Nuprl Lemma : rcos-positive-initially
∀x:{x:ℝ| x ∈ [r0, (r(1570796326)/r(1000000000))]} . (r0 < rcos(x))
Proof
Definitions occuring in Statement :
rcos: rcos(x),
rccint: [l, u],
i-member: r ∈ I,
rdiv: (x/y),
rless: x < y,
int-to-real: r(n),
real: ℝ,
all: ∀x:A. B[x],
set: {x:A| B[x]} ,
natural_number: $n
Definitions unfolded in proof :
so_apply: x[s],
true: True,
less_than': less_than'(a;b),
squash: ↓T,
less_than: a < b,
implies: P ⇒ Q,
rev_implies: P ⇐ Q,
and: P ∧ Q,
iff: P ⇐⇒ Q,
or: P ∨ Q,
guard: {T},
rneq: x ≠ y,
uimplies: b supposing a,
so_lambda: λ2x.t[x],
uall: ∀[x:A]. B[x],
prop: ℙ,
member: t ∈ T,
all: ∀x:A. B[x],
nat_plus: ℕ+,
rfun: I ⟶ℝ,
pi2: snd(t),
pi1: fst(t),
outl: outl(x),
rooint: (l, u),
endpoints: endpoints(I),
left-endpoint: left-endpoint(I),
right-endpoint: right-endpoint(I),
iproper: iproper(I),
top: Top,
subinterval: I ⊆ J ,
rev_uimplies: rev_uimplies(P;Q),
uiff: uiff(P;Q),
rgt: x > y,
rge: x ≥ y,
cand: A c∧ B,
strictly-decreasing-on-interval: f[x] strictly-decreasing for x ∈ I,
sq_stable: SqStable(P),
subtype_rel: A ⊆r B,
sq_exists: ∃x:{A| B[x]},
rless: x < y,
real: ℝ,
int-to-real: r(n)
Lemmas referenced :
rless_wf,
rless-int,
rdiv_wf,
int-to-real_wf,
rccint_wf,
i-member_wf,
real_wf,
set_wf,
rsin-positive-near-11/7,
rcos-positive-near-0,
less_than_wf,
rless-int-fractions3,
rless-cases,
rsin_wf,
rminus_wf,
rcos_wf,
rooint_wf,
derivative-implies-strictly-decreasing,
i-finite_wf,
rless-int-fractions,
deriviative-rcos,
member_riiint_lemma,
member_rooint_lemma,
riiint_wf,
derivative_functionality_wrt_subinterval,
req_wf,
req_weakening,
rsin_functionality,
rminus_functionality,
req_functionality,
function-is-continuous,
rmul-distrib2,
rmul-identity1,
req_inversion,
rminus-as-rmul,
radd_functionality,
req_transitivity,
radd-int,
rmul_functionality,
rmul-zero-both,
radd_comm,
radd-zero-both,
rless_functionality,
rmul_wf,
radd_wf,
radd-preserves-rless,
rleq_weakening_rless,
rleq_weakening_equal,
rless_functionality_wrt_implies,
member_rccint_lemma,
sq_stable__rless
Rules used in proof :
baseClosed,
hypothesisEquality,
imageMemberEquality,
independent_pairFormation,
independent_functionElimination,
productElimination,
because_Cache,
dependent_functionElimination,
inrFormation,
independent_isectElimination,
natural_numberEquality,
lambdaEquality,
sqequalRule,
hypothesis,
thin,
isectElimination,
sqequalHypSubstitution,
extract_by_obid,
introduction,
cut,
lambdaFormation,
sqequalReflexivity,
computationStep,
sqequalTransitivity,
sqequalSubstitution,
unionElimination,
rename,
setElimination,
multiplyEquality,
dependent_set_memberEquality,
setEquality,
productEquality,
voidEquality,
voidElimination,
isect_memberEquality,
levelHypothesis,
addLevel,
addEquality,
minusEquality,
equalitySymmetry,
equalityTransitivity,
imageElimination,
applyEquality,
dependent_set_memberFormation,
computeAll
Latex:
\mforall{}x:\{x:\mBbbR{}| x \mmember{} [r0, (r(1570796326)/r(1000000000))]\} . (r0 < rcos(x))
Date html generated:
2017_10_04-PM-10_22_07
Last ObjectModification:
2017_07_31-PM-06_23_57
Theory : reals_2
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