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comparison agda/fin.agda @ 163:26407ce19d66

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author Shinji KONO <kono@ie.u-ryukyu.ac.jp>
date Wed, 13 Jan 2021 10:52:01 +0900
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162:690a8352c1ad 163:26407ce19d66
1 {-# OPTIONS --allow-unsolved-metas #-}
2
3 module fin where
4
5 open import Data.Fin hiding (_<_ ; _≤_ )
6 open import Data.Fin.Properties hiding ( <-trans )
7 open import Data.Nat
8 open import logic
9 open import nat
10 open import Relation.Binary.PropositionalEquality
11
12
13 -- toℕ<n
14 fin<n : {n : ℕ} {f : Fin n} → toℕ f < n
15 fin<n {_} {zero} = s≤s z≤n
16 fin<n {suc n} {suc f} = s≤s (fin<n {n} {f})
17
18 -- toℕ≤n
19 fin≤n : {n : ℕ} (f : Fin (suc n)) → toℕ f ≤ n
20 fin≤n {_} zero = z≤n
21 fin≤n {suc n} (suc f) = s≤s (fin≤n {n} f)
22
23 pred<n : {n : ℕ} {f : Fin (suc n)} → n > 0 → Data.Nat.pred (toℕ f) < n
24 pred<n {suc n} {zero} (s≤s z≤n) = s≤s z≤n
25 pred<n {suc n} {suc f} (s≤s z≤n) = fin<n
26
27 fin<asa : {n : ℕ} → toℕ (fromℕ< {n} a<sa) ≡ n
28 fin<asa = toℕ-fromℕ< nat.a<sa
29
30 -- fromℕ<-toℕ
31 toℕ→from : {n : ℕ} {x : Fin (suc n)} → toℕ x ≡ n → fromℕ n ≡ x
32 toℕ→from {0} {zero} refl = refl
33 toℕ→from {suc n} {suc x} eq = cong (λ k → suc k ) ( toℕ→from {n} {x} (cong (λ k → Data.Nat.pred k ) eq ))
34
35 0≤fmax : {n : ℕ } → (# 0) Data.Fin.≤ fromℕ< {n} a<sa
36 0≤fmax = subst (λ k → 0 ≤ k ) (sym (toℕ-fromℕ< a<sa)) z≤n
37
38 0<fmax : {n : ℕ } → (# 0) Data.Fin.< fromℕ< {suc n} a<sa
39 0<fmax = subst (λ k → 0 < k ) (sym (toℕ-fromℕ< a<sa)) (s≤s z≤n)
40
41 -- toℕ-injective
42 i=j : {n : ℕ} (i j : Fin n) → toℕ i ≡ toℕ j → i ≡ j
43 i=j {suc n} zero zero refl = refl
44 i=j {suc n} (suc i) (suc j) eq = cong ( λ k → suc k ) ( i=j i j (cong ( λ k → Data.Nat.pred k ) eq) )
45
46 -- raise 1
47 fin+1 : { n : ℕ } → Fin n → Fin (suc n)
48 fin+1 zero = zero
49 fin+1 (suc x) = suc (fin+1 x)
50
51 open import Data.Nat.Properties as NatP hiding ( _≟_ )
52
53 fin+1≤ : { i n : ℕ } → (a : i < n) → fin+1 (fromℕ< a) ≡ fromℕ< (<-trans a a<sa)
54 fin+1≤ {0} {suc i} (s≤s z≤n) = refl
55 fin+1≤ {suc n} {suc (suc i)} (s≤s (s≤s a)) = cong (λ k → suc k ) ( fin+1≤ {n} {suc i} (s≤s a) )
56
57 fin+1-toℕ : { n : ℕ } → { x : Fin n} → toℕ (fin+1 x) ≡ toℕ x
58 fin+1-toℕ {suc n} {zero} = refl
59 fin+1-toℕ {suc n} {suc x} = cong (λ k → suc k ) (fin+1-toℕ {n} {x})
