Nuprl Lemma : iseg-append-one
∀[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,
member: t ∈ T,
prop: ℙ,
rev_implies: P ⇐ Q,
or: P ∨ Q,
guard: {T},
exists: ∃x:A. B[x],
assert: ↑b,
ifthenelse: if b then t else f fi ,
btrue: tt,
less_than: a < b,
squash: ↓T,
less_than': less_than'(a;b),
false: False,
cons: [a / b],
top: Top,
bfalse: ff,
true: True,
subtype_rel: A ⊆r B,
uimplies: b supposing a
Lemmas referenced :
iseg_wf,
append_wf,
cons_wf,
nil_wf,
or_wf,
equal_wf,
list_wf,
iseg_append_iff,
iseg_single,
list-cases,
null_nil_lemma,
length_of_nil_lemma,
product_subtype_list,
null_cons_lemma,
length_of_cons_lemma,
squash_wf,
true_wf,
iff_weakening_equal,
iseg_append,
iseg_weakening
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
lambdaFormation,
independent_pairFormation,
cut,
introduction,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
cumulativity,
hypothesisEquality,
hypothesis,
universeEquality,
dependent_functionElimination,
productElimination,
independent_functionElimination,
unionElimination,
inlFormation,
sqequalRule,
inrFormation,
imageElimination,
voidElimination,
promote_hyp,
hypothesis_subsumption,
isect_memberEquality,
voidEquality,
applyEquality,
lambdaEquality,
equalityTransitivity,
equalitySymmetry,
because_Cache,
natural_numberEquality,
imageMemberEquality,
baseClosed,
independent_isectElimination,
hyp_replacement,
applyLambdaEquality
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_47_37
Last ObjectModification:
2017_02_27-PM-05_04_30
Theory : list_1
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