Nuprl Lemma : factors-prime-product
∀[b:bag(Prime)]. (factors(Π(b)) = b ∈ bag(Prime))
Proof
Definitions occuring in Statement :
factors: factors(n),
Prime: Prime,
int-bag-product: Π(b),
bag: bag(T),
uall: ∀[x:A]. B[x],
equal: s = t ∈ T
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
all: ∀x:A. B[x],
squash: ↓T,
subtype_rel: A ⊆r B,
uimplies: b supposing a,
Prime: Prime,
int_upper: {i...},
prop: ℙ,
so_lambda: λ2x.t[x],
nat_plus: ℕ+,
so_apply: x[s],
implies: P ⇒ Q,
sq_stable: SqStable(P),
exists: ∃x:A. B[x],
factors: factors(n),
empty-bag: {},
bool: 𝔹,
unit: Unit,
it: ⋅,
btrue: tt,
ifthenelse: if b then t else f fi ,
uiff: uiff(P;Q),
and: P ∧ Q,
bfalse: ff,
or: P ∨ Q,
sq_type: SQType(T),
guard: {T},
bnot: ¬bb,
assert: ↑b,
false: False,
nequal: a ≠ b ∈ T ,
decidable: Dec(P),
satisfiable_int_formula: satisfiable_int_formula(fmla),
not: ¬A,
top: Top,
true: True,
iff: P ⇐⇒ Q,
nat: ℕ,
bag: bag(T),
so_lambda: λ2x y.t[x; y],
so_apply: x[s1;s2],
label: ...$L... t,
permutation: permutation(T;L1;L2),
cand: A c∧ B
Lemmas referenced :
bag_wf,
Prime_wf,
bag-product-primes,
bag_to_squash_list,
sq_stable__all,
less_than_wf,
int-bag-product_wf,
subtype_rel_bag,
equal_wf,
factors_wf,
sq_stable__equal,
squash_wf,
list-subtype-bag,
product-factors,
eq_int_wf,
bool_wf,
eqtt_to_assert,
assert_of_eq_int,
nil_wf,
eqff_to_assert,
bool_cases_sqequal,
subtype_base_sq,
bool_subtype_base,
assert-bnot,
neg_assert_of_eq_int,
factorization_wf,
decidable__le,
satisfiable-full-omega-tt,
intformand_wf,
intformnot_wf,
intformle_wf,
itermConstant_wf,
itermVar_wf,
intformless_wf,
intformeq_wf,
int_formula_prop_and_lemma,
int_formula_prop_not_lemma,
int_formula_prop_le_lemma,
int_term_value_constant_lemma,
int_term_value_var_lemma,
int_formula_prop_less_lemma,
int_formula_prop_eq_lemma,
int_formula_prop_wf,
le_wf,
list_wf,
int_upper_wf,
prime_wf,
mul-list_wf,
subtype_rel_list,
true_wf,
mul-list-bag-product,
iff_weakening_equal,
prime-factors-unique,
nat_wf,
subtype_rel_sets,
sq_stable__le,
set_wf,
quotient-member-eq,
permutation_wf,
permutation-equiv,
inject_wf,
int_seg_wf,
length_wf,
permute_list_wf,
list_subtype_base,
int_subtype_base
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
cut,
hypothesis,
introduction,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
dependent_functionElimination,
hypothesisEquality,
imageElimination,
natural_numberEquality,
applyEquality,
intEquality,
independent_isectElimination,
lambdaEquality,
setElimination,
rename,
because_Cache,
sqequalRule,
dependent_set_memberEquality,
independent_functionElimination,
lambdaFormation,
axiomEquality,
productElimination,
promote_hyp,
hyp_replacement,
equalitySymmetry,
applyLambdaEquality,
functionEquality,
imageMemberEquality,
baseClosed,
equalityTransitivity,
unionElimination,
equalityElimination,
dependent_pairFormation,
instantiate,
cumulativity,
voidElimination,
int_eqEquality,
isect_memberEquality,
voidEquality,
independent_pairFormation,
computeAll,
setEquality,
universeEquality,
productEquality,
functionExtensionality
Latex:
\mforall{}[b:bag(Prime)]. (factors(\mPi{}(b)) = b)
Date html generated:
2018_05_21-PM-07_22_57
Last ObjectModification:
2017_07_26-PM-05_06_00
Theory : general
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