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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