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E.: A constructive and formal proof of Lebesgues Dominated Convergence Theorem in the interactive theorem prover Matita
 Journal of Formalized Reasoning
, 2008
"... We present a formalisation of a constructive proof of Lebesgue’s Dominated Convergence Theorem given by Sacerdoti Coen and Zoli in [CSCZ]. The proof is done in the abstract setting of ordered uniformities, also introduced by the two authors as a simplification of Weber’s lattice uniformities given i ..."
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Cited by 7 (4 self)
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We present a formalisation of a constructive proof of Lebesgue’s Dominated Convergence Theorem given by Sacerdoti Coen and Zoli in [CSCZ]. The proof is done in the abstract setting of ordered uniformities, also introduced by the two authors as a simplification of Weber’s lattice uniformities given in [Web91, Web93]. The proof is fully constructive, in the sense that it is done in Bishop’s style and, under certain assumptions, it is also fully predicative. The formalisation is done in the Calculus of (Co)Inductive Constructions using the interactive theorem prover Matita [ASTZ07]. It exploits some peculiar features of Matita and an advanced technique to represent algebraic hierarchies previously introduced by the authors in [ST07]. Moreover, we introduce a new technique to cope with duality to halve the formalisation effort.
MATHEMATICAL LOGIC QUARTERLY
, 2007
"... The axiomofchoice and the law of excluded middle in weak set theories ..."
Contents
"... We present a formalisation of a constructive proof of Lebesgue’s Dominated Convergence Theorem given by Sacerdoti Coen and Zoli in [SZ]. The proof is done in the abstract setting of ordered uniformities, also introduced by the two authors as a simplification of Weber’s lattice uniformities given in ..."
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We present a formalisation of a constructive proof of Lebesgue’s Dominated Convergence Theorem given by Sacerdoti Coen and Zoli in [SZ]. The proof is done in the abstract setting of ordered uniformities, also introduced by the two authors as a simplification of Weber’s lattice uniformities given in [Web91, Web93]. The proof is fully constructive, in the sense that it is done in Bishop’s style and, under certain assumptions, it is also fully predicative. The formalisation is done in the Calculus of (Co)Inductive Constructions using the interactive theorem prover Matita [ASTZ07]. It exploits some peculiar features of Matita and an advanced technique to represent algebraic hierarchies previously introduced by the authors in [ST07]. Moreover, we introduce a new technique to cope with duality to halve the formalisation effort. Both authors were supported by DAMA (Dimostrazione Assistita per la Matematica e
Zermelo's WellOrdering Theorem in Type Theory
"... Abstract. Taking a `set ' to be a type together with an equivalence relation and adding an extensional choice axiom to the logical framework (a restricted version of constructive type theory) it is shown that any `set' can be wellordered. Zermelo's rst proof from 1904 is followed, with a simpli cat ..."
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Abstract. Taking a `set ' to be a type together with an equivalence relation and adding an extensional choice axiom to the logical framework (a restricted version of constructive type theory) it is shown that any `set' can be wellordered. Zermelo's rst proof from 1904 is followed, with a simpli cation to avoid using comparability of wellorderings. The proof has been formalised in the system AgdaLight. 1
The Interpretation of Inuitionistic . . .
, 2008
"... We give an intuitionistic view of Seely’s interpretation of MartinLöf’s intuitionistic type theory in locally cartesian closed categories. The idea is to use MartinLöf type theory itself as metalanguage, and Ecategories, the appropriate notion of categories when working in this metalanguage. As a ..."
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We give an intuitionistic view of Seely’s interpretation of MartinLöf’s intuitionistic type theory in locally cartesian closed categories. The idea is to use MartinLöf type theory itself as metalanguage, and Ecategories, the appropriate notion of categories when working in this metalanguage. As an Ecategorical substitute for the formal system of MartinLöf type theory we use Ecategories with families (Ecwfs). These come in two flavours: groupoidstyle Ecwfs and proofirrelevant Ecwfs. We then analyze Seely’s interpretation as consisting of three parts. The first part is purely categorical: the interpretation of groupoidstyle Ecwfs in Elocally cartesian closed categories. (The key part of this interpretation has been typechecked in the Coq system.) The second is a coherence problem which relates groupoidstyle Ecwfs with proofirrelevant ones. The third is a purely syntactic problem: that proofirrelevant Ecwfs are equivalent to traditional lambda calculus based formulations of MartinLöf type theory.