Mercurial > hg > Members > kono > Proof > ZF-in-agda
annotate ordinal.agda @ 184:65e1b2e415bb
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author | Shinji KONO <kono@ie.u-ryukyu.ac.jp> |
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date | Mon, 22 Jul 2019 17:31:52 +0900 |
parents | de3d87b7494f |
children | ed88384b5102 |
rev | line source |
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34 | 1 {-# OPTIONS --allow-unsolved-metas #-} |
16 | 2 open import Level |
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3 module ordinal where |
3 | 4 |
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5 open import zf |
3 | 6 |
23 | 7 open import Data.Nat renaming ( zero to Zero ; suc to Suc ; ℕ to Nat ; _⊔_ to _n⊔_ ) |
75 | 8 open import Data.Empty |
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9 open import Relation.Binary.PropositionalEquality |
3 | 10 |
24 | 11 data OrdinalD {n : Level} : (lv : Nat) → Set n where |
12 Φ : (lv : Nat) → OrdinalD lv | |
13 OSuc : (lv : Nat) → OrdinalD {n} lv → OrdinalD lv | |
3 | 14 |
24 | 15 record Ordinal {n : Level} : Set n where |
16 | 16 field |
17 lv : Nat | |
24 | 18 ord : OrdinalD {n} lv |
16 | 19 |
24 | 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 | |
17 | 23 |
24 open Ordinal | |
25 | |
27 | 26 _o<_ : {n : Level} ( x y : Ordinal ) → Set n |
17 | 27 _o<_ x y = (lv x < lv y ) ∨ ( ord x d< ord y ) |
3 | 28 |
75 | 29 s<refl : {n : Level } {lx : Nat } { x : OrdinalD {n} lx } → x d< OSuc lx x |
30 s<refl {n} {lv} {Φ lv} = Φ< | |
31 s<refl {n} {lv} {OSuc lv x} = s< s<refl | |
32 | |
33 trio<> : {n : Level} → {lx : Nat} {x : OrdinalD {n} lx } { y : OrdinalD lx } → y d< x → x d< y → ⊥ | |
34 trio<> {n} {lx} {.(OSuc lx _)} {.(OSuc lx _)} (s< s) (s< t) = trio<> s t | |
35 trio<> {n} {lx} {.(OSuc lx _)} {.(Φ lx)} Φ< () | |
36 | |
37 d<→lv : {n : Level} {x y : Ordinal {n}} → ord x d< ord y → lv x ≡ lv y | |
38 d<→lv Φ< = refl | |
39 d<→lv (s< lt) = refl | |
40 | |
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41 o<-subst : {n : Level } {Z X z x : Ordinal {n}} → Z o< X → Z ≡ z → X ≡ x → z o< x |
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42 o<-subst df refl refl = df |
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43 |
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44 open import Data.Nat.Properties |
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problem on Ordinal ( OSuc ℵ )
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45 open import Data.Unit using ( ⊤ ) |
6 | 46 open import Relation.Nullary |
47 | |
48 open import Relation.Binary | |
49 open import Relation.Binary.Core | |
50 | |
