Nuprl Lemma : rcos-nonpositive
∀x:{x:ℝ| x ∈ [π/2, π + π/2]} . (rcos(x) ≤ r0)
Proof
Definitions occuring in Statement :
pi: π,
halfpi: π/2,
rcos: rcos(x),
rccint: [l, u],
i-member: r ∈ I,
rleq: x ≤ y,
radd: a + b,
int-to-real: r(n),
real: ℝ,
all: ∀x:A. B[x],
set: {x:A| B[x]} ,
natural_number: $n
Definitions unfolded in proof :
all: ∀x:A. B[x],
member: t ∈ T,
uall: ∀[x:A]. B[x],
top: Top,
and: P ∧ Q,
cand: A c∧ B,
uiff: uiff(P;Q),
rev_uimplies: rev_uimplies(P;Q),
uimplies: b supposing a,
itermConstant: "const",
req_int_terms: t1 ≡ t2,
false: False,
implies: P ⇒ Q,
not: ¬A,
prop: ℙ,
so_lambda: λ2x.t[x],
so_apply: x[s],
pi: π
Lemmas referenced :
rcos-nonneg,
rsub_wf,
pi_wf,
member_rccint_lemma,
radd-preserves-rleq,
rminus_wf,
halfpi_wf,
rleq-implies-rleq,
radd_wf,
real_term_polynomial,
itermSubtract_wf,
itermAdd_wf,
itermVar_wf,
int-to-real_wf,
real_term_value_const_lemma,
real_term_value_sub_lemma,
real_term_value_add_lemma,
real_term_value_var_lemma,
req-iff-rsub-is-0,
i-member_wf,
rccint_wf,
set_wf,
real_wf,
int-rmul_wf,
rmul_wf,
rleq_functionality,
itermMinus_wf,
real_term_value_minus_lemma,
radd_functionality,
int-rmul-req,
req_weakening,
itermMultiply_wf,
itermConstant_wf,
real_term_value_mul_lemma,
rcos_wf,
rcos_functionality,
rcos-shift-pi
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation,
cut,
introduction,
extract_by_obid,
sqequalHypSubstitution,
dependent_functionElimination,
thin,
setElimination,
rename,
dependent_set_memberEquality,
isectElimination,
hypothesisEquality,
hypothesis,
isect_memberEquality,
voidElimination,
voidEquality,
productElimination,
independent_isectElimination,
independent_pairFormation,
natural_numberEquality,
sqequalRule,
computeAll,
lambdaEquality,
int_eqEquality,
because_Cache,
intEquality
Latex:
\mforall{}x:\{x:\mBbbR{}| x \mmember{} [\mpi{}/2, \mpi{} + \mpi{}/2]\} . (rcos(x) \mleq{} r0)
Date html generated:
2017_10_04-PM-10_25_33
Last ObjectModification:
2017_07_28-AM-08_49_25
Theory : reals_2
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