## TPS: An Interactive and Automatic Tool for Proving Theorems of Type Theory (1994)

Venue: | Higher Order Logic Theorem Proving and Its Applications: 6th International Workshop, HUG '93, volume 780 of Lecture Notes in Computer Science |

Citations: | 1 - 1 self |

### BibTeX

@INPROCEEDINGS{Andrews94tps:an,

author = {Peter B. Andrews and Matthew Bishop and Sunil Issar and Dan Nesmith and Frank Pfenning and Hongwei Xi and Contact Peter and B. Andrews},

title = {TPS: An Interactive and Automatic Tool for Proving Theorems of Type Theory},

booktitle = {Higher Order Logic Theorem Proving and Its Applications: 6th International Workshop, HUG '93, volume 780 of Lecture Notes in Computer Science},

year = {1994},

pages = {366--370},

publisher = {Springer-Verlag}

}

### OpenURL

### Abstract

This is a demonstration of TPS, a theorem proving system for classical type theory (Church's typed l-calculus). TPS can be used interactively or automatically, or in a combination of these modes. An important feature of TPS is the ability to translate between expansion proofs and natural deduction proofs. CATEGORY: Demonstration 1. Introduction This presentation is a demonstration of TPS, a theorem proving system for classical type theory (Church's typed 3 l-calculus [14]) which has been under development at Carnegie Mellon University for a number of years. TPS is based on an approach to automated theorem proving called the mating method [2], which is essentially the same as the connection method developed independently by Bibel [13]. The mating method does not require reduction to clausal form. TPS handles two sorts of proofs, natural deduction proofs and expansion proofs. Natural deduction proofs are human-readable formal proofs. An example of such a proof which was produced aut...

### Citations

854 |
A formulation of a simple theory of types
- Church
- 1940
(Show Context)
Citation Context ...oofs and natural deduction proofs. CATEGORY: Demonstration 1. Introduction This presentation is a demonstration of TPS, a theorem proving system for classical type theory (Church's typed 3 l-calculus =-=[14]-=-) which has been under development at Carnegie Mellon University for a number of years. TPS is based on an approach to automated theorem proving called the mating method [2], which is essentially the ... |

309 |
An Introduction to Mathematical Logic and Type Theory: to Truth through Proof
- Andrews
- 1986
(Show Context)
Citation Context ...n this line is ~q Xsq x ; it was inferred by Rule MP (Modus Ponens) from the wffs in oi i i lines (15) and (13), and the formula asserted in line (10) is the sole hypothesis for this line. See [1] or =-=[4]-=- for more details about this formulation of natural deduction. Expansion proofs are concise Herbrand expansions of theorems of type theory. The structure of an expansion proof, which incorporates a ma... |

149 |
Automated Theorem Proving
- Bibel
- 1987
(Show Context)
Citation Context ...y for a number of years. TPS is based on an approach to automated theorem proving called the mating method [2], which is essentially the same as the connection method developed independently by Bibel =-=[13]-=-. The mating method does not require reduction to clausal form. TPS handles two sorts of proofs, natural deduction proofs and expansion proofs. Natural deduction proofs are human-readable formal proof... |

114 |
Theorem proving via general matings
- Andrews
- 1981
(Show Context)
Citation Context ...urch's typed 3 l-calculus [14]) which has been under development at Carnegie Mellon University for a number of years. TPS is based on an approach to automated theorem proving called the mating method =-=[2]-=-, which is essentially the same as the connection method developed independently by Bibel [13]. The mating method does not require reduction to clausal form. TPS handles two sorts of proofs, natural d... |

71 | On Connections and Higher-Order Logic
- Andrews
- 1989
(Show Context)
Citation Context ... REFLEXIVE r] psp x y i i oii oii Lambda: 32 (34) |- lu lv [u = v] i i =s.lr REFLEXIVE r EquivWffs: 33 oii Expansion proofs were introduced by Miller [23]. A brief explanation of them may be found in =-=[7]-=-, which provides an introduction to many ideas underlying TPS. [23], [24], [25], [27], [28], and [29] contain extensive treatments of expansion proofs, translations between natural deduction proofs an... |

