Mercurial > hg > Members > kono > Proof > ZF-in-agda
changeset 788:c164f4f7cfd1
u<x in UChain again
| author | Shinji KONO <kono@ie.u-ryukyu.ac.jp> |
|---|---|
| date | Tue, 02 Aug 2022 16:09:00 +0900 |
| parents | 56df4675e15a |
| children | a08c456d49d0 |
| files | src/zorn.agda |
| diffstat | 1 files changed, 61 insertions(+), 51 deletions(-) [+] |
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--- a/src/zorn.agda Tue Aug 02 11:34:28 2022 +0900 +++ b/src/zorn.agda Tue Aug 02 16:09:00 2022 +0900 @@ -284,7 +284,7 @@ data UChain ( A : HOD ) ( f : Ordinal → Ordinal ) (mf : ≤-monotonic-f A f) {y : Ordinal } (ay : odef A y ) (supf : Ordinal → Ordinal) (x : Ordinal) : (z : Ordinal) → Set n where ch-init : {z : Ordinal } (fc : FClosure A f y z) → UChain A f mf ay supf x z - ch-is-sup : (u : Ordinal) {z : Ordinal } ( is-sup : ChainP A f mf ay supf u ) + ch-is-sup : (u : Ordinal) {z : Ordinal } (u<x : u o< x) ( is-sup : ChainP A f mf ay supf u ) ( fc : FClosure A f (supf u) z ) → UChain A f mf ay supf x z ∈∧P→o< : {A : HOD } {y : Ordinal} → {P : Set n} → odef A y ∧ P → y o< & A @@ -322,22 +322,24 @@ order {b} {s} {z1} b<z sf<sb fc = zc04 where zc01 : {z1 : Ordinal } → FClosure A f (supf s) z1 → UnionCF A f mf ay supf b ∋ * z1 zc01 (init x refl ) = subst (λ k → odef (UnionCF A f mf ay supf b) k ) (sym &iso) zc03 where - s<z : s o< z - s<z with trio< s z + s<b : s o< b + s<b with trio< s b ... | tri< a ¬b ¬c = a - ... | tri≈ ¬a b ¬c = ⊥-elim ( o<¬≡ (cong supf b) (ordtrans<-≤ sf<sb (supf-mono b<z) )) + ... | tri≈ ¬a b ¬c = ⊥-elim ( o<¬≡ (cong supf b) sf<sb ) ... | tri> ¬a ¬b c with osuc-≡< ( supf-mono c ) - ... | case1 eq = ⊥-elim ( o<¬≡ (sym eq) (ordtrans<-≤ sf<sb (supf-mono b<z) )) - ... | case2 lt = ⊥-elim ( o<> lt (ordtrans<-≤ sf<sb (supf-mono b<z) )) + ... | case1 eq = ⊥-elim ( o<¬≡ (sym eq) sf<sb ) + ... | case2 lt = ⊥-elim ( o<> lt sf<sb ) + s<z : s o< z + s<z = ordtrans s<b b<z zc03 : odef (UnionCF A f mf ay supf b) (supf s) zc03 with csupf (o<→≤ s<z) ... | ⟪ as , ch-init fc ⟫ = ⟪ as , ch-init fc ⟫ - ... | ⟪ as , ch-is-sup u is-sup fc ⟫ = ⟪ as , ch-is-sup u is-sup fc ⟫ + ... | ⟪ as , ch-is-sup u u<x is-sup fc ⟫ = ⟪ as , ch-is-sup u (ordtrans u<x s<b) is-sup fc ⟫ zc01 (fsuc x fc) = subst (λ k → odef (UnionCF A f mf ay supf b) k ) (sym &iso) zc04 where zc04 : odef (UnionCF A f mf ay supf b) (f x) zc04 with subst (λ k → odef (UnionCF A f mf ay supf b) k ) &iso (zc01 fc ) ... | ⟪ as , ch-init fc ⟫ = ⟪ proj2 (mf _ as) , ch-init (fsuc _ fc) ⟫ - ... | ⟪ as , ch-is-sup u is-sup fc ⟫ = ⟪ proj2 (mf _ as) , ch-is-sup u is-sup (fsuc _ fc) ⟫ + ... | ⟪ as , ch-is-sup u u<x is-sup fc ⟫ = ⟪ proj2 (mf _ as) , ch-is-sup u u<x is-sup (fsuc _ fc) ⟫ zc00 : ( * z1 ≡ SUP.sup (sup b<z )) ∨ ( * z1 < SUP.sup ( sup b<z ) ) zc00 = SUP.x<sup (sup b<z) (zc01 fc ) zc04 : (z1 ≡ supf b) ∨ (z1 << supf b) @@ -366,7 +368,7 @@ chain-total A f mf {y} ay supf {xa} {xb} {a} {b} ca cb = ct-ind xa xb ca cb where ct-ind : (xa xb : Ordinal) → {a b : Ordinal} → UChain A f mf ay supf xa a → UChain A f mf ay supf xb b → Tri (* a < * b) (* a ≡ * b) (* b < * a) ct-ind xa xb {a} {b} (ch-init fca) (ch-init fcb) = fcn-cmp y f mf fca fcb - ct-ind xa xb {a} {b} (ch-init fca) (ch-is-sup ub supb fcb) with ChainP.fcy<sup supb fca + ct-ind xa xb {a} {b} (ch-init fca) (ch-is-sup ub u<x supb fcb) with ChainP.fcy<sup supb fca ... | case1 eq with s≤fc (supf ub) f mf fcb ... | case1 eq1 = tri≈ (λ lt → ⊥-elim (<-irr (case1 (sym ct00)) lt)) ct00 (λ lt → ⊥-elim (<-irr (case1 ct00) lt)) where ct00 : * a ≡ * b @@ -374,14 +376,14 @@ ... | case2 lt = tri< ct01 (λ eq → <-irr (case1 (sym eq)) ct01) (λ lt → <-irr (case2 ct01) lt) where ct01 : * a < * b ct01 = subst (λ k → * k < * b ) (sym eq) lt - ct-ind xa xb {a} {b} (ch-init fca) (ch-is-sup ub supb fcb) | case2 lt = tri< ct01 (λ eq → <-irr (case1 (sym eq)) ct01) (λ lt → <-irr (case2 ct01) lt) where + ct-ind xa xb {a} {b} (ch-init fca) (ch-is-sup ub u<x supb fcb) | case2 lt = tri< ct01 (λ eq → <-irr (case1 (sym eq)) ct01) (λ lt → <-irr (case2 ct01) lt) where ct00 : * a < * (supf ub) ct00 = lt ct01 : * a < * b ct01 with s≤fc (supf ub) f mf fcb ... | case1 eq = subst (λ k → * a < k ) eq ct00 ... | case2 lt = IsStrictPartialOrder.trans POO ct00 lt - ct-ind xa xb {a} {b} (ch-is-sup ua supa fca) (ch-init fcb) with ChainP.fcy<sup supa fcb + ct-ind xa xb {a} {b} (ch-is-sup ua u<x supa fca) (ch-init fcb) with ChainP.fcy<sup supa fcb ... | case1 eq with s≤fc (supf ua) f mf fca ... | case1 eq1 = tri≈ (λ lt → ⊥-elim (<-irr (case1 (sym ct00)) lt)) ct00 (λ lt → ⊥-elim (<-irr (case1 ct00) lt)) where ct00 : * a ≡ * b @@ -389,14 +391,14 @@ ... | case2 lt = tri> (λ lt → <-irr (case2 ct01) lt) (λ eq → <-irr (case1 eq) ct01) ct01 where ct01 : * b < * a ct01 = subst (λ k → * k < * a ) (sym eq) lt - ct-ind xa xb {a} {b} (ch-is-sup ua supa fca) (ch-init fcb) | case2 lt = tri> (λ lt → <-irr (case2 ct01) lt) (λ eq → <-irr (case1 eq) ct01) ct01 where + ct-ind