Nuprl Lemma : cos-sin-equation
∀f,g:ℝ ⟶ ℝ.
(((∀x,y:ℝ. ((x = y) ⇒ (f(x) = f(y)))) ∧ (∀x,y:ℝ. ((x = y) ⇒ (g(x) = g(y)))))
∧ (∀x,y:ℝ. (f(x - y) = ((f(x) * f(y)) + (g(x) * g(y)))))
⇐⇒ ¬¬((∃c:ℝ
((r0 ≤ (c - c^2))
∧ (∀x:ℝ. (f(x) = c))
∧ ((∀x:ℝ. (g(x) = rsqrt(c - c^2))) ∨ (∀x:ℝ. (g(x) = -(rsqrt(c - c^2)))))))
∨ (∃a:ℝ. (a ≠ r0 ∧ (∀x:ℝ. (f(x) = rcos(a * x))) ∧ (∀x:ℝ. (g(x) = rsin(a * x)))))))
Proof
Definitions occuring in Statement :
rfun-ap: f(x),
rcos: rcos(x),
rsin: rsin(x),
rsqrt: rsqrt(x),
rneq: x ≠ y,
rleq: x ≤ y,
rnexp: x^k1,
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,
not: ¬A,
implies: P ⇒ Q,
or: P ∨ Q,
and: P ∧ Q,
function: x:A ⟶ B[x],
natural_number: $n
Definitions unfolded in proof :
all: ∀x:A. B[x],
iff: P ⇐⇒ Q,
and: P ∧ Q,
implies: P ⇒ Q,
member: t ∈ T,
uall: ∀[x:A]. B[x],
prop: ℙ,
rev_implies: P ⇐ Q,
not: ¬A,
or: P ∨ Q,
exists: ∃x:A. B[x],
nat: ℕ,
le: A ≤ B,
less_than': less_than'(a;b),
false: False,
subtype_rel: A ⊆r B,
cand: A c∧ B,
stable: Stable{P},
uimplies: b supposing a,
uiff: uiff(P;Q),
req_int_terms: t1 ≡ t2,
top: Top,
rev_uimplies: rev_uimplies(P;Q),
guard: {T},
rdiv: (x/y),
so_lambda: λ2x.t[x],
rneq: x ≠ y,
less_than: a < b,
squash: ↓T,
true: True,
so_apply: x[s],
rat_term_to_real: rat_term_to_real(f;t),
rtermVar: rtermVar(var),
rat_term_ind: rat_term_ind,
pi1: fst(t),
rtermDivide: num "/" denom,
rtermMinus: rtermMinus(num),
rtermConstant: "const",
pi2: snd(t),
int_nzero: ℤ-o,
nequal: a ≠ b ∈ T ,
sq_type: SQType(T),
decidable: Dec(P),
satisfiable_int_formula: satisfiable_int_formula(fmla),
rge: x ≥ y,
rfun-ap: f(x)
Lemmas referenced :
real_wf,
req_wf,
rfun-ap_wf,
rsub_wf,
radd_wf,
rmul_wf,
rleq_wf,
int-to-real_wf,
rnexp_wf,
istype-void,
istype-le,
rsqrt_wf,
rminus_wf,
rneq_wf,
rcos_wf,
rsin_wf,
stable__not,
not_wf,
false_wf,
minimal-double-negation-hyp-elim,
minimal-not-not-excluded-middle,
cos-sin-equation-nc,
DAlembert-equation-iff,
rneq_irreflexivity,
rneq_functionality,
req_weakening,
double-negation-hyp-elim,
cosh_wf,
not-rneq,
itermSubtract_wf,
itermMultiply_wf,
itermConstant_wf,
itermVar_wf,
iff_weakening_uiff,
req_functionality,
rcos_functionality,
rmul_functionality,
req-iff-rsub-is-0,
real_polynomial_null,
istype-int,
real_term_value_sub_lemma,
real_term_value_mul_lemma,
real_term_value_const_lemma,
real_term_value_var_lemma,
rcos0,
cosh_functionality,
cosh0,
rneq-cases,
rdiv_wf,
rmul-rsub-distrib,
radd-preserves-req,
itermAdd_wf,
itermMinus_wf,
rmul_preserves_req,
rinv_wf2,
req_inversion,
radd_functionality,
rcos-rsub,
real_term_value_add_lemma,
real_term_value_minus_lemma,
req_transitivity,
rmul-rinv3,
rmul-identity1,
rneq-by-function,
rless-int,
rless_wf,
assert-rat-term-eq2,
rtermDivide_wf,
rtermMinus_wf,
rtermVar_wf,
rtermConstant_wf,
req-int-fractions2,
subtype_base_sq,
