Nuprl Lemma : blended-real-req
∀[k:ℕ+]. ∀[x,y:ℝ].  blended-real(k;x;y) = y supposing |x - y| ≤ (r1/r(k))
Proof
Definitions occuring in Statement : 
blended-real: blended-real(k;x;y)
, 
rdiv: (x/y)
, 
rleq: x ≤ y
, 
rabs: |x|
, 
rsub: x - y
, 
req: x = y
, 
int-to-real: r(n)
, 
real: ℝ
, 
nat_plus: ℕ+
, 
uimplies: b supposing a
, 
uall: ∀[x:A]. B[x]
, 
natural_number: $n
Definitions unfolded in proof : 
uall: ∀[x:A]. B[x]
, 
member: t ∈ T
, 
uimplies: b supposing a
, 
all: ∀x:A. B[x]
, 
nat_plus: ℕ+
, 
less_than: a < b
, 
squash: ↓T
, 
less_than': less_than'(a;b)
, 
true: True
, 
and: P ∧ Q
, 
prop: ℙ
, 
cand: A c∧ B
, 
real: ℝ
, 
uiff: uiff(P;Q)
, 
guard: {T}
, 
implies: P 
⇒ Q
, 
rneq: x ≠ y
, 
or: P ∨ Q
, 
iff: P 
⇐⇒ Q
, 
rev_implies: P 
⇐ Q
, 
decidable: Dec(P)
, 
not: ¬A
, 
satisfiable_int_formula: satisfiable_int_formula(fmla)
, 
exists: ∃x:A. B[x]
, 
false: False
, 
top: Top
, 
subtype_rel: A ⊆r B
, 
so_lambda: λ2x.t[x]
, 
int_upper: {i...}
, 
le: A ≤ B
, 
so_apply: x[s]
, 
accelerate: accelerate(k;f)
, 
blended-real: blended-real(k;x;y)
, 
blend-seq: blend-seq(k;x;y)
, 
has-value: (a)↓
, 
bool: 𝔹
, 
unit: Unit
, 
it: ⋅
, 
btrue: tt
, 
ifthenelse: if b then t else f fi 
, 
bfalse: ff
, 
sq_type: SQType(T)
, 
bnot: ¬bb
, 
assert: ↑b
, 
nequal: a ≠ b ∈ T 
Lemmas referenced : 
real-regular, 
less_than_wf, 
req-iff-bdd-diff, 
blended-real_wf, 
accelerate_wf, 
regular-int-seq_wf, 
nat_plus_wf, 
accelerate-bdd-diff, 
req_transitivity, 
req_witness, 
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, 
eventually-equal-implies-bdd-diff, 
int_upper_wf, 
all_wf, 
equal_wf, 
less_than_transitivity1, 
value-type-has-value, 
int-value-type, 
lt_int_wf, 
bool_wf, 
eqtt_to_assert, 
assert_of_lt_int, 
int_upper_properties, 
itermMultiply_wf, 
intformle_wf, 
int_term_value_mul_lemma, 
int_formula_prop_le_lemma, 
eqff_to_assert, 
bool_cases_sqequal, 
subtype_base_sq, 
bool_subtype_base, 
assert-bnot, 
false_wf, 
not-lt-2, 
less-iff-le, 
add_functionality_wrt_le, 
add-associates, 
add-zero, 
add-swap, 
add-commutes, 
zero-add, 
le-add-cancel, 
int_subtype_base, 
equal-wf-base, 
true_wf
Rules used in proof : 
sqequalSubstitution, 
sqequalTransitivity, 
computationStep, 
sqequalReflexivity, 
isect_memberFormation, 
introduction, 
cut, 
lambdaFormation, 
extract_by_obid, 
sqequalHypSubstitution, 
isectElimination, 
thin, 
hypothesisEquality, 
dependent_set_memberEquality, 
natural_numberEquality, 
sqequalRule, 
independent_pairFormation, 
imageMemberEquality, 
baseClosed, 
hypothesis, 
because_Cache, 
independent_isectElimination, 
setElimination, 
rename, 
dependent_functionElimination, 
functionExtensionality, 
applyEquality, 
productElimination, 
independent_functionElimination, 
inrFormation, 
unionElimination, 
approximateComputation, 
dependent_pairFormation, 
lambdaEquality, 
int_eqEquality, 
intEquality, 
isect_memberEquality, 
voidElimination, 
voidEquality, 
equalityTransitivity, 
equalitySymmetry, 
callbyvalueReduce, 
sqleReflexivity, 
multiplyEquality, 
equalityElimination, 
promote_hyp, 
instantiate, 
cumulativity, 
divideEquality, 
addEquality, 
addLevel
Latex:
\mforall{}[k:\mBbbN{}\msupplus{}].  \mforall{}[x,y:\mBbbR{}].    blended-real(k;x;y)  =  y  supposing  |x  -  y|  \mleq{}  (r1/r(k))
Date html generated:
2017_10_03-AM-10_08_56
Last ObjectModification:
2017_07_05-PM-04_27_51
Theory : reals
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