blob: 19329b51d695804878c8e50b7cd6dccefafc007f (
plain)
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37
38
39
40
41
42
43
44
45
46
47
48
49
50
51
52
53
54
55
56
57
58
59
60
61
62
63
64
65
66
67
68
69
70
71
72
73
74
75
76
77
78
79
80
81
82
83
84
85
86
87
88
89
90
91
92
93
94
95
96
97
98
99
100
101
102
103
104
105
106
107
108
109
110
111
112
113
114
115
116
|
open import Relation.Binary.Core using (Decidable ; _≡_)
module Precond (Carrier : Set) (deq : Decidable {A = Carrier} _≡_) where
open import Data.Nat using (ℕ)
open import Data.Fin using (Fin ; zero ; suc)
open import Data.Fin.Props using (_≟_)
open import Data.List using (List ; [] ; _∷_)
import Level
import Category.Monad
import Category.Functor
open import Data.Maybe using (Maybe ; nothing ; just ; maybe′)
open Category.Monad.RawMonad {Level.zero} Data.Maybe.monad using (_>>=_)
open Category.Functor.RawFunctor {Level.zero} Data.Maybe.functor using (_<$>_)
open import Data.Vec using (Vec ; [] ; _∷_ ; map ; lookup ; toList ; tabulate)
open import Data.Vec.Properties using (map-cong ; map-∘ ; tabulate-∘)
import Data.List.All
open import Data.List.Any using (here ; there)
open Data.List.Any.Membership-≡ using (_∉_)
open import Data.Maybe using (just)
open import Data.Product using (∃ ; _,_ ; proj₂)
open import Function using (flip ; _∘_ ; id)
open import Relation.Binary.PropositionalEquality using (refl ; cong ; inspect ; [_] ; sym ; decSetoid)
open Relation.Binary.PropositionalEquality.≡-Reasoning using (begin_ ; _≡⟨_⟩_ ; _∎)
open import Relation.Nullary using (yes ; no)
open import Generic using (mapMV ; sequenceV ; sequence-map)
open import FinMap using (FinMapMaybe ; lookupM ; union ; fromFunc ; empty ; insert ; lemma-lookupM-empty ; delete-many ; lemma-tabulate-∘ ; delete ; lemma-lookupM-delete)
import CheckInsert
open CheckInsert (decSetoid deq) using (checkInsert ; lemma-checkInsert-new ; lemma-lookupM-checkInsert-other)
import BFF
open import Bidir (decSetoid deq) using (_in-domain-of_ ; lemma-assoc-domain ; lemma-just-sequence)
open BFF.VecBFF (decSetoid deq) using (get-type ; assoc ; enumerate ; denumerate ; bff)
lemma-lookup-map-just : {n : ℕ} (f : Fin n) {A : Set} (v : Vec A n) → lookup f (map Maybe.just v) ≡ Maybe.just (lookup f v)
lemma-lookup-map-just zero (x ∷ xs) = refl
lemma-lookup-map-just (suc f) (x ∷ xs) = lemma-lookup-map-just f xs
lemma-maybe-just : {A : Set} → (a : A) → (ma : Maybe A) → maybe′ Maybe.just (just a) ma ≡ Maybe.just (maybe′ id a ma)
lemma-maybe-just a (just x) = refl
lemma-maybe-just a nothing = refl
lemma-union-delete-fromFunc : {m n : ℕ} {A : Set} {is : Vec (Fin n) m} {h : FinMapMaybe n A} {g : Vec A n} → (toList is) in-domain-of h → ∃ λ v → union h (delete-many is (map just g)) ≡ map just v
lemma-union-delete-fromFunc {is = []} {h = h} {g = g} p = _ , (begin
union h (map just g)
≡⟨ lemma-tabulate-∘ (λ f → begin
maybe′ just (lookup f (map just g)) (lookup f h)
≡⟨ cong (flip (maybe′ just) (lookup f h)) (lemma-lookup-map-just f g) ⟩
maybe′ just (just (lookup f g)) (lookup f h)
≡⟨ lemma-maybe-just (lookup f g) (lookup f h) ⟩
just (maybe′ id (lookup f g) (lookup f h)) ∎) ⟩
tabulate (λ f → just (maybe′ id (lookup f g) (lookup f h)))
≡⟨ tabulate-∘ just (λ f → maybe′ id (lookup f g) (lookup f h)) ⟩
map just (tabulate (λ f → maybe′ id (lookup f g) (lookup f h))) ∎)
