Nuprl Lemma : qsum-reciprocal-squares-bound
∀[J:ℕ]. Σ1 ≤ n < J. (1/n * n) < 2
Proof
Definitions occuring in Statement :
qsum: Σa ≤ j < b. E[j],
qless: r < s,
qdiv: (r/s),
qmul: r * s,
nat: ℕ,
uall: ∀[x:A]. B[x],
natural_number: $n
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
all: ∀x:A. B[x],
decidable: Dec(P),
or: P ∨ Q,
uimplies: b supposing a,
sq_type: SQType(T),
implies: P ⇒ Q,
guard: {T},
so_lambda: λ2x.t[x],
subtype_rel: A ⊆r B,
int_seg: {i..j-},
nat: ℕ,
so_apply: x[s],
nequal: a ≠ b ∈ T ,
ge: i ≥ j ,
lelt: i ≤ j < k,
and: P ∧ Q,
not: ¬A,
satisfiable_int_formula: satisfiable_int_formula(fmla),
exists: ∃x:A. B[x],
false: False,
top: Top,
prop: ℙ,
uiff: uiff(P;Q),
qsum: Σa ≤ j < b. E[j],
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),
rng_sum: rng_sum,
mon_itop: Π lb ≤ i < ub. E[i],
itop: Π(op,id) lb ≤ i < ub. E[i],
ycomb: Y,
lt_int: i <z j,
bfalse: ff,
grp_id: e,
add_grp_of_rng: r↓+gp,
rng_zero: 0,
qrng: <ℚ+*>,
bor: p ∨bq,
qpositive: qpositive(r),
qsub: r - s,
qadd: r + s,
qmul: r * s,
btrue: tt,
bnot: ¬bb,
qeq: qeq(r;s),
eq_int: (i =z j),
true: True,
squash: ↓T,
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q,
nat_plus: ℕ+,
le: A ≤ B,
less_than': less_than'(a;b),
subtract: n - m,
rev_uimplies: rev_uimplies(P;Q)
Lemmas referenced :
decidable__equal_int,
subtype_base_sq,
int_subtype_base,
qless_witness,
qsum_wf,
qdiv_wf,
qmul_wf,
subtype_rel_set,
rationals_wf,
lelt_wf,
int-subtype-rationals,
qmul-mul,
int_entire_a,
int_seg_properties,
nat_properties,
satisfiable-full-omega-tt,
intformand_wf,
intformeq_wf,
itermVar_wf,
itermConstant_wf,
intformle_wf,
int_formula_prop_and_lemma,
int_formula_prop_eq_lemma,
int_term_value_var_lemma,
int_term_value_constant_lemma,
int_formula_prop_le_lemma,
int_formula_prop_wf,
equal-wf-base,
int-equal-in-rationals,
equal-wf-T-base,
not_wf,
nat_wf,
subtype_rel-equal,
int_seg_wf,
set_wf,
intformless_wf,
int_formula_prop_less_lemma,
qless_wf,
squash_wf,
true_wf,
sum_unroll_base_q,
iff_weakening_equal,
itermSubtract_wf,
intformnot_wf,
int_term_value_subtract_lemma,
int_formula_prop_not_lemma,
subtract_wf,
int-eq-in-rationals,
qsum-reciprocal-squares,
decidable__lt,
false_wf,
not-lt-2,
not-equal-2,
add_functionality_wrt_le,
add-associates,
add-zero,
zero-add,
le-add-cancel,
condition-implies-le,
add-commutes,
minus-add,
minus-zero,
le-add-cancel2,
add-swap,
minus-minus,
minus-one-mul,
minus-one-mul-top,
less_than_wf,
qle_wf,
subtract-add-cancel,
qsub_wf,
qadd_preserves_qless,
qadd_wf,
qmul_preserves_qless,
qless-int,
qadd_ac_1_q,
qadd_comm_q,
qadd_inv_assoc_q,
qinverse_q,
mon_ident_q,
qmul_zero_qrng,
qmul-qdiv-cancel,
qless_transitivity_1_qorder
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
introduction,
cut,
extract_by_obid,
sqequalHypSubstitution,
dependent_functionElimination,
thin,
because_Cache,
natural_numberEquality,
hypothesis,
unionElimination,
instantiate,
isectElimination,
cumulativity,
intEquality,
independent_isectElimination,
independent_functionElimination,
sqequalRule,
lambdaEquality,
applyEquality,
hypothesisEquality,
setElimination,
rename,
productElimination,
lambdaFormation,
dependent_pairFormation,
int_eqEquality,
isect_memberEquality,
voidElimination,
voidEquality,
independent_pairFormation,
computeAll,
equalityTransitivity,
equalitySymmetry,
baseClosed,
addLevel,
impliesFunctionality,
multiplyEquality,
imageElimination,
imageMemberEquality,
universeEquality,
dependent_set_memberEquality,
addEquality,
minusEquality,
hyp_replacement
Latex:
\mforall{}[J:\mBbbN{}]. \mSigma{}1 \mleq{} n < J. (1/n * n) < 2
Date html generated:
2016_10_26-AM-06_34_53
Last ObjectModification:
2016_07_12-AM-07_58_21
Theory : rationals
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