From 14a3c237216d0869eb333e2b73a3b4f76bff6bb3 Mon Sep 17 00:00:00 2001 From: Amamiya Miu Date: Sat, 22 Aug 2026 17:46:01 +0800 Subject: [PATCH] feat(higher): normalize left unitor cells --- AXIOMS.md | 14 +++ BLUEPRINT.md | 4 +- CONJECTURES.md | 14 +-- MODEL_MATRIX.md | 6 +- Ript/Audit/AxiomChecks.lean | 14 +++ .../MarkedZigzagAlignedHammock.lean | 99 ++++++++++++++++--- .../Bicategory/MarkedZigzagLinearHammock.lean | 53 ++++++++++ Ript/Higher/CostExactZigzagMappingSpace.lean | 23 ++++- docs/en/RESEARCH_STATUS.md | 7 +- docs/en/reference/AXIOMS.md | 14 +++ docs/en/reference/BLUEPRINT.md | 4 +- docs/en/reference/CONJECTURES.md | 14 +-- docs/en/reference/MODEL_MATRIX.md | 6 +- docs/eo/RESEARCH_STATUS.md | 6 +- docs/eo/reference/AXIOMS.md | 14 +++ docs/ja/RESEARCH_STATUS.md | 2 +- docs/ja/reference/AXIOMS.md | 14 +++ docs/zh-CN/RESEARCH_STATUS.md | 2 +- docs/zh-CN/reference/AXIOMS.md | 14 +++ 19 files changed, 283 insertions(+), 41 deletions(-) diff --git a/AXIOMS.md b/AXIOMS.md index 3c3c056..b8229b2 100644 --- a/AXIOMS.md +++ b/AXIOMS.md @@ -1034,6 +1034,20 @@ the actual output of `lake env lean Ript/Audit/AxiomChecks.lean`. | `CategoryTheory.Bicategory.MarkedZigzag.HammockPath.markedUnitInv_normalizable` | `[propext, Classical.choice, Quot.sound]` | `Ript/ForMathlib/CategoryTheory/Bicategory/MarkedZigzagAlignedHammock.lean` | | `CategoryTheory.Bicategory.MarkedZigzag.HammockPath.markedCounit_normalizable` | `[propext, Classical.choice, Quot.sound]` | `Ript/ForMathlib/CategoryTheory/Bicategory/MarkedZigzagAlignedHammock.lean` | | `CategoryTheory.Bicategory.MarkedZigzag.HammockPath.markedCounitInv_normalizable` | `[propext, Classical.choice, Quot.sound]` | `Ript/ForMathlib/CategoryTheory/Bicategory/MarkedZigzagAlignedHammock.lean` | +| `CategoryTheory.Bicategory.MarkedZigzag.LinearWord.toWord_nil` | `none` | `Ript/ForMathlib/CategoryTheory/Bicategory/MarkedZigzagLinearHammock.lean` | +| `CategoryTheory.Bicategory.MarkedZigzag.LinearWord.flatten_nil` | `none` | `Ript/ForMathlib/CategoryTheory/Bicategory/MarkedZigzagLinearHammock.lean` | +| `CategoryTheory.Bicategory.MarkedZigzag.LinearWord.append_nil` | `none` | `Ript/ForMathlib/CategoryTheory/Bicategory/MarkedZigzagLinearHammock.lean` | +| `CategoryTheory.Bicategory.MarkedZigzag.LinearWord.append_assoc` | `none` | `Ript/ForMathlib/CategoryTheory/Bicategory/MarkedZigzagLinearHammock.lean` | +| `CategoryTheory.Bicategory.MarkedZigzag.LinearWord.toWordAppendIso_nil` | `[propext, Classical.choice, Quot.sound]` | `Ript/ForMathlib/CategoryTheory/Bicategory/MarkedZigzagLinearHammock.lean` | +| `CategoryTheory.Bicategory.MarkedZigzag.LinearWord.leftUnitor_conjugation` | `[propext, Quot.sound]` | `Ript/ForMathlib/CategoryTheory/Bicategory/MarkedZigzagLinearHammock.lean` | +| `CategoryTheory.Bicategory.MarkedZigzag.LinearWord.leftUnitor_inv_conjugation` | `[propext, Quot.sound]` | `Ript/ForMathlib/CategoryTheory/Bicategory/MarkedZigzagLinearHammock.lean` | +| `CategoryTheory.Bicategory.MarkedZigzag.HammockPath.ofEq` | `none` | `Ript/ForMathlib/CategoryTheory/Bicategory/MarkedZigzagAlignedHammock.lean` | +| `CategoryTheory.Bicategory.MarkedZigzag.HammockPath.toHom_ofEq` | `[propext, Classical.choice, Quot.sound]` | `Ript/ForMathlib/CategoryTheory/Bicategory/MarkedZigzagAlignedHammock.lean` | +| `CategoryTheory.Bicategory.MarkedZigzag.HammockPath.normalizedCellHom_eq_of_rel` | `[propext, Classical.choice, Quot.sound]` | `Ript/ForMathlib/CategoryTheory/Bicategory/MarkedZigzagAlignedHammock.lean` | +| `CategoryTheory.Bicategory.MarkedZigzag.HammockPath.normalizedCellHom_leftUnitor` | `[propext, Classical.choice, Quot.sound]` | `Ript/ForMathlib/CategoryTheory/Bicategory/MarkedZigzagAlignedHammock.lean` | +| `CategoryTheory.Bicategory.MarkedZigzag.HammockPath.normalizedCellHom_leftUnitorInv` | `[propext, Classical.choice, Quot.sound]` | `Ript/ForMathlib/CategoryTheory/Bicategory/MarkedZigzagAlignedHammock.lean` | +| `CategoryTheory.Bicategory.MarkedZigzag.HammockPath.leftUnitor_normalizable` | `[propext, Classical.choice, Quot.sound]` | `Ript/ForMathlib/CategoryTheory/Bicategory/MarkedZigzagAlignedHammock.lean` | +| `CategoryTheory.Bicategory.MarkedZigzag.HammockPath.leftUnitorInv_normalizable` | `[propext, Classical.choice, Quot.sound]` | `Ript/ForMathlib/CategoryTheory/Bicategory/MarkedZigzagAlignedHammock.lean` | | `Ript.Higher.CostExactZigzagMappingSpace.AlignedHammockGrid.linearHammockGridEquiv_simplex` | `[propext, Classical.choice, Quot.sound]` | `Ript/Higher/CostExactZigzagMappingSpace.lean` | | `Ript.Higher.CostExactZigzagMappingSpace.alignedHammockCore` | `[propext, Classical.choice, Quot.sound]` | `Ript/Higher/CostExactZigzagMappingSpace.lean` | | `Ript.Higher.CostExactZigzagMappingSpace.columnRefinementCore` | `[propext, Classical.choice, Quot.sound]` | `Ript/Higher/CostExactZigzagMappingSpace.lean` | diff --git a/BLUEPRINT.md b/BLUEPRINT.md index 4d5ab5f..306722c 100644 --- a/BLUEPRINT.md +++ b/BLUEPRINT.md @@ -390,9 +390,9 @@ Every node in this graph is an existing compiled module. | 12 (zero-truncated common-refinement mapping nerve) | Wrapped rows form a thin common-refinement groupoid; its canonical functor to the discrete row quotient is faithful, full, essentially surjective, and hence a categorical equivalence; the induced nerve map has categorical-nerve equivalence evidence, an explicit simplicial inverse and both homotopies, and exact row-vertex action | PROVED | | 12 (non-thin semantic refinement-path nerve) | Refinement syntax modulo equality of quotient-cell interpretations forms a non-thin groupoid; executable reversal supplies inverses; its semantic functor into the linear mapping category is faithful and essentially surjective on row objects, maps every path to an isomorphism, and induces a nerve map with exact row-vertex and arbitrary-refinement-edge action; the zero-truncation functor to the thin groupoid is full and essentially surjective | PROVED | | 12 (exact refinement-generated semantic image) | Actual quotient 2-cells equipped with existence of an executable refinement generator form a non-thin image subgroupoid of the linear mapping category; inclusion is faithful, the semantic refinement-path functor is full, faithful, essentially surjective and hence an equivalence onto this exact image, and the induced nerve equivalence has an explicit simplicial homotopy inverse, exact generator-edge action, and strict factorization of the original semantic nerve map | PROVED | -| 12 (aligned-cell-augmented hammock paths) | A non-groupoidal generated path syntax alternates executable refinements with arbitrary aligned raw 2-cells and vertical composition, then quotients only by equality of quotient-cell semantics; normalized left/right whiskering enters and leaves binary append through canonical linear-normal-form isomorphisms, horizontal append composes those operations, and semantic equality/image membership are closed under all three; explicit append-isomorphism exchange/cancellation proves normalization naturality for raw left/right whiskering; pure bicategorical coherence identifies two-atomic-step normalization, so source composition/inverse normalize exactly to forward expansion/contraction; generic empty/two-atom formulas normalize marked unit/counit and inverses exactly to pair insertion/deletion; raw identity, original, source structural, marked pair, whiskering, vertical-composite, and equality-transport cells are normalizable; `StructuralGeneratorNormalizable` now lists exactly six associator/unitor obligations and yields all-cell normalization by structural induction | PROVED | +| 12 (aligned-cell-augmented hammock paths) | A non-groupoidal generated path syntax alternates executable refinements with arbitrary aligned raw 2-cells and vertical composition, then quotients only by equality of quotient-cell semantics; normalized left/right whiskering enters and leaves binary append through canonical linear-normal-form isomorphisms, horizontal append composes those operations, and semantic equality/image membership are closed under all three; explicit append-isomorphism exchange/cancellation proves normalization naturality for raw left/right whiskering; pure bicategorical coherence identifies two-atomic-step normalization, so source composition/inverse normalize exactly to forward expansion/contraction; generic empty/two-atom formulas normalize marked unit/counit and inverses exactly to pair insertion/deletion; linear append right-unit/associativity equalities and computable equality paths are available, and left unitor/inverse normalize to identity via arbitrary-iso conjugation; `StructuralGeneratorNormalizable` now lists exactly four associator/right-unitor obligations and yields all-cell normalization by structural induction | PROVED | | 12 (cost-exact two-layer global comparison) | Pseudofunctor-induced functor on homotopy categories; localization-aware relative Rezk map and auxiliary ordinary outer map into the actual marked-zigzag target; explicit source/target outer completeness homotopy equivalences; marked outer arrows factoring through the target actual-equivalence space; packaging with the exact non-groupoidal local nerve comparison; exact vertex, identity, horizontal-composition, associator, and left/right-unitor gluing; arbitrary invertible local 2-cell decoding; explicit pentagon and triangle compatibility | PROVED | -| 12 (global cost-exact complete-Segal/Rezk equivalence) | Discharge the six explicit `StructuralGeneratorNormalizable` fields for associator/inverse and left/right unitors and inverses; deduce every presented quotient 2-cell has an alternating aligned/refinement path and hence semantic fullness, then prove critical-pair coherence and reduced-hammock homotopical invariance (or compare the generated path category to another accepted derived mapping-space construction), connect that comparison to a standard