Nuprl Lemma : icompact-is-subinterval
∀I:Interval. (icompact(I) ⇒ (∃n:ℕ+. I ⊆ i-approx((-∞, ∞);n) ))
Proof
Definitions occuring in Statement :
subinterval: I ⊆ J ,
icompact: icompact(I),
i-approx: i-approx(I;n),
riiint: (-∞, ∞),
interval: Interval,
nat_plus: ℕ+,
all: ∀x:A. B[x],
exists: ∃x:A. B[x],
implies: P ⇒ Q
Definitions unfolded in proof :
all: ∀x:A. B[x],
implies: P ⇒ Q,
i-approx: i-approx(I;n),
riiint: (-∞, ∞),
member: t ∈ T,
uall: ∀[x:A]. B[x],
uimplies: b supposing a,
exists: ∃x:A. B[x],
and: P ∧ Q,
nat_plus: ℕ+,
nat: ℕ,
less_than: a < b,
squash: ↓T,
less_than': less_than'(a;b),
true: True,
prop: ℙ,
guard: {T},
ge: i ≥ j ,
decidable: Dec(P),
or: P ∨ Q,
satisfiable_int_formula: satisfiable_int_formula(fmla),
false: False,
not: ¬A,
top: Top,
icompact: icompact(I),
cand: A c∧ B,
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q,
subinterval: I ⊆ J ,
le: A ≤ B
Lemmas referenced :
r-archimedean,
right-endpoint_wf,
left-endpoint_wf,
imax_wf,
imax_nat_plus,
add_nat_plus,
less_than_wf,
nat_plus_wf,
nat_plus_properties,
nat_properties,
decidable__lt,
satisfiable-full-omega-tt,
intformand_wf,
intformnot_wf,
intformless_wf,
itermConstant_wf,
itermVar_wf,
intformeq_wf,
int_formula_prop_and_lemma,
int_formula_prop_not_lemma,
int_formula_prop_less_lemma,
int_term_value_constant_lemma,
int_term_value_var_lemma,
int_formula_prop_eq_lemma,
int_formula_prop_wf,
equal_wf,
subinterval_wf,
rccint_wf,
int-to-real_wf,
icompact_wf,
interval_wf,
icompact-is-rccint,
imax_ub,
decidable__le,
intformle_wf,
itermAdd_wf,
int_formula_prop_le_lemma,
int_term_value_add_lemma,
le_wf,
member_rccint_lemma,
rleq_wf,
real_wf,
rleq-int,
itermMinus_wf,
int_term_value_minus_lemma,
rleq_transitivity
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation,
sqequalRule,
cut,
introduction,
extract_by_obid,
sqequalHypSubstitution,
dependent_functionElimination,
thin,
isectElimination,
hypothesisEquality,
independent_isectElimination,
hypothesis,
because_Cache,
productElimination,
dependent_pairFormation,
dependent_set_memberEquality,
addEquality,
setElimination,
rename,
natural_numberEquality,
independent_pairFormation,
imageMemberEquality,
baseClosed,
equalityTransitivity,
equalitySymmetry,
applyLambdaEquality,
unionElimination,
lambdaEquality,
int_eqEquality,
intEquality,
isect_memberEquality,
voidElimination,
voidEquality,
computeAll,
independent_functionElimination,
minusEquality,
inlFormation,
inrFormation,
productEquality
Latex:
\mforall{}I:Interval. (icompact(I) {}\mRightarrow{} (\mexists{}n:\mBbbN{}\msupplus{}. I \msubseteq{} i-approx((-\minfty{}, \minfty{});n) ))
Date html generated:
2017_10_03-AM-09_35_24
Last ObjectModification:
2017_07_28-AM-07_52_58
Theory : reals
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