Mercurial > hg > Members > kono > Proof > ZF-in-agda
annotate src/ODUtil.agda @ 1284:45cd80181a29
remove import zf
| author | Shinji KONO <kono@ie.u-ryukyu.ac.jp> |
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| date | Sat, 20 May 2023 09:48:37 +0900 |
| parents | 8a071bf52407 |
| children | 302cfb533201 |
| rev | line source |
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1 {-# OPTIONS --allow-unsolved-metas #-} |
| 431 | 2 open import Level |
| 3 open import Ordinals | |
| 4 module ODUtil {n : Level } (O : Ordinals {n} ) where | |
| 5 | |
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6 open import Data.Nat renaming ( zero to Zero ; suc to Suc ; ℕ to Nat ; _⊔_ to _n⊔_ ) |
| 431 | 7 open import Relation.Binary.PropositionalEquality hiding ( [_] ) |
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8 open import Data.Nat.Properties |
| 431 | 9 open import Data.Empty |
| 10 open import Relation.Nullary | |
| 11 open import Relation.Binary hiding ( _⇔_ ) | |
| 12 | |
| 13 open import logic | |
| 14 open import nat | |
| 15 | |
| 16 open Ordinals.Ordinals O | |
| 17 open Ordinals.IsOrdinals isOrdinal | |
| 18 open Ordinals.IsNext isNext | |
| 19 import OrdUtil | |
| 20 open OrdUtil O | |
| 21 | |
| 22 import OD | |
| 23 open OD O | |
| 24 open OD.OD | |
| 25 open ODAxiom odAxiom | |
| 26 | |
| 27 open HOD | |
| 28 open _∧_ | |
| 29 open _==_ | |
| 30 | |
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31 _⊂_ : ( A B : HOD) → Set n |
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32 _⊂_ A B = ( & A o< & B) ∧ ( A ⊆ B ) |
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33 |
| 1116 | 34 ⊆∩-dist : {a b c : HOD} → a ⊆ b → a ⊆ c → a ⊆ ( b ∩ c ) |
| 35 ⊆∩-dist {a} {b} {c} a<b a<c {z} az = ⟪ a<b az , a<c az ⟫ | |
| 36 | |
| 37 ⊆∩-incl-1 : {a b c : HOD} → a ⊆ c → ( a ∩ b ) ⊆ c | |
| 38 ⊆∩-incl-1 {a} {b} {c} a<c {z} ab = a<c (proj1 ab) | |
| 39 | |
| 40 ⊆∩-incl-2 : {a b c : HOD} → a ⊆ c → ( b ∩ a ) ⊆ c | |
| 41 ⊆∩-incl-2 {a} {b} {c} a<c {z} ab = a<c (proj2 ab) | |
| 42 | |
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43 cseq : HOD → HOD |
| 431 | 44 cseq x = record { od = record { def = λ y → odef x (osuc y) } ; odmax = osuc (odmax x) ; <odmax = lemma } where |
| 45 lemma : {y : Ordinal} → def (od x) (osuc y) → y o< osuc (odmax x) | |
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46 lemma {y} lt = ordtrans <-osuc (ordtrans (<odmax x lt) <-osuc ) |
| 431 | 47 |
| 1150 | 48 ∩-comm : { A B : HOD } → (A ∩ B) ≡ (B ∩ A) |
| 49 ∩-comm {A} {B} = ==→o≡ record { eq← = λ {x} lt → ⟪ proj2 lt , proj1 lt ⟫ ; eq→ = λ {x} lt → ⟪ proj2 lt , proj1 lt ⟫ } | |
| 50 | |
| 51 _∪_ : ( A B : HOD ) → HOD | |
| 52 A ∪ B = record { od = record { def = λ x → odef A x ∨ odef B x } ; | |
| 53 odmax = omax (odmax A) (odmax B) ; <odmax = lemma } where | |
| 54 lemma : {y : Ordinal} → odef A y ∨ odef B y → y o< omax (odmax A) (odmax B) | |
| 55 lemma {y} (case1 a) = ordtrans (<odmax A a) (omax-x _ _) | |
| 56 lemma {y} (case2 b) = ordtrans (<odmax B b) (omax-y _ _) | |
