Nuprl Lemma : product-factors
∀[n:ℕ+]. (Π(factors(n)) = n ∈ ℤ)
Proof
Definitions occuring in Statement :
factors: factors(n),
int-bag-product: Π(b),
nat_plus: ℕ+,
uall: ∀[x:A]. B[x],
int: ℤ,
equal: s = t ∈ T
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
factors: factors(n),
empty-bag: {},
member: t ∈ T,
squash: ↓T,
prop: ℙ,
all: ∀x:A. B[x],
nat_plus: ℕ+,
implies: P ⇒ Q,
bool: 𝔹,
unit: Unit,
it: ⋅,
btrue: tt,
ifthenelse: if b then t else f fi ,
uiff: uiff(P;Q),
and: P ∧ Q,
uimplies: b supposing a,
bfalse: ff,
exists: ∃x:A. B[x],
or: P ∨ Q,
sq_type: SQType(T),
guard: {T},
bnot: ¬bb,
assert: ↑b,
false: False,
int_upper: {i...},
nequal: a ≠ b ∈ T ,
decidable: Dec(P),
satisfiable_int_formula: satisfiable_int_formula(fmla),
not: ¬A,
top: Top,
subtype_rel: A ⊆r B,
so_lambda: λ2x.t[x],
so_apply: x[s],
true: True,
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q
Lemmas referenced :
equal_wf,
squash_wf,
true_wf,
mul-list-bag-product,
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,
nat_plus_properties,
decidable__le,
satisfiable-full-omega-tt,
intformand_wf,
intformnot_wf,
intformle_wf,
itermConstant_wf,
itermVar_wf,
intformeq_wf,
intformless_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_eq_lemma,
int_formula_prop_less_lemma,
int_formula_prop_wf,
le_wf,
subtype_rel_set,
list_wf,
int_upper_wf,
prime_wf,
mul-list_wf,
subtype_rel_list,
iff_weakening_equal,
mul_list_nil_lemma,
nat_plus_wf,
set_wf
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
cut,
applyEquality,
thin,
lambdaEquality,
sqequalHypSubstitution,
imageElimination,
introduction,
extract_by_obid,
isectElimination,
hypothesisEquality,
equalityTransitivity,
hypothesis,
equalitySymmetry,
universeEquality,
intEquality,
dependent_functionElimination,
setElimination,
rename,
because_Cache,
natural_numberEquality,
lambdaFormation,
unionElimination,
equalityElimination,
sqequalRule,
productElimination,
independent_isectElimination,
dependent_pairFormation,
promote_hyp,
instantiate,
cumulativity,
independent_functionElimination,
voidElimination,
dependent_set_memberEquality,
int_eqEquality,
isect_memberEquality,
voidEquality,
independent_pairFormation,
computeAll,
setEquality,
imageMemberEquality,
baseClosed
Latex:
\mforall{}[n:\mBbbN{}\msupplus{}]. (\mPi{}(factors(n)) = n)
Date html generated:
2018_05_21-PM-06_58_17
Last ObjectModification:
2017_07_26-PM-05_00_19
Theory : general
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