Nuprl Lemma : last-not-before
∀[T:Type]. ∀L:T List. (∀x:T. (last(L) before x ∈ L ⇐⇒ False)) supposing (no_repeats(T;L) and (¬↑null(L)))
Proof
Definitions occuring in Statement :
l_before: x before y ∈ l,
last: last(L),
no_repeats: no_repeats(T;l),
null: null(as),
list: T List,
assert: ↑b,
uimplies: b supposing a,
uall: ∀[x:A]. B[x],
all: ∀x:A. B[x],
iff: P ⇐⇒ Q,
not: ¬A,
false: False,
universe: Type
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
all: ∀x:A. B[x],
uimplies: b supposing a,
member: t ∈ T,
not: ¬A,
implies: P ⇒ Q,
false: False,
iff: P ⇐⇒ Q,
and: P ∧ Q,
uiff: uiff(P;Q),
prop: ℙ,
rev_implies: P ⇐ Q
Lemmas referenced :
no_repeats_witness,
no_repeats_iff,
last_wf,
before_last,
l_before_wf,
istype-void,
no_repeats_wf,
istype-assert,
null_wf,
list_wf,
istype-universe,
l_before_member,
l_before_transitivity
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation_alt,
lambdaFormation_alt,
cut,
introduction,
sqequalRule,
sqequalHypSubstitution,
lambdaEquality_alt,
dependent_functionElimination,
thin,
hypothesisEquality,
voidElimination,
functionIsTypeImplies,
inhabitedIsType,
rename,
extract_by_obid,
isectElimination,
independent_functionElimination,
hypothesis,
independent_pairFormation,
because_Cache,
productElimination,
independent_isectElimination,
equalitySymmetry,
equalityIstype,
universeIsType,
functionIsType,
instantiate,
universeEquality
Latex:
\mforall{}[T:Type]
\mforall{}L:T List. (\mforall{}x:T. (last(L) before x \mmember{} L \mLeftarrow{}{}\mRightarrow{} False)) supposing (no\_repeats(T;L) and (\mneg{}\muparrow{}null(L)))
Date html generated:
2019_10_15-AM-10_23_47
Last ObjectModification:
2019_08_05-PM-02_11_52
Theory : list_1
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