Nuprl Lemma : atan_wf
∀[a:{2...}]. ∀[x:ℝ]. atan(a;x) ∈ {y:ℝ| arctangent(x) = y} supposing |x| ≤ (r1/r(a))
Proof
Definitions occuring in Statement :
atan: atan(a;x),
arctangent: arctangent(x),
rdiv: (x/y),
rleq: x ≤ y,
rabs: |x|,
req: x = y,
int-to-real: r(n),
real: ℝ,
int_upper: {i...},
uimplies: b supposing a,
uall: ∀[x:A]. B[x],
member: t ∈ T,
set: {x:A| B[x]} ,
natural_number: $n
Definitions unfolded in proof :
top: Top,
false: False,
exists: ∃x:A. B[x],
satisfiable_int_formula: satisfiable_int_formula(fmla),
not: ¬A,
decidable: Dec(P),
rev_implies: P ⇐ Q,
iff: P ⇐⇒ Q,
or: P ∨ Q,
rneq: x ≠ y,
int_upper: {i...},
atan: atan(a;x),
guard: {T},
all: ∀x:A. B[x],
implies: P ⇒ Q,
prop: ℙ,
and: P ∧ Q,
true: True,
less_than': less_than'(a;b),
squash: ↓T,
less_than: a < b,
nat_plus: ℕ+,
uimplies: b supposing a,
member: t ∈ T,
uall: ∀[x:A]. B[x]
Lemmas referenced :
int_upper_wf,
real_wf,
rless_wf,
int_formula_prop_wf,
int_formula_prop_le_lemma,
int_term_value_var_lemma,
int_term_value_constant_lemma,
int_formula_prop_less_lemma,
int_formula_prop_not_lemma,
int_formula_prop_and_lemma,
intformle_wf,
itermVar_wf,
itermConstant_wf,
intformless_wf,
intformnot_wf,
intformand_wf,
full-omega-unsat,
decidable__lt,
int_upper_properties,
rless-int,
int-to-real_wf,
rdiv_wf,
rabs_wf,
rleq_wf,
req_wf,
regular-int-seq_wf,
accelerate_wf,
req_inversion,
nat_plus_wf,
atan_approx_wf,
arctangent_wf,
less_than_wf,
rational-approx-implies-req,
atan_approx-property
Rules used in proof :
voidEquality,
voidElimination,
isect_memberEquality,
intEquality,
int_eqEquality,
dependent_pairFormation,
approximateComputation,
unionElimination,
dependent_functionElimination,
inrFormation,
because_Cache,
rename,
setElimination,
equalitySymmetry,
equalityTransitivity,
axiomEquality,
productElimination,
independent_isectElimination,
lambdaFormation,
independent_functionElimination,
lambdaEquality,
hypothesis,
baseClosed,
imageMemberEquality,
independent_pairFormation,
sqequalRule,
natural_numberEquality,
dependent_set_memberEquality,
hypothesisEquality,
thin,
isectElimination,
sqequalHypSubstitution,
extract_by_obid,
cut,
introduction,
isect_memberFormation,
sqequalReflexivity,
computationStep,
sqequalTransitivity,
sqequalSubstitution
Latex:
\mforall{}[a:\{2...\}]. \mforall{}[x:\mBbbR{}]. atan(a;x) \mmember{} \{y:\mBbbR{}| arctangent(x) = y\} supposing |x| \mleq{} (r1/r(a))
Date html generated:
2018_05_22-PM-03_05_54
Last ObjectModification:
2018_05_20-PM-11_20_24
Theory : reals_2
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