Nuprl Lemma : rcos-seq-positive
∀n:ℕ. ((r0 < rcos-seq(n)) ∧ (∀t:{t:ℝ| t ∈ [r0, rcos-seq(n)]} . (r0 < rcos(t))))
Proof
Definitions occuring in Statement :
rcos-seq: rcos-seq(n),
rcos: rcos(x),
rccint: [l, u],
i-member: r ∈ I,
rless: x < y,
int-to-real: r(n),
real: ℝ,
nat: ℕ,
all: ∀x:A. B[x],
and: P ∧ Q,
set: {x:A| B[x]} ,
natural_number: $n
Definitions unfolded in proof :
all: ∀x:A. B[x],
member: t ∈ T,
top: Top,
implies: P ⇒ Q,
and: P ∧ Q,
uall: ∀[x:A]. B[x],
nat: ℕ,
decidable: Dec(P),
or: P ∨ Q,
uimplies: b supposing a,
not: ¬A,
satisfiable_int_formula: satisfiable_int_formula(fmla),
exists: ∃x:A. B[x],
false: False,
prop: ℙ,
so_lambda: λ2x.t[x],
so_apply: x[s],
rcos-seq: rcos-seq(n),
int_nzero: ℤ-o,
true: True,
nequal: a ≠ b ∈ T ,
sq_type: SQType(T),
guard: {T},
rneq: x ≠ y,
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q,
less_than: a < b,
squash: ↓T,
less_than': less_than'(a;b),
nat_plus: ℕ+,
sq_stable: SqStable(P),
uiff: uiff(P;Q),
rless: x < y,
sq_exists: ∃x:A [B[x]],
subtype_rel: A ⊆r B,
real: ℝ,
cand: A c∧ B,
rge: x ≥ y,
rgt: x > y,
rfun: I ⟶ℝ,
rev_uimplies: rev_uimplies(P;Q),
rnonneg: rnonneg(x),
rleq: x ≤ y,
ge: i ≥ j ,
riiint: (-∞, ∞),
i-approx: i-approx(I;n),
subtract: n - m,
le: A ≤ B,
continuous: f[x] continuous for x ∈ I,
rdiv: (x/y),
req_int_terms: t1 ≡ t2,
ifun: ifun(f;I),
real-fun: real-fun(f;a;b),
rccint: [l, u],
i-member: r ∈ I
Lemmas referenced :
member_rccint_lemma,
istype-void,
rless_wf,
int-to-real_wf,
rcos-seq_wf,
subtract_wf,
decidable__le,
full-omega-unsat,
intformand_wf,
intformnot_wf,
intformle_wf,
itermConstant_wf,
itermSubtract_wf,
itermVar_wf,
intformless_wf,
istype-int,
int_formula_prop_and_lemma,
int_formula_prop_not_lemma,
int_formula_prop_le_lemma,
int_term_value_constant_lemma,
int_term_value_subtract_lemma,
int_term_value_var_lemma,
int_formula_prop_less_lemma,
int_formula_prop_wf,
istype-le,
real_wf,
i-member_wf,
rccint_wf,
rcos_wf,
istype-less_than,
primrec-wf2,
all_wf,
istype-nat,
primrec0_lemma,
int-rdiv_wf,
subtype_base_sq,
int_subtype_base,
equal-wf-base,
true_wf,
nequal_wf,
rdiv_wf,
rless-int,
rless-int-fractions2,
less_than_wf,
rless_functionality,
req_weakening,
int-rdiv-req,
sq_stable__rless,
rcos-positive-initially,
set_wf,
rleq_wf,
rleq_functionality,
subtract-add-cancel,
rcos-seq-step,
sq_stable__less_than,
nat_plus_properties,
radd_wf,
trivial-rless-radd,
rleq_weakening_equal,
rleq_weakening_rless,
rless_functionality_wrt_implies,
radd_functionality_wrt_rless1,
function-is-continuous,
riiint_wf,
req_functionality,
rcos_functionality,
req_wf,
r-archimedean,
small-reciprocal-real,
squash_wf,
nat_plus_wf,
less_than'_wf,
sq_stable__rleq,
sq_stable__all,
rsub_wf,
rabs_wf,
sq_stable__and,
i-approx_wf,
icompact_wf,
int_term_value_add_lemma,
int_term_value_minus_lemma,
itermAdd_wf,
itermMinus_wf,
nat_properties,
