Nuprl Lemma : rcos-positive-before-half-pi
∀x:{x:ℝ| x ∈ [r0, π/2)} . (r0 < rcos(x))
Proof
Definitions occuring in Statement :
halfpi: π/2,
rcos: rcos(x),
rcoint: [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]} ,
natural_number: $n
Definitions unfolded in proof :
member: t ∈ T,
uall: ∀[x:A]. B[x],
all: ∀x:A. B[x],
sq_stable: SqStable(P),
implies: P ⇒ Q,
top: Top,
and: P ∧ Q,
exists: ∃x:A. B[x],
squash: ↓T,
prop: ℙ,
so_lambda: λ2x.t[x],
so_apply: x[s],
guard: {T},
uimplies: b supposing a,
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q,
itermConstant: "const",
req_int_terms: t1 ≡ t2,
false: False,
not: ¬A,
uiff: uiff(P;Q),
converges-to: lim n→∞.x[n] = y,
sq_exists: ∃x:{A| B[x]},
nat: ℕ,
nat_plus: ℕ+,
rneq: x ≠ y,
or: P ∨ Q,
rless: x < y,
ge: i ≥ j ,
decidable: Dec(P),
satisfiable_int_formula: satisfiable_int_formula(fmla),
rleq: x ≤ y,
rnonneg: rnonneg(x),
le: A ≤ B,
subtype_rel: A ⊆r B,
real: ℝ,
rev_uimplies: rev_uimplies(P;Q),
rge: x ≥ y
Lemmas referenced :
rcos-seq-converges-to-half-pi,
rcos-seq_wf,
nat_wf,
half-pi_wf,
halfpi_wf,
sq_stable__rless,
int-to-real_wf,
rcos_wf,
member_rcoint_lemma,
rcos-seq-positive,
set_wf,
real_wf,
i-member_wf,
rcoint_wf,
member_rccint_lemma,
rleq_wf,
converges-to_functionality,
req_weakening,
req_inversion,
halfpi-half-pi,
radd-preserves-rless,
rsub_wf,
rless_functionality,
radd_wf,
real_term_polynomial,
itermSubtract_wf,
itermAdd_wf,
itermVar_wf,
itermConstant_wf,
real_term_value_const_lemma,
real_term_value_sub_lemma,
real_term_value_add_lemma,
real_term_value_var_lemma,
req-iff-rsub-is-0,
small-reciprocal-real,
rless_wf,
sq_stable__all,
le_wf,
rabs_wf,
rdiv_wf,
rless-int,
nat_properties,
nat_plus_properties,
decidable__lt,
satisfiable-full-omega-tt,
intformand_wf,
intformnot_wf,
intformless_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__rleq,
less_than'_wf,
nat_plus_wf,
squash_wf,
sq_stable__less_than,
decidable__le,
intformle_wf,
int_formula_prop_le_lemma,
rabs-difference-bound-rleq,
rleq_functionality_wrt_implies,
rleq_weakening_equal,
rleq_weakening,
rsub_functionality_wrt_rleq,
rleq_weakening_rless
Rules used in proof :
cut,
introduction,
extract_by_obid,
hypothesis,
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaEquality,
sqequalHypSubstitution,
isectElimination,
thin,
hypothesisEquality,
because_Cache,
lambdaFormation,
setElimination,
rename,
dependent_functionElimination,
natural_numberEquality,
independent_functionElimination,
isect_memberEquality,
voidElimination,
voidEquality,
productElimination,
sqequalRule,
imageMemberEquality,
baseClosed,
imageElimination,
dependent_set_memberEquality,
independent_pairFormation,
productEquality,
independent_isectElimination,
computeAll,
int_eqEquality,
intEquality,
functionEquality,
inrFormation,
unionElimination,
dependent_pairFormation,
independent_pairEquality,
applyEquality,
minusEquality,
axiomEquality,
equalityTransitivity,
equalitySymmetry,
addEquality
Latex:
\mforall{}x:\{x:\mBbbR{}| x \mmember{} [r0, \mpi{}/2)\} . (r0 < rcos(x))
Date html generated:
2017_10_04-PM-10_25_19
Last ObjectModification:
2017_07_28-AM-08_49_05
Theory : reals_2
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