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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 86 87 88 89 90 91 92 93 94 95 96 97 98 99 | import Mathlib
import OAI.Probability.SKBarriers.SpinGlass.Basic
namespace OAI
section
noncomputable section
open scoped BigOperators InnerProductSpace
open MeasureTheory ProbabilityTheory Filter Set
namespace SK.Analytic
theorem finite_unit_packing {E I : Type*} [NormedAddCommGroup E] [InnerProductSpace ℝ E]
[DecidableEq I] (S : Finset I) (u : I → E) (w : E) (b e : ℝ)
(hb : 0<b) (he : 0≤e) (heb : e<b^2/2)
(hw : ⟪w,w⟫_ℝ=1) (hu : ∀ i∈S,⟪u i,u i⟫_ℝ=1)
(hpair : ∀ i∈S,∀ j∈S,i≠j → ⟪u i,u j⟫_ℝ≤e)
(hlarge : ∀ i∈S,b<⟪u i,w⟫_ℝ) : (S.card:ℝ)<2/b^2 := by
by_cases hS : S.Nonempty
· let d : ℝ := S.card
have hd : 0<d := by dsimp [d]; exact_mod_cast hS.card_pos
have hl : d*b<⟪∑ i∈S,u i,w⟫_ℝ := by
rw [sum_inner]
simpa [d] using Finset.sum_lt_sum_of_nonempty hS hlarge
have hu' : ⟪∑ i∈S,u i,∑ i∈S,u i⟫_ℝ≤d+d^2*e := by
rw [sum_inner]
simp_rw [inner_sum]
calc
_ ≤ ∑ i∈S,∑ j∈S,((if i=j then (1:ℝ) else 0)+e) := by
apply Finset.sum_le_sum
intro i hi
apply Finset.sum_le_sum
intro j hj
by_cases hij : i=j
· subst j; rw [ite_eq_left rfl,hu i hi]; linarith only [he]
· simpa only [ite_eq_right hij,zero_add] using hpair i hi j hj hij
_ = _ := by
simp only [Finset.sum_add_distrib,Finset.sum_ite_eq,Finset.sum_const,nsmul_eq_mul]
simp only [Finset.sum_ite_mem,Finset.inter_self]
simp [d]
ring
have hcs := real_inner_mul_inner_self_le (∑ i∈S,u i) w
rw [hw,mul_one] at hcs
have hls : (d*b)^2<⟪∑ i∈S,u i,w⟫_ℝ^2 := by
exact pow_lt_pow_left₀ hl (mul_nonneg hd.le hb.le) (by decide)
have hquad : d^2*b^2<d+d^2*e := by nlinarith only [hls,hcs,hu']
have hdb : d*b^2<2 := by
by_contra h
have hd2 : 2≤d*b^2 := le_of_not_gt h
have hpos : 0<d^2 := sq_pos_of_pos hd
have hs := mul_lt_mul_of_pos_left heb hpos
have hm := mul_le_mul_of_nonneg_left hd2 hd.le
nlinarith only [hquad,hs,hm]
exact (lt_div_iff₀ (sq_pos_of_pos hb)).mpr (by simpa [d] using hdb)
· have hz : S=∅ := Finset.not_nonempty_iff_eq_empty.mp hS
simp only [hz,Finset.card_empty,Nat.cast_zero]
positivity
def spinUnitVector {n : ℕ} (x : Config n) : EuclideanSpace ℝ (Fin n) :=
WithLp.toLp 2 (fun i => spin (x i)/Real.sqrt n)
theorem spinUnitVector_inner {n : ℕ} (x y : Config n) :
⟪spinUnitVector x,spinUnitVector y⟫_ℝ=overlap x y := by
change (∑ i : Fin n,(spin (y i)/Real.sqrt n)*(spin (x i)/Real.sqrt n))=overlap x y
simp only [div_mul_div_comm,← sq,Real.sq_sqrt (Nat.cast_nonneg n),← Finset.sum_div]
exact congrArg (fun z : ℝ => z/(n:ℝ)) (Finset.sum_congr rfl (fun i _ => mul_comm _ _))
@[simp] theorem overlap_flip_first {n : ℕ} (x y : Config n) : overlap (flip x) y = -overlap x y := by
rw [overlap_comm,overlap_flip_right,overlap_comm x y]
theorem overlap_conflict_card {n : ℕ} (hn : 0<n) {I : Type*} [DecidableEq I]
(S : Finset I) (v : I → Config n) (w : Config n) (b e : ℝ)
(hb : 0<b) (he : 0≤e) (heb : e<b^2/2)
(hpair : ∀ i∈S,∀ j∈S,i≠j → |overlap (v i) (v j)|≤e)
(hlarge : ∀ i∈S,b< |overlap (v i) w|) : (S.card:ℝ)<2/b^2 := by
let u (i : I) : Config n := if 0≤overlap (v i) w then v i else flip (v i)
apply finite_unit_packing S (fun i => spinUnitVector (u i)) (spinUnitVector w) b e hb he heb
· rw [spinUnitVector_inner,overlap_self hn]
· intro i hi; rw [spinUnitVector_inner,overlap_self hn]
· intro i hi j hj hij
rw [spinUnitVector_inner]
have h := hpair i hi j hj hij
dsimp only [u]
split_ifs <;> (try simp only [overlap_flip_first,overlap_flip_right,neg_neg]) <;>
linarith only [(abs_le.mp h).1,(abs_le.mp h).2]
· intro i hi
rw [spinUnitVector_inner]
dsimp only [u]
split_ifs with h
· simpa only [abs_of_nonneg h] using hlarge i hi
· simpa only [overlap_flip_first,abs_of_neg (lt_of_not_ge h)] using hlarge i hi
end SK.Analytic
end
end
end OAI
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