Nuprl Lemma : inhabited-covers-reals-implies
∀[A,B:ℝ ⟶ ℙ].
((∃a:ℝ. A[a])
⇒ (∃b:ℝ. B[b])
⇒ (∀r:ℝ. (A[r] ∨ B[r]))
⇒ (∃f,g:ℕ ⟶ ℝ. ∃x:ℝ. ((∀n:ℕ. A[f n]) ∧ (∀n:ℕ. B[g n]) ∧ lim n→∞.f n = x ∧ lim n→∞.g n = x)))
Proof
Definitions occuring in Statement :
converges-to: lim n→∞.x[n] = y,
real: ℝ,
nat: ℕ,
uall: ∀[x:A]. B[x],
prop: ℙ,
so_apply: x[s],
all: ∀x:A. B[x],
exists: ∃x:A. B[x],
implies: P ⇒ Q,
or: P ∨ Q,
and: P ∧ Q,
apply: f a,
function: x:A ⟶ B[x]
Definitions unfolded in proof :
exists: ∃x:A. B[x],
member: t ∈ T,
uall: ∀[x:A]. B[x],
uimplies: b supposing a,
rneq: x ≠ y,
guard: {T},
or: P ∨ Q,
all: ∀x:A. B[x],
iff: P ⇐⇒ Q,
and: P ∧ Q,
rev_implies: P ⇐ Q,
implies: P ⇒ Q,
less_than: a < b,
squash: ↓T,
less_than': less_than'(a;b),
true: True,
prop: ℙ,
cand: A c∧ B,
nat_plus: ℕ+,
uiff: uiff(P;Q),
le: A ≤ B,
false: False,
not: ¬A,
subtype_rel: A ⊆r B,
so_lambda: λ2x.t[x],
so_apply: x[s],
rev_uimplies: rev_uimplies(P;Q),
req_int_terms: t1 ≡ t2,
top: Top,
member-closure: y ∈ closure(A)
Lemmas referenced :
rdiv_wf,
int-to-real_wf,
rless-int,
rless_wf,
rleq-int-fractions2,
less_than_wf,
false_wf,
rless-int-fractions3,
real_wf,
rleq_wf,
all_wf,
or_wf,
uall_wf,
exists_wf,
rsub_wf,
rmul_wf,
ravg-weak-between,
ravg-dist,
ravg_wf,
rleq_weakening_equal,
rabs_wf,
rabs-of-nonneg,
rleq-implies-rleq,
itermSubtract_wf,
itermVar_wf,
itermConstant_wf,
req-iff-rsub-is-0,
req_functionality,
rabs-difference-symmetry,
req_weakening,
real_polynomial_null,
real_term_value_sub_lemma,
real_term_value_var_lemma,
real_term_value_const_lemma,
rleq_functionality,
req_inversion,
rleq_weakening,
itermMultiply_wf,
rmul_functionality,
real_term_value_mul_lemma,
rless-cases,
rleq_weakening_rless,
closures-meet,
nat_wf,
converges-to_wf
Rules used in proof :
cut,
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
dependent_pairFormation,
introduction,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
natural_numberEquality,
hypothesis,
independent_isectElimination,
sqequalRule,
inrFormation,
dependent_functionElimination,
because_Cache,
productElimination,
independent_functionElimination,
independent_pairFormation,
imageMemberEquality,
hypothesisEquality,
baseClosed,
dependent_set_memberEquality,
lambdaFormation,
multiplyEquality,
isect_memberFormation,
productEquality,
applyEquality,
functionExtensionality,
lambdaEquality,
universeEquality,
functionEquality,
cumulativity,
instantiate,
unionElimination,
approximateComputation,
int_eqEquality,
intEquality,
isect_memberEquality,
voidElimination,
voidEquality,
inlFormation,
promote_hyp,
rename
Latex:
\mforall{}[A,B:\mBbbR{} {}\mrightarrow{} \mBbbP{}].
((\mexists{}a:\mBbbR{}. A[a])
{}\mRightarrow{} (\mexists{}b:\mBbbR{}. B[b])
{}\mRightarrow{} (\mforall{}r:\mBbbR{}. (A[r] \mvee{} B[r]))
{}\mRightarrow{} (\mexists{}f,g:\mBbbN{} {}\mrightarrow{} \mBbbR{}. \mexists{}x:\mBbbR{}. ((\mforall{}n:\mBbbN{}. A[f n]) \mwedge{} (\mforall{}n:\mBbbN{}. B[g n]) \mwedge{} lim n\mrightarrow{}\minfty{}.f n = x \mwedge{} lim n\mrightarrow{}\minfty{}.g n = x)))
Date html generated:
2017_10_03-AM-10_04_00
Last ObjectModification:
2017_09_28-PM-06_22_59
Theory : reals
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