Nuprl Lemma : set-equal-cons2
∀[T:Type]
  ∀eq:EqDecider(T). ∀u:T. ∀v,bs:T List.
    (set-equal(T;[u / v];bs) 
⇐⇒ (u ∈ bs) ∧ set-equal(T;filter(λx.(¬b(eq x u));v);filter(λx.(¬b(eq x u));bs)))
Proof
Definitions occuring in Statement : 
set-equal: set-equal(T;x;y)
, 
l_member: (x ∈ l)
, 
filter: filter(P;l)
, 
cons: [a / b]
, 
list: T List
, 
deq: EqDecider(T)
, 
bnot: ¬bb
, 
uall: ∀[x:A]. B[x]
, 
all: ∀x:A. B[x]
, 
iff: P 
⇐⇒ Q
, 
and: P ∧ Q
, 
apply: f a
, 
lambda: λx.A[x]
, 
universe: Type
Definitions unfolded in proof : 
uall: ∀[x:A]. B[x]
, 
all: ∀x:A. B[x]
, 
iff: P 
⇐⇒ Q
, 
and: P ∧ Q
, 
implies: P 
⇒ Q
, 
set-equal: set-equal(T;x;y)
, 
member: t ∈ T
, 
rev_implies: P 
⇐ Q
, 
or: P ∨ Q
, 
prop: ℙ
, 
deq: EqDecider(T)
, 
guard: {T}
, 
not: ¬A
, 
false: False
, 
eqof: eqof(d)
, 
uiff: uiff(P;Q)
, 
uimplies: b supposing a
, 
bool: 𝔹
, 
unit: Unit
, 
it: ⋅
, 
btrue: tt
, 
bfalse: ff
, 
exists: ∃x:A. B[x]
, 
sq_type: SQType(T)
, 
bnot: ¬bb
, 
ifthenelse: if b then t else f fi 
, 
assert: ↑b
, 
cand: A c∧ B
Lemmas referenced : 
cons_member, 
l_member_wf, 
set-equal_wf, 
cons_wf, 
filter_wf5, 
bnot_wf, 
list_wf, 
deq_wf, 
or_wf, 
equal_wf, 
assert_witness, 
assert_wf, 
member_filter, 
iff_wf, 
eqof_wf, 
not_wf, 
iff_transitivity, 
iff_weakening_uiff, 
assert_of_bnot, 
safe-assert-deq, 
bool_wf, 
eqtt_to_assert, 
and_wf, 
eqff_to_assert, 
bool_cases_sqequal, 
subtype_base_sq, 
bool_subtype_base, 
assert-bnot
Rules used in proof : 
sqequalSubstitution, 
sqequalTransitivity, 
computationStep, 
sqequalReflexivity, 
isect_memberFormation, 
lambdaFormation, 
independent_pairFormation, 
sqequalHypSubstitution, 
cut, 
hypothesis, 
dependent_functionElimination, 
thin, 
hypothesisEquality, 
productElimination, 
independent_functionElimination, 
introduction, 
extract_by_obid, 
isectElimination, 
because_Cache, 
inlFormation, 
cumulativity, 
productEquality, 
lambdaEquality, 
applyEquality, 
setElimination, 
rename, 
setEquality, 
universeEquality, 
promote_hyp, 
sqequalRule, 
inrFormation, 
addLevel, 
impliesFunctionality, 
unionElimination, 
voidElimination, 
independent_isectElimination, 
levelHypothesis, 
equalityElimination, 
equalityTransitivity, 
equalitySymmetry, 
hyp_replacement, 
dependent_set_memberEquality, 
applyLambdaEquality, 
dependent_pairFormation, 
instantiate
Latex:
\mforall{}[T:Type]
    \mforall{}eq:EqDecider(T).  \mforall{}u:T.  \mforall{}v,bs:T  List.
        (set-equal(T;[u  /  v];bs)
        \mLeftarrow{}{}\mRightarrow{}  (u  \mmember{}  bs)  \mwedge{}  set-equal(T;filter(\mlambda{}x.(\mneg{}\msubb{}(eq  x  u));v);filter(\mlambda{}x.(\mneg{}\msubb{}(eq  x  u));bs)))
Date html generated:
2017_04_17-AM-07_37_16
Last ObjectModification:
2017_02_27-PM-04_12_15
Theory : list_1
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