### 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

Home Index