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https://github.com/jvlmdr/from_the_book.git | 9fb6080539a2f32bb24719600a9e7531abf2328d | FromTheBook/Ch02/Bertrand/Bertrand.lean | lt4kIff | [143, 1] | [160, 11] | linarith | case hx
n : ℝ
hn : 0 < n
⊢ 0 < 2 * n | no goals |
https://github.com/jvlmdr/from_the_book.git | 9fb6080539a2f32bb24719600a9e7531abf2328d | FromTheBook/Ch02/Bertrand/Bertrand.lean | lt4kIff | [143, 1] | [160, 11] | linarith | n : ℝ
hn : 0 < n
h_nonneg : 0 < (2 * n) ^ (2 / 3)
⊢ 0 ≤ 2 * n | no goals |
https://github.com/jvlmdr/from_the_book.git | 9fb6080539a2f32bb24719600a9e7531abf2328d | FromTheBook/Ch02/Bertrand/Bertrand.lean | lt4kIff | [143, 1] | [160, 11] | linarith | n : ℝ
hn : 0 < n
h_nonneg : 0 < (2 * n) ^ (2 / 3)
⊢ 0 ≤ 20 * (2 * n) ^ (2 / 3) | no goals |
https://github.com/jvlmdr/from_the_book.git | 9fb6080539a2f32bb24719600a9e7531abf2328d | FromTheBook/Ch02/Bertrand/Bertrand.lean | lt4kIff | [143, 1] | [160, 11] | norm_num | n : ℝ
hn : 0 < n
h_nonneg : 0 < (2 * n) ^ (2 / 3)
⊢ 0 < 3 | no goals |
https://github.com/jvlmdr/from_the_book.git | 9fb6080539a2f32bb24719600a9e7531abf2328d | FromTheBook/Ch02/Bertrand/Bertrand.lean | lt4kIff | [143, 1] | [160, 11] | norm_num | n : ℝ
hn : 0 < n
h_nonneg : 0 < (2 * n) ^ (2 / 3)
⊢ 0 ≤ 20 | no goals |
https://github.com/jvlmdr/from_the_book.git | 9fb6080539a2f32bb24719600a9e7531abf2328d | FromTheBook/Ch02/Bertrand/Bertrand.lean | lt4kIff | [143, 1] | [160, 11] | linarith | n : ℝ
hn : 0 < n
h_nonneg : 0 < (2 * n) ^ (2 / 3)
⊢ 0 ≤ (2 * n) ^ (2 / 3) | no goals |
https://github.com/jvlmdr/from_the_book.git | 9fb6080539a2f32bb24719600a9e7531abf2328d | FromTheBook/Ch02/Bertrand/Bertrand.lean | lt4kIff | [143, 1] | [160, 11] | norm_num | n : ℝ
hn : 0 < n
h_nonneg : 0 < (2 * n) ^ (2 / 3)
⊢ 3 = 1 + 2 | no goals |
https://github.com/jvlmdr/from_the_book.git | 9fb6080539a2f32bb24719600a9e7531abf2328d | FromTheBook/Ch02/Bertrand/Bertrand.lean | lt4kIff | [143, 1] | [160, 11] | linarith | n : ℝ
hn : 0 < n
h_nonneg : 0 < (2 * n) ^ (2 / 3)
⊢ 0 < 2 * n | no goals |
https://github.com/jvlmdr/from_the_book.git | 9fb6080539a2f32bb24719600a9e7531abf2328d | FromTheBook/Ch02/Bertrand/Bertrand.lean | lt4kIff | [143, 1] | [160, 11] | norm_num | n : ℝ
hn : 0 < n
h_nonneg : 0 < (2 * n) ^ (2 / 3)
⊢ 0 < 20 | no goals |
https://github.com/jvlmdr/from_the_book.git | 9fb6080539a2f32bb24719600a9e7531abf2328d | FromTheBook/Ch02/Bertrand/Bertrand.lean | lt4kIff | [143, 1] | [160, 11] | apply sq_pos_of_pos | n : ℝ
hn : 0 < n
h_nonneg : 0 < (2 * n) ^ (2 / 3)
⊢ 0 < (2 * n) ^ 2 | case ha
n : ℝ
hn : 0 < n
h_nonneg : 0 < (2 * n) ^ (2 / 3)
⊢ 0 < 2 * n |
https://github.com/jvlmdr/from_the_book.git | 9fb6080539a2f32bb24719600a9e7531abf2328d | FromTheBook/Ch02/Bertrand/Bertrand.lean | lt4kIff | [143, 1] | [160, 11] | linarith | case ha
n : ℝ
hn : 0 < n
h_nonneg : 0 < (2 * n) ^ (2 / 3)
⊢ 0 < 2 * n | no goals |
https://github.com/jvlmdr/from_the_book.git | 9fb6080539a2f32bb24719600a9e7531abf2328d | FromTheBook/Ch02/Bertrand/Bertrand.lean | lt4kIff | [143, 1] | [160, 11] | norm_num | n : ℝ
hn : 0 < n
h_nonneg : 0 < (2 * n) ^ (2 / 3)
⊢ 0 < 2 | no goals |
https://github.com/jvlmdr/from_the_book.git | 9fb6080539a2f32bb24719600a9e7531abf2328d | FromTheBook/Ch02/Bertrand/Bertrand.lean | bertrand | [162, 1] | [184, 13] | intro hn | n : ℕ
⊢ 0 < n → ∃ p, p ∈ Set.Ioc n (2 * n) ∧ Nat.Prime p | n : ℕ
hn : 0 < n
⊢ ∃ p, p ∈ Set.Ioc n (2 * n) ∧ Nat.Prime p |
https://github.com/jvlmdr/from_the_book.git | 9fb6080539a2f32bb24719600a9e7531abf2328d | FromTheBook/Ch02/Bertrand/Bertrand.lean | bertrand | [162, 1] | [184, 13] | cases lt_or_le n 4000 with
| inl hn' => exact landauTrick (by linarith) (by linarith)
| inr hn' =>
by_contra h
simp at h
replace h : ∏ p in primes (Finset.Ioc n (2*n)), p = 1
. apply Finset.prod_eq_one
intro i hi
specialize h i
simp [primes] at *
specialize h hi.left.left hi.left.right hi.right
contradiction
have h' := fourPowLeMulProdPrimes (by linarith : 2 < n)
simp [h] at h'; clear h
replace h' := rpowDivThreeLe (by linarith : 0 < n) h'
have : (81 / 2 : ℝ) < ↑n
. rw [div_lt_iff (by norm_num)]; norm_cast; linarith
replace h' := fourPowLtOf this h'; clear this
rw [lt4kIff (by norm_cast)] at h'
norm_cast at h'
linarith | n : ℕ
hn : 0 < n
⊢ ∃ p, p ∈ Set.Ioc n (2 * n) ∧ Nat.Prime p | no goals |
https://github.com/jvlmdr/from_the_book.git | 9fb6080539a2f32bb24719600a9e7531abf2328d | FromTheBook/Ch02/Bertrand/Bertrand.lean | bertrand | [162, 1] | [184, 13] | exact landauTrick (by linarith) (by linarith) | case inl
n : ℕ
hn : 0 < n
hn' : n < 4000
⊢ ∃ p, p ∈ Set.Ioc n (2 * n) ∧ Nat.Prime p | no goals |
https://github.com/jvlmdr/from_the_book.git | 9fb6080539a2f32bb24719600a9e7531abf2328d | FromTheBook/Ch02/Bertrand/Bertrand.lean | bertrand | [162, 1] | [184, 13] | linarith | n : ℕ
hn : 0 < n
hn' : n < 4000
⊢ 0 < n | no goals |
https://github.com/jvlmdr/from_the_book.git | 9fb6080539a2f32bb24719600a9e7531abf2328d | FromTheBook/Ch02/Bertrand/Bertrand.lean | bertrand | [162, 1] | [184, 13] | linarith | n : ℕ
hn : 0 < n
hn' : n < 4000
⊢ n ≤ 4000 | no goals |
https://github.com/jvlmdr/from_the_book.git | 9fb6080539a2f32bb24719600a9e7531abf2328d | FromTheBook/Ch02/Bertrand/Bertrand.lean | bertrand | [162, 1] | [184, 13] | by_contra h | case inr
n : ℕ
hn : 0 < n
hn' : 4000 ≤ n
⊢ ∃ p, p ∈ Set.Ioc n (2 * n) ∧ Nat.Prime p | case inr
n : ℕ
hn : 0 < n
hn' : 4000 ≤ n
h : ¬∃ p, p ∈ Set.Ioc n (2 * n) ∧ Nat.Prime p
⊢ False |
https://github.com/jvlmdr/from_the_book.git | 9fb6080539a2f32bb24719600a9e7531abf2328d | FromTheBook/Ch02/Bertrand/Bertrand.lean | bertrand | [162, 1] | [184, 13] | simp at h | case inr
n : ℕ
hn : 0 < n
hn' : 4000 ≤ n
