Nuprl Lemma : sv-bag-only-combine
∀[A,B:Type]. ∀[b:bag(A)]. ∀[f:A ⟶ bag(B)].
(sv-bag-only(⋃x∈b.f[x]) = sv-bag-only(f[sv-bag-only(b)]) ∈ B) supposing
((∀a:A. 0 < #(f[a])) and
(∀a:A. single-valued-bag(f[a];B)) and
0 < #(b) and
single-valued-bag(b;A))
Proof
Definitions occuring in Statement :
sv-bag-only: sv-bag-only(b),
single-valued-bag: single-valued-bag(b;T),
bag-combine: ⋃x∈bs.f[x],
bag-size: #(bs),
bag: bag(T),
less_than: a < b,
uimplies: b supposing a,
uall: ∀[x:A]. B[x],
so_apply: x[s],
all: ∀x:A. B[x],
function: x:A ⟶ B[x],
natural_number: $n,
universe: Type,
equal: s = t ∈ T
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
uimplies: b supposing a,
squash: ↓T,
exists: ∃x:A. B[x],
so_apply: x[s],
true: True,
and: P ∧ Q,
cand: A c∧ B,
so_lambda: λ2x.t[x],
all: ∀x:A. B[x],
subtype_rel: A ⊆r B,
nat: ℕ,
decidable: Dec(P),
or: P ∨ Q,
less_than: a < b,
le: A ≤ B,
satisfiable_int_formula: satisfiable_int_formula(fmla),
false: False,
implies: P ⇒ Q,
not: ¬A,
top: Top,
prop: ℙ,
guard: {T},
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q,
single-valued-bag: single-valued-bag(b;T),
sq_or: a ↓∨ b,
similar-bags: similar-bags(A;as;bs),
sq_stable: SqStable(P),
uiff: uiff(P;Q),
rev_uimplies: rev_uimplies(P;Q)
Lemmas referenced :
bag-member-sv-bag-only,
bag-member-implies-hd-append,
sv-bag-only_wf,
single-valued-bag-combine,
bag-combine-size-bound2,
decidable__lt,
bag-size_wf,
bag-combine_wf,
nat_wf,
satisfiable-full-omega-tt,
intformand_wf,
intformnot_wf,
intformless_wf,
itermConstant_wf,
itermVar_wf,
intformle_wf,
int_formula_prop_and_lemma,
int_formula_prop_not_lemma,
int_formula_prop_less_lemma,
int_term_value_constant_lemma,
int_term_value_var_lemma,
int_formula_prop_le_lemma,
int_formula_prop_wf,
single-bag_wf,
single-valued-bag_wf,
squash_wf,
true_wf,
iff_weakening_equal,
bag-member_wf,
bag-combine-single-left,
all_wf,
less_than_wf,
bag_wf,
equal_wf,
bag-combine-append-left,
sv-bag-only-append,
bag-member-append,
bag-size-bag-member,
sq_stable__bag-member,
bag-member-size,
bag-member-combine
Rules used in proof :
cut,
introduction,
extract_by_obid,
sqequalHypSubstitution,
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isectElimination,
thin,
hypothesisEquality,
independent_isectElimination,
hypothesis,
because_Cache,
cumulativity,
imageElimination,
productElimination,
applyEquality,
functionExtensionality,
natural_numberEquality,
sqequalRule,
lambdaEquality,
independent_pairFormation,
dependent_functionElimination,
equalityTransitivity,
equalitySymmetry,
setElimination,
rename,
unionElimination,
dependent_pairFormation,
int_eqEquality,
intEquality,
isect_memberEquality,
voidElimination,
voidEquality,
computeAll,
imageMemberEquality,
baseClosed,
independent_functionElimination,
functionEquality,
universeEquality,
isect_memberFormation,
axiomEquality,
productEquality,
lambdaFormation,
inrFormation,
hyp_replacement,
applyLambdaEquality
Latex:
\mforall{}[A,B:Type]. \mforall{}[b:bag(A)]. \mforall{}[f:A {}\mrightarrow{} bag(B)].
(sv-bag-only(\mcup{}x\mmember{}b.f[x]) = sv-bag-only(f[sv-bag-only(b)])) supposing
((\mforall{}a:A. 0 < \#(f[a])) and
(\mforall{}a:A. single-valued-bag(f[a];B)) and
0 < \#(b) and
single-valued-bag(b;A))
Date html generated:
2017_10_01-AM-08_55_59
Last ObjectModification:
2017_07_26-PM-04_38_01
Theory : bags
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