Mercurial > hg > Members > kono > Proof > ZF-in-agda
changeset 1092:87c2da3811c3
index version
| author | Shinji KONO <kono@ie.u-ryukyu.ac.jp> |
|---|---|
| date | Wed, 21 Dec 2022 07:30:18 +0900 |
| parents | 63c1167b2343 |
| children | 6caa088346f0 |
| files | src/zorn.agda |
| diffstat | 1 files changed, 91 insertions(+), 1287 deletions(-) [+] |
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--- a/src/zorn.agda Tue Dec 20 11:20:52 2022 +0900 +++ b/src/zorn.agda Wed Dec 21 07:30:18 2022 +0900 @@ -5,7 +5,7 @@ open import Relation.Binary.Core open import Relation.Binary.PropositionalEquality import OD hiding ( _⊆_ ) -module zorn {n : Level } (O : Ordinals {n}) (_<_ : (x y : OD.HOD O ) → Set n ) (PO : IsStrictPartialOrder _≡_ _<_ ) where +module zorn1 {n : Level } (O : Ordinals {n}) (_<_ : (x y : OD.HOD O ) → Set n ) (PO : IsStrictPartialOrder _≡_ _<_ ) where -- -- Zorn-lemma : { A : HOD } @@ -108,113 +108,6 @@ <-monotonic-f : (A : HOD) → ( Ordinal → Ordinal ) → Set n <-monotonic-f A f = (x : Ordinal ) → odef A x → ( * x < * (f x) ) ∧ odef A (f x ) -data FClosure (A : HOD) (f : Ordinal → Ordinal ) (s : Ordinal) : Ordinal → Set n where - init : {s1 : Ordinal } → odef A s → s ≡ s1 → FClosure A f s s1 - fsuc : (x : Ordinal) ( p : FClosure A f s x ) → FClosure A f s (f x) - -A∋fc : {A : HOD} (s : Ordinal) {y : Ordinal } (f : Ordinal → Ordinal) (mf : ≤-monotonic-f A f) → (fcy : FClosure A f s y ) → odef A y -A∋fc {A} s f mf (init as refl ) = as -A∋fc {A} s f mf (fsuc y fcy) = proj2 (mf y ( A∋fc {A} s f mf fcy ) ) - -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 - -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 = subst₂ (λ j k → j ≤ k ) refl x=fx (s≤fc 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 (cong (*) s≡x )) refl x<fx ) -... | case2 s<x = case2 ( IsStrictPartialOrder.trans PO s<x x<fx ) - -fcn : {A : HOD} (s : Ordinal) { x : Ordinal} {f : Ordinal → Ordinal} → (mf : ≤-monotonic-f A f) → FClosure A f s x → ℕ -fcn s mf (init as refl) = zero -fcn {A} s {x} {f} mf (fsuc y p) with proj1 (mf y (A∋fc s f mf p)) -... | 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) - → (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 () - fc07 : {x : Ordinal } (cx : FClosure A f s x ) → 0 ≡ fcn s mf cx → * s ≡ * x - fc07 {x} (init as refl) eq = refl - fc07 {.(f x)} (fsuc x cx) eq with proj1 (mf x (A∋fc s f mf cx ) ) - ... | case1 x=fx = subst (λ k → * s ≡ k ) (cong (*) x=fx) ( fc07 cx eq ) - -- ... | case2 x<fx = ? - fc00 : (i j : ℕ ) → i ≡ j → {x y : Ordinal } → (cx : FClosure A f s x ) (cy : FClosure A f s y ) → i ≡ fcn s mf cx → j ≡ fcn s mf cy → * x ≡ * y - fc00 (suc i) (suc j) x cx (init x₃ x₄) x₁ x₂ = ⊥-elim ( fc06 x₄ x₂ ) - fc00 (suc i) (suc j) x (init x₃ x₄) (fsuc x₅ cy) x₁ x₂ = ⊥-elim ( fc06 x₄ x₁ ) - fc00 zero zero refl (init _ refl) (init x₁ refl) i=x i=y = refl - fc00 zero zero refl (init as refl) (fsuc y cy) i=x i=y with proj1 (mf y (A∋fc s f mf cy ) ) - ... | case1 y=fy = subst (λ k → * s ≡ * k ) y=fy (fc07 cy i=y) -- ( fc00 zero zero refl (init as refl) cy i=x i=y ) - fc00 zero zero refl (fsuc x cx) (init as refl) i=x i=y with proj1 (mf x (A∋fc s f mf cx ) ) - ... | case1 x=fx = subst (λ k → * k ≡ * s ) x=fx (sym (fc07 cx i=x)) -- ( fc00 zero zero refl cx (init as refl) i=x i=y ) - fc00 zero zero refl (fsuc x cx) (fsuc y cy) i=x i=y with proj1 (mf x (A∋fc s f mf cx ) ) | proj1 (mf y (A∋fc s f mf cy ) ) - ... | case1 x=fx | case1 y=fy = subst₂ (λ j k → * j ≡ * k ) x=fx y=fy ( fc00 zero zero refl cx cy i=x i=y ) - fc00 (suc i) (suc j) i=j {.(f x)} {.(f y)} (fsuc x cx) (fsuc y cy) i=x j=y with proj1 (mf x (A∋fc s f mf cx ) ) | proj1 (mf y (A∋fc s f mf cy ) ) - ... | case1 x=fx | case1 y=fy = subst₂ (λ j k → * j ≡ * k ) x=fx y=fy ( fc00 (suc i) (suc j) i=j cx cy i=x j=y ) - ... | case1 x=fx | case2 y<fy = subst (λ k → * k ≡ * (f y)) x=fx (fc02 x cx i=x) where - fc02 : (x1 : Ordinal) → (cx1 : FClosure A f s x1 ) → suc i ≡ fcn s mf cx1 → * x1 ≡ * (f y) - fc02 x1 (init x₁ x₂) x = ⊥-elim (fc06 x₂ x) - fc02 .(f x1) (fsuc x1 cx1) i=x1 with proj1 (mf x1 (A∋fc s f mf cx1 ) ) - ... | case1 eq = trans (sym (cong (*) eq )) ( fc02 x1 cx1 i=x1 ) -- derefence while f x ≡ x - ... | case2 lt = subst₂ (λ j k → * (f j) ≡ * (f k )) &iso &iso ( cong (λ k → * ( f (& k ))) fc04) where - fc04 : * x1 ≡ * y - fc04 = fc00 i j (cong pred i=j) cx1 cy (cong pred i=x1) (cong pred j=y) - ... | case2 x<fx | case1 y=fy = subst (λ k → * (f x) ≡ * k ) y=fy (fc03 y cy j=y) where - 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 ) (cong (*) 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) - ... | case2 x₁ | case2 x₂ = subst₂ (λ j k → * (f j) ≡ * (f k) ) &iso &iso (cong (λ k → * (f (& k))) (fc00 i j (cong pred i=j) cx cy (cong pred i=x) (cong pred j=y))) - - -fcn-< : {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 Data.Nat.< fcn s mf cy → * x < * y -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 () - 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 ) (cong (*) 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 = 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 - 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) - → (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 - fc11 : * x < * y - fc11 = fcn-< {A} s {x} {y} {f} mf cx cy a -... | 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 - fc12 : * y < * x - fc12 = fcn-< {A} s {y} {x} {f} mf cy cx c - - - --- open import Relation.Binary.Properties.Poset as Poset - IsTotalOrderSet : ( A : HOD ) → Set (Level.suc n) IsTotalOrderSet A = {a b : HOD} → odef A (& a) → odef A (& b) → Tri (a < b) (a ≡ b) (b < a ) @@ -243,63 +136,6 @@ ax = IsSUP.ax isSUP x≤sup = IsSUP.x≤sup isSUP --- --- Our Proof strategy of the Zorn Lemma --- --- f (f ( ... (supf y))) f (f ( ... (supf z1))) --- / | / | --- / | / | --- supf y < supf z1 < supf z2 --- o< o< --- --- if sup z1 ≡ sup z2, the chain is stopped at sup z1, then f (sup z1) ≡ sup z1 --- this means sup z1 is the Maximal, so f is <-monotonic if we have no Maximal. --- - -fc-stop : ( A : HOD ) ( f : Ordinal → Ordinal ) (mf : ≤-monotonic-f A f) { a b : Ordinal } - → (aa : odef A a ) →( {y : Ordinal} → FClosure A f a y → (y ≡ b ) ∨ (y << b )) → a ≡ b → f a ≡ a -fc-stop A f mf {a} {b} aa x≤sup a=b with x≤sup (fsuc a (init aa refl )) -... | case1 eq = trans eq (sym a=b) -... | case2 lt = ⊥-elim (<-irr (case1 (cong (λ k → * (f k) ) (sym a=b))) (ftrans<-≤ lt fc00 ) ) where - fc00 : b ≤ (f b) - fc00 = proj1 (mf _ (subst (λ k → odef A k) a=b aa )) - -∈∧P→o< : {A : HOD } {y : Ordinal} → {P : Set n} → odef A y ∧ P → y o< & A -∈∧P→o< {A } {y} p = subst (λ k → k o< & A) &iso ( c<→o< (subst (λ k → odef A k ) (sym &iso ) (proj1 p ))) - --- Union of supf z and FClosure A f y - -data UChain { A : HOD } { f : Ordinal → Ordinal } {supf : Ordinal → Ordinal} {y : Ordinal } (ay : odef A y ) - (x : Ordinal) : (z : Ordinal) → Set n where - ch-init : {z : Ordinal } (fc : FClosure A f y z) → UChain ay x z - ch-is-sup : (u : Ordinal) {z : Ordinal } (u<x : u o< x) (supu=u : supf u ≡ u) ( fc : FClosure A f (supf u) z ) → UChain ay x z - -UnionCF : ( A : HOD ) ( f : Ordinal → Ordinal ) {y : Ordinal } (ay : odef A y ) ( supf : Ordinal → Ordinal ) ( x : Ordinal ) → HOD -UnionCF A f ay supf x - = record { od = record { def = λ z → odef A z ∧ UChain {A} {f} {supf} ay x z } ; - odmax = & A ; <odmax = λ {y} sy → ∈∧P→o< sy } - --- Union of chain lower than x - -data IChain {A : HOD} { f : Ordinal → Ordinal } {y : Ordinal } (ay : odef A y ) - {x : Ordinal } (supfz : {z : Ordinal } → z o< x → Ordinal) : (z : Ordinal ) → Set n where - ic-init : {z : Ordinal } (fc : FClosure A f y z) → IChain ay supfz z - ic-isup : {z : Ordinal} (i : Ordinal) (i<x : i o< x) (s<x : supfz i<x o≤ i ) (fc : FClosure A f (supfz i<x) z) → IChain ay supfz z - -UnionIC : ( A : HOD ) ( f : Ordinal → Ordinal ) { x : Ordinal } {y : Ordinal } (ay : odef A y ) (supfz : {z : Ordinal } → z o< x → Ordinal) → HOD -UnionIC A f ay supfz - = record { od = record { def = λ z → odef A z ∧ IChain {A} {f} ay supfz 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} {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 ) -... | tri> ¬a ¬b y<x with osuc-≡< (supf-mono (o<→≤ y<x) ) -... | case1 eq = ⊥-elim ( o<¬≡ (sym eq) sx<sy ) -... | case2 lt = ⊥-elim ( o<> sx<sy lt ) - record IsMinSUP ( A B : HOD ) (sup : Ordinal) : Set n where field as : odef A sup @@ -315,182 +151,32 @@ x≤sup = IsMinSUP.x≤sup isMinSUP minsup = IsMinSUP.minsup isMinSUP +record IChain (A : HOD) ( f : Ordinal → Ordinal ) (x : Ordinal ) : Set n where + field + y : Ordinal + x=fy : x ≡ f y + z09 : {b : Ordinal } { A : HOD } → odef A b → b o< & A z09 {b} {A} ab = subst (λ k → k o< & A) &iso ( c<→o< (subst (λ k → odef A k ) (sym &iso ) ab)) -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 ay supf a) c → odef (UnionCF A f 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 ⟪ ua , ch-is-sup u u<x supu=u fc ⟫ = ⟪ ua , ch-is-sup u (ordtrans<-≤ u<x a≤b) supu=u 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 - field - supf : Ordinal → Ordinal - - supf-mono : {a b : Ordinal } → a o≤ b → supf a o≤ supf b - cfcs : {a b w : Ordinal } → a o< b → b o≤ z → supf a o< b → FClosure A f (supf a) w → odef (UnionCF A f ay supf b) w - asupf : {x : Ordinal } → odef A (supf x) - zo≤sz : {x : Ordinal } → x o≤ z → x o≤ supf x - is-minsup : {x : Ordinal } → (x≤z : x o≤ z) → IsMinSUP A (UnionCF A f ay supf x) (supf x) - - chain : HOD - chain = UnionCF A f ay supf z - chain⊆A : chain ⊆ A - chain⊆A = λ lt → proj1 lt - - chain∋init : {x : Ordinal } → odef (UnionCF A f ay supf x) y - chain∋init {x} = ⟪ ay , ch-init (init ay refl) ⟫ - - mf : ≤-monotonic-f A f - mf x ax = ⟪ case2 mf00 , proj2 (mf< x ax ) ⟫ where - mf00 : * x < * (f x) - mf00 = proj1 ( mf< x ax ) - - f-next : {a z : Ordinal} → odef (UnionCF A f ay supf z) a → odef (UnionCF A f ay supf z) (f a) - f-next {a} ⟪ ua , ch-init fc ⟫ = ⟪ proj2 ( mf _ ua) , ch-init (fsuc _ fc) ⟫ - f-next {a} ⟪ ua , ch-is-sup u su<x su=u fc ⟫ = ⟪ proj2 ( mf _ ua) , ch-is-sup u su<x su=u (fsuc _ fc) ⟫ - - 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 ) - ... | tri> ¬a ¬b y<x with osuc-≡< (supf-mono (o<→≤ y<x) ) - ... | case1 eq = ⊥-elim ( o<¬≡ (sym eq) sx<sy ) - ... | case2 lt = ⊥-elim ( o<> sx<sy lt ) - - csupf : {b : Ordinal } → supf b o< supf z → supf b o< z → odef (UnionCF A f ay supf z) (supf b) -- supf z is not an element of this chain - csupf {b} sb<sz sb<z = cfcs (supf-inject sb<sz) o≤-refl sb<z (init asupf refl) - - minsup : {x : Ordinal } → x o≤ z → MinSUP A (UnionCF A f ay supf x) - minsup {x} x≤z = record { sup = supf x ; isMinSUP = is-minsup x≤z } - - supf-is-minsup : {x : Ordinal } → (x≤z : x o≤ z) → supf x ≡ MinSUP.sup (minsup x≤z) - supf-is-minsup _ = refl - - -- different from order because y o< supf - fcy<sup : {u w : Ordinal } → u o≤ z → FClosure A f y w → (w ≡ supf u ) ∨ ( w << supf u ) - 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 ) - - initial : {x : Ordinal } → x o≤ z → odef (UnionCF A f ay supf x) x → y ≤ x - initial {x} x≤z ⟪ aa , ch-init fc ⟫ = s≤fc y f mf fc - initial {x} x≤z ⟪ aa , ch-is-sup u u<x is-sup fc ⟫ = ≤-ftrans (fcy<sup (ordtrans u<x x≤z) (init ay refl)) (s≤fc _ f mf fc) - - sup=u : {b : Ordinal} → (ab : odef A b) → b o≤ z - → IsSUP A (UnionCF A f ay supf b) b → supf b ≡ b - sup=u {b} ab b≤z is-sup = z50 where - z48 : supf b o≤ b - z48 = IsMinSUP.minsup (is-minsup b≤z) ab (λ ux → IsSUP.x≤sup is-sup ux ) - z50 : supf b ≡ b - z50 with trio< (supf b) b - ... | tri< sb<b ¬b ¬c = ⊥-elim ( o≤> z47 sb<b ) where - z47 : b o≤ supf b - z47 = zo≤sz b≤z - ... | tri≈ ¬a b ¬c = b - ... | tri> ¬a ¬b b<sb = ⊥-elim ( o≤> z48 b<sb ) - - supfeq : {a b : Ordinal } → a o≤ z → b o≤ z → UnionCF A f ay supf a ≡ UnionCF A f ay supf b → supf a ≡ supf b - supfeq {a} {b} a≤z b≤z ua=ub with trio< (supf a) (supf b) - ... | tri< sa<sb ¬b ¬c = ⊥-elim ( o≤> ( - IsMinSUP.minsup (is-minsup b≤z) asupf (λ {z} uzb → IsMinSUP.x≤sup (is-minsup a≤z) (subst (λ k → odef k z) (sym ua=ub) uzb)) ) sa<sb ) - ... | tri≈ ¬a b ¬c = b - ... | tri> ¬a ¬b sb<sa = ⊥-elim ( o≤> ( - IsMinSUP.minsup (is-minsup a≤z) asupf (λ {z} uza → IsMinSUP.x≤sup (is-minsup b≤z) (subst (λ k → odef k z) ua=ub uza)) ) sb<sa ) +-- +-- Our Proof strategy of the Zorn Lemma +-- +-- f (f ( ... (supf y))) f (f ( ... (supf z1))) +-- / | / | +-- / | / | +-- supf y < supf z1 < supf z2 +-- o< o< +-- +-- if sup z1 ≡ sup z2, the chain is stopped at sup z1, then f (sup z1) ≡ sup z1 +-- this means sup z1 is the Maximal, so f is <-monotonic if we have no Maximal. +-- - union-max : {a b : Ordinal } → b o≤ supf a → b o≤ z → supf a o< supf b → UnionCF A f ay supf a ≡ UnionCF A f ay supf b - union-max {a} {b} b≤sa b≤z sa<sb = ==→o≡ record { eq→ = z47 ; eq← = z48 } where - z47 : {w : Ordinal } → odef (UnionCF A f ay supf a ) w → odef ( UnionCF A f ay supf b ) w - z47 {w} ⟪ aw , ch-init fc ⟫ = ⟪ aw , ch-init fc ⟫ - z47 {w} ⟪ aw , ch-is-sup u u<a su=u fc ⟫ = ⟪ aw , ch-is-sup u u<b su=u fc ⟫ where - u<b : u o< b - u<b = ordtrans u<a (supf-inject sa<sb ) - z48 : {w : Ordinal } → odef (UnionCF A f ay supf b ) w → odef ( UnionCF A f ay supf a ) w - z48 {w} ⟪ aw , ch-init fc ⟫ = ⟪ aw , ch-init fc ⟫ - z48 {w} ⟪ aw , ch-is-sup u u<b su=u fc ⟫ = ⟪ aw , ch-is-sup u u<a su=u fc ⟫ where - u<a : u o< a - u<a = supf-inject ( osucprev (begin - osuc (supf u) ≡⟨ cong osuc su=u ⟩ - osuc u ≤⟨ osucc u<b ⟩ - b ≤⟨ b≤sa ⟩ - supf a ∎ )) where open o≤-Reasoning O - - x≤supfx→¬sa<sa : {a b : Ordinal } → b o≤ z → b o≤ supf a → ¬ (supf a o< supf b ) - x≤supfx→¬sa<sa {a} {b} b≤z b≤sa sa<sb = ⊥-elim ( o<¬≡ z27 sa<sb ) where -- x o≤ supf a ∧ supf a o< supf b → ⊥, because it defines the same UnionCF - z27 : supf a ≡ supf b - z27 = supfeq (ordtrans (supf-inject sa<sb) b≤z) b≤z ( union-max b≤sa b≤z sa<sb) - - order : {a b w : Ordinal } → b o≤ z → supf a o< supf b → FClosure A f (supf a) w → w ≤ supf b - order {a} {b} {w} b≤z sa<sb fc = IsMinSUP.x≤sup (is-minsup b≤z) (cfcs (supf-inject sa<sb) b≤z sa<b fc) where - sa<b : supf a o< b - sa<b with x<y∨y≤x (supf a) b - ... | case1 lt = lt - ... | case2 b≤sa = ⊥-elim (x≤supfx→¬sa<sa b≤z b≤sa sa<sb) - - supf-idem : {b : Ordinal } → b o≤ z → supf b o≤ z → supf (supf b) ≡ supf b - supf-idem {b} b≤z sfb≤x = z52 where - z54 : {w : Ordinal} → odef (UnionCF A f ay supf (supf b)) w → (w ≡ supf b) ∨ (w << supf b) - z54 {w} ⟪ aw , ch-init fc ⟫ = fcy<sup b≤z fc - z54 {w} ⟪ aw , ch-is-sup u u<x su=u fc ⟫ = order b≤z (subst (λ k → k o< supf b) (sym su=u) u<x) fc where - u<b : u o< b - u<b = supf-inject (subst (λ k → k o< supf b ) (sym (su=u)) u<x ) - z52 : supf (supf b) ≡ supf b - z52 = sup=u asupf sfb≤x record { ax = asupf ; x≤sup = z54 } - - supf-mono< : {a b : Ordinal } → b o≤ z → supf a o< supf b → supf a << supf b - supf-mono< {a} {b} b≤z sa<sb with order {a} {b} b≤z sa<sb (init asupf refl) - ... | case2 lt = lt - ... | case1 eq = ⊥-elim ( o<¬≡ eq sa<sb ) - f-total : IsTotalOrderSet chain - f-total {a} {b} ⟪ uaa , ch-is-sup ua sua<x sua=ua fca ⟫ ⟪ uab , ch-is-sup ub sub<x sub=ub fcb ⟫ = - subst₂ (λ j k → Tri (j < k) (j ≡ k) (k < j)) *iso *iso fc-total where - fc-total : Tri (* (& a) < * (& b)) (* (& a) ≡ * (& b)) (* (& b) < * (& a) ) - fc-total with trio< ua ub - ... | tri< a₁ ¬b ¬c with ≤-ftrans (order (o<→≤ sub<x) (subst₂ (λ j k → j o< k) (sym sua=ua) (sym sub=ub) a₁) fca ) (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 = cong (*) eq1 - ... | case2 a<b = tri< a<b (λ eq → <-irr (case1 (sym eq)) a<b ) (λ lt → <-irr (case2 a<b ) lt) - fc-total | tri≈ _ refl _ = fcn-cmp _ f mf fca fcb - fc-total | tri> ¬a ¬b c with ≤-ftrans (order (o<→≤ sua<x) (subst₂ (λ j k → j o< k) (sym sub=ub) (sym sua=ua) c) fcb ) (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 = cong (*) (sym eq1) - ... | case2 b<a = tri> (λ lt → <-irr (case2 b<a ) lt) (λ eq → <-irr (case1 eq) b<a ) b<a - f-total {a} {b} ⟪ uaa , ch-init fca ⟫ ⟪ uab , ch-is-sup ub sub<x sub=ub fcb ⟫ = ft00 where - ft01 : (& a) ≤ (& b) → Tri ( a < b) ( a ≡ b) ( b < a ) - ft01 (case1 eq) = tri≈ (λ lt → ⊥-elim (<-irr (case1 (sym a=b)) lt)) a=b (λ lt → ⊥-elim (<-irr (case1 a=b) lt)) where - a=b : a ≡ b - a=b = subst₂ (λ j k → j ≡ k ) *iso *iso (cong (*) eq) - ft01 (case2 lt) = tri< a<b (λ eq → <-irr (case1 (sym eq)) a<b ) (λ lt → <-irr (case2 a<b ) lt) where - a<b : a < b - a<b = subst₂ (λ j k → j < k ) *iso *iso lt - ft00 : Tri ( a < b) ( a ≡ b) ( b < a ) - ft00 = ft01 (≤-ftrans (fcy<sup (o<→≤ sub<x) fca) (s≤fc {A} _ f mf fcb)) - f-total {a} {b} ⟪ uaa , ch-is-sup ua sua<x sua=ua fca ⟫ ⟪ uab , ch-init fcb ⟫ = ft00 where - ft01 : (& b) ≤ (& a) → Tri ( a < b) ( a ≡ b) ( b < a ) - ft01 (case1 eq) = tri≈ (λ lt → ⊥-elim (<-irr (case1 (sym a=b)) lt)) a=b (λ lt → ⊥-elim (<-irr (case1 a=b) lt)) where - a=b : a ≡ b - a=b = subst₂ (λ j k → j ≡ k ) *iso *iso (cong (*) (sym eq)) - ft01 (case2 lt) = tri> (λ lt → <-irr (case2 b<a ) lt) (λ eq → <-irr (case1 eq) b<a ) b<a where - b<a : b < a - b<a = subst₂ (λ j k → j < k ) *iso *iso lt - ft00 : Tri ( a < b) ( a ≡ b) ( b < a ) - ft00 = ft01 (≤-ftrans (fcy<sup (o<→≤ sua<x) fcb) (s≤fc {A} _ f mf fca)) - f-total {a} {b} ⟪ uaa , ch-init fca ⟫ ⟪ uab , ch-init fcb ⟫ = - subst₂ (λ j k → Tri (j < k) (j ≡ k) (k < j)) *iso *iso (fcn-cmp y f mf fca fcb ) +∈∧P→o< : {A : HOD } {y : Ordinal} → {P : Set n} → odef A y ∧ P → y o< & A +∈∧P→o< {A } {y} p = subst (λ k → k o< & A) &iso ( c<→o< (subst (λ k → odef A k ) (sym &iso ) (proj1 p ))) -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 ay supf z) a ) → b o< z → (ab : odef A b) - → HasPrev A (UnionCF A f ay supf z) f b ∨ IsSUP A (UnionCF A f ay supf b) b - → * a < * b → odef ((UnionCF A f ay supf z)) b +-- Union of supf z and FClosure A f y record Maximal ( A : HOD ) : Set (Level.suc n) where field @@ -498,57 +184,6 @@ as : A ∋ maximal ¬maximal<x : {x : HOD} → A ∋ x → ¬ maximal < x -- A is Partial, use negative --- --- supf in TransFinite indution may differ each other, but it is the same because of the minimul sup --- -supf-unique : ( A : HOD ) ( f : Ordinal → Ordinal ) (mf< : <-monotonic-f A f) - {y xa xb : Ordinal} → (ay : odef A y) → (xa o≤ xb ) → (za : ZChain A f mf< ay xa ) (zb : ZChain A f mf< ay xb ) - → {z : Ordinal } → z o≤ xa → ZChain.supf za z ≡ ZChain.supf zb z -supf-unique A f mf< {y} {xa} {xb} ay xa≤xb za zb {z} z≤xa = TransFinite0 {λ z → z o≤ xa → ZChain.supf za z ≡ ZChain.supf zb z } ind z z≤xa where - supfa = ZChain.supf za - supfb = ZChain.supf zb - ind : (x : Ordinal) → ((w : Ordinal) → w o< x → w o≤ xa → supfa w ≡ supfb w) → x o≤ xa → supfa x ≡ supfb x - ind x prev x≤xa = sxa=sxb where - ma = ZChain.minsup za x≤xa - mb = ZChain.minsup zb (OrdTrans x≤xa xa≤xb ) - spa = MinSUP.sup ma - spb = MinSUP.sup mb - sax=spa : supfa x ≡ spa - sax=spa = ZChain.supf-is-minsup za x≤xa - sbx=spb : supfb x ≡ spb - sbx=spb = ZChain.supf-is-minsup zb (OrdTrans x≤xa xa≤xb ) - sxa=sxb : supfa x ≡ supfb x - sxa=sxb with trio< (supfa x) (supfb x) - ... | tri≈ ¬a b ¬c = b - ... | tri< a ¬b ¬c = ⊥-elim ( o≤> ( - begin - supfb x ≡⟨ sbx=spb ⟩ - spb ≤⟨ MinSUP.minsup mb (MinSUP.as ma) (λ {z} uzb → MinSUP.x≤sup ma (z53 uzb)) ⟩ - spa ≡⟨ sym sax=spa ⟩ - supfa x ∎ ) a ) where - open o≤-Reasoning O - z53 : {z : Ordinal } → odef (UnionCF A f ay (ZChain.supf zb) x) z → odef (UnionCF A f ay (ZChain.supf za) x) z - z53 ⟪ as , ch-init fc ⟫ = ⟪ as , ch-init fc ⟫ - z53 {z} ⟪ as , ch-is-sup u u<x su=u fc ⟫ = ⟪ as , ch-is-sup u u<x (trans ua=ub su=u) z55 ⟫ where - ua=ub : supfa u ≡ supfb u - ua=ub = prev u u<x (ordtrans u<x x≤xa ) - z55 : FClosure A f (ZChain.supf za u) z - z55 = subst (λ k → FClosure A f k z ) (sym ua=ub) fc - ... | tri> ¬a ¬b c = ⊥-elim ( o≤> ( - begin - supfa x ≡⟨ sax=spa ⟩ - spa ≤⟨ MinSUP.minsup ma (MinSUP.as mb) (λ uza → MinSUP.x≤sup mb (z53 uza)) ⟩ - spb ≡⟨ sym sbx=spb ⟩ - supfb x ∎ ) c ) where - open o≤-Reasoning O - z53 : {z : Ordinal } → odef (UnionCF A f ay (ZChain.supf za) x) z → odef (UnionCF A f ay (ZChain.supf zb) x) z - z53 ⟪ as , ch-init fc ⟫ = ⟪ as , ch-init fc ⟫ - z53 {z} ⟪ as , ch-is-sup u u<x su=u fc ⟫ = ⟪ as , ch-is-sup u u<x (trans ub=ua su=u) z55 ⟫ where - ub=ua : supfb u ≡ supfa u - ub=ua = sym ( prev u u<x (ordtrans u<x x≤xa )) - z55 : FClosure A f (ZChain.supf zb u) z - z55 = subst (λ k → FClosure A f k z ) (sym ub=ua) fc - Zorn-lemma : { A : HOD } → o∅ o< & A → ( ( B : HOD) → (B⊆A : B ⊆ A) → IsTotalOrderSet B → SUP A B ) -- SUP condition @@ -627,916 +262,86 @@ m02 : MinSUP A B m02 = dont-or (m00 (& A)) m03 - -- Uncountable ascending chain by axiom of choice - cf : ¬ Maximal A → Ordinal → Ordinal - cf nmx x with ODC.∋-p O A (* x) - ... | no _ = o∅ - ... | yes ax with is-o∅ (& ( Gtx ax )) - ... | yes nogt = -- no larger element, so it is maximal - ⊥-elim (no-maximum (z08 nmx) x ⟪ subst (λ k → odef A k) &iso ax , x-is-maximal nmx (subst (λ k → odef A k ) &iso ax) nogt ⟫ ) - ... | no not = & (ODC.minimal O (Gtx ax) (λ eq → not (=od∅→≡o∅ eq))) - is-cf : (nmx : ¬ Maximal A ) → {x : Ordinal} → odef A x → odef A (cf nmx x) ∧ ( * x < * (cf nmx x) ) - is-cf nmx {x} ax with ODC.∋-p O A (* x) - ... | no not = ⊥-elim ( not (subst (λ k → odef A k ) (sym &iso) ax )) - ... | yes ax with is-o∅ (& ( Gtx ax )) - ... | yes nogt = ⊥-elim (no-maximum (z08 nmx) x ⟪ subst (λ k → odef A k) &iso ax , x-is-maximal nmx (subst (λ k → odef A k ) &iso ax) nogt ⟫ ) - ... | no not = ODC.x∋minimal O (Gtx ax) (λ eq → not (=od∅→≡o∅ eq)) - - --- - --- infintie ascention sequence of f - --- - cf-is-<-monotonic : (nmx : ¬ Maximal A ) → (x : Ordinal) → odef A x → ( * x < * (cf nmx x) ) ∧ odef A (cf nmx x ) - cf-is-<-monotonic nmx x ax = ⟪ proj2 (is-cf nmx ax ) , proj1 (is-cf nmx ax ) ⟫ - cf-is-≤-monotonic : (nmx : ¬ Maximal A ) → ≤-monotonic-f A ( cf nmx ) - cf-is-≤-monotonic nmx x ax = ⟪ case2 (proj1 ( cf-is-<-monotonic nmx x ax )) , proj2 ( cf-is-<-monotonic nmx x ax ) ⟫ - - -- - -- maximality of chain - -- - -- supf is fixed for z ≡ & A , we can prove order and is-max - -- we have supf-unique now, it is provable in the first Tranfinte induction - - SZ1 : ( f : Ordinal → Ordinal ) (mf : ≤-monotonic-f A f) (mf< : <-monotonic-f A f) - {init : Ordinal} (ay : odef A init) (zc : ZChain A f mf< ay (& A)) (x : Ordinal) → x o≤ & A → ZChain1 A f mf< ay zc x - SZ1 f mf mf< {y} ay zc x x≤A = zc1 x x≤A where - chain-mono1 : {a b c : Ordinal} → a o≤ b - → odef (UnionCF A f ay (ZChain.supf zc) a) c → odef (UnionCF A f 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 ay (ZChain.supf zc) x) a → (ab : odef A b) - → HasPrev A (UnionCF A f ay (ZChain.supf zc) x) f b - → * a < * b → odef (UnionCF A f 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 su=u fc ⟫ = ⟪ ab , subst (λ k → UChain ay x k ) - (sym (HasPrev.x=fy has-prev)) ( ch-is-sup u u<x su=u (fsuc _ fc)) ⟫ - - supf = ZChain.supf zc - - zc1 : (x : Ordinal ) → x o≤ & A → ZChain1 A f mf< ay zc x - zc1 x x≤A with Oprev-p x - ... | yes op = record { is-max = is-max } where - px = Oprev.oprev op - is-max : {a : Ordinal} {b : Ordinal} → odef (UnionCF A f ay supf x) a → - b o< x → (ab : odef A b) → - HasPrev A (UnionCF A f ay supf x) f b ∨ IsSUP A (UnionCF A f ay supf b) b → - * a < * b → odef (UnionCF A f 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 b<x ab P a<b | case2 is-sup with osuc-≡< (ZChain.supf-mono zc (o<→≤ b<x)) - ... | case2 sb<sx = m10 where - b<A : b o< & A - b<A = z09 ab - m05 : ZChain.supf zc b ≡ b - m05 = ZChain.sup=u zc ab (o<→≤ (z09 ab) ) record { ax = ab ; x≤sup = λ {z} uz → IsSUP.x≤sup (proj2 is-sup) uz } - m10 : odef (UnionCF A f ay supf x) b - m10 = ZChain.cfcs zc b<x x≤A (subst (λ k → k o< x) (sym m05) b<x) (init (ZChain.asupf zc) m05) - ... | case1 sb=sx = ⊥-elim (<-irr (case1 (cong (*) m10)) (proj1 (mf< (supf b) (ZChain.asupf zc)))) where - m17 : MinSUP A (UnionCF A f ay supf x) -- supf z o< supf ( supf x ) - m17 = ZChain.minsup zc x≤A - m18 : supf x ≡ MinSUP.sup m17 - m18 = ZChain.supf-is-minsup zc x≤A - m10 : f (supf b) ≡ supf b - m10 = fc-stop A f mf (ZChain.asupf zc) m11 sb=sx where - m11 : {z : Ordinal} → FClosure A f (supf b) z → (z ≡ ZChain.supf zc x) ∨ (z << ZChain.supf zc x) - m11 {z} fc = subst (λ k → (z ≡ k) ∨ (z << k)) (sym m18) ( MinSUP.x≤sup m17 m13 ) where - m04 : ¬ HasPrev A (UnionCF A f 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 { ax = ab ; x≤sup = λ {z} uz → IsSUP.x≤sup (proj2 is-sup) uz } - m14 : ZChain.supf zc b o< x - m14 = subst (λ k → k o< x ) (sym m05) b<x - m13 : odef (UnionCF A f ay supf x) z - m13 = ZChain.cfcs zc b<x x≤A m14 fc - - ... | no lim = record { is-max = is-max } where - is-max : {a : Ordinal} {b : Ordinal} → odef (UnionCF A f ay supf x) a → - b o< x → (ab : odef A b) → - HasPrev A (UnionCF A f ay supf x) f b ∨ IsSUP A (UnionCF A f ay supf b) b → - * a < * b → odef (UnionCF A f 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 b<x ab P a<b | case2 is-sup with IsSUP.x≤sup (proj2 is-sup) (ZChain.chain∋init zc ) - ... | 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 with osuc-≡< (ZChain.supf-mono zc (o<→≤ b<x)) - ... | case2 sb<sx = m10 where - m09 : b o< & A - m09 = subst (λ k → k o< & A) &iso ( c<→o< (subst (λ k → odef A k ) (sym &iso ) ab)) - m05 : ZChain.supf zc b ≡ b - m05 = ZChain.sup=u zc ab (o<→≤ m09) record { ax = ab ; x≤sup = λ lt → IsSUP.x≤sup (proj2 is-sup) lt } - m10 : odef (UnionCF A f ay supf x) b - m10 = ZChain.cfcs zc b<x x≤A (subst (λ k → k o< x) (sym m05) b<x) (init (ZChain.asupf zc) m05) - ... | case1 sb=sx = ⊥-elim (<-irr (case1 (cong (*) m10)) (proj1 (mf< (supf b) (ZChain.asupf zc)))) where - m17 : MinSUP A (UnionCF A f ay supf x) -- supf z o< supf ( supf x ) - m17 = ZChain.minsup zc x≤A - m18 : supf x ≡ MinSUP.sup m17 - m18 = ZChain.supf-is-minsup zc x≤A - m10 : f (supf b) ≡ supf b - m10 = fc-stop A f mf (ZChain.asupf zc) m11 sb=sx where - m11 : {z : Ordinal} → FClosure A f (supf b) z → (z ≡ ZChain.supf zc x) ∨ (z << ZChain.supf zc x) - m11 {z} fc = subst (λ k → (z ≡ k) ∨ (z << k)) (sym m18) ( MinSUP.x≤sup m17 m13 ) where - m05 = ZChain.sup=u zc ab (o<→≤ (z09 ab) ) record { ax = ab ; x≤sup = λ {z} uz → IsSUP.x≤sup (proj2 is-sup) uz } - m14 : ZChain.supf zc b o< x - m14 = subst (λ k → k o< x ) (sym m05) b<x - m13 : odef (UnionCF A f ay supf x) z - m13 = ZChain.cfcs zc b<x x≤A m14 fc - - 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 = - 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) - → 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 - 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) - → 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) - - -- - -- create all ZChains under o< x - -- - - 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 sup1 where - -- - -- we have previous ordinal to use induction - -- - 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 ) - 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 - - supf0 = ZChain.supf zc - pchain : HOD - pchain = UnionCF A f ay supf0 px - - 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 - ... | 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 ⟫ ) - - mf : ≤-monotonic-f A f - mf x ax = ⟪ case2 mf00 , proj2 (mf< x ax ) ⟫ where - mf00 : * x < * (f x) - mf00 = proj1 ( mf< x ax ) - - -- - -- find the next value of supf - -- - - pchainpx : HOD - pchainpx = record { od = record { def = λ z → (odef A z ∧ UChain ay px z ) - ∨ (FClosure A f (supf0 px) z ∧ (supf0 px o< x)) } ; odmax = & A ; <odmax = zc00 } where - zc00 : {z : Ordinal } → (odef A z ∧ UChain ay px z ) ∨ (FClosure A f (supf0 px) z ∧ (supf0 px o< x) )→ z o< & A - zc00 {z} (case1 lt) = z07 lt - zc00 {z} (case2 fc) = z09 ( A∋fc (supf0 px) f mf (proj1 fc) ) - - zc02 : { a b : Ordinal } → odef A a ∧ UChain ay px a → FClosure A f (supf0 px) b ∧ ( supf0 px o< x) → a ≤ b - zc02 {a} {b} ca fb = zc05 (proj1 fb) where - 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 ) eq (zc05 fb) - ... | case2 lt = ≤-ftrans (zc05 fb) (case2 lt) - zc05 (init b1 refl) = MinSUP.x≤sup (ZChain.minsup zc o≤-refl) ca - - ptotal : IsTotalOrderSet pchainpx - ptotal (case1 a) (case1 b) = ZChain.f-total zc a 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 - eq1 = subst₂ (λ j k → j ≡ k ) *iso *iso (cong (*) eq ) - ... | case2 lt = tri< lt1 (λ eq → <-irr (case1 (sym eq)) lt1) (<-irr (case2 lt1)) where - lt1 : a0 < b0 - lt1 = subst₂ (λ j k → j < k ) *iso *iso lt - ptotal {b0} {a0} (case2 b) (case1 a) with zc02 a b - ... | case1 eq = tri≈ (<-irr (case1 eq1)) (sym eq1) (<-irr (case1 (sym eq1))) where - eq1 : a0 ≡ b0 - eq1 = subst₂ (λ j k → j ≡ k ) *iso *iso (cong (*) eq ) - ... | case2 lt = tri> (<-irr (case2 lt1)) (λ eq → <-irr (case1 eq) lt1) lt1 where - 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 (proj1 a) (proj1 b)) - - pcha : pchainpx ⊆ A - pcha (case1 lt) = proj1 lt - pcha (case2 fc) = A∋fc _ f mf (proj1 fc) - - sup1 : MinSUP A pchainpx - sup1 = minsupP pchainpx pcha ptotal - - -- - -- supf0 px o≤ sp1 - -- - - zc41 : MinSUP A pchainpx → ZChain A f mf< ay x - zc41 sup1 = record { supf = supf1 ; asupf = asupf1 ; zo≤sz = zo≤sz ; is-minsup = is-minsup ; cfcs = cfcs ; supf-mono = supf1-mono } where - - sp1 = MinSUP.sup sup1 - - supf1 : Ordinal → Ordinal - supf1 z with trio< z px - ... | 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 - sf1=sf0 {z} z≤px with trio< z px - ... | tri< a ¬b ¬c = refl - ... | tri≈ ¬a b ¬c = refl - ... | tri> ¬a ¬b c = ⊥-elim ( o≤> z≤px c ) - - sf1=sp1 : {z : Ordinal } → px o< z → supf1 z ≡ sp1 - sf1=sp1 {z} px<z with trio< z px - ... | tri< a ¬b ¬c = ⊥-elim ( o<> px<z a ) - ... | tri≈ ¬a b ¬c = ⊥-elim ( o<¬≡ (sym b) px<z ) - ... | tri> ¬a ¬b c = refl - - sf=eq : { z : Ordinal } → z o< x → supf0 z ≡ supf1 z - sf=eq {z} z<x = sym (sf1=sf0 (subst (λ k → z o< k ) (sym (Oprev.oprev=x op)) z<x )) - - 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 = MinSUP.as sup1 - - 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 - ... | tri< a<px ¬b ¬c = zc19 where - zc21 : MinSUP A (UnionCF A f ay supf0 a) - zc21 = ZChain.minsup zc (o<→≤ a<px) - zc24 : {x₁ : Ordinal} → odef (UnionCF A f 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 = subst (λ k → k o≤ sp1) (sym (ZChain.supf-is-minsup zc (o<→≤ a<px))) ( MinSUP.minsup zc21 (MinSUP.as sup1) zc24 ) - ... | tri≈ ¬a b ¬c = zc18 where - zc21 : MinSUP A (UnionCF A f 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)) - zc24 : {x₁ : Ordinal} → odef (UnionCF A f 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 = subst (λ k → k o≤ sp1) zc20( MinSUP.minsup zc21 (MinSUP.as 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} 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} fc u≤px = subst (λ k → FClosure A f k z ) (sym (sf1=sf0 u≤px)) fc - - -- this is a kind of maximality, so we cannot prove this without <-monotonicity - -- - cfcs : {a b w : Ordinal } - → a o< b → b o≤ x → supf1 a o< b → FClosure A f (supf1 a) w → odef (UnionCF A f ay supf1 b) w - cfcs {a} {b} {w} a<b b≤x sa<b fc with x<y∨y≤x (supf0 a) px - ... | case2 px≤sa = z50 where - a<x : a o< x - a<x = ordtrans<-≤ a<b b≤x - a≤px : a o≤ px - a≤px = subst (λ k → a o< k) (sym (Oprev.oprev=x op)) (ordtrans<-≤ a<b b≤x) - -- supf0 a ≡ px we cannot use previous cfcs, it is in the chain because - -- supf0 a ≡ supf0 (supf0 a) ≡ supf0 px o< x - z50 : odef (UnionCF A f ay supf1 b) w - z50 with osuc-≡< px≤sa - ... | case1 px=sa = ⟪ A∋fc {A} _ f mf fc , cp ⟫ where - sa≤px : supf0 a o≤ px - sa≤px = subst₂ (λ j k → j o< k) px=sa (sym (Oprev.oprev=x op)) px<x - spx=sa : supf0 px ≡ supf0 a - spx=sa = begin - supf0 px ≡⟨ cong supf0 px=sa ⟩ - supf0 (supf0 a) ≡⟨ ZChain.supf-idem zc a≤px sa≤px ⟩ - supf0 a ∎ where open ≡-Reasoning - z51 : supf0 px o< b - z51 = subst (λ k → k o< b ) (sym ( begin supf0 px ≡⟨ spx=sa ⟩ - supf0 a ≡⟨ sym (sf1=sf0 a≤px) ⟩ - supf1 a ∎ )) sa<b where open ≡-Reasoning - z52 : supf1 a ≡ supf1 (supf0 px) - z52 = begin supf1 a ≡⟨ sf1=sf0 a≤px ⟩ - supf0 a ≡⟨ sym (ZChain.supf-idem zc a≤px sa≤px ) ⟩ - supf0 (supf0 a) ≡⟨ sym (sf1=sf0 sa≤px) ⟩ - supf1 (supf0 a) ≡⟨ cong supf1 (sym spx=sa) ⟩ - supf1 (supf0 px) ∎ where open ≡-Reasoning - z53 : supf1 (supf0 px) ≡ supf0 px - z53 = begin - supf1 (supf0 px) ≡⟨ cong supf1 spx=sa ⟩ - supf1 (supf0 a) ≡⟨ sf1=sf0 sa≤px ⟩ - supf0 (supf0 a) ≡⟨ sym ( cong supf0 px=sa ) ⟩ - supf0 px ∎ where open ≡-Reasoning - cp : UChain ay b w - cp = ch-is-sup (supf0 px) z51 z53 (subst (λ k → FClosure A f k w) z52 fc) - ... | case2 px<sa = ⊥-elim ( ¬p<x<op ⟪ px<sa , subst₂ (λ j k → j o< k ) (sf1=sf0 a≤px) (sym (Oprev.oprev=x op)) z53 ⟫ ) where - z53 : supf1 a o< x - z53 = ordtrans<-≤ sa<b b≤x - ... | case1 sa<px with trio< a px - ... | tri< a<px ¬b ¬c = z50 where - z50 : odef (UnionCF A f ay supf1 b) w - z50 with osuc-≡< b≤x - ... | case2 lt with ZChain.cfcs zc a<b (subst (λ k → b o< k) (sym (Oprev.oprev=x op)) lt) sa<b fc - ... | ⟪ az , ch-init fc ⟫ = ⟪ az , ch-init fc ⟫ - ... | ⟪ az , ch-is-sup u u<b su=u fc ⟫ = ⟪ az , ch-is-sup u u<b (trans (sym (sf=eq u<x)) su=u) (fcpu fc u≤px ) ⟫ where - u≤px : u o≤ px - u≤px = subst (λ k → u o< k) (sym (Oprev.oprev=x op)) (ordtrans<-≤ u<b b≤x ) - u<x : u o< x - u<x = ordtrans<-≤ u<b b≤x - z50 | case1 eq with ZChain.cfcs zc a<px o≤-refl sa<px fc - ... | ⟪ az , ch-init fc ⟫ = ⟪ az , ch-init fc ⟫ - ... | ⟪ az , ch-is-sup u u<px su=u fc ⟫ = ⟪ az , ch-is-sup u u<b (trans (sym (sf=eq u<x)) su=u) (fcpu fc (o<→≤ u<px)) ⟫ where -- u o< px → u o< b ? - u<b : u o< b - u<b = subst (λ k → u o< k ) (trans (Oprev.oprev=x op) (sym eq) ) (ordtrans u<px <-osuc ) - u<x : u o< x - u<x = subst (λ k → u o< k ) (Oprev.oprev=x op) ( ordtrans u<px <-osuc ) - ... | tri≈ ¬a a=px ¬c = csupf1 where - -- a ≡ px , b ≡ x, sp o≤ x - px<b : px o< b - px<b = subst₂ (λ j k → j o< k) a=px refl a<b - b=x : b ≡ x - b=x with trio< b x - ... | tri< a ¬b ¬c = ⊥-elim ( ¬p<x<op ⟪ px<b , subst (λ k → b o< k ) (sym (Oprev.oprev=x op)) a ⟫ ) -- px o< b o< x - ... | tri≈ ¬a b ¬c = b - ... | tri> ¬a ¬b c = ⊥-elim ( o≤> b≤x c ) -- x o< b - z51 : FClosure A f (supf1 px) w - z51 = subst (λ k → FClosure A f k w) (sym (trans (cong supf1 (sym a=px)) (sf1=sf0 (o≤-refl0 a=px) ))) fc - z53 : odef A w - z53 = A∋fc {A} _ f mf fc - csupf1 : odef (UnionCF A f ay supf1 b) w - csupf1 with x<y∨y≤x px (supf0 px) - ... | case2 spx≤px = ⟪ z53 , ch-is-sup (supf0 px) z54 z52 fc1 ⟫ where - z54 : supf0 px o< b - z54 = subst (λ k → supf0 px o< k ) (trans (Oprev.oprev=x op) (sym b=x) ) spx≤px - z52 : supf1 (supf0 px) ≡ supf0 px - z52 = trans (sf1=sf0 spx≤px ) ( ZChain.supf-idem zc o≤-refl spx≤px ) - fc1 : FClosure A f (supf1 (supf0 px)) w - fc1 = subst (λ k → FClosure A f k w ) (trans (cong supf0 a=px) (sym z52) ) fc - ... | case1 px<spx = ⊥-elim (¬p<x<op ⟪ px<spx , z54 ⟫ ) where -- supf1 px o≤ spuf1 x → supf1 px ≡ x o< x - z54 : supf0 px o≤ px - z54 = subst₂ (λ j k → supf0 j o< k ) a=px (trans b=x (sym (Oprev.oprev=x op))) sa<b - - ... | tri> ¬a ¬b c = ⊥-elim ( ¬p<x<op ⟪ c , subst (λ k → a o< k ) (sym (Oprev.oprev=x op)) ( ordtrans<-≤ a<b b≤x) ⟫ ) -- px o< a o< b o≤ x - - zc11 : {z : Ordinal} → odef (UnionCF A f 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 u<x su=u fc ⟫ = zc21 fc where - zc21 : {z1 : Ordinal } → FClosure A f (supf1 u) z1 → odef pchainpx z1 - zc21 {z1} (fsuc z2 fc ) with zc21 fc - ... | case1 ⟪ ua1 , ch-init fc₁ ⟫ = case1 ⟪ proj2 ( mf _ ua1) , ch-init (fsuc _ fc₁) ⟫ - ... | case1 ⟪ ua1 , ch-is-sup u u<x su=u fc₁ ⟫ = case1 ⟪ proj2 ( mf _ ua1) , ch-is-sup u u<x su=u (fsuc _ fc₁) ⟫ - ... | case2 fc = case2 ⟪ fsuc _ (proj1 fc) , proj2 fc ⟫ - zc21 (init asp refl ) with trio< (supf0 u) (supf0 px) - ... | tri< a ¬b ¬c = case1 ⟪ asp , ch-is-sup u u<px (trans (sym (sf1=sf0 (o<→≤ u<px))) su=u )(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 - ... | tri≈ ¬a b ¬c = case2 ⟪ (init (subst (λ k → odef A k) b (ZChain.asupf zc) ) - (sym (trans (sf1=sf0 (zc-b<x _ u<x)) b ))) , spx<x ⟫ where - spx<x : supf0 px o< x - spx<x = osucprev ( begin - osuc (supf0 px) ≡⟨ cong osuc (sym b) ⟩ - osuc (supf0 u) ≡⟨ cong osuc (sym (sf1=sf0 (zc-b<x _ u<x) )) ⟩ - osuc (supf1 u) ≡⟨ cong osuc su=u ⟩ - osuc u ≤⟨ osucc u<x ⟩ - x ∎ ) where open o≤-Reasoning O - ... | 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 ⟫ ) - - is-minsup : {z : Ordinal} → z o≤ x → IsMinSUP A (UnionCF A f ay supf1 z) (supf1 z) - is-minsup {z} z≤x with osuc-≡< z≤x - ... | case1 z=x = record { as = zc22 ; x≤sup = z23 ; minsup = z24 } where - px<z : px o< z - px<z = subst (λ k → px o< k) (sym z=x) px<x - zc22 : odef A (supf1 z) - zc22 = subst (λ k → odef A k ) (sym (sf1=sp1 px<z )) ( MinSUP.as sup1 ) - z23 : {w : Ordinal } → odef ( UnionCF A f ay supf1 z ) w → w ≤ supf1 z - z23 {w} uz = subst (λ k → w ≤ k ) (sym (sf1=sp1 px<z)) ( MinSUP.x≤sup sup1 ( - zc11 (subst (λ k → odef (UnionCF A f ay supf1 k) w) z=x uz ))) - z24 : {s : Ordinal } → odef A s → ( {w : Ordinal } → odef ( UnionCF A f ay supf1 z ) w → w ≤ s ) - → supf1 z o≤ s - z24 {s} as sup = subst (λ k → k o≤ s ) (sym (sf1=sp1 px<z)) ( MinSUP.minsup sup1 as z25 ) where - z25 : {w : Ordinal } → odef pchainpx w → w ≤ s - z25 {w} (case2 fc) = sup ⟪ A∋fc _ f mf (proj1 fc) , ch-is-sup (supf0 px) z28 z27 fc1 ⟫ where - -- z=x , supf0 px o< x - z28 : supf0 px o< z -- supf0 px ≡ supf1 px o≤ supf1 x ≡ sp1 o≤ x ≡ z - z28 = subst (λ k → supf0 px o< k) (sym z=x) (proj2 fc) - z29 : supf0 px o≤ px - z29 = zc-b<x _ (proj2 fc) - z27 : supf1 (supf0 px) ≡ supf0 px - z27 = trans (sf1=sf0 z29) ( ZChain.supf-idem zc o≤-refl z29 ) - fc1 : FClosure A f (supf1 (supf0 px)) w - fc1 = subst (λ k → FClosure A f k w) (sym z27) (proj1 fc) - z25 {w} (case1 ⟪ ua , ch-init fc ⟫) = sup ⟪ ua , ch-init fc ⟫ - z25 {w} (case1 ⟪ ua , ch-is-sup u u<x su=u fc ⟫) = sup ⟪ ua , ch-is-sup u u<z - (trans (sf1=sf0 u≤px) su=u) (fcpu fc u≤px) ⟫ where - u≤px : u o< osuc px - u≤px = ordtrans u<x <-osuc - u<z : u o< z - u<z = ordtrans u<x (subst (λ k → px o< k ) (sym z=x) px<x ) - ... | case2 z<x = record { as = zc22 ; x≤sup = z23 ; minsup = z24 } where - z≤px = zc-b<x _ z<x - m = ZChain.is-minsup zc z≤px - zc22 : odef A (supf1 z) - zc22 = subst (λ k → odef A k ) (sym (sf1=sf0 z≤px)) ( IsMinSUP.as m ) - z23 : {w : Ordinal } → odef ( UnionCF A f ay supf1 z ) w → w ≤ supf1 z - z23 {w} ⟪ ua , ch-init fc ⟫ = subst (λ k → w ≤ k ) (sym (sf1=sf0 z≤px)) ( ZChain.fcy<sup zc z≤px fc ) - z23 {w} ⟪ ua , ch-is-sup u u<x su=u fc ⟫ = subst (λ k → w ≤ k ) (sym (sf1=sf0 z≤px)) - (IsMinSUP.x≤sup m ⟪ ua , ch-is-sup u u<x (trans (sym (sf1=sf0 u≤px )) su=u) (fcup fc u≤px ) ⟫ ) where - u≤px : u o≤ px - u≤px = ordtrans u<x z≤px - z24 : {s : Ordinal } → odef A s → ( {w : Ordinal } → odef ( UnionCF A f ay supf1 z ) w → w ≤ s ) - → supf1 z o≤ s - z24 {s} as sup = subst (λ k → k o≤ s ) (sym (sf1=sf0 z≤px)) ( IsMinSUP.minsup m as z25 ) where - z25 : {w : Ordinal } → odef ( UnionCF A f ay supf0 z ) w → w ≤ s - z25 {w} ⟪ ua , ch-init fc ⟫ = sup ⟪ ua , ch-init fc ⟫ - z25 {w} ⟪ ua , ch-is-sup u u<x su=u fc ⟫ = sup ⟪ ua , ch-is-sup u u<x - (trans (sf1=sf0 u≤px) su=u) (fcpu fc u≤px) ⟫ where - u≤px : u o≤ px - u≤px = ordtrans u<x z≤px - - zo≤sz : {z : Ordinal} → z o≤ x → z o≤ supf1 z - zo≤sz {z} z≤x with osuc-≡< z≤x - ... | case2 z<x = subst (λ k → z o≤ k) (sym (sf1=sf0 (zc-b<x _ z<x ))) (ZChain.zo≤sz zc (zc-b<x _ z<x )) - ... | case1 refl with osuc-≡< (supf1-mono (o<→≤ (px<x))) -- px o≤ supf1 px o≤ supf1 x ≡ sp1 → x o≤ sp1 - ... | case2 lt = begin - x ≡⟨ sym (Oprev.oprev=x op) ⟩ - osuc px ≤⟨ osucc (ZChain.zo≤sz zc o≤-refl) ⟩ - osuc (supf0 px) ≡⟨ sym (cong osuc (sf1=sf0 o≤-refl )) ⟩ - osuc (supf1 px) ≤⟨ osucc lt ⟩ - supf1 x ∎ where open o≤-Reasoning O - ... | case1 spx=sx with osuc-≡< ( ZChain.zo≤sz zc o≤-refl ) - ... | case2 lt = begin - x ≡⟨ sym (Oprev.oprev=x op) ⟩ - osuc px ≤⟨ osucc lt ⟩ - supf0 px ≡⟨ sym (sf1=sf0 o≤-refl) ⟩ - supf1 px ≤⟨ supf1-mono (o<→≤ px<x) ⟩ - supf1 x ∎ where open o≤-Reasoning O - ... | case1 px=spx = ⊥-elim ( <<-irr zc40 (proj1 ( mf< (supf0 px) (ZChain.asupf zc))) ) where - zc37 : supf0 px ≡ px - zc37 = sym px=spx - zc39 : supf0 px ≡ sp1 - zc39 = begin - supf0 px ≡⟨ sym (sf1=sf0 o≤-refl) ⟩ - supf1 px ≡⟨ spx=sx ⟩ - supf1 x ≡⟨ sf1=sp1 px<x ⟩ - sp1 ∎ where open ≡-Reasoning - zc40 : f (supf0 px) ≤ supf0 px - zc40 = subst (λ k → f (supf0 px) ≤ k ) (sym zc39) - ( MinSUP.x≤sup sup1 (case2 ⟪ fsuc _ (init (ZChain.asupf zc) refl) , subst (λ k → k o< x) (sym zc37) px<x ⟫ )) - - ... | no lim with trio< x o∅ - ... | tri< a ¬b ¬c = ⊥-elim ( ¬x<0 a ) - ... | tri≈ ¬a x=0 ¬c = record { supf = λ _ → MinSUP.sup (ysup f mf ay) ; asupf = MinSUP.as (ysup f mf ay) ; supf-mono = λ _ → o≤-refl - ; zo≤sz = zo≤sz ; is-minsup = is-minsup ; cfcs = λ a<b b≤0 → ⊥-elim ( ¬x<0 (subst (λ k → _ o< k ) x=0 (ordtrans<-≤ a<b b≤0))) } where - - mf : ≤-monotonic-f A f - mf x ax = ⟪ case2 mf00 , proj2 (mf< x ax ) ⟫ where - mf00 : * x < * (f x) - mf00 = proj1 ( mf< x ax ) - ym = MinSUP.sup (ysup f mf ay) - - zo≤sz : {z : Ordinal} → z o≤ x → z o≤ MinSUP.sup (ysup f mf ay) - zo≤sz {z} z≤x with osuc-≡< z≤x - ... | case1 refl = subst (λ k → k o≤ _) (sym x=0) o∅≤z - ... | case2 lt = ⊥-elim ( ¬x<0 (subst (λ k → z o< k ) x=0 lt ) ) - - is-minsup : {z : Ordinal} → z o≤ x → - IsMinSUP A (UnionCF A f ay (λ _ → MinSUP.sup (ysup f mf ay)) z) (MinSUP.sup (ysup f mf ay)) - is-minsup {z} z≤x with osuc-≡< z≤x - ... | case1 refl = record { as = MinSUP.as (ysup f mf ay) ; x≤sup = λ {w} uw → is00 uw ; minsup = λ {s} as sup → is01 as sup } where - is00 : {w : Ordinal } → odef (UnionCF A f ay (λ _ → MinSUP.sup (ysup f mf ay)) x ) w → w ≤ MinSUP.sup (ysup f mf ay) - is00 {w} ⟪ aw , ch-init fc ⟫ = MinSUP.x≤sup (ysup f mf ay) fc - is00 {w} ⟪ aw , ch-is-sup u u<z su=u fc ⟫ = ⊥-elim (¬x<0 (subst (λ k → u o< k ) x=0 u<z )) - is01 : { s : Ordinal } → odef A s → ( {w : Ordinal } → odef (UnionCF A f ay (λ _ → MinSUP.sup (ysup f mf ay)) x ) w → w ≤ s ) - → ym o≤ s - is01 {s} as sup = MinSUP.minsup (ysup f mf ay) as is02 where - is02 : {w : Ordinal } → odef (uchain f mf ay) w → (w ≡ s) ∨ (w << s) - is02 fc = sup ⟪ A∋fc _ f mf fc , ch-init fc ⟫ - ... | case2 lt = ⊥-elim ( ¬x<0 (subst (λ k → z o< k ) x=0 lt ) ) - - ... | tri> ¬a ¬b 0<x = zc400 usup ssup where - - mf : ≤-monotonic-f