Nuprl Lemma : before-adjacent
∀[T:Type]
∀L:T List. ∀x,y:T.
adjacent(T;L;x;y) ⇒ (∀z:T. (z before y ∈ L ⇒ (z before x ∈ L ∨ (z = x ∈ 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],
member: t ∈ T,
so_lambda: λ2x.t[x],
uimplies: b supposing a,
prop: ℙ,
implies: P ⇒ Q,
or: P ∨ Q,
so_apply: x[s],
false: False,
iff: P ⇐⇒ Q,
and: P ∧ Q,
rev_implies: P ⇐ Q,
guard: {T},
uiff: uiff(P;Q),
squash: ↓T,
true: True,
subtype_rel: A ⊆r B,
not: ¬A,
cand: A c∧ B
Lemmas referenced :
list_induction,
all_wf,
isect_wf,
no_repeats_wf,
adjacent_wf,
l_before_wf,
or_wf,
equal_wf,
list_wf,
no_repeats_witness,
nil_wf,
cons_wf,
adjacent-nil,
adjacent-cons,
cons_before,
l_member_wf,
no_repeats_cons,
hd-before,
l_before_member2,
no_repeats_iff,
not_wf,
squash_wf,
true_wf,
iff_weakening_equal,
l_before_antisymmetry,
adjacent-member
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
lambdaFormation,
cut,
thin,
introduction,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
hypothesisEquality,
sqequalRule,
lambdaEquality,
cumulativity,
because_Cache,
hypothesis,
functionEquality,
independent_functionElimination,
rename,
dependent_functionElimination,
universeEquality,
independent_isectElimination,
voidElimination,
productElimination,
unionElimination,
addLevel,
orFunctionality,
productEquality,
inrFormation,
equalityTransitivity,
equalitySymmetry,
applyEquality,
imageElimination,
natural_numberEquality,
imageMemberEquality,
baseClosed,
hyp_replacement,
applyLambdaEquality,
inlFormation,
independent_pairFormation
Latex:
\mforall{}[T:Type]
\mforall{}L:T List. \mforall{}x,y:T.
adjacent(T;L;x;y) {}\mRightarrow{} (\mforall{}z:T. (z before y \mmember{} L {}\mRightarrow{} (z before x \mmember{} L \mvee{} (z = x))))
supposing no\_repeats(T;L)
Date html generated:
2018_05_21-PM-06_39_31
Last ObjectModification:
2017_07_26-PM-04_53_27
Theory : general
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