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/solutions/Solutions_S03_Infinitely_Many_Primes.lean | C05S03.primes_infinite' | [103, 1] | [128, 23] | 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/solutions/Solutions_S03_Infinitely_Many_Primes.lean | C05S03.primes_infinite' | [103, 1] | [128, 23] | have := Nat.le_of_dvd zero_lt_one this | 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 | 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 : 2 ≤ ∏ i in s', i + 1
p : ℕ
pp : Nat.Prime p
pdvd : p ∣ ∏ i in s', i + 1
this✝¹ : p ∣ ∏ i in s', i
this✝ : p ∣ 1
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✝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/solutions/Solutions_S03_Infinitely_Many_Primes.lean | C05S03.primes_infinite' | [103, 1] | [128, 23] | linarith [pp.two_le] | 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 : 2 ≤ ∏ i in s', i + 1
p : ℕ
pp : Nat.Prime p
pdvd : p ∣ ∏ i in s', i + 1
this✝¹ : p ∣ ∏ i in s', i
this✝ : p ∣ 1
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✝² : 2 ≤ ∏ i in s', i + 1
p : ℕ
pp : Nat.Prime p
pdvd : p ∣ ∏ i in s', i + 1
this✝¹ : p ∣ ∏ i in s', i
this✝ : p ∣ 1
this : p ≤ 1
⊢ False
TACTIC:
|
https://github.com/fpvandoorn/LeanCourse23.git | 7b0a3cf61b802764dc7baee9d9825e9c62cf9c5d | LeanCourse/MIL/C05_Elementary_Number_Theory/solutions/Solutions_S03_Infinitely_Many_Primes.lean | C05S03.primes_infinite' | [103, 1] | [128, 23] | 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/solutions/Solutions_S03_Infinitely_Many_Primes.lean | C05S03.primes_infinite' | [103, 1] | [128, 23] | 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/solutions/Solutions_S03_Infinitely_Many_Primes.lean | C05S03.primes_infinite' | [103, 1] | [128, 23] | 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/solutions/Solutions_S03_Infinitely_Many_Primes.lean | C05S03.primes_infinite' | [103, 1] | [128, 23] | apply Nat.succ_le_succ | 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 | case a
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
⊢ 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
⊢ 2 ≤ ∏ i in s', i + 1
TACTIC:
|
https://github.com/fpvandoorn/LeanCourse23.git | 7b0a3cf61b802764dc7baee9d9825e9c62cf9c5d | LeanCourse/MIL/C05_Elementary_Number_Theory/solutions/Solutions_S03_Infinitely_Many_Primes.lean | C05S03.primes_infinite' | [103, 1] | [128, 23] | apply Nat.succ_le_of_lt | case a
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
⊢ 1 ≤ ∏ i in s', i | case a.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
mem_s' : ∀ {n : ℕ}, n ∈ s' ↔ Nat.Prime n
⊢ 0 < ∏ i in s', i | Please generate a tactic in lean4 to solve the state.
STATE:
case a
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
⊢ 1 ≤ ∏ i in s', i
TACTIC:
|
https://github.com/fpvandoorn/LeanCourse23.git | 7b0a3cf61b802764dc7baee9d9825e9c62cf9c5d | LeanCourse/MIL/C05_Elementary_Number_Theory/solutions/Solutions_S03_Infinitely_Many_Primes.lean | C05S03.primes_infinite' | [103, 1] | [128, 23] | apply Finset.prod_pos | case a.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
mem_s' : ∀ {n : ℕ}, n ∈ s' ↔ Nat.Prime n
⊢ 0 < ∏ i in s', i | case a.h.h0
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
⊢ ∀ i ∈ s', 0 < i | Please generate a tactic in lean4 to solve the state.
STATE:
case a.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
mem_s' : ∀ {n : ℕ}, n ∈ s' ↔ Nat.Prime n
⊢ 0 < ∏ i in s', i
TACTIC:
|
https://github.com/fpvandoorn/LeanCourse23.git | 7b0a3cf61b802764dc7baee9d9825e9c62cf9c5d | LeanCourse/MIL/C05_Elementary_Number_Theory/solutions/Solutions_S03_Infinitely_Many_Primes.lean | C05S03.primes_infinite' | [103, 1] | [128, 23] | intro n ns' | case a.h.h0
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
⊢ ∀ i ∈ s', 0 < i | case a.h.h0
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
n : ℕ
ns' : n ∈ s'
⊢ 0 < n | Please generate a tactic in lean4 to solve the state.
