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 :
segment: as[m..n-],
all: ∀x:A. B[x],
uall: ∀[x:A]. B[x],
member: t ∈ T,
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,
or: P ∨ Q,
int_iseg: {i...j},
cons: [a / b],
decidable: Dec(P),
colength: colength(L),
nil: [],
it: ⋅,
guard: {T},
so_lambda: λ2x.t[x],
so_apply: x[s],
sq_type: SQType(T),
less_than: a < b,
squash: ↓T,
less_than': less_than'(a;b),
so_lambda: λ2x y.t[x; y],
so_apply: x[s1;s2],
subtype_rel: A ⊆r B,
firstn: firstn(n;as),
so_lambda: so_lambda(x,y,z.t[x; y; z]),
so_apply: x[s1;s2;s3],
append: as @ bs,
nth_tl: nth_tl(n;as),
bool: 𝔹,
unit: Unit,
btrue: tt,
uiff: uiff(P;Q),
ifthenelse: if b then t else f fi ,
bfalse: ff,
bnot: ¬bb,
assert: ↑b,
rev_implies: P ⇐ Q,
iff: P ⇐⇒ Q,
le: A ≤ B,
cand: A c∧ B,
subtract: n - m,
le_int: i ≤z j,
lt_int: i <z j,
true: True
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,
list-cases,
int_iseg_wf,
length_wf,
nil_wf,
product_subtype_list,
colength-cons-not-zero,
colength_wf_list,
decidable__le,
intformnot_wf,
int_formula_prop_not_lemma,
istype-le,
subtract-1-ge-0,
subtype_base_sq,
intformeq_wf,
int_formula_prop_eq_lemma,
set_subtype_base,
int_subtype_base,
spread_cons_lemma,
decidable__equal_int,
subtract_wf,
itermSubtract_wf,
itermAdd_wf,
int_term_value_subtract_lemma,
int_term_value_add_lemma,
le_wf,
cons_wf,
istype-nat,
list_wf,
istype-universe,
nth_tl_nil,
list_ind_nil_lemma,
le_int_wf,
eqtt_to_assert,
assert_of_le_int,
reduce_tl_cons_lemma,
eqff_to_assert,
bool_cases_sqequal,
bool_wf,
bool_subtype_base,
assert-bnot,
iff_weakening_uiff,
assert_wf,
length_of_cons_lemma,
non_neg_length,
int_iseg_properties,
add-is-int-iff,
false_wf,
first0,
subtype_rel_list,
top_wf,
firstn_wf,
istype-false,
list_ind_cons_lemma,
lt_int_wf,
assert_of_lt_int,
less_than_wf,
squash_wf,
true_wf,
minus-one-mul,
add-commutes,
minus-add,
minus-minus,
add-associates,
add-swap,
zero-add
Rules used in proof :
sqequalSubstitution,
sqequalRule,
sqequalReflexivity,
sqequalTransitivity,
computationStep,
Error :lambdaFormation_alt,
cut,
thin,
introduction,
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,
independent_pairFormation,
Error :universeIsType,
axiomEquality,
Error :functionIsTypeImplies,
Error :inhabitedIsType,
unionElimination,
voidEquality,
promote_hyp,
hypothesis_subsumption,
productElimination,
Error :equalityIstype,
because_Cache,
Error :dependent_set_memberEquality_alt,
instantiate,
equalityTransitivity,
equalitySymmetry,
applyLambdaEquality,
imageElimination,
baseApply,
closedConclusion,
baseClosed,
applyEquality,
intEquality,
sqequalBase,
universeEquality,
equalityElimination,
cumulativity,
addEquality,
productEquality,
pointwiseFunctionality,
Error :productIsType,
multiplyEquality,
minusEquality,
imageMemberEquality
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:
2019_06_20-PM-01_34_53
Last ObjectModification:
2019_01_02-PM-00_29_18
Theory : list_1
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