annotate agda/delta/monad.agda @ 96:dfe8c67390bd

Unify Levels in delta
author Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
date Tue, 20 Jan 2015 16:25:53 +0900
parents bcd4fe52a504
children a271f3ff1922
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526186c4f298 Split monad-proofs into delta.monad
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
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1 open import basic
526186c4f298 Split monad-proofs into delta.monad
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
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2 open import delta
526186c4f298 Split monad-proofs into delta.monad
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
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3 open import delta.functor
526186c4f298 Split monad-proofs into delta.monad
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
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4 open import nat
526186c4f298 Split monad-proofs into delta.monad
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
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5 open import laws
526186c4f298 Split monad-proofs into delta.monad
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
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6
526186c4f298 Split monad-proofs into delta.monad
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
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7
526186c4f298 Split monad-proofs into delta.monad
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
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8 open import Level
526186c4f298 Split monad-proofs into delta.monad
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
diff changeset
9 open import Relation.Binary.PropositionalEquality
526186c4f298 Split monad-proofs into delta.monad
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
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10 open ≡-Reasoning
526186c4f298 Split monad-proofs into delta.monad
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
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11
526186c4f298 Split monad-proofs into delta.monad
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
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12 module delta.monad where
526186c4f298 Split monad-proofs into delta.monad
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
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13
526186c4f298 Split monad-proofs into delta.monad
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
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14
526186c4f298 Split monad-proofs into delta.monad
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
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15 -- Monad-laws (Category)
526186c4f298 Split monad-proofs into delta.monad
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
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16
526186c4f298 Split monad-proofs into delta.monad
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
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diff changeset
17 monad-law-1-5 : {l : Level} {A : Set l} -> (m : Nat) (n : Nat) -> (ds : Delta (Delta A)) ->
94
bcd4fe52a504 Rewrite monad definitions for delta/deltaM
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 90
diff changeset
18 n-tail n (delta-bind ds (n-tail m)) ≡ delta-bind (n-tail n ds) (n-tail (m + n))
88
526186c4f298 Split monad-proofs into delta.monad
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
diff changeset
19 monad-law-1-5 O O ds = refl
526186c4f298 Split monad-proofs into delta.monad
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
diff changeset
20 monad-law-1-5 O (S n) (mono ds) = begin
94
bcd4fe52a504 Rewrite monad definitions for delta/deltaM
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 90
diff changeset
21 n-tail (S n) (delta-bind (mono ds) (n-tail O)) ≡⟨ refl ⟩
88
526186c4f298 Split monad-proofs into delta.monad
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
diff changeset
22 n-tail (S n) ds ≡⟨ refl ⟩
94
bcd4fe52a504 Rewrite monad definitions for delta/deltaM
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 90
diff changeset
23 delta-bind (mono ds) (n-tail (S n)) ≡⟨ cong (\de -> delta-bind de (n-tail (S n))) (sym (tail-delta-to-mono (S n) ds)) ⟩
bcd4fe52a504 Rewrite monad definitions for delta/deltaM
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 90
diff changeset
24 delta-bind (n-tail (S n) (mono ds)) (n-tail (S n)) ≡⟨ refl ⟩
bcd4fe52a504 Rewrite monad definitions for delta/deltaM
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 90
diff changeset
25 delta-bind (n-tail (S n) (mono ds)) (n-tail (O + S n))
88
526186c4f298 Split monad-proofs into delta.monad
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
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26
526186c4f298 Split monad-proofs into delta.monad
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
diff changeset
27 monad-law-1-5 O (S n) (delta d ds) = begin
94
bcd4fe52a504 Rewrite monad definitions for delta/deltaM
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 90
diff changeset
28 n-tail (S n) (delta-bind (delta d ds) (n-tail O)) ≡⟨ refl ⟩
bcd4fe52a504 Rewrite monad definitions for delta/deltaM
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 90
diff changeset
29 n-tail (S n) (delta (headDelta d) (delta-bind ds tailDelta )) ≡⟨ cong (\t -> t (delta (headDelta d) (delta-bind ds tailDelta ))) (sym (n-tail-plus n)) ⟩
bcd4fe52a504 Rewrite monad definitions for delta/deltaM
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 90
diff changeset
30 ((n-tail n) ∙ tailDelta) (delta (headDelta d) (delta-bind ds tailDelta )) ≡⟨ refl ⟩
bcd4fe52a504 Rewrite monad definitions for delta/deltaM
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 90
diff changeset
31 (n-tail n) (delta-bind ds tailDelta) ≡⟨ monad-law-1-5 (S O) n ds ⟩
bcd4fe52a504 Rewrite monad definitions for delta/deltaM
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 90
diff changeset
32 delta-bind (n-tail n ds) (n-tail (S n)) ≡⟨ refl ⟩
bcd4fe52a504 Rewrite monad definitions for delta/deltaM
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 90
diff changeset
33 delta-bind (((n-tail n) ∙ tailDelta) (delta d ds)) (n-tail (S n)) ≡⟨ cong (\t -> delta-bind (t (delta d ds)) (n-tail (S n))) (n-tail-plus n) ⟩
bcd4fe52a504 Rewrite monad definitions for delta/deltaM
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 90
diff changeset
34 delta-bind (n-tail (S n) (delta d ds)) (n-tail (S n)) ≡⟨ refl ⟩
bcd4fe52a504 Rewrite monad definitions for delta/deltaM
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 90
diff changeset
35 delta-bind (n-tail (S n) (delta d ds)) (n-tail (O + S n))
88
526186c4f298 Split monad-proofs into delta.monad
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
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36
526186c4f298 Split monad-proofs into delta.monad
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
diff changeset
37 monad-law-1-5 (S m) n (mono (mono x)) = begin
94
bcd4fe52a504 Rewrite monad definitions for delta/deltaM
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 90
diff changeset
38 n-tail n (delta-bind (mono (mono x)) (n-tail (S m))) ≡⟨ refl ⟩
88
526186c4f298 Split monad-proofs into delta.monad
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
diff changeset
39 n-tail n (n-tail (S m) (mono x)) ≡⟨ cong (\de -> n-tail n de) (tail-delta-to-mono (S m) x)⟩
526186c4f298 Split monad-proofs into delta.monad
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
diff changeset
40 n-tail n (mono x) ≡⟨ tail-delta-to-mono n x ⟩
526186c4f298 Split monad-proofs into delta.monad
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
diff changeset
41 mono x ≡⟨ sym (tail-delta-to-mono (S m + n) x) ⟩
526186c4f298 Split monad-proofs into delta.monad
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
diff changeset
42 (n-tail (S m + n)) (mono x) ≡⟨ refl ⟩
94
bcd4fe52a504 Rewrite monad definitions for delta/deltaM
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 90
diff changeset
43 delta-bind (mono (mono x)) (n-tail (S m + n)) ≡⟨ cong (\de -> delta-bind de (n-tail (S m + n))) (sym (tail-delta-to-mono n (mono x))) ⟩
bcd4fe52a504 Rewrite monad definitions for delta/deltaM
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 90
diff changeset
44 delta-bind (n-tail n (mono (mono x))) (n-tail (S m + n))
88
526186c4f298 Split monad-proofs into delta.monad
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
diff changeset
45
526186c4f298 Split monad-proofs into delta.monad
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
diff changeset
46 monad-law-1-5 (S m) n (mono (delta x ds)) = begin
94
bcd4fe52a504 Rewrite monad definitions for delta/deltaM
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 90
diff changeset
