Nuprl Lemma : l_contains-cons

∀[T:Type]
  ∀u:T. ∀v,bs:T List.
    ([u / v] ⊆ bs ⇐⇒ ∃cs,ds:T List. ((bs = (cs @ [u / ds]) ∈ (T List)) ∧ v ⊆ cs @ ds)) supposing 
       (no_repeats(T;bs) and 
       no_repeats(T;[u / v]))


Proof



Definitions occuring in Statement :  l_contains: A ⊆ B,  no_repeats: no_repeats(T;l),  append: as @ bs,  cons: [a / b],  list: T List,  uimplies: b supposing a,  uall: ∀[x:A]. B[x],  all: ∀x:A. B[x],  exists: ∃x:A. B[x],  iff: P ⇐⇒ Q,  and: P ∧ Q,  universe: Type,  equal: s = t ∈ T
Definitions unfolded in proof :  uall: ∀[x:A]. B[x],  all: ∀x:A. B[x],  uimplies: b supposing a,  member: t ∈ T,  implies: P ⇒ Q,  iff: P ⇐⇒ Q,  and: P ∧ Q,  prop: ℙ,  rev_implies: P ⇐ Q,  so_lambda: λ2x.t[x],  top: Top,  so_apply: x[s],  l_contains: A ⊆ B,  l_all: (∀x∈L.P[x]),  int_seg: {i..j-},  lelt: i ≤ j < k,  le: A ≤ B,  less_than': less_than'(a;b),  false: False,  not: ¬A,  nat_plus: ℕ+,  less_than: a < b,  squash: ↓T,  true: True,  guard: {T},  decidable: Dec(P),  or: P ∨ Q,  uiff: uiff(P;Q),  satisfiable_int_formula: satisfiable_int_formula(fmla),  exists: ∃x:A. B[x],  append: as @ bs,  so_lambda: so_lambda(x,y,z.t[x; y; z]),  so_apply: x[s1;s2;s3],  cand: A c∧ B,  sq_type: SQType(T),  select: L[n],  cons: [a / b],  ge: i ≥ j 
Lemmas referenced :  no_repeats_witness,  cons_wf,  l_contains_wf,  exists_wf,  list_wf,  equal_wf,  append_wf,  length_wf,  length-append,  no_repeats_wf,  length_of_cons_lemma,  false_wf,  add_nat_plus,  length_wf_nat,  less_than_wf,  nat_plus_wf,  nat_plus_properties,  decidable__lt,  add-is-int-iff,  satisfiable-full-omega-tt,  intformand_wf,  intformnot_wf,  intformless_wf,  itermConstant_wf,  itermVar_wf,  itermAdd_wf,  intformeq_wf,  int_formula_prop_and_lemma,  int_formula_prop_not_lemma,  int_formula_prop_less_lemma,  int_term_value_constant_lemma,  int_term_value_var_lemma,  int_term_value_add_lemma,  int_formula_prop_eq_lemma,  int_formula_prop_wf,  lelt_wf,  select-cons-hd,  l_member_decomp,  list_ind_cons_lemma,  list_ind_nil_lemma,  l_all_iff,  l_member_wf,  cons_member,  nat_wf,  member_append,  or_wf,  no_repeats_cons,  and_wf,  select_wf,  int_seg_properties,  decidable__le,  intformle_wf,  int_formula_prop_le_lemma,  decidable__equal_int,  subtype_base_sq,  int_subtype_base,  int_seg_wf,  all_wf,  select-cons-tl,  subtract_wf,  itermSubtract_wf,  int_term_value_subtract_lemma,  non_neg_length
Rules used in proof :  sqequalSubstitution,  sqequalTransitivity,  computationStep,  sqequalReflexivity,  isect_memberFormation,  lambdaFormation,  cut,  introduction,  extract_by_obid,  sqequalHypSubstitution,  isectElimination,  thin,  hypothesisEquality,  cumulativity,  hypothesis,  independent_functionElimination,  rename,  because_Cache,  independent_pairFormation,  sqequalRule,  lambdaEquality,  productEquality,  applyLambdaEquality,  isect_memberEquality,  voidElimination,  voidEquality,  universeEquality,  dependent_functionElimination,  dependent_set_memberEquality,  natural_numberEquality,  imageMemberEquality,  baseClosed,  equalityTransitivity,  equalitySymmetry,  setElimination,  unionElimination,  pointwiseFunctionality,  promote_hyp,  baseApply,  closedConclusion,  productElimination,  independent_isectElimination,  dependent_pairFormation,  int_eqEquality,  intEquality,  computeAll,  addEquality,  setEquality,  inrFormation,  hyp_replacement,  addLevel,  orFunctionality,  inlFormation,  levelHypothesis,  allFunctionality,  imageElimination,  instantiate,  allLevelFunctionality
Latex:
\mforall{}[T:Type]
    \mforall{}u:T.  \mforall{}v,bs:T  List.
        ([u  /  v]  \msubseteq{}  bs  \mLeftarrow{}{}\mRightarrow{}  \mexists{}cs,ds:T  List.  ((bs  =  (cs  @  [u  /  ds]))  \mwedge{}  v  \msubseteq{}  cs  @  ds))  supposing 
              (no\_repeats(T;bs)  and 
              no\_repeats(T;[u  /  v]))


Date html generated: 2017_04_17-AM-07_29_37
Last ObjectModification: 2017_02_27-PM-04_07_51

Theory : list_1


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