STATE:
case a.h.h0
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
⊢ ∀ i ∈ s', 0 < i
TACTIC:
|
https://github.com/fpvandoorn/LeanCourse23.git | 7b0a3cf61b802764dc7baee9d9825e9c62cf9c5d | LeanCourse/MIL/C05_Elementary_Number_Theory/solutions/Solutions_S03_Infinitely_Many_Primes.lean | C05S03.primes_infinite' | [103, 1] | [128, 23] | apply (mem_s'.mp ns').pos | case a.h.h0
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
n : ℕ
ns' : n ∈ s'
⊢ 0 < n | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case a.h.h0
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
n : ℕ
ns' : n ∈ s'
⊢ 0 < n
TACTIC:
|
https://github.com/fpvandoorn/LeanCourse23.git | 7b0a3cf61b802764dc7baee9d9825e9c62cf9c5d | LeanCourse/MIL/C05_Elementary_Number_Theory/solutions/Solutions_S03_Infinitely_Many_Primes.lean | C05S03.primes_infinite' | [103, 1] | [128, 23] | apply dvd_prod_of_mem | 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 | case ha
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 ∈ 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
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/solutions/Solutions_S03_Infinitely_Many_Primes.lean | C05S03.primes_infinite' | [103, 1] | [128, 23] | rw [mem_s'] | case ha
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 ∈ s' | case ha
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
⊢ Nat.Prime p | Please generate a tactic in lean4 to solve the state.
STATE:
case ha
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 ∈ s'
TACTIC:
|
https://github.com/fpvandoorn/LeanCourse23.git | 7b0a3cf61b802764dc7baee9d9825e9c62cf9c5d | LeanCourse/MIL/C05_Elementary_Number_Theory/solutions/Solutions_S03_Infinitely_Many_Primes.lean | C05S03.primes_infinite' | [103, 1] | [128, 23] | apply pp | case ha
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
⊢ Nat.Prime p | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case ha
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
⊢ Nat.Prime p
TACTIC:
|
https://github.com/fpvandoorn/LeanCourse23.git | 7b0a3cf61b802764dc7baee9d9825e9c62cf9c5d | LeanCourse/MIL/C05_Elementary_Number_Theory/solutions/Solutions_S03_Infinitely_Many_Primes.lean | C05S03.primes_infinite' | [103, 1] | [128, 23] | 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/solutions/Solutions_S03_Infinitely_Many_Primes.lean | C05S03.primes_infinite' | [103, 1] | [128, 23] | 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/solutions/Solutions_S03_Infinitely_Many_Primes.lean | C05S03.bounded_of_ex_finset | [130, 1] | [137, 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/solutions/Solutions_S03_Infinitely_Many_Primes.lean | C05S03.bounded_of_ex_finset | [130, 1] | [137, 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/solutions/Solutions_S03_Infinitely_Many_Primes.lean | C05S03.bounded_of_ex_finset | [130, 1] | [137, 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/solutions/Solutions_S03_Infinitely_Many_Primes.lean | C05S03.bounded_of_ex_finset | [130, 1] | [137, 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/solutions/Solutions_S03_Infinitely_Many_Primes.lean | C05S03.bounded_of_ex_finset | [130, 1] | [137, 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/solutions/Solutions_S03_Infinitely_Many_Primes.lean | C05S03.bounded_of_ex_finset | [130, 1] | [137, 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/solutions/Solutions_S03_Infinitely_Many_Primes.lean | C05S03.ex_finset_of_bounded | [139, 1] | [145, 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/solutions/Solutions_S03_Infinitely_Many_Primes.lean | C05S03.ex_finset_of_bounded | [139, 1] | [145, 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/solutions/Solutions_S03_Infinitely_Many_Primes.lean | C05S03.ex_finset_of_bounded | [139, 1] | [145, 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/solutions/Solutions_S03_Infinitely_Many_Primes.lean | C05S03.ex_finset_of_bounded | [139, 1] | [145, 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/solutions/Solutions_S03_Infinitely_Many_Primes.lean | C05S03.ex_finset_of_bounded | [139, 1] | [145, 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/solutions/Solutions_S03_Infinitely_Many_Primes.lean | C05S03.mod_4_eq_3_or_mod_4_eq_3 | [147, 1] | [153, 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/solutions/Solutions_S03_Infinitely_Many_Primes.lean | C05S03.mod_4_eq_3_or_mod_4_eq_3 | [147, 1] | [153, 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/solutions/Solutions_S03_Infinitely_Many_Primes.lean | C05S03.mod_4_eq_3_or_mod_4_eq_3 | [147, 1] | [153, 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/solutions/Solutions_S03_Infinitely_Many_Primes.lean | C05S03.mod_4_eq_3_or_mod_4_eq_3 | [147, 1] | [153, 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/solutions/Solutions_S03_Infinitely_Many_Primes.lean | C05S03.mod_4_eq_3_or_mod_4_eq_3 | [147, 1] | [153, 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/solutions/Solutions_S03_Infinitely_Many_Primes.lean | C05S03.mod_4_eq_3_or_mod_4_eq_3 | [147, 1] | [153, 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/solutions/Solutions_S03_Infinitely_Many_Primes.lean | C05S03.mod_4_eq_3_or_mod_4_eq_3 | [147, 1] | [153, 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/solutions/Solutions_S03_Infinitely_Many_Primes.lean | C05S03.mod_4_eq_3_or_mod_4_eq_3 | [147, 1] | [153, 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/solutions/Solutions_S03_Infinitely_Many_Primes.lean | C05S03.two_le_of_mod_4_eq_3 | [155, 1] | [159, 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/solutions/Solutions_S03_Infinitely_Many_Primes.lean | C05S03.two_le_of_mod_4_eq_3 | [155, 1] | [159, 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/solutions/Solutions_S03_Infinitely_Many_Primes.lean | C05S03.two_le_of_mod_4_eq_3 | [155, 1] | [159, 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/solutions/Solutions_S03_Infinitely_Many_Primes.lean | C05S03.aux | [161, 1] | [164, 59] | constructor | m n : ℕ
