Mercurial > hg > Members > kono > Proof > ZF-in-agda
changeset 966:39c680188738
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| author | Shinji KONO <kono@ie.u-ryukyu.ac.jp> |
|---|---|
| date | Sat, 05 Nov 2022 13:21:42 +0900 |
| parents | 1c1c6a6ed4fa |
| children | cd0ef83189c5 |
| files | src/zorn.agda |
| diffstat | 1 files changed, 381 insertions(+), 278 deletions(-) [+] |
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--- a/src/zorn.agda Sat Nov 05 10:23:57 2022 +0900 +++ b/src/zorn.agda Sat Nov 05 13:21:42 2022 +0900 @@ -1,30 +1,30 @@ {-# OPTIONS --allow-unsolved-metas #-} open import Level hiding ( suc ; zero ) open import Ordinals -open import Relation.Binary +open import Relation.Binary open import Relation.Binary.Core open import Relation.Binary.PropositionalEquality -import OD +import OD module zorn {n : Level } (O : Ordinals {n}) (_<_ : (x y : OD.HOD O ) → Set n ) (PO : IsStrictPartialOrder _≡_ _<_ ) where -- --- Zorn-lemma : { A : HOD } --- → o∅ o< & A +-- Zorn-lemma : { A : HOD } +-- → o∅ o< & A -- → ( ( B : HOD) → (B⊆A : B ⊆ A) → IsTotalOrderSet B → SUP A B ) -- SUP condition --- → Maximal A +-- → Maximal A -- open import zf open import logic -- open import partfunc {n} O -open import Relation.Nullary -open import Data.Empty -import BAlgbra +open import Relation.Nullary +open import Data.Empty +import BAlgbra open import Data.Nat hiding ( _<_ ; _≤_ ) -open import Data.Nat.Properties -open import nat +open import Data.Nat.Properties +open import nat open inOrdinal O @@ -52,42 +52,42 @@ -- Partial Order on HOD ( possibly limited in A ) -- -_<<_ : (x y : Ordinal ) → Set n +_<<_ : (x y : Ordinal ) → Set n x << y = * x < * y _<=_ : (x y : Ordinal ) → Set n -- we can't use * x ≡ * y, it is Set (Level.suc n). Level (suc n) troubles Chain x <= y = (x ≡ y ) ∨ ( * x < * y ) -POO : IsStrictPartialOrder _≡_ _<<_ -POO = record { isEquivalence = record { refl = refl ; sym = sym ; trans = trans } - ; trans = IsStrictPartialOrder.trans PO +POO : IsStrictPartialOrder _≡_ _<<_ +POO = record { isEquivalence = record { refl = refl ; sym = sym ; trans = trans } + ; trans = IsStrictPartialOrder.trans PO ; irrefl = λ x=y x<y → IsStrictPartialOrder.irrefl PO (cong (*) x=y) x<y - ; <-resp-≈ = record { fst = λ {x} {y} {y1} y=y1 xy1 → subst (λ k → x << k ) y=y1 xy1 ; snd = λ {x} {x1} {y} x=x1 x1y → subst (λ k → k << x ) x=x1 x1y } } - + ; <-resp-≈ = record { fst = λ {x} {y} {y1} y=y1 xy1 → subst (λ k → x << k ) y=y1 xy1 ; snd = λ {x} {x1} {y} x=x1 x1y → subst (λ k → k << x ) x=x1 x1y } } + _≤_ : (x y : HOD) → Set (Level.suc n) x ≤ y = ( x ≡ y ) ∨ ( x < y ) -≤-ftrans : {x y z : HOD} → x ≤ y → y ≤ z → x ≤ z +≤-ftrans : {x y z : HOD} → x ≤ y → y ≤ z → x ≤ z ≤-ftrans {x} {y} {z} (case1 refl ) (case1 refl ) = case1 refl ≤-ftrans {x} {y} {z} (case1 refl ) (case2 y<z) = case2 y<z ≤-ftrans {x} {_} {z} (case2 x<y ) (case1 refl ) = case2 x<y ≤-ftrans {x} {y} {z} (case2 x<y) (case2 y<z) = case2 ( IsStrictPartialOrder.trans PO x<y y<z ) -<=-trans : {x y z : Ordinal } → x <= y → y <= z → x <= z +<=-trans : {x y z : Ordinal } → x <= y → y <= z → x <= z <=-trans {x} {y} {z} (case1 refl ) (case1 refl ) = case1 refl <=-trans {x} {y} {z} (case1 refl ) (case2 y<z) = case2 y<z <=-trans {x} {_} {z} (case2 x<y ) (case1 refl ) = case2 x<y <=-trans {x} {y} {z} (case2 x<y) (case2 y<z) = case2 ( IsStrictPartialOrder.trans PO x<y y<z ) -ftrans<=-< : {x y z : Ordinal } → x <= y → y << z → x << z +ftrans<=-< : {x y z : Ordinal } → x <= y → y << z → x << z ftrans<=-< {x} {y} {z} (case1 eq) y<z = subst (λ k → k < * z) (sym (cong (*) eq)) y<z -ftrans<=-< {x} {y} {z} (case2 lt) y<z = IsStrictPartialOrder.trans PO lt y<z +ftrans<=-< {x} {y} {z} (case2 lt) y<z = IsStrictPartialOrder.trans PO lt y<z -<=to≤ : {x y : Ordinal } → x <= y → * x ≤ * y +<=to≤ : {x y : Ordinal } → x <= y → * x ≤ * y <=to≤ (case1 eq) = case1 (cong (*) eq) <=to≤ (case2 lt) = case2 lt -≤to<= : {x y : Ordinal } → * x ≤ * y → x <= y +≤to<= : {x y : Ordinal } → * x ≤ * y → x <= y ≤to<= (case1 eq) = case1 ( subst₂ (λ j k → j ≡ k ) &iso &iso (cong (&) eq) ) ≤to<= (case2 lt) = case2 lt @@ -101,7 +101,7 @@ open _==_ open _⊆_ --- <-TransFinite : {A x : HOD} → {P : HOD → Set n} → x ∈ A +-- <-TransFinite : {A x : HOD} → {P : HOD → Set n} → x ∈ A -- → ({x : HOD} → A ∋ x → ({y : HOD} → A ∋ y → y < x → P y ) → P x) → P x -- <-TransFinite = ? @@ -122,13 +122,13 @@ A∋fcs : {A : HOD} (s : Ordinal) {y : Ordinal } (f : Ordinal → Ordinal) (mf : ≤-monotonic-f A f) → (fcy : FClosure A f s y ) → odef A s A∋fcs {A} s f mf (init as refl) = as -A∋fcs {A} s f mf (fsuc y fcy) = A∋fcs {A} s f mf fcy +A∋fcs {A} s f mf (fsuc y fcy) = A∋fcs {A} s f mf fcy s≤fc : {A : HOD} (s : Ordinal ) {y : Ordinal } (f : Ordinal → Ordinal) (mf : ≤-monotonic-f A f) → (fcy : FClosure A f s y ) → * s ≤ * y s≤fc {A} s {.s} f mf (init x refl ) = case1 refl s≤fc {A} s {.(f x)} f mf (fsuc x fcy) with proj1 (mf x (A∋fc s f mf fcy ) ) ... | case1 x=fx = subst (λ k → * s ≤ * k ) (*≡*→≡ x=fx) ( s≤fc {A} s f mf fcy ) -... | case2 x<fx with s≤fc {A} s f mf fcy +... | case2 x<fx with s≤fc {A} s f mf fcy ... | case1 s≡x = case2 ( subst₂ (λ j k → j < k ) (sym s≡x) refl x<fx ) ... | case2 s<x = case2 ( IsStrictPartialOrder.trans PO s<x x<fx ) @@ -138,12 +138,12 @@ ... | case1 eq = fcn s mf p ... | case2 y<fy = suc (fcn s mf p ) -fcn-inject : {A : HOD} (s : Ordinal) { x y : Ordinal} {f : Ordinal → Ordinal} → (mf : ≤-monotonic-f A f) +fcn-inject : {A : HOD} (s : Ordinal) { x y : Ordinal} {f : Ordinal → Ordinal} → (mf : ≤-monotonic-f A f) → (cx : FClosure A f s x ) (cy : FClosure A f s y ) → fcn s mf cx ≡ fcn s mf cy → * x ≡ * y 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 fc06 : {y : Ordinal } { as : odef A s } (eq : s ≡ y ) { j : ℕ } → ¬ suc j ≡ fcn {A} s {y} {f} mf (init as eq ) fc06 {x} {y} refl {j} not = fc08 not where - fc08 : {j : ℕ} → ¬ suc j ≡ 0 + fc08 : {j : ℕ} → ¬ suc j ≡ 0 fc08 () fc07 : {x : Ordinal } (cx : FClosure A f s x ) → 0 ≡ fcn s mf cx → * s ≡ * x fc07 {x} (init as refl) eq = refl @@ -174,7 +174,7 @@ fc03 : (y1 : Ordinal) → (cy1 : FClosure A f s y1 ) → suc j ≡ fcn s mf cy1 → * (f x) ≡ * y1 fc03 y1 (init x₁ x₂) x = ⊥-elim (fc06 x₂ x) fc03 .(f y1) (fsuc y1 cy1) j=y1 with proj1 (mf y1 (A∋fc s f mf cy1 ) ) - ... | case1 eq = trans ( fc03 y1 cy1 j=y1 ) eq + ... | case1 eq = trans ( fc03 y1 cy1 j=y1 ) eq ... | case2 lt = subst₂ (λ j k → * (f j) ≡ * (f k )) &iso &iso ( cong (λ k → * ( f (& k ))) fc05) where fc05 : * x ≡ * y1 fc05 = fc00 i j (cong pred i=j) cx cy1 (cong pred i=x) (cong pred j=y1) @@ -186,23 +186,23 @@ fcn-< {A} s {x} {y} {f} mf cx cy x<y = fc01 (fcn s mf cy) cx cy refl x<y where fc06 : {y : Ordinal } { as : odef A s } (eq : s ≡ y ) { j : ℕ } → ¬ suc j ≡ fcn {A} s {y} {f} mf (init as eq ) fc06 {x} {y} refl {j} not = fc08 not where - fc08 : {j : ℕ} → ¬ suc j ≡ 0 + fc08 : {j : ℕ} → ¬ suc j ≡ 0 fc08 () 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 fc01 (suc i) cx (init x₁ x₂) x (s≤s x₃) = ⊥-elim (fc06 x₂ x) fc01 (suc i) {y} cx (fsuc y1 cy) i=y (s≤s x<i) with proj1 (mf y1 (A∋fc s f mf cy ) ) - ... | case1 y=fy = subst (λ k → * x < k ) y=fy ( fc01 (suc i) {y1} cx cy i=y (s≤s x<i) ) + ... | case1 y=fy = subst (λ k → * x < k ) y=fy ( fc01 (suc i) {y1} cx cy i=y (s≤s x<i) ) ... | case2 y<fy with <-cmp (fcn s mf cx ) i ... | tri> ¬a ¬b c = ⊥-elim ( nat-≤> x<i c ) - ... | tri≈ ¬a b ¬c = subst (λ k → k < * (f y1) ) (fcn-inject s mf cy cx (sym (trans b (cong pred i=y) ))) y<fy + ... | tri≈ ¬a b ¬c = subst (λ k → k < * (f y1) ) (fcn-inject s mf cy cx (sym (trans b (cong pred i=y) ))) y<fy ... | tri< a ¬b ¬c = IsStrictPartialOrder.trans PO fc02 y<fy where fc03 : suc i ≡ suc (fcn s mf cy) → i ≡ fcn s mf cy - fc03 eq = cong pred eq - fc02 : * x < * y1 + fc03 eq = cong pred eq + fc02 : * x < * y1 fc02 = fc01 i cx cy (fc03 i=y ) a -fcn-cmp : {A : HOD} (s : Ordinal) { x y : Ordinal } (f : Ordinal → Ordinal) (mf : ≤-monotonic-f A f) +fcn-cmp : {A : HOD} (s : Ordinal) { x y : Ordinal } (f : Ordinal → Ordinal) (mf : ≤-monotonic-f A f) → (cx : FClosure A f s x) → (cy : FClosure A f s y ) → Tri (* x < * y) (* x ≡ * y) (* y < * x ) fcn-cmp {A} s {x} {y} f mf cx cy with <-cmp ( fcn s mf cx ) (fcn s mf cy ) ... | tri< a ¬b ¬c = tri< fc11 (λ eq → <-irr (case1 (sym eq)) fc11) (λ lt → <-irr (case2 fc11) lt) where @@ -211,7 +211,7 @@ ... | tri≈ ¬a b ¬c = tri≈ (λ lt → <-irr (case1 (sym fc10)) lt) fc10 (λ lt → <-irr (case1 fc10) lt) where fc10 : * x ≡ * y fc10 = fcn-inject {A} s {x} {y} {f} mf cx cy b -... | tri> ¬a ¬b c = tri> (λ lt → <-irr (case2 fc12) lt) (λ eq → <-irr (case1 eq) fc12) fc12 where +... | tri> ¬a ¬b c = tri> (λ lt → <-irr (case2 fc12) lt) (λ eq → <-irr (case1 eq) fc12) fc12 where fc12 : * y < * x fc12 = fcn-< {A} s {y} {x} {f} mf cy cx c @@ -238,7 +238,7 @@ ax : odef A x y : Ordinal ay : odef B y - x=fy : x ≡ f y + x=fy : x ≡ f y record IsSUP (A B : HOD) {x : Ordinal } (xa : odef A x) : Set n where field @@ -247,7 +247,7 @@ record IsMinSUP (A B : HOD) ( f : Ordinal → Ordinal ) {x : Ordinal } (xa : odef A x) : Set