Mercurial > hg > Members > kono > Proof > ZF-in-agda
changeset 982:6d29911a9d00
... order supf o< supf is bad?
| author | Shinji KONO <kono@ie.u-ryukyu.ac.jp> |
|---|---|
| date | Fri, 11 Nov 2022 19:27:15 +0900 |
| parents | 6b67e43733ca |
| children | 6101b9bfbe57 |
| files | src/zorn.agda |
| diffstat | 1 files changed, 46 insertions(+), 34 deletions(-) [+] |
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--- a/src/zorn.agda Thu Nov 10 09:07:07 2022 +0900 +++ b/src/zorn.agda Fri Nov 11 19:27:15 2022 +0900 @@ -273,7 +273,7 @@ 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 } (u≤x : supf u o≤ supf x) ( 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 -- @@ -289,7 +289,7 @@ 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 u≤x 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 @@ -297,14 +297,14 @@ ... | 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 u≤x 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 u≤x 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 @@ -312,7 +312,7 @@ ... | 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 u≤x 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 @@ -401,8 +401,8 @@ → odef (UnionCF A f mf ay supf a) c → odef (UnionCF A f mf ay supf b) c chain-mono f mf ay supf supf-mono {a} {b} {c} a≤b ⟪ ua , ch-init fc ⟫ = ⟪ ua , ch-init fc ⟫ -chain-mono f mf ay supf supf-mono {a} {b} {c} a≤b ⟪ uaa , ch-is-sup ua ua≤x is-sup fc ⟫ = - ⟪ uaa , ch-is-sup ua (OrdTrans ua≤x (supf-mono a≤b) ) is-sup fc ⟫ +chain-mono f mf ay supf supf-mono {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 ⟫ record ZChain ( A : HOD ) ( f : Ordinal → Ordinal ) (mf : ≤-monotonic-f A f) {y : Ordinal} (ay : odef A y) ( z : Ordinal ) : Set (Level.suc n) where @@ -435,11 +435,11 @@ chain∋init = ⟪ ay , ch-init (init ay refl) ⟫ f-next : {a z : Ordinal} → odef (UnionCF A f mf ay supf z) a → odef (UnionCF A f mf ay supf z) (f a) f-next {a} ⟪ aa , ch-init fc ⟫ = ⟪ proj2 (mf a aa) , ch-init (fsuc _ fc) ⟫ - f-next {a} ⟪ aa , ch-is-sup u u≤x is-sup fc ⟫ = ⟪ proj2 (mf a aa) , ch-is-sup u u≤x is-sup (fsuc _ fc ) ⟫ + f-next {a} ⟪ aa , ch-is-sup u u<x is-sup fc ⟫ = ⟪ proj2 (mf a aa) , ch-is-sup u u<x is-sup (fsuc _ fc ) ⟫ initial : {z : Ordinal } → odef chain z → * y ≤ * z initial {a} ⟪ aa , ua ⟫ with ua ... | ch-init fc = s≤fc y f mf fc - ... | ch-is-sup u u≤x 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 <= supf u zc7 = ChainP.fcy<sup is-sup (init ay refl) f-total : IsTotalOrderSet chain @@ -617,8 +617,8 @@ → * a < * b → odef (UnionCF A f mf ay (ZChain.supf zc) x) b is-max-hp x {a} {b} ua 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 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 u≤x