Mercurial > hg > Members > kono > Proof > ZF-in-agda

comparison ordinal.agda @ 29:fce60b99dc55

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posturate OD is isomorphic to Ordinal
author Shinji KONO <kono@ie.u-ryukyu.ac.jp>
date Mon, 20 May 2019 18:18:43 +0900
parents constructible-set.agda@f36e40d5d2c3
children 3b0fdb95618e
comparison
equal deleted inserted replaced
28:f36e40d5d2c3 29:fce60b99dc55
1 open import Level
2 module ordinal where
3
4 open import zf
5
6 open import Data.Nat renaming ( zero to Zero ; suc to Suc ; ℕ to Nat ; _⊔_ to _n⊔_ )
7
8 open import Relation.Binary.PropositionalEquality
9
10 data OrdinalD {n : Level} : (lv : Nat) → Set n where
11 Φ : (lv : Nat) → OrdinalD lv
12 OSuc : (lv : Nat) → OrdinalD {n} lv → OrdinalD lv
13 ℵ_ : (lv : Nat) → OrdinalD (Suc lv)
14
15 record Ordinal {n : Level} : Set n where
16 field
17 lv : Nat
18 ord : OrdinalD {n} lv
19
20 data _d<_ {n : Level} : {lx ly : Nat} → OrdinalD {n} lx → OrdinalD {n} ly → Set n where
21 Φ< : {lx : Nat} → {x : OrdinalD {n} lx} → Φ lx d< OSuc lx x
22 s< : {lx : Nat} → {x y : OrdinalD {n} lx} → x d< y → OSuc lx x d< OSuc lx y
23 ℵΦ< : {lx : Nat} → {x : OrdinalD {n} (Suc lx) } → Φ (Suc lx) d< (ℵ lx)
24 ℵ< : {lx : Nat} → {x : OrdinalD {n} (Suc lx) } → OSuc (Suc lx) x d< (ℵ lx)
25
26 open Ordinal
27
28 _o<_ : {n : Level} ( x y : Ordinal ) → Set n
29 _o<_ x y = (lv x < lv y ) ∨ ( ord x d< ord y )
30
31 open import Data.Nat.Properties
32 open import Data.Empty
33 open import Relation.Nullary
34
35 open import Relation.Binary
36 open import Relation.Binary.Core
37
38 o∅ : {n : Level} → Ordinal {n}
39 o∅ = record { lv = Zero ; ord = Φ Zero }
40
41
42 ≡→¬d< : {n : Level} → {lv : Nat} → {x : OrdinalD {n} lv } → x d< x → ⊥
43 ≡→¬d< {n} {lx} {OSuc lx y} (s< t) = ≡→¬d< t
44
45 trio<> : {n : Level} → {lx : Nat} {x : OrdinalD {n} lx } { y : OrdinalD lx } → y d< x → x d< y → ⊥
46 trio<> {n} {lx} {.(OSuc lx _)} {.(OSuc lx _)} (s< s) (s< t) =
47 trio<> s t
48
49 trio<≡ : {n : Level} → {lx : Nat} {x : OrdinalD {n} lx } { y : OrdinalD lx } → x ≡ y → x d< y → ⊥
50 trio<≡ refl = ≡→¬d<
51
52 trio>≡ : {n : Level} → {lx : Nat} {x : OrdinalD {n} lx } { y : OrdinalD lx } → x ≡ y → y d< x → ⊥
53 trio>≡ refl = ≡→¬d<
54
55 triO : {n : Level} → {lx ly : Nat} → OrdinalD {n} lx → OrdinalD {n} ly → Tri (lx < ly) ( lx ≡ ly ) ( lx > ly )
56 triO {n} {lx} {ly} x y = <-cmp lx ly
57
58 triOrdd : {n : Level} → {lx : Nat} → Trichotomous _≡_ ( _d<_ {n} {lx} {lx} )
59 triOrdd {_} {lv} (Φ lv) (Φ lv) = tri≈ ≡→¬d< refl ≡→¬d<
60 triOrdd {_} {Suc lv} (ℵ lv) (ℵ lv) = tri≈ ≡→¬d< refl ≡→¬d<
61 triOrdd {_} {lv} (Φ lv) (OSuc lv y) = tri< Φ< (λ ()) ( λ lt → trio<> lt Φ< )
62 triOrdd {_} {.(Suc lv)} (Φ (Suc lv)) (ℵ lv) = tri< (ℵΦ< {_} {lv} {Φ (Suc lv)} ) (λ ()) ( λ lt → trio<> lt ((ℵΦ< {_} {lv} {Φ (Suc lv)} )) )
63 triOrdd {_} {Suc lv} (ℵ lv) (Φ (Suc lv)) = tri> ( λ lt → trio<> lt (ℵΦ< {_} {lv} {Φ (Suc lv)} ) ) (λ ()) (ℵΦ< {_} {lv} {Φ (Suc lv)} )
