Nuprl Lemma : rcos-nonneg-upto-half-pi
∀x:{x:ℝ| x ∈ [r0, π/2]} . (r0 ≤ rcos(x))
Proof
Definitions occuring in Statement :
halfpi: π/2,
rcos: rcos(x),
rccint: [l, u],
i-member: r ∈ I,
rleq: 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 :
all: ∀x:A. B[x],
uall: ∀[x:A]. B[x],
member: t ∈ T,
sq_stable: SqStable(P),
implies: P ⇒ Q,
uiff: uiff(P;Q),
and: P ∧ Q,
uimplies: b supposing a,
iproper: iproper(I),
top: Top,
prop: ℙ,
so_lambda: λ2x.t[x],
rfun: I ⟶ℝ,
so_apply: x[s],
rev_uimplies: rev_uimplies(P;Q),
squash: ↓T,
continuous: f[x] continuous for x ∈ I,
i-approx: i-approx(I;n),
rccint: [l, u],
nat_plus: ℕ+,
less_than: a < b,
less_than': less_than'(a;b),
true: True,
iff: P ⇐⇒ Q,
guard: {T},
exists: ∃x:A. B[x],
sq_exists: ∃x:{A| B[x]},
rneq: x ≠ y,
or: P ∨ Q,
rev_implies: P ⇐ Q,
rless: x < y,
decidable: Dec(P),
satisfiable_int_formula: satisfiable_int_formula(fmla),
false: False,
not: ¬A,
rleq: x ≤ y,
rnonneg: rnonneg(x),
le: A ≤ B,
subtype_rel: A ⊆r B,
cand: A c∧ B,
i-member: r ∈ I,
rsub: x - y,
rge: x ≥ y,
rgt: x > y
Lemmas referenced :
sq_stable__rleq,
int-to-real_wf,
rcos_wf,
halfpi-positive,
rleq-iff-all-rless,
function-is-continuous,
rccint_wf,
halfpi_wf,
left_endpoint_rccint_lemma,
right_endpoint_rccint_lemma,
i-finite_wf,
real_wf,
i-member_wf,
req_functionality,
rcos_functionality,
req_weakening,
req_wf,
set_wf,
rless_wf,
less_than_wf,
rccint-icompact,
rleq_weakening_rless,
icompact_wf,
member_rccint_lemma,
small-reciprocal-real,
sq_stable__and,
all_wf,
rleq_wf,
rabs_wf,
rsub_wf,
rdiv_wf,
rless-int,
nat_plus_properties,
decidable__lt,
satisfiable-full-omega-tt,
intformand_wf,
intformnot_wf,
intformless_wf,
itermConstant_wf,
itermVar_wf,
int_formula_prop_and_lemma,
int_formula_prop_not_lemma,
int_formula_prop_less_lemma,
int_term_value_constant_lemma,
int_term_value_var_lemma,
int_formula_prop_wf,
sq_stable__rless,
sq_stable__all,
less_than'_wf,
nat_plus_wf,
squash_wf,
rleq_weakening_equal,
rless_transitivity2,
radd_wf,
rminus_wf,
rless_functionality,
rabs_functionality,
rsub_functionality,
rcos-halfpi,
radd_functionality,
rminus-zero,
rleq_functionality,
radd_comm,
radd-zero-both,
rless-cases,
radd-preserves-rless,
req_transitivity,
radd-rminus-both,
radd-rminus-assoc,
radd-ac,
req_inversion,
radd-assoc,
rabs-rleq-iff,
radd-preserves-rleq,
rmul_wf,
uiff_transitivity,
rminus-as-rmul,
rmul-identity1,
rmul-distrib2,
rmul_functionality,
radd-int,
rmul-zero-both,
rleq_functionality_wrt_implies,
rabs-as-rmax,
rleq-rmax,
rcos-positive-before-half-pi,
member_rcoint_lemma,
radd_functionality_wrt_rless2
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation,
setElimination,
thin,
rename,
cut,
introduction,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
natural_numberEquality,
hypothesis,
hypothesisEquality,
independent_functionElimination,
because_Cache,
productElimination,
independent_isectElimination,
dependent_functionElimination,
sqequalRule,
isect_memberEquality,
voidElimination,
voidEquality,
lambdaEquality,
setEquality,
imageMemberEquality,
baseClosed,
imageElimination,
dependent_set_memberEquality,
independent_pairFormation,
functionEquality,
productEquality,
inrFormation,
unionElimination,
dependent_pairFormation,
int_eqEquality,
intEquality,
computeAll,
independent_pairEquality,
applyEquality,
minusEquality,
axiomEquality,
equalityTransitivity,
equalitySymmetry,
promote_hyp,
addLevel,
levelHypothesis,
addEquality
Latex:
\mforall{}x:\{x:\mBbbR{}| x \mmember{} [r0, \mpi{}/2]\} . (r0 \mleq{} rcos(x))
Date html generated:
2016_10_26-PM-00_24_18
Last ObjectModification:
2016_09_12-PM-05_43_31
Theory : reals_2
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