Nuprl Lemma : rcos-seq-differences
∀n:ℕ. (0 < n ⇒ ((rcos-seq(n + 1) - rcos-seq(n)) ≤ ((r1 - rsin(rcos-seq(n - 1))) * (rcos-seq(n) - rcos-seq(n - 1)))))
Proof
Definitions occuring in Statement :
rcos-seq: rcos-seq(n),
rsin: rsin(x),
rleq: x ≤ y,
rsub: x - y,
rmul: a * b,
int-to-real: r(n),
nat: ℕ,
less_than: a < b,
all: ∀x:A. B[x],
implies: P ⇒ Q,
subtract: n - m,
add: n + m,
natural_number: $n
Definitions unfolded in proof :
req_int_terms: t1 ≡ t2,
increasing-on-interval: f[x] increasing for x ∈ I,
rge: x ≥ y,
rev_implies: P ⇐ Q,
true: True,
cand: A c∧ B,
iproper: iproper(I),
real-fun: real-fun(f;a;b),
real: ℝ,
subtype_rel: A ⊆r B,
nat_plus: ℕ+,
sq_exists: ∃x:A [B[x]],
rless: x < y,
guard: {T},
iff: P ⇐⇒ Q,
squash: ↓T,
sq_stable: SqStable(P),
rfun: I ⟶ℝ,
rev_uimplies: rev_uimplies(P;Q),
so_apply: x[s],
so_lambda: λ2x.t[x],
uiff: uiff(P;Q),
prop: ℙ,
and: P ∧ Q,
top: Top,
false: False,
exists: ∃x:A. B[x],
satisfiable_int_formula: satisfiable_int_formula(fmla),
not: ¬A,
uimplies: b supposing a,
or: P ∨ Q,
decidable: Dec(P),
ge: i ≥ j ,
nat: ℕ,
member: t ∈ T,
uall: ∀[x:A]. B[x],
implies: P ⇒ Q,
all: ∀x:A. B[x]
Lemmas referenced :
radd-zero,
rmul_preserves_rleq2,
real_term_value_minus_lemma,
real_term_value_mul_lemma,
itermMinus_wf,
itermMultiply_wf,
radd_comm,
radd-preserves-rleq,
real_term_value_const_lemma,
real_term_value_var_lemma,
real_term_value_add_lemma,
real_term_value_sub_lemma,
real_polynomial_null,
req-iff-rsub-is-0,
rabs_wf,
rabs-difference-bound-rleq,
rleq_functionality_wrt_implies,
rleq_weakening_equal,
rleq_transitivity,
iff_weakening_equal,
subtype_rel_self,
true_wf,
squash_wf,
sq_stable__rleq,
rcos_functionality,
function-is-continuous,
derivative-rsin,
i-finite_wf,
rcos-seq-positive,
right_endpoint_rccint_lemma,
left_endpoint_rccint_lemma,
derivative-implies-increasing,
equal_wf,
derivative-rcos,
subinterval-riiint,
riiint_wf,
derivative_functionality_wrt_subinterval,
rleq_wf,
req_wf,
rsin_functionality,
rminus_functionality,
req_functionality,
member_rccint_lemma,
real-fun-iff-continuous,
nat_plus_properties,
sq_stable__less_than,
rleq_weakening_rless,
real-cont-iff-continuous,
rcos-seq-increasing,
sq_stable__rless,
rminus_wf,
rccint_wf,
i-member_wf,
mean-value-theorem,
req_weakening,
rcos-seq-step,
rsub_functionality,
rleq_functionality,
rcos_wf,
radd_wf,
nat_wf,
less_than_wf,
rless_wf,
real_wf,
set_wf,
subtract-add-cancel,
int_term_value_subtract_lemma,
itermSubtract_wf,
subtract_wf,
rsin_wf,
int-to-real_wf,
rmul_wf,
le_wf,
int_formula_prop_wf,
int_formula_prop_less_lemma,
int_term_value_var_lemma,
int_term_value_add_lemma,
int_term_value_constant_lemma,
int_formula_prop_le_lemma,
int_formula_prop_not_lemma,
int_formula_prop_and_lemma,
intformless_wf,
itermVar_wf,
itermAdd_wf,
itermConstant_wf,
intformle_wf,
intformnot_wf,
intformand_wf,
full-omega-unsat,
decidable__le,
nat_properties,
rcos-seq_wf,
rsub_wf,
rleq-iff-all-rless
Rules used in proof :
universeEquality,
instantiate,
equalitySymmetry,
equalityTransitivity,
productEquality,
applyEquality,
imageElimination,
baseClosed,
imageMemberEquality,
setEquality,
productElimination,
independent_pairFormation,
sqequalRule,
voidEquality,
voidElimination,
isect_memberEquality,
intEquality,
int_eqEquality,
lambdaEquality,
dependent_pairFormation,
independent_functionElimination,
approximateComputation,
independent_isectElimination,
unionElimination,
dependent_functionElimination,
hypothesisEquality,
natural_numberEquality,
hypothesis,
because_Cache,
rename,
setElimination,
addEquality,
dependent_set_memberEquality,
thin,
isectElimination,
sqequalHypSubstitution,
extract_by_obid,
introduction,
cut,
lambdaFormation,
sqequalReflexivity,
computationStep,
sqequalTransitivity,
sqequalSubstitution
Latex:
\mforall{}n:\mBbbN{}
(0 < n
{}\mRightarrow{} ((rcos-seq(n + 1) - rcos-seq(n)) \mleq{} ((r1 - rsin(rcos-seq(n - 1)))
* (rcos-seq(n) - rcos-seq(n - 1)))))
Date html generated:
2018_05_22-PM-02_58_48
Last ObjectModification:
2018_05_20-PM-11_03_31
Theory : reals_2
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