Nuprl Lemma : bag-drop-property2
∀[T:Type]
∀eq:EqDecider(T). ∀x:T. ∀bs:bag(T).
(bag-no-repeats(T;bs)
⇒ x ↓∈ bs
⇒ {(bs = ({x} + bag-drop(eq;bs;x)) ∈ bag(T)) ∧ bag-no-repeats(T;bag-drop(eq;bs;x)) ∧ (¬x ↓∈ bag-drop(eq;bs;x))})
Proof
Definitions occuring in Statement :
bag-drop: bag-drop(eq;bs;a),
bag-member: x ↓∈ bs,
bag-no-repeats: bag-no-repeats(T;bs),
bag-append: as + bs,
single-bag: {x},
bag: bag(T),
deq: EqDecider(T),
uall: ∀[x:A]. B[x],
guard: {T},
all: ∀x:A. B[x],
not: ¬A,
implies: P ⇒ Q,
and: P ∧ Q,
universe: Type,
equal: s = t ∈ T
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
all: ∀x:A. B[x],
implies: P ⇒ Q,
or: P ∨ Q,
guard: {T},
and: P ∧ Q,
cand: A c∧ B,
not: ¬A,
false: False,
bag-no-repeats: bag-no-repeats(T;bs),
squash: ↓T,
prop: ℙ,
uimplies: b supposing a,
true: True,
subtype_rel: A ⊆r B,
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q,
uiff: uiff(P;Q)
Lemmas referenced :
bag-member-single,
single-bag_wf,
bag-no-repeats-append,
iff_weakening_equal,
true_wf,
squash_wf,
bag-drop-no-repeats,
deq_wf,
bag_wf,
bag-no-repeats_wf,
bag-drop_wf,
bag-member_wf,
bag-drop-property
Rules used in proof :
cut,
lemma_by_obid,
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
hypothesis,
sqequalHypSubstitution,
isectElimination,
thin,
hypothesisEquality,
lambdaFormation,
dependent_functionElimination,
introduction,
unionElimination,
independent_pairFormation,
productElimination,
independent_functionElimination,
voidElimination,
sqequalRule,
independent_pairEquality,
axiomEquality,
imageElimination,
imageMemberEquality,
baseClosed,
lambdaEquality,
universeEquality,
independent_isectElimination,
applyEquality,
equalityTransitivity,
equalitySymmetry,
natural_numberEquality,
because_Cache
Latex:
\mforall{}[T:Type]
\mforall{}eq:EqDecider(T). \mforall{}x:T. \mforall{}bs:bag(T).
(bag-no-repeats(T;bs)
{}\mRightarrow{} x \mdownarrow{}\mmember{} bs
{}\mRightarrow{} \{(bs = (\{x\} + bag-drop(eq;bs;x)))
\mwedge{} bag-no-repeats(T;bag-drop(eq;bs;x))
\mwedge{} (\mneg{}x \mdownarrow{}\mmember{} bag-drop(eq;bs;x))\})
Date html generated:
2016_05_15-PM-08_04_41
Last ObjectModification:
2016_01_16-PM-01_29_22
Theory : bags_2
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