Nuprl Lemma : nearby-partition-mesh
∀I:Interval
(icompact(I)
⇒ (∀e:{e:ℝ| r0 < e} . ∀p,p':partition(I).
(nearby-partitions((e/r(2));p;p') ⇒ (partition-mesh(I;p') ≤ (partition-mesh(I;p) + e)))))
Proof
Definitions occuring in Statement :
partition-mesh: partition-mesh(I;p),
nearby-partitions: nearby-partitions(e;p;q),
partition: partition(I),
icompact: icompact(I),
interval: Interval,
rdiv: (x/y),
rleq: x ≤ y,
rless: x < y,
radd: a + b,
int-to-real: r(n),
real: ℝ,
all: ∀x:A. B[x],
implies: P ⇒ Q,
set: {x:A| B[x]} ,
natural_number: $n
Definitions unfolded in proof :
all: ∀x:A. B[x],
implies: P ⇒ Q,
partition-mesh: partition-mesh(I;p),
member: t ∈ T,
prop: ℙ,
uall: ∀[x:A]. B[x],
uimplies: b supposing a,
rneq: x ≠ y,
guard: {T},
or: P ∨ Q,
iff: P ⇐⇒ Q,
and: P ∧ Q,
rev_implies: P ⇐ Q,
less_than: a < b,
squash: ↓T,
less_than': less_than'(a;b),
true: True,
partition: partition(I),
so_lambda: λ2x.t[x],
so_apply: x[s],
nearby-partitions: nearby-partitions(e;p;q),
full-partition: full-partition(I;p),
cand: A c∧ B,
top: Top,
subtype_rel: A ⊆r B,
ge: i ≥ j ,
decidable: Dec(P),
satisfiable_int_formula: satisfiable_int_formula(fmla),
exists: ∃x:A. B[x],
false: False,
not: ¬A,
icompact: icompact(I),
int_seg: {i..j-},
sq_type: SQType(T),
select: L[n],
cons: [a / b],
nat: ℕ,
rsub: x - y,
absval: |i|,
uiff: uiff(P;Q),
rev_uimplies: rev_uimplies(P;Q),
sq_stable: SqStable(P),
lelt: i ≤ j < k,
bool: 𝔹,
unit: Unit,
it: ⋅,
btrue: tt,
ifthenelse: if b then t else f fi ,
bfalse: ff,
bnot: ¬bb,
assert: ↑b
Lemmas referenced :
nearby-partitions_wf,
rdiv_wf,
int-to-real_wf,
rless-int,
rless_wf,
partition_wf,
set_wf,
real_wf,
icompact_wf,
interval_wf,
int_seg_wf,
length_of_cons_lemma,
length_nil,
non_neg_length,
nil_wf,
length_cons,
right-endpoint_wf,
cons_wf,
append_wf,
length_append,
subtype_rel_list,
top_wf,
length-append,
length_of_nil_lemma,
decidable__equal_int,
length_wf,
satisfiable-full-omega-tt,
intformand_wf,
intformnot_wf,
intformeq_wf,
itermAdd_wf,
itermVar_wf,
itermConstant_wf,
int_formula_prop_and_lemma,
int_formula_prop_not_lemma,
int_formula_prop_eq_lemma,
int_term_value_add_lemma,
int_term_value_var_lemma,
int_term_value_constant_lemma,
int_formula_prop_wf,
subtype_base_sq,
int_subtype_base,
rleq_wf,
rabs_wf,
rsub_wf,
left-endpoint_wf,
radd_wf,
rminus_wf,
absval_wf,
nat_wf,
uiff_transitivity,
rleq_functionality,
rabs_functionality,
radd-rminus-both,
req_weakening,
rabs-abs,
minus-zero,
sq_stable__rleq,
rmul_preserves_rleq,
rmul_wf,
rleq_weakening_rless,
rmul-int,
rmul-rdiv-cancel2,
int_seg_properties,
decidable__lt,
intformless_wf,
intformle_wf,
int_formula_prop_less_lemma,
int_formula_prop_le_lemma,
le_wf,
squash_wf,
true_wf,
iff_weakening_equal,
length-singleton,
decidable__le,
and_wf,
equal_wf,
select_cons_tl,
subtract_wf,
itermSubtract_wf,
int_term_value_subtract_lemma,
set_subtype_base,
lt_int_wf,
bool_wf,
eqtt_to_assert,
assert_of_lt_int,
eqff_to_assert,
bool_cases_sqequal,
bool_subtype_base,
assert-bnot,
less_than_wf,
lelt_wf,
select-append,
nearby-frs-mesh,
full-partition_wf,
rmul_preserves_rless,
rless_functionality,
frs-mesh_wf,
radd_functionality,
rmul-rdiv-cancel
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation,
cut,
introduction,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
setElimination,
rename,
because_Cache,
hypothesis,
natural_numberEquality,
independent_isectElimination,
sqequalRule,
inrFormation,
dependent_functionElimination,
productElimination,
independent_functionElimination,
independent_pairFormation,
imageMemberEquality,
hypothesisEquality,
baseClosed,
lambdaEquality,
equalityTransitivity,
equalitySymmetry,
isect_memberEquality,
voidElimination,
voidEquality,
applyEquality,
addEquality,
unionElimination,
dependent_pairFormation,
int_eqEquality,
intEquality,
computeAll,
instantiate,
cumulativity,
minusEquality,
imageElimination,
multiplyEquality,
dependent_set_memberEquality,
applyLambdaEquality,
universeEquality,
equalityElimination,
promote_hyp,
addLevel,
levelHypothesis
Latex:
\mforall{}I:Interval
(icompact(I)
{}\mRightarrow{} (\mforall{}e:\{e:\mBbbR{}| r0 < e\} . \mforall{}p,p':partition(I).
(nearby-partitions((e/r(2));p;p') {}\mRightarrow{} (partition-mesh(I;p') \mleq{} (partition-mesh(I;p) + e)))))
Date html generated:
2017_10_03-PM-00_52_03
Last ObjectModification:
2017_07_28-AM-08_47_20
Theory : reals_2
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