Nuprl Lemma : qsum-reciprocal-squares
∀[J:ℕ+]. (Σ1 ≤ n < J + 1. (1/n * n) ≤ (2 - (1/J)))
Proof
Definitions occuring in Statement :
qsum: Σa ≤ j < b. E[j],
qle: r ≤ s,
qsub: r - s,
qdiv: (r/s),
nat_plus: ℕ+,
uall: ∀[x:A]. B[x],
multiply: n * m,
add: n + m,
natural_number: $n
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
all: ∀x:A. B[x],
nat_plus: ℕ+,
implies: P ⇒ Q,
squash: ↓T,
uimplies: b supposing a,
decidable: Dec(P),
or: P ∨ Q,
satisfiable_int_formula: satisfiable_int_formula(fmla),
exists: ∃x:A. B[x],
false: False,
not: ¬A,
top: Top,
and: P ∧ Q,
prop: ℙ,
so_lambda: λ2x.t[x],
int_seg: {i..j-},
subtype_rel: A ⊆r B,
nequal: a ≠ b ∈ T ,
guard: {T},
lelt: i ≤ j < k,
uiff: uiff(P;Q),
so_apply: x[s],
true: True,
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q,
less_than: a < b,
less_than': less_than'(a;b),
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,
qsub: r - s,
qadd: r + s,
qmul: r * s,
qdiv: (r/s),
qinv: 1/r,
qle: r ≤ s,
grp_leq: a ≤ b,
infix_ap: x f y,
grp_le: ≤b,
pi1: fst(t),
pi2: snd(t),
qadd_grp: <ℚ+>,
q_le: q_le(r;s),
qsum: Σa ≤ j < b. E[j],
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,
subtract: n - m,
grp_id: e,
add_grp_of_rng: r↓+gp,
rng_zero: 0,
qrng: <ℚ+*>,
bor: p ∨bq,
qpositive: qpositive(r),
band: p ∧b q,
rev_uimplies: rev_uimplies(P;Q),
qge: a ≥ b
Lemmas referenced :
nat_plus_properties,
qle_wf,
sum_unroll_hi_q,
decidable__lt,
satisfiable-full-omega-tt,
intformand_wf,
intformnot_wf,
intformless_wf,
itermConstant_wf,
itermAdd_wf,
itermVar_wf,
int_formula_prop_and_lemma,
int_formula_prop_not_lemma,
int_formula_prop_less_lemma,
int_term_value_constant_lemma,
int_term_value_add_lemma,
int_term_value_var_lemma,
int_formula_prop_wf,
qdiv_wf,
int_entire_a,
int_seg_properties,
intformeq_wf,
intformle_wf,
int_formula_prop_eq_lemma,
int_formula_prop_le_lemma,
equal-wf-base,
int_subtype_base,
int-equal-in-rationals,
equal-wf-T-base,
rationals_wf,
int-subtype-rationals,
not_wf,
int_seg_wf,
qsub_wf,
iff_weakening_equal,
add-subtract-cancel,
qsum_wf,
nat_plus_wf,
primrec-wf-nat-plus,
subtype_rel_set,
less_than_wf,
qle_witness,
assert-qeq,
qadd_wf,
squash_wf,
true_wf,
qle_functionality_wrt_implies,
qadd_functionality_wrt_qle,
qle_weakening_eq_qorder,
qmul_preserves_qle,
qless-int,
qmul_wf,
qadd-add,
qmul-mul,
qle-int,
decidable__le,
itermMultiply_wf,
int_term_value_mul_lemma,
qmul_over_plus_qrng,
qmul_over_minus_qrng,
qmul_comm_qrng,
mon_assoc_q,
qadd_comm_q,
qmul-qdiv-cancel5,
qmul-qdiv-cancel,
qadd_ac_1_q,
qmul_ac_1_qrng,
qmul_one_qrng
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
introduction,
cut,
thin,
lambdaFormation,
rename,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
hypothesisEquality,
hypothesis,
setElimination,
applyEquality,
lambdaEquality,
imageElimination,
because_Cache,
natural_numberEquality,
addEquality,
independent_isectElimination,
dependent_functionElimination,
unionElimination,
dependent_pairFormation,
int_eqEquality,
intEquality,
isect_memberEquality,
voidElimination,
voidEquality,
sqequalRule,
independent_pairFormation,
computeAll,
multiplyEquality,
productElimination,
equalityTransitivity,
equalitySymmetry,
baseClosed,
addLevel,
impliesFunctionality,
baseApply,
closedConclusion,
imageMemberEquality,
independent_functionElimination,
minusEquality,
universeEquality
Latex:
\mforall{}[J:\mBbbN{}\msupplus{}]. (\mSigma{}1 \mleq{} n < J + 1. (1/n * n) \mleq{} (2 - (1/J)))
Date html generated:
2018_05_22-AM-00_02_53
Last ObjectModification:
2017_07_26-PM-06_51_08
Theory : rationals
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