Nuprl Lemma : firstn-partition
∀I:Interval
(icompact(I) ⇒ (∀a:ℝ. ∀p:partition(I). ∀i:ℕ||p||. ((a = p[i]) ⇒ (firstn(i;p) ∈ partition([left-endpoint(I), a])))))
Proof
Definitions occuring in Statement :
partition: partition(I),
icompact: icompact(I),
rccint: [l, u],
left-endpoint: left-endpoint(I),
interval: Interval,
req: x = y,
real: ℝ,
firstn: firstn(n;as),
select: L[n],
length: ||as||,
int_seg: {i..j-},
all: ∀x:A. B[x],
implies: P ⇒ Q,
member: t ∈ T,
natural_number: $n
Definitions unfolded in proof :
all: ∀x:A. B[x],
implies: P ⇒ Q,
member: t ∈ T,
partition: partition(I),
uall: ∀[x:A]. B[x],
int_seg: {i..j-},
partitions: partitions(I;p),
and: P ∧ Q,
guard: {T},
lelt: i ≤ j < k,
decidable: Dec(P),
or: P ∨ Q,
less_than: a < b,
squash: ↓T,
uimplies: b supposing a,
satisfiable_int_formula: satisfiable_int_formula(fmla),
exists: ∃x:A. B[x],
false: False,
not: ¬A,
top: Top,
prop: ℙ,
iff: P ⇐⇒ Q,
icompact: icompact(I),
cand: A c∧ B,
frs-non-dec: frs-non-dec(L),
subtype_rel: A ⊆r B,
int_iseg: {i...j},
so_lambda: λ2x.t[x],
so_apply: x[s],
ge: i ≥ j ,
le: A ≤ B,
less_than': less_than'(a;b),
left-endpoint: left-endpoint(I),
endpoints: endpoints(I),
rccint: [l, u],
outl: outl(x),
pi1: fst(t),
right-endpoint: right-endpoint(I),
pi2: snd(t),
assert: ↑b,
ifthenelse: if b then t else f fi ,
btrue: tt,
cons: [a / b],
bfalse: ff,
last: last(L),
rev_implies: P ⇐ Q,
sq_type: SQType(T),
sorted-by: sorted-by(R;L)
Lemmas referenced :
firstn_wf,
real_wf,
int_seg_properties,
length_wf,
decidable__lt,
satisfiable-full-omega-tt,
intformand_wf,
intformnot_wf,
intformless_wf,
itermConstant_wf,
itermVar_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_le_lemma,
int_formula_prop_wf,
partitions_wf,
rccint_wf,
left-endpoint_wf,
rccint-icompact,
req_wf,
select_wf,
decidable__le,
int_seg_wf,
partition_wf,
icompact_wf,
interval_wf,
less_than_wf,
non_neg_length,
length_firstn_eq,
subtype_rel_sets,
lelt_wf,
le_wf,
intformeq_wf,
int_formula_prop_eq_lemma,
subtype_rel_list,
top_wf,
int_seg_subtype_nat,
false_wf,
length_firstn,
select-firstn,
rleq_transitivity,
last_wf,
list_wf,
list-cases,
null_nil_lemma,
length_of_nil_lemma,
left_endpoint_rccint_lemma,
product_subtype_list,
null_cons_lemma,
length_of_cons_lemma,
equal_wf,
req_inversion,
rleq_weakening,
subtract_wf,
itermSubtract_wf,
int_term_value_subtract_lemma,
frs-non-dec-sorted-by,
decidable__equal_int,
subtype_base_sq,
int_subtype_base
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation,
cut,
sqequalHypSubstitution,
setElimination,
thin,
rename,
dependent_set_memberEquality,
introduction,
extract_by_obid,
isectElimination,
hypothesis,
hypothesisEquality,
productElimination,
independent_functionElimination,
natural_numberEquality,
dependent_functionElimination,
unionElimination,
imageElimination,
independent_isectElimination,
dependent_pairFormation,
lambdaEquality,
int_eqEquality,
intEquality,
isect_memberEquality,
voidElimination,
voidEquality,
sqequalRule,
independent_pairFormation,
computeAll,
because_Cache,
applyEquality,
productEquality,
setEquality,
applyLambdaEquality,
equalityTransitivity,
equalitySymmetry,
promote_hyp,
hypothesis_subsumption,
instantiate,
cumulativity
Latex:
\mforall{}I:Interval
(icompact(I)
{}\mRightarrow{} (\mforall{}a:\mBbbR{}. \mforall{}p:partition(I). \mforall{}i:\mBbbN{}||p||.
((a = p[i]) {}\mRightarrow{} (firstn(i;p) \mmember{} partition([left-endpoint(I), a])))))
Date html generated:
2017_10_03-AM-09_42_30
Last ObjectModification:
2017_07_28-AM-07_57_12
Theory : reals
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