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https://github.com/b-mehta/ExponentialRamsey.git | 7e17b629a915a082869f494c8afa56a3e1c7a88d | ExponentialRamsey/Prereq/Mathlib/Data/Nat/Choose/Basic.lean | Nat.choose_le_pow_left | [22, 1] | [28, 17] | exact zero_le' | case inr
s t : ℕ
h : s < t
⊢ 0 ≤ (s + 1 - t) ^ t | no goals |
https://github.com/b-mehta/ExponentialRamsey.git | 7e17b629a915a082869f494c8afa56a3e1c7a88d | ExponentialRamsey/Prereq/Mathlib/Data/Nat/Choose/Basic.lean | Nat.choose_le_pow_left | [22, 1] | [28, 17] | obtain ⟨s, rfl⟩ := exists_add_of_le h | case inl
s t : ℕ
h : t ≤ s
⊢ choose s t ≤ (s + 1 - t) ^ t | case inl.intro
t s : ℕ
h : t ≤ t + s
⊢ choose (t + s) t ≤ (t + s + 1 - t) ^ t |
https://github.com/b-mehta/ExponentialRamsey.git | 7e17b629a915a082869f494c8afa56a3e1c7a88d | ExponentialRamsey/Prereq/Mathlib/Data/Nat/Choose/Basic.lean | Nat.choose_le_pow_left | [22, 1] | [28, 17] | refine (choose_add_le_pow_left t s).trans ?_ | case inl.intro
t s : ℕ
h : t ≤ t + s
⊢ choose (t + s) t ≤ (t + s + 1 - t) ^ t | case inl.intro
t s : ℕ
h : t ≤ t + s
⊢ (s + 1) ^ t ≤ (t + s + 1 - t) ^ t |
https://github.com/b-mehta/ExponentialRamsey.git | 7e17b629a915a082869f494c8afa56a3e1c7a88d | ExponentialRamsey/Prereq/Mathlib/Data/Nat/Choose/Basic.lean | Nat.choose_le_pow_left | [22, 1] | [28, 17] | rw [add_assoc, Nat.add_sub_cancel_left] | case inl.intro
t s : ℕ
h : t ≤ t + s
⊢ (s + 1) ^ t ≤ (t + s + 1 - t) ^ t | no goals |
https://github.com/b-mehta/ExponentialRamsey.git | 7e17b629a915a082869f494c8afa56a3e1c7a88d | ExponentialRamsey/Prereq/Mathlib/Algebra/BigOperators/Ring.lean | sum_tsub | [17, 1] | [23, 77] | rw [← Finset.sum_add_distrib] | α : Type u_1
β : Type u_2
inst✝⁶ : AddCommMonoid β
inst✝⁵ : PartialOrder β
inst✝⁴ : ExistsAddOfLE β
inst✝³ : CovariantClass β β (fun x x_1 => x + x_1) fun x x_1 => x ≤ x_1
inst✝² : ContravariantClass β β (fun x x_1 => x + x_1) fun x x_1 => x ≤ x_1
inst✝¹ : Sub β
inst✝ : OrderedSub β
s : Finset α
f g : α → β
hfg : ∀ x ∈ s, g x ≤ f x
⊢ ∑ x in s, (f x - g x) + ∑ x in s, g x = ∑ x in s, f x | α : Type u_1
β : Type u_2
inst✝⁶ : AddCommMonoid β
inst✝⁵ : PartialOrder β
inst✝⁴ : ExistsAddOfLE β
inst✝³ : CovariantClass β β (fun x x_1 => x + x_1) fun x x_1 => x ≤ x_1
inst✝² : ContravariantClass β β (fun x x_1 => x + x_1) fun x x_1 => x ≤ x_1
inst✝¹ : Sub β
inst✝ : OrderedSub β
s : Finset α
f g : α → β
hfg : ∀ x ∈ s, g x ≤ f x
⊢ ∑ x in s, (f x - g x + g x) = ∑ x in s, f x |
https://github.com/b-mehta/ExponentialRamsey.git | 7e17b629a915a082869f494c8afa56a3e1c7a88d | ExponentialRamsey/Prereq/Mathlib/Algebra/BigOperators/Ring.lean | sum_tsub | [17, 1] | [23, 77] | exact Finset.sum_congr rfl fun x hx => tsub_add_cancel_of_le <| hfg _ hx | α : Type u_1
β : Type u_2
inst✝⁶ : AddCommMonoid β
inst✝⁵ : PartialOrder β
inst✝⁴ : ExistsAddOfLE β
inst✝³ : CovariantClass β β (fun x x_1 => x + x_1) fun x x_1 => x ≤ x_1
inst✝² : ContravariantClass β β (fun x x_1 => x + x_1) fun x x_1 => x ≤ x_1
inst✝¹ : Sub β
inst✝ : OrderedSub β
s : Finset α
f g : α → β
hfg : ∀ x ∈ s, g x ≤ f x
⊢ ∑ x in s, (f x - g x + g x) = ∑ x in s, f x | no goals |
https://github.com/b-mehta/ExponentialRamsey.git | 7e17b629a915a082869f494c8afa56a3e1c7a88d | ExponentialRamsey/Prereq/Mathlib/Data/Nat/Factorial/Basic.lean | Nat.asc_le_pow_mul_factorial | [17, 1] | [23, 7] | induction' s with s ih | s t : ℕ
⊢ ascFactorial t s ≤ s ! * (t + 1) ^ s | case zero
t : ℕ
⊢ ascFactorial t zero ≤ zero ! * (t + 1) ^ zero
case succ
t s : ℕ
ih : ascFactorial t s ≤ s ! * (t + 1) ^ s
⊢ ascFactorial t (succ s) ≤ (succ s)! * (t + 1) ^ succ s |
https://github.com/b-mehta/ExponentialRamsey.git | 7e17b629a915a082869f494c8afa56a3e1c7a88d | ExponentialRamsey/Prereq/Mathlib/Data/Nat/Factorial/Basic.lean | Nat.asc_le_pow_mul_factorial | [17, 1] | [23, 7] | rw [ascFactorial_succ, factorial_succ, pow_succ', mul_mul_mul_comm] | case succ
t s : ℕ
ih : ascFactorial t s ≤ s ! * (t + 1) ^ s
⊢ ascFactorial t (succ s) ≤ (succ s)! * (t + 1) ^ succ s | case succ
t s : ℕ
ih : ascFactorial t s ≤ s ! * (t + 1) ^ s
⊢ (t + s + 1) * ascFactorial t s ≤ (s + 1) * (t + 1) * (s ! * (t + 1) ^ s) |
https://github.com/b-mehta/ExponentialRamsey.git | 7e17b629a915a082869f494c8afa56a3e1c7a88d | ExponentialRamsey/Prereq/Mathlib/Data/Nat/Factorial/Basic.lean | Nat.asc_le_pow_mul_factorial | [17, 1] | [23, 7] | refine' Nat.mul_le_mul _ ih | case succ
t s : ℕ
ih : ascFactorial t s ≤ s ! * (t + 1) ^ s
⊢ (t + s + 1) * ascFactorial t s ≤ (s + 1) * (t + 1) * (s ! * (t + 1) ^ s) | case succ
t s : ℕ
ih : ascFactorial t s ≤ s ! * (t + 1) ^ s
⊢ t + s + 1 ≤ (s + 1) * (t + 1) |
https://github.com/b-mehta/ExponentialRamsey.git | 7e17b629a915a082869f494c8afa56a3e1c7a88d | ExponentialRamsey/Prereq/Mathlib/Data/Nat/Factorial/Basic.lean | Nat.asc_le_pow_mul_factorial | [17, 1] | [23, 7] | rw [add_comm t, add_one_mul s, mul_add_one s, add_assoc] | case succ
t s : ℕ
ih : ascFactorial t s ≤ s ! * (t + 1) ^ s
⊢ t + s + 1 ≤ (s + 1) * (t + 1) | case succ
t s : ℕ
ih : ascFactorial t s ≤ s ! * (t + 1) ^ s
⊢ s + (t + 1) ≤ s * t + s + (t + 1) |
https://github.com/b-mehta/ExponentialRamsey.git | 7e17b629a915a082869f494c8afa56a3e1c7a88d | ExponentialRamsey/Prereq/Mathlib/Data/Nat/Factorial/Basic.lean | Nat.asc_le_pow_mul_factorial | [17, 1] | [23, 7] | simp | case succ
t s : ℕ
ih : ascFactorial t s ≤ s ! * (t + 1) ^ s
⊢ s + (t + 1) ≤ s * t + s + (t + 1) | no goals |
https://github.com/b-mehta/ExponentialRamsey.git | 7e17b629a915a082869f494c8afa56a3e1c7a88d | ExponentialRamsey/Prereq/Mathlib/Data/Nat/Factorial/Basic.lean | Nat.asc_le_pow_mul_factorial | [17, 1] | [23, 7] | simp | case zero
t : ℕ
⊢ ascFactorial t zero ≤ zero ! * (t + 1) ^ zero | no goals |
https://github.com/b-mehta/ExponentialRamsey.git | 7e17b629a915a082869f494c8afa56a3e1c7a88d | ExponentialRamsey/Prereq/Mathlib/Analysis/SpecialFunctions/ExplicitStirling.lean | centralBinom_ratio | [28, 1] | [38, 35] | rw [mul_div_assoc', mul_add, eq_div_iff, ← cast_add_one, div_mul_eq_mul_div, mul_comm, ← cast_mul,
succ_mul_centralBinom_succ, cast_mul, mul_div_cancel] | α : Type u_1
inst✝¹ : Field α
inst✝ : CharZero α
n : ℕ
⊢ ↑(centralBinom (n + 1)) / ↑(centralBinom n) = 4 * ((↑n + 1 / 2) / (↑n + 1)) | α : Type u_1
inst✝¹ : Field α
inst✝ : CharZero α
n : ℕ
⊢ ↑(2 * (2 * n + 1)) = 4 * ↑n + 4 * (1 / 2)
case h
