# Proof Nets and Explicit Substitutions

TL;DR: The simulation technique introduced in [10] is refined to show strong normalization of λ-calculi with explicit substitutions via termination of cut elimination in proof nets and a version of typed λl with named variables is proposed which helps to better understand the complex mechanism of the explicit weakening notation.

Abstract: We refine the simulation technique introduced in [10] to show strong normalization of λ-calculi with explicit substitutions via termination of cut elimination in proof nets [13]. 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. We then show strong normalization of the typed version of the λl-calculus with de Bruijn indices (a calculus with full composition defined in [8]) using a translation from typed λl to proof nets. Finally, we propose a version of typed λl with named variables which helps to better understand the complex mechanism of the explicit weakening notation introduced in the λl-calculus with de Bruijn indices [8].

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##### Citations

71 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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49 citations

48 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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42 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

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2,199 citations

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

...In this paper we refine the simulation technique introduced in [10] to show strong normalization of λ-calculi with explicit substitutions via termination of cut elimination in proof nets [13]....

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...While we refer the interested reader to [13] for more details on linear logic in general, we give here a one-sided presentation of the sequent calculus for MELL:...

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1,480 citations

559 citations

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

...The pioneer calculus with explicit substitutions, λσ, was introduced in [1] as a bridge between the classical λ-calculus and concrete implementations of functional programming languages....

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305 citations

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

...1 Using various translations of the λ-calculus into proof nets, new abstract machines have been proposed, exploiting the Geometry of Interaction and the Dynamic Algebras [14, 2, 5], leading to the works on optimal reduction [15, 17]....

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