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https://github.com/jvlmdr/from_the_book.git | 9fb6080539a2f32bb24719600a9e7531abf2328d | FromTheBook/Ch02/Bertrand/CentralBinom.lean | FactorizationCentralBinom.leOne | [47, 1] | [59, 13] | have hp := Nat.Prime.one_lt hp | n p : ℕ
hp : Nat.Prime p
hnp✝ : 2 * n < p ^ 2
hnp : 2 * n ≤ p ^ 2 - 1
⊢ Nat.log p (p ^ 2 - 1) < Nat.succ 1 | n p : ℕ
hp✝ : Nat.Prime p
hnp✝ : 2 * n < p ^ 2
hnp : 2 * n ≤ p ^ 2 - 1
hp : 1 < p
⊢ Nat.log p (p ^ 2 - 1) < Nat.succ 1 |
https://github.com/jvlmdr/from_the_book.git | 9fb6080539a2f32bb24719600a9e7531abf2328d | FromTheBook/Ch02/Bertrand/CentralBinom.lean | FactorizationCentralBinom.leOne | [47, 1] | [59, 13] | apply Nat.log_lt_of_lt_pow | n p : ℕ
hp✝ : Nat.Prime p
hnp✝ : 2 * n < p ^ 2
hnp : 2 * n ≤ p ^ 2 - 1
hp : 1 < p
⊢ Nat.log p (p ^ 2 - 1) < Nat.succ 1 | case hy
n p : ℕ
hp✝ : Nat.Prime p
hnp✝ : 2 * n < p ^ 2
hnp : 2 * n ≤ p ^ 2 - 1
hp : 1 < p
⊢ p ^ 2 - 1 ≠ 0
case a
n p : ℕ
hp✝ : Nat.Prime p
hnp✝ : 2 * n < p ^ 2
hnp : 2 * n ≤ p ^ 2 - 1
hp : 1 < p
⊢ p ^ 2 - 1 < p ^ Nat.succ 1 |
https://github.com/jvlmdr/from_the_book.git | 9fb6080539a2f32bb24719600a9e7531abf2328d | FromTheBook/Ch02/Bertrand/CentralBinom.lean | FactorizationCentralBinom.leOne | [47, 1] | [59, 13] | . simp [hp] | case hy
n p : ℕ
hp✝ : Nat.Prime p
hnp✝ : 2 * n < p ^ 2
hnp : 2 * n ≤ p ^ 2 - 1
hp : 1 < p
⊢ p ^ 2 - 1 ≠ 0
case a
n p : ℕ
hp✝ : Nat.Prime p
hnp✝ : 2 * n < p ^ 2
hnp : 2 * n ≤ p ^ 2 - 1
hp : 1 < p
⊢ p ^ 2 - 1 < p ^ Nat.succ 1 | case a
n p : ℕ
hp✝ : Nat.Prime p
hnp✝ : 2 * n < p ^ 2
hnp : 2 * n ≤ p ^ 2 - 1
hp : 1 < p
⊢ p ^ 2 - 1 < p ^ Nat.succ 1 |
https://github.com/jvlmdr/from_the_book.git | 9fb6080539a2f32bb24719600a9e7531abf2328d | FromTheBook/Ch02/Bertrand/CentralBinom.lean | FactorizationCentralBinom.leOne | [47, 1] | [59, 13] | . apply Nat.pred_lt
simp [hp]
linarith | case a
n p : ℕ
hp✝ : Nat.Prime p
hnp✝ : 2 * n < p ^ 2
hnp : 2 * n ≤ p ^ 2 - 1
hp : 1 < p
⊢ p ^ 2 - 1 < p ^ Nat.succ 1 | no goals |
https://github.com/jvlmdr/from_the_book.git | 9fb6080539a2f32bb24719600a9e7531abf2328d | FromTheBook/Ch02/Bertrand/CentralBinom.lean | FactorizationCentralBinom.leOne | [47, 1] | [59, 13] | simp [hp] | case hy
n p : ℕ
hp✝ : Nat.Prime p
hnp✝ : 2 * n < p ^ 2
hnp : 2 * n ≤ p ^ 2 - 1
hp : 1 < p
⊢ p ^ 2 - 1 ≠ 0 | no goals |
https://github.com/jvlmdr/from_the_book.git | 9fb6080539a2f32bb24719600a9e7531abf2328d | FromTheBook/Ch02/Bertrand/CentralBinom.lean | FactorizationCentralBinom.leOne | [47, 1] | [59, 13] | apply Nat.pred_lt | case a
n p : ℕ
hp✝ : Nat.Prime p
hnp✝ : 2 * n < p ^ 2
hnp : 2 * n ≤ p ^ 2 - 1
hp : 1 < p
⊢ p ^ 2 - 1 < p ^ Nat.succ 1 | case a.a
n p : ℕ
hp✝ : Nat.Prime p
hnp✝ : 2 * n < p ^ 2
hnp : 2 * n ≤ p ^ 2 - 1
hp : 1 < p
⊢ Nat.sub (p ^ 2) 0 ≠ 0 |
https://github.com/jvlmdr/from_the_book.git | 9fb6080539a2f32bb24719600a9e7531abf2328d | FromTheBook/Ch02/Bertrand/CentralBinom.lean | FactorizationCentralBinom.leOne | [47, 1] | [59, 13] | simp [hp] | case a.a
n p : ℕ
hp✝ : Nat.Prime p
hnp✝ : 2 * n < p ^ 2
hnp : 2 * n ≤ p ^ 2 - 1
hp : 1 < p
⊢ Nat.sub (p ^ 2) 0 ≠ 0 | case a.a
n p : ℕ
hp✝ : Nat.Prime p
hnp✝ : 2 * n < p ^ 2
hnp : 2 * n ≤ p ^ 2 - 1
hp : 1 < p
⊢ ¬p = 0 |
https://github.com/jvlmdr/from_the_book.git | 9fb6080539a2f32bb24719600a9e7531abf2328d | FromTheBook/Ch02/Bertrand/CentralBinom.lean | FactorizationCentralBinom.leOne | [47, 1] | [59, 13] | linarith | case a.a
n p : ℕ
hp✝ : Nat.Prime p
hnp✝ : 2 * n < p ^ 2
hnp : 2 * n ≤ p ^ 2 - 1
hp : 1 < p
⊢ ¬p = 0 | no goals |
https://github.com/jvlmdr/from_the_book.git | 9fb6080539a2f32bb24719600a9e7531abf2328d | FromTheBook/Ch02/Bertrand/CentralBinom.lean | FactorizationCentralBinom.prodIcoPowLePow | [62, 1] | [67, 11] | apply Finset.prod_le_pow_card | n : ℕ
s : Finset ℕ
hn : 0 < n
⊢ ∏ p in s, p ^ ↑(Nat.factorization (Nat.centralBinom n)) p ≤ (2 * n) ^ Finset.card s | case h
n : ℕ
s : Finset ℕ
hn : 0 < n
⊢ ∀ (x : ℕ), x ∈ s → x ^ ↑(Nat.factorization (Nat.centralBinom n)) x ≤ 2 * n |
https://github.com/jvlmdr/from_the_book.git | 9fb6080539a2f32bb24719600a9e7531abf2328d | FromTheBook/Ch02/Bertrand/CentralBinom.lean | FactorizationCentralBinom.prodIcoPowLePow | [62, 1] | [67, 11] | intro x _ | case h
n : ℕ
s : Finset ℕ
hn : 0 < n
⊢ ∀ (x : ℕ), x ∈ s → x ^ ↑(Nat.factorization (Nat.centralBinom n)) x ≤ 2 * n | case h
n : ℕ
s : Finset ℕ
hn : 0 < n
x : ℕ
a✝ : x ∈ s
⊢ x ^ ↑(Nat.factorization (Nat.centralBinom n)) x ≤ 2 * n |
https://github.com/jvlmdr/from_the_book.git | 9fb6080539a2f32bb24719600a9e7531abf2328d | FromTheBook/Ch02/Bertrand/CentralBinom.lean | FactorizationCentralBinom.prodIcoPowLePow | [62, 1] | [67, 11] | apply Nat.pow_factorization_choose_le | case h
n : ℕ
s : Finset ℕ
hn : 0 < n
x : ℕ
a✝ : x ∈ s
⊢ x ^ ↑(Nat.factorization (Nat.centralBinom n)) x ≤ 2 * n | case h.hn
n : ℕ
s : Finset ℕ
hn : 0 < n
x : ℕ
a✝ : x ∈ s
⊢ 0 < 2 * n |
https://github.com/jvlmdr/from_the_book.git | 9fb6080539a2f32bb24719600a9e7531abf2328d | FromTheBook/Ch02/Bertrand/CentralBinom.lean | FactorizationCentralBinom.prodIcoPowLePow | [62, 1] | [67, 11] | linarith | case h.hn
n : ℕ
s : Finset ℕ
hn : 0 < n
x : ℕ
a✝ : x ∈ s
⊢ 0 < 2 * n | no goals |
https://github.com/jvlmdr/from_the_book.git | 9fb6080539a2f32bb24719600a9e7531abf2328d | FromTheBook/Ch02/Bertrand/CentralBinom.lean | FactorizationCentralBinom.prodIcoPowLePow' | [70, 1] | [76, 58] | apply Finset.prod_le_pow_card | n : ℕ
s : Finset ℕ
hs : n = 0 → s = ∅
⊢ ∏ p in s, p ^ ↑(Nat.factorization (Nat.centralBinom n)) p ≤ (2 * n) ^ Finset.card s | case h
n : ℕ
s : Finset ℕ
hs : n = 0 → s = ∅
⊢ ∀ (x : ℕ), x ∈ s → x ^ ↑(Nat.factorization (Nat.centralBinom n)) x ≤ 2 * n |
https://github.com/jvlmdr/from_the_book.git | 9fb6080539a2f32bb24719600a9e7531abf2328d | FromTheBook/Ch02/Bertrand/CentralBinom.lean | FactorizationCentralBinom.prodIcoPowLePow' | [70, 1] | [76, 58] | intro x hx | case h
n : ℕ
s : Finset ℕ
hs : n = 0 → s = ∅
⊢ ∀ (x : ℕ), x ∈ s → x ^ ↑(Nat.factorization (Nat.centralBinom n)) x ≤ 2 * n | case h
n : ℕ
s : Finset ℕ
hs : n = 0 → s = ∅
x : ℕ
hx : x ∈ s
⊢ x ^ ↑(Nat.factorization (Nat.centralBinom n)) x ≤ 2 * n |
https://github.com/jvlmdr/from_the_book.git | 9fb6080539a2f32bb24719600a9e7531abf2328d | FromTheBook/Ch02/Bertrand/CentralBinom.lean | FactorizationCentralBinom.prodIcoPowLePow' | [70, 1] | [76, 58] | cases n with
