Nuprl Lemma : bag-member-splits
∀[T:Type]. ∀[as,bs,cs:bag(T)].  uiff(<as, bs> ↓∈ bag-splits(cs);(as + bs) = cs ∈ bag(T))
Proof
Definitions occuring in Statement : 
bag-splits: bag-splits(b)
, 
bag-member: x ↓∈ bs
, 
bag-append: as + bs
, 
bag: bag(T)
, 
uiff: uiff(P;Q)
, 
uall: ∀[x:A]. B[x]
, 
pair: <a, b>
, 
product: x:A × B[x]
, 
universe: Type
, 
equal: s = t ∈ T
Definitions unfolded in proof : 
uall: ∀[x:A]. B[x]
, 
member: t ∈ T
, 
uiff: uiff(P;Q)
, 
and: P ∧ Q
, 
uimplies: b supposing a
, 
all: ∀x:A. B[x]
, 
prop: ℙ
, 
bag-member: x ↓∈ bs
, 
squash: ↓T
, 
subtype_rel: A ⊆r B
, 
true: True
, 
quotient: x,y:A//B[x; y]
, 
bag: bag(T)
, 
istype: istype(T)
, 
implies: P 
⇒ Q
, 
so_apply: x[s1;s2]
, 
so_lambda: λ2x y.t[x; y]
, 
iff: P 
⇐⇒ Q
, 
guard: {T}
, 
cand: A c∧ B
, 
exists: ∃x:A. B[x]
, 
decidable: Dec(P)
, 
less_than: a < b
, 
sq_type: SQType(T)
, 
so_apply: x[s]
, 
so_lambda: λ2x.t[x]
, 
it: ⋅
, 
nil: []
, 
colength: colength(L)
, 
less_than': less_than'(a;b)
, 
le: A ≤ B
, 
cons: [a / b]
, 
so_apply: x[s1;s2;s3]
, 
so_lambda: so_lambda3, 
bag-splits: bag-splits(b)
, 
or: P ∨ Q
, 
top: Top
, 
satisfiable_int_formula: satisfiable_int_formula(fmla)
, 
not: ¬A
, 
ge: i ≥ j 
, 
false: False
, 
nat: ℕ
, 
pi2: snd(t)
, 
pi1: fst(t)
, 
single-bag: {x}
, 
empty-bag: {}
, 
append: as @ bs
, 
bag-append: as + bs
, 
bag-map: bag-map(f;bs)
, 
rev_implies: P 
⇐ Q
, 
remove-nth: remove-nth(n;L)
, 
lelt: i ≤ j < k
, 
int_seg: {i..j-}
, 
int_iseg: {i...j}
, 
subtract: n - m
, 
tl: tl(l)
, 
lt_int: i <z j
, 
le_int: i ≤z j
, 
nth_tl: nth_tl(n;as)
, 
assert: ↑b
, 
bnot: ¬bb
, 
bfalse: ff
, 
ifthenelse: if b then t else f fi 
, 
btrue: tt
, 
unit: Unit
, 
bool: 𝔹
, 
respects-equality: respects-equality(S;T)
, 
select: L[n]
, 
add-nth: add-nth(n;x;L)
Lemmas referenced : 
bag-member_wf, 
bag_wf, 
bag-splits_wf, 
bag-append_wf, 
istype-universe, 
subtype_rel_self, 
true_wf, 
squash_wf, 
equal_wf, 
permutation_weakening, 
permutation_wf, 
list_wf, 
list-subtype-bag, 
permutation-equiv, 
quotient-member-eq, 
l_member_wf, 
member-permutation, 
bag-splits_wf_list, 
istype-nat, 
cons_wf, 
list_ind_cons_lemma, 
le_wf, 
decidable__le, 
int_term_value_add_lemma, 
int_term_value_subtract_lemma, 
int_formula_prop_not_lemma, 
itermAdd_wf, 
itermSubtract_wf, 
intformnot_wf, 
subtract_wf, 
decidable__equal_int, 
spread_cons_lemma, 
int_subtype_base, 
set_subtype_base, 
int_formula_prop_eq_lemma, 
intformeq_wf, 
subtype_base_sq, 
subtract-1-ge-0, 
istype-le, 
istype-false, 
colength_wf_list, 
colength-cons-not-zero, 
product_subtype_list, 
nil_wf, 
list_ind_nil_lemma, 
list-cases, 
istype-less_than, 
ge_wf, 
int_formula_prop_wf, 
int_formula_prop_less_lemma, 
int_term_value_var_lemma, 
int_term_value_constant_lemma, 
int_formula_prop_le_lemma, 
istype-void, 
int_formula_prop_and_lemma, 
istype-int, 
intformless_wf, 
itermVar_wf, 
itermConstant_wf, 
intformle_wf, 
intformand_wf, 
full-omega-unsat, 
nat_properties, 
empty-bag_wf, 
member_singleton, 
equal-empty-bag, 
empty_bag_append_lemma, 
bag-append-assoc-comm, 
bag-append-assoc, 
