dc7baee9d9825e9c62cf9c5d | LeanCourse/MIL/C05_Elementary_Number_Theory/S03_Infinitely_Many_Primes.lean | C05S03.primes_infinite | [46, 1] | [60, 8] | sorry | n : ℕ
this✝ : 2 ≤ Nat.factorial (n + 1) + 1
p : ℕ
pp : Nat.Prime p
pdvd : p ∣ Nat.factorial (n + 1) + 1
ple : p ≤ n
this : p ∣ Nat.factorial (n + 1)
⊢ p ∣ 1 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
n : ℕ
this✝ : 2 ≤ Nat.factorial (n + 1) + 1
p : ℕ
pp : Nat.Prime p
pdvd : p ∣ Nat.factorial (n + 1) + 1
ple : p ≤ n
this : p ∣ Nat.factorial (n + 1)
⊢ p ∣ 1
TACTIC:
|
https://github.com/fpvandoorn/LeanCourse23.git | 7b0a3cf61b802764dc7baee9d9825e9c62cf9c5d | LeanCourse/MIL/C05_Elementary_Number_Theory/S03_Infinitely_Many_Primes.lean | Nat.Prime.eq_of_dvd_of_prime | [102, 1] | [105, 8] | sorry | p q : ℕ
prime_p : Nat.Prime p
prime_q : Nat.Prime q
h : p ∣ q
⊢ p = q | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
p q : ℕ
prime_p : Nat.Prime p
prime_q : Nat.Prime q
h : p ∣ q
⊢ p = q
TACTIC:
|
https://github.com/fpvandoorn/LeanCourse23.git | 7b0a3cf61b802764dc7baee9d9825e9c62cf9c5d | LeanCourse/MIL/C05_Elementary_Number_Theory/S03_Infinitely_Many_Primes.lean | C05S03.mem_of_dvd_prod_primes | [107, 1] | [115, 8] | intro h₀ h₁ | s : Finset ℕ
p : ℕ
prime_p : Nat.Prime p
⊢ (∀ n ∈ s, Nat.Prime n) → p ∣ ∏ n in s, n → p ∈ s | s : Finset ℕ
p : ℕ
prime_p : Nat.Prime p
h₀ : ∀ n ∈ s, Nat.Prime n
h₁ : p ∣ ∏ n in s, n
⊢ p ∈ s | Please generate a tactic in lean4 to solve the state.
STATE:
s : Finset ℕ
p : ℕ
prime_p : Nat.Prime p
⊢ (∀ n ∈ s, Nat.Prime n) → p ∣ ∏ n in s, n → p ∈ s
TACTIC:
|
https://github.com/fpvandoorn/LeanCourse23.git | 7b0a3cf61b802764dc7baee9d9825e9c62cf9c5d | LeanCourse/MIL/C05_Elementary_Number_Theory/S03_Infinitely_Many_Primes.lean | C05S03.mem_of_dvd_prod_primes | [107, 1] | [115, 8] | induction' s using Finset.induction_on with a s ans ih | s : Finset ℕ
p : ℕ
prime_p : Nat.Prime p
h₀ : ∀ n ∈ s, Nat.Prime n
h₁ : p ∣ ∏ n in s, n
⊢ p ∈ s | case empty
p : ℕ
prime_p : Nat.Prime p
h₀ : ∀ n ∈ ∅, Nat.Prime n
h₁ : p ∣ ∏ n in ∅, n
⊢ p ∈ ∅
case insert
p : ℕ
prime_p : Nat.Prime p
a : ℕ
s : Finset ℕ
ans : a ∉ s
ih : (∀ n ∈ s, Nat.Prime n) → p ∣ ∏ n in s, n → p ∈ s
h₀ : ∀ n ∈ insert a s, Nat.Prime n
h₁ : p ∣ ∏ n in insert a s, n
⊢ p ∈ insert a s | Please generate a tactic in lean4 to solve the state.
STATE:
s : Finset ℕ
p : ℕ
prime_p : Nat.Prime p
h₀ : ∀ n ∈ s, Nat.Prime n
h₁ : p ∣ ∏ n in s, n
⊢ p ∈ s
TACTIC:
|
https://github.com/fpvandoorn/LeanCourse23.git | 7b0a3cf61b802764dc7baee9d9825e9c62cf9c5d | LeanCourse/MIL/C05_Elementary_Number_Theory/S03_Infinitely_Many_Primes.lean | C05S03.mem_of_dvd_prod_primes | [107, 1] | [115, 8] | simp [Finset.prod_insert ans, prime_p.dvd_mul] at h₀ h₁ | case insert
p : ℕ
prime_p : Nat.Prime p
a : ℕ
s : Finset ℕ
ans : a ∉ s
ih : (∀ n ∈ s, Nat.Prime n) → p ∣ ∏ n in s, n → p ∈ s
h₀ : ∀ n ∈ insert a s, Nat.Prime n
h₁ : p ∣ ∏ n in insert a s, n
⊢ p ∈ insert a s | case insert
p : ℕ
prime_p : Nat.Prime p
a : ℕ
s : Finset ℕ
ans : a ∉ s
ih : (∀ n ∈ s, Nat.Prime n) → p ∣ ∏ n in s, n → p ∈ s
h₀ : Nat.Prime a ∧ ∀ a ∈ s, Nat.Prime a
h₁ : p ∣ a ∨ p ∣ ∏ n in s, n
⊢ p ∈ insert a s | Please generate a tactic in lean4 to solve the state.
STATE:
case insert
p : ℕ
prime_p : Nat.Prime p
a : ℕ
s : Finset ℕ
ans : a ∉ s
ih : (∀ n ∈ s, Nat.Prime n) → p ∣ ∏ n in s, n → p ∈ s
h₀ : ∀ n ∈ insert a s, Nat.Prime n
h₁ : p ∣ ∏ n in insert a s, n
⊢ p ∈ insert a s
TACTIC:
|
https://github.com/fpvandoorn/LeanCourse23.git | 7b0a3cf61b802764dc7baee9d9825e9c62cf9c5d | LeanCourse/MIL/C05_Elementary_Number_Theory/S03_Infinitely_Many_Primes.lean | C05S03.mem_of_dvd_prod_primes | [107, 1] | [115, 8] | rw [mem_insert] | case insert
p : ℕ
prime_p : Nat.Prime p
a : ℕ
s : Finset ℕ
ans : a ∉ s
ih : (∀ n ∈ s, Nat.Prime n) → p ∣ ∏ n in s, n → p ∈ s
h₀ : Nat.Prime a ∧ ∀ a ∈ s, Nat.Prime a
h₁ : p ∣ a ∨ p ∣ ∏ n in s, n
⊢ p ∈ insert a s | case insert
p : ℕ
prime_p : Nat.Prime p
a : ℕ
s : Finset ℕ
ans : a ∉ s
ih : (∀ n ∈ s, Nat.Prime n) → p ∣ ∏ n in s, n → p ∈ s
h₀ : Nat.Prime a ∧ ∀ a ∈ s, Nat.Prime a
h₁ : p ∣ a ∨ p ∣ ∏ n in s, n
⊢ p = a ∨ p ∈ s | Please generate a tactic in lean4 to solve the state.
STATE:
case insert
p : ℕ
prime_p : Nat.Prime p
a : ℕ
s : Finset ℕ
ans : a ∉ s
ih : (∀ n ∈ s, Nat.Prime n) → p ∣ ∏ n in s, n → p ∈ s
h₀ : Nat.Prime a ∧ ∀ a ∈ s, Nat.Prime a
h₁ : p ∣ a ∨ p ∣ ∏ n in s, n
⊢ p ∈ insert a s
TACTIC:
|
https://github.com/fpvandoorn/LeanCourse23.git | 7b0a3cf61b802764dc7baee9d9825e9c62cf9c5d | LeanCourse/MIL/C05_Elementary_Number_Theory/S03_Infinitely_Many_Primes.lean | C05S03.mem_of_dvd_prod_primes | [107, 1] | [115, 8] | sorry | case insert
p : ℕ
prime_p : Nat.Prime p
a : ℕ
s : Finset ℕ
ans : a ∉ s
ih : (∀ n ∈ s, Nat.Prime n) → p ∣ ∏ n in s, n → p ∈ s
h₀ : Nat.Prime a ∧ ∀ a ∈ s, Nat.Prime a
h₁ : p ∣ a ∨ p ∣ ∏ n in s, n
⊢ p = a ∨ p ∈ s | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case insert
p : ℕ
prime_p : Nat.Prime p
a : ℕ
s : Finset ℕ
ans : a ∉ s
ih : (∀ n ∈ s, Nat.Prime n) → p ∣ ∏ n in s, n → p ∈ s
h₀ : Nat.Prime a ∧ ∀ a ∈ s, Nat.Prime a
h₁ : p ∣ a ∨ p ∣ ∏ n in s, n
⊢ p = a ∨ p ∈ s
TACTIC:
|
https://github.com/fpvandoorn/LeanCourse23.git | 7b0a3cf61b802764dc7baee9d9825e9c62cf9c5d | LeanCourse/MIL/C05_Elementary_Number_Theory/S03_Infinitely_Many_Primes.lean | C05S03.mem_of_dvd_prod_primes | [107, 1] | [115, 8] | simp at h₁ | case empty
p : ℕ
prime_p : Nat.Prime p
h₀ : ∀ n ∈ ∅, Nat.Prime n
h₁ : p ∣ ∏ n in ∅, n
⊢ p ∈ ∅ | case empty
p : ℕ
prime_p : Nat.Prime p
h₀ : ∀ n ∈ ∅, Nat.Prime n
h₁ : p = 1
⊢ p ∈ ∅ | Please generate a tactic in lean4 to solve the state.
STATE:
case empty
p : ℕ
prime_p : Nat.Prime p
h₀ : ∀ n ∈ ∅, Nat.Prime n
h₁ : p ∣ ∏ n in ∅, n
⊢ p ∈ ∅
TACTIC:
|
https://github.com/fpvandoorn/LeanCourse23.git | 7b0a3cf61b802764dc7baee9d9825e9c62cf9c5d | LeanCourse/MIL/C05_Elementary_Number_Theory/S03_Infinitely_Many_Primes.lean | C05S03.mem_of_dvd_prod_primes | [107, 1] | [115, 8] | linarith [prime_p.two_le] | case empty
p : ℕ
prime_p : Nat.Prime p
h₀ : ∀ n ∈ ∅, Nat.Prime n
h₁ : p = 1
⊢ p ∈ ∅ | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case empty
p : ℕ
prime_p : Nat.Prime p
h₀ : ∀ n ∈ ∅, Nat.Prime n
h₁ : p = 1
⊢ p ∈ ∅
TACTIC:
|
https://github.com/fpvandoorn/LeanCourse23.git | 7b0a3cf61b802764dc7baee9d9825e9c62cf9c5d | LeanCourse/MIL/C05_Elementary_Number_Theory/S03_Infinitely_Many_Primes.lean | C05S03.primes_infinite' | [119, 1] | [137, 8] | intro s | ⊢ ∀ (s : Finset ℕ), ∃ p, Nat.Prime p ∧ p ∉ s | s : Finset ℕ
⊢ ∃ p, Nat.Prime p ∧ p ∉ s | Please generate a tactic in lean4 to solve the state.
STATE:
⊢ ∀ (s : Finset ℕ), ∃ p, Nat.Prime p ∧ p ∉ s
TACTIC:
|
https://github.com/fpvandoorn/LeanCourse23.git | 7b0a3cf61b802764dc7baee9d9825e9c62cf9c5d | LeanCourse/MIL/C05_Elementary_Number_Theory/S03_Infinitely_Many_Primes.lean | C05S03.primes_infinite' | [119, 1] | [137, 8] | by_contra h | s : Finset ℕ
⊢ ∃ p, Nat.Prime p ∧ p ∉ s | s : Finset ℕ
h : ¬∃ p, Nat.Prime p ∧ p ∉ s
⊢ False | Please generate a tactic in lean4 to solve the state.
STATE:
s : Finset ℕ
⊢ ∃ p, Nat.Prime p ∧ p ∉ s
TACTIC:
|
https://github.com/fpvandoorn/LeanCourse23.git | 7b0a3cf61b802764dc7baee9d9825e9c62cf9c5d | LeanCourse/MIL/C05_Elementary_Number_Theory/S03_Infinitely_Many_Primes.lean | C05S03.primes_infinite' | [119, 1] | [137, 8] | push_neg at h | s : Finset ℕ
h : ¬∃ p, Nat.Prime p ∧ p ∉ s
⊢ False | s : Finset ℕ
h : ∀ (p : ℕ), Nat.Prime p → p ∈ s
⊢ False | Please generate a tactic in lean4 to solve the state.
STATE:
s : Finset ℕ
h : ¬∃ p, Nat.Prime p ∧ p ∉ s
⊢ False
TACTIC:
|
https://github.com/fpvandoorn/LeanCourse23.git | 7b0a3cf61b802764dc7baee9d9825e9c62cf9c5d | LeanCourse/MIL/C05_Elementary_Number_Theory/S03_Infinitely_Many_Primes.lean | C05S03.primes_infinite' | [119, 1] | [137, 8] | set s' := s.filter Nat.Prime with s'_def | s : Finset ℕ
h : ∀ (p : ℕ), Nat.Prime p → p ∈ s
⊢ False | s : Finset ℕ
h : ∀ (p : ℕ), Nat.Prime p → p ∈ s
s' : Finset ℕ := filter Nat.Prime s
s'_def : s' = filter Nat.Prime s
⊢ False | Please generate a tactic in lean4 to solve the state.
STATE:
s : Finset ℕ
h : ∀ (p : ℕ), Nat.Prime p → p ∈ s
⊢ False
TACTIC:
|
https://github.com/fpvandoorn/LeanCourse23.git | 7b0a3cf61b802764dc7baee9d9825e9c62cf9c5d | LeanCourse/MIL/C05_Elementary_Number_Theory/S03_Infinitely_Many_Primes.lean | C05S03.primes_infinite' | [119, 1] | [137, 8] | have mem_s' : ∀ {n : ℕ}, n ∈ s' ↔ n.Prime := by
intro n
simp [s'_def]
apply h | s : Finset ℕ
h : ∀ (p : ℕ), Nat.Prime p → p ∈ s
s' : Finset ℕ := filter Nat.Prime s
s'_def : s' = filter Nat.Prime s
⊢ False | s : Finset ℕ
h : ∀ (p : ℕ), Nat.Prime p → p ∈ s
s' : Finset ℕ := filter Nat.Prime s
s'_def : s' = filter Nat.Prime s
mem_s' : ∀ {n : ℕ}, n ∈ s' ↔ Nat.Prime n
⊢ False | Please generate a tactic in lean4 to solve the state.
STATE:
s : Finset ℕ
h : ∀ (p : ℕ), Nat.Prime p → p ∈ s
s' : Finset ℕ := filter Nat.Prime s
s'_def : s' = filter Nat.Prime s
⊢ False
TACTIC:
|
https://github.com/fpvandoorn/LeanCourse23.git | 7b0a3cf61b802764dc7baee9d9825e9c62cf9c5d | LeanCourse/MIL/C05_Elementary_Number_Theory/S03_Infinitely_Many_Primes.lean | C05S03.primes_infinite' | [119, 1] | [137, 8] | have : 2 ≤ (∏ i in s', i) + 1 := by
sorry | s : Finset ℕ
h : ∀ (p : ℕ), Nat.Prime p → p ∈ s
s' : Finset ℕ := filter Nat.Prime s
s'_def : s' = filter Nat.Prime s
mem_s' : ∀ {n : ℕ}, n ∈ s' ↔ Nat.Prime n
⊢ False | s : Finset ℕ
h : ∀ (p : ℕ), Nat.Prime p → p ∈ s
s' : Finset ℕ := filter Nat.Prime s
s'_def : s' = filter Nat.Prime s
mem_s' : ∀ {n : ℕ}, n ∈ s' ↔ Nat.Prime n
this : 2 ≤ ∏ i in s', i + 1
⊢ False | Please generate a tactic in lean4 to solve the state.
