Nuprl Lemma : linearization-value

∀[L:ℤ List List]. ∀[p:iPolynomial()].
  ∀f:ℤ ⟶ ℤ
    (int_term_value(f;ipolynomial-term(p))
    = linearization(p;L) ⋅ map(λvs.accumulate (with value x and list item v):
                                    x * (f v)
                                   over list:
                                     vs
                                   with starting value:
                                    1);L)
    ∈ ℤ) 
  supposing (∀m∈p.(snd(m) ∈ L)) ∧ no_repeats(ℤ List;L)


Proof



Definitions occuring in Statement :  linearization: linearization(p;L),  ipolynomial-term: ipolynomial-term(p),  iPolynomial: iPolynomial(),  int_term_value: int_term_value(f;t),  integer-dot-product: as ⋅ bs,  l_all: (∀x∈L.P[x]),  no_repeats: no_repeats(T;l),  l_member: (x ∈ l),  map: map(f;as),  list_accum: list_accum,  list: T List,  uimplies: b supposing a,  uall: ∀[x:A]. B[x],  pi2: snd(t),  all: ∀x:A. B[x],  and: P ∧ Q,  apply: f a,  lambda: λx.A[x],  function: x:A ⟶ B[x],  multiply: n * m,  natural_number: $n,  int: ℤ,  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 ,  guard: {T},  uimplies: b supposing a,  prop: ℙ,  or: P ∨ Q,  top: Top,  cons: [a / b],  le: A ≤ B,  and: P ∧ Q,  less_than': less_than'(a;b),  not: ¬A,  colength: colength(L),  nil: [],  it: ⋅,  so_lambda: λ2x.t[x],  so_apply: x[s],  sq_type: SQType(T),  less_than: a < b,  squash: ↓T,  so_lambda: λ2x y.t[x; y],  so_apply: x[s1;s2],  sq_stable: SqStable(P),  subtract: n - m,  subtype_rel: A ⊆r B,  iPolynomial: iPolynomial(),  linearization: linearization(p;L),  ipolynomial-term: ipolynomial-term(p),  int_term_value: int_term_value(f;t),  ifthenelse: if b then t else f fi ,  btrue: tt,  itermConstant: "const",  int_term_ind: int_term_ind,  iMonomial: iMonomial(),  pi2: snd(t),  true: True,  iff: P ⇐⇒ Q,  rev_implies: P ⇐ Q,  uiff: uiff(P;Q),  bool: 𝔹,  unit: Unit,  bfalse: ff,  exists: ∃x:A. B[x],  bnot: ¬bb,  assert: ↑b,  bor: p ∨bq,  deq: EqDecider(T),  int_nzero: ℤ-o,  istype: istype(T),  select: L[n],  int_seg: {i..j-},  lelt: i ≤ j < k,  poly-coeff-of: poly-coeff-of(vs;p),  so_lambda: so_lambda(x,y,z.t[x; y; z]),  so_apply: x[s1;s2;s3],  nat_plus: ℕ+,  decidable: Dec(P),  l_all: (∀x∈L.P[x]),  imonomial-less: imonomial-less(m1;m2),  imonomial-term: imonomial-term(m),  cand: A c∧ B,  nequal: a ≠ b ∈ T ,  list_ind: list_ind,  imonomial-le: imonomial-le(m1;m2),  l_member: (x ∈ l)
Lemmas referenced :  nat_properties,  less_than_transitivity1,  less_than_irreflexivity,  ge_wf,  less_than_wf,  no_repeats_wf,  list_wf,  iPolynomial_wf,  list-cases,  deq_member_nil_lemma,  istype-void,  map_nil_lemma,  istype-int,  nil_wf,  product_subtype_list,  colength-cons-not-zero,  nat_wf,  colength_wf_list,  istype-false,  le_wf,  subtract-1-ge-0,  subtype_base_sq,  set_subtype_base,  int_subtype_base,  spread_cons_lemma,  sq_stable__le,  add-associates,  add-commutes,  add-swap,  zero-add,  deq_member_cons_lemma,  map_cons_lemma,  cons_wf,  le_weakening2,  filter-bfalse,  subtype_rel_list,  iMonomial_wf,  top_wf,  null_nil_lemma,  int_dot_nil_left_lemma,  int_dot_cons_lemma,  int_term_value_wf,  ipolynomial-term_wf,  filter_wf5,  l_member_wf,  bor_wf,  list-deq_wf1,  int-deq_wf,  deq-member_wf,  list-deq_wf,  poly-coeff-of_wf,  list_accum_wf,  integer-dot-product_wf,  linearization_wf,  map_wf,  equal_wf,  squash_wf,  true_wf,  add_functionality_wrt_eq,  subtype_rel_self,  iff_weakening_equal,  no_repeats_cons,  filter_nil_lemma,  filter_cons_lemma,  eqtt_to_assert,  assert-deq-member,  eqff_to_assert,  bool_cases_sqequal,  bool_wf,  bool_subtype_base,  assert-bnot,  testxxx_lemma,  list_subtype_base,  assert_wf,  