Nuprl Lemma : last-concat
∀[T:Type]
∀ll:T List List
∃ll1:T List List
∃l1:T List
((concat(ll) = (concat(ll1) @ l1 @ [last(concat(ll))]) ∈ (T List)) ∧ ll1 @ [l1 @ [last(concat(ll))]] ≤ ll)
supposing ¬(concat(ll) = [] ∈ (T List))
Proof
Definitions occuring in Statement :
iseg: l1 ≤ l2,
last: last(L),
concat: concat(ll),
append: as @ bs,
cons: [a / b],
nil: [],
list: T List,
uimplies: b supposing a,
uall: ∀[x:A]. B[x],
all: ∀x:A. B[x],
exists: ∃x:A. B[x],
not: ¬A,
and: P ∧ Q,
universe: Type,
equal: s = t ∈ T
Definitions unfolded in proof :
uimplies: b supposing a,
member: t ∈ T,
not: ¬A,
implies: P ⇒ Q,
false: False,
uall: ∀[x:A]. B[x],
prop: ℙ,
top: Top,
all: ∀x:A. B[x],
decidable: Dec(P),
or: P ∨ Q,
uiff: uiff(P;Q),
and: P ∧ Q,
so_lambda: λ2x.t[x],
so_apply: x[s],
subtype_rel: A ⊆r B,
exists: ∃x:A. B[x],
append: as @ bs,
so_lambda: so_lambda(x,y,z.t[x; y; z]),
so_apply: x[s1;s2;s3],
cand: A c∧ B,
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q,
squash: ↓T,
true: True
Lemmas referenced :
equal-wf-base,
list_wf,
nil_wf,
not_wf,
decidable__assert,
null_wf,
concat_wf,
assert_of_null,
equal-wf-T-base,
list_induction,
isect_wf,
exists_wf,
equal_wf,
append_wf,
cons_wf,
last_wf,
assert_wf,
length_wf,
length-append,
iseg_wf,
concat-nil,
concat-cons,
subtype_rel_list,
top_wf,
append_nil_sq,
list_ind_nil_lemma,
last_lemma,
cons_iseg,
nil_iseg,
band_wf,
length_of_nil_lemma,
null_append,
iff_transitivity,
iff_weakening_uiff,
assert_of_band,
append_assoc_sq,
squash_wf,
true_wf,
last_append,
list_ind_cons_lemma
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
cut,
introduction,
sqequalRule,
sqequalHypSubstitution,
lambdaEquality,
dependent_functionElimination,
thin,
hypothesisEquality,
voidElimination,
extract_by_obid,
isectElimination,
cumulativity,
hypothesis,
baseClosed,
because_Cache,
rename,
independent_functionElimination,
isect_memberEquality,
voidEquality,
unionElimination,
productElimination,
independent_isectElimination,
lambdaFormation,
equalityTransitivity,
equalitySymmetry,
promote_hyp,
productEquality,
addLevel,
impliesFunctionality,
levelHypothesis,
applyLambdaEquality,
universeEquality,
applyEquality,
hyp_replacement,
dependent_pairFormation,
independent_pairFormation,
imageElimination,
natural_numberEquality,
imageMemberEquality
Latex:
\mforall{}[T:Type]
\mforall{}ll:T List List
\mexists{}ll1:T List List
\mexists{}l1:T List
((concat(ll) = (concat(ll1) @ l1 @ [last(concat(ll))]))
\mwedge{} ll1 @ [l1 @ [last(concat(ll))]] \mleq{} ll)
supposing \mneg{}(concat(ll) = [])
Date html generated:
2017_04_17-AM-08_51_28
Last ObjectModification:
2017_02_27-PM-05_09_33
Theory : list_1
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