h₀ : m ∣ n
h₁ : 2 ≤ m
h₂ : m < n
⊢ n / m ∣ n ∧ n / m < n | case left
m n : ℕ
h₀ : m ∣ n
h₁ : 2 ≤ m
h₂ : m < n
⊢ n / m ∣ n
case right
m n : ℕ
h₀ : m ∣ n
h₁ : 2 ≤ m
h₂ : m < n
⊢ n / m < n | 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/solutions/Solutions_S03_Infinitely_Many_Primes.lean | C05S03.aux | [161, 1] | [164, 59] | exact Nat.div_lt_self (lt_of_le_of_lt (zero_le _) h₂) h₁ | case right
m n : ℕ
h₀ : m ∣ n
h₁ : 2 ≤ m
h₂ : m < n
⊢ n / m < n | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case right
m n : ℕ
h₀ : m ∣ n
h₁ : 2 ≤ m
h₂ : m < n
⊢ n / m < n
TACTIC:
|
https://github.com/fpvandoorn/LeanCourse23.git | 7b0a3cf61b802764dc7baee9d9825e9c62cf9c5d | LeanCourse/MIL/C05_Elementary_Number_Theory/solutions/Solutions_S03_Infinitely_Many_Primes.lean | C05S03.aux | [161, 1] | [164, 59] | exact Nat.div_dvd_of_dvd h₀ | case left
m n : ℕ
h₀ : m ∣ n
h₁ : 2 ≤ m
h₂ : m < n
⊢ n / m ∣ n | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case left
m n : ℕ
h₀ : m ∣ n
h₁ : 2 ≤ m
h₂ : m < n
⊢ n / m ∣ n
TACTIC:
|
https://github.com/fpvandoorn/LeanCourse23.git | 7b0a3cf61b802764dc7baee9d9825e9c62cf9c5d | LeanCourse/MIL/C05_Elementary_Number_Theory/solutions/Solutions_S03_Infinitely_Many_Primes.lean | C05S03.exists_prime_factor_mod_4_eq_3 | [166, 1] | [194, 38] | 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/solutions/Solutions_S03_Infinitely_Many_Primes.lean | C05S03.exists_prime_factor_mod_4_eq_3 | [166, 1] | [194, 38] | 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/solutions/Solutions_S03_Infinitely_Many_Primes.lean | C05S03.exists_prime_factor_mod_4_eq_3 | [166, 1] | [194, 38] | 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/solutions/Solutions_S03_Infinitely_Many_Primes.lean | C05S03.exists_prime_factor_mod_4_eq_3 | [166, 1] | [194, 38] | 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/solutions/Solutions_S03_Infinitely_Many_Primes.lean | C05S03.exists_prime_factor_mod_4_eq_3 | [166, 1] | [194, 38] | 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/solutions/Solutions_S03_Infinitely_Many_Primes.lean | C05S03.exists_prime_factor_mod_4_eq_3 | [166, 1] | [194, 38] | 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/solutions/Solutions_S03_Infinitely_Many_Primes.lean | C05S03.exists_prime_factor_mod_4_eq_3 | [166, 1] | [194, 38] | 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/solutions/Solutions_S03_Infinitely_Many_Primes.lean | C05S03.exists_prime_factor_mod_4_eq_3 | [166, 1] | [194, 38] | 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/solutions/Solutions_S03_Infinitely_Many_Primes.lean | C05S03.exists_prime_factor_mod_4_eq_3 | [166, 1] | [194, 38] | 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/solutions/Solutions_S03_Infinitely_Many_Primes.lean | C05S03.exists_prime_factor_mod_4_eq_3 | [166, 1] | [194, 38] | obtain ⟨nmdvdn, nmltn⟩ := aux mdvdn mge2 mltn | 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.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
h1 : n / m % 4 = 3
nmdvdn : n / m ∣ n
nmltn : 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.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/solutions/Solutions_S03_Infinitely_Many_Primes.lean | C05S03.exists_prime_factor_mod_4_eq_3 | [166, 1] | [194, 38] | by_cases nmp : (n / m).Prime | case neg.h.intro.intro.intro.inr.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
h1 : n / m % 4 = 3
nmdvdn : n / m ∣ n
nmltn : n / m < n
⊢ ∃ p, Nat.Prime p ∧ p ∣ n ∧ p % 4 = 3 | case pos
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
nmdvdn : n / m ∣ n
nmltn : n / m < n
nmp : Nat.Prime (n / m)
⊢ ∃ p, Nat.Prime p ∧ p ∣ n ∧ p % 4 = 3
case neg
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
nmdvdn : n / m ∣ n
nmltn : n / m < n
nmp : ¬Nat.Prime (n / 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.inr.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
h1 : n / m % 4 = 3
nmdvdn : n / m ∣ n
nmltn : 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/solutions/Solutions_S03_Infinitely_Many_Primes.lean | C05S03.exists_prime_factor_mod_4_eq_3 | [166, 1] | [194, 38] | rcases ih (n / m) nmltn h1 nmp with ⟨p, pp, pdvd, p4eq⟩ | case neg
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
nmdvdn : n / m ∣ n
nmltn : n / m < n
nmp : ¬Nat.Prime (n / m)
⊢ ∃ p, Nat.Prime p ∧ p ∣ n ∧ p % 4 = 3 | case neg.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
h1 : n / m % 4 = 3
nmdvdn : n / m ∣ n
nmltn : n / m < n
nmp : ¬Nat.Prime (n / m)
p : ℕ
pp : Nat.Prime p
pdvd : p ∣ n / m
p4eq : p % 4 = 3
⊢ ∃ p, Nat.Prime p ∧ p ∣ n ∧ p % 4 = 3 | Please generate a tactic in lean4 to solve the state.