24 | 51 o∅ : {n : Level} → Ordinal {n} |
52 o∅ = record { lv = Zero ; ord = Φ Zero } | |
21 | 53 |
39 | 54 open import Relation.Binary.HeterogeneousEquality using (_≅_;refl) |
55 | |
56 ordinal-cong : {n : Level} {x y : Ordinal {n}} → | |
57 lv x ≡ lv y → ord x ≅ ord y → x ≡ y | |
58 ordinal-cong refl refl = refl | |
21 | 59 |
46 | 60 ordinal-lv : {n : Level} {x y : Ordinal {n}} → x ≡ y → lv x ≡ lv y |
61 ordinal-lv refl = refl | |
62 | |
63 ordinal-d : {n : Level} {x y : Ordinal {n}} → x ≡ y → ord x ≅ ord y | |
64 ordinal-d refl = refl | |
65 | |
24 | 66 ≡→¬d< : {n : Level} → {lv : Nat} → {x : OrdinalD {n} lv } → x d< x → ⊥ |
67 ≡→¬d< {n} {lx} {OSuc lx y} (s< t) = ≡→¬d< t | |
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68 |
24 | 69 trio<≡ : {n : Level} → {lx : Nat} {x : OrdinalD {n} lx } { y : OrdinalD lx } → x ≡ y → x d< y → ⊥ |
17 | 70 trio<≡ refl = ≡→¬d< |
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71 |
24 | 72 trio>≡ : {n : Level} → {lx : Nat} {x : OrdinalD {n} lx } { y : OrdinalD lx } → x ≡ y → y d< x → ⊥ |
17 | 73 trio>≡ refl = ≡→¬d< |
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74 |
24 | 75 triO : {n : Level} → {lx ly : Nat} → OrdinalD {n} lx → OrdinalD {n} ly → Tri (lx < ly) ( lx ≡ ly ) ( lx > ly ) |
76 triO {n} {lx} {ly} x y = <-cmp lx ly | |
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77 |
24 | 78 triOrdd : {n : Level} → {lx : Nat} → Trichotomous _≡_ ( _d<_ {n} {lx} {lx} ) |
79 triOrdd {_} {lv} (Φ lv) (Φ lv) = tri≈ ≡→¬d< refl ≡→¬d< | |
80 triOrdd {_} {lv} (Φ lv) (OSuc lv y) = tri< Φ< (λ ()) ( λ lt → trio<> lt Φ< ) | |
81 triOrdd {_} {lv} (OSuc lv x) (Φ lv) = tri> (λ lt → trio<> lt Φ<) (λ ()) Φ< | |
82 triOrdd {_} {lv} (OSuc lv x) (OSuc lv y) with triOrdd x y | |
83 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) ) | |
84 triOrdd {_} {lv} (OSuc lv x) (OSuc lv x) | tri≈ ¬a refl ¬c = tri≈ ≡→¬d< refl ≡→¬d< | |
85 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) | |
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86 |
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87 osuc : {n : Level} ( x : Ordinal {n} ) → Ordinal {n} |
75 | 88 osuc record { lv = lx ; ord = ox } = record { lv = lx ; ord = OSuc lx ox } |
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89 |
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90 <-osuc : {n : Level} { x : Ordinal {n} } → x o< osuc x |
75 | 91 <-osuc {n} {record { lv = lx ; ord = Φ .lx }} = case2 Φ< |
92 <-osuc {n} {record { lv = lx ; ord = OSuc .lx ox }} = case2 ( s< s<refl ) | |
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93 |