47 n-tail n (delta-bind (mono (delta x ds)) (n-tail (S m))) ≡⟨ refl ⟩
88
526186c4f298 Split monad-proofs into delta.monad
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
diff changeset
48 n-tail n (n-tail (S m) (delta x ds)) ≡⟨ cong (\t -> n-tail n (t (delta x ds))) (sym (n-tail-plus m)) ⟩
526186c4f298 Split monad-proofs into delta.monad
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
diff changeset
49 n-tail n (((n-tail m) ∙ tailDelta) (delta x ds)) ≡⟨ refl ⟩
526186c4f298 Split monad-proofs into delta.monad
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
diff changeset
50 n-tail n ((n-tail m) ds) ≡⟨ cong (\t -> t ds) (n-tail-add {d = ds} n m) ⟩
526186c4f298 Split monad-proofs into delta.monad
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
diff changeset
51 n-tail (n + m) ds ≡⟨ cong (\n -> n-tail n ds) (nat-add-sym n m) ⟩
526186c4f298 Split monad-proofs into delta.monad
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
diff changeset
52 n-tail (m + n) ds ≡⟨ refl ⟩
526186c4f298 Split monad-proofs into delta.monad
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
diff changeset
53 ((n-tail (m + n)) ∙ tailDelta) (delta x ds) ≡⟨ cong (\t -> t (delta x ds)) (n-tail-plus (m + n))⟩
526186c4f298 Split monad-proofs into delta.monad
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
diff changeset
54 n-tail (S (m + n)) (delta x ds) ≡⟨ refl ⟩
526186c4f298 Split monad-proofs into delta.monad
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
diff changeset
55 n-tail (S m + n) (delta x ds) ≡⟨ refl ⟩
94
bcd4fe52a504 Rewrite monad definitions for delta/deltaM
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 90
diff changeset
56 delta-bind (mono (delta x ds)) (n-tail (S m + n)) ≡⟨ cong (\de -> delta-bind de (n-tail (S m + n))) (sym (tail-delta-to-mono n (delta x ds))) ⟩
bcd4fe52a504 Rewrite monad definitions for delta/deltaM
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 90
diff changeset
57 delta-bind (n-tail n (mono (delta x ds))) (n-tail (S m + n))
88
526186c4f298 Split monad-proofs into delta.monad
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
diff changeset
58
526186c4f298 Split monad-proofs into delta.monad
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
diff changeset
59 monad-law-1-5 (S m) O (delta d ds) = begin
94
bcd4fe52a504 Rewrite monad definitions for delta/deltaM
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 90
diff changeset
60 n-tail O (delta-bind (delta d ds) (n-tail (S m))) ≡⟨ refl ⟩
bcd4fe52a504 Rewrite monad definitions for delta/deltaM
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 90
diff changeset
61 (delta-bind (delta d ds) (n-tail (S m))) ≡⟨ refl ⟩
bcd4fe52a504 Rewrite monad definitions for delta/deltaM
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 90
diff changeset
62 delta (headDelta ((n-tail (S m)) d)) (delta-bind ds (tailDelta ∙ (n-tail (S m)))) ≡⟨ refl ⟩
bcd4fe52a504 Rewrite monad definitions for delta/deltaM
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 90
diff changeset
63 delta-bind (delta d ds) (n-tail (S m)) ≡⟨ refl ⟩
bcd4fe52a504 Rewrite monad definitions for delta/deltaM
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 90
diff changeset
64 delta-bind (n-tail O (delta d ds)) (n-tail (S m)) ≡⟨ cong (\n -> delta-bind (n-tail O (delta d ds)) (n-tail n)) (nat-add-right-zero (S m)) ⟩
bcd4fe52a504 Rewrite monad definitions for delta/deltaM
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 90
diff changeset
65 delta-bind (n-tail O (delta d ds)) (n-tail (S m + O))
88
526186c4f298 Split monad-proofs into delta.monad
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
diff changeset
66
526186c4f298 Split monad-proofs into delta.monad
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
diff changeset
67 monad-law-1-5 (S m) (S n) (delta d ds) = begin
94
bcd4fe52a504 Rewrite monad definitions for delta/deltaM
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 90
diff changeset
68 n-tail (S n) (delta-bind (delta d ds) (n-tail (S m))) ≡⟨ cong (\t -> t ((delta-bind (delta d ds) (n-tail (S m))))) (sym (n-tail-plus n)) ⟩
bcd4fe52a504 Rewrite monad definitions for delta/deltaM
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 90
diff changeset
69 ((n-tail n) ∙ tailDelta) (delta-bind (delta d ds) (n-tail (S m))) ≡⟨ refl ⟩
bcd4fe52a504 Rewrite monad definitions for delta/deltaM
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 90
diff changeset
70 ((n-tail n) ∙ tailDelta) (delta (headDelta ((n-tail (S m)) d)) (delta-bind ds (tailDelta ∙ (n-tail (S m))))) ≡⟨ refl ⟩
bcd4fe52a504 Rewrite monad definitions for delta/deltaM
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 90
diff changeset
71 (n-tail n) (delta-bind ds (tailDelta ∙ (n-tail (S m)))) ≡⟨ refl ⟩
bcd4fe52a504 Rewrite monad definitions for delta/deltaM
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 90
diff changeset
72 (n-tail n) (delta-bind ds (n-tail (S (S m)))) ≡⟨ monad-law-1-5 (S (S m)) n ds ⟩
bcd4fe52a504 Rewrite monad definitions for delta/deltaM
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 90
diff changeset
73 delta-bind ((n-tail n) ds) (n-tail (S (S (m + n)))) ≡⟨ cong (\nm -> delta-bind ((n-tail n) ds) (n-tail nm)) (sym (nat-right-increment (S m) n)) ⟩
bcd4fe52a504 Rewrite monad definitions for delta/deltaM
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 90
diff changeset
74 delta-bind ((n-tail n) ds) (n-tail (S m + S n)) ≡⟨ refl ⟩
bcd4fe52a504 Rewrite monad definitions for delta/deltaM
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 90
diff changeset
75 delta-bind (((n-tail n) ∙ tailDelta) (delta d ds)) (n-tail (S m + S n)) ≡⟨ cong (\t -> delta-bind (t (delta d ds)) (n-tail (S m + S n))) (n-tail-plus n) ⟩
bcd4fe52a504 Rewrite monad definitions for delta/deltaM
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 90
diff changeset
76 delta-bind (n-tail (S n) (delta d ds)) (n-tail (S m + S n))
88
526186c4f298 Split monad-proofs into delta.monad
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
diff changeset
77
526186c4f298 Split monad-proofs into delta.monad
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
diff changeset
78
526186c4f298 Split monad-proofs into delta.monad
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
diff changeset
79 monad-law-1-4 : {l : Level} {A : Set l} -> (m n : Nat) -> (dd : Delta (Delta A)) ->
94
bcd4fe52a504 Rewrite monad definitions for delta/deltaM
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 90
diff changeset
80 headDelta ((n-tail n) (delta-bind dd (n-tail m))) ≡ headDelta ((n-tail (m + n)) (headDelta (n-tail n dd)))
88
526186c4f298 Split monad-proofs into delta.monad
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
diff changeset
81 monad-law-1-4 O O (mono dd) = refl
526186c4f298 Split monad-proofs into delta.monad
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
diff changeset
82 monad-law-1-4 O O (delta dd dd₁) = refl
526186c4f298 Split monad-proofs into delta.monad
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
diff changeset
83 monad-law-1-4 O (S n) (mono dd) = begin
94
bcd4fe52a504 Rewrite monad definitions for delta/deltaM
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 90
diff changeset
84 headDelta (n-tail (S n) (delta-bind (mono dd) (n-tail O))) ≡⟨ refl ⟩
88
526186c4f298 Split monad-proofs into delta.monad
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
diff changeset
85 headDelta (n-tail (S n) dd) ≡⟨ refl ⟩
526186c4f298 Split monad-proofs into delta.monad
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
diff changeset
86 headDelta (n-tail (S n) (headDelta (mono dd))) ≡⟨ cong (\de -> headDelta (n-tail (S n) (headDelta de))) (sym (tail-delta-to-mono (S n) dd)) ⟩
526186c4f298 Split monad-proofs into delta.monad
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
diff changeset
87 headDelta (n-tail (S n) (headDelta (n-tail (S n) (mono dd)))) ≡⟨ refl ⟩
526186c4f298 Split monad-proofs into delta.monad
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
diff changeset
88 headDelta (n-tail (O + S n) (headDelta (n-tail (S n) (mono dd))))
526186c4f298 Split monad-proofs into delta.monad
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
diff changeset
89
526186c4f298 Split monad-proofs into delta.monad
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
diff changeset
90 monad-law-1-4 O (S n) (delta d ds) = begin
94
bcd4fe52a504 Rewrite monad definitions for delta/deltaM
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 90
diff changeset
91 headDelta (n-tail (S n) (delta-bind (delta d ds) (n-tail O))) ≡⟨ refl ⟩
bcd4fe52a504 Rewrite monad definitions for delta/deltaM
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 90
diff changeset
92 headDelta (n-tail (S n) (delta-bind (delta d ds) id)) ≡⟨ refl ⟩
bcd4fe52a504 Rewrite monad definitions for delta/deltaM
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 90
diff changeset
93 headDelta (n-tail (S n) (delta (headDelta d) (delta-bind ds tailDelta))) ≡⟨ cong (\t -> headDelta (t (delta (headDelta d) (delta-bind ds tailDelta)))) (sym (n-tail-plus n)) ⟩
