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module Soat.SecondOrder.Algebra.Lift.Initial
import Data.List.Elem
import Data.List.Sublist
import Data.Product
import Data.Setoid.Indexed
import Data.Setoid.Pair
import Data.Setoid.Product
import Soat.FirstOrder.Algebra
import Soat.FirstOrder.Term
import Soat.SecondOrder.Algebra
import Soat.SecondOrder.Algebra.Lift
import Soat.SecondOrder.Signature.Lift
import Syntax.PreorderReasoning.Setoid
%default total
%ambiguity_depth 4
public export
freeAlg : List sig.T -> FirstOrder.Algebra.Algebra sig
freeAlg ctx = FreeAlgebra (irrelevantCast (flip Elem ctx))
public export
Initial : (0 sig : _) -> RawAlgebra (lift sig)
Initial sig = MkRawAlgebra
(\t, ctx => (freeAlg ctx).raw.U t)
(\t, f => bindTerm {a = Free _} (\_ => Done . curry f))
(\_, (MkOp (Op op)) => Call (MkOp op) . unwrap (MkPair []))
Done
(\_, _, t, ts => bindTerm {a = Free _} (\_ => index ts) t)
public export
InitialIsAlgebra : (0 sig : _) -> SecondOrder.Algebra.IsAlgebra (lift sig) (Initial sig)
InitialIsAlgebra sig = MkIsAlgebra
{ equivalence = MkIndexedEquivalence
{ relation = \(t, ctx) => (freeAlg ctx).relation t
, reflexive = \(t, ctx) => (freeAlg ctx).algebra.equivalence.reflexive t
, symmetric = \(t, ctx) => (freeAlg ctx).algebra.equivalence.symmetric t
, transitive = \(t, ctx) => (freeAlg ctx).algebra.equivalence.transitive t
}
, renameCong = \t, f => bindTermCong
{ a = freeAlg _
, env = mate (\_ => Done . curry f)
}
, semCong = \_ , (MkOp (Op op)) => Call (MkOp op) . unwrapIntro
, substCong = \_, _, eq, eqs => bindTermCong'
{ a = freeAlg _
, cong = \_, Refl => index eqs _
, eq
}
, renameId = \t, ctx, tm =>
(freeAlg _).setoid.equivalence.symmetric t _ _ $
bindUnique (mate (\_ => Done . curry reflexive)) id (\i => Done $ sym $ curryUncurry id i) tm
, renameComp = \t, f, g, tm =>
(freeAlg _).setoid.equivalence.symmetric t _ _ $
bindUnique
{ a = freeAlg _
, env = mate (\_ => Done . curry (transitive g f))
, f = bindHomo (mate (\_ => Done . curry f)) . bindHomo (mate (\_ => Done . curry g))
, cong = \i => Done $ sym $ curryUncurry (curry f . curry g) i
, tm
}
, semNat = \f, (MkOp (Op op)), tms =>
Call (MkOp op) $
CalcWith (index (Product (freeAlg _).setoid) _) $
|~ bindTerms {a = Free _} (\_ => Done . curry f) (unwrap (MkPair []) tms)
~~ map (\_ => bindTerm {a = Free _} (\_ => Done . curry f)) (unwrap (MkPair []) tms)
.=.(bindTermsIsMap {a = Free _} _ _)
~~ map (\_ => bindTerm {a = Free _} (\_ => Done . curry (reflexive {x = []} ++ f))) (unwrap (MkPair []) tms)
..<(mapIntro' (\t =>
bindTermCong'
{rel = \_ => Equal}
{a = freeAlg _}
(\_, Refl => Done $ curryUncurry (curry f) _)) $
(Product (freeAlg _).setoid).equivalence.reflexive _ (unwrap (MkPair []) tms))
~~ unwrap (MkPair []) (map (\ty => bindTerm {a = Free _} (\_ => Done . curry (reflexive {x = fst ty} ++ f))) tms)