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94 osuc-lveq : {n : Level} { x : Ordinal {n} } → lv x ≡ lv ( osuc x ) |
75 | 95 osuc-lveq {n} = refl |
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96 |
113 | 97 osucc : {n : Level} {ox oy : Ordinal {n}} → oy o< ox → osuc oy o< osuc ox |
98 osucc {n} {ox} {oy} (case1 x) = case1 x | |
99 osucc {n} {ox} {oy} (case2 x) with d<→lv x | |
100 ... | refl = case2 (s< x) | |
101 | |
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102 nat-<> : { x y : Nat } → x < y → y < x → ⊥ |
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103 nat-<> (s≤s x<y) (s≤s y<x) = nat-<> x<y y<x |
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104 |
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105 nat-<≡ : { x : Nat } → x < x → ⊥ |
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106 nat-<≡ (s≤s lt) = nat-<≡ lt |
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107 |
81 | 108 nat-≡< : { x y : Nat } → x ≡ y → x < y → ⊥ |
109 nat-≡< refl lt = nat-<≡ lt | |
110 | |
75 | 111 ¬a≤a : {la : Nat} → Suc la ≤ la → ⊥ |
112 ¬a≤a (s≤s lt) = ¬a≤a lt | |
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113 |
94 | 114 =→¬< : {x : Nat } → ¬ ( x < x ) |
115 =→¬< {Zero} () | |
116 =→¬< {Suc x} (s≤s lt) = =→¬< lt | |
117 | |
147 | 118 case12-⊥ : {n : Level} {x y : Ordinal {suc n}} → lv x < lv y → ord x d< ord y → ⊥ |
119 case12-⊥ {x} {y} lt1 lt2 with d<→lv lt2 | |
120 ... | refl = nat-≡< refl lt1 | |
121 | |
122 case21-⊥ : {n : Level} {x y : Ordinal {suc n}} → lv x < lv y → ord y d< ord x → ⊥ | |
123 case21-⊥ {x} {y} lt1 lt2 with d<→lv lt2 | |
124 ... | refl = nat-≡< refl lt1 | |
125 | |
111 | 126 o<¬≡ : {n : Level } { ox oy : Ordinal {n}} → ox ≡ oy → ox o< oy → ⊥ |
127 o<¬≡ {_} {ox} {ox} refl (case1 lt) = =→¬< lt | |
128 o<¬≡ {_} {ox} {ox} refl (case2 (s< lt)) = trio<≡ refl lt | |
94 | 129 |
130 ¬x<0 : {n : Level} → { x : Ordinal {suc n} } → ¬ ( x o< o∅ {suc n} ) | |
131 ¬x<0 {n} {x} (case1 ()) | |
132 ¬x<0 {n} {x} (case2 ()) | |
133 | |
81 | 134 o<> : {n : Level} → {x y : Ordinal {n} } → y o< x → x o< y → ⊥ |
135 o<> {n} {x} {y} (case1 x₁) (case1 x₂) = nat-<> x₁ x₂ | |
136 o<> {n} {x} {y} (case1 x₁) (case2 x₂) = nat-≡< (sym (d<→lv x₂)) x₁ | |
137 o<> {n} {x} {y} (case2 x₁) (case1 x₂) = nat-≡< (sym (d<→lv x₁)) x₂ | |
138 o<> {n} {record { lv = lv₁ ; ord = .(OSuc lv₁ _) }} {record { lv = .lv₁ ; ord = .(Φ lv₁) }} (case2 Φ<) (case2 ()) | |
139 o<> {n} {record { lv = lv₁ ; ord = .(OSuc lv₁ _) }} {record { lv = .lv₁ ; ord = .(OSuc lv₁ _) }} (case2 (s< y<x)) (case2 (s< x<y)) = | |
140 o<> (case2 y<x) (case2 x<y) | |
16 | 141 |
24 | 142 orddtrans : {n : Level} {lx : Nat} {x y z : OrdinalD {n} lx } → x d< y → y d< z → x d< z |