| 57 | |
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58 x∪x≡x : { A : HOD } → (A ∪ A) ≡ A |
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59 x∪x≡x {A} = ==→o≡ record { eq← = λ {x} lt → case1 lt ; eq→ = lem00 } where |
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60 lem00 : {x : Ordinal} → odef A x ∨ odef A x → odef A x |
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61 lem00 {x} (case1 ax) = ax |
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62 lem00 {x} (case2 ax) = ax |
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63 |
| 1150 | 64 _＼_ : ( A B : HOD ) → HOD |
| 65 A ＼ B = record { od = record { def = λ x → odef A x ∧ ( ¬ ( odef B x ) ) }; odmax = odmax A ; <odmax = λ y → <odmax A (proj1 y) } | |
| 66 | |
| 67 ¬∅∋ : {x : HOD} → ¬ ( od∅ ∋ x ) | |
| 68 ¬∅∋ {x} = ¬x<0 | |
| 69 | |
| 431 | 70 |
| 71 pair-xx<xy : {x y : HOD} → & (x , x) o< osuc (& (x , y) ) | |
| 72 pair-xx<xy {x} {y} = ⊆→o≤ lemma where | |
| 73 lemma : {z : Ordinal} → def (od (x , x)) z → def (od (x , y)) z | |
| 74 lemma {z} (case1 refl) = case1 refl | |
| 75 lemma {z} (case2 refl) = case1 refl | |
| 76 | |
| 77 pair-<xy : {x y : HOD} → {n : Ordinal} → & x o< next n → & y o< next n → & (x , y) o< next n | |
| 78 pair-<xy {x} {y} {o} x<nn y<nn with trio< (& x) (& y) | inspect (omax (& x)) (& y) | |
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79 ... | tri< a ¬b ¬c | record { eq = eq1 } = next< (subst (λ k → k o< next o ) (sym eq1) (osuc<nx y<nn)) ho< |
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80 ... | tri> ¬a ¬b c | record { eq = eq1 } = next< (subst (λ k → k o< next o ) (sym eq1) (osuc<nx x<nn)) ho< |
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81 ... | tri≈ ¬a b ¬c | record { eq = eq1 } = next< (subst (λ k → k o< next o ) (omax≡ _ _ b) (subst (λ k → osuc k o< next o) b (osuc<nx x<nn))) ho< |
| 431 | 82 |
| 83 -- another form of infinite | |
| 84 -- pair-ord< : {x : Ordinal } → Set n | |
| 85 pair-ord< : {x : HOD } → ( {y : HOD } → & y o< next (odmax y) ) → & ( x , x ) o< next (& x) | |
| 86 pair-ord< {x} ho< = subst (λ k → & (x , x) o< k ) lemmab0 lemmab1 where | |
| 87 lemmab0 : next (odmax (x , x)) ≡ next (& x) | |
| 88 lemmab0 = trans (cong (λ k → next k) (omxx _)) (sym nexto≡) | |
| 89 lemmab1 : & (x , x) o< next ( odmax (x , x)) | |
| 90 lemmab1 = ho< | |
| 91 | |
| 92 trans-⊆ : { A B C : HOD} → A ⊆ B → B ⊆ C → A ⊆ C | |
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93 trans-⊆ A⊆B B⊆C ab = B⊆C (A⊆B ab) |
| 431 | 94 |
| 1126 | 95 trans-⊂ : { A B C : HOD} → A ⊂ B → B ⊂ C → A ⊂ C |
| 96 trans-⊂ ⟪ A<B , A⊆B ⟫ ⟪ B<C , B⊆C ⟫ = ⟪ ordtrans A<B B<C , (λ ab → B⊆C (A⊆B ab)) ⟫ | |
| 97 | |
| 431 | 98 refl-⊆ : {A : HOD} → A ⊆ A |
| 1096 | 99 refl-⊆ x = x |
| 431 | 100 |
| 101 od⊆→o≤ : {x y : HOD } → x ⊆ y → & x o< osuc (& y) | |
| 1096 | 102 od⊆→o≤ {x} {y} lt = ⊆→o≤ {x} {y} (λ {z} x>z → subst (λ k → def (od y) k ) &iso (lt (d→∋ x x>z))) |
| 431 | 103 |
| 480 | 104 ⊆→= : {F U : HOD} → F ⊆ U → U ⊆ F → F =h= U |
