Mercurial > hg > Members > kono > Proof > ZF-in-agda
comparison src/zorn.agda @ 600:71a1ed72cd21
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author | Shinji KONO <kono@ie.u-ryukyu.ac.jp> |
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date | Tue, 14 Jun 2022 06:17:24 +0900 |
parents | d041941a8866 |
children | 8b2a4af84e25 |
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599:d041941a8866 | 600:71a1ed72cd21 |
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87 -- | 87 -- |
88 | 88 |
89 ≤-monotonic-f : (A : HOD) → ( Ordinal → Ordinal ) → Set (Level.suc n) | 89 ≤-monotonic-f : (A : HOD) → ( Ordinal → Ordinal ) → Set (Level.suc n) |
90 ≤-monotonic-f A f = (x : Ordinal ) → odef A x → ( * x ≤ * (f x) ) ∧ odef A (f x ) | 90 ≤-monotonic-f A f = (x : Ordinal ) → odef A x → ( * x ≤ * (f x) ) ∧ odef A (f x ) |
91 | 91 |
92 -- immieate-f : (A : HOD) → ( f : Ordinal → Ordinal ) → Set n | |
93 -- immieate-f A f = { x y : Ordinal } → odef A x → odef A y → ¬ ( ( * x < * y ) ∧ ( * y < * (f x )) ) | |
94 | |
95 data FClosure (A : HOD) (f : Ordinal → Ordinal ) (s : Ordinal) : Ordinal → Set n where | 92 data FClosure (A : HOD) (f : Ordinal → Ordinal ) (s : Ordinal) : Ordinal → Set n where |
96 init : {x : Ordinal} → odef A s → x ≡ s → FClosure A f s x | 93 init : odef A s → FClosure A f s s |
97 fsuc : (x : Ordinal) ( p : FClosure A f s x ) → FClosure A f s (f x) | 94 fsuc : (x : Ordinal) ( p : FClosure A f s x ) → FClosure A f s (f x) |
98 | 95 |
99 A∋fc : {A : HOD} (s : Ordinal) {y : Ordinal } (f : Ordinal → Ordinal) (mf : ≤-monotonic-f A f) → (fcy : FClosure A f s y ) → odef A y | 96 A∋fc : {A : HOD} (s : Ordinal) {y : Ordinal } (f : Ordinal → Ordinal) (mf : ≤-monotonic-f A f) → (fcy : FClosure A f s y ) → odef A y |
100 A∋fc {A} s f mf (init as refl ) = as | 97 A∋fc {A} s f mf (init as) = as |
101 A∋fc {A} s f mf (fsuc y fcy) = proj2 (mf y ( A∋fc {A} s f mf fcy ) ) | 98 A∋fc {A} s f mf (fsuc y fcy) = proj2 (mf y ( A∋fc {A} s f mf fcy ) ) |
102 | 99 |
103 s≤fc : {A : HOD} (s : Ordinal ) {y : Ordinal } (f : Ordinal → Ordinal) (mf : ≤-monotonic-f A f) → (fcy : FClosure A f s y ) → * s ≤ * y | 100 s≤fc : {A : HOD} (s : Ordinal ) {y : Ordinal } (f : Ordinal → Ordinal) (mf : ≤-monotonic-f A f) → (fcy : FClosure A f s y ) → * s ≤ * y |
104 s≤fc {A} s {.s} f mf (init x refl ) = case1 refl | 101 s≤fc {A} s {.s} f mf (init x) = case1 refl |
105 s≤fc {A} s {.(f x)} f mf (fsuc x fcy) with proj1 (mf x (A∋fc s f mf fcy ) ) | 102 s≤fc {A} s {.(f x)} f mf (fsuc x fcy) with proj1 (mf x (A∋fc s f mf fcy ) ) |
106 ... | case1 x=fx = subst (λ k → * s ≤ * k ) (*≡*→≡ x=fx) ( s≤fc {A} s f mf fcy ) | 103 ... | case1 x=fx = subst (λ k → * s ≤ * k ) (*≡*→≡ x=fx) ( s≤fc {A} s f mf fcy ) |
107 ... | case2 x<fx with s≤fc {A} s f mf fcy | 104 ... | case2 x<fx with s≤fc {A} s f mf fcy |
108 ... | case1 s≡x = case2 ( subst₂ (λ j k → j < k ) (sym s≡x) refl x<fx ) | 105 ... | case1 s≡x = case2 ( subst₂ (λ j k → j < k ) (sym s≡x) refl x<fx ) |
109 ... | case2 s<x = case2 ( IsStrictPartialOrder.trans PO s<x x<fx ) | 106 ... | case2 s<x = case2 ( IsStrictPartialOrder.trans PO s<x x<fx ) |
110 | 107 |
111 fcn : {A : HOD} (s : Ordinal) { x : Ordinal} {f : Ordinal → Ordinal} → (mf : ≤-monotonic-f A f) → FClosure A f s x → ℕ | 108 fcn : {A : HOD} (s : Ordinal) { x : Ordinal} {f : Ordinal → Ordinal} → (mf : ≤-monotonic-f A f) → FClosure A f s x → ℕ |
