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Partial Order
author Shinji KONO <kono@ie.u-ryukyu.ac.jp>
date Sat, 09 Apr 2022 07:03:07 +0900
parents 646831f6b06d
children 71436ccbc804
comparison
equal deleted inserted replaced
491:646831f6b06d 492:e28b1da1b58d
45 elm : HOD 45 elm : HOD
46 is-elm : A ∋ elm 46 is-elm : A ∋ elm
47 47
48 open Element 48 open Element
49 49
50 IsPartialOrderSet : ( A : HOD ) → (_<_ : (x y : HOD) → Set n ) → Set (suc n) 50 IsPartialOrderSet : ( A : HOD ) → (_≤_ : (x y : HOD) → Set n ) → Set (suc n)
51 IsPartialOrderSet A _<_ = IsPartialOrder _<A_ _≡A_ where 51 IsPartialOrderSet A _≤_ = IsPartialOrder _≤A_ _≡A_ where
52 _<A_ : (x y : Element A ) → Set n 52 _≤A_ : (x y : Element A ) → Set n
53 x <A y = elm x < elm y 53 x ≤A y = elm x ≤ elm y
54 _≡A_ : (x y : Element A ) → Set (suc n) 54 _≡A_ : (x y : Element A ) → Set (suc n)
55 x ≡A y = elm x ≡ elm y 55 x ≡A y = elm x ≡ elm y
56 56
57 IsTotalOrderSet : ( A : HOD ) → (_<_ : (x y : HOD) → Set n ) → Set (suc n) 57 open _==_
58 IsTotalOrderSet A _<_ = IsTotalOrder _<A_ _≡A_ where 58 open _⊆_
59 _<A_ : (x y : Element A ) → Set n 59
60 x <A y = elm x < elm y 60 ⊆-IsPartialOrderSet : { A B : HOD } → B ⊆ A → {_≤_ : (x y : HOD) → Set n } → IsPartialOrderSet A _≤_ → IsPartialOrderSet B _≤_
61 ⊆-IsPartialOrderSet {A} {B} B⊆A {_≤_} PA = record {
62 isPreorder = record { isEquivalence = record { refl = ? ; sym = {!!} ; trans = {!!} } ; reflexive = {!!} ; trans = {!!} }
63 ; antisym = {!!}
64 }
65
66 IsTotalOrderSet : ( A : HOD ) → (_≤_ : (x y : HOD) → Set n ) → Set (suc n)
67 IsTotalOrderSet A _≤_ = IsTotalOrder _≤A_ _≡A_ where
68 _≤A_ : (x y : Element A ) → Set n
69 x ≤A y = elm x ≤ elm y
61 _≡A_ : (x y : Element A ) → Set (suc n) 70 _≡A_ : (x y : Element A ) → Set (suc n)
62 x ≡A y = elm x ≡ elm y 71 x ≡A y = elm x ≡ elm y
63 72
64 me : { A a : HOD } → A ∋ a → Element A 73 me : { A a : HOD } → A ∋ a → Element A
65 me {A} {a} lt = record { elm = a ; is-elm = lt } 74 me {A} {a} lt = record { elm = a ; is-elm = lt }
66 75
67 record SUP ( A B : HOD ) (_<_ : (x y : HOD) → Set n ) : Set (suc n) where 76 record SUP ( A B : HOD ) (_≤_ : (x y : HOD) → Set n ) : Set (suc n) where
68 field 77 field
69 sup : HOD 78 sup : HOD
70 A∋maximal : A ∋ sup 79 A∋maximal : A ∋ sup
71 x≤sup : {x : HOD} → B ∋ x → (x ≡ sup ) ∨ (x < sup ) -- B is Total, use positive 80 x≤sup : {x : HOD} → B ∋ x → (x ≡ sup ) ∨ (x ≤ sup ) -- B is Total, use positive
72 81
