# A λ-calculus with explicit weakening and explicit substitution

## Summary (1 min read)

### 2. Preliminaries

- The authors give here some de nitions and useful lemmas about rewriting systems.
- The authors also recall the rules for the usual -reduction on -terms with de Bruijn indices and the explicit substitution calculus s e .

### De nition 2.4 (Con uence).

- The second one is a particular case of the rst one.
- The original result is that a rewriting system which is locally con uent, weakly normalizing and increasing (there is a measure which is strictly increased by reduction) is also strongly normalizing.
- The -calculus with de Bruijn indices: the db -calculus.
- This will simplify notations in the next sections and, this is more natural with respect to the typed calculus.
- = ug (the substitution) and (the updating function) are meta functions, i.e. are not in the syntax of the calculus, also known as fi.

### (ax)

- The rst three rules are the usual ones of the db -calculus.
- A label corresponds to a weakening in the proof tree associated with the term.
- This is the motivation of the subscript \w" in the name of the calculus.
- The proof of subject reduction is straightforward.
- Every typed w -terms is strongly normalizable.

### 7. Simulation of the -reduction

- This is a particular case of proposition 6.15 with terms without metavariables.
- Just remark that, on terms without metavariables, the reductions w and b 0 (cf. de nition 6.7) are the same.

### 8. Preservation of strong normalization

- In the subsection 8.1, the authors give the sketch of the proof.
- Sections 8.2 and 8.3 give the de nitions and the main tools used in the proof.

### 8.1. Sketch of the proof

- For the last step (propagation of the substitution), the authors use the projection lemma on an extended syntax of the ws -calculus.
- This syntax allows to keep track of the reducts of the substitution created by reduction of the head redex.

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501 citations

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### Cites background from "A λ-calculus with explicit weakenin..."

...[1, 24, 12]) and more generally for higher-order rewrite systems (e....

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53 citations

### Cites background from "A λ-calculus with explicit weakenin..."

...Since then, different notions of safe composition where introduced, even if PSN becomes more difficult to prove ([8, 14, 1, 29, 31])....

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...λx [44] Yes No Yes Yes Yes No λσ [2] Yes No No No Yes Yes λσ⇑ [23] Yes Yes No No Yes Yes λζ [41] Yes Yes Yes Yes No No λws [14] Yes Yes Yes Yes Yes No λlxr [29] Yes ? Yes Yes Yes Yes...

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...In order to cope with this problem David and Guillaume [14] defined a calculus with labels called λws, which allows controlled composition of ES without losing PSN and SN....

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51 citations

### Cites background from "A λ-calculus with explicit weakenin..."

...Besides the ‚ws-calculus [ DG01 ] and its encoding in linear logic [DCKP00] already mentioned, other computational meanings of logic via the use of operators have already been proposed....

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...In order to cope with this problem David and Guillaume [ DG01 ] deflned a calculus with labels, called ‚ws, which allows controlled composition of explicit substitutions without losing PSN....

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...In calculi like ‚ws [ DG01 ] only cases (1) and (3) are considered by the reduction rules, thus only yielding partial composition....

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##### References

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### "A λ-calculus with explicit weakenin..." refers background in this paper

...The lemma 2.10 is an adaptation of a result given in (Klop, 1992)....

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...10 is an adaptation of a result given in (Klop, 1992)....

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142 citations

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