Nuprl Lemma : KozenSilva-corollary0

∀[r:CRng]. ∀[x,y:Atom].
  ∀[d:ℕ ⟶ ℕ]. ∀[k:ℕ].
    (Moessner(r;x;y;1;λi.if (i =z 0) then 0 else d (i - 1) fi ;k)
    = Π(i∈upto(k)).((((k - i) ⋅r 1)*atom(x)+atom(y)))^(d i)
    ∈ PowerSeries(r)) 
  supposing ¬(x = y ∈ Atom)


Proof



Definitions occuring in Statement :  Moessner: Moessner(r;x;y;h;d;k),  fps-exp: (f)^(n),  fps-scalar-mul: (c)*f,  fps-product: Π(x∈b).f[x],  fps-add: (f+g),  fps-atom: atom(x),  fps-one: 1,  power-series: PowerSeries(X;r),  upto: upto(n),  atom-deq: AtomDeq,  nat: ℕ,  ifthenelse: if b then t else f fi ,  eq_int: (i =z j),  uimplies: b supposing a,  uall: ∀[x:A]. B[x],  not: ¬A,  apply: f a,  lambda: λx.A[x],  function: x:A ⟶ B[x],  subtract: n - m,  natural_number: $n,  atom: Atom,  equal: s = t ∈ T,  rng_nat_op: n ⋅r e,  crng: CRng,  rng_one: 1
Definitions unfolded in proof :  uall: ∀[x:A]. B[x],  member: t ∈ T,  uimplies: b supposing a,  nat: ℕ,  all: ∀x:A. B[x],  implies: P ⇒ Q,  bool: 𝔹,  unit: Unit,  it: ⋅,  btrue: tt,  ifthenelse: if b then t else f fi ,  uiff: uiff(P;Q),  and: P ∧ Q,  le: A ≤ B,  less_than': less_than'(a;b),  false: False,  not: ¬A,  prop: ℙ,  bfalse: ff,  exists: ∃x:A. B[x],  or: P ∨ Q,  sq_type: SQType(T),  guard: {T},  bnot: ¬bb,  assert: ↑b,  ge: i ≥ j ,  int_upper: {i...},  decidable: Dec(P),  satisfiable_int_formula: satisfiable_int_formula(fmla),  top: Top,  eq_int: (i =z j),  subtract: n - m,  squash: ↓T,  true: True,  subtype_rel: A ⊆r B,  so_lambda: λ2x.t[x],  crng: CRng,  int_seg: {i..j-},  lelt: i ≤ j < k,  rng: Rng,  so_apply: x[s],  nequal: a ≠ b ∈ T ,  iff: P ⇐⇒ Q,  rev_implies: P ⇐ Q
Lemmas referenced :  KozenSilva-theorem,  fps-one_wf,  eq_int_wf,  bool_wf,  eqtt_to_assert,  assert_of_eq_int,  false_wf,  le_wf,  eqff_to_assert,  equal_wf,  bool_cases_sqequal,  subtype_base_sq,  bool_subtype_base,  assert-bnot,  neg_assert_of_eq_int,  int_upper_subtype_nat,  nat_properties,  nequal-le-implies,  zero-add,  nat_wf,  subtract_wf,  int_upper_properties,  decidable__le,  satisfiable-full-omega-tt,  intformand_wf,  intformnot_wf,  intformle_wf,  itermConstant_wf,  itermSubtract_wf,  itermVar_wf,  int_formula_prop_and_lemma,  int_formula_prop_not_lemma,  int_formula_prop_le_lemma,  int_term_value_constant_lemma,  int_term_value_subtract_lemma,  int_term_value_var_lemma,  int_formula_prop_wf,  squash_wf,  true_wf,  power-series_wf,  not_wf,  equal-wf-base,  atom_subtype_base,  crng_wf,  upto_wf,  list-subtype-bag,  int_seg_wf,  subtype_rel_self,  bag_wf,  fps-product_wf,  atom-valueall-type,  atom-deq_wf,  fps-exp_wf,  fps-add_wf,  fps-scalar-mul_wf,  rng_nat_op_wf,  int_seg_properties,  intformless_wf,  int_formula_prop_less_lemma,  rng_one_wf,  fps-atom_wf,  int_seg_subtype_nat,  fps-mul_wf,  intformeq_wf,  itermAdd_wf,  int_formula_prop_eq_lemma,  int_term_value_add_lemma,  add-subtract-cancel,  fps-one-slice,  fps-compose_wf,  valueall-type_wf,  deq_wf,  iff_weakening_equal,  fps-compose-one,  mul_one_fps,  add-associates,  add-swap,  add-commutes
Rules used in proof :  sqequalSubstitution,  sqequalTransitivity,  computationStep,  sqequalReflexivity,  isect_memberFormation,  introduction,  cut,  extract_by_obid,  sqequalHypSubstitution,  isectElimination,  thin,  hypothesisEquality,  independent_isectElimination,  hypothesis,  atomEquality,  lambdaEquality,  setElimination,  rename,  because_Cache,  natural_numberEquality,  lambdaFormation,  unionElimination,  equalityElimination,  sqequalRule,  productElimination,  dependent_set_memberEquality,  equalityTransitivity,  equalitySymmetry,  independent_pairFormation,  dependent_pairFormation,  promote_hyp,  dependent_functionElimination,  instantiate,  cumulativity,  independent_functionElimination,  voidElimination,  hypothesis_subsumption,  applyEquality,  functionExtensionality,  int_eqEquality,  intEquality,  isect_memberEquality,  voidEquality,  computeAll,  hyp_replacement,  imageElimination,  universeEquality,  imageMemberEquality,  baseClosed,  axiomEquality,  functionEquality,  applyLambdaEquality,  addEquality,  minusEquality
Latex:
\mforall{}[r:CRng].  \mforall{}[x,y:Atom].
    \mforall{}[d:\mBbbN{}  {}\mrightarrow{}  \mBbbN{}].  \mforall{}[k:\mBbbN{}].
        (Moessner(r;x;y;1;\mlambda{}i.if  (i  =\msubz{}  0)  then  0  else  d  (i  -  1)  fi  ;k)
        =  \mPi{}(i\mmember{}upto(k)).((((k  -  i)  \mcdot{}r  1)*atom(x)+atom(y)))\^{}(d  i)) 
    supposing  \mneg{}(x  =  y)


Date html generated: 2018_05_21-PM-10_14_18
Last ObjectModification: 2017_07_26-PM-06_35_31

Theory : power!series


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