Nuprl Lemma : cons_iseg_not_null
∀[T:Type]. ∀u:T. ∀L,v:T List. (L ≤ [u / v] ⇒ (¬↑null(L)) ⇒ (∃K:T List. ((L = [u / K] ∈ (T List)) ∧ K ≤ v)))
Proof
Definitions occuring in Statement :
iseg: l1 ≤ l2,
null: null(as),
cons: [a / b],
list: T List,
assert: ↑b,
uall: ∀[x:A]. B[x],
all: ∀x:A. B[x],
exists: ∃x:A. B[x],
not: ¬A,
implies: P ⇒ Q,
and: P ∧ Q,
universe: Type,
equal: s = t ∈ T
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
all: ∀x:A. B[x],
implies: P ⇒ Q,
member: t ∈ T,
or: P ∨ Q,
assert: ↑b,
ifthenelse: if b then t else f fi ,
btrue: tt,
not: ¬A,
true: True,
false: False,
cons: [a / b],
top: Top,
bfalse: ff,
iff: P ⇐⇒ Q,
and: P ∧ Q,
exists: ∃x:A. B[x],
cand: A c∧ B,
prop: ℙ
Lemmas referenced :
list-cases,
null_nil_lemma,
product_subtype_list,
null_cons_lemma,
cons_iseg,
and_wf,
equal_wf,
cons_wf,
list_wf,
iseg_wf,
not_wf,
assert_wf,
null_wf
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
lambdaFormation,
cut,
hypothesisEquality,
lemma_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
hypothesis,
dependent_functionElimination,
unionElimination,
sqequalRule,
independent_functionElimination,
natural_numberEquality,
voidElimination,
promote_hyp,
hypothesis_subsumption,
productElimination,
isect_memberEquality,
voidEquality,
because_Cache,
dependent_pairFormation,
dependent_set_memberEquality,
independent_pairFormation,
applyEquality,
lambdaEquality,
setElimination,
rename,
setEquality,
universeEquality
Latex:
\mforall{}[T:Type]. \mforall{}u:T. \mforall{}L,v:T List. (L \mleq{} [u / v] {}\mRightarrow{} (\mneg{}\muparrow{}null(L)) {}\mRightarrow{} (\mexists{}K:T List. ((L = [u / K]) \mwedge{} K \mleq{} v)))
Date html generated:
2016_05_14-PM-01_30_55
Last ObjectModification:
2015_12_26-PM-05_23_50
Theory : list_1
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