Nuprl Lemma : isl-list-index
∀[T:Type]. ∀eq:EqDecider(T). ∀x:T. ∀L:T List. (↑isl(list-index(eq;L;x)) ⇐⇒ (x ∈ L))
Proof
Definitions occuring in Statement :
list-index: list-index(d;L;x),
l_member: (x ∈ l),
list: T List,
deq: EqDecider(T),
assert: ↑b,
isl: isl(x),
uall: ∀[x:A]. B[x],
all: ∀x:A. B[x],
iff: 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],
so_apply: x[s],
implies: P ⇒ Q,
list-index: list-index(d;L;x),
so_lambda: so_lambda(x,y,z.t[x; y; z]),
top: Top,
so_apply: x[s1;s2;s3],
isl: isl(x),
assert: ↑b,
ifthenelse: if b then t else f fi ,
bfalse: ff,
prop: ℙ,
iff: P ⇐⇒ Q,
and: P ∧ Q,
false: False,
rev_implies: P ⇐ Q,
uimplies: b supposing a,
not: ¬A,
btrue: tt,
true: True,
guard: {T},
or: P ∨ Q,
bool: 𝔹,
unit: Unit,
it: ⋅,
eqof: eqof(d),
deq: EqDecider(T),
uiff: uiff(P;Q),
exists: ∃x:A. B[x],
sq_type: SQType(T),
bnot: ¬bb
Lemmas referenced :
list_induction,
iff_wf,
assert_wf,
isl_wf,
int_seg_wf,
length_wf,
top_wf,
list-index_wf,
l_member_wf,
list_wf,
list_ind_nil_lemma,
list_ind_cons_lemma,
deq_wf,
false_wf,
null_nil_lemma,
btrue_wf,
member-implies-null-eq-bfalse,
nil_wf,
btrue_neq_bfalse,
equal_wf,
cons_member,
true_wf,
cons_wf,
eqof_wf,
bool_wf,
eqtt_to_assert,
safe-assert-deq,
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,
cut,
thin,
introduction,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
because_Cache,
sqequalRule,
lambdaEquality,
natural_numberEquality,
cumulativity,
hypothesisEquality,
hypothesis,
independent_functionElimination,
dependent_functionElimination,
isect_memberEquality,
voidElimination,
voidEquality,
rename,
universeEquality,
independent_pairFormation,
independent_isectElimination,
equalityTransitivity,
equalitySymmetry,
unionEquality,
unionElimination,
productElimination,
inrFormation,
applyEquality,
equalityElimination,
setElimination,
inlFormation,
hyp_replacement,
applyLambdaEquality,
dependent_pairFormation,
promote_hyp,
instantiate
Latex:
\mforall{}[T:Type]. \mforall{}eq:EqDecider(T). \mforall{}x:T. \mforall{}L:T List. (\muparrow{}isl(list-index(eq;L;x)) \mLeftarrow{}{}\mRightarrow{} (x \mmember{} L))
Date html generated:
2017_04_17-AM-09_15_07
Last ObjectModification:
2017_02_27-PM-05_20_25
Theory : decidable!equality
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