feat(hott): add file which maps sections of the HoTT book to the HoTT library
This commit is contained in:
Floris van Doorn
Leonardo de Moura
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d111607890
7 changed files with 193 additions and 3 deletions
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hott/book.md
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177
hott/book.md
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HoTT Book in Lean
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=================
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This file lists which sections of the [HoTT book](http://homotopytypetheory.org/book/) have been covered in the Lean [HoTT library](hott.md).
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Summary
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-------
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The rows indicate the chapters, the columns the sections.
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* `+`: completely formalized
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* `¼`, `½` or `¾`: partly formalized
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* `-`: not formalized
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* `.`: no formalizable content
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| | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | 11 | 12 | 13 | 14 | 15 |
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|-------|---|---|---|---|---|---|---|---|---|----|----|----|----|----|----|
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| Ch 1 | . | . | . | . | + | + | + | + | + | . | + | + | | | |
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| Ch 2 | + | + | + | + | . | + | + | + | + | + | + | - | - | + | + |
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| Ch 3 | + | - | + | + | ½ | + | + | - | ½ | . | + | | | | |
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| Ch 4 | - | + | - | + | . | + | - | - | + | | | | | | |
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| Ch 5 | - | . | - | - | - | . | . | ½ | | | | | | | |
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| Ch 6 | . | + | + | + | + | ½ | ½ | ¼ | ¼ | ¼ | ¾ | - | . | | |
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| Ch 7 | + | + | + | - | - | - | - | | | | | | | | |
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| Ch 8 | ¾ | - | - | - | - | - | - | - | - | - | | | | | |
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| Ch 9 | ¾ | + | ¼ | ¼ | ½ | ½ | - | - | - | | | | | | |
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| Ch 10 | - | - | - | - | - | | | | | | | | | | |
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| Ch 11 | - | - | - | - | - | - | | | | | | | | | |
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Things not in the book:
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* 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).
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Chapter 1: Type theory
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---------
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- 1.1 (Type theory versus set theory): no formalizable content.
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- 1.2 (Function types): no formalizable content. Related: [init.function](init/function.hlean)
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- 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).
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- 1.4 (Dependent function types (Π-types)): no formalizable content. Related: [init.function](init/function.hlean)
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- 1.5 (Product types): declaration in [init.datatypes](init/datatypes.hlean), notation in [init.types](init/types.hlean)
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- 1.6 (Dependent pair types (Σ-types)): declaration in [init.datatypes](init/datatypes.hlean), notation in [init.types](init/types.hlean)
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- 1.7 (Coproduct types): declaration in [init.datatypes](init/datatypes.hlean), notation in [init.types](init/types.hlean)
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- 1.8 (The type of booleans): declaration in [init.datatypes](init/datatypes.hlean), notation in [init.bool](init/bool.hlean)
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- 1.9 (The natural numbers): [init.nat](init/nat.hlean) (declaration in [init.datatypes](init/datatypes.hlean))
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- 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).
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- 1.11 (Propositions as types): some logic is in [init.logic](init/logic.hlean) and [init.types](init/types.hlean).
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- 1.12 (Identity types): declaration in [init.datatypes](init/datatypes.hlean), more in [init.logic](init/logic.hlean)
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Chapter 2: Homotopy type theory
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---------
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- 2.1 (Types are higher groupoids): [init.path](init/path.hlean) (pointed types and loop spaces in [types.pointed](types/pointed.hlean))
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- 2.2 (Functions are functors): [init.path](init/path.hlean)
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- 2.3 (Type families are fibrations): [init.path](init/path.hlean)
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- 2.4 (Homotopies and equivalences): homotopies in [init.path](init/path.hlean) and equivalences in [init.equiv](init/equiv.hlean)
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- 2.5 (The higher groupoid structure of type formers): no formalizable content
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- 2.6 (Cartesian product types): [types.prod](types/prod.hlean)
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- 2.7 (Σ-types): [types.sigma](types/sigma.hlean)
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- 2.8 (The unit type): special case of [init.trunc](init/trunc.hlean)
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- 2.9 (Π-types and the function extensionality axiom): [init.funext](init/funext.hlean) and [types.pi](types/pi.hlean)
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- 2.10 (Universes and the univalence axiom): [init.ua](init/ua.hlean)
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- 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)
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- 2.12 (Coproducts): not formalized
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- 2.13 (Natural numbers): not formalized
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- 2.14 (Example: equality of structures): algebra formalized in [algebra.group](algebra/group.hlean).
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- 2.15 (Universal properties): in the corresponding file in the [types](types/types.md) folder.
