Mercurial > hg > Members > kono > Proof > ZF-in-agda
annotate ordinal.agda @ 202:ed88384b5102
ε-induction like loop again
author | Shinji KONO <kono@ie.u-ryukyu.ac.jp> |
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date | Tue, 30 Jul 2019 17:52:15 +0900 |
parents | 65e1b2e415bb |
children | 8edd2a13a7f3 |
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 |
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16 constructor ordinal |
16 | 17 field |
18 lv : Nat | |
24 | 19 ord : OrdinalD {n} lv |
16 | 20 |
24 | 21 data _d<_ {n : Level} : {lx ly : Nat} → OrdinalD {n} lx → OrdinalD {n} ly → Set n where |
22 Φ< : {lx : Nat} → {x : OrdinalD {n} lx} → Φ lx d< OSuc lx x | |
23 s< : {lx : Nat} → {x y : OrdinalD {n} lx} → x d< y → OSuc lx x d< OSuc lx y | |
17 | 24 |
25 open Ordinal | |
26 | |
27 | 27 _o<_ : {n : Level} ( x y : Ordinal ) → Set n |
17 | 28 _o<_ x y = (lv x < lv y ) ∨ ( ord x d< ord y ) |
3 | 29 |
75 | 30 s<refl : {n : Level } {lx : Nat } { x : OrdinalD {n} lx } → x d< OSuc lx x |
31 s<refl {n} {lv} {Φ lv} = Φ< | |
32 s<refl {n} {lv} {OSuc lv x} = s< s<refl | |
33 | |
34 trio<> : {n : Level} → {lx : Nat} {x : OrdinalD {n} lx } { y : OrdinalD lx } → y d< x → x d< y → ⊥ | |
35 trio<> {n} {lx} {.(OSuc lx _)} {.(OSuc lx _)} (s< s) (s< t) = trio<> s t | |
36 trio<> {n} {lx} {.(OSuc lx _)} {.(Φ lx)} Φ< () | |
37 | |
38 d<→lv : {n : Level} {x y : Ordinal {n}} → ord x d< ord y → lv x ≡ lv y | |
39 d<→lv Φ< = refl | |
40 d<→lv (s< lt) = refl | |
41 | |
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42 o<-subst : {n : Level } {Z X z x : Ordinal {n}} → Z o< X → Z ≡ z → X ≡ x → z o< x |
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43 o<-subst df refl refl = df |
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44 |
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45 open import Data.Nat.Properties |
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problem on Ordinal ( OSuc ℵ )
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46 open import Data.Unit using ( ⊤ ) |
6 | 47 open import Relation.Nullary |
48 | |
49 open import Relation.Binary | |
50 open import Relation.Binary.Core | |
51 | |
24 | 52 o∅ : {n : Level} → Ordinal {n} |
53 o∅ = record { lv = Zero ; ord = Φ Zero } | |
21 | 54 |
39 | 55 open import Relation.Binary.HeterogeneousEquality using (_≅_;refl) |
56 | |
57 ordinal-cong : {n : Level} {x y : Ordinal {n}} → | |
58 lv x ≡ lv y → ord x ≅ ord y → x ≡ y | |
59 ordinal-cong refl refl = refl | |
21 | 60 |
46 | 61 ordinal-lv : {n : Level} {x y : Ordinal {n}} → x ≡ y → lv x ≡ lv y |
62 ordinal-lv refl = refl | |
63 | |
64 ordinal-d : {n : Level} {x y : Ordinal {n}} → x ≡ y → ord x ≅ ord y | |
65 ordinal-d refl = refl | |
66 | |
24 | 67 ≡→¬d< : {n : Level} → {lv : Nat} → {x : OrdinalD {n} lv } → x d< x → ⊥ |
68 ≡→¬d< {n} {lx} {OSuc lx y} (s< t) = ≡→¬d< t | |
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69 |
24 | 70 trio<≡ : {n : Level} → {lx : Nat} {x : OrdinalD {n} lx } { y : OrdinalD lx } → x ≡ y → x d< y → ⊥ |
