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Major refactoring of intro.tt + π₁(S¹)

master
Amélia Liao 4 years ago
parent
e52995a27b
commit
ec01798d27
11 changed files with 520 additions and 300 deletions
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  1. +7
    -0
      cubical.cabal
  2. +345
    -185
      intro.tt
  3. +4
    -2
      src/Debug.hs
  4. +14
    -26
      src/Elab.hs
  5. +95
    -69
      src/Elab/Eval.hs
  6. +3
    -2
      src/Elab/Eval/Formula.hs
  7. +4
    -1
      src/Elab/Monad.hs
  8. +31
    -7
      src/Elab/WiredIn.hs
  9. +3
    -3
      src/Main.hs
  10. +11
    -2
      src/Syntax.hs
  11. +3
    -3
      src/Syntax/Pretty.hs

+ 7
- 0
cubical.cabal View File

@ -13,6 +13,11 @@ build-type: Simple
cabal-version: >=2.0 cabal-version: >=2.0
extra-source-files: README.md extra-source-files: README.md
flag release
description: "Release" mode
manual: True
default: False
executable cubical executable cubical
hs-source-dirs: src hs-source-dirs: src
main-is: Main.hs main-is: Main.hs
@ -53,3 +58,5 @@ executable cubical
ghc-options: -Wall -Wextra -Wno-name-shadowing -rtsopts ghc-options: -Wall -Wextra -Wno-name-shadowing -rtsopts
if flag(release)
ghc-options: -XCPP -DRELEASE