| zero => simp at hs; rw [hs] at hx; contradiction
| succ n => apply Nat.pow_factorization_choose_le; simp | case h
n : ℕ
s : Finset ℕ
hs : n = 0 → s = ∅
x : ℕ
hx : x ∈ s
⊢ x ^ ↑(Nat.factorization (Nat.centralBinom n)) x ≤ 2 * n | no goals |
https://github.com/jvlmdr/from_the_book.git | 9fb6080539a2f32bb24719600a9e7531abf2328d | FromTheBook/Ch02/Bertrand/CentralBinom.lean | FactorizationCentralBinom.prodIcoPowLePow' | [70, 1] | [76, 58] | simp at hs | case h.zero
s : Finset ℕ
x : ℕ
hx : x ∈ s
hs : Nat.zero = 0 → s = ∅
⊢ x ^ ↑(Nat.factorization (Nat.centralBinom Nat.zero)) x ≤ 2 * Nat.zero | case h.zero
s : Finset ℕ
x : ℕ
hx : x ∈ s
hs : s = ∅
⊢ x ^ ↑(Nat.factorization (Nat.centralBinom Nat.zero)) x ≤ 2 * Nat.zero |
https://github.com/jvlmdr/from_the_book.git | 9fb6080539a2f32bb24719600a9e7531abf2328d | FromTheBook/Ch02/Bertrand/CentralBinom.lean | FactorizationCentralBinom.prodIcoPowLePow' | [70, 1] | [76, 58] | rw [hs] at hx | case h.zero
s : Finset ℕ
x : ℕ
hx : x ∈ s
hs : s = ∅
⊢ x ^ ↑(Nat.factorization (Nat.centralBinom Nat.zero)) x ≤ 2 * Nat.zero | case h.zero
s : Finset ℕ
x : ℕ
hx : x ∈ ∅
hs : s = ∅
⊢ x ^ ↑(Nat.factorization (Nat.centralBinom Nat.zero)) x ≤ 2 * Nat.zero |
https://github.com/jvlmdr/from_the_book.git | 9fb6080539a2f32bb24719600a9e7531abf2328d | FromTheBook/Ch02/Bertrand/CentralBinom.lean | FactorizationCentralBinom.prodIcoPowLePow' | [70, 1] | [76, 58] | contradiction | case h.zero
s : Finset ℕ
x : ℕ
hx : x ∈ ∅
hs : s = ∅
⊢ x ^ ↑(Nat.factorization (Nat.centralBinom Nat.zero)) x ≤ 2 * Nat.zero | no goals |
https://github.com/jvlmdr/from_the_book.git | 9fb6080539a2f32bb24719600a9e7531abf2328d | FromTheBook/Ch02/Bertrand/CentralBinom.lean | FactorizationCentralBinom.prodIcoPowLePow' | [70, 1] | [76, 58] | apply Nat.pow_factorization_choose_le | case h.succ
s : Finset ℕ
x : ℕ
hx : x ∈ s
n : ℕ
hs : Nat.succ n = 0 → s = ∅
⊢ x ^ ↑(Nat.factorization (Nat.centralBinom (Nat.succ n))) x ≤ 2 * Nat.succ n | case h.succ.hn
s : Finset ℕ
x : ℕ
hx : x ∈ s
n : ℕ
hs : Nat.succ n = 0 → s = ∅
⊢ 0 < 2 * Nat.succ n |
https://github.com/jvlmdr/from_the_book.git | 9fb6080539a2f32bb24719600a9e7531abf2328d | FromTheBook/Ch02/Bertrand/CentralBinom.lean | FactorizationCentralBinom.prodIcoPowLePow' | [70, 1] | [76, 58] | simp | case h.succ.hn
s : Finset ℕ
x : ℕ
hx : x ∈ s
n : ℕ
hs : Nat.succ n = 0 → s = ∅
⊢ 0 < 2 * Nat.succ n | no goals |
https://github.com/jvlmdr/from_the_book.git | 9fb6080539a2f32bb24719600a9e7531abf2328d | FromTheBook/Ch02/Bertrand/CentralBinom.lean | FactorizationCentralBinom.prodRangePowEqProdIcoPowOfLeTwo | [79, 1] | [90, 11] | rw [Finset.range_eq_Ico] | n a b : ℕ
ha : a ≤ 2
⊢ ∏ p in Finset.range b, p ^ ↑(Nat.factorization (Nat.centralBinom n)) p =
∏ p in Finset.Ico a b, p ^ ↑(Nat.factorization (Nat.centralBinom n)) p | n a b : ℕ
ha : a ≤ 2
⊢ ∏ p in Finset.Ico 0 b, p ^ ↑(Nat.factorization (Nat.centralBinom n)) p =
∏ p in Finset.Ico a b, p ^ ↑(Nat.factorization (Nat.centralBinom n)) p |
https://github.com/jvlmdr/from_the_book.git | 9fb6080539a2f32bb24719600a9e7531abf2328d | FromTheBook/Ch02/Bertrand/CentralBinom.lean | FactorizationCentralBinom.prodRangePowEqProdIcoPowOfLeTwo | [79, 1] | [90, 11] | rw [← Finset.prod_subset (Finset.Ico_subset_Ico_left (Nat.zero_le a)) _] | n a b : ℕ
ha : a ≤ 2
⊢ ∏ p in Finset.Ico 0 b, p ^ ↑(Nat.factorization (Nat.centralBinom n)) p =
∏ p in Finset.Ico a b, p ^ ↑(Nat.factorization (Nat.centralBinom n)) p | n a b : ℕ
ha : a ≤ 2
⊢ ∀ (x : ℕ), x ∈ Finset.Ico 0 b → ¬x ∈ Finset.Ico a b → x ^ ↑(Nat.factorization (Nat.centralBinom n)) x = 1 |
https://github.com/jvlmdr/from_the_book.git | 9fb6080539a2f32bb24719600a9e7531abf2328d | FromTheBook/Ch02/Bertrand/CentralBinom.lean | FactorizationCentralBinom.prodRangePowEqProdIcoPowOfLeTwo | [79, 1] | [90, 11] | simp | n a b : ℕ
ha : a ≤ 2
⊢ ∀ (x : ℕ), x ∈ Finset.Ico 0 b → ¬x ∈ Finset.Ico a b → x ^ ↑(Nat.factorization (Nat.centralBinom n)) x = 1 | n a b : ℕ
ha : a ≤ 2
⊢ ∀ (x : ℕ), x < b → (a ≤ x → b ≤ x) → x ^ ↑(Nat.factorization (Nat.centralBinom n)) x = 1 |
https://github.com/jvlmdr/from_the_book.git | 9fb6080539a2f32bb24719600a9e7531abf2328d | FromTheBook/Ch02/Bertrand/CentralBinom.lean | FactorizationCentralBinom.prodRangePowEqProdIcoPowOfLeTwo | [79, 1] | [90, 11] | intro x hxb hbx | n a b : ℕ
ha : a ≤ 2
⊢ ∀ (x : ℕ), x < b → (a ≤ x → b ≤ x) → x ^ ↑(Nat.factorization (Nat.centralBinom n)) x = 1 | n a b : ℕ
ha : a ≤ 2
x : ℕ
hxb : x < b
hbx : a ≤ x → b ≤ x
⊢ x ^ ↑(Nat.factorization (Nat.centralBinom n)) x = 1 |
https://github.com/jvlmdr/from_the_book.git | 9fb6080539a2f32bb24719600a9e7531abf2328d | FromTheBook/Ch02/Bertrand/CentralBinom.lean | FactorizationCentralBinom.prodRangePowEqProdIcoPowOfLeTwo | [79, 1] | [90, 11] | rw [Nat.factorization_eq_zero_of_non_prime] | n a b : ℕ
ha : a ≤ 2
x : ℕ
hxb : x < b
hbx : a ≤ x → b ≤ x
⊢ x ^ ↑(Nat.factorization (Nat.centralBinom n)) x = 1 | n a b : ℕ
ha : a ≤ 2
x : ℕ
hxb : x < b
hbx : a ≤ x → b ≤ x
⊢ x ^ 0 = 1
case hp
n a b : ℕ
ha : a ≤ 2
x : ℕ
hxb : x < b
hbx : a ≤ x → b ≤ x
⊢ ¬Nat.Prime x |
https://github.com/jvlmdr/from_the_book.git | 9fb6080539a2f32bb24719600a9e7531abf2328d | FromTheBook/Ch02/Bertrand/CentralBinom.lean | FactorizationCentralBinom.prodRangePowEqProdIcoPowOfLeTwo | [79, 1] | [90, 11] | simp | n a b : ℕ
ha : a ≤ 2
x : ℕ
hxb : x < b
hbx : a ≤ x → b ≤ x
⊢ x ^ 0 = 1
case hp
n a b : ℕ
ha : a ≤ 2
x : ℕ
hxb : x < b
hbx : a ≤ x → b ≤ x
⊢ ¬Nat.Prime x | case hp
n a b : ℕ
ha : a ≤ 2
x : ℕ
hxb : x < b
hbx : a ≤ x → b ≤ x
⊢ ¬Nat.Prime x |
https://github.com/jvlmdr/from_the_book.git | 9fb6080539a2f32bb24719600a9e7531abf2328d | FromTheBook/Ch02/Bertrand/CentralBinom.lean | FactorizationCentralBinom.prodRangePowEqProdIcoPowOfLeTwo | [79, 1] | [90, 11] | intro hp | case hp
n a b : ℕ
ha : a ≤ 2
x : ℕ
hxb : x < b
hbx : a ≤ x → b ≤ x
⊢ ¬Nat.Prime x | case hp
n a b : ℕ
ha : a ≤ 2
x : ℕ
hxb : x < b
hbx : a ≤ x → b ≤ x
hp : Nat.Prime x
⊢ False |
https://github.com/jvlmdr/from_the_book.git | 9fb6080539a2f32bb24719600a9e7531abf2328d | FromTheBook/Ch02/Bertrand/CentralBinom.lean | FactorizationCentralBinom.prodRangePowEqProdIcoPowOfLeTwo | [79, 1] | [90, 11] | have hp := Nat.Prime.two_le hp | case hp
n a b : ℕ
ha : a ≤ 2
x : ℕ
