From d37984bf8beaec0d0c1ac05c247e1b174a050f5f Mon Sep 17 00:00:00 2001 From: Robb Walters Date: Fri, 8 May 2026 12:48:39 +0300 Subject: [PATCH] =?UTF-8?q?research(cantor-diagonalization-oq-04-oq-01):?= =?UTF-8?q?=20S2=20=E2=80=94=20trivial=20setoid=20+=20refinement-descent?= =?UTF-8?q?=20(build=20pending)?= MIME-Version: 1.0 Content-Type: text/plain; charset=UTF-8 Content-Transfer-Encoding: 8bit Round out the spectrum of coding-feasibility on `Setoid Y`: Part VIII (success side): - `trivialSetoid Y` — collapses Y to a single equivalence class (≈ ≡ True). - `trivial_setoid_codes_endomorphisms` — every inhabited Y vacuously codes its endomorphisms under the trivial setoid (encode = const default, decode = const id). - `bool_trivial_setoid_codes_endomorphisms` — explicit Bool corollary, contrasting with `cannot_code_endomorphisms_bool_setoid` from S1. Demonstrates that coding-feasibility is genuinely setoid-dependent. Part IX (refinement structure): - `IsRefinement s t : Prop` — s.r implies t.r (s finer, t coarser). - `coding_descends_to_coarser` — coding under finer setoid descends to any coarser setoid via the same encode/decode (retract weakens trivially). - `refines_trivial` — every setoid refines trivial; trivial is the canonical top. File: 166 → 221 lines, 8 → 12 theorems, 3 → 5 definitions, 0 sorries unchanged, 0 axioms unchanged. Status remains verified/original. Build pending: worktree's `proofs/.lake` is the recursive self-symlink (per feedback_researcher_lake_symlink_broken.md), so a fresh Mathlib clone (~45 min) is required. Following PRs #16936 and #17008, this opens as draft. Co-Authored-By: Claude Opus 4.7 --- .../Proofs/CantorDiagonalizationOQ04OQ01.lean | 55 ++++++++++++++++++ .../knowledge.md | 58 +++++++++++++++++++ .../meta.json | 42 ++++++++++---- .../cantor-diagonalization-oq-04-oq-01.json | 48 +++++++++------ 4 files changed, 174 insertions(+), 29 deletions(-) diff --git a/proofs/Proofs/CantorDiagonalizationOQ04OQ01.lean b/proofs/Proofs/CantorDiagonalizationOQ04OQ01.lean index 4016c49f2b1..200c58f92ee 100644 --- a/proofs/Proofs/CantorDiagonalizationOQ04OQ01.lean +++ b/proofs/Proofs/CantorDiagonalizationOQ04OQ01.lean @@ -34,6 +34,9 @@ the original theorem exactly. 5. `cannot_code_endomorphisms_bool_setoid` — Bool cannot code its endomorphisms 6. `cantor_setoid_no_surjection` — Cantor diagonal in setoid setting 7. `cannot_code_endomorphisms_nat_parity` — ℕ cannot code endomorphisms up to parity +8. `trivial_setoid_codes_endomorphisms` — every inhabited Y codes under the trivial (one-class) setoid +9. `bool_trivial_setoid_codes_endomorphisms` — Bool codes under trivial setoid (contrast with #5) +10. `coding_descends_to_coarser` — coding under a finer setoid descends to any coarser setoid ## Proof Technique Diagonal construction: g(y) = f(decode(y)(y)). @@ -163,4 +166,56 @@ theorem cannot_code_endomorphisms_nat_parity : CodesEndomorphismsSetoid ℕ paritySetoidN → False := no_coding_setoid_if_fixpoint_free paritySetoidN (fun n : ℕ => n + 1) succ_no_parity_fixpoint +-- ============================================================ +-- Part VIII: Trivial Setoid — Coding Always Possible +-- ============================================================ + +/-- The trivial setoid: every pair is equivalent. Collapses Y to a single class. -/ +def trivialSetoid (Y : Type*) : Setoid Y where + r := fun _ _ => True + iseqv := ⟨fun _ => trivial, fun _ => trivial, fun _ _ => trivial⟩ + +/-- **Setoid-choice sensitivity**: under the trivial (one-class) setoid, every + inhabited type codes its endomorphisms vacuously. The retraction holds because + `s.r a b = True` for all `a, b`. Together with `cannot_code_endomorphisms_bool_setoid` + this shows that coding-feasibility depends genuinely on the setoid structure, + not just on the underlying type. -/ +theorem trivial_setoid_codes_endomorphisms (Y : Type*) [Inhabited Y] : + CodesEndomorphismsSetoid Y (trivialSetoid Y) where + encode := fun _ => default + decode := fun _ => id + retract := fun _ _ => trivial + +/-- Bool DOES code its endomorphisms under the trivial setoid, even though it + fails under the discrete setoid (cf. `cannot_code_endomorphisms_bool_setoid`). + Witnesses that the impossibility result for Bool is setoid-specific. -/ +theorem bool_trivial_setoid_codes_endomorphisms : + CodesEndomorphismsSetoid Bool (trivialSetoid Bool) := + trivial_setoid_codes_endomorphisms Bool + +-- ============================================================ +-- Part IX: Refinement Lemma — Coding Descends to Coarser Setoids +-- ============================================================ + +/-- A setoid `s` **refines** `t` when `s`-equivalence implies `t`-equivalence + (equivalently, `t` is coarser than `s`). The discrete setoid refines every + setoid; every setoid refines the trivial setoid. -/ +def IsRefinement {Y : Type*} (s t : Setoid Y) : Prop := + ∀ a b : Y, s.r a b → t.r a b + +/-- **Refinement-descent**: if `Y` codes its endomorphisms under a finer setoid `s`, + the same encode/decode also code under any coarser setoid `t`. The retraction + transports along refinement because `t`-equivalence is weaker than `s`-equivalence. -/ +theorem coding_descends_to_coarser {Y : Type*} {s t : Setoid Y} + (hst : IsRefinement s t) (c : CodesEndomorphismsSetoid Y s) : + CodesEndomorphismsSetoid Y t where + encode := c.encode + decode := c.decode + retract := fun g y => hst _ _ (c.retract g y) + +/-- Every setoid is a refinement of the trivial setoid (vacuously: the trivial relation + is `True`). This is the canonical "top" of the refinement order on `Setoid Y`. -/ +theorem refines_trivial {Y : Type*} (s : Setoid Y) : IsRefinement s (trivialSetoid Y) := + fun _ _ _ => trivial + end CantorDiagonalizationOQ04OQ01 diff --git a/research/problems/cantor-diagonalization-oq-04-oq-01/knowledge.md b/research/problems/cantor-diagonalization-oq-04-oq-01/knowledge.md index 00f82791cb4..71344efac6b 100644 --- a/research/problems/cantor-diagonalization-oq-04-oq-01/knowledge.md +++ b/research/problems/cantor-diagonalization-oq-04-oq-01/knowledge.md @@ -6,6 +6,64 @@ Generalize Lawvere's retraction FPT from exact Type equality to Setoid equivalen **Parent**: CantorDiagonalizationOQ04 (Lawvere FPT, Type-level retraction version) **OQ**: "Can the retraction version be formalized beyond the Type category?" +## Session 2026-05-08 (Session 2, researcher-3) — ACT (refinement structure) + +**Mode**: REVISIT (RICH, score 24) +**Phase change**: COMPLETED → ACT (re-opened with new structural content) +**Outcome**: Added the *success* side and the *refinement* structure to round out the spectrum of coding-feasibility on `Setoid Y`. + +### What I Did + +Added Parts VIII and IX to `proofs/Proofs/CantorDiagonalizationOQ04OQ01.lean` (166 → 221 lines). + +**Part VIII — Trivial setoid (success side):** +- `trivialSetoid Y : Setoid Y` — collapses Y to one class (`r := fun _ _ => True`). +- `trivial_setoid_codes_endomorphisms` — every inhabited Y vacuously codes its endomorphisms under `trivialSetoid`. Witness: `encode = const default`, `decode = const id`, retract is `trivial`. +- `bool_trivial_setoid_codes_endomorphisms` — explicit Bool corollary, contrasting with the discrete-setoid impossibility (`cannot_code_endomorphisms_bool_setoid`). Demonstrates that *coding-feasibility is genuinely setoid-dependent*, not a fixed property of the underlying type. + +**Part IX — Refinement structure:** +- `IsRefinement s t : Prop` — `s.r a b → t.r a b` for all a, b (s is finer; t is coarser). +- `coding_descends_to_coarser` — `IsRefinement s t → CodesEndomorphismsSetoid Y s → CodesEndomorphismsSetoid Y t`. Same encode/decode work; the retract weakens because `t`-equivalence is implied by `s`-equivalence. +- `refines_trivial` — every setoid refines the trivial setoid. Places the trivial setoid at the canonical *top* of the refinement order on `Setoid Y`. + +### Key Findings (S2) + +1. **Setoid choice matters structurally**: Bool fails coding under discrete setoid (S1), but succeeds under trivial setoid (S2). Same type, different setoid, different coding behavior. +2. **Coding-existence is downward-closed**: in the refinement lattice on `Setoid Y`, the set `{s : CodesEndomorphismsSetoid Y s is inhabited}` is closed under going to coarser setoids. +3. **The discrete and trivial setoids bracket the spectrum**: discrete = strongest condition (often fails), trivial = weakest condition (always succeeds). The interesting structural information lives in setoids strictly between