+ 345
- 185
intro.tt View File

@ -66,12 +66,12 @@ PathP : (A : I -> Type) -> A i0 -> A i1 -> Type
-- Path. -- Path.
Path : {A : Type} -> A -> A -> Type Path : {A : Type} -> A -> A -> Type
Path {A} = PathP (\i -> A)
Path = PathP (\i -> A)
-- reflexivity is given by constant paths -- reflexivity is given by constant paths
refl : {A : Type} {x : A} -> Path x x refl : {A : Type} {x : A} -> Path x x
refl {A} {x} i = x
refl i = x
-- Symmetry (for dpeendent paths) is given by inverting the argument to the path, such that -- Symmetry (for dpeendent paths) is given by inverting the argument to the path, such that
-- sym p i0 = p (inot i0) = p i1 -- sym p i0 = p (inot i0) = p i1
@ -111,7 +111,7 @@ isContr A = (x : A) * ((y : A) -> Path x y)
-- singletons are contracible. Together with transport later on, -- singletons are contracible. Together with transport later on,
-- we get the J elimination principle of paths. -- we get the J elimination principle of paths.
singContr : {A : Type} {a : A} -> isContr (Singl A a) singContr : {A : Type} {a : A} -> isContr (Singl A a)
singContr {A} {a} = ((a, \i -> a), \y i -> (y.2 i, \j -> y.2 (iand i j)))
singContr = ((a, \i -> a), \y i -> (y.2 i, \j -> y.2 (iand i j)))
-- Some more operations on paths. By rearranging parentheses we get a -- Some more operations on paths. By rearranging parentheses we get a
-- proof that the images of equal elements are themselves equal. -- proof that the images of equal elements are themselves equal.
@ -186,6 +186,9 @@ Partial : I -> Type -> Pretype
PartialP : (phi : I) -> Partial phi Type -> Pretype PartialP : (phi : I) -> Partial phi Type -> Pretype
{-# PRIMITIVE PartialP #-} {-# PRIMITIVE PartialP #-}
partialExt : {A : Type} (phi : I) (psi : I) -> Partial phi A -> Partial psi A -> Partial (ior phi psi) A
{-# PRIMITIVE partialExt #-}
-- Why is Partial φ A not just defined as φ -> A? The difference is that -- Why is Partial φ A not just defined as φ -> A? The difference is that
-- Partial φ A has an internal representation which definitionally relates -- Partial φ A has an internal representation which definitionally relates
-- any two partial elements which "agree everywhere", that is, have -- any two partial elements which "agree everywhere", that is, have
@ -202,7 +205,7 @@ PartialP : (phi : I) -> Partial phi Type -> Pretype
-- That is, element of A[phi -> u] is an element of A defined everywhere -- That is, element of A[phi -> u] is an element of A defined everywhere
-- (a total element), which, when IsOne φ, agrees with u. -- (a total element), which, when IsOne φ, agrees with u.
Sub : (A : Type) (phi : I) -> Partial phi A -> Pretype
Sub : (A : Type) (phi : I) -> Partial phi A -> Pretype
{-# PRIMITIVE Sub #-} {-# PRIMITIVE Sub #-}
-- Every total element u : A can be made partial on φ by ignoring the -- Every total element u : A can be made partial on φ by ignoring the
@ -227,7 +230,7 @@ primComp : (A : I -> Type) {phi : I} (u : (i : I) -> Partial phi (A i)) -> Sub (
{-# PRIMITIVE comp primComp #-} {-# PRIMITIVE comp primComp #-}
comp : (A : I -> Type) {phi : I} (u : (i : I) -> Partial phi (A i)) -> Sub (A i0) phi (u i0) -> A i1 comp : (A : I -> Type) {phi : I} (u : (i : I) -> Partial phi (A i)) -> Sub (A i0) phi (u i0) -> A i1
comp A {phi} u a0 = outS (primComp A {phi} u a0)
comp A u a0 = outS (primComp A {phi} u a0)
-- In particular, when φ is a disjunction of the form -- In particular, when φ is a disjunction of the form
-- (j = 0) || (j = 1), we can draw u as being a pair of lines forming a -- (j = 0) || (j = 1), we can draw u as being a pair of lines forming a
@ -257,18 +260,12 @@ comp A {phi} u a0 = outS (primComp A {phi} u a0)
-- x----------y -- x----------y
-- p i -- p i
trans : {A : Type} {x : A} {y : A} {z : A} -> PathP (\i -> A) x y -> PathP (\i -> A) y z -> PathP (\i -> A) x z
trans {A} {x} p q i =
comp (\i -> A)
{ior i (inot i)}
(\j [ (i = i0) -> x, (i = i1) -> q j ])
(inS (p i))
-- In particular when the formula φ = i0 we get the "opposite face" to a -- In particular when the formula φ = i0 we get the "opposite face" to a
-- single point, which corresponds to transport. -- single point, which corresponds to transport.
transp : (A : I -> Type) (x : A i0) -> A i1 transp : (A : I -> Type) (x : A i0) -> A i1
transp A x = comp A {i0} (\i [ ]) (inS x)
transp A x = comp A (\i [ ]) (inS x)
subst : {A : Type} (P : A -> Type) {x : A} {y : A} -> Path x y -> P x -> P y subst : {A : Type} (P : A -> Type) {x : A} {y : A} -> Path x y -> P x -> P y
subst P p x = transp (\i -> P (p i)) x subst P p x = transp (\i -> P (p i)) x
@ -278,19 +275,28 @@ subst P p x = transp (\i -> P (p i)) x
fill : (A : I -> Type) {phi : I} (u : (i : I) -> Partial phi (A i)) (a0 : Sub (A i0) phi (u i0)) fill : (A : I -> Type) {phi : I} (u : (i : I) -> Partial phi (A i)) (a0 : Sub (A i0) phi (u i0))
-> PathP A (outS a0) (comp A {phi} u a0) -> PathP A (outS a0) (comp A {phi} u a0)
fill A {phi} u a0 i =
fill A u a0 i =
comp (\j -> A (iand i j)) comp (\j -> A (iand i j))
{ior phi (inot i)} {ior phi (inot i)}
(\j [ (phi = i1) -> u (iand i j) itIs1 (\j [ (phi = i1) -> u (iand i j) itIs1
, (i = i0) -> outS a0
, (i = i0) -> outS {A i0} {phi} {u i0} a0
]) ])
(inS (outS a0)) (inS (outS a0))
hcomp : {A : Type} {phi : I} (u : (i : I) -> Partial phi A) -> Sub A phi (u i0) -> A hcomp : {A : Type} {phi : I} (u : (i : I) -> Partial phi A) -> Sub A phi (u i0) -> A
hcomp {A} {phi} u a0 = comp (\i -> A) {phi} u a0
hcomp u a0 = comp (\i -> A) {phi} u a0
hfill : {A : Type} {phi : I} (u : (i : I) -> Partial phi A) -> (a0 : Sub A phi (u i0)) -> Path (outS a0) (hcomp u a0) hfill : {A : Type} {phi : I} (u : (i : I) -> Partial phi A) -> (a0 : Sub A phi (u i0)) -> Path (outS a0) (hcomp u a0)
hfill {A} {phi} u a0 i = fill (\i -> A) {phi} u a0 i
hfill u a0 i = fill (\i -> A) {phi} u a0 i
trans : {A : Type} {x : A} {y : A} {z : A} -> PathP (\i -> A) x y -> PathP (\i -> A) y z -> PathP (\i -> A) x z
trans p q i = comp (\j -> A) {ior i (inot i)} (\k [ (i = i0) -> x, (i = i1) -> q k ]) (inS {A} {ior i (inot i)} (p i))
transFiller : {A : Type} {x : A} {y : A} {z : A}
-> (p : Path x y) (q : Path y z)
-> PathP (\i -> Path x (q i)) p (trans {A} {x} {y} {z} p q)
transFiller p q j i = hfill (\k [ (i = i0) -> x, (i = i1) -> q k ]) (inS (p i)) j
-- For instance, the filler of the previous composition square -- For instance, the filler of the previous composition square
-- tells us that trans p refl = p: -- tells us that trans p refl = p:
@ -301,18 +307,18 @@ transRefl p j i = fill (\i -> A) {ior i (inot i)} (\k [ (i = i0) -> x, (i = i1)
rightCancel : {A : Type} {x : A} {y : A} (p : Path x y) -> Path (trans p (sym p)) refl rightCancel : {A : Type} {x : A} {y : A} (p : Path x y) -> Path (trans p (sym p)) refl
rightCancel p j i = cube p i1 j i where rightCancel p j i = cube p i1 j i where
cube : {A : Type} {x : A} {y : A} (p : Path x y) -> I -> I -> I -> A cube : {A : Type} {x : A} {y : A} (p : Path x y) -> I -> I -> I -> A
cube {A} {x} p k j i =
hfill {A} (\ k [ (i = i0) -> x
, (i = i1) -> p (iand (inot k) (inot j))
, (j = i1) -> x
])
(inS (p (iand i (inot j)))) k
cube p k j i =
hfill (\ k [ (i = i0) -> x
, (i = i1) -> p (iand (inot k) (inot j))
, (j = i1) -> x
])
(inS (p (iand i (inot j)))) k
leftCancel : {A : Type} {x : A} {y : A} (p : Path x y) -> Path (trans (sym p) p) refl leftCancel : {A : Type} {x : A} {y : A} (p : Path x y) -> Path (trans (sym p) p) refl
leftCancel p = rightCancel (sym p) leftCancel p = rightCancel (sym p)
transpFill : {A : I -> Type} (x : A i0) -> PathP A x (transp (\i -> A i) x)
transpFill {A} x i = fill (\i -> A i) (\k []) (inS x) i
transpFill : (A : I -> Type) (x : A i0) -> PathP A x (transp (\i -> A i) x)
transpFill A x i = fill (\i -> A i) (\k []) (inS x) i
-- Reduction of composition -- Reduction of composition
--------------------------- ---------------------------
@ -365,7 +371,7 @@ invert : {A : Type} {B : Type} {f : A -> B} -> isEquiv f -> B -> A
invert eqv y = (eqv y) .1 .1 invert eqv y = (eqv y) .1 .1
retract : {A : Type} {B : Type} -> (A -> B) -> (B -> A) -> Type retract : {A : Type} {B : Type} -> (A -> B) -> (B -> A) -> Type
retract {A} {B} f g = (a : A) -> Path (g (f a)) a
retract f g = (a : A) -> Path (g (f a)) a
-- Proving that it's also a retraction is left as an exercise to the -- Proving that it's also a retraction is left as an exercise to the
-- reader. We can package together a function and a proof that it's an -- reader. We can package together a function and a proof that it's an
@ -377,7 +383,7 @@ Equiv A B = (f : A -> B) * isEquiv {A} {B} f
-- The identity function is an equivalence between any type A and -- The identity function is an equivalence between any type A and
-- itself. -- itself.
idEquiv : {A : Type} -> Equiv A A idEquiv : {A : Type} -> Equiv A A
idEquiv {A} = (\x -> x, \y -> ((y, \i -> y), \u i -> (u.2 i, \j -> u.2 (iand i j))))
idEquiv = (\x -> x, \y -> ((y, \i -> y), \u i -> (u.2 i, \j -> u.2 (iand i j))))
-- The glue operation expresses that "extensibility is invariant under -- The glue operation expresses that "extensibility is invariant under
-- equivalence". Less concisely, the Glue type and its constructor, -- equivalence". Less concisely, the Glue type and its constructor,
@ -400,7 +406,7 @@ primGlue : (A : Type) {phi : I}
-- Agreeing with the condition that Glue A [i1 -> (T, e)] = T, -- Agreeing with the condition that Glue A [i1 -> (T, e)] = T,
-- we have that glue {A} {i1} t im => t. -- we have that glue {A} {i1} t im => t.
prim'glue : {A : Type} {phi : I} {T : Partial phi Type} {e : PartialP phi (\o -> Equiv (T o) A)} prim'glue : {A : Type} {phi : I} {T : Partial phi Type} {e : PartialP phi (\o -> Equiv (T o) A)}
-> (t : PartialP phi T)
-> (t : PartialP phi (\o -> T o))
-> (im : Sub A phi (\o -> (e o).1 (t o))) -> (im : Sub A phi (\o -> (e o).1 (t o)))
-> primGlue A T e -> primGlue A T e
@ -410,7 +416,7 @@ glue : {A : Type} {phi : I} {Te : Partial phi ((T : Type) * Equiv T A)}
-> (t : PartialP phi (\o -> (Te o).1)) -> (t : PartialP phi (\o -> (Te o).1))
-> (im : Sub A phi (\o -> (Te o).2.1 (t o))) -> (im : Sub A phi (\o -> (Te o).2.1 (t o)))
-> primGlue A {phi} (\o -> (Te o).1) (\o -> (Te o).2) -> primGlue A {phi} (\o -> (Te o).1) (\o -> (Te o).2)
glue {A} {phi} {Te} t im = prim'glue {A} {phi} {\o -> (Te o).1} {\o -> (Te o).2} t im
glue t im = prim'glue {A} {phi} {\o -> (Te o).1} {\o -> (Te o).2} t im
-- The unglue operation undoes a glueing. Since when φ = i1, -- The unglue operation undoes a glueing. Since when φ = i1,
-- Glue A [φ -> (T, f)] = T, the argument to primUnglue {A} {i1} ... -- Glue A [φ -> (T, f)] = T, the argument to primUnglue {A} {i1} ...
@ -424,7 +430,7 @@ primUnglue : {A : Type} {phi : I} {T : Partial phi Type} {e : PartialP phi (\o -
unglue : {A : Type} (phi : I) {Te : Partial phi ((T : Type) * Equiv T A)} unglue : {A : Type} (phi : I) {Te : Partial phi ((T : Type) * Equiv T A)}
-> primGlue A {phi} (\o -> (Te o).1) (\o -> (Te o).2) -> A -> primGlue A {phi} (\o -> (Te o).1) (\o -> (Te o).2) -> A
unglue {A} phi {Te} = primUnglue {A} {phi} {\o -> (Te o).1} {\o -> (Te o).2}
unglue phi = primUnglue {A} {phi} {\o -> (Te o).1} {\o -> (Te o).2}
-- Diagramatically, i : I |- Glue A [(i \/ ~i) -> (T, e)] can be drawn -- Diagramatically, i : I |- Glue A [(i \/ ~i) -> (T, e)] can be drawn
-- as giving us the dotted line in: -- as giving us the dotted line in:
@ -445,7 +451,7 @@ unglue {A} phi {Te} = primUnglue {A} {phi} {\o -> (Te o).1} {\o -> (Te o).2}
-- equivalences, we can make a line between two types (T i0) and (T i1). -- equivalences, we can make a line between two types (T i0) and (T i1).
Glue : (A : Type) {phi : I} -> Partial phi ((X : Type) * Equiv X A) -> Type Glue : (A : Type) {phi : I} -> Partial phi ((X : Type) * Equiv X A) -> Type
Glue A {phi} u = primGlue A {phi} (\o -> (u o).1) (\o -> (u o).2)
Glue A u = primGlue A {phi} (\o -> (u o).1) (\o -> (u o).2)
-- For example, we can glue together the type A and the type B as long -- For example, we can glue together the type A and the type B as long
-- as there exists an Equiv A B. -- as there exists an Equiv A B.
@ -460,9 +466,9 @@ Glue A {phi} u = primGlue A {phi} (\o -> (u o).1) (\o -> (u o).2)
-- \i → B -- \i → B
-- --
ua : {A : Type} {B : Type} -> Equiv A B -> Path A B ua : {A : Type} {B : Type} -> Equiv A B -> Path A B
ua {A} {B} equiv i =
Glue B (\[ (i = i0) -> (A, equiv),
(i = i1) -> (B, idEquiv) ])
ua equiv i =
Glue B (\[ (i = i0) -> (A, equiv)
, (i = i1) -> (B, idEquiv) ])
lineToEquiv : (A : I -> Type) -> Equiv (A i0) (A i1) lineToEquiv : (A : I -> Type) -> Equiv (A i0) (A i1)
{-# PRIMITIVE lineToEquiv #-} {-# PRIMITIVE lineToEquiv #-}
@ -493,7 +499,7 @@ isEquivTransport A = (lineToEquiv A).2
-- we need to come up with a filler for the Bottom and right faces. -- we need to come up with a filler for the Bottom and right faces.
uaBeta : {A : Type} {B : Type} (f : Equiv A B) -> Path (transp (\i -> ua f i)) f.1 uaBeta : {A : Type} {B : Type} (f : Equiv A B) -> Path (transp (\i -> ua f i)) f.1
uaBeta {A} {B} f i a = transpFill {\i -> B} (f.1 a) (inot i)
uaBeta f i a = transpFill (\i -> B) (f.1 a) (inot i)
-- The terms ua + uaBeta suffice to prove the "full" -- The terms ua + uaBeta suffice to prove the "full"
