Nuprl Lemma : reducible-nat
∀n:ℤ. reducible(n) ⇒ (∃n1:ℕ. (n1 < n ∧ (2 ≤ n1) ∧ (n1 | n))) supposing 2 ≤ n
Proof
Definitions occuring in Statement :
reducible: reducible(a),
divides: b | a,
nat: ℕ,
less_than: a < b,
uimplies: b supposing a,
le: A ≤ B,
all: ∀x:A. B[x],
exists: ∃x:A. B[x],
implies: P ⇒ Q,
and: P ∧ Q,
natural_number: $n,
int: ℤ
Definitions unfolded in proof :
all: ∀x:A. B[x],
uimplies: b supposing a,
member: t ∈ T,
le: A ≤ B,
and: P ∧ Q,
not: ¬A,
implies: P ⇒ Q,
false: False,
uall: ∀[x:A]. B[x],
prop: ℙ,
reducible: reducible(a),
exists: ∃x:A. B[x],
int_nzero: ℤ-o,
decidable: Dec(P),
or: P ∨ Q,
nat: ℕ,
nequal: a ≠ b ∈ T ,
satisfiable_int_formula: satisfiable_int_formula(fmla),
top: Top,
cand: A c∧ B,
iff: P ⇐⇒ Q,
gt: i > j,
rev_implies: P ⇐ Q,
divides: b | a,
guard: {T}
Lemmas referenced :
int_formula_prop_or_lemma,
intformor_wf,
decidable__or,
int_term_value_minus_lemma,
itermMinus_wf,
mul_preserves_le,
equal_wf,
assoced_elim,
decidable__equal_int,
int_formula_prop_eq_lemma,
int_term_value_mul_lemma,
int_formula_prop_less_lemma,
intformeq_wf,
itermMultiply_wf,
intformless_wf,
decidable__lt,
pos_mul_arg_bounds,
divides_wf,
less_than_wf,
and_wf,
int_formula_prop_wf,
int_term_value_var_lemma,
int_term_value_constant_lemma,
int_formula_prop_le_lemma,
int_formula_prop_not_lemma,
int_formula_prop_and_lemma,
itermVar_wf,
itermConstant_wf,
intformle_wf,
intformnot_wf,
intformand_wf,
satisfiable-full-omega-tt,
int_nzero_properties,
decidable__le,
le_wf,
reducible_wf,
less_than'_wf
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation,
isect_memberFormation,
cut,
introduction,
sqequalRule,
sqequalHypSubstitution,
productElimination,
thin,
independent_pairEquality,
lambdaEquality,
dependent_functionElimination,
hypothesisEquality,
voidElimination,
lemma_by_obid,
isectElimination,
natural_numberEquality,
hypothesis,
axiomEquality,
equalityTransitivity,
equalitySymmetry,
rename,
intEquality,
setElimination,
unionElimination,
dependent_pairFormation,
dependent_set_memberEquality,
independent_isectElimination,
int_eqEquality,
isect_memberEquality,
voidEquality,
independent_pairFormation,
computeAll,
independent_functionElimination,
because_Cache,
inlFormation,
minusEquality,
multiplyEquality,
inrFormation
Latex:
\mforall{}n:\mBbbZ{}. reducible(n) {}\mRightarrow{} (\mexists{}n1:\mBbbN{}. (n1 < n \mwedge{} (2 \mleq{} n1) \mwedge{} (n1 | n))) supposing 2 \mleq{} n
Date html generated:
2016_05_15-PM-04_02_33
Last ObjectModification:
2016_01_16-AM-11_02_34
Theory : general
Home
Index