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Chapter 3: Sets and logic
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---------
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- 3.1 (Sets and n-types): [init.trunc](init/trunc.hlean)
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- 3.2 (Propositions as types?): not formalized
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- 3.3 (Mere propositions): [init.trunc](init/trunc.hlean) and [types.hprop_trunc](types/hprop_trunc.hlean) (Lemma 3.3.5).
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- 3.4 (Classical vs. intuitionistic logic): decidable is defined in [init.logic](init/logic.hlean)
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- 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.
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- 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))
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- 3.7 (Propositional truncation): [init.hit](init/hit.hlean) and [hit.trunc](hit/trunc.hlean)
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- 3.8 (The axiom of choice): not formalized
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- 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
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- 3.10 (When are propositions truncated?): no formalizable content
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- 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)
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Chapter 4: Equivalences
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---------
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- 4.1 (Quasi-inverses): not formalized
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- 4.2 (Half adjoint equivalences): [init.equiv](init/equiv.hlean) and [types.equiv](types/equiv.hlean)
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- 4.3 (Bi-invertible maps): not formalized
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- 4.4 (Contractible fibers): [types.equiv](types/equiv.hlean)
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- 4.5 (On the definition of equivalences): no formalizable content
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- 4.6 (Surjections and embeddings): [types.function](types/function.hlean)
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- 4.7 (Closure properties of equivalences): not formalized
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- 4.8 (The object classifier): not formalized
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- 4.9 (Univalence implies function extensionality): [init.funext](init/funext.hlean)
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Chapter 5: Induction
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---------
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- 5.1 (Introduction to inductive types): not formalized
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- 5.2 (Uniqueness of inductive types): no formalizable content
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- 5.3 (W-types): related: [types.W](types/W.hlean)
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- 5.4 (Inductive types are initial algebras): not formalized
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- 5.5 (Homotopy-inductive types): not formalized
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- 5.6 (The general syntax of inductive definitions): no formalizable content
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- 5.7 (Generalizations of inductive types): no formalizable content. Lean has inductive families and mutual induction, but no induction-induction or induction-recursion
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- 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)
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Chapter 6: Higher inductive types
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---------
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- 6.1 (Introduction): no formalizable content
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- 6.2 (Induction principles and dependent paths): dependent paths in [init.pathover](init/pathover.hlean), circle in [hit.circle](hit/circle.hlean)
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- 6.3 (The interval): [hit.interval](hit/interval.hlean)
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- 6.4 (Circles and spheres): [hit.circle](hit/circle.hlean)
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- 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)
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- 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)
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- 6.7 (Hubs and spokes): [hit.two_quotient](hit/two_quotient.hlean) and [hit.torus](hit/torus.hlean) (no dependent eliminator defined yet)
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- 6.8 (Pushouts): [hit.pushout](hit/pushout.hlean) (not everything yet)
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- 6.9 (Truncations): [hit.trunc](hit/trunc.hlean) (not everything yet)
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- 6.10 (Quotients): [hit.set_quotient](hit/set_quotient.hlean), [types.int](types/int/int.md) (folder) (not everything yet)
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- 6.11 (Algebra): [algebra.group](algebra/group.hlean), [algebra.fundamental_group](algebra/fundamental_group.hlean) (no homotopy groups yet)
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- 6.12 (The flattening lemma): not formalized yet
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- 6.13 (The general syntax of higher inductive definitions): no formalizable content
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Chapter 7: Homotopy n-types
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---------
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- 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), [types.hprop_trunc](types/hprop_trunc.hlean) (Theorem 7.1.10)
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- 7.2 (Uniqueness of identity proofs and Hedberg’s theorem): [init.hedberg](init/hedberg.hlean) and [types.trunc](types/trunc.hlean)
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- 7.3 (Truncations): [init.hit](init/hit.hlean), [hit.trunc](hit/trunc.hlean) and [types.trunc](types/trunc.hlean)
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- 7.4 (Colimits of n-types): not formalized
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- 7.5 (Connectedness): not formalized
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- 7.6 (Orthogonal factorization): not formalized
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- 7.7 (Modalities): not formalized, and may be unformalizable in general because it's unclear how to define modalities
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Chapter 8: Homotopy theory
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---------
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- 8.1 (π_1(S^1)): [hit.circle](hit/circle.hlean) (only one of the proofs)
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- 8.2 (Connectedness of suspensions): not formalized
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- 8.3 (πk≤n of an n-connected space and π_{k<n}(S^n)): not formalized
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- 8.4 (Fiber sequences and the long exact sequence): not formalized
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- 8.5 (The Hopf fibration): not formalized
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- 8.6 (The Freudenthal suspension theorem): not formalized
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- 8.7 (The van Kampen theorem): not formalized
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- 8.8 (Whitehead’s theorem and Whitehead’s principle): not formalized
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- 8.9 (A general statement of the encode-decode method): not formalized
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- 8.10 (Additional Results): not formalized
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Chapter 9: Category theory
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---------
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Every file is in the folder [algebra.category](algebra/category/category.md)
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- 9.1 (Categories and precategories): [precategory](algebra/category/precategory.hlean), [iso](algebra/category/iso.hlean), [category](algebra/category/category.hlean), [groupoid](algebra/category/groupoid.hlean) (mostly)
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- 9.2 (Functors and transformations): [functor](algebra/category/functor.hlean), [nat_trans](algebra/category/nat_trans.hlean), [constructions.functor](algebra/category/constructions/functor.hlean)
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- 9.3 (Adjunctions): [adjoint](algebra/category/adjoint.hlean) (only definition)
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- 9.4 (Equivalences): [adjoint](algebra/category/adjoint.hlean) (only definitions)
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- 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)
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- 9.6 (Strict categories): [strict](algebra/category/strict.hlean) (only definition)
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- 9.7 (†-categories): not formalized
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- 9.8 (The structure identity principle): not formalized
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- 9.9 (The Rezk completion): not formalized
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Chapter 10: Set theory
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----------
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Not formalized, and parts may be unformalizable because Lean lacks induction-recursion.
