openAI-math-dataset / lean /OAI /MathematicalPhysics /NavierStokes /ForcedComputation /Programs /FiniteMachineHaltingDischarge.lean
| import OAI.MathematicalPhysics.NavierStokes.ForcedComputation.Programs.FixedInterpreter | |
| import OAI.MathematicalPhysics.NavierStokes.ForcedComputation.Programs.MachineDecisionInput | |
| namespace OAI | |
| /-! Discharge of the finite-table undecidability input, using the fixed | |
| universal interpreter and Mathlib's proved halting theorem. -/ | |
| namespace ForcedComputation | |
| theorem finiteMachineHaltingUndecidable : FiniteMachineHaltingUndecidable := by | |
| rintro ⟨d, hd, hcorrect⟩ | |
| obtain ⟨M, input, hi, hM⟩ := FiniteMachine.exists_fixed_universal_machine | |
| apply ComputablePred.halting_problem 0 | |
| apply ComputablePred.computable_iff.mpr | |
| refine ⟨fun c : Nat.Partrec.Code => d (M, input (Encodable.encode c)), | |
| hd.comp ((Computable.const M).pair (hi.comp Computable.encode)), ?_⟩ | |
| funext c | |
| apply propext | |
| have h := (hcorrect (M, input (Encodable.encode c)) (hM _).1).trans (hM _).2 | |
| simpa only [Denumerable.ofNat_encode] using h.symm | |
| end ForcedComputation | |
| end OAI | |