Nuprl Lemma : period-rcos-is-2pi
∀a:ℝ. ((r0 < a) ⇒ (∀x:ℝ. (rcos(x + a) = rcos(x))) ⇒ (2 * π ≤ a))
Proof
Definitions occuring in Statement :
pi: π,
rcos: rcos(x),
rleq: x ≤ y,
rless: x < y,
int-rmul: k1 * a,
req: x = y,
radd: a + b,
int-to-real: r(n),
real: ℝ,
all: ∀x:A. B[x],
implies: P ⇒ Q,
natural_number: $n
Definitions unfolded in proof :
all: ∀x:A. B[x],
implies: P ⇒ Q,
member: t ∈ T,
prop: ℙ,
uall: ∀[x:A]. B[x],
so_lambda: λ2x.t[x],
so_apply: x[s],
uimplies: b supposing a,
uiff: uiff(P;Q),
and: P ∧ Q,
rev_uimplies: rev_uimplies(P;Q),
guard: {T}
Lemmas referenced :
period-rsin-is-2pi,
real_wf,
all_wf,
req_wf,
rcos_wf,
radd_wf,
rless_wf,
int-to-real_wf,
rminus_wf,
rsin_wf,
halfpi_wf,
req_functionality,
req_inversion,
rcos-shift-half-pi,
req_weakening,
uiff_transitivity,
radd-assoc,
req_transitivity,
radd-ac,
radd_functionality,
radd_comm,
rcos_functionality,
radd-preserves-req,
rminus_functionality,
rsin_functionality,
radd-rminus-assoc,
radd-rminus-both,
radd-zero-both
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation,
cut,
introduction,
extract_by_obid,
sqequalHypSubstitution,
dependent_functionElimination,
thin,
hypothesisEquality,
independent_functionElimination,
hypothesis,
isectElimination,
sqequalRule,
lambdaEquality,
natural_numberEquality,
independent_isectElimination,
productElimination,
because_Cache
Latex:
\mforall{}a:\mBbbR{}. ((r0 < a) {}\mRightarrow{} (\mforall{}x:\mBbbR{}. (rcos(x + a) = rcos(x))) {}\mRightarrow{} (2 * \mpi{} \mleq{} a))
Date html generated:
2016_10_26-PM-00_26_59
Last ObjectModification:
2016_09_12-PM-05_44_08
Theory : reals_2
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