112 fcn s mf (init as refl ) = zero | 109 fcn s mf (init as) = zero |
113 fcn {A} s {x} {f} mf (fsuc y p) with proj1 (mf y (A∋fc s f mf p)) | 110 fcn {A} s {x} {f} mf (fsuc y p) with proj1 (mf y (A∋fc s f mf p)) |
114 ... | case1 eq = fcn s mf p | 111 ... | case1 eq = fcn s mf p |
115 ... | case2 y<fy = suc (fcn s mf p ) | 112 ... | case2 y<fy = suc (fcn s mf p ) |
116 | 113 |
117 fcn-inject : {A : HOD} (s : Ordinal) { x y : Ordinal} {f : Ordinal → Ordinal} → (mf : ≤-monotonic-f A f) | 114 fcn-inject : {A : HOD} (s : Ordinal) { x y : Ordinal} {f : Ordinal → Ordinal} → (mf : ≤-monotonic-f A f) |
118 → (cx : FClosure A f s x ) (cy : FClosure A f s y ) → fcn s mf cx ≡ fcn s mf cy → * x ≡ * y | 115 → (cx : FClosure A f s x ) (cy : FClosure A f s y ) → fcn s mf cx ≡ fcn s mf cy → * x ≡ * y |
119 fcn-inject {A} s {x} {y} {f} mf cx cy eq = fc00 (fcn s mf cx) (fcn s mf cy) eq cx cy refl refl where | 116 fcn-inject {A} s {x} {y} {f} mf cx cy eq = fc00 (fcn s mf cx) (fcn s mf cy) eq cx cy refl refl where |
120 fc00 : (i j : ℕ ) → i ≡ j → {x y : Ordinal } → (cx : FClosure A f s x ) (cy : FClosure A f s y ) → i ≡ fcn s mf cx → j ≡ fcn s mf cy → * x ≡ * y | 117 fc00 : (i j : ℕ ) → i ≡ j → {x y : Ordinal } → (cx : FClosure A f s x ) (cy : FClosure A f s y ) → i ≡ fcn s mf cx → j ≡ fcn s mf cy → * x ≡ * y |
121 fc00 zero zero refl (init _ refl ) (init x₁ refl ) i=x i=y = refl | 118 fc00 zero zero refl (init _) (init x₁) i=x i=y = refl |
122 fc00 zero zero refl (init as refl ) (fsuc y cy) i=x i=y with proj1 (mf y (A∋fc s f mf cy ) ) | 119 fc00 zero zero refl (init as) (fsuc y cy) i=x i=y with proj1 (mf y (A∋fc s f mf cy ) ) |
123 ... | case1 y=fy = subst (λ k → * s ≡ k ) y=fy ( fc00 zero zero refl (init as refl ) cy i=x i=y ) | 120 ... | case1 y=fy = subst (λ k → * s ≡ k ) y=fy ( fc00 zero zero refl (init as) cy i=x i=y ) |
124 fc00 zero zero refl (fsuc x cx) (init as refl ) i=x i=y with proj1 (mf x (A∋fc s f mf cx ) ) | 121 fc00 zero zero refl (fsuc x cx) (init as) i=x i=y with proj1 (mf x (A∋fc s f mf cx ) ) |
125 ... | case1 x=fx = subst (λ k → k ≡ * s ) x=fx ( fc00 zero zero refl cx (init as refl ) i=x i=y ) | 122 ... | case1 x=fx = subst (λ k → k ≡ * s ) x=fx ( fc00 zero zero refl cx (init as) i=x i=y ) |
126 fc00 zero zero refl (fsuc x cx) (fsuc y cy) i=x i=y with proj1 (mf x (A∋fc s f mf cx ) ) | proj1 (mf y (A∋fc s f mf cy ) ) | 123 fc00 zero zero refl (fsuc x cx) (fsuc y cy) i=x i=y with proj1 (mf x (A∋fc s f mf cx ) ) | proj1 (mf y (A∋fc s f mf cy ) ) |
127 ... | case1 x=fx | case1 y=fy = subst₂ (λ j k → j ≡ k ) x=fx y=fy ( fc00 zero zero refl cx cy i=x i=y ) | 124 ... | case1 x=fx | case1 y=fy = subst₂ (λ j k → j ≡ k ) x=fx y=fy ( fc00 zero zero refl cx cy i=x i=y ) |
128 fc00 (suc i) (suc j) i=j {.(f x)} {.(f y)} (fsuc x cx) (fsuc y cy) i=x j=y with proj1 (mf x (A∋fc s f mf cx ) ) | proj1 (mf y (A∋fc s f mf cy ) ) | 125 fc00 (suc i) (suc j) i=j {.(f x)} {.(f y)} (fsuc x cx) (fsuc y cy) i=x j=y with proj1 (mf x (A∋fc s f mf cx ) ) | proj1 (mf y (A∋fc s f mf cy ) ) |
129 ... | case1 x=fx | case1 y=fy = subst₂ (λ j k → j ≡ k ) x=fx y=fy ( fc00 (suc i) (suc j) i=j cx cy i=x j=y ) | 126 ... | case1 x=fx | case1 y=fy = subst₂ (λ j k → j ≡ k ) x=fx y=fy ( fc00 (suc i) (suc j) i=j cx cy i=x j=y ) |
