Nuprl Lemma : qpositive-elim
∀[r:ℚ]. (qpositive(r) ~ if isint(r) then 0 <z r else let p,q = r in (0 <z p ∧b 0 <z q) ∨b(p <z 0 ∧b q <z 0) fi )
Proof
Definitions occuring in Statement :
qpositive: qpositive(r),
rationals: ℚ,
bor: p ∨bq,
band: p ∧b q,
ifthenelse: if b then t else f fi ,
lt_int: i <z j,
bfalse: ff,
btrue: tt,
uall: ∀[x:A]. B[x],
isint: isint def,
spread: spread def,
natural_number: $n,
sqequal: s ~ t
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
qpositive: qpositive(r),
uimplies: b supposing a,
callbyvalueall: callbyvalueall,
has-value: (a)↓,
has-valueall: has-valueall(a)
Lemmas referenced :
valueall-type-has-valueall,
rationals_wf,
rationals-valueall-type,
evalall-reduce
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
introduction,
cut,
sqequalRule,
lemma_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
hypothesis,
independent_isectElimination,
hypothesisEquality,
callbyvalueReduce,
sqequalAxiom
Latex:
\mforall{}[r:\mBbbQ{}]
(qpositive(r) \msim{} if isint(r)
then 0 <z r
else let p,q = r
in (0 <z p \mwedge{}\msubb{} 0 <z q) \mvee{}\msubb{}(p <z 0 \mwedge{}\msubb{} q <z 0)
fi )
Date html generated:
2016_05_15-PM-10_39_37
Last ObjectModification:
2015_12_27-PM-07_58_58
Theory : rationals
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