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Complete sequent calculi for induction and infinite descent
 Proceedings of LICS22
, 2007
"... This paper compares two different styles of reasoning with inductively defined predicates, each style being encapsulated by a corresponding sequent calculus proof system. The first system supports traditional proof by induction, with induction rules formulated as sequent rules for introducing induct ..."
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Cited by 18 (6 self)
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This paper compares two different styles of reasoning with inductively defined predicates, each style being encapsulated by a corresponding sequent calculus proof system. The first system supports traditional proof by induction, with induction rules formulated as sequent rules for introducing inductively defined predicates on the left of sequents. We show this system to be cutfree complete with respect to a natural class of Henkin models; the eliminability of cut follows as a corollary. The second system uses infinite (nonwellfounded) proofs to represent arguments by infinite descent. In this system, the left rules for inductively defined predicates are simple casesplit rules, and an infinitary, global condition on proof trees is required to ensure soundness. We show this system to be cutfree complete with respect to standard models, and again infer the eliminability of cut. The second infinitary system is unsuitable for formal reasoning. However, it has a natural restriction to proofs given by regular trees, i.e. to those proofs representable by finite graphs. This restricted “cyclic ” system subsumes the first system for proof by induction. We conjecture that the two systems are in fact equivalent, i.e., that proof by induction is equivalent to regular proof by infinite descent.
A Generic Modular Data Structure for Proof Attempts Alternating on Ideas and Granularity
 Proceedings of MKM’05, volume 3863 of LNAI, IUB
, 2006
"... Abstract. A practically useful mathematical assistant system requires the sophisticated combination of interaction and automation. Central in such a system is the proof data structure, which has to maintain the current proof state and which has to allow the flexible interplay of various components i ..."
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Cited by 15 (6 self)
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Abstract. A practically useful mathematical assistant system requires the sophisticated combination of interaction and automation. Central in such a system is the proof data structure, which has to maintain the current proof state and which has to allow the flexible interplay of various components including the human user. We describe a parameterized proof data structure for the management of proofs, which includes our experience with the development of two proof assistants. It supports and bridges the gap between abstract level proof explanation and lowlevel proof verification. The proof data structure enables, in particular, the flexible handling of lemmas, the maintenance of different proof alternatives, and the representation of different granularities of proof attempts. 1
A Structured Set of HigherOrder Problems
 Theorem Proving in Higher Order Logics: TPHOLs 2005, LNCS 3603
, 2005
"... Abstract. We present a set of problems that may support the development of calculi and theorem provers for classical higherorder logic. We propose to employ these test problems as quick and easy criteria preceding the formal soundness and completeness analysis of proof systems under development. Ou ..."
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Cited by 8 (5 self)
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Abstract. We present a set of problems that may support the development of calculi and theorem provers for classical higherorder logic. We propose to employ these test problems as quick and easy criteria preceding the formal soundness and completeness analysis of proof systems under development. Our set of problems is structured according to different technical issues and along different notions of semantics (including Henkin semantics) for higherorder logic. Many examples are either theorems or nontheorems depending on the choice of semantics. The examples can thus indicate the deductive strength of a proof system. 1 Motivation: Test Problems for HigherOrder Reasoning Systems Test problems are important for the practical implementation of theorem provers as well as for the preceding theoretical development of calculi, strategies and heuristics. If the test theorems can be proven (resp. the nontheorems cannot) then they ideally provide a strong indication for completeness (resp. soundness). Examples for early publications providing firstorder test problems are [21,29,23]. For more than decade now the TPTP library [28] has been developed as a systematically structured electronic repository of
How to prove inductive theorems? QUODLIBET
 Proceedings of the 19th International Conference on Automated Deduction, volume 2741 of LNCS
, 2003
"... QUODLIBET is a tacticbased inductive theorem proving system that meets today’s standard requirements for theorem provers such as a command interpreter, a sophisticated graphical user interface, and a carefully programmed inference machine kernel that guarantees soundness. In essence, it is the syne ..."
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Cited by 5 (1 self)
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QUODLIBET is a tacticbased inductive theorem proving system that meets today’s standard requirements for theorem provers such as a command interpreter, a sophisticated graphical user interface, and a carefully programmed inference machine kernel that guarantees soundness. In essence, it is the synergetic combination of the features presented in the following sections that makes QUODLIBET a system quite useful in practice; and we hope that it is actually as you like it, which is the Latin “quod libet” translated into English. We start by presenting some of the design goals that have guided the development of QUODLIBET. Note that the system is not intended to pursue the push bottom technology for inductive theorem proving, but to manage more complicated proofs by an effective interplay between interaction and automation. 1.1 Design Goals for Specifications Given algebraic specifications of algorithms in the style of abstract data types, we want to prove theorems even if the specification is not (yet) sufficiently complete. As an example, consider the incomplete specification of the subtraction on the natural numbers E = {∀x. x−0=x, ∀x,y. s(x)−s(y)=x−y} and the conjecture ∀x,y. (x−y=0 ∧ y−x=0 ⇒ x=y).
