Nuprl Lemma : real-from-approx_wf
∀[a:ℝ]. ∀[x:k:ℕ+ ⟶ {v:ℝ| |v - a| ≤ (r1/r(k))} ]. (real-from-approx(n.x[n]) ∈ {b:ℝ| b = a} )
Proof
Definitions occuring in Statement :
real-from-approx: real-from-approx(n.x[n]),
rdiv: (x/y),
rleq: x ≤ y,
rabs: |x|,
rsub: x - y,
req: x = y,
int-to-real: r(n),
real: ℝ,
nat_plus: ℕ+,
uall: ∀[x:A]. B[x],
so_apply: x[s],
member: t ∈ T,
set: {x:A| B[x]} ,
function: x:A ⟶ B[x],
natural_number: $n
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
converges-to: lim n→∞.x[n] = y,
all: ∀x:A. B[x],
sq_exists: ∃x:A [B[x]],
subtype_rel: A ⊆r B,
implies: P ⇒ Q,
so_apply: x[s],
nat_plus: ℕ+,
nat: ℕ,
le: A ≤ B,
and: P ∧ Q,
decidable: Dec(P),
or: P ∨ Q,
iff: P ⇐⇒ Q,
not: ¬A,
rev_implies: P ⇐ Q,
false: False,
prop: ℙ,
uiff: uiff(P;Q),
uimplies: b supposing a,
subtract: n - m,
top: Top,
less_than': less_than'(a;b),
true: True,
so_lambda: λ2x.t[x],
rneq: x ≠ y,
guard: {T},
ge: i ≥ j ,
satisfiable_int_formula: satisfiable_int_formula(fmla),
exists: ∃x:A. B[x],
sq_stable: SqStable(P),
squash: ↓T,
rev_uimplies: rev_uimplies(P;Q),
rge: x ≥ y,
real-from-approx: real-from-approx(n.x[n])
Lemmas referenced :
nat_plus_subtype_nat,
decidable__lt,
false_wf,
not-lt-2,
less-iff-le,
condition-implies-le,
minus-add,
minus-one-mul,
zero-add,
minus-one-mul-top,
add-commutes,
add_functionality_wrt_le,
add-associates,
add-zero,
add-swap,
le-add-cancel,
less_than_wf,
set_wf,
real_wf,
rleq_wf,
rabs_wf,
rsub_wf,
rdiv_wf,
int-to-real_wf,
rless-int,
nat_properties,
nat_plus_properties,
full-omega-unsat,
intformand_wf,
intformnot_wf,
intformless_wf,
itermConstant_wf,
itermAdd_wf,
itermVar_wf,
intformle_wf,
int_formula_prop_and_lemma,
int_formula_prop_not_lemma,
int_formula_prop_less_lemma,
int_term_value_constant_lemma,
int_term_value_add_lemma,
int_term_value_var_lemma,
int_formula_prop_le_lemma,
int_formula_prop_wf,
rless_wf,
sq_stable__rleq,
equal_wf,
le_wf,
nat_wf,
nat_plus_wf,
rleq-int-fractions,
decidable__le,
itermMultiply_wf,
int_term_value_mul_lemma,
rleq_functionality_wrt_implies,
rleq_weakening_equal,
req-from-converges,
req_inversion,
req_wf,
cauchy-limit_wf,
converges-cauchy-witness
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
introduction,
cut,
sqequalRule,
lambdaEquality,
dependent_set_memberEquality,
hypothesisEquality,
applyEquality,
extract_by_obid,
hypothesis,
sqequalHypSubstitution,
lambdaFormation,
functionExtensionality,
because_Cache,
addEquality,
setElimination,
thin,
rename,
natural_numberEquality,
productElimination,
dependent_functionElimination,
unionElimination,
independent_pairFormation,
voidElimination,
independent_functionElimination,
independent_isectElimination,
isectElimination,
isect_memberEquality,
voidEquality,
intEquality,
minusEquality,
inrFormation,
approximateComputation,
dependent_pairFormation,
int_eqEquality,
imageMemberEquality,
baseClosed,
imageElimination,
equalityTransitivity,
equalitySymmetry,
functionEquality,
setEquality,
axiomEquality,
multiplyEquality
Latex:
\mforall{}[a:\mBbbR{}]. \mforall{}[x:k:\mBbbN{}\msupplus{} {}\mrightarrow{} \{v:\mBbbR{}| |v - a| \mleq{} (r1/r(k))\} ]. (real-from-approx(n.x[n]) \mmember{} \{b:\mBbbR{}| b = a\} )
Date html generated:
2018_05_22-PM-01_51_07
Last ObjectModification:
2017_10_25-AM-00_00_51
Theory : reals
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