## Short Proofs of Normalization for the simply-typed λ-calculus, permutative conversions and Gödel's T (1998)

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Venue: | TO APPEAR: ARCHIVE FOR MATHEMATICAL LOGIC |

Citations: | 15 - 1 self |

### BibTeX

@MISC{Joachimski98shortproofs,

author = {Felix Joachimski and Ralph Matthes},

title = {Short Proofs of Normalization for the simply-typed λ-calculus, permutative conversions and Gödel's T },

year = {1998}

}

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### Abstract

Inductive characterizations of the sets of terms, the subset of strongly normalizing terms and normal forms are studied in order to reprove weak and strong normalization for the simplytyped λ-calculus and for an extension by sum types with permutative conversions. The analogous treatment of a new system with generalized applications inspired by von Plato's generalized elimination rules in natural deduction shows the flexibility of the approach which does not use the strong computability/candidate style a la Tait and Girard. It is also shown that the extension of the system with permutative conversions by -rules is still strongly normalizing, and likewise for an extension of the system of generalized applications by a rule of "immediate simplification". By introducing an innitely branching inductive rule the method even extends to Gödel's T.

### Citations

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(Show Context)
Citation Context ...urse of proofs, since we are only dealing with simple types here. 10We will not further comment on typability and type issues for the rest of this paper in order to keep the presentation short. 11See =-=[Bar84]-=- for a discussion of the mentioned concepts and various reduction strategies. 12Let ≻ be a binary relation on the fixed set M. Prog ≻ (X) :⇔ ∀x ∈ M.(∀y ≺ x.y ∈ X) ⇒ x ∈ X, (X is ≻-progressive) WF≻ := ... |

145 |
Über eine bisher noch nicht benützte Erweiterung des finiten Standpunktes
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(Show Context)
Citation Context ...oretical strength we cannot hope to extend the simple normalization proof to Gödel’s T without major modifications since it is well-known that termination of T implies consistency of Peano arithmetic =-=[Göd58]-=- and that this fact is provable in primitive recursive arithmetic. It will turn out that the inductive characterization SN of the strongly normalizing terms can be extended to T while the embedding of... |

60 | Program extraction from normalization proofs
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(Show Context)
Citation Context ...emma (part (i)), using the induction hypothesis for r and s. λxr ↓, since by induction hypothesis r ↓. Qed 2.2. Extracted program. By formalizing this constructive normalization proof in the style of =-=[Ber93]-=- we obtain • the algorithmic content nf : Λ → → NF of the theorem (defined by recursion on the term structure): nf(x) := x nf(λxr) := λx nf(r) nf(rs) := app(nf(r), nf(s)) • the algorithms app : NF × N... |

52 |
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(Show Context)
Citation Context ...hose typed operational semantics [Gog95] has been found independently of [vRS95]. Interestingly, the initial motivation for the work reported here was a kind of reverse engineering of Goguen’s thesis =-=[Gog94]-=- for systems without dependent types. 4 [vR96] stated and proved this characterization correct for pure untyped λ-calculus. 5 [Mat98] employs this approach in an analysis of extensions of Girard’s sys... |

39 |
A Formalization of the Strong Normalization Proof for System F
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(Show Context)
Citation Context ... normalizing iff there are no infinite reduction sequences starting with r. However, the data type of reduction sequences is notoriously problematic for computer-assisted proof development. Following =-=[Alt93]-=- and the tradition in proof theory, we define strong normalizability of a term r (with respect to a binary relation →) inductively by ∀r ′ .r → r ′ ⇒ r ′ ⇓ r ⇓ Thus, when checking strong normalizabili... |

26 |
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(Show Context)
Citation Context ...orollary. r, s ∈ NF =⇒ rs ↓. Proof. Use the lemma for xs and r with new x. Qed 14 This is a rather subtle application of bound variable renaming. It is, of course, possible to make it explicit. 15 In =-=[Dav01]-=-, this idea is made fruitful for the uniform proof of several known and also new characterizations of terms typable in systems of intersection types by normalization properties. 16 Confluence ensures ... |

24 |
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(Show Context)
Citation Context ...ifficulties arise 3 Based on a draft version of the present article, an extension to a logical framework using dependent types has been carried out by Goguen [Gog99] whose typed operational semantics =-=[Gog95]-=- has been found independently of [vRS95]. Interestingly, the initial motivation for the work reported here was a kind of reverse engineering of Goguen’s thesis [Gog94] for systems without dependent ty... |

13 |
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Citation Context ...h r ↓ and r ⇓. 19 This means that the reduction strategy that underlies the definition of WN actually is the standard reduction strategy guaranteed to exist by the standardization theorem (see, e.g., =-=[Dav95]-=- for a short exposition). 7snormal form ↓(r) of a term r ∈ WN can be defined by recursion as follows. 20 3.3. Lemma. (Soundness). SN ⊆ WF→. ↓: WN −→ NF, ↓(x�r ) := x ↓(�r ), ↓(λxr) := λx ↓(r), ↓((λxr)... |