17 | 71 trio<≡ refl = ≡→¬d< |
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72 |
24 | 73 trio>≡ : {n : Level} → {lx : Nat} {x : OrdinalD {n} lx } { y : OrdinalD lx } → x ≡ y → y d< x → ⊥ |
17 | 74 trio>≡ refl = ≡→¬d< |
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75 |
24 | 76 triO : {n : Level} → {lx ly : Nat} → OrdinalD {n} lx → OrdinalD {n} ly → Tri (lx < ly) ( lx ≡ ly ) ( lx > ly ) |
77 triO {n} {lx} {ly} x y = <-cmp lx ly | |
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78 |
24 | 79 triOrdd : {n : Level} → {lx : Nat} → Trichotomous _≡_ ( _d<_ {n} {lx} {lx} ) |
80 triOrdd {_} {lv} (Φ lv) (Φ lv) = tri≈ ≡→¬d< refl ≡→¬d< | |
81 triOrdd {_} {lv} (Φ lv) (OSuc lv y) = tri< Φ< (λ ()) ( λ lt → trio<> lt Φ< ) | |
82 triOrdd {_} {lv} (OSuc lv x) (Φ lv) = tri> (λ lt → trio<> lt Φ<) (λ ()) Φ< | |
83 triOrdd {_} {lv} (OSuc lv x) (OSuc lv y) with triOrdd x y | |
84 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) ) | |
85 triOrdd {_} {lv} (OSuc lv x) (OSuc lv x) | tri≈ ¬a refl ¬c = tri≈ ≡→¬d< refl ≡→¬d< | |
86 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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87 |
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88 osuc : {n : Level} ( x : Ordinal {n} ) → Ordinal {n} |
75 | 89 osuc record { lv = lx ; ord = ox } = record { lv = lx ; ord = OSuc lx ox } |
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90 |
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91 <-osuc : {n : Level} { x : Ordinal {n} } → x o< osuc x |
75 | 92 <-osuc {n} {record { lv = lx ; ord = Φ .lx }} = case2 Φ< |
93 <-osuc {n} {record { lv = lx ; ord = OSuc .lx ox }} = case2 ( s< s<refl ) | |
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94 |
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95 osuc-lveq : {n : Level} { x : Ordinal {n} } → lv x ≡ lv ( osuc x ) |
75 | 96 osuc-lveq {n} = refl |
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97 |
113 | 98 osucc : {n : Level} {ox oy : Ordinal {n}} → oy o< ox → osuc oy o< osuc ox |
99 osucc {n} {ox} {oy} (case1 x) = case1 x | |
100 osucc {n} {ox} {oy} (case2 x) with d<→lv x | |
101 ... | refl = case2 (s< x) | |
102 | |
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103 nat-<> : { x y : Nat } → x < y → y < x → ⊥ |
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104 nat-<> (s≤s x<y) (s≤s y<x) = nat-<> x<y y<x |
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105 |
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106 nat-<≡ : { x : Nat } → x < x → ⊥ |
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107 nat-<≡ (s≤s lt) = nat-<≡ lt |
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108 |
81 | 109 nat-≡< : { x y : Nat } → x ≡ y → x < y → ⊥ |
110 nat-≡< refl lt = nat-<≡ lt | |
111 | |
75 | 112 ¬a≤a : {la : Nat} → Suc la ≤ la → ⊥ |
113 ¬a≤a (s≤s lt) = ¬a≤a lt | |
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114 |
94 | 115 =→¬< : {x : Nat } → ¬ ( x < x ) |
116 =→¬< {Zero} () | |
