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https://github.com/avigad/mathematics_in_lean_source.git | 3fa84e6b3135a3ae41edc6ca195abf0fb1ae3ac3 | MIL/C02_Basics/S02_Proving_Identities_in_Algebraic_Structures.lean | MyRing.eq_neg_of_add_eq_zeroΞ±Ξ± | [313, 1] | [316, 19] | apply neg_eq_of_add_eq_zero | R : Type u_1
instβ : Ring R
a b : R
h : a + b = 0
β’ -b = a | case h
R : Type u_1
instβ : Ring R
a b : R
h : a + b = 0
β’ b + a = 0 |
https://github.com/avigad/mathematics_in_lean_source.git | 3fa84e6b3135a3ae41edc6ca195abf0fb1ae3ac3 | MIL/C02_Basics/S02_Proving_Identities_in_Algebraic_Structures.lean | MyRing.eq_neg_of_add_eq_zeroΞ±Ξ± | [313, 1] | [316, 19] | rw [add_comm, h] | case h
R : Type u_1
instβ : Ring R
a b : R
h : a + b = 0
β’ b + a = 0 | no goals |
https://github.com/avigad/mathematics_in_lean_source.git | 3fa84e6b3135a3ae41edc6ca195abf0fb1ae3ac3 | MIL/C02_Basics/S02_Proving_Identities_in_Algebraic_Structures.lean | MyRing.neg_zeroΞ±Ξ± | [318, 1] | [320, 16] | apply neg_eq_of_add_eq_zero | R : Type u_1
instβ : Ring R
β’ -0 = 0 | case h
R : Type u_1
instβ : Ring R
β’ 0 + 0 = 0 |
https://github.com/avigad/mathematics_in_lean_source.git | 3fa84e6b3135a3ae41edc6ca195abf0fb1ae3ac3 | MIL/C02_Basics/S02_Proving_Identities_in_Algebraic_Structures.lean | MyRing.neg_zeroΞ±Ξ± | [318, 1] | [320, 16] | rw [add_zero] | case h
R : Type u_1
instβ : Ring R
β’ 0 + 0 = 0 | no goals |
https://github.com/avigad/mathematics_in_lean_source.git | 3fa84e6b3135a3ae41edc6ca195abf0fb1ae3ac3 | MIL/C02_Basics/S02_Proving_Identities_in_Algebraic_Structures.lean | MyRing.neg_negΞ±Ξ± | [322, 1] | [324, 20] | apply neg_eq_of_add_eq_zero | R : Type u_1
instβ : Ring R
a : R
β’ - -a = a | case h
R : Type u_1
instβ : Ring R
a : R
β’ -a + a = 0 |
https://github.com/avigad/mathematics_in_lean_source.git | 3fa84e6b3135a3ae41edc6ca195abf0fb1ae3ac3 | MIL/C02_Basics/S02_Proving_Identities_in_Algebraic_Structures.lean | MyRing.neg_negΞ±Ξ± | [322, 1] | [324, 20] | rw [add_left_neg] | case h
R : Type u_1
instβ : Ring R
a : R
β’ -a + a = 0 | no goals |
https://github.com/avigad/mathematics_in_lean_source.git | 3fa84e6b3135a3ae41edc6ca195abf0fb1ae3ac3 | MIL/C02_Basics/S02_Proving_Identities_in_Algebraic_Structures.lean | MyRing.self_sub | [382, 1] | [383, 8] | sorry | R : Type u_1
instβ : Ring R
a : R
β’ a - a = 0 | no goals |
https://github.com/avigad/mathematics_in_lean_source.git | 3fa84e6b3135a3ae41edc6ca195abf0fb1ae3ac3 | MIL/C02_Basics/S02_Proving_Identities_in_Algebraic_Structures.lean | MyRing.self_subΞ±Ξ± | [387, 1] | [388, 37] | rw [sub_eq_add_neg, add_right_neg] | R : Type u_1
instβ : Ring R
a : R
β’ a - a = 0 | no goals |
https://github.com/avigad/mathematics_in_lean_source.git | 3fa84e6b3135a3ae41edc6ca195abf0fb1ae3ac3 | MIL/C02_Basics/S02_Proving_Identities_in_Algebraic_Structures.lean | MyRing.one_add_one_eq_two | [403, 1] | [404, 11] | norm_num | R : Type u_1
instβ : Ring R
β’ 1 + 1 = 2 | no goals |
https://github.com/avigad/mathematics_in_lean_source.git | 3fa84e6b3135a3ae41edc6ca195abf0fb1ae3ac3 | MIL/C02_Basics/S02_Proving_Identities_in_Algebraic_Structures.lean | MyRing.two_mul | [407, 1] | [408, 8] | sorry | R : Type u_1
instβ : Ring R
a : R
β’ 2 * a = a + a | no goals |
https://github.com/avigad/mathematics_in_lean_source.git | 3fa84e6b3135a3ae41edc6ca195abf0fb1ae3ac3 | MIL/C02_Basics/S02_Proving_Identities_in_Algebraic_Structures.lean | MyRing.two_mulΞ±Ξ± | [412, 1] | [413, 46] | rw [β one_add_one_eq_two, add_mul, one_mul] | R : Type u_1
instβ : Ring R
a : R
β’ 2 * a = a + a | no goals |
https://github.com/avigad/mathematics_in_lean_source.git | 3fa84e6b3135a3ae41edc6ca195abf0fb1ae3ac3 | MIL/C02_Basics/S02_Proving_Identities_in_Algebraic_Structures.lean | MyGroup.mul_right_inv | [468, 1] | [469, 8] | sorry | G : Type u_1
instβ : Group G
a : G
β’ a * aβ»ΒΉ = 1 | no goals |
https://github.com/avigad/mathematics_in_lean_source.git | 3fa84e6b3135a3ae41edc6ca195abf0fb1ae3ac3 | MIL/C02_Basics/S02_Proving_Identities_in_Algebraic_Structures.lean | MyGroup.mul_one | [471, 1] | [472, 8] | sorry | G : Type u_1
instβ : Group G
a : G
β’ a * 1 = a | no goals |
https://github.com/avigad/mathematics_in_lean_source.git | 3fa84e6b3135a3ae41edc6ca195abf0fb1ae3ac3 | MIL/C02_Basics/S02_Proving_Identities_in_Algebraic_Structures.lean | MyGroup.mul_inv_rev | [474, 1] | [475, 8] | sorry | G : Type u_1
instβ : Group G
a b : G
β’ (a * b)β»ΒΉ = bβ»ΒΉ * aβ»ΒΉ | no goals |
https://github.com/avigad/mathematics_in_lean_source.git | 3fa84e6b3135a3ae41edc6ca195abf0fb1ae3ac3 | MIL/C02_Basics/S02_Proving_Identities_in_Algebraic_Structures.lean | MyGroup.mul_right_invΞ±Ξ± | [479, 1] | [482, 47] | have h : (a * aβ»ΒΉ)β»ΒΉ * (a * aβ»ΒΉ * (a * aβ»ΒΉ)) = 1 := by
rw [mul_assoc, β mul_assoc aβ»ΒΉ a, mul_left_inv, one_mul, mul_left_inv] | G : Type u_1
instβ : Group G
a : G
β’ a * aβ»ΒΉ = 1 | G : Type u_1
instβ : Group G
a : G
h : (a * aβ»ΒΉ)β»ΒΉ * (a * aβ»ΒΉ * (a * aβ»ΒΉ)) = 1
β’ a * aβ»ΒΉ = 1 |
https://github.com/avigad/mathematics_in_lean_source.git | 3fa84e6b3135a3ae41edc6ca195abf0fb1ae3ac3 | MIL/C02_Basics/S02_Proving_Identities_in_Algebraic_Structures.lean | MyGroup.mul_right_invΞ±Ξ± | [479, 1] | [482, 47] | rw [β h, β mul_assoc, mul_left_inv, one_mul] | G : Type u_1
instβ : Group G
a : G
h : (a * aβ»ΒΉ)β»ΒΉ * (a * aβ»ΒΉ * (a * aβ»ΒΉ)) = 1
β’ a * aβ»ΒΉ = 1 | no goals |
https://github.com/avigad/mathematics_in_lean_source.git | 3fa84e6b3135a3ae41edc6ca195abf0fb1ae3ac3 | MIL/C02_Basics/S02_Proving_Identities_in_Algebraic_Structures.lean | MyGroup.mul_right_invΞ±Ξ± | [479, 1] | [482, 47] | rw [mul_assoc, β mul_assoc aβ»ΒΉ a, mul_left_inv, one_mul, mul_left_inv] | G : Type u_1
instβ : Group G
a : G
β’ (a * aβ»ΒΉ)β»ΒΉ * (a * aβ»ΒΉ * (a * aβ»ΒΉ)) = 1 | no goals |
https://github.com/avigad/mathematics_in_lean_source.git | 3fa84e6b3135a3ae41edc6ca195abf0fb1ae3ac3 | MIL/C02_Basics/S02_Proving_Identities_in_Algebraic_Structures.lean | MyGroup.mul_oneΞ±Ξ± | [484, 1] | [485, 61] | rw [β mul_left_inv a, β mul_assoc, mul_right_inv, one_mul] | G : Type u_1
