Nuprl Lemma : arctangent-bounds
∀x:ℝ. (arctangent(x) ∈ (-(π/2), π/2))
Proof
Definitions occuring in Statement :
arctangent: arctangent(x),
halfpi: π/2,
rooint: (l, u),
i-member: r ∈ I,
rminus: -(x),
real: ℝ,
all: ∀x:A. B[x]
Definitions unfolded in proof :
all: ∀x:A. B[x],
member: t ∈ T,
exists: ∃x:A. B[x],
rtan: rtan(x),
and: P ∧ Q,
so_lambda: λ2x.t[x],
rfun: I ⟶ℝ,
uall: ∀[x:A]. B[x],
prop: ℙ,
so_apply: x[s],
implies: P ⇒ Q,
uimplies: b supposing a,
uiff: uiff(P;Q),
rev_uimplies: rev_uimplies(P;Q),
continuous: f[x] continuous for x ∈ I,
i-approx: i-approx(I;n),
riiint: (-∞, ∞),
nat_plus: ℕ+,
nat: ℕ,
ge: i ≥ j ,
decidable: Dec(P),
or: P ∨ Q,
not: ¬A,
satisfiable_int_formula: satisfiable_int_formula(fmla),
top: Top,
false: False,
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q,
le: A ≤ B,
less_than': less_than'(a;b),
sq_exists: ∃x:A [B[x]],
rneq: x ≠ y,
guard: {T},
rless: x < y,
sq_stable: SqStable(P),
rleq: x ≤ y,
rnonneg: rnonneg(x),
squash: ↓T,
less_than: a < b,
true: True,
cand: A c∧ B,
int-to-real: r(n),
halfpi: π/2,
divide: n ÷ m,
cubic_converge: cubic_converge(b;m),
ifthenelse: if b then t else f fi ,
le_int: i ≤z j,
bnot: ¬bb,
lt_int: i <z j,
bfalse: ff,
btrue: tt,
fastpi: fastpi(n),
primrec: primrec(n;b;c),
primtailrec: primtailrec(n;i;b;f),
subtype_rel: A ⊆r B,
real: ℝ,
req_int_terms: t1 ≡ t2,
rminus: -(x),
rge: x ≥ y,
rgt: x > y,
rdiv: (x/y),
int_nzero: ℤ-o,
nequal: a ≠ b ∈ T ,
sq_type: SQType(T),
subinterval: I ⊆ J ,
i-member: r ∈ I,
rooint: (l, u),
rsub: x - y,
radd: a + b,
accelerate: accelerate(k;f),
reg-seq-list-add: reg-seq-list-add(L),
cbv_list_accum: cbv_list_accum(x,a.f[x; a];y;L),
cons: [a / b],
nil: [],
it: ⋅,
r-ap: f(x),
strictly-increasing-on-interval: f[x] strictly-increasing for x ∈ I
Lemmas referenced :
real_wf,
r-archimedean,
function-is-continuous,
riiint_wf,
rcos_wf,
i-member_wf,
req_functionality,
rcos_functionality,
req_weakening,
req_wf,
nat_properties,
decidable__lt,
full-omega-unsat,
intformnot_wf,
intformless_wf,
itermConstant_wf,
istype-int,
int_formula_prop_not_lemma,
istype-void,
int_formula_prop_less_lemma,
int_term_value_constant_lemma,
int_formula_prop_wf,
istype-less_than,
rccint-icompact,
int-to-real_wf,
rleq-int,
istype-false,
icompact_wf,
rccint_wf,
member_rccint_lemma,
intformand_wf,
itermAdd_wf,
itermMultiply_wf,
itermVar_wf,
intformle_wf,
int_formula_prop_and_lemma,
int_term_value_add_lemma,
int_term_value_mul_lemma,
int_term_value_var_lemma,
int_formula_prop_le_lemma,
sq_stable__and,
rless_wf,
all_wf,
rleq_wf,
rabs_wf,
rsub_wf,
rdiv_wf,
rless-int,
nat_plus_properties,
sq_stable__rless,
sq_stable__all,
sq_stable__rleq,
le_witness_for_triv,
rminus_wf,
halfpi_wf,
rsin_wf,
rsin_functionality,
iff_weakening_uiff,
rleq_functionality,
rabs_functionality,
rsub_functionality,
req_transitivity,
