Nuprl Lemma : filter_functionality_wrt_permutation
∀[A:Type]. ∀L1,L2:A List. ∀p:A ⟶ 𝔹. (permutation(A;L1;L2) ⇒ permutation(A;filter(p;L1);filter(p;L2)))
Proof
Definitions occuring in Statement :
permutation: permutation(T;L1;L2),
filter: filter(P;l),
list: T List,
bool: 𝔹,
uall: ∀[x:A]. B[x],
all: ∀x:A. B[x],
implies: P ⇒ Q,
function: x:A ⟶ B[x],
universe: Type
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
all: ∀x:A. B[x],
implies: P ⇒ Q,
label: ...$L... t,
guard: {T},
permutation: permutation(T;L1;L2),
exists: ∃x:A. B[x],
and: P ∧ Q,
subtype_rel: A ⊆r B,
prop: ℙ,
uimplies: b supposing a,
so_lambda: λ2x.t[x],
so_apply: x[s],
cand: A c∧ B
Lemmas referenced :
permutation-filter,
length_wf_nat,
subtype_rel_list,
assert_wf,
equal_wf,
nat_wf,
filter_wf5,
subtype_rel_dep_function,
bool_wf,
l_member_wf,
subtype_rel_self,
set_wf,
list_wf,
inject_wf,
int_seg_wf,
length_wf,
permute_list_wf,
permutation_wf
Rules used in proof :
cut,
introduction,
extract_by_obid,
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
hypothesis,
sqequalHypSubstitution,
isectElimination,
thin,
hypothesisEquality,
lambdaFormation,
dependent_functionElimination,
independent_functionElimination,
productElimination,
dependent_pairFormation,
independent_pairFormation,
promote_hyp,
dependent_set_memberEquality,
cumulativity,
equalityTransitivity,
equalitySymmetry,
applyEquality,
setEquality,
functionExtensionality,
independent_isectElimination,
lambdaEquality,
setElimination,
rename,
because_Cache,
sqequalRule,
hyp_replacement,
applyLambdaEquality,
productEquality,
natural_numberEquality,
functionEquality,
universeEquality
Latex:
\mforall{}[A:Type]
\mforall{}L1,L2:A List. \mforall{}p:A {}\mrightarrow{} \mBbbB{}. (permutation(A;L1;L2) {}\mRightarrow{} permutation(A;filter(p;L1);filter(p;L2)))
Date html generated:
2017_04_17-AM-08_25_00
Last ObjectModification:
2017_02_27-PM-04_46_33
Theory : list_1
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