Nuprl Lemma : approx-in-interval_wf
∀l:ℝ. ∀u:{u:ℝ| l ≤ u} . ∀x:{x:ℝ| x ∈ [l, u]} . ∀n:ℕ+.
(approx-in-interval(l;u;x;n) ∈ {y:ℝ| (y ∈ [l, u]) ∧ (|x - y| ≤ (r(2)/r(n)))} )
Proof
Definitions occuring in Statement :
approx-in-interval: approx-in-interval(l;u;x;n),
rccint: [l, u],
i-member: r ∈ I,
rdiv: (x/y),
rleq: x ≤ y,
rabs: |x|,
rsub: x - y,
int-to-real: r(n),
real: ℝ,
nat_plus: ℕ+,
all: ∀x:A. B[x],
and: P ∧ Q,
member: t ∈ T,
set: {x:A| B[x]} ,
natural_number: $n
Definitions unfolded in proof :
all: ∀x:A. B[x],
member: t ∈ T,
top: Top,
approx-in-interval: approx-in-interval(l;u;x;n),
has-value: (a)↓,
uall: ∀[x:A]. B[x],
uimplies: b supposing a,
real: ℝ,
nat_plus: ℕ+,
decidable: Dec(P),
or: P ∨ Q,
not: ¬A,
implies: P ⇒ Q,
satisfiable_int_formula: satisfiable_int_formula(fmla),
exists: ∃x:A. B[x],
false: False,
and: P ∧ Q,
prop: ℙ,
bool: 𝔹,
unit: Unit,
it: ⋅,
btrue: tt,
uiff: uiff(P;Q),
less_than: a < b,
less_than': less_than'(a;b),
true: True,
squash: ↓T,
cand: A c∧ B,
sq_stable: SqStable(P),
rneq: x ≠ y,
guard: {T},
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q,
bfalse: ff,
sq_type: SQType(T),
bnot: ¬bb,
ifthenelse: if b then t else f fi ,
assert: ↑b,
int_nzero: ℤ-o,
nequal: a ≠ b ∈ T ,
subtype_rel: A ⊆r B,
rev_uimplies: rev_uimplies(P;Q),
req_int_terms: t1 ≡ t2,
rational-upper-approx: above x within 1/n,
rational-approx: (x within 1/n),
so_lambda: λ2x.t[x],
so_apply: x[s],
rless: x < y,
sq_exists: ∃x:A [B[x]],
rge: x ≥ y,
rational-lower-approx: (below x within 1/n),
le: A ≤ B
Lemmas referenced :
member_rccint_lemma,
istype-void,
value-type-has-value,
int-value-type,
lt_int_wf,
nat_plus_properties,
decidable__lt,
full-omega-unsat,
intformand_wf,
intformnot_wf,
intformless_wf,
itermConstant_wf,
itermMultiply_wf,
itermVar_wf,
istype-int,
int_formula_prop_and_lemma,
int_formula_prop_not_lemma,
int_formula_prop_less_lemma,
int_term_value_constant_lemma,
int_term_value_mul_lemma,
int_term_value_var_lemma,
int_formula_prop_wf,
istype-less_than,
eqtt_to_assert,
assert_of_lt_int,
istype-top,
rleq_weakening_equal,
sq_stable__rleq,
rleq_wf,
rabs_wf,
rsub_wf,
rdiv_wf,
int-to-real_wf,
rless-int,
rless_wf,
eqff_to_assert,
bool_cases_sqequal,
subtype_base_sq,
bool_wf,
bool_subtype_base,
assert-bnot,
iff_weakening_uiff,
assert_wf,
less_than_wf,
subtract_wf,
int-rdiv_wf,
intformeq_wf,
int_formula_prop_eq_lemma,
int_subtype_base,
nequal_wf,
nat_plus_wf,
i-member_wf,
rccint_wf,
real_wf,
radd-preserves-rleq,
radd_wf,
itermSubtract_wf,
itermAdd_wf,
rleq_functionality,
rabs-of-nonneg,
req_weakening,
req-iff-rsub-is-0,
real_polynomial_null,
real_term_value_sub_lemma,
real_term_value_add_lemma,
real_term_value_var_lemma,
real_term_value_const_lemma,
set-value-type,
mul_preserves_lt,
set_subtype_base,
rless_functionality,
nat_plus_inc_int_nzero,
int-rdiv-req,
rless-int-fractions,
add-is-int-iff,
multiply-is-int-iff,
int_term_value_add_lemma,
false_wf,
rational-approx-property,
rational-upper-approx-property,
rational-upper-approx_wf,
rational-approx_wf,
rabs-difference-bound-rleq,
rleq_functionality_wrt_implies,
rsub_functionality_wrt_rleq,
req-int-fractions,
decidable__equal_int,
req_functionality,
req_transitivity,
radd-rdiv,
rdiv_functionality,
radd-int,
rleq_weakening_rless,
radd_functionality_wrt_rleq,
rabs-difference-symmetry,
subtract-is-int-iff,
int_term_value_subtract_lemma,
rational-lower-approx-property,
rational-lower-approx_wf,
rleq_transitivity,
decidable__le,
intformle_wf,
int_formula_prop_le_lemma,
mul_preserves_le,
nat_plus_subtype_nat,
rleq-int-fractions
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation_alt,
cut,
sqequalRule,
introduction,
extract_by_obid,
sqequalHypSubstitution,
dependent_functionElimination,
thin,
isect_memberEquality_alt,
voidElimination,
hypothesis,
callbyvalueReduce,
isectElimination,
intEquality,
independent_isectElimination,
applyEquality,
setElimination,
rename,
hypothesisEquality,
because_Cache,
multiplyEquality,
natural_numberEquality,
addEquality,
dependent_set_memberEquality_alt,
unionElimination,
approximateComputation,
independent_functionElimination,
dependent_pairFormation_alt,
lambdaEquality_alt,
int_eqEquality,
independent_pairFormation,
universeIsType,
inhabitedIsType,
equalityElimination,
equalityTransitivity,
equalitySymmetry,
productElimination,
lessCases,
isect_memberFormation_alt,
axiomSqEquality,
isectIsTypeImplies,
imageMemberEquality,
baseClosed,
imageElimination,
productIsType,
closedConclusion,
inrFormation_alt,
equalityIstype,
promote_hyp,
instantiate,
cumulativity,
baseApply,
sqequalBase,
setIsType,
applyLambdaEquality,
pointwiseFunctionality
Latex:
\mforall{}l:\mBbbR{}. \mforall{}u:\{u:\mBbbR{}| l \mleq{} u\} . \mforall{}x:\{x:\mBbbR{}| x \mmember{} [l, u]\} . \mforall{}n:\mBbbN{}\msupplus{}.
(approx-in-interval(l;u;x;n) \mmember{} \{y:\mBbbR{}| (y \mmember{} [l, u]) \mwedge{} (|x - y| \mleq{} (r(2)/r(n)))\} )
Date html generated:
2019_10_30-AM-09_07_32
Last ObjectModification:
2019_10_10-PM-01_11_54
Theory : reals
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