### Nuprl Lemma : general-fan-theorem-troelstra2

`∀X:n:ℕ ⟶ (ℕn ⟶ 𝔹) ⟶ ℙ. ((∀f:ℕ ⟶ 𝔹. ∃n:ℕ. X[n;f]) `` (∃k:ℕ. ∀f:ℕ ⟶ 𝔹. ∃n:ℕk. X[n;f]))`

Proof

Definitions occuring in Statement :  int_seg: `{i..j-}` nat: `ℕ` bool: `𝔹` prop: `ℙ` so_apply: `x[s1;s2]` all: `∀x:A. B[x]` exists: `∃x:A. B[x]` implies: `P `` Q` function: `x:A ⟶ B[x]` natural_number: `\$n`
Definitions unfolded in proof :  all: `∀x:A. B[x]` implies: `P `` Q` member: `t ∈ T` prop: `ℙ` uall: `∀[x:A]. B[x]` so_lambda: `λ2x.t[x]` so_apply: `x[s1;s2]` subtype_rel: `A ⊆r B` nat: `ℕ` uimplies: `b supposing a` le: `A ≤ B` and: `P ∧ Q` less_than': `less_than'(a;b)` false: `False` not: `¬A` so_apply: `x[s]` exists: `∃x:A. B[x]` ge: `i ≥ j ` decidable: `Dec(P)` or: `P ∨ Q` satisfiable_int_formula: `satisfiable_int_formula(fmla)` top: `Top` int_seg: `{i..j-}` iff: `P `⇐⇒` Q` rev_implies: `P `` Q` uiff: `uiff(P;Q)` subtract: `n - m` sq_stable: `SqStable(P)` lelt: `i ≤ j < k` squash: `↓T` true: `True` bool: `𝔹` unit: `Unit` it: `⋅` btrue: `tt` band: `p ∧b q` ifthenelse: `if b then t else f fi ` outl: `outl(x)` isl: `isl(x)` less_than: `a < b` bfalse: `ff` pi1: `fst(t)` guard: `{T}` rev_uimplies: `rev_uimplies(P;Q)` cand: `A c∧ B` sq_type: `SQType(T)` assert: `↑b` bnot: `¬bb`
Rules used in proof :  sqequalSubstitution sqequalTransitivity computationStep sqequalReflexivity lambdaFormation cut introduction extract_by_obid sqequalHypSubstitution isectElimination thin functionEquality hypothesis sqequalRule lambdaEquality applyEquality hypothesisEquality natural_numberEquality setElimination rename because_Cache independent_isectElimination independent_pairFormation cumulativity universeEquality dependent_functionElimination functionExtensionality instantiate productElimination dependent_pairEquality equalityTransitivity equalitySymmetry independent_functionElimination dependent_set_memberEquality addEquality unionElimination approximateComputation dependent_pairFormation int_eqEquality intEquality isect_memberEquality voidElimination voidEquality minusEquality imageMemberEquality baseClosed imageElimination equalityElimination unionEquality applyLambdaEquality lessCases isect_memberFormation axiomSqEquality inlEquality pointwiseFunctionality promote_hyp baseApply closedConclusion inrFormation inlFormation

Latex:
\mforall{}X:n:\mBbbN{}  {}\mrightarrow{}  (\mBbbN{}n  {}\mrightarrow{}  \mBbbB{})  {}\mrightarrow{}  \mBbbP{}.  ((\mforall{}f:\mBbbN{}  {}\mrightarrow{}  \mBbbB{}.  \mexists{}n:\mBbbN{}.  X[n;f])  {}\mRightarrow{}  (\mexists{}k:\mBbbN{}.  \mforall{}f:\mBbbN{}  {}\mrightarrow{}  \mBbbB{}.  \mexists{}n:\mBbbN{}k.  X[n;f]))

Date html generated: 2019_06_20-PM-03_00_06
Last ObjectModification: 2018_08_20-PM-09_41_05

Theory : continuity

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