Nuprl Lemma : qless-qavg-iff-1
∀[a,b:ℚ]. uiff(a < qavg(a;b);a < b)
Proof
Definitions occuring in Statement :
qavg: qavg(a;b),
qless: r < s,
rationals: ℚ,
uiff: uiff(P;Q),
uall: ∀[x:A]. B[x]
Definitions unfolded in proof :
qavg: qavg(a;b),
uall: ∀[x:A]. B[x],
member: t ∈ T,
uiff: uiff(P;Q),
and: P ∧ Q,
uimplies: b supposing a,
implies: P ⇒ Q,
subtype_rel: A ⊆r B,
not: ¬A,
qeq: qeq(r;s),
callbyvalueall: callbyvalueall,
evalall: evalall(t),
ifthenelse: if b then t else f fi ,
btrue: tt,
eq_int: (i =z j),
bfalse: ff,
assert: ↑b,
false: False,
prop: ℙ,
qless: r < s,
grp_lt: a < b,
set_lt: a <p b,
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),
bor: p ∨bq,
qpositive: qpositive(r),
qsub: r - s,
qadd: r + s,
qmul: r * s,
lt_int: i <z j,
bnot: ¬bb,
true: True,
squash: ↓T,
guard: {T},
iff: P ⇐⇒ Q,
rev_uimplies: rev_uimplies(P;Q),
rev_implies: P ⇐ Q
Lemmas referenced :
qless_witness,
qless_wf,
qdiv_wf,
qadd_wf,
assert-qeq,
rationals_wf,
qmul_preserves_qless,
qmul_wf,
squash_wf,
true_wf,
int-subtype-rationals,
qmul-qdiv-cancel,
subtype_rel_self,
iff_weakening_equal,
uiff_transitivity2,
qadd_preserves_qless,
qadd_ac_1_q,
qadd_inv_assoc_q,
q_distrib,
qmul_one_qrng
Rules used in proof :
sqequalSubstitution,
sqequalRule,
sqequalReflexivity,
sqequalTransitivity,
computationStep,
isect_memberFormation_alt,
introduction,
cut,
independent_pairFormation,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
hypothesisEquality,
independent_functionElimination,
hypothesis,
universeIsType,
natural_numberEquality,
applyEquality,
because_Cache,
independent_isectElimination,
lambdaFormation_alt,
equalityTransitivity,
equalitySymmetry,
productElimination,
voidElimination,
equalityIstype,
inhabitedIsType,
baseClosed,
sqequalBase,
independent_pairEquality,
isect_memberEquality_alt,
isectIsTypeImplies,
lambdaEquality_alt,
imageElimination,
imageMemberEquality,
instantiate,
universeEquality,
minusEquality
Latex:
\mforall{}[a,b:\mBbbQ{}]. uiff(a < qavg(a;b);a < b)
Date html generated:
2020_05_20-AM-09_17_10
Last ObjectModification:
2019_11_01-PM-01_55_31
Theory : rationals
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