Nuprl Lemma : split_tail_rel
∀[A:Type]. ∀[f:A ⟶ 𝔹]. ∀[L:A List]. (((fst(split_tail(L | ∀x.f[x]))) @ (snd(split_tail(L | ∀x.f[x])))) = L ∈ (A List))
Proof
Definitions occuring in Statement :
split_tail: split_tail(L | ∀x.f[x]),
append: as @ bs,
list: T List,
bool: 𝔹,
uall: ∀[x:A]. B[x],
so_apply: x[s],
pi1: fst(t),
pi2: snd(t),
function: x:A ⟶ B[x],
universe: Type,
equal: s = t ∈ T
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
so_lambda: λ2x.t[x],
so_apply: x[s],
all: ∀x:A. B[x],
implies: P ⇒ Q,
prop: ℙ,
split_tail: split_tail(L | ∀x.f[x]),
so_lambda: so_lambda(x,y,z.t[x; y; z]),
top: Top,
so_apply: x[s1;s2;s3],
pi1: fst(t),
pi2: snd(t),
append: as @ bs,
bool: 𝔹,
unit: Unit,
it: ⋅,
btrue: tt,
uiff: uiff(P;Q),
and: P ∧ Q,
uimplies: b supposing a,
ifthenelse: if b then t else f fi ,
bfalse: ff,
exists: ∃x:A. B[x],
or: P ∨ Q,
sq_type: SQType(T),
guard: {T},
bnot: ¬bb,
assert: ↑b,
false: False,
squash: ↓T,
true: True,
subtype_rel: A ⊆r B,
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q
Lemmas referenced :
list_induction,
equal_wf,
list_wf,
append_wf,
split_tail_wf,
pi1_wf,
pi2_wf,
list_ind_nil_lemma,
list_ind_cons_lemma,
bool_wf,
nil_wf,
eqtt_to_assert,
eqff_to_assert,
bool_cases_sqequal,
subtype_base_sq,
bool_subtype_base,
assert-bnot,
list_ind_wf,
cons_wf,
and_wf,
squash_wf,
true_wf,
subtype_rel_self,
iff_weakening_equal
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation_alt,
introduction,
cut,
thin,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
hypothesisEquality,
sqequalRule,
lambdaEquality,
hypothesis,
applyEquality,
productEquality,
lambdaFormation,
equalityTransitivity,
equalitySymmetry,
dependent_functionElimination,
independent_functionElimination,
because_Cache,
isect_memberEquality,
voidElimination,
voidEquality,
rename,
universeIsType,
axiomEquality,
functionIsType,
functionEquality,
universeEquality,
productElimination,
unionElimination,
equalityElimination,
independent_isectElimination,
dependent_pairFormation,
promote_hyp,
instantiate,
cumulativity,
independent_pairEquality,
dependent_set_memberEquality,
independent_pairFormation,
applyLambdaEquality,
setElimination,
imageElimination,
natural_numberEquality,
imageMemberEquality,
baseClosed
Latex:
\mforall{}[A:Type]. \mforall{}[f:A {}\mrightarrow{} \mBbbB{}]. \mforall{}[L:A List].
(((fst(split\_tail(L | \mforall{}x.f[x]))) @ (snd(split\_tail(L | \mforall{}x.f[x])))) = L)
Date html generated:
2019_10_15-AM-10_54_53
Last ObjectModification:
2018_09_27-AM-10_18_49
Theory : list!
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