instβ : Group G
a : G
β’ a * 1 = a | no goals |
https://github.com/avigad/mathematics_in_lean_source.git | 3fa84e6b3135a3ae41edc6ca195abf0fb1ae3ac3 | MIL/C02_Basics/S02_Proving_Identities_in_Algebraic_Structures.lean | MyGroup.mul_inv_revΞ±Ξ± | [487, 1] | [489, 52] | rw [β one_mul (bβ»ΒΉ * aβ»ΒΉ), β mul_left_inv (a * b), mul_assoc, mul_assoc, β mul_assoc b bβ»ΒΉ,
mul_right_inv, one_mul, mul_right_inv, mul_one] | G : Type u_1
instβ : Group G
a b : G
β’ (a * b)β»ΒΉ = bβ»ΒΉ * aβ»ΒΉ | 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 | [38, 1] | [43, 18] | cases m | m : β
h0 : m β 0
h1 : m β 1
β’ 2 β€ m | case zero
h0 : 0 β 0
h1 : 0 β 1
β’ 2 β€ 0
case succ
nβ : β
h0 : nβ + 1 β 0
h1 : nβ + 1 β 1
β’ 2 β€ nβ + 1 |
https://github.com/avigad/mathematics_in_lean_source.git | 3fa84e6b3135a3ae41edc6ca195abf0fb1ae3ac3 | MIL/C05_Elementary_Number_Theory/S03_Infinitely_Many_Primes.lean | C05S03.two_le | [38, 1] | [43, 18] | contradiction | case zero
h0 : 0 β 0
h1 : 0 β 1
β’ 2 β€ 0
case succ
nβ : β
h0 : nβ + 1 β 0
h1 : nβ + 1 β 1
β’ 2 β€ nβ + 1 | case succ
nβ : β
h0 : nβ + 1 β 0
h1 : nβ + 1 β 1
β’ 2 β€ nβ + 1 |
https://github.com/avigad/mathematics_in_lean_source.git | 3fa84e6b3135a3ae41edc6ca195abf0fb1ae3ac3 | MIL/C05_Elementary_Number_Theory/S03_Infinitely_Many_Primes.lean | C05S03.two_le | [38, 1] | [43, 18] | case succ m =>
cases m; contradiction
repeat' apply Nat.succ_le_succ
apply zero_le | m : β
h0 : m + 1 β 0
h1 : m + 1 β 1
β’ 2 β€ m + 1 | 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 | [38, 1] | [43, 18] | cases m | m : β
h0 : m + 1 β 0
h1 : m + 1 β 1
β’ 2 β€ m + 1 | case zero
h0 : 0 + 1 β 0
h1 : 0 + 1 β 1
β’ 2 β€ 0 + 1
case succ
nβ : β
h0 : nβ + 1 + 1 β 0
h1 : nβ + 1 + 1 β 1
β’ 2 β€ nβ + 1 + 1 |
https://github.com/avigad/mathematics_in_lean_source.git | 3fa84e6b3135a3ae41edc6ca195abf0fb1ae3ac3 | MIL/C05_Elementary_Number_Theory/S03_Infinitely_Many_Primes.lean | C05S03.two_le | [38, 1] | [43, 18] | contradiction | case zero
h0 : 0 + 1 β 0
h1 : 0 + 1 β 1
β’ 2 β€ 0 + 1
case succ
nβ : β
h0 : nβ + 1 + 1 β 0
h1 : nβ + 1 + 1 β 1
β’ 2 β€ nβ + 1 + 1 | case succ
nβ : β
h0 : nβ + 1 + 1 β 0
h1 : nβ + 1 + 1 β 1
β’ 2 β€ nβ + 1 + 1 |
https://github.com/avigad/mathematics_in_lean_source.git | 3fa84e6b3135a3ae41edc6ca195abf0fb1ae3ac3 | MIL/C05_Elementary_Number_Theory/S03_Infinitely_Many_Primes.lean | C05S03.two_le | [38, 1] | [43, 18] | repeat' apply Nat.succ_le_succ | case succ
nβ : β
h0 : nβ + 1 + 1 β 0
h1 : nβ + 1 + 1 β 1
β’ 2 β€ nβ + 1 + 1 | case succ.a.a
nβ : β
h0 : nβ + 1 + 1 β 0
h1 : nβ + 1 + 1 β 1
β’ 0 β€ nβ |
https://github.com/avigad/mathematics_in_lean_source.git | 3fa84e6b3135a3ae41edc6ca195abf0fb1ae3ac3 | MIL/C05_Elementary_Number_Theory/S03_Infinitely_Many_Primes.lean | C05S03.two_le | [38, 1] | [43, 18] | apply zero_le | case succ.a.a
nβ : β
h0 : nβ + 1 + 1 β 0
h1 : nβ + 1 + 1 β 1
β’ 0 β€ nβ | 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 | [38, 1] | [43, 18] | apply Nat.succ_le_succ | case succ.a
nβ : β
h0 : nβ + 1 + 1 β 0
h1 : nβ + 1 + 1 β 1
β’ 1 β€ nβ + 1 | case succ.a.a
nβ : β
h0 : nβ + 1 + 1 β 0
h1 : nβ + 1 + 1 β 1
β’ 0 β€ nβ |
https://github.com/avigad/mathematics_in_lean_source.git | 3fa84e6b3135a3ae41edc6ca195abf0fb1ae3ac3 | MIL/C05_Elementary_Number_Theory/S03_Infinitely_Many_Primes.lean | C05S03.exists_prime_factor | [110, 1] | [126, 27] | by_cases np : n.Prime | n : β
h : 2 β€ n
β’ β p, p.Prime β§ p β£ n | case pos
n : β
h : 2 β€ n
np : n.Prime
β’ β p, p.Prime β§ p β£ n
case neg
n : β
h : 2 β€ n
np : Β¬n.Prime
β’ β p, p.Prime β§ p β£ n |
https://github.com/avigad/mathematics_in_lean_source.git | 3fa84e6b3135a3ae41edc6ca195abf0fb1ae3ac3 | MIL/C05_Elementary_Number_Theory/S03_Infinitely_Many_Primes.lean | C05S03.exists_prime_factor | [110, 1] | [126, 27] | induction' n using Nat.strong_induction_on with n ih | case neg
n : β
h : 2 β€ n
np : Β¬n.Prime
β’ β p, p.Prime β§ p β£ n | case neg.h
n : β
ih : β m < n, 2 β€ m β Β¬m.Prime β β p, p.Prime β§ p β£ m
h : 2 β€ n
np : Β¬n.Prime
β’ β p, p.Prime β§ p β£ n |
https://github.com/avigad/mathematics_in_lean_source.git | 3fa84e6b3135a3ae41edc6ca195abf0fb1ae3ac3 | MIL/C05_Elementary_Number_Theory/S03_Infinitely_Many_Primes.lean | C05S03.exists_prime_factor | [110, 1] | [126, 27] | rw [Nat.prime_def_lt] at np | case neg.h
n : β
ih : β m < n, 2 β€ m β Β¬m.Prime β β p, p.Prime β§ p β£ m
h : 2 β€ n
np : Β¬n.Prime
β’ β p, p.Prime β§ p β£ n | case neg.h
n : β
ih : β m < n, 2 β€ m β Β¬m.Prime β β p, p.Prime β§ p β£ m
h : 2 β€ n
np : Β¬(2 β€ n β§ β m < n, m β£ n β m = 1)
β’ β p, p.Prime β§ p β£ n |
https://github.com/avigad/mathematics_in_lean_source.git | 3fa84e6b3135a3ae41edc6ca195abf0fb1ae3ac3 | MIL/C05_Elementary_Number_Theory/S03_Infinitely_Many_Primes.lean | C05S03.exists_prime_factor | [110, 1] | [126, 27] | push_neg at np | case neg.h
n : β
ih : β m < n, 2 β€ m β Β¬m.Prime β β p, p.Prime β§ p β£ m
h : 2 β€ n
np : Β¬(2 β€ n β§ β m < n, m β£ n β m = 1)
β’ β p, p.Prime β§ p β£ n | case neg.h
n : β
ih : β m < n, 2 β€ m β Β¬m.Prime β β p, p.Prime β§ p β£ m
h : 2 β€ n
np : 2 β€ n β β m < n, m β£ n β§ m β 1
β’ β p, p.Prime β§ p β£ n |
https://github.com/avigad/mathematics_in_lean_source.git | 3fa84e6b3135a3ae41edc6ca195abf0fb1ae3ac3 | MIL/C05_Elementary_Number_Theory/S03_Infinitely_Many_Primes.lean | C05S03.exists_prime_factor | [110, 1] | [126, 27] | rcases np h with β¨m, mltn, mdvdn, mne1β© | case neg.h
n : β
ih : β m < n, 2 β€ m β Β¬m.Prime β β p, p.Prime β§ p β£ m
h : 2 β€ n
np : 2 β€ n β β m < n, m β£ n β§ m β 1
β’ β p, p.Prime β§ p β£ n | case neg.h.intro.intro.intro
n : β
ih : β m < n, 2 β€ m β Β¬m.Prime β β p, p.Prime β§ p β£ m
h : 2 β€ n
np : 2 β€ n β β m < n, m β£ n β§ m β 1
m : β
mltn : m < n
mdvdn : m β£ n
mne1 : m β 1
β’ β p, p.Prime β§ p β£ n |
https://github.com/avigad/mathematics_in_lean_source.git | 3fa84e6b3135a3ae41edc6ca195abf0fb1ae3ac3 | MIL/C05_Elementary_Number_Theory/S03_Infinitely_Many_Primes.lean | C05S03.exists_prime_factor | [110, 1] | [126, 27] | have : m β 0 := by
