Nuprl Lemma : qavg-eq-iff-1
∀[a,b:ℚ]. uiff(a = qavg(a;b) ∈ ℚ;a = b ∈ ℚ)
Proof
Definitions occuring in Statement :
qavg: qavg(a;b)
,
rationals: ℚ
,
uiff: uiff(P;Q)
,
uall: ∀[x:A]. B[x]
,
equal: s = t ∈ T
Definitions unfolded in proof :
rev_implies: P
⇐ Q
,
rev_uimplies: rev_uimplies(P;Q)
,
qadd: r + s
,
iff: P
⇐⇒ Q
,
guard: {T}
,
true: True
,
prop: ℙ
,
squash: ↓T
,
false: False
,
assert: ↑b
,
bfalse: ff
,
eq_int: (i =z j)
,
btrue: tt
,
ifthenelse: if b then t else f fi
,
evalall: evalall(t)
,
callbyvalueall: callbyvalueall,
qeq: qeq(r;s)
,
implies: P
⇒ Q
,
not: ¬A
,
subtype_rel: A ⊆r B
,
uimplies: b supposing a
,
and: P ∧ Q
,
uiff: uiff(P;Q)
,
member: t ∈ T
,
uall: ∀[x:A]. B[x]
,
qavg: qavg(a;b)
Lemmas referenced :
qmul_ident,
equal-wf-T-base,
not_wf,
qmul-preserves-eq,
qmul_one_qrng,
q_distrib,
qadd_inv_assoc_q,
qadd_ac_1_q,
iff_weakening_equal,
subtype_rel_self,
int-subtype-rationals,
qmul-qdiv-cancel,
istype-universe,
true_wf,
squash_wf,
equal_wf,
qmul_wf,
rationals_wf,
assert-qeq,
qadd_wf,
qdiv_wf
Rules used in proof :
minusEquality,
independent_functionElimination,
imageMemberEquality,
universeEquality,
instantiate,
imageElimination,
lambdaEquality_alt,
applyLambdaEquality,
universeIsType,
isectIsTypeImplies,
axiomEquality,
isect_memberEquality_alt,
independent_pairEquality,
sqequalBase,
baseClosed,
voidElimination,
productElimination,
equalitySymmetry,
equalityTransitivity,
lambdaFormation_alt,
independent_isectElimination,
because_Cache,
applyEquality,
natural_numberEquality,
thin,
isectElimination,
sqequalHypSubstitution,
extract_by_obid,
hypothesisEquality,
inhabitedIsType,
equalityIstype,
hypothesis,
independent_pairFormation,
cut,
introduction,
isect_memberFormation_alt,
computationStep,
sqequalTransitivity,
sqequalReflexivity,
sqequalRule,
sqequalSubstitution
Latex:
\mforall{}[a,b:\mBbbQ{}]. uiff(a = qavg(a;b);a = b)
Date html generated:
2019_10_29-AM-07_44_31
Last ObjectModification:
2019_10_21-PM-06_50_23
Theory : rationals
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