-- ua axiom of Voevodsky, as can be seen in the paper -- ua axiom of Voevodsky, as can be seen in the paper
@ -513,7 +519,7 @@ JRefl : {A : Type} {x : A}
(P : (y : A) -> Path x y -> Type) (P : (y : A) -> Path x y -> Type)
(d : P x (\i -> x)) (d : P x (\i -> x))
-> Path (J {A} {x} P d {x} (\i -> x)) d -> Path (J {A} {x} P d {x} (\i -> x)) d
JRefl P d i = transpFill {\i -> P x (\j -> x)} d (inot i)
JRefl P d i = transpFill (\i -> P x (\j -> x)) d (inot i)
Jay : {A : Type} {x : A} Jay : {A : Type} {x : A}
(P : ((y : A) * Path x y) -> Type) (P : ((y : A) * Path x y) -> Type)
@ -532,7 +538,7 @@ Jay P d s = transp (\i -> P ((singContr {A} {x}).2 s i)) d
-- definition of equivalence. -- definition of equivalence.
isIso : {A : Type} -> {B : Type} -> (A -> B) -> Type isIso : {A : Type} -> {B : Type} -> (A -> B) -> Type
isIso {A} {B} f = (g : B -> A) * ((y : B) -> Path (f (g y)) y) * ((x : A) -> Path (g (f x)) x)
isIso f = (g : B -> A) * ((y : B) -> Path (f (g y)) y) * ((x : A) -> Path (g (f x)) x)
-- The reason is that the family of types IsIso is not a proposition! -- The reason is that the family of types IsIso is not a proposition!
-- This means that there can be more than one way for a function to be -- This means that there can be more than one way for a function to be
@ -548,7 +554,7 @@ Iso A B = (f : A -> B) * isIso f
-- https://github.com/mortberg/cubicaltt/blob/master/experiments/isoToEquiv.ctt#L7-L55 -- https://github.com/mortberg/cubicaltt/blob/master/experiments/isoToEquiv.ctt#L7-L55
IsoToEquiv : {A : Type} {B : Type} -> Iso A B -> Equiv A B IsoToEquiv : {A : Type} {B : Type} -> Iso A B -> Equiv A B
IsoToEquiv {A} {B} iso = (f, \y -> (fCenter y, fIsCenter y)) where
IsoToEquiv iso = (f, \y -> (fCenter y, fIsCenter y)) where
f = iso.1 f = iso.1
g = iso.2.1 g = iso.2.1
s = iso.2.2.1 s = iso.2.2.1
@ -559,47 +565,47 @@ IsoToEquiv {A} {B} iso = (f, \y -> (fCenter y, fIsCenter y)) where
lemIso y x0 x1 p0 p1 = lemIso y x0 x1 p0 p1 =
let let
rem0 : Path x0 (g y) rem0 : Path x0 (g y)
rem0 i = comp (\i -> A) (\k [ (i = i0) -> t x0 k, (i = i1) -> g y ]) (inS (g (p0 (inot i))))
rem0 i = hcomp (\k [ (i = i0) -> t x0 k, (i = i1) -> g y ]) (inS (g (p0 (inot i))))
rem1 : Path x1 (g y) rem1 : Path x1 (g y)
rem1 i = comp (\i -> A) (\k [ (i = i0) -> t x1 k, (i = i1) -> g y ]) (inS (g (p1 (inot i))))
rem1 i = hcomp (\k [ (i = i0) -> t x1 k, (i = i1) -> g y ]) (inS (g (p1 (inot i))))
p : Path x0 x1 p : Path x0 x1
p i = comp (\i -> A) (\k [ (i = i0) -> rem0 (inot k), (i = i1) -> rem1 (inot k) ]) (inS (g y))
p i = hcomp (\k [ (i = i0) -> rem0 (inot k), (i = i1) -> rem1 (inot k) ]) (inS (g y))
fill0 : I -> I -> A fill0 : I -> I -> A
fill0 i j = comp (\i -> A) (\k [ (i = i0) -> t x0 (iand j k)
, (i = i1) -> g y
, (j = i0) -> g (p0 (inot i))
])
(inS (g (p0 (inot i))))
fill0 i j = hcomp (\k [ (i = i0) -> t x0 (iand j k)
, (i = i1) -> g y
, (j = i0) -> g (p0 (inot i))
])
(inS (g (p0 (inot i))))
fill1 : I -> I -> A fill1 : I -> I -> A
fill1 i j = comp (\i -> A) (\k [ (i = i0) -> t x1 (iand j k)
, (i = i1) -> g y
, (j = i0) -> g (p1 (inot i)) ])
(inS (g (p1 (inot i))))
fill1 i j = hcomp (\k [ (i = i0) -> t x1 (iand j k)
, (i = i1) -> g y
, (j = i0) -> g (p1 (inot i)) ])
(inS (g (p1 (inot i))))
fill2 : I -> I -> A fill2 : I -> I -> A
fill2 i j = comp (\i -> A) (\k [ (i = i0) -> rem0 (ior j (inot k))
, (i = i1) -> rem1 (ior j (inot k))
, (j = i1) -> g y ])
(inS (g y))
fill2 i j = hcomp (\k [ (i = i0) -> rem0 (ior j (inot k))
, (i = i1) -> rem1 (ior j (inot k))
, (j = i1) -> g y ])
(inS (g y))
sq : I -> I -> A sq : I -> I -> A
sq i j = comp (\i -> A) (\k [ (i = i0) -> fill0 j (inot k)
, (i = i1) -> fill1 j (inot k)
, (j = i1) -> g y
, (j = i0) -> t (p i) (inot k) ])
(inS (fill2 i j))
sq i j = hcomp (\k [ (i = i0) -> fill0 j (inot k)
, (i = i1) -> fill1 j (inot k)
, (j = i1) -> g y
, (j = i0) -> t (p i) (inot k) ])
(inS (fill2 i j))
sq1 : I -> I -> B sq1 : I -> I -> B
sq1 i j = comp (\i -> B) (\k [ (i = i0) -> s (p0 (inot j)) k
, (i = i1) -> s (p1 (inot j)) k
, (j = i0) -> s (f (p i)) k
, (j = i1) -> s y k
])
(inS (f (sq i j)))
sq1 i j = hcomp (\k [ (i = i0) -> s (p0 (inot j)) k
, (i = i1) -> s (p1 (inot j)) k
, (j = i0) -> s (f (p i)) k
, (j = i1) -> s y k
])
(inS (f (sq i j)))
in \i -> (p i, \j -> sq1 i (inot j)) in \i -> (p i, \j -> sq1 i (inot j))
fCenter : (y : B) -> fiber f y fCenter : (y : B) -> fiber f y
@ -615,7 +621,7 @@ IsoToId i = ua (IsoToEquiv i)
-- such a function is its own inverse. -- such a function is its own inverse.
involToIso : {A : Type} (f : A -> A) -> ((x : A) -> Path (f (f x)) x) -> isIso f involToIso : {A : Type} (f : A -> A) -> ((x : A) -> Path (f (f x)) x) -> isIso f
involToIso {A} f inv = (f, inv, inv)
involToIso f inv = (f, inv, inv)
-- An example of ua -- An example of ua
--------------------------- ---------------------------
@ -720,7 +726,7 @@ universeNotSet itIs = trueNotFalse (\i -> transp (\j -> itIs Bool Bool notp refl
-- extensionality. The strong version is as follows: -- extensionality. The strong version is as follows:
Hom : {A : Type} {B : A -> Type} (f : (x : A) -> B x) -> (g : (x : A) -> B x) -> Type Hom : {A : Type} {B : A -> Type} (f : (x : A) -> B x) -> (g : (x : A) -> B x) -> Type
Hom {A} f g = (x : A) -> Path (f x) (g x)
Hom f g = (x : A) -> Path (f x) (g x)
happly : {A : Type} {B : A -> Type} {f : (x : A) -> B x} {g : (x : A) -> B x} happly : {A : Type} {B : A -> Type} {f : (x : A) -> B x} {g : (x : A) -> B x}
-> (p : Path f g) -> Hom f g -> (p : Path f g) -> Hom f g
@ -730,11 +736,11 @@ happly p x i = p i x
happlyIsIso : {A : Type} {B : A -> Type} {f : (x : A) -> B x} {g : (x : A) -> B x} happlyIsIso : {A : Type} {B : A -> Type} {f : (x : A) -> B x} {g : (x : A) -> B x}
-> isIso {Path f g} {Hom f g} happly -> isIso {Path f g} {Hom f g} happly
happlyIsIso {A} {B} {f} {g} = (funext {A} {B} {f} {g}, \hom -> refl, \path -> refl)
happlyIsIso = (funext {A} {B} {f} {g}, \hom -> refl, \path -> refl)
pathIsHom : {A : Type} {B : A -> Type} {f : (x : A) -> B x} {g : (x : A) -> B x} pathIsHom : {A : Type} {B : A -> Type} {f : (x : A) -> B x} {g : (x : A) -> B x}
-> Path (Path f g) (Hom f g) -> Path (Path f g) (Hom f g)
pathIsHom {A} {B} {f} {g} =
pathIsHom =
let let
theIso : Iso (Path f g) (Hom f g) theIso : Iso (Path f g) (Hom f g)
theIso = (happly {A} {B} {f} {g}, happlyIsIso {A} {B} {f} {g}) theIso = (happly {A} {B} {f} {g}, happlyIsIso {A} {B} {f} {g})
@ -978,7 +984,7 @@ poSusp : Type -> Type
poSusp A = Pushout {A} {Unit} {Unit} (\x -> tt) (\x -> tt) poSusp A = Pushout {A} {Unit} {Unit} (\x -> tt) (\x -> tt)
Susp_is_poSusp : {A : Type} -> Path (Susp A) (poSusp A) Susp_is_poSusp : {A : Type} -> Path (Susp A) (poSusp A)
Susp_is_poSusp {A} = ua (IsoToEquiv (Susp_to_poSusp {A}, poSusp_to_Susp {A}, poSusp_to_Susp_to_poSusp {A}, Susp_to_poSusp_to_Susp {A})) where
Susp_is_poSusp = ua (IsoToEquiv (Susp_to_poSusp {A}, poSusp_to_Susp {A}, poSusp_to_Susp_to_poSusp {A}, Susp_to_poSusp_to_Susp {A})) where
poSusp_to_Susp : {A : Type} -> poSusp A -> Susp A poSusp_to_Susp : {A : Type} -> poSusp A -> Susp A
poSusp_to_Susp = \case poSusp_to_Susp = \case
inl x -> north inl x -> north
@ -1014,8 +1020,8 @@ data T2 : Type where
(i = i1) -> pathOne j (i = i1) -> pathOne j
] ]
TorusIsTwoCircles : Path T2 (S1 * S1)
TorusIsTwoCircles = ua (IsoToEquiv theIso) where
TorusIsTwoCircles : Equiv T2 (S1 * S1)
TorusIsTwoCircles = IsoToEquiv theIso where
torusToCircs : T2 -> S1 * S1 torusToCircs : T2 -> S1 * S1
torusToCircs = \case torusToCircs = \case
baseT -> (base, base) baseT -> (base, base)
@ -1143,11 +1149,11 @@ isProp_isProp f g i a b = isProp_isSet f a b (f a b) (g a b) i
isProp_isContr : {A : Type} -> isProp (isContr A) isProp_isContr : {A : Type} -> isProp (isContr A)
isProp_isContr {A} z0 z1 j = isProp_isContr {A} z0 z1 j =
( z0.2 z1.1 j ( z0.2 z1.1 j
, \x i -> hcomp {A} (\k [ (i = i0) -> z0.2 z1.1 j
, (i = i1) -> z0.2 x (ior j k)
, (j = i0) -> z0.2 x (iand i k)
, (j = i1) -> z1.2 x i ])
(inS (z0.2 (z1.2 x i) j))
, \x i -> hcomp (\k [ (i = i0) -> z0.2 z1.1 j
, (i = i1) -> z0.2 x (ior j k)
, (j = i0) -> z0.2 x (iand i k)
, (j = i1) -> z1.2 x i ])
(inS (z0.2 (z1.2 x i) j))
) )
isContr_isProp : {A : Type} -> isContr A -> isProp A isContr_isProp : {A : Type} -> isContr A -> isProp A
@ -1170,7 +1176,7 @@ propExt {A} {B} propA propB f g = (f, contract) where
let arg : A let arg : A
arg = g y arg = g y
in ( (arg, propB y (f arg)) in ( (arg, propB y (f arg))
, \fib -> sigmaPath (propA _ _) (isProp_isSet {B} propB y (f fib.1) _ _))
, \fib -> sigmaPath (propA _ _) (isProp_isSet propB y (f fib.1) _ _))
Sq_rec : {A : Type} {B : Type} Sq_rec : {A : Type} {B : Type}
-> isProp B -> isProp B
@ -1185,6 +1191,9 @@ Sq_rec prop f =
work = \case work = \case
inc x -> f x inc x -> f x
Sq_prop : {A : Type} -> isProp (Sq A)
Sq_prop x y i = sq x y i
hitTranspExample : Path (inc false) (inc true) hitTranspExample : Path (inc false) (inc true)
hitTranspExample i = transp (\i -> Sq (notp i)) (sq (inc true) (inc false) i) hitTranspExample i = transp (\i -> Sq (notp i)) (sq (inc true) (inc false) i)
@ -1201,12 +1210,12 @@ S2IsSuspS1 = ua (IsoToEquiv iso) where
loop j -> \i -> surf2 i j loop j -> \i -> surf2 i j
suspSurf : I -> I -> I -> Susp S1 suspSurf : I -> I -> I -> Susp S1
suspSurf i j = hfill {Susp S1} (\k [ (i = i0) -> north
, (i = i1) -> merid base (inot k)
, (j = i0) -> merid base (iand (inot k) i)
, (j = i1) -> merid {S1} base (iand (inot k) i)
])
(inS (merid (loop j) i))
suspSurf i j = hfill (\k [ (i = i0) -> north {S1}
, (i = i1) -> merid {S1} base (inot k)
, (j = i0) -> merid {S1} base (iand (inot k) i)
, (j = i1) -> merid {S1} base (iand (inot k) i)
])
(inS (merid (loop j) i))
fromS2 : S2 -> Susp S1 fromS2 : S2 -> Susp S1
fromS2 = \case { base2 -> north; surf2 i j -> suspSurf i j i1 } fromS2 = \case { base2 -> north; surf2 i j -> suspSurf i j i1 }
@ -1237,14 +1246,14 @@ S3IsSuspS2 = ua (IsoToEquiv iso) where
surf2 j k -> \i -> surf3 i j k surf2 j k -> \i -> surf3 i j k
suspSurf : I -> I -> I -> I -> Susp S2 suspSurf : I -> I -> I -> I -> Susp S2
suspSurf i j k = hfill {Susp S2} (\l [ (i = i0) -> north
, (i = i1) -> merid base2 (inot l)
, (j = i0) -> merid base2 (iand (inot l) i)
, (j = i1) -> merid {S2} base2 (iand (inot l) i)
, (k = i0) -> merid base2 (iand (inot l) i)
, (k = i1) -> merid {S2} base2 (iand (inot l) i)
])
(inS (merid (surf2 j k) i))
suspSurf i j k = hfill (\l [ (i = i0) -> north {S2}
, (i = i1) -> merid {S2} base2 (inot l)
, (j = i0) -> merid {S2} base2 (iand (inot l) i)
, (j = i1) -> merid {S2} base2 (iand (inot l) i)
, (k = i0) -> merid {S2} base2 (iand (inot l) i)
, (k = i1) -> merid {S2} base2 (iand (inot l) i)
])
(inS (merid (surf2 j k) i))
fromS3 : S3 -> Susp S2 fromS3 : S3 -> Susp S2
fromS3 = \case { base3 -> north; surf3 i j k -> suspSurf i j k i1 } fromS3 = \case { base3 -> north; surf3 i j k -> suspSurf i j k i1 }
@ -1263,24 +1272,24 @@ S3IsSuspS2 = ua (IsoToEquiv iso) where
iso = (fromS3, toS3, fromToS3, toFromS3) iso = (fromS3, toS3, fromToS3, toFromS3)
ap_s : {A : Pretype} {B : Pretype} (f : A -> B) {x : A} {y : A} -> Eq_s x y -> Eq_s (f x) (f y) ap_s : {A : Pretype} {B : Pretype} (f : A -> B) {x : A} {y : A} -> Eq_s x y -> Eq_s (f x) (f y)
ap_s {A} {B} f {x} {y} = J_s (\y p -> Eq_s (f x) (f y)) refl_s
ap_s f {x} = J_s (\y p -> Eq_s (f x) (f y)) refl_s
subst_s : {A : Pretype} (P : A -> Pretype) {x : A} {y : A} -> Eq_s x y -> P x -> P y subst_s : {A : Pretype} (P : A -> Pretype) {x : A} {y : A} -> Eq_s x y -> P x -> P y
subst_s {A} P {x} {z} p px = J_s {A} {x} (\y p -> P x -> P y) id p px
subst_s {A} P {x} p px = J_s (\y p -> P x -> P y) id p px
sym_s : {A : Pretype} {x : A} {y : A} -> Eq_s x y -> Eq_s y x sym_s : {A : Pretype} {x : A} {y : A} -> Eq_s x y -> Eq_s y x
sym_s {A} {x} {y} = J_s {A} {x} (\y p -> Eq_s y x) refl_s
sym_s = J_s (\y p -> Eq_s y x) refl_s
UIP : {A : Pretype} {x : A} {y : A} (p : Eq_s x y) (q : Eq_s x y) -> Eq_s p q UIP : {A : Pretype} {x : A} {y : A} (p : Eq_s x y) (q : Eq_s x y) -> Eq_s p q
UIP {A} {x} {y} p q = J_s (\y p -> (q : Eq_s x y) -> Eq_s p q) (uipRefl A x) p q where
UIP p q = J_s (\y p -> (q : Eq_s x y) -> Eq_s p q) (uipRefl A x) p q where
uipRefl : (A : Pretype) (x : A) (p : Eq_s x x) -> Eq_s refl_s p uipRefl : (A : Pretype) (x : A) (p : Eq_s x x) -> Eq_s refl_s p
uipRefl A x p = K_s {A} {x} (\q -> Eq_s refl_s q) refl_s p
uipRefl A x p = K_s (\q -> Eq_s refl_s q) refl_s p
strictEq_pathEq : {A : Type} {x : A} {y : A} -> Eq_s x y -> Path x y strictEq_pathEq : {A : Type} {x : A} {y : A} -> Eq_s x y -> Path x y
strictEq_pathEq {A} {x} {y} eq = J_s {A} {x} (\y p -> Path x y) (\i -> x) {y} eq
strictEq_pathEq eq = J_s {A} {x} (\y p -> Path x y) (\i -> x) {y} eq
seq_pathRefl : {A : Type} {x : A} (p : Eq_s x x) -> Eq_s (strictEq_pathEq p) (refl {A} {x}) seq_pathRefl : {A : Type} {x : A} (p : Eq_s x x) -> Eq_s (strictEq_pathEq p) (refl {A} {x})
seq_pathRefl {A} {x} p = K_s (\p -> Eq_s (strictEq_pathEq {A} {x} {x} p) (refl {A} {x})) refl_s p
seq_pathRefl p = K_s (\p -> Eq_s (strictEq_pathEq {A} {x} {x} p) (refl {A} {x})) refl_s p
Path_nat_strict_nat : (x : Nat) (y : Nat) -> Path x y -> Eq_s x y Path_nat_strict_nat : (x : Nat) (y : Nat) -> Path x y -> Eq_s x y
Path_nat_strict_nat = \case { zero -> zeroCase; succ x -> succCase x } where Path_nat_strict_nat = \case { zero -> zeroCase; succ x -> succCase x } where
