Nuprl Lemma : sub-bag-combine
∀[T,U:Type]. ∀f:T ⟶ bag(U). ∀b1,b2:bag(T). (sub-bag(T;b1;b2) ⇒ sub-bag(U;⋃x∈b1.f[x];⋃x∈b2.f[x]))
Proof
Definitions occuring in Statement :
sub-bag: sub-bag(T;as;bs),
bag-combine: ⋃x∈bs.f[x],
bag: bag(T),
uall: ∀[x:A]. B[x],
so_apply: x[s],
all: ∀x:A. B[x],
implies: P ⇒ Q,
function: x:A ⟶ B[x],
universe: Type
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
all: ∀x:A. B[x],
implies: P ⇒ Q,
sub-bag: sub-bag(T;as;bs),
exists: ∃x:A. B[x],
member: t ∈ T,
squash: ↓T,
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
Lemmas referenced :
bag-combine-append-left,
sub-bag_wf,
squash_wf,
true_wf,
bag_wf,
bag-combine_wf,
iff_weakening_equal,
equal_wf,
bag-append_wf
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
lambdaFormation,
sqequalHypSubstitution,
productElimination,
thin,
cut,
hypothesis,
introduction,
extract_by_obid,
isectElimination,
hypothesisEquality,
applyEquality,
lambdaEquality,
imageElimination,
equalityTransitivity,
equalitySymmetry,
because_Cache,
cumulativity,
sqequalRule,
functionExtensionality,
natural_numberEquality,
imageMemberEquality,
baseClosed,
universeEquality,
independent_isectElimination,
independent_functionElimination,
dependent_pairFormation,
hyp_replacement,
Error :applyLambdaEquality,
functionEquality
Latex:
\mforall{}[T,U:Type]. \mforall{}f:T {}\mrightarrow{} bag(U). \mforall{}b1,b2:bag(T). (sub-bag(T;b1;b2) {}\mRightarrow{} sub-bag(U;\mcup{}x\mmember{}b1.f[x];\mcup{}x\mmember{}b2.f[x]))
Date html generated:
2016_10_25-AM-10_26_28
Last ObjectModification:
2016_07_12-AM-06_42_20
Theory : bags
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