Nuprl Lemma : l_member_decomp
∀[T:Type]. ∀l:T List. ∀x:T. ((x ∈ l) ⇐⇒ ∃l1,l2:T List. (l = (l1 @ [x] @ l2) ∈ (T List)))
Proof
Definitions occuring in Statement :
l_member: (x ∈ l),
append: as @ bs,
cons: [a / b],
nil: [],
list: T List,
uall: ∀[x:A]. B[x],
all: ∀x:A. B[x],
exists: ∃x:A. B[x],
iff: 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],
so_apply: x[s],
implies: P ⇒ Q,
iff: P ⇐⇒ Q,
and: P ∧ Q,
false: False,
prop: ℙ,
rev_implies: P ⇐ Q,
exists: ∃x:A. B[x],
or: P ∨ Q,
so_lambda: so_lambda(x,y,z.t[x; y; z]),
top: Top,
so_apply: x[s1;s2;s3],
cons: [a / b],
uimplies: b supposing a,
sq_type: SQType(T),
guard: {T},
true: True,
not: ¬A,
uiff: uiff(P;Q),
append: as @ bs,
squash: ↓T,
cand: A c∧ B,
nat: ℕ,
ge: i ≥ j ,
decidable: Dec(P),
sq_stable: SqStable(P),
subtract: n - m,
subtype_rel: A ⊆r B,
le: A ≤ B,
less_than': less_than'(a;b)
Lemmas referenced :
list_induction,
all_wf,
iff_wf,
l_member_wf,
exists_wf,
list_wf,
equal_wf,
append_wf,
cons_wf,
nil_wf,
false_wf,
equal-wf-base-T,
nil_member,
or_wf,
cons_member,
equal-wf-T-base,
list-cases,
list_ind_nil_lemma,
product_subtype_list,
list_ind_cons_lemma,
subtype_base_sq,
int_subtype_base,
null_nil_lemma,
btrue_wf,
null_cons_lemma,
bfalse_wf,
and_wf,
null_wf,
btrue_neq_bfalse,
iff_transitivity,
iff_weakening_uiff,
append_is_nil,
squash_wf,
true_wf,
hd_wf,
length_of_nil_lemma,
cons_neq_nil,
length_of_cons_lemma,
length_wf_nat,
nat_wf,
decidable__le,
not-ge-2,
sq_stable__le,
condition-implies-le,
minus-add,
minus-one-mul,
add-swap,
minus-one-mul-top,
add-associates,
add-commutes,
add_functionality_wrt_le,
add-zero,
le-add-cancel2,
reduce_hd_cons_lemma,
tl_wf,
reduce_tl_cons_lemma,
member_append
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
lambdaFormation,
cut,
thin,
introduction,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
hypothesisEquality,
sqequalRule,
lambdaEquality,
cumulativity,
hypothesis,
independent_functionElimination,
independent_pairFormation,
voidElimination,
productElimination,
baseClosed,
because_Cache,
addLevel,
allFunctionality,
impliesFunctionality,
dependent_functionElimination,
rename,
universeEquality,
equalitySymmetry,
equalityTransitivity,
productEquality,
applyLambdaEquality,
unionElimination,
isect_memberEquality,
voidEquality,
natural_numberEquality,
promote_hyp,
hypothesis_subsumption,
instantiate,
intEquality,
independent_isectElimination,
dependent_set_memberEquality,
setElimination,
dependent_pairFormation,
applyEquality,
imageElimination,
imageMemberEquality,
addEquality,
minusEquality,
inlFormation,
inrFormation,
hyp_replacement
Latex:
\mforall{}[T:Type]. \mforall{}l:T List. \mforall{}x:T. ((x \mmember{} l) \mLeftarrow{}{}\mRightarrow{} \mexists{}l1,l2:T List. (l = (l1 @ [x] @ l2)))
Date html generated:
2017_04_14-AM-08_41_14
Last ObjectModification:
2017_02_27-PM-03_31_54
Theory : list_0
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