Nuprl Lemma : select_concat
∀[T:Type]
∀ll:T List List. ∀n:ℕ||concat(ll)||.
∃m:ℕ||ll||
((||concat(firstn(m;ll))|| ≤ n)
c∧ n - ||concat(firstn(m;ll))|| < ||ll[m]||
c∧ (concat(ll)[n] = ll[m][n - ||concat(firstn(m;ll))||] ∈ T))
Proof
Definitions occuring in Statement :
firstn: firstn(n;as),
select: L[n],
length: ||as||,
concat: concat(ll),
list: T List,
int_seg: {i..j-},
less_than: a < b,
uall: ∀[x:A]. B[x],
cand: A c∧ B,
le: A ≤ B,
all: ∀x:A. B[x],
exists: ∃x:A. B[x],
subtract: n - m,
natural_number: $n,
universe: Type,
equal: s = t ∈ T
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
so_lambda: λ2x.t[x],
so_apply: x[s],
implies: P ⇒ Q,
prop: ℙ,
cand: A c∧ B,
int_seg: {i..j-},
uimplies: b supposing a,
guard: {T},
lelt: i ≤ j < k,
and: P ∧ Q,
all: ∀x:A. B[x],
decidable: Dec(P),
or: P ∨ Q,
satisfiable_int_formula: satisfiable_int_formula(fmla),
exists: ∃x:A. B[x],
false: False,
not: ¬A,
top: Top,
less_than: a < b,
squash: ↓T,
le: A ≤ B,
concat: concat(ll),
select: L[n],
nil: [],
it: ⋅,
so_lambda: λ2x y.t[x; y],
so_apply: x[s1;s2],
less_than': less_than'(a;b),
nat_plus: ℕ+,
true: True,
uiff: uiff(P;Q),
subtype_rel: A ⊆r B,
cons: [a / b],
sq_type: SQType(T),
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q,
ge: i ≥ j ,
subtract: n - m,
firstn: firstn(n;as),
so_lambda: so_lambda(x,y,z.t[x; y; z]),
so_apply: x[s1;s2;s3],
bool: 𝔹,
unit: Unit,
btrue: tt,
ifthenelse: if b then t else f fi ,
bfalse: ff
Lemmas referenced :
list_induction,
list_wf,
all_wf,
int_seg_wf,
length_wf,
concat_wf,
exists_wf,
le_wf,
firstn_wf,
less_than_wf,
subtract_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,
equal_wf,
itermSubtract_wf,
int_term_value_subtract_lemma,
reduce_nil_lemma,
length_of_nil_lemma,
stuck-spread,
base_wf,
length_of_cons_lemma,
cons_wf,
reduce_cons_lemma,
false_wf,
add_nat_plus,
length_wf_nat,
nat_plus_wf,
nat_plus_properties,
add-is-int-iff,
itermAdd_wf,
intformeq_wf,
int_term_value_add_lemma,
int_formula_prop_eq_lemma,
lelt_wf,
subtype_rel_list,
top_wf,
subtype_base_sq,
int_subtype_base,
decidable__equal_int,
squash_wf,
true_wf,
select_append_front,
iff_weakening_equal,
first0,
non_neg_length,
append_wf,
length-append,
length_append,
add-member-int_seg2,
lt_int_wf,
bool_wf,
equal-wf-T-base,
assert_wf,
le_int_wf,
bnot_wf,
list_ind_cons_lemma,
uiff_transitivity,
eqtt_to_assert,
assert_of_lt_int,
eqff_to_assert,
assert_functionality_wrt_uiff,
bnot_of_lt_int,
assert_of_le_int,
add-subtract-cancel,
select-cons-tl,
select_append_back,
add-associates,
minus-one-mul,
add-commutes,
minus-one-mul-top,
minus-add
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
cut,
introduction,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
sqequalRule,
independent_functionElimination,
universeEquality,
cumulativity,
hypothesisEquality,
hypothesis,
lambdaEquality,
natural_numberEquality,
because_Cache,
productEquality,
setElimination,
rename,
independent_isectElimination,
productElimination,
dependent_functionElimination,
unionElimination,
dependent_pairFormation,
int_eqEquality,
intEquality,
isect_memberEquality,
voidElimination,
voidEquality,
independent_pairFormation,
computeAll,
imageElimination,
baseClosed,
lambdaFormation,
dependent_set_memberEquality,
imageMemberEquality,
equalityTransitivity,
equalitySymmetry,
applyLambdaEquality,
pointwiseFunctionality,
promote_hyp,
baseApply,
closedConclusion,
addEquality,
applyEquality,
instantiate,
equalityElimination,
hyp_replacement,
minusEquality
Latex:
\mforall{}[T:Type]
\mforall{}ll:T List List. \mforall{}n:\mBbbN{}||concat(ll)||.
\mexists{}m:\mBbbN{}||ll||
((||concat(firstn(m;ll))|| \mleq{} n)
c\mwedge{} n - ||concat(firstn(m;ll))|| < ||ll[m]||
c\mwedge{} (concat(ll)[n] = ll[m][n - ||concat(firstn(m;ll))||]))
Date html generated:
2017_04_14-AM-09_24_20
Last ObjectModification:
2017_02_27-PM-03_59_54
Theory : list_1
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