### Nuprl Lemma : bag-drop-co-restrict

`∀[X:Type]. ∀[eq:EqDecider(X)]. ∀[x:X]. ∀[b:bag(X)].  ((bag-drop(eq;b;x)|¬x) = (b|¬x) ∈ bag(X))`

Proof

Definitions occuring in Statement :  bag-co-restrict: `(b|¬x)` bag-drop: `bag-drop(eq;bs;a)` bag: `bag(T)` deq: `EqDecider(T)` uall: `∀[x:A]. B[x]` 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` squash: `↓T` prop: `ℙ` true: `True` subtype_rel: `A ⊆r B` uimplies: `b supposing a` guard: `{T}` iff: `P `⇐⇒` Q` rev_implies: `P `` Q` implies: `P `` Q` nat: `ℕ` le: `A ≤ B` less_than': `less_than'(a;b)` false: `False` not: `¬A` single-bag: `{x}` cons: `[a / b]` bag-rep: `bag-rep(n;x)` primrec: `primrec(n;b;c)` subtract: `n - m` cons-bag: `x.b` nil: `[]` it: `⋅` empty-bag: `{}`
Lemmas referenced :  bag-drop-property equal_wf squash_wf true_wf bag_wf bag-co-restrict_wf subtype_rel_self iff_weakening_equal bag-co-restrict-append single-bag_wf bag-drop_wf bag-append-ident bag-append_wf bag-co-restrict-rep false_wf le_wf empty-bag_wf
Rules used in proof :  sqequalSubstitution sqequalTransitivity computationStep sqequalReflexivity isect_memberFormation introduction cut extract_by_obid sqequalHypSubstitution isectElimination thin because_Cache dependent_functionElimination hypothesisEquality unionElimination productElimination applyEquality lambdaEquality imageElimination equalityTransitivity hypothesis equalitySymmetry natural_numberEquality sqequalRule imageMemberEquality baseClosed instantiate independent_isectElimination independent_functionElimination isect_memberEquality axiomEquality applyLambdaEquality hyp_replacement dependent_set_memberEquality independent_pairFormation lambdaFormation

Latex:
\mforall{}[X:Type].  \mforall{}[eq:EqDecider(X)].  \mforall{}[x:X].  \mforall{}[b:bag(X)].    ((bag-drop(eq;b;x)|\mneg{}x)  =  (b|\mneg{}x))

Date html generated: 2018_05_21-PM-09_52_56
Last ObjectModification: 2018_05_19-PM-04_21_38

Theory : bags_2

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