73 record Maximal ( A : HOD ) (_<_ : (x y : HOD) → Set n ) : Set (suc n) where 82 record Maximal ( A : HOD ) (_≤_ : (x y : HOD) → Set n ) : Set (suc n) where
74 field 83 field
75 maximal : HOD 84 maximal : HOD
76 A∋maximal : A ∋ maximal 85 A∋maximal : A ∋ maximal
77 ¬maximal<x : {x : HOD} → A ∋ x → ¬ maximal < x -- A is Partial, use negative 86 ¬maximal≤x : {x : HOD} → A ∋ x → ¬ maximal ≤ x -- A is Partial, use negative
78 87
79 open _==_ 88
80 open _⊆_ 89 record ZChain ( A : HOD ) (y : Ordinal) (_≤_ : (x y : HOD) → Set n ) : Set (suc n) where
81
82 record ZChain ( A : HOD ) (y : Ordinal) (_<_ : (x y : HOD) → Set n ) : Set (suc n) where
83 field 90 field
84 fb : (x : Ordinal ) → HOD 91 fb : (x : Ordinal ) → HOD
85 A∋fb : (ox : Ordinal ) → ox o< y → A ∋ fb ox 92 A∋fb : (ox : Ordinal ) → ox o≤ y → A ∋ fb ox
86 total : {ox oz : Ordinal } → (x<y : ox o< y ) → (z<y : oz o< y ) → ( ox ≡ oz ) ∨ ( fb ox < fb oz ) ∨ ( fb oz < fb ox ) 93 total : {ox oz : Ordinal } → (x≤y : ox o≤ y ) → (z≤y : oz o≤ y ) → ( ox ≡ oz ) ∨ ( fb ox ≤ fb oz ) ∨ ( fb oz ≤ fb ox )
87 monotonic : {ox oz : Ordinal } → (x<y : ox o< y ) → (z<y : oz o< y ) → ox o< oz → fb ox < fb oz 94 monotonic : {ox oz : Ordinal } → (x≤y : ox o≤ y ) → (z≤y : oz o≤ y ) → ox o≤ oz → fb ox ≤ fb oz
88 95
89 Zorn-lemma : { A : HOD } → { _<_ : (x y : HOD) → Set n } 96 Zorn-lemma : { A : HOD } → { _≤_ : (x y : HOD) → Set n }
90 → o∅ o< & A 97 → o∅ o< & A
91 → IsPartialOrderSet A _<_ 98 → IsPartialOrderSet A _≤_
92 → ( ( B : HOD) → (B⊆A : B ⊆ A) → IsTotalOrderSet B _<_ → SUP A B _<_ ) -- SUP condition 99 → ( ( B : HOD) → (B⊆A : B ⊆ A) → IsTotalOrderSet B _≤_ → SUP A B _≤_ ) -- SUP condition
93 → Maximal A _<_ 100 → Maximal A _≤_
94 Zorn-lemma {A} {_<_} 0<A PO supP = zorn00 where 101 Zorn-lemma {A} {_≤_} 0<A PO supP = zorn00 where
95 someA : HOD 102 someA : HOD
96 someA = ODC.minimal O A (λ eq → ¬x<0 ( subst (λ k → o∅ o< k ) (=od∅→≡o∅ eq) 0<A )) 103 someA = ODC.minimal O A (λ eq → ¬x<0 ( subst (λ k → o∅ o< k ) (=od∅→≡o∅ eq) 0<A ))
97 isSomeA : A ∋ someA 104 isSomeA : A ∋ someA
98 isSomeA = ODC.x∋minimal O A (λ eq → ¬x<0 ( subst (λ k → o∅ o< k ) (=od∅→≡o∅ eq) 0<A )) 105 isSomeA = ODC.x∋minimal O A (λ eq → ¬x<0 ( subst (λ k → o∅ o< k ) (=od∅→≡o∅ eq) 0<A ))
99 HasMaximal : HOD 106 HasMaximal : HOD
100 HasMaximal = record { od = record { def = λ y → odef A y ∧ ((m : Ordinal) → odef A m → ¬ (* y < * m))} ; odmax = & A ; <odmax = z08 } where 107 HasMaximal = record { od = record { def = λ y → odef A y ∧ ((m : Ordinal) → odef A m → ¬ (* y ≤ * m))} ; odmax = & A ; ≤odmax = z08 } where
