Nuprl Lemma : fastpi-converges
lim n→∞.fastpi(n) = π/2(slower)
Proof
Definitions occuring in Statement :
fastpi: fastpi(n),
half-pi: π/2(slower),
converges-to: lim n→∞.x[n] = y
Definitions unfolded in proof :
converges-to: lim n→∞.x[n] = y,
all: ∀x:A. B[x],
member: t ∈ T,
exists: ∃x:A. B[x],
has-value: (a)↓,
uall: ∀[x:A]. B[x],
uimplies: b supposing a,
nat: ℕ,
le: A ≤ B,
and: P ∧ Q,
less_than': less_than'(a;b),
false: False,
not: ¬A,
implies: P ⇒ Q,
prop: ℙ,
int_upper: {i...},
fastexp: i^n,
efficient-exp-ext,
subtype_rel: A ⊆r B,
so_lambda: λ2x.t[x],
nat_plus: ℕ+,
so_apply: x[s],
sq_stable: SqStable(P),
squash: ↓T,
true: True,
guard: {T},
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q,
sq_type: SQType(T),
sq_exists: ∃x:{A| B[x]},
rneq: x ≠ y,
or: P ∨ Q,
ge: i ≥ j ,
decidable: Dec(P),
satisfiable_int_formula: satisfiable_int_formula(fmla),
top: Top,
less_than: a < b,
int_nzero: ℤ-o,
nequal: a ≠ b ∈ T ,
rev_uimplies: rev_uimplies(P;Q),
rge: x ≥ y,
uiff: uiff(P;Q)
Lemmas referenced :
nat_plus_wf,
value-type-has-value,
int-value-type,
fastexp_wf,
false_wf,
le_wf,
cubic_converge_wf,
nat_plus_subtype_nat,
exp_wf2,
mul_bounds_1a,
exp_wf4,
set_wf,
nat_wf,
equal_wf,
sq_stable__le,
subtype_base_sq,
int_subtype_base,
squash_wf,
true_wf,
exp_mul,
iff_weakening_equal,
exp-fastexp,
all_wf,
rleq_wf,
rabs_wf,
rsub_wf,
fastpi_wf,
half-pi_wf,
rdiv_wf,
int-to-real_wf,
rless-int,
nat_properties,
nat_plus_properties,
decidable__lt,
satisfiable-full-omega-tt,
intformand_wf,
intformnot_wf,
intformless_wf,
itermConstant_wf,
itermVar_wf,
int_formula_prop_and_lemma,
int_formula_prop_not_lemma,
int_formula_prop_less_lemma,
int_term_value_constant_lemma,
int_term_value_var_lemma,
int_formula_prop_wf,
rless_wf,
rneq-int,
not_functionality_wrt_implies,
equal-wf-T-base,
rationals_wf,
exp_wf_nat_plus,
less_than_wf,
equal_functionality_wrt_subtype_rel2,
int-subtype-rationals,
int_nzero-rational,
exp_wf3,
equal-wf-base,
nequal_wf,
rleq_functionality_wrt_implies,
fastpi-property,
rleq_weakening_equal,
rleq-int-fractions,
less_than_transitivity1,
decidable__le,
intformle_wf,
itermMultiply_wf,
int_formula_prop_le_lemma,
int_term_value_mul_lemma,
trivial-int-eq1,
exp_add,
subtract_wf,
itermSubtract_wf,
int_term_value_subtract_lemma,
mul-non-neg1,
mul_preserves_le,
multiply-is-int-iff,
efficient-exp-ext
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation,
cut,
introduction,
extract_by_obid,
hypothesis,
dependent_pairFormation,
sqequalRule,
callbyvalueReduce,
sqequalHypSubstitution,
isectElimination,
thin,
intEquality,
independent_isectElimination,
because_Cache,
dependent_set_memberEquality,
natural_numberEquality,
independent_pairFormation,
hypothesisEquality,
dependent_functionElimination,
applyEquality,
multiplyEquality,
lambdaEquality,
setElimination,
rename,
equalityTransitivity,
equalitySymmetry,
independent_functionElimination,
imageMemberEquality,
baseClosed,
imageElimination,
instantiate,
cumulativity,
universeEquality,
productElimination,
dependent_set_memberFormation,
functionEquality,
inrFormation,
unionElimination,
int_eqEquality,
isect_memberEquality,
voidElimination,
voidEquality,
computeAll,
addLevel,
applyLambdaEquality,
pointwiseFunctionality,
promote_hyp,
baseApply,
closedConclusion
Latex:
lim n\mrightarrow{}\minfty{}.fastpi(n) = \mpi{}/2(slower)
Date html generated:
2017_10_04-PM-10_25_15
Last ObjectModification:
2017_07_28-AM-08_49_03
Theory : reals_2
Home
Index