Nuprl Lemma : bag-count-sqequal
∀[T:Type]. ∀[bs:bag(T)]. ∀[eq:EqDecider(T)]. ∀[x:T]. ((#x in bs) ~ #([y∈bs|eq x y]))
Proof
Definitions occuring in Statement :
bag-count: (#x in bs),
bag-size: #(bs),
bag-filter: [x∈b|p[x]],
bag: bag(T),
deq: EqDecider(T),
uall: ∀[x:A]. B[x],
apply: f a,
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],
squash: ↓T,
exists: ∃x:A. B[x],
bag-filter: [x∈b|p[x]],
bag-size: #(bs),
bag-count: (#x in bs),
deq: EqDecider(T),
subtype_rel: A ⊆r B,
ge: i ≥ j ,
all: ∀x:A. B[x],
decidable: Dec(P),
or: P ∨ Q,
le: A ≤ B,
and: P ∧ Q,
satisfiable_int_formula: satisfiable_int_formula(fmla),
false: False,
implies: P ⇒ Q,
not: ¬A,
top: Top,
prop: ℙ,
sq_type: SQType(T),
guard: {T}
Lemmas referenced :
subtype_base_sq,
nat_wf,
set_subtype_base,
le_wf,
int_subtype_base,
bag_to_squash_list,
count-length-filter,
non_neg_length,
filter_wf5,
decidable__le,
satisfiable-full-omega-tt,
intformand_wf,
intformnot_wf,
intformle_wf,
itermConstant_wf,
itermVar_wf,
int_formula_prop_and_lemma,
int_formula_prop_not_lemma,
int_formula_prop_le_lemma,
int_term_value_constant_lemma,
int_term_value_var_lemma,
int_formula_prop_wf,
filter_functionality,
eta_conv,
bool_wf,
equal_wf,
bag-count_wf,
bag-size_wf,
assert_wf,
bag-filter_wf,
deq_wf,
bag_wf
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
introduction,
cut,
thin,
instantiate,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
cumulativity,
hypothesis,
independent_isectElimination,
sqequalRule,
intEquality,
lambdaEquality,
natural_numberEquality,
hypothesisEquality,
because_Cache,
imageElimination,
productElimination,
promote_hyp,
rename,
applyEquality,
setElimination,
dependent_functionElimination,
unionElimination,
dependent_pairFormation,
int_eqEquality,
isect_memberEquality,
voidElimination,
voidEquality,
independent_pairFormation,
computeAll,
dependent_set_memberEquality,
hyp_replacement,
equalitySymmetry,
Error :applyLambdaEquality,
setEquality,
equalityTransitivity,
independent_functionElimination,
sqequalAxiom,
universeEquality
Latex:
\mforall{}[T:Type]. \mforall{}[bs:bag(T)]. \mforall{}[eq:EqDecider(T)]. \mforall{}[x:T]. ((\#x in bs) \msim{} \#([y\mmember{}bs|eq x y]))
Date html generated:
2016_10_25-AM-11_25_17
Last ObjectModification:
2016_07_12-AM-07_29_33
Theory : bags_2
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