Nuprl Lemma : arctangent-reduction-1
∀x:{x:ℝ| r(-1) < x} . (arctangent(x) = (MachinPi4() + arctangent((x - r1/r1 + x))))
Proof
Definitions occuring in Statement :
MachinPi4: MachinPi4(),
arctangent: arctangent(x),
rdiv: (x/y),
rless: x < y,
rsub: x - y,
req: x = y,
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,
iff: P ⇐⇒ Q,
and: P ∧ Q,
rev_implies: P ⇐ Q,
implies: P ⇒ Q,
less_than: a < b,
squash: ↓T,
less_than': less_than'(a;b),
true: True,
uall: ∀[x:A]. B[x],
prop: ℙ,
so_lambda: λ2x.t[x],
so_apply: x[s],
uimplies: b supposing a,
rneq: x ≠ y,
guard: {T},
or: P ∨ Q,
uiff: uiff(P;Q),
rdiv: (x/y),
req_int_terms: t1 ≡ t2,
false: False,
not: ¬A,
top: Top,
subtype_rel: A ⊆r B,
sq_stable: SqStable(P),
rev_uimplies: rev_uimplies(P;Q)
Lemmas referenced :
arctangent-reduction,
rless-int,
int-to-real_wf,
rless_wf,
set_wf,
real_wf,
rdiv_wf,
rmul_wf,
rinv_wf2,
rminus_wf,
itermSubtract_wf,
itermMultiply_wf,
itermConstant_wf,
itermVar_wf,
itermMinus_wf,
req-iff-rsub-is-0,
rless_functionality,
req_transitivity,
rminus_functionality,
rinv1,
req_weakening,
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,
sq_stable__req,
arctangent_wf,
radd_wf,
MachinPi4_wf,
rsub_wf,
rless-implies-rless,
itermAdd_wf,
real_term_value_add_lemma,
radd_functionality,
req_functionality,
arctangent_functionality,
rmul_functionality,
rinv_functionality2,
rinv-mul-as-rdiv,
rinv-as-rdiv,
pi_wf,
req_wf,
arctangent1,
equal_wf,
req_inversion
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation,
cut,
introduction,
extract_by_obid,
sqequalHypSubstitution,
dependent_functionElimination,
thin,
natural_numberEquality,
productElimination,
independent_functionElimination,
sqequalRule,
independent_pairFormation,
imageMemberEquality,
hypothesisEquality,
baseClosed,
hypothesis,
dependent_set_memberEquality,
isectElimination,
lambdaEquality,
minusEquality,
setElimination,
rename,
because_Cache,
independent_isectElimination,
inrFormation,
approximateComputation,
int_eqEquality,
intEquality,
isect_memberEquality,
voidElimination,
voidEquality,
applyEquality,
imageElimination,
setEquality,
equalityTransitivity,
equalitySymmetry
Latex:
\mforall{}x:\{x:\mBbbR{}| r(-1) < x\} . (arctangent(x) = (MachinPi4() + arctangent((x - r1/r1 + x))))
Date html generated:
2018_05_22-PM-03_06_28
Last ObjectModification:
2017_10_26-PM-07_48_23
Theory : reals_2
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