Nuprl Lemma : real-interval-complete
∀a,b:ℝ. mcomplete({x:ℝ| x ∈ [a, b]} with rmetric())
Proof
Definitions occuring in Statement :
mcomplete: mcomplete(M),
mk-metric-space: X with d,
rmetric: rmetric(),
rccint: [l, u],
i-member: r ∈ I,
real: ℝ,
all: ∀x:A. B[x],
set: {x:A| B[x]}
Definitions unfolded in proof :
all: ∀x:A. B[x],
rmetric: rmetric(),
mcomplete: mcomplete(M),
mk-metric-space: X with d,
member: t ∈ T,
top: Top,
mconverges: x[n]↓ as n→∞,
mcauchy: mcauchy(d;n.x[n]),
mconverges-to: lim n→∞.x[n] = y,
mdist: mdist(d;x;y),
implies: P ⇒ Q,
cauchy: cauchy(n.x[n]),
converges-to: lim n→∞.x[n] = y,
so_lambda: λ2x.t[x],
so_apply: x[s],
subtype_rel: A ⊆r B,
iff: P ⇐⇒ Q,
and: P ∧ Q,
rev_implies: P ⇐ Q,
converges: x[n]↓ as n→∞,
exists: ∃x:A. B[x],
cand: A c∧ B,
uall: ∀[x:A]. B[x],
prop: ℙ,
sq_exists: ∃x:A [B[x]],
nat: ℕ,
nat_plus: ℕ+,
uimplies: b supposing a,
rneq: x ≠ y,
guard: {T},
or: P ∨ Q,
ge: i ≥ j ,
decidable: Dec(P),
not: ¬A,
satisfiable_int_formula: satisfiable_int_formula(fmla),
false: False,
sq_stable: SqStable(P),
squash: ↓T
Lemmas referenced :
converges-iff-cauchy,
member_rccint_lemma,
istype-void,
istype-nat,
rleq_wf,
converges-to_wf,
nat_plus_wf,
istype-le,
rabs_wf,
rsub_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,
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,
real_wf,
constant-rleq-limit,
nat_wf,
sq_stable__rleq,
rleq-limit-constant
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation_alt,
cut,
introduction,
extract_by_obid,
sqequalRule,
sqequalHypSubstitution,
dependent_functionElimination,
thin,
isect_memberEquality_alt,
voidElimination,
hypothesis,
lambdaEquality_alt,
applyEquality,
hypothesisEquality,
because_Cache,
productElimination,
independent_functionElimination,
dependent_pairFormation_alt,
dependent_set_memberEquality_alt,
independent_pairFormation,
productIsType,
universeIsType,
isectElimination,
setElimination,
rename,
inhabitedIsType,
equalityTransitivity,
equalitySymmetry,
functionIsType,
setIsType,
closedConclusion,
natural_numberEquality,
independent_isectElimination,
inrFormation_alt,
unionElimination,
approximateComputation,
int_eqEquality,
functionExtensionality,
imageMemberEquality,
baseClosed,
imageElimination,
equalityIstype
Latex:
\mforall{}a,b:\mBbbR{}. mcomplete(\{x:\mBbbR{}| x \mmember{} [a, b]\} with rmetric())
Date html generated:
2019_10_30-AM-06_44_12
Last ObjectModification:
2019_10_09-PM-05_01_33
Theory : reals
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