Nuprl Lemma : average-qbetween
∀[a,b:ℚ]. a ≤ (a + b/2) ≤ b supposing a ≤ b
Proof
Definitions occuring in Statement :
qbetween: a ≤ b ≤ c,
qle: r ≤ s,
qdiv: (r/s),
qadd: r + s,
rationals: ℚ,
uimplies: b supposing a,
uall: ∀[x:A]. B[x],
natural_number: $n
Definitions unfolded in proof :
qbetween: a ≤ b ≤ c,
uall: ∀[x:A]. B[x],
member: t ∈ T,
uimplies: b supposing a,
and: P ∧ Q,
cand: A c∧ B,
not: ¬A,
implies: P ⇒ Q,
uiff: uiff(P;Q),
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,
subtype_rel: A ⊆r B,
qdiv: (r/s),
true: True,
int_nzero: ℤ-o,
nequal: a ≠ b ∈ T ,
sq_type: SQType(T),
all: ∀x:A. B[x],
guard: {T},
qmul: r * s,
qinv: 1/r,
squash: ↓T,
qadd: r + s,
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,
lt_int: i <z j,
bnot: ¬bb,
rev_uimplies: rev_uimplies(P;Q)
Lemmas referenced :
qmul-ident-div,
qmul_preserves_qle,
qmul-zero-div,
qmul_ident,
q_distrib,
qmul_assoc,
mon_ident_q,
qadd_inv_assoc_q,
qinverse_q,
qadd_comm_q,
qadd_ac_1_q,
mon_assoc_q,
qinv_thru_op_q,
qmul_comm_qrng,
qmul_over_plus_qrng,
squash_wf,
qadd_preserves_qle,
uiff_transitivity2,
uiff_transitivity,
equal-wf-base,
nequal_wf,
true_wf,
int_subtype_base,
subtype_base_sq,
int-subtype-rationals,
qinv_wf,
qadd_wf,
qmul_wf,
false_wf,
qle_wf,
rationals_wf,
equal_wf,
assert-qeq,
qdiv_wf,
qle_witness
Rules used in proof :
sqequalSubstitution,
sqequalRule,
sqequalReflexivity,
sqequalTransitivity,
computationStep,
isect_memberFormation,
introduction,
cut,
independent_pairFormation,
hypothesis,
sqequalHypSubstitution,
productElimination,
thin,
independent_pairEquality,
lemma_by_obid,
isectElimination,
hypothesisEquality,
because_Cache,
independent_isectElimination,
lambdaFormation,
equalityTransitivity,
equalitySymmetry,
voidElimination,
natural_numberEquality,
applyEquality,
independent_functionElimination,
isect_memberEquality,
promote_hyp,
minusEquality,
dependent_set_memberEquality,
addLevel,
instantiate,
cumulativity,
intEquality,
dependent_functionElimination,
baseClosed,
lambdaEquality,
imageElimination,
imageMemberEquality
Latex:
\mforall{}[a,b:\mBbbQ{}]. a \mleq{} (a + b/2) \mleq{} b supposing a \mleq{} b
Date html generated:
2016_05_15-PM-11_19_33
Last ObjectModification:
2016_01_16-PM-09_18_13
Theory : rationals
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