Nuprl Lemma : fastpi-property
∀n:ℕ. (|fastpi(n) - π/2(slower)| ≤ (r1/r(10^(20 * 3^n))))
Proof
Definitions occuring in Statement :
fastpi: fastpi(n),
half-pi: π/2(slower),
rdiv: (x/y),
rleq: x ≤ y,
rabs: |x|,
rsub: x - y,
int-to-real: r(n),
exp: i^n,
nat: ℕ,
all: ∀x:A. B[x],
multiply: n * m,
natural_number: $n
Definitions unfolded in proof :
all: ∀x:A. B[x],
uall: ∀[x:A]. B[x],
member: t ∈ T,
nat: ℕ,
implies: P ⇒ Q,
false: False,
ge: i ≥ j ,
uimplies: b supposing a,
not: ¬A,
satisfiable_int_formula: satisfiable_int_formula(fmla),
exists: ∃x:A. B[x],
top: Top,
and: P ∧ Q,
prop: ℙ,
rleq: x ≤ y,
rnonneg: rnonneg(x),
le: A ≤ B,
squash: ↓T,
real: ℝ,
less_than': less_than'(a;b),
fastpi: fastpi(n),
guard: {T},
sq_type: SQType(T),
nequal: a ≠ b ∈ T ,
true: True,
subtype_rel: A ⊆r B,
int-to-real: r(n),
nat_plus: ℕ+,
has-value: (a)↓,
exp: i^n,
int-rdiv: (a)/k1,
rational-approx: (x within 1/n),
subtract: n - m,
rev_implies: P ⇐ Q,
iff: P ⇐⇒ Q,
primrec: primrec(n;b;c),
less_than: a < b,
or: P ∨ Q,
rneq: x ≠ y,
uiff: uiff(P;Q),
rev_uimplies: rev_uimplies(P;Q),
assert: ↑b,
ifthenelse: if b then t else f fi ,
bfalse: ff,
decidable: Dec(P),
so_apply: x[s],
so_lambda: λ2x.t[x],
rge: x ≥ y,
rdiv: (x/y),
req_int_terms: t1 ≡ t2,
rless: x < y,
sq_exists: ∃x:A [B[x]],
primtailrec: primtailrec(n;i;b;f),
rat_term_to_real: rat_term_to_real(f;t),
rtermDivide: num "/" denom,
rat_term_ind: rat_term_ind,
rtermConstant: "const",
rtermMultiply: left "*" right,
rtermVar: rtermVar(var),
pi1: fst(t),
pi2: snd(t),
sq_stable: SqStable(P)
Lemmas referenced :
nat_properties,
full-omega-unsat,
intformand_wf,
intformle_wf,
itermConstant_wf,
itermVar_wf,
intformless_wf,
istype-int,
int_formula_prop_and_lemma,
istype-void,
int_formula_prop_le_lemma,
int_term_value_constant_lemma,
int_term_value_var_lemma,
int_formula_prop_less_lemma,
int_formula_prop_wf,
ge_wf,
istype-less_than,
le_witness_for_triv,
subtract-1-ge-0,
istype-nat,
regular-int-seq_wf,
le_wf,
false_wf,
fastpi_wf,
nat_plus_wf,
primrec0_lemma,
true_wf,
equal-wf-base,
int_subtype_base,
subtype_base_sq,
mul-commutes,
mul-associates,
int-value-type,
value-type-has-value,
rneq-int,
exp_wf2,
int-to-real_wf,
rdiv_wf,
half-pi_wf,
rleq_wf,
squash_wf,
real_wf,
rabs_wf,
rsub_wf,
subtype_rel_self,
iff_weakening_equal,
less_than_wf,
rational-approx-property,
rless_wf,
rless-int,
exp0_lemma,
rleq_functionality,
rabs-difference-symmetry,
req_weakening,
primrec-unroll,
bool_wf,
bool_subtype_base,
iff_imp_equal_bool,
lt_int_wf,
bfalse_wf,
iff_functionality_wrt_iff,
assert_wf,
iff_weakening_uiff,
assert_of_lt_int,
subtract_wf,
exp-fastexp,
multiply_nat_wf,
istype-le,
exp_wf4,
decidable__le,
intformnot_wf,
itermAdd_wf,
itermSubtract_wf,
int_formula_prop_not_lemma,
int_term_value_add_lemma,
int_term_value_subtract_lemma,
subtract-add-cancel,
set_subtype_base,
mul_bounds_1a,
exp_mul,
exp_wf_nat_plus,
mul_nat_plus,
exp-positive,
nat_wf,
exp_step,
int_term_value_mul_lemma,
int_formula_prop_eq_lemma,
itermMultiply_wf,
intformeq_wf,
decidable__equal_int,
decidable__lt,
nat_plus_properties,
radd_rcos-Taylor,
radd_rcos_wf,
rnexp_wf,
rmul_preserves_rleq,
rmul_wf,
rinv_wf2,
rleq_functionality_wrt_implies,
rnexp_functionality_wrt_rleq,
zero-rleq-rabs,
rleq_weakening_equal,
rleq_weakening,
req-iff-rsub-is-0,
req_transitivity,
rmul-rinv3,
rmul_functionality,
rnexp_functionality,
rmul-identity1,
rinv-as-rdiv,
real_polynomial_null,
real_term_value_sub_lemma,
real_term_value_mul_lemma,
real_term_value_var_lemma,
real_term_value_const_lemma,
mul_bounds_1b,
rless_functionality,
rnexp-int,
req_functionality,
rdiv_functionality,
req_inversion,
rnexp-rdiv,
exp-one,
rneq_functionality,
rmul-int,
assert-rat-term-eq2,
rtermDivide_wf,
rtermConstant_wf,
rtermVar_wf,
rtermMultiply_wf,
rational-approx_wf,
radd_wf,
r-triangle-inequality2,
radd_functionality_wrt_rleq,
sq_stable__rless,
real_term_value_add_lemma,
rleq-int-fractions
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation_alt,
cut,
introduction,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
hypothesisEquality,
hypothesis,
setElimination,
rename,
intWeakElimination,
natural_numberEquality,
independent_isectElimination,
approximateComputation,
independent_functionElimination,
dependent_pairFormation_alt,
lambdaEquality_alt,
int_eqEquality,
dependent_functionElimination,
isect_memberEquality_alt,
voidElimination,
sqequalRule,
independent_pairFormation,
universeIsType,
productElimination,
equalityTransitivity,
equalitySymmetry,
functionIsTypeImplies,
inhabitedIsType,
because_Cache,
imageElimination,
baseClosed,
imageMemberEquality,
applyLambdaEquality,
lambdaFormation,
dependent_set_memberEquality,
voidEquality,
isect_memberEquality,
functionExtensionality,
cumulativity,
instantiate,
addLevel,
divideEquality,
lambdaEquality,
applyEquality,
multiplyEquality,
intEquality,
callbyvalueReduce,
computeAll,
promote_hyp,
levelHypothesis,
universeEquality,
inrFormation,
addEquality,
dependent_set_memberEquality_alt,
unionElimination,
dependent_pairFormation,
closedConclusion,
inrFormation_alt,
equalityIstype,
baseApply,
sqequalBase
Latex:
\mforall{}n:\mBbbN{}. (|fastpi(n) - \mpi{}/2(slower)| \mleq{} (r1/r(10\^{}(20 * 3\^{}n))))
Date html generated:
2019_10_30-AM-11_43_39
Last ObjectModification:
2019_04_09-PM-04_53_55
Theory : reals_2
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