Nuprl Lemma : DAlembert-equation-iff3
∀f:ℝ ⟶ ℝ
((∀x,y:ℝ. ((x = y) ⇒ (f(x) = f(y))))
∧ (∀x,y:ℝ. ((f(x + y) + f(x - y)) = (r(2) * f(x) * f(y))))
∧ (∃u:ℝ
((r0 < u)
∧ (∃g,h:(-(u), u) ⟶ℝ
(d(f(x))/dx = λx.g(x) on (-(u), u)
∧ d(g(x))/dx = λx.h(x) on (-(u), u)
∧ ((h(r0) < r0) ∨ (h(r0) = r0) ∨ (r0 < h(r0)))))))
⇐⇒ (∀x:ℝ. (f(x) = r0))
∨ (∀x:ℝ. (f(x) = r1))
∨ (∃c:{c:ℝ| r0 < c} . ∀x:ℝ. (f(x) = rcos(c * x)))
∨ (∃c:{c:ℝ| r0 < c} . ∀x:ℝ. (f(x) = cosh(c * x))))
Proof
Definitions occuring in Statement :
rfun-ap: f(x),
cosh: cosh(x),
rcos: rcos(x),
derivative: d(f[x])/dx = λz.g[z] on I,
rfun: I ⟶ℝ,
rooint: (l, u),
rless: x < y,
rsub: x - y,
req: x = y,
rmul: a * b,
rminus: -(x),
radd: a + b,
int-to-real: r(n),
real: ℝ,
all: ∀x:A. B[x],
exists: ∃x:A. B[x],
iff: P ⇐⇒ Q,
implies: P ⇒ Q,
or: P ∨ Q,
and: P ∧ Q,
set: {x:A| B[x]} ,
function: x:A ⟶ B[x],
natural_number: $n
Definitions unfolded in proof :
sq_stable: SqStable(P),
stable: Stable{P},
rmul: a * b,
true: True,
less_than': less_than'(a;b),
squash: ↓T,
less_than: a < b,
pi2: snd(t),
pi1: fst(t),
outl: outl(x),
rooint: (l, u),
endpoints: endpoints(I),
left-endpoint: left-endpoint(I),
right-endpoint: right-endpoint(I),
iproper: iproper(I),
r-ap: f(x),
rfun-eq: rfun-eq(I;f;g),
rev_uimplies: rev_uimplies(P;Q),
guard: {T},
not: ¬A,
false: False,
req_int_terms: t1 ≡ t2,
uiff: uiff(P;Q),
rev_implies: P ⇐ Q,
exists: ∃x:A. B[x],
or: P ∨ Q,
cand: A c∧ B,
uimplies: b supposing a,
top: Top,
rfun: I ⟶ℝ,
subtype_rel: A ⊆r B,
label: ...$L... t,
so_apply: x[s],
so_lambda: λ2x.t[x],
uall: ∀[x:A]. B[x],
rfun-ap: f(x),
prop: ℙ,
implies: P ⇒ Q,
and: P ∧ Q,
iff: P ⇐⇒ Q,
member: t ∈ T,
all: ∀x:A. B[x]
Lemmas referenced :
rmul-is-positive,
sq_stable__rless,
derivative-const,
cosh-rminus,
uiff_transitivity,
rleq-implies-rleq,
radd-zero,
radd-preserves-req,
rsqrt0,
rminus-zero,
rleq_weakening_equal,
rless_irreflexivity,
rless_transitivity1,
rleq_antisymmetry,
square-is-zero,
rleq_weakening,
rless_transitivity2,
rmul_over_rminus,
rcos-rminus,
not-rless,
rabs-of-nonpos,
rabs-of-nonneg,
rsqrt_square,
rabs_wf,
rsqrt_functionality,
req_inversion,
square-nonneg,
rminus-rminus,
cosh0,
rcos0,
rmul_functionality,
req_transitivity,
minimal-not-not-excluded-middle,
minimal-double-negation-hyp-elim,
rmul-identity1,
rmul-zero,
rleq_wf,
rleq_weakening_rless,
not_wf,
false_wf,
stable_req,
rsqrt_wf,
rsqrt-positive,
rmul-ac,
derivative-sinh,
cosh_functionality,
derivative-cosh,
sinh_functionality,
sinh_wf,
derivative-minus,
derivative-const-mul,
derivative-rsin,
rcos_functionality,
real_term_value_mul_lemma,
real_term_value_add_lemma,
itermMultiply_wf,
itermAdd_wf,
derivative_unique,
i-finite_wf,
rmul_comm,