them (parity, congruence classes, isomorphism-up-to-coherence). + +### Files Modified + +- `proofs/Proofs/CantorDiagonalizationOQ04OQ01.lean` (166 → 221 lines): +2 defs, +4 theorems, 0 axioms unchanged, 0 sorries unchanged. +- `src/data/proofs/cantor-diagonalization-oq-04-oq-01/meta.json`: lineCount 166→221, theoremCount 8→12, definitionCount 3→5; +4 originalContributions entries; +2 sections for Parts VIII/IX; conclusion summary updated. +- `src/data/research/problems/cantor-diagonalization-oq-04-oq-01.json`: phase COMPLETED→ACT, status completed→active, iteration 1→2; +6 builtItems, +4 insights, +3 nextSteps; lastUpdate 2026-05-07→2026-05-08; leanFiles entry for OQ04OQ01 updated to 221/12/5. +- `research/problems/cantor-diagonalization-oq-04-oq-01/knowledge.md`: this file. + +### Build Status + +**Pending.** The worktree's `proofs/.lake` is the recursive self-symlink (per `feedback_researcher_lake_symlink_broken.md`), forcing every Docker build to fresh-clone Mathlib (~10–15 min) + cache get (~10 min) — total ~45 min. Following the established convention from PRs #16936 (cantor-OQ-01-OQ-01-OQ-02-OQ-01 S5), #17008 (ehrhart-cube-proven-oq-02 S4), this PR opens as **draft**. + +The new theorems use only basic Mathlib primitives already exercised in this file: +- `Setoid` constructor with `Equivalence` proof from `⟨fun _ => trivial, fun _ => trivial, fun _ _ => trivial⟩` (Part VIII) +- `default` (from `Inhabited`), `id`, `trivial` (Part VIII) +- Anonymous constructor of `CodesEndomorphismsSetoid`-records using fields (Parts VIII, IX) + +Risk of build failure is low; review of the proof bodies should be straightforward by inspection. + +### Honesty Assessment + +- **What this is**: a structural rounding-out of S1, not a deep new result. The four new theorems are 1-2 line proofs each, and the trivial-setoid case is genuinely vacuous (the retract holds because `True` is always provable). The refinement-descent lemma is also nearly trivial (function composition). +- **What this is NOT**: a characterization theorem. OQ-04-OQ-01-OQ-02 ("which setoids admit coding?") remains open — S2 only provides the structural backbone (refinement lattice + downward-closure) for a future characterization. +- **What this enables**: future sessions can target the converse of `coding_descends_to_coarser` (when does coding lift to finer setoids?) and the cardinality classification (is `|Y/≈| ≤ 1` the right boundary?). The refinement structure is the right organizing principle. + +### Next Concrete Steps + +1. **S3+ (OQ-04-OQ-01-OQ-02 attack)**: explore the boundary of admissible setoids on a concrete `Y` (try `Bool` with all 5 setoids; classify exactly which admit coding). +2. **S3+ (ℕ spectrum)**: extend `succ_no_parity_fixpoint` to `succ_no_modk_fixpoint` for arbitrary `k ≥ 2`. Find the *finest* setoid on ℕ admitting coding (likely the cofinite-quotient or a similar limit). +3. **Phase-3 (CCC lift)**: target the original OQ-04-OQ-01-OQ-01 (Mathlib `CartesianClosed` typeclass formalization). Setoid layer is the natural intermediate; the next step is replacing `Setoid` with an arbitrary equality-up-to-coherent-equivalence in a CCC. + +--- + ## Session 2026-05-07 (Session 1) — SOLVED **Mode**: FRESH diff --git a/src/data/proofs/cantor-diagonalization-oq-04-oq-01/meta.json b/src/data/proofs/cantor-diagonalization-oq-04-oq-01/meta.json index d3557a4e28b..3aca067dffd 100644 --- a/src/data/proofs/cantor-diagonalization-oq-04-oq-01/meta.json +++ b/src/data/proofs/cantor-diagonalization-oq-04-oq-01/meta.json @@ -21,9 +21,9 @@ ], "axiomCount": 0, "sorries": 0, - "lineCount": 166, - "theoremCount": 8, - "definitionCount": 3, + "lineCount": 221, + "theoremCount": 12, + "definitionCount": 5, "badge": "original", "originalContributions": [ "CodesEndomorphismsSetoid: pointwise-equivalence retract structure", @@ -32,7 +32,11 @@ "typeToSetoidCoding + lawvere_type_from_setoid: discrete-setoid recovers parent Type theorem", "cannot_code_endomorphisms_bool_setoid: Bool cannot code its 4 endomorphisms (not is fixed-point-free)", "cantor_setoid_no_surjection: classical Cantor's theorem in the setoid setting", - "paritySetoidN + cannot_code_endomorphisms_nat_parity: ℕ fails to code its endomorphisms mod 2" + "paritySetoidN + cannot_code_endomorphisms_nat_parity: ℕ fails to code its endomorphisms mod 2", + "trivialSetoid + trivial_setoid_codes_endomorphisms: every inhabited Y trivially codes under the one-class setoid (S2)", + "bool_trivial_setoid_codes_endomorphisms: Bool DOES code under trivial setoid — coding-feasibility is genuinely setoid-dependent (S2)", + "IsRefinement + coding_descends_to_coarser: structural lemma showing coding propagates from finer to coarser setoids (S2)", + "refines_trivial: trivial setoid is the canonical top of the refinement order on Setoid Y (S2)" ], "prerequisites": [ "Setoid equivalence relations (Mathlib.Data.Setoid.Basic)", @@ -67,7 +71,7 @@ "historicalContext": "**Lawvere's fixed-point theorem** (F. William Lawvere, *Diagonal arguments and cartesian closed categories*, La Jolla, 1969) is the categorical heart of every diagonal argument in mathematics. It states: in any cartesian closed category, if some object $Y$ admits a *weakly point-surjective* morphism $T : A \\to Y^A$ (every $g : A \\to Y$ is named by some 'point' of $A$), then every endomorphism $f : Y \\to Y$ has a fixed point. Cantor's theorem (1891), Russell's paradox (1901), Gödel's first incompleteness theorem (1931), Tarski's undefinability of truth (1936), and Turing's halting problem (1936) are all corollaries — each says some particular $Y$ *fails* to be Lawvere-coded in some particular setting.\n\nThe parent proof `cantor-diagonalization-oq-04` formalizes the **retraction version** of Lawvere's theorem in Lean's `Type` category, using exact equality. This file answers the open question OQ-04-OQ-01: **can the retraction version be formalized beyond `Type`, in a topos or general CCC?** The cleanest intermediate step replaces strict Type equality with a `Setoid` equivalence relation. Setoids are the standard model of the *internal language of a topos* without quotient types: a setoid is a type plus an equivalence relation, and one reasons modulo the relation. The discrete setoid (≈ ≡ =) recovers `Type` exactly; richer setoids (parity, congruence classes, etc.) capture quotient-type analogues. Notably, `f` need NOT be a setoid morphism — the fixed point exists for arbitrary functions, a strength uncommon among setoid-flavored generalizations.", "problemStatement": "Let $Y$ be a type with setoid $s : \\text{Setoid}(Y)$ giving an equivalence relation $\\approx$. Define `CodesEndomorphismsSetoid Y s` as a structure with `encode : (Y \\to Y) \\to Y`, `decode : Y \\to (Y \\to Y)`, and the **pointwise setoid retract** $\\forall g \\, y, \\, \\text{decode}(\\text{encode}(g), y) \\approx g(y)$.\n\nProve: for every $f : Y \\to Y$ (not necessarily setoid-respecting), $\\exists p : Y, \\, f(p) \\approx p$. Show:\n\n1. The discrete setoid recovers the Type-level theorem.\n2. $\\text{Bool}$ cannot code its endomorphisms under the discrete setoid.\n3. $\\mathbb{N}$ cannot code its endomorphisms under the parity setoid (mod 2).\n4. There is no point-surjective $h : Y \\to (Y \\to \\text{Prop})$ — Cantor's theorem.", "text": "Lifts Lawvere's fixed-point theorem from Type-level equality to setoid equivalence; recovers the Type version, demonstrates two non-Lawvere examples, and establishes Cantor's theorem in the setoid setting.", - "proofStrategy": "**Three-line core, four-corollary supporting cast.** The diagonal construction $g(y) = f(\\text{decode}(y, y))$ is identical to the Type version; only the final equivalence step uses the setoid relation. (1) Define `CodesEndomorphismsSetoid` with the pointwise-retract field. (2) Prove `lawvere_fixpoint_setoid` in three lines using `Setoid.iseqv.symm`. (3) Derive `no_coding_setoid_if_fixpoint_free` as the contrapositive. (4) Define `discreteSetoid` and `typeToSetoidCoding`; show `lawvere_type_from_setoid` recovers the parent. (5) Two impossibility witnesses: $\\text{Bool}$ via `Bool.not` (closed by `decide`), $\\mathbb{N}$ via successor mod parity (closed by `omega`). (6) Cantor's $h : Y \\to (Y \\to \\text{Prop})$ via Russell's diagonal predicate $D = \\lambda y. \\neg h(y)(y)$, with `iff_of_eq (congr_fun ..)` translating the function-equality to a pointwise iff. Total: 166 lines, 8 theorems, 3 definitions, 0 axioms, 0 sorries.", + "proofStrategy": "**Three-line core, expanding supporting cast.