### Nuprl Lemma : strong-continuity-implies1-sp

`∀[F:(ℕ ⟶ ℕ) ⟶ ℕ]. (↓∃M:n:ℕ ⟶ (ℕn ⟶ ℕ) ⟶ (ℕ?). ∀f:ℕ ⟶ ℕ. (↓∃n:ℕ. ((M n f) = (inl (F f)) ∈ (ℕ?))))`

Proof

Definitions occuring in Statement :  int_seg: `{i..j-}` nat: `ℕ` uall: `∀[x:A]. B[x]` all: `∀x:A. B[x]` exists: `∃x:A. B[x]` squash: `↓T` 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 :  uall: `∀[x:A]. B[x]` member: `t ∈ T` squash: `↓T` exists: `∃x:A. B[x]` all: `∀x:A. B[x]` and: `P ∧ Q` prop: `ℙ` so_lambda: `λ2x.t[x]` subtype_rel: `A ⊆r B` so_apply: `x[s]` nat: `ℕ` uimplies: `b supposing a` le: `A ≤ B` less_than': `less_than'(a;b)` false: `False` not: `¬A` implies: `P `` Q`
Lemmas referenced :  subtype_rel_self false_wf int_seg_subtype_nat int_seg_wf subtype_rel_dep_function unit_wf2 equal_wf exists_wf squash_wf all_wf strong-continuity-implies1 nat_wf
Rules used in proof :  sqequalSubstitution sqequalTransitivity computationStep sqequalReflexivity isect_memberFormation introduction cut hypothesis sqequalHypSubstitution imageElimination sqequalRule imageMemberEquality hypothesisEquality thin baseClosed functionEquality lemma_by_obid isectElimination productElimination dependent_pairFormation lambdaFormation dependent_functionElimination because_Cache lambdaEquality unionEquality applyEquality natural_numberEquality setElimination rename independent_isectElimination independent_pairFormation inlEquality

Latex:
\mforall{}[F:(\mBbbN{}  {}\mrightarrow{}  \mBbbN{})  {}\mrightarrow{}  \mBbbN{}].  (\mdownarrow{}\mexists{}M:n:\mBbbN{}  {}\mrightarrow{}  (\mBbbN{}n  {}\mrightarrow{}  \mBbbN{})  {}\mrightarrow{}  (\mBbbN{}?).  \mforall{}f:\mBbbN{}  {}\mrightarrow{}  \mBbbN{}.  (\mdownarrow{}\mexists{}n:\mBbbN{}.  ((M  n  f)  =  (inl  (F  f)))))

Date html generated: 2016_05_14-PM-09_41_51
Last ObjectModification: 2016_01_15-PM-10_55_49

Theory : continuity

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