Nuprl Lemma : bag-product-no-repeats
∀[A,B:Type]. ∀[as:bag(A)]. ∀[bs:bag(B)].
bag-no-repeats(A × B;as × bs) supposing bag-no-repeats(A;as) ∧ bag-no-repeats(B;bs)
Proof
Definitions occuring in Statement :
bag-no-repeats: bag-no-repeats(T;bs),
bag-product: bs × cs,
bag: bag(T),
uimplies: b supposing a,
uall: ∀[x:A]. B[x],
and: P ∧ Q,
product: x:A × B[x],
universe: Type
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
uimplies: b supposing a,
and: P ∧ Q,
all: ∀x:A. B[x],
implies: P ⇒ Q,
bag-no-repeats: bag-no-repeats(T;bs),
squash: ↓T,
prop: ℙ,
subtype_rel: A ⊆r B,
so_lambda: λ2x.t[x],
so_apply: x[s],
sq_stable: SqStable(P),
exists: ∃x:A. B[x],
empty-bag: {},
top: Top,
cons-bag: x.b,
uiff: uiff(P;Q),
cand: A c∧ B,
not: ¬A,
false: False,
inject: Inj(A;B;f),
pi2: snd(t),
guard: {T}
Lemmas referenced :
bag-no-repeats_wf,
bag_wf,
bag-product_wf,
squash_wf,
nil_wf,
list-subtype-bag,
cons_wf,
all_wf,
bag_to_squash_list,
sq_stable__all,
sq_stable__bag-no-repeats,
list_induction,
list_wf,
bag-product-empty,
empty-bag-no-repeats,
cons-bag-as-append,
bag-product-append,
single-bag_wf,
bag-subtype-list,
subtype_rel_list,
top_wf,
bag-append-no-repeats,
bag-member_wf,
bag-product-single,
bag-map-no-repeats,
and_wf,
equal_wf,
pi2_wf,
bag-member-product,
pi1_wf
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
introduction,
cut,
sqequalHypSubstitution,
productElimination,
thin,
dependent_functionElimination,
hypothesisEquality,
hypothesis,
independent_functionElimination,
sqequalRule,
imageElimination,
imageMemberEquality,
baseClosed,
productEquality,
extract_by_obid,
isectElimination,
cumulativity,
isect_memberEquality,
because_Cache,
equalityTransitivity,
equalitySymmetry,
universeEquality,
lambdaEquality,
functionEquality,
lambdaFormation,
voidEquality,
applyEquality,
independent_isectElimination,
voidElimination,
promote_hyp,
hyp_replacement,
Error :applyLambdaEquality,
rename,
independent_pairFormation,
independent_pairEquality,
dependent_set_memberEquality,
setElimination,
setEquality
Latex:
\mforall{}[A,B:Type]. \mforall{}[as:bag(A)]. \mforall{}[bs:bag(B)].
bag-no-repeats(A \mtimes{} B;as \mtimes{} bs) supposing bag-no-repeats(A;as) \mwedge{} bag-no-repeats(B;bs)
Date html generated:
2016_10_25-AM-10_32_40
Last ObjectModification:
2016_07_12-AM-06_48_44
Theory : bags
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