weak-equivalence interface, and finish the standard Dwyer--Kan/Rezk weak-equivalence and completeness theorem | OPEN_RESEARCH | +| 12 (global cost-exact complete-Segal/Rezk equivalence) | Discharge the four explicit `StructuralGeneratorNormalizable` fields for associator/inverse and right unitor/inverse; deduce every presented quotient 2-cell has an alternating aligned/refinement path and hence semantic fullness, then prove critical-pair coherence and reduced-hammock homotopical invariance (or compare the generated path category to another accepted derived mapping-space construction), connect that comparison to a standard weak-equivalence interface, and finish the standard Dwyer--Kan/Rezk weak-equivalence and completeness theorem | OPEN_RESEARCH | ## Finite deterministic copy-discard theorem records diff --git a/CONJECTURES.md b/CONJECTURES.md index cf0741e..8cd288c 100644 --- a/CONJECTURES.md +++ b/CONJECTURES.md @@ -475,13 +475,15 @@ are normalizable, and normalizability is closed under vertical composition. Normalization naturality for both raw whiskerings is now proved by explicit append-isomorphism exchange and cancellation; source identity and inverse plus equality transport are normalizable too. The exact remaining induction basis -has now shrunk to the six-field `StructuralGeneratorNormalizable` record: -associator and inverse, and both unitors and inverses. Source +has now shrunk to the four-field `StructuralGeneratorNormalizable` record: +associator and inverse plus right unitor and inverse. Source composition/inverse are normalized exactly by executable forward expansion/contraction using the audited two-atomic-step coherence formula; generic empty/two-atom formulas now normalize marked unit/counit and -inverses exactly by pair insertion/deletion. The remaining record already -implies normalization of every raw +inverses exactly by pair insertion/deletion. Computable linear append right- +unit/associativity equalities and equality paths are available; left unitor and +inverse normalize to identity by arbitrary-iso conjugation. The remaining +record already implies normalization of every raw cell by structural induction, but none of its unproved fields is assumed in an unconditional theorem. Competing forward/marked moves, reduced-hammock moves, and their homotopical invariance remain open. @@ -593,8 +595,8 @@ source 2-cell in one-column form, strictly extends refinement paths, and is closed under normalized left/right whiskering and horizontal append. Identity, vertical-composite, original-cell, both whiskering, source-identity/inverse, source-composition/inverse, and equality-transport normalization cases are -proved; all marked unit/counit cases are proved too. The remaining six -structural generator obligations are explicit and +proved; all marked unit/counit and left-unitor cases are proved too. The +remaining four structural generator obligations are explicit and sufficient for the complete induction, but remain unproved; critical-pair coherence and reduced-hammock invariance remain open. diff --git a/MODEL_MATRIX.md b/MODEL_MATRIX.md index 83f3a5c..e9a9595 100644 --- a/MODEL_MATRIX.md +++ b/MODEL_MATRIX.md @@ -564,7 +564,7 @@ or the executable cores. | Presheaf universe | Type-valued presheaves on the internal groupoid | Yoneda is fully faithful; representable transformations/isomorphisms correspond to internal identity/equivalence | Semantic proof layer; Mathlib Yoneda audits with classical choice | | Yoneda envelope | Essential image of representables in the presheaf universe | Groupoid equivalent to the internal groupoid; inclusion factors Yoneda; the restricted Yoneda functor is a Mathlib localization at all internal identities | Noncomputable essential-image witnesses; exact ordinary localization of an already-groupoidal source, not a Rezk completion | | Simplicial interface nerve | Ordinary categorical nerve of the internal groupoid | Complete Kan horn filling, strict Segal, quasicategory, 2-coskeletal; vertices/edges/2-simplices encode interfaces, identities, and composition; homotopy category recovers the groupoid | Semantic proof layer; chosen fillers audit with classical choice; no complete-Segal or Rezk claim | -| Rezk classifying diagram | Outer simplicial category of composable interface strings, followed levelwise by the ordinary nerve | Every vertical level and horizontal row is a groupoid nerve and Kan; every horizontal row is strict Segal; the whole outer diagram is naturally `n ↦ Map(Δ[n], N(M.Object))`; `Map(∂Δ[n], N(M.Object))` is the genuine matching limit; every matching map is a fibration; the actual completeness map has an explicit simplicial homotopy inverse | Semantic proof layer; exact project-local `GroupoidalCompleteSegal` and `HomotopyEquivalenceWitness` evidence proved; the full cost-exact localization, common-universe local comparison, localization-aware all-dimensional relative outer Rezk map, source/target completeness witnesses, arbitrary-2-cell one-skeleton glue, vertical local 2-simplex/composite-diagonal glue, degree-one horizontal compositor squares, degree-two vertical pasting/interchange, the explicit three-tetrahedron degree-two compositor prism, all-degree local prism coherence, arbitrary outer-string vertex/restriction comparison, relative-outer gluing of every all-degree prism source vertex, strict decoded-pair naturality for every restriction, exact side-sensitive outer/local glue for every actual target prism-face vertex, a categorical-nerve equivalence from the presented relative-zigzag mapping nerve to every actual target local nerve, strict all-degree local-map factorization, outer essential surjectivity, target-independent algebraic/simplicial presentation universality, an audited `PresentedDwyerKanCore`, an independent right-associated linear hammock mapping category equivalent to the binary presentation and actual target local nerve, an audited `LinearHammockDwyerKanCore`, an exact arbitrary-height row-grid representation of every linear-hammock simplex, exact quotient/nerve interpretation of the fixed-shape aligned multi-column fragment, an executable elementary forward/marked-pair refinement calculus, an object-level common-refinement quotient sound for semantic isomorphism, a zero-truncated thin refinement-groupoid nerve equivalent to the discrete quotient nerve, a non-thin semantic refinement-path groupoid nerve with exact edge action, its categorical/nerve equivalence to the exact refinement-generated quotient-cell image subgroupoid, and a faithful aligned-cell-augmented non-groupoidal path category containing every source 2-cell in canonical one-column form are proved; normalized whiskering/append have exact three-model formulas, normalization commutes with raw whiskering, identity/original/source structural/marked-pair/transport and closure cases are proved, and a six-field criterion implies all-cell normalization; those six associator/unitor fields, semantic fullness, competing-move coherence, reduced-hammock invariance, standard weak-equivalence packaging, and the final Dwyer--Kan comparison remain open | +| Rezk classifying diagram | Outer simplicial category of composable interface strings, followed levelwise by the ordinary nerve | Every vertical level and horizontal row is a groupoid nerve and Kan; every horizontal row is strict Segal; the whole outer diagram is naturally `n ↦ Map(Δ[n], N(M.Object))`; `Map(∂Δ[n], N(M.Object))` is the genuine matching limit; every matching map is a fibration; the actual completeness map has an explicit simplicial homotopy inverse | Semantic proof layer; exact project-local `GroupoidalCompleteSegal` and `HomotopyEquivalenceWitness` evidence proved; the full cost-exact localization, common-universe local comparison, localization-aware all-dimensional relative outer Rezk map, source/target completeness witnesses, arbitrary-2-cell one-skeleton glue, vertical local 2-simplex/composite-diagonal glue, degree-one horizontal compositor squares, degree-two vertical pasting/interchange, the explicit three-tetrahedron degree-two compositor prism, all-degree local prism coherence, arbitrary outer-string vertex/restriction comparison, relative-outer gluing of every all-degree prism source vertex, strict decoded-pair naturality for every restriction, exact side-sensitive outer/local glue for every actual target prism-face vertex, a categorical-nerve equivalence from the presented relative-zigzag mapping nerve to every actual target local nerve, strict all-degree local-map factorization, outer essential surjectivity, target-independent algebraic/simplicial presentation universality, an audited `PresentedDwyerKanCore`, an independent right-associated linear hammock mapping category equivalent to the binary presentation and actual target local nerve, an audited `LinearHammockDwyerKanCore`, an exact arbitrary-height row-grid representation of every linear-hammock simplex, exact quotient/nerve interpretation of the fixed-shape aligned multi-column fragment, an executable elementary forward/marked-pair refinement calculus, an object-level common-refinement quotient sound for semantic isomorphism, a zero-truncated thin refinement-groupoid nerve equivalent to the discrete quotient nerve, a non-thin semantic refinement-path groupoid nerve with exact edge action, its categorical/nerve equivalence to the exact refinement-generated quotient-cell image subgroupoid, and a faithful aligned-cell-augmented non-groupoidal path category containing every source 2-cell in canonical one-column form are proved; normalized whiskering/append have exact three-model formulas, normalization commutes with raw whiskering, identity/original/source structural/marked-pair/left-unitor/transport and closure cases are proved, and a four-field criterion implies all-cell normalization; those four associator/right-unitor fields, semantic