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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