Nuprl Lemma : permutation-filter
∀[A:Type]. ∀L1,L2:A List. (permutation(A;L1;L2) ⇒ (∀p:A ⟶ 𝔹. permutation({a:A| ↑(p a)} ;filter(p;L1);filter(p;L2))))
Proof
Definitions occuring in Statement :
permutation: permutation(T;L1;L2),
filter: filter(P;l),
list: T List,
assert: ↑b,
bool: 𝔹,
uall: ∀[x:A]. B[x],
all: ∀x:A. B[x],
implies: P ⇒ Q,
set: {x:A| B[x]} ,
apply: f a,
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,
so_apply: x[s],
top: Top,
or: P ∨ Q,
cons: [a / b],
uimplies: b supposing a,
bool: 𝔹,
unit: Unit,
it: ⋅,
btrue: tt,
uiff: uiff(P;Q),
and: P ∧ Q,
ifthenelse: if b then t else f fi ,
ge: i ≥ j ,
le: A ≤ B,
satisfiable_int_formula: satisfiable_int_formula(fmla),
exists: ∃x:A. B[x],
false: False,
not: ¬A,
bfalse: ff,
sq_type: SQType(T),
guard: {T},
bnot: ¬bb,
assert: ↑b,
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q,
cand: A c∧ B
Lemmas referenced :
list_induction,
all_wf,
list_wf,
permutation_wf,
bool_wf,
assert_wf,
filter_type,
filter_nil_lemma,
nil_wf,
filter_cons_lemma,
cons_wf,
list-cases,
product_subtype_list,
permutation_weakening,
permutation-length,
length_of_nil_lemma,
length_of_cons_lemma,
eqtt_to_assert,
non_neg_length,
satisfiable-full-omega-tt,
intformand_wf,
intformle_wf,
itermConstant_wf,
itermVar_wf,
intformeq_wf,
itermAdd_wf,
int_formula_prop_and_lemma,
int_formula_prop_le_lemma,
int_term_value_constant_lemma,
int_term_value_var_lemma,
int_formula_prop_eq_lemma,
int_term_value_add_lemma,
int_formula_prop_wf,
eqff_to_assert,
equal_wf,
bool_cases_sqequal,
subtype_base_sq,
bool_subtype_base,
assert-bnot,
permutation-cons,
filter_append_sq,
append_wf,
length_wf,
length-append,
exists_wf
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
lambdaFormation,
cut,
thin,
introduction,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
hypothesisEquality,
sqequalRule,
lambdaEquality,
cumulativity,
hypothesis,
functionEquality,
setEquality,
applyEquality,
functionExtensionality,
independent_functionElimination,
dependent_functionElimination,
isect_memberEquality,
voidElimination,
voidEquality,
because_Cache,
rename,
universeEquality,
unionElimination,
promote_hyp,
hypothesis_subsumption,
productElimination,
independent_isectElimination,
equalityElimination,
equalityTransitivity,
equalitySymmetry,
natural_numberEquality,
dependent_pairFormation,
int_eqEquality,
intEquality,
independent_pairFormation,
computeAll,
instantiate,
hyp_replacement,
applyLambdaEquality,
dependent_set_memberEquality,
productEquality
Latex:
\mforall{}[A:Type]
\mforall{}L1,L2:A List.
(permutation(A;L1;L2) {}\mRightarrow{} (\mforall{}p:A {}\mrightarrow{} \mBbbB{}. permutation(\{a:A| \muparrow{}(p a)\} ;filter(p;L1);filter(p;L2))))
Date html generated:
2017_04_17-AM-08_24_46
Last ObjectModification:
2017_02_27-PM-04_46_50
Theory : list_1
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