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WorldSystemExamples.agda
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WorldSystemExamples.agda
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module WorldSystemExamples where
-- We note that this code could be shorter if we tried to be clever
-- however, most of this can be automatically generated using Auto,
-- given a few hints
open import Basics hiding (the)
open import Star
open import WorldSystem
module OneWorld where
oneWorld : WorldSystem
oneWorld = record
{ W = One
; _<<_ = \_ _ -> One
; tyW = \_ -> One
; Q = \_ -> One
; st = <>
; _#_ = \_ w -> Just w --\_ _ -> Just <>
; _&_ = \_ _ -> <>
; star& = \_ -> <>
; unst = \w -> Just w
; _&unst&_ = \_ _ -> Real , <>
; <<refl = <>
; <<trans = \_ _ -> <>
; st-def = \w -> w , refl --\_ -> <> , refl
; st-unst = \{refl -> refl}
; defUpset = \{_}{_}{_}{_}{w} _ _ -> w , refl
; op = \_ _ -> <>
; dominateUpset = \_ _ _ _ _ _ -> <>
; stTyW = \_ -> <>
; tyWUpset = \_ _ -> <>
; st&-action = \_ _ -> refl
; &unst&-action = \{ _ (inl _) _ -> refl ; _ (inr _) _ -> refl}
; isActionL = \{_}{_}{w} p -> w , refl , p
; isActionR = \{refl refl -> refl}
; stq = \{ refl refl refl refl -> refl}
; GwqConn = \{ _ _ refl refl refl refl -> inl <> ,- []}
}
module actByReplacement (W' : Set)(pt : W')
(_<_ : W' -> W' -> Set)
(<refl : forall {x} -> x < x)
(<trans : forall {x y z} -> x < y -> y < z -> x < z) where
-- some poset of worlds, we add star at the top, Q=W+{unstar}+{q&unstar | q <- Q}
-- unstar acts by picking some arbitrary (given) point from W
data Top (A : Set) : Set where
top : Top A
the : A -> Top A
W = Top W'
data _<<_ : W -> W -> Set where
<<top : forall u -> u << top
th<<th : forall {u v} -> u < v -> the u << the v
data Q : Sort -> Set where
undef : Q Real
const : W -> Q Real
topTo : W -> Q Real
-- undef or topTo _
topTo? : Maybe W -> Q Real
topTo? (Just w) = topTo w
topTo? Nothing = undef
-- undef or const _
const? : Maybe W -> Q Real
const? (Just w) = const w
const? Nothing = undef
_#_ : {s : Sort} -> Q s -> W -> Maybe W
undef # _ = Nothing
const x # w = Just x
topTo x # top = Just x
topTo x # the w = Nothing
infix 40 _#_
_&_ : Q Real -> Q Real -> Q Real
q & undef = undef
q & const x = const? (q # x)
q & topTo x = topTo? (q # x)
infix 50 _&_
star& : Q Fake -> Q Real
star& ()
unst : W -> Maybe W
unst top = Just (the pt)
unst (the x) = Nothing
_&unst&'_ : Q Real -> One + Q Real -> Q Real
q &unst&' inr undef = undef
q &unst&' inr (const top) = const? (q # the pt)
q &unst&' inr (const (the x)) = undef
q &unst&' inr (topTo top) = topTo? (q # the pt)
q &unst&' inr (topTo (the x)) = undef
q &unst&' inl <> = topTo? (q # the pt)
_&unst&_ : Q Real -> One + Q Real -> Sg Sort Q
q &unst& r = Real , q &unst&' r
<<refl : {w : W} -> w << w
<<refl {top} = <<top top
<<refl {the x} = th<<th <refl
<<trans : {u v w : W} -> u << v -> v << w -> u << w
<<trans u<v (<<top v) = <<top _
<<trans (th<<th u<v) (th<<th v<w) = th<<th (<trans u<v v<w)
st-unst : forall {w usw} -> unst w ~ usw -> top == w
st-unst {top} p = refl
st-unst {the x} ()
def-upset : forall {s}(q : Q s){v qv w} -> q # v ~ qv -> v << w -> Sg W (\qw -> q # w ~ qw)
def-upset undef () v<w
def-upset (const x) qv v<w = x , refl
def-upset (topTo x) {top} refl (<<top .top) = x , refl
def-upset (topTo x) {the y} () v<w
op : forall {s}(q : Q s){v qv' w} -> (qv : q # v ~ qv') -> (v<w : v << w) -> qv' << fst (def-upset q qv v<w)
op undef () v<w
op (const x) refl v<w = <<refl
op (topTo x) {top} refl (<<top .top) = <<refl
op (topTo x) {the x₁} () v<w
-- lift things to One + Q _
-- TODO: is there a way to get these from the record we are in the process of defining?
