Nuprl Lemma : remove-repeats_functionality_wrt_permutation
∀[T:Type]
∀eq:EqDecider(T). ∀L1,L2:T List. (permutation(T;L1;L2) ⇒ permutation(T;remove-repeats(eq;L1);remove-repeats(eq;L2)))
Proof
Definitions occuring in Statement :
remove-repeats: remove-repeats(eq;L),
permutation: permutation(T;L1;L2),
list: T List,
deq: EqDecider(T),
uall: ∀[x:A]. B[x],
all: ∀x:A. B[x],
implies: P ⇒ Q,
universe: Type
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
all: ∀x:A. B[x],
member: t ∈ T,
int_seg: {i..j-},
lelt: i ≤ j < k,
and: P ∧ Q,
uimplies: b supposing a,
not: ¬A,
implies: P ⇒ Q,
satisfiable_int_formula: satisfiable_int_formula(fmla),
exists: ∃x:A. B[x],
false: False,
top: Top,
prop: ℙ,
decidable: Dec(P),
or: P ∨ Q,
subtype_rel: A ⊆r B,
so_lambda: λ2x.t[x],
so_apply: x[s],
guard: {T},
sq_type: SQType(T),
nat: ℕ,
ge: i ≥ j ,
le: A ≤ B,
less_than: a < b,
squash: ↓T,
cons: [a / b],
iff: P ⇐⇒ Q,
remove-repeats: remove-repeats(eq;L),
list_ind: list_ind,
nil: [],
it: ⋅,
deq: EqDecider(T),
rev_implies: P ⇐ Q,
append: as @ bs,
so_lambda: so_lambda(x,y,z.t[x; y; z]),
so_apply: x[s1;s2;s3],
cand: A c∧ B,
label: ...$L... t,
permutation: permutation(T;L1;L2),
bool: 𝔹,
unit: Unit,
btrue: tt,
uiff: uiff(P;Q),
bnot: ¬bb,
ifthenelse: if b then t else f fi ,
bfalse: ff,
assert: ↑b,
eqof: eqof(d),
true: True
Lemmas referenced :
int_seg_properties,
full-omega-unsat,
intformand_wf,
intformless_wf,
itermVar_wf,
itermConstant_wf,
intformle_wf,
istype-int,
int_formula_prop_and_lemma,
istype-void,
int_formula_prop_less_lemma,
int_term_value_var_lemma,
int_term_value_constant_lemma,
int_formula_prop_le_lemma,
int_formula_prop_wf,
int_seg_wf,
decidable__equal_int,
subtract_wf,
subtype_base_sq,
set_subtype_base,
int_subtype_base,
intformnot_wf,
intformeq_wf,
itermSubtract_wf,
int_formula_prop_not_lemma,
int_formula_prop_eq_lemma,
int_term_value_subtract_lemma,
decidable__le,
decidable__lt,
istype-le,
istype-less_than,
subtype_rel_self,
non_neg_length,
nat_properties,
length_wf,
list-cases,
product_subtype_list,
permutation_wf,
list_wf,
le_wf,
remove-repeats_wf,
primrec-wf2,
all_wf,
itermAdd_wf,
int_term_value_add_lemma,
istype-nat,
length_wf_nat,
deq_wf,
istype-universe,
permutation-nil-iff,
remove_repeats_nil_lemma,
permutation-nil,
nil_wf,
permutation-cons,
length_of_cons_lemma,
append_wf,
remove_repeats_cons_lemma,
cons_wf,
filter_wf5,
l_member_wf,
bnot_wf,
list_ind_nil_lemma,
length-append,
permutation-filter,
subtype_rel_list,
assert_wf,
istype-assert,
equal_wf,
inject_wf,
permute_list_wf,
permutation-length,
length_append,
top_wf,
list_ind_cons_lemma,
filter_cons_lemma,
filter_append_sq,
swap-filter-filter,
deq-member_wf,
eqtt_to_assert,
assert-deq-member,
eqff_to_assert,
bool_cases_sqequal,
bool_wf,
bool_subtype_base,
assert-bnot,
remove-repeats-append-sq,
permutation_weakening,
append_functionality_wrt_permutation,
l_member_decomp,
length_of_nil_lemma,
less_than_wf,
permutation_functionality_wrt_permutation,
permutation-rotate,
cons_functionality_wrt_permutation,
filter_functionality_wrt_permutation,
safe-assert-deq,
iff_weakening_uiff,
member_wf,
band-idempotent,
filter-filter,
filter_trivial,
l_all_iff,
iff_transitivity,
eqof_wf,
not_wf,
assert_of_bnot,
member-remove-repeats,
permutation_transitivity
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
Error :isect_memberFormation_alt,
Error :lambdaFormation_alt,
cut,
thin,
introduction,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
natural_numberEquality,
hypothesisEquality,
hypothesis,
setElimination,
rename,
productElimination,
independent_isectElimination,
approximateComputation,
independent_functionElimination,
Error :dependent_pairFormation_alt,
Error :lambdaEquality_alt,
int_eqEquality,
dependent_functionElimination,
Error :isect_memberEquality_alt,
voidElimination,
sqequalRule,
independent_pairFormation,
Error :universeIsType,
unionElimination,
applyEquality,
instantiate,
because_Cache,
equalityTransitivity,
equalitySymmetry,
applyLambdaEquality,
Error :dependent_set_memberEquality_alt,
Error :productIsType,
hypothesis_subsumption,
imageElimination,
promote_hyp,
Error :functionIsType,
functionEquality,
Error :setIsType,
closedConclusion,
addEquality,
universeEquality,
Error :equalityIstype,
Error :inhabitedIsType,
intEquality,
sqequalBase,
hyp_replacement,
setEquality,
equalityElimination,
cumulativity,
imageMemberEquality,
baseClosed,
Error :functionExtensionality_alt
Latex:
\mforall{}[T:Type]
\mforall{}eq:EqDecider(T). \mforall{}L1,L2:T List.
(permutation(T;L1;L2) {}\mRightarrow{} permutation(T;remove-repeats(eq;L1);remove-repeats(eq;L2)))
Date html generated:
2019_06_20-PM-01_55_17
Last ObjectModification:
2018_11_28-PM-05_14_40
Theory : decidable!equality
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