Nuprl Lemma : near-real-implies-real
∀[x:ℕ+ ⟶ ℤ]. ∀[y:ℝ]. x ∈ {x:ℝ| x = y} supposing ∀n:ℕ+. (|(x within 1/n) - y| ≤ (r1/r(n)))
Proof
Definitions occuring in Statement :
rational-approx: (x within 1/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+,
uimplies: b supposing a,
uall: ∀[x:A]. B[x],
all: ∀x:A. B[x],
member: t ∈ T,
set: {x:A| B[x]} ,
function: x:A ⟶ B[x],
natural_number: $n,
int: ℤ
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
uimplies: b supposing a,
all: ∀x:A. B[x],
prop: ℙ,
so_lambda: λ2x.t[x],
nat_plus: ℕ+,
rneq: x ≠ y,
guard: {T},
or: P ∨ Q,
iff: P ⇐⇒ Q,
and: P ∧ Q,
rev_implies: P ⇐ Q,
implies: P ⇒ Q,
decidable: Dec(P),
satisfiable_int_formula: satisfiable_int_formula(fmla),
exists: ∃x:A. B[x],
false: False,
not: ¬A,
top: Top,
so_apply: x[s],
rev_uimplies: rev_uimplies(P;Q),
rge: x ≥ y,
uiff: uiff(P;Q),
less_than: a < b,
squash: ↓T,
less_than': less_than'(a;b),
true: True
Lemmas referenced :
implies-real,
nat_plus_wf,
all_wf,
rleq_wf,
rabs_wf,
rsub_wf,
rational-approx_wf,
rdiv_wf,
int-to-real_wf,
rless-int,
nat_plus_properties,
decidable__lt,
satisfiable-full-omega-tt,
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,
rleq_functionality_wrt_implies,
radd_wf,
rleq_weakening_equal,
r-triangle-inequality2,
radd_functionality_wrt_rleq,
rleq_functionality,
rabs-difference-symmetry,
req_weakening,
req-iff-rabs-rleq,
mul_nat_plus,
less_than_wf,
rational-approx-property,
req_wf,
itermMultiply_wf,
int_term_value_mul_lemma,
rmul_wf,
rleq-int-fractions,
decidable__le,
intformle_wf,
int_formula_prop_le_lemma,
uiff_transitivity,
req_transitivity,
radd_functionality,
req_inversion,
rmul-identity1,
rmul-distrib2,
rmul_functionality,
radd-int,
rmul-int-rdiv
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
introduction,
cut,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
hypothesisEquality,
independent_isectElimination,
lambdaFormation,
hypothesis,
sqequalRule,
axiomEquality,
equalityTransitivity,
equalitySymmetry,
lambdaEquality,
functionExtensionality,
applyEquality,
because_Cache,
natural_numberEquality,
setElimination,
rename,
inrFormation,
dependent_functionElimination,
productElimination,
independent_functionElimination,
unionElimination,
dependent_pairFormation,
int_eqEquality,
intEquality,
isect_memberEquality,
voidElimination,
voidEquality,
independent_pairFormation,
computeAll,
functionEquality,
dependent_set_memberEquality,
imageMemberEquality,
baseClosed,
multiplyEquality,
addEquality
Latex:
\mforall{}[x:\mBbbN{}\msupplus{} {}\mrightarrow{} \mBbbZ{}]. \mforall{}[y:\mBbbR{}]. x \mmember{} \{x:\mBbbR{}| x = y\} supposing \mforall{}n:\mBbbN{}\msupplus{}. (|(x within 1/n) - y| \mleq{} (r1/r(n)))
Date html generated:
2016_10_26-AM-09_18_34
Last ObjectModification:
2016_08_29-PM-00_37_31
Theory : reals
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