Nuprl Lemma : separated-partitions-have-common-refinement
∀I:Interval
∀P,Q:partition(I).
(separated-partitions(P;Q) ⇒ (∃R:partition(I). (frs-increasing(R) ∧ frs-refines(R;P) ∧ frs-refines(R;Q))))
supposing icompact(I)
Proof
Definitions occuring in Statement :
separated-partitions: separated-partitions(P;Q),
partition: partition(I),
frs-increasing: frs-increasing(p),
frs-refines: frs-refines(p;q),
icompact: icompact(I),
interval: Interval,
uimplies: b supposing a,
all: ∀x:A. B[x],
exists: ∃x:A. B[x],
implies: P ⇒ Q,
and: P ∧ Q
Definitions unfolded in proof :
bfalse: ff,
cons: [a / b],
btrue: tt,
ifthenelse: if b then t else f fi ,
assert: ↑b,
last: last(L),
top: Top,
satisfiable_int_formula: satisfiable_int_formula(fmla),
squash: ↓T,
less_than: a < b,
decidable: Dec(P),
rbetween: x≤y≤z,
guard: {T},
or: P ∨ Q,
icompact: icompact(I),
rev_uimplies: rev_uimplies(P;Q),
uiff: uiff(P;Q),
iff: P ⇐⇒ Q,
so_apply: x[s],
so_lambda: λ2x.t[x],
not: ¬A,
false: False,
less_than': less_than'(a;b),
le: A ≤ B,
lelt: i ≤ j < k,
int_seg: {i..j-},
l_all: (∀x∈L.P[x]),
frs-refines: frs-refines(p;q),
separated-partitions: separated-partitions(P;Q),
cand: A c∧ B,
prop: ℙ,
uall: ∀[x:A]. B[x],
partitions: partitions(I;p),
and: P ∧ Q,
exists: ∃x:A. B[x],
partition: partition(I),
member: t ∈ T,
implies: P ⇒ Q,
uimplies: b supposing a,
all: ∀x:A. B[x]
Lemmas referenced :
right-endpoint_wf,
length_of_cons_lemma,
null_cons_lemma,
product_subtype_list,
length_of_nil_lemma,
null_nil_lemma,
list-cases,
last_wf,
decidable__lt,
int_formula_prop_wf,
int_formula_prop_less_lemma,
int_term_value_var_lemma,
int_term_value_subtract_lemma,
int_term_value_constant_lemma,
int_formula_prop_le_lemma,
int_formula_prop_not_lemma,
int_formula_prop_and_lemma,
intformless_wf,
itermVar_wf,
itermSubtract_wf,
itermConstant_wf,
intformle_wf,
intformnot_wf,
intformand_wf,
full-omega-unsat,
subtract_wf,
decidable__le,
i-member-compact,
member_append,
i-member_wf,
l_all_iff,
partition-point-member,
req_weakening,
rleq_functionality,
left-endpoint_wf,
select_wf,
req_wf,
l_member_wf,
append_wf,
l_exists_iff,
lelt_wf,
false_wf,
real_wf,
length_wf,
less_than_wf,
interval_wf,
icompact_wf,
partition_wf,
separated-partitions_wf,
frs-refines_wf,
frs-increasing_wf,
partitions_wf,
frs-increasing-non-dec,
frs-increasing-separated-common-refinement
Rules used in proof :
hypothesis_subsumption,
promote_hyp,
voidEquality,
voidElimination,
isect_memberEquality,
intEquality,
int_eqEquality,
approximateComputation,
imageElimination,
unionElimination,
setEquality,
lambdaEquality,
sqequalRule,
natural_numberEquality,
productEquality,
independent_isectElimination,
because_Cache,
isectElimination,
independent_pairFormation,
dependent_set_memberEquality,
dependent_pairFormation,
productElimination,
independent_functionElimination,
hypothesis,
hypothesisEquality,
rename,
setElimination,
thin,
dependent_functionElimination,
sqequalHypSubstitution,
extract_by_obid,
introduction,
cut,
isect_memberFormation,
lambdaFormation,
sqequalReflexivity,
computationStep,
sqequalTransitivity,
sqequalSubstitution
Latex:
\mforall{}I:Interval
\mforall{}P,Q:partition(I).
(separated-partitions(P;Q)
{}\mRightarrow{} (\mexists{}R:partition(I). (frs-increasing(R) \mwedge{} frs-refines(R;P) \mwedge{} frs-refines(R;Q))))
supposing icompact(I)
Date html generated:
2018_05_22-PM-02_21_54
Last ObjectModification:
2018_05_21-AM-00_41_51
Theory : reals
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