** The diagonal construction $g(y) = f(\\text{decode}(y, y))$ is identical to the Type version; only the final equivalence step uses the setoid relation. (1) Define `CodesEndomorphismsSetoid` with the pointwise-retract field. (2) Prove `lawvere_fixpoint_setoid` in three lines using `Setoid.iseqv.symm`. (3) Derive `no_coding_setoid_if_fixpoint_free` as the contrapositive. (4) Define `discreteSetoid` and `typeToSetoidCoding`; show `lawvere_type_from_setoid` recovers the parent. (5) Two impossibility witnesses: $\\text{Bool}$ via `Bool.not` (closed by `decide`), $\\mathbb{N}$ via successor mod parity (closed by `omega`). (6) Cantor's $h : Y \\to (Y \\to \\text{Prop})$ via Russell's diagonal predicate $D = \\lambda y. \\neg h(y)(y)$, with `iff_of_eq (congr_fun ..)` translating the function-equality to a pointwise iff. (7) **S2 — possibility witness**: `trivialSetoid` (one class, $a \\approx b \\equiv \\top$) and `trivial_setoid_codes_endomorphisms` give vacuous coding for any inhabited $Y$; specialized to $\\text{Bool}$ as `bool_trivial_setoid_codes_endomorphisms`, contrasting with the discrete-setoid impossibility. (8) **S2 — refinement structure**: `IsRefinement s t` (s.r implies t.r) and `coding_descends_to_coarser` show that any encode/decode coding under a finer setoid extends to coarser ones; `refines_trivial` places the trivial setoid at the top. Total: 221 lines, 12 theorems, 5 definitions, 0 axioms, 0 sorries.", "keyInsights": [ "**The diagonal construction is robust to setoid weakening.** $g(y) = f(\\text{decode}(y, y))$ works identically in both the Type and Setoid settings — only the final step changes from `Eq.symm` to `Setoid.iseqv.symm`. This robustness is *why* Lawvere's theorem appears across so many diagonal arguments: the underlying construction depends only on a self-applicative encoding, not on the equality discipline.", "**`f` need not be a setoid morphism.** Many setoid-flavored generalizations require $f$ to respect the equivalence ($a \\approx b \\Rightarrow f(a) \\approx f(b)$). This proof requires *no such hypothesis* — the fixed point exists for arbitrary functions $Y \\to Y$. The reason: the retract is pointwise, so $\\text{decode}(y_0, y_0) \\approx g(y_0)$ holds 'as is' without needing $f$ to preserve $\\approx$. This is genuinely surprising and strictly stronger than the typical CCC formulation.", @@ -150,17 +154,33 @@ "summary": "paritySetoidN: a ≈ b iff a % 2 = b % 2. succ_no_parity_fixpoint (omega) plus no_coding_setoid_if_fixpoint_free yields cannot_code_endomorphisms_nat_parity. Cardinality |ℕ| = |ℕ^ℕ| is irrelevant; the obstruction is structural.", "mathContext": "Generalization: for any $\\mathbb{N}/k\\mathbb{N}$ with $k \\geq 2$, the function $n \\mapsto n+1$ is fixed-point-free mod $k$, so $\\mathbb{N}$ never Lawvere-codes mod $k$." }, + { + "id": "trivial-setoid", + "title": "Part VIII: Trivial Setoid — Coding Always Possible", + "startLine": 169, + "endLine": 194, + "summary": "trivialSetoid Y collapses Y to a single equivalence class (a ≈ b is True for all a, b). trivial_setoid_codes_endomorphisms shows every inhabited Y vacuously codes its endomorphisms under this setoid. bool_trivial_setoid_codes_endomorphisms is the explicit Bool corollary, contrasting with cannot_code_endomorphisms_bool_setoid (Part V) — coding-feasibility is genuinely setoid-dependent.", + "mathContext": "The two extremes — discrete (a ≈ b iff a = b) and trivial (always equivalent) — bracket the lattice of setoids on $Y$. Discrete forces Lawvere to coincide with Type-level coding (failing for Bool, ℕ-mod-parity, etc.). Trivial trivially admits coding. Real mathematical content lives in the intermediate setoids (parity, congruence, isomorphism-up-to-coherence)." + }, + { + "id": "refinement-descent", + "title": "Part IX: Refinement Lemma — Coding Descends to Coarser Setoids", + "startLine": 196, + "endLine": 219, + "summary": "IsRefinement s t : Prop captures `s` is a refinement of `t` (s.r a b → t.r a b, equivalently t is coarser). coding_descends_to_coarser transports a CodesEndomorphismsSetoid Y s structure to CodesEndomorphismsSetoid Y t along refinement (the same encode/decode work; the retract is weakened to t-equivalence). refines_trivial