Nuprl Lemma : absval-implies-rneq
∀x,y:ℝ. ((∃n:ℕ+. 4 < |(x n) - y n|) ⇒ x ≠ y)
Proof
Definitions occuring in Statement :
rneq: x ≠ y,
real: ℝ,
absval: |i|,
nat_plus: ℕ+,
less_than: a < b,
all: ∀x:A. B[x],
exists: ∃x:A. B[x],
implies: P ⇒ Q,
apply: f a,
subtract: n - m,
natural_number: $n
Definitions unfolded in proof :
all: ∀x:A. B[x],
implies: P ⇒ Q,
exists: ∃x:A. B[x],
member: t ∈ T,
prop: ℙ,
uall: ∀[x:A]. B[x],
so_lambda: λ2x.t[x],
real: ℝ,
subtype_rel: A ⊆r B,
nat: ℕ,
so_apply: x[s],
bool: 𝔹,
unit: Unit,
it: ⋅,
btrue: tt,
uiff: uiff(P;Q),
and: P ∧ Q,
uimplies: b supposing a,
less_than: a < b,
less_than': less_than'(a;b),
top: Top,
true: True,
squash: ↓T,
not: ¬A,
false: False,
bfalse: ff,
or: P ∨ Q,
sq_type: SQType(T),
guard: {T},
bnot: ¬bb,
ifthenelse: if b then t else f fi ,
assert: ↑b,
rneq: x ≠ y,
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q,
nat_plus: ℕ+,
decidable: Dec(P),
satisfiable_int_formula: satisfiable_int_formula(fmla)
Lemmas referenced :
exists_wf,
nat_plus_wf,
less_than_wf,
absval_wf,
subtract_wf,
nat_wf,
real_wf,
absval_unfold,
lt_int_wf,
bool_wf,
eqtt_to_assert,
assert_of_lt_int,
top_wf,
eqff_to_assert,
equal_wf,
bool_cases_sqequal,
subtype_base_sq,
bool_subtype_base,
assert-bnot,
rless-iff2,
nat_plus_properties,
decidable__lt,
subtract-is-int-iff,
satisfiable-full-omega-tt,
intformand_wf,
intformnot_wf,
intformless_wf,
itermAdd_wf,
itermVar_wf,
itermConstant_wf,
itermSubtract_wf,
int_formula_prop_and_lemma,
int_formula_prop_not_lemma,
int_formula_prop_less_lemma,
int_term_value_add_lemma,
int_term_value_var_lemma,
int_term_value_constant_lemma,
int_term_value_subtract_lemma,
int_formula_prop_wf,
false_wf,
rless_wf,
minus-is-int-iff,
itermMinus_wf,
int_term_value_minus_lemma
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation,
sqequalHypSubstitution,
productElimination,
thin,
cut,
introduction,
extract_by_obid,
isectElimination,
hypothesis,
sqequalRule,
lambdaEquality,
natural_numberEquality,
applyEquality,
setElimination,
rename,
hypothesisEquality,
minusEquality,
because_Cache,
unionElimination,
equalityElimination,
equalityTransitivity,
equalitySymmetry,
independent_isectElimination,
lessCases,
isect_memberFormation,
sqequalAxiom,
isect_memberEquality,
independent_pairFormation,
voidElimination,
voidEquality,
imageMemberEquality,
baseClosed,
imageElimination,
independent_functionElimination,
dependent_pairFormation,
promote_hyp,
dependent_functionElimination,
instantiate,
cumulativity,
inrFormation,
addEquality,
dependent_set_memberEquality,
pointwiseFunctionality,
baseApply,
closedConclusion,
int_eqEquality,
intEquality,
computeAll,
inlFormation
Latex:
\mforall{}x,y:\mBbbR{}. ((\mexists{}n:\mBbbN{}\msupplus{}. 4 < |(x n) - y n|) {}\mRightarrow{} x \mneq{} y)
Date html generated:
2017_10_03-AM-08_27_11
Last ObjectModification:
2017_07_28-AM-07_24_44
Theory : reals
Home
Index