Nuprl Lemma : rn-metric-complete
∀n:ℕ. mcomplete(ℝ^n with rn-metric(n))
Proof
Definitions occuring in Statement :
rn-metric: rn-metric(n),
real-vec: ℝ^n,
mcomplete: mcomplete(M),
mk-metric-space: X with d,
nat: ℕ,
all: ∀x:A. B[x]
Definitions unfolded in proof :
all: ∀x:A. B[x],
uall: ∀[x:A]. B[x],
member: t ∈ T,
rn-prod-metric: rn-prod-metric(n),
implies: P ⇒ Q,
nat: ℕ,
decidable: Dec(P),
or: P ∨ Q,
uimplies: b supposing a,
sq_type: SQType(T),
guard: {T},
iff: P ⇐⇒ Q,
and: P ∧ Q,
rev_implies: P ⇐ Q,
less_than: a < b,
squash: ↓T,
less_than': less_than'(a;b),
true: True,
prop: ℙ,
cand: A c∧ B,
metric-leq: d1 ≤ d2,
rn-metric: rn-metric(n),
scale-metric: c*d,
prod-metric: prod-metric(k;d),
subtract: n - m,
mdist: mdist(d;x;y),
rmetric: rmetric(),
so_lambda: λ2x.t[x],
top: Top,
so_apply: x[s],
le: A ≤ B,
not: ¬A,
false: False,
subtype_rel: A ⊆r B,
exists: ∃x:A. B[x],
uiff: uiff(P;Q),
rev_uimplies: rev_uimplies(P;Q),
req_int_terms: t1 ≡ t2,
nat_plus: ℕ+,
ge: i ≥ j ,
satisfiable_int_formula: satisfiable_int_formula(fmla),
rneq: x ≠ y,
rless: x < y,
sq_exists: ∃x:A [B[x]],
rat_term_to_real: rat_term_to_real(f;t),
rtermVar: rtermVar(var),
rat_term_ind: rat_term_ind,
pi1: fst(t),
rtermDivide: num "/" denom,
rtermConstant: "const",
pi2: snd(t),
rge: x ≥ y,
label: ...$L... t
Lemmas referenced :
equiv-metrics-preserve-complete,
real-vec_wf,
rn-prod-metric_wf,
rn-metric_wf,
decidable__equal_int,
subtype_base_sq,
int_subtype_base,
mcomplete-rn-prod-metric,
istype-nat,
rless-int,
int-to-real_wf,
rless_wf,
rsum-empty,
istype-void,
istype-le,
rmul_wf,
real-vec-dist_wf,
itermSubtract_wf,
itermMultiply_wf,
itermConstant_wf,
real-vec-dist-nonneg,
itermVar_wf,
metric-leq_wf,
scale-metric_wf,
rleq-int,
istype-false,
rleq_wf,
subtype_rel_sets_simple,
real_wf,
rleq_weakening_rless,
rleq_functionality,
req_weakening,
req-iff-rsub-is-0,
real_polynomial_null,
istype-int,
real_term_value_sub_lemma,
real_term_value_mul_lemma,
real_term_value_const_lemma,
real_term_value_var_lemma,
rleq_weakening_equal,
real-vec-dist-dim0,
rless-int-fractions2,
nat_properties,
decidable__lt,
full-omega-unsat,
intformand_wf,
intformnot_wf,
intformless_wf,
intformeq_wf,
intformle_wf,
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_eq_lemma,
int_formula_prop_le_lemma,
int_formula_prop_wf,
istype-less_than,
int_term_value_mul_lemma,
rdiv_wf,
nat_plus_properties,
rleq-int-fractions2,
decidable__le,
rneq-int,
rn-prod-metric-le-max-metric,
max-metric-leq-rn-metric,
rmul_preserves_rleq2,
mdist_wf,
max-metric_wf,
scale-metric-leq-iff,
assert-rat-term-eq2,
rtermDivide_wf,
rtermConstant_wf,
rtermVar_wf,
rmul_functionality,
rleq_functionality_wrt_implies,
rleq_weakening,
rn-metric-leq-rn-prod-metric,
rleq-implies-rleq,
rsub_wf
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation_alt,
cut,
introduction,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
hypothesisEquality,
hypothesis,
independent_functionElimination,
dependent_functionElimination,
setElimination,
rename,
natural_numberEquality,
unionElimination,
instantiate,
cumulativity,
intEquality,
independent_isectElimination,
because_Cache,
productElimination,
sqequalRule,
equalityTransitivity,
equalitySymmetry,
independent_pairFormation,
imageMemberEquality,
baseClosed,
dependent_set_memberEquality_alt,
universeIsType,
minusEquality,
isect_memberEquality_alt,
voidElimination,
inhabitedIsType,
applyEquality,
lambdaEquality_alt,
productIsType,
dependent_pairFormation_alt,
approximateComputation,
int_eqEquality,
multiplyEquality,
closedConclusion,
inrFormation_alt
Latex:
\mforall{}n:\mBbbN{}. mcomplete(\mBbbR{}\^{}n with rn-metric(n))
Date html generated:
2019_10_30-AM-08_39_10
Last ObjectModification:
2019_10_02-AM-11_04_10
Theory : reals
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