mirror of
https://github.com/idris-lang/Idris2.git
synced 2024-12-01 01:09:03 +03:00
d5c9253e7e
Thanks to the debug info supplied by #2673, I was able to spot which functions were blocking and `public import` the relevant files in `papers/Search/Properties.idr`. As a result the GCL file now type-checks, albeit extremely slowly! I stopped an evaluation of `petersonsCorrect` at the REPL after 15 minutes and ~14GB of RAM consumed.
123 lines
3.6 KiB
Idris
123 lines
3.6 KiB
Idris
||| The content of this module is based on the paper
|
|
||| Applications of Applicative Proof Search
|
|
||| by Liam O'Connor
|
|
||| https://doi.org/10.1145/2976022.2976030
|
|
|
|
module Search.Properties
|
|
|
|
import Data.Fuel
|
|
import Data.List.Lazy
|
|
import Data.List.Lazy.Quantifiers
|
|
import Data.Nat
|
|
import Data.So
|
|
|
|
import public Data.Stream
|
|
import public Data.Colist
|
|
import public Data.Colist1
|
|
|
|
import public Search.Negation
|
|
import public Search.HDecidable
|
|
import public Search.Generator
|
|
|
|
import Decidable.Equality
|
|
|
|
%default total
|
|
|
|
------------------------------------------------------------------------
|
|
-- Type
|
|
|
|
||| Take the product of a list of types
|
|
public export
|
|
Product : List Type -> Type
|
|
Product = foldr Pair ()
|
|
|
|
||| A property amenable to testing
|
|
||| @cs is the list of generators we need (inferrable)
|
|
||| @a is the type we hope is inhabited
|
|
||| NB: the longer the list of generators, the bigger the search space!
|
|
public export
|
|
record Prop (cs : List Type) (a : Type) where
|
|
constructor MkProp
|
|
||| The function trying to find an `a` provided generators for `cs`.
|
|
||| Made total by consuming some fuel along the way.
|
|
runProp : Colist1 (Product cs) -> Fuel -> HDec a
|
|
|
|
------------------------------------------------------------------------
|
|
-- Prop-like structure
|
|
|
|
||| A type constructor satisfying the AProp interface is morally a Prop
|
|
||| It may not make use of all of the powers granted by Prop, hence the
|
|
||| associated `Constraints` list of types.
|
|
public export
|
|
interface AProp (0 t : Type -> Type) where
|
|
0 Constraints : List Type
|
|
toProp : t a -> Prop Constraints a
|
|
|
|
||| Props are trivially AProp
|
|
public export
|
|
AProp (Prop cs) where
|
|
Constraints = cs
|
|
toProp = id
|
|
|
|
||| Half deciders are AProps that do not need any constraints to be satisfied
|
|
public export
|
|
AProp HDec where
|
|
Constraints = []
|
|
toProp = MkProp . const . const
|
|
|
|
||| Deciders are AProps that do not need any constraints to be satisfied
|
|
public export
|
|
AProp Dec where
|
|
Constraints = []
|
|
toProp = MkProp . (const . const . toHDec)
|
|
|
|
||| We can run an AProp to try to generate a value of type a
|
|
public export
|
|
check : (gen : Generator (Product cs)) => (f : Fuel) -> (p : Prop cs a) ->
|
|
{auto pr : So (isTrue (runProp p (generate @{gen}) f))} -> a
|
|
check @{gen} f p @{pr} = evidence (runProp p (generate @{gen}) f) pr
|
|
|
|
||| Provided that we know how to generate candidates of type `a`, we can look
|
|
||| for a witness satisfying a given predicate over `a`.
|
|
public export
|
|
exists : {0 p : a -> Type} -> (aPropt : AProp t) =>
|
|
((x : a) -> t (p x)) ->
|
|
Prop (a :: Constraints @{aPropt}) (DPair a p)
|
|
exists test = MkProp $ \ acs, fuel =>
|
|
let candidates : LazyList a = take fuel (map fst acs) in
|
|
let cs = map snd acs in
|
|
let find = any candidates (\ x => runProp (toProp (test x)) cs fuel) in
|
|
map toDPair find
|
|
|
|
------------------------------------------------------------------------
|
|
-- Examples
|
|
|
|
namespace GT11
|
|
|
|
example : Prop ? (DPair Nat (\ i => Not (i `LTE` 10)))
|
|
example = exists (\ i => not (isLTE 11 i))
|
|
|
|
lemma : DPair Nat (\ i => Not (i `LTE` 10))
|
|
lemma = check (limit 1000) example
|
|
|
|
namespace Pythagoras
|
|
|
|
formula : Type
|
|
formula = DPair Nat $ \ m =>
|
|
DPair Nat $ \ n =>
|
|
DPair Nat $ \ p =>
|
|
(0 `LT` m, 0 `LT` n, 0 `LT` p
|
|
, (m * m + n * n) === (p * p))
|
|
|
|
search : Prop ? Pythagoras.formula
|
|
search = exists $ \ m =>
|
|
exists $ \ n =>
|
|
exists $ \ p =>
|
|
(isLT 0 m && isLT 0 n && isLT 0 p
|
|
&& decEq (m * m + n * n) (p * p))
|
|
|
|
-- This one is quite a bit slower so it's better to run
|
|
-- the compiled version instead
|
|
lemma : HDec Pythagoras.formula
|
|
lemma = runProp search generate (limit 10)
|