Nuprl Lemma : metric-leq-converges-to
∀[X:Type]. ∀[d1,d2:metric(X)]. (d2 ≤ d1 ⇒ (∀x:ℕ ⟶ X. ∀y:X. (lim n→∞.x[n] = y ⇒ lim n→∞.x[n] = y)))
Proof
Definitions occuring in Statement :
mconverges-to: lim n→∞.x[n] = y,
metric-leq: d1 ≤ d2,
metric: metric(X),
nat: ℕ,
uall: ∀[x:A]. B[x],
so_apply: x[s],
all: ∀x:A. B[x],
implies: P ⇒ Q,
function: x:A ⟶ B[x],
universe: Type
Definitions unfolded in proof :
metric-leq: d1 ≤ d2,
uall: ∀[x:A]. B[x],
implies: P ⇒ Q,
all: ∀x:A. B[x],
mconverges-to: lim n→∞.x[n] = y,
member: t ∈ T,
sq_exists: ∃x:A [B[x]],
nat: ℕ,
so_apply: x[s],
nat_plus: 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,
metric: metric(X),
so_lambda: λ2x.t[x],
rev_uimplies: rev_uimplies(P;Q),
rge: x ≥ y
Lemmas referenced :
istype-le,
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,
mconverges-to_wf,
istype-nat,
metric_wf,
istype-universe,
rleq_functionality_wrt_implies,
rleq_weakening_equal
Rules used in proof :
sqequalSubstitution,
sqequalRule,
sqequalReflexivity,
sqequalTransitivity,
computationStep,
isect_memberFormation_alt,
lambdaFormation_alt,
sqequalHypSubstitution,
cut,
hypothesis,
dependent_functionElimination,
thin,
hypothesisEquality,
setElimination,
rename,
introduction,
dependent_set_memberEquality_alt,
independent_functionElimination,
extract_by_obid,
isectElimination,
because_Cache,
functionIsType,
universeIsType,
applyEquality,
closedConclusion,
natural_numberEquality,
independent_isectElimination,
inrFormation_alt,
productElimination,
unionElimination,
approximateComputation,
dependent_pairFormation_alt,
lambdaEquality_alt,
int_eqEquality,
isect_memberEquality_alt,
voidElimination,
independent_pairFormation,
inhabitedIsType,
instantiate,
universeEquality,
equalityTransitivity,
equalitySymmetry
Latex:
\mforall{}[X:Type]. \mforall{}[d1,d2:metric(X)].
(d2 \mleq{} d1 {}\mRightarrow{} (\mforall{}x:\mBbbN{} {}\mrightarrow{} X. \mforall{}y:X. (lim n\mrightarrow{}\minfty{}.x[n] = y {}\mRightarrow{} lim n\mrightarrow{}\minfty{}.x[n] = y)))
Date html generated:
2019_10_30-AM-06_48_39
Last ObjectModification:
2019_10_02-AM-10_59_34
Theory : reals
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