Nuprl Lemma : int-prod-factor
∀[n:ℕ]. ∀[f,g:ℕn ⟶ ℤ]. (Π(f[x] * g[x] | x < n) = (Π(f[x] | x < n) * Π(g[x] | x < n)) ∈ ℤ)
Proof
Definitions occuring in Statement :
int-prod: Π(f[x] | x < k),
int_seg: {i..j-},
nat: ℕ,
uall: ∀[x:A]. B[x],
so_apply: x[s],
function: x:A ⟶ B[x],
multiply: n * m,
natural_number: $n,
int: ℤ,
equal: s = t ∈ T
Definitions unfolded in proof :
member: t ∈ T,
top: Top,
uall: ∀[x:A]. B[x],
so_apply: x[s],
int_seg: {i..j-},
lelt: i ≤ j < k,
and: P ∧ Q,
all: ∀x:A. B[x],
decidable: Dec(P),
or: P ∨ Q,
uimplies: b supposing a,
not: ¬A,
implies: P ⇒ Q,
satisfiable_int_formula: satisfiable_int_formula(fmla),
exists: ∃x:A. B[x],
false: False,
prop: ℙ,
nat: ℕ,
so_lambda: λ2x.t[x],
true: True,
ge: i ≥ j ,
int-prod: Π(f[x] | x < k),
lt_int: i <z j,
subtract: n - m,
ifthenelse: if b then t else f fi ,
btrue: tt,
bool: 𝔹,
unit: Unit,
it: ⋅,
uiff: uiff(P;Q),
bfalse: ff,
sq_type: SQType(T),
guard: {T},
bnot: ¬bb,
assert: ↑b,
squash: ↓T,
subtype_rel: A ⊆r B,
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q,
primrec: primrec(n;b;c),
less_than: a < b,
less_than': less_than'(a;b),
has-value: (a)↓
Lemmas referenced :
int_seg_wf,
decidable__lt,
full-omega-unsat,
intformand_wf,
intformnot_wf,
intformless_wf,
itermVar_wf,
itermSubtract_wf,
itermConstant_wf,
int_formula_prop_and_lemma,
int_formula_prop_not_lemma,
int_formula_prop_less_lemma,
int_term_value_var_lemma,
int_term_value_subtract_lemma,
int_term_value_constant_lemma,
int_formula_prop_wf,
lelt_wf,
subtract_wf,
decidable__equal_int,
intformeq_wf,
int_formula_prop_eq_lemma,
decidable__le,
intformle_wf,
int_formula_prop_le_lemma,
int-prod_wf,
le_wf,
nat_wf,
nat_properties,
ge_wf,
less_than_wf,
primrec-unroll,
lt_int_wf,
bool_wf,
eqtt_to_assert,
assert_of_lt_int,
eqff_to_assert,
equal_wf,
bool_cases_sqequal,
subtype_base_sq,
bool_subtype_base,
assert-bnot,
squash_wf,
true_wf,
subtype_rel_self,
iff_weakening_equal,
top_wf,
itermMultiply_wf,
int_term_value_mul_lemma,
value-type-has-value,
int-value-type,
primrec_wf,
itermAdd_wf,
int_term_value_add_lemma
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
natural_numberEquality,
isect_memberEquality,
voidElimination,
voidEquality,
functionEquality,
cut,
introduction,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
hypothesis,
intEquality,
because_Cache,
hypothesisEquality,
lambdaEquality,
applyEquality,
functionExtensionality,
setElimination,
rename,
dependent_set_memberEquality,
productElimination,
independent_pairFormation,
dependent_functionElimination,
unionElimination,
independent_isectElimination,
approximateComputation,
independent_functionElimination,
dependent_pairFormation,
int_eqEquality,
sqequalRule,
multiplyEquality,
equalityTransitivity,
equalitySymmetry,
isect_memberFormation,
intWeakElimination,
lambdaFormation,
axiomEquality,
equalityElimination,
promote_hyp,
instantiate,
cumulativity,
imageElimination,
universeEquality,
imageMemberEquality,
baseClosed,
addEquality,
minusEquality,
lessCases,
sqequalAxiom,
callbyvalueReduce
Latex:
\mforall{}[n:\mBbbN{}]. \mforall{}[f,g:\mBbbN{}n {}\mrightarrow{} \mBbbZ{}]. (\mPi{}(f[x] * g[x] | x < n) = (\mPi{}(f[x] | x < n) * \mPi{}(g[x] | x < n)))
Date html generated:
2018_05_21-PM-00_29_45
Last ObjectModification:
2018_05_19-AM-06_54_57
Theory : int_2
Home
Index