STATE:
s : Finset ℕ
h : ∀ (p : ℕ), Nat.Prime p → p ∈ s
s' : Finset ℕ := filter Nat.Prime s
s'_def : s' = filter Nat.Prime s
mem_s' : ∀ {n : ℕ}, n ∈ s' ↔ Nat.Prime n
⊢ False
TACTIC:
|
https://github.com/fpvandoorn/LeanCourse23.git | 7b0a3cf61b802764dc7baee9d9825e9c62cf9c5d | LeanCourse/MIL/C05_Elementary_Number_Theory/S03_Infinitely_Many_Primes.lean | C05S03.primes_infinite' | [119, 1] | [137, 8] | rcases exists_prime_factor this with ⟨p, pp, pdvd⟩ | s : Finset ℕ
h : ∀ (p : ℕ), Nat.Prime p → p ∈ s
s' : Finset ℕ := filter Nat.Prime s
s'_def : s' = filter Nat.Prime s
mem_s' : ∀ {n : ℕ}, n ∈ s' ↔ Nat.Prime n
this : 2 ≤ ∏ i in s', i + 1
⊢ False | case intro.intro
s : Finset ℕ
h : ∀ (p : ℕ), Nat.Prime p → p ∈ s
s' : Finset ℕ := filter Nat.Prime s
s'_def : s' = filter Nat.Prime s
mem_s' : ∀ {n : ℕ}, n ∈ s' ↔ Nat.Prime n
this : 2 ≤ ∏ i in s', i + 1
p : ℕ
pp : Nat.Prime p
pdvd : p ∣ ∏ i in s', i + 1
⊢ False | Please generate a tactic in lean4 to solve the state.
STATE:
s : Finset ℕ
h : ∀ (p : ℕ), Nat.Prime p → p ∈ s
s' : Finset ℕ := filter Nat.Prime s
s'_def : s' = filter Nat.Prime s
mem_s' : ∀ {n : ℕ}, n ∈ s' ↔ Nat.Prime n
this : 2 ≤ ∏ i in s', i + 1
⊢ False
TACTIC:
|
https://github.com/fpvandoorn/LeanCourse23.git | 7b0a3cf61b802764dc7baee9d9825e9c62cf9c5d | LeanCourse/MIL/C05_Elementary_Number_Theory/S03_Infinitely_Many_Primes.lean | C05S03.primes_infinite' | [119, 1] | [137, 8] | have : p ∣ ∏ i in s', i := by
sorry | case intro.intro
s : Finset ℕ
h : ∀ (p : ℕ), Nat.Prime p → p ∈ s
s' : Finset ℕ := filter Nat.Prime s
s'_def : s' = filter Nat.Prime s
mem_s' : ∀ {n : ℕ}, n ∈ s' ↔ Nat.Prime n
this : 2 ≤ ∏ i in s', i + 1
p : ℕ
pp : Nat.Prime p
pdvd : p ∣ ∏ i in s', i + 1
⊢ False | case intro.intro
s : Finset ℕ
h : ∀ (p : ℕ), Nat.Prime p → p ∈ s
s' : Finset ℕ := filter Nat.Prime s
s'_def : s' = filter Nat.Prime s
mem_s' : ∀ {n : ℕ}, n ∈ s' ↔ Nat.Prime n
this✝ : 2 ≤ ∏ i in s', i + 1
p : ℕ
pp : Nat.Prime p
pdvd : p ∣ ∏ i in s', i + 1
this : p ∣ ∏ i in s', i
⊢ False | Please generate a tactic in lean4 to solve the state.
STATE:
case intro.intro
s : Finset ℕ
h : ∀ (p : ℕ), Nat.Prime p → p ∈ s
s' : Finset ℕ := filter Nat.Prime s
s'_def : s' = filter Nat.Prime s
mem_s' : ∀ {n : ℕ}, n ∈ s' ↔ Nat.Prime n
this : 2 ≤ ∏ i in s', i + 1
p : ℕ
pp : Nat.Prime p
pdvd : p ∣ ∏ i in s', i + 1
⊢ False
TACTIC:
|
https://github.com/fpvandoorn/LeanCourse23.git | 7b0a3cf61b802764dc7baee9d9825e9c62cf9c5d | LeanCourse/MIL/C05_Elementary_Number_Theory/S03_Infinitely_Many_Primes.lean | C05S03.primes_infinite' | [119, 1] | [137, 8] | have : p ∣ 1 := by
convert Nat.dvd_sub' pdvd this
simp | case intro.intro
s : Finset ℕ
h : ∀ (p : ℕ), Nat.Prime p → p ∈ s
s' : Finset ℕ := filter Nat.Prime s
s'_def : s' = filter Nat.Prime s
mem_s' : ∀ {n : ℕ}, n ∈ s' ↔ Nat.Prime n
this✝ : 2 ≤ ∏ i in s', i + 1
p : ℕ
pp : Nat.Prime p
pdvd : p ∣ ∏ i in s', i + 1
this : p ∣ ∏ i in s', i
⊢ False | case intro.intro
s : Finset ℕ
h : ∀ (p : ℕ), Nat.Prime p → p ∈ s
s' : Finset ℕ := filter Nat.Prime s
s'_def : s' = filter Nat.Prime s
mem_s' : ∀ {n : ℕ}, n ∈ s' ↔ Nat.Prime n
this✝¹ : 2 ≤ ∏ i in s', i + 1
p : ℕ
pp : Nat.Prime p
pdvd : p ∣ ∏ i in s', i + 1
this✝ : p ∣ ∏ i in s', i
this : p ∣ 1
⊢ False | Please generate a tactic in lean4 to solve the state.
STATE:
case intro.intro
s : Finset ℕ
h : ∀ (p : ℕ), Nat.Prime p → p ∈ s
s' : Finset ℕ := filter Nat.Prime s
s'_def : s' = filter Nat.Prime s
mem_s' : ∀ {n : ℕ}, n ∈ s' ↔ Nat.Prime n
this✝ : 2 ≤ ∏ i in s', i + 1
p : ℕ
pp : Nat.Prime p
pdvd : p ∣ ∏ i in s', i + 1
this : p ∣ ∏ i in s', i
⊢ False
TACTIC:
|
https://github.com/fpvandoorn/LeanCourse23.git | 7b0a3cf61b802764dc7baee9d9825e9c62cf9c5d | LeanCourse/MIL/C05_Elementary_Number_Theory/S03_Infinitely_Many_Primes.lean | C05S03.primes_infinite' | [119, 1] | [137, 8] | sorry | case intro.intro
s : Finset ℕ
h : ∀ (p : ℕ), Nat.Prime p → p ∈ s
s' : Finset ℕ := filter Nat.Prime s
s'_def : s' = filter Nat.Prime s
mem_s' : ∀ {n : ℕ}, n ∈ s' ↔ Nat.Prime n
this✝¹ : 2 ≤ ∏ i in s', i + 1
p : ℕ
pp : Nat.Prime p
pdvd : p ∣ ∏ i in s', i + 1
this✝ : p ∣ ∏ i in s', i
this : p ∣ 1
⊢ False | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case intro.intro
s : Finset ℕ
h : ∀ (p : ℕ), Nat.Prime p → p ∈ s
s' : Finset ℕ := filter Nat.Prime s
s'_def : s' = filter Nat.Prime s
mem_s' : ∀ {n : ℕ}, n ∈ s' ↔ Nat.Prime n
this✝¹ : 2 ≤ ∏ i in s', i + 1
p : ℕ
pp : Nat.Prime p
pdvd : p ∣ ∏ i in s', i + 1
this✝ : p ∣ ∏ i in s', i
this : p ∣ 1
⊢ False
TACTIC:
|
https://github.com/fpvandoorn/LeanCourse23.git | 7b0a3cf61b802764dc7baee9d9825e9c62cf9c5d | LeanCourse/MIL/C05_Elementary_Number_Theory/S03_Infinitely_Many_Primes.lean | C05S03.primes_infinite' | [119, 1] | [137, 8] | intro n | s : Finset ℕ
h : ∀ (p : ℕ), Nat.Prime p → p ∈ s
s' : Finset ℕ := filter Nat.Prime s
s'_def : s' = filter Nat.Prime s
⊢ ∀ {n : ℕ}, n ∈ s' ↔ Nat.Prime n | s : Finset ℕ
h : ∀ (p : ℕ), Nat.Prime p → p ∈ s
s' : Finset ℕ := filter Nat.Prime s
s'_def : s' = filter Nat.Prime s
n : ℕ
⊢ n ∈ s' ↔ Nat.Prime n | Please generate a tactic in lean4 to solve the state.
STATE:
s : Finset ℕ
h : ∀ (p : ℕ), Nat.Prime p → p ∈ s
s' : Finset ℕ := filter Nat.Prime s
s'_def : s' = filter Nat.Prime s
⊢ ∀ {n : ℕ}, n ∈ s' ↔ Nat.Prime n
TACTIC:
|
https://github.com/fpvandoorn/LeanCourse23.git | 7b0a3cf61b802764dc7baee9d9825e9c62cf9c5d | LeanCourse/MIL/C05_Elementary_Number_Theory/S03_Infinitely_Many_Primes.lean | C05S03.primes_infinite' | [119, 1] | [137, 8] | simp [s'_def] | s : Finset ℕ
h : ∀ (p : ℕ), Nat.Prime p → p ∈ s
s' : Finset ℕ := filter Nat.Prime s
s'_def : s' = filter Nat.Prime s
n : ℕ
⊢ n ∈ s' ↔ Nat.Prime n | s : Finset ℕ
h : ∀ (p : ℕ), Nat.Prime p → p ∈ s
s' : Finset ℕ := filter Nat.Prime s
s'_def : s' = filter Nat.Prime s
n : ℕ
⊢ Nat.Prime n → n ∈ s | Please generate a tactic in lean4 to solve the state.
STATE:
s : Finset ℕ
h : ∀ (p : ℕ), Nat.Prime p → p ∈ s
s' : Finset ℕ := filter Nat.Prime s
s'_def : s' = filter Nat.Prime s
n : ℕ
⊢ n ∈ s' ↔ Nat.Prime n
TACTIC:
|
https://github.com/fpvandoorn/LeanCourse23.git | 7b0a3cf61b802764dc7baee9d9825e9c62cf9c5d | LeanCourse/MIL/C05_Elementary_Number_Theory/S03_Infinitely_Many_Primes.lean | C05S03.primes_infinite' | [119, 1] | [137, 8] | apply h | s : Finset ℕ
h : ∀ (p : ℕ), Nat.Prime p → p ∈ s
s' : Finset ℕ := filter Nat.Prime s
s'_def : s' = filter Nat.Prime s
n : ℕ
⊢ Nat.Prime n → n ∈ s | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
s : Finset ℕ
h : ∀ (p : ℕ), Nat.Prime p → p ∈ s
s' : Finset ℕ := filter Nat.Prime s
s'_def : s' = filter Nat.Prime s
n : ℕ
⊢ Nat.Prime n → n ∈ s
TACTIC:
|
https://github.com/fpvandoorn/LeanCourse23.git | 7b0a3cf61b802764dc7baee9d9825e9c62cf9c5d | LeanCourse/MIL/C05_Elementary_Number_Theory/S03_Infinitely_Many_Primes.lean | C05S03.primes_infinite' | [119, 1] | [137, 8] | sorry | s : Finset ℕ
h : ∀ (p : ℕ), Nat.Prime p → p ∈ s
s' : Finset ℕ := filter Nat.Prime s
s'_def : s' = filter Nat.Prime s
mem_s' : ∀ {n : ℕ}, n ∈ s' ↔ Nat.Prime n
⊢ 2 ≤ ∏ i in s', i + 1 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
s : Finset ℕ
h : ∀ (p : ℕ), Nat.Prime p → p ∈ s
s' : Finset ℕ := filter Nat.Prime s
s'_def : s' = filter Nat.Prime s
mem_s' : ∀ {n : ℕ}, n ∈ s' ↔ Nat.Prime n
⊢ 2 ≤ ∏ i in s', i + 1
TACTIC:
|
https://github.com/fpvandoorn/LeanCourse23.git | 7b0a3cf61b802764dc7baee9d9825e9c62cf9c5d | LeanCourse/MIL/C05_Elementary_Number_Theory/S03_Infinitely_Many_Primes.lean | C05S03.primes_infinite' | [119, 1] | [137, 8] | sorry | s : Finset ℕ
h : ∀ (p : ℕ), Nat.Prime p → p ∈ s
s' : Finset ℕ := filter Nat.Prime s
s'_def : s' = filter Nat.Prime s
mem_s' : ∀ {n : ℕ}, n ∈ s' ↔ Nat.Prime n
this : 2 ≤ ∏ i in s', i + 1
p : ℕ
pp : Nat.Prime p
pdvd : p ∣ ∏ i in s', i + 1
⊢ p ∣ ∏ i in s', i | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
s : Finset ℕ
h : ∀ (p : ℕ), Nat.Prime p → p ∈ s
s' : Finset ℕ := filter Nat.Prime s
s'_def : s' = filter Nat.Prime s
mem_s' : ∀ {n : ℕ}, n ∈ s' ↔ Nat.Prime n
this : 2 ≤ ∏ i in s', i + 1
p : ℕ
pp : Nat.Prime p
pdvd : p ∣ ∏ i in s', i + 1
⊢ p ∣ ∏ i in s', i
TACTIC:
|
https://github.com/fpvandoorn/LeanCourse23.git | 7b0a3cf61b802764dc7baee9d9825e9c62cf9c5d | LeanCourse/MIL/C05_Elementary_Number_Theory/S03_Infinitely_Many_Primes.lean | C05S03.primes_infinite' | [119, 1] | [137, 8] | convert Nat.dvd_sub' pdvd this | s : Finset ℕ
h : ∀ (p : ℕ), Nat.Prime p → p ∈ s
s' : Finset ℕ := filter Nat.Prime s
s'_def : s' = filter Nat.Prime s
mem_s' : ∀ {n : ℕ}, n ∈ s' ↔ Nat.Prime n
this✝ : 2 ≤ ∏ i in s', i + 1
p : ℕ
pp : Nat.Prime p
pdvd : p ∣ ∏ i in s', i + 1
this : p ∣ ∏ i in s', i
⊢ p ∣ 1 | case h.e'_4
s : Finset ℕ
h : ∀ (p : ℕ), Nat.Prime p → p ∈ s
s' : Finset ℕ := filter Nat.Prime s
s'_def : s' = filter Nat.Prime s
mem_s' : ∀ {n : ℕ}, n ∈ s' ↔ Nat.Prime n
this✝ : 2 ≤ ∏ i in s', i + 1
p : ℕ
pp : Nat.Prime p
pdvd : p ∣ ∏ i in s', i + 1
this : p ∣ ∏ i in s', i
⊢ 1 = ∏ i in s', i + 1 - ∏ i in s', i | Please generate a tactic in lean4 to solve the state.
STATE:
s : Finset ℕ
h : ∀ (p : ℕ), Nat.Prime p → p ∈ s
s' : Finset ℕ := filter Nat.Prime s
s'_def : s' = filter Nat.Prime s
mem_s' : ∀ {n : ℕ}, n ∈ s' ↔ Nat.Prime n
this✝ : 2 ≤ ∏ i in s', i + 1
p : ℕ
pp : Nat.Prime p
pdvd : p ∣ ∏ i in s', i + 1
this : p ∣ ∏ i in s', i
⊢ p ∣ 1
TACTIC:
|
https://github.com/fpvandoorn/LeanCourse23.git | 7b0a3cf61b802764dc7baee9d9825e9c62cf9c5d | LeanCourse/MIL/C05_Elementary_Number_Theory/S03_Infinitely_Many_Primes.lean | C05S03.primes_infinite' | [119, 1] | [137, 8] | simp | case h.e'_4
s : Finset ℕ
h : ∀ (p : ℕ), Nat.Prime p → p ∈ s
s' : Finset ℕ := filter Nat.Prime s
s'_def : s' = filter Nat.Prime s
mem_s' : ∀ {n : ℕ}, n ∈ s' ↔ Nat.Prime n
this✝ : 2 ≤ ∏ i in s', i + 1
p : ℕ
pp : Nat.Prime p
pdvd : p ∣ ∏ i in s', i + 1
this : p ∣ ∏ i in s', i
⊢ 1 = ∏ i in s', i + 1 - ∏ i in s', i | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case h.e'_4
s : Finset ℕ
h : ∀ (p : ℕ), Nat.Prime p → p ∈ s
s' : Finset ℕ := filter Nat.Prime s
s'_def : s' = filter Nat.Prime s
mem_s' : ∀ {n : ℕ}, n ∈ s' ↔ Nat.Prime n
this✝ : 2 ≤ ∏ i in s', i + 1
p : ℕ
pp : Nat.Prime p
pdvd : p ∣ ∏ i in s', i + 1
this : p ∣ ∏ i in s', i
⊢ 1 = ∏ i in s', i + 1 - ∏ i in s', i
TACTIC:
|
https://github.com/fpvandoorn/LeanCourse23.git | 7b0a3cf61b802764dc7baee9d9825e9c62cf9c5d | LeanCourse/MIL/C05_Elementary_Number_Theory/S03_Infinitely_Many_Primes.lean | C05S03.bounded_of_ex_finset | [138, 1] | [145, 25] | rintro ⟨s, hs⟩ | Q : ℕ → Prop
⊢ (∃ s, ∀ (k : ℕ), Q k → k ∈ s) → ∃ n, ∀ (k : ℕ), Q k → k < n | case intro
Q : ℕ → Prop
s : Finset ℕ
hs : ∀ (k : ℕ), Q k → k ∈ s
⊢ ∃ n, ∀ (k : ℕ), Q k → k < n | Please generate a tactic in lean4 to solve the state.