101 z08 : {y : Ordinal} → (odef A y ∧ ((m : Ordinal) → odef A m → ¬ (* y < * m))) → y o< & A 108 z08 : {y : Ordinal} → (odef A y ∧ ((m : Ordinal) → odef A m → ¬ (* y ≤ * m))) → y o< & A
102 z08 {y} P = subst (λ k → k o< & A) &iso (c<→o< {* y} (subst (λ k → odef A k) (sym &iso) (proj1 P))) 109 z08 {y} P = subst (λ k → k o< & A) &iso (c<→o< {* y} (subst (λ k → odef A k) (sym &iso) (proj1 P)))
103 no-maximal : HasMaximal =h= od∅ → (y : Ordinal) → (odef A y ∧ ((m : Ordinal) → odef A m → ¬ (* y < * m))) → ⊥ 110 no-maximal : HasMaximal =h= od∅ → (y : Ordinal) → (odef A y ∧ ((m : Ordinal) → odef A m → ¬ (* y ≤ * m))) → ⊥
104 no-maximal nomx y P = ¬x<0 (eq→ nomx {y} ⟪ proj1 P , (λ m am → (proj2 P) m am ) ⟫ ) 111 no-maximal nomx y P = ¬x<0 (eq→ nomx {y} ⟪ proj1 P , (λ m am → (proj2 P) m am ) ⟫ )
105 Gtx : { x : HOD} → A ∋ x → HOD 112 Gtx : { x : HOD} → A ∋ x → HOD
106 Gtx {x} ax = record { od = record { def = λ y → odef A y ∧ (x < (* y)) } ; odmax = & A ; <odmax = z09 } where 113 Gtx {x} ax = record { od = record { def = λ y → odef A y ∧ (x ≤ (* y)) } ; odmax = & A ; ≤odmax = z09 } where
107 z09 : {y : Ordinal} → (odef A y ∧ (x < (* y))) → y o< & A 114 z09 : {y : Ordinal} → (odef A y ∧ (x ≤ (* y))) → y o< & A
108 z09 {y} P = subst (λ k → k o< & A) &iso (c<→o< {* y} (subst (λ k → odef A k) (sym &iso) (proj1 P))) 115 z09 {y} P = subst (λ k → k o< & A) &iso (c<→o< {* y} (subst (λ k → odef A k) (sym &iso) (proj1 P)))
109 z01 : {a b : HOD} → A ∋ a → A ∋ b → (a ≡ b ) ∨ (a < b ) → b < a → ⊥ 116 z01 : {a b : HOD} → A ∋ a → A ∋ b → (a ≡ b ) ∨ (a ≤ b ) → b ≤ a → ⊥
110 z01 {a} {b} A∋a A∋b (case1 a=b) b<a = {!!} -- proj1 (proj2 (PO (me A∋b) (me A∋a)) (sym a=b)) b<a 117 z01 {a} {b} A∋a A∋b (case1 a=b) b≤a = {!!} -- proj1 (proj2 (PO (me A∋b) (me A∋a)) (sym a=b)) b≤a
111 z01 {a} {b} A∋a A∋b (case2 a<b) b<a = {!!} -- proj1 (proj1 (PO (me A∋b) (me A∋a)) b<a) a<b 118 z01 {a} {b} A∋a A∋b (case2 a≤b) b≤a = {!!} -- proj1 (proj1 (PO (me A∋b) (me A∋a)) b≤a) a≤b
112 -- ZChain is not compatible with the SUP condition 119 -- ZChain is not compatible with the SUP condition
113 record BX (x y : Ordinal) (fb : ( x : Ordinal ) → HOD ) : Set n where 120 record BX (x y : Ordinal) (fb : ( x : Ordinal ) → HOD ) : Set n where
114 field 121 field
115 bx : Ordinal 122 bx : Ordinal
116 bx<y : bx o< y 123 bx≤y : bx o≤ y
117 is-fb : x ≡ & (fb bx ) 124 is-fb : x ≡ & (fb bx )
118 bx<A : (z : ZChain A (& A) _<_ ) → {x : Ordinal } → (bx : BX x (& A) ( ZChain.fb z )) → BX.bx bx o< & A 125 bx≤A : (z : ZChain A (& A) _≤_ ) → {x : Ordinal } → (bx : BX x (& A) ( ZChain.fb z )) → BX.bx bx o≤ & A
