Nuprl Lemma : rprod-split
∀[n,m:ℤ]. ∀[x:{n..m + 1-} ⟶ ℝ]. ∀[i:ℤ].
rprod(n;m;k.x[k]) = (rprod(n;i;k.x[k]) * rprod(i + 1;m;k.x[k])) supposing (i ≤ m) ∧ (n ≤ (i + 1))
Proof
Definitions occuring in Statement :
rprod: rprod(n;m;k.x[k]),
req: x = y,
rmul: a * b,
real: ℝ,
int_seg: {i..j-},
uimplies: b supposing a,
uall: ∀[x:A]. B[x],
so_apply: x[s],
le: A ≤ B,
and: P ∧ Q,
function: x:A ⟶ B[x],
add: n + m,
natural_number: $n,
int: ℤ
Definitions unfolded in proof :
all: ∀x:A. B[x],
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],
top: Top,
and: P ∧ Q,
prop: ℙ,
so_lambda: λ2x.t[x],
so_apply: x[s],
int_seg: {i..j-},
lelt: i ≤ j < k,
decidable: Dec(P),
or: P ∨ Q,
sq_type: SQType(T),
guard: {T},
uiff: uiff(P;Q),
rev_uimplies: rev_uimplies(P;Q),
req_int_terms: t1 ≡ t2,
rprod: rprod(n;m;k.x[k]),
bool: 𝔹,
unit: Unit,
it: ⋅,
btrue: tt,
ifthenelse: if b then t else f fi ,
bfalse: ff,
bnot: ¬bb,
assert: ↑b,
rev_implies: P ⇐ Q,
iff: P ⇐⇒ Q,
subtype_rel: A ⊆r B,
le: A ≤ B,
subtract: n - m,
cand: A c∧ 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,
istype-less_than,
req_witness,
rprod_wf,
int_seg_wf,
rmul_wf,
decidable__lt,
intformnot_wf,
itermAdd_wf,
int_formula_prop_not_lemma,
int_term_value_add_lemma,
istype-le,
decidable__le,
real_wf,
subtract-1-ge-0,
istype-nat,
add-zero,
decidable__equal_int,
subtype_base_sq,
int_subtype_base,
intformeq_wf,
int_formula_prop_eq_lemma,
rprod-empty,
int-to-real_wf,
itermSubtract_wf,
itermMultiply_wf,
req-iff-rsub-is-0,
req_functionality,
rprod-single,
rmul_functionality,
req_weakening,
real_polynomial_null,
real_term_value_sub_lemma,
real_term_value_var_lemma,
real_term_value_mul_lemma,
real_term_value_const_lemma,
lt_int_wf,
eqtt_to_assert,
assert_of_lt_int,
eqff_to_assert,
bool_cases_sqequal,
bool_wf,
bool_subtype_base,
assert-bnot,
iff_weakening_uiff,
assert_wf,
less_than_wf,
int_term_value_subtract_lemma,
subtract_wf,
subtype_rel_function,
int_seg_subtype,
le_reflexive,
add-is-int-iff,
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,
zero-add,
add-commutes,
le-add-cancel,
subtype_rel_self,
rmul_assoc,
trivial-int-eq1
Rules used in proof :
cut,
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation_alt,
introduction,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
hypothesisEquality,
hypothesis,
setElimination,
rename,
intWeakElimination,
natural_numberEquality,
independent_isectElimination,
approximateComputation,
independent_functionElimination,
dependent_pairFormation_alt,
lambdaEquality_alt,
int_eqEquality,
dependent_functionElimination,
isect_memberEquality_alt,
voidElimination,
sqequalRule,
independent_pairFormation,
universeIsType,
productElimination,
isectIsTypeImplies,
inhabitedIsType,
functionIsTypeImplies,
isect_memberFormation_alt,
addEquality,
applyEquality,
dependent_set_memberEquality_alt,
unionElimination,
productIsType,
because_Cache,
closedConclusion,
functionIsType,
instantiate,
cumulativity,
intEquality,
equalityTransitivity,
equalitySymmetry,
equalityElimination,
equalityIstype,
promote_hyp,
baseApply,
baseClosed,
minusEquality,
multiplyEquality
Latex:
\mforall{}[n,m:\mBbbZ{}]. \mforall{}[x:\{n..m + 1\msupminus{}\} {}\mrightarrow{} \mBbbR{}]. \mforall{}[i:\mBbbZ{}].
rprod(n;m;k.x[k]) = (rprod(n;i;k.x[k]) * rprod(i + 1;m;k.x[k])) supposing (i \mleq{} m) \mwedge{} (n \mleq{} (i + 1))
Date html generated:
2019_10_29-AM-10_18_28
Last ObjectModification:
2019_01_15-PM-00_19_28
Theory : reals
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