fullness, competing-move coherence, reduced-hammock invariance, standard weak-equivalence packaging, and the final Dwyer--Kan comparison remain open | The concrete Boolean model proves that `bit tensor unit` and `unit tensor bit` are unequal syntax trees in Lean while tensor symmetry makes them internally @@ -674,8 +674,8 @@ strictly extends the refinement path nerve, and is closed under normalized left/right whiskering and horizontal append. Normalization now commutes with both raw whiskerings; raw identity/original/source-identity/inverse and source-composition/inverse, equality-transport cases plus vertical/whiskering -closure are proved; all marked pair cases are proved too. Six explicit -associator and unitor obligations are +closure are proved; all marked pair and left-unitor cases are proved too. Four +explicit associator and right-unitor obligations are sufficient for the complete structural induction but remain open, together with critical-pair coherence and reduced-hammock invariance. These layers do not add `Equiv α β → α = β` and are not a complete presheaf model. diff --git a/Ript/Audit/AxiomChecks.lean b/Ript/Audit/AxiomChecks.lean index dc434fe..82ff8e7 100644 --- a/Ript/Audit/AxiomChecks.lean +++ b/Ript/Audit/AxiomChecks.lean @@ -1115,6 +1115,20 @@ set_option autoImplicit false #print axioms CategoryTheory.Bicategory.MarkedZigzag.HammockPath.markedUnitInv_normalizable #print axioms CategoryTheory.Bicategory.MarkedZigzag.HammockPath.markedCounit_normalizable #print axioms CategoryTheory.Bicategory.MarkedZigzag.HammockPath.markedCounitInv_normalizable +#print axioms CategoryTheory.Bicategory.MarkedZigzag.LinearWord.toWord_nil +#print axioms CategoryTheory.Bicategory.MarkedZigzag.LinearWord.flatten_nil +#print axioms CategoryTheory.Bicategory.MarkedZigzag.LinearWord.append_nil +#print axioms CategoryTheory.Bicategory.MarkedZigzag.LinearWord.append_assoc +#print axioms CategoryTheory.Bicategory.MarkedZigzag.LinearWord.toWordAppendIso_nil +#print axioms CategoryTheory.Bicategory.MarkedZigzag.LinearWord.leftUnitor_conjugation +#print axioms CategoryTheory.Bicategory.MarkedZigzag.LinearWord.leftUnitor_inv_conjugation +#print axioms CategoryTheory.Bicategory.MarkedZigzag.HammockPath.ofEq +#print axioms CategoryTheory.Bicategory.MarkedZigzag.HammockPath.toHom_ofEq +#print axioms CategoryTheory.Bicategory.MarkedZigzag.HammockPath.normalizedCellHom_eq_of_rel +#print axioms CategoryTheory.Bicategory.MarkedZigzag.HammockPath.normalizedCellHom_leftUnitor +#print axioms CategoryTheory.Bicategory.MarkedZigzag.HammockPath.normalizedCellHom_leftUnitorInv +#print axioms CategoryTheory.Bicategory.MarkedZigzag.HammockPath.leftUnitor_normalizable +#print axioms CategoryTheory.Bicategory.MarkedZigzag.HammockPath.leftUnitorInv_normalizable #print axioms Ript.Higher.CostExactZigzagMappingSpace.AlignedHammockGrid.linearHammockGridEquiv_simplex #print axioms Ript.Higher.CostExactZigzagMappingSpace.alignedHammockCore #print axioms Ript.Higher.CostExactZigzagMappingSpace.columnRefinementCore diff --git a/Ript/ForMathlib/CategoryTheory/Bicategory/MarkedZigzagAlignedHammock.lean b/Ript/ForMathlib/CategoryTheory/Bicategory/MarkedZigzagAlignedHammock.lean index f3d9b53..ad31348 100644 --- a/Ript/ForMathlib/CategoryTheory/Bicategory/MarkedZigzagAlignedHammock.lean +++ b/Ript/ForMathlib/CategoryTheory/Bicategory/MarkedZigzagAlignedHammock.lean @@ -28,9 +28,9 @@ all source 2-cells in one-column form, and is closed under normalized left/right whiskering and horizontal append. Normalization is natural for raw whiskering; identity, original, source-identity/inverse, composition-closure, whiskering, source-composition/inverse, and equality-transport induction branches are -complete; marked unit/counit pairs and inverses are complete too, while six -explicit -structural-generator obligations remain. Coverage of every presented quotient +complete; marked unit/counit pairs and inverses plus left unitor and inverse +are complete too. Four explicit associator/right-unitor obligations remain. +Coverage of every presented quotient 2-cell, critical-pair coherence, and reduced-hammock invariance are still absent, so this is not by itself the classical Dwyer--Kan hammock localization. -/ @@ -1742,6 +1742,14 @@ noncomputable def normalizedCellHom {X Y : B} (LinearWord.toWord W (LinearWord.flatten W second)) := normalizedHom W (Presented.mk W cell) +/-- Presented raw-cell relations induce equality of normalized semantics. -/ +theorem normalizedCellHom_eq_of_rel {X Y : B} + {first second : Word W X Y} {alpha beta : Cell W first second} + (relation : Presented.Rel (W := W) alpha beta) : + normalizedCellHom W alpha = normalizedCellHom W beta := by + unfold normalizedCellHom + rw [show Presented.mk W alpha = Presented.mk W beta from Quot.sound relation] + /-- Normalization commutes exactly with raw left whiskering. -/ theorem normalizedHom_whiskerLeft {X Y Z : B} (pre : Word W X Y) {first second : Word W Y Z} (alpha : Presented.Hom W first second) : @@ -1949,6 +1957,21 @@ def append {X Y Z : B} .vcomp (.whiskerRight first secondSource) (.whiskerLeft firstTarget second) +/-- A propositional equality of linear rows yields the transported identity +generated path. -/ +def ofEq {X Y : B} {first second : LinearWord W X Y} + (equality : first = second) : HammockPath W first second := by + subst second + exact .identity first + +/-- The equality path reduces to the identity after equality elimination. -/ +@[simp] +theorem toHom_ofEq {X Y : B} {first second : LinearWord W X Y} + (equality : first = second) : + toHom W (ofEq W equality) = equality ▸ 𝟙 (LinearWord.toWord W first) := by + subst second + rfl + /-- Exact quotient interpretation of horizontal path append. -/ @[simp] theorem toHom_append {X Y Z : B} @@ -2690,6 +2713,52 @@ theorem normalizedCellHom_markedCounitInv {X Y : B} exact normalizedCellHom_nil_twoAtoms W (.backward f hf) (.forward f) (.markedCounitInv f hf) +/-- Raw left unitor normalizes to the identity of the linear normal form. -/ +@[simp] +theorem normalizedCellHom_leftUnitor {X Y : B} (word : Word W X Y) : + normalizedCellHom W (Cell.leftUnitor (W := W) word) = + 𝟙 (LinearWord.toWord W (LinearWord.flatten W word)) := by + unfold normalizedCellHom normalizedHom + simp only [Word.append_eq_comp, LinearWord.flatten_append] + rw [LinearWord.normalizationIso_append W (.nil X) word] + rw [LinearWord.normalizationIso_nil W X] + simp only [LinearWord.flatten_nil] + rw [LinearWord.toWordAppendIso_nil W (LinearWord.flatten W word)] + simp only [Iso.trans_inv] + unfold LinearWord.appendIso + simp only [Iso.trans_inv, Bicategory.whiskerLeftIso, + Bicategory.whiskerRightIso, LinearWord.toWord_nil] + let iso := LinearWord.normalizationIso W word + let target := LinearWord.toWord W (LinearWord.flatten W word) + change AlignedCell.quotientVcomp W + (AlignedCell.quotientVcomp W + (Presented.wordLeftUnitorIso W target).inv + (AlignedCell.quotientVcomp W + (Presented.whiskerLeftHom W (.nil X) iso.inv) + (Presented.whiskerRightHom W word (𝟙 (Word.nil X))))) + (AlignedCell.quotientVcomp W + (Presented.wordLeftUnitorIso W word).hom iso.hom) = + 𝟙 target + have whiskerIdentity : + Presented.whiskerRightHom W word (𝟙 (Word.nil X)) = + 𝟙 (Word.append (W := W) (.nil X) word) := + Quot.sound (Presented.Rel.whisker_right_id (.nil X) word) + rw [whiskerIdentity, AlignedCell.quotientVcomp_comp_id] + simp only [AlignedCell.quotientVcomp_assoc] + exact LinearWord.leftUnitor_conjugation + (C := Presented.Localization W) iso + +/-- Raw inverse left unitor also normalizes to identity. -/ +@[simp] +theorem normalizedCellHom_leftUnitorInv {X Y : B} (word : Word W X Y) : + normalizedCellHom W (Cell.leftUnitorInv (W := W) word) = + 𝟙 (LinearWord.toWord W (LinearWord.flatten W word)) := by + have relationEquality := normalizedCellHom_eq_of_rel W + (Presented.Rel.left_unitor_inv_hom (W := W) word) + rw [normalizedCellHom_vcomp, normalizedCellHom_leftUnitor, + normalizedCellHom_id] at relationEquality + exact (AlignedCell.quotientVcomp_comp_id W _).symm.trans relationEquality + /-- Equality transport does not change normalized quotient semantics. -/ @[simp] theorem normalizedCellHom_transport {X Y : B} @@ -2860,6 +2929,18 @@ theorem markedCounitInv_normalizable {X Y : B} rw [Normalizable, normalizedCellHom_markedCounitInv] exact refinement_mem_semanticImage W (.insertMarkedCounitPair f hf (.nil Y)) +/-- Left unitor is hammock-normalizable. -/ +theorem leftUnitor_normalizable {X Y : B} (word : Word W X Y) : + Normalizable W (Cell.leftUnitor (W := W) word) := by + rw [Normalizable, normalizedCellHom_leftUnitor] + exact ⟨.identity (LinearWord.flatten W word), rfl⟩ + +/-- Inverse left unitor is hammock-normalizable. -/ +theorem leftUnitorInv_normalizable {X Y : B} (word : Word W X Y) : + Normalizable W (Cell.leftUnitorInv (W := W) word) := by + rw [Normalizable, normalizedCellHom_leftUnitorInv] + exact ⟨.identity (LinearWord.flatten W word), rfl⟩ + /-- Equality transport preserves raw-cell normalizability. -/ theorem transport_normalizable {X Y : B} {first second first' second' : Word W X Y} @@ -2874,7 +2955,7 @@ theorem transport_normalizable {X Y : B} /-- Exact remaining generator obligations for a complete raw-cell normalization induction. Identity, vertical composition, original cells, source identities, source composition, marked pairs, both whiskerings, and -equality transport are already +left unitors, plus equality transport are already discharged separately. -/ structure StructuralGeneratorNormalizable : Prop where /-- Binary associator. -/ @@ -2885,12 +2966,6 @@ structure StructuralGeneratorNormalizable : Prop where associatorInv : ∀ {X Y Z T : B} (first : Word W X Y) (second : Word W Y Z) (third : Word W Z T), Normalizable W (Cell.associatorInv (W := W) first second third) - /-- Binary left unitor. -/ - leftUnitor : ∀ {X Y : B} (word : Word W X Y), - Normalizable