Nuprl Lemma : rat-cube-complex-polyhedron-compact1
∀k:ℕ. ∀K:ℚCube(k) List. (0 < ||K|| ⇒ (∀c∈K.↑Inhabited(c)) ⇒ mcompact(|K|;rn-prod-metric(k)))
Proof
Definitions occuring in Statement :
rat-cube-complex-polyhedron: |K|,
rn-prod-metric: rn-prod-metric(n),
mcompact: mcompact(X;d),
l_all: (∀x∈L.P[x]),
length: ||as||,
list: T List,
nat: ℕ,
assert: ↑b,
less_than: a < b,
all: ∀x:A. B[x],
implies: P ⇒ Q,
natural_number: $n,
inhabited-rat-cube: Inhabited(c),
rational-cube: ℚCube(k)
Definitions unfolded in proof :
l_exists: (∃x∈L. P[x]),
rat-cube-complex-polyhedron: |K|,
ext-eq: A ≡ B,
iff: P ⇐⇒ Q,
stable-union: Error :stable-union,
subtype_rel: A ⊆r B,
l_all: (∀x∈L.P[x]),
so_apply: x[s],
so_lambda: λ2x.t[x],
rev_uimplies: rev_uimplies(P;Q),
uiff: uiff(P;Q),
so_apply: x[s1;s2],
prop: ℙ,
top: Top,
false: False,
exists: ∃x:A. B[x],
satisfiable_int_formula: satisfiable_int_formula(fmla),
not: ¬A,
or: P ∨ Q,
decidable: Dec(P),
ge: i ≥ j ,
nat: ℕ,
squash: ↓T,
less_than: a < b,
le: A ≤ B,
and: P ∧ Q,
lelt: i ≤ j < k,
uimplies: b supposing a,
int_seg: {i..j-},
so_lambda: λ2x y.t[x; y],
member: t ∈ T,
uall: ∀[x:A]. B[x],
implies: P ⇒ Q,
all: ∀x:A. B[x]
Lemmas referenced :
l_exists_wf,
metric-on-subtype,
rat-cube-complex-polyhedron_wf,
Error :stable-union_wf,
mcompact_functionality,
mcompact-rat-cube,
istype-nat,
list_wf,
istype-less_than,
l_member_wf,
inhabited-rat-cube_wf,
assert_wf,
l_all_wf2,
mcomplete-rn-prod-metric,
nsub_finite,
meq_wf,
in-rat-cube_functionality,
int_formula_prop_less_lemma,
intformless_wf,
istype-le,
decidable__lt,
int_formula_prop_wf,
int_term_value_var_lemma,
int_term_value_constant_lemma,
int_formula_prop_le_lemma,
int_formula_prop_not_lemma,
istype-void,
int_formula_prop_and_lemma,
istype-int,
itermVar_wf,
itermConstant_wf,
intformle_wf,
intformnot_wf,
intformand_wf,
full-omega-unsat,
decidable__le,
nat_properties,
int_seg_properties,
select_wf,
in-rat-cube_wf,
rational-cube_wf,
length_wf,
int_seg_wf,
rn-prod-metric_wf,
real-vec_wf,
mcompact-stable-union
Rules used in proof :
functionIsType,
applyEquality,
closedConclusion,
productIsType,
setIsType,
dependent_set_memberEquality_alt,
universeIsType,
independent_pairFormation,
voidElimination,
isect_memberEquality_alt,
int_eqEquality,
dependent_pairFormation_alt,
independent_functionElimination,
approximateComputation,
unionElimination,
imageElimination,
productElimination,
independent_isectElimination,
rename,
setElimination,
because_Cache,
lambdaEquality_alt,
sqequalRule,
natural_numberEquality,
dependent_functionElimination,
hypothesis,
hypothesisEquality,
thin,
isectElimination,
sqequalHypSubstitution,
extract_by_obid,
introduction,
cut,
lambdaFormation_alt,
sqequalReflexivity,
computationStep,
sqequalTransitivity,
sqequalSubstitution
Latex:
\mforall{}k:\mBbbN{}. \mforall{}K:\mBbbQ{}Cube(k) List. (0 < ||K|| {}\mRightarrow{} (\mforall{}c\mmember{}K.\muparrow{}Inhabited(c)) {}\mRightarrow{} mcompact(|K|;rn-prod-metric(k)))
Date html generated:
2019_10_31-AM-06_03_59
Last ObjectModification:
2019_10_30-AM-11_45_41
Theory : real!vectors
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