Nuprl Lemma : bag-partitions-symmetry
∀[T:Type]
∀[eq:EqDecider(T)]. ∀[bs:bag(T)].
(bag-map(λp.<snd(p), fst(p)>;bag-partitions(eq;bs)) = bag-partitions(eq;bs) ∈ bag(bag(T) × bag(T)))
supposing valueall-type(T)
Proof
Definitions occuring in Statement :
bag-partitions: bag-partitions(eq;bs),
bag-map: bag-map(f;bs),
bag: bag(T),
deq: EqDecider(T),
valueall-type: valueall-type(T),
uimplies: b supposing a,
uall: ∀[x:A]. B[x],
pi1: fst(t),
pi2: snd(t),
lambda: λx.A[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,
product-deq: product-deq(A;B;a;b),
so_lambda: λ2x.t[x],
so_apply: x[s],
top: Top,
uiff: uiff(P;Q),
and: P ∧ Q,
cand: A c∧ B,
all: ∀x:A. B[x],
implies: P ⇒ Q,
prop: ℙ,
pi2: snd(t),
pi1: fst(t),
squash: ↓T,
true: True,
subtype_rel: A ⊆r B,
guard: {T},
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q,
bag-partitions: bag-partitions(eq;bs),
callbyvalueall: callbyvalueall,
has-value: (a)↓,
has-valueall: has-valueall(a),
sq_stable: SqStable(P),
exists: ∃x:A. B[x]
Lemmas referenced :
product-deq_wf,
bag_wf,
bag-deq_wf,
deq_wf,
valueall-type_wf,
bag-extensionality2,
bag-partitions_wf,
bag-map_wf,
pi2_wf,
pi1_wf_top,
bag-member_wf,
equal_wf,
squash_wf,
true_wf,
pair_eta_rw,
iff_weakening_equal,
bag-count-map,
valueall-type-has-valueall,
bag-valueall-type,
product-valueall-type,
bag-splits_wf,
evalall-reduce,
equal-count-bag-to-set,
bag-append-comm,
bag-append_wf,
bag-member-splits,
iff_wf,
sq_stable__bag-member,
bag-member-partitions,
bag-member-map
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
introduction,
cut,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
cumulativity,
hypothesisEquality,
hypothesis,
sqequalRule,
isect_memberEquality,
axiomEquality,
because_Cache,
equalityTransitivity,
equalitySymmetry,
productEquality,
independent_isectElimination,
lambdaEquality,
independent_pairEquality,
productElimination,
voidElimination,
voidEquality,
lambdaFormation,
independent_pairFormation,
applyEquality,
imageElimination,
universeEquality,
natural_numberEquality,
imageMemberEquality,
baseClosed,
independent_functionElimination,
sqequalElimination,
dependent_functionElimination,
callbyvalueReduce,
equalityUniverse,
levelHypothesis,
addLevel,
impliesFunctionality,
hyp_replacement,
applyLambdaEquality
Latex:
\mforall{}[T:Type]
\mforall{}[eq:EqDecider(T)]. \mforall{}[bs:bag(T)].
(bag-map(\mlambda{}p.<snd(p), fst(p)>bag-partitions(eq;bs)) = bag-partitions(eq;bs))
supposing valueall-type(T)
Date html generated:
2018_05_21-PM-09_49_42
Last ObjectModification:
2017_07_26-PM-06_31_16
Theory : bags_2
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