Mercurial > hg > Members > kono > Proof > galois

comparison fin.agda @ 72:09fa2ab75703

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add utilties
author Shinji KONO <kono@ie.u-ryukyu.ac.jp>
date Mon, 24 Aug 2020 23:06:10 +0900
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children 69ed81f8e212
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71:da1677fae9ac 72:09fa2ab75703
1 {-# OPTIONS --allow-unsolved-metas #-}
2
3 module fin where
4
5 open import Data.Fin hiding (_<_)
6 open import Data.Nat
7 open import logic
8 open import nat
9 open import Relation.Binary.PropositionalEquality
10
11
12 n≤n : (n : ℕ) → n Data.Nat.≤ n
13 n≤n zero = z≤n
14 n≤n (suc n) = s≤s (n≤n n)
15
16 <→m≤n : {m n : ℕ} → m < n → m Data.Nat.≤ n
17 <→m≤n {zero} lt = z≤n
18 <→m≤n {suc m} {zero} ()
19 <→m≤n {suc m} {suc n} (s≤s lt) = s≤s (<→m≤n lt)
20
21 fin<n : {n : ℕ} {f : Fin n} → toℕ f < n
22 fin<n {_} {zero} = s≤s z≤n
23 fin<n {suc n} {suc f} = s≤s (fin<n {n} {f})
24
25 pred<n : {n : ℕ} {f : Fin (suc n)} → n > 0 → Data.Nat.pred (toℕ f) < n
26 pred<n {suc n} {zero} (s≤s z≤n) = s≤s z≤n
27 pred<n {suc n} {suc f} (s≤s z≤n) = fin<n
28
29 toℕ→from : {n : ℕ} {x : Fin (suc n)} → toℕ x ≡ n → fromℕ n ≡ x
30 toℕ→from {0} {zero} refl = refl
31 toℕ→from {suc n} {suc x} eq = cong (λ k → suc k ) ( toℕ→from {n} {x} (cong (λ k → Data.Nat.pred k ) eq ))
32
33 i=j : {n : ℕ} (i j : Fin n) → toℕ i ≡ toℕ j → i ≡ j
34 i=j {suc n} zero zero refl = refl
35 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) )
36
37 fin+1 : { n : ℕ } → Fin n → Fin (suc n)
38 fin+1 zero = zero
39 fin+1 (suc x) = suc (fin+1 x)
40
41 open import Data.Nat.Properties as NatP hiding ( _≟_ )
42
43 fin+1≤ : { i n : ℕ } → (a : i < n) → fin+1 (fromℕ≤ a) ≡ fromℕ≤ (<-trans a a<sa)
44 fin+1≤ {0} {suc i} (s≤s z≤n) = refl
45 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) )
46
47 fin+1-toℕ : { n : ℕ } → { x : Fin n} → toℕ (fin+1 x) ≡ toℕ x
48 fin+1-toℕ {suc n} {zero} = refl
49 fin+1-toℕ {suc n} {suc x} = cong (λ k → suc k ) (fin+1-toℕ {n} {x})
50
51 open import Relation.Nullary
52 open import Data.Empty
53
54 fin-1 : { n : ℕ } → (x : Fin (suc n)) → ¬ (x ≡ zero ) → Fin n
55 fin-1 zero ne = ⊥-elim (ne refl )
56 fin-1 {n} (suc x) ne = x
57
58 fin-1-sx : { n : ℕ } → (x : Fin n) → fin-1 (suc x) (λ ()) ≡ x
59 fin-1-sx zero = refl
60 fin-1-sx (suc x) = refl
61
62 fin-1-xs : { n : ℕ } → (x : Fin (suc n)) → (ne : ¬ (x ≡ zero )) → suc (fin-1 x ne ) ≡ x
63 fin-1-xs zero ne = ⊥-elim ( ne refl )
64 fin-1-xs (suc x) ne = refl
65
66 -- suc-eq : {n : ℕ } {x y : Fin n} → Fin.suc x ≡ Fin.suc y → x ≡ y
67 -- suc-eq {n} {x} {y} eq = subst₂ (λ j k → j ≡ k ) {!!} {!!} (cong (λ k → Data.Fin.pred k ) eq )
68
69 lemma3 : {a b : ℕ } → (lt : a < b ) → fromℕ≤ (s≤s lt) ≡ suc (fromℕ≤ lt)
70 lemma3 (s≤s lt) = refl
71 lemma12 : {n m : ℕ } → (n<m : n < m ) → (f : Fin m ) → toℕ f ≡ n → f ≡ fromℕ≤ n<m
72 lemma12 {zero} {suc m} (s≤s z≤n) zero refl = refl
73 lemma12 {suc n} {suc m} (s≤s n<m) (suc f) refl = subst ( λ k → suc f ≡ k ) (sym (lemma3 n<m) ) ( cong ( λ k → suc k ) ( lemma12 {n} {m} n<m f refl ) )
74
75 open import Relation.Binary.HeterogeneousEquality as HE using (_≅_ )
76 open import Data.Fin.Properties
77
78 lemma8 : {i j n : ℕ } → ( i ≡ j ) → {i<n : i < n } → {j<n : j < n } → i<n ≅ j<n
79 lemma8 {zero} {zero} {suc n} refl {s≤s z≤n} {s≤s z≤n} = HE.refl
80 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 )
81 lemma10 : {n i j : ℕ } → ( i ≡ j ) → {i<n : i < n } → {j<n : j < n } → fromℕ≤ i<n ≡ fromℕ≤ j<n
82 lemma10 {n} refl = HE.≅-to-≡ (HE.cong (λ k → fromℕ≤ k ) (lemma8 refl ))
83 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
84 lemma31 {a} {b} {c} {a<b} {b<c} {a<c} = HE.≅-to-≡ (lemma8 refl)
85 lemma11 : {n m : ℕ } {x : Fin n } → (n<m : n < m ) → toℕ (fromℕ≤ (NatP.<-trans (toℕ<n x) n<m)) ≡ toℕ x
86 lemma11 {n} {m} {x} n<m = begin
87 toℕ (fromℕ≤ (NatP.<-trans (toℕ<n x) n<m))
88 ≡⟨ toℕ-fromℕ≤ _ ⟩
89 toℕ x
90 ∎ where
91 open ≡-Reasoning
92
93