Nuprl Lemma : cosine-approx-property
∀[x:ℝ]. ∀[k:ℕ]. ∀[N:ℕ+]. ((|x| ≤ (r1/r(2))) ⇒ 1-approx(Σ{-1^i * (x^2 * i)/(2 * i)! | 0≤i≤k};N;cosine-approx(x;k;N)))
Proof
Definitions occuring in Statement :
cosine-approx: cosine-approx(x;k;N),
ireal-approx: j-approx(x;M;z),
rsum: Σ{x[k] | n≤k≤m},
rdiv: (x/y),
rleq: x ≤ y,
rabs: |x|,
rnexp: x^k1,
int-rdiv: (a)/k1,
int-rmul: k1 * a,
int-to-real: r(n),
real: ℝ,
fastexp: i^n,
fact: (n)!,
nat_plus: ℕ+,
nat: ℕ,
uall: ∀[x:A]. B[x],
implies: P ⇒ Q,
multiply: n * m,
minus: -n,
natural_number: $n
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
implies: P ⇒ Q,
nat: ℕ,
true: True,
nequal: a ≠ b ∈ T ,
not: ¬A,
uimplies: b supposing a,
sq_type: SQType(T),
all: ∀x:A. B[x],
guard: {T},
false: False,
bool: 𝔹,
unit: Unit,
it: ⋅,
btrue: tt,
uiff: uiff(P;Q),
and: P ∧ Q,
ifthenelse: if b then t else f fi ,
nat_plus: ℕ+,
ge: i ≥ j ,
decidable: Dec(P),
or: P ∨ Q,
satisfiable_int_formula: satisfiable_int_formula(fmla),
exists: ∃x:A. B[x],
top: Top,
prop: ℙ,
subtype_rel: A ⊆r B,
bfalse: ff,
bnot: ¬bb,
assert: ↑b,
so_lambda: λ2x.t[x],
int_seg: {i..j-},
lelt: i ≤ j < k,
less_than: a < b,
squash: ↓T,
le: A ≤ B,
less_than': less_than'(a;b),
so_apply: x[s],
pointwise-req: x[k] = y[k] for k ∈ [n,m],
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q,
cosine-approx: cosine-approx(x;k;N),
rneq: x ≠ y,
ireal-approx: j-approx(x;M;z),
rleq: x ≤ y,
rnonneg: rnonneg(x),
exp: i^n,
primrec: primrec(n;b;c),
primtailrec: primtailrec(n;i;b;f),
subtract: n - m,
int_nzero: ℤ-o,
rev_uimplies: rev_uimplies(P;Q),
rge: x ≥ y,
req_int_terms: t1 ≡ t2,
rdiv: (x/y)
Lemmas referenced :
poly-approx-property,
subtype_base_sq,
int_subtype_base,
istype-int,
eq_int_wf,
eqtt_to_assert,
assert_of_eq_int,
int-rdiv_wf,
fact_wf,
nat_properties,
nat_plus_properties,
decidable__le,
full-omega-unsat,
intformand_wf,
intformnot_wf,
intformle_wf,
itermConstant_wf,
itermMultiply_wf,
itermVar_wf,
int_formula_prop_and_lemma,
istype-void,
int_formula_prop_not_lemma,
int_formula_prop_le_lemma,
int_term_value_constant_lemma,
int_term_value_mul_lemma,
int_term_value_var_lemma,
int_formula_prop_wf,
istype-le,
eqff_to_assert,
bool_cases_sqequal,
bool_wf,
bool_subtype_base,
assert-bnot,
neg_assert_of_eq_int,
int-to-real_wf,
rnexp_wf,
rsum_functionality,
rmul_wf,
int_seg_properties,
int_seg_subtype_nat,
istype-false,
int_seg_wf,
int-rmul_wf,
fastexp_wf,
equal_wf,
squash_wf,
true_wf,
istype-universe,
exp-minusone,
subtype_rel_self,
iff_weakening_equal,
modulus-is-rem,
ireal-approx_functionality,
cosine-approx_wf,
rsum_wf,
rleq_wf,
rabs_wf,
rdiv_wf,
rless-int,
rless_wf,
le_witness_for_triv,
nat_plus_wf,
istype-nat,
real_wf,
exp_wf2,
req-int-fractions,
nequal_wf,
exp_wf3,
rleq_functionality_wrt_implies,
rnexp_functionality_wrt_rleq,
zero-rleq-rabs,
rleq_weakening_equal,
rleq_weakening,
itermSubtract_wf,
req-iff-rsub-is-0,
nat_plus_inc_int_nzero,
rmul_preserves_req,
decidable__lt,
intformless_wf,
intformeq_wf,
int_formula_prop_less_lemma,
int_formula_prop_eq_lemma,
rinv_wf2,
exp-fastexp,
req_weakening,
rleq_functionality,
rabs-rnexp,
req_functionality,
req_inversion,
rnexp-rdiv,
rneq_functionality,
rnexp-int,
rdiv_functionality,
real_polynomial_null,
real_term_value_sub_lemma,
real_term_value_var_lemma,
real_term_value_const_lemma,
rmul_functionality,
int-rdiv-req,
req_transitivity,
int-rmul-req,
rmul-rinv3,
real_term_value_mul_lemma,
rnexp-mul
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation_alt,
introduction,
cut,
lambdaFormation_alt,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
hypothesisEquality,
lambdaEquality_alt,
remainderEquality,
setElimination,
rename,
because_Cache,
hypothesis,
natural_numberEquality,
instantiate,
cumulativity,
intEquality,
independent_isectElimination,
dependent_functionElimination,
equalityTransitivity,
equalitySymmetry,
independent_functionElimination,
voidElimination,
equalityIstype,
baseClosed,
sqequalBase,
closedConclusion,
inhabitedIsType,
unionElimination,
equalityElimination,
productElimination,
sqequalRule,
dependent_set_memberEquality_alt,
multiplyEquality,
approximateComputation,
dependent_pairFormation_alt,
int_eqEquality,
isect_memberEquality_alt,
independent_pairFormation,
universeIsType,
applyEquality,
promote_hyp,
imageElimination,
addEquality,
minusEquality,
universeEquality,
imageMemberEquality,
inrFormation_alt,
functionIsTypeImplies,
isectIsTypeImplies,
applyLambdaEquality
Latex:
\mforall{}[x:\mBbbR{}]. \mforall{}[k:\mBbbN{}]. \mforall{}[N:\mBbbN{}\msupplus{}].
((|x| \mleq{} (r1/r(2))) {}\mRightarrow{} 1-approx(\mSigma{}\{-1\^{}i * (x\^{}2 * i)/(2 * i)! | 0\mleq{}i\mleq{}k\};N;cosine-approx(x;k;N)))
Date html generated:
2019_10_29-AM-10_36_32
Last ObjectModification:
2019_02_02-AM-11_35_07
Theory : reals
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