Nuprl Lemma : sbhomout-correct

∀[a,b,c,d:ℕ].
  (0 < a + b
  ⇒ 0 < c + d
  ⇒ (∀[L:ℕ2 List]
        (sbhomout(a;b;c;d;L) = let m,n = sbdecode(L) in sbcode((a * m) + (b * n);(c * m) + (d * n)) ∈ (ℕ2 List))))


Proof



Definitions occuring in Statement :  sbhomout: sbhomout(a;b;c;d;L),  sbdecode: sbdecode(L),  sbcode: sbcode(m;n),  list: T List,  int_seg: {i..j-},  nat: ℕ,  less_than: a < b,  uall: ∀[x:A]. B[x],  implies: P ⇒ Q,  spread: spread def,  multiply: n * m,  add: n + m,  natural_number: $n,  equal: s = t ∈ T
Definitions unfolded in proof :  all: ∀x:A. B[x],  implies: P ⇒ Q,  and: P ∧ Q,  cand: A c∧ B,  uall: ∀[x:A]. B[x],  member: t ∈ T,  subtype_rel: A ⊆r B,  nat: ℕ,  decidable: Dec(P),  or: P ∨ Q,  uimplies: b supposing a,  nat_plus: ℕ+,  ge: i ≥ j ,  satisfiable_int_formula: satisfiable_int_formula(fmla),  exists: ∃x:A. B[x],  false: False,  not: ¬A,  top: Top,  prop: ℙ,  so_lambda: λ2x.t[x],  so_apply: x[s],  sbhomout: sbhomout(a;b;c;d;L),  nil: [],  it: ⋅,  sbdecode: sbdecode(L),  reduce: reduce(f;k;as),  list_ind: list_ind,  squash: ↓T,  le: A ≤ B,  uiff: uiff(P;Q),  true: True,  guard: {T},  int_seg: {i..j-},  lelt: i ≤ j < k,  cons: [a / b],  so_lambda: λ2x y.t[x; y],  so_apply: x[s1;s2],  iff: P ⇐⇒ Q,  rev_implies: P ⇐ Q,  bool: 𝔹,  unit: Unit,  btrue: tt,  ifthenelse: if b then t else f fi ,  mtge1: mtge1(a;b;c;d),  bfalse: ff,  sq_type: SQType(T),  bnot: ¬bb,  assert: ↑b,  has-value: (a)↓,  less_than': less_than'(a;b),  less_than: a < b,  sbcode: sbcode(m;n),  subtract: n - m,  nequal: a ≠ b ∈ T 
Lemmas referenced :  mul_bounds_1a,  nat_plus_subtype_nat,  decidable__lt,  mul_preserves_lt,  nat_plus_properties,  nat_properties,  satisfiable-full-omega-tt,  intformand_wf,  intformnot_wf,  intformless_wf,  itermConstant_wf,  itermAdd_wf,  itermMultiply_wf,  itermVar_wf,  intformle_wf,  int_formula_prop_and_lemma,  int_formula_prop_not_lemma,  int_formula_prop_less_lemma,  int_term_value_constant_lemma,  int_term_value_add_lemma,  int_term_value_mul_lemma,  int_term_value_var_lemma,  int_formula_prop_le_lemma,  int_formula_prop_wf,  less_than_wf,  nat_plus_wf,  nat_wf,  list_induction,  int_seg_wf,  all_wf,  uall_wf,  le_wf,  equal_wf,  list_wf,  sbhomout_wf,  sbdecode_wf,  sbcode_wf,  ge_wf,  decidable__le,  subtract_wf,  itermSubtract_wf,  int_term_value_subtract_lemma,  squash_wf,  true_wf,  add-is-int-iff,  int_subtype_base,  mul-commutes,  one-mul,  int_seg_properties,  spread_cons_lemma,  iff_weakening_equal,  mtge1_wf,  bool_wf,  eqtt_to_assert,  eqff_to_assert,  bool_cases_sqequal,  subtype_base_sq,  bool_subtype_base,  assert-bnot,  assert_wf,  bor_wf,  band_wf,  le_int_wf,  lt_int_wf,  or_wf,  iff_transitivity,  iff_weakening_uiff,  assert_of_bor,  assert_of_band,  assert_of_le_int,  assert_of_lt_int,  value-type-has-value,  int-value-type,  decidable__equal_int,  intformeq_wf,  int_formula_prop_eq_lemma,  false_wf,  lelt_wf,  intformor_wf,  int_formula_prop_or_lemma,  cons_wf,  top_wf,  mul_preserves_le,  mul-distributes-right,  add-associates,  minus-add,  minus-one-mul,  mul-associates,  add-swap,  add-commutes,  eq_int_wf,  assert_of_eq_int,  reduce_cons_lemma,  neg_assert_of_eq_int,  add_nat_plus,  multiply_nat_plus
Rules used in proof :  cut,  sqequalSubstitution,  sqequalTransitivity,  computationStep,  sqequalReflexivity,  lambdaFormation,  introduction,  extract_by_obid,  sqequalHypSubstitution,  isectElimination,  thin,  hypothesisEquality,  applyEquality,  hypothesis,  sqequalRule,  independent_pairFormation,  because_Cache,  dependent_functionElimination,  natural_numberEquality,  productElimination,  setElimination,  rename,  unionElimination,  independent_isectElimination,  addEquality,  multiplyEquality,  dependent_pairFormation,  lambdaEquality,  int_eqEquality,  intEquality,  isect_memberEquality,  voidElimination,  voidEquality,  computeAll,  functionEquality,  independent_functionElimination,  productEquality,  dependent_set_memberEquality,  equalityTransitivity,  equalitySymmetry,  intWeakElimination,  axiomEquality,  isect_memberFormation,  imageElimination,  baseApply,  closedConclusion,  baseClosed,  imageMemberEquality,  universeEquality,  equalityElimination,  promote_hyp,  instantiate,  cumulativity,  orFunctionality,  callbyvalueReduce,  lessCases,  sqequalAxiom,  minusEquality,  int_eqReduceTrueSq,  int_eqReduceFalseSq,  applyLambdaEquality
Latex:
\mforall{}[a,b,c,d:\mBbbN{}].
    (0  <  a  +  b
    {}\mRightarrow{}  0  <  c  +  d
    {}\mRightarrow{}  (\mforall{}[L:\mBbbN{}2  List]
                (sbhomout(a;b;c;d;L)
                =  let  m,n  =  sbdecode(L) 
                    in  sbcode((a  *  m)  +  (b  *  n);(c  *  m)  +  (d  *  n)))))


Date html generated: 2018_05_21-PM-11_42_24
Last ObjectModification: 2017_07_26-PM-06_42_52

Theory : rationals


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