Nuprl Lemma : bag-member-ifthenelse
∀[T:Type]. ∀[as,bs:bag(T)]. ∀[x:T]. ∀b:𝔹. uiff(x ↓∈ if b then as else bs fi ;if b then x ↓∈ as else x ↓∈ bs fi )
Proof
Definitions occuring in Statement :
bag-member: x ↓∈ bs,
bag: bag(T),
ifthenelse: if b then t else f fi ,
bool: 𝔹,
uiff: uiff(P;Q),
uall: ∀[x:A]. B[x],
all: ∀x:A. B[x],
universe: Type
Definitions unfolded in proof :
member: t ∈ T,
all: ∀x:A. B[x],
implies: P ⇒ Q,
bool: 𝔹,
unit: Unit,
it: ⋅,
btrue: tt,
uall: ∀[x:A]. B[x],
uiff: uiff(P;Q),
and: P ∧ Q,
uimplies: b supposing a,
ifthenelse: if b then t else f fi ,
bag-member: x ↓∈ bs,
squash: ↓T,
prop: ℙ,
bfalse: ff,
exists: ∃x:A. B[x],
or: P ∨ Q,
sq_type: SQType(T),
guard: {T},
bnot: ¬bb,
assert: ↑b,
false: False
Lemmas referenced :
eqtt_to_assert,
bag-member_wf,
eqff_to_assert,
equal_wf,
bool_wf,
bool_cases_sqequal,
subtype_base_sq,
bool_subtype_base,
assert-bnot,
ifthenelse_wf,
bag_wf
Rules used in proof :
cut,
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
hypothesisEquality,
thin,
because_Cache,
lambdaFormation,
sqequalHypSubstitution,
unionElimination,
equalityElimination,
introduction,
extract_by_obid,
isectElimination,
hypothesis,
productElimination,
independent_isectElimination,
sqequalRule,
independent_pairFormation,
isect_memberFormation,
imageElimination,
imageMemberEquality,
baseClosed,
cumulativity,
dependent_pairFormation,
promote_hyp,
dependent_functionElimination,
instantiate,
equalityTransitivity,
equalitySymmetry,
independent_functionElimination,
voidElimination,
universeEquality,
lambdaEquality,
independent_pairEquality,
isect_memberEquality
Latex:
\mforall{}[T:Type]. \mforall{}[as,bs:bag(T)]. \mforall{}[x:T].
\mforall{}b:\mBbbB{}. uiff(x \mdownarrow{}\mmember{} if b then as else bs fi ;if b then x \mdownarrow{}\mmember{} as else x \mdownarrow{}\mmember{} bs fi )
Date html generated:
2017_10_01-AM-08_54_28
Last ObjectModification:
2017_07_26-PM-04_36_17
Theory : bags
Home
Index