| 1096 | 105 ⊆→= {F} {U} FU UF = record { eq→ = λ {x} lt → subst (λ k → odef U k) &iso (FU (subst (λ k → odef F k) (sym &iso) lt) ) |
| 106 ; eq← = λ {x} lt → subst (λ k → odef F k) &iso (UF (subst (λ k → odef U k) (sym &iso) lt) ) } | |
| 480 | 107 |
| 519 | 108 ¬A∋x→A≡od∅ : (A : HOD) → {x : HOD} → A ∋ x → ¬ ( & A ≡ o∅ ) |
| 109 ¬A∋x→A≡od∅ A {x} ax a=0 = ¬x<0 ( subst (λ k → & x o< k) a=0 (c<→o< ax )) | |
| 110 | |
| 431 | 111 subset-lemma : {A x : HOD } → ( {y : HOD } → x ∋ y → (A ∩ x ) ∋ y ) ⇔ ( x ⊆ A ) |
| 112 subset-lemma {A} {x} = record { | |
| 1096 | 113 proj1 = λ lt x∋z → subst (λ k → odef A k ) &iso ( proj1 (lt (subst (λ k → odef x k) (sym &iso) x∋z ) )) |
| 114 ; proj2 = λ x⊆A lt → ⟪ x⊆A lt , lt ⟫ | |
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115 } |
| 431 | 116 |
| 117 ω<next-o∅ : {y : Ordinal} → infinite-d y → y o< next o∅ | |
| 118 ω<next-o∅ {y} lt = <odmax infinite lt | |
| 119 | |
| 120 nat→ω : Nat → HOD | |
| 121 nat→ω Zero = od∅ | |
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122 nat→ω (Suc y) = Union (nat→ω y , (nat→ω y , nat→ω y)) |
| 431 | 123 |
| 124 ω→nato : {y : Ordinal} → infinite-d y → Nat | |
| 125 ω→nato iφ = Zero | |
| 126 ω→nato (isuc lt) = Suc (ω→nato lt) | |
| 127 | |
| 128 ω→nat : (n : HOD) → infinite ∋ n → Nat | |
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129 ω→nat n = ω→nato |
| 431 | 130 |
| 131 ω∋nat→ω : {n : Nat} → def (od infinite) (& (nat→ω n)) | |
| 132 ω∋nat→ω {Zero} = subst (λ k → def (od infinite) k) (sym ord-od∅) iφ | |
| 133 ω∋nat→ω {Suc n} = subst (λ k → def (od infinite) k) lemma (isuc ( ω∋nat→ω {n})) where | |
| 134 lemma : & (Union (* (& (nat→ω n)) , (* (& (nat→ω n)) , * (& (nat→ω n))))) ≡ & (nat→ω (Suc n)) | |
| 135 lemma = subst (λ k → & (Union (k , ( k , k ))) ≡ & (nat→ω (Suc n))) (sym *iso) refl | |
| 136 | |
| 137 pair1 : { x y : HOD } → (x , y ) ∋ x | |
| 138 pair1 = case1 refl | |
| 139 | |
| 140 pair2 : { x y : HOD } → (x , y ) ∋ y | |
| 141 pair2 = case2 refl | |
| 142 | |
| 143 single : {x y : HOD } → (x , x ) ∋ y → x ≡ y | |
| 144 single (case1 eq) = ==→o≡ ( ord→== (sym eq) ) | |
| 145 single (case2 eq) = ==→o≡ ( ord→== (sym eq) ) | |
| 146 | |
| 1096 | 147 single& : {x y : Ordinal } → odef (* x , * x ) y → x ≡ y |
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148 single& (case1 eq) = sym (trans eq &iso) |
| 1096 | 149 single& (case2 eq) = sym (trans eq &iso) |
| 150 | |
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151 open import Relation.Binary.HeterogeneousEquality as HE using (_≅_ ) |
| 431 | 152 -- postulate f-extensionality : { n m : Level} → HE.Extensionality n m |
| 153 | |
| 1096 | 154 pair=∨ : {a b c : Ordinal } → odef (* a , * b) c → ( a ≡ c ) ∨ ( b ≡ c ) |
| 155 pair=∨ {a} {b} {c} (case1 c=a) = case1 ( sym (trans c=a &iso)) | |
| 156 pair=∨ {a} {b} {c} (case2 c=b) = case2 ( sym (trans c=b &iso)) | |
| 157 | |
| 431 | 158 ω-prev-eq1 : {x y : Ordinal} → & (Union (* y , (* y , * y))) ≡ & (Union (* x , (* x , * x))) → ¬ (x o< y) |
