## Proof synthesis and reflection for linear arithmetic (2006)

Citations: | 6 - 5 self |

### BibTeX

@TECHREPORT{Chaieb06proofsynthesis,

author = {Amine Chaieb and Tobias Nipkow},

title = {Proof synthesis and reflection for linear arithmetic },

institution = {},

year = {2006}

}

### OpenURL

### Abstract

This article presents detailed implementations of quantifier elimination for both integer and real linear arithmetic for theorem provers. The underlying algorithms are those by Cooper (for Z) and by Ferrante and Rackoff (for R). Both algorithms are realized in two entirely different ways: once in tactic style, i.e. by a proof-producing functional program, and once by reflection, i.e. by computations inside the logic rather than in the meta-language. Both formalizations are highly generic because they make only minimal assumptions w.r.t. the underlying logical system and theorem prover. An implementation in Isabelle/HOL shows that the reflective approach is between one and two orders of magnitude faster.

### Citations

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Citation Context ...ng logic, except that we assume it is classical. As evidence of the genericity of our code we can offer that the procedure for Presburger arithmetic is derived from an implementation for Isabelle/HOL =-=[40]-=- by the first author [8], yet is quite close to an implementation in the HOL system [25]. Producing proofs requires no meta-theory but has many disadvantages: (a) the actual implementation requires in... |

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Citation Context ...t of x is now l everywhere. Hence we can view Q(x) as Q ′(l·x) for an appropriate Q ′. 4. Return l | x ∧ Q ′(x) according to the generic theorem unitcoeff : (∃x. Q(l·x)) ↔ (∃x. l | x ∧ Q(x)) See also =-=[18]-=- for good examples of normalization. We restrict the atomic relations to = and > only for the sake of presentation. In practice it is important though to include ≤ and =, which prevent a fatal blow u... |

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Citation Context ...of Gauss-elimination to cope with inequalities. His method has been rediscovered several times [17, 37] and is today referred to as Fourier-Motzkin elimination. Tarski’s result for real closed fields =-=[50]-=- also yields a decision procedure for linear real arithmetic. More efficient quantifier elimination procedures have been developed by Ferrante and Rackoff [19], the work on which we rely, and later by... |

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(Show Context)
Citation Context ...r Arithmetic have been discovered by Presburger [45] and Skolem [49] independently and improved by Cooper [13], Reddy and Loveland [47] and extended to deal with parameters by Weispfenning [52]. Pugh =-=[46]-=- presented an adaptation of Fourier-Motzkin elimination to cope with integers. An automata based method is presented in [55].Linear arithmetic enjoys exciting complexity results. Fischer and Rabin [2... |

373 |
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Citation Context ...ers implement Fourier-Motzkin elimination and only deal with quantifier-free formulae. HOL Light [28] includes full quantifier-elimination for real closed fields [30, 36]. Verifying the CAD algorithm =-=[12]-=- is on-going work [34]. An alternative approach already used in [38] is based on checking certificates for Farkas’s lemma. This technique is very efficient, but is not complete for the integers and do... |

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Citation Context ...de we can offer that the procedure for Presburger arithmetic is derived from an implementation for Isabelle/HOL [40] by the first author [8], yet is quite close to an implementation in the HOL system =-=[25]-=-. Producing proofs requires no meta-theory but has many disadvantages: (a) the actual implementation requires intimate knowledge of the internals of the underlying theorem prover; (b) there is no way ... |

122 |
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Citation Context ... Rackoff [19], the work on which we rely, and later by Weispfenning [51] and Loos and Weispfenning [33]. Quantifier elimination procedures for Presburger Arithmetic have been discovered by Presburger =-=[45]-=- and Skolem [49] independently and improved by Cooper [13], Reddy and Loveland [47] and extended to deal with parameters by Weispfenning [52]. Pugh [46] presented an adaptation of Fourier-Motzkin elim... |

106 |
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Citation Context ...ier-free formulae. HOL Light [28] includes full quantifier-elimination for real closed fields [30, 36]. Verifying the CAD algorithm [12] is on-going work [34]. An alternative approach already used in =-=[38]-=- is based on checking certificates for Farkas’s lemma. This technique is very efficient, but is not complete for the integers and does not allow quantifier elimination, which is the main focus of this... |

