Nuprl Lemma : cosine-approx-lemma
∀a:{2...}. ∀N:ℕ+. (∃k:ℕ [(N ≤ (a^((2 * k) + 2) * ((2 * k) + 2)!))])
Proof
Definitions occuring in Statement :
fact: (n)!,
exp: i^n,
int_upper: {i...},
nat_plus: ℕ+,
nat: ℕ,
le: A ≤ B,
all: ∀x:A. B[x],
sq_exists: ∃x:A [B[x]],
multiply: n * m,
add: n + m,
natural_number: $n
Definitions unfolded in proof :
all: ∀x:A. B[x],
member: t ∈ T,
uall: ∀[x:A]. B[x],
int_upper: {i...},
nat: ℕ,
nat_plus: ℕ+,
decidable: Dec(P),
or: P ∨ Q,
uimplies: b supposing a,
not: ¬A,
implies: P ⇒ Q,
satisfiable_int_formula: satisfiable_int_formula(fmla),
exists: ∃x:A. B[x],
false: False,
top: Top,
and: P ∧ Q,
prop: ℙ,
so_lambda: λ2x.t[x],
so_apply: x[s],
squash: ↓T,
label: ...$L... t,
guard: {T},
true: True,
subtype_rel: A ⊆r B,
iff: P ⇐⇒ Q,
le: A ≤ B,
less_than': less_than'(a;b),
rev_uimplies: rev_uimplies(P;Q),
ge: i ≥ j ,
sq_exists: ∃x:A [B[x]],
sq_type: SQType(T),
rev_implies: P ⇐ Q,
lt_int: i <z j,
bool: 𝔹,
unit: Unit,
it: ⋅,
btrue: tt,
uiff: uiff(P;Q),
ifthenelse: if b then t else f fi ,
bfalse: ff,
bnot: ¬bb,
assert: ↑b,
subtract: n - m,
less_than: a < b,
fact: (n)!,
primrec: primrec(n;b;c),
primtailrec: primtailrec(n;i;b;f)
Lemmas referenced :
int_upper_wf,
int_upper_properties,
mul_preserves_le,
nat_plus_properties,
decidable__le,
full-omega-unsat,
intformand_wf,
intformnot_wf,
intformle_wf,
itermConstant_wf,
itermVar_wf,
istype-int,
int_formula_prop_and_lemma,
istype-void,
int_formula_prop_not_lemma,
int_formula_prop_le_lemma,
int_term_value_constant_lemma,
int_term_value_var_lemma,
int_formula_prop_wf,
istype-le,
itermMultiply_wf,
int_term_value_mul_lemma,
set-value-type,
equal_wf,
le_wf,
int-value-type,
squash_wf,
true_wf,
istype-universe,
exp2,
decidable__equal_int,
intformeq_wf,
int_formula_prop_eq_lemma,
subtype_rel_self,
iff_weakening_equal,
nat_plus_wf,
istype-int_upper,
genfact-inv_wf,
itermAdd_wf,
intformless_wf,
int_term_value_add_lemma,
int_formula_prop_less_lemma,
mul_preserves_le2,
decidable__lt,
istype-less_than,
nat_plus_subtype_nat,
upper_subtype_nat,
istype-false,
less_than_functionality,
le_weakening,
multiply_functionality_wrt_le,
exp_wf2,
nat_properties,
fact_wf,
subtype_base_sq,
set_subtype_base,
int_subtype_base,
ge_wf,
genfact-step,
btrue_wf,
eqtt_to_assert,
assert_of_lt_int,
eqff_to_assert,
lt_int_wf,
bool_cases_sqequal,
bool_wf,
bool_subtype_base,
assert-bnot,
iff_weakening_uiff,
assert_wf,
less_than_wf,
subtract-1-ge-0,
mul-commutes,
fact_unroll,
subtract_wf,
itermSubtract_wf,
int_term_value_subtract_lemma,
istype-nat,
exp-positive,
exp_add
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation_alt,
cut,
introduction,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
natural_numberEquality,
hypothesis,
hypothesisEquality,
setElimination,
rename,
dependent_set_memberEquality_alt,
multiplyEquality,
because_Cache,
dependent_functionElimination,
unionElimination,
independent_isectElimination,
approximateComputation,
independent_functionElimination,
dependent_pairFormation_alt,
lambdaEquality_alt,
int_eqEquality,
isect_memberEquality_alt,
voidElimination,
sqequalRule,
independent_pairFormation,
universeIsType,
cutEval,
equalityTransitivity,
equalitySymmetry,
equalityIstype,
inhabitedIsType,
intEquality,
applyEquality,
imageElimination,
instantiate,
universeEquality,
applyLambdaEquality,
imageMemberEquality,
baseClosed,
productElimination,
addEquality,
cumulativity,
intWeakElimination,
axiomSqEquality,
functionIsTypeImplies,
sqequalIntensionalEquality,
baseApply,
closedConclusion,
equalityElimination,
promote_hyp
Latex:
\mforall{}a:\{2...\}. \mforall{}N:\mBbbN{}\msupplus{}. (\mexists{}k:\mBbbN{} [(N \mleq{} (a\^{}((2 * k) + 2) * ((2 * k) + 2)!))])
Date html generated:
2019_10_29-AM-10_33_29
Last ObjectModification:
2019_02_08-PM-01_37_57
Theory : reals
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