Nuprl Lemma : prime-factors
∀n:{2...}. (∃factors:{m:{2...}| prime(m)} List [(n = Π(factors) ∈ ℤ)])
Proof
Definitions occuring in Statement :
mul-list: Π(ns) ,
prime: prime(a),
list: T List,
int_upper: {i...},
all: ∀x:A. B[x],
sq_exists: ∃x:A [B[x]],
set: {x:A| B[x]} ,
natural_number: $n,
int: ℤ,
equal: s = t ∈ T
Definitions unfolded in proof :
sq_exists: ∃x:A [B[x]],
all: ∀x:A. B[x],
member: t ∈ T,
uall: ∀[x:A]. B[x],
nat: ℕ,
le: A ≤ B,
and: P ∧ Q,
less_than': less_than'(a;b),
false: False,
not: ¬A,
implies: P ⇒ Q,
prop: ℙ,
uimplies: b supposing a,
subtype_rel: A ⊆r B,
cand: A c∧ B,
prime: prime(a),
so_lambda: λ2x.t[x],
so_apply: x[s],
guard: {T},
int_upper: {i...},
ge: i ≥ j ,
decidable: Dec(P),
or: P ∨ Q,
satisfiable_int_formula: satisfiable_int_formula(fmla),
exists: ∃x:A. B[x],
top: Top,
sq_type: SQType(T),
mul-list: Π(ns)
Lemmas referenced :
factorit_wf,
le_wf,
false_wf,
int_upper_wf,
nil_wf,
nat_wf,
prime_wf,
less_than_wf,
subtype_base_sq,
set_subtype_base,
int_subtype_base,
nat_properties,
int_upper_properties,
decidable__equal_int,
satisfiable-full-omega-tt,
intformand_wf,
intformnot_wf,
intformeq_wf,
itermVar_wf,
itermConstant_wf,
intformless_wf,
intformle_wf,
int_formula_prop_and_lemma,
int_formula_prop_not_lemma,
int_formula_prop_eq_lemma,
int_term_value_var_lemma,
int_term_value_constant_lemma,
int_formula_prop_less_lemma,
int_formula_prop_le_lemma,
int_formula_prop_wf,
decidable__le,
null_nil_lemma,
btrue_wf,
member-implies-null-eq-bfalse,
btrue_neq_bfalse,
set_wf,
all_wf,
l_member_wf,
subtype_rel_list,
not_wf,
divides_wf,
assoced_weakening,
reduce_nil_lemma,
list_wf,
equal_wf,
reduce_wf,
mul-list_wf,
subtype_rel_list_set,
itermMultiply_wf,
int_term_value_mul_lemma
Rules used in proof :
sqequalSubstitution,
sqequalRule,
sqequalReflexivity,
sqequalTransitivity,
computationStep,
lambdaFormation,
introduction,
cut,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
because_Cache,
dependent_set_memberEquality,
natural_numberEquality,
independent_pairFormation,
hypothesis,
hypothesisEquality,
independent_isectElimination,
applyEquality,
setEquality,
productEquality,
setElimination,
rename,
productElimination,
independent_functionElimination,
instantiate,
cumulativity,
intEquality,
lambdaEquality,
dependent_functionElimination,
unionElimination,
dependent_pairFormation,
int_eqEquality,
isect_memberEquality,
voidElimination,
voidEquality,
computeAll,
equalityTransitivity,
equalitySymmetry,
functionEquality,
multiplyEquality
Latex:
\mforall{}n:\{2...\}. (\mexists{}factors:\{m:\{2...\}| prime(m)\} List [(n = \mPi{}(factors) )])
Date html generated:
2018_05_21-PM-06_57_49
Last ObjectModification:
2017_07_26-PM-05_00_01
Theory : general
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