α : Type u_1
inst✝¹ : Field α
inst✝ : CharZero α
n : ℕ
⊢ ↑(centralBinom n) ≠ 0
α : Type u_1
inst✝¹ : Field α
inst✝ : CharZero α
n : ℕ
⊢ ↑n + 1 ≠ 0 |
https://github.com/b-mehta/ExponentialRamsey.git | 7e17b629a915a082869f494c8afa56a3e1c7a88d | ExponentialRamsey/Prereq/Mathlib/Analysis/SpecialFunctions/ExplicitStirling.lean | centralBinom_ratio | [28, 1] | [38, 35] | exact Nat.cast_add_one_ne_zero _ | α : Type u_1
inst✝¹ : Field α
inst✝ : CharZero α
n : ℕ
⊢ ↑n + 1 ≠ 0 | no goals |
https://github.com/b-mehta/ExponentialRamsey.git | 7e17b629a915a082869f494c8afa56a3e1c7a88d | ExponentialRamsey/Prereq/Mathlib/Analysis/SpecialFunctions/ExplicitStirling.lean | centralBinom_ratio | [28, 1] | [38, 35] | rw [mul_add_one, ← mul_assoc, cast_add, cast_mul] | α : Type u_1
inst✝¹ : Field α
inst✝ : CharZero α
n : ℕ
⊢ ↑(2 * (2 * n + 1)) = 4 * ↑n + 4 * (1 / 2) | α : Type u_1
inst✝¹ : Field α
inst✝ : CharZero α
n : ℕ
⊢ ↑(2 * 2) * ↑n + ↑2 = 4 * ↑n + 4 * (1 / 2) |
https://github.com/b-mehta/ExponentialRamsey.git | 7e17b629a915a082869f494c8afa56a3e1c7a88d | ExponentialRamsey/Prereq/Mathlib/Analysis/SpecialFunctions/ExplicitStirling.lean | centralBinom_ratio | [28, 1] | [38, 35] | norm_num1 | α : Type u_1
inst✝¹ : Field α
inst✝ : CharZero α
n : ℕ
⊢ ↑(2 * 2) * ↑n + ↑2 = 4 * ↑n + 4 * (1 / 2) | α : Type u_1
inst✝¹ : Field α
inst✝ : CharZero α
n : ℕ
⊢ 4 * ↑n + 2 = 4 * ↑n + 2 |
https://github.com/b-mehta/ExponentialRamsey.git | 7e17b629a915a082869f494c8afa56a3e1c7a88d | ExponentialRamsey/Prereq/Mathlib/Analysis/SpecialFunctions/ExplicitStirling.lean | centralBinom_ratio | [28, 1] | [38, 35] | rfl | α : Type u_1
inst✝¹ : Field α
inst✝ : CharZero α
n : ℕ
⊢ 4 * ↑n + 2 = 4 * ↑n + 2 | no goals |
https://github.com/b-mehta/ExponentialRamsey.git | 7e17b629a915a082869f494c8afa56a3e1c7a88d | ExponentialRamsey/Prereq/Mathlib/Analysis/SpecialFunctions/ExplicitStirling.lean | centralBinom_ratio | [28, 1] | [38, 35] | rw [Nat.cast_ne_zero] | case h
α : Type u_1
inst✝¹ : Field α
inst✝ : CharZero α
n : ℕ
⊢ ↑(centralBinom n) ≠ 0 | case h
α : Type u_1
inst✝¹ : Field α
inst✝ : CharZero α
n : ℕ
⊢ centralBinom n ≠ 0 |
https://github.com/b-mehta/ExponentialRamsey.git | 7e17b629a915a082869f494c8afa56a3e1c7a88d | ExponentialRamsey/Prereq/Mathlib/Analysis/SpecialFunctions/ExplicitStirling.lean | centralBinom_ratio | [28, 1] | [38, 35] | exact centralBinom_ne_zero _ | case h
α : Type u_1
inst✝¹ : Field α
inst✝ : CharZero α
n : ℕ
⊢ centralBinom n ≠ 0 | no goals |
https://github.com/b-mehta/ExponentialRamsey.git | 7e17b629a915a082869f494c8afa56a3e1c7a88d | ExponentialRamsey/Prereq/Mathlib/Analysis/SpecialFunctions/ExplicitStirling.lean | upper_monotone_aux | [42, 1] | [50, 23] | rw [← Real.sqrt_div] | n : ℕ
⊢ (↑n + 1 / 2) / (↑n + 1) ≤ Real.sqrt (↑n + 1 / 3) / Real.sqrt (↑n + 1 + 1 / 3) | n : ℕ
⊢ (↑n + 1 / 2) / (↑n + 1) ≤ Real.sqrt ((↑n + 1 / 3) / (↑n + 1 + 1 / 3))
case hx
n : ℕ
⊢ 0 ≤ ↑n + 1 / 3 |
https://github.com/b-mehta/ExponentialRamsey.git | 7e17b629a915a082869f494c8afa56a3e1c7a88d | ExponentialRamsey/Prereq/Mathlib/Analysis/SpecialFunctions/ExplicitStirling.lean | upper_monotone_aux | [42, 1] | [50, 23] | refine' Real.le_sqrt_of_sq_le _ | n : ℕ
⊢ (↑n + 1 / 2) / (↑n + 1) ≤ Real.sqrt ((↑n + 1 / 3) / (↑n + 1 + 1 / 3))
case hx
n : ℕ
⊢ 0 ≤ ↑n + 1 / 3 | n : ℕ
⊢ ((↑n + 1 / 2) / (↑n + 1)) ^ 2 ≤ (↑n + 1 / 3) / (↑n + 1 + 1 / 3)
case hx
n : ℕ
⊢ 0 ≤ ↑n + 1 / 3 |
https://github.com/b-mehta/ExponentialRamsey.git | 7e17b629a915a082869f494c8afa56a3e1c7a88d | ExponentialRamsey/Prereq/Mathlib/Analysis/SpecialFunctions/ExplicitStirling.lean | upper_monotone_aux | [42, 1] | [50, 23] | rw [div_pow, div_le_div_iff] | n : ℕ
⊢ ((↑n + 1 / 2) / (↑n + 1)) ^ 2 ≤ (↑n + 1 / 3) / (↑n + 1 + 1 / 3)
case hx
n : ℕ
⊢ 0 ≤ ↑n + 1 / 3 | n : ℕ
⊢ (↑n + 1 / 2) ^ 2 * (↑n + 1 + 1 / 3) ≤ (↑n + 1 / 3) * (↑n + 1) ^ 2
case b0
n : ℕ
⊢ 0 < (↑n + 1) ^ 2
case d0
n : ℕ
⊢ 0 < ↑n + 1 + 1 / 3
case hx
n : ℕ
⊢ 0 ≤ ↑n + 1 / 3 |
https://github.com/b-mehta/ExponentialRamsey.git | 7e17b629a915a082869f494c8afa56a3e1c7a88d | ExponentialRamsey/Prereq/Mathlib/Analysis/SpecialFunctions/ExplicitStirling.lean | upper_monotone_aux | [42, 1] | [50, 23] | all_goals positivity | case b0
n : ℕ
⊢ 0 < (↑n + 1) ^ 2
case d0
n : ℕ
⊢ 0 < ↑n + 1 + 1 / 3
case hx
n : ℕ
⊢ 0 ≤ ↑n + 1 / 3 | no goals |
https://github.com/b-mehta/ExponentialRamsey.git | 7e17b629a915a082869f494c8afa56a3e1c7a88d | ExponentialRamsey/Prereq/Mathlib/Analysis/SpecialFunctions/ExplicitStirling.lean | upper_monotone_aux | [42, 1] | [50, 23] | ring_nf | n : ℕ
⊢ (↑n + 1 / 2) ^ 2 * (↑n + 1 + 1 / 3) ≤ (↑n + 1 / 3) * (↑n + 1) ^ 2 | n : ℕ
⊢ ↑(Int.ofNat 1) / ↑3 + ↑n * (↑(Int.ofNat 19) / ↑12) + ↑n ^ 2 * (↑(Int.ofNat 7) / ↑3) + ↑n ^ 3 ≤
↑(Int.ofNat 1) / ↑3 + ↑n * (↑(Int.ofNat 5) / ↑3) + ↑n ^ 2 * (↑(Int.ofNat 7) / ↑3) + ↑n ^ 3 |
https://github.com/b-mehta/ExponentialRamsey.git | 7e17b629a915a082869f494c8afa56a3e1c7a88d | ExponentialRamsey/Prereq/Mathlib/Analysis/SpecialFunctions/ExplicitStirling.lean | upper_monotone_aux | [42, 1] | [50, 23] | gcongr _ + _ * ?_ + _ + _ | n : ℕ
⊢ ↑(Int.ofNat 1) / ↑3 + ↑n * (↑(Int.ofNat 19) / ↑12) + ↑n ^ 2 * (↑(Int.ofNat 7) / ↑3) + ↑n ^ 3 ≤
↑(Int.ofNat 1) / ↑3 + ↑n * (↑(Int.ofNat 5) / ↑3) + ↑n ^ 2 * (↑(Int.ofNat 7) / ↑3) + ↑n ^ 3 | case bc.bc.bc.h
n : ℕ
⊢ ↑(Int.ofNat 19) / ↑12 ≤ ↑(Int.ofNat 5) / ↑3 |
https://github.com/b-mehta/ExponentialRamsey.git | 7e17b629a915a082869f494c8afa56a3e1c7a88d | ExponentialRamsey/Prereq/Mathlib/Analysis/SpecialFunctions/ExplicitStirling.lean | upper_monotone_aux | [42, 1] | [50, 23] | norm_num1 | case bc.bc.bc.h
n : ℕ
⊢ ↑(Int.ofNat 19) / ↑12 ≤ ↑(Int.ofNat 5) / ↑3 | no goals |
https://github.com/b-mehta/ExponentialRamsey.git | 7e17b629a915a082869f494c8afa56a3e1c7a88d | ExponentialRamsey/Prereq/Mathlib/Analysis/SpecialFunctions/ExplicitStirling.lean | upper_monotone_aux | [42, 1] | [50, 23] | positivity | case hx
n : ℕ
⊢ 0 ≤ ↑n + 1 / 3 | no goals |
https://github.com/b-mehta/ExponentialRamsey.git | 7e17b629a915a082869f494c8afa56a3e1c7a88d | ExponentialRamsey/Prereq/Mathlib/Analysis/SpecialFunctions/ExplicitStirling.lean | lower_monotone_aux | [52, 1] | [58, 23] | rw [← Real.sqrt_div, sqrt_le_left, div_pow, div_le_div_iff] | n : ℕ
⊢ Real.sqrt (↑n + 1 / 4) / Real.sqrt (↑n + 1 + 1 / 4) ≤ (↑n + 1 / 2) / (↑n + 1) | n : ℕ
⊢ (↑n + 1 / 4) * (↑n + 1) ^ 2 ≤ (↑n + 1 / 2) ^ 2 * (↑n + 1 + 1 / 4)
