### Nuprl Lemma : cantor2baire2cantor

`∀a:ℕ ⟶ 𝔹. (initF(a) `` (baire2cantor(cantor2baire(a)) = a ∈ (ℕ ⟶ 𝔹)))`

Proof

Definitions occuring in Statement :  initF: `initF(a)` cantor2baire: `cantor2baire(a)` baire2cantor: `baire2cantor(a)` nat: `ℕ` bool: `𝔹` all: `∀x:A. B[x]` implies: `P `` Q` function: `x:A ⟶ B[x]` equal: `s = t ∈ T`
Definitions unfolded in proof :  nequal: `a ≠ b ∈ T ` bnot: `¬bb` uiff: `uiff(P;Q)` it: `⋅` unit: `Unit` bool: `𝔹` subtype_rel: `A ⊆r B` exists: `∃x:A. B[x]` satisfiable_int_formula: `satisfiable_int_formula(fmla)` ge: `i ≥ j ` initF: `initF(a)` rev_implies: `P `` Q` iff: `P `⇐⇒` Q` bfalse: `ff` assert: `↑b` btrue: `tt` ifthenelse: `if b then t else f fi ` eq_int: `(i =z j)` top: `Top` cantor2baire-aux: `cantor2baire-aux(a;n)` nat-pred: `n-1` guard: `{T}` sq_type: `SQType(T)` so_apply: `x[s]` so_lambda: `λ2x.t[x]` uimplies: `b supposing a` or: `P ∨ Q` decidable: `Dec(P)` uall: `∀[x:A]. B[x]` prop: `ℙ` not: `¬A` false: `False` less_than': `less_than'(a;b)` and: `P ∧ Q` le: `A ≤ B` nat: `ℕ` member: `t ∈ T` cantor2baire: `cantor2baire(a)` baire2cantor: `baire2cantor(a)` implies: `P `` Q` all: `∀x:A. B[x]`
Lemmas referenced :  btrue_wf neg_assert_of_eq_int assert-bnot bool_subtype_base bool_cases_sqequal equal_wf eqff_to_assert int_term_value_add_lemma itermAdd_wf assert_of_eq_int nat-pred_wf cantor2baire-aux_wf eq_int_wf eqtt_to_assert equal-wf-base int_formula_prop_wf decidable__equal_int int_formula_prop_le_lemma int_formula_prop_eq_lemma int_term_value_var_lemma int_term_value_constant_lemma int_formula_prop_less_lemma int_formula_prop_not_lemma int_formula_prop_and_lemma intformle_wf intformeq_wf itermVar_wf itermConstant_wf intformless_wf intformnot_wf intformand_wf satisfiable-full-omega-tt decidable__lt nat_properties cantor2baire-aux-pos assert_wf btrue_neq_bfalse assert_elim bfalse_wf iff_imp_equal_bool primrec0_lemma int_subtype_base set_subtype_base subtype_base_sq bool_wf initF_wf nat_wf le_wf false_wf decidable__equal_nat
Rules used in proof :  promote_hyp addEquality productElimination equalityElimination baseClosed computeAll int_eqEquality dependent_pairFormation rename setElimination levelHypothesis because_Cache addLevel voidEquality voidElimination isect_memberEquality independent_functionElimination equalitySymmetry equalityTransitivity lambdaEquality intEquality independent_isectElimination cumulativity instantiate functionEquality applyEquality unionElimination isectElimination hypothesis independent_pairFormation natural_numberEquality dependent_set_memberEquality hypothesisEquality thin dependent_functionElimination sqequalHypSubstitution extract_by_obid introduction sqequalRule functionExtensionality cut lambdaFormation sqequalReflexivity computationStep sqequalTransitivity sqequalSubstitution

Latex:
\mforall{}a:\mBbbN{}  {}\mrightarrow{}  \mBbbB{}.  (initF(a)  {}\mRightarrow{}  (baire2cantor(cantor2baire(a))  =  a))

Date html generated: 2017_04_21-AM-11_22_34
Last ObjectModification: 2017_04_20-PM-03_53_37

Theory : continuity

Home Index