Datasets:
internlm
/
Lean-Github

Dataset card Data Studio Files Community
1
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
https://github.com/avigad/mathematics_in_lean_source.git
3fa84e6b3135a3ae41edc6ca195abf0fb1ae3ac3
MIL/C05_Elementary_Number_Theory/S03_Infinitely_Many_Primes.lean
C05S03.bounded_of_ex_finset
[427, 1]
[434, 25]
use s.sup id + 1
case intro Q : β„• β†’ Prop s : Finset β„• hs : βˆ€ (k : β„•), Q k β†’ k ∈ s ⊒ βˆƒ n, βˆ€ (k : β„•), Q k β†’ k < n
case h Q : β„• β†’ Prop s : Finset β„• hs : βˆ€ (k : β„•), Q k β†’ k ∈ s ⊒ βˆ€ (k : β„•), Q k β†’ k < s.sup id + 1
https://github.com/avigad/mathematics_in_lean_source.git
3fa84e6b3135a3ae41edc6ca195abf0fb1ae3ac3
MIL/C05_Elementary_Number_Theory/S03_Infinitely_Many_Primes.lean
C05S03.bounded_of_ex_finset
[427, 1]
[434, 25]
intro k Qk
case h Q : β„• β†’ Prop s : Finset β„• hs : βˆ€ (k : β„•), Q k β†’ k ∈ s ⊒ βˆ€ (k : β„•), Q k β†’ k < s.sup id + 1
case h Q : β„• β†’ Prop s : Finset β„• hs : βˆ€ (k : β„•), Q k β†’ k ∈ s k : β„• Qk : Q k ⊒ k < s.sup id + 1
https://github.com/avigad/mathematics_in_lean_source.git
3fa84e6b3135a3ae41edc6ca195abf0fb1ae3ac3
MIL/C05_Elementary_Number_Theory/S03_Infinitely_Many_Primes.lean
C05S03.bounded_of_ex_finset
[427, 1]
[434, 25]
apply Nat.lt_succ_of_le
case h Q : β„• β†’ Prop s : Finset β„• hs : βˆ€ (k : β„•), Q k β†’ k ∈ s k : β„• Qk : Q k ⊒ k < s.sup id + 1
case h.a Q : β„• β†’ Prop s : Finset β„• hs : βˆ€ (k : β„•), Q k β†’ k ∈ s k : β„• Qk : Q k ⊒ k ≀ s.sup id
https://github.com/avigad/mathematics_in_lean_source.git
3fa84e6b3135a3ae41edc6ca195abf0fb1ae3ac3
MIL/C05_Elementary_Number_Theory/S03_Infinitely_Many_Primes.lean
C05S03.bounded_of_ex_finset
[427, 1]
[434, 25]
show id k ≀ s.sup id
case h.a Q : β„• β†’ Prop s : Finset β„• hs : βˆ€ (k : β„•), Q k β†’ k ∈ s k : β„• Qk : Q k ⊒ k ≀ s.sup id
case h.a Q : β„• β†’ Prop s : Finset β„• hs : βˆ€ (k : β„•), Q k β†’ k ∈ s k : β„• Qk : Q k ⊒ id k ≀ s.sup id
https://github.com/avigad/mathematics_in_lean_source.git
3fa84e6b3135a3ae41edc6ca195abf0fb1ae3ac3
MIL/C05_Elementary_Number_Theory/S03_Infinitely_Many_Primes.lean
C05S03.bounded_of_ex_finset
[427, 1]
[434, 25]
apply le_sup (hs k Qk)
case h.a Q : β„• β†’ Prop s : Finset β„• hs : βˆ€ (k : β„•), Q k β†’ k ∈ s k : β„• Qk : Q k ⊒ id k ≀ s.sup id
no goals
https://github.com/avigad/mathematics_in_lean_source.git
3fa84e6b3135a3ae41edc6ca195abf0fb1ae3ac3
MIL/C05_Elementary_Number_Theory/S03_Infinitely_Many_Primes.lean
C05S03.ex_finset_of_bounded
[436, 1]
[442, 13]
rintro ⟨n, hn⟩
Q : β„• β†’ Prop inst✝ : DecidablePred Q ⊒ (βˆƒ n, βˆ€ (k : β„•), Q k β†’ k ≀ n) β†’ βˆƒ s, βˆ€ (k : β„•), Q k ↔ k ∈ s
case intro Q : β„• β†’ Prop inst✝ : DecidablePred Q n : β„• hn : βˆ€ (k : β„•), Q k β†’ k ≀ n ⊒ βˆƒ s, βˆ€ (k : β„•), Q k ↔ k ∈ s
https://github.com/avigad/mathematics_in_lean_source.git
3fa84e6b3135a3ae41edc6ca195abf0fb1ae3ac3
MIL/C05_Elementary_Number_Theory/S03_Infinitely_Many_Primes.lean
C05S03.ex_finset_of_bounded
[436, 1]
[442, 13]
use (range (n + 1)).filter Q
case intro Q : β„• β†’ Prop inst✝ : DecidablePred Q n : β„• hn : βˆ€ (k : β„•), Q k β†’ k ≀ n ⊒ βˆƒ s, βˆ€ (k : β„•), Q k ↔ k ∈ s
case h Q : β„• β†’ Prop inst✝ : DecidablePred Q n : β„• hn : βˆ€ (k : β„•), Q k β†’ k ≀ n ⊒ βˆ€ (k : β„•), Q k ↔ k ∈ filter Q (range (n + 1))
https://github.com/avigad/mathematics_in_lean_source.git
3fa84e6b3135a3ae41edc6ca195abf0fb1ae3ac3
MIL/C05_Elementary_Number_Theory/S03_Infinitely_Many_Primes.lean
C05S03.ex_finset_of_bounded
[436, 1]
[442, 13]
intro k
case h Q : β„• β†’ Prop inst✝ : DecidablePred Q n : β„• hn : βˆ€ (k : β„•), Q k β†’ k ≀ n ⊒ βˆ€ (k : β„•), Q k ↔ k ∈ filter Q (range (n + 1))
case h Q : β„• β†’ Prop inst✝ : DecidablePred Q n : β„• hn : βˆ€ (k : β„•), Q k β†’ k ≀ n k : β„• ⊒ Q k ↔ k ∈ filter Q (range (n + 1))
https://github.com/avigad/mathematics_in_lean_source.git
3fa84e6b3135a3ae41edc6ca195abf0fb1ae3ac3
MIL/C05_Elementary_Number_Theory/S03_Infinitely_Many_Primes.lean
C05S03.ex_finset_of_bounded
[436, 1]
[442, 13]
simp [Nat.lt_succ_iff]
case h Q : β„• β†’ Prop inst✝ : DecidablePred Q n : β„• hn : βˆ€ (k : β„•), Q k β†’ k ≀ n k : β„• ⊒ Q k ↔ k ∈ filter Q (range (n + 1))
case h Q : β„• β†’ Prop inst✝ : DecidablePred Q n : β„• hn : βˆ€ (k : β„•), Q k β†’ k ≀ n k : β„• ⊒ Q k β†’ k ≀ n
https://github.com/avigad/mathematics_in_lean_source.git
3fa84e6b3135a3ae41edc6ca195abf0fb1ae3ac3
MIL/C05_Elementary_Number_Theory/S03_Infinitely_Many_Primes.lean
C05S03.ex_finset_of_bounded
[436, 1]
[442, 13]
exact hn k
case h Q : β„• β†’ Prop inst✝ : DecidablePred Q n : β„• hn : βˆ€ (k : β„•), Q k β†’ k ≀ n k : β„• ⊒ Q k β†’ k ≀ n
no goals
https://github.com/avigad/mathematics_in_lean_source.git
3fa84e6b3135a3ae41edc6ca195abf0fb1ae3ac3
MIL/C05_Elementary_Number_Theory/S03_Infinitely_Many_Primes.lean
C05S03.mod_4_eq_3_or_mod_4_eq_3
[495, 1]
[501, 32]
revert h
m n : β„• h : m * n % 4 = 3 ⊒ m % 4 = 3 ∨ n % 4 = 3
m n : β„• ⊒ m * n % 4 = 3 β†’ m % 4 = 3 ∨ n % 4 = 3
https://github.com/avigad/mathematics_in_lean_source.git
3fa84e6b3135a3ae41edc6ca195abf0fb1ae3ac3
MIL/C05_Elementary_Number_Theory/S03_Infinitely_Many_Primes.lean
C05S03.mod_4_eq_3_or_mod_4_eq_3
[495, 1]
[501, 32]
rw [Nat.mul_mod]
m n : β„• ⊒ m * n % 4 = 3 β†’ m % 4 = 3 ∨ n % 4 = 3
m n : β„• ⊒ m % 4 * (n % 4) % 4 = 3 β†’ m % 4 = 3 ∨ n % 4 = 3
https://github.com/avigad/mathematics_in_lean_source.git
3fa84e6b3135a3ae41edc6ca195abf0fb1ae3ac3
MIL/C05_Elementary_Number_Theory/S03_Infinitely_Many_Primes.lean
C05S03.mod_4_eq_3_or_mod_4_eq_3
[495, 1]
[501, 32]
have : m % 4 < 4 := Nat.mod_lt m (by norm_num)
m n : β„• ⊒ m % 4 * (n % 4) % 4 = 3 β†’ m % 4 = 3 ∨ n % 4 = 3
m n : β„• this : m % 4 < 4 ⊒ m % 4 * (n % 4) % 4 = 3 β†’ m % 4 = 3 ∨ n % 4 = 3
https://github.com/avigad/mathematics_in_lean_source.git
3fa84e6b3135a3ae41edc6ca195abf0fb1ae3ac3
MIL/C05_Elementary_Number_Theory/S03_Infinitely_Many_Primes.lean
C05S03.mod_4_eq_3_or_mod_4_eq_3
[495, 1]
[501, 32]
interval_cases m % 4 <;> simp [-Nat.mul_mod_mod]
m n : β„• this : m % 4 < 4 ⊒ m % 4 * (n % 4) % 4 = 3 β†’ m % 4 = 3 ∨ n % 4 = 3
case Β«2Β» m n : β„• this : 2 < 4 ⊒ 2 * (n % 4) % 4 = 3 β†’ n % 4 = 3
https://github.com/avigad/mathematics_in_lean_source.git
3fa84e6b3135a3ae41edc6ca195abf0fb1ae3ac3
MIL/C05_Elementary_Number_Theory/S03_Infinitely_Many_Primes.lean
C05S03.mod_4_eq_3_or_mod_4_eq_3
[495, 1]
[501, 32]
have : n % 4 < 4 := Nat.mod_lt n (by norm_num)
case Β«2Β» m n : β„• this : 2 < 4 ⊒ 2 * (n % 4) % 4 = 3 β†’ n % 4 = 3
case Β«2Β» m n : β„• this✝ : 2 < 4 this : n % 4 < 4 ⊒ 2 * (n % 4) % 4 = 3 β†’ n % 4 = 3
https://github.com/avigad/mathematics_in_lean_source.git
3fa84e6b3135a3ae41edc6ca195abf0fb1ae3ac3
MIL/C05_Elementary_Number_Theory/S03_Infinitely_Many_Primes.lean
C05S03.mod_4_eq_3_or_mod_4_eq_3
[495, 1]
[501, 32]
interval_cases n % 4 <;> simp
case Β«2Β» m n : β„• this✝ : 2 < 4 this : n % 4 < 4 ⊒ 2 * (n % 4) % 4 = 3 β†’ n % 4 = 3
no goals
https://github.com/avigad/mathematics_in_lean_source.git
3fa84e6b3135a3ae41edc6ca195abf0fb1ae3ac3
MIL/C05_Elementary_Number_Theory/S03_Infinitely_Many_Primes.lean
C05S03.mod_4_eq_3_or_mod_4_eq_3
[495, 1]
[501, 32]
norm_num
m n : β„• ⊒ 4 > 0
no goals
https://github.com/avigad/mathematics_in_lean_source.git
3fa84e6b3135a3ae41edc6ca195abf0fb1ae3ac3
MIL/C05_Elementary_Number_Theory/S03_Infinitely_Many_Primes.lean
C05S03.mod_4_eq_3_or_mod_4_eq_3