hxb : x < b
hbx : a ≤ x → b ≤ x
hp : Nat.Prime x
⊢ False | case hp
n a b : ℕ
ha : a ≤ 2
x : ℕ
hxb : x < b
hbx : a ≤ x → b ≤ x
hp✝ : Nat.Prime x
hp : 2 ≤ x
⊢ False |
https://github.com/jvlmdr/from_the_book.git | 9fb6080539a2f32bb24719600a9e7531abf2328d | FromTheBook/Ch02/Bertrand/CentralBinom.lean | FactorizationCentralBinom.prodRangePowEqProdIcoPowOfLeTwo | [79, 1] | [90, 11] | specialize hbx (le_trans ha hp) | case hp
n a b : ℕ
ha : a ≤ 2
x : ℕ
hxb : x < b
hbx : a ≤ x → b ≤ x
hp✝ : Nat.Prime x
hp : 2 ≤ x
⊢ False | case hp
n a b : ℕ
ha : a ≤ 2
x : ℕ
hxb : x < b
hp✝ : Nat.Prime x
hp : 2 ≤ x
hbx : b ≤ x
⊢ False |
https://github.com/jvlmdr/from_the_book.git | 9fb6080539a2f32bb24719600a9e7531abf2328d | FromTheBook/Ch02/Bertrand/CentralBinom.lean | FactorizationCentralBinom.prodRangePowEqProdIcoPowOfLeTwo | [79, 1] | [90, 11] | linarith | case hp
n a b : ℕ
ha : a ≤ 2
x : ℕ
hxb : x < b
hp✝ : Nat.Prime x
hp : 2 ≤ x
hbx : b ≤ x
⊢ False | no goals |
https://github.com/jvlmdr/from_the_book.git | 9fb6080539a2f32bb24719600a9e7531abf2328d | FromTheBook/Ch02/Bertrand/CentralBinom.lean | FactorizationCentralBinom.prodIocSqrtLePowLeProdPrimes | [93, 1] | [107, 39] | rw [primes] | n a b : ℕ
ha : Nat.sqrt (2 * n) ≤ a
⊢ ∏ p in Finset.Ioc a b, p ^ ↑(Nat.factorization (Nat.centralBinom n)) p ≤ ∏ p in primes (Finset.Ioc a b), p | n a b : ℕ
ha : Nat.sqrt (2 * n) ≤ a
⊢ ∏ p in Finset.Ioc a b, p ^ ↑(Nat.factorization (Nat.centralBinom n)) p ≤
∏ p in Finset.filter Nat.Prime (Finset.Ioc a b), p |
https://github.com/jvlmdr/from_the_book.git | 9fb6080539a2f32bb24719600a9e7531abf2328d | FromTheBook/Ch02/Bertrand/CentralBinom.lean | FactorizationCentralBinom.prodIocSqrtLePowLeProdPrimes | [93, 1] | [107, 39] | rw [← @Finset.prod_filter_of_ne _ _ _ _ _ Nat.Prime] | n a b : ℕ
ha : Nat.sqrt (2 * n) ≤ a
⊢ ∏ p in Finset.Ioc a b, p ^ ↑(Nat.factorization (Nat.centralBinom n)) p ≤
∏ p in Finset.filter Nat.Prime (Finset.Ioc a b), p | n a b : ℕ
ha : Nat.sqrt (2 * n) ≤ a
⊢ ∏ x in Finset.filter Nat.Prime (Finset.Ioc a b), x ^ ↑(Nat.factorization (Nat.centralBinom n)) x ≤
∏ p in Finset.filter Nat.Prime (Finset.Ioc a b), p
case hp
n a b : ℕ
ha : Nat.sqrt (2 * n) ≤ a
⊢ ∀ (x : ℕ), x ∈ Finset.Ioc a b → x ^ ↑(Nat.factorization (Nat.centralBinom n)) x ≠ 1 → Nat.Prime x |
https://github.com/jvlmdr/from_the_book.git | 9fb6080539a2f32bb24719600a9e7531abf2328d | FromTheBook/Ch02/Bertrand/CentralBinom.lean | FactorizationCentralBinom.prodIocSqrtLePowLeProdPrimes | [93, 1] | [107, 39] | . apply Finset.prod_le_prod'
simp
intro p hbp _ hp
replace hbp := lt_of_le_of_lt ha hbp
conv => rhs; rw [← Nat.pow_one p]
rw [pow_le_pow_iff (Nat.Prime.one_lt hp)]
rw [Nat.sqrt_lt'] at hbp
exact leOne hp hbp | n a b : ℕ
ha : Nat.sqrt (2 * n) ≤ a
⊢ ∏ x in Finset.filter Nat.Prime (Finset.Ioc a b), x ^ ↑(Nat.factorization (Nat.centralBinom n)) x ≤
∏ p in Finset.filter Nat.Prime (Finset.Ioc a b), p
case hp
n a b : ℕ
ha : Nat.sqrt (2 * n) ≤ a
⊢ ∀ (x : ℕ), x ∈ Finset.Ioc a b → x ^ ↑(Nat.factorization (Nat.centralBinom n)) x ≠ 1 → Nat.Prime x | case hp
n a b : ℕ
ha : Nat.sqrt (2 * n) ≤ a
⊢ ∀ (x : ℕ), x ∈ Finset.Ioc a b → x ^ ↑(Nat.factorization (Nat.centralBinom n)) x ≠ 1 → Nat.Prime x |
https://github.com/jvlmdr/from_the_book.git | 9fb6080539a2f32bb24719600a9e7531abf2328d | FromTheBook/Ch02/Bertrand/CentralBinom.lean | FactorizationCentralBinom.prodIocSqrtLePowLeProdPrimes | [93, 1] | [107, 39] | . intro p _
apply primeOfPowFactorizationNeOne | case hp
n a b : ℕ
ha : Nat.sqrt (2 * n) ≤ a
⊢ ∀ (x : ℕ), x ∈ Finset.Ioc a b → x ^ ↑(Nat.factorization (Nat.centralBinom n)) x ≠ 1 → Nat.Prime x | no goals |
https://github.com/jvlmdr/from_the_book.git | 9fb6080539a2f32bb24719600a9e7531abf2328d | FromTheBook/Ch02/Bertrand/CentralBinom.lean | FactorizationCentralBinom.prodIocSqrtLePowLeProdPrimes | [93, 1] | [107, 39] | apply Finset.prod_le_prod' | n a b : ℕ
ha : Nat.sqrt (2 * n) ≤ a
⊢ ∏ x in Finset.filter Nat.Prime (Finset.Ioc a b), x ^ ↑(Nat.factorization (Nat.centralBinom n)) x ≤
∏ p in Finset.filter Nat.Prime (Finset.Ioc a b), p | case h
n a b : ℕ
ha : Nat.sqrt (2 * n) ≤ a
⊢ ∀ (i : ℕ), i ∈ Finset.filter Nat.Prime (Finset.Ioc a b) → i ^ ↑(Nat.factorization (Nat.centralBinom n)) i ≤ i |
https://github.com/jvlmdr/from_the_book.git | 9fb6080539a2f32bb24719600a9e7531abf2328d | FromTheBook/Ch02/Bertrand/CentralBinom.lean | FactorizationCentralBinom.prodIocSqrtLePowLeProdPrimes | [93, 1] | [107, 39] | simp | case h
n a b : ℕ
ha : Nat.sqrt (2 * n) ≤ a
⊢ ∀ (i : ℕ), i ∈ Finset.filter Nat.Prime (Finset.Ioc a b) → i ^ ↑(Nat.factorization (Nat.centralBinom n)) i ≤ i | case h
n a b : ℕ
ha : Nat.sqrt (2 * n) ≤ a
⊢ ∀ (i : ℕ), a < i → i ≤ b → Nat.Prime i → i ^ ↑(Nat.factorization (Nat.centralBinom n)) i ≤ i |
https://github.com/jvlmdr/from_the_book.git | 9fb6080539a2f32bb24719600a9e7531abf2328d | FromTheBook/Ch02/Bertrand/CentralBinom.lean | FactorizationCentralBinom.prodIocSqrtLePowLeProdPrimes | [93, 1] | [107, 39] | intro p hbp _ hp | case h
n a b : ℕ
ha : Nat.sqrt (2 * n) ≤ a
⊢ ∀ (i : ℕ), a < i → i ≤ b → Nat.Prime i → i ^ ↑(Nat.factorization (Nat.centralBinom n)) i ≤ i | case h
n a b : ℕ
ha : Nat.sqrt (2 * n) ≤ a
p : ℕ
hbp : a < p
a✝ : p ≤ b
hp : Nat.Prime p
⊢ p ^ ↑(Nat.factorization (Nat.centralBinom n)) p ≤ p |
https://github.com/jvlmdr/from_the_book.git | 9fb6080539a2f32bb24719600a9e7531abf2328d | FromTheBook/Ch02/Bertrand/CentralBinom.lean | FactorizationCentralBinom.prodIocSqrtLePowLeProdPrimes | [93, 1] | [107, 39] | replace hbp := lt_of_le_of_lt ha hbp | case h
n a b : ℕ
ha : Nat.sqrt (2 * n) ≤ a
p : ℕ
hbp : a < p
a✝ : p ≤ b
hp : Nat.Prime p
⊢ p ^ ↑(Nat.factorization (Nat.centralBinom n)) p ≤ p | case h
n a b : ℕ
ha : Nat.sqrt (2 * n) ≤ a
p : ℕ
a✝ : p ≤ b
hp : Nat.Prime p
hbp : Nat.sqrt (2 * n) < p
⊢ p ^ ↑(Nat.factorization (Nat.centralBinom n)) p ≤ p |
https://github.com/jvlmdr/from_the_book.git | 9fb6080539a2f32bb24719600a9e7531abf2328d | FromTheBook/Ch02/Bertrand/CentralBinom.lean | FactorizationCentralBinom.prodIocSqrtLePowLeProdPrimes | [93, 1] | [107, 39] | conv => rhs; rw [← Nat.pow_one p] | case h
n a b : ℕ
ha : Nat.sqrt (2 * n) ≤ a
p : ℕ
a✝ : p ≤ b
hp : Nat.Prime p
hbp : Nat.sqrt (2 * n) < p
⊢ p ^ ↑(Nat.factorization (Nat.centralBinom n)) p ≤ p | case h
n a b : ℕ
ha : Nat.sqrt (2 * n) ≤ a