intro mz
rw [mz, zero_dvd_iff] at mdvdn
linarith | case neg.h.intro.intro.intro
n : β
ih : β m < n, 2 β€ m β Β¬m.Prime β β p, p.Prime β§ p β£ m
h : 2 β€ n
np : 2 β€ n β β m < n, m β£ n β§ m β 1
m : β
mltn : m < n
mdvdn : m β£ n
mne1 : m β 1
β’ β p, p.Prime β§ p β£ n | case neg.h.intro.intro.intro
n : β
ih : β m < n, 2 β€ m β Β¬m.Prime β β p, p.Prime β§ p β£ m
h : 2 β€ n
np : 2 β€ n β β m < n, m β£ n β§ m β 1
m : β
mltn : m < n
mdvdn : m β£ n
mne1 : m β 1
this : m β 0
β’ β p, p.Prime β§ p β£ n |
https://github.com/avigad/mathematics_in_lean_source.git | 3fa84e6b3135a3ae41edc6ca195abf0fb1ae3ac3 | MIL/C05_Elementary_Number_Theory/S03_Infinitely_Many_Primes.lean | C05S03.exists_prime_factor | [110, 1] | [126, 27] | have mgt2 : 2 β€ m := two_le this mne1 | case neg.h.intro.intro.intro
n : β
ih : β m < n, 2 β€ m β Β¬m.Prime β β p, p.Prime β§ p β£ m
h : 2 β€ n
np : 2 β€ n β β m < n, m β£ n β§ m β 1
m : β
mltn : m < n
mdvdn : m β£ n
mne1 : m β 1
this : m β 0
β’ β p, p.Prime β§ p β£ n | case neg.h.intro.intro.intro
n : β
ih : β m < n, 2 β€ m β Β¬m.Prime β β p, p.Prime β§ p β£ m
h : 2 β€ n
np : 2 β€ n β β m < n, m β£ n β§ m β 1
m : β
mltn : m < n
mdvdn : m β£ n
mne1 : m β 1
this : m β 0
mgt2 : 2 β€ m
β’ β p, p.Prime β§ p β£ n |
https://github.com/avigad/mathematics_in_lean_source.git | 3fa84e6b3135a3ae41edc6ca195abf0fb1ae3ac3 | MIL/C05_Elementary_Number_Theory/S03_Infinitely_Many_Primes.lean | C05S03.exists_prime_factor | [110, 1] | [126, 27] | by_cases mp : m.Prime | case neg.h.intro.intro.intro
n : β
ih : β m < n, 2 β€ m β Β¬m.Prime β β p, p.Prime β§ p β£ m
h : 2 β€ n
np : 2 β€ n β β m < n, m β£ n β§ m β 1
m : β
mltn : m < n
mdvdn : m β£ n
mne1 : m β 1
this : m β 0
mgt2 : 2 β€ m
β’ β p, p.Prime β§ p β£ n | case pos
n : β
ih : β m < n, 2 β€ m β Β¬m.Prime β β p, p.Prime β§ p β£ m
h : 2 β€ n
np : 2 β€ n β β m < n, m β£ n β§ m β 1
m : β
mltn : m < n
mdvdn : m β£ n
mne1 : m β 1
this : m β 0
mgt2 : 2 β€ m
mp : m.Prime
β’ β p, p.Prime β§ p β£ n
case neg
n : β
ih : β m < n, 2 β€ m β Β¬m.Prime β β p, p.Prime β§ p β£ m
h : 2 β€ n
np : 2 β€ n β β m < n, m β£ n β§ m β 1
m : β
mltn : m < n
mdvdn : m β£ n
mne1 : m β 1
this : m β 0
mgt2 : 2 β€ m
mp : Β¬m.Prime
β’ β p, p.Prime β§ p β£ n |
https://github.com/avigad/mathematics_in_lean_source.git | 3fa84e6b3135a3ae41edc6ca195abf0fb1ae3ac3 | MIL/C05_Elementary_Number_Theory/S03_Infinitely_Many_Primes.lean | C05S03.exists_prime_factor | [110, 1] | [126, 27] | . rcases ih m mltn mgt2 mp with β¨p, pp, pdvdβ©
use p, pp
apply pdvd.trans mdvdn | case neg
n : β
ih : β m < n, 2 β€ m β Β¬m.Prime β β p, p.Prime β§ p β£ m
h : 2 β€ n
np : 2 β€ n β β m < n, m β£ n β§ m β 1
m : β
mltn : m < n
mdvdn : m β£ n
mne1 : m β 1
this : m β 0
mgt2 : 2 β€ m
mp : Β¬m.Prime
β’ β p, p.Prime β§ p β£ 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 | [110, 1] | [126, 27] | use n, np | case pos
n : β
h : 2 β€ n
np : n.Prime
β’ β p, p.Prime β§ p β£ 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 | [110, 1] | [126, 27] | intro mz | n : β
ih : β m < n, 2 β€ m β Β¬m.Prime β β p, p.Prime β§ p β£ m
h : 2 β€ n
np : 2 β€ n β β m < n, m β£ n β§ m β 1
m : β
mltn : m < n
mdvdn : m β£ n
mne1 : m β 1
β’ m β 0 | n : β
ih : β m < n, 2 β€ m β Β¬m.Prime β β p, p.Prime β§ p β£ m
h : 2 β€ n
np : 2 β€ n β β m < n, m β£ n β§ m β 1
m : β
mltn : m < n
mdvdn : m β£ n
mne1 : m β 1
mz : m = 0
β’ 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 | [110, 1] | [126, 27] | rw [mz, zero_dvd_iff] at mdvdn | n : β
ih : β m < n, 2 β€ m β Β¬m.Prime β β p, p.Prime β§ p β£ m
h : 2 β€ n
np : 2 β€ n β β m < n, m β£ n β§ m β 1
m : β
mltn : m < n
mdvdn : m β£ n
mne1 : m β 1
mz : m = 0
β’ False | n : β
ih : β m < n, 2 β€ m β Β¬m.Prime β β p, p.Prime β§ p β£ m
h : 2 β€ n
np : 2 β€ n β β m < n, m β£ n β§ m β 1
m : β
mltn : m < n
mdvdn : n = 0
mne1 : m β 1
mz : m = 0
β’ 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 | [110, 1] | [126, 27] | linarith | n : β
ih : β m < n, 2 β€ m β Β¬m.Prime β β p, p.Prime β§ p β£ m
h : 2 β€ n
np : 2 β€ n β β m < n, m β£ n β§ m β 1
m : β
mltn : m < n
mdvdn : n = 0
mne1 : m β 1
mz : m = 0
β’ 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 | [110, 1] | [126, 27] | use m, mp | case pos
n : β
ih : β m < n, 2 β€ m β Β¬m.Prime β β p, p.Prime β§ p β£ m
h : 2 β€ n
np : 2 β€ n β β m < n, m β£ n β§ m β 1
m : β
mltn : m < n
mdvdn : m β£ n
mne1 : m β 1
this : m β 0
mgt2 : 2 β€ m
mp : m.Prime
β’ β p, p.Prime β§ p β£ 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 | [110, 1] | [126, 27] | rcases ih m mltn mgt2 mp with β¨p, pp, pdvdβ© | case neg
n : β
ih : β m < n, 2 β€ m β Β¬m.Prime β β p, p.Prime β§ p β£ m
h : 2 β€ n
np : 2 β€ n β β m < n, m β£ n β§ m β 1
m : β
mltn : m < n
mdvdn : m β£ n
mne1 : m β 1
this : m β 0
mgt2 : 2 β€ m
mp : Β¬m.Prime
β’ β p, p.Prime β§ p β£ n | case neg.intro.intro
n : β
ih : β m < n, 2 β€ m β Β¬m.Prime β β p, p.Prime β§ p β£ m
h : 2 β€ n
np : 2 β€ n β β m < n, m β£ n β§ m β 1
m : β
mltn : m < n
mdvdn : m β£ n
mne1 : m β 1
this : m β 0
mgt2 : 2 β€ m
mp : Β¬m.Prime
p : β
pp : p.Prime
pdvd : p β£ m
β’ β p, p.Prime β§ p β£ n |
https://github.com/avigad/mathematics_in_lean_source.git | 3fa84e6b3135a3ae41edc6ca195abf0fb1ae3ac3 | MIL/C05_Elementary_Number_Theory/S03_Infinitely_Many_Primes.lean | C05S03.exists_prime_factor | [110, 1] | [126, 27] | use p, pp | case neg.intro.intro
n : β
ih : β m < n, 2 β€ m β Β¬m.Prime β β p, p.Prime β§ p β£ m
h : 2 β€ n
np : 2 β€ n β β m < n, m β£ n β§ m β 1
m : β
mltn : m < n
mdvdn : m β£ n
mne1 : m β 1
this : m β 0
mgt2 : 2 β€ m
mp : Β¬m.Prime
p : β
pp : p.Prime
pdvd : p β£ m
β’ β p, p.Prime β§ p β£ n | case right
n : β
ih : β m < n, 2 β€ m β Β¬m.Prime β β p, p.Prime β§ p β£ m
h : 2 β€ n
np : 2 β€ n β β m < n, m β£ n β§ m β 1
m : β
mltn : m < n
mdvdn : m β£ n
mne1 : m β 1
this : m β 0
mgt2 : 2 β€ m
mp : Β¬m.Prime
p : β
pp : p.Prime
pdvd : p β£ m
β’ p β£ n |
