## Automating Proofs in Category Theory

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Citations: | 5 - 0 self |

### BibTeX

@MISC{Kozen_automatingproofs,

author = {Dexter Kozen and Christoph Kreitz},

title = {Automating Proofs in Category Theory},

year = {}

}

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### Abstract

Abstract. We introduce a semi-automated proof system for basic category-theoretic reasoning. It is based on a first-order sequent calculus that captures the basic properties of categories, functors and natural transformations as well as a small set of proof tactics that automate proof search in this calculus. We demonstrate our approach by automating the proof that the functor categories Fun[C × D,E] and Fun[C,Fun[D,E]] are naturally isomorphic. 1

### Citations

759 | Notions of computation and monads
- Moggi
- 1991
(Show Context)
Citation Context ...= ψA Γ ⊢ ϕ = ψ Finally, we also allow equations on types and substitution of equals for equals in type expressions. Any such equation α = β takes the form of a rule (13) (14) (15) (16) (17) (18) (19) =-=(20)-=- Γ ⊢ A : α . (21) Γ ⊢ A : β Other Rules There are also various rules for products, weakening, and other structural rules for manipulation of sequents. These are all quite standard and do not bear expl... |

509 |
word problems in universal algebras
- Knuth, Bendix
- 1970
(Show Context)
Citation Context ...ve rewrite system is incomplete, as it cannot prove the equality of some terms that can be shown equal with the inference rules. We have used the superposition-based Knuth-Bendix completion procedure =-=[15]-=- to generate the following additional typed rewrites. Rewrite Type Rules F 2 <1A, 1X> ↦→ 1 F 1 <A,X> E(F 1 <A,X>,F 1 <A,X>) (02),(08) F 2 <g, k>◦F 2 <f, h> ↦→ F 2 <g◦f, k◦h> E(F 1 <A,X>,F 1 <C,X>) (01... |

500 |
Categories for the Working Mathematician
- MacLane
- 1971
(Show Context)
Citation Context ...tegories, which are then applied to the category of sets. The authors formulate the theory of functors including Freyd’s adjoint functor theorem, i.e., their work covers nearly all of chapters I–V of =-=[18]-=-. The formalization of Saïbi and Huet is directly based on the constructive type theory of Coq. Simpson [26], on the other hand, makes only indirect use of it. Instead, his formalization is set up in ... |

417 |
Category theory for computing science
- Barr, Wells
- 1995
(Show Context)
Citation Context ... relies mostly on the underlying logic. 3s2 An Axiomatization of Elementary Category Theory 2.1 Notational Conventions We assume familiarity with the basic definitions and notation of category theory =-=[6, 18]-=-. To simplify notation, we will adhere to the following conventions. – Symbols in sans serif, such as C, always denote categories. The categories Set and Cat are the categories of sets and set functio... |

158 | Implementing Mathematics with the Nuprl Proof Development System
- Constable
- 1987
(Show Context)
Citation Context ...tomation in Section 2. This axiomatization is a slight modification of a system presented in [16]. We then describe an implementation of this calculus within the proof environment of the Nuprl system =-=[7, 1]-=- in Section 3 and strategies for automated proof search in Section 4. These strategies attempt to capture the general patterns of formal reasoning that we have observed in hand-constructed proofs usin... |

103 | A categorical manifesto
- Goguen
- 1991
(Show Context)
Citation Context ..., B), the following diagram commutes: ϕA F 1 A F 2 g ✲ F 1 B ❄ 2 ❄ G g✲ 1 G B G 1 A Composition and identities are defined by ϕB (5) (6) (7) (8) (9) (10) (ϕ ◦ ψ)A def = ϕA ◦ ψA (11) 1F A def = 1F 1A. =-=(12)-=- 5sThe property (10), along with the typing of ϕ, are captured in the following rules. Analysis Synthesis Γ ⊢ ϕ : Fun[C, D](F, G) Γ ⊢ F, G : Fun[C, D] Γ ⊢ ϕ : Fun[C, D](F, G), Γ ⊢ A : C Γ ⊢ ϕA : D(F 1... |

