Nuprl Lemma : bag-bind-assoc
∀[A,B,C:Type]. ∀[f:A ⟶ bag(B)]. ∀[g:B ⟶ bag(C)]. ∀[bs:bag(A)].
(bag-bind(bag-bind(bs;f);g) = bag-bind(bs;λa.bag-bind(f a;g)) ∈ bag(C))
Proof
Definitions occuring in Statement :
bag-bind: bag-bind(bs;f),
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-bind: bag-bind(bs;f),
bag: bag(T),
quotient: x,y:A//B[x; y],
and: P ∧ Q,
squash: ↓T,
prop: ℙ,
subtype_rel: A ⊆r B,
true: True,
uimplies: b supposing a,
guard: {T},
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q,
implies: P ⇒ Q,
all: ∀x:A. B[x],
so_apply: x[s],
so_lambda: λ2x.t[x],
empty-bag: {},
concat: concat(ll),
top: Top,
bag-union: bag-union(bbs),
bag-map: bag-map(f;bs),
bag-append: as + bs
Lemmas referenced :
bag_wf,
equal_wf,
squash_wf,
true_wf,
istype-universe,
bag-union_wf,
bag-map_wf,
subtype_rel_self,
list-subtype-bag,
iff_weakening_equal,
list_wf,
permutation_wf,
list_induction,
empty-bag_wf,
reduce_nil_lemma,
map_nil_lemma,
reduce_cons_lemma,
map_cons_lemma,
map_append_sq,
bag-append_wf,
bag-append-union
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation_alt,
introduction,
cut,
sqequalRule,
sqequalHypSubstitution,
pointwiseFunctionalityForEquality,
extract_by_obid,
isectElimination,
thin,
hypothesisEquality,
hypothesis,
pertypeElimination,
productElimination,
rename,
applyEquality,
lambdaEquality_alt,
imageElimination,
equalityTransitivity,
equalitySymmetry,
universeIsType,
inhabitedIsType,
universeEquality,
because_Cache,
functionIsType,
natural_numberEquality,
imageMemberEquality,
baseClosed,
independent_isectElimination,
instantiate,
independent_functionElimination,
hyp_replacement,
applyLambdaEquality,
productIsType,
equalityIsType4,
isect_memberEquality_alt,
axiomEquality,
functionExtensionality,
lambdaEquality,
cumulativity,
dependent_functionElimination,
lambdaFormation,
voidEquality,
voidElimination,
isect_memberEquality,
levelHypothesis,
equalityUniverse
Latex:
\mforall{}[A,B,C:Type]. \mforall{}[f:A {}\mrightarrow{} bag(B)]. \mforall{}[g:B {}\mrightarrow{} bag(C)]. \mforall{}[bs:bag(A)].
(bag-bind(bag-bind(bs;f);g) = bag-bind(bs;\mlambda{}a.bag-bind(f a;g)))
Date html generated:
2019_10_15-AM-11_05_44
Last ObjectModification:
2018_10_09-AM-10_52_35
Theory : bags
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