### 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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