Nuprl Lemma : adjacent-full-partition-points
∀[I:Interval]
∀[p:partition(I)]
∀i:ℕ||full-partition(I;p)|| - 1. r0≤full-partition(I;p)[i + 1] - full-partition(I;p)[i]≤partition-mesh(I;p)
supposing icompact(I)
Proof
Definitions occuring in Statement :
partition-mesh: partition-mesh(I;p),
full-partition: full-partition(I;p),
partition: partition(I),
icompact: icompact(I),
interval: Interval,
rbetween: x≤y≤z,
rsub: x - y,
int-to-real: r(n),
select: L[n],
length: ||as||,
int_seg: {i..j-},
uimplies: b supposing a,
uall: ∀[x:A]. B[x],
all: ∀x:A. B[x],
subtract: n - m,
add: n + m,
natural_number: $n
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
uimplies: b supposing a,
all: ∀x:A. B[x],
rbetween: x≤y≤z,
and: P ∧ Q,
int_seg: {i..j-},
guard: {T},
lelt: i ≤ j < k,
decidable: Dec(P),
or: P ∨ Q,
satisfiable_int_formula: satisfiable_int_formula(fmla),
exists: ∃x:A. B[x],
false: False,
implies: P ⇒ Q,
not: ¬A,
top: Top,
prop: ℙ,
less_than: a < b,
squash: ↓T,
uiff: uiff(P;Q),
sq_stable: SqStable(P),
rleq: x ≤ y,
rnonneg: rnonneg(x),
le: A ≤ B,
subtype_rel: A ⊆r B,
partition: partition(I),
full-partition: full-partition(I;p),
sq_type: SQType(T),
select: L[n],
cons: [a / b],
subtract: n - m,
less_than': less_than'(a;b),
true: True,
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q,
icompact: icompact(I),
nat: ℕ,
bool: 𝔹,
unit: Unit,
it: ⋅,
btrue: tt,
ifthenelse: if b then t else f fi ,
bfalse: ff,
bnot: ¬bb,
assert: ↑b,
last: last(L),
append: as @ bs,
list_ind: list_ind,
nil: [],
right-endpoint: right-endpoint(I),
pi2: snd(t),
endpoints: endpoints(I),
left-endpoint: left-endpoint(I),
pi1: fst(t),
ge: i ≥ j
Lemmas referenced :
adjacent-partition-points,
sq_stable__and,
rleq_wf,
int-to-real_wf,
rsub_wf,
select_wf,
real_wf,
full-partition_wf,
int_seg_properties,
subtract_wf,
length_wf,
decidable__le,
satisfiable-full-omega-tt,
intformand_wf,
intformnot_wf,
intformle_wf,
itermConstant_wf,
itermAdd_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_add_lemma,
int_term_value_var_lemma,
int_formula_prop_wf,
decidable__lt,
subtract-is-int-iff,
intformless_wf,
itermSubtract_wf,
int_formula_prop_less_lemma,
int_term_value_subtract_lemma,
false_wf,
partition-mesh_wf,
sq_stable__rleq,
less_than'_wf,
nat_plus_wf,
squash_wf,
length_of_cons_lemma,
length-append,
length_of_nil_lemma,
add-subtract-cancel,
int_seg_wf,
partition_wf,
icompact_wf,
interval_wf,
decidable__equal_int,
subtype_base_sq,
int_subtype_base,
cons_wf,
right-endpoint_wf,
nil_wf,
lelt_wf,
left-endpoint_wf,
rbetween_wf,
true_wf,
select_append_front,
iff_weakening_equal,
append_wf,
le_wf,
length_append,
subtype_rel_list,
top_wf,
length-singleton,
intformeq_wf,
int_formula_prop_eq_lemma,
select_cons_tl,
general_arith_equation1,
select-append,
lt_int_wf,
bool_wf,
eqtt_to_assert,
assert_of_lt_int,
eqff_to_assert,
equal_wf,
bool_cases_sqequal,
bool_subtype_base,
assert-bnot,
less_than_wf,
add-is-int-iff,
list-cases,
product_subtype_list,
non_neg_length
Rules used in proof :
cut,
introduction,
extract_by_obid,
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
hypothesis,
sqequalHypSubstitution,
isectElimination,
thin,
hypothesisEquality,
lambdaFormation,
productElimination,
natural_numberEquality,
independent_isectElimination,
addEquality,
setElimination,
rename,
because_Cache,
dependent_functionElimination,
unionElimination,
dependent_pairFormation,
lambdaEquality,
int_eqEquality,
intEquality,
isect_memberEquality,
voidElimination,
voidEquality,
sqequalRule,
independent_pairFormation,
computeAll,
pointwiseFunctionality,
equalityTransitivity,
equalitySymmetry,
promote_hyp,
imageElimination,
baseApply,
closedConclusion,
baseClosed,
independent_functionElimination,
independent_pairEquality,
applyEquality,
minusEquality,
axiomEquality,
imageMemberEquality,
instantiate,
cumulativity,
dependent_set_memberEquality,
universeEquality,
equalityElimination,
hypothesis_subsumption
Latex:
\mforall{}[I:Interval]
\mforall{}[p:partition(I)]
\mforall{}i:\mBbbN{}||full-partition(I;p)|| - 1
r0\mleq{}full-partition(I;p)[i + 1] - full-partition(I;p)[i]\mleq{}partition-mesh(I;p)
supposing icompact(I)
Date html generated:
2017_10_03-AM-09_41_17
Last ObjectModification:
2017_07_28-AM-07_56_28
Theory : reals
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