119 bx<A z {x} bx = BX.bx<y bx 126 bx≤A z {x} bx = BX.bx≤y bx
120 B : (z : ZChain A (& A) _<_ ) → HOD 127 B : (z : ZChain A (& A) _≤_ ) → HOD
121 B z = record { od = record { def = λ x → BX x (& A) ( ZChain.fb z ) } ; odmax = & A ; <odmax = {!!} } 128 B z = record { od = record { def = λ x → BX x (& A) ( ZChain.fb z ) } ; odmax = & A ; ≤odmax = {!!} }
122 z11 : (z : ZChain A (& A) _<_ ) → (x : Element (B z)) → elm x ≡ ZChain.fb z ? 129 z11 : (z : ZChain A (& A) _≤_ ) → (x : Element (B z)) → elm x ≡ ZChain.fb z (BX.bx (is-elm x))
123 z11 z x = subst₂ (λ j k → j ≡ k ) *iso *iso ( cong (*) (BX.is-fb (is-elm x)) ) 130 z11 z x = subst₂ (λ j k → j ≡ k ) *iso *iso ( cong (*) (BX.is-fb (is-elm x)) )
124 obx : (z : ZChain A (& A) _<_ ) → {x : HOD} → B z ∋ x → Ordinal 131 obx : (z : ZChain A (& A) _≤_ ) → {x : HOD} → B z ∋ x → Ordinal
125 obx z {x} bx = BX.bx bx 132 obx z {x} bx = BX.bx bx
126 obx=fb : (z : ZChain A (& A) _<_ ) → {x : HOD} → (bx : B z ∋ x ) → x ≡ ZChain.fb z {!!} 133 obx=fb : (z : ZChain A (& A) _≤_ ) → {x : HOD} → (bx : B z ∋ x ) → x ≡ ZChain.fb z (BX.bx bx)
127 obx=fb z {x} bx = subst₂ (λ j k → j ≡ k ) *iso *iso (cong (*) (BX.is-fb bx)) 134 obx=fb z {x} bx = subst₂ (λ j k → j ≡ k ) *iso *iso (cong (*) (BX.is-fb bx))
128 B⊆A : (z : ZChain A (& A) _<_ ) → B z ⊆ A 135 B⊆A : (z : ZChain A (& A) _≤_ ) → B z ⊆ A
129 B⊆A z = record { incl = λ {x} bx → subst (λ k → odef A k ) (sym (BX.is-fb bx)) (ZChain.A∋fb z (BX.bx bx) (BX.bx<y bx) ) } 136 B⊆A z = record { incl = λ {x} bx → subst (λ k → odef A k ) (sym (BX.is-fb bx)) (ZChain.A∋fb z (BX.bx bx) (BX.bx≤y bx) ) }
130 PO-B : (z : ZChain A (& A) _<_ ) → IsPartialOrderSet (B z) _<_ 137 PO-B : (z : ZChain A (& A) _≤_ ) → IsPartialOrderSet (B z) _≤_
131 PO-B z = ? -- a b = {!!} -- PO record { elm = elm a ; is-elm = incl ( B⊆A z) (is-elm a) } record { elm = elm b ; is-elm = incl ( B⊆A z) (is-elm b) } 138 PO-B z = subst₂ (λ j k → IsPartialOrder j k ) {!!} {!!} {!!} where
132 bx-monotonic : (z : ZChain A (& A) _<_ ) → {x y : Element (B z)} → obx z (is-elm x) o< obx z (is-elm y) → elm x < elm y 139 _≤B_ = {!!}
133 bx-monotonic z {x} {y} a = subst₂ (λ j k → j < k ) (sym (z11 z x)) (sym (z11 z y)) (ZChain.monotonic z (bx<A z (is-elm x)) (bx<A z (is-elm y)) a ) 140 _≡B_ = {!!}
141 -- a b = {!!} -- PO record { elm = elm a ; is-elm = incl ( B⊆A z) (is-elm a) } record { elm = elm b ; is-elm = incl ( B⊆A z) (is-elm b) }
