Nuprl Lemma : bag-union-is-single
∀[T:Type]. ∀[x:T].
∀bbs:bag(bag(T))
uiff(bag-union(bbs) = {x} ∈ bag(T);↓∃bbs':bag(bag(T))
((bbs = {x}.bbs' ∈ bag(bag(T))) ∧ (bag-union(bbs') = {} ∈ bag(T))))
Proof
Definitions occuring in Statement :
bag-union: bag-union(bbs),
cons-bag: x.b,
single-bag: {x},
empty-bag: {},
bag: bag(T),
uiff: uiff(P;Q),
uall: ∀[x:A]. B[x],
all: ∀x:A. B[x],
exists: ∃x:A. B[x],
squash: ↓T,
and: P ∧ Q,
universe: Type,
equal: s = t ∈ T
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
all: ∀x:A. B[x],
uiff: uiff(P;Q),
and: P ∧ Q,
uimplies: b supposing a,
squash: ↓T,
prop: ℙ,
exists: ∃x:A. B[x],
nat: ℕ,
implies: P ⇒ Q,
false: False,
ge: i ≥ j ,
not: ¬A,
satisfiable_int_formula: satisfiable_int_formula(fmla),
top: Top,
or: P ∨ Q,
cons: [a / b],
le: A ≤ B,
less_than': less_than'(a;b),
colength: colength(L),
nil: [],
it: ⋅,
guard: {T},
so_lambda: λ2x.t[x],
so_apply: x[s],
sq_type: SQType(T),
less_than: a < b,
so_lambda: λ2x y.t[x; y],
so_apply: x[s1;s2],
decidable: Dec(P),
subtype_rel: A ⊆r B,
single-bag: {x},
bag-union: bag-union(bbs),
concat: concat(ll),
bag: bag(T),
quotient: x,y:A//B[x; y],
cand: A c∧ B,
true: True,
bag-append: as + bs,
cons-bag: x.b,
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q,
respects-equality: respects-equality(S;T),
reduce: reduce(f;k;as),
list_ind: list_ind,
append: as @ bs,
empty-bag: {},
so_lambda: so_lambda(x,y,z.t[x; y; z]),
so_apply: x[s1;s2;s3]
Lemmas referenced :
bag_wf,
bag-union_wf,
single-bag_wf,
squash_wf,
equal_wf,
cons-bag_wf,
equal-wf-T-base,
istype-universe,
bag_to_squash_list,
nat_properties,
full-omega-unsat,
intformand_wf,
intformle_wf,
itermConstant_wf,
itermVar_wf,
intformless_wf,
istype-int,
int_formula_prop_and_lemma,
istype-void,
int_formula_prop_le_lemma,
int_term_value_constant_lemma,
int_term_value_var_lemma,
int_formula_prop_less_lemma,
int_formula_prop_wf,
ge_wf,
istype-less_than,
list-cases,
product_subtype_list,
colength-cons-not-zero,
colength_wf_list,
istype-false,
istype-le,
list_wf,
subtract-1-ge-0,
subtype_base_sq,
intformeq_wf,
int_formula_prop_eq_lemma,
set_subtype_base,
int_subtype_base,
spread_cons_lemma,
decidable__equal_int,
subtract_wf,
intformnot_wf,
itermSubtract_wf,
itermAdd_wf,
int_formula_prop_not_lemma,
int_term_value_subtract_lemma,
int_term_value_add_lemma,
decidable__le,
le_wf,
cons_wf,
list-subtype-bag,
istype-nat,
reduce_nil_lemma,
permutation-length,
length_of_nil_lemma,
length_of_cons_lemma,
permutation_wf,
concat-cons2,
bag-append-is-single-iff,
true_wf,
subtype_rel_self,
iff_weakening_equal,
subtype-respects-equality,
list_ind_cons_lemma,
list_ind_nil_lemma,
bag-append-assoc-comm,
bag-append-empty,
bag-subtype-list,
bag-append_wf,
nil_wf
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation_alt,
introduction,
cut,
lambdaFormation_alt,
independent_pairFormation,
hypothesis,
sqequalHypSubstitution,
imageElimination,
sqequalRule,
imageMemberEquality,
hypothesisEquality,
thin,
baseClosed,
equalityIstype,
universeIsType,
extract_by_obid,
isectElimination,
dependent_functionElimination,
productEquality,
lambdaEquality_alt,
productElimination,
independent_pairEquality,
isect_memberEquality_alt,
isectIsTypeImplies,
inhabitedIsType,
axiomEquality,
functionIsTypeImplies,
instantiate,
universeEquality,
promote_hyp,
equalitySymmetry,
hyp_replacement,
applyLambdaEquality,
equalityTransitivity,
rename,
setElimination,
intWeakElimination,
natural_numberEquality,
independent_isectElimination,
approximateComputation,
independent_functionElimination,
dependent_pairFormation_alt,
int_eqEquality,
voidElimination,
unionElimination,
sqequalBase,
hypothesis_subsumption,
because_Cache,
dependent_set_memberEquality_alt,
baseApply,
closedConclusion,
applyEquality,
intEquality,
pertypeElimination,
cumulativity,
productIsType
Latex:
\mforall{}[T:Type]. \mforall{}[x:T].
\mforall{}bbs:bag(bag(T))
uiff(bag-union(bbs) = \{x\};\mdownarrow{}\mexists{}bbs':bag(bag(T)). ((bbs = \{x\}.bbs') \mwedge{} (bag-union(bbs') = \{\})))
Date html generated:
2019_10_15-AM-11_00_22
Last ObjectModification:
2018_11_30-AM-09_57_39
Theory : bags
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