* One major difference is that in this library we heavily use pathovers, so 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](types/cubical/square.hlean), [squareovers](types/cubical/squareover.hlean), and the rudiments of [cubes](types/cubical/cube.hlean) and [cubeovers](types/cubical/cubeover.hlean).
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
- 2.8 (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 [types.cubical.square](types/cubical/square.hlean) (characterization of pathovers in equality types)
- 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)
- 3.2 (Propositions as types?): not formalized
- 3.3 (Mere propositions): [init.trunc](init/trunc.hlean) and [types.hprop_trunc](types/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 is equiv_trunc in [hit.trunc](hit/trunc.hlean), Lemma 9.3.2 is not formalized
- 3.10 (When are propositions truncated?): no formalizable content
- 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 [hit.circle](hit/circle.hlean)
- 6.3 (The interval): [hit.interval](hit/interval.hlean)
- 6.4 (Circles and spheres): [hit.circle](hit/circle.hlean)
- 6.5 (Suspensions): [hit.suspension](hit/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 [hit.circle](hit/circle.hlean)) and a bit in [hit.sphere](hit/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 [hit.torus](hit/torus.hlean) (no dependent eliminator defined yet)
- 6.7 (Hubs and spokes): [hit.two_quotient](hit/two_quotient.hlean) and [hit.torus](hit/torus.hlean) (no dependent eliminator defined yet)
- 9.5 (The Yoneda lemma): [constructions.opposite](algebra/category/constructions/opposite.hlean), [constructions.product](algebra/category/constructions/product.hlean), [yoneda](algebra/category/yoneda.hlean) (up to definition of Yoneda embedding)