https://github.com/avigad/mathematics_in_lean_source.git | 3fa84e6b3135a3ae41edc6ca195abf0fb1ae3ac3 | MIL/C05_Elementary_Number_Theory/S03_Infinitely_Many_Primes.lean | C05S03.exists_prime_factor | [110, 1] | [126, 27] | apply pdvd.trans mdvdn | case right
n : β
ih : β m < n, 2 β€ m β Β¬m.Prime β β p, p.Prime β§ p β£ m
h : 2 β€ n
np : 2 β€ n β β m < n, m β£ n β§ m β 1
m : β
mltn : m < n
mdvdn : m β£ n
mne1 : m β 1
this : m β 0
mgt2 : 2 β€ m
mp : Β¬m.Prime
p : β
pp : p.Prime
pdvd : p β£ m
β’ p β£ n | no goals |
https://github.com/avigad/mathematics_in_lean_source.git | 3fa84e6b3135a3ae41edc6ca195abf0fb1ae3ac3 | MIL/C05_Elementary_Number_Theory/S03_Infinitely_Many_Primes.lean | C05S03.primes_infinite | [136, 1] | [170, 23] | intro n | β’ β (n : β), β p > n, p.Prime | n : β
β’ β p > n, p.Prime |
https://github.com/avigad/mathematics_in_lean_source.git | 3fa84e6b3135a3ae41edc6ca195abf0fb1ae3ac3 | MIL/C05_Elementary_Number_Theory/S03_Infinitely_Many_Primes.lean | C05S03.primes_infinite | [136, 1] | [170, 23] | have : 2 β€ Nat.factorial (n + 1) + 1 := by
apply Nat.succ_le_succ
exact Nat.succ_le_of_lt (Nat.factorial_pos _) | n : β
β’ β p > n, p.Prime | n : β
this : 2 β€ (n + 1).factorial + 1
β’ β p > n, p.Prime |
https://github.com/avigad/mathematics_in_lean_source.git | 3fa84e6b3135a3ae41edc6ca195abf0fb1ae3ac3 | MIL/C05_Elementary_Number_Theory/S03_Infinitely_Many_Primes.lean | C05S03.primes_infinite | [136, 1] | [170, 23] | rcases exists_prime_factor this with β¨p, pp, pdvdβ© | n : β
this : 2 β€ (n + 1).factorial + 1
β’ β p > n, p.Prime | case intro.intro
n : β
this : 2 β€ (n + 1).factorial + 1
p : β
pp : p.Prime
pdvd : p β£ (n + 1).factorial + 1
β’ β p > n, p.Prime |
https://github.com/avigad/mathematics_in_lean_source.git | 3fa84e6b3135a3ae41edc6ca195abf0fb1ae3ac3 | MIL/C05_Elementary_Number_Theory/S03_Infinitely_Many_Primes.lean | C05S03.primes_infinite | [136, 1] | [170, 23] | refine' β¨p, _, ppβ© | case intro.intro
n : β
this : 2 β€ (n + 1).factorial + 1
p : β
pp : p.Prime
pdvd : p β£ (n + 1).factorial + 1
β’ β p > n, p.Prime | case intro.intro
n : β
this : 2 β€ (n + 1).factorial + 1
p : β
pp : p.Prime
pdvd : p β£ (n + 1).factorial + 1
β’ p > n |
https://github.com/avigad/mathematics_in_lean_source.git | 3fa84e6b3135a3ae41edc6ca195abf0fb1ae3ac3 | MIL/C05_Elementary_Number_Theory/S03_Infinitely_Many_Primes.lean | C05S03.primes_infinite | [136, 1] | [170, 23] | by_contra ple | case intro.intro
n : β
this : 2 β€ (n + 1).factorial + 1
p : β
pp : p.Prime
pdvd : p β£ (n + 1).factorial + 1
β’ p > n | case intro.intro
n : β
this : 2 β€ (n + 1).factorial + 1
p : β
pp : p.Prime
pdvd : p β£ (n + 1).factorial + 1
ple : Β¬p > n
β’ False |
https://github.com/avigad/mathematics_in_lean_source.git | 3fa84e6b3135a3ae41edc6ca195abf0fb1ae3ac3 | MIL/C05_Elementary_Number_Theory/S03_Infinitely_Many_Primes.lean | C05S03.primes_infinite | [136, 1] | [170, 23] | push_neg at ple | case intro.intro
n : β
this : 2 β€ (n + 1).factorial + 1
p : β
pp : p.Prime
pdvd : p β£ (n + 1).factorial + 1
ple : Β¬p > n
β’ False | case intro.intro
n : β
this : 2 β€ (n + 1).factorial + 1
p : β
pp : p.Prime
pdvd : p β£ (n + 1).factorial + 1
ple : p β€ n
β’ False |
https://github.com/avigad/mathematics_in_lean_source.git | 3fa84e6b3135a3ae41edc6ca195abf0fb1ae3ac3 | MIL/C05_Elementary_Number_Theory/S03_Infinitely_Many_Primes.lean | C05S03.primes_infinite | [136, 1] | [170, 23] | have : p β£ Nat.factorial (n + 1) := by
apply Nat.dvd_factorial
apply pp.pos
linarith | case intro.intro
n : β
this : 2 β€ (n + 1).factorial + 1
p : β
pp : p.Prime
pdvd : p β£ (n + 1).factorial + 1
ple : p β€ n
β’ False | case intro.intro
n : β
thisβ : 2 β€ (n + 1).factorial + 1
p : β
pp : p.Prime
pdvd : p β£ (n + 1).factorial + 1
ple : p β€ n
this : p β£ (n + 1).factorial
β’ False |
https://github.com/avigad/mathematics_in_lean_source.git | 3fa84e6b3135a3ae41edc6ca195abf0fb1ae3ac3 | MIL/C05_Elementary_Number_Theory/S03_Infinitely_Many_Primes.lean | C05S03.primes_infinite | [136, 1] | [170, 23] | have : p β£ 1 := by
convert Nat.dvd_sub' pdvd this
simp | case intro.intro
n : β
thisβ : 2 β€ (n + 1).factorial + 1
p : β
pp : p.Prime
pdvd : p β£ (n + 1).factorial + 1
ple : p β€ n
this : p β£ (n + 1).factorial
β’ False | case intro.intro
n : β
thisβΒΉ : 2 β€ (n + 1).factorial + 1
p : β
pp : p.Prime
pdvd : p β£ (n + 1).factorial + 1
ple : p β€ n
thisβ : p β£ (n + 1).factorial
this : p β£ 1
β’ False |
https://github.com/avigad/mathematics_in_lean_source.git | 3fa84e6b3135a3ae41edc6ca195abf0fb1ae3ac3 | MIL/C05_Elementary_Number_Theory/S03_Infinitely_Many_Primes.lean | C05S03.primes_infinite | [136, 1] | [170, 23] | have := Nat.le_of_dvd zero_lt_one this | case intro.intro
n : β
thisβΒΉ : 2 β€ (n + 1).factorial + 1
p : β
pp : p.Prime
pdvd : p β£ (n + 1).factorial + 1
ple : p β€ n
thisβ : p β£ (n + 1).factorial
this : p β£ 1
β’ False | case intro.intro
n : β
thisβΒ² : 2 β€ (n + 1).factorial + 1
p : β
pp : p.Prime
pdvd : p β£ (n + 1).factorial + 1
ple : p β€ n
thisβΒΉ : p β£ (n + 1).factorial
thisβ : p β£ 1
this : p β€ 1
β’ False |
https://github.com/avigad/mathematics_in_lean_source.git | 3fa84e6b3135a3ae41edc6ca195abf0fb1ae3ac3 | MIL/C05_Elementary_Number_Theory/S03_Infinitely_Many_Primes.lean | C05S03.primes_infinite | [136, 1] | [170, 23] | linarith [pp.two_le] | case intro.intro
n : β
thisβΒ² : 2 β€ (n + 1).factorial + 1
p : β
pp : p.Prime
pdvd : p β£ (n + 1).factorial + 1
ple : p β€ n
thisβΒΉ : p β£ (n + 1).factorial
thisβ : p β£ 1
this : p β€ 1
β’ 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_infinite | [136, 1] | [170, 23] | apply Nat.succ_le_succ | n : β
β’ 2 β€ (n + 1).factorial + 1 | case a
n : β
β’ 1 β€ (n + 1).factorial |
https://github.com/avigad/mathematics_in_lean_source.git | 3fa84e6b3135a3ae41edc6ca195abf0fb1ae3ac3 | MIL/C05_Elementary_Number_Theory/S03_Infinitely_Many_Primes.lean | C05S03.primes_infinite | [136, 1] | [170, 23] | exact Nat.succ_le_of_lt (Nat.factorial_pos _) | case a
n : β
