### Nuprl Lemma : simple_general_fan_theorem

`∀[T:Type]`
`  (Bounded(T)`
`  `` (∀[X:n:ℕ ⟶ (ℕn ⟶ T) ⟶ ℙ]`
`        (∀n:ℕ. ∀s:ℕn ⟶ T.  Dec(X[n;s])) `` (∃k:ℕ [(∀f:ℕ ⟶ T. ∃n:ℕk. X[n;f])]) supposing ∀f:ℕ ⟶ T. (↓∃n:ℕ. X[n;f])))`

Proof

Definitions occuring in Statement :  bounded-type: `Bounded(T)` int_seg: `{i..j-}` nat: `ℕ` decidable: `Dec(P)` uimplies: `b supposing a` uall: `∀[x:A]. B[x]` prop: `ℙ` so_apply: `x[s1;s2]` all: `∀x:A. B[x]` sq_exists: `∃x:A [B[x]]` exists: `∃x:A. B[x]` squash: `↓T` implies: `P `` Q` function: `x:A ⟶ B[x]` natural_number: `\$n` universe: `Type`
Definitions unfolded in proof :  nat_plus: `ℕ+` ge: `i ≥ j ` seq-adjoin: `s++t` assert: `↑b` ifthenelse: `if b then t else f fi ` bnot: `¬bb` bfalse: `ff` cand: `A c∧ B` btrue: `tt` it: `⋅` unit: `Unit` bool: `𝔹` seq-append: `seq-append(n;m;s1;s2)` istype: `istype(T)` sq_type: `SQType(T)` less_than: `a < b` true: `True` top: `Top` subtract: `n - m` uiff: `uiff(P;Q)` rev_implies: `P `` Q` iff: `P `⇐⇒` Q` or: `P ∨ Q` decidable: `Dec(P)` exists: `∃x:A. B[x]` sq_exists: `∃x:A [B[x]]` prop: `ℙ` so_apply: `x[s]` not: `¬A` false: `False` less_than': `less_than'(a;b)` subtype_rel: `A ⊆r B` le: `A ≤ B` and: `P ∧ Q` lelt: `i ≤ j < k` sq_stable: `SqStable(P)` guard: `{T}` int_seg: `{i..j-}` so_apply: `x[s1;s2]` nat: `ℕ` so_lambda: `λ2x.t[x]` so_lambda: `λ2x y.t[x; y]` squash: `↓T` all: `∀x:A. B[x]` member: `t ∈ T` uimplies: `b supposing a` implies: `P `` Q` uall: `∀[x:A]. B[x]` bounded-type: `Bounded(T)`
Rules used in proof :  baseApply multiplyEquality functionExtensionality promote_hyp isectIsTypeImplies axiomSqEquality lessCases equalityElimination functionExtensionality_alt intEquality cumulativity hyp_replacement dependent_pairFormation_alt dependent_set_memberFormation_alt universeEquality productEquality minusEquality isect_memberEquality_alt unionElimination voidElimination productIsType setIsType instantiate functionIsType universeIsType independent_pairFormation independent_isectElimination equalitySymmetry equalityTransitivity equalityIstype productElimination independent_functionElimination addEquality dependent_set_memberEquality_alt applyEquality setElimination natural_numberEquality closedConclusion because_Cache functionEquality isectElimination extract_by_obid rename inhabitedIsType functionIsTypeImplies baseClosed imageMemberEquality hypothesis imageElimination hypothesisEquality thin dependent_functionElimination lambdaEquality_alt sqequalHypSubstitution introduction cut lambdaFormation_alt isect_memberFormation_alt computationStep sqequalTransitivity sqequalReflexivity sqequalRule sqequalSubstitution

Latex:
\mforall{}[T:Type]
(Bounded(T)
{}\mRightarrow{}  (\mforall{}[X:n:\mBbbN{}  {}\mrightarrow{}  (\mBbbN{}n  {}\mrightarrow{}  T)  {}\mrightarrow{}  \mBbbP{}]
(\mforall{}n:\mBbbN{}.  \mforall{}s:\mBbbN{}n  {}\mrightarrow{}  T.    Dec(X[n;s]))  {}\mRightarrow{}  (\mexists{}k:\mBbbN{}  [(\mforall{}f:\mBbbN{}  {}\mrightarrow{}  T.  \mexists{}n:\mBbbN{}k.  X[n;f])])
supposing  \mforall{}f:\mBbbN{}  {}\mrightarrow{}  T.  (\mdownarrow{}\mexists{}n:\mBbbN{}.  X[n;f])))

Date html generated: 2019_10_15-AM-10_20_24
Last ObjectModification: 2019_10_07-PM-04_44_24

Theory : bar-induction

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