rmul-zero-both,
rless-int,
rmul_preserves_rless,
radd-rminus,
rless_functionality,
radd-preserves-rless,
derivative_functionality,
subinterval-riiint,
riiint_wf,
derivative_functionality_wrt_subinterval,
derivative-rcos,
req_weakening,
rsin_functionality,
rminus_functionality,
req_functionality,
rsin_wf,
derivative-function-rmul-const,
rfun-ap_wf,
real_term_value_minus_lemma,
real_term_value_const_lemma,
real_term_value_var_lemma,
real_term_value_sub_lemma,
real_polynomial_null,
req-iff-rsub-is-0,
itermMinus_wf,
itermConstant_wf,
itermVar_wf,
itermSubtract_wf,
cosh_wf,
rcos_wf,
rless-implies-rless,
or_wf,
i-member_wf,
set_wf,
subtype_rel_self,
subtype_rel_dep_function,
member_rooint_lemma,
derivative_wf,
rminus_wf,
rooint_wf,
rfun_wf,
rless_wf,
exists_wf,
int-to-real_wf,
rmul_wf,
rsub_wf,
radd_wf,
req_wf,
real_wf,
all_wf,
DAlembert-equation-iff
Rules used in proof :
imageElimination,
orLevelFunctionality,
orFunctionality,
addLevel,
dependent_pairFormation,
baseClosed,
imageMemberEquality,
inrFormation,
inlFormation,
unionElimination,
independent_functionElimination,
intEquality,
int_eqEquality,
approximateComputation,
dependent_set_memberEquality,
rename,
setElimination,
independent_isectElimination,
setEquality,
voidEquality,
voidElimination,
isect_memberEquality,
natural_numberEquality,
functionExtensionality,
applyEquality,
functionEquality,
because_Cache,
lambdaEquality,
isectElimination,
productEquality,
productElimination,
sqequalRule,
independent_pairFormation,
hypothesisEquality,
thin,
dependent_functionElimination,
sqequalHypSubstitution,
hypothesis,
lambdaFormation,
sqequalReflexivity,
computationStep,
sqequalTransitivity,
sqequalSubstitution,
extract_by_obid,
introduction,
cut
Latex:
\mforall{}f:\mBbbR{} {}\mrightarrow{} \mBbbR{}
((\mforall{}x,y:\mBbbR{}. ((x = y) {}\mRightarrow{} (f(x) = f(y))))
\mwedge{} (\mforall{}x,y:\mBbbR{}. ((f(x + y) + f(x - y)) = (r(2) * f(x) * f(y))))
\mwedge{} (\mexists{}u:\mBbbR{}
((r0 < u)
\mwedge{} (\mexists{}g,h:(-(u), u) {}\mrightarrow{}\mBbbR{}
(d(f(x))/dx = \mlambda{}x.g(x) on (-(u), u)
\mwedge{} d(g(x))/dx = \mlambda{}x.h(x) on (-(u), u)
\mwedge{} ((h(r0) < r0) \mvee{} (h(r0) = r0) \mvee{} (r0 < h(r0)))))))
\mLeftarrow{}{}\mRightarrow{} (\mforall{}x:\mBbbR{}. (f(x) = r0))
\mvee{} (\mforall{}x:\mBbbR{}. (f(x) = r1))
\mvee{} (\mexists{}c:\{c:\mBbbR{}| r0 < c\} . \mforall{}x:\mBbbR{}. (f(x) = rcos(c * x)))
\mvee{} (\mexists{}c:\{c:\mBbbR{}| r0 < c\} . \mforall{}x:\mBbbR{}. (f(x) = cosh(c * x))))
Date html generated:
2018_05_22-PM-03_09_42
Last ObjectModification:
2018_05_20-PM-11_53_01
Theory : reals_2
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