Mercurial > hg > Members > kono > Proof > ZF-in-agda

comparison LEMC.agda @ 284:a8f9c8a27e8d

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minimal from LEM
author Shinji KONO <kono@ie.u-ryukyu.ac.jp>
date Sun, 10 May 2020 09:19:32 +0900
parents 2d77b6d12a22
children 313140ae5e3d
comparison
equal deleted inserted replaced
283:2d77b6d12a22 284:a8f9c8a27e8d
29 field 29 field
30 a-choice : OD 30 a-choice : OD
31 is-in : X ∋ a-choice 31 is-in : X ∋ a-choice
32 32
33 open choiced 33 open choiced
34
35 double-neg-eilm : {A : Set (suc n)} → ¬ ¬ A → A -- we don't have this in intutionistic logic
36 double-neg-eilm {A} notnot with p∨¬p A -- assuming axiom of choice
37 ... | case1 p = p
38 ... | case2 ¬p = ⊥-elim ( notnot ¬p )
39
34 40
35 OD→ZFC : ZFC 41 OD→ZFC : ZFC
36 OD→ZFC = record { 42 OD→ZFC = record {
37 ZFSet = OD 43 ZFSet = OD
38 ; _∋_ = _∋_ 44 ; _∋_ = _∋_
89 min : OD 95 min : OD
90 x∋min : x ∋ min 96 x∋min : x ∋ min
91 min-empty : (y : OD ) → ¬ ( min ∋ y) ∧ (x ∋ y) 97 min-empty : (y : OD ) → ¬ ( min ∋ y) ∧ (x ∋ y)
92 open Minimal 98 open Minimal
93 open _∧_ 99 open _∧_
94 induction : {x : OD} → ({y : OD} → x ∋ y → (ne : ¬ (y == od∅ )) → Minimal y ne ) → (ne : ¬ (x == od∅ )) → Minimal x ne 100 --
95 induction {x} prev ne = c2 101 -- from https://math.stackexchange.com/questions/2973777/is-it-possible-to-prove-regularity-with-transfinite-induction-only
96 where 102 --
97 ch : choiced x 103 induction : {x : OD} → ({y : OD} → x ∋ y → (u : OD ) → (u∋x : u ∋ y) → Minimal u (∅< u∋x))
98 ch = choice-func x ne 104 → (u : OD ) → (u∋x : u ∋ x) → Minimal u (∅< u∋x)
99 c1 : OD 105 induction {x} prev u u∋x with p∨¬p ((y : OD) → ¬ (x ∋ y) ∧ (u ∋ y))
100 c1 = a-choice ch -- x ∋ c1 106 ... | case1 P =
101 c2 : Minimal x ne 107 record { min = x
102 c2 with p∨¬p ( (y : OD ) → ¬ ( def c1 (od→ord y) ∧ (def x (od→ord y))) ) 108 ; x∋min = u∋x
103 c2 | case1 Yes = record { 109 ; min-empty = P
104 min = c1 110 }
105 ; x∋min = is-in ch 111 ... | case2 NP =
106 ; min-empty = Yes 112 record { min = min min2
107 } 113 ; x∋min = x∋min min2
108 c2 | case2 No = c3 where 114 ; min-empty = min-empty min2
109 y1 : OD 115
110 y1 = record { def = λ y → ( def c1 y ∧ def x y) } 116 } where
111 noty1 : ¬ (y1 == od∅ ) 117 p : OD
112 noty1 not = ⊥-elim ( No (λ z n → ∅< n not ) ) 118 p = record { def = λ y1 → def x y1 ∧ def u y1 }
113 ch1 : choiced y1 -- a-choice ch1 ∈ c1 , a-choice ch1 ∈ x 119 np : ¬ (p == od∅)
114 ch1 = choice-func y1 noty1 120 np p∅ = NP (λ y p∋y → ∅< p∋y p∅ )
115 c3 : Minimal x ne 121 y1choice : choiced p
116 c3 with is-o∅ (od→ord (a-choice ch1)) 122 y1choice = choice-func p np
