Nuprl Lemma : partition-choice-ap_wf
∀I:Interval
(icompact(I) ⇒ (∀p:partition(I). ∀x:partition-choice(full-partition(I;p)). ∀i:ℕ||p|| + 1. (x[i] ∈ {x:ℝ| x ∈ I} )))
Proof
Definitions occuring in Statement :
partition-choice-ap: x[i],
partition-choice: partition-choice(p),
full-partition: full-partition(I;p),
partition: partition(I),
icompact: icompact(I),
i-member: r ∈ I,
interval: Interval,
real: ℝ,
length: ||as||,
int_seg: {i..j-},
all: ∀x:A. B[x],
implies: P ⇒ Q,
member: t ∈ T,
set: {x:A| B[x]} ,
add: n + m,
natural_number: $n
Definitions unfolded in proof :
all: ∀x:A. B[x],
member: t ∈ T,
implies: P ⇒ Q,
uall: ∀[x:A]. B[x],
uimplies: b supposing a,
prop: ℙ,
full-partition: full-partition(I;p),
top: Top,
decidable: Dec(P),
or: P ∨ Q,
partition: partition(I),
satisfiable_int_formula: satisfiable_int_formula(fmla),
exists: ∃x:A. B[x],
false: False,
not: ¬A,
sq_type: SQType(T),
guard: {T},
partition-choice-ap: x[i]
Lemmas referenced :
partition-choice-member,
partition-choice_wf,
full-partition_wf,
partition_wf,
icompact_wf,
interval_wf,
length_of_cons_lemma,
subtype_base_sq,
int_subtype_base,
length-append,
length_of_nil_lemma,
decidable__equal_int,
length_wf,
real_wf,
satisfiable-full-omega-tt,
intformnot_wf,
intformeq_wf,
itermAdd_wf,
itermVar_wf,
itermConstant_wf,
itermSubtract_wf,
int_formula_prop_not_lemma,
int_formula_prop_eq_lemma,
int_term_value_add_lemma,
int_term_value_var_lemma,
int_term_value_constant_lemma,
int_term_value_subtract_lemma,
int_formula_prop_wf
Rules used in proof :
cut,
introduction,
extract_by_obid,
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation,
hypothesis,
sqequalHypSubstitution,
dependent_functionElimination,
thin,
hypothesisEquality,
independent_functionElimination,
sqequalRule,
isectElimination,
independent_isectElimination,
isect_memberEquality,
voidElimination,
voidEquality,
instantiate,
cumulativity,
intEquality,
because_Cache,
unionElimination,
setElimination,
rename,
natural_numberEquality,
dependent_pairFormation,
lambdaEquality,
int_eqEquality,
computeAll,
equalityTransitivity,
equalitySymmetry
Latex:
\mforall{}I:Interval
(icompact(I)
{}\mRightarrow{} (\mforall{}p:partition(I). \mforall{}x:partition-choice(full-partition(I;p)). \mforall{}i:\mBbbN{}||p|| + 1.
(x[i] \mmember{} \{x:\mBbbR{}| x \mmember{} I\} )))
Date html generated:
2016_10_26-AM-09_41_13
Last ObjectModification:
2016_08_15-PM-00_26_25
Theory : reals
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