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159 ω-prev-eq1 {x} {y} eq x<y with eq→ (ord→== eq) record { owner = & (* y , * y) ; ao = case2 refl |
| 1096 | 160 ; ox = subst (λ k → odef k (& (* y))) (sym *iso) (case1 refl) } -- (* x , (* x , * x)) ∋ * y |
| 161 ... | record { owner = u ; ao = xxx∋u ; ox = uy } with xxx∋u | |
| 162 ... | case1 u=x = ⊥-elim ( o<> x<y (osucprev (begin | |
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163 osuc y ≡⟨ sym (cong osuc &iso) ⟩ |
| 1096 | 164 osuc (& (* y)) ≤⟨ osucc (c<→o< {* y} {* u} uy) ⟩ -- * x ≡ * u ∋ * y |
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165 & (* u) ≡⟨ &iso ⟩ |
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166 u ≡⟨ u=x ⟩ |
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167 & (* x) ≡⟨ &iso ⟩ |
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168 x ∎ ))) where open o≤-Reasoning O |
| 1096 | 169 ... | case2 u=xx = ⊥-elim (o<¬≡ ( begin |
| 170 x ≡⟨ single& (subst₂ (λ j k → odef j k ) (begin | |
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171 * u ≡⟨ cong (*) u=xx ⟩ |
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172 * (& (* x , * x)) ≡⟨ *iso ⟩ |
| 1096 | 173 (* x , * x ) ∎ ) &iso uy ) ⟩ -- (* x , * x ) ∋ * y |
| 174 y ∎ ) x<y) where open ≡-Reasoning | |
| 431 | 175 |
| 176 ω-prev-eq : {x y : Ordinal} → & (Union (* y , (* y , * y))) ≡ & (Union (* x , (* x , * x))) → x ≡ y | |
| 177 ω-prev-eq {x} {y} eq with trio< x y | |
| 178 ω-prev-eq {x} {y} eq | tri< a ¬b ¬c = ⊥-elim (ω-prev-eq1 eq a) | |
| 179 ω-prev-eq {x} {y} eq | tri≈ ¬a b ¬c = b | |
| 180 ω-prev-eq {x} {y} eq | tri> ¬a ¬b c = ⊥-elim (ω-prev-eq1 (sym eq) c) | |
| 181 | |
| 182 ω-∈s : (x : HOD) → Union ( x , (x , x)) ∋ x | |
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183 ω-∈s x = record { owner = & ( x , x ) ; ao = case2 refl ; ox = subst₂ (λ j k → odef j k ) (sym *iso) refl (case2 refl) } |
| 431 | 184 |
| 185 ωs≠0 : (x : HOD) → ¬ ( Union ( x , (x , x)) ≡ od∅ ) | |
| 186 ωs≠0 y eq = ⊥-elim ( ¬x<0 (subst (λ k → & y o< k ) ord-od∅ (c<→o< (subst (λ k → odef k (& y )) eq (ω-∈s y) ))) ) | |
| 187 | |
| 188 nat→ω-iso : {i : HOD} → (lt : infinite ∋ i ) → nat→ω ( ω→nat i lt ) ≡ i | |
| 189 nat→ω-iso {i} = ε-induction {λ i → (lt : infinite ∋ i ) → nat→ω ( ω→nat i lt ) ≡ i } ind i where | |
| 190 ind : {x : HOD} → ({y : HOD} → x ∋ y → (lt : infinite ∋ y) → nat→ω (ω→nat y lt) ≡ y) → | |
| 191 (lt : infinite ∋ x) → nat→ω (ω→nat x lt) ≡ x | |
| 192 ind {x} prev lt = ind1 lt *iso where | |
| 193 ind1 : {ox : Ordinal } → (ltd : infinite-d ox ) → * ox ≡ x → nat→ω (ω→nato ltd) ≡ x | |
| 194 ind1 {o∅} iφ refl = sym o∅≡od∅ | |
| 195 ind1 (isuc {x₁} ltd) ox=x = begin | |
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196 nat→ω (ω→nato (isuc ltd) ) |
| 431 | 197 ≡⟨⟩ |
| 198 Union (nat→ω (ω→nato ltd) , (nat→ω (ω→nato ltd) , nat→ω (ω→nato ltd))) | |
| 199 ≡⟨ cong (λ k → Union (k , (k , k ))) lemma ⟩ | |
| 200 Union (* x₁ , (* x₁ , * x₁)) | |
| 201 ≡⟨ trans ( sym *iso) ox=x ⟩ | |
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202 x |
| 431 | 203 ∎ where |
