Nuprl Lemma : bag-filter-is-nil
∀[T:Type]. ∀[p:T ⟶ 𝔹]. ∀[bs:bag(T)]. ([x∈bs|p[x]] ~ []) supposing ∀x:T. (¬↑p[x])
Proof
Definitions occuring in Statement :
bag-filter: [x∈b|p[x]],
bag: bag(T),
nil: [],
assert: ↑b,
bool: 𝔹,
uimplies: b supposing a,
uall: ∀[x:A]. B[x],
so_apply: x[s],
all: ∀x:A. B[x],
not: ¬A,
function: x:A ⟶ B[x],
universe: Type,
sqequal: s ~ t
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
uimplies: b supposing a,
prop: ℙ,
so_lambda: λ2x.t[x],
so_apply: x[s],
bag: bag(T),
quotient: x,y:A//B[x; y],
and: P ∧ Q,
bag-filter: [x∈b|p[x]],
l_all: (∀x∈L.P[x]),
all: ∀x:A. B[x],
not: ¬A,
implies: P ⇒ Q,
int_seg: {i..j-},
guard: {T},
lelt: i ≤ j < k,
decidable: Dec(P),
or: P ∨ Q,
satisfiable_int_formula: satisfiable_int_formula(fmla),
exists: ∃x:A. B[x],
false: False,
top: Top,
less_than: a < b,
squash: ↓T,
cons: [a / b]
Lemmas referenced :
bag_wf,
all_wf,
not_wf,
assert_wf,
bool_wf,
list_wf,
filter_is_nil,
nil_wf,
equal-wf-base,
permutation_wf,
select_wf,
int_seg_properties,
length_wf,
decidable__le,
satisfiable-full-omega-tt,
intformand_wf,
intformnot_wf,
intformle_wf,
itermConstant_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_var_lemma,
int_formula_prop_wf,
decidable__lt,
intformless_wf,
int_formula_prop_less_lemma,
int_seg_wf,
list-cases,
product_subtype_list,
null_nil_lemma,
btrue_wf,
null_cons_lemma,
bfalse_wf,
and_wf,
equal_wf,
null_wf,
btrue_neq_bfalse,
equal-wf-T-base
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
introduction,
cut,
hypothesis,
sqequalAxiom,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
cumulativity,
hypothesisEquality,
sqequalRule,
isect_memberEquality,
because_Cache,
lambdaEquality,
applyEquality,
functionExtensionality,
equalityTransitivity,
equalitySymmetry,
functionEquality,
universeEquality,
pointwiseFunctionalityForEquality,
pertypeElimination,
productElimination,
independent_isectElimination,
productEquality,
lambdaFormation,
setElimination,
rename,
natural_numberEquality,
dependent_functionElimination,
unionElimination,
dependent_pairFormation,
int_eqEquality,
intEquality,
voidElimination,
voidEquality,
independent_pairFormation,
computeAll,
imageElimination,
independent_functionElimination,
promote_hyp,
hypothesis_subsumption,
dependent_set_memberEquality,
applyLambdaEquality,
baseClosed
Latex:
\mforall{}[T:Type]. \mforall{}[p:T {}\mrightarrow{} \mBbbB{}]. \mforall{}[bs:bag(T)]. ([x\mmember{}bs|p[x]] \msim{} []) supposing \mforall{}x:T. (\mneg{}\muparrow{}p[x])
Date html generated:
2017_10_01-AM-08_45_20
Last ObjectModification:
2017_07_26-PM-04_30_40
Theory : bags
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