Goal:
  
  A : Type
  B : Type
  P : B → A → A → Type
  b0 : B
  bs : list B
  hd : A
  hd' : A
  tl : list A
  p : P b0 hd hd'
  e : #|bs| = #|tl|
  b : B
  bs0 : list B
  hd0 : A
  hd'0 : A
  tl0 : list A
  p0 : P b hd0 hd'0
  e0 : #|bs0| = #|tl0|
  H : {|
        pr1 := b0;
        pr2 :=
          {|
            pr1 := bs;
            pr2 :=
              {|
                pr1 := hd;
                pr2 :=
                  {|
                    pr1 := hd';
                    pr2 := {| pr1 := tl; pr2 := {| pr1 := p; pr2 := e |} |}
                  |}
              |}
          |}
      |} =
      {|
        pr1 := b;
        pr2 :=
          {|
            pr1 := bs0;
            pr2 :=
              {|
                pr1 := hd0;
                pr2 :=
                  {|
                    pr1 := hd'0;
                    pr2 :=
                      {| pr1 := tl0; pr2 := {| pr1 := p0; pr2 := e0 |} |}
                  |}
              |}
          |}
      |}
  ============================
   ({|
      pr1 :=
        {| pr1 := b0 :: bs; pr2 := {| pr1 := hd :: tl; pr2 := hd' :: tl |} |};
      pr2 := OnOne2All_hd P b0 bs hd hd' tl p e
    |} =
    {|
      pr1 :=
        {|
          pr1 := b :: bs0; pr2 := {| pr1 := hd0 :: tl0; pr2 := hd'0 :: tl0 |}
        |};
      pr2 := OnOne2All_hd P b bs0 hd0 hd'0 tl0 p0 e0
    |})


DEBUG:Going to execute:
set (rhs := y) in *; pattern sigma rhs; clearbody rhs; destruct H;
            simpl