https://github.com/yiyunliu/takahashi-factorization
A proof of factorization using Takahashi's method
https://github.com/yiyunliu/takahashi-factorization
coq-formalization lambda-calculus
Last synced: 7 months ago
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A proof of factorization using Takahashi's method
- Host: GitHub
- URL: https://github.com/yiyunliu/takahashi-factorization
- Owner: yiyunliu
- Created: 2024-06-20T20:57:48.000Z (almost 2 years ago)
- Default Branch: main
- Last Pushed: 2024-06-24T00:27:31.000Z (almost 2 years ago)
- Last Synced: 2025-02-15T22:28:33.729Z (over 1 year ago)
- Topics: coq-formalization, lambda-calculus
- Language: Coq
- Homepage:
- Size: 21.5 KB
- Stars: 1
- Watchers: 1
- Forks: 1
- Open Issues: 0
-
Metadata Files:
- Readme: README.md
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README
# A Coq mechanization of the factorization theorem for the untyped lambda calculus
This repository contains mechanized proofs Hindley's postponement
theorem for the weak head reduction
([normalization_h.v](theories/normalization_h.v))
and leftmost-outermost reduction
([normalization_lo.v](theories/normalization_lo.v)) strategies.
Note that [normalization_h.v](theories/normalization_h.v) also
includes a proof that recovers the standardization theorem from
factorization theorem for weak head reduction.
The proof is based on
[Takahashi's
proof](https://www.sciencedirect.com/science/article/pii/S0890540185710577)
and structured in a way similar to [Accattoli et
al. 2019](https://link.springer.com/chapter/10.1007/978-3-030-34175-6_9).
I managed to prove
postponement for leftmost-outermost reduction directly without indexed
parallel reduction, a proof device introduced by Accattoli to work
around the limitation of Takahashi's method.
Indexed parallel reduction is nice on paper, but is painful to
mechanize, especially when the development relies on simultaneous
substitution for its metatheory; the generalized substitution property
(morphing lemma) would require adding up the number reductions for
each pair of reductions to substitute. The development avoids indexing
altogether and instead uses Takahashi's * operator to bound the
reduction steps.
Despite Accattoli's observation that Takahashi's method fails to work
on leftmost-outermost reduction, it's possible to revise her statement
of Lemma 2.4 to a more generous statement (`ipar_starseq_morphing` in
the development) to remove the dependency on the left-substitutivity
of the essential reduction relation. This revised statement not only
holds for both head and lo reductions, but is also much easier to
prove.
Still, I find it more intuitive to recover leftmost-outermost
standardization from the factorization theorem for weak head
reduction, a useful result on its own.