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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