89 |
General theory of natural equivalences
- Eilenberg, Lane
- 1945
(Show Context)
Citation Context ... isomorphic. 1 Introduction Category theory is a popular framework for expressing abstract properties of mathematical structures. Since its invention in 1945 by Samuel Eilenberg and Saunders Mac Lane =-=[10]-=-, it has had a wide impact in many areas of mathematics and computer science. The beauty of category theory is that it allows one to be completely precise about otherwise informal concepts. Abstract a... |

44 | The Nuprl open logical environment
- Allen, Constable, et al.
(Show Context)
Citation Context ...l quite standard and do not bear explicit mention. 6s3 Implementation of the Formal Theory As platform for the implementation of our proof calculus we have selected the Nuprl proof development system =-=[7, 2, 17, 1]-=-. Nuprl is an environment for the development of formalized mathematical knowledge that supports interactive and tactic-based reasoning, decision procedures, language extensions through user-defined c... |

42 |
A calculus of functions for program derivation
- Bird
- 1990
(Show Context)
Citation Context ...mposable arrows. This problem is solved by the introduction of an error object, which is never a member of any set of arrows. Since his interests lie in a formalization of the Bird-Meertens formalism =-=[7]-=-, there are no attempts to improve automation beyond Isabelle’s generic prover. Another formalization of category theory in Isabelle is O’Keefe’s work described in [22]. His main focus is on the reada... |

28 | Constructive category theory
- Huet, Säıbi
- 2000
(Show Context)
Citation Context ...n automation, but unfortunately neither a description nor the sources have been published. In the Coq library there are two contributions concerning category theory. The development of Saïbi and Huet =-=[14, 25]-=- contains definitions and constructions up to cartesian closed categories, which are then applied to the category of sets. The authors formulate the theory of functors including Freyd’s adjoint functo... |

25 |
Innovations in computational type theory using Nuprl
- Allen, Bickford, et al.
- 2006
(Show Context)
Citation Context ...tomation in Section 2. This axiomatization is a slight modification of a system presented in [16]. We then describe an implementation of this calculus within the proof environment of the Nuprl system =-=[7, 1]-=- in Section 3 and strategies for automated proof search in Section 4. These strategies attempt to capture the general patterns of formal reasoning that we have observed in hand-constructed proofs usin... |

24 | Formalized Mathematics
- Harrison
- 1996
(Show Context)
Citation Context ...rd group consists of formalizations of category theory in interactive proof systems. In these formalizations, practical issues like feasibility, automation and elegance of the design (in the sense of =-=[13]-=-) play an important role. There are at least two formalizations of category theory in Isabelle/HOL that should be mentioned here. Glimming’s 2001 master thesis [11] describes a development of basic ca... |

6 |
Towards a readable formalisation of category theory
- O’Keefe
- 2004
(Show Context)
Citation Context ...alization of Bird-Meertens formalism, there are no attempts to improve automation beyond Isabelle’s generic prover. Another formalization of category theory in Isabelle is O’Keefe’s work described in =-=[21]-=-. His main focus is on the readability of the proofs, aiming at a representation close to one in a mathematical textbook. Therefore he uses a sectioning concept provided by Isabelle. This saves a lot ... |

5 | Concrete categories
- Bancerek
- 1996
(Show Context)
Citation Context ... of categories, which are essentially the rules of typed monoids. These rules include typing rules for composition and identities Γ ⊢ A, B, C : C, Γ ⊢ f : C(A, B), Γ ⊢ g : C(B, C) Γ ⊢ g ◦ f : C(A, C) =-=(3)-=- Γ ⊢ A : C , (4) Γ ⊢ 1A : C(A, A) as well as equational rules for associativity and two-sided identity. Functors A functor F from C to D is determined by its object and arrow components F 1 and F 2 . ... |

5 | Miscellaneous facts about functors
- Bancerek
- 2001
(Show Context)
Citation Context ...which are essentially the rules of typed monoids. These rules include typing rules for composition and identities Γ ⊢ A, B, C : C, Γ ⊢ f : C(A, B), Γ ⊢ g : C(B, C) Γ ⊢ g ◦ f : C(A, C) (3) Γ ⊢ A : C , =-=(4)-=- Γ ⊢ 1A : C(A, A) as well as equational rules for associativity and two-sided identity. Functors A functor F from C to D is determined by its object and arrow components F 1 and F 2 . The components m... |

