Nuprl Lemma : q-not-limit-zero-diverges
∀f:ℕ ⟶ ℚ
(∃q:ℚ. (0 < q ∧ (∀n:ℕ. ∃m:ℕ. ((n ≤ m) ∧ (q ≤ f[m]))))) ⇒ (∀B:ℚ. ∃n:ℕ. (B ≤ Σ0 ≤ 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: ℕ,
uimplies: b supposing a,
so_apply: x[s],
le: A ≤ B,
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,
exists: ∃x:A. B[x],
and: P ∧ Q,
prop: ℙ,
so_lambda: λ2x.t[x],
nat: ℕ,
not: ¬A,
guard: {T},
false: False,
true: True,
squash: ↓T,
iff: P ⇐⇒ Q,
ge: i ≥ j ,
decidable: Dec(P),
or: P ∨ Q,
satisfiable_int_formula: satisfiable_int_formula(fmla),
top: Top,
cand: A c∧ B,
pi1: fst(t),
le: A ≤ B,
less_than': less_than'(a;b),
int_seg: {i..j-},
lelt: i ≤ j < k,
compose: f o g,
sq_type: SQType(T),
less_than: a < b
Lemmas referenced :
qle_witness,
int-subtype-rationals,
nat_wf,
rationals_wf,
exists_wf,
qless_wf,
all_wf,
le_wf,
qle_wf,
q-archimedean,
qdiv_wf,
qless_transitivity_2_qorder,
qle_weakening_eq_qorder,
qless_irreflexivity,
equal-wf-T-base,
qmul_preserves_qle2,
subtype_rel_set,
qle_weakening_lt_qorder,
qmul_wf,
squash_wf,
true_wf,
qmul_comm_qrng,
qmul-qdiv-cancel,
iff_weakening_equal,
nat_properties,
decidable__le,
satisfiable-full-omega-tt,
intformand_wf,
intformnot_wf,
intformle_wf,
itermConstant_wf,
itermAdd_wf,
itermVar_wf,
int_formula_prop_and_lemma,
int_formula_prop_not_lemma,
int_formula_prop_le_lemma,
int_term_value_constant_lemma,
int_term_value_add_lemma,
int_term_value_var_lemma,
int_formula_prop_wf,
pi1_wf_top,
equal_wf,
decidable__lt,
intformless_wf,
int_formula_prop_less_lemma,
less_than_wf,
fun_exp_wf,
false_wf,
int_seg_wf,
int_seg_subtype_nat,
ge_wf,
member-less_than,
int_seg_properties,
subtract-add-cancel,
fun_exp1_lemma,
subtract_wf,
itermSubtract_wf,
int_term_value_subtract_lemma,
fun_exp_add1,
decidable__equal_int,
intformeq_wf,
int_formula_prop_eq_lemma,
subtype_base_sq,
int_subtype_base,
lelt_wf,
qsum_wf,
qsum-qle,
qsum-const,
qle_transitivity_qorder,
qsum-subsequence-qle,
subtype_rel_dep_function,
subtype_rel_self
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,
productElimination,
productEquality,
because_Cache,
setElimination,
functionEquality,
independent_isectElimination,
voidElimination,
baseClosed,
intEquality,
dependent_pairFormation,
imageElimination,
equalityTransitivity,
equalitySymmetry,
imageMemberEquality,
universeEquality,
dependent_set_memberEquality,
addEquality,
unionElimination,
int_eqEquality,
isect_memberEquality,
voidEquality,
independent_pairFormation,
computeAll,
independent_pairEquality,
comment,
intWeakElimination,
hyp_replacement,
Error :applyLambdaEquality,
instantiate,
cumulativity
Latex:
\mforall{}f:\mBbbN{} {}\mrightarrow{} \mBbbQ{}
(\mexists{}q:\mBbbQ{}. (0 < q \mwedge{} (\mforall{}n:\mBbbN{}. \mexists{}m:\mBbbN{}. ((n \mleq{} m) \mwedge{} (q \mleq{} f[m]))))) {}\mRightarrow{} (\mforall{}B:\mBbbQ{}. \mexists{}n:\mBbbN{}. (B \mleq{} \mSigma{}0 \mleq{} i < n. f[i]))
supposing \mforall{}n:\mBbbN{}. (0 \mleq{} f[n])
Date html generated:
2016_10_26-AM-06_37_32
Last ObjectModification:
2016_07_12-AM-08_00_11
Theory : rationals
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