_#?_ : forall {s} -> One + Q s -> W -> Maybe W
_#?_ {s} = either (\_ -> Just) _#_
infix 40 _#?_
def-upset? : forall {s}(q : One + Q s){v qv w} -> q #? v ~ qv -> v << w -> Sg W (q #? w ~_)
def-upset? {s} = either {C = \q -> forall {v qv w} -> q #? v ~ qv -> v << w -> Sg W (q #? w ~_)}
(\_ {_}{_}{w} _ _ -> w , refl) (\q qv v<w -> def-upset q qv v<w)
op? : forall {s}(q : One + Q s){v qv' w} -> (qv : q #? v ~ qv') -> (v<w : v << w)
-> qv' << fst (def-upset? q qv v<w)
op? = either {C = \q -> forall {v qv' w} -> (qv : q #? v ~ qv') -> (v<w : v << w)
-> qv' << fst (def-upset? q qv v<w)}
(\{_ refl v<w -> v<w}) (\q qv v<w -> op q qv v<w)
dominateUpset : forall {s}(q : Q s)(r : One + Q Real){v qv' rv' w}
-> (qv : q # v ~ qv')(rv : r #? v ~ rv')
-> qv' << rv' -> (v<w : v << w)
-> fst (def-upset q qv v<w) << fst (def-upset? r rv v<w)
dominateUpset undef r () rv qv<rv v<w
dominateUpset (const x) r refl rv qv<rv v<w = <<trans qv<rv (op? r rv v<w)
dominateUpset (topTo x) r {top} refl rv qv<rv (<<top .top) = <<trans qv<rv (op? r rv (<<top top))
dominateUpset (topTo x) r {the _} () rv qv<rv v<w
&unst&-action : forall q r w
-> snd (q &unst& r) # w == ((q #_) =<< (unst =<< (r #? w)))
&unst&-action q (inr undef) w = refl
&unst&-action undef (inr (const top)) w = refl
&unst&-action (const x) (inr (const top)) w = refl
&unst&-action (topTo x) (inr (const top)) top = refl
&unst&-action (topTo x) (inr (const top)) (the w) = refl
&unst&-action q (inr (const (the x))) w = refl
&unst&-action undef (inr (topTo top)) top = refl
&unst&-action (const x) (inr (topTo top)) top = refl
&unst&-action (topTo x) (inr (topTo top)) top = refl
&unst&-action q (inr (topTo (the x))) top = refl
&unst&-action undef (inr (topTo top)) (the w) = refl
&unst&-action (const x) (inr (topTo top)) (the w) = refl
&unst&-action (topTo x) (inr (topTo top)) (the w) = refl
&unst&-action q (inr (topTo (the x))) (the w) = refl
&unst&-action undef (inl <>) top = refl
&unst&-action undef (inl <>) (the w) = refl
&unst&-action (const x) (inl <>) top = refl
&unst&-action (const x) (inl <>) (the w) = refl
&unst&-action (topTo x) (inl <>) top = refl
&unst&-action (topTo x) (inl <>) (the w) = refl
isActionL : forall q r {w qrw'} -> q & r # w ~ qrw'
-> Sg W (\rw' -> (r # w ~ rw') * (q # rw') ~ qrw')
isActionL q undef ()
isActionL undef (const y) ()
isActionL (const x) (const y) qrw = y , refl , qrw
isActionL (topTo x) (const top) qrw = top , refl , qrw
isActionL (topTo x) (const (the y)) ()
isActionL undef (topTo y) ()
isActionL (const x) (topTo y) {top} refl = y , refl , refl
isActionL (const x) (topTo y) {the _} ()
isActionL (topTo x) (topTo top) {top} refl = top , refl , refl
isActionL (topTo x) (topTo top) {the _} ()
isActionL (topTo x) (topTo (the y)) ()
isActionR : forall q r {w rw' qrw'}
-> r # w ~ rw'
-> q # rw' ~ qrw'
-> q & r # w ~ qrw'
isActionR q undef () qrw
isActionR undef (const y) rw ()
isActionR (const x) (const y) rw qrw = qrw
isActionR (topTo x) (const top) refl qrw = qrw
isActionR (topTo x) (const (the y)) refl ()
isActionR undef (topTo y) rw ()
isActionR (const x) (topTo y) {top} refl qrw = qrw
isActionR (const x) (topTo y) {the x₁} () qrw
isActionR (topTo x) (topTo y) {the x₁} () qrw
isActionR (topTo x) (topTo top) {top} refl qrw = qrw
isActionR (topTo x) (topTo (the y)) {top} refl ()
GwqConn : forall (q : Q Real) (w : W) {u qu'} su=w qu {v qv'} sv=w qv
-> Star {Sg W (\u -> top == w * Sg W \qu -> q # u ~ qu)}
(\{(u , _ , qu' , _) (v , _ , qv' , _) -> (u << v) + (qv' << qu')})
(u , su=w , qu' , qu) (v , sv=w , qv' , qv)
GwqConn undef .top refl () refl ()
GwqConn (const x) .top refl refl refl refl = inr <<refl ,- []
GwqConn (topTo x) .top {top} refl refl {top} refl refl = []
GwqConn (topTo x) .top {top} refl refl {the _} refl ()
GwqConn (topTo x) .top {the _} refl () refl qv
replacement : WorldSystem
replacement = record
{ W = W
; _<<_ = _<<_
; tyW = top ==_
; Q = Q
; st = const top
; _#_ = _#_
; _&_ = _&_
; star& = star&
; unst = unst
; _&unst&_ = _&unst&_
; <<refl = <<refl
; <<trans = <<trans