witnesses that the trivial setoid is the canonical top of the refinement order on Setoid Y.", + "mathContext": "Refinement-monotonicity is the structural counterpart of the discrete-to-trivial spectrum: coding gets *easier* as the setoid coarsens. This justifies the strategy 'pick the right setoid for the question being asked' — finer setoids extract sharper structural information; coarser setoids admit more codings vacuously. The result also shows coding-existence is a *downward-closed* predicate on the refinement lattice." + }, { "id": "namespace-end", "title": "Namespace End", - "startLine": 166, - "endLine": 166, + "startLine": 221, + "endLine": 221, "summary": "Closes the CantorDiagonalizationOQ04OQ01 namespace.", "mathContext": "Setoid layer = bridge from Lean's Type-level Lawvere to a fully categorical Lawvere in any CCC. The next step (OQ-XX) is reformulation in Mathlib's CartesianClosed." } ], "conclusion": { - "summary": "Lawvere's fixed-point theorem extends from Type to Setoid with the diagonal construction unchanged: only the final equivalence step uses `Setoid.iseqv.symm` instead of `Eq.symm`. The retraction is **pointwise** in the setoid relation, and **f need not be a setoid morphism** — a strength uncommon among setoid-flavored generalizations. The discrete setoid recovers the parent Type-level theorem exactly, proving this is a strict generalization. Two concrete non-Lawvere examples ($\\text{Bool}$ via `not`, $\\mathbb{N}$ mod parity via `succ`) and Cantor's theorem in the setoid setting round out the file. 166 lines, 8 theorems, 3 definitions, 0 axioms, 0 sorries.", + "summary": "Lawvere's fixed-point theorem extends from Type to Setoid with the diagonal construction unchanged: only the final equivalence step uses `Setoid.iseqv.symm` instead of `Eq.symm`. The retraction is **pointwise** in the setoid relation, and **f need not be a setoid morphism** — a strength uncommon among setoid-flavored generalizations. The discrete setoid recovers the parent Type-level theorem exactly, proving this is a strict generalization. Two concrete non-Lawvere examples ($\\text{Bool}$ via `not`, $\\mathbb{N}$ mod parity via `succ`) and Cantor's theorem in the setoid setting establish the failure side. **Session 2 adds the success side and the refinement structure**: the *trivial* (one-class) setoid always admits coding (`trivial_setoid_codes_endomorphisms`), so $\\text{Bool}$ — which fails under the discrete setoid — succeeds under the trivial one (`bool_trivial_setoid_codes_endomorphisms`). The structural lemma `coding_descends_to_coarser` (with `IsRefinement` and `refines_trivial`) shows that coding-existence is downward-closed on the refinement lattice of setoids, so the discrete-trivial pair brackets the full coding spectrum. 221 lines, 12 theorems, 5 definitions, 0 axioms, 0 sorries.", "implications": "The setoid layer is the **maximal Lawvere FPT achievable purely in Lean's term language**. A fully categorical reformulation in Mathlib's `CartesianClosed` typeclass would require reasoning with internal-homs ($\\text{ihom}$), evaluation morphisms, and currying-as-morphisms — none of which are natively expressible as Lean functions. The setoid version captures the **mathematical content** of equality-up-to-coherent-equivalence (the topos-theoretic / CCC notion) while staying in Lean's term language, so it is the right intermediate goal between the parent Type-level proof and a future categorical one. The strength that **f need not be a setoid morphism** is mathematically interesting in its own right: it suggests that Lawvere's theorem is *more* than just a categorical fact — it is an absolute combinatorial principle about self-applicative encodings, robust to the precise notion of equality used. Pedagogically, the file demonstrates that **setoids are a useful intermediate** in any 'lift from Type to topos' formalization plan.", "openQuestions": [ "Reformulate `lawvere_fixpoint_setoid` using Mathlib's `CartesianClosed` typeclass (with `ihom`, evaluation, and global elements as morphisms from the terminal object). The setoid proof provides the mathematical content; the challenge is expressing it in Mathlib's category-theory framework, which currently lacks an internal-language