# HG changeset patch
# User Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
# Date 1417354010 -32400
# Node ID 0ad0ae7a3cbef50329576d815b576d054134d398
# Parent  e95f15af3f8b28cf8f9eddc93bf385d5452542ba
Proving monad-law-1

diff -r e95f15af3f8b -r 0ad0ae7a3cbe agda/delta.agda
--- a/agda/delta.agda	Sun Nov 30 19:00:32 2014 +0900
+++ b/agda/delta.agda	Sun Nov 30 22:26:50 2014 +0900
@@ -91,9 +91,25 @@
 
 n-tail : {l : Level} {A : Set l} -> Int -> ((Delta A) -> (Delta A))
 n-tail O = id
-n-tail (S n) =  (n-tail n) ∙ tailDelta
+n-tail (S n) =  tailDelta ∙ (n-tail n)
+
+flip  : {l : Level} {A : Set l} -> (f : A -> A) -> f ∙ (f ∙ f) ≡ (f ∙ f) ∙ f
+flip f = refl
 
-postulate n-tail-plus : (n : Int) -> (tailDelta ∙ (n-tail n)) ≡ n-tail (S n)
+n-tail-plus : {l : Level} {A : Set l} -> (n : Int) -> ((n-tail {l} {A} n) ∙ tailDelta) ≡ n-tail (S n)
+n-tail-plus O = refl
+n-tail-plus (S n) = begin
+  n-tail (S n) ∙ tailDelta ≡⟨ refl ⟩
+  (tailDelta ∙ (n-tail n)) ∙ tailDelta ≡⟨ refl ⟩
+  tailDelta ∙ ((n-tail n) ∙ tailDelta) ≡⟨ cong (\t -> tailDelta ∙ t) (n-tail-plus n) ⟩
+  n-tail (S (S n))
+  ∎
+
+postulate  n-tail-add : {l : Level} {A : Set l} -> (n m : Int) -> (n-tail {l} {A} n) ∙ (n-tail m) ≡ n-tail (n + m)
+postulate int-add-assoc : (n m : Int) -> n + m ≡ m + n
+postulate int-add-right-zero : (n : Int) -> n  ≡ n + O
+postulate int-add-right : (n m : Int) -> S n + S m ≡ S (S (n + m))
+
 
 
 
@@ -102,21 +118,97 @@
 tail-delta-to-mono : {l : Level} {A : Set l} -> (n : Int) -> (x : A) ->
   (n-tail n) (mono x) ≡ (mono x)
 tail-delta-to-mono O x = refl
-tail-delta-to-mono (S n) x = begin 
-  n-tail (S n) (mono x)  ≡⟨ refl ⟩
-  ((n-tail n) ∙ tailDelta) (mono x) ≡⟨ refl ⟩
-  (n-tail n) (tailDelta (mono x))  ≡⟨ refl ⟩
-  (n-tail n) (mono x)  ≡⟨ tail-delta-to-mono n x ⟩
-  mono x
-  ∎
-{- begin
+tail-delta-to-mono (S n) x =  begin
   n-tail (S n) (mono x)           ≡⟨ refl ⟩
   tailDelta (n-tail n (mono x))   ≡⟨ refl ⟩
   tailDelta (n-tail n (mono x))   ≡⟨ cong (\t -> tailDelta t) (tail-delta-to-mono n x) ⟩
   tailDelta (mono x)              ≡⟨ refl ⟩
   mono x
   ∎
--}
+
+monad-law-1-5 : {l : Level} {A : Set l} -> (m : Int) (n : Int) -> (ds : Delta (Delta A)) ->
+  n-tail n (bind ds (n-tail m))  ≡ bind (n-tail n ds) (n-tail (m + n)) 
+monad-law-1-5 O O ds = refl
+monad-law-1-5 O (S n) (mono ds) = begin
+  n-tail (S n) (bind (mono ds) (n-tail O)) ≡⟨ refl ⟩
+  n-tail (S n) ds ≡⟨ refl ⟩
+  bind (mono ds) (n-tail (S n)) ≡⟨ cong (\de -> bind de (n-tail (S n))) (sym (tail-delta-to-mono (S n) ds)) ⟩
+  bind (n-tail (S n) (mono ds)) (n-tail (S n)) ≡⟨ refl ⟩
+  bind (n-tail (S n) (mono ds)) (n-tail (O + S n))
+  ∎
