Nuprl Lemma : l_disjoint-representatives
∀[T:Type]
((∀x,y:T. Dec(x = y ∈ T))
⇒ (∀L:T List List
∃reps:T List List
(reps ⊆ L ∧ (∀as∈L.(∃rep∈reps. ¬l_disjoint(T;as;rep))) ∧ (∀rep1,rep2∈reps. l_disjoint(T;rep1;rep2)))
supposing (∀as∈L.0 < ||as||)))
Proof
Definitions occuring in Statement :
pairwise: (∀x,y∈L. P[x; y]),
l_disjoint: l_disjoint(T;l1;l2),
l_contains: A ⊆ B,
l_exists: (∃x∈L. P[x]),
l_all: (∀x∈L.P[x]),
length: ||as||,
list: T List,
less_than: a < b,
decidable: Dec(P),
uimplies: b supposing a,
uall: ∀[x:A]. B[x],
all: ∀x:A. B[x],
exists: ∃x:A. B[x],
not: ¬A,
implies: P ⇒ Q,
and: P ∧ Q,
natural_number: $n,
universe: Type,
equal: s = t ∈ T
Definitions unfolded in proof :
pairwise: (∀x,y∈L. P[x; y]),
uall: ∀[x:A]. B[x],
implies: P ⇒ Q,
all: ∀x:A. B[x],
member: t ∈ T,
so_lambda: λ2x.t[x],
uimplies: b supposing a,
prop: ℙ,
so_apply: x[s],
and: P ∧ Q,
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,
top: Top,
less_than: a < b,
squash: ↓T,
l_all: (∀x∈L.P[x]),
cand: A c∧ B,
l_disjoint: l_disjoint(T;l1;l2),
subtype_rel: A ⊆r B,
uiff: uiff(P;Q),
iff: P ⇐⇒ Q,
append: as @ bs,
so_lambda: so_lambda(x,y,z.t[x; y; z]),
so_apply: x[s1;s2;s3],
rev_implies: P ⇐ Q,
l_exists: (∃x∈L. P[x]),
le: A ≤ B,
less_than': less_than'(a;b),
nat_plus: ℕ+,
true: True,
select: L[n],
cons: [a / b],
subtract: n - m,
ge: i ≥ j ,
so_lambda: λ2x y.t[x; y],
so_apply: x[s1;s2]
Lemmas referenced :
list_induction,
list_wf,
isect_wf,
l_all_wf,
l_member_wf,
less_than_wf,
length_wf,
exists_wf,
l_contains_wf,
l_exists_wf,
not_wf,
l_disjoint_wf,
all_wf,
int_seg_wf,
select_wf,
int_seg_properties,
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,
member-less_than,
nil_wf,
length_of_nil_lemma,
l_contains_nil,
l_all_nil,
l_all_wf_nil,
cons_wf,
length_of_cons_lemma,
add-is-int-iff,
itermAdd_wf,
int_term_value_add_lemma,
false_wf,
decidable_wf,
equal_wf,
l_all_cons,
decidable__l_exists,
decidable__not,
decidable__l_disjoint,
l_contains_transitivity,
list_ind_cons_lemma,
list_ind_nil_lemma,
l_contains_append2,
l_contains_cons,
cons_member,
add_nat_plus,
length_wf_nat,
nat_plus_wf,
nat_plus_properties,
intformeq_wf,
int_formula_prop_eq_lemma,
lelt_wf,
list-cases,
product_subtype_list,
add-member-int_seg2,
subtract_wf,
itermSubtract_wf,
int_term_value_subtract_lemma,
non_neg_length,
select-cons-tl,
add-subtract-cancel,
pairwise-cons
Rules used in proof :
sqequalSubstitution,
sqequalRule,
sqequalReflexivity,
sqequalTransitivity,
computationStep,
isect_memberFormation,
lambdaFormation,
cut,
thin,
introduction,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
cumulativity,
hypothesisEquality,
hypothesis,
lambdaEquality,
setElimination,
rename,
natural_numberEquality,
because_Cache,
setEquality,
productEquality,
independent_isectElimination,
productElimination,
dependent_functionElimination,
unionElimination,
dependent_pairFormation,
int_eqEquality,
intEquality,
isect_memberEquality,
voidElimination,
voidEquality,
independent_pairFormation,
computeAll,
imageElimination,
independent_functionElimination,
applyEquality,
addEquality,
pointwiseFunctionality,
equalityTransitivity,
equalitySymmetry,
promote_hyp,
baseApply,
closedConclusion,
baseClosed,
universeEquality,
inlFormation,
dependent_set_memberEquality,
imageMemberEquality,
applyLambdaEquality,
hypothesis_subsumption,
instantiate
Latex:
\mforall{}[T:Type]
((\mforall{}x,y:T. Dec(x = y))
{}\mRightarrow{} (\mforall{}L:T List List
\mexists{}reps:T List List
(reps \msubseteq{} L
\mwedge{} (\mforall{}as\mmember{}L.(\mexists{}rep\mmember{}reps. \mneg{}l\_disjoint(T;as;rep)))
\mwedge{} (\mforall{}rep1,rep2\mmember{}reps. l\_disjoint(T;rep1;rep2)))
supposing (\mforall{}as\mmember{}L.0 < ||as||)))
Date html generated:
2017_04_17-AM-08_13_51
Last ObjectModification:
2017_02_27-PM-04_40_08
Theory : list_1
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