bcd4fe52a504 Rewrite monad definitions for delta/deltaM
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 90
diff changeset
94 headDelta (((n-tail n) ∙ tailDelta) (delta (headDelta d) (delta-bind ds tailDelta))) ≡⟨ refl ⟩
bcd4fe52a504 Rewrite monad definitions for delta/deltaM
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 90
diff changeset
95 headDelta (n-tail n (delta-bind ds tailDelta)) ≡⟨ monad-law-1-4 (S O) n ds ⟩
bcd4fe52a504 Rewrite monad definitions for delta/deltaM
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 90
diff changeset
96 headDelta (n-tail (S n) (headDelta (n-tail n ds))) ≡⟨ refl ⟩
bcd4fe52a504 Rewrite monad definitions for delta/deltaM
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 90
diff changeset
97 headDelta (n-tail (S n) (headDelta (((n-tail n) ∙ tailDelta) (delta d ds)))) ≡⟨ cong (\t -> headDelta (n-tail (S n) (headDelta (t (delta d ds))))) (n-tail-plus n) ⟩
bcd4fe52a504 Rewrite monad definitions for delta/deltaM
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 90
diff changeset
98 headDelta (n-tail (S n) (headDelta (n-tail (S n) (delta d ds)))) ≡⟨ refl ⟩
88
526186c4f298 Split monad-proofs into delta.monad
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
diff changeset
99 headDelta (n-tail (O + S n) (headDelta (n-tail (S n) (delta d ds))))
526186c4f298 Split monad-proofs into delta.monad
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
diff changeset
100
526186c4f298 Split monad-proofs into delta.monad
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
diff changeset
101 monad-law-1-4 (S m) n (mono dd) = begin
94
bcd4fe52a504 Rewrite monad definitions for delta/deltaM
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 90
diff changeset
102 headDelta (n-tail n (delta-bind (mono dd) (n-tail (S m)))) ≡⟨ refl ⟩
bcd4fe52a504 Rewrite monad definitions for delta/deltaM
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 90
diff changeset
103 headDelta (n-tail n ((n-tail (S m)) dd)) ≡⟨ cong (\t -> headDelta (t dd)) (n-tail-add {d = dd} n (S m)) ⟩
bcd4fe52a504 Rewrite monad definitions for delta/deltaM
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 90
diff changeset
104 headDelta (n-tail (n + S m) dd) ≡⟨ cong (\n -> headDelta ((n-tail n) dd)) (nat-add-sym n (S m)) ⟩
bcd4fe52a504 Rewrite monad definitions for delta/deltaM
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 90
diff changeset
105 headDelta (n-tail (S m + n) dd) ≡⟨ refl ⟩
bcd4fe52a504 Rewrite monad definitions for delta/deltaM
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 90
diff changeset
106 headDelta (n-tail (S m + n) (headDelta (mono dd))) ≡⟨ cong (\de -> headDelta (n-tail (S m + n) (headDelta de))) (sym (tail-delta-to-mono n dd)) ⟩
88
526186c4f298 Split monad-proofs into delta.monad
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
diff changeset
107 headDelta (n-tail (S m + n) (headDelta (n-tail n (mono dd))))
526186c4f298 Split monad-proofs into delta.monad
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
diff changeset
108
526186c4f298 Split monad-proofs into delta.monad
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
diff changeset
109 monad-law-1-4 (S m) O (delta d ds) = begin
94
bcd4fe52a504 Rewrite monad definitions for delta/deltaM
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 90
diff changeset
110 headDelta (n-tail O (delta-bind (delta d ds) (n-tail (S m)))) ≡⟨ refl ⟩
bcd4fe52a504 Rewrite monad definitions for delta/deltaM
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 90
diff changeset
111 headDelta (delta-bind (delta d ds) (n-tail (S m))) ≡⟨ refl ⟩
bcd4fe52a504 Rewrite monad definitions for delta/deltaM
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 90
diff changeset
112 headDelta (delta (headDelta ((n-tail (S m) d))) (delta-bind ds (tailDelta ∙ (n-tail (S m))))) ≡⟨ refl ⟩
88
526186c4f298 Split monad-proofs into delta.monad
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
diff changeset
113 headDelta (n-tail (S m) d) ≡⟨ cong (\n -> headDelta ((n-tail n) d)) (nat-add-right-zero (S m)) ⟩
526186c4f298 Split monad-proofs into delta.monad
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
diff changeset
114 headDelta (n-tail (S m + O) d) ≡⟨ refl ⟩
526186c4f298 Split monad-proofs into delta.monad
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
diff changeset
115 headDelta (n-tail (S m + O) (headDelta (delta d ds))) ≡⟨ refl ⟩
526186c4f298 Split monad-proofs into delta.monad
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
diff changeset
116 headDelta (n-tail (S m + O) (headDelta (n-tail O (delta d ds))))
526186c4f298 Split monad-proofs into delta.monad
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
diff changeset
117
526186c4f298 Split monad-proofs into delta.monad
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
diff changeset
118 monad-law-1-4 (S m) (S n) (delta d ds) = begin
94
bcd4fe52a504 Rewrite monad definitions for delta/deltaM
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 90
diff changeset
119 headDelta (n-tail (S n) (delta-bind (delta d ds) (n-tail (S m)))) ≡⟨ refl ⟩
bcd4fe52a504 Rewrite monad definitions for delta/deltaM
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 90
diff changeset
120 headDelta (n-tail (S n) (delta (headDelta ((n-tail (S m)) d)) (delta-bind ds (tailDelta ∙ (n-tail (S m)))))) ≡⟨ cong (\t -> headDelta (t (delta (headDelta ((n-tail (S m)) d)) (delta-bind ds (tailDelta ∙ (n-tail (S m))))))) (sym (n-tail-plus n)) ⟩
bcd4fe52a504 Rewrite monad definitions for delta/deltaM
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 90
diff changeset
121 headDelta ((((n-tail n) ∙ tailDelta) (delta (headDelta ((n-tail (S m)) d)) (delta-bind ds (tailDelta ∙ (n-tail (S m))))))) ≡⟨ refl ⟩
bcd4fe52a504 Rewrite monad definitions for delta/deltaM
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 90
diff changeset
122 headDelta (n-tail n (delta-bind ds (tailDelta ∙ (n-tail (S m))))) ≡⟨ refl ⟩
bcd4fe52a504 Rewrite monad definitions for delta/deltaM
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 90
diff changeset
123 headDelta (n-tail n (delta-bind ds (n-tail (S (S m))))) ≡⟨ monad-law-1-4 (S (S m)) n ds ⟩
88
526186c4f298 Split monad-proofs into delta.monad
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
diff changeset
124 headDelta (n-tail ((S (S m) + n)) (headDelta (n-tail n ds))) ≡⟨ cong (\nm -> headDelta ((n-tail nm) (headDelta (n-tail n ds)))) (sym (nat-right-increment (S m) n)) ⟩
526186c4f298 Split monad-proofs into delta.monad
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
diff changeset
125 headDelta (n-tail (S m + S n) (headDelta (n-tail n ds))) ≡⟨ refl ⟩
526186c4f298 Split monad-proofs into delta.monad
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
diff changeset
126 headDelta (n-tail (S m + S n) (headDelta (((n-tail n) ∙ tailDelta) (delta d ds)))) ≡⟨ cong (\t -> headDelta (n-tail (S m + S n) (headDelta (t (delta d ds))))) (n-tail-plus n) ⟩
526186c4f298 Split monad-proofs into delta.monad
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
diff changeset
127 headDelta (n-tail (S m + S n) (headDelta (n-tail (S n) (delta d ds))))
526186c4f298 Split monad-proofs into delta.monad
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
diff changeset
128
526186c4f298 Split monad-proofs into delta.monad
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
diff changeset
129
94
bcd4fe52a504 Rewrite monad definitions for delta/deltaM
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 90
diff changeset
130 monad-law-1-2 : {l : Level} {A : Set l} -> (d : Delta (Delta A)) -> headDelta (delta-mu d) ≡ (headDelta (headDelta d))
96
dfe8c67390bd Unify Levels in delta
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 94
diff changeset
131 monad-law-1-2 (mono _) = refl
88
526186c4f298 Split monad-proofs into delta.monad
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
diff changeset
132 monad-law-1-2 (delta _ _) = refl
526186c4f298 Split monad-proofs into delta.monad
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
diff changeset
133
526186c4f298 Split monad-proofs into delta.monad
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
diff changeset
134 monad-law-1-3 : {l : Level} {A : Set l} -> (n : Nat) -> (d : Delta (Delta (Delta A))) ->
94
bcd4fe52a504 Rewrite monad definitions for delta/deltaM
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 90
diff changeset
135 delta-bind (delta-fmap delta-mu d) (n-tail n) ≡ delta-bind (delta-bind d (n-tail n)) (n-tail n)
88
526186c4f298 Split monad-proofs into delta.monad
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
diff changeset
136 monad-law-1-3 O (mono d) = refl
526186c4f298 Split monad-proofs into delta.monad
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
diff changeset
137 monad-law-1-3 O (delta d ds) = begin
94
bcd4fe52a504 Rewrite monad definitions for delta/deltaM