27 | The TPS Theorem Proving System
- Andrews, Issar, et al.
- 1990
(Show Context)
Citation Context ...2546 and CCR-9201893. 2 Figure 1-1: A Natural Deduction Proof 1 (1) 1 |- X = x Hyp i i 1 (2) 1 |- "q .q Xsq x Equality: 1 oi i i (3) 3 |- "X p X X Hyp i oii 1 1 1 (4) 3 |- p X X UI: X 3 oii =-=i 1 1 1 1 (5) 1 |- p X Xsp X x UI: [p -=-X ] 2 oii i i 1 (6) 1,3 |- p X x RuleP: 4 5 oii i i 1 (7) 1 |- "X p X Xsp X x Deduct: 6 i oii i i 1 (8) 1 |- "p ."X p X Xsp X x UGen: p 7 oii i i i 1 1 (9) |- X = xs"p ."X p X... |

22 | Proof Transformations in Higher-Order Logic
- Pfenning
- 1987
(Show Context)
Citation Context ...quivWffs: 33 oii Expansion proofs were introduced by Miller [23]. A brief explanation of them may be found in [7], which provides an introduction to many ideas underlying TPS. [23], [24], [25], [27], =-=[28]-=-, and [29] contain extensive treatments of expansion proofs, translations between natural deduction proofs and expansion proofs, and ways of improving the natural deduction proofs obtained by these tr... |

16 |
Expansion tree proofs and their conversion to natural deduction proofs
- Miller
- 1984
(Show Context)
Citation Context ...s.lr REFLEXIVE r EquivWffs: 33 oii Expansion proofs were introduced by Miller [23]. A brief explanation of them may be found in [7], which provides an introduction to many ideas underlying TPS. [23], =-=[24]-=-, [25], [27], [28], and [29] contain extensive treatments of expansion proofs, translations between natural deduction proofs and expansion proofs, and ways of improving the natural deduction proofs ob... |

15 | Presenting Intuitive Deductions via Symmetric Simplification
- Pfenning, Nesmith
- 1990
(Show Context)
Citation Context ...33 oii Expansion proofs were introduced by Miller [23]. A brief explanation of them may be found in [7], which provides an introduction to many ideas underlying TPS. [23], [24], [25], [27], [28], and =-=[29]-=- contain extensive treatments of expansion proofs, translations between natural deduction proofs and expansion proofs, and ways of improving the natural deduction proofs obtained by these translations... |

13 |
A Compact Representation of Proofs. Studia Logica 46/4
- Miller
(Show Context)
Citation Context ...EFLEXIVE r EquivWffs: 33 oii Expansion proofs were introduced by Miller [23]. A brief explanation of them may be found in [7], which provides an introduction to many ideas underlying TPS. [23], [24], =-=[25]-=-, [27], [28], and [29] contain extensive treatments of expansion proofs, translations between natural deduction proofs and expansion proofs, and ways of improving the natural deduction proofs obtained... |

11 |
A look at TPS
- Miller, Cohen, et al.
- 1982
(Show Context)
Citation Context ...RuleP: 11 16 oi i 1 (18) 10 |- q Xsq x Deduct: 17 oi i i 1 (19) 10 |- "q .q Xsq x UGen: q 18 oi i i 1 (20) 10 |- X = x Equality: 19 i i 1 1 (21) |- "p ["X p X Xsp X x ]sX = x Deduct: 20=-= oii i i i 1 1 (22) |- [X = xs"p ."X p X Xsp X x]-=- i i oii i 1 1s."p ["X p X Xsp X x]sX = x RuleP: 9 21 1 1 (23) |- X = xs"p ."X p X Xsp X x ImpEquiv: 22 i i oii i 1 1 (24) |- X = x = "p ."X p X Xsp X x Ext=: 23 i i oii ... |

8 |
Path-Focused Duplication: A Search Procedure for General Matings
- Issar
- 1990
(Show Context)
Citation Context ... out by searching for an expansion proof, and then translates this into a natural deduction proof. Search procedures using outermost quantifier duplication [2] and path-focused quantifier duplication =-=[18]-=- [19] are implemented in TPS. TPS uses Huet's higherorder unification algorithm [17], and applies primitive substitutions [7] to introduce connectives and quantifiers in substitution terms for set var... |