102 |
Metafunctions: proving them correct and using them efficiently as new proof proceedures
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(Show Context)
Citation Context ...t complete for the integers and does not allow quantifier elimination, which is the main focus of this paper. The method of reflection goes back at least to the meta-functions used by Boyer and Moore =-=[7]-=- and later became popular in theorem provers based on type theory [5]. It has been studied by several researchers [29, 1]. Our use of reflection is rather computational and has nothing to do with “log... |

95 | Super-exponential complexity of Presburger arithmetic
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(Show Context)
Citation Context ... v0 + t d d | l d ∤ v0 + t d d | l 1 True Fig. 5. Dp and alldvdφ for normalized φ-formulae As in §3.3 we prove the (reflected) premises of cooper −∞: isnormZ+ p → ∃z.∀x.x < z → (⦇p⦈x·vs = ⦇p−⦈ x·vs ) =-=(20)-=- isnormZ+ isnormZ+ p → ∀x, k.⦇p−⦈ x·vs = ⦇p−⦈ (x−k·Dp)·vs (21) p → ∀x.¬(∃j ∈ {1..Dp }.∃b ∈ {⦇t⦈ i·vs τ | t ∈ {B p }}.⦇p⦈ (b+j)·vs ) → ⦇p⦈ x·vs → ⦇p⦈ (x−δp)·vs (22) All these theorems are proved by str... |

92 |
Theorem proving in arithmetic without multiplication
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(Show Context)
Citation Context ...spfenning [51] and Loos and Weispfenning [33]. Quantifier elimination procedures for Presburger Arithmetic have been discovered by Presburger [45] and Skolem [49] independently and improved by Cooper =-=[13]-=-, Reddy and Loveland [47] and extended to deal with parameters by Weispfenning [52]. Pugh [46] presented an adaptation of Fourier-Motzkin elimination to cope with integers. An automata based method is... |

87 |
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(Show Context)
Citation Context ...ms are formulae to be proved, solutions are theorems, and termination will be guaranteed because all decompositions yield smaller terms. This style of theorem proving was invented with the LCF system =-=[24, 44]-=-, where decomp is called a tactic. Hence we refer to it as tactic-style theorem proving. Note that the interface of our theorem data type is quite abstract. Theorems may be implemented as full-blownp... |

86 | Theorem Proving with the Real Numbers
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- 1998
(Show Context)
Citation Context ...plete. For the reals most theorem provers implement Fourier-Motzkin elimination and only deal with quantifier-free formulae. HOL Light [28] includes full quantifier-elimination for real closed fields =-=[30, 36]-=-. Verifying the CAD algorithm [12] is on-going work [34]. An alternative approach already used in [38] is based on checking certificates for Farkas’s lemma. This technique is very efficient, but is no... |

81 |
The complexity of linear problems in fields
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(Show Context)
Citation Context ... decision procedure for linear real arithmetic. More efficient quantifier elimination procedures have been developed by Ferrante and Rackoff [19], the work on which we rely, and later by Weispfenning =-=[51]-=- and Loos and Weispfenning [33]. Quantifier elimination procedures for Presburger Arithmetic have been discovered by Presburger [45] and Skolem [49] independently and improved by Cooper [13], Reddy an... |

69 | A compiled implementation of strong reduction
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(Show Context)
Citation Context ... p. The fact that the executable code is again ML is a coincidence. A similar approach is adopted in PVS [15] and in ACL2 [7]. Other approaches include the the use of an internal λ-calculus evaluator =-=[27]-=- as in Coq [5]. One could also perform the evaluation step by rewriting inside the theorem prover, but the performance penalty is usually prohibitive. There is also the practical issue of where reify ... |

