Nuprl Lemma : non-empty-bag-implies-non-void
∀[T:Type]. ∀[bs:bag(T)]. ((¬(bs = {} ∈ bag(T))) ⇒ (↓T))
Proof
Definitions occuring in Statement :
empty-bag: {},
bag: bag(T),
uall: ∀[x:A]. B[x],
not: ¬A,
squash: ↓T,
implies: P ⇒ Q,
universe: Type,
equal: s = t ∈ T
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
implies: P ⇒ Q,
not: ¬A,
uiff: uiff(P;Q),
and: P ∧ Q,
uimplies: b supposing a,
subtype_rel: A ⊆r B,
nat: ℕ,
prop: ℙ,
all: ∀x:A. B[x],
decidable: Dec(P),
or: P ∨ Q,
satisfiable_int_formula: satisfiable_int_formula(fmla),
exists: ∃x:A. B[x],
false: False,
top: Top,
squash: ↓T
Lemmas referenced :
empty-bag_wf,
bag_wf,
equal_wf,
not_wf,
bag-member-iff-size,
int_formula_prop_wf,
int_formula_prop_le_lemma,
int_term_value_var_lemma,
int_term_value_constant_lemma,
int_formula_prop_less_lemma,
int_formula_prop_not_lemma,
int_formula_prop_and_lemma,
intformle_wf,
itermVar_wf,
itermConstant_wf,
intformless_wf,
intformnot_wf,
intformand_wf,
satisfiable-full-omega-tt,
decidable__lt,
nat_wf,
bag-size_wf,
le_wf,
empty-bag-iff-size
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
introduction,
cut,
lambdaFormation,
hypothesis,
addLevel,
sqequalHypSubstitution,
impliesFunctionality,
lemma_by_obid,
isectElimination,
thin,
hypothesisEquality,
productElimination,
independent_isectElimination,
applyEquality,
lambdaEquality,
setElimination,
rename,
sqequalRule,
natural_numberEquality,
levelHypothesis,
promote_hyp,
dependent_functionElimination,
unionElimination,
because_Cache,
dependent_pairFormation,
int_eqEquality,
intEquality,
isect_memberEquality,
voidElimination,
voidEquality,
independent_pairFormation,
computeAll,
imageElimination,
imageMemberEquality,
baseClosed,
impliesLevelFunctionality,
universeEquality
Latex:
\mforall{}[T:Type]. \mforall{}[bs:bag(T)]. ((\mneg{}(bs = \{\})) {}\mRightarrow{} (\mdownarrow{}T))
Date html generated:
2016_05_15-PM-02_37_20
Last ObjectModification:
2016_01_16-AM-08_49_50
Theory : bags
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