我正在尝试使用 Lean 4 将停止问题简化为双曲几何，以证明双曲几何的不可判定性，在图灵机的对角化方面存在问题，还缺少双曲几何的一些公理。
不幸的是，我收到了意外的令牌错误，我不知道这意味着什么以及如何解决它。我试过在几个地方去掉逗号，但所有这些地方都在那一行，并没有解决问题。
有人可以帮我解决精益的对角化问题吗？
我正在使用以下代码：
import Mathlib.Tactic
import Mathlib.Data.Real.Basic
import Mathlib.Data.List.Basic
import Mathlib.Data.Finset.Basic
-- Suppress linter warnings for unused variables
set_option linter.unusedVariables false
open Classical -- Use classical logic for decidability
-- Define a Turing Machine
structure TuringMachine (states symbols : Type) where
transition : states → symbols → states × symbols × Bool -- Transition function
startState : states -- Start state
haltState : states -- Halt state
-- Define finite states and symbols
inductive TMState
| q0 | q1 | halt
deriving DecidableEq, Fintype
inductive TMSymbol
| zero | one | blank
deriving DecidableEq, Fintype
-- Example Turing Machine
def tm_example : TuringMachine TMState TMSymbol := {
transition := fun state symbol =>
match state, symbol with
| TMState.q0, TMSymbol.zero => (TMState.q1, TMSymbol.one, false)
| TMState.q1, TMSymbol.one => (TMState.halt, TMSymbol.one, true)
| _, _ => (TMState.halt, TMSymbol.blank, true),
startState := TMState.q0,
haltState := TMState.halt
}
-- Execution of a Turing Machine for a number of steps
noncomputable def TuringMachine.execute {states symbols : Type} [Fintype states] [Fintype symbols]
[DecidableEq states] [DecidableEq symbols]
(tm : TuringMachine states symbols) (steps : ℕ) (input : List symbols) :
Option (states × List symbols × Bool) :=
let rec step (currentState : states) (tape : List symbols) (remainingSteps : ℕ) :
Option (states × List symbols × Bool) :=
match remainingSteps with
| 0 => none
| n + 1 =>
match tape with
| [] => none
| symbol :: rest =>
let (newState, newSymbol, halt) := tm.transition currentState symbol
let newTape := newSymbol :: rest
if halt then
some (newState, newTape, true)
else
step newState newTape n
step tm.startState input steps
-- Input for the Turing Machine
def input_example : List TMSymbol := [TMSymbol.zero, TMSymbol.one]
-- Explicit Definitions for States and Symbols
def orderedStates : List TMState := [TMState.q0, TMState.q1, TMState.halt]
def orderedSymbols : List TMSymbol := [TMSymbol.zero, TMSymbol.one, TMSymbol.blank]
-- Poincaré Disk Model
structure PoincareDisk where
x : ℝ
y : ℝ
hyp_condition : x^2 + y^2 < 1 -- Point lies strictly inside the unit disk
namespace PoincareDisk
-- Geodesics
structure Geodesic where
center : ℝ × ℝ -- Center of the defining circle
radius : ℝ -- Radius of the circle
is_diameter : Bool -- True for diameters, False for orthogonal arcs
-- Membership of a point on a geodesic
def onGeodesic (p : PoincareDisk) (g : Geodesic) : Prop :=
if g.is_diameter then
p.x = g.center.1 -- Diameter case
else
(p.x - g.center.1)^2 + (p.y - g.center.2)^2 = g.radius^2 -- Orthogonal arc case
end PoincareDisk
-- Encode TuringMachine states and symbols as hyperbolic geometry configurations
noncomputable def TuringMachine.encodeAsGeometry (tm : TuringMachine TMState TMSymbol) (input : List TMSymbol) :
List PoincareDisk × List PoincareDisk.Geodesic :=
-- Map states and symbols to unique ℝ values
let stateToReal (state : TMState) : ℝ :=
orderedStates.indexOf state + 1
let symbolToReal (symbol : TMSymbol) : ℝ :=
orderedSymbols.indexOf symbol + 1
