Nuprl Lemma : strong-continuity2-no-inner-squash-cantor4

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


Proof




Definitions occuring in Statement :  quotient: x,y:A//B[x; y] int_seg: {i..j-} nat: assert: b isl: isl(x) bool: 𝔹 uimplies: supposing a all: x:A. B[x] exists: x:A. B[x] and: P ∧ Q true: True unit: Unit apply: a function: x:A ⟶ B[x] inl: inl x union: left right natural_number: $n equal: t ∈ T
Definitions unfolded in proof :  true: True squash: T less_than: a < b nat_plus: + prop: implies:  Q not: ¬A false: False less_than': less_than'(a;b) and: P ∧ Q le: A ≤ B uimplies: supposing a uall: [x:A]. B[x] member: t ∈ T all: x:A. B[x]
Lemmas referenced :  biject-bool-nsub2 surject-nat-bool less_than_wf retraction-nat-nsub false_wf int_seg_subtype_nat int_seg_wf bool_wf nat_wf strong-continuity2-half-squash-surject-biject
Rules used in proof :  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 independent_functionElimination

Latex:
\mforall{}F:(\mBbbN{}  {}\mrightarrow{}  \mBbbB{})  {}\mrightarrow{}  \mBbbB{}
    \00D9(\mexists{}M:n:\mBbbN{}  {}\mrightarrow{}  (\mBbbN{}n  {}\mrightarrow{}  \mBbbB{})  {}\mrightarrow{}  (\mBbbB{}?)
          \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: 2018_05_21-PM-01_19_05
Last ObjectModification: 2018_05_18-PM-04_17_46

Theory : continuity


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