Nuprl Lemma : null-bag-size
∀[T:Type]. ∀[x:bag(T)]. (bag-null(x) ~ (#(x) =z 0))
Proof
Definitions occuring in Statement :
bag-size: #(bs),
bag-null: bag-null(bs),
bag: bag(T),
eq_int: (i =z j),
uall: ∀[x:A]. B[x],
natural_number: $n,
universe: Type,
sqequal: s ~ t
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
uimplies: b supposing a,
sq_type: SQType(T),
all: ∀x:A. B[x],
implies: P ⇒ Q,
guard: {T},
subtype_rel: A ⊆r B,
bool: 𝔹,
unit: Unit,
it: ⋅,
btrue: tt,
uiff: uiff(P;Q),
and: P ∧ Q,
nat: ℕ,
bfalse: ff,
exists: ∃x:A. B[x],
prop: ℙ,
or: P ∨ Q,
bnot: ¬bb,
ifthenelse: if b then t else f fi ,
assert: ↑b,
false: False,
decidable: Dec(P),
satisfiable_int_formula: satisfiable_int_formula(fmla),
not: ¬A,
top: Top,
empty-bag: {},
bag-null: bag-null(bs),
nequal: a ≠ b ∈ T
Lemmas referenced :
subtype_base_sq,
bool_wf,
bool_subtype_base,
bag_wf,
eq_int_wf,
bag-size_wf,
eqtt_to_assert,
assert_of_eq_int,
nat_wf,
eqff_to_assert,
equal_wf,
bool_cases_sqequal,
assert-bnot,
neg_assert_of_eq_int,
bag-size-zero,
decidable__le,
satisfiable-full-omega-tt,
intformand_wf,
intformnot_wf,
intformle_wf,
itermVar_wf,
itermConstant_wf,
intformeq_wf,
int_formula_prop_and_lemma,
int_formula_prop_not_lemma,
int_formula_prop_le_lemma,
int_term_value_var_lemma,
int_term_value_constant_lemma,
int_formula_prop_eq_lemma,
int_formula_prop_wf,
null_nil_lemma,
btrue_wf,
bag-null_wf,
assert-bag-null,
equal-wf-T-base,
bfalse_wf,
bag_size_empty_lemma
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
introduction,
cut,
thin,
instantiate,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
cumulativity,
hypothesis,
independent_isectElimination,
dependent_functionElimination,
equalityTransitivity,
equalitySymmetry,
independent_functionElimination,
sqequalAxiom,
hypothesisEquality,
sqequalRule,
isect_memberEquality,
because_Cache,
universeEquality,
applyEquality,
natural_numberEquality,
lambdaFormation,
unionElimination,
equalityElimination,
productElimination,
lambdaEquality,
setElimination,
rename,
dependent_pairFormation,
promote_hyp,
voidElimination,
int_eqEquality,
intEquality,
voidEquality,
independent_pairFormation,
computeAll,
baseClosed,
hyp_replacement,
applyLambdaEquality
Latex:
\mforall{}[T:Type]. \mforall{}[x:bag(T)]. (bag-null(x) \msim{} (\#(x) =\msubz{} 0))
Date html generated:
2017_10_01-AM-08_45_56
Last ObjectModification:
2017_07_26-PM-04_31_01
Theory : bags
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