Nuprl Lemma : no_repeats-concat-iff
∀[T:Type]. ∀[ll:T List List].
uiff(no_repeats(T;concat(ll));(∀l∈ll.no_repeats(T;l)) ∧ (∀l1,l2∈ll. l_disjoint(T;l1;l2)))
Proof
Definitions occuring in Statement :
pairwise: (∀x,y∈L. P[x; y]),
l_disjoint: l_disjoint(T;l1;l2),
l_all: (∀x∈L.P[x]),
no_repeats: no_repeats(T;l),
concat: concat(ll),
list: T List,
uiff: uiff(P;Q),
uall: ∀[x:A]. B[x],
and: P ∧ Q,
universe: Type
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
so_lambda: λ2x.t[x],
uiff: uiff(P;Q),
and: P ∧ Q,
prop: ℙ,
uimplies: b supposing a,
all: ∀x:A. B[x],
so_apply: x[s],
so_lambda: λ2x y.t[x; y],
so_apply: x[s1;s2],
implies: P ⇒ Q,
concat: concat(ll),
top: Top,
true: True,
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q,
l_all: (∀x∈L.P[x]),
int_seg: {i..j-},
guard: {T},
lelt: i ≤ j < k,
decidable: Dec(P),
or: P ∨ Q,
satisfiable_int_formula: satisfiable_int_formula(fmla),
exists: ∃x:A. B[x],
false: False,
not: ¬A,
pairwise: (∀x,y∈L. P[x; y]),
l_disjoint: l_disjoint(T;l1;l2),
subtype_rel: A ⊆r B,
less_than: a < b,
squash: ↓T,
cand: A c∧ B
Lemmas referenced :
false_wf,
int_term_value_add_lemma,
itermAdd_wf,
add-is-int-iff,
length_of_cons_lemma,
pairwise-cons,
cons_wf,
l_all-cons,
concat-cons,
no_repeats_append_iff,
append_wf,
iff_weakening_uiff,
l_disjoint-concat,
uiff_wf,
l_all_wf_nil,
length_wf,
int_seg_wf,
int_formula_prop_less_lemma,
intformless_wf,
decidable__lt,
int_formula_prop_wf,
int_term_value_var_lemma,
int_term_value_constant_lemma,
int_formula_prop_le_lemma,
int_formula_prop_not_lemma,
int_formula_prop_and_lemma,
itermVar_wf,
itermConstant_wf,
intformle_wf,
intformnot_wf,
intformand_wf,
satisfiable-full-omega-tt,
decidable__le,
int_seg_properties,
length_of_nil_lemma,
select_wf,
pairwise-nil,
l_all-nil,
true_wf,
and_wf,
no_repeats_witness,
no_repeats_nil,
nil_wf,
reduce_nil_lemma,
l_disjoint_wf,
pairwise_wf2,
l_member_wf,
l_all_wf,
concat_wf,
no_repeats_wf,
list_wf,
list_induction
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
introduction,
cut,
thin,
lemma_by_obid,
sqequalHypSubstitution,
isectElimination,
hypothesisEquality,
hypothesis,
sqequalRule,
lambdaEquality,
productEquality,
isectEquality,
cumulativity,
because_Cache,
lambdaFormation,
setElimination,
rename,
setEquality,
independent_functionElimination,
dependent_functionElimination,
isect_memberEquality,
voidElimination,
voidEquality,
independent_pairFormation,
natural_numberEquality,
productElimination,
independent_pairEquality,
axiomEquality,
equalityTransitivity,
equalitySymmetry,
addLevel,
independent_isectElimination,
unionElimination,
dependent_pairFormation,
int_eqEquality,
intEquality,
computeAll,
applyEquality,
universeEquality,
instantiate,
imageElimination,
addEquality,
pointwiseFunctionality,
promote_hyp,
baseApply,
closedConclusion,
baseClosed
Latex:
\mforall{}[T:Type]. \mforall{}[ll:T List List].
uiff(no\_repeats(T;concat(ll));(\mforall{}l\mmember{}ll.no\_repeats(T;l)) \mwedge{} (\mforall{}l1,l2\mmember{}ll. l\_disjoint(T;l1;l2)))
Date html generated:
2016_05_14-PM-02_55_06
Last ObjectModification:
2016_01_15-AM-07_33_09
Theory : list_1
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