Nuprl Lemma : split_tail_lemma
∀[A:Type]
∀f:A ⟶ 𝔹. ∀L:A List.
(∀a∈L.∃L1,L2:A List. (((L = (L1 @ L2) ∈ (A List)) ∧ (a ∈ L2) ∧ (∀b∈L2.↑f[b])) ∧ ¬↑f[last(L1)] supposing ¬↑null(L1))
supposing (∀b≥a∈L.↑f[b]))
Proof
Definitions occuring in Statement :
l_all_since: (∀x≥a∈L.P[x]),
l_all: (∀x∈L.P[x]),
last: last(L),
l_member: (x ∈ l),
null: null(as),
append: as @ bs,
list: T List,
assert: ↑b,
bool: 𝔹,
uimplies: b supposing a,
uall: ∀[x:A]. B[x],
so_apply: x[s],
all: ∀x:A. B[x],
exists: ∃x:A. B[x],
not: ¬A,
and: P ∧ Q,
function: x:A ⟶ B[x],
universe: Type,
equal: s = t ∈ T
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
all: ∀x:A. B[x],
member: t ∈ T,
so_lambda: λ2x.t[x],
prop: ℙ,
uimplies: b supposing a,
so_apply: x[s],
and: P ∧ Q,
top: Top,
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q,
implies: P ⇒ Q,
l_all_since: (∀x≥a∈L.P[x]),
exists: ∃x:A. B[x],
cand: A c∧ B,
squash: ↓T,
true: True,
subtype_rel: A ⊆r B,
guard: {T},
not: ¬A,
false: False,
split_tail: split_tail(L | ∀x.f[x]),
so_lambda: so_lambda(x,y,z.t[x; y; z]),
so_apply: x[s1;s2;s3],
pi1: fst(t),
assert: ↑b,
ifthenelse: if b then t else f fi ,
btrue: tt,
bool: 𝔹,
unit: Unit,
it: ⋅,
uiff: uiff(P;Q),
bfalse: ff,
or: P ∨ Q,
sq_type: SQType(T),
bnot: ¬bb
Lemmas referenced :
l_all_iff,
l_member_wf,
isect_wf,
l_all_since_wf,
assert_wf,
exists_wf,
list_wf,
equal_wf,
append_wf,
length_wf,
length-append,
l_all_wf,
not_wf,
null_wf,
last_wf,
assert_witness,
l_before_wf,
bool_wf,
split_tail_wf,
pi1_wf,
pi2_wf,
squash_wf,
true_wf,
split_tail_rel,
iff_weakening_equal,
all_wf,
split_tail_max,
split_tail_correct,
l_all_fwd,
list_induction,
list_ind_nil_lemma,
null_nil_lemma,
list_ind_cons_lemma,
eqtt_to_assert,
eqff_to_assert,
bool_cases_sqequal,
subtype_base_sq,
bool_subtype_base,
assert-bnot,
list_ind_wf,
nil_wf,
cons_wf,
null_cons_lemma,
false_wf,
assert_functionality_wrt_uiff,
assert_elim,
bfalse_wf,
btrue_neq_bfalse,
last_cons,
last_singleton
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
lambdaFormation,
cut,
introduction,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
hypothesisEquality,
dependent_functionElimination,
sqequalRule,
lambdaEquality,
cumulativity,
hypothesis,
setElimination,
rename,
applyEquality,
functionExtensionality,
because_Cache,
productEquality,
applyLambdaEquality,
isect_memberEquality,
voidElimination,
voidEquality,
setEquality,
isectEquality,
independent_isectElimination,
productElimination,
independent_functionElimination,
independent_pairEquality,
functionEquality,
universeEquality,
dependent_pairFormation,
equalityTransitivity,
equalitySymmetry,
imageElimination,
natural_numberEquality,
imageMemberEquality,
baseClosed,
independent_pairFormation,
addLevel,
existsFunctionality,
andLevelFunctionality,
existsLevelFunctionality,
unionElimination,
equalityElimination,
promote_hyp,
instantiate,
impliesFunctionality,
levelHypothesis
Latex:
\mforall{}[A:Type]
\mforall{}f:A {}\mrightarrow{} \mBbbB{}. \mforall{}L:A List.
(\mforall{}a\mmember{}L.\mexists{}L1,L2:A List
(((L = (L1 @ L2)) \mwedge{} (a \mmember{} L2) \mwedge{} (\mforall{}b\mmember{}L2.\muparrow{}f[b])) \mwedge{} \mneg{}\muparrow{}f[last(L1)] supposing \mneg{}\muparrow{}null(L1))
supposing (\mforall{}b\mgeq{}a\mmember{}L.\muparrow{}f[b]))
Date html generated:
2017_10_01-AM-08_34_59
Last ObjectModification:
2017_07_26-PM-04_25_36
Theory : list!
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