.=.(mapUnwrap (MkPair []) tms)
, varNat = \_, _ => Done Refl
, substNat = \t, f, tm, tms => bindUnique
{ a = freeAlg _
, env = mate (\_ => index $ map (\_ => bindTerm {a = Free _} (\_ => Done . curry f)) tms)
, f = bindHomo (mate (\_ => Done . curry f)) . bindHomo (mate (\_ => index tms))
, cong = \i =>
reflect (index (freeAlg _).setoid _) $
sym $
indexMap tms i
, tm
}
, substExnat = \t, ctx, f, tm, tms => bindUnique
{ a = freeAlg _
, env = mate (\_ => index $ shuffle f tms)
, f = bindHomo (mate (\_ => index tms)) . bindHomo (mate (\_ => Done . curry f))
, cong = \i =>
reflect (index (freeAlg _).setoid _) $
sym $
indexShuffle f i
, tm
}
, substComm = \t, ctx, tm, tms, tms' => bindUnique
{ a = freeAlg _
, env = mate (\_ => index $ map (\_ => bindTerm {a = Free _} (\_ => index tms')) tms)
, f = bindHomo (mate (\_ => index tms')) . bindHomo (mate (\_ => index tms))
, cong = \i =>
reflect (index (freeAlg _).setoid _) $
sym $
indexMap tms i
, tm
}
, substVarL = \_, _, _ => (freeAlg _).setoid.equivalence.reflexive _ _
, substVarR = \t, ctx, tm =>
(freeAlg _).setoid.equivalence.symmetric t _ _ $
bindUnique
{ v = irrelevantCast (flip Elem ctx)
, a = freeAlg ctx
, env = mate (\_ => index $ tabulate (Done {sig = sig, v = flip Elem ctx}))
, f = id
, cong = \i =>
reflect (index (freeAlg ctx).setoid _) $
sym $
indexTabulate Done i
, tm
}
, substCompat = \ctx, (MkOp (Op op)), tms, tms' =>
Call (MkOp op) $
CalcWith (index (Product (freeAlg _).setoid) _) $
|~ bindTerms {a = Free _} (\_ => index tms') (unwrap (MkPair []) tms)
~~ map (\_ => bindTerm {a = Free _} (\_ => index tms')) (unwrap (MkPair []) tms)
.=.(bindTermsIsMap {a = Free _} _ _)
~~ map (\t => (Initial sig).extendSubst ctx tms' ([], t)) (unwrap (MkPair []) tms)
..<(mapIntro' (\t => bindTermCong'
{rel = \_ => Equal}
{a = freeAlg _}
(\t, Refl => CalcWith (index (freeAlg _).setoid _) $
|~ index (map (\_ => bindTerm {a = Free _} (\_ => Done . curry ([] {ys = []} ++ reflexive))) tms') _
~~ bindTerm {a = Free _} (\_ => Done . curry ([] {ys = []} ++ reflexive)) (index tms' _)
.=.(indexMap tms' _)
~~ index tms' _
..<(bindUnique
{ env = mate (\_ => Done . curry ([] {ys = []} ++ reflexive))
, f = id
, cong = \i => Done $ sym $ trans (curryUncurry _ i) (curryUncurry id i)
, tm = index tms' _
}))) $
(Product (freeAlg _).setoid).equivalence.reflexive _ (unwrap (MkPair []) tms))
~~ unwrap (MkPair []) (map ((Initial sig).extendSubst ctx tms') tms)
.=.(mapUnwrap (MkPair []) tms)
}
public export
InitialAlgebra : (0 sig : _) -> SecondOrder.Algebra.Algebra (lift sig)
InitialAlgebra sig = MkAlgebra (Initial sig) (InitialIsAlgebra sig)
public export
freeToInitialHomo : (0 sig : _) -> (ctx : List sig.T)
-> FreeAlgebra (irrelevantCast (flip Elem ctx)) ~>
projectAlgebra sig (InitialAlgebra sig) ctx
freeToInitialHomo sig ctx = MkHomomorphism
