Nuprl Lemma : select_append_back
∀[T:Type]. ∀[as,bs:T List]. ∀[i:{||as||..||as|| + ||bs||-}]. (as @ bs[i] = bs[i - ||as||] ∈ T)
Proof
Definitions occuring in Statement :
select: L[n],
length: ||as||,
append: as @ bs,
list: T List,
int_seg: {i..j-},
uall: ∀[x:A]. B[x],
subtract: n - m,
add: n + m,
universe: Type,
equal: s = t ∈ T
Definitions unfolded in proof :
member: t ∈ T,
uall: ∀[x:A]. B[x],
so_lambda: λ2x.t[x],
int_seg: {i..j-},
uimplies: b supposing a,
all: ∀x:A. B[x],
implies: P ⇒ Q,
exists: ∃x:A. B[x],
subtype_rel: A ⊆r B,
nat: ℕ,
so_apply: x[s],
prop: ℙ,
top: Top,
lelt: i ≤ j < k,
and: P ∧ Q,
append: as @ bs,
so_lambda: so_lambda(x,y,z.t[x; y; z]),
so_apply: x[s1;s2;s3],
le: A ≤ B,
ge: i ≥ j ,
subtract: n - m,
nat_plus: ℕ+,
less_than: a < b,
squash: ↓T,
less_than': less_than'(a;b),
true: True,
uiff: uiff(P;Q),
false: False,
not: ¬A,
decidable: Dec(P),
or: P ∨ Q,
cand: A c∧ B,
sq_stable: SqStable(P),
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q,
guard: {T}
Lemmas referenced :
int_seg_wf,
length_wf,
list_wf,
list_induction,
all_wf,
equal_wf,
select_wf,
append_wf,
non_neg_length,
length_wf_nat,
nat_wf,
set_subtype_base,
le_wf,
int_subtype_base,
length-append,
subtract_wf,
length_of_nil_lemma,
list_ind_nil_lemma,
length_of_cons_lemma,
list_ind_cons_lemma,
add_functionality_wrt_le,
le_reflexive,
minus-one-mul,
zero-add,
one-mul,
add-mul-special,
add-associates,
two-mul,
add-commutes,
mul-distributes-right,
zero-mul,
add-swap,
omega-shadow,
less_than_wf,
mul-commutes,
mul-associates,
not-le-2,
false_wf,
minus-zero,
add-zero,
minus-one-mul-top,
mul-distributes,
le-add-cancel,
less-iff-le,
minus-add,
not-lt-2,
mul-swap,
int_seg_properties,
nat_properties,
decidable__le,
decidable__lt,
squash_wf,
sq_stable__le,
condition-implies-le,
true_wf,
select_cons_tl,
length_append,
subtype_rel_list,
top_wf,
iff_weakening_equal,
le-add-cancel2,
minus-minus,
lelt_wf,
le-add-cancel-alt,
add-is-int-iff
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
cut,
introduction,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
cumulativity,
hypothesisEquality,
hypothesis,
addEquality,
because_Cache,
universeEquality,
isect_memberFormation,
sqequalRule,
isect_memberEquality,
axiomEquality,
lambdaEquality,
setElimination,
rename,
independent_isectElimination,
lambdaFormation,
dependent_pairFormation,
sqequalIntensionalEquality,
applyEquality,
intEquality,
natural_numberEquality,
equalityTransitivity,
equalitySymmetry,
dependent_functionElimination,
independent_functionElimination,
productElimination,
promote_hyp,
voidElimination,
voidEquality,
multiplyEquality,
minusEquality,
dependent_set_memberEquality,
independent_pairFormation,
imageMemberEquality,
baseClosed,
unionElimination,
imageElimination,
productEquality,
baseApply,
closedConclusion
Latex:
\mforall{}[T:Type]. \mforall{}[as,bs:T List]. \mforall{}[i:\{||as||..||as|| + ||bs||\msupminus{}\}]. (as @ bs[i] = bs[i - ||as||])
Date html generated:
2017_04_14-AM-08_38_00
Last ObjectModification:
2017_02_27-PM-03_30_08
Theory : list_0
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