PostTuringMachine.lean`

Explanation legend Lean4 Domain Explanation Language (L4DEL)

L4DEL explanations are a separate layer. They do not change the Lean source. Intuition is explanatory prose; numbered items are a reading path; sources link the wording back to evidence.

L4DEL: Lean4 Domain Explanation Language

Annotation provenance. This overlay contains 10 curated AI-written annotations and 7190 mechanical source-derived annotations. They have not been human-reviewed as explanations; treat them as unverified and not mathematical authority. Retained generation history: The 71 original curated annotations were produced in a Claude Sonnet 5.5 session at medium effort; Claude Opus 5.5 is recorded in that session as the advisor model, not the direct author. The bulk source-derived expansion was generated mechanically from pinned source comments by tooling developed in a GPT-5.6 Sol session at ultra reasoning; GPT-5.6 Sol did not originate the quoted source prose. On 2026-10-17 the mechanically generated source-comment annotations of the Formal Conjectures, Carleson, FLT, PFR, CompFiles, Mathematics in Lean and Theorem Proving in Lean overlays were regenerated by the same generator after three corrections made in a Claude Sonnet 5.5 session: titles no longer cut a sentence inside mathematics or end on a dangling word, workflow markers are removed from the quoted prose, and declaration names follow the scanner's namespace resolution. The boilerplate step line was dropped, and comments that were only a row of dashes no longer yield annotations. No new prose was written and no human review is claimed.

Credit: The upstream authors and contributors retain credit for the project and mathematics. lean4-explainer only renders their pinned source and adds the separately disclosed explanation layer.

Project sources: Pinned project documentation, Formal Conjectures paper, arXiv v1

6 declarations, 0 contain sorry, 0 declared as axioms

Imports (2)

Imported by 2 files

theorem Part.get_eq_get line 23 source
standard axioms: Quot.sound, propext what this means
Statement uses: Part, Part.Dom, Membership.mem, Part.get, Eq
Proof uses: Iff.mpr, Exists, Part.dom_iff_mem, Exists.intro, Eq.mp, congrArg, propext, Part.eq_iff_of_dom, Eq.symm, Part.eq_get_iff_mem
2 instances, coercions and auxiliary constants left out.

Tactic trace: 2 steps
Open to load the checked before-and-after goal states.
lemma StateTransition.dom_of_apply_eq_none line 29 source
standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Option, Eq, Option.none, Membership.mem, Part, StateTransition.eval
Proof uses: PFun.fix_stop, Part.some, Sum, Option.elim, Sum.inl, Sum.inr, of_eq_true, Eq.trans, True, congrFun', congrArg, eq_self
3 instances, coercions and auxiliary constants left out.

Tactic trace: 2 steps
Open to load the checked before-and-after goal states.
theorem StateTransition.apply_get_eval line 34 source
standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Option, Part.Dom, StateTransition.eval, Eq, Part.get, Option.none
Proof uses: Membership.mem, Part, Part.get_mem, And.right, StateTransition.Reaches, Eq.mp, And, congrArg, propext, StateTransition.mem_eval
1 instances, coercions and auxiliary constants left out.

Tactic trace: 3 steps
Open to load the checked before-and-after goal states.
theorem StateTransition.eval_get_eval TODO(Paul-Lez): also prove this for PFun.fix/golf using the PFun.fix API line 41 source
Source comment
TODO(Paul-Lez): also prove this for `PFun.fix`/golf using the `PFun.fix` API

standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Option, Part.Dom, StateTransition.eval, Eq, Part, Part.get
Proof uses: Eq.symm, Part.get_eq_get, StateTransition.dom_of_apply_eq_none, of_eq_true, Option.none, Eq.trans, True, congrFun', congrArg, StateTransition.apply_get_eval, eq_self

Tactic trace: 3 steps
Open to load the checked before-and-after goal states.
```
theorem eval_get_eval {σ : Type*} {f : σ → Option σ} {s : σ} (H : (eval f s).Dom) :
    eval f ((eval f s).get H) = eval f s := by
…
```
theorem StateTransition.eval_eq_eval TODO(Paul-Lez): also prove this for PFun.fix/golf using the PFun.fix API line 48 source
Source comment
TODO(Paul-Lez): also prove this for `PFun.fix`/golf using the `PFun.fix` API

standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Option, Eq, Option.some, Part, StateTransition.eval
Proof uses: StateTransition.reaches_eval, Relation.ReflTransGen.single, Membership.mem, Eq.mpr, id, congrArg, Eq.refl
1 instances, coercions and auxiliary constants left out.

Tactic trace: 4 steps
Open to load the checked before-and-after goal states.
```
theorem eval_eq_eval {σ : Type*} {f : σ → Option σ} {a a' : σ} (H : f a = some a'):
    eval f a = eval f a' := by
…
```
theorem StateTransition.eval_dom_iff TODO(lezeau): this should be generalized to PFun.fix line 56 source
Source comment
TODO(lezeau): this should be generalized to `PFun.fix`

standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Option, Iff, Exists, Nat, Eq, Nat.iterate, Option.bind, HAdd.hAdd, OfNat.ofNat, Option.some, Option.none, Part.Dom, StateTransition.eval
Proof uses: Iff.intro, Nat.recAux, Eq.mpr, Membership.mem, Part, id, Exists.intro, StateTransition.dom_of_apply_eq_none, Eq.mp, congrArg, Option.bind_some, Function.iterate_one, zero_add, Or, Option.eq_none_or_eq_some, congrFun', Option.bind_congr', Eq.refl, StateTransition.eval_eq_eval, StateTransition.evalInduction and 9 more
18 instances, coercions and auxiliary constants left out.

Tactic trace: 27 steps
Open to load the checked before-and-after goal states.
```
theorem eval_dom_iff {σ : Type*} {f : σ → Option σ} {s : σ} :
    (∃ n, ((Option.bind · f)^[n+1] s) = none) ↔ (eval f s).Dom := by
…
```