Nuprl Lemma : log-contraction-Taylor
∀[a,x:ℝ]. |log-contraction(a;x) - rlog(a)| ≤ ((r1/r(4)) * |x - rlog(a)|^3) supposing (r0 < a) ∧ (|x - rlog(a)| ≤ r1)
Proof
Definitions occuring in Statement :
log-contraction: log-contraction(a;x),
rlog: rlog(x),
rdiv: (x/y),
rleq: x ≤ y,
rless: x < y,
rabs: |x|,
rnexp: x^k1,
rsub: x - y,
rmul: a * b,
int-to-real: r(n),
real: ℝ,
uimplies: b supposing a,
uall: ∀[x:A]. B[x],
and: P ∧ Q,
natural_number: $n
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
uimplies: b supposing a,
and: P ∧ Q,
all: ∀x:A. B[x],
implies: P ⇒ Q,
subtype_rel: A ⊆r B,
rleq: x ≤ y,
rnonneg: rnonneg(x),
le: A ≤ B,
not: ¬A,
false: False,
rneq: x ≠ y,
guard: {T},
or: P ∨ Q,
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q,
less_than: a < b,
squash: ↓T,
less_than': less_than'(a;b),
true: True,
prop: ℙ,
nat: ℕ,
rge: x ≥ y,
rgt: x > y,
uiff: uiff(P;Q),
nat_plus: ℕ+,
so_lambda: λ2x y.t[x; y],
rfun: I ⟶ℝ,
int_seg: {i..j-},
bool: 𝔹,
unit: Unit,
it: ⋅,
btrue: tt,
ifthenelse: if b then t else f fi ,
bfalse: ff,
exists: ∃x:A. B[x],
sq_type: SQType(T),
bnot: ¬bb,
assert: ↑b,
so_apply: x[s1;s2],
top: Top,
so_lambda: λ2x.t[x],
so_apply: x[s],
sq_stable: SqStable(P),
decidable: Dec(P),
eq_int: (i =z j),
lelt: i ≤ j < k,
satisfiable_int_formula: satisfiable_int_formula(fmla),
rev_uimplies: rev_uimplies(P;Q),
log-contraction: log-contraction(a;x),
finite-deriv-seq: finite-deriv-seq(I;k;i,x.F[i; x]),
Taylor-remainder: Taylor-remainder(I;n;b;a;i,x.F[i; x]),
Taylor-approx: Taylor-approx(n;a;b;i,x.F[i; x]),
rsub: x - y,
rsum: Σ{x[k] | n≤k≤m},
from-upto: [n, m),
lt_int: i <z j,
fact: (n)!,
primrec: primrec(n;b;c),
subtract: n - m,
callbyvalueall: callbyvalueall,
has-value: (a)↓,
has-valueall: has-valueall(a),
rdiv: (x/y),
label: ...$L... t,
cand: A c∧ B,
rless: x < y,
sq_exists: ∃x:{A| B[x]}
Lemmas referenced :
rnexp-positive,
radd_wf,
rexp_wf,
nat_plus_subtype_nat,
nat_plus_wf,
less_than'_wf,
rsub_wf,
rmul_wf,
rdiv_wf,
int-to-real_wf,
rless-int,
rless_wf,
rnexp_wf,
false_wf,
le_wf,
rabs_wf,
rlog_wf,
log-contraction_wf,
rleq_wf,
real_wf,
rless_functionality_wrt_implies,
rleq_weakening_equal,
rleq_weakening_rless,
radd_functionality_wrt_rless1,
rexp-positive,
rless_functionality,
req_weakening,
radd-zero-both,
radd_comm,
rleq-iff-all-rless,
Taylor-theorem,
riiint_wf,
iproper-riiint,
less_than_wf,
eq_int_wf,
bool_wf,
eqtt_to_assert,
assert_of_eq_int,
eqff_to_assert,
equal_wf,
bool_cases_sqequal,
subtype_base_sq,
bool_subtype_base,
assert-bnot,
neg_assert_of_eq_int,
i-member_wf,
int_seg_wf,
member_riiint_lemma,
true_wf,
req_wf,
set_wf,
sq_stable__rless,
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,
req_functionality,
rnexp_functionality,
rdiv_functionality,
rsub_functionality,
rexp_functionality,
radd_functionality,
rmul_functionality,
derivative-log-contraction,
