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... term t and unit type U, if Γ ` t@U : T , then there exists a type V such that Γ ` t : ∀X .V and V [U/X ]≡ T . The following lemma is standard in proofs of subject reduction for System F-like systems =-=[13, 5]-=-. It ensures that well-typedness is preserved under substitution on type and term variables. Lemma 3.2 (Substitution lemma) For any term t, basis term b, context Γ, unit type U and type T , 1. If Γ ` ...

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...ion 4 shows an abstract interpretation of λCA into Additive, the additive fragment of λlin. Finally, section 5 concludes. 2 The Calculus We introduce the calculus λCA, which extends explicit System F =-=[16]-=- with linear combinations of λ - terms. Figure 1 shows the abstract syntax of types and terms of the calculus, where the terms are based on those of λlin [4]. Our choice of explicit System F instead o...

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...ifferential operator. In recent years, there has been growing research interest in these two calculi and their variants, as they could provide an explicit link between linear logic and linear algebra =-=[1, 2, 3, 6, 7, 8, 9, 10, 11, 14, 15, 17, 19]-=-. The two languages, λlin and λalg, are rather similar: they both merge the untyped λ -calculus –higherorder computation in its simplest and most general form– with linear algebraic constructions –sum...

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... term t and unit type U, if Γ ` t@U : T , then there exists a type V such that Γ ` t : ∀X .V and V [U/X ]≡ T . The following lemma is standard in proofs of subject reduction for System F-like systems =-=[13, 5]-=-. It ensures that well-typedness is preserved under substitution on type and term variables. Lemma 3.2 (Substitution lemma) For any term t, basis term b, context Γ, unit type U and type T , 1. If Γ ` ...

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... term t and unit type U, if Γ ⊢ t@U : T , then there exists a type V such that Γ ⊢ t : ∀X .V and V [U/X ]≡ T . The following lemma is standard in proofs of subject reduction for System F-like systems =-=[12, 4]-=-. It ensures that well-typedness is preserved under substitution on type and term variables. Lemma 3.2 (Substitution lemma) For any term t, basis term b, context Γ, unit type U and type T , 1. If Γ ⊢ ...

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...retation in System F. 1 Introduction Two algebraic versions of λ -calculus arose independently in different contexts: the linear-algebraic λ - calculus (λlin) [4] and the algebraic λ -calculus (λalg) =-=[20]-=-. The former was first introduced as a candidate λ -calculus for quantum computation; a linear combination of terms reflect the phenomenon of superposition, i.e. the capacity for a quantum system to b...

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...ifferential operator. In recent years, there has been growing research interest in these two calculi and their variants, as they could provide an explicit link between linear logic and linear algebra =-=[1, 2, 3, 6, 7, 8, 9, 10, 11, 14, 15, 17, 19]-=-. The two languages, λlin and λalg, are rather similar: they both merge the untyped λ -calculus –higherorder computation in its simplest and most general form– with linear algebraic constructions –sum...

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...ifferential operator. In recent years, there has been growing research interest in these two calculi and their variants, as they could provide an explicit link between linear logic and linear algebra =-=[1, 2, 3, 6, 7, 8, 9, 10, 11, 14, 15, 17, 19]-=-. The two languages, λlin and λalg, are rather similar: they both merge the untyped λ -calculus –higherorder computation in its simplest and most general form– with linear algebraic constructions –sum...

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...rposition, i.e. the capacity for a quantum system to be in two or more states at the same time. The latter was introduced in the context of linear logic, as a fragment of the differential λ -calculus =-=[11]-=-, an extension to λ -calculus with a differential operator making the resource-aware behaviour explicit. This extension produces a calculus where superposition of terms may happen. Then λalg can be se...

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...ifferential operator. In recent years, there has been growing research interest in these two calculi and their variants, as they could provide an explicit link between linear logic and linear algebra =-=[1, 2, 3, 6, 7, 8, 9, 10, 11, 14, 15, 17, 19]-=-. The two languages, λlin and λalg, are rather similar: they both merge the untyped λ -calculus –higherorder computation in its simplest and most general form– with linear algebraic constructions –sum...

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...ifferential operator. In recent years, there has been growing research interest in these two calculi and their variants, as they could provide an explicit link between linear logic and linear algebra =-=[1, 2, 5, 6, 7, 8, 9, 10, 13, 14, 16, 18]-=-. The two languages, λlin and λalg, are rather similar: they both merge the untyped λ -calculus –higherorder computation in its simplest and most general form– with linear algebraic constructions –sum...

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...ifferential operator. In recent years, there has been growing research interest in these two calculi and their variants, as they could provide an explicit link between linear logic and linear algebra =-=[1, 2, 3, 6, 7, 8, 9, 10, 11, 14, 15, 17, 19]-=-. The two languages, λlin and λalg, are rather similar: they both merge the untyped λ -calculus –higherorder computation in its simplest and most general form– with linear algebraic constructions –sum...

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...ifferential operator. In recent years, there has been growing research interest in these two calculi and their variants, as they could provide an explicit link between linear logic and linear algebra =-=[1, 2, 3, 6, 7, 8, 9, 10, 11, 14, 15, 17, 19]-=-. The two languages, λlin and λalg, are rather similar: they both merge the untyped λ -calculus –higherorder computation in its simplest and most general form– with linear algebraic constructions –sum...

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...ifferential operator. In recent years, there has been growing research interest in these two calculi and their variants, as they could provide an explicit link between linear logic and linear algebra =-=[1, 2, 3, 6, 7, 8, 9, 10, 11, 14, 15, 17, 19]-=-. The two languages, λlin and λalg, are rather similar: they both merge the untyped λ -calculus –higherorder computation in its simplest and most general form– with linear algebraic constructions –sum...

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...ifferential operator. In recent years, there has been growing research interest in these two calculi and their variants, as they could provide an explicit link between linear logic and linear algebra =-=[1, 2, 3, 6, 7, 8, 9, 10, 11, 14, 15, 17, 19]-=-. The two languages, λlin and λalg, are rather similar: they both merge the untyped λ -calculus –higherorder computation in its simplest and most general form– with linear algebraic constructions –sum...

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...ifferential operator. In recent years, there has been growing research interest in these two calculi and their variants, as they could provide an explicit link between linear logic and linear algebra =-=[1, 2, 3, 6, 7, 8, 9, 10, 11, 14, 15, 17, 19]-=-. The two languages, λlin and λalg, are rather similar: they both merge the untyped λ -calculus –higherorder computation in its simplest and most general form– with linear algebraic constructions –sum...

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...ifferential operator. In recent years, there has been growing research interest in these two calculi and their variants, as they could provide an explicit link between linear logic and linear algebra =-=[1, 2, 3, 6, 7, 8, 9, 10, 11, 14, 15, 17, 19]-=-. The two languages, λlin and λalg, are rather similar: they both merge the untyped λ -calculus –higherorder computation in its simplest and most general form– with linear algebraic constructions –sum...