117 =→¬< {Suc x} (s≤s lt) = =→¬< lt | |
118 | |
147 | 119 case12-⊥ : {n : Level} {x y : Ordinal {suc n}} → lv x < lv y → ord x d< ord y → ⊥ |
120 case12-⊥ {x} {y} lt1 lt2 with d<→lv lt2 | |
121 ... | refl = nat-≡< refl lt1 | |
122 | |
123 case21-⊥ : {n : Level} {x y : Ordinal {suc n}} → lv x < lv y → ord y d< ord x → ⊥ | |
124 case21-⊥ {x} {y} lt1 lt2 with d<→lv lt2 | |
125 ... | refl = nat-≡< refl lt1 | |
126 | |
111 | 127 o<¬≡ : {n : Level } { ox oy : Ordinal {n}} → ox ≡ oy → ox o< oy → ⊥ |
128 o<¬≡ {_} {ox} {ox} refl (case1 lt) = =→¬< lt | |
129 o<¬≡ {_} {ox} {ox} refl (case2 (s< lt)) = trio<≡ refl lt | |
94 | 130 |
131 ¬x<0 : {n : Level} → { x : Ordinal {suc n} } → ¬ ( x o< o∅ {suc n} ) | |
132 ¬x<0 {n} {x} (case1 ()) | |
133 ¬x<0 {n} {x} (case2 ()) | |
134 | |
81 | 135 o<> : {n : Level} → {x y : Ordinal {n} } → y o< x → x o< y → ⊥ |
136 o<> {n} {x} {y} (case1 x₁) (case1 x₂) = nat-<> x₁ x₂ | |
137 o<> {n} {x} {y} (case1 x₁) (case2 x₂) = nat-≡< (sym (d<→lv x₂)) x₁ | |
138 o<> {n} {x} {y} (case2 x₁) (case1 x₂) = nat-≡< (sym (d<→lv x₁)) x₂ | |
139 o<> {n} {record { lv = lv₁ ; ord = .(OSuc lv₁ _) }} {record { lv = .lv₁ ; ord = .(Φ lv₁) }} (case2 Φ<) (case2 ()) | |
140 o<> {n} {record { lv = lv₁ ; ord = .(OSuc lv₁ _) }} {record { lv = .lv₁ ; ord = .(OSuc lv₁ _) }} (case2 (s< y<x)) (case2 (s< x<y)) = | |
141 o<> (case2 y<x) (case2 x<y) | |
16 | 142 |
24 | 143 orddtrans : {n : Level} {lx : Nat} {x y z : OrdinalD {n} lx } → x d< y → y d< z → x d< z |
144 orddtrans {_} {lx} {.(Φ lx)} {.(OSuc lx _)} {.(OSuc lx _)} Φ< (s< y<z) = Φ< | |
145 orddtrans {_} {lx} {.(OSuc lx _)} {.(OSuc lx _)} {.(OSuc lx _)} (s< x<y) (s< y<z) = s< ( orddtrans x<y y<z ) | |
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146 |
75 | 147 osuc-≡< : {n : Level} { a x : Ordinal {n} } → x o< osuc a → (x ≡ a ) ∨ (x o< a) |
148 osuc-≡< {n} {a} {x} (case1 lt) = case2 (case1 lt) | |
149 osuc-≡< {n} {record { lv = lv₁ ; ord = Φ .lv₁ }} {record { lv = .lv₁ ; ord = .(Φ lv₁) }} (case2 Φ<) = case1 refl | |
150 osuc-≡< {n} {record { lv = lv₁ ; ord = OSuc .lv₁ ord₁ }} {record { lv = .lv₁ ; ord = .(Φ lv₁) }} (case2 Φ<) = case2 (case2 Φ<) | |
151 osuc-≡< {n} {record { lv = lv₁ ; ord = Φ .lv₁ }} {record { lv = .lv₁ ; ord = .(OSuc lv₁ _) }} (case2 (s< ())) | |
152 osuc-≡< {n} {record { lv = la ; ord = OSuc la oa }} {record { lv = la ; ord = (OSuc la ox) }} (case2 (s< lt)) with | |
153 osuc-≡< {n} {record { lv = la ; ord = oa }} {record { lv = la ; ord = ox }} (case2 lt ) | |
154 ... | case1 refl = case1 refl | |
155 ... | case2 (case2 x) = case2 (case2( s< x) ) | |
156 ... | case2 (case1 x) = ⊥-elim (¬a≤a x) where | |
157 | |
158 osuc-< : {n : Level} { x y : Ordinal {n} } → y o< osuc x → x o< y → ⊥ | |
159 osuc-< {n} {x} {y} y<ox x<y with osuc-≡< y<ox | |
160 osuc-< {n} {x} {x} y<ox (case1 x₁) | case1 refl = ⊥-elim (¬a≤a x₁) | |
161 osuc-< {n} {x} {x} (case1 x₂) (case2 x₁) | case1 refl = ⊥-elim (¬a≤a x₂) | |
162 osuc-< {n} {x} {x} (case2 x₂) (case2 x₁) | case1 refl = ≡→¬d< x₁ | |
