Nuprl Lemma : atan-small_wf
∀[x:ℝ]. atan-small(x) ∈ {y:ℝ| arctangent(x) = y} supposing |x| ≤ (r1/r(2))
Proof
Definitions occuring in Statement :
atan-small: atan-small(x),
arctangent: arctangent(x),
rdiv: (x/y),
rleq: x ≤ y,
rabs: |x|,
req: x = y,
int-to-real: r(n),
real: ℝ,
uimplies: b supposing a,
uall: ∀[x:A]. B[x],
member: t ∈ T,
set: {x:A| B[x]} ,
natural_number: $n
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
uimplies: b supposing a,
atan-small: atan-small(x),
rneq: x ≠ y,
guard: {T},
or: P ∨ Q,
all: ∀x:A. B[x],
iff: P ⇐⇒ Q,
and: P ∧ Q,
rev_implies: P ⇐ Q,
implies: P ⇒ Q,
less_than: a < b,
squash: ↓T,
less_than': less_than'(a;b),
true: True,
prop: ℙ,
subtype_rel: A ⊆r B,
int_upper: {i...},
decidable: Dec(P),
not: ¬A,
satisfiable_int_formula: satisfiable_int_formula(fmla),
exists: ∃x:A. B[x],
false: False,
top: Top,
so_lambda: λ2x.t[x],
so_apply: x[s],
sq_stable: SqStable(P)
Lemmas referenced :
atan_wf,
atan-size-bound_wf,
rleq_wf,
rabs_wf,
rdiv_wf,
rless-int,
rless_wf,
int-to-real_wf,
int_upper_properties,
decidable__lt,
full-omega-unsat,
intformand_wf,
intformnot_wf,
intformless_wf,
itermConstant_wf,
itermVar_wf,
intformle_wf,
int_formula_prop_and_lemma,
int_formula_prop_not_lemma,
int_formula_prop_less_lemma,
int_term_value_constant_lemma,
int_term_value_var_lemma,
int_formula_prop_le_lemma,
int_formula_prop_wf,
real_wf,
set_wf,
int_upper_wf,
sq_stable__rleq,
equal_wf
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
introduction,
cut,
sqequalRule,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
dependent_set_memberEquality,
hypothesisEquality,
hypothesis,
because_Cache,
independent_isectElimination,
inrFormation,
dependent_functionElimination,
productElimination,
independent_functionElimination,
independent_pairFormation,
natural_numberEquality,
imageMemberEquality,
baseClosed,
applyEquality,
lambdaEquality,
setElimination,
rename,
setEquality,
unionElimination,
approximateComputation,
dependent_pairFormation,
int_eqEquality,
intEquality,
isect_memberEquality,
voidElimination,
voidEquality,
axiomEquality,
equalityTransitivity,
equalitySymmetry,
lambdaFormation,
imageElimination
Latex:
\mforall{}[x:\mBbbR{}]. atan-small(x) \mmember{} \{y:\mBbbR{}| arctangent(x) = y\} supposing |x| \mleq{} (r1/r(2))
Date html generated:
2018_05_22-PM-03_06_49
Last ObjectModification:
2017_10_26-PM-10_45_32
Theory : reals_2
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