### Nuprl Lemma : two-factorizations_wf

`∀[n:ℕ]. (two-factorizations(n) ∈ {p:ℤ × ℤ| ((1 ≤ (fst(p))) ∧ ((fst(p)) ≤ n)) ∧ (((fst(p)) * (snd(p))) = n ∈ ℤ)}  List)`

Proof

Definitions occuring in Statement :  two-factorizations: `two-factorizations(n)` list: `T List` nat: `ℕ` uall: `∀[x:A]. B[x]` pi1: `fst(t)` pi2: `snd(t)` le: `A ≤ B` and: `P ∧ Q` member: `t ∈ T` set: `{x:A| B[x]} ` product: `x:A × B[x]` multiply: `n * m` natural_number: `\$n` int: `ℤ` equal: `s = t ∈ T`
Definitions unfolded in proof :  uall: `∀[x:A]. B[x]` member: `t ∈ T` two-factorizations: `two-factorizations(n)` nat: `ℕ` and: `P ∧ Q` prop: `ℙ` all: `∀x:A. B[x]` implies: `P `` Q` nequal: `a ≠ b ∈ T ` ge: `i ≥ j ` not: `¬A` uimplies: `b supposing a` satisfiable_int_formula: `satisfiable_int_formula(fmla)` exists: `∃x:A. B[x]` false: `False` top: `Top` subtype_rel: `A ⊆r B` pi1: `fst(t)` pi2: `snd(t)` guard: `{T}` so_lambda: `λ2x.t[x]` so_apply: `x[s]` cand: `A c∧ B` sq_stable: `SqStable(P)` squash: `↓T` decidable: `Dec(P)` or: `P ∨ Q` uiff: `uiff(P;Q)` int_nzero: `ℤ-o` sq_type: `SQType(T)`
Lemmas referenced :  from-upto_wf list_wf le_wf less_than_wf mapfilter_wf eq_int_wf nat_properties satisfiable-full-omega-tt intformand_wf intformeq_wf itermVar_wf itermConstant_wf intformle_wf int_formula_prop_and_lemma int_formula_prop_eq_lemma int_term_value_var_lemma int_term_value_constant_lemma int_formula_prop_le_lemma int_formula_prop_wf equal-wf-base int_subtype_base equal_wf assert_wf equal-wf-T-base set_wf sq_stable__and sq_stable__le sq_stable__less_than member-less_than squash_wf decidable__le intformnot_wf int_formula_prop_not_lemma intformless_wf itermAdd_wf int_formula_prop_less_lemma int_term_value_add_lemma assert_of_eq_int div_rem_sum nequal_wf subtype_base_sq decidable__equal_int add-is-int-iff multiply-is-int-iff itermMultiply_wf int_term_value_mul_lemma false_wf nat_wf
Rules used in proof :  sqequalSubstitution sqequalTransitivity computationStep sqequalReflexivity isect_memberFormation introduction cut extract_by_obid sqequalHypSubstitution isectElimination thin natural_numberEquality addEquality setElimination rename hypothesisEquality hypothesis setEquality intEquality productEquality because_Cache lambdaFormation lambdaEquality remainderEquality productElimination independent_isectElimination dependent_pairFormation int_eqEquality dependent_functionElimination isect_memberEquality voidElimination voidEquality sqequalRule independent_pairFormation computeAll applyEquality baseClosed multiplyEquality independent_functionElimination dependent_set_memberEquality independent_pairEquality divideEquality imageMemberEquality imageElimination unionElimination instantiate cumulativity equalityTransitivity equalitySymmetry pointwiseFunctionality promote_hyp baseApply closedConclusion axiomEquality

Latex:
\mforall{}[n:\mBbbN{}]
(two-factorizations(n)  \mmember{}  \{p:\mBbbZ{}  \mtimes{}  \mBbbZ{}|
((1  \mleq{}  (fst(p)))  \mwedge{}  ((fst(p))  \mleq{}  n))  \mwedge{}  (((fst(p))  *  (snd(p)))  =  n)\}    List)

Date html generated: 2018_05_21-PM-09_05_51
Last ObjectModification: 2017_07_26-PM-06_28_41

Theory : general

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