130 ... | case1 x=fx | case2 y<fy = subst (λ k → k ≡ * (f y)) x=fx (fc02 x cx i=x) where | 127 ... | case1 x=fx | case2 y<fy = subst (λ k → k ≡ * (f y)) x=fx (fc02 x cx i=x) where |
141 ... | case2 lt = subst₂ (λ j k → * (f j) ≡ * (f k )) &iso &iso ( cong (λ k → * ( f (& k ))) fc05) where | 138 ... | case2 lt = subst₂ (λ j k → * (f j) ≡ * (f k )) &iso &iso ( cong (λ k → * ( f (& k ))) fc05) where |
142 fc05 : * x ≡ * y1 | 139 fc05 : * x ≡ * y1 |
143 fc05 = fc00 i j (cong pred i=j) cx cy1 (cong pred i=x) (cong pred j=y1) | 140 fc05 = fc00 i j (cong pred i=j) cx cy1 (cong pred i=x) (cong pred j=y1) |
144 ... | case2 x₁ | case2 x₂ = subst₂ (λ j k → * (f j) ≡ * (f k) ) &iso &iso (cong (λ k → * (f (& k))) (fc00 i j (cong pred i=j) cx cy (cong pred i=x) (cong pred j=y))) | 141 ... | case2 x₁ | case2 x₂ = subst₂ (λ j k → * (f j) ≡ * (f k) ) &iso &iso (cong (λ k → * (f (& k))) (fc00 i j (cong pred i=j) cx cy (cong pred i=x) (cong pred j=y))) |
145 | 142 |
143 | |
146 fcn-< : {A : HOD} (s : Ordinal ) { x y : Ordinal} {f : Ordinal → Ordinal} → (mf : ≤-monotonic-f A f) | 144 fcn-< : {A : HOD} (s : Ordinal ) { x y : Ordinal} {f : Ordinal → Ordinal} → (mf : ≤-monotonic-f A f) |
147 → (cx : FClosure A f s x ) (cy : FClosure A f s y ) → fcn s mf cx Data.Nat.< fcn s mf cy → * x < * y | 145 → (cx : FClosure A f s x ) (cy : FClosure A f s y ) → fcn s mf cx Data.Nat.< fcn s mf cy → * x < * y |
148 fcn-< {A} s {x} {y} {f} mf cx cy x<y = fc01 (fcn s mf cy) cx cy refl x<y where | 146 fcn-< {A} s {x} {y} {f} mf cx cy x<y = fc01 (fcn s mf cy) cx cy refl x<y where |
149 fc01 : (i : ℕ ) → {y : Ordinal } → (cx : FClosure A f s x ) (cy : FClosure A f s y ) → (i ≡ fcn s mf cy ) → fcn s mf cx Data.Nat.< i → * x < * y | 147 fc01 : (i : ℕ ) → {y : Ordinal } → (cx : FClosure A f s x ) (cy : FClosure A f s y ) → (i ≡ fcn s mf cy ) → fcn s mf cx Data.Nat.< i → * x < * y |
150 fc01 (suc i) {y} cx (fsuc y1 cy) i=y (s≤s x<i) with proj1 (mf y1 (A∋fc s f mf cy ) ) | 148 fc01 (suc i) {y} cx (fsuc y1 cy) i=y (s≤s x<i) with proj1 (mf y1 (A∋fc s f mf cy ) ) |
168 fc10 : * x ≡ * y | 166 fc10 : * x ≡ * y |
169 fc10 = fcn-inject {A} s {x} {y} {f} mf cx cy b | 167 fc10 = fcn-inject {A} s {x} {y} {f} mf cx cy b |
170 ... | tri> ¬a ¬b c = tri> (λ lt → <-irr (case2 fc12) lt) (λ eq → <-irr (case1 eq) fc12) fc12 where | 168 ... | tri> ¬a ¬b c = tri> (λ lt → <-irr (case2 fc12) lt) (λ eq → <-irr (case1 eq) fc12) fc12 where |
171 fc12 : * y < * x | 169 fc12 : * y < * x |
172 fc12 = fcn-< {A} s {y} {x} {f} mf cy cx c | 170 fc12 = fcn-< {A} s {y} {x} {f} mf cy cx c |
171 | |
173 | 172 |
174 fcn-imm : {A : HOD} (s : Ordinal) { x y : Ordinal } (f : Ordinal → Ordinal) (mf : ≤-monotonic-f A f) | 173 fcn-imm : {A : HOD} (s : Ordinal) { x y : Ordinal } (f : Ordinal → Ordinal) (mf : ≤-monotonic-f A f) |
175 → (cx : FClosure A f s x) → (cy : FClosure A f s y ) → ¬ ( ( * x < * y ) ∧ ( * y < * (f x )) ) | 174 → (cx : FClosure A f s x) → (cy : FClosure A f s y ) → ¬ ( ( * x < * y ) ∧ ( * y < * (f x )) ) |
176 fcn-imm {A} s {x} {y} f mf cx cy ⟪ x<y , y<fx ⟫ = fc21 where | 175 fcn-imm {A} s {x} {y} f mf cx cy ⟪ x<y , y<fx ⟫ = fc21 where |
177 fc20 : fcn s mf cy Data.Nat.< suc (fcn s mf cx) → (fcn s mf cy ≡ fcn s mf cx) ∨ ( fcn s mf cy Data.Nat.< fcn s mf cx ) | 176 fc20 : fcn s mf cy Data.Nat.< suc (fcn s mf cx) → (fcn s mf cy ≡ fcn s mf cx) ∨ ( fcn s mf cy Data.Nat.< fcn s mf cx ) |
