A Generic Modular Data Structure for Proof Attempts Alternating on Ideas and Granularity

Autexier, Serge; Benzmüller, Christoph; Dietrich, Dominik; Meier, Andreas; Wirth, Claus-Peter

doi:10.1007/11618027_9

Cited by 13 publications

(9 citation statements)

References 9 publications

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“…§ 5.4. Moreover, as our proof trees are customary AND-trees (and no AND/OR-trees that admit alternative proof attempts as in [5,6]), Lemma 4.1 means that the whole proof attempt is failed for a γ-multiplicity of 1.…”

Section: Lemma 41 Using the Reductive Rules Ofmentioning

confidence: 99%

, , and Non-Permutability of -Steps

Wirth

2012

Journal of Symbolic Computation

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Using a human-oriented formal example proof of the (lim +) theorem, i.e. that the sum of limits is the limit of the sum, which is of value for reference on its own, we exhibit a nonpermutability of β-steps and δ + -steps (according to Smullyan's classification), which is not visible with non-liberalized δ-rules and not serious with further liberalized δ-rules, such as the δ + + -rule. Besides a careful presentation of the search for a proof of (lim +) with several pedagogical intentions, the main subject is to explain why the order of β-steps plays such a practically important role in some calculi.

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Section: Lemma 41 Using the Reductive Rules Ofmentioning

confidence: 99%

, , and Non-Permutability of -Steps

Wirth

2012

Journal of Symbolic Computation

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“…We have already argued that dynamic, context sensitive proof step evaluation requires support from a sophisticated and ideally cognitively adequate mathematical domain reasoner 4 . The ΩMEGA system, with its various support tools for human-oriented, abstract level proof representation and proof construction, has therefore been chosen as the domain reasoner of choice in the DIALOG project (see the paper by [44] in this volume).…”

Section: Dynamic Proof Step Evaluation With ωMegamentioning

confidence: 99%

“…Proofs are represented in ΩMEGA's proof data structure (PDS) which allows the shared representation of several (ongoing) proof variants [4]. The PDS is a collection of proof trees (with nodes as multi-conclusion sequents), which can be linked to one another (in order to express the dependency of one proof on another one whose proof task is treated as a lemma).…”

Section: Proof Management Correctness Analysis and Content Underspecmentioning

confidence: 99%

Resource-Bounded Modelling and Analysis of Human-Level Interactive Proofs

Benzmüller

Schiller

Siekmann

2010

Resource-Adaptive Cognitive Processes

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“…Users are usually presented with an extract of the fully formalised proof, but can explore the proof on different levels of abstractions (e.g. see the proof plan data structure PDS [7,22] in Ωmega [4,56] or high-level proofs in [13]), or request the system to print all steps, declarations, and definitions.…”

Section: An Environment For Correctness and Understandingmentioning

confidence: 99%

“…Still, no universal alternative is available. Although a formal proof cannot guarantee 100% correctness because, for example, one cannot prove the correctness of the formal prover itself (a well-known result in computability theory, [24]) the certainty achieved is close to "certain" 4 . The advent, in the last twenty years, of proof assistants was followed by an impressive record of deep mathematical theorems formally proved.…”

Section: Introductionmentioning

confidence: 99%

Formal Proof: Reconciling Correctness and Understanding

Calude

Müller

2009

Lecture Notes in Computer Science

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A good proof is a proof that makes us wiser. Manin [41, p. 209].Abstract. Hilbert's concept of formal proof is an ideal of rigour for mathematics which has important applications in mathematical logic, but seems irrelevant for the practice of mathematics. The advent, in the last twenty years, of proof assistants was followed by an impressive record of deep mathematical theorems formally proved. Formal proof is practically achievable. With formal proof, correctness reaches a standard that no pen-and-paper proof can match, but an essential component of mathematics -the insight and understanding -seems to be in short supply. So, what makes a proof understandable? To answer this question we first suggest a list of symptoms of understanding. We then propose a vision of an environment in which users can write and check formal proofs as well as query them with reference to the symptoms of understanding. In this way, the environment reconciles the main features of proof: correctness and understanding.

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A Generic Modular Data Structure for Proof Attempts Alternating on Ideas and Granularity

Cited by 13 publications

References 9 publications

, , and Non-Permutability of -Steps

, , and Non-Permutability of -Steps

Resource-Bounded Modelling and Analysis of Human-Level Interactive Proofs

Formal Proof: Reconciling Correctness and Understanding

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