Nuprl Lemma : longer-list-not-member
∀[T:Type]
∀eq:∀x,y:T. Dec(x = y ∈ T). ∀L1,L2:T List.
(no_repeats(T;L1) ⇒ no_repeats(T;L2) ⇒ (||L2|| > ||L1||) ⇒ (∃x:T. ((x ∈ L2) ∧ (¬(x ∈ L1)))))
Proof
Definitions occuring in Statement :
no_repeats: no_repeats(T;l),
l_member: (x ∈ l),
length: ||as||,
list: T List,
decidable: Dec(P),
uall: ∀[x:A]. B[x],
gt: i > j,
all: ∀x:A. B[x],
exists: ∃x:A. B[x],
not: ¬A,
implies: P ⇒ Q,
and: P ∧ Q,
universe: Type,
equal: s = t ∈ T
Lemmas :
l_member_decidable,
remove-first_wf,
l_member_wf,
isl_wf,
not_wf,
equal_wf,
all_wf,
decidable_wf,
subtype_rel_self,
no_repeats_cons,
cons_member,
cons_wf,
no_repeats-remove-first,
length-remove-first,
lelt_wf,
assert_wf,
btrue_wf,
bfalse_wf,
and_wf,
list_induction,
null_nil_lemma,
member-implies-null-eq-bfalse,
nil_wf,
btrue_neq_bfalse,
remove-first-cons,
list_ind_cons_lemma,
list_ind_nil_lemma,
member_append,
member_singleton,
bool_wf,
eqtt_to_assert,
assert_elim
Latex:
\mforall{}[T:Type]
\mforall{}eq:\mforall{}x,y:T. Dec(x = y). \mforall{}L1,L2:T List.
(no\_repeats(T;L1) {}\mRightarrow{} no\_repeats(T;L2) {}\mRightarrow{} (||L2|| > ||L1||) {}\mRightarrow{} (\mexists{}x:T. ((x \mmember{} L2) \mwedge{} (\mneg{}(x \mmember{} L1)))))
Date html generated:
2015_07_22-PM-00_17_17
Last ObjectModification:
2015_01_28-AM-10_46_05
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