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 90
diff changeset
138 delta-bind (delta-fmap delta-mu (delta d ds)) (n-tail O) ≡⟨ refl ⟩
bcd4fe52a504 Rewrite monad definitions for delta/deltaM
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 90
diff changeset
139 delta-bind (delta (delta-mu d) (delta-fmap delta-mu ds)) (n-tail O) ≡⟨ refl ⟩
bcd4fe52a504 Rewrite monad definitions for delta/deltaM
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 90
diff changeset
140 delta (headDelta (delta-mu d)) (delta-bind (delta-fmap delta-mu ds) tailDelta) ≡⟨ cong (\dx -> delta dx (delta-bind (delta-fmap delta-mu ds) tailDelta)) (monad-law-1-2 d) ⟩
bcd4fe52a504 Rewrite monad definitions for delta/deltaM
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 90
diff changeset
141 delta (headDelta (headDelta d)) (delta-bind (delta-fmap delta-mu ds) tailDelta) ≡⟨ cong (\dx -> delta (headDelta (headDelta d)) dx) (monad-law-1-3 (S O) ds) ⟩
bcd4fe52a504 Rewrite monad definitions for delta/deltaM
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 90
diff changeset
142 delta (headDelta (headDelta d)) (delta-bind (delta-bind ds tailDelta) tailDelta) ≡⟨ refl ⟩
bcd4fe52a504 Rewrite monad definitions for delta/deltaM
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 90
diff changeset
143 delta-bind (delta (headDelta d) (delta-bind ds tailDelta)) (n-tail O) ≡⟨ refl ⟩
bcd4fe52a504 Rewrite monad definitions for delta/deltaM
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 90
diff changeset
144 delta-bind (delta-bind (delta d ds) (n-tail O)) (n-tail O)
88
526186c4f298 Split monad-proofs into delta.monad
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
diff changeset
145
526186c4f298 Split monad-proofs into delta.monad
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
diff changeset
146 monad-law-1-3 (S n) (mono (mono d)) = begin
94
bcd4fe52a504 Rewrite monad definitions for delta/deltaM
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 90
diff changeset
147 delta-bind (delta-fmap delta-mu (mono (mono d))) (n-tail (S n)) ≡⟨ refl ⟩
bcd4fe52a504 Rewrite monad definitions for delta/deltaM
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 90
diff changeset
148 delta-bind (mono d) (n-tail (S n)) ≡⟨ refl ⟩
88
526186c4f298 Split monad-proofs into delta.monad
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
diff changeset
149 (n-tail (S n)) d ≡⟨ refl ⟩
94
bcd4fe52a504 Rewrite monad definitions for delta/deltaM
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 90
diff changeset
150 delta-bind (mono d) (n-tail (S n)) ≡⟨ cong (\t -> delta-bind t (n-tail (S n))) (sym (tail-delta-to-mono (S n) d))⟩
bcd4fe52a504 Rewrite monad definitions for delta/deltaM
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 90
diff changeset
151 delta-bind (n-tail (S n) (mono d)) (n-tail (S n)) ≡⟨ refl ⟩
bcd4fe52a504 Rewrite monad definitions for delta/deltaM
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 90
diff changeset
152 delta-bind (n-tail (S n) (mono d)) (n-tail (S n)) ≡⟨ refl ⟩
bcd4fe52a504 Rewrite monad definitions for delta/deltaM
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 90
diff changeset
153 delta-bind (delta-bind (mono (mono d)) (n-tail (S n))) (n-tail (S n))
88
526186c4f298 Split monad-proofs into delta.monad
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
diff changeset
154
526186c4f298 Split monad-proofs into delta.monad
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
diff changeset
155 monad-law-1-3 (S n) (mono (delta d ds)) = begin
94
bcd4fe52a504 Rewrite monad definitions for delta/deltaM
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 90
diff changeset
156 delta-bind (delta-fmap delta-mu (mono (delta d ds))) (n-tail (S n)) ≡⟨ refl ⟩
bcd4fe52a504 Rewrite monad definitions for delta/deltaM
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 90
diff changeset
157 delta-bind (mono (delta-mu (delta d ds))) (n-tail (S n)) ≡⟨ refl ⟩
bcd4fe52a504 Rewrite monad definitions for delta/deltaM
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 90
diff changeset
158 n-tail (S n) (delta-mu (delta d ds)) ≡⟨ refl ⟩
bcd4fe52a504 Rewrite monad definitions for delta/deltaM
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 90
diff changeset
159 n-tail (S n) (delta (headDelta d) (delta-bind ds tailDelta)) ≡⟨ cong (\t -> t (delta (headDelta d) (delta-bind ds tailDelta))) (sym (n-tail-plus n)) ⟩
bcd4fe52a504 Rewrite monad definitions for delta/deltaM
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 90
diff changeset
160 (n-tail n ∙ tailDelta) (delta (headDelta d) (delta-bind ds tailDelta)) ≡⟨ refl ⟩
bcd4fe52a504 Rewrite monad definitions for delta/deltaM
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 90
diff changeset
161 n-tail n (delta-bind ds tailDelta) ≡⟨ monad-law-1-5 (S O) n ds ⟩
bcd4fe52a504 Rewrite monad definitions for delta/deltaM
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 90
diff changeset
162 delta-bind (n-tail n ds) (n-tail (S n)) ≡⟨ refl ⟩
bcd4fe52a504 Rewrite monad definitions for delta/deltaM
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 90
diff changeset
163 delta-bind (((n-tail n) ∙ tailDelta) (delta d ds)) (n-tail (S n)) ≡⟨ cong (\t -> (delta-bind (t (delta d ds)) (n-tail (S n)))) (n-tail-plus n) ⟩
bcd4fe52a504 Rewrite monad definitions for delta/deltaM
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 90
diff changeset
164 delta-bind (n-tail (S n) (delta d ds)) (n-tail (S n)) ≡⟨ refl ⟩
bcd4fe52a504 Rewrite monad definitions for delta/deltaM
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 90
diff changeset
165 delta-bind (delta-bind (mono (delta d ds)) (n-tail (S n))) (n-tail (S n))
88
526186c4f298 Split monad-proofs into delta.monad
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
diff changeset
166
526186c4f298 Split monad-proofs into delta.monad
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
diff changeset
167 monad-law-1-3 (S n) (delta (mono d) ds) = begin
94
bcd4fe52a504 Rewrite monad definitions for delta/deltaM
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 90
diff changeset
168 delta-bind (delta-fmap delta-mu (delta (mono d) ds)) (n-tail (S n)) ≡⟨ refl ⟩
bcd4fe52a504 Rewrite monad definitions for delta/deltaM
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 90
diff changeset
169 delta-bind (delta (delta-mu (mono d)) (delta-fmap delta-mu ds)) (n-tail (S n)) ≡⟨ refl ⟩
bcd4fe52a504 Rewrite monad definitions for delta/deltaM
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 90
diff changeset
170 delta-bind (delta d (delta-fmap delta-mu ds)) (n-tail (S n)) ≡⟨ refl ⟩
bcd4fe52a504 Rewrite monad definitions for delta/deltaM
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 90
diff changeset
171 delta (headDelta ((n-tail (S n)) d)) (delta-bind (delta-fmap delta-mu ds) (tailDelta ∙ (n-tail (S n)))) ≡⟨ refl ⟩
bcd4fe52a504 Rewrite monad definitions for delta/deltaM
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 90
diff changeset
172 delta (headDelta ((n-tail (S n)) d)) (delta-bind (delta-fmap delta-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) ⟩
bcd4fe52a504 Rewrite monad definitions for delta/deltaM
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 90
diff changeset
173 delta (headDelta ((n-tail (S n)) d)) (delta-bind (delta-bind ds (n-tail (S (S n)))) (n-tail (S (S n)))) ≡⟨ refl ⟩
bcd4fe52a504 Rewrite monad definitions for delta/deltaM
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 90
diff changeset
174 delta (headDelta ((n-tail (S n)) d)) (delta-bind (delta-bind ds (tailDelta ∙ (n-tail (S n)))) (n-tail (S (S n)))) ≡⟨ refl ⟩
bcd4fe52a504 Rewrite monad definitions for delta/deltaM
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 90
diff changeset
175 delta (headDelta ((n-tail (S n)) d)) (delta-bind (delta-bind ds (tailDelta ∙ (n-tail (S n)))) (tailDelta ∙ (n-tail (S n)))) ≡⟨ refl ⟩
bcd4fe52a504 Rewrite monad definitions for delta/deltaM
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 90
diff changeset
176 delta (headDelta ((n-tail (S n)) (headDelta (mono d)))) (delta-bind (delta-bind ds (tailDelta ∙ (n-tail (S n)))) (tailDelta ∙ (n-tail (S n)))) ≡⟨ cong (\de -> delta (headDelta ((n-tail (S n)) (headDelta de))) (delta-bind (delta-bind ds (tailDelta ∙ (n-tail (S n)))) (tailDelta ∙ (n-tail (S n))))) (sym (tail-delta-to-mono (S n) d)) ⟩
bcd4fe52a504 Rewrite monad definitions for delta/deltaM
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 90
diff changeset
177 delta (headDelta ((n-tail (S n)) (headDelta ((n-tail (S n)) (mono d))))) (delta-bind (delta-bind ds (tailDelta ∙ (n-tail (S n)))) (tailDelta ∙ (n-tail (S n)))) ≡⟨ refl ⟩
bcd4fe52a504 Rewrite monad definitions for delta/deltaM
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 90
diff changeset
178 delta-bind (delta (headDelta ((n-tail (S n)) (mono d))) (delta-bind ds (tailDelta ∙ (n-tail (S n))))) (n-tail (S n)) ≡⟨ refl ⟩
bcd4fe52a504 Rewrite monad definitions for delta/deltaM