143 orddtrans {_} {lx} {.(Φ lx)} {.(OSuc lx _)} {.(OSuc lx _)} Φ< (s< y<z) = Φ< | |
144 orddtrans {_} {lx} {.(OSuc lx _)} {.(OSuc lx _)} {.(OSuc lx _)} (s< x<y) (s< y<z) = s< ( orddtrans x<y y<z ) | |
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145 |
75 | 146 osuc-≡< : {n : Level} { a x : Ordinal {n} } → x o< osuc a → (x ≡ a ) ∨ (x o< a) |
147 osuc-≡< {n} {a} {x} (case1 lt) = case2 (case1 lt) | |
148 osuc-≡< {n} {record { lv = lv₁ ; ord = Φ .lv₁ }} {record { lv = .lv₁ ; ord = .(Φ lv₁) }} (case2 Φ<) = case1 refl | |
149 osuc-≡< {n} {record { lv = lv₁ ; ord = OSuc .lv₁ ord₁ }} {record { lv = .lv₁ ; ord = .(Φ lv₁) }} (case2 Φ<) = case2 (case2 Φ<) | |
150 osuc-≡< {n} {record { lv = lv₁ ; ord = Φ .lv₁ }} {record { lv = .lv₁ ; ord = .(OSuc lv₁ _) }} (case2 (s< ())) | |
151 osuc-≡< {n} {record { lv = la ; ord = OSuc la oa }} {record { lv = la ; ord = (OSuc la ox) }} (case2 (s< lt)) with | |
152 osuc-≡< {n} {record { lv = la ; ord = oa }} {record { lv = la ; ord = ox }} (case2 lt ) | |
153 ... | case1 refl = case1 refl | |
154 ... | case2 (case2 x) = case2 (case2( s< x) ) | |
155 ... | case2 (case1 x) = ⊥-elim (¬a≤a x) where | |
156 | |
157 osuc-< : {n : Level} { x y : Ordinal {n} } → y o< osuc x → x o< y → ⊥ | |
158 osuc-< {n} {x} {y} y<ox x<y with osuc-≡< y<ox | |
159 osuc-< {n} {x} {x} y<ox (case1 x₁) | case1 refl = ⊥-elim (¬a≤a x₁) | |
160 osuc-< {n} {x} {x} (case1 x₂) (case2 x₁) | case1 refl = ⊥-elim (¬a≤a x₂) | |
161 osuc-< {n} {x} {x} (case2 x₂) (case2 x₁) | case1 refl = ≡→¬d< x₁ | |
81 | 162 osuc-< {n} {x} {y} y<ox (case1 x₂) | case2 (case1 x₁) = nat-<> x₂ x₁ |
163 osuc-< {n} {x} {y} y<ox (case1 x₂) | case2 (case2 x₁) = nat-≡< (sym (d<→lv x₁)) x₂ | |
164 osuc-< {n} {x} {y} y<ox (case2 x<y) | case2 y<x = o<> (case2 x<y) y<x | |
75 | 165 |
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166 max : (x y : Nat) → Nat |
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167 max Zero Zero = Zero |
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168 max Zero (Suc x) = (Suc x) |
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169 max (Suc x) Zero = (Suc x) |
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170 max (Suc x) (Suc y) = Suc ( max x y ) |
3 | 171 |
24 | 172 maxαd : {n : Level} → { lx : Nat } → OrdinalD {n} lx → OrdinalD lx → OrdinalD lx |
17 | 173 maxαd x y with triOrdd x y |
174 maxαd x y | tri< a ¬b ¬c = y | |
175 maxαd x y | tri≈ ¬a b ¬c = x | |
176 maxαd x y | tri> ¬a ¬b c = x | |
6 | 177 |
127 | 178 minαd : {n : Level} → { lx : Nat } → OrdinalD {n} lx → OrdinalD lx → OrdinalD lx |
179 minαd x y with triOrdd x y | |
180 minαd x y | tri< a ¬b ¬c = x | |
