Nuprl Lemma : atan-approx_wf
∀[k:ℕ]. ∀[x:ℝ]. ∀[N:ℕ+]. (atan-approx(k;x;N) ∈ ℤ)
Proof
Definitions occuring in Statement :
atan-approx: atan-approx(k;x;N),
real: ℝ,
nat_plus: ℕ+,
nat: ℕ,
uall: ∀[x:A]. B[x],
member: t ∈ T,
int: ℤ
Definitions unfolded in proof :
le: A ≤ B,
real: ℝ,
decidable: Dec(P),
less_than': less_than'(a;b),
squash: ↓T,
less_than: a < b,
assert: ↑b,
bnot: ¬bb,
or: P ∨ Q,
bfalse: ff,
and: P ∧ Q,
uiff: uiff(P;Q),
ifthenelse: if b then t else f fi ,
btrue: tt,
it: ⋅,
unit: Unit,
bool: 𝔹,
guard: {T},
sq_type: SQType(T),
true: True,
has-value: (a)↓,
subtype_rel: A ⊆r B,
prop: ℙ,
top: Top,
all: ∀x:A. B[x],
false: False,
exists: ∃x:A. B[x],
satisfiable_int_formula: satisfiable_int_formula(fmla),
uimplies: b supposing a,
implies: P ⇒ Q,
not: ¬A,
ge: i ≥ j ,
nat_plus: ℕ+,
nequal: a ≠ b ∈ T ,
nat: ℕ,
int_nzero: ℤ-o,
atan-approx: atan-approx(k;x;N),
member: t ∈ T,
uall: ∀[x:A]. B[x]
Lemmas referenced :
real_wf,
equal-wf-T-base,
int_entire_a,
false_wf,
int_formula_prop_less_lemma,
intformless_wf,
add-is-int-iff,
decidable__lt,
nat_plus_wf,
le_wf,
int_formula_prop_not_lemma,
int_formula_prop_le_lemma,
int_formula_prop_and_lemma,
intformnot_wf,
intformle_wf,
intformand_wf,
decidable__le,
absval_wf,
divide_wf,
add_nat_plus,
less_than_wf,
mul_nat_plus,
neg_assert_of_eq_int,
assert-bnot,
bool_subtype_base,
bool_cases_sqequal,
equal_wf,
eqff_to_assert,
assert_of_eq_int,
eqtt_to_assert,
bool_wf,
true_wf,
subtype_base_sq,
eq_int_wf,
int-value-type,
value-type-has-value,
rmul_wf,
nat_wf,
int-to-real_wf,
nequal_wf,
int_subtype_base,
equal-wf-base,
int_formula_prop_wf,
int_term_value_var_lemma,
int_term_value_constant_lemma,
int_term_value_mul_lemma,
int_term_value_add_lemma,
int_formula_prop_eq_lemma,
itermVar_wf,
itermConstant_wf,
itermMultiply_wf,
itermAdd_wf,
intformeq_wf,
full-omega-unsat,
nat_plus_properties,
nat_properties,
int-rdiv_wf,
poly-approx_wf
Rules used in proof :
axiomEquality,
pointwiseFunctionality,
divideEquality,
applyLambdaEquality,
imageMemberEquality,
independent_pairFormation,
minusEquality,
promote_hyp,
productElimination,
equalityElimination,
unionElimination,
equalitySymmetry,
equalityTransitivity,
cumulativity,
instantiate,
addLevel,
remainderEquality,
callbyvalueReduce,
because_Cache,
applyEquality,
baseClosed,
closedConclusion,
baseApply,
sqequalRule,
voidEquality,
voidElimination,
isect_memberEquality,
dependent_functionElimination,
intEquality,
int_eqEquality,
dependent_pairFormation,
independent_functionElimination,
approximateComputation,
independent_isectElimination,
lambdaFormation,
hypothesis,
rename,
setElimination,
natural_numberEquality,
multiplyEquality,
addEquality,
dependent_set_memberEquality,
lambdaEquality,
hypothesisEquality,
thin,
isectElimination,
sqequalHypSubstitution,
extract_by_obid,
cut,
introduction,
isect_memberFormation,
sqequalReflexivity,
computationStep,
sqequalTransitivity,
sqequalSubstitution
Latex:
\mforall{}[k:\mBbbN{}]. \mforall{}[x:\mBbbR{}]. \mforall{}[N:\mBbbN{}\msupplus{}]. (atan-approx(k;x;N) \mmember{} \mBbbZ{})
Date html generated:
2018_05_22-PM-03_04_58
Last ObjectModification:
2018_05_20-PM-11_15_48
Theory : reals_2
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