Nuprl Lemma : rcos-strictly-decreasing
rcos(x) strictly-decreasing for x ∈ [r0, π]
Proof
Definitions occuring in Statement :
pi: π,
rcos: rcos(x),
strictly-decreasing-on-interval: f[x] strictly-decreasing for x ∈ I,
rccint: [l, u],
int-to-real: r(n),
natural_number: $n
Definitions unfolded in proof :
iproper: iproper(I),
all: ∀x:A. B[x],
member: t ∈ T,
top: Top,
implies: P ⇒ Q,
prop: ℙ,
uall: ∀[x:A]. B[x],
so_lambda: λ2x.t[x],
rfun: I ⟶ℝ,
so_apply: x[s],
subinterval: I ⊆ J ,
true: True,
and: P ∧ Q,
ifun: ifun(f;I),
real-fun: real-fun(f;a;b),
uimplies: b supposing a,
i-finite: i-finite(I),
rccint: [l, u],
isl: isl(x),
assert: ↑b,
ifthenelse: if b then t else f fi ,
btrue: tt,
uiff: uiff(P;Q),
rev_uimplies: rev_uimplies(P;Q),
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q
Lemmas referenced :
left_endpoint_rccint_lemma,
right_endpoint_rccint_lemma,
pi-positive,
i-finite_wf,
rccint_wf,
int-to-real_wf,
pi_wf,
derivative-implies-strictly-decreasing-closed,
rless_wf,
rcos_wf,
real_wf,
i-member_wf,
rminus_wf,
rsin_wf,
set_wf,
rooint_wf,
derivative_functionality_wrt_subinterval,
riiint_wf,
member_rccint_lemma,
member_riiint_lemma,
rleq_wf,
deriviative-rcos,
req_wf,
left-endpoint_wf,
right-endpoint_wf,
req_weakening,
req_functionality,
rminus_functionality,
rsin_functionality,
radd-preserves-rleq,
radd_wf,
rmul_wf,
uiff_transitivity,
rleq_functionality,
req_transitivity,
radd_functionality,
rminus-as-rmul,
req_inversion,
rmul-identity1,
rmul-distrib2,
rmul_functionality,
radd-int,
rmul-zero-both,
radd_comm,
radd-zero-both,
rsin-nonneg,
radd-preserves-rless,
rsin-positive,
rless_functionality
Rules used in proof :
cut,
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
sqequalRule,
introduction,
extract_by_obid,
sqequalHypSubstitution,
dependent_functionElimination,
thin,
isect_memberEquality,
voidElimination,
voidEquality,
hypothesis,
lambdaFormation,
isectElimination,
natural_numberEquality,
dependent_set_memberEquality,
hypothesisEquality,
lambdaEquality,
setElimination,
rename,
setEquality,
because_Cache,
independent_functionElimination,
productEquality,
independent_isectElimination,
independent_pairFormation,
productElimination,
minusEquality,
addEquality,
addLevel,
levelHypothesis
Latex:
rcos(x) strictly-decreasing for x \mmember{} [r0, \mpi{}]
Date html generated:
2016_10_26-PM-00_26_09
Last ObjectModification:
2016_10_12-PM-03_51_57
Theory : reals_2
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