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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 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 124 125 126 127 128 129 130 131 132 133 134 135 | import OAI.Combinatorics.HadwigerCounterexample.IndependentColumnBatches
import OAI.Combinatorics.HadwigerCounterexample.CollisionInequality
namespace OAI
/-! # Bounded Query Templates -/
noncomputable section
namespace HadwigerCounterexample
open scoped BigOperators
open Matrix Module LinearMap Function
variable {p : Shape} {n N : ℕ} {I : Type} [Fintype I] [DecidableEq I]
structure Template (p : Shape) (I : Type) where
aim : I→Bool
t : Tagging p I
s : I→Label p.b
f : I→Flavor p.g
abbrev Inputs (p : Shape) (n : ℕ) (I : Type) := I → Knobs p.g n
namespace Template
variable (v : Template p I)
def draw (y : Inputs p n I) : Volley p n I where
obj := v.aim
sel := v.s
par i := lit (v.f i) (v.t i) (y i)
def state (q : I→F) (n : ℕ) : Wgt (Inputs p n I) :=
prodW (fun i : I => scopedBit p n (v.t i) (v.f i) (q i))
omit [Fintype I] [DecidableEq I] in
theorem draw_legal (y : Inputs p n I) : (v.draw y).legal v.t :=
fun _ => lit_allows _ _ _
end Template
abbrev KeySlots (p : Shape) (t : Tagging p I) := Σ i, TagStar (t i)
def scatter (t : Tagging p I) (N : ℕ) : Keys (p:=p) t N ≃ Rack (KeySlots p t) N where
toFun x := (Matrix.of (fun j i => x i.1 i.2 true j),Matrix.of (fun j i => x i.1 i.2 false j))
invFun x := fun i j b => if b then x.1.col ⟨i,j⟩ else x.2.col ⟨i,j⟩
left_inv x := by funext i j b; cases b <;> rfl
right_inv x := by apply Prod.ext <;> rfl
def globPort (p : Shape) (t : Tagging p I) (q : I→Bool) :
Route (KeySlots p t) (Ports p) :=
fun x => (q x.1,x.2.val)
def nomAsk (t : Tagging p I) (v : Volley p n I) :
NomQuery p n (globPort p t v.obj) :=
fun _ => Matrix.of (fun i j => v.rayAt j.val.1 i)
omit [Fintype I] [DecidableEq I] in
theorem peek_keys (t : Tagging p I) (v : Volley p n I) (x : UnitType p n N) :
peekR p (globPort p t v.obj) (nomAsk t v) (nomAsk t v) (mergeR p x) =
scatter t N (v.keys t x) := by
apply Prod.ext
· ext i j; rfl
· ext i j; rfl
theorem raw_keys (t : Tagging p I) (v : Volley p n I) (m : Mixer p n) {N : ℕ}
(hN : 2*num (Full p n)<N) (ha : ∀ i, Injective (nomAsk t v i).mulVec) :
let r := raw m N
mapW (pairW r r) ((scatter t N) ∘ v.keys t) = rackStation (database m (nomAsk t v) (nomAsk t v)) N := by
classical
let g := globPort p t v.obj
let A := nomAsk t v
have hh := law_peekR m hN ha ha
rw [← law_mergeR m N, mapW_nested] at hh
refine Eq.trans ?_ hh
congr 1
def ownPort (t : Tagging p I) : Route (KeySlots p t) (Comp p.g) := fun x => x.2.val
def ownMap (t : Tagging p I) (q : I→F) : GeoM (ownPort (p:=p) t) :=
fun _ => Matrix.diagonal (fun i => q i.val.1)
def ownPart (t : Tagging p I) (q : I→F) {N : ℕ}
(x : Keys (p:=p) t N) :=
gramPart (ownMap (p:=p) t q) (scatter t N x)
def lawNu (p : Shape) (N : ℕ) (t : Tagging p I) (q : I→F) :
Wgt (Keys (p:=p) t N) := unifOn (ownPart (p:=p) t q)
theorem lawNu_rack (t : Tagging p I) (q : I→F) :
mapW (lawNu p N t q) (scatter t N) = rackWave (ownMap t q) N :=
on_equiv ..
def ownOne (p : Shape) {N : ℕ} (l : Tag p.g) (q : F) (x : AtomKey (p:=p) l N) :=
∀ i,dotProduct (x i true) (x i false) = q
def lawOmega (p : Shape) (N : ℕ) (l : Tag p.g) (q : F) :
Wgt (AtomKey (p:=p) l N) := unifOn (ownOne p l q)
omit [Fintype I] in
theorem ownPart_def (t : Tagging p I) (q : I→F) {N : ℕ} (x : Keys (p:=p) t N) :
ownPart (p:=p) t q x ↔
(∀ i, ownOne p (t i) (q i) (x i)) ∧
∀ (i j : I) (e : Comp p.g) (hi : inc e (t i)) (hj : inc e (t j)), i≠j →
dotProduct (x i ⟨e,hi⟩ true) (x j ⟨e,hj⟩ false)=0 := by
classical
let Q := ownPort (p:=p) t
unfold ownPart
rw [part_scores]
have he i j (e : Comp p.g) (hi : inc e (t i)) (hj : inc e (t j)) :
let a : Port Q e := ⟨⟨i,⟨e,hi⟩⟩,rfl⟩
let b : Port Q e := ⟨⟨j,⟨e,hj⟩⟩,rfl⟩
scoreR (scatter t N x) a b=dotProduct (x i ⟨e,hi⟩ true) (x j ⟨e,hj⟩ false) := rfl
constructor
· intro h
constructor
· rintro i ⟨e,hi⟩
let a : Port Q e := ⟨⟨i,⟨e,hi⟩⟩,rfl⟩
exact (he i i e hi hi).symm.trans ((h e a a).trans (Matrix.diagonal_apply_eq ..))
intro i j e hi hj heq
let a : Port Q e := ⟨⟨i,⟨e,hi⟩⟩,rfl⟩
let b : Port Q e := ⟨⟨j,⟨e,hj⟩⟩,rfl⟩
have hh : a≠b := fun h => heq (congrArg (fun x:Port Q e => x.val.1) h)
exact (he i j e hi hj).symm.trans ((h e a b).trans (Matrix.diagonal_apply_ne _ hh))
· rintro ⟨h,h'⟩ e ⟨⟨i,v⟩,ha⟩ ⟨⟨j,w⟩,hb⟩
rcases v with ⟨v,hv⟩; rcases w with ⟨w,hw⟩
change v = e at ha; change w=e at hb
subst v; subst w
let a : Port Q e := ⟨⟨i,⟨e,hv⟩⟩,rfl⟩
let b : Port Q e := ⟨⟨j,⟨e,hw⟩⟩,rfl⟩
by_cases he : i=j
· subst j
refine (show scoreR (scatter t N x) a.val a.val=_ from h i ⟨e,hv⟩).trans ?_
exact (Matrix.diagonal_apply_eq (fun j : Port Q e => q j.val.1) a).symm
have hh : a≠b := fun h => he (congrArg (fun x:Port Q e => x.val.1) h)
refine Eq.trans (h' i j e hv hw he) ?_
exact (Matrix.diagonal_apply_ne (fun j : Port Q e => q j.val.1) hh).symm
end HadwigerCounterexample
end
end OAI
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