Nuprl Lemma : rng_sum-int
∀[a,b:ℤ]. ∀[f:{a..b-} ⟶ ℤ]. (Σ(ℤ-rng) a ≤ i < b. f[i]) = Σ(f[a + i] | i < b - a) ∈ ℤ supposing a ≤ b
Proof
Definitions occuring in Statement :
sum: Σ(f[x] | x < k),
int_seg: {i..j-},
uimplies: b supposing a,
uall: ∀[x:A]. B[x],
so_apply: x[s],
le: A ≤ B,
function: x:A ⟶ B[x],
subtract: n - m,
add: n + m,
int: ℤ,
equal: s = t ∈ T,
int_ring: ℤ-rng,
rng_sum: rng_sum
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
uimplies: b supposing a,
rng_sum: rng_sum,
mon_itop: Π lb ≤ i < ub. E[i],
add_grp_of_rng: r↓+gp,
grp_op: *,
pi2: snd(t),
pi1: fst(t),
grp_id: e,
int_ring: ℤ-rng,
rng_plus: +r,
rng_zero: 0,
nat: ℕ,
all: ∀x:A. B[x],
decidable: Dec(P),
or: P ∨ Q,
satisfiable_int_formula: satisfiable_int_formula(fmla),
exists: ∃x:A. B[x],
false: False,
implies: P ⇒ Q,
not: ¬A,
top: Top,
and: P ∧ Q,
prop: ℙ,
so_lambda: λ2x.t[x],
so_apply: x[s],
int_seg: {i..j-},
uiff: uiff(P;Q),
lelt: i ≤ j < k,
ge: i ≥ j ,
itop: Π(op,id) lb ≤ i < ub. E[i],
ycomb: Y,
bool: 𝔹,
unit: Unit,
it: ⋅,
btrue: tt,
ifthenelse: if b then t else f fi ,
infix_ap: x f y,
bfalse: ff,
sq_type: SQType(T),
guard: {T},
bnot: ¬bb,
assert: ↑b,
nequal: a ≠ b ∈ T ,
true: True,
le: A ≤ B,
subtract: n - m,
subtype_rel: A ⊆r B,
squash: ↓T,
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q
Lemmas referenced :
sum-as-primrec,
decidable__le,
subtract_wf,
satisfiable-full-omega-tt,
intformand_wf,
intformnot_wf,
intformle_wf,
itermConstant_wf,
itermSubtract_wf,
itermVar_wf,
int_formula_prop_and_lemma,
int_formula_prop_not_lemma,
int_formula_prop_le_lemma,
int_term_value_constant_lemma,
int_term_value_subtract_lemma,
int_term_value_var_lemma,
int_formula_prop_wf,
le_wf,
int_seg_wf,
add-member-int_seg1,
lelt_wf,
nat_properties,
intformless_wf,
int_formula_prop_less_lemma,
ge_wf,
less_than_wf,
lt_int_wf,
bool_wf,
eqtt_to_assert,
assert_of_lt_int,
itermAdd_wf,
int_term_value_add_lemma,
eqff_to_assert,
equal_wf,
bool_cases_sqequal,
subtype_base_sq,
bool_subtype_base,
assert-bnot,
primrec-unroll,
eq_int_wf,
assert_of_eq_int,
neg_assert_of_eq_int,
intformeq_wf,
int_formula_prop_eq_lemma,
nat_wf,
itop_wf,
decidable__lt,
primrec_wf,
decidable__equal_int,
add-associates,
minus-one-mul,
add-swap,
minus-one-mul-top,
add-commutes,
add-mul-special,
zero-mul,
zero-add,
itermMinus_wf,
int_term_value_minus_lemma,
add_functionality_wrt_eq,
iff_weakening_equal
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
introduction,
cut,
sqequalRule,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
dependent_set_memberEquality,
because_Cache,
dependent_functionElimination,
natural_numberEquality,
hypothesisEquality,
hypothesis,
unionElimination,
independent_isectElimination,
dependent_pairFormation,
lambdaEquality,
int_eqEquality,
intEquality,
isect_memberEquality,
voidElimination,
voidEquality,
independent_pairFormation,
computeAll,
applyEquality,
functionExtensionality,
setElimination,
rename,
productElimination,
lambdaFormation,
intWeakElimination,
independent_functionElimination,
axiomEquality,
functionEquality,
addEquality,
equalityElimination,
equalityTransitivity,
equalitySymmetry,
promote_hyp,
instantiate,
cumulativity,
minusEquality,
imageElimination,
imageMemberEquality,
baseClosed
Latex:
\mforall{}[a,b:\mBbbZ{}]. \mforall{}[f:\{a..b\msupminus{}\} {}\mrightarrow{} \mBbbZ{}]. (\mSigma{}(\mBbbZ{}-rng) a \mleq{} i < b. f[i]) = \mSigma{}(f[a + i] | i < b - a) supposing a \mleq{} b
Date html generated:
2018_05_21-PM-08_27_23
Last ObjectModification:
2017_07_26-PM-05_54_57
Theory : general
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