p : ℕ
a✝ : p ≤ b
hp : Nat.Prime p
hbp : Nat.sqrt (2 * n) < p
⊢ p ^ ↑(Nat.factorization (Nat.centralBinom n)) p ≤ p ^ 1 |
https://github.com/jvlmdr/from_the_book.git | 9fb6080539a2f32bb24719600a9e7531abf2328d | FromTheBook/Ch02/Bertrand/CentralBinom.lean | FactorizationCentralBinom.prodIocSqrtLePowLeProdPrimes | [93, 1] | [107, 39] | rw [pow_le_pow_iff (Nat.Prime.one_lt hp)] | case h
n a b : ℕ
ha : Nat.sqrt (2 * n) ≤ a
p : ℕ
a✝ : p ≤ b
hp : Nat.Prime p
hbp : Nat.sqrt (2 * n) < p
⊢ p ^ ↑(Nat.factorization (Nat.centralBinom n)) p ≤ p ^ 1 | case h
n a b : ℕ
ha : Nat.sqrt (2 * n) ≤ a
p : ℕ
a✝ : p ≤ b
hp : Nat.Prime p
hbp : Nat.sqrt (2 * n) < p
⊢ ↑(Nat.factorization (Nat.centralBinom n)) p ≤ 1 |
https://github.com/jvlmdr/from_the_book.git | 9fb6080539a2f32bb24719600a9e7531abf2328d | FromTheBook/Ch02/Bertrand/CentralBinom.lean | FactorizationCentralBinom.prodIocSqrtLePowLeProdPrimes | [93, 1] | [107, 39] | rw [Nat.sqrt_lt'] at hbp | case h
n a b : ℕ
ha : Nat.sqrt (2 * n) ≤ a
p : ℕ
a✝ : p ≤ b
hp : Nat.Prime p
hbp : Nat.sqrt (2 * n) < p
⊢ ↑(Nat.factorization (Nat.centralBinom n)) p ≤ 1 | case h
n a b : ℕ
ha : Nat.sqrt (2 * n) ≤ a
p : ℕ
a✝ : p ≤ b
hp : Nat.Prime p
hbp : 2 * n < p ^ 2
⊢ ↑(Nat.factorization (Nat.centralBinom n)) p ≤ 1 |
https://github.com/jvlmdr/from_the_book.git | 9fb6080539a2f32bb24719600a9e7531abf2328d | FromTheBook/Ch02/Bertrand/CentralBinom.lean | FactorizationCentralBinom.prodIocSqrtLePowLeProdPrimes | [93, 1] | [107, 39] | exact leOne hp hbp | case h
n a b : ℕ
ha : Nat.sqrt (2 * n) ≤ a
p : ℕ
a✝ : p ≤ b
hp : Nat.Prime p
hbp : 2 * n < p ^ 2
⊢ ↑(Nat.factorization (Nat.centralBinom n)) p ≤ 1 | no goals |
https://github.com/jvlmdr/from_the_book.git | 9fb6080539a2f32bb24719600a9e7531abf2328d | FromTheBook/Ch02/Bertrand/CentralBinom.lean | FactorizationCentralBinom.prodIocSqrtLePowLeProdPrimes | [93, 1] | [107, 39] | intro p _ | case hp
n a b : ℕ
ha : Nat.sqrt (2 * n) ≤ a
⊢ ∀ (x : ℕ), x ∈ Finset.Ioc a b → x ^ ↑(Nat.factorization (Nat.centralBinom n)) x ≠ 1 → Nat.Prime x | case hp
n a b : ℕ
ha : Nat.sqrt (2 * n) ≤ a
p : ℕ
a✝ : p ∈ Finset.Ioc a b
⊢ p ^ ↑(Nat.factorization (Nat.centralBinom n)) p ≠ 1 → Nat.Prime p |
https://github.com/jvlmdr/from_the_book.git | 9fb6080539a2f32bb24719600a9e7531abf2328d | FromTheBook/Ch02/Bertrand/CentralBinom.lean | FactorizationCentralBinom.prodIocSqrtLePowLeProdPrimes | [93, 1] | [107, 39] | apply primeOfPowFactorizationNeOne | case hp
n a b : ℕ
ha : Nat.sqrt (2 * n) ≤ a
p : ℕ
a✝ : p ∈ Finset.Ioc a b
⊢ p ^ ↑(Nat.factorization (Nat.centralBinom n)) p ≠ 1 → Nat.Prime p | no goals |
https://github.com/jvlmdr/from_the_book.git | 9fb6080539a2f32bb24719600a9e7531abf2328d | FromTheBook/Ch02/Bertrand/CentralBinom.lean | sqrtTwoMulLeSelf | [117, 1] | [120, 34] | conv => rhs; rw [← Nat.sqrt_eq n] | n : ℕ
hn : 2 ≤ n
⊢ Nat.sqrt (2 * n) ≤ n | n : ℕ
hn : 2 ≤ n
⊢ Nat.sqrt (2 * n) ≤ Nat.sqrt (n * n) |
https://github.com/jvlmdr/from_the_book.git | 9fb6080539a2f32bb24719600a9e7531abf2328d | FromTheBook/Ch02/Bertrand/CentralBinom.lean | sqrtTwoMulLeSelf | [117, 1] | [120, 34] | apply Nat.sqrt_le_sqrt | n : ℕ
hn : 2 ≤ n
⊢ Nat.sqrt (2 * n) ≤ Nat.sqrt (n * n) | case h
n : ℕ
hn : 2 ≤ n
⊢ 2 * n ≤ n * n |
https://github.com/jvlmdr/from_the_book.git | 9fb6080539a2f32bb24719600a9e7531abf2328d | FromTheBook/Ch02/Bertrand/CentralBinom.lean | sqrtTwoMulLeSelf | [117, 1] | [120, 34] | exact Nat.mul_le_mul_right _ hn | case h
n : ℕ
hn : 2 ≤ n
⊢ 2 * n ≤ n * n | no goals |
https://github.com/jvlmdr/from_the_book.git | 9fb6080539a2f32bb24719600a9e7531abf2328d | FromTheBook/Ch02/Bertrand/CentralBinom.lean | prodRangePowFactorization | [122, 1] | [129, 9] | cases b with
| zero => simp
| succ b =>
rw [← Finset.prod_range_mul_prod_Ico _ (Nat.zero_lt_succ _)]
simp | b n : ℕ
⊢ ∏ p in Finset.range b, p ^ ↑(Nat.factorization n) p = ∏ p in Finset.Ico 1 b, p ^ ↑(Nat.factorization n) p | no goals |
https://github.com/jvlmdr/from_the_book.git | 9fb6080539a2f32bb24719600a9e7531abf2328d | FromTheBook/Ch02/Bertrand/CentralBinom.lean | prodRangePowFactorization | [122, 1] | [129, 9] | simp | case zero
n : ℕ
⊢ ∏ p in Finset.range Nat.zero, p ^ ↑(Nat.factorization n) p =
∏ p in Finset.Ico 1 Nat.zero, p ^ ↑(Nat.factorization n) p | no goals |
https://github.com/jvlmdr/from_the_book.git | 9fb6080539a2f32bb24719600a9e7531abf2328d | FromTheBook/Ch02/Bertrand/CentralBinom.lean | prodRangePowFactorization | [122, 1] | [129, 9] | rw [← Finset.prod_range_mul_prod_Ico _ (Nat.zero_lt_succ _)] | case succ
n b : ℕ
⊢ ∏ p in Finset.range (Nat.succ b), p ^ ↑(Nat.factorization n) p =
∏ p in Finset.Ico 1 (Nat.succ b), p ^ ↑(Nat.factorization n) p | case succ
n b : ℕ
⊢ (∏ k in Finset.range (Nat.succ 0), k ^ ↑(Nat.factorization n) k) *
∏ k in Finset.Ico (Nat.succ 0) (Nat.succ b), k ^ ↑(Nat.factorization n) k =
∏ p in Finset.Ico 1 (Nat.succ b), p ^ ↑(Nat.factorization n) p |
https://github.com/jvlmdr/from_the_book.git | 9fb6080539a2f32bb24719600a9e7531abf2328d | FromTheBook/Ch02/Bertrand/CentralBinom.lean | prodRangePowFactorization | [122, 1] | [129, 9] | simp | case succ
n b : ℕ
⊢ (∏ k in Finset.range (Nat.succ 0), k ^ ↑(Nat.factorization n) k) *
∏ k in Finset.Ico (Nat.succ 0) (Nat.succ b), k ^ ↑(Nat.factorization n) k =
∏ p in Finset.Ico 1 (Nat.succ b), p ^ ↑(Nat.factorization n) p | no goals |
https://github.com/jvlmdr/from_the_book.git | 9fb6080539a2f32bb24719600a9e7531abf2328d | FromTheBook/Ch02/Bertrand/CentralBinom.lean | powFactorizationPos | [131, 1] | [134, 60] | cases p with
| zero => simp
| succ p => exact Nat.one_le_pow _ _ (Nat.zero_lt_succ _) | n p : ℕ
⊢ 0 < p ^ ↑(Nat.factorization n) p | no goals |
https://github.com/jvlmdr/from_the_book.git | 9fb6080539a2f32bb24719600a9e7531abf2328d | FromTheBook/Ch02/Bertrand/CentralBinom.lean | powFactorizationPos | [131, 1] | [134, 60] | simp | case zero
n : ℕ
⊢ 0 < Nat.zero ^ ↑(Nat.factorization n) Nat.zero | no goals |
https://github.com/jvlmdr/from_the_book.git | 9fb6080539a2f32bb24719600a9e7531abf2328d | FromTheBook/Ch02/Bertrand/CentralBinom.lean | powFactorizationPos | [131, 1] | [134, 60] | exact Nat.one_le_pow _ _ (Nat.zero_lt_succ _) | case succ
n p : ℕ
⊢ 0 < Nat.succ p ^ ↑(Nat.factorization n) (Nat.succ p) | no goals |