layer.", @@ -237,10 +257,10 @@ ], "opens": [], "namespace": "CantorDiagonalizationOQ04OQ01", - "lineCount": 166, + "lineCount": 221, "axiomCount": 0, - "theoremCount": 8, - "definitionCount": 3, + "theoremCount": 12, + "definitionCount": 5, "path": "Proofs/CantorDiagonalizationOQ04OQ01.lean", "sorries": 0 }, diff --git a/src/data/research/problems/cantor-diagonalization-oq-04-oq-01.json b/src/data/research/problems/cantor-diagonalization-oq-04-oq-01.json index c4ea6a62ede..9da5829b926 100644 --- a/src/data/research/problems/cantor-diagonalization-oq-04-oq-01.json +++ b/src/data/research/problems/cantor-diagonalization-oq-04-oq-01.json @@ -1,8 +1,8 @@ { "slug": "cantor-diagonalization-oq-04-oq-01", "title": "Lawvere Fixed-Point Theorem for Setoids (CCC Generalization)", - "phase": "COMPLETED", - "status": "completed", + "phase": "ACT", + "status": "active", "tier": "B", "path": "full", "problemStatement": { @@ -21,20 +21,20 @@ "goal": "Proved: Lawvere FPT for Setoids, with full gallery entry" }, "currentState": { - "phase": "DONE", - "since": "2026-05-07T17:00:00Z", - "iteration": 1, - "focus": "Lawvere FPT for Setoids: 8 theorems, 3 definitions, 0 sorries, 0 axioms in CantorDiagonalizationOQ04OQ01.lean (166 lines). Gallery status=verified, badge=original. Merged in PR #16393.", + "phase": "ACT", + "since": "2026-05-08T08:30:00Z", + "iteration": 2, + "focus": "S2 (researcher-3, 2026-05-08): added the success/refinement structure to CantorDiagonalizationOQ04OQ01.lean. trivialSetoid + trivial_setoid_codes_endomorphisms (vacuous coding for any inhabited Y), bool_trivial_setoid_codes_endomorphisms (Bool succeeds under trivial — contrast with discrete-setoid failure), IsRefinement + coding_descends_to_coarser + refines_trivial (refinement-monotonicity of coding-existence). File 166→221 lines, 8→12 theorems, 3→5 definitions, 0 sorries, 0 axioms unchanged.", "blockers": [], - "nextAction": "None — entry complete. Optional follow-on directions tracked in nextSteps (CCC typeclass lift, characterization of admissible setoids).", + "nextAction": "S3+: pursue OQ-04-OQ-01-OQ-02 (characterization of admissible setoids) by exploring the converse direction — when does refinement-descent's converse hold? E.g., are there structural setoid invariants (e.g., quotient cardinality |Y/≈| ≤ 1, or additivity in the lattice) that classify exactly when CodesEndomorphismsSetoid Y s is inhabited?", "attemptCounts": { - "total": 0, - "currentApproach": 0, - "approachesTried": 0 + "total": 1, + "currentApproach": 1, + "approachesTried": 1 } }, "knowledge": { - "progressSummary": "COMPLETED: Lawvere FPT proved for Setoids. CantorDiagonalizationOQ04OQ01.lean has 8 theorems, 3 definitions, 0 sorries, 0 axioms (166 lines). Gallery status=verified, badge=original; merged in PR #16393. The diagonal construction g(y) = f(decode(y)(y)) carries over from the Type-level proof; the retraction condition only needs to hold up to setoid equivalence (decode(encode(f))(y) ≈ f(y)).", + "progressSummary": "S2 (researcher-3, 2026-05-08, ACT): rounded out the success side of the Lawvere-setoid spectrum. trivialSetoid (collapses Y to one class, ≈ ≡ True) admits coding vacuously for any inhabited Y (trivial_setoid_codes_endomorphisms); the explicit Bool corollary bool_trivial_setoid_codes_endomorphisms contrasts with the discrete-setoid impossibility from S1. The structural lemma coding_descends_to_coarser (built on IsRefinement s t : Prop) shows that coding-existence is downward-closed on the refinement lattice — coarser setoids inherit coding from finer ones. refines_trivial places the trivial setoid at the top. CantorDiagonalizationOQ04OQ01.lean: 166→221 lines, 8→12 theorems, 3→5 definitions, 0 sorries, 0 axioms unchanged. Status remains verified/original.\n\nS1 (2026-05-07, COMPLETED): proved Lawvere FPT for Setoids. 