Nuprl Lemma : descending-append

∀[A:Type]. ∀[<:A ⟶ A ⟶ ℙ].
  ∀L1,L2:A List.
    (descending(a,b.<[a;b];L1 @ L2)
    ⇐⇒ descending(a,b.<[a;b];L1)
        ∧ descending(a,b.<[a;b];L2)
        ∧ (<[hd(L2);last(L1)]) supposing (0 < ||L2|| and 0 < ||L1||))


Proof



Definitions occuring in Statement :  descending: descending(a,b.<[a; b];L),  last: last(L),  length: ||as||,  append: as @ bs,  hd: hd(l),  list: T List,  less_than: a < b,  uimplies: b supposing a,  uall: ∀[x:A]. B[x],  prop: ℙ,  so_apply: x[s1;s2],  all: ∀x:A. B[x],  iff: P ⇐⇒ Q,  and: P ∧ Q,  function: x:A ⟶ B[x],  natural_number: $n,  universe: Type
Definitions unfolded in proof :  uall: ∀[x:A]. B[x],  all: ∀x:A. B[x],  iff: P ⇐⇒ Q,  and: P ∧ Q,  implies: P ⇒ Q,  member: t ∈ T,  so_lambda: λ2x y.t[x; y],  so_apply: x[s1;s2],  prop: ℙ,  rev_implies: P ⇐ Q,  uimplies: b supposing a,  ge: i ≥ j ,  decidable: Dec(P),  or: P ∨ Q,  less_than: a < b,  squash: ↓T,  not: ¬A,  satisfiable_int_formula: satisfiable_int_formula(fmla),  exists: ∃x:A. B[x],  false: False,  assert: ↑b,  ifthenelse: if b then t else f fi ,  btrue: tt,  less_than': less_than'(a;b),  cons: [a / b],  bfalse: ff,  subtype_rel: A ⊆r B,  descending: descending(a,b.<[a; b];L),  cand: A c∧ B,  true: True,  guard: {T},  subtract: n - m,  uiff: uiff(P;Q),  top: Top,  le: A ≤ B,  lelt: i ≤ j < k,  int_seg: {i..j-},  nat: ℕ,  last: last(L),  sq_type: SQType(T),  it: ⋅,  nil: [],  select: L[n],  so_apply: x[s1;s2;s3],  so_lambda: so_lambda3,  append: as @ bs,  nat_plus: ℕ+
Lemmas referenced :  descending_wf,  append_wf,  istype-less_than,  length_wf,  hd_wf,  decidable__le,  full-omega-unsat,  intformand_wf,  intformnot_wf,  intformle_wf,  itermConstant_wf,  itermVar_wf,  intformless_wf,  istype-int,  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_less_lemma,  int_formula_prop_wf,  last_wf,  list-cases,  null_nil_lemma,  length_of_nil_lemma,  product_subtype_list,  null_cons_lemma,  length_of_cons_lemma,  istype-void,  list_wf,  istype-universe,  length-append,  int_seg_wf,  subtract_wf,  member-less_than,  iff_weakening_equal,  subtype_rel_self,  false_wf,  subtract-is-int-iff,  int_seg_properties,  select_wf,  add-member-int_seg2,  select_append_front,  true_wf,  squash_wf,  equal_wf,  istype-le,  int_term_value_add_lemma,  int_term_value_subtract_lemma,  itermAdd_wf,  itermSubtract_wf,  decidable__lt,  non_neg_length,  less_than_wf,  le_wf,  add-zero,  zero-mul,  add-mul-special,  add-swap,  minus-one-mul,  add-associates,  add-commutes,  select_append_back,  zero-add,  select-nthtl,  subtype_rel_list,  top_wf,  nth_tl_append,  add-is-int-iff,  decidable__equal_int,  int_subtype_base,  subtype_base_sq,  istype-base,  stuck-spread,  length_wf_nat,  int_formula_prop_eq_lemma,  intformeq_wf,  general_arith_equation1,  list_ind_cons_lemma,  list_ind_nil_lemma,  nat_plus_properties,  add_nat_plus
Rules used in proof :  sqequalSubstitution,  sqequalTransitivity,  computationStep,  sqequalReflexivity,  isect_memberFormation_alt,  lambdaFormation_alt,  independent_pairFormation,  universeIsType,  cut,  introduction,  extract_by_obid,  sqequalHypSubstitution,  isectElimination,  thin,  hypothesisEquality,  sqequalRule,  lambdaEquality_alt,  applyEquality,  inhabitedIsType,  hypothesis,  productIsType,  isectIsType,  natural_numberEquality,  because_Cache,  independent_isectElimination,  dependent_functionElimination,  unionElimination,  imageElimination,  productElimination,  approximateComputation,  independent_functionElimination,  dependent_pairFormation_alt,  int_eqEquality,  Error :memTop,  voidElimination,  promote_hyp,  hypothesis_subsumption,  functionIsType,  universeEquality,  instantiate,  rename,  imageMemberEquality,  baseClosed,  baseApply,  pointwiseFunctionality,  closedConclusion,  cumulativity,  functionEquality,  equalitySymmetry,  equalityTransitivity,  isect_memberEquality_alt,  addEquality,  dependent_set_memberEquality_alt,  setElimination,  productEquality,  hyp_replacement,  minusEquality,  intEquality,  equalityIsType1,  applyLambdaEquality
Latex:
\mforall{}[A:Type].  \mforall{}[<:A  {}\mrightarrow{}  A  {}\mrightarrow{}  \mBbbP{}].
    \mforall{}L1,L2:A  List.
        (descending(a,b.<[a;b];L1  @  L2)
        \mLeftarrow{}{}\mRightarrow{}  descending(a,b.<[a;b];L1)
                \mwedge{}  descending(a,b.<[a;b];L2)
                \mwedge{}  (<[hd(L2);last(L1)])  supposing  (0  <  ||L2||  and  0  <  ||L1||))


Date html generated: 2020_05_20-AM-08_07_28
Last ObjectModification: 2019_12_31-PM-06_30_35

Theory : general


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