Nuprl Lemma : frs-increasing-separated-common-refinement
∀p,q:ℝ List.
(frs-increasing(p)
⇒ frs-increasing(q)
⇒ frs-separated(p;q)
⇒ (∃r:ℝ List. (frs-increasing(r) ∧ frs-refines(r;p) ∧ frs-refines(r;q) ∧ frs-refines(p @ q;r))))
Proof
Definitions occuring in Statement :
frs-separated: frs-separated(p;q),
frs-increasing: frs-increasing(p),
frs-refines: frs-refines(p;q),
real: ℝ,
append: as @ bs,
list: T List,
all: ∀x:A. B[x],
exists: ∃x:A. B[x],
implies: P ⇒ Q,
and: P ∧ Q
Definitions unfolded in proof :
all: ∀x:A. B[x],
implies: P ⇒ Q,
uall: ∀[x:A]. B[x],
member: t ∈ T,
so_lambda: λ2x y.t[x; y],
trans: Trans(T;x,y.E[x; y]),
guard: {T},
uimplies: b supposing a,
prop: ℙ,
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q,
and: P ∧ Q,
frs-separated: frs-separated(p;q),
so_lambda: λ2x.t[x],
so_apply: x[s],
rneq: x ≠ y,
exists: ∃x:A. B[x],
cand: A c∧ B,
frs-refines: frs-refines(p;q),
l_all: (∀x∈L.P[x]),
l_contains: A ⊆ B,
l_member: (x ∈ l),
l_exists: (∃x∈L. P[x]),
nat: ℕ,
int_seg: {i..j-},
lelt: i ≤ j < k,
le: A ≤ B,
ge: i ≥ j ,
decidable: Dec(P),
or: P ∨ Q,
satisfiable_int_formula: satisfiable_int_formula(fmla),
false: False,
not: ¬A,
top: Top,
less_than: a < b,
squash: ↓T,
uiff: uiff(P;Q)
Lemmas referenced :
merge-strict-exists,
real_wf,
rless_wf,
rless_transitivity2,
rleq_weakening_rless,
l_member_wf,
frs-increasing-sorted-by,
frs-separated_wf,
frs-increasing_wf,
list_wf,
l_all_iff,
l_all_wf2,
rneq_wf,
all_wf,
lelt_wf,
length_wf,
req_weakening,
req_wf,
select_wf,
int_seg_properties,
nat_properties,
decidable__le,
satisfiable-full-omega-tt,
intformand_wf,
intformnot_wf,
intformle_wf,
itermConstant_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_var_lemma,
int_formula_prop_wf,
decidable__lt,
intformless_wf,
int_formula_prop_less_lemma,
int_seg_wf,
length-append,
append_wf,
add-is-int-iff,
itermAdd_wf,
int_term_value_add_lemma,
false_wf,
frs-refines_wf,
le_wf,
and_wf,
equal_wf,
nat_wf,
less_than_wf
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation,
cut,
introduction,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
hypothesis,
sqequalRule,
lambdaEquality,
hypothesisEquality,
independent_functionElimination,
dependent_functionElimination,
independent_isectElimination,
because_Cache,
productElimination,
independent_pairFormation,
addLevel,
setElimination,
rename,
setEquality,
allFunctionality,
levelHypothesis,
promote_hyp,
functionEquality,
dependent_pairFormation,
dependent_set_memberEquality,
equalityTransitivity,
equalitySymmetry,
natural_numberEquality,
unionElimination,
int_eqEquality,
intEquality,
isect_memberEquality,
voidElimination,
voidEquality,
computeAll,
imageElimination,
addEquality,
pointwiseFunctionality,
baseApply,
closedConclusion,
baseClosed,
productEquality,
hyp_replacement,
applyEquality
Latex:
\mforall{}p,q:\mBbbR{} List.
(frs-increasing(p)
{}\mRightarrow{} frs-increasing(q)
{}\mRightarrow{} frs-separated(p;q)
{}\mRightarrow{} (\mexists{}r:\mBbbR{} List. (frs-increasing(r) \mwedge{} frs-refines(r;p) \mwedge{} frs-refines(r;q) \mwedge{} frs-refines(p @ q;r))))
Date html generated:
2016_10_26-AM-09_33_11
Last ObjectModification:
2016_07_12-AM-08_20_23
Theory : reals
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