From afa5efb6d3096dcf073b8d0fbc0478dc81efc5d9 Mon Sep 17 00:00:00 2001 From: Amamiya Miu Date: Sat, 22 Aug 2026 17:00:50 +0800 Subject: [PATCH] feat(higher): normalize marked pair cells --- AXIOMS.md | 10 + BLUEPRINT.md | 4 +- CONJECTURES.md | 13 +- MODEL_MATRIX.md | 5 +- Ript/Audit/AxiomChecks.lean | 10 + .../MarkedZigzagAlignedHammock.lean | 249 ++++++++++++++++-- Ript/Higher/CostExactZigzagMappingSpace.lean | 45 +++- docs/en/RESEARCH_STATUS.md | 6 +- docs/en/reference/AXIOMS.md | 10 + docs/en/reference/BLUEPRINT.md | 4 +- docs/en/reference/CONJECTURES.md | 13 +- docs/en/reference/MODEL_MATRIX.md | 5 +- docs/eo/RESEARCH_STATUS.md | 7 +- docs/eo/reference/AXIOMS.md | 10 + docs/ja/RESEARCH_STATUS.md | 2 +- docs/ja/reference/AXIOMS.md | 10 + docs/zh-CN/RESEARCH_STATUS.md | 2 +- docs/zh-CN/reference/AXIOMS.md | 10 + 18 files changed, 369 insertions(+), 46 deletions(-) diff --git a/AXIOMS.md b/AXIOMS.md index 38c1eab..3c3c056 100644 --- a/AXIOMS.md +++ b/AXIOMS.md @@ -1024,6 +1024,16 @@ the actual output of `lake env lean Ript/Audit/AxiomChecks.lean`. | `CategoryTheory.Bicategory.MarkedZigzag.HammockPath.normalizedCellHom_sourceCompInv` | `[propext, Classical.choice, Quot.sound]` | `Ript/ForMathlib/CategoryTheory/Bicategory/MarkedZigzagAlignedHammock.lean` | | `CategoryTheory.Bicategory.MarkedZigzag.HammockPath.sourceComp_normalizable` | `[propext, Classical.choice, Quot.sound]` | `Ript/ForMathlib/CategoryTheory/Bicategory/MarkedZigzagAlignedHammock.lean` | | `CategoryTheory.Bicategory.MarkedZigzag.HammockPath.sourceCompInv_normalizable` | `[propext, Classical.choice, Quot.sound]` | `Ript/ForMathlib/CategoryTheory/Bicategory/MarkedZigzagAlignedHammock.lean` | +| `CategoryTheory.Bicategory.MarkedZigzag.HammockPath.normalizedCellHom_nil_twoAtoms` | `[propext, Classical.choice, Quot.sound]` | `Ript/ForMathlib/CategoryTheory/Bicategory/MarkedZigzagAlignedHammock.lean` | +| `CategoryTheory.Bicategory.MarkedZigzag.HammockPath.normalizedCellHom_twoAtoms_nil` | `[propext, Classical.choice, Quot.sound]` | `Ript/ForMathlib/CategoryTheory/Bicategory/MarkedZigzagAlignedHammock.lean` | +| `CategoryTheory.Bicategory.MarkedZigzag.HammockPath.normalizedCellHom_markedUnit` | `[propext, Classical.choice, Quot.sound]` | `Ript/ForMathlib/CategoryTheory/Bicategory/MarkedZigzagAlignedHammock.lean` | +| `CategoryTheory.Bicategory.MarkedZigzag.HammockPath.normalizedCellHom_markedUnitInv` | `[propext, Classical.choice, Quot.sound]` | `Ript/ForMathlib/CategoryTheory/Bicategory/MarkedZigzagAlignedHammock.lean` | +| `CategoryTheory.Bicategory.MarkedZigzag.HammockPath.normalizedCellHom_markedCounit` | `[propext, Classical.choice, Quot.sound]` | `Ript/ForMathlib/CategoryTheory/Bicategory/MarkedZigzagAlignedHammock.lean` | +| `CategoryTheory.Bicategory.MarkedZigzag.HammockPath.normalizedCellHom_markedCounitInv` | `[propext, Classical.choice, Quot.sound]` | `Ript/ForMathlib/CategoryTheory/Bicategory/MarkedZigzagAlignedHammock.lean` | +| `CategoryTheory.Bicategory.MarkedZigzag.HammockPath.markedUnit_normalizable` | `[propext, Classical.choice, Quot.sound]` | `Ript/ForMathlib/CategoryTheory/Bicategory/MarkedZigzagAlignedHammock.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` | | `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 ff1c30a..4d5ab5f 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; raw identity, original, source-identity/inverse, source-composition/inverse, whiskering, vertical-composite, and equality-transport cells are normalizable; `StructuralGeneratorNormalizable` now lists exactly ten remaining marked pair, associator, and 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; 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 (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 ten explicit `StructuralGeneratorNormalizable` fields for marked unit/counit and inverses, 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 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 | ## Finite deterministic copy-discard theorem records diff --git a/CONJECTURES.md b/CONJECTURES.md index 524dc8f..cf0741e 100644 --- a/CONJECTURES.md +++ b/CONJECTURES.md @@ -475,11 +475,13 @@ 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 ten-field `StructuralGeneratorNormalizable` record: -marked unit/counit and inverses, associator and inverse, and both unitors and -inverses. Source composition/inverse are now normalized exactly by executable +has now shrunk to the six-field `StructuralGeneratorNormalizable` record: +associator and inverse, and