Nuprl Lemma : arctangent-rleq
∀x:ℝ. arctangent(x) ≤ x supposing r0 ≤ x
Proof
Definitions occuring in Statement :
arctangent: arctangent(x),
rleq: x ≤ y,
int-to-real: r(n),
real: ℝ,
uimplies: b supposing a,
all: ∀x:A. B[x],
natural_number: $n
Definitions unfolded in proof :
increasing-on-interval: f[x] increasing for x ∈ I,
top: Top,
req_int_terms: t1 ≡ t2,
rdiv: (x/y),
rev_uimplies: rev_uimplies(P;Q),
r-ap: f(x),
rfun-eq: rfun-eq(I;f;g),
rge: x ≥ y,
true: True,
squash: ↓T,
less_than: a < b,
rev_implies: P ⇐ Q,
iff: P ⇐⇒ Q,
uiff: uiff(P;Q),
real: ℝ,
subtype_rel: A ⊆r B,
rnonneg: rnonneg(x),
rleq: x ≤ y,
or: P ∨ Q,
guard: {T},
rneq: x ≠ y,
not: ¬A,
less_than': less_than'(a;b),
le: A ≤ B,
nat: ℕ,
so_apply: x[s],
rfun: I ⟶ℝ,
so_lambda: λ2x.t[x],
prop: ℙ,
false: False,
and: P ∧ Q,
bfalse: ff,
btrue: tt,
ifthenelse: if b then t else f fi ,
assert: ↑b,
isl: isl(x),
rciint: [l, ∞),
i-finite: i-finite(I),
iproper: iproper(I),
implies: P ⇒ Q,
uall: ∀[x:A]. B[x],
member: t ∈ T,
uimplies: b supposing a,
all: ∀x:A. B[x]
Lemmas referenced :
arctangent0,
rleq-implies-rleq,
member_rciint_lemma,
real_term_value_minus_lemma,
real_term_value_add_lemma,
real_term_value_var_lemma,
real_term_value_const_lemma,
real_term_value_mul_lemma,
real_term_value_sub_lemma,
real_polynomial_null,
rmul-rinv,
rminus_functionality,
req_transitivity,
rleq_functionality,
req-iff-rsub-is-0,
itermMinus_wf,
itermAdd_wf,
itermVar_wf,
itermConstant_wf,
itermMultiply_wf,
itermSubtract_wf,
rminus_wf,
rinv_wf2,
rmul-zero-both,
rmul_wf,
rmul_preserves_rleq,
req_wf,
rnexp_functionality,
radd_functionality,
rdiv_functionality,
rsub_functionality,
req_functionality,
function-is-continuous,
derivative_functionality2,
subinterval-riiint,
req_weakening,
riiint_wf,
derivative-id,
derivative-sub,
derivative-arctangent,
radd_functionality_wrt_rleq,
rleq_weakening_equal,
rless_functionality_wrt_implies,
rless-int,
trivial-rless-radd,
rleq_wf,
nat_plus_wf,
less_than'_wf,
set_wf,
rless_wf,
le_wf,
rnexp_wf,
radd_wf,
rdiv_wf,
i-member_wf,
arctangent_wf,
rsub_wf,
false_wf,
true_wf,
int-to-real_wf,
rciint_wf,
derivative-implies-increasing,
real_wf,
rnexp2-nonneg
Rules used in proof :
voidEquality,
isect_memberEquality,
intEquality,
int_eqEquality,
approximateComputation,
baseClosed,
imageMemberEquality,
equalitySymmetry,
equalityTransitivity,
axiomEquality,
minusEquality,
applyEquality,
independent_pairEquality,
inrFormation,
independent_isectElimination,
independent_pairFormation,
dependent_set_memberEquality,
setEquality,
because_Cache,
rename,
setElimination,
lambdaEquality,
productEquality,
voidElimination,
productElimination,
sqequalRule,
independent_functionElimination,
natural_numberEquality,
isectElimination,
hypothesis,
hypothesisEquality,
thin,
dependent_functionElimination,
sqequalHypSubstitution,
extract_by_obid,
cut,
introduction,
isect_memberFormation,
lambdaFormation,
sqequalReflexivity,
computationStep,
sqequalTransitivity,
sqequalSubstitution
Latex:
\mforall{}x:\mBbbR{}. arctangent(x) \mleq{} x supposing r0 \mleq{} x
Date html generated:
2018_05_22-PM-03_03_13
Last ObjectModification:
2018_05_20-PM-11_10_11
Theory : reals_2
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