n where field x≤sup : {y : Ordinal} → odef B y → (y ≡ x ) ∨ (y << x ) - minsup : { sup1 : Ordinal } → odef A sup1 + minsup : { sup1 : Ordinal } → odef A sup1 → ( {z : Ordinal } → odef B z → (z ≡ sup1 ) ∨ (z << sup1 )) → x o≤ sup1 not-hp : ¬ ( HasPrev A B f x ) @@ -261,18 +261,19 @@ -- sup and its fclosure is in a chain HOD -- chain HOD is sorted by sup as Ordinal and <-ordered -- whole chain is a union of separated Chain --- minimum index is sup of y not ϕ +-- minimum index is sup of y not ϕ -- record ChainP (A : HOD ) ( f : Ordinal → Ordinal ) (mf : ≤-monotonic-f A f) {y : Ordinal} (ay : odef A y) (supf : Ordinal → Ordinal) (u : Ordinal) : Set n where field - fcy<sup : {z : Ordinal } → FClosure A f y z → (z ≡ supf u) ∨ ( z << supf u ) + fcy<sup : {z : Ordinal } → FClosure A f y z → (z ≡ supf u) ∨ ( z << supf u ) order : {s z1 : Ordinal} → (lt : supf s o< supf u ) → FClosure A f (supf s ) z1 → (z1 ≡ supf u ) ∨ ( z1 << supf u ) supu=u : supf u ≡ u 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-is-sup : (u : Ordinal) {z : Ordinal } (u<x : supf u o< supf x) ( is-sup : ChainP A f mf ay supf u ) + (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 ) ( fc : FClosure A f (supf u) z ) → UChain A f mf ay supf x z -- @@ -287,36 +288,67 @@ {s s1 a b : Ordinal } ( ca : UChain A f mf ay supf s a ) ( cb : UChain A f mf ay supf s1 b ) → Tri (* a < * b) (* a ≡ * b) (* b < * a ) 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 + ... | 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 + ct00 = trans (cong (*) eq) eq1 + ... | 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 + 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 + ... | 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 + ct00 = sym (trans (cong (*) eq) eq1 ) + ... | 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 + 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 ua<x supa fca) (ch-is-sup ub ub<x supb fcb) with trio< ua ub - ... | tri< a₁ ¬b ¬c with ChainP.order supb (subst₂ (λ j k → j o< k ) (sym (ChainP.supu=u supa )) (sym (ChainP.supu=u supb )) a₁) fca - ... | case1 eq with s≤fc (supf ub) f mf fcb + ... | tri< a₁ ¬b ¬c with ChainP.order supb (subst₂ (λ j k → j o< k ) (sym (ChainP.supu=u supa )) (sym (ChainP.supu=u 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 ct00 : * a ≡ * b ct00 = trans (cong (*) eq) eq1 ... | case2 lt = tri< ct02 (λ eq → <-irr (case1 (sym eq)) ct02) (λ lt → <-irr (case2 ct02) lt) where - ct02 : * a < * b + ct02 : * a < * b ct02 = subst (λ k → * k < * b ) (sym eq) lt 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 : * 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 ua<x supa fca) (ch-is-sup ub ub<x 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 ) (cong supf (sym eq)) 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 (subst₂ (λ j k → j o< k ) (sym (ChainP.supu=u supb )) (sym (ChainP.supu=u supa )) c) fcb - ... | case1 eq with s≤fc (supf ua) f mf fca + 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 (subst₂ (λ j k → j o< k ) (sym (ChainP.supu=u supb )) (sym (ChainP.supu=u 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 ct00 = sym (trans (cong (*) eq) eq1) ... | case2 lt = tri> (λ lt → <-irr (case2 ct02) lt) (λ eq → <-irr (case1 eq) ct02) ct02 where - ct02 : * b < * a + ct02 : * b < * a ct02 = subst (λ k → * k < * a ) (sym eq) lt 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 + ct04 : * b < * a ct04 with s≤fc (supf ua) f mf fca ... | case1 eq = subst (λ k → * b < k ) eq ct05 ... | case2 lt = IsStrictPartialOrder.trans POO ct05 lt @@ -326,13 +358,13 @@ -- Union of supf z which o< x -- -UnionCF : ( A : HOD ) ( f : Ordinal → Ordinal ) (mf : ≤-monotonic-f A f) {y : Ordinal } (ay : odef A y ) +UnionCF : ( A : HOD ) ( f : Ordinal → Ordinal ) (mf : ≤-monotonic-f A f) {y : Ordinal } (ay : odef A y ) ( supf : Ordinal → Ordinal ) ( x : Ordinal ) → HOD UnionCF A f mf ay supf x = record { od = record { def = λ z → odef A z ∧ UChain A f mf ay supf x z } ; odmax = & A ; <odmax = λ {y} sy → ∈∧P→o< sy } -supf-inject0 : {x y : Ordinal } {supf : Ordinal → Ordinal } → (supf-mono : {x y : Ordinal } → x o≤ y → supf x o≤ supf y ) - → supf x o< supf y → x o< y +supf-inject0 : {x y : Ordinal } {supf : Ordinal → Ordinal } → (supf-mono : {x y : Ordinal } → x o≤ y → supf x o≤ supf y ) + → supf x o< supf y → x o< y supf-inject0 {x} {y} {supf} supf-mono sx<sy with trio< x y ... | tri< a ¬b ¬c = a ... | tri≈ ¬a refl ¬c = ⊥-elim ( o<¬≡ (cong supf refl) sx<sy ) @@ -344,8 +376,8 @@ field sup : Ordinal asm : odef A sup - x≤sup : {x : Ordinal } → odef B x → (x ≡ sup ) ∨ (x << sup ) - minsup : { sup1 : Ordinal } → odef A sup1 + x≤sup : {x : Ordinal } → odef B x → (x ≡ sup ) ∨ (x << sup ) + minsup : { sup1 : Ordinal } → odef A sup1 → ( {x : Ordinal } → odef B x → (x ≡ sup1 ) ∨ (x << sup1 )) → sup o≤ sup1 z09 : {b : Ordinal } { A : HOD } → odef A b → b o< & A @@ -353,37 +385,38 @@ M→S : { A : HOD } { f : Ordinal → Ordinal } {mf : ≤-monotonic-f A f} {y : Ordinal} {ay : odef A y} { x : Ordinal } → (supf : Ordinal → Ordinal ) - → MinSUP A (UnionCF A f mf ay supf x) - → SUP A (UnionCF A f mf ay supf x) -M→S {A} {f} {mf} {y} {ay} {x} supf ms = record { sup = * (MinSUP.sup ms) + → MinSUP A (UnionCF A f mf ay supf x) + → SUP A (UnionCF A f mf ay supf x) +M→S {A} {f} {mf} {y} {ay} {x} supf ms = record { sup = * (MinSUP.sup ms) ; as = subst (λ k → odef A k) (sym &iso) (MinSUP.asm ms) ; x≤sup = ms00 } where msup = MinSUP.sup ms ms00 : {z : HOD} → UnionCF A f mf ay supf x ∋ z → (z ≡ * msup) ∨ (z < * msup) - ms00 {z} uz with MinSUP.x≤sup ms uz + ms00 {z} uz with MinSUP.x≤sup ms uz ... | case1 eq = case1 (subst (λ k → k ≡ _) *iso ( cong (*) eq)) ... | case2 lt = case2 (subst₂ (λ j k → j < k ) *iso refl lt ) -chain-mono : {A : HOD} ( f : Ordinal → Ordinal ) → (mf : ≤-monotonic-f A f ) {y : Ordinal} (ay : odef A y) (supf : Ordinal → Ordinal ) +chain-mono : {A : HOD} ( f : Ordinal → Ordinal ) → (mf : ≤-monotonic-f A f ) {y : Ordinal} (ay : odef A y) (supf : Ordinal → Ordinal ) (supf-mono : {x y : Ordinal } → x o≤ y → supf x o≤ supf y ) {a b c : Ordinal} → a o≤ b → 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 ⟫ + ⟪ uaa , ch-is-sup ua (ordtrans<-≤ ua<x (supf-mono a≤b ) ) is-sup fc ⟫ -record ZChain ( A : HOD ) ( f : Ordinal → Ordinal ) (mf : ≤-monotonic-f A f) +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 field - supf : Ordinal → Ordinal - sup=u : {b : Ordinal} → (ab : odef A b) → b o≤ z - → IsSUP A (UnionCF A f mf ay supf b) ab ∧ (¬ HasPrev A (UnionCF A f mf ay supf b) f b ) → supf b ≡ b + supf : Ordinal → Ordinal + sup=u : {b : Ordinal} → (ab : odef A b) → b o≤ z + → IsSUP A (UnionCF A f mf ay supf b) ab ∧ (¬ HasPrev A (UnionCF A f mf ay supf b) f b ) → supf b ≡ b asupf : {x : Ordinal } → odef A (supf x) supf-mono : {x y : Ordinal } → x o≤ y → supf x o≤ supf y supf-< : {x y : Ordinal } → supf x o< supf y → supf x << supf y supfmax : {x : Ordinal } → z o< x → supf x ≡ supf z - chain∋init : odef (UnionCF A f mf ay supf z) y - minsup : {x : Ordinal } → x o≤ z → MinSUP A (UnionCF A f mf ay supf x) + minsup : {x : Ordinal } → x o≤ z → MinSUP A (UnionCF A f mf ay supf x) supf-is-minsup : {x : Ordinal } → (x≤z : x o≤ z) → supf x ≡ MinSUP.sup ( minsup x≤z ) csupf : {b : Ordinal } → supf b o< z → odef (UnionCF A f mf ay supf z) (supf b) -- supf z is not an element of this chain @@ -392,23 +425,27 @@ chain⊆A : chain ⊆' A chain⊆A = λ lt → proj1 lt - sup : {x : Ordinal } → x o≤ z → SUP A (UnionCF A f mf ay supf x) - sup {x} x≤z = M→S supf (minsup x≤z) + sup : {x : Ordinal } → x o≤ z → SUP A (UnionCF A f mf ay supf x) + sup {x} x≤z = M→S supf (minsup x≤z) s=ms : {x : Ordinal } → (x≤z : x o≤ z ) → & (SUP.sup (sup x≤z)) ≡ MinSUP.sup (minsup x≤z) s=ms {x} x≤z = &iso + chain∋init : odef chain y + 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 ) ⟫ 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 - zc7 : y <= supf u + zc7 : y <= supf u zc7 = ChainP.fcy<sup is-sup (init ay refl) f-total : IsTotalOrderSet chain - f-total {a} {b} ca cb = subst₂ (λ j k → Tri (j < k) (j ≡ k) (k < j)) *iso *iso uz01 where + f-total {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) ) - uz01 = chain-total A f mf ay supf ( (proj2 ca)) ( (proj2 cb)) + uz01 = chain-total A f mf ay supf ( (proj2 ca)) ( (proj2 cb)) supf-<= : {x y : Ordinal } → supf x <= supf y → supf x o≤ supf y supf-<= {x} {y} (case1 sx=sy) = o≤-refl0 sx=sy @@ -417,7 +454,7 @@ ... | tri≈ ¬a b ¬c = o≤-refl0 b ... | tri> ¬a ¬b c = ⊥-elim (<-irr (case2 sx<sy ) (supf-< c) ) - supf-inject : {x y : Ordinal } → supf x o< supf y → x o< y + supf-inject : {x y : Ordinal } → supf x o< supf y → x o< y supf-inject {x} {y} sx<sy with trio< x y ... | tri< a ¬b ¬c = a ... | tri≈ ¬a refl ¬c = ⊥-elim ( o<¬≡ (cong supf refl) sx<sy ) @@ -425,36 +462,68 @@ ... | case1 eq = ⊥-elim ( o<¬≡ (sym eq) sx<sy ) ... | case2 lt = ⊥-elim ( o<> sx<sy lt ) - fcy<sup : {u w : Ordinal } → u o≤ z → FClosure A f y w → (w ≡ supf u ) ∨ ( w << supf u ) -- different from order because y o< supf - fcy<sup {u} {w} u≤z fc with chain∋init - ... | ⟪ ay1 , ch-is-sup uy uy<x is-sup fcy ⟫ = <=-trans (ChainP.fcy<sup is-sup fc) ? + fcy<sup : {u w : Ordinal } → u o≤ z → FClosure A f y w → (w ≡ supf u ) ∨ ( w << supf u ) -- different from order because y o< supf + fcy<sup {u} {w} u≤z fc with MinSUP.x≤sup (minsup u≤z) ⟪ subst (λ k → odef A k ) (sym &iso) (A∋fc {A} y f mf fc) + , ch-init (subst (λ k → FClosure A f y k) (sym &iso) fc ) ⟫ + ... | case1 eq = case1 (subst (λ k → k ≡ supf u ) &iso (trans eq (sym (supf-is-minsup u≤z ) ) )) + ... | case2 lt = case2 (subst₂ (λ j k → j << k ) &iso (sym (supf-is-minsup u≤z )) lt ) - -- with MinSUP.x≤sup (minsup u≤z) ⟪ subst (λ k → odef A k ) (sym &iso) (A∋fc {A} y f mf fc) - -- , ? ⟫ --ch-init (subst (λ k → FClosure A f y k) (sym &iso) fc ) ⟫ - -- ... | case1 eq = case1 (subst (λ k → k ≡ supf u ) &iso (trans eq (sym (supf-is-minsup u≤z ) ) )) - -- ... | case2 lt = case2 (subst₂ (λ j k → j << k ) &iso (sym (supf-is-minsup u≤z )) lt ) + -- ordering is not proved here but in ZChain1 - -- ordering is not proved here but in ZChain1 - - IsMinSUP→NotHasPrev : {x sp : Ordinal } → odef A sp + IsMinSUP→NotHasPrev : {x sp : Ordinal } → odef A sp → ({y : Ordinal} → odef (UnionCF A f mf ay supf x) y → (y ≡ sp ) ∨ (y << sp )) → ( {a : Ordinal } → a << f a ) → ¬ ( HasPrev A (UnionCF A f mf ay supf x) f sp ) IsMinSUP→NotHasPrev {x} {sp} asp is-sup <-mono-f hp = ⊥-elim (<-irr ( <=to≤ fsp≤sp) sp<fsp ) where sp<fsp : sp << f sp - sp<fsp = <-mono-f - pr = HasPrev.y hp + sp<fsp = <-mono-f + pr = HasPrev.y hp im00 : f (f pr) <= sp im00 = is-sup ( f-next (f-next (HasPrev.ay hp))) fsp≤sp : f sp <= sp fsp≤sp = subst (λ k → f k <= sp ) (sym (HasPrev.x=fy hp)) im00 -record ZChain1 ( A : HOD ) ( f : Ordinal → Ordinal ) (mf : ≤-monotonic-f A f) + UChain⊆ : { supf1 : Ordinal → Ordinal } + → ( { x : Ordinal } → x o< z → supf x ≡ supf1 x) + → ( { x : Ordinal } → z o≤ x → supf z o≤ supf1 x) + → UnionCF A f mf ay supf z ⊆' UnionCF A f mf ay supf1 z + UChain⊆ {supf1} eq<x lex ⟪ az , ch-init fc ⟫ = ⟪ az , ch-init fc ⟫ + UChain⊆ {supf1} eq<x lex ⟪ az , ch-is-sup u {x} u<x is-sup fc ⟫ = ⟪ az , ch-is-sup u u<x1 cp1 fc1 ⟫ where + u<x0 : u o< z + u<x0 = supf-inject u<x + u<x1 : supf1 u o< supf1 z + u<x1 = subst (λ k → k o< supf1 z ) (eq<x u<x0) (ordtrans<-≤ u<x (lex o≤-refl ) ) + fc1 : FClosure A f (supf1 u) x + fc1 = subst (λ k → FClosure A f k x ) (eq<x u<x0) fc + uc01 : {s : Ordinal } → supf1 s o< supf1 u → s o< z + uc01 {s} s<u with trio< s z + ... | tri< a ¬b ¬c = a + ... | tri≈ ¬a b ¬c = ⊥-elim ( o≤> uc02 s<u ) where -- (supf-mono (o<→≤ u<x0)) + uc02 : supf1 u o≤ supf1 s + uc02 = begin + supf1 u <⟨ u<x1 ⟩ + supf1 z ≡⟨ cong supf1 (sym b) ⟩ + supf1 s ∎ where open o≤-Reasoning O + ... | tri> ¬a ¬b c = ⊥-elim ( o≤> uc03 s<u ) where + uc03 : supf1 u o≤ supf1 s + uc03 = begin + supf1 u ≡⟨ sym (eq<x u<x0) ⟩ + supf u <⟨ u<x ⟩ + supf z ≤⟨ lex (o<→≤ c) ⟩ + supf1 s ∎ where open o≤-Reasoning O + cp1 : ChainP A f mf ay supf1 u + cp1 = record { fcy<sup = λ {z} fc → subst (λ k → (z ≡ k) ∨ ( z << k ) ) (eq<x u<x0) (ChainP.fcy<sup is-sup fc ) + ; order = λ {s} {z} s<u fc → subst (λ k → (z ≡ k) ∨ ( z << k ) ) (eq<x u<x0) + (ChainP.order is-sup (subst₂ (λ j k → j o< k ) (sym (eq<x (uc01 s<u) )) (sym (eq<x u<x0)) s<u) + (subst (λ k → FClosure A f k z ) (sym (eq<x (uc01 s<u) )) fc )) + ; supu=u = trans (sym (eq<x u<x0)) (ChainP.supu=u is-sup) } + +record ZChain1 ( A : HOD ) ( f : Ordinal → Ordinal ) (mf : ≤-monotonic-f A f) {y : Ordinal} (ay : odef A y) (zc : ZChain A f mf ay (& A)) ( z : Ordinal ) : Set (Level.suc n) where supf = ZChain.supf zc field - is-max : {a b : Ordinal } → (ca : odef (UnionCF A f mf ay supf z) a ) → supf b o< supf z → (ab : odef A b) - → HasPrev A (UnionCF A f mf ay supf z) f b ∨ IsSUP A (UnionCF A f mf ay supf b) ab + is-max : {a b : Ordinal } → (ca : odef (UnionCF A f mf ay supf z) a ) → supf b o< supf z → (ab : odef A b) + → HasPrev A (UnionCF A f mf ay supf z) f b ∨ IsSUP A (UnionCF A f mf ay supf b) ab → * a < * b → odef ((UnionCF A f mf ay supf z)) b 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) @@ -464,17 +533,21 @@ as : A ∋ maximal ¬maximal<x : {x : HOD} → A ∋ x → ¬ maximal < x -- A is Partial, use negative -Zorn-lemma : { A : HOD } - → o∅ o< & A +init-uchain : (A : HOD) ( f : Ordinal → Ordinal ) (mf : ≤-monotonic-f A f) {y : Ordinal } → (ay : odef A y ) + { supf : Ordinal → Ordinal } { x : Ordinal } → odef (UnionCF A f mf ay supf x) y +init-uchain A f mf ay = ⟪ ay , ch-init (init ay refl) ⟫ + +Zorn-lemma : { A : HOD } + → o∅ o< & A → ( ( B : HOD) → (B⊆A : B ⊆' A) → IsTotalOrderSet B → SUP A B ) -- SUP condition - → Maximal A + → Maximal A 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} {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 ))) s : HOD - s = ODC.minimal O A (λ eq → ¬x<0 ( subst (λ k → o∅ o< k ) (=od∅→≡o∅ eq) 0<A )) + s = ODC.minimal O A (λ eq → ¬x<0 ( subst (λ k → o∅ o< k ) (=od∅→≡o∅ eq) 0<A )) as : A ∋ * ( & s ) as = subst (λ k → odef A (& k) ) (sym *iso) ( ODC.x∋minimal O A (λ eq → ¬x<0 ( subst (λ k → o∅ o< k ) (=od∅→≡o∅ eq) 0<A )) ) as0 : odef A (& s ) @@ -482,20 +555,20 @@ s<A : & s o< & A s<A = c<→o< (subst (λ k → odef A (& k) ) *iso as ) HasMaximal : HOD - HasMaximal = record { od = record { def = λ x → odef A x ∧ ( (m : Ordinal) → odef A m → ¬ (* x < * m)) } ; odmax = & A ; <odmax = z07 } + HasMaximal = record { od = record { def = λ x → odef A x ∧ ( (m : Ordinal) → odef A m → ¬ (* x < * m)) } ; odmax = & A ; <odmax = z07 } no-maximum : HasMaximal =h= od∅ → (x : Ordinal) → odef A x ∧ ((m : Ordinal) → odef A m → odef A x ∧ (¬ (* x < * m) )) → ⊥ - no-maximum nomx x P = ¬x<0 (eq→ nomx {x} ⟪ proj1 P , (λ m ma p → proj2 ( proj2 P m ma ) p ) ⟫ ) + no-maximum nomx x P = ¬x<0 (eq→ nomx {x} ⟪ proj1 P , (λ m ma p → proj2 ( proj2 P m ma ) p ) ⟫ ) Gtx : { x : HOD} → A ∋ x → HOD - Gtx {x} ax = record { od = record { def = λ y → odef A y ∧ (x < (* y)) } ; odmax = & A ; <odmax = z07 } + Gtx {x} ax = record { od = record { def = λ y → odef A y ∧ (x < (* y)) } ; odmax = & A ; <odmax = z07 } z08 : ¬ Maximal A → HasMaximal =h= od∅ z08 nmx = record { eq→ = λ {x} lt → ⊥-elim ( nmx record {maximal = * x ; as = subst (λ k → odef A k) (sym &iso) (proj1 lt) ; ¬maximal<x = λ {y} ay → subst (λ k → ¬ (* x < k)) *iso (proj2 lt (& y) ay) } ) ; eq← = λ {y} lt → ⊥-elim ( ¬x<0 lt )} x-is-maximal : ¬ Maximal A → {x : Ordinal} → (ax : odef A x) → & (Gtx (subst (λ k → odef A k ) (sym &iso) ax)) ≡ o∅ → (m : Ordinal) → odef A m → odef A x ∧ (¬ (* x < * m)) x-is-maximal nmx {x} ax nogt m am = ⟪ subst (λ k → odef A k) &iso (subst (λ k → odef A k ) (sym &iso) ax) , ¬x<m ⟫ where ¬x<m : ¬ (* x < * m) - ¬x<m x<m = ∅< {Gtx (subst (λ k → odef A k ) (sym &iso) ax)} {* m} ⟪ subst (λ k → odef A k) (sym &iso) am , subst (λ k → * x < k ) (cong (*) (sym &iso)) x<m ⟫ (≡o∅→=od∅ nogt) + ¬x<m x<m = ∅< {Gtx (subst (λ k → odef A k ) (sym &iso) ax)} {* m} ⟪ subst (λ k → odef A k) (sym &iso) am , subst (λ k → * x < k ) (cong (*) (sym &iso)) x<m ⟫ (≡o∅→=od∅ nogt) - minsupP : ( B : HOD) → (B⊆A : B ⊆' A) → IsTotalOrderSet B → MinSUP A B + minsupP : ( B : HOD) → (B⊆A : B ⊆' A) → IsTotalOrderSet B → MinSUP A B minsupP B B⊆A total = m02 where xsup : (sup : Ordinal ) → Set n xsup sup = {w : Ordinal } → odef B w → (w ≡ sup ) ∨ (w << sup ) @@ -536,7 +609,7 @@ m05 {w} bw with SUP.x≤sup S {* w} (subst (λ k → odef B k) (sym &iso) bw ) ... | case1 eq = case1 ( subst₂ (λ j k → j ≡ k ) &iso refl (cong (&) eq) ) ... | case2 lt = case2 ( subst (λ k → _ < k ) (sym *iso) lt ) - m02 : MinSUP A B + m02 : MinSUP A B m02 = dont-or (m00 (& A)) m03 -- Uncountable ascending chain by axiom of choice @@ -568,18 +641,19 @@ -- supf is fixed for z ≡ & A , we can prove order and is-max -- - SZ1 : ( f : Ordinal → Ordinal ) (mf : ≤-monotonic-f A f) + SZ1 : ( 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 f mf {y} ay zc x = zc1 x where chain-mono1 : {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 {a} {b} {c} a≤b = chain-mono f mf ay (ZChain.supf zc) (ZChain.supf-mono zc) a≤b - is-max-hp : (x : Ordinal) {a : Ordinal} {b : Ordinal} → odef (UnionCF A f mf ay (ZChain.supf zc) x) a → (ab : odef A b) - → HasPrev A (UnionCF A f mf ay (ZChain.supf zc) x) f b + is-max-hp : (x : Ordinal) {a : Ordinal} {b : Ordinal} → odef (UnionCF A f mf ay (ZChain.supf zc) x) a → (ab : odef A b) + → HasPrev A (UnionCF A f mf ay (ZChain.supf zc) x) f b → * 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)) ⟫ + (sym (HasPrev.x=fy has-prev)) ( ch-is-sup u u<x is-sup (fsuc _ fc)) ⟫ supf = ZChain.supf zc @@ -593,17 +667,19 @@ zc04 = ZChain.csupf zc (z09 (ZChain.asupf zc)) 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 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 zc06 = ChainP.supu=u is-sup - zc08 : supf u o< supf b + zc08 : supf u o< supf b zc08 = ordtrans≤-< (ZChain.supf-<= zc (≤to<= ( s≤fc _ f mf fc ))) ss<sb csupf-fc {b} {s} {z1} b<z ss≤sb (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 (csupf-fc b<z ss≤sb 