Nuprl Lemma : cantor-interval-converges-ext
∀a,b:ℝ. ∀f:ℕ ⟶ 𝔹. fst(cantor-interval(a;b;f;n))↓ as n→∞ supposing a ≤ b
Proof
Definitions occuring in Statement :
cantor-interval: cantor-interval(a;b;f;n),
converges: x[n]↓ as n→∞,
rleq: x ≤ y,
real: ℝ,
nat: ℕ,
bool: 𝔹,
uimplies: b supposing a,
pi1: fst(t),
all: ∀x:A. B[x],
function: x:A ⟶ B[x]
Definitions unfolded in proof :
member: t ∈ T,
pi1: fst(t),
so_lambda: λ2x.t[x],
so_apply: x[s],
accelerate: accelerate(k;f),
cantor-interval-converges,
converges-iff-cauchy,
cantor-interval-cauchy-ext,
uall: ∀[x:A]. B[x],
so_lambda: so_lambda(x,y,z,w.t[x; y; z; w]),
so_apply: x[s1;s2;s3;s4],
top: Top,
uimplies: b supposing a,
so_lambda: λ2x y.t[x; y],
so_apply: x[s1;s2]
Lemmas referenced :
cantor-interval-converges,
lifting-strict-spread,
istype-void,
strict4-apply,
strict4-divide,
converges-iff-cauchy,
cantor-interval-cauchy-ext
Rules used in proof :
introduction,
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
cut,
instantiate,
extract_by_obid,
hypothesis,
sqequalRule,
thin,
sqequalHypSubstitution,
equalityTransitivity,
equalitySymmetry,
isectElimination,
baseClosed,
isect_memberEquality_alt,
voidElimination,
independent_isectElimination
Latex:
\mforall{}a,b:\mBbbR{}. \mforall{}f:\mBbbN{} {}\mrightarrow{} \mBbbB{}. fst(cantor-interval(a;b;f;n))\mdownarrow{} as n\mrightarrow{}\minfty{} supposing a \mleq{} b
Date html generated:
2019_10_30-AM-07_39_19
Last ObjectModification:
2019_04_02-AM-10_55_46
Theory : reals
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