Nuprl Lemma : sublist-if-no_repeats
∀[A:Type]
∀R:A ⟶ A ⟶ 𝔹. ∀as,bs:A List.
(StAntiSym(A;x,y.↑R[x;y])
⇒ Irrefl(A;x,y.↑R[x;y])
⇒ Trans(A;x,y.↑R[x;y])
⇒ no_repeats(A;as)
⇒ l-ordered(A;x,y.↑R[x;y];as)
⇒ l-ordered(A;x,y.↑R[x;y];bs)
⇒ (∀a∈as.(a ∈ bs))
⇒ as ⊆ bs)
Proof
Definitions occuring in Statement :
l-ordered: l-ordered(T;x,y.R[x; y];L),
sublist: L1 ⊆ L2,
l_all: (∀x∈L.P[x]),
no_repeats: no_repeats(T;l),
l_member: (x ∈ l),
list: T List,
irrefl: Irrefl(T;x,y.E[x; y]),
st_anti_sym: StAntiSym(T;x,y.R[x; y]),
trans: Trans(T;x,y.E[x; y]),
assert: ↑b,
bool: 𝔹,
uall: ∀[x:A]. B[x],
so_apply: x[s1;s2],
all: ∀x:A. B[x],
implies: P ⇒ Q,
function: x:A ⟶ B[x],
universe: Type
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
all: ∀x:A. B[x],
member: t ∈ T,
so_lambda: λ2x.t[x],
implies: P ⇒ Q,
prop: ℙ,
so_lambda: λ2x y.t[x; y],
so_apply: x[s1;s2],
so_apply: x[s],
top: Top,
uiff: uiff(P;Q),
and: P ∧ Q,
uimplies: b supposing a,
iff: P ⇐⇒ Q,
subtype_rel: A ⊆r B,
not: ¬A,
false: False,
append: as @ bs,
so_lambda: so_lambda(x,y,z.t[x; y; z]),
so_apply: x[s1;s2;s3],
or: P ∨ Q,
cand: A c∧ B,
rev_implies: P ⇐ Q,
st_anti_sym: StAntiSym(T;x,y.R[x; y]),
guard: {T}
Lemmas referenced :
list_induction,
all_wf,
list_wf,
st_anti_sym_wf,
assert_wf,
irrefl_wf,
trans_wf,
no_repeats_wf,
l-ordered_wf,
l_all_wf2,
l_member_wf,
sublist_wf,
nil-sublist,
true_wf,
no_repeats_nil_uiff,
l-ordered-nil-true,
l_all_nil_iff,
nil_wf,
null_nil_lemma,
btrue_wf,
member-implies-null-eq-bfalse,
btrue_neq_bfalse,
l-ordered-decomp2,
sublist_append,
cons_wf,
filter_wf5,
list_ind_nil_lemma,
not_wf,
no_repeats_cons,
l-ordered-cons,
l_all_cons,
bool_wf,
cons_sublist_cons,
l-ordered-append,
l_all_iff,
append_wf,
member_append,
member_filter_2,
member_filter,
cons_member,
and_wf,
equal_wf
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
lambdaFormation,
cut,
thin,
introduction,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
hypothesisEquality,
sqequalRule,
lambdaEquality,
cumulativity,
hypothesis,
functionEquality,
because_Cache,
applyEquality,
functionExtensionality,
setElimination,
rename,
setEquality,
independent_functionElimination,
isect_memberEquality,
voidElimination,
voidEquality,
addLevel,
allFunctionality,
impliesFunctionality,
productElimination,
independent_isectElimination,
equalityTransitivity,
equalitySymmetry,
dependent_functionElimination,
hyp_replacement,
Error :applyLambdaEquality,
productEquality,
universeEquality,
inlFormation,
independent_pairFormation,
levelHypothesis,
promote_hyp,
unionElimination,
allLevelFunctionality,
impliesLevelFunctionality,
dependent_set_memberEquality
Latex:
\mforall{}[A:Type]
\mforall{}R:A {}\mrightarrow{} A {}\mrightarrow{} \mBbbB{}. \mforall{}as,bs:A List.
(StAntiSym(A;x,y.\muparrow{}R[x;y])
{}\mRightarrow{} Irrefl(A;x,y.\muparrow{}R[x;y])
{}\mRightarrow{} Trans(A;x,y.\muparrow{}R[x;y])
{}\mRightarrow{} no\_repeats(A;as)
{}\mRightarrow{} l-ordered(A;x,y.\muparrow{}R[x;y];as)
{}\mRightarrow{} l-ordered(A;x,y.\muparrow{}R[x;y];bs)
{}\mRightarrow{} (\mforall{}a\mmember{}as.(a \mmember{} bs))
{}\mRightarrow{} as \msubseteq{} bs)
Date html generated:
2016_10_25-AM-10_57_32
Last ObjectModification:
2016_07_12-AM-07_05_08
Theory : general
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