Nuprl Lemma : truncate-rational_wf
∀q:ℚ. ∀e:{e:ℚ| 0 < e} . (truncate-rational(q;e) ∈ ∃q':ℚ [(|q - 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],
sq_exists: ∃x:A [B[x]],
member: t ∈ T,
set: {x:A| B[x]} ,
natural_number: $n
Definitions unfolded in proof :
all: ∀x:A. B[x],
member: t ∈ T,
subtype_rel: A ⊆r B,
uall: ∀[x:A]. B[x],
prop: ℙ,
so_lambda: λ2x.t[x],
so_apply: x[s],
sq_exists: ∃x:A [B[x]],
rational-truncate,
ifthenelse: if b then t else f fi ,
rational-truncate1,
truncate-rational: truncate-rational(q;e)
Lemmas referenced :
rational-truncate,
subtype_rel_self,
rationals_wf,
all_wf,
qless_wf,
sq_exists_wf,
qle_wf,
qabs_wf,
qsub_wf,
set_wf,
int-subtype-rationals,
rational-truncate1
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation,
cut,
applyEquality,
thin,
instantiate,
extract_by_obid,
hypothesis,
sqequalRule,
introduction,
sqequalHypSubstitution,
isectElimination,
functionEquality,
setEquality,
natural_numberEquality,
because_Cache,
hypothesisEquality,
lambdaEquality,
setElimination,
rename
Latex:
\mforall{}q:\mBbbQ{}. \mforall{}e:\{e:\mBbbQ{}| 0 < e\} . (truncate-rational(q;e) \mmember{} \mexists{}q':\mBbbQ{} [(|q - q'| \mleq{} e)])
Date html generated:
2018_05_22-AM-00_31_28
Last ObjectModification:
2018_05_19-PM-04_10_18
Theory : rationals
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