second-derivative-log-contraction,
third-derivative-log-contraction,
rminus_wf,
rexp-rlog,
radd-rminus-both,
map_cons_lemma,
fact0_redex_lemma,
rnexp_zero_lemma,
map_nil_lemma,
valueall-type-has-valueall,
list_wf,
valueall-type-real-list,
cons_wf,
fact_wf,
nil_wf,
evalall-reduce,
radd-list_wf-bag,
list-subtype-bag,
subtype_rel_self,
req_transitivity,
radd-list-cons,
set_subtype_base,
iff_weakening_equal,
squash_wf,
rneq_wf,
uiff_transitivity,
rdiv-zero,
rmul-int,
radd_list_nil_lemma,
rinv_wf2,
rmul-zero,
rmul-zero-both,
rnexp2,
rmul-rdiv-cancel2,
Taylor-remainder_wf,
rleq_functionality,
rabs_functionality,
rabs-difference-bound-rleq,
rmin_wf,
rmin_ub,
trivial-rsub-rleq,
zero-rleq-rabs,
rmax_wf,
rmax_lb,
trivial-rleq-radd,
rleq_functionality_wrt_implies,
radd-preserves-rleq,
rabs-bounds,
rabs-difference-symmetry,
req_inversion,
radd-assoc,
radd-ac,
radd-rminus-assoc,
rminus-as-rmul,
rmul-identity1,
rmul-distrib2,
radd-int,
third-derivative-log-contraction-bound,
third-derivative-log-contraction-nonneg,
rleq_transitivity,
r-triangle-inequality,
radd_functionality_wrt_rleq,
rabs-rmul,
rminus_functionality,
radd-rmax,
rmax_functionality,
rabs-rminus,
rminus-radd,
rminus-rminus,
radd-rmin,
rmin_functionality,
rnexp-rleq-iff,
rabs-rnexp,
rmul_preserves_rleq,
rleq-int-fractions,
rabs-of-nonneg,
rmul_comm,
rmul-int-rdiv,
rmul-nonneg-case1,
rneq-int,
nat_plus_properties,
intformeq_wf,
int_formula_prop_eq_lemma,
rleq-int-fractions2,
rmul_functionality_wrt_rleq2,
rnexp_step,
rmul-assoc,
rmul-ac
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
introduction,
cut,
sqequalHypSubstitution,
productElimination,
thin,
lambdaFormation,
because_Cache,
extract_by_obid,
dependent_functionElimination,
isectElimination,
hypothesisEquality,
hypothesis,
independent_functionElimination,
applyEquality,
sqequalRule,
lambdaEquality,
independent_pairEquality,
natural_numberEquality,
independent_isectElimination,
inrFormation,
independent_pairFormation,
imageMemberEquality,
baseClosed,
dependent_set_memberEquality,
minusEquality,
axiomEquality,
equalityTransitivity,
equalitySymmetry,
productEquality,
isect_memberEquality,
voidElimination,
addLevel,
setElimination,
rename,
unionElimination,
equalityElimination,
dependent_pairFormation,
promote_hyp,
instantiate,
cumulativity,
setEquality,
voidEquality,
imageElimination,
intEquality,
hypothesis_subsumption,
addEquality,
int_eqEquality,
computeAll,
callbyvalueReduce,
sqleReflexivity,
multiplyEquality,
universeEquality,
inlFormation
Latex:
\mforall{}[a,x:\mBbbR{}].
|log-contraction(a;x) - rlog(a)| \mleq{} ((r1/r(4)) * |x - rlog(a)|\^{}3)
supposing (r0 < a) \mwedge{} (|x - rlog(a)| \mleq{} r1)
Date html generated:
2017_10_04-PM-10_33_02
Last ObjectModification:
2017_07_28-AM-08_50_28
Theory : reals_2
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