Nuprl Lemma : int-prod-split
∀[n:ℕ]. ∀[f:ℕn ⟶ ℤ]. ∀[m:ℕn + 1]. (Π(f[x] | x < n) = (Π(f[x] | x < m) * Π(f[x + m] | x < n - m)) ∈ 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,
subtract: n - m,
add: n + m,
natural_number: $n,
int: ℤ,
equal: s = t ∈ T
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
nat: ℕ,
implies: P ⇒ Q,
false: False,
ge: i ≥ j ,
uimplies: b supposing a,
not: ¬A,
satisfiable_int_formula: satisfiable_int_formula(fmla),
exists: ∃x:A. B[x],
all: ∀x:A. B[x],
top: Top,
and: P ∧ Q,
prop: ℙ,
so_lambda: λ2x.t[x],
so_apply: x[s],
int_seg: {i..j-},
decidable: Dec(P),
or: P ∨ Q,
sq_type: SQType(T),
guard: {T},
subtract: n - m,
lelt: i ≤ j < k,
int-prod: Π(f[x] | x < k),
bool: 𝔹,
unit: Unit,
it: ⋅,
btrue: tt,
uiff: uiff(P;Q),
bfalse: ff,
subtype_rel: A ⊆r B,
bnot: ¬bb,
ifthenelse: if b then t else f fi ,
assert: ↑b,
rev_implies: P ⇐ Q,
iff: P ⇐⇒ Q,
lt_int: i <z j,
le: A ≤ B,
less_than': less_than'(a;b),
true: True
Lemmas referenced :
nat_properties,
full-omega-unsat,
intformand_wf,
intformle_wf,
itermConstant_wf,
itermVar_wf,
intformless_wf,
istype-int,
int_formula_prop_and_lemma,
istype-void,
int_formula_prop_le_lemma,
int_term_value_constant_lemma,
int_term_value_var_lemma,
int_formula_prop_less_lemma,
int_formula_prop_wf,
ge_wf,
less_than_wf,
int_prod0_lemma,
int_seg_wf,
subtract-1-ge-0,
nat_wf,
decidable__equal_int,
subtype_base_sq,
int_subtype_base,
int_seg_properties,
int_seg_subtype_special,
int_seg_cases,
subtract_wf,
lt_int_wf,
eqtt_to_assert,
assert_of_lt_int,
eqff_to_assert,
bool_subtype_base,
bool_cases_sqequal,
bool_wf,
assert-bnot,
iff_weakening_uiff,
assert_wf,
primrec-unroll,
intformnot_wf,
intformeq_wf,
itermSubtract_wf,
int_formula_prop_not_lemma,
int_formula_prop_eq_lemma,
int_term_value_subtract_lemma,
int-prod_wf,
decidable__le,
le_wf,
decidable__lt,
one-mul,
itermAdd_wf,
int_term_value_add_lemma,
zero-add,
primrec_wf,
itermMultiply_wf,
int_term_value_mul_lemma,
set_subtype_base,
lelt_wf,
subtype_rel_function,
int_seg_subtype,
istype-false,
not-le-2,
condition-implies-le,
add-associates,
minus-add,
minus-one-mul,
add-swap,
minus-one-mul-top,
add-mul-special,
zero-mul,
add-zero,
add-commutes,
le-add-cancel2,
subtype_rel_self,
subtract-add-cancel,
int_seg_subtype_nat,
add-member-int_seg1,
mul-associates,
mul-swap
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
Error :isect_memberFormation_alt,
introduction,
cut,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
hypothesisEquality,
hypothesis,
setElimination,
rename,
intWeakElimination,
Error :lambdaFormation_alt,
natural_numberEquality,
independent_isectElimination,
approximateComputation,
independent_functionElimination,
Error :dependent_pairFormation_alt,
Error :lambdaEquality_alt,
int_eqEquality,
dependent_functionElimination,
Error :isect_memberEquality_alt,
voidElimination,
sqequalRule,
independent_pairFormation,
Error :universeIsType,
axiomEquality,
Error :isectIsTypeImplies,
Error :inhabitedIsType,
Error :functionIsTypeImplies,
addEquality,
Error :functionIsType,
unionElimination,
instantiate,
cumulativity,
intEquality,
because_Cache,
equalityTransitivity,
equalitySymmetry,
hypothesis_subsumption,
productElimination,
equalityElimination,
Error :equalityIsType4,
baseApply,
closedConclusion,
baseClosed,
applyEquality,
promote_hyp,
Error :equalityIsType1,
multiplyEquality,
Error :dependent_set_memberEquality_alt,
Error :productIsType,
minusEquality
Latex:
\mforall{}[n:\mBbbN{}]. \mforall{}[f:\mBbbN{}n {}\mrightarrow{} \mBbbZ{}]. \mforall{}[m:\mBbbN{}n + 1]. (\mPi{}(f[x] | x < n) = (\mPi{}(f[x] | x < m) * \mPi{}(f[x + m] | x < n - m)))
Date html generated:
2019_06_20-PM-01_18_37
Last ObjectModification:
2018_10_15-PM-02_14_19
Theory : int_2
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