62 | Applying Linear Quantifier Elimination
- Loos, Weispfenning
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(Show Context)
Citation Context ...real arithmetic. More efficient quantifier elimination procedures have been developed by Ferrante and Rackoff [19], the work on which we rely, and later by Weispfenning [51] and Loos and Weispfenning =-=[33]-=-. Quantifier elimination procedures for Presburger Arithmetic have been discovered by Presburger [45] and Skolem [49] independently and improved by Cooper [13], Reddy and Loveland [47] and extended to... |

61 | Functional unification of higher-order patterns - Nipkow - 1993 |

53 | Metatheory and reflection in theorem proving: A survey and critique
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(Show Context)
Citation Context ...hod of reflection goes back at least to the meta-functions used by Boyer and Moore [7] and later became popular in theorem provers based on type theory [5]. It has been studied by several researchers =-=[29, 1]-=-. Our use of reflection is rather computational and has nothing to do with “logical reflection” [29]. Laurent Théry verified Presburger’s original algorithm in Coq (see the Coq home page). We also ver... |

48 | Executing higher order logic
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(Show Context)
Citation Context ...e theorem θ(A). The free variables in a theorem th can be instantiated from left to right with terms t1, . . . , tn by writing th[t1, . . . , tn]. For example, if th is the theorem m ≤ m + n·n then th=-=[1, 2]-=- is the theorem 1 ≤ 1 + 2·2. Function gen performs ∀-introduction: it takes a variable x and a theorem P (x) and returns the theorem ∀x.P (x). Note that we assume that fwd performs higher-order matchi... |

46 | An automata-theoretic approach to Presburger arithmetic constraints
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(Show Context)
Citation Context ...veland [47] and extended to deal with parameters by Weispfenning [52]. Pugh [46] presented an adaptation of Fourier-Motzkin elimination to cope with integers. An automata based method is presented in =-=[55]-=-.Linear arithmetic enjoys exciting complexity results. Fischer and Rabin [20] proved lower bounds which were later refined [3, 4, 23] using alternation [11]. Upper bounds can be found in [43, 19, 51,... |

45 |
A decision procedure for the first order theory of real addition with order
- Ferrante, Rackoff
- 1975
(Show Context)
Citation Context .... Tarski’s result for real closed fields [50] also yields a decision procedure for linear real arithmetic. More efficient quantifier elimination procedures have been developed by Ferrante and Rackoff =-=[19]-=-, the work on which we rely, and later by Weispfenning [51] and Loos and Weispfenning [33]. Quantifier elimination procedures for Presburger Arithmetic have been discovered by Presburger [45] and Skol... |

40 |
Logic and Computation
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- 1982
(Show Context)
Citation Context ...ms are formulae to be proved, solutions are theorems, and termination will be guaranteed because all decompositions yield smaller terms. This style of theorem proving was invented with the LCF system =-=[24, 44]-=-, where decomp is called a tactic. Hence we refer to it as tactic-style theorem proving. Note that the interface of our theorem data type is quite abstract. Theorems may be implemented as full-blownp... |

39 |
The complexity of logical theories
- Berman
- 1980
(Show Context)
Citation Context ...ination to cope with integers. An automata based method is presented in [55].Linear arithmetic enjoys exciting complexity results. Fischer and Rabin [20] proved lower bounds which were later refined =-=[3, 4, 23]-=- using alternation [11]. Upper bounds can be found in [43, 19, 51, 33]. The complexity of subclasses of Presburger arithmetic is also well studied [48, 26]. Weispfenning [53, 52, 51, 33] provides prec... |

30 |
Beiträge zur Theorie der Linearen Ungleichungen
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- 1936
(Show Context)
Citation Context ...ge (§3) and then formalized and verified in logic (§4). Related work Fourier [21] presented an extension of Gauss-elimination to cope with inequalities. His method has been rediscovered several times =-=[17, 37]-=- and is today referred to as Fourier-Motzkin elimination. Tarski’s result for real closed fields [50] also yields a decision procedure for linear real arithmetic. More efficient quantifier elimination... |

