Nuprl Lemma : rng_prod_plus

∀[r:CRng]. ∀[n:ℕ]. ∀[F,G:ℕn ⟶ |r|].
  ((Π(r) 0 
         ≤ i 
         < n
     F[i] +r G[i])
  = Σ{r} p ∈ functions-list(n;2). (Π(r) 0 
                                        ≤ i 
                                        < n
                                    if (p i =z 0) then F[i] else G[i] fi )
  ∈ |r|)


Proof



Definitions occuring in Statement :  rng_lsum: Σ{r} x ∈ as. f[x],  functions-list: functions-list(n;b),  int_seg: {i..j-},  nat: ℕ,  ifthenelse: if b then t else f fi ,  eq_int: (i =z j),  uall: ∀[x:A]. B[x],  infix_ap: x f y,  so_apply: x[s],  apply: f a,  function: x:A ⟶ B[x],  natural_number: $n,  equal: s = t ∈ T,  rng_prod: rng_prod,  crng: CRng,  rng_plus: +r,  rng_car: |r|
Definitions unfolded in proof :  uall: ∀[x:A]. B[x],  member: t ∈ T,  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],  all: ∀x:A. B[x],  top: Top,  and: P ∧ Q,  prop: ℙ,  so_lambda: λ2x.t[x],  so_apply: x[s],  guard: {T},  int_seg: {i..j-},  lelt: i ≤ j < k,  crng: CRng,  rng: Rng,  le: A ≤ B,  less_than': less_than'(a;b),  true: True,  nat_plus: ℕ+,  subtype_rel: A ⊆r B,  bool: 𝔹,  unit: Unit,  it: ⋅,  btrue: tt,  uiff: uiff(P;Q),  ifthenelse: if b then t else f fi ,  bfalse: ff,  or: P ∨ Q,  sq_type: SQType(T),  bnot: ¬bb,  assert: ↑b,  decidable: Dec(P),  squash: ↓T,  infix_ap: x f y,  nequal: a ≠ b ∈ T ,  iff: P ⇐⇒ Q,  rev_implies: P ⇐ Q,  less_than: a < b,  sq_stable: SqStable(P),  cand: A c∧ B,  rev_uimplies: rev_uimplies(P;Q),  subtract: n - m,  inject: Inj(A;B;f)
Lemmas referenced :  nat_properties,  full-omega-unsat,  intformand_wf,  intformle_wf,  itermConstant_wf,  itermVar_wf,  intformless_wf,  istype-int,  int_formula_prop_and_lemma,  istype-void,  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,  less_than_wf,  rng_prod_empty_lemma,  int_seg_wf,  int_seg_properties,  subtract-1-ge-0,  rng_car_wf,  nat_wf,  crng_wf,  rng_one_wf,  istype-false,  le_wf,  rng_lsum_cons_lemma,  rng_lsum_nil_lemma,  rng_plus_zero,  infix_ap_wf,  rng_plus_wf,  eq_int_wf,  eqtt_to_assert,  assert_of_eq_int,  eqff_to_assert,  bool_cases_sqequal,  subtype_base_sq,  bool_wf,  bool_subtype_base,  assert-bnot,  neg_assert_of_eq_int,  functions-list_wf,  decidable__le,  intformnot_wf,  int_formula_prop_not_lemma,  equal_wf,  istype-universe,  rng_times_wf,  decidable__lt,  itermSubtract_wf,  int_term_value_subtract_lemma,  subtract_wf,  rng_lsum_wf,  rng_prod_wf,  iff_weakening_equal,  squash_wf,  true_wf,  list_wf,  rng_wf,  functions-list-0,  subtype_rel_self,  rng_prod_unroll_hi,  rng_times_over_plus,  rng_lsum-split,  filter_wf5,  l_member_wf,  rng_times_lsum_r,  rng_lsum_map,  int_subtype_base,  intformeq_wf,  int_formula_prop_eq_lemma,  map_wf,  rng_lsum_functionality_wrt_permutation,  set_subtype_base,  lelt_wf,  sq_stable__no_repeats,  permutation-when-no_repeats,  no_repeats_filter,  member_filter,  member-functions-list,  member-map,  decidable__equal_int,  subtype_rel_function,  int_seg_subtype,  not-le-2,  condition-implies-le,  add-associates,  minus-add,  minus-one-mul,  add-swap,  minus-one-mul-top,  add-mul-special,  zero-mul,  add-zero,  add-commutes,  le-add-cancel2,  all_wf,  no_repeats_wf,  no_repeats-functions-list,  false_wf,  no_repeats_map,  set_wf,  equal-wf-T-base,  not_wf,  bnot_wf,  assert_wf,  bool_cases,  iff_transitivity,  iff_weakening_uiff,  assert_of_bnot,  filter_type,  equal-wf-base,  btrue_neq_bfalse,  eq_int_eq_true,  and_wf,  bfalse_wf,  assert_elim
Rules used in proof :  sqequalSubstitution,  sqequalTransitivity,  computationStep,  sqequalReflexivity,  isect_memberFormation_alt,  introduction,  cut,  extract_by_obid,  sqequalHypSubstitution,  isectElimination,  thin,  hypothesisEquality,  hypothesis,  setElimination,  rename,  intWeakElimination,  lambdaFormation_alt,  natural_numberEquality,  independent_isectElimination,  approximateComputation,  independent_functionElimination,  dependent_pairFormation_alt,  lambdaEquality_alt,  int_eqEquality,  dependent_functionElimination,  isect_memberEquality_alt,  voidElimination,  sqequalRule,  independent_pairFormation,  universeIsType,  axiomEquality,  inhabitedIsType,  functionIsTypeImplies,  functionIsType,  productElimination,  because_Cache,  functionEquality,  dependent_set_memberEquality_alt,  equalitySymmetry,  applyEquality,  unionElimination,  equalityElimination,  equalityTransitivity,  equalityIsType1,  promote_hyp,  instantiate,  cumulativity,  imageElimination,  universeEquality,  productIsType,  imageMemberEquality,  baseClosed,  setIsType,  equalityIsType2,  baseApply,  closedConclusion,  hyp_replacement,  intEquality,  addEquality,  minusEquality,  multiplyEquality,  functionExtensionality_alt,  applyLambdaEquality,  productEquality,  setEquality,  functionExtensionality,  lambdaFormation,  voidEquality,  isect_memberEquality,  lambdaEquality,  dependent_pairFormation,  dependent_set_memberEquality,  impliesFunctionality,  levelHypothesis,  addLevel
Latex:
\mforall{}[r:CRng].  \mforall{}[n:\mBbbN{}].  \mforall{}[F,G:\mBbbN{}n  {}\mrightarrow{}  |r|].
    ((\mPi{}(r)  0 
                  \mleq{}  i 
                  <  n
          F[i]  +r  G[i])
    =  \mSigma{}\{r\}  p  \mmember{}  functions-list(n;2).  (\mPi{}(r)  0 
                                                                                \mleq{}  i 
                                                                                <  n
                                                                        if  (p  i  =\msubz{}  0)  then  F[i]  else  G[i]  fi  ))


Date html generated: 2019_10_16-AM-11_27_08
Last ObjectModification: 2018_10_10-AM-10_15_55

Theory : matrices


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