Nuprl Lemma : s-insert-no-repeats
∀[T:Type]. ∀[x:T]. ∀[L:T List]. (no_repeats(T;s-insert(x;L))) supposing (no_repeats(T;L) and sorted(L)) supposing T ⊆r \000Cℤ
Proof
Definitions occuring in Statement :
s-insert: s-insert(x;l),
no_repeats: no_repeats(T;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],
s-insert: s-insert(x;l),
no_repeats: no_repeats(T;l),
sorted: sorted(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],
not: ¬A,
false: False,
nat: ℕ,
ge: i ≥ j ,
satisfiable_int_formula: satisfiable_int_formula(fmla),
exists: ∃x:A. B[x],
and: P ∧ Q,
decidable: Dec(P),
or: P ∨ Q,
le: A ≤ B,
subtype_rel: A ⊆r B,
int_seg: {i..j-},
guard: {T},
lelt: i ≤ j < k,
uiff: uiff(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,
label: ...$L... t
Lemmas referenced :
list_induction,
isect_wf,
sorted_wf,
no_repeats_wf,
s-insert_wf,
list_wf,
no_repeats_witness,
subtype_rel_wf,
length_of_nil_lemma,
stuck-spread,
base_wf,
list_ind_nil_lemma,
length_of_cons_lemma,
nat_properties,
satisfiable-full-omega-tt,
intformand_wf,
intformnot_wf,
intformeq_wf,
itermVar_wf,
intformless_wf,
itermConstant_wf,
intformle_wf,
int_formula_prop_and_lemma,
int_formula_prop_not_lemma,
int_formula_prop_eq_lemma,
int_term_value_var_lemma,
int_formula_prop_less_lemma,
int_term_value_constant_lemma,
int_formula_prop_le_lemma,
decidable__equal_int,
int_formula_prop_wf,
le_wf,
equal-wf-base,
int_subtype_base,
equal_wf,
select_wf,
cons_wf,
nil_wf,
not_wf,
nat_wf,
less_than_wf,
uall_wf,
all_wf,
int_seg_wf,
int_seg_properties,
list_ind_cons_lemma,
ifthenelse_wf,
eq_int_wf,
lt_int_wf,
bool_wf,
equal-wf-T-base,
assert_wf,
bnot_wf,
no_repeats_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,
sorted-cons,
l_all_iff,
subtype_rel_transitivity,
l_member_wf,
cons_member,
or_wf,
equal_functionality_wrt_subtype_rel2,
member-s-insert
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
introduction,
cut,
thin,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
because_Cache,
sqequalRule,
lambdaEquality,
cumulativity,
hypothesisEquality,
independent_isectElimination,
hypothesis,
independent_functionElimination,
lambdaFormation,
rename,
dependent_functionElimination,
isect_memberEquality,
equalityTransitivity,
equalitySymmetry,
intEquality,
universeEquality,
baseClosed,
voidElimination,
voidEquality,
setElimination,
natural_numberEquality,
dependent_pairFormation,
int_eqEquality,
independent_pairFormation,
unionElimination,
computeAll,
dependent_set_memberEquality,
productElimination,
applyEquality,
equalityElimination,
impliesFunctionality,
setEquality,
promote_hyp,
addLevel
Latex:
\mforall{}[T:Type]
\mforall{}[x:T]. \mforall{}[L:T List]. (no\_repeats(T;s-insert(x;L))) supposing (no\_repeats(T;L) and sorted(L))
supposing T \msubseteq{}r \mBbbZ{}
Date html generated:
2017_04_17-AM-08_32_18
Last ObjectModification:
2017_02_27-PM-04_53_04
Theory : list_1
Home
Index