Goal:
  
  A : Type
  P : nat → A → A → Type
  i : nat
  hd : A
  hd' : A
  tl : list A
  p : P i hd hd'
  i0 : nat
  hd0 : A
  hd'0 : A
  tl0 : list A
  p0 : P i0 hd0 hd'0
  ============================
   ({|
      pr1 := {| pr1 := i; pr2 := {| pr1 := hd :: tl; pr2 := hd' :: tl |} |};
      pr2 := OnOne2i_hd P i hd hd' tl p
    |} =
    {|
      pr1 :=
        {|
          pr1 :=
            pr1
              {|
                pr1 := i;
                pr2 :=
                  {|
                    pr1 := hd;
                    pr2 := {| pr1 := hd'; pr2 := {| pr1 := tl; pr2 := p |} |}
                  |}
              |};
          pr2 :=
            {|
              pr1 :=
                pr1
                  (pr2
                     {|
                       pr1 := i;
                       pr2 :=
                         {|
                           pr1 := hd;
                           pr2 :=
                             {|
                               pr1 := hd'; pr2 := {| pr1 := tl; pr2 := p |}
                             |}
                         |}
                     |})
                :: pr1
                     (pr2
                        (pr2
                           (pr2
                              {|
                                pr1 := i;
                                pr2 :=
                                  {|
                                    pr1 := hd;
                                    pr2 :=
                                      {|
                                        pr1 := hd';
                                        pr2 := {| pr1 := tl; pr2 := p |}
                                      |}
                                  |}
                              |})));
              pr2 :=
                pr1
                  (pr2
                     (pr2
                        {|
                          pr1 := i;
                          pr2 :=
                            {|
                              pr1 := hd;
                              pr2 :=
                                {|
                                  pr1 := hd';
                                  pr2 := {| pr1 := tl; pr2 := p |}
                                |}
                            |}
                        |}))
                :: pr1
                     (pr2
                        (pr2
                           (pr2
                              {|
                                pr1 := i;
                                pr2 :=
                                  {|
                                    pr1 := hd;
                                    pr2 :=
                                      {|
                                        pr1 := hd';
                                        pr2 := {| pr1 := tl; pr2 := p |}
                                      |}
                                  |}
                              |})))
            |}
        |};
      pr2 :=
        OnOne2i_hd P
          (pr1
             {|
               pr1 := i;
               pr2 :=
                 {|
                   pr1 := hd;
                   pr2 := {| pr1 := hd'; pr2 := {| pr1 := tl; pr2 := p |} |}
                 |}
             |})
          (pr1
             (pr2
                {|
                  pr1 := i;
                  pr2 :=
                    {|
                      pr1 := hd;
                      pr2 :=
                        {| pr1 := hd'; pr2 := {| pr1 := tl; pr2 := p |} |}
                    |}
                |}))
          (pr1
             (pr2
                (pr2
                   {|
                     pr1 := i;
                     pr2 :=
                       {|
                         pr1 := hd;
                         pr2 :=
                           {| pr1 := hd'; pr2 := {| pr1 := tl; pr2 := p |} |}
                       |}
                   |})))
          (pr1
             (pr2
                (pr2
                   (pr2
                      {|
                        pr1 := i;
                        pr2 :=
                          {|
                            pr1 := hd;
                            pr2 :=
                              {|
                                pr1 := hd'; pr2 := {| pr1 := tl; pr2 := p |}
                              |}
                          |}
                      |}))))
          (pr2
             (pr2
                (pr2
                   (pr2
                      {|
                        pr1 := i;
                        pr2 :=
                          {|
                            pr1 := hd;
                            pr2 :=
                              {|
                                pr1 := hd'; pr2 := {| pr1 := tl; pr2 := p |}
                              |}
                          |}
                      |}))))
    |})


DEBUG:Going to execute:
simpl