; st-def = \w -> top , refl
; st-unst = st-unst
; defUpset = \{_}{q} -> def-upset q
; op = \{_}{q} -> op q
; dominateUpset = dominateUpset
; stTyW = \ _ -> refl
; tyWUpset = \{ (<<top u) q → refl ; (th<<th x) ()}
; st&-action = \()
; &unst&-action = &unst&-action
; isActionL = \{q}{r} -> isActionL q r
; isActionR = \{q}{r} -> isActionR q r
; stq = \_ _ _ _ -> refl
; GwqConn = GwqConn
}
module twoWorlds where
-- Heaven above earth
data W : Set where H E : W
data _<<_ : W -> W -> Set where
E<E : E << E
H<H : H << H
E<H : E << H
data Q : Sort -> Set where
-- C stands for "current"
-- H: constantly returns H
-- HH: H->H, undefined on E
C H HH undef : Q Real
-- HE: H->E, undefined on E
-- E: constantly returns E
HE E : Q Fake
_#_ : forall {s} -> Q s -> W -> Maybe W
C # w = Just w
H # w = Just H
E # w = Just E
HE # H = Just E
HE # E = Nothing
HH # H = Just H
HH # E = Nothing
undef # w = Nothing
infix 40 _#_
_&_ : Q Real -> Q Real -> Q Real
q & C = q
q & undef = undef
C & q = q
undef & q = undef
H & H = H
HH & H = H
H & HH = HH
HH & HH = HH
infix 50 _&_
star& : Q Fake -> Q Real
star& HE = HH
star& E = H
unst : W -> Maybe W
unst H = Just E
unst E = Nothing
_&unst&'_ : Q Real -> Q Real -> Sg Sort Q
q &unst&' undef = Real , undef
HH &unst&' r = Real , undef
undef &unst&' r = Real , undef
C &unst&' C = Fake , HE
C &unst&' H = Fake , E
C &unst&' HH = Fake , HE
H &unst&' C = Real , HH
H &unst&' H = Real , H
H &unst&' HH = Real , HH
_&unst&_ : Q Real -> One + Q Real -> Sg Sort Q
q &unst& inl <> = q &unst&' C
q &unst& inr r = q &unst&' r
<<refl : {w : W} -> w << w
<<refl {H} = H<H
<<refl {E} = E<E
<<trans : {u v w : W} → u << v → v << w → u << w
<<trans {_} {H} u<v H<H = u<v
<<trans {_} {E} E<E v<w = v<w
st-unst : forall {w usw} -> unst w ~ usw -> H == w
st-unst {H} _ = refl
st-unst {E} ()
def-upset : forall {s}(q : Q s){v qv' w} -> q # v == Just qv' -> v << w -> Sg W (\qw -> q # w == Just qw)
def-upset q qv E<E = _ , qv
def-upset q qv H<H = _ , qv
def-upset C qv E<H = H , refl
def-upset H qv E<H = H , refl
def-upset E qv E<H = E , refl
def-upset HE qv E<H = E , refl
def-upset HH qv E<H = H , refl
def-upset undef () E<H
op : forall {s}(q : Q s){v qv' w} -> (qv : q # v == Just qv') -> (v<w : v << w) -> qv' << fst (def-upset q qv v<w)
op q qv E<E = <<refl
op q qv H<H = <<refl
op C refl E<H = E<H
op H refl E<H = H<H
op HH () E<H
op undef () E<H
op HE () E<H
op E refl E<H = E<E
_#?_ : forall {s} -> One + Q s -> W -> Maybe W
_#?_ {s} = either (\_ -> Just) _#_
infix 40 _#?_
def-upset? : forall {s}(q : One + Q s){v qv w} -> q #? v ~ qv -> v << w -> Sg W (q #? w ~_)
def-upset? {s} = either {C = \q -> forall {v qv w} -> q #? v ~ qv -> v << w -> Sg W (q #? w ~_)}
(\_ {_}{_}{w} _ _ -> w , refl) (\q qv v<w -> def-upset q qv v<w)
op? : forall {s}(q : One + Q s){v qv' w} -> (qv : q #? v ~ qv') -> (v<w : v << w)
-> qv' << fst (def-upset? q qv v<w)
op? = either {C = \q -> forall {v qv' w} -> (qv : q #? v ~ qv') -> (v<w : v << w)
-> qv' << fst (def-upset? q qv v<w)}
(\{_ refl v<w -> v<w}) (\q qv v<w -> op q qv v<w)
dominateUpset : forall {s}(q : Q s)(r : One + Q Real){v qv' rv' w}
-> (qv : q # v ~ qv')(rv : r #? v ~ rv')
-> qv' << rv' -> (v<w : v << w)
-> fst (def-upset q qv v<w) << fst (def-upset? r rv v<w)
dominateUpset q (inl <>) qv refl qv<rv E<E = qv<rv
dominateUpset q (inl <>) qv refl qv<rv H<H = qv<rv
dominateUpset C (inl <>) refl refl qv<rv E<H = H<H
dominateUpset H (inl <>) refl refl qv<rv E<H = H<H
dominateUpset HH (inl <>) () refl qv<rv E<H
dominateUpset undef (inl <>) () refl qv<rv E<H
dominateUpset HE (inl <>) () refl qv<rv E<H
dominateUpset E (inl <>) refl refl qv<rv E<H = E<H
dominateUpset q (inr r) qv rv qv<rv E<E = qv<rv
dominateUpset q (inr r) qv rv qv<rv H<H = qv<rv
dominateUpset C (inr C) refl refl qv<rv E<H = H<H
dominateUpset C (inr H) refl refl qv<rv E<H = H<H
dominateUpset C (inr HH) refl () qv<rv E<H
dominateUpset C (inr undef) refl () qv<rv E<H
dominateUpset H (inr C) refl refl qv<rv E<H = H<H
dominateUpset H (inr H) refl refl qv<rv E<H = qv<rv