+monad-law-1-5 O (S n) (delta d ds) = begin
+  n-tail (S n) (bind (delta d ds) (n-tail O)) ≡⟨ refl ⟩
+  n-tail (S n) (delta (headDelta d) (bind ds tailDelta )) ≡⟨ cong (\t -> t  (delta (headDelta d) (bind ds tailDelta ))) (sym (n-tail-plus n)) ⟩
+  ((n-tail n) ∙ tailDelta) (delta (headDelta d) (bind ds tailDelta )) ≡⟨ refl ⟩
+  (n-tail n) (bind ds tailDelta) ≡⟨ monad-law-1-5 (S O) n ds ⟩
+  bind (n-tail n ds) (n-tail  (S n)) ≡⟨ refl ⟩
+  bind (((n-tail n) ∙ tailDelta) (delta d ds)) (n-tail  (S n)) ≡⟨ cong (\t -> bind (t (delta d ds)) (n-tail  (S n))) (n-tail-plus n) ⟩
+  bind (n-tail (S n) (delta d ds)) (n-tail  (S n)) ≡⟨ refl ⟩
+  bind (n-tail (S n) (delta d ds)) (n-tail (O + S n))
+  ∎
+monad-law-1-5 (S m) n (mono (mono x)) = begin
+  n-tail n (bind (mono (mono x)) (n-tail (S m))) ≡⟨ refl ⟩
+  n-tail n (n-tail (S m) (mono x)) ≡⟨ cong (\de -> n-tail n de) (tail-delta-to-mono (S m) x)⟩
+  n-tail n (mono x) ≡⟨ tail-delta-to-mono n x ⟩
+  mono x  ≡⟨ sym (tail-delta-to-mono (S m + n) x) ⟩
+  (n-tail (S m + n)) (mono x) ≡⟨ refl ⟩
+  bind (mono (mono x)) (n-tail (S m + n)) ≡⟨ cong (\de -> bind de (n-tail (S m + n))) (sym (tail-delta-to-mono n (mono x))) ⟩
+  bind (n-tail n (mono (mono x))) (n-tail (S m + n))
+  ∎
+monad-law-1-5 (S m) n (mono (delta x ds)) = begin
+  n-tail n (bind (mono (delta x ds)) (n-tail (S m))) ≡⟨ refl ⟩
+  n-tail n (n-tail (S m) (delta x ds)) ≡⟨ cong (\t -> n-tail n (t (delta x ds))) (sym (n-tail-plus m)) ⟩
+  n-tail n (((n-tail m) ∙ tailDelta) (delta x ds)) ≡⟨ refl ⟩
+  n-tail n ((n-tail m) ds) ≡⟨ cong (\t -> t ds) (n-tail-add n m)  ⟩
+  n-tail (n + m) ds  ≡⟨ cong (\n -> n-tail n ds) (int-add-assoc n m) ⟩
+  n-tail (m + n) ds  ≡⟨ refl ⟩
+  ((n-tail (m + n)) ∙ tailDelta) (delta x ds)  ≡⟨ cong (\t -> t (delta x ds)) (n-tail-plus (m + n))⟩
+  n-tail (S (m + n)) (delta x ds)  ≡⟨ refl ⟩
+  n-tail (S m + n) (delta x ds)  ≡⟨ refl ⟩
+  bind (mono (delta x ds)) (n-tail (S m + n)) ≡⟨ cong (\de -> bind de (n-tail (S m + n))) (sym (tail-delta-to-mono n (delta x ds))) ⟩
+  bind (n-tail n (mono (delta x ds))) (n-tail (S m + n))
+  ∎
+monad-law-1-5 (S m) O (delta d ds) = begin
+  n-tail O (bind (delta d ds) (n-tail (S m))) ≡⟨ refl ⟩
+  (bind (delta d ds) (n-tail (S m))) ≡⟨ refl ⟩
+  delta (headDelta ((n-tail (S m)) d)) (bind ds (tailDelta ∙ (n-tail (S m)))) ≡⟨ refl ⟩
+  bind (delta d ds) (n-tail (S m)) ≡⟨ refl ⟩
+  bind (n-tail O (delta d ds)) (n-tail (S m)) ≡⟨ cong (\n -> bind (n-tail O (delta d ds)) (n-tail n)) (int-add-right-zero (S m)) ⟩
+  bind (n-tail O (delta d ds)) (n-tail (S m + O))
+  ∎
+monad-law-1-5 (S m) (S n) (delta d ds) = begin
+  n-tail (S n) (bind (delta d ds) (n-tail (S m))) ≡⟨ cong (\t -> t ((bind (delta d ds) (n-tail (S m))))) (sym (n-tail-plus n)) ⟩