12 |
Soundness of the Logical Framework for Its Typed Operational Semantics, Typed Lambda Calculi and
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(Show Context)
Citation Context ...tem with sum types and a case-construct. Difficulties arise 3 Based on a draft version of the present article, an extension to a logical framework using dependent types has been carried out by Goguen =-=[Gog99]-=- whose typed operational semantics [Gog95] has been found independently of [vRS95]. Interestingly, the initial motivation for the work reported here was a kind of reverse engineering of Goguen’s thesi... |

12 |
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(Show Context)
Citation Context ... general notion of introduction/elimination systems which incorporate the underlying concept. 6 “The type of any subterm of r : ρ is subtype of either the type of a free variable or of ρ.” See, e.g., =-=[Hin97]-=- for a good exposition. 2swith so-called critical eliminations r(x.s, y.t) : τ of a term r of type ρ + σ where τ need neither be a subtype of ρ nor of σ. This has led to the invention of endsegments a... |

11 |
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(Show Context)
Citation Context ...nd written either r�x[�s ] or r[�x := �s ]. Terms which differ only in names of bound variables are identified, i.e., α-equal terms are equal. 7 Other approaches can be found for instance in [Dil68], =-=[How80]-=- and [Sch93]. 8 Nipkow verified lemmata 3.3, 3.4 and 4 with the theorem prover Isabelle and gave worthwhile hints for improvement. 3s1.2. Inductive characterization and normal forms. The set of terms ... |

6 |
An early proof of normalization
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(Show Context)
Citation Context ...s enjoy continuous interest [Gog95, vRS95, vRSSX99] in the literature although the main results of weak and strong normalization have been established quite early by Turing (around 1941, published in =-=[Gan80]-=-) and Sanchis [San67]. This article employs a proof method that allows to show strong normalization for all typed terms without recourse to inclusive predicates such as strong computability [Tai67] or... |

6 |
Reduction Properties of ΠIE-Systems
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- 2001
(Show Context)
Citation Context ...is characterization correct for pure untyped λ-calculus. 5 [Mat98] employs this approach in an analysis of extensions of Girard’s system F by various forms of monotone inductive types. More recently, =-=[Joa01]-=- even gives a precise definition and analysis of the general notion of introduction/elimination systems which incorporate the underlying concept. 6 “The type of any subterm of r : ρ is subtype of eith... |

4 | On the strong normalisation of natural deduction with permutationconversions
- Groote
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(Show Context)
Citation Context ... these last clauses we obtain the grammar of normal forms NF ∋ r, s, t ::= x | x(s, z.t) | λxr. 29 Concerning the choice of variable names, see footnote 14. 30 The double-negation-translation used in =-=[dG99]-=- for establishing strong normalization of → does not cover →η. 15s6.3. Types and type assignment. By using the types and type assignments for variables from 1.3, the typable terms and their unique typ... |

3 | Operational aspects of normalization by evaluation
- Aehlig, Joachimski
- 2001
(Show Context)
Citation Context ...part (λ) of the proof when showing that λx.sx ⇓ follows from sx ⇓, but clearly s ⇓, since the set {r | r ⇓} = WF→βη of strongly normalizing terms is closed under subterms. It has been shown elsewhere =-=[AJ01]-=- that strong normalization (and confluence) of combined βreduction and η-expansion follows from the same property of β-reduction alone. As a consequence SN also exactly characterizes the set of strong... |

3 |
Zur Berechenbarkeit primitiv-rekursiver Funktionale endlicher Typen
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- 1968
(Show Context)
Citation Context ...s usual and written either r�x[�s ] or r[�x := �s ]. Terms which differ only in names of bound variables are identified, i.e., α-equal terms are equal. 7 Other approaches can be found for instance in =-=[Dil68]-=-, [How80] and [Sch93]. 8 Nipkow verified lemmata 3.3, 3.4 and 4 with the theorem prover Isabelle and gave worthwhile hints for improvement. 3s1.2. Inductive characterization and normal forms. The set ... |

3 | Ordinal analysis of terms of type - Howard - 1980 |

2 |
Strong normalization for arithmetic (variations on a theme of Prawitz
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(Show Context)
Citation Context ...d neither be a subtype of ρ nor of σ. This has led to the invention of endsegments and validity predicates (see [Pra71] for a graphical and [vdP96] for a term-based definition) or improper reductions =-=[Lei75]-=-. In section 5 we use the vector notation together with an only marginal modification of the basic normalization argument and come up with a short and easily formalizable proof which even covers the η... |

1 | expansion in Godel's T, pure type systems and calculi with permutative conversions. Unpublished draft - Joachimski - 1997 |