Nuprl Lemma : append_segment
∀T:Type. ∀as:T List. ∀i:{0...||as||}. ∀j:{i...||as||}. ∀k:{j...||as||}.
  (((as[i..j-]) @ (as[j..k-])) = (as[i..k-]) ∈ (T List))
Proof
Definitions occuring in Statement : 
segment: as[m..n-]
, 
length: ||as||
, 
append: as @ bs
, 
list: T List
, 
int_iseg: {i...j}
, 
all: ∀x:A. B[x]
, 
natural_number: $n
, 
universe: Type
, 
equal: s = t ∈ T
Definitions unfolded in proof : 
all: ∀x:A. B[x]
, 
member: t ∈ T
, 
uall: ∀[x:A]. B[x]
, 
int_iseg: {i...j}
, 
subtype_rel: A ⊆r B
, 
so_lambda: λ2x.t[x]
, 
prop: ℙ
, 
and: P ∧ Q
, 
so_apply: x[s]
, 
uimplies: b supposing a
, 
implies: P 
⇒ Q
, 
cand: A c∧ B
, 
guard: {T}
, 
decidable: Dec(P)
, 
or: P ∨ Q
, 
satisfiable_int_formula: satisfiable_int_formula(fmla)
, 
exists: ∃x:A. B[x]
, 
false: False
, 
not: ¬A
, 
top: Top
, 
true: True
, 
squash: ↓T
, 
iff: P 
⇐⇒ Q
, 
rev_implies: P 
⇐ Q
, 
int_seg: {i..j-}
, 
lelt: i ≤ j < k
, 
le: A ≤ B
, 
lapp_imon: <T List,@>
, 
grp_car: |g|
, 
pi1: fst(t)
, 
imon: IMonoid
, 
grp_op: *
, 
pi2: snd(t)
, 
infix_ap: x f y
Lemmas referenced : 
int_iseg_wf, 
length_wf, 
list_wf, 
subtype_rel_sets, 
le_wf, 
int_iseg_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, 
equal_wf, 
squash_wf, 
true_wf, 
append_wf, 
segment_factor, 
iff_weakening_equal, 
mon_itop_wf, 
lapp_imon_wf, 
cons_wf, 
select_wf, 
int_seg_properties, 
decidable__lt, 
intformless_wf, 
int_formula_prop_less_lemma, 
nil_wf, 
int_seg_wf, 
grp_car_wf, 
imon_wf, 
mon_itop_split
Rules used in proof : 
sqequalSubstitution, 
sqequalTransitivity, 
computationStep, 
sqequalReflexivity, 
lambdaFormation, 
cut, 
hypothesis, 
introduction, 
extract_by_obid, 
sqequalHypSubstitution, 
isectElimination, 
thin, 
setElimination, 
rename, 
hypothesisEquality, 
cumulativity, 
natural_numberEquality, 
universeEquality, 
because_Cache, 
applyEquality, 
sqequalRule, 
intEquality, 
lambdaEquality, 
productEquality, 
independent_isectElimination, 
setEquality, 
productElimination, 
independent_pairFormation, 
applyLambdaEquality, 
equalityTransitivity, 
equalitySymmetry, 
dependent_functionElimination, 
unionElimination, 
dependent_pairFormation, 
int_eqEquality, 
isect_memberEquality, 
voidElimination, 
voidEquality, 
computeAll, 
imageElimination, 
imageMemberEquality, 
baseClosed, 
independent_functionElimination
Latex:
\mforall{}T:Type.  \mforall{}as:T  List.  \mforall{}i:\{0...||as||\}.  \mforall{}j:\{i...||as||\}.  \mforall{}k:\{j...||as||\}.
    (((as[i..j\msupminus{}])  @  (as[j..k\msupminus{}]))  =  (as[i..k\msupminus{}]))
Date html generated:
2017_10_01-AM-09_57_44
Last ObjectModification:
2017_03_03-PM-00_59_19
Theory : list_2
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