Nuprl Lemma : compact-subinterval
∀I:Interval. ∀J:{J:Interval| icompact(J)} . (J ⊆ I ⇒ (∃n:{n:ℕ+| icompact(i-approx(I;n))} . J ⊆ i-approx(I;n) ))
Proof
Definitions occuring in Statement :
subinterval: I ⊆ J ,
icompact: icompact(I),
i-approx: i-approx(I;n),
interval: Interval,
nat_plus: ℕ+,
all: ∀x:A. B[x],
exists: ∃x:A. B[x],
implies: P ⇒ Q,
set: {x:A| B[x]}
Definitions unfolded in proof :
all: ∀x:A. B[x],
implies: P ⇒ Q,
member: t ∈ T,
sq_stable: SqStable(P),
squash: ↓T,
prop: ℙ,
uall: ∀[x:A]. B[x],
so_lambda: λ2x.t[x],
so_apply: x[s],
and: P ∧ Q,
uimplies: b supposing a,
iff: P ⇐⇒ Q,
exists: ∃x:A. B[x],
guard: {T},
icompact: icompact(I),
subinterval: I ⊆ J ,
nat_plus: ℕ+,
rev_implies: P ⇐ Q,
decidable: Dec(P),
or: P ∨ Q,
satisfiable_int_formula: satisfiable_int_formula(fmla),
false: False,
not: ¬A,
top: Top,
cand: A c∧ B,
i-nonvoid: i-nonvoid(I),
rbetween: x≤y≤z
Lemmas referenced :
sq_stable__icompact,
i-member-compact,
icompact-endpoints,
i-member-iff,
subinterval_wf,
set_wf,
interval_wf,
icompact_wf,
left-endpoint_wf,
right-endpoint_wf,
i-approx_wf,
less_than_wf,
imax_wf,
imax_nat_plus,
nat_plus_wf,
nat_plus_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,
i-approx-closed,
i-approx-finite,
i-member_wf,
i-approx-monotonic,
imax_ub,
decidable__le,
intformle_wf,
int_formula_prop_le_lemma,
le_wf,
real_wf,
i-member-between
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation,
cut,
setElimination,
thin,
rename,
introduction,
extract_by_obid,
sqequalHypSubstitution,
dependent_functionElimination,
hypothesisEquality,
hypothesis,
independent_functionElimination,
sqequalRule,
imageMemberEquality,
baseClosed,
imageElimination,
because_Cache,
isectElimination,
lambdaEquality,
productElimination,
independent_isectElimination,
dependent_pairFormation,
dependent_set_memberEquality,
natural_numberEquality,
equalityTransitivity,
equalitySymmetry,
applyLambdaEquality,
unionElimination,
int_eqEquality,
intEquality,
isect_memberEquality,
voidElimination,
voidEquality,
independent_pairFormation,
computeAll,
inrFormation,
inlFormation
Latex:
\mforall{}I:Interval. \mforall{}J:\{J:Interval| icompact(J)\} .
(J \msubseteq{} I {}\mRightarrow{} (\mexists{}n:\{n:\mBbbN{}\msupplus{}| icompact(i-approx(I;n))\} . J \msubseteq{} i-approx(I;n) ))
Date html generated:
2017_10_03-AM-09_34_55
Last ObjectModification:
2017_07_28-AM-07_52_37
Theory : reals
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