Nuprl Lemma : bag-count-drop
∀[T:Type]
∀eq:EqDecider(T). ∀x:T. ∀bs:bag(T).
((x ↓∈ bs ∧ ((#x in bag-drop(eq;bs;x)) = ((#x in bs) - 1) ∈ ℕ))
∨ ((¬x ↓∈ bs) ∧ ((#x in bag-drop(eq;bs;x)) = (#x in bs) ∈ ℕ)))
Proof
Definitions occuring in Statement :
bag-drop: bag-drop(eq;bs;a),
bag-count: (#x in bs),
bag-member: x ↓∈ bs,
bag: bag(T),
deq: EqDecider(T),
nat: ℕ,
uall: ∀[x:A]. B[x],
all: ∀x:A. B[x],
not: ¬A,
or: P ∨ Q,
and: P ∧ Q,
subtract: n - m,
natural_number: $n,
universe: Type,
equal: s = t ∈ T
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
all: ∀x:A. B[x],
member: t ∈ T,
or: P ∨ Q,
exists: ∃x:A. B[x],
and: P ∧ Q,
cand: A c∧ B,
sq_stable: SqStable(P),
implies: P ⇒ Q,
squash: ↓T,
prop: ℙ,
guard: {T},
not: ¬A,
false: False,
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q,
sq_or: a ↓∨ b,
uiff: uiff(P;Q),
uimplies: b supposing a,
subtype_rel: A ⊆r B,
sq_type: SQType(T),
true: True,
nat: ℕ,
single-bag: {x},
deq: EqDecider(T),
bool: 𝔹,
unit: Unit,
it: ⋅,
btrue: tt,
eqof: eqof(d),
ifthenelse: if b then t else f fi ,
decidable: Dec(P),
satisfiable_int_formula: satisfiable_int_formula(fmla),
top: Top,
bfalse: ff,
bnot: ¬bb,
assert: ↑b,
bag-drop: bag-drop(eq;bs;a)
Lemmas referenced :
bag-remove1-property,
sq_stable__bag-member,
not_wf,
bag-member_wf,
equal_wf,
nat_wf,
bag-count_wf,
bag-drop_wf,
bag_wf,
deq_wf,
bag-member-append,
single-bag_wf,
bag-member-single,
subtype_base_sq,
int_subtype_base,
subtract_wf,
cons_wf,
nil_wf,
list-subtype-bag,
ifthenelse_wf,
bool_wf,
eqtt_to_assert,
safe-assert-deq,
decidable__equal_int,
full-omega-unsat,
intformnot_wf,
intformeq_wf,
itermSubtract_wf,
itermAdd_wf,
itermConstant_wf,
itermVar_wf,
int_formula_prop_not_lemma,
int_formula_prop_eq_lemma,
int_term_value_subtract_lemma,
int_term_value_add_lemma,
int_term_value_constant_lemma,
int_term_value_var_lemma,
int_formula_prop_wf,
eqff_to_assert,
bool_cases_sqequal,
bool_subtype_base,
assert-bnot,
member_wf,
squash_wf,
true_wf,
bag-count-append,
subtype_rel_self,
iff_weakening_equal,
add_functionality_wrt_eq,
bag-count-single,
unit_wf2
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
lambdaFormation,
cut,
introduction,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
because_Cache,
dependent_functionElimination,
hypothesisEquality,
unionElimination,
inlFormation,
productElimination,
hypothesis,
independent_functionElimination,
sqequalRule,
imageMemberEquality,
baseClosed,
imageElimination,
independent_pairFormation,
productEquality,
inrFormation,
cumulativity,
voidElimination,
universeEquality,
hyp_replacement,
equalitySymmetry,
applyLambdaEquality,
independent_isectElimination,
instantiate,
intEquality,
applyEquality,
natural_numberEquality,
equalityTransitivity,
lambdaEquality,
setElimination,
rename,
equalityElimination,
approximateComputation,
dependent_pairFormation,
int_eqEquality,
isect_memberEquality,
voidEquality,
promote_hyp,
unionEquality
Latex:
\mforall{}[T:Type]
\mforall{}eq:EqDecider(T). \mforall{}x:T. \mforall{}bs:bag(T).
((x \mdownarrow{}\mmember{} bs \mwedge{} ((\#x in bag-drop(eq;bs;x)) = ((\#x in bs) - 1)))
\mvee{} ((\mneg{}x \mdownarrow{}\mmember{} bs) \mwedge{} ((\#x in bag-drop(eq;bs;x)) = (\#x in bs))))
Date html generated:
2019_10_16-AM-11_31_24
Last ObjectModification:
2018_08_21-PM-01_59_33
Theory : bags_2
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