| import Mathlib |
| import OAI.Computability.MinUncut.Machines.Machine |
|
|
| namespace OAI |
|
|
| section |
| namespace MinUncutGames.Foundations.Hastad.SourceFieldArray |
|
|
| open Turing |
| open MinUncutGames.Foundations.Complexity SourceMachine |
|
|
| variable {K Λ σ : Type} [DecidableEq K] |
|
|
| def finish (source : K) (exit : Option Λ) : |
| TM2.Stmt (Alphabet (K := K)) Λ (σ × Option Bool) := |
| .load (fun state => (state.1, none)) (Reduction.MachineTransfer.exitAt source exit) |
|
|
| theorem finishStep (source : K) (label : Λ) (exit : Option Λ) |
| (program : Λ → TM2.Stmt (Alphabet (K := K)) Λ (σ × Option Bool)) |
| (atFinish : program label = finish source exit) |
| (base : K → List Bool) (ambient : σ) (register : Option Bool) : |
| TM2.step program ⟨some label, (ambient, register), base⟩ = |
| some ⟨exit, (ambient, none), base⟩ := by |
| change some (TM2.stepAux (program label) (ambient, register) base) = _ |
| rw [atFinish] |
| cases exit <;> simp [finish, Reduction.MachineTransfer.exitAt, TM2.stepAux] |
|
|
| def sequenceTapes (source : K) (destination : Nat → K) (base : K → List Bool) |
| (offset : Nat) (values : List Nat) (suffix : List Bool) : K → List Bool := |
| match values with |
| | [] => base |
| | n :: rest => sequenceTapes source destination |
| (afterField source (destination offset) base n (encodeWords rest ++ suffix)) |
| (offset + 1) rest suffix |
|
|
| theorem sequenceTrace (source : K) (destination : Nat → K) |
| (start loopLabel : Nat → Λ) (exit : Option Λ) |
| (program : Λ → TM2.Stmt (Alphabet (K := K)) Λ (σ × Option Bool)) |
| (base : K → List Bool) (offset : Nat) (values : List Nat) (suffix : List Bool) |
| (hsep : ∀ r, r < values.length → source ≠ destination (offset + r)) |
| (atStart : ∀ r, r < values.length → |
| program (start (offset + r)) = fieldStart (destination (offset + r)) (loopLabel (offset + r))) |
| (atLoop : ∀ r, r < values.length → program (loopLabel (offset + r)) = |
| fieldLoop source (destination (offset + r)) (loopLabel (offset + r)) |
| (some (start (offset + r + 1)))) |
| (atDone : program (start (offset + values.length)) = finish source exit) |
| (hinput : base source = encodeWords values ++ suffix) |
| (ambient : σ) (register : Option Bool) : |
| (MachineComposition.advance (TM2.step program))^[values.sum + 2 * values.length + 1] |
| (some ⟨some (start offset), (ambient, register), base⟩) = |
| some ⟨exit, (ambient, none), sequenceTapes source destination base offset values suffix⟩ := by |
| induction values generalizing offset base register with |
| | nil => |
| simpa only [List.sum_nil, List.length_nil, Nat.mul_zero, Nat.zero_add, |
| Nat.add_zero, Function.iterate_one, MachineComposition.advance_some, sequenceTapes] |
| using finishStep source (start offset) exit program atDone base ambient register |
| | cons n rest ih => |
| have hr0 : 0 < (n :: rest).length := by simp |
| have hsep0 : source ≠ destination offset := by simpa using hsep 0 hr0 |
| have hs0 : program (start offset) = fieldStart (destination offset) (loopLabel offset) := by |
| simpa using atStart 0 hr0 |
| have hl0 : program (loopLabel offset) = fieldLoop source (destination offset) |
| (loopLabel offset) (some (start (offset + 1))) := by simpa using atLoop 0 hr0 |
| have hin : base source = encodeWord n ++ (encodeWords rest ++ suffix) := by |
| simpa only [encodeWords, List.append_assoc] using hinput |
| let nextBase := afterField source (destination offset) base n (encodeWords rest ++ suffix) |
