### Nuprl Lemma : cantor2baire-aux-pos

`∀[a:ℕ ⟶ 𝔹]. ∀[n:ℕ].`
`  cantor2baire-aux(a;n) ~ if a n then cantor2baire-aux(a;n-1) + 1 else cantor2baire-aux(a;n-1) fi  supposing 0 < n`

Proof

Definitions occuring in Statement :  cantor2baire-aux: `cantor2baire-aux(a;n)` nat-pred: `n-1` nat: `ℕ` ifthenelse: `if b then t else f fi ` bool: `𝔹` less_than: `a < b` uimplies: `b supposing a` uall: `∀[x:A]. B[x]` apply: `f a` function: `x:A ⟶ B[x]` add: `n + m` natural_number: `\$n` sqequal: `s ~ t`
Definitions unfolded in proof :  uall: `∀[x:A]. B[x]` member: `t ∈ T` uimplies: `b supposing a` cantor2baire-aux: `cantor2baire-aux(a;n)` nat: `ℕ` top: `Top` all: `∀x:A. B[x]` implies: `P `` Q` bool: `𝔹` unit: `Unit` it: `⋅` btrue: `tt` uiff: `uiff(P;Q)` and: `P ∧ Q` ifthenelse: `if b then t else f fi ` ge: `i ≥ j ` not: `¬A` satisfiable_int_formula: `satisfiable_int_formula(fmla)` exists: `∃x:A. B[x]` false: `False` prop: `ℙ` bfalse: `ff` or: `P ∨ Q` sq_type: `SQType(T)` guard: `{T}` bnot: `¬bb` assert: `↑b` rev_implies: `P `` Q` iff: `P `⇐⇒` Q` squash: `↓T` decidable: `Dec(P)` subtype_rel: `A ⊆r B` true: `True` so_lambda: `λ2x.t[x]` so_apply: `x[s]`
Lemmas referenced :  primrec-unroll istype-void subtract-add-cancel lt_int_wf eqtt_to_assert assert_of_lt_int nat_properties full-omega-unsat intformand_wf intformless_wf itermVar_wf itermConstant_wf istype-int int_formula_prop_and_lemma int_formula_prop_less_lemma int_term_value_var_lemma int_term_value_constant_lemma int_formula_prop_wf eqff_to_assert bool_cases_sqequal subtype_base_sq bool_wf bool_subtype_base assert-bnot iff_weakening_uiff assert_wf less_than_wf int_subtype_base equal_wf cantor2baire-aux_wf subtract_wf decidable__le intformnot_wf intformle_wf itermSubtract_wf int_formula_prop_not_lemma int_formula_prop_le_lemma int_term_value_subtract_lemma le_wf add_functionality_wrt_eq nat-pred_wf nat-pred-as-sub iff_weakening_equal nat_wf set_subtype_base
Rules used in proof :  sqequalSubstitution sqequalTransitivity computationStep sqequalReflexivity Error :isect_memberFormation_alt,  introduction cut sqequalRule extract_by_obid sqequalHypSubstitution isectElimination thin setElimination rename because_Cache hypothesis Error :isect_memberEquality_alt,  voidElimination natural_numberEquality Error :inhabitedIsType,  Error :lambdaFormation_alt,  unionElimination equalityElimination equalityTransitivity equalitySymmetry productElimination independent_isectElimination hypothesisEquality approximateComputation independent_functionElimination Error :dependent_pairFormation_alt,  Error :lambdaEquality_alt,  int_eqEquality dependent_functionElimination independent_pairFormation Error :universeIsType,  Error :equalityIsType1,  promote_hyp instantiate cumulativity applyEquality imageElimination universeEquality intEquality addEquality Error :dependent_set_memberEquality_alt,  Error :functionIsType,  imageMemberEquality baseClosed axiomSqEquality

Latex:
\mforall{}[a:\mBbbN{}  {}\mrightarrow{}  \mBbbB{}].  \mforall{}[n:\mBbbN{}].
cantor2baire-aux(a;n)  \msim{}  if  a  n  then  cantor2baire-aux(a;n-1)  +  1  else  cantor2baire-aux(a;n-1)  fi
supposing  0  <  n

Date html generated: 2019_06_20-PM-03_07_37
Last ObjectModification: 2018_10_03-PM-00_24_16

Theory : continuity

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