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import OAI.Combinatorics.IndependentSets.Expansion.ExpanderTables
namespace OAI
namespace IndependentSetsGames.Foundations.PCP.ExpanderRowControl
open ExpanderTables
abbrev degree (d : Nat) : Nat := d * d
abbrev cloudSize (d : Nat) : Nat := degree d * degree d
abbrev rowFactor (d : Nat) : Nat := cloudSize d * degree d
structure Control (d : Nat) where
firstH : Fin (cloudSize d) × Fin d
lastInput : Fin d
firstG : Fin (degree d)
secondG : Fin (degree d)
lastH : Fin (cloudSize d) × Fin d
deriving DecidableEq, Fintype
def squarePorts {d : Nat} (s : Control d) : Fin (degree d) × Fin (degree d) :=
(rowIndex (degree d) (degree d)).symm s.firstH.1
def start {d : Nat} (H : Table (cloudSize d) d)
(cloud : Fin (cloudSize d)) (port : Fin (degree d)) : Control d :=
let ports := (rowIndex d d).symm port
let first := lookup H (cloud, ports.1)
let square := (rowIndex (degree d) (degree d)).symm first.1
{ firstH := first
lastInput := ports.2
firstG := square.1
secondG := square.2
lastH := first }
def receiveFirst {d : Nat} (s : Control d) (returnedPort : Fin (degree d)) : Control d :=
{ s with firstG := returnedPort }
def receiveSecond {d : Nat} (H : Table (cloudSize d) d)
(s : Control d) (returnedPort : Fin (degree d)) : Control d :=
{ s with
secondG := returnedPort
lastH := lookup H
(rowIndex (degree d) (degree d) (returnedPort,s.firstG),s.lastInput) }
def firstOffset {d : Nat} (s : Control d) : Fin (degree d) := (squarePorts s).1
def secondOffset {d : Nat} (s : Control d) : Fin (degree d) := (squarePorts s).2
def outputPort {d : Nat} (s : Control d) : Fin (degree d) :=
rowIndex d d (s.lastH.2,s.firstH.2)
def outputOffset {d : Nat} (s : Control d) : Fin (rowFactor d) :=
rowIndex (cloudSize d) (degree d) (s.lastH.1,outputPort s)
def firstAddress {d : Nat} (vertex : Nat) (s : Control d) : Nat :=
degree d * vertex + (firstOffset s).val
def secondAddress {d : Nat} (vertex : Nat) (s : Control d) : Nat :=
degree d * vertex + (secondOffset s).val
def outputAddress {d : Nat} (vertex : Nat) (s : Control d) : Nat :=
rowFactor d * vertex + (outputOffset s).val
@[simp] theorem firstOffset_receiveFirst {d : Nat} (s : Control d) (r : Fin (degree d)) :
firstOffset (receiveFirst s r) = firstOffset s := rfl
@[simp] theorem secondOffset_receiveFirst {d : Nat} (s : Control d) (r : Fin (degree d)) :
secondOffset (receiveFirst s r) = secondOffset s := rfl
@[simp] theorem receiveFirst_port {d : Nat} (s : Control d) (r : Fin (degree d)) :
(receiveFirst s r).firstG = r := rfl
@[simp] theorem receiveSecond_port {d : Nat} (H : Table (cloudSize d) d)
(s : Control d) (r : Fin (degree d)) : (receiveSecond H s r).secondG = r := rfl
theorem firstAddress_eq_rowIndex {v d : Nat} (vertex : Fin v) (s : Control d) :
firstAddress vertex.val s =
(rowIndex v (degree d) (vertex,firstOffset s)).val := by
rw [rowIndex_val]
exact Nat.add_comm _ _
theorem secondAddress_eq_rowIndex {v d : Nat} (vertex : Fin v) (s : Control d) :
secondAddress vertex.val s =
(rowIndex v (degree d) (vertex,secondOffset s)).val := by
rw [rowIndex_val]
exact Nat.add_comm _ _
theorem firstAddress_lt {v d : Nat} (vertex : Fin v) (s : Control d) :
firstAddress vertex.val s < v * degree d := by
rw [firstAddress_eq_rowIndex]
exact (rowIndex v (degree d) (vertex,firstOffset s)).isLt
theorem secondAddress_lt {v d : Nat} (vertex : Fin v) (s : Control d) :
