Nuprl Lemma : last-concat-non-null
∀[T:Type]. ∀[ll:T List List].
((¬↑null(concat(ll))) ∧ (last(concat(ll)) = last(last(ll)) ∈ T)) supposing ((¬↑null(last(ll))) and (¬↑null(ll)))
Proof
Definitions occuring in Statement :
last: last(L),
null: null(as),
concat: concat(ll),
list: T List,
assert: ↑b,
uimplies: b supposing a,
uall: ∀[x:A]. B[x],
not: ¬A,
and: P ∧ Q,
universe: Type,
equal: s = t ∈ T
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
so_lambda: λ2x.t[x],
uimplies: b supposing a,
prop: ℙ,
and: P ∧ Q,
so_apply: x[s],
implies: P ⇒ Q,
concat: concat(ll),
all: ∀x:A. B[x],
top: Top,
assert: ↑b,
ifthenelse: if b then t else f fi ,
btrue: tt,
bfalse: ff,
not: ¬A,
false: False,
true: True,
or: P ∨ Q,
cons: [a / b],
last: last(L),
subtract: n - m,
select: L[n],
subtype_rel: A ⊆r B,
cand: A c∧ B,
squash: ↓T,
guard: {T},
uiff: uiff(P;Q),
rev_uimplies: rev_uimplies(P;Q),
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q
Lemmas referenced :
list_induction,
list_wf,
isect_wf,
not_wf,
assert_wf,
null_wf,
last_wf,
concat_wf,
equal_wf,
null_nil_lemma,
reduce_nil_lemma,
null_cons_lemma,
true_wf,
list-cases,
product_subtype_list,
reduce_cons_lemma,
length_of_cons_lemma,
length_of_nil_lemma,
subtype_rel_list,
top_wf,
false_wf,
append-nil,
assert_functionality_wrt_uiff,
cons_wf,
assert_elim,
bfalse_wf,
btrue_neq_bfalse,
squash_wf,
last_cons,
concat-cons,
null_append,
assert_of_null,
append_wf,
band_wf,
equal-wf-T-base,
length_wf,
iff_transitivity,
iff_weakening_uiff,
assert_of_band,
last_append,
iff_weakening_equal
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
introduction,
cut,
thin,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
cumulativity,
hypothesisEquality,
hypothesis,
sqequalRule,
lambdaEquality,
because_Cache,
independent_isectElimination,
productEquality,
independent_functionElimination,
dependent_functionElimination,
isect_memberEquality,
voidElimination,
voidEquality,
lambdaFormation,
rename,
productElimination,
independent_pairEquality,
axiomEquality,
equalityTransitivity,
equalitySymmetry,
universeEquality,
natural_numberEquality,
unionElimination,
promote_hyp,
hypothesis_subsumption,
applyEquality,
independent_pairFormation,
addLevel,
levelHypothesis,
imageElimination,
imageMemberEquality,
baseClosed,
applyLambdaEquality,
hyp_replacement,
impliesFunctionality
Latex:
\mforall{}[T:Type]. \mforall{}[ll:T List List].
((\mneg{}\muparrow{}null(concat(ll))) \mwedge{} (last(concat(ll)) = last(last(ll)))) supposing
((\mneg{}\muparrow{}null(last(ll))) and
(\mneg{}\muparrow{}null(ll)))
Date html generated:
2017_04_17-AM-08_51_40
Last ObjectModification:
2017_02_27-PM-05_10_00
Theory : list_1
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