W (Cell.leftUnitor (W := W) word) - /-- Inverse binary left unitor. -/ - leftUnitorInv : ∀ {X Y : B} (word : Word W X Y), - Normalizable W (Cell.leftUnitorInv (W := W) word) /-- Binary right unitor. -/ rightUnitor : ∀ {X Y : B} (word : Word W X Y), Normalizable W (Cell.rightUnitor (W := W) word) @@ -2928,8 +3003,8 @@ theorem normalizable_of_structuralGenerators exact generators.associator first second third | associatorInv first second third => exact generators.associatorInv first second third - | leftUnitor word => exact generators.leftUnitor word - | leftUnitorInv word => exact generators.leftUnitorInv word + | leftUnitor word => exact leftUnitor_normalizable W word + | leftUnitorInv word => exact leftUnitorInv_normalizable W word | rightUnitor word => exact generators.rightUnitor word | rightUnitorInv word => exact generators.rightUnitorInv word | transport sourceEquality targetEquality cell member => diff --git a/Ript/ForMathlib/CategoryTheory/Bicategory/MarkedZigzagLinearHammock.lean b/Ript/ForMathlib/CategoryTheory/Bicategory/MarkedZigzagLinearHammock.lean index 6333266..1b4b008 100644 --- a/Ript/ForMathlib/CategoryTheory/Bicategory/MarkedZigzagLinearHammock.lean +++ b/Ript/ForMathlib/CategoryTheory/Bicategory/MarkedZigzagLinearHammock.lean @@ -52,6 +52,28 @@ def append {X Y Z : B} : | .nil _, second => second | .cons step rest, second => .cons step (append rest second) +/-- Appending an empty row on the right returns the original linear row. -/ +@[simp] +theorem append_nil {X Y : B} (word : LinearWord W X Y) : + append W word (.nil Y) = word := by + induction word with + | nil => rfl + | cons step rest ih => + simp only [append] + rw [ih] + +/-- Linear row append is associative. -/ +theorem append_assoc {X Y Z T : B} + (first : LinearWord W X Y) (second : LinearWord W Y Z) + (third : LinearWord W Z T) : + append W (append W first second) third = + append W first (append W second third) := by + induction first with + | nil => rfl + | cons step rest ih => + simp only [append] + rw [ih] + /-- Convert a right-associated linear word to the binary marked-zigzag syntax. -/ def toWord {X Y : B} : LinearWord W X Y → Word W X Y @@ -59,12 +81,22 @@ def toWord {X Y : B} : LinearWord W X Y → Word W X Y | .cons step rest => Word.append (W := W) (.atom step) (toWord rest) +/-- The empty linear row converts to the empty binary word. -/ +@[simp] +theorem toWord_nil (X : B) : toWord W (.nil X) = .nil X := + rfl + /-- Flatten a binary marked-zigzag composition tree to a linear word. -/ def flatten {X Y : B} : Word W X Y → LinearWord W X Y | .atom step => .cons step (.nil _) | .nil X => .nil X | .comp first second => append W (flatten first) (flatten second) +/-- The empty binary word flattens to the empty linear row. -/ +@[simp] +theorem flatten_nil (X : B) : flatten W (.nil X) = .nil X := + rfl + /-- Flattening a one-step forward word gives the corresponding singleton linear row. -/ @[simp] @@ -146,6 +178,12 @@ noncomputable def toWordAppendIso {X Y Z : B} (Presented.wordAssociatorIso W (Word.atom step) (toWord W rest) (toWord W second)).symm +/-- The append comparison with an empty first row is inverse left unitor. -/ +theorem toWordAppendIso_nil {X Y : B} (second : LinearWord W X Y) : + toWordAppendIso W (.nil X) second = + (Presented.wordLeftUnitorIso W (toWord W second)).symm := + rfl + /-- The append comparison for a singleton first row is the left-unitor comparison on the second row followed by inverse associativity. -/ theorem toWordAppendIso_singleton {X Y Z : B} (step : Step W X Y) @@ -234,6 +272,21 @@ theorem rightAssociatedPair_coherence {C : Type u} [Bicategory.{w, v} C] (α_ first second (𝟙 Z)).hom := by bicategory_coherence +/-- Conjugating a left unitor along an arbitrary isomorphism cancels. -/ +theorem leftUnitor_conjugation {C : Type u} [Bicategory.{w, v} C] + {X Y : C} {first second : X ⟶ Y} (iso : first ≅ second) : + (λ_ second).inv ≫ (𝟙 X ◁ iso.inv) ≫ + (λ_ first).hom ≫ iso.hom = 𝟙 second := by + simp + +/-- Inverse left-unitor conjugation along an arbitrary isomorphism also +cancels. -/ +theorem leftUnitor_inv_conjugation {C : Type u} [Bicategory.{w, v} C] + {X Y : C} {first second : X ⟶ Y} (iso : first ≅ second) : + iso.inv ≫ (λ_ first).inv ≫ (𝟙 X ◁ iso.hom) ≫ + (λ_ second).hom = 𝟙 second := by + simp + /-- Normalizing a binary word of two atomic steps is exactly inverse right unitor followed by the associator into the canonical right-associated row. -/ theorem normalizationIso_twoAtoms_hom {X Y Z : B} diff --git a/Ript/Higher/CostExactZigzagMappingSpace.lean b/Ript/Higher/CostExactZigzagMappingSpace.lean index 9566e4b..0e47709 100644 --- a/Ript/Higher/CostExactZigzagMappingSpace.lean +++ b/Ript/Higher/CostExactZigzagMappingSpace.lean @@ -36,7 +36,8 @@ normalized left/right whiskering and horizontal append, with exact three-model nerve formulas. A raw-cell normalization core records the completed identity, original, source-identity/inverse, source-composition/inverse, vertical, whiskering, marked-pair/inverse, and transport branches, plus a conditional -all-cell induction from six remaining structural generators. Coverage of all +left-unitor/inverse and transport branches, plus a conditional all-cell +induction from four remaining structural generators. Coverage of all presented quotient 2-cells, competing-move coherence, and reduced-hammock invariance are still absent, so these results are not by themselves the final Dwyer--Kan theorem. @@ -1432,6 +1433,20 @@ structure HammockRawCellNormalizationCore : Prop where (costExactArrows R) (Bicategory.MarkedZigzag.Cell.markedCounitInv (W := costExactArrows R) f hf) + /-- Left unitor is normalizable. -/ + leftUnitor : ∀ {M N : ProcessModel.{u, v, w} R} + (word : CostExactZigzag.Word (R := R) M N), + Bicategory.MarkedZigzag.HammockPath.Normalizable + (costExactArrows R) + (Bicategory.MarkedZigzag.Cell.leftUnitor + (W := costExactArrows R) word) + /-- Inverse left unitor is normalizable. -/ + leftUnitorInv : ∀ {M N : ProcessModel.{u, v, w} R} + (word : CostExactZigzag.Word (R := R) M N), + Bicategory.MarkedZigzag.HammockPath.Normalizable + (costExactArrows R) + (Bicategory.MarkedZigzag.Cell.leftUnitorInv + (W := costExactArrows R) word) /-- Raw left whiskering preserves normalizability. -/ whiskerLeft : ∀ (M N P : ProcessModel.{u, v, w} R) (pre : CostExactZigzag.Word (R := R) M N) @@ -1515,6 +1530,12 @@ theorem hammockRawCellNormalizationCore : markedCounitInv := fun f hf => Bicategory.MarkedZigzag.HammockPath.markedCounitInv_normalizable (costExactArrows R) f hf + leftUnitor := fun word => + Bicategory.MarkedZigzag.HammockPath.leftUnitor_normalizable + (costExactArrows R) word + leftUnitorInv := fun word => + Bicategory.MarkedZigzag.HammockPath.leftUnitorInv_normalizable + (costExactArrows R) word whiskerLeft := fun _ _ _ pre {_ _} {_} member => Bicategory.MarkedZigzag.HammockPath.whiskerLeft_normalizable (costExactArrows R) pre member diff --git a/docs/en/RESEARCH_STATUS.md b/docs/en/RESEARCH_STATUS.md index 14ecbb6..edd125e 100644 --- a/docs/en/RESEARCH_STATUS.md +++ b/docs/en/RESEARCH_STATUS.md @@ -399,8 +399,11 @@ exchange and cancellation; source identity/inverse and equality transport are normalizable as well. Source composition/inverse now normalize exactly to executable forward expansion/contraction using a pure bicategorical two-atom coherence formula. Generic empty/two-atom formulas now normalize marked -unit/counit and inverses exactly to pair insertion/deletion. The remaining six -associator/unitor obligations are recorded exactly and already imply all-cell +unit/counit and inverses exactly to pair insertion/deletion. These structural +obligations are further reduced: computable linear append +right-unit/associativity equalities and equality paths are available, and left +unitor/inverse normalize to identity by arbitrary-iso conjugation. The +remaining four associator/right-unitor obligations already imply all-cell normalization conditionally, but are not assumed unconditionally. Semantic fullness, competing-move coherence, reduced-hammock invariance, standard weak-equivalence packaging, and diff --git a/docs/en/reference/AXIOMS.md b/docs/en/reference/AXIOMS.md index ccfe883..7b01d15 100644 --- a/docs/en/reference/AXIOMS.md +++ b/docs/en/reference/AXIOMS.md @@ -1034,6 +1034,20 @@ the actual output of `lake env lean Ript/Audit/AxiomChecks.lean`. | `CategoryTheory.Bicategory.MarkedZigzag.HammockPath.markedUnitInv_normalizable` | `[propext, Classical.choice, Quot.sound]` | `Ript/ForMathlib/CategoryTheory/Bicategory/MarkedZigzagAlignedHammock.lean` | | `CategoryTheory.Bicategory.MarkedZigzag.HammockPath.markedCounit_normalizable` | `[propext, Classical.choice, Quot.sound]` | `Ript/ForMathlib/CategoryTheory/Bicategory/MarkedZigzagAlignedHammock.lean` | | `CategoryTheory.Bicategory.MarkedZigzag.HammockPath.markedCounitInv_normalizable` | `[propext, Classical.choice, Quot.sound]` | `Ript/ForMathlib/CategoryTheory/Bicategory/MarkedZigzagAlignedHammock.lean` | +| `CategoryTheory.Bicategory.MarkedZigzag.LinearWord.toWord_nil` | `none` | `Ript/ForMathlib/CategoryTheory/Bicategory/MarkedZigzagLinearHammock.lean` | +| `CategoryTheory.Bicategory.MarkedZigzag.LinearWord.flatten_nil` | `none` | `Ript/ForMathlib/CategoryTheory/Bicategory/MarkedZigzagLinearHammock.lean` | +| `CategoryTheory.Bicategory.MarkedZigzag.LinearWord.append_nil` | `none` | `Ript/ForMathlib/CategoryTheory/Bicategory/MarkedZigzagLinearHammock.lean` | +| `CategoryTheory.Bicategory.MarkedZigzag.LinearWord.append_assoc` | `none` | `Ript/ForMathlib/CategoryTheory/Bicategory/MarkedZigzagLinearHammock.lean` | +| `CategoryTheory.Bicategory.MarkedZigzag.LinearWord.toWordAppendIso_nil` | `[propext, Classical.choice, Quot.sound]` | `Ript/ForMathlib/CategoryTheory/Bicategory/MarkedZigzagLinearHammock.lean` | +| `CategoryTheory.Bicategory.MarkedZigzag.LinearWord.leftUnitor_conjugation` | `[