Nuprl Lemma : rcos-positive-near-0
∀x:{x:ℝ| x ∈ (r(-1), r1)} . (r0 < rcos(x))
Proof
Definitions occuring in Statement :
rcos: rcos(x),
rooint: (l, u),
i-member: r ∈ I,
rless: x < y,
int-to-real: r(n),
real: ℝ,
all: ∀x:A. B[x],
set: {x:A| B[x]} ,
minus: -n,
natural_number: $n
Definitions unfolded in proof :
all: ∀x:A. B[x],
member: t ∈ T,
uall: ∀[x:A]. B[x],
prop: ℙ,
so_lambda: λ2x.t[x],
so_apply: x[s],
uimplies: b supposing a,
uiff: uiff(P;Q),
and: P ∧ Q,
sq_stable: SqStable(P),
implies: P ⇒ Q,
top: Top,
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q,
cand: A c∧ B,
itermConstant: "const",
req_int_terms: t1 ≡ t2,
false: False,
not: ¬A,
squash: ↓T,
guard: {T}
Lemmas referenced :
rabs-difference-rcos-rleq,
int-to-real_wf,
set_wf,
real_wf,
i-member_wf,
rooint_wf,
rabs_wf,
rsub_wf,
rcos_wf,
rleq_functionality,
rabs_functionality,
rsub_functionality,
req_weakening,
rcos0,
sq_stable__rless,
member_rooint_lemma,
rabs-difference-bound-iff,
rless-implies-rless,
real_term_polynomial,
itermSubtract_wf,
itermVar_wf,
itermConstant_wf,
real_term_value_const_lemma,
real_term_value_sub_lemma,
real_term_value_var_lemma,
req-iff-rsub-is-0,
radd_wf,
itermAdd_wf,
real_term_value_add_lemma,
rless_transitivity2
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation,
cut,
introduction,
extract_by_obid,
sqequalHypSubstitution,
dependent_functionElimination,
thin,
setElimination,
rename,
hypothesisEquality,
hypothesis,
isectElimination,
natural_numberEquality,
sqequalRule,
lambdaEquality,
minusEquality,
because_Cache,
independent_isectElimination,
productElimination,
independent_functionElimination,
isect_memberEquality,
voidElimination,
voidEquality,
computeAll,
int_eqEquality,
intEquality,
independent_pairFormation,
imageMemberEquality,
baseClosed,
imageElimination
Latex:
\mforall{}x:\{x:\mBbbR{}| x \mmember{} (r(-1), r1)\} . (r0 < rcos(x))
Date html generated:
2017_10_04-PM-10_22_00
Last ObjectModification:
2017_07_28-AM-08_48_22
Theory : reals_2
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