Nuprl Lemma : bag-function
∀[T,A:Type]. ∀[f:(T List) ⟶ bag(A)].
f ∈ bag(T) ⟶ bag(A) supposing ∀as,bs:T List. (f[as @ bs] = (f[as] + f[bs]) ∈ bag(A))
Proof
Definitions occuring in Statement :
bag-append: as + bs,
bag: bag(T),
append: as @ bs,
list: T List,
uimplies: b supposing a,
uall: ∀[x:A]. B[x],
so_apply: x[s],
all: ∀x:A. B[x],
member: t ∈ T,
function: x:A ⟶ B[x],
universe: Type,
equal: s = t ∈ T
Definitions unfolded in proof :
member: t ∈ T,
squash: ↓T,
uall: ∀[x:A]. B[x],
prop: ℙ,
so_lambda: λ2x.t[x],
so_apply: x[s],
all: ∀x:A. B[x],
true: True,
subtype_rel: A ⊆r B,
uimplies: b supposing a,
guard: {T},
iff: P ⇐⇒ Q,
and: P ∧ Q,
rev_implies: P ⇐ Q,
implies: P ⇒ Q,
append: as @ bs,
so_lambda: so_lambda(x,y,z.t[x; y; z]),
top: Top,
so_apply: x[s1;s2;s3],
bag: bag(T),
quotient: x,y:A//B[x; y],
so_lambda: λ2x y.t[x; y],
so_apply: x[s1;s2],
trans: Trans(T;x,y.E[x; y]),
refl: Refl(T;x,y.E[x; y])
Lemmas referenced :
all_wf,
squash_wf,
true_wf,
list_wf,
equal_wf,
bag_wf,
cons_wf,
nil_wf,
iff_weakening_equal,
list_ind_cons_lemma,
list_ind_nil_lemma,
permutation-invariant2,
bag-append_wf,
bag-append-assoc-comm,
equal-wf-base,
permutation_wf,
bag-append-comm,
append_wf
Rules used in proof :
cut,
applyEquality,
thin,
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaEquality,
sqequalHypSubstitution,
imageElimination,
introduction,
extract_by_obid,
isectElimination,
hypothesisEquality,
equalityTransitivity,
hypothesis,
equalitySymmetry,
functionEquality,
cumulativity,
universeEquality,
because_Cache,
sqequalRule,
functionExtensionality,
dependent_functionElimination,
natural_numberEquality,
imageMemberEquality,
baseClosed,
independent_isectElimination,
productElimination,
independent_functionElimination,
lambdaFormation,
isect_memberEquality,
voidElimination,
voidEquality,
pointwiseFunctionalityForEquality,
pertypeElimination,
productEquality,
isect_memberFormation,
axiomEquality
Latex:
\mforall{}[T,A:Type]. \mforall{}[f:(T List) {}\mrightarrow{} bag(A)].
f \mmember{} bag(T) {}\mrightarrow{} bag(A) supposing \mforall{}as,bs:T List. (f[as @ bs] = (f[as] + f[bs]))
Date html generated:
2017_10_01-AM-08_45_09
Last ObjectModification:
2017_07_26-PM-04_30_34
Theory : bags
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