Nuprl Lemma : qsum-delta
∀[a,b:ℤ]. ∀[E:{a..b-} ⟶ ℚ]. ∀[i:ℤ]. (Σa ≤ j < b. E[j] * delta(i;j) = if a ≤z i ∧b i <z b then E[i] else 0 fi ∈ ℚ)
Proof
Definitions occuring in Statement :
delta: delta(i;j),
qsum: Σa ≤ j < b. E[j],
qmul: r * s,
rationals: ℚ,
band: p ∧b q,
int_seg: {i..j-},
le_int: i ≤z j,
ifthenelse: if b then t else f fi ,
lt_int: i <z j,
uall: ∀[x:A]. B[x],
so_apply: x[s],
function: x:A ⟶ B[x],
natural_number: $n,
int: ℤ,
equal: s = t ∈ T
Definitions unfolded in proof :
all: ∀x:A. B[x],
member: t ∈ T,
uall: ∀[x:A]. B[x],
int_upper: {i...},
so_lambda: λ2x.t[x],
implies: P ⇒ Q,
bool: 𝔹,
unit: Unit,
it: ⋅,
btrue: tt,
uiff: uiff(P;Q),
and: P ∧ Q,
uimplies: b supposing a,
band: p ∧b q,
ifthenelse: if b then t else f fi ,
so_apply: x[s],
int_seg: {i..j-},
subtype_rel: A ⊆r B,
lelt: i ≤ j < k,
prop: ℙ,
bfalse: ff,
exists: ∃x:A. B[x],
or: P ∨ Q,
sq_type: SQType(T),
guard: {T},
bnot: ¬bb,
assert: ↑b,
false: False,
not: ¬A,
rev_implies: P ⇐ Q,
iff: P ⇐⇒ Q,
satisfiable_int_formula: satisfiable_int_formula(fmla),
top: Top,
squash: ↓T,
true: True,
decidable: Dec(P),
subtract: n - m,
le: A ≤ B,
less_than': less_than'(a;b),
delta: delta(i;j),
label: ...$L... t,
infix_ap: x f y,
ycomb: Y,
itop: Π(op,id) lb ≤ i < ub. E[i],
rng_zero: 0,
rng_plus: +r,
qrng: <ℚ+*>,
grp_id: e,
pi1: fst(t),
pi2: snd(t),
grp_op: *,
add_grp_of_rng: r↓+gp,
mon_itop: Π lb ≤ i < ub. E[i],
rng_sum: rng_sum,
qsum: Σa ≤ j < b. E[j]
Lemmas referenced :
istype-int,
all_wf,
int_seg_wf,
rationals_wf,
eqtt_to_assert,
assert_of_le_int,
assert_of_lt_int,
equal_wf,
qsum_wf,
qmul_wf,
delta_wf,
le_wf,
less_than_wf,
eqff_to_assert,
int_subtype_base,
bool_cases_sqequal,
subtype_base_sq,
bool_wf,
bool_subtype_base,
assert-bnot,
iff_weakening_uiff,
assert_wf,
equal-wf-T-base,
int_upper_wf,
le_int_wf,
lt_int_wf,
full-omega-unsat,
intformand_wf,
intformless_wf,
itermVar_wf,
intformle_wf,
int_formula_prop_and_lemma,
istype-void,
int_formula_prop_less_lemma,
int_term_value_var_lemma,
int_formula_prop_le_lemma,
int_formula_prop_wf,
squash_wf,
true_wf,
istype-universe,
sum_unroll_base_q,
int-subtype-rationals,
subtype_rel_self,
iff_weakening_equal,
int_seg_properties,
subtype_rel_function,
subtract_wf,
int_seg_subtype,
le_reflexive,
decidable__le,
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,
set_subtype_base,
int_le_to_int_upper,
int_upper_ind,
bfalse_wf,
int_upper_properties,
itermSubtract_wf,
itermConstant_wf,
intformnot_wf,
int_term_value_subtract_lemma,
int_term_value_constant_lemma,
int_formula_prop_not_lemma,
btrue_wf,
equal-wf-base,
sum_unroll_hi_q,
decidable__lt,
qadd_wf,
int_seg_inc,
eq_int_wf,
intformeq_wf,
int_formula_prop_eq_lemma,
bnot_wf,
not_wf,
bor_wf,
or_wf,
intformor_wf,
int_formula_prop_or_lemma,
decidable__equal_int,
itermAdd_wf,
int_term_value_add_lemma,
bool_cases,
iff_transitivity,
assert_of_band,
assert_of_bnot,
uiff_transitivity,
assert_functionality_wrt_uiff,
bnot_thru_band,
bnot_of_le_int,
bnot_of_lt_int,
assert_of_bor,
assert_of_eq_int,
qmul_zero_qrng,
qadd_comm_q,
mon_ident_q,
qmul_one_qrng,
lelt_wf,
satisfiable-full-omega-tt
Rules used in proof :
cut,
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation_alt,
introduction,
extract_by_obid,
hypothesis,
hypothesisEquality,
lambdaEquality_alt,
sqequalHypSubstitution,
isectElimination,
thin,
functionEquality,
setElimination,
rename,
because_Cache,
closedConclusion,
sqequalRule,
intEquality,
inhabitedIsType,
unionElimination,
equalityElimination,
productElimination,
independent_isectElimination,
applyEquality,
universeIsType,
dependent_set_memberEquality_alt,
independent_pairFormation,
productIsType,
equalityTransitivity,
equalitySymmetry,
dependent_pairFormation_alt,
equalityIsType2,
baseApply,
baseClosed,
promote_hyp,
dependent_functionElimination,
instantiate,
cumulativity,
independent_functionElimination,
voidElimination,
equalityIsType1,
functionIsType,
natural_numberEquality,
approximateComputation,
int_eqEquality,
isect_memberEquality_alt,
imageElimination,
universeEquality,
imageMemberEquality,
addEquality,
minusEquality,
multiplyEquality,
productEquality,
hyp_replacement,
applyLambdaEquality,
equalityIsType4,
unionIsType,
inlFormation_alt,
inrFormation_alt,
axiomEquality,
isect_memberEquality,
dependent_set_memberEquality,
dependent_pairFormation,
lambdaFormation,
functionExtensionality,
lambdaEquality,
isect_memberFormation,
computeAll,
voidEquality
Latex:
\mforall{}[a,b:\mBbbZ{}]. \mforall{}[E:\{a..b\msupminus{}\} {}\mrightarrow{} \mBbbQ{}]. \mforall{}[i:\mBbbZ{}].
(\mSigma{}a \mleq{} j < b. E[j] * delta(i;j) = if a \mleq{}z i \mwedge{}\msubb{} i <z b then E[i] else 0 fi )
Date html generated:
2019_10_16-PM-00_31_54
Last ObjectModification:
2018_10_11-PM-11_46_17
Theory : rationals
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