Nuprl Lemma : r-bound-property
∀[x:ℝ]. ((r(-r-bound(x)) ≤ x) ∧ (x ≤ r(r-bound(x))))
Proof
Definitions occuring in Statement :
r-bound: r-bound(x),
rleq: x ≤ y,
int-to-real: r(n),
real: ℝ,
uall: ∀[x:A]. B[x],
and: P ∧ Q,
minus: -n
Definitions unfolded in proof :
guard: {T},
rge: x ≥ y,
rev_uimplies: rev_uimplies(P;Q),
top: Top,
req_int_terms: t1 ≡ t2,
uiff: uiff(P;Q),
less_than': less_than'(a;b),
rev_implies: P ⇐ Q,
iff: P ⇐⇒ Q,
uimplies: b supposing a,
real: ℝ,
false: False,
not: ¬A,
le: A ≤ B,
rnonneg: rnonneg(x),
rleq: x ≤ y,
pi1: fst(t),
implies: P ⇒ Q,
prop: ℙ,
exists: ∃x:A. B[x],
so_apply: x[s],
nat_plus: ℕ+,
so_lambda: λ2x.t[x],
all: ∀x:A. B[x],
subtype_rel: A ⊆r B,
r-bound: r-bound(x),
cand: A c∧ B,
and: P ∧ Q,
member: t ∈ T,
uall: ∀[x:A]. B[x]
Lemmas referenced :
rleq_transitivity,
rleq_weakening_equal,
rleq_functionality_wrt_implies,
rabs-bounds,
real_term_value_minus_lemma,
real_term_value_const_lemma,
real_term_value_var_lemma,
real_term_value_mul_lemma,
real_term_value_sub_lemma,
real_polynomial_null,
rminus-int,
req_weakening,
req_transitivity,
rleq_functionality,
req-iff-rsub-is-0,
itermMinus_wf,
itermConstant_wf,
itermVar_wf,
itermMultiply_wf,
itermSubtract_wf,
rminus_wf,
rmul_wf,
false_wf,
rleq-int,
rmul_reverses_rleq,
r-bound_wf,
rsub_wf,
less_than'_wf,
equal_wf,
int-to-real_wf,
rabs_wf,
rleq_wf,
nat_plus_wf,
exists_wf,
real_wf,
subtype_rel_self,
integer-bound
Rules used in proof :
voidEquality,
isect_memberEquality,
intEquality,
int_eqEquality,
approximateComputation,
independent_isectElimination,
axiomEquality,
natural_numberEquality,
minusEquality,
voidElimination,
independent_pairEquality,
independent_pairFormation,
independent_functionElimination,
dependent_functionElimination,
equalitySymmetry,
equalityTransitivity,
productElimination,
lambdaFormation,
because_Cache,
rename,
setElimination,
hypothesisEquality,
lambdaEquality,
functionEquality,
isectElimination,
sqequalHypSubstitution,
sqequalRule,
hypothesis,
extract_by_obid,
instantiate,
thin,
applyEquality,
cut,
introduction,
isect_memberFormation,
sqequalReflexivity,
computationStep,
sqequalTransitivity,
sqequalSubstitution
Latex:
\mforall{}[x:\mBbbR{}]. ((r(-r-bound(x)) \mleq{} x) \mwedge{} (x \mleq{} r(r-bound(x))))
Date html generated:
2018_05_22-PM-01_50_40
Last ObjectModification:
2018_05_21-AM-00_09_25
Theory : reals
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