h : ¬∃ p, p ∈ Set.Ioc n (2 * n) ∧ Nat.Prime p
⊢ False | case inr
n : ℕ
hn : 0 < n
hn' : 4000 ≤ n
h : ∀ (x : ℕ), n < x → x ≤ 2 * n → ¬Nat.Prime x
⊢ False |
https://github.com/jvlmdr/from_the_book.git | 9fb6080539a2f32bb24719600a9e7531abf2328d | FromTheBook/Ch02/Bertrand/Bertrand.lean | bertrand | [162, 1] | [184, 13] | replace h : ∏ p in primes (Finset.Ioc n (2*n)), p = 1 | case inr
n : ℕ
hn : 0 < n
hn' : 4000 ≤ n
h : ∀ (x : ℕ), n < x → x ≤ 2 * n → ¬Nat.Prime x
⊢ False | case h
n : ℕ
hn : 0 < n
hn' : 4000 ≤ n
h : ∀ (x : ℕ), n < x → x ≤ 2 * n → ¬Nat.Prime x
⊢ ∏ p in primes (Finset.Ioc n (2 * n)), p = 1
case inr
n : ℕ
hn : 0 < n
hn' : 4000 ≤ n
h : ∏ p in primes (Finset.Ioc n (2 * n)), p = 1
⊢ False |
https://github.com/jvlmdr/from_the_book.git | 9fb6080539a2f32bb24719600a9e7531abf2328d | FromTheBook/Ch02/Bertrand/Bertrand.lean | bertrand | [162, 1] | [184, 13] | . apply Finset.prod_eq_one
intro i hi
specialize h i
simp [primes] at *
specialize h hi.left.left hi.left.right hi.right
contradiction | case h
n : ℕ
hn : 0 < n
hn' : 4000 ≤ n
h : ∀ (x : ℕ), n < x → x ≤ 2 * n → ¬Nat.Prime x
⊢ ∏ p in primes (Finset.Ioc n (2 * n)), p = 1
case inr
n : ℕ
hn : 0 < n
hn' : 4000 ≤ n
h : ∏ p in primes (Finset.Ioc n (2 * n)), p = 1
⊢ False | case inr
n : ℕ
hn : 0 < n
hn' : 4000 ≤ n
h : ∏ p in primes (Finset.Ioc n (2 * n)), p = 1
⊢ False |
https://github.com/jvlmdr/from_the_book.git | 9fb6080539a2f32bb24719600a9e7531abf2328d | FromTheBook/Ch02/Bertrand/Bertrand.lean | bertrand | [162, 1] | [184, 13] | have h' := fourPowLeMulProdPrimes (by linarith : 2 < n) | case inr
n : ℕ
hn : 0 < n
hn' : 4000 ≤ n
h : ∏ p in primes (Finset.Ioc n (2 * n)), p = 1
⊢ False | case inr
n : ℕ
hn : 0 < n
hn' : 4000 ≤ n
h : ∏ p in primes (Finset.Ioc n (2 * n)), p = 1
h' : 4 ^ n ≤ (2 * n) ^ (1 + Nat.sqrt (2 * n)) * 4 ^ Nat.pred (2 * n / 3) * ∏ p in primes (Finset.Ioc n (2 * n)), p
⊢ False |
https://github.com/jvlmdr/from_the_book.git | 9fb6080539a2f32bb24719600a9e7531abf2328d | FromTheBook/Ch02/Bertrand/Bertrand.lean | bertrand | [162, 1] | [184, 13] | simp [h] at h' | case inr
n : ℕ
hn : 0 < n
hn' : 4000 ≤ n
h : ∏ p in primes (Finset.Ioc n (2 * n)), p = 1
h' : 4 ^ n ≤ (2 * n) ^ (1 + Nat.sqrt (2 * n)) * 4 ^ Nat.pred (2 * n / 3) * ∏ p in primes (Finset.Ioc n (2 * n)), p
⊢ False | case inr
n : ℕ
hn : 0 < n
hn' : 4000 ≤ n
h : ∏ p in primes (Finset.Ioc n (2 * n)), p = 1
h' : 4 ^ n ≤ (2 * n) ^ (1 + Nat.sqrt (2 * n)) * 4 ^ Nat.pred (2 * n / 3)
⊢ False |
https://github.com/jvlmdr/from_the_book.git | 9fb6080539a2f32bb24719600a9e7531abf2328d | FromTheBook/Ch02/Bertrand/Bertrand.lean | bertrand | [162, 1] | [184, 13] | clear h | case inr
n : ℕ
hn : 0 < n
hn' : 4000 ≤ n
h : ∏ p in primes (Finset.Ioc n (2 * n)), p = 1
h' : 4 ^ n ≤ (2 * n) ^ (1 + Nat.sqrt (2 * n)) * 4 ^ Nat.pred (2 * n / 3)
⊢ False | case inr
n : ℕ
hn : 0 < n
hn' : 4000 ≤ n
h' : 4 ^ n ≤ (2 * n) ^ (1 + Nat.sqrt (2 * n)) * 4 ^ Nat.pred (2 * n / 3)
⊢ False |
https://github.com/jvlmdr/from_the_book.git | 9fb6080539a2f32bb24719600a9e7531abf2328d | FromTheBook/Ch02/Bertrand/Bertrand.lean | bertrand | [162, 1] | [184, 13] | replace h' := rpowDivThreeLe (by linarith : 0 < n) h' | case inr
n : ℕ
hn : 0 < n
hn' : 4000 ≤ n
h' : 4 ^ n ≤ (2 * n) ^ (1 + Nat.sqrt (2 * n)) * 4 ^ Nat.pred (2 * n / 3)
⊢ False | case inr
n : ℕ
hn : 0 < n
hn' : 4000 ≤ n
h' : 4 ^ (↑n / 3) ≤ (2 * ↑n) ^ (1 + Real.sqrt (2 * ↑n))
⊢ False |
https://github.com/jvlmdr/from_the_book.git | 9fb6080539a2f32bb24719600a9e7531abf2328d | FromTheBook/Ch02/Bertrand/Bertrand.lean | bertrand | [162, 1] | [184, 13] | have : (81 / 2 : ℝ) < ↑n | case inr
n : ℕ
hn : 0 < n
hn' : 4000 ≤ n
h' : 4 ^ (↑n / 3) ≤ (2 * ↑n) ^ (1 + Real.sqrt (2 * ↑n))
⊢ False | case this
n : ℕ
hn : 0 < n
hn' : 4000 ≤ n
h' : 4 ^ (↑n / 3) ≤ (2 * ↑n) ^ (1 + Real.sqrt (2 * ↑n))
⊢ 81 / 2 < ↑n
case inr
n : ℕ
hn : 0 < n
hn' : 4000 ≤ n
h' : 4 ^ (↑n / 3) ≤ (2 * ↑n) ^ (1 + Real.sqrt (2 * ↑n))
this : 81 / 2 < ↑n
⊢ False |
https://github.com/jvlmdr/from_the_book.git | 9fb6080539a2f32bb24719600a9e7531abf2328d | FromTheBook/Ch02/Bertrand/Bertrand.lean | bertrand | [162, 1] | [184, 13] | . rw [div_lt_iff (by norm_num)]; norm_cast; linarith | case this
n : ℕ
hn : 0 < n
hn' : 4000 ≤ n
h' : 4 ^ (↑n / 3) ≤ (2 * ↑n) ^ (1 + Real.sqrt (2 * ↑n))
⊢ 81 / 2 < ↑n
case inr
n : ℕ
hn : 0 < n
hn' : 4000 ≤ n
h' : 4 ^ (↑n / 3) ≤ (2 * ↑n) ^ (1 + Real.sqrt (2 * ↑n))
this : 81 / 2 < ↑n
⊢ False | case inr
n : ℕ
hn : 0 < n
hn' : 4000 ≤ n
h' : 4 ^ (↑n / 3) ≤ (2 * ↑n) ^ (1 + Real.sqrt (2 * ↑n))
this : 81 / 2 < ↑n
⊢ False |
https://github.com/jvlmdr/from_the_book.git | 9fb6080539a2f32bb24719600a9e7531abf2328d | FromTheBook/Ch02/Bertrand/Bertrand.lean | bertrand | [162, 1] | [184, 13] | replace h' := fourPowLtOf this h' | case inr
n : ℕ
hn : 0 < n
hn' : 4000 ≤ n
h' : 4 ^ (↑n / 3) ≤ (2 * ↑n) ^ (1 + Real.sqrt (2 * ↑n))
this : 81 / 2 < ↑n
⊢ False | case inr
n : ℕ
hn : 0 < n
hn' : 4000 ≤ n
this : 81 / 2 < ↑n
h' : 4 ^ ↑n < 2 ^ (20 * (2 * ↑n) ^ (2 / 3))
⊢ False |
https://github.com/jvlmdr/from_the_book.git | 9fb6080539a2f32bb24719600a9e7531abf2328d | FromTheBook/Ch02/Bertrand/Bertrand.lean | bertrand | [162, 1] | [184, 13] | clear this | case inr
n : ℕ
hn : 0 < n
hn' : 4000 ≤ n
this : 81 / 2 < ↑n
h' : 4 ^ ↑n < 2 ^ (20 * (2 * ↑n) ^ (2 / 3))
⊢ False | case inr
n : ℕ
hn : 0 < n
hn' : 4000 ≤ n
h' : 4 ^ ↑n < 2 ^ (20 * (2 * ↑n) ^ (2 / 3))
⊢ False |
https://github.com/jvlmdr/from_the_book.git | 9fb6080539a2f32bb24719600a9e7531abf2328d | FromTheBook/Ch02/Bertrand/Bertrand.lean | bertrand | [162, 1] | [184, 13] | rw [lt4kIff (by norm_cast)] at h' | case inr
n : ℕ
hn : 0 < n
hn' : 4000 ≤ n
h' : 4 ^ ↑n < 2 ^ (20 * (2 * ↑n) ^ (2 / 3))
⊢ False | case inr
n : ℕ
hn : 0 < n
hn' : 4000 ≤ n
