Nuprl Lemma : bag-no-repeats-cons
∀[T:Type]. ∀[b:bag(T)]. ∀[x:T]. uiff(bag-no-repeats(T;x.b);bag-no-repeats(T;b) ∧ (¬x ↓∈ b))
Proof
Definitions occuring in Statement :
bag-member: x ↓∈ bs,
bag-no-repeats: bag-no-repeats(T;bs),
cons-bag: x.b,
bag: bag(T),
uiff: uiff(P;Q),
uall: ∀[x:A]. B[x],
not: ¬A,
and: P ∧ Q,
universe: Type
Definitions unfolded in proof :
empty-bag: {},
cons-bag: x.b,
bag-append: as + bs,
append: as @ bs,
all: ∀x:A. B[x],
so_lambda: so_lambda(x,y,z.t[x; y; z]),
member: t ∈ T,
top: Top,
so_apply: x[s1;s2;s3],
uall: ∀[x:A]. B[x],
prop: ℙ,
and: P ∧ Q,
uiff: uiff(P;Q),
uimplies: b supposing a,
bag-no-repeats: bag-no-repeats(T;bs),
squash: ↓T,
not: ¬A,
implies: P ⇒ Q,
false: False,
cand: A c∧ B,
so_lambda: λ2x.t[x],
so_apply: x[s],
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q
Lemmas referenced :
list_ind_cons_lemma,
list_ind_nil_lemma,
bag-member_wf,
bag-no-repeats_wf,
cons-bag_wf,
not_wf,
bag_wf,
cons_bag_empty_lemma,
bag-member-single,
single-bag_wf,
all_wf,
bag-single-no-repeats,
iff_weakening_uiff,
bag-append_wf,
empty-bag_wf,
bag-no-repeats-append,
rev_implies_wf,
uiff_wf,
and_wf,
equal_wf
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
sqequalRule,
cut,
introduction,
extract_by_obid,
sqequalHypSubstitution,
dependent_functionElimination,
thin,
isect_memberEquality,
voidElimination,
voidEquality,
hypothesis,
isectElimination,
cumulativity,
hypothesisEquality,
productEquality,
because_Cache,
universeEquality,
isect_memberFormation,
productElimination,
independent_pairEquality,
imageElimination,
imageMemberEquality,
baseClosed,
lambdaEquality,
equalityTransitivity,
equalitySymmetry,
independent_pairFormation,
independent_functionElimination,
independent_isectElimination,
functionEquality,
lambdaFormation,
addLevel,
hyp_replacement,
dependent_set_memberEquality,
applyEquality,
setElimination,
rename,
setEquality
Latex:
\mforall{}[T:Type]. \mforall{}[b:bag(T)]. \mforall{}[x:T]. uiff(bag-no-repeats(T;x.b);bag-no-repeats(T;b) \mwedge{} (\mneg{}x \mdownarrow{}\mmember{} b))
Date html generated:
2016_10_25-AM-10_35_11
Last ObjectModification:
2016_07_12-AM-06_49_55
Theory : bags
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