| import Mathlib |
| import OAI.Probability.SKBarriers.Replicas.TriplePerturbation |
|
|
| namespace OAI |
|
|
| section |
|
|
| noncomputable section |
| open scoped BigOperators Matrix |
| open MeasureTheory ProbabilityTheory Set |
| namespace SK.Analytic |
|
|
| def tripleClockMatrix (a b c r q : ℝ) : Fin 3 → Fin 3 → ℝ := |
| !![a,r,r |
|
|
| def tripleClockPerturbed (a b c r q δ g G : ℝ) : Fin 3 → Fin 3 → ℝ := |
| fun i j => tripleClockMatrix a b c r q i j+δ*g*tripleImbalance i j+ |
| δ^2*G*tripleQuadraticCorrection i j |
| |
| theorem tripleClockMatrix_square (a b c r q : ℝ) : |
| matrixSquare 3 (tripleClockMatrix a b c r q)=a^2+b^2+c^2+4*r^2+2*q^2 := by |
| norm_num [matrixSquare,tripleClockMatrix,Fin.sum_univ_succ] |
| ring |
|
|
| theorem tripleClockPerturbed_square (a b c r q δ g G : ℝ) : |
| matrixSquare 3 (tripleClockPerturbed a b c r q δ g G)- |
| matrixSquare 3 (tripleClockMatrix a b c r q)= |
| 4*δ^2*g^2+2*δ^2*G*(b+c-2*q)+4*δ^4*G^2 := by |
| norm_num [matrixSquare,tripleClockMatrix,tripleClockPerturbed,tripleImbalance, |
| tripleQuadraticCorrection,Fin.sum_univ_succ] |
| ring |
|
|
| theorem tripleClockPerturbed_symmetric (a b r q δ g G : ℝ) : |
| tripleClockPerturbed a b b r q δ g G=triplePerturbedMatrix a b r q δ g G := rfl |
|
|
| theorem tripleClockPerturbed_singleton (a b c r q δ g G v w : ℝ) : |
| (fun i j => tripleClockPerturbed a b c r q δ g G i j+ |
| (![v,δ*w,-δ*w] : Fin 3 → ℝ) i*(![v,δ*w,-δ*w] : Fin 3 → ℝ) j)= |
| tripleClockPerturbed (a+v^2) b c r q δ (g+v*w) (G+w^2) := by |
| funext i j |
| fin_cases i < |
| norm_num [tripleClockPerturbed,tripleClockMatrix,tripleImbalance,tripleQuadraticCorrection] |
| all_goals ring |
|
|
| theorem tripleClockPerturbed_pair (a b c r q δ g G v : ℝ) : |
| (fun i j => tripleClockPerturbed a b c r q δ g G i j+ |
| (![0,v,v] : Fin 3 → ℝ) i*(![0,v,v] : Fin 3 → ℝ) j)= |
| tripleClockPerturbed a (b+v^2) (c+v^2) r (q+v^2) δ g G := by |
| funext i j |
| fin_cases i < |
| norm_num [tripleClockPerturbed,tripleClockMatrix,tripleImbalance,tripleQuadraticCorrection] |
| all_goals ring |
|
|
| theorem tripleClockPerturbed_tail0 (a b c r q δ g G v : ℝ) : |
| (fun i j => tripleClockPerturbed a b c r q δ g G i j+ |
| (![v,0,0] : Fin 3 → ℝ) i*(![v,0,0] : Fin 3 → ℝ) j)= |
| tripleClockPerturbed (a+v^2) b c r q δ g G := by |
| funext i j |
| fin_cases i < |
| norm_num [tripleClockPerturbed,tripleClockMatrix,tripleImbalance,tripleQuadraticCorrection] |
| all_goals ring |
|
|
| theorem tripleClockPerturbed_tail1 (a b c r q δ g G v : ℝ) : |
| (fun i j => tripleClockPerturbed a b c r q δ g G i j+ |
| (![0,v,0] : Fin 3 → ℝ) i*(![0,v,0] : Fin 3 → ℝ) j)= |
| tripleClockPerturbed a (b+v^2) c r q δ g G := by |
| funext i j |
| fin_cases i < |
| norm_num [tripleClockPerturbed,tripleClockMatrix,tripleImbalance,tripleQuadraticCorrection] |
| all_goals ring |
|
|
| theorem tripleClockPerturbed_tail2 (a b c r q δ g G v : ℝ) : |
| (fun i j => tripleClockPerturbed a b c r q δ g G i j+ |
| (![0,0,v] : Fin 3 → ℝ) i*(![0,0,v] : Fin 3 → ℝ) j)= |
| tripleClockPerturbed a b (c+v^2) r q δ g G := by |
| funext i j |
| fin_cases i < |
| norm_num [tripleClockPerturbed,tripleClockMatrix,tripleImbalance,tripleQuadraticCorrection] |
| all_goals ring |
|
|
| theorem tripleClockPerturbed_common (r v : ℝ) : |
| (fun i j => tripleClockMatrix r r r r r i j+ |
| (![v,v,v] : Fin 3 → ℝ) i*(![v,v,v] : Fin 3 → ℝ) j)= |
| tripleClockMatrix (r+v^2) (r+v^2) (r+v^2) (r+v^2) (r+v^2) := by |
| funext i j |
| fin_cases i < |
|
|
| theorem tripleClockPerturbed_tail_penalty (a b c r q δ g G m v : ℝ) : |
| m*(matrixSquare 3 (tripleClockPerturbed (a+v^2) (b+v^2) (c+v^2) r q δ g G)- |
| matrixSquare 3 (tripleClockPerturbed a b c r q δ g G))= |
| m*((a+v^2)^2-a^2+(b+v^2)^2-b^2+(c+v^2)^2-c^2)+4*m*δ^2*G*v^2 := by |
| have H := tripleClockPerturbed_square a b c r q δ g G |
| have H' := tripleClockPerturbed_square (a+v^2) (b+v^2) (c+v^2) r q δ g G |
| rw [tripleClockMatrix_square] at H H' |
| linear_combination m*H'-m*H |
| |
| theorem tripleClockPerturbed_middle_penalty (a b r δ g G m v w : ℝ) : |
| m*(matrixSquare 3 (tripleClockPerturbed (a+v^2) b b r b δ (g+v*w) (G+w^2))- |
| matrixSquare 3 (tripleClockPerturbed a b b r b δ g G))= |
| m*((a+v^2)^2-a^2)+4*m*δ^2*((g+v*w)^2-g^2)+4*m*δ^4*((G+w^2)^2-G^2) := by |
| have H := tripleClockPerturbed_square a b b r b δ g G |
| have H' := tripleClockPerturbed_square (a+v^2) b b r b δ (g+v*w) (G+w^2) |
| rw [tripleClockMatrix_square] at H H' |
| linear_combination m*H'-m*H |
|
|
| theorem tripleClockPerturbed_pair_penalty (a b r δ g G m v : ℝ) : |
| m/2*(matrixSquare 3 (tripleClockPerturbed a (b+v^2) (b+v^2) r (b+v^2) δ g G)- |
| matrixSquare 3 (tripleClockPerturbed a b b r b δ g G))= |
| 2*m*((b+v^2)^2-b^2) := by |
| have H := tripleClockPerturbed_square a b b r b δ g G |
| have H' := tripleClockPerturbed_square a (b+v^2) (b+v^2) r (b+v^2) δ g G |
| rw [tripleClockMatrix_square] at H H' |
| linear_combination m/2*H'-m/2*H |
| |
| end SK.Analytic |
| |
| end |
| end |
| |
| end OAI |
| |
