Nuprl Lemma : small-rexp-remainder
∀[x:{x:ℝ| |x| ≤ (r1/r(4))} ]. ∀[n:ℕ]. (|e^x - Σ{(x^k/r((k)!)) | 0≤k≤n}| ≤ (r1/r(4^n * 3 * (n)!)))
Proof
Definitions occuring in Statement :
rexp: e^x,
rsum: Σ{x[k] | n≤k≤m},
rdiv: (x/y),
rleq: x ≤ y,
rabs: |x|,
rnexp: x^k1,
rsub: x - y,
int-to-real: r(n),
real: ℝ,
fact: (n)!,
exp: i^n,
nat: ℕ,
uall: ∀[x:A]. B[x],
set: {x:A| B[x]} ,
multiply: n * m,
natural_number: $n
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
all: ∀x:A. B[x],
nat: ℕ,
decidable: Dec(P),
or: P ∨ Q,
uimplies: b supposing a,
sq_type: SQType(T),
implies: P ⇒ Q,
guard: {T},
rleq: x ≤ y,
rnonneg: rnonneg(x),
le: A ≤ B,
and: P ∧ Q,
rneq: x ≠ y,
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q,
less_than: a < b,
squash: ↓T,
less_than': less_than'(a;b),
true: True,
prop: ℙ,
exp: i^n,
primrec: primrec(n;b;c),
primtailrec: primtailrec(n;i;b;f),
fact: (n)!,
so_lambda: λ2x.t[x],
subtype_rel: A ⊆r B,
false: False,
not: ¬A,
int_seg: {i..j-},
lelt: i ≤ j < k,
ge: i ≥ j ,
satisfiable_int_formula: satisfiable_int_formula(fmla),
exists: ∃x:A. B[x],
top: Top,
nat_plus: ℕ+,
so_apply: x[s],
uiff: uiff(P;Q),
rev_uimplies: rev_uimplies(P;Q),
rdiv: (x/y),
req_int_terms: t1 ≡ t2,
stable: Stable{P},
sq_stable: SqStable(P),
rge: x ≥ y,
rless: x < y,
sq_exists: ∃x:A [B[x]],
rmul: a * b,
int-to-real: r(n),
rinv: rinv(x),
mu-ge: mu-ge(f;n),
ifthenelse: if b then t else f fi ,
lt_int: i <z j,
absval: |i|,
btrue: tt,
eq_int: (i =z j),
accelerate: accelerate(k;f),
imax: imax(a;b),
reg-seq-inv: reg-seq-inv(x),
le_int: i ≤z j,
bnot: ¬bb,
bfalse: ff,
reg-seq-mul: reg-seq-mul(x;y),
so_lambda: λ2x y.t[x; y],
rfun: I ⟶ℝ,
so_apply: x[s1;s2],
finite-deriv-seq: finite-deriv-seq(I;k;i,x.F[i; x]),
Taylor-remainder: Taylor-remainder(I;n;b;a;i,x.F[i; x]),
Taylor-approx: Taylor-approx(n;a;b;i,x.F[i; x]),
pointwise-req: x[k] = y[k] for k ∈ [n,m],
label: ...$L... t,
rgt: x > y,
int_nzero: ℤ-o,
nequal: a ≠ b ∈ T ,
cand: A c∧ B
Lemmas referenced :
decidable__equal_int,
subtype_base_sq,
int_subtype_base,
le_witness_for_triv,
istype-nat,
real_wf,
rleq_wf,
rabs_wf,
rdiv_wf,
int-to-real_wf,
rless-int,
rless_wf,
rsub_wf,
rexp_wf,
rsum_wf,
rnexp_wf,
int_seg_subtype_nat,
istype-false,
fact_wf,
int_seg_properties,
nat_properties,
decidable__lt,
decidable__le,
full-omega-unsat,
intformand_wf,
intformnot_wf,
intformle_wf,
itermConstant_wf,
itermVar_wf,
istype-int,
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_var_lemma,
int_formula_prop_wf,
istype-le,
nat_plus_properties,
intformless_wf,
int_formula_prop_less_lemma,
int_seg_wf,
fact0_redex_lemma,
equal-wf-base,
intformeq_wf,
int_formula_prop_eq_lemma,
rleq_functionality,
rabs_functionality,
rsub_functionality,
req_weakening,
rsum-single,
rnexp_zero_lemma,
rmul_wf,
rinv_wf2,
itermSubtract_wf,
itermMultiply_wf,
rexp0,
req_functionality,
req_transitivity,
rinv1,
req-iff-rsub-is-0,
real_polynomial_null,
real_term_value_sub_lemma,
real_term_value_mul_lemma,
real_term_value_const_lemma,
real_term_value_var_lemma,
req_inversion,
stable__rleq,
false_wf,
not_wf,
minimal-double-negation-hyp-elim,
minimal-not-not-excluded-middle,
rexp-increasing,
rminus_wf,
radd-preserves-rleq,
radd_wf,
itermAdd_wf,
