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import OAI.Computability.BinPacking.Machines.PackingSetupGraph
namespace OAI
noncomputable section
namespace BinPackingGap.GraphPackingRegisters
open PackingItemExpression PackingInventoryDescriptors
open PackingMachineLayout PackingRegisterFrame
def parameters (fixed : InventoryData) (K : Nat) : PackingSetupExpression.Parameters :=
⟨fixed.P, K, fixed.d, fixed.tPlus, fixed.tMinus⟩
inductive Work
| leftEndpoint | rightEndpoint | leftPower | rightPower
| counter | one | temporary | reverse | geometryFactor | header
deriving DecidableEq
instance : Fintype Work where
elems := {.leftEndpoint, .rightEndpoint, .leftPower, .rightPower, .counter,
.one, .temporary, .reverse, .geometryFactor, .header}
complete w := by cases w <;> simp
inductive Extra
| input | scanScratch | scanTemporary | threshold | endpoints | vertices | edges
| vertexCounter | edgeCounter | repetitionCounter | poolCounter | powerCounter
| decodeScratch | reverseOutput | output
deriving DecidableEq
instance : Fintype Extra where
elems := {.input, .scanScratch, .scanTemporary, .threshold, .endpoints,
.vertices, .edges, .vertexCounter, .edgeCounter, .repetitionCounter,
.poolCounter, .powerCounter, .decodeScratch, .reverseOutput, .output}
complete e := by cases e <;> simp
abbrev Numerator (fixed : InventoryData) := PackingDescriptorExpression.numerator fixed
abbrev Other (fixed : InventoryData) (K : Nat) := PackingSetupMachine.Reg (parameters fixed K) ⊕ Work
abbrev Register (fixed : InventoryData) (K : Nat) :=
PackingMachineLayout.Reg (Numerator fixed) denominator (Other fixed K)
abbrev Tape (fixed : InventoryData) (K : Nat) :=
PackingMachineLayout.Tape (Numerator fixed) denominator (Other fixed K) Extra
def slots (fixed : InventoryData) (K : Nat) :
(Register fixed K ⊕ Fin 6) ↪ Tape fixed K := registers (Numerator fixed) denominator
def item (fixed : InventoryData) (K : Nat) : Variable ↪ Register fixed K :=
inputs (Numerator fixed) denominator
def setup (fixed : InventoryData) (K : Nat) :
PackingSetupMachine.Reg (parameters fixed K) ↪ Register fixed K :=
(Function.Embedding.inl : PackingSetupMachine.Reg (parameters fixed K) ↪ Other fixed K).trans
(others (Numerator fixed) denominator)
def work (fixed : InventoryData) (K : Nat) : Work ↪ Register fixed K :=
(Function.Embedding.inr : Work ↪ Other fixed K).trans
(others (Numerator fixed) denominator)
def input (fixed : InventoryData) (K : Nat) (i : PackingSetupExpression.Input) : Register fixed K :=
setup fixed K (PackingSetupMachine.input (parameters fixed K) i)
def output (fixed : InventoryData) (K : Nat) (o : PackingSetupExpression.Output) : Register fixed K :=
setup fixed K (PackingSetupMachine.output (parameters fixed K) o)
def setupSlots (fixed : InventoryData) (K : Nat) :
(PackingSetupMachine.Reg (parameters fixed K) ⊕ Fin 6) ↪ Tape fixed K :=
localSlots (slots fixed K) (setup fixed K)
def graphSlots (fixed : InventoryData) (K : Nat) : Fin 7 ↪ Tape fixed K where
toFun := ![.inr .input, .inr .scanScratch, .inr .scanTemporary,
.inr .threshold, .inr .endpoints, .inr .vertices, .inr .edges]
inj' := by
intro i j h
fin_cases i <;> fin_cases j <;> simp_all
inductive CountSource
| input (i : PackingSetupExpression.Input)
| output (o : PackingSetupExpression.Output)
| endpoint (right : Bool)
deriving DecidableEq, Fintype
def countRegister (fixed : InventoryData) (K : Nat) : CountSource → Register fixed K
| .input i => input fixed K i
| .output o => output fixed K o
| .endpoint right => work fixed K (if right then .rightEndpoint else .leftEndpoint)
def tallyRegisters (fixed : InventoryData) (K : Nat) (source : CountSource) :
BinaryToTallyMachine.Reg ↪ Register fixed K where
toFun
| .source => countRegister fixed K source
| .counter => work fixed K .counter
| .one => work fixed K .one
| .temporary => work fixed K .temporary