STATE:
Q : ℕ → Prop
⊢ (∃ s, ∀ (k : ℕ), Q k → k ∈ s) → ∃ n, ∀ (k : ℕ), Q k → k < n
TACTIC:
|
https://github.com/fpvandoorn/LeanCourse23.git | 7b0a3cf61b802764dc7baee9d9825e9c62cf9c5d | LeanCourse/MIL/C05_Elementary_Number_Theory/S03_Infinitely_Many_Primes.lean | C05S03.bounded_of_ex_finset | [138, 1] | [145, 25] | use s.sup id + 1 | case intro
Q : ℕ → Prop
s : Finset ℕ
hs : ∀ (k : ℕ), Q k → k ∈ s
⊢ ∃ n, ∀ (k : ℕ), Q k → k < n | case h
Q : ℕ → Prop
s : Finset ℕ
hs : ∀ (k : ℕ), Q k → k ∈ s
⊢ ∀ (k : ℕ), Q k → k < sup s id + 1 | Please generate a tactic in lean4 to solve the state.
STATE:
case intro
Q : ℕ → Prop
s : Finset ℕ
hs : ∀ (k : ℕ), Q k → k ∈ s
⊢ ∃ n, ∀ (k : ℕ), Q k → k < n
TACTIC:
|
https://github.com/fpvandoorn/LeanCourse23.git | 7b0a3cf61b802764dc7baee9d9825e9c62cf9c5d | LeanCourse/MIL/C05_Elementary_Number_Theory/S03_Infinitely_Many_Primes.lean | C05S03.bounded_of_ex_finset | [138, 1] | [145, 25] | intro k Qk | case h
Q : ℕ → Prop
s : Finset ℕ
hs : ∀ (k : ℕ), Q k → k ∈ s
⊢ ∀ (k : ℕ), Q k → k < sup s id + 1 | case h
Q : ℕ → Prop
s : Finset ℕ
hs : ∀ (k : ℕ), Q k → k ∈ s
k : ℕ
Qk : Q k
⊢ k < sup s id + 1 | Please generate a tactic in lean4 to solve the state.
STATE:
case h
Q : ℕ → Prop
s : Finset ℕ
hs : ∀ (k : ℕ), Q k → k ∈ s
⊢ ∀ (k : ℕ), Q k → k < sup s id + 1
TACTIC:
|
https://github.com/fpvandoorn/LeanCourse23.git | 7b0a3cf61b802764dc7baee9d9825e9c62cf9c5d | LeanCourse/MIL/C05_Elementary_Number_Theory/S03_Infinitely_Many_Primes.lean | C05S03.bounded_of_ex_finset | [138, 1] | [145, 25] | apply Nat.lt_succ_of_le | case h
Q : ℕ → Prop
s : Finset ℕ
hs : ∀ (k : ℕ), Q k → k ∈ s
k : ℕ
Qk : Q k
⊢ k < sup s id + 1 | case h.a
Q : ℕ → Prop
s : Finset ℕ
hs : ∀ (k : ℕ), Q k → k ∈ s
k : ℕ
Qk : Q k
⊢ k ≤ sup s id | Please generate a tactic in lean4 to solve the state.
STATE:
case h
Q : ℕ → Prop
s : Finset ℕ
hs : ∀ (k : ℕ), Q k → k ∈ s
k : ℕ
Qk : Q k
⊢ k < sup s id + 1
TACTIC:
|
https://github.com/fpvandoorn/LeanCourse23.git | 7b0a3cf61b802764dc7baee9d9825e9c62cf9c5d | LeanCourse/MIL/C05_Elementary_Number_Theory/S03_Infinitely_Many_Primes.lean | C05S03.bounded_of_ex_finset | [138, 1] | [145, 25] | show id k ≤ s.sup id | case h.a
Q : ℕ → Prop
s : Finset ℕ
hs : ∀ (k : ℕ), Q k → k ∈ s
k : ℕ
Qk : Q k
⊢ k ≤ sup s id | case h.a
Q : ℕ → Prop
s : Finset ℕ
hs : ∀ (k : ℕ), Q k → k ∈ s
k : ℕ
Qk : Q k
⊢ id k ≤ sup s id | Please generate a tactic in lean4 to solve the state.
STATE:
case h.a
Q : ℕ → Prop
s : Finset ℕ
hs : ∀ (k : ℕ), Q k → k ∈ s
k : ℕ
Qk : Q k
⊢ k ≤ sup s id
TACTIC:
|
https://github.com/fpvandoorn/LeanCourse23.git | 7b0a3cf61b802764dc7baee9d9825e9c62cf9c5d | LeanCourse/MIL/C05_Elementary_Number_Theory/S03_Infinitely_Many_Primes.lean | C05S03.bounded_of_ex_finset | [138, 1] | [145, 25] | apply le_sup (hs k Qk) | case h.a
Q : ℕ → Prop
s : Finset ℕ
hs : ∀ (k : ℕ), Q k → k ∈ s
k : ℕ
Qk : Q k
⊢ id k ≤ sup s id | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case h.a
Q : ℕ → Prop
s : Finset ℕ
hs : ∀ (k : ℕ), Q k → k ∈ s
k : ℕ
Qk : Q k
⊢ id k ≤ sup s id
TACTIC:
|
https://github.com/fpvandoorn/LeanCourse23.git | 7b0a3cf61b802764dc7baee9d9825e9c62cf9c5d | LeanCourse/MIL/C05_Elementary_Number_Theory/S03_Infinitely_Many_Primes.lean | C05S03.ex_finset_of_bounded | [147, 1] | [153, 13] | rintro ⟨n, hn⟩ | Q : ℕ → Prop
inst✝ : DecidablePred Q
⊢ (∃ n, ∀ (k : ℕ), Q k → k ≤ n) → ∃ s, ∀ (k : ℕ), Q k ↔ k ∈ s | case intro
Q : ℕ → Prop
inst✝ : DecidablePred Q
n : ℕ
hn : ∀ (k : ℕ), Q k → k ≤ n
⊢ ∃ s, ∀ (k : ℕ), Q k ↔ k ∈ s | Please generate a tactic in lean4 to solve the state.
STATE:
Q : ℕ → Prop
inst✝ : DecidablePred Q
⊢ (∃ n, ∀ (k : ℕ), Q k → k ≤ n) → ∃ s, ∀ (k : ℕ), Q k ↔ k ∈ s
TACTIC:
|
https://github.com/fpvandoorn/LeanCourse23.git | 7b0a3cf61b802764dc7baee9d9825e9c62cf9c5d | LeanCourse/MIL/C05_Elementary_Number_Theory/S03_Infinitely_Many_Primes.lean | C05S03.ex_finset_of_bounded | [147, 1] | [153, 13] | use (range (n + 1)).filter Q | case intro
Q : ℕ → Prop
inst✝ : DecidablePred Q
n : ℕ
hn : ∀ (k : ℕ), Q k → k ≤ n
⊢ ∃ s, ∀ (k : ℕ), Q k ↔ k ∈ s | case h
Q : ℕ → Prop
inst✝ : DecidablePred Q
n : ℕ
hn : ∀ (k : ℕ), Q k → k ≤ n
⊢ ∀ (k : ℕ), Q k ↔ k ∈ filter Q (range (n + 1)) | Please generate a tactic in lean4 to solve the state.
STATE:
case intro
Q : ℕ → Prop
inst✝ : DecidablePred Q
n : ℕ
hn : ∀ (k : ℕ), Q k → k ≤ n
⊢ ∃ s, ∀ (k : ℕ), Q k ↔ k ∈ s
TACTIC:
|
https://github.com/fpvandoorn/LeanCourse23.git | 7b0a3cf61b802764dc7baee9d9825e9c62cf9c5d | LeanCourse/MIL/C05_Elementary_Number_Theory/S03_Infinitely_Many_Primes.lean | C05S03.ex_finset_of_bounded | [147, 1] | [153, 13] | intro k | case h
Q : ℕ → Prop
inst✝ : DecidablePred Q
n : ℕ
hn : ∀ (k : ℕ), Q k → k ≤ n
⊢ ∀ (k : ℕ), Q k ↔ k ∈ filter Q (range (n + 1)) | case h
Q : ℕ → Prop
inst✝ : DecidablePred Q
n : ℕ
hn : ∀ (k : ℕ), Q k → k ≤ n
k : ℕ
⊢ Q k ↔ k ∈ filter Q (range (n + 1)) | Please generate a tactic in lean4 to solve the state.
STATE:
case h
Q : ℕ → Prop
inst✝ : DecidablePred Q
n : ℕ
hn : ∀ (k : ℕ), Q k → k ≤ n
⊢ ∀ (k : ℕ), Q k ↔ k ∈ filter Q (range (n + 1))
TACTIC:
|
https://github.com/fpvandoorn/LeanCourse23.git | 7b0a3cf61b802764dc7baee9d9825e9c62cf9c5d | LeanCourse/MIL/C05_Elementary_Number_Theory/S03_Infinitely_Many_Primes.lean | C05S03.ex_finset_of_bounded | [147, 1] | [153, 13] | simp [Nat.lt_succ_iff] | case h
Q : ℕ → Prop
inst✝ : DecidablePred Q
n : ℕ
hn : ∀ (k : ℕ), Q k → k ≤ n
k : ℕ
⊢ Q k ↔ k ∈ filter Q (range (n + 1)) | case h
Q : ℕ → Prop
inst✝ : DecidablePred Q
n : ℕ
hn : ∀ (k : ℕ), Q k → k ≤ n
k : ℕ
⊢ Q k → k ≤ n | Please generate a tactic in lean4 to solve the state.
STATE:
case h
Q : ℕ → Prop
inst✝ : DecidablePred Q
n : ℕ
hn : ∀ (k : ℕ), Q k → k ≤ n
k : ℕ
⊢ Q k ↔ k ∈ filter Q (range (n + 1))
TACTIC:
|
https://github.com/fpvandoorn/LeanCourse23.git | 7b0a3cf61b802764dc7baee9d9825e9c62cf9c5d | LeanCourse/MIL/C05_Elementary_Number_Theory/S03_Infinitely_Many_Primes.lean | C05S03.ex_finset_of_bounded | [147, 1] | [153, 13] | exact hn k | case h
Q : ℕ → Prop
inst✝ : DecidablePred Q
n : ℕ
hn : ∀ (k : ℕ), Q k → k ≤ n
k : ℕ
⊢ Q k → k ≤ n | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case h
Q : ℕ → Prop
inst✝ : DecidablePred Q
n : ℕ
hn : ∀ (k : ℕ), Q k → k ≤ n
k : ℕ
⊢ Q k → k ≤ n
TACTIC:
|
https://github.com/fpvandoorn/LeanCourse23.git | 7b0a3cf61b802764dc7baee9d9825e9c62cf9c5d | LeanCourse/MIL/C05_Elementary_Number_Theory/S03_Infinitely_Many_Primes.lean | C05S03.mod_4_eq_3_or_mod_4_eq_3 | [161, 1] | [167, 42] | revert h | m n : ℕ
h : m * n % 4 = 3
⊢ m % 4 = 3 ∨ n % 4 = 3 | m n : ℕ
⊢ m * n % 4 = 3 → m % 4 = 3 ∨ n % 4 = 3 | Please generate a tactic in lean4 to solve the state.
STATE:
m n : ℕ
h : m * n % 4 = 3
⊢ m % 4 = 3 ∨ n % 4 = 3
TACTIC:
|
https://github.com/fpvandoorn/LeanCourse23.git | 7b0a3cf61b802764dc7baee9d9825e9c62cf9c5d | LeanCourse/MIL/C05_Elementary_Number_Theory/S03_Infinitely_Many_Primes.lean | C05S03.mod_4_eq_3_or_mod_4_eq_3 | [161, 1] | [167, 42] | rw [Nat.mul_mod] | m n : ℕ
⊢ m * n % 4 = 3 → m % 4 = 3 ∨ n % 4 = 3 | m n : ℕ
⊢ m % 4 * (n % 4) % 4 = 3 → m % 4 = 3 ∨ n % 4 = 3 | Please generate a tactic in lean4 to solve the state.
STATE:
m n : ℕ
⊢ m * n % 4 = 3 → m % 4 = 3 ∨ n % 4 = 3
TACTIC:
|
https://github.com/fpvandoorn/LeanCourse23.git | 7b0a3cf61b802764dc7baee9d9825e9c62cf9c5d | LeanCourse/MIL/C05_Elementary_Number_Theory/S03_Infinitely_Many_Primes.lean | C05S03.mod_4_eq_3_or_mod_4_eq_3 | [161, 1] | [167, 42] | have : m % 4 < 4 := Nat.mod_lt m (by norm_num) | m n : ℕ
⊢ m % 4 * (n % 4) % 4 = 3 → m % 4 = 3 ∨ n % 4 = 3 | m n : ℕ
this : m % 4 < 4
⊢ m % 4 * (n % 4) % 4 = 3 → m % 4 = 3 ∨ n % 4 = 3 | Please generate a tactic in lean4 to solve the state.
STATE:
m n : ℕ
⊢ m % 4 * (n % 4) % 4 = 3 → m % 4 = 3 ∨ n % 4 = 3
TACTIC:
|
https://github.com/fpvandoorn/LeanCourse23.git | 7b0a3cf61b802764dc7baee9d9825e9c62cf9c5d | LeanCourse/MIL/C05_Elementary_Number_Theory/S03_Infinitely_Many_Primes.lean | C05S03.mod_4_eq_3_or_mod_4_eq_3 | [161, 1] | [167, 42] | interval_cases hm : m % 4 <;> simp [hm] | m n : ℕ
this : m % 4 < 4
⊢ m % 4 * (n % 4) % 4 = 3 → m % 4 = 3 ∨ n % 4 = 3 | case «2»
m n : ℕ
hm : m % 4 = 2
this : 2 < 4
⊢ 2 * (n % 4) % 4 = 3 → n % 4 = 3 | Please generate a tactic in lean4 to solve the state.
STATE:
m n : ℕ
this : m % 4 < 4
⊢ m % 4 * (n % 4) % 4 = 3 → m % 4 = 3 ∨ n % 4 = 3
TACTIC:
|
https://github.com/fpvandoorn/LeanCourse23.git | 7b0a3cf61b802764dc7baee9d9825e9c62cf9c5d | LeanCourse/MIL/C05_Elementary_Number_Theory/S03_Infinitely_Many_Primes.lean | C05S03.mod_4_eq_3_or_mod_4_eq_3 | [161, 1] | [167, 42] | have : n % 4 < 4 := Nat.mod_lt n (by norm_num) | case «2»
m n : ℕ
hm : m % 4 = 2
this : 2 < 4
⊢ 2 * (n % 4) % 4 = 3 → n % 4 = 3 | case «2»
m n : ℕ
hm : m % 4 = 2
this✝ : 2 < 4
this : n % 4 < 4
⊢ 2 * (n % 4) % 4 = 3 → n % 4 = 3 | Please generate a tactic in lean4 to solve the state.