xa xb {a} {b} (ch-is-sup ua u<x supa fca) (ch-init fcb) | case2 lt = tri> (λ lt → <-irr (case2 ct01) lt) (λ eq → <-irr (case1 eq) ct01) ct01 where ct00 : * b < * (supf ua) ct00 = lt ct01 : * b < * a ct01 with s≤fc (supf ua) f mf fca ... | case1 eq = subst (λ k → * b < k ) eq ct00 ... | case2 lt = IsStrictPartialOrder.trans POO ct00 lt - ct-ind xa xb {a} {b} (ch-is-sup ua supa fca) (ch-is-sup ub supb fcb) with trio< (supf ua) (supf ub) + ct-ind xa xb {a} {b} (ch-is-sup ua ua<x supa fca) (ch-is-sup ub ub<x supb fcb) with trio< (supf ua) (supf ub) ... | tri< a₁ ¬b ¬c with ChainP.order supb a₁ fca ... | case1 eq with s≤fc (supf ub) f mf fcb ... | case1 eq1 = tri≈ (λ lt → ⊥-elim (<-irr (case1 (sym ct00)) lt)) ct00 (λ lt → ⊥-elim (<-irr (case1 ct00) lt)) where @@ -405,16 +407,16 @@ ... | case2 lt = tri< ct02 (λ eq → <-irr (case1 (sym eq)) ct02) (λ lt → <-irr (case2 ct02) lt) where ct02 : * a < * b ct02 = subst (λ k → * k < * b ) (sym eq) lt - ct-ind xa xb {a} {b} (ch-is-sup ua supa fca) (ch-is-sup ub supb fcb) | tri< a₁ ¬b ¬c | case2 lt = tri< ct02 (λ eq → <-irr (case1 (sym eq)) ct02) (λ lt → <-irr (case2 ct02) lt) where + ct-ind xa xb {a} {b} (ch-is-sup ua ua<x supa fca) (ch-is-sup ub ub<x supb fcb) | tri< a₁ ¬b ¬c | case2 lt = tri< ct02 (λ eq → <-irr (case1 (sym eq)) ct02) (λ lt → <-irr (case2 ct02) lt) where ct03 : * a < * (supf ub) ct03 = lt ct02 : * a < * b ct02 with s≤fc (supf ub) f mf fcb ... | case1 eq = subst (λ k → * a < k ) eq ct03 ... | case2 lt = IsStrictPartialOrder.trans POO ct03 lt - ct-ind xa xb {a} {b} (ch-is-sup ua supa fca) (ch-is-sup ub supb fcb) | tri≈ ¬a eq ¬c + ct-ind xa xb {a} {b} (ch-is-sup ua ua<x supa fca) (ch-is-sup ub ub<x supb fcb) | tri≈ ¬a eq ¬c = fcn-cmp (supf ua) f mf fca (subst (λ k → FClosure A f k b ) (sym eq) fcb ) - ct-ind xa xb {a} {b} (ch-is-sup ua supa fca) (ch-is-sup ub supb fcb) | tri> ¬a ¬b c with ChainP.order supa c fcb + ct-ind xa xb {a} {b} (ch-is-sup ua ua<x supa fca) (ch-is-sup ub ub<x supb fcb) | tri> ¬a ¬b c with ChainP.order supa c fcb ... | case1 eq with s≤fc (supf ua) f mf fca ... | case1 eq1 = tri≈ (λ lt → ⊥-elim (<-irr (case1 (sym ct00)) lt)) ct00 (λ lt → ⊥-elim (<-irr (case1 ct00) lt)) where ct00 : * a ≡ * b @@ -422,7 +424,7 @@ ... | case2 lt = tri> (λ lt → <-irr (case2 ct02) lt) (λ eq → <-irr (case1 eq) ct02) ct02 where ct02 : * b < * a ct02 = subst (λ k → * k < * a ) (sym eq) lt - ct-ind xa xb {a} {b} (ch-is-sup ua supa fca) (ch-is-sup ub supb fcb) | tri> ¬a ¬b c | case2 lt = tri> (λ lt → <-irr (case2 ct04) lt) (λ eq → <-irr (case1 (eq)) ct04) ct04 where + ct-ind xa xb {a} {b} (ch-is-sup ua ua<x supa fca) (ch-is-sup ub ub<x supb fcb) | tri> ¬a ¬b c | case2 lt = tri> (λ lt → <-irr (case2 ct04) lt) (λ eq → <-irr (case1 (eq)) ct04) ct04 where ct05 : * b < * (supf ua) ct05 = lt ct04 : * b < * a @@ -441,8 +443,8 @@ Zorn-lemma {A} 0<A supP = zorn00 where <-irr0 : {a b : HOD} → A ∋ a → A ∋ b → (a ≡ b ) ∨ (a < b ) → b < a → ⊥ <-irr0 {a} {b} A∋a A∋b = <-irr - z07 : {y : Ordinal} → {P : Set n} → odef A y ∧ P → y o< & A - z07 {y} p = subst (λ k → k o< & A) &iso ( c<→o< (subst (λ k → odef A k ) (sym &iso ) (proj1 p ))) + z07 : {y : Ordinal} {A : HOD } → {P : Set n} → odef A y ∧ P → y o< & A + z07 {y} {A} p = subst (λ k → k o< & A) &iso ( c<→o< (subst (λ k → odef A k ) (sym &iso ) (proj1 p ))) z09 : {b : Ordinal } { A : HOD } → odef A b → b o< & A z09 {b} {A} ab = subst (λ k → k o< & A) &iso ( c<→o< (subst (λ k → odef A k ) (sym &iso ) ab)) s : HOD @@ -499,25 +501,29 @@ SZ1 :( A : HOD ) ( f : Ordinal → Ordinal ) (mf : ≤-monotonic-f A f) {init : Ordinal} (ay : odef A init) (zc : ZChain A f mf ay (& A)) (x : Ordinal) → ZChain1 A f mf ay zc x SZ1 A f mf {y} ay zc x = TransFinite { λ x → ZChain1 A f mf ay zc x } zc1 x where - chain-mono1 : (x : Ordinal) {a b c : Ordinal} → - odef (UnionCF A f mf ay (ZChain.supf zc) a) c → odef (UnionCF A f mf ay (ZChain.supf zc) b) c - chain-mono1 x {a} {b} {c} ⟪ ua , ch-init fc ⟫ = + chain-mono1 : (x : Ordinal) {a b c : Ordinal} → a o≤ b + → odef (UnionCF A f mf ay (ZChain.supf zc) a) c → odef (UnionCF A f mf ay (ZChain.supf zc) b) c + chain-mono1 x {a} {b} {c} a≤b ⟪ ua , ch-init fc ⟫ = ⟪ ua , ch-init fc ⟫ - chain-mono1 x {a} {b} {c} ⟪ uaa , ch-is-sup ua is-sup fc ⟫ = - ⟪ uaa , ch-is-sup ua is-sup fc ⟫ + chain-mono1 x {a} {b} {c} a≤b ⟪ uaa , ch-is-sup ua ua<x is-sup fc ⟫ = + ⟪ uaa , ch-is-sup ua (ordtrans<-≤ ua<x a≤b ) is-sup fc ⟫ chain<ZA : {x : Ordinal } → UnionCF A f mf ay (ZChain.supf zc) x ⊆' UnionCF A f mf ay (ZChain.supf zc) (& A) chain<ZA {x} ux with proj2 ux ... | ch-init fc = ⟪ proj1 ux , ch-init fc ⟫ - ... | ch-is-sup u is-sup fc = ⟪ proj1 ux , ch-is-sup u is-sup fc ⟫ + ... | ch-is-sup u pu<x is-sup fc = ⟪ proj1 ux , ch-is-sup u u<x is-sup fc ⟫ where + u<A : (& ( * ( ZChain.supf zc u))) o< & A + u<A = c<→o< (subst (λ k → odef A k ) (sym &iso) (A∋fcs _ f mf fc) ) + u<x : u o< & A + u<x = subst (λ k → k o< & A ) (trans &iso ?) u<A