propext, Quot.sound]` | `Ript/ForMathlib/CategoryTheory/Bicategory/MarkedZigzagLinearHammock.lean` | +| `CategoryTheory.Bicategory.MarkedZigzag.LinearWord.leftUnitor_inv_conjugation` | `[propext, Quot.sound]` | `Ript/ForMathlib/CategoryTheory/Bicategory/MarkedZigzagLinearHammock.lean` | +| `CategoryTheory.Bicategory.MarkedZigzag.HammockPath.ofEq` | `none` | `Ript/ForMathlib/CategoryTheory/Bicategory/MarkedZigzagAlignedHammock.lean` | +| `CategoryTheory.Bicategory.MarkedZigzag.HammockPath.toHom_ofEq` | `[propext, Classical.choice, Quot.sound]` | `Ript/ForMathlib/CategoryTheory/Bicategory/MarkedZigzagAlignedHammock.lean` | +| `CategoryTheory.Bicategory.MarkedZigzag.HammockPath.normalizedCellHom_eq_of_rel` | `[propext, Classical.choice, Quot.sound]` | `Ript/ForMathlib/CategoryTheory/Bicategory/MarkedZigzagAlignedHammock.lean` | +| `CategoryTheory.Bicategory.MarkedZigzag.HammockPath.normalizedCellHom_leftUnitor` | `[propext, Classical.choice, Quot.sound]` | `Ript/ForMathlib/CategoryTheory/Bicategory/MarkedZigzagAlignedHammock.lean` | +| `CategoryTheory.Bicategory.MarkedZigzag.HammockPath.normalizedCellHom_leftUnitorInv` | `[propext, Classical.choice, Quot.sound]` | `Ript/ForMathlib/CategoryTheory/Bicategory/MarkedZigzagAlignedHammock.lean` | +| `CategoryTheory.Bicategory.MarkedZigzag.HammockPath.leftUnitor_normalizable` | `[propext, Classical.choice, Quot.sound]` | `Ript/ForMathlib/CategoryTheory/Bicategory/MarkedZigzagAlignedHammock.lean` | +| `CategoryTheory.Bicategory.MarkedZigzag.HammockPath.leftUnitorInv_normalizable` | `[propext, Classical.choice, Quot.sound]` | `Ript/ForMathlib/CategoryTheory/Bicategory/MarkedZigzagAlignedHammock.lean` | | `Ript.Higher.CostExactZigzagMappingSpace.AlignedHammockGrid.linearHammockGridEquiv_simplex` | `[propext, Classical.choice, Quot.sound]` | `Ript/Higher/CostExactZigzagMappingSpace.lean` | | `Ript.Higher.CostExactZigzagMappingSpace.alignedHammockCore` | `[propext, Classical.choice, Quot.sound]` | `Ript/Higher/CostExactZigzagMappingSpace.lean` | | `Ript.Higher.CostExactZigzagMappingSpace.columnRefinementCore` | `[propext, Classical.choice, Quot.sound]` | `Ript/Higher/CostExactZigzagMappingSpace.lean` | diff --git a/docs/en/reference/BLUEPRINT.md b/docs/en/reference/BLUEPRINT.md index c3c2636..3da79ab 100644 --- a/docs/en/reference/BLUEPRINT.md +++ b/docs/en/reference/BLUEPRINT.md @@ -390,9 +390,9 @@ Every node in this graph is an existing compiled module. | 12 (zero-truncated common-refinement mapping nerve) | Wrapped rows form a thin common-refinement groupoid; its canonical functor to the discrete row quotient is faithful, full, essentially surjective, and hence a categorical equivalence; the induced nerve map has categorical-nerve equivalence evidence, an explicit simplicial inverse and both homotopies, and exact row-vertex action | PROVED | | 12 (non-thin semantic refinement-path nerve) | Refinement syntax modulo equality of quotient-cell interpretations forms a non-thin groupoid; executable reversal supplies inverses; its semantic functor into the linear mapping category is faithful and essentially surjective on row objects, maps every path to an isomorphism, and induces a nerve map with exact row-vertex and arbitrary-refinement-edge action; the zero-truncation functor to the thin groupoid is full and essentially surjective | PROVED | | 12 (exact refinement-generated semantic image) | Actual quotient 2-cells equipped with existence of an executable refinement generator form a non-thin image subgroupoid of the linear mapping category; inclusion is faithful, the semantic refinement-path functor is full, faithful, essentially surjective and hence an equivalence onto this exact image, and the induced nerve equivalence has an explicit simplicial homotopy inverse, exact generator-edge action, and strict factorization of the original semantic nerve map | PROVED | -| 12 (aligned-cell-augmented hammock paths) | A non-groupoidal generated path syntax alternates executable refinements with arbitrary aligned raw 2-cells and vertical composition, then quotients only by equality of quotient-cell semantics; normalized left/right whiskering enters and leaves binary append through canonical linear-normal-form isomorphisms, horizontal append composes those operations, and semantic equality/image membership are closed under all three; explicit append-isomorphism exchange/cancellation proves normalization naturality for raw left/right whiskering; pure bicategorical coherence identifies two-atomic-step normalization, so source composition/inverse normalize exactly to forward expansion/contraction; generic empty/two-atom formulas normalize marked unit/counit and inverses exactly to pair insertion/deletion; raw identity, original, source structural, marked pair, whiskering, vertical-composite, and equality-transport cells are normalizable; `StructuralGeneratorNormalizable` now lists exactly six associator/unitor obligations and yields all-cell normalization by structural induction | PROVED | +| 12 (aligned-cell-augmented hammock paths) | A non-groupoidal generated path syntax alternates executable refinements with arbitrary aligned raw 2-cells and vertical composition, then quotients only by equality of quotient-cell semantics; normalized left/right whiskering enters and leaves binary append through canonical linear-normal-form isomorphisms, horizontal append composes those operations, and semantic equality/image membership are closed under all three; explicit append-isomorphism exchange/cancellation proves normalization naturality for raw left/right whiskering; pure bicategorical coherence identifies two-atomic-step normalization, so source composition/inverse normalize exactly to forward expansion/contraction; generic empty/two-atom formulas normalize marked unit/counit and inverses exactly to pair insertion/deletion; linear append right-unit/associativity equalities and computable equality paths are available, and left unitor/inverse normalize to identity via arbitrary-iso conjugation; `StructuralGeneratorNormalizable` now lists exactly four associator/right-unitor obligations and yields all-cell normalization by structural induction | PROVED | | 12 (cost-exact two-layer global comparison) | Pseudofunctor-induced functor on homotopy categories; localization-aware relative Rezk map and auxiliary ordinary outer map into the actual marked-zigzag target; explicit source/target outer completeness homotopy equivalences; marked outer arrows factoring through the target actual-equivalence space; packaging with the exact non-groupoidal local nerve comparison; exact vertex, identity, horizontal-composition, associator, and left/right-unitor gluing; arbitrary invertible local 2-cell decoding; explicit pentagon and triangle compatibility | PROVED | -| 12 (global cost-exact complete-Segal/Rezk equivalence) | Discharge the six explicit `StructuralGeneratorNormalizable` fields for associator/inverse and left/right unitors and inverses; deduce every presented quotient 2-cell has an alternating aligned/refinement path and hence semantic fullness, then prove critical-pair coherence and reduced-hammock homotopical invariance (or compare the generated path category to another accepted derived mapping-space construction), connect that comparison to a standard weak-equivalence interface, and finish the standard Dwyer--Kan/Rezk weak-equivalence and completeness theorem | OPEN_RESEARCH | +| 12 (global cost-exact complete-Segal/Rezk equivalence) | Discharge the four explicit `StructuralGeneratorNormalizable` fields for associator/inverse and right unitor/inverse; deduce every presented quotient 2-cell has an alternating aligned/refinement path and hence semantic fullness, then prove critical-pair coherence and reduced-hammock homotopical invariance (or compare the generated path category to another accepted derived mapping-space construction), connect that comparison to a standard weak-equivalence interface, and finish the standard Dwyer--Kan/Rezk weak-equivalence and completeness theorem | OPEN_RESEARCH | ## Finite deterministic copy-discard theorem records diff --git a/docs/en/reference/CONJECTURES.md b/docs/en/reference/CONJECTURES.md index eee2480..2d3d0cc 100644 --- a/docs/en/reference/CONJECTURES.md +++ b/docs/en/reference/CONJECTURES.md @@ -475,13 +475,15 @@ are normalizable, and normalizability is closed under vertical composition. Normalization naturality for both raw whiskerings is now proved by explicit append-isomorphism exchange and cancellation; source identity and inverse plus equality transport are normalizable too. The exact remaining induction basis -has now shrunk to the six-field `StructuralGeneratorNormalizable` record: -associator and inverse, and both unitors and inverses. Source +has now shrunk to the four-field `StructuralGeneratorNormalizable` record: +associator and inverse plus right unitor and inverse. Source composition/inverse are normalized exactly by executable forward expansion/contraction using the audited two-atomic-step coherence formula; generic empty/two-atom formulas now normalize marked unit/counit and -inverses exactly by pair insertion/deletion. The remaining record already -implies normalization of every raw +inverses exactly by pair insertion/deletion. Computable linear append right- +unit/associativity equalities and equality paths are available; left unitor and +inverse normalize to identity by arbitrary-iso conjugation. The remaining +record already implies normalization of every raw cell by structural induction, but none of its unproved fields is assumed in an unconditional theorem. Competing forward/marked moves, reduced-hammock moves, and their homotopical invariance remain open. @@ -593,8 +595,8 @@ source 2-cell in one-column form, strictly extends refinement paths, and is closed under normalized left/right whiskering and horizontal