Nuprl Lemma : qavg-between
∀[a,b:ℚ]. a < qavg(a;b) < b supposing a < b
Proof
Definitions occuring in Statement :
qavg: qavg(a;b),
q-between: a < b < c,
qless: r < s,
rationals: ℚ,
uimplies: b supposing a,
uall: ∀[x:A]. B[x]
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
uimplies: b supposing a,
qavg: qavg(a;b),
q-between: a < b < c,
and: P ∧ Q,
implies: P ⇒ Q,
prop: ℙ,
subtype_rel: A ⊆r B,
qless: r < s,
grp_lt: a < b,
set_lt: a <p b,
assert: ↑b,
ifthenelse: if b then t else f fi ,
set_blt: a <b b,
band: p ∧b q,
infix_ap: x f y,
set_le: ≤b,
pi2: snd(t),
oset_of_ocmon: g↓oset,
dset_of_mon: g↓set,
grp_le: ≤b,
pi1: fst(t),
qadd_grp: <ℚ+>,
q_le: q_le(r;s),
callbyvalueall: callbyvalueall,
evalall: evalall(t),
bor: p ∨bq,
qpositive: qpositive(r),
qsub: r - s,
qadd: r + s,
qmul: r * s,
btrue: tt,
lt_int: i <z j,
bnot: ¬bb,
bfalse: ff,
qeq: qeq(r;s),
eq_int: (i =z j),
true: True,
uiff: uiff(P;Q),
cand: A c∧ B,
rev_uimplies: rev_uimplies(P;Q),
squash: ↓T,
not: ¬A,
false: False,
guard: {T},
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q
Lemmas referenced :
qadd_comm_q,
q_distrib,
qmul_ident,
qadd_preserves_qless,
iff_weakening_equal,
equal_wf,
assert-qeq,
qmul-qdiv-cancel,
true_wf,
squash_wf,
qadd_wf,
qdiv_wf,
qmul_wf,
int-subtype-rationals,
qmul_preserves_qless,
rationals_wf,
qless_wf,
qavg_wf,
qless_witness
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
introduction,
cut,
sqequalRule,
sqequalHypSubstitution,
productElimination,
thin,
independent_pairEquality,
lemma_by_obid,
isectElimination,
hypothesisEquality,
hypothesis,
independent_functionElimination,
because_Cache,
isect_memberEquality,
equalityTransitivity,
equalitySymmetry,
natural_numberEquality,
applyEquality,
independent_isectElimination,
independent_pairFormation,
lambdaEquality,
imageElimination,
lambdaFormation,
voidElimination,
imageMemberEquality,
baseClosed,
universeEquality
Latex:
\mforall{}[a,b:\mBbbQ{}]. a < qavg(a;b) < b supposing a < b
Date html generated:
2016_05_15-PM-11_06_06
Last ObjectModification:
2016_01_16-PM-09_28_21
Theory : rationals
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