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135052a | 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 | import Mathlib
import OAI.Probability.SKBarriers.Replicas.TripleClockAlgebra
namespace OAI
section
noncomputable section
open scoped BigOperators Matrix
open MeasureTheory ProbabilityTheory Set
namespace SK.Analytic
def narrowClockPerturbed (a b c r q δ g y K : ℝ) : Fin 3 → Fin 3 → ℝ :=
fun i j => tripleClockMatrix a b c r q i j+
δ*(!![0,g,-g;g,2*y,0;-g,0,-2*y] : Fin 3 → Fin 3 → ℝ) i j+
δ^2*K*tripleQuadraticCorrection i j
theorem narrowClockPerturbed_pair_zero (a b c r q δ g K : ℝ) :
narrowClockPerturbed a b c r q δ g 0 K=tripleClockPerturbed a b c r q δ g K := by
funext i j
fin_cases i <;> fin_cases j <;>
norm_num [narrowClockPerturbed,tripleClockPerturbed,tripleImbalance]
theorem narrowClockPerturbed_square (a b c r q δ g y K : ℝ) :
matrixSquare 3 (narrowClockPerturbed a b c r q δ g y K)=
matrixSquare 3 (tripleClockMatrix a b c r q)+4*δ*y*(b-c)+
δ^2*(4*g^2+8*y^2+2*K*(b+c-2*q))+4*δ^4*K^2 := by
norm_num [matrixSquare,Fin.sum_univ_succ,narrowClockPerturbed,tripleClockMatrix,tripleQuadraticCorrection]
ring
theorem narrowClockPerturbed_common (r δ g K v w : ℝ) :
(fun i j => narrowClockPerturbed r r r r r δ g g K i j+
(![v,v+δ*w,v-δ*w] : Fin 3 → ℝ) i*(![v,v+δ*w,v-δ*w] : Fin 3 → ℝ) j)=
narrowClockPerturbed (r+v^2) (r+v^2) (r+v^2) (r+v^2) (r+v^2) δ (g+v*w) (g+v*w) (K+w^2) := by
funext i j
fin_cases i <;> fin_cases j <;>
norm_num [narrowClockPerturbed,tripleClockMatrix,tripleQuadraticCorrection] <;> ring
theorem narrowClockPerturbed_singleton (a b r δ g y K v w : ℝ) :
(fun i j => narrowClockPerturbed a b b r b δ g y K i j+
(![v,δ*w,-δ*w] : Fin 3 → ℝ) i*(![v,δ*w,-δ*w] : Fin 3 → ℝ) j)=
narrowClockPerturbed (a+v^2) b b r b δ (g+v*w) y (K+w^2) := by
funext i j
fin_cases i <;> fin_cases j <;>
norm_num [narrowClockPerturbed,tripleClockMatrix,tripleQuadraticCorrection] <;> ring
theorem narrowClockPerturbed_pair (a b r δ g y K v w : ℝ) :
(fun i j => narrowClockPerturbed a b b r b δ g y K i j+
(![0,v+δ*w,v-δ*w] : Fin 3 → ℝ) i*(![0,v+δ*w,v-δ*w] : Fin 3 → ℝ) j)=
narrowClockPerturbed a (b+v^2) (b+v^2) r (b+v^2) δ g (y+v*w) (K+w^2) := by
funext i j
fin_cases i <;> fin_cases j <;>
norm_num [narrowClockPerturbed,tripleClockMatrix,tripleQuadraticCorrection] <;> ring
theorem narrowClockPerturbed_common_penalty (r δ g K m v w : ℝ) :
m/3*(matrixSquare 3 (narrowClockPerturbed (r+v^2) (r+v^2) (r+v^2) (r+v^2) (r+v^2)
δ (g+v*w) (g+v*w) (K+w^2))-matrixSquare 3 (narrowClockPerturbed r r r r r δ g g K))=
3*m*((r+v^2)^2-r^2)+4*m*δ^2*((g+v*w)^2-g^2)+4*(m/3)*δ^4*((K+w^2)^2-K^2) := by
rw [narrowClockPerturbed_square,narrowClockPerturbed_square,
tripleClockMatrix_square,tripleClockMatrix_square]
ring
theorem narrowClockPerturbed_singleton_penalty (a b r δ g y K m v w : ℝ) :
m*(matrixSquare 3 (narrowClockPerturbed (a+v^2) b b r b δ (g+v*w) y (K+w^2))-
matrixSquare 3 (narrowClockPerturbed a b b r b δ g y K))=
m*((a+v^2)^2-a^2)+4*m*δ^2*((g+v*w)^2-g^2)+4*m*δ^4*((K+w^2)^2-K^2) := by
rw [narrowClockPerturbed_square,narrowClockPerturbed_square,
tripleClockMatrix_square,tripleClockMatrix_square]
ring
theorem narrowClockPerturbed_pair_penalty (a b r δ g y K m v w : ℝ) :
m/2*(matrixSquare 3 (narrowClockPerturbed a (b+v^2) (b+v^2) r (b+v^2) δ g (y+v*w) (K+w^2))-
matrixSquare 3 (narrowClockPerturbed a b b r b δ g y K))=
2*m*((b+v^2)^2-b^2)+4*m*δ^2*((y+v*w)^2-y^2)+4*(m/2)*δ^4*((K+w^2)^2-K^2) := by
rw [narrowClockPerturbed_square,narrowClockPerturbed_square,
tripleClockMatrix_square,tripleClockMatrix_square]
ring
end SK.Analytic
end
end
end OAI
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