HoTT Book in Lean ================= This file lists which sections of the [HoTT book](http://homotopytypetheory.org/book/) have been covered in the Lean [HoTT library](hott.md). Summary ------- The rows indicate the chapters, the columns the sections. * `+`: completely formalized * `¼`, `½` or `¾`: partly formalized * `-`: not formalized * `.`: no formalizable content | | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | 11 | 12 | 13 | 14 | 15 | |-------|---|---|---|---|---|---|---|---|---|----|----|----|----|----|----| | Ch 1 | . | . | . | . | + | + | + | + | + | . | + | + | | | | | Ch 2 | + | + | + | + | . | + | + | + | + | + | + | + | + | + | + | | Ch 3 | + | + | + | + | ½ | + | + | - | + | . | + | | | | | | Ch 4 | - | + | + | + | . | + | ½ | + | + | | | | | | | | Ch 5 | - | . | ½ | - | - | . | . | ½ | | | | | | | | | Ch 6 | . | + | + | + | + | ½ | ½ | + | ¾ | ¼ | ¾ | + | . | | | | Ch 7 | + | + | + | - | ¾ | - | - | | | | | | | | | | Ch 8 | ¾ | ¾ | - | - | - | - | - | - | - | - | | | | | | | Ch 9 | ¾ | + | + | ½ | ¾ | ½ | - | - | - | | | | | | | | Ch 10 | - | - | - | - | - | | | | | | | | | | | | Ch 11 | - | - | - | - | - | - | | | | | | | | | | Theorems and definitions in the library which are not in the book: * A major difference is that in this library we heavily use pathovers [D. Licata, G. Brunerie. A Cubical Approach to Synthetic Homotopy Theory]. This means that we need less theorems about transports, but instead corresponding theorems about pathovers. These are in [init.pathover](init/pathover.hlean). For higher paths there are [squares](cubical/square.hlean), [squareovers](cubical/squareover.hlean), and the rudiments of [cubes](cubical/cube.hlean) and [cubeovers](cubical/cubeover.hlean). * The category theory library is more extensive than what is presented in the book. For example, we have [limits](algebra/category/limits/limits.md). Chapter 1: Type theory --------- - 1.1 (Type theory versus set theory): no formalizable content. - 1.2 (Function types): no formalizable content. Related: [init.function](init/function.hlean) - 1.3 (Universes and families): no formalizable content (Lean also has a hierarchy of universes `Type.{i} : Type.{i + 1}`, but they are *not* cumulative). - 1.4 (Dependent function types (Π-types)): no formalizable content. Related: [init.function](init/function.hlean) - 1.5 (Product types): declaration in [init.datatypes](init/datatypes.hlean), notation in [init.types](init/types.hlean) - 1.6 (Dependent pair types (Σ-types)): declaration in [init.datatypes](init/datatypes.hlean), notation in [init.types](init/types.hlean) - 1.7 (Coproduct types): declaration in [init.datatypes](init/datatypes.hlean), notation in [init.types](init/types.hlean) - 1.8 (The type of booleans): declaration in [init.datatypes](init/datatypes.hlean), notation in [init.bool](init/bool.hlean) - 1.9 (The natural numbers): [init.nat](init/nat.hlean) (declaration in [init.datatypes](init/datatypes.hlean)) - 1.10 (Pattern matching and recursion): no formalizable content (we can use the "pattern matching" notation using the function definition package, which are reduced to applying recursors). - 1.11 (Propositions as types): some logic is in [init.logic](init/logic.hlean) and [init.types](init/types.hlean). - 1.12 (Identity types): declaration in [init.datatypes](init/datatypes.hlean), more in [init.logic](init/logic.hlean) Chapter 2: Homotopy type theory --------- - 2.1 (Types are higher groupoids): [init.path](init/path.hlean) (pointed types and loop spaces in [types.pointed](types/pointed.hlean)) - 2.2 (Functions are functors): [init.path](init/path.hlean) - 2.3 (Type families are