142 bx-monotonic : (z : ZChain A (& A) _≤_ ) → {x y : Element (B z)} → obx z (is-elm x) o≤ obx z (is-elm y) → elm x ≤ elm y
143 bx-monotonic z {x} {y} a = subst₂ (λ j k → j ≤ k ) (sym (z11 z x)) (sym (z11 z y)) (ZChain.monotonic z (bx≤A z (is-elm x)) (bx≤A z (is-elm y)) a )
134 open import Relation.Binary.HeterogeneousEquality as HE using (_≅_ ) 144 open import Relation.Binary.HeterogeneousEquality as HE using (_≅_ )
135 z12 : (z : ZChain A (& A) _<_ ) → {a b : HOD } → (x : BX (& a) (& A) (ZChain.fb z)) (y : BX (& b) (& A) (ZChain.fb z)) 145 z12 : (z : ZChain A (& A) _≤_ ) → {a b : HOD } → (x : BX (& a) (& A) (ZChain.fb z)) (y : BX (& b) (& A) (ZChain.fb z))
136 → obx z x ≡ obx z y → bx<A z x ≅ bx<A z y 146 → obx z x ≡ obx z y → bx≤A z x ≅ bx≤A z y
137 z12 z {a} {b} x y eq = {!!} 147 z12 z {a} {b} x y eq = {!!}
138 bx-inject : (z : ZChain A (& A) _<_ ) → {x y : Element (B z)} → BX.bx (is-elm x) ≡ BX.bx (is-elm y) → elm x ≡ elm y 148 bx-inject : (z : ZChain A (& A) _≤_ ) → {x y : Element (B z)} → BX.bx (is-elm x) ≡ BX.bx (is-elm y) → elm x ≡ elm y
139 bx-inject z {x} {y} eq = begin 149 bx-inject z {x} {y} eq = begin
140 elm x ≡⟨ {!!} ⟩ 150 elm x ≡⟨ {!!} ⟩
141 {!!} ≡⟨ cong (λ k → {!!} ) {!!} ⟩ 151 {!!} ≡⟨ cong (λ k → {!!} ) {!!} ⟩
142 {!!} ≡⟨ {!!} ⟩ 152 {!!} ≡⟨ {!!} ⟩
143 elm y ∎ where open ≡-Reasoning 153 elm y ∎ where open ≡-Reasoning
144 B-is-total : (z : ZChain A (& A) _<_ ) → IsTotalOrderSet (B z) _<_ 154 B-is-total : (z : ZChain A (& A) _≤_ ) → IsTotalOrderSet (B z) _≤_
145 B-is-total = ? 155 B-is-total = {!!}
146 B-Tri : (z : ZChain A (& A) _<_ ) → Trichotomous (λ (x : Element A) y → elm x ≡ elm y ) (λ x y → elm x < elm y ) 156 B-Tri : (z : ZChain A (& A) _≤_ ) → Trichotomous (λ (x : Element A) y → elm x ≡ elm y ) (λ x y → elm x ≤ elm y )
147 B-Tri z x y with trio< (obx z ?) (obx z ?) 157 B-Tri z x y with trio< (obx z {!!}) (obx z {!!})
148 ... | tri< a ¬b ¬c = ? where -- tri< z10 (λ eq → proj1 (proj2 (PO-B z x y) eq ) z10) (λ ¬c → proj1 (proj1 (PO-B z y x) ¬c ) z10) where 158 ... | tri< a ¬b ¬c = {!!} where -- tri≤ z10 (λ eq → proj1 (proj2 (PO-B z x y) eq ) z10) (λ ¬c → proj1 (proj1 (PO-B z y x) ¬c ) z10) where
149 z10 : elm x < elm y 159 z10 : elm x ≤ elm y
150 z10 = ? -- bx-monotonic z {x} {y} a 160 z10 = {!!} -- bx-monotonic z {x} {y} a
151 ... | tri≈ ¬a b ¬c = ? -- tri≈ {!!} (bx-inject z {x} {y} b) {!!} 161 ... | tri≈ ¬a b ¬c = {!!} -- tri≈ {!!} (bx-inject z {x} {y} b) {!!}
