Mercurial > hg > Members > kono > Proof > ZF-in-agda
changeset 536:c43375ade2c5
remove unsed in zorn
| author | Shinji KONO <kono@ie.u-ryukyu.ac.jp> |
|---|---|
| date | Sat, 23 Apr 2022 18:39:07 +0900 |
| parents | b83dde5dbd33 |
| children | e12add1519ec |
| files | src/zorn.agda |
| diffstat | 1 files changed, 0 insertions(+), 279 deletions(-) [+] |
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--- a/src/zorn.agda Sat Apr 23 18:35:20 2022 +0900 +++ b/src/zorn.agda Sat Apr 23 18:39:07 2022 +0900 @@ -143,191 +143,6 @@ -- OS<-cmp : {x : HOD} → Trichotomous {_} {IsTotalOrderSet x} _OS≈_ _OS<_ -- OS<-cmp A B = {!!} --- tree structure -data IChain (A : HOD) : Ordinal → Set n where - ifirst : {ox : Ordinal} → odef A ox → IChain A ox - inext : {ox oy : Ordinal} → odef A oy → * ox < * oy → IChain A ox → IChain A oy - --- * ox < .. < * oy -ic-connect : {A : HOD} {oy : Ordinal} → (ox : Ordinal) → (iy : IChain A oy) → Set n -ic-connect {A} ox (ifirst {oy} ay) = Lift n ⊥ -ic-connect {A} ox (inext {oy} {oz} ay y<z iz) = (ox ≡ oy) ∨ ic-connect ox iz - -ic→odef : {A : HOD} {ox : Ordinal} → IChain A ox → odef A ox -ic→odef {A} {ox} (ifirst ax) = ax -ic→odef {A} {ox} (inext ax x<y ic) = ax - -ic→< : {A : HOD} → IsPartialOrderSet A → (x : Ordinal) → odef A x → {y : Ordinal} → (iy : IChain A y) → ic-connect {A} {y} x iy → * x < * y -ic→< {A} PO x ax {y} (ifirst ay) () -ic→< {A} PO x ax {y} (inext ay x<y iy) (case1 refl) = x<y -ic→< {A} PO x ax {y} (inext {oy} ay x<y iy) (case2 ic) = IsStrictPartialOrder.trans PO - {me (subst (λ k → odef A k) (sym &iso) ax )} {me (subst (λ k → odef A k) (sym &iso) (ic→odef {A} {oy} iy) ) } {me (subst (λ k → odef A k) (sym &iso) ay) } - (ic→< {A} PO x ax iy ic ) x<y - -record IChained (A : HOD) (x y : Ordinal) : Set n where - field - iy : IChain A y - ic : ic-connect x iy - --- --- all tree from x --- -IChainSet : (A : HOD) {x : Ordinal} → odef A x → HOD -IChainSet A {x} ax = record { od = record { def = λ y → odef A y ∧ IChained A x y } - ; odmax = & A ; <odmax = λ {y} lt → subst (λ k → k o< & A) &iso (c<→o< (subst (λ k → odef A k) (sym &iso) (proj1 lt))) } - -IChainSet⊆A : {A : HOD} → {x : Ordinal } → (ax : odef A x ) → IChainSet A ax ⊆ A -IChainSet⊆A {A} x = record { incl = λ {oy} y → proj1 y } - -¬IChained-refl : (A : HOD) {x : Ordinal} → IsPartialOrderSet A → ¬ IChained A x x -¬IChained-refl A {x} PO record { iy = iy ; ic = ic } = IsStrictPartialOrder.irrefl PO - {me (subst (λ k → odef A k ) (sym &iso) ic0) } {me (subst (λ k → odef A k ) (sym &iso) ic0) } refl (ic→< {A} PO x ic0 iy ic ) where - ic0 : odef A x - ic0 = ic→odef {A} iy - --- there is a y, & y > & x - -record OSup> (A : HOD) {x : Ordinal} (ax : A ∋ * x) : Set n where - field - y : Ordinal - icy : odef (IChainSet A ax ) y - y>x : x o< y - -record IChainSup> (A : HOD) {x : Ordinal} (ax : A ∋ * x) : Set n where - field - y : Ordinal - A∋y : odef A y - y>x : * x < * y - --- finite IChain --- --- tree structured - -ic→A∋y : (A : HOD) {x y : Ordinal} (ax : A ∋ * x) → odef (IChainSet A ax) y → A ∋ * y -ic→A∋y A {x} {y} ax ⟪ ay , _ ⟫ = subst (λ k → odef A k) (sym &iso) ay - -record InfiniteChain (A : HOD) (max : Ordinal) {x : Ordinal} (ax : A ∋ * x) : Set n where - field - chain<x : (y : Ordinal ) → odef (IChainSet A ax) y → y o< max - c-infinite : (y : Ordinal ) → (cy : odef (IChainSet A ax) y ) - → IChainSup> A (ic→A∋y A ax cy) - -open import Data.Nat hiding (_<_ ; _≤_ ) -import Data.Nat.Properties as NP -open import nat - -data Chain (A : HOD) (s : Ordinal) (next : Ordinal → Ordinal ) : ( x : Ordinal ) → Set n where - cfirst : odef A s → Chain A s next s - csuc : (x : Ordinal ) → (ax : odef A x ) → Chain A s next x → odef A (next x) → Chain A s next (next x ) - -ct∈A : (A : HOD ) (s : Ordinal) → (next : Ordinal → Ordinal ) → {x : Ordinal} → Chain A s next x → odef A x -ct∈A A s next {x} (cfirst x₁) = x₁ -ct∈A A s next {.(next x )} (csuc x ax t anx) = anx - --- --- extract single chain from countable infinite chains --- -TransitiveClosure : (A : HOD) (s : Ordinal) → (next : Ordinal → Ordinal ) → HOD -TransitiveClosure A s next = record { od = record { def = λ x → Chain A s next x } ; odmax = & A ; <odmax = cc01 } where - cc01 : {y : Ordinal} → Chain A s next y → y o< & A - cc01 {y} cy = subst (λ k → k o< & A ) &iso ( c<→o< (subst (λ k → odef A k ) (sym &iso) ( ct∈A A s next cy ) ) ) - -cton0 : (A : HOD ) (s : Ordinal) → (next : Ordinal → Ordinal ) {y : Ordinal } → Chain A s next y → ℕ -cton0 A s next (cfirst _) = zero -cton0 A s next (csuc x ax z _) = suc (cton0 A s next z) -cton : (A : HOD ) (s : Ordinal) → (next : Ordinal → Ordinal ) → Element (TransitiveClosure A s next) → ℕ -cton A s next y = cton0 A s next (is-elm y) - -cinext : (A : HOD) {x max : Ordinal } → (ax : A ∋ * x ) → (ifc : InfiniteChain A max ax ) → Ordinal → Ordinal -cinext A ax ifc y with ODC.∋-p O (IChainSet A ax) (* y) -... | yes ics-y = IChainSup>.y ( InfiniteChain.c-infinite ifc y (subst (λ k → odef (IChainSet A ax) k) &iso ics-y )) -... | no _ = o∅ - -InFCSet : (A : HOD) → {x max : Ordinal} (ax : A ∋ * x) - → (ifc : InfiniteChain A max ax ) → HOD -InFCSet A {x} ax ifc = TransitiveClosure (IChainSet A ax) x (cinext A ax ifc ) - -InFCSet⊆A : (A : HOD) → {x max : Ordinal} (ax : A ∋ * x) → (ifc : InfiniteChain A max ax ) → InFCSet A ax ifc ⊆ A -InFCSet⊆A A {x} ax ifc = record { incl = λ {y} lt → incl (IChainSet⊆A ax) ( - ct∈A (IChainSet A ax) x (cinext A ax ifc) lt ) } - -cinext→IChainSup : (A : HOD) {x max : Ordinal } → (ax : A ∋ * x ) → (ifc : InfiniteChain A max ax ) → (y : Ordinal ) - → (ay1 : IChainSet A ax ∋ * y ) → IChainSup> A (subst (odef A) (sym &iso) (proj1 (subst (λ k → odef (IChainSet A ax) k) &iso ay1))) -cinext→IChainSup A {x} ax ifc y ay with ODC.