rleq-int,
rccint-icompact,
le-add-cancel,
add-zero,
add-associates,
add_functionality_wrt_le,
add-commutes,
minus-one-mul-top,
zero-add,
minus-one-mul,
minus-add,
condition-implies-le,
not-lt-2,
false_wf,
decidable__lt,
rabs-difference-bound-rleq,
rmin_ub,
rmin_strict_ub,
rmin_wf,
rless_transitivity2,
rless_transitivity1,
rleq_functionality_wrt_implies,
rsub_functionality_wrt_rleq,
int_term_value_mul_lemma,
rinv_wf2,
rminus_wf,
itermMultiply_wf,
rmul_wf,
rmul_preserves_rleq,
req_transitivity,
rmul-int,
req_inversion,
rmul_functionality,
rminus-int,
rminus_functionality,
radd-int,
radd_functionality,
int-rinv-cancel2,
req-iff-rsub-is-0,
real_polynomial_null,
real_term_value_sub_lemma,
real_term_value_mul_lemma,
real_term_value_var_lemma,
real_term_value_const_lemma,
real_term_value_minus_lemma,
real_term_value_add_lemma,
radd_functionality_wrt_rleq,
rmul-rinv,
rless-cases,
rsin_wf,
rmax_wf,
left_endpoint_rccint_lemma,
right_endpoint_rccint_lemma,
rsin_functionality,
ifun_wf,
rmin-rleq-rmax,
integral_wf,
rsub_functionality,
ftc-total-integral,
derivative-minus-minus,
derivative-rcos,
square-rless-1-iff,
rleq_weakening,
radd-preserves-rless,
radd-preserves-req,
rsin-rcos-pythag,
exp_wf2,
le_wf,
rnexp_wf,
rnexp-rless,
rnexp-int,
exp-zero,
Riemann-integral_wf,
integral-is-Riemann,
rabs-bounds,
Riemann-integral-rless,
rleq_transitivity,
rabs-rsin-rleq,
equal_wf,
trivial-rsub-rless,
rinv-as-rdiv,
subtype_rel_sets
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation_alt,
cut,
thin,
sqequalRule,
introduction,
extract_by_obid,
sqequalHypSubstitution,
dependent_functionElimination,
isect_memberEquality_alt,
voidElimination,
hypothesis,
rename,
setElimination,
productIsType,
universeIsType,
isectElimination,
natural_numberEquality,
dependent_set_memberEquality_alt,
hypothesisEquality,
unionElimination,
independent_isectElimination,
approximateComputation,
independent_functionElimination,
dependent_pairFormation_alt,
lambdaEquality_alt,
int_eqEquality,
independent_pairFormation,
functionIsType,
setIsType,
closedConclusion,
because_Cache,
productElimination,
productEquality,
setEquality,
inhabitedIsType,
isect_memberEquality,
voidEquality,
dependent_set_memberEquality,
addLevel,
lambdaFormation,
instantiate,
cumulativity,
intEquality,
equalityTransitivity,
equalitySymmetry,
baseClosed,
inrFormation,
imageMemberEquality,
multiplyEquality,
imageElimination,
lambdaEquality,
addEquality,
applyEquality,
equalityIsType1,
axiomEquality,
independent_pairEquality,
functionEquality,
dependent_pairFormation,
minusEquality,
promote_hyp
Latex:
\mforall{}n:\mBbbN{}. ((r0 < rcos-seq(n)) \mwedge{} (\mforall{}t:\{t:\mBbbR{}| t \mmember{} [r0, rcos-seq(n)]\} . (r0 < rcos(t))))
Date html generated:
2019_10_30-AM-11_43_06
Last ObjectModification:
2018_11_08-PM-05_58_27
Theory : reals_2
Home
Index