dominateUpset H (inr HH) refl () qv<rv E<H
dominateUpset H (inr undef) refl () qv<rv E<H
dominateUpset HH (inr r) () rv qv<rv E<H
dominateUpset undef (inr r) () rv qv<rv E<H
dominateUpset HE (inr r) () rv qv<rv E<H
dominateUpset E (inr C) refl refl qv<rv E<H = E<H
dominateUpset E (inr H) refl refl qv<rv E<H = qv<rv
dominateUpset E (inr HH) refl () qv<rv E<H
dominateUpset E (inr undef) refl () qv<rv E<H
st&-action : forall q w
-> star& q # w == ((H #_) =<< q # w)
st&-action HE H = refl
st&-action HE E = refl
st&-action E w = refl
&unst&-action : forall q r w
-> snd (q &unst& r) # w == ((q #_) =<< (unst =<< (r #? w)))
&unst&-action C (inl <>) H = refl
&unst&-action C (inl <>) E = refl
&unst&-action H (inl <>) H = refl
&unst&-action H (inl <>) E = refl
&unst&-action HH (inl <>) H = refl
&unst&-action HH (inl <>) E = refl
&unst&-action undef (inl <>) H = refl
&unst&-action undef (inl <>) E = refl
&unst&-action C (inr C) H = refl
&unst&-action C (inr C) E = refl
&unst&-action H (inr C) H = refl
&unst&-action H (inr C) E = refl
&unst&-action HH (inr C) H = refl
&unst&-action HH (inr C) E = refl
&unst&-action undef (inr C) H = refl
&unst&-action undef (inr C) E = refl
&unst&-action C (inr H) w = refl
&unst&-action H (inr H) w = refl
&unst&-action HH (inr H) w = refl
&unst&-action undef (inr H) w = refl
&unst&-action C (inr HH) H = refl
&unst&-action C (inr HH) E = refl
&unst&-action H (inr HH) H = refl
&unst&-action H (inr HH) E = refl
&unst&-action HH (inr HH) H = refl
&unst&-action HH (inr HH) E = refl
&unst&-action undef (inr HH) H = refl
&unst&-action undef (inr HH) E = refl
&unst&-action q (inr undef) w = refl
isActionL : forall q r {w qrw'} -> q & r # w ~ qrw'
-> Sg W (\rw' -> (r # w ~ rw') * (q # rw') ~ qrw')
isActionL q C qrw = _ , refl , qrw
isActionL C H qrw = _ , qrw , refl
isActionL H H qrw = H , refl , qrw
isActionL HH H qrw = H , refl , qrw
isActionL undef H ()
isActionL C HH qrw = _ , qrw , refl
isActionL H HH {H} refl = H , refl , refl
isActionL H HH {E} ()
isActionL HH HH {H} qrw = H , refl , qrw
isActionL HH HH {E} ()
isActionL undef HH ()
isActionL q undef ()
isActionR : forall q r {w rw' qrw'}
-> r # w ~ rw'
-> q # rw' ~ qrw'
-> q & r # w ~ qrw'
isActionR q C refl qrw = qrw
isActionR C H refl qrw = qrw
isActionR H H refl qrw = qrw
isActionR HH H refl qrw = qrw
isActionR undef H refl ()
isActionR C HH rw refl = rw
isActionR H HH {H} refl refl = refl
isActionR H HH {E} () refl
isActionR HH HH {H} refl qrw = qrw
isActionR HH HH {E} () qrw
isActionR undef HH rw ()
isActionR q undef () qrw
owoto : forall {u v} -> (u << v) + (v << u)
owoto {H} {H} = inl H<H
owoto {H} {E} = inr E<H
owoto {E} {H} = inl E<H
owoto {E} {E} = inl E<E
GwqConn : forall {s}(q : Q s)(w : W) {u qu'} su=w qu {v qv'} sv=w qv
-> Star {Sg W (\u -> H == w * Sg W \qu -> q # u ~ qu)}
(\{(u , _ , qu' , _) (v , _ , qv' , _) -> (u << v) + (qv' << qu')})
(u , su=w , qu' , qu) (v , sv=w , qv' , qv)
GwqConn C .H refl refl refl refl = owoto ,- []
GwqConn H .H refl refl refl refl = inr H<H ,- []
GwqConn HH .H {H} refl refl {H} refl refl = inl H<H ,- []
GwqConn HH .H {H} refl refl {E} refl ()
GwqConn HH .H {E} refl () refl qv
GwqConn undef .H refl () refl ()
GwqConn HE .H {H} refl refl {H} refl refl = inl H<H ,- []
GwqConn HE .H {H} refl refl {E} refl ()
GwqConn HE .H {E} refl () refl qv
GwqConn E .H refl refl refl refl = inr E<E ,- []
twoWorlds : WorldSystem
twoWorlds = record
{ W = W
; _<<_ = _<<_
; tyW = H ==_
; Q = Q
; st = H
; _#_ = _#_
; _&_ = _&_
; star& = star&
; unst = unst
; _&unst&_ = _&unst&_
; <<refl = <<refl
; <<trans = <<trans
; st-def = \w -> H , refl
; st-unst = st-unst
; defUpset = \{_}{q} -> def-upset q
; op = \{_}{q} -> op q
; dominateUpset = dominateUpset
; stTyW = \_ -> refl
; tyWUpset = \{ H<H refl -> refl}
; st&-action = st&-action
; &unst&-action = &unst&-action
; isActionL = \{q}{r} -> isActionL q r
; isActionR = \{q}{r} -> isActionR q r
; stq = \_ _ _ _ -> refl
; GwqConn = GwqConn
}
-- TODO: Compare this to actual system F
-- My first attempt was wrong: had type operators!
