Nuprl Lemma : bag-sum_wf
∀[A:Type]. ∀[f:A ⟶ ℤ]. ∀[ba:bag(A)]. (bag-sum(ba;x.f[x]) ∈ ℤ)
Proof
Definitions occuring in Statement :
bag-sum: bag-sum(ba;x.f[x]),
bag: bag(T),
uall: ∀[x:A]. B[x],
so_apply: x[s],
member: t ∈ T,
function: x:A ⟶ B[x],
int: ℤ,
universe: Type
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
bag: bag(T),
quotient: x,y:A//B[x; y],
and: P ∧ Q,
all: ∀x:A. B[x],
implies: P ⇒ Q,
bag-sum: bag-sum(ba;x.f[x]),
so_lambda: λ2x.t[x],
so_lambda: λ2x y.t[x; y],
so_apply: x[s],
so_apply: x[s1;s2],
prop: ℙ,
squash: ↓T,
true: True,
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q,
decidable: Dec(P),
or: P ∨ Q,
uimplies: b supposing a,
satisfiable_int_formula: satisfiable_int_formula(fmla),
exists: ∃x:A. B[x],
false: False,
not: ¬A,
top: Top,
append: as @ bs,
so_lambda: so_lambda(x,y,z.t[x; y; z]),
so_apply: x[s1;s2;s3],
subtype_rel: A ⊆r B,
guard: {T}
Lemmas referenced :
list_wf,
permutation-invariant,
equal_wf,
list_accum_wf,
squash_wf,
true_wf,
cons_wf,
decidable__equal_int,
satisfiable-full-omega-tt,
intformand_wf,
intformnot_wf,
intformeq_wf,
itermVar_wf,
intformimplies_wf,
int_formula_prop_and_lemma,
int_formula_prop_not_lemma,
int_formula_prop_eq_lemma,
int_term_value_var_lemma,
int_formual_prop_imp_lemma,
int_formula_prop_wf,
permutation_wf,
equal-wf-base,
bag_wf,
list_induction,
all_wf,
append_wf,
nil_wf,
list_ind_nil_lemma,
list_accum_cons_lemma,
list_accum_nil_lemma,
list_ind_cons_lemma,
iff_weakening_equal,
itermAdd_wf,
int_term_value_add_lemma,
itermConstant_wf,
int_term_value_constant_lemma
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
introduction,
cut,
sqequalHypSubstitution,
pointwiseFunctionalityForEquality,
intEquality,
sqequalRule,
pertypeElimination,
productElimination,
thin,
equalityTransitivity,
hypothesis,
equalitySymmetry,
extract_by_obid,
isectElimination,
cumulativity,
hypothesisEquality,
lambdaFormation,
because_Cache,
rename,
lambdaEquality,
natural_numberEquality,
addEquality,
applyEquality,
functionExtensionality,
independent_functionElimination,
addLevel,
hyp_replacement,
imageElimination,
equalityUniverse,
levelHypothesis,
imageMemberEquality,
baseClosed,
dependent_functionElimination,
unionElimination,
independent_isectElimination,
dependent_pairFormation,
int_eqEquality,
isect_memberEquality,
voidElimination,
voidEquality,
independent_pairFormation,
computeAll,
productEquality,
axiomEquality,
functionEquality,
universeEquality
Latex:
\mforall{}[A:Type]. \mforall{}[f:A {}\mrightarrow{} \mBbbZ{}]. \mforall{}[ba:bag(A)]. (bag-sum(ba;x.f[x]) \mmember{} \mBbbZ{})
Date html generated:
2017_10_01-AM-08_47_52
Last ObjectModification:
2017_07_26-PM-04_32_12
Theory : bags
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