both unitors and inverses. Source +composition/inverse are normalized exactly by executable forward expansion/contraction using the audited two-atomic-step coherence -formula. The remaining record already implies normalization of every raw +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 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. @@ -591,7 +593,8 @@ 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. The remaining ten structural generator obligations are explicit and +proved; all marked unit/counit cases are proved too. The remaining six +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 e526717..83f3a5c 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-identity/source-composition/transport and closure cases are proved, and a ten-field criterion implies all-cell normalization; those ten marked-pair/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/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 | 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 @@ -674,7 +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. Ten explicit marked-pair, associator, and unitor obligations are +closure are proved; all marked pair cases are proved too. Six explicit +associator and 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 dbdcd36..dc434fe 100644 --- a/Ript/Audit/AxiomChecks.lean +++ b/Ript/Audit/AxiomChecks.lean @@ -1105,6 +1105,16 @@ set_option autoImplicit false #print axioms CategoryTheory.Bicategory.MarkedZigzag.HammockPath.normalizedCellHom_sourceCompInv #print axioms CategoryTheory.Bicategory.MarkedZigzag.HammockPath.sourceComp_normalizable #print axioms CategoryTheory.Bicategory.MarkedZigzag.HammockPath.sourceCompInv_normalizable +#print axioms CategoryTheory.Bicategory.MarkedZigzag.HammockPath.normalizedCellHom_nil_twoAtoms +#print axioms CategoryTheory.Bicategory.MarkedZigzag.HammockPath.normalizedCellHom_twoAtoms_nil +#print axioms CategoryTheory.Bicategory.MarkedZigzag.HammockPath.normalizedCellHom_markedUnit +#print axioms CategoryTheory.Bicategory.MarkedZigzag.HammockPath.normalizedCellHom_markedUnitInv +#print axioms CategoryTheory.Bicategory.MarkedZigzag.HammockPath.normalizedCellHom_markedCounit +#print axioms CategoryTheory.Bicategory.MarkedZigzag.HammockPath.normalizedCellHom_markedCounitInv +#print axioms CategoryTheory.Bicategory.MarkedZigzag.HammockPath.markedUnit_normalizable +#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 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 c46c4d5..f3d9b53 100644 --- a/Ript/ForMathlib/CategoryTheory/Bicategory/MarkedZigzagAlignedHammock.lean +++ b/Ript/ForMathlib/CategoryTheory/Bicategory/MarkedZigzagAlignedHammock.lean @@ -28,7 +28,8 @@ 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, while ten explicit +complete; marked unit/counit pairs and inverses are complete too, while six +explicit structural-generator 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. @@ -2495,6 +2496,200 @@ theorem normalizedCellHom_sourceCompInv {X Y Z : B} (Cell.sourceCompInv (W := W) f g)) rw [whiskerEquality] +/-- Generic normalization of a raw cell from the empty word to a closed +two-atomic-step word. -/ +theorem normalizedCellHom_nil_twoAtoms {X Y : B} + (first : Step W X Y) (second : Step W Y X) + (cell : Cell W (.nil X) + (Word.append (W := W) (.atom first) (.atom second))) : + normalizedCellHom W cell = + AlignedCell.quotientVcomp W + (Presented.wordLeftUnitorIso W (.nil X)).inv + (AlignedCell.quotientVcomp W + (Presented.whiskerRightHom W (.nil X) (Presented.mk W cell)) + (Presented.wordAssociatorIso W + (.atom first) (.atom second) (.nil X)).hom) := by + unfold normalizedCellHom normalizedHom + simp only [Word.append_eq_comp, LinearWord.flatten_append, + LinearWord.flatten] + rw [LinearWord.normalizationIso_nil W X] + rw [LinearWord.normalizationIso_twoAtoms_hom W first second] + simp only [Iso.refl_inv] + change AlignedCell.quotientVcomp W (𝟙 (Word.nil X)) + (AlignedCell.quotientVcomp W (Presented.mk W cell) + (AlignedCell.quotientVcomp W + (Presented.wordRightUnitorIso W + (.comp (.atom first) (.atom second))).inv + (Presented.wordAssociatorIso W + (.atom first) (.atom second) (.nil X)).hom)) = + AlignedCell.quotientVcomp W + (Presented.wordLeftUnitorIso W (.nil X)).inv + (AlignedCell.quotientVcomp W + (Presented.whiskerRightHom W (.nil X) (Presented.mk W cell)) + (Presented.wordAssociatorIso W + (.atom first) (.atom second) (.nil X)).hom) + rw [AlignedCell.quotientVcomp_id_comp] + have whiskerEquality : + Presented.whiskerRightHom W (.nil X) (Presented.mk W cell) = + AlignedCell.quotientVcomp W + (Presented.wordRightUnitorIso W (.nil X)).hom + (AlignedCell.quotientVcomp W (Presented.mk W cell) + (Presented.wordRightUnitorIso W + (Word.append (W := W) (.atom first) (.atom second))).inv) := + Quot.sound (Presented.Rel.whisker_right_id_word cell) + rw [whiskerEquality] + simp only [AlignedCell.quotientVcomp_assoc] + rw [← AlignedCell.quotientVcomp_assoc W + (Presented.wordLeftUnitorIso W (.nil X)).inv + (Presented.wordRightUnitorIso W (.nil X)).hom] + have unitCancellation : + AlignedCell.quotientVcomp W + (Presented.wordLeftUnitorIso W (.nil X)).inv + (Presented.wordRightUnitorIso W (.nil X)).hom = + 𝟙 (Word.nil X) := by + change (λ_ (𝟙 (⟨X⟩ : Presented.Localization W))).inv ≫ + (ρ_ (𝟙 (⟨X⟩ : Presented.Localization W))).hom = _ + rw [unitors_inv_equal] + exact (Presented.wordRightUnitorIso W (.nil X)).inv_hom_id + rw [unitCancellation, AlignedCell.quotientVcomp_id_comp] + rfl + +/-- Generic normalization of a raw cell from a closed two-atomic-step word +to the empty word. -/ +theorem normalizedCellHom_twoAtoms_nil {X Y : B} + (first : Step W X Y) (second : Step W Y X) + (cell : Cell W + (Word.append (W := W) (.atom first) (.atom second)) (.nil X)) : + normalizedCellHom W cell = + AlignedCell.quotientVcomp W + (Presented.wordAssociatorIso W + (.atom first) (.atom second) (.nil X)).inv + (AlignedCell.quotientVcomp W + (Presented.whiskerRightHom W (.nil X) (Presented.mk W cell)) + (Presented.wordLeftUnitorIso W (.nil X)).hom) := by + unfold normalizedCellHom normalizedHom + simp only [Word.append_eq_comp, LinearWord.flatten_append, + LinearWord.flatten] + rw [LinearWord.normalizationIso_twoAtoms_inv W first second] + rw [LinearWord.normalizationIso_nil W X] + simp only [Iso.refl_hom] + change AlignedCell.quotientVcomp W + (AlignedCell.quotientVcomp W + (Presented.wordAssociatorIso W + (.atom first) (.atom second) (.nil X)).inv + (Presented.wordRightUnitorIso W + (Word.append (W := W) (.atom first) (.atom second))).hom) + (AlignedCell.quotientVcomp W (Presented.mk W cell) (𝟙 (Word.nil X))) = + AlignedCell.quotientVcomp W + (Presented.wordAssociatorIso W + (.atom first) (.atom second) (.nil X)).inv + (AlignedCell.quotientVcomp W + (Presented.whiskerRightHom W (.nil X) (Presented.mk W cell)) + (Presented.wordLeftUnitorIso W (.nil X)).hom) + rw [AlignedCell.quotientVcomp_comp_id] + rw [AlignedCell.quotientVcomp_assoc] + have whiskerEquality : + Presented.whiskerRightHom W (.nil X) (Presented.mk W cell) = + AlignedCell.quotientVcomp W + (Presented.wordRightUnitorIso W + (Word.append (W := W) (.atom first) (.atom second))).hom + (AlignedCell.quotientVcomp W (Presented.mk W cell) + (Presented.wordRightUnitorIso W (.nil X)).inv) := + Quot.sound (Presented.Rel.whisker_right_id_word cell) + rw [whiskerEquality] + rw [AlignedCell.quotientVcomp_assoc W + (Presented.wordRightUnitorIso W + (Word.append (W := W) (.atom first) (.atom second))).hom + (AlignedCell.quotientVcomp W (Presented.mk W cell) + (Presented.wordRightUnitorIso W (.nil X)).inv) + (Presented.wordLeftUnitorIso W (.nil X)).hom] + rw [AlignedCell.quotientVcomp_assoc W + (Presented.mk W cell) + (Presented.wordRightUnitorIso W (.nil X)).inv + (Presented.wordLeftUnitorIso W (.nil X)).hom] + have unitCancellation : + AlignedCell.quotientVcomp W + (Presented.wordRightUnitorIso W (.nil X)).inv + (Presented.wordLeftUnitorIso W (.nil X)).hom = + 𝟙 (Word.nil X) := by + change (ρ_ (𝟙 (⟨X⟩ : Presented.Localization W))).inv ≫ + (λ_ (𝟙 (⟨X⟩ : Presented.Localization W))).hom = _ + rw [← unitors_inv_equal] + exact (Presented.wordLeftUnitorIso W (.nil X)).inv_hom_id + rw [unitCancellation, AlignedCell.quotientVcomp_comp_id] + +/-- Marked unit normalizes to executable insertion of its forward/reverse +pair. -/ +@[simp] +theorem normalizedCellHom_markedUnit {X Y : B} + (f : X ⟶ Y) (hf : W f) : + normalizedCellHom W (Cell.markedUnit (W := W) f hf) = + ColumnRefinement.toHom W (.insertMarkedUnitPair f hf (.nil X)) := by + change normalizedCellHom W (Cell.markedUnit (W := W) f hf) = + AlignedCell.quotientVcomp W + (Presented.wordLeftUnitorIso W (.nil X)).inv + (AlignedCell.quotientVcomp W + (Presented.whiskerRightHom W (.nil X) + (Presented.mk W (Cell.markedUnit (W := W) f hf))) + (Presented.wordAssociatorIso W (Word.forward W f) + (Word.backward W f hf) (.nil X)).hom) + exact normalizedCellHom_nil_twoAtoms W + (.forward f) (.backward f hf) (.markedUnit f hf) + +/-- Inverse marked unit normalizes to executable