A f - mf x ax = ⟪ case2 mf00 , proj2 (mf< x ax ) ⟫ where - mf00 : * x < * (f x) - mf00 = proj1 ( mf< x ax ) - - pzc : {z : Ordinal} → z o< x → ZChain A f mf< ay z - pzc {z} z<x = prev z z<x - - ysp = MinSUP.sup (ysup f mf ay) - - supfz : {z : Ordinal } → z o< x → Ordinal - supfz {z} z<x = ZChain.supf (pzc (ob<x lim z<x)) z - - pchainU : HOD - pchainU = UnionIC A f ay supfz - - zeq : {xa xb z : Ordinal } - → (xa<x : xa o< x) → (xb<x : xb o< x) → xa o≤ xb → z o≤ xa - → ZChain.supf (pzc xa<x) z ≡ ZChain.supf (pzc xb<x) z - zeq {xa} {xb} {z} xa<x xb<x xa≤xb z≤xa = supf-unique A f mf< ay xa≤xb - (pzc xa<x) (pzc xb<x) z≤xa - - iceq : {ix iy : Ordinal } → ix ≡ iy → {i<x : ix o< x } {i<y : iy o< x } → supfz i<x ≡ supfz i<y - iceq refl = cong supfz o<-irr - - IChain-i : {z : Ordinal } → IChain ay supfz z → Ordinal - IChain-i (ic-init fc) = o∅ - IChain-i (ic-isup ia ia<x sa<x fca) = ia - - pic<x : {z : Ordinal } → (ic : IChain ay supfz z ) → osuc (IChain-i ic) o< x - pic<x {z} (ic-init fc) = ob<x lim 0<x -- 0<x ∧ lim x → osuc o∅ o< x - pic<x {z} (ic-isup ia ia<x sa<x fca) = ob<x lim ia<x - - pchainU⊆chain : {z : Ordinal } → (pz : odef pchainU z) → odef (ZChain.chain (pzc (pic<x (proj2 pz)))) z - pchainU⊆chain {z} ⟪ aw , ic-init fc ⟫ = ⟪ aw , ch-init fc ⟫ - pchainU⊆chain {z} ⟪ aw , (ic-isup ia ia<x sa<x fca) ⟫ = ZChain.cfcs (pzc (ob<x lim ia<x) ) <-osuc o≤-refl uz03 fca where - uz02 : FClosure A f (ZChain.supf (pzc (ob<x lim ia<x)) ia ) z - uz02 = fca - uz03 : ZChain.supf (pzc (ob<x lim ia<x)) ia o≤ ia - uz03 = sa<x - - chain⊆pchainU : {z w : Ordinal } → (z<x : z o< x) → odef (UnionCF A f ay (ZChain.supf (pzc (ob<x lim z<x))) z) w → odef pchainU w - chain⊆pchainU {z} {w} z<x ⟪ aw , ch-init fc ⟫ = ⟪ aw , ic-init fc ⟫ - chain⊆pchainU {z} {w} z<x ⟪ aw , ch-is-sup u u<oz su=u fc ⟫ - = ⟪ aw , ic-isup u u<x (o≤-refl0 su≡u) (subst (λ k → FClosure A f k w ) su=su fc) ⟫ where - u<x : u o< x - u<x = ordtrans u<oz z<x - su=su : ZChain.supf (pzc (ob<x lim z<x)) u ≡ supfz u<x - su=su = sym ( zeq _ _ (o<→≤ (osucc u<oz)) (o<→≤ <-osuc) ) - su≡u : supfz u<x ≡ u - su≡u = begin - ZChain.supf (pzc (ob<x lim u<x )) u ≡⟨ sym su=su ⟩ - ZChain.supf (pzc (ob<x lim z<x)) u ≡⟨ su=u ⟩ - u ∎ where open ≡-Reasoning - - IC⊆ : {a b : Ordinal } (ia : IChain ay supfz a ) (ib : IChain ay supfz b ) - → IChain-i ia o< IChain-i ib → odef (ZChain.chain (pzc (pic<x ib))) a - IC⊆ {a} {b} (ic-init fc ) ib ia<ib = ⟪ A∋fc _ f mf fc , ch-init fc ⟫ - IC⊆ {a} {b} (ic-isup i i<x sa<x fc ) (ic-init fcb ) ia<ib = ⊥-elim ( ¬x<0 ia<ib ) - IC⊆ {a} {b} (ic-isup i i<x sa<x fc ) (ic-isup j j<x sb<x fcb ) ia<ib - = ZChain.cfcs (pzc (ob<x lim j<x) ) (o<→≤ ia<ib) o≤-refl (OrdTrans (ZChain.supf-mono (pzc (ob<x lim j<x)) (o<→≤ ia<ib)) sb<x) - (subst (λ k → FClosure A f k a) (zeq _ _ (osucc (o<→≤ ia<ib)) (o<→≤ <-osuc)) fc ) - - ptotalU : IsTotalOrderSet pchainU - ptotalU {a} {b} ia ib with trio< (IChain-i (proj2 ia)) (IChain-i (proj2 ib)) - ... | tri< ia<ib ¬b ¬c = ZChain.f-total (pzc (pic<x (proj2 ib))) (IC⊆ (proj2 ia) (proj2 ib) ia<ib) (pchainU⊆chain ib) - ... | tri≈ ¬a ia=ib ¬c = subst₂ (λ j k → Tri (j < k) (j ≡ k) (k < j)) *iso *iso ( pcmp (proj2 ia) (proj2 ib) ia=ib ) where - pcmp : (ia : IChain ay supfz (& a)) → (ib : IChain ay supfz (& b)) → IChain-i ia ≡ IChain-i ib - → Tri (* (& a) < * (& b)) (* (& a) ≡ * (& b)) (* (& b) < * (& a) ) - pcmp (ic-init fca) (ic-init fcb) eq = fcn-cmp _ f mf fca fcb - pcmp (ic-init fca) (ic-isup i i<x s<x fcb) eq with ZChain.fcy<sup (pzc i<x) o≤-refl fca - ... | case1 eq1 = ct22 where - ct22 : Tri (* (& a) < * (& b)) (* (& a) ≡ * (& b)) (* (& b) < * (& a) ) - ct22 with subst (λ k → k ≤ & b) (sym (zeq _ _ (o<→≤ <-osuc) o≤-refl )) (s≤fc _ f mf fcb ) - ... | case1 eq2 = tri≈ (λ lt → ⊥-elim (<-irr (case1 (sym ct00)) lt)) ct00 (λ lt → ⊥-elim (<-irr (case1 ct00) lt)) where - ct00 : * (& a) ≡ * (& b) - ct00 = cong (*) (trans eq1 eq2) - ... | case2 lt = tri< fc11 (λ eq → <-irr (case1 (sym eq)) fc11) (λ lt → <-irr (case2 fc11) lt) where - fc11 : * (& a) < * (& b) - fc11 = subst (λ k → k < * (& b) ) (cong (*) (sym eq1)) lt - ... | case2 lt = tri< fc11 (λ eq → <-irr (case1 (sym eq)) fc11) (λ lt → <-irr (case2 fc11) lt) where - fc11 : * (& a) < * (& b) - fc11 = ftrans<-≤ lt (subst (λ k → k ≤ & b) (sym (zeq _ _ (o<→≤ <-osuc) o≤-refl )) (s≤fc _ f mf fcb ) ) - pcmp (ic-isup i i<x s<x fca) (ic-init fcb) eq with ZChain.fcy<sup (pzc i<x) o≤-refl fcb - ... | case1 eq1 = ct22 where - ct22 : Tri (* (& a) < * (& b)) (* (& a) ≡ * (& b)) (* (& b) < * (& a) ) - ct22 with subst (λ k → k ≤ & a) (sym (zeq _ _ (o<→≤ <-osuc) o≤-refl )) (s≤fc _ f mf fca ) - ... | case1 eq2 = tri≈ (λ lt → ⊥-elim (<-irr (case1 (sym ct00)) lt)) ct00 (λ lt → ⊥-elim (<-irr (case1 ct00) lt)) where - ct00 : * (& a) ≡ * (& b) - ct00 = cong (*) (sym (trans eq1 eq2)) - ... | case2 lt = tri> (λ lt → <-irr (case2 fc11) lt) (λ eq → <-irr (case1 eq) fc11) fc11 where - fc11 : * (& b) < * (& a) - fc11 = subst (λ k → k < * (& a) ) (cong (*) (sym eq1)) lt - ... | case2 lt = tri> (λ lt → <-irr (case2 fc12) lt) (λ eq → <-irr (case1 eq) fc12) fc12 where - fc12 : * (& b) < * (& a) - fc12 = ftrans<-≤ lt (subst (λ k → k ≤ & a) (sym (zeq _ _ (o<→≤ <-osuc) o≤-refl )) (s≤fc _ f mf fca ) ) - pcmp (ic-isup i i<x s<x fca) (ic-isup i i<y s<y fcb) refl = fcn-cmp _ f mf fca (subst (λ k → FClosure A f k (& b)) pc01 fcb ) where - pc01 : supfz i<y ≡ supfz i<x - pc01 = cong supfz o<-irr - ... | tri> ¬a ¬b ib<ia = ZChain.f-total (pzc (pic<x (proj2 ia))) (pchainU⊆chain ia) (IC⊆ (proj2 ib) (proj2 ia) ib<ia) - - - usup : MinSUP A pchainU - usup = minsupP pchainU (λ ic → proj1 ic ) ptotalU - spu0 = MinSUP.sup usup - - - pchainS : HOD - pchainS = record { od = record { def = λ z → (odef A z ∧ IChain ay supfz z ) - ∨ (FClosure A f spu0 z ∧ (spu0 o< x)) } ; odmax = & A ; <odmax = zc00 } where - zc00 : {z : Ordinal } → (odef A z ∧ IChain ay supfz z ) ∨ (FClosure A f spu0 z ∧ (spu0 o< x) )→ z o< & A - zc00 {z} (case1 lt) = z07 lt - zc00 {z} (case2 fc) = z09 ( A∋fc spu0 f mf (proj1 fc) ) - - zc02 : { a b : Ordinal } → odef A a ∧ IChain ay supfz a → FClosure A f spu0 b ∧ ( spu0 o< x) → a ≤ b - zc02 {a} {b} ca fb = zc05 (proj1 fb) where - zc05 : {b : Ordinal } → FClosure A f spu0 b → a ≤ b - zc05 (fsuc b1 fb ) with proj1 ( mf b1 (A∋fc spu0 f mf fb )) - ... | case1 eq = subst (λ k → a ≤ k ) eq (zc05 fb) - ... | case2 lt = ≤-ftrans (zc05 fb) (case2 lt) - zc05 (init b1 refl) = MinSUP.x≤sup usup ca - - ptotalS : IsTotalOrderSet pchainS - ptotalS (case1 a) (case1 b) = ptotalU a b - ptotalS {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 - eq1 = subst₂ (λ j k → j ≡ k ) *iso *iso (cong (*) eq ) - ... | case2 lt = tri< lt1 (λ eq → <-irr (case1 (sym eq)) lt1) (<-irr (case2 lt1)) where - lt1 : a0 < b0 - lt1 = subst₂ (λ j k → j < k ) *iso *iso lt - ptotalS {b0} {a0} (case2 b) (case1 a) with zc02 a b - ... | case1 eq = tri≈ (<-irr (case1 eq1)) (sym eq1) (<-irr (case1 (sym eq1))) where - eq1 : a0 ≡ b0 - eq1 = subst₂ (λ j k → j ≡ k ) *iso *iso (cong (*) eq ) - ... | case2 lt = tri> (<-irr (case2 lt1)) (λ eq → <-irr (case1 eq) lt1) lt1 where - lt1 : a0 < b0 - lt1 = subst₂ (λ j k → j < k ) *iso *iso lt - ptotalS (case2 a) (case2 b) = subst₂ (λ j k → Tri (j < k) (j ≡ k) (k < j)) *iso *iso (fcn-cmp spu0 f mf (proj1 a) (proj1 b)) - - S⊆A : pchainS ⊆ A - S⊆A (case1 lt) = proj1 lt - S⊆A (case2 fc) = A∋fc _ f mf (proj1 fc) - - ssup : MinSUP A pchainS - ssup = minsupP pchainS S⊆A ptotalS - - zc400 : MinSUP A pchainU → MinSUP A pchainS → ZChain A f mf< ay x - zc400 usup ssup = record { supf = supf1 ; asupf = asupf ; zo≤sz = zo≤sz ; is-minsup = is-minsup ; cfcs = cfcs ; supf-mono = supf-mono } where - - spu = MinSUP.sup usup - sps = MinSUP.sup ssup - - supf1 : Ordinal → Ordinal - supf1 z with trio< z x - ... | tri< a ¬b ¬c = ZChain.supf (pzc (ob<x lim a)) z -- each sup o< x - ... | tri≈ ¬a b ¬c = spu -- sup of all sup o< x - ... | tri> ¬a ¬b c = sps -- sup of spu which o< x - -- if x o< spu, spu is not included in UnionCF x - -- the chain - - pchain : HOD - pchain = UnionCF A f ay supf1 x - - -- pchain ⊆ pchainU ⊆ pchianS - - sf1=sf : {z : Ordinal } → (a : z o< x ) → supf1 z ≡ ZChain.supf (pzc (ob<x lim a)) z - sf1=sf {z} z<x with trio< z x - ... | tri< a ¬b ¬c = cong ( λ k → ZChain.supf (pzc (ob<x lim k)) z) o<-irr - ... | tri≈ ¬a b ¬c = ⊥-elim (¬a z<x) - ... | tri> ¬a ¬b c = ⊥-elim (¬a z<x) - - sf1=spu : {z : Ordinal } → x ≡ z → supf1 z ≡ spu - sf1=spu {z} eq with trio< z x - ... | tri< a ¬b ¬c = ⊥-elim (¬b (sym eq)) - ... | tri≈ ¬a b ¬c = refl - ... | tri> ¬a ¬b c = ⊥-elim (¬b (sym eq)) - - sf1=sps : {z : Ordinal } → (a : x o< z ) → supf1 z ≡ sps - sf1=sps {z} x<z with trio< z x - ... | tri< a ¬b ¬c = ⊥-elim (o<> x<z a) - ... | tri≈ ¬a b ¬c = ⊥-elim (¬c x<z ) - ... | tri> ¬a ¬b c = refl - - asupf : {z : Ordinal } → odef A (supf1 z) - asupf {z} with trio< z x - ... | tri< a ¬b ¬c = ZChain.asupf (pzc (ob<x lim a)) - ... | tri≈ ¬a b ¬c = MinSUP.as usup - ... | tri> ¬a ¬b c = MinSUP.as ssup - - supf-mono : {z y : Ordinal } → z o≤ y → supf1 z o≤ supf1 y - supf-mono {z} {y} z≤y with trio< y x - ... | tri< y<x ¬b ¬c = zc01 where - open o≤-Reasoning O - zc01 : supf1 z o≤ ZChain.supf (pzc (ob<x lim y<x)) y - zc01 = begin - supf1 z ≡⟨ sf1=sf (ordtrans≤-< z≤y y<x) ⟩ - ZChain.supf (pzc (ob<x lim (ordtrans≤-< z≤y y<x))) z ≡⟨ zeq _ _ (osucc z≤y) (o<→≤ <-osuc) ⟩ - ZChain.supf (pzc (ob<x lim y<x)) z ≤⟨ ZChain.supf-mono (pzc (ob<x lim y<x)) z≤y ⟩ - ZChain.supf (pzc (ob<x lim y<x)) y ∎ - ... | tri≈ ¬a b ¬c = zc01 where -- supf1 z o≤ spu - open o≤-Reasoning O - zc01 : supf1 z o≤ spu - zc01 with osuc-≡< (subst (λ k → z o≤ k) b z≤y) - ... | case1 z=x = o≤-refl0 (sf1=spu (sym z=x)) - ... | case2 z<x = subst (λ k → k o≤ spu ) (sym (sf1=sf z<x)) ( IsMinSUP.minsup (ZChain.is-minsup (pzc (ob<x lim z<x)) (o<→≤ <-osuc) ) - (MinSUP.as usup) (λ uw → MinSUP.x≤sup usup (chain⊆pchainU z<x uw)) ) - ... | tri> ¬a ¬b c = zc01 where -- supf1 z o≤ sps - zc01 : supf1 z o≤ sps - zc01 with trio< z x - ... | tri< z<x ¬b ¬c = IsMinSUP.minsup (ZChain.is-minsup (pzc (ob<x lim z<x)) (o<→≤ <-osuc) ) - (MinSUP.as ssup) (λ uw → MinSUP.x≤sup ssup (case1 (chain⊆pchainU z<x uw)) ) - ... | tri≈ ¬a z=x ¬c = MinSUP.minsup usup (MinSUP.as ssup) (λ uw → MinSUP.x≤sup ssup (case1 uw) ) - ... | tri> ¬a ¬b c = o≤-refl -- (sf1=sps c) - - is-minsup : {z : Ordinal} → z o≤ x → IsMinSUP A (UnionCF A f ay supf1 z) (supf1 z) - is-minsup {z} z≤x with osuc-≡< z≤x - ... | case1 z=x = record { as = asupf ; x≤sup = zm00 ; minsup = zm01 } where - zm00 : {w : Ordinal } → odef (UnionCF A f ay supf1 z) w → w ≤ supf1 z - zm00 {w} ⟪ az , ch-init fc ⟫ = subst (λ k → w ≤ k ) (sym (sf1=spu (sym z=x))) ( MinSUP.x≤sup usup ⟪ az , ic-init fc ⟫ ) - zm00 {w} ⟪ az , ch-is-sup u u<b su=u fc ⟫ = subst (λ k → w ≤ k ) (sym (sf1=spu (sym z=x))) - ( MinSUP.x≤sup usup ⟪ az , ic-isup u u<x (o≤-refl0 zm05) (subst (λ k → FClosure A f k w) (sym zm06) fc) ⟫ ) where - u<x : u o< x - u<x = subst (λ k → u o< k) z=x u<b - zm06 : supfz (subst (λ k → u o< k) z=x u<b) ≡ supf1 u - zm06 = trans (zeq _ _ o≤-refl (o<→≤ <-osuc) ) (sym (sf1=sf u<x )) - zm05 : supfz (subst (λ k → u o< k) z=x u<b) ≡ u - zm05 = trans zm06 su=u - zm01 : { s : Ordinal } → odef A s → ( {x : Ordinal } → odef (UnionCF A f ay supf1 z) x → x ≤ s ) → supf1 z o≤ s - zm01 {s} as sup = subst (λ k → k o≤ s ) (sym (sf1=spu (sym z=x))) ( MinSUP.minsup usup as zm02 ) where - zm02 : {w : Ordinal } → odef pchainU w → w ≤ s - zm02 {w} uw with pchainU⊆chain uw - ... | ⟪ az , ch-init fc ⟫ = sup ⟪ az , ch-init fc ⟫ - ... | ⟪ az , ch-is-sup u1 u<b su=u fc ⟫ = sup ⟪ az , ch-is-sup u1 (ordtrans u<b zm05) (trans zm03 su=u) zm04 ⟫ where - zm05 : osuc (IChain-i (proj2 uw)) o< z - zm05 = subst (λ k → osuc (IChain-i (proj2 uw)) o< k) (sym z=x) ( pic<x (proj2 uw) ) - u<x : u1 o< x - u<x = subst (λ k → u1 o< k) z=x ( ordtrans u<b zm05 ) - zm03 : supf1 u1 ≡ ZChain.supf (prev (osuc (IChain-i (proj2 uw))) (pic<x (proj2 uw))) u1 - zm03 = trans (sf1=sf u<x) (zeq _ _ (osucc u<b) (o<→≤ <-osuc) ) - zm04 : FClosure A f (supf1 u1) w - zm04 = subst (λ k → FClosure A f k w) (sym zm03) fc - ... | case2 z<x = record { as = asupf ; x≤sup = zm00 ; minsup = zm01 } where - supf0 = ZChain.supf (pzc (ob<x lim z<x)) - msup : IsMinSUP A (UnionCF A f ay supf0 z) (supf0 z) - msup = ZChain.is-minsup (pzc (ob<x lim z<x)) (o<→≤ <-osuc) - s1=0 : {u : Ordinal } → u o< z → supf1 u ≡ supf0 u - s1=0 {u} u<z = trans (sf1=sf (ordtrans u<z z<x)) (zeq _ _ (o<→≤ (osucc u<z)) (o<→≤ <-osuc) ) - zm00 : {w : Ordinal } → odef (UnionCF A f ay supf1 z) w → w ≤ supf1 z - zm00 {w} ⟪ az , ch-init fc ⟫ = subst (λ k → w ≤ k ) (sym (sf1=sf z<x)) ( IsMinSUP.x≤sup msup ⟪ az , ch-init fc ⟫ ) - zm00 {w} ⟪ az , ch-is-sup u u<b su=u fc ⟫ = subst (λ k → w ≤ k ) (sym (sf1=sf z<x)) - ( IsMinSUP.x≤sup msup ⟪ az , ch-is-sup u u<b (trans (sym (s1=0 u<b)) su=u) (subst (λ k → FClosure A f k w) (s1=0 u<b) fc) ⟫ ) - zm01 : { s : Ordinal } → odef A s → ( {x : Ordinal } → odef (UnionCF A f ay supf1 z) x → x ≤ s ) → supf1 z o≤ s - zm01 {s} as sup = subst (λ k → k o≤ s ) (sym (sf1=sf z<x)) ( IsMinSUP.minsup msup as zm02 ) where - zm02 : {w : Ordinal } → odef (UnionCF A f ay supf0 z) w → w ≤ s - zm02 {w} ⟪ az , ch-init fc ⟫ = sup ⟪ az , ch-init fc ⟫ - zm02 {w} ⟪ az , ch-is-sup u u<b su=u fc ⟫ = sup - ⟪ az , ch-is-sup u u<b (trans (s1=0 u<b) su=u) (subst (λ k → FClosure A f k w) (sym (s1=0 u<b)) fc) ⟫ - - - cfcs : {a b w : Ordinal } → a o< b → b o≤ x → supf1 a o< b → FClosure A f (supf1 a) w → odef (UnionCF A f ay supf1 b) w - cfcs {a} {b} {w} a<b b≤x sa<b fc with osuc-≡< b≤x - ... | case1 b=x with trio< a x - ... | tri< a<x ¬b ¬c = zc40 where - sa = ZChain.supf (pzc (ob<x lim a<x)) a - m = omax a sa -- x is limit ordinal, so we have sa o< m o< x - m<x : m o< x - m<x with trio< a sa | inspect (omax a) sa - ... | tri< a<sa ¬b ¬c | record { eq = eq } = ob<x lim (ordtrans<-≤ sa<b b≤x ) - ... | tri≈ ¬a a=sa ¬c | record { eq = eq } = subst (λ k → k o< x) eq zc41 where - zc41 : omax a sa o< x - zc41 = osucprev ( begin - osuc ( omax a sa ) ≡⟨ cong (λ k → osuc (omax a k)) (sym a=sa) ⟩ - osuc ( omax a a ) ≡⟨ cong osuc (omxx _) ⟩ - osuc ( osuc a ) ≤⟨ o<→≤ (ob<x lim (ob<x lim a<x)) ⟩ - x ∎ ) where open o≤-Reasoning O - ... | tri> ¬a ¬b c | record { eq = eq } = ob<x lim a<x - sam = ZChain.supf (pzc (ob<x lim m<x)) a - zc42 : osuc a o≤ osuc m - zc42 = osucc (o<→≤ ( omax-x _ _ ) ) - sam<m : sam o< m - sam<m = subst (λ k → k o< m ) (supf-unique A f mf< ay zc42 (pzc (ob<x lim a<x)) (pzc (ob<x lim m<x)) (o<→≤ <-osuc)) ( omax-y _ _ ) - fcm : FClosure A f (ZChain.supf (pzc (ob<x lim m<x)) a) w - fcm = subst (λ k → FClosure A f k w ) (zeq (ob<x lim a<x) (ob<x lim m<x) zc42 (o<→≤ <-osuc) ) fc - zcm : odef (UnionCF A f ay (ZChain.supf (pzc (ob<x lim m<x))) (osuc (omax a sa))) w - zcm = ZChain.cfcs (pzc (ob<x lim m<x)) (o<→≤ (omax-x _ _)) o≤-refl (o<→≤ sam<m) fcm - zc40 : odef (UnionCF A f ay supf1 b) w - zc40 with ZChain.cfcs (pzc (ob<x lim m<x)) (o<→≤ (omax-x _ _)) o≤-refl (o<→≤ sam<m) fcm - ... | ⟪ az , ch-init fc ⟫ = ⟪ az , ch-init fc ⟫ - ... | ⟪ az , ch-is-sup u u<x su=u fc ⟫ = ⟪ az , ch-is-sup u u<b (trans zc45 su=u) zc44 ⟫ where - u<b : u o< b - u<b = osucprev ( begin - osuc u ≤⟨ osucc u<x ⟩ - osuc m ≤⟨ osucc m<x ⟩ - x ≡⟨ sym b=x ⟩ - b ∎ ) where open o≤-Reasoning O - zc45 : supf1 u ≡ ZChain.supf (pzc (ob<x lim m<x)) u - zc45 = begin - supf1 u ≡⟨ sf1=sf (subst (λ k → u o< k) b=x u<b ) ⟩ - ZChain.supf (pzc (ob<x lim (subst (λ k → u o< k) b=x u<b ))) u ≡⟨ zeq _ _ (osucc u<x) (o<→≤ <-osuc) ⟩ - ZChain.supf (pzc (ob<x lim m<x)) u ∎ where open ≡-Reasoning - zc44 : FClosure A f (supf1 u) w - zc44 = subst (λ k → FClosure A f k w ) (sym zc45) fc - ... | tri≈ ¬a b ¬c = ⊥-elim ( ¬a (ordtrans<-≤ a<b b≤x)) - ... | tri> ¬a ¬b c = ⊥-elim ( ¬a (ordtrans<-≤ a<b b≤x)) - cfcs {a} {b} {w} a<b b≤x sa<b fc | case2 b<x = zc40 where - supfb = ZChain.supf (pzc (ob<x lim b<x)) - sb=sa : {a : Ordinal } → a o< b → supf1 a ≡ ZChain.supf (pzc (ob<x lim b<x)) a - sb=sa {a} a<b = trans (sf1=sf (ordtrans<-≤ a<b b≤x)) (zeq _ _ (o<→≤ (osucc a<b)) (o<→≤ <-osuc) ) - fcb : FClosure A f (supfb a) w - fcb = subst (λ k → FClosure A f k w) (sb=sa a<b) fc - -- supfb a o< b assures it is in Union b - zcb : odef (UnionCF A f ay supfb b) w - zcb = ZChain.cfcs (pzc (ob<x lim b<x)) a<b (o<→≤ <-osuc) (subst (λ k → k o< b) (sb=sa a<b) sa<b) fcb - zc40 : odef (UnionCF A f ay supf1 b) w - zc40 with zcb - ... | ⟪ az , ch-init fc ⟫ = ⟪ az , ch-init fc ⟫ - ... | ⟪ az , ch-is-sup u u<x su=u fc ⟫ = ⟪ az , ch-is-sup u u<x (trans zc45 su=u) zc44 ⟫ where - zc45 : supf1 u ≡ ZChain.supf (pzc (ob<x lim b<x)) u - zc45 = begin - supf1 u ≡⟨ sf1=sf (ordtrans u<x b<x) ⟩ - ZChain.supf (pzc (ob<x lim (ordtrans u<x b<x) )) u ≡⟨ zeq _ _ (o<→≤ (osucc u<x)) (o<→≤ <-osuc) ⟩ - ZChain.supf (pzc (ob<x lim b<x )) u ∎ where