Nuprl Lemma : sv-bag-only-append
∀[A:Type]. ∀[as,bs:bag(A)].
(sv-bag-only(as + bs) = sv-bag-only(as) ∈ A) supposing
(similar-bags(A;as;bs) and
0 < #(as) and
single-valued-bag(as;A) and
single-valued-bag(bs;A))
Proof
Definitions occuring in Statement :
sv-bag-only: sv-bag-only(b),
single-valued-bag: single-valued-bag(b;T),
similar-bags: similar-bags(A;as;bs),
bag-size: #(bs),
bag-append: as + bs,
bag: bag(T),
less_than: a < b,
uimplies: b supposing a,
uall: ∀[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],
implies: P ⇒ Q,
prop: ℙ,
subtype_rel: A ⊆r B,
nat: ℕ,
true: True,
and: P ∧ Q,
cand: A c∧ B,
top: Top,
all: ∀x:A. B[x],
decidable: Dec(P),
or: P ∨ Q,
guard: {T},
ge: i ≥ j ,
satisfiable_int_formula: satisfiable_int_formula(fmla),
false: False,
not: ¬A,
sv-bag-only: sv-bag-only(b),
bag-append: as + bs,
bag-size: #(bs),
listp: A List+,
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q
Lemmas referenced :
bag_to_squash_list,
similar-bags_wf,
less_than_wf,
bag-size_wf,
single-valued-bag_wf,
bag_wf,
nat_wf,
single-valued-bag-append,
bag-size-append,
decidable__lt,
nat_properties,
satisfiable-full-omega-tt,
intformand_wf,
intformle_wf,
itermConstant_wf,
itermVar_wf,
intformnot_wf,
intformless_wf,
itermAdd_wf,
int_formula_prop_and_lemma,
int_formula_prop_le_lemma,
int_term_value_constant_lemma,
int_term_value_var_lemma,
int_formula_prop_not_lemma,
int_formula_prop_less_lemma,
int_term_value_add_lemma,
int_formula_prop_wf,
hd-append-sq,
subtype_rel_list,
top_wf,
length_wf,
hd_wf,
listp_properties,
list-subtype-bag,
squash_wf,
true_wf,
equal_wf,
sv-bag-only_wf,
bag-append_wf,
iff_weakening_equal
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
introduction,
cut,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
hypothesisEquality,
imageElimination,
productElimination,
independent_functionElimination,
hypothesis,
cumulativity,
sqequalRule,
isect_memberEquality,
axiomEquality,
because_Cache,
equalityTransitivity,
equalitySymmetry,
natural_numberEquality,
applyEquality,
lambdaEquality,
setElimination,
rename,
universeEquality,
independent_isectElimination,
independent_pairFormation,
voidElimination,
voidEquality,
dependent_functionElimination,
addEquality,
unionElimination,
applyLambdaEquality,
dependent_pairFormation,
int_eqEquality,
intEquality,
computeAll,
lambdaFormation,
dependent_set_memberEquality,
functionEquality,
imageMemberEquality,
baseClosed,
productEquality
Latex:
\mforall{}[A:Type]. \mforall{}[as,bs:bag(A)].
(sv-bag-only(as + bs) = sv-bag-only(as)) supposing
(similar-bags(A;as;bs) and
0 < \#(as) and
single-valued-bag(as;A) and
single-valued-bag(bs;A))
Date html generated:
2017_10_01-AM-08_55_52
Last ObjectModification:
2017_07_26-PM-04_37_57
Theory : bags
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