### Nuprl Lemma : regularize_wf

`∀[k:ℕ+]. ∀[f:ℕ+ ⟶ ℤ].  (regularize(k;f) ∈ ℕ+ ⟶ ℤ)`

Proof

Definitions occuring in Statement :  regularize: `regularize(k;f)` nat_plus: `ℕ+` uall: `∀[x:A]. B[x]` member: `t ∈ T` function: `x:A ⟶ B[x]` int: `ℤ`
Definitions unfolded in proof :  uall: `∀[x:A]. B[x]` regularize: `regularize(k;f)` member: `t ∈ T` subtype_rel: `A ⊆r B` all: `∀x:A. B[x]` implies: `P `` Q` bool: `𝔹` unit: `Unit` it: `⋅` btrue: `tt` uiff: `uiff(P;Q)` and: `P ∧ Q` uimplies: `b supposing a` ifthenelse: `if b then t else f fi ` bfalse: `ff` exists: `∃x:A. B[x]` prop: `ℙ` or: `P ∨ Q` sq_type: `SQType(T)` guard: `{T}` bnot: `¬bb` assert: `↑b` false: `False` so_lambda: `λ2x.t[x]` so_apply: `x[s]` nat: `ℕ` decidable: `Dec(P)` not: `¬A` regular-upto: `regular-upto(k;n;f)` top: `Top` true: `True` le: `A ≤ B` less_than': `less_than'(a;b)` int_seg: `{i..j-}` nat_plus: `ℕ+` iff: `P `⇐⇒` Q` rev_implies: `P `` Q` lelt: `i ≤ j < k` subtract: `n - m` rev_uimplies: `rev_uimplies(P;Q)` ge: `i ≥ j ` less_than: `a < b` squash: `↓T` satisfiable_int_formula: `satisfiable_int_formula(fmla)` absval: `|i|` has-value: `(a)↓` int_nzero: `ℤ-o` nequal: `a ≠ b ∈ T `
Lemmas referenced :  regular-upto_wf nat_plus_wf bool_wf eqtt_to_assert eqff_to_assert nat_plus_subtype_nat equal_wf bool_cases_sqequal subtype_base_sq bool_subtype_base assert-bnot not_wf assert_wf assert_of_bnot bnot_wf exists_wf nat_wf mu-property mu_wf uall_wf isect_wf less_than_wf decidable__equal_int int_subtype_base bdd_all_zero_lemma assert-bdd-all false_wf le_wf le_int_wf absval_wf subtract_wf decidable__lt not-lt-2 condition-implies-le minus-add minus-one-mul zero-add minus-one-mul-top add-commutes add_functionality_wrt_le add-associates add-zero le-add-cancel int_seg_wf bdd-all_wf all_wf assert_of_le_int int_seg_properties nat_properties nat_plus_properties full-omega-unsat intformnot_wf intformeq_wf itermSubtract_wf itermMultiply_wf itermConstant_wf itermVar_wf int_formula_prop_not_lemma int_formula_prop_eq_lemma int_term_value_subtract_lemma int_term_value_mul_lemma int_term_value_constant_lemma int_term_value_var_lemma int_formula_prop_wf int_seg_cases int_seg_subtype intformand_wf intformless_wf intformle_wf int_formula_prop_and_lemma int_formula_prop_less_lemma int_formula_prop_le_lemma decidable__le value-type-has-value int-value-type set-value-type seq-min-upper_wf mul_nzero subtype_rel_sets nequal_wf equal-wf-base
Rules used in proof :  sqequalSubstitution sqequalTransitivity computationStep sqequalReflexivity isect_memberFormation lambdaEquality cut introduction extract_by_obid sqequalHypSubstitution isectElimination thin hypothesisEquality applyEquality because_Cache hypothesis sqequalRule functionExtensionality lambdaFormation unionElimination equalityElimination equalityTransitivity equalitySymmetry productElimination independent_isectElimination dependent_pairFormation promote_hyp dependent_functionElimination instantiate cumulativity independent_functionElimination voidElimination functionEquality intEquality addLevel existsFunctionality productEquality setElimination rename natural_numberEquality isect_memberEquality voidEquality allFunctionality dependent_set_memberEquality independent_pairFormation multiplyEquality addEquality minusEquality levelHypothesis allLevelFunctionality applyLambdaEquality imageMemberEquality baseClosed approximateComputation int_eqEquality hypothesis_subsumption callbyvalueReduce divideEquality setEquality

Latex:
\mforall{}[k:\mBbbN{}\msupplus{}].  \mforall{}[f:\mBbbN{}\msupplus{}  {}\mrightarrow{}  \mBbbZ{}].    (regularize(k;f)  \mmember{}  \mBbbN{}\msupplus{}  {}\mrightarrow{}  \mBbbZ{})

Date html generated: 2017_10_03-AM-09_07_30
Last ObjectModification: 2017_09_11-PM-01_40_53

Theory : reals

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