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204 open ≡-Reasoning |
| 431 | 205 lemma0 : x ∋ * x₁ |
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206 lemma0 = subst (λ k → odef k (& (* x₁))) (trans (sym *iso) ox=x) |
| 1096 | 207 record { owner = & ( * x₁ , * x₁ ) ; ao = case2 refl ; ox = subst (λ k → odef k (& (* x₁))) (sym *iso) (case1 refl) } |
| 431 | 208 lemma1 : infinite ∋ * x₁ |
| 209 lemma1 = subst (λ k → odef infinite k) (sym &iso) ltd | |
| 210 lemma3 : {x y : Ordinal} → (ltd : infinite-d x ) (ltd1 : infinite-d y ) → y ≡ x → ltd ≅ ltd1 | |
| 211 lemma3 iφ iφ refl = HE.refl | |
| 212 lemma3 iφ (isuc {y} ltd1) eq = ⊥-elim ( ¬x<0 (subst₂ (λ j k → j o< k ) &iso eq (c<→o< (ω-∈s (* y)) ))) | |
| 213 lemma3 (isuc {y} ltd) iφ eq = ⊥-elim ( ¬x<0 (subst₂ (λ j k → j o< k ) &iso (sym eq) (c<→o< (ω-∈s (* y)) ))) | |
| 214 lemma3 (isuc {x} ltd) (isuc {y} ltd1) eq with lemma3 ltd ltd1 (ω-prev-eq (sym eq)) | |
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215 ... | t = HE.cong₂ (λ j k → isuc {j} k ) (HE.≡-to-≅ (ω-prev-eq eq)) t |
| 431 | 216 lemma2 : {x y : Ordinal} → (ltd : infinite-d x ) (ltd1 : infinite-d y ) → y ≡ x → ω→nato ltd ≡ ω→nato ltd1 |
| 217 lemma2 {x} {y} ltd ltd1 eq = lemma6 eq (lemma3 {x} {y} ltd ltd1 eq) where | |
| 218 lemma6 : {x y : Ordinal} → {ltd : infinite-d x } {ltd1 : infinite-d y } → y ≡ x → ltd ≅ ltd1 → ω→nato ltd ≡ ω→nato ltd1 | |
| 219 lemma6 refl HE.refl = refl | |
| 220 lemma : nat→ω (ω→nato ltd) ≡ * x₁ | |
| 221 lemma = trans (cong (λ k → nat→ω k) (lemma2 {x₁} {_} ltd (subst (λ k → infinite-d k ) (sym &iso) ltd) &iso ) ) ( prev {* x₁} lemma0 lemma1 ) | |
| 222 | |
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223 |
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224 ω→nat-iso0 : (x : Nat) → {ox : Ordinal } → (ltd : infinite-d ox) → * ox ≡ nat→ω x → ω→nato ltd ≡ x |
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225 ω→nat-iso0 Zero iφ eq = refl |
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226 ω→nat-iso0 (Suc x) iφ eq = ⊥-elim ( ωs≠0 _ (trans (sym eq) o∅≡od∅ )) |
| 1100 | 227 ω→nat-iso0 Zero (isuc ltd) eq = ⊥-elim ( ωs≠0 _ (subst (λ k → k ≡ od∅ ) *iso eq )) |
| 228 ω→nat-iso0 (Suc i) (isuc {x} ltd) eq = cong Suc ( ω→nat-iso0 i ltd (lemma1 eq) ) where | |
| 229 lemma1 : * (& (Union (* x , (* x , * x)))) ≡ Union (nat→ω i , (nat→ω i , nat→ω i)) → * x ≡ nat→ω i | |
| 230 lemma1 eq = subst (λ k → * x ≡ k ) *iso (cong (λ k → * k) | |
| 231 ( ω-prev-eq (subst (λ k → _ ≡ k ) &iso (cong (λ k → & k ) (sym | |
| 232 (subst (λ k → _ ≡ Union ( k , ( k , k ))) (sym *iso ) eq )))))) | |
| 233 | |
| 431 | 234 ω→nat-iso : {i : Nat} → ω→nat ( nat→ω i ) (ω∋nat→ω {i}) ≡ i |
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235 ω→nat-iso {i} = ω→nat-iso0 i (ω∋nat→ω {i}) *iso |
| 431 | 236 |
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237 Repl⊆ : {A B : HOD} (A⊆B : A ⊆ B) → { ψa : ( x : HOD) → A ∋ x → HOD } { ψb : ( x : HOD) → B ∋ x → HOD } |
| 1106 | 238 → ( {z : Ordinal } → (az : odef A z ) → (ψa (* z) (subst (odef A) (sym &iso) az) ≡ ψb (* z) (subst (odef B) (sym &iso) (A⊆B az)))) |