STATE:
case neg
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
nmdvdn : n / m ∣ n
nmltn : n / m < n
nmp : ¬Nat.Prime (n / m)
⊢ ∃ p, Nat.Prime p ∧ p ∣ n ∧ p % 4 = 3
TACTIC:
|
https://github.com/fpvandoorn/LeanCourse23.git | 7b0a3cf61b802764dc7baee9d9825e9c62cf9c5d | LeanCourse/MIL/C05_Elementary_Number_Theory/solutions/Solutions_S03_Infinitely_Many_Primes.lean | C05S03.exists_prime_factor_mod_4_eq_3 | [166, 1] | [194, 38] | use p | case neg.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
h1 : n / m % 4 = 3
nmdvdn : n / m ∣ n
nmltn : n / m < n
nmp : ¬Nat.Prime (n / m)
p : ℕ
pp : Nat.Prime p
pdvd : p ∣ n / m
p4eq : p % 4 = 3
⊢ ∃ p, Nat.Prime p ∧ p ∣ n ∧ p % 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
h1 : n / m % 4 = 3
nmdvdn : n / m ∣ n
nmltn : n / m < n
nmp : ¬Nat.Prime (n / m)
p : ℕ
pp : Nat.Prime p
pdvd : p ∣ n / m
p4eq : p % 4 = 3
⊢ Nat.Prime p ∧ p ∣ n ∧ p % 4 = 3 | Please generate a tactic in lean4 to solve the state.
STATE:
case neg.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
h1 : n / m % 4 = 3
nmdvdn : n / m ∣ n
nmltn : n / m < n
nmp : ¬Nat.Prime (n / m)
p : ℕ
pp : Nat.Prime p
pdvd : p ∣ n / m
p4eq : p % 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/solutions/Solutions_S03_Infinitely_Many_Primes.lean | C05S03.exists_prime_factor_mod_4_eq_3 | [166, 1] | [194, 38] | exact ⟨pp, pdvd.trans nmdvdn, p4eq⟩ | 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
h1 : n / m % 4 = 3
nmdvdn : n / m ∣ n
nmltn : n / m < n
nmp : ¬Nat.Prime (n / m)
p : ℕ
pp : Nat.Prime p
pdvd : p ∣ n / m
p4eq : p % 4 = 3
⊢ Nat.Prime p ∧ p ∣ n ∧ p % 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
h1 : n / m % 4 = 3
nmdvdn : n / m ∣ n
nmltn : n / m < n
nmp : ¬Nat.Prime (n / m)
p : ℕ
pp : Nat.Prime p
pdvd : p ∣ n / m
p4eq : p % 4 = 3
⊢ Nat.Prime p ∧ p ∣ n ∧ p % 4 = 3
TACTIC:
|
https://github.com/fpvandoorn/LeanCourse23.git | 7b0a3cf61b802764dc7baee9d9825e9c62cf9c5d | LeanCourse/MIL/C05_Elementary_Number_Theory/solutions/Solutions_S03_Infinitely_Many_Primes.lean | C05S03.exists_prime_factor_mod_4_eq_3 | [166, 1] | [194, 38] | 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/solutions/Solutions_S03_Infinitely_Many_Primes.lean | C05S03.exists_prime_factor_mod_4_eq_3 | [166, 1] | [194, 38] | 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/solutions/Solutions_S03_Infinitely_Many_Primes.lean | C05S03.exists_prime_factor_mod_4_eq_3 | [166, 1] | [194, 38] | 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/solutions/Solutions_S03_Infinitely_Many_Primes.lean | C05S03.exists_prime_factor_mod_4_eq_3 | [166, 1] | [194, 38] | 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/solutions/Solutions_S03_Infinitely_Many_Primes.lean | C05S03.exists_prime_factor_mod_4_eq_3 | [166, 1] | [194, 38] | 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/solutions/Solutions_S03_Infinitely_Many_Primes.lean | C05S03.exists_prime_factor_mod_4_eq_3 | [166, 1] | [194, 38] | 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/solutions/Solutions_S03_Infinitely_Many_Primes.lean | C05S03.exists_prime_factor_mod_4_eq_3 | [166, 1] | [194, 38] | 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/solutions/Solutions_S03_Infinitely_Many_Primes.lean | C05S03.exists_prime_factor_mod_4_eq_3 | [166, 1] | [194, 38] | by_cases mp : m.Prime | 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 pos
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
mp : Nat.Prime m
⊢ ∃ p, Nat.Prime p ∧ p ∣ n ∧ p % 4 = 3
case neg
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
mp : ¬Nat.Prime 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.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/solutions/Solutions_S03_Infinitely_Many_Primes.lean | C05S03.exists_prime_factor_mod_4_eq_3 | [166, 1] | [194, 38] | rcases ih m mltn h1 mp with ⟨p, pp, pdvd, p4eq⟩ | case neg
