Nuprl Lemma : assert-bag-deq-member
∀[A:Type]. ∀[eq:EqDecider(A)]. ∀[b:bag(A)]. ∀[x:A]. uiff(↑bag-deq-member(eq;x;b);x ↓∈ b)
Proof
Definitions occuring in Statement :
bag-deq-member: bag-deq-member(eq;x;b),
bag-member: x ↓∈ bs,
bag: bag(T),
deq: EqDecider(T),
assert: ↑b,
uiff: uiff(P;Q),
uall: ∀[x:A]. B[x],
universe: Type
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
uiff: uiff(P;Q),
and: P ∧ Q,
uimplies: b supposing a,
bag: bag(T),
implies: P ⇒ Q,
prop: ℙ,
quotient: x,y:A//B[x; y],
bag-deq-member: bag-deq-member(eq;x;b),
bag-member: x ↓∈ bs,
squash: ↓T,
exists: ∃x:A. B[x],
all: ∀x:A. B[x],
iff: P ⇐⇒ Q,
cand: A c∧ B,
subtype_rel: A ⊆r B,
rev_implies: P ⇐ Q
Lemmas referenced :
assert_wf,
bag-deq-member_wf,
bag-member_wf,
equal-wf-base,
list_wf,
permutation_wf,
assert_witness,
bag_wf,
deq_wf,
assert-deq-member,
list-subtype-bag,
equal_wf,
l_member_wf,
deq-member_wf
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
introduction,
cut,
independent_pairFormation,
thin,
sqequalHypSubstitution,
pointwiseFunctionalityForEquality,
functionEquality,
extract_by_obid,
isectElimination,
cumulativity,
hypothesisEquality,
sqequalRule,
hypothesis,
pertypeElimination,
productElimination,
lambdaEquality,
imageMemberEquality,
baseClosed,
because_Cache,
productEquality,
independent_functionElimination,
imageElimination,
independent_pairEquality,
isect_memberEquality,
equalityTransitivity,
equalitySymmetry,
universeEquality,
dependent_functionElimination,
dependent_pairFormation,
applyEquality,
independent_isectElimination,
hyp_replacement,
applyLambdaEquality
Latex:
\mforall{}[A:Type]. \mforall{}[eq:EqDecider(A)]. \mforall{}[b:bag(A)]. \mforall{}[x:A]. uiff(\muparrow{}bag-deq-member(eq;x;b);x \mdownarrow{}\mmember{} b)
Date html generated:
2018_05_21-PM-09_47_17
Last ObjectModification:
2017_07_26-PM-06_30_09
Theory : bags_2
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