Nuprl Lemma : list-diff-property
∀[T:Type]
∀eq:EqDecider(T). ∀as,bs:T List.
((∀x:T. ((x ∈ as-bs) ⇐⇒ (x ∈ as) ∧ (¬(x ∈ bs)))) ∧ no_repeats(T;as-bs) supposing no_repeats(T;as))
Proof
Definitions occuring in Statement :
list-diff: as-bs,
no_repeats: no_repeats(T;l),
l_member: (x ∈ l),
list: T List,
deq: EqDecider(T),
uimplies: b supposing a,
uall: ∀[x:A]. B[x],
all: ∀x:A. B[x],
iff: P ⇐⇒ Q,
not: ¬A,
and: P ∧ Q,
universe: Type
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
all: ∀x:A. B[x],
and: P ∧ Q,
member: t ∈ T,
list-diff: as-bs,
iff: P ⇐⇒ Q,
implies: P ⇒ Q,
prop: ℙ,
rev_implies: P ⇐ Q,
so_lambda: λ2x.t[x],
so_apply: x[s],
not: ¬A,
false: False,
uimplies: b supposing a
Lemmas referenced :
list_wf,
deq_wf,
member_filter,
bnot_wf,
deq-member_wf,
l_member_wf,
filter_wf5,
all_wf,
iff_wf,
and_wf,
assert_wf,
not_wf,
iff_transitivity,
iff_weakening_uiff,
assert_of_bnot,
assert-deq-member,
no_repeats_filter,
no_repeats_witness,
list-diff_wf,
no_repeats_wf
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
lambdaFormation,
independent_pairFormation,
cut,
lemma_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
hypothesisEquality,
hypothesis,
universeEquality,
sqequalRule,
addLevel,
allFunctionality,
productElimination,
impliesFunctionality,
dependent_functionElimination,
lambdaEquality,
independent_functionElimination,
cumulativity,
because_Cache,
setElimination,
rename,
setEquality,
voidElimination,
introduction,
andLevelFunctionality,
independent_isectElimination
Latex:
\mforall{}[T:Type]
\mforall{}eq:EqDecider(T). \mforall{}as,bs:T List.
((\mforall{}x:T. ((x \mmember{} as-bs) \mLeftarrow{}{}\mRightarrow{} (x \mmember{} as) \mwedge{} (\mneg{}(x \mmember{} bs))))
\mwedge{} no\_repeats(T;as-bs) supposing no\_repeats(T;as))
Date html generated:
2016_05_14-PM-03_29_47
Last ObjectModification:
2015_12_26-PM-06_03_11
Theory : decidable!equality
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