### Nuprl Lemma : proper-divisor_wf

`∀[n:ℕ+]. (proper-divisor(n) ∈ Dec(∃n1:ℤ [(n1 < n ∧ (2 ≤ n1) ∧ (n1 | n))]))`

Proof

Definitions occuring in Statement :  proper-divisor: `proper-divisor(n)` divides: `b | a` nat_plus: `ℕ+` less_than: `a < b` decidable: `Dec(P)` uall: `∀[x:A]. B[x]` le: `A ≤ B` sq_exists: `∃x:A [B[x]]` and: `P ∧ Q` member: `t ∈ T` natural_number: `\$n` int: `ℤ`
Definitions unfolded in proof :  uall: `∀[x:A]. B[x]` member: `t ∈ T` proper-divisor: `proper-divisor(n)` int_upper: `{i...}` le: `A ≤ B` and: `P ∧ Q` less_than': `less_than'(a;b)` false: `False` not: `¬A` implies: `P `` Q` prop: `ℙ` so_lambda: `λ2x.t[x]` nat_plus: `ℕ+` so_apply: `x[s]` sq_exists: `∃x:A [B[x]]` all: `∀x:A. B[x]` decidable: `Dec(P)` or: `P ∨ Q` bool: `𝔹` unit: `Unit` it: `⋅` btrue: `tt` uiff: `uiff(P;Q)` uimplies: `b supposing a` ifthenelse: `if b then t else f fi ` bfalse: `ff` iff: `P `⇐⇒` Q` rev_implies: `P `` Q` guard: `{T}` nat: `ℕ` satisfiable_int_formula: `satisfiable_int_formula(fmla)` exists: `∃x:A. B[x]` top: `Top` cand: `A c∧ B` divides: `b | a` subtype_rel: `A ⊆r B` sq_type: `SQType(T)` int_nzero: `ℤ-o` true: `True` nequal: `a ≠ b ∈ T ` has-value: `(a)↓` less_than: `a < b` squash: `↓T` ge: `i ≥ j ` exp: `i^n` eq_int: `(i =z j)` subtract: `n - m`
Lemmas referenced :  trial-division_wf cons_wf int_upper_wf false_wf le_wf nil_wf sq_exists_wf less_than_wf divides_wf top_wf equal_wf nat_plus_wf not_wf lt_int_wf bool_wf uiff_transitivity equal-wf-T-base assert_wf eqtt_to_assert assert_of_lt_int eq_int_wf assert_of_eq_int iff_transitivity bnot_wf iff_weakening_uiff eqff_to_assert assert_of_bnot le_int_wf assert_functionality_wrt_uiff bnot_of_lt_int assert_of_le_int proper-divisor-aux_wf nat_plus_properties decidable__lt satisfiable-full-omega-tt intformand_wf intformnot_wf intformless_wf itermVar_wf itermMultiply_wf itermConstant_wf int_formula_prop_and_lemma int_formula_prop_not_lemma int_formula_prop_less_lemma int_term_value_var_lemma int_term_value_mul_lemma int_term_value_constant_lemma int_formula_prop_wf intformle_wf int_formula_prop_le_lemma intformeq_wf int_formula_prop_eq_lemma decidable__equal_int int_subtype_base set_wf subtype_base_sq equal-wf-base divides_iff_div_exact true_wf nequal_wf value-type-has-value iroot-property nat_plus_subtype_nat decidable__le iroot_wf nat_wf nat_properties decidable__or or_wf intformor_wf int_formula_prop_or_lemma primrec-unroll primrec1_lemma itermAdd_wf int_term_value_add_lemma exp_wf2 mul-distributes mul-distributes-right add-associates mul-commutes mul-swap one-mul add-swap add-commutes two-mul mul_preserves_lt not-lt-2 less-iff-le add_functionality_wrt_le zero-add le-add-cancel
Rules used in proof :  sqequalSubstitution sqequalTransitivity computationStep sqequalReflexivity isect_memberFormation introduction cut extract_by_obid sqequalHypSubstitution isectElimination thin hypothesisEquality natural_numberEquality hypothesis dependent_set_memberEquality sqequalRule independent_pairFormation lambdaFormation because_Cache unionEquality intEquality lambdaEquality productEquality setElimination rename unionElimination equalityTransitivity equalitySymmetry dependent_functionElimination independent_functionElimination axiomEquality inlEquality equalityElimination baseClosed productElimination independent_isectElimination impliesFunctionality multiplyEquality dependent_pairFormation int_eqEquality isect_memberEquality voidElimination voidEquality computeAll baseApply closedConclusion applyEquality inrEquality setEquality instantiate cumulativity promote_hyp addLevel callbyvalueReduce imageMemberEquality applyLambdaEquality addEquality imageElimination

Latex:
\mforall{}[n:\mBbbN{}\msupplus{}].  (proper-divisor(n)  \mmember{}  Dec(\mexists{}n1:\mBbbZ{}  [(n1  <  n  \mwedge{}  (2  \mleq{}  n1)  \mwedge{}  (n1  |  n))]))

Date html generated: 2018_05_21-PM-08_16_08
Last ObjectModification: 2017_07_26-PM-05_50_20

Theory : general

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