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4 k кантора.Proof of Nuprl Lemma : cantor-to-fb_wf Step * 1 1 1 of Lemma cantor-to-fb_wf. 1. b : ℕ ─→ ℕ + 2. g : ℕ ─→ 𝔹 3. n : ℕ 4. k : ℕ @i 5. Σ (b j | j < n) = k ∈ ℕ @i 6. bb : ℕ @i 7. (b n) = bb ∈ ℕ @i 8. ↑ (bb - 2 < z mu(λ i.(bb - 2 < z i ∨ b (g (k + i)))) ∨ b (g (k + mu(λ i.(bb - 2 < z i ∨ b (g (k + i))))))) 9. v : ℕ @i 10. mu(λ i.(bb - 2 < z i ∨ b (g (k + i)))) = v ∈ ℕ @i ⊢ (∀ [i:ℕ ]. ¬↑ (bb - 2 < z i ∨ b (g (k + i))) supposing i < v) ⇒ (v ∈ ℕ bb) BY. 1 1. b : ℕ ─→ ℕ + 2. g : ℕ ─→ 𝔹 3. n : ℕ 4. k : ℕ @i 5. Σ (b j | j < n) = k ∈ ℕ @i 6. bb : ℕ @i 7. (b n) = bb ∈ ℕ @i 8. ↑ (bb - 2 < z mu(λ i.(bb - 2 < z i ∨ b (g (k + i)))) ∨ b (g (k + mu(λ i.(bb - 2 < z i ∨ b (g (k + i))))))) 9. v : ℤ @i 10. 0 ≤ v@i 11. mu(λ i.(bb - 2 < z i ∨ b (g (k + i)))) = v ∈ ℕ @i 12. ∀ [i:ℕ ]. ¬↑ (bb - 2 < z i ∨ b (g (k + i))) supposing i < v@i ⊢ v < bb. 1. b : \mBbbN<> <>\mrightarrow<> \mBbbN<>\msupplus<> 2. g : \mBbbN<> <>\mrightarrow<> \mBbbB<> 3. n : \mBbbN<> 4. k : \mBbbN<>@i 5. \mSigma<>(b j | j < n) = k@i 6. bb : \mBbbN<>@i 7. (b n) = bb@i 8. \muparrow<>(bb - 2 <z mu(\mlambda<>i.(bb - 2 <z i \mvee<>\msubb<>(g (k + i)))) \mvee<>\msubb<>(g (k + mu(\mlambda<>i.(bb - 2 <z i \mvee<>\msubb<>(g (k + i))))))) 9. v : \mBbbN<>@i 10. mu(\mlambda<>i.(bb - 2 <z i \mvee<>\msubb<>(g (k + i)))) = v@i \mvdash<> (\mforall<>[i:\mBbbN<>]. \mneg<>\muparrow<>(bb - 2 <z i \mvee<>\msubb<>(g (k + i))) supposing i < v) <>\mRightarrow<> (v \mmember<> \mBbbN<>bb) | |
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