Nuprl Lemma : integral-by-parts
∀I:Interval. ∀u,v,u',v':{h:I ⟶ℝ| ∀x,y:{t:ℝ| t ∈ I} . ((x = y) ⇒ ((h x) = (h y)))} .
(d(u[t])/dt = λt.u'[t] on I
⇒ d(v[t])/dt = λt.v'[t] on I
⇒ iproper(I)
⇒ (∀a,b:{a:ℝ| a ∈ I} . (a_∫-b u[t] * v'[t] dt = ((u[b] * v[b]) - u[a] * v[a] - a_∫-b u'[t] * v[t] dt))))
Proof
Definitions occuring in Statement :
integral: a_∫-b f[x] dx,
derivative: d(f[x])/dx = λz.g[z] on I,
rfun: I ⟶ℝ,
i-member: r ∈ I,
iproper: iproper(I),
interval: Interval,
rsub: x - y,
req: x = y,
rmul: a * b,
real: ℝ,
so_apply: x[s],
all: ∀x:A. B[x],
implies: P ⇒ Q,
set: {x:A| B[x]} ,
apply: f a
Definitions unfolded in proof :
all: ∀x:A. B[x],
member: t ∈ T,
implies: P ⇒ Q,
prop: ℙ,
uall: ∀[x:A]. B[x],
so_lambda: λ2x.t[x],
so_apply: x[s],
label: ...$L... t,
rfun: I ⟶ℝ,
sq_stable: SqStable(P),
squash: ↓T,
subtype_rel: A ⊆r B,
uimplies: b supposing a,
top: Top,
guard: {T},
subinterval: I ⊆ J ,
ifun: ifun(f;I),
real-fun: real-fun(f;a;b),
and: P ∧ Q,
cand: A c∧ B,
iff: P ⇐⇒ Q,
or: P ∨ Q,
uiff: uiff(P;Q),
req_int_terms: t1 ≡ t2,
false: False,
not: ¬A
Lemmas referenced :
integration-by-parts,
set_wf,
real_wf,
i-member_wf,
iproper_wf,
derivative_wf,
rfun_wf,
all_wf,
req_wf,
interval_wf,
rmin-rmax-subinterval,
sq_stable__i-member,
rmul_wf,
subtype_rel_sets,
rccint_wf,
rmin_wf,
rmax_wf,
member_rccint_lemma,
left_endpoint_rccint_lemma,
right_endpoint_rccint_lemma,
rleq_wf,
rmul_functionality,
sq_stable__req,
ifun_wf,
rccint-icompact,
rmin-rleq-rmax,
integral_wf,
derivative-of-integral,
ftc-integral,
rsub_wf,
rmin-rleq,
rleq-rmax,
ifun_subtype_3,
rmin_ub,
rmin_lb,
rleq_weakening_equal,
rmax_lb,
rmax_ub,
int-to-real_wf,
req-implies-req,
itermSubtract_wf,
itermMultiply_wf,
itermVar_wf,
itermConstant_wf,
req-iff-rsub-is-0,
req_functionality,
req_weakening,
rsub_functionality,
integral-same-endpoints,
real_polynomial_null,
real_term_value_sub_lemma,
real_term_value_mul_lemma,
real_term_value_var_lemma,
real_term_value_const_lemma
Rules used in proof :
cut,
introduction,
extract_by_obid,
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation,
hypothesis,
sqequalHypSubstitution,
dependent_functionElimination,
thin,
hypothesisEquality,
setElimination,
rename,
isectElimination,
sqequalRule,
lambdaEquality,
applyEquality,
setEquality,
because_Cache,
functionEquality,
independent_functionElimination,
imageMemberEquality,
baseClosed,
imageElimination,
dependent_set_memberEquality,
independent_isectElimination,
isect_memberEquality,
voidElimination,
voidEquality,
productEquality,
productElimination,
independent_pairFormation,
equalityTransitivity,
equalitySymmetry,
inlFormation,
natural_numberEquality,
approximateComputation,
int_eqEquality,
intEquality
Latex:
\mforall{}I:Interval. \mforall{}u,v,u',v':\{h:I {}\mrightarrow{}\mBbbR{}| \mforall{}x,y:\{t:\mBbbR{}| t \mmember{} I\} . ((x = y) {}\mRightarrow{} ((h x) = (h y)))\} .
(d(u[t])/dt = \mlambda{}t.u'[t] on I
{}\mRightarrow{} d(v[t])/dt = \mlambda{}t.v'[t] on I
{}\mRightarrow{} iproper(I)
{}\mRightarrow{} (\mforall{}a,b:\{a:\mBbbR{}| a \mmember{} I\} .
(a\_\mint{}\msupminus{}b u[t] * v'[t] dt = ((u[b] * v[b]) - u[a] * v[a] - a\_\mint{}\msupminus{}b u'[t] * v[t] dt))))
Date html generated:
2019_10_31-AM-06_17_11
Last ObjectModification:
2018_08_27-PM-00_08_13
Theory : reals_2
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