Nuprl Lemma : cantor_ivl-converges
∀a,b:ℝ. ∀f:ℕ ⟶ 𝔹. fst(cantor_ivl(a;b;f;n))↓ as n→∞ supposing a ≤ b
Proof
Definitions occuring in Statement :
cantor_ivl: cantor_ivl(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],
member: t ∈ T,
uimplies: b supposing a,
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,
converges: x[n]↓ as n→∞,
exists: ∃x:A. B[x],
so_lambda: λ2x.t[x],
top: Top,
so_apply: x[s],
nat: ℕ,
less_than': less_than'(a;b),
guard: {T}
Lemmas referenced :
cantor-interval-converges,
less_than'_wf,
rsub_wf,
real_wf,
nat_plus_wf,
converges-to_wf,
cantor_ivl_wf,
nat_wf,
pi1_wf_top,
equal_wf,
bool_wf,
rleq_wf,
cantor-interval_wf,
subtype_rel_dep_function,
int_seg_wf,
int_seg_subtype_nat,
false_wf,
subtype_rel_self,
converges-to_functionality,
cantor-interval-req,
req_weakening
Rules used in proof :
cut,
introduction,
extract_by_obid,
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation,
hypothesis,
sqequalHypSubstitution,
dependent_functionElimination,
thin,
hypothesisEquality,
isect_memberFormation,
sqequalRule,
lambdaEquality,
productElimination,
independent_pairEquality,
voidElimination,
isectElimination,
applyEquality,
setElimination,
rename,
minusEquality,
natural_numberEquality,
axiomEquality,
equalityTransitivity,
equalitySymmetry,
independent_isectElimination,
dependent_pairFormation,
functionExtensionality,
productEquality,
isect_memberEquality,
voidEquality,
independent_functionElimination,
functionEquality,
because_Cache,
independent_pairFormation
Latex:
\mforall{}a,b:\mBbbR{}. \mforall{}f:\mBbbN{} {}\mrightarrow{} \mBbbB{}. fst(cantor\_ivl(a;b;f;n))\mdownarrow{} as n\mrightarrow{}\minfty{} supposing a \mleq{} b
Date html generated:
2017_10_03-AM-09_53_53
Last ObjectModification:
2017_07_28-AM-08_02_53
Theory : reals
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