Nuprl Lemma : rcos-seq-converges-to-half-pi
lim n→∞.rcos-seq(n) = π/2(slower)
Proof
Definitions occuring in Statement :
half-pi: π/2(slower),
rcos-seq: rcos-seq(n),
converges-to: lim n→∞.x[n] = y
Definitions unfolded in proof :
real: ℝ,
iff: P ⇐⇒ Q,
rcos-seq-converges-ext,
pi1: fst(t),
pi2: snd(t),
converges: x[n]↓ as n→∞,
so_apply: x[s],
so_lambda: λ2x.t[x],
sq_type: SQType(T),
nequal: a ≠ b ∈ T ,
uiff: uiff(P;Q),
ge: i ≥ j ,
guard: {T},
subtype_rel: A ⊆r B,
true: True,
squash: ↓T,
less_than: a < b,
top: Top,
exists: ∃x:A. B[x],
satisfiable_int_formula: satisfiable_int_formula(fmla),
or: P ∨ Q,
decidable: Dec(P),
nat: ℕ,
prop: ℙ,
implies: P ⇒ Q,
not: ¬A,
false: False,
less_than': less_than'(a;b),
and: P ∧ Q,
le: A ≤ B,
int_upper: {i...},
all: ∀x:A. B[x],
has-value: (a)↓,
nat_plus: ℕ+,
uimplies: b supposing a,
uall: ∀[x:A]. B[x],
member: t ∈ T,
half-pi: π/2(slower)
Lemmas referenced :
regular-int-seq_wf,
half-pi_wf,
real-regular,
iff_weakening_equal,
subtype_rel_self,
real_wf,
squash_wf,
converges-to_wf,
converges_wf,
rcos-seq-converges-ext,
true_wf,
equal-wf-base,
int_subtype_base,
subtype_base_sq,
mul_nat_plus,
equal_wf,
int_formula_prop_eq_lemma,
int_term_value_add_lemma,
intformeq_wf,
itermAdd_wf,
add-is-int-iff,
nat_properties,
nat_wf,
add_nat_wf,
rcos-seq_wf,
less_than_wf,
int_formula_prop_wf,
int_formula_prop_less_lemma,
int_term_value_var_lemma,
int_term_value_mul_lemma,
int_term_value_constant_lemma,
int_formula_prop_le_lemma,
int_formula_prop_not_lemma,
int_formula_prop_and_lemma,
intformless_wf,
itermVar_wf,
itermMultiply_wf,
itermConstant_wf,
intformle_wf,
intformnot_wf,
intformand_wf,
full-omega-unsat,
decidable__le,
nat_plus_properties,
le_wf,
false_wf,
exp-ratio_wf2,
int-value-type,
value-type-has-value,
nat_plus_wf
Rules used in proof :
universeEquality,
functionEquality,
imageElimination,
cumulativity,
instantiate,
addLevel,
productElimination,
closedConclusion,
baseApply,
promote_hyp,
pointwiseFunctionality,
applyLambdaEquality,
equalitySymmetry,
equalityTransitivity,
divideEquality,
applyEquality,
baseClosed,
imageMemberEquality,
voidEquality,
voidElimination,
isect_memberEquality,
int_eqEquality,
lambdaEquality,
dependent_pairFormation,
independent_functionElimination,
approximateComputation,
unionElimination,
lambdaFormation,
independent_pairFormation,
dependent_set_memberEquality,
dependent_functionElimination,
addEquality,
because_Cache,
callbyvalueReduce,
hypothesisEquality,
rename,
setElimination,
natural_numberEquality,
multiplyEquality,
independent_isectElimination,
intEquality,
thin,
isectElimination,
sqequalHypSubstitution,
hypothesis,
extract_by_obid,
introduction,
sqequalRule,
sqequalReflexivity,
computationStep,
sqequalTransitivity,
sqequalSubstitution,
functionExtensionality,
cut
Latex:
lim n\mrightarrow{}\minfty{}.rcos-seq(n) = \mpi{}/2(slower)
Date html generated:
2018_05_22-PM-02_58_52
Last ObjectModification:
2018_05_20-PM-11_03_42
Theory : reals_2
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