181 minαd x y | tri≈ ¬a b ¬c = y | |
182 minαd x y | tri> ¬a ¬b c = x | |
183 | |
24 | 184 _o≤_ : {n : Level} → Ordinal → Ordinal → Set (suc n) |
23 | 185 a o≤ b = (a ≡ b) ∨ ( a o< b ) |
186 | |
27 | 187 ordtrans : {n : Level} {x y z : Ordinal {n} } → x o< y → y o< z → x o< z |
188 ordtrans {n} {x} {y} {z} (case1 x₁) (case1 x₂) = case1 ( <-trans x₁ x₂ ) | |
81 | 189 ordtrans {n} {x} {y} {z} (case1 x₁) (case2 x₂) with d<→lv x₂ |
27 | 190 ... | refl = case1 x₁ |
81 | 191 ordtrans {n} {x} {y} {z} (case2 x₁) (case1 x₂) with d<→lv x₁ |
27 | 192 ... | refl = case1 x₂ |
193 ordtrans {n} {x} {y} {z} (case2 x₁) (case2 x₂) with d<→lv x₁ | d<→lv x₂ | |
194 ... | refl | refl = case2 ( orddtrans x₁ x₂ ) | |
195 | |
24 | 196 trio< : {n : Level } → Trichotomous {suc n} _≡_ _o<_ |
23 | 197 trio< a b with <-cmp (lv a) (lv b) |
24 | 198 trio< a b | tri< a₁ ¬b ¬c = tri< (case1 a₁) (λ refl → ¬b (cong ( λ x → lv x ) refl ) ) lemma1 where |
199 lemma1 : ¬ (Suc (lv b) ≤ lv a) ∨ (ord b d< ord a) | |
200 lemma1 (case1 x) = ¬c x | |
81 | 201 lemma1 (case2 x) = ⊥-elim (nat-≡< (sym ( d<→lv x )) a₁ ) |
24 | 202 trio< a b | tri> ¬a ¬b c = tri> lemma1 (λ refl → ¬b (cong ( λ x → lv x ) refl ) ) (case1 c) where |
203 lemma1 : ¬ (Suc (lv a) ≤ lv b) ∨ (ord a d< ord b) | |
204 lemma1 (case1 x) = ¬a x | |
81 | 205 lemma1 (case2 x) = ⊥-elim (nat-≡< (sym ( d<→lv x )) c ) |
23 | 206 trio< a b | tri≈ ¬a refl ¬c with triOrdd ( ord a ) ( ord b ) |
24 | 207 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 |
208 lemma1 : (record { lv = _ ; ord = x }) ≡ b → x ≡ ord b | |
209 lemma1 refl = refl | |
210 lemma2 : ¬ (Suc (lv b) ≤ lv b) ∨ (ord b d< x) | |
211 lemma2 (case1 x) = ¬a x | |
212 lemma2 (case2 x) = trio<> x a | |
213 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 | |
214 lemma1 : (record { lv = _ ; ord = x }) ≡ b → x ≡ ord b | |
215 lemma1 refl = refl | |
216 lemma2 : ¬ (Suc (lv b) ≤ lv b) ∨ (x d< ord b) | |
217 lemma2 (case1 x) = ¬a x | |
218 lemma2 (case2 x) = trio<> x c | |
219 trio< record { lv = .(lv b) ; ord = x } b | tri≈ ¬a refl ¬c | tri≈ ¬a₁ refl ¬c₁ = tri≈ lemma1 refl lemma1 where | |
220 lemma1 : ¬ (Suc (lv b) ≤ lv b) ∨ (ord b d< ord b) | |
221 lemma1 (case1 x) = ¬a x | |
222 lemma1 (case2 x) = ≡→¬d< x | |
23 | 223 |
180 | 224 xo<ab : {n : Level} {oa ob : Ordinal {suc n}} → ( {ox : Ordinal {suc n}} → ox o< oa → ox o< ob ) → oa o< osuc ob |
225 xo<ab {n} {oa} {ob} a→b with trio< oa ob | |
226 xo<ab {n} {oa} {ob} a→b | tri< a ¬b ¬c = ordtrans a <-osuc | |
227 xo<ab {n} {oa} {ob} a→b | tri≈ ¬a refl ¬c = <-osuc | |
228 xo<ab {n} {oa} {ob} a→b | tri> ¬a ¬b c = ⊥-elim ( o<¬≡ refl (a→b c ) ) | |
229 | |
