open import Level
open import Relation.Binary.PropositionalEquality

module systemF {l : Level}  where

-- Bool

Bool = \{l : Level} -> {X : Set l} -> X -> X -> X

T : Bool
T = \{X : Set} -> \(x : X) -> \y -> x

F : Bool
F = \{X : Set} -> \x -> \(y : X) -> y

D : {X : Set} -> (U V : X) -> Bool -> X
D {X} u v bool = bool {X} u v

lemma-bool-t : {X : Set} -> {u v : X} -> D {X} u v T ≡ u
lemma-bool-t = refl

lemma-bool-f : {X : Set} -> {u v : X} -> D {X} u v F ≡ v
lemma-bool-f = refl

-- Product

_×_ : {i j k : Level}  -> Set i -> Set j -> Set (i ⊔ j ⊔ (suc k))
_×_  {i} {j} {k} U V = {X : Set k} -> (U -> V -> X) -> X

<_,_> : {i j k : Level} -> {U : Set i} -> {V : Set j} -> U -> V -> (U × V)
<_,_> {i} {j} {k} {U} {V} u v = \{X : Set k} -> \(x : U  -> V -> X) -> x u v

π1 : {i j : Level} -> {U : Set i} -> {V : Set j} -> (U × V) -> U
π1  {i} {j} {U} {V} t = t {U} \(x : U) -> \(y : V) -> x

π2 : {i j : Level} ->  {U : Set i} -> {V : Set j} -> (U × V) -> V
π2  {i} {j} {U} {V} t = t {V} \(x : U) -> \(y : V) -> y

lemma-product-pi1 : {U V : Set l} -> {u : U} -> {v : V} -> π1 (< u , v >) ≡ u
lemma-product-pi1 = refl

lemma-product-pi2 : {U V : Set l} -> {u : U} -> {v : V} -> π2 (< u , v >) ≡ v
lemma-product-pi2 = refl

-- Empty


-- Sum

_+_ : {l : Level} ->  Set l -> Set l -> Set (suc l)
_+_ {l} U V = {X : Set  l} -> (U -> X) -> (V -> X) -> X

ι1 :  {U V : Set l} -> U -> (U + V)
ι1  {U} {V} u =  \{X : Set l} -> \(x : U -> X) -> \(y : V -> X) -> x u

ι2 :  {U V : Set l} -> V -> (U + V)
ι2  {U} {V} v =  \{X : Set l} -> \(x : U -> X) -> \(y : V -> X) -> y v

δ : {l : Level} -> {U V : Set l} -> {X : Set l} -> (U -> X) -> (V -> X) -> (U + V) -> X
δ {l} {U} {V} {X} u v t = t {X} u v

lemma-sum-iota1 : {U V X R : Set l} -> {u : U} -> {ux : (U -> X)} -> {vx : (V -> X)} -> δ ux vx (ι1 u) ≡ ux u
lemma-sum-iota1 = refl

lemma-sum-iota2 : {U V X R : Set l} -> {v : V} -> {ux : (U -> X)} -> {vx : (V -> X)} -> δ ux vx (ι2 v) ≡ vx v
lemma-sum-iota2 = refl


-- Existential

data V {l} (X : Set l) : Set l
  where
    v : X -> V X

Σ_,_ : (X : Set l) -> V X -> Set (suc l)
Σ_,_ U u = {Y : Set l} -> ({X : Set l} -> (V X) -> Y) -> Y

⟨_,_⟩ : (U : Set l) -> (u : V U) -> Σ U , u
⟨_,_⟩ U u = \{Y : Set l} -> \(x : {X : Set l} -> (V X) -> Y) -> x {U} u

∇_,_,_ :  {W : Set l} -> (X : Set l) -> { u : V X } -> X -> W -> Σ X , u  -> W
∇_,_,_ {W} X {u} x w t = t {W} (\{X : Set l} -> \(x : V X) -> w)

{-
  lemma-nabla on proofs and types
  (∇ X , x , w ) ⟨ U , u ⟩ ≡ w
  w[U/X][u/x^[U/X]]
-}

lemma-nabla : {X W : Set l} -> {x : X} -> {w : W} -> (∇_,_,_ {W} X {v x} x w) ⟨ X , (v x) ⟩ ≡ w
lemma-nabla = refl


-- Int

Int : {l : Level} -> Set (suc l)
Int {l} = {X : Set l} -> X -> (X -> X) -> X

O : {l : Level} -> Int
O {l} = \{X : Set l} -> \(x : X) -> \(y : X -> X) -> x

S : {l : Level} -> Int -> Int
S {l} t = \{X : Set l} -> \(x : X) -> \(y : X -> X) -> y (t {X} x y)

It : {U : Set l} -> (u : U) -> (U -> U) -> Int -> U
It {U} u f t = t {U} u f

lemma-it-o : {U : Set l} -> {u : U} -> {f : U -> U} -> It u f O ≡ u
lemma-it-o = refl

lemma-it-s-o : {U : Set l} -> {u : U} -> {f : U -> U} -> {t : Int} -> It u f (S t) ≡ f (It u f t)
lemma-it-s-o = refl

-- U only
--g : {i : Level} -> {U : Set (suc i)} -> {f : U -> Int {i} -> U} -> (U × (Int {i})) -> U
--g {i} {U} {f} = \(x : (U × Int {i})) -> (f (π1 x) (π2 x))

-- Int Only
--g : {i : Level} -> {U : Set (suc i)} -> {f : U -> Int {i} -> U} -> (U × (Int {i})) -> Int {i}
--g {i} {U} {f} = \(x : (U × Int {i})) -> (π2 x)

g : {i : Level} -> {U : Set (suc i)} -> {f : U -> Int {i} -> U} -> (U × (Int {i})) -> U × Int {i}
g {i} {U} {f} = \(x : (U × Int {i})) -> < (f (π1 x)  (π2 x)) , (π2 x) >