Nuprl Lemma : sorted-cons
∀[T:Type]. ∀[x:T]. ∀[L:T List]. uiff(sorted([x / L]);sorted(L) ∧ (∀z∈L.x ≤ z)) supposing T ⊆r ℤ
Proof
Definitions occuring in Statement :
l_all: (∀x∈L.P[x]),
sorted: sorted(L),
cons: [a / b],
list: T List,
uiff: uiff(P;Q),
uimplies: b supposing a,
subtype_rel: A ⊆r B,
uall: ∀[x:A]. B[x],
le: A ≤ B,
and: P ∧ Q,
int: ℤ,
universe: Type
Definitions unfolded in proof :
sorted: sorted(L),
all: ∀x:A. B[x],
member: t ∈ T,
top: Top,
uall: ∀[x:A]. B[x],
uimplies: b supposing a,
uiff: uiff(P;Q),
and: P ∧ Q,
int_seg: {i..j-},
le: A ≤ B,
not: ¬A,
implies: P ⇒ Q,
false: False,
subtype_rel: A ⊆r B,
lelt: i ≤ j < k,
guard: {T},
prop: ℙ,
l_all: (∀x∈L.P[x]),
sq_stable: SqStable(P),
squash: ↓T,
so_lambda: λ2x.t[x],
exists: ∃x:A. B[x],
nat: ℕ,
so_apply: x[s],
subtract: n - m,
nat_plus: ℕ+,
less_than: a < b,
less_than': less_than'(a;b),
true: True,
decidable: Dec(P),
or: P ∨ Q,
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q,
select: L[n],
cons: [a / b],
sq_type: SQType(T)
Lemmas referenced :
length_of_cons_lemma,
int_seg_wf,
length_wf,
less_than'_wf,
select_wf,
less_than_transitivity2,
le_weakening2,
sq_stable__le,
all_wf,
le_wf,
non_neg_length,
length_wf_nat,
nat_wf,
set_subtype_base,
int_subtype_base,
equal_wf,
l_all_wf,
l_member_wf,
list_wf,
subtype_rel_wf,
add-commutes,
less-iff-le,
add_functionality_wrt_le,
subtract_wf,
le_reflexive,
add-associates,
minus-add,
minus-one-mul,
one-mul,
add-swap,
add-mul-special,
two-mul,
mul-distributes-right,
zero-add,
zero-mul,
add-zero,
not-lt-2,
omega-shadow,
less_than_wf,
mul-distributes,
mul-associates,
mul-commutes,
minus-one-mul-top,
int_seg_properties,
nat_properties,
decidable__lt,
add-member-int_seg2,
decidable__le,
false_wf,
not-le-2,
condition-implies-le,
le-add-cancel2,
lelt_wf,
subtype_rel_list,
le-add-cancel,
squash_wf,
true_wf,
select_cons_tl,
iff_weakening_equal,
select-cons-tl,
decidable__equal_int,
subtype_base_sq,
minus-minus,
le_antisymmetry_iff,
not-equal-2,
minus-zero,
le-add-cancel-alt
Rules used in proof :
sqequalSubstitution,
sqequalRule,
sqequalReflexivity,
sqequalTransitivity,
computationStep,
cut,
introduction,
extract_by_obid,
sqequalHypSubstitution,
dependent_functionElimination,
thin,
isect_memberEquality,
voidElimination,
voidEquality,
hypothesis,
isect_memberFormation,
independent_pairFormation,
lambdaFormation,
isectElimination,
natural_numberEquality,
setElimination,
rename,
hypothesisEquality,
cumulativity,
productElimination,
independent_pairEquality,
lambdaEquality,
because_Cache,
applyEquality,
independent_isectElimination,
axiomEquality,
equalityTransitivity,
equalitySymmetry,
independent_functionElimination,
imageMemberEquality,
baseClosed,
imageElimination,
addEquality,
dependent_pairFormation,
sqequalIntensionalEquality,
intEquality,
promote_hyp,
productEquality,
setEquality,
universeEquality,
multiplyEquality,
minusEquality,
dependent_set_memberEquality,
unionElimination,
instantiate
Latex:
\mforall{}[T:Type]. \mforall{}[x:T]. \mforall{}[L:T List]. uiff(sorted([x / L]);sorted(L) \mwedge{} (\mforall{}z\mmember{}L.x \mleq{} z)) supposing T \msubseteq{}r \mBbbZ{}
Date html generated:
2017_04_14-AM-08_44_06
Last ObjectModification:
2017_02_27-PM-03_32_54
Theory : list_0
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