Nuprl Lemma : bag-drop-property
∀[T:Type]
  ∀eq:EqDecider(T). ∀x:T. ∀bs:bag(T).
    ((bs = ({x} + bag-drop(eq;bs;x)) ∈ bag(T)) ∨ ((¬x ↓∈ bs) ∧ (bs = bag-drop(eq;bs;x) ∈ bag(T))))
Proof
Definitions occuring in Statement : 
bag-drop: bag-drop(eq;bs;a)
, 
bag-member: x ↓∈ bs
, 
bag-append: as + bs
, 
single-bag: {x}
, 
bag: bag(T)
, 
deq: EqDecider(T)
, 
uall: ∀[x:A]. B[x]
, 
all: ∀x:A. B[x]
, 
not: ¬A
, 
or: 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]
, 
bag-drop: bag-drop(eq;bs;a)
, 
implies: P 
⇒ Q
, 
or: P ∨ Q
, 
exists: ∃x:A. B[x]
, 
and: P ∧ Q
, 
outl: outl(x)
, 
prop: ℙ
, 
uimplies: b supposing a
, 
isl: isl(x)
, 
assert: ↑b
, 
ifthenelse: if b then t else f fi 
, 
btrue: tt
, 
true: True
, 
not: ¬A
, 
false: False
, 
guard: {T}
, 
cand: A c∧ B
, 
so_lambda: λ2x.t[x]
, 
so_apply: x[s]
Lemmas referenced : 
bag-remove1-property, 
bag-remove1_wf, 
bag_wf, 
unit_wf2, 
and_wf, 
equal_wf, 
outl_wf, 
assert_wf, 
isl_wf, 
bag-append_wf, 
single-bag_wf, 
btrue_wf, 
bfalse_wf, 
btrue_neq_bfalse, 
not_wf, 
bag-member_wf, 
or_wf, 
exists_wf, 
equal-wf-T-base, 
deq_wf
Rules used in proof : 
cut, 
introduction, 
extract_by_obid, 
sqequalSubstitution, 
sqequalTransitivity, 
computationStep, 
sqequalReflexivity, 
isect_memberFormation, 
hypothesis, 
sqequalHypSubstitution, 
isectElimination, 
thin, 
hypothesisEquality, 
lambdaFormation, 
dependent_functionElimination, 
unionEquality, 
unionElimination, 
inlFormation, 
productElimination, 
sqequalRule, 
equalitySymmetry, 
dependent_set_memberEquality, 
independent_pairFormation, 
equalityTransitivity, 
applyLambdaEquality, 
setElimination, 
rename, 
independent_isectElimination, 
promote_hyp, 
hyp_replacement, 
natural_numberEquality, 
independent_functionElimination, 
voidElimination, 
productEquality, 
inrFormation, 
lambdaEquality, 
inlEquality, 
baseClosed, 
universeEquality
Latex:
\mforall{}[T:Type]
    \mforall{}eq:EqDecider(T).  \mforall{}x:T.  \mforall{}bs:bag(T).
        ((bs  =  (\{x\}  +  bag-drop(eq;bs;x)))  \mvee{}  ((\mneg{}x  \mdownarrow{}\mmember{}  bs)  \mwedge{}  (bs  =  bag-drop(eq;bs;x))))
Date html generated:
2019_10_16-AM-11_31_19
Last ObjectModification:
2018_08_21-PM-01_59_20
Theory : bags_2
Home
Index