Definition eq_sind_r :
    forall (A:Type) (x:A) (P:A -> SProp), P x -> forall y:A, y = x -> P y.Definition eq_ind_r :
    forall (A:Type) (x:A) (P:A -> Prop), P x -> forall y:A, y = x -> P y.Definition eq_rec_r :
    forall (A:Type) (x:A) (P:A -> Set), P x -> forall y:A, y = x -> P y.Definition eq_rect_r :
    forall (A:Type) (x:A) (P:A -> Type), P x -> forall y:A, y = x -> P y.Notation "'rew' H 'in' H'" := (eq_rect _ _ H' _ H)
    (at level 10, H' at level 10,
     format "'[' 'rew'  H  in  '/' H' ']'").Notation "'rew' [ P ] H 'in' H'" := (eq_rect _ P H' _ H)
    (at level 10, H' at level 10,
     format "'[' 'rew'  [ P ]  '/    ' H  in  '/' H' ']'").Notation "'rew' <- H 'in' H'" := (eq_rect_r _ H' H)
    (at level 10, H' at level 10,
     format "'[' 'rew'  <-  H  in  '/' H' ']'").Notation "'rew' <- [ P ] H 'in' H'" := (eq_rect_r P H' H)
    (at level 10, H' at level 10,
     format "'[' 'rew'  <-  [ P ]  '/    ' H  in  '/' H' ']'").Notation "'rew' -> H 'in' H'" := (eq_rect _ _ H' _ H)
    (at level 10, H' at level 10, only parsing).Notation "'rew' -> [ P ] H 'in' H'" := (eq_rect _ P H' _ H)
    (at level 10, H' at level 10, only parsing).Notation "'rew' 'dependent' H 'in' H'"
    := (match H with
        | eq_refl => H'
        end)
         (at level 10, H' at level 10,
          format "'[' 'rew'  'dependent'  '/    ' H  in  '/' H' ']'").Notation "'rew' 'dependent' -> H 'in' H'"
    := (match H with
        | eq_refl => H'
        end)
         (at level 10, H' at level 10, only parsing).Notation "'rew' 'dependent' <- H 'in' H'"
    := (match eq_sym H with
        | eq_refl => H'
        end)
         (at level 10, H' at level 10,
          format "'[' 'rew'  'dependent'  <-  '/    ' H  in  '/' H' ']'").Notation "'rew' 'dependent' [ 'fun' y p => P ] H 'in' H'"
    := (match H as p in (_ = y) return P with
        | eq_refl => H'
        end)
         (at level 10, H' at level 10, y name, p name,
          format "'[' 'rew'  'dependent'  [ 'fun'  y  p  =>  P ]  '/    ' H  in  '/' H' ']'").Notation "'rew' 'dependent' -> [ 'fun' y p => P ] H 'in' H'"
    := (match H as p in (_ = y) return P with
        | eq_refl => H'
        end)
         (at level 10, H' at level 10, y name, p name, only parsing).Notation "'rew' 'dependent' <- [ 'fun' y p => P ] H 'in' H'"
    := (match eq_sym H as p in (_ = y) return P with
        | eq_refl => H'
        end)
         (at level 10, H' at level 10, y name, p name,
          format "'[' 'rew'  'dependent'  <-  [ 'fun'  y  p  =>  P ]  '/    ' H  in  '/' H' ']'").Notation "'rew' 'dependent' [ P ] H 'in' H'"
    := (match H as p in (_ = y) return P y p with
        | eq_refl => H'
        end)
         (at level 10, H' at level 10,
          format "'[' 'rew'  'dependent'  [ P ]  '/    ' H  in  '/' H' ']'").Notation "'rew' 'dependent' -> [ P ] H 'in' H'"
    := (match H as p in (_ = y) return P y p with
        | eq_refl => H'
        end)
         (at level 10, H' at level 10,
          only parsing).Notation "'rew' 'dependent' <- [ P ] H 'in' H'"
    := (match eq_sym H as p in (_ = y) return P y p with
        | eq_refl => H'
        end)
         (at level 10, H' at level 10,
          format "'[' 'rew'  'dependent'  <-  [ P ]  '/    ' H  in  '/' H' ']'").

Theorem eq_trans_map {A B} {x1 x2 x3:A} {y1:B x1} {y2:B x2} {y3:B x3} (H1:x1=x2) (H2:x2=x3) (H1': rew H1 in y1 = y2) (H2': rew H2 in y2 = y3) : rew eq_trans H1 H2 in y1 = y3.