lemma-union-delete-fromFunc {n = n} {is = i ∷ is} {h = h} {g = g} ((x , px) Data.List.All.∷ ps) = _ , (begin
union h (delete i (delete-many is (map just g)))
≡⟨ lemma-tabulate-∘ inner ⟩
union h (delete-many is (map just g))
≡⟨ proj₂ (lemma-union-delete-fromFunc ps) ⟩
map just _ ∎)
where inner : (f : Fin n) → maybe′ just (lookupM f (delete i (delete-many is (map just g)))) (lookup f h) ≡ maybe′ just (lookupM f (delete-many is (map just g))) (lookup f h)
inner f with f ≟ i
inner .i | yes refl = begin
maybe′ just (lookupM i (delete i (delete-many is (map just g)))) (lookup i h)
≡⟨ cong (maybe′ just _) px ⟩
just x
≡⟨ cong (maybe′ just _) (sym px) ⟩
maybe′ just (lookupM i (delete-many is (map just g))) (lookup i h) ∎
inner f | no f≢i = cong (flip (maybe′ just) (lookup f h)) (lemma-lookupM-delete (delete-many is (map just g)) f≢i)
assoc-enough : {getlen : ℕ → ℕ} (get : get-type getlen) → {m : ℕ} → (s : Vec Carrier m) → (v : Vec Carrier (getlen m)) → ∃ (λ h → assoc (get (enumerate s)) v ≡ just h) → ∃ λ u → bff get s v ≡ just u
assoc-enough get s v (h , p) = let w , pw = lemma-union-delete-fromFunc (lemma-assoc-domain (get s′) v h p) in _ , (begin
bff get s v
≡⟨ cong (flip _>>=_ (flip mapMV s′ ∘ flip lookupM) ∘ _<$>_ (flip union g′)) p ⟩
mapMV (flip lookupM (union h g′)) s′
≡⟨ sym (sequence-map (flip lookupM (union h g′)) s′) ⟩
sequenceV (map (flip lookupM (union h g′)) s′)
≡⟨ cong (sequenceV ∘ flip map s′ ∘ flip lookupM) pw ⟩
sequenceV (map (flip lookupM (map just w)) s′)
≡⟨ cong sequenceV (map-cong (λ f → lemma-lookup-map-just f w) s′) ⟩
sequenceV (map (Maybe.just ∘ flip lookup w) s′)
≡⟨ cong sequenceV (map-∘ just (flip lookup w) s′) ⟩
sequenceV (map Maybe.just (map (flip lookup w) s′))
≡⟨ lemma-just-sequence (map (flip lookup w) s′) ⟩
just (map (flip lookup w) s′) ∎)
where s′ = enumerate s
g = fromFunc (denumerate s)
g′ = delete-many (get s′) g
data All-different {A : Set} : List A → Set where
different-[] : All-different []
different-∷ : {x : A} {xs : List A} → x ∉ xs → All-different xs → All-different (x ∷ xs)
lemma-∉-lookupM-assoc : {m n : ℕ} → (i : Fin n) → (is : Vec (Fin n) m) → (xs : Vec Carrier m) → (h : FinMapMaybe n Carrier) → assoc is xs ≡ just h → (i ∉ toList is) → lookupM i h ≡ nothing
lemma-∉-lookupM-assoc i [] [] .empty refl i∉is = lemma-lookupM-empty i
lemma-∉-lookupM-assoc i (i' ∷ is') (x' ∷ xs') h ph i∉is with assoc is' xs' | inspect (assoc is') xs'
lemma-∉-lookupM-assoc i (i' ∷ is') (x' ∷ xs') h () i∉is | nothing | [ ph' ]
lemma-∉-lookupM-assoc i (i' ∷ is') (x' ∷ xs') h ph i∉is | just h' | [ ph' ] = begin
lookupM i h
≡⟨ sym (lemma-lookupM-checkInsert-other i i' (i∉is ∘ here) x' h' h ph) ⟩
lookupM i h'
≡⟨ lemma-∉-lookupM-assoc i is' xs' h' ph' (i∉is ∘ there) ⟩
nothing ∎
different-assoc : {m n : ℕ} → (u : Vec (Fin n) m) → (v : Vec Carrier m) → All-different (toList u) → ∃ λ h → assoc u v ≡ just h
different-assoc [] [] p = empty , refl
different-assoc (u ∷ us) (v ∷ vs) (different-∷ u∉us diff-us) with different-assoc us vs diff-us
different-assoc (u ∷ us) (v ∷ vs) (different-∷ u∉us diff-us) | h , p' = insert u v h , (begin
assoc (u ∷ us) (v ∷ vs)
≡⟨ refl ⟩
(assoc us vs >>= checkInsert u v)
≡⟨ cong (flip _>>=_ (checkInsert u v)) p' ⟩
checkInsert u v h
≡⟨ lemma-checkInsert-new u v h (lemma-∉-lookupM-assoc u us vs h p' u∉us) ⟩
just (insert u v h) ∎)
|