@ -1296,7 +1305,7 @@ Path_nat_strict_nat = \case { zero -> zeroCase; succ x -> succCase x } where
pathToEqS_K : {A : Type} {x : A} pathToEqS_K : {A : Type} {x : A}
-> (s : {x : A} {y : A} -> Path x y -> Eq_s x y) -> (s : {x : A} {y : A} -> Path x y -> Eq_s x y)
-> (P : Path x x -> Type) -> P refl -> (p : Path x x) -> P p -> (P : Path x x -> Type) -> P refl -> (p : Path x x) -> P p
pathToEqS_K {A} {x} p_to_s P pr loop = transp (\i -> P (inv x loop i)) psloop where
pathToEqS_K p_to_s P pr loop = transp (\i -> P (inv x loop i)) psloop where
psloop : P (strictEq_pathEq (p_to_s loop)) psloop : P (strictEq_pathEq (p_to_s loop))
psloop = K_s (\l -> P (strictEq_pathEq {A} {x} {x} l)) pr (p_to_s {x} {x} loop) psloop = K_s (\l -> P (strictEq_pathEq {A} {x} {x} l)) pr (p_to_s {x} {x} loop)
@ -1306,7 +1315,7 @@ pathToEqS_K {A} {x} p_to_s P pr loop = transp (\i -> P (inv x loop i)) psloop wh
aux = seq_pathRefl (p_to_s (\i -> x)) aux = seq_pathRefl (p_to_s (\i -> x))
pathToEq_isSet : {A : Type} -> ({x : A} {y : A} -> Path x y -> Eq_s x y) -> isHSet A pathToEq_isSet : {A : Type} -> ({x : A} {y : A} -> Path x y -> Eq_s x y) -> isHSet A
pathToEq_isSet {A} p_to_s = axK_to_isSet {A} (\{x} -> pathToEqS_K {A} {x} p_to_s) where
pathToEq_isSet p_to_s = axK_to_isSet {A} (\{x} -> pathToEqS_K {A} {x} p_to_s) where
axK_to_isSet : {A : Type} -> ({x : A} -> (P : Path x x -> Type) -> P refl -> (p : Path x x) -> P p) -> isHSet A axK_to_isSet : {A : Type} -> ({x : A} -> (P : Path x x -> Type) -> P refl -> (p : Path x x) -> P p) -> isHSet A
axK_to_isSet K x y p q = J (\y p -> (q : Path x y) -> Path p q) (uipRefl x) p q where axK_to_isSet K x y p q = J (\y p -> (q : Path x y) -> Path p q) (uipRefl x) p q where
uipRefl : (x : A) (p : Path x x) -> Path refl p uipRefl : (x : A) (p : Path x x) -> Path refl p
@ -1334,34 +1343,33 @@ equivCtrPath : {A : Type} {B : Type} (e : Equiv A B) (y : B)
equivCtrPath e y = (e.2 y).2 equivCtrPath e y = (e.2 y).2
contr : {A : Type} {phi : I} -> isContr A -> (u : Partial phi A) -> Sub A phi u contr : {A : Type} {phi : I} -> isContr A -> (u : Partial phi A) -> Sub A phi u
contr {A} {phi} p u = primComp (\i -> A) (\i [ (phi = i1) -> p.2 (u itIs1) i ]) (inS p.1)
contr p u = primComp (\i -> A) (\i [ (phi = i1) -> p.2 (u itIs1) i ]) (inS p.1)
contr' : {A : Type} -> ({phi : I} -> (u : Partial phi A) -> Sub A phi u) -> isContr A contr' : {A : Type} -> ({phi : I} -> (u : Partial phi A) -> Sub A phi u) -> isContr A
contr' {A} contr = (x, \y i -> outS (contr (\ [ (i = i0) -> x, (i = i1) -> y ])) ) where
contr' contr = (x, \y i -> outS (contr (\ [ (i = i0) -> x, (i = i1) -> y ])) ) where
x : A x : A
x = outS (contr (\ [])) x = outS (contr (\ []))
leftIsOne : {a : I} {b : I} -> Eq_s a i1 -> Eq_s (ior a b) i1 leftIsOne : {a : I} {b : I} -> Eq_s a i1 -> Eq_s (ior a b) i1
leftIsOne {a} {b} p = J_s {I} {i1} (\i p -> IsOne (ior i b)) refl_s (sym_s p)
leftIsOne p = J_s {I} {i1} (\i p -> IsOne (ior i b)) refl_s (sym_s p)
rightIsOne : {a : I} {b : I} -> Eq_s b i1 -> Eq_s (ior a b) i1 rightIsOne : {a : I} {b : I} -> Eq_s b i1 -> Eq_s (ior a b) i1
rightIsOne {a} {b} p = J_s {I} {i1} (\i p -> IsOne (ior a i)) refl_s (sym_s p)
rightIsOne p = J_s {I} {i1} (\i p -> IsOne (ior a i)) refl_s (sym_s p)
bothAreOne : {a : I} {b : I} -> Eq_s a i1 -> Eq_s b i1 -> Eq_s (iand a b) i1 bothAreOne : {a : I} {b : I} -> Eq_s a i1 -> Eq_s b i1 -> Eq_s (iand a b) i1
bothAreOne {a} {b} p q = J_s {I} {i1} (\i p -> IsOne (iand i b)) q (sym_s p)
bothAreOne p q = J_s {I} {i1} (\i p -> IsOne (iand i b)) q (sym_s p)
S1Map_to_baseLoop : {X : Type} -> (S1 -> X) -> (a : X) * Path a a S1Map_to_baseLoop : {X : Type} -> (S1 -> X) -> (a : X) * Path a a
S1Map_to_baseLoop {X} f = (f base, \i -> f (loop i))
S1Map_to_baseLoop f = (f base, \i -> f (loop i))
baseLoop_to_S1Map : {X : Type} -> ((a : X) * Path a a) -> S1 -> X
baseLoop_to_S1Map {X} p = \case
base -> p.1
loop i -> p.2 i
S1_univ : {X : Type} -> Path (S1 -> X) ((a : X) * Path a a) S1_univ : {X : Type} -> Path (S1 -> X) ((a : X) * Path a a)
S1_univ = IsoToId {S1 -> X} {(a : X) * Path a a} (S1Map_to_baseLoop, baseLoop_to_S1Map, ret, sec) where
S1_univ = IsoToId {S1 -> X} {(a : X) * Path a a} (S1Map_to_baseLoop, fro, ret, sec) where
to = S1Map_to_baseLoop to = S1Map_to_baseLoop
fro = baseLoop_to_S1Map
fro : {X : Type} -> ((a : X) * Path a a) -> S1 -> X
fro p = \case
base -> p.1
loop i -> p.2 i
sec : {X : Type} -> (f : S1 -> X) -> Path (fro (to f)) f sec : {X : Type} -> (f : S1 -> X) -> Path (fro (to f)) f
sec {X} f = funext {S1} {\s -> X} {\x -> fro (to f) x} {f} h where sec {X} f = funext {S1} {\s -> X} {\x -> fro (to f) x} {f} h where
@ -1386,7 +1394,7 @@ encodeDecode code c0 decode encDec based = IsoToId (encode {a0}, decode _, encDe
encode alpha = transp (\i -> code (alpha i)) c0 encode alpha = transp (\i -> code (alpha i)) c0
encodeRefl : Path (encode refl) c0 encodeRefl : Path (encode refl) c0
encodeRefl = sym (transpFill {\i -> code a0} c0)
encodeRefl = sym (transpFill (\i -> code a0) c0)
decEnc : {x : A} (p : Path a0 x) -> Path (decode _ (encode p)) p decEnc : {x : A} (p : Path a0 x) -> Path (decode _ (encode p)) p
decEnc p = J (\x p -> Path (decode _ (encode p)) p) q p where decEnc p = J (\x p -> Path (decode _ (encode p)) p) q p where
@ -1402,7 +1410,7 @@ S1_elim P pb pq = \case
loop i -> pq i loop i -> pq i
PathP_is_Path : (P : I -> Type) (p : P i0) (q : P i1) -> Path (PathP P p q) (Path {P i1} (transp (\i -> P i) p) q) PathP_is_Path : (P : I -> Type) (p : P i0) (q : P i1) -> Path (PathP P p q) (Path {P i1} (transp (\i -> P i) p) q)
PathP_is_Path P p q i = PathP (\j -> P (ior i j)) (transpFill {\j -> P j} p i) q
PathP_is_Path P p q i = PathP (\j -> P (ior i j)) (transpFill (\j -> P j) p i) q
Constant : {A : Type} {B : Type} -> (A -> B) -> Type Constant : {A : Type} {B : Type} -> (A -> B) -> Type
Constant f = (y : B) * (x : A) -> Path y (f x) Constant f = (y : B) * (x : A) -> Path y (f x)
@ -1417,7 +1425,7 @@ Constant_weakly : {A : Type} {B : Type} (f : A -> B) -> Constant f -> Weakly f
Constant_weakly f p x y = trans (sym (p.2 x)) (p.2 y) Constant_weakly f p x y = trans (sym (p.2 x)) (p.2 y)
Constant_conditionally : {A : Type} {B : Type} -> (f : A -> B) -> Constant f -> Conditionally f Constant_conditionally : {A : Type} {B : Type} -> (f : A -> B) -> Constant f -> Conditionally f
Constant_conditionally {A} {B} f p = transp (\i -> Conditionally (c_const (inot i))) (const_c p.1) where
Constant_conditionally f p = transp (\i -> Conditionally (c_const (inot i))) (const_c p.1) where
c_const : Path f (\y -> p.1) c_const : Path f (\y -> p.1)
c_const i x = p.2 x (inot i) c_const i x = p.2 x (inot i)
@ -1441,7 +1449,7 @@ S1_connected f = (f'.1, p) where
(Bool_isSet _ _ rr (transp (\i -> P (loop i)) rr)) (Bool_isSet _ _ rr (transp (\i -> P (loop i)) rr))
isProp_isEquiv : {A : Type} {B : Type} {f : A -> B} -> isProp (isEquiv f) isProp_isEquiv : {A : Type} {B : Type} {f : A -> B} -> isProp (isEquiv f)
isProp_isEquiv {f} p q i y =
isProp_isEquiv p q i y =
let let
p2 = (p y).2 p2 = (p y).2
q2 = (q y).2 q2 = (q y).2
@ -1462,10 +1470,10 @@ isProp_EqvSq x y = sigmaPath x1_is_y1 (isProp_isEquiv {P} {Sq P} {y.1} (transp (
equivLemma : {A : Type} {B : Type} {e : Equiv A B} {e' : Equiv A B} equivLemma : {A : Type} {B : Type} {e : Equiv A B} {e' : Equiv A B}
-> Path e.1 e'.1 -> Path e.1 e'.1
-> Path e e' -> Path e e'
equivLemma {A} {B} {e} {e'} p = sigmaPath p (isProp_isEquiv {A} {B} {e'.1} (transp (\i -> isEquiv (p i)) e.2) e'.2)
equivLemma p = sigmaPath p (isProp_isEquiv {A} {B} {e'.1} (transp (\i -> isEquiv (p i)) e.2) e'.2)
isProp_to_Sq_equiv : {P : Type} -> isProp P -> Equiv P (Sq P) isProp_to_Sq_equiv : {P : Type} -> isProp P -> Equiv P (Sq P)
isProp_to_Sq_equiv {P} prop = propExt prop sqProp inc proj where
isProp_to_Sq_equiv prop = propExt prop sqProp inc proj where
proj : Sq P -> P proj : Sq P -> P
proj = Sq_rec prop (\x -> x) proj = Sq_rec prop (\x -> x)
@ -1476,69 +1484,58 @@ Sq_equiv_to_isProp : {P : Type} -> Equiv P (Sq P) -> isProp P
Sq_equiv_to_isProp eqv = transp (\i -> isProp (ua eqv (inot i))) (\x y i -> sq x y i) Sq_equiv_to_isProp eqv = transp (\i -> isProp (ua eqv (inot i))) (\x y i -> sq x y i)
exercise_3_21 : {P : Type} -> Equiv (isProp P) (Equiv P (Sq P)) exercise_3_21 : {P : Type} -> Equiv (isProp P) (Equiv P (Sq P))
exercise_3_21 {P} = propExt (isProp_isProp {P}) (isProp_EqvSq {P}) isProp_to_Sq_equiv Sq_equiv_to_isProp
exercise_3_21 = propExt (isProp_isProp {P}) (isProp_EqvSq {P}) isProp_to_Sq_equiv Sq_equiv_to_isProp
uaret : {A : Type} {B : Type} -> retract {Equiv A B} {Path A B} (ua {A} {B}) (idToEquiv {A} {B}) uaret : {A : Type} {B : Type} -> retract {Equiv A B} {Path A B} (ua {A} {B}) (idToEquiv {A} {B})
uaret eqv = equivLemma (uaBeta eqv) uaret eqv = equivLemma (uaBeta eqv)
f1 : {A : Type} -> (p : (B : Type) * Equiv A B) -> (B : Type) * Path A B
f1 {A} p = (p.1, ua p.2)
f2 : {A : Type} -> (p : (B : Type) * Path A B) -> (B : Type) * Equiv A B
f2 {A} p = (p.1, idToEquiv p.2)
uaretSig : {A : Type} (a : (B : Type) * Equiv A B) -> Path (f2 (f1 a)) a
uaretSig {A} p = sigmaPath (\i -> p.1) (uaret {A} {p.1} p.2)
isContrRetract : {A : Type} {B : Type} isContrRetract : {A : Type} {B : Type}
-> (f : A -> B) -> (g : B -> A) -> (f : A -> B) -> (g : B -> A)
-> (h : retract f g) -> (h : retract f g)
-> isContr B -> isContr A -> isContr B -> isContr A
isContrRetract {A} {B} f g h v = (g b, p) where
isContrRetract f g h v = (g b, p) where
b = v.1 b = v.1
p : (x : A) -> Path (g b) x p : (x : A) -> Path (g b) x
p x i = comp (\i -> A) (\j [ (i = i0) -> g b, (i = i1) -> h x j ]) (inS (g (v.2 (f x) i))) p x i = comp (\i -> A) (\j [ (i = i0) -> g b, (i = i1) -> h x j ]) (inS (g (v.2 (f x) i)))
contrEquivSingl : {A : Type} -> isContr ((B : Type) * Equiv A B)
contrEquivSingl = isContrRetract f1 f2 uaretSig singContr where
f1 : {A : Type} -> (p : (B : Type) * Equiv A B) -> (B : Type) * Path A B
f1 p = (p.1, ua p.2)
f2 : {A : Type} -> (p : (B : Type) * Path A B) -> (B : Type) * Equiv A B
f2 p = (p.1, idToEquiv p.2)
uaretSig : {A : Type} (a : (B : Type) * Equiv A B) -> Path (f2 (f1 a)) a
uaretSig {A} p = sigmaPath (\i -> p.1) (uaret {A} {p.1} p.2)
curry : {A : Type} {B : A -> Type} {C : (x : A) -> B x -> Type} curry : {A : Type} {B : A -> Type} {C : (x : A) -> B x -> Type}
-> Path ((x : A) (y : B x) -> C x y) ((p : (x : A) * B x) -> C p.1 p.2) -> Path ((x : A) (y : B x) -> C x y) ((p : (x : A) * B x) -> C p.1 p.2)
curry {A} {B} {C} = IsoToId (to, from, \f -> refl, \g -> refl) where
curry = IsoToId (to, from, \f -> refl, \g -> refl) where
to : ((x : A) (y : B x) -> C x y) -> (p : (x : A) * B x) -> C p.1 p.2 to : ((x : A) (y : B x) -> C x y) -> (p : (x : A) * B x) -> C p.1 p.2
to f p = f p.1 p.2 to f p = f p.1 p.2
from : ((p : (x : A) * B x) -> C p.1 p.2) -> (x : A) (y : B x) -> C x y from : ((p : (x : A) * B x) -> C p.1 p.2) -> (x : A) (y : B x) -> C x y
from f x y = f (x, y) from f x y = f (x, y)
ContractibleIfInhabited : {A : Type} -> Path (A -> isContr A) (isProp A)
ContractibleIfInhabited {A} = IsoToId (to, from, toFrom, fromTo) where
to : (A -> isContr A) -> isProp A
to cont x y = trans (sym (p.2 x)) (p.2 y) where
p = cont x
contrToProp : {A : Type} -> (A -> isContr A) -> isProp A
contrToProp cont x y = trans (sym (p.2 x)) (p.2 y) where
p = cont x
ContractibleIfInhabited : {A : Type} -> isEquiv contrToProp
ContractibleIfInhabited = (IsoToEquiv (contrToProp, from, toFrom, fromTo)).2 where
from : isProp A -> A -> isContr A from : isProp A -> A -> isContr A
from prop x = (x, prop x) from prop x = (x, prop x)
toFrom : (y : isProp A) -> Path (to (from y)) y
toFrom y = isProp_isProp {A} (to (from y)) y
fromTo : (y : A -> isContr A) -> Path (from (to y)) y
fromTo y i a = isProp_isContr {A} (from (to y) a) (y a) i
toFrom : (y : isProp A) -> Path (contrToProp (from y)) y
toFrom y = isProp_isProp {A} (contrToProp (from y)) y
data cone (A : Type) : Type where
point : cone A
base : A -> cone A
side i : (x : A) -> cone A [ (i = i0) -> point, (i = i1) -> base x ]
ConeA_contr : {A : Type} -> isContr (cone A)
ConeA_contr {A} = (point, contr) where
contr : (y : cone A) -> Path point y
contr = \case
point -> refl
base x -> \i -> side x i
side x i -> \j -> side x (iand i j)
fromTo : (y : A -> isContr A) -> Path (from (contrToProp y)) y
fromTo y i a = isProp_isContr {A} (from (contrToProp y) a) (y a) i
contrSinglEquiv : {B : Type} -> isContr ((A : Type) * Equiv A B) contrSinglEquiv : {B : Type} -> isContr ((A : Type) * Equiv A B)
contrSinglEquiv {B} = (center, contract) where
contrSinglEquiv = (center, contract) where
center : (A : Type) * Equiv A B center : (A : Type) * Equiv A B
center = (B, idEquiv) center = (B, idEquiv)
@ -1564,51 +1561,75 @@ contrSinglEquiv {B} = (center, contract) where
contr : (v : fiber unglueB b) -> Path ctr v contr : (v : fiber unglueB b) -> Path ctr v
contr v j = ( glue {B} {ior (inot i) i} {sys} contr v j = ( glue {B} {ior (inot i) i} {sys}
(\[ (i = i0) -> v.2 j, (i = i1) -> ((w.2.2 b).2 v j).1 ]) (\[ (i = i0) -> v.2 j, (i = i1) -> ((w.2.2 b).2 v j).1 ])
(inS (comp (\i -> B)
(\k [ (i = i0) -> v.2 (iand j k)
, (i = i1) -> ((w.2.2 b).2 v j).2 k
, (j = i0) -> fill (\j -> B) (\k [ (i = i0) -> b, (i = i1) -> (w.2.2 b).1.2 k ]) (inS b) k