β’ 1 β€ (n + 1).factorial | no goals |
https://github.com/avigad/mathematics_in_lean_source.git | 3fa84e6b3135a3ae41edc6ca195abf0fb1ae3ac3 | MIL/C05_Elementary_Number_Theory/S03_Infinitely_Many_Primes.lean | C05S03.primes_infinite | [136, 1] | [170, 23] | apply Nat.dvd_factorial | n : β
this : 2 β€ (n + 1).factorial + 1
p : β
pp : p.Prime
pdvd : p β£ (n + 1).factorial + 1
ple : p β€ n
β’ p β£ (n + 1).factorial | case a
n : β
this : 2 β€ (n + 1).factorial + 1
p : β
pp : p.Prime
pdvd : p β£ (n + 1).factorial + 1
ple : p β€ n
β’ 0 < p
case a
n : β
this : 2 β€ (n + 1).factorial + 1
p : β
pp : p.Prime
pdvd : p β£ (n + 1).factorial + 1
ple : p β€ n
β’ p β€ n + 1 |
https://github.com/avigad/mathematics_in_lean_source.git | 3fa84e6b3135a3ae41edc6ca195abf0fb1ae3ac3 | MIL/C05_Elementary_Number_Theory/S03_Infinitely_Many_Primes.lean | C05S03.primes_infinite | [136, 1] | [170, 23] | apply pp.pos | case a
n : β
this : 2 β€ (n + 1).factorial + 1
p : β
pp : p.Prime
pdvd : p β£ (n + 1).factorial + 1
ple : p β€ n
β’ 0 < p
case a
n : β
this : 2 β€ (n + 1).factorial + 1
p : β
pp : p.Prime
pdvd : p β£ (n + 1).factorial + 1
ple : p β€ n
β’ p β€ n + 1 | case a
n : β
this : 2 β€ (n + 1).factorial + 1
p : β
pp : p.Prime
pdvd : p β£ (n + 1).factorial + 1
ple : p β€ n
β’ p β€ n + 1 |
https://github.com/avigad/mathematics_in_lean_source.git | 3fa84e6b3135a3ae41edc6ca195abf0fb1ae3ac3 | MIL/C05_Elementary_Number_Theory/S03_Infinitely_Many_Primes.lean | C05S03.primes_infinite | [136, 1] | [170, 23] | linarith | case a
n : β
this : 2 β€ (n + 1).factorial + 1
p : β
pp : p.Prime
pdvd : p β£ (n + 1).factorial + 1
ple : p β€ n
β’ p β€ n + 1 | no goals |
https://github.com/avigad/mathematics_in_lean_source.git | 3fa84e6b3135a3ae41edc6ca195abf0fb1ae3ac3 | MIL/C05_Elementary_Number_Theory/S03_Infinitely_Many_Primes.lean | C05S03.primes_infinite | [136, 1] | [170, 23] | convert Nat.dvd_sub' pdvd this | n : β
thisβ : 2 β€ (n + 1).factorial + 1
p : β
pp : p.Prime
pdvd : p β£ (n + 1).factorial + 1
ple : p β€ n
this : p β£ (n + 1).factorial
β’ p β£ 1 | case h.e'_4
n : β
thisβ : 2 β€ (n + 1).factorial + 1
p : β
pp : p.Prime
pdvd : p β£ (n + 1).factorial + 1
ple : p β€ n
this : p β£ (n + 1).factorial
β’ 1 = (n + 1).factorial + 1 - (n + 1).factorial |
https://github.com/avigad/mathematics_in_lean_source.git | 3fa84e6b3135a3ae41edc6ca195abf0fb1ae3ac3 | MIL/C05_Elementary_Number_Theory/S03_Infinitely_Many_Primes.lean | C05S03.primes_infinite | [136, 1] | [170, 23] | simp | case h.e'_4
n : β
thisβ : 2 β€ (n + 1).factorial + 1
p : β
pp : p.Prime
pdvd : p β£ (n + 1).factorial + 1
ple : p β€ n
this : p β£ (n + 1).factorial
β’ 1 = (n + 1).factorial + 1 - (n + 1).factorial | no goals |
https://github.com/avigad/mathematics_in_lean_source.git | 3fa84e6b3135a3ae41edc6ca195abf0fb1ae3ac3 | MIL/C05_Elementary_Number_Theory/S03_Infinitely_Many_Primes.lean | Nat.Prime.eq_of_dvd_of_prime | [293, 1] | [301, 13] | cases prime_q.eq_one_or_self_of_dvd _ h | p q : β
prime_p : p.Prime
prime_q : q.Prime
h : p β£ q
β’ p = q | case inl
p q : β
prime_p : p.Prime
prime_q : q.Prime
h : p β£ q
hβ : p = 1
β’ p = q
case inr
p q : β
prime_p : p.Prime
prime_q : q.Prime
h : p β£ q
hβ : p = q
β’ p = q |
https://github.com/avigad/mathematics_in_lean_source.git | 3fa84e6b3135a3ae41edc6ca195abf0fb1ae3ac3 | MIL/C05_Elementary_Number_Theory/S03_Infinitely_Many_Primes.lean | Nat.Prime.eq_of_dvd_of_prime | [293, 1] | [301, 13] | assumption | case inr
p q : β
prime_p : p.Prime
prime_q : q.Prime
h : p β£ q
hβ : p = q
β’ p = q | no goals |
https://github.com/avigad/mathematics_in_lean_source.git | 3fa84e6b3135a3ae41edc6ca195abf0fb1ae3ac3 | MIL/C05_Elementary_Number_Theory/S03_Infinitely_Many_Primes.lean | Nat.Prime.eq_of_dvd_of_prime | [293, 1] | [301, 13] | linarith [prime_p.two_le] | case inl
p q : β
prime_p : p.Prime
prime_q : q.Prime
h : p β£ q
hβ : p = 1
β’ p = q | no goals |
https://github.com/avigad/mathematics_in_lean_source.git | 3fa84e6b3135a3ae41edc6ca195abf0fb1ae3ac3 | MIL/C05_Elementary_Number_Theory/S03_Infinitely_Many_Primes.lean | C05S03.mem_of_dvd_prod_primes | [324, 1] | [339, 19] | intro hβ hβ | s : Finset β
p : β
prime_p : p.Prime
β’ (β n β s, n.Prime) β p β£ β n β s, n β p β s | s : Finset β
p : β
prime_p : p.Prime
hβ : β n β s, n.Prime
hβ : p β£ β n β s, n
β’ p β s |
https://github.com/avigad/mathematics_in_lean_source.git | 3fa84e6b3135a3ae41edc6ca195abf0fb1ae3ac3 | MIL/C05_Elementary_Number_Theory/S03_Infinitely_Many_Primes.lean | C05S03.mem_of_dvd_prod_primes | [324, 1] | [339, 19] | induction' s using Finset.induction_on with a s ans ih | s : Finset β
p : β
prime_p : p.Prime
hβ : β n β s, n.Prime
hβ : p β£ β n β s, n
β’ p β s | case empty
p : β
prime_p : p.Prime
hβ : β n β β
, n.Prime
hβ : p β£ β n β β
, n
β’ p β β
case insert
p : β
prime_p : p.Prime
a : β
s : Finset β
ans : a β s
ih : (β n β s, n.Prime) β p β£ β n β s, n β p β s
hβ : β n β insert a s, n.Prime
hβ : p β£ β n β insert a s, n
β’ p β insert a s |
https://github.com/avigad/mathematics_in_lean_source.git | 3fa84e6b3135a3ae41edc6ca195abf0fb1ae3ac3 | MIL/C05_Elementary_Number_Theory/S03_Infinitely_Many_Primes.lean | C05S03.mem_of_dvd_prod_primes | [324, 1] | [339, 19] | simp [Finset.prod_insert ans, prime_p.dvd_mul] at hβ hβ | case insert
p : β
prime_p : p.Prime
a : β
s : Finset β
ans : a β s
ih : (β n β s, n.Prime) β p β£ β n β s, n β p β s
hβ : β n β insert a s, n.Prime
hβ : p β£ β n β insert a s, n
β’ p β insert a s | case insert
p : β
prime_p : p.Prime
a : β
s : Finset β
ans : a β s
ih : (β n β s, n.Prime) β p β£ β n β s, n β p β s
hβ : a.Prime β§ β a β s, a.Prime
hβ : p β£ a β¨ p β£ β n β s, n
β’ p β insert a s |
https://github.com/avigad/mathematics_in_lean_source.git | 3fa84e6b3135a3ae41edc6ca195abf0fb1ae3ac3 | MIL/C05_Elementary_Number_Theory/S03_Infinitely_Many_Primes.lean | C05S03.mem_of_dvd_prod_primes | [324, 1] | [339, 19] | rw [mem_insert] | case insert
p : β
prime_p : p.Prime
a : β
s : Finset β
ans : a β s
ih : (β n β s, n.Prime) β p β£ β n β s, n β p β s
