blob: 33a6d8ed9f3e695221b1fb6a202356a10916db03 (
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
117
118
119
120
121
122
123
124
125
126
127
128
129
130
131
132
133
134
135
136
137
138
139
140
141
142
143
144
145
146
147
148
149
150
151
152
153
154
155
156
157
158
159
160
161
162
|
{-# OPTIONS --without-K --safe #-}
module Data.Subset.Functional where
open import Data.Bool.Base using (Bool; _∨_; _∧_; if_then_else_)
open import Data.Bool.Properties using (if-float)
open import Data.Fin.Base using (Fin)
open import Data.Fin.Permutation using (Permutation′; _⟨$⟩ʳ_; _⟨$⟩ˡ_; inverseˡ; inverseʳ)
open import Data.Fin.Properties using (suc-injective; 0≢1+n)
open import Data.Nat.Base using (ℕ)
open import Function.Base using (∣_⟩-_; _∘_)
open import Function.Definitions using (Injective)
open import Relation.Binary.PropositionalEquality as ≡ using (_≡_; _≗_; _≢_)
open import Data.Vector as V using (Vector; head; tail)
open Bool
open Fin
open ℕ
Subset : ℕ → Set
Subset = Vector Bool
private
variable n A : ℕ
variable B C : Set
⊥ : Subset n
⊥ _ = false
_∪_ : Subset n → Subset n → Subset n
(A ∪ B) k = A k ∨ B k
_∩_ : Subset n → Subset n → Subset n
(A ∩ B) k = A k ∧ B k
⁅_⁆ : Fin n → Subset n
⁅_⁆ zero zero = true
⁅_⁆ zero (suc _) = false
⁅_⁆ (suc k) zero = false
⁅_⁆ (suc k) (suc i) = ⁅ k ⁆ i
⁅⁆-refl : (k : Fin n) → ⁅ k ⁆ k ≡ true
⁅⁆-refl zero = ≡.refl
⁅⁆-refl (suc k) = ⁅⁆-refl k
⁅x⁆y≡true
: (x y : Fin n)
→ .(⁅ x ⁆ y ≡ true)
→ x ≡ y
⁅x⁆y≡true zero zero prf = ≡.refl
⁅x⁆y≡true (suc x) (suc y) prf = ≡.cong suc (⁅x⁆y≡true x y prf)
⁅x⁆y≡false
: (x y : Fin n)
→ .(⁅ x ⁆ y ≡ false)
→ x ≢ y
⁅x⁆y≡false zero (suc y) prf = 0≢1+n
⁅x⁆y≡false (suc x) zero prf = 0≢1+n ∘ ≡.sym
⁅x⁆y≡false (suc x) (suc y) prf = ⁅x⁆y≡false x y prf ∘ suc-injective
f-⁅⁆
: {n m : ℕ}
(f : Fin n → Fin m)
→ Injective _≡_ _≡_ f
→ (x y : Fin n)
→ ⁅ x ⁆ y ≡ ⁅ f x ⁆ (f y)
f-⁅⁆ f f-inj zero zero = ≡.sym (⁅⁆-refl (f zero))
f-⁅⁆ f f-inj zero (suc y) with ⁅ f zero ⁆ (f (suc y)) in eq
... | true with () ← f-inj (⁅x⁆y≡true (f zero) (f (suc y)) eq)
... | false = ≡.refl
f-⁅⁆ f f-inj (suc x) zero with ⁅ f (suc x) ⁆ (f zero) in eq
... | true with () ← f-inj (⁅x⁆y≡true (f (suc x)) (f zero) eq)
... | false = ≡.refl
f-⁅⁆ f f-inj (suc x) (suc y) = f-⁅⁆ (f ∘ suc) (suc-injective ∘ f-inj) x y
⁅⁆∘ρ
: (ρ : Permutation′ (suc n))
