### Nuprl Lemma : fan_theorem-ext

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

Proof

Definitions occuring in Statement :  int_seg: `{i..j-}` nat: `ℕ` bool: `𝔹` decidable: `Dec(P)` uimplies: `b supposing a` uall: `∀[x:A]. B[x]` prop: `ℙ` so_apply: `x[s1;s2]` all: `∀x:A. B[x]` exists: `∃x:A. B[x]` squash: `↓T` implies: `P `` Q` function: `x:A ⟶ B[x]` natural_number: `\$n`
Definitions unfolded in proof :  member: `t ∈ T` fan_theorem simple_fan_theorem'-ext
Lemmas referenced :  fan_theorem simple_fan_theorem'-ext
Rules used in proof :  introduction sqequalSubstitution sqequalTransitivity computationStep sqequalReflexivity cut instantiate extract_by_obid hypothesis sqequalRule thin sqequalHypSubstitution equalityTransitivity equalitySymmetry

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

Date html generated: 2019_06_20-PM-01_15_32
Last ObjectModification: 2019_03_12-PM-04_25_58

Theory : int_2

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