Nuprl Lemma : vdf-wf+

∀[A,B:Type]. ∀[C:A ⟶ B ⟶ Type].
  ∀n:ℕ
    ((vdf(A;B;a,b.C[a;b];n) ∈ Type)
    ∧ (∀f:vdf(A;B;a,b.C[a;b];n). ∀L:(a:A × b:B × C[a;b]) List.
         ((||L|| ≤ (n + 1))
         ⇒ ((vdf-eq(A;f;L) ∈ ℙ)
            ∧ (vdf-eq(A;f;L) ⊆r (∀[i:ℕ||L||]. ((fst(L[i])) = (f firstn(i;L) (fst(snd(L[i])))) ∈ A)))))))


Proof



Definitions occuring in Statement :  vdf: vdf(A;B;a,b.C[a; b];n),  vdf-eq: vdf-eq(A;f;L),  firstn: firstn(n;as),  select: L[n],  length: ||as||,  list: T List,  int_seg: {i..j-},  nat: ℕ,  subtype_rel: A ⊆r B,  uall: ∀[x:A]. B[x],  prop: ℙ,  so_apply: x[s1;s2],  pi1: fst(t),  pi2: snd(t),  le: A ≤ B,  all: ∀x:A. B[x],  implies: P ⇒ Q,  and: P ∧ Q,  member: t ∈ T,  apply: f a,  function: x:A ⟶ B[x],  product: x:A × B[x],  add: n + m,  natural_number: $n,  universe: Type,  equal: s = t ∈ T
Definitions unfolded in proof :  uall: ∀[x:A]. B[x],  member: t ∈ T,  all: ∀x:A. B[x],  nat: ℕ,  implies: P ⇒ Q,  false: False,  ge: i ≥ j ,  uimplies: b supposing a,  not: ¬A,  satisfiable_int_formula: satisfiable_int_formula(fmla),  exists: ∃x:A. B[x],  and: P ∧ Q,  prop: ℙ,  subtype_rel: A ⊆r B,  vdf: vdf(A;B;a,b.C[a; b];n),  lt_int: i <z j,  subtract: n - m,  ifthenelse: if b then t else f fi ,  btrue: tt,  cand: A c∧ B,  so_apply: x[s1;s2],  so_lambda: λ2x.t[x],  so_apply: x[s],  or: P ∨ Q,  vdf-eq: vdf-eq(A;f;L),  select: L[n],  nil: [],  it: ⋅,  so_lambda: λ2x y.t[x; y],  dep-all: dep-all(n;i.P[i]),  cons: [a / b],  spreadn: spread3,  firstn: firstn(n;as),  so_lambda: so_lambda3,  so_apply: x[s1;s2;s3],  bfalse: ff,  le: A ≤ B,  decidable: Dec(P),  true: True,  int_seg: {i..j-},  lelt: i ≤ j < k,  sq_type: SQType(T),  guard: {T},  pi1: fst(t),  pi2: snd(t),  bool: 𝔹,  unit: Unit,  uiff: uiff(P;Q),  bnot: ¬bb,  assert: ↑b,  rev_implies: P ⇐ Q,  iff: P ⇐⇒ Q,  less_than: a < b,  squash: ↓T,  less_than': less_than'(a;b),  int_iseg: {i...j}
Lemmas referenced :  nat_properties,  full-omega-unsat,  intformand_wf,  intformle_wf,  itermConstant_wf,  itermVar_wf,  intformless_wf,  istype-int,  int_formula_prop_and_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,  ge_wf,  istype-less_than,  subtract-1-ge-0,  istype-nat,  istype-universe,  list_wf,  equal-wf-base,  length_wf_nat,  set_subtype_base,  le_wf,  int_subtype_base,  istype-le,  length_wf,  list-cases,  length_of_nil_lemma,  stuck-spread,  istype-base,  true_wf,  product_subtype_list,  length_of_cons_lemma,  list_ind_cons_lemma,  dep-isect-wf,  equal_wf,  nil_wf,  istype-true,  le_weakening2,  non_neg_length,  decidable__lt,  itermAdd_wf,  int_term_value_add_lemma,  int_seg_properties,  int_seg_wf,  first0,  cons_wf,  subtype_rel_list,  top_wf,  decidable__equal_int,  subtype_base_sq,  int_seg_subtype_special,  int_seg_cases,  lt_int_wf,  eqtt_to_assert,  assert_of_lt_int,  eqff_to_assert,  bool_cases_sqequal,  bool_wf,  bool_subtype_base,  assert-bnot,  iff_weakening_uiff,  assert_wf,  less_than_wf,  subtract-add-cancel,  decidable__le,  intformnot_wf,  int_formula_prop_not_lemma,  subtract_wf,  itermSubtract_wf,  int_term_value_subtract_lemma,  istype-top,  select-firstn,  firstn-firstn,  firstn_wf,  squash_wf,  length_firstn_eq,  subtype_rel_self,  iff_weakening_equal,  length_firstn,  select_wf,  lelt_wf,  intformeq_wf,  int_formula_prop_eq_lemma
Rules used in proof :  sqequalSubstitution,  sqequalTransitivity,  computationStep,  sqequalReflexivity,  isect_memberFormation_alt,  introduction,  cut,  lambdaFormation_alt,  extract_by_obid,  sqequalHypSubstitution,  isectElimination,  thin,  hypothesisEquality,  hypothesis,  setElimination,  rename,  sqequalRule,  intWeakElimination,  natural_numberEquality,  independent_isectElimination,  approximateComputation,  independent_functionElimination,  dependent_pairFormation_alt,  lambdaEquality_alt,  int_eqEquality,  dependent_functionElimination,  Error :memTop,  independent_pairFormation,  universeIsType,  voidElimination,  productElimination,  independent_pairEquality,  axiomEquality,  equalityTransitivity,  equalitySymmetry,  functionIsTypeImplies,  inhabitedIsType,  functionIsType,  because_Cache,  isect_memberEquality_alt,  isectIsTypeImplies,  instantiate,  universeEquality,  functionEquality,  setEquality,  productEquality,  applyEquality,  intEquality,  baseClosed,  unionElimination,  promote_hyp,  hypothesis_subsumption,  dependent_set_memberEquality_alt,  equalityIstype,  sqequalBase,  productIsType,  dependentIntersectionElimination,  cumulativity,  depIsectIsType,  addEquality,  equalityElimination,  axiomSqEquality,  imageElimination,  lessCases,  imageMemberEquality,  closedConclusion
Latex:
\mforall{}[A,B:Type].  \mforall{}[C:A  {}\mrightarrow{}  B  {}\mrightarrow{}  Type].
    \mforall{}n:\mBbbN{}
        ((vdf(A;B;a,b.C[a;b];n)  \mmember{}  Type)
        \mwedge{}  (\mforall{}f:vdf(A;B;a,b.C[a;b];n).  \mforall{}L:(a:A  \mtimes{}  b:B  \mtimes{}  C[a;b])  List.
                  ((||L||  \mleq{}  (n  +  1))
                  {}\mRightarrow{}  ((vdf-eq(A;f;L)  \mmember{}  \mBbbP{})
                        \mwedge{}  (vdf-eq(A;f;L)  \msubseteq{}r  (\mforall{}[i:\mBbbN{}||L||].  ((fst(L[i]))  =  (f  firstn(i;L)  (fst(snd(L[i])))))))))))


Date html generated: 2020_05_19-PM-09_40_11
Last ObjectModification: 2020_03_05-PM-04_39_08

Theory : co-recursion-2


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