STATE:
case «2»
m n : ℕ
hm : m % 4 = 2
this : 2 < 4
⊢ 2 * (n % 4) % 4 = 3 → n % 4 = 3
TACTIC:
|
https://github.com/fpvandoorn/LeanCourse23.git | 7b0a3cf61b802764dc7baee9d9825e9c62cf9c5d | LeanCourse/MIL/C05_Elementary_Number_Theory/S03_Infinitely_Many_Primes.lean | C05S03.mod_4_eq_3_or_mod_4_eq_3 | [161, 1] | [167, 42] | interval_cases hn : n % 4 <;> simp [hn] | case «2»
m n : ℕ
hm : m % 4 = 2
this✝ : 2 < 4
this : n % 4 < 4
⊢ 2 * (n % 4) % 4 = 3 → n % 4 = 3 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case «2»
m n : ℕ
hm : m % 4 = 2
this✝ : 2 < 4
this : n % 4 < 4
⊢ 2 * (n % 4) % 4 = 3 → n % 4 = 3
TACTIC:
|
https://github.com/fpvandoorn/LeanCourse23.git | 7b0a3cf61b802764dc7baee9d9825e9c62cf9c5d | LeanCourse/MIL/C05_Elementary_Number_Theory/S03_Infinitely_Many_Primes.lean | C05S03.mod_4_eq_3_or_mod_4_eq_3 | [161, 1] | [167, 42] | norm_num | m n : ℕ
⊢ 4 > 0 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
m n : ℕ
⊢ 4 > 0
TACTIC:
|
https://github.com/fpvandoorn/LeanCourse23.git | 7b0a3cf61b802764dc7baee9d9825e9c62cf9c5d | LeanCourse/MIL/C05_Elementary_Number_Theory/S03_Infinitely_Many_Primes.lean | C05S03.mod_4_eq_3_or_mod_4_eq_3 | [161, 1] | [167, 42] | norm_num | m n : ℕ
hm : m % 4 = 2
this : 2 < 4
⊢ 4 > 0 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
m n : ℕ
hm : m % 4 = 2
this : 2 < 4
⊢ 4 > 0
TACTIC:
|
https://github.com/fpvandoorn/LeanCourse23.git | 7b0a3cf61b802764dc7baee9d9825e9c62cf9c5d | LeanCourse/MIL/C05_Elementary_Number_Theory/S03_Infinitely_Many_Primes.lean | C05S03.two_le_of_mod_4_eq_3 | [169, 1] | [173, 20] | intro neq | case h1
n : ℕ
h : n % 4 = 3
⊢ n ≠ 1 | case h1
n : ℕ
h : n % 4 = 3
neq : n = 1
⊢ False | Please generate a tactic in lean4 to solve the state.
STATE:
case h1
n : ℕ
h : n % 4 = 3
⊢ n ≠ 1
TACTIC:
|
https://github.com/fpvandoorn/LeanCourse23.git | 7b0a3cf61b802764dc7baee9d9825e9c62cf9c5d | LeanCourse/MIL/C05_Elementary_Number_Theory/S03_Infinitely_Many_Primes.lean | C05S03.two_le_of_mod_4_eq_3 | [169, 1] | [173, 20] | rw [neq] at h | case h1
n : ℕ
h : n % 4 = 3
neq : n = 1
⊢ False | case h1
n : ℕ
h : 1 % 4 = 3
neq : n = 1
⊢ False | Please generate a tactic in lean4 to solve the state.
STATE:
case h1
n : ℕ
h : n % 4 = 3
neq : n = 1
⊢ False
TACTIC:
|
https://github.com/fpvandoorn/LeanCourse23.git | 7b0a3cf61b802764dc7baee9d9825e9c62cf9c5d | LeanCourse/MIL/C05_Elementary_Number_Theory/S03_Infinitely_Many_Primes.lean | C05S03.two_le_of_mod_4_eq_3 | [169, 1] | [173, 20] | norm_num at h | case h1
n : ℕ
h : 1 % 4 = 3
neq : n = 1
⊢ False | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case h1
n : ℕ
h : 1 % 4 = 3
neq : n = 1
⊢ False
TACTIC:
|
https://github.com/fpvandoorn/LeanCourse23.git | 7b0a3cf61b802764dc7baee9d9825e9c62cf9c5d | LeanCourse/MIL/C05_Elementary_Number_Theory/S03_Infinitely_Many_Primes.lean | C05S03.aux | [175, 1] | [176, 8] | sorry | m n : ℕ
h₀ : m ∣ n
h₁ : 2 ≤ m
h₂ : m < n
⊢ n / m ∣ n ∧ n / m < n | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
m n : ℕ
h₀ : m ∣ n
h₁ : 2 ≤ m
h₂ : m < n
⊢ n / m ∣ n ∧ n / m < n
TACTIC:
|
https://github.com/fpvandoorn/LeanCourse23.git | 7b0a3cf61b802764dc7baee9d9825e9c62cf9c5d | LeanCourse/MIL/C05_Elementary_Number_Theory/S03_Infinitely_Many_Primes.lean | C05S03.exists_prime_factor_mod_4_eq_3 | [177, 1] | [196, 10] | by_cases np : n.Prime | n : ℕ
h : n % 4 = 3
⊢ ∃ p, Nat.Prime p ∧ p ∣ n ∧ p % 4 = 3 | case pos
n : ℕ
h : n % 4 = 3
np : Nat.Prime n
⊢ ∃ p, Nat.Prime p ∧ p ∣ n ∧ p % 4 = 3
case neg
n : ℕ
h : n % 4 = 3
np : ¬Nat.Prime n
⊢ ∃ p, Nat.Prime p ∧ p ∣ n ∧ p % 4 = 3 | Please generate a tactic in lean4 to solve the state.
STATE:
n : ℕ
h : n % 4 = 3
⊢ ∃ p, Nat.Prime p ∧ p ∣ n ∧ p % 4 = 3
TACTIC:
|
https://github.com/fpvandoorn/LeanCourse23.git | 7b0a3cf61b802764dc7baee9d9825e9c62cf9c5d | LeanCourse/MIL/C05_Elementary_Number_Theory/S03_Infinitely_Many_Primes.lean | C05S03.exists_prime_factor_mod_4_eq_3 | [177, 1] | [196, 10] | induction' n using Nat.strong_induction_on with n ih | case neg
n : ℕ
h : n % 4 = 3
np : ¬Nat.Prime n
⊢ ∃ p, Nat.Prime p ∧ p ∣ n ∧ p % 4 = 3 | case neg.h
n : ℕ
ih : ∀ m < n, m % 4 = 3 → ¬Nat.Prime m → ∃ p, Nat.Prime p ∧ p ∣ m ∧ p % 4 = 3
h : n % 4 = 3
np : ¬Nat.Prime n
⊢ ∃ p, Nat.Prime p ∧ p ∣ n ∧ p % 4 = 3 | Please generate a tactic in lean4 to solve the state.
STATE:
case neg
n : ℕ
h : n % 4 = 3
np : ¬Nat.Prime n
⊢ ∃ p, Nat.Prime p ∧ p ∣ n ∧ p % 4 = 3
TACTIC:
|
https://github.com/fpvandoorn/LeanCourse23.git | 7b0a3cf61b802764dc7baee9d9825e9c62cf9c5d | LeanCourse/MIL/C05_Elementary_Number_Theory/S03_Infinitely_Many_Primes.lean | C05S03.exists_prime_factor_mod_4_eq_3 | [177, 1] | [196, 10] | rw [Nat.prime_def_lt] at np | case neg.h
n : ℕ
ih : ∀ m < n, m % 4 = 3 → ¬Nat.Prime m → ∃ p, Nat.Prime p ∧ p ∣ m ∧ p % 4 = 3
h : n % 4 = 3
np : ¬Nat.Prime n
⊢ ∃ p, Nat.Prime p ∧ p ∣ n ∧ p % 4 = 3 | case neg.h
n : ℕ
ih : ∀ m < n, m % 4 = 3 → ¬Nat.Prime m → ∃ p, Nat.Prime p ∧ p ∣ m ∧ p % 4 = 3
h : n % 4 = 3
np : ¬(2 ≤ n ∧ ∀ m < n, m ∣ n → m = 1)
⊢ ∃ p, Nat.Prime p ∧ p ∣ n ∧ p % 4 = 3 | Please generate a tactic in lean4 to solve the state.
STATE:
case neg.h
n : ℕ
ih : ∀ m < n, m % 4 = 3 → ¬Nat.Prime m → ∃ p, Nat.Prime p ∧ p ∣ m ∧ p % 4 = 3
h : n % 4 = 3
np : ¬Nat.Prime n
⊢ ∃ p, Nat.Prime p ∧ p ∣ n ∧ p % 4 = 3
TACTIC:
|
https://github.com/fpvandoorn/LeanCourse23.git | 7b0a3cf61b802764dc7baee9d9825e9c62cf9c5d | LeanCourse/MIL/C05_Elementary_Number_Theory/S03_Infinitely_Many_Primes.lean | C05S03.exists_prime_factor_mod_4_eq_3 | [177, 1] | [196, 10] | push_neg at np | case neg.h
n : ℕ
ih : ∀ m < n, m % 4 = 3 → ¬Nat.Prime m → ∃ p, Nat.Prime p ∧ p ∣ m ∧ p % 4 = 3
h : n % 4 = 3
np : ¬(2 ≤ n ∧ ∀ m < n, m ∣ n → m = 1)
⊢ ∃ p, Nat.Prime p ∧ p ∣ n ∧ p % 4 = 3 | case neg.h
n : ℕ
ih : ∀ m < n, m % 4 = 3 → ¬Nat.Prime m → ∃ p, Nat.Prime p ∧ p ∣ m ∧ p % 4 = 3
h : n % 4 = 3
np : 2 ≤ n → ∃ m < n, m ∣ n ∧ m ≠ 1
⊢ ∃ p, Nat.Prime p ∧ p ∣ n ∧ p % 4 = 3 | Please generate a tactic in lean4 to solve the state.
STATE:
case neg.h
n : ℕ
ih : ∀ m < n, m % 4 = 3 → ¬Nat.Prime m → ∃ p, Nat.Prime p ∧ p ∣ m ∧ p % 4 = 3
h : n % 4 = 3
np : ¬(2 ≤ n ∧ ∀ m < n, m ∣ n → m = 1)
⊢ ∃ p, Nat.Prime p ∧ p ∣ n ∧ p % 4 = 3
TACTIC:
|
https://github.com/fpvandoorn/LeanCourse23.git | 7b0a3cf61b802764dc7baee9d9825e9c62cf9c5d | LeanCourse/MIL/C05_Elementary_Number_Theory/S03_Infinitely_Many_Primes.lean | C05S03.exists_prime_factor_mod_4_eq_3 | [177, 1] | [196, 10] | rcases np (two_le_of_mod_4_eq_3 h) with ⟨m, mltn, mdvdn, mne1⟩ | case neg.h
n : ℕ
ih : ∀ m < n, m % 4 = 3 → ¬Nat.Prime m → ∃ p, Nat.Prime p ∧ p ∣ m ∧ p % 4 = 3
h : n % 4 = 3
np : 2 ≤ n → ∃ m < n, m ∣ n ∧ m ≠ 1
⊢ ∃ p, Nat.Prime p ∧ p ∣ n ∧ p % 4 = 3 | case neg.h.intro.intro.intro
n : ℕ
ih : ∀ m < n, m % 4 = 3 → ¬Nat.Prime m → ∃ p, Nat.Prime p ∧ p ∣ m ∧ p % 4 = 3
h : n % 4 = 3
np : 2 ≤ n → ∃ m < n, m ∣ n ∧ m ≠ 1
m : ℕ
mltn : m < n
mdvdn : m ∣ n
mne1 : m ≠ 1
⊢ ∃ p, Nat.Prime p ∧ p ∣ n ∧ p % 4 = 3 | Please generate a tactic in lean4 to solve the state.
STATE:
case neg.h
n : ℕ
ih : ∀ m < n, m % 4 = 3 → ¬Nat.Prime m → ∃ p, Nat.Prime p ∧ p ∣ m ∧ p % 4 = 3
h : n % 4 = 3
np : 2 ≤ n → ∃ m < n, m ∣ n ∧ m ≠ 1
⊢ ∃ p, Nat.Prime p ∧ p ∣ n ∧ p % 4 = 3
TACTIC:
|
https://github.com/fpvandoorn/LeanCourse23.git | 7b0a3cf61b802764dc7baee9d9825e9c62cf9c5d | LeanCourse/MIL/C05_Elementary_Number_Theory/S03_Infinitely_Many_Primes.lean | C05S03.exists_prime_factor_mod_4_eq_3 | [177, 1] | [196, 10] | have mge2 : 2 ≤ m := by
apply two_le _ mne1
intro mz
rw [mz, zero_dvd_iff] at mdvdn
linarith | case neg.h.intro.intro.intro
n : ℕ
ih : ∀ m < n, m % 4 = 3 → ¬Nat.Prime m → ∃ p, Nat.Prime p ∧ p ∣ m ∧ p % 4 = 3
h : n % 4 = 3
np : 2 ≤ n → ∃ m < n, m ∣ n ∧ m ≠ 1
m : ℕ
mltn : m < n
mdvdn : m ∣ n
mne1 : m ≠ 1
⊢ ∃ p, Nat.Prime p ∧ p ∣ n ∧ p % 4 = 3 | case neg.h.intro.intro.intro
n : ℕ
ih : ∀ m < n, m % 4 = 3 → ¬Nat.Prime m → ∃ p, Nat.Prime p ∧ p ∣ m ∧ p % 4 = 3
h : n % 4 = 3
np : 2 ≤ n → ∃ m < n, m ∣ n ∧ m ≠ 1
m : ℕ
mltn : m < n
mdvdn : m ∣ n
mne1 : m ≠ 1
mge2 : 2 ≤ m
⊢ ∃ p, Nat.Prime p ∧ p ∣ n ∧ p % 4 = 3 | Please generate a tactic in lean4 to solve the state.
STATE:
case neg.h.intro.intro.intro
n : ℕ
ih : ∀ m < n, m % 4 = 3 → ¬Nat.Prime m → ∃ p, Nat.Prime p ∧ p ∣ m ∧ p % 4 = 3
h : n % 4 = 3
np : 2 ≤ n → ∃ m < n, m ∣ n ∧ m ≠ 1
m : ℕ
mltn : m < n
mdvdn : m ∣ n
mne1 : m ≠ 1
⊢ ∃ p, Nat.Prime p ∧ p ∣ n ∧ p % 4 = 3
TACTIC:
|
https://github.com/fpvandoorn/LeanCourse23.git | 7b0a3cf61b802764dc7baee9d9825e9c62cf9c5d | LeanCourse/MIL/C05_Elementary_Number_Theory/S03_Infinitely_Many_Primes.lean | C05S03.exists_prime_factor_mod_4_eq_3 | [177, 1] | [196, 10] | have neq : m * (n / m) = n := Nat.mul_div_cancel' mdvdn | case neg.h.intro.intro.intro
n : ℕ
ih : ∀ m < n, m % 4 = 3 → ¬Nat.Prime m → ∃ p, Nat.Prime p ∧ p ∣ m ∧ p % 4 = 3
h : n % 4 = 3
np : 2 ≤ n → ∃ m < n, m ∣ n ∧ m ≠ 1
m : ℕ
mltn : m < n
mdvdn : m ∣ n
mne1 : m ≠ 1
mge2 : 2 ≤ m
⊢ ∃ p, Nat.Prime p ∧ p ∣ n ∧ p % 4 = 3 | case neg.h.intro.intro.intro
n : ℕ
ih : ∀ m < n, m % 4 = 3 → ¬Nat.Prime m → ∃ p, Nat.Prime p ∧ p ∣ m ∧ p % 4 = 3
h : n % 4 = 3
np : 2 ≤ n → ∃ m < n, m ∣ n ∧ m ≠ 1
m : ℕ
mltn : m < n
mdvdn : m ∣ n
mne1 : m ≠ 1
mge2 : 2 ≤ m
neq : m * (n / m) = n
⊢ ∃ p, Nat.Prime p ∧ p ∣ n ∧ p % 4 = 3 | Please generate a tactic in lean4 to solve the state.