{ func = MkIndexedSetoidHomomorphism
{ H = \_ => id
, homomorphic = \_, _, _ => id
}
, semHomo = \(MkOp op), tms =>
Call (MkOp op) $
reflect (index (Product ((InitialAlgebra sig).setoidAt ctx)) _) $
sym $
trans (unwrapWrap (extend (Initial sig).U ctx) _) (mapId tms)
}
public export
fromInitial : (a : Algebra (lift sig)) -> (InitialAlgebra sig).setoid ~> a.setoid
fromInitial a = bundle (\(t, ctx) =>
index (bindFunc {a = projectAlgebra sig a ctx} (reindex (flip MkPair ctx) a.varFunc)) t)
public export
fromInitialHomo : (a : Algebra (lift sig)) -> InitialAlgebra sig ~> a
fromInitialHomo a = MkHomomorphism
{ func = fromInitial a
, renameHomo = \t, f => bindUnique'
{v = irrelevantCast (flip Elem _)}
{a = projectAlgebra sig a _}
(bindHomo (reindex (flip MkPair _) a.varFunc) . bindHomo (mate (\_ => Done . curry f)))
(a.renameHomo f . bindHomo (reindex (flip MkPair _) a.varFunc))
(\i => a.algebra.equivalence.symmetric _ _ _ $ a.algebra.varNat f i)
, semHomo = \ctx, (MkOp (Op op)), tms =>
a.algebra.semCong ctx (MkOp (Op op)) $
CalcWith (index (Product (reindex (\ty => (snd ty, fst ty ++ ctx)) a.setoid)) _) $
|~ wrap (MkPair []) (bindTerms {a = project a.raw ctx} (\_ => a.raw.var) (unwrap (MkPair []) tms))
~~ wrap (MkPair []) (map (\t => (fromInitial a).H (t, ctx)) (unwrap (MkPair []) tms))
.=.(cong (wrap _) $ bindTermsIsMap {a = project a.raw ctx} _ _)
~~ wrap (MkPair []) (unwrap (MkPair []) (map (\ty => (fromInitial a).H (snd ty, fst ty ++ ctx)) tms))
.=.(cong (wrap _) $ mapUnwrap (MkPair []) tms)
~~ map (\ty => (fromInitial a).H (snd ty, fst ty ++ ctx)) tms
.=.(wrapUnwrap _)
, varHomo = \i => a.algebra.equivalence.reflexive _ $ a.raw.var i
, substHomo = \t, ctx, tm, tms => bindUnique'
{v = irrelevantCast (flip Elem _)}
{a = projectAlgebra sig a _}
(bindHomo (reindex (flip MkPair _) a.varFunc) . bindHomo (mate (\_ => index tms)))
(a.substHomo1 ctx _ . bindHomo (reindex (flip MkPair _) a.varFunc))
(\i => CalcWith (index a.setoid _) $
|~ bindTerm {a = project a.raw _} (\_ => a.raw.var) (index tms i)
~~ index (map (\_ => bindTerm {a = project a.raw _} (\_ => a.raw.var)) tms) i
.=<(indexMap {f = \_ => bindTerm {a = project a.raw _} (\_ => a.raw.var)} tms i)
~~ a.raw.subst _ ctx (a.raw.var i) (map (\_ => bindTerm {a = project a.raw _} (\_ => a.raw.var)) tms)
..<(a.algebra.substVarL ctx i _))
tm
}
public export
fromInitialUnique : {a : SecondOrder.Algebra.Algebra (lift sig)}
-> (f : InitialAlgebra sig ~> a)
-> (t : sig.T) -> (ctx : List sig.T) -> (tm : Term sig (flip Elem ctx) t)
-> a.relation (t, ctx) (f.func.H (t, ctx) tm) ((fromInitial a).H (t, ctx) tm)
fromInitialUnique {sig = sig} {a = a} f t ctx = bindUnique
{v = irrelevantCast (flip Elem _)}
{a = projectAlgebra sig a ctx}
(reindex (flip MkPair ctx) a.varFunc)
(projectHomo f ctx . freeToInitialHomo sig ctx)
f.varHomo
|
