Nuprl Lemma : average-q-between
∀[a,b:ℚ]. a < (a + b/2) < b supposing a < b
Proof
Definitions occuring in Statement :
q-between: a < b < c,
qless: 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 :
q-between: 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,
qinv: 1/r,
qmul: r * s,
guard: {T},
all: ∀x:A. B[x],
sq_type: SQType(T),
nequal: a ≠ b ∈ T ,
int_nzero: ℤ-o,
true: True,
subtype_rel: A ⊆r B,
qdiv: (r/s),
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),
rev_implies: P ⇐ Q,
iff: P ⇐⇒ Q
Lemmas referenced :
qless_witness,
qdiv_wf,
assert-qeq,
qless_wf,
rationals_wf,
istype-void,
qmul_wf,
qadd_wf,
qinv_wf,
int-subtype-rationals,
iff_weakening_uiff,
assert_wf,
qeq_wf2,
equal-wf-base,
subtype_base_sq,
int_subtype_base,
istype-int,
nequal_wf,
uiff_transitivity,
uiff_transitivity2,
qadd_preserves_qless,
squash_wf,
true_wf,
qmul_over_plus_qrng,
qmul_comm_qrng,
qinv_thru_op_q,
mon_assoc_q,
qadd_ac_1_q,
qadd_comm_q,
qinverse_q,
qadd_inv_assoc_q,
mon_ident_q,
qmul_assoc,
qmul_ident,
q_distrib,
qmul-zero-div,
qmul_preserves_qless,
qmul-ident-div,
false_wf,
equal_wf
Rules used in proof :
sqequalSubstitution,
sqequalRule,
sqequalReflexivity,
sqequalTransitivity,
computationStep,
isect_memberFormation_alt,
introduction,
cut,
independent_pairFormation,
hypothesis,
sqequalHypSubstitution,
productElimination,
thin,
independent_pairEquality,
extract_by_obid,
isectElimination,
hypothesisEquality,
because_Cache,
independent_isectElimination,
lambdaFormation_alt,
equalityTransitivity,
equalitySymmetry,
voidElimination,
equalityIsType4,
inhabitedIsType,
baseClosed,
independent_functionElimination,
universeIsType,
isect_memberEquality_alt,
dependent_functionElimination,
intEquality,
cumulativity,
instantiate,
addLevel,
dependent_set_memberEquality,
minusEquality,
promote_hyp,
applyEquality,
natural_numberEquality,
lemma_by_obid,
lambdaFormation,
lambdaEquality,
imageElimination,
imageMemberEquality,
dependent_set_memberEquality_alt,
lambdaEquality_alt
Latex:
\mforall{}[a,b:\mBbbQ{}]. a < (a + b/2) < b supposing a < b
Date html generated:
2019_10_16-PM-00_33_52
Last ObjectModification:
2018_10_10-AM-11_05_03
Theory : rationals
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