Nuprl Lemma : bag-bind-append2
∀[A,B:Type]. ∀[F,G:A ⟶ bag(B)]. ∀[ba:bag(A)].
(bag-bind(ba;λa.((F a) + (G a))) = (bag-bind(ba;F) + bag-bind(ba;G)) ∈ 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],
apply: f a,
lambda: λx.A[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,
subtype_rel: A ⊆r B,
squash: ↓T,
true: True,
prop: ℙ,
uimplies: b supposing a,
implies: P ⇒ Q,
all: ∀x:A. B[x],
empty-bag: {},
so_apply: x[s1;s2;s3],
so_lambda: so_lambda(x,y,z.t[x; y; z]),
append: as @ bs,
concat: concat(ll),
top: Top,
bag-union: bag-union(bbs),
bag-map: bag-map(f;bs),
bag-append: as + bs,
bag-bind: bag-bind(bs;f),
so_apply: x[s],
so_lambda: λ2x.t[x],
rev_implies: P ⇐ Q,
iff: P ⇐⇒ Q,
guard: {T}
Lemmas referenced :
bag_wf,
bag-bind_wf,
subtype_rel_self,
bag-append_wf,
equal_wf,
list-subtype-bag,
istype-universe,
list_wf,
permutation_wf,
reduce_cons_lemma,
map_cons_lemma,
empty-bag_wf,
list_ind_nil_lemma,
reduce_nil_lemma,
map_nil_lemma,
list_induction,
iff_weakening_equal,
bag-append-ac,
true_wf,
squash_wf,
bag-append-assoc2
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation_alt,
introduction,
cut,
sqequalHypSubstitution,
pointwiseFunctionalityForEquality,
extract_by_obid,
isectElimination,
thin,
hypothesisEquality,
hypothesis,
sqequalRule,
pertypeElimination,
productElimination,
rename,
equalitySymmetry,
dependent_set_memberEquality_alt,
independent_pairFormation,
equalityTransitivity,
productIsType,
equalityIsType1,
inhabitedIsType,
applyLambdaEquality,
setElimination,
applyEquality,
because_Cache,
lambdaEquality_alt,
imageElimination,
universeIsType,
universeEquality,
natural_numberEquality,
imageMemberEquality,
baseClosed,
hyp_replacement,
independent_isectElimination,
equalityIsType4,
isect_memberEquality_alt,
axiomEquality,
functionIsType,
independent_functionElimination,
dependent_functionElimination,
lambdaFormation,
cumulativity,
voidEquality,
voidElimination,
isect_memberEquality,
functionExtensionality,
lambdaEquality,
levelHypothesis,
equalityUniverse
Latex:
\mforall{}[A,B:Type]. \mforall{}[F,G:A {}\mrightarrow{} bag(B)]. \mforall{}[ba:bag(A)].
(bag-bind(ba;\mlambda{}a.((F a) + (G a))) = (bag-bind(ba;F) + bag-bind(ba;G)))
Date html generated:
2019_10_15-AM-11_05_53
Last ObjectModification:
2018_10_09-AM-10_52_22
Theory : bags
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