Nuprl Lemma : sum-ite
∀[k:ℕ]. ∀[f,g:ℕk ⟶ ℤ]. ∀[p:ℕk ⟶ 𝔹].
(Σ(if p[i] then f[i] + g[i] else f[i] fi | i < k) = (Σ(f[i] | i < k) + Σ(if p[i] then g[i] else 0 fi | i < k)) ∈ ℤ)
Proof
Definitions occuring in Statement :
sum: Σ(f[x] | x < k),
int_seg: {i..j-},
nat: ℕ,
ifthenelse: if b then t else f fi ,
bool: 𝔹,
uall: ∀[x:A]. B[x],
so_apply: x[s],
function: x:A ⟶ B[x],
add: n + m,
natural_number: $n,
int: ℤ,
equal: s = t ∈ T
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
so_lambda: λ2x.t[x],
so_apply: x[s],
nat: ℕ,
all: ∀x:A. B[x],
implies: P ⇒ Q,
false: False,
ge: i ≥ j ,
uimplies: b supposing a,
satisfiable_int_formula: satisfiable_int_formula(fmla),
exists: ∃x:A. B[x],
not: ¬A,
top: Top,
and: P ∧ Q,
prop: ℙ,
decidable: Dec(P),
or: P ∨ Q,
subtype_rel: A ⊆r B,
le: A ≤ B,
less_than': less_than'(a;b),
bool: 𝔹,
unit: Unit,
it: ⋅,
btrue: tt,
uiff: uiff(P;Q),
ifthenelse: if b then t else f fi ,
bfalse: ff,
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q,
int_seg: {i..j-},
lelt: i ≤ j < k,
sq_type: SQType(T),
guard: {T},
bnot: ¬bb,
assert: ↑b
Lemmas referenced :
sum-as-primrec,
ifthenelse_wf,
int_seg_wf,
nat_properties,
satisfiable-full-omega-tt,
intformand_wf,
intformle_wf,
itermConstant_wf,
itermVar_wf,
intformless_wf,
int_formula_prop_and_lemma,
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,
bool_wf,
primrec0_lemma,
decidable__le,
subtract_wf,
intformnot_wf,
itermSubtract_wf,
int_formula_prop_not_lemma,
int_term_value_subtract_lemma,
subtype_rel_dep_function,
int_seg_subtype,
false_wf,
subtype_rel_self,
primrec-unroll,
eq_int_wf,
uiff_transitivity,
equal-wf-base,
int_subtype_base,
assert_wf,
eqtt_to_assert,
assert_of_eq_int,
intformeq_wf,
int_formula_prop_eq_lemma,
iff_transitivity,
bnot_wf,
not_wf,
iff_weakening_uiff,
eqff_to_assert,
assert_of_bnot,
decidable__lt,
lelt_wf,
decidable__equal_int,
add-is-int-iff,
itermAdd_wf,
int_term_value_add_lemma,
equal_wf,
bool_cases_sqequal,
subtype_base_sq,
bool_subtype_base,
assert-bnot,
nat_wf
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
introduction,
cut,
sqequalRule,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
hypothesisEquality,
lambdaEquality,
applyEquality,
functionExtensionality,
natural_numberEquality,
setElimination,
rename,
because_Cache,
hypothesis,
intEquality,
addEquality,
lambdaFormation,
intWeakElimination,
independent_isectElimination,
dependent_pairFormation,
int_eqEquality,
dependent_functionElimination,
isect_memberEquality,
voidElimination,
voidEquality,
independent_pairFormation,
computeAll,
independent_functionElimination,
axiomEquality,
functionEquality,
unionElimination,
equalityElimination,
baseApply,
closedConclusion,
baseClosed,
productElimination,
impliesFunctionality,
dependent_set_memberEquality,
equalityTransitivity,
equalitySymmetry,
pointwiseFunctionality,
promote_hyp,
instantiate,
cumulativity
Latex:
\mforall{}[k:\mBbbN{}]. \mforall{}[f,g:\mBbbN{}k {}\mrightarrow{} \mBbbZ{}]. \mforall{}[p:\mBbbN{}k {}\mrightarrow{} \mBbbB{}].
(\mSigma{}(if p[i] then f[i] + g[i] else f[i] fi | i < k)
= (\mSigma{}(f[i] | i < k) + \mSigma{}(if p[i] then g[i] else 0 fi | i < k)))
Date html generated:
2017_04_14-AM-09_20_36
Last ObjectModification:
2017_02_27-PM-03_57_06
Theory : int_2
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