, (j = i1) -> v.2 k
])
(inS b)))
, fill (\j -> B) (\k [ (i = i0) -> v.2 (iand j k)
, (i = i1) -> ((w.2.2 b).2 v j).2 k
, (j = i0) -> fill (\j -> B) (\k [ (i = i0) -> b, (i = i1) -> (w.2.2 b).1.2 k ]) (inS b) k
, (j = i1) -> v.2 k
])
(inS b)
(inS (hcomp (\k [ (i = i0) -> v.2 (iand j k)
, (i = i1) -> ((w.2.2 b).2 v j).2 k
, (j = i0) -> fill (\j -> B)
{ior i (inot i)}
(\k [ (i = i0) -> b
, (i = i1) -> (w.2.2 b).1.2 k ])
(inS {B} {ior i (inot i)} b) k
, (j = i1) -> v.2 k
])
(inS b)))
, hfill (\k [ (i = i0) -> v.2 (iand j k)
, (i = i1) -> ((w.2.2 b).2 v j).2 k
, (j = i0) -> fill (\j -> B)
{ior i (inot i)}
(\k [ (i = i0) -> b
, (i = i1) -> (w.2.2 b).1.2 k ])
(inS {B} {ior i (inot i)} b) k
, (j = i1) -> v.2 k
])
(inS b)
) )
in (ctr, contr) in (ctr, contr)
in (GlueB, unglueB, unglueEquiv) in (GlueB, unglueB, unglueEquiv)
uaIdEquiv : {A : Type} -> Path (ua idEquiv) (\i -> A) uaIdEquiv : {A : Type} -> Path (ua idEquiv) (\i -> A)
uaIdEquiv {A} i j = Glue A {ior i (ior (inot j) j)} (\o -> (A, idEquiv))
uaIdEquiv i j = Glue A {ior i (ior (inot j) j)} (\o -> (A, idEquiv))
EquivJ : (P : (X : Type) (Y : Type) -> Equiv X Y -> Type) EquivJ : (P : (X : Type) (Y : Type) -> Equiv X Y -> Type)
-> ((X : Type) -> P X X idEquiv) -> ((X : Type) -> P X X idEquiv)
-> {X : Type} {Y : Type} (E : Equiv X Y) -> {X : Type} {Y : Type} (E : Equiv X Y)
-> P X Y E -> P X Y E
EquivJ P p {X} {Y} E =
EquivJ P p E =
subst {(X : Type) * Equiv X Y} subst {(X : Type) * Equiv X Y}
(\x -> P x.1 Y x.2) (\x -> P x.1 Y x.2)
(\i -> isContr_isProp contrSinglEquiv (Y, idEquiv) (X, E) i) (\i -> isContr_isProp contrSinglEquiv (Y, idEquiv) (X, E) i)
(p Y) (p Y)
EquivJ_domain : {Y : Type} (P : (X : Type) -> Equiv X Y -> Type)
-> P Y idEquiv
-> {X : Type} (E : Equiv X Y)
-> P X E
EquivJ_domain P p E = subst {(X : Type) * Equiv X Y} (\x -> P x.1 x.2) q p where
q : Path {(X : Type) * Equiv X Y} (Y, idEquiv) (X, E)
q = isContr_isProp contrSinglEquiv (Y, idEquiv) (X, E)
EquivJ_range : {X : Type} (P : (Y : Type) -> Equiv X Y -> Type)
-> P X idEquiv
-> {Y : Type} (E : Equiv X Y)
-> P Y E
EquivJ_range P p E = subst {(Y : Type) * Equiv X Y} (\x -> P x.1 x.2) q p
where
q : Path {(Y : Type) * Equiv X Y} (X, idEquiv) (Y, E)
q = isContr_isProp {(Y : Type) * Equiv X Y} (contrEquivSingl {X}) (X, idEquiv) (Y, E)
pathToEquiv : {A : Type} {B : Type} -> Path A B -> Equiv A B pathToEquiv : {A : Type} {B : Type} -> Path A B -> Equiv A B
pathToEquiv {A} {B} = J {Type} {A} (\B p -> Equiv A B) idEquiv
pathToEquiv = J {Type} {A} (\B p -> Equiv A B) idEquiv
univalence : {A : Type} {B : Type} -> Equiv (Path A B) (Equiv A B) univalence : {A : Type} {B : Type} -> Equiv (Path A B) (Equiv A B)
univalence = IsoToEquiv (pathToEquiv, ua, pathToEquiv_ua, ua_pathToEquiv) where univalence = IsoToEquiv (pathToEquiv, ua, pathToEquiv_ua, ua_pathToEquiv) where
pathToEquiv_refl : {A : Type} -> Path (pathToEquiv (refl {Type} {A})) idEquiv pathToEquiv_refl : {A : Type} -> Path (pathToEquiv (refl {Type} {A})) idEquiv
pathToEquiv_refl {A} = JRefl (\B p -> Equiv A B) idEquiv
pathToEquiv_refl = JRefl (\B p -> Equiv A B) idEquiv
ua_pathToEquiv : {A : Type} {B : Type} (p : Path A B) -> Path (ua (pathToEquiv p)) p ua_pathToEquiv : {A : Type} {B : Type} (p : Path A B) -> Path (ua (pathToEquiv p)) p
ua_pathToEquiv {A} {B} p = J {Type} {A} (\B p -> Path (ua {A} {B} (pathToEquiv {A} {B} p)) p) lemma p where
ua_pathToEquiv p = J {Type} {A} (\B p -> Path (ua {A} {B} (pathToEquiv {A} {B} p)) p) lemma p where
lemma : Path (ua (pathToEquiv (\i -> A))) (\i -> A) lemma : Path (ua (pathToEquiv (\i -> A))) (\i -> A)
lemma = transp (\i -> Path (ua (pathToEquiv_refl {A} (inot i))) (\i -> A)) uaIdEquiv lemma = transp (\i -> Path (ua (pathToEquiv_refl {A} (inot i))) (\i -> A)) uaIdEquiv
pathToEquiv_ua : {A : Type} {B : Type} (p : Equiv A B) -> Path (pathToEquiv (ua p)) p pathToEquiv_ua : {A : Type} {B : Type} (p : Equiv A B) -> Path (pathToEquiv (ua p)) p
pathToEquiv_ua {A} {B} p = EquivJ (\A B e -> Path (pathToEquiv (ua e)) e) lemma p where
pathToEquiv_ua p = EquivJ (\A B e -> Path (pathToEquiv (ua e)) e) lemma p where
lemma : (A : Type) -> Path (pathToEquiv (ua idEquiv)) idEquiv lemma : (A : Type) -> Path (pathToEquiv (ua idEquiv)) idEquiv
lemma A = transp (\i -> Path (pathToEquiv (uaIdEquiv {A} (inot i))) idEquiv) pathToEquiv_refl lemma A = transp (\i -> Path (pathToEquiv (uaIdEquiv {A} (inot i))) idEquiv) pathToEquiv_refl
@ -1621,19 +1642,18 @@ total_fibers : {A : Type} {P : A -> Type} {Q : A -> Type}
-> {xv : (a : A) * Q a} -> {xv : (a : A) * Q a}
-> (f : (el : A) -> P el -> Q el) -> (f : (el : A) -> P el -> Q el)
-> Iso (fiber (total f) xv) (fiber (f xv.1) xv.2) -> Iso (fiber (total f) xv) (fiber (f xv.1) xv.2)
total_fibers {A} {P} {Q} {xv} f = (to, fro, toFro {xv}, froTo) where
total_fibers f = (to, fro, toFro {xv}, froTo) where
to : {xv : (a : A) * Q a} -> fiber (total f) xv -> fiber (f xv.1) xv.2 to : {xv : (a : A) * Q a} -> fiber (total f) xv -> fiber (f xv.1) xv.2
to peq = J {(a : A) * Q a} {_} (\xv eq -> fiber (f xv.1) xv.2) (peq.1.2, refl) (sym peq.2)
to peq = J {(a : A) * Q a} (\xv eq -> fiber (f xv.1) xv.2) (peq.1.2, refl) (sym peq.2)
fro : {xv : (a : A) * Q a} -> fiber (f xv.1) xv.2 -> fiber (total f) xv fro : {xv : (a : A) * Q a} -> fiber (f xv.1) xv.2 -> fiber (total f) xv
fro peq = ((xv.1, peq.1), \i -> (_, peq.2 i)) fro peq = ((xv.1, peq.1), \i -> (_, peq.2 i))
toFro : {xv : (a : A) * Q a} -> (y : fiber (f xv.1) xv.2) -> Path (to (fro y)) y toFro : {xv : (a : A) * Q a} -> (y : fiber (f xv.1) xv.2) -> Path (to (fro y)) y
toFro {xv} peq =
J {Q xv.1} {f xv.1 p}
toFro peq =
J {_} {f xv.1 p}
(\xv1 eq1 -> Path (to {(xv.1, xv1)} (fro (p, sym eq1))) (p, sym eq1)) (\xv1 eq1 -> Path (to {(xv.1, xv1)} (fro (p, sym eq1))) (p, sym eq1))
(JRefl {(a : A) * Q a} {(xv.1, f xv.1 peq.1)}
(\xv1 eq1 -> fiber (f xv1.1) xv1.2) (p, refl))
(JRefl {(a : A) * Q a} {(_, _)} (\xv1 eq1 -> fiber (f xv1.1) xv1.2) (p, refl))
(sym eq) (sym eq)
where p : P xv.1 where p : P xv.1
p = peq.1 p = peq.1
@ -1642,9 +1662,9 @@ total_fibers {A} {P} {Q} {xv} f = (to, fro, toFro {xv}, froTo) where
eq = peq.2 eq = peq.2
froTo : {xv : (a : A) * Q a} -> (y : fiber (total f) xv) -> Path (fro {xv} (to {xv} y)) y froTo : {xv : (a : A) * Q a} -> (y : fiber (total f) xv) -> Path (fro {xv} (to {xv} y)) y
froTo {xv} apeq =
froTo apeq =
J {(a : A) * Q a} {total f (a, p)} J {(a : A) * Q a} {total f (a, p)}
(\xv1 eq1 -> Path (fro {_} (to {_} ((a, p), sym eq1))) ((a, p), sym eq1))
(\xv1 eq1 -> Path (fro (to ((a, p), sym eq1))) ((a, p), sym eq1))
(ap fro (JRefl {(a : A) * Q a} {(a, _)} (ap fro (JRefl {(a : A) * Q a} {(a, _)}
(\xv1 eq1 -> fiber (f xv1.1) xv1.2) (p, refl))) (\xv1 eq1 -> fiber (f xv1.1) xv1.2) (p, refl)))
(sym eq) (sym eq)
@ -1669,22 +1689,162 @@ totalEquiv : {A : Type} {P : A -> Type} {Q : A -> Type}
-> (f : (el : A) -> P el -> Q el) -> (f : (el : A) -> P el -> Q el)
-> ((x : A) -> isEquiv (f x)) -> ((x : A) -> isEquiv (f x))
-> isEquiv (total f) -> isEquiv (total f)
totalEquiv f iseqv xv = isContrRetract {fiber (total f) xv} {fiber (f xv.1) xv.2} eqv.1 eqv.2.1 eqv.2.2.2 (iseqv xv.1 xv.2) where
totalEquiv f iseqv xv = isContrRetract eqv.1 eqv.2.1 eqv.2.2.2 (iseqv xv.1 xv.2) where
eqv : Iso (fiber (total f) xv) (fiber (f xv.1) xv.2) eqv : Iso (fiber (total f) xv) (fiber (f xv.1) xv.2)
eqv = total_fibers f eqv = total_fibers f
contrIsEquiv : {A : Type} {B : Type} -> isContr A -> isContr B contrIsEquiv : {A : Type} {B : Type} -> isContr A -> isContr B
-> (f : A -> B) -> isEquiv f -> (f : A -> B) -> isEquiv f
contrIsEquiv cA cB f y = ((cA.1, isContr_isProp cB _ _), \v -> sigmaPath (isContr_isProp cA _ _) (isProp_isSet (isContr_isProp cB) _ _ _ v.2))
contrIsEquiv cA cB f y =
( (cA.1, isContr_isProp cB _ _)
, \v -> sigmaPath (isContr_isProp cA _ _)
(isProp_isSet (isContr_isProp cB) _ _ _ v.2)
)
theorem722 : {A : Type} {R : A -> A -> Type} theorem722 : {A : Type} {R : A -> A -> Type}
-> ((x : A) (y : A) -> isProp (R x y)) -> ((x : A) (y : A) -> isProp (R x y))
-> ({x : A} -> R x x) -> ({x : A} -> R x x)
-> (f : (x : A) (y : A) -> R x y -> Path x y) -> (f : (x : A) (y : A) -> R x y -> Path x y)
-> {x : A} {y : A} -> isEquiv {R x y} {Path x y} (f x y) -> {x : A} {y : A} -> isEquiv {R x y} {Path x y} (f x y)
theorem722 {A} {R} prop rho toId {x} {y} = fiberEquiv {A} {R x} {Path x} (toId x) (totalEquiv x) y where
theorem722 prop rho toId {x} {y} = fiberEquiv (toId x) (totalEquiv x) y where
rContr : (x : A) -> isContr ((y : A) * R x y) rContr : (x : A) -> isContr ((y : A) * R x y)
rContr x = ((x, rho {x}), \y -> sigmaPath (toId _ _ y.2) (prop _ _ _ y.2)) rContr x = ((x, rho {x}), \y -> sigmaPath (toId _ _ y.2) (prop _ _ _ y.2))
totalEquiv : (x : A) -> isEquiv (total {A} {R x} {Path x} (toId x))
totalEquiv x = contrIsEquiv (rContr x) singContr (total {A} {R x} {Path x} (toId x))
totalEquiv : (x : A) -> isEquiv (total (toId x))
totalEquiv x = contrIsEquiv (rContr x) singContr (total (toId x))
lemma492 : {A : Type} {B : Type} {X : Type}
-> (e : Equiv A B)
-> isEquiv {X -> A} {X -> B} (\f x -> e.1 (f x))
lemma492 =
EquivJ (\A B e -> isEquiv {X -> _} {X -> B} (\f x -> e.1 (f x)))
(\R -> (idEquiv {X -> R}).2)
twoOutOfThree_1 : {A : Type} {B : Type} {C : Type} {f : A -> B} {g : B -> C}
-> isEquiv f
-> isEquiv g
-> isEquiv (\x -> g (f x))
twoOutOfThree_1 feq geq =
EquivJ_range (\_ g -> isEquiv (\x -> g.1 (f x))) feq (g, geq)
twoOutOfThree_2 : {A : Type} {B : Type} {C : Type} {f : A -> B} {g : B -> C}
-> isEquiv f
-> isEquiv (\x -> g (f x))
-> isEquiv g
twoOutOfThree_2 feq gofeq =
EquivJ_domain (\_ f -> isEquiv (\x -> g (f.1 x)) -> isEquiv g) id (f, feq) gofeq
twoOutOfThree_3 : {A : Type} {B : Type} {C : Type} {f : A -> B} {g : B -> C}
-> isEquiv g
-> isEquiv (\x -> g (f x))
-> isEquiv f
twoOutOfThree_3 geq gofeq =
EquivJ_range (\_ g -> isEquiv (\x -> g.1 (f x)) -> isEquiv f) id (g, geq) gofeq
equivalence_isEmbedding : {A : Type} {B : Type}
-> {f : A -> B}
-> isEquiv f
-> {x : A} {y : A}
-> isEquiv (ap f {x} {y})
equivalence_isEmbedding feqv {x} {y} =
EquivJ (\A B eq -> (x : A) (y : A) -> isEquiv {_} {Path (eq.1 x) (eq.1 y)} (ap eq.1)) (\X x y -> (idEquiv {Path _ _}).2) (f, feqv) x y
isOfHLevel : Type -> Nat -> Type
isOfHLevel A = \case
zero -> (a : A) (b : A) -> Path a b
succ n -> (a : A) (b : A) -> isOfHLevel (Path a b) n
Sphere : Nat -> Type
Sphere = \case
zero -> Bottom
succ n -> Susp (Sphere n)
sphereFull : {A : Type} {n : Nat} (f : Sphere n -> A) -> Type
sphereFull f = (top : A) * (x : Sphere n) -> Path top (f x)
spheresFull : {n : Nat} -> Type -> Type
spheresFull A = (f : Sphere n -> A) -> sphereFull f
spheresFull_hLevel : {A : Type} (n : Nat) -> spheresFull {succ n} A -> isOfHLevel A n
spheresFull_hLevel =
\case
zero -> \full a b ->
let f : Sphere (succ zero) -> A
f = \case
north -> a
south -> b
merid u i -> absurd u
p = (full f).2
in trans (sym (p north)) (p south)
succ n -> \h x y -> spheresFull_hLevel n (helper h x y)
where
helper : {n : Nat} -> spheresFull {succ (succ n)} A -> (x : A) (y : A) -> spheresFull {succ n} (Path x y)
helper h x y f = (trans p q, r (transFiller p q)) where
f' : Susp (Sphere (succ n)) -> A
f' = \case
north -> x
south -> y
merid u i -> f u i
p : Path x (h f').1
p i = (h f').2 north (inot i)
q : Path (h f').1 y
q = (h f').2 south
r : (fillpq : PathP (\i -> Path x (q i)) p (trans p q))
(s : Sphere (succ n))
-> Path (fillpq i1) (f s)
r fillpq s i j = hcomp (\k [ (i = i0) -> fillpq k j
, (i = i1) -> (h f').2 (merid s j) k
, (j = i0) -> p (iand (inot k) i)
, (j = i1) -> q k ])
(inS (p (ior i j)))
isOfHLevel_hasSpheres : {A : Type} (n : Nat) -> isOfHLevel A n -> spheresFull {succ n} A
isOfHLevel_hasSpheres =
\case
zero -> \prop f -> (f north, \x -> prop (f north) (f x))
succ n -> \h -> helper {n} (\x y -> isOfHLevel_hasSpheres n (h x y))
where
helper : {n : Nat} -> ((a : A) (b : A) -> spheresFull {succ n} (Path a b))
-> spheresFull {succ (succ n)} A
helper {n} h f = (f north, r) where
north' = north {Sphere (succ n)}
south' = south {Sphere (succ n)}
merid' = merid {Sphere (succ n)}
r : (x : Sphere (succ (succ n))) -> Path (f north) (f x)
r = \case
north -> refl
south -> (h (f north') (f south') (\x i -> f (merid x i))).1
merid x i -> \j ->
hcomp (\k [ (i = i0) -> f north'
, (i = i1) -> (h (f north') (f south') (\x i -> f (merid' x i))).2 x (inot k) j
, (j = i0) -> f north'
, (j = i1) -> f (merid' x i) ])
(inS (f (merid' x (iand i j))))
data nTrunc (A : Type) (n : Nat) : Type where
incn : A -> nTrunc A n
hub : (f : Sphere (succ n) -> nTrunc A n) -> nTrunc A n
spoke i : (f : Sphere (succ n) -> nTrunc A n) (x : Sphere (succ n)) -> nTrunc A n [ (i = i0) -> hub f, (i = i1) -> f x ]
nTrunc_isOfHLevel : {A : Type} {n : Nat} -> isOfHLevel (nTrunc A n) n
nTrunc_isOfHLevel = spheresFull_hLevel {nTrunc A n} n (\f -> (hub f, \x i -> spoke f x i))
nTrunc_rec : {A : Type} {n : Nat} {B : Type}
-> isOfHLevel B n
-> (A -> B)
-> nTrunc A n -> B
nTrunc_rec bofhl f = go (isOfHLevel_hasSpheres n bofhl) where
work : (p : spheresFull {succ n} B) -> nTrunc A n -> B
work p = \case
hub sph -> (p (\x -> work p (sph x))).1
incn x -> f x
go : (p : spheresFull {succ n} B) -> nTrunc A n -> B
go p = \case
incn x -> f x
hub sph -> (p (\x -> work p (sph x))).1
spoke sph cell i -> (p (\x -> work p (sph x))).2 cell i

