Nuprl Lemma : prime-factors2
∀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 :
all: ∀x:A. B[x],
implies: P ⇒ Q,
member: t ∈ T,
uall: ∀[x:A]. B[x],
int_upper: {i...},
sq_exists: ∃x:A [B[x]],
int_seg: {i..j-},
lelt: i ≤ j < k,
and: P ∧ Q,
le: A ≤ B,
prop: ℙ,
subtype_rel: A ⊆r B,
so_lambda: λ2x.t[x],
less_than: a < b,
squash: ↓T,
so_apply: x[s],
uimplies: b supposing a,
so_lambda: λ2x y.t[x; y],
so_apply: x[s1;s2],
decidable: Dec(P),
or: P ∨ Q,
int_nzero: ℤ-o,
nequal: a ≠ b ∈ T ,
not: ¬A,
satisfiable_int_formula: satisfiable_int_formula(fmla),
exists: ∃x:A. B[x],
false: False,
top: Top,
iff: P ⇐⇒ Q,
sq_type: SQType(T),
guard: {T},
less_than': less_than'(a;b),
nat_plus: ℕ+,
nat: ℕ,
true: True,
rev_implies: P ⇐ Q,
uiff: uiff(P;Q),
atomic: atomic(a),
cand: A c∧ B,
mul-list: Π(ns)
Lemmas referenced :
int_seg_wf,
list_wf,
int_upper_wf,
prime_wf,
istype-int,
set_subtype_base,
lelt_wf,
int_subtype_base,
list_subtype_base,
le_wf,
istype-int_upper,
natrec_wf_intseg,
sq_exists_wf,
equal-wf-base,
subtype_rel_function,
subtype_rel_self,
decidable__proper_divisor,
divide_wfa,
int_upper_properties,
full-omega-unsat,
intformand_wf,
intformeq_wf,
itermVar_wf,
itermConstant_wf,
intformle_wf,
int_formula_prop_and_lemma,
istype-void,
int_formula_prop_eq_lemma,
int_term_value_var_lemma,
int_term_value_constant_lemma,
int_formula_prop_le_lemma,
int_formula_prop_wf,
nequal_wf,
set-value-type,
equal_wf,
int-value-type,
istype-le,
istype-less_than,
divides_iff_div_exact,
subtype_base_sq,
div_bounds_1,
upper_subtype_nat,
istype-false,
decidable__lt,
intformnot_wf,
intformless_wf,
int_formula_prop_not_lemma,
int_formula_prop_less_lemma,
decidable__equal_int,
itermMultiply_wf,
int_term_value_mul_lemma,
decidable__le,
mul_preserves_le,
merge-int_wf,
squash_wf,
true_wf,
istype-universe,
mul-list-merge,
subtype_rel_list,
iff_weakening_equal,
mul-list_wf,
multiply-is-int-iff,
false_wf,
cons_wf,
nil_wf,
atomic_imp_prime,
assoced_wf,
reducible_wf,
assoced_nelim,
reducible-nat,
less_than_wf,
divides_wf,
reduce_cons_lemma,
reduce_nil_lemma
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation_alt,
cut,
thin,
sqequalRule,
functionIsType,
universeIsType,
introduction,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
natural_numberEquality,
setElimination,
rename,
hypothesisEquality,
hypothesis,
setIsType,
setEquality,
productElimination,
equalityIstype,
applyEquality,
intEquality,
lambdaEquality_alt,
imageElimination,
independent_isectElimination,
baseApply,
closedConclusion,
baseClosed,
inhabitedIsType,
sqequalBase,
equalitySymmetry,
because_Cache,
functionExtensionality,
dependent_functionElimination,
unionElimination,
dependent_set_memberEquality_alt,
approximateComputation,
independent_functionElimination,
dependent_pairFormation_alt,
int_eqEquality,
isect_memberEquality_alt,
voidElimination,
independent_pairFormation,
cutEval,
equalityTransitivity,
productIsType,
instantiate,
cumulativity,
promote_hyp,
dependent_set_memberFormation_alt,
equalityIsType4,
universeEquality,
imageMemberEquality,
multiplyEquality,
divideEquality,
pointwiseFunctionality
Latex:
\mforall{}n:\{2...\}. (\mexists{}factors:\{m:\{2...\}| prime(m)\} List [(n = \mPi{}(factors) )])
Date html generated:
2020_05_20-AM-08_14_23
Last ObjectModification:
2019_11_27-PM-01_46_06
Theory : general
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