Nuprl Lemma : atan-log_wf
∀[a:{2...}]. ∀[M:ℤ]. (atan-log(a;M) ∈ {k:ℕ| M ≤ (((2 * k) + 3) * a^((2 * k) + 3))} )
Proof
Definitions occuring in Statement :
atan-log: atan-log(a;M),
exp: i^n,
int_upper: {i...},
nat: ℕ,
uall: ∀[x:A]. B[x],
le: A ≤ B,
member: t ∈ T,
set: {x:A| B[x]} ,
multiply: n * m,
add: n + m,
natural_number: $n,
int: ℤ
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
less_than': less_than'(a;b),
le: A ≤ B,
nat: ℕ,
subtype_rel: A ⊆r B,
prop: ℙ,
and: P ∧ Q,
top: Top,
false: False,
exists: ∃x:A. B[x],
satisfiable_int_formula: satisfiable_int_formula(fmla),
implies: P ⇒ Q,
not: ¬A,
uimplies: b supposing a,
or: P ∨ Q,
decidable: Dec(P),
all: ∀x:A. B[x],
int_upper: {i...},
guard: {T},
atan-log: atan-log(a;M),
nat_plus: ℕ+,
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q,
uiff: uiff(P;Q),
true: True,
less_than: a < b,
squash: ↓T,
so_lambda: λ2x.t[x],
so_apply: x[s],
ge: i ≥ j ,
subtract: n - m
Lemmas referenced :
istype-int,
istype-int_upper,
exp-greater,
exp_wf2,
le_weakening2,
le_wf,
false_wf,
int_upper_subtype_nat,
mul_preserves_le,
int_formula_prop_wf,
int_term_value_var_lemma,
int_term_value_constant_lemma,
int_formula_prop_le_lemma,
int_formula_prop_not_lemma,
int_formula_prop_and_lemma,
itermVar_wf,
itermConstant_wf,
intformle_wf,
intformnot_wf,
intformand_wf,
full-omega-unsat,
decidable__le,
int_upper_properties,
gen_log_aux_wf,
exp_wf_nat_plus,
istype-false,
istype-le,
decidable__lt,
not-lt-2,
add_functionality_wrt_le,
add-commutes,
istype-void,
zero-add,
le-add-cancel,
istype-less_than,
subtype_rel_sets,
int_upper_wf,
nat_wf,
subtract_wf,
itermSubtract_wf,
int_term_value_subtract_lemma,
nat_properties,
itermAdd_wf,
itermMultiply_wf,
int_term_value_add_lemma,
int_term_value_mul_lemma,
istype-nat,
subtype_rel_set,
upper_subtype_nat,
squash_wf,
true_wf,
exp_add,
subtype_rel_self,
iff_weakening_equal,
exp_mul,
mul-commutes,
minus-zero,
add-zero
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation_alt,
introduction,
cut,
sqequalHypSubstitution,
hypothesis,
sqequalRule,
axiomEquality,
equalityTransitivity,
equalitySymmetry,
extract_by_obid,
isect_memberEquality_alt,
isectElimination,
thin,
hypothesisEquality,
isectIsTypeImplies,
inhabitedIsType,
natural_numberEquality,
because_Cache,
lambdaFormation,
dependent_set_memberEquality,
applyEquality,
independent_pairFormation,
voidEquality,
voidElimination,
isect_memberEquality,
intEquality,
int_eqEquality,
lambdaEquality,
dependent_pairFormation,
independent_functionElimination,
approximateComputation,
independent_isectElimination,
unionElimination,
dependent_functionElimination,
rename,
setElimination,
dependent_set_memberEquality_alt,
lambdaFormation_alt,
productElimination,
lambdaEquality_alt,
closedConclusion,
imageMemberEquality,
baseClosed,
multiplyEquality,
addEquality,
dependent_pairFormation_alt,
universeIsType,
imageElimination,
instantiate,
universeEquality
Latex:
\mforall{}[a:\{2...\}]. \mforall{}[M:\mBbbZ{}]. (atan-log(a;M) \mmember{} \{k:\mBbbN{}| M \mleq{} (((2 * k) + 3) * a\^{}((2 * k) + 3))\} )
Date html generated:
2019_10_31-AM-06_05_51
Last ObjectModification:
2018_11_08-PM-05_57_25
Theory : reals_2
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