rsin-rminus,
rminus_functionality,
rsin-halfpi,
rmin_wf,
rmin_strict_ub,
rmin_lb,
rmin-rleq,
rmin_strict_lb,
member_rooint_lemma,
rless-implies-rless,
rmul_wf,
itermSubtract_wf,
itermMinus_wf,
req-iff-rsub-is-0,
real_polynomial_null,
real_term_value_sub_lemma,
real_term_value_var_lemma,
real_term_value_const_lemma,
real_term_value_minus_lemma,
rcos-rminus,
rcos-halfpi,
rleq_weakening_rless,
radd-preserves-rless,
radd_wf,
rless_functionality,
real_term_value_add_lemma,
rleq_functionality_wrt_implies,
rleq_weakening_equal,
rabs-rleq-iff,
rleq-implies-rleq,
radd-preserves-rleq,
rless_transitivity2,
squash_wf,
true_wf,
rabs-rminus,
subtype_rel_self,
iff_weakening_equal,
rmul_preserves_rleq2,
sq_stable__less_than,
decidable__le,
rinv_wf2,
rmul_functionality,
req_inversion,
radd-int,
radd_functionality,
rmul-int,
rmul-rinv,
real_term_value_mul_lemma,
rless_functionality_wrt_implies,
rcos-positive,
rooint_wf,
halfpi-positive,
trivial-rless-radd,
rmul_assoc,
rabs-difference-bound-rleq,
rinv-as-rdiv,
rmul_preserves_rleq,
minus-one-mul-top,
subtype_base_sq,
int_subtype_base,
nequal_wf,
int-rinv-cancel2,
subtype_rel_sets_simple,
rmul_preserves_rless,
rmul-rinv3,
rminus-int,
rmul_reverses_rless,
trivial-rsub-rless,
rsub_functionality_wrt_rleq,
rleq_weakening,
rcos-positive-before-half-pi,
member_rcoint_lemma,
radd_comm_eq,
IVT-strictly-increasing-open,
rtan_wf,
rless_transitivity1,
rtan_functionality,
req_witness,
rtan_functionality_wrt_rless,
i-member_functionality,
arctangent_wf,
arctangent_functionality,
arctangent-rtan
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation_alt,
cut,
universeIsType,
introduction,
extract_by_obid,
hypothesis,
sqequalHypSubstitution,
dependent_functionElimination,
thin,
hypothesisEquality,
productElimination,
sqequalRule,
independent_pairFormation,
lambdaEquality_alt,
isectElimination,
setElimination,
rename,
setIsType,
inhabitedIsType,
independent_functionElimination,
because_Cache,
independent_isectElimination,
dependent_set_memberEquality_alt,
natural_numberEquality,
unionElimination,
approximateComputation,
dependent_pairFormation_alt,
isect_memberEquality_alt,
voidElimination,
minusEquality,
addEquality,
multiplyEquality,
int_eqEquality,
functionEquality,
productEquality,
closedConclusion,
inrFormation_alt,
productIsType,
equalityTransitivity,
equalitySymmetry,
functionIsTypeImplies,
imageMemberEquality,
baseClosed,
imageElimination,
inlFormation_alt,
applyEquality,
equalityIstype,
instantiate,
universeEquality,
promote_hyp,
cumulativity,
intEquality,
sqequalBase
Latex:
\mforall{}x:\mBbbR{}. (arctangent(x) \mmember{} (-(\mpi{}/2), \mpi{}/2))
Date html generated:
2019_10_30-AM-11_44_59
Last ObjectModification:
2019_04_03-AM-00_28_31
Theory : reals_2
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