Nuprl Lemma : bag-parts'_wf2
∀[T:Type]. ∀[eq:EqDecider(T)]. ∀[x:T]. ∀[bs:bag(T)]. (bag-parts'(eq;bs;x) ∈ bag({L:bag(T) List+| ¬x ↓∈ hd(L)} )) suppos\000Cing valueall-type(T)
Proof
Definitions occuring in Statement :
bag-parts': bag-parts'(eq;bs;x),
bag-member: x ↓∈ bs,
bag: bag(T),
listp: A List+,
hd: hd(l),
deq: EqDecider(T),
valueall-type: valueall-type(T),
uimplies: b supposing a,
uall: ∀[x:A]. B[x],
not: ¬A,
member: t ∈ T,
set: {x:A| B[x]} ,
universe: Type
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
uimplies: b supposing a,
subtype_rel: A ⊆r B,
all: ∀x:A. B[x],
listp: A List+,
prop: ℙ,
so_lambda: λ2x.t[x],
so_apply: x[s],
implies: P ⇒ Q,
not: ¬A,
false: False,
uiff: uiff(P;Q),
and: P ∧ Q
Lemmas referenced :
bag_wf,
deq_wf,
valueall-type_wf,
bag-subtype,
listp_wf,
bag-parts'_wf,
not_wf,
bag-member_wf,
hd_wf,
listp_properties,
subtype_rel_bag,
subtype_rel_sets,
bag-member-parts'
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
introduction,
cut,
sqequalHypSubstitution,
hypothesis,
sqequalRule,
axiomEquality,
equalityTransitivity,
equalitySymmetry,
lemma_by_obid,
isectElimination,
thin,
hypothesisEquality,
isect_memberEquality,
because_Cache,
universeEquality,
applyEquality,
dependent_functionElimination,
independent_isectElimination,
setEquality,
cumulativity,
setElimination,
rename,
lambdaEquality,
lambdaFormation,
independent_functionElimination,
voidElimination,
productElimination
Latex:
\mforall{}[T:Type]
\mforall{}[eq:EqDecider(T)]. \mforall{}[x:T]. \mforall{}[bs:bag(T)].
(bag-parts'(eq;bs;x) \mmember{} bag(\{L:bag(T) List\msupplus{}| \mneg{}x \mdownarrow{}\mmember{} hd(L)\} ))
supposing valueall-type(T)
Date html generated:
2016_05_15-PM-08_08_01
Last ObjectModification:
2015_12_27-PM-04_13_19
Theory : bags_2
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