Nuprl Lemma : partition-choice-indep-funtype
∀[I:Interval]
∀[p:partition(I)]. (partition-choice(full-partition(I;p)) ⊆r (ℕ||p|| + 1 ⟶ {x:ℝ| x ∈ I} )) supposing icompact(I)
Proof
Definitions occuring in Statement :
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-},
uimplies: b supposing a,
subtype_rel: A ⊆r B,
uall: ∀[x:A]. B[x],
set: {x:A| B[x]} ,
function: x:A ⟶ B[x],
add: n + m,
natural_number: $n
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
uimplies: b supposing a,
full-partition: full-partition(I;p),
all: ∀x:A. B[x],
top: Top,
partition: partition(I),
subtype_rel: A ⊆r B,
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,
prop: ℙ,
icompact: icompact(I),
partition-choice: partition-choice(p),
int_seg: {i..j-},
lelt: i ≤ j < k,
uiff: uiff(P;Q),
less_than: a < b,
l_all: (∀x∈L.P[x]),
subtract: n - m,
i-member: r ∈ I,
rccint: [l, u],
guard: {T}
Lemmas referenced :
length_of_cons_lemma,
length_nil,
non_neg_length,
nil_wf,
length_cons,
real_wf,
right-endpoint_wf,
cons_wf,
append_wf,
length_append,
subtype_rel_list,
top_wf,
length-append,
length_of_nil_lemma,
decidable__equal_int,
length_wf,
satisfiable-full-omega-tt,
intformnot_wf,
intformeq_wf,
itermAdd_wf,
itermVar_wf,
itermConstant_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_formula_prop_wf,
partition-choice_wf,
full-partition_wf,
partition_wf,
icompact_wf,
interval_wf,
decidable__lt,
subtract_wf,
add-is-int-iff,
intformand_wf,
intformless_wf,
itermSubtract_wf,
int_formula_prop_and_lemma,
int_formula_prop_less_lemma,
int_term_value_subtract_lemma,
false_wf,
lelt_wf,
full-partition-point-member,
add-member-int_seg2,
decidable__le,
intformle_wf,
int_formula_prop_le_lemma,
select_wf,
i-member-between,
i-member_wf,
rccint_wf,
equal_wf,
int_seg_properties,
int_seg_wf
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
introduction,
cut,
sqequalRule,
extract_by_obid,
sqequalHypSubstitution,
dependent_functionElimination,
thin,
isect_memberEquality,
voidElimination,
voidEquality,
hypothesis,
isectElimination,
hypothesisEquality,
independent_isectElimination,
because_Cache,
setElimination,
rename,
applyEquality,
lambdaEquality,
addEquality,
natural_numberEquality,
unionElimination,
productElimination,
dependent_pairFormation,
int_eqEquality,
intEquality,
equalityTransitivity,
equalitySymmetry,
computeAll,
axiomEquality,
functionExtensionality,
dependent_set_memberEquality,
independent_pairFormation,
pointwiseFunctionality,
promote_hyp,
baseApply,
closedConclusion,
baseClosed,
independent_functionElimination,
lambdaFormation,
setEquality
Latex:
\mforall{}[I:Interval]
\mforall{}[p:partition(I)]. (partition-choice(full-partition(I;p)) \msubseteq{}r (\mBbbN{}||p|| + 1 {}\mrightarrow{} \{x:\mBbbR{}| x \mmember{} I\} ))
supposing icompact(I)
Date html generated:
2017_10_03-AM-09_44_54
Last ObjectModification:
2017_07_28-AM-07_58_43
Theory : reals
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