module SystemF where
data W : Set where
tm ty box : W
_<<_ : W -> W -> Set
_<<_ = _==_
tyW : W -> Set
tyW tm = Zero
tyW ty = One
tyW box = One
-- Question: do we want to have a quant . which
-- a) takes tm -> tm, undefined else (currently have this - forces having tmTo : W -> Q Real,
-- and then tyTo ty , tyTo box : Q Real)
-- b) acts as the identity (can probably remove tmTo ty and tmTo box)
--
-- Turns out still need tmTo st and tmTo box in case (b)
-- because id &unst& st = tmTo tm : Q Fake
-- and st& tmTo tm = tmTo ty : Q Real
-- thus st & tmTo ty = tmTo box
--
-- So we wouldn't actually gain much in plan (b)
data Q : Sort -> Set where
tmTo : W -> Q Real
tyTo : W -> Q Real
st box : Q Real
undef : Q Real
_#_ : forall {s} -> Q s -> W -> Maybe W
tmTo w # tm = Just w
tmTo w # _ = Nothing
st # tm = Just ty
st # ty = Just box
st # box = Just box
box # w = Just box
undef # w = Nothing
tyTo w # ty = Just w
tyTo w # _ = Nothing
infix 40 _#_
st-def : forall w -> Sg W (st # w ~_)
st-def tm = ty , refl
st-def ty = box , refl
st-def box = box , refl
st-act : W -> W
st-act w = fst (st-def w)
_&_ : Q Real -> Q Real -> Q Real
tmTo v & tmTo tm = tmTo v
tmTo v & tyTo tm = tyTo v
tmTo v & _ = undef
tyTo v & tmTo ty = tmTo v
tyTo v & tyTo ty = tyTo v
tyTo v & st = tmTo v
tyTo v & _ = undef
st & tmTo w = tmTo (st-act w)
st & tyTo w = tyTo (st-act w)
st & st = box
st & box = box
box & tmTo w = tmTo box
box & tyTo w = tyTo box
box & st = box
box & box = box
_ & undef = undef
undef & _ = undef
infix 50 _&_
unst : W -> Maybe W
unst ty = Just tm
unst _ = Nothing
_&unst&_ : Q Real -> One + Q Real -> Sg Sort Q
q &unst& inl <> with q # tm
... | Just w = _ , tyTo w
... | Nothing = _ , undef
q &unst& inr (tmTo ty) = _ , q & tmTo tm
q &unst& inr (tmTo _) = _ , undef
q &unst& inr (tyTo ty) = _ , q & tyTo tm
q &unst& inr (tyTo _) = _ , undef
q &unst& inr st = _ , q & tmTo tm
q &unst& inr box = _ , undef
q &unst& inr undef = _ , undef
<<trans : forall {u v w} -> u << v -> v << w -> u << w
<<trans refl refl = refl
st-unst : forall {w usw} -> unst w ~ usw -> st-act usw == w
st-unst {tm} ()
st-unst {ty} refl = refl
st-unst {box} ()
def-upset : forall {s}(q : Q s){v qv w} -> q # v ~ qv -> v << w -> Sg W (\qw -> q # w ~ qw)
def-upset q qv refl = _ , qv
op : forall {s}(q : Q s){v qv' w} -> (qv : q # v ~ qv') -> (v<w : v << w) -> qv' << fst (def-upset q qv v<w)
op q qv refl = refl
_#?_ : forall {s} -> One + Q s -> W -> Maybe W
_#?_ {s} = either (\_ -> Just) _#_
infix 40 _#?_
def-upset? : forall {s}(q : One + Q s){v qv w} -> q #? v ~ qv -> v << w -> Sg W (q #? w ~_)
def-upset? {s} = either {C = \q -> forall {v qv w} -> q #? v ~ qv -> v << w -> Sg W (q #? w ~_)}
(\_ {_}{_}{w} _ _ -> w , refl) def-upset
dominateUpset : forall {s}(q : Q s)(r : One + Q Real){v qv' rv' w}
-> (qv : q # v ~ qv')(rv : r #? v ~ rv')
-> qv' << rv' -> (v<w : v << w)
-> fst (def-upset q qv v<w) << fst (def-upset? r rv v<w)
dominateUpset _ (inl <>) qv refl refl refl = refl
dominateUpset q (inr r) qv rv refl refl = refl
stTyW : forall w -> tyW (st-act w)
stTyW tm = <>
stTyW ty = <>
stTyW box = <>
&unst&-action : forall q r w
-> snd (q &unst& r) # w == ((q #_) =<< (unst =<< (r #? w)))
&unst&-action q (inl <>) tm with q # tm
&unst&-action q (inl <>) tm | Just x = refl
&unst&-action q (inl <>) tm | Nothing = refl
&unst&-action q (inl <>) ty with q # tm
&unst&-action q (inl <>) ty | Just x = refl
&unst&-action q (inl <>) ty | Nothing = refl
&unst&-action q (inl <>) box with q # tm
&unst&-action q (inl <>) box | Just x = refl
&unst&-action q (inl <>) box | Nothing = refl
&unst&-action q (inr (tmTo tm)) tm = refl
&unst&-action q (inr (tmTo tm)) ty = refl
&unst&-action q (inr (tmTo tm)) box = refl
&unst&-action (tmTo x) (inr (tmTo ty)) tm = refl
&unst&-action (tmTo x) (inr (tmTo ty)) ty = refl
&unst&-action (tmTo x) (inr (tmTo ty)) box = refl
&unst&-action (tyTo x) (inr (tmTo ty)) tm = refl
&unst&-action (tyTo x) (inr (tmTo ty)) ty = refl
&unst&-action (tyTo x) (inr (tmTo ty)) box = refl
