Nuprl Lemma : m-unique-limit
∀[X:Type]. ∀d:metric(X). ∀x:ℕ ⟶ X. ∀[y1,y2:X]. (y1 ≡ y2) supposing (lim n→∞.x[n] = y1 and lim n→∞.x[n] = y2)
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],
so_apply: x[s],
all: ∀x:A. B[x],
function: x:A ⟶ B[x],
universe: Type
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
all: ∀x:A. B[x],
uimplies: b supposing a,
mconverges-to: lim n→∞.x[n] = y,
meq: x ≡ y,
metric: metric(X),
implies: P ⇒ Q,
so_lambda: λ2x.t[x],
so_apply: x[s],
prop: ℙ,
uiff: uiff(P;Q),
and: P ∧ Q,
nat_plus: ℕ+,
decidable: Dec(P),
or: P ∨ Q,
not: ¬A,
satisfiable_int_formula: satisfiable_int_formula(fmla),
exists: ∃x:A. B[x],
false: False,
top: Top,
sq_exists: ∃x:A [B[x]],
rev_uimplies: rev_uimplies(P;Q),
nat: ℕ,
guard: {T},
ge: i ≥ j ,
rneq: x ≠ y,
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q,
rge: x ≥ y,
rleq: x ≤ y,
rnonneg: rnonneg(x),
le: A ≤ B,
req_int_terms: t1 ≡ t2,
mdist: mdist(d;x;y),
less_than: a < b,
squash: ↓T,
less_than': less_than'(a;b),
true: True,
subtype_rel: A ⊆r B,
rdiv: (x/y)
Lemmas referenced :
req_witness,
int-to-real_wf,
mconverges-to_wf,
istype-nat,
metric_wf,
istype-universe,
infinitesmal-difference,
nat_plus_properties,
decidable__lt,
full-omega-unsat,
intformand_wf,
intformnot_wf,
intformless_wf,
itermConstant_wf,
itermMultiply_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_mul_lemma,
int_term_value_var_lemma,
int_formula_prop_wf,
istype-less_than,
rleq_functionality_wrt_implies,
mdist_wf,
imax_wf,
imax_nat,
nat_properties,
decidable__le,
intformle_wf,
intformeq_wf,
int_formula_prop_le_lemma,
int_formula_prop_eq_lemma,
istype-le,
rdiv_wf,
rless-int,
rless_wf,
imax_ub,
rleq_weakening_equal,
rleq_weakening,
le_witness_for_triv,
nat_plus_wf,
itermSubtract_wf,
req-iff-rsub-is-0,
real_polynomial_null,
real_term_value_sub_lemma,
real_term_value_var_lemma,
real_term_value_const_lemma,
radd_wf,
rabs_wf,
rsub_wf,
mdist-nonneg,
mdist-triangle-inequality,
rleq_functionality,
rabs_functionality,
req_weakening,
rabs-of-nonneg,
radd_functionality_wrt_rleq,
radd_functionality,
mdist-symm,
rmul_wf,
rinv_wf2,
rneq_functionality,
rmul-int,
rneq-int,
set_subtype_base,
less_than_wf,
int_subtype_base,
itermAdd_wf,
req_transitivity,
rmul_functionality,
rinv_functionality2,
req_inversion,
rinv-of-rmul,
rmul-rinv3,
rinv-as-rdiv,
real_term_value_add_lemma,
real_term_value_mul_lemma
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation_alt,
introduction,
cut,
lambdaFormation_alt,
sqequalHypSubstitution,
sqequalRule,
extract_by_obid,
isectElimination,
thin,
applyEquality,
setElimination,
rename,
hypothesisEquality,
hypothesis,
natural_numberEquality,
independent_functionElimination,
universeIsType,
lambdaEquality_alt,
isect_memberEquality_alt,
because_Cache,
isectIsTypeImplies,
inhabitedIsType,
functionIsType,
dependent_functionElimination,
functionIsTypeImplies,
instantiate,
universeEquality,
productElimination,
independent_isectElimination,
dependent_set_memberEquality_alt,
multiplyEquality,
unionElimination,
approximateComputation,
dependent_pairFormation_alt,
int_eqEquality,
voidElimination,
independent_pairFormation,
equalityTransitivity,
equalitySymmetry,
applyLambdaEquality,
equalityIstype,
closedConclusion,
inrFormation_alt,
inlFormation_alt,
imageMemberEquality,
baseClosed,
baseApply,
intEquality,
sqequalBase
Latex:
\mforall{}[X:Type]
\mforall{}d:metric(X). \mforall{}x:\mBbbN{} {}\mrightarrow{} X.
\mforall{}[y1,y2:X]. (y1 \mequiv{} y2) supposing (lim n\mrightarrow{}\minfty{}.x[n] = y1 and lim n\mrightarrow{}\minfty{}.x[n] = y2)
Date html generated:
2019_10_30-AM-06_39_28
Last ObjectModification:
2019_10_02-AM-10_52_19
Theory : reals
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