∋-p O (IChainSet A ax) (* y) -... | no not = ⊥-elim ( not ay ) -... | yes ay1 = InfiniteChain.c-infinite ifc y (subst (λ k → odef (IChainSet A ax) k) &iso ay ) - -TransitiveClosure-is-total : (A : HOD) → {x max : Ordinal} (ax : A ∋ * x) - → IsPartialOrderSet A - → (ifc : InfiniteChain A max ax ) - → IsTotalOrderSet ( InFCSet A ax ifc ) -TransitiveClosure-is-total A {x} ax PO ifc = record { isEquivalence = IsStrictPartialOrder.isEquivalence IPO - ; trans = λ {x} {y} {z} → IsStrictPartialOrder.trans IPO {x} {y} {z} ; compare = cmp } where - IPO : IsPartialOrderSet (InFCSet A ax ifc ) - IPO = ⊆-IsPartialOrderSet record { incl = λ {y} lt → incl (InFCSet⊆A A {x} ax ifc) lt} PO - B = IChainSet A ax - cnext = cinext A ax ifc - ct02 : {oy : Ordinal} → (y : Chain B x cnext oy ) → A ∋ * oy - ct02 y = incl (IChainSet⊆A {A} ax) (subst (λ k → odef (IChainSet A ax) k) (sym &iso) (ct∈A B x cnext y) ) - ct-inject : {ox oy : Ordinal} → (x1 : Chain B x cnext ox ) → (y : Chain B x cnext oy ) - → (cton0 B x cnext x1) ≡ (cton0 B x cnext y) → ox ≡ oy - ct-inject {ox} {ox} (cfirst x) (cfirst x₁) refl = refl - ct-inject {.(cnext x₀ )} {.(cnext x₃ )} (csuc x₀ ax x₁ x₂) (csuc x₃ ax₁ y x₄) eq = cong cnext ct05 where - ct06 : {x y : ℕ} → suc x ≡ suc y → x ≡ y - ct06 refl = refl - ct05 : x₀ ≡ x₃ - ct05 = ct-inject x₁ y (ct06 eq) - ct-monotonic : {ox oy : Ordinal} → (x1 : Chain B x cnext ox ) → (y : Chain B x cnext oy ) - → (cton0 B x cnext x1) Data.Nat.< (cton0 B x cnext y) → * ox < * oy - ct-monotonic {ox} {oy} x1 (csuc oy1 ay y _) (s≤s lt) with NP.<-cmp ( cton0 B x cnext x1 ) ( cton0 B x cnext y ) - ... | tri< a ¬b ¬c = ct07 where - ct07 : * ox < * (cnext oy1) - ct07 with ODC.∋-p O (IChainSet A ax) (* oy1) - ... | no not = ⊥-elim ( not (subst (λ k → odef (IChainSet A ax) k ) (sym &iso) ay ) ) - ... | yes ay1 = IsStrictPartialOrder.trans PO {me (ct02 x1) } {me (ct02 y)} {me ct031 } (ct-monotonic x1 y a ) ct011 where - ct031 : A ∋ * (IChainSup>.y (InfiniteChain.c-infinite ifc oy1 (subst (λ k → odef (IChainSet A ax) k) &iso ay1 ) )) - ct031 = subst (λ k → odef A k ) (sym &iso) ( - IChainSup>.A∋y (InfiniteChain.c-infinite ifc oy1 (subst (λ k → odef (IChainSet A ax) k) &iso ay1 )) ) - ct011 : * oy1 < * ( IChainSup>.y (InfiniteChain.c-infinite ifc oy1 (subst (λ k → odef (IChainSet A ax) k) &iso ay1 )) ) - ct011 = IChainSup>.y>x (InfiniteChain.c-infinite ifc oy1 (subst (λ k → odef (IChainSet A ax) k) &iso ay1 )) - ... | tri≈ ¬a b ¬c = ct11 where - ct11 : * ox < * (cnext oy1) - ct11 with ODC.