File size: 4,076 Bytes
d5b7741 | 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 | import OAI.Combinatorics.ContingencyTables.Transport.BoundedProfileWalk
import OAI.Combinatorics.ContingencyTables.Transport.ListedBooleanStep
namespace OAI
/-! # The bounded binary loop preserves the physical profile walk -/
namespace ContingencyTables.SmallGraphProfiles
open ProfilePrograms PhysicalCompletionFibres FirstPaperPhysicalMarginal CompletionCounts
open SmallContextCoordinates FirstSuccess FairBits
open scoped BigOperators Classical
noncomputable section
variable {m n q : ℕ} (r : Fin m → ℕ) (c : Fin n → ℕ)
variable (e : Fin q ≃ Cells (paperSmall r c))
local notation "d" => dimensionAllowance (I := Fin m) (J := Fin n)
lemma stateCode_bounded (x : PaperStates r c) (a : Cells (paperSmall r c)) (side : Bool) :
physicalProfile (paperSmall r c) (stateCapacity r c) r c (d^12) x (a,side)≤d^20 := by
have he := (physicalProfile_feasible r c x).bounded (a,side)
exact he.trans_eq (stateCapacity_small r c _ _ a.property)
lemma capViews_stateCode (x y : PaperStates r c) :
capViews (stateCodeData r c e x,stateCode r c e y)=stateCode r c e y := by
have hb (side : Bool) :
capView ((q,d^20),viewList (paperSmall r c) e
(physicalProfile (paperSmall r c) (stateCapacity r c) r c (d^12) y) side)=
viewList (paperSmall r c) e
(physicalProfile (paperSmall r c) (stateCapacity r c) r c (d^12) y) side := by
apply capView_eq
· simp only [viewList,List.length_ofFn]
· intro a ha
obtain ⟨i,rfl⟩ := List.mem_ofFn.mp ha
exact stateCode_bounded r c y (e i) side
simpa only [capViews,stateCodeData,codeCheckData,cellCatalog,List.length_ofFn,stateCode,profileCode]
using Prod.ext (hb false) (hb true)
variable (hr : 0 < (largeIndices ((dimensionAllowance (I := Fin m) (J := Fin n))^20,List.ofFn r)).length)
variable (hc : 0 < (largeIndices ((dimensionAllowance (I := Fin m) (J := Fin n))^20,List.ofFn c)).length)
variable (b h : ℕ) (hb : Nat.clog 2 (stateBound r c+2) ≤ b)
local notation "s" => smallProposalBits (I := Fin m) (J := Fin n)
local notation "f" => 25*(d+b+h+1)^62
def encodedConfiguration (initial current : PaperStates r c) : ProfileConfiguration :=
(((stateCodeData r c e initial,d^12),(d,h)),stateCode r c e current)
theorem advanceProfile_encoded (htotal : (∑ i,r i)=∑ j,c j)
(initial x : PaperStates r c) (bits : Fin (s+f) → Bool) :
advanceProfile ((List.ofFn (splitWordEquiv s f bits).1,List.ofFn (splitWordEquiv s f bits).2),
encodedConfiguration r c e h initial x)=
encodedConfiguration r c e h initial (listedBitStep r c e hr hc b h hb x bits) := by
apply Prod.ext
· rfl
· change capViews (stateCodeData r c e initial,
profileStep (encodedStepInput r c e b h x
(splitWordEquiv s f bits).1 (splitWordEquiv s f bits).2))=_
rw [profileStep_listedBitStep r c e hr hc b h hb htotal]
exact capViews_stateCode r c e initial _
theorem profileWalk_ofFn (htotal : (∑ i,r i)=∑ j,c j)
(initial x : PaperStates r c) (steps : ℕ) (bits : Fin steps → Fin (s+f) → Bool) :
profileWalk (encodedConfiguration r c e h initial x,
List.ofFn (fun i => (List.ofFn (splitWordEquiv s f (bits i)).1,
List.ofFn (splitWordEquiv s f (bits i)).2)))=
encodedConfiguration r c e h initial
(wordWalk (listedBitStep r c e hr hc b h hb) x steps bits) := by
induction steps with
| zero => rfl
| succ steps ih =>
unfold profileWalk at ih ⊢
rw [List.ofFn_succ',List.concat_eq_append,List.foldl_append]
change advanceProfile ((List.ofFn (splitWordEquiv s f (bits (Fin.last steps))).1,
List.ofFn (splitWordEquiv s f (bits (Fin.last steps))).2),
(List.ofFn (fun i => (List.ofFn (splitWordEquiv s f ((Fin.init bits) i)).1,
List.ofFn (splitWordEquiv s f ((Fin.init bits) i)).2))).foldl
(fun state word => advanceProfile (word,state))
(encodedConfiguration r c e h initial x))=_
rw [ih (Fin.init bits)]
exact advanceProfile_encoded r c e hr hc b h hb htotal initial _ _
end
end ContingencyTables.SmallGraphProfiles
end OAI
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