| import Mathlib |
| import OAI.Probability.SKGap.Localization.ResidualInduction |
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| namespace OAI |
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| section |
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| noncomputable section |
| open scoped BigOperators |
| namespace SKGapCutoff.Primary |
| open Matrix |
| variable {n : ℕ} |
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| lemma terminal_field_identity (j : ℝ) (J : Interaction n) (h : Fin n→ℝ) |
| (k : ℕ) (x : Spin n) (i : Fin n) : |
| fld j J h (k+1) x i-h i-J.mulVec (mag j J h (k+2) x) i+ |
| j*onsager j J h (k+1) x*mag j J h (k+2) x i = |
| J.mulVec (residual j J h (k+1) x) i- |
| j*onsager j J h (k+1) x*(residual j J h k x i+residual j J h (k+1) x i) := by |
| rw [fld_eq] |
| simp only [Nat.add_sub_cancel,residual,Matrix.mulVec,dotProduct,mul_sub,Finset.sum_sub_distrib] |
| ring |
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| theorem terminal_residual_identity (hn : 0<n) (j : ℝ) (J : Interaction n) |
| (hJ : J.IsSymm) (h : Fin n→ℝ) (k : ℕ) (x : Spin n) |
| (w y : Fin n→ℝ) (χ c : ℝ) |
| (hy : ∀i,y i=J.mulVec w i-j*χ*w i-j*c*mag j J h (k+1) x i) : |
| (∑i,(fld j J h (k+1) x i-h i-J.mulVec (mag j J h (k+2) x) i+ |
| j*onsager j J h (k+1) x*mag j J h (k+2) x i)*w i)- |
| j*(onsager j J h (k+2) x-onsager j J h (k+1) x)*(n:ℝ)*c = |
| (∑i,residual j J h (k+1) x i*y i)+ |
| j*(χ-onsager j J h (k+1) x)*(∑i,residual j J h (k+1) x i*w i)- |
| j*onsager j J h (k+1) x*(∑i,residual j J h k x i*w i)- |
| j*c*(∑i,residual j J h (k+1) x i*mag j J h (k+2) x i) := by |
| have hsym:=mulVec_pair_symmetric J hJ (residual j J h (k+1) x) w |
| have hinner:=residual_inner_product hn j J h (k+1) x |
| have hfield : (∑i,residual j J h (k+1) x i*J.mulVec w i)= |
| (∑i,residual j J h (k+1) x i*y i)+ |
| j*χ*(∑i,residual j J h (k+1) x i*w i)+ |
| j*c*(∑i,residual j J h (k+1) x i*mag j J h (k+1) x i) := by |
| simp only [hy,Finset.mul_sum,←Finset.sum_add_distrib] |
| apply Finset.sum_congr rfl |
| intro i _ |
| ring |
| have ht : (∑i,(fld j J h (k+1) x i-h i-J.mulVec (mag j J h (k+2) x) i+ |
| j*onsager j J h (k+1) x*mag j J h (k+2) x i)*w i)= |
| (∑i,residual j J h (k+1) x i*J.mulVec w i)- |
| j*onsager j J h (k+1) x*(∑i,residual j J h k x i*w i)- |
| j*onsager j J h (k+1) x*(∑i,residual j J h (k+1) x i*w i) := by |
| simp_rw [terminal_field_identity] |
| simp only [sub_mul,mul_add,add_mul,Finset.sum_sub_distrib,Finset.sum_add_distrib] |
| rw [show (∑i,J.mulVec (residual j J h (k+1) x) i*w i)= |
| ∑i,residual j J h (k+1) x i*J.mulVec w i from hsym] |
| simp only [Finset.mul_sum] |
| ring_nf |
| rw [ht,hfield,hinner] |
| ring |
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| end SKGapCutoff.Primary |
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| end |
| end |
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| end OAI |
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