∋-p O (IChainSet A ax) (* oy1) - ... | no not = ⊥-elim ( not (subst (λ k → odef (IChainSet A ax) k ) (sym &iso) ay ) ) - ... | yes ay1 = subst (λ k → * k < _) (sym (ct-inject _ _ b)) ct011 where - ct011 : * oy1 < * ( IChainSup>.y (InfiniteChain.c-infinite ifc oy1 (subst (λ k → odef (IChainSet A ax) k) &iso ay1 )) ) - ct011 = IChainSup>.y>x (InfiniteChain.c-infinite ifc oy1 (subst (λ k → odef (IChainSet A ax) k) &iso ay1 )) - ... | tri> ¬a ¬b c = ⊥-elim ( nat-≤> lt c ) - ct12 : {y z : Element (TransitiveClosure B x cnext) } → elm y ≡ elm z → elm y < elm z → ⊥ - ct12 {y} {z} y=z y<z = IsStrictPartialOrder.irrefl IPO {y} {z} y=z y<z - ct13 : {y z : Element (TransitiveClosure B x cnext) } → elm y < elm z → elm z < elm y → ⊥ - ct13 {y} {z} y<z y>z = IsStrictPartialOrder.irrefl IPO {y} {y} refl ( IsStrictPartialOrder.trans IPO {y} {z} {y} y<z y>z ) - ct17 : (x1 : Element (TransitiveClosure B x cnext)) → Chain B x cnext (& (elm x1)) - ct17 x1 = is-elm x1 - cmp : Trichotomous _ _ - cmp x1 y with NP.<-cmp (cton B x cnext x1) (cton B x cnext y) - ... | tri< a ¬b ¬c = tri< ct04 ct14 ct15 where - ct04 : elm x1 < elm y - ct04 = subst₂ (λ j k → j < k ) *iso *iso (ct-monotonic (is-elm x1) (is-elm y) a) - ct14 : ¬ elm x1 ≡ elm y - ct14 eq = ct12 {x1} {y} eq (subst₂ (λ j k → j < k ) *iso *iso (ct-monotonic (is-elm x1) (is-elm y) a ) ) - ct15 : ¬ (elm y < elm x1) - ct15 lt = ct13 {y} {x1} lt (subst₂ (λ j k → j < k ) *iso *iso (ct-monotonic (is-elm x1) (is-elm y) a ) ) - ... | tri≈ ¬a b ¬c = tri≈ (ct12 {x1} {y} ct16) ct16 (ct12 {y} {x1} (sym ct16)) where - ct16 : elm x1 ≡ elm y - ct16 = subst₂ (λ j k → j ≡ k ) *iso *iso (cong (*) (ct-inject {& (elm x1)} {& (elm y)} (is-elm x1) (is-elm y) b )) - ... | tri> ¬a ¬b c = tri> ct15 ct14 ct04 where - ct04 : elm y < elm x1 - ct04 = subst₂ (λ j k → j < k ) *iso *iso (ct-monotonic (is-elm y) (is-elm x1) c) - ct14 : ¬ elm x1 ≡ elm y - ct14 eq = ct12 {y} {x1} (sym eq) (subst₂ (λ j k → j < k ) *iso *iso (ct-monotonic (is-elm y) (is-elm x1) c ) ) - ct15 : ¬ (elm x1 < elm y) - ct15 lt = ct13 {x1} {y} lt (subst₂ (λ j k → j < k ) *iso *iso (ct-monotonic (is-elm y) (is-elm x1) c ) ) - -record IsFC (A : HOD) {x : Ordinal} (ax : A ∋ * x) (y : Ordinal) : Set n where - field - icy : odef (IChainSet A ax) y - c-finite : ¬ IChainSup> A (subst (λ k → odef A k ) (sym &iso) (proj1 icy) ) record Maximal ( A : HOD ) : Set (Level.suc n) where field @@ -335,100 +150,6 @@ A∋maximal : A ∋ maximal ¬maximal<x : {x : HOD} → A ∋ x → ¬ maximal < x -- A is Partial, use negative --- --- possible three cases in a limit ordinal step --- --- case 1) < goes x o< --- case 2) no > x in some chain ( maximal ) --- case 3) countably infinite chain below x --- -Zorn-lemma-3case : { A : HOD } - → o∅ o< & A - → IsPartialOrderSet A - → (x : Ordinal ) → (ax : odef A x) → OSup> A (d→∋ A ax) ∨ Maximal A ∨ InfiniteChain A x (d→∋ A ax) -Zorn-lemma-3case {A} 0<A PO x ax = zc2 where - Gtx : HOD - Gtx = record { od = record { def = λ y → odef ( IChainSet A ax ) y ∧ ( x o< y ) } ; odmax = & A - ; <odmax = λ lt → subst (λ k → k o< & A) &iso (c<→o< (d→∋ A (proj1 (proj1 lt)))) } - HG : HOD - HG = record { od = record { def = λ y → odef A y ∧ IsFC A (d→∋ A ax ) y } ; odmax = & A - ; <odmax = λ lt → subst (λ k → k o< & A) &iso (c<→o< (d→∋ A (proj1 lt) )) } - zc2 : OSup> A (d→∋ A ax) ∨ Maximal A ∨ InfiniteChain A x (d→∋ A ax ) - zc2 with is-o∅ (& Gtx) - ... | no not = case1 record { y = & y ; icy = zc4 ; y>x = proj2 zc3 } where - y : HOD - y = ODC.minimal O Gtx (λ eq → not (=od∅→≡o∅ eq)) - zc3 : odef ( IChainSet A ax ) (& y) ∧ ( x o< (& y )) - zc3 = ODC.x∋minimal O Gtx (λ eq → not (=od∅→≡o∅ eq)) - zc4 : odef (IChainSet A (subst (OD.def (od A)) (sym &iso) ax)) (& y) - zc4 = ⟪ proj1 (proj1 zc3) , (subst (λ k → IChained A k (& y)) (sym &iso) (proj2 (proj1 zc3))) ⟫ - ... | yes nogt with is-o∅ (& HG) - ... | no finite-chain = case2 (case1 record { maximal = y ; A∋maximal = proj1 zc3 ; ¬maximal<x = zc4 } ) where - y : HOD - y = ODC.minimal O HG (λ eq → finite-chain (=od∅→≡o∅ eq)) - zc3 : odef A (& y) ∧ IsFC A (d→∋ A ax ) (& y) - zc3 = ODC.x∋minimal O HG (λ eq → finite-chain (=od∅→≡o∅ eq)) - zc4 : {z : HOD} → A ∋ z → ¬ (y < z) - zc4 {z} az y<z = IsFC.c-finite (proj2 zc3) record { y = & z ; A∋y = az ; y>x = subst₂ (λ j k → j < k ) (sym *iso) (sym *iso) y<z } - ... | yes inifite = case2 (case2 record { c-infinite = zc91 ; chain<x = zc10 } ) where - B : HOD - B = IChainSet A ax -- (me (subst (OD.def (od A)) (sym &iso) (is-elm x))) - B1 : HOD - B1 = IChainSet A (subst (OD.def (od A)) (sym &iso) ax) - Nx : (y : Ordinal) → odef A y → HOD - Nx y ay = record { od = record { def = λ x → odef A x ∧ ( * y < * x ) } ; odmax = & A - ; <odmax = λ lt → subst (λ k → k o< & A) &iso (c<→o< (d→∋ A (proj1 lt))) } - zc10 : (y : Ordinal) → odef (IChainSet A (subst (OD.def (od A)) (sym &iso) ax)) y → y o< x - zc10 oy icsy = zc21 where - zc20 : (y : HOD) → (IChainSet A ax) ∋ y → x o< & y → ⊥ - zc20 y icsy lt = ¬A∋x→A≡od∅ Gtx ⟪ icsy , lt ⟫ nogt - zc22 : IChainSet A ax ∋ * oy - zc22 = ⟪ subst (λ k → odef A k) (sym &iso) (proj1 icsy) , subst₂ (λ j k → IChained A j k ) &iso (sym &iso) (proj2 icsy) ⟫ - zc21 : oy o< x - zc21 with