&unst&-action st (inr (tmTo ty)) tm = refl
&unst&-action st (inr (tmTo ty)) ty = refl
&unst&-action st (inr (tmTo ty)) box = refl
&unst&-action box (inr (tmTo ty)) tm = refl
&unst&-action box (inr (tmTo ty)) ty = refl
&unst&-action box (inr (tmTo ty)) box = refl
&unst&-action undef (inr (tmTo ty)) tm = refl
&unst&-action undef (inr (tmTo ty)) ty = refl
&unst&-action undef (inr (tmTo ty)) box = refl
&unst&-action q (inr (tmTo box)) tm = refl
&unst&-action q (inr (tmTo box)) ty = refl
&unst&-action q (inr (tmTo box)) box = refl
&unst&-action q (inr (tyTo tm)) tm = refl
&unst&-action q (inr (tyTo tm)) ty = refl
&unst&-action q (inr (tyTo tm)) box = refl
&unst&-action (tmTo x) (inr (tyTo ty)) tm = refl
&unst&-action (tmTo x) (inr (tyTo ty)) ty = refl
&unst&-action (tmTo x) (inr (tyTo ty)) box = refl
&unst&-action (tyTo x) (inr (tyTo ty)) tm = refl
&unst&-action (tyTo x) (inr (tyTo ty)) ty = refl
&unst&-action (tyTo x) (inr (tyTo ty)) box = refl
&unst&-action st (inr (tyTo ty)) tm = refl
&unst&-action st (inr (tyTo ty)) ty = refl
&unst&-action st (inr (tyTo ty)) box = refl
&unst&-action box (inr (tyTo ty)) tm = refl
&unst&-action box (inr (tyTo ty)) ty = refl
&unst&-action box (inr (tyTo ty)) box = refl
&unst&-action undef (inr (tyTo ty)) tm = refl
&unst&-action undef (inr (tyTo ty)) ty = refl
&unst&-action undef (inr (tyTo ty)) box = refl
&unst&-action q (inr (tyTo box)) tm = refl
&unst&-action q (inr (tyTo box)) ty = refl
&unst&-action q (inr (tyTo box)) box = refl
&unst&-action (tmTo x) (inr st) tm = refl
&unst&-action (tmTo x) (inr st) ty = refl
&unst&-action (tmTo x) (inr st) box = refl
&unst&-action (tyTo x) (inr st) tm = refl
&unst&-action (tyTo x) (inr st) ty = refl
&unst&-action (tyTo x) (inr st) box = refl
&unst&-action st (inr st) tm = refl
&unst&-action st (inr st) ty = refl
&unst&-action st (inr st) box = refl
&unst&-action box (inr st) tm = refl
&unst&-action box (inr st) ty = refl
&unst&-action box (inr st) box = refl
&unst&-action undef (inr st) tm = refl
&unst&-action undef (inr st) ty = refl
&unst&-action undef (inr st) box = refl
&unst&-action q (inr box) w = refl
&unst&-action q (inr undef) w = refl
isActionL : forall q r {w qrw'} -> q & r # w ~ qrw'
-> Sg W (\rw' -> (r # w ~ rw') * (q # rw') ~ qrw')
isActionL (tmTo x) (tmTo tm) {tm} refl = tm , refl , refl
isActionL (tmTo x) (tmTo tm) {ty} ()
isActionL (tmTo x) (tmTo tm) {box} ()
isActionL (tmTo x) (tmTo ty) ()
isActionL (tmTo x) (tmTo box) ()
isActionL (tyTo x) (tmTo tm) ()
isActionL (tyTo x) (tmTo ty) {tm} refl = ty , refl , refl
isActionL (tyTo x) (tmTo ty) {ty} ()
isActionL (tyTo x) (tmTo ty) {box} ()
isActionL (tyTo x) (tmTo box) ()
isActionL st (tmTo y) {tm} refl = y , refl , snd (st-def y)
isActionL st (tmTo y) {ty} ()
isActionL st (tmTo y) {box} ()
isActionL box (tmTo y) {tm} refl = y , refl , refl
isActionL box (tmTo y) {ty} ()
isActionL box (tmTo y) {box} ()
isActionL undef (tmTo y) ()
isActionL (tmTo x) (tyTo tm) {tm} ()
isActionL (tmTo x) (tyTo tm) {ty} refl = tm , refl , refl
isActionL (tmTo x) (tyTo tm) {box} ()
isActionL (tmTo x) (tyTo ty) ()
isActionL (tmTo x) (tyTo box) ()
isActionL (tyTo x) (tyTo tm) ()
isActionL (tyTo x) (tyTo ty) {tm} ()
isActionL (tyTo x) (tyTo ty) {ty} refl = ty , refl , refl
isActionL (tyTo x) (tyTo ty) {box} ()
isActionL (tyTo x) (tyTo box) ()
isActionL st (tyTo y) {tm} ()
isActionL st (tyTo y) {ty} refl = y , refl , snd (st-def y)
isActionL st (tyTo y) {box} ()
isActionL box (tyTo y) {tm} ()
isActionL box (tyTo y) {ty} refl = y , refl , refl
isActionL box (tyTo y) {box} ()
isActionL undef (tyTo y) ()
isActionL (tmTo x) st ()
isActionL (tyTo x) st {tm} refl = ty , refl , refl
isActionL (tyTo x) st {ty} ()
isActionL (tyTo x) st {box} ()
isActionL st st {tm} refl = ty , refl , refl
isActionL st st {ty} refl = box , refl , refl
isActionL st st {box} refl = box , refl , refl
isActionL box st {w} refl = st-act w , snd (st-def w) , refl
isActionL undef st ()
isActionL (tmTo x) box ()