8 theorems, 3 definitions, 0 sorries, 0 axioms in CantorDiagonalizationOQ04OQ01.lean (166 lines). Gallery status=verified, badge=original; merged in PR #16393.", "builtItems": [ "CodesEndomorphismsSetoid structure (encode, decode, setoid retraction)", "lawvere_fixpoint_setoid: main theorem in CantorDiagonalizationOQ04OQ01.lean", @@ -44,19 +44,31 @@ "cannot_code_endomorphisms_bool_setoid: Bool impossibility", "cannot_code_endomorphisms_nat_parity: \u2115/parity impossibility", "cantor_setoid_no_surjection: Cantor diagonal in setoid setting", - "paritySetoidN: concrete non-trivial setoid example" + "paritySetoidN: concrete non-trivial setoid example", + "S2 (researcher-3, 2026-05-08): trivialSetoid Y — every pair equivalent (one-class collapse)", + "S2 (researcher-3, 2026-05-08): trivial_setoid_codes_endomorphisms — every inhabited Y codes vacuously under trivial setoid", + "S2 (researcher-3, 2026-05-08): bool_trivial_setoid_codes_endomorphisms — Bool succeeds under trivial (vs. discrete failure in S1)", + "S2 (researcher-3, 2026-05-08): IsRefinement s t — definition of setoid refinement (s.r implies t.r)", + "S2 (researcher-3, 2026-05-08): coding_descends_to_coarser — coding-existence is monotone-downward in the refinement lattice", + "S2 (researcher-3, 2026-05-08): refines_trivial — trivial setoid is the canonical top of the refinement order" ], "insights": [ "The diagonal construction g(y) = f(decode(y)(y)) works unchanged in the setoid setting", "f need NOT be a setoid morphism \u2014 fixed point exists for all f : Y \u2192 Y", "Discrete setoid recovers the Type-level theorem exactly (type reduction to special case)", "Setoid framework models CCC equality-up-to-isomorphism within Lean type theory", - "Retraction condition decode(encode(g))(y) \u2248 g(y) (pointwise setoid) is the correct weakening" + "Retraction condition decode(encode(g))(y) \u2248 g(y) (pointwise setoid) is the correct weakening", + "S2 \u2014 the trivial (one-class) setoid always admits coding vacuously: choose any constant encode and id-decode; \u2248 \u2261 True absorbs the retract. So Bool *can* code its endomorphisms \u2014 it just depends on the setoid choice.", + "S2 \u2014 coding-existence is monotone-DOWNWARD in the refinement lattice on Setoid Y: finer setoid admits coding \u21d2 every coarser one does too (same encode/decode, weaker retract). So discrete-setoid coding is the strongest form, trivial-setoid coding the weakest.", + "S2 \u2014 the discrete and trivial setoids bracket the full coding spectrum: between them lie the interesting setoids (parity, congruence classes, isomorphism-up-to-coherence) where coding-feasibility carries genuine information.", + "S2 \u2014 counterexample direction: knowing Bool cannot code under discrete (S1) does *not* preclude coding under coarser setoids. Conversely, knowing Bool codes under trivial (S2) does *not* lift to finer setoids. The refinement structure is the right organizing principle for OQ-04-OQ-01-OQ-02 (characterization)." ], "mathlibGaps": [], "nextSteps": [ "Lift to Mathlib CartesianClosed typeclass: formalize in abstract CCC with terminal object", - "Characterize which setoids Y admit CodesEndomorphismsSetoid structures" + "Characterize which setoids Y admit CodesEndomorphismsSetoid structures (OQ-04-OQ-01-OQ-02): the refinement lattice provides the structural backbone \u2014 admissible setoids form a downward-closed subset. Open: is the boundary characterized by a quotient-cardinality condition |Y/\u2248| \u2264 1, by an algebraic invariant of the relation, or by a Lawvere-style fixed-point criterion?", + "Bracket the spectrum on \u2115: parity (mod 2) fails, mod-k for k\u22652 fails (extend succ_no_parity_fixpoint), but trivial succeeds. Find the finest setoid on \u2115 admitting coding.", + "Pursue the converse of coding_descends_to_coarser: when does coding under coarser t lift to coding under finer s? Likely never in full generality, but specific algebraic conditions on the refinement could give partial converses." ] }, "tags": [ @@ -92,7 +104,7 @@ "mathlib": [] }, "started": "2026-05-06T14:45:58.508Z", - "lastUpdate": "2026-05-07T17:00:00Z", + "lastUpdate": "2026-05-08T08:30:00Z", "significance": 5, "tractability": 4, "leanFiles": [ @@ -297,10 +309,10 @@ { "path": "Proofs/CantorDiagonalizationOQ04OQ01.lean", "filename": "CantorDiagonalizationOQ04OQ01.lean", - "lineCount": 167, - "theoremCount": 8, + "lineCount": 221, + "theoremCount": 12, "axiomCount": 0, - "defCount": 3, + "defCount": 5, "sorryCount": 0, "isAristotle": false, "githubUrl": "https://github.com/rjwalters/lean-genius/blob/main/proofs/Proofs/CantorDiagonalizationOQ04OQ01.lean"