Mercurial > hg > Members > kono > Proof > ZF-in-agda
diff src/PFOD.agda @ 1100:c90eec304cfa
PFOD
author | Shinji KONO <kono@ie.u-ryukyu.ac.jp> |
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date | Mon, 26 Dec 2022 14:00:57 +0900 |
parents | c2501d308c95 |
children | d122d0c1b094 |
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--- a/src/PFOD.agda Sun Dec 25 10:12:56 2022 +0900 +++ b/src/PFOD.agda Mon Dec 26 14:00:57 2022 +0900 @@ -138,13 +138,33 @@ le02 : infinite ∋ * x le02 = power→ infinite _ lt (subst (λ k → odef X k) (sym &iso) Xx) le01 : (wx : odef infinite (& (* x))) → ω2→f X lt (ω→nat (* x) wx) ≡ i1 - le01 wx = ? + le01 wx with ODC.∋-p O X (nat→ω (ω→nat _ wx) ) + ... | yes p = refl + ... | no ¬p = ⊥-elim ( ¬p (subst (λ k → odef X k ) le03 Xx )) where + le03 : x ≡ & (nat→ω (ω→nato wx)) + le03 = subst₂ (λ j k → j ≡ k ) &iso refl (cong (&) (sym ( nat→ω-iso wx ) ) ) + +¬i0≡i1 : ¬ ( i0 ≡ i1 ) +¬i0≡i1 () + +¬i0→i1 : {x : Two} → ¬ (x ≡ i0 ) → x ≡ i1 +¬i0→i1 {i0} ne = ⊥-elim ( ne refl ) +¬i0→i1 {i1} ne = refl + +¬i1→i0 : {x : Two} → ¬ (x ≡ i1 ) → x ≡ i0 +¬i1→i0 {i0} ne = refl +¬i1→i0 {i1} ne = ⊥-elim ( ne refl ) fω→2-iso : (f : Nat → Two) → ω2→f ( fω→2 f ) (ω2∋f f) ≡ f -fω→2-iso f = f-extensionality le01 where +fω→2-iso f = f-extensionality (λ x → le01 x ) where le01 : (x : Nat) → ω2→f (fω→2 f) (ω2∋f f) x ≡ f x - le01 Zero = ? - le01 (Suc x) = ? + le01 x with ODC.∋-p O (fω→2 f) (nat→ω x) + 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 + le02 : infinite-d (& (* (& (nat→ω x)))) + le02 = proj1 (proj2 p ) + le01 x | no ¬p = sym ( ¬i1→i0 le04 ) where + le04 : ¬ f x ≡ i1 + le04 fx=1 = ¬p ⟪ ω∋nat→ω {x} , ⟪ subst (λ k → infinite-d k) (sym &iso) (ω∋nat→ω {x}) , le05 ⟫ ⟫ where + le05 : (lt : infinite-d (& (* (& (nat→ω x))))) → f (ω→nato lt) ≡ i1 + le05 lt = trans (cong f (ω→nat-iso0 x lt (trans *iso *iso))) fx=1 - -