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**1 - 2**of**2**### Intuitionistic Logic of Proofs

, 2009

"... The logic of proofs LP was introduced in [3] and thoroughly studied in [1]. LP is a natural extension of the propositional calculus in the language representing proofs as formal objects. Proof expressing terms are constructed using constants, variables, and symbols of natural operations on derivatio ..."

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The logic of proofs LP was introduced in [3] and thoroughly studied in [1]. LP is a natural extension of the propositional calculus in the language representing proofs as formal objects. Proof expressing terms are constructed using constants, variables, and symbols of natural operations on derivations. Then

### Intuitionistic Logic of Proofs with dependent proof terms

"... The basic logic of proofs extends the usual propositional language by expressions of the form “s is a proof of A”, for any proposition A. In this paper we explore the extension of its intuitionistic fragment to a language including expressions of the form “t is a proof of B, dependent from s being a ..."

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The basic logic of proofs extends the usual propositional language by expressions of the form “s is a proof of A”, for any proposition A. In this paper we explore the extension of its intuitionistic fragment to a language including expressions of the form “t is a proof of B, dependent from s being a proof of A”. We aim at laying down a ground comparison with equivalent constructions present in theories of dependent types, especially those similarly based on the Brouwer-Heyting-Kolmogorov semantics. We further translate this extended language to a natural deduction calculus which allows for a double interpretation of the construction on which a proof term may depend: as actually proven, or valid assumption, or as possibly proven, locally true assumption. We show meta-theoretical properties for this calculus and explain normalisation to a language with only unconditional proofs. We conclude by stating the characterization of our calculus with standard intutionistic logic of proofs.