# Proof Nets and Explicit Substitutions

##### Citations

77 citations

### Cites background from "Proof Nets and Explicit Substitutio..."

...Explicit substitutions (ES) have been connected to linear logic by Kesner and co-authors in a sequence of works [26, 32, 33], culminating in the linear substitution calculus (LSC), a new formalism with ES behaviorally isomorphic to proof nets (introduced in [6], developed in [1, 3, 4, 7, 10], and bearing similarities with calculi by De Bruijn [25], Nederpelt [42], and Milner [41])....

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53 citations

### Cites background from "Proof Nets and Explicit Substitutio..."

...But the λws-calculus has a complicated syntax and its named version [13] is even less intelligible....

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...However, the strong normalisation proof for λws given in [13] reveals a natural semantics for composition of ES via Linear Logic’s proof-nets [19], suggesting that weakening (explicit erasure) and contraction (explicit duplication) can be added to the calculus without losing strong normalisation....

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49 citations

44 citations

### Cites background from "Proof Nets and Explicit Substitutio..."

...Solutions proposed in LL include – adopting a syntax which identifies contractions made at several exponential depths, as in [23] – for now it seems hard to apply it in differential nets with boxes, we will see how the rule of codereliction against box introduces many difficulties; – using such an identification as an equivalence relation, as hinted in [12] for DINs and investigated in [3,8] for LL proof nets – an elegant solution, though it is less so with respect to freely moving around weakenings, as it may generate infinite trees with weakened leaves; – using it as a set of reductions, as in [4] – which is is the way we are adopting here....

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37 citations

##### References

44 citations

### "Proof Nets and Explicit Substitutio..." refers background in this paper

...Finally, in [10], the first two authors of this work showed for the first time that explicit substitutions could be tightly related to linear logic’s proof nets, by providing a translation into a variant of proof nets from λx [19, 4], a simple calculus with explicit substitutions and named variables, but no composition....

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15 citations

### "Proof Nets and Explicit Substitutio..." refers background or methods in this paper

...We first remind the following result from [9]: Lemma 1 (Termination of R E )....

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...Extended reduction modulo an equivalence relation Unfortunately, the original notion of reduction on PN is not well adapted to simulate neither the β rule of λ-calculus, nor the rules dealing with propagation of substitution in explicit substitution calculi: too many inessential details on the order of application of the rules are still present, and to make abstraction from them, one is naturally led to define an equivalence relation on PN , as is done in [9], where the following two equivalences are introduced:...

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...We first propose a notion of equivalence relation for proof nets that extends the one in [9], and we show that cut elimination modulo this equivalence relation is terminating....

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...3 Termination of RE We know from [9] that R E is terminating, and we can show easily that wc is terminating too, so if we could show that the wc-rule can be postponed with respect to all the other rules of R E , we would be easily done using a well-known abstract lemma....

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...Finally, besides the equivalence relation defined in [9], for the sake of simulating explicit substitutions, we will also need an extra reduction rule allowing to remove unneeded weakening links:...

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6 citations

### "Proof Nets and Explicit Substitutio..." refers background in this paper

...As expected the λl-calculus enjoys the subject reduction property (see [16] for a detailed proof)....

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5 citations

### "Proof Nets and Explicit Substitutio..." refers methods in this paper

...To achieve this result, we use the following abstract theorem (see for example [12]) : Theorem 4....

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