Nuprl Lemma : bag-remove-size-member-no-repeats
∀[T:Type]
∀eq:EqDecider(T). ∀bs:bag(T). ∀x:T. (#(bs - x) = (#(bs) - 1) ∈ ℤ) supposing (x ↓∈ bs and bag-no-repeats(T;bs))
Proof
Definitions occuring in Statement :
bag-remove: bs - x,
bag-member: x ↓∈ bs,
bag-no-repeats: bag-no-repeats(T;bs),
bag-size: #(bs),
bag: bag(T),
deq: EqDecider(T),
uimplies: b supposing a,
uall: ∀[x:A]. B[x],
all: ∀x:A. B[x],
subtract: n - m,
natural_number: $n,
int: ℤ,
universe: Type,
equal: s = t ∈ T
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
all: ∀x:A. B[x],
or: P ∨ Q,
and: P ∧ Q,
uimplies: b supposing a,
squash: ↓T,
prop: ℙ,
subtype_rel: A ⊆r B,
nat: ℕ,
true: True,
guard: {T},
iff: P ⇐⇒ Q,
implies: P ⇒ Q,
not: ¬A,
false: False
Lemmas referenced :
bag-remove-size,
equal_wf,
squash_wf,
true_wf,
subtract_wf,
bag-size_wf,
nat_wf,
bag-count-member-no-repeats,
iff_weakening_equal,
bag-member_wf,
bag-no-repeats_wf,
bag_wf,
deq_wf
Rules used in proof :
cut,
introduction,
extract_by_obid,
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
hypothesis,
sqequalHypSubstitution,
isectElimination,
thin,
hypothesisEquality,
lambdaFormation,
dependent_functionElimination,
unionElimination,
productElimination,
applyEquality,
lambdaEquality,
imageElimination,
equalityTransitivity,
equalitySymmetry,
universeEquality,
intEquality,
cumulativity,
setElimination,
rename,
sqequalRule,
independent_isectElimination,
natural_numberEquality,
imageMemberEquality,
baseClosed,
because_Cache,
independent_functionElimination,
isect_memberEquality,
axiomEquality,
voidElimination
Latex:
\mforall{}[T:Type]
\mforall{}eq:EqDecider(T). \mforall{}bs:bag(T). \mforall{}x:T.
(\#(bs - x) = (\#(bs) - 1)) supposing (x \mdownarrow{}\mmember{} bs and bag-no-repeats(T;bs))
Date html generated:
2018_05_21-PM-09_47_54
Last ObjectModification:
2017_07_26-PM-06_30_29
Theory : bags_2
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