Nuprl Lemma : bag-drop-commutes
∀[T:Type]. ∀[eq:EqDecider(T)]. ∀[bs:bag(T)]. ∀[x,y:T].
(bag-drop(eq;bag-drop(eq;bs;x);y) = bag-drop(eq;bag-drop(eq;bs;y);x) ∈ bag(T))
Proof
Definitions occuring in Statement :
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],
implies: P ⇒ Q,
decidable: Dec(P),
or: P ∨ Q,
prop: ℙ,
bag-drop: bag-drop(eq;bs;a),
exists: ∃x:A. B[x],
and: P ∧ Q,
so_lambda: λ2x.t[x],
so_apply: x[s],
top: Top,
true: True,
squash: ↓T,
subtype_rel: A ⊆r B,
uimplies: b supposing a,
guard: {T},
iff: P ⇐⇒ Q,
uiff: uiff(P;Q),
rev_implies: P ⇐ Q,
sq_or: a ↓∨ b,
not: ¬A,
false: False,
isl: isl(x),
outl: outl(x),
assert: ↑b,
ifthenelse: if b then t else f fi ,
btrue: tt
Lemmas referenced :
decidable-equal-deq,
bag-drop_wf,
equal_wf,
bag_wf,
bag-remove1-property,
bag-remove1_wf,
unit_wf2,
bag-append_wf,
single-bag_wf,
all_wf,
or_wf,
exists_wf,
not_wf,
bag-member_wf,
equal-wf-T-base,
bag-append-assoc,
squash_wf,
true_wf,
bag-append-comm,
subtype_rel_self,
iff_weakening_equal,
bag-append-cancel,
bag-member-append,
bag-member-single,
bfalse_wf,
and_wf,
outl_wf,
assert_wf,
isl_wf,
deq_wf,
btrue_wf,
btrue_neq_bfalse,
member_wf
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
introduction,
cut,
lambdaFormation,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
because_Cache,
independent_functionElimination,
hypothesis,
dependent_functionElimination,
hypothesisEquality,
unionElimination,
hyp_replacement,
equalitySymmetry,
applyLambdaEquality,
productElimination,
sqequalRule,
equalityTransitivity,
unionEquality,
isect_memberEquality,
axiomEquality,
promote_hyp,
inlEquality,
lambdaEquality,
productEquality,
baseClosed,
voidElimination,
voidEquality,
natural_numberEquality,
applyEquality,
imageElimination,
universeEquality,
imageMemberEquality,
instantiate,
independent_isectElimination,
inlFormation,
dependent_set_memberEquality,
independent_pairFormation,
setElimination,
rename,
inrFormation
Latex:
\mforall{}[T:Type]. \mforall{}[eq:EqDecider(T)]. \mforall{}[bs:bag(T)]. \mforall{}[x,y:T].
(bag-drop(eq;bag-drop(eq;bs;x);y) = bag-drop(eq;bag-drop(eq;bs;y);x))
Date html generated:
2019_10_16-AM-11_31_35
Last ObjectModification:
2018_08_21-PM-01_59_37
Theory : bags_2
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