191 fc21 with <-cmp (suc ( fcn s mf cx )) (fcn s mf cy ) | 190 fc21 with <-cmp (suc ( fcn s mf cx )) (fcn s mf cy ) |
192 ... | tri< a ¬b ¬c = <-irr (case2 y<fx) (fc22 a) where -- suc ncx < ncy | 191 ... | tri< a ¬b ¬c = <-irr (case2 y<fx) (fc22 a) where -- suc ncx < ncy |
193 cxx : FClosure A f s (f x) | 192 cxx : FClosure A f s (f x) |
194 cxx = fsuc x cx | 193 cxx = fsuc x cx |
195 fc16 : (x : Ordinal ) → (cx : FClosure A f s x) → (fcn s mf cx ≡ fcn s mf (fsuc x cx)) ∨ ( suc (fcn s mf cx ) ≡ fcn s mf (fsuc x cx)) | 194 fc16 : (x : Ordinal ) → (cx : FClosure A f s x) → (fcn s mf cx ≡ fcn s mf (fsuc x cx)) ∨ ( suc (fcn s mf cx ) ≡ fcn s mf (fsuc x cx)) |
196 fc16 x (init as refl ) with proj1 (mf s as ) | 195 fc16 x (init as) with proj1 (mf s as ) |
197 ... | case1 _ = case1 refl | 196 ... | case1 _ = case1 refl |
198 ... | case2 _ = case2 refl | 197 ... | case2 _ = case2 refl |
199 fc16 .(f x) (fsuc x cx ) with proj1 (mf (f x) (A∋fc s f mf (fsuc x cx)) ) | 198 fc16 .(f x) (fsuc x cx ) with proj1 (mf (f x) (A∋fc s f mf (fsuc x cx)) ) |
200 ... | case1 _ = case1 refl | 199 ... | case1 _ = case1 refl |
201 ... | case2 _ = case2 refl | 200 ... | case2 _ = case2 refl |
206 ... | tri≈ ¬a b ¬c = <-irr (case1 (fc17 cx cy b)) y<fx | 205 ... | tri≈ ¬a b ¬c = <-irr (case1 (fc17 cx cy b)) y<fx |
207 ... | tri> ¬a ¬b c with fc20 c -- ncy < suc ncx | 206 ... | tri> ¬a ¬b c with fc20 c -- ncy < suc ncx |
208 ... | case1 y=x = <-irr (case1 ( fcn-inject s mf cy cx y=x )) x<y | 207 ... | case1 y=x = <-irr (case1 ( fcn-inject s mf cy cx y=x )) x<y |
209 ... | case2 y<x = <-irr (case2 x<y) (fcn-< s mf cy cx y<x ) | 208 ... | case2 y<x = <-irr (case2 x<y) (fcn-< s mf cy cx y<x ) |
210 | 209 |
211 | |
212 -- open import Relation.Binary.Properties.Poset as Poset | 210 -- open import Relation.Binary.Properties.Poset as Poset |
213 | 211 |
214 IsTotalOrderSet : ( A : HOD ) → Set (Level.suc n) | 212 IsTotalOrderSet : ( A : HOD ) → Set (Level.suc n) |
215 IsTotalOrderSet A = {a b : HOD} → odef A (& a) → odef A (& b) → Tri (a < b) (a ≡ b) (b < a ) | 213 IsTotalOrderSet A = {a b : HOD} → odef A (& a) → odef A (& b) → Tri (a < b) (a ≡ b) (b < a ) |
216 | 214 |
233 | 231 |
234 record IsSup (A B : HOD) {x : Ordinal } (xa : odef A x) : Set n where | 232 record IsSup (A B : HOD) {x : Ordinal } (xa : odef A x) : Set n where |
235 field | 233 field |
236 x<sup : {y : Ordinal} → odef B y → (y ≡ x ) ∨ (y << x ) | 234 x<sup : {y : Ordinal} → odef B y → (y ≡ x ) ∨ (y << x ) |
237 | 235 |
238 y1 : (A : HOD) → (y : Ordinal) → odef A y → * (& (* y , * y)) ⊆' A | |
239 y1 A y ay {x} lt with subst (λ k → odef k x) *iso lt | |
240 ... | case1 eq = subst (λ k → odef A k ) (sym (trans eq &iso)) ay | |
241 ... | case2 eq = subst (λ k → odef A k ) (sym (trans eq &iso)) ay | |
242 | |
243 record FChain ( A : HOD ) ( f : Ordinal → Ordinal ) (p c : Ordinal) ( x : Ordinal ) : Set n where | 236 record FChain ( A : HOD ) ( f : Ordinal → Ordinal ) (p c : Ordinal) ( x : Ordinal ) : Set n where |
244 field | 237 field |
245 fc∨sup : FClosure A f p x | 238 fc∨sup : FClosure A f p x |
246 chain∋p : odef (* c) p | 239 chain∋p : odef (* c) p |
247 | 240 |
241 record FSup ( A : HOD ) ( f : Ordinal → Ordinal ) (p c : Ordinal) ( x : Ordinal ) : Set n where | |