+  ((n-tail n) ∙ tailDelta) (bind (delta d ds) (n-tail (S m))) ≡⟨ refl ⟩
+  ((n-tail n) ∙ tailDelta) (delta (headDelta ((n-tail (S m)) d)) (bind ds (tailDelta ∙  (n-tail (S m))))) ≡⟨ refl ⟩
+  (n-tail n) (bind ds (tailDelta ∙ (n-tail (S m)))) ≡⟨ refl ⟩
+  (n-tail n) (bind ds (n-tail (S (S m)))) ≡⟨ monad-law-1-5 (S (S m)) n ds ⟩
+  bind ((n-tail n) ds) (n-tail (S (S (m + n)))) ≡⟨ cong (\nm -> bind ((n-tail n) ds) (n-tail nm))  (sym (int-add-right m n)) ⟩
+  bind ((n-tail n) ds) (n-tail (S m + S n)) ≡⟨ refl ⟩
+  bind (((n-tail n) ∙ tailDelta) (delta d ds)) (n-tail (S m + S n)) ≡⟨ cong (\t -> bind (t (delta d ds)) (n-tail (S m + S n))) (n-tail-plus n) ⟩
+  bind (n-tail (S n) (delta d ds)) (n-tail (S m + S n))
+  ∎
+
+monad-law-1-4 : {l : Level} {A : Set l} -> (n : Int) -> (dd : Delta (Delta A)) ->
+  headDelta ((n-tail n) (bind dd tailDelta)) ≡ headDelta ((n-tail (S n)) (headDelta (n-tail n dd))) 
+monad-law-1-4 O (mono dd) = refl
+monad-law-1-4 O (delta dd dd₁) = refl
+monad-law-1-4 (S n) (mono dd) = begin
+  headDelta (n-tail (S n) (bind (mono dd) tailDelta)) ≡⟨ refl ⟩
+  headDelta (n-tail (S n) (tailDelta dd)) ≡⟨ cong (\t -> headDelta (t dd)) (n-tail-plus (S n)) ⟩
+  headDelta (n-tail (S (S n)) dd) ≡⟨ refl ⟩
+  headDelta (n-tail (S (S n)) (headDelta (mono dd))) ≡⟨ cong (\de -> headDelta (n-tail (S (S n)) (headDelta de))) (sym (tail-delta-to-mono (S n) dd)) ⟩
+  headDelta (n-tail (S (S n)) (headDelta (n-tail (S n) (mono dd))))
+  ∎
+monad-law-1-4 (S n) (delta d ds) = begin
+  headDelta (n-tail (S n) (bind (delta d ds) tailDelta)) ≡⟨ refl ⟩
+  headDelta (n-tail (S n) (delta (headDelta (tailDelta d)) (bind ds (tailDelta ∙  tailDelta)))) ≡⟨ cong (\t -> headDelta (t (delta (headDelta (tailDelta d)) (bind ds (tailDelta ∙  tailDelta))))) (sym (n-tail-plus n)) ⟩
+  headDelta (((n-tail n) ∙ tailDelta) (delta (headDelta (tailDelta d)) (bind ds (tailDelta ∙  tailDelta)))) ≡⟨ refl ⟩
+  headDelta (n-tail n (bind ds (tailDelta ∙  tailDelta))) ≡⟨ {!!} ⟩
+  headDelta (n-tail (S (S n)) (headDelta ((n-tail n ds)))) ≡⟨ refl ⟩
+  headDelta (n-tail (S (S n)) (headDelta ((n-tail n ∙ tailDelta) (delta d ds)))) ≡⟨ cong (\t ->  headDelta (n-tail (S (S n)) (headDelta (t (delta d ds))))) (n-tail-plus n) ⟩
+  headDelta (n-tail (S (S n)) (headDelta (n-tail (S n) (delta d ds))))
+  ∎
+
 monad-law-1-2 : {l : Level} {A : Set l} -> (d : Delta (Delta A)) -> headDelta (mu d) ≡ (headDelta (headDelta d))
 monad-law-1-2 (mono _) = refl
 monad-law-1-2 (delta _ _) = refl
@@ -146,9 +238,11 @@
   bind (fmap mu (mono (delta d ds))) (n-tail (S n)) ≡⟨ refl ⟩
   bind (mono (mu (delta d ds))) (n-tail (S n)) ≡⟨ refl ⟩