Diophantus’ 20th problem and fermat’s last theorem for n=4  Formalization of . . .
, 2005
"... We present the proof of Diophantus’ 20th problem (book VI of Diophantus’ Arithmetica), which consists in wondering if there exist right triangles whose sides may be measured as integers and whose surface may be a square. This problem was negatively solved by Fermat in the 17th century, who used the ..."
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Cited by 2 (0 self)
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We present the proof of Diophantus’ 20th problem (book VI of Diophantus’ Arithmetica), which consists in wondering if there exist right triangles whose sides may be measured as integers and whose surface may be a square. This problem was negatively solved by Fermat in the 17th century, who used the wonderful method (ipse dixit Fermat) of infinite descent. This method, which is, historically, the first use of induction, consists in producing smaller and smaller nonnegative integer solutions assuming that one exists; this naturally leads to a reductio ad absurdum reasoning because we are bounded by zero. We describe the formalization of this proof which has been carried out in the Coq proof assistant. Moreover, as a direct and no less historical application, we also provide the proof (by Fermat) of Fermat’s last theorem for n = 4, as well as the corresponding formalization made in Coq.
JACQUES HERBRAND: LIFE, LOGIC, AND AUTOMATED DEDUCTION
"... The lives of mathematical prodigies who passed away very early after groundbreaking work invoke a fascination for later generations: The early death of Niels Henrik Abel (1802–1829) from ill health after a sled trip to visit his fiancé for Christmas; the obscure circumstances of Evariste Galois ’ (1 ..."
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The lives of mathematical prodigies who passed away very early after groundbreaking work invoke a fascination for later generations: The early death of Niels Henrik Abel (1802–1829) from ill health after a sled trip to visit his fiancé for Christmas; the obscure circumstances of Evariste Galois ’ (1811–1832) duel; the deaths of consumption of Gotthold Eisenstein (1823–1852) (who sometimes lectured his few students from his bedside) and of Gustav Roch (1839–1866) in Venice; the drowning of the topologist Pavel Samuilovich Urysohn (1898–1924) on vacation; the burial of Raymond Paley (1907–1933) in an avalanche at Deception Pass in the Rocky Mountains; as well as the fatal imprisonment of Gerhard Gentzen (1909–1945) in Prague1 — these are tales most scholars of logic and mathematics have heard in their student days. Jacques Herbrand, a young prodigy admitted to the École Normale Supérieure as the best student of the year1925, when he was17, died only six years later in a mountaineering accident in La Bérarde (Isère) in France. He left a legacy in logic and mathematics that is outstanding.
A SelfContained and Easily . . .
, 2010
"... We present the only proof of PIERRE FERMAT by descente infinie that is known to exist today. As the text of its Latin original requires active mathematical interpretation, it is more a proof sketch than a proper mathematical proof. We discuss descente infinie from the mathematical, logical, historic ..."
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We present the only proof of PIERRE FERMAT by descente infinie that is known to exist today. As the text of its Latin original requires active mathematical interpretation, it is more a proof sketch than a proper mathematical proof. We discuss descente infinie from the mathematical, logical, historical, linguistic, and refined logichistorical points of view. We provide the required preliminaries from number theory and develop a selfcontained proof in a modern form, which nevertheless is intended to follow FERMAT’s ideas closely. We then annotate an English translation of FERMAT’s original proof with terms from the modern proof. Including all important facts, we present a concise and selfcontained discussion of FERMAT’s proof sketch, which is easily accessible to laymen in number theory as well as to laymen in the history of mathematics, and which provides new clarification of the Method of Descente
Easily Accessible Discussion of the Method of Descente Infinie and Fermat’s Only Explicitly Known Proof
, 902
"... We present the only proof of Pierre Fermat by descente infinie that is known to exist today. We discuss descente infinie from the mathematical, logical, historical, linguistic, and refined logichistorical points of view. We provide the required preliminaries from number theory and present a selfco ..."
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We present the only proof of Pierre Fermat by descente infinie that is known to exist today. We discuss descente infinie from the mathematical, logical, historical, linguistic, and refined logichistorical points of view. We provide the required preliminaries from number theory and present a selfcontained proof in a modern form, which nevertheless is intended to follow Fermat’s ideas as interpreted into the cited Latin original. We then annotate an English translation of Fermat’s original proof with terms from the modern proof. Although the paper consists of reviews, compilations, and simple number theory with mainly pedagogical intentions, its gestalt is original and it fills a gap regarding the easy accessibility of the subject. 2