Nuprl Lemma : Machin-formula
π = ((r(16) * arctangent((r1/r(5)))) - r(4) * arctangent((r1/r(239))))
Proof
Definitions occuring in Statement :
arctangent: arctangent(x),
pi: π,
rdiv: (x/y),
rsub: x - y,
req: x = y,
rmul: a * b,
int-to-real: r(n),
natural_number: $n
Definitions unfolded in proof :
member: t ∈ T,
uall: ∀[x:A]. B[x],
uimplies: b supposing a,
rneq: x ≠ y,
guard: {T},
or: P ∨ Q,
all: ∀x:A. B[x],
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,
prop: ℙ,
uiff: uiff(P;Q),
rdiv: (x/y),
req_int_terms: t1 ≡ t2,
false: False,
not: ¬A,
top: Top
Lemmas referenced :
Machin-lemma,
arctangent_wf,
rdiv_wf,
int-to-real_wf,
rless-int,
rless_wf,
real_wf,
req_wf,
pi_wf,
rsub_wf,
rmul_wf,
equal_wf,
rmul_preserves_req,
radd_wf,
rinv_wf2,
itermSubtract_wf,
itermMultiply_wf,
itermVar_wf,
itermConstant_wf,
req-iff-rsub-is-0,
itermAdd_wf,
req-implies-req,
req_functionality,
req_transitivity,
rmul-rinv3,
real_polynomial_null,
real_term_value_sub_lemma,
real_term_value_mul_lemma,
real_term_value_var_lemma,
real_term_value_const_lemma,
real_term_value_add_lemma
Rules used in proof :
cut,
introduction,
extract_by_obid,
thin,
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
sqequalHypSubstitution,
isectElimination,
natural_numberEquality,
hypothesis,
independent_isectElimination,
sqequalRule,
inrFormation,
dependent_functionElimination,
because_Cache,
productElimination,
independent_functionElimination,
independent_pairFormation,
imageMemberEquality,
hypothesisEquality,
baseClosed,
lambdaFormation,
equalityTransitivity,
equalitySymmetry,
minusEquality,
approximateComputation,
lambdaEquality,
int_eqEquality,
intEquality,
isect_memberEquality,
voidElimination,
voidEquality
Latex:
\mpi{} = ((r(16) * arctangent((r1/r(5)))) - r(4) * arctangent((r1/r(239))))
Date html generated:
2018_05_22-PM-03_03_51
Last ObjectModification:
2017_10_22-PM-05_56_36
Theory : reals_2
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