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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