Nuprl Lemma : bag-member-parts'
∀[T:Type]
∀[eq:EqDecider(T)]. ∀[x:T]. ∀[bs:bag(T)]. ∀[L:bag(T) List+].
uiff(L ↓∈ bag-parts'(eq;bs;x);(¬x ↓∈ hd(L)) ∧ (∀x∈tl(L).¬(x = {} ∈ bag(T))) ∧ (bag-union(L) = bs ∈ bag(T)))
supposing valueall-type(T)
Proof
Definitions occuring in Statement :
bag-parts': bag-parts'(eq;bs;x),
bag-member: x ↓∈ bs,
bag-union: bag-union(bbs),
empty-bag: {},
bag: bag(T),
l_all: (∀x∈L.P[x]),
listp: A List+,
hd: hd(l),
tl: tl(l),
deq: EqDecider(T),
valueall-type: valueall-type(T),
uiff: uiff(P;Q),
uimplies: b supposing a,
uall: ∀[x:A]. B[x],
not: ¬A,
and: P ∧ Q,
universe: Type,
equal: s = t ∈ T
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
uimplies: b supposing a,
bag-parts': bag-parts'(eq;bs;x),
all: ∀x:A. B[x],
implies: P ⇒ Q,
bool: 𝔹,
unit: Unit,
it: ⋅,
btrue: tt,
uiff: uiff(P;Q),
and: P ∧ Q,
ifthenelse: if b then t else f fi ,
bfalse: ff,
exists: ∃x:A. B[x],
prop: ℙ,
or: P ∨ Q,
sq_type: SQType(T),
guard: {T},
bnot: ¬bb,
assert: ↑b,
false: False,
not: ¬A,
listp: A List+,
l_all: (∀x∈L.P[x]),
int_seg: {i..j-},
lelt: i ≤ j < k,
decidable: Dec(P),
satisfiable_int_formula: satisfiable_int_formula(fmla),
top: Top,
less_than: a < b,
squash: ↓T,
bag-member: x ↓∈ bs,
so_lambda: λ2x.t[x],
so_apply: x[s],
subtype_rel: A ⊆r B,
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q,
ge: i ≥ j ,
le: A ≤ B,
less_than': less_than'(a;b),
true: True,
cons: [a / b],
cand: A c∧ B,
bag-union: bag-union(bbs),
concat: concat(ll),
reduce: reduce(f;k;as),
list_ind: list_ind,
append: as @ bs,
empty-bag: {},
nil: [],
length: ||as||,
bag-append: as + bs,
nat: ℕ,
callbyvalueall: callbyvalueall,
has-value: (a)↓,
has-valueall: has-valueall(a),
sq_or: a ↓∨ b,
sq_stable: SqStable(P),
subtract: n - m,
cons-bag: x.b,
rev_uimplies: rev_uimplies(P;Q),
tl: tl(l),
pi2: snd(t)
Lemmas referenced :
bag-null_wf,
bool_wf,
eqtt_to_assert,
assert-bag-null,
eqff_to_assert,
equal_wf,
bool_cases_sqequal,
subtype_base_sq,
bool_subtype_base,
assert-bnot,
equal-wf-T-base,
bag_wf,
bag-member_wf,
hd_wf,
listp_properties,
select_wf,
int_seg_properties,
decidable__le,
satisfiable-full-omega-tt,
intformand_wf,
intformnot_wf,
intformle_wf,
itermConstant_wf,
itermVar_wf,
int_formula_prop_and_lemma,
int_formula_prop_not_lemma,
int_formula_prop_le_lemma,
int_term_value_constant_lemma,
int_term_value_var_lemma,
int_formula_prop_wf,
decidable__lt,
length_wf,
tl_wf,
intformless_wf,
int_formula_prop_less_lemma,
int_seg_wf,
listp_wf,
bag-parts'_wf,
not_wf,
l_all_wf2,
l_member_wf,
bag-union_wf,
subtype_rel_set,
list_wf,
less_than_wf,
list-subtype-bag,
deq_wf,
valueall-type_wf,
equal-empty-bag,
iff_weakening_uiff,
single-bag_wf,
cons_wf_listp,
nil_wf,
bag-member-single,
uiff_wf,
list-cases,
length_of_nil_lemma,
product_subtype_list,
length_of_cons_lemma,
