Nuprl Lemma : assert-deq-sub-bag
∀[T:Type]. ∀eq:EqDecider(T). ∀as,bs:bag(T). (↑deq-sub-bag(eq;as;bs) ⇐⇒ sub-bag(T;as;bs))
Proof
Definitions occuring in Statement :
deq-sub-bag: deq-sub-bag(eq;as;bs),
sub-bag: sub-bag(T;as;bs),
bag: bag(T),
deq: EqDecider(T),
assert: ↑b,
uall: ∀[x:A]. B[x],
all: ∀x:A. B[x],
iff: P ⇐⇒ Q,
universe: Type
Definitions unfolded in proof :
member: t ∈ T,
deq-sub-bag: deq-sub-bag(eq;as;bs),
all: ∀x:A. B[x],
uall: ∀[x:A]. B[x],
decidable__sub-bag,
deq-exists,
decidable_functionality,
iff_preserves_decidability,
sub-bag-iff,
decidable__assert,
not: ¬A,
false: False,
bfalse: ff,
true: True,
rev_implies: P ⇐ Q,
and: P ∧ Q,
iff: P ⇐⇒ Q,
btrue: tt,
ifthenelse: if b then t else f fi ,
assert: ↑b,
or: P ∨ Q,
decidable: Dec(P),
prop: ℙ,
implies: P ⇒ Q,
subtype_rel: A ⊆r B
Lemmas referenced :
deq_wf,
bag_wf,
false_wf,
true_wf,
deq-witness_wf,
sub-bag_wf,
equal_wf,
decidable_wf,
decidable__sub-bag,
deq-exists,
decidable_functionality,
iff_preserves_decidability,
sub-bag-iff,
decidable__assert
Rules used in proof :
universeEquality,
hypothesis,
hypothesisEquality,
thin,
isectElimination,
sqequalHypSubstitution,
extract_by_obid,
introduction,
cut,
lambdaFormation,
isect_memberFormation,
sqequalReflexivity,
computationStep,
sqequalTransitivity,
sqequalSubstitution,
sqequalRule,
dependent_functionElimination,
independent_functionElimination,
voidElimination,
natural_numberEquality,
independent_pairFormation,
unionElimination,
because_Cache,
functionEquality,
cumulativity,
isectEquality,
equalitySymmetry,
equalityTransitivity,
lambdaEquality,
instantiate,
applyEquality
Latex:
\mforall{}[T:Type]. \mforall{}eq:EqDecider(T). \mforall{}as,bs:bag(T). (\muparrow{}deq-sub-bag(eq;as;bs) \mLeftarrow{}{}\mRightarrow{} sub-bag(T;as;bs))
Date html generated:
2020_05_20-AM-09_04_48
Last ObjectModification:
2020_01_27-PM-04_02_53
Theory : bags_2
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