242 field | |
243 sup : (z : Ordinal) → FClosure A f p z → * z < * x | |
244 min : ( x1 : Ordinal) → ((z : Ordinal) → FClosure A f p z → * z < * x1 ) → ( x ≡ x1 ) ∨ ( * x < * x1 ) | |
245 chain∋x : odef (* c) x | |
246 chain∋p : odef (* c) p | |
247 | |
248 data Fc∨sup (A : HOD) {y : Ordinal} (ay : odef A y) ( f : Ordinal → Ordinal ) (c : Ordinal) : (x : Ordinal) → Set n where | 248 data Fc∨sup (A : HOD) {y : Ordinal} (ay : odef A y) ( f : Ordinal → Ordinal ) (c : Ordinal) : (x : Ordinal) → Set n where |
249 Finit : {z : Ordinal} → z ≡ y → Fc∨sup A ay f c z | 249 Finit : {z : Ordinal} → z ≡ y → Fc∨sup A ay f c z |
250 Fc : {p x : Ordinal} → p o< x → Fc∨sup A ay f c p → FChain A f p c x → Fc∨sup A ay f c x | 250 Fsup : {p x : Ordinal} → p o< x → Fc∨sup A ay f c p → FSup A f p c x → Fc∨sup A ay f c x |
251 Fc : {p x : Ordinal} → p o< x → Fc∨sup A ay f c p → FChain A f p c x → Fc∨sup A ay f c x | |
251 | 252 |
252 record ZChain ( A : HOD ) (x : Ordinal) ( f : Ordinal → Ordinal ) ( z : Ordinal ) : Set (Level.suc n) where | 253 record ZChain ( A : HOD ) (x : Ordinal) ( f : Ordinal → Ordinal ) ( z : Ordinal ) : Set (Level.suc n) where |
253 field | 254 field |
254 chain : HOD | 255 chain : HOD |
255 chain⊆A : chain ⊆' A | 256 chain⊆A : chain ⊆' A |
421 zc4 : ZChain A y f x | 422 zc4 : ZChain A y f x |
422 zc4 with ODC.∋-p O A (* x) | 423 zc4 with ODC.∋-p O A (* x) |
423 ... | no noax = -- ¬ A ∋ p, just skip | 424 ... | no noax = -- ¬ A ∋ p, just skip |
424 record { chain = ZChain.chain zc0 ; chain⊆A = ZChain.chain⊆A zc0 ; initial = ZChain.initial zc0 | 425 record { chain = ZChain.chain zc0 ; chain⊆A = ZChain.chain⊆A zc0 ; initial = ZChain.initial zc0 |
425 ; f-total = ZChain.f-total zc0 ; f-next = ZChain.f-next zc0 | 426 ; f-total = ZChain.f-total zc0 ; f-next = ZChain.f-next zc0 |
426 ; f-immediate = ZChain.f-immediate zc0 ; chain∋x = ZChain.chain∋x zc0 ; is-max = zc11 ; fc∨sup = {!!} } where -- no extention | 427 ; f-immediate = ZChain.f-immediate zc0 ; chain∋x = ZChain.chain∋x zc0 ; is-max = zc11 ; fc∨sup = ZChain.fc∨sup zc0 } where -- no extention |
427 zc11 : {a b : Ordinal} → odef (ZChain.chain zc0) a → b o< osuc x → (ab : odef A b) → | 428 zc11 : {a b : Ordinal} → odef (ZChain.chain zc0) a → b o< osuc x → (ab : odef A b) → |
428 HasPrev A (ZChain.chain zc0) ab f ∨ IsSup A (ZChain.chain zc0) ab → | 429 HasPrev A (ZChain.chain zc0) ab f ∨ IsSup A (ZChain.chain zc0) ab → |
429 * a < * b → odef (ZChain.chain zc0) b | 430 * a < * b → odef (ZChain.chain zc0) b |
430 zc11 {a} {b} za b<ox ab P a<b with osuc-≡< b<ox | 431 zc11 {a} {b} za b<ox ab P a<b with osuc-≡< b<ox |
431 ... | case1 eq = ⊥-elim ( noax (subst (λ k → odef A k) (trans eq (sym &iso)) ab ) ) | 432 ... | case1 eq = ⊥-elim ( noax (subst (λ k → odef A k) (trans eq (sym &iso)) ab ) ) |
439 zc17 {a} {b} za b<ox ab P a<b with osuc-≡< b<ox | 440 zc17 {a} {b} za b<ox ab P a<b with osuc-≡< b<ox |
440 ... | case2 lt = ZChain.is-max zc0 za (zc0-b<x b lt) ab P a<b | 441 ... | case2 lt = ZChain.is-max zc0 za (zc0-b<x b lt) ab P a<b |
441 ... | case1 b=x = subst (λ k → odef chain k ) (trans (sym (HasPrev.x=fy pr )) (trans &iso (sym b=x)) ) ( ZChain.f-next zc0 (HasPrev.ay pr)) | 442 ... | case1 b=x = subst (λ k → odef chain k ) (trans (sym (HasPrev.x=fy pr )) (trans &iso (sym b=x)) ) ( ZChain.f-next zc0 (HasPrev.ay pr)) |