case b0
n : ℕ
⊢ 0 < ↑n + 1 + 1 / 4
case d0
n : ℕ
⊢ 0 < (↑n + 1) ^ 2
n : ℕ
⊢ 0 ≤ (↑n + 1 / 2) / (↑n + 1)
case hx
n : ℕ
⊢ 0 ≤ ↑n + 1 / 4 |
https://github.com/b-mehta/ExponentialRamsey.git | 7e17b629a915a082869f494c8afa56a3e1c7a88d | ExponentialRamsey/Prereq/Mathlib/Analysis/SpecialFunctions/ExplicitStirling.lean | lower_monotone_aux | [52, 1] | [58, 23] | all_goals positivity | case b0
n : ℕ
⊢ 0 < ↑n + 1 + 1 / 4
case d0
n : ℕ
⊢ 0 < (↑n + 1) ^ 2
n : ℕ
⊢ 0 ≤ (↑n + 1 / 2) / (↑n + 1)
case hx
n : ℕ
⊢ 0 ≤ ↑n + 1 / 4 | no goals |
https://github.com/b-mehta/ExponentialRamsey.git | 7e17b629a915a082869f494c8afa56a3e1c7a88d | ExponentialRamsey/Prereq/Mathlib/Analysis/SpecialFunctions/ExplicitStirling.lean | lower_monotone_aux | [52, 1] | [58, 23] | ring_nf | n : ℕ
⊢ (↑n + 1 / 4) * (↑n + 1) ^ 2 ≤ (↑n + 1 / 2) ^ 2 * (↑n + 1 + 1 / 4) | n : ℕ
⊢ ↑(Int.ofNat 1) / ↑4 + ↑n * (↑(Int.ofNat 3) / ↑2) + ↑n ^ 2 * (↑(Int.ofNat 9) / ↑4) + ↑n ^ 3 ≤
↑(Int.ofNat 5) / ↑16 + ↑n * (↑(Int.ofNat 3) / ↑2) + ↑n ^ 2 * (↑(Int.ofNat 9) / ↑4) + ↑n ^ 3 |
https://github.com/b-mehta/ExponentialRamsey.git | 7e17b629a915a082869f494c8afa56a3e1c7a88d | ExponentialRamsey/Prereq/Mathlib/Analysis/SpecialFunctions/ExplicitStirling.lean | lower_monotone_aux | [52, 1] | [58, 23] | gcongr ?_ + _ + _ + _ | n : ℕ
⊢ ↑(Int.ofNat 1) / ↑4 + ↑n * (↑(Int.ofNat 3) / ↑2) + ↑n ^ 2 * (↑(Int.ofNat 9) / ↑4) + ↑n ^ 3 ≤
↑(Int.ofNat 5) / ↑16 + ↑n * (↑(Int.ofNat 3) / ↑2) + ↑n ^ 2 * (↑(Int.ofNat 9) / ↑4) + ↑n ^ 3 | case bc.bc.bc
n : ℕ
⊢ ↑(Int.ofNat 1) / ↑4 ≤ ↑(Int.ofNat 5) / ↑16 |
https://github.com/b-mehta/ExponentialRamsey.git | 7e17b629a915a082869f494c8afa56a3e1c7a88d | ExponentialRamsey/Prereq/Mathlib/Analysis/SpecialFunctions/ExplicitStirling.lean | lower_monotone_aux | [52, 1] | [58, 23] | norm_num1 | case bc.bc.bc
n : ℕ
⊢ ↑(Int.ofNat 1) / ↑4 ≤ ↑(Int.ofNat 5) / ↑16 | no goals |
https://github.com/b-mehta/ExponentialRamsey.git | 7e17b629a915a082869f494c8afa56a3e1c7a88d | ExponentialRamsey/Prereq/Mathlib/Analysis/SpecialFunctions/ExplicitStirling.lean | lower_monotone_aux | [52, 1] | [58, 23] | positivity | case hx
n : ℕ
⊢ 0 ≤ ↑n + 1 / 4 | no goals |
https://github.com/b-mehta/ExponentialRamsey.git | 7e17b629a915a082869f494c8afa56a3e1c7a88d | ExponentialRamsey/Prereq/Mathlib/Analysis/SpecialFunctions/ExplicitStirling.lean | centralBinomialLower_monotone | [66, 1] | [77, 15] | refine' monotone_nat_of_le_succ _ | ⊢ Monotone centralBinomialLower | ⊢ ∀ (n : ℕ), centralBinomialLower n ≤ centralBinomialLower (n + 1) |
https://github.com/b-mehta/ExponentialRamsey.git | 7e17b629a915a082869f494c8afa56a3e1c7a88d | ExponentialRamsey/Prereq/Mathlib/Analysis/SpecialFunctions/ExplicitStirling.lean | centralBinomialLower_monotone | [66, 1] | [77, 15] | intro n | ⊢ ∀ (n : ℕ), centralBinomialLower n ≤ centralBinomialLower (n + 1) | n : ℕ
⊢ centralBinomialLower n ≤ centralBinomialLower (n + 1) |
https://github.com/b-mehta/ExponentialRamsey.git | 7e17b629a915a082869f494c8afa56a3e1c7a88d | ExponentialRamsey/Prereq/Mathlib/Analysis/SpecialFunctions/ExplicitStirling.lean | centralBinomialLower_monotone | [66, 1] | [77, 15] | rw [centralBinomialLower, centralBinomialLower, _root_.pow_succ, ←div_div] | n : ℕ
⊢ centralBinomialLower n ≤ centralBinomialLower (n + 1) | n : ℕ
⊢ ↑(centralBinom n) * Real.sqrt (↑n + 1 / 4) / 4 ^ n ≤
↑(centralBinom (n + 1)) * Real.sqrt (↑(n + 1) + 1 / 4) / 4 / 4 ^ n |
https://github.com/b-mehta/ExponentialRamsey.git | 7e17b629a915a082869f494c8afa56a3e1c7a88d | ExponentialRamsey/Prereq/Mathlib/Analysis/SpecialFunctions/ExplicitStirling.lean | centralBinomialLower_monotone | [66, 1] | [77, 15] | refine' div_le_div_of_le (by positivity) _ | n : ℕ
⊢ ↑(centralBinom n) * Real.sqrt (↑n + 1 / 4) / 4 ^ n ≤
↑(centralBinom (n + 1)) * Real.sqrt (↑(n + 1) + 1 / 4) / 4 / 4 ^ n | n : ℕ
⊢ ↑(centralBinom n) * Real.sqrt (↑n + 1 / 4) ≤ ↑(centralBinom (n + 1)) * Real.sqrt (↑(n + 1) + 1 / 4) / 4 |
https://github.com/b-mehta/ExponentialRamsey.git | 7e17b629a915a082869f494c8afa56a3e1c7a88d | ExponentialRamsey/Prereq/Mathlib/Analysis/SpecialFunctions/ExplicitStirling.lean | centralBinomialLower_monotone | [66, 1] | [77, 15] | rw [le_div_iff, mul_assoc, mul_comm, ← div_le_div_iff, centralBinom_ratio, mul_comm,
mul_div_assoc, Nat.cast_add_one] | n : ℕ
⊢ ↑(centralBinom n) * Real.sqrt (↑n + 1 / 4) ≤ ↑(centralBinom (n + 1)) * Real.sqrt (↑(n + 1) + 1 / 4) / 4 | n : ℕ
⊢ 4 * (Real.sqrt (↑n + 1 / 4) / Real.sqrt (↑n + 1 + 1 / 4)) ≤ 4 * ((↑n + 1 / 2) / (↑n + 1))
case b0
n : ℕ
⊢ 0 < Real.sqrt (↑(n + 1) + 1 / 4)
case d0
n : ℕ
⊢ 0 < ↑(centralBinom n)
n : ℕ
⊢ 0 < 4 |
https://github.com/b-mehta/ExponentialRamsey.git | 7e17b629a915a082869f494c8afa56a3e1c7a88d | ExponentialRamsey/Prereq/Mathlib/Analysis/SpecialFunctions/ExplicitStirling.lean | centralBinomialLower_monotone | [66, 1] | [77, 15] | refine' mul_le_mul_of_nonneg_left (lower_monotone_aux n) (by positivity) | n : ℕ
⊢ 4 * (Real.sqrt (↑n + 1 / 4) / Real.sqrt (↑n + 1 + 1 / 4)) ≤ 4 * ((↑n + 1 / 2) / (↑n + 1))
case b0
n : ℕ
⊢ 0 < Real.sqrt (↑(n + 1) + 1 / 4)
case d0
n : ℕ
⊢ 0 < ↑(centralBinom n)
n : ℕ
⊢ 0 < 4 | case b0
n : ℕ
⊢ 0 < Real.sqrt (↑(n + 1) + 1 / 4)
case d0
n : ℕ
⊢ 0 < ↑(centralBinom n)
n : ℕ
⊢ 0 < 4 |
https://github.com/b-mehta/ExponentialRamsey.git | 7e17b629a915a082869f494c8afa56a3e1c7a88d | ExponentialRamsey/Prereq/Mathlib/Analysis/SpecialFunctions/ExplicitStirling.lean | centralBinomialLower_monotone | [66, 1] | [77, 15] | positivity | n : ℕ
⊢ 0 ≤ 4 ^ n | no goals |
https://github.com/b-mehta/ExponentialRamsey.git | 7e17b629a915a082869f494c8afa56a3e1c7a88d | ExponentialRamsey/Prereq/Mathlib/Analysis/SpecialFunctions/ExplicitStirling.lean | centralBinomialLower_monotone | [66, 1] | [77, 15] | positivity | n : ℕ
⊢ 0 ≤ 4 | no goals |
https://github.com/b-mehta/ExponentialRamsey.git | 7e17b629a915a082869f494c8afa56a3e1c7a88d | ExponentialRamsey/Prereq/Mathlib/Analysis/SpecialFunctions/ExplicitStirling.lean | centralBinomialLower_monotone | [66, 1] | [77, 15] | positivity | case b0
n : ℕ
⊢ 0 < Real.sqrt (↑(n + 1) + 1 / 4) | no goals |
https://github.com/b-mehta/ExponentialRamsey.git | 7e17b629a915a082869f494c8afa56a3e1c7a88d | ExponentialRamsey/Prereq/Mathlib/Analysis/SpecialFunctions/ExplicitStirling.lean | centralBinomialLower_monotone | [66, 1] | [77, 15] | rw [Nat.cast_pos] | case d0
n : ℕ
