Nuprl Lemma : sort-int-trivial
∀T:Type. ∀bs:T List. ((T ⊆r ℤ) ⇒ sorted(bs) ⇒ (sort-int(bs) = bs ∈ (T List)))
Proof
Definitions occuring in Statement :
sort-int: sort-int(as),
sorted: sorted(L),
list: T List,
subtype_rel: A ⊆r B,
all: ∀x:A. B[x],
implies: P ⇒ Q,
int: ℤ,
universe: Type,
equal: s = t ∈ T
Definitions unfolded in proof :
sort-int: sort-int(as),
all: ∀x:A. B[x],
uall: ∀[x:A]. B[x],
member: t ∈ T,
so_lambda: λ2x.t[x],
implies: P ⇒ Q,
prop: ℙ,
uimplies: b supposing a,
so_apply: x[s],
merge-int: merge-int(as;bs),
reduce: reduce(f;k;as),
list_ind: list_ind,
nil: [],
it: ⋅,
subtype_rel: A ⊆r B,
top: Top,
uiff: uiff(P;Q),
and: P ∧ Q,
or: P ∨ Q,
cons: [a / b],
bool: 𝔹,
unit: Unit,
btrue: tt,
ifthenelse: if b then t else f fi ,
bfalse: ff,
exists: ∃x:A. B[x],
sq_type: SQType(T),
guard: {T},
bnot: ¬bb,
assert: ↑b,
false: False,
not: ¬A,
l_all: (∀x∈L.P[x]),
int_seg: {i..j-},
lelt: i ≤ j < k,
le: A ≤ B,
less_than': less_than'(a;b),
nat_plus: ℕ+,
less_than: a < b,
squash: ↓T,
true: True,
select: L[n]
Lemmas referenced :
list_induction,
subtype_rel_wf,
sorted_wf,
equal_wf,
list_wf,
merge-int_wf,
nil_wf,
reduce_cons_lemma,
sorted-cons,
length_wf_nat,
nat_wf,
insert-int_wf,
cons_wf,
list-cases,
insert_int_nil_lemma,
product_subtype_list,
insert-int-cons,
subtype_rel_list,
lt_int_wf,
bool_wf,
eqtt_to_assert,
assert_of_lt_int,
eqff_to_assert,
bool_cases_sqequal,
subtype_base_sq,
bool_subtype_base,
assert-bnot,
less_than_wf,
length_of_cons_lemma,
false_wf,
add_nat_plus,
nat_plus_wf,
lelt_wf,
length_wf,
less_than_transitivity1,
less_than_irreflexivity
Rules used in proof :
sqequalSubstitution,
sqequalRule,
sqequalReflexivity,
sqequalTransitivity,
computationStep,
lambdaFormation,
cut,
thin,
introduction,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
hypothesisEquality,
lambdaEquality,
functionEquality,
cumulativity,
intEquality,
hypothesis,
independent_isectElimination,
independent_functionElimination,
voidEquality,
voidElimination,
because_Cache,
rename,
dependent_functionElimination,
isect_memberEquality,
productElimination,
dependent_set_memberEquality,
equalityTransitivity,
equalitySymmetry,
hyp_replacement,
applyLambdaEquality,
setElimination,
universeEquality,
unionElimination,
promote_hyp,
hypothesis_subsumption,
applyEquality,
equalityElimination,
dependent_pairFormation,
instantiate,
natural_numberEquality,
independent_pairFormation,
imageMemberEquality,
baseClosed,
addEquality
Latex:
\mforall{}T:Type. \mforall{}bs:T List. ((T \msubseteq{}r \mBbbZ{}) {}\mRightarrow{} sorted(bs) {}\mRightarrow{} (sort-int(bs) = bs))
Date html generated:
2017_09_29-PM-05_49_52
Last ObjectModification:
2017_07_26-PM-01_38_55
Theory : list_0
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