STATE:
case neg.h.intro.intro.intro
n : ℕ
ih : ∀ m < n, m % 4 = 3 → ¬Nat.Prime m → ∃ p, Nat.Prime p ∧ p ∣ m ∧ p % 4 = 3
h : n % 4 = 3
np : 2 ≤ n → ∃ m < n, m ∣ n ∧ m ≠ 1
m : ℕ
mltn : m < n
mdvdn : m ∣ n
mne1 : m ≠ 1
mge2 : 2 ≤ m
⊢ ∃ p, Nat.Prime p ∧ p ∣ n ∧ p % 4 = 3
TACTIC:
|
https://github.com/fpvandoorn/LeanCourse23.git | 7b0a3cf61b802764dc7baee9d9825e9c62cf9c5d | LeanCourse/MIL/C05_Elementary_Number_Theory/S03_Infinitely_Many_Primes.lean | C05S03.exists_prime_factor_mod_4_eq_3 | [177, 1] | [196, 10] | have : m % 4 = 3 ∨ n / m % 4 = 3 := by
apply mod_4_eq_3_or_mod_4_eq_3
rw [neq, h] | case neg.h.intro.intro.intro
n : ℕ
ih : ∀ m < n, m % 4 = 3 → ¬Nat.Prime m → ∃ p, Nat.Prime p ∧ p ∣ m ∧ p % 4 = 3
h : n % 4 = 3
np : 2 ≤ n → ∃ m < n, m ∣ n ∧ m ≠ 1
m : ℕ
mltn : m < n
mdvdn : m ∣ n
mne1 : m ≠ 1
mge2 : 2 ≤ m
neq : m * (n / m) = n
⊢ ∃ p, Nat.Prime p ∧ p ∣ n ∧ p % 4 = 3 | case neg.h.intro.intro.intro
n : ℕ
ih : ∀ m < n, m % 4 = 3 → ¬Nat.Prime m → ∃ p, Nat.Prime p ∧ p ∣ m ∧ p % 4 = 3
h : n % 4 = 3
np : 2 ≤ n → ∃ m < n, m ∣ n ∧ m ≠ 1
m : ℕ
mltn : m < n
mdvdn : m ∣ n
mne1 : m ≠ 1
mge2 : 2 ≤ m
neq : m * (n / m) = n
this : m % 4 = 3 ∨ n / m % 4 = 3
⊢ ∃ p, Nat.Prime p ∧ p ∣ n ∧ p % 4 = 3 | Please generate a tactic in lean4 to solve the state.
STATE:
case neg.h.intro.intro.intro
n : ℕ
ih : ∀ m < n, m % 4 = 3 → ¬Nat.Prime m → ∃ p, Nat.Prime p ∧ p ∣ m ∧ p % 4 = 3
h : n % 4 = 3
np : 2 ≤ n → ∃ m < n, m ∣ n ∧ m ≠ 1
m : ℕ
mltn : m < n
mdvdn : m ∣ n
mne1 : m ≠ 1
mge2 : 2 ≤ m
neq : m * (n / m) = n
⊢ ∃ p, Nat.Prime p ∧ p ∣ n ∧ p % 4 = 3
TACTIC:
|
https://github.com/fpvandoorn/LeanCourse23.git | 7b0a3cf61b802764dc7baee9d9825e9c62cf9c5d | LeanCourse/MIL/C05_Elementary_Number_Theory/S03_Infinitely_Many_Primes.lean | C05S03.exists_prime_factor_mod_4_eq_3 | [177, 1] | [196, 10] | rcases this with h1 | h1 | case neg.h.intro.intro.intro
n : ℕ
ih : ∀ m < n, m % 4 = 3 → ¬Nat.Prime m → ∃ p, Nat.Prime p ∧ p ∣ m ∧ p % 4 = 3
h : n % 4 = 3
np : 2 ≤ n → ∃ m < n, m ∣ n ∧ m ≠ 1
m : ℕ
mltn : m < n
mdvdn : m ∣ n
mne1 : m ≠ 1
mge2 : 2 ≤ m
neq : m * (n / m) = n
this : m % 4 = 3 ∨ n / m % 4 = 3
⊢ ∃ p, Nat.Prime p ∧ p ∣ n ∧ p % 4 = 3 | case neg.h.intro.intro.intro.inl
n : ℕ
ih : ∀ m < n, m % 4 = 3 → ¬Nat.Prime m → ∃ p, Nat.Prime p ∧ p ∣ m ∧ p % 4 = 3
h : n % 4 = 3
np : 2 ≤ n → ∃ m < n, m ∣ n ∧ m ≠ 1
m : ℕ
mltn : m < n
mdvdn : m ∣ n
mne1 : m ≠ 1
mge2 : 2 ≤ m
neq : m * (n / m) = n
h1 : m % 4 = 3
⊢ ∃ p, Nat.Prime p ∧ p ∣ n ∧ p % 4 = 3
case neg.h.intro.intro.intro.inr
n : ℕ
ih : ∀ m < n, m % 4 = 3 → ¬Nat.Prime m → ∃ p, Nat.Prime p ∧ p ∣ m ∧ p % 4 = 3
h : n % 4 = 3
np : 2 ≤ n → ∃ m < n, m ∣ n ∧ m ≠ 1
m : ℕ
mltn : m < n
mdvdn : m ∣ n
mne1 : m ≠ 1
mge2 : 2 ≤ m
neq : m * (n / m) = n
h1 : n / m % 4 = 3
⊢ ∃ p, Nat.Prime p ∧ p ∣ n ∧ p % 4 = 3 | Please generate a tactic in lean4 to solve the state.
STATE:
case neg.h.intro.intro.intro
n : ℕ
ih : ∀ m < n, m % 4 = 3 → ¬Nat.Prime m → ∃ p, Nat.Prime p ∧ p ∣ m ∧ p % 4 = 3
h : n % 4 = 3
np : 2 ≤ n → ∃ m < n, m ∣ n ∧ m ≠ 1
m : ℕ
mltn : m < n
mdvdn : m ∣ n
mne1 : m ≠ 1
mge2 : 2 ≤ m
neq : m * (n / m) = n
this : m % 4 = 3 ∨ n / m % 4 = 3
⊢ ∃ p, Nat.Prime p ∧ p ∣ n ∧ p % 4 = 3
TACTIC:
|
https://github.com/fpvandoorn/LeanCourse23.git | 7b0a3cf61b802764dc7baee9d9825e9c62cf9c5d | LeanCourse/MIL/C05_Elementary_Number_Theory/S03_Infinitely_Many_Primes.lean | C05S03.exists_prime_factor_mod_4_eq_3 | [177, 1] | [196, 10] | . sorry | case neg.h.intro.intro.intro.inl
n : ℕ
ih : ∀ m < n, m % 4 = 3 → ¬Nat.Prime m → ∃ p, Nat.Prime p ∧ p ∣ m ∧ p % 4 = 3
h : n % 4 = 3
np : 2 ≤ n → ∃ m < n, m ∣ n ∧ m ≠ 1
m : ℕ
mltn : m < n
mdvdn : m ∣ n
mne1 : m ≠ 1
mge2 : 2 ≤ m
neq : m * (n / m) = n
h1 : m % 4 = 3
⊢ ∃ p, Nat.Prime p ∧ p ∣ n ∧ p % 4 = 3
case neg.h.intro.intro.intro.inr
n : ℕ
ih : ∀ m < n, m % 4 = 3 → ¬Nat.Prime m → ∃ p, Nat.Prime p ∧ p ∣ m ∧ p % 4 = 3
h : n % 4 = 3
np : 2 ≤ n → ∃ m < n, m ∣ n ∧ m ≠ 1
m : ℕ
mltn : m < n
mdvdn : m ∣ n
mne1 : m ≠ 1
mge2 : 2 ≤ m
neq : m * (n / m) = n
h1 : n / m % 4 = 3
⊢ ∃ p, Nat.Prime p ∧ p ∣ n ∧ p % 4 = 3 | case neg.h.intro.intro.intro.inr
n : ℕ
ih : ∀ m < n, m % 4 = 3 → ¬Nat.Prime m → ∃ p, Nat.Prime p ∧ p ∣ m ∧ p % 4 = 3
h : n % 4 = 3
np : 2 ≤ n → ∃ m < n, m ∣ n ∧ m ≠ 1
m : ℕ
mltn : m < n
mdvdn : m ∣ n
mne1 : m ≠ 1
mge2 : 2 ≤ m
neq : m * (n / m) = n
h1 : n / m % 4 = 3
⊢ ∃ p, Nat.Prime p ∧ p ∣ n ∧ p % 4 = 3 | Please generate a tactic in lean4 to solve the state.
STATE:
case neg.h.intro.intro.intro.inl
n : ℕ
ih : ∀ m < n, m % 4 = 3 → ¬Nat.Prime m → ∃ p, Nat.Prime p ∧ p ∣ m ∧ p % 4 = 3
h : n % 4 = 3
np : 2 ≤ n → ∃ m < n, m ∣ n ∧ m ≠ 1
m : ℕ
mltn : m < n
mdvdn : m ∣ n
mne1 : m ≠ 1
mge2 : 2 ≤ m
neq : m * (n / m) = n
h1 : m % 4 = 3
⊢ ∃ p, Nat.Prime p ∧ p ∣ n ∧ p % 4 = 3
case neg.h.intro.intro.intro.inr
n : ℕ
ih : ∀ m < n, m % 4 = 3 → ¬Nat.Prime m → ∃ p, Nat.Prime p ∧ p ∣ m ∧ p % 4 = 3
h : n % 4 = 3
np : 2 ≤ n → ∃ m < n, m ∣ n ∧ m ≠ 1
m : ℕ
mltn : m < n
mdvdn : m ∣ n
mne1 : m ≠ 1
mge2 : 2 ≤ m
neq : m * (n / m) = n
h1 : n / m % 4 = 3
⊢ ∃ p, Nat.Prime p ∧ p ∣ n ∧ p % 4 = 3
TACTIC:
|
https://github.com/fpvandoorn/LeanCourse23.git | 7b0a3cf61b802764dc7baee9d9825e9c62cf9c5d | LeanCourse/MIL/C05_Elementary_Number_Theory/S03_Infinitely_Many_Primes.lean | C05S03.exists_prime_factor_mod_4_eq_3 | [177, 1] | [196, 10] | . sorry | case neg.h.intro.intro.intro.inr
n : ℕ
ih : ∀ m < n, m % 4 = 3 → ¬Nat.Prime m → ∃ p, Nat.Prime p ∧ p ∣ m ∧ p % 4 = 3
h : n % 4 = 3
np : 2 ≤ n → ∃ m < n, m ∣ n ∧ m ≠ 1
m : ℕ
mltn : m < n
mdvdn : m ∣ n
mne1 : m ≠ 1
mge2 : 2 ≤ m
neq : m * (n / m) = n
h1 : n / m % 4 = 3
⊢ ∃ p, Nat.Prime p ∧ p ∣ n ∧ p % 4 = 3 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case neg.h.intro.intro.intro.inr
n : ℕ
ih : ∀ m < n, m % 4 = 3 → ¬Nat.Prime m → ∃ p, Nat.Prime p ∧ p ∣ m ∧ p % 4 = 3
h : n % 4 = 3
np : 2 ≤ n → ∃ m < n, m ∣ n ∧ m ≠ 1
m : ℕ
mltn : m < n
mdvdn : m ∣ n
mne1 : m ≠ 1
mge2 : 2 ≤ m
neq : m * (n / m) = n
h1 : n / m % 4 = 3
⊢ ∃ p, Nat.Prime p ∧ p ∣ n ∧ p % 4 = 3
TACTIC:
|
https://github.com/fpvandoorn/LeanCourse23.git | 7b0a3cf61b802764dc7baee9d9825e9c62cf9c5d | LeanCourse/MIL/C05_Elementary_Number_Theory/S03_Infinitely_Many_Primes.lean | C05S03.exists_prime_factor_mod_4_eq_3 | [177, 1] | [196, 10] | use n | case pos
n : ℕ
h : n % 4 = 3
np : Nat.Prime n
⊢ ∃ p, Nat.Prime p ∧ p ∣ n ∧ p % 4 = 3 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case pos
n : ℕ
h : n % 4 = 3
np : Nat.Prime n
⊢ ∃ p, Nat.Prime p ∧ p ∣ n ∧ p % 4 = 3
TACTIC:
|
https://github.com/fpvandoorn/LeanCourse23.git | 7b0a3cf61b802764dc7baee9d9825e9c62cf9c5d | LeanCourse/MIL/C05_Elementary_Number_Theory/S03_Infinitely_Many_Primes.lean | C05S03.exists_prime_factor_mod_4_eq_3 | [177, 1] | [196, 10] | apply two_le _ mne1 | n : ℕ
ih : ∀ m < n, m % 4 = 3 → ¬Nat.Prime m → ∃ p, Nat.Prime p ∧ p ∣ m ∧ p % 4 = 3
h : n % 4 = 3
np : 2 ≤ n → ∃ m < n, m ∣ n ∧ m ≠ 1
m : ℕ
mltn : m < n
mdvdn : m ∣ n
mne1 : m ≠ 1
⊢ 2 ≤ m | n : ℕ
ih : ∀ m < n, m % 4 = 3 → ¬Nat.Prime m → ∃ p, Nat.Prime p ∧ p ∣ m ∧ p % 4 = 3
h : n % 4 = 3
np : 2 ≤ n → ∃ m < n, m ∣ n ∧ m ≠ 1
m : ℕ
mltn : m < n
mdvdn : m ∣ n
mne1 : m ≠ 1
⊢ m ≠ 0 | Please generate a tactic in lean4 to solve the state.
STATE:
n : ℕ
ih : ∀ m < n, m % 4 = 3 → ¬Nat.Prime m → ∃ p, Nat.Prime p ∧ p ∣ m ∧ p % 4 = 3
h : n % 4 = 3
np : 2 ≤ n → ∃ m < n, m ∣ n ∧ m ≠ 1
m : ℕ
mltn : m < n
mdvdn : m ∣ n
mne1 : m ≠ 1
⊢ 2 ≤ m
TACTIC:
|
https://github.com/fpvandoorn/LeanCourse23.git | 7b0a3cf61b802764dc7baee9d9825e9c62cf9c5d | LeanCourse/MIL/C05_Elementary_Number_Theory/S03_Infinitely_Many_Primes.lean | C05S03.exists_prime_factor_mod_4_eq_3 | [177, 1] | [196, 10] | intro mz | n : ℕ
ih : ∀ m < n, m % 4 = 3 → ¬Nat.Prime m → ∃ p, Nat.Prime p ∧ p ∣ m ∧ p % 4 = 3
h : n % 4 = 3
np : 2 ≤ n → ∃ m < n, m ∣ n ∧ m ≠ 1
m : ℕ
mltn : m < n
mdvdn : m ∣ n
mne1 : m ≠ 1
⊢ m ≠ 0 | n : ℕ
ih : ∀ m < n, m % 4 = 3 → ¬Nat.Prime m → ∃ p, Nat.Prime p ∧ p ∣ m ∧ p % 4 = 3
h : n % 4 = 3
np : 2 ≤ n → ∃ m < n, m ∣ n ∧ m ≠ 1
m : ℕ
mltn : m < n
mdvdn : m ∣ n
mne1 : m ≠ 1
mz : m = 0
⊢ False | Please generate a tactic in lean4 to solve the state.
STATE:
n : ℕ
ih : ∀ m < n, m % 4 = 3 → ¬Nat.Prime m → ∃ p, Nat.Prime p ∧ p ∣ m ∧ p % 4 = 3
h : n % 4 = 3
np : 2 ≤ n → ∃ m < n, m ∣ n ∧ m ≠ 1
m : ℕ
mltn : m < n
mdvdn : m ∣ n
mne1 : m ≠ 1
⊢ m ≠ 0
TACTIC:
|
https://github.com/fpvandoorn/LeanCourse23.git | 7b0a3cf61b802764dc7baee9d9825e9c62cf9c5d | LeanCourse/MIL/C05_Elementary_Number_Theory/S03_Infinitely_Many_Primes.lean | C05S03.exists_prime_factor_mod_4_eq_3 | [177, 1] | [196, 10] | rw [mz, zero_dvd_iff] at mdvdn | n : ℕ
ih : ∀ m < n, m % 4 = 3 → ¬Nat.Prime m → ∃ p, Nat.Prime p ∧ p ∣ m ∧ p % 4 = 3
h : n % 4 = 3
np : 2 ≤ n → ∃ m < n, m ∣ n ∧ m ≠ 1
m : ℕ
mltn : m < n
mdvdn : m ∣ n
mne1 : m ≠ 1
mz : m = 0
⊢ False | n : ℕ
ih : ∀ m < n, m % 4 = 3 → ¬Nat.Prime m → ∃ p, Nat.Prime p ∧ p ∣ m ∧ p % 4 = 3
h : n % 4 = 3
np : 2 ≤ n → ∃ m < n, m ∣ n ∧ m ≠ 1
m : ℕ
mltn : m < n
mdvdn : n = 0
mne1 : m ≠ 1
mz : m = 0
⊢ False | Please generate a tactic in lean4 to solve the state.