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 90
diff changeset
179 delta-bind (delta-bind (delta (mono d) ds) (n-tail (S n))) (n-tail (S n))
88
526186c4f298 Split monad-proofs into delta.monad
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
diff changeset
180
526186c4f298 Split monad-proofs into delta.monad
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
diff changeset
181 monad-law-1-3 (S n) (delta (delta d dd) ds) = begin
94
bcd4fe52a504 Rewrite monad definitions for delta/deltaM
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 90
diff changeset
182 delta-bind (delta-fmap delta-mu (delta (delta d dd) ds)) (n-tail (S n)) ≡⟨ refl ⟩
bcd4fe52a504 Rewrite monad definitions for delta/deltaM
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 90
diff changeset
183 delta-bind (delta (delta-mu (delta d dd)) (delta-fmap delta-mu ds)) (n-tail (S n)) ≡⟨ refl ⟩
bcd4fe52a504 Rewrite monad definitions for delta/deltaM
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 90
diff changeset
184 delta (headDelta ((n-tail (S n)) (delta-mu (delta d dd)))) (delta-bind (delta-fmap delta-mu ds) (tailDelta ∙ (n-tail (S n)))) ≡⟨ refl ⟩
bcd4fe52a504 Rewrite monad definitions for delta/deltaM
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 90
diff changeset
185 delta (headDelta ((n-tail (S n)) (delta (headDelta d) (delta-bind dd tailDelta)))) (delta-bind (delta-fmap delta-mu ds) (tailDelta ∙ (n-tail (S n)))) ≡⟨ cong (\t -> delta (headDelta (t (delta (headDelta d) (delta-bind dd tailDelta)))) (delta-bind (delta-fmap delta-mu ds) (tailDelta ∙ (n-tail (S n)))))(sym (n-tail-plus n)) ⟩
bcd4fe52a504 Rewrite monad definitions for delta/deltaM
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 90
diff changeset
186 delta (headDelta (((n-tail n) ∙ tailDelta) (delta (headDelta d) (delta-bind dd tailDelta)))) (delta-bind (delta-fmap delta-mu ds) (tailDelta ∙ (n-tail (S n)))) ≡⟨ refl ⟩
bcd4fe52a504 Rewrite monad definitions for delta/deltaM
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 90
diff changeset
187 delta (headDelta ((n-tail n) (delta-bind dd tailDelta))) (delta-bind (delta-fmap delta-mu ds) (tailDelta ∙ (n-tail (S n)))) ≡⟨ refl ⟩
bcd4fe52a504 Rewrite monad definitions for delta/deltaM
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 90
diff changeset
188 delta (headDelta ((n-tail n) (delta-bind dd tailDelta))) (delta-bind (delta-fmap delta-mu ds) (n-tail (S (S n)))) ≡⟨ cong (\de -> delta (headDelta ((n-tail n) (delta-bind dd tailDelta))) de) (monad-law-1-3 (S (S n)) ds) ⟩
bcd4fe52a504 Rewrite monad definitions for delta/deltaM
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 90
diff changeset
189 delta (headDelta ((n-tail n) (delta-bind dd tailDelta))) (delta-bind (delta-bind ds (n-tail (S (S n)))) (n-tail (S (S n)))) ≡⟨ cong (\de -> delta de ( (delta-bind (delta-bind ds (n-tail (S (S n)))) (n-tail (S (S n)))))) (monad-law-1-4 (S O) n dd) ⟩
bcd4fe52a504 Rewrite monad definitions for delta/deltaM
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 90
diff changeset
190 delta (headDelta ((n-tail (S n)) (headDelta (n-tail n dd)))) (delta-bind (delta-bind ds (n-tail (S (S n)))) (n-tail (S (S n)))) ≡⟨ refl ⟩
bcd4fe52a504 Rewrite monad definitions for delta/deltaM
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 90
diff changeset
191 delta (headDelta ((n-tail (S n)) (headDelta (n-tail n dd)))) (delta-bind (delta-bind ds (n-tail (S (S n)))) (tailDelta ∙ (n-tail (S n)))) ≡⟨ refl ⟩
bcd4fe52a504 Rewrite monad definitions for delta/deltaM
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 90
diff changeset
192 delta (headDelta ((n-tail (S n)) (headDelta (n-tail n dd)))) (delta-bind (delta-bind ds (tailDelta ∙ (n-tail (S n)))) (tailDelta ∙ (n-tail (S n)))) ≡⟨ refl ⟩
bcd4fe52a504 Rewrite monad definitions for delta/deltaM
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 90
diff changeset
193 delta (headDelta ((n-tail (S n)) (headDelta (((n-tail n) ∙ tailDelta) (delta d dd))))) (delta-bind (delta-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))))) (delta-bind (delta-bind ds (tailDelta ∙ (n-tail (S n)))) (tailDelta ∙ (n-tail (S n))))) (n-tail-plus n) ⟩
bcd4fe52a504 Rewrite monad definitions for delta/deltaM
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 90
diff changeset
194 delta (headDelta ((n-tail (S n)) (headDelta ((n-tail (S n)) (delta d dd))))) (delta-bind (delta-bind ds (tailDelta ∙ (n-tail (S n)))) (tailDelta ∙ (n-tail (S n)))) ≡⟨ refl ⟩
bcd4fe52a504 Rewrite monad definitions for delta/deltaM
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 90
diff changeset
195 delta-bind (delta (headDelta ((n-tail (S n)) (delta d dd))) (delta-bind ds (tailDelta ∙ (n-tail (S n))))) (n-tail (S n)) ≡⟨ refl ⟩
bcd4fe52a504 Rewrite monad definitions for delta/deltaM
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 90
diff changeset
196 delta-bind (delta-bind (delta (delta d dd) ds) (n-tail (S n))) (n-tail (S n))
88
526186c4f298 Split monad-proofs into delta.monad
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parents:
diff changeset
197
526186c4f298 Split monad-proofs into delta.monad
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
diff changeset
198
526186c4f298 Split monad-proofs into delta.monad
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
diff changeset
199
90
55d11ce7e223 Unify levels on data type. only use suc to proofs
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 88
diff changeset
200 -- monad-law-1 : join . delta-fmap join = join . join
94
bcd4fe52a504 Rewrite monad definitions for delta/deltaM
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 90
diff changeset
201 monad-law-1 : {l : Level} {A : Set l} -> (d : Delta (Delta (Delta A))) -> ((delta-mu ∙ (delta-fmap delta-mu)) d) ≡ ((delta-mu ∙ delta-mu) d)
88
526186c4f298 Split monad-proofs into delta.monad
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
diff changeset
202 monad-law-1 (mono d) = refl
526186c4f298 Split monad-proofs into delta.monad
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
diff changeset
203 monad-law-1 (delta x d) = begin
94
bcd4fe52a504 Rewrite monad definitions for delta/deltaM
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 90
diff changeset
204 (delta-mu ∙ delta-fmap delta-mu) (delta x d) ≡⟨ refl ⟩
bcd4fe52a504 Rewrite monad definitions for delta/deltaM
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 90
diff changeset
205 delta-mu (delta-fmap delta-mu (delta x d)) ≡⟨ refl ⟩
bcd4fe52a504 Rewrite monad definitions for delta/deltaM
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 90
diff changeset
206 delta-mu (delta (delta-mu x) (delta-fmap delta-mu d)) ≡⟨ refl ⟩
bcd4fe52a504 Rewrite monad definitions for delta/deltaM
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 90
diff changeset
207 delta (headDelta (delta-mu x)) (delta-bind (delta-fmap delta-mu d) tailDelta) ≡⟨ cong (\x -> delta x (delta-bind (delta-fmap delta-mu d) tailDelta)) (monad-law-1-2 x) ⟩
bcd4fe52a504 Rewrite monad definitions for delta/deltaM
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 90
diff changeset
208 delta (headDelta (headDelta x)) (delta-bind (delta-fmap delta-mu d) tailDelta) ≡⟨ cong (\d -> delta (headDelta (headDelta x)) d) (monad-law-1-3 (S O) d) ⟩
bcd4fe52a504 Rewrite monad definitions for delta/deltaM
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 90
diff changeset
209 delta (headDelta (headDelta x)) (delta-bind (delta-bind d tailDelta) tailDelta) ≡⟨ refl ⟩
bcd4fe52a504 Rewrite monad definitions for delta/deltaM
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 90
diff changeset
210 delta-mu (delta (headDelta x) (delta-bind d tailDelta)) ≡⟨ refl ⟩
bcd4fe52a504 Rewrite monad definitions for delta/deltaM
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 90
diff changeset
211 delta-mu (delta-mu (delta x d)) ≡⟨ refl ⟩
bcd4fe52a504 Rewrite monad definitions for delta/deltaM
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 90
diff changeset
212 (delta-mu ∙ delta-mu) (delta x d)
88
526186c4f298 Split monad-proofs into delta.monad
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
diff changeset
213
526186c4f298 Split monad-proofs into delta.monad
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
diff changeset
214
526186c4f298 Split monad-proofs into delta.monad
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
diff changeset
215
94
bcd4fe52a504 Rewrite monad definitions for delta/deltaM
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 90
diff changeset
216 monad-law-2-1 : {l : Level} {A : Set l} -> (n : Nat) -> (d : Delta A) -> (delta-bind (delta-fmap delta-eta d) (n-tail n)) ≡ d
88
526186c4f298 Split monad-proofs into delta.monad
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
diff changeset
217 monad-law-2-1 O (mono x) = refl
526186c4f298 Split monad-proofs into delta.monad
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
diff changeset
218 monad-law-2-1 O (delta x d) = begin
94