| have hfirst := (fieldInTime source (destination offset) hsep0 (start offset) |
| (loopLabel offset) (some (start (offset + 1))) program hs0 hl0 base n |
| (encodeWords rest ++ suffix) hin ambient register).evals_in_steps |
| change (MachineComposition.advance (TM2.step program))^[n + 2] |
| (some ⟨some (start offset), (ambient, register), base⟩) = |
| some ⟨some (start (offset + 1)), (ambient, none), nextBase⟩ at hfirst |
| have hsep' : ∀ r, r < rest.length → source ≠ destination (offset + 1 + r) := by |
| intro r hr |
| simpa [Nat.add_assoc, Nat.add_comm, Nat.add_left_comm] using hsep (r + 1) (by simp; omega) |
| have hs' : ∀ r, r < rest.length → program (start (offset + 1 + r)) = |
| fieldStart (destination (offset + 1 + r)) (loopLabel (offset + 1 + r)) := by |
| intro r hr |
| simpa [Nat.add_assoc, Nat.add_comm, Nat.add_left_comm] using atStart (r + 1) (by simp; omega) |
| have hl' : ∀ r, r < rest.length → program (loopLabel (offset + 1 + r)) = |
| fieldLoop source (destination (offset + 1 + r)) (loopLabel (offset + 1 + r)) |
| (some (start (offset + 1 + r + 1))) := by |
| intro r hr |
| simpa [Nat.add_assoc, Nat.add_comm, Nat.add_left_comm] using atLoop (r + 1) (by simp; omega) |
| have hd' : program (start (offset + 1 + rest.length)) = finish source exit := by |
| simpa [Nat.add_assoc, Nat.add_comm, Nat.add_left_comm] using atDone |
| have hin' : nextBase source = encodeWords rest ++ suffix := by |
| simp [nextBase, afterField, hsep0] |
| rw [show (n :: rest).sum + 2 * (n :: rest).length + 1 = |
| (rest.sum + 2 * rest.length + 1) + (n + 2) by simp; omega] |
| rw [Function.iterate_add_apply, hfirst, |
| ih nextBase (offset + 1) hsep' hs' hl' hd' hin' none] |
| rfl |
|
|
| def sequenceInTime (source : K) (destination : Nat → K) |
| (start loopLabel : Nat → Λ) (exit : Option Λ) |
| (program : Λ → TM2.Stmt (Alphabet (K := K)) Λ (σ × Option Bool)) |
| (base : K → List Bool) (offset : Nat) (values : List Nat) (suffix : List Bool) |
| (hsep : ∀ r, r < values.length → source ≠ destination (offset + r)) |
| (atStart : ∀ r, r < values.length → |
| program (start (offset + r)) = fieldStart (destination (offset + r)) (loopLabel (offset + r))) |
| (atLoop : ∀ r, r < values.length → program (loopLabel (offset + r)) = |
| fieldLoop source (destination (offset + r)) (loopLabel (offset + r)) |
| (some (start (offset + r + 1)))) |
| (atDone : program (start (offset + values.length)) = finish source exit) |
| (hinput : base source = encodeWords values ++ suffix) |
| (ambient : σ) (register : Option Bool) : |
| StateTransition.EvalsToInTime (TM2.step program) |
| ⟨some (start offset), (ambient, register), base⟩ |
| (some ⟨exit, (ambient, none), sequenceTapes source destination base offset values suffix⟩) |
| (values.sum + 2 * values.length + 1) where |
| steps := values.sum + 2 * values.length + 1 |
| evals_in_steps := sequenceTrace source destination start loopLabel exit program base offset |
| values suffix hsep atStart atLoop atDone hinput ambient register |
| steps_le_m := Nat.le_refl _ |
|
|
| theorem sequence_time_eq_length (values : List Nat) : |
| values.sum + 2 * values.length + 1 = (encodeWords values).length + values.length + 1 := by |
| rw [encodeWords_length] |
| omega |
|
|
| theorem sequenceTapes_other (source : K) (destination : Nat → K) |
| (p : K) (hpS : p ≠ source) (base : K → List Bool) (offset : Nat) |
| (values : List Nat) (suffix : List Bool) |
| (hpD : ∀ r, r < values.length → p ≠ destination (offset + r)) : |