https://github.com/jvlmdr/from_the_book.git | 9fb6080539a2f32bb24719600a9e7531abf2328d | FromTheBook/Ch02/Bertrand/CentralBinom.lean | prodLeProdMulProdOfUnionOfOneLe | [136, 1] | [144, 13] | rw [← Finset.prod_union_inter] | s t₁ t₂ : Finset ℕ
f : ℕ → ℕ
hs : s ⊆ t₁ ∪ t₂
hf : ∀ (y : ℕ), 1 ≤ f y
⊢ ∏ x in s, f x ≤ (∏ x in t₁, f x) * ∏ x in t₂, f x | s t₁ t₂ : Finset ℕ
f : ℕ → ℕ
hs : s ⊆ t₁ ∪ t₂
hf : ∀ (y : ℕ), 1 ≤ f y
⊢ ∏ x in s, f x ≤ (∏ x in t₁ ∪ t₂, f x) * ∏ x in t₁ ∩ t₂, f x |
https://github.com/jvlmdr/from_the_book.git | 9fb6080539a2f32bb24719600a9e7531abf2328d | FromTheBook/Ch02/Bertrand/CentralBinom.lean | prodLeProdMulProdOfUnionOfOneLe | [136, 1] | [144, 13] | apply le_trans _ (Nat.le_mul_of_pos_right _) | s t₁ t₂ : Finset ℕ
f : ℕ → ℕ
hs : s ⊆ t₁ ∪ t₂
hf : ∀ (y : ℕ), 1 ≤ f y
⊢ ∏ x in s, f x ≤ (∏ x in t₁ ∪ t₂, f x) * ∏ x in t₁ ∩ t₂, f x | s t₁ t₂ : Finset ℕ
f : ℕ → ℕ
hs : s ⊆ t₁ ∪ t₂
hf : ∀ (y : ℕ), 1 ≤ f y
⊢ ∏ x in s, f x ≤ ∏ x in t₁ ∪ t₂, f x
s t₁ t₂ : Finset ℕ
f : ℕ → ℕ
hs : s ⊆ t₁ ∪ t₂
hf : ∀ (y : ℕ), 1 ≤ f y
⊢ 0 < ∏ x in t₁ ∩ t₂, f x |
https://github.com/jvlmdr/from_the_book.git | 9fb6080539a2f32bb24719600a9e7531abf2328d | FromTheBook/Ch02/Bertrand/CentralBinom.lean | prodLeProdMulProdOfUnionOfOneLe | [136, 1] | [144, 13] | . exact Finset.prod_mono_set_of_one_le' hf hs | s t₁ t₂ : Finset ℕ
f : ℕ → ℕ
hs : s ⊆ t₁ ∪ t₂
hf : ∀ (y : ℕ), 1 ≤ f y
⊢ ∏ x in s, f x ≤ ∏ x in t₁ ∪ t₂, f x
s t₁ t₂ : Finset ℕ
f : ℕ → ℕ
hs : s ⊆ t₁ ∪ t₂
hf : ∀ (y : ℕ), 1 ≤ f y
⊢ 0 < ∏ x in t₁ ∩ t₂, f x | s t₁ t₂ : Finset ℕ
f : ℕ → ℕ
hs : s ⊆ t₁ ∪ t₂
hf : ∀ (y : ℕ), 1 ≤ f y
⊢ 0 < ∏ x in t₁ ∩ t₂, f x |
https://github.com/jvlmdr/from_the_book.git | 9fb6080539a2f32bb24719600a9e7531abf2328d | FromTheBook/Ch02/Bertrand/CentralBinom.lean | prodLeProdMulProdOfUnionOfOneLe | [136, 1] | [144, 13] | . apply Finset.prod_pos
intro x _
apply hf | s t₁ t₂ : Finset ℕ
f : ℕ → ℕ
hs : s ⊆ t₁ ∪ t₂
hf : ∀ (y : ℕ), 1 ≤ f y
⊢ 0 < ∏ x in t₁ ∩ t₂, f x | no goals |
https://github.com/jvlmdr/from_the_book.git | 9fb6080539a2f32bb24719600a9e7531abf2328d | FromTheBook/Ch02/Bertrand/CentralBinom.lean | prodLeProdMulProdOfUnionOfOneLe | [136, 1] | [144, 13] | exact Finset.prod_mono_set_of_one_le' hf hs | s t₁ t₂ : Finset ℕ
f : ℕ → ℕ
hs : s ⊆ t₁ ∪ t₂
hf : ∀ (y : ℕ), 1 ≤ f y
⊢ ∏ x in s, f x ≤ ∏ x in t₁ ∪ t₂, f x | no goals |
https://github.com/jvlmdr/from_the_book.git | 9fb6080539a2f32bb24719600a9e7531abf2328d | FromTheBook/Ch02/Bertrand/CentralBinom.lean | prodLeProdMulProdOfUnionOfOneLe | [136, 1] | [144, 13] | apply Finset.prod_pos | s t₁ t₂ : Finset ℕ
f : ℕ → ℕ
hs : s ⊆ t₁ ∪ t₂
hf : ∀ (y : ℕ), 1 ≤ f y
⊢ 0 < ∏ x in t₁ ∩ t₂, f x | case h0
s t₁ t₂ : Finset ℕ
f : ℕ → ℕ
hs : s ⊆ t₁ ∪ t₂
hf : ∀ (y : ℕ), 1 ≤ f y
⊢ ∀ (i : ℕ), i ∈ t₁ ∩ t₂ → 0 < f i |
https://github.com/jvlmdr/from_the_book.git | 9fb6080539a2f32bb24719600a9e7531abf2328d | FromTheBook/Ch02/Bertrand/CentralBinom.lean | prodLeProdMulProdOfUnionOfOneLe | [136, 1] | [144, 13] | intro x _ | case h0
s t₁ t₂ : Finset ℕ
f : ℕ → ℕ
hs : s ⊆ t₁ ∪ t₂
hf : ∀ (y : ℕ), 1 ≤ f y
⊢ ∀ (i : ℕ), i ∈ t₁ ∩ t₂ → 0 < f i | case h0
s t₁ t₂ : Finset ℕ
f : ℕ → ℕ
hs : s ⊆ t₁ ∪ t₂
hf : ∀ (y : ℕ), 1 ≤ f y
x : ℕ
a✝ : x ∈ t₁ ∩ t₂
⊢ 0 < f x |
https://github.com/jvlmdr/from_the_book.git | 9fb6080539a2f32bb24719600a9e7531abf2328d | FromTheBook/Ch02/Bertrand/CentralBinom.lean | prodLeProdMulProdOfUnionOfOneLe | [136, 1] | [144, 13] | apply hf | case h0
s t₁ t₂ : Finset ℕ
f : ℕ → ℕ
hs : s ⊆ t₁ ∪ t₂
hf : ∀ (y : ℕ), 1 ≤ f y
x : ℕ
a✝ : x ∈ t₁ ∩ t₂
⊢ 0 < f x | no goals |
https://github.com/jvlmdr/from_the_book.git | 9fb6080539a2f32bb24719600a9e7531abf2328d | FromTheBook/Ch02/Bertrand/CentralBinom.lean | mulDivLeSelf | [146, 1] | [152, 37] | cases b with
| zero => simp
| succ b =>
conv => rhs; rw [← Nat.mul_div_right n (Nat.zero_lt_succ b)]
apply Nat.div_le_div_right
exact Nat.mul_le_mul_right _ hab | n a b : ℕ
hab : a ≤ b
⊢ a * n / b ≤ n | no goals |
https://github.com/jvlmdr/from_the_book.git | 9fb6080539a2f32bb24719600a9e7531abf2328d | FromTheBook/Ch02/Bertrand/CentralBinom.lean | mulDivLeSelf | [146, 1] | [152, 37] | simp | case zero
n a : ℕ
hab : a ≤ Nat.zero
⊢ a * n / Nat.zero ≤ n | no goals |
https://github.com/jvlmdr/from_the_book.git | 9fb6080539a2f32bb24719600a9e7531abf2328d | FromTheBook/Ch02/Bertrand/CentralBinom.lean | mulDivLeSelf | [146, 1] | [152, 37] | conv => rhs; rw [← Nat.mul_div_right n (Nat.zero_lt_succ b)] | case succ
n a b : ℕ
hab : a ≤ Nat.succ b
⊢ a * n / Nat.succ b ≤ n | case succ
n a b : ℕ
hab : a ≤ Nat.succ b
⊢ a * n / Nat.succ b ≤ Nat.succ b * n / Nat.succ b |
https://github.com/jvlmdr/from_the_book.git | 9fb6080539a2f32bb24719600a9e7531abf2328d | FromTheBook/Ch02/Bertrand/CentralBinom.lean | mulDivLeSelf | [146, 1] | [152, 37] | apply Nat.div_le_div_right | case succ
n a b : ℕ
hab : a ≤ Nat.succ b
⊢ a * n / Nat.succ b ≤ Nat.succ b * n / Nat.succ b | case succ.h
n a b : ℕ
hab : a ≤ Nat.succ b
⊢ a * n ≤ Nat.succ b * n |
https://github.com/jvlmdr/from_the_book.git | 9fb6080539a2f32bb24719600a9e7531abf2328d | FromTheBook/Ch02/Bertrand/CentralBinom.lean | mulDivLeSelf | [146, 1] | [152, 37] | exact Nat.mul_le_mul_right _ hab | case succ.h
n a b : ℕ
hab : a ≤ Nat.succ b
⊢ a * n ≤ Nat.succ b * n | no goals |
https://github.com/jvlmdr/from_the_book.git | 9fb6080539a2f32bb24719600a9e7531abf2328d | FromTheBook/Ch02/Bertrand/CentralBinom.lean | mulRightLeOfLeOne | [154, 1] | [156, 28] | rw [← Nat.one_mul z] | x y z : ℕ
h1 : x ≤ 1
h : y ≤ z
⊢ x * y ≤ z | x y z : ℕ
h1 : x ≤ 1
h : y ≤ z
⊢ x * y ≤ 1 * z |
https://github.com/jvlmdr/from_the_book.git | 9fb6080539a2f32bb24719600a9e7531abf2328d | FromTheBook/Ch02/Bertrand/CentralBinom.lean | mulRightLeOfLeOne | [154, 1] | [156, 28] | apply Nat.mul_le_mul h1 h | x y z : ℕ
h1 : x ≤ 1
h : y ≤ z
⊢ x * y ≤ 1 * z | no goals |
https://github.com/jvlmdr/from_the_book.git | 9fb6080539a2f32bb24719600a9e7531abf2328d | FromTheBook/Ch02/Bertrand/CentralBinom.lean | mulLeftLeOfLeOne | [158, 1] | [160, 28] | rw [← Nat.mul_one z] | x y z : ℕ
h1 : y ≤ 1
h : x ≤ z
⊢ x * y ≤ z | x y z : ℕ
h1 : y ≤ 1
h : x ≤ z
⊢ x * y ≤ z * 1 |
https://github.com/jvlmdr/from_the_book.git | 9fb6080539a2f32bb24719600a9e7531abf2328d | FromTheBook/Ch02/Bertrand/CentralBinom.lean | mulLeftLeOfLeOne | [158, 1] | [160, 28] | apply Nat.mul_le_mul h h1 | x y z : ℕ
