quox/lib/Quox/Syntax/Shift.idr

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module Quox.Syntax.Shift
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import public Quox.Var
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import Data.Nat
import Data.So
import Data.Singleton
import Syntax.PreorderReasoning
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%default total
||| represents the difference between a smaller scope and a larger one.
public export
data Shift : (from, to : Nat) -> Type where
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SZ : Shift from from
SS : Shift from to -> Shift from (S to)
%name Shift by, bz
%builtin Natural Shift
public export
(.nat) : Shift from to -> Nat
(SZ).nat = Z
(SS by).nat = S by.nat
%transform "Shift.(.nat)" Shift.(.nat) = believe_me
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public export Cast (Shift from to) Nat where cast = (.nat)
public export Cast (Shift from to) Integer where cast = cast . cast {to = Nat}
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export Eq (Shift from to) where (==) = (==) `on` (.nat)
export Ord (Shift from to) where compare = compare `on` (.nat)
||| shift equivalence, ignoring indices
public export
data Eqv : Shift from1 to1 -> Shift from2 to2 -> Type where
EqSZ : SZ `Eqv` SZ
EqSS : by `Eqv` bz -> SS by `Eqv` SS bz
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%name Shift.Eqv e
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using (by : Shift from to, bz : Shift from to)
||| two equivalent shifts are equal if they have the same indices.
export
0 fromEqv : by `Eqv` bz -> by = bz
fromEqv EqSZ = Refl
fromEqv (EqSS e) = cong SS $ fromEqv e
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||| two equal shifts are equivalent.
export
0 toEqv : by = bz -> by `Eqv` bz
toEqv Refl {by = SZ} = EqSZ
toEqv Refl {by = (SS by)} = EqSS $ toEqv Refl
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export
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cmpLen : Shift from1 to -> Shift from2 to -> Either Ordering (from1 = from2)
cmpLen SZ SZ = Right Refl
cmpLen SZ (SS by) = Left LT
cmpLen (SS by) SZ = Left GT
cmpLen (SS by) (SS bz) = cmpLen by bz
export
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0 shiftDiff : (by : Shift from to) -> to = by.nat + from
shiftDiff SZ = Refl
shiftDiff (SS by) = cong S $ shiftDiff by
export
0 shiftVarLT : (by : Shift from to) -> (i : Var from) ->
by.nat + i.nat `LT` to
shiftVarLT by i =
rewrite plusSuccRightSucc by.nat i.nat in
transitive
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(plusLteMonotoneLeft by.nat (S i.nat) from (toNatLT i))
(replace {p=(`LTE` to)} (shiftDiff by) reflexive)
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public export
fromNat : (by : Nat) -> Shift from (by + from)
fromNat Z = SZ
fromNat (S by) = SS $ fromNat by
%transform "Shift.fromNat" Shift.fromNat x = believe_me x
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public export
fromNat0 : (by : Nat) -> Shift 0 by
fromNat0 by = rewrite sym $ plusZeroRightNeutral by in fromNat by
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export
0 fromToNat : (by : Shift from to) -> by `Eqv` fromNat by.nat {from}
fromToNat SZ = EqSZ
fromToNat (SS by) = EqSS $ fromToNat by
export
0 toFromNat : (from, by : Nat) -> by = (fromNat by {from}).nat
toFromNat from 0 = Refl
toFromNat from (S k) = cong S $ toFromNat from k
export
0 toNatInj' : (by : Shift from1 to1) -> (bz : Shift from2 to2) ->
by.nat = bz.nat -> by `Eqv` bz
toNatInj' SZ SZ prf = EqSZ
toNatInj' (SS by) (SS bz) prf = EqSS $ toNatInj' by bz $ injective prf
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toNatInj' (SS by) SZ Refl impossible
export
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0 toNatInj : {by, bz : Shift from to} -> by.nat = bz.nat -> by = bz
toNatInj {by, bz} e = fromEqv $ toNatInj' by bz e
export %inline
Injective Shift.(.nat) where injective eq = irrelevantEq $ toNatInj eq
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public export
ssDown : Shift (S from) to -> Shift from to
ssDown SZ = SS SZ
ssDown (SS by) = SS (ssDown by)
export
0 ssDownEqv : (by : Shift (S from) to) -> ssDown by `Eqv` SS by
ssDownEqv SZ = EqSS EqSZ
ssDownEqv (SS by) = EqSS $ ssDownEqv by
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private %inline