[495, 1]
[501, 32]
norm_num
m n : β„• this : 2 < 4 ⊒ 4 > 0
no goals
https://github.com/avigad/mathematics_in_lean_source.git
3fa84e6b3135a3ae41edc6ca195abf0fb1ae3ac3
MIL/C05_Elementary_Number_Theory/S03_Infinitely_Many_Primes.lean
C05S03.two_le_of_mod_4_eq_3
[503, 1]
[507, 20]
intro neq
case h1 n : β„• h : n % 4 = 3 ⊒ n β‰  1
case h1 n : β„• h : n % 4 = 3 neq : n = 1 ⊒ False
https://github.com/avigad/mathematics_in_lean_source.git
3fa84e6b3135a3ae41edc6ca195abf0fb1ae3ac3
MIL/C05_Elementary_Number_Theory/S03_Infinitely_Many_Primes.lean
C05S03.two_le_of_mod_4_eq_3
[503, 1]
[507, 20]
rw [neq] at h
case h1 n : β„• h : n % 4 = 3 neq : n = 1 ⊒ False
case h1 n : β„• h : 1 % 4 = 3 neq : n = 1 ⊒ False
https://github.com/avigad/mathematics_in_lean_source.git
3fa84e6b3135a3ae41edc6ca195abf0fb1ae3ac3
MIL/C05_Elementary_Number_Theory/S03_Infinitely_Many_Primes.lean
C05S03.two_le_of_mod_4_eq_3
[503, 1]
[507, 20]
norm_num at h
case h1 n : β„• h : 1 % 4 = 3 neq : n = 1 ⊒ False
no goals
https://github.com/avigad/mathematics_in_lean_source.git
3fa84e6b3135a3ae41edc6ca195abf0fb1ae3ac3
MIL/C05_Elementary_Number_Theory/S03_Infinitely_Many_Primes.lean
C05S03.aux
[517, 1]
[523, 59]
constructor
m n : β„• hβ‚€ : m ∣ n h₁ : 2 ≀ m hβ‚‚ : m < n ⊒ n / m ∣ n ∧ n / m < n
case left m n : β„• hβ‚€ : m ∣ n h₁ : 2 ≀ m hβ‚‚ : m < n ⊒ n / m ∣ n case right m n : β„• hβ‚€ : m ∣ n h₁ : 2 ≀ m hβ‚‚ : m < n ⊒ n / m < n
https://github.com/avigad/mathematics_in_lean_source.git
3fa84e6b3135a3ae41edc6ca195abf0fb1ae3ac3
MIL/C05_Elementary_Number_Theory/S03_Infinitely_Many_Primes.lean
C05S03.aux
[517, 1]
[523, 59]
exact Nat.div_lt_self (lt_of_le_of_lt (zero_le _) hβ‚‚) h₁
case right m n : β„• hβ‚€ : m ∣ n h₁ : 2 ≀ m hβ‚‚ : m < n ⊒ n / m < n
no goals
https://github.com/avigad/mathematics_in_lean_source.git
3fa84e6b3135a3ae41edc6ca195abf0fb1ae3ac3
MIL/C05_Elementary_Number_Theory/S03_Infinitely_Many_Primes.lean
C05S03.aux
[517, 1]
[523, 59]
exact Nat.div_dvd_of_dvd hβ‚€
case left m n : β„• hβ‚€ : m ∣ n h₁ : 2 ≀ m hβ‚‚ : m < n ⊒ n / m ∣ n
no goals
https://github.com/avigad/mathematics_in_lean_source.git
3fa84e6b3135a3ae41edc6ca195abf0fb1ae3ac3
MIL/C05_Elementary_Number_Theory/S03_Infinitely_Many_Primes.lean
C05S03.exists_prime_factor_mod_4_eq_3
[533, 1]
[565, 38]
by_cases np : n.Prime
n : β„• h : n % 4 = 3 ⊒ βˆƒ p, p.Prime ∧ p ∣ n ∧ p % 4 = 3
case pos n : β„• h : n % 4 = 3 np : n.Prime ⊒ βˆƒ p, p.Prime ∧ p ∣ n ∧ p % 4 = 3 case neg n : β„• h : n % 4 = 3 np : Β¬n.Prime ⊒ βˆƒ p, p.Prime ∧ p ∣ n ∧ p % 4 = 3
https://github.com/avigad/mathematics_in_lean_source.git
3fa84e6b3135a3ae41edc6ca195abf0fb1ae3ac3
MIL/C05_Elementary_Number_Theory/S03_Infinitely_Many_Primes.lean
C05S03.exists_prime_factor_mod_4_eq_3
[533, 1]
[565, 38]
induction' n using Nat.strong_induction_on with n ih
case neg n : β„• h : n % 4 = 3 np : Β¬n.Prime ⊒ βˆƒ p, p.Prime ∧ p ∣ n ∧ p % 4 = 3
case neg.h n : β„• ih : βˆ€ m < n, m % 4 = 3 β†’ Β¬m.Prime β†’ βˆƒ p, p.Prime ∧ p ∣ m ∧ p % 4 = 3 h : n % 4 = 3 np : Β¬n.Prime ⊒ βˆƒ p, p.Prime ∧ p ∣ n ∧ p % 4 = 3
https://github.com/avigad/mathematics_in_lean_source.git
3fa84e6b3135a3ae41edc6ca195abf0fb1ae3ac3
MIL/C05_Elementary_Number_Theory/S03_Infinitely_Many_Primes.lean
C05S03.exists_prime_factor_mod_4_eq_3
[533, 1]
[565, 38]
rw [Nat.prime_def_lt] at np
case neg.h n : β„• ih : βˆ€ m < n, m % 4 = 3 β†’ Β¬m.Prime β†’ βˆƒ p, p.Prime ∧ p ∣ m ∧ p % 4 = 3 h : n % 4 = 3 np : Β¬n.Prime ⊒ βˆƒ p, p.Prime ∧ p ∣ n ∧ p % 4 = 3
case neg.h n : β„• ih : βˆ€ m < n, m % 4 = 3 β†’ Β¬m.Prime β†’ βˆƒ p, p.Prime ∧ p ∣ m ∧ p % 4 = 3 h : n % 4 = 3 np : Β¬(2 ≀ n ∧ βˆ€ m < n, m ∣ n β†’ m = 1) ⊒ βˆƒ p, p.Prime ∧ p ∣ n ∧ p % 4 = 3
https://github.com/avigad/mathematics_in_lean_source.git
3fa84e6b3135a3ae41edc6ca195abf0fb1ae3ac3
MIL/C05_Elementary_Number_Theory/S03_Infinitely_Many_Primes.lean
C05S03.exists_prime_factor_mod_4_eq_3
[533, 1]
[565, 38]
push_neg at np
case neg.h n : β„• ih : βˆ€ m < n, m % 4 = 3 β†’ Β¬m.Prime β†’ βˆƒ p, p.Prime ∧ p ∣ m ∧ p % 4 = 3 h : n % 4 = 3 np : Β¬(2 ≀ n ∧ βˆ€ m < n, m ∣ n β†’ m = 1) ⊒ βˆƒ p, p.Prime ∧ p ∣ n ∧ p % 4 = 3
case neg.h n : β„• ih : βˆ€ m < n, m % 4 = 3 β†’ Β¬m.Prime β†’ βˆƒ p, p.Prime ∧ p ∣ m ∧ p % 4 = 3 h : n % 4 = 3 np : 2 ≀ n β†’ βˆƒ m < n, m ∣ n ∧ m β‰  1 ⊒ βˆƒ p, p.Prime ∧ p ∣ n ∧ p % 4 = 3
https://github.com/avigad/mathematics_in_lean_source.git
3fa84e6b3135a3ae41edc6ca195abf0fb1ae3ac3
MIL/C05_Elementary_Number_Theory/S03_Infinitely_Many_Primes.lean
C05S03.exists_prime_factor_mod_4_eq_3
[533, 1]
[565, 38]
rcases np (two_le_of_mod_4_eq_3 h) with ⟨m, mltn, mdvdn, mne1⟩
case neg.h n : β„• ih : βˆ€ m < n, m % 4 = 3 β†’ Β¬m.Prime β†’ βˆƒ p, p.Prime ∧ p ∣ m ∧ p % 4 = 3 h : n % 4 = 3 np : 2 ≀ n β†’ βˆƒ m < n, m ∣ n ∧ m β‰  1 ⊒ βˆƒ p, p.Prime ∧ p ∣ n ∧ p % 4 = 3
case neg.h.intro.intro.intro n : β„• ih : βˆ€ m < n, m % 4 = 3 β†’ Β¬m.Prime β†’ βˆƒ p, p.Prime ∧ p ∣ m ∧ p % 4 = 3 h : n % 4 = 3 np : 2 ≀ n β†’ βˆƒ m < n, m ∣ n ∧ m β‰  1 m : β„• mltn : m < n mdvdn : m ∣ n mne1 : m β‰  1 ⊒ βˆƒ p, p.Prime ∧ p ∣ n ∧ p % 4 = 3
https://github.com/avigad/mathematics_in_lean_source.git
3fa84e6b3135a3ae41edc6ca195abf0fb1ae3ac3
MIL/C05_Elementary_Number_Theory/S03_Infinitely_Many_Primes.lean
C05S03.exists_prime_factor_mod_4_eq_3
[533, 1]
[565, 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 : β„• ih : βˆ€ m < n, m % 4 = 3 β†’ Β¬m.Prime β†’ βˆƒ p, p.Prime ∧ p ∣ m ∧ p % 4 = 3 h : n % 4 = 3 np : 2 ≀ n β†’ βˆƒ m < n, m ∣ n ∧ m β‰  1 m : β„• mltn : m < n mdvdn : m ∣ n mne1 : m β‰  1 ⊒ βˆƒ p, p.Prime ∧ p ∣ n ∧ p % 4 = 3
case neg.h.intro.intro.intro n : β„• ih : βˆ€ m < n, m % 4 = 3 β†’ Β¬m.Prime β†’ βˆƒ p, p.Prime ∧ p ∣ m ∧ p % 4 = 3 h : n % 4 = 3 np : 2 ≀ n β†’ βˆƒ m < n, m ∣ n ∧ m β‰  1 m : β„• mltn : m < n mdvdn : m ∣ n mne1 : m β‰  1 mge2 : 2 ≀ m ⊒ βˆƒ p, p.Prime ∧ p ∣ n ∧ p % 4 = 3
https://github.com/avigad/mathematics_in_lean_source.git
3fa84e6b3135a3ae41edc6ca195abf0fb1ae3ac3
MIL/C05_Elementary_Number_Theory/S03_Infinitely_Many_Primes.lean
C05S03.exists_prime_factor_mod_4_eq_3
[533, 1]
[565, 38]
have neq : m * (n / m) = n := Nat.mul_div_cancel' mdvdn
case neg.h.intro.intro.intro n : β„• ih : βˆ€ m < n, m % 4 = 3 β†’ Β¬m.Prime β†’ βˆƒ p, p.Prime ∧ p ∣ m ∧ p % 4 = 3 h : n % 4 = 3 np : 2 ≀ n β†’ βˆƒ m < n, m ∣ n ∧ m β‰  1 m : β„• mltn : m < n mdvdn : m ∣ n mne1 : m β‰  1 mge2 : 2 ≀ m ⊒ βˆƒ p, p.Prime ∧ p ∣ n ∧ p % 4 = 3
case neg.h.intro.intro.intro n : β„• ih : βˆ€ m < n, m % 4 = 3 β†’ Β¬m.Prime β†’ βˆƒ p, p.Prime ∧ p ∣ m ∧ p % 4 = 3 h : n % 4 = 3 np : 2 ≀ n β†’ βˆƒ m < n, m ∣ n ∧ m β‰  1 m : β„• mltn : m < n mdvdn : m ∣ n mne1 : m β‰  1 mge2 : 2 ≀ m neq : m * (n / m) = n ⊒ βˆƒ p, p.Prime ∧ p ∣ n ∧ p % 4 = 3
https://github.com/avigad/mathematics_in_lean_source.git
3fa84e6b3135a3ae41edc6ca195abf0fb1ae3ac3
MIL/C05_Elementary_Number_Theory/S03_Infinitely_Many_Primes.lean
C05S03.exists_prime_factor_mod_4_eq_3
[533, 1]
[565, 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 : β„• ih : βˆ€ m < n, m % 4 = 3 β†’ Β¬m.Prime β†’ βˆƒ p, p.Prime ∧ p ∣ m ∧ p % 4 = 3 h : n % 4 = 3 np : 2 ≀ n β†’ βˆƒ m < n, m ∣ n ∧ m β‰  1 m : β„• mltn : m < n mdvdn : m ∣ n mne1 : m β‰  1 mge2 : 2 ≀ m neq : m * (n / m) = n ⊒ βˆƒ p, p.Prime ∧ p ∣ n ∧ p % 4 = 3
