Nuprl Lemma : partition-exists
∀I:Interval. (icompact(I) ⇒ (∀e:ℝ. ((r0 < e) ⇒ (∃p:partition(I). (partition-mesh(I;p) ≤ e)))))
Proof
Definitions occuring in Statement :
partition-mesh: partition-mesh(I;p),
partition: partition(I),
icompact: icompact(I),
interval: Interval,
rleq: x ≤ y,
rless: x < y,
int-to-real: r(n),
real: ℝ,
all: ∀x:A. B[x],
exists: ∃x:A. B[x],
implies: P ⇒ Q,
natural_number: $n
Definitions unfolded in proof :
all: ∀x:A. B[x],
implies: P ⇒ Q,
uall: ∀[x:A]. B[x],
member: t ∈ T,
uimplies: b supposing a,
prop: ℙ,
rneq: x ≠ y,
guard: {T},
or: P ∨ Q,
icompact: icompact(I),
and: P ∧ Q,
exists: ∃x:A. B[x],
nat_plus: ℕ+,
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q,
rless: x < y,
sq_exists: ∃x:{A| B[x]},
decidable: Dec(P),
satisfiable_int_formula: satisfiable_int_formula(fmla),
false: False,
not: ¬A,
top: Top,
uiff: uiff(P;Q),
rev_uimplies: rev_uimplies(P;Q),
rleq: x ≤ y,
rnonneg: rnonneg(x),
le: A ≤ B,
subtype_rel: A ⊆r B
Lemmas referenced :
rmul_comm,
uiff_transitivity,
rmul_preserves_rleq,
rmul-rdiv-cancel2,
nat_plus_wf,
rmul_wf,
rsub_wf,
less_than'_wf,
rleq_weakening_rless,
rmul_preserves_rleq2,
req_weakening,
mesh-uniform-partition,
rleq_functionality,
int_formula_prop_wf,
int_term_value_var_lemma,
int_term_value_constant_lemma,
int_formula_prop_less_lemma,
int_formula_prop_not_lemma,
int_formula_prop_and_lemma,
itermVar_wf,
itermConstant_wf,
intformless_wf,
intformnot_wf,
intformand_wf,
satisfiable-full-omega-tt,
decidable__lt,
nat_plus_properties,
rless-int,
partition-mesh_wf,
uniform-partition_wf,
rabs_wf,
rleq_transitivity,
rabs-bounds,
rleq_wf,
i-length_wf,
rdiv_wf,
integer-bound,
interval_wf,
icompact_wf,
real_wf,
int-to-real_wf,
rless_wf,
icompact-length-nonneg
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation,
cut,
lemma_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
hypothesisEquality,
independent_isectElimination,
hypothesis,
natural_numberEquality,
dependent_functionElimination,
sqequalRule,
inrFormation,
productElimination,
dependent_pairFormation,
setElimination,
rename,
because_Cache,
independent_functionElimination,
unionElimination,
lambdaEquality,
int_eqEquality,
intEquality,
isect_memberEquality,
voidElimination,
voidEquality,
independent_pairFormation,
computeAll,
isect_memberFormation,
introduction,
independent_pairEquality,
applyEquality,
minusEquality,
axiomEquality,
equalityTransitivity,
equalitySymmetry
Latex:
\mforall{}I:Interval. (icompact(I) {}\mRightarrow{} (\mforall{}e:\mBbbR{}. ((r0 < e) {}\mRightarrow{} (\mexists{}p:partition(I). (partition-mesh(I;p) \mleq{} e)))))
Date html generated:
2016_05_18-AM-09_02_28
Last ObjectModification:
2016_01_17-AM-02_32_35
Theory : reals
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