Nuprl Lemma : q-not-limit-zero-diverges-2
∀f:ℕ+ ⟶ ℚ
(∃q:ℚ. (0 < q ∧ (∀n:ℕ. ∃m:ℕ. (n < m ∧ (q ≤ f[m]))))) ⇒ (∀B:ℚ. ∃n:ℕ. (B ≤ Σ1 ≤ i < n. f[i]))
supposing ∀n:ℕ+. (0 ≤ f[n])
Proof
Definitions occuring in Statement :
qsum: Σa ≤ j < b. E[j],
qle: r ≤ s,
qless: r < s,
rationals: ℚ,
nat_plus: ℕ+,
nat: ℕ,
less_than: a < b,
uimplies: b supposing a,
so_apply: x[s],
all: ∀x:A. B[x],
exists: ∃x:A. B[x],
implies: P ⇒ Q,
and: P ∧ Q,
function: x:A ⟶ B[x],
natural_number: $n
Definitions unfolded in proof :
all: ∀x:A. B[x],
uimplies: b supposing a,
member: t ∈ T,
uall: ∀[x:A]. B[x],
subtype_rel: A ⊆r B,
so_apply: x[s],
implies: P ⇒ Q,
so_lambda: λ2x.t[x],
nat: ℕ,
bool: 𝔹,
unit: Unit,
it: ⋅,
btrue: tt,
ifthenelse: if b then t else f fi ,
uiff: uiff(P;Q),
and: P ∧ Q,
bfalse: ff,
exists: ∃x:A. B[x],
prop: ℙ,
or: P ∨ Q,
sq_type: SQType(T),
guard: {T},
bnot: ¬bb,
assert: ↑b,
false: False,
le: A ≤ B,
less_than': less_than'(a;b),
not: ¬A,
ge: i ≥ j ,
int_upper: {i...},
nat_plus: ℕ+,
decidable: Dec(P),
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q,
top: Top,
true: True,
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),
callbyvalueall: callbyvalueall,
evalall: evalall(t),
bor: p ∨bq,
qpositive: qpositive(r),
qsub: r - s,
qadd: r + s,
qmul: r * s,
lt_int: i <z j,
qeq: qeq(r;s),
eq_int: (i =z j),
nequal: a ≠ b ∈ T ,
subtract: n - m,
satisfiable_int_formula: satisfiable_int_formula(fmla),
squash: ↓T,
int_seg: {i..j-},
lelt: i ≤ j < k,
sq_stable: SqStable(P),
qsum: Σa ≤ j < b. E[j],
rng_sum: rng_sum,
mon_itop: Π lb ≤ i < ub. E[i],
add_grp_of_rng: r↓+gp,
grp_op: *,
grp_id: e,
qrng: <ℚ+*>,
rng_plus: +r,
rng_zero: 0,
itop: Π(op,id) lb ≤ i < ub. E[i],
ycomb: Y
Lemmas referenced :
qle_witness,
int-subtype-rationals,
nat_plus_wf,
q-not-limit-zero-diverges,
eq_int_wf,
bool_wf,
eqtt_to_assert,
assert_of_eq_int,
eqff_to_assert,
equal_wf,
bool_cases_sqequal,
subtype_base_sq,
bool_subtype_base,
assert-bnot,
neg_assert_of_eq_int,
int_upper_subtype_nat,
false_wf,
le_wf,
nat_properties,
nequal-le-implies,
zero-add,
decidable__lt,
not-lt-2,
add_functionality_wrt_le,
add-commutes,
le-add-cancel,
less_than_wf,
nat_wf,
rationals_wf,
exists_wf,
qless_wf,
all_wf,
qle_wf,
less_than_transitivity2,
not-equal-2,
add-associates,
add-zero,
condition-implies-le,
minus-add,
minus-zero,
decidable__le,
satisfiable-full-omega-tt,
intformand_wf,
intformnot_wf,
intformle_wf,
itermVar_wf,
intformless_wf,
int_formula_prop_and_lemma,
int_formula_prop_not_lemma,
int_formula_prop_le_lemma,
int_term_value_var_lemma,
int_formula_prop_less_lemma,
int_formula_prop_wf,
intformeq_wf,
itermConstant_wf,
int_formula_prop_eq_lemma,
int_term_value_constant_lemma,
squash_wf,
true_wf,
qsum_wf,
int_seg_wf,
decidable__equal_int,
int_subtype_base,
sq_stable__le,
less-iff-le,
int_seg_properties,
less_than_transitivity1,
less_than_irreflexivity,
sum_unroll_base_q,
iff_weakening_equal,
sum_unroll_lo_q,
mon_ident_q
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation,
isect_memberFormation,
cut,
introduction,
sqequalRule,
sqequalHypSubstitution,
lambdaEquality,
dependent_functionElimination,
thin,
hypothesisEquality,
extract_by_obid,
isectElimination,
natural_numberEquality,
hypothesis,
applyEquality,
functionExtensionality,
independent_functionElimination,
rename,
setElimination,
because_Cache,
unionElimination,
equalityElimination,
productElimination,
independent_isectElimination,
equalityTransitivity,
equalitySymmetry,
dependent_pairFormation,
promote_hyp,
instantiate,
cumulativity,
voidElimination,
hypothesis_subsumption,
dependent_set_memberEquality,
independent_pairFormation,
isect_memberEquality,
voidEquality,
intEquality,
productEquality,
functionEquality,
addEquality,
minusEquality,
int_eqEquality,
computeAll,
hyp_replacement,
imageElimination,
imageMemberEquality,
baseClosed,
universeEquality
Latex:
\mforall{}f:\mBbbN{}\msupplus{} {}\mrightarrow{} \mBbbQ{}
(\mexists{}q:\mBbbQ{}. (0 < q \mwedge{} (\mforall{}n:\mBbbN{}. \mexists{}m:\mBbbN{}. (n < m \mwedge{} (q \mleq{} f[m]))))) {}\mRightarrow{} (\mforall{}B:\mBbbQ{}. \mexists{}n:\mBbbN{}. (B \mleq{} \mSigma{}1 \mleq{} i < n. f[i]))
supposing \mforall{}n:\mBbbN{}\msupplus{}. (0 \mleq{} f[n])
Date html generated:
2018_05_22-AM-00_16_43
Last ObjectModification:
2017_07_26-PM-06_52_59
Theory : rationals
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