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-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) ⟫ 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) ) ) @@ -625,24 +701,24 @@ HasPrev A (UnionCF A f mf ay supf x) f b ∨ IsSUP A (UnionCF A f mf ay supf b) ab → * a < * b → odef (UnionCF A f mf ay supf x) b 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 + 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 sb<sx 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 b<x = ZChain.supf-inject zc sb<sx - m04 : ¬ HasPrev A (UnionCF A f mf ay supf b) f b - m04 nhp = proj1 is-sup ( record { ax = HasPrev.ax nhp ; y = HasPrev.y nhp ; ay = + m04 : ¬ HasPrev A (UnionCF A f mf ay supf b) f b + m04 nhp = proj1 is-sup ( record { ax = HasPrev.ax nhp ; y = HasPrev.y nhp ; ay = chain-mono1 (o<→≤ b<x) (HasPrev.ay nhp) ; x=fy = HasPrev.x=fy nhp } ) m05 : ZChain.supf zc b ≡ b - m05 = ZChain.sup=u zc ab (o<→≤ (z09 ab) ) ⟪ record { x≤sup = λ {z} uz → IsSUP.x≤sup (proj2 is-sup) uz } , m04 ⟫ + m05 = ZChain.sup=u zc ab (o<→≤ (z09 ab) ) ⟪ record { x≤sup = λ {z} uz → IsSUP.x≤sup (proj2 is-sup) uz } , m04 ⟫ m08 : {z : Ordinal} → (fcz : FClosure A f y z ) → z <= ZChain.supf zc b m08 {z} fcz = ZChain.fcy<sup zc (o<→≤ b<A) fcz - m09 : {s z1 : Ordinal} → ZChain.supf zc s o< ZChain.supf zc b + m09 : {s z1 : Ordinal} → ZChain.supf zc s o< ZChain.supf zc b → FClosure A f (ZChain.supf zc s) z1 → z1 <= ZChain.supf zc b - m09 {s} {z} s<b fcz = order b<A s<b fcz - m06 : ChainP A f mf ay supf b + m09 {s} {z} s<b fcz = order b<A s<b fcz + m06 : ChainP A f mf ay supf b m06 = record { fcy<sup = m08 ; order = m09 ; supu=u = m05 } ... | no lim = record { is-max = is-max ; order = order } where is-max : {a : Ordinal} {b : Ordinal} → odef (UnionCF A f mf ay supf x) a → @@ -650,46 +726,42 @@ HasPrev A (UnionCF A f mf ay supf x) f b ∨ IsSUP A (UnionCF A f mf ay supf b) ab → * a < * b → odef (UnionCF A f mf ay supf x) b is-max {a} {b} ua sb<sx ab P a<b with ODC.or-exclude O P - 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 - = ⟪ ab , ch-is-sup b sb<sx m06 (subst (λ k → FClosure A f k b) (sym m05) (init ab refl)) ⟫ where + 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 sb<sx 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 + b<x : b o< x b<x = ZChain.supf-inject zc sb<sx m07 : {z : Ordinal} → FClosure A f y z → z <= ZChain.supf zc b m07 {z} fc = ZChain.fcy<sup zc (o<→≤ m09) fc - m08 : {s z1 : Ordinal} → ZChain.supf zc s o< ZChain.supf zc b + m08 : {s z1 : Ordinal} → ZChain.supf zc s o< ZChain.supf zc b → FClosure A f (ZChain.supf zc s) z1 → z1 <= ZChain.supf zc b m08 {s} {z1} s<b fc = order m09 s<b fc m04 : ¬ HasPrev A (UnionCF A f mf ay supf b) f b - m04 nhp = proj1 is-sup ( record { ax = HasPrev.ax nhp ; y = HasPrev.y nhp ; ay = - chain-mono1 (o<→≤ b<x) (HasPrev.ay nhp) + m04 nhp = proj1 is-sup ( record { ax = HasPrev.ax nhp ; y = HasPrev.y nhp ; ay = + chain-mono1 (o<→≤ b<x) (HasPrev.ay nhp) ; x=fy = HasPrev.x=fy nhp } ) m05 : ZChain.supf zc b ≡ b m05 = ZChain.sup=u zc ab (o<→≤ m09) ⟪ record { x≤sup = λ lt → IsSUP.x≤sup (proj2 is-sup) lt } , m04 ⟫ -- ZChain on x - m06 : ChainP A f mf ay supf b + m06 : ChainP A f mf ay supf b m06 = record { fcy<sup = m07 ; order = m08 ; supu=u = m05 } uchain : ( f : Ordinal → Ordinal ) → (mf : ≤-monotonic-f A f ) {y : Ordinal} (ay : odef A y) → HOD - uchain f mf {y} ay = record { od = record { def = λ x → FClosure A f y x } ; odmax = & A ; <odmax = + uchain f mf {y} ay = record { od = record { def = λ x → FClosure A f y x } ; odmax = & A ; <odmax = λ {z} cz → subst (λ k → k o< & A) &iso ( c<→o< (subst (λ k → odef A k ) (sym &iso ) (A∋fc y f mf cz ))) } - utotal : ( f : Ordinal → Ordinal ) → (mf : ≤-monotonic-f A f ) {y : Ordinal} (ay : odef A y) + utotal : ( f : Ordinal → Ordinal ) → (mf : ≤-monotonic-f A f ) {y : Ordinal} (ay : odef A y) → IsTotalOrderSet (uchain f mf ay) - utotal f mf {y} ay {a} {b} ca cb = subst₂ (λ j k → Tri (j < k) (j ≡ k) (k < j)) *iso *iso uz01 where + utotal f mf {y} ay {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) ) uz01 = fcn-cmp y f mf ca cb - ysup : ( f : Ordinal → Ordinal ) → (mf : ≤-monotonic-f A f ) {y : Ordinal} (ay : odef A y) + ysup : ( f : Ordinal → Ordinal ) → (mf : ≤-monotonic-f A f ) {y : Ordinal} (ay : odef A y) → MinSUP A (uchain f mf ay) - ysup f mf {y} ay = minsupP (uchain f mf ay) (λ lt → A∋fc y f mf lt) (utotal f mf ay) + ysup f mf {y} ay = minsupP (uchain f mf ay) (λ lt → A∋fc y f mf lt) (utotal f mf ay) - UChainPreserve : {x z : Ordinal } { f : Ordinal → Ordinal } → {mf : ≤-monotonic-f A f } {y : Ordinal} {ay : odef A y} - → { supf0 supf1 : Ordinal → Ordinal } - → ( ( z : Ordinal ) → z o< x → supf0 z ≡ supf1 z) - → UnionCF A f mf ay supf0 x ≡ UnionCF A f mf ay supf1 x - UChainPreserve {x} {f} {mf} {y} {ay} {supf0} {supf1} <x→eq = ? SUP⊆ : { B C : HOD } → B ⊆' C → SUP A C → SUP A B SUP⊆ {B} {C} B⊆C sup = record { sup = SUP.sup sup ; as = SUP.as sup ; x≤sup = λ lt → SUP.x≤sup sup (B⊆C lt) } @@ -703,7 +775,7 @@ -- create all ZChains under o< x -- - ind : ( f : Ordinal → Ordinal ) → (mf : ≤-monotonic-f A f ) {y : Ordinal} (ay : odef A y) → (x : Ordinal) + ind : ( f : Ordinal → Ordinal ) → (mf : ≤-monotonic-f A f ) {y : Ordinal} (ay : odef A y) → (x : Ordinal) → ((z : Ordinal) → z o< x → ZChain A f mf ay z) → ZChain A f mf ay x ind f mf {y} ay x prev with Oprev-p x ... | yes op = zc41 where @@ -712,39 +784,39 @@ -- px = Oprev.oprev op zc : ZChain A f mf ay (Oprev.oprev op) - zc = prev px (subst (λ k → px o< k) (Oprev.oprev=x op) <-osuc ) + zc = prev px (subst (λ k → px o< k) (Oprev.oprev=x op) <-osuc ) px<x : px o< x px<x = subst (λ k → px o< k) (Oprev.oprev=x op) <-osuc opx=x : osuc px ≡ x opx=x = Oprev.oprev=x op zc-b<x : (b : Ordinal ) → b o< x → b o< osuc px - zc-b<x b lt = subst (λ k → b o< k ) (sym (Oprev.oprev=x op)) lt + zc-b<x b lt = subst (λ k → b o< k ) (sym (Oprev.oprev=x op)) lt supf0 = ZChain.supf zc pchain : HOD pchain = UnionCF A f mf ay supf0 px - supf-mono : {a b : Ordinal } → a o≤ b → supf0 a o≤ supf0 b + supf-mono : {a b : Ordinal } → a o≤ b → supf0 a o≤ supf0 b supf-mono = ZChain.supf-mono zc zc04 : {b : Ordinal} → b o≤ x → (b o≤ px ) ∨ (b ≡ x ) - zc04 {b} b≤x with trio< b px + zc04 {b} b≤x with trio< b px ... | tri< a ¬b ¬c = case1 (o<→≤ a) ... | tri≈ ¬a b ¬c = case1 (o≤-refl0 b) ... | tri> ¬a ¬b px<b with osuc-≡< b≤x ... | case1 eq = case2 eq - ... | case2 b<x = ⊥-elim ( ¬p<x<op ⟪ px<b , subst (λ k → b o< k ) (sym (Oprev.oprev=x op)) b<x ⟫ ) + ... | case2 b<x = ⊥-elim ( ¬p<x<op ⟪ px<b , subst (λ k → b o< k ) (sym (Oprev.oprev=x op)) b<x ⟫ ) -- -- find the next value of supf -- pchainpx : HOD - pchainpx = record { od = record { def = λ z → (odef A z ∧ UChain A f mf ay supf0 px z ) + pchainpx = record { od = record { def = λ z → (odef A z ∧ UChain A f mf ay supf0 px z ) ∨ FClosure A f (supf0 px) z } ; odmax = & A ; <odmax = zc00 } where zc00 : {z : Ordinal } → (odef A z ∧ UChain A f mf ay supf0 px z ) ∨ FClosure A f (supf0 px) z → z o< & A - zc00 {z} (case1 lt) = z07 lt + zc00 {z} (case1 lt) = z07 lt zc00 {z} (case2 fc) = z09 ( A∋fc (supf0 px) f mf fc ) zc02 : { a b : Ordinal } → odef A a ∧ UChain A f mf ay supf0 px a → FClosure A f (supf0 px) b → a <= b @@ -754,15 +826,15 @@ zc05 : {b : Ordinal } → FClosure A f (supf0 px) b → a <= b zc05 (fsuc b1 fb ) with proj1 ( mf b1 (A∋fc (supf0 px) f mf fb )) ... | case1 eq = subst (λ k → a <= k ) (subst₂ (λ j k → j ≡ k) &iso &iso (cong (&) eq)) (zc05 fb) - ... | case2 lt = <=-trans (zc05 fb) (case2 lt) - zc05 (init b1 refl) with MinSUP.x≤sup (ZChain.minsup zc o≤-refl) + ... | case2 lt = <=-trans (zc05 fb) (case2 lt) + zc05 (init b1 refl) with MinSUP.x≤sup (ZChain.minsup zc o≤-refl) (subst (λ k → odef A k ∧ UChain A f mf ay supf0 px k) (sym &iso) ca ) ... | case1 eq = case1 (subst₂ (λ j k → j ≡ k ) &iso zc06 eq ) - ... | case2 lt = case2 (subst₂ (λ j k → j < k ) *iso (cong (*) zc06) lt ) - + ... | case2 lt = case2 (subst₂ (λ j k → j < k ) *iso (cong (*) zc06) lt ) + ptotal : IsTotalOrderSet pchainpx - ptotal (case1 a) (case1 b) = subst₂ (λ j k → Tri (j < k) (j ≡ k) (k < j)) *iso *iso - (chain-total A f mf ay supf0 (proj2 a) (proj2 b)) + ptotal (case1 a) (case1 b) = subst₂ (λ j k → Tri (j < k) (j ≡ k) (k < j)) *iso *iso + (chain-total A f mf ay supf0 (proj2 a) (proj2 b)) ptotal {a0} {b0} (case1 a) (case2 b) with zc02 a b ... | case1 eq = tri≈ (<-irr (case1 (sym eq1))) eq1 (<-irr (case1 eq1)) where eq1 : a0 ≡ b0 @@ -778,31 +850,31 @@ lt1 : a0 < b0 lt1 = subst₂ (λ j k → j < k ) *iso *iso lt ptotal (case2 a) (case2 b) = subst₂ (λ j k → Tri (j < k) (j ≡ k) (k < j)) *iso *iso (fcn-cmp (supf0 px) f mf a b) - + pcha : pchainpx ⊆' A pcha (case1 lt) = proj1 lt pcha (case2 fc) = A∋fc _ f mf fc - - sup1 : MinSUP A pchainpx + + sup1 : MinSUP A pchainpx sup1 = minsupP pchainpx pcha ptotal sp1 = MinSUP.sup sup1 -- -- supf0 px o≤ sp1 - -- - - zc41 : ZChain A f mf ay x + -- + + zc41 : ZChain A f mf ay x zc41 with MinSUP.x≤sup sup1 (case2 (init (ZChain.asupf zc {px}) refl )) zc41 | (case2 sfpx<sp1) = record { supf = supf1 ; sup=u = ? ; asupf = ? ; supf-mono = supf1-mono ; supf-< = ? ; supfmax = ? ; minsup = ? ; supf-is-minsup = ? ; csupf = csupf1 } where -- supf0 px is included by the chain of sp1 - -- ( UnionCF A f mf ay supf0 px ∪ FClosure (supf0 px) ) ≡ UnionCF supf1 x + -- ( UnionCF A f mf ay supf0 px ∪ FClosure (supf0 px) ) ≡ UnionCF supf1 x -- supf1 x ≡ sp1, which is not included now supf1 : Ordinal → Ordinal - supf1 z with trio< z px + supf1 z with trio< z px ... | tri< a ¬b ¬c = supf0 z - ... | tri≈ ¬a b ¬c = supf0 z + ... | tri≈ ¬a b ¬c = supf0 z ... | tri> ¬a ¬b c = sp1 sf1=sf0 : {z : Ordinal } → z o≤ px → supf1 z ≡ supf0 z @@ -819,12 +891,12 @@ asupf1 : {z : Ordinal } → odef A (supf1 z) asupf1 {z} with trio< z px - ... | tri< a ¬b ¬c = ZChain.asupf zc - ... | tri≈ ¬a b ¬c = ZChain.asupf zc + ... | tri< a ¬b ¬c = ZChain.asupf zc + ... | tri≈ ¬a b ¬c = ZChain.asupf zc ... | tri> ¬a ¬b c = MinSUP.asm sup1 - supf1-mono : {a b : Ordinal } → a o≤ b → supf1 a o≤ supf1 b - supf1-mono {a} {b} a≤b with trio< b px + supf1-mono : {a b : Ordinal } → a o≤ b → supf1 a o≤ supf1 b + supf1-mono {a} {b} a≤b with trio< b px ... | tri< a ¬b ¬c = subst₂ (λ j k → j o≤ k ) (sym (sf1=sf0 (o<→≤ (ordtrans≤-< a≤b a)))) refl ( supf-mono a≤b ) ... | tri≈ ¬a b ¬c = subst₂ (λ j k → j o≤ k ) (sym (sf1=sf0 (subst (λ k → a o≤ k) b a≤b))) refl ( supf-mono a≤b ) supf1-mono {a} {b} a≤b | tri> ¬a ¬b c with trio< a px @@ -833,25 +905,26 @@ zc21 = ZChain.minsup zc (o<→≤ a<px) zc24 : {x₁ : Ordinal} → odef (UnionCF A f mf ay supf0 a) x₁ → (x₁ ≡ sp1) ∨ (x₁ << sp1) zc24 {x₁} ux = MinSUP.x≤sup sup1 (case1 (chain-mono f mf ay supf0 (ZChain.supf-mono zc) (o<→≤ a<px) ux ) ) - zc19 : supf0 a o≤ sp1 + zc19 : supf0 a o≤ sp1 zc19 = subst (λ k → k o≤ sp1) (sym (ZChain.supf-is-minsup zc (o<→≤ a<px))) ( MinSUP.minsup zc21 (MinSUP.asm sup1) zc24 ) ... | tri≈ ¬a b ¬c = zc18 where zc21 : MinSUP A (UnionCF A f mf ay supf0 a) zc21 = ZChain.minsup zc (o≤-refl0 b) zc20 : MinSUP.sup zc21 ≡ supf0 a - zc20 = sym (ZChain.supf-is-minsup zc (o≤-refl0 b)) + zc20 = sym (ZChain.supf-is-minsup zc (o≤-refl0 b)) zc24 : {x₁ : Ordinal} → odef (UnionCF A f mf ay supf0 a) x₁ → (x₁ ≡ sp1) ∨ (x₁ << sp1) zc24 {x₁} ux = MinSUP.x≤sup sup1 (case1 (chain-mono f mf ay supf0 (ZChain.supf-mono zc) (o≤-refl0 b) ux ) ) - zc18 : supf0 a o≤ sp1 + zc18 : supf0 a o≤ sp1 zc18 = subst (λ k → k o≤ sp1) zc20( MinSUP.minsup zc21 (MinSUP.asm sup1) zc24 ) ... | tri> ¬a ¬b c = o≤-refl - fcup : {u z : Ordinal } → FClosure A f (supf1 u) z → u o≤ px → FClosure A f (supf0 u) z + fcup : {u z : Ordinal } → FClosure A f (supf1 u) z → u o≤ px → FClosure A f (supf0 u) z fcup {u} {z} fc u≤px = subst (λ k → FClosure A f k z ) (sf1=sf0 u≤px) fc - fcpu : {u z : Ordinal } → FClosure A f (supf0 u) z → u o≤ px → FClosure A f (supf1 u) z + fcpu : {u z : Ordinal } → FClosure A f (supf0 u) z → u o≤ px → FClosure A f (supf1 u) z fcpu {u} {z} fc u≤px = subst (λ k → FClosure A f k z ) (sym (sf1=sf0 u≤px)) fc zc11 : {z : Ordinal} → odef (UnionCF A f mf ay supf1 x) z → odef pchainpx z + zc11 {z} ⟪ az , ch-init fc ⟫ = case1 ⟪ az , ch-init fc ⟫ zc11 {z} ⟪ az , ch-is-sup u su<sx is-sup fc ⟫ = zc21 fc where u<x : u o< x u<x = supf-inject0 supf1-mono su<sx @@ -859,18 +932,19 @@ u≤px = zc-b<x _ u<x zc21 : {z1 : Ordinal } → FClosure A f (supf1 u) z1 → odef pchainpx z1 zc21 {z1} (fsuc z2 fc ) with zc21 fc - ... | case1 ⟪ ua1 , ch-is-sup u u<x u1-is-sup fc₁ ⟫ = case1 ⟪ proj2 ( mf _ ua1) , ch-is-sup u u<x u1-is-sup (fsuc _ fc₁) ⟫ - ... | case2 fc = case2 (fsuc _ fc) + ... | case1 ⟪ ua1 , ch-init fc₁ ⟫ = case1 ⟪ proj2 ( mf _ ua1) , ch-init (fsuc _ fc₁) ⟫ + ... | case1 ⟪ ua1 , ch-is-sup u u<x u1-is-sup fc₁ ⟫ = case1 ⟪ proj2 ( mf _ ua1) , ch-is-sup u u<x u1-is-sup (fsuc _ fc₁) ⟫ + ... | case2 fc = case2 (fsuc _ fc) zc21 (init asp refl ) with trio< (supf0 u) (supf0 px) | inspect supf1 u ... | tri< a ¬b ¬c | _ = case1 ⟪ asp , ch-is-sup u a record {fcy<sup = zc13 ; order = zc17 ; supu=u = trans (sym (sf1=sf0 (o<→≤ u<px))) (ChainP.supu=u is-sup) } (init asp0 (sym (sf1=sf0 (o<→≤ u<px))) ) ⟫ where u<px : u o< px u<px = ZChain.supf-inject zc a asp0 : odef A (supf0 u) - asp0 = ZChain.asupf zc + asp0 = ZChain.asupf zc zc17 : {s : Ordinal} {z1 : Ordinal} → supf0 s o< supf0 u → FClosure A f (supf0 s) z1 → (z1 ≡ supf0 u) ∨ (z1 << supf0 u) - zc17 {s} {z1} ss<spx fc = subst (λ k → (z1 ≡ k) ∨ (z1 << k)) ((sf1=sf0 u≤px)) ( ChainP.order is-sup + zc17 {s} {z1} ss<spx fc = subst (λ k → (z1 ≡ k) ∨ (z1 << k)) ((sf1=sf0 u≤px)) ( ChainP.order is-sup (subst₂ (λ j k → j o< k ) (sym (sf1=sf0 zc18)) (sym (sf1=sf0 u≤px)) ss<spx) (fcpu fc zc18) ) where zc18 : s o≤ px zc18 = ordtrans (ZChain.supf-inject zc ss<spx) u≤px @@ -879,6 +953,7 @@ ... | tri≈ ¬a b ¬c | _ = case2 (init (subst (λ k → odef A k) b (ZChain.asupf zc) ) (sym (trans (sf1=sf0 u≤px) b ))) ... | tri> ¬a ¬b c | _ = ⊥-elim ( ¬p<x<op ⟪ ZChain.supf-inject zc c , subst (λ k → u o< k ) (sym (Oprev.oprev=x op)) u<x ⟫ ) zc12 : {z : Ordinal} → odef pchainpx z → odef (UnionCF A f mf ay supf1 x) z + zc12 {z} (case1 ⟪ az , ch-init fc ⟫ ) = ⟪ az , ch-init fc ⟫ zc12 {z} (case1 ⟪ az , ch-is-sup u su<sx is-sup fc ⟫ ) = zc21 fc where u<px : u o< px u<px = ZChain.supf-inject zc su<sx @@ -887,17 +962,18 @@ u≤px : u o≤ px u≤px = o<→≤ u<px s1u<s1x : supf1 u o< supf1 x - s1u<s1x = ordtrans<-≤ (subst₂ (λ j k → j o< k ) (sym (sf1=sf0 u≤px )) (sym (sf1=sf0 o≤-refl)) su<sx) (supf1-mono (o<→≤ px<x) ) + s1u<s1x = ordtrans<-≤ (subst₂ (λ j k → j o< k ) (sym (sf1=sf0 u≤px )) (sym (sf1=sf0 o≤-refl)) su<sx) (supf1-mono (o<→≤ px<x) ) zc21 : {z1 : Ordinal } → FClosure A f (supf0 u) z1 → odef (UnionCF A f mf ay supf1 x) z1 zc21 {z1} (fsuc z2 fc ) with zc21 fc - ... | ⟪ ua1 , ch-is-sup u u<x u1-is-sup fc₁ ⟫ = ⟪ proj2 ( mf _ ua1) , ch-is-sup u u<x u1-is-sup (fsuc _ fc₁) ⟫ + ... | ⟪ ua1 , ch-init fc₁ ⟫ = ⟪ proj2 ( mf _ ua1) , ch-init (fsuc _ fc₁) ⟫ + ... | ⟪ ua1 , ch-is-sup u u<x u1-is-sup fc₁ ⟫ = ⟪ proj2 ( mf _ ua1) , ch-is-sup u u<x u1-is-sup (fsuc _ fc₁) ⟫ zc21 (init asp refl ) with trio< u px | inspect supf1 u - ... | tri< a ¬b ¬c | _ = ⟪ asp , ch-is-sup u - s1u<s1x + ... | tri< a ¬b ¬c | _ = ⟪ asp , ch-is-sup u + s1u<s1x record {fcy<sup = zc13 ; order = zc17 ; supu=u = trans (sf1=sf0 u≤px ) (ChainP.supu=u is-sup) } zc14 ⟫ where zc17 : {s : Ordinal} {z1 : Ordinal} → supf1 s o< supf1 u → FClosure A f (supf1 s) z1 → (z1 ≡ supf1 u) ∨ (z1 << supf1 u) - zc17 {s} {z1} ss<su fc = subst (λ k → (z1 ≡ k) ∨ (z1 << k)) (sym (sf1=sf0 (o<→≤ u<px))) ( ChainP.order is-sup + zc17 {s} {z1} ss<su fc = subst (λ k → (z1 ≡ k) ∨ (z1 << k)) (sym (sf1=sf0 (o<→≤ u<px))) ( ChainP.order is-sup (subst₂ (λ j k → j o< k ) (sf1=sf0 s≤px) (sf1=sf0 (o<→≤ u<px)) ss<su) (fcup fc s≤px) ) where s≤px : s o≤ px -- ss<su : supf1 s o< supf1 u s≤px = ordtrans ( supf-inject0 supf1-mono ss<su ) (o<→≤ u<px) @@ -905,7 +981,7 @@ zc14 = init (subst (λ k → odef A k ) (sym (sf1=sf0 u≤px)) asp) (sf1=sf0 u≤px) zc13 : {z : Ordinal } → FClosure A f y z → (z ≡ supf1 u) ∨ ( z << supf1 u ) zc13 {z} fc = subst (λ k → (z ≡ k) ∨ ( z << k )) (sym (sf1=sf0 u≤px )) ( ChainP.fcy<sup is-sup fc ) - ... | tri≈ ¬a b ¬c | _ = ⟪ asp , ch-is-sup px (subst (λ k → supf1 k o< supf1 x) b s1u<s1x) record { fcy<sup = zc13 + ... | tri≈ ¬a b ¬c | _ = ⟪ asp , ch-is-sup px (subst (λ k → supf1 k o< supf1 x) b s1u<s1x) record { fcy<sup = zc13 ; order = zc17 ; supu=u = zc18 } (init asupf1 (trans (sf1=sf0 o≤-refl ) (cong supf0 (sym b))) ) ⟫ where zc13 : {z : Ordinal } → FClosure A f y z → (z ≡ supf1 px) ∨ ( z << supf1 px ) zc13 {z} fc = subst (λ k → (z ≡ k) ∨ (z << k)) (trans (cong supf0 b) (sym (sf1=sf0 o≤-refl))) (ChainP.fcy<sup is-sup fc ) @@ -931,9 +1007,10 @@ zc20 = ? zc21 : {z1 : Ordinal } → FClosure A f (supf0 px) z1 → odef (UnionCF A f mf ay supf1 x) z1 zc21 {z1} (fsuc z2 fc ) with zc21 fc - ... | ⟪ ua1 , ch-is-sup u u<x u1-is-sup fc₁ ⟫ = ⟪ proj2 ( mf _ ua1) , ch-is-sup u u<x u1-is-sup (fsuc _ fc₁) ⟫ + ... | ⟪ ua1 , ch-init fc₁ ⟫ = ⟪ proj2 ( mf _ ua1) , ch-init (fsuc _ fc₁) ⟫ + ... | ⟪ ua1 , ch-is-sup u u<x u1-is-sup fc₁ ⟫ = ⟪ proj2 ( mf _ ua1) , ch-is-sup u u<x u1-is-sup (fsuc _ fc₁) ⟫ zc21 (init asp refl ) with zc20 - ... | case1 sfpx=px = ⟪ asp , ch-is-sup px zc18 + ... | case1 sfpx=px = ⟪ asp , ch-is-sup px zc18 record {fcy<sup = zc13 ; order = zc17 ; supu=u = zc15 } zc14 ⟫ where zc15 : supf1 px ≡ px zc15 = trans (sf1=sf0 o≤-refl ) (sfpx=px) @@ -945,28 +1022,29 @@ zc13 {z} fc = subst (λ k → (z ≡ k) ∨ ( z << k )) (sym (sf1=sf0 o≤-refl)) ( ZChain.fcy<sup zc o≤-refl fc ) zc17 : {s : Ordinal} {z1 : Ordinal} → supf1 s o< supf1 px → FClosure A f (supf1 s) z1 → (z1 ≡ supf1 px) ∨ (z1 << supf1 px) - zc17 {s} {z1} ss<spx fc = subst (λ k → (z1 ≡ k) ∨ (z1 << k )) mins-is-spx + zc17 {s} {z1} ss<spx fc = subst (λ k → (z1 ≡ k) ∨ (z1 << k )) mins-is-spx (MinSUP.x≤sup mins (csupf17 (fcup fc (o<→≤ s<px) )) ) where mins : MinSUP A (UnionCF A f mf ay supf0 px) mins = ZChain.minsup zc o≤-refl - mins-is-spx : MinSUP.sup mins ≡ supf1 px + mins-is-spx : MinSUP.sup mins ≡ supf1 px mins-is-spx = trans (sym ( ZChain.supf-is-minsup zc o≤-refl ) ) (sym (sf1=sf0 o≤-refl )) s<px : s o< px s<px = supf-inject0 supf1-mono ss<spx sf<px : supf0 s o< px sf<px = subst₂ (λ j k → j o< k ) (sf1=sf0 (o<→≤ s<px)) (trans (sf1=sf0 o≤-refl) (sfpx=px)) ss<spx csupf17 : {z1 : Ordinal } → FClosure A f (supf0 s) z1 → odef (UnionCF A f mf ay supf0 px) z1 - csupf17 (init as refl ) = ZChain.csupf zc sf<px + csupf17 (init as refl ) = ZChain.csupf zc sf<px csupf17 (fsuc x fc) = ZChain.f-next zc (csupf17 fc) ... | case2 sfp<px with ZChain.csupf zc sfp<px -- odef (UnionCF A f mf ay supf0 px) (supf0 px) - ... | ⟪ ua1 , ch-is-sup u u<x is-sup fc₁ ⟫ = ⟪ ua1 , ch-is-sup u zc18 + ... | ⟪ ua1 , ch-init fc₁ ⟫ = ⟪ ua1 , ch-init fc₁ ⟫ + ... | ⟪ ua1 , ch-is-sup u u<x is-sup fc₁ ⟫ = ⟪ ua1 , ch-is-sup u zc18 record { fcy<sup = z10 ; order = z11 ; supu=u = z12 } (fcpu fc₁ ? ) ⟫ where - z10 : {z : Ordinal } → FClosure A f y z → (z ≡ supf1 u) ∨ ( z << supf1 u ) + z10 : {z : Ordinal } → FClosure A f y z → (z ≡ supf1 u) ∨ ( z << supf1 u ) z10 {z} fc = subst (λ k → (z ≡ k) ∨ ( z << k )) (sym (sf1=sf0 ? )) (ChainP.fcy<sup is-sup fc) - z11 : {s z1 : Ordinal} → (lt : supf1 s o< supf1 u ) → FClosure A f (supf1 s ) z1 + z11 : {s z1 : Ordinal} → (lt : supf1 s o< supf1 u ) → FClosure A f (supf1 s ) z1 → (z1 ≡ supf1 u ) ∨ ( z1 << supf1 u ) - z11 {s} {z} lt fc = subst (λ k → (z ≡ k) ∨ ( z << k )) (sym (sf1=sf0 ? )) + z11 {s} {z} lt fc = subst (λ k → (z ≡ k) ∨ ( z << k )) (sym (sf1=sf0 ? )) (ChainP.order is-sup lt0 (fcup fc s≤px )) where s<u : s o< u s<u = supf-inject0 supf1-mono lt @@ -984,11 +1062,11 @@ record STMP {z : Ordinal} (z≤x : z o≤ x ) : Set (Level.suc n) where field tsup : MinSUP A (UnionCF A f mf ay supf1 z) - tsup=sup : supf1 z ≡ MinSUP.sup tsup + tsup=sup : supf1 z ≡ MinSUP.sup tsup sup : {z : Ordinal} → (z≤x : z o≤ x ) → STMP z≤x - sup {z} z≤x with trio< z px - ... | tri< a ¬b ¬c = record { tsup = record { sup = MinSUP.sup m ; asm = MinSUP.asm m + sup {z} z≤x with trio< z px + ... | tri< a ¬b ¬c = record { tsup = record { sup = MinSUP.sup m ; asm = MinSUP.asm m ; x≤sup = ms00 ; minsup = ms01 } ; tsup=sup = trans (sf1=sf0 (o<→≤ a) ) (ZChain.supf-is-minsup zc (o<→≤ a)) } where m = ZChain.minsup zc (o<→≤ a) ms00 : {x : Ordinal} → odef (UnionCF A f mf ay supf1 z) x → (x ≡ MinSUP.sup m) ∨ (x << MinSUP.sup m) @@ -996,7 +1074,7 @@ ms01 : {sup2 : Ordinal} → odef A sup2 → ({x : Ordinal} → odef (UnionCF A f mf ay supf1 z) x → (x ≡ sup2) ∨ (x << sup2)) → MinSUP.sup m o≤ sup2 ms01 {sup2} us P = MinSUP.minsup m ? ? - ... | tri≈ ¬a b ¬c = record { tsup = record { sup = MinSUP.sup m ; asm = MinSUP.asm m + ... | tri≈ ¬a b ¬c = record { tsup = record { sup = MinSUP.sup m ; asm = MinSUP.asm m ; x≤sup = ms00 ; minsup = ms01 } ; tsup=sup = trans (sf1=sf0 (o≤-refl0 b) ) (ZChain.supf-is-minsup zc (o≤-refl0 b)) } where m = ZChain.minsup zc (o≤-refl0 b) ms00 : {x : Ordinal} → odef (UnionCF A f mf ay supf1 z) x → (x ≡ MinSUP.sup m) ∨ (x << MinSUP.sup m) @@ -1013,14 +1091,14 @@ odef (UnionCF A f mf ay supf1 z) x → (x ≡ sup2) ∨ (x << sup2)) → MinSUP.sup m o≤ sup2 ms01 {sup2} us P = MinSUP.minsup m ? ? - csupf1 : {z1 : Ordinal } → supf1 z1 o< x → odef (UnionCF A f mf ay supf1 x) (supf1 z1) + csupf1 : {z1 : Ordinal } → supf1 z1 o< x → odef (UnionCF A f mf ay supf1 x) (supf1 z1) csupf1 {z1} sz1<x = csupf2 where -- z1 o< px → supf1 z1 ≡ supf0 z1 -- z1 ≡ px , supf0 px o< px .. px o< z1, x o≤ z1 ... supf1 z1 ≡ sp1 -- z1 ≡ px , supf0 px ≡ px psz1≤px : supf1 z1 o≤ px - psz1≤px = subst (λ k → supf1 z1 o< k ) (sym opx=x) sz1<x - csupf2 : odef (UnionCF A f mf ay supf1 x) (supf1 z1) + psz1≤px = subst (λ k → supf1 z1 o< k ) (sym opx=x) sz1<x + csupf2 : odef (UnionCF A f mf ay supf1 x) (supf1 z1) csupf2 with trio< z1 px | inspect supf1 z1 csupf2 | tri< a ¬b ¬c | record { eq = eq1 } with osuc-≡< psz1≤px ... | case2 lt = zc12 (case1 cs03) where @@ -1030,7 +1108,7 @@ ... | case1 sfz=sfpx = zc12 (case2 (init (ZChain.asupf zc) cs04 )) where supu=u : supf1 (supf1 z1) ≡ supf1 z1 supu=u = trans (cong supf1 sfz=px) (sym sfz=sfpx) - cs04 : supf0 px ≡ supf0 z1 + cs04 : supf0 px ≡ supf0 z1 cs04 = begin supf0 px ≡⟨ sym (sf1=sf0 o≤-refl) ⟩ supf1 px ≡⟨ sym sfz=sfpx ⟩ @@ -1043,7 +1121,7 @@ cs06 : supf0 px o< osuc px cs06 = subst (λ k → supf0 px o< k ) (sym opx=x) ? csupf2 | tri≈ ¬a b ¬c | record { eq = eq1 } = zc12 (case2 (init (ZChain.asupf zc) (cong supf0 (sym b)))) - csupf2 | tri> ¬a ¬b px<z1 | record { eq = eq1 } = ? + csupf2 | tri> ¬a ¬b px<z1 | record { eq = eq1 } = ? -- ⊥-elim ( ¬p<x<op ⟪ px<z1 , subst (λ k → z1 o< k) (sym opx=x) z1<x ⟫ ) @@ -1054,7 +1132,7 @@ -- supf1 x ≡ supf0 px because of supfmax supf1 : Ordinal → Ordinal - supf1 z with trio< z px + supf1 z with trio< z px ... | tri< a ¬b ¬c = supf0 z ... | tri≈ ¬a b ¬c = supf0 px ... | tri> ¬a ¬b c = supf0 px @@ -1069,7 +1147,7 @@ zc17 = ? -- px o< z, px o< supf0 px supf-mono1 : {z w : Ordinal } → z o≤ w → supf1 z o≤ supf1 w - supf-mono1 {z} {w} z≤w with trio< w px + supf-mono1 {z} {w} z≤w with trio< w px ... | tri< a ¬b ¬c = subst₂ (λ j k → j o≤ k ) (sym (sf1=sf0 (ordtrans≤-< z≤w a))) refl ( supf-mono z≤w ) ... | tri≈ ¬a refl ¬c with trio< z px ... | tri< a ¬b ¬c = zc17 @@ -1084,22 +1162,25 @@ pchain1 = UnionCF A f mf ay supf1 x 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 u1 u1<x u1-is-sup fc ⟫ = ⟪ az , ch-is-sup u1 ? ? ? ⟫ - zc111 : {z : Ordinal} → z o< px → OD.def (od pchain1) z → OD.def (od pchain) z + zc111 : {z : Ordinal} → z o< px → OD.def (od pchain1) z → OD.def (od pchain) z + zc111 {z} z<px ⟪ az , ch-init fc ⟫ = ⟪ az , ch-init fc ⟫ zc111 {z} z<px ⟪ az , ch-is-sup u1 u1<x u1-is-sup fc ⟫ = ⟪ az , ch-is-sup u1 ? ? ? ⟫ zc11 : (¬ xSUP (UnionCF A f mf ay supf0 px) f x ) ∨ (HasPrev A pchain f x ) → {z : Ordinal} → OD.def (od pchain1) z → OD.def (od pchain) z ∨ ( (supf0 px ≡ px) ∧ FClosure A f px z ) - zc11 P {z} ⟪ az , ch-is-sup u1 u1<x u1-is-sup fc ⟫ with trio< u1 px - ... | tri< u1<px ¬b ¬c = case1 ⟪ az , ch-is-sup u1 ? ? fc ⟫ + zc11 P {z} ⟪ az , ch-init fc ⟫ = case1 ⟪ az , ch-init fc ⟫ + zc11 P {z} ⟪ az , ch-is-sup u1 u1<x u1-is-sup fc ⟫ with trio< u1 px + ... | tri< u1<px ¬b ¬c = case1 ⟪ az , ch-is-sup u1 ? ? fc ⟫ ... | tri≈ ¬a eq ¬c = case2 ⟪ subst (λ k → supf0 k ≡ k) eq s1u=u , subst (λ k → FClosure A f k z) zc12 ? ⟫ where s1u=u : supf0 u1 ≡ u1 s1u=u = ? -- ChainP.supu=u u1-is-sup - zc12 : supf0 u1 ≡ px + zc12 : supf0 u1 ≡ px zc12 = trans s1u=u eq zc11 (case1 ¬sp=x) {z} ⟪ az , ch-is-sup u1 u1<x u1-is-sup fc ⟫ | tri> ¬a ¬b px<u = ⊥-elim (¬sp=x zcsup) where - eq : u1 ≡ x + eq : u1 ≡ x eq with trio< u1 x ... | tri< a ¬b ¬c = ⊥-elim ( ¬p<x<op ⟪ px<u , subst (λ k → u1 o< k ) (sym (Oprev.oprev=x op )) a ⟫ ) ... | tri≈ ¬a b ¬c = b @@ -1107,25 +1188,26 @@ s1u=x : supf0 u1 ≡ x s1u=x = trans ? eq zc13 : osuc px o< osuc u1 - zc13 = o≤-refl0 ( trans (Oprev.oprev=x op) (sym eq ) ) + 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) ? )) ... | case1 eq1 = ⊥-elim ( o<¬≡ zc14 ? ) where zc14 : u ≡ osuc px zc14 = begin - u ≡⟨ sym ( ChainP.supu=u is-sup) ⟩ - supf0 u ≡⟨ ? ⟩ - supf0 u1 ≡⟨ s1u=x ⟩ - x ≡⟨ sym (Oprev.oprev=x op) ⟩ + u ≡⟨ sym ( ChainP.supu=u is-sup) ⟩ + supf0 u ≡⟨ ? ⟩ + supf0 u1 ≡⟨ s1u=x ⟩ + x ≡⟨ sym (Oprev.oprev=x op) ⟩ osuc px ∎ where open ≡-Reasoning ... | case2 lt = subst (λ k → (w ≡ k) ∨ (w << k)) s1u=x ? zc12 : supf0 x ≡ u1 zc12 = subst (λ k → supf0 k ≡ u1) eq ? - zcsup : xSUP (UnionCF A f mf ay supf0 px) f x + zcsup : xSUP (UnionCF A f mf ay supf0 px) f x zcsup = record { ax = subst (λ k → odef A k) (trans zc12 eq) (ZChain.asupf zc) ; is-sup = record { x≤sup = x≤sup ; minsup = ? } } zc11 (case2 hp) {z} ⟪ az , ch-is-sup u1 u1<x u1-is-sup fc ⟫ | tri> ¬a ¬b px<u = case1 ? where - eq : u1 ≡ x + eq : u1 ≡ x eq with trio< u1 x ... | tri< a ¬b ¬c = ⊥-elim ( ¬p<x<op ⟪ px<u , subst (λ k → u1 o< k ) (sym (Oprev.oprev=x op )) a ⟫ ) ... | tri≈ ¬a b ¬c = b @@ -1145,10 +1227,10 @@ record STMP {z : Ordinal} (z≤x : z o≤ x ) : Set (Level.suc n) where field tsup : MinSUP A (UnionCF A f mf ay supf1 z) - tsup=sup : supf1 z ≡ MinSUP.sup tsup + tsup=sup : supf1 z ≡ MinSUP.sup tsup sup : {z : Ordinal} → (z≤x : z o≤ x ) → STMP z≤x - sup {z} z≤x with trio< z px + sup {z} z≤x with trio< z px ... | tri< a ¬b ¬c = ? -- jrecord { tsup = ZChain.minsup zc (o<→≤ a) ; tsup=sup = ZChain.supf-is-minsup zc (o<→≤ a) } ... | tri≈ ¬a b ¬c = ? -- record { tsup = ZChain.minsup zc (o≤-refl0 b) ; tsup=sup = ZChain.supf-is-minsup zc (o≤-refl0 b) } ... | tri> ¬a ¬b px<z = zc35 where @@ -1156,15 +1238,15 @@ zc30 with osuc-≡< z≤x ... | case1 eq = eq ... | case2 z<x = ⊥-elim (¬p<x<op ⟪ px<z , subst (λ k → z o< k ) (sym (Oprev.oprev=x op)) z<x ⟫ ) - zc32 = ZChain.sup zc o≤-refl + zc32 = ZChain.sup zc o≤-refl zc34 : ¬ (supf0 px ≡ px) → {w : HOD} → UnionCF A f mf ay supf0 z ∋ w → (w ≡ SUP.sup zc32) ∨ (w < SUP.sup zc32) zc34 ne {w} lt with zc11 ? ⟪ proj1 lt , ? ⟫ - ... | case1 lt = SUP.x≤sup zc32 lt + ... | case1 lt = SUP.x≤sup zc32 lt ... | case2 ⟪ spx=px , fc ⟫ = ⊥-elim ( ne spx=px ) zc33 : supf0 z ≡ & (SUP.sup zc32) zc33 = ? -- trans (sym (supfx (o≤-refl0 (sym zc30)))) ( ZChain.supf-is-minsup zc o≤-refl ) zc36 : ¬ (supf0 px ≡ px) → STMP z≤x - zc36 ne = ? -- record { tsup = record { sup = SUP.sup zc32 ; as = SUP.as zc32 ; x≤sup = zc34 ne } ; tsup=sup = zc33 } + zc36 ne = ? -- record { tsup = record { sup = SUP.sup zc32 ; as = SUP.as zc32 ; x≤sup = zc34 ne } ; tsup=sup = zc33 } zc35 : STMP z≤x zc35 with trio< (supf0 px) px ... | tri< a ¬b ¬c = zc36 ¬b @@ -1175,23 +1257,23 @@ sup=u : {b : Ordinal} (ab : odef A b) → b o≤ x → IsMinSUP A (UnionCF A f mf ay supf0 b) supf0 ab ∧ (¬ HasPrev A (UnionCF A f mf ay supf0 b) f b ) → supf0 b ≡ b sup=u {b} ab b≤x is-sup with trio< b px - ... | tri< a ¬b ¬c = ZChain.sup=u zc ab (o<→≤ a) ⟪ record { x≤sup = λ lt → IsMinSUP.x≤sup (proj1 is-sup) lt } , proj2 is-sup ⟫ - ... | tri≈ ¬a b ¬c = ZChain.sup=u zc ab (o≤-refl0 b) ⟪ record { x≤sup = λ lt → IsMinSUP.x≤sup (proj1 is-sup) lt } , proj2 is-sup ⟫ + ... | tri< a ¬b ¬c = ZChain.sup=u zc ab (o<→≤ a) ⟪ record { x≤sup = λ lt → IsMinSUP.x≤sup (proj1 is-sup) lt } , proj2 is-sup ⟫ + ... | tri≈ ¬a b ¬c = ZChain.sup=u zc ab (o≤-refl0 b) ⟪ record { x≤sup = λ lt → IsMinSUP.x≤sup (proj1 is-sup) lt } , proj2 is-sup ⟫ ... | tri> ¬a ¬b px<b = zc31 ? where zc30 : x ≡ b zc30 with osuc-≡< b≤x ... | case1 eq = sym (eq) ... | case2 b<x = ⊥-elim (¬p<x<op ⟪ px<b , subst (λ k → b o< k ) (sym (Oprev.oprev=x op)) b<x ⟫ ) - zcsup : xSUP (UnionCF A f mf ay supf0 px) supf0 x + zcsup : xSUP (UnionCF A f mf ay supf0 px) supf0 x zcsup with zc30 - ... | refl = record { ax = ab ; is-sup = record { x≤sup = λ {w} lt → - IsMinSUP.x≤sup (proj1 is-sup) ? ; minsup = ? } } + ... | refl = record { ax = ab ; is-sup = record { x≤sup = λ {w} lt → + IsMinSUP.x≤sup (proj1 is-sup) ? ; minsup = ? } } zc31 : ( (¬ xSUP (UnionCF A f mf ay supf0 px) supf0 x ) ∨ HasPrev A (UnionCF A f mf ay supf0 px) f x ) → supf0 b ≡ b zc31 (case1 ¬sp=x) with zc30 ... | refl = ⊥-elim (¬sp=x zcsup ) zc31 (case2 hasPrev ) with zc30 - ... | refl = ⊥-elim ( proj2 is-sup record { ax = HasPrev.ax hasPrev ; y = HasPrev.y hasPrev - ; ay = ? ; x=fy = HasPrev.x=fy hasPrev } ) + ... | refl = ⊥-elim ( proj2 is-sup record { ax = HasPrev.ax hasPrev ; y = HasPrev.y hasPrev + ; ay = ? ; x=fy = HasPrev.x=fy hasPrev } ) ... | no lim = zc5 where @@ -1210,10 +1292,10 @@ pchain = UnionCF A f mf ay supf0 x ptotal0 : IsTotalOrderSet pchain - ptotal0 {a} {b} ca cb = subst₂ (λ j k → Tri (j < k) (j ≡ k) (k < j)) *iso *iso uz01 where + ptotal0 {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) ) - uz01 = chain-total A f mf ay supf0 ( (proj2 ca)) ( (proj2 cb)) - + uz01 = chain-total A f mf ay supf0 ( (proj2 ca)) ( (proj2 cb)) + usup : MinSUP A pchain usup = minsupP pchain (λ lt → proj1 lt) ptotal0 spu = MinSUP.sup usup @@ -1229,66 +1311,71 @@ is-max-hp : (supf : Ordinal → Ordinal) (x : Ordinal) {a : Ordinal} {b : Ordinal} → odef (UnionCF A f mf ay supf x) a → b o< x → (ab : odef A b) → - HasPrev A (UnionCF A f mf ay supf x) f b → + HasPrev A (UnionCF A f mf ay supf x) f b → * 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-is-sup u u<x is-sup fc ⟫ = ? -- ⟪ ab , + ... | ⟪ 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 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 : HasPrev A pchain f x → ¬ xSUP pchain f x zc70 pr xsup = ? no-extension : ¬ ( xSUP (UnionCF A f mf ay supf0 x) supf0 x ) → ZChain A f mf ay x - no-extension ¬sp=x = ? where -- record { supf = supf1 ; sup=u = sup=u + no-extension ¬sp=x = ? where -- record { supf = supf1 ; sup=u = sup=u -- ; sup = sup ; supf-is-sup = sis ; supf-mono = {!!} ; asupf = ? } where supfu : {u : Ordinal } → ( a : u o< x ) → (z : Ordinal) → Ordinal supfu {u} a z = ZChain.supf (pzc (osuc u) (ob<x lim a)) z pchain0=1 : pchain ≡ pchain1 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 zc12 : {z : Ordinal} → FClosure A f (supf0 u) z → odef pchain1 z zc12 (fsuc x fc) with zc12 fc - ... | ⟪ 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 ? ? ) ⟫ + ... | ⟪ 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) ⟫ + 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 zc13 : {z : Ordinal} → FClosure A f (supf1 u) z → odef pchain z zc13 (fsuc x fc) with zc13 fc - ... | ⟪ 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-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) ⟫ zc13 (init asu su=z ) with trio< u x - ... | tri< a ¬b ¬c = ⟪ ? , ch-is-sup u ? ? (init ? ? ) ⟫ + ... | tri< a ¬b ¬c = ⟪ ? , ch-is-sup u ? ? (init ? ? ) ⟫ ... | tri≈ ¬a b ¬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) {!!}) {!!} ) ... | tri≈ ¬a b ¬c = {!!} - ... | tri> ¬a ¬b x<z = ⊥-elim (o<¬≡ refl (ordtrans<-≤ x<z z≤x )) + ... | tri> ¬a ¬b x<z = ⊥-elim (o<¬≡ refl (ordtrans<-≤ x<z z≤x )) sis : {z : Ordinal} (x≤z : z o≤ x) → supf1 z ≡ & (SUP.sup (sup {!!})) sis {z} z≤x with trio< z x ... | tri< a ¬b ¬c = {!!} where zc8 = ZChain.supf-is-minsup (pzc z a) {!!} ... | tri≈ ¬a b ¬c = {!!} - ... | tri> ¬a ¬b x<z = ⊥-elim (o<¬≡ refl (ordtrans<-≤ x<z z≤x )) + ... | tri> ¬a ¬b x<z = ⊥-elim (o<¬≡ refl (ordtrans<-≤ x<z z≤x )) sup=u : {b : Ordinal} (ab : odef A b) → b o≤ x → IsMinSUP A (UnionCF A f mf ay supf1 b) f ab ∧ (¬ HasPrev A (UnionCF A f mf ay supf1 b) f b ) → supf1 b ≡ b sup=u {z} ab z≤x is-sup with trio< z x ... | tri< a ¬b ¬c = ? -- ZChain.sup=u (pzc (osuc b) (ob<x lim a)) ab {!!} record { x≤sup = {!!} } ... | tri≈ ¬a b ¬c = {!!} -- ZChain.sup=u (pzc (osuc ?) ?) ab {!!} record { x≤sup = {!!} } - ... | tri> ¬a ¬b x<z = ⊥-elim (o<¬≡ refl (ordtrans<-≤ x<z z≤x )) + ... | tri> ¬a ¬b x<z = ⊥-elim (o<¬≡ refl (ordtrans<-≤ x<z z≤x )) - zc5 : ZChain A f mf ay x + zc5 : ZChain A f mf ay x zc5 with ODC.∋-p O A (* x) ... | no noax = no-extension {!!} -- ¬ A ∋ p, just skip - ... | yes ax with ODC.p∨¬p O ( HasPrev A pchain f x ) + ... | yes ax with ODC.p∨¬p O ( HasPrev A pchain f x ) -- we have to check adding x preserve is-max ZChain A y f mf x - ... | case1 pr = no-extension {!!} + ... | case1 pr = no-extension {!!} ... | case2 ¬fy<x with ODC.p∨¬p O (IsMinSUP A pchain f ax ) - ... | case1 is-sup = ? -- record { supf = supf1 ; sup=u = {!!} - -- ; sup = {!!} ; supf-is-sup = {!!} ; supf-mono = {!!}; asupf = {!!} } -- where -- x is a sup of (zc ?) + ... | case1 is-sup = ? -- record { supf = supf1 ; sup=u = {!!} + -- ; sup = {!!} ; supf-is-sup = {!!} ; supf-mono = {!!}; asupf = {!!} } -- where -- x is a sup of (zc ?) ... | case2 ¬x=sup = no-extension {!!} -- x is not f y' nor sup of former ZChain from y -- no extention - + --- --- the maximum chain has fix point of any ≤-monotonic function --- @@ -1296,13 +1383,13 @@ SZ : ( f : Ordinal → Ordinal ) → (mf : ≤-monotonic-f A f ) → {y : Ordinal} (ay : odef A y) → (x : Ordinal) → ZChain A f mf ay x SZ f mf {y} ay x = TransFinite {λ z → ZChain A f mf ay z } (λ x → ind f mf ay x ) x - msp0 : ( f : Ordinal → Ordinal ) → (mf : ≤-monotonic-f A f ) {x y : Ordinal} (ay : odef A y) - → (zc : ZChain A f mf ay x ) + msp0 : ( f : Ordinal → Ordinal ) → (mf : ≤-monotonic-f A f ) {x y : Ordinal} (ay : odef A y) + → (zc : ZChain A f mf ay x ) → MinSUP A (UnionCF A f mf ay (ZChain.supf zc) x) msp0 f mf {x} ay zc = minsupP (UnionCF A f mf ay (ZChain.supf zc) x) (ZChain.chain⊆A zc) (ZChain.f-total zc) fixpoint : ( f : Ordinal → Ordinal ) → (mf : ≤-monotonic-f A f ) (zc : ZChain A f mf as0 (& A) ) - → (sp1 : MinSUP A (ZChain.chain zc)) + → (sp1 : MinSUP A (ZChain.chain zc)) → (ssp<as : ZChain.supf zc (MinSUP.sup sp1) o< ZChain.supf zc (& A)) → f (MinSUP.sup sp1) ≡ MinSUP.sup sp1 fixpoint f mf zc sp1 ssp<as = z14 where @@ -1312,7 +1399,7 @@ sp = MinSUP.sup sp1 asp : odef A sp asp = MinSUP.asm sp1 - z10 : {a b : Ordinal } → (ca : odef chain a ) → supf b o< supf (& A) → (ab : odef A b ) + z10 : {a b : Ordinal } → (ca : odef chain a ) → supf b o< supf (& A) → (ab : odef A b ) → HasPrev A chain f b ∨ IsSUP A (UnionCF A f mf as0 (ZChain.supf zc) b) ab → * a < * b → odef chain b z10 = ZChain1.is-max (SZ1 f mf as0 zc (& A) ) @@ -1325,22 +1412,22 @@ z13 : * (& s) < * sp z13 with MinSUP.x≤sup sp1 ( ZChain.chain∋init zc ) ... | case1 eq = ⊥-elim ( ne eq ) - ... | case2 lt = lt + ... | case2 lt = lt z19 : IsSUP A (UnionCF A f mf as0 (ZChain.supf zc) sp) asp z19 = record { x≤sup = z20 } where z20 : {y : Ordinal} → odef (UnionCF A f mf as0 (ZChain.supf zc) sp) y → (y ≡ sp) ∨ (y << sp) - z20 {y} zy with MinSUP.x≤sup sp1 + z20 {y} zy with MinSUP.x≤sup sp1 (subst (λ k → odef chain k ) (sym &iso) (chain-mono f mf as0 supf (ZChain.supf-mono zc) (o<→≤ z22) zy )) - ... | case1 y=p = case1 (subst (λ k → k ≡ _ ) &iso y=p ) + ... | case1 y=p = case1 (subst (λ k → k ≡ _ ) &iso y=p ) ... | case2 y<p = case2 (subst (λ k → * k < _ ) &iso y<p ) z14 : f sp ≡ sp z14 with ZChain.f-total zc (subst (λ k → odef chain k) (sym &iso) (ZChain.f-next zc z12 )) (subst (λ k → odef chain k) (sym &iso) z12 ) ... | tri< a ¬b ¬c = ⊥-elim z16 where z16 : ⊥ z16 with proj1 (mf (( MinSUP.sup sp1)) ( MinSUP.asm sp1 )) - ... | case1 eq = ⊥-elim (¬b (sym eq) ) - ... | case2 lt = ⊥-elim (¬c lt ) - ... | tri≈ ¬a b ¬c = subst₂ (λ j k → j ≡ k ) &iso &iso ( cong (&) b ) + ... | case1 eq = ⊥-elim (¬b (sym eq) ) + ... | case2 lt = ⊥-elim (¬c lt ) + ... | tri≈ ¬a b ¬c = subst₂ (λ j