## A Basic Extended Simple Type Theory (2001)

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@MISC{Farmer01abasic,

author = {William M. Farmer},

title = {A Basic Extended Simple Type Theory},

year = {2001}

}

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### Abstract

This paper presents an extended version of Church's simple type theory called Basic Extended Simple Type Theory (bestt). By adding type variables and support for reasoning with tuples, lists, and sets to simple type theory, it is intended to be a practical logic for formalized mathematics. 1

### Citations

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A formulation of the simple theory of types
- Church
- 1940
(Show Context)
Citation Context ... and F. Ramsey [15] suggested in the 1920s a simplified formulation of the ramified theory of types called the simple theory of types or, more briefly, simple type theory. A. Church presented in 1940 =-=[4]-=- a version of simple type theory that included lambda-notation. Church’s simple type theory, which we will denote as cstt, can be viewed as a “function theory”: functions are the basic objects and rea... |

500 |
T.: Introduction to HOL: A Theorem Proving Environment for Higher Order Logic: Cambridge
- Melham
- 1993
(Show Context)
Citation Context ...oning with functions, is a more practical reasoning system than traditional Zermelo-Fraenkel set theory. cstt is the basis of the logics used in several computer theorem proving systems including hol =-=[10]-=-, imps [8, 9], Isabelle [14], ProofPower [12], pvs [13], and tps [3]. This paper presents an extended version of cstt called Basic Extended Simple Type Theory (bestt). It adds the following facilities... |

420 | Isabelle: A generic theorem prover
- Paulson
- 1994
(Show Context)
Citation Context ...ore practical reasoning system than traditional Zermelo-Fraenkel set theory. cstt is the basis of the logics used in several computer theorem proving systems including hol [10], imps [8, 9], Isabelle =-=[14]-=-, ProofPower [12], pvs [13], and tps [3]. This paper presents an extended version of cstt called Basic Extended Simple Type Theory (bestt). It adds the following facilities to cstt: (1) Type variables... |

304 |
An Introduction To Mathematical Logic and Type Theory: To Truth Through Proof
- Andrews
- 1986
(Show Context)
Citation Context ..., and equality (first shown by L. Henkin in [11] and improved by P. Andrews in [1]). cstt has been widely influential in formalized mathematics. Many people have argued (e.g., see Andrews’ remarks in =-=[2]-=-) that cstt, with its strong support for reasoning with functions, is a more practical reasoning system than traditional Zermelo-Fraenkel set theory. cstt is the basis of the logics used in several co... |

206 | PVS: Combining specification, proof checking, and model checking
- Owre, Rajan, et al.
- 1996
(Show Context)
Citation Context ...tem than traditional Zermelo-Fraenkel set theory. cstt is the basis of the logics used in several computer theorem proving systems including hol [10], imps [8, 9], Isabelle [14], ProofPower [12], pvs =-=[13]-=-, and tps [3]. This paper presents an extended version of cstt called Basic Extended Simple Type Theory (bestt). It adds the following facilities to cstt: (1) Type variables for forming polymorphic ty... |

117 |
Mathematical logic as based on the theory of types
- Russell
- 1908
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Citation Context ...s and support for reasoning with tuples, lists, and sets to simple type theory, it is intended to be a practical logic for formalized mathematics. 1 Introduction B. Russell introduced a logic in 1908 =-=[16]-=- now called the ramified theory of types to serve as a foundation for mathematics. It included a hierarchy of types to avoid the set-theoretic and semantic paradoxes that troubled mathematicians and p... |

99 |
Principia Mathematica
- Whitehead, Russell
- 1957
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Citation Context ...oid the set-theoretic and semantic paradoxes that troubled mathematicians and philosophers in the early 1900s and was employed as the logic of Whitehead and Russell’s monumental Principia Mathematica =-=[18]-=-. Being an overly complex system, L. Chwistek [5] and F. Ramsey [15] suggested in the 1920s a simplified formulation of the ramified theory of types called the simple theory of types or, more briefly,... |

78 | IMPS: An interactive mathematical proof system
- Farmer, Guttman, et al.
- 1990
(Show Context)
Citation Context ...functions, is a more practical reasoning system than traditional Zermelo-Fraenkel set theory. cstt is the basis of the logics used in several computer theorem proving systems including hol [10], imps =-=[8, 9]-=-, Isabelle [14], ProofPower [12], pvs [13], and tps [3]. This paper presents an extended version of cstt called Basic Extended Simple Type Theory (bestt). It adds the following facilities to cstt: (1)... |

