https://github.com/phadej/ncill

Agda mechanization of cut elimination in Non-Commutative Intuitionistic Linear Logic
https://github.com/phadej/ncill

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Agda mechanization of cut elimination in Non-Commutative Intuitionistic Linear Logic

• Host: GitHub
• URL: https://github.com/phadej/ncill
• Owner: phadej
• Created: 2018-08-30T11:05:43.000Z (over 6 years ago)
• Default Branch: master
• Last Pushed: 2019-06-26T11:12:49.000Z (over 5 years ago)
• Last Synced: 2024-11-07T03:48:49.118Z (2 months ago)
• Language: Agda
• Size: 25.4 KB
• Stars: 6
• Watchers: 3
• Forks: 0
• Open Issues: 0
• Metadata Files:
  • Readme: README

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README

        
-- Non-Commutative Linear Lambda Calculus

--

-- Relations and non-commutative linear logic

-- by Carolyn Brown, Doug Gurr

-- https://www.sciencedirect.com/science/article/pii/0022404994001472

--

-- Frank Pfenning's lectures on logic influeced the proof making.

-- The proof is essentially the same as his proof for STLC,

-- yet in Agda (not informal / Twelf) and for NCILL.

-- Cut-Elimination proof for NCILL is easy to mechanize, as

-- contexts are lists, not sets (STLC) or multisets (ILL).

--

-- Rendered HTML docs are at http://oleg.fi/ncill/

-- Source code is at https://github.com/phadej/ncill

--

module NCILL.README where

-- Non-commutative Intuitinistic Linear Logic

------------------------------------------------------------------------

--

-- ∙ Types

import NCILL.Types

-- ∙ Context is just a set of types: Ctx = List Ty

import NCILL.Ctx

-- ∙ Sqnt are sequents

import NCILL.Sequent

-- ∙ Sqnt⁺ are sequents with Cut

import NCILL.SequentPlus

-- ∙ We prove Cut Elimination Theorem (Soundness), Completeness is trivial

import NCILL.Cut

-- ⋯ using Admissibility of Cut lemma

import NCILL.Admit

-- ∙ For development, you can

import NCILL

-- Examples

------------------------------------------------------------------------

-- ∙ We show terms for many equivalences in NCILL:

--   ∘ Quantale structure: Monoids of ⊗ and ♯1; ⊕ and ♯0; & and ♯⊤,

--     distributivity

--   ∘ More equivalences from Girard's Linear Logic (only ⊗-comm doesn't hold!)

--   ∘ Example of difference between ⊸ and ⊸ʳ

--

import NCILL.Examples.Equivalences

-- ∙ Though CUT is admissible in NCILL, η-reductions aren't performed.

--   This example shows ⊗ η and β -reductions.

--   ∘ original and η-reduced term aren't the same, yet logically they are 

--   ∘ β involves CUT, and terms are definitionally equal

--   ∘ we compare that to Agda's own behaviour

--

import NCILL.Examples.EtaBeta

-- Extras

------------------------------------------------------------------------

-- ∙ there are some extras to write the proofs

import ListExtras          -- few additional list related lemmas

import Simple-list-solver  -- solver for list concatenations (simpler to use than monoid-solver)

import AssocProofs         -- associativity lemmas up to 5 arguments "pre-proved"