Section assoc.
Variable K V : Type.
Variable K_eq_dec : forall k k' : K, {k = k'} + {k <> k'}.
Fixpoint assoc (l : list (K * V)) (k : K) : option V :=
match l with
| [] => None
| (k', v) :: l' =>
if K_eq_dec k k' then
Some v
else
assoc l' k
end.
Definition assoc_default (l : list (K * V)) (k : K) (default : V) : V :=
match (assoc l k) with
| Some x => x
| None => default
end.
Fixpoint assoc_set (l : list (K * V)) (k : K) (v : V) : list (K * V) :=
match l with
| [] => [(k, v)]
| (k', v') :: l' =>
if K_eq_dec k k' then
(k, v) :: l'
else
(k', v') :: (assoc_set l' k v)
end.
Fixpoint assoc_del (l : list (K * V)) (k : K) : list (K * V) :=
match l with
| [] => []
| (k', v') :: l' =>
if K_eq_dec k k' then
assoc_del l' k
else
(k', v') :: (assoc_del l' k)
end.
Proof using.
induction l; intros; simpl; repeat (break_match; simpl); subst; congruence.
Qed.
Proof using.
intros. subst. auto using get_set_same.
Qed.
Proof using.
induction l; intros; simpl; repeat (break_match; simpl); subst; try congruence; auto.
Qed.
Proof using.
intros.
induction l.
- auto.
- simpl in *. repeat break_match; intuition.
subst. simpl in *. congruence.
Qed.
Proof using.
induction l; intros; simpl in *.
- auto.
- repeat break_match; subst; simpl in *; auto.
break_if; try congruence.
Qed.
Proof using.
induction l; intros; simpl in *.
- auto.
- repeat (break_match; simpl); subst; try congruence; auto.
Qed.
Proof using.
unfold assoc_default.
intros.
repeat break_match; auto;
rewrite get_set_diff in * by auto; congruence.
Qed.
Proof using.
unfold assoc_default.
intros.
repeat break_match; auto;
rewrite get_set_same in *; congruence.
Qed.
Proof using.
intros. unfold assoc_default.
break_match; congruence.
Qed.
Proof using.
intros. unfold assoc_default.
break_match; congruence.
Qed.
Proof using.
intros. induction l; simpl in *; auto; try congruence.
repeat break_match; simpl in *; intuition.
- subst. find_inversion. auto.
- repeat find_rewrite. auto.
Qed.
Proof using.
intros. unfold assoc_default.
rewrite get_set_same. auto.
Qed.
Proof using.
induction l; intros; simpl in *; repeat break_match; simpl in *; subst; try congruence; eauto;
break_if; congruence.
Qed.
Definition a_equiv (l1 : list (K * V)) l2 :=
forall k,
assoc l1 k = assoc l2 k.
Proof using.
intros. unfold a_equiv. auto.
Qed.
Proof using.
unfold a_equiv. intros. auto.
Qed.
Proof using.
unfold a_equiv in *.
intros. repeat find_higher_order_rewrite.
auto.
Qed.
Ltac assoc_destruct :=
match goal with
| [ |- context [assoc (assoc_set _ ?k0' _) ?k0 ] ] =>
destruct (K_eq_dec k0 k0'); [subst k0'; rewrite get_set_same with (k := k0)|
rewrite get_set_diff with (k' := k0) by auto]
end.
Ltac assoc_rewrite :=
match goal with
| [  |- context [assoc (assoc_set _ ?k0' _) ?k0 ] ] =>
first [rewrite get_set_same with (k := k0) by auto |
rewrite get_set_diff with (k' := k0) by auto ]
end.
Proof using.
unfold a_equiv.
intros.
assoc_destruct.
- now repeat assoc_rewrite.
- assoc_destruct.
+ now repeat assoc_rewrite.
+ now repeat assoc_rewrite.
Qed.
Proof using.
intros.
destruct l; auto.
unfold a_equiv in *. simpl in *.
destruct p.
specialize (H k).
break_if; try congruence.
Qed.
Proof using.
unfold a_equiv.
intros.
assoc_destruct; assoc_rewrite; auto.
Qed.

Lemma assoc_default_a_equiv :
forall l l' (k : K) (v : V),
a_equiv l l' ->
assoc_default l k v = assoc_default l' k v.