Nuprl Lemma : empty-bag-iff-size
∀[T:Type]. ∀[bs:bag(T)]. uiff(bs = {} ∈ bag(T);#(bs) ≤ 0)
Proof
Definitions occuring in Statement :
bag-size: #(bs),
empty-bag: {},
bag: bag(T),
uiff: uiff(P;Q),
uall: ∀[x:A]. B[x],
le: A ≤ B,
natural_number: $n,
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,
le: A ≤ B,
not: ¬A,
implies: P ⇒ Q,
false: False,
prop: ℙ,
subtype_rel: A ⊆r B,
nat: ℕ,
so_lambda: λ2x.t[x],
so_apply: x[s],
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q,
all: ∀x:A. B[x],
decidable: Dec(P),
or: P ∨ Q,
satisfiable_int_formula: satisfiable_int_formula(fmla),
exists: ∃x:A. B[x],
top: Top,
squash: ↓T,
guard: {T},
less_than: a < b
Lemmas referenced :
bag_wf,
less_than'_wf,
bag-size_wf,
nat_wf,
equal-wf-T-base,
le_wf,
iff_weakening_uiff,
all_wf,
not_wf,
bag-member_wf,
empty-bag-iff-no-member,
uiff_wf,
decidable__le,
decidable__lt,
satisfiable-full-omega-tt,
intformand_wf,
intformnot_wf,
intformless_wf,
itermConstant_wf,
itermVar_wf,
intformle_wf,
int_formula_prop_and_lemma,
int_formula_prop_not_lemma,
int_formula_prop_less_lemma,
int_term_value_constant_lemma,
int_term_value_var_lemma,
int_formula_prop_le_lemma,
int_formula_prop_wf,
bag-member-iff-size
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
introduction,
cut,
sqequalRule,
sqequalHypSubstitution,
productElimination,
thin,
independent_pairEquality,
isect_memberEquality,
isectElimination,
hypothesisEquality,
lambdaEquality,
dependent_functionElimination,
because_Cache,
axiomEquality,
equalityTransitivity,
hypothesis,
equalitySymmetry,
extract_by_obid,
cumulativity,
voidElimination,
natural_numberEquality,
applyEquality,
setElimination,
rename,
baseClosed,
universeEquality,
addLevel,
independent_pairFormation,
independent_isectElimination,
independent_functionElimination,
instantiate,
lambdaFormation,
unionElimination,
dependent_pairFormation,
int_eqEquality,
intEquality,
voidEquality,
computeAll,
imageElimination,
imageMemberEquality
Latex:
\mforall{}[T:Type]. \mforall{}[bs:bag(T)]. uiff(bs = \{\};\#(bs) \mleq{} 0)
Date html generated:
2017_10_01-AM-08_53_26
Last ObjectModification:
2017_07_26-PM-04_35_07
Theory : bags
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