Nuprl Lemma : dfan-iff-twkl!
∀T:Type. ((∃k:ℕ. T ~ ℕk) ⇒ (Fan_d(T) ⇐⇒ WKL!(T)))
Proof
Definitions occuring in Statement :
twkl!: WKL!(T),
dfan: Fan_d(T),
equipollent: A ~ B,
int_seg: {i..j-},
nat: ℕ,
all: ∀x:A. B[x],
exists: ∃x:A. B[x],
iff: P ⇐⇒ Q,
implies: P ⇒ Q,
natural_number: $n,
universe: Type
Definitions unfolded in proof :
all: ∀x:A. B[x],
implies: P ⇒ Q,
iff: P ⇐⇒ Q,
and: P ∧ Q,
uall: ∀[x:A]. B[x],
member: t ∈ T,
prop: ℙ,
rev_implies: P ⇐ Q,
so_lambda: λ2x.t[x],
nat: ℕ,
so_apply: x[s]
Lemmas referenced :
dfan-implies-twkl!,
dfan_wf,
twkl!-implies-dfan,
twkl!_wf,
exists_wf,
nat_wf,
equipollent_wf,
int_seg_wf
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation,
independent_pairFormation,
cut,
lemma_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
hypothesisEquality,
independent_functionElimination,
hypothesis,
dependent_functionElimination,
sqequalRule,
lambdaEquality,
natural_numberEquality,
setElimination,
rename,
universeEquality
Latex:
\mforall{}T:Type. ((\mexists{}k:\mBbbN{}. T \msim{} \mBbbN{}k) {}\mRightarrow{} (Fan\_d(T) \mLeftarrow{}{}\mRightarrow{} WKL!(T)))
Date html generated:
2016_05_14-PM-04_12_06
Last ObjectModification:
2015_12_26-PM-07_54_10
Theory : fan-theorem
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