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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