module system-t where
open  import  Relation.Binary.PropositionalEquality

record _×_ ( U : Set ) ( V : Set )   :  Set where
    field
       π1 : U
       π2 : V

<_,_> : {U V : Set} -> U -> V -> U × V
< u , v >  = record {π1 = u ; π2 = v }

open _×_

postulate U : Set
postulate V : Set

postulate v : V
postulate u : U

f : U -> V
f = \u -> v


UV : Set
UV = U × V

uv : U × V
uv  = < u , v >

lemma01 : π1  < u , v >   ≡ u
lemma01 = refl

lemma02 : π2  < u , v >   ≡ v
lemma02 = refl

lemma03 : (uv : U × V ) → < π1  uv , π2 uv >   ≡ uv
lemma03 uv = refl

lemma04 : (λ x →  f x ) u ≡  f u
lemma04 = refl

lemma05 : (λ  x → f x ) ≡  f
lemma05 = refl

nn = λ (x : U ) →  u
n1 = λ ( x : U ) →  f x

data Bool : Set where
  T : Bool 
  F : Bool

D : { U : Set } -> U -> U -> Bool -> U 
D u v T = u 
D u v F = v

data Int : Set where 
    zero : Int
    S : Int →  Int

pred : Int -> Int
pred zero = zero
pred (S t) = t

R : { U : Set } -> U -> ( U -> Int -> U ) -> Int -> U
R u v zero = u 
R u v ( S t ) = v (R u v t) t

null : Int -> Bool
null zero = T
null (S _)  = F

It : { T : Set } -> T -> (T -> T) -> Int -> T
It u v zero = u
It u v ( S t ) = v (It u v t )

R' : { T : Set } -> T -> ( T -> Int -> T ) -> Int -> T
R' u v t = π1 ( It ( < u , zero > ) (λ x →  < v (π1 x) (π2 x) , S (π2 x) > ) t )

sum : Int -> Int -> Int
sum x y = R y ( λ z -> λ w -> S z ) x

mul : Int -> Int -> Int
mul x y = R zero ( λ z -> λ w -> sum y z ) x 

sum' : Int -> Int -> Int
sum' x y = R' y ( λ z -> λ w -> S z ) x

mul' : Int -> Int -> Int
mul' x y = R' zero ( λ z -> λ w -> sum y z ) x

fact : Int -> Int
fact  n = R  (S zero) (λ z -> λ w -> mul (S w) z ) n

fact' : Int -> Int
fact' n = R' (S zero) (λ z -> λ w -> mul (S w) z ) n

f3 = fact (S (S (S zero)))
f3' = fact' (S (S (S zero)))

lemma21 : f3 ≡ f3'
lemma21 = refl

lemma07 : { U : Set } -> ( u : U ) -> ( v : U -> Int -> U ) ->( t :  Int ) -> 
        (π2 (It < u , zero > (λ x → < v (π1 x) (π2 x) , S (π2 x) >) t )) ≡  t
lemma07 u v zero   =  refl
lemma07 u v (S t)  =  cong ( \x -> S x ) ( lemma07 u v t )

lemma06 : { U : Set } -> ( u : U ) -> ( v : U -> Int -> U ) ->( t :  Int ) -> ( (R u v t) ≡  (R' u v t ))
lemma06 u v zero = refl 
lemma06 u v (S t) =  trans  ( cong ( \x ->  v x t )  ( lemma06 u v t ) ) 
                            ( cong ( \y ->  v (R' u v t) y ) (sym ( lemma07 u v t ) ) )

lemma08 : ( n m : Int ) -> ( sum' n m ≡ sum n m )
lemma08 zero _ = refl 
lemma08 (S t) y =  cong ( \x -> S x ) ( lemma08 t y ) 

lemma09 : ( n m : Int ) -> ( mul' n m ≡ mul n m )
lemma09 zero _ = refl 
lemma09 (S t) y =  cong ( \x -> sum y x) ( lemma09 t y ) 

lemma10 : ( n : Int ) -> ( fact n  ≡ fact n )
lemma10 zero = refl 
lemma10 (S t) =  cong ( \x -> mul (S t) x ) ( lemma10 t ) 

lemma11 : ( n : Int ) -> ( fact n  ≡ fact' n )
lemma11 n = lemma06 (S zero) (λ z -> λ w -> mul (S w) z ) n

lemma06' : { U : Set } -> ( u : U ) -> ( v : U -> Int -> U ) ->( t :  Int ) -> ( (R u v t) ≡  (R' u v t ))
lemma06' u v zero = refl
lemma06' u v (S t) = let open ≡-Reasoning in 
   begin 
       R u v (S t)
   ≡⟨⟩
       v (R u v t) t
   ≡⟨ cong (\x -> v x t ) ( lemma06' u v t )  ⟩
       v (R' u v t) t
   ≡⟨ cong (\x -> v (R' u v t) x ) ( sym ( lemma07 u v t )) ⟩
       v (R' u v t) (π2 (It < u , zero > (λ x → < v (π1 x) (π2 x) , S (π2 x) >) t))
   ≡⟨⟩
       R' u v (S t)
   ∎