Nuprl Lemma : bag-double-summation
∀[R:Type]. ∀[add:R ⟶ R ⟶ R]. ∀[zero:R].
∀[T,A,B:Type]. ∀[f:T ⟶ bag(A)]. ∀[g:T ⟶ B]. ∀[h:B ⟶ A ⟶ R]. ∀[b:bag(T)].
(Σ(x∈b). Σ(y∈f[x]). h[g[x];y] = Σ(p∈⋃x∈b.bag-map(λy.<g[x], y>;f[x])). h[fst(p);snd(p)] ∈ R)
supposing IsMonoid(R;add;zero) ∧ Comm(R;add)
Proof
Definitions occuring in Statement :
bag-summation: Σ(x∈b). f[x],
bag-combine: ⋃x∈bs.f[x],
bag-map: bag-map(f;bs),
bag: bag(T),
comm: Comm(T;op),
uimplies: b supposing a,
uall: ∀[x:A]. B[x],
so_apply: x[s1;s2],
so_apply: x[s],
pi1: fst(t),
pi2: snd(t),
and: P ∧ Q,
lambda: λx.A[x],
function: x:A ⟶ B[x],
pair: <a, b>,
universe: Type,
equal: s = t ∈ T,
monoid_p: IsMonoid(T;op;id)
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
uimplies: b supposing a,
and: P ∧ Q,
bag: bag(T),
quotient: x,y:A//B[x; y],
all: ∀x:A. B[x],
implies: P ⇒ Q,
so_lambda: λ2x y.t[x; y],
so_apply: x[s1;s2],
nat: ℕ,
false: False,
ge: i ≥ j ,
not: ¬A,
satisfiable_int_formula: satisfiable_int_formula(fmla),
exists: ∃x:A. B[x],
top: Top,
prop: ℙ,
guard: {T},
int_seg: {i..j-},
lelt: i ≤ j < k,
le: A ≤ B,
decidable: Dec(P),
or: P ∨ Q,
subtype_rel: A ⊆r B,
so_lambda: λ2x.t[x],
so_apply: x[s],
sq_type: SQType(T),
less_than: a < b,
squash: ↓T,
empty-bag: {},
cons: [a / b],
assert: ↑b,
ifthenelse: if b then t else f fi ,
bfalse: ff,
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q,
uiff: uiff(P;Q),
int_iseg: {i...j},
cand: A c∧ B,
single-bag: {x},
bag-append: as + bs,
pi1: fst(t),
pi2: snd(t),
true: True,
infix_ap: x f y,
monoid_p: IsMonoid(T;op;id)
Lemmas referenced :
quotient-member-eq,
list_wf,
permutation_wf,
permutation-equiv,
nat_properties,
full-omega-unsat,
intformand_wf,
intformle_wf,
itermConstant_wf,
itermVar_wf,
intformless_wf,
istype-int,
int_formula_prop_and_lemma,
istype-void,
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,
istype-less_than,
int_seg_properties,
int_seg_wf,
subtract-1-ge-0,
decidable__equal_int,
subtract_wf,
subtype_base_sq,
set_subtype_base,
lelt_wf,
int_subtype_base,
intformnot_wf,
intformeq_wf,
itermSubtract_wf,
int_formula_prop_not_lemma,
int_formula_prop_eq_lemma,
int_term_value_subtract_lemma,
decidable__le,
decidable__lt,
istype-le,
subtype_rel_self,
non_neg_length,
length_wf,
decidable__assert,
null_wf,
list-cases,
product_subtype_list,
null_cons_lemma,
last-lemma-sq,
pos_length,
iff_transitivity,
not_wf,
equal-wf-T-base,
assert_wf,
bnot_wf,
iff_weakening_uiff,
assert_of_null,
istype-assert,
nil_wf,
length_of_nil_lemma,
assert_of_bnot,
firstn_wf,
length_firstn,
itermAdd_wf,
int_term_value_add_lemma,
istype-nat,
length_wf_nat,
equal_wf,
bag-summation_wf,
bag-combine_wf,
list-subtype-bag,
bag-map_wf,
bag_wf,
monoid_p_wf,
comm_wf,
bag-append_wf,
single-bag_wf,
last_wf,
pi1_wf,
pi2_wf,
infix_ap_wf,
bag-summation-empty,
bag-combine-empty-left,
squash_wf,
true_wf,
bag-combine-append-left,
iff_weakening_equal,
bag-summation-append,
bag-subtype-list,
bag-summation-map,
istype-universe,
bag-summation-single,
bag-combine-single-left
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation_alt,
introduction,
cut,
sqequalHypSubstitution,
productElimination,
thin,
pointwiseFunctionalityForEquality,
hypothesisEquality,
hypothesis,
sqequalRule,
pertypeElimination,
promote_hyp,
equalityTransitivity,
equalitySymmetry,
inhabitedIsType,
lambdaFormation_alt,
rename,
extract_by_obid,
isectElimination,
lambdaEquality_alt,
independent_isectElimination,
dependent_functionElimination,
independent_functionElimination,
setElimination,
intWeakElimination,
natural_numberEquality,
approximateComputation,
dependent_pairFormation_alt,
int_eqEquality,
isect_memberEquality_alt,
voidElimination,
independent_pairFormation,
universeIsType,
axiomEquality,
functionIsTypeImplies,
unionElimination,
applyEquality,
instantiate,
cumulativity,
intEquality,
applyLambdaEquality,
dependent_set_memberEquality_alt,
because_Cache,
productIsType,
hypothesis_subsumption,
imageElimination,
baseClosed,
functionIsType,
equalityIstype,
addEquality,
hyp_replacement,
productEquality,
independent_pairEquality,
sqequalBase,
isectIsTypeImplies,
closedConclusion,
imageMemberEquality,
universeEquality
Latex:
\mforall{}[R:Type]. \mforall{}[add:R {}\mrightarrow{} R {}\mrightarrow{} R]. \mforall{}[zero:R].
\mforall{}[T,A,B:Type]. \mforall{}[f:T {}\mrightarrow{} bag(A)]. \mforall{}[g:T {}\mrightarrow{} B]. \mforall{}[h:B {}\mrightarrow{} A {}\mrightarrow{} R]. \mforall{}[b:bag(T)].
(\mSigma{}(x\mmember{}b). \mSigma{}(y\mmember{}f[x]). h[g[x];y] = \mSigma{}(p\mmember{}\mcup{}x\mmember{}b.bag-map(\mlambda{}y.<g[x], y>f[x])). h[fst(p);snd(p)])
supposing IsMonoid(R;add;zero) \mwedge{} Comm(R;add)
Date html generated:
2019_10_15-AM-11_00_57
Last ObjectModification:
2019_08_08-PM-06_16_03
Theory : bags
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