Nuprl Lemma : iseg_append_single
∀[T:Type]. ∀l1,l2:T List. ∀x:T. (l1 ≤ l2 @ [x] ⇐⇒ l1 ≤ l2 ∨ (l1 = (l2 @ [x]) ∈ (T List)))
Proof
Definitions occuring in Statement :
iseg: l1 ≤ l2,
append: as @ bs,
cons: [a / b],
nil: [],
list: T List,
uall: ∀[x:A]. B[x],
all: ∀x:A. B[x],
iff: P ⇐⇒ Q,
or: P ∨ Q,
universe: Type,
equal: s = t ∈ T
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
all: ∀x:A. B[x],
iff: P ⇐⇒ Q,
and: P ∧ Q,
implies: P ⇒ Q,
or: P ∨ Q,
member: t ∈ T,
prop: ℙ,
guard: {T},
exists: ∃x:A. B[x],
so_lambda: λ2x.t[x],
top: Top,
so_apply: x[s],
rev_implies: P ⇐ Q,
squash: ↓T,
true: True,
less_than: a < b,
less_than': less_than'(a;b),
false: False,
cons: [a / b],
assert: ↑b,
ifthenelse: if b then t else f fi ,
btrue: tt,
bfalse: ff,
cand: A c∧ B
Lemmas referenced :
equal_wf,
list_wf,
append_wf,
cons_wf,
nil_wf,
iseg_wf,
or_wf,
exists_wf,
less_than_wf,
length_wf,
length-append,
iseg_append_iff,
iff_wf,
list-cases,
length_of_nil_lemma,
product_subtype_list,
length_of_cons_lemma,
cons_iseg,
iseg_nil,
null_nil_lemma,
null_cons_lemma,
length-singleton,
iseg_weakening
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
lambdaFormation,
cut,
independent_pairFormation,
sqequalHypSubstitution,
unionElimination,
thin,
inlFormation,
hypothesis,
introduction,
extract_by_obid,
isectElimination,
cumulativity,
hypothesisEquality,
sqequalRule,
inrFormation,
productElimination,
lambdaEquality,
productEquality,
natural_numberEquality,
applyLambdaEquality,
isect_memberEquality,
voidElimination,
voidEquality,
addLevel,
impliesFunctionality,
because_Cache,
dependent_functionElimination,
independent_functionElimination,
universeEquality,
applyEquality,
imageElimination,
imageMemberEquality,
baseClosed,
hyp_replacement,
equalitySymmetry,
promote_hyp,
hypothesis_subsumption,
rename,
dependent_pairFormation
Latex:
\mforall{}[T:Type]. \mforall{}l1,l2:T List. \mforall{}x:T. (l1 \mleq{} l2 @ [x] \mLeftarrow{}{}\mRightarrow{} l1 \mleq{} l2 \mvee{} (l1 = (l2 @ [x])))
Date html generated:
2017_04_17-AM-08_46_08
Last ObjectModification:
2017_02_27-PM-05_04_42
Theory : list_1
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