is-max-hp : (x : Ordinal) {a : Ordinal} {b : Ordinal} → odef (UnionCF A f mf ay (ZChain.supf zc) x) a → b o< x → (ab : odef A b) → HasPrev A (UnionCF A f mf ay (ZChain.supf zc) x) ab f → * a < * b → odef (UnionCF A f mf ay (ZChain.supf zc) x) b is-max-hp x {a} {b} ua b<x ab has-prev a<b with HasPrev.ay has-prev ... | ⟪ ab0 , ch-init fc ⟫ = ⟪ ab , ch-init ( subst (λ k → FClosure A f y k) (sym (HasPrev.x=fy has-prev)) (fsuc _ fc )) ⟫ - ... | ⟪ ab0 , ch-is-sup u is-sup fc ⟫ = ⟪ ab , + ... | ⟪ ab0 , ch-is-sup u u<x is-sup fc ⟫ = ⟪ ab , subst (λ k → UChain A f mf ay (ZChain.supf zc) x k ) - (sym (HasPrev.x=fy has-prev)) ( ch-is-sup u is-sup (fsuc _ fc)) ⟫ + (sym (HasPrev.x=fy has-prev)) ( ch-is-sup u u<x is-sup (fsuc _ fc)) ⟫ zc1 : (x : Ordinal) → ((y₁ : Ordinal) → y₁ o< x → ZChain1 A f mf ay zc y₁) → ZChain1 A f mf ay zc x zc1 x prev with Oprev-p x ... | yes op = record { is-max = is-max } where @@ -537,20 +543,23 @@ m01 : odef (UnionCF A f mf ay (ZChain.supf zc) x) b m01 with trio< b px --- px < b < x ... | tri> ¬a ¬b c = ⊥-elim (¬p<x<op ⟪ c , subst (λ k → b o< k ) (sym (Oprev.oprev=x op)) b<x ⟫) - ... | tri< b<px ¬b ¬c = chain-mono1 x m04 where + ... | tri< b<px ¬b ¬c = chain-mono1 x (ordtrans px<x ? ) m04 where m03 : odef (UnionCF A f mf ay (ZChain.supf zc) px) a -- if a ∈ chain of px, is-max of px can be used m03 with proj2 ua ... | ch-init fc = ⟪ proj1 ua , ch-init fc ⟫ - ... | ch-is-sup u is-sup-a fc = ⟪ proj1 ua , ch-is-sup u is-sup-a fc ⟫ + ... | ch-is-sup u u<x is-sup fc with trio< u px + ... | tri< a ¬b ¬c = ⟪ proj1 ua , ch-is-sup u a is-sup fc ⟫ + ... | tri≈ ¬a u=px ¬c = ? --- supf u < a < b , + ... | tri> ¬a ¬b c = ⊥-elim ( ¬p<x<op ⟪ c , subst (λ k → u o< k ) (sym (Oprev.oprev=x op)) u<x ⟫ ) m04 : odef (UnionCF A f mf ay (ZChain.supf zc) px) b m04 = ZChain1.is-max (prev px px<x) m03 b<px ab - (case2 record {x<sup = λ {z} lt → IsSup.x<sup is-sup (chain-mono1 x lt) } ) a<b - ... | tri≈ ¬a b=px ¬c = ⟪ ab , ch-is-sup b m06 (subst (λ k → FClosure A f k b) m05 (init ab refl)) ⟫ where + (case2 record {x<sup = λ {z} lt → IsSup.x<sup is-sup (chain-mono1 x ? lt) } ) a<b + ... | tri≈ ¬a b=px ¬c = ⟪ ab , ch-is-sup b b<x m06 (subst (λ k → FClosure A f k b) m05 (init ab refl)) ⟫ where b<A : b o< & A b<A = z09 ab m05 : b ≡ ZChain.supf zc b m05 = sym ( ZChain.sup=u zc ab (z09 ab) - record { x<sup = λ {z} uz → IsSup.x<sup is-sup (chain-mono1 x uz ) } ) + record { x<sup = λ {z} uz → IsSup.x<sup is-sup (chain-mono1 x ? uz ) } ) m08 : {z : Ordinal} → (fcz : FClosure A f y z ) → z <= ZChain.supf zc b m08 {z} fcz = ZChain.fcy<sup zc b<A fcz m09 : {sup1 z1 : Ordinal} → (ZChain.supf zc sup1) o< (ZChain.supf zc b) @@ -565,9 +574,9 @@ * a < * b → odef (UnionCF A f mf ay (ZChain.supf zc) x) b is-max {a} {b} ua b<x ab (case1 has-prev) a<b = is-max-hp x {a} {b} ua b<x ab has-prev a<b is-max {a} {b} ua b<x ab (case2 is-sup) a<b with IsSup.x<sup is-sup (init-uchain A f mf ay ) - ... | case1 b=y = ⊥-elim ( <-irr ( ZChain.initial zc (chain<ZA {x} (chain-mono1 (osuc x) ua )) ) + ... | case1 b=y = ⊥-elim ( <-irr ( ZChain.initial zc (chain<ZA {x} (chain-mono1 (osuc x) ? ua )) ) (subst (λ k → * a < * k ) (sym b=y) a<b ) ) - ... | case2 y<b = chain-mono1 x m04 where + ... | case2 y<b = chain-mono1 x ? m04 where m09 : b o< & A m09 = subst (λ k → k o< & A) &iso ( c<→o< (subst (λ k → odef A k ) (sym &iso ) ab)) m07 : {z : Ordinal} → FClosure A f y z → z <= ZChain.supf zc b @@ -577,11 +586,11 @@ m08 {sup1} {z1} s<b fc = ZChain.order zc m09 s<b fc m05 : b ≡ ZChain.supf zc b m05 = sym (ZChain.sup=u zc ab m09 - record { x<sup = λ lt → IsSup.x<sup is-sup (chain-mono1 x lt )} ) -- ZChain on x + record { x<sup = λ lt → IsSup.x<sup is-sup (chain-mono1 x ? lt )} ) -- ZChain on x m06 : ChainP A f mf ay (ZChain.supf zc) b m06 = record { fcy<sup = m07 ; order = m08 } m04 : odef (UnionCF A f mf ay (ZChain.supf zc) (osuc b)) b - m04 = ⟪ ab , ch-is-sup b m06 (subst (λ k → FClosure A f k b) m05 (init ab refl)) ⟫ + m04 = ⟪ ab , ch-is-sup b ? m06 (subst (λ k → FClosure A f k b) m05 (init ab refl)) ⟫ --- --- the maximum chain has fix point of any ≤-monotonic function @@ -677,11 +686,11 @@ isy : {z : Ordinal } → odef (UnionCF A f mf ay isupf o∅) z → * y ≤ * z isy {z} ⟪ az , uz ⟫ with uz ... | ch-init fc = s≤fc y f mf fc - ... | ch-is-sup u is-sup fc = ≤-ftrans (subst (λ k → * y ≤ k) (sym *iso) y<sup) (s≤fc (& (SUP.sup (ysup f mf ay))) f mf fc ) + ... | ch-is-sup u u<x is-sup fc = ≤-ftrans (subst (λ k → * y ≤ k) (sym *iso) y<sup) (s≤fc (& (SUP.sup (ysup f mf ay))) f mf fc ) inext : {a : Ordinal} → odef (UnionCF A f mf ay isupf o∅) a → odef (UnionCF A f mf ay isupf o∅) (f a) inext {a} ua with (proj2 ua) ... | ch-init fc = ⟪ proj2 (mf _ (proj1 ua)) , ch-init (fsuc _ fc ) ⟫ - ... | ch-is-sup u is-sup fc = ⟪ proj2 (mf _ (proj1 ua)) , ch-is-sup u is-sup (fsuc _ fc) ⟫ + ... | ch-is-sup u u<x is-sup fc = ⟪ proj2 (mf _ (proj1 ua)) , ch-is-sup u u<x is-sup (fsuc _ fc) ⟫ itotal : IsTotalOrderSet (UnionCF A f mf ay isupf o∅) itotal {a} {b} ca cb = subst₂ (λ j k → Tri (j < k) (j ≡ k) (k < j)) *iso *iso uz01 where uz01 : Tri (* (& a) < * (& b)) (* (& a) ≡ * (& b)) (* (& b) < * (& a) ) @@ -689,7 +698,7 @@ mono : {x : Ordinal} {z : Ordinal} → x o< z → isupf x o≤ isupf z mono {x} {z} x<z = o≤-refl csupf : {z : Ordinal} → z o≤ o∅ → odef (UnionCF A f mf ay isupf z ) (isupf z) - csupf {z} z≤0 = ⟪ asi , ch-is-sup spi uz02 (init asi refl) ⟫ where + csupf {z} z≤0 = ⟪ asi , ch-is-sup spi ? uz02 (init asi refl) ⟫ where uz03 : {z : Ordinal } → FClosure A f y z → (z ≡ isupf spi) ∨ (z << isupf spi) uz03 {z} fc with SUP.x<sup sp (subst (λ k → FClosure A f y k ) (sym &iso) fc ) ... | case1 eq = case1 ( begin @@ -732,11 +741,11 @@ pchain⊆A {y} ny = proj1 ny pnext : {a : Ordinal} → odef pchain a → odef pchain (f a) pnext {a} ⟪ aa , ch-init fc ⟫ = ⟪ proj2 (mf a aa) , ch-init (fsuc _ fc) ⟫ - pnext {a} ⟪ aa , ch-is-sup u is-sup fc ⟫ = ⟪ proj2 (mf a aa) , ch-is-sup u is-sup (fsuc _ fc ) ⟫ + pnext {a} ⟪ aa , ch-is-sup u u<x is-sup fc ⟫ = ⟪ proj2 (mf a aa) , ch-is-sup u ? is-sup (fsuc _ fc ) ⟫ pinit : {y₁ : Ordinal} → odef pchain y₁ → * y ≤ * y₁ pinit {a} ⟪ aa , ua ⟫ with ua ... | ch-init fc = s≤fc y f mf fc - ... | ch-is-sup u is-sup fc = ≤-ftrans (<=to≤ zc7) (s≤fc _ f mf fc) where + ... | ch-is-sup u u<x is-sup fc = ≤-ftrans (<=to≤ zc7) (s≤fc _ f mf fc) where zc7 : y <= (ZChain.supf zc) u zc7 = ChainP.fcy<sup is-sup (init ay refl) pcy : odef pchain y @@ -754,13 +763,14 @@ sup {z} z<x with trio< z px ... | tri< a ¬b ¬c = ZChain.sup zc a ... | tri> ¬a ¬b c = ⊥-elim (¬p<x<op ⟪ c , subst (λ k → z o< k) (sym (Oprev.oprev=x op)) z<x ⟫ ) -- px < z < x - ... | tri≈ ¬a b ¬c = record { sup = * (supf0 px) ; A∋maximal = subst (λ k → odef A k) (sym &iso) (proj1 (ZChain.csupf zc o≤-refl )) - ; x<sup = x<sup } where + ... | tri≈ ¬a b ¬c = record { sup = * (supf0 px) ; A∋maximal = subst (λ k → odef A k ) (sym &iso) (proj1 zc8) ; x<sup = x<sup } where + zc9 : SUP A (UnionCF A f mf ay supf0 x) + zc9 = supP pchain pchain⊆A ptotal zc8 : odef (UnionCF A f mf ay supf0 z) (supf0 px) zc8 = subst (λ k → odef (UnionCF A f mf ay supf0 z) k ) (cong supf0 b) (ZChain.csupf zc (subst (λ k → z o≤ k) b o≤-refl )) - x<sup : {w : HOD} → UnionCF A f mf ay supf0 z ∋ w → (w ≡ * (supf0 px)) ∨ (w < * (supf0 px)) + x<sup : {w : HOD} → UnionCF A f mf ay supf0 z ∋ w → (w ≡ * (supf0 px) ) ∨ (w < * (supf0 px) ) x<sup {w} ⟪ aw , ch-init fc ⟫ = ? - x<sup {w} ⟪ aw , ch-is-sup u is-sup fc ⟫ = ? + x<sup {w} ⟪ aw , ch-is-sup u u<x is-sup fc ⟫ = ? zc4 : ZChain A f mf ay x zc4 with ODC.