deletion of its +forward/reverse pair. -/ +@[simp] +theorem normalizedCellHom_markedUnitInv {X Y : B} + (f : X ⟶ Y) (hf : W f) : + normalizedCellHom W (Cell.markedUnitInv (W := W) f hf) = + ColumnRefinement.toHom W (.deleteMarkedUnitPair f hf (.nil X)) := by + change normalizedCellHom W (Cell.markedUnitInv (W := W) f hf) = + AlignedCell.quotientVcomp W + (Presented.wordAssociatorIso W (Word.forward W f) + (Word.backward W f hf) (.nil X)).inv + (AlignedCell.quotientVcomp W + (Presented.whiskerRightHom W (.nil X) + (Presented.mk W (Cell.markedUnitInv (W := W) f hf))) + (Presented.wordLeftUnitorIso W (.nil X)).hom) + exact normalizedCellHom_twoAtoms_nil W + (.forward f) (.backward f hf) (.markedUnitInv f hf) + +/-- Marked counit normalizes to executable deletion of its reverse/forward +pair. -/ +@[simp] +theorem normalizedCellHom_markedCounit {X Y : B} + (f : X ⟶ Y) (hf : W f) : + normalizedCellHom W (Cell.markedCounit (W := W) f hf) = + ColumnRefinement.toHom W (.deleteMarkedCounitPair f hf (.nil Y)) := by + change normalizedCellHom W (Cell.markedCounit (W := W) f hf) = + AlignedCell.quotientVcomp W + (Presented.wordAssociatorIso W (Word.backward W f hf) + (Word.forward W f) (.nil Y)).inv + (AlignedCell.quotientVcomp W + (Presented.whiskerRightHom W (.nil Y) + (Presented.mk W (Cell.markedCounit (W := W) f hf))) + (Presented.wordLeftUnitorIso W (.nil Y)).hom) + exact normalizedCellHom_twoAtoms_nil W + (.backward f hf) (.forward f) (.markedCounit f hf) + +/-- Inverse marked counit normalizes to executable insertion of its +reverse/forward pair. -/ +@[simp] +theorem normalizedCellHom_markedCounitInv {X Y : B} + (f : X ⟶ Y) (hf : W f) : + normalizedCellHom W (Cell.markedCounitInv (W := W) f hf) = + ColumnRefinement.toHom W (.insertMarkedCounitPair f hf (.nil Y)) := by + change normalizedCellHom W (Cell.markedCounitInv (W := W) f hf) = + AlignedCell.quotientVcomp W + (Presented.wordLeftUnitorIso W (.nil Y)).inv + (AlignedCell.quotientVcomp W + (Presented.whiskerRightHom W (.nil Y) + (Presented.mk W (Cell.markedCounitInv (W := W) f hf))) + (Presented.wordAssociatorIso W (Word.backward W f hf) + (Word.forward W f) (.nil Y)).hom) + exact normalizedCellHom_nil_twoAtoms W + (.backward f hf) (.forward f) (.markedCounitInv f hf) + /-- Equality transport does not change normalized quotient semantics. -/ @[simp] theorem normalizedCellHom_transport {X Y : B} @@ -2637,6 +2832,34 @@ theorem sourceCompInv_normalizable {X Y Z : B} rw [Normalizable, normalizedCellHom_sourceCompInv] exact refinement_mem_semanticImage W (.contractForward f g (.nil Z)) +/-- Marked unit is hammock-normalizable. -/ +theorem markedUnit_normalizable {X Y : B} + (f : X ⟶ Y) (hf : W f) : + Normalizable W (Cell.markedUnit (W := W) f hf) := by + rw [Normalizable, normalizedCellHom_markedUnit] + exact refinement_mem_semanticImage W (.insertMarkedUnitPair f hf (.nil X)) + +/-- Inverse marked unit is hammock-normalizable. -/ +theorem markedUnitInv_normalizable {X Y : B} + (f : X ⟶ Y) (hf : W f) : + Normalizable W (Cell.markedUnitInv (W := W) f hf) := by + rw [Normalizable, normalizedCellHom_markedUnitInv] + exact refinement_mem_semanticImage W (.deleteMarkedUnitPair f hf (.nil X)) + +/-- Marked counit is hammock-normalizable. -/ +theorem markedCounit_normalizable {X Y : B} + (f : X ⟶ Y) (hf : W f) : + Normalizable W (Cell.markedCounit (W := W) f hf) := by + rw [Normalizable, normalizedCellHom_markedCounit] + exact refinement_mem_semanticImage W (.deleteMarkedCounitPair f hf (.nil Y)) + +/-- Inverse marked counit is hammock-normalizable. -/ +theorem markedCounitInv_normalizable {X Y : B} + (f : X ⟶ Y) (hf : W f) : + Normalizable W (Cell.markedCounitInv (W := W) f hf) := by + rw [Normalizable, normalizedCellHom_markedCounitInv] + exact refinement_mem_semanticImage W (.insertMarkedCounitPair f hf (.nil Y)) + /-- Equality transport preserves raw-cell normalizability. -/ theorem transport_normalizable {X Y : B} {first second first' second' : Word W X Y} @@ -2650,22 +2873,10 @@ 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, both whiskerings, and equality transport -are already +source identities, source composition, marked pairs, both whiskerings, and +equality transport are already discharged separately. -/ structure StructuralGeneratorNormalizable : Prop where - /-- Marked unit. -/ - markedUnit : ∀ {X Y : B} (f : X ⟶ Y) (hf : W f), - Normalizable W (Cell.markedUnit (W := W) f hf) - /-- Inverse marked unit. -/ - markedUnitInv : ∀ {X Y : B} (f : X ⟶ Y) (hf : W