rleq_weakening_rless,
rabs-of-nonpos,
real_term_value_add_lemma,
itermMinus_wf,
real_term_value_minus_lemma,
sq_stable__rleq,
rabs-rleq-iff,
rexp-non-decreasing,
rleq_functionality_wrt_implies,
radd_functionality_wrt_rleq,
rleq_weakening_equal,
rminus_functionality_wrt_rleq,
istype-less_than,
sqle-mono-implies-equal,
int-mono,
not-rless,
rleq-implies-rleq,
rabs-of-nonneg,
rsub_functionality_wrt_rleq,
rleq-iff-all-rless,
exp_wf2,
multiply_nat_plus,
multiply-is-int-iff,
int_term_value_mul_lemma,
Taylor-theorem,
riiint_wf,
iproper-riiint,
i-member_wf,
member_riiint_lemma,
true_wf,
rexp_functionality,
req_wf,
derivative-rexp,
sq_stable__rless,
rsum_functionality,
rmul_functionality,
rdiv_functionality,
rnexp_functionality,
rinv-mul-as-rdiv,
mul_bounds_1b,
exp_wf_nat_plus,
mul_nat_plus,
Taylor-remainder_wf,
rneq-int,
fact-non-zero,
rabs-rmul,
rmul_preserves_rleq,
rmul-rinv3,
rexp-positive,
rabs-rnexp,
rnexp-rleq,
zero-rleq-rabs,
rmax-req2,
rmin-req,
rmin_wf,
rleq_transitivity,
rleq_weakening,
rmax_wf,
rmax-req,
rmin-req2,
trivial-rsub-rleq,
int_term_value_add_lemma,
set_subtype_base,
le_wf,
rneq_functionality,
rmul-int,
exp-positive,
rnexp-rdiv,
rless_functionality,
rnexp-int,
req-int-fractions,
exp_wf3,
nequal_wf,
equal_wf,
squash_wf,
istype-universe,
exp-one,
subtype_rel_self,
iff_weakening_equal,
rmax_lb,
rabs-bounds,
rmul_preserves_rleq2,
rmul-identity1,
rinv-as-rdiv,
rless_transitivity2,
int_entire_a,
exp_step,
mul_nzero,
subtract_wf,
int_term_value_subtract_lemma,
rinv-of-rmul,
rinv_functionality2,
rmul_assoc,
rmul_preserves_rleq3,
rmul-nonneg-case1,
rmul_preserves_rless,
rless-implies-rless,
rmul_preserves_rneq_iff2,
r-triangle-inequality2,
iff_weakening_uiff,
radd_functionality
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation_alt,
introduction,
cut,
extract_by_obid,
sqequalHypSubstitution,
dependent_functionElimination,
thin,
setElimination,
rename,
hypothesisEquality,
hypothesis,
natural_numberEquality,
unionElimination,
instantiate,
isectElimination,
cumulativity,
intEquality,
independent_isectElimination,
because_Cache,
independent_functionElimination,
sqequalRule,
lambdaEquality_alt,
productElimination,
equalityTransitivity,
equalitySymmetry,
functionIsTypeImplies,
inhabitedIsType,
isect_memberEquality_alt,
isectIsTypeImplies,
setIsType,
universeIsType,
closedConclusion,
inrFormation_alt,
independent_pairFormation,
imageMemberEquality,
baseClosed,
applyEquality,
addEquality,
lambdaFormation_alt,
imageElimination,
dependent_set_memberEquality_alt,
approximateComputation,
dependent_pairFormation_alt,
int_eqEquality,
voidElimination,
applyLambdaEquality,
equalityIstype,
sqequalBase,
unionEquality,
functionEquality,
functionIsType,
unionIsType,
dependent_set_memberFormation_alt,
multiplyEquality,
pointwiseFunctionality,
promote_hyp,
baseApply,
universeEquality
Latex:
\mforall{}[x:\{x:\mBbbR{}| |x| \mleq{} (r1/r(4))\} ]. \mforall{}[n:\mBbbN{}]. (|e\^{}x - \mSigma{}\{(x\^{}k/r((k)!)) | 0\mleq{}k\mleq{}n\}| \mleq{} (r1/r(4\^{}n * 3 * (n)!)))
Date html generated:
2019_10_30-AM-11_40_37
Last ObjectModification:
2019_02_04-AM-10_12_35
Theory : reals_2
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