case neg.h.intro.intro.intro n : β„• ih : βˆ€ m < n, m % 4 = 3 β†’ Β¬m.Prime β†’ βˆƒ p, p.Prime ∧ p ∣ m ∧ p % 4 = 3 h : n % 4 = 3 np : 2 ≀ n β†’ βˆƒ m < n, m ∣ n ∧ m β‰  1 m : β„• mltn : m < n mdvdn : m ∣ n mne1 : m β‰  1 mge2 : 2 ≀ m neq : m * (n / m) = n this : m % 4 = 3 ∨ n / m % 4 = 3 ⊒ βˆƒ p, p.Prime ∧ p ∣ n ∧ p % 4 = 3
https://github.com/avigad/mathematics_in_lean_source.git
3fa84e6b3135a3ae41edc6ca195abf0fb1ae3ac3
MIL/C05_Elementary_Number_Theory/S03_Infinitely_Many_Primes.lean
C05S03.exists_prime_factor_mod_4_eq_3
[533, 1]
[565, 38]
rcases this with h1 | h1
case neg.h.intro.intro.intro n : β„• ih : βˆ€ m < n, m % 4 = 3 β†’ Β¬m.Prime β†’ βˆƒ p, p.Prime ∧ p ∣ m ∧ p % 4 = 3 h : n % 4 = 3 np : 2 ≀ n β†’ βˆƒ m < n, m ∣ n ∧ m β‰  1 m : β„• mltn : m < n mdvdn : m ∣ n mne1 : m β‰  1 mge2 : 2 ≀ m neq : m * (n / m) = n this : m % 4 = 3 ∨ n / m % 4 = 3 ⊒ βˆƒ p, p.Prime ∧ p ∣ n ∧ p % 4 = 3
case neg.h.intro.intro.intro.inl n : β„• ih : βˆ€ m < n, m % 4 = 3 β†’ Β¬m.Prime β†’ βˆƒ p, p.Prime ∧ p ∣ m ∧ p % 4 = 3 h : n % 4 = 3 np : 2 ≀ n β†’ βˆƒ m < n, m ∣ n ∧ m β‰  1 m : β„• mltn : m < n mdvdn : m ∣ n mne1 : m β‰  1 mge2 : 2 ≀ m neq : m * (n / m) = n h1 : m % 4 = 3 ⊒ βˆƒ p, p.Prime ∧ p ∣ n ∧ p % 4 = 3 case neg.h.intro.intro.intro.inr n : β„• ih : βˆ€ m < n, m % 4 = 3 β†’ Β¬m.Prime β†’ βˆƒ p, p.Prime ∧ p ∣ m ∧ p % 4 = 3 h : n % 4 = 3 np : 2 ≀ n β†’ βˆƒ m < n, m ∣ n ∧ m β‰  1 m : β„• mltn : m < n mdvdn : m ∣ n mne1 : m β‰  1 mge2 : 2 ≀ m neq : m * (n / m) = n h1 : n / m % 4 = 3 ⊒ βˆƒ p, p.Prime ∧ p ∣ n ∧ p % 4 = 3
https://github.com/avigad/mathematics_in_lean_source.git
3fa84e6b3135a3ae41edc6ca195abf0fb1ae3ac3
MIL/C05_Elementary_Number_Theory/S03_Infinitely_Many_Primes.lean
C05S03.exists_prime_factor_mod_4_eq_3
[533, 1]
[565, 38]
obtain ⟨nmdvdn, nmltn⟩ := aux mdvdn mge2 mltn
case neg.h.intro.intro.intro.inr n : β„• ih : βˆ€ m < n, m % 4 = 3 β†’ Β¬m.Prime β†’ βˆƒ p, p.Prime ∧ p ∣ m ∧ p % 4 = 3 h : n % 4 = 3 np : 2 ≀ n β†’ βˆƒ m < n, m ∣ n ∧ m β‰  1 m : β„• mltn : m < n mdvdn : m ∣ n mne1 : m β‰  1 mge2 : 2 ≀ m neq : m * (n / m) = n h1 : n / m % 4 = 3 ⊒ βˆƒ p, p.Prime ∧ p ∣ n ∧ p % 4 = 3
case neg.h.intro.intro.intro.inr.intro n : β„• ih : βˆ€ m < n, m % 4 = 3 β†’ Β¬m.Prime β†’ βˆƒ p, p.Prime ∧ p ∣ m ∧ p % 4 = 3 h : n % 4 = 3 np : 2 ≀ n β†’ βˆƒ m < n, m ∣ n ∧ m β‰  1 m : β„• mltn : m < n mdvdn : m ∣ n mne1 : m β‰  1 mge2 : 2 ≀ m neq : m * (n / m) = n h1 : n / m % 4 = 3 nmdvdn : n / m ∣ n nmltn : n / m < n ⊒ βˆƒ p, p.Prime ∧ p ∣ n ∧ p % 4 = 3
https://github.com/avigad/mathematics_in_lean_source.git
3fa84e6b3135a3ae41edc6ca195abf0fb1ae3ac3
MIL/C05_Elementary_Number_Theory/S03_Infinitely_Many_Primes.lean
C05S03.exists_prime_factor_mod_4_eq_3
[533, 1]
[565, 38]
by_cases nmp : (n / m).Prime
case neg.h.intro.intro.intro.inr.intro n : β„• ih : βˆ€ m < n, m % 4 = 3 β†’ Β¬m.Prime β†’ βˆƒ p, p.Prime ∧ p ∣ m ∧ p % 4 = 3 h : n % 4 = 3 np : 2 ≀ n β†’ βˆƒ m < n, m ∣ n ∧ m β‰  1 m : β„• mltn : m < n mdvdn : m ∣ n mne1 : m β‰  1 mge2 : 2 ≀ m neq : m * (n / m) = n h1 : n / m % 4 = 3 nmdvdn : n / m ∣ n nmltn : n / m < n ⊒ βˆƒ p, p.Prime ∧ p ∣ n ∧ p % 4 = 3
case pos n : β„• ih : βˆ€ m < n, m % 4 = 3 β†’ Β¬m.Prime β†’ βˆƒ p, p.Prime ∧ p ∣ m ∧ p % 4 = 3 h : n % 4 = 3 np : 2 ≀ n β†’ βˆƒ m < n, m ∣ n ∧ m β‰  1 m : β„• mltn : m < n mdvdn : m ∣ n mne1 : m β‰  1 mge2 : 2 ≀ m neq : m * (n / m) = n h1 : n / m % 4 = 3 nmdvdn : n / m ∣ n nmltn : n / m < n nmp : (n / m).Prime ⊒ βˆƒ p, p.Prime ∧ p ∣ n ∧ p % 4 = 3 case neg n : β„• ih : βˆ€ m < n, m % 4 = 3 β†’ Β¬m.Prime β†’ βˆƒ p, p.Prime ∧ p ∣ m ∧ p % 4 = 3 h : n % 4 = 3 np : 2 ≀ n β†’ βˆƒ m < n, m ∣ n ∧ m β‰  1 m : β„• mltn : m < n mdvdn : m ∣ n mne1 : m β‰  1 mge2 : 2 ≀ m neq : m * (n / m) = n h1 : n / m % 4 = 3 nmdvdn : n / m ∣ n nmltn : n / m < n nmp : Β¬(n / m).Prime ⊒ βˆƒ p, p.Prime ∧ p ∣ n ∧ p % 4 = 3
https://github.com/avigad/mathematics_in_lean_source.git
3fa84e6b3135a3ae41edc6ca195abf0fb1ae3ac3
MIL/C05_Elementary_Number_Theory/S03_Infinitely_Many_Primes.lean
C05S03.exists_prime_factor_mod_4_eq_3
[533, 1]
[565, 38]
rcases ih (n / m) nmltn h1 nmp with ⟨p, pp, pdvd, p4eq⟩
case neg n : β„• ih : βˆ€ m < n, m % 4 = 3 β†’ Β¬m.Prime β†’ βˆƒ p, p.Prime ∧ p ∣ m ∧ p % 4 = 3 h : n % 4 = 3 np : 2 ≀ n β†’ βˆƒ m < n, m ∣ n ∧ m β‰  1 m : β„• mltn : m < n mdvdn : m ∣ n mne1 : m β‰  1 mge2 : 2 ≀ m neq : m * (n / m) = n h1 : n / m % 4 = 3 nmdvdn : n / m ∣ n nmltn : n / m < n nmp : Β¬(n / m).Prime ⊒ βˆƒ p, p.Prime ∧ p ∣ n ∧ p % 4 = 3
case neg.intro.intro.intro n : β„• ih : βˆ€ m < n, m % 4 = 3 β†’ Β¬m.Prime β†’ βˆƒ p, p.Prime ∧ p ∣ m ∧ p % 4 = 3 h : n % 4 = 3 np : 2 ≀ n β†’ βˆƒ m < n, m ∣ n ∧ m β‰  1 m : β„• mltn : m < n mdvdn : m ∣ n mne1 : m β‰  1 mge2 : 2 ≀ m neq : m * (n / m) = n h1 : n / m % 4 = 3 nmdvdn : n / m ∣ n nmltn : n / m < n nmp : Β¬(n / m).Prime p : β„• pp : p.Prime pdvd : p ∣ n / m p4eq : p % 4 = 3 ⊒ βˆƒ p, p.Prime ∧ p ∣ n ∧ p % 4 = 3
https://github.com/avigad/mathematics_in_lean_source.git
3fa84e6b3135a3ae41edc6ca195abf0fb1ae3ac3
MIL/C05_Elementary_Number_Theory/S03_Infinitely_Many_Primes.lean
C05S03.exists_prime_factor_mod_4_eq_3
[533, 1]
[565, 38]
use p
case neg.intro.intro.intro n : β„• ih : βˆ€ m < n, m % 4 = 3 β†’ Β¬m.Prime β†’ βˆƒ p, p.Prime ∧ p ∣ m ∧ p % 4 = 3 h : n % 4 = 3 np : 2 ≀ n β†’ βˆƒ m < n, m ∣ n ∧ m β‰  1 m : β„• mltn : m < n mdvdn : m ∣ n mne1 : m β‰  1 mge2 : 2 ≀ m neq : m * (n / m) = n h1 : n / m % 4 = 3 nmdvdn : n / m ∣ n nmltn : n / m < n nmp : Β¬(n / m).Prime p : β„• pp : p.Prime pdvd : p ∣ n / m p4eq : p % 4 = 3 ⊒ βˆƒ p, p.Prime ∧ p ∣ n ∧ p % 4 = 3
case h n : β„• ih : βˆ€ m < n, m % 4 = 3 β†’ Β¬m.Prime β†’ βˆƒ p, p.Prime ∧ p ∣ m ∧ p % 4 = 3 h : n % 4 = 3 np : 2 ≀ n β†’ βˆƒ m < n, m ∣ n ∧ m β‰  1 m : β„• mltn : m < n mdvdn : m ∣ n mne1 : m β‰  1 mge2 : 2 ≀ m neq : m * (n / m) = n h1 : n / m % 4 = 3 nmdvdn : n / m ∣ n nmltn : n / m < n nmp : Β¬(n / m).Prime p : β„• pp : p.Prime pdvd : p ∣ n / m p4eq : p % 4 = 3 ⊒ p.Prime ∧ p ∣ n ∧ p % 4 = 3
https://github.com/avigad/mathematics_in_lean_source.git
3fa84e6b3135a3ae41edc6ca195abf0fb1ae3ac3
MIL/C05_Elementary_Number_Theory/S03_Infinitely_Many_Primes.lean
C05S03.exists_prime_factor_mod_4_eq_3
[533, 1]
[565, 38]
exact ⟨pp, pdvd.trans nmdvdn, p4eq⟩
case h n : β„• ih : βˆ€ m < n, m % 4 = 3 β†’ Β¬m.Prime β†’ βˆƒ p, p.Prime ∧ p ∣ m ∧ p % 4 = 3 h : n % 4 = 3 np : 2 ≀ n β†’ βˆƒ m < n, m ∣ n ∧ m β‰  1 m : β„• mltn : m < n mdvdn : m ∣ n mne1 : m β‰  1 mge2 : 2 ≀ m neq : m * (n / m) = n h1 : n / m % 4 = 3 nmdvdn : n / m ∣ n nmltn : n / m < n nmp : Β¬(n / m).Prime p : β„• pp : p.Prime pdvd : p ∣ n / m p4eq : p % 4 = 3 ⊒ p.Prime ∧ p ∣ n ∧ p % 4 = 3
no goals
https://github.com/avigad/mathematics_in_lean_source.git
3fa84e6b3135a3ae41edc6ca195abf0fb1ae3ac3
MIL/C05_Elementary_Number_Theory/S03_Infinitely_Many_Primes.lean
C05S03.exists_prime_factor_mod_4_eq_3
[533, 1]
[565, 38]
use n
case pos n : β„• h : n % 4 = 3 np : n.Prime ⊒ βˆƒ p, p.Prime ∧ p ∣ n ∧ p % 4 = 3
no goals
https://github.com/avigad/mathematics_in_lean_source.git
3fa84e6b3135a3ae41edc6ca195abf0fb1ae3ac3
MIL/C05_Elementary_Number_Theory/S03_Infinitely_Many_Primes.lean
C05S03.exists_prime_factor_mod_4_eq_3
[533, 1]
[565, 38]
apply two_le _ mne1
n : β„• ih : βˆ€ m < n, m % 4 = 3 β†’ Β¬m.Prime β†’ βˆƒ p, p.Prime ∧ p ∣ m ∧ p % 4 = 3 h : n % 4 = 3 np : 2 ≀ n β†’ βˆƒ m < n, m ∣ n ∧ m β‰  1 m : β„• mltn : m < n mdvdn : m ∣ n mne1 : m β‰  1 ⊒ 2 ≀ m
n : β„• ih : βˆ€ m < n, m % 4 = 3 β†’ Β¬m.Prime β†’ βˆƒ p, p.Prime ∧ p ∣ m ∧ p % 4 = 3 h : n % 4 = 3 np : 2 ≀ n β†’ βˆƒ m < n, m ∣ n ∧ m β‰  1 m : β„• mltn : m < n mdvdn : m ∣ n mne1 : m β‰  1 ⊒ m β‰  0
