Nuprl Lemma : bag-combine-eq-out
∀[A,B,C:Type]. ∀[as:bag(A)]. ∀[bs:bag(B)]. ∀[f:A ⟶ bag(C)]. ∀[g:B ⟶ bag(C)]. ∀[h:A ⟶ B].
(⋃a∈as.f[a] = ⋃b∈bs.g[b] ∈ bag(C)) supposing
((∀a:A. (a ↓∈ as ⇒ (g[h[a]] = f[a] ∈ bag(C)))) and
(bs = bag-map(h;as) ∈ bag(B)))
Proof
Definitions occuring in Statement :
bag-member: x ↓∈ bs,
bag-combine: ⋃x∈bs.f[x],
bag-map: bag-map(f;bs),
bag: bag(T),
uimplies: b supposing a,
uall: ∀[x:A]. B[x],
so_apply: x[s],
all: ∀x:A. B[x],
implies: P ⇒ Q,
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],
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,
all: ∀x:A. B[x],
sq_stable: SqStable(P)
Lemmas referenced :
equal_wf,
squash_wf,
true_wf,
bag_wf,
bag-combine_wf,
bag-combine-map,
iff_weakening_equal,
set_wf,
bag-member_wf,
bag-subtype,
sq_stable__bag-member,
all_wf,
bag-map_wf
Rules used in proof :
cut,
hypothesis,
thin,
applyEquality,
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaEquality,
sqequalHypSubstitution,
imageElimination,
introduction,
extract_by_obid,
isectElimination,
hypothesisEquality,
equalityTransitivity,
equalitySymmetry,
because_Cache,
cumulativity,
sqequalRule,
functionExtensionality,
natural_numberEquality,
imageMemberEquality,
baseClosed,
universeEquality,
independent_isectElimination,
productElimination,
independent_functionElimination,
functionEquality,
dependent_functionElimination,
setElimination,
rename,
hyp_replacement,
applyLambdaEquality,
isect_memberFormation,
isect_memberEquality,
axiomEquality
Latex:
\mforall{}[A,B,C:Type]. \mforall{}[as:bag(A)]. \mforall{}[bs:bag(B)]. \mforall{}[f:A {}\mrightarrow{} bag(C)]. \mforall{}[g:B {}\mrightarrow{} bag(C)]. \mforall{}[h:A {}\mrightarrow{} B].
(\mcup{}a\mmember{}as.f[a] = \mcup{}b\mmember{}bs.g[b]) supposing
((\mforall{}a:A. (a \mdownarrow{}\mmember{} as {}\mRightarrow{} (g[h[a]] = f[a]))) and
(bs = bag-map(h;as)))
Date html generated:
2017_10_01-AM-08_57_13
Last ObjectModification:
2017_07_26-PM-04_39_21
Theory : bags
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