⊢ 0 < ↑(centralBinom n) | case d0
n : ℕ
⊢ 0 < centralBinom n |
https://github.com/b-mehta/ExponentialRamsey.git | 7e17b629a915a082869f494c8afa56a3e1c7a88d | ExponentialRamsey/Prereq/Mathlib/Analysis/SpecialFunctions/ExplicitStirling.lean | centralBinomialLower_monotone | [66, 1] | [77, 15] | exact centralBinom_pos _ | case d0
n : ℕ
⊢ 0 < centralBinom n | no goals |
https://github.com/b-mehta/ExponentialRamsey.git | 7e17b629a915a082869f494c8afa56a3e1c7a88d | ExponentialRamsey/Prereq/Mathlib/Analysis/SpecialFunctions/ExplicitStirling.lean | centralBinomialLower_monotone | [66, 1] | [77, 15] | positivity | n : ℕ
⊢ 0 < 4 | no goals |
https://github.com/b-mehta/ExponentialRamsey.git | 7e17b629a915a082869f494c8afa56a3e1c7a88d | ExponentialRamsey/Prereq/Mathlib/Analysis/SpecialFunctions/ExplicitStirling.lean | centralBinomialUpper_monotone | [85, 1] | [98, 15] | refine' antitone_nat_of_succ_le _ | ⊢ Antitone centralBinomialUpper | ⊢ ∀ (n : ℕ), centralBinomialUpper (n + 1) ≤ centralBinomialUpper n |
https://github.com/b-mehta/ExponentialRamsey.git | 7e17b629a915a082869f494c8afa56a3e1c7a88d | ExponentialRamsey/Prereq/Mathlib/Analysis/SpecialFunctions/ExplicitStirling.lean | centralBinomialUpper_monotone | [85, 1] | [98, 15] | intro n | ⊢ ∀ (n : ℕ), centralBinomialUpper (n + 1) ≤ centralBinomialUpper n | n : ℕ
⊢ centralBinomialUpper (n + 1) ≤ centralBinomialUpper n |
https://github.com/b-mehta/ExponentialRamsey.git | 7e17b629a915a082869f494c8afa56a3e1c7a88d | ExponentialRamsey/Prereq/Mathlib/Analysis/SpecialFunctions/ExplicitStirling.lean | centralBinomialUpper_monotone | [85, 1] | [98, 15] | rw [centralBinomialUpper, centralBinomialUpper, _root_.pow_succ, ← div_div] | n : ℕ
⊢ centralBinomialUpper (n + 1) ≤ centralBinomialUpper n | n : ℕ
⊢ ↑(centralBinom (n + 1)) * Real.sqrt (↑(n + 1) + 1 / 3) / 4 / 4 ^ n ≤
↑(centralBinom n) * Real.sqrt (↑n + 1 / 3) / 4 ^ n |
https://github.com/b-mehta/ExponentialRamsey.git | 7e17b629a915a082869f494c8afa56a3e1c7a88d | ExponentialRamsey/Prereq/Mathlib/Analysis/SpecialFunctions/ExplicitStirling.lean | centralBinomialUpper_monotone | [85, 1] | [98, 15] | refine' div_le_div_of_le (by positivity) _ | n : ℕ
⊢ ↑(centralBinom (n + 1)) * Real.sqrt (↑(n + 1) + 1 / 3) / 4 / 4 ^ n ≤
↑(centralBinom n) * Real.sqrt (↑n + 1 / 3) / 4 ^ n | n : ℕ
⊢ ↑(centralBinom (n + 1)) * Real.sqrt (↑(n + 1) + 1 / 3) / 4 ≤ ↑(centralBinom n) * Real.sqrt (↑n + 1 / 3) |
https://github.com/b-mehta/ExponentialRamsey.git | 7e17b629a915a082869f494c8afa56a3e1c7a88d | ExponentialRamsey/Prereq/Mathlib/Analysis/SpecialFunctions/ExplicitStirling.lean | centralBinomialUpper_monotone | [85, 1] | [98, 15] | rw [div_le_iff, mul_assoc, mul_comm _ (_ * _), ← div_le_div_iff, mul_comm, mul_div_assoc,
centralBinom_ratio, Nat.cast_add_one] | n : ℕ
⊢ ↑(centralBinom (n + 1)) * Real.sqrt (↑(n + 1) + 1 / 3) / 4 ≤ ↑(centralBinom n) * Real.sqrt (↑n + 1 / 3) | n : ℕ
⊢ 4 * ((↑n + 1 / 2) / (↑n + 1)) ≤ 4 * (Real.sqrt (↑n + 1 / 3) / Real.sqrt (↑n + 1 + 1 / 3))
case b0
n : ℕ
⊢ 0 < ↑(centralBinom n)
case d0
n : ℕ
⊢ 0 < Real.sqrt (↑(n + 1) + 1 / 3)
n : ℕ
⊢ 0 < 4 |
https://github.com/b-mehta/ExponentialRamsey.git | 7e17b629a915a082869f494c8afa56a3e1c7a88d | ExponentialRamsey/Prereq/Mathlib/Analysis/SpecialFunctions/ExplicitStirling.lean | centralBinomialUpper_monotone | [85, 1] | [98, 15] | refine' mul_le_mul_of_nonneg_left (upper_monotone_aux _) (by positivity) | n : ℕ
⊢ 4 * ((↑n + 1 / 2) / (↑n + 1)) ≤ 4 * (Real.sqrt (↑n + 1 / 3) / Real.sqrt (↑n + 1 + 1 / 3))
case b0
n : ℕ
⊢ 0 < ↑(centralBinom n)
case d0
n : ℕ
⊢ 0 < Real.sqrt (↑(n + 1) + 1 / 3)
n : ℕ
⊢ 0 < 4 | case b0
n : ℕ
⊢ 0 < ↑(centralBinom n)
case d0
n : ℕ
⊢ 0 < Real.sqrt (↑(n + 1) + 1 / 3)
n : ℕ
⊢ 0 < 4 |
https://github.com/b-mehta/ExponentialRamsey.git | 7e17b629a915a082869f494c8afa56a3e1c7a88d | ExponentialRamsey/Prereq/Mathlib/Analysis/SpecialFunctions/ExplicitStirling.lean | centralBinomialUpper_monotone | [85, 1] | [98, 15] | positivity | n : ℕ
⊢ 0 ≤ 4 ^ n | no goals |
https://github.com/b-mehta/ExponentialRamsey.git | 7e17b629a915a082869f494c8afa56a3e1c7a88d | ExponentialRamsey/Prereq/Mathlib/Analysis/SpecialFunctions/ExplicitStirling.lean | centralBinomialUpper_monotone | [85, 1] | [98, 15] | positivity | n : ℕ
⊢ 0 ≤ 4 | no goals |
https://github.com/b-mehta/ExponentialRamsey.git | 7e17b629a915a082869f494c8afa56a3e1c7a88d | ExponentialRamsey/Prereq/Mathlib/Analysis/SpecialFunctions/ExplicitStirling.lean | centralBinomialUpper_monotone | [85, 1] | [98, 15] | rw [Nat.cast_pos] | case b0
n : ℕ
⊢ 0 < ↑(centralBinom n) | case b0
n : ℕ
⊢ 0 < centralBinom n |
https://github.com/b-mehta/ExponentialRamsey.git | 7e17b629a915a082869f494c8afa56a3e1c7a88d | ExponentialRamsey/Prereq/Mathlib/Analysis/SpecialFunctions/ExplicitStirling.lean | centralBinomialUpper_monotone | [85, 1] | [98, 15] | exact centralBinom_pos _ | case b0
n : ℕ
⊢ 0 < centralBinom n | no goals |
https://github.com/b-mehta/ExponentialRamsey.git | 7e17b629a915a082869f494c8afa56a3e1c7a88d | ExponentialRamsey/Prereq/Mathlib/Analysis/SpecialFunctions/ExplicitStirling.lean | centralBinomialUpper_monotone | [85, 1] | [98, 15] | positivity | case d0
n : ℕ
⊢ 0 < Real.sqrt (↑(n + 1) + 1 / 3) | no goals |
https://github.com/b-mehta/ExponentialRamsey.git | 7e17b629a915a082869f494c8afa56a3e1c7a88d | ExponentialRamsey/Prereq/Mathlib/Analysis/SpecialFunctions/ExplicitStirling.lean | centralBinomialUpper_monotone | [85, 1] | [98, 15] | positivity | n : ℕ
⊢ 0 < 4 | no goals |
https://github.com/b-mehta/ExponentialRamsey.git | 7e17b629a915a082869f494c8afa56a3e1c7a88d | ExponentialRamsey/Prereq/Mathlib/Analysis/SpecialFunctions/ExplicitStirling.lean | centralBinom_limit | [100, 1] | [119, 6] | have := Real.pi_pos | ⊢ Tendsto (fun n => ↑(centralBinom n) * Real.sqrt ↑n / 4 ^ n) atTop (𝓝 (Real.sqrt π)⁻¹) | this : 0 < π
⊢ Tendsto (fun n => ↑(centralBinom n) * Real.sqrt ↑n / 4 ^ n) atTop (𝓝 (Real.sqrt π)⁻¹) |
https://github.com/b-mehta/ExponentialRamsey.git | 7e17b629a915a082869f494c8afa56a3e1c7a88d | ExponentialRamsey/Prereq/Mathlib/Analysis/SpecialFunctions/ExplicitStirling.lean | centralBinom_limit | [100, 1] | [119, 6] | have : (sqrt π)⁻¹ = sqrt π / sqrt π ^ 2 := by
rw [inv_eq_one_div, sq, ← div_div, div_self]
positivity | this : 0 < π
⊢ Tendsto (fun n => ↑(centralBinom n) * Real.sqrt ↑n / 4 ^ n) atTop (𝓝 (Real.sqrt π)⁻¹) | this✝ : 0 < π
