Nuprl Lemma : nth_tl-partition
∀I:Interval
(icompact(I)
⇒ (∀a:ℝ. ∀p:partition(I). ∀i:ℕ||p||. ((a = p[i]) ⇒ (nth_tl(i + 1;p) ∈ partition([a, right-endpoint(I)])))))
Proof
Definitions occuring in Statement :
partition: partition(I),
icompact: icompact(I),
rccint: [l, u],
right-endpoint: right-endpoint(I),
interval: Interval,
req: x = y,
real: ℝ,
select: L[n],
length: ||as||,
nth_tl: nth_tl(n;as),
int_seg: {i..j-},
all: ∀x:A. B[x],
implies: P ⇒ Q,
member: t ∈ T,
add: n + m,
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),
int_iseg: {i...j},
true: True,
subtype_rel: A ⊆r B,
le: A ≤ B,
uiff: uiff(P;Q),
rev_implies: P ⇐ Q,
left-endpoint: left-endpoint(I),
endpoints: endpoints(I),
rccint: [l, u],
outl: outl(x),
pi1: fst(t),
nat: ℕ,
less_than': less_than'(a;b),
subtract: n - m,
right-endpoint: right-endpoint(I),
pi2: snd(t),
last: last(L),
sq_type: SQType(T),
select: L[n],
nil: [],
it: ⋅,
so_lambda: λ2x y.t[x; y],
so_apply: x[s1;s2],
assert: ↑b,
ifthenelse: if b then t else f fi ,
btrue: tt,
cons: [a / b],
bfalse: ff,
sorted-by: sorted-by(R;L)
Lemmas referenced :
nth_tl_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,
right-endpoint_wf,
rccint-icompact,
req_wf,
select_wf,
decidable__le,
int_seg_wf,
partition_wf,
icompact_wf,
interval_wf,
less_than_wf,
squash_wf,
true_wf,
length_nth_tl,
itermAdd_wf,
int_term_value_add_lemma,
le_wf,
iff_weakening_equal,
itermSubtract_wf,
int_term_value_subtract_lemma,
lelt_wf,
subtract_wf,
add-member-int_seg2,
rleq_wf,
select_nth_tl,
select-nth_tl,
false_wf,
subtype_rel_list,
top_wf,
add-zero,
rleq_transitivity,
rleq_weakening,
subtype_base_sq,
int_subtype_base,
decidable__equal_int,
intformeq_wf,
int_formula_prop_eq_lemma,
equal_wf,
add_functionality_wrt_eq,
frs-non-dec-sorted-by,
list-cases,
stuck-spread,
base_wf,
length_of_nil_lemma,
null_nil_lemma,
product_subtype_list,
length_of_cons_lemma,
null_cons_lemma,
last_wf
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation,
cut,
sqequalHypSubstitution,
setElimination,
thin,
rename,
dependent_set_memberEquality,
introduction,
extract_by_obid,
isectElimination,
hypothesis,
hypothesisEquality,
addEquality,
natural_numberEquality,
productElimination,
independent_functionElimination,
dependent_functionElimination,
unionElimination,
imageElimination,
independent_isectElimination,
dependent_pairFormation,
lambdaEquality,
int_eqEquality,
intEquality,
isect_memberEquality,
voidElimination,
voidEquality,
sqequalRule,
independent_pairFormation,
computeAll,
because_Cache,
applyEquality,
equalityTransitivity,
equalitySymmetry,
productEquality,
imageMemberEquality,
baseClosed,
universeEquality,
instantiate,
cumulativity,
promote_hyp,
hypothesis_subsumption
Latex:
\mforall{}I:Interval
(icompact(I)
{}\mRightarrow{} (\mforall{}a:\mBbbR{}. \mforall{}p:partition(I). \mforall{}i:\mBbbN{}||p||.
((a = p[i]) {}\mRightarrow{} (nth\_tl(i + 1;p) \mmember{} partition([a, right-endpoint(I)])))))
Date html generated:
2017_10_03-AM-09_42_48
Last ObjectModification:
2017_07_28-AM-07_57_22
Theory : reals
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