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239 → Replace' A ψa ⊆ Replace' B ψb |
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240 Repl⊆ {A} {B} A⊆B {ψa} {ψb} eq record { z = z ; az = az ; x=ψz = x=ψz } = record { z = z ; az = A⊆B az |
| 1106 | 241 ; x=ψz = trans x=ψz (cong (&) (eq az) ) } |
| 242 | |
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243 PPP : {P : HOD} → Power P ∋ P |
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244 PPP {P} z pz = subst (λ k → odef k z ) *iso pz |
| 1106 | 245 |
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246 UPower⊆Q : {P Q : HOD} → P ⊆ Q → Union (Power P) ⊆ Q |
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247 UPower⊆Q {P} {Q} P⊆Q {z} record { owner = y ; ao = ppy ; ox = yz } = P⊆Q (ppy _ yz) |
| 1106 | 248 |
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249 UPower∩ : {P : HOD} → ({ p q : HOD } → P ∋ p → P ∋ q → P ∋ (p ∩ q)) |
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250 → { p q : HOD } → Union (Power P) ∋ p → Union (Power P) ∋ q → Union (Power P) ∋ (p ∩ q) |
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251 UPower∩ {P} each {p} {q} record { owner = x ; ao = ppx ; ox = xz } record { owner = y ; ao = ppy ; ox = yz } |
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252 = record { owner = & P ; ao = PPP ; ox = lem03 } where |
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253 lem03 : odef (* (& P)) (& (p ∩ q)) |
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254 lem03 = subst (λ k → odef k (& (p ∩ q))) (sym *iso) ( each (ppx _ xz) (ppy _ yz) ) |
| 1150 | 255 |
| 256 -- ∋-irr : {X x : HOD} → (a b : X ∋ x ) → a ≡ b | |
| 257 -- ∋-irr {X} {x} _ _ = refl | |
| 258 -- is requed in | |
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259 Replace∩ : {X P Q : HOD} → {ψ : (x : HOD) → X ∋ x → HOD} → (P⊆X : P ⊆ X) → (Q⊆X : Q ⊆ X ) |
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260 → ( {x : HOD} → (a b : X ∋ x ) → ψ x a ≡ ψ x b ) |
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261 → (Replace' (P ∩ Q) (λ _ pq → ψ _ (P⊆X (proj1 pq )))) ⊆ ( Replace' P (λ _ p → ψ _ (P⊆X p)) ∩ Replace' Q (λ _ q → ψ _ (Q⊆X q))) |
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262 Replace∩ {X} {P} {Q} {ψ} P⊆X Q⊆X ψ-irr = lem04 where |
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263 lem04 : {x : Ordinal} → OD.def (od (Replace' (P ∩ Q) (λ z pq → ψ z (P⊆X (proj1 pq)) ))) x |
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264 → OD.def (od (Replace' P (λ z p → ψ z (P⊆X p)) ∩ Replace' Q (λ z q → ψ z (Q⊆X q)))) x |
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265 lem04 {x} record { z = z ; az = az ; x=ψz = x=ψz } = ⟪ |
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266 record { z = _ ; az = proj1 az ; x=ψz = trans x=ψz (cong (&)(ψ-irr _ _)) } , |
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267 record { z = _ ; az = proj2 az ; x=ψz = trans x=ψz (cong (&)(ψ-irr _ _)) } ⟫ |
| 1150 | 268 |