Nuprl Lemma : list-partition-permutation
∀T:Type. ∀L:T List. ∀f:ℕ||L|| ⟶ 𝔹. let as,bs = list-partition(f;L) in permutation(T;L;as @ bs)
Proof
Definitions occuring in Statement :
list-partition: list-partition(f;L),
permutation: permutation(T;L1;L2),
length: ||as||,
append: as @ bs,
list: T List,
int_seg: {i..j-},
bool: 𝔹,
all: ∀x:A. B[x],
function: x:A ⟶ B[x],
spread: spread def,
natural_number: $n,
universe: Type
Definitions unfolded in proof :
all: ∀x:A. B[x],
uall: ∀[x:A]. B[x],
member: t ∈ T,
so_lambda: λ2x.t[x],
implies: P ⇒ Q,
so_apply: x[s],
list-partition: list-partition(f;L),
list_ind: list_ind,
nil: [],
it: ⋅,
append: as @ bs,
so_lambda: so_lambda(x,y,z.t[x; y; z]),
top: Top,
so_apply: x[s1;s2;s3],
int_seg: {i..j-},
lelt: i ≤ j < k,
and: P ∧ Q,
uimplies: b supposing a,
not: ¬A,
satisfiable_int_formula: satisfiable_int_formula(fmla),
exists: ∃x:A. B[x],
false: False,
prop: ℙ,
subtype_rel: A ⊆r B,
le: A ≤ B,
less_than': less_than'(a;b),
decidable: Dec(P),
or: P ∨ Q,
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q,
uiff: uiff(P;Q),
subtract: n - m,
true: True,
ge: i ≥ j ,
bool: 𝔹,
unit: Unit,
btrue: tt,
ifthenelse: if b then t else f fi ,
bfalse: ff,
sq_type: SQType(T),
guard: {T},
bnot: ¬bb,
assert: ↑b,
cand: A c∧ B,
cons: [a / b]
Lemmas referenced :
list_induction,
all_wf,
int_seg_wf,
length_wf,
bool_wf,
length_of_nil_lemma,
list_ind_nil_lemma,
istype-void,
permutation-nil,
int_seg_properties,
full-omega-unsat,
intformand_wf,
intformless_wf,
itermVar_wf,
itermConstant_wf,
intformle_wf,
istype-int,
int_formula_prop_and_lemma,
int_formula_prop_less_lemma,
int_term_value_var_lemma,
int_term_value_constant_lemma,
int_formula_prop_le_lemma,
int_formula_prop_wf,
length_of_cons_lemma,
list_ind_cons_lemma,
list-partition_wf,
permutation_wf,
append_wf,
list_wf,
istype-universe,
subtype_rel_function,
int_seg_subtype,
istype-false,
decidable__le,
not-le-2,
condition-implies-le,
minus-add,
minus-one-mul,
add-swap,
minus-one-mul-top,
add-mul-special,
zero-mul,
add-zero,
add-associates,
add-commutes,
le-add-cancel,
subtype_rel_self,
non_neg_length,
intformnot_wf,
int_formula_prop_not_lemma,
decidable__lt,
itermAdd_wf,
int_term_value_add_lemma,
istype-le,
istype-less_than,
eqtt_to_assert,
permutation-cons,
cons_wf,
eqff_to_assert,
bool_cases_sqequal,
subtype_base_sq,
bool_subtype_base,
assert-bnot,
length-append,
nil_wf
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
Error :lambdaFormation_alt,
cut,
thin,
introduction,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
hypothesisEquality,
sqequalRule,
Error :lambdaEquality_alt,
functionEquality,
closedConclusion,
natural_numberEquality,
hypothesis,
because_Cache,
Error :inhabitedIsType,
productElimination,
Error :equalityIstype,
equalityTransitivity,
equalitySymmetry,
dependent_functionElimination,
independent_functionElimination,
Error :functionIsType,
Error :universeIsType,
Error :isect_memberEquality_alt,
voidElimination,
setElimination,
rename,
independent_isectElimination,
approximateComputation,
Error :dependent_pairFormation_alt,
int_eqEquality,
independent_pairFormation,
addEquality,
instantiate,
universeEquality,
applyEquality,
unionElimination,
minusEquality,
multiplyEquality,
Error :dependent_set_memberEquality_alt,
Error :productIsType,
equalityElimination,
promote_hyp,
cumulativity,
applyLambdaEquality
Latex:
\mforall{}T:Type. \mforall{}L:T List. \mforall{}f:\mBbbN{}||L|| {}\mrightarrow{} \mBbbB{}. let as,bs = list-partition(f;L) in permutation(T;L;as @ bs)
Date html generated:
2019_06_20-PM-01_48_26
Last ObjectModification:
2018_11_28-PM-01_11_59
Theory : list_1
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