https://github.com/avigad/mathematics_in_lean_source.git
3fa84e6b3135a3ae41edc6ca195abf0fb1ae3ac3
MIL/C05_Elementary_Number_Theory/S03_Infinitely_Many_Primes.lean
C05S03.exists_prime_factor_mod_4_eq_3
[533, 1]
[565, 38]
intro mz
n : β„• ih : βˆ€ m < n, m % 4 = 3 β†’ Β¬m.Prime β†’ βˆƒ p, p.Prime ∧ p ∣ m ∧ p % 4 = 3 h : n % 4 = 3 np : 2 ≀ n β†’ βˆƒ m < n, m ∣ n ∧ m β‰  1 m : β„• mltn : m < n mdvdn : m ∣ n mne1 : m β‰  1 ⊒ m β‰  0
n : β„• ih : βˆ€ m < n, m % 4 = 3 β†’ Β¬m.Prime β†’ βˆƒ p, p.Prime ∧ p ∣ m ∧ p % 4 = 3 h : n % 4 = 3 np : 2 ≀ n β†’ βˆƒ m < n, m ∣ n ∧ m β‰  1 m : β„• mltn : m < n mdvdn : m ∣ n mne1 : m β‰  1 mz : m = 0 ⊒ False
https://github.com/avigad/mathematics_in_lean_source.git
3fa84e6b3135a3ae41edc6ca195abf0fb1ae3ac3
MIL/C05_Elementary_Number_Theory/S03_Infinitely_Many_Primes.lean
C05S03.exists_prime_factor_mod_4_eq_3
[533, 1]
[565, 38]
rw [mz, zero_dvd_iff] at mdvdn
n : β„• ih : βˆ€ m < n, m % 4 = 3 β†’ Β¬m.Prime β†’ βˆƒ p, p.Prime ∧ p ∣ m ∧ p % 4 = 3 h : n % 4 = 3 np : 2 ≀ n β†’ βˆƒ m < n, m ∣ n ∧ m β‰  1 m : β„• mltn : m < n mdvdn : m ∣ n mne1 : m β‰  1 mz : m = 0 ⊒ False
n : β„• ih : βˆ€ m < n, m % 4 = 3 β†’ Β¬m.Prime β†’ βˆƒ p, p.Prime ∧ p ∣ m ∧ p % 4 = 3 h : n % 4 = 3 np : 2 ≀ n β†’ βˆƒ m < n, m ∣ n ∧ m β‰  1 m : β„• mltn : m < n mdvdn : n = 0 mne1 : m β‰  1 mz : m = 0 ⊒ False
https://github.com/avigad/mathematics_in_lean_source.git
3fa84e6b3135a3ae41edc6ca195abf0fb1ae3ac3
MIL/C05_Elementary_Number_Theory/S03_Infinitely_Many_Primes.lean
C05S03.exists_prime_factor_mod_4_eq_3
[533, 1]
[565, 38]
linarith
n : β„• ih : βˆ€ m < n, m % 4 = 3 β†’ Β¬m.Prime β†’ βˆƒ p, p.Prime ∧ p ∣ m ∧ p % 4 = 3 h : n % 4 = 3 np : 2 ≀ n β†’ βˆƒ m < n, m ∣ n ∧ m β‰  1 m : β„• mltn : m < n mdvdn : n = 0 mne1 : m β‰  1 mz : m = 0 ⊒ False
no goals
https://github.com/avigad/mathematics_in_lean_source.git
3fa84e6b3135a3ae41edc6ca195abf0fb1ae3ac3
MIL/C05_Elementary_Number_Theory/S03_Infinitely_Many_Primes.lean
C05S03.exists_prime_factor_mod_4_eq_3
[533, 1]
[565, 38]
apply mod_4_eq_3_or_mod_4_eq_3
n : β„• ih : βˆ€ m < n, m % 4 = 3 β†’ Β¬m.Prime β†’ βˆƒ p, p.Prime ∧ p ∣ m ∧ p % 4 = 3 h : n % 4 = 3 np : 2 ≀ n β†’ βˆƒ m < n, m ∣ n ∧ m β‰  1 m : β„• mltn : m < n mdvdn : m ∣ n mne1 : m β‰  1 mge2 : 2 ≀ m neq : m * (n / m) = n ⊒ m % 4 = 3 ∨ n / m % 4 = 3
case h n : β„• ih : βˆ€ m < n, m % 4 = 3 β†’ Β¬m.Prime β†’ βˆƒ p, p.Prime ∧ p ∣ m ∧ p % 4 = 3 h : n % 4 = 3 np : 2 ≀ n β†’ βˆƒ m < n, m ∣ n ∧ m β‰  1 m : β„• mltn : m < n mdvdn : m ∣ n mne1 : m β‰  1 mge2 : 2 ≀ m neq : m * (n / m) = n ⊒ m * (n / m) % 4 = 3
https://github.com/avigad/mathematics_in_lean_source.git
3fa84e6b3135a3ae41edc6ca195abf0fb1ae3ac3
MIL/C05_Elementary_Number_Theory/S03_Infinitely_Many_Primes.lean
C05S03.exists_prime_factor_mod_4_eq_3
[533, 1]
[565, 38]
rw [neq, h]
case h n : β„• ih : βˆ€ m < n, m % 4 = 3 β†’ Β¬m.Prime β†’ βˆƒ p, p.Prime ∧ p ∣ m ∧ p % 4 = 3 h : n % 4 = 3 np : 2 ≀ n β†’ βˆƒ m < n, m ∣ n ∧ m β‰  1 m : β„• mltn : m < n mdvdn : m ∣ n mne1 : m β‰  1 mge2 : 2 ≀ m neq : m * (n / m) = n ⊒ m * (n / m) % 4 = 3
no goals
https://github.com/avigad/mathematics_in_lean_source.git
3fa84e6b3135a3ae41edc6ca195abf0fb1ae3ac3
MIL/C05_Elementary_Number_Theory/S03_Infinitely_Many_Primes.lean
C05S03.exists_prime_factor_mod_4_eq_3
[533, 1]
[565, 38]
by_cases mp : m.Prime
case neg.h.intro.intro.intro.inl n : β„• ih : βˆ€ m < n, m % 4 = 3 β†’ Β¬m.Prime β†’ βˆƒ p, p.Prime ∧ p ∣ m ∧ p % 4 = 3 h : n % 4 = 3 np : 2 ≀ n β†’ βˆƒ m < n, m ∣ n ∧ m β‰  1 m : β„• mltn : m < n mdvdn : m ∣ n mne1 : m β‰  1 mge2 : 2 ≀ m neq : m * (n / m) = n h1 : m % 4 = 3 ⊒ βˆƒ p, p.Prime ∧ p ∣ n ∧ p % 4 = 3
case pos n : β„• ih : βˆ€ m < n, m % 4 = 3 β†’ Β¬m.Prime β†’ βˆƒ p, p.Prime ∧ p ∣ m ∧ p % 4 = 3 h : n % 4 = 3 np : 2 ≀ n β†’ βˆƒ m < n, m ∣ n ∧ m β‰  1 m : β„• mltn : m < n mdvdn : m ∣ n mne1 : m β‰  1 mge2 : 2 ≀ m neq : m * (n / m) = n h1 : m % 4 = 3 mp : m.Prime ⊒ βˆƒ p, p.Prime ∧ p ∣ n ∧ p % 4 = 3 case neg n : β„• ih : βˆ€ m < n, m % 4 = 3 β†’ Β¬m.Prime β†’ βˆƒ p, p.Prime ∧ p ∣ m ∧ p % 4 = 3 h : n % 4 = 3 np : 2 ≀ n β†’ βˆƒ m < n, m ∣ n ∧ m β‰  1 m : β„• mltn : m < n mdvdn : m ∣ n mne1 : m β‰  1 mge2 : 2 ≀ m neq : m * (n / m) = n h1 : m % 4 = 3 mp : Β¬m.Prime ⊒ βˆƒ p, p.Prime ∧ p ∣ n ∧ p % 4 = 3
https://github.com/avigad/mathematics_in_lean_source.git
3fa84e6b3135a3ae41edc6ca195abf0fb1ae3ac3
MIL/C05_Elementary_Number_Theory/S03_Infinitely_Many_Primes.lean
C05S03.exists_prime_factor_mod_4_eq_3
[533, 1]
[565, 38]
rcases ih m mltn h1 mp with ⟨p, pp, pdvd, p4eq⟩
case neg n : β„• ih : βˆ€ m < n, m % 4 = 3 β†’ Β¬m.Prime β†’ βˆƒ p, p.Prime ∧ p ∣ m ∧ p % 4 = 3 h : n % 4 = 3 np : 2 ≀ n β†’ βˆƒ m < n, m ∣ n ∧ m β‰  1 m : β„• mltn : m < n mdvdn : m ∣ n mne1 : m β‰  1 mge2 : 2 ≀ m neq : m * (n / m) = n h1 : m % 4 = 3 mp : Β¬m.Prime ⊒ βˆƒ p, p.Prime ∧ p ∣ n ∧ p % 4 = 3
case neg.intro.intro.intro n : β„• ih : βˆ€ m < n, m % 4 = 3 β†’ Β¬m.Prime β†’ βˆƒ p, p.Prime ∧ p ∣ m ∧ p % 4 = 3 h : n % 4 = 3 np : 2 ≀ n β†’ βˆƒ m < n, m ∣ n ∧ m β‰  1 m : β„• mltn : m < n mdvdn : m ∣ n mne1 : m β‰  1 mge2 : 2 ≀ m neq : m * (n / m) = n h1 : m % 4 = 3 mp : Β¬m.Prime p : β„• pp : p.Prime pdvd : p ∣ m p4eq : p % 4 = 3 ⊒ βˆƒ p, p.Prime ∧ p ∣ n ∧ p % 4 = 3
https://github.com/avigad/mathematics_in_lean_source.git
3fa84e6b3135a3ae41edc6ca195abf0fb1ae3ac3
MIL/C05_Elementary_Number_Theory/S03_Infinitely_Many_Primes.lean
C05S03.exists_prime_factor_mod_4_eq_3
[533, 1]
[565, 38]
use p
case neg.intro.intro.intro n : β„• ih : βˆ€ m < n, m % 4 = 3 β†’ Β¬m.Prime β†’ βˆƒ p, p.Prime ∧ p ∣ m ∧ p % 4 = 3 h : n % 4 = 3 np : 2 ≀ n β†’ βˆƒ m < n, m ∣ n ∧ m β‰  1 m : β„• mltn : m < n mdvdn : m ∣ n mne1 : m β‰  1 mge2 : 2 ≀ m neq : m * (n / m) = n h1 : m % 4 = 3 mp : Β¬m.Prime p : β„• pp : p.Prime pdvd : p ∣ m p4eq : p % 4 = 3 ⊒ βˆƒ p, p.Prime ∧ p ∣ n ∧ p % 4 = 3
case h n : β„• ih : βˆ€ m < n, m % 4 = 3 β†’ Β¬m.Prime β†’ βˆƒ p, p.Prime ∧ p ∣ m ∧ p % 4 = 3 h : n % 4 = 3 np : 2 ≀ n β†’ βˆƒ m < n, m ∣ n ∧ m β‰  1 m : β„• mltn : m < n mdvdn : m ∣ n mne1 : m β‰  1 mge2 : 2 ≀ m neq : m * (n / m) = n h1 : m % 4 = 3 mp : Β¬m.Prime p : β„• pp : p.Prime pdvd : p ∣ m p4eq : p % 4 = 3 ⊒ p.Prime ∧ p ∣ n ∧ p % 4 = 3
https://github.com/avigad/mathematics_in_lean_source.git
3fa84e6b3135a3ae41edc6ca195abf0fb1ae3ac3
MIL/C05_Elementary_Number_Theory/S03_Infinitely_Many_Primes.lean
C05S03.exists_prime_factor_mod_4_eq_3
[533, 1]
[565, 38]
exact ⟨pp, pdvd.trans mdvdn, p4eq⟩
case h n : β„• ih : βˆ€ m < n, m % 4 = 3 β†’ Β¬m.Prime β†’ βˆƒ p, p.Prime ∧ p ∣ m ∧ p % 4 = 3 h : n % 4 = 3 np : 2 ≀ n β†’ βˆƒ m < n, m ∣ n ∧ m β‰  1 m : β„• mltn : m < n mdvdn : m ∣ n mne1 : m β‰  1 mge2 : 2 ≀ m neq : m * (n / m) = n h1 : m % 4 = 3 mp : Β¬m.Prime p : β„• pp : p.Prime pdvd : p ∣ m p4eq : p % 4 = 3 ⊒ p.Prime ∧ p ∣ n ∧ p % 4 = 3
no goals
https://github.com/avigad/mathematics_in_lean_source.git
3fa84e6b3135a3ae41edc6ca195abf0fb1ae3ac3
MIL/C05_Elementary_Number_Theory/S03_Infinitely_Many_Primes.lean
C05S03.exists_prime_factor_mod_4_eq_3
[533, 1]
[565, 38]
use m
case pos n : β„• ih : βˆ€ m < n, m % 4 = 3 β†’ Β¬m.Prime β†’ βˆƒ p, p.Prime ∧ p ∣ m ∧ p % 4 = 3 h : n % 4 = 3 np : 2 ≀ n β†’ βˆƒ m < n, m ∣ n ∧ m β‰  1 m : β„• mltn : m < n mdvdn : m ∣ n mne1 : m β‰  1 mge2 : 2 ≀ m neq : m * (n / m) = n h1 : m % 4 = 3 mp : m.Prime ⊒ βˆƒ p, p.Prime ∧ p ∣ n ∧ p % 4 = 3
