Step
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1
1
1
1
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of Lemma
unsquashed-continuity-false-troelstra
1. ∀F:(ℕ ⟶ ℕ) ⟶ ℕ. ∀a:ℕ ⟶ ℕ. ∃n:ℕ. ∀b:ℕ ⟶ ℕ. ((a = b ∈ (ℕn ⟶ ℕ)) ⇒ ((F a) = (F b) ∈ ℕ))
2. ∀a:ℕ ⟶ ℕ. ∀F:(ℕ ⟶ ℕ) ⟶ ℕ. ∃n:ℕ. ∀b:ℕ ⟶ ℕ. ((a = b ∈ (ℕn ⟶ ℕ)) ⇒ ((F a) = (F b) ∈ ℕ))
3. ∀F:(ℕ ⟶ ℕ) ⟶ ℕ. ∃n:ℕ. ∀b:ℕ ⟶ ℕ. ((0s = b ∈ (ℕn ⟶ ℕ)) ⇒ ((F 0s) = (F b) ∈ ℕ))
4. Phi : F:((ℕ ⟶ ℕ) ⟶ ℕ) ⟶ ℕ
5. ∀F:(ℕ ⟶ ℕ) ⟶ ℕ. ∀b:ℕ ⟶ ℕ. ((0s = b ∈ (ℕPhi F ⟶ ℕ)) ⇒ ((F 0s) = (F b) ∈ ℕ))
6. n0 : Phi* (λf.0) ~ 0s^(Phi (λf.0))
7. ∀beta,a:ℕ ⟶ ℕ.
∃y,x:ℕ
((↑isl(gamma-neighbourhood(beta;Phi* (λf.0)) a^(x)))
∧ (y = outl(gamma-neighbourhood(beta;Phi* (λf.0)) a^(x)) ∈ ℕ)
∧ (∀a,b:finite-nat-seq().
((↑isl(gamma-neighbourhood(beta;Phi* (λf.0)) a))
⇒ ((gamma-neighbourhood(beta;Phi* (λf.0)) a) = (gamma-neighbourhood(beta;Phi* (λf.0)) a**b) ∈ (ℕ?)))))
8. Psi : beta:(ℕ ⟶ ℕ) ⟶ a:(ℕ ⟶ ℕ) ⟶ ℕ
9. ∀beta,a:ℕ ⟶ ℕ.
∃x:ℕ
((↑isl(gamma-neighbourhood(beta;Phi* (λf.0)) a^(x)))
∧ ((Psi beta a) = outl(gamma-neighbourhood(beta;Phi* (λf.0)) a^(x)) ∈ ℕ)
∧ (∀a,b:finite-nat-seq().
((↑isl(gamma-neighbourhood(beta;Phi* (λf.0)) a))
⇒ ((gamma-neighbourhood(beta;Phi* (λf.0)) a) = (gamma-neighbourhood(beta;Phi* (λf.0)) a**b) ∈ (ℕ?)))))
⊢ False
BY
{ Assert ⌜∀beta:ℕ ⟶ ℕ. ((∀x:ℕ. ((beta x) = 0 ∈ ℤ)) ⇒ ((Psi beta) = (λf.0) ∈ ((ℕ ⟶ ℕ) ⟶ ℕ)))⌝⋅ }
1
.....assertion.....
1. ∀F:(ℕ ⟶ ℕ) ⟶ ℕ. ∀a:ℕ ⟶ ℕ. ∃n:ℕ. ∀b:ℕ ⟶ ℕ. ((a = b ∈ (ℕn ⟶ ℕ)) ⇒ ((F a) = (F b) ∈ ℕ))
2. ∀a:ℕ ⟶ ℕ. ∀F:(ℕ ⟶ ℕ) ⟶ ℕ. ∃n:ℕ. ∀b:ℕ ⟶ ℕ. ((a = b ∈ (ℕn ⟶ ℕ)) ⇒ ((F a) = (F b) ∈ ℕ))
3. ∀F:(ℕ ⟶ ℕ) ⟶ ℕ. ∃n:ℕ. ∀b:ℕ ⟶ ℕ. ((0s = b ∈ (ℕn ⟶ ℕ)) ⇒ ((F 0s) = (F b) ∈ ℕ))
4. Phi : F:((ℕ ⟶ ℕ) ⟶ ℕ) ⟶ ℕ
5. ∀F:(ℕ ⟶ ℕ) ⟶ ℕ. ∀b:ℕ ⟶ ℕ. ((0s = b ∈ (ℕPhi F ⟶ ℕ)) ⇒ ((F 0s) = (F b) ∈ ℕ))
6. n0 : Phi* (λf.0) ~ 0s^(Phi (λf.0))
7. ∀beta,a:ℕ ⟶ ℕ.
