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
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