f), - Normalizable W (Cell.markedUnitInv (W := W) f hf) - /-- Marked counit. -/ - markedCounit : ∀ {X Y : B} (f : X ⟶ Y) (hf : W f), - Normalizable W (Cell.markedCounit (W := W) f hf) - /-- Inverse marked counit. -/ - markedCounitInv : ∀ {X Y : B} (f : X ⟶ Y) (hf : W f), - Normalizable W (Cell.markedCounitInv (W := W) f hf) /-- Binary associator. -/ associator : ∀ {X Y Z T : B} (first : Word W X Y) (second : Word W Y Z) (third : Word W Z T), @@ -2705,10 +2916,10 @@ theorem normalizable_of_structuralGenerators | sourceIdInv => exact sourceIdInv_normalizable W | sourceComp f g => exact sourceComp_normalizable W f g | sourceCompInv f g => exact sourceCompInv_normalizable W f g - | markedUnit f hf => exact generators.markedUnit f hf - | markedUnitInv f hf => exact generators.markedUnitInv f hf - | markedCounit f hf => exact generators.markedCounit f hf - | markedCounitInv f hf => exact generators.markedCounitInv f hf + | markedUnit f hf => exact markedUnit_normalizable W f hf + | markedUnitInv f hf => exact markedUnitInv_normalizable W f hf + | markedCounit f hf => exact markedCounit_normalizable W f hf + | markedCounitInv f hf => exact markedCounitInv_normalizable W f hf | whiskerLeft pre cell member => exact whiskerLeft_normalizable W pre member | whiskerRight cell post member => diff --git a/Ript/Higher/CostExactZigzagMappingSpace.lean b/Ript/Higher/CostExactZigzagMappingSpace.lean index 65e51e8..9566e4b 100644 --- a/Ript/Higher/CostExactZigzagMappingSpace.lean +++ b/Ript/Higher/CostExactZigzagMappingSpace.lean @@ -35,8 +35,9 @@ canonical one-column representation. Generated paths are now also closed under 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, and transport branches, plus a conditional all-cell induction from -ten remaining structural generators. Coverage of all presented quotient 2-cells, competing-move +whiskering, marked-pair/inverse, and transport branches, plus a conditional +all-cell induction from six 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. -/ @@ -1403,6 +1404,34 @@ structure HammockRawCellNormalizationCore : Prop where (costExactArrows R) (Bicategory.MarkedZigzag.Cell.sourceCompInv (W := costExactArrows R) f g) + /-- Marked unit is normalizable. -/ + markedUnit : ∀ {M N : ProcessModel.{u, v, w} R} + (f : M ⟶ N) (hf : costExactArrows R f), + Bicategory.MarkedZigzag.HammockPath.Normalizable + (costExactArrows R) + (Bicategory.MarkedZigzag.Cell.markedUnit + (W := costExactArrows R) f hf) + /-- Inverse marked unit is normalizable. -/ + markedUnitInv : ∀ {M N : ProcessModel.{u, v, w} R} + (f : M ⟶ N) (hf : costExactArrows R f), + Bicategory.MarkedZigzag.HammockPath.Normalizable + (costExactArrows R) + (Bicategory.MarkedZigzag.Cell.markedUnitInv + (W := costExactArrows R) f hf) + /-- Marked counit is normalizable. -/ + markedCounit : ∀ {M N : ProcessModel.{u, v, w} R} + (f : M ⟶ N) (hf : costExactArrows R f), + Bicategory.MarkedZigzag.HammockPath.Normalizable + (costExactArrows R) + (Bicategory.MarkedZigzag.Cell.markedCounit + (W := costExactArrows R) f hf) + /-- Inverse marked counit is normalizable. -/ + markedCounitInv : ∀ {M N : ProcessModel.{u, v, w} R} + (f : M ⟶ N) (hf : costExactArrows R f), + Bicategory.MarkedZigzag.HammockPath.Normalizable + (costExactArrows R) + (Bicategory.MarkedZigzag.Cell.markedCounitInv + (W := costExactArrows R) f hf) /-- Raw left whiskering preserves normalizability. -/ whiskerLeft : ∀ (M N P : ProcessModel.{u, v, w} R) (pre : CostExactZigzag.Word (R := R) M N) @@ -1474,6 +1503,18 @@ theorem hammockRawCellNormalizationCore : sourceCompInv := fun f g => Bicategory.MarkedZigzag.HammockPath.sourceCompInv_normalizable (costExactArrows R) f g + markedUnit := fun f hf => + Bicategory.MarkedZigzag.HammockPath.markedUnit_normalizable + (costExactArrows R) f hf + markedUnitInv := fun f hf => + Bicategory.MarkedZigzag.HammockPath.markedUnitInv_normalizable + (costExactArrows R) f hf + markedCounit := fun f hf => + Bicategory.MarkedZigzag.HammockPath.markedCounit_normalizable + (costExactArrows R) f hf + markedCounitInv := fun f hf => + Bicategory.MarkedZigzag.HammockPath.markedCounitInv_normalizable + (costExactArrows R) f hf 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 265ee2b..14ecbb6 100644 --- a/docs/en/RESEARCH_STATUS.md +++ b/docs/en/RESEARCH_STATUS.md @@ -398,8 +398,10 @@ naturality for raw left/right whiskering is now proved by explicit append-iso 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. The remaining ten marked-pair/associator/unitor obligations -are recorded exactly and already imply all-cell normalization conditionally, +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 +normalization conditionally, but are not assumed unconditionally. Semantic fullness, competing-move coherence, reduced-hammock invariance, standard weak-equivalence packaging, and the global Dwyer--Kan/Rezk theorem remain. diff --git a/docs/en/reference/AXIOMS.md b/docs/en/reference/AXIOMS.md index 97eca55..ccfe883 100644 --- a/docs/en/reference/AXIOMS.md +++ b/docs/en/reference/AXIOMS.md @@ -1024,6 +1024,16 @@ the actual output of `lake env lean Ript/Audit/AxiomChecks.lean`. | `CategoryTheory.Bicategory.MarkedZigzag.HammockPath.normalizedCellHom_sourceCompInv` | `[propext, Classical.choice, Quot.sound]` | `Ript/ForMathlib/CategoryTheory/Bicategory/MarkedZigzagAlignedHammock.lean` | | `CategoryTheory.Bicategory.MarkedZigzag.HammockPath.sourceComp_normalizable` | `[propext, Classical.choice, Quot.sound]` | `Ript/ForMathlib/CategoryTheory/Bicategory/MarkedZigzagAlignedHammock.lean` | | `CategoryTheory.Bicategory.MarkedZigzag.HammockPath.sourceCompInv_normalizable` | `[propext, Classical.choice, Quot.sound]` | `Ript/ForMathlib/CategoryTheory/Bicategory/MarkedZigzagAlignedHammock.lean` | +| `CategoryTheory.Bicategory.MarkedZigzag.HammockPath.normalizedCellHom_nil_twoAtoms` | `[propext, Classical.choice, Quot.sound]` | `Ript/ForMathlib/CategoryTheory/Bicategory/MarkedZigzagAlignedHammock.lean` | +| `CategoryTheory.Bicategory.MarkedZigzag.HammockPath.normalizedCellHom_twoAtoms_nil` | `[propext, Classical.choice, Quot.sound]` | `Ript/ForMathlib/CategoryTheory/Bicategory/MarkedZigzagAlignedHammock.lean` | +| `CategoryTheory.Bicategory.MarkedZigzag.HammockPath.normalizedCellHom_markedUnit` | `[propext, Classical.choice, Quot.sound]` | `Ript/ForMathlib/CategoryTheory/Bicategory/MarkedZigzagAlignedHammock.lean` | +| `CategoryTheory.Bicategory.MarkedZigzag.HammockPath.normalizedCellHom_markedUnitInv` | `[propext, Classical.choice, Quot.sound]` | `Ript/ForMathlib/CategoryTheory/Bicategory/MarkedZigzagAlignedHammock.lean` | +| `CategoryTheory.Bicategory.MarkedZigzag.HammockPath.normalizedCellHom_markedCounit` | `[propext, Classical.choice, Quot.sound]` | `Ript/ForMathlib/CategoryTheory/Bicategory/MarkedZigzagAlignedHammock.lean` | +| `CategoryTheory.Bicategory.MarkedZigzag.HammockPath.normalizedCellHom_markedCounitInv` | `[propext, Classical.choice, Quot.sound]` | `Ript/ForMathlib/CategoryTheory/Bicategory/MarkedZigzagAlignedHammock.lean` | +| `CategoryTheory.Bicategory.MarkedZigzag.HammockPath.markedUnit_normalizable` | `[propext, Classical.choice, Quot.sound]` | `Ript/ForMathlib/CategoryTheory/Bicategory/MarkedZigzagAlignedHammock.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` | | `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 53238b2..c3c2636 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; raw identity, original, source-identity/inverse, source-composition/inverse, whiskering, vertical-composite, and equality-transport cells are normalizable; `StructuralGeneratorNormalizable` now lists exactly ten remaining marked pair, associator, and 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; 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 (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 ten explicit `StructuralGeneratorNormalizable` fields for marked unit/counit and inverses, 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 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 | ## Finite deterministic copy-discard theorem records diff --git a/docs/en/reference/CONJECTURES.md b/docs/en/reference/CONJECTURES.md index f5d1897..eee2480 100644 --- a/docs/en/reference/CONJECTURES.md +++ b/docs/en/reference/CONJECTURES.md @@ -475,11 +475,13 @@ 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 ten-field `StructuralGeneratorNormalizable` record: -marked unit/counit and inverses, associator and inverse, and both unitors and -inverses. Source composition/inverse are now normalized exactly by executable +has now shrunk to the six-field `StructuralGeneratorNormalizable` record: +associator and inverse, and both unitors and inverses. Source +composition/inverse are normalized exactly by executable forward expansion/contraction using the audited two-atomic-step coherence -formula. The remaining record already implies normalization of every raw +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 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. @@ -591,7 +593,8 @@ 