this : (Real.sqrt π)⁻¹ = Real.sqrt π / Real.sqrt π ^ 2
⊢ Tendsto (fun n => ↑(centralBinom n) * Real.sqrt ↑n / 4 ^ n) atTop (𝓝 (Real.sqrt π)⁻¹) |
https://github.com/b-mehta/ExponentialRamsey.git | 7e17b629a915a082869f494c8afa56a3e1c7a88d | ExponentialRamsey/Prereq/Mathlib/Analysis/SpecialFunctions/ExplicitStirling.lean | centralBinom_limit | [100, 1] | [119, 6] | rw [this] | this✝ : 0 < π
this : (Real.sqrt π)⁻¹ = Real.sqrt π / Real.sqrt π ^ 2
⊢ Tendsto (fun n => ↑(centralBinom n) * Real.sqrt ↑n / 4 ^ n) atTop (𝓝 (Real.sqrt π)⁻¹) | this✝ : 0 < π
this : (Real.sqrt π)⁻¹ = Real.sqrt π / Real.sqrt π ^ 2
⊢ Tendsto (fun n => ↑(centralBinom n) * Real.sqrt ↑n / 4 ^ n) atTop (𝓝 (Real.sqrt π / Real.sqrt π ^ 2)) |
https://github.com/b-mehta/ExponentialRamsey.git | 7e17b629a915a082869f494c8afa56a3e1c7a88d | ExponentialRamsey/Prereq/Mathlib/Analysis/SpecialFunctions/ExplicitStirling.lean | centralBinom_limit | [100, 1] | [119, 6] | have : Tendsto (fun n => Stirling.stirlingSeq (2 * n)) atTop (𝓝 (sqrt π)) := by
refine' Tendsto.comp Stirling.tendsto_stirlingSeq_sqrt_pi _
exact tendsto_id.const_mul_atTop' two_pos | this✝ : 0 < π
this : (Real.sqrt π)⁻¹ = Real.sqrt π / Real.sqrt π ^ 2
⊢ Tendsto (fun n => ↑(centralBinom n) * Real.sqrt ↑n / 4 ^ n) atTop (𝓝 (Real.sqrt π / Real.sqrt π ^ 2)) | this✝¹ : 0 < π
this✝ : (Real.sqrt π)⁻¹ = Real.sqrt π / Real.sqrt π ^ 2
this : Tendsto (fun n => Stirling.stirlingSeq (2 * n)) atTop (𝓝 (Real.sqrt π))
⊢ Tendsto (fun n => ↑(centralBinom n) * Real.sqrt ↑n / 4 ^ n) atTop (𝓝 (Real.sqrt π / Real.sqrt π ^ 2)) |
https://github.com/b-mehta/ExponentialRamsey.git | 7e17b629a915a082869f494c8afa56a3e1c7a88d | ExponentialRamsey/Prereq/Mathlib/Analysis/SpecialFunctions/ExplicitStirling.lean | centralBinom_limit | [100, 1] | [119, 6] | refine' (this.div (Stirling.tendsto_stirlingSeq_sqrt_pi.pow 2) (by positivity)).congr' _ | this✝¹ : 0 < π
this✝ : (Real.sqrt π)⁻¹ = Real.sqrt π / Real.sqrt π ^ 2
this : Tendsto (fun n => Stirling.stirlingSeq (2 * n)) atTop (𝓝 (Real.sqrt π))
⊢ Tendsto (fun n => ↑(centralBinom n) * Real.sqrt ↑n / 4 ^ n) atTop (𝓝 (Real.sqrt π / Real.sqrt π ^ 2)) | this✝¹ : 0 < π
this✝ : (Real.sqrt π)⁻¹ = Real.sqrt π / Real.sqrt π ^ 2
this : Tendsto (fun n => Stirling.stirlingSeq (2 * n)) atTop (𝓝 (Real.sqrt π))
⊢ ((fun n => Stirling.stirlingSeq (2 * n)) / fun x => Stirling.stirlingSeq x ^ 2) =ᶠ[atTop] fun n =>
↑(centralBinom n) * Real.sqrt ↑n / 4 ^ n |
https://github.com/b-mehta/ExponentialRamsey.git | 7e17b629a915a082869f494c8afa56a3e1c7a88d | ExponentialRamsey/Prereq/Mathlib/Analysis/SpecialFunctions/ExplicitStirling.lean | centralBinom_limit | [100, 1] | [119, 6] | filter_upwards [eventually_gt_atTop (0 : ℕ)] with n hn | this✝¹ : 0 < π
this✝ : (Real.sqrt π)⁻¹ = Real.sqrt π / Real.sqrt π ^ 2
this : Tendsto (fun n => Stirling.stirlingSeq (2 * n)) atTop (𝓝 (Real.sqrt π))
⊢ ((fun n => Stirling.stirlingSeq (2 * n)) / fun x => Stirling.stirlingSeq x ^ 2) =ᶠ[atTop] fun n =>
↑(centralBinom n) * Real.sqrt ↑n / 4 ^ n | case h
this✝¹ : 0 < π
this✝ : (Real.sqrt π)⁻¹ = Real.sqrt π / Real.sqrt π ^ 2
this : Tendsto (fun n => Stirling.stirlingSeq (2 * n)) atTop (𝓝 (Real.sqrt π))
n : ℕ
hn : 0 < n
⊢ ((fun n => Stirling.stirlingSeq (2 * n)) / fun x => Stirling.stirlingSeq x ^ 2) n =
↑(centralBinom n) * Real.sqrt ↑n / 4 ^ n |
https://github.com/b-mehta/ExponentialRamsey.git | 7e17b629a915a082869f494c8afa56a3e1c7a88d | ExponentialRamsey/Prereq/Mathlib/Analysis/SpecialFunctions/ExplicitStirling.lean | centralBinom_limit | [100, 1] | [119, 6] | dsimp | case h
this✝¹ : 0 < π
this✝ : (Real.sqrt π)⁻¹ = Real.sqrt π / Real.sqrt π ^ 2
this : Tendsto (fun n => Stirling.stirlingSeq (2 * n)) atTop (𝓝 (Real.sqrt π))
n : ℕ
hn : 0 < n
⊢ ((fun n => Stirling.stirlingSeq (2 * n)) / fun x => Stirling.stirlingSeq x ^ 2) n =
↑(centralBinom n) * Real.sqrt ↑n / 4 ^ n | case h
this✝¹ : 0 < π
this✝ : (Real.sqrt π)⁻¹ = Real.sqrt π / Real.sqrt π ^ 2
this : Tendsto (fun n => Stirling.stirlingSeq (2 * n)) atTop (𝓝 (Real.sqrt π))
n : ℕ
hn : 0 < n
⊢ Stirling.stirlingSeq (2 * n) / Stirling.stirlingSeq n ^ 2 = ↑(centralBinom n) * Real.sqrt ↑n / 4 ^ n |
https://github.com/b-mehta/ExponentialRamsey.git | 7e17b629a915a082869f494c8afa56a3e1c7a88d | ExponentialRamsey/Prereq/Mathlib/Analysis/SpecialFunctions/ExplicitStirling.lean | centralBinom_limit | [100, 1] | [119, 6] | rw [Stirling.stirlingSeq, Stirling.stirlingSeq, centralBinom, two_mul n, cast_add_choose,
←two_mul] | case h
this✝¹ : 0 < π
this✝ : (Real.sqrt π)⁻¹ = Real.sqrt π / Real.sqrt π ^ 2
this : Tendsto (fun n => Stirling.stirlingSeq (2 * n)) atTop (𝓝 (Real.sqrt π))
n : ℕ
hn : 0 < n
⊢ Stirling.stirlingSeq (2 * n) / Stirling.stirlingSeq n ^ 2 = ↑(centralBinom n) * Real.sqrt ↑n / 4 ^ n | case h
this✝¹ : 0 < π
this✝ : (Real.sqrt π)⁻¹ = Real.sqrt π / Real.sqrt π ^ 2
this : Tendsto (fun n => Stirling.stirlingSeq (2 * n)) atTop (𝓝 (Real.sqrt π))
n : ℕ
hn : 0 < n
⊢ ↑(2 * n)! / (Real.sqrt (2 * ↑(2 * n)) * (↑(2 * n) / rexp 1) ^ (2 * n)) /
(↑n ! / (Real.sqrt (2 * ↑n) * (↑n / rexp 1) ^ n)) ^ 2 =
↑(2 * n)! / (↑n ! * ↑n !) * Real.sqrt ↑n / 4 ^ n |
https://github.com/b-mehta/ExponentialRamsey.git | 7e17b629a915a082869f494c8afa56a3e1c7a88d | ExponentialRamsey/Prereq/Mathlib/Analysis/SpecialFunctions/ExplicitStirling.lean | centralBinom_limit | [100, 1] | [119, 6] | field_simp | case h
this✝¹ : 0 < π
this✝ : (Real.sqrt π)⁻¹ = Real.sqrt π / Real.sqrt π ^ 2
this : Tendsto (fun n => Stirling.stirlingSeq (2 * n)) atTop (𝓝 (Real.sqrt π))
n : ℕ
hn : 0 < n
⊢ ↑(2 * n)! / (Real.sqrt (2 * ↑(2 * n)) * (↑(2 * n) / rexp 1) ^ (2 * n)) /
(↑n ! / (Real.sqrt (2 * ↑n) * (↑n / rexp 1) ^ n)) ^ 2 =
↑(2 * n)! / (↑n ! * ↑n !) * Real.sqrt ↑n / 4 ^ n | case h
this✝¹ : 0 < π
this✝ : (Real.sqrt π)⁻¹ = Real.sqrt π / Real.sqrt π ^ 2
this : Tendsto (fun n => Stirling.stirlingSeq (2 * n)) atTop (𝓝 (Real.sqrt π))
n : ℕ
hn : 0 < n
⊢ ↑(2 * n)! * rexp 1 ^ (2 * n) * (Real.sqrt 2 * Real.sqrt ↑n * ↑n ^ n) ^ 2 * (↑n ! * ↑n ! * 4 ^ n) =
↑(2 * n)! * Real.sqrt ↑n *
(Real.sqrt 2 * (Real.sqrt 2 * Real.sqrt ↑n) * (2 * ↑n) ^ (2 * n) * (↑n ! * rexp 1 ^ n) ^ 2) |
https://github.com/b-mehta/ExponentialRamsey.git | 7e17b629a915a082869f494c8afa56a3e1c7a88d | ExponentialRamsey/Prereq/Mathlib/Analysis/SpecialFunctions/ExplicitStirling.lean | centralBinom_limit | [100, 1] | [119, 6] | ring_nf | case h