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ssDownViaNat : Shift (S from) to -> Shift from to
ssDownViaNat by =
rewrite shiftDiff by in
rewrite sym $ plusSuccRightSucc by.nat from in
fromNat $ S by.nat
%transform "Shift.ssDown" ssDown = ssDownViaNat
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public export
weak : (s : Nat) -> Shift from to -> Shift (s + from) (s + to)
weak s SZ = SZ
weak s (SS by) {to = S to} =
rewrite sym $ plusSuccRightSucc s to in
SS $ weak s by
private
weakViaNat : (s : Nat) -> Shift from to -> Shift (s + from) (s + to)
weakViaNat s by =
rewrite shiftDiff by in
rewrite plusAssociative s by.nat from in
rewrite plusCommutative s by.nat in
rewrite sym $ plusAssociative by.nat s from in
fromNat by.nat
%transform "Shift.weak" Shift.weak = weakViaNat
export
getFrom : {to : Nat} -> Shift from to -> Singleton from
getFrom SZ = Val to
getFrom (SS by) = getFrom by
private
0 getFromViaNatProof : (by : Shift from to) -> from = to `minus` by.nat
getFromViaNatProof by = Calc $
|~ from
~~ minus (by.nat + from) by.nat ..<(minusPlus {})
~~ minus to by.nat ..<(cong (flip minus by.nat) (shiftDiff by))
private
getFromViaNat : {to : Nat} -> Shift from to -> Singleton from
getFromViaNat by = rewrite getFromViaNatProof by in Val _
%transform "Shift.getFrom" Shift.getFrom = getFromViaNat
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public export
shift : Shift from to -> Var from -> Var to
shift SZ i = i
shift (SS by) i = VS $ shift by i
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private %inline
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shiftViaNat' : (by : Shift from to) -> (i : Var from) ->
(0 p : by.nat + i.nat `LT` to) -> Var to
shiftViaNat' by i p = V $ by.nat + i.nat
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private %inline
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shiftViaNat : Shift from to -> Var from -> Var to
shiftViaNat by i = shiftViaNat' by i $ shiftVarLT by i
private
0 shiftViaNatCorrect : (by : Shift from to) -> (i : Var from) ->
(0 p : by.nat + i.nat `LT` to) ->
shiftViaNat' by i p = shift by i
shiftViaNatCorrect SZ i (LTESucc p) = fromToNat i _
shiftViaNatCorrect (SS by) i (LTESucc p) = cong VS $ shiftViaNatCorrect by i p
%transform "Shift.shift" shift = shiftViaNat
public export
(.) : Shift from mid -> Shift mid to -> Shift from to
by . SZ = by
by . SS bz = SS $ by . bz
private
0 compNatProof : (by : Shift from mid) -> (bz : Shift mid to) ->
to = by.nat + bz.nat + from
compNatProof by bz = Calc $
|~ to
~~ bz.nat + mid ...(shiftDiff {})
~~ bz.nat + (by.nat + from) ...(cong (bz.nat +) (shiftDiff {}))
~~ bz.nat + by.nat + from ...(plusAssociative {})
~~ by.nat + bz.nat + from ...(cong (+ from) (plusCommutative {}))
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private %inline
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compViaNat' : (by : Shift from mid) -> (bz : Shift mid to) ->
Shift from (by.nat + bz.nat + from)
compViaNat' by bz = fromNat $ by.nat + bz.nat
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private %inline
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compViaNat : (by : Shift from mid) -> (bz : Shift mid to) -> Shift from to
compViaNat by bz = rewrite compNatProof by bz in compViaNat' by bz
private
0 compViaNatCorrect : (by : Shift from mid) -> (bz : Shift mid to) ->
by . bz `Eqv` compViaNat' by bz
compViaNatCorrect by SZ =
rewrite plusZeroRightNeutral by.nat in fromToNat by
compViaNatCorrect by (SS bz) =
rewrite sym $ plusSuccRightSucc by.nat bz.nat in
EqSS $ compViaNatCorrect by bz
%transform "Shift.(.)" Shift.(.) = compViaNat
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export infixl 8 //
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public export
interface CanShift f where
(//) : f from -> Shift from to -> f to
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export %inline
CanShift Var where i // by = shift by i
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namespace CanShift
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public export %inline
[Map] (Functor f, CanShift tm) => CanShift (f . tm) where
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x // by = map (// by) x
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public export %inline
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[Const] CanShift (\_ => a) where x // _ = x