h' : ↑n < 4000
⊢ False |
https://github.com/jvlmdr/from_the_book.git | 9fb6080539a2f32bb24719600a9e7531abf2328d | FromTheBook/Ch02/Bertrand/Bertrand.lean | bertrand | [162, 1] | [184, 13] | norm_cast at h' | case inr
n : ℕ
hn : 0 < n
hn' : 4000 ≤ n
h' : ↑n < 4000
⊢ False | case inr
n : ℕ
hn : 0 < n
hn' : 4000 ≤ n
h' : n < 4000
⊢ False |
https://github.com/jvlmdr/from_the_book.git | 9fb6080539a2f32bb24719600a9e7531abf2328d | FromTheBook/Ch02/Bertrand/Bertrand.lean | bertrand | [162, 1] | [184, 13] | linarith | case inr
n : ℕ
hn : 0 < n
hn' : 4000 ≤ n
h' : n < 4000
⊢ False | no goals |
https://github.com/jvlmdr/from_the_book.git | 9fb6080539a2f32bb24719600a9e7531abf2328d | FromTheBook/Ch02/Bertrand/Bertrand.lean | bertrand | [162, 1] | [184, 13] | apply Finset.prod_eq_one | case h
n : ℕ
hn : 0 < n
hn' : 4000 ≤ n
h : ∀ (x : ℕ), n < x → x ≤ 2 * n → ¬Nat.Prime x
⊢ ∏ p in primes (Finset.Ioc n (2 * n)), p = 1 | case h.h
n : ℕ
hn : 0 < n
hn' : 4000 ≤ n
h : ∀ (x : ℕ), n < x → x ≤ 2 * n → ¬Nat.Prime x
⊢ ∀ (x : ℕ), x ∈ primes (Finset.Ioc n (2 * n)) → x = 1 |
https://github.com/jvlmdr/from_the_book.git | 9fb6080539a2f32bb24719600a9e7531abf2328d | FromTheBook/Ch02/Bertrand/Bertrand.lean | bertrand | [162, 1] | [184, 13] | intro i hi | case h.h
n : ℕ
hn : 0 < n
hn' : 4000 ≤ n
h : ∀ (x : ℕ), n < x → x ≤ 2 * n → ¬Nat.Prime x
⊢ ∀ (x : ℕ), x ∈ primes (Finset.Ioc n (2 * n)) → x = 1 | case h.h
n : ℕ
hn : 0 < n
hn' : 4000 ≤ n
h : ∀ (x : ℕ), n < x → x ≤ 2 * n → ¬Nat.Prime x
i : ℕ
hi : i ∈ primes (Finset.Ioc n (2 * n))
⊢ i = 1 |
https://github.com/jvlmdr/from_the_book.git | 9fb6080539a2f32bb24719600a9e7531abf2328d | FromTheBook/Ch02/Bertrand/Bertrand.lean | bertrand | [162, 1] | [184, 13] | specialize h i | case h.h
n : ℕ
hn : 0 < n
hn' : 4000 ≤ n
h : ∀ (x : ℕ), n < x → x ≤ 2 * n → ¬Nat.Prime x
i : ℕ
hi : i ∈ primes (Finset.Ioc n (2 * n))
⊢ i = 1 | case h.h
n : ℕ
hn : 0 < n
hn' : 4000 ≤ n
i : ℕ
hi : i ∈ primes (Finset.Ioc n (2 * n))
h : n < i → i ≤ 2 * n → ¬Nat.Prime i
⊢ i = 1 |
https://github.com/jvlmdr/from_the_book.git | 9fb6080539a2f32bb24719600a9e7531abf2328d | FromTheBook/Ch02/Bertrand/Bertrand.lean | bertrand | [162, 1] | [184, 13] | simp [primes] at * | case h.h
n : ℕ
hn : 0 < n
hn' : 4000 ≤ n
i : ℕ
hi : i ∈ primes (Finset.Ioc n (2 * n))
h : n < i → i ≤ 2 * n → ¬Nat.Prime i
⊢ i = 1 | case h.h
n : ℕ
hn : 0 < n
hn' : 4000 ≤ n
i : ℕ
h : n < i → i ≤ 2 * n → ¬Nat.Prime i
hi : (n < i ∧ i ≤ 2 * n) ∧ Nat.Prime i
⊢ i = 1 |
https://github.com/jvlmdr/from_the_book.git | 9fb6080539a2f32bb24719600a9e7531abf2328d | FromTheBook/Ch02/Bertrand/Bertrand.lean | bertrand | [162, 1] | [184, 13] | specialize h hi.left.left hi.left.right hi.right | case h.h
n : ℕ
hn : 0 < n
hn' : 4000 ≤ n
i : ℕ
h : n < i → i ≤ 2 * n → ¬Nat.Prime i
hi : (n < i ∧ i ≤ 2 * n) ∧ Nat.Prime i
⊢ i = 1 | case h.h
n : ℕ
hn : 0 < n
hn' : 4000 ≤ n
i : ℕ
hi : (n < i ∧ i ≤ 2 * n) ∧ Nat.Prime i
h : False
⊢ i = 1 |
https://github.com/jvlmdr/from_the_book.git | 9fb6080539a2f32bb24719600a9e7531abf2328d | FromTheBook/Ch02/Bertrand/Bertrand.lean | bertrand | [162, 1] | [184, 13] | contradiction | case h.h
n : ℕ
hn : 0 < n
hn' : 4000 ≤ n
i : ℕ
hi : (n < i ∧ i ≤ 2 * n) ∧ Nat.Prime i
h : False
⊢ i = 1 | no goals |
https://github.com/jvlmdr/from_the_book.git | 9fb6080539a2f32bb24719600a9e7531abf2328d | FromTheBook/Ch02/Bertrand/Bertrand.lean | bertrand | [162, 1] | [184, 13] | linarith | n : ℕ
hn : 0 < n
hn' : 4000 ≤ n
h : ∏ p in primes (Finset.Ioc n (2 * n)), p = 1
⊢ 2 < n | no goals |
https://github.com/jvlmdr/from_the_book.git | 9fb6080539a2f32bb24719600a9e7531abf2328d | FromTheBook/Ch02/Bertrand/Bertrand.lean | bertrand | [162, 1] | [184, 13] | linarith | n : ℕ
hn : 0 < n
hn' : 4000 ≤ n
h' : 4 ^ n ≤ (2 * n) ^ (1 + Nat.sqrt (2 * n)) * 4 ^ Nat.pred (2 * n / 3)
⊢ 0 < n | no goals |
https://github.com/jvlmdr/from_the_book.git | 9fb6080539a2f32bb24719600a9e7531abf2328d | FromTheBook/Ch02/Bertrand/Bertrand.lean | bertrand | [162, 1] | [184, 13] | rw [div_lt_iff (by norm_num)] | case this
n : ℕ
hn : 0 < n
hn' : 4000 ≤ n
h' : 4 ^ (↑n / 3) ≤ (2 * ↑n) ^ (1 + Real.sqrt (2 * ↑n))
⊢ 81 / 2 < ↑n | case this
n : ℕ
hn : 0 < n
hn' : 4000 ≤ n
h' : 4 ^ (↑n / 3) ≤ (2 * ↑n) ^ (1 + Real.sqrt (2 * ↑n))
⊢ 81 < ↑n * 2 |
https://github.com/jvlmdr/from_the_book.git | 9fb6080539a2f32bb24719600a9e7531abf2328d | FromTheBook/Ch02/Bertrand/Bertrand.lean | bertrand | [162, 1] | [184, 13] | norm_cast | case this
n : ℕ
hn : 0 < n
hn' : 4000 ≤ n
h' : 4 ^ (↑n / 3) ≤ (2 * ↑n) ^ (1 + Real.sqrt (2 * ↑n))
⊢ 81 < ↑n * 2 | case this
n : ℕ
hn : 0 < n
hn' : 4000 ≤ n
h' : 4 ^ (↑n / 3) ≤ (2 * ↑n) ^ (1 + Real.sqrt (2 * ↑n))
⊢ 81 < n * 2 |
https://github.com/jvlmdr/from_the_book.git | 9fb6080539a2f32bb24719600a9e7531abf2328d | FromTheBook/Ch02/Bertrand/Bertrand.lean | bertrand | [162, 1] | [184, 13] | linarith | case this
n : ℕ
hn : 0 < n
hn' : 4000 ≤ n
h' : 4 ^ (↑n / 3) ≤ (2 * ↑n) ^ (1 + Real.sqrt (2 * ↑n))
⊢ 81 < n * 2 | no goals |
https://github.com/jvlmdr/from_the_book.git | 9fb6080539a2f32bb24719600a9e7531abf2328d | FromTheBook/Ch02/Bertrand/Bertrand.lean | bertrand | [162, 1] | [184, 13] | norm_num | n : ℕ
hn : 0 < n
hn' : 4000 ≤ n
h' : 4 ^ (↑n / 3) ≤ (2 * ↑n) ^ (1 + Real.sqrt (2 * ↑n))
⊢ 0 < 2 | no goals |
https://github.com/jvlmdr/from_the_book.git | 9fb6080539a2f32bb24719600a9e7531abf2328d | FromTheBook/Ch02/Bertrand/Bertrand.lean | bertrand | [162, 1] | [184, 13] | norm_cast | n : ℕ
hn : 0 < n
hn' : 4000 ≤ n
h' : 4 ^ ↑n < 2 ^ (20 * (2 * ↑n) ^ (2 / 3))
⊢ 0 < ↑n | no goals |
https://github.com/jvlmdr/from_the_book.git | 9fb6080539a2f32bb24719600a9e7531abf2328d | FromTheBook/Ch02/Bertrand/ProdPrimes.lean | chooseHalfLePowFour | [21, 1] | [28, 46] | rw [Nat.Ico_succ_singleton] | m : ℕ
