Nuprl Lemma : remove-repeats_property
∀[T:Type]
∀eq:EqDecider(T). ∀L:T List. (no_repeats(T;remove-repeats(eq;L)) ∧ (∀a:T. ((a ∈ remove-repeats(eq;L)) ⇐⇒ (a ∈ L))))
Proof
Definitions occuring in Statement :
remove-repeats: remove-repeats(eq;L),
no_repeats: no_repeats(T;l),
l_member: (x ∈ l),
list: T List,
deq: EqDecider(T),
uall: ∀[x:A]. B[x],
all: ∀x:A. B[x],
iff: P ⇐⇒ Q,
and: P ∧ Q,
universe: Type
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
all: ∀x:A. B[x],
member: t ∈ T,
so_lambda: λ2x.t[x],
prop: ℙ,
and: P ∧ Q,
so_apply: x[s],
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q,
implies: P ⇒ Q,
remove-repeats: remove-repeats(eq;L),
so_lambda: so_lambda(x,y,z.t[x; y; z]),
top: Top,
so_apply: x[s1;s2;s3],
cand: A c∧ B,
deq: EqDecider(T),
uiff: uiff(P;Q),
uimplies: b supposing a,
not: ¬A,
false: False,
eqof: eqof(d),
or: P ∨ Q,
guard: {T},
decidable: Dec(P)
Lemmas referenced :
list_induction,
no_repeats_wf,
remove-repeats_wf,
all_wf,
iff_wf,
l_member_wf,
list_wf,
list_ind_nil_lemma,
no_repeats_nil,
nil_wf,
list_ind_cons_lemma,
no_repeats_cons,
filter_wf5,
bnot_wf,
no_repeats_filter,
cons_wf,
deq_wf,
member_filter,
not_wf,
assert_wf,
member_wf,
eqof_wf,
equal_wf,
iff_transitivity,
iff_weakening_uiff,
assert_of_bnot,
safe-assert-deq,
cons_member,
decidable-equal-deq
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
lambdaFormation,
cut,
thin,
introduction,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
hypothesisEquality,
sqequalRule,
lambdaEquality,
productEquality,
cumulativity,
hypothesis,
independent_functionElimination,
dependent_functionElimination,
isect_memberEquality,
voidElimination,
voidEquality,
independent_pairFormation,
because_Cache,
rename,
productElimination,
applyEquality,
setElimination,
setEquality,
independent_isectElimination,
universeEquality,
addLevel,
impliesFunctionality,
andLevelFunctionality,
levelHypothesis,
impliesLevelFunctionality,
unionElimination,
inlFormation,
inrFormation
Latex:
\mforall{}[T:Type]
\mforall{}eq:EqDecider(T). \mforall{}L:T List.
(no\_repeats(T;remove-repeats(eq;L)) \mwedge{} (\mforall{}a:T. ((a \mmember{} remove-repeats(eq;L)) \mLeftarrow{}{}\mRightarrow{} (a \mmember{} L))))
Date html generated:
2017_04_17-AM-09_10_16
Last ObjectModification:
2017_02_27-PM-05_18_28
Theory : decidable!equality
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