Use new API (Trace is now an image).

Lebesgue/Set_theory/Set_system/Set_system_base_any.v

@@ -223,7 +223,7 @@ assert (HA2 : forall i, exists B, P B /\ A i = trace V B).
   intros i; induction (HA1 i) as [B HB]; exists B; easy.
 destruct (choice _ HA2) as [B HB].
 replace (interf_any A) with (interf_any (tracef_map_any V B)).
-rewrite <- trace_interf_any; apply Tr, HP, HB; easy.
+rewrite <- trace_interf_any; apply Im, HP, HB; easy.
 apply interf_any_ext; intros i; symmetry; apply HB.
 Qed.
 
@@ -234,7 +234,7 @@ assert (HA2 : forall i, exists B, P B /\ A i = trace V B).
   intros i; induction (HA1 i) as [B HB]; exists B; easy.
 destruct (choice _ HA2) as [B HB].
 replace (unionf_any A) with (unionf_any (tracef_map_any V B)).
-rewrite <- trace_unionf_any; apply Tr, HP, HB; easy.
+rewrite <- trace_unionf_any; apply Im, HP, HB; easy.
 apply unionf_any_ext; intros i; symmetry; apply HB.
 Qed.
 

Lebesgue/Set_theory/Set_system/Set_system_base_seq.v

@@ -362,7 +362,7 @@ assert (HA2 : forall n, exists B, P B /\ A n = trace V B).
   intros n; induction (HA1 n) as [B HB]; exists B; easy.
 destruct (choice _ HA2) as [B HB].
 replace (inter_seq A) with (inter_seq (tracem_seq V B)).
-rewrite <- trace_inter_seq; apply Tr, HP, HB.
+rewrite <- trace_inter_seq; apply Im, HP, HB.
 apply inter_seq_ext; intros n; symmetry; apply HB.
 Qed.
 
@@ -373,7 +373,7 @@ assert (HA2 : forall n, exists B, P B /\ A n = trace V B).
   intros n; induction (HA1 n) as [B HB]; exists B; easy.
 destruct (choice _ HA2) as [B HB].
 replace (union_seq A) with (union_seq (tracem_seq V B)).
-rewrite <- trace_union_seq; apply Tr, HP, HB.
+rewrite <- trace_union_seq; apply Im, HP, HB.
 apply union_seq_ext; intros n; symmetry; apply HB.
 Qed.
 
@@ -387,7 +387,7 @@ destruct (choice _ HA3) as [B HB1].
 assert (HB2 : decr_seq B).
   intros n. admit.
 replace (inter_seq A) with (inter_seq (tracem_seq V B)).
-rewrite <- trace_inter_seq; apply Tr, HP; try apply HB1; easy.
+rewrite <- trace_inter_seq; apply Im, HP; try apply HB1; easy.
 apply inter_seq_ext; intros n; symmetry; apply HB1.
 Admitted.
 
@@ -401,7 +401,7 @@ destruct (choice _ HA3) as [B HB1].
 assert (HB2 : incr_seq B).
   intros n. admit.
 replace (union_seq A) with (union_seq (tracem_seq V B)).
-rewrite <- trace_union_seq; apply Tr, HP; try apply HB1; easy.
+rewrite <- trace_union_seq; apply Im, HP; try apply HB1; easy.
 apply union_seq_ext; intros n; symmetry; apply HB1.
 Admitted.
 
@@ -417,7 +417,7 @@ assert (HB2 : disj_seq B).
 
  admit.
 replace (union_seq A) with (union_seq (tracem_seq V B)).
-rewrite <- trace_union_seq; apply Tr, HP; try apply HB1; easy.
+rewrite <- trace_union_seq; apply Im, HP; try apply HB1; easy.
 apply union_seq_ext; intros n; symmetry; apply HB1.
 Admitted.