k → j ≡ k ) &iso &iso ( cong (&) b ) ... | tri> ¬a ¬b c = ⊥-elim z17 where z15 : (f sp ≡ MinSUP.sup sp1) ∨ (* (f sp) < * (MinSUP.sup sp1) ) z15 = MinSUP.x≤sup sp1 (ZChain.f-next zc z12 ) @@ -1367,37 +1454,37 @@ -- z04 : (nmx : ¬ Maximal A ) → (zc : ZChain A (cf nmx) (cf-is-≤-monotonic nmx) as0 (& A)) → ⊥ - z04 nmx zc = <-irr0 {* (cf nmx c)} {* c} - (subst (λ k → odef A k ) (sym &iso) (proj1 (is-cf nmx (MinSUP.asm msp1 )))) + z04 nmx zc = <-irr0 {* (cf nmx c)} {* c} + (subst (λ k → odef A k ) (sym &iso) (proj1 (is-cf nmx (MinSUP.asm msp1 )))) (subst (λ k → odef A k) (sym &iso) (MinSUP.asm msp1) ) (case1 ( cong (*)( fixpoint (cf nmx) (cf-is-≤-monotonic nmx ) zc msp1 ss<sa ))) -- x ≡ f x ̄ (proj1 (cf-is-<-monotonic nmx c (MinSUP.asm msp1 ))) where -- x < f x supf = ZChain.supf zc msp1 : MinSUP A (ZChain.chain zc) - msp1 = msp0 (cf nmx) (cf-is-≤-monotonic nmx) as0 zc - c : Ordinal - c = MinSUP.sup msp1 + msp1 = msp0 (cf nmx) (cf-is-≤-monotonic nmx) as0 zc + c : Ordinal + c = MinSUP.sup msp1 mc = c mc<A : mc o< & A mc<A = ∈∧P→o< ⟪ MinSUP.asm msp1 , lift true ⟫ c=mc : c ≡ mc c=mc = refl - z20 : mc << cf nmx mc + z20 : mc << cf nmx mc z20 = proj1 (cf-is-<-monotonic nmx mc (MinSUP.asm msp1) ) asc : odef A (supf mc) asc = ZChain.asupf zc - spd : MinSUP A (uchain (cf nmx) (cf-is-≤-monotonic nmx) asc ) + spd : MinSUP A (uchain (cf nmx) (cf-is-≤-monotonic nmx) asc ) spd = ysup (cf nmx) (cf-is-≤-monotonic nmx) asc d = MinSUP.sup spd d<A : d o< & A d<A = ∈∧P→o< ⟪ MinSUP.asm spd , lift true ⟫ msup : MinSUP A (UnionCF A (cf nmx) (cf-is-≤-monotonic nmx) as0 supf d) - msup = ZChain.minsup zc (o<→≤ d<A) + msup = ZChain.minsup zc (o<→≤ d<A) sd=ms : supf d ≡ MinSUP.sup ( ZChain.minsup zc (o<→≤ d<A) ) sd=ms = ZChain.supf-is-minsup zc (o<→≤ d<A) - sc<<d : {mc : Ordinal } → (asc : odef A (supf mc)) → (spd : MinSUP A (uchain (cf nmx) (cf-is-≤-monotonic nmx) asc )) + sc<<d : {mc : Ordinal } → (asc : odef A (supf mc)) → (spd : MinSUP A (uchain (cf nmx) (cf-is-≤-monotonic nmx) asc )) → supf mc << MinSUP.sup spd sc<<d {mc} asc spd = z25 where d1 : Ordinal @@ -1417,10 +1504,10 @@ z32 = MinSUP.x≤sup spd (fsuc _ (init asc refl)) z29 : (* (cf nmx d1) ≡ * d1) ∨ (* (cf nmx d1) < * d1) z29 with z32 - ... | case1 eq1 = case1 (cong (*) (trans (cong (cf nmx) (sym eq)) eq1) ) - ... | case2 lt = case2 (subst (λ k → * k < * d1 ) (cong (cf nmx) eq) lt) + ... | case1 eq1 = case1 (cong (*) (trans (cong (cf nmx) (sym eq)) eq1) ) + ... | case2 lt = case2 (subst (λ k → * k < * d1 ) (cong (cf nmx) eq) lt) - fsc<<d : {mc z : Ordinal } → (asc : odef A (supf mc)) → (spd : MinSUP A (uchain (cf nmx) (cf-is-≤-monotonic nmx) asc )) + fsc<<d : {mc z : Ordinal } → (asc : odef A (supf mc)) → (spd : MinSUP A (uchain (cf nmx) (cf-is-≤-monotonic nmx) asc )) → (fc : FClosure A (cf nmx) (supf mc) z) → z << MinSUP.sup spd fsc<<d {mc} {z} asc spd fc = z25 where d1 : Ordinal @@ -1440,18 +1527,24 @@ z32 = MinSUP.x≤sup spd (fsuc _ fc) z29 : (* (cf nmx d1) ≡ * d1) ∨ (* (cf nmx d1) < * d1) z29 with z32 - ... | case1 eq1 = case1 (cong (*) (trans (cong (cf nmx) (sym eq)) eq1) ) - ... | case2 lt = case2 (subst (λ k → * k < * d1 ) (cong (cf nmx) eq) lt) + ... | case1 eq1 = case1 (cong (*) (trans (cong (cf nmx) (sym eq)) eq1) ) + ... | case2 lt = case2 (subst (λ k → * k < * d1 ) (cong (cf nmx) eq) lt) smc<<d : supf mc << d smc<<d = sc<<d asc spd - sz<<c : {z : Ordinal } → z o< & A → supf z <= mc + 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) )) 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 + not-hasprev record { ax = ax ; y = y ; ay = ⟪ ua1 , ch-init fc ⟫ ; x=fy = x=fy } = ⊥-elim ( <-irr z31 z30 ) where + z30 : * mc < * (cf nmx mc) + z30 = proj1 (cf-is-<-monotonic nmx mc (MinSUP.asm msp1)) + 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 z30 : * mc < * (cf nmx mc) z30 = proj1 (cf-is-<-monotonic nmx mc (MinSUP.asm msp1)) @@ -1466,6 +1559,13 @@ not-hasprev : ¬ HasPrev A (UnionCF A (cf nmx) (cf-is-≤-monotonic nmx) as0 supf d) (cf nmx) d + not-hasprev record { ax = ax ; y = y ; ay = ⟪ ua1 , ch-init fc ⟫ ; x=fy = x=fy } = ⊥-elim ( <-irr z31 z30 ) where + z30 : * d < * (cf nmx d) + z30 = proj1 (cf-is-<-monotonic nmx d (MinSUP.asm spd)) + z32 : ( cf nmx (cf nmx y) ≡ supf mc ) ∨ ( * (cf nmx (cf nmx y)) < * (supf mc) ) + 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 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 @@ -1474,7 +1574,7 @@ z53 : supf u o< supf (& A) z53 = ordtrans<-≤ u<x (ZChain.supf-mono zc (o<→≤ d<A) ) 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) + 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 z53 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 @@ -1483,15 +1583,15 @@ z57 : u << mc → supf u o≤ supf mc z57 lt = ZChain.supf-<= zc (case2 z58) where z58 : supf u << supf mc -- supf u o< supf d -- supf u << supf d - z58 = subst₂ ( λ j k → j << k ) (sym (ChainP.supu=u is-sup1)) (sym sc=c) lt + z58 = subst₂ ( λ j k → j << k ) (sym (ChainP.supu=u is-sup1)) (sym sc=c) lt 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 u<x (fsuc _ ( fsuc _ fc )) - 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 + 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 - z47 {mc} {d1} {asc} spd (case1 eq) smc<d = subst (λ k → k < * d1 ) (sym (cong (*) eq)) smc<d + z47 {mc} {d1} {asc} spd (case1 eq) smc<d = subst (λ k → k < * d1 ) (sym (cong (*) eq)) smc<d z47 {mc} {d1} {asc} spd (case2 lt) smc<d = IsStrictPartialOrder.trans PO lt smc<d z30 : * d < * (cf nmx d) z30 = proj1 (cf-is-<-monotonic nmx d (MinSUP.asm spd)) @@ -1511,23 +1611,26 @@ z23 lt = MinSUP.x≤sup spd lt z22 : {y : Ordinal} → odef (UnionCF A (cf nmx) (cf-is-≤-monotonic nmx) as0 (ZChain.supf zc) d) y → (y ≡ MinSUP.sup spd) ∨ (y << MinSUP.sup spd) + 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) 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 (ZChain.supf-mono zc (o<→≤ d<A) ) 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) + 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 z53 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 z57 lt = ZChain.supf-<= zc (case2 z58) where z58 : supf u << supf mc -- supf u o< supf d -- supf u << supf d - z58 = subst₂ ( λ j k → j << k ) (sym (ChainP.supu=u is-sup1)) (sym sc=c) lt + z58 = subst₂ ( λ j k → j << k ) (sym (ChainP.supu=u is-sup1)) (sym sc=c) lt z51 : supf u o≤ supf mc - z51 = or z52 (λ le → o≤-refl0 (z56 le) ) z57 + z51 = or z52 (λ le → o≤-refl0 (z56 le) ) z57 z60 : u o< supf mc → (a ≡ d ) ∨ ( * a < * d ) - z60 u<smc = case2 ( ftrans<=-< (ZChain1.order (SZ1 (cf nmx) (cf-is-≤-monotonic nmx) as0 zc (& A)) mc<A + z60 u<smc = case2 ( ftrans<=-< (ZChain1.order (SZ1 (cf nmx) (cf-is-≤-monotonic nmx) as0 zc (& A)) mc<A (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 ) ) @@ -1545,27 +1648,27 @@ sms<sa : supf mc o< supf (& A) sms<sa with osuc-≡< ( ZChain.supf-mono zc (o<→≤ ( ∈∧P→o< ⟪ MinSUP.asm msp1 , lift true ⟫) )) ... | case2 lt = lt - ... | case1 eq = ⊥-elim ( o<¬≡ eq ( ordtrans<-≤ (sc<sd (subst (λ k → supf mc << k ) (sym sd=d) (sc<<d {mc} asc spd)) ) + ... | case1 eq = ⊥-elim ( o<¬≡ eq ( ordtrans<-≤ (sc<sd (subst (λ k → supf mc << k ) (sym sd=d) (sc<<d {mc} asc spd)) ) ( ZChain.supf-mono zc (o<→≤ d<A )))) ss<sa : supf c o< supf (& A) ss<sa = subst (λ k → supf k o< supf (& A)) (sym c=mc) sms<sa - zorn00 : Maximal A - zorn00 with is-o∅ ( & HasMaximal ) -- we have no Level (suc n) LEM + zorn00 : Maximal A + zorn00 with is-o∅ ( & HasMaximal ) -- we have no Level (suc n) LEM ... | no not = record { maximal = ODC.minimal O HasMaximal (λ eq → not (=od∅→≡o∅ eq)) ; as = zorn01 ; ¬maximal<x = zorn02 } where -- yes we have the maximal zorn03 : odef HasMaximal ( & ( ODC.minimal O HasMaximal (λ eq → not (=od∅→≡o∅ eq)) ) ) zorn03 = ODC.x∋minimal O HasMaximal (λ eq → not (=od∅→≡o∅ eq)) -- Axiom of choice zorn01 : A ∋ ODC.minimal O HasMaximal (λ eq → not (=od∅→≡o∅ eq)) - zorn01 = proj1 zorn03 + zorn01 = proj1 zorn03 zorn02 : {x : HOD} → A ∋ x → ¬ (ODC.minimal O HasMaximal (λ eq → not (=od∅→≡o∅ eq)) < x) zorn02 {x} ax m<x = proj2 zorn03 (& x) ax (subst₂ (λ j k → j < k) (sym *iso) (sym *iso) m<x ) ... | yes ¬Maximal = ⊥-elim ( z04 nmx (SZ (cf nmx) (cf-is-≤-monotonic nmx) as0 (& A) )) where -- if we have no maximal, make ZChain, which contradict SUP condition - nmx : ¬ Maximal A + nmx : ¬ Maximal A nmx mx = ∅< {HasMaximal} zc5 ( ≡o∅→=od∅ ¬Maximal ) where - zc5 : odef A (& (Maximal.maximal mx)) ∧ (( y : Ordinal ) → odef A y → ¬ (* (& (Maximal.maximal mx)) < * y)) + zc5 : odef A (& (Maximal.maximal mx)) ∧ (( y : Ordinal ) → odef A y → ¬ (* (& (Maximal.maximal mx)) < * y)) zc5 = ⟪ Maximal.as mx , (λ y ay mx<y → Maximal.¬maximal<x mx (subst (λ k → odef A k ) (sym &iso) ay) (subst (λ k → k < * y) *iso mx<y) ) ⟫ -- usage (see filter.agda ) @@ -1573,7 +1676,7 @@ -- _⊆'_ : ( A B : HOD ) → Set n -- _⊆'_ A B = (x : Ordinal ) → odef A x → odef B x --- MaximumSubset : {L P : HOD} +-- MaximumSubset : {L P : HOD} -- → o∅ o< & L → o∅ o< & P → P ⊆ L -- → IsPartialOrderSet P _⊆'_ -- → ( (B : HOD) → B ⊆ P → IsTotalOrderSet B _⊆'_ → SUP P B _⊆'_ )