Nuprl Lemma : bag-drop-property
∀[T:Type]
∀eq:EqDecider(T). ∀x:T. ∀bs:bag(T).
((bs = ({x} + bag-drop(eq;bs;x)) ∈ bag(T)) ∨ ((¬x ↓∈ bs) ∧ (bs = bag-drop(eq;bs;x) ∈ bag(T))))
Proof
Definitions occuring in Statement :
bag-drop: bag-drop(eq;bs;a),
bag-member: x ↓∈ bs,
bag-append: as + bs,
single-bag: {x},
bag: bag(T),
deq: EqDecider(T),
uall: ∀[x:A]. B[x],
all: ∀x:A. B[x],
not: ¬A,
or: P ∨ Q,
and: P ∧ Q,
universe: Type,
equal: s = t ∈ T
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
all: ∀x:A. B[x],
bag-drop: bag-drop(eq;bs;a),
implies: P ⇒ Q,
or: P ∨ Q,
exists: ∃x:A. B[x],
and: P ∧ Q,
outl: outl(x),
prop: ℙ,
uimplies: b supposing a,
isl: isl(x),
assert: ↑b,
ifthenelse: if b then t else f fi ,
btrue: tt,
true: True,
not: ¬A,
false: False,
guard: {T},
cand: A c∧ B,
so_lambda: λ2x.t[x],
so_apply: x[s]
Lemmas referenced :
bag-remove1-property,
bag-remove1_wf,
bag_wf,
unit_wf2,
and_wf,
equal_wf,
outl_wf,
assert_wf,
isl_wf,
bag-append_wf,
single-bag_wf,
btrue_wf,
bfalse_wf,
btrue_neq_bfalse,
not_wf,
bag-member_wf,
or_wf,
exists_wf,
equal-wf-T-base,
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,
unionEquality,
unionElimination,
inlFormation,
productElimination,
sqequalRule,
equalitySymmetry,
dependent_set_memberEquality,
independent_pairFormation,
equalityTransitivity,
applyLambdaEquality,
setElimination,
rename,
independent_isectElimination,
promote_hyp,
hyp_replacement,
natural_numberEquality,
independent_functionElimination,
voidElimination,
productEquality,
inrFormation,
lambdaEquality,
inlEquality,
baseClosed,
universeEquality
Latex:
\mforall{}[T:Type]
\mforall{}eq:EqDecider(T). \mforall{}x:T. \mforall{}bs:bag(T).
((bs = (\{x\} + bag-drop(eq;bs;x))) \mvee{} ((\mneg{}x \mdownarrow{}\mmember{} bs) \mwedge{} (bs = bag-drop(eq;bs;x))))
Date html generated:
2019_10_16-AM-11_31_19
Last ObjectModification:
2018_08_21-PM-01_59_20
Theory : bags_2
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