Nuprl Lemma : bag-member-partitions
∀[T:Type]
∀[eq:EqDecider(T)]. ∀[as,bs,cs:bag(T)]. uiff(<as, bs> ↓∈ bag-partitions(eq;cs);(as + bs) = cs ∈ bag(T))
supposing valueall-type(T)
Proof
Definitions occuring in Statement :
bag-partitions: bag-partitions(eq;bs),
bag-member: x ↓∈ bs,
bag-append: as + bs,
bag: bag(T),
deq: EqDecider(T),
valueall-type: valueall-type(T),
uiff: uiff(P;Q),
uimplies: b supposing a,
uall: ∀[x:A]. B[x],
pair: <a, b>,
product: x:A × B[x],
universe: Type,
equal: s = t ∈ T
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
uimplies: b supposing a,
bag-partitions: bag-partitions(eq;bs),
product-deq: product-deq(A;B;a;b),
so_lambda: λ2x.t[x],
so_apply: x[s],
implies: P ⇒ Q,
all: ∀x:A. B[x],
callbyvalueall: callbyvalueall,
has-value: (a)↓,
has-valueall: has-valueall(a),
uiff: uiff(P;Q),
and: P ∧ Q,
prop: ℙ,
bag-member: x ↓∈ bs,
squash: ↓T,
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q
Lemmas referenced :
product-deq_wf,
bag_wf,
bag-deq_wf,
valueall-type-has-valueall,
bag-valueall-type,
product-valueall-type,
bag-splits_wf,
evalall-reduce,
bag-member_wf,
bag-partitions_wf,
equal_wf,
bag-append_wf,
deq_wf,
valueall-type_wf,
bag-member-splits,
iff_weakening_uiff,
bag-to-set_wf,
member-bag-to-set,
uiff_wf
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
introduction,
cut,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
cumulativity,
hypothesisEquality,
hypothesis,
sqequalRule,
productEquality,
independent_isectElimination,
lambdaEquality,
independent_functionElimination,
lambdaFormation,
because_Cache,
dependent_functionElimination,
callbyvalueReduce,
productElimination,
independent_pairEquality,
isect_memberEquality,
axiomEquality,
equalityTransitivity,
equalitySymmetry,
imageElimination,
imageMemberEquality,
baseClosed,
universeEquality,
addLevel,
independent_pairFormation
Latex:
\mforall{}[T:Type]
\mforall{}[eq:EqDecider(T)]. \mforall{}[as,bs,cs:bag(T)]. uiff(<as, bs> \mdownarrow{}\mmember{} bag-partitions(eq;cs);(as + bs) = cs)
supposing valueall-type(T)
Date html generated:
2018_05_21-PM-09_49_34
Last ObjectModification:
2017_07_26-PM-06_31_12
Theory : bags_2
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