Nuprl Lemma : atan_approx_wf
∀[a:{2...}]. ∀[x:ℝ]. ∀[N:ℕ+]. (atan_approx(a;x;N) ∈ ℤ)
Proof
Definitions occuring in Statement :
atan_approx: atan_approx(a;x;M),
real: ℝ,
int_upper: {i...},
nat_plus: ℕ+,
uall: ∀[x:A]. B[x],
member: t ∈ T,
natural_number: $n,
int: ℤ
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
atan_approx: atan_approx(a;x;M),
has-value: (a)↓,
uimplies: b supposing a,
nat_plus: ℕ+,
subtype_rel: A ⊆r B,
nat: ℕ,
guard: {T},
int_upper: {i...},
ge: i ≥ j ,
all: ∀x:A. B[x],
decidable: Dec(P),
or: P ∨ Q,
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,
less_than: a < b,
squash: ↓T,
less_than': less_than'(a;b),
true: True
Lemmas referenced :
value-type-has-value,
int-value-type,
atan-approx_wf,
atan-log_wf,
nat_wf,
le_wf,
exp_wf2,
nat_properties,
nat_plus_properties,
int_upper_properties,
decidable__le,
full-omega-unsat,
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,
mul_nat_plus,
less_than_wf,
nat_plus_wf,
real_wf,
int_upper_wf
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
introduction,
cut,
sqequalRule,
callbyvalueReduce,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
intEquality,
independent_isectElimination,
hypothesis,
multiplyEquality,
natural_numberEquality,
setElimination,
rename,
hypothesisEquality,
applyEquality,
lambdaEquality,
setEquality,
because_Cache,
addEquality,
dependent_set_memberEquality,
dependent_functionElimination,
unionElimination,
approximateComputation,
independent_functionElimination,
dependent_pairFormation,
int_eqEquality,
isect_memberEquality,
voidElimination,
voidEquality,
independent_pairFormation,
imageMemberEquality,
baseClosed,
axiomEquality,
equalityTransitivity,
equalitySymmetry
Latex:
\mforall{}[a:\{2...\}]. \mforall{}[x:\mBbbR{}]. \mforall{}[N:\mBbbN{}\msupplus{}]. (atan\_approx(a;x;N) \mmember{} \mBbbZ{})
Date html generated:
2018_05_22-PM-03_05_40
Last ObjectModification:
2017_10_25-PM-11_05_40
Theory : reals_2
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