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
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_apply: x[s],
false: False,
ge: i ≥ j ,
gt: i > j,
le: A ≤ B,
and: P ∧ Q,
uimplies: b supposing a,
satisfiable_int_formula: satisfiable_int_formula(fmla),
exists: ∃x:A. B[x],
not: ¬A,
top: Top,
decidable: Dec(P),
or: P ∨ Q,
less_than: a < b,
squash: ↓T,
uiff: uiff(P;Q),
subtype_rel: A ⊆r B,
cand: A c∧ B,
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q,
guard: {T},
l_member: (x ∈ l),
l_all: (∀x∈L.P[x]),
nat: ℕ,
int_seg: {i..j-},
lelt: i ≤ j < k,
isl: isl(x),
assert: ↑b,
ifthenelse: if b then t else f fi ,
btrue: tt,
true: True,
bfalse: ff,
append: as @ bs,
so_lambda: so_lambda(x,y,z.t[x; y; z]),
so_apply: x[s1;s2;s3],
bool: 𝔹,
unit: Unit,
it: ⋅
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:
2016_05_17-AM-09_31_13
Last ObjectModification:
2016_01_17-PM-11_11_36
Theory : classrel!lemmas
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