Nuprl Lemma : nearby-cases-ext
∀n:ℕ+. ∀x,y:ℝ. ((x < y) ∨ (y < x) ∨ (|x - y| ≤ (r1/r(n))))
Proof
Definitions occuring in Statement :
rdiv: (x/y),
rleq: x ≤ y,
rless: x < y,
rabs: |x|,
rsub: x - y,
int-to-real: r(n),
real: ℝ,
nat_plus: ℕ+,
all: ∀x:A. B[x],
or: P ∨ Q,
natural_number: $n
Definitions unfolded in proof :
member: t ∈ T,
nearby-cases,
decidable__lt,
decidable__squash,
uall: ∀[x:A]. B[x],
so_lambda: so_lambda(x,y,z,w.t[x; y; z; w]),
so_apply: x[s1;s2;s3;s4],
top: Top,
uimplies: b supposing a,
strict4: strict4(F),
and: P ∧ Q,
all: ∀x:A. B[x],
implies: P ⇒ Q,
has-value: (a)↓,
prop: ℙ,
guard: {T},
or: P ∨ Q,
squash: ↓T,
so_lambda: λ2x y.t[x; y],
so_apply: x[s1;s2],
so_lambda: λ2x.t[x],
so_apply: x[s],
decidable_functionality,
iff_preserves_decidability,
decidable__and,
decidable__less_than',
rleq_functionality_wrt_implies
Lemmas referenced :
nearby-cases,
lifting-strict-spread,
has-value_wf_base,
base_wf,
is-exception_wf,
top_wf,
equal_wf,
lifting-strict-decide,
lifting-strict-less,
decidable__lt,
decidable__squash,
decidable_functionality,
iff_preserves_decidability,
decidable__and,
decidable__less_than',
rleq_functionality_wrt_implies
Rules used in proof :
introduction,
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
cut,
instantiate,
extract_by_obid,
hypothesis,
sqequalRule,
thin,
sqequalHypSubstitution,
isectElimination,
baseClosed,
isect_memberEquality,
voidElimination,
voidEquality,
independent_isectElimination,
independent_pairFormation,
lambdaFormation,
callbyvalueApply,
baseApply,
closedConclusion,
hypothesisEquality,
applyExceptionCases,
inrFormation,
imageMemberEquality,
imageElimination,
exceptionSqequal,
inlFormation,
callbyvalueDecide,
equalityTransitivity,
equalitySymmetry,
unionEquality,
unionElimination,
sqleReflexivity,
dependent_functionElimination,
independent_functionElimination,
decideExceptionCases,
because_Cache
Latex:
\mforall{}n:\mBbbN{}\msupplus{}. \mforall{}x,y:\mBbbR{}. ((x < y) \mvee{} (y < x) \mvee{} (|x - y| \mleq{} (r1/r(n))))
Date html generated:
2017_10_03-AM-08_48_16
Last ObjectModification:
2017_07_28-AM-07_33_18
Theory : reals
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