open ≡-Reasoning - zc44 : FClosure A f (supf1 u) w - zc44 = subst (λ k → FClosure A f k w ) (sym zc45) fc - - zo≤sz : {z : Ordinal} → z o≤ x → z o≤ supf1 z - zo≤sz {z} z≤x with osuc-≡< z≤x - ... | case2 z<x = subst (λ k → z o≤ k) (sym (trans (sf1=sf z<x) (sym (zeq _ _ (o<→≤ <-osuc) o≤-refl)))) ( ZChain.zo≤sz (pzc z<x) o≤-refl ) - ... | case1 refl with x<y∨y≤x (supf1 spu) x - ... | case2 x≤ssp = z40 where - z40 : z o≤ supf1 z - z40 with x<y∨y≤x z spu - ... | case1 z<spu = o<→≤ ( subst (λ k → z o< k ) (sym (sf1=spu refl)) z<spu ) - ... | case2 spu≤z = begin -- x ≡ supf1 spu ≡ spu ≡ supf1 x - x ≤⟨ x≤ssp ⟩ - supf1 spu ≤⟨ supf-mono spu≤z ⟩ - supf1 x ∎ where open o≤-Reasoning O - ... | case1 ssp<x = subst (λ k → x o≤ k) (sym (sf1=spu refl)) z47 where - z47 : x o≤ spu - z47 with x<y∨y≤x spu x - ... | case2 lt = lt - ... | case1 spu<x = ⊥-elim ( <<-irr (MinSUP.x≤sup usup z48) (proj1 ( mf< spu (MinSUP.as usup)))) where - z70 : odef (UnionCF A f ay supf1 z) (supf1 spu) - z70 = cfcs spu<x o≤-refl ssp<x (init asupf refl ) - z73 : IsSUP A (UnionCF A f ay (ZChain.supf (pzc (ob<x lim spu<x))) spu) spu - z73 = record { ax = MinSUP.as usup ; x≤sup = λ uw → MinSUP.x≤sup usup (chain⊆pchainU spu<x uw ) } - z49 : supfz spu<x ≡ spu - z49 = begin - supfz spu<x ≡⟨ ZChain.sup=u (pzc (ob<x lim spu<x)) (MinSUP.as usup) (o<→≤ <-osuc) z73 ⟩ - spu ∎ where open ≡-Reasoning - z50 : supfz spu<x o≤ spu - z50 = o≤-refl0 z49 - z48 : odef pchainU (f spu) - z48 = ⟪ proj2 (mf _ (MinSUP.as usup) ) , ic-isup _ (subst (λ k → k o< x) refl spu<x) z50 - (fsuc _ (init (ZChain.asupf (pzc (ob<x lim spu<x))) z49)) ⟫ - +-- -- Uncountable ascending chain by axiom of choice +-- cf : ¬ Maximal A → Ordinal → Ordinal +-- cf nmx x with ODC.∋-p O A (* x) +-- ... | no _ = o∅ +-- ... | yes ax with is-o∅ (& ( Gtx ax )) +-- ... | yes nogt = -- no larger element, so it is maximal +-- ⊥-elim (no-maximum (z08 nmx) x ⟪ subst (λ k → odef A k) &iso ax , x-is-maximal nmx (subst (λ k → odef A k ) &iso ax) nogt ⟫ ) +-- ... | no not = & (ODC.minimal O (Gtx ax) (λ eq → not (=od∅→≡o∅ eq))) +-- is-cf : (nmx : ¬ Maximal A ) → {x : Ordinal} → odef A x → odef A (cf nmx x) ∧ ( * x < * (cf nmx x) ) +-- is-cf nmx {x} ax with ODC.∋-p O A (* x) +-- ... | no not = ⊥-elim ( not (subst (λ k → odef A k ) (sym &iso) ax )) +-- ... | yes ax with is-o∅ (& ( Gtx ax )) +-- ... | yes nogt = ⊥-elim (no-maximum (z08 nmx) x ⟪ subst (λ k → odef A k) &iso ax , x-is-maximal nmx (subst (λ k → odef A k ) &iso ax) nogt ⟫ ) +-- ... | no not = ODC.x∋minimal O (Gtx ax) (λ eq → not (=od∅→≡o∅ eq)) +-- +-- --- +-- --- infintie ascention sequence of f +-- --- +-- cf-is-<-monotonic : (nmx : ¬ Maximal A ) → (x : Ordinal) → odef A x → ( * x < * (cf nmx x) ) ∧ odef A (cf nmx x ) +-- cf-is-<-monotonic nmx x ax = ⟪ proj2 (is-cf nmx ax ) , proj1 (is-cf nmx ax ) ⟫ +-- cf-is-≤-monotonic : (nmx : ¬ Maximal A ) → ≤-monotonic-f A ( cf nmx ) +-- cf-is-≤-monotonic nmx x ax = ⟪ case2 (proj1 ( cf-is-<-monotonic nmx x ax )) , proj2 ( cf-is-<-monotonic nmx x ax ) ⟫ --- --- the maximum chain has fix point of any ≤-monotonic function --- - 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 + record ZChain ( A : HOD ) {y : Ordinal} (ay : odef A y) (x : Ordinal) : Set (Level.suc n) where + field + chain : HOD + chain⊆A : chain ⊆ A + f-total : IsTotalOrderSet chain + cf : Ordinal → Ordinal + is-cf : {x : Ordinal} → odef A x → odef A (cf x) ∧ ( * x < * (cf x) ) + f-next : {x : Ordinal } → odef chain x → odef chain (cf x) + fixpoint : (sp1 : MinSUP A chain ) → odef chain (MinSUP.sup sp1) + cf-is-<-monotonic : <-monotonic-f A cf + cf-is-<-monotonic x ax = ⟪ proj2 (is-cf ax ) , proj1 (is-cf ax ) ⟫ + cf-is-≤-monotonic : ≤-monotonic-f A cf + cf-is-≤-monotonic x ax = ⟪ case2 (proj1 ( cf-is-<-monotonic x ax )) , proj2 ( cf-is-<-monotonic x ax ) ⟫ - 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 ay (ZChain.supf zc) x) - msp0 f mf< {x} ay zc = minsupP (UnionCF A f ay (ZChain.supf zc) x) (ZChain.chain⊆A zc) (ZChain.f-total zc) + SZ : ¬ Maximal A → {y : Ordinal} (ay : odef A y) → (x : Ordinal) → ZChain A ay x + SZ nmx {y} ay x = TransFinite {λ z → ZChain A ay z } (λ x → ind x ) x where + ind : (x : Ordinal) → ((z : Ordinal) → z o< x → ZChain A ay z) → ZChain A ay x + ind x prev = ? -- with Oprev-p x + +-- record { +-- chain = record { od = record { def = λ x → odef A x ∧ IChain A f x } ; odmax = & A ; <odmax = λ lt → z09 (proj1 lt) } ; +-- chain⊆A = λ cx → proj1 cx ; +-- f-total = λ ia ib → subst₂ (λ j k → Tri (j < k) (j ≡ k) (k < j)) *iso *iso (f-total (proj2 ia) (proj2 ib)) ; +-- f-next = λ ix → ⟪ ? , f-next (proj2 ix) ⟫ ; +-- fixpoint = λ sp1 → ⟪ ? , ? ⟫ +-- } where +-- f-total : {a b : Ordinal } → IChain A f a → IChain A f b → Tri (a << b) (* a ≡ * b) (b << a) +-- f-total = ? +-- f-next : {a : Ordinal } → IChain A f a → IChain A f (f a) +-- f-next record { y = y ; x=fy = x=fy } = record { y = f y ; x=fy = cong f x=fy } + + msp0 : ( f : Ordinal → Ordinal ) → (mf< : <-monotonic-f A f ) {y : Ordinal} (ay : odef A y) + → (zc : ZChain A ay (& A) ) + → MinSUP A (ZChain.chain zc) + msp0 f mf< {x} ay zc = minsupP (ZChain.chain zc) (ZChain.chain⊆A zc) (ZChain.f-total zc) -- f eventualy stop -- we can prove contradict here, it is here for a historical reason -- - fixpoint : ( f : Ordinal → Ordinal ) → (mf : ≤-monotonic-f A f ) (mf< : <-monotonic-f A f ) (zc : ZChain A f mf< as0 (& A) ) + fixpoint : (zc : ZChain A as0 (& A)) → (sp1 : MinSUP A (ZChain.chain zc)) - → f (MinSUP.sup sp1) ≡ MinSUP.sup sp1 - fixpoint f mf mf< zc sp1 = z14 where + → ZChain.cf zc (MinSUP.sup sp1) ≡ MinSUP.sup sp1 + fixpoint zc sp1 = z14 where chain = ZChain.chain zc - supf = ZChain.supf zc sp : Ordinal sp = MinSUP.sup sp1 asp : odef A sp asp = MinSUP.as sp1 - ay = as0 - z10 : {a b : Ordinal } → (ca : odef chain a ) → b o< (& A) → (ab : odef A b ) - → HasPrev A chain f b ∨ IsSUP A (UnionCF A f ay (ZChain.supf zc) b) b - → * a < * b → odef chain b - z10 = ZChain1.is-max (SZ1 f mf mf< as0 zc (& A) o≤-refl ) - z22 : sp o< & A - z22 = z09 asp + f = ZChain.cf zc + mf : ≤-monotonic-f A f + mf = ZChain.cf-is-≤-monotonic zc z12 : odef chain sp - z12 with o≡? (& s) sp - ... | yes eq = subst (λ k → odef chain k) eq ( ZChain.chain∋init zc ) - ... | no ne = ZChain1.is-max (SZ1 f mf mf< as0 zc (& A) o≤-refl) {& s} {sp} ( ZChain.chain∋init zc ) (z09 asp) asp (case2 z19 ) z13 where - z13 : * (& s) < * sp - z13 with MinSUP.x≤sup sp1 ( ZChain.chain∋init zc ) - ... | case1 eq = ⊥-elim ( ne eq ) - ... | case2 lt = lt - z19 : IsSUP A (UnionCF A f ay (ZChain.supf zc) sp) sp - z19 = record { ax = asp ; x≤sup = z20 } where - z20 : {y : Ordinal} → odef (UnionCF A f ay (ZChain.supf zc) sp) y → (y ≡ sp) ∨ (y << sp) - 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 ) - ... | case2 y<p = case2 (subst (λ k → * k < _ ) &iso y<p ) + z12 = ZChain.fixpoint zc sp1 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 @@ -1559,16 +364,15 @@ -- ¬ Maximal create cf which is a <-monotonic function by axiom of choice. This contradicts fix point of ZChain -- - ¬Maximal→¬cf-mono : (nmx : ¬ Maximal A ) → (zc : ZChain A (cf nmx) (cf-is-<-monotonic nmx) as0 (& A)) → ⊥ - ¬Maximal→¬cf-mono nmx zc = <-irr0 {* (cf nmx c)} {* c} - (subst (λ k → odef A k ) (sym &iso) (proj1 (is-cf nmx (MinSUP.as msp1 )))) + ¬Maximal→¬cf-mono : (nmx : ¬ Maximal A ) → (zc : ZChain A as0 (& A)) → ⊥ + ¬Maximal→¬cf-mono nmx zc = <-irr0 {* (ZChain.cf zc c)} {* c} + (subst (λ k → odef A k ) (sym &iso) (proj1 (ZChain.is-cf zc (MinSUP.as msp1 )))) (subst (λ k → odef A k) (sym &iso) (MinSUP.as msp1) ) - (case1 ( cong (*)( fixpoint (cf nmx) (cf-is-≤-monotonic nmx ) (cf-is-<-monotonic nmx ) zc msp1 ))) -- x ≡ f x ̄ - (proj1 (cf-is-<-monotonic nmx c (MinSUP.as msp1 ))) where -- x < f x + (case1 ( cong (*)( fixpoint zc msp1 ))) -- x ≡ f x ̄ + (proj1 (ZChain.cf-is-<-monotonic zc c (MinSUP.as 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 + msp1 = msp0 (ZChain.cf zc) (ZChain.cf-is-<-monotonic zc) as0 zc c : Ordinal c = MinSUP.sup msp1 @@ -1582,7 +386,7 @@ 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 ( ¬Maximal→¬cf-mono nmx (SZ (cf nmx) (cf-is-<-monotonic nmx) as0 (& A) )) where + ... | yes ¬Maximal = ⊥-elim ( ¬Maximal→¬cf-mono nmx (SZ nmx as0 (& A) )) where -- if we have no maximal, make ZChain, which contradict SUP condition nmx : ¬ Maximal A nmx mx = ∅< {HasMaximal} zc5 ( ≡o∅→=od∅ ¬Maximal ) where