### Nuprl Lemma : strong-continuity-implies1

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

Proof

Definitions occuring in Statement :  int_seg: `{i..j-}` nat: `ℕ` assert: `↑b` isl: `isl(x)` uimplies: `b supposing a` uall: `∀[x:A]. B[x]` all: `∀x:A. B[x]` exists: `∃x:A. B[x]` squash: `↓T` and: `P ∧ Q` 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 :  cand: `A c∧ B` squash: `↓T` exists: `∃x:A. B[x]` implies: `P `` Q` not: `¬A` false: `False` less_than': `less_than'(a;b)` le: `A ≤ B` uimplies: `b supposing a` so_apply: `x[s]` subtype_rel: `A ⊆r B` and: `P ∧ Q` prop: `ℙ` so_lambda: `λ2x.t[x]` nat: `ℕ` all: `∀x:A. B[x]` member: `t ∈ T` uall: `∀[x:A]. B[x]` guard: `{T}`
Lemmas referenced :  isl_wf assert_wf isect_wf subtype_rel_self false_wf int_seg_subtype_nat subtype_rel_dep_function equal_wf squash_wf all_wf unit_wf2 int_seg_wf nat_wf exists_wf squash-from-quotient strong-continuity2-no-inner-squash implies-quotient-true
Rules used in proof :  promote_hyp dependent_pairFormation productElimination baseClosed imageMemberEquality imageElimination independent_functionElimination inlEquality lambdaFormation independent_pairFormation independent_isectElimination functionExtensionality applyEquality productEquality lambdaEquality sqequalRule unionEquality because_Cache rename setElimination natural_numberEquality hypothesis functionEquality isectElimination hypothesisEquality thin dependent_functionElimination sqequalHypSubstitution extract_by_obid cut introduction isect_memberFormation sqequalReflexivity computationStep sqequalTransitivity sqequalSubstitution

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))))  \mwedge{}  (\mforall{}n:\mBbbN{}.  (M  n  f)  =  (inl  (F  f))  supposing  \muparrow{}isl(M  n  f))))

Date html generated: 2018_05_21-PM-01_17_47
Last ObjectModification: 2018_05_18-PM-04_03_51

Theory : continuity

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