Nuprl Lemma : append_firstn_lastn
∀[T:Type]. ∀[L:T List]. ∀[n:{0...||L||}]. ((firstn(n;L) @ nth_tl(n;L)) = L ∈ (T List))
Proof
Definitions occuring in Statement :
firstn: firstn(n;as),
length: ||as||,
nth_tl: nth_tl(n;as),
append: as @ bs,
list: T List,
int_iseg: {i...j},
uall: ∀[x:A]. B[x],
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],
int_iseg: {i...j},
so_apply: x[s],
implies: P ⇒ Q,
firstn: firstn(n;as),
nth_tl: nth_tl(n;as),
all: ∀x:A. B[x],
so_lambda: so_lambda(x,y,z.t[x; y; z]),
top: Top,
so_apply: x[s1;s2;s3],
append: as @ bs,
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 ,
bfalse: ff,
exists: ∃x:A. B[x],
prop: ℙ,
or: P ∨ Q,
sq_type: SQType(T),
guard: {T},
bnot: ¬bb,
assert: ↑b,
false: False,
not: ¬A,
satisfiable_int_formula: satisfiable_int_formula(fmla),
squash: ↓T,
cand: A c∧ B,
decidable: Dec(P),
le: A ≤ B,
true: True,
subtype_rel: A ⊆r B,
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q
Lemmas referenced :
list_induction,
uall_wf,
int_iseg_wf,
length_wf,
equal_wf,
list_wf,
append_wf,
firstn_wf,
nth_tl_wf,
length_of_nil_lemma,
list_ind_nil_lemma,
reduce_tl_nil_lemma,
nth_tl_nil,
subtract_wf,
le_int_wf,
bool_wf,
eqtt_to_assert,
assert_of_le_int,
nil_wf,
eqff_to_assert,
bool_cases_sqequal,
subtype_base_sq,
bool_subtype_base,
assert-bnot,
le_wf,
length_of_cons_lemma,
list_ind_cons_lemma,
reduce_tl_cons_lemma,
lt_int_wf,
assert_of_lt_int,
int_iseg_properties,
satisfiable-full-omega-tt,
intformand_wf,
intformle_wf,
itermVar_wf,
itermConstant_wf,
intformless_wf,
int_formula_prop_and_lemma,
int_formula_prop_le_lemma,
int_term_value_var_lemma,
int_term_value_constant_lemma,
int_formula_prop_less_lemma,
int_formula_prop_wf,
cons_wf,
decidable__le,
intformnot_wf,
itermSubtract_wf,
int_formula_prop_not_lemma,
int_term_value_subtract_lemma,
add-is-int-iff,
itermAdd_wf,
int_term_value_add_lemma,
false_wf,
iff_weakening_equal,
less_than_wf
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
introduction,
cut,
thin,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
hypothesisEquality,
sqequalRule,
lambdaEquality,
natural_numberEquality,
cumulativity,
hypothesis,
setElimination,
rename,
because_Cache,
independent_functionElimination,
dependent_functionElimination,
isect_memberEquality,
voidElimination,
voidEquality,
lambdaFormation,
unionElimination,
equalityElimination,
productElimination,
independent_isectElimination,
dependent_pairFormation,
equalityTransitivity,
equalitySymmetry,
promote_hyp,
instantiate,
int_eqEquality,
intEquality,
independent_pairFormation,
computeAll,
applyEquality,
imageElimination,
dependent_set_memberEquality,
addEquality,
pointwiseFunctionality,
baseApply,
closedConclusion,
baseClosed,
productEquality,
imageMemberEquality,
axiomEquality,
universeEquality
Latex:
\mforall{}[T:Type]. \mforall{}[L:T List]. \mforall{}[n:\{0...||L||\}]. ((firstn(n;L) @ nth\_tl(n;L)) = L)
Date html generated:
2017_04_14-AM-09_25_18
Last ObjectModification:
2017_02_27-PM-03_59_38
Theory : list_1
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