117 ... | yes eq = record { 123 y1 : OD
118 min = od∅ 124 y1 = a-choice y1choice
119 ; x∋min = def-subst {x} {od→ord (a-choice ch1)} {x} (proj2 (is-in ch1)) refl (trans eq (sym ord-od∅) ) 125 y1prop : (x ∋ y1) ∧ (u ∋ y1)
120 ; min-empty = λ z p → ⊥-elim ( ¬x<0 (proj1 p) ) 126 y1prop = is-in y1choice
121 } 127 min2 : Minimal u (∅< (proj2 y1prop))
122 ... | no n = record { 128 min2 = prev (proj1 y1prop) u (proj2 y1prop)
123 min = min min3 129 Min2 : (x : OD) → (u : OD ) → (u∋x : u ∋ x) → Minimal u (∅< u∋x)
124 ; x∋min = x∋min3 (x∋min min3) 130 Min2 x u u∋x = (ε-induction {λ y → (u : OD ) → (u∋x : u ∋ y) → Minimal u (∅< u∋x) } induction x u u∋x )
125 ; min-empty = min3-empty -- λ y p → min3-empty min3 y p -- p : (min min3 ∋ y) ∧ (x ∋ y) 131 cx : {x : OD} → ¬ (x == od∅ ) → choiced x
126 } where 132 cx {x} nx = choice-func x nx
127 lemma : (a-choice ch1 == od∅ ) → od→ord (a-choice ch1) ≡ o∅
128 lemma eq = begin
129 od→ord (a-choice ch1)
130 ≡⟨ cong (λ k → od→ord k ) (==→o≡ eq ) ⟩
131 od→ord od∅
132 ≡⟨ ord-od∅ ⟩
133 o∅
134 ∎ where open ≡-Reasoning
135 -- Minimal (a-choice ch1) ch1not
136 -- min ∈ a-choice ch1 , min ∩ a-choice ch1 ≡ ∅
137 ch1not : ¬ (a-choice ch1 == od∅)
138 ch1not ch1=0 = n (lemma ch1=0)
139 min3 : Minimal (a-choice ch1) ch1not
140 min3 = prev (proj2 (is-in ch1)) (λ ch1=0 → n (lemma ch1=0))
141 x∋min3 : a-choice ch1 ∋ min min3 → x ∋ min min3
142 x∋min3 lt = {!!}
143 min3-empty : (y : OD ) → ¬ ((min min3 ∋ y) ∧ (x ∋ y))
144 min3-empty y p = min-empty min3 y record { proj1 = proj1 p ; proj2 = ? } -- (min min3 ∋ y) ∧ (a-choice ch1 ∋ y)
145 -- p : (min min3 ∋ y) ∧ (x ∋ y)
146
147
148 Min1 : (x : OD) → (ne : ¬ (x == od∅ )) → Minimal x ne
149 Min1 x ne = (ε-induction {λ y → (ne : ¬ (y == od∅ ) ) → Minimal y ne } induction x ne )
150 minimal : (x : OD ) → ¬ (x == od∅ ) → OD 133 minimal : (x : OD ) → ¬ (x == od∅ ) → OD
151 minimal x not = min (Min1 x not ) 134 minimal x not = min (Min2 (a-choice (cx not) ) x (is-in (cx not)))
152 x∋minimal : (x : OD ) → ( ne : ¬ (x == od∅ ) ) → def x ( od→ord ( minimal x ne ) ) 135 x∋minimal : (x : OD ) → ( ne : ¬ (x == od∅ ) ) → def x ( od→ord ( minimal x ne ) )
153 x∋minimal x ne = x∋min (Min1 x ne) 136 x∋minimal x ne = x∋min (Min2 (a-choice (cx ne) ) x (is-in (cx ne)))
154 minimal-1 : (x : OD ) → ( ne : ¬ (x == od∅ ) ) → (y : OD ) → ¬ ( def (minimal x ne) (od→ord y)) ∧ (def x (od→ord y) ) 137 minimal-1 : (x : OD ) → ( ne : ¬ (x == od∅ ) ) → (y : OD ) → ¬ ( def (minimal x ne) (od→ord y)) ∧ (def x (od→ord y) )
155 minimal-1 x ne y = min-empty (Min1 x ne ) y 138 minimal-1 x ne y = min-empty (Min2 (a-choice (cx ne) ) x (is-in (cx ne))) y
156 139
157 140
158 141
159 142