Nuprl Lemma : bag-bind-append
∀[A,B:Type]. ∀[ba,bb:bag(A)]. ∀[f:A ⟶ bag(B)]. (bag-bind(ba + bb;f) = (bag-bind(ba;f) + bag-bind(bb;f)) ∈ bag(B))
Proof
Definitions occuring in Statement :
bag-bind: bag-bind(bs;f),
bag-append: as + bs,
bag: bag(T),
uall: ∀[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,
bag: bag(T),
quotient: x,y:A//B[x; y],
and: P ∧ Q,
prop: ℙ,
all: ∀x:A. B[x],
implies: P ⇒ Q,
so_lambda: λ2x.t[x],
subtype_rel: A ⊆r B,
uimplies: b supposing a,
so_apply: x[s],
bag-bind: bag-bind(bs;f),
bag-append: as + bs,
append: as @ bs,
so_lambda: so_lambda(x,y,z.t[x; y; z]),
top: Top,
so_apply: x[s1;s2;s3],
bag-map: bag-map(f;bs),
bag-union: bag-union(bbs),
concat: concat(ll),
true: True,
squash: ↓T,
guard: {T},
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q,
so_lambda: λ2x y.t[x; y],
so_apply: x[s1;s2]
Lemmas referenced :
bag_wf,
equal-wf-base,
list_wf,
permutation_wf,
equal_wf,
list_induction,
bag-bind_wf,
bag-append_wf,
list-subtype-bag,
list_ind_nil_lemma,
map_nil_lemma,
reduce_nil_lemma,
list_ind_cons_lemma,
map_cons_lemma,
reduce_cons_lemma,
bag-append-assoc2,
squash_wf,
true_wf,
iff_weakening_equal,
quotient-member-eq,
permutation-equiv,
and_wf
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
introduction,
cut,
sqequalHypSubstitution,
pointwiseFunctionalityForEquality,
extract_by_obid,
isectElimination,
thin,
cumulativity,
hypothesisEquality,
hypothesis,
sqequalRule,
pertypeElimination,
productElimination,
rename,
hyp_replacement,
equalitySymmetry,
applyLambdaEquality,
productEquality,
because_Cache,
functionEquality,
isect_memberEquality,
axiomEquality,
universeEquality,
equalityTransitivity,
lambdaFormation,
dependent_functionElimination,
independent_functionElimination,
lambdaEquality,
applyEquality,
independent_isectElimination,
functionExtensionality,
voidElimination,
voidEquality,
equalityUniverse,
levelHypothesis,
natural_numberEquality,
imageElimination,
imageMemberEquality,
baseClosed,
dependent_set_memberEquality,
independent_pairFormation,
setElimination
Latex:
\mforall{}[A,B:Type]. \mforall{}[ba,bb:bag(A)]. \mforall{}[f:A {}\mrightarrow{} bag(B)].
(bag-bind(ba + bb;f) = (bag-bind(ba;f) + bag-bind(bb;f)))
Date html generated:
2017_10_01-AM-09_05_55
Last ObjectModification:
2017_07_26-PM-04_46_05
Theory : bags
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