∃y,x:ℕ
((↑isl(gamma-neighbourhood(beta;Phi* (λf.0)) a^(x)))
∧ (y = outl(gamma-neighbourhood(beta;Phi* (λf.0)) a^(x)) ∈ ℕ)
∧ (∀a,b:finite-nat-seq().
((↑isl(gamma-neighbourhood(beta;Phi* (λf.0)) a))
⇒ ((gamma-neighbourhood(beta;Phi* (λf.0)) a) = (gamma-neighbourhood(beta;Phi* (λf.0)) a**b) ∈ (ℕ?)))))
8. Psi : beta:(ℕ ⟶ ℕ) ⟶ a:(ℕ ⟶ ℕ) ⟶ ℕ
9. ∀beta,a:ℕ ⟶ ℕ.
∃x:ℕ
((↑isl(gamma-neighbourhood(beta;Phi* (λf.0)) a^(x)))
∧ ((Psi beta a) = outl(gamma-neighbourhood(beta;Phi* (λf.0)) a^(x)) ∈ ℕ)
∧ (∀a,b:finite-nat-seq().
((↑isl(gamma-neighbourhood(beta;Phi* (λf.0)) a))
⇒ ((gamma-neighbourhood(beta;Phi* (λf.0)) a) = (gamma-neighbourhood(beta;Phi* (λf.0)) a**b) ∈ (ℕ?)))))
⊢ ∀beta:ℕ ⟶ ℕ. ((∀x:ℕ. ((beta x) = 0 ∈ ℤ)) ⇒ ((Psi beta) = (λf.0) ∈ ((ℕ ⟶ ℕ) ⟶ ℕ)))
2
1. ∀F:(ℕ ⟶ ℕ) ⟶ ℕ. ∀a:ℕ ⟶ ℕ. ∃n:ℕ. ∀b:ℕ ⟶ ℕ. ((a = b ∈ (ℕn ⟶ ℕ)) ⇒ ((F a) = (F b) ∈ ℕ))
2. ∀a:ℕ ⟶ ℕ. ∀F:(ℕ ⟶ ℕ) ⟶ ℕ. ∃n:ℕ. ∀b:ℕ ⟶ ℕ. ((a = b ∈ (ℕn ⟶ ℕ)) ⇒ ((F a) = (F b) ∈ ℕ))
3. ∀F:(ℕ ⟶ ℕ) ⟶ ℕ. ∃n:ℕ. ∀b:ℕ ⟶ ℕ. ((0s = b ∈ (ℕn ⟶ ℕ)) ⇒ ((F 0s) = (F b) ∈ ℕ))
4. Phi : F:((ℕ ⟶ ℕ) ⟶ ℕ) ⟶ ℕ
5. ∀F:(ℕ ⟶ ℕ) ⟶ ℕ. ∀b:ℕ ⟶ ℕ. ((0s = b ∈ (ℕPhi F ⟶ ℕ)) ⇒ ((F 0s) = (F b) ∈ ℕ))
6. n0 : Phi* (λf.0) ~ 0s^(Phi (λf.0))
7. ∀beta,a:ℕ ⟶ ℕ.
∃y,x:ℕ
((↑isl(gamma-neighbourhood(beta;Phi* (λf.0)) a^(x)))
∧ (y = outl(gamma-neighbourhood(beta;Phi* (λf.0)) a^(x)) ∈ ℕ)
∧ (∀a,b:finite-nat-seq().
((↑isl(gamma-neighbourhood(beta;Phi* (λf.0)) a))
⇒ ((gamma-neighbourhood(beta;Phi* (λf.0)) a) = (gamma-neighbourhood(beta;Phi* (λf.0)) a**b) ∈ (ℕ?)))))
8. Psi : beta:(ℕ ⟶ ℕ) ⟶ a:(ℕ ⟶ ℕ) ⟶ ℕ
9. ∀beta,a:ℕ ⟶ ℕ.
∃x:ℕ
((↑isl(gamma-neighbourhood(beta;Phi* (λf.0)) a^(x)))
∧ ((Psi beta a) = outl(gamma-neighbourhood(beta;Phi* (λf.0)) a^(x)) ∈ ℕ)
∧ (∀a,b:finite-nat-seq().
((↑isl(gamma-neighbourhood(beta;Phi* (λf.0)) a))
⇒ ((gamma-neighbourhood(beta;Phi* (λf.0)) a) = (gamma-neighbourhood(beta;Phi* (λf.0)) a**b) ∈ (ℕ?)))))
