Nuprl Lemma : sorted-by-filter
∀[T:Type]. ∀[R:T ⟶ T ⟶ ℙ]. ∀P:T ⟶ 𝔹. ∀L:T List. (sorted-by(R;L) ⇒ sorted-by(R;filter(P;L)))
Proof
Definitions occuring in Statement :
sorted-by: sorted-by(R;L),
filter: filter(P;l),
list: T List,
bool: 𝔹,
uall: ∀[x:A]. B[x],
prop: ℙ,
all: ∀x:A. B[x],
implies: P ⇒ Q,
function: x:A ⟶ B[x],
universe: Type
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
all: ∀x:A. B[x],
member: t ∈ T,
so_lambda: λ2x.t[x],
implies: P ⇒ Q,
prop: ℙ,
subtype_rel: A ⊆r B,
so_apply: x[s],
uimplies: b supposing a,
top: Top,
sorted-by: sorted-by(R;L),
int_seg: {i..j-},
guard: {T},
lelt: i ≤ j < k,
and: P ∧ Q,
decidable: Dec(P),
or: P ∨ Q,
satisfiable_int_formula: satisfiable_int_formula(fmla),
exists: ∃x:A. B[x],
false: False,
not: ¬A,
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q,
cand: A c∧ B,
bool: 𝔹,
unit: Unit,
it: ⋅,
btrue: tt,
uiff: uiff(P;Q),
ifthenelse: if b then t else f fi ,
bfalse: ff
Lemmas referenced :
list_induction,
sorted-by_wf,
subtype_rel_dep_function,
l_member_wf,
subtype_rel_self,
set_wf,
filter_wf5,
bool_wf,
list_wf,
filter_nil_lemma,
filter_cons_lemma,
int_seg_wf,
length_wf,
nil_wf,
all_wf,
select_wf,
length_of_nil_lemma,
int_seg_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,
decidable__lt,
intformless_wf,
int_formula_prop_less_lemma,
sorted-by-cons,
cons_wf,
equal-wf-T-base,
assert_wf,
bnot_wf,
not_wf,
eqtt_to_assert,
uiff_transitivity,
eqff_to_assert,
assert_of_bnot,
equal_wf,
l_all_iff,
member_filter
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
lambdaFormation,
cut,
thin,
introduction,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
hypothesisEquality,
sqequalRule,
lambdaEquality,
functionEquality,
cumulativity,
applyEquality,
instantiate,
universeEquality,
setEquality,
hypothesis,
independent_isectElimination,
setElimination,
rename,
because_Cache,
dependent_functionElimination,
independent_functionElimination,
isect_memberEquality,
voidElimination,
voidEquality,
natural_numberEquality,
functionExtensionality,
productElimination,
unionElimination,
dependent_pairFormation,
int_eqEquality,
intEquality,
independent_pairFormation,
computeAll,
equalityTransitivity,
equalitySymmetry,
baseClosed,
equalityElimination
Latex:
\mforall{}[T:Type]. \mforall{}[R:T {}\mrightarrow{} T {}\mrightarrow{} \mBbbP{}]. \mforall{}P:T {}\mrightarrow{} \mBbbB{}. \mforall{}L:T List. (sorted-by(R;L) {}\mRightarrow{} sorted-by(R;filter(P;L)))
Date html generated:
2017_04_17-AM-07_43_36
Last ObjectModification:
2017_02_27-PM-04_16_24
Theory : list_1
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