Nuprl Lemma : r-strict-bound-property
∀x:ℝ. ((r(-(r-bound(x) + 1)) < x) ∧ (x < r(r-bound(x) + 1)))
Proof
Definitions occuring in Statement :
r-bound: r-bound(x),
rless: x < y,
int-to-real: r(n),
real: ℝ,
all: ∀x:A. B[x],
and: P ∧ Q,
add: n + m,
minus: -n,
natural_number: $n
Definitions unfolded in proof :
all: ∀x:A. B[x],
uall: ∀[x:A]. B[x],
member: t ∈ T,
and: P ∧ Q,
cand: A c∧ B,
top: Top,
subtype_rel: A ⊆r B,
nat_plus: ℕ+,
rev_implies: P ⇐ Q,
uimplies: b supposing a,
rge: x ≥ y,
guard: {T},
implies: P ⇒ Q,
prop: ℙ,
iff: P ⇐⇒ Q,
less_than: a < b,
squash: ↓T,
less_than': less_than'(a;b),
true: True,
uiff: uiff(P;Q)
Lemmas referenced :
r-bound-property,
real_wf,
minus-add,
int-to-real_wf,
r-bound_wf,
nat_plus_wf,
radd_wf,
rless_functionality_wrt_implies,
rleq_weakening,
req_inversion,
radd-int,
equal_wf,
radd-preserves-rless,
rminus_wf,
rless-int,
rmul_wf,
rless_wf,
rless_functionality,
radd-zero-both,
req_weakening,
radd_comm,
radd_functionality,
rmul-zero-both,
rmul_functionality,
req_transitivity,
rminus-as-rmul,
radd-assoc,
rmul-identity1,
rmul-distrib2,
trivial-rless-radd
Rules used in proof :
cut,
introduction,
extract_by_obid,
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation,
hypothesis,
sqequalHypSubstitution,
isectElimination,
thin,
hypothesisEquality,
productElimination,
independent_pairFormation,
isect_memberEquality,
voidElimination,
voidEquality,
addEquality,
minusEquality,
applyEquality,
lambdaEquality,
setElimination,
rename,
sqequalRule,
natural_numberEquality,
because_Cache,
dependent_functionElimination,
independent_isectElimination,
independent_functionElimination,
equalityTransitivity,
equalitySymmetry,
imageMemberEquality,
baseClosed,
addLevel,
levelHypothesis
Latex:
\mforall{}x:\mBbbR{}. ((r(-(r-bound(x) + 1)) < x) \mwedge{} (x < r(r-bound(x) + 1)))
Date html generated:
2017_10_03-AM-08_53_12
Last ObjectModification:
2017_07_28-AM-07_35_49
Theory : reals
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