STATE:
n : ℕ
ih : ∀ m < n, m % 4 = 3 → ¬Nat.Prime m → ∃ p, Nat.Prime p ∧ p ∣ m ∧ p % 4 = 3
h : n % 4 = 3
np : 2 ≤ n → ∃ m < n, m ∣ n ∧ m ≠ 1
m : ℕ
mltn : m < n
mdvdn : m ∣ n
mne1 : m ≠ 1
mz : m = 0
⊢ False
TACTIC:
|
https://github.com/fpvandoorn/LeanCourse23.git | 7b0a3cf61b802764dc7baee9d9825e9c62cf9c5d | LeanCourse/MIL/C05_Elementary_Number_Theory/S03_Infinitely_Many_Primes.lean | C05S03.exists_prime_factor_mod_4_eq_3 | [177, 1] | [196, 10] | linarith | n : ℕ
ih : ∀ m < n, m % 4 = 3 → ¬Nat.Prime m → ∃ p, Nat.Prime p ∧ p ∣ m ∧ p % 4 = 3
h : n % 4 = 3
np : 2 ≤ n → ∃ m < n, m ∣ n ∧ m ≠ 1
m : ℕ
mltn : m < n
mdvdn : n = 0
mne1 : m ≠ 1
mz : m = 0
⊢ False | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
n : ℕ
ih : ∀ m < n, m % 4 = 3 → ¬Nat.Prime m → ∃ p, Nat.Prime p ∧ p ∣ m ∧ p % 4 = 3
h : n % 4 = 3
np : 2 ≤ n → ∃ m < n, m ∣ n ∧ m ≠ 1
m : ℕ
mltn : m < n
mdvdn : n = 0
mne1 : m ≠ 1
mz : m = 0
⊢ False
TACTIC:
|
https://github.com/fpvandoorn/LeanCourse23.git | 7b0a3cf61b802764dc7baee9d9825e9c62cf9c5d | LeanCourse/MIL/C05_Elementary_Number_Theory/S03_Infinitely_Many_Primes.lean | C05S03.exists_prime_factor_mod_4_eq_3 | [177, 1] | [196, 10] | apply mod_4_eq_3_or_mod_4_eq_3 | n : ℕ
ih : ∀ m < n, m % 4 = 3 → ¬Nat.Prime m → ∃ p, Nat.Prime p ∧ p ∣ m ∧ p % 4 = 3
h : n % 4 = 3
np : 2 ≤ n → ∃ m < n, m ∣ n ∧ m ≠ 1
m : ℕ
mltn : m < n
mdvdn : m ∣ n
mne1 : m ≠ 1
mge2 : 2 ≤ m
neq : m * (n / m) = n
⊢ m % 4 = 3 ∨ n / m % 4 = 3 | case h
n : ℕ
ih : ∀ m < n, m % 4 = 3 → ¬Nat.Prime m → ∃ p, Nat.Prime p ∧ p ∣ m ∧ p % 4 = 3
h : n % 4 = 3
np : 2 ≤ n → ∃ m < n, m ∣ n ∧ m ≠ 1
m : ℕ
mltn : m < n
mdvdn : m ∣ n
mne1 : m ≠ 1
mge2 : 2 ≤ m
neq : m * (n / m) = n
⊢ m * (n / m) % 4 = 3 | Please generate a tactic in lean4 to solve the state.
STATE:
n : ℕ
ih : ∀ m < n, m % 4 = 3 → ¬Nat.Prime m → ∃ p, Nat.Prime p ∧ p ∣ m ∧ p % 4 = 3
h : n % 4 = 3
np : 2 ≤ n → ∃ m < n, m ∣ n ∧ m ≠ 1
m : ℕ
mltn : m < n
mdvdn : m ∣ n
mne1 : m ≠ 1
mge2 : 2 ≤ m
neq : m * (n / m) = n
⊢ m % 4 = 3 ∨ n / m % 4 = 3
TACTIC:
|
https://github.com/fpvandoorn/LeanCourse23.git | 7b0a3cf61b802764dc7baee9d9825e9c62cf9c5d | LeanCourse/MIL/C05_Elementary_Number_Theory/S03_Infinitely_Many_Primes.lean | C05S03.exists_prime_factor_mod_4_eq_3 | [177, 1] | [196, 10] | rw [neq, h] | case h
n : ℕ
ih : ∀ m < n, m % 4 = 3 → ¬Nat.Prime m → ∃ p, Nat.Prime p ∧ p ∣ m ∧ p % 4 = 3
h : n % 4 = 3
np : 2 ≤ n → ∃ m < n, m ∣ n ∧ m ≠ 1
m : ℕ
mltn : m < n
mdvdn : m ∣ n
mne1 : m ≠ 1
mge2 : 2 ≤ m
neq : m * (n / m) = n
⊢ m * (n / m) % 4 = 3 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case h
n : ℕ
ih : ∀ m < n, m % 4 = 3 → ¬Nat.Prime m → ∃ p, Nat.Prime p ∧ p ∣ m ∧ p % 4 = 3
h : n % 4 = 3
np : 2 ≤ n → ∃ m < n, m ∣ n ∧ m ≠ 1
m : ℕ
mltn : m < n
mdvdn : m ∣ n
mne1 : m ≠ 1
mge2 : 2 ≤ m
neq : m * (n / m) = n
⊢ m * (n / m) % 4 = 3
TACTIC:
|
https://github.com/fpvandoorn/LeanCourse23.git | 7b0a3cf61b802764dc7baee9d9825e9c62cf9c5d | LeanCourse/MIL/C05_Elementary_Number_Theory/S03_Infinitely_Many_Primes.lean | C05S03.exists_prime_factor_mod_4_eq_3 | [177, 1] | [196, 10] | sorry | case neg.h.intro.intro.intro.inl
n : ℕ
ih : ∀ m < n, m % 4 = 3 → ¬Nat.Prime m → ∃ p, Nat.Prime p ∧ p ∣ m ∧ p % 4 = 3
h : n % 4 = 3
np : 2 ≤ n → ∃ m < n, m ∣ n ∧ m ≠ 1
m : ℕ
mltn : m < n
mdvdn : m ∣ n
mne1 : m ≠ 1
mge2 : 2 ≤ m
neq : m * (n / m) = n
h1 : m % 4 = 3
⊢ ∃ p, Nat.Prime p ∧ p ∣ n ∧ p % 4 = 3 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case neg.h.intro.intro.intro.inl
n : ℕ
ih : ∀ m < n, m % 4 = 3 → ¬Nat.Prime m → ∃ p, Nat.Prime p ∧ p ∣ m ∧ p % 4 = 3
h : n % 4 = 3
np : 2 ≤ n → ∃ m < n, m ∣ n ∧ m ≠ 1
m : ℕ
mltn : m < n
mdvdn : m ∣ n
mne1 : m ≠ 1
mge2 : 2 ≤ m
neq : m * (n / m) = n
h1 : m % 4 = 3
⊢ ∃ p, Nat.Prime p ∧ p ∣ n ∧ p % 4 = 3
TACTIC:
|
https://github.com/fpvandoorn/LeanCourse23.git | 7b0a3cf61b802764dc7baee9d9825e9c62cf9c5d | LeanCourse/MIL/C05_Elementary_Number_Theory/S03_Infinitely_Many_Primes.lean | C05S03.exists_prime_factor_mod_4_eq_3 | [177, 1] | [196, 10] | sorry | case neg.h.intro.intro.intro.inr
n : ℕ
ih : ∀ m < n, m % 4 = 3 → ¬Nat.Prime m → ∃ p, Nat.Prime p ∧ p ∣ m ∧ p % 4 = 3
h : n % 4 = 3
np : 2 ≤ n → ∃ m < n, m ∣ n ∧ m ≠ 1
m : ℕ
mltn : m < n
mdvdn : m ∣ n
mne1 : m ≠ 1
mge2 : 2 ≤ m
neq : m * (n / m) = n
h1 : n / m % 4 = 3
⊢ ∃ p, Nat.Prime p ∧ p ∣ n ∧ p % 4 = 3 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case neg.h.intro.intro.intro.inr
n : ℕ
ih : ∀ m < n, m % 4 = 3 → ¬Nat.Prime m → ∃ p, Nat.Prime p ∧ p ∣ m ∧ p % 4 = 3
h : n % 4 = 3
np : 2 ≤ n → ∃ m < n, m ∣ n ∧ m ≠ 1
m : ℕ
mltn : m < n
mdvdn : m ∣ n
mne1 : m ≠ 1
mge2 : 2 ≤ m
neq : m * (n / m) = n
h1 : n / m % 4 = 3
⊢ ∃ p, Nat.Prime p ∧ p ∣ n ∧ p % 4 = 3
TACTIC:
|
https://github.com/fpvandoorn/LeanCourse23.git | 7b0a3cf61b802764dc7baee9d9825e9c62cf9c5d | LeanCourse/MIL/C05_Elementary_Number_Theory/S03_Infinitely_Many_Primes.lean | C05S03.primes_mod_4_eq_3_infinite | [204, 1] | [228, 16] | by_contra h | ⊢ ∀ (n : ℕ), ∃ p > n, Nat.Prime p ∧ p % 4 = 3 | h : ¬∀ (n : ℕ), ∃ p > n, Nat.Prime p ∧ p % 4 = 3
⊢ False | Please generate a tactic in lean4 to solve the state.
STATE:
⊢ ∀ (n : ℕ), ∃ p > n, Nat.Prime p ∧ p % 4 = 3
TACTIC:
|
https://github.com/fpvandoorn/LeanCourse23.git | 7b0a3cf61b802764dc7baee9d9825e9c62cf9c5d | LeanCourse/MIL/C05_Elementary_Number_Theory/S03_Infinitely_Many_Primes.lean | C05S03.primes_mod_4_eq_3_infinite | [204, 1] | [228, 16] | push_neg at h | h : ¬∀ (n : ℕ), ∃ p > n, Nat.Prime p ∧ p % 4 = 3
⊢ False | h : ∃ n, ∀ p > n, Nat.Prime p → p % 4 ≠ 3
⊢ False | Please generate a tactic in lean4 to solve the state.
STATE:
h : ¬∀ (n : ℕ), ∃ p > n, Nat.Prime p ∧ p % 4 = 3
⊢ False
TACTIC:
|
https://github.com/fpvandoorn/LeanCourse23.git | 7b0a3cf61b802764dc7baee9d9825e9c62cf9c5d | LeanCourse/MIL/C05_Elementary_Number_Theory/S03_Infinitely_Many_Primes.lean | C05S03.primes_mod_4_eq_3_infinite | [204, 1] | [228, 16] | rcases h with ⟨n, hn⟩ | h : ∃ n, ∀ p > n, Nat.Prime p → p % 4 ≠ 3
⊢ False | case intro
n : ℕ
hn : ∀ p > n, Nat.Prime p → p % 4 ≠ 3
⊢ False | Please generate a tactic in lean4 to solve the state.
STATE:
h : ∃ n, ∀ p > n, Nat.Prime p → p % 4 ≠ 3
⊢ False
TACTIC:
|
https://github.com/fpvandoorn/LeanCourse23.git | 7b0a3cf61b802764dc7baee9d9825e9c62cf9c5d | LeanCourse/MIL/C05_Elementary_Number_Theory/S03_Infinitely_Many_Primes.lean | C05S03.primes_mod_4_eq_3_infinite | [204, 1] | [228, 16] | have : ∃ s : Finset Nat, ∀ p : ℕ, p.Prime ∧ p % 4 = 3 ↔ p ∈ s := by
apply ex_finset_of_bounded
use n
contrapose! hn
rcases hn with ⟨p, ⟨pp, p4⟩, pltn⟩
exact ⟨p, pltn, pp, p4⟩ | case intro
n : ℕ
hn : ∀ p > n, Nat.Prime p → p % 4 ≠ 3
⊢ False | case intro
n : ℕ
hn : ∀ p > n, Nat.Prime p → p % 4 ≠ 3
this : ∃ s, ∀ (p : ℕ), Nat.Prime p ∧ p % 4 = 3 ↔ p ∈ s
⊢ False | Please generate a tactic in lean4 to solve the state.
STATE:
case intro
n : ℕ
hn : ∀ p > n, Nat.Prime p → p % 4 ≠ 3
⊢ False
TACTIC:
|
https://github.com/fpvandoorn/LeanCourse23.git | 7b0a3cf61b802764dc7baee9d9825e9c62cf9c5d | LeanCourse/MIL/C05_Elementary_Number_Theory/S03_Infinitely_Many_Primes.lean | C05S03.primes_mod_4_eq_3_infinite | [204, 1] | [228, 16] | rcases this with ⟨s, hs⟩ | case intro
n : ℕ
hn : ∀ p > n, Nat.Prime p → p % 4 ≠ 3
this : ∃ s, ∀ (p : ℕ), Nat.Prime p ∧ p % 4 = 3 ↔ p ∈ s
⊢ False | case intro.intro
n : ℕ
hn : ∀ p > n, Nat.Prime p → p % 4 ≠ 3
s : Finset ℕ
hs : ∀ (p : ℕ), Nat.Prime p ∧ p % 4 = 3 ↔ p ∈ s
⊢ False | Please generate a tactic in lean4 to solve the state.
STATE:
case intro
n : ℕ
hn : ∀ p > n, Nat.Prime p → p % 4 ≠ 3
this : ∃ s, ∀ (p : ℕ), Nat.Prime p ∧ p % 4 = 3 ↔ p ∈ s
⊢ False
TACTIC:
|
https://github.com/fpvandoorn/LeanCourse23.git | 7b0a3cf61b802764dc7baee9d9825e9c62cf9c5d | LeanCourse/MIL/C05_Elementary_Number_Theory/S03_Infinitely_Many_Primes.lean | C05S03.primes_mod_4_eq_3_infinite | [204, 1] | [228, 16] | have h₁ : ((4 * ∏ i in erase s 3, i) + 3) % 4 = 3 := by
sorry | case intro.intro
n : ℕ
hn : ∀ p > n, Nat.Prime p → p % 4 ≠ 3
s : Finset ℕ
hs : ∀ (p : ℕ), Nat.Prime p ∧ p % 4 = 3 ↔ p ∈ s
⊢ False | case intro.intro
n : ℕ
hn : ∀ p > n, Nat.Prime p → p % 4 ≠ 3
s : Finset ℕ
hs : ∀ (p : ℕ), Nat.Prime p ∧ p % 4 = 3 ↔ p ∈ s
h₁ : (4 * ∏ i in erase s 3, i + 3) % 4 = 3
⊢ False | Please generate a tactic in lean4 to solve the state.
STATE:
case intro.intro
n : ℕ
hn : ∀ p > n, Nat.Prime p → p % 4 ≠ 3
s : Finset ℕ
hs : ∀ (p : ℕ), Nat.Prime p ∧ p % 4 = 3 ↔ p ∈ s
⊢ False
TACTIC:
|
https://github.com/fpvandoorn/LeanCourse23.git | 7b0a3cf61b802764dc7baee9d9825e9c62cf9c5d | LeanCourse/MIL/C05_Elementary_Number_Theory/S03_Infinitely_Many_Primes.lean | C05S03.primes_mod_4_eq_3_infinite | [204, 1] | [228, 16] | rcases exists_prime_factor_mod_4_eq_3 h₁ with ⟨p, pp, pdvd, p4eq⟩ | case intro.intro
n : ℕ
hn : ∀ p > n, Nat.Prime p → p % 4 ≠ 3
s : Finset ℕ
hs : ∀ (p : ℕ), Nat.Prime p ∧ p % 4 = 3 ↔ p ∈ s
h₁ : (4 * ∏ i in erase s 3, i + 3) % 4 = 3
⊢ False | case intro.intro.intro.intro.intro
n : ℕ
hn : ∀ p > n, Nat.Prime p → p % 4 ≠ 3
s : Finset ℕ
hs : ∀ (p : ℕ), Nat.Prime p ∧ p % 4 = 3 ↔ p ∈ s
h₁ : (4 * ∏ i in erase s 3, i + 3) % 4 = 3
p : ℕ
pp : Nat.Prime p
pdvd : p ∣ 4 * ∏ i in erase s 3, i + 3
p4eq : p % 4 = 3
⊢ False | Please generate a tactic in lean4 to solve the state.