26 | Mixed real-integer linear quantifier elimination
- Weispfenning
- 1999
(Show Context)
Citation Context ...the Coq home page). We also verified Cooper’s algorithm in Isabelle [10]. Finally note that the first-order theory of linear arithmetic over both reals and integers also admits quantifier elimination =-=[54]-=-. We verified this algorithm in Isabelle [9]. An automata based algorithm [6] has also been presented to solve the decision problem (not the quantifier elimination problem).2 Quantifier elimination f... |

25 |
Presburger arithmetic with bounded quantifier alternation
- Reddy, Loveland
- 1978
(Show Context)
Citation Context ...nd Weispfenning [33]. Quantifier elimination procedures for Presburger Arithmetic have been discovered by Presburger [45] and Skolem [49] independently and improved by Cooper [13], Reddy and Loveland =-=[47]-=- and extended to deal with parameters by Weispfenning [52]. Pugh [46] presented an adaptation of Fourier-Motzkin elimination to cope with integers. An automata based method is presented in [55].Linea... |

24 | An effective decision procedure for linear arithmetic over the integers and reals
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- 2005
(Show Context)
Citation Context ...lly note that the first-order theory of linear arithmetic over both reals and integers also admits quantifier elimination [54]. We verified this algorithm in Isabelle [9]. An automata based algorithm =-=[6]-=- has also been presented to solve the decision problem (not the quantifier elimination problem).2 Quantifier elimination for linear arithmetic This section provides an informal introduction to linear... |

24 | A proof-producing decision procedure for real arithmetic
- McLaughlin, Harrison
- 2005
(Show Context)
Citation Context ...plete. For the reals most theorem provers implement Fourier-Motzkin elimination and only deal with quantifier-free formulae. HOL Light [28] includes full quantifier-elimination for real closed fields =-=[30, 36]-=-. Verifying the CAD algorithm [12] is on-going work [34]. An alternative approach already used in [38] is based on checking certificates for Farkas’s lemma. This technique is very efficient, but is no... |

23 |
Solution d’une question particulière du calcul des inegalités. Nouveau Bulletin des Sciences par la Scociété Philomatique de Paris
- Fourier
(Show Context)
Citation Context ... to the two quantifier elimination algorithms (§2) they are first formalized as proofproducing functions in the meta-language (§3) and then formalized and verified in logic (§4). Related work Fourier =-=[21]-=- presented an extension of Gauss-elimination to cope with inequalities. His method has been rediscovered several times [17, 37] and is today referred to as Fourier-Motzkin elimination. Tarski’s result... |

21 | Autarkic computations in formal proofs
- Barendregt, Barendsen
(Show Context)
Citation Context ...hod of reflection goes back at least to the meta-functions used by Boyer and Moore [7] and later became popular in theorem provers based on type theory [5]. It has been studied by several researchers =-=[29, 1]-=-. Our use of reflection is rather computational and has nothing to do with “logical reflection” [29]. Laurent Théry verified Presburger’s original algorithm in Coq (see the Coq home page). We also ver... |

18 |
Precise bounds for Presburger arithmetic and the reals with addition: Preliminary report
- Berman
- 1977
(Show Context)
Citation Context ...ination to cope with integers. An automata based method is presented in [55].Linear arithmetic enjoys exciting complexity results. Fischer and Rabin [20] proved lower bounds which were later refined =-=[3, 4, 23]-=- using alternation [11]. Upper bounds can be found in [43, 19, 51, 33]. The complexity of subclasses of Presburger arithmetic is also well studied [48, 26]. Weispfenning [53, 52, 51, 33] provides prec... |

13 |
A Computer Program for Presburger’s Algorithm. In: Summary of talks presented at the
- Davis
- 1957
(Show Context)
Citation Context ... parameters. A bound on the automata size is due to Klaedtke [31]. The first implementation of a decision procedure for Presburger arithmetic, and in fact the first theorem prover, dates back to 1957 =-=[16]-=-. But in the following we mostly focus on those papers that are concerned with proof-producing decision procedures. Norrish [41] discusses proof-producing implementations in HOL [25] of Cooper’s algor... |

