Nuprl Lemma : bag-member-bag-diff
∀[T:Type]. ∀eq:EqDecider(T). ∀x:T. ∀bs:bag(T). uiff(x ↓∈ bs;↑isl(bag-diff(eq;bs;{x})))
Proof
Definitions occuring in Statement :
bag-diff: bag-diff(eq;bs;as),
bag-member: x ↓∈ bs,
single-bag: {x},
bag: bag(T),
deq: EqDecider(T),
assert: ↑b,
isl: isl(x),
uiff: uiff(P;Q),
uall: ∀[x:A]. B[x],
all: ∀x:A. B[x],
universe: Type
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
all: ∀x:A. B[x],
implies: P ⇒ Q,
isl: isl(x),
assert: ↑b,
ifthenelse: if b then t else f fi ,
btrue: tt,
uiff: uiff(P;Q),
and: P ∧ Q,
uimplies: b supposing a,
true: True,
prop: ℙ,
exists: ∃x:A. B[x],
squash: ↓T,
bfalse: ff,
false: False,
so_lambda: λ2x.t[x],
so_apply: x[s],
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q,
bag-member: x ↓∈ bs,
not: ¬A
Lemmas referenced :
bag-diff-property,
single-bag_wf,
bag-diff_wf,
bag_wf,
unit_wf2,
equal_wf,
bag-append_wf,
true_wf,
false_wf,
all_wf,
not_wf,
iff_weakening_uiff,
bag-member_wf,
squash_wf,
exists_wf,
bag-member-iff,
assert_witness,
isl_wf,
assert_wf,
uiff_wf,
deq_wf
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
introduction,
cut,
lambdaFormation,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
because_Cache,
dependent_functionElimination,
hypothesisEquality,
cumulativity,
hypothesis,
unionEquality,
unionElimination,
sqequalRule,
independent_pairFormation,
natural_numberEquality,
axiomEquality,
equalityTransitivity,
equalitySymmetry,
dependent_pairFormation,
imageMemberEquality,
baseClosed,
imageElimination,
voidElimination,
lambdaEquality,
independent_functionElimination,
addLevel,
productElimination,
independent_isectElimination,
independent_pairEquality,
isect_memberEquality,
universeEquality
Latex:
\mforall{}[T:Type]. \mforall{}eq:EqDecider(T). \mforall{}x:T. \mforall{}bs:bag(T). uiff(x \mdownarrow{}\mmember{} bs;\muparrow{}isl(bag-diff(eq;bs;\{x\})))
Date html generated:
2018_05_21-PM-09_49_20
Last ObjectModification:
2017_07_26-PM-06_31_00
Theory : bags_2
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