Nuprl Lemma : qavg-qle-iff-2
∀[a,b:ℚ]. uiff(qavg(b;a) ≤ b;a ≤ b)
Proof
Definitions occuring in Statement :
qavg: qavg(a;b),
qle: r ≤ s,
rationals: ℚ,
uiff: uiff(P;Q),
uall: ∀[x:A]. B[x]
Definitions unfolded in proof :
rev_implies: P ⇐ Q,
rev_uimplies: rev_uimplies(P;Q),
bnot: ¬bb,
dset_of_mon: g↓set,
oset_of_ocmon: g↓oset,
set_le: ≤b,
band: p ∧b q,
set_blt: a <b b,
set_lt: a <p b,
grp_lt: a < b,
qless: r < s,
iff: P ⇐⇒ Q,
squash: ↓T,
guard: {T},
true: True,
lt_int: i <z j,
qmul: r * s,
qadd: r + s,
qsub: r - s,
qpositive: qpositive(r),
bor: p ∨bq,
q_le: q_le(r;s),
qadd_grp: <ℚ+>,
pi2: snd(t),
pi1: fst(t),
grp_le: ≤b,
infix_ap: x f y,
grp_leq: a ≤ b,
qle: r ≤ s,
prop: ℙ,
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),
not: ¬A,
subtype_rel: A ⊆r B,
implies: P ⇒ Q,
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_preserves_qle,
iff_weakening_equal,
qmul-qdiv-cancel,
subtype_rel_self,
equal-wf-T-base,
not_wf,
true_wf,
squash_wf,
qmul_one_qrng,
q_distrib,
mon_ident_q,
qinverse_q,
qadd_comm_q,
qadd_ac_1_q,
qadd_preserves_qle,
uiff_transitivity2,
int-subtype-rationals,
qmul_wf,
qmul_preserves_qle2,
rationals_wf,
assert-qeq,
qadd_wf,
qdiv_wf,
qle_wf,
qle_witness
Rules used in proof :
universeEquality,
instantiate,
imageMemberEquality,
imageElimination,
lambdaEquality_alt,
minusEquality,
isectIsTypeImplies,
isect_memberEquality_alt,
independent_pairEquality,
sqequalBase,
baseClosed,
inhabitedIsType,
equalityIstype,
voidElimination,
productElimination,
equalitySymmetry,
equalityTransitivity,
lambdaFormation_alt,
independent_isectElimination,
because_Cache,
applyEquality,
natural_numberEquality,
universeIsType,
hypothesis,
independent_functionElimination,
hypothesisEquality,
thin,
isectElimination,
sqequalHypSubstitution,
extract_by_obid,
independent_pairFormation,
cut,
introduction,
isect_memberFormation_alt,
computationStep,
sqequalTransitivity,
sqequalReflexivity,
sqequalRule,
sqequalSubstitution
Latex:
\mforall{}[a,b:\mBbbQ{}]. uiff(qavg(b;a) \mleq{} b;a \mleq{} b)
Date html generated:
2019_10_29-AM-07_46_03
Last ObjectModification:
2019_10_21-PM-08_23_31
Theory : rationals
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