∋-p O A (* px) @@ -831,7 +841,7 @@ zc12 : odef A (ZChain.supf ozc z) ∧ UChain A f mf ay (ZChain.supf ozc) (osuc z) (ZChain.supf ozc z) zc12 = ? zc11 : odef A (ZChain.supf ozc z) ∧ UChain A f mf ay psupf x (ZChain.supf ozc z) - zc11 = ⟪ az , ch-is-sup z cp1 (subst (λ k → FClosure A f k _) (sym eq1) (init az refl) ) ⟫ where + zc11 = ⟪ az , ch-is-sup z z<x cp1 (subst (λ k → FClosure A f k _) (sym eq1) (init az refl) ) ⟫ where az : odef A ( ZChain.supf ozc z ) az = proj1 zc12 zc20 : {z1 : Ordinal} → FClosure A f y z1 → (z1 ≡ psupf z) ∨ (z1 << psupf z) @@ -841,7 +851,7 @@ cp1 : ChainP A f mf ay psupf z cp1 = record { fcy<sup = zc20 ; order = ? } --- u = supf u = supf z - ... | tri≈ ¬a b ¬c | record { eq = eq1 } = ⟪ sa , ch-is-sup {!!} {!!} {!!} ⟫ where + ... | tri≈ ¬a b ¬c | record { eq = eq1 } = ⟪ sa , ch-is-sup ? {!!} {!!} {!!} ⟫ where sa = SUP.A∋maximal usup ... | tri> ¬a ¬b c | record { eq = eq1 } = {!!} @@ -859,11 +869,11 @@ pchain⊆A {y} ny = proj1 ny pnext : {a : Ordinal} → odef pchain a → odef pchain (f a) pnext {a} ⟪ aa , ch-init fc ⟫ = ⟪ proj2 ( mf a aa ) , ch-init (fsuc _ fc) ⟫ - pnext {a} ⟪ aa , ch-is-sup u is-sup fc ⟫ = ⟪ proj2 ( mf a aa ) , ch-is-sup u is-sup (fsuc _ fc) ⟫ + pnext {a} ⟪ aa , ch-is-sup u u<x is-sup fc ⟫ = ⟪ proj2 ( mf a aa ) , ch-is-sup u ? is-sup (fsuc _ fc) ⟫ pinit : {y₁ : Ordinal} → odef pchain y₁ → * y ≤ * y₁ pinit {a} ⟪ aa , ua ⟫ with ua ... | ch-init fc = s≤fc y f mf fc - ... | ch-is-sup u is-sup fc = ≤-ftrans (<=to≤ zc7) (s≤fc _ f mf fc) where + ... | ch-is-sup u u<x is-sup fc = ≤-ftrans (<=to≤ zc7) (s≤fc _ f mf fc) where zc7 : y <= psupf _ zc7 = ChainP.fcy<sup is-sup (init ay refl) pcy : odef pchain y @@ -879,9 +889,9 @@ * a < * b → odef (UnionCF A f mf ay supf x) b is-max-hp supf x {a} {b} ua b<x ab has-prev a<b with HasPrev.ay has-prev ... | ⟪ ab0 , ch-init fc ⟫ = ⟪ ab , ch-init ( subst (λ k → FClosure A f y k) (sym (HasPrev.x=fy has-prev)) (fsuc _ fc )) ⟫ - ... | ⟪ ab0 , ch-is-sup u is-sup fc ⟫ = ⟪ ab , + ... | ⟪ ab0 , ch-is-sup u u<x is-sup fc ⟫ = ⟪ ab , subst (λ k → UChain A f mf ay supf x k ) - (sym (HasPrev.x=fy has-prev)) ( ch-is-sup u is-sup (fsuc _ fc)) ⟫ + (sym (HasPrev.x=fy has-prev)) ( ch-is-sup u ? is-sup (fsuc _ fc)) ⟫ no-extension : ZChain A f mf ay x no-extension = record { initial = pinit ; chain∋init = pcy ; supf = psupf ; sup=u = {!!}