bcd4fe52a504 Rewrite monad definitions for delta/deltaM
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 90
diff changeset
219 delta-bind (delta-fmap delta-eta (delta x d)) (n-tail O) ≡⟨ refl ⟩
bcd4fe52a504 Rewrite monad definitions for delta/deltaM
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 90
diff changeset
220 delta-bind (delta (delta-eta x) (delta-fmap delta-eta d)) id ≡⟨ refl ⟩
bcd4fe52a504 Rewrite monad definitions for delta/deltaM
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 90
diff changeset
221 delta (headDelta (delta-eta x)) (delta-bind (delta-fmap delta-eta d) tailDelta) ≡⟨ refl ⟩
bcd4fe52a504 Rewrite monad definitions for delta/deltaM
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 90
diff changeset
222 delta x (delta-bind (delta-fmap delta-eta d) tailDelta) ≡⟨ cong (\de -> delta x de) (monad-law-2-1 (S O) d) ⟩
88
526186c4f298 Split monad-proofs into delta.monad
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
diff changeset
223 delta x d ∎
526186c4f298 Split monad-proofs into delta.monad
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
diff changeset
224 monad-law-2-1 (S n) (mono x) = begin
94
bcd4fe52a504 Rewrite monad definitions for delta/deltaM
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 90
diff changeset
225 delta-bind (delta-fmap delta-eta (mono x)) (n-tail (S n)) ≡⟨ refl ⟩
bcd4fe52a504 Rewrite monad definitions for delta/deltaM
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 90
diff changeset
226 delta-bind (mono (mono x)) (n-tail (S n)) ≡⟨ refl ⟩
88
526186c4f298 Split monad-proofs into delta.monad
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
diff changeset
227 n-tail (S n) (mono x) ≡⟨ tail-delta-to-mono (S n) x ⟩
526186c4f298 Split monad-proofs into delta.monad
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
diff changeset
228 mono x ∎
526186c4f298 Split monad-proofs into delta.monad
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
diff changeset
229 monad-law-2-1 (S n) (delta x d) = begin
94
bcd4fe52a504 Rewrite monad definitions for delta/deltaM
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 90
diff changeset
230 delta-bind (delta-fmap delta-eta (delta x d)) (n-tail (S n)) ≡⟨ refl ⟩
bcd4fe52a504 Rewrite monad definitions for delta/deltaM
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 90
diff changeset
231 delta-bind (delta (delta-eta x) (delta-fmap delta-eta d)) (n-tail (S n)) ≡⟨ refl ⟩
bcd4fe52a504 Rewrite monad definitions for delta/deltaM
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 90
diff changeset
232 delta (headDelta ((n-tail (S n) (delta-eta x)))) (delta-bind (delta-fmap delta-eta d) (tailDelta ∙ (n-tail (S n)))) ≡⟨ refl ⟩
bcd4fe52a504 Rewrite monad definitions for delta/deltaM
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 90
diff changeset
233 delta (headDelta ((n-tail (S n) (delta-eta x)))) (delta-bind (delta-fmap delta-eta d) (n-tail (S (S n)))) ≡⟨ cong (\de -> delta (headDelta (de)) (delta-bind (delta-fmap delta-eta d) (n-tail (S (S n))))) (tail-delta-to-mono (S n) x) ⟩
bcd4fe52a504 Rewrite monad definitions for delta/deltaM
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 90
diff changeset
234 delta (headDelta (delta-eta x)) (delta-bind (delta-fmap delta-eta d) (n-tail (S (S n)))) ≡⟨ refl ⟩
bcd4fe52a504 Rewrite monad definitions for delta/deltaM
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 90
diff changeset
235 delta x (delta-bind (delta-fmap delta-eta d) (n-tail (S (S n)))) ≡⟨ cong (\d -> delta x d) (monad-law-2-1 (S (S n)) d) ⟩
88
526186c4f298 Split monad-proofs into delta.monad
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
diff changeset
236 delta x d
526186c4f298 Split monad-proofs into delta.monad
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
diff changeset
237
526186c4f298 Split monad-proofs into delta.monad
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
diff changeset
238
526186c4f298 Split monad-proofs into delta.monad
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
diff changeset
239
90
55d11ce7e223 Unify levels on data type. only use suc to proofs
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 88
diff changeset
240 -- monad-law-2 : join . delta-fmap return = join . return = id
55d11ce7e223 Unify levels on data type. only use suc to proofs
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 88
diff changeset
241 -- monad-law-2 join . delta-fmap return = join . return
88
526186c4f298 Split monad-proofs into delta.monad
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
diff changeset
242 monad-law-2 : {l : Level} {A : Set l} -> (d : Delta A) ->
94
bcd4fe52a504 Rewrite monad definitions for delta/deltaM
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 90
diff changeset
243 (delta-mu ∙ delta-fmap delta-eta) d ≡ (delta-mu ∙ delta-eta) d
88
526186c4f298 Split monad-proofs into delta.monad
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
diff changeset
244 monad-law-2 (mono x) = refl
526186c4f298 Split monad-proofs into delta.monad
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
diff changeset
245 monad-law-2 (delta x d) = begin
94
bcd4fe52a504 Rewrite monad definitions for delta/deltaM
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 90
diff changeset
246 (delta-mu ∙ delta-fmap delta-eta) (delta x d) ≡⟨ refl ⟩
bcd4fe52a504 Rewrite monad definitions for delta/deltaM
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 90
diff changeset
247 delta-mu (delta-fmap delta-eta (delta x d)) ≡⟨ refl ⟩
bcd4fe52a504 Rewrite monad definitions for delta/deltaM
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 90
diff changeset
248 delta-mu (delta (mono x) (delta-fmap delta-eta d)) ≡⟨ refl ⟩
bcd4fe52a504 Rewrite monad definitions for delta/deltaM
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 90
diff changeset
249 delta (headDelta (mono x)) (delta-bind (delta-fmap delta-eta d) tailDelta) ≡⟨ refl ⟩
bcd4fe52a504 Rewrite monad definitions for delta/deltaM
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 90
diff changeset
250 delta x (delta-bind (delta-fmap delta-eta d) tailDelta) ≡⟨ cong (\d -> delta x d) (monad-law-2-1 (S O) d) ⟩
88
526186c4f298 Split monad-proofs into delta.monad
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
diff changeset
251 (delta x d) ≡⟨ refl ⟩
94
bcd4fe52a504 Rewrite monad definitions for delta/deltaM
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 90
diff changeset
252 delta-mu (mono (delta x d)) ≡⟨ refl ⟩
bcd4fe52a504 Rewrite monad definitions for delta/deltaM
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 90
diff changeset
253 delta-mu (delta-eta (delta x d)) ≡⟨ refl ⟩
bcd4fe52a504 Rewrite monad definitions for delta/deltaM
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 90
diff changeset
254 (delta-mu ∙ delta-eta) (delta x d)
88
526186c4f298 Split monad-proofs into delta.monad
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
diff changeset
255
526186c4f298 Split monad-proofs into delta.monad
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
diff changeset
256
526186c4f298 Split monad-proofs into delta.monad
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
diff changeset
257
526186c4f298 Split monad-proofs into delta.monad
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
diff changeset
258 -- monad-law-2' : join . return = id
94
bcd4fe52a504 Rewrite monad definitions for delta/deltaM
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 90
diff changeset
259 monad-law-2' : {l : Level} {A : Set l} -> (d : Delta A) -> (delta-mu ∙ delta-eta) d ≡ id d
88
526186c4f298 Split monad-proofs into delta.monad
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
diff changeset
260 monad-law-2' d = refl
526186c4f298 Split monad-proofs into delta.monad
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
diff changeset
261
526186c4f298 Split monad-proofs into delta.monad
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
diff changeset
262
90
55d11ce7e223 Unify levels on data type. only use suc to proofs
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 88
diff changeset
263 -- monad-law-3 : return . f = delta-fmap f . return
94
bcd4fe52a504 Rewrite monad definitions for delta/deltaM
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 90
diff changeset
264 monad-law-3 : {l : Level} {A B : Set l} (f : A -> B) (x : A) -> (delta-eta ∙ f) x ≡ (delta-fmap f ∙ delta-eta) x
88
526186c4f298 Split monad-proofs into delta.monad
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
diff changeset
265 monad-law-3 f x = refl
526186c4f298 Split monad-proofs into delta.monad
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
diff changeset
266
526186c4f298 Split monad-proofs into delta.monad
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
diff changeset
267
96
dfe8c67390bd Unify Levels in delta
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 94
diff changeset
268 monad-law-4-1 : {l : Level} {A B : Set l} -> (n : Nat) -> (f : A -> B) -> (ds : Delta (Delta A)) ->
