Nuprl Lemma : bag-count-is-zero
∀[T:Type]. ∀[eq:EqDecider(T)]. ∀[x:T]. ∀[bs:bag(T)]. (#x in bs) ~ 0 supposing ¬x ↓∈ bs
Proof
Definitions occuring in Statement :
bag-count: (#x in bs),
bag-member: x ↓∈ bs,
bag: bag(T),
deq: EqDecider(T),
uimplies: b supposing a,
uall: ∀[x:A]. B[x],
not: ¬A,
natural_number: $n,
universe: Type,
sqequal: s ~ t
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
uimplies: b supposing a,
nat: ℕ,
so_lambda: λ2x.t[x],
so_apply: x[s],
sq_type: SQType(T),
all: ∀x:A. B[x],
implies: P ⇒ Q,
guard: {T},
prop: ℙ,
not: ¬A,
uiff: uiff(P;Q),
and: P ∧ Q,
subtype_rel: A ⊆r B,
decidable: Dec(P),
or: P ∨ Q,
ge: i ≥ j ,
satisfiable_int_formula: satisfiable_int_formula(fmla),
exists: ∃x:A. B[x],
false: False,
top: Top,
le: A ≤ B
Lemmas referenced :
int_formula_prop_eq_lemma,
intformeq_wf,
decidable__equal_int,
int_formula_prop_wf,
int_formula_prop_not_lemma,
int_term_value_var_lemma,
int_term_value_constant_lemma,
int_formula_prop_le_lemma,
int_formula_prop_and_lemma,
intformnot_wf,
itermVar_wf,
itermConstant_wf,
intformle_wf,
intformand_wf,
satisfiable-full-omega-tt,
nat_properties,
decidable__le,
nat_wf,
bag-count_wf,
le_wf,
deq_wf,
bag_wf,
bag-member_wf,
not_wf,
bag-member-count,
int_subtype_base,
set_subtype_base,
subtype_base_sq
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
introduction,
cut,
thin,
instantiate,
lemma_by_obid,
sqequalHypSubstitution,
isectElimination,
because_Cache,
independent_isectElimination,
sqequalRule,
hypothesis,
hypothesisEquality,
dependent_functionElimination,
equalityTransitivity,
equalitySymmetry,
independent_functionElimination,
sqequalAxiom,
isect_memberEquality,
universeEquality,
lambdaFormation,
productElimination,
natural_numberEquality,
applyEquality,
lambdaEquality,
setElimination,
rename,
promote_hyp,
unionElimination,
setEquality,
intEquality,
dependent_pairFormation,
int_eqEquality,
voidElimination,
voidEquality,
independent_pairFormation,
computeAll,
dependent_set_memberEquality
Latex:
\mforall{}[T:Type]. \mforall{}[eq:EqDecider(T)]. \mforall{}[x:T]. \mforall{}[bs:bag(T)]. (\#x in bs) \msim{} 0 supposing \mneg{}x \mdownarrow{}\mmember{} bs
Date html generated:
2016_05_15-PM-07_58_06
Last ObjectModification:
2016_01_16-PM-01_31_07
Theory : bags_2
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