5 |
Logic and automation for algebra of programming
- Glimming
- 2001
(Show Context)
Citation Context ...ance of the design (in the sense of [13]) play an important role. There are at least two formalizations of category theory in Isabelle/HOL that should be mentioned here. Glimming’s 2001 master thesis =-=[11]-=- describes a development of basic category theory and a couple of concrete categories. As HOL does not admit the definition of partial functions, Glimming had to address the problem of the composition... |

4 |
G.: A Higher-Order Calculus for Categories
- Cáccamo, Winskel
- 2001
(Show Context)
Citation Context ...tegory theory to use its machinery in several domains of computer science (for example denotational semantics). One of these approaches is Caccamo’s and Winskel’s Higher order calculus for categories =-=[8]-=-. The authors present a second order calculus for a fragment of category theory. Their approach is at a level higher than ours. Their basic types are (small) categories and the syntactic judgments des... |

3 | Categorial background for duality theory
- Bancerek
(Show Context)
Citation Context ... 1 A), called the component of ϕ at A, such that for all arrows g : C(A, B), the following diagram commutes: ϕA F 1 A F 2 g ✲ F 1 B ❄ 2 ❄ G g✲ 1 G B G 1 A Composition and identities are defined by ϕB =-=(5)-=- (6) (7) (8) (9) (10) (ϕ ◦ ψ)A def = ϕA ◦ ψA (11) 1F A def = 1F 1A. (12) 5sThe property (10), along with the typing of ϕ, are captured in the following rules. Analysis Synthesis Γ ⊢ ϕ : Fun[C, D](F, G... |

3 |
A graphical approach to monad compositions
- Eklund
- 2001
(Show Context)
Citation Context ...e component of ϕ at A, such that for all arrows g : C(A, B), the following diagram commutes: ϕA F 1 A F 2 g ✲ F 1 B ❄ 2 ❄ G g✲ 1 G B G 1 A Composition and identities are defined by ϕB (5) (6) (7) (8) =-=(9)-=- (10) (ϕ ◦ ψ)A def = ϕA ◦ ψA (11) 1F A def = 1F 1A. (12) 5sThe property (10), along with the typing of ϕ, are captured in the following rules. Analysis Synthesis Γ ⊢ ϕ : Fun[C, D](F, G) Γ ⊢ F, G : Fun... |

3 |
M.Barr & C.Wells, Category Theory for Computing Science
- Barr
- 1990
(Show Context)
Citation Context ...sAutomating Proofs in Category Theory 395 2 An Axiomatization of Elementary Category Theory 2.1 Notational Conventions We assume familiarity with the basic definitions and notation of category theory =-=[6,20]-=-. To simplify notation, we will adhere to the following conventions. – Symbols in sans serif, such as C, always denote categories. The categories Set and Cat are the categories of sets and set functio... |

2 | Toward the Automation of Category Theory
- Kozen
- 2004
(Show Context)
Citation Context ...ing. We first give a formal first-order axiomatization of elementary category theory that is amenable to automation in Section 2. This axiomatization is a slight modification of a system presented in =-=[16]-=-. We then describe an implementation of this calculus within the proof environment of the Nuprl system [7, 1] in Section 3 and strategies for automated proof search in Section 4. These strategies atte... |

1 |
The Nuprl Proof Development System, V5: Reference Manual and User’s Guide
- Kreitz
- 2002
(Show Context)
Citation Context ...l quite standard and do not bear explicit mention. 6s3 Implementation of the Formal Theory As platform for the implementation of our proof calculus we have selected the Nuprl proof development system =-=[7, 2, 17, 1]-=-. Nuprl is an environment for the development of formalized mathematical knowledge that supports interactive and tactic-based reasoning, decision procedures, language extensions through user-defined c... |

1 |
Innovations in computational type theory using Nuprl
- al
(Show Context)
Citation Context ...tomation in Section 2. This axiomatization is a slight modification of a system presented in [18]. We then describe an implementation of this calculus within the proof environment of the Nuprl system =-=[9,1]-=- in Section 3 and strategies for automated proof search in Section 4. These strategies attempt to capture the general patterns of formal reasoning that we have observed in hand-constructed proofs usin... |