Nuprl Lemma : qadd-elim
∀[r,s:ℚ].
(r + s ~ if isint(r)
then if isint(s) then r + s else let i,j = s in <(r * j) + i, j> fi
else let p,q = r
in if isint(s) then <p + (s * q), q> else let i,j = s in <(p * j) + (i * q), q * j> fi
fi )
Proof
Definitions occuring in Statement :
qadd: r + s,
rationals: ℚ,
ifthenelse: if b then t else f fi ,
bfalse: ff,
btrue: tt,
uall: ∀[x:A]. B[x],
isint: isint def,
spread: spread def,
pair: <a, b>,
multiply: n * m,
add: n + m,
sqequal: s ~ t
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
qadd: 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{}].
(r + s \msim{} if isint(r)
then if isint(s) then r + s else let i,j = s in <(r * j) + i, j> fi
else let p,q = r
in if isint(s) then <p + (s * q), q> else let i,j = s in <(p * j) + (i * q), q * j> fi
fi )
Date html generated:
2016_05_15-PM-10_39_42
Last ObjectModification:
2015_12_27-PM-07_58_53
Theory : rationals
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