no goals
https://github.com/avigad/mathematics_in_lean_source.git
3fa84e6b3135a3ae41edc6ca195abf0fb1ae3ac3
MIL/C05_Elementary_Number_Theory/S03_Infinitely_Many_Primes.lean
C05S03.exists_prime_factor_mod_4_eq_3
[533, 1]
[565, 38]
use n / m
case pos n : β„• ih : βˆ€ m < n, m % 4 = 3 β†’ Β¬m.Prime β†’ βˆƒ p, p.Prime ∧ p ∣ m ∧ p % 4 = 3 h : n % 4 = 3 np : 2 ≀ n β†’ βˆƒ m < n, m ∣ n ∧ m β‰  1 m : β„• mltn : m < n mdvdn : m ∣ n mne1 : m β‰  1 mge2 : 2 ≀ m neq : m * (n / m) = n h1 : n / m % 4 = 3 nmdvdn : n / m ∣ n nmltn : n / m < n nmp : (n / m).Prime ⊒ βˆƒ p, p.Prime ∧ p ∣ n ∧ p % 4 = 3
no goals
https://github.com/avigad/mathematics_in_lean_source.git
3fa84e6b3135a3ae41edc6ca195abf0fb1ae3ac3
MIL/C05_Elementary_Number_Theory/S03_Infinitely_Many_Primes.lean
C05S03.primes_mod_4_eq_3_infinite
[591, 1]
[653, 16]
by_contra h
⊒ βˆ€ (n : β„•), βˆƒ p > n, p.Prime ∧ p % 4 = 3
h : Β¬βˆ€ (n : β„•), βˆƒ p > n, p.Prime ∧ p % 4 = 3 ⊒ False
https://github.com/avigad/mathematics_in_lean_source.git
3fa84e6b3135a3ae41edc6ca195abf0fb1ae3ac3
MIL/C05_Elementary_Number_Theory/S03_Infinitely_Many_Primes.lean
C05S03.primes_mod_4_eq_3_infinite
[591, 1]
[653, 16]
push_neg at h
h : Β¬βˆ€ (n : β„•), βˆƒ p > n, p.Prime ∧ p % 4 = 3 ⊒ False
h : βˆƒ n, βˆ€ p > n, p.Prime β†’ p % 4 β‰  3 ⊒ False
https://github.com/avigad/mathematics_in_lean_source.git
3fa84e6b3135a3ae41edc6ca195abf0fb1ae3ac3
MIL/C05_Elementary_Number_Theory/S03_Infinitely_Many_Primes.lean
C05S03.primes_mod_4_eq_3_infinite
[591, 1]
[653, 16]
rcases h with ⟨n, hn⟩
h : βˆƒ n, βˆ€ p > n, p.Prime β†’ p % 4 β‰  3 ⊒ False
case intro n : β„• hn : βˆ€ p > n, p.Prime β†’ p % 4 β‰  3 ⊒ False
https://github.com/avigad/mathematics_in_lean_source.git
3fa84e6b3135a3ae41edc6ca195abf0fb1ae3ac3
MIL/C05_Elementary_Number_Theory/S03_Infinitely_Many_Primes.lean
C05S03.primes_mod_4_eq_3_infinite
[591, 1]
[653, 16]
have : βˆƒ s : Finset Nat, βˆ€ p : β„•, 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 : β„• hn : βˆ€ p > n, p.Prime β†’ p % 4 β‰  3 ⊒ False
case intro n : β„• hn : βˆ€ p > n, p.Prime β†’ p % 4 β‰  3 this : βˆƒ s, βˆ€ (p : β„•), p.Prime ∧ p % 4 = 3 ↔ p ∈ s ⊒ False
https://github.com/avigad/mathematics_in_lean_source.git
3fa84e6b3135a3ae41edc6ca195abf0fb1ae3ac3
MIL/C05_Elementary_Number_Theory/S03_Infinitely_Many_Primes.lean
C05S03.primes_mod_4_eq_3_infinite
[591, 1]
[653, 16]
rcases this with ⟨s, hs⟩
case intro n : β„• hn : βˆ€ p > n, p.Prime β†’ p % 4 β‰  3 this : βˆƒ s, βˆ€ (p : β„•), p.Prime ∧ p % 4 = 3 ↔ p ∈ s ⊒ False
case intro.intro n : β„• hn : βˆ€ p > n, p.Prime β†’ p % 4 β‰  3 s : Finset β„• hs : βˆ€ (p : β„•), p.Prime ∧ p % 4 = 3 ↔ p ∈ s ⊒ False
https://github.com/avigad/mathematics_in_lean_source.git
3fa84e6b3135a3ae41edc6ca195abf0fb1ae3ac3
MIL/C05_Elementary_Number_Theory/S03_Infinitely_Many_Primes.lean
C05S03.primes_mod_4_eq_3_infinite
[591, 1]
[653, 16]
have h₁ : ((4 * ∏ i in erase s 3, i) + 3) % 4 = 3 := by rw [add_comm, Nat.add_mul_mod_self_left]
case intro.intro n : β„• hn : βˆ€ p > n, p.Prime β†’ p % 4 β‰  3 s : Finset β„• hs : βˆ€ (p : β„•), p.Prime ∧ p % 4 = 3 ↔ p ∈ s ⊒ False
case intro.intro n : β„• hn : βˆ€ p > n, p.Prime β†’ p % 4 β‰  3 s : Finset β„• hs : βˆ€ (p : β„•), p.Prime ∧ p % 4 = 3 ↔ p ∈ s h₁ : (4 * ∏ i ∈ s.erase 3, i + 3) % 4 = 3 ⊒ False
https://github.com/avigad/mathematics_in_lean_source.git
3fa84e6b3135a3ae41edc6ca195abf0fb1ae3ac3
MIL/C05_Elementary_Number_Theory/S03_Infinitely_Many_Primes.lean
C05S03.primes_mod_4_eq_3_infinite
[591, 1]
[653, 16]
rcases exists_prime_factor_mod_4_eq_3 h₁ with ⟨p, pp, pdvd, p4eq⟩
case intro.intro n : β„• hn : βˆ€ p > n, p.Prime β†’ p % 4 β‰  3 s : Finset β„• hs : βˆ€ (p : β„•), p.Prime ∧ p % 4 = 3 ↔ p ∈ s h₁ : (4 * ∏ i ∈ s.erase 3, i + 3) % 4 = 3 ⊒ False
case intro.intro.intro.intro.intro n : β„• hn : βˆ€ p > n, p.Prime β†’ p % 4 β‰  3 s : Finset β„• hs : βˆ€ (p : β„•), p.Prime ∧ p % 4 = 3 ↔ p ∈ s h₁ : (4 * ∏ i ∈ s.erase 3, i + 3) % 4 = 3 p : β„• pp : p.Prime pdvd : p ∣ 4 * ∏ i ∈ s.erase 3, i + 3 p4eq : p % 4 = 3 ⊒ False
https://github.com/avigad/mathematics_in_lean_source.git
3fa84e6b3135a3ae41edc6ca195abf0fb1ae3ac3
MIL/C05_Elementary_Number_Theory/S03_Infinitely_Many_Primes.lean
C05S03.primes_mod_4_eq_3_infinite
[591, 1]
[653, 16]
have ps : p ∈ s := by rw [← hs p] exact ⟨pp, p4eq⟩
case intro.intro.intro.intro.intro n : β„• hn : βˆ€ p > n, p.Prime β†’ p % 4 β‰  3 s : Finset β„• hs : βˆ€ (p : β„•), p.Prime ∧ p % 4 = 3 ↔ p ∈ s h₁ : (4 * ∏ i ∈ s.erase 3, i + 3) % 4 = 3 p : β„• pp : p.Prime pdvd : p ∣ 4 * ∏ i ∈ s.erase 3, i + 3 p4eq : p % 4 = 3 ⊒ False
case intro.intro.intro.intro.intro n : β„• hn : βˆ€ p > n, p.Prime β†’ p % 4 β‰  3 s : Finset β„• hs : βˆ€ (p : β„•), p.Prime ∧ p % 4 = 3 ↔ p ∈ s h₁ : (4 * ∏ i ∈ s.erase 3, i + 3) % 4 = 3 p : β„• pp : p.Prime pdvd : p ∣ 4 * ∏ i ∈ s.erase 3, i + 3 p4eq : p % 4 = 3 ps : p ∈ s ⊒ False
https://github.com/avigad/mathematics_in_lean_source.git
3fa84e6b3135a3ae41edc6ca195abf0fb1ae3ac3
MIL/C05_Elementary_Number_Theory/S03_Infinitely_Many_Primes.lean
C05S03.primes_mod_4_eq_3_infinite
[591, 1]
[653, 16]
have pne3 : p β‰  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 : β„• hn : βˆ€ p > n, p.Prime β†’ p % 4 β‰  3 s : Finset β„• hs : βˆ€ (p : β„•), p.Prime ∧ p % 4 = 3 ↔ p ∈ s h₁ : (4 * ∏ i ∈ s.erase 3, i + 3) % 4 = 3 p : β„• pp : p.Prime pdvd : p ∣ 4 * ∏ i ∈ s.erase 3, i + 3 p4eq : p % 4 = 3 ps : p ∈ s ⊒ False
case intro.intro.intro.intro.intro n : β„• hn : βˆ€ p > n, p.Prime β†’ p % 4 β‰  3 s : Finset β„• hs : βˆ€ (p : β„•), p.Prime ∧ p % 4 = 3 ↔ p ∈ s h₁ : (4 * ∏ i ∈ s.erase 3, i + 3) % 4 = 3 p : β„• pp : p.Prime pdvd : p ∣ 4 * ∏ i ∈ s.erase 3, i + 3 p4eq : p % 4 = 3 ps : p ∈ s pne3 : p β‰  3 ⊒ False
https://github.com/avigad/mathematics_in_lean_source.git
3fa84e6b3135a3ae41edc6ca195abf0fb1ae3ac3
MIL/C05_Elementary_Number_Theory/S03_Infinitely_Many_Primes.lean
C05S03.primes_mod_4_eq_3_infinite
[591, 1]
[653, 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 : β„• hn : βˆ€ p > n, p.Prime β†’ p % 4 β‰  3 s : Finset β„• hs : βˆ€ (p : β„•), p.Prime ∧ p % 4 = 3 ↔ p ∈ s h₁ : (4 * ∏ i ∈ s.erase 3, i + 3) % 4 = 3 p : β„• pp : p.Prime pdvd : p ∣ 4 * ∏ i ∈ s.erase 3, i + 3 p4eq : p % 4 = 3 ps : p ∈ s pne3 : p β‰  3 ⊒ False
case intro.intro.intro.intro.intro n : β„• hn : βˆ€ p > n, p.Prime β†’ p % 4 β‰  3 s : Finset β„• hs : βˆ€ (p : β„•), p.Prime ∧ p % 4 = 3 ↔ p ∈ s h₁ : (4 * ∏ i ∈ s.erase 3, i + 3) % 4 = 3 p : β„• pp : p.Prime pdvd : p ∣ 4 * ∏ i ∈ s.erase 3, i + 3 p4eq : p % 4 = 3 ps : p ∈ s pne3 : p β‰  3 this : p ∣ 4 * ∏ i ∈ s.erase 3, i ⊒ False
https://github.com/avigad/mathematics_in_lean_source.git
3fa84e6b3135a3ae41edc6ca195abf0fb1ae3ac3
MIL/C05_Elementary_Number_Theory/S03_Infinitely_Many_Primes.lean
C05S03.primes_mod_4_eq_3_infinite
[591, 1]
[653, 16]
have : p ∣ 3 := by convert Nat.dvd_sub' pdvd this simp
case intro.intro.intro.intro.intro n : β„• hn : βˆ€ p > n, p.Prime β†’ p % 4 β‰  3 s : Finset β„• hs : βˆ€ (p : β„•), p.Prime ∧ p % 4 = 3 ↔ p ∈ s h₁ : (4 * ∏ i ∈ s.erase 3, i + 3) % 4 = 3 p : β„• pp : p.Prime pdvd : p ∣ 4 * ∏ i ∈ s.erase 3, i + 3 p4eq : p % 4 = 3 ps : p ∈ s pne3 : p β‰  3 this : p ∣ 4 * ∏ i ∈ s.erase 3, i ⊒ False
