Nuprl Lemma : remove-repeats-fun-as-filter
∀[A,B:Type]. ∀[eq:EqDecider(B)]. ∀[f:A ⟶ B]. ∀[R:A ⟶ A ⟶ 𝔹]. ∀[L:A List].
(remove-repeats-fun(eq;f;L) ~ filter(λa.(¬b(∃x∈L.R[x;a] ∧b (eq (f x) (f a)))_b);L)) supposing
(sorted-by(λx,y. (↑R[x;y]);L) and
StAntiSym(A;x,y.↑R[x;y]) and
Irrefl(A;x,y.↑R[x;y]))
Proof
Definitions occuring in Statement :
remove-repeats-fun: remove-repeats-fun(eq;f;L),
bl-exists: (∃x∈L.P[x])_b,
sorted-by: sorted-by(R;L),
filter: filter(P;l),
list: T List,
deq: EqDecider(T),
irrefl: Irrefl(T;x,y.E[x; y]),
st_anti_sym: StAntiSym(T;x,y.R[x; y]),
band: p ∧b q,
bnot: ¬bb,
assert: ↑b,
bool: 𝔹,
uimplies: b supposing a,
uall: ∀[x:A]. B[x],
so_apply: x[s1;s2],
apply: f a,
lambda: λx.A[x],
function: x:A ⟶ B[x],
universe: Type,
sqequal: s ~ t
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
all: ∀x:A. B[x],
nat: ℕ,
implies: P ⇒ Q,
false: False,
ge: i ≥ j ,
uimplies: b supposing a,
satisfiable_int_formula: satisfiable_int_formula(fmla),
exists: ∃x:A. B[x],
not: ¬A,
top: Top,
and: P ∧ Q,
prop: ℙ,
so_apply: x[s1;s2],
so_lambda: λ2x y.t[x; y],
subtype_rel: A ⊆r B,
guard: {T},
or: P ∨ Q,
remove-repeats-fun: remove-repeats-fun(eq;f;L),
so_lambda: so_lambda(x,y,z.t[x; y; z]),
so_apply: x[s1;s2;s3],
cons: [a / b],
colength: colength(L),
decidable: Dec(P),
nil: [],
it: ⋅,
so_lambda: λ2x.t[x],
so_apply: x[s],
sq_type: SQType(T),
less_than: a < b,
squash: ↓T,
less_than': less_than'(a;b),
iff: P ⇐⇒ Q,
deq: EqDecider(T),
bool: 𝔹,
unit: Unit,
btrue: tt,
uiff: uiff(P;Q),
eqof: eqof(d),
rev_implies: P ⇐ Q,
bnot: ¬bb,
ifthenelse: if b then t else f fi ,
bfalse: ff,
assert: ↑b,
irrefl: Irrefl(T;x,y.E[x; y]),
st_anti_sym: StAntiSym(T;x,y.R[x; y]),
cand: A c∧ B,
band: p ∧b q,
label: ...$L... t
Lemmas referenced :
nat_properties,
satisfiable-full-omega-tt,
intformand_wf,
intformle_wf,
itermConstant_wf,
itermVar_wf,
intformless_wf,
int_formula_prop_and_lemma,
int_formula_prop_le_lemma,
int_term_value_constant_lemma,
int_term_value_var_lemma,
int_formula_prop_less_lemma,
int_formula_prop_wf,
ge_wf,
less_than_wf,
sorted-by_wf,
assert_wf,
l_member_wf,
st_anti_sym_wf,
irrefl_wf,
equal-wf-T-base,
nat_wf,
colength_wf_list,
less_than_transitivity1,
less_than_irreflexivity,
list-cases,
filter_nil_lemma,
list_ind_nil_lemma,
sorted-by_wf_nil,
product_subtype_list,
spread_cons_lemma,
intformeq_wf,
itermAdd_wf,
int_formula_prop_eq_lemma,
int_term_value_add_lemma,
decidable__le,
intformnot_wf,
int_formula_prop_not_lemma,
le_wf,
equal_wf,
subtract_wf,
itermSubtract_wf,
int_term_value_subtract_lemma,
subtype_base_sq,
set_subtype_base,
int_subtype_base,
decidable__equal_int,
filter_cons_lemma,
list_ind_cons_lemma,
sorted-by-cons,
filter-filter,
bl-exists_wf,
cons_wf,
band_wf,
bool_wf,
eqtt_to_assert,
assert-bl-exists,
l_exists_functionality,
iff_transitivity,
iff_weakening_uiff,
assert_of_band,
safe-assert-deq,
set_wf,
eqff_to_assert,
bool_cases_sqequal,
bool_subtype_base,
assert-bnot,
l_exists_wf,
list_wf,
deq_wf,
l_exists_iff,
cons_member,
and_wf,
l_all_iff,
filter-sq,
bnot_wf,
eqof_wf,
not_wf,
assert_of_bnot,
l_exists_cons,
l_all_fwd
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
introduction,
cut,
thin,
lambdaFormation,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
hypothesisEquality,
hypothesis,
setElimination,
rename,
intWeakElimination,
natural_numberEquality,
independent_isectElimination,
dependent_pairFormation,
lambdaEquality,
int_eqEquality,
intEquality,
dependent_functionElimination,
isect_memberEquality,
voidElimination,
voidEquality,
sqequalRule,
independent_pairFormation,
computeAll,
independent_functionElimination,
sqequalAxiom,
cumulativity,
applyEquality,
functionExtensionality,
setEquality,
equalityTransitivity,
equalitySymmetry,
because_Cache,
unionElimination,
promote_hyp,
hypothesis_subsumption,
productElimination,
applyLambdaEquality,
dependent_set_memberEquality,
addEquality,
baseClosed,
instantiate,
imageElimination,
equalityElimination,
productEquality,
functionEquality,
universeEquality,
hyp_replacement,
addLevel,
impliesFunctionality,
levelHypothesis,
inrFormation,
inlFormation
Latex:
\mforall{}[A,B:Type]. \mforall{}[eq:EqDecider(B)]. \mforall{}[f:A {}\mrightarrow{} B]. \mforall{}[R:A {}\mrightarrow{} A {}\mrightarrow{} \mBbbB{}]. \mforall{}[L:A List].
(remove-repeats-fun(eq;f;L) \msim{} filter(\mlambda{}a.(\mneg{}\msubb{}(\mexists{}x\mmember{}L.R[x;a] \mwedge{}\msubb{} (eq (f x) (f a)))\_b);L)) supposing
(sorted-by(\mlambda{}x,y. (\muparrow{}R[x;y]);L) and
StAntiSym(A;x,y.\muparrow{}R[x;y]) and
Irrefl(A;x,y.\muparrow{}R[x;y]))
Date html generated:
2017_04_17-AM-09_12_33
Last ObjectModification:
2017_02_27-PM-05_20_42
Theory : decidable!equality
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