h1 : y ≤ 1
h : x ≤ z
⊢ x * y ≤ z * 1 | no goals |
https://github.com/jvlmdr/from_the_book.git | 9fb6080539a2f32bb24719600a9e7531abf2328d | FromTheBook/Ch02/Bertrand/CentralBinom.lean | fourPowLeMulProdPrimes | [162, 1] | [216, 45] | apply le_trans (Nat.four_pow_le_two_mul_self_mul_centralBinom _ (by linarith)) | n : ℕ
hn : 2 < n
⊢ 4 ^ n ≤ (2 * n) ^ (1 + Nat.sqrt (2 * n)) * 4 ^ Nat.pred (2 * n / 3) * ∏ p in primes (Finset.Ioc n (2 * n)), p | n : ℕ
hn : 2 < n
⊢ 2 * n * Nat.centralBinom n ≤
(2 * n) ^ (1 + Nat.sqrt (2 * n)) * 4 ^ Nat.pred (2 * n / 3) * ∏ p in primes (Finset.Ioc n (2 * n)), p |
https://github.com/jvlmdr/from_the_book.git | 9fb6080539a2f32bb24719600a9e7531abf2328d | FromTheBook/Ch02/Bertrand/CentralBinom.lean | fourPowLeMulProdPrimes | [162, 1] | [216, 45] | rw [Nat.one_add, Nat.pow_succ] | n : ℕ
hn : 2 < n
⊢ 2 * n * Nat.centralBinom n ≤
(2 * n) ^ (1 + Nat.sqrt (2 * n)) * 4 ^ Nat.pred (2 * n / 3) * ∏ p in primes (Finset.Ioc n (2 * n)), p | n : ℕ
hn : 2 < n
⊢ 2 * n * Nat.centralBinom n ≤
(2 * n) ^ Nat.sqrt (2 * n) * (2 * n) * 4 ^ Nat.pred (2 * n / 3) * ∏ p in primes (Finset.Ioc n (2 * n)), p |
https://github.com/jvlmdr/from_the_book.git | 9fb6080539a2f32bb24719600a9e7531abf2328d | FromTheBook/Ch02/Bertrand/CentralBinom.lean | fourPowLeMulProdPrimes | [162, 1] | [216, 45] | have : ∀ {x y z : ℕ}, x * (2*n) * y * z = (2*n) * (x * y * z) := by intro x y z; linarith | n : ℕ
hn : 2 < n
⊢ 2 * n * Nat.centralBinom n ≤
(2 * n) ^ Nat.sqrt (2 * n) * (2 * n) * 4 ^ Nat.pred (2 * n / 3) * ∏ p in primes (Finset.Ioc n (2 * n)), p | n : ℕ
hn : 2 < n
this : ∀ {x y z : ℕ}, x * (2 * n) * y * z = 2 * n * (x * y * z)
⊢ 2 * n * Nat.centralBinom n ≤
(2 * n) ^ Nat.sqrt (2 * n) * (2 * n) * 4 ^ Nat.pred (2 * n / 3) * ∏ p in primes (Finset.Ioc n (2 * n)), p |
https://github.com/jvlmdr/from_the_book.git | 9fb6080539a2f32bb24719600a9e7531abf2328d | FromTheBook/Ch02/Bertrand/CentralBinom.lean | fourPowLeMulProdPrimes | [162, 1] | [216, 45] | rw [this] | n : ℕ
hn : 2 < n
this : ∀ {x y z : ℕ}, x * (2 * n) * y * z = 2 * n * (x * y * z)
⊢ 2 * n * Nat.centralBinom n ≤
(2 * n) ^ Nat.sqrt (2 * n) * (2 * n) * 4 ^ Nat.pred (2 * n / 3) * ∏ p in primes (Finset.Ioc n (2 * n)), p | n : ℕ
hn : 2 < n
this : ∀ {x y z : ℕ}, x * (2 * n) * y * z = 2 * n * (x * y * z)
⊢ 2 * n * Nat.centralBinom n ≤
2 * n * ((2 * n) ^ Nat.sqrt (2 * n) * 4 ^ Nat.pred (2 * n / 3) * ∏ p in primes (Finset.Ioc n (2 * n)), p) |
https://github.com/jvlmdr/from_the_book.git | 9fb6080539a2f32bb24719600a9e7531abf2328d | FromTheBook/Ch02/Bertrand/CentralBinom.lean | fourPowLeMulProdPrimes | [162, 1] | [216, 45] | clear this | n : ℕ
hn : 2 < n
this : ∀ {x y z : ℕ}, x * (2 * n) * y * z = 2 * n * (x * y * z)
⊢ 2 * n * Nat.centralBinom n ≤
2 * n * ((2 * n) ^ Nat.sqrt (2 * n) * 4 ^ Nat.pred (2 * n / 3) * ∏ p in primes (Finset.Ioc n (2 * n)), p) | n : ℕ
hn : 2 < n
⊢ 2 * n * Nat.centralBinom n ≤
2 * n * ((2 * n) ^ Nat.sqrt (2 * n) * 4 ^ Nat.pred (2 * n / 3) * ∏ p in primes (Finset.Ioc n (2 * n)), p) |
https://github.com/jvlmdr/from_the_book.git | 9fb6080539a2f32bb24719600a9e7531abf2328d | FromTheBook/Ch02/Bertrand/CentralBinom.lean | fourPowLeMulProdPrimes | [162, 1] | [216, 45] | apply Nat.mul_le_mul_left | n : ℕ
hn : 2 < n
⊢ 2 * n * Nat.centralBinom n ≤
2 * n * ((2 * n) ^ Nat.sqrt (2 * n) * 4 ^ Nat.pred (2 * n / 3) * ∏ p in primes (Finset.Ioc n (2 * n)), p) | case h
n : ℕ
hn : 2 < n
⊢ Nat.centralBinom n ≤ (2 * n) ^ Nat.sqrt (2 * n) * 4 ^ Nat.pred (2 * n / 3) * ∏ p in primes (Finset.Ioc n (2 * n)), p |
https://github.com/jvlmdr/from_the_book.git | 9fb6080539a2f32bb24719600a9e7531abf2328d | FromTheBook/Ch02/Bertrand/CentralBinom.lean | fourPowLeMulProdPrimes | [162, 1] | [216, 45] | rw [← Nat.prod_pow_factorization_centralBinom] | case h
n : ℕ
hn : 2 < n
⊢ Nat.centralBinom n ≤ (2 * n) ^ Nat.sqrt (2 * n) * 4 ^ Nat.pred (2 * n / 3) * ∏ p in primes (Finset.Ioc n (2 * n)), p | case h
n : ℕ
hn : 2 < n
⊢ ∏ p in Finset.range (2 * n + 1), p ^ ↑(Nat.factorization (Nat.centralBinom n)) p ≤
(2 * n) ^ Nat.sqrt (2 * n) * 4 ^ Nat.pred (2 * n / 3) * ∏ p in primes (Finset.Ioc n (2 * n)), p |
https://github.com/jvlmdr/from_the_book.git | 9fb6080539a2f32bb24719600a9e7531abf2328d | FromTheBook/Ch02/Bertrand/CentralBinom.lean | fourPowLeMulProdPrimes | [162, 1] | [216, 45] | rw [Finset.range_eq_Ico] | case h
n : ℕ
hn : 2 < n
⊢ ∏ p in Finset.range (2 * n + 1), p ^ ↑(Nat.factorization (Nat.centralBinom n)) p ≤
(2 * n) ^ Nat.sqrt (2 * n) * 4 ^ Nat.pred (2 * n / 3) * ∏ p in primes (Finset.Ioc n (2 * n)), p | case h
n : ℕ
hn : 2 < n
⊢ ∏ p in Finset.Ico 0 (2 * n + 1), p ^ ↑(Nat.factorization (Nat.centralBinom n)) p ≤
(2 * n) ^ Nat.sqrt (2 * n) * 4 ^ Nat.pred (2 * n / 3) * ∏ p in primes (Finset.Ioc n (2 * n)), p |
https://github.com/jvlmdr/from_the_book.git | 9fb6080539a2f32bb24719600a9e7531abf2328d | FromTheBook/Ch02/Bertrand/CentralBinom.lean | fourPowLeMulProdPrimes | [162, 1] | [216, 45] | conv => lhs; rw [← Finset.prod_Ico_consecutive _ (Nat.zero_le _) (Nat.succ_le_succ (Nat.sqrt_le_self _))] | case h
n : ℕ
hn : 2 < n
⊢ ∏ p in Finset.Ico 0 (2 * n + 1), p ^ ↑(Nat.factorization (Nat.centralBinom n)) p ≤
(2 * n) ^ Nat.sqrt (2 * n) * 4 ^ Nat.pred (2 * n / 3) * ∏ p in primes (Finset.Ioc n (2 * n)), p | case h
n : ℕ
hn : 2 < n
⊢ (∏ i in Finset.Ico 0 (Nat.succ (Nat.sqrt (2 * n))), i ^ ↑(Nat.factorization (Nat.centralBinom n)) i) *
∏ i in Finset.Ico (Nat.succ (Nat.sqrt (2 * n))) (Nat.succ (2 * n)),
i ^ ↑(Nat.factorization (Nat.centralBinom n)) i ≤
(2 * n) ^ Nat.sqrt (2 * n) * 4 ^ Nat.pred (2 * n / 3) * ∏ p in primes (Finset.Ioc n (2 * n)), p |
https://github.com/jvlmdr/from_the_book.git | 9fb6080539a2f32bb24719600a9e7531abf2328d | FromTheBook/Ch02/Bertrand/CentralBinom.lean | fourPowLeMulProdPrimes | [162, 1] | [216, 45] | simp [Nat.Ico_succ_succ] | case h
n : ℕ
hn : 2 < n
⊢ (∏ i in Finset.Ico 0 (Nat.succ (Nat.sqrt (2 * n))), i ^ ↑(Nat.factorization (Nat.centralBinom n)) i) *
∏ i in Finset.Ico (Nat.succ (Nat.sqrt (2 * n))) (Nat.succ (2 * n)),
i ^ ↑(Nat.factorization (Nat.centralBinom n)) i ≤
(2 * n) ^ Nat.sqrt (2 * n) * 4 ^ Nat.pred (2 * n / 3) * ∏ p in primes (Finset.Ioc n (2 * n)), p | case h
n : ℕ
hn : 2 < n
⊢ (∏ x in Finset.range (Nat.succ (Nat.sqrt (2 * n))), x ^ ↑(Nat.factorization (Nat.centralBinom n)) x) *
