Section RaftLinearizableProofs.
Context {orig_base_params : BaseParams}.
Context {one_node_params : OneNodeParams orig_base_params}.
Context {raft_params : RaftParams orig_base_params}.
Context {oiai : output_implies_applied_interface}.
Context {aiii : applied_implies_input_interface}.
Context {copi : causal_order_preserved_interface}.
Context {iboi : input_before_output_interface}.
Context {oci : output_correct_interface}.
Context {ogii : output_greatest_id_interface}.
Definition op_eq_dec : forall x y : op key, {x = y} + {x <> y}.
Proof using.
decide equality; auto using key_eq_dec.
Qed.
Fixpoint import (tr : list (name * (raft_input + list raft_output)))
: list (op key) :=
match tr with
| [] => []
| (_, (inl (ClientRequest c id cmd))) :: xs =>
I (c, id) :: remove op_eq_dec (I (c, id)) (import xs)
| (_, (inr l)) :: xs =>
let os := dedup op_eq_dec
(filterMap (fun x =>
match x with
| ClientResponse c id cmd => Some (O (c, id))
| _ => None
end) l)
in os ++ remove_all op_eq_dec os (import xs)
| _ :: xs => import xs
end.
Inductive exported (env_i : key -> option input) (env_o : key -> option output) :
list (IR key) -> list (input * output) -> Prop :=
| exported_nil : exported env_i env_o nil nil
| exported_IO : forall k i o l tr,
env_i k = Some i ->
env_o k = Some o ->
exported env_i env_o l tr ->
exported env_i env_o (IRI k :: IRO k :: l) ((i, o) :: tr)
| exported_IU : forall k i o l tr,
env_i k = Some i ->
exported env_i env_o l tr ->
exported env_i env_o (IRI k :: IRU k :: l) ((i, o) :: tr).
Fixpoint get_input (tr : list (name * (raft_input + list raft_output))) (k : key)
: option input :=
match tr with
| [] => None
| (_, (inl (ClientRequest c id cmd))) :: xs =>
if (sumbool_and _ _ _ _
(clientId_eq_dec c (fst k))
(eq_nat_dec id (snd k))) then
Some cmd
else
get_input xs k
| _ :: xs => get_input xs k
end.
Fixpoint get_output' (os : list raft_output) (k : key) : option output :=
match os with
| [] => None
| ClientResponse c id o :: xs =>
if (sumbool_and _ _ _ _
(clientId_eq_dec c (fst k))
(eq_nat_dec id (snd k))) then
Some o
else
get_output' xs k
| _ :: xs => get_output' xs k
end.
Fixpoint get_output (tr : list (name * (raft_input + list raft_output))) (k : key)
: option output :=
match tr with
| [] => None
| (_, (inr os)) :: xs => (match get_output' os k with
| Some o => Some o
| None => get_output xs k
end)
| _ :: xs => get_output xs k
end.
Proof using.
unfold has_key.
intros.
destruct e.
simpl.
break_if; repeat (do_bool; intuition).
Qed.

Lemma has_key_intro' :
forall e c i,
eClient e = c ->
eId e = i ->
has_key c i e = true.