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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