Nuprl Lemma : converges-to-cosine
∀x:ℝ. lim n→∞.if n=0 then r0 else (cosine((x within 1/n)) within 1/n) = cosine(x)
Proof
Definitions occuring in Statement :
cosine: cosine(x),
converges-to: lim n→∞.x[n] = y,
rational-approx: (x within 1/n),
int-to-real: r(n),
real: ℝ,
all: ∀x:A. B[x],
int_eq: if a=b then c else d,
natural_number: $n
Definitions unfolded in proof :
all: ∀x:A. B[x],
converges-to: lim n→∞.x[n] = y,
sq_exists: ∃x:{A| B[x]},
member: t ∈ T,
nat: ℕ,
nat_plus: ℕ+,
uall: ∀[x:A]. B[x],
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,
so_lambda: λ2x.t[x],
real: ℝ,
le: A ≤ B,
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q,
uiff: uiff(P;Q),
subtype_rel: A ⊆r B,
less_than': less_than'(a;b),
true: True,
subtract: n - m,
rneq: x ≠ y,
guard: {T},
ge: i ≥ j ,
so_apply: x[s],
rev_uimplies: rev_uimplies(P;Q),
rge: x ≥ y,
bool: 𝔹,
unit: Unit,
it: ⋅,
btrue: tt,
bfalse: ff,
sq_type: SQType(T),
bnot: ¬bb,
ifthenelse: if b then t else f fi ,
assert: ↑b,
nequal: a ≠ b ∈ T ,
int_upper: {i...}
Lemmas referenced :
nat_plus_properties,
decidable__le,
satisfiable-full-omega-tt,
intformand_wf,
intformnot_wf,
intformle_wf,
itermConstant_wf,
itermMultiply_wf,
itermVar_wf,
intformless_wf,
int_formula_prop_and_lemma,
int_formula_prop_not_lemma,
int_formula_prop_le_lemma,
int_term_value_constant_lemma,
int_term_value_mul_lemma,
int_term_value_var_lemma,
int_formula_prop_less_lemma,
int_formula_prop_wf,
le_wf,
nat_wf,
all_wf,
rleq_wf,
rabs_wf,
rsub_wf,
int-to-real_wf,
rational-approx_wf,
cosine_wf,
decidable__lt,
false_wf,
not-lt-2,
not-equal-2,
add_functionality_wrt_le,
add-associates,
add-zero,
add-commutes,
zero-add,
le-add-cancel,
condition-implies-le,
minus-add,
minus-zero,
less_than_wf,
rdiv_wf,
rless-int,
nat_properties,
rless_wf,
nat_plus_wf,
real_wf,
rational-approx-property,
equal_wf,
rleq_functionality,
rabs-difference-symmetry,
req_weakening,
rabs-difference-cosine-rleq,
rleq_functionality_wrt_implies,
rleq_weakening_equal,
eq_int_wf,
bool_wf,
eqtt_to_assert,
assert_of_eq_int,
intformeq_wf,
int_formula_prop_eq_lemma,
eqff_to_assert,
bool_cases_sqequal,
subtype_base_sq,
bool_subtype_base,
assert-bnot,
neg_assert_of_eq_int,
int_upper_subtype_nat,
nequal-le-implies,
radd_wf,
r-triangle-inequality2,
int_upper_properties,
radd_functionality_wrt_rleq,
rmul_wf,
rleq-int-fractions,
uiff_transitivity,
req_transitivity,
radd_functionality,
req_inversion,
rmul-identity1,
rmul-distrib2,
rmul_functionality,
radd-int,
rmul-int-rdiv
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation,
dependent_set_memberFormation,
dependent_set_memberEquality,
multiplyEquality,
natural_numberEquality,
sqequalHypSubstitution,
setElimination,
thin,
rename,
cut,
hypothesisEquality,
hypothesis,
introduction,
extract_by_obid,
isectElimination,
dependent_functionElimination,
unionElimination,
independent_isectElimination,
dependent_pairFormation,
lambdaEquality,
int_eqEquality,
intEquality,
isect_memberEquality,
voidElimination,
voidEquality,
sqequalRule,
independent_pairFormation,
computeAll,
because_Cache,
functionEquality,
productElimination,
independent_functionElimination,
addEquality,
applyEquality,
minusEquality,
inrFormation,
equalityTransitivity,
equalitySymmetry,
equalityElimination,
int_eqReduceTrueSq,
promote_hyp,
instantiate,
cumulativity,
int_eqReduceFalseSq,
hypothesis_subsumption
Latex:
\mforall{}x:\mBbbR{}. lim n\mrightarrow{}\minfty{}.if n=0 then r0 else (cosine((x within 1/n)) within 1/n) = cosine(x)
Date html generated:
2017_10_04-PM-10_21_01
Last ObjectModification:
2017_07_28-AM-08_48_15
Theory : reals_2
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