⊢ ∑ k in {m}, Nat.choose (2 * m + 1) k = ∑ k in Finset.Ico m (m + 1), Nat.choose (2 * m + 1) k | no goals |
https://github.com/jvlmdr/from_the_book.git | 9fb6080539a2f32bb24719600a9e7531abf2328d | FromTheBook/Ch02/Bertrand/ProdPrimes.lean | chooseHalfLePowFour | [21, 1] | [28, 46] | rw [Finset.range_eq_Ico] | m : ℕ
⊢ ∑ k in Finset.Ico 0 (m + 1), Nat.choose (2 * m + 1) k = ∑ k in Finset.range (m + 1), Nat.choose (2 * m + 1) k | no goals |
https://github.com/jvlmdr/from_the_book.git | 9fb6080539a2f32bb24719600a9e7531abf2328d | FromTheBook/Ch02/Bertrand/ProdPrimes.lean | prodPrimesIocLeChoose | [30, 1] | [39, 49] | rw [primes] | m : ℕ
⊢ ∏ p in primes (Finset.Ioc (m + 1) (2 * m + 1)), p ≤ Nat.choose (2 * m + 1) m | m : ℕ
⊢ ∏ p in Finset.filter Nat.Prime (Finset.Ioc (m + 1) (2 * m + 1)), p ≤ Nat.choose (2 * m + 1) m |
https://github.com/jvlmdr/from_the_book.git | 9fb6080539a2f32bb24719600a9e7531abf2328d | FromTheBook/Ch02/Bertrand/ProdPrimes.lean | prodPrimesIocLeChoose | [30, 1] | [39, 49] | apply Nat.le_of_dvd | m : ℕ
⊢ ∏ p in Finset.filter Nat.Prime (Finset.Ioc (m + 1) (2 * m + 1)), p ≤ Nat.choose (2 * m + 1) m | case h
m : ℕ
⊢ 0 < Nat.choose (2 * m + 1) m
case a
m : ℕ
⊢ ∏ p in Finset.filter Nat.Prime (Finset.Ioc (m + 1) (2 * m + 1)), p ∣ Nat.choose (2 * m + 1) m |
https://github.com/jvlmdr/from_the_book.git | 9fb6080539a2f32bb24719600a9e7531abf2328d | FromTheBook/Ch02/Bertrand/ProdPrimes.lean | prodPrimesIocLeChoose | [30, 1] | [39, 49] | . apply Nat.choose_pos; linarith | case h
m : ℕ
⊢ 0 < Nat.choose (2 * m + 1) m
case a
m : ℕ
⊢ ∏ p in Finset.filter Nat.Prime (Finset.Ioc (m + 1) (2 * m + 1)), p ∣ Nat.choose (2 * m + 1) m | case a
m : ℕ
⊢ ∏ p in Finset.filter Nat.Prime (Finset.Ioc (m + 1) (2 * m + 1)), p ∣ Nat.choose (2 * m + 1) m |
https://github.com/jvlmdr/from_the_book.git | 9fb6080539a2f32bb24719600a9e7531abf2328d | FromTheBook/Ch02/Bertrand/ProdPrimes.lean | prodPrimesIocLeChoose | [30, 1] | [39, 49] | apply Finset.prod_primes_dvd <;> simp [← Nat.prime_iff] | case a
m : ℕ
⊢ ∏ p in Finset.filter Nat.Prime (Finset.Ioc (m + 1) (2 * m + 1)), p ∣ Nat.choose (2 * m + 1) m | case a.div
m : ℕ
⊢ ∀ (a : ℕ), m + 1 < a → a ≤ 2 * m + 1 → Nat.Prime a → a ∣ Nat.choose (2 * m + 1) m |
https://github.com/jvlmdr/from_the_book.git | 9fb6080539a2f32bb24719600a9e7531abf2328d | FromTheBook/Ch02/Bertrand/ProdPrimes.lean | prodPrimesIocLeChoose | [30, 1] | [39, 49] | intro p ha hb hp | case a.div
m : ℕ
⊢ ∀ (a : ℕ), m + 1 < a → a ≤ 2 * m + 1 → Nat.Prime a → a ∣ Nat.choose (2 * m + 1) m | case a.div
m p : ℕ
ha : m + 1 < p
hb : p ≤ 2 * m + 1
hp : Nat.Prime p
⊢ p ∣ Nat.choose (2 * m + 1) m |
https://github.com/jvlmdr/from_the_book.git | 9fb6080539a2f32bb24719600a9e7531abf2328d | FromTheBook/Ch02/Bertrand/ProdPrimes.lean | prodPrimesIocLeChoose | [30, 1] | [39, 49] | rw [two_mul, add_assoc] | case a.div
m p : ℕ
ha : m + 1 < p
hb : p ≤ 2 * m + 1
hp : Nat.Prime p
⊢ p ∣ Nat.choose (2 * m + 1) m | case a.div
m p : ℕ
ha : m + 1 < p
hb : p ≤ 2 * m + 1
hp : Nat.Prime p
⊢ p ∣ Nat.choose (m + (m + 1)) m |
https://github.com/jvlmdr/from_the_book.git | 9fb6080539a2f32bb24719600a9e7531abf2328d | FromTheBook/Ch02/Bertrand/ProdPrimes.lean | prodPrimesIocLeChoose | [30, 1] | [39, 49] | apply Nat.Prime.dvd_choose_add hp <;> linarith | case a.div
m p : ℕ
ha : m + 1 < p
hb : p ≤ 2 * m + 1
hp : Nat.Prime p
⊢ p ∣ Nat.choose (m + (m + 1)) m | no goals |
https://github.com/jvlmdr/from_the_book.git | 9fb6080539a2f32bb24719600a9e7531abf2328d | FromTheBook/Ch02/Bertrand/ProdPrimes.lean | prodPrimesIocLeChoose | [30, 1] | [39, 49] | apply Nat.choose_pos | case h
m : ℕ
⊢ 0 < Nat.choose (2 * m + 1) m | case h.a
m : ℕ
⊢ m ≤ 2 * m + 1 |
https://github.com/jvlmdr/from_the_book.git | 9fb6080539a2f32bb24719600a9e7531abf2328d | FromTheBook/Ch02/Bertrand/ProdPrimes.lean | prodPrimesIocLeChoose | [30, 1] | [39, 49] | linarith | case h.a
m : ℕ
⊢ m ≤ 2 * m + 1 | no goals |
https://github.com/jvlmdr/from_the_book.git | 9fb6080539a2f32bb24719600a9e7531abf2328d | FromTheBook/Ch02/Bertrand/ProdPrimes.lean | primesSuccEqPrimesSelf | [41, 1] | [46, 13] | simp [primes] | n : ℕ
hn : ¬Nat.Prime (Nat.succ n)
⊢ primes (Finset.range (Nat.succ (Nat.succ n))) = primes (Finset.range (Nat.succ n)) | n : ℕ
hn : ¬Nat.Prime (Nat.succ n)
⊢ Finset.filter Nat.Prime (Finset.range (Nat.succ (Nat.succ n))) = Finset.filter Nat.Prime (Finset.range (Nat.succ n)) |
https://github.com/jvlmdr/from_the_book.git | 9fb6080539a2f32bb24719600a9e7531abf2328d | FromTheBook/Ch02/Bertrand/ProdPrimes.lean | primesSuccEqPrimesSelf | [41, 1] | [46, 13] | rw [Finset.range_succ] | n : ℕ
hn : ¬Nat.Prime (Nat.succ n)
⊢ Finset.filter Nat.Prime (Finset.range (Nat.succ (Nat.succ n))) = Finset.filter Nat.Prime (Finset.range (Nat.succ n)) | n : ℕ
hn : ¬Nat.Prime (Nat.succ n)
⊢ Finset.filter Nat.Prime (insert (n + 1) (Finset.range (n + 1))) = Finset.filter Nat.Prime (Finset.range (Nat.succ n)) |
https://github.com/jvlmdr/from_the_book.git | 9fb6080539a2f32bb24719600a9e7531abf2328d | FromTheBook/Ch02/Bertrand/ProdPrimes.lean | primesSuccEqPrimesSelf | [41, 1] | [46, 13] | simp [Finset.filter_insert] | n : ℕ
hn : ¬Nat.Prime (Nat.succ n)
⊢ Finset.filter Nat.Prime (insert (n + 1) (Finset.range (n + 1))) = Finset.filter Nat.Prime (Finset.range (Nat.succ n)) | n : ℕ
hn : ¬Nat.Prime (Nat.succ n)
⊢ Nat.Prime (n + 1) → False |
https://github.com/jvlmdr/from_the_book.git | 9fb6080539a2f32bb24719600a9e7531abf2328d | FromTheBook/Ch02/Bertrand/ProdPrimes.lean | primesSuccEqPrimesSelf | [41, 1] | [46, 13] | assumption | n : ℕ
hn : ¬Nat.Prime (Nat.succ n)
⊢ Nat.Prime (n + 1) → False | no goals |