152 ... | tri> ¬a ¬b c = ? -- tri> (λ ¬a → proj1 (proj1 (PO-B z x y) ¬a ) (bx-monotonic z {y} {x} c) ) (λ eq → proj2 (proj2 (PO-B z x y) eq ) (bx-monotonic z {y} {x} c)) (bx-monotonic z {y} {x} c) 162 ... | tri> ¬a ¬b c = {!!} -- tri> (λ ¬a → proj1 (proj1 (PO-B z x y) ¬a ) (bx-monotonic z {y} {x} c) ) (λ eq → proj2 (proj2 (PO-B z x y) eq ) (bx-monotonic z {y} {x} c)) (bx-monotonic z {y} {x} c)
153 ZChain→¬SUP : (z : ZChain A (& A) _<_ ) → ¬ (SUP A (B z) _<_ ) 163 ZChain→¬SUP : (z : ZChain A (& A) _≤_ ) → ¬ (SUP A (B z) _≤_ )
154 ZChain→¬SUP z sp = ⊥-elim {!!} where 164 ZChain→¬SUP z sp = ⊥-elim {!!} where
155 z03 : & (SUP.sup sp) o< osuc (& A) 165 z03 : & (SUP.sup sp) o< osuc (& A)
156 z03 = ordtrans (c<→o< (SUP.A∋maximal sp)) <-osuc 166 z03 = ordtrans (c<→o< (SUP.A∋maximal sp)) <-osuc
157 z02 : (x : HOD) → B z ∋ x → SUP.sup sp < x → ⊥ 167 z02 : (x : HOD) → B z ∋ x → SUP.sup sp ≤ x → ⊥
158 z02 x xe s<x = z01 (incl (B⊆A z) xe) (SUP.A∋maximal sp) (SUP.x≤sup sp xe) s<x 168 z02 x xe s≤x = z01 (incl (B⊆A z) xe) (SUP.A∋maximal sp) (SUP.x≤sup sp xe) s≤x
159 ind : HasMaximal =h= od∅ 169 ind : HasMaximal =h= od∅
160 → (x : Ordinal) → ((y : Ordinal) → y o< x → ZChain A y _<_ ) 170 → (x : Ordinal) → ((y : Ordinal) → y o< x → ZChain A y _≤_ )
161 → ZChain A x _<_ 171 → ZChain A x _≤_
162 ind nomx x prev with Oprev-p x 172 ind nomx x prev with Oprev-p x
163 ... | yes op with ODC.∋-p O A (* x) 173 ... | yes op with ODC.∋-p O A (* x)
164 ... | no ¬Ax = {!!} where 174 ... | no ¬Ax = {!!} where
165 -- we have previous ordinal and ¬ A ∋ x, use previous Zchain 175 -- we have previous ordinal and ¬ A ∋ x, use previous Zchain
166 px = Oprev.oprev op 176 px = Oprev.oprev op
167 zc1 : ZChain A px _<_ 177 zc1 : ZChain A px _≤_
168 zc1 = prev px (subst (λ k → px o< k) (Oprev.oprev=x op) <-osuc) 178 zc1 = prev px (subst (λ k → px o< k) (Oprev.oprev=x op) <-osuc)
169 z04 : {!!} -- (sup : HOD) (as : A ∋ sup) → & sup o< osuc x → sup < ZChain.fb zc1 as 179 z04 : {!!} -- (sup : HOD) (as : A ∋ sup) → & sup o≤ osuc x → sup ≤ ZChain.fb zc1 as
170 z04 sup as s<x with trio< (& sup) x 180 z04 sup as s≤x with trio< (& sup) x
171 ... | tri≈ ¬a b ¬c = ⊥-elim (¬Ax (subst (λ k → odef A k) (trans b (sym &iso)) as) ) 181 ... | tri≈ ¬a b ¬c = ⊥-elim (¬Ax (subst (λ k → odef A k) (trans b (sym &iso)) as) )
172 ... | tri< a ¬b ¬c = {!!} -- ZChain.¬x≤sup zc1 _ as ( subst (λ k → & sup o< k ) (sym (Oprev.oprev=x op)) a ) 182 ... | tri< a ¬b ¬c = {!!} -- ZChain.¬x≤sup zc1 _ as ( subst (λ k → & sup o≤ k ) (sym (Oprev.oprev=x op)) a )
173 ... | tri> ¬a ¬b c with osuc-≡< s<x 183 ... | tri> ¬a ¬b c with osuc-≡< s≤x