442 zc9 : ZChain A y f x | 443 zc9 : ZChain A y f x |
443 zc9 = record { chain = ZChain.chain zc0 ; chain⊆A = ZChain.chain⊆A zc0 ; f-total = ZChain.f-total zc0 ; f-next = ZChain.f-next zc0 -- no extention | 444 zc9 = record { chain = ZChain.chain zc0 ; chain⊆A = ZChain.chain⊆A zc0 ; f-total = ZChain.f-total zc0 ; f-next = ZChain.f-next zc0 -- no extention |
444 ; initial = ZChain.initial zc0 ; f-immediate = ZChain.f-immediate zc0 ; chain∋x = ZChain.chain∋x zc0 ; is-max = zc17 ; fc∨sup = {!!}} | 445 ; initial = ZChain.initial zc0 ; f-immediate = ZChain.f-immediate zc0 ; chain∋x = ZChain.chain∋x zc0 ; is-max = zc17 ; fc∨sup = ZChain.fc∨sup zc0 } |
445 ... | case2 ¬fy<x with ODC.p∨¬p O (IsSup A (ZChain.chain zc0) ax ) | 446 ... | case2 ¬fy<x with ODC.p∨¬p O (IsSup A (ZChain.chain zc0) ax ) |
446 ... | case1 is-sup = -- x is a sup of zc0 | 447 ... | case1 is-sup = -- x is a sup of zc0 |
447 record { chain = schain ; chain⊆A = s⊆A ; f-total = scmp ; f-next = scnext | 448 record { chain = schain ; chain⊆A = s⊆A ; f-total = scmp ; f-next = scnext |
448 ; initial = scinit ; f-immediate = simm ; chain∋x = case1 (ZChain.chain∋x zc0) ; is-max = s-ismax ; fc∨sup = {!!}} where | 449 ; initial = scinit ; f-immediate = simm ; chain∋x = case1 (ZChain.chain∋x zc0) ; is-max = s-ismax ; fc∨sup = s-fc∨sup} where |
449 sup0 : SUP A (ZChain.chain zc0) | 450 sup0 : SUP A (ZChain.chain zc0) |
450 sup0 = record { sup = * x ; A∋maximal = ax ; x<sup = x21 } where | 451 sup0 = record { sup = * x ; A∋maximal = ax ; x<sup = x21 } where |
451 x21 : {y : HOD} → ZChain.chain zc0 ∋ y → (y ≡ * x) ∨ (y < * x) | 452 x21 : {y : HOD} → ZChain.chain zc0 ∋ y → (y ≡ * x) ∨ (y < * x) |
452 x21 {y} zy with IsSup.x<sup is-sup zy | 453 x21 {y} zy with IsSup.x<sup is-sup zy |
453 ... | case1 y=x = case1 ( subst₂ (λ j k → j ≡ * k ) *iso &iso ( cong (*) y=x) ) | 454 ... | case1 y=x = case1 ( subst₂ (λ j k → j ≡ * k ) *iso &iso ( cong (*) y=x) ) |
466 A∋schain (case1 zx ) = ZChain.chain⊆A zc0 zx | 467 A∋schain (case1 zx ) = ZChain.chain⊆A zc0 zx |
467 A∋schain {y} (case2 fx ) = A∋fc (& sp) f mf fx | 468 A∋schain {y} (case2 fx ) = A∋fc (& sp) f mf fx |
468 s⊆A : schain ⊆' A | 469 s⊆A : schain ⊆' A |
469 s⊆A {x} (case1 zx) = ZChain.chain⊆A zc0 zx | 470 s⊆A {x} (case1 zx) = ZChain.chain⊆A zc0 zx |
470 s⊆A {x} (case2 fx) = A∋fc (& sp) f mf fx | 471 s⊆A {x} (case2 fx) = A∋fc (& sp) f mf fx |
472 s-fc∨sup : {c : Ordinal} → odef schain c → Fc∨sup A (s⊆A (case1 (ZChain.chain∋x zc0))) f (& schain) c | |
473 s-fc∨sup {c} (case1 cx) = {!!} | |
474 s-fc∨sup {c} (case2 fc) = {!!} | |
471 cmp : {a b : HOD} (za : odef chain (& a)) (fb : FClosure A f (& sp) (& b)) → Tri (a < b) (a ≡ b) (b < a ) | 475 cmp : {a b : HOD} (za : odef chain (& a)) (fb : FClosure A f (& sp) (& b)) → Tri (a < b) (a ≡ b) (b < a ) |
472 cmp {a} {b} za fb with SUP.x<sup sup0 za | s≤fc (& sp) f mf fb | 476 cmp {a} {b} za fb with SUP.x<sup sup0 za | s≤fc (& sp) f mf fb |
473 ... | case1 sp=a | case1 sp=b = tri≈ (λ lt → <-irr (case1 (sym eq)) lt ) eq (λ lt → <-irr (case1 eq) lt ) where | 477 ... | case1 sp=a | case1 sp=b = tri≈ (λ lt → <-irr (case1 (sym eq)) lt ) eq (λ lt → <-irr (case1 eq) lt ) where |
474 eq : a ≡ b | 478 eq : a ≡ b |
