Nuprl Lemma : derivative-radd_rcos
d(radd_rcos(x))/dx = λx.r1 - rsin(x) on (-∞, ∞)
Proof
Definitions occuring in Statement :
radd_rcos: radd_rcos(x),
rsin: rsin(x),
derivative: d(f[x])/dx = λz.g[z] on I,
riiint: (-∞, ∞),
rsub: x - y,
int-to-real: r(n),
natural_number: $n
Definitions unfolded in proof :
member: t ∈ T,
rfun: I ⟶ℝ,
uall: ∀[x:A]. B[x],
prop: ℙ,
subtype_rel: A ⊆r B,
so_lambda: λ2x.t[x],
so_apply: x[s],
all: ∀x:A. B[x],
itermConstant: "const",
req_int_terms: t1 ≡ t2,
false: False,
implies: P ⇒ Q,
not: ¬A,
top: Top,
uiff: uiff(P;Q),
and: P ∧ Q,
uimplies: b supposing a,
rfun-eq: rfun-eq(I;f;g),
r-ap: f(x),
squash: ↓T,
guard: {T},
sq_stable: SqStable(P)
Lemmas referenced :
riiint_wf,
real_wf,
i-member_wf,
rcos_wf,
int-to-real_wf,
rminus_wf,
rsin_wf,
radd_wf,
radd_rcos_wf,
req_wf,
rsub_wf,
set_wf,
real_term_polynomial,
itermSubtract_wf,
itermAdd_wf,
itermConstant_wf,
itermMinus_wf,
itermVar_wf,
real_term_value_const_lemma,
real_term_value_sub_lemma,
real_term_value_add_lemma,
real_term_value_minus_lemma,
real_term_value_var_lemma,
req-iff-rsub-is-0,
derivative-add,
derivative-id,
deriviative-rcos,
derivative_functionality,
equal_wf,
req_inversion,
sq_stable__req
Rules used in proof :
cut,
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
introduction,
extract_by_obid,
hypothesis,
sqequalRule,
lambdaEquality,
setElimination,
thin,
rename,
hypothesisEquality,
sqequalHypSubstitution,
setEquality,
isectElimination,
because_Cache,
natural_numberEquality,
applyEquality,
dependent_functionElimination,
computeAll,
int_eqEquality,
intEquality,
isect_memberEquality,
voidElimination,
voidEquality,
productElimination,
independent_isectElimination,
independent_functionElimination,
lambdaFormation,
equalitySymmetry,
equalityTransitivity,
imageElimination,
baseClosed,
imageMemberEquality
Latex:
d(radd\_rcos(x))/dx = \mlambda{}x.r1 - rsin(x) on (-\minfty{}, \minfty{})
Date html generated:
2017_10_04-PM-10_22_29
Last ObjectModification:
2017_07_28-AM-08_48_31
Theory : reals_2
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