### Nuprl Lemma : baire2cantor2baire

`∀a:ℕ ⟶ ℕ. (init0(a) `` increasing-sequence(a) `` (cantor2baire(baire2cantor(a)) = a ∈ (ℕ ⟶ ℕ)))`

Proof

Definitions occuring in Statement :  init0: `init0(a)` cantor2baire: `cantor2baire(a)` baire2cantor: `baire2cantor(a)` increasing-sequence: `increasing-sequence(a)` nat: `ℕ` all: `∀x:A. B[x]` implies: `P `` Q` function: `x:A ⟶ B[x]` equal: `s = t ∈ T`
Definitions unfolded in proof :  increasing-sequence: `increasing-sequence(a)` rev_implies: `P `` Q` iff: `P `⇐⇒` Q` squash: `↓T` true: `True` label: `...\$L... t` nequal: `a ≠ b ∈ T ` baire2cantor: `baire2cantor(a)` assert: `↑b` bnot: `¬bb` sq_type: `SQType(T)` bfalse: `ff` guard: `{T}` ifthenelse: `if b then t else f fi ` uiff: `uiff(P;Q)` btrue: `tt` it: `⋅` unit: `Unit` bool: `𝔹` subtype_rel: `A ⊆r B` init0: `init0(a)` less_than': `less_than'(a;b)` le: `A ≤ B` cantor2baire-aux: `cantor2baire-aux(a;n)` cantor2baire: `cantor2baire(a)` or: `P ∨ Q` decidable: `Dec(P)` prop: `ℙ` and: `P ∧ Q` top: `Top` not: `¬A` exists: `∃x:A. B[x]` satisfiable_int_formula: `satisfiable_int_formula(fmla)` uimplies: `b supposing a` ge: `i ≥ j ` false: `False` nat: `ℕ` member: `t ∈ T` uall: `∀[x:A]. B[x]` implies: `P `` Q` all: `∀x:A. B[x]`
Lemmas referenced :  nat-pred-as-sub iff_weakening_equal ifthenelse_wf true_wf squash_wf add_nat_wf int_term_value_add_lemma itermAdd_wf btrue_wf bfalse_wf nat-pred_wf subtract-add-cancel neg_assert_of_eq_int assert-bnot bool_subtype_base subtype_base_sq bool_cases_sqequal equal_wf eqff_to_assert int_formula_prop_eq_lemma intformeq_wf decidable__equal_int assert_of_eq_int eqtt_to_assert bool_wf eq_int_wf primrec-unroll le_wf false_wf primrec0_lemma init0_wf increasing-sequence_wf nat_wf int_term_value_subtract_lemma int_formula_prop_not_lemma itermSubtract_wf intformnot_wf subtract_wf decidable__le less_than_wf ge_wf int_formula_prop_wf int_formula_prop_less_lemma int_term_value_var_lemma int_term_value_constant_lemma int_formula_prop_le_lemma int_formula_prop_and_lemma intformless_wf itermVar_wf itermConstant_wf intformle_wf intformand_wf satisfiable-full-omega-tt nat_properties
Rules used in proof :  baseClosed imageMemberEquality universeEquality imageElimination addEquality cumulativity instantiate promote_hyp applyLambdaEquality because_Cache productElimination equalityTransitivity equalityElimination equalitySymmetry levelHypothesis addLevel dependent_set_memberEquality functionEquality applyEquality unionElimination axiomEquality independent_functionElimination computeAll independent_pairFormation sqequalRule voidEquality voidElimination isect_memberEquality dependent_functionElimination intEquality int_eqEquality lambdaEquality dependent_pairFormation independent_isectElimination natural_numberEquality intWeakElimination rename setElimination hypothesis hypothesisEquality thin isectElimination sqequalHypSubstitution extract_by_obid introduction functionExtensionality cut lambdaFormation sqequalReflexivity computationStep sqequalTransitivity sqequalSubstitution

Latex:
\mforall{}a:\mBbbN{}  {}\mrightarrow{}  \mBbbN{}.  (init0(a)  {}\mRightarrow{}  increasing-sequence(a)  {}\mRightarrow{}  (cantor2baire(baire2cantor(a))  =  a))

Date html generated: 2017_04_21-AM-11_22_42
Last ObjectModification: 2017_04_20-PM-03_57_46

Theory : continuity

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