trio< oy x - ... | tri< a ¬b ¬c = a - ... | tri≈ ¬a b ¬c = ⊥-elim (¬IChained-refl A PO (subst₂ (λ j k → IChained A j k ) &iso b (proj2 icsy)) ) - ... | tri> ¬a ¬b c = ⊥-elim ( zc20 (* oy) zc22 (subst (λ k → x o< k) (sym &iso) c )) - zc91 : (y : Ordinal) (cy : odef B1 y) → IChainSup> A (ic→A∋y A (subst (OD.def (od A)) (sym &iso) ax) cy) - zc91 y cy with is-o∅ (& (Nx y (proj1 cy) )) - ... | yes no-next = ⊥-elim zc16 where - zc18 : ¬ IChainSup> A (subst (odef A) (sym &iso) (proj1 (subst (λ k → odef (IChainSet A (d→∋ A ax)) k) (sym &iso) cy))) - zc18 ics = ¬A∋x→A≡od∅ (Nx y (proj1 cy) ) ⟪ subst (λ k → odef A k ) (sym &iso) (IChainSup>.A∋y ics) - , subst₂ (λ j k → j < k ) *iso (cong (*) (sym &iso))( IChainSup>.y>x ics) ⟫ no-next - zc17 : IsFC A {x} (d→∋ A ax) (& (* y)) - zc17 = record { icy = subst (λ k → odef (IChainSet A (d→∋ A ax)) k) (sym &iso) cy ; c-finite = zc18 } - zc16 : ⊥ - zc16 = ¬A∋x→A≡od∅ HG ⟪ subst (λ k → odef A k ) (sym &iso) (proj1 cy ) , zc17 ⟫ inifite - ... | no not = record { y = & zc13 ; A∋y = proj1 zc12 ; y>x = proj2 zc12 } where - zc13 = ODC.minimal O (Nx y (proj1 cy)) (λ eq → not (=od∅→≡o∅ eq )) - zc12 : odef A (& zc13 ) ∧ ( * y < * ( & zc13 )) - zc12 = ODC.x∋minimal O (Nx y (proj1 cy)) (λ eq → not (=od∅→≡o∅ eq )) - -all-climb-case : { A : HOD } → (0<A : o∅ o< & A) → IsPartialOrderSet A - → (( x : Ordinal ) → (ax : odef A (& (* x))) → OSup> A ax ) - → (x : HOD) ( ax : A ∋ x ) - → InfiniteChain A (& A) (d→∋ A ax) -all-climb-case {A} 0<A PO climb x ax = record { c-infinite = ac00 ; chain<x = ac01 } where - B = IChainSet A ax - ac01 : (y : Ordinal) → odef (IChainSet A (d→∋ A ax)) y → y o< & A - ac01 y ⟪ ay , _ ⟫ = subst (λ k → k o< & A ) &iso (c<→o< (subst (λ k → odef A k ) (sym &iso) ay) ) - ac00 : (y : Ordinal) (cy : odef (IChainSet A (d→∋ A ax)) y) → IChainSup> A (ic→A∋y A (d→∋ A ax) cy) - ac00 y cy = record { y = z ; A∋y = az ; y>x = y<z} where - ay : odef A (& (* y)) - ay = subst (λ k → odef A k) (sym &iso) (proj1 cy) - z : Ordinal - z = OSup>.y ( climb y ay) - az : odef A z - az = subst (λ k → odef A k) &iso ( incl (IChainSet⊆A {A} ay ) (subst (λ k → odef (IChainSet A ay) k ) (sym &iso) (OSup>.icy ( climb y ay)))) - icy : odef (IChainSet A ay ) z - icy = OSup>.icy ( climb y ay ) - y<z : * y < * z - y<z = ic→< {A} PO y (subst (λ k → odef A k) &iso ay) (IChained.iy (proj2 icy)) - (subst (λ k → ic-connect k (IChained.iy (proj2 icy))) &iso (IChained.ic (proj2 icy))) - --- <-TransFinite : ( A : HOD ) → IsTotalOrderSet A --- → ( (x : Ordinal) → ((y : Ordinal) → y o< x → ZChain A y) → ZChain A x ) → (x : Ordinal ) → ZChain A x --- <-TransFinite A TA ind x = TransFinite {ZChain A} ind x -- -- inductive maxmum tree from x