this✝¹ : 0 < π
this✝ : (Real.sqrt π)⁻¹ = Real.sqrt π / Real.sqrt π ^ 2
this : Tendsto (fun n => Stirling.stirlingSeq (2 * n)) atTop (𝓝 (Real.sqrt π))
n : ℕ
hn : 0 < n
⊢ ↑(2 * n)! * rexp 1 ^ (2 * n) * (Real.sqrt 2 * Real.sqrt ↑n * ↑n ^ n) ^ 2 * (↑n ! * ↑n ! * 4 ^ n) =
↑(2 * n)! * Real.sqrt ↑n *
(Real.sqrt 2 * (Real.sqrt 2 * Real.sqrt ↑n) * (2 * ↑n) ^ (2 * n) * (↑n ! * rexp 1 ^ n) ^ 2) | case h
this✝¹ : 0 < π
this✝ : (Real.sqrt π)⁻¹ = Real.sqrt π / Real.sqrt π ^ 2
this : Tendsto (fun n => Stirling.stirlingSeq (2 * n)) atTop (𝓝 (Real.sqrt π))
n : ℕ
hn : 0 < n
⊢ ↑(n * 2)! * rexp 1 ^ (n * 2) * Real.sqrt 2 ^ 2 * Real.sqrt ↑n ^ 2 * ↑n ^ (n * 2) * ↑n ! ^ 2 * 4 ^ n =
↑(n * 2)! * rexp 1 ^ (n * 2) * Real.sqrt 2 ^ 2 * Real.sqrt ↑n ^ 2 * ↑n ^ (n * 2) * ↑n ! ^ 2 * 2 ^ (n * 2) |
https://github.com/b-mehta/ExponentialRamsey.git | 7e17b629a915a082869f494c8afa56a3e1c7a88d | ExponentialRamsey/Prereq/Mathlib/Analysis/SpecialFunctions/ExplicitStirling.lean | centralBinom_limit | [100, 1] | [119, 6] | rw [mul_comm n 2, pow_mul (2 : ℝ)] | case h
this✝¹ : 0 < π
this✝ : (Real.sqrt π)⁻¹ = Real.sqrt π / Real.sqrt π ^ 2
this : Tendsto (fun n => Stirling.stirlingSeq (2 * n)) atTop (𝓝 (Real.sqrt π))
n : ℕ
hn : 0 < n
⊢ ↑(n * 2)! * rexp 1 ^ (n * 2) * Real.sqrt 2 ^ 2 * Real.sqrt ↑n ^ 2 * ↑n ^ (n * 2) * ↑n ! ^ 2 * 4 ^ n =
↑(n * 2)! * rexp 1 ^ (n * 2) * Real.sqrt 2 ^ 2 * Real.sqrt ↑n ^ 2 * ↑n ^ (n * 2) * ↑n ! ^ 2 * 2 ^ (n * 2) | case h
this✝¹ : 0 < π
this✝ : (Real.sqrt π)⁻¹ = Real.sqrt π / Real.sqrt π ^ 2
this : Tendsto (fun n => Stirling.stirlingSeq (2 * n)) atTop (𝓝 (Real.sqrt π))
n : ℕ
hn : 0 < n
⊢ ↑(2 * n)! * rexp 1 ^ (2 * n) * Real.sqrt 2 ^ 2 * Real.sqrt ↑n ^ 2 * ↑n ^ (2 * n) * ↑n ! ^ 2 * 4 ^ n =
↑(2 * n)! * rexp 1 ^ (2 * n) * Real.sqrt 2 ^ 2 * Real.sqrt ↑n ^ 2 * ↑n ^ (2 * n) * ↑n ! ^ 2 * (2 ^ 2) ^ n |
https://github.com/b-mehta/ExponentialRamsey.git | 7e17b629a915a082869f494c8afa56a3e1c7a88d | ExponentialRamsey/Prereq/Mathlib/Analysis/SpecialFunctions/ExplicitStirling.lean | centralBinom_limit | [100, 1] | [119, 6] | norm_num1 | case h
this✝¹ : 0 < π
this✝ : (Real.sqrt π)⁻¹ = Real.sqrt π / Real.sqrt π ^ 2
this : Tendsto (fun n => Stirling.stirlingSeq (2 * n)) atTop (𝓝 (Real.sqrt π))
n : ℕ
hn : 0 < n
⊢ ↑(2 * n)! * rexp 1 ^ (2 * n) * Real.sqrt 2 ^ 2 * Real.sqrt ↑n ^ 2 * ↑n ^ (2 * n) * ↑n ! ^ 2 * 4 ^ n =
↑(2 * n)! * rexp 1 ^ (2 * n) * Real.sqrt 2 ^ 2 * Real.sqrt ↑n ^ 2 * ↑n ^ (2 * n) * ↑n ! ^ 2 * (2 ^ 2) ^ n | case h
this✝¹ : 0 < π
this✝ : (Real.sqrt π)⁻¹ = Real.sqrt π / Real.sqrt π ^ 2
this : Tendsto (fun n => Stirling.stirlingSeq (2 * n)) atTop (𝓝 (Real.sqrt π))
n : ℕ
hn : 0 < n
⊢ ↑(2 * n)! * rexp 1 ^ (2 * n) * Real.sqrt 2 ^ 2 * Real.sqrt ↑n ^ 2 * ↑n ^ (2 * n) * ↑n ! ^ 2 * 4 ^ n =
↑(2 * n)! * rexp 1 ^ (2 * n) * Real.sqrt 2 ^ 2 * Real.sqrt ↑n ^ 2 * ↑n ^ (2 * n) * ↑n ! ^ 2 * 4 ^ n |
https://github.com/b-mehta/ExponentialRamsey.git | 7e17b629a915a082869f494c8afa56a3e1c7a88d | ExponentialRamsey/Prereq/Mathlib/Analysis/SpecialFunctions/ExplicitStirling.lean | centralBinom_limit | [100, 1] | [119, 6] | rfl | case h
this✝¹ : 0 < π
this✝ : (Real.sqrt π)⁻¹ = Real.sqrt π / Real.sqrt π ^ 2
this : Tendsto (fun n => Stirling.stirlingSeq (2 * n)) atTop (𝓝 (Real.sqrt π))
n : ℕ
hn : 0 < n
⊢ ↑(2 * n)! * rexp 1 ^ (2 * n) * Real.sqrt 2 ^ 2 * Real.sqrt ↑n ^ 2 * ↑n ^ (2 * n) * ↑n ! ^ 2 * 4 ^ n =
↑(2 * n)! * rexp 1 ^ (2 * n) * Real.sqrt 2 ^ 2 * Real.sqrt ↑n ^ 2 * ↑n ^ (2 * n) * ↑n ! ^ 2 * 4 ^ n | no goals |
https://github.com/b-mehta/ExponentialRamsey.git | 7e17b629a915a082869f494c8afa56a3e1c7a88d | ExponentialRamsey/Prereq/Mathlib/Analysis/SpecialFunctions/ExplicitStirling.lean | centralBinom_limit | [100, 1] | [119, 6] | rw [inv_eq_one_div, sq, ← div_div, div_self] | this : 0 < π
⊢ (Real.sqrt π)⁻¹ = Real.sqrt π / Real.sqrt π ^ 2 | this : 0 < π
⊢ Real.sqrt π ≠ 0 |
https://github.com/b-mehta/ExponentialRamsey.git | 7e17b629a915a082869f494c8afa56a3e1c7a88d | ExponentialRamsey/Prereq/Mathlib/Analysis/SpecialFunctions/ExplicitStirling.lean | centralBinom_limit | [100, 1] | [119, 6] | positivity | this : 0 < π
⊢ Real.sqrt π ≠ 0 | no goals |
https://github.com/b-mehta/ExponentialRamsey.git | 7e17b629a915a082869f494c8afa56a3e1c7a88d | ExponentialRamsey/Prereq/Mathlib/Analysis/SpecialFunctions/ExplicitStirling.lean | centralBinom_limit | [100, 1] | [119, 6] | refine' Tendsto.comp Stirling.tendsto_stirlingSeq_sqrt_pi _ | this✝ : 0 < π
this : (Real.sqrt π)⁻¹ = Real.sqrt π / Real.sqrt π ^ 2
⊢ Tendsto (fun n => Stirling.stirlingSeq (2 * n)) atTop (𝓝 (Real.sqrt π)) | this✝ : 0 < π
this : (Real.sqrt π)⁻¹ = Real.sqrt π / Real.sqrt π ^ 2
⊢ Tendsto (fun n => 2 * n) atTop atTop |
https://github.com/b-mehta/ExponentialRamsey.git | 7e17b629a915a082869f494c8afa56a3e1c7a88d | ExponentialRamsey/Prereq/Mathlib/Analysis/SpecialFunctions/ExplicitStirling.lean | centralBinom_limit | [100, 1] | [119, 6] | exact tendsto_id.const_mul_atTop' two_pos | this✝ : 0 < π
this : (Real.sqrt π)⁻¹ = Real.sqrt π / Real.sqrt π ^ 2
⊢ Tendsto (fun n => 2 * n) atTop atTop | no goals |
https://github.com/b-mehta/ExponentialRamsey.git | 7e17b629a915a082869f494c8afa56a3e1c7a88d | ExponentialRamsey/Prereq/Mathlib/Analysis/SpecialFunctions/ExplicitStirling.lean | centralBinom_limit | [100, 1] | [119, 6] | positivity | this✝¹ : 0 < π
this✝ : (Real.sqrt π)⁻¹ = Real.sqrt π / Real.sqrt π ^ 2
this : Tendsto (fun n => Stirling.stirlingSeq (2 * n)) atTop (𝓝 (Real.sqrt π))
⊢ Real.sqrt π ^ 2 ≠ 0 | no goals |
https://github.com/b-mehta/ExponentialRamsey.git | 7e17b629a915a082869f494c8afa56a3e1c7a88d | ExponentialRamsey/Prereq/Mathlib/Analysis/SpecialFunctions/ExplicitStirling.lean | centralBinomialUpper_limit | [121, 1] | [131, 15] | have : (sqrt π)⁻¹ = (sqrt π)⁻¹ / Real.sqrt 1 := by rw [Real.sqrt_one, div_one] | ⊢ Tendsto centralBinomialUpper atTop (𝓝 (Real.sqrt π)⁻¹) | this : (Real.sqrt π)⁻¹ = (Real.sqrt π)⁻¹ / Real.sqrt 1
⊢ Tendsto centralBinomialUpper atTop (𝓝 (Real.sqrt π)⁻¹) |
https://github.com/b-mehta/ExponentialRamsey.git | 7e17b629a915a082869f494c8afa56a3e1c7a88d | ExponentialRamsey/Prereq/Mathlib/Analysis/SpecialFunctions/ExplicitStirling.lean | centralBinomialUpper_limit | [121, 1] | [131, 15] | have h : Real.sqrt 1 ≠ 0 := sqrt_ne_zero'.2 zero_lt_one | this : (Real.sqrt π)⁻¹ = (Real.sqrt π)⁻¹ / Real.sqrt 1
