Nuprl Lemma : select_append
∀[T:Type]. ∀[as,bs:T List]. ∀[i:ℕ||as|| + ||bs||]. (as @ bs[i] = if i <z ||as|| then as[i] else bs[i - ||as||] fi ∈ T)
Proof
Definitions occuring in Statement :
select: L[n],
length: ||as||,
append: as @ bs,
list: T List,
int_seg: {i..j-},
ifthenelse: if b then t else f fi ,
lt_int: i <z j,
uall: ∀[x:A]. B[x],
subtract: 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,
int_seg: {i..j-},
all: ∀x:A. B[x],
implies: P ⇒ Q,
bool: 𝔹,
unit: Unit,
it: ⋅,
btrue: tt,
uiff: uiff(P;Q),
and: P ∧ Q,
uimplies: b supposing a,
ifthenelse: if b then t else f fi ,
lelt: i ≤ j < k,
prop: ℙ,
bfalse: ff,
exists: ∃x:A. B[x],
or: P ∨ Q,
sq_type: SQType(T),
guard: {T},
bnot: ¬bb,
assert: ↑b,
false: False,
decidable: Dec(P),
less_than: a < b,
squash: ↓T,
not: ¬A,
satisfiable_int_formula: satisfiable_int_formula(fmla),
top: Top
Lemmas referenced :
lt_int_wf,
length_wf,
bool_wf,
eqtt_to_assert,
assert_of_lt_int,
select_append_front,
lelt_wf,
eqff_to_assert,
equal_wf,
bool_cases_sqequal,
subtype_base_sq,
bool_subtype_base,
assert-bnot,
not_functionality_wrt_uiff,
assert_wf,
less_than_wf,
select_append_back,
decidable__le,
add-is-int-iff,
full-omega-unsat,
intformand_wf,
intformnot_wf,
intformle_wf,
itermVar_wf,
intformless_wf,
int_formula_prop_and_lemma,
int_formula_prop_not_lemma,
int_formula_prop_le_lemma,
int_term_value_var_lemma,
int_formula_prop_less_lemma,
int_formula_prop_wf,
false_wf,
int_seg_wf,
list_wf
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
Error :isect_memberFormation_alt,
introduction,
cut,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
setElimination,
rename,
hypothesisEquality,
hypothesis,
lambdaFormation,
unionElimination,
equalityElimination,
because_Cache,
productElimination,
independent_isectElimination,
sqequalRule,
dependent_set_memberEquality,
independent_pairFormation,
natural_numberEquality,
cumulativity,
dependent_pairFormation,
equalityTransitivity,
equalitySymmetry,
promote_hyp,
dependent_functionElimination,
instantiate,
independent_functionElimination,
voidElimination,
pointwiseFunctionality,
imageElimination,
baseApply,
closedConclusion,
baseClosed,
approximateComputation,
lambdaEquality,
int_eqEquality,
intEquality,
isect_memberEquality,
voidEquality,
addEquality,
Error :universeIsType,
axiomEquality,
Error :inhabitedIsType,
universeEquality
Latex:
\mforall{}[T:Type]. \mforall{}[as,bs:T List]. \mforall{}[i:\mBbbN{}||as|| + ||bs||].
(as @ bs[i] = if i <z ||as|| then as[i] else bs[i - ||as||] fi )
Date html generated:
2019_06_20-PM-01_19_45
Last ObjectModification:
2018_09_26-PM-05_20_43
Theory : list_1
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