Nuprl Lemma : split-by-indices
∀[T:Type]
∀L:T List. ∀ids:ℕ List.
∃L1,L2:T List. (permutation(T;L;L1 @ L2) ∧ (∃f:ℕ||L1|| ⟶ {i:ℕ||L||| (i ∈ ids)} . (Bij(ℕ||L1||;{i:ℕ||L||| (i ∈ ids)}\000C ;f) ∧ (∀j:ℕ||L1||. (L1[j] = L[f j] ∈ T)))))
Proof
Definitions occuring in Statement :
permutation: permutation(T;L1;L2),
l_member: (x ∈ l),
select: L[n],
length: ||as||,
append: as @ bs,
list: T List,
biject: Bij(A;B;f),
int_seg: {i..j-},
nat: ℕ,
uall: ∀[x:A]. B[x],
all: ∀x:A. B[x],
exists: ∃x:A. B[x],
and: P ∧ Q,
set: {x:A| B[x]} ,
apply: f a,
function: x:A ⟶ B[x],
natural_number: $n,
int: ℤ,
universe: Type,
equal: s = t ∈ T
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
all: ∀x:A. B[x],
member: t ∈ T,
exists: ∃x:A. B[x],
implies: P ⇒ Q,
top: Top,
prop: ℙ,
so_lambda: λ2x.t[x],
so_apply: x[s],
and: P ∧ Q,
cand: A c∧ B,
let: let,
int_seg: {i..j-},
subtype_rel: A ⊆r B,
uimplies: b supposing a,
nat: ℕ,
guard: {T},
lelt: i ≤ j < k,
decidable: Dec(P),
or: P ∨ Q,
satisfiable_int_formula: satisfiable_int_formula(fmla),
false: False,
not: ¬A,
less_than: a < b,
squash: ↓T,
index-split: index-split(L;idxs),
pi1: fst(t),
pi2: snd(t)
Lemmas referenced :
index-split_property,
index-split_wf,
list_wf,
pi1_wf_top,
equal_wf,
pi2_wf,
permutation_wf,
append_wf,
exists_wf,
int_seg_wf,
length_wf,
l_member_wf,
subtype_rel_list,
nat_wf,
biject_wf,
all_wf,
select_wf,
int_seg_properties,
decidable__le,
satisfiable-full-omega-tt,
intformand_wf,
intformnot_wf,
intformle_wf,
itermConstant_wf,
itermVar_wf,
int_formula_prop_and_lemma,
int_formula_prop_not_lemma,
int_formula_prop_le_lemma,
int_term_value_constant_lemma,
int_term_value_var_lemma,
int_formula_prop_wf,
decidable__lt,
intformless_wf,
int_formula_prop_less_lemma,
zero-le-nat,
set_wf,
index-split-permutation,
firstn_wf,
permute-to-front_wf,
filter_wf5,
upto_wf,
int-list-member_wf,
nth_tl_wf
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
lambdaFormation,
cut,
introduction,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
hypothesisEquality,
dependent_functionElimination,
dependent_pairFormation,
cumulativity,
hypothesis,
productEquality,
productElimination,
independent_pairEquality,
isect_memberEquality,
voidElimination,
voidEquality,
equalityTransitivity,
equalitySymmetry,
independent_functionElimination,
because_Cache,
sqequalRule,
lambdaEquality,
independent_pairFormation,
functionEquality,
natural_numberEquality,
setEquality,
intEquality,
setElimination,
rename,
applyEquality,
independent_isectElimination,
functionExtensionality,
unionElimination,
int_eqEquality,
computeAll,
imageElimination,
universeEquality
Latex:
\mforall{}[T:Type]
\mforall{}L:T List. \mforall{}ids:\mBbbN{} List.
\mexists{}L1,L2:T List
(permutation(T;L;L1 @ L2)
\mwedge{} (\mexists{}f:\mBbbN{}||L1|| {}\mrightarrow{} \{i:\mBbbN{}||L||| (i \mmember{} ids)\} . (Bij(\mBbbN{}||L1||;\{i:\mBbbN{}||L||| (i \mmember{} ids)\} ;f) \mwedge{} (\mforall{}j:\mBbbN{}||L1||. \000C(L1[j] = L[f j])))))
Date html generated:
2018_05_21-PM-07_33_06
Last ObjectModification:
2017_07_26-PM-05_08_02
Theory : general
Home
Index