Nuprl Lemma : bag-append-is-empty
∀[T:Type]. ∀as,bs:bag(T). uiff((as + bs) = {} ∈ bag(T);(as = {} ∈ bag(T)) ∧ (bs = {} ∈ bag(T)))
Proof
Definitions occuring in Statement :
bag-append: as + bs,
empty-bag: {},
bag: bag(T),
uiff: uiff(P;Q),
uall: ∀[x:A]. B[x],
all: ∀x:A. B[x],
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,
true: True,
subtype_rel: A ⊆r B,
top: Top,
bag-size: #(bs),
exists: ∃x:A. B[x],
prop: ℙ,
nat: ℕ,
or: P ∨ Q,
empty-bag: {},
cons: [a / b],
ge: i ≥ j ,
le: A ≤ B,
satisfiable_int_formula: satisfiable_int_formula(fmla),
false: False,
implies: P ⇒ Q,
not: ¬A
Lemmas referenced :
bag-size_wf,
bag_size_empty_lemma,
bag-size-append,
bag_to_squash_list,
equal-wf-T-base,
nat_wf,
bag_wf,
bag-append_wf,
list-subtype-bag,
empty_bag_append_lemma,
list-cases,
empty-bag_wf,
product_subtype_list,
length_of_cons_lemma,
non_neg_length,
satisfiable-full-omega-tt,
intformand_wf,
intformle_wf,
itermConstant_wf,
itermVar_wf,
intformeq_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_eq_lemma,
int_term_value_add_lemma,
int_formula_prop_wf
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
introduction,
cut,
lambdaFormation,
independent_pairFormation,
applyEquality,
thin,
lambdaEquality,
sqequalHypSubstitution,
imageElimination,
extract_by_obid,
isectElimination,
because_Cache,
hypothesis,
natural_numberEquality,
sqequalRule,
imageMemberEquality,
hypothesisEquality,
baseClosed,
isect_memberEquality,
voidElimination,
voidEquality,
productElimination,
promote_hyp,
equalitySymmetry,
hyp_replacement,
Error :applyLambdaEquality,
intEquality,
addEquality,
cumulativity,
setElimination,
rename,
independent_isectElimination,
independent_pairEquality,
axiomEquality,
dependent_functionElimination,
equalityTransitivity,
productEquality,
universeEquality,
unionElimination,
hypothesis_subsumption,
dependent_pairFormation,
int_eqEquality,
computeAll
Latex:
\mforall{}[T:Type]. \mforall{}as,bs:bag(T). uiff((as + bs) = \{\};(as = \{\}) \mwedge{} (bs = \{\}))
Date html generated:
2016_10_25-AM-10_23_20
Last ObjectModification:
2016_07_12-AM-06_40_10
Theory : bags
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