Nuprl Lemma : split_tail_max
∀[A:Type]
∀f:A ⟶ 𝔹. ∀L:A List. ∀a:A.
((a ∈ L) ⇒ ((a ∈ snd(split_tail(L | ∀x.f[x])))) supposing ((∀b:A. (a before b ∈ L ⇒ (↑f[b]))) and (↑f[a])))
Proof
Definitions occuring in Statement :
split_tail: split_tail(L | ∀x.f[x]),
l_before: x before y ∈ l,
l_member: (x ∈ l),
list: T List,
assert: ↑b,
bool: 𝔹,
uimplies: b supposing a,
uall: ∀[x:A]. B[x],
so_apply: x[s],
pi2: snd(t),
all: ∀x:A. B[x],
implies: P ⇒ Q,
function: x:A ⟶ B[x],
universe: Type
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
all: ∀x:A. B[x],
member: t ∈ T,
so_lambda: λ2x.t[x],
implies: P ⇒ Q,
prop: ℙ,
uimplies: b supposing a,
so_apply: x[s],
split_tail: split_tail(L | ∀x.f[x]),
so_lambda: so_lambda(x,y,z.t[x; y; z]),
top: Top,
so_apply: x[s1;s2;s3],
pi2: snd(t),
l_member: (x ∈ l),
select: L[n],
nil: [],
it: ⋅,
so_lambda: λ2x y.t[x; y],
so_apply: x[s1;s2],
cand: A c∧ B,
nat: ℕ,
iff: P ⇐⇒ Q,
and: P ∧ Q,
or: P ∨ Q,
rev_implies: P ⇐ Q,
not: ¬A,
false: False,
bool: 𝔹,
unit: Unit,
btrue: tt,
uiff: uiff(P;Q),
ifthenelse: if b then t else f fi ,
bfalse: ff,
guard: {T}
Lemmas referenced :
list_induction,
all_wf,
l_member_wf,
isect_wf,
assert_wf,
l_before_wf,
split_tail_wf,
list_wf,
pi2_wf,
equal_wf,
list_ind_nil_lemma,
list_ind_cons_lemma,
bool_wf,
length_of_nil_lemma,
stuck-spread,
base_wf,
assert_witness,
nil_wf,
exists_wf,
nat_wf,
less_than_wf,
equal-wf-T-base,
cons_wf,
cons_member,
split_tail_trivial,
cons_before,
bnot_wf,
not_wf,
assert_elim,
not_assert_elim,
and_wf,
btrue_neq_bfalse,
eqtt_to_assert,
uiff_transitivity,
eqff_to_assert,
assert_of_bnot,
list_ind_wf,
ifthenelse_wf
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation_alt,
lambdaFormation,
cut,
thin,
introduction,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
hypothesisEquality,
sqequalRule,
lambdaEquality,
functionEquality,
hypothesis,
applyEquality,
productEquality,
equalityTransitivity,
equalitySymmetry,
dependent_functionElimination,
independent_functionElimination,
isect_memberEquality,
voidElimination,
voidEquality,
rename,
because_Cache,
universeIsType,
universeEquality,
baseClosed,
independent_isectElimination,
functionIsType,
inhabitedIsType,
setElimination,
natural_numberEquality,
productElimination,
unionElimination,
hyp_replacement,
applyLambdaEquality,
inlFormation,
independent_pairFormation,
dependent_set_memberEquality,
equalityElimination,
spreadEquality,
dependent_pairEquality,
inrFormation,
independent_pairEquality
Latex:
\mforall{}[A:Type]
\mforall{}f:A {}\mrightarrow{} \mBbbB{}. \mforall{}L:A List. \mforall{}a:A.
((a \mmember{} L)
{}\mRightarrow{} ((a \mmember{} snd(split\_tail(L | \mforall{}x.f[x])))) supposing
((\mforall{}b:A. (a before b \mmember{} L {}\mRightarrow{} (\muparrow{}f[b]))) and
(\muparrow{}f[a])))
Date html generated:
2019_10_15-AM-10_54_46
Last ObjectModification:
2018_09_27-AM-10_18_51
Theory : list!
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