## Logic-free reasoning in Isabelle/Isar (2008)

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@MISC{Berghofer08logic-freereasoning,

author = {Stefan Berghofer and Makarius Wenzel},

title = {Logic-free reasoning in Isabelle/Isar},

year = {2008}

}

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### Abstract

Traditionally a rigorous mathematical document consists of a sequence of definition – statement – proof. Taking this basic outline as starting point we investigate how these three categories of text can be represented adequately in the formal language of Isabelle/Isar. Proofs represented in human-readable form have been the initial motivation of Isar language design 10 years ago. The principles developed here allow to turn deductions of the Isabelle logical framework into a format that transcends the raw logical calculus, with more direct description of reasoning using pseudo-natural language elements. Statements describe the main result of a theorem in an open format as a reasoning scheme, saying that in the context of certain parameters and assumptions certain conclusions can be derived. This idea of turning Isar context elements into rule statements has been recently refined to support the dual form of elimination rules as well. Definitions in their primitive form merely name existing elements of the logical environment, by stating a suitable equation or logical equivalence. Inductive definitions provide a convenient derived principle to describe a new predicate as the closure of given natural deduction rules. Again there is a direct connection to Isar principles, rules stemming from an inductive characterization are immediately available in structured reasoning. All three sub-categories benefit from replacing raw logical encodings by native Isar language elements. The overall formality in the presented mathematical text is reduced. Instead of manipulating auxiliary logical connectives and quantifiers, the mathematical concepts are emphasized.

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(Show Context)
Citation Context ... definitions, statements, and proofs by an example about well-founded multiset ordering. 2 Natural Deduction Revisited About 75 years ago Gentzen introduced a logical calculus for “natural deduction” =-=[3]-=- that was intended to formalize the way mathematical reasoning actually works, unlike earlier calculi due to Hilbert and Russel. Since we share the motivation to approximate mathematical reasoning, we... |

433 | Isabelle: A Generic Theorem Prover
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(Show Context)
Citation Context ...f trees, this provides a “propositions-as-types” and “proofs-as-programs” view of natural deduction, which underlies systems for constructive type-theory like Coq [2, 16]. The Isabelle/Pure framework =-=[8, 9]-=- implements a generic version of higherorder natural deduction, without presupposing any constructive reading. Natural deduction rules are represented in Isabelle as propositions of the “meta-logic”, ... |

187 |
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(Show Context)
Citation Context ...f trees, this provides a “propositions-as-types” and “proofs-as-programs” view of natural deduction, which underlies systems for constructive type-theory like Coq [2, 16]. The Isabelle/Pure framework =-=[8, 9]-=- implements a generic version of higherorder natural deduction, without presupposing any constructive reading. Natural deduction rules are represented in Isabelle as propositions of the “meta-logic”, ... |

174 | Inductive definitions in the system Coq – rules and properties
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(Show Context)
Citation Context ...ed on the Knaster-Tarski fixpoint theorem. The Coq system is based on the Calculus of Inductive Constructions introduced by Paulin-Mohring, which contains inductive definitions as a primitive concept =-=[7]-=-. Inductive definitions in Isabelle were first introduced by Paulson [10], using fixpoints over the lattice of sets. Our refined version works on generic lattices, which subsume predicates in HOL. Man... |

87 | An overview of the MIZAR project
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(Show Context)
Citation Context ...s, the mathematical concepts are emphasized. 1 Introduction Isabelle/Isar [13, 14, 15] enables to produce formal mathematical documents with full proof checking. Similar in spirit to the Mizar system =-=[12, 11]-=-, the user writes text in a formal language that is checked by the machine. As a side-effect of this, Isabelle/Isar produces high-quality documents using existing L ATEX technology — the present paper... |

85 | Isar - A generic interpretative approach to readable formal proof documents
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(Show Context)
Citation Context ... formality in the presented mathematical text is reduced. Instead of manipulating auxiliary logical connectives and quantifiers, the mathematical concepts are emphasized. 1 Introduction Isabelle/Isar =-=[13, 14, 15]-=- enables to produce formal mathematical documents with full proof checking. Similar in spirit to the Mizar system [12, 11], the user writes text in a formal language that is checked by the machine. As... |

