Nuprl Lemma : s-insert-sorted
∀[T:Type]. ∀[x:T]. ∀[L:T List]. sorted(s-insert(x;L)) supposing sorted(L) supposing T ⊆r ℤ
Proof
Definitions occuring in Statement :
s-insert: s-insert(x;l),
sorted: sorted(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,
so_lambda: λ2x.t[x],
prop: ℙ,
so_apply: x[s],
implies: P ⇒ Q,
all: ∀x:A. B[x],
sorted: sorted(L),
le: A ≤ B,
and: P ∧ Q,
int_seg: {i..j-},
s-insert: s-insert(x;l),
select: L[n],
nil: [],
it: ⋅,
so_lambda: λ2x y.t[x; y],
top: Top,
so_apply: x[s1;s2],
so_lambda: so_lambda(x,y,z.t[x; y; z]),
so_apply: x[s1;s2;s3],
sq_stable: SqStable(P),
lelt: i ≤ j < k,
uiff: uiff(P;Q),
squash: ↓T,
subtype_rel: A ⊆r B,
not: ¬A,
less_than': less_than'(a;b),
true: True,
false: False,
guard: {T},
rev_uimplies: rev_uimplies(P;Q),
cand: A c∧ B,
bool: 𝔹,
unit: Unit,
btrue: tt,
ifthenelse: if b then t else f fi ,
bfalse: ff,
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q,
l_all: (∀x∈L.P[x]),
or: P ∨ Q
Lemmas referenced :
list_induction,
isect_wf,
sorted_wf,
s-insert_wf,
le_witness_for_triv,
list_wf,
subtype_rel_wf,
int_seg_wf,
length_wf,
length_of_nil_lemma,
stuck-spread,
istype-base,
istype-void,
list_ind_nil_lemma,
length_of_cons_lemma,
sq_stable__le,
less-iff-le,
add_functionality_wrt_le,
add-associates,
istype-int,
zero-add,
add-swap,
add-commutes,
le-add-cancel2,
le_wf,
less_than_transitivity1,
less_than_irreflexivity,
list_ind_cons_lemma,
ifthenelse_wf,
eq_int_wf,
cons_wf,
lt_int_wf,
equal-wf-T-base,
bool_wf,
assert_wf,
equal_wf,
bnot_wf,
not_wf,
less_than_wf,
sorted-cons,
le_int_wf,
uiff_transitivity,
eqtt_to_assert,
assert_of_eq_int,
iff_transitivity,
iff_weakening_uiff,
eqff_to_assert,
assert_of_bnot,
assert_of_lt_int,
assert_functionality_wrt_uiff,
bnot_of_lt_int,
assert_of_le_int,
l_all_cons,
le_weakening2,
l_all_iff,
l_member_wf,
member-s-insert
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
Error :isect_memberFormation_alt,
introduction,
cut,
thin,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
hypothesisEquality,
sqequalRule,
Error :lambdaEquality_alt,
cumulativity,
independent_isectElimination,
hypothesis,
because_Cache,
Error :universeIsType,
independent_functionElimination,
Error :lambdaFormation_alt,
rename,
Error :isectIsType,
Error :inhabitedIsType,
dependent_functionElimination,
Error :isect_memberEquality_alt,
productElimination,
equalityTransitivity,
equalitySymmetry,
intEquality,
natural_numberEquality,
setElimination,
universeEquality,
baseClosed,
voidElimination,
addEquality,
imageMemberEquality,
imageElimination,
applyEquality,
Error :functionIsType,
Error :equalityIsType1,
independent_pairFormation,
unionElimination,
equalityElimination,
Error :setIsType,
hyp_replacement,
Error :dependent_set_memberEquality_alt,
Error :productIsType,
applyLambdaEquality,
Error :unionIsType
Latex:
\mforall{}[T:Type]. \mforall{}[x:T]. \mforall{}[L:T List]. sorted(s-insert(x;L)) supposing sorted(L) supposing T \msubseteq{}r \mBbbZ{}
Date html generated:
2019_06_20-PM-00_42_27
Last ObjectModification:
2018_10_02-AM-10_06_17
Theory : list_0
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