single-bag_wf, 
map_wf, 
member_append, 
member_map, 
bag_to_squash_list, 
append_wf, 
equal-wf-base, 
permutation_inversion, 
list_induction, 
append_is_nil, 
permutation-nil-iff, 
iff_weakening_equal, 
trivial-equal, 
permutation-cons, 
pi1_wf, 
pi2_wf, 
false_wf, 
add-is-int-iff, 
add_nat_wf, 
decidable__lt, 
non_neg_length, 
length-append, 
remove-nth_wf, 
length_wf_nat, 
length_of_cons_lemma, 
top_wf, 
subtype_rel_list, 
length_append, 
length_wf, 
firstn-append, 
nth_tl-append, 
bfalse_wf, 
btrue_wf, 
le_weakening2, 
length_firstn, 
length_firstn_eq, 
nth_tl_wf, 
firstn_wf, 
ifthenelse_wf, 
general-append-cancellation, 
firstn_all, 
le_int_wf, 
less_than_wf, 
assert_wf, 
bool_wf, 
iff_weakening_uiff, 
assert-bnot, 
bool_subtype_base, 
bool_cases_sqequal, 
eqff_to_assert, 
assert_of_le_int, 
assert_of_lt_int, 
eqtt_to_assert, 
lt_int_wf, 
respects-equality-product, 
respects-equality-list-bag, 
respects-equality-trivial, 
istype-base, 
bnot_wf, 
not_wf, 
istype-assert, 
bool_cases, 
iff_transitivity, 
assert_of_bnot, 
append_assoc_sq, 
nil-append, 
nth_tl_is_nil, 
minus-one-mul, 
add-mul-special, 
zero-mul, 
bag-append-comm, 
select_wf, 
select-append, 
add-remove-nth, 
firstn_append, 
firstn_firstn, 
add-zero, 
add-commutes, 
add-associates, 
add-swap, 
nth_tl_nth_tl, 
subtype-respects-equality, 
general_arith_equation1, 
first0, 
append-nil, 
zero-add, 
minus-minus, 
minus-add
Rules used in proof : 
sqequalSubstitution, 
sqequalTransitivity, 
computationStep, 
sqequalReflexivity, 
isect_memberFormation_alt, 
introduction, 
cut, 
independent_pairFormation, 
hypothesis, 
universeIsType, 
extract_by_obid, 
sqequalHypSubstitution, 
isectElimination, 
thin, 
productEquality, 
hypothesisEquality, 
independent_pairEquality, 
dependent_functionElimination, 
sqequalRule, 
imageElimination, 
imageMemberEquality, 
baseClosed, 
equalityIstype, 
inhabitedIsType, 
productElimination, 
isect_memberEquality_alt, 
axiomEquality, 
isectIsTypeImplies, 
instantiate, 
universeEquality, 
natural_numberEquality, 
equalityTransitivity, 
lambdaEquality_alt, 
applyEquality, 
equalitySymmetry, 
sqequalBase, 
productIsType, 
pertypeElimination, 
pointwiseFunctionality, 
independent_isectElimination, 
because_Cache, 
lambdaFormation_alt, 
promote_hyp, 
applyLambdaEquality, 
hyp_replacement, 
independent_functionElimination, 
intEquality, 
closedConclusion, 
baseApply, 
dependent_set_memberEquality_alt, 
hypothesis_subsumption, 
unionElimination, 
functionIsTypeImplies, 
voidElimination, 
int_eqEquality, 
dependent_pairFormation_alt, 
approximateComputation, 
intWeakElimination, 
rename, 
setElimination, 
functionEquality, 
Error :memTop, 
functionIsType, 
voidEquality, 
inlFormation_alt, 
inrFormation_alt, 
addEquality, 
cumulativity, 
equalityElimination, 
multiplyEquality, 
minusEquality
Latex:
\mforall{}[T:Type].  \mforall{}[as,bs,cs:bag(T)].    uiff(<as,  bs>  \mdownarrow{}\mmember{}  bag-splits(cs);(as  +  bs)  =  cs)
Date html generated:
2020_05_20-AM-08_02_34
Last ObjectModification:
2019_12_31-PM-10_16_27
Theory : bags
Home
Index