Nuprl Lemma : sorted-by-strict-no_repeats
∀[T:Type]. ∀[R:T ⟶ T ⟶ ℙ]. ∀[L:T List]. (no_repeats(T;L)) supposing (sorted-by(R;L) and (∀a:T. (¬(R a a))))
Proof
Definitions occuring in Statement :
sorted-by: sorted-by(R;L),
no_repeats: no_repeats(T;l),
list: T List,
uimplies: b supposing a,
uall: ∀[x:A]. B[x],
prop: ℙ,
all: ∀x:A. B[x],
not: ¬A,
apply: f a,
function: x:A ⟶ B[x],
universe: Type
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
uimplies: b supposing a,
no_repeats: no_repeats(T;l),
not: ¬A,
implies: P ⇒ Q,
false: False,
prop: ℙ,
nat: ℕ,
ge: i ≥ j ,
all: ∀x:A. B[x],
decidable: Dec(P),
or: P ∨ Q,
satisfiable_int_formula: satisfiable_int_formula(fmla),
exists: ∃x:A. B[x],
top: Top,
and: P ∧ Q,
subtype_rel: A ⊆r B,
so_lambda: λ2x.t[x],
so_apply: x[s],
guard: {T},
int_seg: {i..j-},
lelt: i ≤ j < k,
le: A ≤ B,
sorted-by: sorted-by(R;L)
Lemmas referenced :
equal_wf,
select_wf,
nat_properties,
decidable__le,
satisfiable-full-omega-tt,
intformand_wf,
intformnot_wf,
intformle_wf,
itermConstant_wf,
itermVar_wf,
int_formula_prop_and_lemma,
int_formula_prop_not_lemma,
int_formula_prop_le_lemma,
int_term_value_constant_lemma,
int_term_value_var_lemma,
int_formula_prop_wf,
not_wf,
nat_wf,
less_than_wf,
length_wf,
no_repeats_witness,
sorted-by_wf,
subtype_rel_dep_function,
l_member_wf,
subtype_rel_self,
set_wf,
all_wf,
list_wf,
decidable__lt,
lelt_wf,
decidable__equal_int,
intformeq_wf,
intformless_wf,
int_formula_prop_eq_lemma,
int_formula_prop_less_lemma,
le_wf
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
introduction,
cut,
lambdaFormation,
thin,
sqequalHypSubstitution,
independent_functionElimination,
voidElimination,
because_Cache,
hypothesis,
extract_by_obid,
isectElimination,
setElimination,
rename,
independent_isectElimination,
hypothesisEquality,
dependent_functionElimination,
natural_numberEquality,
unionElimination,
dependent_pairFormation,
lambdaEquality,
int_eqEquality,
intEquality,
isect_memberEquality,
voidEquality,
sqequalRule,
independent_pairFormation,
computeAll,
equalityTransitivity,
equalitySymmetry,
cumulativity,
applyEquality,
instantiate,
functionEquality,
universeEquality,
setEquality,
functionExtensionality,
hyp_replacement,
applyLambdaEquality,
dependent_set_memberEquality,
productElimination
Latex:
\mforall{}[T:Type]. \mforall{}[R:T {}\mrightarrow{} T {}\mrightarrow{} \mBbbP{}]. \mforall{}[L:T List].
(no\_repeats(T;L)) supposing (sorted-by(R;L) and (\mforall{}a:T. (\mneg{}(R a a))))
Date html generated:
2017_04_17-AM-07_43_46
Last ObjectModification:
2017_02_27-PM-04_16_18
Theory : list_1
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