Nuprl Lemma : member-iseg-sorted-by

∀[T:Type]. ∀[R:T ⟶ T ⟶ ℙ].
  ∀sa,sb:T List.
    (sa ≤ sb
       ⇒ sorted-by(R;sb)
       ⇒ (∀x:T. ((x ∈ sa) ⇐⇒ (¬↑null(sa)) ∧ (x ∈ sb) ∧ ((x = last(sa) ∈ T) ∨ (R x last(sa)))))) supposing 
       (AntiSym(T;x,y.R x y) and 
       Irrefl(T;x,y.R x y))


Proof



Definitions occuring in Statement :  sorted-by: sorted-by(R;L),  iseg: l1 ≤ l2,  last: last(L),  l_member: (x ∈ l),  null: null(as),  list: T List,  irrefl: Irrefl(T;x,y.E[x; y]),  anti_sym: AntiSym(T;x,y.R[x; y]),  assert: ↑b,  uimplies: b supposing a,  uall: ∀[x:A]. B[x],  prop: ℙ,  all: ∀x:A. B[x],  iff: P ⇐⇒ Q,  not: ¬A,  implies: P ⇒ Q,  or: P ∨ Q,  and: P ∧ Q,  apply: f a,  function: x:A ⟶ B[x],  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,  irrefl: Irrefl(T;x,y.E[x; y]),  not: ¬A,  implies: P ⇒ Q,  false: False,  anti_sym: AntiSym(T;x,y.R[x; y]),  iff: P ⇐⇒ Q,  and: P ∧ Q,  prop: ℙ,  rev_implies: P ⇐ Q,  or: P ∨ Q,  subtype_rel: A ⊆r B,  so_lambda: λ2x.t[x],  so_apply: x[s],  istype: istype(T),  so_lambda: λ2x y.t[x; y],  so_apply: x[s1;s2],  last: last(L),  lelt: i ≤ j < k,  int_seg: {i..j-},  top: Top,  satisfiable_int_formula: satisfiable_int_formula(fmla),  squash: ↓T,  less_than: a < b,  ge: i ≥ j ,  guard: {T},  sq_type: SQType(T),  decidable: Dec(P),  nat: ℕ,  cand: A c∧ B,  exists: ∃x:A. B[x],  l_member: (x ∈ l),  sorted-by: sorted-by(R;L),  true: True,  assert: ↑b,  ifthenelse: if b then t else f fi ,  btrue: tt,  select: L[n],  nil: [],  it: ⋅,  cons: [a / b],  bfalse: ff,  nat_plus: ℕ+,  uiff: uiff(P;Q)
Lemmas referenced :  assert_elim,  null_wf,  member-implies-null-eq-bfalse,  btrue_neq_bfalse,  istype-assert,  iseg_member,  l_member_wf,  istype-void,  last_wf,  subtype_rel_self,  sorted-by_wf,  subtype_rel_dep_function,  iseg_wf,  anti_sym_wf,  irrefl_wf,  list_wf,  istype-universe,  iseg-sorted-by,  istype-less_than,  istype-le,  int_formula_prop_less_lemma,  intformless_wf,  decidable__lt,  int_formula_prop_wf,  int_formula_prop_eq_lemma,  int_term_value_var_lemma,  int_term_value_subtract_lemma,  int_term_value_constant_lemma,  int_formula_prop_le_lemma,  int_formula_prop_not_lemma,  int_formula_prop_and_lemma,  istype-int,  intformeq_wf,  itermVar_wf,  itermSubtract_wf,  itermConstant_wf,  intformle_wf,  intformnot_wf,  intformand_wf,  full-omega-unsat,  decidable__le,  nat_properties,  select_wf,  int_subtype_base,  subtype_base_sq,  length_wf,  subtract_wf,  decidable__equal_int,  last_member,  iseg_select,  iff_weakening_equal,  true_wf,  squash_wf,  equal_wf,  list-cases,  length_of_nil_lemma,  null_nil_lemma,  stuck-spread,  istype-base,  product_subtype_list,  length_of_cons_lemma,  null_cons_lemma,  add_nat_plus,  length_wf_nat,  nat_plus_properties,  add-is-int-iff,  itermAdd_wf,  int_term_value_add_lemma,  false_wf
Rules used in proof :  sqequalSubstitution,  sqequalTransitivity,  computationStep,  sqequalReflexivity,  isect_memberFormation_alt,  lambdaFormation_alt,  cut,  introduction,  sqequalRule,  sqequalHypSubstitution,  isect_memberEquality_alt,  isectElimination,  thin,  hypothesisEquality,  lambdaEquality_alt,  dependent_functionElimination,  voidElimination,  functionIsTypeImplies,  inhabitedIsType,  isectIsTypeImplies,  rename,  axiomEquality,  hypothesis,  independent_pairFormation,  extract_by_obid,  independent_isectElimination,  equalityTransitivity,  equalitySymmetry,  independent_functionElimination,  universeIsType,  productElimination,  productIsType,  functionIsType,  unionIsType,  equalityIstype,  applyEquality,  instantiate,  universeEquality,  cumulativity,  functionEquality,  setEquality,  setIsType,  setElimination,  because_Cache,  dependent_set_memberEquality_alt,  closedConclusion,  inrFormation_alt,  int_eqEquality,  dependent_pairFormation_alt,  approximateComputation,  imageElimination,  intEquality,  inlFormation_alt,  unionElimination,  natural_numberEquality,  hyp_replacement,  applyLambdaEquality,  baseClosed,  imageMemberEquality,  Error :memTop,  promote_hyp,  hypothesis_subsumption,  pointwiseFunctionality,  baseApply
Latex:
\mforall{}[T:Type].  \mforall{}[R:T  {}\mrightarrow{}  T  {}\mrightarrow{}  \mBbbP{}].
    \mforall{}sa,sb:T  List.
        (sa  \mleq{}  sb
              {}\mRightarrow{}  sorted-by(R;sb)
              {}\mRightarrow{}  (\mforall{}x:T
                          ((x  \mmember{}  sa)  \mLeftarrow{}{}\mRightarrow{}  (\mneg{}\muparrow{}null(sa))  \mwedge{}  (x  \mmember{}  sb)  \mwedge{}  ((x  =  last(sa))  \mvee{}  (R  x  last(sa))))))  supposing 
              (AntiSym(T;x,y.R  x  y)  and 
              Irrefl(T;x,y.R  x  y))


Date html generated: 2020_05_19-PM-09_43_12
Last ObjectModification: 2019_12_31-PM-00_12_56

Theory : list_1


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