Nuprl Lemma : length-eq-lists-diff-elts
∀[T:Type]
∀eq:∀x,y:T. Dec(x = y ∈ T). ∀L1,L2:T List.
(no_repeats(T;L1) ⇒ (||L1|| ≥ ||L2|| ) ⇒ (∃x:T. ((x ∈ L2) ∧ (¬(x ∈ L1)))) ⇒ (∃x:T. ((x ∈ L1) ∧ (¬(x ∈ L2)))))
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],
ge: 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],
implies: P ⇒ Q,
member: t ∈ T,
so_lambda: λ2x.t[x],
so_apply: x[s],
decidable: Dec(P),
or: P ∨ Q,
prop: ℙ,
iff: P ⇐⇒ Q,
and: P ∧ Q,
exists: ∃x:A. B[x],
rev_implies: P ⇐ Q,
int_seg: {i..j-},
uimplies: b supposing a,
ge: i ≥ j ,
guard: {T},
lelt: i ≤ j < k,
satisfiable_int_formula: satisfiable_int_formula(fmla),
false: False,
not: ¬A,
top: Top,
less_than: a < b,
squash: ↓T,
deq: EqDecider(T),
l_member: (x ∈ l),
cand: A c∧ B,
nat: ℕ,
le: A ≤ B,
subtype_rel: A ⊆r B,
less_than': less_than'(a;b),
uiff: uiff(P;Q),
eqof: eqof(d),
inject: Inj(A;B;f),
pi1: fst(t),
no_repeats: no_repeats(T;l)
Latex:
\mforall{}[T:Type]
\mforall{}eq:\mforall{}x,y:T. Dec(x = y). \mforall{}L1,L2:T List.
(no\_repeats(T;L1)
{}\mRightarrow{} (||L1|| \mgeq{} ||L2|| )
{}\mRightarrow{} (\mexists{}x:T. ((x \mmember{} L2) \mwedge{} (\mneg{}(x \mmember{} L1))))
{}\mRightarrow{} (\mexists{}x:T. ((x \mmember{} L1) \mwedge{} (\mneg{}(x \mmember{} L2)))))
Date html generated:
2016_05_17-AM-09_31_24
Last ObjectModification:
2016_01_17-PM-11_10_25
Theory : classrel!lemmas
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