   n-tail (S n) (mu (delta d ds))  ≡⟨ refl ⟩
-  n-tail (S n) (delta (headDelta d) (bind ds tailDelta))  ≡⟨ refl ⟩
-  n-tail n (bind ds tailDelta)  ≡⟨ {!!} ⟩
+  n-tail (S n) (delta (headDelta d) (bind ds tailDelta))  ≡⟨ cong (\t -> t (delta (headDelta d) (bind ds tailDelta))) (sym (n-tail-plus n)) ⟩
+  (n-tail n ∙ tailDelta) (delta (headDelta d) (bind ds tailDelta))  ≡⟨ refl ⟩
+  n-tail n (bind ds tailDelta)  ≡⟨ monad-law-1-5 (S O) n ds ⟩
   bind (n-tail n ds) (n-tail (S n)) ≡⟨ refl ⟩
+  bind (((n-tail n) ∙ tailDelta) (delta d ds)) (n-tail (S n)) ≡⟨ cong (\t -> (bind (t (delta d ds)) (n-tail (S n))))  (n-tail-plus n) ⟩
   bind (n-tail (S n) (delta d ds)) (n-tail (S n)) ≡⟨ refl ⟩
   bind (bind (mono (delta d ds)) (n-tail (S n))) (n-tail (S n))
   ∎
@@ -156,8 +250,10 @@
   bind (fmap mu (delta (mono d) ds)) (n-tail (S n)) ≡⟨ refl ⟩
   bind (delta (mu (mono d)) (fmap mu ds)) (n-tail (S n)) ≡⟨ refl ⟩
   bind (delta d (fmap mu ds)) (n-tail (S n)) ≡⟨ refl ⟩
-  delta (headDelta ((n-tail (S n)) d)) (bind (fmap mu ds) (tailDelta ∙ (n-tail (S n)))) ≡⟨ {!!} ⟩
-  delta (headDelta ((n-tail (S n)) d)) (bind (fmap mu ds) (n-tail (S (S n)))) ≡⟨ {!!} ⟩
+  delta (headDelta ((n-tail (S n)) d)) (bind (fmap mu ds) (tailDelta ∙ (n-tail (S n)))) ≡⟨ refl ⟩
+  delta (headDelta ((n-tail (S n)) d)) (bind (fmap mu ds) (n-tail (S (S n)))) ≡⟨ cong (\de -> delta (headDelta ((n-tail (S n)) d)) de) (monad-law-1-3 (S (S n)) ds) ⟩
+  delta (headDelta ((n-tail (S n)) d)) (bind (bind ds (n-tail (S (S n)))) (n-tail (S (S n)))) ≡⟨ refl ⟩
+  delta (headDelta ((n-tail (S n)) d)) (bind (bind ds (tailDelta ∙ (n-tail (S n))))  (n-tail (S (S n)))) ≡⟨ refl ⟩
   delta (headDelta ((n-tail (S n)) d)) (bind (bind ds (tailDelta ∙ (n-tail (S n)))) (tailDelta ∙  (n-tail (S n)))) ≡⟨ refl ⟩
   delta (headDelta ((n-tail (S n)) (headDelta (mono d)))) (bind (bind ds (tailDelta ∙ (n-tail (S n)))) (tailDelta ∙  (n-tail (S n)))) ≡⟨ cong (\de -> delta (headDelta ((n-tail (S n)) (headDelta de))) (bind (bind ds (tailDelta ∙ (n-tail (S n)))) (tailDelta ∙  (n-tail (S n))))) (sym (tail-delta-to-mono (S n) d)) ⟩
   delta (headDelta ((n-tail (S n)) (headDelta ((n-tail (S n)) (mono d))))) (bind (bind ds (tailDelta ∙ (n-tail (S n)))) (tailDelta ∙  (n-tail (S n)))) ≡⟨ refl ⟩
@@ -168,10 +264,15 @@
   bind (fmap mu (delta (delta d dd) ds)) (n-tail (S n)) ≡⟨ refl ⟩
   bind (delta (mu (delta d dd)) (fmap mu ds)) (n-tail (S n)) ≡⟨ refl ⟩
   delta (headDelta ((n-tail (S n)) (mu (delta d dd)))) (bind (fmap mu ds) (tailDelta ∙ (n-tail (S n)))) ≡⟨ refl ⟩
-  delta (headDelta ((n-tail (S n)) (delta (headDelta d) (bind dd tailDelta)))) (bind (fmap mu ds) (tailDelta ∙ (n-tail (S n)))) ≡⟨ refl ⟩