∏ x in Finset.Ioc (Nat.sqrt (2 * n)) (2 * n), x ^ ↑(Nat.factorization (Nat.centralBinom n)) x ≤
(2 * n) ^ Nat.sqrt (2 * n) * 4 ^ Nat.pred (2 * n / 3) * ∏ p in primes (Finset.Ioc n (2 * n)), p |
https://github.com/jvlmdr/from_the_book.git | 9fb6080539a2f32bb24719600a9e7531abf2328d | FromTheBook/Ch02/Bertrand/CentralBinom.lean | fourPowLeMulProdPrimes | [162, 1] | [216, 45] | conv => rhs; rw [mul_assoc] | case h
n : ℕ
hn : 2 < n
⊢ (∏ x in Finset.range (Nat.succ (Nat.sqrt (2 * n))), x ^ ↑(Nat.factorization (Nat.centralBinom n)) x) *
∏ x in Finset.Ioc (Nat.sqrt (2 * n)) (2 * n), x ^ ↑(Nat.factorization (Nat.centralBinom n)) x ≤
(2 * n) ^ Nat.sqrt (2 * n) * 4 ^ Nat.pred (2 * n / 3) * ∏ p in primes (Finset.Ioc n (2 * n)), p | case h
n : ℕ
hn : 2 < n
⊢ (∏ x in Finset.range (Nat.succ (Nat.sqrt (2 * n))), x ^ ↑(Nat.factorization (Nat.centralBinom n)) x) *
∏ x in Finset.Ioc (Nat.sqrt (2 * n)) (2 * n), x ^ ↑(Nat.factorization (Nat.centralBinom n)) x ≤
(2 * n) ^ Nat.sqrt (2 * n) * (4 ^ Nat.pred (2 * n / 3) * ∏ p in primes (Finset.Ioc n (2 * n)), p) |
https://github.com/jvlmdr/from_the_book.git | 9fb6080539a2f32bb24719600a9e7531abf2328d | FromTheBook/Ch02/Bertrand/CentralBinom.lean | fourPowLeMulProdPrimes | [162, 1] | [216, 45] | apply Nat.mul_le_mul | case h
n : ℕ
hn : 2 < n
⊢ (∏ x in Finset.range (Nat.succ (Nat.sqrt (2 * n))), x ^ ↑(Nat.factorization (Nat.centralBinom n)) x) *
∏ x in Finset.Ioc (Nat.sqrt (2 * n)) (2 * n), x ^ ↑(Nat.factorization (Nat.centralBinom n)) x ≤
(2 * n) ^ Nat.sqrt (2 * n) * (4 ^ Nat.pred (2 * n / 3) * ∏ p in primes (Finset.Ioc n (2 * n)), p) | case h.h₁
n : ℕ
hn : 2 < n
⊢ ∏ x in Finset.range (Nat.succ (Nat.sqrt (2 * n))), x ^ ↑(Nat.factorization (Nat.centralBinom n)) x ≤
(2 * n) ^ Nat.sqrt (2 * n)
case h.h₂
n : ℕ
hn : 2 < n
⊢ ∏ x in Finset.Ioc (Nat.sqrt (2 * n)) (2 * n), x ^ ↑(Nat.factorization (Nat.centralBinom n)) x ≤
4 ^ Nat.pred (2 * n / 3) * ∏ p in primes (Finset.Ioc n (2 * n)), p |
https://github.com/jvlmdr/from_the_book.git | 9fb6080539a2f32bb24719600a9e7531abf2328d | FromTheBook/Ch02/Bertrand/CentralBinom.lean | fourPowLeMulProdPrimes | [162, 1] | [216, 45] | . rw [FactorizationCentralBinom.prodRangePowEqProdIcoPowOfLeTwo (Nat.le_succ _ : 1 ≤ 2)]
apply le_of_le_of_eq (FactorizationCentralBinom.prodIcoPowLePow' _)
. norm_num
. intro hn; rw [hn]; norm_num | case h.h₁
n : ℕ
hn : 2 < n
⊢ ∏ x in Finset.range (Nat.succ (Nat.sqrt (2 * n))), x ^ ↑(Nat.factorization (Nat.centralBinom n)) x ≤
(2 * n) ^ Nat.sqrt (2 * n)
case h.h₂
n : ℕ
hn : 2 < n
⊢ ∏ x in Finset.Ioc (Nat.sqrt (2 * n)) (2 * n), x ^ ↑(Nat.factorization (Nat.centralBinom n)) x ≤
4 ^ Nat.pred (2 * n / 3) * ∏ p in primes (Finset.Ioc n (2 * n)), p | case h.h₂
n : ℕ
hn : 2 < n
⊢ ∏ x in Finset.Ioc (Nat.sqrt (2 * n)) (2 * n), x ^ ↑(Nat.factorization (Nat.centralBinom n)) x ≤
4 ^ Nat.pred (2 * n / 3) * ∏ p in primes (Finset.Ioc n (2 * n)), p |
https://github.com/jvlmdr/from_the_book.git | 9fb6080539a2f32bb24719600a9e7531abf2328d | FromTheBook/Ch02/Bertrand/CentralBinom.lean | fourPowLeMulProdPrimes | [162, 1] | [216, 45] | . have hs : Finset.Ioc (Nat.sqrt (2*n)) (2*n) ⊆ Finset.Ioc (Nat.sqrt (2*n)) (2*n/3) ∪ Finset.Ioc (2*n/3) (2*n)
. repeat rw [← Nat.Ico_succ_succ]
apply Finset.Ico_subset_Ico_union_Ico
apply le_trans (prodLeProdMulProdOfUnionOfOneLe hs (powFactorizationPos _))
clear hs
apply Nat.mul_le_mul
. apply le_trans FactorizationCentralBinom.prodIocSqrtPowLeProdPrimes
apply le_trans _ (@prodPrimesLePowFour (2*n/3))
rw [prodPrimes, Finset.range_eq_Ico, Nat.Ico_succ_right]
apply Finset.prod_le_prod_of_subset_of_one_le' <;> rw [primes]
. apply Finset.monotone_filter_left
exact Finset.Ioc_subset_Iic_self
. intro p hp _
simp at hp
replace hp := Nat.Prime.one_lt hp.right
linarith
. conv => lhs; rw [← Finset.prod_Ioc_consecutive _ (mulDivLeSelf (by norm_num)) (by linarith : n ≤ 2*n)]
apply mulRightLeOfLeOne
. apply le_of_eq
apply Finset.prod_eq_one
intro x hx
simp at hx
rw [Nat.factorization_centralBinom_of_two_mul_self_lt_three_mul hn hx.right]
. simp
. apply @Nat.lt_of_div_lt_div _ _ 3
simp
exact hx.left
. apply FactorizationCentralBinom.prodIocSqrtLePowLeProdPrimes
exact sqrtTwoMulLeSelf (le_of_lt hn) | case h.h₂
n : ℕ
hn : 2 < n
⊢ ∏ x in Finset.Ioc (Nat.sqrt (2 * n)) (2 * n), x ^ ↑(Nat.factorization (Nat.centralBinom n)) x ≤
4 ^ Nat.pred (2 * n / 3) * ∏ p in primes (Finset.Ioc n (2 * n)), p | no goals |
https://github.com/jvlmdr/from_the_book.git | 9fb6080539a2f32bb24719600a9e7531abf2328d | FromTheBook/Ch02/Bertrand/CentralBinom.lean | fourPowLeMulProdPrimes | [162, 1] | [216, 45] | linarith | n : ℕ
hn : 2 < n
⊢ 0 < n | no goals |
https://github.com/jvlmdr/from_the_book.git | 9fb6080539a2f32bb24719600a9e7531abf2328d | FromTheBook/Ch02/Bertrand/CentralBinom.lean | fourPowLeMulProdPrimes | [162, 1] | [216, 45] | intro x y z | n : ℕ
hn : 2 < n
⊢ ∀ {x y z : ℕ}, x * (2 * n) * y * z = 2 * n * (x * y * z) | n : ℕ
hn : 2 < n
x y z : ℕ
⊢ x * (2 * n) * y * z = 2 * n * (x * y * z) |
https://github.com/jvlmdr/from_the_book.git | 9fb6080539a2f32bb24719600a9e7531abf2328d | FromTheBook/Ch02/Bertrand/CentralBinom.lean | fourPowLeMulProdPrimes | [162, 1] | [216, 45] | linarith | n : ℕ
hn : 2 < n
x y z : ℕ
⊢ x * (2 * n) * y * z = 2 * n * (x * y * z) | no goals |
https://github.com/jvlmdr/from_the_book.git | 9fb6080539a2f32bb24719600a9e7531abf2328d | FromTheBook/Ch02/Bertrand/CentralBinom.lean | fourPowLeMulProdPrimes | [162, 1] | [216, 45] | rw [FactorizationCentralBinom.prodRangePowEqProdIcoPowOfLeTwo (Nat.le_succ _ : 1 ≤ 2)] | case h.h₁
n : ℕ
hn : 2 < n
⊢ ∏ x in Finset.range (Nat.succ (Nat.sqrt (2 * n))), x ^ ↑(Nat.factorization (Nat.centralBinom n)) x ≤
(2 * n) ^ Nat.sqrt (2 * n) | case h.h₁
n : ℕ
hn : 2 < n
⊢ ∏ p in Finset.Ico 1 (Nat.succ (Nat.sqrt (2 * n))), p ^ ↑(Nat.factorization (Nat.centralBinom n)) p ≤
(2 * n) ^ Nat.sqrt (2 * n) |
https://github.com/jvlmdr/from_the_book.git | 9fb6080539a2f32bb24719600a9e7531abf2328d | FromTheBook/Ch02/Bertrand/CentralBinom.lean | fourPowLeMulProdPrimes | [162, 1] | [216, 45] | apply le_of_le_of_eq (FactorizationCentralBinom.prodIcoPowLePow' _) | case h.h₁
n : ℕ
hn : 2 < n
⊢ ∏ p in Finset.Ico 1 (Nat.succ (Nat.sqrt (2 * n))), p ^ ↑(Nat.factorization (Nat.centralBinom n)) p ≤
