Nuprl Lemma : concat-is-nil
∀[T:Type]. ∀[LL:T List List]. uiff(concat(LL) = [] ∈ (T List);(∀L∈LL.L = [] ∈ (T List)))
Proof
Definitions occuring in Statement :
l_all: (∀x∈L.P[x]),
concat: concat(ll),
nil: [],
list: T List,
uiff: uiff(P;Q),
uall: ∀[x:A]. B[x],
universe: Type,
equal: s = t ∈ T
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
so_lambda: λ2x.t[x],
prop: ℙ,
so_apply: x[s],
implies: P ⇒ Q,
concat: concat(ll),
all: ∀x:A. B[x],
top: Top,
uiff: uiff(P;Q),
and: P ∧ Q,
uimplies: b supposing a,
rev_implies: P ⇐ Q,
iff: P ⇐⇒ Q,
l_all: (∀x∈L.P[x])
Lemmas referenced :
list_induction,
list_wf,
uiff_wf,
equal-wf-T-base,
concat_wf,
l_all_wf,
l_member_wf,
reduce_nil_lemma,
reduce_cons_lemma,
l_all_cons,
int_seg_wf,
length_wf,
cons_wf,
length_of_nil_lemma,
l_all_nil,
nil_wf,
equal-wf-base,
l_all_wf_nil,
iff_weakening_uiff,
append_wf,
append_is_nil
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
introduction,
cut,
thin,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
cumulativity,
hypothesisEquality,
hypothesis,
sqequalRule,
lambdaEquality,
baseClosed,
because_Cache,
setElimination,
rename,
setEquality,
independent_functionElimination,
dependent_functionElimination,
isect_memberEquality,
voidElimination,
voidEquality,
lambdaFormation,
addLevel,
productElimination,
independent_pairFormation,
independent_isectElimination,
axiomEquality,
natural_numberEquality,
instantiate,
productEquality,
applyLambdaEquality,
independent_pairEquality,
equalityTransitivity,
equalitySymmetry,
universeEquality
Latex:
\mforall{}[T:Type]. \mforall{}[LL:T List List]. uiff(concat(LL) = [];(\mforall{}L\mmember{}LL.L = []))
Date html generated:
2017_04_17-AM-08_51_58
Last ObjectModification:
2017_02_27-PM-05_08_34
Theory : list_1
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