int_subtype_base,
nequal_wf,
decidable__equal_int,
full-omega-unsat,
intformnot_wf,
intformeq_wf,
int_formula_prop_not_lemma,
int_formula_prop_eq_lemma,
int_term_value_constant_lemma,
int_term_value_mul_lemma,
int_formula_prop_wf,
rdiv_functionality,
rmul_over_rminus,
rcos-rminus,
rsin_functionality,
rsin-rminus,
rnexp_functionality,
rnexp-rmul,
rsin-rcos-pythag,
rmul_preserves_rneq_iff2,
rmul_comm,
rnexp2,
square-is-one,
squash_wf,
true_wf,
rminus-int,
subtype_rel_self,
iff_weakening_equal,
rabs_wf,
rleq_functionality,
rabs_functionality,
rless_transitivity1,
rless_irreflexivity,
rless_functionality,
rabs-of-nonneg,
rleq-int,
rleq_functionality_wrt_implies,
rleq_weakening_equal,
cosh-ge-1,
cosh-gt-1,
rfun-ap_functionality,
req-implies-req,
square-nonneg,
rsub_functionality,
stable_req,
rmul-zero,
square-is-zero,
rsqrt-unique2,
stable__and,
stable__all,
rsqrt_squared
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation_alt,
independent_pairFormation,
sqequalRule,
productIsType,
functionIsType,
universeIsType,
cut,
introduction,
extract_by_obid,
hypothesis,
because_Cache,
sqequalHypSubstitution,
isectElimination,
thin,
hypothesisEquality,
unionIsType,
natural_numberEquality,
dependent_set_memberEquality_alt,
voidElimination,
applyEquality,
lambdaEquality_alt,
setElimination,
rename,
inhabitedIsType,
equalityTransitivity,
equalitySymmetry,
productElimination,
unionEquality,
productEquality,
functionEquality,
independent_isectElimination,
independent_functionElimination,
unionElimination,
dependent_functionElimination,
approximateComputation,
int_eqEquality,
isect_memberEquality_alt,
dependent_pairFormation_alt,
minusEquality,
inrFormation_alt,
closedConclusion,
inlFormation_alt,
imageMemberEquality,
baseClosed,
instantiate,
cumulativity,
intEquality,
equalityIstype,
sqequalBase,
imageElimination,
universeEquality,
lambdaFormation,
promote_hyp
Latex:
\mforall{}f,g:\mBbbR{} {}\mrightarrow{} \mBbbR{}.
(((\mforall{}x,y:\mBbbR{}. ((x = y) {}\mRightarrow{} (f(x) = f(y)))) \mwedge{} (\mforall{}x,y:\mBbbR{}. ((x = y) {}\mRightarrow{} (g(x) = g(y)))))
\mwedge{} (\mforall{}x,y:\mBbbR{}. (f(x - y) = ((f(x) * f(y)) + (g(x) * g(y)))))
\mLeftarrow{}{}\mRightarrow{} \mneg{}\mneg{}((\mexists{}c:\mBbbR{}
((r0 \mleq{} (c - c\^{}2))
\mwedge{} (\mforall{}x:\mBbbR{}. (f(x) = c))
\mwedge{} ((\mforall{}x:\mBbbR{}. (g(x) = rsqrt(c - c\^{}2))) \mvee{} (\mforall{}x:\mBbbR{}. (g(x) = -(rsqrt(c - c\^{}2)))))))
\mvee{} (\mexists{}a:\mBbbR{}. (a \mneq{} r0 \mwedge{} (\mforall{}x:\mBbbR{}. (f(x) = rcos(a * x))) \mwedge{} (\mforall{}x:\mBbbR{}. (g(x) = rsin(a * x)))))))
Date html generated:
2019_10_31-AM-06_24_42
Last ObjectModification:
2019_04_10-AM-11_20_38
Theory : reals_2
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