### Nuprl Lemma : least-factor_wf

`∀[n:ℤ]. least-factor(n) ∈ {p:ℕ| 1 < p ∧ prime(p) ∧ (p | n) ∧ (∀q:ℕ. (prime(q) `` (q | n) `` (p ≤ q)))}  supposing 1 < |n\000C|`

Proof

Definitions occuring in Statement :  least-factor: `least-factor(n)` prime: `prime(a)` divides: `b | a` absval: `|i|` nat: `ℕ` less_than: `a < b` uimplies: `b supposing a` uall: `∀[x:A]. B[x]` le: `A ≤ B` all: `∀x:A. B[x]` implies: `P `` Q` and: `P ∧ Q` member: `t ∈ T` set: `{x:A| B[x]} ` natural_number: `\$n` int: `ℤ`
Definitions unfolded in proof :  uall: `∀[x:A]. B[x]` member: `t ∈ T` uimplies: `b supposing a` prop: `ℙ` subtype_rel: `A ⊆r B` nat: `ℕ` all: `∀x:A. B[x]` int_nzero: `ℤ-o` nequal: `a ≠ b ∈ T ` not: `¬A` implies: `P `` Q` satisfiable_int_formula: `satisfiable_int_formula(fmla)` exists: `∃x:A. B[x]` false: `False` top: `Top` and: `P ∧ Q` iff: `P `⇐⇒` Q` rev_implies: `P `` Q` assoced: `a ~ b` int_seg: `{i..j-}` lelt: `i ≤ j < k` decidable: `Dec(P)` or: `P ∨ Q` less_than: `a < b` squash: `↓T` uiff: `uiff(P;Q)` rev_uimplies: `rev_uimplies(P;Q)` guard: `{T}` le: `A ≤ B` less_than': `less_than'(a;b)` ge: `i ≥ j ` least-factor: `least-factor(n)` subtract: `n - m` cand: `A c∧ B` so_lambda: `λ2x.t[x]` so_apply: `x[s]` atomic: `atomic(a)` reducible: `reducible(a)` bool: `𝔹` unit: `Unit` it: `⋅` btrue: `tt` ifthenelse: `if b then t else f fi ` bfalse: `ff` gt: `i > j` sq_type: `SQType(T)` nat_plus: `ℕ+` true: `True` divides: `b | a` prime: `prime(a)`
Lemmas referenced :  less_than_wf absval_wf nat_wf divides_iff_rem_zero satisfiable-full-omega-tt intformand_wf intformeq_wf itermVar_wf itermConstant_wf intformless_wf int_formula_prop_and_lemma int_formula_prop_eq_lemma int_term_value_var_lemma int_term_value_constant_lemma int_formula_prop_less_lemma int_formula_prop_wf equal-wf-base int_subtype_base nequal_wf absval_assoced decidable__le subtract_wf intformnot_wf intformle_wf itermSubtract_wf int_formula_prop_not_lemma int_formula_prop_le_lemma int_term_value_subtract_lemma decidable__lt lelt_wf assert_of_eq_int itermAdd_wf int_term_value_add_lemma subtract-add-cancel assert_wf eq_int_wf int_seg_properties int_seg_subtype_nat false_wf nat_properties mu_wf mu-property add-nat le_wf add-subtract-cancel add-associates add-swap add-commutes zero-add divides_wf equal-wf-T-base not_wf prime_wf all_wf atomic_imp_prime assoced_wf reducible_wf assoced_nelim absval_ifthenelse int_nzero_properties int_nzero_wf lt_int_wf bool_wf le_int_wf bnot_wf itermMinus_wf int_term_value_minus_lemma assoced_functionality_wrt_assoced assoced_weakening uiff_transitivity eqtt_to_assert assert_of_lt_int eqff_to_assert assert_functionality_wrt_uiff bnot_of_lt_int assert_of_le_int equal_wf itermMultiply_wf int_term_value_mul_lemma subtype_base_sq neg_mul_arg_bounds gt_wf decidable__equal_int mul_preserves_lt not-lt-2 less-iff-le add_functionality_wrt_le le-add-cancel divides_reflexivity divides_functionality_wrt_assoced equal-wf-base-T divides_transitivity
Rules used in proof :  sqequalSubstitution sqequalTransitivity computationStep sqequalReflexivity isect_memberFormation introduction cut sqequalHypSubstitution hypothesis sqequalRule axiomEquality equalityTransitivity equalitySymmetry extract_by_obid isectElimination thin natural_numberEquality hypothesisEquality applyEquality lambdaEquality setElimination rename isect_memberEquality because_Cache intEquality dependent_functionElimination dependent_set_memberEquality lambdaFormation independent_isectElimination dependent_pairFormation int_eqEquality voidElimination voidEquality independent_pairFormation computeAll baseApply closedConclusion baseClosed productElimination independent_functionElimination unionElimination imageElimination remainderEquality addEquality applyLambdaEquality promote_hyp addLevel impliesFunctionality productEquality functionEquality minusEquality levelHypothesis equalityElimination instantiate cumulativity inrFormation inlFormation multiplyEquality impliesLevelFunctionality

Latex:
\mforall{}[n:\mBbbZ{}]
least-factor(n)  \mmember{}  \{p:\mBbbN{}|  1  <  p  \mwedge{}  prime(p)  \mwedge{}  (p  |  n)  \mwedge{}  (\mforall{}q:\mBbbN{}.  (prime(q)  {}\mRightarrow{}  (q  |  n)  {}\mRightarrow{}  (p  \mleq{}  q)))\}    su\000Cpposing  1  <  |n|

Date html generated: 2018_05_21-PM-06_58_55
Last ObjectModification: 2017_07_26-PM-05_00_24

Theory : general

Home Index