Nuprl Lemma : iproper-approx
∀I:Interval
  (iproper(I) 
⇒ (∀n:ℕ+. (icompact(i-approx(I;n)) 
⇒ (icompact(i-approx(I;n + 1)) ∧ iproper(i-approx(I;n + 1))))))
Proof
Definitions occuring in Statement : 
icompact: icompact(I)
, 
i-approx: i-approx(I;n)
, 
iproper: iproper(I)
, 
interval: Interval
, 
nat_plus: ℕ+
, 
all: ∀x:A. B[x]
, 
implies: P 
⇒ Q
, 
and: P ∧ Q
, 
add: n + m
, 
natural_number: $n
Definitions unfolded in proof : 
all: ∀x:A. B[x]
, 
implies: P 
⇒ Q
, 
and: P ∧ Q
, 
cand: A c∧ B
, 
member: t ∈ T
, 
prop: ℙ
, 
uall: ∀[x:A]. B[x]
, 
true: True
, 
less_than': less_than'(a;b)
, 
le: A ≤ B
, 
top: Top
, 
subtype_rel: A ⊆r B
, 
subtract: n - m
, 
uimplies: b supposing a
, 
uiff: uiff(P;Q)
, 
false: False
, 
rev_implies: P 
⇐ Q
, 
not: ¬A
, 
iff: P 
⇐⇒ Q
, 
or: P ∨ Q
, 
decidable: Dec(P)
, 
nat_plus: ℕ+
, 
exists: ∃x:A. B[x]
, 
i-nonvoid: i-nonvoid(I)
, 
icompact: icompact(I)
, 
guard: {T}
, 
satisfiable_int_formula: satisfiable_int_formula(fmla)
, 
rless: x < y
, 
sq_exists: ∃x:{A| B[x]}
, 
iproper: iproper(I)
, 
interval: Interval
, 
i-approx: i-approx(I;n)
, 
i-finite: i-finite(I)
, 
rccint: [l, u]
, 
isl: isl(x)
, 
assert: ↑b
, 
ifthenelse: if b then t else f fi 
, 
btrue: tt
, 
right-endpoint: right-endpoint(I)
, 
left-endpoint: left-endpoint(I)
, 
endpoints: endpoints(I)
, 
outl: outl(x)
, 
pi1: fst(t)
, 
pi2: snd(t)
, 
real: ℝ
, 
bfalse: ff
, 
rge: x ≥ y
, 
rneq: x ≠ y
, 
rsub: x - y
, 
squash: ↓T
, 
less_than: a < b
, 
itermConstant: "const"
, 
req_int_terms: t1 ≡ t2
, 
rdiv: (x/y)
Lemmas referenced : 
icompact_wf, 
i-approx_wf, 
nat_plus_wf, 
iproper_wf, 
less_than_wf, 
le-add-cancel, 
add-zero, 
add-associates, 
add_functionality_wrt_le, 
add-commutes, 
minus-one-mul-top, 
zero-add, 
minus-one-mul, 
minus-add, 
condition-implies-le, 
less-iff-le, 
not-lt-2, 
false_wf, 
decidable__lt, 
i-approx-compact, 
int_formula_prop_wf, 
int_term_value_constant_lemma, 
int_term_value_add_lemma, 
int_term_value_var_lemma, 
int_formula_prop_le_lemma, 
int_formula_prop_not_lemma, 
itermConstant_wf, 
itermAdd_wf, 
itermVar_wf, 
intformle_wf, 
intformnot_wf, 
satisfiable-full-omega-tt, 
decidable__le, 
nat_plus_properties, 
i-approx-monotonic, 
icompact-endpoints-rleq, 
rless-int-fractions, 
intformless_wf, 
itermMultiply_wf, 
int_formula_prop_less_lemma, 
int_term_value_mul_lemma, 
rless-int, 
left_endpoint_rccint_lemma, 
right_endpoint_rccint_lemma, 
intformand_wf, 
itermMinus_wf, 
int_formula_prop_and_lemma, 
int_term_value_minus_lemma, 
i-finite_wf, 
rleq_weakening_equal, 
rless_functionality_wrt_implies, 
rless_wf, 
int-to-real_wf, 
rdiv_wf, 
radd_wf, 
radd_comm, 
rless_functionality, 
equal_wf, 
radd-preserves-rless, 
real_wf, 
rsub_wf, 
radd-assoc, 
req_inversion, 
radd-ac, 
radd-rminus-assoc, 
req_weakening, 
radd-rminus-both, 
req_transitivity, 
radd_functionality, 
radd-zero-both, 
rminus_wf, 
rleq_wf, 
rsub_functionality_wrt_rleq, 
radd-int, 
rmul_functionality, 
rmul-distrib2, 
rmul-identity1, 
rleq_functionality, 
uiff_transitivity, 
rmul_wf, 
radd-preserves-rleq, 
rmul_comm, 
rmul_preserves_rless, 
regular-int-seq_wf, 
rleq_weakening_rless, 
rless_transitivity2, 
rless_transitivity1, 
rinv_wf2, 
rinv-as-rdiv, 
real_term_polynomial, 
itermSubtract_wf, 
real_term_value_const_lemma, 
real_term_value_sub_lemma, 
real_term_value_add_lemma, 
real_term_value_minus_lemma, 
real_term_value_var_lemma, 
real_term_value_mul_lemma, 
req-iff-rsub-is-0
Rules used in proof : 
sqequalSubstitution, 
sqequalTransitivity, 
computationStep, 
sqequalReflexivity, 
lambdaFormation, 
cut, 
independent_pairFormation, 
hypothesis, 
introduction, 
extract_by_obid, 
sqequalHypSubstitution, 
isectElimination, 
thin, 
hypothesisEquality, 
because_Cache, 
minusEquality, 
intEquality, 
voidEquality, 
isect_memberEquality, 
lambdaEquality, 
applyEquality, 
sqequalRule, 
independent_isectElimination, 
independent_functionElimination, 
voidElimination, 
unionElimination, 
natural_numberEquality, 
rename, 
setElimination, 
addEquality, 
dependent_set_memberEquality, 
dependent_functionElimination, 
productElimination, 
computeAll, 
int_eqEquality, 
dependent_pairFormation, 
multiplyEquality, 
inrFormation, 
equalitySymmetry, 
equalityTransitivity, 
levelHypothesis, 
addLevel, 
baseClosed, 
imageMemberEquality, 
functionExtensionality
Latex:
\mforall{}I:Interval
    (iproper(I)
    {}\mRightarrow{}  (\mforall{}n:\mBbbN{}\msupplus{}
                (icompact(i-approx(I;n))  {}\mRightarrow{}  (icompact(i-approx(I;n  +  1))  \mwedge{}  iproper(i-approx(I;n  +  1))))))
Date html generated:
2017_10_03-AM-09_34_12
Last ObjectModification:
2017_07_28-AM-07_52_06
Theory : reals
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