| sequenceTapes source destination base offset values suffix p = base p := by |
| induction values generalizing base offset with |
| | nil => rfl |
| | cons n rest ih => |
| have hp0 : p ≠ destination offset := by simpa using hpD 0 (by simp) |
| have htail : ∀ r, r < rest.length → p ≠ destination (offset + 1 + r) := by |
| intro r hr |
| simpa [Nat.add_assoc, Nat.add_comm, Nat.add_left_comm] using hpD (r + 1) (by simp; omega) |
| rw [sequenceTapes, ih _ (offset + 1) htail] |
| exact fieldTapes_other source (destination offset) p hpS hp0 _ _ _ |
|
|
| theorem sequenceTapes_source (source : K) (destination : Nat → K) |
| (base : K → List Bool) (offset : Nat) (values : List Nat) (suffix : List Bool) |
| (hsep : ∀ r, r < values.length → source ≠ destination (offset + r)) |
| (hinput : base source = encodeWords values ++ suffix) : |
| sequenceTapes source destination base offset values suffix source = suffix := by |
| induction values generalizing base offset with |
| | nil => simpa [sequenceTapes, encodeWords] using hinput |
| | cons n rest ih => |
| have hsep0 : source ≠ destination offset := by simpa using hsep 0 (by simp) |
| have htail : ∀ r, r < rest.length → source ≠ destination (offset + 1 + r) := by |
| intro r hr |
| simpa [Nat.add_assoc, Nat.add_comm, Nat.add_left_comm] using hsep (r + 1) (by simp; omega) |
| apply ih _ (offset + 1) htail |
| simp [afterField, hsep0] |
|
|
| theorem sequenceTapes_selected (source : K) (destination : Nat → K) |
| (base : K → List Bool) (offset : Nat) (values : List Nat) (suffix : List Bool) |
| (j : Nat) (hj : j < values.length) |
| (hsep : ∀ r, r < values.length → source ≠ destination (offset + r)) |
| (hdist : ∀ r s, r < values.length → s < values.length → |
| destination (offset + r) = destination (offset + s) → r = s) : |
| sequenceTapes source destination base offset values suffix (destination (offset + j)) = |
| encodeWord (values[j]'hj) ++ base (destination (offset + j)) := by |
| induction values generalizing base offset j with |
| | nil => simp at hj |
| | cons n rest ih => |
| have hsep0 : source ≠ destination offset := by simpa using hsep 0 (by simp) |
| let nextBase := afterField source (destination offset) base n (encodeWords rest ++ suffix) |
| cases j with |
| | zero => |
| have hframe : ∀ r, r < rest.length → destination offset ≠ destination (offset + 1 + r) := by |
| intro r hr heq |
| have hh := hdist 0 (r + 1) (by simp) (by simp; omega) |
| (by simpa [Nat.add_assoc, Nat.add_comm, Nat.add_left_comm] using heq) |
| omega |
| simp only [sequenceTapes, Nat.add_zero, List.getElem_cons_zero] |
| rw [sequenceTapes_other source destination (destination offset) hsep0.symm _ |
| (offset + 1) rest suffix hframe] |
| simp [afterField] |
| | succ j => |
| have hj' : j < rest.length := by simpa using hj |
| have hsep' : ∀ r, r < rest.length → source ≠ destination (offset + 1 + r) := by |
| intro r hr |
| simpa [Nat.add_assoc, Nat.add_comm, Nat.add_left_comm] using hsep (r + 1) (by simp; omega) |
| have hdist' : ∀ r s, r < rest.length → s < rest.length → |
| destination (offset + 1 + r) = destination (offset + 1 + s) → r = s := by |
| intro r s hr hs heq |
| have hh := hdist (r + 1) (s + 1) (by simp; omega) (by simp; omega) |
| (by simpa [Nat.add_assoc, Nat.add_comm, Nat.add_left_comm] using heq) |
| omega |
| have hsrc : destination (offset + 1 + j) ≠ source := (hsep' j hj').symm |
| have hdst : destination (offset + 1 + j) ≠ destination offset := by |
| intro heq |
| have hh := hdist (j + 1) 0 hj (by simp) |