⊢ Tendsto centralBinomialUpper atTop (𝓝 (Real.sqrt π)⁻¹) | this : (Real.sqrt π)⁻¹ = (Real.sqrt π)⁻¹ / Real.sqrt 1
h : Real.sqrt 1 ≠ 0
⊢ Tendsto centralBinomialUpper atTop (𝓝 (Real.sqrt π)⁻¹) |
https://github.com/b-mehta/ExponentialRamsey.git | 7e17b629a915a082869f494c8afa56a3e1c7a88d | ExponentialRamsey/Prereq/Mathlib/Analysis/SpecialFunctions/ExplicitStirling.lean | centralBinomialUpper_limit | [121, 1] | [131, 15] | rw [this] | this : (Real.sqrt π)⁻¹ = (Real.sqrt π)⁻¹ / Real.sqrt 1
h : Real.sqrt 1 ≠ 0
⊢ Tendsto centralBinomialUpper atTop (𝓝 (Real.sqrt π)⁻¹) | this : (Real.sqrt π)⁻¹ = (Real.sqrt π)⁻¹ / Real.sqrt 1
h : Real.sqrt 1 ≠ 0
⊢ Tendsto centralBinomialUpper atTop (𝓝 ((Real.sqrt π)⁻¹ / Real.sqrt 1)) |
https://github.com/b-mehta/ExponentialRamsey.git | 7e17b629a915a082869f494c8afa56a3e1c7a88d | ExponentialRamsey/Prereq/Mathlib/Analysis/SpecialFunctions/ExplicitStirling.lean | centralBinomialUpper_limit | [121, 1] | [131, 15] | refine' (centralBinom_limit.div (tendsto_coe_nat_div_add_atTop (1 / 3 : ℝ)).sqrt h).congr' _ | this : (Real.sqrt π)⁻¹ = (Real.sqrt π)⁻¹ / Real.sqrt 1
h : Real.sqrt 1 ≠ 0
⊢ Tendsto centralBinomialUpper atTop (𝓝 ((Real.sqrt π)⁻¹ / Real.sqrt 1)) | this : (Real.sqrt π)⁻¹ = (Real.sqrt π)⁻¹ / Real.sqrt 1
h : Real.sqrt 1 ≠ 0
⊢ ((fun n => ↑(centralBinom n) * Real.sqrt ↑n / 4 ^ n) / fun x => Real.sqrt (↑x / (↑x + 1 / 3))) =ᶠ[atTop]
centralBinomialUpper |
https://github.com/b-mehta/ExponentialRamsey.git | 7e17b629a915a082869f494c8afa56a3e1c7a88d | ExponentialRamsey/Prereq/Mathlib/Analysis/SpecialFunctions/ExplicitStirling.lean | centralBinomialUpper_limit | [121, 1] | [131, 15] | filter_upwards [eventually_gt_atTop 0] with n hn | this : (Real.sqrt π)⁻¹ = (Real.sqrt π)⁻¹ / Real.sqrt 1
h : Real.sqrt 1 ≠ 0
⊢ ((fun n => ↑(centralBinom n) * Real.sqrt ↑n / 4 ^ n) / fun x => Real.sqrt (↑x / (↑x + 1 / 3))) =ᶠ[atTop]
centralBinomialUpper | case h
this : (Real.sqrt π)⁻¹ = (Real.sqrt π)⁻¹ / Real.sqrt 1
h : Real.sqrt 1 ≠ 0
n : ℕ
hn : 0 < n
⊢ ((fun n => ↑(centralBinom n) * Real.sqrt ↑n / 4 ^ n) / fun x => Real.sqrt (↑x / (↑x + 1 / 3))) n =
centralBinomialUpper n |
https://github.com/b-mehta/ExponentialRamsey.git | 7e17b629a915a082869f494c8afa56a3e1c7a88d | ExponentialRamsey/Prereq/Mathlib/Analysis/SpecialFunctions/ExplicitStirling.lean | centralBinomialUpper_limit | [121, 1] | [131, 15] | dsimp | case h
this : (Real.sqrt π)⁻¹ = (Real.sqrt π)⁻¹ / Real.sqrt 1
h : Real.sqrt 1 ≠ 0
n : ℕ
hn : 0 < n
⊢ ((fun n => ↑(centralBinom n) * Real.sqrt ↑n / 4 ^ n) / fun x => Real.sqrt (↑x / (↑x + 1 / 3))) n =
centralBinomialUpper n | case h
this : (Real.sqrt π)⁻¹ = (Real.sqrt π)⁻¹ / Real.sqrt 1
h : Real.sqrt 1 ≠ 0
n : ℕ
hn : 0 < n
⊢ ↑(centralBinom n) * Real.sqrt ↑n / 4 ^ n / Real.sqrt (↑n / (↑n + 1 / 3)) = centralBinomialUpper n |
https://github.com/b-mehta/ExponentialRamsey.git | 7e17b629a915a082869f494c8afa56a3e1c7a88d | ExponentialRamsey/Prereq/Mathlib/Analysis/SpecialFunctions/ExplicitStirling.lean | centralBinomialUpper_limit | [121, 1] | [131, 15] | rw [sqrt_div (Nat.cast_nonneg _), centralBinomialUpper, div_div, mul_div_assoc',
div_div_eq_mul_div, mul_right_comm, mul_div_mul_right] | case h
this : (Real.sqrt π)⁻¹ = (Real.sqrt π)⁻¹ / Real.sqrt 1
h : Real.sqrt 1 ≠ 0
n : ℕ
hn : 0 < n
⊢ ↑(centralBinom n) * Real.sqrt ↑n / 4 ^ n / Real.sqrt (↑n / (↑n + 1 / 3)) = centralBinomialUpper n | case h.hc
this : (Real.sqrt π)⁻¹ = (Real.sqrt π)⁻¹ / Real.sqrt 1
h : Real.sqrt 1 ≠ 0
n : ℕ
hn : 0 < n
⊢ Real.sqrt ↑n ≠ 0 |
https://github.com/b-mehta/ExponentialRamsey.git | 7e17b629a915a082869f494c8afa56a3e1c7a88d | ExponentialRamsey/Prereq/Mathlib/Analysis/SpecialFunctions/ExplicitStirling.lean | centralBinomialUpper_limit | [121, 1] | [131, 15] | rw [Real.sqrt_one, div_one] | ⊢ (Real.sqrt π)⁻¹ = (Real.sqrt π)⁻¹ / Real.sqrt 1 | no goals |
https://github.com/b-mehta/ExponentialRamsey.git | 7e17b629a915a082869f494c8afa56a3e1c7a88d | ExponentialRamsey/Prereq/Mathlib/Analysis/SpecialFunctions/ExplicitStirling.lean | centralBinomialUpper_limit | [121, 1] | [131, 15] | positivity | case h.hc
this : (Real.sqrt π)⁻¹ = (Real.sqrt π)⁻¹ / Real.sqrt 1
h : Real.sqrt 1 ≠ 0
n : ℕ
hn : 0 < n
⊢ Real.sqrt ↑n ≠ 0 | no goals |
https://github.com/b-mehta/ExponentialRamsey.git | 7e17b629a915a082869f494c8afa56a3e1c7a88d | ExponentialRamsey/Prereq/Mathlib/Analysis/SpecialFunctions/ExplicitStirling.lean | centralBinomialLower_limit | [133, 1] | [142, 15] | have : (sqrt π)⁻¹ = (sqrt π)⁻¹ / Real.sqrt 1 := by rw [Real.sqrt_one, div_one] | ⊢ Tendsto centralBinomialLower atTop (𝓝 (Real.sqrt π)⁻¹) | this : (Real.sqrt π)⁻¹ = (Real.sqrt π)⁻¹ / Real.sqrt 1
⊢ Tendsto centralBinomialLower atTop (𝓝 (Real.sqrt π)⁻¹) |
https://github.com/b-mehta/ExponentialRamsey.git | 7e17b629a915a082869f494c8afa56a3e1c7a88d | ExponentialRamsey/Prereq/Mathlib/Analysis/SpecialFunctions/ExplicitStirling.lean | centralBinomialLower_limit | [133, 1] | [142, 15] | have h : Real.sqrt 1 ≠ 0 := sqrt_ne_zero'.2 zero_lt_one | this : (Real.sqrt π)⁻¹ = (Real.sqrt π)⁻¹ / Real.sqrt 1
⊢ Tendsto centralBinomialLower atTop (𝓝 (Real.sqrt π)⁻¹) | this : (Real.sqrt π)⁻¹ = (Real.sqrt π)⁻¹ / Real.sqrt 1
h : Real.sqrt 1 ≠ 0
⊢ Tendsto centralBinomialLower atTop (𝓝 (Real.sqrt π)⁻¹) |
https://github.com/b-mehta/ExponentialRamsey.git | 7e17b629a915a082869f494c8afa56a3e1c7a88d | ExponentialRamsey/Prereq/Mathlib/Analysis/SpecialFunctions/ExplicitStirling.lean | centralBinomialLower_limit | [133, 1] | [142, 15] | rw [this] | this : (Real.sqrt π)⁻¹ = (Real.sqrt π)⁻¹ / Real.sqrt 1
h : Real.sqrt 1 ≠ 0
⊢ Tendsto centralBinomialLower atTop (𝓝 (Real.sqrt π)⁻¹) | this : (Real.sqrt π)⁻¹ = (Real.sqrt π)⁻¹ / Real.sqrt 1
h : Real.sqrt 1 ≠ 0
⊢ Tendsto centralBinomialLower atTop (𝓝 ((Real.sqrt π)⁻¹ / Real.sqrt 1)) |
https://github.com/b-mehta/ExponentialRamsey.git | 7e17b629a915a082869f494c8afa56a3e1c7a88d | ExponentialRamsey/Prereq/Mathlib/Analysis/SpecialFunctions/ExplicitStirling.lean | centralBinomialLower_limit | [133, 1] | [142, 15] | refine' (centralBinom_limit.div (tendsto_coe_nat_div_add_atTop (1 / 4 : ℝ)).sqrt h).congr' _ | this : (Real.sqrt π)⁻¹ = (Real.sqrt π)⁻¹ / Real.sqrt 1
h : Real.sqrt 1 ≠ 0
⊢ Tendsto centralBinomialLower atTop (𝓝 ((Real.sqrt π)⁻¹ / Real.sqrt 1)) | this : (Real.sqrt π)⁻¹ = (Real.sqrt π)⁻¹ / Real.sqrt 1
h : Real.sqrt 1 ≠ 0
⊢ ((fun n => ↑(centralBinom n) * Real.sqrt ↑n / 4 ^ n) / fun x => Real.sqrt (↑x / (↑x + 1 / 4))) =ᶠ[atTop]