64 triOrdd {_} {Suc lv} (ℵ lv) (OSuc (Suc lv) y) = tri> ( λ lt → trio<> lt (ℵ< {_} {lv} {y} ) ) (λ ()) (ℵ< {_} {lv} {y} )
65 triOrdd {_} {lv} (OSuc lv x) (Φ lv) = tri> (λ lt → trio<> lt Φ<) (λ ()) Φ<
66 triOrdd {_} {.(Suc lv)} (OSuc (Suc lv) x) (ℵ lv) = tri< ℵ< (λ ()) (λ lt → trio<> lt ℵ< )
67 triOrdd {_} {lv} (OSuc lv x) (OSuc lv y) with triOrdd x y
68 triOrdd {_} {lv} (OSuc lv x) (OSuc lv y) | tri< a ¬b ¬c = tri< (s< a) (λ tx=ty → trio<≡ tx=ty (s< a) ) ( λ lt → trio<> lt (s< a) )
69 triOrdd {_} {lv} (OSuc lv x) (OSuc lv x) | tri≈ ¬a refl ¬c = tri≈ ≡→¬d< refl ≡→¬d<
70 triOrdd {_} {lv} (OSuc lv x) (OSuc lv y) | tri> ¬a ¬b c = tri> ( λ lt → trio<> lt (s< c) ) (λ tx=ty → trio>≡ tx=ty (s< c) ) (s< c)
71
72 d<→lv : {n : Level} {x y : Ordinal {n}} → ord x d< ord y → lv x ≡ lv y
73 d<→lv Φ< = refl
74 d<→lv (s< lt) = refl
75 d<→lv ℵΦ< = refl
76 d<→lv ℵ< = refl
77
78 orddtrans : {n : Level} {lx : Nat} {x y z : OrdinalD {n} lx } → x d< y → y d< z → x d< z
79 orddtrans {_} {lx} {.(Φ lx)} {.(OSuc lx _)} {.(OSuc lx _)} Φ< (s< y<z) = Φ<
80 orddtrans {_} {Suc lx} {Φ (Suc lx)} {OSuc (Suc lx) y} {ℵ lx} Φ< ℵ< = ℵΦ< {_} {lx} {y}
81 orddtrans {_} {lx} {.(OSuc lx _)} {.(OSuc lx _)} {.(OSuc lx _)} (s< x<y) (s< y<z) = s< ( orddtrans x<y y<z )
82 orddtrans {_} {Suc lx} {.(OSuc (Suc lx) _)} {.(OSuc (Suc lx) _)} {.(ℵ _)} (s< x<y) ℵ< = ℵ<
83 orddtrans {_} {Suc lx} {Φ (Suc lx)} {.(ℵ _)} {z} ℵΦ< ()
84 orddtrans {_} {Suc lx} {OSuc (Suc lx) _} {.(ℵ _)} {z} ℵ< ()
85
86 max : (x y : Nat) → Nat
87 max Zero Zero = Zero
88 max Zero (Suc x) = (Suc x)
89 max (Suc x) Zero = (Suc x)
90 max (Suc x) (Suc y) = Suc ( max x y )
91
92 maxαd : {n : Level} → { lx : Nat } → OrdinalD {n} lx → OrdinalD lx → OrdinalD lx
93 maxαd x y with triOrdd x y
94 maxαd x y | tri< a ¬b ¬c = y
95 maxαd x y | tri≈ ¬a b ¬c = x
96 maxαd x y | tri> ¬a ¬b c = x
97
98 maxα : {n : Level} → Ordinal {n} → Ordinal → Ordinal
99 maxα x y with <-cmp (lv x) (lv y)
100 maxα x y | tri< a ¬b ¬c = x
101 maxα x y | tri> ¬a ¬b c = y
102 maxα x y | tri≈ ¬a refl ¬c = record { lv = lv x ; ord = maxαd (ord x) (ord y) }
103
104 _o≤_ : {n : Level} → Ordinal → Ordinal → Set (suc n)
105 a o≤ b = (a ≡ b) ∨ ( a o< b )
106
107 ordtrans : {n : Level} {x y z : Ordinal {n} } → x o< y → y o< z → x o< z
108 ordtrans {n} {x} {y} {z} (case1 x₁) (case1 x₂) = case1 ( <-trans x₁ x₂ )
109 ordtrans {n} {x} {y} {z} (case1 x₁) (case2 x₂) with d<→lv x₂
110 ... | refl = case1 x₁
111 ordtrans {n} {x} {y} {z} (case2 x₁) (case1 x₂) with d<→lv x₁
112 ... | refl = case1 x₂
113 ordtrans {n} {x} {y} {z} (case2 x₁) (case2 x₂) with d<→lv x₁ | d<→lv x₂
114 ... | refl | refl = case2 ( orddtrans x₁ x₂ )
115
116
117 trio< : {n : Level } → Trichotomous {suc n} _≡_ _o<_
118 trio< a b with <-cmp (lv a) (lv b)
119 trio< a b | tri< a₁ ¬b ¬c = tri< (case1 a₁) (λ refl → ¬b (cong ( λ x → lv x ) refl ) ) lemma1 where
120 lemma1 : ¬ (Suc (lv b) ≤ lv a) ∨ (ord b d< ord a)
121 lemma1 (case1 x) = ¬c x
122 lemma1 (case2 x) with d<→lv x
123 lemma1 (case2 x) | refl = ¬b refl
