feat(hott): add file which maps sections of the HoTT book to the HoTT library

hott/book.md

@@ -0,0 +1,177 @@
+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 | + | + | - | ½ | .  | +  |    |    |    |    |
+| Ch 4  | - | + | - | + | . | + | - | - | + |    |    |    |    |    |    |
+| Ch 5  | - | . | - | - | - | . | . | ½ |   |    |    |    |    |    |    |
+| Ch 6  | . | + | + | + | + | ½ | ½ | ¼ | ¼ | ¼  | ¾  | -  | .  |    |    |
+| Ch 7  | + | + | + | - | - | - | - |   |   |    |    |    |    |    |    |
+| Ch 8  | ¾ | - | - | - | - | - | - | - | - | -  |    |    |    |    |    |
+| Ch 9  | ¾ | + | ¼ | ¼ | ½ | ½ | - | - | - |    |    |    |    |    |    |
+| Ch 10 | - | - | - | - | - |   |   |   |   |    |    |    |    |    |    |
+| Ch 11 | - | - | - | - | - | - |   |   |   |    |    |    |    |    |    |
+
+Things not in the book:
+
+* 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
+- 2.6 (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 [types.cubical.square](types/cubical/square.hlean) (characterization of pathovers in equality types)
+- 2.12 (Coproducts): not formalized
+- 2.13 (Natural numbers): not formalized
+- 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
+- 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): not formalized
+- 4.4 (Contractible fibers): [types.equiv](types/equiv.hlean)
+- 4.5 (On the definition of equivalences): no formalizable content
+- 4.6 (Surjections and embeddings): [types.function](types/function.hlean)
+- 4.7 (Closure properties of equivalences): not formalized
+- 4.8 (The object classifier): not formalized
+- 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): related: [types.W](types/W.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)
+
+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)
+- 6.8 (Pushouts): [hit.pushout](hit/pushout.hlean) (not everything yet)
+- 6.9 (Truncations): [hit.trunc](hit/trunc.hlean) (not everything yet)
+- 6.10 (Quotients): [hit.set_quotient](hit/set_quotient.hlean), [types.int](types/int/int.md) (folder) (not everything yet)
+- 6.11 (Algebra): [algebra.group](algebra/group.hlean), [algebra.fundamental_group](algebra/fundamental_group.hlean) (no homotopy groups yet)
+- 6.12 (The flattening lemma): not formalized yet
+- 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), [types.hprop_trunc](types/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): not formalized
+- 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
+---------
+
+- 8.1 (π_1(S^1)): [hit.circle](hit/circle.hlean) (only one of the proofs)
+- 8.2 (Connectedness of suspensions): not formalized
+- 8.3 (πk≤n of an n-connected space and π_{k<n}(S^n)): not formalized
+- 8.4 (Fiber sequences and the long exact sequence): not formalized 
+- 8.5 (The Hopf fibration): not formalized 
+- 8.6 (The Freudenthal suspension theorem): not formalized 
+- 8.7 (The van Kampen theorem): not formalized 
+- 8.8 (Whitehead’s theorem and Whitehead’s principle): not formalized 
+- 8.9 (A general statement of the encode-decode method): not formalized 
+- 8.10 (Additional Results): not formalized 
+
+Chapter 9: Category theory
+---------
+
+Every file is in the folder [algebra.category](algebra/category/category.md)
+
+- 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)
+- 9.2 (Functors and transformations): [functor](algebra/category/functor.hlean), [nat_trans](algebra/category/nat_trans.hlean), [constructions.functor](algebra/category/constructions/functor.hlean)
+- 9.3 (Adjunctions): [adjoint](algebra/category/adjoint.hlean) (only definition)
+- 9.4 (Equivalences): [adjoint](algebra/category/adjoint.hlean) (only definitions)
+- 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)
+- 9.6 (Strict categories): [strict](algebra/category/strict.hlean) (only definition)
+- 9.7 (†-categories): not formalized
+- 9.8 (The structure identity principle): not formalized
+- 9.9 (The Rezk completion): not formalized
+
+Chapter 10: Set theory
+----------
+
+Not formalized, and parts may be unformalizable because Lean lacks induction-recursion.
+
+
+Chapter 11: Real numbers
+----------
+
+- 11.1 (The field of rational numbers): To be ported from the standard library.
+
+The rest is not formalized, and parts may be unformalizable because Lean lacks induction-induction

