Nuprl Lemma : no_repeats_member
∀[T:Type]. ∀L:T List. ∀x:T. (x ∈ L) ⇒ (x ∈! L) supposing no_repeats(T;L)
Proof
Definitions occuring in Statement :
l_member!: (x ∈! 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],
implies: P ⇒ Q,
universe: Type
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,
so_apply: x[s],
not: ¬A,
false: False,
iff: P ⇐⇒ Q,
and: P ∧ Q,
rev_implies: P ⇐ Q,
uiff: uiff(P;Q),
or: P ∨ Q,
cand: A c∧ B,
guard: {T}
Lemmas referenced :
list_induction,
all_wf,
isect_wf,
no_repeats_wf,
l_member_wf,
l_member!_wf,
list_wf,
no_repeats_witness,
nil_wf,
null_nil_lemma,
btrue_wf,
member-implies-null-eq-bfalse,
btrue_neq_bfalse,
cons_wf,
cons_member,
cons_member!,
no_repeats_cons,
not_wf,
equal_wf,
and_wf
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
lambdaFormation,
cut,
thin,
introduction,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
hypothesisEquality,
sqequalRule,
lambdaEquality,
cumulativity,
hypothesis,
functionEquality,
independent_functionElimination,
because_Cache,
rename,
independent_isectElimination,
equalityTransitivity,
equalitySymmetry,
voidElimination,
dependent_functionElimination,
universeEquality,
productElimination,
unionElimination,
inlFormation,
independent_pairFormation,
productEquality,
inrFormation,
hyp_replacement,
dependent_set_memberEquality,
applyLambdaEquality,
setElimination
Latex:
\mforall{}[T:Type]. \mforall{}L:T List. \mforall{}x:T. (x \mmember{} L) {}\mRightarrow{} (x \mmember{}! L) supposing no\_repeats(T;L)
Date html generated:
2017_04_17-AM-07_50_29
Last ObjectModification:
2017_02_27-PM-04_23_34
Theory : list_1
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