74 |
A Partial Functions Version of Church’s Simple Theory of Types
- Farmer
- 1990
(Show Context)
Citation Context ... . A∗) is the unique value x of type α satisfying A∗ if it exists and is an unspecified member of type α otherwise. The partial semantics for bestt is based on the partial semantics for cstt given in =-=[6, 7]-=-. This semantics can be straightforwardly extended to handle type variables (following the semantics in [10]) and the machinery in bestt for tuples, lists, and sets. The definedness of expressions is ... |

57 |
The foundations of mathematics
- Ramsey
- 1926
(Show Context)
Citation Context ...cians and philosophers in the early 1900s and was employed as the logic of Whitehead and Russell’s monumental Principia Mathematica [18]. Being an overly complex system, L. Chwistek [5] and F. Ramsey =-=[15]-=- suggested in the 1920s a simplified formulation of the ramified theory of types called the simple theory of types or, more briefly, simple type theory. A. Church presented in 1940 [4] a version of si... |

29 |
A Theory of Propositional Types
- Henkin
- 1963
(Show Context)
Citation Context ...aster.ca. 1smade exceptionally small. The full machinery of predicate calculus can be developed in cstt from just function application, function abstraction, and equality (first shown by L. Henkin in =-=[11]-=- and improved by P. Andrews in [1]). cstt has been widely influential in formalized mathematics. Many people have argued (e.g., see Andrews’ remarks in [2]) that cstt, with its strong support for reas... |

29 | Elements of ML Programming - Ullman - 1998 |

23 | PVS: Combining speci proof checking, and model checking - Owre, Rajan, et al. - 1996 |

14 | System description: Tps: A theorem proving system for type theory
- Andrews, Bishop, et al.
- 2000
(Show Context)
Citation Context ...tional Zermelo-Fraenkel set theory. cstt is the basis of the logics used in several computer theorem proving systems including hol [10], imps [8, 9], Isabelle [14], ProofPower [12], pvs [13], and tps =-=[3]-=-. This paper presents an extended version of cstt called Basic Extended Simple Type Theory (bestt). It adds the following facilities to cstt: (1) Type variables for forming polymorphic types and expre... |

11 |
A reduction of the axioms for the theory of propositional types
- Andrews
- 1963
(Show Context)
Citation Context ...l. The full machinery of predicate calculus can be developed in cstt from just function application, function abstraction, and equality (first shown by L. Henkin in [11] and improved by P. Andrews in =-=[1]-=-). cstt has been widely influential in formalized mathematics. Many people have argued (e.g., see Andrews’ remarks in [2]) that cstt, with its strong support for reasoning with functions, is a more pr... |

8 | Formalizing Undefinedness Arising in Calculus
- Farmer
(Show Context)
Citation Context ... . A∗) is the unique value x of type α satisfying A∗ if it exists and is an unspecified member of type α otherwise. The partial semantics for bestt is based on the partial semantics for cstt given in =-=[6, 7]-=-. This semantics can be straightforwardly extended to handle type variables (following the semantics in [10]) and the machinery in bestt for tuples, lists, and sets. The definedness of expressions is ... |

8 |
Thayer Fábrega. imps: An updated system description
- Farmer, Guttman, et al.
- 1996
(Show Context)
Citation Context ...functions, is a more practical reasoning system than traditional Zermelo-Fraenkel set theory. cstt is the basis of the logics used in several computer theorem proving systems including hol [10], imps =-=[8, 9]-=-, Isabelle [14], ProofPower [12], pvs [13], and tps [3]. This paper presents an extended version of cstt called Basic Extended Simple Type Theory (bestt). It adds the following facilities to cstt: (1)... |

2 |
Antynomje logikiformalnej
- Chwistek
- 1921
(Show Context)
Citation Context ...troubled mathematicians and philosophers in the early 1900s and was employed as the logic of Whitehead and Russell’s monumental Principia Mathematica [18]. Being an overly complex system, L. Chwistek =-=[5]-=- and F. Ramsey [15] suggested in the 1920s a simplified formulation of the ramified theory of types called the simple theory of types or, more briefly, simple type theory. A. Church presented in 1940 ... |