129 | 230 maxα : {n : Level} → Ordinal {suc n} → Ordinal → Ordinal |
231 maxα x y with trio< x y | |
127 | 232 maxα x y | tri< a ¬b ¬c = y |
233 maxα x y | tri> ¬a ¬b c = x | |
129 | 234 maxα x y | tri≈ ¬a refl ¬c = x |
84 | 235 |
129 | 236 minα : {n : Level} → Ordinal {suc n} → Ordinal → Ordinal |
237 minα {n} x y with trio< {n} x y | |
127 | 238 minα x y | tri< a ¬b ¬c = x |
239 minα x y | tri> ¬a ¬b c = y | |
129 | 240 minα x y | tri≈ ¬a refl ¬c = x |
241 | |
242 min1 : {n : Level} → {x y z : Ordinal {suc n} } → z o< x → z o< y → z o< minα x y | |
243 min1 {n} {x} {y} {z} z<x z<y with trio< {n} x y | |
244 min1 {n} {x} {y} {z} z<x z<y | tri< a ¬b ¬c = z<x | |
245 min1 {n} {x} {y} {z} z<x z<y | tri≈ ¬a refl ¬c = z<x | |
246 min1 {n} {x} {y} {z} z<x z<y | tri> ¬a ¬b c = z<y | |
247 | |
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248 -- |
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249 -- max ( osuc x , osuc y ) |
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250 -- |
88 | 251 |
84 | 252 omax : {n : Level} ( x y : Ordinal {suc n} ) → Ordinal {suc n} |
88 | 253 omax {n} x y with trio< x y |
84 | 254 omax {n} x y | tri< a ¬b ¬c = osuc y |
255 omax {n} x y | tri> ¬a ¬b c = osuc x | |
88 | 256 omax {n} x y | tri≈ ¬a refl ¬c = osuc x |
84 | 257 |
258 omax< : {n : Level} ( x y : Ordinal {suc n} ) → x o< y → osuc y ≡ omax x y | |
88 | 259 omax< {n} x y lt with trio< x y |
84 | 260 omax< {n} x y lt | tri< a ¬b ¬c = refl |
88 | 261 omax< {n} x y lt | tri≈ ¬a b ¬c = ⊥-elim (¬a lt ) |
262 omax< {n} x y lt | tri> ¬a ¬b c = ⊥-elim (¬a lt ) | |
263 | |
264 omax≡ : {n : Level} ( x y : Ordinal {suc n} ) → x ≡ y → osuc y ≡ omax x y | |
265 omax≡ {n} x y eq with trio< x y | |
266 omax≡ {n} x y eq | tri< a ¬b ¬c = ⊥-elim (¬b eq ) | |
267 omax≡ {n} x y eq | tri≈ ¬a refl ¬c = refl | |
268 omax≡ {n} x y eq | tri> ¬a ¬b c = ⊥-elim (¬b eq ) | |
84 | 269 |
86 | 270 omax-x : {n : Level} ( x y : Ordinal {suc n} ) → x o< omax x y |
88 | 271 omax-x {n} x y with trio< x y |
272 omax-x {n} x y | tri< a ¬b ¬c = ordtrans a <-osuc | |
86 | 273 omax-x {n} x y | tri> ¬a ¬b c = <-osuc |
88 | 274 omax-x {n} x y | tri≈ ¬a refl ¬c = <-osuc |
86 | 275 |
276 omax-y : {n : Level} ( x y : Ordinal {suc n} ) → y o< omax x y | |
88 | 277 omax-y {n} x y with trio< x y |
86 | 278 omax-y {n} x y | tri< a ¬b ¬c = <-osuc |
88 | 279 omax-y {n} x y | tri> ¬a ¬b c = ordtrans c <-osuc |
280 omax-y {n} x y | tri≈ ¬a refl ¬c = <-osuc | |
86 | 281 |
88 | 282 omxx : {n : Level} ( x : Ordinal {suc n} ) → omax x x ≡ osuc x |
283 omxx {n} x with trio< x x | |
284 omxx {n} x | tri< a ¬b ¬c = ⊥-elim (¬b refl ) | |
285 omxx {n} x | tri> ¬a ¬b c = ⊥-elim (¬b refl ) | |
