* 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).
- 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
- 2.11 (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)
- 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)
- 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)
- 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)
- 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
- 6.10 (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)
