Nuprl Lemma : Machin-lemma
(π/r(4)) = ((r(4) * arctangent((r1/r(5)))) - 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 :
uall: ∀[x:A]. B[x],
member: t ∈ T,
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,
nat_plus: ℕ+,
decidable: Dec(P),
not: ¬A,
satisfiable_int_formula: satisfiable_int_formula(fmla),
exists: ∃x:A. B[x],
top: Top,
false: False,
uiff: uiff(P;Q),
le: A ≤ B,
cand: A c∧ B,
i-member: r ∈ I,
rooint: (l, u),
rdiv: (x/y),
req_int_terms: t1 ≡ t2,
rminus: -(x),
halfpi: π/2,
divide: n ÷ m,
cubic_converge: cubic_converge(b;m),
ifthenelse: if b then t else f fi ,
le_int: i ≤z j,
bnot: ¬bb,
lt_int: i <z j,
bfalse: ff,
btrue: tt,
fastpi: fastpi(n),
primrec: primrec(n;b;c),
primtailrec: primtailrec(n;i;b;f),
int-to-real: r(n),
rless: x < y,
sq_exists: ∃x:A [B[x]],
subtype_rel: A ⊆r B,
real: ℝ,
rmul: a * b,
rinv: rinv(x),
mu-ge: mu-ge(f;n),
absval: |i|,
eq_int: (i =z j),
accelerate: accelerate(k;f),
imax: imax(a;b),
reg-seq-inv: reg-seq-inv(x),
reg-seq-mul: reg-seq-mul(x;y),
rge: x ≥ y,
rgt: x > y,
nat: ℕ,
rev_uimplies: rev_uimplies(P;Q),
exp: i^n,
subtract: n - m,
so_lambda: λ2x.t[x],
so_apply: x[s],
sq_type: SQType(T),
int_nzero: ℤ-o,
nequal: a ≠ b ∈ T ,
rat_term_to_real: rat_term_to_real(f;t),
rtermDivide: num "/" denom,
rat_term_ind: rat_term_ind,
rtermConstant: "const",
pi1: fst(t),
rtermMultiply: left "*" right,
pi2: snd(t),
radd: a + b,
reg-seq-list-add: reg-seq-list-add(L),
cbv_list_accum: cbv_list_accum(x,a.f[x; a];y;L),
cons: [a / b],
pi: π,
int-rmul: k1 * a,
nil: [],
it: ⋅,
sq_stable: SqStable(P)
Lemmas referenced :
rtan-arctangent,
rdiv_wf,
int-to-real_wf,
rless-int,
rless_wf,
arctangent-rleq,
rleq-int-fractions2,
decidable__lt,
full-omega-unsat,
intformnot_wf,
intformless_wf,
itermConstant_wf,
istype-int,
int_formula_prop_not_lemma,
istype-void,
int_formula_prop_less_lemma,
int_term_value_constant_lemma,
int_formula_prop_wf,
istype-less_than,
istype-false,
arctangent_functionality_wrt_rless,
rless-int-fractions2,
arctangent_wf,
member_rooint_lemma,
arctangent-bounds,
rminus_wf,
halfpi_wf,
rleq_wf,
req_wf,
rtan_wf,
rless_functionality,
arctangent0,
req_weakening,
rmul_preserves_rleq2,
rleq-int,
rmul_preserves_rless,
rmul_wf,
itermSubtract_wf,
itermMultiply_wf,
itermVar_wf,
rinv_wf2,
rleq_functionality,
req-iff-rsub-is-0,
real_polynomial_null,
real_term_value_sub_lemma,
real_term_value_mul_lemma,
real_term_value_var_lemma,
real_term_value_const_lemma,
rtan-double,
nat_plus_properties,
rless_functionality_wrt_implies,
rleq_weakening_equal,
rleq_weakening_rless,
rsub_wf,
rnexp_wf,
decidable__le,
intformle_wf,
int_formula_prop_le_lemma,
istype-le,
req_functionality,
rsub_functionality,
rnexp_functionality,
exp_wf2,
req_inversion,
rnexp-rdiv,
rneq_functionality,
rnexp-int,
rdiv_functionality,
subtype_base_sq,
nat_plus_wf,
set_subtype_base,
less_than_wf,
int_subtype_base,
exp-one,
rmul_preserves_req,
radd_wf,
itermAdd_wf,
minus-one-mul-top,
nequal_wf,
decidable__equal_int,
intformeq_wf,
int_formula_prop_eq_lemma,
int_term_value_mul_lemma,
req_transitivity,
radd_functionality,
int-rinv-cancel2,
rmul-int,
real_term_value_add_lemma,
i-member_wf,
rooint_wf,
rneq-int,
assert-rat-term-eq2,
rtermDivide_wf,
rtermMultiply_wf,
rtermConstant_wf,
rmul_functionality,
rless_transitivity1,
rleq_weakening,
rless_transitivity2,
rless-implies-rless,
itermMinus_wf,
subtype_rel_self,
real_wf,
real_term_value_minus_lemma,
rtan_functionality,
pi_wf,
radd-preserves-rless,
rinv-mul-as-rdiv,
sq_stable__i-member,
rtan-pi-over-4,
rtan-rsub,
rmul-rinv,
req-int-fractions,
rdiv-int-fractions,
arctangent-rtan,
arctangent_functionality,
req-implies-req
Rules used in proof :
cut,
introduction,
extract_by_obid,
sqequalHypSubstitution,
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isectElimination,
thin,
closedConclusion,
natural_numberEquality,
hypothesis,
independent_isectElimination,
sqequalRule,
inrFormation_alt,
dependent_functionElimination,
because_Cache,
productElimination,
independent_functionElimination,
independent_pairFormation,
imageMemberEquality,
hypothesisEquality,
baseClosed,
universeIsType,
dependent_set_memberEquality_alt,
unionElimination,
approximateComputation,
dependent_pairFormation_alt,
lambdaEquality_alt,
isect_memberEquality_alt,
voidElimination,
lambdaFormation_alt,
productIsType,
inhabitedIsType,
setElimination,
rename,
equalityIstype,
equalityTransitivity,
equalitySymmetry,
promote_hyp,
int_eqEquality,
minusEquality,
addEquality,
applyEquality,
instantiate,
cumulativity,
intEquality,
sqequalBase,
setEquality,
applyLambdaEquality,
imageElimination,
multiplyEquality
Latex:
(\mpi{}/r(4)) = ((r(4) * arctangent((r1/r(5)))) - arctangent((r1/r(239))))
Date html generated:
2019_10_31-AM-06_04_58
Last ObjectModification:
2019_04_03-AM-00_28_48
Theory : reals_2
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