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
mp : ¬Nat.Prime m
⊢ ∃ p, Nat.Prime p ∧ p ∣ n ∧ p % 4 = 3 | case neg.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
h1 : m % 4 = 3
mp : ¬Nat.Prime m
p : ℕ
pp : Nat.Prime p
pdvd : p ∣ m
p4eq : p % 4 = 3
⊢ ∃ p, Nat.Prime p ∧ p ∣ n ∧ p % 4 = 3 | Please generate a tactic in lean4 to solve the state.
STATE:
case neg
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
mp : ¬Nat.Prime m
⊢ ∃ p, Nat.Prime p ∧ p ∣ n ∧ p % 4 = 3
TACTIC:
|
https://github.com/fpvandoorn/LeanCourse23.git | 7b0a3cf61b802764dc7baee9d9825e9c62cf9c5d | LeanCourse/MIL/C05_Elementary_Number_Theory/solutions/Solutions_S03_Infinitely_Many_Primes.lean | C05S03.exists_prime_factor_mod_4_eq_3 | [166, 1] | [194, 38] | use p | case neg.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
h1 : m % 4 = 3
mp : ¬Nat.Prime m
p : ℕ
pp : Nat.Prime p
pdvd : p ∣ m
p4eq : p % 4 = 3
⊢ ∃ p, Nat.Prime p ∧ p ∣ n ∧ p % 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
h1 : m % 4 = 3
mp : ¬Nat.Prime m
p : ℕ
pp : Nat.Prime p
pdvd : p ∣ m
p4eq : p % 4 = 3
⊢ Nat.Prime p ∧ p ∣ n ∧ p % 4 = 3 | Please generate a tactic in lean4 to solve the state.
STATE:
case neg.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
h1 : m % 4 = 3
mp : ¬Nat.Prime m
p : ℕ
pp : Nat.Prime p
pdvd : p ∣ m
p4eq : p % 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/solutions/Solutions_S03_Infinitely_Many_Primes.lean | C05S03.exists_prime_factor_mod_4_eq_3 | [166, 1] | [194, 38] | exact ⟨pp, pdvd.trans mdvdn, p4eq⟩ | 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
h1 : m % 4 = 3
mp : ¬Nat.Prime m
p : ℕ
pp : Nat.Prime p
pdvd : p ∣ m
p4eq : p % 4 = 3
⊢ Nat.Prime p ∧ p ∣ n ∧ p % 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
h1 : m % 4 = 3
mp : ¬Nat.Prime m
p : ℕ
pp : Nat.Prime p
pdvd : p ∣ m
p4eq : p % 4 = 3
⊢ Nat.Prime p ∧ p ∣ n ∧ p % 4 = 3
TACTIC:
|
https://github.com/fpvandoorn/LeanCourse23.git | 7b0a3cf61b802764dc7baee9d9825e9c62cf9c5d | LeanCourse/MIL/C05_Elementary_Number_Theory/solutions/Solutions_S03_Infinitely_Many_Primes.lean | C05S03.exists_prime_factor_mod_4_eq_3 | [166, 1] | [194, 38] | use m | case pos
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
mp : Nat.Prime m
⊢ ∃ 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
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
mp : Nat.Prime m
⊢ ∃ p, Nat.Prime p ∧ p ∣ n ∧ p % 4 = 3
TACTIC:
|
https://github.com/fpvandoorn/LeanCourse23.git | 7b0a3cf61b802764dc7baee9d9825e9c62cf9c5d | LeanCourse/MIL/C05_Elementary_Number_Theory/solutions/Solutions_S03_Infinitely_Many_Primes.lean | C05S03.exists_prime_factor_mod_4_eq_3 | [166, 1] | [194, 38] | use n / m | case pos
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
nmdvdn : n / m ∣ n
nmltn : n / m < n
nmp : Nat.Prime (n / m)
⊢ ∃ 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
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
nmdvdn : n / m ∣ n
nmltn : n / m < n
nmp : Nat.Prime (n / m)
⊢ ∃ p, Nat.Prime p ∧ p ∣ n ∧ p % 4 = 3
TACTIC:
|
https://github.com/fpvandoorn/LeanCourse23.git | 7b0a3cf61b802764dc7baee9d9825e9c62cf9c5d | LeanCourse/MIL/C05_Elementary_Number_Theory/solutions/Solutions_S03_Infinitely_Many_Primes.lean | C05S03.primes_mod_4_eq_3_infinite | [196, 1] | [235, 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/solutions/Solutions_S03_Infinitely_Many_Primes.lean | C05S03.primes_mod_4_eq_3_infinite | [196, 1] | [235, 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/solutions/Solutions_S03_Infinitely_Many_Primes.lean | C05S03.primes_mod_4_eq_3_infinite | [196, 1] | [235, 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/solutions/Solutions_S03_Infinitely_Many_Primes.lean | C05S03.primes_mod_4_eq_3_infinite | [196, 1] | [235, 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/solutions/Solutions_S03_Infinitely_Many_Primes.lean | C05S03.primes_mod_4_eq_3_infinite | [196, 1] | [235, 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/solutions/Solutions_S03_Infinitely_Many_Primes.lean | C05S03.primes_mod_4_eq_3_infinite | [196, 1] | [235, 16] | have h₁ : ((4 * ∏ i in erase s 3, i) + 3) % 4 = 3 := by