case intro.intro.intro.intro.intro n : β„• hn : βˆ€ p > n, p.Prime β†’ p % 4 β‰  3 s : Finset β„• hs : βˆ€ (p : β„•), p.Prime ∧ p % 4 = 3 ↔ p ∈ s h₁ : (4 * ∏ i ∈ s.erase 3, i + 3) % 4 = 3 p : β„• pp : p.Prime pdvd : p ∣ 4 * ∏ i ∈ s.erase 3, i + 3 p4eq : p % 4 = 3 ps : p ∈ s pne3 : p β‰  3 this✝ : p ∣ 4 * ∏ i ∈ s.erase 3, i this : p ∣ 3 ⊒ False
https://github.com/avigad/mathematics_in_lean_source.git
3fa84e6b3135a3ae41edc6ca195abf0fb1ae3ac3
MIL/C05_Elementary_Number_Theory/S03_Infinitely_Many_Primes.lean
C05S03.primes_mod_4_eq_3_infinite
[591, 1]
[653, 16]
have : p = 3 := by apply pp.eq_of_dvd_of_prime Nat.prime_three this
case intro.intro.intro.intro.intro n : β„• hn : βˆ€ p > n, p.Prime β†’ p % 4 β‰  3 s : Finset β„• hs : βˆ€ (p : β„•), p.Prime ∧ p % 4 = 3 ↔ p ∈ s h₁ : (4 * ∏ i ∈ s.erase 3, i + 3) % 4 = 3 p : β„• pp : p.Prime pdvd : p ∣ 4 * ∏ i ∈ s.erase 3, i + 3 p4eq : p % 4 = 3 ps : p ∈ s pne3 : p β‰  3 this✝ : p ∣ 4 * ∏ i ∈ s.erase 3, i this : p ∣ 3 ⊒ False
case intro.intro.intro.intro.intro n : β„• hn : βˆ€ p > n, p.Prime β†’ p % 4 β‰  3 s : Finset β„• hs : βˆ€ (p : β„•), p.Prime ∧ p % 4 = 3 ↔ p ∈ s h₁ : (4 * ∏ i ∈ s.erase 3, i + 3) % 4 = 3 p : β„• pp : p.Prime pdvd : p ∣ 4 * ∏ i ∈ s.erase 3, i + 3 p4eq : p % 4 = 3 ps : p ∈ s pne3 : p β‰  3 this✝¹ : p ∣ 4 * ∏ i ∈ s.erase 3, i this✝ : p ∣ 3 this : p = 3 ⊒ False
https://github.com/avigad/mathematics_in_lean_source.git
3fa84e6b3135a3ae41edc6ca195abf0fb1ae3ac3
MIL/C05_Elementary_Number_Theory/S03_Infinitely_Many_Primes.lean
C05S03.primes_mod_4_eq_3_infinite
[591, 1]
[653, 16]
contradiction
case intro.intro.intro.intro.intro n : β„• hn : βˆ€ p > n, p.Prime β†’ p % 4 β‰  3 s : Finset β„• hs : βˆ€ (p : β„•), p.Prime ∧ p % 4 = 3 ↔ p ∈ s h₁ : (4 * ∏ i ∈ s.erase 3, i + 3) % 4 = 3 p : β„• pp : p.Prime pdvd : p ∣ 4 * ∏ i ∈ s.erase 3, i + 3 p4eq : p % 4 = 3 ps : p ∈ s pne3 : p β‰  3 this✝¹ : p ∣ 4 * ∏ i ∈ s.erase 3, i this✝ : p ∣ 3 this : p = 3 ⊒ False
no goals
https://github.com/avigad/mathematics_in_lean_source.git
3fa84e6b3135a3ae41edc6ca195abf0fb1ae3ac3
MIL/C05_Elementary_Number_Theory/S03_Infinitely_Many_Primes.lean
C05S03.primes_mod_4_eq_3_infinite
[591, 1]
[653, 16]
apply ex_finset_of_bounded
n : β„• hn : βˆ€ p > n, p.Prime β†’ p % 4 β‰  3 ⊒ βˆƒ s, βˆ€ (p : β„•), p.Prime ∧ p % 4 = 3 ↔ p ∈ s
case a n : β„• hn : βˆ€ p > n, p.Prime β†’ p % 4 β‰  3 ⊒ βˆƒ n, βˆ€ (k : β„•), k.Prime ∧ k % 4 = 3 β†’ k ≀ n
https://github.com/avigad/mathematics_in_lean_source.git
3fa84e6b3135a3ae41edc6ca195abf0fb1ae3ac3
MIL/C05_Elementary_Number_Theory/S03_Infinitely_Many_Primes.lean
C05S03.primes_mod_4_eq_3_infinite
[591, 1]
[653, 16]
use n
case a n : β„• hn : βˆ€ p > n, p.Prime β†’ p % 4 β‰  3 ⊒ βˆƒ n, βˆ€ (k : β„•), k.Prime ∧ k % 4 = 3 β†’ k ≀ n
case h n : β„• hn : βˆ€ p > n, p.Prime β†’ p % 4 β‰  3 ⊒ βˆ€ (k : β„•), k.Prime ∧ k % 4 = 3 β†’ k ≀ n
https://github.com/avigad/mathematics_in_lean_source.git
3fa84e6b3135a3ae41edc6ca195abf0fb1ae3ac3
MIL/C05_Elementary_Number_Theory/S03_Infinitely_Many_Primes.lean
C05S03.primes_mod_4_eq_3_infinite
[591, 1]
[653, 16]
contrapose! hn
case h n : β„• hn : βˆ€ p > n, p.Prime β†’ p % 4 β‰  3 ⊒ βˆ€ (k : β„•), k.Prime ∧ k % 4 = 3 β†’ k ≀ n
case h n : β„• hn : βˆƒ k, (k.Prime ∧ k % 4 = 3) ∧ n < k ⊒ βˆƒ p > n, p.Prime ∧ p % 4 = 3
https://github.com/avigad/mathematics_in_lean_source.git
3fa84e6b3135a3ae41edc6ca195abf0fb1ae3ac3
MIL/C05_Elementary_Number_Theory/S03_Infinitely_Many_Primes.lean
C05S03.primes_mod_4_eq_3_infinite
[591, 1]
[653, 16]
rcases hn with ⟨p, ⟨pp, p4⟩, pltn⟩
case h n : β„• hn : βˆƒ k, (k.Prime ∧ k % 4 = 3) ∧ n < k ⊒ βˆƒ p > n, p.Prime ∧ p % 4 = 3
case h.intro.intro.intro n p : β„• pltn : n < p pp : p.Prime p4 : p % 4 = 3 ⊒ βˆƒ p > n, p.Prime ∧ p % 4 = 3
https://github.com/avigad/mathematics_in_lean_source.git
3fa84e6b3135a3ae41edc6ca195abf0fb1ae3ac3
MIL/C05_Elementary_Number_Theory/S03_Infinitely_Many_Primes.lean
C05S03.primes_mod_4_eq_3_infinite
[591, 1]
[653, 16]
exact ⟨p, pltn, pp, p4⟩
case h.intro.intro.intro n p : β„• pltn : n < p pp : p.Prime p4 : p % 4 = 3 ⊒ βˆƒ p > n, p.Prime ∧ p % 4 = 3
no goals
https://github.com/avigad/mathematics_in_lean_source.git
3fa84e6b3135a3ae41edc6ca195abf0fb1ae3ac3
MIL/C05_Elementary_Number_Theory/S03_Infinitely_Many_Primes.lean
C05S03.primes_mod_4_eq_3_infinite
[591, 1]
[653, 16]
rw [add_comm, Nat.add_mul_mod_self_left]
n : β„• hn : βˆ€ p > n, p.Prime β†’ p % 4 β‰  3 s : Finset β„• hs : βˆ€ (p : β„•), p.Prime ∧ p % 4 = 3 ↔ p ∈ s ⊒ (4 * ∏ i ∈ s.erase 3, i + 3) % 4 = 3
no goals
https://github.com/avigad/mathematics_in_lean_source.git
3fa84e6b3135a3ae41edc6ca195abf0fb1ae3ac3
MIL/C05_Elementary_Number_Theory/S03_Infinitely_Many_Primes.lean
C05S03.primes_mod_4_eq_3_infinite
[591, 1]
[653, 16]
rw [← hs p]
n : β„• hn : βˆ€ p > n, p.Prime β†’ p % 4 β‰  3 s : Finset β„• hs : βˆ€ (p : β„•), p.Prime ∧ p % 4 = 3 ↔ p ∈ s h₁ : (4 * ∏ i ∈ s.erase 3, i + 3) % 4 = 3 p : β„• pp : p.Prime pdvd : p ∣ 4 * ∏ i ∈ s.erase 3, i + 3 p4eq : p % 4 = 3 ⊒ p ∈ s
n : β„• hn : βˆ€ p > n, p.Prime β†’ p % 4 β‰  3 s : Finset β„• hs : βˆ€ (p : β„•), p.Prime ∧ p % 4 = 3 ↔ p ∈ s h₁ : (4 * ∏ i ∈ s.erase 3, i + 3) % 4 = 3 p : β„• pp : p.Prime pdvd : p ∣ 4 * ∏ i ∈ s.erase 3, i + 3 p4eq : p % 4 = 3 ⊒ p.Prime ∧ p % 4 = 3
https://github.com/avigad/mathematics_in_lean_source.git
3fa84e6b3135a3ae41edc6ca195abf0fb1ae3ac3
MIL/C05_Elementary_Number_Theory/S03_Infinitely_Many_Primes.lean
C05S03.primes_mod_4_eq_3_infinite
[591, 1]
[653, 16]
exact ⟨pp, p4eq⟩
n : β„• hn : βˆ€ p > n, p.Prime β†’ p % 4 β‰  3 s : Finset β„• hs : βˆ€ (p : β„•), p.Prime ∧ p % 4 = 3 ↔ p ∈ s h₁ : (4 * ∏ i ∈ s.erase 3, i + 3) % 4 = 3 p : β„• pp : p.Prime pdvd : p ∣ 4 * ∏ i ∈ s.erase 3, i + 3 p4eq : p % 4 = 3 ⊒ p.Prime ∧ p % 4 = 3
no goals
https://github.com/avigad/mathematics_in_lean_source.git
3fa84e6b3135a3ae41edc6ca195abf0fb1ae3ac3
MIL/C05_Elementary_Number_Theory/S03_Infinitely_Many_Primes.lean
C05S03.primes_mod_4_eq_3_infinite
[591, 1]
[653, 16]
intro peq
n : β„• hn : βˆ€ p > n, p.Prime β†’ p % 4 β‰  3 s : Finset β„• hs : βˆ€ (p : β„•), p.Prime ∧ p % 4 = 3 ↔ p ∈ s h₁ : (4 * ∏ i ∈ s.erase 3, i + 3) % 4 = 3 p : β„• pp : p.Prime pdvd : p ∣ 4 * ∏ i ∈ s.erase 3, i + 3 p4eq : p % 4 = 3 ps : p ∈ s ⊒ p β‰  3
n : β„• hn : βˆ€ p > n, p.Prime β†’ p % 4 β‰  3 s : Finset β„• hs : βˆ€ (p : β„•), p.Prime ∧ p % 4 = 3 ↔ p ∈ s h₁ : (4 * ∏ i ∈ s.erase 3, i + 3) % 4 = 3 p : β„• pp : p.Prime pdvd : p ∣ 4 * ∏ i ∈ s.erase 3, i + 3 p4eq : p % 4 = 3 ps : p ∈ s peq : p = 3 ⊒ False
https://github.com/avigad/mathematics_in_lean_source.git
3fa84e6b3135a3ae41edc6ca195abf0fb1ae3ac3
MIL/C05_Elementary_Number_Theory/S03_Infinitely_Many_Primes.lean
C05S03.primes_mod_4_eq_3_infinite
[591, 1]
[653, 16]
rw [peq, ← Nat.dvd_add_iff_left (dvd_refl 3)] at pdvd
n : β„• hn : βˆ€ p > n, p.Prime β†’ p % 4 β‰  3 s : Finset β„• hs : βˆ€ (p : β„•), p.Prime ∧ p % 4 = 3 ↔ p ∈ s h₁ : (4 * ∏ i ∈ s.erase 3, i + 3) % 4 = 3 p : β„• pp : p.Prime pdvd : p ∣ 4 * ∏ i ∈ s.erase 3, i + 3 p4eq : p % 4 = 3 ps : p ∈ s peq : p = 3 ⊒ False
n : β„• hn : βˆ€ p > n, p.Prime β†’ p % 4 β‰  3 s : Finset β„• hs : βˆ€ (p : β„•), p.Prime ∧ p % 4 = 3 ↔ p ∈ s h₁ : (4 * ∏ i ∈ s.erase 3, i + 3) % 4 = 3 p : β„• pp : p.Prime pdvd : 3 ∣ 4 * ∏ i ∈ s.erase 3, i p4eq : p % 4 = 3 ps : p ∈ s peq : p = 3 ⊒ False
https://github.com/avigad/mathematics_in_lean_source.git
3fa84e6b3135a3ae41edc6ca195abf0fb1ae3ac3
MIL/C05_Elementary_Number_Theory/S03_Infinitely_Many_Primes.lean
C05S03.primes_mod_4_eq_3_infinite
[591, 1]
[653, 16]
rw [Nat.prime_three.dvd_mul] at pdvd