475 eq = trans sp=a (subst₂ (λ j k → j ≡ k ) *iso *iso sp=b ) | 479 eq = trans sp=a (subst₂ (λ j k → j ≡ k ) *iso *iso sp=b ) |
523 simm {a} {b} (case2 sa) (case2 sb) p = fcn-imm {A} (& sp) {a} {b} f mf sa sb p | 527 simm {a} {b} (case2 sa) (case2 sb) p = fcn-imm {A} (& sp) {a} {b} f mf sa sb p |
524 s-ismax : {a b : Ordinal} → odef schain a → b o< osuc x → (ab : odef A b) | 528 s-ismax : {a b : Ordinal} → odef schain a → b o< osuc x → (ab : odef A b) |
525 → HasPrev A schain ab f ∨ IsSup A schain ab | 529 → HasPrev A schain ab f ∨ IsSup A schain ab |
526 → * a < * b → odef schain b | 530 → * a < * b → odef schain b |
527 s-ismax {a} {b} sa b<ox ab p a<b with osuc-≡< b<ox -- b is x? | 531 s-ismax {a} {b} sa b<ox ab p a<b with osuc-≡< b<ox -- b is x? |
528 ... | case1 b=x = case2 (subst (λ k → FClosure A f (& sp) k ) (sym (trans b=x x=sup )) (init (SUP.A∋maximal sup0) refl )) | 532 ... | case1 b=x = case2 (subst (λ k → FClosure A f (& sp) k ) (sym (trans b=x x=sup )) (init (SUP.A∋maximal sup0) )) |
529 s-ismax {a} {b} (case1 za) b<ox ab (case1 p) a<b | case2 b<x = z21 p where -- has previous | 533 s-ismax {a} {b} (case1 za) b<ox ab (case1 p) a<b | case2 b<x = z21 p where -- has previous |
530 z21 : HasPrev A schain ab f → odef schain b | 534 z21 : HasPrev A schain ab f → odef schain b |
531 z21 record { y = y ; ay = (case1 zy) ; x=fy = x=fy } = | 535 z21 record { y = y ; ay = (case1 zy) ; x=fy = x=fy } = |
532 case1 (ZChain.is-max zc0 za (zc0-b<x b b<x) ab (case1 record { y = y ; ay = zy ; x=fy = x=fy }) a<b ) | 536 case1 (ZChain.is-max zc0 za (zc0-b<x b b<x) ab (case1 record { y = y ; ay = zy ; x=fy = x=fy }) a<b ) |
533 z21 record { y = y ; ay = (case2 sy) ; x=fy = x=fy } = subst (λ k → odef schain k) (sym x=fy) (case2 (fsuc y sy) ) | 537 z21 record { y = y ; ay = (case2 sy) ; x=fy = x=fy } = subst (λ k → odef schain k) (sym x=fy) (case2 (fsuc y sy) ) |
544 z23 with IsSup.x<sup (z24 p) ( ZChain.chain∋x zc0 ) | 548 z23 with IsSup.x<sup (z24 p) ( ZChain.chain∋x zc0 ) |
545 ... | case1 y=b = subst (λ k → odef chain k ) y=b ( ZChain.chain∋x zc0 ) | 549 ... | case1 y=b = subst (λ k → odef chain k ) y=b ( ZChain.chain∋x zc0 ) |
546 ... | case2 y<b = ZChain.is-max zc0 (ZChain.chain∋x zc0 ) (zc0-b<x b b<x) ab (case2 (z24 p)) y<b | 550 ... | case2 y<b = ZChain.is-max zc0 (ZChain.chain∋x zc0 ) (zc0-b<x b b<x) ab (case2 (z24 p)) y<b |
547 ... | case2 ¬x=sup = -- x is not f y' nor sup of former ZChain from y | 551 ... | case2 ¬x=sup = -- x is not f y' nor sup of former ZChain from y |
548 record { chain = ZChain.chain zc0 ; chain⊆A = ZChain.chain⊆A zc0 ; f-total = ZChain.f-total zc0 ; f-next = ZChain.f-next zc0 | 552 record { chain = ZChain.chain zc0 ; chain⊆A = ZChain.chain⊆A zc0 ; f-total = ZChain.f-total zc0 ; f-next = ZChain.f-next zc0 |
549 ; initial = ZChain.initial zc0 ; f-immediate = ZChain.f-immediate zc0 ; chain∋x = ZChain.chain∋x zc0 ; is-max = z18 ; fc∨sup = {!!} } where | 553 ; initial = ZChain.initial zc0 ; f-immediate = ZChain.f-immediate zc0 ; chain∋x = ZChain.chain∋x zc0 ; is-max = z18 ; fc∨sup = ZChain.fc∨sup zc0 } where |
550 -- no extention | 554 -- no extention |
551 z18 : {a b : Ordinal} → odef (ZChain.chain zc0) a → b o< osuc x → (ab : odef A b) → | 555 z18 : {a b : Ordinal} → odef (ZChain.chain zc0) a → b o< osuc x → (ab : odef A b) → |