rw [add_comm, Nat.add_mul_mod_self_left]
norm_num | 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/solutions/Solutions_S03_Infinitely_Many_Primes.lean | C05S03.primes_mod_4_eq_3_infinite | [196, 1] | [235, 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/solutions/Solutions_S03_Infinitely_Many_Primes.lean | C05S03.primes_mod_4_eq_3_infinite | [196, 1] | [235, 16] | have ps : p ∈ s := by
rw [← hs p]
exact ⟨pp, p4eq⟩ | 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/solutions/Solutions_S03_Infinitely_Many_Primes.lean | C05S03.primes_mod_4_eq_3_infinite | [196, 1] | [235, 16] | have pne3 : p ≠ 3 := by
intro peq
rw [peq, ← Nat.dvd_add_iff_left (dvd_refl 3)] at pdvd
rw [Nat.prime_three.dvd_mul] at pdvd
norm_num at pdvd
have : 3 ∈ s.erase 3 := by
apply mem_of_dvd_prod_primes Nat.prime_three _ pdvd
intro n
simp [← hs n]
tauto
simp at this | 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/solutions/Solutions_S03_Infinitely_Many_Primes.lean | C05S03.primes_mod_4_eq_3_infinite | [196, 1] | [235, 16] | have : p ∣ 4 * ∏ i in erase s 3, i := by
apply dvd_trans _ (dvd_mul_left _ _)
apply dvd_prod_of_mem
simp
constructor <;> assumption | 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/solutions/Solutions_S03_Infinitely_Many_Primes.lean | C05S03.primes_mod_4_eq_3_infinite | [196, 1] | [235, 16] | have : p ∣ 3 := by
convert Nat.dvd_sub' pdvd this
simp | 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/solutions/Solutions_S03_Infinitely_Many_Primes.lean | C05S03.primes_mod_4_eq_3_infinite | [196, 1] | [235, 16] | have : p = 3 := by
apply pp.eq_of_dvd_of_prime Nat.prime_three this | 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/solutions/Solutions_S03_Infinitely_Many_Primes.lean | C05S03.primes_mod_4_eq_3_infinite | [196, 1] | [235, 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/solutions/Solutions_S03_Infinitely_Many_Primes.lean | C05S03.primes_mod_4_eq_3_infinite | [196, 1] | [235, 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/solutions/Solutions_S03_Infinitely_Many_Primes.lean | C05S03.primes_mod_4_eq_3_infinite | [196, 1] | [235, 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/solutions/Solutions_S03_Infinitely_Many_Primes.lean | C05S03.primes_mod_4_eq_3_infinite | [196, 1] | [235, 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/solutions/Solutions_S03_Infinitely_Many_Primes.lean | C05S03.primes_mod_4_eq_3_infinite | [196, 1] | [235, 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/solutions/Solutions_S03_Infinitely_Many_Primes.lean | C05S03.primes_mod_4_eq_3_infinite | [196, 1] | [235, 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/solutions/Solutions_S03_Infinitely_Many_Primes.lean | C05S03.primes_mod_4_eq_3_infinite | [196, 1] | [235, 16] | rw [add_comm, Nat.add_mul_mod_self_left] | 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 | 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
⊢ 3 % 4 = 3 | 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/solutions/Solutions_S03_Infinitely_Many_Primes.lean | C05S03.primes_mod_4_eq_3_infinite | [196, 1] | [235, 16] | norm_num | 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
⊢ 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
⊢ 3 % 4 = 3