(x : Fin (suc n))
→ ⁅ ρ ⟨$⟩ʳ x ⁆ ≗ ⁅ x ⁆ ∘ (ρ ⟨$⟩ˡ_)
⁅⁆∘ρ {n} ρ x y = begin
⁅ ρ ⟨$⟩ʳ x ⁆ y ≡⟨ f-⁅⁆ (ρ ⟨$⟩ˡ_) ρˡ-inj (ρ ⟨$⟩ʳ x) y ⟩
⁅ ρ ⟨$⟩ˡ (ρ ⟨$⟩ʳ x) ⁆ (ρ ⟨$⟩ˡ y) ≡⟨ ≡.cong (λ h → ⁅ h ⁆ (ρ ⟨$⟩ˡ y)) (inverseˡ ρ) ⟩
⁅ x ⁆ (ρ ⟨$⟩ˡ y) ∎
where
open ≡.≡-Reasoning
ρˡ-inj : {x y : Fin (suc n)} → ρ ⟨$⟩ˡ x ≡ ρ ⟨$⟩ˡ y → x ≡ y
ρˡ-inj {x} {y} ρˡx≡ρˡy = begin
x ≡⟨ inverseʳ ρ ⟨
ρ ⟨$⟩ʳ (ρ ⟨$⟩ˡ x) ≡⟨ ≡.cong (ρ ⟨$⟩ʳ_) ρˡx≡ρˡy ⟩
ρ ⟨$⟩ʳ (ρ ⟨$⟩ˡ y) ≡⟨ inverseʳ ρ ⟩
y ∎
opaque
-- TODO dependent fold
foldl : (B → Fin A → B) → B → Subset A → B
foldl {B = B} f = V.foldl (λ _ → B) (λ { {k} acc b → if b then f acc k else acc })
foldl-cong₁
: {f g : B → Fin A → B}
→ (∀ x y → f x y ≡ g x y)
→ (e : B)
→ (S : Subset A)
→ foldl f e S ≡ foldl g e S
foldl-cong₁ {B = B} f≗g e S = V.foldl-cong (λ _ → B) (λ { {k} x y → ≡.cong (if y then_else x) (f≗g x k) }) e S
foldl-cong₂
: (f : B → Fin A → B)
(e : B)
{S₁ S₂ : Subset A}
→ (S₁ ≗ S₂)
→ foldl f e S₁ ≡ foldl f e S₂
foldl-cong₂ {B = B} f e S₁≗S₂ = V.foldl-cong-arg (λ _ → B) (λ {n} acc b → if b then f acc n else acc) e S₁≗S₂
foldl-suc
: (f : B → Fin (suc A) → B)
→ (e : B)
→ (S : Subset (suc A))
→ foldl f e S ≡ foldl (∣ f ⟩- suc) (if head S then f e zero else e) (tail S)
foldl-suc f e S = ≡.refl
foldl-⊥
: {A : ℕ}
{B : Set}
(f : B → Fin A → B)
(e : B)
→ foldl f e ⊥ ≡ e
foldl-⊥ {zero} _ _ = ≡.refl
foldl-⊥ {suc A} f e = foldl-⊥ (∣ f ⟩- suc) e
foldl-⁅⁆
: (f : B → Fin A → B)
(e : B)
(k : Fin A)
→ foldl f e ⁅ k ⁆ ≡ f e k
foldl-⁅⁆ f e zero = foldl-⊥ f (f e zero)
foldl-⁅⁆ f e (suc k) = foldl-⁅⁆ (∣ f ⟩- suc) e k
foldl-fusion
: (h : C → B)
{f : C → Fin A → C}
{g : B → Fin A → B}
→ (∀ x n → h (f x n) ≡ g (h x) n)
→ (e : C)
→ h ∘ foldl f e ≗ foldl g (h e)
foldl-fusion {C = C} {A = A} h {f} {g} fuse e = V.foldl-fusion h ≡.refl fuse′
where
open ≡.≡-Reasoning
fuse′
: {k : Fin A}
(acc : C)
(b : Bool)
→ h (if b then f acc k else acc) ≡ (if b then g (h acc) k else h acc)
fuse′ {k} acc b = begin
h (if b then f acc k else acc) ≡⟨ if-float h b ⟩
(if b then h (f acc k) else h acc) ≡⟨ ≡.cong (if b then_else h acc) (fuse acc k) ⟩
(if b then g (h acc) k else h acc) ∎
Subset0≗⊥ : (S : Subset 0) → S ≗ ⊥
Subset0≗⊥ S ()
foldl-[] : (f : B → Fin 0 → B) (e : B) (S : Subset 0) → foldl f e S ≡ e
foldl-[] f e S = ≡.trans (foldl-cong₂ f e (Subset0≗⊥ S)) (foldl-⊥ f e)
|