is-sup (fsuc _ fc)) ⟫ + ... | ⟪ 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 u<x is-sup (fsuc _ fc)) ⟫ supf = ZChain.supf zc @@ -633,7 +633,7 @@ zc05 : odef (UnionCF A f mf ay supf b) (supf s) zc05 with zc04 ... | ⟪ as , ch-init fc ⟫ = ⟪ as , ch-init fc ⟫ - ... | ⟪ as , ch-is-sup u u≤x is-sup fc ⟫ = ⟪ as , ch-is-sup u (o<→≤ zc08) is-sup fc ⟫ where + ... | ⟪ as , ch-is-sup u u<x is-sup fc ⟫ = ⟪ as , ch-is-sup u (ZChain.supf-inject zc zc08) is-sup fc ⟫ where zc07 : FClosure A f (supf u) (supf s) -- supf u ≤ supf s → supf u o≤ supf s zc07 = fc zc06 : supf u ≡ u @@ -644,7 +644,7 @@ 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 (csupf-fc b<z ss≤sb fc ) ... | ⟪ as , ch-init fc ⟫ = ⟪ proj2 (mf _ as) , ch-init (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) ⟫ + ... | ⟪ as , ch-is-sup u u<x is-sup fc ⟫ = ⟪ proj2 (mf _ as) , ch-is-sup u u<x is-sup (fsuc _ fc) ⟫ order : {b s z1 : Ordinal} → b o< & A → supf s o< supf b → FClosure A f (supf s) z1 → (z1 ≡ supf b) ∨ (z1 << supf b) order {b} {s} {z1} b<z ss<sb fc = zc04 where zc00 : ( z1 ≡ MinSUP.sup (ZChain.minsup zc (o<→≤ b<z) )) ∨ ( z1 << MinSUP.sup ( ZChain.minsup zc (o<→≤ b<z) ) ) @@ -668,7 +668,7 @@ is-max {a} {b} ua b<x ab P a<b with ODC.or-exclude O P is-max {a} {b} ua b<x ab P a<b | case1 has-prev = is-max-hp x {a} {b} ua ab has-prev a<b is-max {a} {b} ua sb<sx ab P a<b | case2 is-sup - = ⟪ ab , ch-is-sup b (ZChain.supf-mono zc (o<→≤ b<x)) m06 (subst (λ k → FClosure A f k b) (sym m05) (init ab refl)) ⟫ where + = ⟪ ab , ch-is-sup b b<x m06 (subst (λ k → FClosure A f k b) (sym m05) (init ab refl)) ⟫ where b<A : b o< & A b<A = z09 ab b<x : b o< x @@ -694,7 +694,7 @@ is-max {a} {b} ua sb<sx ab P a<b | case1 has-prev = is-max-hp x {a} {b} ua ab has-prev a<b is-max {a} {b} ua sb<sx ab P a<b | case2 is-sup with IsSUP.x≤sup (proj2 is-sup) (init-uchain A f mf ay ) ... | case1 b=y = ⟪ subst (λ k → odef A k ) b=y ay , ch-init (subst (λ k → FClosure A f y k ) b=y (init ay refl )) ⟫ - ... | case2 y<b = ⟪ ab , ch-is-sup b (ZChain.supf-mono zc (o<→≤ b<x)) m06 (subst (λ k → FClosure A f k b) (sym m05) (init ab refl)) ⟫ where + ... | case2 y<b = ⟪ ab , ch-is-sup b b<x m06 (subst (λ k → FClosure A f k b) (sym m05) (init ab refl)) ⟫ where m09 : b o< & A m09 = subst (λ k → k o< & A) &iso ( c<→o< (subst (λ k → odef A k ) (sym &iso ) ab)) b<x : b o< x @@ -1051,7 +1051,7 @@ zc13 = o≤-refl0 ( trans (Oprev.oprev=x op) (sym eq ) ) x≤sup : {w : Ordinal} → odef (UnionCF A f mf ay supf0 px) w → (w ≡ x) ∨ (w << x) x≤sup {w} ⟪ az , ch-init {w} fc ⟫ = subst (λ k → (w ≡ k) ∨ (w << k)) s1u=x ? - x≤sup {w} ⟪ az , ch-is-sup u u≤x is-sup fc ⟫ with osuc-≡< ( supf-mono (ordtrans (o<→≤ u≤x) ? )) + x≤sup {w} ⟪ az , ch-is-sup u u<x is-sup fc ⟫ with osuc-≡< ( supf-mono (ordtrans (o<→≤ u<x) ? )) ... | case1 eq1 = ⊥-elim ( o<¬≡ zc14 ? ) where zc14 : u ≡ osuc px zc14 = begin @@ -1175,9 +1175,9 @@ * 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 u≤x 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 u≤x is-sup (fsuc _ fc)) ⟫ + -- (sym (HasPrev.x=fy has-prev)) ( ch-is-sup u u<x is-sup (fsuc _ fc)) ⟫ zc70 : HasPrev A pchain f x → ¬ xSUP pchain f x zc70 pr xsup = ? @@ -1191,23 +1191,23 @@ pchain0=1 = ==→o≡ record { eq→ = zc10 ; eq← = zc11 } where zc10 : {z : Ordinal} → OD.def (od pchain) z → OD.def (od pchain1) z zc10 {z} ⟪ az , ch-init fc ⟫ = ⟪ az , ch-init fc ⟫ - zc10 {z} ⟪ az , ch-is-sup u u≤x is-sup fc ⟫ = zc12 fc where + zc10 {z} ⟪ az , ch-is-sup u u<x is-sup fc ⟫ = zc12 fc where zc12 : {z : Ordinal} → FClosure A f (supf0 u) z → odef pchain1 z zc12 (fsuc x fc) with zc12 fc ... | ⟪ ua1 , ch-init fc₁ ⟫ = ⟪ proj2 ( mf _ ua1) , ch-init (fsuc _ fc₁) ⟫ - ... | ⟪ ua1 , ch-is-sup u u≤x is-sup fc₁ ⟫ = ⟪ proj2 ( mf _ ua1) , ch-is-sup u u≤x is-sup (fsuc _ fc₁) ⟫ + ... | ⟪ ua1 , ch-is-sup u u<x is-sup fc₁ ⟫ = ⟪ proj2 ( mf _ ua1) , ch-is-sup u u<x is-sup (fsuc _ fc₁) ⟫ zc12 (init asu su=z ) = ⟪ subst (λ k → odef A k) su=z asu , ch-is-sup u ? ? (init ? ? ) ⟫ zc11 : {z : Ordinal} → OD.def (od pchain1) z → OD.def (od pchain) z zc11 {z} ⟪ az , ch-init fc ⟫ = ⟪ az , ch-init fc ⟫ - zc11 {z} ⟪ az , ch-is-sup u u≤x is-sup fc ⟫ = zc13 fc where + zc11 {z} ⟪ az , ch-is-sup u u<x is-sup fc ⟫ = zc13 fc where zc13 : {z : Ordinal} → FClosure A f (supf1 u) z → odef pchain z zc13 (fsuc x fc) with zc13 fc ... | ⟪ ua1 , ch-init fc₁ ⟫ = ⟪ proj2 ( mf _ ua1) , ch-init (fsuc _ fc₁) ⟫ - ... | ⟪ ua1 , ch-is-sup u u≤x is-sup fc₁ ⟫ = ⟪ proj2 ( mf _ ua1) , ch-is-sup u u≤x is-sup (fsuc _ fc₁) ⟫ + ... | ⟪ ua1 , ch-is-sup u u<x is-sup fc₁ ⟫ = ⟪ proj2 ( mf _ ua1) , ch-is-sup u u<x is-sup (fsuc _ fc₁) ⟫ zc13 (init asu su=z ) with trio< u x ... | tri< a ¬b ¬c = ⟪ ? , ch-is-sup u ? ? (init ? ? ) ⟫ ... | tri≈ ¬a b ¬c = ? - ... | tri> ¬a ¬b c = ? -- ⊥-elim ( o≤> u≤x c ) + ... | tri> ¬a ¬b c = ? -- ⊥-elim ( o≤> u<x c ) sup : {z : Ordinal} → z o≤ x → SUP A (UnionCF A f mf ay supf1 z) sup {z} z≤x with trio< z x ... | tri< a ¬b ¬c = SUP⊆ ? (ZChain.sup (pzc (osuc z) {!!}) {!!} ) @@ -1396,6 +1396,18 @@ sz<<c : {z : Ordinal } → z o< & A → supf z <= mc sz<<c z<A = MinSUP.x≤sup msp1 (ZChain.csupf zc (z09 (ZChain.asupf zc) )) + z511 : {u y mc : Ordinal} (u<x : u o< mc ) (is-sup1 : ChainP A (cf nmx) (cf-is-≤-monotonic nmx) as0 (ZChain.supf zc) u) + (fc : FClosure A (cf nmx) (ZChain.supf zc u) y) (x=fy : mc ≡ cf nmx y) → + ( * (cf nmx (cf nmx y)) ≡ * (supf mc)) ∨ (* (cf nmx (cf nmx y)) < * (supf mc) ) + z511 {u} {y} {mc} u<x is-sup fc x=fy with osuc-≡< (ZChain.supf-mono zc (o<→≤ u<x)) + ... | case2 lt = <=to≤ (ZChain1.order (SZ1 (cf nmx) (cf-is-≤-monotonic nmx) as0 zc (& A)) ? lt (fsuc _ ( fsuc _ fc ))) + ... | case1 eq = ? + -- u ≡ supf u ≡ supf mc ≡ supf (cf nmx y) + -- supf u << cf nmx (cf nmx ( ... (supf u )) <= spuf mc ≡ u + -- y ≡ supf u + -- y ≡ cf nmx (supf u) + -- y ≡ cf nmx (cf nmx (supf u)) + sc=c : supf mc ≡ mc sc=c = ZChain.sup=u zc (MinSUP.asm msp1) (o<→≤ (∈∧P→o< ⟪ MinSUP.asm msp1 , lift true ⟫ )) ⟪ is-sup , not-hasprev ⟫ where not-hasprev : ¬ HasPrev A (UnionCF A (cf nmx) (cf-is-≤-monotonic nmx) as0 (ZChain.supf zc) mc) (cf nmx) mc @@ -1405,7 +1417,7 @@ z31 : ( * (cf nmx mc) ≡ * mc ) ∨ ( * (cf nmx mc) < * mc ) z31 = <=to≤ ( MinSUP.x≤sup msp1 (subst (λ k → odef (ZChain.chain zc) (cf nmx k)) (sym x=fy) ⟪ proj2 (cf-is-≤-monotonic nmx _ (proj2 (cf-is-≤-monotonic nmx _ ua1 ) )) , ch-init (fsuc _ (fsuc _ fc)) ⟫ )) - not-hasprev record { ax = ax ; y = y ; ay = ⟪ ua1 , ch-is-sup u u≤x is-sup1 fc ⟫; x=fy = x=fy } = ⊥-elim ( <-irr z48 z32 ) where + not-hasprev record { ax = ax ; y = y ; ay = ⟪ ua1 , ch-is-sup u u<x is-sup1 fc ⟫; x=fy = x=fy } = ⊥-elim ( <-irr z48 z32 ) where z30 : * mc < * (cf nmx mc) z30 = proj1 (cf-is-<-monotonic nmx mc (MinSUP.asm msp1)) z31 : ( supf mc ≡ mc ) ∨ ( * (supf mc) < * mc ) @@ -1413,7 +1425,7 @@ z32 : * (supf mc) < * (cf nmx (cf nmx y)) z32 = ftrans<=-< z31 (subst (λ k → * mc < * k ) (cong (cf nmx) x=fy) z30 ) z48 : ( * (cf nmx (cf nmx y)) ≡ * (supf mc)) ∨ (* (cf nmx (cf nmx y)) < * (supf mc) ) - z48 = <=to≤ (ZChain1.order (SZ1 (cf nmx) (cf-is-≤-monotonic nmx) as0 zc (& A)) mc<A ? (fsuc _ ( fsuc _ fc ))) + z48 = z511 u<x is-sup1 fc x=fy is-sup : IsSUP A (UnionCF A (cf nmx) (cf-is-≤-monotonic nmx) as0 (ZChain.supf zc) mc) (MinSUP.asm msp1) is-sup = record { x≤sup = λ zy → MinSUP.x≤sup msp1 (chain-mono (cf nmx) (cf-is-≤-monotonic nmx) as0 supf (ZChain.supf-mono zc) (o<→≤ mc<A) zy ) } @@ -1426,16 +1438,17 @@ z32 = ZChain.fcy<sup zc (o<→≤ mc<A) (fsuc _ (fsuc _ fc)) z31 : ( * (cf nmx d) ≡ * d ) ∨ ( * (cf nmx d) < * d ) z31 = case2 ( subst (λ k → * (cf nmx k) < * d ) (sym x=fy) ( ftrans<=-< z32 ( sc<<d {mc} asc spd ) )) - not-hasprev record { ax = ax ; y = y ; ay = ⟪ ua1 , ch-is-sup u u≤x is-sup1 fc ⟫; x=fy = x=fy } = ⊥-elim ( <-irr z46 z30 ) where + not-hasprev record { ax = ax ; y = y ; ay = ⟪ ua1 , ch-is-sup u u<x is-sup1 fc ⟫; x=fy = x=fy } = ⊥-elim ( <-irr z46 z30 ) where z45 : (* (cf nmx (cf nmx y)) ≡ * d) ∨ (* (cf nmx (cf nmx y)) < * d) → (* (cf nmx d) ≡ * d) ∨ (* (cf nmx d) < * d) z45 p = subst (λ k → (* (cf nmx k) ≡ * d) ∨ (* (cf nmx k) < * d)) (sym x=fy) p z48 : supf mc << d z48 = sc<<d {mc} asc spd z53 : supf u o≤ supf (& A) - z53 = OrdTrans u≤x ? + z53 = ZChain.supf-mono zc (o<→≤ (z09 (subst (λ k → odef A k) (ChainP.supu=u is-sup1) (ZChain.asupf zc) ))) -- OrdTrans u<x ? z52 : ( u ≡ mc ) ∨ ( u << mc ) z52 = MinSUP.x≤sup msp1 ⟪ subst (λ k → odef A k ) (ChainP.supu=u is-sup1) (A∋fcs _ _ (cf-is-≤-monotonic nmx) fc) - , ch-is-sup u ? is-sup1 (init (ZChain.asupf zc) (ChainP.supu=u is-sup1)) ⟫ + , ch-is-sup u (z09 (subst (λ k → odef A k) (ChainP.supu=u is-sup1) (ZChain.asupf zc) )) + is-sup1 (init (ZChain.asupf zc) (ChainP.supu=u is-sup1)) ⟫ z51 : supf u o≤ supf mc z51 = or z52 (λ le → o≤-refl0 (z56 le) ) z57 where z56 : u ≡ mc → supf u ≡ supf mc @@ -1447,7 +1460,7 @@ z49 : supf u o< supf mc → (cf nmx (cf nmx y) ≡ supf mc) ∨ (* (cf nmx (cf nmx y)) < * (supf mc) ) z49 su<smc = ZChain1.order (SZ1 (cf nmx) (cf-is-≤-monotonic nmx) as0 zc (& A)) mc<A su<smc (fsuc _ ( fsuc _ fc )) z50 : (cf nmx (cf nmx y) ≡ supf d) ∨ (* (cf nmx (cf nmx y)) < * (supf d) ) - z50 = ZChain1.order (SZ1 (cf nmx) (cf-is-≤-monotonic nmx) as0 zc (& A)) d<A ? (fsuc _ ( fsuc _ fc )) + z50 = ≤to<= ( z511 u<x is-sup1 fc x=fy ) z47 : {mc d1 : Ordinal } {asc : odef A (supf mc)} (spd : MinSUP A (uchain (cf nmx) (cf-is-≤-monotonic nmx) asc )) → (cf nmx (cf nmx y) ≡ supf mc) ∨ (* (cf nmx (cf nmx y)) < * (supf mc) ) → supf mc << d1 → * (cf nmx (cf nmx y)) < * d1 @@ -1474,13 +1487,12 @@ z22 {a} ⟪ aa , ch-init fc ⟫ = case2 ( ( ftrans<=-< z32 ( sc<<d {mc} asc spd ) )) where z32 : ( a ≡ supf mc ) ∨ ( * a < * (supf mc) ) z32 = ZChain.fcy<sup zc (o<→≤ mc<A) fc - z22 {a} ⟪ aa , ch-is-sup u u≤x is-sup1 fc ⟫ = tri u (supf mc) + z22 {a} ⟪ aa , ch-is-sup u u<x is-sup1 fc ⟫ = tri u (supf mc) z60 z61 ( λ sc<u → ⊥-elim ( o≤> ( subst (λ k → k o≤ supf mc) (ChainP.supu=u is-sup1) z51) sc<u )) where - z53 : supf u o< supf (& A) - z53 = ordtrans<-≤ u≤x ? z52 : ( u ≡ mc ) ∨ ( u << mc ) z52 = MinSUP.x≤sup msp1 ⟪ subst (λ k → odef A k ) (ChainP.supu=u is-sup1) (A∋fcs _ _ (cf-is-≤-monotonic nmx) fc) - , ch-is-sup u ? is-sup1 (init (ZChain.asupf zc) (ChainP.supu=u is-sup1)) ⟫ + , ch-is-sup u (z09 (subst (λ k → odef A k) (ChainP.supu=u is-sup1) (ZChain.asupf zc) )) + is-sup1 (init (ZChain.asupf zc) (ChainP.supu=u is-sup1)) ⟫ z56 : u ≡ mc → supf u ≡ supf mc z56 eq = cong supf eq z57 : u << mc → supf u o≤ supf mc @@ -1494,7 +1506,7 @@ (subst (λ k → k o< supf mc) (sym (ChainP.supu=u is-sup1)) u<smc) fc ) smc<<d ) z61 : u ≡ supf mc → (a ≡ d ) ∨ ( * a < * d ) z61 u=sc = case2 (fsc<<d {mc} asc spd (subst (λ k → FClosure A (cf nmx) k a) (trans (ChainP.supu=u is-sup1) u=sc) fc ) ) - -- u≤x : ZChain.supf zc u o< ZChain.supf zc d + -- u<x : ZChain.supf zc u o< ZChain.supf zc d -- supf u o< supf c → order sd=d : supf d ≡ d