STATE:
case intro.intro
n : ℕ
hn : ∀ p > n, Nat.Prime p → p % 4 ≠ 3
s : Finset ℕ
hs : ∀ (p : ℕ), Nat.Prime p ∧ p % 4 = 3 ↔ p ∈ s
h₁ : (4 * ∏ i in erase s 3, i + 3) % 4 = 3
⊢ False
TACTIC:
|
https://github.com/fpvandoorn/LeanCourse23.git | 7b0a3cf61b802764dc7baee9d9825e9c62cf9c5d | LeanCourse/MIL/C05_Elementary_Number_Theory/S03_Infinitely_Many_Primes.lean | C05S03.primes_mod_4_eq_3_infinite | [204, 1] | [228, 16] | have ps : p ∈ s := by
sorry | case intro.intro.intro.intro.intro
n : ℕ
hn : ∀ p > n, Nat.Prime p → p % 4 ≠ 3
s : Finset ℕ
hs : ∀ (p : ℕ), Nat.Prime p ∧ p % 4 = 3 ↔ p ∈ s
h₁ : (4 * ∏ i in erase s 3, i + 3) % 4 = 3
p : ℕ
pp : Nat.Prime p
pdvd : p ∣ 4 * ∏ i in erase s 3, i + 3
p4eq : p % 4 = 3
⊢ False | case intro.intro.intro.intro.intro
n : ℕ
hn : ∀ p > n, Nat.Prime p → p % 4 ≠ 3
s : Finset ℕ
hs : ∀ (p : ℕ), Nat.Prime p ∧ p % 4 = 3 ↔ p ∈ s
h₁ : (4 * ∏ i in erase s 3, i + 3) % 4 = 3
p : ℕ
pp : Nat.Prime p
pdvd : p ∣ 4 * ∏ i in erase s 3, i + 3
p4eq : p % 4 = 3
ps : p ∈ s
⊢ False | Please generate a tactic in lean4 to solve the state.
STATE:
case intro.intro.intro.intro.intro
n : ℕ
hn : ∀ p > n, Nat.Prime p → p % 4 ≠ 3
s : Finset ℕ
hs : ∀ (p : ℕ), Nat.Prime p ∧ p % 4 = 3 ↔ p ∈ s
h₁ : (4 * ∏ i in erase s 3, i + 3) % 4 = 3
p : ℕ
pp : Nat.Prime p
pdvd : p ∣ 4 * ∏ i in erase s 3, i + 3
p4eq : p % 4 = 3
⊢ False
TACTIC:
|
https://github.com/fpvandoorn/LeanCourse23.git | 7b0a3cf61b802764dc7baee9d9825e9c62cf9c5d | LeanCourse/MIL/C05_Elementary_Number_Theory/S03_Infinitely_Many_Primes.lean | C05S03.primes_mod_4_eq_3_infinite | [204, 1] | [228, 16] | have pne3 : p ≠ 3 := by
sorry | case intro.intro.intro.intro.intro
n : ℕ
hn : ∀ p > n, Nat.Prime p → p % 4 ≠ 3
s : Finset ℕ
hs : ∀ (p : ℕ), Nat.Prime p ∧ p % 4 = 3 ↔ p ∈ s
h₁ : (4 * ∏ i in erase s 3, i + 3) % 4 = 3
p : ℕ
pp : Nat.Prime p
pdvd : p ∣ 4 * ∏ i in erase s 3, i + 3
p4eq : p % 4 = 3
ps : p ∈ s
⊢ False | case intro.intro.intro.intro.intro
n : ℕ
hn : ∀ p > n, Nat.Prime p → p % 4 ≠ 3
s : Finset ℕ
hs : ∀ (p : ℕ), Nat.Prime p ∧ p % 4 = 3 ↔ p ∈ s
h₁ : (4 * ∏ i in erase s 3, i + 3) % 4 = 3
p : ℕ
pp : Nat.Prime p
pdvd : p ∣ 4 * ∏ i in erase s 3, i + 3
p4eq : p % 4 = 3
ps : p ∈ s
pne3 : p ≠ 3
⊢ False | Please generate a tactic in lean4 to solve the state.
STATE:
case intro.intro.intro.intro.intro
n : ℕ
hn : ∀ p > n, Nat.Prime p → p % 4 ≠ 3
s : Finset ℕ
hs : ∀ (p : ℕ), Nat.Prime p ∧ p % 4 = 3 ↔ p ∈ s
h₁ : (4 * ∏ i in erase s 3, i + 3) % 4 = 3
p : ℕ
pp : Nat.Prime p
pdvd : p ∣ 4 * ∏ i in erase s 3, i + 3
p4eq : p % 4 = 3
ps : p ∈ s
⊢ False
TACTIC:
|
https://github.com/fpvandoorn/LeanCourse23.git | 7b0a3cf61b802764dc7baee9d9825e9c62cf9c5d | LeanCourse/MIL/C05_Elementary_Number_Theory/S03_Infinitely_Many_Primes.lean | C05S03.primes_mod_4_eq_3_infinite | [204, 1] | [228, 16] | have : p ∣ 4 * ∏ i in erase s 3, i := by
sorry | case intro.intro.intro.intro.intro
n : ℕ
hn : ∀ p > n, Nat.Prime p → p % 4 ≠ 3
s : Finset ℕ
hs : ∀ (p : ℕ), Nat.Prime p ∧ p % 4 = 3 ↔ p ∈ s
h₁ : (4 * ∏ i in erase s 3, i + 3) % 4 = 3
p : ℕ
pp : Nat.Prime p
pdvd : p ∣ 4 * ∏ i in erase s 3, i + 3
p4eq : p % 4 = 3
ps : p ∈ s
pne3 : p ≠ 3
⊢ False | case intro.intro.intro.intro.intro
n : ℕ
hn : ∀ p > n, Nat.Prime p → p % 4 ≠ 3
s : Finset ℕ
hs : ∀ (p : ℕ), Nat.Prime p ∧ p % 4 = 3 ↔ p ∈ s
h₁ : (4 * ∏ i in erase s 3, i + 3) % 4 = 3
p : ℕ
pp : Nat.Prime p
pdvd : p ∣ 4 * ∏ i in erase s 3, i + 3
p4eq : p % 4 = 3
ps : p ∈ s
pne3 : p ≠ 3
this : p ∣ 4 * ∏ i in erase s 3, i
⊢ False | Please generate a tactic in lean4 to solve the state.
STATE:
case intro.intro.intro.intro.intro
n : ℕ
hn : ∀ p > n, Nat.Prime p → p % 4 ≠ 3
s : Finset ℕ
hs : ∀ (p : ℕ), Nat.Prime p ∧ p % 4 = 3 ↔ p ∈ s
h₁ : (4 * ∏ i in erase s 3, i + 3) % 4 = 3
p : ℕ
pp : Nat.Prime p
pdvd : p ∣ 4 * ∏ i in erase s 3, i + 3
p4eq : p % 4 = 3
ps : p ∈ s
pne3 : p ≠ 3
⊢ False
TACTIC:
|
https://github.com/fpvandoorn/LeanCourse23.git | 7b0a3cf61b802764dc7baee9d9825e9c62cf9c5d | LeanCourse/MIL/C05_Elementary_Number_Theory/S03_Infinitely_Many_Primes.lean | C05S03.primes_mod_4_eq_3_infinite | [204, 1] | [228, 16] | have : p ∣ 3 := by
sorry | case intro.intro.intro.intro.intro
n : ℕ
hn : ∀ p > n, Nat.Prime p → p % 4 ≠ 3
s : Finset ℕ
hs : ∀ (p : ℕ), Nat.Prime p ∧ p % 4 = 3 ↔ p ∈ s
h₁ : (4 * ∏ i in erase s 3, i + 3) % 4 = 3
p : ℕ
pp : Nat.Prime p
pdvd : p ∣ 4 * ∏ i in erase s 3, i + 3
p4eq : p % 4 = 3
ps : p ∈ s
pne3 : p ≠ 3
this : p ∣ 4 * ∏ i in erase s 3, i
⊢ False | case intro.intro.intro.intro.intro
n : ℕ
hn : ∀ p > n, Nat.Prime p → p % 4 ≠ 3
s : Finset ℕ
hs : ∀ (p : ℕ), Nat.Prime p ∧ p % 4 = 3 ↔ p ∈ s
h₁ : (4 * ∏ i in erase s 3, i + 3) % 4 = 3
p : ℕ
pp : Nat.Prime p
pdvd : p ∣ 4 * ∏ i in erase s 3, i + 3
p4eq : p % 4 = 3
ps : p ∈ s
pne3 : p ≠ 3
this✝ : p ∣ 4 * ∏ i in erase s 3, i
this : p ∣ 3
⊢ False | Please generate a tactic in lean4 to solve the state.
STATE:
case intro.intro.intro.intro.intro
n : ℕ
hn : ∀ p > n, Nat.Prime p → p % 4 ≠ 3
s : Finset ℕ
hs : ∀ (p : ℕ), Nat.Prime p ∧ p % 4 = 3 ↔ p ∈ s
h₁ : (4 * ∏ i in erase s 3, i + 3) % 4 = 3
p : ℕ
pp : Nat.Prime p
pdvd : p ∣ 4 * ∏ i in erase s 3, i + 3
p4eq : p % 4 = 3
ps : p ∈ s
pne3 : p ≠ 3
this : p ∣ 4 * ∏ i in erase s 3, i
⊢ False
TACTIC:
|
https://github.com/fpvandoorn/LeanCourse23.git | 7b0a3cf61b802764dc7baee9d9825e9c62cf9c5d | LeanCourse/MIL/C05_Elementary_Number_Theory/S03_Infinitely_Many_Primes.lean | C05S03.primes_mod_4_eq_3_infinite | [204, 1] | [228, 16] | have : p = 3 := by
sorry | case intro.intro.intro.intro.intro
n : ℕ
hn : ∀ p > n, Nat.Prime p → p % 4 ≠ 3
s : Finset ℕ
hs : ∀ (p : ℕ), Nat.Prime p ∧ p % 4 = 3 ↔ p ∈ s
h₁ : (4 * ∏ i in erase s 3, i + 3) % 4 = 3
p : ℕ
pp : Nat.Prime p
pdvd : p ∣ 4 * ∏ i in erase s 3, i + 3
p4eq : p % 4 = 3
ps : p ∈ s
pne3 : p ≠ 3
this✝ : p ∣ 4 * ∏ i in erase s 3, i
this : p ∣ 3
⊢ False | case intro.intro.intro.intro.intro
n : ℕ
hn : ∀ p > n, Nat.Prime p → p % 4 ≠ 3
s : Finset ℕ
hs : ∀ (p : ℕ), Nat.Prime p ∧ p % 4 = 3 ↔ p ∈ s
h₁ : (4 * ∏ i in erase s 3, i + 3) % 4 = 3
p : ℕ
pp : Nat.Prime p
pdvd : p ∣ 4 * ∏ i in erase s 3, i + 3
p4eq : p % 4 = 3
ps : p ∈ s
pne3 : p ≠ 3
this✝¹ : p ∣ 4 * ∏ i in erase s 3, i
this✝ : p ∣ 3
this : p = 3
⊢ False | Please generate a tactic in lean4 to solve the state.
STATE:
case intro.intro.intro.intro.intro
n : ℕ
hn : ∀ p > n, Nat.Prime p → p % 4 ≠ 3
s : Finset ℕ
hs : ∀ (p : ℕ), Nat.Prime p ∧ p % 4 = 3 ↔ p ∈ s
h₁ : (4 * ∏ i in erase s 3, i + 3) % 4 = 3
p : ℕ
pp : Nat.Prime p
pdvd : p ∣ 4 * ∏ i in erase s 3, i + 3
p4eq : p % 4 = 3
ps : p ∈ s
pne3 : p ≠ 3
this✝ : p ∣ 4 * ∏ i in erase s 3, i
this : p ∣ 3
⊢ False
TACTIC:
|
https://github.com/fpvandoorn/LeanCourse23.git | 7b0a3cf61b802764dc7baee9d9825e9c62cf9c5d | LeanCourse/MIL/C05_Elementary_Number_Theory/S03_Infinitely_Many_Primes.lean | C05S03.primes_mod_4_eq_3_infinite | [204, 1] | [228, 16] | contradiction | case intro.intro.intro.intro.intro
n : ℕ
hn : ∀ p > n, Nat.Prime p → p % 4 ≠ 3
s : Finset ℕ
hs : ∀ (p : ℕ), Nat.Prime p ∧ p % 4 = 3 ↔ p ∈ s
h₁ : (4 * ∏ i in erase s 3, i + 3) % 4 = 3
p : ℕ
pp : Nat.Prime p
pdvd : p ∣ 4 * ∏ i in erase s 3, i + 3
p4eq : p % 4 = 3
ps : p ∈ s
pne3 : p ≠ 3
this✝¹ : p ∣ 4 * ∏ i in erase s 3, i
this✝ : p ∣ 3
this : p = 3
⊢ False | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case intro.intro.intro.intro.intro
n : ℕ
hn : ∀ p > n, Nat.Prime p → p % 4 ≠ 3
s : Finset ℕ
hs : ∀ (p : ℕ), Nat.Prime p ∧ p % 4 = 3 ↔ p ∈ s
h₁ : (4 * ∏ i in erase s 3, i + 3) % 4 = 3
p : ℕ
pp : Nat.Prime p
pdvd : p ∣ 4 * ∏ i in erase s 3, i + 3
p4eq : p % 4 = 3
ps : p ∈ s
pne3 : p ≠ 3
this✝¹ : p ∣ 4 * ∏ i in erase s 3, i
this✝ : p ∣ 3
this : p = 3
⊢ False
TACTIC:
|
https://github.com/fpvandoorn/LeanCourse23.git | 7b0a3cf61b802764dc7baee9d9825e9c62cf9c5d | LeanCourse/MIL/C05_Elementary_Number_Theory/S03_Infinitely_Many_Primes.lean | C05S03.primes_mod_4_eq_3_infinite | [204, 1] | [228, 16] | apply ex_finset_of_bounded | n : ℕ
hn : ∀ p > n, Nat.Prime p → p % 4 ≠ 3
⊢ ∃ s, ∀ (p : ℕ), Nat.Prime p ∧ p % 4 = 3 ↔ p ∈ s | case a
n : ℕ
hn : ∀ p > n, Nat.Prime p → p % 4 ≠ 3
⊢ ∃ n, ∀ (k : ℕ), Nat.Prime k ∧ k % 4 = 3 → k ≤ n | Please generate a tactic in lean4 to solve the state.
STATE:
n : ℕ
hn : ∀ p > n, Nat.Prime p → p % 4 ≠ 3
⊢ ∃ s, ∀ (p : ℕ), Nat.Prime p ∧ p % 4 = 3 ↔ p ∈ s
TACTIC:
|
https://github.com/fpvandoorn/LeanCourse23.git | 7b0a3cf61b802764dc7baee9d9825e9c62cf9c5d | LeanCourse/MIL/C05_Elementary_Number_Theory/S03_Infinitely_Many_Primes.lean | C05S03.primes_mod_4_eq_3_infinite | [204, 1] | [228, 16] | use n | case a
n : ℕ
hn : ∀ p > n, Nat.Prime p → p % 4 ≠ 3
⊢ ∃ n, ∀ (k : ℕ), Nat.Prime k ∧ k % 4 = 3 → k ≤ n | case h
n : ℕ
hn : ∀ p > n, Nat.Prime p → p % 4 ≠ 3
⊢ ∀ (k : ℕ), Nat.Prime k ∧ k % 4 = 3 → k ≤ n | Please generate a tactic in lean4 to solve the state.
STATE:
case a
n : ℕ
hn : ∀ p > n, Nat.Prime p → p % 4 ≠ 3
⊢ ∃ n, ∀ (k : ℕ), Nat.Prime k ∧ k % 4 = 3 → k ≤ n
TACTIC:
|
https://github.com/fpvandoorn/LeanCourse23.git | 7b0a3cf61b802764dc7baee9d9825e9c62cf9c5d | LeanCourse/MIL/C05_Elementary_Number_Theory/S03_Infinitely_Many_Primes.lean | C05S03.primes_mod_4_eq_3_infinite | [204, 1] | [228, 16] | contrapose! hn | case h
n : ℕ
hn : ∀ p > n, Nat.Prime p → p % 4 ≠ 3
⊢ ∀ (k : ℕ), Nat.Prime k ∧ k % 4 = 3 → k ≤ n | case h
n : ℕ
hn : ∃ k, (Nat.Prime k ∧ k % 4 = 3) ∧ n < k
⊢ ∃ p > n, Nat.Prime p ∧ p % 4 = 3 | Please generate a tactic in lean4 to solve the state.
