Nuprl Lemma : index-split_property
∀[T:Type]
∀L:T List. ∀idxs:ℕ List.
let L1 = fst(index-split(L;idxs)) in
∃f:ℕ||L1|| ⟶ {i:ℕ||L||| (i ∈ idxs)} . (Bij(ℕ||L1||;{i:ℕ||L||| (i ∈ idxs)} ;f) ∧ (∀j:ℕ||L1||. (L1[j] = L[f j] ∈ \000CT)))
Proof
Definitions occuring in Statement :
index-split: index-split(L;idxs),
l_member: (x ∈ l),
select: L[n],
length: ||as||,
list: T List,
biject: Bij(A;B;f),
int_seg: {i..j-},
nat: ℕ,
let: let,
uall: ∀[x:A]. B[x],
pi1: fst(t),
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,
uimplies: b supposing a,
sq_type: SQType(T),
implies: P ⇒ Q,
guard: {T},
int_seg: {i..j-},
subtype_rel: A ⊆r B,
nat: ℕ,
index-split: index-split(L;idxs),
let: let,
pi1: fst(t),
prop: ℙ,
so_lambda: λ2x.t[x],
so_apply: x[s],
iff: P ⇐⇒ Q,
and: P ∧ Q,
int_iseg: {i...j},
cand: A c∧ B,
ge: i ≥ j ,
decidable: Dec(P),
or: P ∨ Q,
false: False,
le: A ≤ B,
not: ¬A,
satisfiable_int_formula: satisfiable_int_formula(fmla),
exists: ∃x:A. B[x],
top: Top,
lelt: i ≤ j < k,
less_than: a < b,
squash: ↓T,
true: True,
rev_implies: P ⇐ Q,
less_than': less_than'(a;b),
biject: Bij(A;B;f),
uiff: uiff(P;Q),
no_repeats: no_repeats(T;l),
surject: Surj(A;B;f),
sq_stable: SqStable(P),
l_member: (x ∈ l),
permute-to-front: permute-to-front(L;idxs)
Lemmas referenced :
permute-to-front-permutation,
permutation-length,
permute-to-front_wf,
subtype_base_sq,
int_subtype_base,
length_upto,
length_wf_nat,
length-filter,
int_seg_wf,
length_wf,
int-list-member_wf,
subtype_rel_list,
nat_wf,
istype-nat,
upto_wf,
list_wf,
istype-universe,
filter_type,
assert_wf,
l_member_wf,
subtype_rel_sets_simple,
assert-int-list-member,
istype-assert,
list_subtype_base,
set_subtype_base,
lelt_wf,
non_neg_length,
decidable__le,
full-omega-unsat,
intformand_wf,
intformnot_wf,
intformle_wf,
itermConstant_wf,
itermVar_wf,
istype-int,
int_formula_prop_and_lemma,
istype-void,
int_formula_prop_not_lemma,
int_formula_prop_le_lemma,
int_term_value_constant_lemma,
int_term_value_var_lemma,
int_formula_prop_wf,
istype-le,
length_firstn,
select_wf,
int_seg_properties,
decidable__lt,
intformless_wf,
int_formula_prop_less_lemma,
biject_wf,
firstn_wf,
less_than_wf,
length_firstn_eq,
iff_weakening_equal,
zero-le-nat,
subtype_rel_set,
int_seg_subtype_nat,
istype-false,
no_repeats_inject,
no_repeats_upto,
no_repeats_filter,
istype-less_than,
le_wf,
decidable__equal_int_seg,
decidable__equal_set,
sq_stable__l_member,
member_filter,
member_upto,
int_formula_prop_eq_lemma,
intformeq_wf,
decidable__equal_int,
nat_properties,
equal_wf,
squash_wf,
true_wf,
select_firstn,
subtype_rel_self,
permute_list_select,
append_wf,
filter_wf5,
bnot_wf,
length-append,
filter-split-length,
select_append_front
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation_alt,
lambdaFormation_alt,
cut,
introduction,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
hypothesisEquality,
dependent_functionElimination,
because_Cache,
hypothesis,
independent_isectElimination,
instantiate,
cumulativity,
intEquality,
equalityTransitivity,
equalitySymmetry,
independent_functionElimination,
sqequalRule,
natural_numberEquality,
lambdaEquality_alt,
setElimination,
rename,
applyEquality,
universeIsType,
universeEquality,
setEquality,
productElimination,
inhabitedIsType,
equalityIstype,
setIsType,
dependent_set_memberEquality_alt,
unionElimination,
approximateComputation,
dependent_pairFormation_alt,
int_eqEquality,
isect_memberEquality_alt,
voidElimination,
independent_pairFormation,
productIsType,
closedConclusion,
imageElimination,
functionIsType,
imageMemberEquality,
baseClosed,
isectIsTypeImplies,
functionIsTypeImplies,
sqequalBase,
baseApply,
applyLambdaEquality,
promote_hyp
Latex:
\mforall{}[T:Type]
\mforall{}L:T List. \mforall{}idxs:\mBbbN{} List.
let L1 = fst(index-split(L;idxs)) in
\mexists{}f:\mBbbN{}||L1|| {}\mrightarrow{} \{i:\mBbbN{}||L||| (i \mmember{} idxs)\} . (Bij(\mBbbN{}||L1||;\{i:\mBbbN{}||L||| (i \mmember{} idxs)\} ;f) \mwedge{} (\mforall{}j:\mBbbN{}||L1||\000C. (L1[j] = L[f j])))
Date html generated:
2019_10_15-AM-11_13_01
Last ObjectModification:
2019_06_25-PM-01_22_43
Theory : general
Home
Index