Nuprl Lemma : rat2real-qavg-2
∀[a,b:ℚ]. ((r(2) * rat2real(qavg(a;b))) = (rat2real(a) + rat2real(b)))
Proof
Definitions occuring in Statement :
rat2real: rat2real(q),
req: x = y,
rmul: a * b,
radd: a + b,
int-to-real: r(n),
uall: ∀[x:A]. B[x],
natural_number: $n,
qavg: qavg(a;b),
rationals: ℚ
Definitions unfolded in proof :
rev_uimplies: rev_uimplies(P;Q),
uiff: uiff(P;Q),
pi2: snd(t),
rtermDivide: num "/" denom,
rtermConstant: "const",
rtermMultiply: left "*" right,
pi1: fst(t),
rtermVar: rtermVar(var),
rat_term_ind: rat_term_ind,
rtermAdd: left "+" right,
rat_term_to_real: rat_term_to_real(f;t),
not: ¬A,
false: False,
prop: ℙ,
true: True,
less_than': less_than'(a;b),
squash: ↓T,
less_than: a < b,
rev_implies: P ⇐ Q,
and: P ∧ Q,
iff: P ⇐⇒ Q,
all: ∀x:A. B[x],
or: P ∨ Q,
guard: {T},
rneq: x ≠ y,
uimplies: b supposing a,
implies: P ⇒ Q,
member: t ∈ T,
uall: ∀[x:A]. B[x]
Lemmas referenced :
rat2real-qavg,
req_weakening,
rmul_functionality,
req_functionality,
istype-int,
rtermVar_wf,
rtermAdd_wf,
rtermDivide_wf,
rtermConstant_wf,
rtermMultiply_wf,
assert-rat-term-eq2,
rless_wf,
rless-int,
rdiv_wf,
rationals_wf,
radd_wf,
qavg_wf,
rat2real_wf,
int-to-real_wf,
rmul_wf,
req_witness
Rules used in proof :
approximateComputation,
int_eqEquality,
lambdaEquality_alt,
baseClosed,
imageMemberEquality,
independent_pairFormation,
productElimination,
dependent_functionElimination,
inrFormation_alt,
independent_isectElimination,
closedConclusion,
universeIsType,
isectIsTypeImplies,
because_Cache,
isect_memberEquality_alt,
sqequalRule,
inhabitedIsType,
independent_functionElimination,
hypothesisEquality,
hypothesis,
natural_numberEquality,
thin,
isectElimination,
sqequalHypSubstitution,
extract_by_obid,
cut,
introduction,
isect_memberFormation_alt,
sqequalReflexivity,
computationStep,
sqequalTransitivity,
sqequalSubstitution
Latex:
\mforall{}[a,b:\mBbbQ{}]. ((r(2) * rat2real(qavg(a;b))) = (rat2real(a) + rat2real(b)))
Date html generated:
2019_10_31-AM-05_57_40
Last ObjectModification:
2019_10_30-PM-02_58_18
Theory : reals
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