7 |
A Unification Algorithm for Typed l-Calculus
- Huet
- 1975
(Show Context)
Citation Context ...duction proof. Search procedures using outermost quantifier duplication [2] and path-focused quantifier duplication [18] [19] are implemented in TPS. TPS uses Huet's higherorder unification algorithm =-=[17]-=-, and applies primitive substitutions [7] to introduce connectives and quantifiers in substitution terms for set variables. Much must be done to explore these search procedures more thoroughly and to ... |

6 |
Using Programs to Teach Logic to Computer Scientists
- Goldson, Reeves
- 1993
(Show Context)
Citation Context ...he textbook [4]. Students quickly learn to use ETPS by reading the manual [30] (which contains several complete examples of how to construct proofs) and doing assigned exercises. ETPS was reviewed in =-=[15]-=-. The basic tools in TPS for automatically applying rules of inference to construct natural deduction proofs are tactics, which can be combined using tacticals [16]. A tactic applies rules of inferenc... |

6 |
Operational Issues in Automated Theorem Proving Using Matings
- Issar
- 1991
(Show Context)
Citation Context ...by searching for an expansion proof, and then translates this into a natural deduction proof. Search procedures using outermost quantifier duplication [2] and path-focused quantifier duplication [18] =-=[19]-=- are implemented in TPS. TPS uses Huet's higherorder unification algorithm [17], and applies primitive substitutions [7] to introduce connectives and quantifiers in substitution terms for set variable... |

3 |
Eve Longini Cohen, Frank Pfenning, "Automating HigherOrder Logic," in Automated Theorem Proving: After 25 Years, edited by
- Andrews, Miller
- 1984
(Show Context)
Citation Context ... work supported by the National Science Foundation under grants CCR-9002546 and CCR-9201893. 2 Figure 1-1: A Natural Deduction Proof 1 (1) 1 |- X = x Hyp i i 1 (2) 1 |- "q .q Xsq x Equality: 1 oi=-= i i (3) 3 |- &quo-=-t;X p X X Hyp i oii 1 1 1 (4) 3 |- p X X UI: X 3 oii i 1 1 1 1 (5) 1 |- p X Xsp X x UI: [p X ] 2 oii i i 1 (6) 1,3 |- p X x RuleP: 4 5 oii i i 1 (7) 1 |- "X p X Xsp X x Deduct: 6 i oii i i 1 (8) ... |

3 |
TPS3 Facilities Guide for Users
- Andrews, Issar, et al.
- 1997
(Show Context)
Citation Context ... 3.5 megabytes. The compiled core image for a Decstation 3100 occupies about 14.4 megabytes using Allegro Common Lisp, and about 31.4 megabytes using CMU Common Lisp. Considerable documentation [11], =-=[12]-=-, [20], [21], [26], [30] has been written, though more is needed. The Facilities Guides [11] [12] are produced automatically. An earlier version of TPS, which contributed much to the present version, ... |

2 |
Transforming Matings into Natural Deduction Proofs," in 5th Conference on Automated Deduction, edited by
- Andrews
- 1980
(Show Context)
Citation Context ...erted in this line is ~q Xsq x ; it was inferred by Rule MP (Modus Ponens) from the wffs in oi i i lines (15) and (13), and the formula asserted in line (10) is the sole hypothesis for this line. See =-=[1]-=- or [4] for more details about this formulation of natural deduction. Expansion proofs are concise Herbrand expansions of theorems of type theory. The structure of an expansion proof, which incorporat... |

2 |
Analytic and Non-analytic Proofs," in 7th International Conference on Automated Deduction, edited by
- Pfenning
- 1984
(Show Context)
Citation Context ...VE r EquivWffs: 33 oii Expansion proofs were introduced by Miller [23]. A brief explanation of them may be found in [7], which provides an introduction to many ideas underlying TPS. [23], [24], [25], =-=[27]-=-, [28], and [29] contain extensive treatments of expansion proofs, translations between natural deduction proofs and expansion proofs, and ways of improving the natural deduction proofs obtained by th... |