Nuprl Lemma : bag-remove-size
∀[T:Type]
∀eq:EqDecider(T). ∀bs:bag(T). ∀x:T.
((x ↓∈ bs ∧ (#(bs - x) = (#(bs) - (#x in bs)) ∈ ℤ)) ∨ ((¬x ↓∈ bs) ∧ (#(bs - x) = #(bs) ∈ ℤ)))
Proof
Definitions occuring in Statement :
bag-remove: bs - x,
bag-count: (#x in bs),
bag-member: x ↓∈ bs,
bag-size: #(bs),
bag: bag(T),
deq: EqDecider(T),
uall: ∀[x:A]. B[x],
all: ∀x:A. B[x],
not: ¬A,
or: P ∨ Q,
and: P ∧ Q,
subtract: n - m,
int: ℤ,
universe: Type,
equal: s = t ∈ T
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
all: ∀x:A. B[x],
member: t ∈ T,
implies: P ⇒ Q,
decidable: Dec(P),
or: P ∨ Q,
and: P ∧ Q,
cand: A c∧ B,
prop: ℙ,
subtype_rel: A ⊆r B,
nat: ℕ,
guard: {T},
squash: ↓T,
uimplies: b supposing a,
true: True,
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q,
exists: ∃x:A. B[x],
bag-count: (#x in bs),
bag-size: #(bs),
bag-remove: bs - x,
bag-filter: [x∈b|p[x]],
count: count(P;L),
so_lambda: λ2x.t[x],
deq: EqDecider(T),
bool: 𝔹,
unit: Unit,
it: ⋅,
btrue: tt,
ifthenelse: if b then t else f fi ,
uiff: uiff(P;Q),
eqof: eqof(d),
bfalse: ff,
sq_type: SQType(T),
bnot: ¬bb,
assert: ↑b,
false: False,
not: ¬A,
so_apply: x[s],
top: Top,
subtract: n - m,
satisfiable_int_formula: satisfiable_int_formula(fmla)
Lemmas referenced :
bag_wf,
deq_wf,
decidable__bag-member,
decidable-equal-deq,
not_wf,
bag-member_wf,
equal_wf,
bag-size_wf,
bag-remove_wf,
nat_wf,
squash_wf,
true_wf,
bag-remove-trivial,
iff_weakening_equal,
subtract_wf,
bag-count_wf,
bag_to_squash_list,
bag-member-list,
list_induction,
all_wf,
length_wf,
filter_wf5,
l_member_wf,
bnot_wf,
reduce_wf,
bool_wf,
eqtt_to_assert,
safe-assert-deq,
eqff_to_assert,
bool_cases_sqequal,
subtype_base_sq,
bool_subtype_base,
assert-bnot,
list_wf,
filter_nil_lemma,
length_of_nil_lemma,
reduce_nil_lemma,
filter_cons_lemma,
length_of_cons_lemma,
reduce_cons_lemma,
int_subtype_base,
decidable__equal_int,
ifthenelse_wf,
satisfiable-full-omega-tt,
intformnot_wf,
intformeq_wf,
itermSubtract_wf,
itermAdd_wf,
itermVar_wf,
itermConstant_wf,
int_formula_prop_not_lemma,
int_formula_prop_eq_lemma,
int_term_value_subtract_lemma,
int_term_value_add_lemma,
int_term_value_var_lemma,
int_term_value_constant_lemma,
int_formula_prop_wf,
add_functionality_wrt_eq
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
lambdaFormation,
hypothesisEquality,
cut,
introduction,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
cumulativity,
hypothesis,
universeEquality,
independent_functionElimination,
because_Cache,
dependent_functionElimination,
unionElimination,
inlFormation,
independent_pairFormation,
productEquality,
intEquality,
applyEquality,
lambdaEquality,
setElimination,
rename,
sqequalRule,
inrFormation,
imageElimination,
equalityTransitivity,
equalitySymmetry,
independent_isectElimination,
natural_numberEquality,
imageMemberEquality,
baseClosed,
productElimination,
promote_hyp,
hyp_replacement,
applyLambdaEquality,
setEquality,
addEquality,
equalityElimination,
dependent_pairFormation,
instantiate,
voidElimination,
isect_memberEquality,
voidEquality,
int_eqEquality,
computeAll
Latex:
\mforall{}[T:Type]
\mforall{}eq:EqDecider(T). \mforall{}bs:bag(T). \mforall{}x:T.
((x \mdownarrow{}\mmember{} bs \mwedge{} (\#(bs - x) = (\#(bs) - (\#x in bs)))) \mvee{} ((\mneg{}x \mdownarrow{}\mmember{} bs) \mwedge{} (\#(bs - x) = \#(bs))))
Date html generated:
2018_05_21-PM-09_47_52
Last ObjectModification:
2017_07_26-PM-06_30_27
Theory : bags_2
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