TACTIC:
|
https://github.com/fpvandoorn/LeanCourse23.git | 7b0a3cf61b802764dc7baee9d9825e9c62cf9c5d | LeanCourse/MIL/C05_Elementary_Number_Theory/solutions/Solutions_S03_Infinitely_Many_Primes.lean | C05S03.primes_mod_4_eq_3_infinite | [196, 1] | [235, 16] | rw [← hs p] | 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 | 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
⊢ Nat.Prime p ∧ p % 4 = 3 | 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/solutions/Solutions_S03_Infinitely_Many_Primes.lean | C05S03.primes_mod_4_eq_3_infinite | [196, 1] | [235, 16] | exact ⟨pp, p4eq⟩ | 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
⊢ Nat.Prime p ∧ p % 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
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
⊢ Nat.Prime p ∧ p % 4 = 3
TACTIC:
|
https://github.com/fpvandoorn/LeanCourse23.git | 7b0a3cf61b802764dc7baee9d9825e9c62cf9c5d | LeanCourse/MIL/C05_Elementary_Number_Theory/solutions/Solutions_S03_Infinitely_Many_Primes.lean | C05S03.primes_mod_4_eq_3_infinite | [196, 1] | [235, 16] | intro peq | 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 | 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
peq : p = 3
⊢ False | 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/solutions/Solutions_S03_Infinitely_Many_Primes.lean | C05S03.primes_mod_4_eq_3_infinite | [196, 1] | [235, 16] | rw [peq, ← Nat.dvd_add_iff_left (dvd_refl 3)] at pdvd | 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
peq : p = 3
⊢ False | 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 : 3 ∣ 4 * ∏ i in erase s 3, i
p4eq : p % 4 = 3
ps : p ∈ s
peq : p = 3
⊢ False | 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
peq : p = 3
⊢ False
TACTIC:
|
https://github.com/fpvandoorn/LeanCourse23.git | 7b0a3cf61b802764dc7baee9d9825e9c62cf9c5d | LeanCourse/MIL/C05_Elementary_Number_Theory/solutions/Solutions_S03_Infinitely_Many_Primes.lean | C05S03.primes_mod_4_eq_3_infinite | [196, 1] | [235, 16] | rw [Nat.prime_three.dvd_mul] at pdvd | 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 : 3 ∣ 4 * ∏ i in erase s 3, i
p4eq : p % 4 = 3
ps : p ∈ s
peq : p = 3
⊢ False | 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 : 3 ∣ 4 ∨ 3 ∣ ∏ i in erase s 3, i
p4eq : p % 4 = 3
ps : p ∈ s
peq : p = 3
⊢ False | 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 : 3 ∣ 4 * ∏ i in erase s 3, i
p4eq : p % 4 = 3
ps : p ∈ s
peq : p = 3
⊢ False
TACTIC:
|
https://github.com/fpvandoorn/LeanCourse23.git | 7b0a3cf61b802764dc7baee9d9825e9c62cf9c5d | LeanCourse/MIL/C05_Elementary_Number_Theory/solutions/Solutions_S03_Infinitely_Many_Primes.lean | C05S03.primes_mod_4_eq_3_infinite | [196, 1] | [235, 16] | norm_num at pdvd | 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 : 3 ∣ 4 ∨ 3 ∣ ∏ i in erase s 3, i
p4eq : p % 4 = 3
ps : p ∈ s
peq : p = 3
⊢ False | 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
p4eq : p % 4 = 3
ps : p ∈ s
peq : p = 3
pdvd : 3 ∣ ∏ i in erase s 3, i
⊢ False | 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 : 3 ∣ 4 ∨ 3 ∣ ∏ i in erase s 3, i
p4eq : p % 4 = 3
ps : p ∈ s
peq : p = 3
⊢ False
TACTIC:
|
https://github.com/fpvandoorn/LeanCourse23.git | 7b0a3cf61b802764dc7baee9d9825e9c62cf9c5d | LeanCourse/MIL/C05_Elementary_Number_Theory/solutions/Solutions_S03_Infinitely_Many_Primes.lean | C05S03.primes_mod_4_eq_3_infinite | [196, 1] | [235, 16] | have : 3 ∈ s.erase 3 := by
apply mem_of_dvd_prod_primes Nat.prime_three _ pdvd
intro n
simp [← hs n]
tauto | 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
p4eq : p % 4 = 3
ps : p ∈ s
peq : p = 3
pdvd : 3 ∣ ∏ i in erase s 3, i
⊢ False | 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
p4eq : p % 4 = 3
ps : p ∈ s
peq : p = 3
pdvd : 3 ∣ ∏ i in erase s 3, i
