Nuprl Lemma : split_tail_trivial
∀[A:Type]. ∀[f:A ⟶ 𝔹]. ∀[L:A List].
split_tail(L | ∀x.f[x]) = <[], L> ∈ (A List × (A List)) supposing ∀b:A. ((b ∈ L) ⇒ (↑f[b]))
Proof
Definitions occuring in Statement :
split_tail: split_tail(L | ∀x.f[x]),
l_member: (x ∈ l),
nil: [],
list: T List,
assert: ↑b,
bool: 𝔹,
uimplies: b supposing a,
uall: ∀[x:A]. B[x],
so_apply: x[s],
all: ∀x:A. B[x],
implies: P ⇒ Q,
function: x:A ⟶ B[x],
pair: <a, b>,
product: 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],
uimplies: b supposing a,
implies: P ⇒ Q,
prop: ℙ,
so_apply: x[s],
all: ∀x:A. B[x],
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],
iff: P ⇐⇒ Q,
and: P ∧ Q,
rev_implies: P ⇐ Q,
guard: {T},
or: P ∨ Q,
bool: 𝔹,
unit: Unit,
it: ⋅,
btrue: tt,
uiff: uiff(P;Q),
ifthenelse: if b then t else f fi ,
bfalse: ff,
not: ¬A,
false: False
Lemmas referenced :
list_induction,
isect_wf,
all_wf,
l_member_wf,
assert_wf,
equal_wf,
list_wf,
split_tail_wf,
nil_wf,
list_ind_nil_lemma,
list_ind_cons_lemma,
bool_wf,
cons_wf,
cons_member,
equal-wf-T-base,
bnot_wf,
not_wf,
eqtt_to_assert,
uiff_transitivity,
eqff_to_assert,
assert_of_bnot,
list_ind_wf,
ifthenelse_wf
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation_alt,
introduction,
cut,
thin,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
hypothesisEquality,
sqequalRule,
lambdaEquality,
functionEquality,
hypothesis,
applyEquality,
productEquality,
independent_pairEquality,
independent_functionElimination,
dependent_functionElimination,
isect_memberEquality,
voidElimination,
voidEquality,
lambdaFormation,
rename,
because_Cache,
axiomEquality,
equalityTransitivity,
equalitySymmetry,
universeIsType,
functionIsType,
universeEquality,
independent_isectElimination,
inhabitedIsType,
lambdaFormation_alt,
productElimination,
inrFormation,
baseClosed,
unionElimination,
equalityElimination,
hyp_replacement,
applyLambdaEquality,
spreadEquality,
inlFormation
Latex:
\mforall{}[A:Type]. \mforall{}[f:A {}\mrightarrow{} \mBbbB{}]. \mforall{}[L:A List].
split\_tail(L | \mforall{}x.f[x]) = <[], L> supposing \mforall{}b:A. ((b \mmember{} L) {}\mRightarrow{} (\muparrow{}f[b]))
Date html generated:
2019_10_15-AM-10_54_41
Last ObjectModification:
2018_09_27-AM-10_18_53
Theory : list!
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