append. Identity, vertical-composite, original-cell, both whiskering, source-identity/inverse, source-composition/inverse, and equality-transport normalization cases are -proved; all marked unit/counit cases are proved too. The remaining six -structural generator obligations are explicit and +proved; all marked unit/counit and left-unitor cases are proved too. The +remaining four structural generator obligations are explicit and sufficient for the complete induction, but remain unproved; critical-pair coherence and reduced-hammock invariance remain open. diff --git a/docs/en/reference/MODEL_MATRIX.md b/docs/en/reference/MODEL_MATRIX.md index 9cd9107..df45c43 100644 --- a/docs/en/reference/MODEL_MATRIX.md +++ b/docs/en/reference/MODEL_MATRIX.md @@ -564,7 +564,7 @@ or the executable cores. | Presheaf universe | Type-valued presheaves on the internal groupoid | Yoneda is fully faithful; representable transformations/isomorphisms correspond to internal identity/equivalence | Semantic proof layer; Mathlib Yoneda audits with classical choice | | Yoneda envelope | Essential image of representables in the presheaf universe | Groupoid equivalent to the internal groupoid; inclusion factors Yoneda; the restricted Yoneda functor is a Mathlib localization at all internal identities | Noncomputable essential-image witnesses; exact ordinary localization of an already-groupoidal source, not a Rezk completion | | Simplicial interface nerve | Ordinary categorical nerve of the internal groupoid | Complete Kan horn filling, strict Segal, quasicategory, 2-coskeletal; vertices/edges/2-simplices encode interfaces, identities, and composition; homotopy category recovers the groupoid | Semantic proof layer; chosen fillers audit with classical choice; no complete-Segal or Rezk claim | -| Rezk classifying diagram | Outer simplicial category of composable interface strings, followed levelwise by the ordinary nerve | Every vertical level and horizontal row is a groupoid nerve and Kan; every horizontal row is strict Segal; the whole outer diagram is naturally `n ↦ Map(Δ[n], N(M.Object))`; `Map(∂Δ[n], N(M.Object))` is the genuine matching limit; every matching map is a fibration; the actual completeness map has an explicit simplicial homotopy inverse | Semantic proof layer; exact project-local `GroupoidalCompleteSegal` and `HomotopyEquivalenceWitness` evidence proved; the full cost-exact localization, common-universe local comparison, localization-aware all-dimensional relative outer Rezk map, source/target completeness witnesses, arbitrary-2-cell one-skeleton glue, vertical local 2-simplex/composite-diagonal glue, degree-one horizontal compositor squares, degree-two vertical pasting/interchange, the explicit three-tetrahedron degree-two compositor prism, all-degree local prism coherence, arbitrary outer-string vertex/restriction comparison, relative-outer gluing of every all-degree prism source vertex, strict decoded-pair naturality for every restriction, exact side-sensitive outer/local glue for every actual target prism-face vertex, a categorical-nerve equivalence from the presented relative-zigzag mapping nerve to every actual target local nerve, strict all-degree local-map factorization, outer essential surjectivity, target-independent algebraic/simplicial presentation universality, an audited `PresentedDwyerKanCore`, an independent right-associated linear hammock mapping category equivalent to the binary presentation and actual target local nerve, an audited `LinearHammockDwyerKanCore`, an exact arbitrary-height row-grid representation of every linear-hammock simplex, exact quotient/nerve interpretation of the fixed-shape aligned multi-column fragment, an executable elementary forward/marked-pair refinement calculus, an object-level common-refinement quotient sound for semantic isomorphism, a zero-truncated thin refinement-groupoid nerve equivalent to the discrete quotient nerve, a non-thin semantic refinement-path groupoid nerve with exact edge action, its categorical/nerve equivalence to the exact refinement-generated quotient-cell image subgroupoid, and a faithful aligned-cell-augmented non-groupoidal path category containing every source 2-cell in canonical one-column form are proved; normalized whiskering/append have exact three-model formulas, normalization commutes with raw whiskering, identity/original/source structural/marked-pair/transport and closure cases are proved, and a six-field criterion implies all-cell normalization; those six associator/unitor fields, semantic fullness, competing-move coherence, reduced-hammock invariance, standard weak-equivalence packaging, and the final Dwyer--Kan comparison remain open | +| Rezk classifying diagram | Outer simplicial category of composable interface strings, followed levelwise by the ordinary nerve | Every vertical level and horizontal row is a groupoid nerve and Kan; every horizontal row is strict Segal; the whole outer diagram is naturally `n ↦ Map(Δ[n], N(M.Object))`; `Map(∂Δ[n], N(M.Object))` is the genuine matching limit; every matching map is a fibration; the actual completeness map has an explicit simplicial homotopy inverse | Semantic proof layer; exact project-local `GroupoidalCompleteSegal` and `HomotopyEquivalenceWitness` evidence proved; the full cost-exact localization, common-universe local comparison, localization-aware all-dimensional relative outer Rezk map, source/target completeness witnesses, arbitrary-2-cell one-skeleton glue, vertical local 2-simplex/composite-diagonal glue, degree-one horizontal compositor squares, degree-two vertical pasting/interchange, the explicit three-tetrahedron degree-two compositor prism, all-degree local prism coherence, arbitrary outer-string vertex/restriction comparison, relative-outer gluing of every all-degree prism source vertex, strict decoded-pair naturality for every restriction, exact side-sensitive outer/local glue for every actual target prism-face vertex, a categorical-nerve equivalence from the presented relative-zigzag mapping nerve to every actual target local nerve, strict all-degree local-map factorization, outer essential surjectivity, target-independent algebraic/simplicial presentation universality, an audited `PresentedDwyerKanCore`, an independent right-associated linear hammock mapping category equivalent to the binary presentation and actual target local nerve, an audited `LinearHammockDwyerKanCore`, an exact arbitrary-height row-grid representation of every linear-hammock simplex, exact quotient/nerve interpretation of the fixed-shape aligned multi-column fragment, an executable elementary forward/marked-pair refinement calculus, an object-level common-refinement quotient sound for semantic isomorphism, a zero-truncated thin refinement-groupoid nerve equivalent to the discrete quotient nerve, a non-thin semantic refinement-path groupoid nerve with exact edge action, its categorical/nerve equivalence to the exact refinement-generated quotient-cell image subgroupoid, and a faithful aligned-cell-augmented non-groupoidal path category containing every source 2-cell in canonical one-column form are proved; normalized whiskering/append have exact three-model formulas, normalization commutes with raw whiskering, identity/original/source structural/marked-pair/left-unitor/transport and closure cases are proved, and a four-field criterion implies all-cell normalization; those four associator/right-unitor fields, semantic fullness, competing-move coherence, reduced-hammock invariance, standard weak-equivalence packaging, and the final Dwyer--Kan comparison remain open | The concrete Boolean model proves that `bit tensor unit` and `unit tensor bit` are unequal syntax trees in Lean while tensor symmetry makes them internally @@ -674,8 +674,8 @@ strictly extends the refinement path nerve, and is closed under normalized left/right whiskering and horizontal append. Normalization now commutes with both raw whiskerings; raw identity/original/source-identity/inverse and source-composition/inverse, equality-transport cases plus vertical/whiskering -closure are proved; all marked pair cases are proved too. Six explicit -associator and unitor obligations are +closure are proved; all marked pair and left-unitor cases are proved too. Four +explicit associator and right-unitor obligations are sufficient for the complete structural induction but remain open, together with critical-pair coherence and reduced-hammock invariance. These layers do not add `Equiv α β → α = β` and are not a complete presheaf model. diff --git a/docs/eo/RESEARCH_STATUS.md b/docs/eo/RESEARCH_STATUS.md index 95881f3..5ed0384 100644 --- a/docs/eo/RESEARCH_STATUS.md +++ b/docs/eo/RESEARCH_STATUS.md @@ -370,8 +370,10 @@ font-identecaj/inversaj kaj transportaj kazoj, kune kun vertikala/whiskering fermo, estas pruvitaj. Font-kompona/inversa estas pruvitaj per pura bikategoria du-atoma kohereco kaj ekzakta forward expand/contract. Ĝeneralaj malplena- vorto/du-atoma-ciklo formuloj normaligas markitajn unit/counit kaj inversojn -per ekzakta para enmeto/forigo. La ceteraj ses asociatoraj kaj unuitoraj -generatoraj kampoj estas eksplicitaj kaj kondiĉe +per ekzakta para enmeto/forigo. Linearaj append-right-unit/asocieco kaj +komputeblaj egalecvojoj estas aldonitaj; left-unitor kaj inverso normaligas al +identeco per arbitra-izomorfia konjugacio. La ceteraj kvar asociatoraj kaj +right-unitoraj generatoraj kampoj estas eksplicitaj kaj kondiĉe implicas normaligon de ĉiu kruda ĉelo, sed ne estas senkondiĉe pruvitaj. Semantika pleneco, kohereco de kritikaj paroj, reduktita-hammock invariant eco kaj norma malfort-ekvivalenta pako restas malfermitaj. diff --git a/docs/eo/reference/AXIOMS.md b/docs/eo/reference/AXIOMS.md index dd9d1bb..17c2050 100644 --- a/docs/eo/reference/AXIOMS.md +++ b/docs/eo/reference/AXIOMS.md @@ -1036,6 +1036,20 @@ per `scripts/sync-doc-reference-tables.sh` kaj