Nuprl Lemma : radd_rcos-Taylor
∀b:ℝ. (|radd_rcos(b) - π/2(slower)| ≤ (|b - π/2(slower)|^3/r(2)))
Proof
Definitions occuring in Statement :
half-pi: π/2(slower),
radd_rcos: radd_rcos(x),
rdiv: (x/y),
rleq: x ≤ y,
rabs: |x|,
rnexp: x^k1,
rsub: x - y,
int-to-real: r(n),
real: ℝ,
all: ∀x:A. B[x],
natural_number: $n
Definitions unfolded in proof :
all: ∀x:A. B[x],
uall: ∀[x:A]. B[x],
member: t ∈ T,
subtype_rel: A ⊆r B,
prop: ℙ,
nat: ℕ,
le: A ≤ B,
and: P ∧ Q,
less_than': less_than'(a;b),
false: False,
not: ¬A,
implies: P ⇒ Q,
uimplies: b supposing a,
rneq: x ≠ y,
guard: {T},
or: P ∨ Q,
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q,
less_than: a < b,
squash: ↓T,
true: True,
uiff: uiff(P;Q),
nat_plus: ℕ+,
so_lambda: λ2x y.t[x; y],
int_seg: {i..j-},
bool: 𝔹,
unit: Unit,
it: ⋅,
btrue: tt,
ifthenelse: if b then t else f fi ,
rfun: I ⟶ℝ,
bfalse: ff,
exists: ∃x:A. B[x],
sq_type: SQType(T),
bnot: ¬bb,
assert: ↑b,
so_apply: x[s1;s2],
top: Top,
decidable: Dec(P),
eq_int: (i =z j),
lelt: i ≤ j < k,
satisfiable_int_formula: satisfiable_int_formula(fmla),
so_lambda: λ2x.t[x],
so_apply: x[s],
sq_stable: SqStable(P),
rev_uimplies: rev_uimplies(P;Q),
Taylor-remainder: Taylor-remainder(I;n;b;a;i,x.F[i; x]),
Taylor-approx: Taylor-approx(n;a;b;i,x.F[i; x]),
fact: (n)!,
primrec: primrec(n;b;c),
subtract: n - m,
rsum: Σ{x[k] | n≤k≤m},
callbyvalueall: callbyvalueall,
evalall: evalall(t),
map: map(f;as),
list_ind: list_ind,
from-upto: [n, m),
lt_int: i <z j,
nil: [],
rdiv: (x/y),
itermConstant: "const",
req_int_terms: t1 ≡ t2,
rsub: x - y,
label: ...$L... t,
cand: A c∧ B,
rge: x ≥ y,
absval: |i|
Lemmas referenced :
rleq-iff-all-rless,
rabs_wf,
rsub_wf,
radd_rcos_wf,
real_wf,
req_wf,
radd_wf,
rcos_wf,
half-pi_wf,
rdiv_wf,
rnexp_wf,
false_wf,
le_wf,
int-to-real_wf,
rless-int,
rless_wf,
Taylor-theorem,
riiint_wf,
iproper-riiint,
less_than_wf,
eq_int_wf,
bool_wf,
eqtt_to_assert,
assert_of_eq_int,
i-member_wf,
eqff_to_assert,
equal_wf,
bool_cases_sqequal,
subtype_base_sq,
bool_subtype_base,
assert-bnot,
neg_assert_of_eq_int,
rsin_wf,
rminus_wf,
int_seg_wf,
member_riiint_lemma,
true_wf,
decidable__equal_int,
int_subtype_base,
int_seg_properties,
int_seg_subtype,
int_seg_cases,
satisfiable-full-omega-tt,
intformand_wf,
intformless_wf,
itermVar_wf,
itermConstant_wf,
intformle_wf,
int_formula_prop_and_lemma,
int_formula_prop_less_lemma,
int_term_value_var_lemma,
int_term_value_constant_lemma,
int_formula_prop_le_lemma,
int_formula_prop_wf,
set_wf,
radd_rcos-deriv-seq,
sq_stable__rless,
req_weakening,
req_functionality,
radd_rcos_functionality,
rsub_functionality,
rsin_functionality,
rminus_functionality,
rcos_functionality,
rsum_wf,
rmul_wf,
fact_wf,
int_seg_subtype_nat,
nat_plus_wf,
decidable__lt,
nat_plus_properties,
intformnot_wf,
int_formula_prop_not_lemma,
