Nuprl Lemma : decidable-finite-cantor-ext
∀[T:Type]. ∀[R:T ⟶ T ⟶ ℙ]. ((∀x,y:T. Dec(R[x;y])) ⇒ (∀n:ℕ. ∀F:(ℕn ⟶ 𝔹) ⟶ T. Dec(∃f,g:ℕn ⟶ 𝔹. R[F f;F g])))
Proof
Definitions occuring in Statement :
int_seg: {i..j-},
nat: ℕ,
bool: 𝔹,
decidable: Dec(P),
uall: ∀[x:A]. B[x],
prop: ℙ,
so_apply: x[s1;s2],
all: ∀x:A. B[x],
exists: ∃x:A. B[x],
implies: P ⇒ Q,
apply: f a,
function: x:A ⟶ B[x],
natural_number: $n,
universe: Type
Definitions unfolded in proof :
member: t ∈ T,
it: ⋅,
ifthenelse: if b then t else f fi ,
finite-cantor-decider: finite-cantor-decider(dcdr;n;F),
decidable-finite-cantor,
sq_stable_from_decidable,
sq_stable__from_stable,
stable__from_decidable,
any: any x
Lemmas referenced :
decidable-finite-cantor,
sq_stable_from_decidable,
sq_stable__from_stable,
stable__from_decidable
Rules used in proof :
introduction,
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
cut,
instantiate,
extract_by_obid,
hypothesis,
sqequalRule,
thin,
sqequalHypSubstitution,
equalityTransitivity,
equalitySymmetry
Latex:
\mforall{}[T:Type]. \mforall{}[R:T {}\mrightarrow{} T {}\mrightarrow{} \mBbbP{}].
((\mforall{}x,y:T. Dec(R[x;y])) {}\mRightarrow{} (\mforall{}n:\mBbbN{}. \mforall{}F:(\mBbbN{}n {}\mrightarrow{} \mBbbB{}) {}\mrightarrow{} T. Dec(\mexists{}f,g:\mBbbN{}n {}\mrightarrow{} \mBbbB{}. R[F f;F g])))
Date html generated:
2018_05_21-PM-01_17_19
Last ObjectModification:
2018_05_19-AM-06_32_33
Theory : continuity
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