Nuprl Lemma : implies-eq-upto-baire2cantor

∀a,b:ℕ ⟶ ℕ. ∀n:ℕ.  ((a = b ∈ (ℕn ⟶ ℕ)) ⇒ (baire2cantor(a) = baire2cantor(b) ∈ (ℕn ⟶ 𝔹)))


Proof



Definitions occuring in Statement :  baire2cantor: baire2cantor(a),  int_seg: {i..j-},  nat: ℕ,  bool: 𝔹,  all: ∀x:A. B[x],  implies: P ⇒ Q,  function: x:A ⟶ B[x],  natural_number: $n,  equal: s = t ∈ T
Definitions unfolded in proof :  all: ∀x:A. B[x],  implies: P ⇒ Q,  baire2cantor: baire2cantor(a),  nat-pred: n-1,  member: t ∈ T,  uall: ∀[x:A]. B[x],  int_seg: {i..j-},  bool: 𝔹,  unit: Unit,  it: ⋅,  btrue: tt,  uiff: uiff(P;Q),  and: P ∧ Q,  uimplies: b supposing a,  lelt: i ≤ j < k,  le: A ≤ B,  less_than': less_than'(a;b),  false: False,  not: ¬A,  guard: {T},  nat: ℕ,  ge: i ≥ j ,  decidable: Dec(P),  or: P ∨ Q,  satisfiable_int_formula: satisfiable_int_formula(fmla),  exists: ∃x:A. B[x],  top: Top,  prop: ℙ,  bfalse: ff,  subtype_rel: A ⊆r B,  so_lambda: λ2x.t[x],  so_apply: x[s],  sq_type: SQType(T),  bnot: ¬bb,  ifthenelse: if b then t else f fi ,  assert: ↑b,  nequal: a ≠ b ∈ T ,  squash: ↓T,  true: True
Lemmas referenced :  eq_int_wf,  eqtt_to_assert,  assert_of_eq_int,  istype-false,  int_seg_properties,  nat_properties,  decidable__lt,  full-omega-unsat,  intformand_wf,  intformnot_wf,  intformless_wf,  itermConstant_wf,  itermVar_wf,  intformeq_wf,  istype-int,  int_formula_prop_and_lemma,  istype-void,  int_formula_prop_not_lemma,  int_formula_prop_less_lemma,  int_term_value_constant_lemma,  int_term_value_var_lemma,  int_formula_prop_eq_lemma,  int_formula_prop_wf,  istype-le,  istype-less_than,  eqff_to_assert,  set_subtype_base,  lelt_wf,  int_subtype_base,  bool_subtype_base,  bool_cases_sqequal,  subtype_base_sq,  bool_wf,  assert-bnot,  neg_assert_of_eq_int,  subtract_wf,  decidable__le,  intformle_wf,  itermSubtract_wf,  int_formula_prop_le_lemma,  int_term_value_subtract_lemma,  ifthenelse_wf,  squash_wf,  true_wf,  istype-universe,  bfalse_wf,  btrue_wf,  int_seg_wf,  subtype_rel_function,  nat_wf,  int_seg_subtype_nat,  subtype_rel_self,  istype-nat
Rules used in proof :  sqequalSubstitution,  sqequalTransitivity,  computationStep,  sqequalReflexivity,  Error :lambdaFormation_alt,  cut,  Error :functionExtensionality_alt,  sqequalRule,  thin,  introduction,  extract_by_obid,  sqequalHypSubstitution,  isectElimination,  setElimination,  rename,  hypothesisEquality,  hypothesis,  natural_numberEquality,  Error :inhabitedIsType,  unionElimination,  equalityElimination,  equalityTransitivity,  equalitySymmetry,  productElimination,  independent_isectElimination,  int_eqReduceTrueSq,  Error :dependent_set_memberEquality_alt,  independent_pairFormation,  dependent_functionElimination,  approximateComputation,  independent_functionElimination,  Error :dependent_pairFormation_alt,  Error :lambdaEquality_alt,  int_eqEquality,  Error :isect_memberEquality_alt,  voidElimination,  Error :universeIsType,  Error :productIsType,  because_Cache,  Error :equalityIsType4,  baseApply,  closedConclusion,  baseClosed,  applyEquality,  intEquality,  promote_hyp,  instantiate,  cumulativity,  int_eqReduceFalseSq,  Error :equalityIsType1,  imageElimination,  universeEquality,  applyLambdaEquality,  imageMemberEquality,  Error :functionIsType
Latex:
\mforall{}a,b:\mBbbN{}  {}\mrightarrow{}  \mBbbN{}.  \mforall{}n:\mBbbN{}.    ((a  =  b)  {}\mRightarrow{}  (baire2cantor(a)  =  baire2cantor(b)))


Date html generated: 2019_06_20-PM-03_07_42
Last ObjectModification: 2018_10_30-PM-02_07_53

Theory : continuity


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