Nuprl Lemma : bag-subtract-member-if-no-repeats
∀[T:Type]. ∀[eq:EqDecider(T)]. ∀[bs,as:bag(T)]. ∀[x:T].
uiff(x ↓∈ bag-subtract(eq;bs;as);x ↓∈ bs ∧ (¬x ↓∈ as)) supposing bag-no-repeats(T;bs)
Proof
Definitions occuring in Statement :
bag-subtract: bag-subtract(eq;bs;as),
bag-member: x ↓∈ bs,
bag-no-repeats: bag-no-repeats(T;bs),
bag: bag(T),
deq: EqDecider(T),
uiff: uiff(P;Q),
uimplies: b supposing a,
uall: ∀[x:A]. B[x],
not: ¬A,
and: P ∧ Q,
universe: Type
Definitions unfolded in proof :
member: t ∈ T,
uall: ∀[x:A]. B[x],
prop: ℙ,
and: P ∧ Q,
uimplies: b supposing a,
uiff: uiff(P;Q),
bag-member: x ↓∈ bs,
squash: ↓T,
not: ¬A,
implies: P ⇒ Q,
false: False,
all: ∀x:A. B[x],
so_lambda: λ2x.t[x],
so_apply: x[s],
sq_stable: SqStable(P),
exists: ∃x:A. B[x],
subtype_rel: A ⊆r B,
bag-subtract: bag-subtract(eq;bs;as),
bag-accum: bag-accum(v,x.f[v; x];init;bs),
top: Top,
so_lambda: λ2x y.t[x; y],
so_apply: x[s1;s2],
empty-bag: {},
sq_or: a ↓∨ b,
guard: {T},
or: P ∨ Q,
cons-bag: x.b,
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q,
true: True
Lemmas referenced :
bag-member_wf,
bag-subtract_wf,
not_wf,
bag-no-repeats_wf,
bag_wf,
deq_wf,
uiff_wf,
squash_wf,
all_wf,
bag_to_squash_list,
sq_stable__all,
sq_stable__uiff,
sq_stable__bag-member,
sq_stable__and,
sq_stable__not,
list_induction,
list-subtype-bag,
list_wf,
list_accum_nil_lemma,
bag-member-empty-iff,
nil_wf,
list_accum_cons_lemma,
bag-drop_wf,
bag-drop-no-repeats,
sq_or_wf,
equal_wf,
bag-member-cons,
cons_wf,
iff_weakening_uiff,
bag-drop-property,
true_wf,
iff_weakening_equal,
bag-member-append,
single-bag_wf,
bag-no-repeats-append,
bag-member-single
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
cut,
introduction,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
cumulativity,
hypothesisEquality,
hypothesis,
productEquality,
because_Cache,
universeEquality,
isect_memberFormation,
sqequalRule,
productElimination,
independent_pairEquality,
isect_memberEquality,
imageElimination,
imageMemberEquality,
baseClosed,
lambdaEquality,
dependent_functionElimination,
equalityTransitivity,
equalitySymmetry,
voidElimination,
independent_functionElimination,
functionEquality,
lambdaFormation,
promote_hyp,
rename,
applyEquality,
independent_isectElimination,
hyp_replacement,
Error :applyLambdaEquality,
voidEquality,
independent_pairFormation,
inrFormation,
addLevel,
unionElimination,
natural_numberEquality,
inlFormation
Latex:
\mforall{}[T:Type]. \mforall{}[eq:EqDecider(T)]. \mforall{}[bs,as:bag(T)]. \mforall{}[x:T].
uiff(x \mdownarrow{}\mmember{} bag-subtract(eq;bs;as);x \mdownarrow{}\mmember{} bs \mwedge{} (\mneg{}x \mdownarrow{}\mmember{} as)) supposing bag-no-repeats(T;bs)
Date html generated:
2016_10_25-AM-11_28_31
Last ObjectModification:
2016_07_12-AM-07_35_09
Theory : bags_2
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