openAI-math-dataset / lean /OAI /Computability /PerfectCompleteness /Machines /DescriptorProjectionMachine.lean
| import OAI.Computability.PerfectCompleteness.Reduction.SignedCompletionSchedule | |
| import OAI.Computability.UniqueGames.Machines.MachineFiniteTable | |
| namespace OAI | |
| namespace PerfectCompleteness.DescriptorProjectionMachine | |
| open Turing | |
| open UniqueGamesTheorem.Foundations.Complexity | |
| open SignedCompletionSchedule | |
| open scoped Classical | |
| noncomputable section | |
| variable {branch : Nat → Nat} {n t q : Nat} {rows repeats : Nat → Nat} | |
| (hq : 0 < q) | |
| (large : CanonicalKeyEncoding.partitionWidth (TreeCanonical.locationCount branch n t) ≤ q) | |
| def projection (descriptor : Descriptor (rows := rows) (repeats := repeats) hq large) : | |
| ProjectionTable q := | |
| LocalCompletionSerialization.projection | |
| (FiniteConstraintMachine.result_correct hq large ⟨descriptor.1, descriptor.2.1⟩) | |
| descriptor.2.2 | |
| theorem projection_apply | |
| (descriptor : Descriptor (rows := rows) (repeats := repeats) hq large) | |
| (a : Fin (2 * q)) : | |
| (projection hq large descriptor).images[a] = descriptorTable hq large descriptor a := | |
| (descriptorTable_eq_parsed hq large descriptor.1 descriptor.2 a).symm | |
| def outputWords (descriptor : Descriptor (rows := rows) (repeats := repeats) hq large) : | |
| List Nat := Encoding.tableWords (projection hq large descriptor) | |
| theorem outputWords_eq | |
| (descriptor : Descriptor (rows := rows) (repeats := repeats) hq large) : | |
| outputWords hq large descriptor = | |
| List.ofFn (fun a : Fin (2 * q) => (descriptorTable hq large descriptor a).val) := | |
| (LocalCompletionSerialization.rawTableWords_eq | |
| (FiniteConstraintMachine.result_correct hq large ⟨descriptor.1, descriptor.2.1⟩) | |
| descriptor.2.2).symm | |
| @[simp] theorem outputWords_length | |
| (descriptor : Descriptor (rows := rows) (repeats := repeats) hq large) : | |
| (outputWords hq large descriptor).length = 2 * q := | |
| Encoding.tableWords_length _ | |
| def outputBits (descriptor : Descriptor (rows := rows) (repeats := repeats) hq large) : | |
| List Bool := encodeWords (outputWords hq large descriptor) | |
| theorem outputBits_length_le | |
| (descriptor : Descriptor (rows := rows) (repeats := repeats) hq large) : | |
| (outputBits hq large descriptor).length ≤ (2 * q) * (q + 1) := by | |
| have hword : ∀ value ∈ outputWords hq large descriptor, value ≤ q := by | |
| intro value hvalue | |
| rw [outputWords_eq] at hvalue | |
| obtain ⟨a, ha⟩ := List.mem_ofFn.mp hvalue | |
| have hrange := (descriptorTable hq large descriptor a).isLt | |
| omega | |
| simpa only [outputBits, outputWords_length] using | |
| encodeWords_length_le (outputWords hq large descriptor) q hword | |
| theorem outputBits_parse | |
| (descriptor : Descriptor (rows := rows) (repeats := repeats) hq large) : | |
| decodeWords (outputBits hq large descriptor) >>= Encoding.parseProjection q = | |
| some (projection hq large descriptor) := by | |
| rw [outputBits, decodeWords_encodeWords] | |
| exact Encoding.parseProjection_encoded _ | |
| variable {σ : Type} [Fintype σ] [DecidableEq σ] | |
| (select : σ → Descriptor (rows := rows) (repeats := repeats) hq large) | |
| def keys (σ : Type) [Fintype σ] : List σ := (Finset.univ : Finset σ).toList | |
| omit [DecidableEq σ] in | |
| theorem keys_complete (state : σ) : state ∈ keys σ := by | |
| simp [keys] | |
| def pushBound : Nat := | |
| ((keys σ).map (fun state => (outputBits hq large (select state)).length)).sum | |
| def statement {K Λ : Type} (dst : K) (exit : Λ) : | |
| TM2.Stmt (fun _ : K => Bool) Λ σ := | |
| MachineFiniteTable.emit dst (fun state => outputBits hq large (select state)) (keys σ) | |
| (.goto (fun _ => exit)) | |
| theorem statement_trace {K Λ : Type} [DecidableEq K] | |
| (dst : K) (exit : Λ) (state : σ) (tapes : K → List Bool) : | |
| TM2.stepAux (statement hq large select dst exit) state tapes = { | |
| l := some exit | |
| var := state | |
| stk := Function.update tapes dst (outputBits hq large (select state) ++ tapes dst) | |
| } := by | |
