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comparison agda/delta.agda @ 131:d205ff1e406f InfiniteDeltaWithMonad

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Cleanup proofs
author Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
date Tue, 03 Feb 2015 12:57:13 +0900
parents e6499a50ccbd
children 2bf1fa6d2006
comparison
equal deleted inserted replaced
130:ac45d065cbf2 131:d205ff1e406f
1 open import Level
2 open import Relation.Binary.PropositionalEquality
3 open ≡-Reasoning
4
1 open import list 5 open import list
2 open import basic 6 open import basic
3 open import nat 7 open import nat
4 open import laws 8 open import laws
5
6 open import Level
7 open import Relation.Binary.PropositionalEquality
8 open ≡-Reasoning
9 9
10 module delta where 10 module delta where
11 11
12 data Delta {l : Level} (A : Set l) : (Nat -> (Set l)) where 12 data Delta {l : Level} (A : Set l) : (Nat -> (Set l)) where
13 mono : A -> Delta A (S O) 13 mono : A -> Delta A (S O)
36 -- Monad (Category) 36 -- Monad (Category)
37 delta-eta : {l : Level} {A : Set l} {n : Nat} -> A -> Delta A (S n) 37 delta-eta : {l : Level} {A : Set l} {n : Nat} -> A -> Delta A (S n)
38 delta-eta {n = O} x = mono x 38 delta-eta {n = O} x = mono x
39 delta-eta {n = (S n)} x = delta x (delta-eta {n = n} x) 39 delta-eta {n = (S n)} x = delta x (delta-eta {n = n} x)
40 40
41
42
43
44 delta-mu : {l : Level} {A : Set l} {n : Nat} -> (Delta (Delta A (S n)) (S n)) -> Delta A (S n) 41 delta-mu : {l : Level} {A : Set l} {n : Nat} -> (Delta (Delta A (S n)) (S n)) -> Delta A (S n)
45 delta-mu (mono x) = x 42 delta-mu (mono x) = x
46 delta-mu (delta x d) = delta (headDelta x) (delta-mu (delta-fmap tailDelta d)) 43 delta-mu (delta x d) = delta (headDelta x) (delta-mu (delta-fmap tailDelta d))
47 44
48 delta-bind : {l : Level} {A B : Set l} {n : Nat} -> (Delta A (S n)) -> (A -> Delta B (S n)) -> Delta B (S n) 45 delta-bind : {l : Level} {A B : Set l} {n : Nat} -> (Delta A (S n)) -> (A -> Delta B (S n)) -> Delta B (S n)
49 delta-bind d f = delta-mu (delta-fmap f d) 46 delta-bind d f = delta-mu (delta-fmap f d)
50
51 --delta-bind (mono x) f = f x
52 --delta-bind (delta x d) f = delta (headDelta (f x)) (tailDelta (f x))
53
54 47
55 {- 48 {-
56 -- Monad (Haskell) 49 -- Monad (Haskell)
57 delta-return : {l : Level} {A : Set l} -> A -> Delta A (S O) 50 delta-return : {l : Level} {A : Set l} -> A -> Delta A (S O)
58 delta-return = delta-eta 51 delta-return = delta-eta
60 _>>=_ : {l : Level} {A B : Set l} {n : Nat} -> 53 _>>=_ : {l : Level} {A B : Set l} {n : Nat} ->
61 (x : Delta A n) -> (f : A -> (Delta B n)) -> (Delta B n) 54 (x : Delta A n) -> (f : A -> (Delta B n)) -> (Delta B n)
62 d >>= f = delta-bind d f 55 d >>= f = delta-bind d f
63 56
64 -} 57 -}
65
66 {-
67 -- proofs
68
69 -- sub-proofs
70