174 ... | case1 eq = ⊥-elim (¬Ax (subst (λ k → odef A k) (trans eq (sym &iso)) as) ) 184 ... | case1 eq = ⊥-elim (¬Ax (subst (λ k → odef A k) (trans eq (sym &iso)) as) )
175 ... | case2 lt = ⊥-elim (¬a lt ) 185 ... | case2 lt = ⊥-elim (¬a lt )
176 ... | yes ax = z06 where -- we have previous ordinal and A ∋ x 186 ... | yes ax = z06 where -- we have previous ordinal and A ∋ x
177 px = Oprev.oprev op 187 px = Oprev.oprev op
178 zc1 : ZChain A px _<_ 188 zc1 : ZChain A px _≤_
179 zc1 = prev px (subst (λ k → px o< k) (Oprev.oprev=x op) <-osuc) 189 zc1 = prev px (subst (λ k → px o< k) (Oprev.oprev=x op) <-osuc)
180 z06 : ZChain A x _<_ 190 z06 : ZChain A x _≤_
181 z06 with is-o∅ (& (Gtx ax)) 191 z06 with is-o∅ (& (Gtx ax))
182 ... | yes nogt = ⊥-elim (no-maximal nomx x ⟪ subst (λ k → odef A k) &iso ax , x-is-maximal ⟫ ) where -- no larger element, so it is maximal 192 ... | yes nogt = ⊥-elim (no-maximal nomx x ⟪ subst (λ k → odef A k) &iso ax , x-is-maximal ⟫ ) where -- no larger element, so it is maximal
183 x-is-maximal : (m : Ordinal) → odef A m → ¬ (* x < * m) 193 x-is-maximal : (m : Ordinal) → odef A m → ¬ (* x ≤ * m)
184 x-is-maximal m am = ¬x<m where 194 x-is-maximal m am = ¬x≤m where
185 ¬x<m : ¬ (* x < * m) 195 ¬x≤m : ¬ (* x ≤ * m)
186 ¬x<m x<m = ∅< {Gtx ax} {* m} ⟪ subst (λ k → odef A k) (sym &iso) am , subst (λ k → * x < k ) (cong (*) (sym &iso)) x<m ⟫ (≡o∅→=od∅ nogt) 196 ¬x≤m x≤m = ∅< {Gtx ax} {* m} ⟪ subst (λ k → odef A k) (sym &iso) am , subst (λ k → * x ≤ k ) (cong (*) (sym &iso)) x≤m ⟫ (≡o∅→=od∅ nogt)
187 ... | no not = {!!} where 197 ... | no not = {!!} where
188 ind nomx x prev | no ¬ox with trio< (& A) x --- limit ordinal case 198 ind nomx x prev | no ¬ox with trio< (& A) x --- limit ordinal case
189 ... | tri< a ¬b ¬c = {!!} where 199 ... | tri< a ¬b ¬c = {!!} where
190 zc1 : ZChain A (& A) _<_ 200 zc1 : ZChain A (& A) _≤_
191 zc1 = prev (& A) a 201 zc1 = prev (& A) a
192 ... | tri≈ ¬a b ¬c = {!!} where 202 ... | tri≈ ¬a b ¬c = {!!} where
193 ... | tri> ¬a ¬b c with ODC.∋-p O A (* x) 203 ... | tri> ¬a ¬b c with ODC.∋-p O A (* x)
194 ... | no ¬Ax = {!!} where 204 ... | no ¬Ax = {!!} where
195 ... | yes ax with is-o∅ (& (Gtx ax)) 205 ... | yes ax with is-o∅ (& (Gtx ax))
196 ... | yes nogt = ⊥-elim (no-maximal nomx x ⟪ subst (λ k → odef A k) &iso ax , x-is-maximal ⟫ ) where -- no larger element, so it is maximal 206 ... | yes nogt = ⊥-elim (no-maximal nomx x ⟪ subst (λ k → odef A k) &iso ax , x-is-maximal ⟫ ) where -- no larger element, so it is maximal