https://github.com/jvlmdr/from_the_book.git | 9fb6080539a2f32bb24719600a9e7531abf2328d | FromTheBook/Ch02/Bertrand/ProdPrimes.lean | caseStrongRecOnOddPrimes | [48, 1] | [70, 43] | intro n | p : ℕ → Prop
zero : p 0
two : p 2
not_prime : ∀ (n : ℕ), ¬Nat.Prime (Nat.succ n) → (∀ (x : ℕ), x ≤ n → p x) → p (Nat.succ n)
odd_prime : ∀ (n : ℕ), Nat.Prime (Nat.succ (2 * n)) → (∀ (x : ℕ), x ≤ 2 * n → p x) → p (Nat.succ (2 * n))
⊢ ∀ (n : ℕ), p n | p : ℕ → Prop
zero : p 0
two : p 2
not_prime : ∀ (n : ℕ), ¬Nat.Prime (Nat.succ n) → (∀ (x : ℕ), x ≤ n → p x) → p (Nat.succ n)
odd_prime : ∀ (n : ℕ), Nat.Prime (Nat.succ (2 * n)) → (∀ (x : ℕ), x ≤ 2 * n → p x) → p (Nat.succ (2 * n))
n : ℕ
⊢ p n |
https://github.com/jvlmdr/from_the_book.git | 9fb6080539a2f32bb24719600a9e7531abf2328d | FromTheBook/Ch02/Bertrand/ProdPrimes.lean | caseStrongRecOnOddPrimes | [48, 1] | [70, 43] | induction n using Nat.caseStrongInductionOn with
| zero => exact zero
| ind k h_ind =>
cases dec_em (Nat.Prime k.succ) with
| inr hk_prime =>
apply not_prime _ hk_prime
exact h_ind
| inl hk_prime =>
cases Nat.Prime.eq_two_or_odd' hk_prime with
| inl hk_two => rw [hk_two]; exact two
| inr hk_two =>
cases hk_two with
| intro m hm =>
simp [Nat.add_one] at hm
simp [hm] at *
apply odd_prime _ hk_prime h_ind | p : ℕ → Prop
zero : p 0
two : p 2
not_prime : ∀ (n : ℕ), ¬Nat.Prime (Nat.succ n) → (∀ (x : ℕ), x ≤ n → p x) → p (Nat.succ n)
odd_prime : ∀ (n : ℕ), Nat.Prime (Nat.succ (2 * n)) → (∀ (x : ℕ), x ≤ 2 * n → p x) → p (Nat.succ (2 * n))
n : ℕ
⊢ p n | no goals |
https://github.com/jvlmdr/from_the_book.git | 9fb6080539a2f32bb24719600a9e7531abf2328d | FromTheBook/Ch02/Bertrand/ProdPrimes.lean | caseStrongRecOnOddPrimes | [48, 1] | [70, 43] | exact zero | case zero
p : ℕ → Prop
zero : p 0
two : p 2
not_prime : ∀ (n : ℕ), ¬Nat.Prime (Nat.succ n) → (∀ (x : ℕ), x ≤ n → p x) → p (Nat.succ n)
odd_prime : ∀ (n : ℕ), Nat.Prime (Nat.succ (2 * n)) → (∀ (x : ℕ), x ≤ 2 * n → p x) → p (Nat.succ (2 * n))
⊢ p 0 | no goals |
https://github.com/jvlmdr/from_the_book.git | 9fb6080539a2f32bb24719600a9e7531abf2328d | FromTheBook/Ch02/Bertrand/ProdPrimes.lean | caseStrongRecOnOddPrimes | [48, 1] | [70, 43] | cases dec_em (Nat.Prime k.succ) with
| inr hk_prime =>
apply not_prime _ hk_prime
exact h_ind
| inl hk_prime =>
cases Nat.Prime.eq_two_or_odd' hk_prime with
| inl hk_two => rw [hk_two]; exact two
| inr hk_two =>
cases hk_two with
| intro m hm =>
simp [Nat.add_one] at hm
simp [hm] at *
apply odd_prime _ hk_prime h_ind | case ind
p : ℕ → Prop
zero : p 0
two : p 2
not_prime : ∀ (n : ℕ), ¬Nat.Prime (Nat.succ n) → (∀ (x : ℕ), x ≤ n → p x) → p (Nat.succ n)
odd_prime : ∀ (n : ℕ), Nat.Prime (Nat.succ (2 * n)) → (∀ (x : ℕ), x ≤ 2 * n → p x) → p (Nat.succ (2 * n))
k : ℕ
h_ind : ∀ (m : ℕ), m ≤ k → p m
⊢ p (Nat.succ k) | no goals |
https://github.com/jvlmdr/from_the_book.git | 9fb6080539a2f32bb24719600a9e7531abf2328d | FromTheBook/Ch02/Bertrand/ProdPrimes.lean | caseStrongRecOnOddPrimes | [48, 1] | [70, 43] | apply not_prime _ hk_prime | case ind.inr
p : ℕ → Prop
zero : p 0
two : p 2
not_prime : ∀ (n : ℕ), ¬Nat.Prime (Nat.succ n) → (∀ (x : ℕ), x ≤ n → p x) → p (Nat.succ n)
odd_prime : ∀ (n : ℕ), Nat.Prime (Nat.succ (2 * n)) → (∀ (x : ℕ), x ≤ 2 * n → p x) → p (Nat.succ (2 * n))
k : ℕ
h_ind : ∀ (m : ℕ), m ≤ k → p m
hk_prime : ¬Nat.Prime (Nat.succ k)
⊢ p (Nat.succ k) | case ind.inr
p : ℕ → Prop
zero : p 0
two : p 2
not_prime : ∀ (n : ℕ), ¬Nat.Prime (Nat.succ n) → (∀ (x : ℕ), x ≤ n → p x) → p (Nat.succ n)
odd_prime : ∀ (n : ℕ), Nat.Prime (Nat.succ (2 * n)) → (∀ (x : ℕ), x ≤ 2 * n → p x) → p (Nat.succ (2 * n))
k : ℕ
h_ind : ∀ (m : ℕ), m ≤ k → p m
hk_prime : ¬Nat.Prime (Nat.succ k)
⊢ ∀ (x : ℕ), x ≤ k → p x |
https://github.com/jvlmdr/from_the_book.git | 9fb6080539a2f32bb24719600a9e7531abf2328d | FromTheBook/Ch02/Bertrand/ProdPrimes.lean | caseStrongRecOnOddPrimes | [48, 1] | [70, 43] | exact h_ind | case ind.inr
p : ℕ → Prop
zero : p 0
two : p 2
not_prime : ∀ (n : ℕ), ¬Nat.Prime (Nat.succ n) → (∀ (x : ℕ), x ≤ n → p x) → p (Nat.succ n)
odd_prime : ∀ (n : ℕ), Nat.Prime (Nat.succ (2 * n)) → (∀ (x : ℕ), x ≤ 2 * n → p x) → p (Nat.succ (2 * n))
k : ℕ
h_ind : ∀ (m : ℕ), m ≤ k → p m
hk_prime : ¬Nat.Prime (Nat.succ k)
⊢ ∀ (x : ℕ), x ≤ k → p x | no goals |
https://github.com/jvlmdr/from_the_book.git | 9fb6080539a2f32bb24719600a9e7531abf2328d | FromTheBook/Ch02/Bertrand/ProdPrimes.lean | caseStrongRecOnOddPrimes | [48, 1] | [70, 43] | cases Nat.Prime.eq_two_or_odd' hk_prime with
| inl hk_two => rw [hk_two]; exact two
| inr hk_two =>
cases hk_two with
| intro m hm =>
simp [Nat.add_one] at hm
simp [hm] at *
apply odd_prime _ hk_prime h_ind | case ind.inl
p : ℕ → Prop
zero : p 0
two : p 2
not_prime : ∀ (n : ℕ), ¬Nat.Prime (Nat.succ n) → (∀ (x : ℕ), x ≤ n → p x) → p (Nat.succ n)
odd_prime : ∀ (n : ℕ), Nat.Prime (Nat.succ (2 * n)) → (∀ (x : ℕ), x ≤ 2 * n → p x) → p (Nat.succ (2 * n))
k : ℕ
h_ind : ∀ (m : ℕ), m ≤ k → p m
hk_prime : Nat.Prime (Nat.succ k)
⊢ p (Nat.succ k) | no goals |
https://github.com/jvlmdr/from_the_book.git | 9fb6080539a2f32bb24719600a9e7531abf2328d | FromTheBook/Ch02/Bertrand/ProdPrimes.lean | caseStrongRecOnOddPrimes | [48, 1] | [70, 43] | rw [hk_two] | case ind.inl.inl
p : ℕ → Prop
zero : p 0
two : p 2
not_prime : ∀ (n : ℕ), ¬Nat.Prime (Nat.succ n) → (∀ (x : ℕ), x ≤ n → p x) → p (Nat.succ n)
odd_prime : ∀ (n : ℕ), Nat.Prime (Nat.succ (2 * n)) → (∀ (x : ℕ), x ≤ 2 * n → p x) → p (Nat.succ (2 * n))
k : ℕ
h_ind : ∀ (m : ℕ), m ≤ k → p m
hk_prime : Nat.Prime (Nat.succ k)
hk_two : Nat.succ k = 2
⊢ p (Nat.succ k) | case ind.inl.inl
p : ℕ → Prop
zero : p 0
two : p 2
not_prime : ∀ (n : ℕ), ¬Nat.Prime (Nat.succ n) → (∀ (x : ℕ), x ≤ n → p x) → p (Nat.succ n)