secondAddress vertex.val s < v * degree d := by
rw [secondAddress_eq_rowIndex]
exact (rowIndex v (degree d) (vertex,secondOffset s)).isLt
theorem lookup_vertex_div {v q : Nat} (G : Table v q) (x : Fin v × Fin q) :
(lookup G x).1.val = (reverseIndex G (rowIndex v q x)).val / q := rfl
theorem lookup_port_mod {v q : Nat} (G : Table v q) (x : Fin v × Fin q) :
(lookup G x).2.val = (reverseIndex G (rowIndex v q x)).val % q := rfl
theorem lookup_port_eq_remainder {v q : Nat} (G : Table v q) (x : Fin v × Fin q)
(r : Fin q) (hr : r.val = (reverseIndex G (rowIndex v q x)).val % q) :
(lookup G x).2 = r := Fin.ext ((lookup_port_mod G x).trans hr.symm)
theorem rowIndex_lookup {v q : Nat} (G : Table v q) (x : Fin v × Fin q) :
rowIndex v q (lookup G x) = reverseIndex G (rowIndex v q x) := by
simp only [lookup, Equiv.apply_symm_apply]
def outputPair {v d : Nat} (vertex : Fin v) (s : Control d) :
Fin (v * cloudSize d) × Fin (degree d) :=
(rowIndex v (cloudSize d) (vertex,s.lastH.1),outputPort s)
theorem outputAddress_eq_rowIndex {v d : Nat} (vertex : Fin v) (s : Control d) :
outputAddress vertex.val s =
(rowIndex (v * cloudSize d) (degree d) (outputPair vertex s)).val := by
simp only [outputAddress, outputOffset, outputPair, rowIndex_val, rowFactor]
ring
theorem outputAddress_lt {v d : Nat} (vertex : Fin v) (s : Control d) :
outputAddress vertex.val s < (v * cloudSize d) * degree d := by
rw [outputAddress_eq_rowIndex]
exact (rowIndex (v * cloudSize d) (degree d) (outputPair vertex s)).isLt
def evaluateRow {v d : Nat} (G : Table v (degree d)) (H : Table (cloudSize d) d)
(vertex : Fin v) (cloud : Fin (cloudSize d)) (port : Fin (degree d)) :
Fin v × Control d :=
let s₀ := start H cloud port
let g₁ := lookup G (vertex,firstOffset s₀)
let s₁ := receiveFirst s₀ g₁.2
let g₂ := lookup G (g₁.1,secondOffset s₁)
(g₂.1, receiveSecond H s₁ g₂.2)
theorem evaluateRow_eq_lookup_step {v d : Nat}
(G : Table v (degree d)) (H : Table (cloudSize d) d)
(vertex : Fin v) (cloud : Fin (cloudSize d)) (port : Fin (degree d)) :
outputPair (evaluateRow G H vertex cloud port).1 (evaluateRow G H vertex cloud port).2 =
lookup (step G H) (rowIndex v (cloudSize d) (vertex,cloud),port) := by
rw [lookup_step]
simp only [evaluateRow, start, receiveFirst, receiveSecond, firstOffset, secondOffset,
squarePorts, outputPair, outputPort, stepLookup, Equiv.symm_apply_apply]
theorem outputAddress_eq_step_row {v d : Nat}
(G : Table v (degree d)) (H : Table (cloudSize d) d)
(vertex : Fin v) (cloud : Fin (cloudSize d)) (port : Fin (degree d)) :
outputAddress (evaluateRow G H vertex cloud port).1.val (evaluateRow G H vertex cloud port).2 =
(rowIndex (v * cloudSize d) (degree d)
(lookup (step G H) (rowIndex v (cloudSize d) (vertex,cloud),port))).val := by
rw [outputAddress_eq_rowIndex, evaluateRow_eq_lookup_step]
theorem outputAddress_eq_step_reverseIndex {v d : Nat}
(G : Table v (degree d)) (H : Table (cloudSize d) d)
(vertex : Fin v) (cloud : Fin (cloudSize d)) (port : Fin (degree d)) :
outputAddress (evaluateRow G H vertex cloud port).1.val (evaluateRow G H vertex cloud port).2 =
(reverseIndex (step G H) (rowIndex (v * cloudSize d) (degree d)
(rowIndex v (cloudSize d) (vertex,cloud),port))).val := by
rw [outputAddress_eq_step_row, rowIndex_lookup]
end IndependentSetsGames.Foundations.PCP.ExpanderRowControl
end OAI
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