Nuprl Lemma : sine-is-limit
∀x:ℝ. Σi.-1^i * (x^(2 * i) + 1)/((2 * i) + 1)! = sine(x)
Proof
Definitions occuring in Statement :
sine: sine(x),
series-sum: Σn.x[n] = a,
rnexp: x^k1,
int-rdiv: (a)/k1,
int-rmul: k1 * a,
real: ℝ,
all: ∀x:A. B[x],
multiply: n * m,
add: n + m,
minus: -n,
natural_number: $n,
fastexp: i^n,
fact: (n)!
Definitions unfolded in proof :
all: ∀x:A. B[x],
sine: sine(x),
member: t ∈ T,
subtype_rel: A ⊆r B,
uall: ∀[x:A]. B[x],
so_lambda: λ2x.t[x],
nat: ℕ,
ge: i ≥ j ,
decidable: Dec(P),
or: P ∨ Q,
uimplies: b supposing a,
satisfiable_int_formula: satisfiable_int_formula(fmla),
exists: ∃x:A. B[x],
false: False,
implies: P ⇒ Q,
not: ¬A,
top: Top,
and: P ∧ Q,
prop: ℙ,
nat_plus: ℕ+,
int_nzero: ℤ-o,
so_apply: x[s],
nequal: a ≠ b ∈ T ,
guard: {T},
pi1: fst(t)
Lemmas referenced :
sine-exists-ext,
all_wf,
exists_wf,
series-sum_wf,
int-rmul_wf,
int-rdiv_wf,
fact_wf,
nat_properties,
decidable__le,
satisfiable-full-omega-tt,
intformand_wf,
intformnot_wf,
intformle_wf,
itermConstant_wf,
itermAdd_wf,
itermMultiply_wf,
itermVar_wf,
int_formula_prop_and_lemma,
int_formula_prop_not_lemma,
int_formula_prop_le_lemma,
int_term_value_constant_lemma,
int_term_value_add_lemma,
int_term_value_mul_lemma,
int_term_value_var_lemma,
int_formula_prop_wf,
le_wf,
subtype_rel_sets,
less_than_wf,
nequal_wf,
nat_plus_properties,
intformeq_wf,
intformless_wf,
int_formula_prop_eq_lemma,
int_formula_prop_less_lemma,
equal-wf-base,
int_subtype_base,
rnexp_wf,
nat_wf,
equal_wf,
real_wf
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation,
cut,
applyEquality,
thin,
instantiate,
extract_by_obid,
hypothesis,
lambdaEquality,
sqequalHypSubstitution,
sqequalRule,
hypothesisEquality,
introduction,
isectElimination,
because_Cache,
dependent_set_memberEquality,
addEquality,
multiplyEquality,
natural_numberEquality,
setElimination,
rename,
dependent_functionElimination,
unionElimination,
independent_isectElimination,
dependent_pairFormation,
int_eqEquality,
intEquality,
isect_memberEquality,
voidElimination,
voidEquality,
independent_pairFormation,
computeAll,
setEquality,
equalityTransitivity,
equalitySymmetry,
Error :applyLambdaEquality,
baseClosed,
independent_functionElimination,
productElimination
Latex:
\mforall{}x:\mBbbR{}. \mSigma{}i.-1\^{}i * (x\^{}(2 * i) + 1)/((2 * i) + 1)! = sine(x)
Date html generated:
2016_10_26-AM-09_24_22
Last ObjectModification:
2016_08_25-PM-11_38_00
Theory : reals
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