odd_prime : ∀ (n : ℕ), Nat.Prime (Nat.succ (2 * n)) → (∀ (x : ℕ), x ≤ 2 * n → p x) → p (Nat.succ (2 * n))
k : ℕ
h_ind : ∀ (m : ℕ), m ≤ k → p m
hk_prime : Nat.Prime (Nat.succ k)
hk_two : Nat.succ k = 2
⊢ p 2 |
https://github.com/jvlmdr/from_the_book.git | 9fb6080539a2f32bb24719600a9e7531abf2328d | FromTheBook/Ch02/Bertrand/ProdPrimes.lean | caseStrongRecOnOddPrimes | [48, 1] | [70, 43] | exact two | case ind.inl.inl
p : ℕ → Prop
zero : p 0
two : p 2
not_prime : ∀ (n : ℕ), ¬Nat.Prime (Nat.succ n) → (∀ (x : ℕ), x ≤ n → p x) → p (Nat.succ n)
odd_prime : ∀ (n : ℕ), Nat.Prime (Nat.succ (2 * n)) → (∀ (x : ℕ), x ≤ 2 * n → p x) → p (Nat.succ (2 * n))
k : ℕ
h_ind : ∀ (m : ℕ), m ≤ k → p m
hk_prime : Nat.Prime (Nat.succ k)
hk_two : Nat.succ k = 2
⊢ p 2 | no goals |
https://github.com/jvlmdr/from_the_book.git | 9fb6080539a2f32bb24719600a9e7531abf2328d | FromTheBook/Ch02/Bertrand/ProdPrimes.lean | caseStrongRecOnOddPrimes | [48, 1] | [70, 43] | cases hk_two with
| intro m hm =>
simp [Nat.add_one] at hm
simp [hm] at *
apply odd_prime _ hk_prime h_ind | case ind.inl.inr
p : ℕ → Prop
zero : p 0
two : p 2
not_prime : ∀ (n : ℕ), ¬Nat.Prime (Nat.succ n) → (∀ (x : ℕ), x ≤ n → p x) → p (Nat.succ n)
odd_prime : ∀ (n : ℕ), Nat.Prime (Nat.succ (2 * n)) → (∀ (x : ℕ), x ≤ 2 * n → p x) → p (Nat.succ (2 * n))
k : ℕ
h_ind : ∀ (m : ℕ), m ≤ k → p m
hk_prime : Nat.Prime (Nat.succ k)
hk_two : Odd (Nat.succ k)
⊢ p (Nat.succ k) | no goals |
https://github.com/jvlmdr/from_the_book.git | 9fb6080539a2f32bb24719600a9e7531abf2328d | FromTheBook/Ch02/Bertrand/ProdPrimes.lean | caseStrongRecOnOddPrimes | [48, 1] | [70, 43] | simp [Nat.add_one] at hm | case ind.inl.inr.intro
p : ℕ → Prop
zero : p 0
two : p 2
not_prime : ∀ (n : ℕ), ¬Nat.Prime (Nat.succ n) → (∀ (x : ℕ), x ≤ n → p x) → p (Nat.succ n)
odd_prime : ∀ (n : ℕ), Nat.Prime (Nat.succ (2 * n)) → (∀ (x : ℕ), x ≤ 2 * n → p x) → p (Nat.succ (2 * n))
k : ℕ
h_ind : ∀ (m : ℕ), m ≤ k → p m
hk_prime : Nat.Prime (Nat.succ k)
m : ℕ
hm : Nat.succ k = 2 * m + 1
⊢ p (Nat.succ k) | case ind.inl.inr.intro
p : ℕ → Prop
zero : p 0
two : p 2
not_prime : ∀ (n : ℕ), ¬Nat.Prime (Nat.succ n) → (∀ (x : ℕ), x ≤ n → p x) → p (Nat.succ n)
odd_prime : ∀ (n : ℕ), Nat.Prime (Nat.succ (2 * n)) → (∀ (x : ℕ), x ≤ 2 * n → p x) → p (Nat.succ (2 * n))
k : ℕ
h_ind : ∀ (m : ℕ), m ≤ k → p m
hk_prime : Nat.Prime (Nat.succ k)
m : ℕ
hm : k = 2 * m
⊢ p (Nat.succ k) |
https://github.com/jvlmdr/from_the_book.git | 9fb6080539a2f32bb24719600a9e7531abf2328d | FromTheBook/Ch02/Bertrand/ProdPrimes.lean | caseStrongRecOnOddPrimes | [48, 1] | [70, 43] | simp [hm] at * | case ind.inl.inr.intro
p : ℕ → Prop
zero : p 0
two : p 2
not_prime : ∀ (n : ℕ), ¬Nat.Prime (Nat.succ n) → (∀ (x : ℕ), x ≤ n → p x) → p (Nat.succ n)
odd_prime : ∀ (n : ℕ), Nat.Prime (Nat.succ (2 * n)) → (∀ (x : ℕ), x ≤ 2 * n → p x) → p (Nat.succ (2 * n))
k : ℕ
h_ind : ∀ (m : ℕ), m ≤ k → p m
hk_prime : Nat.Prime (Nat.succ k)
m : ℕ
hm : k = 2 * m
⊢ p (Nat.succ k) | case ind.inl.inr.intro
p : ℕ → Prop
zero : p 0
two : p 2
not_prime : ∀ (n : ℕ), ¬Nat.Prime (Nat.succ n) → (∀ (x : ℕ), x ≤ n → p x) → p (Nat.succ n)
odd_prime : ∀ (n : ℕ), Nat.Prime (Nat.succ (2 * n)) → (∀ (x : ℕ), x ≤ 2 * n → p x) → p (Nat.succ (2 * n))
k m : ℕ
h_ind : ∀ (m_1 : ℕ), m_1 ≤ 2 * m → p m_1
hk_prime : Nat.Prime (Nat.succ (2 * m))
hm : True
⊢ p (Nat.succ (2 * m)) |
https://github.com/jvlmdr/from_the_book.git | 9fb6080539a2f32bb24719600a9e7531abf2328d | FromTheBook/Ch02/Bertrand/ProdPrimes.lean | caseStrongRecOnOddPrimes | [48, 1] | [70, 43] | apply odd_prime _ hk_prime h_ind | case ind.inl.inr.intro
p : ℕ → Prop
zero : p 0
two : p 2
not_prime : ∀ (n : ℕ), ¬Nat.Prime (Nat.succ n) → (∀ (x : ℕ), x ≤ n → p x) → p (Nat.succ n)
odd_prime : ∀ (n : ℕ), Nat.Prime (Nat.succ (2 * n)) → (∀ (x : ℕ), x ≤ 2 * n → p x) → p (Nat.succ (2 * n))
k m : ℕ
h_ind : ∀ (m_1 : ℕ), m_1 ≤ 2 * m → p m_1
hk_prime : Nat.Prime (Nat.succ (2 * m))
hm : True
⊢ p (Nat.succ (2 * m)) | no goals |
https://github.com/jvlmdr/from_the_book.git | 9fb6080539a2f32bb24719600a9e7531abf2328d | FromTheBook/Ch02/Bertrand/ProdPrimes.lean | prodPrimesLePowFour | [77, 1] | [104, 32] | induction n using caseStrongRecOnOddPrimes with
| zero => simp
| two => simp
| not_prime n hp hi =>
specialize hi n
simp at hi
cases n with
| zero => simp
| succ n =>
simp only [Nat.pred_succ] at *
simp [prodPrimes]
rw [primesSuccEqPrimesSelf hp, Nat.add_one]
rw [← prodPrimes, Nat.add_one]
apply le_trans hi
simp [pow_succ]
| odd_prime n hp hi =>
simp
conv => rhs; rw [two_mul, pow_add]
specialize hi n.succ _
. cases n with
| zero => contradiction | succ n => linarith
simp at hi
rw [prodPrimesSplit (by linarith : n.succ ≤ (2*n).succ)]
apply Nat.mul_le_mul hi
apply le_trans _ chooseHalfLePowFour
exact prodPrimesIocLeChoose | n : ℕ
⊢ prodPrimes n ≤ 4 ^ Nat.pred n | no goals |
https://github.com/jvlmdr/from_the_book.git | 9fb6080539a2f32bb24719600a9e7531abf2328d | FromTheBook/Ch02/Bertrand/ProdPrimes.lean | prodPrimesLePowFour | [77, 1] | [104, 32] | simp | case zero
⊢ prodPrimes 0 ≤ 4 ^ Nat.pred 0 | no goals |
https://github.com/jvlmdr/from_the_book.git | 9fb6080539a2f32bb24719600a9e7531abf2328d | FromTheBook/Ch02/Bertrand/ProdPrimes.lean | prodPrimesLePowFour | [77, 1] | [104, 32] | simp | case two
⊢ prodPrimes 2 ≤ 4 ^ Nat.pred 2 | no goals |
https://github.com/jvlmdr/from_the_book.git | 9fb6080539a2f32bb24719600a9e7531abf2328d | FromTheBook/Ch02/Bertrand/ProdPrimes.lean | prodPrimesLePowFour | [77, 1] | [104, 32] | specialize hi n | case not_prime
n : ℕ
hp : ¬Nat.Prime (Nat.succ n)
hi : ∀ (x : ℕ), x ≤ n → prodPrimes x ≤ 4 ^ Nat.pred x
⊢ prodPrimes (Nat.succ n) ≤ 4 ^ Nat.pred (Nat.succ n) | case not_prime
n : ℕ
hp : ¬Nat.Prime (Nat.succ n)
hi : n ≤ n → prodPrimes n ≤ 4 ^ Nat.pred n