+ 4
- 2
src/Debug.hs View File

@ -1,18 +1,20 @@
{-# LANGUAGE CPP #-} {-# LANGUAGE CPP #-}
{-# LANGUAGE ConstraintKinds #-} {-# LANGUAGE ConstraintKinds #-}
{-# LANGUAGE KindSignatures #-} {-# LANGUAGE KindSignatures #-}
{-# LANGUAGE BangPatterns #-}
#undef RELEASE
module Debug where module Debug where
import qualified Debug.Trace as D import qualified Debug.Trace as D
#if defined(RELEASE) #if defined(RELEASE)
import GHC.Exts import GHC.Exts
#else
#endif
import GHC.Stack import GHC.Stack
import Prettyprinter import Prettyprinter
import qualified Data.Text.Lazy as T import qualified Data.Text.Lazy as T
import Data.Text.Prettyprint.Doc.Render.Text (renderLazy) import Data.Text.Prettyprint.Doc.Render.Text (renderLazy)
#endif
traceDoc :: Doc a -> b -> b traceDoc :: Doc a -> b -> b


+ 14
- 26
src/Elab.hs View File

@ -33,7 +33,6 @@ import Prettyprinter
import Syntax.Pretty import Syntax.Pretty
import Syntax import Syntax
import Debug (traceM, traceDocM)
infer :: P.Expr -> ElabM (Term, NFType) infer :: P.Expr -> ElabM (Term, NFType)
infer (P.Span ex a b) = withSpan a b $ infer ex infer (P.Span ex a b) = withSpan a b $ infer ex
@ -146,47 +145,36 @@ check (P.Let items body) ty = do
check (P.LamSystem bs) ty = do check (P.LamSystem bs) ty = do
(extent, dom) <- isPartialType ty (extent, dom) <- isPartialType ty
eqns <- for (zip [(0 :: Int)..] bs) $ \(n, (formula, rhs)) -> do
env <- ask
eqns <- for bs \(formula, rhs) -> do
(phi, fv) <- checkFormula formula (phi, fv) <- checkFormula formula
env <- ask
n <- newName n <- newName
rhses <- for (truthAssignments phi (getEnv env)) $ \e -> do rhses <- for (truthAssignments phi (getEnv env)) $ \e -> do
let env' = env{ getEnv = e } let env' = env{ getEnv = e }
local (const env') $ local (const env') $
(Nothing,) <$> check rhs (substitute (snd <$> Map.restrictKeys e fv) (dom (VVar n)))
check rhs (substitute (snd <$> Map.restrictKeys e fv) (dom (VVar n)))
pure (n, (phi, head rhses)) pure (n, (phi, head rhses))
unify extent (foldl ior VI0 (map (fst . snd) eqns)) unify extent (foldl ior VI0 (map (fst . snd) eqns))
for_ eqns $ \(n, (formula, (binding, rhs))) -> do
let
k = case binding of
Just v -> assume v (VEqStrict VI formula VI1) . const
Nothing -> id
k $ for_ eqns $ \(n', (formula', (binding, rhs'))) -> do
for_ eqns \(n, (formula, rhs)) -> do
for_ eqns $ \(n', (formula', rhs')) -> do
let let
k = case binding of
Just v -> assume v (VEqStrict VI formula VI1) . const
Nothing -> id
truth = possible mempty (iand formula formula') truth = possible mempty (iand formula formula')
add [] = id
add ((~(HVar x), True):xs) = redefine x VI VI1 . add xs
add ((~(HVar x), False):xs) = redefine x VI VI0 . add xs
k $ when ((n /= n') && fst truth) . add (Map.toList (snd truth)) $ do
vl <- eval rhs
vl' <- eval rhs'
when ((n /= n') && fst truth) . for_ (truthAssignments (iand formula formula') (getEnv env)) $ \e -> do
let env' = env { getEnv = e }
vl = eval' env' rhs
vl' = eval' env' rhs'
unify vl vl' unify vl vl'
`withNote` vsep [ pretty "These two cases must agree because they are both possible:" `withNote` vsep [ pretty "These two cases must agree because they are both possible:"
, indent 2 $ pretty '*' <+> prettyTm (quote formula) <+> operator (pretty "=>") <+> prettyTm rhs
, indent 2 $ pretty '*' <+> prettyTm (quote formula') <+> operator (pretty "=>") <+> prettyTm rhs'
, indent 2 $ pretty '*' <+> prettyTm (quote formula) <+> operator (pretty "=>") <+> pretty vl
, indent 2 $ pretty '*' <+> prettyTm (quote formula') <+> operator (pretty "=>") <+> pretty (zonk vl')
] ]
`withNote` (pretty "Consider this face, where both are true:" <+> showFace False (snd truth)) `withNote` (pretty "Consider this face, where both are true:" <+> showFace False (snd truth))
name <- newName name <- newName
let
mkB name (Just v, b) = App P.Ex (Lam P.Ex v b) (Ref name)
mkB _ (Nothing, b) = b
pure (Lam P.Ex name (System (Map.fromList (map (\(_, (x, y)) -> (quote x, mkB name y)) eqns))))
pure (Lam P.Ex name (System (Map.fromList (map (\(_, (x, y)) -> (quote x, y)) eqns))))
check (P.LamCase pats) ty = check (P.LamCase pats) ty =
do do
@ -510,7 +498,7 @@ checkStatement (P.ReplNf e) k = do
checkStatement (P.ReplTy e) k = do checkStatement (P.ReplTy e) k = do
(t, ty) <- infer e (t, ty) <- infer e
h <- asks commHook h <- asks commHook
liftIO (h (prettyTm t <+> colon <+> align (prettyVl ty)))
liftIO (h (prettyTm t <+> colon <+> align (prettyVl (zonk ty))))
k k
checkStatement (P.Data name tele retk constrs) k = checkStatement (P.Data name tele retk constrs) k =


