Nuprl Lemma : rcos-seq-converges
rcos-seq(n)↓ as n→∞
Proof
Definitions occuring in Statement :
rcos-seq: rcos-seq(n),
converges: x[n]↓ as n→∞
Definitions unfolded in proof :
all: ∀x:A. B[x],
member: t ∈ T,
uall: ∀[x:A]. B[x],
implies: P ⇒ Q,
exists: ∃x:A. B[x],
int_upper: {i...},
le: A ≤ B,
and: P ∧ Q,
less_than': less_than'(a;b),
false: False,
not: ¬A,
prop: ℙ,
nat_plus: ℕ+,
less_than: a < b,
squash: ↓T,
true: True,
cand: A c∧ B,
so_lambda: λ2x.t[x],
nat: ℕ,
decidable: Dec(P),
or: P ∨ Q,
uimplies: b supposing a,
satisfiable_int_formula: satisfiable_int_formula(fmla),
top: Top,
subtype_rel: A ⊆r B,
rneq: x ≠ y,
guard: {T},
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q,
so_apply: x[s],
subtract: n - m,
rev_uimplies: rev_uimplies(P;Q),
rge: x ≥ y,
rsub: x - y,
rless: x < y,
sq_exists: ∃x:{A| B[x]},
real: ℝ,
rdiv: (x/y),
rmul: a * b,
int-to-real: r(n),
rinv: rinv(x),
mu-ge: mu-ge(f;n),
ifthenelse: if b then t else f fi ,
lt_int: i <z j,
absval: |i|,
btrue: tt,
eq_int: (i =z j),
accelerate: accelerate(k;f),
imax: imax(a;b),
canonical-bound: canonical-bound(r),
reg-seq-inv: reg-seq-inv(x),
le_int: i ≤z j,
bnot: ¬bb,
bfalse: ff,
reg-seq-mul: reg-seq-mul(x;y),
uiff: uiff(P;Q),
sq_stable: SqStable(P),
sq_type: SQType(T),
rfun: I ⟶ℝ,
subinterval: I ⊆ J ,
ifun: ifun(f;I),
real-fun: real-fun(f;a;b),
rcos-seq: rcos-seq(n),
int_nzero: ℤ-o,
nequal: a ≠ b ∈ T ,
increasing-on-interval: f[x] increasing for x ∈ I,
fastexp: i^n,
efficient-exp-ext
Lemmas referenced :
increasing-sequence-converges,
rcos-seq_wf,
nat_wf,
rcos-seq-increasing,
false_wf,
le_wf,
less_than_wf,
nat_plus_wf,
all_wf,
rleq_wf,
rsub_wf,
nat_plus_properties,
decidable__le,
satisfiable-full-omega-tt,
intformand_wf,
intformnot_wf,
intformle_wf,
itermConstant_wf,
itermAdd_wf,
itermVar_wf,
intformless_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_less_lemma,
int_formula_prop_wf,
nat_plus_subtype_nat,
rmul_wf,
rdiv_wf,
int-to-real_wf,
rless-int,
rless_wf,
subtract_wf,
itermSubtract_wf,
int_term_value_subtract_lemma,
decidable__lt,
exists_wf,
int_upper_properties,
rcos-seq-differences,
rsin_wf,
rleq_functionality_wrt_implies,
rleq_weakening_equal,
subtract-add-cancel,
radd-preserves-rless,
real_wf,
equal_wf,
radd_wf,
rminus_wf,
rless_functionality,
radd-zero-both,
req_weakening,
radd-rminus-assoc,
radd_comm,
radd_functionality,
radd-preserves-rleq,
sq_stable__less_than,
uiff_transitivity,
rleq_functionality,
req_transitivity,
req_inversion,
radd-assoc,
radd-ac,
radd-rminus-both,
decidable__equal_int,
subtype_base_sq,
int_subtype_base,
not-lt-2,
not-equal-2,
less-iff-le,
add_functionality_wrt_le,
add-associates,
add-zero,
add-commutes,
le-add-cancel2,
condition-implies-le,
minus-add,
add-swap,
minus-minus,
minus-one-mul,
zero-add,
minus-one-mul-top,
le-add-cancel,
primrec-wf-nat-plus,
rless_functionality_wrt_implies,
rleq_weakening_rless,
derivative-implies-increasing-simple,
i-member_wf,
rccint_wf,
rcos_wf,
set_wf,
derivative_functionality_wrt_subinterval,
riiint_wf,
member_rccint_lemma,
member_riiint_lemma,
deriviative-rsin,
left_endpoint_rccint_lemma,
right_endpoint_rccint_lemma,
req_functionality,
rcos_functionality,
req_wf,
rcos-seq-positive,
primrec0_lemma,
int-rdiv_wf,
equal-wf-base,
true_wf,
nequal_wf,
rleq-int-fractions2,
int-rdiv-req,
rmul_preserves_rless,
rmul_comm,
fastexp_wf,
efficient-exp-ext
Rules used in proof :
cut,
introduction,
extract_by_obid,
sqequalHypSubstitution,
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
dependent_functionElimination,
thin,
lambdaEquality,
isectElimination,
hypothesisEquality,
hypothesis,
independent_functionElimination,
sqequalRule,
lambdaFormation,
dependent_pairFormation,
dependent_set_memberEquality,
natural_numberEquality,
independent_pairFormation,
imageMemberEquality,
baseClosed,
productEquality,
addEquality,
setElimination,
rename,
unionElimination,
independent_isectElimination,
int_eqEquality,
intEquality,
isect_memberEquality,
voidElimination,
voidEquality,
computeAll,
applyEquality,
inrFormation,
because_Cache,
productElimination,
equalityTransitivity,
equalitySymmetry,
addLevel,
levelHypothesis,
dependent_set_memberFormation,
imageElimination,
instantiate,
cumulativity,
minusEquality,
setEquality,
multiplyEquality
Latex:
rcos-seq(n)\mdownarrow{} as n\mrightarrow{}\minfty{}
Date html generated:
2017_10_04-PM-10_24_00
Last ObjectModification:
2017_07_28-AM-08_48_43
Theory : reals_2
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