Nuprl Lemma : list-decomp-no_repeats

[T:Type]. ∀[l1,l2,l3,l4:T List]. ∀[x:T].
  ((l1 l3 ∈ (T List)) ∧ (l2 l4 ∈ (T List))) supposing 
     ((((l1 [x]) l2) ((l3 [x]) l4) ∈ (T List)) and 
     no_repeats(T;(l1 [x]) l2))


Definitions occuring in Statement :  no_repeats: no_repeats(T;l) append: as bs cons: [a b] nil: [] list: List uimplies: supposing a uall: [x:A]. B[x] and: P ∧ Q universe: Type equal: t ∈ T
Definitions unfolded in proof :  uall: [x:A]. B[x] member: t ∈ T uimplies: supposing a and: P ∧ Q cand: c∧ B prop: iff: ⇐⇒ Q implies:  Q rev_implies:  Q all: x:A. B[x] subtype_rel: A ⊆B le: A ≤ B less_than': less_than'(a;b) false: False not: ¬A squash: T top: Top true: True guard: {T} int_seg: {i..j-} lelt: i ≤ j < k decidable: Dec(P) or: P ∨ Q uiff: uiff(P;Q) satisfiable_int_formula: satisfiable_int_formula(fmla) exists: x:A. B[x] ge: i ≥  sq_type: SQType(T) select: L[n] cons: [a b] less_than: a < b no_repeats: no_repeats(T;l) nat: subtract: m
Lemmas referenced :  equal_wf list_wf append_wf cons_wf nil_wf no_repeats_wf list_extensionality_iff int_seg_subtype length_wf false_wf le_wf squash_wf true_wf add_functionality_wrt_eq length_append subtype_rel_list top_wf iff_weakening_equal length-singleton length-append length_of_cons_lemma length_of_nil_lemma int_seg_properties decidable__le add-is-int-iff satisfiable-full-omega-tt intformnot_wf intformle_wf itermAdd_wf itermVar_wf itermConstant_wf int_formula_prop_not_lemma int_formula_prop_le_lemma int_term_value_add_lemma int_term_value_var_lemma int_term_value_constant_lemma int_formula_prop_wf int_seg_wf decidable__equal_int non_neg_length intformand_wf int_formula_prop_and_lemma decidable__lt intformless_wf int_formula_prop_less_lemma lelt_wf select_append_front select_append_back subtype_base_sq int_subtype_base intformeq_wf itermSubtract_wf int_formula_prop_eq_lemma int_term_value_subtract_lemma decidable__or less_than_wf intformor_wf int_formula_prop_or_lemma length_wf_nat nat_properties nat_wf le_weakening2 select_wf not_wf add-member-int_seg1 subtract_wf and_wf add-associates minus-add minus-one-mul add-swap add-mul-special add-commutes zero-add zero-mul add-zero
Rules used in proof :  sqequalSubstitution sqequalTransitivity computationStep sqequalReflexivity isect_memberFormation introduction cut independent_pairFormation hypothesis sqequalRule sqequalHypSubstitution productElimination thin independent_pairEquality axiomEquality extract_by_obid isectElimination cumulativity hypothesisEquality isect_memberEquality because_Cache equalityTransitivity equalitySymmetry universeEquality independent_functionElimination lambdaFormation dependent_functionElimination applyEquality natural_numberEquality addEquality independent_isectElimination lambdaEquality imageElimination intEquality voidElimination voidEquality imageMemberEquality baseClosed applyLambdaEquality setElimination rename unionElimination pointwiseFunctionality promote_hyp baseApply closedConclusion dependent_pairFormation int_eqEquality computeAll dependent_set_memberEquality instantiate hyp_replacement multiplyEquality productEquality

\mforall{}[T:Type].  \mforall{}[l1,l2,l3,l4:T  List].  \mforall{}[x:T].
    ((l1  =  l3)  \mwedge{}  (l2  =  l4))  supposing 
          ((((l1  @  [x])  @  l2)  =  ((l3  @  [x])  @  l4))  and 
          no\_repeats(T;(l1  @  [x])  @  l2))

Date html generated: 2017_10_01-AM-08_39_11
Last ObjectModification: 2017_07_26-PM-04_27_23

Theory : list!

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