Nuprl Lemma : qeq-elim
∀[r,s:ℚ].
(qeq(r;s) ~ if isint(r)
then if isint(s) then (r =z s) else let i,j = s in (r * j =z i) fi
else let p,q = r
in if isint(s) then (p =z s * q) else let i,j = s in (p * j =z i * q) fi
fi )
Proof
Definitions occuring in Statement :
rationals: ℚ,
qeq: qeq(r;s),
ifthenelse: if b then t else f fi ,
eq_int: (i =z j),
bfalse: ff,
btrue: tt,
uall: ∀[x:A]. B[x],
isint: isint def,
spread: spread def,
multiply: n * m,
sqequal: s ~ t
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
qeq: qeq(r;s),
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,
because_Cache,
sqequalAxiom,
isect_memberEquality
Latex:
\mforall{}[r,s:\mBbbQ{}].
(qeq(r;s) \msim{} if isint(r)
then if isint(s) then (r =\msubz{} s) else let i,j = s in (r * j =\msubz{} i) fi
else let p,q = r
in if isint(s) then (p =\msubz{} s * q) else let i,j = s in (p * j =\msubz{} i * q) fi
fi )
Date html generated:
2016_05_15-PM-10_39_48
Last ObjectModification:
2015_12_27-PM-07_58_48
Theory : rationals
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