Mercurial > hg > Members > atton > delta_monad

annotate agda/delta/functor.agda @ 105:e6499a50ccbd

Find changesets by keywords (author, files, the commit message), revision number or hash, or revset expression.
Retrying prove monad-laws for delta
author Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
date Tue, 27 Jan 2015 17:49:25 +0900
parents ebd0d6e2772c
children 0a3b6cb91a05
Ignore whitespace changes - Everywhere: Within whitespace: At end of lines:
rev   line source
87
6789c65a75bc Split functor-proofs into delta.functor
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
diff changeset
1 open import Level
6789c65a75bc Split functor-proofs into delta.functor
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
diff changeset
2 open import Relation.Binary.PropositionalEquality
6789c65a75bc Split functor-proofs into delta.functor
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
diff changeset
3
104
ebd0d6e2772c Trying redenition Delta with length constraints
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 97
diff changeset
4 open import basic
ebd0d6e2772c Trying redenition Delta with length constraints
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 97
diff changeset
5 open import delta
ebd0d6e2772c Trying redenition Delta with length constraints
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 97
diff changeset
6 open import laws
ebd0d6e2772c Trying redenition Delta with length constraints
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 97
diff changeset
7 open import nat
ebd0d6e2772c Trying redenition Delta with length constraints
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 97
diff changeset
8
87
6789c65a75bc Split functor-proofs into delta.functor
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
diff changeset
9 module delta.functor where
6789c65a75bc Split functor-proofs into delta.functor
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
diff changeset
10
6789c65a75bc Split functor-proofs into delta.functor
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
diff changeset
11 -- Functor-laws
6789c65a75bc Split functor-proofs into delta.functor
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
diff changeset
12
6789c65a75bc Split functor-proofs into delta.functor
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
diff changeset
13 -- Functor-law-1 : T(id) = id'
105
e6499a50ccbd Retrying prove monad-laws for delta
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 104
diff changeset
14 functor-law-1 : {l : Level} {A : Set l} {n : Nat} -> (d : Delta A (S n)) -> (delta-fmap id) d ≡ id d
87
6789c65a75bc Split functor-proofs into delta.functor
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
diff changeset
15 functor-law-1 (mono x) = refl
6789c65a75bc Split functor-proofs into delta.functor
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
diff changeset
16 functor-law-1 (delta x d) = cong (delta x) (functor-law-1 d)
6789c65a75bc Split functor-proofs into delta.functor
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
diff changeset
17
6789c65a75bc Split functor-proofs into delta.functor
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
diff changeset
18 -- Functor-law-2 : T(f . g) = T(f) . T(g)
105
e6499a50ccbd Retrying prove monad-laws for delta
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 104
diff changeset
19 functor-law-2 : {l : Level} {n : Nat} {A B C : Set l} ->
e6499a50ccbd Retrying prove monad-laws for delta
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 104
diff changeset
20 (f : B -> C) -> (g : A -> B) -> (d : Delta A (S n)) ->
93
8d92ed54a94f Prove functor-laws for deltaM
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 90
diff changeset
21 (delta-fmap (f ∙ g)) d ≡ ((delta-fmap f) ∙ (delta-fmap g)) d
87
6789c65a75bc Split functor-proofs into delta.functor
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
diff changeset
22 functor-law-2 f g (mono x) = refl
6789c65a75bc Split functor-proofs into delta.functor
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
diff changeset
23 functor-law-2 f g (delta x d) = cong (delta (f (g x))) (functor-law-2 f g d)
6789c65a75bc Split functor-proofs into delta.functor
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
diff changeset
24
105
e6499a50ccbd Retrying prove monad-laws for delta
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 104
diff changeset
25
e6499a50ccbd Retrying prove monad-laws for delta
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 104
diff changeset
26
e6499a50ccbd Retrying prove monad-laws for delta
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 104
diff changeset
27 delta-is-functor : {l : Level} {n : Nat} -> Functor {l} (\A -> Delta A (S n))
89
5411ce26d525 Defining DeltaM in Agda...
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 87
diff changeset
28 delta-is-functor = record { fmap = delta-fmap ;
87
6789c65a75bc Split functor-proofs into delta.functor
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
diff changeset
29 preserve-id = functor-law-1;
6789c65a75bc Split functor-proofs into delta.functor
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
diff changeset
30 covariant = \f g -> functor-law-2 g f}
105
e6499a50ccbd Retrying prove monad-laws for delta
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 104
diff changeset
31
e6499a50ccbd Retrying prove monad-laws for delta
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 104
diff changeset
32
e6499a50ccbd Retrying prove monad-laws for delta
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 104
diff changeset
33 open ≡-Reasoning
e6499a50ccbd Retrying prove monad-laws for delta
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 104
diff changeset
34 delta-fmap-equiv : {l : Level} {A B : Set l} {n : Nat}
e6499a50ccbd Retrying prove monad-laws for delta
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 104
diff changeset
35 (f g : A -> B) (eq : f ≡ g) (d : Delta A (S n)) ->
e6499a50ccbd Retrying prove monad-laws for delta
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 104
diff changeset
36 delta-fmap f d ≡ delta-fmap g d
e6499a50ccbd Retrying prove monad-laws for delta
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 104
diff changeset
37 delta-fmap-equiv f g eq (mono x) = begin
e6499a50ccbd Retrying prove monad-laws for delta
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 104
diff changeset
38 mono (f x) ≡⟨ cong (\h -> (mono (h x))) eq ⟩
e6499a50ccbd Retrying prove monad-laws for delta
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 104
diff changeset
39 mono (g x) ∎
e6499a50ccbd Retrying prove monad-laws for delta
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 104
diff changeset
40 delta-fmap-equiv f g eq (delta x d) = begin
e6499a50ccbd Retrying prove monad-laws for delta
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 104
diff changeset
41 delta (f x) (delta-fmap f d) ≡⟨ cong (\h -> (delta (h x) (delta-fmap f d))) eq ⟩
e6499a50ccbd Retrying prove monad-laws for delta
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 104
diff changeset
42 delta (g x) (delta-fmap f d) ≡⟨ cong (\fx -> (delta (g x) fx)) (delta-fmap-equiv f g eq d) ⟩
e6499a50ccbd Retrying prove monad-laws for delta
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 104
diff changeset
43 delta (g x) (delta-fmap g d) ∎