Nuprl Lemma : DAlembert-equation-iff
∀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))))
⇐⇒ (∀x:ℝ. (f(x) = r0)) ∨ (¬¬((∃c:ℝ. ∀x:ℝ. (f(x) = rcos(c * x))) ∨ (∃c:ℝ. ∀x:ℝ. (f(x) = cosh(c * x))))))
Proof
Definitions occuring in Statement :
rfun-ap: f(x),
cosh: cosh(x),
rcos: rcos(x),
rsub: x - y,
req: x = y,
rmul: a * b,
radd: a + b,
int-to-real: r(n),
real: ℝ,
all: ∀x:A. B[x],
exists: ∃x:A. B[x],
iff: P ⇐⇒ Q,
not: ¬A,
implies: P ⇒ Q,
or: P ∨ Q,
and: P ∧ Q,
function: x:A ⟶ B[x],
natural_number: $n
Definitions unfolded in proof :
top: Top,
not: ¬A,
false: False,
req_int_terms: t1 ≡ t2,
so_apply: x[s],
so_lambda: λ2x.t[x],
prop: ℙ,
rev_uimplies: rev_uimplies(P;Q),
uiff: uiff(P;Q),
rfun-ap: f(x),
exists: ∃x:A. B[x],
or: P ∨ Q,
uimplies: b supposing a,
member: t ∈ T,
uall: ∀[x:A]. B[x],
cand: A c∧ B,
and: P ∧ Q,
implies: P ⇒ Q,
all: ∀x:A. B[x],
rdiv: (x/y),
true: True,
less_than': less_than'(a;b),
squash: ↓T,
less_than: a < b,
rev_implies: P ⇐ Q,
iff: P ⇐⇒ Q,
guard: {T},
rneq: x ≠ y,
rfun: I ⟶ℝ,
subtype_rel: A ⊆r B,
sq_stable: SqStable(P),
sq_exists: ∃x:{A| B[x]},
le: A ≤ B,
riiint: (-∞, ∞),
i-approx: i-approx(I;n),
nat_plus: ℕ+,
continuous: f[x] continuous for x ∈ I,
rge: x ≥ y,
stable: Stable{P},
rgt: x > y,
rooint: (l, u),
i-member: r ∈ I,
nat: ℕ
Lemmas referenced :
cosh-rminus,
sinh-rminus,
cosh-radd,
rsin-rminus,
rcos-rminus,
rsub_functionality,
rcos-radd,
uiff_transitivity,
real_term_value_const_lemma,
real_term_value_minus_lemma,
real_term_value_add_lemma,
real_term_value_var_lemma,
real_term_value_mul_lemma,
real_term_value_sub_lemma,
real_polynomial_null,
radd_functionality,
cosh0,
rcos0,
rmul-zero,
sinh_wf,
itermConstant_wf,
rsin_wf,
req-iff-rsub-is-0,
itermMinus_wf,
itermAdd_wf,
itermVar_wf,
itermMultiply_wf,
itermSubtract_wf,
rminus_wf,
rmul-distrib1,
int-to-real_wf,
rsub_wf,
radd_wf,
all_wf,
exists_wf,
or_wf,
req_wf,
cosh_functionality,
cosh_wf,
req_weakening,
rmul_functionality,
rcos_functionality,
rmul_wf,
rcos_wf,
rfun-ap_functionality,
real_wf,
rfun-ap_wf,
req_functionality,
req_transitivity,
req-implies-req,
radd_comm,
radd-zero-both,
rsub-int,
radd-int,
req_inversion,
subtract_wf,
rmul-rinv,
rmul-rinv3,
rinv-mul-as-rdiv,
rmul_comm,
rinv_wf2,
rless_wf,
rless-int,
rdiv_wf,
rmul_preserves_req,
not_wf,
radd-zero,
square-req-self-iff,
i-member_wf,
set_wf,
subtype_rel_self,
true_wf,
subtype_rel_dep_function,
member_riiint_lemma,
riiint_wf,
function-is-continuous,
rccint_wf,
rmin_strict_ub,
sq_stable__rless,
rmin_wf,
member_rccint_lemma,
i-approx_wf,
icompact_wf,
false_wf,
rleq-int,
rccint-icompact,
less_than_wf,
rleq_weakening_equal,
rleq_functionality_wrt_implies,
rmin-rleq,
rleq_functionality,
rmul_reverses_rleq_iff,
rabs_functionality,
rabs_wf,
rabs-rleq-iff,
rless_functionality_wrt_implies,
rabs-difference-bound-rleq,
rinv-as-rdiv,
rless_functionality,
radd-preserves-rless,
rless-int-fractions3,
minimal-not-not-excluded-middle,
minimal-double-negation-hyp-elim,
rleq_weakening_rless,
rminus_functionality_wrt_rleq,
stable__not,
rleq_weakening,
rleq_wf,
cosh-inv-cosh,
inv-cosh_wf,
DAlembert-equation-lemma,
cosh-ge-1,
halfpi_wf,
rcoint_wf,
not-rless,
rooint_wf,
rcos-positive,
rmul-zero-both,
rmul_preserves_rleq2,
rabs-of-nonneg,
rmul_preserves_rleq,
member_rcoint_lemma,
rabs-rmul,
rless_transitivity2,
rless-implies-rless,
member_rooint_lemma,
rabs-rless-iff,
equal_wf,
stable__false,
trivial-rsub-rless,
rmul_preserves_rless,
arcsine-root-bounds,
rsqrt1,
rless_transitivity1,
rsqrt_wf,
rsqrt_functionality_wrt_rless,
rsqrt-positive,
arcsine-bounds,
rsqrt_nonneg,
arcsine-nonneg,
arcsine_wf,
rcos-nonneg-upto-half-pi,
square-req-iff,
rsin-rcos-pythag,
rsin-arcsine,
rnexp_functionality,
le_wf,
rnexp_wf,
rsqrt_squared,
rnexp2,
rleq_antisymmetry,
halfpi-positive,
stable_req
Rules used in proof :
independent_functionElimination,
voidEquality,
voidElimination,
isect_memberEquality,
intEquality,
int_eqEquality,
approximateComputation,
natural_numberEquality,
functionEquality,
lambdaEquality,
sqequalRule,
independent_pairFormation,
dependent_functionElimination,
productElimination,
unionElimination,
independent_isectElimination,
because_Cache,
hypothesis,
hypothesisEquality,
applyEquality,
functionExtensionality,
thin,
isectElimination,
sqequalHypSubstitution,
extract_by_obid,
introduction,
lambdaFormation,
sqequalReflexivity,
computationStep,
sqequalTransitivity,
sqequalSubstitution,
cut,
productEquality,
addEquality,
baseClosed,
imageMemberEquality,
inrFormation,
inlFormation,
rename,
setElimination,
setEquality,
imageElimination,
dependent_pairFormation,
minusEquality,
dependent_set_memberEquality,
equalitySymmetry,
equalityTransitivity,
multiplyEquality
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))))
\mLeftarrow{}{}\mRightarrow{} (\mforall{}x:\mBbbR{}. (f(x) = r0))
\mvee{} (\mneg{}\mneg{}((\mexists{}c:\mBbbR{}. \mforall{}x:\mBbbR{}. (f(x) = rcos(c * x))) \mvee{} (\mexists{}c:\mBbbR{}. \mforall{}x:\mBbbR{}. (f(x) = cosh(c * x))))))
Date html generated:
2017_10_04-PM-11_04_24
Last ObjectModification:
2017_08_01-PM-10_09_49
Theory : reals_2
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