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**1 - 5**of**5**### Formally Proving a Compiler Transformation Safe

"... Abstract We prove that the Call Arity analysis and transformation, as implemented in the Haskell compiler GHC, is safe, i.e. does not impede the performance of the program. We formalized syntax, semantics, the analysis and the transformation in the interactive theorem prover Isabelle to obtain a ma ..."

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Abstract We prove that the Call Arity analysis and transformation, as implemented in the Haskell compiler GHC, is safe, i.e. does not impede the performance of the program. We formalized syntax, semantics, the analysis and the transformation in the interactive theorem prover Isabelle to obtain a machine-checked proof and hence a level of rigor rarely obtained for compiler optimization safety theorems. The proof is modular and introduces trace trees as a suitable abstraction in abstract cardinality analyses. We discuss the breadth of the formalization gap.

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, 2013

"... In his dissertation [3], Olin Shivers introduces a concept of control flow graphs for functional languages, provides an algorithm to statically derive a safe approximation of the control flow graph and proves this algorithm correct. In this research project [1], Shivers ’ algorithms and proofs are f ..."

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In his dissertation [3], Olin Shivers introduces a concept of control flow graphs for functional languages, provides an algorithm to statically derive a safe approximation of the control flow graph and proves this algorithm correct. In this research project [1], Shivers ’ algorithms and proofs are formalized using the HOLCF extension of the logic HOL in the theorem prover Isabelle.

### Reasoning about Constants in Nominal Isabelle, or how to Formalize the Second Fixed Point Theorem

"... Abstract. Nominal Isabelle is a framework for reasoning about pro-gramming languages with named bound variables (as opposed to de Bruijn indices). It is a definitional extension of the HOL object logic of the Isabelle theorem prover. Nominal Isabelle supports the definition of term languages of calc ..."

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Abstract. Nominal Isabelle is a framework for reasoning about pro-gramming languages with named bound variables (as opposed to de Bruijn indices). It is a definitional extension of the HOL object logic of the Isabelle theorem prover. Nominal Isabelle supports the definition of term languages of calculi with bindings, functions on the terms of these calculi and provides mechanisms that automatically rename binders. Functions defined in Nominal Isabelle can be defined with assumptions: The binders can be assumed fresh for any arguments of the functions. Defining functions is often one of the more complicated part of reasoning with Nominal Isabelle, and together with analysing freshness is the part that differs most from paper proofs. In this paper we show how to define terms from λ-calculus and reason about them without having to carry around the freshness conditions. As a case study we formalize the second fixed point theorem of the λ-calculus. 1

### Term-Generic Logic

"... We introduce term-generic logic (TGL), a first-order logic parameterized with terms defined axiomatically (rather than constructively), by requiring terms to only provide free variable and substitution operators satisfying some reasonable axioms. TGL has a notion of model that generalizes both first ..."

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We introduce term-generic logic (TGL), a first-order logic parameterized with terms defined axiomatically (rather than constructively), by requiring terms to only provide free variable and substitution operators satisfying some reasonable axioms. TGL has a notion of model that generalizes both first-order models and Henkin models of the λ-calculus. The abstract notions of term syntax and model are shown to be sufficient for obtaining the completeness theorem of a Gentzen system generalizing that of first-order logic. Various systems featuring bindings and contextual reasoning, ranging from pure type systems to the pi-calculus, are captured as theories inside TGL. For two particular, but rather typical instances—untyped λ-calculus and System F—the general-purpose TGL models are shown to be equivalent with standard ad hoc models.

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, 2014

"... HAL is a multi-disciplinary open access archive for the deposit and dissemination of sci-entific research documents, whether they are pub-lished or not. The documents may come from teaching and research institutions in France or abroad, or from public or private research centers. L’archive ouverte p ..."

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HAL is a multi-disciplinary open access archive for the deposit and dissemination of sci-entific research documents, whether they are pub-lished or not. The documents may come from teaching and research institutions in France or abroad, or from public or private research centers. L’archive ouverte pluridisciplinaire HAL, est destinée au dépôt et a ̀ la diffusion de documents scientifiques de niveau recherche, publiés ou non, émanant des établissements d’enseignement et de recherche français ou étrangers, des laboratoires publics ou privés. HOCore in Coq