inj' := by
intro a b h
cases source with
| input i =>
cases a <;> cases b <;>
simp_all [countRegister, input, setup, work, others, PackingSetupMachine.input]
| output o =>
cases a <;> cases b <;>
simp_all [countRegister, output, setup, work, others, PackingSetupMachine.output]
| endpoint right =>
cases right <;> cases a <;> cases b <;>
simp_all [countRegister, work, others]
def tallySlots (fixed : InventoryData) (K : Nat) (source : CountSource) :
(BinaryToTallyMachine.Reg ⊕ Fin 6) ↪ Tape fixed K :=
localSlots (slots fixed K) (tallyRegisters fixed K source)
theorem tally_outside (fixed : InventoryData) (K : Nat) (source : CountSource)
(tape : Extra) (i : BinaryToTallyMachine.Reg ⊕ Fin 6) :
tallySlots fixed K source i ≠ Sum.inr tape := by
simp [tallySlots, localSlots, slots, registers]
def endpointRegisters (fixed : InventoryData) (K : Nat) (right : Bool) :
Fin 7 ↪ Register fixed K where
toFun := ![countRegister fixed K (.endpoint right), output fixed K .three,
work fixed K (if right then .rightPower else .leftPower), work fixed K .counter,
work fixed K .one, work fixed K .temporary, work fixed K .reverse]
inj' := by
intro i j h
cases right <;> fin_cases i <;> fin_cases j <;>
simp_all [countRegister, output, setup, work, others, PackingSetupMachine.output]
def endpointSlots (fixed : InventoryData) (K : Nat) (right : Bool) :
(Fin 7 ⊕ Fin 6) ↪ Tape fixed K :=
localSlots (slots fixed K) (endpointRegisters fixed K right)
theorem endpoint_outside (fixed : InventoryData) (K : Nat) (right : Bool)
(tape : Extra) (i : Fin 7 ⊕ Fin 6) :
endpointSlots fixed K right i ≠ Sum.inr tape := by
simp [endpointSlots, localSlots, slots, registers]
def powerSlots (fixed : InventoryData) (K : Nat) (source : CountSource)
(destination : Variable) : Fin 11 ↪ Tape fixed K where
toFun := ![slots fixed K (.inl (countRegister fixed K source)),
slots fixed K (.inl (item fixed K destination)),
slots fixed K (.inl (work fixed K .temporary)),
slots fixed K (.inr 0), slots fixed K (.inr 1), slots fixed K (.inr 2),
slots fixed K (.inr 3), slots fixed K (.inr 4), slots fixed K (.inr 5),
.inr .powerCounter, slots fixed K (.inl (work fixed K .reverse))]
inj' := by
intro i j h
cases source with
| input a =>
fin_cases i <;> fin_cases j <;>
simp_all [countRegister, input, setup, work, item, inputs, others,
slots, registers, PackingSetupMachine.input]
all_goals
first
| change (Sum.inl (Sum.inr _) : Register fixed K) = Sum.inl (Sum.inl _) at h
| change (Sum.inl (Sum.inl _) : Register fixed K) = Sum.inl (Sum.inr _) at h
simp at h
| output a =>
fin_cases i <;> fin_cases j <;>
simp_all [countRegister, output, setup, work, item, inputs, others,
slots, registers, PackingSetupMachine.output]
all_goals
first
| change (Sum.inl (Sum.inr _) : Register fixed K) = Sum.inl (Sum.inl _) at h
| change (Sum.inl (Sum.inl _) : Register fixed K) = Sum.inl (Sum.inr _) at h
simp at h
| endpoint right =>
cases right <;> fin_cases i <;> fin_cases j <;>
simp_all [countRegister, work, item, inputs, others, slots, registers]
all_goals
first
| change (Sum.inl (Sum.inr _) : Register fixed K) = Sum.inl (Sum.inl _) at h
| change (Sum.inl (Sum.inl _) : Register fixed K) = Sum.inl (Sum.inr _) at h
simp at h
def decoderSlots (fixed : InventoryData) (K : Nat) (right : Bool) : Fin 3 ↪ Tape fixed K where
toFun := ![.inr .endpoints, .inr .decodeScratch,
slots fixed K (.inl (countRegister fixed K (.endpoint right)))]
inj' := by
intro i j h
fin_cases i <;> fin_cases j <;> simp_all [slots, registers]
def thresholdSlots (fixed : InventoryData) (K : Nat) : Fin 3 ↪ Tape fixed K where
toFun := ![.inr .threshold, .inr .decodeScratch, slots fixed K (.inl (input fixed K .k))]
inj' := by
intro i j h
fin_cases i <;> fin_cases j <;> simp_all [slots, registers]
end BinPackingGap.GraphPackingRegisters
end
end OAI
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