+ 95
- 69
src/Elab/Eval.hs View File

@ -36,6 +36,8 @@ import Syntax
import System.IO.Unsafe ( unsafePerformIO ) import System.IO.Unsafe ( unsafePerformIO )
import {-# SOURCE #-} Elab.WiredIn import {-# SOURCE #-} Elab.WiredIn
import Debug (traceM, traceDocM)
import Prettyprinter (pretty, (<+>))
eval :: HasCallStack => Term -> ElabM Value eval :: HasCallStack => Term -> ElabM Value
eval t = asks (flip eval' t) eval t = asks (flip eval' t)
@ -214,37 +216,51 @@ evalCase env rng (val@(VNe (HPCon _ _ x) sp)) ((Con x', _, k):xs)
| x == x' = foldl applProj (eval' env k) sp | x == x' = foldl applProj (eval' env k) sp
| otherwise = evalCase env rng val xs | otherwise = evalCase env rng val xs
evalCase _ _ (VVar ((== trueCaseSentinel) -> True)) _ = VI1
evalCase env rng sc xs = VCase (getEnv env) (fun rng) sc xs evalCase env rng sc xs = VCase (getEnv env) (fun rng) sc xs
-- This is a great big HACK; When we see a system [ case x of ... => p
-- ], we somehow need to make the 'case x of ...' become VI1. The way we
-- do this is by substituting x/trueCaseSentinel in truthAssignments,
-- and then making case trueCaseSentinel of ... => VI1 always.
trueCaseSentinel :: Name
trueCaseSentinel = Bound (T.pack "sentinel for true cases") (-1000)
evalFix' :: HasCallStack => ElabEnv -> Name -> NFType -> Term -> Value evalFix' :: HasCallStack => ElabEnv -> Name -> NFType -> Term -> Value
evalFix' env name nft term = fix $ \val -> eval' env{ getEnv = Map.insert name (nft, GluedVl (HVar name) mempty val) (getEnv env) } term evalFix' env name nft term = fix $ \val -> eval' env{ getEnv = Map.insert name (nft, GluedVl (HVar name) mempty val) (getEnv env) } term
evalFix :: HasCallStack => Name -> NFType -> Term -> ElabM Value evalFix :: HasCallStack => Name -> NFType -> Term -> ElabM Value
evalFix name nft term = do evalFix name nft term = do
t <- ask t <- ask
pure (evalFix' t name nft term)
pure (evalFix' t name (GluedVl (HVar name) mempty nft) term)
data NotEqual = NotEqual Value Value data NotEqual = NotEqual Value Value
deriving (Show, Typeable, Exception) deriving (Show, Typeable, Exception)
unify' :: HasCallStack => Value -> Value -> ElabM ()
unify' (GluedVl h sp a) (GluedVl h' sp' b)
unify' :: HasCallStack => Bool -> Value -> Value -> ElabM ()
unify' cs topa@(GluedVl h sp a) topb@(GluedVl h' sp' b)
| h == h', length sp == length sp' = | h == h', length sp == length sp' =
traverse_ (uncurry unify'Spine) (Seq.zip sp sp')
`catchElab` \(_ :: SomeException) -> unify' a b
unify' topa topb = join $ go <$> forceIO topa <*> forceIO topb where
traverse_ (uncurry (unify'Spine cs topa topb)) (Seq.zip sp sp')
`catchElab` \(_ :: SomeException) -> unify' cs a b
unify' canSwitch topa topb = join $ go <$> forceIO topa <*> forceIO topb where
go (VNe (HMeta mv) sp) rhs = solveMeta mv sp rhs go (VNe (HMeta mv) sp) rhs = solveMeta mv sp rhs
go rhs (VNe (HMeta mv) sp) = solveMeta mv sp rhs go rhs (VNe (HMeta mv) sp) = solveMeta mv sp rhs
go (VNe (HPCon _ _ x) sp) (VNe (HPCon _ _ y) sp')
| x == y = traverse_ (uncurry unify'Spine) (Seq.zip sp sp')
go topa@(VNe (HPCon _ _ x) sp) topb@(VNe (HPCon _ _ y) sp')
| x == y = traverse_ (uncurry (unify'Spine canSwitch topa topb)) (Seq.zip sp sp')
go (VNe (HPCon s _ _) _) rhs | Just v <- trivialSystem s = go v rhs go (VNe (HPCon s _ _) _) rhs | Just v <- trivialSystem s = go v rhs
go lhs (VNe (HPCon s _ _) _) | Just v <- trivialSystem s = go lhs v go lhs (VNe (HPCon s _ _) _) | Just v <- trivialSystem s = go lhs v
go (VCase e _ _ b) (VCase e' _ _ b') = do go (VCase e _ _ b) (VCase e' _ _ b') = do
env <- ask env <- ask
let let
go (_, _, a) (_, _, b) = unify' (eval' env{getEnv=moreDefinedFrom e e' <$> e} a) (eval' env{getEnv=moreDefinedFrom e e' <$> e'} b)
go (_, _, a) (_, _, b)
| a == b = pure ()
| otherwise = unify' canSwitch (eval' env{getEnv=moreDefinedFrom e e' <$> e} a) (eval' env{getEnv=moreDefinedFrom e e' <$> e'} b)
zipWithM_ go (sortOn (\(x, _, _) -> x) b) (sortOn (\(x, _, _) -> x) b') zipWithM_ go (sortOn (\(x, _, _) -> x) b) (sortOn (\(x, _, _) -> x) b')
go (VCase e _ _ b) y = do go (VCase e _ _ b) y = do
@ -253,33 +269,33 @@ unify' topa topb = join $ go <$> forceIO topa <*> forceIO topb where
go (_, n, a') = do go (_, n, a') = do
ns <- replicateM n (VVar <$> newName) ns <- replicateM n (VVar <$> newName)
let a = foldl (vApp Ex) (eval' env{getEnv=e} a') ns let a = foldl (vApp Ex) (eval' env{getEnv=e} a') ns
unify' a y
unify' canSwitch a y
traverse_ go b traverse_ go b
go (VNe x a) (VNe x' a')
go topa@(VNe x a) topb@(VNe x' a')
| x == x', length a == length a' = | x == x', length a == length a' =
traverse_ (uncurry unify'Spine) (Seq.zip a a')
traverse_ (uncurry (unify'Spine canSwitch topa topb)) (Seq.zip a a')
go (VLam p (Closure n k)) vl = do go (VLam p (Closure n k)) vl = do
t <- VVar <$> newName' n t <- VVar <$> newName' n
unify' (k t) (vApp p vl t)
unify' canSwitch (k t) (vApp p vl t)
go vl (VLam p (Closure n k)) = do go vl (VLam p (Closure n k)) = do
t <- VVar <$> newName' n t <- VVar <$> newName' n
unify' (vApp p vl t) (k t)
unify' canSwitch (vApp p vl t) (k t)
go (VPair a b) vl = unify' a (vProj1 vl) *> unify' b (vProj2 vl)
go vl (VPair a b) = unify' (vProj1 vl) a *> unify' (vProj2 vl) b
go (VPair a b) vl = unify' canSwitch a (vProj1 vl) *> unify' canSwitch b (vProj2 vl)
go vl (VPair a b) = unify' canSwitch (vProj1 vl) a *> unify' canSwitch (vProj2 vl) b
go (VPi p d (Closure n k)) (VPi p' d' (Closure _ k')) | p == p' = do go (VPi p d (Closure n k)) (VPi p' d' (Closure _ k')) | p == p' = do
t <- VVar <$> newName' n t <- VVar <$> newName' n
unify' d d'
unify' (k t) (k' t)
unify' canSwitch d d'
unify' canSwitch (k t) (k' t)
go (VSigma d (Closure n k)) (VSigma d' (Closure _ k')) = do go (VSigma d (Closure n k)) (VSigma d' (Closure _ k')) = do
t <- VVar <$> newName' n t <- VVar <$> newName' n
unify' d d'
unify' (k t) (k' t)
unify' canSwitch d d'
unify' canSwitch (k t) (k' t)
go VType VType = pure () go VType VType = pure ()
go VTypeω VTypeω = pure () go VTypeω VTypeω = pure ()
@ -287,56 +303,63 @@ unify' topa topb = join $ go <$> forceIO topa <*> forceIO topb where
go VI VI = pure () go VI VI = pure ()
go (VPath l x y) (VPath l' x' y') = do go (VPath l x y) (VPath l' x' y') = do
unify' l l'
unify' x x'
unify' y y'
unify' canSwitch l l'
unify' canSwitch x x'
unify' canSwitch y y'
go (VLine l x y p) p' = do go (VLine l x y p) p' = do
n <- VVar <$> newName' (Bound (T.singleton 'i') (- 1)) n <- VVar <$> newName' (Bound (T.singleton 'i') (- 1))
unify' (p @@ n) (ielim l x y p' n)
unify' canSwitch (p @@ n) (ielim l x y p' n)
go p' (VLine l x y p) = do go p' (VLine l x y p) = do
n <- VVar <$> newName n <- VVar <$> newName
unify' (ielim l x y p' n) (p @@ n)
unify' canSwitch (ielim l x y p' n) (p @@ n)
go (VPartial phi r) (VPartial phi' r') = unify' phi phi' *> unify' r r'
go (VPartialP phi r) (VPartialP phi' r') = unify' phi phi' *> unify' r r'
go (VPartial phi r) (VPartial phi' r') = unify' canSwitch phi phi' *> unify' canSwitch r r'
go (VPartialP phi r) (VPartialP phi' r') = unify' canSwitch phi phi' *> unify' canSwitch r r'
go (VSub a phi u) (VSub a' phi' u') = traverse_ (uncurry unify') [(a, a'), (phi, phi'), (u, u')]
go (VInc a phi u) (VInc a' phi' u') = traverse_ (uncurry unify') [(a, a'), (phi, phi'), (u, u')]
go (VSub a phi u) (VSub a' phi' u') = traverse_ (uncurry (unify' canSwitch)) [(a, a'), (phi, phi'), (u, u')]
go (VInc a phi u) (VInc a' phi' u') = traverse_ (uncurry (unify' canSwitch)) [(a, a'), (phi, phi'), (u, u')]
go (VComp a phi u a0) (VComp a' phi' u' a0') = go (VComp a phi u a0) (VComp a' phi' u' a0') =
traverse_ (uncurry unify') [(a, a'), (phi, phi'), (u, u'), (a0, a0')]
traverse_ (uncurry (unify' canSwitch)) [(a, a'), (phi, phi'), (u, u'), (a0, a0')]
go (VHComp a phi u a0) (VHComp a' phi' u' a0') = go (VHComp a phi u a0) (VHComp a' phi' u' a0') =
traverse_ (uncurry unify') [(a, a'), (phi, phi'), (u, u'), (a0, a0')]
traverse_ (uncurry (unify' canSwitch)) [(a, a'), (phi, phi'), (u, u'), (a0, a0')]
go (VGlueTy _ (force -> VI1) u _0) rhs = unify' (u @@ VReflStrict VI VI1) rhs
go lhs (VGlueTy _ (force -> VI1) u _0) = unify' lhs (u @@ VReflStrict VI VI1)
go (VGlueTy _ (force -> VI1) u _0) rhs = unify' canSwitch (u @@ VReflStrict VI VI1) rhs
go lhs (VGlueTy _ (force -> VI1) u _0) = unify' canSwitch lhs (u @@ VReflStrict VI VI1)
go (VGlueTy a phi u a0) (VGlueTy a' phi' u' a0') = go (VGlueTy a phi u a0) (VGlueTy a' phi' u' a0') =
traverse_ (uncurry unify') [(a, a'), (phi, phi'), (u, u'), (a0, a0')]
traverse_ (uncurry (unify' canSwitch)) [(a, a'), (phi, phi'), (u, u'), (a0, a0')]
go (VGlue a phi u a0 t x) (VGlue a' phi' u' a0' t' x') = go (VGlue a phi u a0 t x) (VGlue a' phi' u' a0' t' x') =
traverse_ (uncurry unify') [(a, a'), (phi, phi'), (u, u'), (a0, a0'), (t, t'), (x, x')]
traverse_ (uncurry (unify' canSwitch)) [(a, a'), (phi, phi'), (u, u'), (a0, a0'), (t, t'), (x, x')]
go (VUnglue a phi u a0 x) (VUnglue a' phi' u' a0' x') = go (VUnglue a phi u a0 x) (VUnglue a' phi' u' a0' x') =
traverse_ (uncurry unify') [(a, a'), (phi, phi'), (u, u'), (a0, a0'), (x, x')]
traverse_ (uncurry (unify' canSwitch)) [(a, a'), (phi, phi'), (u, u'), (a0, a0'), (x, x')]
go (VSystem sys) rhs = goSystem unify' sys rhs
go rhs (VSystem sys) = goSystem (flip unify') sys rhs
go (VSystem sys) rhs = goSystem (unify' canSwitch) sys rhs
go rhs (VSystem sys) = goSystem (flip (unify' canSwitch)) sys rhs
go (VEqStrict a x y) (VEqStrict a' x' y') = traverse_ (uncurry unify') [(a, a'), (x, x'), (y, y')]
go (VReflStrict a x) (VReflStrict a' x') = traverse_ (uncurry unify') [(a, a'), (x, x')]
go (VEqStrict a x y) (VEqStrict a' x' y') = traverse_ (uncurry (unify' canSwitch)) [(a, a'), (x, x'), (y, y')]
go (VReflStrict a x) (VReflStrict a' x') = traverse_ (uncurry (unify' canSwitch)) [(a, a'), (x, x')]
go _ VReflStrict{} = pure () go _ VReflStrict{} = pure ()
go VReflStrict{} _ = pure () go VReflStrict{} _ = pure ()
go x y
| x == y = pure ()
| otherwise =
case (toDnf x, toDnf y) of
(Just xs, Just ys) -> unify'Formula xs ys
_ -> fail
go (VINot x) (VINot y) = unify' canSwitch x y
go x y =
case (toDnf x, toDnf y) of
(Just xs, Just ys) -> unify'Formula xs ys
_ ->
if canSwitch
then goDumb x y
else fail
goDumb (VIOr a b) (VIOr a' b') = unify' canSwitch a a' *> goDumb b b'
goDumb (VIAnd a b) (VIAnd a' b') = unify' canSwitch a a' *> goDumb b b'
goDumb x y = switch $ unify' False x y
goSystem :: (Value -> Value -> ElabM ()) -> Map.Map Value Value -> Value -> ElabM () goSystem :: (Value -> Value -> ElabM ()) -> Map.Map Value Value -> Value -> ElabM ()
goSystem k sys rhs = do goSystem k sys rhs = do
@ -348,7 +371,6 @@ unify' topa topb = join $ go <$> forceIO topa <*> forceIO topb where
k (eval' env{getEnv = e} i_q) (eval' env{getEnv = e} rhs_q) k (eval' env{getEnv = e} i_q) (eval' env{getEnv = e} rhs_q)
fail = throwElab $ NotEqual topa topb fail = throwElab $ NotEqual topa topb
unify'Formula x y unify'Formula x y
| compareDNFs x y = pure () | compareDNFs x y = pure ()
| otherwise = fail | otherwise = fail
@ -371,31 +393,32 @@ trivialSystem = go . force where
go VSystem{} = Nothing go VSystem{} = Nothing
go x = Just x go x = Just x
unify'Spine :: Projection -> Projection -> ElabM ()
unify'Spine (PApp a v) (PApp a' v')
| a == a' = unify' v v'
unify'Spine :: Bool -> Value -> Value -> Projection -> Projection -> ElabM ()
unify'Spine cs _ _ (PApp a v) (PApp a' v')
| a == a' = unify' cs v v'
unify'Spine PProj1 PProj1 = pure ()
unify'Spine PProj2 PProj2 = pure ()
unify'Spine _ _ _ PProj1 PProj1 = pure ()
unify'Spine _ _ _ PProj2 PProj2 = pure ()
unify'Spine (PIElim _ _ _ i) (PIElim _ _ _ j) = unify' i j
unify'Spine (POuc a phi u) (POuc a' phi' u') =
traverse_ (uncurry unify') [(a, a'), (phi, phi'), (u, u')]
unify'Spine cs _ _ (PIElim _ _ _ i) (PIElim _ _ _ j) = unify' cs i j
unify'Spine cs _ _ (POuc a phi u) (POuc a' phi' u') =
traverse_ (uncurry (unify' cs)) [(a, a'), (phi, phi'), (u, u')]
unify'Spine (PK a x p pr) (PK a' x' p' pr') =
traverse_ (uncurry unify') [(a, a'), (x, x'), (p, p'), (pr, pr')]
unify'Spine cs _ _ (PK a x p pr) (PK a' x' p' pr') =
traverse_ (uncurry (unify' cs)) [(a, a'), (x, x'), (p, p'), (pr, pr')]
unify'Spine (PJ a x p pr y) (PJ a' x' p' pr' y') =
traverse_ (uncurry unify') [(a, a'), (x, x'), (p, p'), (pr, pr'), (y, y')]
unify'Spine cs _ _ (PJ a x p pr y) (PJ a' x' p' pr' y') =
traverse_ (uncurry (unify' cs)) [(a, a'), (x, x'), (p, p'), (pr, pr'), (y, y')]
unify'Spine _ _ = throwElab (NotEqual undefined undefined)
unify'Spine _ x y _ _ = throwElab (NotEqual x y)
unify :: HasCallStack => Value -> Value -> ElabM () unify :: HasCallStack => Value -> Value -> ElabM ()
unify (GluedVl h sp a) (GluedVl h' sp' b)
| h == h', length sp == length sp' =
traverse_ (uncurry unify'Spine) (Seq.zip sp sp')
`catchElab` \(_ :: SomeException) -> unify' a b
unify a b = unify' a b `catchElab` \(_ :: SomeException) -> liftIO $ throwIO (NotEqual a b)
unify x y = shallowly $ go x y where
go topa@(GluedVl h sp a) topb@(GluedVl h' sp' b)
| h == h', length sp == length sp' =
traverse_ (uncurry (unify'Spine True topa topb)) (Seq.zip sp sp')
`catchElab` \(_ :: SomeException) -> unify' True a b
go a b = unify' True a b `catchElab` \(_ :: SomeException) -> liftIO $ throwIO (NotEqual a b)
isConvertibleTo :: Value -> Value -> ElabM (Term -> Term) isConvertibleTo :: Value -> Value -> ElabM (Term -> Term)
isConvertibleTo a b = isConvertibleTo (force a) (force b) where isConvertibleTo a b = isConvertibleTo (force a) (force b) where
@ -420,7 +443,7 @@ isConvertibleTo a b = isConvertibleTo (force a) (force b) where
pure (\f -> Lam Ex nm (wp (IElim (quote a) (quote x) (quote y) f (Ref nm)))) pure (\f -> Lam Ex nm (wp (IElim (quote a) (quote x) (quote y) f (Ref nm))))
isConvertibleTo a b = do isConvertibleTo a b = do
unify' a b
unify' True a b
pure id pure id
newMeta' :: Bool -> Value -> ElabM Value newMeta' :: Bool -> Value -> ElabM Value
@ -495,7 +518,7 @@ checkScope mv scope (VNe h sp) =
case h of case h of
HVar v@Bound{} -> HVar v@Bound{} ->
unless (v `Set.member` scope) . throwElab $ unless (v `Set.member` scope) . throwElab $
NotInScope v
ScopeCheckingFail v
HVar{} -> pure () HVar{} -> pure ()
HCon{} -> pure () HCon{} -> pure ()
HPCon{} -> pure () HPCon{} -> pure ()
@ -566,7 +589,10 @@ checkSpine scope (PApp Ex (VVar n@Bound{}) Seq.:<| xs)
checkSpine _ (p Seq.:<| _) = throwElab $ SpineProj p checkSpine _ (p Seq.:<| _) = throwElab $ SpineProj p
checkSpine _ Seq.Empty = pure [] checkSpine _ Seq.Empty = pure []
data MetaException = NonLinearSpine { getDupeName :: Name } | SpineProj { getSpineProjection :: Projection } | CircularSolution { getCycle :: MV }
data MetaException = NonLinearSpine { getDupeName :: Name }
| SpineProj { getSpineProjection :: Projection }
| CircularSolution { getCycle :: MV }
| ScopeCheckingFail { outOfScope :: Name }
deriving (Show, Typeable, Exception) deriving (Show, Typeable, Exception)
substituteIO :: Map.Map Name Value -> Value -> IO Value substituteIO :: Map.Map Name Value -> Value -> IO Value
@ -582,7 +608,7 @@ substituteIO sub = substituteIO . force where
HVar v -> HVar v ->
case Map.lookup v sub of case Map.lookup v sub of
Just vl -> substituteIO $ foldl applProj vl sp' Just vl -> substituteIO $ foldl applProj vl sp'
Nothing -> pure $ VNe hd sp'
Nothing -> pure $ foldl applProj (VNe hd mempty) sp'
hd -> pure $ VNe hd sp' hd -> pure $ VNe hd sp'
substituteIO (GluedVl h sp vl) = GluedVl h <$> traverse (substituteSp sub) sp <*> substituteIO vl substituteIO (GluedVl h sp vl) = GluedVl h <$> traverse (substituteSp sub) sp <*> substituteIO vl


+ 3
- 2
src/Elab/Eval/Formula.hs View File

@ -9,7 +9,7 @@ import Data.Set (Set)
import Syntax import Syntax
import {-# SOURCE #-} Elab.WiredIn (inot, ior, iand) import {-# SOURCE #-} Elab.WiredIn (inot, ior, iand)
import Elab.Eval (substitute)
import Elab.Eval (substitute, trueCaseSentinel)
toDnf :: Value -> Maybe Value toDnf :: Value -> Maybe Value
toDnf = fmap (dnf2Val . normalise) . val2Dnf where toDnf = fmap (dnf2Val . normalise) . val2Dnf where
@ -77,7 +77,8 @@ truthAssignments VI1 m = pure m
truthAssignments (VIOr x y) m = truthAssignments x m ++ truthAssignments y m truthAssignments (VIOr x y) m = truthAssignments x m ++ truthAssignments y m
truthAssignments (VIAnd x y) m = truthAssignments x =<< truthAssignments y m truthAssignments (VIAnd x y) m = truthAssignments x =<< truthAssignments y m
truthAssignments (VNe (HVar x) Seq.Empty) m = pure (Map.insert x (VI, VI1) (sub x VI1 <$> m)) truthAssignments (VNe (HVar x) Seq.Empty) m = pure (Map.insert x (VI, VI1) (sub x VI1 <$> m))
truthAssignments (VINot (VNe (HVar x) Seq.Empty)) m = pure (Map.insert x (VI, VI0) m)
truthAssignments (VINot (VNe (HVar x) Seq.Empty)) m = pure (Map.insert x (VI, VI0) (sub x VI0 <$> m))
truthAssignments (VCase _ _ (VNe (HVar x) _) _) m = pure (Map.insert x (VI, VVar trueCaseSentinel) m)
truthAssignments _ m = pure m truthAssignments _ m = pure m
sub :: Name -> Value -> (NFType, NFEndp) -> (Value, Value) sub :: Name -> Value -> (NFType, NFEndp) -> (Value, Value)


