Nuprl Lemma : before-adjacent2
∀[T:Type]
∀L:T List. ∀x,y:T.
adjacent(T;L;x;y) ⇒ (∀z:T. (x before z ∈ L ⇒ (y before z ∈ L ∨ (z = y ∈ T)))) supposing no_repeats(T;L)
Proof
Definitions occuring in Statement :
adjacent: adjacent(T;L;x;y),
l_before: x before y ∈ l,
no_repeats: no_repeats(T;l),
list: T List,
uimplies: b supposing a,
uall: ∀[x:A]. B[x],
all: ∀x:A. B[x],
implies: 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],
uimplies: b supposing a,
member: t ∈ T,
implies: P ⇒ Q,
guard: {T},
and: P ∧ Q,
or: P ∨ Q,
prop: ℙ,
uiff: uiff(P;Q),
not: ¬A,
false: False,
squash: ↓T,
true: True
Lemmas referenced :
no_repeats_witness,
adjacent-member,
l_before_member,
l_tricotomy,
l_before_wf,
equal_wf,
adjacent_wf,
no_repeats_wf,
list_wf,
before-adjacent,
l_before_antisymmetry,
no_repeats_iff,
squash_wf,
true_wf
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
lambdaFormation,
cut,
introduction,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
hypothesisEquality,
independent_functionElimination,
hypothesis,
rename,
because_Cache,
dependent_functionElimination,
productElimination,
unionElimination,
sqequalRule,
inrFormation,
cumulativity,
inlFormation,
universeEquality,
independent_isectElimination,
voidElimination,
equalitySymmetry,
hyp_replacement,
applyEquality,
lambdaEquality,
imageElimination,
equalityTransitivity,
natural_numberEquality,
imageMemberEquality,
baseClosed
Latex:
\mforall{}[T:Type]
\mforall{}L:T List. \mforall{}x,y:T.
adjacent(T;L;x;y) {}\mRightarrow{} (\mforall{}z:T. (x before z \mmember{} L {}\mRightarrow{} (y before z \mmember{} L \mvee{} (z = y))))
supposing no\_repeats(T;L)
Date html generated:
2016_10_25-AM-10_46_25
Last ObjectModification:
2016_07_12-AM-06_55_53
Theory : general
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