imonomial-term_wf,  subtype_rel_product,  int_nzero_wf,  sorted_wf,  ipolynomial-term-cons-value,  length_of_nil_lemma,  stuck-spread,  istype-base,  int_seg_wf,  imonomial-less_wf,  length_of_cons_lemma,  length_wf,  select_wf,  non_neg_length,  length_wf_nat,  list_ind_wf,  intlex_wf,  mul-commutes,  less-iff-le,  add_functionality_wrt_le,  subtract_wf,  le_reflexive,  minus-add,  minus-one-mul,  one-mul,  add-mul-special,  two-mul,  mul-distributes-right,  zero-mul,  add-zero,  not-lt-2,  omega-shadow,  mul-distributes,  mul-associates,  minus-one-mul-top,  int_seg_properties,  decidable__lt,  list_ind_nil_lemma,  assert-list-deq,  iff_weakening_uiff,  equal-wf-base-T,  less_than_transitivity2,  filter_is_nil,  add-member-int_seg2,  decidable__le,  not-le-2,  condition-implies-le,  le-add-cancel2,  le-add-cancel,  select_cons_tl,  add-subtract-cancel,  list_ind_cons_lemma,  intlex-reflexive,  btrue_wf,  int_nzero_properties,  itermConstant_wf,  list_accum_nil_lemma,  int_term_wf,  list_accum_cons_lemma,  itermMultiply_wf,  itermVar_wf,  int_term_value_mul_lemma,  int_term_value_var_lemma,  mul-swap,  intlex-antisym,  not-equal-2,  intlex-total,  l_all_wf_nil,  l_all_wf,  add_nat_plus,  iff_imp_equal_bool,  select-cons-tl,  not-equal-implies-less,  filter_trivial,  istype-top
Rules used in proof :  cut,  sqequalSubstitution,  sqequalTransitivity,  computationStep,  sqequalReflexivity,  Error :isect_memberFormation_alt,  introduction,  thin,  Error :lambdaFormation_alt,  extract_by_obid,  sqequalHypSubstitution,  isectElimination,  hypothesisEquality,  hypothesis,  setElimination,  rename,  intWeakElimination,  natural_numberEquality,  independent_isectElimination,  independent_functionElimination,  voidElimination,  Error :universeIsType,  sqequalRule,  Error :lambdaEquality_alt,  dependent_functionElimination,  Error :isect_memberEquality_alt,  axiomEquality,  Error :functionIsTypeImplies,  Error :inhabitedIsType,  intEquality,  equalityTransitivity,  equalitySymmetry,  unionElimination,  Error :functionIsType,  because_Cache,  promote_hyp,  hypothesis_subsumption,  productElimination,  Error :equalityIsType1,  Error :dependent_set_memberEquality_alt,  independent_pairFormation,  instantiate,  cumulativity,  imageElimination,  imageMemberEquality,  baseClosed,  applyLambdaEquality,  applyEquality,  minusEquality,  Error :equalityIsType4,  addEquality,  functionExtensionality,  Error :setIsType,  multiplyEquality,  universeEquality,  equalityElimination,  Error :dependent_pairFormation_alt,  independent_pairEquality,  setEquality,  sqequalIntensionalEquality,  Error :productIsType,  hyp_replacement,  productEquality
Latex:
\mforall{}[L:\mBbbZ{}  List  List].  \mforall{}[p:iPolynomial()].
    \mforall{}f:\mBbbZ{}  {}\mrightarrow{}  \mBbbZ{}
        (int\_term\_value(f;ipolynomial-term(p))
        =  linearization(p;L)  \mcdot{}  map(\mlambda{}vs.accumulate  (with  value  x  and  list  item  v):
                                                                        x  *  (f  v)
                                                                      over  list:
                                                                          vs
                                                                      with  starting  value:
                                                                        1);L)) 
    supposing  (\mforall{}m\mmember{}p.(snd(m)  \mmember{}  L))  \mwedge{}  no\_repeats(\mBbbZ{}  List;L)


Date html generated: 2019_06_20-PM-00_47_26
Last ObjectModification: 2018_10_04-PM-02_39_59

Theory : omega


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