Nuprl Lemma : l_index_hd
∀[T:Type]. ∀[dT:EqDecider(T)]. ∀[L:T List]. index(L;hd(L)) ~ 0 supposing ¬↑null(L)
Proof
Definitions occuring in Statement :
l_index: index(L;x),
hd: hd(l),
null: null(as),
list: T List,
deq: EqDecider(T),
assert: ↑b,
uimplies: b supposing a,
uall: ∀[x:A]. B[x],
not: ¬A,
natural_number: $n,
universe: Type,
sqequal: s ~ t
Definitions unfolded in proof :
member: t ∈ T,
uall: ∀[x:A]. B[x],
all: ∀x:A. B[x],
or: P ∨ Q,
assert: ↑b,
ifthenelse: if b then t else f fi ,
btrue: tt,
not: ¬A,
implies: P ⇒ Q,
true: True,
false: False,
cons: [a / b],
top: Top,
bfalse: ff,
l_index: index(L;x),
mu: mu(f),
uimplies: b supposing a,
prop: ℙ,
mu-ge: mu-ge(f;n),
select: L[n],
deq: EqDecider(T),
bool: 𝔹,
unit: Unit,
it: ⋅,
eqof: eqof(d),
uiff: uiff(P;Q),
and: P ∧ Q,
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q
Lemmas referenced :
list-cases,
null_nil_lemma,
product_subtype_list,
null_cons_lemma,
reduce_hd_cons_lemma,
not_wf,
assert_wf,
null_wf,
list_wf,
deq_wf,
bool_wf,
equal-wf-T-base,
equal_wf,
bnot_wf,
eqof_wf,
uiff_transitivity,
eqtt_to_assert,
safe-assert-deq,
iff_transitivity,
iff_weakening_uiff,
eqff_to_assert,
assert_of_bnot
Rules used in proof :
cut,
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
hypothesisEquality,
introduction,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
hypothesis,
dependent_functionElimination,
unionElimination,
sqequalRule,
independent_functionElimination,
natural_numberEquality,
voidElimination,
promote_hyp,
hypothesis_subsumption,
productElimination,
isect_memberEquality,
voidEquality,
cumulativity,
because_Cache,
universeEquality,
isect_memberFormation,
sqequalAxiom,
equalityTransitivity,
equalitySymmetry,
callbyvalueReduce,
sqleReflexivity,
applyEquality,
setElimination,
rename,
baseClosed,
lambdaFormation,
equalityElimination,
independent_isectElimination,
independent_pairFormation,
addLevel,
impliesFunctionality,
levelHypothesis
Latex:
\mforall{}[T:Type]. \mforall{}[dT:EqDecider(T)]. \mforall{}[L:T List]. index(L;hd(L)) \msim{} 0 supposing \mneg{}\muparrow{}null(L)
Date html generated:
2017_04_17-AM-09_15_53
Last ObjectModification:
2017_02_27-PM-05_21_17
Theory : decidable!equality
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