Nuprl Lemma : merge-int-comm
∀[T:Type]
∀[as,bs:T List]. merge-int(as;bs) = merge-int(bs;as) ∈ (T List) supposing sorted(as) ∧ sorted(bs) supposing T ⊆r ℤ
Proof
Definitions occuring in Statement :
sorted: sorted(L),
merge-int: merge-int(as;bs),
list: T List,
uimplies: b supposing a,
subtype_rel: A ⊆r B,
uall: ∀[x:A]. B[x],
and: P ∧ Q,
int: ℤ,
universe: Type,
equal: s = t ∈ T
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
uimplies: b supposing a,
so_lambda: λ2x.t[x],
and: P ∧ Q,
prop: ℙ,
so_apply: x[s],
implies: P ⇒ Q,
merge-int: merge-int(as;bs),
all: ∀x:A. B[x],
top: Top,
subtype_rel: A ⊆r B,
uiff: uiff(P;Q),
rev_implies: P ⇐ Q,
iff: P ⇐⇒ Q,
guard: {T},
true: True,
squash: ↓T,
sort-int: sort-int(as),
cand: A c∧ B,
or: P ∨ Q,
cons: [a / b],
bool: 𝔹,
unit: Unit,
it: ⋅,
btrue: tt,
ifthenelse: if b then t else f fi ,
bfalse: ff,
exists: ∃x:A. B[x],
sq_type: SQType(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,
select: L[n]
Lemmas referenced :
list_induction,
uall_wf,
list_wf,
isect_wf,
sorted_wf,
equal_wf,
merge-int_wf,
reduce_nil_lemma,
nil_wf,
reduce_cons_lemma,
sorted-cons,
cons_wf,
subtype_rel_wf,
iff_weakening_equal,
sort-int-trivial,
true_wf,
squash_wf,
insert-int_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,
length_wf_nat,
nat_plus_wf,
lelt_wf,
length_wf,
less_than_transitivity1,
less_than_irreflexivity,
insert-int-comm,
nat_wf
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
introduction,
cut,
thin,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
hypothesisEquality,
sqequalRule,
lambdaEquality,
cumulativity,
hypothesis,
productEquality,
independent_isectElimination,
independent_functionElimination,
dependent_functionElimination,
isect_memberEquality,
voidElimination,
voidEquality,
productElimination,
because_Cache,
axiomEquality,
equalityTransitivity,
equalitySymmetry,
lambdaFormation,
rename,
intEquality,
universeEquality,
baseClosed,
imageMemberEquality,
natural_numberEquality,
imageElimination,
applyEquality,
independent_pairFormation,
unionElimination,
promote_hyp,
hypothesis_subsumption,
equalityElimination,
dependent_pairFormation,
instantiate,
dependent_set_memberEquality,
setElimination,
addEquality,
applyLambdaEquality,
hyp_replacement,
levelHypothesis,
equalityUniverse
Latex:
\mforall{}[T:Type]
\mforall{}[as,bs:T List]. merge-int(as;bs) = merge-int(bs;as) supposing sorted(as) \mwedge{} sorted(bs)
supposing T \msubseteq{}r \mBbbZ{}
Date html generated:
2017_09_29-PM-05_49_59
Last ObjectModification:
2017_07_26-PM-01_38_58
Theory : list_0
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