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f205175 | 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 124 | import Mathlib
namespace OAI
section
section
open Set MeasureTheory ProbabilityTheory
open Real
open Set Real
open scoped Topology
open scoped BigOperators
open Set
open Set MeasureTheory
open scoped Classical
open Set MeasureTheory Real
open scoped Interval
namespace CriticalPercolation.FiniteEntropy
lemma integral_polynomial_sum (n : ℕ) (c : ℕ → ℝ) :
(∫ t in (0:ℝ)..1, ∑ i ∈ Finset.range n, c i * t^i) =
∑ i ∈ Finset.range n, c i / ((i:ℝ)+1) := by
rw [intervalIntegral.integral_finsetSum]
· apply Finset.sum_congr rfl
intro i hi
rw [intervalIntegral.integral_const_mul, integral_pow]
simp [div_eq_mul_inv]
· intro i hi
exact (by fun_prop : Continuous (fun t : ℝ => c i * t^i)).intervalIntegrable 0 1
lemma integral_cubic_square (a b c d : ℝ) :
(∫ t in (0:ℝ)..1, (a + b*t + c*t^2 + d*t^3)^2) =
a^2 + a*b + (b^2+2*a*c)/3 + (2*a*d+2*b*c)/4 +
(c^2+2*b*d)/5 + (2*c*d)/6 + d^2/7 := by
have hpoly (t : ℝ) : (a+b*t+c*t^2+d*t^3)^2 =
∑ i ∈ Finset.range 7, (![a^2, 2*a*b, b^2+2*a*c, 2*a*d+2*b*c,
c^2+2*b*d, 2*c*d, d^2] : Fin 7 → ℝ) ⟨i % 7, Nat.mod_lt _ (by norm_num)⟩ * t^i := by
simp [Finset.sum_range_succ]
ring
simp_rw [hpoly]
rw [integral_polynomial_sum]
simp [Finset.sum_range_succ]
ring
theorem cubic_endpoint (a b c d : ℝ) :
a^2 ≤ 16 * ∫ t in (0:ℝ)..1, (a+b*t+c*t^2+d*t^3)^2 := by
rw [integral_cubic_square]
let z₀ := a+b/2+c/3+d/4
let z₁ := -(b/2+c/2+9*d/20)/3
let z₂ := (c/6+d/4)/5
let z₃ := -(d/20)/7
have hid : 16*(a^2+a*b+(b^2+2*a*c)/3+(2*a*d+2*b*c)/4+
(c^2+2*b*d)/5+(2*c*d)/6+d^2/7)-a^2 =
3*(z₀-z₁)^2+5*(z₀-z₂)^2+7*(z₀-z₃)^2+
15*(z₁-z₂)^2+21*(z₁-z₃)^2+35*(z₂-z₃)^2 := by
dsimp [z₀, z₁, z₂, z₃]
ring
have : 0 ≤ 16*(a^2+a*b+(b^2+2*a*c)/3+(2*a*d+2*b*c)/4+
(c^2+2*b*d)/5+(2*c*d)/6+d^2/7)-a^2 := by
rw [hid]
positivity
linarith
end CriticalPercolation.FiniteEntropy
namespace CriticalPercolation.FiniteEntropy
def endpointKernel (t : ℝ) : ℝ := 16 - 120*t + 240*t^2 - 140*t^3
lemma continuous_endpointKernel : Continuous endpointKernel := by
unfold endpointKernel
fun_prop
lemma integral_endpointKernel_sq :
(∫ t in (0:ℝ)..1, (endpointKernel t)^2) = 16 := by
have he (t : ℝ) : endpointKernel t = 16 + (-120)*t + 240*t^2 + (-140)*t^3 := by
unfold endpointKernel; ring
simp_rw [he]
rw [integral_cubic_square]
norm_num
lemma integral_endpointKernel_power {k : ℕ} (hk : k ≤ 3) :
(∫ t in (0:ℝ)..1, endpointKernel t * (1-t)^k) = 1 := by
interval_cases k
· let c : Fin 7 → ℝ := ![16, -120, 240, -140, 0, 0, 0]
have ht (t : ℝ) : endpointKernel t * (1-t)^0 =
∑ j ∈ Finset.range 7, c ⟨j % 7, Nat.mod_lt _ (by norm_num)⟩ * t^j := by
simp [c, endpointKernel, Finset.sum_range_succ]
ring
simp_rw [ht]
rw [integral_polynomial_sum]
norm_num [c, Finset.sum_range_succ]
· let c : Fin 7 → ℝ := ![16, -136, 360, -380, 140, 0, 0]
have ht (t : ℝ) : endpointKernel t * (1-t)^1 =
∑ j ∈ Finset.range 7, c ⟨j % 7, Nat.mod_lt _ (by norm_num)⟩ * t^j := by
simp [c, endpointKernel, Finset.sum_range_succ]
ring
simp_rw [ht]
rw [integral_polynomial_sum]
norm_num [c, Finset.sum_range_succ]
· let c : Fin 7 → ℝ := ![16, -152, 496, -740, 520, -140, 0]
have ht (t : ℝ) : endpointKernel t * (1-t)^2 =
∑ j ∈ Finset.range 7, c ⟨j % 7, Nat.mod_lt _ (by norm_num)⟩ * t^j := by
simp [c, endpointKernel, Finset.sum_range_succ]
ring
simp_rw [ht]
rw [integral_polynomial_sum]
norm_num [c, Finset.sum_range_succ]
· let c : Fin 7 → ℝ := ![16, -168, 648, -1236, 1260, -660, 140]
have ht (t : ℝ) : endpointKernel t * (1-t)^3 =
∑ j ∈ Finset.range 7, c ⟨j % 7, Nat.mod_lt _ (by norm_num)⟩ * t^j := by
simp [c, endpointKernel, Finset.sum_range_succ]
ring
simp_rw [ht]
rw [integral_polynomial_sum]
norm_num [c, Finset.sum_range_succ]
end CriticalPercolation.FiniteEntropy
end
end
end OAI
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