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

comparison src/PFOD.agda @ 1174:375615f9d60f

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Func and Funcs
author Shinji KONO <kono@ie.u-ryukyu.ac.jp>
date Sat, 18 Feb 2023 11:51:22 +0900
parents d122d0c1b094
children 362e43a1477c
comparison
equal deleted inserted replaced
1173:4b2f49a5a1d5 1174:375615f9d60f
142 ... | yes p = refl 142 ... | yes p = refl
143 ... | no ¬p = ⊥-elim ( ¬p (subst (λ k → odef X k ) le03 Xx )) where 143 ... | no ¬p = ⊥-elim ( ¬p (subst (λ k → odef X k ) le03 Xx )) where
144 le03 : x ≡ & (nat→ω (ω→nato wx)) 144 le03 : x ≡ & (nat→ω (ω→nato wx))
145 le03 = subst₂ (λ j k → j ≡ k ) &iso refl (cong (&) (sym ( nat→ω-iso wx ) ) ) 145 le03 = subst₂ (λ j k → j ≡ k ) &iso refl (cong (&) (sym ( nat→ω-iso wx ) ) )
146 146
147 ¬i0≡i1 : ¬ ( i0 ≡ i1 )
148 ¬i0≡i1 ()
149
150 ¬i0→i1 : {x : Two} → ¬ (x ≡ i0 ) → x ≡ i1
151 ¬i0→i1 {i0} ne = ⊥-elim ( ne refl )
152 ¬i0→i1 {i1} ne = refl
153
154 ¬i1→i0 : {x : Two} → ¬ (x ≡ i1 ) → x ≡ i0
155 ¬i1→i0 {i0} ne = refl
156 ¬i1→i0 {i1} ne = ⊥-elim ( ne refl )
157
158 fω→2-iso : (f : Nat → Two) → ω2→f ( fω→2 f ) (ω2∋f f) ≡ f 147 fω→2-iso : (f : Nat → Two) → ω2→f ( fω→2 f ) (ω2∋f f) ≡ f
159 fω→2-iso f = f-extensionality (λ x → le01 x ) where 148 fω→2-iso f = f-extensionality (λ x → le01 x ) where
160 le01 : (x : Nat) → ω2→f (fω→2 f) (ω2∋f f) x ≡ f x 149 le01 : (x : Nat) → ω2→f (fω→2 f) (ω2∋f f) x ≡ f x
161 le01 x with ODC.∋-p O (fω→2 f) (nat→ω x) 150 le01 x with ODC.∋-p O (fω→2 f) (nat→ω x)
162 le01 x | yes p = subst (λ k → i1 ≡ f k ) (ω→nat-iso0 x (proj1 (proj2 p)) (trans *iso *iso)) (sym ((proj2 (proj2 p)) le02)) where 151 le01 x | yes p = subst (λ k → i1 ≡ f k ) (ω→nat-iso0 x (proj1 (proj2 p)) (trans *iso *iso)) (sym ((proj2 (proj2 p)) le02)) where