71 n-tail-plus : {l : Level} {A : Set l} -> (n : Nat) -> ((n-tail {l} {A} n) ∙ tailDelta) ≡ n-tail (S n)
72 n-tail-plus O = refl
73 n-tail-plus (S n) = begin
74 n-tail (S n) ∙ tailDelta ≡⟨ refl ⟩
75 (tailDelta ∙ (n-tail n)) ∙ tailDelta ≡⟨ refl ⟩
76 tailDelta ∙ ((n-tail n) ∙ tailDelta) ≡⟨ cong (\t -> tailDelta ∙ t) (n-tail-plus n) ⟩
77 n-tail (S (S n))
78 ∎
79
80 n-tail-add : {l : Level} {A : Set l} {d : Delta A} -> (n m : Nat) -> (n-tail {l} {A} n) ∙ (n-tail m) ≡ n-tail (n + m)
81 n-tail-add O m = refl
82 n-tail-add (S n) O = begin
83 n-tail (S n) ∙ n-tail O ≡⟨ refl ⟩
84 n-tail (S n) ≡⟨ cong (\n -> n-tail n) (nat-add-right-zero (S n))⟩
85 n-tail (S n + O)
86 ∎
87 n-tail-add {l} {A} {d} (S n) (S m) = begin
88 n-tail (S n) ∙ n-tail (S m) ≡⟨ refl ⟩
89 (tailDelta ∙ (n-tail n)) ∙ n-tail (S m) ≡⟨ refl ⟩
90 tailDelta ∙ ((n-tail n) ∙ n-tail (S m)) ≡⟨ cong (\t -> tailDelta ∙ t) (n-tail-add {l} {A} {d} n (S m)) ⟩
91 tailDelta ∙ (n-tail (n + (S m))) ≡⟨ refl ⟩
92 n-tail (S (n + S m)) ≡⟨ refl ⟩
93 n-tail (S n + S m) ∎
94
95 tail-delta-to-mono : {l : Level} {A : Set l} -> (n : Nat) -> (x : A) ->
96 (n-tail n) (mono x) ≡ (mono x)
97 tail-delta-to-mono O x = refl
98 tail-delta-to-mono (S n) x = begin
99 n-tail (S n) (mono x) ≡⟨ refl ⟩
100 tailDelta (n-tail n (mono x)) ≡⟨ refl ⟩
101 tailDelta (n-tail n (mono x)) ≡⟨ cong (\t -> tailDelta t) (tail-delta-to-mono n x) ⟩
102 tailDelta (mono x) ≡⟨ refl ⟩
103 mono x ∎
104
105 head-delta-natural-transformation : {l : Level} {A B : Set l}
106 -> (f : A -> B) -> (d : Delta A) -> headDelta (delta-fmap f d) ≡ f (headDelta d)
107 head-delta-natural-transformation f (mono x) = refl
108 head-delta-natural-transformation f (delta x d) = refl
109
110 n-tail-natural-transformation : {l : Level} {A B : Set l}
111 -> (n : Nat) -> (f : A -> B) -> (d : Delta A) -> n-tail n (delta-fmap f d) ≡ delta-fmap f (n-tail n d)
112 n-tail-natural-transformation O f d = refl
113 n-tail-natural-transformation (S n) f (mono x) = begin
114 n-tail (S n) (delta-fmap f (mono x)) ≡⟨ refl ⟩
115 n-tail (S n) (mono (f x)) ≡⟨ tail-delta-to-mono (S n) (f x) ⟩
116 (mono (f x)) ≡⟨ refl ⟩
117 delta-fmap f (mono x) ≡⟨ cong (\d -> delta-fmap f d) (sym (tail-delta-to-mono (S n) x)) ⟩
118 delta-fmap f (n-tail (S n) (mono x)) ∎
119 n-tail-natural-transformation (S n) f (delta x d) = begin
120 n-tail (S n) (delta-fmap f (delta x d)) ≡⟨ refl ⟩
121 n-tail (S n) (delta (f x) (delta-fmap f d)) ≡⟨ cong (\t -> t (delta (f x) (delta-fmap f d))) (sym (n-tail-plus n)) ⟩
122 ((n-tail n) ∙ tailDelta) (delta (f x) (delta-fmap f d)) ≡⟨ refl ⟩
123 n-tail n (delta-fmap f d) ≡⟨ n-tail-natural-transformation n f d ⟩
124 delta-fmap f (n-tail n d) ≡⟨ refl ⟩
125 delta-fmap f (((n-tail n) ∙ tailDelta) (delta x d)) ≡⟨ cong (\t -> delta-fmap f (t (delta x d))) (n-tail-plus n) ⟩
126 delta-fmap f (n-tail (S n) (delta x d)) ∎
127 -}