STATE:
case h
n : ℕ
hn : ∀ p > n, Nat.Prime p → p % 4 ≠ 3
⊢ ∀ (k : ℕ), Nat.Prime k ∧ k % 4 = 3 → k ≤ n
TACTIC:
|
https://github.com/fpvandoorn/LeanCourse23.git | 7b0a3cf61b802764dc7baee9d9825e9c62cf9c5d | LeanCourse/MIL/C05_Elementary_Number_Theory/S03_Infinitely_Many_Primes.lean | C05S03.primes_mod_4_eq_3_infinite | [204, 1] | [228, 16] | rcases hn with ⟨p, ⟨pp, p4⟩, pltn⟩ | case h
n : ℕ
hn : ∃ k, (Nat.Prime k ∧ k % 4 = 3) ∧ n < k
⊢ ∃ p > n, Nat.Prime p ∧ p % 4 = 3 | case h.intro.intro.intro
n p : ℕ
pltn : n < p
pp : Nat.Prime p
p4 : p % 4 = 3
⊢ ∃ p > n, Nat.Prime p ∧ p % 4 = 3 | Please generate a tactic in lean4 to solve the state.
STATE:
case h
n : ℕ
hn : ∃ k, (Nat.Prime k ∧ k % 4 = 3) ∧ n < k
⊢ ∃ p > n, Nat.Prime p ∧ p % 4 = 3
TACTIC:
|
https://github.com/fpvandoorn/LeanCourse23.git | 7b0a3cf61b802764dc7baee9d9825e9c62cf9c5d | LeanCourse/MIL/C05_Elementary_Number_Theory/S03_Infinitely_Many_Primes.lean | C05S03.primes_mod_4_eq_3_infinite | [204, 1] | [228, 16] | exact ⟨p, pltn, pp, p4⟩ | case h.intro.intro.intro
n p : ℕ
pltn : n < p
pp : Nat.Prime p
p4 : p % 4 = 3
⊢ ∃ p > n, Nat.Prime p ∧ p % 4 = 3 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case h.intro.intro.intro
n p : ℕ
pltn : n < p
pp : Nat.Prime p
p4 : p % 4 = 3
⊢ ∃ p > n, Nat.Prime p ∧ p % 4 = 3
TACTIC:
|
https://github.com/fpvandoorn/LeanCourse23.git | 7b0a3cf61b802764dc7baee9d9825e9c62cf9c5d | LeanCourse/MIL/C05_Elementary_Number_Theory/S03_Infinitely_Many_Primes.lean | C05S03.primes_mod_4_eq_3_infinite | [204, 1] | [228, 16] | sorry | n : ℕ
hn : ∀ p > n, Nat.Prime p → p % 4 ≠ 3
s : Finset ℕ
hs : ∀ (p : ℕ), Nat.Prime p ∧ p % 4 = 3 ↔ p ∈ s
⊢ (4 * ∏ i in erase s 3, i + 3) % 4 = 3 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
n : ℕ
hn : ∀ p > n, Nat.Prime p → p % 4 ≠ 3
s : Finset ℕ
hs : ∀ (p : ℕ), Nat.Prime p ∧ p % 4 = 3 ↔ p ∈ s
⊢ (4 * ∏ i in erase s 3, i + 3) % 4 = 3
TACTIC:
|
https://github.com/fpvandoorn/LeanCourse23.git | 7b0a3cf61b802764dc7baee9d9825e9c62cf9c5d | LeanCourse/MIL/C05_Elementary_Number_Theory/S03_Infinitely_Many_Primes.lean | C05S03.primes_mod_4_eq_3_infinite | [204, 1] | [228, 16] | sorry | n : ℕ
hn : ∀ p > n, Nat.Prime p → p % 4 ≠ 3
s : Finset ℕ
hs : ∀ (p : ℕ), Nat.Prime p ∧ p % 4 = 3 ↔ p ∈ s
h₁ : (4 * ∏ i in erase s 3, i + 3) % 4 = 3
p : ℕ
pp : Nat.Prime p
pdvd : p ∣ 4 * ∏ i in erase s 3, i + 3
p4eq : p % 4 = 3
⊢ p ∈ s | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
n : ℕ
hn : ∀ p > n, Nat.Prime p → p % 4 ≠ 3
s : Finset ℕ
hs : ∀ (p : ℕ), Nat.Prime p ∧ p % 4 = 3 ↔ p ∈ s
h₁ : (4 * ∏ i in erase s 3, i + 3) % 4 = 3
p : ℕ
pp : Nat.Prime p
pdvd : p ∣ 4 * ∏ i in erase s 3, i + 3
p4eq : p % 4 = 3
⊢ p ∈ s
TACTIC:
|
https://github.com/fpvandoorn/LeanCourse23.git | 7b0a3cf61b802764dc7baee9d9825e9c62cf9c5d | LeanCourse/MIL/C05_Elementary_Number_Theory/S03_Infinitely_Many_Primes.lean | C05S03.primes_mod_4_eq_3_infinite | [204, 1] | [228, 16] | sorry | n : ℕ
hn : ∀ p > n, Nat.Prime p → p % 4 ≠ 3
s : Finset ℕ
hs : ∀ (p : ℕ), Nat.Prime p ∧ p % 4 = 3 ↔ p ∈ s
h₁ : (4 * ∏ i in erase s 3, i + 3) % 4 = 3
p : ℕ
pp : Nat.Prime p
pdvd : p ∣ 4 * ∏ i in erase s 3, i + 3
p4eq : p % 4 = 3
ps : p ∈ s
⊢ p ≠ 3 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
n : ℕ
hn : ∀ p > n, Nat.Prime p → p % 4 ≠ 3
s : Finset ℕ
hs : ∀ (p : ℕ), Nat.Prime p ∧ p % 4 = 3 ↔ p ∈ s
h₁ : (4 * ∏ i in erase s 3, i + 3) % 4 = 3
p : ℕ
pp : Nat.Prime p
pdvd : p ∣ 4 * ∏ i in erase s 3, i + 3
p4eq : p % 4 = 3
ps : p ∈ s
⊢ p ≠ 3
TACTIC:
|
https://github.com/fpvandoorn/LeanCourse23.git | 7b0a3cf61b802764dc7baee9d9825e9c62cf9c5d | LeanCourse/MIL/C05_Elementary_Number_Theory/S03_Infinitely_Many_Primes.lean | C05S03.primes_mod_4_eq_3_infinite | [204, 1] | [228, 16] | sorry | n : ℕ
hn : ∀ p > n, Nat.Prime p → p % 4 ≠ 3
s : Finset ℕ
hs : ∀ (p : ℕ), Nat.Prime p ∧ p % 4 = 3 ↔ p ∈ s
h₁ : (4 * ∏ i in erase s 3, i + 3) % 4 = 3
p : ℕ
pp : Nat.Prime p
pdvd : p ∣ 4 * ∏ i in erase s 3, i + 3
p4eq : p % 4 = 3
ps : p ∈ s
pne3 : p ≠ 3
⊢ p ∣ 4 * ∏ i in erase s 3, i | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
n : ℕ
hn : ∀ p > n, Nat.Prime p → p % 4 ≠ 3
s : Finset ℕ
hs : ∀ (p : ℕ), Nat.Prime p ∧ p % 4 = 3 ↔ p ∈ s
h₁ : (4 * ∏ i in erase s 3, i + 3) % 4 = 3
p : ℕ
pp : Nat.Prime p
pdvd : p ∣ 4 * ∏ i in erase s 3, i + 3
p4eq : p % 4 = 3
ps : p ∈ s
pne3 : p ≠ 3
⊢ p ∣ 4 * ∏ i in erase s 3, i
TACTIC:
|
https://github.com/fpvandoorn/LeanCourse23.git | 7b0a3cf61b802764dc7baee9d9825e9c62cf9c5d | LeanCourse/MIL/C05_Elementary_Number_Theory/S03_Infinitely_Many_Primes.lean | C05S03.primes_mod_4_eq_3_infinite | [204, 1] | [228, 16] | sorry | n : ℕ
hn : ∀ p > n, Nat.Prime p → p % 4 ≠ 3
s : Finset ℕ
hs : ∀ (p : ℕ), Nat.Prime p ∧ p % 4 = 3 ↔ p ∈ s
h₁ : (4 * ∏ i in erase s 3, i + 3) % 4 = 3
p : ℕ
pp : Nat.Prime p
pdvd : p ∣ 4 * ∏ i in erase s 3, i + 3
p4eq : p % 4 = 3
ps : p ∈ s
pne3 : p ≠ 3
this : p ∣ 4 * ∏ i in erase s 3, i
⊢ p ∣ 3 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
n : ℕ
hn : ∀ p > n, Nat.Prime p → p % 4 ≠ 3
s : Finset ℕ
hs : ∀ (p : ℕ), Nat.Prime p ∧ p % 4 = 3 ↔ p ∈ s
h₁ : (4 * ∏ i in erase s 3, i + 3) % 4 = 3
p : ℕ
pp : Nat.Prime p
pdvd : p ∣ 4 * ∏ i in erase s 3, i + 3
p4eq : p % 4 = 3
ps : p ∈ s
pne3 : p ≠ 3
this : p ∣ 4 * ∏ i in erase s 3, i
⊢ p ∣ 3
TACTIC:
|
https://github.com/fpvandoorn/LeanCourse23.git | 7b0a3cf61b802764dc7baee9d9825e9c62cf9c5d | LeanCourse/MIL/C05_Elementary_Number_Theory/S03_Infinitely_Many_Primes.lean | C05S03.primes_mod_4_eq_3_infinite | [204, 1] | [228, 16] | sorry | n : ℕ
hn : ∀ p > n, Nat.Prime p → p % 4 ≠ 3
s : Finset ℕ
hs : ∀ (p : ℕ), Nat.Prime p ∧ p % 4 = 3 ↔ p ∈ s
h₁ : (4 * ∏ i in erase s 3, i + 3) % 4 = 3
p : ℕ
pp : Nat.Prime p
pdvd : p ∣ 4 * ∏ i in erase s 3, i + 3
p4eq : p % 4 = 3
ps : p ∈ s
pne3 : p ≠ 3
this✝ : p ∣ 4 * ∏ i in erase s 3, i
this : p ∣ 3
⊢ p = 3 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
n : ℕ
hn : ∀ p > n, Nat.Prime p → p % 4 ≠ 3
s : Finset ℕ
hs : ∀ (p : ℕ), Nat.Prime p ∧ p % 4 = 3 ↔ p ∈ s
h₁ : (4 * ∏ i in erase s 3, i + 3) % 4 = 3
p : ℕ
pp : Nat.Prime p
pdvd : p ∣ 4 * ∏ i in erase s 3, i + 3
p4eq : p % 4 = 3
ps : p ∈ s
pne3 : p ≠ 3
this✝ : p ∣ 4 * ∏ i in erase s 3, i
this : p ∣ 3
⊢ p = 3
TACTIC:
|
https://github.com/fpvandoorn/LeanCourse23.git | 7b0a3cf61b802764dc7baee9d9825e9c62cf9c5d | LeanCourse/MIL/C05_Elementary_Number_Theory/solutions/Solutions_S01_Irrational_Roots.lean | even_of_even_sqr | [5, 1] | [7, 25] | rw [pow_two, Nat.prime_two.dvd_mul] at h | m : ℕ
h : 2 ∣ m ^ 2
⊢ 2 ∣ m | m : ℕ
h : 2 ∣ m ∨ 2 ∣ m
⊢ 2 ∣ m | Please generate a tactic in lean4 to solve the state.
STATE:
m : ℕ
h : 2 ∣ m ^ 2
⊢ 2 ∣ m
TACTIC:
|
https://github.com/fpvandoorn/LeanCourse23.git | 7b0a3cf61b802764dc7baee9d9825e9c62cf9c5d | LeanCourse/MIL/C05_Elementary_Number_Theory/solutions/Solutions_S01_Irrational_Roots.lean | even_of_even_sqr | [5, 1] | [7, 25] | cases h <;> assumption | m : ℕ
h : 2 ∣ m ∨ 2 ∣ m
⊢ 2 ∣ m | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
m : ℕ
h : 2 ∣ m ∨ 2 ∣ m
⊢ 2 ∣ m
TACTIC:
|
https://github.com/fpvandoorn/LeanCourse23.git | 7b0a3cf61b802764dc7baee9d9825e9c62cf9c5d | LeanCourse/MIL/C05_Elementary_Number_Theory/solutions/Solutions_S01_Irrational_Roots.lean | factorization_mul' | [61, 1] | [64, 6] | rw [Nat.factorization_mul mnez nnez] | m n : ℕ
mnez : m ≠ 0
nnez : n ≠ 0
p : ℕ
⊢ ↑(Nat.factorization (m * n)) p = ↑(Nat.factorization m) p + ↑(Nat.factorization n) p | m n : ℕ
mnez : m ≠ 0
nnez : n ≠ 0
p : ℕ
⊢ ↑(Nat.factorization m + Nat.factorization n) p = ↑(Nat.factorization m) p + ↑(Nat.factorization n) p | Please generate a tactic in lean4 to solve the state.
STATE:
m n : ℕ
mnez : m ≠ 0
nnez : n ≠ 0
p : ℕ
⊢ ↑(Nat.factorization (m * n)) p = ↑(Nat.factorization m) p + ↑(Nat.factorization n) p
TACTIC:
|
https://github.com/fpvandoorn/LeanCourse23.git | 7b0a3cf61b802764dc7baee9d9825e9c62cf9c5d | LeanCourse/MIL/C05_Elementary_Number_Theory/solutions/Solutions_S01_Irrational_Roots.lean | factorization_mul' | [61, 1] | [64, 6] | rfl | m n : ℕ
mnez : m ≠ 0
nnez : n ≠ 0
p : ℕ
⊢ ↑(Nat.factorization m + Nat.factorization n) p = ↑(Nat.factorization m) p + ↑(Nat.factorization n) p | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
m n : ℕ
mnez : m ≠ 0
nnez : n ≠ 0
p : ℕ
⊢ ↑(Nat.factorization m + Nat.factorization n) p = ↑(Nat.factorization m) p + ↑(Nat.factorization n) p
TACTIC:
|
https://github.com/fpvandoorn/LeanCourse23.git | 7b0a3cf61b802764dc7baee9d9825e9c62cf9c5d | LeanCourse/MIL/C05_Elementary_Number_Theory/solutions/Solutions_S01_Irrational_Roots.lean | factorization_pow' | [66, 1] | [69, 6] | rw [Nat.factorization_pow] | n k p : ℕ
⊢ ↑(Nat.factorization (n ^ k)) p = k * ↑(Nat.factorization n) p | n k p : ℕ
⊢ ↑(k • Nat.factorization n) p = k * ↑(Nat.factorization n) p | Please generate a tactic in lean4 to solve the state.
STATE:
n k p : ℕ
⊢ ↑(Nat.factorization (n ^ k)) p = k * ↑(Nat.factorization n) p
TACTIC:
|
https://github.com/fpvandoorn/LeanCourse23.git | 7b0a3cf61b802764dc7baee9d9825e9c62cf9c5d | LeanCourse/MIL/C05_Elementary_Number_Theory/solutions/Solutions_S01_Irrational_Roots.lean | factorization_pow' | [66, 1] | [69, 6] | rfl | n k p : ℕ
⊢ ↑(k • Nat.factorization n) p = k * ↑(Nat.factorization n) p | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
n k p : ℕ
⊢ ↑(k • Nat.factorization n) p = k * ↑(Nat.factorization n) p
TACTIC:
|
https://github.com/fpvandoorn/LeanCourse23.git | 7b0a3cf61b802764dc7baee9d9825e9c62cf9c5d | LeanCourse/MIL/C05_Elementary_Number_Theory/solutions/Solutions_S01_Irrational_Roots.lean | Nat.Prime.factorization' | [71, 1] | [74, 7] | rw [prime_p.factorization] | p : ℕ
prime_p : Prime p
⊢ ↑(Nat.factorization p) p = 1 | p : ℕ
prime_p : Prime p
⊢ (↑fun₀ | p => 1) p = 1 | Please generate a tactic in lean4 to solve the state.
STATE:
p : ℕ
prime_p : Prime p
⊢ ↑(Nat.factorization p) p = 1
TACTIC:
|
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