# HG changeset patch
# User Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
# Date 1412661236 -32400
# Node ID 742e62fc63e427c2d8e59e26dff2bb03107304d0
# Parent  5ba82f107a95336b2787b0a0e7604169c5e70962
Define Monad-law 1-4

diff -r 5ba82f107a95 -r 742e62fc63e4 agda/similar.agda
--- a/agda/similar.agda	Tue Oct 07 14:43:33 2014 +0900
+++ b/agda/similar.agda	Tue Oct 07 14:53:56 2014 +0900
@@ -1,7 +1,12 @@
 open import list
+open import Relation.Binary.PropositionalEquality
+open ≡-Reasoning
 
 module similar where
 
+id : {A : Set} -> A -> A
+id x = x
+
 postulate String : Set
 postulate show   : {A : Set} -> A -> String
 
@@ -16,6 +21,9 @@
 mu : {A : Set} -> Similar (Similar A) -> Similar A
 mu (similar lx (similar llx x _ _) ly (similar _ _ lly y)) = similar (lx ++ llx) x (ly ++ lly) y
 
+return : {A : Set} -> A -> Similar A
+return x = similar [] x [] x
+
 returnS : {A : Set} -> A -> Similar A
 returnS x = similar [[ (show x) ]] x [[ (show x) ]] x
 
@@ -23,3 +31,21 @@
 returnSS x y = similar [[ (show x) ]] x [[ (show y) ]] y
 
 
+_∙_ : {A B C : Set} -> (A -> B) -> (B -> C) -> (A -> C)
+f ∙ g = \x -> g (f x)
+
+monad-law-1 : mu ∙ (fmap mu) ≡ mu ∙ mu
+monad-law-1 = {!!} 
+
+--monad-law-2 : mu ∙ fmap return ≡ mu ∙ return ≡id
+monad-law-2-1 : mu ∙ fmap return ≡ mu ∙ return
+monad-law-2-1 = {!!}
+
+monad-law-2-2 : mu ∙ return ≡ id
+monad-law-2-2 = {!!}
+
+monad-law-3 : ∀{f} -> return ∙ f ≡ fmap f ∙ return
+monad-law-3 = {!!} 
+
+monad-law-4 : ∀{f} -> mu ∙ fmap (fmap f) ≡ fmap f ∙ mu
+monad-law-4 = {!!}