81 | 163 osuc-< {n} {x} {y} y<ox (case1 x₂) | case2 (case1 x₁) = nat-<> x₂ x₁ |
164 osuc-< {n} {x} {y} y<ox (case1 x₂) | case2 (case2 x₁) = nat-≡< (sym (d<→lv x₁)) x₂ | |
165 osuc-< {n} {x} {y} y<ox (case2 x<y) | case2 y<x = o<> (case2 x<y) y<x | |
75 | 166 |
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167 max : (x y : Nat) → Nat |
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168 max Zero Zero = Zero |
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169 max Zero (Suc x) = (Suc x) |
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170 max (Suc x) Zero = (Suc x) |
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171 max (Suc x) (Suc y) = Suc ( max x y ) |
3 | 172 |
24 | 173 maxαd : {n : Level} → { lx : Nat } → OrdinalD {n} lx → OrdinalD lx → OrdinalD lx |
17 | 174 maxαd x y with triOrdd x y |
175 maxαd x y | tri< a ¬b ¬c = y | |
176 maxαd x y | tri≈ ¬a b ¬c = x | |
177 maxαd x y | tri> ¬a ¬b c = x | |
6 | 178 |
127 | 179 minαd : {n : Level} → { lx : Nat } → OrdinalD {n} lx → OrdinalD lx → OrdinalD lx |
180 minαd x y with triOrdd x y | |
181 minαd x y | tri< a ¬b ¬c = x | |
182 minαd x y | tri≈ ¬a b ¬c = y | |
183 minαd x y | tri> ¬a ¬b c = x | |
184 | |
24 | 185 _o≤_ : {n : Level} → Ordinal → Ordinal → Set (suc n) |
23 | 186 a o≤ b = (a ≡ b) ∨ ( a o< b ) |
187 | |
27 | 188 ordtrans : {n : Level} {x y z : Ordinal {n} } → x o< y → y o< z → x o< z |
189 ordtrans {n} {x} {y} {z} (case1 x₁) (case1 x₂) = case1 ( <-trans x₁ x₂ ) | |
81 | 190 ordtrans {n} {x} {y} {z} (case1 x₁) (case2 x₂) with d<→lv x₂ |
27 | 191 ... | refl = case1 x₁ |
81 | 192 ordtrans {n} {x} {y} {z} (case2 x₁) (case1 x₂) with d<→lv x₁ |
27 | 193 ... | refl = case1 x₂ |
194 ordtrans {n} {x} {y} {z} (case2 x₁) (case2 x₂) with d<→lv x₁ | d<→lv x₂ | |
195 ... | refl | refl = case2 ( orddtrans x₁ x₂ ) | |
196 | |
24 | 197 trio< : {n : Level } → Trichotomous {suc n} _≡_ _o<_ |
23 | 198 trio< a b with <-cmp (lv a) (lv b) |
24 | 199 trio< a b | tri< a₁ ¬b ¬c = tri< (case1 a₁) (λ refl → ¬b (cong ( λ x → lv x ) refl ) ) lemma1 where |
200 lemma1 : ¬ (Suc (lv b) ≤ lv a) ∨ (ord b d< ord a) | |
201 lemma1 (case1 x) = ¬c x | |
81 | 202 lemma1 (case2 x) = ⊥-elim (nat-≡< (sym ( d<→lv x )) a₁ ) |
24 | 203 trio< a b | tri> ¬a ¬b c = tri> lemma1 (λ refl → ¬b (cong ( λ x → lv x ) refl ) ) (case1 c) where |
204 lemma1 : ¬ (Suc (lv a) ≤ lv b) ∨ (ord a d< ord b) | |
205 lemma1 (case1 x) = ¬a x | |
81 | 206 lemma1 (case2 x) = ⊥-elim (nat-≡< (sym ( d<→lv x )) c ) |
23 | 207 trio< a b | tri≈ ¬a refl ¬c with triOrdd ( ord a ) ( ord b ) |
24 | 208 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 |
209 lemma1 : (record { lv = _ ; ord = x }) ≡ b → x ≡ ord b | |
210 lemma1 refl = refl | |
211 lemma2 : ¬ (Suc (lv b) ≤ lv b) ∨ (ord b d< x) | |
212 lemma2 (case1 x) = ¬a x | |
213 lemma2 (case2 x) = trio<> x a | |
214 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 | |
215 lemma1 : (record { lv = _ ; ord = x }) ≡ b → x ≡ ord b | |
216 lemma1 refl = refl | |
217 lemma2 : ¬ (Suc (lv b) ≤ lv b) ∨ (x d< ord b) | |
