| import Mathlib |
| import OAI.Probability.SKBarriers.Replicas.TripleClockAlgebra |
|
|
| namespace OAI |
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|
| section |
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|
| 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 |
| 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 < |
| 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 < |
| norm_num [narrowClockPerturbed,tripleClockMatrix,tripleQuadraticCorrection] < |
|
|
| 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 < |
| norm_num [narrowClockPerturbed,tripleClockMatrix,tripleQuadraticCorrection] < |
|
|
| 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 < |
| norm_num [narrowClockPerturbed,tripleClockMatrix,tripleQuadraticCorrection] < |
|
|
| 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 |
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|
| end |
| end |
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| end OAI |
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|