fibrations): [init.path](init/path.hlean) - 2.4 (Homotopies and equivalences): homotopies in [init.path](init/path.hlean) and equivalences in [init.equiv](init/equiv.hlean) - 2.5 (The higher groupoid structure of type formers): no formalizable content - 2.6 (Cartesian product types): [types.prod](types/prod.hlean) - 2.7 (Σ-types): [types.sigma](types/sigma.hlean) - 2.8 (The unit type): special case of [init.trunc](init/trunc.hlean) - 2.9 (Π-types and the function extensionality axiom): [init.funext](init/funext.hlean) and [types.pi](types/pi.hlean) - 2.10 (Universes and the univalence axiom): [init.ua](init/ua.hlean) - 2.11 (Identity type): [init.equiv](init/equiv.hlean) (ap is equivalence), [types.eq](types/eq.hlean) and [cubical.square](cubical/square.hlean) (characterization of pathovers in equality types) - 2.12 (Coproducts): [types.sum](types/sum.hlean) - 2.13 (Natural numbers): [types.nat.hott](types/nat/hott.hlean) - 2.14 (Example: equality of structures): algebra formalized in [algebra.group](algebra/group.hlean). - 2.15 (Universal properties): in the corresponding file in the [types](types/types.md) folder. Chapter 3: Sets and logic --------- - 3.1 (Sets and n-types): [init.trunc](init/trunc.hlean). Example 3.1.9 in [types.univ](types/univ.hlean) - 3.2 (Propositions as types?): [types.univ](types/univ.hlean) - 3.3 (Mere propositions): [init.trunc](init/trunc.hlean) and [hprop_trunc](hprop_trunc.hlean) (Lemma 3.3.5). - 3.4 (Classical vs. intuitionistic logic): decidable is defined in [init.logic](init/logic.hlean) - 3.5 (Subsets and propositional resizing): Lemma 3.5.1 is subtype_eq in [types.sigma](types/sigma.hlean), we don't have propositional resizing as axiom yet. - 3.6 (The logic of mere propositions): in the corresponding file in the [types](types/types.md) folder. (is_trunc_prod is defined in [types.sigma](types/sigma.hlean)) - 3.7 (Propositional truncation): [init.hit](init/hit.hlean) and [hit.trunc](hit/trunc.hlean) - 3.8 (The axiom of choice): not formalized - 3.9 (The principle of unique choice): Lemma 9.3.1 in [hit.trunc](hit/trunc.hlean), Lemma 9.3.2 in [types.trunc](types/trunc.hlean) - 3.10 (When are propositions truncated?): no formalizable content - 3.11 (Contractibility): [init.trunc](init/trunc.hlean) (mostly), [types.pi](types/pi.hlean) (Lemma 3.11.6), [types.trunc](types/trunc.hlean) (Lemma 3.11.7), [types.sigma](types/sigma.hlean) (Lemma 3.11.9) Chapter 4: Equivalences --------- - 4.1 (Quasi-inverses): not formalized - 4.2 (Half adjoint equivalences): [init.equiv](init/equiv.hlean) and [types.equiv](types/equiv.hlean) - 4.3 (Bi-invertible maps): [function](function.hlean) ("biinv f" is "is_retraction f × is_section f") - 4.4 (Contractible fibers): [types.equiv](types/equiv.hlean) - 4.5 (On the definition of equivalences): no formalizable content - 4.6 (Surjections and embeddings): [function](function.hlean) - 4.7 (Closure properties of equivalences): 4.7.1 in [init.equiv](init/equiv.hlean); 4.7.2 in [function](function.hlean); 4.7.5 and 4.7.7 in [types.sigma](types/sigma.hlean) (sigma_functor is a generalization of total(f)); and 4.7.6 in 4.7.6 in [types.fiber](types/fiber.hlean). - 4.8 (The object classifier): 4.8.1 and 4.8.2 in [types.fiber](types/fiber.hlean); 4.8.3 and 4.8.4 in [types.univ](types/univ.hlean) - 4.9 (Univalence implies function extensionality): [init.funext](init/funext.hlean) Chapter 5: Induction --------- - 5.1 (Introduction to inductive types): not formalized - 5.2 (Uniqueness of inductive types): no formalizable content - 5.3 (W-types): [types.W](types/W.hlean) defines W-types. - 5.4 (Inductive types are initial algebras): not formalized - 5.5 (Homotopy-inductive types): not formalized - 5.6 (The general syntax of inductive definitions): no formalizable content - 5.7 (Generalizations of inductive types): no formalizable content. Lean has inductive families and mutual induction, but no induction-induction or induction-recursion - 5.8 (Identity types and identity systems): 5.8.1-5.8.4 not formalized, 5.8.5 in [init.ua](init/ua.hlean) and 5.8.6 in [init.funext](init/funext.hlean) Chapter 6: Higher inductive types --------- - 6.1 (Introduction): no formalizable content - 6.2 (Induction principles and dependent paths): dependent paths in [init.pathover](init/pathover.hlean), circle in [homotopy.circle](homotopy/circle.hlean) - 6.3 (The interval): [homotopy.interval](homotopy/interval.hlean) - 6.4 (Circles and spheres): [homotopy.sphere](homotopy/sphere.hlean) and [homotopy.circle](homotopy/circle.hlean) - 6.5 (Suspensions): [homotopy.suspension](homotopy/suspension.hlean) (we define the circle to be the suspension of bool, but Lemma 6.5.1 is similar to proving the ordinary induction principle for the circle in [homotopy.circle](homotopy/circle.hlean)) and a bit in [homotopy.sphere](homotopy/sphere.hlean) and [types.pointed](types/pointed.hlean) - 6.6 (Cell complexes): we define the torus using the quotient, see [hit.two_quotient](hit/two_quotient.hlean) and [homotopy.torus](homotopy/torus.hlean) (no dependent eliminator defined yet) - 6.7 (Hubs and spokes): [hit.two_quotient](hit/two_quotient.hlean) and [homotopy.torus](homotopy/torus.hlean) (no dependent eliminator defined yet) - 6.8 (Pushouts): [hit.pushout](hit/pushout.hlean). Some of the "standard homotopy-theoretic constructions" have separate files, although not all of them have been defined explicitly yet - 6.9 (Truncations): [hit.trunc](hit/trunc.hlean) (except Lemma 6.9.3) - 6.10 (Quotients): [hit.set_quotient](hit/set_quotient.hlean) (up to 6.10.3). We define integers differently, to make them compute, in the folder [types.int](types/int/int.md). 6.10.13 is in [types.int.hott](types/int/hott.hlean) - 6.11 (Algebra): [algebra.group](algebra/group.hlean), [algebra.fundamental_group](algebra/fundamental_group.hlean) (no homotopy groups yet) - 6.12 (The flattening lemma): [hit.quotient](hit/quotient.hlean) (for quotients instead of homotopy coequalizers, but these are practically the same) - 6.13 (The general syntax of higher inductive definitions): no formalizable content Chapter 7: Homotopy n-types --------- - 7.1 (Definition of n-types): [init.trunc](init/trunc.hlean), [types.trunc](types/trunc.hlean), [types.sigma](types/sigma.hlean) (Theorem 7.1.8), [types.pi](types/pi.hlean) (Theorem 7.1.9), [hprop_trunc](hprop_trunc.hlean) (Theorem 7.1.10) - 7.2 (Uniqueness of identity proofs and Hedberg’s theorem): [init.hedberg](init/hedberg.hlean) and [types.trunc](types/trunc.hlean) - 7.3 (Truncations): [init.hit](init/hit.hlean), [hit.trunc](hit/trunc.hlean) and [types.trunc](types/trunc.hlean) - 7.4 (Colimits of n-types): not formalized - 7.5 (Connectedness): [homotopy.connectedness](homotopy/connectedness.hlean) (the main "induction principle" Lemma 7.5.7) - 7.6 (Orthogonal factorization): not formalized - 7.7 (Modalities): not formalized, and may be unformalizable in general because it's unclear how to define modalities Chapter 8: Homotopy theory --------- Unless otherwise noted, the files are in the folder [homotopy](homotopy/homotopy.md) - 8.1 (π_1(S^1)): [homotopy.circle](homotopy/circle.hlean) (only one of the proofs) - 8.2 (Connectedness of suspensions): [homotopy.connectedness](homotopy/connectedness.hlean) (different proof) - 8.3 (πk≤n of an n-connected space and π_{k