Nuprl Lemma : nearby-partition-choice
∀I:Interval
(icompact(I)
⇒ (∀p,q:partition(I). ∀e:{e:ℝ| r0 < e} .
(nearby-partitions(e;p;q)
⇒ (∀x:partition-choice(full-partition(I;p))
∃y:partition-choice(full-partition(I;q)). ∀i:ℕ||p|| + 1. (|x[i] - y[i]| ≤ e)))))
Proof
Definitions occuring in Statement :
partition-choice-ap: x[i],
partition-choice: partition-choice(p),
full-partition: full-partition(I;p),
nearby-partitions: nearby-partitions(e;p;q),
partition: partition(I),
icompact: icompact(I),
interval: Interval,
rleq: x ≤ y,
rless: x < y,
rabs: |x|,
rsub: x - y,
int-to-real: r(n),
real: ℝ,
length: ||as||,
int_seg: {i..j-},
all: ∀x:A. B[x],
exists: ∃x:A. B[x],
implies: P ⇒ Q,
set: {x:A| B[x]} ,
add: n + m,
natural_number: $n
Definitions unfolded in proof :
all: ∀x:A. B[x],
implies: P ⇒ Q,
member: t ∈ T,
uall: ∀[x:A]. B[x],
uimplies: b supposing a,
prop: ℙ,
partition: partition(I),
so_lambda: λ2x.t[x],
so_apply: x[s],
top: Top,
full-partition: full-partition(I;p),
cand: A c∧ B,
and: P ∧ Q,
nearby-partitions: nearby-partitions(e;p;q),
decidable: Dec(P),
or: P ∨ Q,
satisfiable_int_formula: satisfiable_int_formula(fmla),
exists: ∃x:A. B[x],
false: False,
not: ¬A,
cons: [a / b],
select: L[n],
guard: {T},
sq_type: SQType(T),
int_seg: {i..j-},
less_than': less_than'(a;b),
le: A ≤ B,
rev_implies: P ⇐ Q,
iff: P ⇐⇒ Q,
squash: ↓T,
sq_stable: SqStable(P),
true: True,
nat: ℕ,
subtype_rel: A ⊆r B,
absval: |i|,
itermConstant: "const",
req_int_terms: t1 ≡ t2,
uiff: uiff(P;Q),
rev_uimplies: rev_uimplies(P;Q),
icompact: icompact(I),
rge: x ≥ y,
rgt: x > y,
lelt: i ≤ j < k,
assert: ↑b,
bnot: ¬bb,
bfalse: ff,
ifthenelse: if b then t else f fi ,
btrue: tt,
it: ⋅,
unit: Unit,
bool: 𝔹,
partition-choice-ap: x[i],
partition-choice: partition-choice(p),
ge: i ≥ j ,
less_than: a < b,
pi1: fst(t),
subtract: n - m,
frs-non-dec: frs-non-dec(L),
rless: x < y,
sq_exists: ∃x:A [B[x]],
nat_plus: ℕ+,
real: ℝ
Lemmas referenced :
partition-choice_wf,
full-partition_wf,
nearby-partitions_wf,
set_wf,
real_wf,
rless_wf,
int-to-real_wf,
partition_wf,
icompact_wf,
interval_wf,
length_wf,
int_seg_wf,
length_of_cons_lemma,
length-append,
length_of_nil_lemma,
decidable__equal_int,
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,
int_subtype_base,
subtype_base_sq,
false_wf,
rleq-int,
sq_stable__rleq,
nat_wf,
absval_wf,
left-endpoint_wf,
rsub_wf,
rabs_wf,
rleq_wf,
uiff_transitivity2,
rleq_functionality,
rabs_functionality,
real_term_polynomial,
itermSubtract_wf,
real_term_value_const_lemma,
real_term_value_sub_lemma,
real_term_value_var_lemma,
req-iff-rsub-is-0,
req_weakening,
squash_wf,
true_wf,
rabs-int,
rleq_functionality_wrt_implies,
rleq_weakening_equal,
rleq_weakening_rless,
int_seg_properties,
decidable__le,
length-singleton,
iff_weakening_equal,
top_wf,
subtype_rel_list,
length_append,
le_wf,
int_formula_prop_le_lemma,
int_formula_prop_less_lemma,
intformle_wf,
intformless_wf,
decidable__lt,
nil_wf,
right-endpoint_wf,
cons_wf,
append_wf,
select_cons_tl,
lelt_wf,
less_than_wf,
assert-bnot,
bool_subtype_base,
bool_cases_sqequal,
equal_wf,
eqff_to_assert,
assert_of_lt_int,
eqtt_to_assert,
bool_wf,
lt_int_wf,
set_subtype_base,
int_term_value_subtract_lemma,
subtract_wf,
select-append,
full-partition-non-dec,
list_wf,
member_rccint_lemma,
full-omega-unsat,
non_neg_length,
frs-non-dec_wf,
all_wf,
select_wf,
subtract-is-int-iff,
exists_wf,
rleq_weakening,
add-member-int_seg2,
real_polynomial_null,
rless-cases,
radd_wf,
trivial-rless-radd,
sq_stable__rless,
sq_stable__less_than,
nat_plus_properties,
trivial-rsub-rless,
rabs-difference-bound-rleq,
rless_transitivity2,
req_transitivity
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation,
cut,
introduction,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
hypothesisEquality,
independent_isectElimination,
hypothesis,
setElimination,
rename,
sqequalRule,
lambdaEquality,
natural_numberEquality,
independent_pairFormation,
voidEquality,
voidElimination,
isect_memberEquality,
dependent_functionElimination,
productElimination,
because_Cache,
unionElimination,
equalityTransitivity,
equalitySymmetry,
dependent_pairFormation,
int_eqEquality,
intEquality,
computeAll,
independent_functionElimination,
cumulativity,
instantiate,
imageElimination,
baseClosed,
imageMemberEquality,
minusEquality,
applyEquality,
addEquality,
universeEquality,
promote_hyp,
equalityElimination,
dependent_set_memberEquality,
approximateComputation,
productEquality,
functionEquality,
setEquality,
pointwiseFunctionality,
baseApply,
closedConclusion,
functionExtensionality
Latex:
\mforall{}I:Interval
(icompact(I)
{}\mRightarrow{} (\mforall{}p,q:partition(I). \mforall{}e:\{e:\mBbbR{}| r0 < e\} .
(nearby-partitions(e;p;q)
{}\mRightarrow{} (\mforall{}x:partition-choice(full-partition(I;p))
\mexists{}y:partition-choice(full-partition(I;q)). \mforall{}i:\mBbbN{}||p|| + 1. (|x[i] - y[i]| \mleq{} e)))))
Date html generated:
2019_10_30-AM-07_52_07
Last ObjectModification:
2018_08_23-AM-11_22_23
Theory : reals
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