Nuprl Lemma : truncate-rational-property
∀q:ℚ. ∀e:{e:ℚ| 0 < e} . ((|q - truncate-rational(q;e)| ≤ e) ∧ (|truncate-rational(q;e) - q| ≤ e))
Proof
Definitions occuring in Statement :
truncate-rational: truncate-rational(q;e),
qabs: |r|,
qle: r ≤ s,
qless: r < s,
qsub: r - s,
rationals: ℚ,
all: ∀x:A. B[x],
and: P ∧ Q,
set: {x:A| B[x]} ,
natural_number: $n
Definitions unfolded in proof :
all: ∀x:A. B[x],
and: P ∧ Q,
cand: A c∧ B,
member: t ∈ T,
squash: ↓T,
uall: ∀[x:A]. B[x],
prop: ℙ,
subtype_rel: A ⊆r B,
sq_exists: ∃x:{A| B[x]},
so_lambda: λ2x.t[x],
so_apply: x[s],
true: True,
uimplies: b supposing a,
guard: {T},
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q,
implies: P ⇒ Q,
sq_stable: SqStable(P)
Lemmas referenced :
decidable__qle,
sq_stable_from_decidable,
equal_wf,
int-subtype-rationals,
qless_wf,
set_wf,
qminus-qsub,
iff_weakening_equal,
qabs_wf,
sq_exists_wf,
truncate-rational_wf,
qsub_wf,
qabs-qminus,
rationals_wf,
true_wf,
squash_wf,
qle_wf
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation,
cut,
independent_pairFormation,
hypothesis,
applyEquality,
thin,
lambdaEquality,
sqequalHypSubstitution,
imageElimination,
lemma_by_obid,
isectElimination,
hypothesisEquality,
equalityTransitivity,
equalitySymmetry,
dependent_functionElimination,
setElimination,
rename,
sqequalRule,
natural_numberEquality,
imageMemberEquality,
baseClosed,
universeEquality,
independent_isectElimination,
productElimination,
independent_functionElimination,
because_Cache,
introduction
Latex:
\mforall{}q:\mBbbQ{}. \mforall{}e:\{e:\mBbbQ{}| 0 < e\} . ((|q - truncate-rational(q;e)| \mleq{} e) \mwedge{} (|truncate-rational(q;e) - q| \mleq{} e))
Date html generated:
2016_05_15-PM-11_43_10
Last ObjectModification:
2016_01_16-PM-09_11_21
Theory : rationals
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