(2 * n) ^ Nat.sqrt (2 * n) | case h.h₁
n : ℕ
hn : 2 < n
⊢ (2 * n) ^ Finset.card (Finset.Ico 1 (Nat.succ (Nat.sqrt (2 * n)))) = (2 * n) ^ Nat.sqrt (2 * n)
n : ℕ
hn : 2 < n
⊢ n = 0 → Finset.Ico 1 (Nat.succ (Nat.sqrt (2 * n))) = ∅ |
https://github.com/jvlmdr/from_the_book.git | 9fb6080539a2f32bb24719600a9e7531abf2328d | FromTheBook/Ch02/Bertrand/CentralBinom.lean | fourPowLeMulProdPrimes | [162, 1] | [216, 45] | . norm_num | case h.h₁
n : ℕ
hn : 2 < n
⊢ (2 * n) ^ Finset.card (Finset.Ico 1 (Nat.succ (Nat.sqrt (2 * n)))) = (2 * n) ^ Nat.sqrt (2 * n)
n : ℕ
hn : 2 < n
⊢ n = 0 → Finset.Ico 1 (Nat.succ (Nat.sqrt (2 * n))) = ∅ | n : ℕ
hn : 2 < n
⊢ n = 0 → Finset.Ico 1 (Nat.succ (Nat.sqrt (2 * n))) = ∅ |
https://github.com/jvlmdr/from_the_book.git | 9fb6080539a2f32bb24719600a9e7531abf2328d | FromTheBook/Ch02/Bertrand/CentralBinom.lean | fourPowLeMulProdPrimes | [162, 1] | [216, 45] | . intro hn; rw [hn]; norm_num | n : ℕ
hn : 2 < n
⊢ n = 0 → Finset.Ico 1 (Nat.succ (Nat.sqrt (2 * n))) = ∅ | no goals |
https://github.com/jvlmdr/from_the_book.git | 9fb6080539a2f32bb24719600a9e7531abf2328d | FromTheBook/Ch02/Bertrand/CentralBinom.lean | fourPowLeMulProdPrimes | [162, 1] | [216, 45] | norm_num | case h.h₁
n : ℕ
hn : 2 < n
⊢ (2 * n) ^ Finset.card (Finset.Ico 1 (Nat.succ (Nat.sqrt (2 * n)))) = (2 * n) ^ Nat.sqrt (2 * n) | no goals |
https://github.com/jvlmdr/from_the_book.git | 9fb6080539a2f32bb24719600a9e7531abf2328d | FromTheBook/Ch02/Bertrand/CentralBinom.lean | fourPowLeMulProdPrimes | [162, 1] | [216, 45] | intro hn | n : ℕ
hn : 2 < n
⊢ n = 0 → Finset.Ico 1 (Nat.succ (Nat.sqrt (2 * n))) = ∅ | n : ℕ
hn✝ : 2 < n
hn : n = 0
⊢ Finset.Ico 1 (Nat.succ (Nat.sqrt (2 * n))) = ∅ |
https://github.com/jvlmdr/from_the_book.git | 9fb6080539a2f32bb24719600a9e7531abf2328d | FromTheBook/Ch02/Bertrand/CentralBinom.lean | fourPowLeMulProdPrimes | [162, 1] | [216, 45] | rw [hn] | n : ℕ
hn✝ : 2 < n
hn : n = 0
⊢ Finset.Ico 1 (Nat.succ (Nat.sqrt (2 * n))) = ∅ | n : ℕ
hn✝ : 2 < n
hn : n = 0
⊢ Finset.Ico 1 (Nat.succ (Nat.sqrt (2 * 0))) = ∅ |
https://github.com/jvlmdr/from_the_book.git | 9fb6080539a2f32bb24719600a9e7531abf2328d | FromTheBook/Ch02/Bertrand/CentralBinom.lean | fourPowLeMulProdPrimes | [162, 1] | [216, 45] | norm_num | n : ℕ
hn✝ : 2 < n
hn : n = 0
⊢ Finset.Ico 1 (Nat.succ (Nat.sqrt (2 * 0))) = ∅ | no goals |
https://github.com/jvlmdr/from_the_book.git | 9fb6080539a2f32bb24719600a9e7531abf2328d | FromTheBook/Ch02/Bertrand/CentralBinom.lean | fourPowLeMulProdPrimes | [162, 1] | [216, 45] | have hs : Finset.Ioc (Nat.sqrt (2*n)) (2*n) ⊆ Finset.Ioc (Nat.sqrt (2*n)) (2*n/3) ∪ Finset.Ioc (2*n/3) (2*n) | case h.h₂
n : ℕ
hn : 2 < n
⊢ ∏ x in Finset.Ioc (Nat.sqrt (2 * n)) (2 * n), x ^ ↑(Nat.factorization (Nat.centralBinom n)) x ≤
4 ^ Nat.pred (2 * n / 3) * ∏ p in primes (Finset.Ioc n (2 * n)), p | case hs
n : ℕ
hn : 2 < n
⊢ Finset.Ioc (Nat.sqrt (2 * n)) (2 * n) ⊆ Finset.Ioc (Nat.sqrt (2 * n)) (2 * n / 3) ∪ Finset.Ioc (2 * n / 3) (2 * n)
case h.h₂
n : ℕ
hn : 2 < n
hs : Finset.Ioc (Nat.sqrt (2 * n)) (2 * n) ⊆ Finset.Ioc (Nat.sqrt (2 * n)) (2 * n / 3) ∪ Finset.Ioc (2 * n / 3) (2 * n)
⊢ ∏ x in Finset.Ioc (Nat.sqrt (2 * n)) (2 * n), x ^ ↑(Nat.factorization (Nat.centralBinom n)) x ≤
4 ^ Nat.pred (2 * n / 3) * ∏ p in primes (Finset.Ioc n (2 * n)), p |
https://github.com/jvlmdr/from_the_book.git | 9fb6080539a2f32bb24719600a9e7531abf2328d | FromTheBook/Ch02/Bertrand/CentralBinom.lean | fourPowLeMulProdPrimes | [162, 1] | [216, 45] | . repeat rw [← Nat.Ico_succ_succ]
apply Finset.Ico_subset_Ico_union_Ico | case hs
n : ℕ
hn : 2 < n
⊢ Finset.Ioc (Nat.sqrt (2 * n)) (2 * n) ⊆ Finset.Ioc (Nat.sqrt (2 * n)) (2 * n / 3) ∪ Finset.Ioc (2 * n / 3) (2 * n)
case h.h₂
n : ℕ
hn : 2 < n
hs : Finset.Ioc (Nat.sqrt (2 * n)) (2 * n) ⊆ Finset.Ioc (Nat.sqrt (2 * n)) (2 * n / 3) ∪ Finset.Ioc (2 * n / 3) (2 * n)
⊢ ∏ x in Finset.Ioc (Nat.sqrt (2 * n)) (2 * n), x ^ ↑(Nat.factorization (Nat.centralBinom n)) x ≤
4 ^ Nat.pred (2 * n / 3) * ∏ p in primes (Finset.Ioc n (2 * n)), p | case h.h₂
n : ℕ
hn : 2 < n
hs : Finset.Ioc (Nat.sqrt (2 * n)) (2 * n) ⊆ Finset.Ioc (Nat.sqrt (2 * n)) (2 * n / 3) ∪ Finset.Ioc (2 * n / 3) (2 * n)
⊢ ∏ x in Finset.Ioc (Nat.sqrt (2 * n)) (2 * n), x ^ ↑(Nat.factorization (Nat.centralBinom n)) x ≤
4 ^ Nat.pred (2 * n / 3) * ∏ p in primes (Finset.Ioc n (2 * n)), p |
https://github.com/jvlmdr/from_the_book.git | 9fb6080539a2f32bb24719600a9e7531abf2328d | FromTheBook/Ch02/Bertrand/CentralBinom.lean | fourPowLeMulProdPrimes | [162, 1] | [216, 45] | apply le_trans (prodLeProdMulProdOfUnionOfOneLe hs (powFactorizationPos _)) | case h.h₂
n : ℕ
hn : 2 < n
hs : Finset.Ioc (Nat.sqrt (2 * n)) (2 * n) ⊆ Finset.Ioc (Nat.sqrt (2 * n)) (2 * n / 3) ∪ Finset.Ioc (2 * n / 3) (2 * n)
⊢ ∏ x in Finset.Ioc (Nat.sqrt (2 * n)) (2 * n), x ^ ↑(Nat.factorization (Nat.centralBinom n)) x ≤
4 ^ Nat.pred (2 * n / 3) * ∏ p in primes (Finset.Ioc n (2 * n)), p | case h.h₂
n : ℕ
hn : 2 < n
hs : Finset.Ioc (Nat.sqrt (2 * n)) (2 * n) ⊆ Finset.Ioc (Nat.sqrt (2 * n)) (2 * n / 3) ∪ Finset.Ioc (2 * n / 3) (2 * n)
⊢ (∏ x in Finset.Ioc (Nat.sqrt (2 * n)) (2 * n / 3), x ^ ↑(Nat.factorization (Nat.centralBinom n)) x) *
∏ x in Finset.Ioc (2 * n / 3) (2 * n), x ^ ↑(Nat.factorization (Nat.centralBinom n)) x ≤
4 ^ Nat.pred (2 * n / 3) * ∏ p in primes (Finset.Ioc n (2 * n)), p |
https://github.com/jvlmdr/from_the_book.git | 9fb6080539a2f32bb24719600a9e7531abf2328d | FromTheBook/Ch02/Bertrand/CentralBinom.lean | fourPowLeMulProdPrimes | [162, 1] | [216, 45] | clear hs | case h.h₂
n : ℕ
hn : 2 < n
hs : Finset.Ioc (Nat.sqrt (2 * n)) (2 * n) ⊆ Finset.Ioc (Nat.sqrt (2 * n)) (2 * n / 3) ∪ Finset.Ioc (2 * n / 3) (2 * n)
⊢ (∏ x in Finset.Ioc (Nat.sqrt (2 * n)) (2 * n / 3), x ^ ↑(Nat.factorization (Nat.centralBinom n)) x) *
∏ x in Finset.Ioc (2 * n / 3) (2 * n), x ^ ↑(Nat.factorization (Nat.centralBinom n)) x ≤
4 ^ Nat.pred (2 * n / 3) * ∏ p in primes (Finset.Ioc n (2 * n)), p | case h.h₂
n : ℕ
hn : 2 < n
⊢ (∏ x in Finset.Ioc (Nat.sqrt (2 * n)) (2 * n / 3), x ^ ↑(Nat.factorization (Nat.centralBinom n)) x) *
∏ x in Finset.Ioc (2 * n / 3) (2 * n), x ^ ↑(Nat.factorization (Nat.centralBinom n)) x ≤
4 ^ Nat.pred (2 * n / 3) * ∏ p in primes (Finset.Ioc n (2 * n)), p |
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