Nuprl Lemma : insert-int-sorted
∀[T:Type]. ∀[x:T]. ∀[l:T List]. sorted(insert-int(x;l)) supposing sorted(l) supposing T ⊆r ℤ
Proof
Definitions occuring in Statement :
sorted: sorted(L),
insert-int: insert-int(x;l),
list: T List,
uimplies: b supposing a,
subtype_rel: A ⊆r B,
uall: ∀[x:A]. B[x],
int: ℤ,
universe: Type
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
uimplies: b supposing a,
all: ∀x:A. B[x],
nat: ℕ,
implies: P ⇒ Q,
false: False,
and: P ∧ Q,
ge: i ≥ j ,
le: A ≤ B,
cand: A c∧ B,
less_than: a < b,
squash: ↓T,
guard: {T},
prop: ℙ,
sorted: sorted(L),
or: P ∨ Q,
cons: [a / b],
less_than': less_than'(a;b),
not: ¬A,
colength: colength(L),
nil: [],
it: ⋅,
so_lambda: λ2x.t[x],
so_apply: x[s],
sq_type: SQType(T),
so_lambda: λ2x y.t[x; y],
so_apply: x[s1;s2],
sq_stable: SqStable(P),
decidable: Dec(P),
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q,
uiff: uiff(P;Q),
subtract: n - m,
true: True,
subtype_rel: A ⊆r B,
insert-int: insert-int(x;l),
so_lambda: so_lambda3,
top: Top,
so_apply: x[s1;s2;s3],
int_seg: {i..j-},
lelt: i ≤ j < k,
exists: ∃x:A. B[x],
nat_plus: ℕ+,
bool: 𝔹,
unit: Unit,
btrue: tt,
ifthenelse: if b then t else f fi ,
bfalse: ff,
bnot: ¬bb,
assert: ↑b,
l_member: (x ∈ l),
select: L[n],
l_all: (∀x∈L.P[x]),
gt: i > j
Lemmas referenced :
nat_properties,
less_than_transitivity1,
less_than_irreflexivity,
ge_wf,
istype-less_than,
le_witness_for_triv,
list-cases,
product_subtype_list,
colength-cons-not-zero,
colength_wf_list,
istype-void,
istype-le,
subtract-1-ge-0,
subtype_base_sq,
nat_wf,
set_subtype_base,
le_wf,
int_subtype_base,
spread_cons_lemma,
sq_stable__le,
decidable__equal_int,
subtract_wf,
istype-false,
not-equal-2,
condition-implies-le,
add-associates,
minus-add,
minus-one-mul,
add-swap,
minus-one-mul-top,
le_antisymmetry_iff,
add_functionality_wrt_le,
add-commutes,
zero-add,
le-add-cancel,
minus-minus,
le_weakening2,
istype-nat,
list_wf,
subtype_rel_wf,
istype-universe,
list_ind_nil_lemma,
select_wf,
cons_wf,
nil_wf,
less_than_transitivity2,
length_wf,
length_of_cons_lemma,
length_of_nil_lemma,
less-iff-le,
le-add-cancel2,
subtype_rel_transitivity,
base_wf,
equal_wf,
int_seg_wf,
less_than'_wf,
sorted_wf,
le_reflexive,
one-mul,
add-mul-special,
two-mul,
mul-distributes-right,
zero-mul,
false_wf,
omega-shadow,
less_than_wf,
mul-distributes,
mul-associates,
mul-commutes,
add-zero,
int_seg_properties,
sorted-cons,
insert-int-cons,
subtype_rel_list,
lt_int_wf,
eqtt_to_assert,
assert_of_lt_int,
insert-int_wf,
eqff_to_assert,
bool_cases_sqequal,
bool_wf,
bool_subtype_base,
assert-bnot,
iff_weakening_uiff,
assert_wf,
l_all_iff,
l_member_wf,
member-insert-int,
decidable__le,
not-le-2,
decidable__lt,
not-lt-2,
squash_wf,
true_wf,
istype-int,
subtype_rel_self,
iff_weakening_equal,
select-cons-tl,
not-gt-2,
minus-zero
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation_alt,
introduction,
cut,
thin,
lambdaFormation_alt,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
hypothesisEquality,
hypothesis,
setElimination,
rename,
intWeakElimination,
independent_pairFormation,
productElimination,
imageElimination,
natural_numberEquality,
independent_isectElimination,
independent_functionElimination,
voidElimination,
universeIsType,
sqequalRule,
lambdaEquality_alt,
dependent_functionElimination,
isect_memberEquality_alt,
equalityTransitivity,
equalitySymmetry,
functionIsTypeImplies,
inhabitedIsType,
isectIsTypeImplies,
unionElimination,
promote_hyp,
hypothesis_subsumption,
Error :memTop,
equalityIstype,
because_Cache,
dependent_set_memberEquality_alt,
instantiate,
cumulativity,
intEquality,
imageMemberEquality,
baseClosed,
applyLambdaEquality,
addEquality,
minusEquality,
baseApply,
closedConclusion,
applyEquality,
sqequalBase,
universeEquality,
isect_memberEquality,
voidEquality,
isect_memberFormation,
lambdaFormation,
lambdaEquality,
dependent_pairFormation,
sqequalIntensionalEquality,
independent_pairEquality,
axiomEquality,
multiplyEquality,
dependent_set_memberEquality,
equalityElimination,
dependent_pairFormation_alt,
setEquality,
setIsType,
hyp_replacement,
productIsType
Latex:
\mforall{}[T:Type]. \mforall{}[x:T]. \mforall{}[l:T List]. sorted(insert-int(x;l)) supposing sorted(l) supposing T \msubseteq{}r \mBbbZ{}
Date html generated:
2020_05_19-PM-09_37_22
Last ObjectModification:
2019_12_31-PM-00_59_34
Theory : list_0
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