Nuprl Lemma : not-ni-eventually-equal-inf
∀[x:ℕ∞]. (¬ni-eventually-equal(x;0∞) ⇐⇒ x = ∞ ∈ ℕ∞)
Proof
Definitions occuring in Statement :
ni-eventually-equal: ni-eventually-equal(f;g),
nat-inf-infinity: ∞,
nat2inf: n∞,
nat-inf: ℕ∞,
uall: ∀[x:A]. B[x],
iff: P ⇐⇒ Q,
not: ¬A,
natural_number: $n,
equal: s = t ∈ T
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
iff: P ⇐⇒ Q,
and: P ∧ Q,
implies: P ⇒ Q,
prop: ℙ,
nat: ℕ,
le: A ≤ B,
less_than': less_than'(a;b),
false: False,
not: ¬A,
rev_implies: P ⇐ Q,
nat-inf: ℕ∞,
so_lambda: λ2x.t[x],
ge: i ≥ j ,
all: ∀x:A. B[x],
decidable: Dec(P),
or: P ∨ Q,
uimplies: b supposing a,
satisfiable_int_formula: satisfiable_int_formula(fmla),
exists: ∃x:A. B[x],
top: Top,
so_apply: x[s],
nat-inf-infinity: ∞,
bool: 𝔹,
unit: Unit,
it: ⋅,
btrue: tt,
uiff: uiff(P;Q),
bfalse: ff,
sq_type: SQType(T),
guard: {T},
bnot: ¬bb,
ifthenelse: if b then t else f fi ,
assert: ↑b,
ni-eventually-equal: ni-eventually-equal(f;g),
subtype_rel: A ⊆r B,
nat2inf: n∞,
int_upper: {i...},
subtract: n - m
Lemmas referenced :
not_wf,
ni-eventually-equal_wf,
nat2inf_wf,
false_wf,
le_wf,
equal-wf-T-base,
nat-inf_wf,
nat_wf,
all_wf,
assert_wf,
nat_properties,
decidable__le,
satisfiable-full-omega-tt,
intformand_wf,
intformnot_wf,
intformle_wf,
itermConstant_wf,
itermAdd_wf,
itermVar_wf,
int_formula_prop_and_lemma,
int_formula_prop_not_lemma,
int_formula_prop_le_lemma,
int_term_value_constant_lemma,
int_term_value_add_lemma,
int_term_value_var_lemma,
int_formula_prop_wf,
bool_wf,
eqtt_to_assert,
btrue_wf,
eqff_to_assert,
equal_wf,
bool_cases_sqequal,
subtype_base_sq,
bool_subtype_base,
assert-bnot,
int_upper_wf,
int_upper_subtype_nat,
lt_int_wf,
assert_of_lt_int,
int_upper_properties,
intformless_wf,
int_formula_prop_less_lemma,
less_than_wf,
bfalse_wf,
ge_wf,
add-zero,
subtract_wf,
itermSubtract_wf,
int_term_value_subtract_lemma,
add-associates,
subtract-add-cancel,
assert_elim,
minus-one-mul,
add-commutes,
add-mul-special,
zero-mul,
zero-add,
not_assert_elim,
int_subtype_base,
btrue_neq_bfalse,
equal-wf-base
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
introduction,
cut,
independent_pairFormation,
lambdaFormation,
hypothesis,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
hypothesisEquality,
dependent_set_memberEquality,
natural_numberEquality,
sqequalRule,
independent_functionElimination,
voidElimination,
because_Cache,
baseClosed,
productElimination,
independent_pairEquality,
lambdaEquality,
dependent_functionElimination,
axiomEquality,
setElimination,
rename,
functionExtensionality,
functionEquality,
applyEquality,
addEquality,
unionElimination,
independent_isectElimination,
dependent_pairFormation,
int_eqEquality,
intEquality,
isect_memberEquality,
voidEquality,
computeAll,
equalityElimination,
equalityTransitivity,
equalitySymmetry,
promote_hyp,
instantiate,
cumulativity,
intWeakElimination,
hyp_replacement,
applyLambdaEquality
Latex:
\mforall{}[x:\mBbbN{}\minfty{}]. (\mneg{}ni-eventually-equal(x;0\minfty{}) \mLeftarrow{}{}\mRightarrow{} x = \minfty{})
Date html generated:
2017_10_01-AM-08_30_20
Last ObjectModification:
2017_07_26-PM-04_24_31
Theory : basic
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