isActionL (tyTo x) box ()
isActionL st box refl = box , refl , refl
isActionL box box refl = box , refl , refl
isActionL undef box ()
isActionL (tmTo x) undef ()
isActionL (tyTo x) undef ()
isActionL st undef ()
isActionL box undef ()
isActionL undef undef ()
isActionR : forall q r {w rw' qrw'}
-> r # w ~ rw'
-> q # rw' ~ qrw'
-> q & r # w ~ qrw'
isActionR (tmTo x) (tmTo tm) {tm} refl refl = refl
isActionR (tmTo x) (tmTo ty) {tm} refl ()
isActionR (tmTo x) (tmTo box) {tm} refl ()
isActionR (tyTo x) (tmTo tm) {tm} refl ()
isActionR (tyTo x) (tmTo ty) {tm} refl refl = refl
isActionR (tyTo x) (tmTo box) {tm} refl ()
isActionR st (tmTo tm) {tm} refl refl = refl
isActionR st (tmTo ty) {tm} refl refl = refl
isActionR st (tmTo box) {tm} refl refl = refl
isActionR box (tmTo tm) {tm} refl refl = refl
isActionR box (tmTo ty) {tm} refl refl = refl
isActionR box (tmTo box) {tm} refl refl = refl
isActionR undef (tmTo tm) {tm} refl ()
isActionR undef (tmTo ty) {tm} refl ()
isActionR undef (tmTo box) {tm} refl ()
isActionR q (tmTo x) {ty} () q-rw
isActionR q (tmTo x) {box} () q-rw
isActionR q (tyTo x) {tm} () q-rw
isActionR (tmTo x) (tyTo tm) {ty} refl refl = refl
isActionR (tmTo x) (tyTo ty) {ty} refl ()
isActionR (tmTo x) (tyTo box) {ty} refl ()
isActionR (tyTo x) (tyTo tm) {ty} refl ()
isActionR (tyTo x) (tyTo ty) {ty} refl refl = refl
isActionR (tyTo x) (tyTo box) {ty} refl ()
isActionR st (tyTo tm) {ty} refl refl = refl
isActionR st (tyTo ty) {ty} refl refl = refl
isActionR st (tyTo box) {ty} refl refl = refl
isActionR box (tyTo tm) {ty} refl refl = refl
isActionR box (tyTo ty) {ty} refl refl = refl
isActionR box (tyTo box) {ty} refl refl = refl
isActionR undef (tyTo tm) {ty} refl ()
isActionR undef (tyTo ty) {ty} refl ()
isActionR undef (tyTo box) {ty} refl ()
isActionR q (tyTo x) {box} () q-rw
isActionR (tmTo x) st {tm} refl ()
isActionR (tyTo x) st {tm} refl refl = refl
isActionR st st {tm} refl refl = refl
isActionR box st {tm} refl refl = refl
isActionR undef st {tm} refl ()
isActionR (tmTo x) st {ty} refl ()
isActionR (tyTo x) st {ty} refl ()
isActionR st st {ty} refl refl = refl
isActionR box st {ty} refl refl = refl
isActionR undef st {ty} refl ()
isActionR (tmTo x) st {box} refl ()
isActionR (tyTo x) st {box} refl ()
isActionR st st {box} refl refl = refl
isActionR box st {box} refl refl = refl
isActionR undef st {box} refl ()
isActionR (tmTo x) box {tm} refl ()
isActionR (tyTo x) box {tm} refl ()
isActionR st box {tm} refl refl = refl
isActionR box box {tm} refl refl = refl
isActionR undef box {tm} refl ()
isActionR (tmTo x) box {ty} refl ()
isActionR (tyTo x) box {ty} refl ()
isActionR st box {ty} refl refl = refl
isActionR box box {ty} refl refl = refl
isActionR undef box {ty} refl ()
isActionR (tmTo x) box {box} refl ()
isActionR (tyTo x) box {box} refl ()
isActionR st box {box} refl refl = refl
isActionR box box {box} refl refl = refl
isActionR undef box {box} refl ()
isActionR q undef {tm} () q-rw
isActionR q undef {ty} () q-rw
isActionR q undef {box} () q-rw
#functional : forall {s}(q : Q s){w qw1 qw2} -> q # w ~ qw1 -> q # w ~ qw2 -> qw1 == qw2
#functional q {w} qw1 rewrite qw1 = \{refl -> refl}
stq : forall {u su} -> st-act u == su
-> forall {v} -> st-act v == su
-> forall {q : Q Real}{qu} -> q # u ~ qu
-> forall {qv} -> q # v ~ qv
-> st-act qu == st-act qv
stq {tm} refl {tm} refl {q} q-u q-v rewrite #functional q q-u q-v = refl
stq {tm} refl {ty} () q-u q-v
stq {tm} refl {box} () q-u q-v
stq {ty} refl {tm} () q-u q-v
stq {ty} refl {ty} refl {q} q-u q-v rewrite #functional q q-u q-v = refl
stq {ty} refl {box} refl {tmTo x} () ()
stq {ty} refl {box} refl {tyTo x} q-u ()
stq {ty} refl {box} refl {st} refl refl = refl
stq {ty} refl {box} refl {box} refl refl = refl
stq {ty} refl {box} refl {undef} () ()
stq {box} refl {tm} () q-u q-v
stq {box} refl {ty} refl {tmTo x} () ()
stq {box} refl {ty} refl {tyTo x} () q-v
stq {box} refl {ty} refl {st} refl refl = refl
stq {box} refl {ty} refl {box} refl refl = refl
