Nuprl Lemma : partition-split-cons
∀[I:Interval]
∀[a:ℝ]. ∀[bs:ℝ List].
(partitions(I;[a / bs]) ⇒ (partitions([left-endpoint(I), a];[]) ∧ partitions([a, right-endpoint(I)];bs)))
supposing icompact(I)
Proof
Definitions occuring in Statement :
partitions: partitions(I;p),
icompact: icompact(I),
rccint: [l, u],
right-endpoint: right-endpoint(I),
left-endpoint: left-endpoint(I),
interval: Interval,
real: ℝ,
cons: [a / b],
nil: [],
list: T List,
uimplies: b supposing a,
uall: ∀[x:A]. B[x],
implies: P ⇒ Q,
and: P ∧ Q
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
uimplies: b supposing a,
implies: P ⇒ Q,
and: P ∧ Q,
cand: A c∧ B,
prop: ℙ,
partitions: partitions(I;p),
frs-non-dec: frs-non-dec(L),
all: ∀x:A. B[x],
rleq: x ≤ y,
rnonneg: rnonneg(x),
le: A ≤ B,
not: ¬A,
false: False,
int_seg: {i..j-},
nat_plus: ℕ+,
guard: {T},
lelt: i ≤ j < k,
decidable: Dec(P),
or: P ∨ Q,
satisfiable_int_formula: satisfiable_int_formula(fmla),
exists: ∃x:A. B[x],
top: Top,
subtype_rel: A ⊆r B,
less_than: a < b,
squash: ↓T,
less_than': less_than'(a;b),
length: ||as||,
list_ind: list_ind,
nil: [],
it: ⋅,
i-finite: i-finite(I),
rccint: [l, u],
isl: isl(x),
assert: ↑b,
ifthenelse: if b then t else f fi ,
btrue: tt,
true: True,
select: L[n],
so_lambda: λ2x y.t[x; y],
so_apply: x[s1;s2],
cons: [a / b],
bfalse: ff,
icompact: icompact(I),
uiff: uiff(P;Q),
subtract: n - m,
ge: i ≥ j ,
sq_type: SQType(T),
right-endpoint: right-endpoint(I),
left-endpoint: left-endpoint(I),
endpoints: endpoints(I),
outl: outl(x),
pi1: fst(t),
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q
Lemmas referenced :
partitions_wf,
cons_wf,
real_wf,
less_than'_wf,
rsub_wf,
select_wf,
nil_wf,
length_of_nil_lemma,
nat_plus_properties,
int_seg_properties,
decidable__le,
satisfiable-full-omega-tt,
intformand_wf,
intformnot_wf,
intformle_wf,
itermConstant_wf,
itermVar_wf,
int_formula_prop_and_lemma,
int_formula_prop_not_lemma,
int_formula_prop_le_lemma,
int_term_value_constant_lemma,
int_term_value_var_lemma,
int_formula_prop_wf,
decidable__lt,
intformless_wf,
int_formula_prop_less_lemma,
nat_plus_wf,
le_wf,
int_seg_wf,
length_wf,
less_than_wf,
false_wf,
left-endpoint_wf,
rccint_wf,
right-endpoint_wf,
last_wf,
list-cases,
null_nil_lemma,
right_endpoint_rccint_lemma,
left_endpoint_rccint_lemma,
stuck-spread,
base_wf,
product_subtype_list,
null_cons_lemma,
length_of_cons_lemma,
list_wf,
icompact_wf,
interval_wf,
add-member-int_seg2,
subtract_wf,
itermSubtract_wf,
int_term_value_subtract_lemma,
non_neg_length,
itermAdd_wf,
int_term_value_add_lemma,
lelt_wf,
add-associates,
add-swap,
add-commutes,
zero-add,
squash_wf,
add-subtract-cancel,
subtype_base_sq,
int_subtype_base,
rleq_wf,
select-cons-tl,
true_wf,
equal_wf,
last_cons,
iff_weakening_equal
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
introduction,
cut,
lambdaFormation,
independent_pairFormation,
hypothesis,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
hypothesisEquality,
independent_isectElimination,
sqequalRule,
lambdaEquality,
dependent_functionElimination,
productElimination,
independent_pairEquality,
because_Cache,
applyEquality,
setElimination,
rename,
natural_numberEquality,
unionElimination,
dependent_pairFormation,
int_eqEquality,
intEquality,
isect_memberEquality,
voidElimination,
voidEquality,
computeAll,
minusEquality,
axiomEquality,
equalityTransitivity,
equalitySymmetry,
imageElimination,
baseClosed,
promote_hyp,
hypothesis_subsumption,
independent_functionElimination,
dependent_set_memberEquality,
addEquality,
hyp_replacement,
productEquality,
cumulativity,
universeEquality,
imageMemberEquality,
instantiate,
addLevel,
levelHypothesis
Latex:
\mforall{}[I:Interval]
\mforall{}[a:\mBbbR{}]. \mforall{}[bs:\mBbbR{} List].
(partitions(I;[a / bs])
{}\mRightarrow{} (partitions([left-endpoint(I), a];[]) \mwedge{} partitions([a, right-endpoint(I)];bs)))
supposing icompact(I)
Date html generated:
2017_10_03-AM-09_41_32
Last ObjectModification:
2017_07_28-AM-07_56_40
Theory : reals
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