Nuprl Lemma : blend-close-reals
∀[k:ℕ+]. ∀[x,y:ℝ]. ((|x - y| ≤ (r1/r(k))) ⇒ 3-regular-seq(blend-seq(k;x;y)))
Proof
Definitions occuring in Statement :
blend-seq: blend-seq(k;x;y),
rdiv: (x/y),
rleq: x ≤ y,
rabs: |x|,
rsub: x - y,
int-to-real: r(n),
real: ℝ,
regular-int-seq: k-regular-seq(f),
nat_plus: ℕ+,
uall: ∀[x:A]. B[x],
implies: P ⇒ Q,
natural_number: $n
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
all: ∀x:A. B[x],
implies: P ⇒ Q,
regular-int-seq: k-regular-seq(f),
le: A ≤ B,
and: P ∧ Q,
not: ¬A,
false: False,
nat_plus: ℕ+,
real: ℝ,
subtype_rel: A ⊆r B,
prop: ℙ,
uimplies: b supposing a,
rneq: x ≠ y,
guard: {T},
or: P ∨ Q,
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q,
decidable: Dec(P),
satisfiable_int_formula: satisfiable_int_formula(fmla),
exists: ∃x:A. B[x],
top: Top,
nat: ℕ,
uiff: uiff(P;Q),
rev_uimplies: rev_uimplies(P;Q),
ge: i ≥ j ,
sq_type: SQType(T),
squash: ↓T,
true: True,
so_lambda: λ2x.t[x],
so_apply: x[s],
less_than: a < b,
less_than': less_than'(a;b),
sq_stable: SqStable(P),
blend-seq: blend-seq(k;x;y),
bool: 𝔹,
unit: Unit,
it: ⋅,
btrue: tt,
ifthenelse: if b then t else f fi ,
bfalse: ff,
bnot: ¬bb,
assert: ↑b
Lemmas referenced :
nat_plus_wf,
less_than'_wf,
absval_wf,
subtract_wf,
le_wf,
less_than_wf,
regular-int-seq_wf,
rleq_wf,
rabs_wf,
rsub_wf,
rdiv_wf,
int-to-real_wf,
rless-int,
nat_plus_properties,
decidable__lt,
full-omega-unsat,
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,
rless_wf,
real_wf,
nat_wf,
decidable__le,
intformle_wf,
int_formula_prop_le_lemma,
decidable__equal_int,
intformeq_wf,
itermAdd_wf,
itermSubtract_wf,
itermMultiply_wf,
itermMinus_wf,
int_formula_prop_eq_lemma,
int_term_value_add_lemma,
int_term_value_subtract_lemma,
int_term_value_mul_lemma,
int_term_value_minus_lemma,
add-is-int-iff,
subtract-is-int-iff,
multiply-is-int-iff,
false_wf,
and_wf,
equal_wf,
le_functionality,
le_weakening,
int-triangle-inequality,
add_functionality_wrt_le,
subtype_base_sq,
int_subtype_base,
squash_wf,
true_wf,
absval_mul,
iff_weakening_equal,
set_subtype_base,
absval-non-neg,
absval_pos,
nat_plus_subtype_nat,
close-reals-iff,
mul_preserves_le,
mul_cancel_in_le,
sq_stable__regular-int-seq,
lt_int_wf,
bool_wf,
eqtt_to_assert,
assert_of_lt_int,
eqff_to_assert,
bool_cases_sqequal,
bool_subtype_base,
assert-bnot,
blend-seq_wf,
absval-minus
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
cut,
introduction,
extract_by_obid,
hypothesis,
lambdaFormation,
sqequalHypSubstitution,
sqequalRule,
productElimination,
thin,
independent_pairEquality,
lambdaEquality,
dependent_functionElimination,
hypothesisEquality,
voidElimination,
isectElimination,
multiplyEquality,
natural_numberEquality,
addEquality,
setElimination,
rename,
because_Cache,
applyEquality,
axiomEquality,
equalityTransitivity,
equalitySymmetry,
independent_isectElimination,
inrFormation,
independent_functionElimination,
unionElimination,
approximateComputation,
dependent_pairFormation,
int_eqEquality,
intEquality,
isect_memberEquality,
voidEquality,
independent_pairFormation,
dependent_set_memberEquality,
hyp_replacement,
pointwiseFunctionality,
promote_hyp,
baseApply,
closedConclusion,
baseClosed,
applyLambdaEquality,
instantiate,
cumulativity,
imageElimination,
imageMemberEquality,
universeEquality,
equalityElimination
Latex:
\mforall{}[k:\mBbbN{}\msupplus{}]. \mforall{}[x,y:\mBbbR{}]. ((|x - y| \mleq{} (r1/r(k))) {}\mRightarrow{} 3-regular-seq(blend-seq(k;x;y)))
Date html generated:
2017_10_03-AM-10_08_17
Last ObjectModification:
2017_07_05-PM-03_25_37
Theory : reals
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