⊢ prodPrimes (Nat.succ n) ≤ 4 ^ Nat.pred (Nat.succ n) |
https://github.com/jvlmdr/from_the_book.git | 9fb6080539a2f32bb24719600a9e7531abf2328d | FromTheBook/Ch02/Bertrand/ProdPrimes.lean | prodPrimesLePowFour | [77, 1] | [104, 32] | simp at hi | case not_prime
n : ℕ
hp : ¬Nat.Prime (Nat.succ n)
hi : n ≤ n → prodPrimes n ≤ 4 ^ Nat.pred n
⊢ prodPrimes (Nat.succ n) ≤ 4 ^ Nat.pred (Nat.succ n) | case not_prime
n : ℕ
hp : ¬Nat.Prime (Nat.succ n)
hi : prodPrimes n ≤ 4 ^ Nat.pred n
⊢ prodPrimes (Nat.succ n) ≤ 4 ^ Nat.pred (Nat.succ n) |
https://github.com/jvlmdr/from_the_book.git | 9fb6080539a2f32bb24719600a9e7531abf2328d | FromTheBook/Ch02/Bertrand/ProdPrimes.lean | prodPrimesLePowFour | [77, 1] | [104, 32] | cases n with
| zero => simp
| succ n =>
simp only [Nat.pred_succ] at *
simp [prodPrimes]
rw [primesSuccEqPrimesSelf hp, Nat.add_one]
rw [← prodPrimes, Nat.add_one]
apply le_trans hi
simp [pow_succ] | case not_prime
n : ℕ
hp : ¬Nat.Prime (Nat.succ n)
hi : prodPrimes n ≤ 4 ^ Nat.pred n
⊢ prodPrimes (Nat.succ n) ≤ 4 ^ Nat.pred (Nat.succ n) | no goals |
https://github.com/jvlmdr/from_the_book.git | 9fb6080539a2f32bb24719600a9e7531abf2328d | FromTheBook/Ch02/Bertrand/ProdPrimes.lean | prodPrimesLePowFour | [77, 1] | [104, 32] | simp | case not_prime.zero
hp : ¬Nat.Prime (Nat.succ Nat.zero)
hi : prodPrimes Nat.zero ≤ 4 ^ Nat.pred Nat.zero
⊢ prodPrimes (Nat.succ Nat.zero) ≤ 4 ^ Nat.pred (Nat.succ Nat.zero) | no goals |
https://github.com/jvlmdr/from_the_book.git | 9fb6080539a2f32bb24719600a9e7531abf2328d | FromTheBook/Ch02/Bertrand/ProdPrimes.lean | prodPrimesLePowFour | [77, 1] | [104, 32] | simp only [Nat.pred_succ] at * | case not_prime.succ
n : ℕ
hp : ¬Nat.Prime (Nat.succ (Nat.succ n))
hi : prodPrimes (Nat.succ n) ≤ 4 ^ Nat.pred (Nat.succ n)
⊢ prodPrimes (Nat.succ (Nat.succ n)) ≤ 4 ^ Nat.pred (Nat.succ (Nat.succ n)) | case not_prime.succ
n : ℕ
hp : ¬Nat.Prime (Nat.succ (Nat.succ n))
hi : prodPrimes (Nat.succ n) ≤ 4 ^ n
⊢ prodPrimes (Nat.succ (Nat.succ n)) ≤ 4 ^ (n + 1) |
https://github.com/jvlmdr/from_the_book.git | 9fb6080539a2f32bb24719600a9e7531abf2328d | FromTheBook/Ch02/Bertrand/ProdPrimes.lean | prodPrimesLePowFour | [77, 1] | [104, 32] | simp [prodPrimes] | case not_prime.succ
n : ℕ
hp : ¬Nat.Prime (Nat.succ (Nat.succ n))
hi : prodPrimes (Nat.succ n) ≤ 4 ^ n
⊢ prodPrimes (Nat.succ (Nat.succ n)) ≤ 4 ^ (n + 1) | case not_prime.succ
n : ℕ
hp : ¬Nat.Prime (Nat.succ (Nat.succ n))
hi : prodPrimes (Nat.succ n) ≤ 4 ^ n
⊢ ∏ p in primes (Finset.range (Nat.succ (Nat.succ (Nat.succ n)))), p ≤ 4 ^ (n + 1) |
https://github.com/jvlmdr/from_the_book.git | 9fb6080539a2f32bb24719600a9e7531abf2328d | FromTheBook/Ch02/Bertrand/ProdPrimes.lean | prodPrimesLePowFour | [77, 1] | [104, 32] | rw [primesSuccEqPrimesSelf hp, Nat.add_one] | case not_prime.succ
n : ℕ
hp : ¬Nat.Prime (Nat.succ (Nat.succ n))
hi : prodPrimes (Nat.succ n) ≤ 4 ^ n
⊢ ∏ p in primes (Finset.range (Nat.succ (Nat.succ (Nat.succ n)))), p ≤ 4 ^ (n + 1) | case not_prime.succ
n : ℕ
hp : ¬Nat.Prime (Nat.succ (Nat.succ n))
hi : prodPrimes (Nat.succ n) ≤ 4 ^ n
⊢ ∏ p in primes (Finset.range (Nat.succ (Nat.succ n))), p ≤ 4 ^ Nat.succ n |
https://github.com/jvlmdr/from_the_book.git | 9fb6080539a2f32bb24719600a9e7531abf2328d | FromTheBook/Ch02/Bertrand/ProdPrimes.lean | prodPrimesLePowFour | [77, 1] | [104, 32] | rw [← prodPrimes, Nat.add_one] | case not_prime.succ
n : ℕ
hp : ¬Nat.Prime (Nat.succ (Nat.succ n))
hi : prodPrimes (Nat.succ n) ≤ 4 ^ n
⊢ ∏ p in primes (Finset.range (Nat.succ (Nat.succ n))), p ≤ 4 ^ Nat.succ n | case not_prime.succ
n : ℕ
hp : ¬Nat.Prime (Nat.succ (Nat.succ n))
hi : prodPrimes (Nat.succ n) ≤ 4 ^ n
⊢ prodPrimes (Nat.succ n) ≤ 4 ^ Nat.succ n |
https://github.com/jvlmdr/from_the_book.git | 9fb6080539a2f32bb24719600a9e7531abf2328d | FromTheBook/Ch02/Bertrand/ProdPrimes.lean | prodPrimesLePowFour | [77, 1] | [104, 32] | apply le_trans hi | case not_prime.succ
n : ℕ
hp : ¬Nat.Prime (Nat.succ (Nat.succ n))
hi : prodPrimes (Nat.succ n) ≤ 4 ^ n
⊢ prodPrimes (Nat.succ n) ≤ 4 ^ Nat.succ n | case not_prime.succ
n : ℕ
hp : ¬Nat.Prime (Nat.succ (Nat.succ n))
hi : prodPrimes (Nat.succ n) ≤ 4 ^ n
⊢ 4 ^ n ≤ 4 ^ Nat.succ n |
https://github.com/jvlmdr/from_the_book.git | 9fb6080539a2f32bb24719600a9e7531abf2328d | FromTheBook/Ch02/Bertrand/ProdPrimes.lean | prodPrimesLePowFour | [77, 1] | [104, 32] | simp [pow_succ] | case not_prime.succ
n : ℕ
hp : ¬Nat.Prime (Nat.succ (Nat.succ n))
hi : prodPrimes (Nat.succ n) ≤ 4 ^ n
⊢ 4 ^ n ≤ 4 ^ Nat.succ n | no goals |
https://github.com/jvlmdr/from_the_book.git | 9fb6080539a2f32bb24719600a9e7531abf2328d | FromTheBook/Ch02/Bertrand/ProdPrimes.lean | prodPrimesLePowFour | [77, 1] | [104, 32] | simp | case odd_prime
n : ℕ
hp : Nat.Prime (Nat.succ (2 * n))
hi : ∀ (x : ℕ), x ≤ 2 * n → prodPrimes x ≤ 4 ^ Nat.pred x
⊢ prodPrimes (Nat.succ (2 * n)) ≤ 4 ^ Nat.pred (Nat.succ (2 * n)) | case odd_prime
n : ℕ
hp : Nat.Prime (Nat.succ (2 * n))
hi : ∀ (x : ℕ), x ≤ 2 * n → prodPrimes x ≤ 4 ^ Nat.pred x
⊢ prodPrimes (Nat.succ (2 * n)) ≤ 4 ^ (2 * n) |
https://github.com/jvlmdr/from_the_book.git | 9fb6080539a2f32bb24719600a9e7531abf2328d | FromTheBook/Ch02/Bertrand/ProdPrimes.lean | prodPrimesLePowFour | [77, 1] | [104, 32] | conv => rhs; rw [two_mul, pow_add] | case odd_prime
n : ℕ
hp : Nat.Prime (Nat.succ (2 * n))
hi : ∀ (x : ℕ), x ≤ 2 * n → prodPrimes x ≤ 4 ^ Nat.pred x
⊢ prodPrimes (Nat.succ (2 * n)) ≤ 4 ^ (2 * n) | case odd_prime
n : ℕ
hp : Nat.Prime (Nat.succ (2 * n))
hi : ∀ (x : ℕ), x ≤ 2 * n → prodPrimes x ≤ 4 ^ Nat.pred x
⊢ prodPrimes (Nat.succ (2 * n)) ≤ 4 ^ n * 4 ^ n |
