Nuprl Lemma : cantor-interval-converges
∀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 :
all: ∀x:A. B[x],
uimplies: b supposing a,
member: t ∈ T,
rleq: x ≤ y,
rnonneg: rnonneg(x),
le: A ≤ B,
and: P ∧ Q,
not: ¬A,
implies: P ⇒ Q,
false: False,
uall: ∀[x:A]. B[x],
subtype_rel: A ⊆r B,
real: ℝ,
prop: ℙ,
so_lambda: λ2x.t[x],
so_apply: x[s],
nat: ℕ,
less_than': less_than'(a;b),
top: Top,
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q
Lemmas referenced :
less_than'_wf,
rsub_wf,
real_wf,
nat_plus_wf,
converges-iff-cauchy,
cantor-interval_wf,
subtype_rel_dep_function,
nat_wf,
bool_wf,
int_seg_wf,
int_seg_subtype_nat,
false_wf,
subtype_rel_self,
pi1_wf_top,
equal_wf,
cantor-interval-cauchy-ext,
rleq_wf
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation,
isect_memberFormation,
cut,
introduction,
sqequalRule,
sqequalHypSubstitution,
lambdaEquality,
dependent_functionElimination,
thin,
hypothesisEquality,
productElimination,
independent_pairEquality,
voidElimination,
extract_by_obid,
isectElimination,
applyEquality,
hypothesis,
setElimination,
rename,
minusEquality,
natural_numberEquality,
axiomEquality,
equalityTransitivity,
equalitySymmetry,
because_Cache,
independent_isectElimination,
independent_pairFormation,
productEquality,
isect_memberEquality,
voidEquality,
independent_functionElimination,
functionEquality
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:
2017_10_03-AM-09_52_38
Last ObjectModification:
2017_07_28-AM-08_02_27
Theory : reals
Home
Index