Nuprl Lemma : bag-sum_wf_nat
∀[A:Type]. ∀[f:A ⟶ ℕ]. ∀[ba:bag(A)]. (bag-sum(ba;x.f[x]) ∈ ℕ)
Proof
Definitions occuring in Statement :
bag-sum: bag-sum(ba;x.f[x]),
bag: bag(T),
nat: ℕ,
uall: ∀[x:A]. B[x],
so_apply: x[s],
member: t ∈ T,
function: x:A ⟶ B[x],
universe: Type
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
bag: bag(T),
so_lambda: λ2x.t[x],
so_apply: x[s],
subtype_rel: A ⊆r B,
prop: ℙ,
quotient: x,y:A//B[x; y],
and: P ∧ Q,
all: ∀x:A. B[x],
implies: P ⇒ Q,
squash: ↓T,
nat: ℕ,
bag-sum: bag-sum(ba;x.f[x]),
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,
guard: {T},
le: A ≤ B,
so_lambda: λ2x y.t[x; y],
so_apply: x[s1;s2],
int_seg: {i..j-},
lelt: i ≤ j < k,
decidable: Dec(P),
or: P ∨ Q,
less_than': less_than'(a;b),
less_than: a < b,
cons: [a / b],
assert: ↑b,
ifthenelse: if b then t else f fi ,
bfalse: ff,
iff: P ⇐⇒ Q,
uiff: uiff(P;Q),
rev_implies: P ⇐ Q,
int_iseg: {i...j},
cand: A c∧ B
Lemmas referenced :
squash_wf,
le_wf,
bag-sum_wf,
list_wf,
permutation_wf,
equal_wf,
equal-wf-base,
bag_wf,
nat_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,
less_than'_wf,
list_accum_wf,
length_wf,
int_seg_wf,
int_seg_properties,
decidable__le,
subtract_wf,
intformnot_wf,
itermSubtract_wf,
int_formula_prop_not_lemma,
int_term_value_subtract_lemma,
decidable__equal_int,
int_seg_subtype,
false_wf,
intformeq_wf,
int_formula_prop_eq_lemma,
non_neg_length,
decidable__lt,
lelt_wf,
decidable__assert,
null_wf,
list-cases,
list_accum_nil_lemma,
product_subtype_list,
null_cons_lemma,
last-lemma-sq,
pos_length,
iff_transitivity,
not_wf,
equal-wf-T-base,
assert_wf,
bnot_wf,
assert_of_null,
iff_weakening_uiff,
assert_of_bnot,
firstn_wf,
length_firstn,
zero-le-nat,
append_wf,
cons_wf,
last_wf,
nil_wf,
add_nat_wf,
add-is-int-iff,
itermAdd_wf,
int_term_value_add_lemma,
length_wf_nat
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
introduction,
cut,
sqequalHypSubstitution,
pointwiseFunctionalityForEquality,
extract_by_obid,
isectElimination,
thin,
natural_numberEquality,
cumulativity,
hypothesisEquality,
sqequalRule,
lambdaEquality,
applyEquality,
functionExtensionality,
hypothesis,
because_Cache,
pertypeElimination,
productElimination,
equalityTransitivity,
equalitySymmetry,
lambdaFormation,
rename,
imageMemberEquality,
baseClosed,
dependent_functionElimination,
independent_functionElimination,
productEquality,
imageElimination,
dependent_set_memberEquality,
axiomEquality,
isect_memberEquality,
functionEquality,
universeEquality,
setElimination,
intWeakElimination,
independent_isectElimination,
dependent_pairFormation,
int_eqEquality,
intEquality,
voidElimination,
voidEquality,
independent_pairFormation,
computeAll,
independent_pairEquality,
addEquality,
unionElimination,
applyLambdaEquality,
hypothesis_subsumption,
promote_hyp,
impliesFunctionality,
pointwiseFunctionality,
baseApply,
closedConclusion
Latex:
\mforall{}[A:Type]. \mforall{}[f:A {}\mrightarrow{} \mBbbN{}]. \mforall{}[ba:bag(A)]. (bag-sum(ba;x.f[x]) \mmember{} \mBbbN{})
Date html generated:
2017_10_01-AM-08_47_56
Last ObjectModification:
2017_07_26-PM-04_32_15
Theory : bags
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