74 | Isabelle/Isar – a versatile environment for human-readable formal proof documents
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(Show Context)
Citation Context ... formality in the presented mathematical text is reduced. Instead of manipulating auxiliary logical connectives and quantifiers, the mathematical concepts are emphasized. 1 Introduction Isabelle/Isar =-=[13, 14, 15]-=- enables to produce formal mathematical documents with full proof checking. Similar in spirit to the Mizar system [12, 11], the user writes text in a formal language that is checked by the machine. As... |

30 | A package for inductive relation definitions in HOL
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(Show Context)
Citation Context ... predicates provide a convenient way to define concepts by specifying a collection of characteristic introduction rules. Support for inductive definitions is available in many theorem provers. Melham =-=[6]-=- describes a version for the HOL system using an impredicative encoding, meaning that the definition involves universal quantification over predicate variables, whereas Harrison’s inductive definition... |

24 | Inductive definitions: Automation and application - Harrison - 1995 |

23 | Interpretation of locales in Isabelle: Theories and proof contexts
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(Show Context)
Citation Context ...ions and a technical lemma required for the well-foundedness proof. 1 Our development refers to a locally fixed less relation, which is introduced by commencing the following locale context (see also =-=[1]-=-). locale less-relation = fixes less :: α ⇒ α ⇒ bool (infix ≺ 50) begin The locale already contributes to the “logic-free” approach, since it avoids explicit abstraction or quantification over that pa... |

22 | A fixedpoint approach to (co)inductive and (co)datatype definitions
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(Show Context)
Citation Context ...e Calculus of Inductive Constructions introduced by Paulin-Mohring, which contains inductive definitions as a primitive concept [7]. Inductive definitions in Isabelle were first introduced by Paulson =-=[10]-=-, using fixpoints over the lattice of sets. Our refined version works on generic lattices, which subsume predicates in HOL. Many well-known concepts of mathematics can be viewed as an inductive predic... |

21 | Isabelle/Isar – a generic framework for human-readable proof documents
- Wenzel
- 2007
(Show Context)
Citation Context ... formality in the presented mathematical text is reduced. Instead of manipulating auxiliary logical connectives and quantifiers, the mathematical concepts are emphasized. 1 Introduction Isabelle/Isar =-=[13, 14, 15]-=- enables to produce formal mathematical documents with full proof checking. Similar in spirit to the Mizar system [12, 11], the user writes text in a formal language that is checked by the machine. As... |

9 | A comparison of the mathematical proof languages mizar and isar
- Wenzel, Wiedijk
- 2003
(Show Context)
Citation Context ...belle/Isar shares the mission of formal reasoning that approximates traditional mathematical style with the pioneering Mizar system [12, 11]. There are many similarities and dissimilarities, see also =-=[17, 16]-=- for some comparison. Concerning the logical foundations, Isar uses the Isabelle/Pure framework [8, 9] which implements a generic higher-order version of Gentzen’s natural deduction calculus [3]. In c... |

7 |
On the rules of suppositions
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- 1934
(Show Context)
Citation Context ...ntzen’s natural deduction calculus [3]. In contrast, Mizar works specifically with classical first-order logic, and the style of reasoning is modeled after the “supposition calculus” due to Jaskowski =-=[5]-=-. The basic paradigm of structured proof composition in Mizar is quite different from Isar. Where Isar revolves around natural deduction rules that emerge from local proof bodies and refine pending go... |

4 | Some Features of the Mizar Language
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- 1993
(Show Context)
Citation Context ...s, the mathematical concepts are emphasized. 1 Introduction Isabelle/Isar [13, 14, 15] enables to produce formal mathematical documents with full proof checking. Similar in spirit to the Mizar system =-=[12, 11]-=-, the user writes text in a formal language that is checked by the machine. As a side-effect of this, Isabelle/Isar produces high-quality documents using existing L ATEX technology — the present paper... |

1 |
et al.: The Coq Proof Assistant Reference Manual. INRIA
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(Show Context)
Citation Context ...ether with certain reductions on proof trees, this provides a “propositions-as-types” and “proofs-as-programs” view of natural deduction, which underlies systems for constructive type-theory like Coq =-=[2, 16]-=-. The Isabelle/Pure framework [8, 9] implements a generic version of higherorder natural deduction, without presupposing any constructive reading. Natural deduction rules are represented in Isabelle a... |