[Agda] Modelling term-rewriting systems in Agda

Twan van Laarhoven twanvl at gmail.com
Wed Oct 23 12:08:30 CEST 2013

```I threw it up on github, https://gist.github.com/twanvl/7115932
I have some more proofs, such as
confluent-by-sn : LocalConfluent R → StronglyNormalizing R → Confluent (* R)

but these are scattered over several files, so I didn't include them.

Twan

On 22/10/13 16:22, Jacques Carette wrote:
> Thank you, this was exactly the kind of example I was looking for.  Is your
> complete example available somewhere?  Or would you mind sharing it?
>
> Jacques
>
> On 2013-10-22 11:13 AM, Twan van Laarhoven wrote:
>>
>> As a somewhat related data point, a while ago I proved confluence of beta
>> reduction for untyped lambda calculus (as I am sure many others have as well).
>> To encode the rewrite rules I used a data type to represent the reduction
>> relation. Here is single step beta reduction
>>
>>   data _-B->_ {V} : Term V → Term V → Set where
>>     beta      : ∀ {x y} → (ap (lam x) y) -B-> (subst y x)
>>     under-ap₁ : ∀ {x y x'} → (x -B-> x') → (ap x y -B-> ap x' y)
>>     under-ap₂ : ∀ {x y y'} → (y -B-> y') → (ap x y -B-> ap x y')
>>     under-lam : ∀ {x x'} → (x -B-> x') → (lam x -B-> lam x')
>>
>> And here is the parallel beta reduction:
>>
>>   data _-PB->_ {V} : Term V → Term V → Set where
>>     var       : ∀ {v} → var v -PB-> var v -- no reduction in leaf
>>     beta      : ∀ {x y x' y'} → (x -PB-> x') → (y -PB-> y')
>>                                 → (ap (lam x) y) -PB-> (subst y x)
>>     under-ap  : ∀ {x y x' y'} → (x -PB-> x') → (y -PB-> y')
>>                                 → (ap x y -PB-> ap x' y')
>>     under-lam : ∀ {x x'} → (x -PB-> x') → (lam x -PB-> lam x')
>>
>> In the generic case you might try to split the `under` constructors into
>> another data type.
>>
>> For confluence it turned out to be useful to define a datatype:
>>
>>    data CommonReduct {a r s} {A : Set a} (R : Rel A r) (S : Rel A s)
>>                        (x y : A) : Set (a ⊔ r ⊔ s) where
>>       _||_ : {z : A} → R x z → S y z → CommonReduct R S x y
>>
>> then confluence can be expressed as
>>
>>     Confluent : ∀ {a r A} → (R : Rel A r) → Set (a ⊔ r)
>>     Confluent R = ∀ {x y z} → R x y → R x z → CommonReduct R R y z
>>
>> And using Data.Star you can prove lemma's like strong normalization & local
>> confluence imply global confluence.
>>
>>
>> Twan
>>
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>

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