https://github.com/jvlmdr/from_the_book.git | 9fb6080539a2f32bb24719600a9e7531abf2328d | FromTheBook/Ch02/Bertrand/ProdPrimes.lean | prodPrimesLePowFour | [77, 1] | [104, 32] | specialize hi n.succ _ | case odd_prime
n : ℕ
hp : Nat.Prime (Nat.succ (2 * n))
hi : ∀ (x : ℕ), x ≤ 2 * n → prodPrimes x ≤ 4 ^ Nat.pred x
⊢ prodPrimes (Nat.succ (2 * n)) ≤ 4 ^ n * 4 ^ n | case odd_prime
n : ℕ
hp : Nat.Prime (Nat.succ (2 * n))
hi : ∀ (x : ℕ), x ≤ 2 * n → prodPrimes x ≤ 4 ^ Nat.pred x
⊢ Nat.succ n ≤ 2 * n
case odd_prime
n : ℕ
hp : Nat.Prime (Nat.succ (2 * n))
hi : prodPrimes (Nat.succ n) ≤ 4 ^ Nat.pred (Nat.succ n)
⊢ prodPrimes (Nat.succ (2 * n)) ≤ 4 ^ n * 4 ^ n |
https://github.com/jvlmdr/from_the_book.git | 9fb6080539a2f32bb24719600a9e7531abf2328d | FromTheBook/Ch02/Bertrand/ProdPrimes.lean | prodPrimesLePowFour | [77, 1] | [104, 32] | . cases n with
| zero => contradiction | succ n => linarith | case odd_prime
n : ℕ
hp : Nat.Prime (Nat.succ (2 * n))
hi : ∀ (x : ℕ), x ≤ 2 * n → prodPrimes x ≤ 4 ^ Nat.pred x
⊢ Nat.succ n ≤ 2 * n
case odd_prime
n : ℕ
hp : Nat.Prime (Nat.succ (2 * n))
hi : prodPrimes (Nat.succ n) ≤ 4 ^ Nat.pred (Nat.succ n)
⊢ prodPrimes (Nat.succ (2 * n)) ≤ 4 ^ n * 4 ^ n | case odd_prime
n : ℕ
hp : Nat.Prime (Nat.succ (2 * n))
hi : prodPrimes (Nat.succ n) ≤ 4 ^ Nat.pred (Nat.succ n)
⊢ prodPrimes (Nat.succ (2 * n)) ≤ 4 ^ n * 4 ^ n |
https://github.com/jvlmdr/from_the_book.git | 9fb6080539a2f32bb24719600a9e7531abf2328d | FromTheBook/Ch02/Bertrand/ProdPrimes.lean | prodPrimesLePowFour | [77, 1] | [104, 32] | simp at hi | case odd_prime
n : ℕ
hp : Nat.Prime (Nat.succ (2 * n))
hi : prodPrimes (Nat.succ n) ≤ 4 ^ Nat.pred (Nat.succ n)
⊢ prodPrimes (Nat.succ (2 * n)) ≤ 4 ^ n * 4 ^ n | case odd_prime
n : ℕ
hp : Nat.Prime (Nat.succ (2 * n))
hi : prodPrimes (Nat.succ n) ≤ 4 ^ n
⊢ prodPrimes (Nat.succ (2 * n)) ≤ 4 ^ n * 4 ^ n |
https://github.com/jvlmdr/from_the_book.git | 9fb6080539a2f32bb24719600a9e7531abf2328d | FromTheBook/Ch02/Bertrand/ProdPrimes.lean | prodPrimesLePowFour | [77, 1] | [104, 32] | rw [prodPrimesSplit (by linarith : n.succ ≤ (2*n).succ)] | case odd_prime
n : ℕ
hp : Nat.Prime (Nat.succ (2 * n))
hi : prodPrimes (Nat.succ n) ≤ 4 ^ n
⊢ prodPrimes (Nat.succ (2 * n)) ≤ 4 ^ n * 4 ^ n | case odd_prime
n : ℕ
hp : Nat.Prime (Nat.succ (2 * n))
hi : prodPrimes (Nat.succ n) ≤ 4 ^ n
⊢ prodPrimes (Nat.succ n) * ∏ p in primes (Finset.Ioc (Nat.succ n) (Nat.succ (2 * n))), p ≤ 4 ^ n * 4 ^ n |
https://github.com/jvlmdr/from_the_book.git | 9fb6080539a2f32bb24719600a9e7531abf2328d | FromTheBook/Ch02/Bertrand/ProdPrimes.lean | prodPrimesLePowFour | [77, 1] | [104, 32] | apply Nat.mul_le_mul hi | case odd_prime
n : ℕ
hp : Nat.Prime (Nat.succ (2 * n))
hi : prodPrimes (Nat.succ n) ≤ 4 ^ n
⊢ prodPrimes (Nat.succ n) * ∏ p in primes (Finset.Ioc (Nat.succ n) (Nat.succ (2 * n))), p ≤ 4 ^ n * 4 ^ n | case odd_prime
n : ℕ
hp : Nat.Prime (Nat.succ (2 * n))
hi : prodPrimes (Nat.succ n) ≤ 4 ^ n
⊢ ∏ p in primes (Finset.Ioc (Nat.succ n) (Nat.succ (2 * n))), p ≤ 4 ^ n |
https://github.com/jvlmdr/from_the_book.git | 9fb6080539a2f32bb24719600a9e7531abf2328d | FromTheBook/Ch02/Bertrand/ProdPrimes.lean | prodPrimesLePowFour | [77, 1] | [104, 32] | apply le_trans _ chooseHalfLePowFour | case odd_prime
n : ℕ
hp : Nat.Prime (Nat.succ (2 * n))
hi : prodPrimes (Nat.succ n) ≤ 4 ^ n
⊢ ∏ p in primes (Finset.Ioc (Nat.succ n) (Nat.succ (2 * n))), p ≤ 4 ^ n | n : ℕ
hp : Nat.Prime (Nat.succ (2 * n))
hi : prodPrimes (Nat.succ n) ≤ 4 ^ n
⊢ ∏ p in primes (Finset.Ioc (Nat.succ n) (Nat.succ (2 * n))), p ≤ Nat.choose (2 * n + 1) n |
https://github.com/jvlmdr/from_the_book.git | 9fb6080539a2f32bb24719600a9e7531abf2328d | FromTheBook/Ch02/Bertrand/ProdPrimes.lean | prodPrimesLePowFour | [77, 1] | [104, 32] | exact prodPrimesIocLeChoose | n : ℕ
hp : Nat.Prime (Nat.succ (2 * n))
hi : prodPrimes (Nat.succ n) ≤ 4 ^ n
⊢ ∏ p in primes (Finset.Ioc (Nat.succ n) (Nat.succ (2 * n))), p ≤ Nat.choose (2 * n + 1) n | no goals |
https://github.com/jvlmdr/from_the_book.git | 9fb6080539a2f32bb24719600a9e7531abf2328d | FromTheBook/Ch02/Bertrand/ProdPrimes.lean | prodPrimesLePowFour | [77, 1] | [104, 32] | cases n with
| zero => contradiction | succ n => linarith | case odd_prime
n : ℕ
hp : Nat.Prime (Nat.succ (2 * n))
hi : ∀ (x : ℕ), x ≤ 2 * n → prodPrimes x ≤ 4 ^ Nat.pred x
⊢ Nat.succ n ≤ 2 * n | no goals |
https://github.com/jvlmdr/from_the_book.git | 9fb6080539a2f32bb24719600a9e7531abf2328d | FromTheBook/Ch02/Bertrand/ProdPrimes.lean | prodPrimesLePowFour | [77, 1] | [104, 32] | contradiction | case odd_prime.zero
hp : Nat.Prime (Nat.succ (2 * Nat.zero))
hi : ∀ (x : ℕ), x ≤ 2 * Nat.zero → prodPrimes x ≤ 4 ^ Nat.pred x
⊢ Nat.succ Nat.zero ≤ 2 * Nat.zero | no goals |
https://github.com/jvlmdr/from_the_book.git | 9fb6080539a2f32bb24719600a9e7531abf2328d | FromTheBook/Ch02/Bertrand/ProdPrimes.lean | prodPrimesLePowFour | [77, 1] | [104, 32] | linarith | case odd_prime.succ
n : ℕ
hp : Nat.Prime (Nat.succ (2 * Nat.succ n))
hi : ∀ (x : ℕ), x ≤ 2 * Nat.succ n → prodPrimes x ≤ 4 ^ Nat.pred x
⊢ Nat.succ (Nat.succ n) ≤ 2 * Nat.succ n | no goals |
https://github.com/jvlmdr/from_the_book.git | 9fb6080539a2f32bb24719600a9e7531abf2328d | FromTheBook/Ch02/Bertrand/ProdPrimes.lean | prodPrimesLePowFour | [77, 1] | [104, 32] | linarith | n : ℕ
hp : Nat.Prime (Nat.succ (2 * n))
hi : prodPrimes (Nat.succ n) ≤ 4 ^ n
⊢ Nat.succ n ≤ Nat.succ (2 * n) | no goals |
https://github.com/jvlmdr/from_the_book.git | 9fb6080539a2f32bb24719600a9e7531abf2328d | FromTheBook/Ch02/Bertrand/ProdPrimes.lean | prodPrimesRealLePowFour | [106, 1] | [116, 15] | norm_cast | x : ℝ
hx : 1 ≤ x
⊢ ↑(prodPrimes ⌊x⌋₊) ≤ ↑(4 ^ (⌊x⌋₊ - 1)) | x : ℝ
hx : 1 ≤ x
⊢ prodPrimes ⌊x⌋₊ ≤ 4 ^ (⌊x⌋₊ - 1) |
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