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. The remaining ten structural generator obligations are explicit and +proved; all marked unit/counit cases are proved too. The remaining six +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 38ba5d6..9cd9107 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-identity/source-composition/transport and closure cases are proved, and a ten-field criterion implies all-cell normalization; those ten marked-pair/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/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 | 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 @@ -674,7 +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. Ten explicit marked-pair, associator, and unitor obligations are +closure are proved; all marked pair cases are proved too. Six explicit +associator and 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 443b2dd..95881f3 100644 --- a/docs/eo/RESEARCH_STATUS.md +++ b/docs/eo/RESEARCH_STATUS.md @@ -368,9 +368,10 @@ kaj vertikala kunmeto konservas normaligeblecon. Normaligo nun estas nature kongrua kun ambaŭ krudaj whiskering-operacioj; krudaj identecaj, originalaj, 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. La ceteraj dek -markitaj-paraj, asociatoraj kaj -unuitoraj generatoraj kampoj estas eksplicitaj kaj kondiĉe +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 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 4151901..dd9d1bb 100644 --- a/docs/eo/reference/AXIOMS.md +++ b/docs/eo/reference/AXIOMS.md @@ -1026,6 +1026,16 @@ per `scripts/sync-doc-reference-tables.sh` kaj ne estu mane redaktataj. | `CategoryTheory.Bicategory.MarkedZigzag.HammockPath.normalizedCellHom_sourceCompInv` | `[propext, Classical.choice, Quot.sound]` | `Ript/ForMathlib/CategoryTheory/Bicategory/MarkedZigzagAlignedHammock.lean` | | `CategoryTheory.Bicategory.MarkedZigzag.HammockPath.sourceComp_normalizable` | `[propext, Classical.choice, Quot.sound]` | `Ript/ForMathlib/CategoryTheory/Bicategory/MarkedZigzagAlignedHammock.lean` | | `CategoryTheory.Bicategory.MarkedZigzag.HammockPath.sourceCompInv_normalizable` | `[propext, Classical.choice, Quot.sound]` | `Ript/ForMathlib/CategoryTheory/Bicategory/MarkedZigzagAlignedHammock.lean` | +| `CategoryTheory.Bicategory.MarkedZigzag.HammockPath.normalizedCellHom_nil_twoAtoms` | `[propext, Classical.choice, Quot.sound]` | `Ript/ForMathlib/CategoryTheory/Bicategory/MarkedZigzagAlignedHammock.lean` | +| `CategoryTheory.Bicategory.MarkedZigzag.HammockPath.normalizedCellHom_twoAtoms_nil` | `[propext, Classical.choice, Quot.sound]` | `Ript/ForMathlib/CategoryTheory/Bicategory/MarkedZigzagAlignedHammock.lean` | +| `CategoryTheory.Bicategory.MarkedZigzag.HammockPath.normalizedCellHom_markedUnit` | `[propext, Classical.choice, Quot.sound]` | `Ript/ForMathlib/CategoryTheory/Bicategory/MarkedZigzagAlignedHammock.lean` | +| `CategoryTheory.Bicategory.MarkedZigzag.HammockPath.normalizedCellHom_markedUnitInv` | `[propext, Classical.choice, Quot.sound]` | `Ript/ForMathlib/CategoryTheory/Bicategory/MarkedZigzagAlignedHammock.lean` | +| `CategoryTheory.Bicategory.MarkedZigzag.HammockPath.normalizedCellHom_markedCounit` | `[propext, Classical.choice, Quot.sound]` | `Ript/ForMathlib/CategoryTheory/Bicategory/MarkedZigzagAlignedHammock.lean` | +| `CategoryTheory.Bicategory.MarkedZigzag.HammockPath.normalizedCellHom_markedCounitInv` | `[propext, Classical.choice, Quot.sound]` | `Ript/ForMathlib/CategoryTheory/Bicategory/MarkedZigzagAlignedHammock.lean` | +| `CategoryTheory.Bicategory.MarkedZigzag.HammockPath.markedUnit_normalizable` | `[propext, Classical.choice, Quot.sound]` | `Ript/ForMathlib/CategoryTheory/Bicategory/MarkedZigzagAlignedHammock.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` | | `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 43bcc04..c778610 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 と厳密に一致します。残る十の marked pair、associator、unitor 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。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。 