Nuprl Lemma : cosine-approx_wf
∀[x:ℝ]. ∀[k:ℕ]. ∀[N:ℕ+]. (cosine-approx(x;k;N) ∈ ℤ)
Proof
Definitions occuring in Statement :
cosine-approx: cosine-approx(x;k;N),
real: ℝ,
nat_plus: ℕ+,
nat: ℕ,
uall: ∀[x:A]. B[x],
member: t ∈ T,
int: ℤ
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
cosine-approx: cosine-approx(x;k;N),
nat: ℕ,
true: True,
nequal: a ≠ b ∈ T ,
not: ¬A,
implies: P ⇒ Q,
uimplies: b supposing a,
sq_type: SQType(T),
all: ∀x:A. B[x],
guard: {T},
false: False,
bool: 𝔹,
unit: Unit,
it: ⋅,
btrue: tt,
uiff: uiff(P;Q),
and: P ∧ Q,
ifthenelse: if b then t else f fi ,
nat_plus: ℕ+,
ge: i ≥ j ,
decidable: Dec(P),
or: P ∨ Q,
satisfiable_int_formula: satisfiable_int_formula(fmla),
exists: ∃x:A. B[x],
top: Top,
prop: ℙ,
subtype_rel: A ⊆r B,
bfalse: ff,
bnot: ¬bb,
assert: ↑b
Lemmas referenced :
poly-approx_wf,
subtype_base_sq,
int_subtype_base,
istype-int,
eq_int_wf,
eqtt_to_assert,
assert_of_eq_int,
int-rdiv_wf,
fact_wf,
nat_properties,
nat_plus_properties,
decidable__le,
full-omega-unsat,
intformand_wf,
intformnot_wf,
intformle_wf,
itermConstant_wf,
itermMultiply_wf,
itermVar_wf,
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_mul_lemma,
int_term_value_var_lemma,
int_formula_prop_wf,
istype-le,
int-to-real_wf,
eqff_to_assert,
bool_cases_sqequal,
bool_wf,
bool_subtype_base,
assert-bnot,
neg_assert_of_eq_int,
rnexp_wf,
decidable__lt,
intformless_wf,
int_formula_prop_less_lemma,
istype-less_than,
nat_plus_wf,
istype-nat,
real_wf
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation_alt,
introduction,
cut,
sqequalRule,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
hypothesisEquality,
lambdaEquality_alt,
remainderEquality,
setElimination,
rename,
because_Cache,
hypothesis,
closedConclusion,
natural_numberEquality,
lambdaFormation_alt,
instantiate,
cumulativity,
intEquality,
independent_isectElimination,
dependent_functionElimination,
equalityTransitivity,
equalitySymmetry,
independent_functionElimination,
voidElimination,
equalityIstype,
baseClosed,
sqequalBase,
inhabitedIsType,
unionElimination,
equalityElimination,
productElimination,
dependent_set_memberEquality_alt,
multiplyEquality,
approximateComputation,
dependent_pairFormation_alt,
int_eqEquality,
isect_memberEquality_alt,
independent_pairFormation,
universeIsType,
applyEquality,
promote_hyp,
minusEquality,
axiomEquality,
isectIsTypeImplies
Latex:
\mforall{}[x:\mBbbR{}]. \mforall{}[k:\mBbbN{}]. \mforall{}[N:\mBbbN{}\msupplus{}]. (cosine-approx(x;k;N) \mmember{} \mBbbZ{})
Date html generated:
2019_10_29-AM-10_36_16
Last ObjectModification:
2019_02_02-AM-11_22_18
Theory : reals
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