hβ : a.Prime β§ β a β s, a.Prime
hβ : p β£ a β¨ p β£ β n β s, n
β’ p β insert a s | case insert
p : β
prime_p : p.Prime
a : β
s : Finset β
ans : a β s
ih : (β n β s, n.Prime) β p β£ β n β s, n β p β s
hβ : a.Prime β§ β a β s, a.Prime
hβ : p β£ a β¨ p β£ β n β s, n
β’ p = a β¨ p β s |
https://github.com/avigad/mathematics_in_lean_source.git | 3fa84e6b3135a3ae41edc6ca195abf0fb1ae3ac3 | MIL/C05_Elementary_Number_Theory/S03_Infinitely_Many_Primes.lean | C05S03.mem_of_dvd_prod_primes | [324, 1] | [339, 19] | rcases hβ with hβ | hβ | case insert
p : β
prime_p : p.Prime
a : β
s : Finset β
ans : a β s
ih : (β n β s, n.Prime) β p β£ β n β s, n β p β s
hβ : a.Prime β§ β a β s, a.Prime
hβ : p β£ a β¨ p β£ β n β s, n
β’ p = a β¨ p β s | case insert.inl
p : β
prime_p : p.Prime
a : β
s : Finset β
ans : a β s
ih : (β n β s, n.Prime) β p β£ β n β s, n β p β s
hβ : a.Prime β§ β a β s, a.Prime
hβ : p β£ a
β’ p = a β¨ p β s
case insert.inr
p : β
prime_p : p.Prime
a : β
s : Finset β
ans : a β s
ih : (β n β s, n.Prime) β p β£ β n β s, n β p β s
hβ : a.Prime β§ β a β s, a.Prime
hβ : p β£ β n β s, n
β’ p = a β¨ p β s |
https://github.com/avigad/mathematics_in_lean_source.git | 3fa84e6b3135a3ae41edc6ca195abf0fb1ae3ac3 | MIL/C05_Elementary_Number_Theory/S03_Infinitely_Many_Primes.lean | C05S03.mem_of_dvd_prod_primes | [324, 1] | [339, 19] | right | case insert.inr
p : β
prime_p : p.Prime
a : β
s : Finset β
ans : a β s
ih : (β n β s, n.Prime) β p β£ β n β s, n β p β s
hβ : a.Prime β§ β a β s, a.Prime
hβ : p β£ β n β s, n
β’ p = a β¨ p β s | case insert.inr.h
p : β
prime_p : p.Prime
a : β
s : Finset β
ans : a β s
ih : (β n β s, n.Prime) β p β£ β n β s, n β p β s
hβ : a.Prime β§ β a β s, a.Prime
hβ : p β£ β n β s, n
β’ p β s |
https://github.com/avigad/mathematics_in_lean_source.git | 3fa84e6b3135a3ae41edc6ca195abf0fb1ae3ac3 | MIL/C05_Elementary_Number_Theory/S03_Infinitely_Many_Primes.lean | C05S03.mem_of_dvd_prod_primes | [324, 1] | [339, 19] | exact ih hβ.2 hβ | case insert.inr.h
p : β
prime_p : p.Prime
a : β
s : Finset β
ans : a β s
ih : (β n β s, n.Prime) β p β£ β n β s, n β p β s
hβ : a.Prime β§ β a β s, a.Prime
hβ : p β£ β n β s, n
β’ p β s | no goals |
https://github.com/avigad/mathematics_in_lean_source.git | 3fa84e6b3135a3ae41edc6ca195abf0fb1ae3ac3 | MIL/C05_Elementary_Number_Theory/S03_Infinitely_Many_Primes.lean | C05S03.mem_of_dvd_prod_primes | [324, 1] | [339, 19] | simp at hβ | case empty
p : β
prime_p : p.Prime
hβ : β n β β
, n.Prime
hβ : p β£ β n β β
, n
β’ p β β
| case empty
p : β
prime_p : p.Prime
hβ : β n β β
, n.Prime
hβ : p = 1
β’ p β β
|
https://github.com/avigad/mathematics_in_lean_source.git | 3fa84e6b3135a3ae41edc6ca195abf0fb1ae3ac3 | MIL/C05_Elementary_Number_Theory/S03_Infinitely_Many_Primes.lean | C05S03.mem_of_dvd_prod_primes | [324, 1] | [339, 19] | linarith [prime_p.two_le] | case empty
p : β
prime_p : p.Prime
hβ : β n β β
, n.Prime
hβ : p = 1
β’ p β β
| no goals |
https://github.com/avigad/mathematics_in_lean_source.git | 3fa84e6b3135a3ae41edc6ca195abf0fb1ae3ac3 | MIL/C05_Elementary_Number_Theory/S03_Infinitely_Many_Primes.lean | C05S03.mem_of_dvd_prod_primes | [324, 1] | [339, 19] | left | case insert.inl
p : β
prime_p : p.Prime
a : β
s : Finset β
ans : a β s
ih : (β n β s, n.Prime) β p β£ β n β s, n β p β s
hβ : a.Prime β§ β a β s, a.Prime
hβ : p β£ a
β’ p = a β¨ p β s | case insert.inl.h
p : β
prime_p : p.Prime
a : β
s : Finset β
ans : a β s
ih : (β n β s, n.Prime) β p β£ β n β s, n β p β s
hβ : a.Prime β§ β a β s, a.Prime
hβ : p β£ a
β’ p = a |
https://github.com/avigad/mathematics_in_lean_source.git | 3fa84e6b3135a3ae41edc6ca195abf0fb1ae3ac3 | MIL/C05_Elementary_Number_Theory/S03_Infinitely_Many_Primes.lean | C05S03.mem_of_dvd_prod_primes | [324, 1] | [339, 19] | exact prime_p.eq_of_dvd_of_prime hβ.1 hβ | case insert.inl.h
p : β
prime_p : p.Prime
a : β
s : Finset β
ans : a β s
ih : (β n β s, n.Prime) β p β£ β n β s, n β p β s
hβ : a.Prime β§ β a β s, a.Prime
hβ : p β£ a
β’ p = a | no goals |
https://github.com/avigad/mathematics_in_lean_source.git | 3fa84e6b3135a3ae41edc6ca195abf0fb1ae3ac3 | MIL/C05_Elementary_Number_Theory/S03_Infinitely_Many_Primes.lean | C05S03.primes_infinite' | [370, 1] | [406, 23] | intro s | β’ β (s : Finset β), β p, p.Prime β§ p β s | s : Finset β
β’ β p, p.Prime β§ p β s |
https://github.com/avigad/mathematics_in_lean_source.git | 3fa84e6b3135a3ae41edc6ca195abf0fb1ae3ac3 | MIL/C05_Elementary_Number_Theory/S03_Infinitely_Many_Primes.lean | C05S03.primes_infinite' | [370, 1] | [406, 23] | by_contra h | s : Finset β
β’ β p, p.Prime β§ p β s | s : Finset β
h : Β¬β p, p.Prime β§ p β s
β’ False |
https://github.com/avigad/mathematics_in_lean_source.git | 3fa84e6b3135a3ae41edc6ca195abf0fb1ae3ac3 | MIL/C05_Elementary_Number_Theory/S03_Infinitely_Many_Primes.lean | C05S03.primes_infinite' | [370, 1] | [406, 23] | push_neg at h | s : Finset β
h : Β¬β p, p.Prime β§ p β s
β’ False | s : Finset β
h : β (p : β), p.Prime β p β s
β’ False |
https://github.com/avigad/mathematics_in_lean_source.git | 3fa84e6b3135a3ae41edc6ca195abf0fb1ae3ac3 | MIL/C05_Elementary_Number_Theory/S03_Infinitely_Many_Primes.lean | C05S03.primes_infinite' | [370, 1] | [406, 23] | set s' := s.filter Nat.Prime with s'_def | s : Finset β
h : β (p : β), p.Prime β p β s
β’ False | s : Finset β
h : β (p : β), p.Prime β p β s
s' : Finset β := filter Nat.Prime s
s'_def : s' = filter Nat.Prime s
β’ False |
https://github.com/avigad/mathematics_in_lean_source.git | 3fa84e6b3135a3ae41edc6ca195abf0fb1ae3ac3 | MIL/C05_Elementary_Number_Theory/S03_Infinitely_Many_Primes.lean | C05S03.primes_infinite' | [370, 1] | [406, 23] | have mem_s' : β {n : β}, n β s' β n.Prime := by
intro n
simp [s'_def]
apply h | s : Finset β
h : β (p : β), p.Prime β p β s
s' : Finset β := filter Nat.Prime s
s'_def : s' = filter Nat.Prime s
β’ False | s : Finset β
h : β (p : β), p.Prime β p β s
s' : Finset β := filter Nat.Prime s
s'_def : s' = filter Nat.Prime s