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 7ab5f59..8fe2f53 100644 --- a/docs/ja/reference/AXIOMS.md +++ b/docs/ja/reference/AXIOMS.md @@ -1024,6 +1024,16 @@ | `CategoryTheory.Bicategory.MarkedZigzag.HammockPath.normalizedCellHom_sourceCompInv` | `[propext, Classical.choice, Quot.sound]` | `Ript/ForMathlib/CategoryTheory/Bicategory/MarkedZigzagAlignedHammock.lean` | | `CategoryTheory.Bicategory.MarkedZigzag.HammockPath.sourceComp_normalizable` | `[propext, Classical.choice, Quot.sound]` | `Ript/ForMathlib/CategoryTheory/Bicategory/MarkedZigzagAlignedHammock.lean` | | `CategoryTheory.Bicategory.MarkedZigzag.HammockPath.sourceCompInv_normalizable` | `[propext, Classical.choice, Quot.sound]` | `Ript/ForMathlib/CategoryTheory/Bicategory/MarkedZigzagAlignedHammock.lean` | +| `CategoryTheory.Bicategory.MarkedZigzag.HammockPath.normalizedCellHom_nil_twoAtoms` | `[propext, Classical.choice, Quot.sound]` | `Ript/ForMathlib/CategoryTheory/Bicategory/MarkedZigzagAlignedHammock.lean` | +| `CategoryTheory.Bicategory.MarkedZigzag.HammockPath.normalizedCellHom_twoAtoms_nil` | `[propext, Classical.choice, Quot.sound]` | `Ript/ForMathlib/CategoryTheory/Bicategory/MarkedZigzagAlignedHammock.lean` | +| `CategoryTheory.Bicategory.MarkedZigzag.HammockPath.normalizedCellHom_markedUnit` | `[propext, Classical.choice, Quot.sound]` | `Ript/ForMathlib/CategoryTheory/Bicategory/MarkedZigzagAlignedHammock.lean` | +| `CategoryTheory.Bicategory.MarkedZigzag.HammockPath.normalizedCellHom_markedUnitInv` | `[propext, Classical.choice, Quot.sound]` | `Ript/ForMathlib/CategoryTheory/Bicategory/MarkedZigzagAlignedHammock.lean` | +| `CategoryTheory.Bicategory.MarkedZigzag.HammockPath.normalizedCellHom_markedCounit` | `[propext, Classical.choice, Quot.sound]` | `Ript/ForMathlib/CategoryTheory/Bicategory/MarkedZigzagAlignedHammock.lean` | +| `CategoryTheory.Bicategory.MarkedZigzag.HammockPath.normalizedCellHom_markedCounitInv` | `[propext, Classical.choice, Quot.sound]` | `Ript/ForMathlib/CategoryTheory/Bicategory/MarkedZigzagAlignedHammock.lean` | +| `CategoryTheory.Bicategory.MarkedZigzag.HammockPath.markedUnit_normalizable` | `[propext, Classical.choice, Quot.sound]` | `Ript/ForMathlib/CategoryTheory/Bicategory/MarkedZigzagAlignedHammock.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` | | `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 1e8f24f..0a359c8 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。剩余十个 marked pair、associator 和 unitor 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。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。 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 c67739b..72319e7 100644 --- a/docs/zh-CN/reference/AXIOMS.md +++ b/docs/zh-CN/reference/AXIOMS.md @@ -1024,6 +1024,16 @@ | `CategoryTheory.Bicategory.MarkedZigzag.HammockPath.normalizedCellHom_sourceCompInv` | `[propext, Classical.choice, Quot.sound]` | `Ript/ForMathlib/CategoryTheory/Bicategory/MarkedZigzagAlignedHammock.lean` | | `CategoryTheory.Bicategory.MarkedZigzag.HammockPath.sourceComp_normalizable` | `[propext, Classical.choice, Quot.sound]` | `Ript/ForMathlib/CategoryTheory/Bicategory/MarkedZigzagAlignedHammock.lean` | | `CategoryTheory.Bicategory.MarkedZigzag.HammockPath.sourceCompInv_normalizable` | `[propext, Classical.choice, Quot.sound]` | `Ript/ForMathlib/CategoryTheory/Bicategory/MarkedZigzagAlignedHammock.lean` | +| `CategoryTheory.Bicategory.MarkedZigzag.HammockPath.normalizedCellHom_nil_twoAtoms` | `[propext, Classical.choice, Quot.sound]` | `Ript/ForMathlib/CategoryTheory/Bicategory/MarkedZigzagAlignedHammock.lean` | +| `CategoryTheory.Bicategory.MarkedZigzag.HammockPath.normalizedCellHom_twoAtoms_nil` | `[propext, Classical.choice, Quot.sound]` | `Ript/ForMathlib/CategoryTheory/Bicategory/MarkedZigzagAlignedHammock.lean` | +| `CategoryTheory.Bicategory.MarkedZigzag.HammockPath.normalizedCellHom_markedUnit` | `[propext, Classical.choice, Quot.sound]` | `Ript/ForMathlib/CategoryTheory/Bicategory/MarkedZigzagAlignedHammock.lean` | +| `CategoryTheory.Bicategory.MarkedZigzag.HammockPath.normalizedCellHom_markedUnitInv` | `[propext, Classical.choice, Quot.sound]` | `Ript/ForMathlib/CategoryTheory/Bicategory/MarkedZigzagAlignedHammock.lean` | +| `CategoryTheory.Bicategory.MarkedZigzag.HammockPath.normalizedCellHom_markedCounit` | `[propext, Classical.choice, Quot.sound]` | `Ript/ForMathlib/CategoryTheory/Bicategory/MarkedZigzagAlignedHammock.lean` | +| `CategoryTheory.Bicategory.MarkedZigzag.HammockPath.normalizedCellHom_markedCounitInv` | `[propext, Classical.choice, Quot.sound]` | `Ript/ForMathlib/CategoryTheory/Bicategory/MarkedZigzagAlignedHammock.lean` | +| `CategoryTheory.Bicategory.MarkedZigzag.HammockPath.markedUnit_normalizable` | `[propext, Classical.choice, Quot.sound]` | `Ript/ForMathlib/CategoryTheory/Bicategory/MarkedZigzagAlignedHammock.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` | | `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` |