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
Home
Index