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Chapter 11: Real numbers
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----------
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- 11.1 (The field of rational numbers): To be ported from the standard library.
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The rest is not formalized, and parts may be unformalizable because Lean lacks induction-induction
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@ -62,6 +62,10 @@ namespace interval
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definition is_contr_interval [instance] [priority 900] : is_contr interval :=
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is_contr.mk zero (λx, interval.rec_on x idp seg !pathover_eq_r_idp)
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open is_equiv equiv
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definition naive_funext_of_interval : naive_funext :=
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λA P f g p, ap (λ(i : interval) (x : A), interval.elim_on i (f x) (g x) (p x)) seg
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end interval open interval
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definition cube : ℕ → Type₀
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@ -67,6 +67,9 @@ namespace trunc
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(λyy, trunc.rec_on yy (λy, ap tr !right_inv))
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(λxx, trunc.rec_on xx (λx, ap tr !left_inv))
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definition trunc_homotopy {f g : X → Y} (p : f ~ g) : trunc_functor n f ~ trunc_functor n g :=
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λxx, trunc.rec_on xx (λx, ap tr (p x))
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section
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open equiv.ops
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definition trunc_equiv_trunc (f : X ≃ Y) : trunc n X ≃ trunc n Y :=
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@ -10,11 +10,14 @@ files and directories:
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* [algebra](algebra/algebra.md) : algebraic structures
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* [arity](arity.hlean) : a file containing theorems about functions with arity 2 or higher
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See [book.md](book.md) for an overview of the sections of the [HoTT book](http://homotopytypetheory.org/book/) which have been covered.
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Lean's homotopy type theory kernel is a version of Martin-Löf Type Theory with:
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* universe polymorphism
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* a non-cumulative hierarchy of universes, `Type 0`, `Type 1`, ...
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* inductively defined types
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* [Two HITs](init/hit.hlean): `n`-truncation and quotients.
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Note that there is no proof-irrelevant or impredicative universe.
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@ -7,6 +7,8 @@ Authors: Leonardo de Moura
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prelude
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import init.reserved_notation
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-- this is not in init.types, because that file depends on init.num,
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-- which depends on these definitions
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namespace bool
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definition cond {A : Type} (b : bool) (t e : A) :=
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bool.rec_on b e t
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@ -7,7 +7,7 @@ Ported from Coq HoTT
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-/
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prelude
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import .function .datatypes .relation .tactic
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import .function .tactic
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open function eq
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@ -557,10 +557,10 @@ namespace eq
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/- Whiskering -/
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definition whisker_left (p : x = y) {q r : y = z} (h : q = r) : p ⬝ q = p ⬝ r :=
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definition whisker_left [unfold 8] (p : x = y) {q r : y = z} (h : q = r) : p ⬝ q = p ⬝ r :=
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idp ◾ h
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definition whisker_right {p q : x = y} (h : p = q) (r : y = z) : p ⬝ r = q ⬝ r :=
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definition whisker_right [unfold 7] {p q : x = y} (h : p = q) (r : y = z) : p ⬝ r = q ⬝ r :=
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h ◾ idp
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-- Unwhiskering, a.k.a. cancelling
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@ -21,6 +21,7 @@ Lean's default logical framework is a version of the Calculus of Constructions w
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* universe polymorphism
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* a non-cumulative hierarchy of universes, `Type 1`, `Type 2`, ... above `Prop`
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* inductively defined types
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* quotient types
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The `standard` library does not rely on any axioms beyond this framework, and is
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hence constructive. It includes theories of the natural numbers, integers,
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