-- Encode state as a PoincareDisk point
let encodeState (state : TMState) : PoincareDisk :=
{ x := 0.9 * Real.sin (stateToReal state * 0.1),
y := 0.9 * Real.cos (stateToReal state * 0.1),
hyp_condition := by
have h : Real.sin (stateToReal state * 0.1)^2 + Real.cos (stateToReal state * 0.1)^2 = 1 :=
Real.sin_sq_add_cos_sq (stateToReal state * 0.1)
linarith [h] }
-- Encode symbol as a geodesic
let encodeSymbol (symbol : TMSymbol) : PoincareDisk.Geodesic :=
{ center := (0, 0),
radius := 1 / (symbolToReal symbol + 2),
is_diameter := false }
let states := orderedStates.map encodeState
let geodesics := input.map encodeSymbol
(states, geodesics)
-- Undecidability Proof for Turing Machine Execution
theorem TuringMachineExecutionUndecidable :
¬(∀ (tm : TuringMachine TMState TMSymbol) (input : List TMSymbol),
∃ steps, tm.execute steps input = some (tm.haltState, [], true)) := by
{
-- Assume decidability of halting for contradiction
intro h_decidable,
-- Define a diagonal Turing Machine that alternates indefinitely
let diagonal_tm : TuringMachine TMState TMSymbol := {
transition := fun state symbol =>
match state, symbol with
| TMState.q0, TMSymbol.zero => (TMState.q1, TMSymbol.one, false) -- Move to q1
| TMState.q1, TMSymbol.one => (TMState.q0, TMSymbol.zero, false) -- Back to q0
| _, _ => (TMState.halt, TMSymbol.blank, true), -- Default to halt
startState := TMState.q0,
haltState := TMState.halt
}
-- Input for the diagonal machine
let input := [TMSymbol.zero, TMSymbol.one]
-- Assume the diagonal machine halts (contradiction setup)
have h_halts : ∃ steps, diagonal_tm.execute steps input = some (diagonal_tm.haltState, [], true),
{ exact h_decidable diagonal_tm input }, -- From assumption of decidability
-- Extract steps and result from halting assumption
cases h_halts with steps h_steps
-- Prove that the diagonal machine cannot halt
have h_contradiction : diagonal_tm.execute steps input ≠ some (diagonal_tm.haltState, [], true),
{
induction steps with
| zero =>
-- Base case: 0 steps mean no execution, so the machine cannot halt
simp [TuringMachine.execute] at h_steps,
contradiction
| succ n ih =>
-- Recursive case: analyze the machine's step-by-step execution
simp [TuringMachine.execute] at h_steps,
cases input with
| [] =>
-- If the input is empty, the machine cannot proceed, so it cannot halt
simp [TuringMachine.execute] at h_steps,
contradiction
| symbol :: rest =>
-- Simulate the first step of the machine
cases diagonal_tm.transition diagonal_tm.startState symbol with
| (new_state, new_symbol, halt) =>
by_cases halt
-- Case 1: Halt = true leads to contradiction, as diagonal_tm loops indefinitely
{ simp [*] at h_steps, contradiction }
-- Case 2: Halt = false, continue recursion using IH
{ apply ih, assumption }
},
-- Conclude contradiction
exact h_contradiction h_steps,
}
-- Test the Geometry Encoding
noncomputable def encoded_geometry := TuringMachine.encodeAsGeometry tm_example input_example
我遇到的错误是这些
错误：
mathlib-stable.lean:120:0
unsolved goals
h_decidable : ∀ (tm : TuringMachine TMState TMSymbol) (input : List TMSymbol),
∃ steps, tm.execute steps input = some (tm.haltState, [], true)
⊢ False
mathlib-stable.lean:122:19
unexpected token ','; expected '}'
unexpected token ','unexpected token ','unknown tacticcasescasescases'