rneq-int,
fact-non-zero,
fact0_redex_lemma,
rnexp_zero_lemma,
rsum-split-first,
itermAdd_wf,
int_term_value_add_lemma,
radd_functionality,
map_nil_lemma,
radd_list_nil_lemma,
rmul_functionality,
rdiv_functionality,
rcos-half-pi,
rsin-half-pi,
rinv_wf2,
real_term_polynomial,
itermSubtract_wf,
itermMultiply_wf,
itermMinus_wf,
real_term_value_const_lemma,
real_term_value_sub_lemma,
real_term_value_add_lemma,
real_term_value_mul_lemma,
real_term_value_var_lemma,
real_term_value_minus_lemma,
req-iff-rsub-is-0,
radd-zero-both,
sq_stable__req,
req_inversion,
rminus-as-rmul,
rmul-int,
radd_comm,
radd-ac,
req_transitivity,
rminus-radd,
rmul-rdiv-cancel2,
uiff_transitivity,
Taylor-remainder_wf,
ifthenelse_wf,
rleq_functionality,
rabs_functionality,
set_subtype_base,
rmul_preserves_rleq,
rleq_wf,
rnexp_functionality,
rmul-rinv3,
rinv-mul-as-rdiv,
rabs-of-nonneg,
rleq-int,
rless_transitivity1,
rleq_weakening,
rless_functionality,
rabs-rdiv,
rabs-rmul,
rmul_comm,
zero-rleq-rabs,
rleq_functionality_wrt_implies,
rmul_functionality_wrt_rleq2,
rabs-rsin-rleq,
rleq_weakening_equal,
rabs-rnexp,
rnexp-rleq,
rabs-difference-bound-rleq,
squash_wf,
rabs-int,
iff_weakening_equal,
radd-preserves-rleq,
rmax_wf,
radd_functionality_wrt_rleq,
radd-rmax,
rmax_functionality,
rmax_lb,
rabs-rminus,
rabs-bounds,
rmin_wf,
rmin_ub,
radd-rmin,
rmin_functionality,
rmul-rdiv-cancel,
rmul-ac,
rmul-assoc,
rmul_preserves_req,
rnexp_step,
rabs-rleq-iff,
rmul-zero-both,
radd-int,
rmul-distrib2,
rmul-identity1,
radd-rminus-assoc,
radd-assoc,
rmul_over_rminus,
rmul-distrib,
radd-rminus-both
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation,
cut,
introduction,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
hypothesisEquality,
hypothesis,
applyEquality,
lambdaEquality,
setElimination,
rename,
setEquality,
sqequalRule,
dependent_set_memberEquality,
natural_numberEquality,
independent_pairFormation,
independent_isectElimination,
inrFormation,
dependent_functionElimination,
because_Cache,
productElimination,
independent_functionElimination,
imageMemberEquality,
baseClosed,
unionElimination,
equalityElimination,
dependent_pairFormation,
equalityTransitivity,
equalitySymmetry,
promote_hyp,
instantiate,
cumulativity,
voidElimination,
isect_memberEquality,
voidEquality,
intEquality,
hypothesis_subsumption,
addEquality,
int_eqEquality,
computeAll,
imageElimination,
applyLambdaEquality,
callbyvalueReduce,
sqleReflexivity,
multiplyEquality,
minusEquality,
inlFormation,
productEquality,
universeEquality
Latex:
\mforall{}b:\mBbbR{}. (|radd\_rcos(b) - \mpi{}/2(slower)| \mleq{} (|b - \mpi{}/2(slower)|\^{}3/r(2)))
Date html generated:
2017_10_04-PM-10_24_30
Last ObjectModification:
2017_07_28-AM-08_48_50
Theory : reals_2
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