Nuprl Lemma : bag-filter-trivial
∀[T:Type]. ∀[p:T ⟶ 𝔹]. ∀[bs:bag(T)]. ([x∈bs|p[x]] = bs ∈ bag(T)) supposing ∀x:T. (↑p[x])
Proof
Definitions occuring in Statement :
bag-filter: [x∈b|p[x]],
bag: bag(T),
assert: ↑b,
bool: 𝔹,
uimplies: b supposing a,
uall: ∀[x:A]. B[x],
so_apply: x[s],
all: ∀x:A. B[x],
function: x:A ⟶ B[x],
universe: Type,
equal: s = t ∈ T
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
uimplies: b supposing a,
bag-filter: [x∈b|p[x]],
bag: bag(T),
quotient: x,y:A//B[x; y],
and: P ∧ Q,
all: ∀x:A. B[x],
implies: P ⇒ Q,
so_lambda: λ2x y.t[x; y],
so_apply: x[s1;s2],
so_apply: x[s],
prop: ℙ,
so_lambda: λ2x.t[x],
squash: ↓T,
true: True,
l_all: (∀x∈L.P[x]),
guard: {T},
int_seg: {i..j-},
lelt: i ≤ j < k,
decidable: Dec(P),
or: P ∨ Q,
satisfiable_int_formula: satisfiable_int_formula(fmla),
exists: ∃x:A. B[x],
false: False,
not: ¬A,
top: Top,
less_than: a < b
Lemmas referenced :
list_wf,
quotient-member-eq,
permutation_wf,
permutation-equiv,
filter_wf5,
l_member_wf,
equal_wf,
equal-wf-base,
bag_wf,
all_wf,
assert_wf,
bool_wf,
squash_wf,
true_wf,
filter_trivial,
int_seg_wf,
length_wf,
select_wf,
int_seg_properties,
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
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
introduction,
cut,
sqequalHypSubstitution,
pointwiseFunctionalityForEquality,
because_Cache,
sqequalRule,
pertypeElimination,
productElimination,
thin,
equalityTransitivity,
hypothesis,
equalitySymmetry,
extract_by_obid,
isectElimination,
cumulativity,
hypothesisEquality,
lambdaFormation,
rename,
lambdaEquality,
independent_isectElimination,
dependent_functionElimination,
applyEquality,
functionExtensionality,
setElimination,
setEquality,
independent_functionElimination,
productEquality,
isect_memberEquality,
axiomEquality,
functionEquality,
universeEquality,
addLevel,
hyp_replacement,
imageElimination,
natural_numberEquality,
imageMemberEquality,
baseClosed,
levelHypothesis,
unionElimination,
dependent_pairFormation,
int_eqEquality,
intEquality,
voidElimination,
voidEquality,
independent_pairFormation,
computeAll
Latex:
\mforall{}[T:Type]. \mforall{}[p:T {}\mrightarrow{} \mBbbB{}]. \mforall{}[bs:bag(T)]. ([x\mmember{}bs|p[x]] = bs) supposing \mforall{}x:T. (\muparrow{}p[x])
Date html generated:
2017_10_01-AM-08_45_23
Last ObjectModification:
2017_07_26-PM-04_30_41
Theory : bags
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