### Nuprl Lemma : strong-continuity2-no-inner-squash-cantor3

`∀F:(ℕ ⟶ 𝔹) ⟶ ℕ2`
`  ⇃(∃M:n:ℕ ⟶ (ℕn ⟶ 𝔹) ⟶ (ℕ2?)`
`     ∀f:ℕ ⟶ 𝔹. ((∃n:ℕ. ((M n f) = (inl (F f)) ∈ (ℕ2?))) ∧ (∀n:ℕ. (M n f) = (inl (F f)) ∈ (ℕ2?) supposing ↑isl(M n f))))`

Proof

Definitions occuring in Statement :  quotient: `x,y:A//B[x; y]` int_seg: `{i..j-}` nat: `ℕ` assert: `↑b` isl: `isl(x)` bool: `𝔹` uimplies: `b supposing a` all: `∀x:A. B[x]` exists: `∃x:A. B[x]` and: `P ∧ Q` true: `True` unit: `Unit` apply: `f a` function: `x:A ⟶ B[x]` inl: `inl x` union: `left + right` natural_number: `\$n` equal: `s = t ∈ T`
Definitions unfolded in proof :  true: `True` squash: `↓T` less_than: `a < b` nat_plus: `ℕ+` prop: `ℙ` implies: `P `` Q` not: `¬A` false: `False` less_than': `less_than'(a;b)` and: `P ∧ Q` le: `A ≤ B` uimplies: `b supposing a` uall: `∀[x:A]. B[x]` member: `t ∈ T` all: `∀x:A. B[x]`
Lemmas referenced :  strong-continuity2-half-squash-surject-biject trivial-biject-exists surject-nat-bool less_than_wf retraction-nat-nsub false_wf int_seg_subtype_nat int_seg_wf bool_wf nat_wf
Rules used in proof :  independent_functionElimination baseClosed hypothesisEquality imageMemberEquality dependent_set_memberEquality dependent_functionElimination independent_pairFormation sqequalRule independent_isectElimination natural_numberEquality thin isectElimination sqequalHypSubstitution hypothesis extract_by_obid introduction cut functionEquality lambdaFormation sqequalReflexivity computationStep sqequalTransitivity sqequalSubstitution

Latex:
\mforall{}F:(\mBbbN{}  {}\mrightarrow{}  \mBbbB{})  {}\mrightarrow{}  \mBbbN{}2
\00D9(\mexists{}M:n:\mBbbN{}  {}\mrightarrow{}  (\mBbbN{}n  {}\mrightarrow{}  \mBbbB{})  {}\mrightarrow{}  (\mBbbN{}2?)
\mforall{}f:\mBbbN{}  {}\mrightarrow{}  \mBbbB{}
((\mexists{}n:\mBbbN{}.  ((M  n  f)  =  (inl  (F  f))))  \mwedge{}  (\mforall{}n:\mBbbN{}.  (M  n  f)  =  (inl  (F  f))  supposing  \muparrow{}isl(M  n  f))))

Date html generated: 2017_09_29-PM-06_06_55
Last ObjectModification: 2017_09_04-AM-10_57_18

Theory : continuity

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