Nuprl Lemma : rcos-positive
∀x:{x:ℝ| x ∈ (-(π/2), π/2)} . (r0 < rcos(x))
Proof
Definitions occuring in Statement :
halfpi: π/2,
rcos: rcos(x),
rooint: (l, u),
i-member: r ∈ I,
rless: x < y,
rminus: -(x),
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],
uimplies: b supposing a,
rneq: x ≠ y,
guard: {T},
or: P ∨ Q,
iff: P ⇐⇒ Q,
and: P ∧ Q,
rev_implies: P ⇐ Q,
implies: P ⇒ Q,
less_than: a < b,
squash: ↓T,
less_than': less_than'(a;b),
true: True,
prop: ℙ,
nat_plus: ℕ+,
so_lambda: λ2x.t[x],
so_apply: x[s],
sq_stable: SqStable(P),
cand: A c∧ B,
top: Top,
rsub: x - y,
rev_uimplies: rev_uimplies(P;Q),
uiff: uiff(P;Q),
rge: x ≥ y,
subtype_rel: A ⊆r B,
itermConstant: "const",
req_int_terms: t1 ≡ t2,
false: False,
not: ¬A
Lemmas referenced :
rless-cases,
int-to-real_wf,
rdiv_wf,
rless-int,
rless_wf,
rless-int-fractions2,
less_than_wf,
set_wf,
real_wf,
i-member_wf,
rooint_wf,
rminus_wf,
halfpi_wf,
rleq_wf,
sq_stable__rless,
member_rooint_lemma,
rleq_weakening_rless,
member_rcoint_lemma,
rcos-positive-before-half-pi,
rless-int-fractions3,
rminus-zero,
radd_functionality,
radd_comm,
req_weakening,
radd-zero-both,
rabs_functionality,
rless_functionality,
radd_wf,
rabs_wf,
rabs-rless-iff,
rminus-int,
true_wf,
squash_wf,
rmul-rdiv-cancel,
uiff_transitivity2,
rmul_comm,
rminus_functionality,
rmul_over_rminus,
rmul-rdiv-cancel2,
req_functionality,
uiff_transitivity,
rmul_wf,
req_wf,
rmul_preserves_req,
rleq_weakening,
rless_transitivity2,
rabs-difference-rcos-rleq,
rcos0,
rsub_functionality,
rleq_functionality,
rleq_weakening_equal,
rleq_functionality_wrt_implies,
rcos_wf,
rsub_wf,
rabs-difference-bound-rleq,
rless_functionality_wrt_implies,
rmul-int,
rmul-distrib,
req_transitivity,
rmul-one-both,
radd-int,
iff_weakening_equal,
iff_transitivity,
rmul_preserves_rless,
rminus_functionality_wrt_rleq,
real_term_polynomial,
itermSubtract_wf,
itermConstant_wf,
itermMinus_wf,
real_term_value_const_lemma,
real_term_value_sub_lemma,
real_term_value_minus_lemma,
req-iff-rsub-is-0,
rless-implies-rless,
itermVar_wf,
real_term_value_var_lemma,
req_inversion,
rcos-rminus
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation,
cut,
introduction,
extract_by_obid,
sqequalHypSubstitution,
dependent_functionElimination,
thin,
isectElimination,
natural_numberEquality,
hypothesis,
independent_isectElimination,
sqequalRule,
inrFormation,
because_Cache,
productElimination,
independent_functionElimination,
independent_pairFormation,
imageMemberEquality,
hypothesisEquality,
baseClosed,
dependent_set_memberEquality,
multiplyEquality,
setElimination,
rename,
unionElimination,
lambdaEquality,
productEquality,
imageElimination,
voidEquality,
voidElimination,
isect_memberEquality,
minusEquality,
levelHypothesis,
addLevel,
equalitySymmetry,
equalityTransitivity,
applyEquality,
universeEquality,
addEquality,
computeAll,
intEquality,
int_eqEquality
Latex:
\mforall{}x:\{x:\mBbbR{}| x \mmember{} (-(\mpi{}/2), \mpi{}/2)\} . (r0 < rcos(x))
Date html generated:
2017_10_04-PM-10_25_39
Last ObjectModification:
2017_07_28-AM-08_49_28
Theory : reals_2
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