hott/hit/interval.hlean

@@ -62,6 +62,10 @@ namespace interval
   definition is_contr_interval [instance] [priority 900] : is_contr interval :=
   is_contr.mk zero (λx, interval.rec_on x idp seg !pathover_eq_r_idp)
 
+  open is_equiv equiv
+  definition naive_funext_of_interval : naive_funext :=
+  λA P f g p, ap (λ(i : interval) (x : A), interval.elim_on i (f x) (g x) (p x)) seg
+
 end interval open interval
 
 definition cube : ℕ → Type₀

hott/hit/trunc.hlean

@@ -67,6 +67,9 @@ namespace trunc
              (λyy, trunc.rec_on yy (λy, ap tr !right_inv))
              (λxx, trunc.rec_on xx (λx, ap tr !left_inv))
 
+  definition trunc_homotopy {f g : X → Y} (p : f ~ g) : trunc_functor n f ~ trunc_functor n g :=
+  λxx, trunc.rec_on xx (λx, ap tr (p x))
+
   section
     open equiv.ops
     definition trunc_equiv_trunc (f : X ≃ Y) : trunc n X ≃ trunc n Y :=

hott/hott.md

@@ -10,11 +10,14 @@ files and directories:
 * [algebra](algebra/algebra.md) : algebraic structures
 * [arity](arity.hlean) : a file containing theorems about functions with arity 2 or higher
 
+See [book.md](book.md) for an overview of the sections of the [HoTT book](http://homotopytypetheory.org/book/) which have been covered.
+
 Lean's homotopy type theory kernel is a version of Martin-Löf Type Theory with:
 
 * universe polymorphism
 * a non-cumulative hierarchy of universes, `Type 0`, `Type 1`, ...
 * inductively defined types
+* [Two HITs](init/hit.hlean): `n`-truncation and quotients.
 
 Note that there is no proof-irrelevant or impredicative universe.
 

hott/init/bool.hlean

@@ -7,6 +7,8 @@ Authors: Leonardo de Moura
 prelude
 import init.reserved_notation
 
+-- this is not in init.types, because that file depends on init.num,
+-- which depends on these definitions
 namespace bool
   definition cond {A : Type} (b : bool) (t e : A) :=
   bool.rec_on b e t

hott/init/path.hlean

@@ -7,7 +7,7 @@ Ported from Coq HoTT
 -/
 
 prelude
-import .function .datatypes .relation .tactic
+import .function .tactic
 
 open function eq
 
@@ -557,10 +557,10 @@ namespace eq
 
   /- Whiskering -/
 
-  definition whisker_left (p : x = y) {q r : y = z} (h : q = r) : p ⬝ q = p ⬝ r :=
+  definition whisker_left [unfold 8] (p : x = y) {q r : y = z} (h : q = r) : p ⬝ q = p ⬝ r :=
   idp ◾ h
 
-  definition whisker_right {p q : x = y} (h : p = q) (r : y = z) : p ⬝ r = q ⬝ r :=
+  definition whisker_right [unfold 7] {p q : x = y} (h : p = q) (r : y = z) : p ⬝ r = q ⬝ r :=
   h ◾ idp
 
   -- Unwhiskering, a.k.a. cancelling

library/library.md

@@ -21,6 +21,7 @@ Lean's default logical framework is a version of the Calculus of Constructions w
 * universe polymorphism
 * a non-cumulative hierarchy of universes, `Type 1`, `Type 2`, ... above `Prop`
 * inductively defined types
+* quotient types
 
 The `standard` library does not rely on any axioms beyond this framework, and is
 hence constructive. It includes theories of the natural numbers, integers,