Nuprl Lemma : inhabited-covers-real-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 :
uall: ∀[x:A]. B[x],
implies: P ⇒ Q,
or: P ∨ Q,
all: ∀x:A. B[x],
exists: ∃x:A. B[x],
member: t ∈ T,
prop: ℙ,
so_lambda: λ2x.t[x],
so_apply: x[s],
and: P ∧ Q,
subtype_rel: A ⊆r B,
nat: ℕ,
ge: i ≥ j ,
decidable: Dec(P),
uimplies: b supposing a,
not: ¬A,
satisfiable_int_formula: satisfiable_int_formula(fmla),
false: False,
top: Top,
rneq: x ≠ y,
guard: {T},
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q,
less_than: a < b,
squash: ↓T,
less_than': less_than'(a;b),
true: True,
pi1: fst(t),
pi2: snd(t),
cand: A c∧ B
Lemmas referenced :
all_wf,
real_wf,
or_wf,
exists_wf,
cover-seq-property-ext,
cover-seq_wf,
nat_wf,
equal_wf,
nat_properties,
decidable__le,
full-omega-unsat,
intformand_wf,
intformnot_wf,
intformle_wf,
itermConstant_wf,
itermAdd_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_add_lemma,
int_term_value_var_lemma,
int_formula_prop_wf,
le_wf,
rdiv_wf,
radd_wf,
int-to-real_wf,
rless-int,
rless_wf,
common-limit-midpoints,
pi1_wf_top,
pi2_wf,
req_witness,
req_wf,
req_weakening,
squash_wf,
true_wf,
top_wf,
subtype_rel_product,
iff_weakening_equal,
converges-to_wf
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
lambdaFormation,
rename,
sqequalHypSubstitution,
sqequalRule,
productElimination,
thin,
cut,
introduction,
extract_by_obid,
isectElimination,
hypothesis,
lambdaEquality,
applyEquality,
functionExtensionality,
hypothesisEquality,
functionEquality,
cumulativity,
universeEquality,
dependent_functionElimination,
independent_functionElimination,
dependent_pairFormation,
productEquality,
because_Cache,
equalityTransitivity,
equalitySymmetry,
dependent_set_memberEquality,
addEquality,
setElimination,
natural_numberEquality,
unionElimination,
independent_isectElimination,
approximateComputation,
int_eqEquality,
intEquality,
isect_memberEquality,
voidElimination,
voidEquality,
independent_pairFormation,
independent_pairEquality,
inrFormation,
imageMemberEquality,
baseClosed,
inlFormation,
imageElimination
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_16
Last ObjectModification:
2017_07_06-AM-11_21_57
Theory : reals
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