-  delta (headDelta ((n-tail n) (bind dd tailDelta))) (bind (fmap mu ds) (tailDelta ∙ (n-tail (S n)))) ≡⟨ {!!} ⟩
-
+  delta (headDelta ((n-tail (S n)) (delta (headDelta d) (bind dd tailDelta)))) (bind (fmap mu ds) (tailDelta ∙ (n-tail (S n)))) ≡⟨ cong (\t -> delta (headDelta (t (delta (headDelta d) (bind dd tailDelta)))) (bind (fmap mu ds) (tailDelta ∙ (n-tail (S n)))))(sym (n-tail-plus n)) ⟩
+  delta (headDelta (((n-tail n) ∙ tailDelta) (delta (headDelta d) (bind dd tailDelta)))) (bind (fmap mu ds) (tailDelta ∙ (n-tail (S n)))) ≡⟨ refl ⟩
+  delta (headDelta ((n-tail n) (bind dd tailDelta))) (bind (fmap mu ds) (tailDelta ∙ (n-tail (S n)))) ≡⟨ refl ⟩
+  delta (headDelta ((n-tail n) (bind dd tailDelta))) (bind (fmap mu ds) (n-tail (S (S n)))) ≡⟨ cong (\de -> delta (headDelta ((n-tail n) (bind dd tailDelta))) de) (monad-law-1-3 (S (S n)) ds) ⟩
+  delta (headDelta ((n-tail n) (bind dd tailDelta))) (bind (bind  ds (n-tail (S (S n))))  (n-tail (S (S n)))) ≡⟨ cong (\de -> delta de ( (bind (bind  ds (n-tail (S (S n))))  (n-tail (S (S n)))))) (monad-law-1-4 n dd) ⟩
+  delta (headDelta ((n-tail (S n)) (headDelta (n-tail n dd)))) (bind (bind  ds (n-tail (S (S n))))  (n-tail (S (S n)))) ≡⟨ refl ⟩
+  delta (headDelta ((n-tail (S n)) (headDelta (n-tail n dd)))) (bind (bind  ds (n-tail (S (S n)))) (tailDelta ∙ (n-tail (S n)))) ≡⟨ refl ⟩
   delta (headDelta ((n-tail (S n)) (headDelta (n-tail n dd)))) (bind (bind  ds (tailDelta ∙ (n-tail (S n)))) (tailDelta ∙ (n-tail (S n)))) ≡⟨ refl ⟩
+  delta (headDelta ((n-tail (S n)) (headDelta (((n-tail n) ∙ tailDelta) (delta d dd))))) (bind (bind  ds (tailDelta ∙ (n-tail (S n)))) (tailDelta ∙ (n-tail (S n)))) ≡⟨ cong (\t -> delta (headDelta ((n-tail (S n)) (headDelta (t (delta d dd))))) (bind (bind  ds (tailDelta ∙ (n-tail (S n)))) (tailDelta ∙ (n-tail (S n))))) (n-tail-plus n) ⟩
   delta (headDelta ((n-tail (S n)) (headDelta ((n-tail (S n)) (delta d dd))))) (bind (bind  ds (tailDelta ∙ (n-tail (S n)))) (tailDelta ∙ (n-tail (S n)))) ≡⟨ refl ⟩
   bind (delta (headDelta ((n-tail (S n)) (delta d dd))) (bind  ds (tailDelta ∙ (n-tail (S n))))) (n-tail (S n)) ≡⟨ refl ⟩
   bind (bind (delta (delta d dd) ds) (n-tail (S n))) (n-tail (S n))
diff -r e95f15af3f8b -r 0ad0ae7a3cbe delta.hs
--- a/delta.hs	Sun Nov 30 19:00:32 2014 +0900
+++ b/delta.hs	Sun Nov 30 22:26:50 2014 +0900
@@ -5,6 +5,11 @@
 
 data Delta a = Mono a | Delta a (Delta a) deriving Show
 
+instance (Eq a) => Eq (Delta a) where
+    (Mono x) == (Mono y)         = x == y
+    (Mono _) == (Delta _ _)      = False
+    (Delta x xs) == (Delta y ys) = (x == y) && (xs == ys)
+
 -- basic functions
 
 deltaAppend :: Delta a -> Delta a -> Delta a