197 x-is-maximal : (m : Ordinal) → odef A m → ¬ (* x < * m) 207 x-is-maximal : (m : Ordinal) → odef A m → ¬ (* x ≤ * m)
198 x-is-maximal m am = ¬x<m where 208 x-is-maximal m am = ¬x≤m where
199 ¬x<m : ¬ (* x < * m) 209 ¬x≤m : ¬ (* x ≤ * m)
200 ¬x<m x<m = ∅< {Gtx ax} {* m} ⟪ subst (λ k → odef A k) (sym &iso) am , subst (λ k → * x < k ) (cong (*) (sym &iso)) x<m ⟫ (≡o∅→=od∅ nogt) 210 ¬x≤m x≤m = ∅< {Gtx ax} {* m} ⟪ subst (λ k → odef A k) (sym &iso) am , subst (λ k → * x ≤ k ) (cong (*) (sym &iso)) x≤m ⟫ (≡o∅→=od∅ nogt)
201 ... | no not = {!!} where 211 ... | no not = {!!} where
202 zorn00 : Maximal A _<_ 212 zorn00 : Maximal A _≤_
203 zorn00 with is-o∅ ( & HasMaximal ) 213 zorn00 with is-o∅ ( & HasMaximal )
204 ... | no not = record { maximal = ODC.minimal O HasMaximal (λ eq → not (=od∅→≡o∅ eq)) ; A∋maximal = zorn01 ; ¬maximal<x = zorn02 } where 214 ... | no not = record { maximal = ODC.minimal O HasMaximal (λ eq → not (=od∅→≡o∅ eq)) ; A∋maximal = zorn01 ; ¬maximal≤x = zorn02 } where
205 -- yes we have the maximal 215 -- yes we have the maximal
206 hasm : odef HasMaximal ( & ( ODC.minimal O HasMaximal (λ eq → not (=od∅→≡o∅ eq)) ) ) 216 hasm : odef HasMaximal ( & ( ODC.minimal O HasMaximal (λ eq → not (=od∅→≡o∅ eq)) ) )
207 hasm = ODC.x∋minimal O HasMaximal (λ eq → not (=od∅→≡o∅ eq)) 217 hasm = ODC.x∋minimal O HasMaximal (λ eq → not (=od∅→≡o∅ eq))
208 zorn01 : A ∋ ODC.minimal O HasMaximal (λ eq → not (=od∅→≡o∅ eq)) 218 zorn01 : A ∋ ODC.minimal O HasMaximal (λ eq → not (=od∅→≡o∅ eq))
209 zorn01 = proj1 hasm 219 zorn01 = proj1 hasm
210 zorn02 : {x : HOD} → A ∋ x → ¬ (ODC.minimal O HasMaximal (λ eq → not (=od∅→≡o∅ eq)) < x) 220 zorn02 : {x : HOD} → A ∋ x → ¬ (ODC.minimal O HasMaximal (λ eq → not (=od∅→≡o∅ eq)) ≤ x)
211 zorn02 {x} ax m<x = ((proj2 hasm) (& x) ax) (subst₂ (λ j k → j < k) (sym *iso) (sym *iso) m<x ) 221 zorn02 {x} ax m≤x = ((proj2 hasm) (& x) ax) (subst₂ (λ j k → j ≤ k) (sym *iso) (sym *iso) m≤x )
212 ... | yes ¬Maximal = ⊥-elim {!!} where 222 ... | yes ¬Maximal = ⊥-elim {!!} where
213 -- if we have no maximal, make ZChain, which contradict SUP condition 223 -- if we have no maximal, make ZChain, which contradict SUP condition
214 z : (x : Ordinal) → HasMaximal =h= od∅ → ZChain A x _<_ 224 z : (x : Ordinal) → HasMaximal =h= od∅ → ZChain A x _≤_
215 z x nomx = TransFinite (ind nomx) x 225 z x nomx = TransFinite (ind nomx) x
216 226
217 _⊆'_ : ( A B : HOD ) → Set n 227 _⊆'_ : ( A B : HOD ) → Set n
218 _⊆'_ A B = (x : Ordinal ) → odef A x → odef B x 228 _⊆'_ A B = (x : Ordinal ) → odef A x → odef B x
219 229