552 HasPrev A (ZChain.chain zc0) ab f ∨ IsSup A (ZChain.chain zc0) ab → | 556 HasPrev A (ZChain.chain zc0) ab f ∨ IsSup A (ZChain.chain zc0) ab → |
553 * a < * b → odef (ZChain.chain zc0) b | 557 * a < * b → odef (ZChain.chain zc0) b |
554 z18 {a} {b} za b<x ab p a<b with osuc-≡< b<x | 558 z18 {a} {b} za b<x ab p a<b with osuc-≡< b<x |
559 x<sup = λ {y} zy → subst (λ k → (y ≡ k) ∨ (y << k)) (trans b=x (sym &iso)) (IsSup.x<sup b=sup zy) } ) | 563 x<sup = λ {y} zy → subst (λ k → (y ≡ k) ∨ (y << k)) (trans b=x (sym &iso)) (IsSup.x<sup b=sup zy) } ) |
560 ... | no ¬ox with trio< x y | 564 ... | no ¬ox with trio< x y |
561 ... | tri< a ¬b ¬c = record { chain = {!!} ; chain⊆A = {!!} ; f-total = {!!} ; f-next = {!!} | 565 ... | tri< a ¬b ¬c = record { chain = {!!} ; chain⊆A = {!!} ; f-total = {!!} ; f-next = {!!} |
562 ; initial = {!!} ; f-immediate = {!!} ; chain∋x = {!!} ; is-max = {!!} ; fc∨sup = {!!} } | 566 ; initial = {!!} ; f-immediate = {!!} ; chain∋x = {!!} ; is-max = {!!} ; fc∨sup = {!!} } |
563 ... | tri≈ ¬a b ¬c = {!!} | 567 ... | tri≈ ¬a b ¬c = {!!} |
564 ... | tri> ¬a ¬b y<x = {!!} where | 568 ... | tri> ¬a ¬b y<x = UnionZ where |
565 UnionZ : ZChain A y f x | 569 UnionZ : ZChain A y f x |
566 UnionZ = record { chain = Uz ; chain⊆A = Uz⊆A ; f-total = u-total ; f-next = u-next | 570 UnionZ = record { chain = Uz ; chain⊆A = Uz⊆A ; f-total = u-total ; f-next = u-next |
567 ; initial = u-initial ; f-immediate = {!!} ; chain∋x = {!!} ; is-max = {!!} ; fc∨sup = {!!} } where --- limit ordinal case | 571 ; initial = u-initial ; f-immediate = {!!} ; chain∋x = {!!} ; is-max = {!!} ; fc∨sup = {!!} } where --- limit ordinal case |
568 record UZFChain (z : Ordinal) : Set n where -- Union of ZFChain from y which has maximality o< x | 572 record UZFChain (z : Ordinal) : Set n where -- Union of ZFChain from y which has maximality o< x |
569 field | 573 field |
593 FC : Fc∨sup A (chain⊆A za (chain∋x za)) f (& (chain za)) i | 597 FC : Fc∨sup A (chain⊆A za (chain∋x za)) f (& (chain za)) i |
594 FC = fc∨sup za zai | 598 FC = fc∨sup za zai |
595 um01 : odef (chain zb) i | 599 um01 : odef (chain zb) i |
596 um01 with FC | 600 um01 with FC |
597 ... | Finit i=y = subst (λ k → odef (chain zb) k ) (sym i=y) ( chain∋x zb ) | 601 ... | Finit i=y = subst (λ k → odef (chain zb) k ) (sym i=y) ( chain∋x zb ) |
602 ... | Fsup {p} {i} p<i pFC sup = ? | |
598 ... | Fc {p} {i} p<i pFC FC with (FChain.fc∨sup FC) | 603 ... | Fc {p} {i} p<i pFC FC with (FChain.fc∨sup FC) |
599 ... | fc = um02 i fc where | 604 ... | fc = um02 i fc where |
600 um02 : (i2 : Ordinal) → FClosure A f p i2 → odef (chain zb) i2 | 605 um02 : (i2 : Ordinal) → FClosure A f p i2 → odef (chain zb) i2 |
601 um02 i2 (init ap i2=p ) = subst (λ k → odef (chain zb) k ) (sym i2=p) (previ p p<i um04 ) where | 606 um02 i2 (init ap ) = previ p p<i um04 where |
602 um04 : odef (chain za) p | 607 um04 : odef (chain za) p |
603 um04 = subst (λ k → odef k p ) *iso ( FChain.chain∋p FC ) | 608 um04 = subst (λ k → odef k p ) *iso ( FChain.chain∋p FC ) |
604 um02 i (fsuc j fc) = f-next zb ( um02 j fc ) | 609 um02 i (fsuc j fc) = f-next zb ( um02 j fc ) |
605 u-total : IsTotalOrderSet Uz | 610 u-total : IsTotalOrderSet Uz |
606 u-total {x} {y} ux uy with trio< (UZFChain.u ux) (UZFChain.u uy) | 611 u-total {x} {y} ux uy with trio< (UZFChain.u ux) (UZFChain.u uy) |