Nuprl Lemma : remove_leading_property
∀[T:Type]. ∀L:T List. ∀P:T ⟶ 𝔹. ∃xs:{x:T| ↑P[x]} List. (L = (xs @ remove_leading(x.P[x];L)) ∈ (T List))
Proof
Definitions occuring in Statement :
remove_leading: remove_leading(a.P[a];L),
append: as @ bs,
list: T List,
assert: ↑b,
bool: 𝔹,
uall: ∀[x:A]. B[x],
so_apply: x[s],
all: ∀x:A. B[x],
exists: ∃x:A. B[x],
set: {x:A| B[x]} ,
function: x:A ⟶ B[x],
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],
prop: ℙ,
subtype_rel: A ⊆r B,
uimplies: b supposing a,
implies: P ⇒ Q,
top: Top,
or: P ∨ Q,
assert: ↑b,
ifthenelse: if b then t else f fi ,
btrue: tt,
not: ¬A,
true: True,
false: False,
cons: [a / b],
bfalse: ff,
guard: {T},
nat: ℕ,
le: A ≤ B,
and: P ∧ Q,
decidable: Dec(P),
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q,
uiff: uiff(P;Q),
subtract: n - m,
less_than': less_than'(a;b),
listp: A List+,
remove_leading: remove_leading(a.P[a];L),
so_lambda: so_lambda(x,y,z.t[x; y; z]),
so_apply: x[s1;s2;s3],
exists: ∃x:A. B[x],
bool: 𝔹,
unit: Unit,
it: ⋅,
sq_type: SQType(T),
bnot: ¬bb,
append: as @ bs,
squash: ↓T
Lemmas referenced :
list_induction,
all_wf,
bool_wf,
exists_wf,
list_wf,
assert_wf,
equal_wf,
append_wf,
subtype_rel_list,
remove_leading_wf,
not_wf,
null_wf3,
top_wf,
hd_wf,
listp_properties,
list-cases,
length_of_nil_lemma,
null_nil_lemma,
product_subtype_list,
length_of_cons_lemma,
null_cons_lemma,
length_wf_nat,
nat_wf,
decidable__lt,
false_wf,
not-lt-2,
condition-implies-le,
minus-add,
minus-one-mul,
zero-add,
minus-one-mul-top,
add-commutes,
add_functionality_wrt_le,
add-associates,
add-zero,
le-add-cancel,
less_than_wf,
length_wf,
list_ind_nil_lemma,
nil_wf,
append_back_nil,
equal-wf-base-T,
list_ind_cons_lemma,
eqtt_to_assert,
eqff_to_assert,
bool_cases_sqequal,
subtype_base_sq,
bool_subtype_base,
assert-bnot,
cons_wf,
squash_wf,
true_wf,
iff_weakening_equal,
nil-append
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
lambdaFormation,
cut,
thin,
introduction,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
hypothesisEquality,
sqequalRule,
lambdaEquality,
functionEquality,
cumulativity,
hypothesis,
setEquality,
because_Cache,
applyEquality,
functionExtensionality,
independent_isectElimination,
setElimination,
rename,
isect_memberEquality,
voidElimination,
voidEquality,
dependent_functionElimination,
unionElimination,
independent_functionElimination,
natural_numberEquality,
promote_hyp,
hypothesis_subsumption,
productElimination,
addEquality,
independent_pairFormation,
intEquality,
minusEquality,
equalityTransitivity,
equalitySymmetry,
dependent_set_memberEquality,
universeEquality,
dependent_pairFormation,
baseClosed,
equalityElimination,
instantiate,
imageElimination,
imageMemberEquality
Latex:
\mforall{}[T:Type]. \mforall{}L:T List. \mforall{}P:T {}\mrightarrow{} \mBbbB{}. \mexists{}xs:\{x:T| \muparrow{}P[x]\} List. (L = (xs @ remove\_leading(x.P[x];L)))
Date html generated:
2018_05_21-PM-06_43_31
Last ObjectModification:
2017_07_26-PM-04_54_41
Theory : general
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