Nuprl Lemma : list_accum_invariant3

∀[T,A:Type].
  ∀f:A ⟶ T ⟶ A
    ∀[P:A ⟶ (T List) ⟶ ℙ]
      ∀L:T List. ∀a:A.
        (P[a;[]]
        ⇒ (∀a:A. ∀x:T. ∀L':T List.  (L' @ [x] ≤ L ⇒ P[a;L'] ⇒ P[f[a;x];L' @ [x]]))
        ⇒ P[accumulate (with value a and list item x):
              f[a;x]
             over list:
               L
             with starting value:
              a);L])


Proof



Definitions occuring in Statement :  iseg: l1 ≤ l2,  append: as @ bs,  list_accum: list_accum,  cons: [a / b],  nil: [],  list: T List,  uall: ∀[x:A]. B[x],  prop: ℙ,  so_apply: x[s1;s2],  all: ∀x:A. B[x],  implies: P ⇒ Q,  function: x:A ⟶ B[x],  universe: Type
Definitions unfolded in proof :  uall: ∀[x:A]. B[x],  all: ∀x:A. B[x],  implies: P ⇒ Q,  exists: ∃x:A. B[x],  member: t ∈ T,  prop: ℙ,  nat: ℕ,  so_lambda: λ2x.t[x],  so_apply: x[s1;s2],  subtype_rel: A ⊆r B,  so_apply: x[s],  so_lambda: λ2x y.t[x; y],  uiff: uiff(P;Q),  and: P ∧ Q,  uimplies: b supposing a,  top: Top,  not: ¬A,  satisfiable_int_formula: satisfiable_int_formula(fmla),  false: False,  squash: ↓T,  true: True,  guard: {T},  iff: P ⇐⇒ Q,  rev_implies: P ⇐ Q,  decidable: Dec(P),  or: P ∨ Q
Lemmas referenced :  length_wf_nat,  length_wf,  equal_wf,  equal-wf-base-T,  all_wf,  list_wf,  iseg_wf,  append_wf,  cons_wf,  nil_wf,  int_subtype_base,  list_accum_wf,  set_wf,  less_than_wf,  primrec-wf2,  nat_wf,  length_zero,  list_accum_nil_lemma,  last_lemma,  assert_of_null,  length_of_nil_lemma,  satisfiable-full-omega-tt,  intformand_wf,  intformeq_wf,  itermVar_wf,  itermConstant_wf,  intformless_wf,  int_formula_prop_and_lemma,  int_formula_prop_eq_lemma,  int_term_value_var_lemma,  int_term_value_constant_lemma,  int_formula_prop_less_lemma,  int_formula_prop_wf,  assert_wf,  null_wf,  list_accum_append,  subtype_rel_list,  top_wf,  list_accum_cons_lemma,  last_wf,  squash_wf,  true_wf,  iff_weakening_equal,  iseg_weakening,  iseg_append,  length-append,  length_of_cons_lemma,  decidable__equal_int,  add-is-int-iff,  intformnot_wf,  itermSubtract_wf,  itermAdd_wf,  int_formula_prop_not_lemma,  int_term_value_subtract_lemma,  int_term_value_add_lemma,  false_wf
Rules used in proof :  sqequalSubstitution,  sqequalTransitivity,  computationStep,  sqequalReflexivity,  isect_memberFormation,  lambdaFormation,  cut,  dependent_pairFormation,  introduction,  extract_by_obid,  sqequalHypSubstitution,  isectElimination,  thin,  cumulativity,  hypothesisEquality,  hypothesis,  intEquality,  setElimination,  rename,  productElimination,  baseClosed,  sqequalRule,  lambdaEquality,  because_Cache,  functionEquality,  applyEquality,  functionExtensionality,  baseApply,  closedConclusion,  natural_numberEquality,  dependent_functionElimination,  independent_functionElimination,  universeEquality,  equalitySymmetry,  independent_isectElimination,  dependent_set_memberEquality,  isect_memberEquality,  voidElimination,  voidEquality,  hyp_replacement,  applyLambdaEquality,  int_eqEquality,  independent_pairFormation,  computeAll,  imageElimination,  equalityTransitivity,  imageMemberEquality,  equalityUniverse,  levelHypothesis,  unionElimination,  pointwiseFunctionality,  promote_hyp
Latex:
\mforall{}[T,A:Type].
    \mforall{}f:A  {}\mrightarrow{}  T  {}\mrightarrow{}  A
        \mforall{}[P:A  {}\mrightarrow{}  (T  List)  {}\mrightarrow{}  \mBbbP{}]
            \mforall{}L:T  List.  \mforall{}a:A.
                (P[a;[]]
                {}\mRightarrow{}  (\mforall{}a:A.  \mforall{}x:T.  \mforall{}L':T  List.    (L'  @  [x]  \mleq{}  L  {}\mRightarrow{}  P[a;L']  {}\mRightarrow{}  P[f[a;x];L'  @  [x]]))
                {}\mRightarrow{}  P[accumulate  (with  value  a  and  list  item  x):
                            f[a;x]
                          over  list:
                              L
                          with  starting  value:
                            a);L])


Date html generated: 2017_04_17-AM-07_38_43
Last ObjectModification: 2017_02_27-PM-04_13_27

Theory : list_1


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