Nuprl Lemma : bag-bind-com
∀[A,B,C:Type]. ∀[f:A ⟶ B ⟶ bag(C)]. ∀[ba:bag(A)]. ∀[bb:bag(B)].
(bag-bind(ba;λa.bag-bind(bb;λb.f[a;b])) = bag-bind(bb;λb.bag-bind(ba;λa.f[a;b])) ∈ bag(C))
Proof
Definitions occuring in Statement :
bag-bind: bag-bind(bs;f),
bag: bag(T),
uall: ∀[x:A]. B[x],
so_apply: x[s1;s2],
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,
prop: ℙ,
all: ∀x:A. B[x],
implies: P ⇒ Q,
so_apply: x[s1;s2],
subtype_rel: A ⊆r B,
uimplies: b supposing a,
so_lambda: λ2x.t[x],
so_apply: x[s],
bag-map: bag-map(f;bs),
bag-union: bag-union(bbs),
top: Top,
concat: concat(ll),
empty-bag: {},
bag-append: as + bs,
true: True,
squash: ↓T,
guard: {T},
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q,
so_lambda: λ2x y.t[x; y]
Lemmas referenced :
bag_wf,
equal-wf-base,
list_wf,
permutation_wf,
equal_wf,
bag-union_wf,
bag-map_wf,
list-subtype-bag,
list_induction,
map_nil_lemma,
reduce_nil_lemma,
bag-bind-empty-right,
empty-bag_wf,
map_cons_lemma,
reduce_cons_lemma,
bag-append_wf,
squash_wf,
true_wf,
iff_weakening_equal,
bag-bind-append2,
bag-bind_wf,
quotient-member-eq,
permutation-equiv,
and_wf
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
introduction,
cut,
sqequalRule,
sqequalHypSubstitution,
pointwiseFunctionalityForEquality,
extract_by_obid,
isectElimination,
thin,
cumulativity,
hypothesisEquality,
hypothesis,
pertypeElimination,
productElimination,
productEquality,
because_Cache,
isect_memberEquality,
axiomEquality,
functionEquality,
equalityTransitivity,
equalitySymmetry,
lambdaFormation,
rename,
dependent_functionElimination,
independent_functionElimination,
hyp_replacement,
applyLambdaEquality,
lambdaEquality,
applyEquality,
functionExtensionality,
independent_isectElimination,
voidElimination,
voidEquality,
equalityUniverse,
levelHypothesis,
natural_numberEquality,
imageElimination,
universeEquality,
imageMemberEquality,
baseClosed,
dependent_set_memberEquality,
independent_pairFormation,
setElimination
Latex:
\mforall{}[A,B,C:Type]. \mforall{}[f:A {}\mrightarrow{} B {}\mrightarrow{} bag(C)]. \mforall{}[ba:bag(A)]. \mforall{}[bb:bag(B)].
(bag-bind(ba;\mlambda{}a.bag-bind(bb;\mlambda{}b.f[a;b])) = bag-bind(bb;\mlambda{}b.bag-bind(ba;\mlambda{}a.f[a;b])))
Date html generated:
2017_10_01-AM-09_06_10
Last ObjectModification:
2017_07_26-PM-04_46_20
Theory : bags
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