60
61 open import Relation.Nullary
62 open import Data.Empty
63
64 fin-1 : { n : ℕ } → (x : Fin (suc n)) → ¬ (x ≡ zero ) → Fin n
65 fin-1 zero ne = ⊥-elim (ne refl )
66 fin-1 {n} (suc x) ne = x
67
68 fin-1-sx : { n : ℕ } → (x : Fin n) → fin-1 (suc x) (λ ()) ≡ x
69 fin-1-sx zero = refl
70 fin-1-sx (suc x) = refl
71
72 fin-1-xs : { n : ℕ } → (x : Fin (suc n)) → (ne : ¬ (x ≡ zero )) → suc (fin-1 x ne ) ≡ x
73 fin-1-xs zero ne = ⊥-elim ( ne refl )
74 fin-1-xs (suc x) ne = refl
75
76 -- suc-injective
77 -- suc-eq : {n : ℕ } {x y : Fin n} → Fin.suc x ≡ Fin.suc y → x ≡ y
78 -- suc-eq {n} {x} {y} eq = subst₂ (λ j k → j ≡ k ) {!!} {!!} (cong (λ k → Data.Fin.pred k ) eq )
79
80 -- this is refl
81 lemma3 : {a b : ℕ } → (lt : a < b ) → fromℕ< (s≤s lt) ≡ suc (fromℕ< lt)
82 lemma3 (s≤s lt) = refl
83
84 -- fromℕ<-toℕ
85 lemma12 : {n m : ℕ } → (n<m : n < m ) → (f : Fin m ) → toℕ f ≡ n → f ≡ fromℕ< n<m
86 lemma12 {zero} {suc m} (s≤s z≤n) zero refl = refl
87 lemma12 {suc n} {suc m} (s≤s n<m) (suc f) refl = cong suc ( lemma12 {n} {m} n<m f refl )
88
89 open import Relation.Binary.HeterogeneousEquality as HE using (_≅_ )
90 open import Data.Fin.Properties
91
92 -- <-irrelevant
93 <-nat=irr : {i j n : ℕ } → ( i ≡ j ) → {i<n : i < n } → {j<n : j < n } → i<n ≅ j<n
94 <-nat=irr {zero} {zero} {suc n} refl {s≤s z≤n} {s≤s z≤n} = HE.refl
95 <-nat=irr {suc i} {suc i} {suc n} refl {s≤s i<n} {s≤s j<n} = HE.cong (λ k → s≤s k ) ( <-nat=irr {i} {i} {n} refl )
96
97 lemma8 : {i j n : ℕ } → ( i ≡ j ) → {i<n : i < n } → {j<n : j < n } → i<n ≅ j<n
98 lemma8 {zero} {zero} {suc n} refl {s≤s z≤n} {s≤s z≤n} = HE.refl
99 lemma8 {suc i} {suc i} {suc n} refl {s≤s i<n} {s≤s j<n} = HE.cong (λ k → s≤s k ) ( lemma8 {i} {i} {n} refl )
100
101 -- fromℕ<-irrelevant
102 lemma10 : {n i j : ℕ } → ( i ≡ j ) → {i<n : i < n } → {j<n : j < n } → fromℕ< i<n ≡ fromℕ< j<n
103 lemma10 {n} refl = HE.≅-to-≡ (HE.cong (λ k → fromℕ< k ) (lemma8 refl ))
104
105 lemma31 : {a b c : ℕ } → { a<b : a < b } { b<c : b < c } { a<c : a < c } → NatP.<-trans a<b b<c ≡ a<c
106 lemma31 {a} {b} {c} {a<b} {b<c} {a<c} = HE.≅-to-≡ (lemma8 refl)
107
108 -- toℕ-fromℕ<
109 lemma11 : {n m : ℕ } {x : Fin n } → (n<m : n < m ) → toℕ (fromℕ< (NatP.<-trans (toℕ<n x) n<m)) ≡ toℕ x
110 lemma11 {n} {m} {x} n<m = begin
111 toℕ (fromℕ< (NatP.<-trans (toℕ<n x) n<m))
112 ≡⟨ toℕ-fromℕ< _ ⟩
113 toℕ x
114 ∎ where
115 open ≡-Reasoning
116
117