Nuprl Lemma : implies-close-reals
∀[x,y:ℝ]. ∀[m:ℕ+]. ∀[k:ℕ]. ((|(x m) - y m| ≤ (2 * k)) ⇒ (|x - y| ≤ (r(2 + k)/r(m))))
Proof
Definitions occuring in Statement :
rdiv: (x/y),
rleq: x ≤ y,
rabs: |x|,
rsub: x - y,
int-to-real: r(n),
real: ℝ,
absval: |i|,
nat_plus: ℕ+,
nat: ℕ,
uall: ∀[x:A]. B[x],
le: A ≤ B,
implies: P ⇒ Q,
apply: f a,
multiply: n * m,
subtract: n - m,
add: n + m,
natural_number: $n
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
implies: P ⇒ Q,
all: ∀x:A. B[x],
rev_uimplies: rev_uimplies(P;Q),
real: ℝ,
uimplies: b supposing a,
rge: x ≥ y,
guard: {T},
nat: ℕ,
nat_plus: ℕ+,
rneq: x ≠ y,
or: P ∨ Q,
iff: P ⇐⇒ Q,
and: P ∧ Q,
rev_implies: P ⇐ Q,
ge: i ≥ j ,
decidable: Dec(P),
not: ¬A,
satisfiable_int_formula: satisfiable_int_formula(fmla),
exists: ∃x:A. B[x],
false: False,
top: Top,
prop: ℙ,
uiff: uiff(P;Q),
subtype_rel: A ⊆r B,
rleq: x ≤ y,
rnonneg: rnonneg(x),
le: A ≤ B,
squash: ↓T,
true: True,
rational-approx: (x within 1/n),
int_nzero: ℤ-o,
nequal: a ≠ b ∈ T ,
rless: x < y,
sq_exists: ∃x:A [B[x]],
so_lambda: λ2x.t[x],
so_apply: x[s],
sq_type: SQType(T),
sq_stable: SqStable(P),
rdiv: (x/y),
req_int_terms: t1 ≡ t2
Lemmas referenced :
rational-approx-property,
uimplies_transitivity,
rleq_wf,
rabs_wf,
rsub_wf,
radd_wf,
rational-approx_wf,
rleq_functionality_wrt_implies,
rleq_weakening_equal,
r-triangle-inequality2,
radd_functionality_wrt_rleq,
rdiv_wf,
int-to-real_wf,
rless-int,
nat_properties,
nat_plus_properties,
decidable__lt,
full-omega-unsat,
intformand_wf,
intformnot_wf,
intformless_wf,
itermConstant_wf,
itermVar_wf,
istype-int,
int_formula_prop_and_lemma,
istype-void,
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,
rleq_functionality,
rabs-difference-symmetry,
req_weakening,
istype-le,
absval_wf,
subtract_wf,
le_witness_for_triv,
istype-nat,
nat_plus_wf,
real_wf,
itermMultiply_wf,
int_term_value_mul_lemma,
absval_pos,
decidable__le,
intformle_wf,
int_formula_prop_le_lemma,
squash_wf,
true_wf,
rabs-int,
subtype_rel_self,
iff_weakening_equal,
int-rdiv_wf,
intformeq_wf,
int_formula_prop_eq_lemma,
int_subtype_base,
set_subtype_base,
less_than_wf,
nequal_wf,
uiff_transitivity,
rabs_functionality,
rsub_functionality,
int-rdiv-req,
req_transitivity,
rsub-rdiv,
rabs-rdiv,
uiff_transitivity2,
rdiv_functionality,
rsub-int,
rneq_wf,
subtype_base_sq,
rleq-int-fractions,
istype-less_than,
mul_preserves_le,
nat_plus_subtype_nat,
sq_stable__less_than,
rmul_preserves_rleq,
rmul_wf,
rinv_wf2,
itermSubtract_wf,
itermAdd_wf,
radd_functionality,
rmul_functionality,
rmul-rinv,
rmul-int,
req_inversion,
radd-int,
req-iff-rsub-is-0,
real_polynomial_null,
real_term_value_sub_lemma,
real_term_value_mul_lemma,
real_term_value_add_lemma,
real_term_value_const_lemma,
real_term_value_var_lemma
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation_alt,
introduction,
cut,
lambdaFormation_alt,
extract_by_obid,
sqequalHypSubstitution,
dependent_functionElimination,
thin,
hypothesisEquality,
because_Cache,
isectElimination,
hypothesis,
setElimination,
rename,
independent_functionElimination,
independent_isectElimination,
equalityTransitivity,
equalitySymmetry,
addEquality,
closedConclusion,
natural_numberEquality,
sqequalRule,
inrFormation_alt,
productElimination,
unionElimination,
approximateComputation,
dependent_pairFormation_alt,
lambdaEquality_alt,
int_eqEquality,
isect_memberEquality_alt,
voidElimination,
independent_pairFormation,
universeIsType,
applyEquality,
inhabitedIsType,
multiplyEquality,
functionIsTypeImplies,
isectIsTypeImplies,
dependent_set_memberEquality_alt,
imageElimination,
imageMemberEquality,
baseClosed,
instantiate,
universeEquality,
equalityIstype,
baseApply,
sqequalBase,
intEquality,
cumulativity,
applyLambdaEquality
Latex:
\mforall{}[x,y:\mBbbR{}]. \mforall{}[m:\mBbbN{}\msupplus{}]. \mforall{}[k:\mBbbN{}]. ((|(x m) - y m| \mleq{} (2 * k)) {}\mRightarrow{} (|x - y| \mleq{} (r(2 + k)/r(m))))
Date html generated:
2019_10_29-AM-10_03_15
Last ObjectModification:
2019_04_23-PM-11_27_06
Theory : reals
Home
Index