n : β„• hn : βˆ€ p > n, p.Prime β†’ p % 4 β‰  3 s : Finset β„• hs : βˆ€ (p : β„•), p.Prime ∧ p % 4 = 3 ↔ p ∈ s h₁ : (4 * ∏ i ∈ s.erase 3, i + 3) % 4 = 3 p : β„• pp : p.Prime pdvd : 3 ∣ 4 * ∏ i ∈ s.erase 3, i p4eq : p % 4 = 3 ps : p ∈ s peq : p = 3 ⊒ False
n : β„• hn : βˆ€ p > n, p.Prime β†’ p % 4 β‰  3 s : Finset β„• hs : βˆ€ (p : β„•), p.Prime ∧ p % 4 = 3 ↔ p ∈ s h₁ : (4 * ∏ i ∈ s.erase 3, i + 3) % 4 = 3 p : β„• pp : p.Prime pdvd : 3 ∣ 4 ∨ 3 ∣ ∏ i ∈ s.erase 3, i p4eq : p % 4 = 3 ps : p ∈ s peq : p = 3 ⊒ False
https://github.com/avigad/mathematics_in_lean_source.git
3fa84e6b3135a3ae41edc6ca195abf0fb1ae3ac3
MIL/C05_Elementary_Number_Theory/S03_Infinitely_Many_Primes.lean
C05S03.primes_mod_4_eq_3_infinite
[591, 1]
[653, 16]
norm_num at pdvd
n : β„• hn : βˆ€ p > n, p.Prime β†’ p % 4 β‰  3 s : Finset β„• hs : βˆ€ (p : β„•), p.Prime ∧ p % 4 = 3 ↔ p ∈ s h₁ : (4 * ∏ i ∈ s.erase 3, i + 3) % 4 = 3 p : β„• pp : p.Prime pdvd : 3 ∣ 4 ∨ 3 ∣ ∏ i ∈ s.erase 3, i p4eq : p % 4 = 3 ps : p ∈ s peq : p = 3 ⊒ False
n : β„• hn : βˆ€ p > n, p.Prime β†’ p % 4 β‰  3 s : Finset β„• hs : βˆ€ (p : β„•), p.Prime ∧ p % 4 = 3 ↔ p ∈ s h₁ : (4 * ∏ i ∈ s.erase 3, i + 3) % 4 = 3 p : β„• pp : p.Prime p4eq : p % 4 = 3 ps : p ∈ s peq : p = 3 pdvd : 3 ∣ ∏ i ∈ s.erase 3, i ⊒ False
https://github.com/avigad/mathematics_in_lean_source.git
3fa84e6b3135a3ae41edc6ca195abf0fb1ae3ac3
MIL/C05_Elementary_Number_Theory/S03_Infinitely_Many_Primes.lean
C05S03.primes_mod_4_eq_3_infinite
[591, 1]
[653, 16]
have : 3 ∈ s.erase 3 := by apply mem_of_dvd_prod_primes Nat.prime_three _ pdvd intro n simp [← hs n] tauto
n : β„• hn : βˆ€ p > n, p.Prime β†’ p % 4 β‰  3 s : Finset β„• hs : βˆ€ (p : β„•), p.Prime ∧ p % 4 = 3 ↔ p ∈ s h₁ : (4 * ∏ i ∈ s.erase 3, i + 3) % 4 = 3 p : β„• pp : p.Prime p4eq : p % 4 = 3 ps : p ∈ s peq : p = 3 pdvd : 3 ∣ ∏ i ∈ s.erase 3, i ⊒ False
n : β„• hn : βˆ€ p > n, p.Prime β†’ p % 4 β‰  3 s : Finset β„• hs : βˆ€ (p : β„•), p.Prime ∧ p % 4 = 3 ↔ p ∈ s h₁ : (4 * ∏ i ∈ s.erase 3, i + 3) % 4 = 3 p : β„• pp : p.Prime p4eq : p % 4 = 3 ps : p ∈ s peq : p = 3 pdvd : 3 ∣ ∏ i ∈ s.erase 3, i this : 3 ∈ s.erase 3 ⊒ False
https://github.com/avigad/mathematics_in_lean_source.git
3fa84e6b3135a3ae41edc6ca195abf0fb1ae3ac3
MIL/C05_Elementary_Number_Theory/S03_Infinitely_Many_Primes.lean
C05S03.primes_mod_4_eq_3_infinite
[591, 1]
[653, 16]
simp at this
n : β„• hn : βˆ€ p > n, p.Prime β†’ p % 4 β‰  3 s : Finset β„• hs : βˆ€ (p : β„•), p.Prime ∧ p % 4 = 3 ↔ p ∈ s h₁ : (4 * ∏ i ∈ s.erase 3, i + 3) % 4 = 3 p : β„• pp : p.Prime p4eq : p % 4 = 3 ps : p ∈ s peq : p = 3 pdvd : 3 ∣ ∏ i ∈ s.erase 3, i this : 3 ∈ s.erase 3 ⊒ False
no goals
https://github.com/avigad/mathematics_in_lean_source.git
3fa84e6b3135a3ae41edc6ca195abf0fb1ae3ac3
MIL/C05_Elementary_Number_Theory/S03_Infinitely_Many_Primes.lean
C05S03.primes_mod_4_eq_3_infinite
[591, 1]
[653, 16]
apply mem_of_dvd_prod_primes Nat.prime_three _ pdvd
n : β„• hn : βˆ€ p > n, p.Prime β†’ p % 4 β‰  3 s : Finset β„• hs : βˆ€ (p : β„•), p.Prime ∧ p % 4 = 3 ↔ p ∈ s h₁ : (4 * ∏ i ∈ s.erase 3, i + 3) % 4 = 3 p : β„• pp : p.Prime p4eq : p % 4 = 3 ps : p ∈ s peq : p = 3 pdvd : 3 ∣ ∏ i ∈ s.erase 3, i ⊒ 3 ∈ s.erase 3
n : β„• hn : βˆ€ p > n, p.Prime β†’ p % 4 β‰  3 s : Finset β„• hs : βˆ€ (p : β„•), p.Prime ∧ p % 4 = 3 ↔ p ∈ s h₁ : (4 * ∏ i ∈ s.erase 3, i + 3) % 4 = 3 p : β„• pp : p.Prime p4eq : p % 4 = 3 ps : p ∈ s peq : p = 3 pdvd : 3 ∣ ∏ i ∈ s.erase 3, i ⊒ βˆ€ n ∈ s.erase 3, n.Prime
https://github.com/avigad/mathematics_in_lean_source.git
3fa84e6b3135a3ae41edc6ca195abf0fb1ae3ac3
MIL/C05_Elementary_Number_Theory/S03_Infinitely_Many_Primes.lean
C05S03.primes_mod_4_eq_3_infinite
[591, 1]
[653, 16]
intro n
n : β„• hn : βˆ€ p > n, p.Prime β†’ p % 4 β‰  3 s : Finset β„• hs : βˆ€ (p : β„•), p.Prime ∧ p % 4 = 3 ↔ p ∈ s h₁ : (4 * ∏ i ∈ s.erase 3, i + 3) % 4 = 3 p : β„• pp : p.Prime p4eq : p % 4 = 3 ps : p ∈ s peq : p = 3 pdvd : 3 ∣ ∏ i ∈ s.erase 3, i ⊒ βˆ€ n ∈ s.erase 3, n.Prime
n✝ : β„• hn : βˆ€ p > n✝, p.Prime β†’ p % 4 β‰  3 s : Finset β„• hs : βˆ€ (p : β„•), p.Prime ∧ p % 4 = 3 ↔ p ∈ s h₁ : (4 * ∏ i ∈ s.erase 3, i + 3) % 4 = 3 p : β„• pp : p.Prime p4eq : p % 4 = 3 ps : p ∈ s peq : p = 3 pdvd : 3 ∣ ∏ i ∈ s.erase 3, i n : β„• ⊒ n ∈ s.erase 3 β†’ n.Prime
https://github.com/avigad/mathematics_in_lean_source.git
3fa84e6b3135a3ae41edc6ca195abf0fb1ae3ac3
MIL/C05_Elementary_Number_Theory/S03_Infinitely_Many_Primes.lean
C05S03.primes_mod_4_eq_3_infinite
[591, 1]
[653, 16]
simp [← hs n]
n✝ : β„• hn : βˆ€ p > n✝, p.Prime β†’ p % 4 β‰  3 s : Finset β„• hs : βˆ€ (p : β„•), p.Prime ∧ p % 4 = 3 ↔ p ∈ s h₁ : (4 * ∏ i ∈ s.erase 3, i + 3) % 4 = 3 p : β„• pp : p.Prime p4eq : p % 4 = 3 ps : p ∈ s peq : p = 3 pdvd : 3 ∣ ∏ i ∈ s.erase 3, i n : β„• ⊒ n ∈ s.erase 3 β†’ n.Prime
n✝ : β„• hn : βˆ€ p > n✝, p.Prime β†’ p % 4 β‰  3 s : Finset β„• hs : βˆ€ (p : β„•), p.Prime ∧ p % 4 = 3 ↔ p ∈ s h₁ : (4 * ∏ i ∈ s.erase 3, i + 3) % 4 = 3 p : β„• pp : p.Prime p4eq : p % 4 = 3 ps : p ∈ s peq : p = 3 pdvd : 3 ∣ ∏ i ∈ s.erase 3, i n : β„• ⊒ Β¬n = 3 β†’ n.Prime β†’ n % 4 = 3 β†’ n.Prime
https://github.com/avigad/mathematics_in_lean_source.git
3fa84e6b3135a3ae41edc6ca195abf0fb1ae3ac3
MIL/C05_Elementary_Number_Theory/S03_Infinitely_Many_Primes.lean
C05S03.primes_mod_4_eq_3_infinite
[591, 1]
[653, 16]
tauto
n✝ : β„• hn : βˆ€ p > n✝, p.Prime β†’ p % 4 β‰  3 s : Finset β„• hs : βˆ€ (p : β„•), p.Prime ∧ p % 4 = 3 ↔ p ∈ s h₁ : (4 * ∏ i ∈ s.erase 3, i + 3) % 4 = 3 p : β„• pp : p.Prime p4eq : p % 4 = 3 ps : p ∈ s peq : p = 3 pdvd : 3 ∣ ∏ i ∈ s.erase 3, i n : β„• ⊒ Β¬n = 3 β†’ n.Prime β†’ n % 4 = 3 β†’ n.Prime
no goals
https://github.com/avigad/mathematics_in_lean_source.git
3fa84e6b3135a3ae41edc6ca195abf0fb1ae3ac3
MIL/C05_Elementary_Number_Theory/S03_Infinitely_Many_Primes.lean
C05S03.primes_mod_4_eq_3_infinite
[591, 1]
[653, 16]
apply dvd_trans _ (dvd_mul_left _ _)
n : β„• hn : βˆ€ p > n, p.Prime β†’ p % 4 β‰  3 s : Finset β„• hs : βˆ€ (p : β„•), p.Prime ∧ p % 4 = 3 ↔ p ∈ s h₁ : (4 * ∏ i ∈ s.erase 3, i + 3) % 4 = 3 p : β„• pp : p.Prime pdvd : p ∣ 4 * ∏ i ∈ s.erase 3, i + 3 p4eq : p % 4 = 3 ps : p ∈ s pne3 : p β‰  3 ⊒ p ∣ 4 * ∏ i ∈ s.erase 3, i
n : β„• hn : βˆ€ p > n, p.Prime β†’ p % 4 β‰  3 s : Finset β„• hs : βˆ€ (p : β„•), p.Prime ∧ p % 4 = 3 ↔ p ∈ s h₁ : (4 * ∏ i ∈ s.erase 3, i + 3) % 4 = 3 p : β„• pp : p.Prime pdvd : p ∣ 4 * ∏ i ∈ s.erase 3, i + 3 p4eq : p % 4 = 3 ps : p ∈ s pne3 : p β‰  3 ⊒ p ∣ ∏ i ∈ s.erase 3, i
https://github.com/avigad/mathematics_in_lean_source.git
3fa84e6b3135a3ae41edc6ca195abf0fb1ae3ac3
MIL/C05_Elementary_Number_Theory/S03_Infinitely_Many_Primes.lean
C05S03.primes_mod_4_eq_3_infinite
[591, 1]
[653, 16]
apply dvd_prod_of_mem
n : β„• hn : βˆ€ p > n, p.Prime β†’ p % 4 β‰  3 s : Finset β„• hs : βˆ€ (p : β„•), p.Prime ∧ p % 4 = 3 ↔ p ∈ s h₁ : (4 * ∏ i ∈ s.erase 3, i + 3) % 4 = 3 p : β„• pp : p.Prime pdvd : p ∣ 4 * ∏ i ∈ s.erase 3, i + 3 p4eq : p % 4 = 3 ps : p ∈ s pne3 : p β‰  3 ⊒ p ∣ ∏ i ∈ s.erase 3, i
case ha n : β„• hn : βˆ€ p > n, p.Prime β†’ p % 4 β‰  3 s : Finset β„• hs : βˆ€ (p : β„•), p.Prime ∧ p % 4 = 3 ↔ p ∈ s h₁ : (4 * ∏ i ∈ s.erase 3, i + 3) % 4 = 3 p : β„• pp : p.Prime pdvd : p ∣ 4 * ∏ i ∈ s.erase 3, i + 3 p4eq : p % 4 = 3 ps : p ∈ s pne3 : p β‰  3 ⊒ p ∈ s.erase 3
https://github.com/avigad/mathematics_in_lean_source.git
3fa84e6b3135a3ae41edc6ca195abf0fb1ae3ac3
