Nuprl Lemma : remove-repeats-fun-member
∀[A,B:Type].
∀eq:EqDecider(B). ∀f:A ⟶ B. ∀L:A List. ∀a:A.
((a ∈ remove-repeats-fun(eq;f;L)) ⇐⇒ ∃i:ℕ||L||. ((L[i] = a ∈ A) ∧ (∀j:ℕi. (¬((f L[j]) = (f a) ∈ B)))))
Proof
Definitions occuring in Statement :
remove-repeats-fun: remove-repeats-fun(eq;f;L),
l_member: (x ∈ l),
select: L[n],
length: ||as||,
list: T List,
deq: EqDecider(T),
int_seg: {i..j-},
uall: ∀[x:A]. B[x],
all: ∀x:A. B[x],
exists: ∃x:A. B[x],
iff: P ⇐⇒ Q,
not: ¬A,
and: P ∧ Q,
apply: f a,
function: x:A ⟶ B[x],
natural_number: $n,
universe: Type,
equal: s = t ∈ T
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
all: ∀x:A. B[x],
member: t ∈ T,
so_lambda: λ2x.t[x],
prop: ℙ,
and: P ∧ Q,
int_seg: {i..j-},
uimplies: b supposing a,
guard: {T},
lelt: i ≤ j < k,
decidable: Dec(P),
or: P ∨ Q,
satisfiable_int_formula: satisfiable_int_formula(fmla),
exists: ∃x:A. B[x],
false: False,
implies: P ⇒ Q,
not: ¬A,
top: Top,
less_than: a < b,
squash: ↓T,
so_apply: x[s],
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],
select: L[n],
nil: [],
it: ⋅,
so_lambda: λ2x y.t[x; y],
so_apply: x[s1;s2],
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q,
deq: EqDecider(T),
le: A ≤ B,
less_than': less_than'(a;b),
nat_plus: ℕ+,
true: True,
uiff: uiff(P;Q),
cons: [a / b],
cand: A c∧ B,
ge: i ≥ j ,
eqof: eqof(d),
subtract: n - m,
subtype_rel: A ⊆r B
Lemmas referenced :
list_induction,
all_wf,
iff_wf,
l_member_wf,
remove-repeats-fun_wf,
exists_wf,
int_seg_wf,
length_wf,
equal_wf,
select_wf,
int_seg_properties,
decidable__le,
satisfiable-full-omega-tt,
intformand_wf,
intformnot_wf,
intformle_wf,
itermConstant_wf,
itermVar_wf,
int_formula_prop_and_lemma,
int_formula_prop_not_lemma,
int_formula_prop_le_lemma,
int_term_value_constant_lemma,
int_term_value_var_lemma,
int_formula_prop_wf,
decidable__lt,
intformless_wf,
int_formula_prop_less_lemma,
not_wf,
list_wf,
list_ind_nil_lemma,
length_of_nil_lemma,
stuck-spread,
base_wf,
null_nil_lemma,
btrue_wf,
member-implies-null-eq-bfalse,
nil_wf,
btrue_neq_bfalse,
equal-wf-base-T,
list_ind_cons_lemma,
length_of_cons_lemma,
cons_wf,
filter_wf5,
bnot_wf,
deq_wf,
cons_member,
false_wf,
add_nat_plus,
length_wf_nat,
less_than_wf,
nat_plus_wf,
nat_plus_properties,
add-is-int-iff,
itermAdd_wf,
intformeq_wf,
int_term_value_add_lemma,
int_formula_prop_eq_lemma,
lelt_wf,
non_neg_length,
member_filter_2,
assert_wf,
eqof_wf,
iff_transitivity,
iff_weakening_uiff,
assert_of_bnot,
safe-assert-deq,
add-member-int_seg2,
subtract_wf,
itermSubtract_wf,
int_term_value_subtract_lemma,
select-cons-tl,
add-subtract-cancel,
decidable__equal_int,
squash_wf,
le_wf,
iff_weakening_equal,
select_cons_tl,
true_wf,
member_filter
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
lambdaFormation,
cut,
thin,
introduction,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
hypothesisEquality,
sqequalRule,
lambdaEquality,
cumulativity,
because_Cache,
functionExtensionality,
applyEquality,
hypothesis,
natural_numberEquality,
productEquality,
setElimination,
rename,
independent_isectElimination,
productElimination,
dependent_functionElimination,
unionElimination,
dependent_pairFormation,
int_eqEquality,
intEquality,
isect_memberEquality,
voidElimination,
voidEquality,
independent_pairFormation,
computeAll,
imageElimination,
independent_functionElimination,
baseClosed,
equalityTransitivity,
equalitySymmetry,
setEquality,
functionEquality,
universeEquality,
dependent_set_memberEquality,
imageMemberEquality,
applyLambdaEquality,
pointwiseFunctionality,
promote_hyp,
baseApply,
closedConclusion,
addEquality,
impliesFunctionality,
inlFormation,
inrFormation
Latex:
\mforall{}[A,B:Type].
\mforall{}eq:EqDecider(B). \mforall{}f:A {}\mrightarrow{} B. \mforall{}L:A List. \mforall{}a:A.
((a \mmember{} remove-repeats-fun(eq;f;L)) \mLeftarrow{}{}\mRightarrow{} \mexists{}i:\mBbbN{}||L||. ((L[i] = a) \mwedge{} (\mforall{}j:\mBbbN{}i. (\mneg{}((f L[j]) = (f a))))))
Date html generated:
2017_04_17-AM-09_12_07
Last ObjectModification:
2017_02_27-PM-05_20_47
Theory : decidable!equality
Home
Index