+ 4
- 1
src/Elab/Monad.hs View File

@ -137,10 +137,13 @@ switch :: ElabM a -> ElabM a
switch k = switch k =
do do
depth <- asks pingPong depth <- asks pingPong
when (depth >= 128) $ throwElab StackOverflow
when (depth >= 128) $ error "stack overflow"
local go k local go k
where go e = e { pingPong = pingPong e + 1 } where go e = e { pingPong = pingPong e + 1 }
shallowly :: ElabM a -> ElabM a
shallowly = local (\e -> e { pingPong = 0 })
newtype NotInScope = NotInScope { nameNotInScope :: Name } newtype NotInScope = NotInScope { nameNotInScope :: Name }
deriving (Show, Typeable) deriving (Show, Typeable)
deriving anyclass (Exception) deriving anyclass (Exception)


+ 31
- 7
src/Elab/WiredIn.hs View File

@ -7,7 +7,31 @@
{-# LANGUAGE CPP #-} {-# LANGUAGE CPP #-}
{-# LANGUAGE ConstraintKinds #-} {-# LANGUAGE ConstraintKinds #-}
{-# LANGUAGE KindSignatures #-} {-# LANGUAGE KindSignatures #-}
module Elab.WiredIn where
module Elab.WiredIn
( wiType
, wiValue
, wiredInNames
, NoSuchPrimitive(..)
, iand
, ior
, inot
, ielim
, incS
, outS
, comp
, fill
, hComp
, glueType
, glueElem
, unglue
, fun
, system
, strictK
, strictJ
, projIntoCase
)
where
import Control.Exception ( assert, Exception ) import Control.Exception ( assert, Exception )
@ -31,7 +55,6 @@ import Syntax.Pretty (prettyTm, prettyVl)
import Syntax import Syntax
import System.IO.Unsafe import System.IO.Unsafe
import Data.Text.Prettyprint.Doc (viaShow)
wiType :: WiredIn -> NFType wiType :: WiredIn -> NFType
wiType WiType = VType wiType WiType = VType
@ -48,6 +71,7 @@ wiType WiPathP = dprod (VI ~> VType) \a -> a @@ VI0 ~> a @@ VI1 ~> VType
wiType WiPartial = VI ~> VType ~> VTypeω wiType WiPartial = VI ~> VType ~> VTypeω
wiType WiPartialP = dprod VI \x -> VPartial x VType ~> VTypeω wiType WiPartialP = dprod VI \x -> VPartial x VType ~> VTypeω
wiType WiPOr = forAll VType \a -> dprod VI \phi -> dprod VI \psi -> VPartial phi a ~> VPartial psi a ~> VPartial (ior phi psi) a
wiType WiSub = dprod VType \a -> dprod VI \phi -> VPartial phi a ~> VTypeω wiType WiSub = dprod VType \a -> dprod VI \phi -> VPartial phi a ~> VTypeω
wiType WiInS = forAll VType \a -> forAll VI \phi -> dprod a \u -> VSub a phi (fun (const u)) wiType WiInS = forAll VType \a -> forAll VI \phi -> dprod a \u -> VSub a phi (fun (const u))
@ -82,6 +106,8 @@ wiValue WiPathP = functions [(Ex, "A"), (Ex, "x"), (Ex, "y")] \[a, x, y] ->
wiValue WiPartial = functions [(Ex, "phi"), (Ex, "A")] \[phi, a] -> VPartial phi a wiValue WiPartial = functions [(Ex, "phi"), (Ex, "A")] \[phi, a] -> VPartial phi a
wiValue WiPartialP = functions [(Ex, "phi"), (Ex, "A")] \[phi, a] -> VPartialP phi a wiValue WiPartialP = functions [(Ex, "phi"), (Ex, "A")] \[phi, a] -> VPartialP phi a
wiValue WiPOr = functions [(Im, "A"), (Ex, "phi"), (Ex, "psi"), (Ex, "a"), (Ex, "b")] \[_, phi, psi, a, b] -> mkVSystem (Map.fromList [(phi, a), (psi, b)])
wiValue WiSub = functions [(Ex, "A"), (Ex, "phi"), (Ex, "u")] \[a, phi, u] -> VSub a phi u wiValue WiSub = functions [(Ex, "A"), (Ex, "phi"), (Ex, "u")] \[a, phi, u] -> VSub a phi u
wiValue WiInS = functions [(Im, "A"), (Im, "phi"), (Ex, "u")] \[a, phi, u] -> incS a phi u wiValue WiInS = functions [(Im, "A"), (Im, "phi"), (Ex, "u")] \[a, phi, u] -> incS a phi u
wiValue WiOutS = functions [(Im, "A"), (Im, "phi"), (Im, "u"), (Ex, "u0")] \[a, phi, u, x] -> outS a phi u x wiValue WiOutS = functions [(Im, "A"), (Im, "phi"), (Im, "u"), (Ex, "u0")] \[a, phi, u, x] -> outS a phi u x
@ -153,6 +179,7 @@ wiredInNames = Map.fromList
, ("Partial", WiPartial) , ("Partial", WiPartial)
, ("PartialP", WiPartialP) , ("PartialP", WiPartialP)
, ("partialExt", WiPOr)
, ("Sub", WiSub) , ("Sub", WiSub)
, ("inS", WiInS) , ("inS", WiInS)
, ("outS", WiOutS) , ("outS", WiOutS)
@ -244,7 +271,7 @@ outS a phi u (VSystem fs) = mkVSystem (fmap (outS a phi u) fs)
outS _ _ _ v = error $ "can't outS " ++ show (prettyTm (quote v)) outS _ _ _ v = error $ "can't outS " ++ show (prettyTm (quote v))
comp :: DebugCallStack => NFLine -> NFEndp -> Value -> Value -> Value comp :: DebugCallStack => NFLine -> NFEndp -> Value -> Value -> Value
comp _a VI1 u _a0 = u @@ VI1 @@ VReflStrict VI VI1
comp _a (force -> VI1) u _a0 = u @@ VI1 @@ VReflStrict VI VI1
comp a psi@phi u incA0@(outS (a @@ VI0) phi (u @@ VI0) -> a0) = comp a psi@phi u incA0@(outS (a @@ VI0) phi (u @@ VI0) -> a0) =
case force (a @@ VVar name) of case force (a @@ VVar name) of
VPi{} -> VPi{} ->
@ -418,7 +445,7 @@ nthArg i xs = error $ "can't get " ++ show i ++ "th argument of " ++ show (prett
system :: (Value -> Value -> Value) -> Value system :: (Value -> Value -> Value) -> Value
system k = VLam P.Ex $ Closure (Bound "i" 0) \i -> VLam P.Ex $ Closure (Bound "[i]" 0) \isone -> k i isone system k = VLam P.Ex $ Closure (Bound "i" 0) \i -> VLam P.Ex $ Closure (Bound "[i]" 0) \isone -> k i isone
fill :: HasCallStack => NFLine -> NFEndp -> Value -> Value -> NFEndp -> Value
fill :: DebugCallStack => NFLine -> NFEndp -> Value -> Value -> NFEndp -> Value
fill a phi u a0 j = fill a phi u a0 j =
comp (line \i -> a @@ (i `iand` j)) comp (line \i -> a @@ (i `iand` j))
(phi `ior` inot j) (phi `ior` inot j)
@ -505,9 +532,6 @@ transp line a0 = comp (fun line) VI0 (system \_ _ -> VSystem mempty) (incS (line
gtrans :: (NFEndp -> Value) -> NFEndp -> Value -> Value gtrans :: (NFEndp -> Value) -> NFEndp -> Value -> Value
gtrans line phi a0 = comp (fun line) phi (system \_ _ -> mkVSystem (Map.singleton phi a0)) (incS (line VI0) VI0 a0) gtrans line phi a0 = comp (fun line) phi (system \_ _ -> mkVSystem (Map.singleton phi a0)) (incS (line VI0) VI0 a0)
forward :: (Value -> Value) -> Value -> Value -> Value
forward line phi a0 = gtrans (\i -> line (ior phi i)) phi a0
transHit :: Name -> (NFEndp -> Value) -> NFEndp -> Value -> Value transHit :: Name -> (NFEndp -> Value) -> NFEndp -> Value -> Value
transHit name line phi x = transHit name line phi (force x) where transHit name line phi x = transHit name line phi (force x) where
transHit name line phi (VHComp _ psi u u0) = VHComp (line VI1) psi (system \i j -> transHit name line phi (u @@ i @@ j)) (transHit name line phi (outS (line VI0) phi u u0)) transHit name line phi (VHComp _ psi u u0) = VHComp (line VI1) psi (system \i j -> transHit name line phi (u @@ i @@ j)) (transHit name line phi (outS (line VI0) phi u u0))


+ 3
- 3
src/Main.hs View File

@ -66,7 +66,7 @@ evalArgExpr env str =
Right e -> Right e ->
flip runElab env (do flip runElab env (do
(e, _) <- infer e (e, _) <- infer e
liftIO . Lt.putStrLn . render . prettyTm . quote . zonk =<< Elab.Eval.eval e)
liftIO . print . prettyTm . quote =<< Elab.Eval.eval e)
`catch` \e -> do `catch` \e -> do
displayExceptions' inp (e :: SomeException) displayExceptions' inp (e :: SomeException)
Left e -> liftIO $ print e Left e -> liftIO $ print e
@ -121,7 +121,7 @@ enterReplIn env =
newe <- liftIO $ try $ mkrepl <$> checkFiles (reverse loadedFiles) newe <- liftIO $ try $ mkrepl <$> checkFiles (reverse loadedFiles)
case newe of case newe of
Left e -> do Left e -> do
liftIO $ displayExceptions' ":r" (e :: SomeException)
liftIO $ do displayExceptions' ":r" (e :: SomeException)
loop env envref loop env envref
Right newe -> do Right newe -> do
liftIO $ writeIORef envref newe liftIO $ writeIORef envref newe
@ -156,7 +156,7 @@ checkFiles files = runElab (go 1 files ask) =<< emptyEnv where
liftIO . putStrLn $ "[" ++ pad (show n) ++ "/" ++ show size ++ "] Loading " ++ x liftIO . putStrLn $ "[" ++ pad (show n) ++ "/" ++ show size ++ "] Loading " ++ x
code <- liftIO $ Bsl.readFile x code <- liftIO $ Bsl.readFile x
case runAlex (code <> Bsl.singleton 10) parseProg of case runAlex (code <> Bsl.singleton 10) parseProg of
Left e -> liftIO $ print e *> error (show e)
Left e -> liftIO (print e) *> k
Right prog -> Right prog ->
local (\e -> e { currentFile = Just (T.pack x), loadedFiles = x:loadedFiles e }) (checkProgram prog (go (n + 1 :: Int) xs k)) local (\e -> e { currentFile = Just (T.pack x), loadedFiles = x:loadedFiles e }) (checkProgram prog (go (n + 1 :: Int) xs k))
`catchElab` \e -> liftIO $ displayAndDie (T.decodeUtf8 (Bsl.toStrict code)) (e :: SomeException) `catchElab` \e -> liftIO $ displayAndDie (T.decodeUtf8 (Bsl.toStrict code)) (e :: SomeException)


+ 11
- 2
src/Syntax.hs View File

@ -31,6 +31,7 @@ data WiredIn
| WiPartial -- (φ : I) -> Type -> Typeω | WiPartial -- (φ : I) -> Type -> Typeω
| WiPartialP -- (φ : I) -> Partial r Type -> Typeω | WiPartialP -- (φ : I) -> Partial r Type -> Typeω
| WiPOr -- (A : Type) (φ ψ : I) -> Partial φ A -> Partial ψ A -> Partial (ior φ ψ) A
| WiSub -- (A : Type) (φ : I) -> Partial φ A -> Typeω | WiSub -- (A : Type) (φ : I) -> Partial φ A -> Typeω
| WiInS -- {A : Type} {φ : I} (u : A) -> Sub A φ (λ x -> u) | WiInS -- {A : Type} {φ : I} (u : A) -> Sub A φ (λ x -> u)
@ -228,7 +229,7 @@ quoteWith names (GluedVl h sp (VLine ty x y (VLam p (Closure n k)))) =
quoteWith names (GluedVl h sp vl) quoteWith names (GluedVl h sp vl)
| GluedVl _ _ inner <- vl = quoteWith names (GluedVl h sp inner) | GluedVl _ _ inner <- vl = quoteWith names (GluedVl h sp inner)
| alwaysShort vl = quoteWith names vl
| True || alwaysShort vl = quoteWith names vl
| _ Seq.:|> PIElim _ x y i <- sp = | _ Seq.:|> PIElim _ x y i <- sp =
case i of case i of
VI0 -> quoteWith names x VI0 -> quoteWith names x
@ -321,7 +322,15 @@ data Head
| HPCon Value Value Name | HPCon Value Value Name
| HMeta MV | HMeta MV
| HData Bool Name | HData Bool Name
deriving (Eq, Ord, Show)
deriving (Ord, Show)
instance Eq Head where
HVar x == HVar y = x == y
HCon _ x == HCon _ y = x == y
HPCon _ _ x == HPCon _ _ y = x == y
HMeta x == HMeta y = x == y
HData x y == HData x' y' = x == x' && y == y'
_ == _ = False
data Projection data Projection
= PApp Plicity Value = PApp Plicity Value


+ 3
- 3
src/Syntax/Pretty.hs View File

@ -102,10 +102,10 @@ prettyTm' implicits = go True 0 where
I0 -> keyword (pretty "i0") I0 -> keyword (pretty "i0")
I1 -> keyword (pretty "i1") I1 -> keyword (pretty "i1")
IAnd x y -> parenIf (p > and_prec) $
IAnd x y -> parenIf (True || p > and_prec) $
go False and_prec x <+> operator (pretty "/\\") <+> go False and_prec y go False and_prec x <+> operator (pretty "/\\") <+> go False and_prec y
IOr x y -> parenIf (p > or_prec) $
IOr x y -> parenIf (True || p > or_prec) $
go False or_prec x <+> operator (pretty "\\/") <+> go False or_prec y go False or_prec x <+> operator (pretty "\\/") <+> go False or_prec y
INot x -> operator (pretty "~") <> go False p x INot x -> operator (pretty "~") <> go False p x
@ -213,7 +213,7 @@ printImplicits :: Bool
#if defined(RELEASE) #if defined(RELEASE)
printImplicits = False printImplicits = False
#else #else
printImplicits = False
printImplicits = True
#endif #endif
render :: Doc AnsiStyle -> Lazy.Text render :: Doc AnsiStyle -> Lazy.Text


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