Nuprl Lemma : mconverges-to_wf
∀[X:Type]. ∀[d:X ⟶ X ⟶ ℝ]. ∀[x:ℕ ⟶ X]. ∀[y:X]. (lim n→∞.x[n] = y ∈ ℙ)
Proof
Definitions occuring in Statement :
mconverges-to: lim n→∞.x[n] = y
,
real: ℝ
,
nat: ℕ
,
uall: ∀[x:A]. B[x]
,
prop: ℙ
,
so_apply: x[s]
,
member: t ∈ T
,
function: x:A ⟶ B[x]
,
universe: Type
Definitions unfolded in proof :
mconverges-to: lim n→∞.x[n] = y
,
mdist: mdist(d;x;y)
,
uall: ∀[x:A]. B[x]
,
member: t ∈ T
,
so_lambda: λ2x.t[x]
,
implies: P
⇒ Q
,
prop: ℙ
,
nat: ℕ
,
so_apply: x[s]
,
nat_plus: ℕ+
,
uimplies: b supposing a
,
rneq: x ≠ y
,
guard: {T}
,
or: P ∨ Q
,
all: ∀x:A. B[x]
,
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
Lemmas referenced :
all_wf,
nat_plus_wf,
sq_exists_wf,
nat_wf,
le_wf,
rleq_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,
istype-nat,
real_wf,
istype-universe
Rules used in proof :
sqequalSubstitution,
sqequalRule,
sqequalReflexivity,
sqequalTransitivity,
computationStep,
isect_memberFormation_alt,
introduction,
cut,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
hypothesis,
lambdaEquality_alt,
because_Cache,
functionEquality,
setElimination,
rename,
applyEquality,
hypothesisEquality,
closedConclusion,
natural_numberEquality,
independent_isectElimination,
inrFormation_alt,
dependent_functionElimination,
productElimination,
independent_functionElimination,
unionElimination,
approximateComputation,
dependent_pairFormation_alt,
int_eqEquality,
isect_memberEquality_alt,
voidElimination,
independent_pairFormation,
universeIsType,
inhabitedIsType,
axiomEquality,
equalityTransitivity,
equalitySymmetry,
isectIsTypeImplies,
functionIsType,
instantiate,
universeEquality
Latex:
\mforall{}[X:Type]. \mforall{}[d:X {}\mrightarrow{} X {}\mrightarrow{} \mBbbR{}]. \mforall{}[x:\mBbbN{} {}\mrightarrow{} X]. \mforall{}[y:X]. (lim n\mrightarrow{}\minfty{}.x[n] = y \mmember{} \mBbbP{})
Date html generated:
2019_10_30-AM-06_38_06
Last ObjectModification:
2019_10_02-AM-10_51_07
Theory : reals
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