modular_definitions: remove unused definitions

Signed-off-by: Alban Gruin <alban.gruin@irit.fr>

src/modular_definitions.v

@@ -3,13 +3,10 @@ Import ListNotations.
 
 Module Type Sig.
   Parameter opcode : Set.
-  Axiom opcode_beq : opcode -> opcode -> bool.
-  Axiom opcode_eq_dec : forall (o o' : opcode), {o = o'} + {o <> o'}.
 
   Parameter stage : Set.
   Axiom first_stage : stage.
   Axiom stage_beq : stage -> stage -> bool.
-  Axiom stage_eq_dec : forall (s s' : stage), {s = s'} + {s <> s'}.
   Axiom stage_beq_eq_true : forall s, stage_beq s s = true.
 
   Axiom leb : stage -> stage -> bool.
@@ -41,17 +38,17 @@ Module Pipeline (Def : Sig).
     end.
 
   Definition next_id (trace : trace_kind) :=
-    match trace with
+    let filtered := List.filter (fun (elt : instr_kind) =>
+                                   match elt with
+                                   | (_, (st, _)) => stage_beq st first_stage
+                                   end) trace in
+    let mapped := List.map (fun (elt : instr_kind) =>
+                              match elt with
+                              | (opc, _) => idx opc
+                              end) filtered in
+    match mapped with
     | [] => 0
-    | (opc, _) :: r =>
-        let id := idx opc in
-        List.fold_left (fun cid (elt : instr_kind) =>
-                          let (opc, st) := elt in
-                          let (s, _) := st in
-                          if stage_beq s first_stage then
-                            Nat.min cid (idx opc)
-                          else
-                            cid) r id
+    | id :: r => List.fold_left Nat.min r id
     end.
 
   Lemma state_leb_eq_true : forall st, state_leb st st = true.
@@ -62,13 +59,6 @@ Module Pipeline (Def : Sig).
     now rewrite stage_beq_eq_true, Nat.leb_le.
   Qed.
 
-  Lemma instr_kind_is_comparable : forall (e e' : instr_kind), {e = e'} + {e <> e'}.
-  Proof.
-    repeat decide equality.
-    - apply stage_eq_dec.
-    - apply opcode_eq_dec.
-  Qed.
-
   Definition compare_two (e1 e2 : option instr_kind) :=
     match e1, e2 with
     | None, _