ne estu mane redaktataj. | `CategoryTheory.Bicategory.MarkedZigzag.HammockPath.markedUnitInv_normalizable` | `[propext, Classical.choice, Quot.sound]` | `Ript/ForMathlib/CategoryTheory/Bicategory/MarkedZigzagAlignedHammock.lean` | | `CategoryTheory.Bicategory.MarkedZigzag.HammockPath.markedCounit_normalizable` | `[propext, Classical.choice, Quot.sound]` | `Ript/ForMathlib/CategoryTheory/Bicategory/MarkedZigzagAlignedHammock.lean` | | `CategoryTheory.Bicategory.MarkedZigzag.HammockPath.markedCounitInv_normalizable` | `[propext, Classical.choice, Quot.sound]` | `Ript/ForMathlib/CategoryTheory/Bicategory/MarkedZigzagAlignedHammock.lean` | +| `CategoryTheory.Bicategory.MarkedZigzag.LinearWord.toWord_nil` | `none` | `Ript/ForMathlib/CategoryTheory/Bicategory/MarkedZigzagLinearHammock.lean` | +| `CategoryTheory.Bicategory.MarkedZigzag.LinearWord.flatten_nil` | `none` | `Ript/ForMathlib/CategoryTheory/Bicategory/MarkedZigzagLinearHammock.lean` | +| `CategoryTheory.Bicategory.MarkedZigzag.LinearWord.append_nil` | `none` | `Ript/ForMathlib/CategoryTheory/Bicategory/MarkedZigzagLinearHammock.lean` | +| `CategoryTheory.Bicategory.MarkedZigzag.LinearWord.append_assoc` | `none` | `Ript/ForMathlib/CategoryTheory/Bicategory/MarkedZigzagLinearHammock.lean` | +| `CategoryTheory.Bicategory.MarkedZigzag.LinearWord.toWordAppendIso_nil` | `[propext, Classical.choice, Quot.sound]` | `Ript/ForMathlib/CategoryTheory/Bicategory/MarkedZigzagLinearHammock.lean` | +| `CategoryTheory.Bicategory.MarkedZigzag.LinearWord.leftUnitor_conjugation` | `[propext, Quot.sound]` | `Ript/ForMathlib/CategoryTheory/Bicategory/MarkedZigzagLinearHammock.lean` | +| `CategoryTheory.Bicategory.MarkedZigzag.LinearWord.leftUnitor_inv_conjugation` | `[propext, Quot.sound]` | `Ript/ForMathlib/CategoryTheory/Bicategory/MarkedZigzagLinearHammock.lean` | +| `CategoryTheory.Bicategory.MarkedZigzag.HammockPath.ofEq` | `none` | `Ript/ForMathlib/CategoryTheory/Bicategory/MarkedZigzagAlignedHammock.lean` | +| `CategoryTheory.Bicategory.MarkedZigzag.HammockPath.toHom_ofEq` | `[propext, Classical.choice, Quot.sound]` | `Ript/ForMathlib/CategoryTheory/Bicategory/MarkedZigzagAlignedHammock.lean` | +| `CategoryTheory.Bicategory.MarkedZigzag.HammockPath.normalizedCellHom_eq_of_rel` | `[propext, Classical.choice, Quot.sound]` | `Ript/ForMathlib/CategoryTheory/Bicategory/MarkedZigzagAlignedHammock.lean` | +| `CategoryTheory.Bicategory.MarkedZigzag.HammockPath.normalizedCellHom_leftUnitor` | `[propext, Classical.choice, Quot.sound]` | `Ript/ForMathlib/CategoryTheory/Bicategory/MarkedZigzagAlignedHammock.lean` | +| `CategoryTheory.Bicategory.MarkedZigzag.HammockPath.normalizedCellHom_leftUnitorInv` | `[propext, Classical.choice, Quot.sound]` | `Ript/ForMathlib/CategoryTheory/Bicategory/MarkedZigzagAlignedHammock.lean` | +| `CategoryTheory.Bicategory.MarkedZigzag.HammockPath.leftUnitor_normalizable` | `[propext, Classical.choice, Quot.sound]` | `Ript/ForMathlib/CategoryTheory/Bicategory/MarkedZigzagAlignedHammock.lean` | +| `CategoryTheory.Bicategory.MarkedZigzag.HammockPath.leftUnitorInv_normalizable` | `[propext, Classical.choice, Quot.sound]` | `Ript/ForMathlib/CategoryTheory/Bicategory/MarkedZigzagAlignedHammock.lean` | | `Ript.Higher.CostExactZigzagMappingSpace.AlignedHammockGrid.linearHammockGridEquiv_simplex` | `[propext, Classical.choice, Quot.sound]` | `Ript/Higher/CostExactZigzagMappingSpace.lean` | | `Ript.Higher.CostExactZigzagMappingSpace.alignedHammockCore` | `[propext, Classical.choice, Quot.sound]` | `Ript/Higher/CostExactZigzagMappingSpace.lean` | | `Ript.Higher.CostExactZigzagMappingSpace.columnRefinementCore` | `[propext, Classical.choice, Quot.sound]` | `Ript/Higher/CostExactZigzagMappingSpace.lean` | diff --git a/docs/ja/RESEARCH_STATUS.md b/docs/ja/RESEARCH_STATUS.md index c778610..a953ea8 100644 --- a/docs/ja/RESEARCH_STATUS.md +++ b/docs/ja/RESEARCH_STATUS.md @@ -151,7 +151,7 @@ fold は一意な解釈です。生成合同は全代数で健全で、木項モ 制限付きながら真に独立した右結合 linear hammock 対象モデルもできました。typed step 列は二分 words と相互変換・平坦化され、長さを厳密に保存し、同値な mapping category、nerve の明示的ホモトピー逆、実際の対象 local nerve への直接比較を与えます。`LinearHammockDwyerKanCore` はこれを outer essential surjectivity と統合します。古典的な任意グリッド hammock または他の受理された derived 構成との比較は未解決です。 -任意高さの垂直 grid も明示化されました。`n`-grid は `n + 1` 行の linear hammocks、`n` 本の隣接商 2-cell 辺、全端点方程式を持ち、strict-Segal 再構成により linear hammock nerve の `n`-simplex と同値です。行、辺、復号、双方向 round trip は厳密に証明されています。固定形状の水平多列部分も形式化されました。同形の行は各共通列に一つの raw atomic 2-cell を持ち、幅と実行可能な水平 append は厳密で、商解釈は interchange により列ごとの恒等と垂直合成を保存し、任意高さの aligned grid は行と解釈済み辺を厳密に保つ genuine simplex を再構成します。基本的な前向き列 refinement も実行可能です。恒等列の挿入/削除、合成列の展開/縮約、任意の共通 prefix 下での move、推移合成、符号付き幅変化、商意味論での双方向 cancellation が証明されています。marked reverse 構造にも unit pair `f ; f⁻¹` と counit pair `f⁻¹ ; f` の実行可能な挿入/削除が加わり、符号付き幅 `±2`、厳密な意味同型、双方向 round trip、任意 prefix 安定性が証明されました。各 refinement は実行可能な逆と統一意味同型を持ちます。二本脚 common-refinement span は同値関係と行商を構成し、商等式は common-refinability と同値で、対象等式を仮定せず意味同型を与えます。0-切断 mapping 層も完成しました。包装された行は薄い common-refinement 群胚を構成し、離散行商圏と圏同値で、nerve 比較には明示的単体逆と両ホモトピーがあります。非薄意味 refinement-path 群胚は quotient-cell 意味で異なる paths を保持し、linear mapping nerve への nerve map は faithful、行対象上 essentially surjective、全 path を同型へ写し、厳密な頂点/辺式を持ちます。薄群胚への 0-切断も full かつ essentially surjective です。実際の商 2-cell と「実行可能な refinement が生成する」という存在証明は厳密な意味像群胚を構成します。path 群胚はこれと圏同値で、nerve 比較は明示的単体ホモトピー逆を持ち、完全な linear mapping category への像包含は faithful、元の意味 nerve map は厳密にこの像を経由します。さらに大きい非群胚生成 path 圏は refinement と任意の aligned raw 2-cell を交互に合成できます。意味関手と refinement 埋め込みはいずれも faithful で、旧 nerve map は厳密にこれを経由し、各始域 2-cell は right-unitor で共役された元の商 2-cell に等しい正準一列辺を持ちます。正規化された左右 whiskering と水平 append は意味同値と像所属を保存し、厳密な三モデル nerve 公式を持ちます。正規化は raw 左右 whiskering と自然に可換することが証明され、raw identity、original、source identity/逆、source composition/逆、transport、および垂直/whiskering 閉包分岐が完成しました。source composition の正逆は純双圏二原子コヒーレンス式により forward expand/contract と厳密に一致し、一般の空 word/二原子閉路式により marked unit/counit とその逆も pair insert/delete と厳密に一致します。残る六の associator と左右 unitor 正逆構造生成元フィールドは明示され、条件付きで全 raw Cell 正規化を導きますが、まだ無条件には証明されていません。semantic fullness、critical-pair coherence、reduced-hammock 不変性、標準弱同値 packaging は未解決です。 +任意高さの垂直 grid も明示化されました。`n`-grid は `n + 1` 行の linear hammocks、`n` 本の隣接商 2-cell 辺、全端点方程式を持ち、strict-Segal 再構成により linear hammock nerve の `n`-simplex と同値です。行、辺、復号、双方向 round trip は厳密に証明されています。固定形状の水平多列部分も形式化されました。同形の行は各共通列に一つの raw atomic 2-cell を持ち、幅と実行可能な水平 append は厳密で、商解釈は interchange により列ごとの恒等と垂直合成を保存し、任意高さの aligned grid は行と解釈済み辺を厳密に保つ genuine simplex を再構成します。基本的な前向き列 refinement も実行可能です。恒等列の挿入/削除、合成列の展開/縮約、任意の共通 prefix 下での move、推移合成、符号付き幅変化、商意味論での双方向 cancellation が証明されています。marked reverse 構造にも unit pair `f ; f⁻¹` と counit pair `f⁻¹ ; f` の実行可能な挿入/削除が加わり、符号付き幅 `±2`、厳密な意味同型、双方向 round trip、任意 prefix 安定性が証明されました。各 refinement は実行可能な逆と統一意味同型を持ちます。二本脚 common-refinement span は同値関係と行商を構成し、商等式は common-refinability と同値で、対象等式を仮定せず意味同型を与えます。0-切断 mapping 層も完成しました。包装された行は薄い common-refinement 群胚を構成し、離散行商圏と圏同値で、nerve 比較には明示的単体逆と両ホモトピーがあります。非薄意味 refinement-path 群胚は quotient-cell 意味で異なる paths を保持し、linear mapping nerve への nerve map は faithful、行対象上 essentially surjective、全 path を同型へ写し、厳密な頂点/辺式を持ちます。薄群胚への 0-切断も full かつ essentially surjective です。実際の商 2-cell と「実行可能な refinement が生成する」という存在証明は厳密な意味像群胚を構成します。path 群胚はこれと圏同値で、nerve 比較は明示的単体ホモトピー逆を持ち、完全な linear mapping category への像包含は faithful、元の意味 nerve map は厳密にこの像を経由します。さらに大きい非群胚生成 path 圏は refinement と任意の aligned raw 2-cell を交互に合成できます。意味関手と refinement 埋め込みはいずれも faithful で、旧 nerve map は厳密にこれを経由し、各始域 2-cell は right-unitor で共役された元の商 2-cell に等しい正準一列辺を持ちます。正規化された左右 whiskering と水平 append は意味同値と像所属を保存し、厳密な三モデル nerve 公式を持ちます。正規化は raw 左右 whiskering と自然に可換することが証明され、raw identity、original、source identity/逆、source composition/逆、transport、および垂直/whiskering 閉包分岐が完成しました。source composition の正逆は純双圏二原子コヒーレンス式により forward expand/contract と厳密に一致し、一般の空 word/二原子閉路式により marked unit/counit とその逆も pair insert/delete と厳密に一致します。linear append の right-unit/associativity 等式と計算可能な equality path が追加され、left unitor とその逆は任意同型の共役消去により恒等へ正規化されます。残る四の associator 正逆と right unitor 正逆構造生成元フィールドは明示され、条件付きで全 raw Cell 正規化を導きますが、まだ無条件には証明されていません。semantic fullness、critical-pair coherence、reduced-hammock 不変性、標準弱同値 packaging は未解決です。 異なる資源代数のモデルは順序付き加法準同型で比較できます。直列、並列、構造、予算則が再添字 付けされ、異種強モデル射は資源写像とともに合成します。これらは、資源代数とモデルを対象、 diff --git a/docs/ja/reference/AXIOMS.md b/docs/ja/reference/AXIOMS.md index 8fe2f53..ddf5d9f 100644 --- a/docs/ja/reference/AXIOMS.md +++ b/docs/ja/reference/AXIOMS.md @@ -1034,6 +1034,20 @@ | `CategoryTheory.Bicategory.MarkedZigzag.HammockPath.markedUnitInv_normalizable` | `[propext, Classical.choice, Quot.sound]` | `Ript/ForMathlib/CategoryTheory/Bicategory/MarkedZigzagAlignedHammock.lean` | | `CategoryTheory.Bicategory.MarkedZigzag.HammockPath.markedCounit_normalizable` | `[propext, Classical.choice, Quot.sound]` | `Ript/ForMathlib/CategoryTheory/Bicategory/MarkedZigzagAlignedHammock.lean` | | `CategoryTheory.Bicategory.MarkedZigzag.HammockPath.markedCounitInv_normalizable` | `[propext, Classical.choice, Quot.sound]` | `Ript/ForMathlib/CategoryTheory/Bicategory/MarkedZigzagAlignedHammock.lean` | +| `CategoryTheory.Bicategory.MarkedZigzag.LinearWord.toWord_nil` | `none` | `Ript/ForMathlib/CategoryTheory/Bicategory/MarkedZigzagLinearHammock.lean` | +| `CategoryTheory.Bicategory.MarkedZigzag.LinearWord.flatten_nil` | `none` | `Ript/ForMathlib/CategoryTheory/Bicategory/MarkedZigzagLinearHammock.lean` | +| `CategoryTheory.Bicategory.MarkedZigzag.LinearWord.append_nil` | `none` | `Ript/ForMathlib/CategoryTheory/Bicategory/MarkedZigzagLinearHammock.lean` | +| `CategoryTheory.Bicategory.MarkedZigzag.LinearWord.append_assoc` | `none` | `Ript/ForMathlib/CategoryTheory/Bicategory/MarkedZigzagLinearHammock.lean` | +| `CategoryTheory.Bicategory.MarkedZigzag.LinearWord.toWordAppendIso_nil` | `[propext, Classical.choice, Quot.sound]` | `Ript/ForMathlib/CategoryTheory/Bicategory/MarkedZigzagLinearHammock.lean` | +| `CategoryTheory.Bicategory.MarkedZigzag.LinearWord.leftUnitor_conjugation` | `[propext, Quot.sound]` | `Ript/ForMathlib/CategoryTheory/Bicategory/MarkedZigzagLinearHammock.lean` | +| `CategoryTheory.Bicategory.MarkedZigzag.LinearWord.leftUnitor_inv_conjugation` | `[propext, Quot.sound]` | `Ript/ForMathlib/CategoryTheory/Bicategory/MarkedZigzagLinearHammock.lean` | +| `CategoryTheory.Bicategory.MarkedZigzag.HammockPath.ofEq` | `none` | `Ript/ForMathlib/CategoryTheory/Bicategory/MarkedZigzagAlignedHammock.lean` | +| `CategoryTheory.Bicategory.MarkedZigzag.HammockPath.toHom_ofEq` | `[propext, Classical.choice, Quot.sound]` | `Ript/ForMathlib/CategoryTheory/Bicategory/MarkedZigzagAlignedHammock.lean` | +| `CategoryTheory.Bicategory.MarkedZigzag.HammockPath.normalizedCellHom_eq_of_rel` | `[propext, Classical.choice, Quot.sound]` | `Ript/ForMathlib/CategoryTheory/Bicategory/MarkedZigzagAlignedHammock.lean` | +| `CategoryTheory.Bicategory.MarkedZigzag.HammockPath.normalizedCellHom_leftUnitor` | `[propext, Classical.choice, Quot.sound]` | `Ript/ForMathlib/CategoryTheory/Bicategory/MarkedZigzagAlignedHammock.lean` | +| `CategoryTheory.Bicategory.MarkedZigzag.HammockPath.normalizedCellHom_leftUnitorInv` | `[propext, Classical.choice, Quot.sound]` | `Ript/ForMathlib/CategoryTheory/Bicategory/MarkedZigzagAlignedHammock.lean` | +| `CategoryTheory.Bicategory.MarkedZigzag.HammockPath.leftUnitor_normalizable` | `[propext, Classical.choice, Quot.sound]` | `Ript/ForMathlib/CategoryTheory/Bicategory/MarkedZigzagAlignedHammock.lean` | +| `CategoryTheory.Bicategory.MarkedZigzag.HammockPath.leftUnitorInv_normalizable` | `[propext, Classical.choice, Quot.sound]` | `Ript/ForMathlib/CategoryTheory/Bicategory/MarkedZigzagAlignedHammock.lean` | | `Ript.Higher.CostExactZigzagMappingSpace.AlignedHammockGrid.linearHammockGridEquiv_simplex` | `[propext, Classical.choice, Quot.sound]` | `Ript/Higher/CostExactZigzagMappingSpace.lean` | | `Ript.Higher.CostExactZigzagMappingSpace.alignedHammockCore` | `[propext, Classical.choice, Quot.sound]` | `Ript/Higher/CostExactZigzagMappingSpace.lean` | | `Ript.Higher.CostExactZigzagMappingSpace.columnRefinementCore` | `[propext, Classical.choice, Quot.sound]` | `Ript/Higher/CostExactZigzagMappingSpace.lean` | diff --git a/docs/zh-CN/RESEARCH_STATUS.md b/docs/zh-CN/RESEARCH_STATUS.md index 0a359c8..6df708c 100644 --- a/docs/zh-CN/RESEARCH_STATUS.md +++ b/docs/zh-CN/RESEARCH_STATUS.md @@ -144,7 +144,7 @@ quarter/half-flip 树分别实现为概率、保留相干的随机酉量子仪 现已有受限但真正独立的右结合 linear hammock 对象模型：typed step 列表与二叉 words 相互转换和扁平化，精确保留长度，并给出等价 mapping category、nerve 显式同伦逆以及到实际 target local nerve 的直接比较。`LinearHammockDwyerKanCore` 将其与 outer essential surjectivity 组合。尚缺与经典任意网格 hammock 或其他认可 derived 构造的比较。 -任意高度的纵向 grid 现也已显式化：`n`-grid 包含 `n + 1` 行 linear hammocks、`n` 条相邻商 2-胞腔边和全部端点方程，strict-Segal 重构将其与 linear hammock nerve 的 `n`-simplices 等价，并精确证明行、边、解码和双向 round trip。固定形状的横向多列片段也已形式化：等形行的每个公共列含一个原始原子 2-胞腔，宽度与可执行横向拼接精确，商解释通过 interchange 保持逐列恒等和纵向合成，任意高度 aligned grid 重构为具有精确行和解释边的真实 simplex。基本前向列细化也已可执行：恒等列可插入/删除，复合列可展开/收缩，move 可在任意公共前缀下提升并传递复合；带符号宽度变化精确，两个生成器对在商语义中双向抵消。marked reverse 结构现在也包含可执行的 unit pair `f ; f⁻¹` 与 counit pair `f⁻¹ ; f` 插入/删除，其带符号宽度为 `±2`，语义同构、双向 round trip 和任意前缀稳定性均已证明。每个 refinement 现有可执行逆向和统一语义同构；双腿 common-refinement span 构成等价关系及行商，商相等精确等价于可共同细化，并只推出语义同构而非对象相等。0-截断 mapping 层也已完成：包装行构成薄 common-refinement 群胚，并与离散行商范畴等价；nerve 比较具有显式单纯逆和双向同伦。非薄语义 refinement-path 群胚现保留按 quotient-cell 语义区分的路径；其 nerve map 到 linear mapping nerve 对态射 faithful、对行对象本质满、将全部路径映为同构并具有精确顶点/边公式，且到薄群胚的 0-截断 full 且本质满。实际商 2-胞腔连同“由可执行 refinement 生成”的存在见证现构成精确语义像群胚；路径群胚与其范畴等价，nerve 比较具有显式单纯同伦逆，像包含到完整 linear mapping category 忠实，原语义 nerve map 严格经过它。更大的非群胚生成路径范畴现可交替复合 refinement 与任意 aligned raw 2-胞腔；其语义及 refinement 嵌入均 faithful，旧 nerve map 严格经过它，并且每个源 2-胞腔都有一个规范单列边，其商语义是由左右规范 right-unitor 共轭的原 2-胞腔。规范左右 whiskering 与横向 append 现保持语义等价和像成员资格，并具有精确三模型 nerve 公式。正规化现已证明与 raw 左右 whiskering 自然相容；raw 恒等、original、source identity/逆、source composition/逆、transport 及纵向/whiskering 闭包分支均已完成。source composition 正反方向通过纯双范畴两原子协调公式精确对应 forward expand/contract；通用空 word/两原子闭环公式又将 marked unit/counit 及其逆精确对应 pair insert/delete。剩余六个 associator 与左右 unitor 正反结构生成元字段被精确登记，并在条件下推出全部 raw Cell 正规化，但尚未无条件证明；semantic fullness、critical-pair 协调、约化 hammock 不变性及标准弱等价封装仍开放。 +任意高度的纵向 grid 现也已显式化：`n`-grid 包含 `n + 1` 行 linear hammocks、`n` 条相邻商 2-胞腔边和全部端点方程，strict-Segal 重构将其与 linear hammock nerve 的 `n`-simplices 等价，并精确证明行、边、解码和双向 round trip。固定形状的横向多列片段也已形式化：等形行的每个公共列含一个原始原子 2-胞腔，宽度与可执行横向拼接精确，商解释通过 interchange 保持逐列恒等和纵向合成，任意高度 aligned grid 重构为具有精确行和解释边的真实 simplex。基本前向列细化也已可执行：恒等列可插入/删除，复合列可展开/收缩，move 可在任意公共前缀下提升并传递复合；带符号宽度变化精确，两个生成器对在商语义中双向抵消。marked reverse 结构现在也包含可执行的 unit pair `f ; f⁻¹` 与 counit pair `f⁻¹ ; f` 插入/删除，其带符号宽度为 `±2`，语义同构、双向 round trip 和任意前缀稳定性均已证明。每个 refinement 现有可执行逆向和统一语义同构；双腿 common-refinement span 构成等价关系及行商，商相等精确等价于可共同细化，并只推出语义同构而非对象相等。0-截断 mapping 层也已完成：包装行构成薄 common-refinement 群胚，并与离散行商范畴等价；nerve 比较具有显式单纯逆和双向同伦。非薄语义 refinement-path 群胚现保留按 quotient-cell 语义区分的路径；其 nerve map 到 linear mapping nerve 对态射 faithful、对行对象本质满、将全部路径映为同构并具有精确顶点/边公式，且到薄群胚的 0-截断 full 且本质满。实际商 2-胞腔连同“由可执行 refinement 生成”的存在见证现构成精确语义像群胚；路径群胚与其范畴等价，nerve 比较具有显式单纯同伦逆，像包含到完整 linear mapping category 忠实，原语义 nerve map 严格经过它。更大的非群胚生成路径范畴现可交替复合 refinement 与任意 aligned raw 2-胞腔；其语义及 refinement 嵌入均 faithful，旧 nerve map 严格经过它，并且每个源 2-胞腔都有一个规范单列边，其商语义是由左右规范 right-unitor 共轭的原 2-胞腔。规范左右 whiskering 与横向 append 现保持语义等价和像成员资格，并具有精确三模型 nerve 公式。正规化现已证明与 raw 左右 whiskering 自然相容；raw 恒等、original、source identity/逆、source composition/逆、transport 及纵向/whiskering 闭包分支均已完成。source composition 正反方向通过纯双范畴两原子协调公式精确对应 forward expand/contract；通用空 word/两原子闭环公式又将 marked unit/counit 及其逆精确对应 pair insert/delete。线性 append 的右单位/结合等式与可计算 equality path 已建立，left unitor 及其逆通过任意同构共轭精确正规化为恒等。剩余四个 associator 正反与 right unitor 正反结构生成元字段被精确登记，并在条件下推出全部 raw Cell 正规化，但尚未无条件证明；semantic fullness、critical-pair 协调、约化 hammock 不变性及标准弱等价封装仍开放。 模型比较不再要求全局使用同一资源代数。有序加法同态重索引串行、并行、结构和预算律；跨资源 代数的强辫模型态射随同态复合，并在每个固定资源映射上形成单子自然变换的局部范畴。四维计算 diff --git a/docs/zh-CN/reference/AXIOMS.md b/docs/zh-CN/reference/AXIOMS.md index 72319e7..714ff97 100644 --- a/docs/zh-CN/reference/AXIOMS.md +++ b/docs/zh-CN/reference/AXIOMS.md @@ -1034,6 +1034,20 @@ | `CategoryTheory.Bicategory.MarkedZigzag.HammockPath.markedUnitInv_normalizable` | `[propext, Classical.choice, Quot.sound]` | `Ript/ForMathlib/CategoryTheory/Bicategory/MarkedZigzagAlignedHammock.lean` | | `CategoryTheory.Bicategory.MarkedZigzag.HammockPath.markedCounit_normalizable` | `[propext, Classical.choice, Quot.sound]` | `Ript/ForMathlib/CategoryTheory/Bicategory/MarkedZigzagAlignedHammock.lean` | | `CategoryTheory.Bicategory.MarkedZigzag.HammockPath.markedCounitInv_normalizable` | `[propext, Classical.choice, Quot.sound]` | `Ript/ForMathlib/CategoryTheory/Bicategory/MarkedZigzagAlignedHammock.lean` | +| `CategoryTheory.Bicategory.MarkedZigzag.LinearWord.toWord_nil` | `none` | `Ript/ForMathlib/CategoryTheory/Bicategory/MarkedZigzagLinearHammock.lean` | +| `CategoryTheory.Bicategory.MarkedZigzag.LinearWord.flatten_nil` | `none` | `Ript/ForMathlib/CategoryTheory/Bicategory/MarkedZigzagLinearHammock.lean` | +| `CategoryTheory.Bicategory.MarkedZigzag.LinearWord.append_nil` | `none` | `Ript/ForMathlib/CategoryTheory/Bicategory/MarkedZigzagLinearHammock.lean` | +| `CategoryTheory.Bicategory.MarkedZigzag.LinearWord.append_assoc` | `none` | `Ript/ForMathlib/CategoryTheory/Bicategory/MarkedZigzagLinearHammock.lean` | +| `CategoryTheory.Bicategory.MarkedZigzag.LinearWord.toWordAppendIso_nil` | `[propext, Classical.choice, Quot.sound]` | `Ript/ForMathlib/CategoryTheory/Bicategory/MarkedZigzagLinearHammock.lean` | +| `CategoryTheory.Bicategory.MarkedZigzag.LinearWord.leftUnitor_conjugation` | `[propext, Quot.sound]` | `Ript/ForMathlib/CategoryTheory/Bicategory/MarkedZigzagLinearHammock.lean` | +| `CategoryTheory.Bicategory.MarkedZigzag.LinearWord.leftUnitor_inv_conjugation` | `[propext, Quot.sound]` | `Ript/ForMathlib/CategoryTheory/Bicategory/MarkedZigzagLinearHammock.lean` | +| `CategoryTheory.Bicategory.MarkedZigzag.HammockPath.ofEq` | `none` | `Ript/ForMathlib/CategoryTheory/Bicategory/MarkedZigzagAlignedHammock.lean` | +| `CategoryTheory.Bicategory.MarkedZigzag.HammockPath.toHom_ofEq` | `[propext, Classical.choice, Quot.sound]` | `Ript/ForMathlib/CategoryTheory/Bicategory/MarkedZigzagAlignedHammock.lean` | +| `CategoryTheory.Bicategory.MarkedZigzag.HammockPath.normalizedCellHom_eq_of_rel` | `[propext, Classical.choice, Quot.sound]` | `Ript/ForMathlib/CategoryTheory/Bicategory/MarkedZigzagAlignedHammock.lean` | +| `CategoryTheory.Bicategory.MarkedZigzag.HammockPath.normalizedCellHom_leftUnitor` | `[propext, Classical.choice, Quot.sound]` | `Ript/ForMathlib/CategoryTheory/Bicategory/MarkedZigzagAlignedHammock.lean` | +| `CategoryTheory.Bicategory.MarkedZigzag.HammockPath.normalizedCellHom_leftUnitorInv` | `[propext, Classical.choice, Quot.sound]` | `Ript/ForMathlib/CategoryTheory/Bicategory/MarkedZigzagAlignedHammock.lean` | +| `CategoryTheory.Bicategory.MarkedZigzag.HammockPath.leftUnitor_normalizable` | `[propext, Classical.choice, Quot.sound]` | `Ript/ForMathlib/CategoryTheory/Bicategory/MarkedZigzagAlignedHammock.lean` | +| `CategoryTheory.Bicategory.MarkedZigzag.HammockPath.leftUnitorInv_normalizable` | `[propext, Classical.choice, Quot.sound]` | `Ript/ForMathlib/CategoryTheory/Bicategory/MarkedZigzagAlignedHammock.lean` | | `Ript.Higher.CostExactZigzagMappingSpace.AlignedHammockGrid.linearHammockGridEquiv_simplex` | `[propext, Classical.choice, Quot.sound]` | `Ript/Higher/CostExactZigzagMappingSpace.lean` | | `Ript.Higher.CostExactZigzagMappingSpace.alignedHammockCore` | `[propext, Classical.choice, Quot.sound]` | `Ript/Higher/CostExactZigzagMappingSpace.lean` | | `Ript.Higher.CostExactZigzagMappingSpace.columnRefinementCore` | `[propext, Classical.choice, Quot.sound]` | `Ript/Higher/CostExactZigzagMappingSpace.lean` |