mem_s' : β {n : β}, n β s' β n.Prime
β’ False |
https://github.com/avigad/mathematics_in_lean_source.git | 3fa84e6b3135a3ae41edc6ca195abf0fb1ae3ac3 | MIL/C05_Elementary_Number_Theory/S03_Infinitely_Many_Primes.lean | C05S03.primes_infinite' | [370, 1] | [406, 23] | have : 2 β€ (β i in s', i) + 1 := by
apply Nat.succ_le_succ
apply Nat.succ_le_of_lt
apply Finset.prod_pos
intro n ns'
apply (mem_s'.mp ns').pos | s : Finset β
h : β (p : β), p.Prime β p β s
s' : Finset β := filter Nat.Prime s
s'_def : s' = filter Nat.Prime s
mem_s' : β {n : β}, n β s' β n.Prime
β’ False | s : Finset β
h : β (p : β), p.Prime β p β s
s' : Finset β := filter Nat.Prime s
s'_def : s' = filter Nat.Prime s
mem_s' : β {n : β}, n β s' β n.Prime
this : 2 β€ β i β s', i + 1
β’ False |
https://github.com/avigad/mathematics_in_lean_source.git | 3fa84e6b3135a3ae41edc6ca195abf0fb1ae3ac3 | MIL/C05_Elementary_Number_Theory/S03_Infinitely_Many_Primes.lean | C05S03.primes_infinite' | [370, 1] | [406, 23] | rcases exists_prime_factor this with β¨p, pp, pdvdβ© | s : Finset β
h : β (p : β), p.Prime β p β s
s' : Finset β := filter Nat.Prime s
s'_def : s' = filter Nat.Prime s
mem_s' : β {n : β}, n β s' β n.Prime
this : 2 β€ β i β s', i + 1
β’ False | case intro.intro
s : Finset β
h : β (p : β), p.Prime β p β s
s' : Finset β := filter Nat.Prime s
s'_def : s' = filter Nat.Prime s
mem_s' : β {n : β}, n β s' β n.Prime
this : 2 β€ β i β s', i + 1
p : β
pp : p.Prime
pdvd : p β£ β i β s', i + 1
β’ False |
https://github.com/avigad/mathematics_in_lean_source.git | 3fa84e6b3135a3ae41edc6ca195abf0fb1ae3ac3 | MIL/C05_Elementary_Number_Theory/S03_Infinitely_Many_Primes.lean | C05S03.primes_infinite' | [370, 1] | [406, 23] | have : p β£ β i in s', i := by
apply dvd_prod_of_mem
rw [mem_s']
apply pp | case intro.intro
s : Finset β
h : β (p : β), p.Prime β p β s
s' : Finset β := filter Nat.Prime s
s'_def : s' = filter Nat.Prime s
mem_s' : β {n : β}, n β s' β n.Prime
this : 2 β€ β i β s', i + 1
p : β
pp : p.Prime
pdvd : p β£ β i β s', i + 1
β’ False | case intro.intro
s : Finset β
h : β (p : β), p.Prime β p β s
s' : Finset β := filter Nat.Prime s
s'_def : s' = filter Nat.Prime s
mem_s' : β {n : β}, n β s' β n.Prime
thisβ : 2 β€ β i β s', i + 1
p : β
pp : p.Prime
pdvd : p β£ β i β s', i + 1
this : p β£ β i β s', i
β’ False |
https://github.com/avigad/mathematics_in_lean_source.git | 3fa84e6b3135a3ae41edc6ca195abf0fb1ae3ac3 | MIL/C05_Elementary_Number_Theory/S03_Infinitely_Many_Primes.lean | C05S03.primes_infinite' | [370, 1] | [406, 23] | have : p β£ 1 := by
convert Nat.dvd_sub' pdvd this
simp | case intro.intro
s : Finset β
h : β (p : β), p.Prime β p β s
s' : Finset β := filter Nat.Prime s
s'_def : s' = filter Nat.Prime s
mem_s' : β {n : β}, n β s' β n.Prime
thisβ : 2 β€ β i β s', i + 1
p : β
pp : p.Prime
pdvd : p β£ β i β s', i + 1
this : p β£ β i β s', i
β’ False | case intro.intro
s : Finset β
h : β (p : β), p.Prime β p β s
s' : Finset β := filter Nat.Prime s
s'_def : s' = filter Nat.Prime s
mem_s' : β {n : β}, n β s' β n.Prime
thisβΒΉ : 2 β€ β i β s', i + 1
p : β
pp : p.Prime
pdvd : p β£ β i β s', i + 1
thisβ : p β£ β i β s', i
this : p β£ 1
β’ False |
https://github.com/avigad/mathematics_in_lean_source.git | 3fa84e6b3135a3ae41edc6ca195abf0fb1ae3ac3 | MIL/C05_Elementary_Number_Theory/S03_Infinitely_Many_Primes.lean | C05S03.primes_infinite' | [370, 1] | [406, 23] | have := Nat.le_of_dvd zero_lt_one this | case intro.intro
s : Finset β
h : β (p : β), p.Prime β p β s
s' : Finset β := filter Nat.Prime s
s'_def : s' = filter Nat.Prime s
mem_s' : β {n : β}, n β s' β n.Prime
thisβΒΉ : 2 β€ β i β s', i + 1
p : β
pp : p.Prime
pdvd : p β£ β i β s', i + 1
thisβ : p β£ β i β s', i
this : p β£ 1
β’ False | case intro.intro
s : Finset β
h : β (p : β), p.Prime β p β s
s' : Finset β := filter Nat.Prime s
s'_def : s' = filter Nat.Prime s
mem_s' : β {n : β}, n β s' β n.Prime
thisβΒ² : 2 β€ β i β s', i + 1
p : β
pp : p.Prime
pdvd : p β£ β i β s', i + 1
thisβΒΉ : p β£ β i β s', i
thisβ : p β£ 1
this : p β€ 1
β’ False |
https://github.com/avigad/mathematics_in_lean_source.git | 3fa84e6b3135a3ae41edc6ca195abf0fb1ae3ac3 | MIL/C05_Elementary_Number_Theory/S03_Infinitely_Many_Primes.lean | C05S03.primes_infinite' | [370, 1] | [406, 23] | linarith [pp.two_le] | case intro.intro
s : Finset β
h : β (p : β), p.Prime β p β s
s' : Finset β := filter Nat.Prime s
s'_def : s' = filter Nat.Prime s
mem_s' : β {n : β}, n β s' β n.Prime
thisβΒ² : 2 β€ β i β s', i + 1
p : β
pp : p.Prime
pdvd : p β£ β i β s', i + 1
thisβΒΉ : p β£ β i β s', i
thisβ : p β£ 1
this : p β€ 1
β’ 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_infinite' | [370, 1] | [406, 23] | intro n | s : Finset β
h : β (p : β), p.Prime β p β s
s' : Finset β := filter Nat.Prime s
s'_def : s' = filter Nat.Prime s
β’ β {n : β}, n β s' β n.Prime | s : Finset β
h : β (p : β), p.Prime β p β s
s' : Finset β := filter Nat.Prime s
s'_def : s' = filter Nat.Prime s
n : β
β’ n β s' 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_infinite' | [370, 1] | [406, 23] | simp [s'_def] | s : Finset β
h : β (p : β), p.Prime β p β s
s' : Finset β := filter Nat.Prime s
s'_def : s' = filter Nat.Prime s
n : β
β’ n β s' β n.Prime | s : Finset β
h : β (p : β), p.Prime β p β s
s' : Finset β := filter Nat.Prime s
s'_def : s' = filter Nat.Prime s
n : β
β’ n.Prime β n β s |
https://github.com/avigad/mathematics_in_lean_source.git | 3fa84e6b3135a3ae41edc6ca195abf0fb1ae3ac3 | MIL/C05_Elementary_Number_Theory/S03_Infinitely_Many_Primes.lean | C05S03.primes_infinite' | [370, 1] | [406, 23] | apply h | s : Finset β
h : β (p : β), p.Prime β p β s
s' : Finset β := filter Nat.Prime s
s'_def : s' = filter Nat.Prime s
n : β
β’ n.Prime β n β 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_infinite' | [370, 1] | [406, 23] | apply Nat.succ_le_succ | s : Finset β
h : β (p : β), p.Prime β p β s
s' : Finset β := filter Nat.Prime s
s'_def : s' = filter Nat.Prime s
