Nuprl Lemma : dense-in-interval-implies
∀I:Interval
∀[X:{a:ℝ| a ∈ I} ⟶ ℙ]
(dense-in-interval(I;X)
⇒ (∃u,v:{a:ℝ| a ∈ I} . u ≠ v)
⇒ (∀a:{a:ℝ| a ∈ I} . ∃x:ℕ ⟶ {a:ℝ| a ∈ I} . ((∀n:ℕ. (X (x n))) ∧ lim n→∞.x n = a)))
Proof
Definitions occuring in Statement :
dense-in-interval: dense-in-interval(I;X),
i-member: r ∈ I,
interval: Interval,
converges-to: lim n→∞.x[n] = y,
rneq: x ≠ y,
real: ℝ,
nat: ℕ,
uall: ∀[x:A]. B[x],
prop: ℙ,
all: ∀x:A. B[x],
exists: ∃x:A. B[x],
implies: P ⇒ Q,
and: P ∧ Q,
set: {x:A| B[x]} ,
apply: f a,
function: x:A ⟶ B[x]
Definitions unfolded in proof :
so_apply: x[s],
so_lambda: λ2x.t[x],
subtype_rel: A ⊆r B,
cand: A c∧ B,
uimplies: b supposing a,
guard: {T},
squash: ↓T,
sq_stable: SqStable(P),
and: P ∧ Q,
dense-in-interval: dense-in-interval(I;X),
rneq: x ≠ y,
prop: ℙ,
or: P ∨ Q,
member: t ∈ T,
exists: ∃x:A. B[x],
implies: P ⇒ Q,
uall: ∀[x:A]. B[x],
all: ∀x:A. B[x],
true: True,
less_than': less_than'(a;b),
less_than: a < b,
rev_implies: P ⇐ Q,
iff: P ⇐⇒ Q,
rev_uimplies: rev_uimplies(P;Q),
uiff: uiff(P;Q),
top: Top,
not: ¬A,
false: False,
req_int_terms: t1 ≡ t2,
rge: x ≥ y,
pi1: fst(t),
nat: ℕ,
nequal: a ≠ b ∈ T ,
satisfiable_int_formula: satisfiable_int_formula(fmla),
decidable: Dec(P),
assert: ↑b,
bnot: ¬bb,
sq_type: SQType(T),
bfalse: ff,
it: ⋅,
unit: Unit,
bool: 𝔹,
btrue: tt,
ifthenelse: if b then t else f fi ,
subtract: n - m,
eq_int: (i =z j),
primrec: primrec(n;b;c),
exp: i^n,
nat_plus: ℕ+,
le: A ≤ B,
rnonneg: rnonneg(x),
rleq: x ≤ y,
ge: i ≥ j ,
rdiv: (x/y),
int_nzero: ℤ-o
Lemmas referenced :
interval_wf,
dense-in-interval_wf,
exists_wf,
set_wf,
real_wf,
rless_wf,
i-member_wf,
rleq_weakening_rless,
sq_stable__i-member,
i-member-between,
rneq_wf,
rneq-cases,
rless-int,
int-to-real_wf,
rdiv_wf,
rmul_wf,
rsub_wf,
rabs_wf,
rleq_wf,
ravg_wf,
ravg-between,
sq_stable_rneq,
ravg_comm,
rsub_functionality,
rabs_functionality,
req_inversion,
req_weakening,
rleq_functionality,
ravg-dist,
real_term_value_var_lemma,
real_term_value_const_lemma,
real_term_value_sub_lemma,
real_polynomial_null,
rabs-of-nonneg,
rabs-difference-symmetry,
req-iff-rsub-is-0,
itermVar_wf,
itermConstant_wf,
itermSubtract_wf,
rleq_weakening,
rsub_functionality_wrt_rleq,
rleq_weakening_equal,
rleq_functionality_wrt_implies,
real_term_value_add_lemma,
itermAdd_wf,
radd-zero,
radd_wf,
radd-preserves-rleq,
subtype_rel_sets,
all_wf,
equal_wf,
converges-to_wf,
nat_wf,
int_seg_wf,
primrec_wf,
primrec-wf2,
less_than_wf,
le_wf,
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,
intformle_wf,
intformnot_wf,
intformand_wf,
full-omega-unsat,
decidable__le,
subtract_wf,
neg_assert_of_eq_int,
assert-bnot,
bool_subtype_base,
subtype_base_sq,
bool_cases_sqequal,
eqff_to_assert,
assert_of_eq_int,
eqtt_to_assert,
bool_wf,
eq_int_wf,
primrec-unroll,
exp-positive,
exp0_lemma,
nat_plus_wf,
nat_plus_properties,
exp_wf2,
less_than'_wf,
ge_wf,
nat_properties,
real_term_value_mul_lemma,
real_term_value_minus_lemma,
rmul-identity1,
rinv1,
rmul_functionality,
req_transitivity,
itermMultiply_wf,
rinv_wf2,
itermMinus_wf,
rminus_wf,
int_formula_prop_eq_lemma,
intformeq_wf,
nequal_wf,
true_wf,
exp_wf3,
int_nzero-rational,
int-subtype-rationals,
equal_functionality_wrt_subtype_rel2,
int_subtype_base,
rationals_wf,
equal-wf-base,
not_functionality_wrt_implies,
rneq-int,
rinv-mul-as-rdiv,
rmul-rinv3,
rmul_preserves_rleq,
exp_wf_nat_plus,
req-int-fractions,
one-mul,
exp_step,
exp-positive-stronger,
rmul_assoc,
simple-converges-to
Rules used in proof :
cumulativity,
functionEquality,
inrFormation,
universeEquality,
lambdaEquality,
setEquality,
functionExtensionality,
applyEquality,
productEquality,
inlFormation,
independent_pairFormation,
dependent_set_memberEquality,
independent_isectElimination,
imageElimination,
baseClosed,
imageMemberEquality,
sqequalRule,
isectElimination,
dependent_pairFormation,
unionElimination,
hypothesisEquality,
independent_functionElimination,
hypothesis,
because_Cache,
rename,
setElimination,
dependent_functionElimination,
extract_by_obid,
introduction,
thin,
productElimination,
sqequalHypSubstitution,
cut,
isect_memberFormation,
lambdaFormation,
sqequalReflexivity,
computationStep,
sqequalTransitivity,
sqequalSubstitution,
natural_numberEquality,
voidEquality,
voidElimination,
isect_memberEquality,
intEquality,
int_eqEquality,
approximateComputation,
equalitySymmetry,
equalityTransitivity,
promote_hyp,
instantiate,
equalityElimination,
axiomEquality,
minusEquality,
independent_pairEquality,
intWeakElimination,
addLevel,
closedConclusion,
baseApply
Latex:
\mforall{}I:Interval
\mforall{}[X:\{a:\mBbbR{}| a \mmember{} I\} {}\mrightarrow{} \mBbbP{}]
(dense-in-interval(I;X)
{}\mRightarrow{} (\mexists{}u,v:\{a:\mBbbR{}| a \mmember{} I\} . u \mneq{} v)
{}\mRightarrow{} (\mforall{}a:\{a:\mBbbR{}| a \mmember{} I\} . \mexists{}x:\mBbbN{} {}\mrightarrow{} \{a:\mBbbR{}| a \mmember{} I\} . ((\mforall{}n:\mBbbN{}. (X (x n))) \mwedge{} lim n\mrightarrow{}\minfty{}.x n = a)))
Date html generated:
2017_10_03-AM-10_19_34
Last ObjectModification:
2017_07_31-AM-11_30_20
Theory : reals
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