Nuprl Lemma : arctangent-reduction
∀B:{B:ℝ| r0 < B} . ∀x:{x:ℝ| (r(-1)/B) < x} . (arctangent(x) = (arctangent(B) + arctangent((x - B/r1 + (x * B)))))
Proof
Definitions occuring in Statement :
arctangent: arctangent(x),
rdiv: (x/y),
rless: x < y,
rsub: x - y,
req: x = y,
rmul: a * b,
radd: a + b,
int-to-real: r(n),
real: ℝ,
all: ∀x:A. B[x],
set: {x:A| B[x]} ,
minus: -n,
natural_number: $n
Definitions unfolded in proof :
all: ∀x:A. B[x],
member: t ∈ T,
uall: ∀[x:A]. B[x],
uimplies: b supposing a,
rneq: x ≠ y,
guard: {T},
or: P ∨ Q,
sq_stable: SqStable(P),
implies: P ⇒ Q,
squash: ↓T,
prop: ℙ,
iff: P ⇐⇒ Q,
and: P ∧ Q,
uiff: uiff(P;Q),
nat: ℕ,
decidable: Dec(P),
not: ¬A,
satisfiable_int_formula: satisfiable_int_formula(fmla),
exists: ∃x:A. B[x],
top: Top,
false: False,
rev_implies: P ⇐ Q,
less_than: a < b,
less_than': less_than'(a;b),
true: True,
rdiv: (x/y),
req_int_terms: t1 ≡ t2,
rge: x ≥ y,
so_lambda: λ2x.t[x],
rfun: I ⟶ℝ,
so_apply: x[s],
rfun-eq: rfun-eq(I;f;g),
r-ap: f(x),
rev_uimplies: rev_uimplies(P;Q),
subtype_rel: A ⊆r B,
rless: x < y,
sq_exists: ∃x:A [B[x]],
i-member: r ∈ I,
roiint: (l, ∞),
concave-on: concave-on(I;x.f[x]),
cand: A c∧ B,
nat_plus: ℕ+,
rat_term_to_real: rat_term_to_real(f;t),
rtermDivide: num "/" denom,
rat_term_ind: rat_term_ind,
rtermConstant: "const",
rtermVar: rtermVar(var),
pi1: fst(t),
rtermMultiply: left "*" right,
pi2: snd(t)
Lemmas referenced :
real_wf,
rless_wf,
rdiv_wf,
int-to-real_wf,
sq_stable__rless,
radd_wf,
rmul_wf,
rmul_preserves_rless,
rnexp2-nonneg,
derivative-arctangent,
rinv_wf2,
rminus_wf,
itermSubtract_wf,
itermMultiply_wf,
itermConstant_wf,
itermVar_wf,
itermMinus_wf,
rless-implies-rless,
rsub_wf,
itermAdd_wf,
req-iff-rsub-is-0,
rnexp_wf,
decidable__le,
full-omega-unsat,
intformnot_wf,
intformle_wf,
istype-int,
int_formula_prop_not_lemma,
istype-void,
int_formula_prop_le_lemma,
int_term_value_constant_lemma,
int_formula_prop_wf,
istype-le,
trivial-rless-radd,
rless-int,
rless_functionality,
req_transitivity,
rminus_functionality,
rmul-rinv,
real_polynomial_null,
real_term_value_sub_lemma,
real_term_value_mul_lemma,
real_term_value_const_lemma,
real_term_value_var_lemma,
real_term_value_minus_lemma,
real_term_value_add_lemma,
rless_functionality_wrt_implies,
rleq_weakening_equal,
radd_functionality_wrt_rleq,
antiderivatives-equal,
roiint_wf,
iproper-roiint,
i-member_wf,
arctangent_wf,
member_roiint_lemma,
riiint_wf,
req_weakening,
subinterval-riiint,
derivative_functionality2,
rnexp-positive,
arctangent-chain-rule,
req_functionality,
rdiv_functionality,
rnexp_functionality,
radd_functionality,
rmul_functionality,
req_wf,
derivative-rdiv,
istype-top,
subtype_rel_dep_function,
top_wf,
concave-positive-nonzero-on,
rleq_weakening,
rccint_wf,
member_riiint_lemma,
true_wf,
istype-true,
derivative-sub,
derivative-id,
derivative-const,
derivative-add,
derivative-const-mul2,
rmul-is-positive,
nat_plus_properties,
sq_stable__rneq,
rmul_comm,
derivative_functionality,
rnexp2,
derivative-add-const,
rdiv-rdiv,
rmul-rinv3,
squash_wf,
subtype_rel_self,
iff_weakening_equal,
req_inversion,
rless_transitivity1,
assert-rat-term-eq2,
rtermDivide_wf,
rtermVar_wf,
rtermMultiply_wf,
rtermConstant_wf,
square-nonneg,
rmul_preserves_req,
uiff_transitivity,
arctangent_functionality,
arctangent0
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation_alt,
setIsType,
universeIsType,
cut,
introduction,
extract_by_obid,
hypothesis,
sqequalHypSubstitution,
isectElimination,
thin,
closedConclusion,
minusEquality,
natural_numberEquality,
setElimination,
rename,
because_Cache,
independent_isectElimination,
sqequalRule,
inrFormation_alt,
dependent_functionElimination,
hypothesisEquality,
independent_functionElimination,
imageMemberEquality,
baseClosed,
imageElimination,
productElimination,
inhabitedIsType,
dependent_set_memberEquality_alt,
unionElimination,
approximateComputation,
dependent_pairFormation_alt,
lambdaEquality_alt,
isect_memberEquality_alt,
voidElimination,
independent_pairFormation,
int_eqEquality,
equalityTransitivity,
equalitySymmetry,
applyEquality,
setEquality,
inlFormation_alt,
productIsType,
equalityIstype,
instantiate,
universeEquality
Latex:
\mforall{}B:\{B:\mBbbR{}| r0 < B\} . \mforall{}x:\{x:\mBbbR{}| (r(-1)/B) < x\} .
(arctangent(x) = (arctangent(B) + arctangent((x - B/r1 + (x * B)))))
Date html generated:
2019_10_31-AM-06_08_00
Last ObjectModification:
2019_04_03-AM-00_28_15
Theory : reals_2
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