13 |
Elementary bounds for Presburger Arithmetic
- Oppen
- 1973
(Show Context)
Citation Context ...ecause one has to go through the inference rules in the kernel; (d) if the prover is based on proof objects this can lead to excessive space consumption (proofs may require (super-) exponential space =-=[20, 43]-=-). These shortcomings can be overcome by defining and verifying the decision procedure inside the logic, provided one can perform the compu-tations thus expressed in an efficient manner, typically by... |

13 | Complexity and uniformity of elimination in Presburger arithmetic
- Weispfenning
- 1997
(Show Context)
Citation Context ... were later refined [3, 4, 23] using alternation [11]. Upper bounds can be found in [43, 19, 51, 33]. The complexity of subclasses of Presburger arithmetic is also well studied [48, 26]. Weispfenning =-=[53, 52, 51, 33]-=- provides precise bounds for the quantifier elimination problem allowing (non linear) parameters. A bound on the automata size is due to Klaedtke [31]. The first implementation of a decision procedure... |

13 | Evaluating, testing, and animating PVS specifications
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- 2001
(Show Context)
Citation Context ... termevaluator [2] which generates executable code from the definition of simp and “runs” simp p. The fact that the executable code is again ML is a coincidence. A similar approach is adopted in PVS =-=[15]-=- and in ACL2 [7]. Other approaches include the the use of an internal λ-calculus evaluator [27] as in Coq [5]. One could also perform the evaluation step by rewriting inside the theorem prover, but th... |

11 | Verifying and reflecting quantifier elimination for Presburger arithmetic
- Chaieb, Nipkow
- 2005
(Show Context)
Citation Context ...putational and has nothing to do with “logical reflection” [29]. Laurent Théry verified Presburger’s original algorithm in Coq (see the Coq home page). We also verified Cooper’s algorithm in Isabelle =-=[10]-=-. Finally note that the first-order theory of linear arithmetic over both reals and integers also admits quantifier elimination [54]. We verified this algorithm in Isabelle [9]. An automata based algo... |

11 | Systems of linear inequalities - Dines - 1919 |

11 |
The complexity of almost linear Diophantine problems
- Weispfenning
- 1990
(Show Context)
Citation Context ...r Presburger Arithmetic have been discovered by Presburger [45] and Skolem [49] independently and improved by Cooper [13], Reddy and Loveland [47] and extended to deal with parameters by Weispfenning =-=[52]-=-. Pugh [46] presented an adaptation of Fourier-Motzkin elimination to cope with integers. An automata based method is presented in [55].Linear arithmetic enjoys exciting complexity results. Fischer a... |

10 | Validated Proof-Producing Decision Procedures
- Klapper, Stump
- 2004
(Show Context)
Citation Context ...urk in the code for a long time because they are not caught by a standard static type system which cannot express the precise form the theorem produced or expected by some function must have (but see =-=[32]-=-). This problem is exacerbated by the fact that decision procedures are re-implemented time and again for different systems, and that in the literature these implementations are only sketched if discu... |

10 |
Importing HOL into Isabelle/HOL
- Obua, Skalberg
- 2006
(Show Context)
Citation Context ... surprises at runtime. The explicitness of reflection pays off even more during maintenance, where tactics can be awkward to modify. Due to the progress in sharing theorems with other theorem provers =-=[35, 42]-=-, reflected decision procedures are ultimately shared for free. A final advantage of reflection is that it allows to formalize notions like duality, cf. §4.3, which reduces the size of the background ... |

9 | HOL Light Tutorial (for version 2.20
- Harrison
- 2007
(Show Context)
Citation Context ...ith quantifier-free Presburger arithmetic and is even there incomplete. For the reals most theorem provers implement Fourier-Motzkin elimination and only deal with quantifier-free formulae. HOL Light =-=[28]-=- includes full quantifier-elimination for real closed fields [30, 36]. Verifying the CAD algorithm [12] is on-going work [34]. An alternative approach already used in [38] is based on checking certifi... |