218 lemma2 (case1 x) = ¬a x | |
219 lemma2 (case2 x) = trio<> x c | |
220 trio< record { lv = .(lv b) ; ord = x } b | tri≈ ¬a refl ¬c | tri≈ ¬a₁ refl ¬c₁ = tri≈ lemma1 refl lemma1 where | |
221 lemma1 : ¬ (Suc (lv b) ≤ lv b) ∨ (ord b d< ord b) | |
222 lemma1 (case1 x) = ¬a x | |
223 lemma1 (case2 x) = ≡→¬d< x | |
23 | 224 |
180 | 225 xo<ab : {n : Level} {oa ob : Ordinal {suc n}} → ( {ox : Ordinal {suc n}} → ox o< oa → ox o< ob ) → oa o< osuc ob |
226 xo<ab {n} {oa} {ob} a→b with trio< oa ob | |
227 xo<ab {n} {oa} {ob} a→b | tri< a ¬b ¬c = ordtrans a <-osuc | |
228 xo<ab {n} {oa} {ob} a→b | tri≈ ¬a refl ¬c = <-osuc | |
229 xo<ab {n} {oa} {ob} a→b | tri> ¬a ¬b c = ⊥-elim ( o<¬≡ refl (a→b c ) ) | |
230 | |
129 | 231 maxα : {n : Level} → Ordinal {suc n} → Ordinal → Ordinal |
232 maxα x y with trio< x y | |
127 | 233 maxα x y | tri< a ¬b ¬c = y |
234 maxα x y | tri> ¬a ¬b c = x | |
129 | 235 maxα x y | tri≈ ¬a refl ¬c = x |
84 | 236 |
129 | 237 minα : {n : Level} → Ordinal {suc n} → Ordinal → Ordinal |
238 minα {n} x y with trio< {n} x y | |
127 | 239 minα x y | tri< a ¬b ¬c = x |
240 minα x y | tri> ¬a ¬b c = y | |
129 | 241 minα x y | tri≈ ¬a refl ¬c = x |
242 | |
243 min1 : {n : Level} → {x y z : Ordinal {suc n} } → z o< x → z o< y → z o< minα x y | |
244 min1 {n} {x} {y} {z} z<x z<y with trio< {n} x y | |
245 min1 {n} {x} {y} {z} z<x z<y | tri< a ¬b ¬c = z<x | |
246 min1 {n} {x} {y} {z} z<x z<y | tri≈ ¬a refl ¬c = z<x | |
247 min1 {n} {x} {y} {z} z<x z<y | tri> ¬a ¬b c = z<y | |
248 | |
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249 -- |
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250 -- max ( osuc x , osuc y ) |
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251 -- |
88 | 252 |
84 | 253 omax : {n : Level} ( x y : Ordinal {suc n} ) → Ordinal {suc n} |
88 | 254 omax {n} x y with trio< x y |
84 | 255 omax {n} x y | tri< a ¬b ¬c = osuc y |
256 omax {n} x y | tri> ¬a ¬b c = osuc x | |
88 | 257 omax {n} x y | tri≈ ¬a refl ¬c = osuc x |
84 | 258 |
259 omax< : {n : Level} ( x y : Ordinal {suc n} ) → x o< y → osuc y ≡ omax x y | |
88 | 260 omax< {n} x y lt with trio< x y |
84 | 261 omax< {n} x y lt | tri< a ¬b ¬c = refl |
88 | 262 omax< {n} x y lt | tri≈ ¬a b ¬c = ⊥-elim (¬a lt ) |
263 omax< {n} x y lt | tri> ¬a ¬b c = ⊥-elim (¬a lt ) | |
264 | |
265 omax≡ : {n : Level} ( x y : Ordinal {suc n} ) → x ≡ y → osuc y ≡ omax x y | |
266 omax≡ {n} x y eq with trio< x y | |
267 omax≡ {n} x y eq | tri< a ¬b ¬c = ⊥-elim (¬b eq ) | |
268 omax≡ {n} x y eq | tri≈ ¬a refl ¬c = refl | |
269 omax≡ {n} x y eq | tri> ¬a ¬b c = ⊥-elim (¬b eq ) | |
84 | 270 |
86 | 271 omax-x : {n : Level} ( x y : Ordinal {suc n} ) → x o< omax x y |
88 | 272 omax-x {n} x y with trio< x y |
273 omax-x {n} x y | tri< a ¬b ¬c = ordtrans a <-osuc | |
86 | 274 omax-x {n} x y | tri> ¬a ¬b c = <-osuc |
88 | 275 omax-x {n} x y | tri≈ ¬a refl ¬c = <-osuc |
86 | 276 |
277 omax-y : {n : Level} ( x y : Ordinal {suc n} ) → y o< omax x y | |
88 | 278 omax-y {n} x y with trio< x y |
