Nuprl Lemma : bag-summation-constant-int
∀[T:Type]. ∀[a:ℤ]. ∀[bs:bag(T)]. (Σ(x∈bs). a = (#(bs) * a) ∈ ℤ)
Proof
Definitions occuring in Statement :
bag-summation: Σ(x∈b). f[x],
bag-size: #(bs),
bag: bag(T),
uall: ∀[x:A]. B[x],
lambda: λx.A[x],
multiply: n * m,
add: n + m,
natural_number: $n,
int: ℤ,
universe: Type,
equal: s = t ∈ T
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
subtype_rel: A ⊆r B,
integ_dom: IntegDom{i},
crng: CRng,
all: ∀x:A. B[x],
int_ring: ℤ-rng,
rng_car: |r|,
pi1: fst(t),
prop: ℙ,
squash: ↓T,
rng: Rng,
nat: ℕ,
true: True,
and: P ∧ Q,
uimplies: b supposing a,
cand: A c∧ B,
rng_plus: +r,
pi2: snd(t),
rng_zero: 0,
so_lambda: λ2x.t[x],
assoc: Assoc(T;op),
infix_ap: x f y,
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,
comm: Comm(T;op),
so_apply: x[s]
Lemmas referenced :
bag-summation-constant,
int_ring_wf,
integ_dom_wf,
bag_wf,
equal_wf,
squash_wf,
true_wf,
rng_car_wf,
rng_nat_op-int,
bag-size_wf,
nat_wf,
bag-summation_wf,
assoc_wf,
comm_wf,
rng_zero_wf,
decidable__equal_int,
satisfiable-full-omega-tt,
intformnot_wf,
intformeq_wf,
itermAdd_wf,
itermVar_wf,
int_formula_prop_not_lemma,
int_formula_prop_eq_lemma,
int_term_value_add_lemma,
int_term_value_var_lemma,
int_formula_prop_wf
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
introduction,
cut,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
hypothesisEquality,
hypothesis,
applyEquality,
lambdaEquality,
setElimination,
rename,
sqequalRule,
dependent_functionElimination,
cumulativity,
isect_memberEquality,
axiomEquality,
because_Cache,
intEquality,
universeEquality,
hyp_replacement,
equalitySymmetry,
imageElimination,
equalityTransitivity,
multiplyEquality,
natural_numberEquality,
imageMemberEquality,
baseClosed,
productElimination,
independent_isectElimination,
independent_pairFormation,
productEquality,
functionExtensionality,
functionEquality,
addEquality,
unionElimination,
dependent_pairFormation,
int_eqEquality,
voidElimination,
voidEquality,
computeAll
Latex:
\mforall{}[T:Type]. \mforall{}[a:\mBbbZ{}]. \mforall{}[bs:bag(T)]. (\mSigma{}(x\mmember{}bs). a = (\#(bs) * a))
Date html generated:
2016_10_25-AM-11_27_53
Last ObjectModification:
2016_07_12-AM-07_34_37
Theory : bags_2
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