Nuprl Lemma : member-count-repeats1
∀[T:Type]. ∀eq:EqDecider(T). ∀x:T. ∀L:T List.  ((x ∈ map(λp.(fst(p));count-repeats(L,eq))) 
⇐⇒ (x ∈ L))
Proof
Definitions occuring in Statement : 
count-repeats: count-repeats(L,eq)
, 
l_member: (x ∈ l)
, 
map: map(f;as)
, 
list: T List
, 
deq: EqDecider(T)
, 
uall: ∀[x:A]. B[x]
, 
pi1: fst(t)
, 
all: ∀x:A. B[x]
, 
iff: P 
⇐⇒ Q
, 
lambda: λx.A[x]
, 
universe: Type
Definitions unfolded in proof : 
uall: ∀[x:A]. B[x]
, 
member: t ∈ T
, 
all: ∀x:A. B[x]
, 
iff: P 
⇐⇒ Q
, 
and: P ∧ Q
, 
implies: P 
⇒ Q
, 
prop: ℙ
, 
so_lambda: λ2x.t[x]
, 
so_apply: x[s]
, 
rev_implies: P 
⇐ Q
, 
not: ¬A
, 
false: False
, 
or: P ∨ Q
, 
uimplies: b supposing a
, 
sq_type: SQType(T)
, 
guard: {T}
, 
uiff: uiff(P;Q)
, 
ifthenelse: if b then t else f fi 
, 
btrue: tt
, 
bfalse: ff
, 
isl: isl(x)
, 
assert: ↑b
, 
true: True
Lemmas referenced : 
apply-alist-count-repeats, 
list_wf, 
deq_wf, 
deq-member_wf, 
l_member_wf, 
map_wf, 
nat_plus_wf, 
pi1_wf, 
count-repeats_wf, 
assert_wf, 
bnot_wf, 
not_wf, 
bool_cases, 
subtype_base_sq, 
bool_wf, 
bool_subtype_base, 
eqtt_to_assert, 
assert-deq-member, 
eqff_to_assert, 
iff_transitivity, 
iff_weakening_uiff, 
assert_of_bnot, 
isl-apply-alist, 
and_wf, 
equal_wf, 
unit_wf2, 
isl_wf, 
btrue_wf, 
assert_elim, 
apply-alist_wf, 
bfalse_wf, 
btrue_neq_bfalse
Rules used in proof : 
cut, 
lemma_by_obid, 
sqequalSubstitution, 
sqequalTransitivity, 
computationStep, 
sqequalReflexivity, 
isect_memberFormation, 
hypothesis, 
sqequalHypSubstitution, 
isectElimination, 
thin, 
hypothesisEquality, 
lambdaFormation, 
universeEquality, 
equalityTransitivity, 
equalitySymmetry, 
independent_pairFormation, 
productEquality, 
lambdaEquality, 
sqequalRule, 
independent_functionElimination, 
voidElimination, 
dependent_functionElimination, 
unionElimination, 
instantiate, 
cumulativity, 
independent_isectElimination, 
productElimination, 
impliesFunctionality, 
dependent_set_memberEquality, 
unionEquality, 
applyEquality, 
setElimination, 
rename, 
setEquality, 
natural_numberEquality
Latex:
\mforall{}[T:Type]
    \mforall{}eq:EqDecider(T).  \mforall{}x:T.  \mforall{}L:T  List.    ((x  \mmember{}  map(\mlambda{}p.(fst(p));count-repeats(L,eq)))  \mLeftarrow{}{}\mRightarrow{}  (x  \mmember{}  L))
Date html generated:
2016_05_14-PM-03_23_04
Last ObjectModification:
2015_12_26-PM-06_21_15
Theory : decidable!equality
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