this : 3 ∈ erase s 3
⊢ False | 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
p4eq : p % 4 = 3
ps : p ∈ s
peq : p = 3
pdvd : 3 ∣ ∏ i in erase s 3, i
⊢ False
TACTIC:
|
https://github.com/fpvandoorn/LeanCourse23.git | 7b0a3cf61b802764dc7baee9d9825e9c62cf9c5d | LeanCourse/MIL/C05_Elementary_Number_Theory/solutions/Solutions_S03_Infinitely_Many_Primes.lean | C05S03.primes_mod_4_eq_3_infinite | [196, 1] | [235, 16] | simp at this | 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
p4eq : p % 4 = 3
ps : p ∈ s
peq : p = 3
pdvd : 3 ∣ ∏ i in erase s 3, i
this : 3 ∈ erase s 3
⊢ False | 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
p4eq : p % 4 = 3
ps : p ∈ s
peq : p = 3
pdvd : 3 ∣ ∏ i in erase s 3, i
this : 3 ∈ erase s 3
⊢ False
TACTIC:
|
https://github.com/fpvandoorn/LeanCourse23.git | 7b0a3cf61b802764dc7baee9d9825e9c62cf9c5d | LeanCourse/MIL/C05_Elementary_Number_Theory/solutions/Solutions_S03_Infinitely_Many_Primes.lean | C05S03.primes_mod_4_eq_3_infinite | [196, 1] | [235, 16] | apply mem_of_dvd_prod_primes Nat.prime_three _ pdvd | 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
p4eq : p % 4 = 3
ps : p ∈ s
peq : p = 3
pdvd : 3 ∣ ∏ i in erase s 3, i
⊢ 3 ∈ erase s 3 | 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
p4eq : p % 4 = 3
ps : p ∈ s
peq : p = 3
pdvd : 3 ∣ ∏ i in erase s 3, i
⊢ ∀ n ∈ erase s 3, Nat.Prime n | 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
p4eq : p % 4 = 3
ps : p ∈ s
peq : p = 3
pdvd : 3 ∣ ∏ i in erase s 3, i
⊢ 3 ∈ erase s 3
TACTIC:
|
https://github.com/fpvandoorn/LeanCourse23.git | 7b0a3cf61b802764dc7baee9d9825e9c62cf9c5d | LeanCourse/MIL/C05_Elementary_Number_Theory/solutions/Solutions_S03_Infinitely_Many_Primes.lean | C05S03.primes_mod_4_eq_3_infinite | [196, 1] | [235, 16] | intro n | 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
p4eq : p % 4 = 3
ps : p ∈ s
peq : p = 3
pdvd : 3 ∣ ∏ i in erase s 3, i
⊢ ∀ n ∈ erase s 3, Nat.Prime n | 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
p4eq : p % 4 = 3
ps : p ∈ s
peq : p = 3
pdvd : 3 ∣ ∏ i in erase s 3, i
n : ℕ
⊢ n ∈ erase s 3 → Nat.Prime n | 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
p4eq : p % 4 = 3
ps : p ∈ s
peq : p = 3
pdvd : 3 ∣ ∏ i in erase s 3, i
⊢ ∀ n ∈ erase s 3, Nat.Prime n
TACTIC:
|
https://github.com/fpvandoorn/LeanCourse23.git | 7b0a3cf61b802764dc7baee9d9825e9c62cf9c5d | LeanCourse/MIL/C05_Elementary_Number_Theory/solutions/Solutions_S03_Infinitely_Many_Primes.lean | C05S03.primes_mod_4_eq_3_infinite | [196, 1] | [235, 16] | simp [← hs n] | 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
p4eq : p % 4 = 3
ps : p ∈ s
peq : p = 3
pdvd : 3 ∣ ∏ i in erase s 3, i
n : ℕ
⊢ n ∈ erase s 3 → Nat.Prime n | 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
p4eq : p % 4 = 3
ps : p ∈ s
peq : p = 3
pdvd : 3 ∣ ∏ i in erase s 3, i
n : ℕ
⊢ ¬n = 3 → Nat.Prime n → n % 4 = 3 → Nat.Prime n | 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
p4eq : p % 4 = 3
ps : p ∈ s
peq : p = 3
pdvd : 3 ∣ ∏ i in erase s 3, i
n : ℕ
⊢ n ∈ erase s 3 → Nat.Prime n
TACTIC:
|
https://github.com/fpvandoorn/LeanCourse23.git | 7b0a3cf61b802764dc7baee9d9825e9c62cf9c5d | LeanCourse/MIL/C05_Elementary_Number_Theory/solutions/Solutions_S03_Infinitely_Many_Primes.lean | C05S03.primes_mod_4_eq_3_infinite | [196, 1] | [235, 16] | tauto | 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
p4eq : p % 4 = 3
ps : p ∈ s
peq : p = 3
pdvd : 3 ∣ ∏ i in erase s 3, i
n : ℕ
⊢ ¬n = 3 → Nat.Prime n → n % 4 = 3 → Nat.Prime n | 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
p4eq : p % 4 = 3
ps : p ∈ s
peq : p = 3
pdvd : 3 ∣ ∏ i in erase s 3, i
n : ℕ
⊢ ¬n = 3 → Nat.Prime n → n % 4 = 3 → Nat.Prime n
TACTIC:
|
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