| (by simpa [Nat.add_assoc, Nat.add_comm, Nat.add_left_comm] using heq) |
| omega |
| have hkeep : nextBase (destination (offset + 1 + j)) = base (destination (offset + 1 + j)) := |
| fieldTapes_other source (destination offset) _ hsrc hdst _ _ _ |
| simp only [sequenceTapes, List.getElem_cons_succ] |
| rw [show offset + (j + 1) = offset + 1 + j by omega] |
| rw [ih _ (offset + 1) j hj' hsep' hdist'] |
| exact congrArg (encodeWord rest[j] ++ ·) hkeep |
| |
| def boundedIndex (q j : Nat) : Fin (q + 1) := ⟨min j q, by omega⟩ |
| |
| @[simp] theorem boundedIndex_val {q j : Nat} (hj : j ≤ q) : |
| (boundedIndex q j).val = j := Nat.min_eq_left hj |
| |
| def destination (q j : Nat) : Fin (q + 1) := boundedIndex q (j + 1) |
| def startLabel (q j : Nat) : Fin (q + 1) × Bool := (boundedIndex q j, false) |
| def loopLabel (q j : Nat) : Fin (q + 1) × Bool := (boundedIndex q j, true) |
| |
| theorem destination_ne_source (q r : Nat) (hr : r < q) : |
| (0 : Fin (q + 1)) ≠ destination q r := by |
| intro heq |
| have hh := congrArg Fin.val heq |
| simp only [destination, boundedIndex_val (Nat.succ_le_iff.mpr hr)] at hh |
| change 0 = r + 1 at hh |
| omega |
| |
| theorem destination_injective_below (q r s : Nat) (hr : r < q) (hs : s < q) |
| (heq : destination q r = destination q s) : r = s := by |
| have hh := congrArg Fin.val heq |
| simp only [destination, boundedIndex_val (Nat.succ_le_iff.mpr hr), |
| boundedIndex_val (Nat.succ_le_iff.mpr hs)] at hh |
| omega |
| |
| @[simp] theorem destination_eq_succ {q : Nat} (j : Fin q) : destination q j.val = j.succ := by |
| apply Fin.ext |
| exact boundedIndex_val (Nat.succ_le_iff.mpr j.isLt) |
| |
| theorem output_source (q : Nat) (values : List Nat) (hlen : values.length = q) |
| (base : Fin (q + 1) → List Bool) (suffix : List Bool) |
| (hinput : base 0 = encodeWords values ++ suffix) : |
| sequenceTapes (0 : Fin (q + 1)) (destination q) base 0 values suffix 0 = suffix := by |
| apply sequenceTapes_source _ _ _ _ _ _ _ hinput |
| intro r hr |
| simpa using destination_ne_source q r (by omega) |
| |
| theorem output_field (q : Nat) (values : List Nat) (hlen : values.length = q) |
| (base : Fin (q + 1) → List Bool) (suffix : List Bool) (j : Fin q) : |
| sequenceTapes (0 : Fin (q + 1)) (destination q) base 0 values suffix j.succ = |
| encodeWord (values[j.val]'(by omega)) ++ base j.succ := by |
| have h := sequenceTapes_selected (0 : Fin (q + 1)) (destination q) base 0 values suffix |
| j.val (by omega) |
| (by intro r hr; simpa using destination_ne_source q r (by omega)) |
| (by |
| intro r s hr hs heq |
| exact destination_injective_below q r s (by omega) (by omega) |
| (by simpa only [Nat.zero_add] using heq)) |
| simpa only [Nat.zero_add, destination_eq_succ] using h |
|
|
| def program (q : Nat) (label : Fin (q + 1) × Bool) : |
| TM2.Stmt (fun _ : Fin (q + 1) => Bool) (Fin (q + 1) × Bool) (Unit × Option Bool) := |
| if label.1.val < q then |
| if label.2 then |
| fieldLoop 0 (destination q label.1.val) (label.1, true) |
| (some (startLabel q (label.1.val + 1))) |
| else fieldStart (destination q label.1.val) (label.1, true) |
| else finish 0 none |
|
|
| def machine (q : Nat) : FinTM2 where |
| K := Fin (q + 1) |
| k₀ := 0 |
| k₁ := 0 |
| Γ _ := Bool |
| Λ := Fin (q + 1) × Bool |
| main := startLabel q 0 |
| σ := Unit × Option Bool |
| initialState := ((), none) |
| m := program q |
|
|
| theorem atStart (q r : Nat) (hr : r < q) : |
| program q (startLabel q r) = fieldStart (destination q r) (loopLabel q r) := by |
