Nuprl Lemma : bag-diff-equal-inl
∀[T:Type]
∀eq:EqDecider(T). ∀as,bs:bag(T).
∀[cs:bag(T)]. uiff(bag-diff(eq;bs;as) = (inl cs) ∈ (bag(T)?);bs = (as + cs) ∈ bag(T))
Proof
Definitions occuring in Statement :
bag-diff: bag-diff(eq;bs;as),
bag-append: as + bs,
bag: bag(T),
deq: EqDecider(T),
uiff: uiff(P;Q),
uall: ∀[x:A]. B[x],
all: ∀x:A. B[x],
unit: Unit,
inl: inl x,
union: left + right,
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,
uiff: uiff(P;Q),
and: P ∧ Q,
uimplies: b supposing a,
outl: outl(x),
prop: ℙ,
isl: isl(x),
assert: ↑b,
ifthenelse: if b then t else f fi ,
btrue: tt,
true: True,
not: ¬A,
false: False,
so_lambda: λ2x.t[x],
so_apply: x[s]
Lemmas referenced :
bag-diff-property,
bag-diff_wf,
bag_wf,
unit_wf2,
and_wf,
equal_wf,
outl_wf,
assert_wf,
isl_wf,
bag-append_wf,
btrue_wf,
bfalse_wf,
btrue_neq_bfalse,
all_wf,
not_wf,
deq_wf,
bag-append-cancel
Rules used in proof :
cut,
introduction,
extract_by_obid,
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
hypothesis,
sqequalHypSubstitution,
isectElimination,
thin,
hypothesisEquality,
lambdaFormation,
dependent_functionElimination,
cumulativity,
unionEquality,
unionElimination,
sqequalRule,
independent_pairFormation,
equalitySymmetry,
dependent_set_memberEquality,
equalityTransitivity,
applyLambdaEquality,
setElimination,
rename,
productElimination,
independent_isectElimination,
promote_hyp,
hyp_replacement,
natural_numberEquality,
inlEquality,
independent_pairEquality,
isect_memberEquality,
axiomEquality,
because_Cache,
independent_functionElimination,
voidElimination,
inrEquality,
lambdaEquality,
universeEquality
Latex:
\mforall{}[T:Type]
\mforall{}eq:EqDecider(T). \mforall{}as,bs:bag(T). \mforall{}[cs:bag(T)]. uiff(bag-diff(eq;bs;as) = (inl cs);bs = (as + cs))
Date html generated:
2018_05_21-PM-09_49_17
Last ObjectModification:
2017_07_26-PM-06_30_58
Theory : bags_2
Home
Index