Nuprl Lemma : select-unshuffle
∀[T:Type]. ∀as:T List. ∀i:ℕ||unshuffle(as)||. (unshuffle(as)[i] = <as[2 * i], as[(2 * i) + 1]> ∈ (T × T))
Proof
Definitions occuring in Statement :
unshuffle: unshuffle(L),
select: L[n],
length: ||as||,
list: T List,
int_seg: {i..j-},
uall: ∀[x:A]. B[x],
all: ∀x:A. B[x],
pair: <a, b>,
product: x:A × B[x],
multiply: n * m,
add: n + m,
natural_number: $n,
universe: Type,
equal: s = t ∈ T
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
all: ∀x:A. B[x],
nat: ℕ,
implies: P ⇒ Q,
false: False,
ge: i ≥ j ,
uimplies: b supposing a,
not: ¬A,
satisfiable_int_formula: satisfiable_int_formula(fmla),
exists: ∃x:A. B[x],
top: Top,
and: P ∧ Q,
prop: ℙ,
guard: {T},
int_seg: {i..j-},
lelt: i ≤ j < k,
decidable: Dec(P),
or: P ∨ Q,
subtype_rel: A ⊆r B,
so_lambda: λ2x.t[x],
so_apply: x[s],
sq_type: SQType(T),
le: A ≤ B,
less_than: a < b,
squash: ↓T,
unshuffle: unshuffle(L),
lt_int: i <z j,
ifthenelse: if b then t else f fi ,
btrue: tt,
cons: [a / b],
true: True,
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q,
uiff: uiff(P;Q),
bfalse: ff,
bool: 𝔹,
unit: Unit,
it: ⋅,
select: L[n],
subtract: n - m,
less_than': less_than'(a;b),
nat_plus: ℕ+,
nequal: a ≠ b ∈ T ,
int_nzero: ℤ-o
Lemmas referenced :
nat_properties,
full-omega-unsat,
intformand_wf,
intformle_wf,
itermConstant_wf,
itermVar_wf,
intformless_wf,
istype-int,
int_formula_prop_and_lemma,
istype-void,
int_formula_prop_le_lemma,
int_term_value_constant_lemma,
int_term_value_var_lemma,
int_formula_prop_less_lemma,
int_formula_prop_wf,
ge_wf,
istype-less_than,
int_seg_properties,
int_seg_wf,
subtract-1-ge-0,
decidable__equal_int,
subtract_wf,
subtype_base_sq,
set_subtype_base,
int_subtype_base,
intformnot_wf,
intformeq_wf,
itermSubtract_wf,
int_formula_prop_not_lemma,
int_formula_prop_eq_lemma,
int_term_value_subtract_lemma,
decidable__le,
decidable__lt,
istype-le,
subtype_rel_self,
istype-nat,
non_neg_length,
length_wf,
unshuffle_wf,
itermAdd_wf,
int_term_value_add_lemma,
length_wf_nat,
list_wf,
istype-universe,
list-cases,
length_of_nil_lemma,
reduce_tl_nil_lemma,
product_subtype_list,
length_of_cons_lemma,
reduce_hd_cons_lemma,
reduce_tl_cons_lemma,
lt_int_wf,
cons_wf,
equal_wf,
length-unshuffle,
length-nil,
select_wf,
itermMultiply_wf,
int_term_value_mul_lemma,
iff_weakening_equal,
assert_wf,
bnot_wf,
not_wf,
less_than_wf,
istype-assert,
equal-wf-T-base,
bool_wf,
le_int_wf,
le_wf,
tl_wf,
add-is-int-iff,
false_wf,
bool_cases,
bool_subtype_base,
eqtt_to_assert,
assert_of_lt_int,
eqff_to_assert,
iff_transitivity,
iff_weakening_uiff,
assert_of_bnot,
uiff_transitivity,
assert_functionality_wrt_uiff,
bnot_of_lt_int,
assert_of_le_int,
rem_bounds_1,
nequal_wf,
div_rem_sum,
le_weakening2,
mul-distributes,
add-associates,
mul-commutes,
add-commutes,
add-swap,
select-cons-tl,
squash_wf,
true_wf,
select_cons_tl
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
Error :isect_memberFormation_alt,
introduction,
cut,
Error :lambdaFormation_alt,
thin,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
hypothesisEquality,
hypothesis,
setElimination,
rename,
intWeakElimination,
natural_numberEquality,
independent_isectElimination,
approximateComputation,
independent_functionElimination,
Error :dependent_pairFormation_alt,
Error :lambdaEquality_alt,
int_eqEquality,
dependent_functionElimination,
Error :isect_memberEquality_alt,
voidElimination,
sqequalRule,
independent_pairFormation,
Error :universeIsType,
axiomEquality,
Error :functionIsTypeImplies,
Error :inhabitedIsType,
productElimination,
unionElimination,
applyEquality,
instantiate,
because_Cache,
equalityTransitivity,
equalitySymmetry,
applyLambdaEquality,
Error :dependent_set_memberEquality_alt,
Error :productIsType,
hypothesis_subsumption,
imageElimination,
productEquality,
addEquality,
universeEquality,
promote_hyp,
independent_pairEquality,
multiplyEquality,
closedConclusion,
imageMemberEquality,
baseClosed,
Error :functionIsType,
pointwiseFunctionality,
baseApply,
cumulativity,
equalityElimination,
Error :equalityIsType1,
intEquality,
Error :equalityIsType4,
minusEquality
Latex:
\mforall{}[T:Type]. \mforall{}as:T List. \mforall{}i:\mBbbN{}||unshuffle(as)||. (unshuffle(as)[i] = <as[2 * i], as[(2 * i) + 1]>)
Date html generated:
2019_06_20-PM-01_48_02
Last ObjectModification:
2018_10_18-PM-00_42_06
Theory : list_1
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