Nuprl Lemma : derivative-arctangent
d(arctangent(x))/dx = λx.(r1/r1 + x^2) on (-∞, ∞)
Proof
Definitions occuring in Statement :
arctangent: arctangent(x),
derivative: d(f[x])/dx = λz.g[z] on I,
riiint: (-∞, ∞),
rdiv: (x/y),
rnexp: x^k1,
radd: a + b,
int-to-real: r(n),
natural_number: $n
Definitions unfolded in proof :
all: ∀x:A. B[x],
member: t ∈ T,
arctangent: arctangent(x),
top: Top,
uall: ∀[x:A]. B[x],
true: True,
prop: ℙ,
so_lambda: λ2x.t[x],
so_apply: x[s],
nat: ℕ,
le: A ≤ B,
and: P ∧ Q,
less_than': less_than'(a;b),
false: False,
not: ¬A,
implies: P ⇒ Q,
uiff: uiff(P;Q),
uimplies: b supposing a,
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q,
less_than: a < b,
squash: ↓T,
rge: x ≥ y,
guard: {T},
rfun: I ⟶ℝ,
rneq: x ≠ y,
or: P ∨ Q,
rev_uimplies: rev_uimplies(P;Q)
Lemmas referenced :
rnexp2-nonneg,
real_wf,
derivative-of-integral,
riiint_wf,
member_riiint_lemma,
int-to-real_wf,
true_wf,
radd_wf,
rnexp_wf,
false_wf,
le_wf,
trivial-rless-radd,
rless-int,
rless_functionality_wrt_implies,
rleq_weakening_equal,
radd_functionality_wrt_rleq,
rdiv_wf,
rless_wf,
i-member_wf,
req_functionality,
rdiv_functionality,
req_weakening,
radd_functionality,
rnexp_functionality,
req_wf,
set_wf,
all_wf
Rules used in proof :
cut,
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation,
introduction,
extract_by_obid,
sqequalHypSubstitution,
dependent_functionElimination,
thin,
hypothesisEquality,
hypothesis,
sqequalRule,
isect_memberEquality,
voidElimination,
voidEquality,
dependent_set_memberEquality,
isectElimination,
natural_numberEquality,
independent_pairFormation,
because_Cache,
productElimination,
independent_isectElimination,
independent_functionElimination,
imageMemberEquality,
baseClosed,
equalityTransitivity,
equalitySymmetry,
lambdaEquality,
setElimination,
rename,
inrFormation,
setEquality,
functionEquality,
applyEquality
Latex:
d(arctangent(x))/dx = \mlambda{}x.(r1/r1 + x\^{}2) on (-\minfty{}, \minfty{})
Date html generated:
2018_05_22-PM-03_01_58
Last ObjectModification:
2017_10_21-PM-11_17_56
Theory : reals_2
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