Nuprl Lemma : rat2real-qavg
∀[a,b:ℚ]. (rat2real(qavg(a;b)) = (rat2real(a) + rat2real(b)/r(2)))
Proof
Definitions occuring in Statement :
rat2real: rat2real(q),
rdiv: (x/y),
req: x = y,
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),
false: False,
sq_type: SQType(T),
not: ¬A,
nequal: a ≠ b ∈ T ,
int_nzero: ℤ-o,
subtype_rel: A ⊆r B,
prop: ℙ,
true: True,
less_than': less_than'(a;b),
squash: ↓T,
less_than: a < b,
implies: P ⇒ Q,
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,
qavg: qavg(a;b),
member: t ∈ T,
uall: ∀[x:A]. B[x]
Lemmas referenced :
rat2real-qadd,
rdiv_functionality,
rat2real-qdiv,
req_transitivity,
req_functionality,
req_weakening,
nequal_wf,
istype-int,
int_subtype_base,
subtype_base_sq,
int_nzero-rational,
qadd_wf,
qdiv_wf,
rationals_wf,
rless_wf,
rless-int,
int-to-real_wf,
radd_wf,
rdiv_wf,
qavg_wf,
rat2real_wf,
req_witness
Rules used in proof :
sqequalBase,
equalityIstype,
voidElimination,
equalitySymmetry,
equalityTransitivity,
intEquality,
cumulativity,
instantiate,
lambdaFormation_alt,
dependent_set_memberEquality_alt,
applyEquality,
isectIsTypeImplies,
isect_memberEquality_alt,
inhabitedIsType,
universeIsType,
baseClosed,
imageMemberEquality,
independent_pairFormation,
independent_functionElimination,
productElimination,
because_Cache,
dependent_functionElimination,
inrFormation_alt,
sqequalRule,
independent_isectElimination,
natural_numberEquality,
closedConclusion,
hypothesis,
hypothesisEquality,
thin,
isectElimination,
sqequalHypSubstitution,
extract_by_obid,
cut,
introduction,
isect_memberFormation_alt,
sqequalReflexivity,
computationStep,
sqequalTransitivity,
sqequalSubstitution
Latex:
\mforall{}[a,b:\mBbbQ{}]. (rat2real(qavg(a;b)) = (rat2real(a) + rat2real(b)/r(2)))
Date html generated:
2019_10_31-AM-05_57_28
Last ObjectModification:
2019_10_30-PM-02_56_59
Theory : reals
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