File size: 4,604 Bytes
fcab081 | 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 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 | import OAI.Combinatorics.HadwigerCounterexample.OrientationRankBounds
namespace OAI
/-! # Basis Conditioning -/
noncomputable section
namespace HadwigerCounterexample
open Matrix Module LinearMap Finset Filter Function
open scoped BigOperators
variable {p : Shape} {n N : ℕ} {I : Type} [Fintype I] [DecidableEq I]
theorem totalRank_pos {x : Profile p n} (h : x≠0) :
0<totalRank x := by
obtain ⟨i,hi⟩ := Function.ne_iff.mp h
apply lt_of_lt_of_le _ (Finset.single_le_sum (fun i _=>Nat.zero_le ((x i).rank))
(Finset.mem_univ i))
apply Nat.pos_of_ne_zero; intro h
obtain ⟨A,B,hh⟩ := mul_factor (x i) 0 h.le
apply hi; ext a b; rw [←hh,Matrix.mul_apply]; simp
def glintRate (K : ℕ) := bp (-((K^2:ℕ):ℝ)-3)
theorem glint_within {K d : ℕ} (h : d≤ K^2) :
glintRate K ≤ (1/8:ℝ)*bp (- (d:ℝ)) := by
unfold glintRate
have hh : bp (-3)=1/8 := by norm_num [bp]
rw [←hh,←bp_add]
exact bp_le (by linarith only [(show (d:ℝ) ≤ ((K^2:ℕ):ℝ) from by exact_mod_cast h)])
namespace Template
variable (v : Solo p)
def stains (m : Mixer p n) (f : Vec I →ₗ[F] Profile p n) (z : Knobs p.g n) : Vec I :=
fun i=> Tsym m (f (Pi.single i 1)) (useAtom p (v.t ()) (v.s ()) (v.f ()) z)
theorem staining (m : Mixer p n) (f : Vec I→ₗ[F] Profile p n) (z : Knobs p.g n) (i : Vec I) :
vectorPhase i (stains v m f z)=shanty v m (f i) z := by
classical
let h := LinearMap.flip (Tsym m) (useAtom p (v.t ()) (v.s ()) (v.f ()) z) ∘ₗ f
have ht : h i=h (∑ j: I,i j • (Pi.single j 1 : Vec I)) := by
have he (j:I) : i j • (Pi.single j 1 : Vec I)=Pi.single j (i j) := by
funext k; by_cases hh:k=j <;> simp [hh]
simp_rw [he]
rw [Finset.univ_sum_single]
change wsign (∑ j:I,i j*h (Pi.single j 1))=wsign (h i)
rw [ht,map_sum]; simp_rw [map_smul]; rfl
def luminous (v : Solo p) (m : Mixer p n) (w : Wgt (Knobs p.g n)) (D : AtomKey (p:=p) (v.t ())
N →Prop) (K : ℕ) (i : Orient p n N) :=
let P := offBeat v D i
0<pr w P ∧ ∀ (d : ℕ), d≤ K^2 → ∀ f : Vec (Fin d)→ₗ[F] Profile p n,
Injective f → (∀ x,totalRank (f x)≤K) → ∀ x,
(7/8:ℝ)*bp (-(d:ℝ)) ≤ pr (onW w P) (fun z=> stains v m f z=x)
set_option maxHeartbeats 800000 in
theorem glow_luminous (v : Solo p) (m : Mixer p n) (w : Wgt (Knobs p.g n))
(hw : IsLaw w)
(S : AtomKey (p:=p) (v.t ()) N → Prop)
(K : ℕ) (i : Orient p n N) (δ : ℝ) (hδ : 0<δ) (he : δ ≤ pr w (offBeat v S i))
(h : ∀ x, 0<totalRank x → totalRank x≤K → ¬ glow v m w S (glintRate K) δ x i) :
luminous v m w S K i := by
classical
let P := offBeat v S i
have hP : 0<pr w P := hδ.trans_le he
let u := onW w P; have hu : IsLaw u := on_law hw.nz hP
refine ⟨hP,fun d hd f hf hg x => ?_⟩
let s := stains v m f
have heq :
pr u (fun z => s z=x) = avg unif (fun i => vectorPhase i x*
avg u (fun z => vectorPhase i (s z))) := by
unfold pr
simp_rw [vector_expand]
rw [avg_avg]
apply avg_congr
intro i hi
rw [avg_mul]; ring
let b := bp (-(d:ℝ))
let Q (k : Vec (Fin d)) := vectorPhase k x * avg u (fun z => vectorPhase k (s z))
have hb : 0 < b := bp_pos _
have hh k : ind (k=0)-b/8 ≤ Q k := by
by_cases hk : k=0
· subst k
have hQ : Q 0=1 := by unfold Q; simp only [vectorPhase,zero_dotProduct,wsign_zero,one_mul,
hu.avg_const]
rw [ind_one (show (0:Vec (Fin d))=0 from rfl),hQ]; linarith
let k' := f k
have hT := h k' (totalRank_pos (hf.ne hk |>.trans_eq (map_zero f))) (hg k)
have hz : |zsumPeak p w (v.shanty m k') P| ≤ (glintRate K)*pr w P :=
le_of_lt (lt_of_not_ge (fun h => hT ⟨he,h⟩))
have ht :
avg u (fun z => vectorPhase k (s z))=
zsumPeak p w (v.shanty m k') P/pr w P := by
unfold s u onW; simp_rw [staining]; rw [avg_norm,subW_tot]
congr 1; simp [zsumPeak,avg,subW,mul_assoc,k']
have h1 : |Q k|≤b/8 := by
unfold Q; rw [ht,abs_mul,vectorPhase,wsign_abs,one_mul,abs_div,abs_of_pos hP]
refine le_trans ((div_le_iff₀ hP).mpr hz) ((glint_within hd).trans_eq (by unfold b; ring))
rw [ind_zero hk,zero_sub]; exact (abs_le.mp h1).1
have h0 : unif (0:Vec (Fin d))=b := by
unfold b; rw [unif_apply,card_space,dim_vec,bp_neg,Fintype.card_fin]
change _ ≤ pr u (fun z => s z=x)
rw [heq]
refine le_trans ?_ (avg_mono unif_law.nz hh)
rw [avg_sub,unif_law.avg_const,
show avg unif (fun i : Vec (Fin d)=> ind (i=0))=b from (pr_point' unif _).trans h0]
change 7/8*b≤_
linarith
end Template
end HadwigerCounterexample
end
end OAI
|