Nuprl Lemma : partition-endpoints
∀[I:Interval]
∀[p:partition(I)]. ∀[x:ℝ]. full-partition(I;p)[0]≤x≤full-partition(I;p)[||full-partition(I;p)|| - 1] supposing x ∈ I
supposing icompact(I)
Proof
Definitions occuring in Statement :
full-partition: full-partition(I;p),
partition: partition(I),
icompact: icompact(I),
i-member: r ∈ I,
interval: Interval,
rbetween: x≤y≤z,
real: ℝ,
select: L[n],
length: ||as||,
uimplies: b supposing a,
uall: ∀[x:A]. B[x],
subtract: n - m,
natural_number: $n
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
uimplies: b supposing a,
full-partition: full-partition(I;p),
select: L[n],
cons: [a / b],
all: ∀x:A. B[x],
member: t ∈ T,
top: Top,
squash: ↓T,
prop: ℙ,
partition: partition(I),
ge: i ≥ j ,
decidable: Dec(P),
or: P ∨ Q,
le: A ≤ B,
and: P ∧ Q,
satisfiable_int_formula: satisfiable_int_formula(fmla),
exists: ∃x:A. B[x],
false: False,
implies: P ⇒ Q,
not: ¬A,
subtype_rel: A ⊆r B,
true: True,
guard: {T},
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q,
int_seg: {i..j-},
lelt: i ≤ j < k,
rbetween: x≤y≤z,
cand: A c∧ B,
icompact: icompact(I),
subtract: n - m,
sq_stable: SqStable(P),
rleq: x ≤ y,
rnonneg: rnonneg(x)
Lemmas referenced :
sq_stable__rleq,
sq_stable__and,
zero-mul,
zero-add,
add-mul-special,
add-swap,
minus-one-mul,
add-associates,
i-member-compact,
nat_plus_wf,
rsub_wf,
less_than'_wf,
rleq_wf,
interval_wf,
icompact_wf,
partition_wf,
i-member_wf,
lelt_wf,
select_append_back,
decidable__le,
length-singleton,
iff_weakening_equal,
top_wf,
subtype_rel_list,
length_append,
le_wf,
int_formula_prop_wf,
int_formula_prop_le_lemma,
int_term_value_var_lemma,
int_term_value_add_lemma,
int_term_value_subtract_lemma,
int_term_value_constant_lemma,
int_formula_prop_less_lemma,
int_formula_prop_not_lemma,
int_formula_prop_and_lemma,
intformle_wf,
itermVar_wf,
itermAdd_wf,
itermSubtract_wf,
itermConstant_wf,
intformless_wf,
intformnot_wf,
intformand_wf,
satisfiable-full-omega-tt,
decidable__lt,
non_neg_length,
length_wf,
subtract_wf,
nil_wf,
right-endpoint_wf,
cons_wf,
append_wf,
select_cons_tl,
left-endpoint_wf,
real_wf,
true_wf,
squash_wf,
rbetween_wf,
length_of_nil_lemma,
length-append,
length_of_cons_lemma
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
sqequalRule,
cut,
lemma_by_obid,
sqequalHypSubstitution,
dependent_functionElimination,
thin,
isect_memberEquality,
voidElimination,
voidEquality,
hypothesis,
isectElimination,
applyEquality,
lambdaEquality,
imageElimination,
hypothesisEquality,
equalityTransitivity,
equalitySymmetry,
independent_isectElimination,
because_Cache,
setElimination,
rename,
addEquality,
natural_numberEquality,
unionElimination,
productElimination,
dependent_pairFormation,
int_eqEquality,
intEquality,
independent_pairFormation,
computeAll,
imageMemberEquality,
baseClosed,
independent_functionElimination,
universeEquality,
dependent_set_memberEquality,
multiplyEquality,
minusEquality,
lambdaFormation,
introduction,
independent_pairEquality,
axiomEquality
Latex:
\mforall{}[I:Interval]
\mforall{}[p:partition(I)]. \mforall{}[x:\mBbbR{}].
full-partition(I;p)[0]\mleq{}x\mleq{}full-partition(I;p)[||full-partition(I;p)|| - 1] supposing x \mmember{} I
supposing icompact(I)
Date html generated:
2016_05_18-AM-08_57_33
Last ObjectModification:
2016_01_17-AM-02_31_16
Theory : reals
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