MIL/C05_Elementary_Number_Theory/S03_Infinitely_Many_Primes.lean
C05S03.primes_mod_4_eq_3_infinite
[591, 1]
[653, 16]
simp
case ha n : β„• hn : βˆ€ p > n, p.Prime β†’ p % 4 β‰  3 s : Finset β„• hs : βˆ€ (p : β„•), p.Prime ∧ p % 4 = 3 ↔ p ∈ s h₁ : (4 * ∏ i ∈ s.erase 3, i + 3) % 4 = 3 p : β„• pp : p.Prime pdvd : p ∣ 4 * ∏ i ∈ s.erase 3, i + 3 p4eq : p % 4 = 3 ps : p ∈ s pne3 : p β‰  3 ⊒ p ∈ s.erase 3
case ha n : β„• hn : βˆ€ p > n, p.Prime β†’ p % 4 β‰  3 s : Finset β„• hs : βˆ€ (p : β„•), p.Prime ∧ p % 4 = 3 ↔ p ∈ s h₁ : (4 * ∏ i ∈ s.erase 3, i + 3) % 4 = 3 p : β„• pp : p.Prime pdvd : p ∣ 4 * ∏ i ∈ s.erase 3, i + 3 p4eq : p % 4 = 3 ps : p ∈ s pne3 : p β‰  3 ⊒ Β¬p = 3 ∧ p ∈ s
https://github.com/avigad/mathematics_in_lean_source.git
3fa84e6b3135a3ae41edc6ca195abf0fb1ae3ac3
MIL/C05_Elementary_Number_Theory/S03_Infinitely_Many_Primes.lean
C05S03.primes_mod_4_eq_3_infinite
[591, 1]
[653, 16]
constructor <;> assumption
case ha n : β„• hn : βˆ€ p > n, p.Prime β†’ p % 4 β‰  3 s : Finset β„• hs : βˆ€ (p : β„•), p.Prime ∧ p % 4 = 3 ↔ p ∈ s h₁ : (4 * ∏ i ∈ s.erase 3, i + 3) % 4 = 3 p : β„• pp : p.Prime pdvd : p ∣ 4 * ∏ i ∈ s.erase 3, i + 3 p4eq : p % 4 = 3 ps : p ∈ s pne3 : p β‰  3 ⊒ Β¬p = 3 ∧ p ∈ s
no goals
https://github.com/avigad/mathematics_in_lean_source.git
3fa84e6b3135a3ae41edc6ca195abf0fb1ae3ac3
MIL/C05_Elementary_Number_Theory/S03_Infinitely_Many_Primes.lean
C05S03.primes_mod_4_eq_3_infinite
[591, 1]
[653, 16]
convert Nat.dvd_sub' pdvd this
n : β„• hn : βˆ€ p > n, p.Prime β†’ p % 4 β‰  3 s : Finset β„• hs : βˆ€ (p : β„•), p.Prime ∧ p % 4 = 3 ↔ p ∈ s h₁ : (4 * ∏ i ∈ s.erase 3, i + 3) % 4 = 3 p : β„• pp : p.Prime pdvd : p ∣ 4 * ∏ i ∈ s.erase 3, i + 3 p4eq : p % 4 = 3 ps : p ∈ s pne3 : p β‰  3 this : p ∣ 4 * ∏ i ∈ s.erase 3, i ⊒ p ∣ 3
case h.e'_4 n : β„• hn : βˆ€ p > n, p.Prime β†’ p % 4 β‰  3 s : Finset β„• hs : βˆ€ (p : β„•), p.Prime ∧ p % 4 = 3 ↔ p ∈ s h₁ : (4 * ∏ i ∈ s.erase 3, i + 3) % 4 = 3 p : β„• pp : p.Prime pdvd : p ∣ 4 * ∏ i ∈ s.erase 3, i + 3 p4eq : p % 4 = 3 ps : p ∈ s pne3 : p β‰  3 this : p ∣ 4 * ∏ i ∈ s.erase 3, i ⊒ 3 = 4 * ∏ i ∈ s.erase 3, i + 3 - 4 * ∏ i ∈ s.erase 3, i
https://github.com/avigad/mathematics_in_lean_source.git
3fa84e6b3135a3ae41edc6ca195abf0fb1ae3ac3
MIL/C05_Elementary_Number_Theory/S03_Infinitely_Many_Primes.lean
C05S03.primes_mod_4_eq_3_infinite
[591, 1]
[653, 16]
simp
case h.e'_4 n : β„• hn : βˆ€ p > n, p.Prime β†’ p % 4 β‰  3 s : Finset β„• hs : βˆ€ (p : β„•), p.Prime ∧ p % 4 = 3 ↔ p ∈ s h₁ : (4 * ∏ i ∈ s.erase 3, i + 3) % 4 = 3 p : β„• pp : p.Prime pdvd : p ∣ 4 * ∏ i ∈ s.erase 3, i + 3 p4eq : p % 4 = 3 ps : p ∈ s pne3 : p β‰  3 this : p ∣ 4 * ∏ i ∈ s.erase 3, i ⊒ 3 = 4 * ∏ i ∈ s.erase 3, i + 3 - 4 * ∏ i ∈ s.erase 3, i
no goals
https://github.com/avigad/mathematics_in_lean_source.git
3fa84e6b3135a3ae41edc6ca195abf0fb1ae3ac3
MIL/C05_Elementary_Number_Theory/S03_Infinitely_Many_Primes.lean
C05S03.primes_mod_4_eq_3_infinite
[591, 1]
[653, 16]
apply pp.eq_of_dvd_of_prime Nat.prime_three this
n : β„• hn : βˆ€ p > n, p.Prime β†’ p % 4 β‰  3 s : Finset β„• hs : βˆ€ (p : β„•), p.Prime ∧ p % 4 = 3 ↔ p ∈ s h₁ : (4 * ∏ i ∈ s.erase 3, i + 3) % 4 = 3 p : β„• pp : p.Prime pdvd : p ∣ 4 * ∏ i ∈ s.erase 3, i + 3 p4eq : p % 4 = 3 ps : p ∈ s pne3 : p β‰  3 this✝ : p ∣ 4 * ∏ i ∈ s.erase 3, i this : p ∣ 3 ⊒ p = 3
no goals
https://github.com/avigad/mathematics_in_lean_source.git
3fa84e6b3135a3ae41edc6ca195abf0fb1ae3ac3
MIL/C03_Logic/S04_Conjunction_and_Iff.lean
C03S04.aux
[246, 1]
[248, 17]
sorry
x y : ℝ h : x ^ 2 + y ^ 2 = 0 ⊒ x ^ 2 = 0
no goals
https://github.com/avigad/mathematics_in_lean_source.git
3fa84e6b3135a3ae41edc6ca195abf0fb1ae3ac3
MIL/C03_Logic/S04_Conjunction_and_Iff.lean
C03S04.auxΞ±Ξ±
[255, 1]
[257, 17]
linarith [pow_two_nonneg x, pow_two_nonneg y]
x y : ℝ h : x ^ 2 + y ^ 2 = 0 ⊒ x ^ 2 = 0
no goals
https://github.com/avigad/mathematics_in_lean_source.git
3fa84e6b3135a3ae41edc6ca195abf0fb1ae3ac3
MIL/C03_Logic/S04_Conjunction_and_Iff.lean
C03S04.not_monotone_iff
[307, 1]
[310, 6]
rw [Monotone]
f : ℝ β†’ ℝ ⊒ Β¬Monotone f ↔ βˆƒ x y, x ≀ y ∧ f x > f y
f : ℝ β†’ ℝ ⊒ (Β¬βˆ€ ⦃a b : ℝ⦄, a ≀ b β†’ f a ≀ f b) ↔ βˆƒ x y, x ≀ y ∧ f x > f y
https://github.com/avigad/mathematics_in_lean_source.git
3fa84e6b3135a3ae41edc6ca195abf0fb1ae3ac3
MIL/C03_Logic/S04_Conjunction_and_Iff.lean
C03S04.not_monotone_iff
[307, 1]
[310, 6]
push_neg
f : ℝ β†’ ℝ ⊒ (Β¬βˆ€ ⦃a b : ℝ⦄, a ≀ b β†’ f a ≀ f b) ↔ βˆƒ x y, x ≀ y ∧ f x > f y
f : ℝ β†’ ℝ ⊒ (βˆƒ a b, a ≀ b ∧ f b < f a) ↔ βˆƒ x y, x ≀ y ∧ f x > f y
https://github.com/avigad/mathematics_in_lean_source.git
3fa84e6b3135a3ae41edc6ca195abf0fb1ae3ac3
MIL/C03_Logic/S04_Conjunction_and_Iff.lean
C03S04.not_monotone_iff
[307, 1]
[310, 6]
rfl
f : ℝ β†’ ℝ ⊒ (βˆƒ a b, a ≀ b ∧ f b < f a) ↔ βˆƒ x y, x ≀ y ∧ f x > f y
no goals
https://github.com/opencompl/egg-tactic-code.git
8b4aa748047a43213fc2c0dfca6b7af4a475f785
Evaluation/GroupsKnuthBendixSimp.lean
inv_mul_cancel_left
[15, 1]
[27, 53]
simp [assocMul, invLeft, mulOne, oneMul, invRight]
G : Type inv : G β†’ G mul : G β†’ G β†’ G one : G assocMul : βˆ€ (a b c : G), mul a (mul b c) = mul (mul a b) c invLeft : βˆ€ (a : G), mul (inv a) a = one mulOne : βˆ€ (a : G), mul a one = a oneMul : βˆ€ (a : G), mul one a = a invRight : βˆ€ (a : G), mul a (inv a) = one x y : G ⊒ mul (inv x) (mul x y) = y
no goals
https://github.com/opencompl/egg-tactic-code.git
8b4aa748047a43213fc2c0dfca6b7af4a475f785
Evaluation/GroupsKnuthBendixSimp.lean
mul_inv_cancel_left
[30, 1]
[42, 53]
simp [assocMul, invLeft, mulOne, oneMul, invRight]
G : Type inv : G β†’ G mul : G β†’ G β†’ G one : G assocMul : βˆ€ (a b c : G), mul a (mul b c) = mul (mul a b) c invLeft : βˆ€ (a : G), mul (inv a) a = one mulOne : βˆ€ (a : G), mul a one = a oneMul : βˆ€ (a : G), mul one a = a invRight : βˆ€ (a : G), mul a (inv a) = one x y : G ⊒ mul x (mul (inv x) y) = y
no goals
https://github.com/opencompl/egg-tactic-code.git
8b4aa748047a43213fc2c0dfca6b7af4a475f785
Evaluation/FunctionalProgramming.lean
Test.append_assoc
[30, 1]
[33, 62]
induction as with | nil => eggxplosion [append_nil, append_cons] | cons a as ih => eggxplosion [ih, append_nil, append_cons]
α : Type u_1 as bs cs : List α ⊒ as ++ bs ++ cs = as ++ (bs ++ cs)
no goals
https://github.com/opencompl/egg-tactic-code.git
8b4aa748047a43213fc2c0dfca6b7af4a475f785
Evaluation/FunctionalProgramming.lean
Test.reverseAux_eq_append
[56, 1]
[60, 64]
induction as generalizing bs with | nil => eggxplosion [reverseAux_nil, reverseAux_cons] | cons a as ih => eggxplosion [reverseAux_nil, reverseAux_cons, append_assoc]
α : Type u_1 as bs : List α ⊒ reverseAux as bs = reverseAux as [] ++ bs
no goals
https://github.com/opencompl/egg-tactic-code.git
8b4aa748047a43213fc2c0dfca6b7af4a475f785
Evaluation/FunctionalProgramming.lean
Test.reverse_append
[66, 1]
[69, 51]
induction as generalizing bs with | nil => eggxplosion [] | cons a as ih => eggxplosion [ih, append_assoc]
α : Type u_1 as bs : List α ⊒ List.reverse (as ++ bs) = List.reverse bs ++ List.reverse as
no goals
https://github.com/opencompl/egg-tactic-code.git
8b4aa748047a43213fc2c0dfca6b7af4a475f785
Evaluation/FunctionalProgramming.lean
Test.all_deforestation
[87, 1]
[93, 48]
intros p xs
Ξ± : Sort u_1 ⊒ βˆ€ (p : Ξ± β†’ Bool) (xs : List Ξ±), all p xs = all' p xs
Ξ± : Sort u_1 p : Ξ± β†’ Bool xs : List Ξ± ⊒ all p xs = all' p xs