reduce_hd_cons_lemma,
ge_wf,
squash_wf,
true_wf,
iff_weakening_equal,
reduce_tl_cons_lemma,
bag-member-empty-iff,
l_all_nil,
empty-bag_wf,
reduce_cons_lemma,
reduce_nil_lemma,
bag-append-empty,
bag-subtype-list,
bag-size_wf,
bag-size-append,
subtype_rel_self,
bag_size_empty_lemma,
nat_wf,
non_neg_length,
itermAdd_wf,
int_term_value_add_lemma,
length_wf_nat,
nat_properties,
intformeq_wf,
int_formula_prop_eq_lemma,
bag-size-zero,
l_all_cons,
valueall-type-has-valueall,
bag-valueall-type,
set-valueall-type,
list-valueall-type,
bag-parts_wf,
evalall-reduce,
bag-member-append,
bag-map_wf,
bag-filter_wf,
eq_int_wf,
bag-count_wf,
subtype_rel_bag,
assert_wf,
bag-append_wf,
sq_or_wf,
exists_wf,
or_wf,
bag-member-map,
l_all_iff,
all_wf,
sq_stable__and,
sq_stable__not,
sq_stable__all,
sq_stable__equal,
bag-member-parts,
bag-member-filter,
assert_of_eq_int,
bag-member-count,
reduce_tl_nil_lemma,
decidable__assert,
null_wf3,
subtype_rel_list,
top_wf,
assert_of_null,
false_wf,
not-lt-2,
condition-implies-le,
minus-add,
minus-one-mul,
zero-add,
minus-one-mul-top,
add-commutes,
add_functionality_wrt_le,
add-associates,
add-zero,
le-add-cancel,
cons_wf,
bag_union_cons_lemma,
empty_bag_append_lemma,
bag-count-is-zero
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
introduction,
cut,
extract_by_obid,
sqequalHypSubstitution,
dependent_functionElimination,
thin,
cumulativity,
hypothesisEquality,
hypothesis,
lambdaFormation,
unionElimination,
equalityElimination,
isectElimination,
equalityTransitivity,
equalitySymmetry,
productElimination,
independent_isectElimination,
because_Cache,
sqequalRule,
dependent_pairFormation,
promote_hyp,
instantiate,
independent_functionElimination,
voidElimination,
baseClosed,
independent_pairEquality,
isect_memberEquality,
lambdaEquality,
setElimination,
rename,
natural_numberEquality,
int_eqEquality,
intEquality,
voidEquality,
independent_pairFormation,
computeAll,
imageElimination,
axiomEquality,
imageMemberEquality,
productEquality,
setEquality,
applyEquality,
universeEquality,
addLevel,
hypothesis_subsumption,
applyLambdaEquality,
dependent_set_memberEquality,
hyp_replacement,
addEquality,
callbyvalueReduce,
orFunctionality,
functionEquality,
inlFormation,
minusEquality,
inrFormation
Latex:
\mforall{}[T:Type]
\mforall{}[eq:EqDecider(T)]. \mforall{}[x:T]. \mforall{}[bs:bag(T)]. \mforall{}[L:bag(T) List\msupplus{}].
uiff(L \mdownarrow{}\mmember{} bag-parts'(eq;bs;x);(\mneg{}x \mdownarrow{}\mmember{} hd(L)) \mwedge{} (\mforall{}x\mmember{}tl(L).\mneg{}(x = \{\})) \mwedge{} (bag-union(L) = bs))
supposing valueall-type(T)
Date html generated:
2018_05_21-PM-09_51_27
Last ObjectModification:
2017_07_26-PM-06_31_30
Theory : bags_2
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