Nuprl Lemma : insert-by-no-repeats
∀[T:Type]. ∀[eq,r:T ⟶ T ⟶ 𝔹].
(∀[x:T]. ∀[L:T List].
(no_repeats(T;insert-by(eq;r;x;L))) supposing (no_repeats(T;L) and sorted-by(λx,y. (↑(r x y));L))) supposing
(Linorder(T;a,b.↑(r a b)) and
(∀a,b:T. (↑(eq a b) ⇐⇒ a = b ∈ T)))
Proof
Definitions occuring in Statement :
insert-by: insert-by(eq;r;x;l),
sorted-by: sorted-by(R;L),
no_repeats: no_repeats(T;l),
list: T List,
linorder: Linorder(T;x,y.R[x; y]),
assert: ↑b,
bool: 𝔹,
uimplies: b supposing a,
uall: ∀[x:A]. B[x],
all: ∀x:A. B[x],
iff: P ⇐⇒ Q,
apply: f a,
lambda: λx.A[x],
function: x:A ⟶ B[x],
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],
all: ∀x:A. B[x],
prop: ℙ,
so_apply: x[s],
implies: P ⇒ Q,
so_lambda: λ2x y.t[x; y],
so_apply: x[s1;s2],
insert-by: insert-by(eq;r;x;l),
no_repeats: no_repeats(T;l),
sorted-by: sorted-by(R;L),
select: L[n],
nil: [],
it: ⋅,
top: Top,
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 ,
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q,
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,
linorder: Linorder(T;x,y.R[x; y]),
order: Order(T;x,y.R[x; y]),
anti_sym: AntiSym(T;x,y.R[x; y])
Lemmas referenced :
list_induction,
isect_wf,
sorted-by_wf,
l_member_wf,
assert_wf,
no_repeats_wf,
insert-by_wf,
list_wf,
no_repeats_witness,
linorder_wf,
all_wf,
iff_wf,
equal_wf,
bool_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,
select_wf,
cons_wf,
nil_wf,
not_wf,
nat_wf,
less_than_wf,
uall_wf,
int_seg_wf,
int_seg_properties,
list_ind_cons_lemma,
ifthenelse_wf,
equal-wf-T-base,
bnot_wf,
no_repeats_cons,
eqtt_to_assert,
uiff_transitivity,
eqff_to_assert,
assert_of_bnot,
sorted-by-cons,
cons_member,
l_all_iff,
member-insert-by
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
introduction,
cut,
thin,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
because_Cache,
sqequalRule,
lambdaEquality,
cumulativity,
hypothesisEquality,
lambdaFormation,
hypothesis,
setElimination,
rename,
applyEquality,
functionExtensionality,
setEquality,
dependent_functionElimination,
independent_functionElimination,
isect_memberEquality,
equalityTransitivity,
equalitySymmetry,
functionEquality,
universeEquality,
baseClosed,
independent_isectElimination,
voidElimination,
voidEquality,
natural_numberEquality,
dependent_pairFormation,
int_eqEquality,
intEquality,
independent_pairFormation,
unionElimination,
computeAll,
dependent_set_memberEquality,
productElimination,
equalityElimination
Latex:
\mforall{}[T:Type]. \mforall{}[eq,r:T {}\mrightarrow{} T {}\mrightarrow{} \mBbbB{}].
(\mforall{}[x:T]. \mforall{}[L:T List].
(no\_repeats(T;insert-by(eq;r;x;L))) supposing
(no\_repeats(T;L) and
sorted-by(\mlambda{}x,y. (\muparrow{}(r x y));L))) supposing
(Linorder(T;a,b.\muparrow{}(r a b)) and
(\mforall{}a,b:T. (\muparrow{}(eq a b) \mLeftarrow{}{}\mRightarrow{} a = b)))
Date html generated:
2017_04_17-AM-08_32_37
Last ObjectModification:
2017_02_27-PM-04_54_25
Theory : list_1
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