mem_s' : β {n : β}, n β s' β n.Prime
β’ 2 β€ β i β s', i + 1 | case a
s : Finset β
h : β (p : β), p.Prime β p β s
s' : Finset β := filter Nat.Prime s
s'_def : s' = filter Nat.Prime s
mem_s' : β {n : β}, n β s' β n.Prime
β’ 1 β€ β i β s', i |
https://github.com/avigad/mathematics_in_lean_source.git | 3fa84e6b3135a3ae41edc6ca195abf0fb1ae3ac3 | MIL/C05_Elementary_Number_Theory/S03_Infinitely_Many_Primes.lean | C05S03.primes_infinite' | [370, 1] | [406, 23] | apply Nat.succ_le_of_lt | case a
s : Finset β
h : β (p : β), p.Prime β p β s
s' : Finset β := filter Nat.Prime s
s'_def : s' = filter Nat.Prime s
mem_s' : β {n : β}, n β s' β n.Prime
β’ 1 β€ β i β s', i | case a.h
s : Finset β
h : β (p : β), p.Prime β p β s
s' : Finset β := filter Nat.Prime s
s'_def : s' = filter Nat.Prime s
mem_s' : β {n : β}, n β s' β n.Prime
β’ 0 < β i β s', i |
https://github.com/avigad/mathematics_in_lean_source.git | 3fa84e6b3135a3ae41edc6ca195abf0fb1ae3ac3 | MIL/C05_Elementary_Number_Theory/S03_Infinitely_Many_Primes.lean | C05S03.primes_infinite' | [370, 1] | [406, 23] | apply Finset.prod_pos | case a.h
s : Finset β
h : β (p : β), p.Prime β p β s
s' : Finset β := filter Nat.Prime s
s'_def : s' = filter Nat.Prime s
mem_s' : β {n : β}, n β s' β n.Prime
β’ 0 < β i β s', i | case a.h.h0
s : Finset β
h : β (p : β), p.Prime β p β s
s' : Finset β := filter Nat.Prime s
s'_def : s' = filter Nat.Prime s
mem_s' : β {n : β}, n β s' β n.Prime
β’ β i β s', 0 < i |
https://github.com/avigad/mathematics_in_lean_source.git | 3fa84e6b3135a3ae41edc6ca195abf0fb1ae3ac3 | MIL/C05_Elementary_Number_Theory/S03_Infinitely_Many_Primes.lean | C05S03.primes_infinite' | [370, 1] | [406, 23] | intro n ns' | case a.h.h0
s : Finset β
h : β (p : β), p.Prime β p β s
s' : Finset β := filter Nat.Prime s
s'_def : s' = filter Nat.Prime s
mem_s' : β {n : β}, n β s' β n.Prime
β’ β i β s', 0 < i | case a.h.h0
s : Finset β
h : β (p : β), p.Prime β p β s
s' : Finset β := filter Nat.Prime s
s'_def : s' = filter Nat.Prime s
mem_s' : β {n : β}, n β s' β n.Prime
n : β
ns' : n β s'
β’ 0 < n |
https://github.com/avigad/mathematics_in_lean_source.git | 3fa84e6b3135a3ae41edc6ca195abf0fb1ae3ac3 | MIL/C05_Elementary_Number_Theory/S03_Infinitely_Many_Primes.lean | C05S03.primes_infinite' | [370, 1] | [406, 23] | apply (mem_s'.mp ns').pos | case a.h.h0
s : Finset β
h : β (p : β), p.Prime β p β s
s' : Finset β := filter Nat.Prime s
s'_def : s' = filter Nat.Prime s
mem_s' : β {n : β}, n β s' β n.Prime
n : β
ns' : n β s'
β’ 0 < n | no goals |
https://github.com/avigad/mathematics_in_lean_source.git | 3fa84e6b3135a3ae41edc6ca195abf0fb1ae3ac3 | MIL/C05_Elementary_Number_Theory/S03_Infinitely_Many_Primes.lean | C05S03.primes_infinite' | [370, 1] | [406, 23] | apply dvd_prod_of_mem | s : Finset β
h : β (p : β), p.Prime β p β s
s' : Finset β := filter Nat.Prime s
s'_def : s' = filter Nat.Prime s
mem_s' : β {n : β}, n β s' β n.Prime
this : 2 β€ β i β s', i + 1
p : β
pp : p.Prime
pdvd : p β£ β i β s', i + 1
β’ p β£ β i β s', i | case ha
s : Finset β
h : β (p : β), p.Prime β p β s
s' : Finset β := filter Nat.Prime s
s'_def : s' = filter Nat.Prime s
mem_s' : β {n : β}, n β s' β n.Prime
this : 2 β€ β i β s', i + 1
p : β
pp : p.Prime
pdvd : p β£ β i β s', i + 1
β’ p β s' |
https://github.com/avigad/mathematics_in_lean_source.git | 3fa84e6b3135a3ae41edc6ca195abf0fb1ae3ac3 | MIL/C05_Elementary_Number_Theory/S03_Infinitely_Many_Primes.lean | C05S03.primes_infinite' | [370, 1] | [406, 23] | rw [mem_s'] | case ha
s : Finset β
h : β (p : β), p.Prime β p β s
s' : Finset β := filter Nat.Prime s
s'_def : s' = filter Nat.Prime s
mem_s' : β {n : β}, n β s' β n.Prime
this : 2 β€ β i β s', i + 1
p : β
pp : p.Prime
pdvd : p β£ β i β s', i + 1
β’ p β s' | case ha
s : Finset β
h : β (p : β), p.Prime β p β s
s' : Finset β := filter Nat.Prime s
s'_def : s' = filter Nat.Prime s
mem_s' : β {n : β}, n β s' β n.Prime
this : 2 β€ β i β s', i + 1
p : β
pp : p.Prime
pdvd : p β£ β i β s', i + 1
β’ p.Prime |
https://github.com/avigad/mathematics_in_lean_source.git | 3fa84e6b3135a3ae41edc6ca195abf0fb1ae3ac3 | MIL/C05_Elementary_Number_Theory/S03_Infinitely_Many_Primes.lean | C05S03.primes_infinite' | [370, 1] | [406, 23] | apply pp | case ha
s : Finset β
h : β (p : β), p.Prime β p β s
s' : Finset β := filter Nat.Prime s
s'_def : s' = filter Nat.Prime s
mem_s' : β {n : β}, n β s' β n.Prime
this : 2 β€ β i β s', i + 1
p : β
pp : p.Prime
pdvd : p β£ β i β s', i + 1
β’ p.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_infinite' | [370, 1] | [406, 23] | convert Nat.dvd_sub' pdvd this | s : Finset β
h : β (p : β), p.Prime β p β s
s' : Finset β := filter Nat.Prime s
s'_def : s' = filter Nat.Prime s
mem_s' : β {n : β}, n β s' β n.Prime
thisβ : 2 β€ β i β s', i + 1
p : β
pp : p.Prime
pdvd : p β£ β i β s', i + 1
this : p β£ β i β s', i
β’ p β£ 1 | case h.e'_4
s : Finset β
h : β (p : β), p.Prime β p β s
s' : Finset β := filter Nat.Prime s
s'_def : s' = filter Nat.Prime s
mem_s' : β {n : β}, n β s' β n.Prime
thisβ : 2 β€ β i β s', i + 1
p : β
pp : p.Prime
pdvd : p β£ β i β s', i + 1
this : p β£ β i β s', i
β’ 1 = β i β s', i + 1 - β i β s', i |
https://github.com/avigad/mathematics_in_lean_source.git | 3fa84e6b3135a3ae41edc6ca195abf0fb1ae3ac3 | MIL/C05_Elementary_Number_Theory/S03_Infinitely_Many_Primes.lean | C05S03.primes_infinite' | [370, 1] | [406, 23] | simp | case h.e'_4
s : Finset β
h : β (p : β), p.Prime β p β s
s' : Finset β := filter Nat.Prime s
s'_def : s' = filter Nat.Prime s
mem_s' : β {n : β}, n β s' β n.Prime
thisβ : 2 β€ β i β s', i + 1
p : β
pp : p.Prime
pdvd : p β£ β i β s', i + 1
this : p β£ β i β s', i
β’ 1 = β i β s', i + 1 - β i β s', i | no goals |
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] | rintro β¨s, hsβ© | Q : β β Prop
β’ (β s, β (k : β), Q k β k β s) β β n, β (k : β), Q k β k < n | case intro
Q : β β Prop
s : Finset β
hs : β (k : β), Q k β k β s
β’ β n, β (k : β), Q k β k < n |
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