Nuprl Lemma : divides-iff-factors
∀n,m:ℕ+. (n | m ⇐⇒ sub-bag(Prime;factors(n);factors(m)))
Proof
Definitions occuring in Statement :
factors: factors(n),
Prime: Prime,
sub-bag: sub-bag(T;as;bs),
divides: b | a,
nat_plus: ℕ+,
all: ∀x:A. B[x],
iff: P ⇐⇒ Q
Definitions unfolded in proof :
all: ∀x:A. B[x],
iff: P ⇐⇒ Q,
and: P ∧ Q,
implies: P ⇒ Q,
member: t ∈ T,
prop: ℙ,
uall: ∀[x:A]. B[x],
nat_plus: ℕ+,
rev_implies: P ⇐ Q,
divides: b | a,
exists: ∃x:A. B[x],
gt: i > j,
decidable: Dec(P),
or: P ∨ Q,
uimplies: b supposing a,
satisfiable_int_formula: satisfiable_int_formula(fmla),
false: False,
not: ¬A,
top: Top,
sub-bag: sub-bag(T;as;bs),
sq_type: SQType(T),
guard: {T},
squash: ↓T,
true: True,
subtype_rel: A ⊆r B,
Prime: Prime,
int_upper: {i...}
Lemmas referenced :
divides_wf,
sub-bag_wf,
Prime_wf,
factors_wf,
nat_plus_wf,
nat_plus_properties,
decidable__lt,
satisfiable-full-omega-tt,
intformand_wf,
intformnot_wf,
intformless_wf,
itermConstant_wf,
itermMultiply_wf,
itermVar_wf,
intformeq_wf,
int_formula_prop_and_lemma,
int_formula_prop_not_lemma,
int_formula_prop_less_lemma,
int_term_value_constant_lemma,
int_term_value_mul_lemma,
int_term_value_var_lemma,
int_formula_prop_eq_lemma,
int_formula_prop_wf,
pos_mul_arg_bounds,
less_than_wf,
subtype_base_sq,
int_subtype_base,
equal_wf,
squash_wf,
true_wf,
bag_wf,
append-factors,
bag-append_wf,
iff_weakening_equal,
int-bag-product_wf,
subtype_rel_bag,
product-factors,
int-bag-product-append
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation,
independent_pairFormation,
cut,
introduction,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
setElimination,
rename,
hypothesisEquality,
hypothesis,
productElimination,
dependent_functionElimination,
natural_numberEquality,
equalityTransitivity,
equalitySymmetry,
unionElimination,
independent_isectElimination,
dependent_pairFormation,
lambdaEquality,
int_eqEquality,
intEquality,
isect_memberEquality,
voidElimination,
voidEquality,
sqequalRule,
computeAll,
independent_functionElimination,
dependent_set_memberEquality,
instantiate,
cumulativity,
applyEquality,
imageElimination,
universeEquality,
because_Cache,
imageMemberEquality,
baseClosed,
applyLambdaEquality,
multiplyEquality
Latex:
\mforall{}n,m:\mBbbN{}\msupplus{}. (n | m \mLeftarrow{}{}\mRightarrow{} sub-bag(Prime;factors(n);factors(m)))
Date html generated:
2018_05_21-PM-07_31_16
Last ObjectModification:
2017_07_26-PM-05_06_31
Theory : general
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