286 omxx {n} x | tri≈ ¬a refl ¬c = refl | |
86 | 287 |
288 omxxx : {n : Level} ( x : Ordinal {suc n} ) → omax x (omax x x ) ≡ osuc (osuc x) | |
88 | 289 omxxx {n} x = trans ( cong ( λ k → omax x k ) (omxx x )) (sym ( omax< x (osuc x) <-osuc )) |
86 | 290 |
91 | 291 open _∧_ |
292 | |
293 osuc2 : {n : Level} ( x y : Ordinal {suc n} ) → ( osuc x o< osuc (osuc y )) ⇔ (x o< osuc y) | |
294 proj1 (osuc2 {n} x y) (case1 lt) = case1 lt | |
295 proj1 (osuc2 {n} x y) (case2 (s< lt)) = case2 lt | |
296 proj2 (osuc2 {n} x y) (case1 lt) = case1 lt | |
297 proj2 (osuc2 {n} x y) (case2 lt) with d<→lv lt | |
298 ... | refl = case2 (s< lt) | |
299 | |
24 | 300 OrdTrans : {n : Level} → Transitive {suc n} _o≤_ |
16 | 301 OrdTrans (case1 refl) (case1 refl) = case1 refl |
302 OrdTrans (case1 refl) (case2 lt2) = case2 lt2 | |
303 OrdTrans (case2 lt1) (case1 refl) = case2 lt1 | |
81 | 304 OrdTrans (case2 x) (case2 y) = case2 (ordtrans x y) |
16 | 305 |
24 | 306 OrdPreorder : {n : Level} → Preorder (suc n) (suc n) (suc n) |
307 OrdPreorder {n} = record { Carrier = Ordinal | |
16 | 308 ; _≈_ = _≡_ |
23 | 309 ; _∼_ = _o≤_ |
16 | 310 ; isPreorder = record { |
311 isEquivalence = record { refl = refl ; sym = sym ; trans = trans } | |
312 ; reflexive = case1 | |
24 | 313 ; trans = OrdTrans |
16 | 314 } |
315 } | |
316 | |
167 | 317 TransFinite : {n m : Level} → { ψ : Ordinal {n} → Set m } |
24 | 318 → ( ∀ (lx : Nat ) → ψ ( record { lv = lx ; ord = Φ lx } ) ) |
319 → ( ∀ (lx : Nat ) → (x : OrdinalD lx ) → ψ ( record { lv = lx ; ord = x } ) → ψ ( record { lv = lx ; ord = OSuc lx x } ) ) | |
22 | 320 → ∀ (x : Ordinal) → ψ x |
81 | 321 TransFinite caseΦ caseOSuc record { lv = lv ; ord = (Φ (lv)) } = caseΦ lv |
322 TransFinite caseΦ caseOSuc record { lv = lx ; ord = (OSuc lx ox) } = | |
323 caseOSuc lx ox (TransFinite caseΦ caseOSuc record { lv = lx ; ord = ox }) | |
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324 |
184 | 325 -- we cannot prove this in intutionistic logic |
142 | 326 -- (¬ (∀ y → ¬ ( ψ y ))) → (ψ y → p ) → p |
166 | 327 -- postulate |
328 -- TransFiniteExists : {n m l : Level} → ( ψ : Ordinal {n} → Set m ) | |
329 -- → (exists : ¬ (∀ y → ¬ ( ψ y ) )) | |
330 -- → {p : Set l} ( P : { y : Ordinal {n} } → ψ y → p ) | |
331 -- → p | |
332 -- | |
333 -- Instead we prove | |
334 -- | |
335 TransFiniteExists : {n m l : Level} → ( ψ : Ordinal {n} → Set m ) | |
165 | 336 → {p : Set l} ( P : { y : Ordinal {n} } → ψ y → ¬ p ) |
337 → (exists : ¬ (∀ y → ¬ ( ψ y ) )) | |
338 → ¬ p | |
166 | 339 TransFiniteExists {n} {m} {l} ψ {p} P = contra-position ( λ p y ψy → P {y} ψy p ) |
165 | 340 |