94
bcd4fe52a504 Rewrite monad definitions for delta/deltaM
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 90
diff changeset
269 delta-bind (delta-fmap (delta-fmap f) ds) (n-tail n) ≡ delta-fmap f (delta-bind ds (n-tail n))
88
526186c4f298 Split monad-proofs into delta.monad
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
diff changeset
270 monad-law-4-1 O f (mono d) = refl
526186c4f298 Split monad-proofs into delta.monad
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
diff changeset
271 monad-law-4-1 O f (delta d ds) = begin
94
bcd4fe52a504 Rewrite monad definitions for delta/deltaM
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 90
diff changeset
272 delta-bind (delta-fmap (delta-fmap f) (delta d ds)) (n-tail O) ≡⟨ refl ⟩
bcd4fe52a504 Rewrite monad definitions for delta/deltaM
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 90
diff changeset
273 delta-bind (delta (delta-fmap f d) (delta-fmap (delta-fmap f) ds)) (n-tail O) ≡⟨ refl ⟩
bcd4fe52a504 Rewrite monad definitions for delta/deltaM
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 90
diff changeset
274 delta (headDelta (delta-fmap f d)) (delta-bind (delta-fmap (delta-fmap f) ds) tailDelta) ≡⟨ cong (\de -> delta de (delta-bind (delta-fmap (delta-fmap f) ds) tailDelta)) (head-delta-natural-transformation f d) ⟩
bcd4fe52a504 Rewrite monad definitions for delta/deltaM
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 90
diff changeset
275 delta (f (headDelta d)) (delta-bind (delta-fmap (delta-fmap f) ds) tailDelta) ≡⟨ cong (\de -> delta (f (headDelta d)) de) (monad-law-4-1 (S O) f ds) ⟩
bcd4fe52a504 Rewrite monad definitions for delta/deltaM
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 90
diff changeset
276 delta (f (headDelta d)) (delta-fmap f (delta-bind ds tailDelta)) ≡⟨ refl ⟩
bcd4fe52a504 Rewrite monad definitions for delta/deltaM
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 90
diff changeset
277 delta-fmap f (delta (headDelta d) (delta-bind ds tailDelta)) ≡⟨ refl ⟩
bcd4fe52a504 Rewrite monad definitions for delta/deltaM
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 90
diff changeset
278 delta-fmap f (delta-bind (delta d ds) (n-tail O)) ∎
88
526186c4f298 Split monad-proofs into delta.monad
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
diff changeset
279 monad-law-4-1 (S n) f (mono d) = begin
94
bcd4fe52a504 Rewrite monad definitions for delta/deltaM
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 90
diff changeset
280 delta-bind (delta-fmap (delta-fmap f) (mono d)) (n-tail (S n)) ≡⟨ refl ⟩
bcd4fe52a504 Rewrite monad definitions for delta/deltaM
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 90
diff changeset
281 delta-bind (mono (delta-fmap f d)) (n-tail (S n)) ≡⟨ refl ⟩
90
55d11ce7e223 Unify levels on data type. only use suc to proofs
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 88
diff changeset
282 n-tail (S n) (delta-fmap f d) ≡⟨ n-tail-natural-transformation (S n) f d ⟩
55d11ce7e223 Unify levels on data type. only use suc to proofs
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 88
diff changeset
283 delta-fmap f (n-tail (S n) d) ≡⟨ refl ⟩
94
bcd4fe52a504 Rewrite monad definitions for delta/deltaM
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 90
diff changeset
284 delta-fmap f (delta-bind (mono d) (n-tail (S n)))
88
526186c4f298 Split monad-proofs into delta.monad
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
diff changeset
285
526186c4f298 Split monad-proofs into delta.monad
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
diff changeset
286 monad-law-4-1 (S n) f (delta d ds) = begin
94
bcd4fe52a504 Rewrite monad definitions for delta/deltaM
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 90
diff changeset
287 delta-bind (delta-fmap (delta-fmap f) (delta d ds)) (n-tail (S n)) ≡⟨ refl ⟩
bcd4fe52a504 Rewrite monad definitions for delta/deltaM
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 90
diff changeset
288 delta (headDelta (n-tail (S n) (delta-fmap f d))) (delta-bind (delta-fmap (delta-fmap f) ds) (tailDelta ∙ (n-tail (S n)))) ≡⟨ refl ⟩
bcd4fe52a504 Rewrite monad definitions for delta/deltaM
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 90
diff changeset
289 delta (headDelta (n-tail (S n) (delta-fmap f d))) (delta-bind (delta-fmap (delta-fmap f) ds) (n-tail (S (S n)))) ≡⟨ cong (\de -> delta (headDelta de) (delta-bind (delta-fmap (delta-fmap f) ds) (n-tail (S (S n))))) (n-tail-natural-transformation (S n) f d) ⟩
bcd4fe52a504 Rewrite monad definitions for delta/deltaM
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 90
diff changeset
290 delta (headDelta (delta-fmap f ((n-tail (S n) d)))) (delta-bind (delta-fmap (delta-fmap f) ds) (n-tail (S (S n)))) ≡⟨ cong (\de -> delta de (delta-bind (delta-fmap (delta-fmap f) ds) (n-tail (S (S n))))) (head-delta-natural-transformation f (n-tail (S n) d)) ⟩
bcd4fe52a504 Rewrite monad definitions for delta/deltaM
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 90
diff changeset
291 delta (f (headDelta (n-tail (S n) d))) (delta-bind (delta-fmap (delta-fmap f) ds) (n-tail (S (S n)))) ≡⟨ cong (\de -> delta (f (headDelta (n-tail (S n) d))) de) (monad-law-4-1 (S (S n)) f ds) ⟩
bcd4fe52a504 Rewrite monad definitions for delta/deltaM
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 90
diff changeset
292 delta (f (headDelta (n-tail (S n) d))) (delta-fmap f (delta-bind ds (n-tail (S (S n))))) ≡⟨ refl ⟩
bcd4fe52a504 Rewrite monad definitions for delta/deltaM
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 90
diff changeset
293 delta-fmap f (delta (headDelta (n-tail (S n) d)) (delta-bind ds (n-tail (S (S n))))) ≡⟨ refl ⟩
bcd4fe52a504 Rewrite monad definitions for delta/deltaM
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 90
diff changeset
294 delta-fmap f (delta (headDelta (n-tail (S n) d)) (delta-bind ds (tailDelta ∙ (n-tail (S n))))) ≡⟨ refl ⟩
bcd4fe52a504 Rewrite monad definitions for delta/deltaM
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 90
diff changeset
295 delta-fmap f (delta-bind (delta d ds) (n-tail (S n))) ∎
88
526186c4f298 Split monad-proofs into delta.monad
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
diff changeset
296
526186c4f298 Split monad-proofs into delta.monad
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
diff changeset
297
90
55d11ce7e223 Unify levels on data type. only use suc to proofs
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 88
diff changeset
298 -- monad-law-4 : join . delta-fmap (delta-fmap f) = delta-fmap f . join
55d11ce7e223 Unify levels on data type. only use suc to proofs
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 88
diff changeset
299 monad-law-4 : {l : Level} {A B : Set l} (f : A -> B) (d : Delta (Delta A)) ->
94
bcd4fe52a504 Rewrite monad definitions for delta/deltaM
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 90
diff changeset
300 (delta-mu ∙ delta-fmap (delta-fmap f)) d ≡ (delta-fmap f ∙ delta-mu) d
88
526186c4f298 Split monad-proofs into delta.monad
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
diff changeset
301 monad-law-4 f (mono d) = refl
526186c4f298 Split monad-proofs into delta.monad
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
diff changeset
302 monad-law-4 f (delta (mono x) ds) = begin
94
bcd4fe52a504 Rewrite monad definitions for delta/deltaM
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 90
diff changeset
303 (delta-mu ∙ delta-fmap (delta-fmap f)) (delta (mono x) ds) ≡⟨ refl ⟩
bcd4fe52a504 Rewrite monad definitions for delta/deltaM
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 90
diff changeset
304 delta-mu ( delta-fmap (delta-fmap f) (delta (mono x) ds)) ≡⟨ refl ⟩
bcd4fe52a504 Rewrite monad definitions for delta/deltaM
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 90
diff changeset
305 delta-mu (delta (mono (f x)) (delta-fmap (delta-fmap f) ds)) ≡⟨ refl ⟩
bcd4fe52a504 Rewrite monad definitions for delta/deltaM
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 90
diff changeset
306 delta (headDelta (mono (f x))) (delta-bind (delta-fmap (delta-fmap f) ds) tailDelta) ≡⟨ refl ⟩
bcd4fe52a504 Rewrite monad definitions for delta/deltaM
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 90
diff changeset
307 delta (f x) (delta-bind (delta-fmap (delta-fmap f) ds) tailDelta) ≡⟨ cong (\de -> delta (f x) de) (monad-law-4-1 (S O) f ds) ⟩
bcd4fe52a504 Rewrite monad definitions for delta/deltaM
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 90
diff changeset
308 delta (f x) (delta-fmap f (delta-bind ds tailDelta)) ≡⟨ refl ⟩
bcd4fe52a504 Rewrite monad definitions for delta/deltaM
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 90
diff changeset
309 delta-fmap f (delta x (delta-bind ds tailDelta)) ≡⟨ refl ⟩
bcd4fe52a504 Rewrite monad definitions for delta/deltaM
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 90
diff changeset
