Nuprl Lemma : C_Array-elem_vs_DVALp

∀store:C_STOREp(). ∀ctyp:C_TYPE(). ∀env:C_TYPE_env(). ∀dval:C_DVALUEp(). ∀n:ℤ.
  (C_STOREp-welltyped(env;store)
  ⇒ (↑C_Array?(ctyp))
  ⇒ (0 ≤ n)
  ⇒ n < C_Array-length(ctyp)
  ⇒ (↑(C_TYPE_vs_DVALp(env;ctyp) dval))
  ⇒ (↑(C_TYPE_vs_DVALp(env;C_Array-elems(ctyp)) (DVp_Array-arr(dval) (DVp_Array-lower(dval) + n)))))


Proof



Definitions occuring in Statement :  C_STOREp-welltyped: C_STOREp-welltyped(env;store),  C_STOREp: C_STOREp(),  C_TYPE_vs_DVALp: C_TYPE_vs_DVALp(env;ctyp),  DVp_Array-arr: DVp_Array-arr(v),  DVp_Array-lower: DVp_Array-lower(v),  C_DVALUEp: C_DVALUEp(),  C_TYPE_env: C_TYPE_env(),  C_Array-elems: C_Array-elems(v),  C_Array-length: C_Array-length(v),  C_Array?: C_Array?(v),  C_TYPE: C_TYPE(),  assert: ↑b,  less_than: a < b,  le: A ≤ B,  all: ∀x:A. B[x],  implies: P ⇒ Q,  apply: f a,  add: n + m,  natural_number: $n,  int: ℤ
Definitions unfolded in proof :  all: ∀x:A. B[x],  implies: P ⇒ Q,  ext-eq: A ≡ B,  and: P ∧ Q,  member: t ∈ T,  subtype_rel: A ⊆r B,  uall: ∀[x:A]. B[x],  bool: 𝔹,  unit: Unit,  it: ⋅,  btrue: tt,  uiff: uiff(P;Q),  uimplies: b supposing a,  sq_type: SQType(T),  guard: {T},  eq_atom: x =a y,  ifthenelse: if b then t else f fi ,  C_Void: C_Void(),  C_Array-length: C_Array-length(v),  pi2: snd(t),  C_Array?: C_Array?(v),  pi1: fst(t),  assert: ↑b,  bfalse: ff,  C_Array-elems: C_Array-elems(v),  false: False,  exists: ∃x:A. B[x],  prop: ℙ,  or: P ∨ Q,  bnot: ¬bb,  C_Int: C_Int(),  C_Struct: C_Struct(fields),  C_Array: C_Array(length;elems),  C_Pointer: C_Pointer(to),  nat: ℕ,  C_TYPE_vs_DVALp: C_TYPE_vs_DVALp(env;ctyp),  C_TYPE_ind: C_TYPE_ind,  DVp_Null: DVp_Null(x),  DVp_Array?: DVp_Array?(v),  DVp_Array-lower: DVp_Array-lower(v),  DVp_Array-upper: DVp_Array-upper(v),  DVp_Array-arr: DVp_Array-arr(v),  DVp_Int: DVp_Int(int),  DVp_Pointer: DVp_Pointer(ptr),  DVp_Array: DVp_Array(lower;upper;arr),  DVp_Struct: DVp_Struct(lbls;struct),  let: let,  band: p ∧b q,  le: A ≤ B,  so_lambda: λ2x.t[x],  int_seg: {i..j-},  lelt: i ≤ j < k,  ge: i ≥ j ,  decidable: Dec(P),  satisfiable_int_formula: satisfiable_int_formula(fmla),  not: ¬A,  top: Top,  so_apply: x[s],  iff: P ⇐⇒ Q,  rev_implies: P ⇐ Q,  l_all: (∀x∈L.P[x])
Lemmas referenced :  select-upto,  length_upto,  assert-bl-all,  assert_of_band,  iff_weakening_uiff,  iff_transitivity,  l_all_wf2,  int_subtype_base,  equal-wf-T-base,  lelt_wf,  int_formula_prop_eq_lemma,  int_formula_prop_less_lemma,  intformeq_wf,  intformless_wf,  decidable__lt,  int_formula_prop_wf,  int_term_value_constant_lemma,  int_term_value_var_lemma,  int_term_value_subtract_lemma,  int_formula_prop_le_lemma,  int_formula_prop_not_lemma,  int_formula_prop_and_lemma,  itermConstant_wf,  itermVar_wf,  itermSubtract_wf,  intformle_wf,  intformnot_wf,  intformand_wf,  satisfiable-full-omega-tt,  decidable__le,  nat_properties,  add-member-int_seg1,  l_member_wf,  upto_wf,  int_seg_wf,  bl-all_wf,  assert_of_eq_int,  subtract_wf,  eq_int_wf,  C_DVALUEp-ext,  C_STOREp_wf,  C_TYPE_wf,  C_TYPE_env_wf,  C_DVALUEp_wf,  C_STOREp-welltyped_wf,  C_Array?_wf,  le_wf,  nat_wf,  C_Array-length_wf,  less_than_wf,  C_TYPE_vs_DVALp_wf,  assert_wf,  neg_assert_of_eq_atom,  assert-bnot,  bool_subtype_base,  bool_cases_sqequal,  equal_wf,  eqff_to_assert,  it_wf,  unit_subtype_base,  atom_subtype_base,  subtype_base_sq,  assert_of_eq_atom,  eqtt_to_assert,  bool_wf,  eq_atom_wf,  C_TYPE-ext
Rules used in proof :  sqequalSubstitution,  sqequalTransitivity,  computationStep,  sqequalReflexivity,  lambdaFormation,  cut,  lemma_by_obid,  promote_hyp,  sqequalHypSubstitution,  productElimination,  thin,  hypothesis_subsumption,  hypothesis,  hypothesisEquality,  applyEquality,  sqequalRule,  isectElimination,  tokenEquality,  unionElimination,  equalityElimination,  equalityTransitivity,  equalitySymmetry,  independent_isectElimination,  instantiate,  cumulativity,  atomEquality,  dependent_functionElimination,  independent_functionElimination,  because_Cache,  voidElimination,  dependent_pairFormation,  equalityEquality,  lambdaEquality,  setElimination,  rename,  natural_numberEquality,  intEquality,  dependent_set_memberEquality,  independent_pairFormation,  int_eqEquality,  isect_memberEquality,  voidEquality,  computeAll,  setEquality,  productEquality,  baseApply,  closedConclusion,  baseClosed
Latex:
\mforall{}store:C\_STOREp().  \mforall{}ctyp:C\_TYPE().  \mforall{}env:C\_TYPE\_env().  \mforall{}dval:C\_DVALUEp().  \mforall{}n:\mBbbZ{}.
    (C\_STOREp-welltyped(env;store)
    {}\mRightarrow{}  (\muparrow{}C\_Array?(ctyp))
    {}\mRightarrow{}  (0  \mleq{}  n)
    {}\mRightarrow{}  n  <  C\_Array-length(ctyp)
    {}\mRightarrow{}  (\muparrow{}(C\_TYPE\_vs\_DVALp(env;ctyp)  dval))
    {}\mRightarrow{}  (\muparrow{}(C\_TYPE\_vs\_DVALp(env;C\_Array-elems(ctyp)) 
                (DVp\_Array-arr(dval)  (DVp\_Array-lower(dval)  +  n)))))


Date html generated: 2016_05_16-AM-08_51_42
Last ObjectModification: 2016_01_17-AM-09_43_38

Theory : C-semantics


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