9 | On the automata size for Presburger arithmetic
- Klaedtke
- 2004
(Show Context)
Citation Context ...o well studied [48, 26]. Weispfenning [53, 52, 51, 33] provides precise bounds for the quantifier elimination problem allowing (non linear) parameters. A bound on the automata size is due to Klaedtke =-=[31]-=-. The first implementation of a decision procedure for Presburger arithmetic, and in fact the first theorem prover, dates back to 1957 [16]. But in the following we mostly focus on those papers that a... |

9 | Complete integer decision procedures as derived rules in HOL
- Norrish
(Show Context)
Citation Context ...er arithmetic, and in fact the first theorem prover, dates back to 1957 [16]. But in the following we mostly focus on those papers that are concerned with proof-producing decision procedures. Norrish =-=[41]-=- discusses proof-producing implementations in HOL [25] of Cooper’s algorithm (in tactic-style) and Pugh’s algorithm (by reflection of a proof trace found on the meta-level). The key difference to our ... |

8 | Verifying mixed real-integer quantifier elimination
- Chaieb
- 2006
(Show Context)
Citation Context ... algorithm in Isabelle [10]. Finally note that the first-order theory of linear arithmetic over both reals and integers also admits quantifier elimination [54]. We verified this algorithm in Isabelle =-=[9]-=-. An automata based algorithm [6] has also been presented to solve the decision problem (not the quantifier elimination problem).2 Quantifier elimination for linear arithmetic This section provides a... |

7 |
The complexity of Presburger arithmetic with bounded quantifier alternation depth,” Theor
- Fürer
- 1982
(Show Context)
Citation Context ...ination to cope with integers. An automata based method is presented in [55].Linear arithmetic enjoys exciting complexity results. Fischer and Rabin [20] proved lower bounds which were later refined =-=[3, 4, 23]-=- using alternation [11]. Upper bounds can be found in [43, 19, 51, 33]. The complexity of subclasses of Presburger arithmetic is also well studied [48, 26]. Weispfenning [53, 52, 51, 33] provides prec... |

7 |
Subclasses of Presburger arithmetic and the polynomial-time hierarchy
- Grädel
- 1988
(Show Context)
Citation Context ...oved lower bounds which were later refined [3, 4, 23] using alternation [11]. Upper bounds can be found in [43, 19, 51, 33]. The complexity of subclasses of Presburger arithmetic is also well studied =-=[48, 26]-=-. Weispfenning [53, 52, 51, 33] provides precise bounds for the quantifier elimination problem allowing (non linear) parameters. A bound on the automata size is due to Klaedtke [31]. The first impleme... |

7 |
Über einige Satzfunktionen in der Arithmetik
- Skolem
- 1970
(Show Context)
Citation Context ...he work on which we rely, and later by Weispfenning [51] and Loos and Weispfenning [33]. Quantifier elimination procedures for Presburger Arithmetic have been discovered by Presburger [45] and Skolem =-=[49]-=- independently and improved by Cooper [13], Reddy and Loveland [47] and extended to deal with parameters by Weispfenning [52]. Pugh [46] presented an adaptation of Fourier-Motzkin elimination to cope ... |

5 |
Une procédure de décision réflexive pour un fragment de l’arithmétique de Presburger
- Crégut
- 2004
(Show Context)
Citation Context ... design principles on an abstract level but omits the details of proof synthesis. For that he refers to the actual code, which we would argue is much too system specific to be easily portable. Crégut =-=[14]-=- presents an implementation of Pugh’s method for Coq [5], using the same technique as Norrish. His implementation only deals with quantifier-free Presburger arithmetic and is even there incomplete. Fo... |

4 |
Coq’Art: The Calculus of Inductive Constructions, volume XXV of Text in theor. comp. science: an EATCS series
- Bertot, Castéran
- 2004
(Show Context)
Citation Context ...ails of proof synthesis. For that he refers to the actual code, which we would argue is much too system specific to be easily portable. Crégut [14] presents an implementation of Pugh’s method for Coq =-=[5]-=-, using the same technique as Norrish. His implementation only deals with quantifier-free Presburger arithmetic and is even there incomplete. For the reals most theorem provers implement Fourier-Motzk... |

4 | Contributions à la certification des calculs sur R : théorie, preuves, programmation - Mahboubi - 2006 |