310 delta-fmap f (delta (headDelta (mono x)) (delta-bind ds tailDelta)) ≡⟨ refl ⟩
bcd4fe52a504 Rewrite monad definitions for delta/deltaM
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 90
diff changeset
311 delta-fmap f (delta-mu (delta (mono x) ds)) ≡⟨ refl ⟩
bcd4fe52a504 Rewrite monad definitions for delta/deltaM
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 90
diff changeset
312 (delta-fmap f ∙ delta-mu) (delta (mono x) ds) ∎
88
526186c4f298 Split monad-proofs into delta.monad
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
diff changeset
313 monad-law-4 f (delta (delta x d) ds) = begin
94
bcd4fe52a504 Rewrite monad definitions for delta/deltaM
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 90
diff changeset
314 (delta-mu ∙ delta-fmap (delta-fmap f)) (delta (delta x d) ds) ≡⟨ refl ⟩
bcd4fe52a504 Rewrite monad definitions for delta/deltaM
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 90
diff changeset
315 delta-mu (delta-fmap (delta-fmap f) (delta (delta x d) ds)) ≡⟨ refl ⟩
bcd4fe52a504 Rewrite monad definitions for delta/deltaM
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 90
diff changeset
316 delta-mu (delta (delta (f x) (delta-fmap f d)) (delta-fmap (delta-fmap f) ds)) ≡⟨ refl ⟩
bcd4fe52a504 Rewrite monad definitions for delta/deltaM
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 90
diff changeset
317 delta (headDelta (delta (f x) (delta-fmap f d))) (delta-bind (delta-fmap (delta-fmap f) ds) tailDelta) ≡⟨ refl ⟩
bcd4fe52a504 Rewrite monad definitions for delta/deltaM
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 90
diff changeset
318 delta (f x) (delta-bind (delta-fmap (delta-fmap f) ds) tailDelta) ≡⟨ cong (\de -> delta (f x) de) (monad-law-4-1 (S O) f ds) ⟩
bcd4fe52a504 Rewrite monad definitions for delta/deltaM
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 90
diff changeset
319 delta (f x) (delta-fmap f (delta-bind ds tailDelta)) ≡⟨ refl ⟩
bcd4fe52a504 Rewrite monad definitions for delta/deltaM
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 90
diff changeset
320 delta-fmap f (delta x (delta-bind ds tailDelta)) ≡⟨ refl ⟩
bcd4fe52a504 Rewrite monad definitions for delta/deltaM
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 90
diff changeset
321 delta-fmap f (delta (headDelta (delta x d)) (delta-bind ds tailDelta)) ≡⟨ refl ⟩
bcd4fe52a504 Rewrite monad definitions for delta/deltaM
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 90
diff changeset
322 delta-fmap f (delta-mu (delta (delta x d) ds)) ≡⟨ refl ⟩
bcd4fe52a504 Rewrite monad definitions for delta/deltaM
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 90
diff changeset
323 (delta-fmap f ∙ delta-mu) (delta (delta x d) ds) ∎
88
526186c4f298 Split monad-proofs into delta.monad
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
diff changeset
324
526186c4f298 Split monad-proofs into delta.monad
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
diff changeset
325 delta-is-monad : {l : Level} {A : Set l} -> Monad {l} {A} Delta delta-is-functor
94
bcd4fe52a504 Rewrite monad definitions for delta/deltaM
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 90
diff changeset
326 delta-is-monad = record { eta = delta-eta;
bcd4fe52a504 Rewrite monad definitions for delta/deltaM
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 90
diff changeset
327 mu = delta-mu;
bcd4fe52a504 Rewrite monad definitions for delta/deltaM
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 90
diff changeset
328 return = delta-eta;
bcd4fe52a504 Rewrite monad definitions for delta/deltaM
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 90
diff changeset
329 bind = delta-bind;
88
526186c4f298 Split monad-proofs into delta.monad
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
diff changeset
330 association-law = monad-law-1;
526186c4f298 Split monad-proofs into delta.monad
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
diff changeset
331 left-unity-law = monad-law-2;
526186c4f298 Split monad-proofs into delta.monad
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
diff changeset
332 right-unity-law = monad-law-2' }
526186c4f298 Split monad-proofs into delta.monad
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
diff changeset
333
526186c4f298 Split monad-proofs into delta.monad
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
diff changeset
334
96
dfe8c67390bd Unify Levels in delta
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 94
diff changeset
335
88
526186c4f298 Split monad-proofs into delta.monad
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
diff changeset
336 {-
96
dfe8c67390bd Unify Levels in delta
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 94
diff changeset
337
88
526186c4f298 Split monad-proofs into delta.monad
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
diff changeset
338 -- Monad-laws (Haskell)
526186c4f298 Split monad-proofs into delta.monad
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
diff changeset
339 -- monad-law-h-1 : return a >>= k = k a
96
dfe8c67390bd Unify Levels in delta
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 94
diff changeset
340 monad-law-h-1 : {l : Level} {A B : Set l} ->
88
526186c4f298 Split monad-proofs into delta.monad
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
diff changeset
341 (a : A) -> (k : A -> (Delta B)) ->
96
dfe8c67390bd Unify Levels in delta
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 94
diff changeset
342 (delta-return a >>= k) ≡ (k a)
88
526186c4f298 Split monad-proofs into delta.monad
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
diff changeset
343 monad-law-h-1 a k = refl
526186c4f298 Split monad-proofs into delta.monad
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
diff changeset
344
526186c4f298 Split monad-proofs into delta.monad
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
diff changeset
345
526186c4f298 Split monad-proofs into delta.monad
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
diff changeset
346
526186c4f298 Split monad-proofs into delta.monad
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
diff changeset
347 -- monad-law-h-2 : m >>= return = m
96
dfe8c67390bd Unify Levels in delta
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 94
diff changeset
348 monad-law-h-2 : {l : Level}{A : Set l} -> (m : Delta A) -> (m >>= delta-return) ≡ m
88
526186c4f298 Split monad-proofs into delta.monad
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
diff changeset
349 monad-law-h-2 (mono x) = refl
526186c4f298 Split monad-proofs into delta.monad
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
diff changeset
350 monad-law-h-2 (delta x d) = cong (delta x) (monad-law-h-2 d)
526186c4f298 Split monad-proofs into delta.monad
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
diff changeset
351
526186c4f298 Split monad-proofs into delta.monad
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
diff changeset
352
526186c4f298 Split monad-proofs into delta.monad
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
diff changeset
353
96
dfe8c67390bd Unify Levels in delta
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 94
diff changeset
354 -- monad-law-h-3 : m >>= (\x -> f x >>= g) = (m >>= f) >>= g
dfe8c67390bd Unify Levels in delta
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 94
diff changeset
355 monad-law-h-3 : {l : Level} {A B C : Set l} ->
dfe8c67390bd Unify Levels in delta
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 94
diff changeset
356 (m : Delta A) -> (f : A -> (Delta B)) -> (g : B -> (Delta C)) ->
dfe8c67390bd Unify Levels in delta
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 94
diff changeset
357 (delta-bind m (\x -> delta-bind (f x) g)) ≡ (delta-bind (delta-bind m f) g)
dfe8c67390bd Unify Levels in delta
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 94
diff changeset
358 monad-law-h-3 (mono x) f g = refl
dfe8c67390bd Unify Levels in delta
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 94
diff changeset
359 monad-law-h-3 (delta x d) f g = begin
dfe8c67390bd Unify Levels in delta
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 94
diff changeset
360 (delta-bind (delta x d) (\x -> delta-bind (f x) g)) ≡⟨ {!!} ⟩
dfe8c67390bd Unify Levels in delta
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 94
diff changeset
361 (delta-bind (delta-bind (delta x d) f) g) ∎
88
526186c4f298 Split monad-proofs into delta.monad
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
diff changeset
362
96
dfe8c67390bd Unify Levels in delta
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 94
diff changeset
363
dfe8c67390bd Unify Levels in delta
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 94
diff changeset
364
dfe8c67390bd Unify Levels in delta
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 94
diff changeset
365
dfe8c67390bd Unify Levels in delta
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 94
diff changeset
366 -}