Nuprl Lemma : split_rel_last
∀[A:Type]
∀r:A ⟶ A ⟶ 𝔹. ∀L:A List.
(∃L1,L2:A List
(((L = (L1 @ L2) ∈ (A List)) ∧ (¬↑null(L2)) ∧ (∀b∈L2.↑r[b;last(L)]))
∧ ¬↑r[last(L1);last(L)] supposing ¬↑null(L1))) supposing
((¬↑null(L)) and
(∀a:A. (↑r[a;a])))
Proof
Definitions occuring in Statement :
l_all: (∀x∈L.P[x]),
last: last(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[s1;s2],
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 :
implies: P ⇒ Q,
top: Top,
and: P ∧ Q,
prop: ℙ,
so_apply: x[s],
so_apply: x[s1;s2],
uimplies: b supposing a,
so_lambda: λ2x.t[x],
member: t ∈ T,
all: ∀x:A. B[x],
uall: ∀[x:A]. B[x],
assert: ↑b,
ifthenelse: if b then t else f fi ,
btrue: tt,
not: ¬A,
false: False,
true: True,
decidable: Dec(P),
or: P ∨ Q,
exists: ∃x:A. B[x],
cons: [a / b],
bfalse: ff,
guard: {T},
select: L[n],
subtract: n - m,
last: last(L),
so_apply: x[s1;s2;s3],
so_lambda: so_lambda(x,y,z.t[x; y; z]),
rev_implies: P ⇐ Q,
iff: P ⇐⇒ Q,
it: ⋅,
nil: [],
list_ind: list_ind,
append: as @ bs,
cand: A c∧ B,
uiff: uiff(P;Q),
squash: ↓T,
subtype_rel: A ⊆r B,
sq_type: SQType(T)
Lemmas referenced :
bool_wf,
l_member_wf,
last_wf,
l_all_wf,
length-append,
length_wf,
append_wf,
equal_wf,
list_wf,
exists_wf,
null_wf,
not_wf,
assert_wf,
all_wf,
isect_wf,
list_induction,
null_nil_lemma,
assert_witness,
true_wf,
decidable__assert,
cons_wf,
product_subtype_list,
null_cons_lemma,
list-cases,
null_wf2,
not_assert_elim,
btrue_wf,
length_of_nil_lemma,
length_of_cons_lemma,
list_ind_nil_lemma,
l_all_single,
btrue_neq_bfalse,
bfalse_wf,
assert_elim,
nil_wf,
false_wf,
assert_of_null,
nat_wf,
length_wf_nat,
squash_wf,
last_cons,
subtype_rel_self,
iff_weakening_equal,
l_all_cons,
top_wf,
subtype_rel_list,
last_cons2,
and_wf,
iff_weakening_uiff,
equal-wf-T-base,
iff_functionality_wrt_iff,
iff_imp_equal_bool,
bool_subtype_base,
subtype_base_sq,
list_ind_cons_lemma
Rules used in proof :
universeEquality,
universeIsType,
functionEquality,
dependent_functionElimination,
because_Cache,
independent_functionElimination,
isectEquality,
setEquality,
independent_isectElimination,
rename,
setElimination,
voidEquality,
voidElimination,
isect_memberEquality,
applyLambdaEquality,
productEquality,
hypothesis,
applyEquality,
lambdaEquality,
sqequalRule,
hypothesisEquality,
isectElimination,
sqequalHypSubstitution,
extract_by_obid,
introduction,
thin,
cut,
lambdaFormation,
isect_memberFormation_alt,
sqequalReflexivity,
computationStep,
sqequalTransitivity,
sqequalSubstitution,
natural_numberEquality,
functionIsType,
inhabitedIsType,
unionElimination,
productElimination,
hypothesis_subsumption,
promote_hyp,
functionExtensionality,
cumulativity,
equalitySymmetry,
equalityTransitivity,
independent_pairFormation,
dependent_pairFormation,
hyp_replacement,
dependent_set_memberEquality,
imageElimination,
imageMemberEquality,
baseClosed,
instantiate
Latex:
\mforall{}[A:Type]
\mforall{}r:A {}\mrightarrow{} A {}\mrightarrow{} \mBbbB{}. \mforall{}L:A List.
(\mexists{}L1,L2:A List
(((L = (L1 @ L2)) \mwedge{} (\mneg{}\muparrow{}null(L2)) \mwedge{} (\mforall{}b\mmember{}L2.\muparrow{}r[b;last(L)]))
\mwedge{} \mneg{}\muparrow{}r[last(L1);last(L)] supposing \mneg{}\muparrow{}null(L1))) supposing
((\mneg{}\muparrow{}null(L)) and
(\mforall{}a:A. (\muparrow{}r[a;a])))
Date html generated:
2019_10_15-AM-10_54_33
Last ObjectModification:
2018_09_27-AM-10_47_46
Theory : list!
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