Nuprl Lemma : arctangent-rinv
∀[x:{x:ℝ| x ∈ (r0, ∞)} ]. (arctangent(rinv(x)) = (π/2 - arctangent(x)))
Proof
Definitions occuring in Statement :
arctangent: arctangent(x),
halfpi: π/2,
roiint: (l, ∞),
i-member: r ∈ I,
rsub: x - y,
rinv: rinv(x),
req: x = y,
int-to-real: r(n),
real: ℝ,
uall: ∀[x:A]. B[x],
set: {x:A| B[x]} ,
natural_number: $n
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
uiff: uiff(P;Q),
and: P ∧ Q,
rev_uimplies: rev_uimplies(P;Q),
uimplies: b supposing a,
implies: P ⇒ Q,
rneq: x ≠ y,
guard: {T},
or: P ∨ Q,
all: ∀x:A. B[x],
top: Top,
sq_stable: SqStable(P),
squash: ↓T,
prop: ℙ,
so_lambda: λ2x.t[x],
rfun: I ⟶ℝ,
so_apply: x[s],
req_int_terms: t1 ≡ t2,
false: False,
not: ¬A,
nat: ℕ,
decidable: Dec(P),
satisfiable_int_formula: satisfiable_int_formula(fmla),
exists: ∃x:A. B[x],
rless: x < y,
sq_exists: ∃x:A [B[x]],
nat_plus: ℕ+,
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q,
less_than: a < b,
less_than': less_than'(a;b),
true: True,
rge: x ≥ y,
rgt: x > y,
rfun-eq: rfun-eq(I;f;g),
r-ap: f(x),
rdiv: (x/y),
cand: A c∧ B,
rat_term_to_real: rat_term_to_real(f;t),
rtermConstant: "const",
rat_term_ind: rat_term_ind,
pi1: fst(t),
rtermAdd: left "+" right,
rtermDivide: num "/" denom,
rtermVar: rtermVar(var),
pi2: snd(t),
nequal: a ≠ b ∈ T ,
int_nzero: ℤ-o,
sq_type: SQType(T),
pi: π
Lemmas referenced :
radd-preserves-req,
rsub_wf,
halfpi_wf,
arctangent_wf,
req_witness,
rinv_wf2,
member_roiint_lemma,
istype-void,
sq_stable__rless,
int-to-real_wf,
rless_wf,
real_wf,
i-member_wf,
roiint_wf,
radd_wf,
rdiv_wf,
itermSubtract_wf,
itermAdd_wf,
itermVar_wf,
antiderivatives-equal,
iproper-roiint,
derivative-const,
req_functionality,
req_transitivity,
radd_functionality,
req_weakening,
arctangent_functionality,
rinv-as-rdiv,
req-iff-rsub-is-0,
real_polynomial_null,
istype-int,
real_term_value_sub_lemma,
real_term_value_add_lemma,
real_term_value_var_lemma,
real_term_value_const_lemma,
rnexp_wf,
decidable__le,
full-omega-unsat,
intformnot_wf,
intformle_wf,
itermConstant_wf,
int_formula_prop_not_lemma,
int_formula_prop_le_lemma,
int_term_value_constant_lemma,
int_formula_prop_wf,
istype-le,
rnexp-positive,
nat_plus_properties,
trivial-rless-radd,
rless-int,
rless_functionality_wrt_implies,
rleq_weakening_equal,
rleq_weakening_rless,
derivative-arctangent,
rnexp2-nonneg,
riiint_wf,
subinterval-riiint,
radd_functionality_wrt_rleq,
derivative_functionality2,
arctangent-chain-rule,
rdiv_functionality,
rnexp_functionality,
req_wf,
derivative-rinv-basic,
rmul_preserves_rless,
rmul_wf,
itermMultiply_wf,
rless_functionality,
rmul-rinv,
real_term_value_mul_lemma,
derivative-add,
derivative_functionality,
req_inversion,
rnexp-rdiv,
rnexp-one,
rmul-is-positive,
rdiv-rdiv,
assert-rat-term-eq2,
rtermAdd_wf,
rtermDivide_wf,
rtermConstant_wf,
rtermVar_wf,
radd_comm,
rmul_functionality,
rinv1,
rmul-identity1,
pi_wf,
arctangent1,
rmul_comm,
nequal_wf,
true_wf,
equal-wf-base,
int_term_value_mul_lemma,
int_formula_prop_eq_lemma,
intformeq_wf,
decidable__equal_int,
int_subtype_base,
subtype_base_sq,
int-rmul_wf,
rmul_preserves_req,
int-rinv-cancel,
int-rmul-req
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation_alt,
introduction,
cut,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
because_Cache,
hypothesis,
setElimination,
rename,
productElimination,
independent_isectElimination,
independent_functionElimination,
sqequalRule,
inrFormation_alt,
dependent_functionElimination,
isect_memberEquality_alt,
voidElimination,
natural_numberEquality,
hypothesisEquality,
imageMemberEquality,
baseClosed,
imageElimination,
universeIsType,
setIsType,
closedConclusion,
lambdaEquality_alt,
inhabitedIsType,
approximateComputation,
int_eqEquality,
equalityTransitivity,
equalitySymmetry,
lambdaFormation_alt,
dependent_set_memberEquality_alt,
unionElimination,
dependent_pairFormation_alt,
independent_pairFormation,
minusEquality,
equalityIstype,
inlFormation_alt,
productIsType,
inrFormation,
dependent_set_memberEquality,
voidEquality,
isect_memberEquality,
dependent_pairFormation,
lambdaEquality,
intEquality,
lambdaFormation,
addLevel,
cumulativity,
instantiate
Latex:
\mforall{}[x:\{x:\mBbbR{}| x \mmember{} (r0, \minfty{})\} ]. (arctangent(rinv(x)) = (\mpi{}/2 - arctangent(x)))
Date html generated:
2019_10_31-AM-06_05_16
Last ObjectModification:
2019_04_03-AM-00_28_42
Theory : reals_2
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