Nuprl Lemma : mtb-seq-mtb-point-cantor-mconverges-to
∀[X:Type]. ∀d:metric(X). ∀mtb:m-TB(X;d). ∀x:X. lim n→∞.mtb-seq(mtb;mtb-point-cantor(mtb;x)) n = x
Proof
Definitions occuring in Statement :
mtb-point-cantor: mtb-point-cantor(mtb;p),
mtb-seq: mtb-seq(mtb;s),
m-TB: m-TB(X;d),
mconverges-to: lim n→∞.x[n] = y,
metric: metric(X),
uall: ∀[x:A]. B[x],
all: ∀x:A. B[x],
apply: f a,
universe: Type
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
all: ∀x:A. B[x],
m-TB: m-TB(X;d),
mtb-point-cantor: mtb-point-cantor(mtb;p),
mtb-seq: mtb-seq(mtb;s),
spreadn: spread3,
mconverges-to: lim n→∞.x[n] = y,
sq_exists: ∃x:A [B[x]],
member: t ∈ T,
subtype_rel: A ⊆r B,
implies: P ⇒ Q,
nat_plus: ℕ+,
nat: ℕ,
uimplies: b supposing a,
rneq: x ≠ y,
guard: {T},
or: P ∨ Q,
iff: P ⇐⇒ Q,
and: P ∧ Q,
rev_implies: P ⇐ Q,
ge: i ≥ j ,
decidable: Dec(P),
not: ¬A,
satisfiable_int_formula: satisfiable_int_formula(fmla),
exists: ∃x:A. B[x],
false: False,
top: Top,
prop: ℙ,
uiff: uiff(P;Q),
rev_uimplies: rev_uimplies(P;Q),
rge: x ≥ y
Lemmas referenced :
nat_plus_subtype_nat,
istype-le,
istype-nat,
rleq_wf,
mdist_wf,
rdiv_wf,
int-to-real_wf,
rless-int,
nat_properties,
nat_plus_properties,
decidable__lt,
full-omega-unsat,
intformand_wf,
intformnot_wf,
intformless_wf,
itermConstant_wf,
itermVar_wf,
istype-int,
int_formula_prop_and_lemma,
istype-void,
int_formula_prop_not_lemma,
int_formula_prop_less_lemma,
int_term_value_constant_lemma,
int_term_value_var_lemma,
int_formula_prop_wf,
rless_wf,
nat_plus_wf,
m-TB_wf,
metric_wf,
istype-universe,
itermAdd_wf,
intformle_wf,
int_term_value_add_lemma,
int_formula_prop_le_lemma,
rleq-int-fractions,
istype-less_than,
decidable__le,
itermMultiply_wf,
int_term_value_mul_lemma,
rleq_functionality,
mdist-symm,
req_weakening,
rleq_functionality_wrt_implies,
rleq_weakening_equal
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation_alt,
lambdaFormation_alt,
sqequalHypSubstitution,
setElimination,
thin,
rename,
productElimination,
sqequalRule,
dependent_set_memberFormation_alt,
cut,
hypothesisEquality,
applyEquality,
introduction,
extract_by_obid,
hypothesis,
isectElimination,
functionIsType,
because_Cache,
universeIsType,
closedConclusion,
natural_numberEquality,
independent_isectElimination,
inrFormation_alt,
dependent_functionElimination,
independent_functionElimination,
unionElimination,
approximateComputation,
dependent_pairFormation_alt,
lambdaEquality_alt,
int_eqEquality,
isect_memberEquality_alt,
voidElimination,
independent_pairFormation,
instantiate,
universeEquality,
addEquality,
dependent_set_memberEquality_alt,
multiplyEquality,
equalityTransitivity,
equalitySymmetry
Latex:
\mforall{}[X:Type]. \mforall{}d:metric(X). \mforall{}mtb:m-TB(X;d). \mforall{}x:X. lim n\mrightarrow{}\minfty{}.mtb-seq(mtb;mtb-point-cantor(mtb;x)) n = x
Date html generated:
2019_10_30-AM-06_57_43
Last ObjectModification:
2019_10_03-PM-03_45_26
Theory : reals
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