stq {box} refl {ty} refl {undef} () ()
stq {box} refl {box} refl {q} q-u q-v rewrite #functional q q-u q-v = refl
GwqConn : forall {s}(q : Q s)(w : W) {u qu'} su=w qu {v qv'} sv=w qv
-> Star {Sg W (\u -> st-act u == w * Sg W \qu -> q # u ~ qu)}
(\{(u , _ , qu' , _) (v , _ , qv' , _) -> (u << v) + (qv' << qu')})
(u , su=w , qu' , qu) (v , sv=w , qv' , qv)
GwqConn q tm {tm} () q-u {v} s-v q-v
GwqConn q tm {ty} () q-u {v} s-v q-v
GwqConn q tm {box} () q-u {v} s-v q-v
GwqConn q ty {tm} refl q-u {tm} refl q-v = inl refl ,- []
GwqConn q ty {tm} refl q-u {ty} () q-v
GwqConn q ty {tm} refl q-u {box} () q-v
GwqConn q ty {ty} () q-u {v} s-v q-v
GwqConn q ty {box} () q-u {v} s-v q-v
GwqConn q box {tm} () q-u {v} s-v q-v
GwqConn q box {ty} refl q-u {tm} () q-v
GwqConn q box {ty} refl q-u {ty} refl q-v = inl refl ,- []
GwqConn (tmTo x) box {ty} refl () {box} refl q-v
GwqConn (tyTo x) box {ty} refl refl {box} refl ()
GwqConn st box {ty} refl refl {box} refl refl = inr refl ,- []
GwqConn box box {ty} refl refl {box} refl refl = inr refl ,- []
GwqConn undef box {ty} refl () {box} refl q-v
GwqConn q box {box} refl q-u {tm} () q-v
GwqConn (tmTo x) box {box} refl () {ty} refl q-v
GwqConn (tyTo x) box {box} refl () {ty} refl q-v
GwqConn st box {box} refl refl {ty} refl refl = inr refl ,- []
GwqConn box box {box} refl refl {ty} refl refl = inr refl ,- []
GwqConn undef box {box} refl () {ty} refl q-v
GwqConn q box {box} refl q-u {box} refl q-v = inl refl ,- []
systemF : WorldSystem
systemF = record
{ W = W
; _<<_ = _<<_
; tyW = tyW
; Q = Q
; st = st
; _#_ = _#_
; _&_ = _&_
; star& = \()
; unst = unst
; _&unst&_ = _&unst&_
; <<refl = refl
; <<trans = <<trans
; st-def = st-def
; st-unst = st-unst
; defUpset = \{_}{q} -> def-upset q
; op = \{_}{q} -> op q
; dominateUpset = dominateUpset
; stTyW = stTyW
; tyWUpset = \{refl tyV -> tyV}
; st&-action = \()
; &unst&-action = &unst&-action
; isActionL = \{q}{r} -> isActionL q r
; isActionR = \{q}{r} -> isActionR q r
; stq = stq
; GwqConn = GwqConn
}
module SystemFw where
data W : Set where
tm ty : W
_<<_ : W -> W -> Set
_<<_ = _==_
tyW : W -> Set
tyW tm = Zero
tyW ty = One
data Q : Sort -> Set where
tmTo : W -> Q Real
st tyToTy undef : Q Real
unstar constTm : Q Fake
_#_ : forall {s} -> Q s -> W -> Maybe W
tmTo w # tm = Just w
tmTo w # ty = Nothing
st # w = Just ty
undef # w = Nothing
unstar # tm = Nothing
unstar # ty = Just tm
tyToTy # tm = Nothing
tyToTy # ty = Just ty
constTm # w = Just tm
infix 40 _#_
st-def : forall w -> Sg W (st # w ~_)
st-def tm = ty , refl
st-def ty = ty , refl
st-act : W -> W
st-act w = fst (st-def w)
_&_ : Q Real -> Q Real -> Q Real
tmTo w & tmTo tm = tmTo w
tmTo w & _ = undef
st & tmTo w = tmTo ty
st & st = st
st & tyToTy = tyToTy
st & undef = undef
tyToTy & tmTo tm = undef
tyToTy & tmTo ty = tmTo ty
tyToTy & st = st
tyToTy & tyToTy = tyToTy
tyToTy & undef = undef
undef & r = undef
infix 50 _&_
star& : Q Fake → Q Real
star& unstar = tyToTy
star& constTm = st
unst : W -> Maybe W
unst ty = Just tm
unst _ = Nothing
_&unst&_ : Q Real -> One + Q Real -> Sg Sort Q
q &unst& inl <> with q # tm
... | Just tm = _ , unstar
... | Just ty = _ , tyToTy
... | Nothing = _ , undef
q &unst& inr (tmTo tm) = _ , undef
q &unst& inr (tmTo ty) with q # tm
... | Just w = _ , tmTo w
... | Nothing = _ , undef
q &unst& inr st with q # tm
... | Just tm = _ , constTm
... | Just ty = _ , st
... | Nothing = _ , undef
q &unst& inr tyToTy with q # tm
... | Just tm = _ , unstar
... | Just ty = _ , tyToTy
... | Nothing = _ , undef
q &unst& inr undef = _ , undef
<<trans : forall {u v w} -> u << v -> v << w -> u << w
<<trans refl refl = refl
st-unst : forall {w usw} -> unst w ~ usw -> st-act usw == w
st-unst {tm} ()
st-unst {ty} refl = refl
def-upset : forall {s}(q : Q s){v qv w} -> q # v ~ qv -> v << w -> Sg W (\qw -> q # w ~ qw)
def-upset q qv refl = _ , qv
op : forall {s}(q : Q s){v qv' w} -> (qv : q # v ~ qv') -> (v<w : v << w) -> qv' << fst (def-upset q qv v<w)