124 trio< a b | tri> ¬a ¬b c = tri> lemma1 (λ refl → ¬b (cong ( λ x → lv x ) refl ) ) (case1 c) where
125 lemma1 : ¬ (Suc (lv a) ≤ lv b) ∨ (ord a d< ord b)
126 lemma1 (case1 x) = ¬a x
127 lemma1 (case2 x) with d<→lv x
128 lemma1 (case2 x) | refl = ¬b refl
129 trio< a b | tri≈ ¬a refl ¬c with triOrdd ( ord a ) ( ord b )
130 trio< record { lv = .(lv b) ; ord = x } b | tri≈ ¬a refl ¬c | tri< a ¬b ¬c₁ = tri< (case2 a) (λ refl → ¬b (lemma1 refl )) lemma2 where
131 lemma1 : (record { lv = _ ; ord = x }) ≡ b → x ≡ ord b
132 lemma1 refl = refl
133 lemma2 : ¬ (Suc (lv b) ≤ lv b) ∨ (ord b d< x)
134 lemma2 (case1 x) = ¬a x
135 lemma2 (case2 x) = trio<> x a
136 trio< record { lv = .(lv b) ; ord = x } b | tri≈ ¬a refl ¬c | tri> ¬a₁ ¬b c = tri> lemma2 (λ refl → ¬b (lemma1 refl )) (case2 c) where
137 lemma1 : (record { lv = _ ; ord = x }) ≡ b → x ≡ ord b
138 lemma1 refl = refl
139 lemma2 : ¬ (Suc (lv b) ≤ lv b) ∨ (x d< ord b)
140 lemma2 (case1 x) = ¬a x
141 lemma2 (case2 x) = trio<> x c
142 trio< record { lv = .(lv b) ; ord = x } b | tri≈ ¬a refl ¬c | tri≈ ¬a₁ refl ¬c₁ = tri≈ lemma1 refl lemma1 where
143 lemma1 : ¬ (Suc (lv b) ≤ lv b) ∨ (ord b d< ord b)
144 lemma1 (case1 x) = ¬a x
145 lemma1 (case2 x) = ≡→¬d< x
146
147 OrdTrans : {n : Level} → Transitive {suc n} _o≤_
148 OrdTrans (case1 refl) (case1 refl) = case1 refl
149 OrdTrans (case1 refl) (case2 lt2) = case2 lt2
150 OrdTrans (case2 lt1) (case1 refl) = case2 lt1
151 OrdTrans (case2 (case1 x)) (case2 (case1 y)) = case2 (case1 ( <-trans x y ) )
152 OrdTrans (case2 (case1 x)) (case2 (case2 y)) with d<→lv y
153 OrdTrans (case2 (case1 x)) (case2 (case2 y)) | refl = case2 (case1 x )
154 OrdTrans (case2 (case2 x)) (case2 (case1 y)) with d<→lv x
155 OrdTrans (case2 (case2 x)) (case2 (case1 y)) | refl = case2 (case1 y)
156 OrdTrans (case2 (case2 x)) (case2 (case2 y)) with d<→lv x | d<→lv y
157 OrdTrans (case2 (case2 x)) (case2 (case2 y)) | refl | refl = case2 (case2 (orddtrans x y ))
158
159 OrdPreorder : {n : Level} → Preorder (suc n) (suc n) (suc n)
160 OrdPreorder {n} = record { Carrier = Ordinal
161 ; _≈_ = _≡_
162 ; _∼_ = _o≤_
163 ; isPreorder = record {
164 isEquivalence = record { refl = refl ; sym = sym ; trans = trans }
165 ; reflexive = case1
166 ; trans = OrdTrans
167 }
168 }
169
170 TransFinite : {n : Level} → ( ψ : Ordinal {n} → Set n )
171 → ( ∀ (lx : Nat ) → ψ ( record { lv = Suc lx ; ord = ℵ lx } ))
172 → ( ∀ (lx : Nat ) → ψ ( record { lv = lx ; ord = Φ lx } ) )
173 → ( ∀ (lx : Nat ) → (x : OrdinalD lx ) → ψ ( record { lv = lx ; ord = x } ) → ψ ( record { lv = lx ; ord = OSuc lx x } ) )
174 → ∀ (x : Ordinal) → ψ x
175 TransFinite ψ caseℵ caseΦ caseOSuc record { lv = lv ; ord = Φ lv } = caseΦ lv
176 TransFinite ψ caseℵ caseΦ caseOSuc record { lv = lv ; ord = OSuc lv ord₁ } = caseOSuc lv ord₁
177 ( TransFinite ψ caseℵ caseΦ caseOSuc (record { lv = lv ; ord = ord₁ } ))
178 TransFinite ψ caseℵ caseΦ caseOSuc record { lv = Suc lv₁ ; ord = ℵ lv₁ } = caseℵ lv₁
179