centralBinomialLower |
https://github.com/b-mehta/ExponentialRamsey.git | 7e17b629a915a082869f494c8afa56a3e1c7a88d | ExponentialRamsey/Prereq/Mathlib/Analysis/SpecialFunctions/ExplicitStirling.lean | centralBinomialLower_limit | [133, 1] | [142, 15] | filter_upwards [eventually_gt_atTop 0] with n hn | this : (Real.sqrt π)⁻¹ = (Real.sqrt π)⁻¹ / Real.sqrt 1
h : Real.sqrt 1 ≠ 0
⊢ ((fun n => ↑(centralBinom n) * Real.sqrt ↑n / 4 ^ n) / fun x => Real.sqrt (↑x / (↑x + 1 / 4))) =ᶠ[atTop]
centralBinomialLower | case h
this : (Real.sqrt π)⁻¹ = (Real.sqrt π)⁻¹ / Real.sqrt 1
h : Real.sqrt 1 ≠ 0
n : ℕ
hn : 0 < n
⊢ ((fun n => ↑(centralBinom n) * Real.sqrt ↑n / 4 ^ n) / fun x => Real.sqrt (↑x / (↑x + 1 / 4))) n =
centralBinomialLower n |
https://github.com/b-mehta/ExponentialRamsey.git | 7e17b629a915a082869f494c8afa56a3e1c7a88d | ExponentialRamsey/Prereq/Mathlib/Analysis/SpecialFunctions/ExplicitStirling.lean | centralBinomialLower_limit | [133, 1] | [142, 15] | dsimp | case h
this : (Real.sqrt π)⁻¹ = (Real.sqrt π)⁻¹ / Real.sqrt 1
h : Real.sqrt 1 ≠ 0
n : ℕ
hn : 0 < n
⊢ ((fun n => ↑(centralBinom n) * Real.sqrt ↑n / 4 ^ n) / fun x => Real.sqrt (↑x / (↑x + 1 / 4))) n =
centralBinomialLower n | case h
this : (Real.sqrt π)⁻¹ = (Real.sqrt π)⁻¹ / Real.sqrt 1
h : Real.sqrt 1 ≠ 0
n : ℕ
hn : 0 < n
⊢ ↑(centralBinom n) * Real.sqrt ↑n / 4 ^ n / Real.sqrt (↑n / (↑n + 1 / 4)) = centralBinomialLower n |
https://github.com/b-mehta/ExponentialRamsey.git | 7e17b629a915a082869f494c8afa56a3e1c7a88d | ExponentialRamsey/Prereq/Mathlib/Analysis/SpecialFunctions/ExplicitStirling.lean | centralBinomialLower_limit | [133, 1] | [142, 15] | rw [sqrt_div (Nat.cast_nonneg _), centralBinomialLower, div_div, mul_div_assoc',
div_div_eq_mul_div, mul_right_comm, mul_div_mul_right] | case h
this : (Real.sqrt π)⁻¹ = (Real.sqrt π)⁻¹ / Real.sqrt 1
h : Real.sqrt 1 ≠ 0
n : ℕ
hn : 0 < n
⊢ ↑(centralBinom n) * Real.sqrt ↑n / 4 ^ n / Real.sqrt (↑n / (↑n + 1 / 4)) = centralBinomialLower n | case h.hc
this : (Real.sqrt π)⁻¹ = (Real.sqrt π)⁻¹ / Real.sqrt 1
h : Real.sqrt 1 ≠ 0
n : ℕ
hn : 0 < n
⊢ Real.sqrt ↑n ≠ 0 |
https://github.com/b-mehta/ExponentialRamsey.git | 7e17b629a915a082869f494c8afa56a3e1c7a88d | ExponentialRamsey/Prereq/Mathlib/Analysis/SpecialFunctions/ExplicitStirling.lean | centralBinomialLower_limit | [133, 1] | [142, 15] | rw [Real.sqrt_one, div_one] | ⊢ (Real.sqrt π)⁻¹ = (Real.sqrt π)⁻¹ / Real.sqrt 1 | no goals |
https://github.com/b-mehta/ExponentialRamsey.git | 7e17b629a915a082869f494c8afa56a3e1c7a88d | ExponentialRamsey/Prereq/Mathlib/Analysis/SpecialFunctions/ExplicitStirling.lean | centralBinomialLower_limit | [133, 1] | [142, 15] | positivity | case h.hc
this : (Real.sqrt π)⁻¹ = (Real.sqrt π)⁻¹ / Real.sqrt 1
h : Real.sqrt 1 ≠ 0
n : ℕ
hn : 0 < n
⊢ Real.sqrt ↑n ≠ 0 | no goals |
https://github.com/b-mehta/ExponentialRamsey.git | 7e17b629a915a082869f494c8afa56a3e1c7a88d | ExponentialRamsey/Prereq/Mathlib/Analysis/SpecialFunctions/ExplicitStirling.lean | centralBinomialUpper_bound | [144, 1] | [150, 23] | have := pi_pos | n : ℕ
⊢ ↑(centralBinom n) ≤ 4 ^ n / Real.sqrt (π * (↑n + 1 / 4)) | n : ℕ
this : 0 < π
⊢ ↑(centralBinom n) ≤ 4 ^ n / Real.sqrt (π * (↑n + 1 / 4)) |
https://github.com/b-mehta/ExponentialRamsey.git | 7e17b629a915a082869f494c8afa56a3e1c7a88d | ExponentialRamsey/Prereq/Mathlib/Analysis/SpecialFunctions/ExplicitStirling.lean | centralBinomialUpper_bound | [144, 1] | [150, 23] | have := centralBinomialLower_monotone.ge_of_tendsto centralBinomialLower_limit n | n : ℕ
this : 0 < π
⊢ ↑(centralBinom n) ≤ 4 ^ n / Real.sqrt (π * (↑n + 1 / 4)) | n : ℕ
this✝ : 0 < π
this : centralBinomialLower n ≤ (Real.sqrt π)⁻¹
⊢ ↑(centralBinom n) ≤ 4 ^ n / Real.sqrt (π * (↑n + 1 / 4)) |
https://github.com/b-mehta/ExponentialRamsey.git | 7e17b629a915a082869f494c8afa56a3e1c7a88d | ExponentialRamsey/Prereq/Mathlib/Analysis/SpecialFunctions/ExplicitStirling.lean | centralBinomialUpper_bound | [144, 1] | [150, 23] | rwa [sqrt_mul, ← div_div, le_div_iff, div_eq_mul_one_div ((4 : ℝ) ^ n : ℝ), ← div_le_iff',
one_div (sqrt π)] | n : ℕ
this✝ : 0 < π
this : centralBinomialLower n ≤ (Real.sqrt π)⁻¹
⊢ ↑(centralBinom n) ≤ 4 ^ n / Real.sqrt (π * (↑n + 1 / 4)) | n : ℕ
this✝ : 0 < π
this : centralBinomialLower n ≤ (Real.sqrt π)⁻¹
⊢ 0 < 4 ^ n
n : ℕ
this✝ : 0 < π
this : centralBinomialLower n ≤ (Real.sqrt π)⁻¹
⊢ 0 < Real.sqrt (↑n + 1 / 4)
case hx
n : ℕ
this✝ : 0 < π
this : centralBinomialLower n ≤ (Real.sqrt π)⁻¹
⊢ 0 ≤ π |
https://github.com/b-mehta/ExponentialRamsey.git | 7e17b629a915a082869f494c8afa56a3e1c7a88d | ExponentialRamsey/Prereq/Mathlib/Analysis/SpecialFunctions/ExplicitStirling.lean | centralBinomialUpper_bound | [144, 1] | [150, 23] | all_goals positivity | n : ℕ
this✝ : 0 < π
this : centralBinomialLower n ≤ (Real.sqrt π)⁻¹
⊢ 0 < 4 ^ n
n : ℕ
this✝ : 0 < π
this : centralBinomialLower n ≤ (Real.sqrt π)⁻¹
⊢ 0 < Real.sqrt (↑n + 1 / 4)
case hx
n : ℕ
this✝ : 0 < π
this : centralBinomialLower n ≤ (Real.sqrt π)⁻¹
⊢ 0 ≤ π | no goals |
https://github.com/b-mehta/ExponentialRamsey.git | 7e17b629a915a082869f494c8afa56a3e1c7a88d | ExponentialRamsey/Prereq/Mathlib/Analysis/SpecialFunctions/ExplicitStirling.lean | centralBinomialUpper_bound | [144, 1] | [150, 23] | positivity | case hx
n : ℕ
this✝ : 0 < π
this : centralBinomialLower n ≤ (Real.sqrt π)⁻¹
⊢ 0 ≤ π | no goals |
https://github.com/b-mehta/ExponentialRamsey.git | 7e17b629a915a082869f494c8afa56a3e1c7a88d | ExponentialRamsey/Prereq/Mathlib/Analysis/SpecialFunctions/ExplicitStirling.lean | centralBinomialLower_bound | [152, 1] | [157, 23] | have := pi_pos | n : ℕ
⊢ 4 ^ n / Real.sqrt (π * (↑n + 1 / 3)) ≤ ↑(centralBinom n) | n : ℕ
this : 0 < π
⊢ 4 ^ n / Real.sqrt (π * (↑n + 1 / 3)) ≤ ↑(centralBinom n) |
https://github.com/b-mehta/ExponentialRamsey.git | 7e17b629a915a082869f494c8afa56a3e1c7a88d | ExponentialRamsey/Prereq/Mathlib/Analysis/SpecialFunctions/ExplicitStirling.lean | centralBinomialLower_bound | [152, 1] | [157, 23] | have := centralBinomialUpper_monotone.le_of_tendsto centralBinomialUpper_limit n | n : ℕ
this : 0 < π
⊢ 4 ^ n / Real.sqrt (π * (↑n + 1 / 3)) ≤ ↑(centralBinom n) | n : ℕ
this✝ : 0 < π
this : (Real.sqrt π)⁻¹ ≤ centralBinomialUpper n
⊢ 4 ^ n / Real.sqrt (π * (↑n + 1 / 3)) ≤ ↑(centralBinom n) |
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