Nuprl Lemma : no_repeats-before-equality
∀[T:Type]
∀as,bs:T List.
(as = bs ∈ (T List)
⇐⇒ (∀x:T. ((x ∈ as) ⇐⇒ (x ∈ bs))) ∧ (∀x,y:T. (x before y ∈ as ⇐⇒ x before y ∈ bs))) supposing
(no_repeats(T;bs) and
no_repeats(T;as))
Proof
Definitions occuring in Statement :
l_before: x before y ∈ l,
no_repeats: no_repeats(T;l),
l_member: (x ∈ l),
list: T List,
uimplies: b supposing a,
uall: ∀[x:A]. B[x],
all: ∀x:A. B[x],
iff: P ⇐⇒ Q,
and: 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,
iff: P ⇐⇒ Q,
and: P ∧ Q,
prop: ℙ,
rev_implies: P ⇐ Q,
so_lambda: λ2x.t[x],
so_apply: x[s],
or: P ∨ Q,
not: ¬A,
false: False,
uiff: uiff(P;Q),
guard: {T},
cand: A c∧ B,
squash: ↓T,
true: True
Lemmas referenced :
no_repeats_witness,
and_wf,
equal_wf,
list_wf,
l_member_wf,
l_before_wf,
all_wf,
iff_wf,
no_repeats_wf,
list_induction,
nil_wf,
equal-wf-base-T,
cons_wf,
cons_member,
null_nil_lemma,
btrue_wf,
member-implies-null-eq-bfalse,
btrue_neq_bfalse,
nil_member,
no_repeats_cons,
cons_before,
or_wf,
l_before_member,
l_before_member2,
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,
independent_pairFormation,
addLevel,
hyp_replacement,
equalitySymmetry,
sqequalRule,
dependent_set_memberEquality,
applyLambdaEquality,
setElimination,
productElimination,
levelHypothesis,
cumulativity,
productEquality,
lambdaEquality,
functionEquality,
baseClosed,
dependent_functionElimination,
inlFormation,
independent_isectElimination,
equalityTransitivity,
voidElimination,
impliesFunctionality,
allFunctionality,
allLevelFunctionality,
andLevelFunctionality,
impliesLevelFunctionality,
promote_hyp,
inrFormation,
unionElimination,
applyEquality,
imageElimination,
universeEquality,
natural_numberEquality,
imageMemberEquality
Latex:
\mforall{}[T:Type]
\mforall{}as,bs:T List.
(as = bs
\mLeftarrow{}{}\mRightarrow{} (\mforall{}x:T. ((x \mmember{} as) \mLeftarrow{}{}\mRightarrow{} (x \mmember{} bs)))
\mwedge{} (\mforall{}x,y:T. (x before y \mmember{} as \mLeftarrow{}{}\mRightarrow{} x before y \mmember{} bs))) supposing
(no\_repeats(T;bs) and
no\_repeats(T;as))
Date html generated:
2018_05_21-PM-07_39_27
Last ObjectModification:
2017_07_26-PM-05_13_44
Theory : general
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