| rw [statement, MachineFiniteTable.stepAux_emit dst _ (keys σ) (keys_complete)] | |
| rfl | |
| def emitInTime {K Λ : Type} [DecidableEq K] | |
| (dst : K) (program : Λ → TM2.Stmt (fun _ : K => Bool) Λ σ) | |
| (label exit : Λ) | |
| (atLabel : program label = statement hq large select dst exit) | |
| (state : σ) (tapes : K → List Bool) : | |
| StateTransition.EvalsToInTime (TM2.step program) | |
| ⟨some label, state, tapes⟩ | |
| (some ⟨some exit, state, | |
| Function.update tapes dst (outputBits hq large (select state) ++ tapes dst)⟩) 1 := | |
| MachineFiniteTable.emitInTime (Γ := fun _ : K => Bool) | |
| dst (fun state => outputBits hq large (select state)) | |
| (keys σ) keys_complete program label exit | |
| (by simpa only [statement] using atLabel) state tapes | |
| omit [Fintype σ] [DecidableEq σ] in | |
| theorem output_frame {K : Type} [DecidableEq K] | |
| (dst : K) (state : σ) (tapes : K → List Bool) (k : K) (hne : k ≠ dst) : | |
| (Function.update tapes dst (outputBits hq large (select state) ++ tapes dst)) k = | |
| tapes k := | |
| MachineFiniteTable.output_frame dst (fun state => outputBits hq large (select state)) | |
| state tapes k hne | |
| theorem statement_pushBound {K Λ : Type} (dst : K) (exit : Λ) : | |
| Runtime.statementPushBound (statement hq large select dst exit) ≤ | |
| pushBound hq large select := by | |
| have h := MachineFiniteTable.emit_push_bound (Γ := fun _ : K => Bool) | |
| dst (fun state => outputBits hq large (select state)) (keys σ) | |
| (.goto (fun _ => exit)) | |
| simpa only [statement, pushBound, MachineFiniteTable.tableBound, | |
| Runtime.statementPushBound, Nat.add_zero] using h | |
| def reverseStatement {K Λ : Type} (dst : K) (exit : Λ) : | |
| TM2.Stmt (fun _ : K => Bool) Λ σ := | |
| MachineFiniteTable.emit dst (fun state => (outputBits hq large (select state)).reverse) | |
| (keys σ) (.goto (fun _ => exit)) | |
| theorem reverse_statement_trace {K Λ : Type} [DecidableEq K] | |
| (dst : K) (exit : Λ) (state : σ) (tapes : K → List Bool) : | |
| TM2.stepAux (reverseStatement hq large select dst exit) state tapes = { | |
| l := some exit | |
| var := state | |
| stk := Function.update tapes dst | |
| ((outputBits hq large (select state)).reverse ++ tapes dst) | |
| } := by | |
| rw [reverseStatement, MachineFiniteTable.stepAux_emit dst _ (keys σ) keys_complete] | |
| rfl | |
| def reverseEmitInTime {K Λ : Type} [DecidableEq K] | |
| (dst : K) (program : Λ → TM2.Stmt (fun _ : K => Bool) Λ σ) | |
| (label exit : Λ) | |
| (atLabel : program label = reverseStatement hq large select dst exit) | |
| (state : σ) (tapes : K → List Bool) : | |
| StateTransition.EvalsToInTime (TM2.step program) | |
| ⟨some label, state, tapes⟩ | |
| (some ⟨some exit, state, | |
| Function.update tapes dst | |
| ((outputBits hq large (select state)).reverse ++ tapes dst)⟩) 1 := | |
| MachineFiniteTable.emitInTime (Γ := fun _ : K => Bool) dst | |
| (fun state => (outputBits hq large (select state)).reverse) | |
| (keys σ) keys_complete program label exit | |
| (by simpa only [reverseStatement] using atLabel) state tapes | |
| omit [Fintype σ] [DecidableEq σ] in | |
| theorem reverse_output_frame {K : Type} [DecidableEq K] | |
| (dst : K) (state : σ) (tapes : K → List Bool) (k : K) (hne : k ≠ dst) : | |
| (Function.update tapes dst | |
| ((outputBits hq large (select state)).reverse ++ tapes dst)) k = tapes k := | |
| MachineFiniteTable.output_frame dst | |
| (fun state => (outputBits hq large (select state)).reverse) state tapes k hne | |
| theorem reverse_statement_pushBound {K Λ : Type} (dst : K) (exit : Λ) : | |
| Runtime.statementPushBound (reverseStatement hq large select dst exit) ≤ | |
| pushBound hq large select := by | |
| have h := MachineFiniteTable.emit_push_bound (Γ := fun _ : K => Bool) | |
| dst (fun state => (outputBits hq large (select state)).reverse) (keys σ) | |
| (.goto (fun _ => exit)) | |
| simpa only [reverseStatement, pushBound, MachineFiniteTable.tableBound, | |
| List.length_reverse, Runtime.statementPushBound, Nat.add_zero] using h | |
| end | |
| end PerfectCompleteness.DescriptorProjectionMachine | |
| end OAI | |