42 lines
2.2 KiB
Org Mode
42 lines
2.2 KiB
Org Mode
* Church Rosser, surjective pairing, and strong normalization
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This repository contains a mechanized proof that the lambda calculus
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with beta and eta rules for functions and pairs is in fact confluent
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for the subset of terms that are "strongly $\beta$-normalizing".
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Our notion of $\beta$-strong normalization, based on Abel's POPLMark
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Reloaded survey, is inductively defined
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and captures a strict subset of the usual notion of strong
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normalization in the sense that ill-formed terms such as $\pi_1
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\lambda x . x$ are rejected.
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* Joinability: A transitive equational relation
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The joinability relation $a \Leftrightarrow b$, which is true exactly
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when $\exists c, a \leadsto_{\beta\eta} c \wedge b
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\leadsto_{\beta\eta} c$, is transitive on the set of strongly
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normalizing terms thanks to the Church-Rosser theorem proven in this
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repository.
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** Joinability and logical predicates
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When building a logical predicate where adequacy ensures beta strong
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normalization, we can then show that two types $A \Leftrightarrow B$
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share the same meaning if both are semantically well-formed. The
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strong normalization constraint can be extracted from adequacy so we
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have transitivity of $A \Leftrightarrow B$ available in the proof that
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the logical predicate is functional over joinable terms.
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** Joinability and typed equality
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For systems with a type-directed equality, the same joinability
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relation can be used to model the equality. The fundamental theorem
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for the judgmental equality would take the following form: $\vdash a
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\equiv b \in A$ implies $\vDash a, b \in A$ and $a \Leftrightarrow b$.
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Note that such a result does not immediately give us the decidability
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of type conversion because the algorithm of beta-eta normalization
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may identify more terms when eta is not type-preserving. However, I
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believe Coquand's algorithm can be proven correct using the method
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described in [[https://www.researchgate.net/publication/226663076_Justifying_Algorithms_for_be-Conversion][Goguen 2005]]. We have all the ingredients needed:
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- Strong normalization of beta (needed for the termination metric)
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- Church-Rosser for $\beta$-SN terms (the disciplined expansion of
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Coquand's algorithm preserves both SN and typing)
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