| simp [program, startLabel, loopLabel, boundedIndex_val hr.le, hr] |
|
|
| theorem atLoop (q r : Nat) (hr : r < q) : |
| program q (loopLabel q r) = fieldLoop (0 : Fin (q + 1)) (destination q r) |
| (loopLabel q r) (some (startLabel q (r + 1))) := by |
| simp [program, startLabel, loopLabel, boundedIndex_val hr.le, hr] |
|
|
| theorem atDone (q : Nat) : program q (startLabel q q) = finish (0 : Fin (q + 1)) none := by |
| simp [program, startLabel] |
|
|
| def machineInTime (q : Nat) (values : List Nat) (hlen : values.length = q) |
| (base : Fin (q + 1) → List Bool) (suffix : List Bool) |
| (hinput : base 0 = encodeWords values ++ suffix) (register : Option Bool) : |
| StateTransition.EvalsToInTime (machine q).step |
| ⟨some (startLabel q 0), ((), register), base⟩ |
| (some ⟨none, ((), none), sequenceTapes (0 : Fin (q + 1)) (destination q) base 0 values suffix⟩) |
| (values.sum + 2 * q + 1) := by |
| have hsep : ∀ r, r < values.length → (0 : Fin (q + 1)) ≠ destination q (0 + r) := by |
| intro r hr heq |
| have hr' : r + 1 ≤ q := by omega |
| have hh := congrArg Fin.val heq |
| simp only [Nat.zero_add, destination, boundedIndex_val hr'] at hh |
| change 0 = r + 1 at hh |
| omega |
| have hs : ∀ r, r < values.length → program q (startLabel q (0 + r)) = |
| fieldStart (destination q (0 + r)) (loopLabel q (0 + r)) := by |
| intro r hr |
| simpa only [Nat.zero_add] using atStart q r (by omega) |
| have hl : ∀ r, r < values.length → program q (loopLabel q (0 + r)) = |
| fieldLoop (0 : Fin (q + 1)) (destination q (0 + r)) (loopLabel q (0 + r)) |
| (some (startLabel q (0 + r + 1))) := by |
| intro r hr |
| simpa only [Nat.zero_add] using atLoop q r (by omega) |
| have hd : program q (startLabel q (0 + values.length)) = finish (0 : Fin (q + 1)) none := by |
| simpa [hlen] using atDone q |
| simpa only [machine, FinTM2.Cfg, FinTM2.step, hlen] using! sequenceInTime (0 : Fin (q + 1)) (destination q) |
| (startLabel q) (loopLabel q) none (program q) base 0 values suffix hsep hs hl hd hinput () register |
|
|
| def machineInTime_inputBound (q : Nat) (values : List Nat) (hlen : values.length = q) |
| (base : Fin (q + 1) → List Bool) (suffix : List Bool) |
| (hinput : base 0 = encodeWords values ++ suffix) (register : Option Bool) : |
| StateTransition.EvalsToInTime (machine q).step |
| ⟨some (startLabel q 0), ((), register), base⟩ |
| (some ⟨none, ((), none), sequenceTapes (0 : Fin (q + 1)) (destination q) base 0 values suffix⟩) |
| ((base 0).length + q + 1) := by |
| let run := machineInTime q values hlen base suffix hinput register |
| refine { toEvalsTo := run.toEvalsTo, steps_le_m := ?_ } |
| have hb := run.steps_le_m |
| have hi : (base 0).length = values.sum + q + suffix.length := by |
| rw [hinput, List.length_append, encodeWords_length, hlen] |
| omega |
|
|
| def sixFieldsInTime (base : Fin 7 → List Bool) (a b c d e f : Nat) (suffix : List Bool) |
| (hinput : base 0 = encodeWords [a, b, c, d, e, f] ++ suffix) (register : Option Bool) : |
| StateTransition.EvalsToInTime (machine 6).step |
| ⟨some (startLabel 6 0), ((), register), base⟩ |
| (some ⟨none, ((), none), |
| sequenceTapes (0 : Fin 7) (destination 6) base 0 [a, b, c, d, e, f] suffix⟩) |
| (a + b + c + d + e + f + 13) := by |
| have h := machineInTime 6 [a, b, c, d, e, f] rfl base suffix hinput register |
| convert h using 1 |
| simp |
| omega |
|
|
| end MinUncutGames.Foundations.Hastad.SourceFieldArray |
|
|
| end |
|
|
| end OAI |
|
|
