Nuprl Lemma : mconverges-to_functionality
∀[X:Type]. ∀d:metric(X). ∀x1,x2:ℕ ⟶ X. ∀[y:X]. {lim n→∞.x1[n] = y ⇒ lim n→∞.x2[n] = y} supposing ∀n:ℕ. x1[n] ≡ x2[n]
Proof
Definitions occuring in Statement :
mconverges-to: lim n→∞.x[n] = y,
meq: x ≡ y,
metric: metric(X),
nat: ℕ,
uimplies: b supposing a,
uall: ∀[x:A]. B[x],
guard: {T},
so_apply: x[s],
all: ∀x:A. B[x],
implies: P ⇒ Q,
function: x:A ⟶ B[x],
universe: Type
Definitions unfolded in proof :
guard: {T},
uall: ∀[x:A]. B[x],
all: ∀x:A. B[x],
uimplies: b supposing a,
member: t ∈ T,
meq: x ≡ y,
metric: metric(X),
so_apply: x[s],
implies: P ⇒ Q,
mconverges-to: lim n→∞.x[n] = y,
nat_plus: ℕ+,
rneq: x ≠ y,
or: P ∨ Q,
iff: P ⇐⇒ Q,
and: P ∧ Q,
rev_implies: P ⇐ Q,
sq_exists: ∃x:A [B[x]],
nat: ℕ,
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,
so_lambda: λ2x.t[x]
Lemmas referenced :
req_witness,
int-to-real_wf,
rdiv_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,
req_inversion,
mdist_wf,
rleq_transitivity,
rleq_weakening,
istype-le,
rleq_wf,
nat_plus_wf,
mconverges-to_wf,
istype-nat,
meq_wf,
metric_wf,
istype-universe,
meq-same,
mdist_functionality
Rules used in proof :
sqequalSubstitution,
sqequalRule,
sqequalReflexivity,
sqequalTransitivity,
computationStep,
isect_memberFormation_alt,
lambdaFormation_alt,
cut,
introduction,
sqequalHypSubstitution,
lambdaEquality_alt,
dependent_functionElimination,
thin,
hypothesisEquality,
extract_by_obid,
isectElimination,
applyEquality,
setElimination,
rename,
hypothesis,
natural_numberEquality,
independent_functionElimination,
functionIsTypeImplies,
inhabitedIsType,
closedConclusion,
because_Cache,
independent_isectElimination,
inrFormation_alt,
productElimination,
unionElimination,
approximateComputation,
dependent_pairFormation_alt,
int_eqEquality,
isect_memberEquality_alt,
voidElimination,
independent_pairFormation,
universeIsType,
promote_hyp,
dependent_set_memberEquality_alt,
equalityTransitivity,
equalitySymmetry,
functionIsType,
instantiate,
universeEquality
Latex:
\mforall{}[X:Type]
\mforall{}d:metric(X). \mforall{}x1,x2:\mBbbN{} {}\mrightarrow{} X.
\mforall{}[y:X]. \{lim n\mrightarrow{}\minfty{}.x1[n] = y {}\mRightarrow{} lim n\mrightarrow{}\minfty{}.x2[n] = y\} supposing \mforall{}n:\mBbbN{}. x1[n] \mequiv{} x2[n]
Date html generated:
2019_10_30-AM-06_38_26
Last ObjectModification:
2019_10_02-AM-10_51_25
Theory : reals
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