10. ∀beta:ℕ ⟶ ℕ. ((∀x:ℕ. ((beta x) = 0 ∈ ℤ)) ⇒ ((Psi beta) = (λf.0) ∈ ((ℕ ⟶ ℕ) ⟶ ℕ)))
⊢ False
Latex:
Latex:
1. \mforall{}F:(\mBbbN{} {}\mrightarrow{} \mBbbN{}) {}\mrightarrow{} \mBbbN{}. \mforall{}a:\mBbbN{} {}\mrightarrow{} \mBbbN{}. \mexists{}n:\mBbbN{}. \mforall{}b:\mBbbN{} {}\mrightarrow{} \mBbbN{}. ((a = b) {}\mRightarrow{} ((F a) = (F b)))
2. \mforall{}a:\mBbbN{} {}\mrightarrow{} \mBbbN{}. \mforall{}F:(\mBbbN{} {}\mrightarrow{} \mBbbN{}) {}\mrightarrow{} \mBbbN{}. \mexists{}n:\mBbbN{}. \mforall{}b:\mBbbN{} {}\mrightarrow{} \mBbbN{}. ((a = b) {}\mRightarrow{} ((F a) = (F b)))
3. \mforall{}F:(\mBbbN{} {}\mrightarrow{} \mBbbN{}) {}\mrightarrow{} \mBbbN{}. \mexists{}n:\mBbbN{}. \mforall{}b:\mBbbN{} {}\mrightarrow{} \mBbbN{}. ((0s = b) {}\mRightarrow{} ((F 0s) = (F b)))
4. Phi : F:((\mBbbN{} {}\mrightarrow{} \mBbbN{}) {}\mrightarrow{} \mBbbN{}) {}\mrightarrow{} \mBbbN{}
5. \mforall{}F:(\mBbbN{} {}\mrightarrow{} \mBbbN{}) {}\mrightarrow{} \mBbbN{}. \mforall{}b:\mBbbN{} {}\mrightarrow{} \mBbbN{}. ((0s = b) {}\mRightarrow{} ((F 0s) = (F b)))
6. n0 : Phi* (\mlambda{}f.0) \msim{} 0s\^{}(Phi (\mlambda{}f.0))
7. \mforall{}beta,a:\mBbbN{} {}\mrightarrow{} \mBbbN{}.
\mexists{}y,x:\mBbbN{}
((\muparrow{}isl(gamma-neighbourhood(beta;Phi* (\mlambda{}f.0)) a\^{}(x)))
\mwedge{} (y = outl(gamma-neighbourhood(beta;Phi* (\mlambda{}f.0)) a\^{}(x)))
\mwedge{} (\mforall{}a,b:finite-nat-seq().
((\muparrow{}isl(gamma-neighbourhood(beta;Phi* (\mlambda{}f.0)) a))
{}\mRightarrow{} ((gamma-neighbourhood(beta;Phi* (\mlambda{}f.0)) a)
= (gamma-neighbourhood(beta;Phi* (\mlambda{}f.0)) a**b)))))
8. Psi : beta:(\mBbbN{} {}\mrightarrow{} \mBbbN{}) {}\mrightarrow{} a:(\mBbbN{} {}\mrightarrow{} \mBbbN{}) {}\mrightarrow{} \mBbbN{}
9. \mforall{}beta,a:\mBbbN{} {}\mrightarrow{} \mBbbN{}.
\mexists{}x:\mBbbN{}
((\muparrow{}isl(gamma-neighbourhood(beta;Phi* (\mlambda{}f.0)) a\^{}(x)))
\mwedge{} ((Psi beta a) = outl(gamma-neighbourhood(beta;Phi* (\mlambda{}f.0)) a\^{}(x)))
\mwedge{} (\mforall{}a,b:finite-nat-seq().
((\muparrow{}isl(gamma-neighbourhood(beta;Phi* (\mlambda{}f.0)) a))
{}\mRightarrow{} ((gamma-neighbourhood(beta;Phi* (\mlambda{}f.0)) a)
= (gamma-neighbourhood(beta;Phi* (\mlambda{}f.0)) a**b)))))
\mvdash{} False
By
Latex:
Assert \mkleeneopen{}\mforall{}beta:\mBbbN{} {}\mrightarrow{} \mBbbN{}. ((\mforall{}x:\mBbbN{}. ((beta x) = 0)) {}\mRightarrow{} ((Psi beta) = (\mlambda{}f.0)))\mkleeneclose{}\mcdot{}
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