Nuprl Lemma : no_repeats_concat
∀[T:Type]. ∀[ll:T List List].
uiff(no_repeats(T;concat(ll));∀i:ℕ||ll||
(no_repeats(T;ll[i])
∧ (∀j:{j:ℕ||ll||| ¬(i = j ∈ ℤ)} . ∀k:ℕ||ll[i]||. (¬(ll[i][k] ∈ ll[j])))))
Proof
Definitions occuring in Statement :
no_repeats: no_repeats(T;l),
l_member: (x ∈ l),
select: L[n],
length: ||as||,
concat: concat(ll),
list: T List,
int_seg: {i..j-},
uiff: uiff(P;Q),
uall: ∀[x:A]. B[x],
all: ∀x:A. B[x],
not: ¬A,
and: P ∧ Q,
set: {x:A| B[x]} ,
natural_number: $n,
int: ℤ,
universe: Type,
equal: s = t ∈ T
Definitions unfolded in proof :
uiff: uiff(P;Q),
and: P ∧ Q,
uimplies: b supposing a,
member: t ∈ T,
all: ∀x:A. B[x],
not: ¬A,
implies: P ⇒ Q,
false: False,
prop: ℙ,
uall: ∀[x:A]. B[x],
int_seg: {i..j-},
guard: {T},
sq_stable: SqStable(P),
squash: ↓T,
lelt: i ≤ j < k,
decidable: Dec(P),
or: P ∨ Q,
satisfiable_int_formula: satisfiable_int_formula(fmla),
exists: ∃x:A. B[x],
top: Top,
less_than: a < b,
so_lambda: λ2x.t[x],
so_apply: x[s],
so_lambda: λ2x y.t[x; y],
so_apply: x[s1;s2],
l_all: (∀x∈L.P[x]),
pairwise: (∀x,y∈L. P[x; y]),
l_disjoint: l_disjoint(T;l1;l2),
iff: P ⇐⇒ Q,
subtype_rel: A ⊆r B,
rev_implies: P ⇐ Q,
le: A ≤ B,
cand: A c∧ B,
l_member: (x ∈ l),
nat: ℕ,
ge: i ≥ j ,
true: True
Lemmas referenced :
l_member_wf,
select_wf,
list_wf,
int_seg_properties,
length_wf,
sq_stable__not,
equal_wf,
decidable__le,
satisfiable-full-omega-tt,
intformand_wf,
intformnot_wf,
intformle_wf,
itermConstant_wf,
itermVar_wf,
int_formula_prop_and_lemma,
int_formula_prop_not_lemma,
int_formula_prop_le_lemma,
int_term_value_constant_lemma,
int_term_value_var_lemma,
int_formula_prop_wf,
decidable__lt,
intformless_wf,
int_formula_prop_less_lemma,
int_seg_wf,
set_wf,
not_wf,
no_repeats_witness,
l_all_wf,
no_repeats_wf,
pairwise_wf2,
l_disjoint_wf,
all_wf,
iff_weakening_uiff,
concat_wf,
no_repeats-concat-iff,
uiff_wf,
lelt_wf,
select_member,
intformeq_wf,
int_formula_prop_eq_lemma,
subtype_rel_sets,
equal-wf-T-base,
set_subtype_base,
int_subtype_base,
less_than_transitivity2,
le_weakening2,
nat_properties,
squash_wf,
true_wf
Rules used in proof :
cut,
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
independent_pairFormation,
isect_memberFormation,
introduction,
lambdaFormation,
sqequalHypSubstitution,
productElimination,
thin,
hypothesis,
independent_functionElimination,
voidElimination,
extract_by_obid,
isectElimination,
cumulativity,
hypothesisEquality,
because_Cache,
setElimination,
rename,
independent_isectElimination,
natural_numberEquality,
intEquality,
sqequalRule,
imageMemberEquality,
baseClosed,
imageElimination,
dependent_functionElimination,
unionElimination,
dependent_pairFormation,
lambdaEquality,
int_eqEquality,
isect_memberEquality,
voidEquality,
computeAll,
independent_pairEquality,
setEquality,
productEquality,
instantiate,
addLevel,
applyEquality,
universeEquality,
equalityTransitivity,
equalitySymmetry,
dependent_set_memberEquality,
hyp_replacement
Latex:
\mforall{}[T:Type]. \mforall{}[ll:T List List].
uiff(no\_repeats(T;concat(ll));\mforall{}i:\mBbbN{}||ll||
(no\_repeats(T;ll[i])
\mwedge{} (\mforall{}j:\{j:\mBbbN{}||ll||| \mneg{}(i = j)\} . \mforall{}k:\mBbbN{}||ll[i]||.
(\mneg{}(ll[i][k] \mmember{} ll[j])))))
Date html generated:
2016_10_21-AM-10_30_22
Last ObjectModification:
2016_07_12-AM-05_44_35
Theory : list_1
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