Nuprl Lemma : ni-selector-property
∀p:ℕ∞ ⟶ 𝔹. (∃x:ℕ∞. p x = ff ⇐⇒ p ni-selector(p) = ff)
Proof
Definitions occuring in Statement :
ni-selector: ni-selector(p),
nat-inf: ℕ∞,
bfalse: ff,
bool: 𝔹,
all: ∀x:A. B[x],
exists: ∃x:A. B[x],
iff: P ⇐⇒ Q,
apply: f a,
function: x:A ⟶ B[x],
equal: s = t ∈ T
Definitions unfolded in proof :
all: ∀x:A. B[x],
iff: P ⇐⇒ Q,
and: P ∧ Q,
implies: P ⇒ Q,
member: t ∈ T,
uall: ∀[x:A]. B[x],
bool: 𝔹,
unit: Unit,
it: ⋅,
btrue: tt,
bfalse: ff,
prop: ℙ,
so_lambda: λ2x.t[x],
so_apply: x[s],
rev_implies: P ⇐ Q,
exists: ∃x:A. B[x],
nat: ℕ,
false: False,
ge: i ≥ j ,
uimplies: b supposing a,
satisfiable_int_formula: satisfiable_int_formula(fmla),
not: ¬A,
top: Top,
guard: {T},
subtype_rel: A ⊆r B,
int_seg: {i..j-},
lelt: i ≤ j < k,
le: A ≤ B,
less_than': less_than'(a;b),
decidable: Dec(P),
or: P ∨ Q,
uiff: uiff(P;Q),
assert: ↑b,
ifthenelse: if b then t else f fi ,
true: True,
sq_type: SQType(T),
bnot: ¬bb,
less_than: a < b,
nat-inf: ℕ∞,
ni-selector: ni-selector(p),
nat2inf: n∞,
rev_uimplies: rev_uimplies(P;Q),
nat-inf-infinity: ∞
Lemmas referenced :
nat-inf_wf,
ni-selector_wf,
bool_wf,
equal_wf,
exists_wf,
equal-wf-T-base,
nat_properties,
satisfiable-full-omega-tt,
intformand_wf,
intformle_wf,
itermConstant_wf,
itermVar_wf,
intformless_wf,
int_formula_prop_and_lemma,
int_formula_prop_le_lemma,
int_term_value_constant_lemma,
int_term_value_var_lemma,
int_formula_prop_less_lemma,
int_formula_prop_wf,
ge_wf,
less_than_wf,
assert_witness,
nat2inf_wf,
less_than_transitivity1,
less_than_irreflexivity,
int_seg_wf,
int_seg_properties,
int_seg_subtype_nat,
false_wf,
decidable__le,
subtract_wf,
intformnot_wf,
itermSubtract_wf,
int_formula_prop_not_lemma,
int_term_value_subtract_lemma,
decidable__equal_int,
int_seg_subtype,
intformeq_wf,
int_formula_prop_eq_lemma,
le_wf,
eqtt_to_assert,
eqff_to_assert,
bool_cases_sqequal,
subtype_base_sq,
bool_subtype_base,
assert-bnot,
decidable__lt,
lelt_wf,
itermAdd_wf,
int_term_value_add_lemma,
nat_wf,
not_wf,
assert_wf,
not_assert_elim,
btrue_neq_bfalse,
assert_of_lt_int,
lt_int_wf,
all_wf,
iff_imp_equal_bool,
bnot_wf,
b-exists_wf,
assert-b-exists,
iff_wf,
assert_of_bnot,
true_wf,
equal-nat-inf-infinity,
and_wf,
assert_elim
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation,
independent_pairFormation,
cut,
applyEquality,
functionExtensionality,
hypothesisEquality,
introduction,
extract_by_obid,
hypothesis,
sqequalHypSubstitution,
isectElimination,
thin,
unionElimination,
equalityElimination,
equalityTransitivity,
equalitySymmetry,
dependent_functionElimination,
independent_functionElimination,
sqequalRule,
lambdaEquality,
baseClosed,
dependent_pairFormation,
functionEquality,
setElimination,
rename,
intWeakElimination,
natural_numberEquality,
independent_isectElimination,
int_eqEquality,
intEquality,
isect_memberEquality,
voidElimination,
voidEquality,
computeAll,
because_Cache,
productElimination,
applyLambdaEquality,
hypothesis_subsumption,
dependent_set_memberEquality,
promote_hyp,
instantiate,
cumulativity,
addEquality,
hyp_replacement,
addLevel,
impliesFunctionality,
impliesLevelFunctionality,
existsFunctionality,
existsLevelFunctionality
Latex:
\mforall{}p:\mBbbN{}\minfty{} {}\mrightarrow{} \mBbbB{}. (\mexists{}x:\mBbbN{}\minfty{}. p x = ff \mLeftarrow{}{}\mRightarrow{} p ni-selector(p) = ff)
Date html generated:
2017_10_01-AM-08_29_29
Last ObjectModification:
2017_07_26-PM-04_24_01
Theory : basic
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