Nuprl Lemma : sub-bag-list-if-bag-no-repeats-sq
∀[A:Type]. ∀as:bag(A). ∀bs:A List. (bag-no-repeats(A;as) ⇒ b_all(A;as;x.(x ∈ bs)) ⇒ (↓sub-bag(A;as;bs)))
Proof
Definitions occuring in Statement :
b_all: b_all(T;b;x.P[x]),
sub-bag: sub-bag(T;as;bs),
bag-no-repeats: bag-no-repeats(T;bs),
bag: bag(T),
l_member: (x ∈ l),
list: T List,
uall: ∀[x:A]. B[x],
all: ∀x:A. B[x],
squash: ↓T,
implies: P ⇒ Q,
universe: Type
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
all: ∀x:A. B[x],
implies: P ⇒ Q,
squash: ↓T,
exists: ∃x:A. B[x],
prop: ℙ,
so_lambda: λ2x.t[x],
so_apply: x[s],
bag-no-repeats: bag-no-repeats(T;bs),
sq_stable: SqStable(P),
and: P ∧ Q,
uimplies: b supposing a,
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q,
b_all: b_all(T;b;x.P[x]),
subtype_rel: A ⊆r B,
empty-bag: {},
not: ¬A,
false: False,
uiff: uiff(P;Q),
or: P ∨ Q,
guard: {T},
sub-bag: sub-bag(T;as;bs),
bag-append: as + bs,
true: True,
append: as @ bs,
so_lambda: so_lambda(x,y,z.t[x; y; z]),
top: Top,
so_apply: x[s1;s2;s3],
cons-bag: x.b
Lemmas referenced :
bag_to_squash_list,
b_all_wf,
l_member_wf,
bag-no-repeats_wf,
sq_stable__no_repeats,
no_repeats-bag,
l_all_iff,
list-member-bag-member,
list_induction,
all_wf,
list_wf,
no_repeats_wf,
l_all_wf2,
squash_wf,
sub-bag_wf,
list-subtype-bag,
empty-sub-bag,
nil_wf,
null_nil_lemma,
btrue_wf,
member-implies-null-eq-bfalse,
btrue_neq_bfalse,
l_all_cons,
no_repeats_cons,
l_member_decomp,
append_wf,
cons_wf,
bag_wf,
member_append,
member_singleton,
and_wf,
equal_wf,
exists_wf,
true_wf,
bag-append-comm-assoc,
bag-append_wf,
iff_weakening_equal,
bag-append-comm,
list_ind_cons_lemma,
list_ind_nil_lemma,
cons-bag_wf
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
introduction,
cut,
lambdaFormation,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
hypothesisEquality,
imageElimination,
productElimination,
promote_hyp,
hypothesis,
equalitySymmetry,
hyp_replacement,
applyLambdaEquality,
cumulativity,
sqequalRule,
lambdaEquality,
rename,
independent_functionElimination,
independent_isectElimination,
imageMemberEquality,
baseClosed,
dependent_functionElimination,
setElimination,
setEquality,
because_Cache,
functionEquality,
applyEquality,
voidEquality,
voidElimination,
equalityTransitivity,
universeEquality,
addLevel,
allFunctionality,
impliesFunctionality,
levelHypothesis,
unionElimination,
inlFormation,
inrFormation,
allLevelFunctionality,
impliesLevelFunctionality,
dependent_set_memberEquality,
independent_pairFormation,
natural_numberEquality,
dependent_pairFormation,
isect_memberEquality
Latex:
\mforall{}[A:Type]
\mforall{}as:bag(A). \mforall{}bs:A List. (bag-no-repeats(A;as) {}\mRightarrow{} b\_all(A;as;x.(x \mmember{} bs)) {}\mRightarrow{} (\mdownarrow{}sub-bag(A;as;bs)))
Date html generated:
2018_05_21-PM-07_39_04
Last ObjectModification:
2017_07_26-PM-05_13_25
Theory : general
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