86 | 279 omax-y {n} x y | tri< a ¬b ¬c = <-osuc |
88 | 280 omax-y {n} x y | tri> ¬a ¬b c = ordtrans c <-osuc |
281 omax-y {n} x y | tri≈ ¬a refl ¬c = <-osuc | |
86 | 282 |
88 | 283 omxx : {n : Level} ( x : Ordinal {suc n} ) → omax x x ≡ osuc x |
284 omxx {n} x with trio< x x | |
285 omxx {n} x | tri< a ¬b ¬c = ⊥-elim (¬b refl ) | |
286 omxx {n} x | tri> ¬a ¬b c = ⊥-elim (¬b refl ) | |
287 omxx {n} x | tri≈ ¬a refl ¬c = refl | |
86 | 288 |
289 omxxx : {n : Level} ( x : Ordinal {suc n} ) → omax x (omax x x ) ≡ osuc (osuc x) | |
88 | 290 omxxx {n} x = trans ( cong ( λ k → omax x k ) (omxx x )) (sym ( omax< x (osuc x) <-osuc )) |
86 | 291 |
91 | 292 open _∧_ |
293 | |
294 osuc2 : {n : Level} ( x y : Ordinal {suc n} ) → ( osuc x o< osuc (osuc y )) ⇔ (x o< osuc y) | |
295 proj1 (osuc2 {n} x y) (case1 lt) = case1 lt | |
296 proj1 (osuc2 {n} x y) (case2 (s< lt)) = case2 lt | |
297 proj2 (osuc2 {n} x y) (case1 lt) = case1 lt | |
298 proj2 (osuc2 {n} x y) (case2 lt) with d<→lv lt | |
299 ... | refl = case2 (s< lt) | |
300 | |
24 | 301 OrdTrans : {n : Level} → Transitive {suc n} _o≤_ |
16 | 302 OrdTrans (case1 refl) (case1 refl) = case1 refl |
303 OrdTrans (case1 refl) (case2 lt2) = case2 lt2 | |
304 OrdTrans (case2 lt1) (case1 refl) = case2 lt1 | |
81 | 305 OrdTrans (case2 x) (case2 y) = case2 (ordtrans x y) |
16 | 306 |
24 | 307 OrdPreorder : {n : Level} → Preorder (suc n) (suc n) (suc n) |
308 OrdPreorder {n} = record { Carrier = Ordinal | |
16 | 309 ; _≈_ = _≡_ |
23 | 310 ; _∼_ = _o≤_ |
16 | 311 ; isPreorder = record { |
312 isEquivalence = record { refl = refl ; sym = sym ; trans = trans } | |
313 ; reflexive = case1 | |
24 | 314 ; trans = OrdTrans |
16 | 315 } |
316 } | |
317 | |
167 | 318 TransFinite : {n m : Level} → { ψ : Ordinal {n} → Set m } |
24 | 319 → ( ∀ (lx : Nat ) → ψ ( record { lv = lx ; ord = Φ lx } ) ) |
320 → ( ∀ (lx : Nat ) → (x : OrdinalD lx ) → ψ ( record { lv = lx ; ord = x } ) → ψ ( record { lv = lx ; ord = OSuc lx x } ) ) | |
22 | 321 → ∀ (x : Ordinal) → ψ x |
81 | 322 TransFinite caseΦ caseOSuc record { lv = lv ; ord = (Φ (lv)) } = caseΦ lv |
323 TransFinite caseΦ caseOSuc record { lv = lx ; ord = (OSuc lx ox) } = | |
324 caseOSuc lx ox (TransFinite caseΦ caseOSuc record { lv = lx ; ord = ox }) | |
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325 |
184 | 326 -- we cannot prove this in intutionistic logic |
142 | 327 -- (¬ (∀ y → ¬ ( ψ y ))) → (ψ y → p ) → p |
166 | 328 -- postulate |
329 -- TransFiniteExists : {n m l : Level} → ( ψ : Ordinal {n} → Set m ) | |
330 -- → (exists : ¬ (∀ y → ¬ ( ψ y ) )) | |
331 -- → {p : Set l} ( P : { y : Ordinal {n} } → ψ y → p ) | |
332 -- → p | |
333 -- | |
334 -- Instead we prove | |
335 -- | |
336 TransFiniteExists : {n m l : Level} → ( ψ : Ordinal {n} → Set m ) | |
165 | 337 → {p : Set l} ( P : { y : Ordinal {n} } → ψ y → ¬ p ) |
338 → (exists : ¬ (∀ y → ¬ ( ψ y ) )) | |
339 → ¬ p | |
166 | 340 TransFiniteExists {n} {m} {l} ψ {p} P = contra-position ( λ p y ψy → P {y} ψy p ) |
165 | 341 |